Properties

Label 4.43.b.a
Level $4$
Weight $43$
Character orbit 4.b
Analytic conductor $44.691$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 4 = 2^{2} \)
Weight: \( k \) \(=\) \( 43 \)
Character orbit: \([\chi]\) \(=\) 4.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(44.6910828688\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Defining polynomial: \(x^{20} - 10 x^{19} + 5706153082570973323 x^{18} - 51355377743138759622 x^{17} + \)\(13\!\cdots\!54\)\( x^{16} - \)\(10\!\cdots\!92\)\( x^{15} + \)\(16\!\cdots\!26\)\( x^{14} - \)\(11\!\cdots\!24\)\( x^{13} + \)\(12\!\cdots\!57\)\( x^{12} - \)\(73\!\cdots\!94\)\( x^{11} + \)\(55\!\cdots\!15\)\( x^{10} - \)\(27\!\cdots\!66\)\( x^{9} + \)\(15\!\cdots\!08\)\( x^{8} - \)\(60\!\cdots\!68\)\( x^{7} + \)\(24\!\cdots\!96\)\( x^{6} - \)\(74\!\cdots\!44\)\( x^{5} + \)\(21\!\cdots\!84\)\( x^{4} - \)\(43\!\cdots\!04\)\( x^{3} + \)\(79\!\cdots\!60\)\( x^{2} - \)\(79\!\cdots\!00\)\( x + \)\(58\!\cdots\!00\)\(\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: multiple of \( 2^{380}\cdot 3^{41}\cdot 5^{10}\cdot 7^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 40108 + \beta_{1} ) q^{2} + ( 16 - 78 \beta_{1} - \beta_{2} ) q^{3} + ( 65558433009 + 39523 \beta_{1} - 9 \beta_{2} + \beta_{3} ) q^{4} + ( 6961784165605 + 421261 \beta_{1} - 2 \beta_{2} + 16 \beta_{3} + \beta_{4} ) q^{5} + ( 341935932572220 - 3142930 \beta_{1} - 104610 \beta_{2} - 88 \beta_{3} + 6 \beta_{4} + \beta_{5} ) q^{6} + ( -699431483 + 3491944317 \beta_{1} + 2610895 \beta_{2} + 4342 \beta_{3} - 5 \beta_{4} - 3 \beta_{5} + \beta_{7} ) q^{7} + ( -1383070785180405223 + 78238901807 \beta_{1} + 173248969 \beta_{2} + 56033 \beta_{3} - 430 \beta_{4} + 16 \beta_{5} - 5 \beta_{6} + \beta_{8} ) q^{8} + ( -36658529353719209214 - 2142931497418 \beta_{1} + 4873091 \beta_{2} + 2493609 \beta_{3} - 13058 \beta_{4} + 640 \beta_{5} + 15 \beta_{6} + 3 \beta_{7} + \beta_{9} ) q^{9} +O(q^{10})\) \( q +(40108 + \beta_{1}) q^{2} +(16 - 78 \beta_{1} - \beta_{2}) q^{3} +(65558433009 + 39523 \beta_{1} - 9 \beta_{2} + \beta_{3}) q^{4} +(6961784165605 + 421261 \beta_{1} - 2 \beta_{2} + 16 \beta_{3} + \beta_{4}) q^{5} +(341935932572220 - 3142930 \beta_{1} - 104610 \beta_{2} - 88 \beta_{3} + 6 \beta_{4} + \beta_{5}) q^{6} +(-699431483 + 3491944317 \beta_{1} + 2610895 \beta_{2} + 4342 \beta_{3} - 5 \beta_{4} - 3 \beta_{5} + \beta_{7}) q^{7} +(-1383070785180405223 + 78238901807 \beta_{1} + 173248969 \beta_{2} + 56033 \beta_{3} - 430 \beta_{4} + 16 \beta_{5} - 5 \beta_{6} + \beta_{8}) q^{8} +(-36658529353719209214 - 2142931497418 \beta_{1} + 4873091 \beta_{2} + 2493609 \beta_{3} - 13058 \beta_{4} + 640 \beta_{5} + 15 \beta_{6} + 3 \beta_{7} + \beta_{9}) q^{9} +(2131257874551723860 + 6947379622816 \beta_{1} + 11104405622 \beta_{2} + 1233614 \beta_{3} - 170645 \beta_{4} + 908 \beta_{5} - 1104 \beta_{6} - 48 \beta_{7} + 12 \beta_{8} + \beta_{10}) q^{10} +(-105177802260826 + 526025487384599 \beta_{1} - 68143417585 \beta_{2} + 94588072 \beta_{3} - 105159 \beta_{4} - 40474 \beta_{5} + 7714 \beta_{6} + 1828 \beta_{7} - 9 \beta_{8} + \beta_{9} + \beta_{13}) q^{11} +(-\)\(12\!\cdots\!26\)\( + 353910158829357 \beta_{1} + 260155155886 \beta_{2} + 16808434 \beta_{3} + 5207009 \beta_{4} + 115137 \beta_{5} - 86603 \beta_{6} - 2421 \beta_{7} - 314 \beta_{8} - 39 \beta_{9} + 5 \beta_{10} - \beta_{12}) q^{12} +(-\)\(72\!\cdots\!59\)\( + 3545162838377344 \beta_{1} - 8213652798 \beta_{2} - 1521982093 \beta_{3} + 29036622 \beta_{4} + 140043 \beta_{5} - 43267 \beta_{6} + 1137 \beta_{7} + 129 \beta_{8} + 262 \beta_{9} - 7 \beta_{10} + \beta_{11} + \beta_{12} + 2 \beta_{14}) q^{13} +(-\)\(15\!\cdots\!16\)\( + 143041859695653 \beta_{1} + 13412810995724 \beta_{2} + 4434149109 \beta_{3} + 42160686 \beta_{4} - 2484096 \beta_{5} + 265364 \beta_{6} - 260703 \beta_{7} + 1043 \beta_{8} + 897 \beta_{9} - 16 \beta_{10} + 2 \beta_{11} + 3 \beta_{12} - 18 \beta_{13} + 10 \beta_{14} + \beta_{15}) q^{14} +(73897410208353715 - 369439009244321405 \beta_{1} - 24091187304308 \beta_{2} - 70294400757 \beta_{3} + 84437472 \beta_{4} - 117605 \beta_{5} - 5242206 \beta_{6} + 465444 \beta_{7} - 4873 \beta_{8} + 72 \beta_{9} - 382 \beta_{10} + 4 \beta_{11} - 65 \beta_{12} - 45 \beta_{13} - 28 \beta_{14} - 4 \beta_{15} + \beta_{16} + \beta_{17}) q^{15} +(\)\(13\!\cdots\!61\)\( - 1396236095435349923 \beta_{1} + 79819189856348 \beta_{2} + 79875726520 \beta_{3} - 2735260635 \beta_{4} - 171651264 \beta_{5} + 20038364 \beta_{6} - 443536 \beta_{7} + 95519 \beta_{8} - 11492 \beta_{9} - 265 \beta_{10} - 40 \beta_{11} - 28 \beta_{12} + 908 \beta_{13} - 83 \beta_{14} + 4 \beta_{15} + 4 \beta_{16} + 4 \beta_{17} + \beta_{18}) q^{16} +(\)\(40\!\cdots\!68\)\( - 4992752125758806751 \beta_{1} + 11718590633585 \beta_{2} - 483072009595 \beta_{3} - 18411917581 \beta_{4} - 409723540 \beta_{5} + 76612196 \beta_{6} - 2329627 \beta_{7} + 389944 \beta_{8} + 66921 \beta_{9} + 17418 \beta_{10} + 2 \beta_{11} + 753 \beta_{12} - 182 \beta_{13} - 248 \beta_{14} + 16 \beta_{16} + 16 \beta_{17} - 4 \beta_{18} + \beta_{19}) q^{17} +(-\)\(10\!\cdots\!40\)\( - 36571998176495876096 \beta_{1} + 2598277024973304 \beta_{2} - 1939285224080 \beta_{3} + 3799178278 \beta_{4} + 79124960 \beta_{5} - 154323148 \beta_{6} - 43185461 \beta_{7} + 1505938 \beta_{8} + 554543 \beta_{9} - 17308 \beta_{10} + 288 \beta_{11} - 675 \beta_{12} - 19598 \beta_{13} + 198 \beta_{14} - 80 \beta_{15} + 49 \beta_{16} + 48 \beta_{17} + 28 \beta_{18} + 4 \beta_{19}) q^{18} +(-1899013063583619112 + 9494170848487345719 \beta_{1} + 449064276965328 \beta_{2} + 1828588740898 \beta_{3} - 1892492855 \beta_{4} - 1169956430 \beta_{5} + 144097738 \beta_{6} - 54536151 \beta_{7} + 1843943 \beta_{8} + 20645 \beta_{9} - 307420 \beta_{10} + 184 \beta_{11} - 4570 \beta_{12} - 1147 \beta_{13} + 753 \beta_{14} + 328 \beta_{15} + 42 \beta_{16} + 46 \beta_{17} - 96 \beta_{18} + 40 \beta_{19}) q^{19} +(\)\(31\!\cdots\!26\)\( - 759873564126601908 \beta_{1} + 9519918777125989 \beta_{2} + 6371473772321 \beta_{3} + 426767313654 \beta_{4} - 12436474032 \beta_{5} + 440744865 \beta_{6} + 427018264 \beta_{7} + 5940940 \beta_{8} - 7260060 \beta_{9} - 168884 \beta_{10} + 3616 \beta_{11} + 708 \beta_{12} + 132024 \beta_{13} + 15192 \beta_{14} - 336 \beta_{15} + 36 \beta_{16} - 16 \beta_{17} + 316 \beta_{18} + 144 \beta_{19}) q^{20} +(\)\(38\!\cdots\!78\)\( + \)\(44\!\cdots\!93\)\( \beta_{1} - 1037881884947580 \beta_{2} + 49356482523815 \beta_{3} - 593112960793 \beta_{4} + 28447314075 \beta_{5} - 7023540973 \beta_{6} + 220862077 \beta_{7} - 175779719 \beta_{8} - 3061386 \beta_{9} + 4098693 \beta_{10} - 899 \beta_{11} - 30397 \beta_{12} + 3460 \beta_{13} + 5922 \beta_{14} - 800 \beta_{16} - 608 \beta_{17} - 872 \beta_{18} + 730 \beta_{19}) q^{21} +(-\)\(23\!\cdots\!92\)\( + 21135505270666176880 \beta_{1} + 164204629157339878 \beta_{2} + 545142258863874 \beta_{3} + 296287871678 \beta_{4} + 73359705659 \beta_{5} + 3521479364 \beta_{6} - 3644709146 \beta_{7} + 5614698 \beta_{8} + 72908566 \beta_{9} - 250768 \beta_{10} - 22020 \beta_{11} + 26978 \beta_{12} + 113588 \beta_{13} + 85068 \beta_{14} + 3070 \beta_{15} - 1144 \beta_{16} - 2432 \beta_{17} + 1568 \beta_{18} + 2336 \beta_{19}) q^{22} +(-\)\(30\!\cdots\!21\)\( + \)\(15\!\cdots\!11\)\( \beta_{1} + 166287618471516364 \beta_{2} + 802984995644445 \beta_{3} - 976382742950 \beta_{4} + 37627018503 \beta_{5} + 19249954214 \beta_{6} + 7287426932 \beta_{7} + 1837696461 \beta_{8} - 4684292 \beta_{9} - 10194058 \beta_{10} - 16756 \beta_{11} + 349077 \beta_{12} + 73825 \beta_{13} + 102338 \beta_{14} - 12940 \beta_{15} - 8565 \beta_{16} - 4189 \beta_{17} - 2624 \beta_{18} + 7920 \beta_{19}) q^{23} +(\)\(25\!\cdots\!08\)\( - \)\(15\!\cdots\!00\)\( \beta_{1} + 1515400612496511348 \beta_{2} + 372016493743140 \beta_{3} + 1080725122292 \beta_{4} - 237372387520 \beta_{5} - 38523674804 \beta_{6} + 34335332064 \beta_{7} - 67222516 \beta_{8} - 531271776 \beta_{9} + 3498548 \beta_{10} - 139232 \beta_{11} + 47744 \beta_{12} - 5787600 \beta_{13} + 290572 \beta_{14} + 13616 \beta_{15} + 13568 \beta_{16} - 6608 \beta_{17} - 372 \beta_{18} + 22080 \beta_{19}) q^{24} +(\)\(42\!\cdots\!06\)\( - \)\(12\!\cdots\!17\)\( \beta_{1} + 29173527356127304 \beta_{2} + 5208789440722850 \beta_{3} + 31130389912857 \beta_{4} + 406235814948 \beta_{5} + 153346626679 \beta_{6} + 8396566748 \beta_{7} - 14728459328 \beta_{8} + 18328622 \beta_{9} - 52893566 \beta_{10} + 10490 \beta_{11} + 151953 \beta_{12} - 14406 \beta_{13} + 1967064 \beta_{14} - 57328 \beta_{16} + 5520 \beta_{17} + 12956 \beta_{18} + 55129 \beta_{19}) q^{25} +(\)\(15\!\cdots\!52\)\( - \)\(73\!\cdots\!02\)\( \beta_{1} + 917626536336879046 \beta_{2} + 3571215587101006 \beta_{3} - 32314695965225 \beta_{4} + 84467666700 \beta_{5} + 94371475960 \beta_{6} - 375386780770 \beta_{7} - 974443584 \beta_{8} + 968154166 \beta_{9} + 22305657 \beta_{10} + 696128 \beta_{11} - 428862 \beta_{12} + 15092660 \beta_{13} + 1227900 \beta_{14} - 75040 \beta_{15} + 276874 \beta_{16} + 53216 \beta_{17} - 45416 \beta_{18} + 128680 \beta_{19}) q^{26} +(\)\(18\!\cdots\!44\)\( - \)\(91\!\cdots\!45\)\( \beta_{1} + 44905878171609133513 \beta_{2} + 42954460153508822 \beta_{3} - 48116750493161 \beta_{4} - 13437896822178 \beta_{5} - 1536421525418 \beta_{6} - 114983343505 \beta_{7} + 75099874601 \beta_{8} + 137411283 \beta_{9} + 539525068 \beta_{10} + 475944 \beta_{11} - 11015838 \beta_{12} - 296101 \beta_{13} + 14307087 \beta_{14} + 326360 \beta_{15} - 517042 \beta_{16} + 118986 \beta_{17} + 128096 \beta_{18} + 238808 \beta_{19}) q^{27} +(-\)\(81\!\cdots\!20\)\( - \)\(14\!\cdots\!94\)\( \beta_{1} + 7237738336087212924 \beta_{2} + 1149477911872212 \beta_{3} - 659407770602378 \beta_{4} - 11762809885594 \beta_{5} + 549558591862 \beta_{6} + 2608573112482 \beta_{7} - 4859692348 \beta_{8} + 6508931798 \beta_{9} + 65346270 \beta_{10} + 3135744 \beta_{11} - 1138950 \beta_{12} + 58564672 \beta_{13} - 2472320 \beta_{14} - 353920 \beta_{15} + 2114144 \beta_{16} + 255872 \beta_{17} - 198688 \beta_{18} + 415104 \beta_{19}) q^{28} +(\)\(42\!\cdots\!03\)\( - \)\(20\!\cdots\!89\)\( \beta_{1} + 475349385199370654 \beta_{2} + 124965803343236928 \beta_{3} + 1103316453626287 \beta_{4} + 74098372203948 \beta_{5} + 2048123476058 \beta_{6} + 446875014688 \beta_{7} - 118794559604 \beta_{8} + 1900422152 \beta_{9} - 957372704 \beta_{10} + 213488 \beta_{11} + 1893510 \beta_{12} + 24409444 \beta_{13} + 50125256 \beta_{14} - 4650784 \beta_{16} + 147616 \beta_{17} + 236504 \beta_{18} + 459530 \beta_{19}) q^{29} +(\)\(16\!\cdots\!32\)\( - \)\(14\!\cdots\!77\)\( \beta_{1} + 256472497597036684 \beta_{2} - 375851875557442657 \beta_{3} + 1946108845290266 \beta_{4} + 18585070551000 \beta_{5} - 2071106642908 \beta_{6} - 13265980237637 \beta_{7} - 13020954287 \beta_{8} - 51953026053 \beta_{9} - 101068752 \beta_{10} - 13536554 \beta_{11} + 16321841 \beta_{12} - 593090342 \beta_{13} - 60429714 \beta_{14} + 1307499 \beta_{15} + 11340208 \beta_{16} - 594176 \beta_{17} + 79552 \beta_{18} + 96448 \beta_{19}) q^{30} +(-\)\(19\!\cdots\!96\)\( + \)\(95\!\cdots\!68\)\( \beta_{1} - \)\(31\!\cdots\!77\)\( \beta_{2} + 528921746818692323 \beta_{3} - 544219033908355 \beta_{4} - 391967448271796 \beta_{5} + 13913199364934 \beta_{6} + 5421267605401 \beta_{7} - 459640860627 \beta_{8} + 8044706836 \beta_{9} - 5430144394 \beta_{10} - 7379124 \beta_{11} + 133373813 \beta_{12} + 215840193 \beta_{13} + 136732118 \beta_{14} - 5894988 \beta_{15} - 29619989 \beta_{16} - 1844781 \beta_{17} - 1402304 \beta_{18} - 1415920 \beta_{19}) q^{31} +(-\)\(58\!\cdots\!84\)\( + \)\(14\!\cdots\!12\)\( \beta_{1} - \)\(12\!\cdots\!72\)\( \beta_{2} - 1379145407559794928 \beta_{3} - 19086342294825436 \beta_{4} - 33422066464768 \beta_{5} + 12454716688768 \beta_{6} + 22506256913216 \beta_{7} - 34417819380 \beta_{8} + 48772728304 \beta_{9} - 2710826196 \beta_{10} - 46492704 \beta_{11} + 123918480 \beta_{12} - 276839696 \beta_{13} - 255085724 \beta_{14} + 6615760 \beta_{15} + 54970128 \beta_{16} - 5044528 \beta_{17} + 3450292 \beta_{18} - 5874432 \beta_{19}) q^{32} +(-\)\(97\!\cdots\!87\)\( + \)\(55\!\cdots\!24\)\( \beta_{1} - 13048750733504525639 \beta_{2} + 1069899738125998279 \beta_{3} + 13768448262967324 \beta_{4} + 2537595629624848 \beta_{5} - 97948447120501 \beta_{6} + 11965356828705 \beta_{7} + 3869375762616 \beta_{8} + 19071744279 \beta_{9} + 35856739308 \beta_{10} - 6687780 \beta_{11} - 373148398 \beta_{12} + 887310372 \beta_{13} + 785900512 \beta_{14} - 129861856 \beta_{16} - 5436768 \beta_{17} - 7580328 \beta_{18} - 13492566 \beta_{19}) q^{33} +(-\)\(21\!\cdots\!32\)\( + \)\(42\!\cdots\!89\)\( \beta_{1} - \)\(19\!\cdots\!64\)\( \beta_{2} - 5084538914257330592 \beta_{3} + 78652495819531454 \beta_{4} - 265746980479424 \beta_{5} + 67528172183476 \beta_{6} - 28099088601877 \beta_{7} - 11007754574 \beta_{8} + 253201888079 \beta_{9} - 13125250324 \beta_{10} + 184843552 \beta_{11} + 267103933 \beta_{12} + 5852378930 \beta_{13} - 440781818 \beta_{14} - 17204816 \beta_{15} + 236817169 \beta_{16} + 1845808 \beta_{17} + 9312412 \beta_{18} - 27647100 \beta_{19}) q^{34} +(\)\(55\!\cdots\!18\)\( - \)\(27\!\cdots\!32\)\( \beta_{1} - \)\(91\!\cdots\!29\)\( \beta_{2} + 9025581414578970634 \beta_{3} - 9082978742217448 \beta_{4} - 7294335900271588 \beta_{5} - 49828975449736 \beta_{6} + 10326092815377 \beta_{7} - 10289537510024 \beta_{8} + 154325821924 \beta_{9} + 4045506180 \beta_{10} + 67082296 \beta_{11} - 1916861226 \beta_{12} + 2473319772 \beta_{13} + 4969719525 \beta_{14} + 80908744 \beta_{15} - 407646790 \beta_{16} + 16770574 \beta_{17} - 2402272 \beta_{18} - 41616568 \beta_{19}) q^{35} +(\)\(45\!\cdots\!33\)\( - \)\(11\!\cdots\!13\)\( \beta_{1} - \)\(41\!\cdots\!83\)\( \beta_{2} - 36847498745727835481 \beta_{3} - 481998585982759860 \beta_{4} - 1516132270933088 \beta_{5} - 69442279690798 \beta_{6} - 322455899460176 \beta_{7} + 312273730648 \beta_{8} - 1794149033720 \beta_{9} - 15658836136 \beta_{10} + 472575552 \beta_{11} - 88161336 \beta_{12} - 17439510800 \beta_{13} + 1308159024 \beta_{14} - 94465952 \beta_{15} + 753202312 \beta_{16} + 64112352 \beta_{17} - 16365512 \beta_{18} - 45379040 \beta_{19}) q^{36} +(\)\(16\!\cdots\!93\)\( - \)\(57\!\cdots\!68\)\( \beta_{1} + \)\(13\!\cdots\!70\)\( \beta_{2} - 386002140944314529 \beta_{3} + 191880167840738938 \beta_{4} + 13940928495821991 \beta_{5} + 787048865452001 \beta_{6} + 75646666028741 \beta_{7} + 7657176629013 \beta_{8} + 672743268766 \beta_{9} - 338827790451 \beta_{10} + 56948661 \beta_{11} + 3217623653 \beta_{12} + 5289540384 \beta_{13} + 23418944426 \beta_{14} - 935793152 \beta_{16} + 88626432 \beta_{17} + 76618432 \beta_{18} - 19652272 \beta_{19}) q^{37} +(-\)\(41\!\cdots\!76\)\( + \)\(38\!\cdots\!80\)\( \beta_{1} + \)\(50\!\cdots\!26\)\( \beta_{2} + 9467534133593577030 \beta_{3} + 1301972561476853050 \beta_{4} - 3721175582710671 \beta_{5} + 148733759479868 \beta_{6} + 993453420444546 \beta_{7} - 239456244722 \beta_{8} + 1926121301970 \beta_{9} + 67681784208 \beta_{10} - 1889650604 \beta_{11} + 695422038 \beta_{12} - 59430329860 \beta_{13} + 26625672772 \beta_{14} + 176438570 \beta_{15} + 1356004280 \beta_{16} + 48147840 \beta_{17} - 139487520 \beta_{18} + 104123360 \beta_{19}) q^{38} +(\)\(59\!\cdots\!95\)\( - \)\(29\!\cdots\!29\)\( \beta_{1} + \)\(36\!\cdots\!59\)\( \beta_{2} + \)\(14\!\cdots\!38\)\( \beta_{3} - 168625185311889399 \beta_{4} - 20149575794040069 \beta_{5} - 1041990265625560 \beta_{6} + 292278224054921 \beta_{7} + 110311209472548 \beta_{8} - 34679075604 \beta_{9} + 649258951544 \beta_{10} - 267164464 \beta_{11} + 6222456372 \beta_{12} + 4991895956 \beta_{13} + 97434413490 \beta_{14} - 872775888 \beta_{15} - 1282670740 \beta_{16} - 66791116 \beta_{17} + 190350656 \beta_{18} + 322907600 \beta_{19}) q^{39} +(-\)\(10\!\cdots\!78\)\( + \)\(31\!\cdots\!74\)\( \beta_{1} + \)\(31\!\cdots\!46\)\( \beta_{2} + 2117789016956075834 \beta_{3} - 5807509435777847724 \beta_{4} + 5225791368783776 \beta_{5} + 2042087993415326 \beta_{6} - 2571692987923456 \beta_{7} + 1717720111418 \beta_{8} + 5473807606528 \beta_{9} + 193081760704 \beta_{10} - 3189381632 \beta_{11} + 7775913216 \beta_{12} + 195988030208 \beta_{13} + 194875185472 \beta_{14} + 1067575552 \beta_{15} - 273105664 \beta_{16} - 563533568 \beta_{17} - 163476928 \beta_{18} + 742658048 \beta_{19}) q^{40} +(\)\(61\!\cdots\!67\)\( + \)\(43\!\cdots\!85\)\( \beta_{1} - \)\(98\!\cdots\!04\)\( \beta_{2} - \)\(36\!\cdots\!78\)\( \beta_{3} - 333453717689967309 \beta_{4} - 41186134650428076 \beta_{5} - 4563167427441647 \beta_{6} + 643131526144 \beta_{7} - 317590887199016 \beta_{8} - 5929449690970 \beta_{9} + 1284310926094 \beta_{10} + 407166326 \beta_{11} - 41594263389 \beta_{12} - 13807436258 \beta_{13} + 378134109560 \beta_{14} + 1826174768 \beta_{16} - 925677008 \beta_{17} - 253747852 \beta_{18} + 1246462115 \beta_{19}) q^{41} +(\)\(19\!\cdots\!48\)\( + \)\(36\!\cdots\!30\)\( \beta_{1} + \)\(11\!\cdots\!80\)\( \beta_{2} + \)\(43\!\cdots\!72\)\( \beta_{3} + 16909096359629050212 \beta_{4} - 40690547669877152 \beta_{5} - 3919595835695928 \beta_{6} + 3235505180836302 \beta_{7} + 73520397970324 \beta_{8} - 40442931758154 \beta_{9} - 156336400544 \beta_{10} + 14947856192 \beta_{11} - 6147542206 \beta_{12} + 151621739700 \beta_{13} + 850688275324 \beta_{14} - 1430900000 \beta_{15} - 9814760950 \beta_{16} - 919869472 \beta_{17} + 823991448 \beta_{18} + 1548613800 \beta_{19}) q^{42} +(-\)\(24\!\cdots\!28\)\( + \)\(12\!\cdots\!52\)\( \beta_{1} - \)\(59\!\cdots\!77\)\( \beta_{2} + \)\(10\!\cdots\!92\)\( \beta_{3} - 1295494186219282430 \beta_{4} + 262319443475495348 \beta_{5} + 12763049131916820 \beta_{6} - 2535029505483118 \beta_{7} + 311670337077198 \beta_{8} - 9325368178446 \beta_{9} - 6314537065640 \beta_{10} - 1752729328 \beta_{11} + 30290919556 \beta_{12} - 82305617894 \beta_{13} + 1283199661610 \beta_{14} + 7537318640 \beta_{15} + 20193394332 \beta_{16} - 438182332 \beta_{17} - 1894291392 \beta_{18} + 1468740752 \beta_{19}) q^{43} +(-\)\(20\!\cdots\!34\)\( - \)\(22\!\cdots\!25\)\( \beta_{1} + \)\(14\!\cdots\!66\)\( \beta_{2} + 50132905202713234846 \beta_{3} - 32065560845272377297 \beta_{4} - 75957395771774225 \beta_{5} - 14325814057776373 \beta_{6} + 15411820706510725 \beta_{7} + 403247882357658 \beta_{8} + 65885815264151 \beta_{9} - 847624669013 \beta_{10} + 10986180096 \beta_{11} - 99513334383 \beta_{12} - 1340881051776 \beta_{13} + 2760520864512 \beta_{14} - 9752670976 \beta_{15} - 35603059904 \beta_{16} + 3427430656 \beta_{17} + 2865602624 \beta_{18} - 482405120 \beta_{19}) q^{44} +(-\)\(28\!\cdots\!57\)\( + \)\(85\!\cdots\!28\)\( \beta_{1} + \)\(21\!\cdots\!74\)\( \beta_{2} - \)\(71\!\cdots\!71\)\( \beta_{3} - 25843044674730080210 \beta_{4} - 1472970134717730371 \beta_{5} + 39636334001093697 \beta_{6} - 7398320331675261 \beta_{7} + 1689000203296671 \beta_{8} - 2977133151014 \beta_{9} + 6425639124027 \beta_{10} - 16821126045 \beta_{11} + 366076866249 \beta_{12} - 435892518588 \beta_{13} + 3996015987822 \beta_{14} + 71214526176 \beta_{16} + 6685787040 \beta_{17} - 1811809512 \beta_{18} - 4137262278 \beta_{19}) q^{45} +(-\)\(66\!\cdots\!40\)\( + \)\(60\!\cdots\!79\)\( \beta_{1} - \)\(13\!\cdots\!24\)\( \beta_{2} + \)\(15\!\cdots\!19\)\( \beta_{3} + 34968459072629888178 \beta_{4} - 270332900946906936 \beta_{5} + 52023244464053348 \beta_{6} - 52632344781611729 \beta_{7} + 1028007618953933 \beta_{8} + 162301519883695 \beta_{9} + 767043546160 \beta_{10} - 92547899010 \beta_{11} - 69254550323 \beta_{12} + 2231483284274 \beta_{13} + 8269233958198 \beta_{14} + 9168332479 \beta_{15} - 86872823856 \beta_{16} + 8915110144 \beta_{17} - 126677184 \beta_{18} - 11081589440 \beta_{19}) q^{46} +(-\)\(27\!\cdots\!10\)\( + \)\(13\!\cdots\!94\)\( \beta_{1} - \)\(22\!\cdots\!03\)\( \beta_{2} + \)\(85\!\cdots\!93\)\( \beta_{3} - 10612532063070478579 \beta_{4} + 1345415797629487870 \beta_{5} - 26608005882757934 \beta_{6} - 9182407869909805 \beta_{7} - 3691880700713281 \beta_{8} - 55426785755688 \beta_{9} + 19084182637106 \beta_{10} + 38115286756 \beta_{11} - 435937120857 \beta_{12} - 1424733151493 \beta_{13} + 12664295763404 \beta_{14} - 52441943780 \beta_{15} + 129185998489 \beta_{16} + 9528821689 \beta_{17} + 7851271424 \beta_{18} - 20799141056 \beta_{19}) q^{47} +(\)\(11\!\cdots\!32\)\( + \)\(25\!\cdots\!56\)\( \beta_{1} - \)\(30\!\cdots\!32\)\( \beta_{2} - \)\(17\!\cdots\!56\)\( \beta_{3} - 25288705332084652088 \beta_{4} - 1569783890144033280 \beta_{5} + 83182080582540576 \beta_{6} + 124104137676963712 \beta_{7} + 1579547613086552 \beta_{8} - 641300123360928 \beta_{9} + 6255484206104 \beta_{10} + 35103778240 \beta_{11} + 244672195744 \beta_{12} + 7543988494816 \beta_{13} + 23896053670536 \beta_{14} + 72806843040 \beta_{15} - 138497203808 \beta_{16} - 13018664288 \beta_{17} - 17564619608 \beta_{18} - 28027040768 \beta_{19}) q^{48} +(-\)\(38\!\cdots\!19\)\( + \)\(21\!\cdots\!08\)\( \beta_{1} - \)\(46\!\cdots\!40\)\( \beta_{2} - \)\(46\!\cdots\!24\)\( \beta_{3} + 4514362396143931580 \beta_{4} - 1930206642406678920 \beta_{5} - 74736632809632104 \beta_{6} - 4881763352444932 \beta_{7} + 10074870929374040 \beta_{8} - 165375095070892 \beta_{9} - 111423147798096 \beta_{10} + 237582893328 \beta_{11} - 1202927402076 \beta_{12} + 142628001720 \beta_{13} + 38319320413520 \beta_{14} + 36789595200 \beta_{16} - 32836885824 \beta_{17} + 23459286864 \beta_{18} - 33196857300 \beta_{19}) q^{49} +(-\)\(53\!\cdots\!76\)\( + \)\(42\!\cdots\!73\)\( \beta_{1} + \)\(21\!\cdots\!52\)\( \beta_{2} - \)\(12\!\cdots\!64\)\( \beta_{3} - \)\(28\!\cdots\!48\)\( \beta_{4} + 839400707230435744 \beta_{5} - 311513791982556616 \beta_{6} - 34955734114206526 \beta_{7} + 3117337049337548 \beta_{8} + 528309549600410 \beta_{9} + 14998795752928 \beta_{10} + 442156078784 \beta_{11} - 650329208658 \beta_{12} - 18842122473044 \beta_{13} + 59965346316708 \beta_{14} - 45339183584 \beta_{15} + 23924870054 \beta_{16} - 56278812384 \beta_{17} - 30379374936 \beta_{18} - 16732231144 \beta_{19}) q^{50} +(-\)\(12\!\cdots\!34\)\( + \)\(63\!\cdots\!41\)\( \beta_{1} + \)\(30\!\cdots\!52\)\( \beta_{2} + \)\(10\!\cdots\!24\)\( \beta_{3} - \)\(12\!\cdots\!43\)\( \beta_{4} - 10047134750565501626 \beta_{5} + 489966477988287586 \beta_{6} + 16104063480701652 \beta_{7} + 14987502284883351 \beta_{8} - 78790239098415 \beta_{9} + 53655420180384 \beta_{10} - 331314670272 \beta_{11} + 5090338979632 \beta_{12} + 11182227072513 \beta_{13} + 97173067330016 \beta_{14} + 291802704576 \beta_{15} - 473363353904 \beta_{16} - 82828667568 \beta_{17} + 4692845568 \beta_{18} + 21525623040 \beta_{19}) q^{51} +(-\)\(29\!\cdots\!58\)\( + \)\(15\!\cdots\!92\)\( \beta_{1} - \)\(18\!\cdots\!15\)\( \beta_{2} - \)\(88\!\cdots\!27\)\( \beta_{3} + \)\(66\!\cdots\!34\)\( \beta_{4} - 3134327027826707312 \beta_{5} + 78441370418225701 \beta_{6} - 501645650487179144 \beta_{7} + 11834115018873340 \beta_{8} + 780306311608820 \beta_{9} + 8438093657468 \beta_{10} - 789292448096 \beta_{11} - 644688056876 \beta_{12} - 9359444233576 \beta_{13} + 129510268367032 \beta_{14} - 444765103760 \beta_{15} + 935026475188 \beta_{16} + 13379641264 \beta_{17} + 33904217644 \beta_{18} + 96753451728 \beta_{19}) q^{52} +(\)\(17\!\cdots\!03\)\( + \)\(15\!\cdots\!24\)\( \beta_{1} - \)\(25\!\cdots\!42\)\( \beta_{2} - \)\(17\!\cdots\!75\)\( \beta_{3} - \)\(17\!\cdots\!22\)\( \beta_{4} + 18655870076401381529 \beta_{5} + 470343775178855805 \beta_{6} + 160060338366004743 \beta_{7} - 32041221588685549 \beta_{8} - 52984637831374 \beta_{9} + 498185395725823 \beta_{10} - 2178642046457 \beta_{11} + 219696956853 \beta_{12} + 7683101986676 \beta_{13} + 203201927542422 \beta_{14} - 1576061374368 \beta_{16} + 91173197600 \beta_{17} - 95565400520 \beta_{18} + 219566711922 \beta_{19}) q^{53} +(\)\(40\!\cdots\!80\)\( - \)\(36\!\cdots\!46\)\( \beta_{1} + \)\(64\!\cdots\!96\)\( \beta_{2} - \)\(87\!\cdots\!98\)\( \beta_{3} - \)\(28\!\cdots\!28\)\( \beta_{4} - 37458043506597845778 \beta_{5} + 2876121462370527716 \beta_{6} + 1420606058224135534 \beta_{7} + 32858290909389442 \beta_{8} - 8220682533495138 \beta_{9} - 158135251029520 \beta_{10} - 1519632306420 \beta_{11} + 11889027981306 \beta_{12} + 75841659925540 \beta_{13} + 259892437686684 \beta_{14} + 160162999942 \beta_{15} + 2899749336712 \beta_{16} + 230097217152 \beta_{17} + 197793215008 \beta_{18} + 334046958880 \beta_{19}) q^{54} +(\)\(10\!\cdots\!89\)\( - \)\(54\!\cdots\!91\)\( \beta_{1} + \)\(28\!\cdots\!25\)\( \beta_{2} + \)\(44\!\cdots\!30\)\( \beta_{3} - \)\(51\!\cdots\!37\)\( \beta_{4} - 70376641140691551539 \beta_{5} - 2551139095878876264 \beta_{6} + 448007403160411275 \beta_{7} + 135927467901293740 \beta_{8} + 194721044513964 \beta_{9} - 756113993330392 \beta_{10} + 1799135116592 \beta_{11} - 34944247485988 \beta_{12} - 55409190198404 \beta_{13} + 372663316775362 \beta_{14} - 1259190417712 \beta_{15} - 3178137789244 \beta_{16} + 449783779148 \beta_{17} - 207578696896 \beta_{18} + 460546176336 \beta_{19}) q^{55} +(-\)\(20\!\cdots\!92\)\( - \)\(79\!\cdots\!56\)\( \beta_{1} - \)\(87\!\cdots\!36\)\( \beta_{2} - \)\(21\!\cdots\!64\)\( \beta_{3} + \)\(33\!\cdots\!20\)\( \beta_{4} - 18486626937776843904 \beta_{5} + 1462358618482519112 \beta_{6} - 3523314271444591296 \beta_{7} + 40929280137941704 \beta_{8} + 8518214533349824 \beta_{9} - 470149948219720 \beta_{10} + 5957973212352 \beta_{11} + 8880274597632 \beta_{12} - 169906567042272 \beta_{13} + 449823096068296 \beta_{14} + 2199033313568 \beta_{15} + 4558705948160 \beta_{16} + 178959875872 \beta_{17} + 213478587848 \beta_{18} + 420045495680 \beta_{19}) q^{56} +(\)\(68\!\cdots\!03\)\( + \)\(16\!\cdots\!80\)\( \beta_{1} - \)\(35\!\cdots\!37\)\( \beta_{2} - \)\(51\!\cdots\!55\)\( \beta_{3} + \)\(52\!\cdots\!40\)\( \beta_{4} + 51397642913485373496 \beta_{5} - 278162999287429575 \beta_{6} + 403800275497341243 \beta_{7} + 5137781457742560 \beta_{8} + 2817857371360197 \beta_{9} + 83702924687820 \beta_{10} + 14559756730908 \beta_{11} + 12739338038646 \beta_{12} + 23103348063804 \beta_{13} + 561951510553680 \beta_{14} - 4080745905312 \beta_{16} + 53178258528 \beta_{17} - 20965532952 \beta_{18} + 185872679238 \beta_{19}) q^{57} +(-\)\(88\!\cdots\!60\)\( + \)\(43\!\cdots\!44\)\( \beta_{1} + \)\(33\!\cdots\!34\)\( \beta_{2} - \)\(17\!\cdots\!70\)\( \beta_{3} - \)\(45\!\cdots\!89\)\( \beta_{4} + 21307081718446545004 \beta_{5} - 8201354552402790320 \beta_{6} - 954715588984775480 \beta_{7} - 6736524237570756 \beta_{8} + 7908287598189848 \beta_{9} + 509049655339977 \beta_{10} + 2693587219712 \beta_{11} - 75834370835512 \beta_{12} - 202231300107696 \beta_{13} + 491188127794288 \beta_{14} - 287454134400 \beta_{15} + 1708629514728 \beta_{16} - 444607286400 \beta_{17} - 399252638880 \beta_{18} - 391223993952 \beta_{19}) q^{58} +(-\)\(15\!\cdots\!26\)\( + \)\(75\!\cdots\!48\)\( \beta_{1} - \)\(95\!\cdots\!94\)\( \beta_{2} + \)\(35\!\cdots\!30\)\( \beta_{3} - \)\(42\!\cdots\!06\)\( \beta_{4} + 8518114096872853288 \beta_{5} + 9702136876702240484 \beta_{6} - 134399657720174305 \beta_{7} - 277956487961553046 \beta_{8} - 691084675386990 \beta_{9} + 3215173766098028 \beta_{10} - 6030537756248 \beta_{11} + 108585684529074 \beta_{12} + 497317992203458 \beta_{13} + 401342431561019 \beta_{14} + 3834548315224 \beta_{15} + 1005635746750 \beta_{16} - 1507634439062 \beta_{17} + 845352054752 \beta_{18} - 1488207481160 \beta_{19}) q^{59} +(\)\(13\!\cdots\!84\)\( + \)\(16\!\cdots\!38\)\( \beta_{1} - \)\(11\!\cdots\!20\)\( \beta_{2} - \)\(98\!\cdots\!32\)\( \beta_{3} - \)\(30\!\cdots\!10\)\( \beta_{4} - 21827711431361650566 \beta_{5} - 380006757814435830 \beta_{6} + 15789248597885691390 \beta_{7} - 23433368118849284 \beta_{8} - 69926921081322102 \beta_{9} + 3321737705189890 \beta_{10} - 27608018341120 \beta_{11} - 22543481644378 \beta_{12} + 976354734734016 \beta_{13} - 27686475810944 \beta_{14} - 8504993240960 \beta_{15} - 12678508675552 \beta_{16} - 1236907212160 \beta_{17} - 1622843024736 \beta_{18} - 2715941553024 \beta_{19}) q^{60} +(-\)\(36\!\cdots\!41\)\( + \)\(43\!\cdots\!30\)\( \beta_{1} - \)\(10\!\cdots\!34\)\( \beta_{2} + \)\(58\!\cdots\!95\)\( \beta_{3} - \)\(24\!\cdots\!00\)\( \beta_{4} - \)\(35\!\cdots\!81\)\( \beta_{5} - 4307892265215049065 \beta_{6} - 2442420622223724503 \beta_{7} + 930869414083053461 \beta_{8} - 18854291320777218 \beta_{9} - 8607008498907663 \beta_{10} - 73130937188343 \beta_{11} - 132202113569161 \beta_{12} - 98806441860108 \beta_{13} - 883361654204006 \beta_{14} + 22399523134048 \beta_{16} - 1866245174496 \beta_{17} + 1829731785016 \beta_{18} - 4376578511950 \beta_{19}) q^{61} +(-\)\(41\!\cdots\!12\)\( + \)\(38\!\cdots\!68\)\( \beta_{1} + \)\(16\!\cdots\!92\)\( \beta_{2} + \)\(10\!\cdots\!64\)\( \beta_{3} + \)\(27\!\cdots\!72\)\( \beta_{4} + \)\(28\!\cdots\!76\)\( \beta_{5} + 30726137725587528536 \beta_{6} - 22652294629850468648 \beta_{7} + 57061075505768440 \beta_{8} + 120232546642951992 \beta_{9} - 1296235508771680 \beta_{10} + 6990632537744 \beta_{11} + 433380063571176 \beta_{12} - 326171549762928 \beta_{13} - 1548529427129200 \beta_{14} - 763537691320 \beta_{15} - 43763125408176 \beta_{16} - 1226942541568 \beta_{17} - 1655181825728 \beta_{18} - 5087409445056 \beta_{19}) q^{62} +(\)\(66\!\cdots\!93\)\( - \)\(33\!\cdots\!15\)\( \beta_{1} - \)\(39\!\cdots\!00\)\( \beta_{2} - \)\(57\!\cdots\!23\)\( \beta_{3} + \)\(66\!\cdots\!56\)\( \beta_{4} + \)\(12\!\cdots\!09\)\( \beta_{5} - 28740059054078358562 \beta_{6} - 10600369447624178456 \beta_{7} - 1592152951673230055 \beta_{8} - 12253983267623520 \beta_{9} - 1507043861705698 \beta_{10} + 6929021459580 \beta_{11} - 51008665599855 \beta_{12} - 3195294212726147 \beta_{13} - 4011364730620632 \beta_{14} - 5151420384380 \beta_{15} + 63919542127471 \beta_{16} + 1732255364895 \beta_{17} + 154113066880 \beta_{18} - 4733449581344 \beta_{19}) q^{63} +(-\)\(25\!\cdots\!00\)\( - \)\(58\!\cdots\!60\)\( \beta_{1} - \)\(38\!\cdots\!72\)\( \beta_{2} + \)\(12\!\cdots\!12\)\( \beta_{3} - \)\(52\!\cdots\!92\)\( \beta_{4} + \)\(17\!\cdots\!24\)\( \beta_{5} + 8145102913385576192 \beta_{6} + 28965609193232145664 \beta_{7} - 467248890406476368 \beta_{8} + 127210767521053120 \beta_{9} - 16987676395096528 \beta_{10} + 76089077050240 \beta_{11} - 40792853866432 \beta_{12} - 1723317488401984 \beta_{13} - 5440061080827376 \beta_{14} + 23275857081664 \beta_{15} - 77816719465920 \beta_{16} + 3367431210304 \beta_{17} + 3093213081936 \beta_{18} - 1967589678080 \beta_{19}) q^{64} +(\)\(71\!\cdots\!27\)\( - \)\(17\!\cdots\!73\)\( \beta_{1} + \)\(35\!\cdots\!96\)\( \beta_{2} + \)\(88\!\cdots\!06\)\( \beta_{3} - \)\(10\!\cdots\!35\)\( \beta_{4} - \)\(14\!\cdots\!04\)\( \beta_{5} - 9947204517965696037 \beta_{6} - 9122513411493636744 \beta_{7} - 1737385460265747016 \beta_{8} + 7713906209575274 \beta_{9} + 26929252292160538 \beta_{10} + 271491084263890 \beta_{11} + 769837737016761 \beta_{12} - 671630545941862 \beta_{13} - 10165099989942072 \beta_{14} + 99632261483664 \beta_{16} + 8776239589520 \beta_{17} - 6675762801188 \beta_{18} + 4992629848713 \beta_{19}) q^{65} +(\)\(23\!\cdots\!00\)\( - \)\(99\!\cdots\!39\)\( \beta_{1} + \)\(11\!\cdots\!84\)\( \beta_{2} + \)\(61\!\cdots\!60\)\( \beta_{3} + \)\(16\!\cdots\!30\)\( \beta_{4} - \)\(15\!\cdots\!16\)\( \beta_{5} - 99635294319953210900 \beta_{6} + 20893184746955167149 \beta_{7} - 1874128729077798530 \beta_{8} - 450650903245132311 \beta_{9} + 7580576083716044 \beta_{10} - 78396059462688 \beta_{11} - 2240641251673237 \beta_{12} + 6020880923971806 \beta_{13} - 14322281092592214 \beta_{14} + 8380707385296 \beta_{15} - 66443435693385 \beta_{16} + 12839210279760 \beta_{17} + 12270427885380 \beta_{18} + 13893578513820 \beta_{19}) q^{66} +(\)\(13\!\cdots\!10\)\( - \)\(67\!\cdots\!95\)\( \beta_{1} + \)\(72\!\cdots\!31\)\( \beta_{2} - \)\(19\!\cdots\!48\)\( \beta_{3} + \)\(23\!\cdots\!67\)\( \beta_{4} + \)\(89\!\cdots\!90\)\( \beta_{5} - 12472035540912926994 \beta_{6} + 27154161366329832666 \beta_{7} + 4145671387197595097 \beta_{8} + 34692240490003311 \beta_{9} - 46504385665979736 \beta_{10} + 47598404782448 \beta_{11} - 838666908540996 \beta_{12} + 13241368949383735 \beta_{13} - 21535346081404434 \beta_{14} - 22648042253680 \beta_{15} + 49956280764580 \beta_{16} + 11899601195612 \beta_{17} - 12891728283968 \beta_{18} + 28398422808880 \beta_{19}) q^{67} +(-\)\(10\!\cdots\!22\)\( - \)\(21\!\cdots\!22\)\( \beta_{1} - \)\(14\!\cdots\!24\)\( \beta_{2} + \)\(34\!\cdots\!48\)\( \beta_{3} - \)\(14\!\cdots\!36\)\( \beta_{4} + \)\(19\!\cdots\!04\)\( \beta_{5} - 67205677969737367142 \beta_{6} - \)\(15\!\cdots\!68\)\( \beta_{7} - 1225696825241553992 \beta_{8} + 707743393173774120 \beta_{9} + 87951433901056696 \beta_{10} - 39843851290816 \beta_{11} - 88038901390104 \beta_{12} - 3031655973542224 \beta_{13} - 28433587248817936 \beta_{14} - 26856240567840 \beta_{15} + 60331351722152 \beta_{16} + 2316517530208 \beta_{17} + 11536475965208 \beta_{18} + 39831806075552 \beta_{19}) q^{68} +(\)\(24\!\cdots\!70\)\( - \)\(48\!\cdots\!93\)\( \beta_{1} + \)\(83\!\cdots\!60\)\( \beta_{2} + \)\(52\!\cdots\!29\)\( \beta_{3} + \)\(69\!\cdots\!77\)\( \beta_{4} + \)\(55\!\cdots\!61\)\( \beta_{5} - \)\(20\!\cdots\!55\)\( \beta_{6} + 26500594620619627071 \beta_{7} - 16236407197716518181 \beta_{8} + 422934385391296674 \beta_{9} + 23415706305365943 \beta_{10} - 672051779416257 \beta_{11} - 1650391379010539 \beta_{12} + 1102660206190308 \beta_{13} - 35771434609605274 \beta_{14} - 219508176850208 \beta_{16} - 15439856517984 \beta_{17} - 2033472095016 \beta_{18} + 48598022916810 \beta_{19}) q^{69} +(\)\(12\!\cdots\!76\)\( - \)\(10\!\cdots\!84\)\( \beta_{1} + \)\(31\!\cdots\!32\)\( \beta_{2} - \)\(69\!\cdots\!92\)\( \beta_{3} + \)\(34\!\cdots\!44\)\( \beta_{4} + \)\(15\!\cdots\!64\)\( \beta_{5} + \)\(54\!\cdots\!88\)\( \beta_{6} + \)\(30\!\cdots\!04\)\( \beta_{7} - 206608420578853388 \beta_{8} + 170528163567452908 \beta_{9} - 29931539352108640 \beta_{10} + 320925212750392 \beta_{11} + 7746450120068228 \beta_{12} - 17801521832997656 \beta_{13} - 45204324095492520 \beta_{14} - 33125603022052 \beta_{15} + 380867460421360 \beta_{16} - 44158584524032 \beta_{17} - 20599755166784 \beta_{18} + 45620755916224 \beta_{19}) q^{70} +(\)\(11\!\cdots\!21\)\( - \)\(58\!\cdots\!27\)\( \beta_{1} + \)\(12\!\cdots\!84\)\( \beta_{2} - \)\(31\!\cdots\!77\)\( \beta_{3} + \)\(39\!\cdots\!14\)\( \beta_{4} - \)\(64\!\cdots\!71\)\( \beta_{5} - \)\(70\!\cdots\!34\)\( \beta_{6} - 44717508555122892028 \beta_{7} + 5479879746190255627 \beta_{8} + 226513437708435004 \beta_{9} + 168525319514323162 \beta_{10} - 308783154159916 \beta_{11} + 4688297101081251 \beta_{12} - 51009307490607849 \beta_{13} - 42171036506732078 \beta_{14} + 183334193036588 \beta_{15} - 765283622713987 \beta_{16} - 77195788539979 \beta_{17} + 38097523251904 \beta_{18} + 17688203213360 \beta_{19}) q^{71} +(\)\(69\!\cdots\!65\)\( + \)\(44\!\cdots\!75\)\( \beta_{1} - \)\(67\!\cdots\!27\)\( \beta_{2} - \)\(16\!\cdots\!87\)\( \beta_{3} + \)\(43\!\cdots\!10\)\( \beta_{4} + \)\(12\!\cdots\!44\)\( \beta_{5} + \)\(11\!\cdots\!79\)\( \beta_{6} - \)\(28\!\cdots\!96\)\( \beta_{7} - 13638679291616915223 \beta_{8} - 2738938890525870592 \beta_{9} - 400956306981022848 \beta_{10} - 700572128302080 \beta_{11} + 1121398121028096 \beta_{12} + 30287471376250368 \beta_{13} - 57879269540518272 \beta_{14} - 128815727255040 \beta_{15} + 919040585550336 \beta_{16} - 53108017700352 \beta_{17} - 73598010091392 \beta_{18} - 26399009071104 \beta_{19}) q^{72} +(\)\(33\!\cdots\!27\)\( + \)\(45\!\cdots\!46\)\( \beta_{1} - \)\(10\!\cdots\!61\)\( \beta_{2} + \)\(52\!\cdots\!53\)\( \beta_{3} + \)\(45\!\cdots\!02\)\( \beta_{4} + \)\(22\!\cdots\!92\)\( \beta_{5} - \)\(98\!\cdots\!89\)\( \beta_{6} + \)\(10\!\cdots\!71\)\( \beta_{7} + 23873138243693237768 \beta_{8} - 2382521769470663835 \beta_{9} - 374604290758468208 \beta_{10} + 556616474311088 \beta_{11} - 2516986325283540 \beta_{12} + 8463841095170472 \beta_{13} - 30210716595790352 \beta_{14} - 1260457216902976 \beta_{16} - 41921606013888 \beta_{17} + 84019329723632 \beta_{18} - 115248293947196 \beta_{19}) q^{73} +(-\)\(25\!\cdots\!32\)\( + \)\(18\!\cdots\!06\)\( \beta_{1} + \)\(62\!\cdots\!46\)\( \beta_{2} - \)\(51\!\cdots\!74\)\( \beta_{3} - \)\(14\!\cdots\!81\)\( \beta_{4} + \)\(52\!\cdots\!88\)\( \beta_{5} + 52655245040002261656 \beta_{6} - \)\(38\!\cdots\!10\)\( \beta_{7} - 27476932952154999120 \beta_{8} + 3701195059974847950 \beta_{9} + 80874235672987601 \beta_{10} - 606106717666240 \beta_{11} - 16082470284222966 \beta_{12} - 15810530451571452 \beta_{13} - 29365523145264660 \beta_{14} + 57007763145312 \beta_{15} + 1329924024944370 \beta_{16} + 39159317015392 \beta_{17} - 73316970281992 \beta_{18} - 205492202045880 \beta_{19}) q^{74} +(\)\(14\!\cdots\!42\)\( - \)\(74\!\cdots\!18\)\( \beta_{1} - \)\(37\!\cdots\!86\)\( \beta_{2} + \)\(25\!\cdots\!86\)\( \beta_{3} - \)\(23\!\cdots\!72\)\( \beta_{4} - \)\(27\!\cdots\!72\)\( \beta_{5} - \)\(11\!\cdots\!24\)\( \beta_{6} + \)\(35\!\cdots\!03\)\( \beta_{7} - 76774425663310061076 \beta_{8} + 652373671486573776 \beta_{9} + 76959268555290860 \beta_{10} + 795280493016104 \beta_{11} - 14229521197888974 \beta_{12} + 192836886488961488 \beta_{13} + 26930263060215075 \beta_{14} - 635708262239784 \beta_{15} - 942189913368130 \beta_{16} + 198820123254026 \beta_{17} + 40476171153632 \beta_{18} - 313513542573832 \beta_{19}) q^{75} +(-\)\(36\!\cdots\!78\)\( - \)\(41\!\cdots\!43\)\( \beta_{1} - \)\(67\!\cdots\!02\)\( \beta_{2} - \)\(48\!\cdots\!26\)\( \beta_{3} + \)\(20\!\cdots\!57\)\( \beta_{4} - \)\(12\!\cdots\!47\)\( \beta_{5} + \)\(68\!\cdots\!45\)\( \beta_{6} + \)\(89\!\cdots\!75\)\( \beta_{7} + 34991944545445069838 \beta_{8} - 1505871755185094235 \beta_{9} + 1545898925311976289 \beta_{10} + 3443694310201856 \beta_{11} + 2058809139241683 \beta_{12} - 99233353995818368 \beta_{13} + 103615624284270848 \beta_{14} + 932703807303424 \beta_{15} + 668485192569792 \beta_{16} + 175746478717696 \beta_{17} + 83270476135104 \beta_{18} - 382355875969280 \beta_{19}) q^{76} +(-\)\(69\!\cdots\!18\)\( + \)\(29\!\cdots\!47\)\( \beta_{1} - \)\(67\!\cdots\!64\)\( \beta_{2} - \)\(28\!\cdots\!83\)\( \beta_{3} - \)\(98\!\cdots\!63\)\( \beta_{4} - \)\(15\!\cdots\!95\)\( \beta_{5} - \)\(29\!\cdots\!43\)\( \beta_{6} - 87577354784708879437 \beta_{7} + \)\(12\!\cdots\!83\)\( \beta_{8} + 14669333046597116474 \beta_{9} + 818992684854848779 \beta_{10} + 3844417834980723 \beta_{11} + 20511358867220097 \beta_{12} - 5335899091587660 \beta_{13} + 197340983095773166 \beta_{14} + 963131269763680 \beta_{16} + 333736548213024 \beta_{17} - 199733250591816 \beta_{18} - 351928544676590 \beta_{19}) q^{77} +(\)\(13\!\cdots\!36\)\( - \)\(11\!\cdots\!57\)\( \beta_{1} + \)\(95\!\cdots\!56\)\( \beta_{2} - \)\(22\!\cdots\!37\)\( \beta_{3} - \)\(34\!\cdots\!06\)\( \beta_{4} - \)\(27\!\cdots\!88\)\( \beta_{5} + \)\(93\!\cdots\!72\)\( \beta_{6} - \)\(27\!\cdots\!09\)\( \beta_{7} + \)\(14\!\cdots\!65\)\( \beta_{8} - 6804229731783271017 \beta_{9} - 271375220047324016 \beta_{10} - 907471698900946 \beta_{11} + 16920894733112133 \beta_{12} + 196157510676352098 \beta_{13} + 368883431988407526 \beta_{14} + 113824532513815 \beta_{15} - 2187119333977344 \beta_{16} + 305642964021248 \beta_{17} + 389424198601728 \beta_{18} - 244135933154304 \beta_{19}) q^{78} +(\)\(90\!\cdots\!50\)\( - \)\(45\!\cdots\!62\)\( \beta_{1} - \)\(17\!\cdots\!20\)\( \beta_{2} + \)\(34\!\cdots\!88\)\( \beta_{3} - \)\(42\!\cdots\!30\)\( \beta_{4} + \)\(74\!\cdots\!98\)\( \beta_{5} - \)\(88\!\cdots\!20\)\( \beta_{6} - \)\(76\!\cdots\!72\)\( \beta_{7} + 35967782287503344672 \beta_{8} - 3091956708890106908 \beta_{9} - 1748850431822087232 \beta_{10} - 85584451370720 \beta_{11} + 3607945134576624 \beta_{12} - 519613990162574656 \beta_{13} + 513367504433027974 \beta_{14} + 958267915414240 \beta_{15} + 4943074237969904 \beta_{16} - 21396112842680 \beta_{17} - 488858545268800 \beta_{18} + 129222158879600 \beta_{19}) q^{79} +(-\)\(70\!\cdots\!18\)\( - \)\(97\!\cdots\!50\)\( \beta_{1} - \)\(49\!\cdots\!72\)\( \beta_{2} + \)\(31\!\cdots\!60\)\( \beta_{3} + \)\(23\!\cdots\!46\)\( \beta_{4} - \)\(13\!\cdots\!88\)\( \beta_{5} + \)\(17\!\cdots\!40\)\( \beta_{6} + \)\(47\!\cdots\!80\)\( \beta_{7} + 21238025279159429318 \beta_{8} + 22669370896781724504 \beta_{9} - 5472449135594149450 \beta_{10} - 7031929061298960 \beta_{11} - 17442001795456344 \beta_{12} + 12795444580348088 \beta_{13} + 813227249989197938 \beta_{14} - 3052282735566360 \beta_{15} - 7936896062950936 \beta_{16} - 50413844307480 \beta_{17} + 485218245867642 \beta_{18} + 568634772635648 \beta_{19}) q^{80} +(\)\(25\!\cdots\!90\)\( + \)\(18\!\cdots\!16\)\( \beta_{1} - \)\(42\!\cdots\!23\)\( \beta_{2} - \)\(14\!\cdots\!45\)\( \beta_{3} - \)\(26\!\cdots\!12\)\( \beta_{4} - \)\(25\!\cdots\!80\)\( \beta_{5} - \)\(19\!\cdots\!09\)\( \beta_{6} - \)\(97\!\cdots\!59\)\( \beta_{7} - \)\(27\!\cdots\!88\)\( \beta_{8} - 63831209243521432029 \beta_{9} + 1520685693584269524 \beta_{10} - 21585938351374332 \beta_{11} - 48086311043100006 \beta_{12} - 65506368662364060 \beta_{13} + 954634745933085744 \beta_{14} + 10293443442525600 \beta_{16} - 772303162983264 \beta_{17} - 306065042030376 \beta_{18} + 1267870637021130 \beta_{19}) q^{81} +(\)\(19\!\cdots\!60\)\( + \)\(59\!\cdots\!40\)\( \beta_{1} - \)\(18\!\cdots\!56\)\( \beta_{2} + \)\(41\!\cdots\!36\)\( \beta_{3} + \)\(41\!\cdots\!40\)\( \beta_{4} - \)\(18\!\cdots\!04\)\( \beta_{5} + \)\(22\!\cdots\!96\)\( \beta_{6} + \)\(25\!\cdots\!42\)\( \beta_{7} - \)\(27\!\cdots\!16\)\( \beta_{8} - 25742223542757149998 \beta_{9} + 301413903170048912 \beta_{10} + 10208008516454336 \beta_{11} + 39674180624943062 \beta_{12} - 371130848542729284 \beta_{13} + 1451989219524653908 \beta_{14} - 1045626059157600 \beta_{15} - 14821697447679378 \beta_{16} - 1447732613237088 \beta_{17} - 269153397036408 \beta_{18} + 1952873540267832 \beta_{19}) q^{82} +(\)\(20\!\cdots\!70\)\( - \)\(10\!\cdots\!00\)\( \beta_{1} + \)\(11\!\cdots\!48\)\( \beta_{2} + \)\(14\!\cdots\!54\)\( \beta_{3} - \)\(17\!\cdots\!14\)\( \beta_{4} + \)\(15\!\cdots\!84\)\( \beta_{5} - \)\(23\!\cdots\!28\)\( \beta_{6} - \)\(24\!\cdots\!93\)\( \beta_{7} + \)\(81\!\cdots\!26\)\( \beta_{8} - 8287771876154788922 \beta_{9} + 2839296802015631084 \beta_{10} - 6542631063999896 \beta_{11} + 131946397754370642 \beta_{12} + 806421326264020958 \beta_{13} + 1540904238447694271 \beta_{14} + 2084488206971800 \beta_{15} + 13038666501130846 \beta_{16} - 1635657765999974 \beta_{17} + 746943900654176 \beta_{18} + 2420109873842776 \beta_{19}) q^{83} +(\)\(10\!\cdots\!48\)\( + \)\(19\!\cdots\!64\)\( \beta_{1} + \)\(82\!\cdots\!88\)\( \beta_{2} + \)\(42\!\cdots\!92\)\( \beta_{3} - \)\(19\!\cdots\!40\)\( \beta_{4} - \)\(48\!\cdots\!28\)\( \beta_{5} + \)\(40\!\cdots\!04\)\( \beta_{6} - \)\(92\!\cdots\!92\)\( \beta_{7} + \)\(27\!\cdots\!72\)\( \beta_{8} - 15943064638599109584 \beta_{9} + 16423725475600123408 \beta_{10} - 3262411249810048 \beta_{11} + 13839553942041776 \beta_{12} + 970162535519223200 \beta_{13} + 1839627045664140576 \beta_{14} + 4705517764627008 \beta_{15} - 14301436632504016 \beta_{16} - 1562568810958528 \beta_{17} - 1790248779570352 \beta_{18} + 2705689275081920 \beta_{19}) q^{84} +(-\)\(50\!\cdots\!38\)\( + \)\(43\!\cdots\!15\)\( \beta_{1} - \)\(10\!\cdots\!00\)\( \beta_{2} - \)\(19\!\cdots\!91\)\( \beta_{3} + \)\(23\!\cdots\!33\)\( \beta_{4} - \)\(11\!\cdots\!59\)\( \beta_{5} - \)\(54\!\cdots\!37\)\( \beta_{6} - 61693919935137819369 \beta_{7} - \)\(45\!\cdots\!41\)\( \beta_{8} + \)\(16\!\cdots\!94\)\( \beta_{9} - 10723653454326582417 \beta_{10} + 57633049042109895 \beta_{11} + 39466024635751571 \beta_{12} - 50813256981810552 \beta_{13} + 1866040426985385438 \beta_{14} + 7008576613760704 \beta_{16} - 506376543848640 \beta_{17} + 2278762495957552 \beta_{18} + 1882280003081588 \beta_{19}) q^{85} +(-\)\(54\!\cdots\!44\)\( + \)\(49\!\cdots\!42\)\( \beta_{1} - \)\(11\!\cdots\!10\)\( \beta_{2} + \)\(11\!\cdots\!04\)\( \beta_{3} + \)\(24\!\cdots\!98\)\( \beta_{4} - \)\(39\!\cdots\!01\)\( \beta_{5} + \)\(65\!\cdots\!92\)\( \beta_{6} + \)\(21\!\cdots\!68\)\( \beta_{7} + \)\(14\!\cdots\!32\)\( \beta_{8} + 78436511857525249236 \beta_{9} + 3077328451950143136 \beta_{10} - 30897340439690872 \beta_{11} - 362202163459335876 \beta_{12} - 309176716111617896 \beta_{13} + 1326263787582497768 \beta_{14} + 2922404457134788 \beta_{15} + 5330979818747824 \beta_{16} + 2136274630758144 \beta_{17} - 2419078871978304 \beta_{18} + 647802177384640 \beta_{19}) q^{86} +(\)\(22\!\cdots\!95\)\( - \)\(11\!\cdots\!77\)\( \beta_{1} + \)\(53\!\cdots\!82\)\( \beta_{2} + \)\(22\!\cdots\!71\)\( \beta_{3} - \)\(17\!\cdots\!88\)\( \beta_{4} - \)\(39\!\cdots\!29\)\( \beta_{5} - \)\(16\!\cdots\!78\)\( \beta_{6} + \)\(13\!\cdots\!66\)\( \beta_{7} - \)\(28\!\cdots\!49\)\( \beta_{8} + 5745561097436176788 \beta_{9} + 7447018791630869282 \beta_{10} + 21555120455342884 \beta_{11} - 382263228902226593 \beta_{12} - 377646946774169565 \beta_{13} + 837353710876539878 \beta_{14} - 16678090918983460 \beta_{15} - 14211835442109695 \beta_{16} + 5388780113835721 \beta_{17} + 2442700537763136 \beta_{18} - 2239182662118960 \beta_{19}) q^{87} +(\)\(30\!\cdots\!24\)\( - \)\(20\!\cdots\!24\)\( \beta_{1} + \)\(24\!\cdots\!76\)\( \beta_{2} - \)\(21\!\cdots\!96\)\( \beta_{3} - \)\(40\!\cdots\!92\)\( \beta_{4} - \)\(24\!\cdots\!24\)\( \beta_{5} + \)\(20\!\cdots\!32\)\( \beta_{6} - \)\(35\!\cdots\!84\)\( \beta_{7} - \)\(36\!\cdots\!48\)\( \beta_{8} - \)\(14\!\cdots\!36\)\( \beta_{9} - 34287678641048034932 \beta_{10} + 61706106859632096 \beta_{11} + 19770174987160960 \beta_{12} - 2235308954717810480 \beta_{13} - 535991519843731148 \beta_{14} + 6572460752165840 \beta_{15} + 46255368066599680 \beta_{16} + 5204935628675792 \beta_{17} - 10500532936268 \beta_{18} - 5609030107449920 \beta_{19}) q^{88} +(\)\(14\!\cdots\!11\)\( + \)\(14\!\cdots\!94\)\( \beta_{1} - \)\(34\!\cdots\!29\)\( \beta_{2} + \)\(50\!\cdots\!01\)\( \beta_{3} + \)\(10\!\cdots\!62\)\( \beta_{4} + \)\(83\!\cdots\!48\)\( \beta_{5} - \)\(22\!\cdots\!21\)\( \beta_{6} + \)\(29\!\cdots\!55\)\( \beta_{7} + \)\(27\!\cdots\!44\)\( \beta_{8} - \)\(42\!\cdots\!79\)\( \beta_{9} + 8959755593933086280 \beta_{10} - 68006592395429304 \beta_{11} + 250882063510876944 \beta_{12} + 378325241699379504 \beta_{13} - 2050255667151509008 \beta_{14} - 60193925238187264 \beta_{16} + 8033216357995392 \beta_{17} - 1900768962731232 \beta_{18} - 9554628676705480 \beta_{19}) q^{89} +(\)\(26\!\cdots\!12\)\( - \)\(28\!\cdots\!62\)\( \beta_{1} - \)\(65\!\cdots\!06\)\( \beta_{2} - \)\(37\!\cdots\!46\)\( \beta_{3} + \)\(33\!\cdots\!51\)\( \beta_{4} - \)\(33\!\cdots\!16\)\( \beta_{5} + \)\(48\!\cdots\!16\)\( \beta_{6} - \)\(27\!\cdots\!90\)\( \beta_{7} - \)\(49\!\cdots\!08\)\( \beta_{8} + \)\(12\!\cdots\!30\)\( \beta_{9} - 11736879595837411927 \beta_{10} + 19299079035978432 \beta_{11} + 1173116270529530766 \beta_{12} + 3332488081185356908 \beta_{13} - 5196791837900511516 \beta_{14} - 1307529525151712 \beta_{15} + 98896245839351302 \beta_{16} + 4629469272967968 \beta_{17} + 6083767570436392 \beta_{18} - 13740173822788712 \beta_{19}) q^{90} +(\)\(58\!\cdots\!22\)\( - \)\(29\!\cdots\!88\)\( \beta_{1} - \)\(11\!\cdots\!37\)\( \beta_{2} - \)\(87\!\cdots\!78\)\( \beta_{3} + \)\(10\!\cdots\!68\)\( \beta_{4} - \)\(13\!\cdots\!56\)\( \beta_{5} - \)\(39\!\cdots\!68\)\( \beta_{6} - \)\(46\!\cdots\!35\)\( \beta_{7} - \)\(52\!\cdots\!72\)\( \beta_{8} + 56219895967203127472 \beta_{9} - 31229252718368305980 \beta_{10} - 17856926170114952 \beta_{11} + 106725102429423510 \beta_{12} - 3276440654344058320 \beta_{13} - 7195039475674159063 \beta_{14} + 44009221973925256 \beta_{15} - 124362237800996806 \beta_{16} - 4464231542528738 \beta_{17} - 10028784684856672 \beta_{18} - 14904945834954520 \beta_{19}) q^{91} +(-\)\(15\!\cdots\!24\)\( - \)\(65\!\cdots\!22\)\( \beta_{1} + \)\(69\!\cdots\!52\)\( \beta_{2} + \)\(58\!\cdots\!96\)\( \beta_{3} - \)\(16\!\cdots\!22\)\( \beta_{4} + \)\(24\!\cdots\!82\)\( \beta_{5} + \)\(48\!\cdots\!58\)\( \beta_{6} + \)\(57\!\cdots\!90\)\( \beta_{7} - \)\(37\!\cdots\!24\)\( \beta_{8} + \)\(18\!\cdots\!06\)\( \beta_{9} + 39169117054902503562 \beta_{10} - 156895205110585600 \beta_{11} - 31752002994686274 \beta_{12} - 2441756007037841472 \beta_{13} - 13259394293582472832 \beta_{14} - 61085796860819840 \beta_{15} + 129593421279995296 \beta_{16} - 1745520450258816 \beta_{17} + 11880799019978784 \beta_{18} - 15606503003633024 \beta_{19}) q^{92} +(-\)\(45\!\cdots\!92\)\( + \)\(55\!\cdots\!72\)\( \beta_{1} - \)\(13\!\cdots\!36\)\( \beta_{2} + \)\(20\!\cdots\!76\)\( \beta_{3} + \)\(22\!\cdots\!00\)\( \beta_{4} + \)\(44\!\cdots\!48\)\( \beta_{5} - \)\(93\!\cdots\!92\)\( \beta_{6} + \)\(18\!\cdots\!36\)\( \beta_{7} + \)\(94\!\cdots\!04\)\( \beta_{8} + \)\(11\!\cdots\!96\)\( \beta_{9} + 59516135301698336964 \beta_{10} - 117586360743879676 \beta_{11} - 1132490044568521232 \beta_{12} + 1027225491621906248 \beta_{13} - 16136078158705839144 \beta_{14} - 146368390988294464 \beta_{16} - 19075540579039936 \beta_{17} - 12308726285977936 \beta_{18} - 8939573799489964 \beta_{19}) q^{93} +(-\)\(60\!\cdots\!88\)\( + \)\(53\!\cdots\!78\)\( \beta_{1} - \)\(58\!\cdots\!08\)\( \beta_{2} + \)\(10\!\cdots\!98\)\( \beta_{3} - \)\(62\!\cdots\!88\)\( \beta_{4} + \)\(22\!\cdots\!72\)\( \beta_{5} + \)\(56\!\cdots\!28\)\( \beta_{6} - \)\(16\!\cdots\!66\)\( \beta_{7} + \)\(12\!\cdots\!98\)\( \beta_{8} - \)\(77\!\cdots\!98\)\( \beta_{9} + 5020849158908347712 \beta_{10} + 194125665712510676 \beta_{11} - 1724480703260266066 \beta_{12} - 6608609209774502420 \beta_{13} - 21472002881490754460 \beta_{14} - 19329549927189142 \beta_{15} + 65600710313503280 \beta_{16} - 27507628869495040 \beta_{17} + 5792879910729920 \beta_{18} + 309618967423680 \beta_{19}) q^{94} +(\)\(15\!\cdots\!51\)\( - \)\(78\!\cdots\!49\)\( \beta_{1} + \)\(56\!\cdots\!61\)\( \beta_{2} - \)\(27\!\cdots\!26\)\( \beta_{3} + \)\(33\!\cdots\!53\)\( \beta_{4} - \)\(15\!\cdots\!21\)\( \beta_{5} - \)\(10\!\cdots\!44\)\( \beta_{6} + \)\(20\!\cdots\!47\)\( \beta_{7} + \)\(18\!\cdots\!56\)\( \beta_{8} + 63091911301728066992 \beta_{9} + 1490285385884113736 \beta_{10} - 84142147861058800 \beta_{11} + 1654047049179059868 \beta_{12} + 19829656114089683564 \beta_{13} - 26395758353758232056 \beta_{14} - 24181601244543760 \beta_{15} - 42946339680657948 \beta_{16} - 21035536965264700 \beta_{17} + 802399615388416 \beta_{18} + 16323298164560064 \beta_{19}) q^{95} +(-\)\(18\!\cdots\!28\)\( + \)\(11\!\cdots\!76\)\( \beta_{1} + \)\(11\!\cdots\!96\)\( \beta_{2} + \)\(25\!\cdots\!40\)\( \beta_{3} + \)\(31\!\cdots\!16\)\( \beta_{4} + \)\(22\!\cdots\!32\)\( \beta_{5} + \)\(17\!\cdots\!76\)\( \beta_{6} + \)\(20\!\cdots\!20\)\( \beta_{7} - \)\(37\!\cdots\!16\)\( \beta_{8} + \)\(85\!\cdots\!96\)\( \beta_{9} + 6742456975082513376 \beta_{10} + 48242420202382080 \beta_{11} + 721868152639513216 \beta_{12} + 16580363984871807360 \beta_{13} - 28899150619927056736 \beta_{14} + 168969579578457216 \beta_{15} - 122501130933205376 \beta_{16} - 35368789066376064 \beta_{17} - 18567773221968096 \beta_{18} + 37816210345297920 \beta_{19}) q^{96} +(\)\(13\!\cdots\!08\)\( + \)\(12\!\cdots\!73\)\( \beta_{1} - \)\(30\!\cdots\!19\)\( \beta_{2} + \)\(22\!\cdots\!65\)\( \beta_{3} + \)\(12\!\cdots\!35\)\( \beta_{4} - \)\(50\!\cdots\!44\)\( \beta_{5} - \)\(19\!\cdots\!36\)\( \beta_{6} - \)\(25\!\cdots\!67\)\( \beta_{7} - \)\(19\!\cdots\!36\)\( \beta_{8} - \)\(22\!\cdots\!35\)\( \beta_{9} - \)\(13\!\cdots\!14\)\( \beta_{10} + 777062850950897754 \beta_{11} + 703959196682727105 \beta_{12} - 1766300852755264614 \beta_{13} - 32458272954864345256 \beta_{14} + 242843165526315792 \beta_{16} - 5645690602026864 \beta_{17} + 29086678661617116 \beta_{18} + 56245052101250697 \beta_{19}) q^{97} +(\)\(91\!\cdots\!80\)\( - \)\(39\!\cdots\!31\)\( \beta_{1} - \)\(78\!\cdots\!92\)\( \beta_{2} + \)\(20\!\cdots\!92\)\( \beta_{3} - \)\(45\!\cdots\!56\)\( \beta_{4} + \)\(50\!\cdots\!08\)\( \beta_{5} + \)\(29\!\cdots\!28\)\( \beta_{6} - \)\(59\!\cdots\!28\)\( \beta_{7} - \)\(43\!\cdots\!32\)\( \beta_{8} + \)\(18\!\cdots\!00\)\( \beta_{9} + 34886874220498191216 \beta_{10} - 755985539630358912 \beta_{11} + 295814834278717572 \beta_{12} - 9258576093538132632 \beta_{13} - 32823934170928446216 \beta_{14} + 70492845274269120 \beta_{15} - 390107418164109804 \beta_{16} + 41326783851704256 \beta_{17} - 42148341489921744 \beta_{18} + 78812107872123216 \beta_{19}) q^{98} +(\)\(61\!\cdots\!42\)\( - \)\(30\!\cdots\!60\)\( \beta_{1} + \)\(12\!\cdots\!10\)\( \beta_{2} - \)\(63\!\cdots\!58\)\( \beta_{3} + \)\(73\!\cdots\!10\)\( \beta_{4} + \)\(14\!\cdots\!00\)\( \beta_{5} - \)\(44\!\cdots\!68\)\( \beta_{6} - \)\(63\!\cdots\!89\)\( \beta_{7} + \)\(23\!\cdots\!18\)\( \beta_{8} - \)\(21\!\cdots\!50\)\( \beta_{9} + \)\(18\!\cdots\!64\)\( \beta_{10} + 296408608591538664 \beta_{11} - 4438007932253340798 \beta_{12} - 56584524943593657574 \beta_{13} - 33494256585226149549 \beta_{14} - 232764583938221032 \beta_{15} + 725771410087198574 \beta_{16} + 74102152147884666 \beta_{17} + 57818119955658464 \beta_{18} + 82105226620198520 \beta_{19}) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q + 802164q^{2} + 1311168818192q^{4} + 139235684997000q^{5} + 6838718638036512q^{6} - 27661415389266953664q^{8} - 733170595646091132876q^{9} + O(q^{10}) \) \( 20q + 802164q^{2} + 1311168818192q^{4} + 139235684997000q^{5} + 6838718638036512q^{6} - 27661415389266953664q^{8} - \)\(73\!\cdots\!76\)\(q^{9} + 42625185369378343240q^{10} - \)\(25\!\cdots\!20\)\(q^{12} - \)\(14\!\cdots\!48\)\(q^{13} - \)\(30\!\cdots\!28\)\(q^{14} + \)\(27\!\cdots\!00\)\(q^{16} + \)\(81\!\cdots\!28\)\(q^{17} - \)\(21\!\cdots\!24\)\(q^{18} + \)\(63\!\cdots\!20\)\(q^{20} + \)\(76\!\cdots\!24\)\(q^{21} - \)\(46\!\cdots\!00\)\(q^{22} + \)\(50\!\cdots\!72\)\(q^{24} + \)\(84\!\cdots\!20\)\(q^{25} + \)\(30\!\cdots\!88\)\(q^{26} - \)\(16\!\cdots\!00\)\(q^{28} + \)\(84\!\cdots\!04\)\(q^{29} + \)\(32\!\cdots\!60\)\(q^{30} - \)\(11\!\cdots\!36\)\(q^{32} - \)\(19\!\cdots\!80\)\(q^{33} - \)\(43\!\cdots\!48\)\(q^{34} + \)\(90\!\cdots\!52\)\(q^{36} + \)\(32\!\cdots\!48\)\(q^{37} - \)\(83\!\cdots\!60\)\(q^{38} - \)\(20\!\cdots\!60\)\(q^{40} + \)\(12\!\cdots\!80\)\(q^{41} + \)\(39\!\cdots\!80\)\(q^{42} - \)\(40\!\cdots\!00\)\(q^{44} - \)\(57\!\cdots\!40\)\(q^{45} - \)\(13\!\cdots\!28\)\(q^{46} + \)\(22\!\cdots\!00\)\(q^{48} - \)\(76\!\cdots\!16\)\(q^{49} - \)\(10\!\cdots\!60\)\(q^{50} - \)\(58\!\cdots\!92\)\(q^{52} + \)\(35\!\cdots\!72\)\(q^{53} + \)\(80\!\cdots\!24\)\(q^{54} - \)\(40\!\cdots\!68\)\(q^{56} + \)\(13\!\cdots\!60\)\(q^{57} - \)\(17\!\cdots\!68\)\(q^{58} + \)\(27\!\cdots\!40\)\(q^{60} - \)\(72\!\cdots\!40\)\(q^{61} - \)\(83\!\cdots\!60\)\(q^{62} - \)\(51\!\cdots\!68\)\(q^{64} + \)\(14\!\cdots\!40\)\(q^{65} + \)\(47\!\cdots\!60\)\(q^{66} - \)\(20\!\cdots\!48\)\(q^{68} + \)\(49\!\cdots\!84\)\(q^{69} + \)\(24\!\cdots\!00\)\(q^{70} + \)\(13\!\cdots\!24\)\(q^{72} + \)\(67\!\cdots\!52\)\(q^{73} - \)\(50\!\cdots\!68\)\(q^{74} - \)\(72\!\cdots\!20\)\(q^{76} - \)\(13\!\cdots\!80\)\(q^{77} + \)\(26\!\cdots\!60\)\(q^{78} - \)\(14\!\cdots\!80\)\(q^{80} + \)\(50\!\cdots\!84\)\(q^{81} + \)\(38\!\cdots\!08\)\(q^{82} + \)\(21\!\cdots\!64\)\(q^{84} - \)\(10\!\cdots\!60\)\(q^{85} - \)\(10\!\cdots\!28\)\(q^{86} + \)\(61\!\cdots\!00\)\(q^{88} + \)\(29\!\cdots\!44\)\(q^{89} + \)\(52\!\cdots\!80\)\(q^{90} - \)\(31\!\cdots\!40\)\(q^{92} - \)\(90\!\cdots\!00\)\(q^{93} - \)\(12\!\cdots\!68\)\(q^{94} - \)\(36\!\cdots\!68\)\(q^{96} + \)\(27\!\cdots\!48\)\(q^{97} + \)\(18\!\cdots\!96\)\(q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{20} - 10 x^{19} + 5706153082570973323 x^{18} - 51355377743138759622 x^{17} + \)\(13\!\cdots\!54\)\( x^{16} - \)\(10\!\cdots\!92\)\( x^{15} + \)\(16\!\cdots\!26\)\( x^{14} - \)\(11\!\cdots\!24\)\( x^{13} + \)\(12\!\cdots\!57\)\( x^{12} - \)\(73\!\cdots\!94\)\( x^{11} + \)\(55\!\cdots\!15\)\( x^{10} - \)\(27\!\cdots\!66\)\( x^{9} + \)\(15\!\cdots\!08\)\( x^{8} - \)\(60\!\cdots\!68\)\( x^{7} + \)\(24\!\cdots\!96\)\( x^{6} - \)\(74\!\cdots\!44\)\( x^{5} + \)\(21\!\cdots\!84\)\( x^{4} - \)\(43\!\cdots\!04\)\( x^{3} + \)\(79\!\cdots\!60\)\( x^{2} - \)\(79\!\cdots\!00\)\( x + \)\(58\!\cdots\!00\)\(\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(\)\(35\!\cdots\!75\)\( \nu^{19} + \)\(97\!\cdots\!52\)\( \nu^{18} + \)\(19\!\cdots\!57\)\( \nu^{17} + \)\(55\!\cdots\!00\)\( \nu^{16} + \)\(41\!\cdots\!58\)\( \nu^{15} + \)\(13\!\cdots\!16\)\( \nu^{14} + \)\(46\!\cdots\!70\)\( \nu^{13} + \)\(16\!\cdots\!48\)\( \nu^{12} + \)\(29\!\cdots\!95\)\( \nu^{11} + \)\(11\!\cdots\!12\)\( \nu^{10} + \)\(11\!\cdots\!69\)\( \nu^{9} + \)\(46\!\cdots\!56\)\( \nu^{8} + \)\(22\!\cdots\!08\)\( \nu^{7} + \)\(10\!\cdots\!40\)\( \nu^{6} + \)\(25\!\cdots\!92\)\( \nu^{5} + \)\(12\!\cdots\!24\)\( \nu^{4} + \)\(12\!\cdots\!48\)\( \nu^{3} + \)\(58\!\cdots\!80\)\( \nu^{2} + \)\(20\!\cdots\!00\)\( \nu + \)\(44\!\cdots\!00\)\(\)\()/ \)\(25\!\cdots\!00\)\( \)
\(\beta_{2}\)\(=\)\((\)\(-\)\(35\!\cdots\!75\)\( \nu^{19} - \)\(97\!\cdots\!52\)\( \nu^{18} - \)\(19\!\cdots\!57\)\( \nu^{17} - \)\(55\!\cdots\!00\)\( \nu^{16} - \)\(41\!\cdots\!58\)\( \nu^{15} - \)\(13\!\cdots\!16\)\( \nu^{14} - \)\(46\!\cdots\!70\)\( \nu^{13} - \)\(16\!\cdots\!48\)\( \nu^{12} - \)\(29\!\cdots\!95\)\( \nu^{11} - \)\(11\!\cdots\!12\)\( \nu^{10} - \)\(11\!\cdots\!69\)\( \nu^{9} - \)\(46\!\cdots\!56\)\( \nu^{8} - \)\(22\!\cdots\!08\)\( \nu^{7} - \)\(10\!\cdots\!40\)\( \nu^{6} - \)\(25\!\cdots\!92\)\( \nu^{5} - \)\(12\!\cdots\!24\)\( \nu^{4} - \)\(12\!\cdots\!48\)\( \nu^{3} - \)\(58\!\cdots\!80\)\( \nu^{2} + \)\(31\!\cdots\!00\)\( \nu - \)\(44\!\cdots\!00\)\(\)\()/ \)\(32\!\cdots\!00\)\( \)
\(\beta_{3}\)\(=\)\((\)\(-\)\(31\!\cdots\!19\)\( \nu^{19} + \)\(34\!\cdots\!68\)\( \nu^{18} - \)\(17\!\cdots\!73\)\( \nu^{17} + \)\(19\!\cdots\!24\)\( \nu^{16} - \)\(37\!\cdots\!14\)\( \nu^{15} + \)\(41\!\cdots\!36\)\( \nu^{14} - \)\(43\!\cdots\!86\)\( \nu^{13} + \)\(47\!\cdots\!88\)\( \nu^{12} - \)\(28\!\cdots\!19\)\( \nu^{11} + \)\(30\!\cdots\!64\)\( \nu^{10} - \)\(10\!\cdots\!73\)\( \nu^{9} + \)\(11\!\cdots\!40\)\( \nu^{8} - \)\(22\!\cdots\!92\)\( \nu^{7} + \)\(23\!\cdots\!16\)\( \nu^{6} - \)\(24\!\cdots\!16\)\( \nu^{5} + \)\(24\!\cdots\!08\)\( \nu^{4} - \)\(10\!\cdots\!32\)\( \nu^{3} + \)\(10\!\cdots\!80\)\( \nu^{2} - \)\(52\!\cdots\!00\)\( \nu - \)\(34\!\cdots\!00\)\(\)\()/ \)\(25\!\cdots\!00\)\( \)
\(\beta_{4}\)\(=\)\((\)\(\)\(71\!\cdots\!47\)\( \nu^{19} - \)\(14\!\cdots\!96\)\( \nu^{18} + \)\(39\!\cdots\!57\)\( \nu^{17} - \)\(70\!\cdots\!12\)\( \nu^{16} + \)\(86\!\cdots\!34\)\( \nu^{15} - \)\(13\!\cdots\!64\)\( \nu^{14} + \)\(98\!\cdots\!98\)\( \nu^{13} - \)\(11\!\cdots\!32\)\( \nu^{12} + \)\(64\!\cdots\!27\)\( \nu^{11} - \)\(44\!\cdots\!04\)\( \nu^{10} + \)\(24\!\cdots\!85\)\( \nu^{9} - \)\(15\!\cdots\!56\)\( \nu^{8} + \)\(51\!\cdots\!48\)\( \nu^{7} + \)\(35\!\cdots\!52\)\( \nu^{6} + \)\(56\!\cdots\!56\)\( \nu^{5} + \)\(86\!\cdots\!52\)\( \nu^{4} + \)\(24\!\cdots\!28\)\( \nu^{3} + \)\(60\!\cdots\!80\)\( \nu^{2} + \)\(10\!\cdots\!00\)\( \nu + \)\(25\!\cdots\!00\)\(\)\()/ \)\(35\!\cdots\!00\)\( \)
\(\beta_{5}\)\(=\)\((\)\(-\)\(53\!\cdots\!07\)\( \nu^{19} + \)\(42\!\cdots\!80\)\( \nu^{18} - \)\(30\!\cdots\!53\)\( \nu^{17} + \)\(22\!\cdots\!72\)\( \nu^{16} - \)\(69\!\cdots\!38\)\( \nu^{15} + \)\(49\!\cdots\!16\)\( \nu^{14} - \)\(84\!\cdots\!98\)\( \nu^{13} + \)\(54\!\cdots\!88\)\( \nu^{12} - \)\(58\!\cdots\!47\)\( \nu^{11} + \)\(34\!\cdots\!48\)\( \nu^{10} - \)\(23\!\cdots\!97\)\( \nu^{9} + \)\(12\!\cdots\!48\)\( \nu^{8} - \)\(53\!\cdots\!72\)\( \nu^{7} + \)\(24\!\cdots\!68\)\( \nu^{6} - \)\(62\!\cdots\!52\)\( \nu^{5} + \)\(25\!\cdots\!36\)\( \nu^{4} - \)\(28\!\cdots\!72\)\( \nu^{3} + \)\(99\!\cdots\!80\)\( \nu^{2} - \)\(19\!\cdots\!00\)\( \nu + \)\(52\!\cdots\!00\)\(\)\()/ \)\(62\!\cdots\!00\)\( \)
\(\beta_{6}\)\(=\)\((\)\(\)\(40\!\cdots\!43\)\( \nu^{19} - \)\(84\!\cdots\!80\)\( \nu^{18} + \)\(21\!\cdots\!37\)\( \nu^{17} - \)\(47\!\cdots\!28\)\( \nu^{16} + \)\(47\!\cdots\!22\)\( \nu^{15} - \)\(10\!\cdots\!64\)\( \nu^{14} + \)\(54\!\cdots\!02\)\( \nu^{13} - \)\(13\!\cdots\!52\)\( \nu^{12} + \)\(35\!\cdots\!03\)\( \nu^{11} - \)\(92\!\cdots\!12\)\( \nu^{10} + \)\(13\!\cdots\!33\)\( \nu^{9} - \)\(37\!\cdots\!32\)\( \nu^{8} + \)\(27\!\cdots\!88\)\( \nu^{7} - \)\(85\!\cdots\!32\)\( \nu^{6} + \)\(30\!\cdots\!88\)\( \nu^{5} - \)\(99\!\cdots\!84\)\( \nu^{4} + \)\(14\!\cdots\!88\)\( \nu^{3} - \)\(44\!\cdots\!20\)\( \nu^{2} + \)\(14\!\cdots\!00\)\( \nu - \)\(27\!\cdots\!00\)\(\)\()/ \)\(25\!\cdots\!00\)\( \)
\(\beta_{7}\)\(=\)\((\)\(\)\(30\!\cdots\!07\)\( \nu^{19} + \)\(19\!\cdots\!40\)\( \nu^{18} + \)\(16\!\cdots\!73\)\( \nu^{17} + \)\(10\!\cdots\!28\)\( \nu^{16} + \)\(36\!\cdots\!18\)\( \nu^{15} + \)\(22\!\cdots\!44\)\( \nu^{14} + \)\(41\!\cdots\!98\)\( \nu^{13} + \)\(24\!\cdots\!92\)\( \nu^{12} + \)\(26\!\cdots\!47\)\( \nu^{11} + \)\(15\!\cdots\!72\)\( \nu^{10} + \)\(98\!\cdots\!37\)\( \nu^{9} + \)\(52\!\cdots\!12\)\( \nu^{8} + \)\(20\!\cdots\!52\)\( \nu^{7} + \)\(99\!\cdots\!32\)\( \nu^{6} + \)\(22\!\cdots\!72\)\( \nu^{5} + \)\(95\!\cdots\!04\)\( \nu^{4} + \)\(96\!\cdots\!52\)\( \nu^{3} + \)\(35\!\cdots\!20\)\( \nu^{2} + \)\(26\!\cdots\!00\)\( \nu + \)\(44\!\cdots\!00\)\(\)\()/ \)\(15\!\cdots\!00\)\( \)
\(\beta_{8}\)\(=\)\((\)\(\)\(27\!\cdots\!15\)\( \nu^{19} - \)\(32\!\cdots\!24\)\( \nu^{18} + \)\(14\!\cdots\!01\)\( \nu^{17} - \)\(17\!\cdots\!40\)\( \nu^{16} + \)\(32\!\cdots\!14\)\( \nu^{15} - \)\(38\!\cdots\!12\)\( \nu^{14} + \)\(36\!\cdots\!70\)\( \nu^{13} - \)\(43\!\cdots\!36\)\( \nu^{12} + \)\(22\!\cdots\!75\)\( \nu^{11} - \)\(27\!\cdots\!04\)\( \nu^{10} + \)\(81\!\cdots\!37\)\( \nu^{9} - \)\(10\!\cdots\!32\)\( \nu^{8} + \)\(16\!\cdots\!44\)\( \nu^{7} - \)\(21\!\cdots\!40\)\( \nu^{6} + \)\(15\!\cdots\!36\)\( \nu^{5} - \)\(23\!\cdots\!08\)\( \nu^{4} + \)\(53\!\cdots\!64\)\( \nu^{3} - \)\(96\!\cdots\!60\)\( \nu^{2} - \)\(76\!\cdots\!00\)\( \nu - \)\(68\!\cdots\!00\)\(\)\()/ \)\(25\!\cdots\!00\)\( \)
\(\beta_{9}\)\(=\)\((\)\(\)\(60\!\cdots\!09\)\( \nu^{19} - \)\(12\!\cdots\!64\)\( \nu^{18} + \)\(33\!\cdots\!47\)\( \nu^{17} - \)\(66\!\cdots\!64\)\( \nu^{16} + \)\(74\!\cdots\!90\)\( \nu^{15} - \)\(14\!\cdots\!24\)\( \nu^{14} + \)\(86\!\cdots\!86\)\( \nu^{13} - \)\(15\!\cdots\!52\)\( \nu^{12} + \)\(56\!\cdots\!49\)\( \nu^{11} - \)\(93\!\cdots\!00\)\( \nu^{10} + \)\(21\!\cdots\!51\)\( \nu^{9} - \)\(31\!\cdots\!88\)\( \nu^{8} + \)\(46\!\cdots\!48\)\( \nu^{7} - \)\(59\!\cdots\!96\)\( \nu^{6} + \)\(51\!\cdots\!40\)\( \nu^{5} - \)\(54\!\cdots\!80\)\( \nu^{4} + \)\(22\!\cdots\!68\)\( \nu^{3} - \)\(18\!\cdots\!20\)\( \nu^{2} + \)\(20\!\cdots\!00\)\( \nu + \)\(28\!\cdots\!00\)\(\)\()/ \)\(17\!\cdots\!00\)\( \)
\(\beta_{10}\)\(=\)\((\)\(-\)\(17\!\cdots\!19\)\( \nu^{19} + \)\(34\!\cdots\!20\)\( \nu^{18} - \)\(95\!\cdots\!41\)\( \nu^{17} + \)\(18\!\cdots\!24\)\( \nu^{16} - \)\(20\!\cdots\!06\)\( \nu^{15} + \)\(40\!\cdots\!52\)\( \nu^{14} - \)\(23\!\cdots\!66\)\( \nu^{13} + \)\(46\!\cdots\!36\)\( \nu^{12} - \)\(14\!\cdots\!99\)\( \nu^{11} + \)\(29\!\cdots\!76\)\( \nu^{10} - \)\(53\!\cdots\!29\)\( \nu^{9} + \)\(10\!\cdots\!96\)\( \nu^{8} - \)\(10\!\cdots\!84\)\( \nu^{7} + \)\(22\!\cdots\!56\)\( \nu^{6} - \)\(95\!\cdots\!24\)\( \nu^{5} + \)\(24\!\cdots\!32\)\( \nu^{4} - \)\(18\!\cdots\!84\)\( \nu^{3} + \)\(10\!\cdots\!60\)\( \nu^{2} + \)\(13\!\cdots\!00\)\( \nu + \)\(80\!\cdots\!00\)\(\)\()/ \)\(27\!\cdots\!00\)\( \)
\(\beta_{11}\)\(=\)\((\)\(-\)\(43\!\cdots\!16\)\( \nu^{19} + \)\(15\!\cdots\!89\)\( \nu^{18} - \)\(23\!\cdots\!55\)\( \nu^{17} + \)\(92\!\cdots\!36\)\( \nu^{16} - \)\(51\!\cdots\!48\)\( \nu^{15} + \)\(22\!\cdots\!50\)\( \nu^{14} - \)\(57\!\cdots\!34\)\( \nu^{13} + \)\(29\!\cdots\!20\)\( \nu^{12} - \)\(36\!\cdots\!96\)\( \nu^{11} + \)\(22\!\cdots\!93\)\( \nu^{10} - \)\(13\!\cdots\!83\)\( \nu^{9} + \)\(10\!\cdots\!96\)\( \nu^{8} - \)\(24\!\cdots\!40\)\( \nu^{7} + \)\(25\!\cdots\!64\)\( \nu^{6} - \)\(20\!\cdots\!72\)\( \nu^{5} + \)\(32\!\cdots\!56\)\( \nu^{4} + \)\(79\!\cdots\!40\)\( \nu^{3} + \)\(14\!\cdots\!00\)\( \nu^{2} + \)\(79\!\cdots\!00\)\( \nu - \)\(56\!\cdots\!00\)\(\)\()/ \)\(15\!\cdots\!00\)\( \)
\(\beta_{12}\)\(=\)\((\)\(-\)\(99\!\cdots\!17\)\( \nu^{19} + \)\(16\!\cdots\!25\)\( \nu^{18} - \)\(54\!\cdots\!98\)\( \nu^{17} + \)\(89\!\cdots\!32\)\( \nu^{16} - \)\(11\!\cdots\!98\)\( \nu^{15} + \)\(19\!\cdots\!06\)\( \nu^{14} - \)\(13\!\cdots\!88\)\( \nu^{13} + \)\(22\!\cdots\!08\)\( \nu^{12} - \)\(85\!\cdots\!57\)\( \nu^{11} + \)\(14\!\cdots\!33\)\( \nu^{10} - \)\(31\!\cdots\!42\)\( \nu^{9} + \)\(55\!\cdots\!48\)\( \nu^{8} - \)\(64\!\cdots\!52\)\( \nu^{7} + \)\(11\!\cdots\!08\)\( \nu^{6} - \)\(69\!\cdots\!92\)\( \nu^{5} + \)\(12\!\cdots\!56\)\( \nu^{4} - \)\(32\!\cdots\!52\)\( \nu^{3} + \)\(55\!\cdots\!80\)\( \nu^{2} - \)\(51\!\cdots\!00\)\( \nu + \)\(38\!\cdots\!00\)\(\)\()/ \)\(26\!\cdots\!00\)\( \)
\(\beta_{13}\)\(=\)\((\)\(-\)\(33\!\cdots\!99\)\( \nu^{19} + \)\(14\!\cdots\!72\)\( \nu^{18} - \)\(18\!\cdots\!29\)\( \nu^{17} + \)\(74\!\cdots\!04\)\( \nu^{16} - \)\(40\!\cdots\!18\)\( \nu^{15} + \)\(15\!\cdots\!08\)\( \nu^{14} - \)\(46\!\cdots\!66\)\( \nu^{13} + \)\(16\!\cdots\!04\)\( \nu^{12} - \)\(30\!\cdots\!59\)\( \nu^{11} + \)\(94\!\cdots\!08\)\( \nu^{10} - \)\(11\!\cdots\!65\)\( \nu^{9} + \)\(29\!\cdots\!72\)\( \nu^{8} - \)\(24\!\cdots\!56\)\( \nu^{7} + \)\(47\!\cdots\!16\)\( \nu^{6} - \)\(27\!\cdots\!12\)\( \nu^{5} + \)\(32\!\cdots\!96\)\( \nu^{4} - \)\(12\!\cdots\!16\)\( \nu^{3} + \)\(56\!\cdots\!40\)\( \nu^{2} - \)\(97\!\cdots\!00\)\( \nu - \)\(31\!\cdots\!00\)\(\)\()/ \)\(25\!\cdots\!00\)\( \)
\(\beta_{14}\)\(=\)\((\)\(-\)\(20\!\cdots\!87\)\( \nu^{19} - \)\(20\!\cdots\!80\)\( \nu^{18} - \)\(10\!\cdots\!33\)\( \nu^{17} - \)\(11\!\cdots\!48\)\( \nu^{16} - \)\(24\!\cdots\!98\)\( \nu^{15} - \)\(25\!\cdots\!24\)\( \nu^{14} - \)\(27\!\cdots\!18\)\( \nu^{13} - \)\(30\!\cdots\!32\)\( \nu^{12} - \)\(18\!\cdots\!27\)\( \nu^{11} - \)\(20\!\cdots\!92\)\( \nu^{10} - \)\(67\!\cdots\!97\)\( \nu^{9} - \)\(78\!\cdots\!12\)\( \nu^{8} - \)\(14\!\cdots\!92\)\( \nu^{7} - \)\(17\!\cdots\!12\)\( \nu^{6} - \)\(15\!\cdots\!92\)\( \nu^{5} - \)\(18\!\cdots\!44\)\( \nu^{4} - \)\(67\!\cdots\!92\)\( \nu^{3} - \)\(78\!\cdots\!20\)\( \nu^{2} - \)\(36\!\cdots\!00\)\( \nu + \)\(59\!\cdots\!00\)\(\)\()/ \)\(12\!\cdots\!00\)\( \)
\(\beta_{15}\)\(=\)\((\)\(\)\(29\!\cdots\!45\)\( \nu^{19} + \)\(18\!\cdots\!96\)\( \nu^{18} + \)\(17\!\cdots\!91\)\( \nu^{17} + \)\(10\!\cdots\!80\)\( \nu^{16} + \)\(44\!\cdots\!14\)\( \nu^{15} + \)\(23\!\cdots\!08\)\( \nu^{14} + \)\(60\!\cdots\!90\)\( \nu^{13} + \)\(27\!\cdots\!24\)\( \nu^{12} + \)\(46\!\cdots\!05\)\( \nu^{11} + \)\(17\!\cdots\!96\)\( \nu^{10} + \)\(21\!\cdots\!07\)\( \nu^{9} + \)\(68\!\cdots\!08\)\( \nu^{8} + \)\(54\!\cdots\!04\)\( \nu^{7} + \)\(14\!\cdots\!40\)\( \nu^{6} + \)\(69\!\cdots\!36\)\( \nu^{5} + \)\(16\!\cdots\!92\)\( \nu^{4} + \)\(33\!\cdots\!24\)\( \nu^{3} + \)\(70\!\cdots\!40\)\( \nu^{2} - \)\(53\!\cdots\!00\)\( \nu + \)\(40\!\cdots\!00\)\(\)\()/ \)\(25\!\cdots\!00\)\( \)
\(\beta_{16}\)\(=\)\((\)\(-\)\(31\!\cdots\!17\)\( \nu^{19} - \)\(10\!\cdots\!64\)\( \nu^{18} - \)\(17\!\cdots\!47\)\( \nu^{17} - \)\(57\!\cdots\!68\)\( \nu^{16} - \)\(39\!\cdots\!54\)\( \nu^{15} - \)\(12\!\cdots\!56\)\( \nu^{14} - \)\(46\!\cdots\!78\)\( \nu^{13} - \)\(14\!\cdots\!28\)\( \nu^{12} - \)\(31\!\cdots\!97\)\( \nu^{11} - \)\(90\!\cdots\!76\)\( \nu^{10} - \)\(12\!\cdots\!75\)\( \nu^{9} - \)\(33\!\cdots\!44\)\( \nu^{8} - \)\(26\!\cdots\!08\)\( \nu^{7} - \)\(67\!\cdots\!72\)\( \nu^{6} - \)\(30\!\cdots\!36\)\( \nu^{5} - \)\(70\!\cdots\!12\)\( \nu^{4} - \)\(13\!\cdots\!88\)\( \nu^{3} - \)\(28\!\cdots\!80\)\( \nu^{2} - \)\(18\!\cdots\!00\)\( \nu - \)\(19\!\cdots\!00\)\(\)\()/ \)\(83\!\cdots\!00\)\( \)
\(\beta_{17}\)\(=\)\((\)\(-\)\(20\!\cdots\!55\)\( \nu^{19} + \)\(12\!\cdots\!36\)\( \nu^{18} - \)\(11\!\cdots\!69\)\( \nu^{17} + \)\(66\!\cdots\!80\)\( \nu^{16} - \)\(24\!\cdots\!26\)\( \nu^{15} + \)\(14\!\cdots\!28\)\( \nu^{14} - \)\(28\!\cdots\!10\)\( \nu^{13} + \)\(16\!\cdots\!84\)\( \nu^{12} - \)\(18\!\cdots\!95\)\( \nu^{11} + \)\(10\!\cdots\!36\)\( \nu^{10} - \)\(70\!\cdots\!13\)\( \nu^{9} + \)\(40\!\cdots\!28\)\( \nu^{8} - \)\(14\!\cdots\!36\)\( \nu^{7} + \)\(86\!\cdots\!40\)\( \nu^{6} - \)\(15\!\cdots\!24\)\( \nu^{5} + \)\(93\!\cdots\!72\)\( \nu^{4} - \)\(63\!\cdots\!16\)\( \nu^{3} + \)\(40\!\cdots\!40\)\( \nu^{2} + \)\(42\!\cdots\!00\)\( \nu + \)\(32\!\cdots\!00\)\(\)\()/ \)\(25\!\cdots\!00\)\( \)
\(\beta_{18}\)\(=\)\((\)\(-\)\(49\!\cdots\!05\)\( \nu^{19} - \)\(69\!\cdots\!88\)\( \nu^{18} - \)\(27\!\cdots\!03\)\( \nu^{17} - \)\(37\!\cdots\!20\)\( \nu^{16} - \)\(59\!\cdots\!22\)\( \nu^{15} - \)\(82\!\cdots\!64\)\( \nu^{14} - \)\(67\!\cdots\!50\)\( \nu^{13} - \)\(94\!\cdots\!92\)\( \nu^{12} - \)\(42\!\cdots\!85\)\( \nu^{11} - \)\(60\!\cdots\!08\)\( \nu^{10} - \)\(15\!\cdots\!91\)\( \nu^{9} - \)\(22\!\cdots\!44\)\( \nu^{8} - \)\(30\!\cdots\!32\)\( \nu^{7} - \)\(46\!\cdots\!40\)\( \nu^{6} - \)\(27\!\cdots\!28\)\( \nu^{5} - \)\(49\!\cdots\!16\)\( \nu^{4} - \)\(48\!\cdots\!92\)\( \nu^{3} - \)\(21\!\cdots\!20\)\( \nu^{2} + \)\(43\!\cdots\!00\)\( \nu - \)\(16\!\cdots\!00\)\(\)\()/ \)\(20\!\cdots\!00\)\( \)
\(\beta_{19}\)\(=\)\((\)\(\)\(10\!\cdots\!47\)\( \nu^{19} + \)\(86\!\cdots\!68\)\( \nu^{18} + \)\(57\!\cdots\!81\)\( \nu^{17} + \)\(47\!\cdots\!88\)\( \nu^{16} + \)\(12\!\cdots\!90\)\( \nu^{15} + \)\(10\!\cdots\!48\)\( \nu^{14} + \)\(14\!\cdots\!38\)\( \nu^{13} + \)\(11\!\cdots\!04\)\( \nu^{12} + \)\(89\!\cdots\!67\)\( \nu^{11} + \)\(75\!\cdots\!80\)\( \nu^{10} + \)\(32\!\cdots\!93\)\( \nu^{9} + \)\(27\!\cdots\!36\)\( \nu^{8} + \)\(65\!\cdots\!04\)\( \nu^{7} + \)\(57\!\cdots\!32\)\( \nu^{6} + \)\(66\!\cdots\!00\)\( \nu^{5} + \)\(62\!\cdots\!20\)\( \nu^{4} + \)\(23\!\cdots\!64\)\( \nu^{3} + \)\(26\!\cdots\!40\)\( \nu^{2} - \)\(23\!\cdots\!00\)\( \nu + \)\(19\!\cdots\!00\)\(\)\()/ \)\(27\!\cdots\!00\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} + 78 \beta_{1} - 8\)\()/16\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{9} + 3 \beta_{7} + 15 \beta_{6} + 640 \beta_{5} - 13058 \beta_{4} + 2493609 \beta_{3} + 4873107 \beta_{2} - 2142931496170 \beta_{1} - 146077518485231568615\)\()/256\)
\(\nu^{3}\)\(=\)\((\)\(-238808 \beta_{19} - 128096 \beta_{18} - 118986 \beta_{17} + 517042 \beta_{16} - 326360 \beta_{15} - 14307087 \beta_{14} + 296101 \beta_{13} + 11015838 \beta_{12} - 475944 \beta_{11} - 539525068 \beta_{10} - 137411259 \beta_{9} - 75099874601 \beta_{8} + 114983343577 \beta_{7} + 1536421525778 \beta_{6} + 13437896837538 \beta_{5} + 48116750179769 \beta_{4} - 42954460093662206 \beta_{3} - 263743856434516897555 \beta_{2} + 74897996911503621604485 \beta_{1} - 18379979481416517865968\)\()/4096\)
\(\nu^{4}\)\(=\)\((\)\(633935314689637 \beta_{19} - 153032523064724 \beta_{18} - 386151583395408 \beta_{17} + 5146721729535472 \beta_{16} - 5221760 \beta_{15} + 477317372737629480 \beta_{14} - 32753184326444414 \beta_{13} - 24043155345296595 \beta_{12} - 10792969183302270 \beta_{11} + 760342838159733674 \beta_{10} - 196044088321227835164 \beta_{9} - 135214822978226807360 \beta_{8} - 979632149568717165914 \beta_{7} - 12062801819498938666339 \beta_{6} - 232041847312708775004252 \beta_{5} + 809779576513881588315017 \beta_{4} - 1155692270402890756529571368 \beta_{3} - 2934655760078670939597772590 \beta_{2} + 1282167060374461276165090229366823 \beta_{1} + 19247384244858229046223201539605136998217\)\()/32768\)
\(\nu^{5}\)\(=\)\((\)\(\)\(52\!\cdots\!80\)\( \beta_{19} + \)\(29\!\cdots\!60\)\( \beta_{18} + \)\(32\!\cdots\!30\)\( \beta_{17} - \)\(15\!\cdots\!10\)\( \beta_{16} + \)\(46\!\cdots\!20\)\( \beta_{15} + \)\(63\!\cdots\!75\)\( \beta_{14} - \)\(39\!\cdots\!90\)\( \beta_{13} - \)\(28\!\cdots\!50\)\( \beta_{12} + \)\(13\!\cdots\!00\)\( \beta_{11} + \)\(12\!\cdots\!40\)\( \beta_{10} + \)\(34\!\cdots\!70\)\( \beta_{9} + \)\(17\!\cdots\!50\)\( \beta_{8} + \)\(11\!\cdots\!63\)\( \beta_{7} - \)\(55\!\cdots\!64\)\( \beta_{6} - \)\(37\!\cdots\!44\)\( \beta_{5} - \)\(13\!\cdots\!86\)\( \beta_{4} + \)\(12\!\cdots\!22\)\( \beta_{3} + \)\(42\!\cdots\!61\)\( \beta_{2} - \)\(31\!\cdots\!86\)\( \beta_{1} + \)\(70\!\cdots\!66\)\(\)\()/524288\)
\(\nu^{6}\)\(=\)\((\)\(-\)\(41\!\cdots\!05\)\( \beta_{19} + \)\(13\!\cdots\!92\)\( \beta_{18} + \)\(26\!\cdots\!68\)\( \beta_{17} - \)\(47\!\cdots\!08\)\( \beta_{16} + \)\(27\!\cdots\!20\)\( \beta_{15} - \)\(28\!\cdots\!16\)\( \beta_{14} + \)\(29\!\cdots\!18\)\( \beta_{13} + \)\(14\!\cdots\!34\)\( \beta_{12} + \)\(68\!\cdots\!89\)\( \beta_{11} - \)\(61\!\cdots\!19\)\( \beta_{10} + \)\(91\!\cdots\!04\)\( \beta_{9} + \)\(90\!\cdots\!63\)\( \beta_{8} + \)\(65\!\cdots\!45\)\( \beta_{7} + \)\(81\!\cdots\!33\)\( \beta_{6} + \)\(15\!\cdots\!37\)\( \beta_{5} + \)\(10\!\cdots\!48\)\( \beta_{4} + \)\(68\!\cdots\!78\)\( \beta_{3} + \)\(18\!\cdots\!21\)\( \beta_{2} - \)\(82\!\cdots\!32\)\( \beta_{1} - \)\(78\!\cdots\!63\)\(\)\()/1048576\)
\(\nu^{7}\)\(=\)\((\)\(-\)\(25\!\cdots\!24\)\( \beta_{19} - \)\(15\!\cdots\!24\)\( \beta_{18} - \)\(18\!\cdots\!83\)\( \beta_{17} + \)\(84\!\cdots\!33\)\( \beta_{16} - \)\(11\!\cdots\!20\)\( \beta_{15} - \)\(36\!\cdots\!11\)\( \beta_{14} + \)\(32\!\cdots\!21\)\( \beta_{13} + \)\(16\!\cdots\!67\)\( \beta_{12} - \)\(75\!\cdots\!40\)\( \beta_{11} - \)\(60\!\cdots\!14\)\( \beta_{10} - \)\(18\!\cdots\!16\)\( \beta_{9} - \)\(83\!\cdots\!83\)\( \beta_{8} - \)\(14\!\cdots\!40\)\( \beta_{7} + \)\(36\!\cdots\!26\)\( \beta_{6} + \)\(20\!\cdots\!36\)\( \beta_{5} + \)\(70\!\cdots\!61\)\( \beta_{4} - \)\(62\!\cdots\!91\)\( \beta_{3} - \)\(18\!\cdots\!44\)\( \beta_{2} + \)\(21\!\cdots\!64\)\( \beta_{1} - \)\(46\!\cdots\!30\)\(\)\()/16777216\)
\(\nu^{8}\)\(=\)\((\)\(\)\(21\!\cdots\!15\)\( \beta_{19} - \)\(75\!\cdots\!44\)\( \beta_{18} - \)\(12\!\cdots\!04\)\( \beta_{17} + \)\(30\!\cdots\!08\)\( \beta_{16} - \)\(89\!\cdots\!40\)\( \beta_{15} + \)\(15\!\cdots\!04\)\( \beta_{14} - \)\(19\!\cdots\!18\)\( \beta_{13} - \)\(66\!\cdots\!75\)\( \beta_{12} - \)\(36\!\cdots\!56\)\( \beta_{11} + \)\(34\!\cdots\!68\)\( \beta_{10} - \)\(42\!\cdots\!48\)\( \beta_{9} - \)\(47\!\cdots\!82\)\( \beta_{8} - \)\(38\!\cdots\!72\)\( \beta_{7} - \)\(42\!\cdots\!75\)\( \beta_{6} - \)\(86\!\cdots\!94\)\( \beta_{5} - \)\(10\!\cdots\!71\)\( \beta_{4} - \)\(37\!\cdots\!58\)\( \beta_{3} - \)\(99\!\cdots\!74\)\( \beta_{2} + \)\(43\!\cdots\!27\)\( \beta_{1} + \)\(34\!\cdots\!75\)\(\)\()/33554432\)
\(\nu^{9}\)\(=\)\((\)\(\)\(12\!\cdots\!60\)\( \beta_{19} + \)\(75\!\cdots\!40\)\( \beta_{18} + \)\(10\!\cdots\!40\)\( \beta_{17} - \)\(40\!\cdots\!68\)\( \beta_{16} + \)\(88\!\cdots\!60\)\( \beta_{15} + \)\(17\!\cdots\!73\)\( \beta_{14} - \)\(21\!\cdots\!92\)\( \beta_{13} - \)\(85\!\cdots\!08\)\( \beta_{12} + \)\(40\!\cdots\!00\)\( \beta_{11} + \)\(29\!\cdots\!04\)\( \beta_{10} + \)\(89\!\cdots\!70\)\( \beta_{9} + \)\(40\!\cdots\!20\)\( \beta_{8} + \)\(77\!\cdots\!43\)\( \beta_{7} - \)\(20\!\cdots\!36\)\( \beta_{6} - \)\(99\!\cdots\!72\)\( \beta_{5} - \)\(31\!\cdots\!84\)\( \beta_{4} + \)\(28\!\cdots\!72\)\( \beta_{3} + \)\(87\!\cdots\!49\)\( \beta_{2} - \)\(12\!\cdots\!46\)\( \beta_{1} + \)\(27\!\cdots\!74\)\(\)\()/ 536870912 \)
\(\nu^{10}\)\(=\)\((\)\(-\)\(10\!\cdots\!60\)\( \beta_{19} + \)\(36\!\cdots\!44\)\( \beta_{18} + \)\(52\!\cdots\!76\)\( \beta_{17} - \)\(17\!\cdots\!48\)\( \beta_{16} + \)\(88\!\cdots\!60\)\( \beta_{15} - \)\(77\!\cdots\!66\)\( \beta_{14} + \)\(11\!\cdots\!08\)\( \beta_{13} + \)\(25\!\cdots\!16\)\( \beta_{12} + \)\(17\!\cdots\!48\)\( \beta_{11} - \)\(15\!\cdots\!92\)\( \beta_{10} + \)\(20\!\cdots\!88\)\( \beta_{9} + \)\(22\!\cdots\!88\)\( \beta_{8} + \)\(21\!\cdots\!10\)\( \beta_{7} + \)\(19\!\cdots\!13\)\( \beta_{6} + \)\(47\!\cdots\!64\)\( \beta_{5} + \)\(75\!\cdots\!66\)\( \beta_{4} + \)\(19\!\cdots\!95\)\( \beta_{3} + \)\(48\!\cdots\!65\)\( \beta_{2} - \)\(21\!\cdots\!14\)\( \beta_{1} - \)\(15\!\cdots\!68\)\(\)\()/ 1073741824 \)
\(\nu^{11}\)\(=\)\((\)\(-\)\(18\!\cdots\!68\)\( \beta_{19} - \)\(11\!\cdots\!88\)\( \beta_{18} - \)\(17\!\cdots\!48\)\( \beta_{17} + \)\(58\!\cdots\!04\)\( \beta_{16} + \)\(40\!\cdots\!80\)\( \beta_{15} - \)\(24\!\cdots\!29\)\( \beta_{14} + \)\(38\!\cdots\!44\)\( \beta_{13} + \)\(13\!\cdots\!76\)\( \beta_{12} - \)\(68\!\cdots\!32\)\( \beta_{11} - \)\(45\!\cdots\!48\)\( \beta_{10} - \)\(13\!\cdots\!14\)\( \beta_{9} - \)\(60\!\cdots\!00\)\( \beta_{8} - \)\(10\!\cdots\!99\)\( \beta_{7} + \)\(35\!\cdots\!40\)\( \beta_{6} + \)\(14\!\cdots\!92\)\( \beta_{5} + \)\(42\!\cdots\!44\)\( \beta_{4} - \)\(38\!\cdots\!64\)\( \beta_{3} - \)\(12\!\cdots\!09\)\( \beta_{2} + \)\(21\!\cdots\!78\)\( \beta_{1} - \)\(47\!\cdots\!74\)\(\)\()/ 536870912 \)
\(\nu^{12}\)\(=\)\((\)\(\)\(16\!\cdots\!68\)\( \beta_{19} - \)\(50\!\cdots\!08\)\( \beta_{18} - \)\(57\!\cdots\!32\)\( \beta_{17} + \)\(30\!\cdots\!20\)\( \beta_{16} + \)\(48\!\cdots\!00\)\( \beta_{15} + \)\(12\!\cdots\!78\)\( \beta_{14} - \)\(19\!\cdots\!08\)\( \beta_{13} - \)\(24\!\cdots\!16\)\( \beta_{12} - \)\(27\!\cdots\!96\)\( \beta_{11} + \)\(21\!\cdots\!72\)\( \beta_{10} - \)\(30\!\cdots\!00\)\( \beta_{9} - \)\(33\!\cdots\!72\)\( \beta_{8} - \)\(35\!\cdots\!10\)\( \beta_{7} - \)\(27\!\cdots\!29\)\( \beta_{6} - \)\(78\!\cdots\!96\)\( \beta_{5} - \)\(14\!\cdots\!86\)\( \beta_{4} - \)\(31\!\cdots\!83\)\( \beta_{3} - \)\(70\!\cdots\!81\)\( \beta_{2} + \)\(30\!\cdots\!86\)\( \beta_{1} + \)\(23\!\cdots\!64\)\(\)\()/ 1073741824 \)
\(\nu^{13}\)\(=\)\((\)\(\)\(13\!\cdots\!68\)\( \beta_{19} + \)\(90\!\cdots\!72\)\( \beta_{18} + \)\(14\!\cdots\!52\)\( \beta_{17} - \)\(41\!\cdots\!36\)\( \beta_{16} - \)\(64\!\cdots\!40\)\( \beta_{15} + \)\(16\!\cdots\!27\)\( \beta_{14} - \)\(34\!\cdots\!92\)\( \beta_{13} - \)\(11\!\cdots\!60\)\( \beta_{12} + \)\(56\!\cdots\!28\)\( \beta_{11} + \)\(34\!\cdots\!12\)\( \beta_{10} + \)\(94\!\cdots\!62\)\( \beta_{9} + \)\(45\!\cdots\!56\)\( \beta_{8} + \)\(52\!\cdots\!37\)\( \beta_{7} - \)\(28\!\cdots\!24\)\( \beta_{6} - \)\(10\!\cdots\!56\)\( \beta_{5} - \)\(26\!\cdots\!80\)\( \beta_{4} + \)\(24\!\cdots\!44\)\( \beta_{3} + \)\(96\!\cdots\!91\)\( \beta_{2} - \)\(18\!\cdots\!62\)\( \beta_{1} + \)\(39\!\cdots\!02\)\(\)\()/ 268435456 \)
\(\nu^{14}\)\(=\)\((\)\(-\)\(12\!\cdots\!00\)\( \beta_{19} + \)\(33\!\cdots\!40\)\( \beta_{18} + \)\(25\!\cdots\!28\)\( \beta_{17} - \)\(25\!\cdots\!32\)\( \beta_{16} - \)\(90\!\cdots\!60\)\( \beta_{15} - \)\(99\!\cdots\!30\)\( \beta_{14} + \)\(16\!\cdots\!92\)\( \beta_{13} + \)\(73\!\cdots\!32\)\( \beta_{12} + \)\(20\!\cdots\!08\)\( \beta_{11} - \)\(13\!\cdots\!80\)\( \beta_{10} + \)\(22\!\cdots\!44\)\( \beta_{9} + \)\(24\!\cdots\!52\)\( \beta_{8} + \)\(29\!\cdots\!02\)\( \beta_{7} + \)\(18\!\cdots\!49\)\( \beta_{6} + \)\(64\!\cdots\!52\)\( \beta_{5} + \)\(12\!\cdots\!14\)\( \beta_{4} + \)\(25\!\cdots\!31\)\( \beta_{3} + \)\(50\!\cdots\!25\)\( \beta_{2} - \)\(22\!\cdots\!86\)\( \beta_{1} - \)\(17\!\cdots\!00\)\(\)\()/ 536870912 \)
\(\nu^{15}\)\(=\)\((\)\(-\)\(20\!\cdots\!04\)\( \beta_{19} - \)\(13\!\cdots\!68\)\( \beta_{18} - \)\(22\!\cdots\!16\)\( \beta_{17} + \)\(56\!\cdots\!08\)\( \beta_{16} + \)\(14\!\cdots\!80\)\( \beta_{15} - \)\(21\!\cdots\!39\)\( \beta_{14} + \)\(58\!\cdots\!76\)\( \beta_{13} + \)\(18\!\cdots\!32\)\( \beta_{12} - \)\(91\!\cdots\!20\)\( \beta_{11} - \)\(53\!\cdots\!24\)\( \beta_{10} - \)\(13\!\cdots\!98\)\( \beta_{9} - \)\(68\!\cdots\!24\)\( \beta_{8} - \)\(37\!\cdots\!65\)\( \beta_{7} + \)\(47\!\cdots\!12\)\( \beta_{6} + \)\(14\!\cdots\!72\)\( \beta_{5} + \)\(32\!\cdots\!88\)\( \beta_{4} - \)\(30\!\cdots\!44\)\( \beta_{3} - \)\(14\!\cdots\!95\)\( \beta_{2} + \)\(29\!\cdots\!78\)\( \beta_{1} - \)\(66\!\cdots\!42\)\(\)\()/ 268435456 \)
\(\nu^{16}\)\(=\)\((\)\(\)\(18\!\cdots\!24\)\( \beta_{19} - \)\(41\!\cdots\!16\)\( \beta_{18} - \)\(10\!\cdots\!52\)\( \beta_{17} + \)\(42\!\cdots\!00\)\( \beta_{16} + \)\(22\!\cdots\!40\)\( \beta_{15} + \)\(15\!\cdots\!42\)\( \beta_{14} - \)\(26\!\cdots\!24\)\( \beta_{13} + \)\(56\!\cdots\!16\)\( \beta_{12} - \)\(30\!\cdots\!80\)\( \beta_{11} + \)\(15\!\cdots\!60\)\( \beta_{10} - \)\(34\!\cdots\!16\)\( \beta_{9} - \)\(36\!\cdots\!40\)\( \beta_{8} - \)\(48\!\cdots\!78\)\( \beta_{7} - \)\(23\!\cdots\!63\)\( \beta_{6} - \)\(10\!\cdots\!40\)\( \beta_{5} - \)\(21\!\cdots\!46\)\( \beta_{4} - \)\(40\!\cdots\!21\)\( \beta_{3} - \)\(72\!\cdots\!91\)\( \beta_{2} + \)\(31\!\cdots\!30\)\( \beta_{1} + \)\(26\!\cdots\!24\)\(\)\()/ 536870912 \)
\(\nu^{17}\)\(=\)\((\)\(\)\(61\!\cdots\!28\)\( \beta_{19} + \)\(43\!\cdots\!52\)\( \beta_{18} + \)\(73\!\cdots\!68\)\( \beta_{17} - \)\(15\!\cdots\!20\)\( \beta_{16} - \)\(54\!\cdots\!60\)\( \beta_{15} + \)\(51\!\cdots\!03\)\( \beta_{14} - \)\(19\!\cdots\!68\)\( \beta_{13} - \)\(58\!\cdots\!56\)\( \beta_{12} + \)\(29\!\cdots\!04\)\( \beta_{11} + \)\(16\!\cdots\!12\)\( \beta_{10} + \)\(38\!\cdots\!38\)\( \beta_{9} + \)\(20\!\cdots\!28\)\( \beta_{8} - \)\(22\!\cdots\!51\)\( \beta_{7} - \)\(15\!\cdots\!44\)\( \beta_{6} - \)\(41\!\cdots\!96\)\( \beta_{5} - \)\(74\!\cdots\!60\)\( \beta_{4} + \)\(71\!\cdots\!64\)\( \beta_{3} + \)\(43\!\cdots\!95\)\( \beta_{2} - \)\(97\!\cdots\!14\)\( \beta_{1} + \)\(21\!\cdots\!14\)\(\)\()/ 536870912 \)
\(\nu^{18}\)\(=\)\((\)\(-\)\(55\!\cdots\!84\)\( \beta_{19} + \)\(10\!\cdots\!20\)\( \beta_{18} - \)\(49\!\cdots\!40\)\( \beta_{17} - \)\(13\!\cdots\!16\)\( \beta_{16} - \)\(97\!\cdots\!80\)\( \beta_{15} - \)\(50\!\cdots\!26\)\( \beta_{14} + \)\(87\!\cdots\!80\)\( \beta_{13} - \)\(66\!\cdots\!92\)\( \beta_{12} + \)\(90\!\cdots\!80\)\( \beta_{11} - \)\(32\!\cdots\!12\)\( \beta_{10} + \)\(10\!\cdots\!32\)\( \beta_{9} + \)\(10\!\cdots\!32\)\( \beta_{8} + \)\(15\!\cdots\!58\)\( \beta_{7} + \)\(60\!\cdots\!91\)\( \beta_{6} + \)\(33\!\cdots\!80\)\( \beta_{5} + \)\(72\!\cdots\!14\)\( \beta_{4} + \)\(12\!\cdots\!41\)\( \beta_{3} + \)\(20\!\cdots\!43\)\( \beta_{2} - \)\(91\!\cdots\!66\)\( \beta_{1} - \)\(79\!\cdots\!32\)\(\)\()/ 1073741824 \)
\(\nu^{19}\)\(=\)\((\)\(-\)\(93\!\cdots\!08\)\( \beta_{19} - \)\(67\!\cdots\!24\)\( \beta_{18} - \)\(11\!\cdots\!92\)\( \beta_{17} + \)\(20\!\cdots\!08\)\( \beta_{16} + \)\(98\!\cdots\!20\)\( \beta_{15} - \)\(61\!\cdots\!95\)\( \beta_{14} + \)\(32\!\cdots\!40\)\( \beta_{13} + \)\(92\!\cdots\!24\)\( \beta_{12} - \)\(47\!\cdots\!84\)\( \beta_{11} - \)\(25\!\cdots\!76\)\( \beta_{10} - \)\(53\!\cdots\!46\)\( \beta_{9} - \)\(31\!\cdots\!56\)\( \beta_{8} + \)\(23\!\cdots\!35\)\( \beta_{7} + \)\(24\!\cdots\!24\)\( \beta_{6} + \)\(58\!\cdots\!92\)\( \beta_{5} + \)\(78\!\cdots\!40\)\( \beta_{4} - \)\(78\!\cdots\!96\)\( \beta_{3} - \)\(66\!\cdots\!55\)\( \beta_{2} + \)\(15\!\cdots\!66\)\( \beta_{1} - \)\(35\!\cdots\!78\)\(\)\()/ 536870912 \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(-1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
0.500000 9.03374e8i
0.500000 + 9.03374e8i
0.500000 5.41536e8i
0.500000 + 5.41536e8i
0.500000 + 4.95367e8i
0.500000 4.95367e8i
0.500000 8.71475e8i
0.500000 + 8.71475e8i
0.500000 2.73023e7i
0.500000 + 2.73023e7i
0.500000 + 1.25309e9i
0.500000 1.25309e9i
0.500000 3.38044e8i
0.500000 + 3.38044e8i
0.500000 + 4.69874e8i
0.500000 4.69874e8i
0.500000 1.16651e9i
0.500000 + 1.16651e9i
0.500000 + 5.70227e8i
0.500000 5.70227e8i
−2.05493e6 418685.i 1.44540e10i 4.04745e12 + 1.72074e12i 8.07741e14 6.05166e15 2.97020e16i 7.29328e17i −7.59680e18 5.23061e18i −9.94985e19 −1.65985e21 3.38189e20i
3.2 −2.05493e6 + 418685.i 1.44540e10i 4.04745e12 1.72074e12i 8.07741e14 6.05166e15 + 2.97020e16i 7.29328e17i −7.59680e18 + 5.23061e18i −9.94985e19 −1.65985e21 + 3.38189e20i
3.3 −2.05360e6 425178.i 8.66457e9i 4.03649e12 + 1.74629e12i −5.87785e14 3.68399e15 1.77936e16i 1.08401e18i −7.54686e18 5.30241e18i 3.43442e19 1.20707e21 + 2.49913e20i
3.4 −2.05360e6 + 425178.i 8.66457e9i 4.03649e12 1.74629e12i −5.87785e14 3.68399e15 + 1.77936e16i 1.08401e18i −7.54686e18 + 5.30241e18i 3.43442e19 1.20707e21 2.49913e20i
3.5 −1.51618e6 1.44888e6i 7.92587e9i 1.99543e11 + 4.39352e12i 1.59403e13 −1.14836e16 + 1.20170e16i 6.57146e16i 6.06313e18 6.95047e18i 4.65995e19 −2.41683e19 2.30956e19i
3.6 −1.51618e6 + 1.44888e6i 7.92587e9i 1.99543e11 4.39352e12i 1.59403e13 −1.14836e16 1.20170e16i 6.57146e16i 6.06313e18 + 6.95047e18i 4.65995e19 −2.41683e19 + 2.30956e19i
3.7 −766250. 1.95215e6i 1.39436e10i −3.22377e12 + 2.99168e12i −2.28561e14 2.72200e16 1.06843e16i 3.10753e17i 8.31043e18 + 4.00092e18i −8.50048e19 1.75135e20 + 4.46186e20i
3.8 −766250. + 1.95215e6i 1.39436e10i −3.22377e12 2.99168e12i −2.28561e14 2.72200e16 + 1.06843e16i 3.10753e17i 8.31043e18 4.00092e18i −8.50048e19 1.75135e20 4.46186e20i
3.9 85368.2 2.09541e6i 4.36838e8i −4.38347e12 3.57763e11i 5.12233e14 9.15355e14 + 3.72920e13i 3.44914e17i −1.12387e18 + 9.15464e18i 1.09228e20 4.37284e19 1.07334e21i
3.10 85368.2 + 2.09541e6i 4.36838e8i −4.38347e12 + 3.57763e11i 5.12233e14 9.15355e14 3.72920e13i 3.44914e17i −1.12387e18 9.15464e18i 1.09228e20 4.37284e19 + 1.07334e21i
3.11 232318. 2.08424e6i 2.00495e10i −4.29010e12 9.68414e11i −5.01114e14 −4.17880e16 4.65785e15i 9.07936e16i −3.01508e18 + 8.71664e18i −2.92563e20 −1.16418e20 + 1.04444e21i
3.12 232318. + 2.08424e6i 2.00495e10i −4.29010e12 + 9.68414e11i −5.01114e14 −4.17880e16 + 4.65785e15i 9.07936e16i −3.01508e18 8.71664e18i −2.92563e20 −1.16418e20 1.04444e21i
3.13 1.12655e6 1.76888e6i 5.40870e9i −1.85982e12 3.98546e12i −7.68266e14 9.56734e15 + 6.09318e15i 1.45472e17i −9.14497e18 1.20003e18i 8.01649e19 −8.65491e20 + 1.35897e21i
3.14 1.12655e6 + 1.76888e6i 5.40870e9i −1.85982e12 + 3.98546e12i −7.68266e14 9.56734e15 6.09318e15i 1.45472e17i −9.14497e18 + 1.20003e18i 8.01649e19 −8.65491e20 1.35897e21i
3.15 1.57912e6 1.38001e6i 7.51798e9i 5.89205e11 4.35840e12i 5.33275e14 −1.03749e16 1.18718e16i 7.46884e17i −5.08420e18 7.69555e18i 5.28989e19 8.42107e20 7.35924e20i
3.16 1.57912e6 + 1.38001e6i 7.51798e9i 5.89205e11 + 4.35840e12i 5.33275e14 −1.03749e16 + 1.18718e16i 7.46884e17i −5.08420e18 + 7.69555e18i 5.28989e19 8.42107e20 + 7.35924e20i
3.17 1.70189e6 1.22541e6i 1.86642e10i 1.39480e12 4.17101e12i 4.71964e14 2.28712e16 + 3.17644e16i 6.63278e17i −2.73739e18 8.80780e18i −2.38933e20 8.03231e20 5.78348e20i
3.18 1.70189e6 + 1.22541e6i 1.86642e10i 1.39480e12 + 4.17101e12i 4.71964e14 2.28712e16 3.17644e16i 6.63278e17i −2.73739e18 + 8.80780e18i −2.38933e20 8.03231e20 + 5.78348e20i
3.19 2.06680e6 355528.i 9.12363e9i 4.14525e12 1.46961e12i −1.85810e14 −3.24370e15 1.88567e16i 7.39963e17i 8.04489e18 4.51113e18i 2.61784e19 −3.84032e20 + 6.60607e19i
3.20 2.06680e6 + 355528.i 9.12363e9i 4.14525e12 + 1.46961e12i −1.85810e14 −3.24370e15 + 1.88567e16i 7.39963e17i 8.04489e18 + 4.51113e18i 2.61784e19 −3.84032e20 6.60607e19i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4.43.b.a 20
4.b odd 2 1 inner 4.43.b.a 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4.43.b.a 20 1.a even 1 1 trivial
4.43.b.a 20 4.b odd 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{43}^{\mathrm{new}}(4, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( \)\(27\!\cdots\!76\)\( - \)\(49\!\cdots\!16\)\( T - \)\(46\!\cdots\!28\)\( T^{2} + \)\(30\!\cdots\!72\)\( T^{3} - \)\(10\!\cdots\!08\)\( T^{4} + \)\(42\!\cdots\!00\)\( T^{5} + \)\(14\!\cdots\!92\)\( T^{6} - \)\(85\!\cdots\!72\)\( T^{7} + \)\(47\!\cdots\!92\)\( T^{8} - \)\(11\!\cdots\!44\)\( T^{9} + \)\(44\!\cdots\!52\)\( T^{10} - \)\(25\!\cdots\!36\)\( T^{11} + \)\(24\!\cdots\!12\)\( T^{12} - \)\(10\!\cdots\!48\)\( T^{13} + \)\(38\!\cdots\!32\)\( T^{14} + \)\(25\!\cdots\!00\)\( T^{15} - \)\(14\!\cdots\!48\)\( T^{16} + 9660330320179703808 T^{17} - 333850867648 T^{18} - 802164 T^{19} + T^{20} \)
$3$ \( \)\(70\!\cdots\!00\)\( + \)\(37\!\cdots\!00\)\( T^{2} + \)\(40\!\cdots\!00\)\( T^{4} + \)\(17\!\cdots\!00\)\( T^{6} + \)\(42\!\cdots\!00\)\( T^{8} + \)\(60\!\cdots\!40\)\( T^{10} + \)\(52\!\cdots\!00\)\( T^{12} + \)\(28\!\cdots\!56\)\( T^{14} + \)\(87\!\cdots\!48\)\( T^{16} + \)\(14\!\cdots\!28\)\( T^{18} + T^{20} \)
$5$ \( ( -\)\(15\!\cdots\!00\)\( + \)\(87\!\cdots\!00\)\( T + \)\(76\!\cdots\!00\)\( T^{2} - \)\(45\!\cdots\!00\)\( T^{3} - \)\(11\!\cdots\!00\)\( T^{4} - \)\(53\!\cdots\!00\)\( T^{5} + \)\(61\!\cdots\!00\)\( T^{6} + \)\(54\!\cdots\!00\)\( T^{7} - \)\(13\!\cdots\!80\)\( T^{8} - 69617842498500 T^{9} + T^{10} )^{2} \)
$7$ \( \)\(72\!\cdots\!00\)\( + \)\(31\!\cdots\!00\)\( T^{2} + \)\(40\!\cdots\!00\)\( T^{4} + \)\(19\!\cdots\!00\)\( T^{6} + \)\(35\!\cdots\!00\)\( T^{8} + \)\(31\!\cdots\!00\)\( T^{10} + \)\(14\!\cdots\!00\)\( T^{12} + \)\(35\!\cdots\!96\)\( T^{14} + \)\(48\!\cdots\!68\)\( T^{16} + \)\(35\!\cdots\!48\)\( T^{18} + T^{20} \)
$11$ \( \)\(33\!\cdots\!00\)\( + \)\(76\!\cdots\!00\)\( T^{2} + \)\(39\!\cdots\!00\)\( T^{4} + \)\(63\!\cdots\!00\)\( T^{6} + \)\(40\!\cdots\!00\)\( T^{8} + \)\(13\!\cdots\!00\)\( T^{10} + \)\(25\!\cdots\!00\)\( T^{12} + \)\(28\!\cdots\!00\)\( T^{14} + \)\(18\!\cdots\!40\)\( T^{16} + \)\(66\!\cdots\!20\)\( T^{18} + T^{20} \)
$13$ \( ( -\)\(18\!\cdots\!00\)\( + \)\(42\!\cdots\!00\)\( T + \)\(48\!\cdots\!40\)\( T^{2} - \)\(65\!\cdots\!84\)\( T^{3} - \)\(47\!\cdots\!96\)\( T^{4} + \)\(29\!\cdots\!32\)\( T^{5} + \)\(19\!\cdots\!08\)\( T^{6} - \)\(39\!\cdots\!72\)\( T^{7} - \)\(28\!\cdots\!28\)\( T^{8} + \)\(72\!\cdots\!24\)\( T^{9} + T^{10} )^{2} \)
$17$ \( ( \)\(11\!\cdots\!00\)\( - \)\(63\!\cdots\!00\)\( T - \)\(45\!\cdots\!80\)\( T^{2} + \)\(33\!\cdots\!04\)\( T^{3} - \)\(35\!\cdots\!56\)\( T^{4} - \)\(11\!\cdots\!12\)\( T^{5} + \)\(17\!\cdots\!88\)\( T^{6} + \)\(12\!\cdots\!72\)\( T^{7} - \)\(24\!\cdots\!28\)\( T^{8} - \)\(40\!\cdots\!64\)\( T^{9} + T^{10} )^{2} \)
$19$ \( \)\(14\!\cdots\!00\)\( + \)\(10\!\cdots\!00\)\( T^{2} + \)\(16\!\cdots\!00\)\( T^{4} + \)\(92\!\cdots\!00\)\( T^{6} + \)\(20\!\cdots\!00\)\( T^{8} + \)\(14\!\cdots\!00\)\( T^{10} + \)\(47\!\cdots\!00\)\( T^{12} + \)\(85\!\cdots\!00\)\( T^{14} + \)\(86\!\cdots\!40\)\( T^{16} + \)\(45\!\cdots\!40\)\( T^{18} + T^{20} \)
$23$ \( \)\(58\!\cdots\!00\)\( + \)\(59\!\cdots\!00\)\( T^{2} + \)\(26\!\cdots\!00\)\( T^{4} + \)\(67\!\cdots\!00\)\( T^{6} + \)\(10\!\cdots\!00\)\( T^{8} + \)\(11\!\cdots\!40\)\( T^{10} + \)\(81\!\cdots\!80\)\( T^{12} + \)\(37\!\cdots\!56\)\( T^{14} + \)\(10\!\cdots\!08\)\( T^{16} + \)\(16\!\cdots\!68\)\( T^{18} + T^{20} \)
$29$ \( ( \)\(19\!\cdots\!96\)\( - \)\(66\!\cdots\!08\)\( T + \)\(28\!\cdots\!72\)\( T^{2} + \)\(72\!\cdots\!44\)\( T^{3} - \)\(32\!\cdots\!24\)\( T^{4} - \)\(23\!\cdots\!40\)\( T^{5} + \)\(82\!\cdots\!96\)\( T^{6} + \)\(19\!\cdots\!56\)\( T^{7} - \)\(53\!\cdots\!28\)\( T^{8} - \)\(42\!\cdots\!52\)\( T^{9} + T^{10} )^{2} \)
$31$ \( \)\(65\!\cdots\!00\)\( + \)\(13\!\cdots\!00\)\( T^{2} + \)\(95\!\cdots\!00\)\( T^{4} + \)\(28\!\cdots\!00\)\( T^{6} + \)\(32\!\cdots\!00\)\( T^{8} + \)\(17\!\cdots\!00\)\( T^{10} + \)\(55\!\cdots\!00\)\( T^{12} + \)\(98\!\cdots\!00\)\( T^{14} + \)\(97\!\cdots\!40\)\( T^{16} + \)\(49\!\cdots\!40\)\( T^{18} + T^{20} \)
$37$ \( ( -\)\(89\!\cdots\!00\)\( + \)\(38\!\cdots\!00\)\( T + \)\(14\!\cdots\!60\)\( T^{2} - \)\(44\!\cdots\!36\)\( T^{3} - \)\(11\!\cdots\!16\)\( T^{4} + \)\(11\!\cdots\!48\)\( T^{5} + \)\(35\!\cdots\!28\)\( T^{6} - \)\(43\!\cdots\!88\)\( T^{7} - \)\(32\!\cdots\!48\)\( T^{8} - \)\(16\!\cdots\!24\)\( T^{9} + T^{10} )^{2} \)
$41$ \( ( -\)\(13\!\cdots\!76\)\( + \)\(39\!\cdots\!60\)\( T - \)\(34\!\cdots\!00\)\( T^{2} + \)\(98\!\cdots\!00\)\( T^{3} + \)\(10\!\cdots\!80\)\( T^{4} - \)\(89\!\cdots\!64\)\( T^{5} + \)\(75\!\cdots\!20\)\( T^{6} + \)\(14\!\cdots\!00\)\( T^{7} - \)\(18\!\cdots\!00\)\( T^{8} - \)\(61\!\cdots\!40\)\( T^{9} + T^{10} )^{2} \)
$43$ \( \)\(33\!\cdots\!00\)\( + \)\(65\!\cdots\!00\)\( T^{2} + \)\(33\!\cdots\!00\)\( T^{4} + \)\(21\!\cdots\!00\)\( T^{6} + \)\(42\!\cdots\!00\)\( T^{8} + \)\(35\!\cdots\!00\)\( T^{10} + \)\(15\!\cdots\!00\)\( T^{12} + \)\(36\!\cdots\!76\)\( T^{14} + \)\(49\!\cdots\!28\)\( T^{16} + \)\(34\!\cdots\!28\)\( T^{18} + T^{20} \)
$47$ \( \)\(32\!\cdots\!00\)\( + \)\(28\!\cdots\!00\)\( T^{2} + \)\(92\!\cdots\!00\)\( T^{4} + \)\(14\!\cdots\!00\)\( T^{6} + \)\(10\!\cdots\!00\)\( T^{8} + \)\(36\!\cdots\!40\)\( T^{10} + \)\(53\!\cdots\!40\)\( T^{12} + \)\(35\!\cdots\!76\)\( T^{14} + \)\(11\!\cdots\!68\)\( T^{16} + \)\(17\!\cdots\!68\)\( T^{18} + T^{20} \)
$53$ \( ( \)\(51\!\cdots\!00\)\( - \)\(28\!\cdots\!00\)\( T - \)\(41\!\cdots\!80\)\( T^{2} + \)\(16\!\cdots\!16\)\( T^{3} - \)\(57\!\cdots\!16\)\( T^{4} - \)\(10\!\cdots\!68\)\( T^{5} + \)\(56\!\cdots\!48\)\( T^{6} + \)\(23\!\cdots\!88\)\( T^{7} - \)\(13\!\cdots\!28\)\( T^{8} - \)\(17\!\cdots\!36\)\( T^{9} + T^{10} )^{2} \)
$59$ \( \)\(97\!\cdots\!00\)\( + \)\(45\!\cdots\!00\)\( T^{2} + \)\(76\!\cdots\!00\)\( T^{4} + \)\(56\!\cdots\!00\)\( T^{6} + \)\(19\!\cdots\!00\)\( T^{8} + \)\(34\!\cdots\!00\)\( T^{10} + \)\(35\!\cdots\!00\)\( T^{12} + \)\(21\!\cdots\!00\)\( T^{14} + \)\(74\!\cdots\!40\)\( T^{16} + \)\(13\!\cdots\!00\)\( T^{18} + T^{20} \)
$61$ \( ( \)\(33\!\cdots\!04\)\( + \)\(30\!\cdots\!00\)\( T + \)\(25\!\cdots\!20\)\( T^{2} - \)\(29\!\cdots\!40\)\( T^{3} - \)\(18\!\cdots\!40\)\( T^{4} + \)\(89\!\cdots\!96\)\( T^{5} + \)\(40\!\cdots\!00\)\( T^{6} - \)\(10\!\cdots\!60\)\( T^{7} - \)\(35\!\cdots\!80\)\( T^{8} + \)\(36\!\cdots\!20\)\( T^{9} + T^{10} )^{2} \)
$67$ \( \)\(10\!\cdots\!00\)\( + \)\(21\!\cdots\!00\)\( T^{2} + \)\(15\!\cdots\!00\)\( T^{4} + \)\(52\!\cdots\!00\)\( T^{6} + \)\(78\!\cdots\!00\)\( T^{8} + \)\(60\!\cdots\!40\)\( T^{10} + \)\(25\!\cdots\!00\)\( T^{12} + \)\(57\!\cdots\!76\)\( T^{14} + \)\(69\!\cdots\!88\)\( T^{16} + \)\(42\!\cdots\!48\)\( T^{18} + T^{20} \)
$71$ \( \)\(26\!\cdots\!00\)\( + \)\(77\!\cdots\!00\)\( T^{2} + \)\(98\!\cdots\!00\)\( T^{4} + \)\(70\!\cdots\!00\)\( T^{6} + \)\(31\!\cdots\!00\)\( T^{8} + \)\(89\!\cdots\!00\)\( T^{10} + \)\(16\!\cdots\!00\)\( T^{12} + \)\(19\!\cdots\!00\)\( T^{14} + \)\(14\!\cdots\!40\)\( T^{16} + \)\(58\!\cdots\!20\)\( T^{18} + T^{20} \)
$73$ \( ( -\)\(19\!\cdots\!00\)\( - \)\(14\!\cdots\!00\)\( T + \)\(51\!\cdots\!60\)\( T^{2} + \)\(41\!\cdots\!36\)\( T^{3} - \)\(13\!\cdots\!76\)\( T^{4} - \)\(44\!\cdots\!08\)\( T^{5} + \)\(14\!\cdots\!08\)\( T^{6} + \)\(20\!\cdots\!48\)\( T^{7} - \)\(62\!\cdots\!68\)\( T^{8} - \)\(33\!\cdots\!76\)\( T^{9} + T^{10} )^{2} \)
$79$ \( \)\(15\!\cdots\!00\)\( + \)\(60\!\cdots\!00\)\( T^{2} + \)\(46\!\cdots\!00\)\( T^{4} + \)\(13\!\cdots\!00\)\( T^{6} + \)\(16\!\cdots\!00\)\( T^{8} + \)\(99\!\cdots\!00\)\( T^{10} + \)\(33\!\cdots\!00\)\( T^{12} + \)\(64\!\cdots\!00\)\( T^{14} + \)\(71\!\cdots\!40\)\( T^{16} + \)\(42\!\cdots\!40\)\( T^{18} + T^{20} \)
$83$ \( \)\(49\!\cdots\!00\)\( + \)\(54\!\cdots\!00\)\( T^{2} + \)\(18\!\cdots\!00\)\( T^{4} + \)\(28\!\cdots\!00\)\( T^{6} + \)\(22\!\cdots\!00\)\( T^{8} + \)\(10\!\cdots\!40\)\( T^{10} + \)\(30\!\cdots\!20\)\( T^{12} + \)\(55\!\cdots\!96\)\( T^{14} + \)\(62\!\cdots\!68\)\( T^{16} + \)\(38\!\cdots\!28\)\( T^{18} + T^{20} \)
$89$ \( ( \)\(94\!\cdots\!56\)\( + \)\(11\!\cdots\!32\)\( T + \)\(37\!\cdots\!52\)\( T^{2} - \)\(19\!\cdots\!96\)\( T^{3} - \)\(27\!\cdots\!64\)\( T^{4} - \)\(22\!\cdots\!60\)\( T^{5} + \)\(43\!\cdots\!16\)\( T^{6} + \)\(37\!\cdots\!96\)\( T^{7} - \)\(32\!\cdots\!48\)\( T^{8} - \)\(14\!\cdots\!72\)\( T^{9} + T^{10} )^{2} \)
$97$ \( ( \)\(12\!\cdots\!00\)\( - \)\(37\!\cdots\!00\)\( T - \)\(10\!\cdots\!20\)\( T^{2} + \)\(39\!\cdots\!64\)\( T^{3} + \)\(20\!\cdots\!64\)\( T^{4} - \)\(14\!\cdots\!92\)\( T^{5} + \)\(29\!\cdots\!68\)\( T^{6} + \)\(23\!\cdots\!52\)\( T^{7} - \)\(12\!\cdots\!88\)\( T^{8} - \)\(13\!\cdots\!24\)\( T^{9} + T^{10} )^{2} \)
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