# Properties

 Label 4.39.b.a Level 4 Weight 39 Character orbit 4.b Analytic conductor 36.585 Analytic rank 0 Dimension 18 CM No Inner twists 2

# Learn more about

## Newspace parameters

 Level: $$N$$ = $$4 = 2^{2}$$ Weight: $$k$$ = $$39$$ Character orbit: $$[\chi]$$ = 4.b (of order $$2$$ and degree $$1$$)

## Newform invariants

 Self dual: No Analytic conductor: $$36.5853876134$$ Analytic rank: $$0$$ Dimension: $$18$$ Coefficient field: $$\mathbb{Q}[x]/(x^{18} - \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: multiple of $$2^{306}\cdot 3^{34}\cdot 5^{10}\cdot 19^{2}$$ Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{17}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q$$ $$+ ( 20235 - \beta_{1} ) q^{2}$$ $$+ ( -18 + 165 \beta_{1} - \beta_{2} ) q^{3}$$ $$+ ( 11142631926 - 19441 \beta_{1} + 6 \beta_{2} - \beta_{3} ) q^{4}$$ $$+ ( -499516043420 + 637220 \beta_{1} + 14 \beta_{2} - 10 \beta_{3} - \beta_{4} ) q^{5}$$ $$+ ( 45422189524262 + 3306023 \beta_{1} - 86952 \beta_{2} + 157 \beta_{3} + 5 \beta_{4} - \beta_{5} ) q^{6}$$ $$+ ( 169124764 - 1521663324 \beta_{1} - 153215 \beta_{2} - 99 \beta_{3} - 2 \beta_{5} - \beta_{7} ) q^{7}$$ $$+ ( 5022823634250437 - 10796968461 \beta_{1} - 16299533 \beta_{2} - 20951 \beta_{3} - 510 \beta_{4} + 11 \beta_{5} - 6 \beta_{6} + \beta_{9} ) q^{8}$$ $$+ ( -418320708331751700 + 428473847650 \beta_{1} + 11133454 \beta_{2} + 338301 \beta_{3} - 787 \beta_{4} + 184 \beta_{5} + 93 \beta_{6} - \beta_{7} + \beta_{8} ) q^{9}$$ $$+O(q^{10})$$ $$q$$ $$+(20235 - \beta_{1}) q^{2}$$ $$+(-18 + 165 \beta_{1} - \beta_{2}) q^{3}$$ $$+(11142631926 - 19441 \beta_{1} + 6 \beta_{2} - \beta_{3}) q^{4}$$ $$+(-499516043420 + 637220 \beta_{1} + 14 \beta_{2} - 10 \beta_{3} - \beta_{4}) q^{5}$$ $$+(45422189524262 + 3306023 \beta_{1} - 86952 \beta_{2} + 157 \beta_{3} + 5 \beta_{4} - \beta_{5}) q^{6}$$ $$+(169124764 - 1521663324 \beta_{1} - 153215 \beta_{2} - 99 \beta_{3} - 2 \beta_{5} - \beta_{7}) q^{7}$$ $$+(5022823634250437 - 10796968461 \beta_{1} - 16299533 \beta_{2} - 20951 \beta_{3} - 510 \beta_{4} + 11 \beta_{5} - 6 \beta_{6} + \beta_{9}) q^{8}$$ $$+(-418320708331751700 + 428473847650 \beta_{1} + 11133454 \beta_{2} + 338301 \beta_{3} - 787 \beta_{4} + 184 \beta_{5} + 93 \beta_{6} - \beta_{7} + \beta_{8}) q^{9}$$ $$+(-185003626252549811 + 486059740176 \beta_{1} - 1144261360 \beta_{2} + 457368 \beta_{3} - 23811 \beta_{4} + 342 \beta_{5} - 609 \beta_{6} + 66 \beta_{7} + 8 \beta_{9} + \beta_{11}) q^{10}$$ $$+(1100028363112 - 9897051783583 \beta_{1} - 1070214124 \beta_{2} - 7145146 \beta_{3} + 11672 \beta_{4} - 7292 \beta_{5} + 2386 \beta_{6} - 371 \beta_{7} + 5 \beta_{9} - \beta_{11} - \beta_{13}) q^{11}$$ $$+(11899819413177209049 - 44626411930100 \beta_{1} - 28002804835 \beta_{2} + 24131 \beta_{3} + 1677598 \beta_{4} - 76732 \beta_{5} + 25873 \beta_{6} - 1823 \beta_{7} - 3 \beta_{8} - 145 \beta_{9} - 2 \beta_{10} - 5 \beta_{11} - \beta_{13} + \beta_{15}) q^{12}$$ $$+(-75094253412177337830 - 290682474468001 \beta_{1} - 7677365216 \beta_{2} - 725451255 \beta_{3} - 12466063 \beta_{4} + 103594 \beta_{5} - 65303 \beta_{6} - 802 \beta_{7} - 39 \beta_{8} + 562 \beta_{9} + 7 \beta_{10} + 33 \beta_{11} - \beta_{13} - \beta_{14} - 2 \beta_{15}) q^{13}$$ $$+(-$$$$41\!\cdots\!91$$$$- 31008164938796 \beta_{1} - 444415710254 \beta_{2} - 1639069196 \beta_{3} - 23250314 \beta_{4} - 148304 \beta_{5} + 32390 \beta_{6} + 16516 \beta_{7} + 721 \beta_{8} - 359 \beta_{9} + 9 \beta_{10} + 19 \beta_{11} + 52 \beta_{13} - 3 \beta_{14} + 4 \beta_{15} + \beta_{16}) q^{14}$$ $$+(824925218764200 - 7425468328414239 \beta_{1} + 377772966543 \beta_{2} - 8014619120 \beta_{3} + 14237526 \beta_{4} - 2744090 \beta_{5} + 1753829 \beta_{6} - 241474 \beta_{7} - 378 \beta_{8} + 3146 \beta_{9} - 93 \beta_{10} - 1328 \beta_{11} + 2 \beta_{13} - 12 \beta_{14} + 48 \beta_{15} - 4 \beta_{16} - \beta_{17}) q^{15}$$ $$+($$$$81\!\cdots\!78$$$$- 4433892621923524 \beta_{1} - 7537364496198 \beta_{2} - 10752267706 \beta_{3} - 48421129 \beta_{4} - 14025777 \beta_{5} - 1405283 \beta_{6} - 181784 \beta_{7} - 19333 \beta_{8} - 326 \beta_{9} + 234 \beta_{10} - 140 \beta_{11} + \beta_{12} - 693 \beta_{13} + 4 \beta_{14} - 44 \beta_{15} + 20 \beta_{16} - 4 \beta_{17}) q^{16}$$ $$+(-$$$$60\!\cdots\!57$$$$- 7783692088776690 \beta_{1} - 193875139485 \beta_{2} + 27062526690 \beta_{3} + 414964298 \beta_{4} - 7156852 \beta_{5} - 1741570 \beta_{6} + 183214 \beta_{7} - 5676 \beta_{8} - 359191 \beta_{9} - 683 \beta_{10} + 13442 \beta_{11} - 4 \beta_{12} - 206 \beta_{13} - 18 \beta_{14} + 92 \beta_{15} - 64 \beta_{16} - 32 \beta_{17}) q^{17}$$ $$+(-$$$$12\!\cdots\!32$$$$+ 409626720548130028 \beta_{1} - 48986024991614 \beta_{2} + 404197015580 \beta_{3} + 2904151632 \beta_{4} + 30549632 \beta_{5} + 27862023 \beta_{6} + 14593520 \beta_{7} + 344840 \beta_{8} - 228468 \beta_{9} + 228 \beta_{10} + 3570 \beta_{11} + 44 \beta_{12} + 1428 \beta_{13} + 384 \beta_{14} - 192 \beta_{15} + 128 \beta_{16} - 112 \beta_{17}) q^{18}$$ $$+(-23203251851933308 + 208655344933424357 \beta_{1} + 58057421087652 \beta_{2} + 250368027240 \beta_{3} - 457377796 \beta_{4} + 420481652 \beta_{5} - 53964656 \beta_{6} + 4057997 \beta_{7} - 18228 \beta_{8} + 6181171 \beta_{9} + 2690 \beta_{10} - 54851 \beta_{11} - 160 \beta_{12} - 7 \beta_{13} + 280 \beta_{14} - 3168 \beta_{15} - 248 \beta_{16} - 446 \beta_{17}) q^{19}$$ $$+($$$$77\!\cdots\!40$$$$+ 242544292911793242 \beta_{1} - 360578985237332 \beta_{2} + 535701179006 \beta_{3} + 22863901500 \beta_{4} - 944066460 \beta_{5} - 188993584 \beta_{6} - 122089872 \beta_{7} - 1940044 \beta_{8} - 1594888 \beta_{9} - 4568 \beta_{10} + 24784 \beta_{11} + 908 \beta_{12} + 37284 \beta_{13} + 1008 \beta_{14} + 592 \beta_{15} - 80 \beta_{16} - 1328 \beta_{17}) q^{20}$$ $$+(-$$$$20\!\cdots\!50$$$$+ 2829939314534471579 \beta_{1} + 74156683807684 \beta_{2} + 4746989929677 \beta_{3} + 96398258060 \beta_{4} - 5142327262 \beta_{5} + 672448219 \beta_{6} + 57718740 \beta_{7} + 208191 \beta_{8} - 65917924 \beta_{9} + 961 \beta_{10} - 343379 \beta_{11} - 2984 \beta_{12} + 7931 \beta_{13} + 723 \beta_{14} - 1370 \beta_{15} + 1408 \beta_{16} - 3392 \beta_{17}) q^{21}$$ $$+(-$$$$27\!\cdots\!44$$$$- 201187048436173955 \beta_{1} - 1888738789507212 \beta_{2} - 10515938131881 \beta_{3} - 145558675557 \beta_{4} - 986382827 \beta_{5} + 1054472748 \beta_{6} + 1302658648 \beta_{7} + 2297198 \beta_{8} + 2260222 \beta_{9} + 3294 \beta_{10} - 211958 \beta_{11} + 11616 \beta_{12} - 162440 \beta_{13} - 12458 \beta_{14} + 3128 \beta_{15} - 4722 \beta_{16} - 8064 \beta_{17}) q^{22}$$ $$+(-499358830024923704 + 4529652121525211427 \beta_{1} - 11805428852731309 \beta_{2} + 6396170373766 \beta_{3} - 13401027614 \beta_{4} + 28865043926 \beta_{5} - 553923129 \beta_{6} + 589875636 \beta_{7} + 937858 \beta_{8} + 213456070 \beta_{9} - 190775 \beta_{10} + 2888600 \beta_{11} - 34240 \beta_{12} + 48910 \beta_{13} - 3300 \beta_{14} + 99216 \beta_{15} + 13236 \beta_{16} - 13331 \beta_{17}) q^{23}$$ $$+(-$$$$28\!\cdots\!28$$$$- 14003804577478644104 \beta_{1} - 12433870707827120 \beta_{2} - 50694431677200 \beta_{3} + 169571590516 \beta_{4} - 31659688900 \beta_{5} + 7705105036 \beta_{6} - 10377341472 \beta_{7} + 49581492 \beta_{8} + 57840736 \beta_{9} + 112728 \beta_{10} - 730384 \beta_{11} + 102364 \beta_{12} - 989516 \beta_{13} - 51600 \beta_{14} + 6384 \beta_{15} - 16336 \beta_{16} - 19312 \beta_{17}) q^{24}$$ $$+($$$$24\!\cdots\!95$$$$- 3107357034461976592 \beta_{1} - 34329945040095 \beta_{2} + 182177504602611 \beta_{3} + 2800412323075 \beta_{4} - 255721479660 \beta_{5} + 1086681215 \beta_{6} + 1618540849 \beta_{7} - 10111207 \beta_{8} - 254168767 \beta_{9} - 797771 \beta_{10} - 2738534 \beta_{11} - 267812 \beta_{12} + 293306 \beta_{13} + 2646 \beta_{14} - 9940 \beta_{15} + 7616 \beta_{16} - 4384 \beta_{17}) q^{25}$$ $$+($$$$78\!\cdots\!99$$$$+ 80954532806266401794 \beta_{1} - 43741278616008828 \beta_{2} - 281870210187536 \beta_{3} - 9633316629219 \beta_{4} - 11164084650 \beta_{5} - 32999007051 \beta_{6} + 32564433058 \beta_{7} - 200613936 \beta_{8} + 348562368 \beta_{9} + 165288 \beta_{10} + 7300565 \beta_{11} + 650232 \beta_{12} + 3637640 \beta_{13} + 194304 \beta_{14} + 13440 \beta_{15} + 48384 \beta_{16} + 65184 \beta_{17}) q^{26}$$ $$+(17469251620786166110 -$$$$15\!\cdots\!38$$$$\beta_{1} + 597208061187366709 \beta_{2} + 216021984723136 \beta_{3} - 529762584236 \beta_{4} + 1922669241612 \beta_{5} + 2841325560 \beta_{6} + 2049249299 \beta_{7} - 103242300 \beta_{8} - 4878603483 \beta_{9} - 4200570 \beta_{10} - 59625861 \beta_{11} - 1485152 \beta_{12} + 1207407 \beta_{13} + 87240 \beta_{14} - 1952544 \beta_{15} - 238184 \beta_{16} + 229606 \beta_{17}) q^{27}$$ $$+(-$$$$13\!\cdots\!94$$$$+$$$$41\!\cdots\!60$$$$\beta_{1} - 203440897511593634 \beta_{2} - 81700121390942 \beta_{3} + 43002855342324 \beta_{4} - 237573279304 \beta_{5} - 59988453738 \beta_{6} - 49712582858 \beta_{7} + 160693118 \beta_{8} + 1970978154 \beta_{9} - 1388588 \beta_{10} + 33268530 \beta_{11} + 2998304 \beta_{12} + 8018410 \beta_{13} + 1148544 \beta_{14} - 386570 \beta_{15} + 426112 \beta_{16} + 632704 \beta_{17}) q^{28}$$ $$+($$$$66\!\cdots\!92$$$$-$$$$11\!\cdots\!72$$$$\beta_{1} - 29648743545745832 \beta_{2} + 4241272603003468 \beta_{3} + 38857739724689 \beta_{4} - 4695414685024 \beta_{5} - 240244281498 \beta_{6} + 20171329150 \beta_{7} - 353007046 \beta_{8} + 17984313986 \beta_{9} - 41308026 \beta_{10} + 125564112 \beta_{11} - 5790696 \beta_{12} + 3566712 \beta_{13} - 259056 \beta_{14} + 791840 \beta_{15} - 654976 \beta_{16} + 1093824 \beta_{17}) q^{29}$$ $$+(-$$$$20\!\cdots\!73$$$$-$$$$15\!\cdots\!52$$$$\beta_{1} - 594484361087531002 \beta_{2} - 7654471692136524 \beta_{3} - 357027734136118 \beta_{4} - 1218867081544 \beta_{5} + 376946474674 \beta_{6} - 358609314900 \beta_{7} + 3744499659 \beta_{8} + 8073562019 \beta_{9} - 31649549 \beta_{10} - 593087 \beta_{11} + 9587776 \beta_{12} - 78791620 \beta_{13} - 1619601 \beta_{14} - 1607828 \beta_{15} + 290011 \beta_{16} + 1577728 \beta_{17}) q^{30}$$ $$+($$$$20\!\cdots\!16$$$$-$$$$18\!\cdots\!53$$$$\beta_{1} + 1267718719346456748 \beta_{2} - 7537903382467477 \beta_{3} + 14630956431010 \beta_{4} + 13903548066308 \beta_{5} + 375994652183 \beta_{6} + 82052632795 \beta_{7} - 247804894 \beta_{8} - 22866869930 \beta_{9} - 215867703 \beta_{10} + 354623464 \beta_{11} - 14335552 \beta_{12} + 17285150 \beta_{13} - 2506340 \beta_{14} + 26913168 \beta_{15} + 1979188 \beta_{16} + 1203661 \beta_{17}) q^{31}$$ $$+($$$$22\!\cdots\!84$$$$-$$$$79\!\cdots\!76$$$$\beta_{1} - 1204971140218928216 \beta_{2} - 4836092332222760 \beta_{3} + 1105670371819484 \beta_{4} - 2285407691140 \beta_{5} - 231927725772 \beta_{6} + 1383047181728 \beta_{7} - 8243714580 \beta_{8} + 10807207592 \beta_{9} - 348163928 \beta_{10} - 119308592 \beta_{11} + 16866308 \beta_{12} - 80625108 \beta_{13} - 15422448 \beta_{14} + 7730768 \beta_{15} - 4973488 \beta_{16} - 2105360 \beta_{17}) q^{32}$$ $$+(-$$$$14\!\cdots\!31$$$$+$$$$11\!\cdots\!22$$$$\beta_{1} + 327439840616691812 \beta_{2} + 74673997741928181 \beta_{3} + 373600269785901 \beta_{4} - 19942296773472 \beta_{5} + 2594924707741 \beta_{6} + 191040754943 \beta_{7} + 2012483277 \beta_{8} - 190476478838 \beta_{9} - 757358934 \beta_{10} - 2443821412 \beta_{11} - 10018216 \beta_{12} + 45960940 \beta_{13} + 2588868 \beta_{14} - 17058168 \beta_{15} + 11117952 \beta_{16} - 8375616 \beta_{17}) q^{33}$$ $$+($$$$20\!\cdots\!55$$$$+$$$$61\!\cdots\!27$$$$\beta_{1} + 2537672280645717282 \beta_{2} - 5527231459209060 \beta_{3} - 3419477285478664 \beta_{4} - 16051879606864 \beta_{5} - 201033182033 \beta_{6} - 3417898427264 \beta_{7} - 12231222392 \beta_{8} - 48629642612 \beta_{9} - 2084833500 \beta_{10} + 90994650 \beta_{11} - 15543572 \beta_{12} + 751931092 \beta_{13} + 5298560 \beta_{14} + 37352256 \beta_{15} - 12859264 \beta_{16} - 22792048 \beta_{17}) q^{34}$$ $$+(-$$$$41\!\cdots\!76$$$$+$$$$37\!\cdots\!14$$$$\beta_{1} - 21576560590078248806 \beta_{2} - 221413665309505614 \beta_{3} + 474276803517540 \beta_{4} - 100446144583088 \beta_{5} - 6424292499778 \beta_{6} + 479069898952 \beta_{7} + 8921770092 \beta_{8} + 625375613366 \beta_{9} - 3300684030 \beta_{10} + 758689142 \beta_{11} + 88276704 \beta_{12} - 133128334 \beta_{13} + 43042200 \beta_{14} - 273464928 \beta_{15} - 2535288 \beta_{16} - 38304606 \beta_{17}) q^{35}$$ $$+(-$$$$32\!\cdots\!86$$$$+$$$$12\!\cdots\!35$$$$\beta_{1} - 854759346952245642 \beta_{2} + 399882236720041847 \beta_{3} + 13626825913273336 \beta_{4} + 1217905971016 \beta_{5} + 12524854897056 \beta_{6} + 2675867169504 \beta_{7} + 113048427496 \beta_{8} - 230624026896 \beta_{9} - 8235199152 \beta_{10} - 85945440 \beta_{11} - 219001192 \beta_{12} + 6433224 \beta_{13} + 140197344 \beta_{14} - 97950048 \beta_{15} + 27495776 \beta_{16} - 46237792 \beta_{17}) q^{36}$$ $$+(-$$$$70\!\cdots\!14$$$$+$$$$45\!\cdots\!31$$$$\beta_{1} + 1402261299036499944 \beta_{2} + 910290482765093013 \beta_{3} - 1718024754139103 \beta_{4} + 364074560049618 \beta_{5} + 3113959698077 \beta_{6} - 2234421554010 \beta_{7} + 27367721069 \beta_{8} - 135606093942 \beta_{9} - 15409524717 \beta_{10} + 21845882213 \beta_{11} + 450277568 \beta_{12} - 741192325 \beta_{13} + 13818427 \beta_{14} + 230792310 \beta_{15} - 101577728 \beta_{16} - 36846080 \beta_{17}) q^{37}$$ $$+($$$$57\!\cdots\!44$$$$+$$$$42\!\cdots\!55$$$$\beta_{1} +$$$$11\!\cdots\!00$$$$\beta_{2} + 239281723961441749 \beta_{3} - 11893476460335487 \beta_{4} - 5823045341585 \beta_{5} - 9083341022396 \beta_{6} + 36491379303112 \beta_{7} - 116692417638 \beta_{8} - 392583877302 \beta_{9} - 27263579414 \beta_{10} - 2814913522 \beta_{11} - 767325280 \beta_{12} - 4597856984 \beta_{13} + 31431954 \beta_{14} - 537879064 \beta_{15} + 158095610 \beta_{16} + 54709120 \beta_{17}) q^{38}$$ $$+($$$$32\!\cdots\!64$$$$-$$$$29\!\cdots\!44$$$$\beta_{1} - 50331345966529175793 \beta_{2} - 3619239654503481121 \beta_{3} + 7458304063808120 \beta_{4} - 1565635326517374 \beta_{5} + 92019267886428 \beta_{6} - 16336472405819 \beta_{7} + 75931344744 \beta_{8} - 2919846382464 \beta_{9} - 51608818836 \beta_{10} - 35672103912 \beta_{11} + 1099604672 \beta_{12} - 119732736 \beta_{13} - 465567600 \beta_{14} + 2089610688 \beta_{15} - 117299152 \beta_{16} + 197621516 \beta_{17}) q^{39}$$ $$+($$$$20\!\cdots\!30$$$$-$$$$76\!\cdots\!42$$$$\beta_{1} - 25190137682793374722 \beta_{2} + 260782185446047002 \beta_{3} + 11968123280754964 \beta_{4} - 334037260707090 \beta_{5} - 41077286259772 \beta_{6} - 95304406845440 \beta_{7} - 503933926400 \beta_{8} - 467687906422 \beta_{9} - 88545154048 \beta_{10} - 7565855744 \beta_{11} - 1357108224 \beta_{12} + 9260353536 \beta_{13} - 911204352 \beta_{14} + 888278016 \beta_{15} + 19636224 \beta_{16} + 452628480 \beta_{17}) q^{40}$$ $$+(-$$$$23\!\cdots\!14$$$$-$$$$86\!\cdots\!80$$$$\beta_{1} + 61721338873378899 \beta_{2} + 9079542921606030613 \beta_{3} - 12873720165668371 \beta_{4} + 823464176154668 \beta_{5} - 66482857881367 \beta_{6} - 10885303031777 \beta_{7} + 163509372119 \beta_{8} + 11986154100971 \beta_{9} - 132341210529 \beta_{10} - 80523644202 \beta_{11} + 1044632244 \beta_{12} - 38575482 \beta_{13} - 593579142 \beta_{14} - 2253582220 \beta_{15} + 533211968 \beta_{16} + 747492768 \beta_{17}) q^{41}$$ $$+(-$$$$78\!\cdots\!30$$$$+$$$$14\!\cdots\!78$$$$\beta_{1} +$$$$14\!\cdots\!04$$$$\beta_{2} + 2882732360875279384 \beta_{3} + 121783100795972152 \beta_{4} + 20841029228112 \beta_{5} + 287633111646526 \beta_{6} + 74452411372624 \beta_{7} + 1895245299408 \beta_{8} - 449487779336 \beta_{9} - 251749252824 \beta_{10} - 12455381620 \beta_{11} + 62062200 \beta_{12} + 25279543048 \beta_{13} - 449952000 \beta_{14} + 5617011840 \beta_{15} - 1075565312 \beta_{16} + 748956832 \beta_{17}) q^{42}$$ $$+(-$$$$19\!\cdots\!66$$$$+$$$$17\!\cdots\!47$$$$\beta_{1} +$$$$69\!\cdots\!63$$$$\beta_{2} - 18826104805648524568 \beta_{3} + 39606334928761128 \beta_{4} + 2660671914866664 \beta_{5} - 353603492473800 \beta_{6} + 86269610297146 \beta_{7} + 353701704392 \beta_{8} - 7403192590642 \beta_{9} - 341790884740 \beta_{10} + 399297225026 \beta_{11} - 2971999808 \beta_{12} - 13124973302 \beta_{13} + 3328016720 \beta_{14} - 11900572992 \beta_{15} + 1344587888 \beta_{16} + 566002460 \beta_{17}) q^{43}$$ $$+(-$$$$57\!\cdots\!05$$$$+$$$$26\!\cdots\!60$$$$\beta_{1} -$$$$31\!\cdots\!41$$$$\beta_{2} - 590644221563166179 \beta_{3} - 319018532314595998 \beta_{4} - 2233039205171332 \beta_{5} + 182085193220623 \beta_{6} + 239956517506751 \beta_{7} + 179431402595 \beta_{8} + 3320106383217 \beta_{9} - 529096055486 \beta_{10} + 70026579941 \beta_{11} + 8103940544 \beta_{12} - 88559047007 \beta_{13} + 4365196032 \beta_{14} - 5982103649 \beta_{15} - 1553053952 \beta_{16} - 738465536 \beta_{17}) q^{44}$$ $$+($$$$33\!\cdots\!10$$$$+$$$$47\!\cdots\!09$$$$\beta_{1} +$$$$13\!\cdots\!26$$$$\beta_{2} + 72028486914174380853 \beta_{3} + 387568768440730511 \beta_{4} - 19420396106878850 \beta_{5} + 978799363651905 \beta_{6} + 134110399290152 \beta_{7} + 734322269509 \beta_{8} - 61631460098136 \beta_{9} - 842826997893 \beta_{10} - 93857198877 \beta_{11} - 15608425896 \beta_{12} + 19542990213 \beta_{13} + 7233273693 \beta_{14} + 16665686970 \beta_{15} - 1099569792 \beta_{16} - 2839616832 \beta_{17}) q^{45}$$ $$+($$$$12\!\cdots\!23$$$$+$$$$94\!\cdots\!04$$$$\beta_{1} +$$$$63\!\cdots\!30$$$$\beta_{2} + 5222986174119757620 \beta_{3} + 923940289084709394 \beta_{4} - 11002914962565112 \beta_{5} - 1061215706150742 \beta_{6} - 1274026863401668 \beta_{7} - 5804694080049 \beta_{8} + 12353530225991 \beta_{9} - 815848573097 \beta_{10} + 270888212429 \beta_{11} + 25899839936 \beta_{12} - 76828404084 \beta_{13} + 2082380579 \beta_{14} - 45023763844 \beta_{15} + 3935662495 \beta_{16} - 5484456704 \beta_{17}) q^{46}$$ $$+($$$$75\!\cdots\!44$$$$-$$$$68\!\cdots\!47$$$$\beta_{1} +$$$$57\!\cdots\!30$$$$\beta_{2} - 71119224455381267173 \beta_{3} + 144503050500041126 \beta_{4} + 34348091863583656 \beta_{5} + 1580480666365277 \beta_{6} + 3866443158215 \beta_{7} - 671349652202 \beta_{8} + 95670961586426 \beta_{9} - 1376635767605 \beta_{10} - 2245890936048 \beta_{11} - 34565575936 \beta_{12} + 217523938226 \beta_{13} - 15289350060 \beta_{14} + 47506759344 \beta_{15} - 7371556836 \beta_{16} - 9226882553 \beta_{17}) q^{47}$$ $$+(-$$$$13\!\cdots\!00$$$$+$$$$27\!\cdots\!64$$$$\beta_{1} -$$$$37\!\cdots\!92$$$$\beta_{2} - 20587606885765633072 \beta_{3} - 1498150713395007288 \beta_{4} - 7636271367943928 \beta_{5} + 3227936310204824 \beta_{6} + 2176104048575680 \beta_{7} + 23996571290280 \beta_{8} - 5326299224528 \beta_{9} - 1112007956560 \beta_{10} - 508594706848 \beta_{11} + 40442282872 \beta_{12} + 445109520424 \beta_{13} - 15786058272 \beta_{14} + 29671357280 \beta_{15} + 12561700192 \beta_{16} - 9245380064 \beta_{17}) q^{48}$$ $$+(-$$$$17\!\cdots\!67$$$$-$$$$16\!\cdots\!64$$$$\beta_{1} -$$$$40\!\cdots\!60$$$$\beta_{2} + 64974680729460751704 \beta_{3} - 73490545575153280 \beta_{4} - 31988968369587712 \beta_{5} - 3672340699144712 \beta_{6} + 224485664234496 \beta_{7} - 6844615205344 \beta_{8} - 102659743379844 \beta_{9} - 843684690716 \beta_{10} + 2676310235088 \beta_{11} - 31691723952 \beta_{12} - 37268939232 \beta_{13} - 52646368656 \beta_{14} - 94416540960 \beta_{15} - 5438098176 \beta_{16} - 7270843776 \beta_{17}) q^{49}$$ $$+($$$$13\!\cdots\!11$$$$-$$$$24\!\cdots\!93$$$$\beta_{1} +$$$$71\!\cdots\!56$$$$\beta_{2} + 6954395125908078696 \beta_{3} + 967688382245178760 \beta_{4} - 20077920486330832 \beta_{5} + 9159578856488098 \beta_{6} - 91013269457104 \beta_{7} + 3852097120304 \beta_{8} - 57946816669560 \beta_{9} - 896867681192 \beta_{10} - 1405040255500 \beta_{11} + 1387298312 \beta_{12} - 280172186504 \beta_{13} - 1269274368 \beta_{14} + 283313089408 \beta_{15} - 1179802880 \beta_{16} + 3217422688 \beta_{17}) q^{50}$$ $$+(-$$$$16\!\cdots\!42$$$$+$$$$15\!\cdots\!04$$$$\beta_{1} -$$$$96\!\cdots\!87$$$$\beta_{2} + 74350807330685564822 \beta_{3} - 147328051602280328 \beta_{4} - 71623686010947260 \beta_{5} - 3405101145701470 \beta_{6} - 219485792507651 \beta_{7} + 2160538855392 \beta_{8} - 273105057816187 \beta_{9} + 1714893213552 \beta_{10} + 4247614020127 \beta_{11} + 61622828032 \beta_{12} - 1566392894785 \beta_{13} + 34735095360 \beta_{14} - 100902041856 \beta_{15} + 17918088384 \beta_{16} + 24263435568 \beta_{17}) q^{51}$$ $$+($$$$55\!\cdots\!12$$$$-$$$$74\!\cdots\!78$$$$\beta_{1} -$$$$15\!\cdots\!96$$$$\beta_{2} + 63701368919034732910 \beta_{3} + 4740565962005927276 \beta_{4} + 27701204174576308 \beta_{5} - 11124863132619248 \beta_{6} - 10603998111349456 \beta_{7} - 109124019438524 \beta_{8} + 61597144876120 \beta_{9} + 3724908956232 \beta_{10} + 6356479238928 \beta_{11} - 173691171332 \beta_{12} - 1606984614668 \beta_{13} + 43864406192 \beta_{14} - 100206718320 \beta_{15} - 50207230096 \beta_{16} + 48146359568 \beta_{17}) q^{52}$$ $$+($$$$50\!\cdots\!30$$$$+$$$$77\!\cdots\!93$$$$\beta_{1} + 7862693683702622494 \beta_{2} -$$$$76\!\cdots\!79$$$$\beta_{3} - 2694882912894519145 \beta_{4} + 349832891147275270 \beta_{5} + 1710846347245597 \beta_{6} - 2575578271029784 \beta_{7} + 55402384470161 \beta_{8} + 1399967176641928 \beta_{9} + 8967043260751 \beta_{10} - 16742299770985 \beta_{11} + 315805845752 \beta_{12} + 85435197745 \beta_{13} + 250591478697 \beta_{14} + 400749914194 \beta_{15} + 50216521600 \beta_{16} + 84694341568 \beta_{17}) q^{53}$$ $$+(-$$$$43\!\cdots\!34$$$$-$$$$29\!\cdots\!90$$$$\beta_{1} +$$$$55\!\cdots\!04$$$$\beta_{2} - 75438307313786400274 \beta_{3} - 20190112483427392390 \beta_{4} + 371297319495838962 \beta_{5} - 24894944960250276 \beta_{6} + 28540168789410424 \beta_{7} + 101615919588342 \beta_{8} + 309758384452038 \beta_{9} + 18400710282534 \beta_{10} + 6042898056354 \beta_{11} - 503050916512 \beta_{12} + 3590004508440 \beta_{13} - 33969025218 \beta_{14} - 1400383206312 \beta_{15} - 67090911850 \beta_{16} + 97484267648 \beta_{17}) q^{54}$$ $$+(-$$$$26\!\cdots\!48$$$$+$$$$23\!\cdots\!48$$$$\beta_{1} -$$$$10\!\cdots\!35$$$$\beta_{2} +$$$$15\!\cdots\!93$$$$\beta_{3} - 3217270878887245944 \beta_{4} - 574527595360309074 \beta_{5} - 57928089747897900 \beta_{6} - 1055621143733053 \beta_{7} - 157130880392 \beta_{8} - 881031136941296 \beta_{9} + 29530471311412 \beta_{10} + 23074831539448 \beta_{11} + 649583529152 \beta_{12} + 8626122419600 \beta_{13} + 65636534128 \beta_{14} - 177575472576 \beta_{15} + 36040622032 \beta_{16} + 90984256756 \beta_{17}) q^{55}$$ $$+($$$$64\!\cdots\!12$$$$+$$$$18\!\cdots\!64$$$$\beta_{1} -$$$$78\!\cdots\!92$$$$\beta_{2} +$$$$29\!\cdots\!64$$$$\beta_{3} + 33111256571474917240 \beta_{4} + 177131352351020968 \beta_{5} + 59075192110332040 \beta_{6} - 19304286066576064 \beta_{7} - 12462936601352 \beta_{8} - 715352617270208 \beta_{9} + 40573314785424 \beta_{10} - 54930251359584 \beta_{11} - 721533893784 \beta_{12} + 3191788237560 \beta_{13} - 82837739616 \beta_{14} + 150791696032 \beta_{15} + 60096587296 \beta_{16} + 23924992608 \beta_{17}) q^{56}$$ $$+($$$$92\!\cdots\!55$$$$-$$$$71\!\cdots\!38$$$$\beta_{1} -$$$$19\!\cdots\!20$$$$\beta_{2} -$$$$51\!\cdots\!05$$$$\beta_{3} - 19414992901743179649 \beta_{4} + 737544835493478192 \beta_{5} - 147044270823965385 \beta_{6} - 3527307477873627 \beta_{7} - 368268471090921 \beta_{8} - 952153277654154 \beta_{9} + 62828226632910 \beta_{10} + 54289942330332 \beta_{11} + 589365643368 \beta_{12} - 1735622459940 \beta_{13} - 720789540156 \beta_{14} - 1157934229368 \beta_{15} - 141822425472 \beta_{16} - 130891644096 \beta_{17}) q^{57}$$ $$+($$$$34\!\cdots\!61$$$$-$$$$64\!\cdots\!40$$$$\beta_{1} +$$$$13\!\cdots\!08$$$$\beta_{2} -$$$$93\!\cdots\!36$$$$\beta_{3} - 38149435196399470187 \beta_{4} - 615277234680043866 \beta_{5} + 202694562527203391 \beta_{6} - 49004097137559630 \beta_{7} - 703051142785728 \beta_{8} - 2689337310009432 \beta_{9} + 67636448993696 \beta_{10} - 31456559427895 \beta_{11} - 87052353312 \beta_{12} - 14103929387744 \beta_{13} + 162506963968 \beta_{14} + 5294268039680 \beta_{15} + 337952363520 \beta_{16} - 321694931328 \beta_{17}) q^{58}$$ $$+($$$$16\!\cdots\!74$$$$-$$$$15\!\cdots\!79$$$$\beta_{1} -$$$$12\!\cdots\!71$$$$\beta_{2} +$$$$48\!\cdots\!94$$$$\beta_{3} - 10199451579769257540 \beta_{4} + 811626646360051048 \beta_{5} + 9408066911525766 \beta_{6} - 19034033083738354 \beta_{7} - 146364913994588 \beta_{8} + 8347600045367720 \beta_{9} + 84627934243654 \beta_{10} - 183314725223820 \beta_{11} - 790271148256 \beta_{12} - 42639834971608 \beta_{13} - 858985035960 \beta_{14} + 2485152451296 \beta_{15} - 444792960744 \beta_{16} - 626479982138 \beta_{17}) q^{59}$$ $$+(-$$$$64\!\cdots\!54$$$$+$$$$19\!\cdots\!96$$$$\beta_{1} -$$$$36\!\cdots\!22$$$$\beta_{2} -$$$$75\!\cdots\!34$$$$\beta_{3} - 24961238496938643316 \beta_{4} + 469761730363040968 \beta_{5} + 200259360224049514 \beta_{6} + 201441894688136650 \beta_{7} + 1199528926028802 \beta_{8} + 2746898544807574 \beta_{9} + 108810445856556 \beta_{10} + 301008313397198 \beta_{11} + 2351274498912 \beta_{12} + 10478411486614 \beta_{13} - 74767986816 \beta_{14} + 580681091850 \beta_{15} + 445858770304 \beta_{16} - 781457995136 \beta_{17}) q^{60}$$ $$+(-$$$$22\!\cdots\!62$$$$-$$$$91\!\cdots\!69$$$$\beta_{1} -$$$$24\!\cdots\!58$$$$\beta_{2} -$$$$31\!\cdots\!61$$$$\beta_{3} + 50775547794005176179 \beta_{4} - 3775540108001220742 \beta_{5} - 151997575342133025 \beta_{6} + 30262371250074276 \beta_{7} + 1544001010500979 \beta_{8} - 17007507193561044 \beta_{9} + 81455685138925 \beta_{10} - 7149730998871 \beta_{11} - 4211229107080 \beta_{12} + 4253287995359 \beta_{13} + 506183818711 \beta_{14} + 1321784257710 \beta_{15} - 154708310144 \beta_{16} - 860207303744 \beta_{17}) q^{61}$$ $$+(-$$$$49\!\cdots\!16$$$$-$$$$34\!\cdots\!72$$$$\beta_{1} +$$$$37\!\cdots\!00$$$$\beta_{2} -$$$$13\!\cdots\!88$$$$\beta_{3} + 76486709024835073784 \beta_{4} + 247964894966126904 \beta_{5} + 255618671463241376 \beta_{6} - 283441260103720864 \beta_{7} + 456531930291016 \beta_{8} + 15830232992420808 \beta_{9} + 101937812317384 \beta_{10} + 100773353939160 \beta_{11} + 6507512368064 \beta_{12} + 25540115131616 \beta_{13} - 105599529816 \beta_{14} - 13944018506976 \beta_{15} - 486306952248 \beta_{16} - 573815780096 \beta_{17}) q^{62}$$ $$+(-$$$$81\!\cdots\!16$$$$+$$$$72\!\cdots\!27$$$$\beta_{1} +$$$$12\!\cdots\!69$$$$\beta_{2} +$$$$45\!\cdots\!28$$$$\beta_{3} - 7192209052705503958 \beta_{4} + 11251277234439036906 \beta_{5} - 1870010343227356245 \beta_{6} + 112686880441195258 \beta_{7} - 218401374858726 \beta_{8} - 6618921688108506 \beta_{9} - 1778374874115 \beta_{10} + 539791821018528 \beta_{11} - 8428197322624 \beta_{12} + 160687710333294 \beta_{13} + 3033804442380 \beta_{14} - 10029746261040 \beta_{15} + 1362180065540 \beta_{16} + 324261371969 \beta_{17}) q^{63}$$ $$+($$$$13\!\cdots\!04$$$$-$$$$21\!\cdots\!56$$$$\beta_{1} -$$$$27\!\cdots\!00$$$$\beta_{2} -$$$$80\!\cdots\!28$$$$\beta_{3} -$$$$39\!\cdots\!32$$$$\beta_{4} - 2015726386994104272 \beta_{5} + 2017490550551110544 \beta_{6} - 144481969475253120 \beta_{7} - 4111638123662608 \beta_{8} - 1852548061874144 \beta_{9} - 130070289050592 \beta_{10} - 1230263430203840 \beta_{11} + 9164578618832 \beta_{12} - 99687424192528 \beta_{13} + 1895848339264 \beta_{14} - 4860268069824 \beta_{15} - 2603843288000 \beta_{16} + 1381405640896 \beta_{17}) q^{64}$$ $$+($$$$52\!\cdots\!00$$$$-$$$$13\!\cdots\!64$$$$\beta_{1} -$$$$34\!\cdots\!11$$$$\beta_{2} +$$$$26\!\cdots\!87$$$$\beta_{3} +$$$$14\!\cdots\!99$$$$\beta_{4} - 9490850304682947740 \beta_{5} - 3066035622506155085 \beta_{6} + 34716638629155413 \beta_{7} - 8323573942563139 \beta_{8} + 44112256991048401 \beta_{9} - 357975607153187 \beta_{10} - 699955026695278 \beta_{11} - 8113516544164 \beta_{12} + 21269998253122 \beta_{13} + 5900035117662 \beta_{14} + 6962439174460 \beta_{15} + 2418815530432 \beta_{16} + 3439761095392 \beta_{17}) q^{65}$$ $$+(-$$$$33\!\cdots\!29$$$$+$$$$11\!\cdots\!75$$$$\beta_{1} +$$$$60\!\cdots\!82$$$$\beta_{2} +$$$$12\!\cdots\!92$$$$\beta_{3} +$$$$75\!\cdots\!16$$$$\beta_{4} + 998394265821327584 \beta_{5} + 4892591199655828945 \beta_{6} + 644175508684991152 \beta_{7} + 7317952220365752 \beta_{8} - 52496693935579372 \beta_{9} - 650596759547012 \beta_{10} - 40768057196210 \beta_{11} + 2820301500660 \beta_{12} + 17354326074188 \beta_{13} - 1741812317568 \beta_{14} + 15716911324864 \beta_{15} - 2310396875904 \beta_{16} + 4911379165680 \beta_{17}) q^{66}$$ $$+(-$$$$17\!\cdots\!84$$$$+$$$$16\!\cdots\!47$$$$\beta_{1} +$$$$13\!\cdots\!24$$$$\beta_{2} -$$$$67\!\cdots\!26$$$$\beta_{3} +$$$$14\!\cdots\!16$$$$\beta_{4} - 12943294563462872932 \beta_{5} - 3558866273979843750 \beta_{6} + 101413758858653875 \beta_{7} + 966846409202616 \beta_{8} - 91898431586623305 \beta_{9} - 996999679165980 \beta_{10} - 352588151531715 \beta_{11} + 5786779579200 \beta_{12} - 376740700717515 \beta_{13} - 2456834669520 \beta_{14} + 15536065511232 \beta_{15} + 132509582352 \beta_{16} + 6164813671428 \beta_{17}) q^{67}$$ $$+(-$$$$20\!\cdots\!88$$$$-$$$$21\!\cdots\!38$$$$\beta_{1} +$$$$32\!\cdots\!12$$$$\beta_{2} +$$$$66\!\cdots\!42$$$$\beta_{3} -$$$$67\!\cdots\!88$$$$\beta_{4} - 2245681079579641368 \beta_{5} + 5319164060745924128 \beta_{6} - 1495602055098431392 \beta_{7} + 806417220130568 \beta_{8} + 21296217463130032 \beta_{9} - 1451974055310960 \beta_{10} + 4136344565944352 \beta_{11} - 20924669921928 \beta_{12} + 254909721146984 \beta_{13} - 10440007381152 \beta_{14} + 15784317337376 \beta_{15} + 4750604043488 \beta_{16} + 6068930741280 \beta_{17}) q^{68}$$ $$+(-$$$$21\!\cdots\!50$$$$-$$$$42\!\cdots\!71$$$$\beta_{1} -$$$$10\!\cdots\!28$$$$\beta_{2} +$$$$10\!\cdots\!87$$$$\beta_{3} -$$$$54\!\cdots\!16$$$$\beta_{4} + 41670633804094900086 \beta_{5} - 10457625291449823907 \beta_{6} - 298015499980325312 \beta_{7} + 36283648415432577 \beta_{8} + 94806600223253744 \beta_{9} - 1750797021953505 \beta_{10} + 2443650538297711 \beta_{11} + 39291410747800 \beta_{12} - 88146520139719 \beta_{13} - 30005418615471 \beta_{14} - 47444376713310 \beta_{15} - 6283230258816 \beta_{16} + 2421184699584 \beta_{17}) q^{69}$$ $$+($$$$10\!\cdots\!68$$$$+$$$$74\!\cdots\!88$$$$\beta_{1} -$$$$27\!\cdots\!72$$$$\beta_{2} +$$$$34\!\cdots\!12$$$$\beta_{3} +$$$$41\!\cdots\!40$$$$\beta_{4} - 11054442542960194676 \beta_{5} + 16132127575626777144 \beta_{6} + 2500160843648070064 \beta_{7} - 19770390526245316 \beta_{8} + 200416211905243612 \beta_{9} - 2652501813298020 \beta_{10} - 685557513297356 \beta_{11} - 60602886551232 \beta_{12} - 352623548910608 \beta_{13} + 6325200896140 \beta_{14} + 63578722598384 \beta_{15} + 13084016499324 \beta_{16} - 2612170920192 \beta_{17}) q^{70}$$ $$+(-$$$$99\!\cdots\!08$$$$+$$$$89\!\cdots\!53$$$$\beta_{1} -$$$$14\!\cdots\!15$$$$\beta_{2} -$$$$21\!\cdots\!54$$$$\beta_{3} +$$$$47\!\cdots\!38$$$$\beta_{4} -$$$$10\!\cdots\!66$$$$\beta_{5} - 20708289586439865919 \beta_{6} - 2512966885636840480 \beta_{7} + 5390785385870510 \beta_{8} + 201904879348847322 \beta_{9} - 2886616185887233 \beta_{10} - 3405712499720296 \beta_{11} + 81080607804736 \beta_{12} + 346990140524178 \beta_{13} - 21840067140540 \beta_{14} + 45843104497392 \beta_{15} - 14199549724308 \beta_{16} - 13468696915621 \beta_{17}) q^{71}$$ $$+(-$$$$14\!\cdots\!55$$$$+$$$$31\!\cdots\!19$$$$\beta_{1} +$$$$81\!\cdots\!55$$$$\beta_{2} +$$$$13\!\cdots\!61$$$$\beta_{3} +$$$$38\!\cdots\!22$$$$\beta_{4} - 15925860667379295853 \beta_{5} + 44810704634946087018 \beta_{6} + 1144759537853624320 \beta_{7} + 41462144531597312 \beta_{8} - 178138908773293671 \beta_{9} - 3267593478481920 \beta_{10} - 10654884910798848 \beta_{11} - 89943181830144 \beta_{12} + 96381707194368 \beta_{13} + 27650933932032 \beta_{14} - 16501471733760 \beta_{15} + 8937689530368 \beta_{16} - 23240802361344 \beta_{17}) q^{72}$$ $$+($$$$42\!\cdots\!97$$$$-$$$$20\!\cdots\!94$$$$\beta_{1} -$$$$52\!\cdots\!70$$$$\beta_{2} +$$$$18\!\cdots\!21$$$$\beta_{3} +$$$$98\!\cdots\!21$$$$\beta_{4} + 25237712538580887160 \beta_{5} - 47269719179402881715 \beta_{6} + 53498161889733847 \beta_{7} - 104626588484335095 \beta_{8} - 582534664131800332 \beta_{9} - 3208453142694228 \beta_{10} - 719818608831248 \beta_{11} + 88096456939504 \beta_{12} - 42516116930848 \beta_{13} + 63845696444880 \beta_{14} + 139868297180576 \beta_{15} - 6088452145408 \beta_{16} - 34556781928576 \beta_{17}) q^{73}$$ $$+(-$$$$13\!\cdots\!61$$$$+$$$$69\!\cdots\!54$$$$\beta_{1} -$$$$90\!\cdots\!00$$$$\beta_{2} +$$$$33\!\cdots\!28$$$$\beta_{3} -$$$$55\!\cdots\!67$$$$\beta_{4} + 998525906148864518 \beta_{5} + 58483735499350415013 \beta_{6} - 5973963670094041806 \beta_{7} - 14536224324512240 \beta_{8} - 1025424401821564576 \beta_{9} - 1367183588676664 \beta_{10} + 2204545733503085 \beta_{11} - 51966563476904 \beta_{12} + 1361837331190056 \beta_{13} + 1405916750592 \beta_{14} - 431217545126272 \beta_{15} - 17785516384000 \beta_{16} - 39944070514912 \beta_{17}) q^{74}$$ $$+(-$$$$51\!\cdots\!34$$$$+$$$$46\!\cdots\!61$$$$\beta_{1} -$$$$35\!\cdots\!09$$$$\beta_{2} +$$$$46\!\cdots\!34$$$$\beta_{3} + 57184061496926020940 \beta_{4} + 46766757170931030048 \beta_{5} -$$$$10\!\cdots\!82$$$$\beta_{6} + 10775042133607046788 \beta_{7} + 5789247694426788 \beta_{8} + 587178053418802254 \beta_{9} + 529341575172390 \beta_{10} + 13403168397487878 \beta_{11} - 9261530113504 \beta_{12} + 1064717597377914 \beta_{13} + 95873769472200 \beta_{14} - 333779833527072 \beta_{15} + 40243797217688 \beta_{16} - 31701627158554 \beta_{17}) q^{75}$$ $$+($$$$19\!\cdots\!65$$$$-$$$$55\!\cdots\!64$$$$\beta_{1} +$$$$22\!\cdots\!65$$$$\beta_{2} +$$$$35\!\cdots\!47$$$$\beta_{3} +$$$$79\!\cdots\!62$$$$\beta_{4} + 77148203541963614516 \beta_{5} +$$$$10\!\cdots\!69$$$$\beta_{6} + 14189311443038312045 \beta_{7} - 88998524613471591 \beta_{8} + 234088576620934947 \beta_{9} + 2728165526575974 \beta_{10} + 16228988549455551 \beta_{11} + 117007335391040 \beta_{12} - 2152537772613197 \beta_{13} + 5300035967232 \beta_{14} - 83437357193459 \beta_{15} - 64431481094912 \beta_{16} - 18732218445056 \beta_{17}) q^{76}$$ $$+(-$$$$62\!\cdots\!78$$$$-$$$$60\!\cdots\!67$$$$\beta_{1} -$$$$15\!\cdots\!84$$$$\beta_{2} -$$$$35\!\cdots\!17$$$$\beta_{3} +$$$$16\!\cdots\!44$$$$\beta_{4} -$$$$15\!\cdots\!94$$$$\beta_{5} -$$$$13\!\cdots\!47$$$$\beta_{6} + 975935802224923632 \beta_{7} + 300514311187558925 \beta_{8} + 131291475325607232 \beta_{9} + 7650116977321043 \beta_{10} - 16415527419393869 \beta_{11} - 259061860012040 \beta_{12} + 488909480872085 \beta_{13} + 2385969543693 \beta_{14} - 146869161790182 \beta_{15} + 75820550438784 \beta_{16} + 27808217518016 \beta_{17}) q^{77}$$ $$+(-$$$$80\!\cdots\!81$$$$-$$$$61\!\cdots\!32$$$$\beta_{1} -$$$$39\!\cdots\!10$$$$\beta_{2} -$$$$35\!\cdots\!76$$$$\beta_{3} -$$$$13\!\cdots\!38$$$$\beta_{4} - 5683830245117703248 \beta_{5} +$$$$23\!\cdots\!34$$$$\beta_{6} - 19919234838748490020 \beta_{7} + 189077452254028935 \beta_{8} + 3584918445173730687 \beta_{9} + 11247619924712079 \beta_{10} - 4829822659091211 \beta_{11} + 418978260591616 \beta_{12} - 1250750288907348 \beta_{13} - 63191667947397 \beta_{14} + 1278137468837724 \beta_{15} - 63230937471017 \beta_{16} + 70299852845056 \beta_{17}) q^{78}$$ $$+(-$$$$10\!\cdots\!08$$$$+$$$$93\!\cdots\!12$$$$\beta_{1} +$$$$83\!\cdots\!22$$$$\beta_{2} +$$$$11\!\cdots\!66$$$$\beta_{3} -$$$$20\!\cdots\!24$$$$\beta_{4} +$$$$61\!\cdots\!60$$$$\beta_{5} -$$$$25\!\cdots\!92$$$$\beta_{6} - 32560385956787059970 \beta_{7} - 51219948171521600 \beta_{8} - 2307362245537835400 \beta_{9} + 18379746723036680 \beta_{10} - 10879546500823992 \beta_{11} - 597100690078656 \beta_{12} - 6826611721641192 \beta_{13} - 120745368877920 \beta_{14} + 767880073952640 \beta_{15} + 7234643447520 \beta_{16} + 135230111115960 \beta_{17}) q^{79}$$ $$+($$$$21\!\cdots\!68$$$$-$$$$20\!\cdots\!72$$$$\beta_{1} +$$$$33\!\cdots\!68$$$$\beta_{2} -$$$$70\!\cdots\!52$$$$\beta_{3} -$$$$13\!\cdots\!94$$$$\beta_{4} -$$$$21\!\cdots\!66$$$$\beta_{5} +$$$$34\!\cdots\!02$$$$\beta_{6} - 15132732747182510320 \beta_{7} - 138187505980220594 \beta_{8} + 637679266351988292 \beta_{9} + 31698224898767908 \beta_{10} + 1924459154935944 \beta_{11} + 702159644550026 \beta_{12} + 5470911858838382 \beta_{13} - 320738232529368 \beta_{14} + 438503521379400 \beta_{15} + 98917897979592 \beta_{16} + 196548603310552 \beta_{17}) q^{80}$$ $$+($$$$49\!\cdots\!06$$$$-$$$$28\!\cdots\!90$$$$\beta_{1} -$$$$75\!\cdots\!44$$$$\beta_{2} -$$$$32\!\cdots\!71$$$$\beta_{3} -$$$$61\!\cdots\!59$$$$\beta_{4} -$$$$50\!\cdots\!00$$$$\beta_{5} -$$$$63\!\cdots\!15$$$$\beta_{6} + 1587946726186772115 \beta_{7} - 780695832184605039 \beta_{8} + 3956588750450247354 \beta_{9} + 38184402946964130 \beta_{10} + 41108586244617348 \beta_{11} - 762111064562472 \beta_{12} - 87417311195772 \beta_{13} - 424079188047396 \beta_{14} - 564525584287560 \beta_{15} - 141816395903616 \beta_{16} + 205312088827584 \beta_{17}) q^{81}$$ $$+($$$$19\!\cdots\!88$$$$+$$$$23\!\cdots\!12$$$$\beta_{1} -$$$$16\!\cdots\!04$$$$\beta_{2} -$$$$11\!\cdots\!68$$$$\beta_{3} +$$$$18\!\cdots\!76$$$$\beta_{4} +$$$$10\!\cdots\!08$$$$\beta_{5} +$$$$61\!\cdots\!26$$$$\beta_{6} + 52472112857610921168 \beta_{7} - 235463705559423888 \beta_{8} - 8862359746164901016 \beta_{9} + 53787317087411576 \beta_{10} + 8956017315601300 \beta_{11} + 619977510362472 \beta_{12} - 7956520404671720 \beta_{13} + 124178621838592 \beta_{14} - 1735988236008064 \beta_{15} + 298133502650112 \beta_{16} + 209078248283616 \beta_{17}) q^{82}$$ $$+(-$$$$37\!\cdots\!74$$$$+$$$$33\!\cdots\!35$$$$\beta_{1} +$$$$17\!\cdots\!79$$$$\beta_{2} +$$$$29\!\cdots\!30$$$$\beta_{3} -$$$$50\!\cdots\!12$$$$\beta_{4} +$$$$97\!\cdots\!20$$$$\beta_{5} -$$$$80\!\cdots\!34$$$$\beta_{6} + 99448401974222575578 \beta_{7} - 106521009913177628 \beta_{8} - 2109006422015903036 \beta_{9} + 50658945475513750 \beta_{10} - 61259818012126072 \beta_{11} - 315433324910944 \beta_{12} + 22386682299592924 \beta_{13} - 347917711645560 \beta_{14} + 292339206583776 \beta_{15} - 299194510548264 \beta_{16} + 79012450722358 \beta_{17}) q^{83}$$ $$+(-$$$$37\!\cdots\!64$$$$+$$$$75\!\cdots\!32$$$$\beta_{1} +$$$$23\!\cdots\!28$$$$\beta_{2} +$$$$15\!\cdots\!96$$$$\beta_{3} -$$$$67\!\cdots\!64$$$$\beta_{4} +$$$$10\!\cdots\!60$$$$\beta_{5} +$$$$11\!\cdots\!84$$$$\beta_{6} - 92649489190446168256 \beta_{7} + 745659866458031088 \beta_{8} - 1011192858175456096 \beta_{9} + 54491107014815968 \beta_{10} - 87606320801290304 \beta_{11} - 256816230415600 \beta_{12} - 3057443386041040 \beta_{13} + 1069348074748224 \beta_{14} - 806365148460608 \beta_{15} + 186612646047296 \beta_{16} - 74804657799232 \beta_{17}) q^{84}$$ $$+(-$$$$17\!\cdots\!70$$$$-$$$$47\!\cdots\!39$$$$\beta_{1} -$$$$12\!\cdots\!14$$$$\beta_{2} -$$$$30\!\cdots\!63$$$$\beta_{3} +$$$$13\!\cdots\!56$$$$\beta_{4} +$$$$11\!\cdots\!50$$$$\beta_{5} -$$$$10\!\cdots\!05$$$$\beta_{6} - 4076554429741528722 \beta_{7} + 1012405759995324051 \beta_{8} - 5301517447135396254 \beta_{9} + 40491885929817773 \beta_{10} + 11644456555744247 \beta_{11} + 1116010682292656 \beta_{12} - 607466528257543 \beta_{13} + 1253100763550377 \beta_{14} + 3002572600110930 \beta_{15} - 248185536505088 \beta_{16} - 349518403883648 \beta_{17}) q^{85}$$ $$+($$$$48\!\cdots\!82$$$$+$$$$35\!\cdots\!53$$$$\beta_{1} +$$$$82\!\cdots\!28$$$$\beta_{2} +$$$$19\!\cdots\!47$$$$\beta_{3} +$$$$90\!\cdots\!35$$$$\beta_{4} +$$$$84\!\cdots\!85$$$$\beta_{5} +$$$$12\!\cdots\!68$$$$\beta_{6} +$$$$10\!\cdots\!28$$$$\beta_{7} - 981683284644459804 \beta_{8} + 22614042167298310340 \beta_{9} + 32390783330225924 \beta_{10} + 14885286608509228 \beta_{11} - 2102290690625984 \beta_{12} + 26565441329948432 \beta_{13} + 262511788029972 \beta_{14} - 2369798506107248 \beta_{15} - 256879037140316 \beta_{16} - 628919355921664 \beta_{17}) q^{86}$$ $$+(-$$$$94\!\cdots\!72$$$$+$$$$84\!\cdots\!97$$$$\beta_{1} -$$$$14\!\cdots\!17$$$$\beta_{2} +$$$$20\!\cdots\!80$$$$\beta_{3} +$$$$30\!\cdots\!70$$$$\beta_{4} -$$$$44\!\cdots\!02$$$$\beta_{5} -$$$$20\!\cdots\!51$$$$\beta_{6} -$$$$29\!\cdots\!58$$$$\beta_{7} + 240499209666021318 \beta_{8} + 16898879336886029810 \beta_{9} + 26908228929779835 \beta_{10} + 160029477586770280 \beta_{11} + 3355770824760000 \beta_{12} - 41091020754078070 \beta_{13} + 1723985693129940 \beta_{14} - 6913890923709264 \beta_{15} + 571670539178396 \beta_{16} - 858421086473881 \beta_{17}) q^{87}$$ $$+(-$$$$45\!\cdots\!24$$$$+$$$$57\!\cdots\!88$$$$\beta_{1} -$$$$23\!\cdots\!80$$$$\beta_{2} +$$$$23\!\cdots\!56$$$$\beta_{3} -$$$$51\!\cdots\!28$$$$\beta_{4} -$$$$28\!\cdots\!80$$$$\beta_{5} +$$$$21\!\cdots\!24$$$$\beta_{6} + 95684883987577853472 \beta_{7} - 423364896682609972 \beta_{8} - 66448335055870048 \beta_{9} - 50090779438260568 \beta_{10} + 333162078345165584 \beta_{11} - 4321971019437660 \beta_{12} - 24320386206402100 \beta_{13} - 652014553460848 \beta_{14} - 614175917015792 \beta_{15} - 900941899522608 \beta_{16} - 1098262988855952 \beta_{17}) q^{88}$$ $$+(-$$$$34\!\cdots\!71$$$$-$$$$11\!\cdots\!30$$$$\beta_{1} -$$$$28\!\cdots\!22$$$$\beta_{2} +$$$$24\!\cdots\!61$$$$\beta_{3} -$$$$65\!\cdots\!95$$$$\beta_{4} +$$$$37\!\cdots\!08$$$$\beta_{5} -$$$$25\!\cdots\!95$$$$\beta_{6} - 16030478154197728917 \beta_{7} + 262418964555464749 \beta_{8} - 16714079773950666456 \beta_{9} - 59399338912643648 \beta_{10} - 247779763810762360 \beta_{11} + 5195414571567392 \beta_{12} - 2064803048362280 \beta_{13} - 1439079523024392 \beta_{14} - 5924619743535120 \beta_{15} + 1523230348743168 \beta_{16} - 885342535538432 \beta_{17}) q^{89}$$ $$+(-$$$$12\!\cdots\!81$$$$-$$$$42\!\cdots\!30$$$$\beta_{1} +$$$$59\!\cdots\!28$$$$\beta_{2} +$$$$53\!\cdots\!80$$$$\beta_{3} +$$$$82\!\cdots\!41$$$$\beta_{4} -$$$$20\!\cdots\!58$$$$\beta_{5} +$$$$42\!\cdots\!33$$$$\beta_{6} -$$$$32\!\cdots\!18$$$$\beta_{7} + 2252114286936554672 \beta_{8} - 56307000918760424688 \beta_{9} - 232322912782166376 \beta_{10} - 129245053225597491 \beta_{11} - 5193715376000824 \beta_{12} - 4469095089431112 \beta_{13} - 1391531168112384 \beta_{14} + 18569203153313664 \beta_{15} - 1291853556839680 \beta_{16} - 771975659556256 \beta_{17}) q^{90}$$ $$+(-$$$$17\!\cdots\!92$$$$+$$$$15\!\cdots\!90$$$$\beta_{1} -$$$$17\!\cdots\!26$$$$\beta_{2} -$$$$19\!\cdots\!66$$$$\beta_{3} +$$$$46\!\cdots\!08$$$$\beta_{4} -$$$$56\!\cdots\!92$$$$\beta_{5} -$$$$33\!\cdots\!42$$$$\beta_{6} +$$$$81\!\cdots\!80$$$$\beta_{7} + 554033524706523500 \beta_{8} + 1622486446496594154 \beta_{9} - 288842681783621966 \beta_{10} + 110171452338950034 \beta_{11} + 4345228997251680 \beta_{12} + 8367030887852142 \beta_{13} - 2339948084240040 \beta_{14} + 17772456052273824 \beta_{15} + 622127924472264 \beta_{16} + 57314865704178 \beta_{17}) q^{91}$$ $$+($$$$63\!\cdots\!02$$$$-$$$$12\!\cdots\!76$$$$\beta_{1} -$$$$47\!\cdots\!74$$$$\beta_{2} -$$$$59\!\cdots\!86$$$$\beta_{3} +$$$$19\!\cdots\!12$$$$\beta_{4} +$$$$90\!\cdots\!12$$$$\beta_{5} +$$$$44\!\cdots\!70$$$$\beta_{6} +$$$$47\!\cdots\!50$$$$\beta_{7} - 3806165330946894678 \beta_{8} - 8605580908910434098 \beta_{9} - 378076584725359204 \beta_{10} - 925834819413156250 \beta_{11} - 2205187695332384 \beta_{12} + 83905969758927054 \beta_{13} - 6155179765659264 \beta_{14} + 6932127911550802 \beta_{15} + 508625234644864 \beta_{16} + 1077686197389440 \beta_{17}) q^{92}$$ $$+($$$$23\!\cdots\!12$$$$-$$$$24\!\cdots\!60$$$$\beta_{1} -$$$$61\!\cdots\!64$$$$\beta_{2} +$$$$33\!\cdots\!76$$$$\beta_{3} +$$$$13\!\cdots\!48$$$$\beta_{4} -$$$$25\!\cdots\!76$$$$\beta_{5} -$$$$54\!\cdots\!52$$$$\beta_{6} - 5832006066216111084 \beta_{7} - 4240330917040857960 \beta_{8} + 39647065339347692620 \beta_{9} - 504452239546289080 \beta_{10} + 361843209406157252 \beta_{11} - 1707568984959184 \beta_{12} - 4333645435719092 \beta_{13} - 1561974299840388 \beta_{14} - 137788519122184 \beta_{15} - 1493080040279296 \beta_{16} + 2090000872380800 \beta_{17}) q^{93}$$ $$+(-$$$$18\!\cdots\!70$$$$-$$$$13\!\cdots\!80$$$$\beta_{1} +$$$$98\!\cdots\!16$$$$\beta_{2} -$$$$53\!\cdots\!36$$$$\beta_{3} -$$$$63\!\cdots\!28$$$$\beta_{4} -$$$$30\!\cdots\!88$$$$\beta_{5} +$$$$39\!\cdots\!00$$$$\beta_{6} -$$$$45\!\cdots\!48$$$$\beta_{7} + 3066736262196322090 \beta_{8} + 92368539708268698298 \beta_{9} - 461880669528329318 \beta_{10} + 185093451085963198 \beta_{11} + 6566633566675520 \beta_{12} - 132784372218524728 \beta_{13} + 444438194599906 \beta_{14} - 45339982517091288 \beta_{15} + 3912730575095818 \beta_{16} + 3765351353424640 \beta_{17}) q^{94}$$ $$+(-$$$$17\!\cdots\!32$$$$+$$$$15\!\cdots\!80$$$$\beta_{1} +$$$$67\!\cdots\!39$$$$\beta_{2} -$$$$40\!\cdots\!53$$$$\beta_{3} +$$$$90\!\cdots\!12$$$$\beta_{4} +$$$$11\!\cdots\!94$$$$\beta_{5} -$$$$47\!\cdots\!68$$$$\beta_{6} -$$$$20\!\cdots\!39$$$$\beta_{7} - 571817846590611752 \beta_{8} - 84843377493485481336 \beta_{9} - 717158498100520436 \beta_{10} - 881310178827948512 \beta_{11} - 13561839902931712 \beta_{12} + 164333653851000936 \beta_{13} - 3462883422687024 \beta_{14} - 1617695920653120 \beta_{15} - 3732499409462544 \beta_{16} + 4264037866132156 \beta_{17}) q^{95}$$ $$+(-$$$$17\!\cdots\!96$$$$+$$$$12\!\cdots\!36$$$$\beta_{1} -$$$$15\!\cdots\!68$$$$\beta_{2} +$$$$24\!\cdots\!80$$$$\beta_{3} +$$$$95\!\cdots\!80$$$$\beta_{4} -$$$$30\!\cdots\!60$$$$\beta_{5} +$$$$48\!\cdots\!80$$$$\beta_{6} - 92522079112930579712 \beta_{7} + 7024088842986167712 \beta_{8} + 22161778729088491712 \beta_{9} - 664761868027920192 \beta_{10} + 1666886744407340416 \beta_{11} + 20056542547179232 \beta_{12} - 52561315134837856 \beta_{13} + 20308860961946496 \beta_{14} - 14819339743733376 \beta_{15} + 3312332014872960 \beta_{16} + 4836186672833664 \beta_{17}) q^{96}$$ $$+(-$$$$34\!\cdots\!13$$$$-$$$$16\!\cdots\!38$$$$\beta_{1} -$$$$41\!\cdots\!33$$$$\beta_{2} +$$$$71\!\cdots\!98$$$$\beta_{3} +$$$$92\!\cdots\!46$$$$\beta_{4} -$$$$22\!\cdots\!40$$$$\beta_{5} -$$$$39\!\cdots\!14$$$$\beta_{6} +$$$$10\!\cdots\!46$$$$\beta_{7} + 16369584114194284400 \beta_{8} + 61056367787629243809 \beta_{9} - 855753865671960299 \beta_{10} + 775643183816144346 \beta_{11} - 27201641550141348 \beta_{12} + 19812310202691066 \beta_{13} + 10265902011139350 \beta_{14} + 32301959594671788 \beta_{15} - 5885077786196544 \beta_{16} + 3778108000035552 \beta_{17}) q^{97}$$ $$+($$$$44\!\cdots\!79$$$$+$$$$20\!\cdots\!23$$$$\beta_{1} +$$$$88\!\cdots\!92$$$$\beta_{2} -$$$$14\!\cdots\!92$$$$\beta_{3} -$$$$67\!\cdots\!64$$$$\beta_{4} -$$$$54\!\cdots\!52$$$$\beta_{5} +$$$$42\!\cdots\!40$$$$\beta_{6} +$$$$12\!\cdots\!12$$$$\beta_{7} - 11137556872419736416 \beta_{8} - 63127411849680732176 \beta_{9} - 558948744436419120 \beta_{10} + 210225868422239880 \beta_{11} + 31652252823613296 \beta_{12} + 262169884376286096 \beta_{13} + 8423548381011456 \beta_{14} + 41145602004955392 \beta_{15} - 664475195241984 \beta_{16} + 2178000917268288 \beta_{17}) q^{98}$$ $$+(-$$$$28\!\cdots\!54$$$$+$$$$25\!\cdots\!73$$$$\beta_{1} +$$$$74\!\cdots\!65$$$$\beta_{2} +$$$$26\!\cdots\!10$$$$\beta_{3} +$$$$97\!\cdots\!16$$$$\beta_{4} +$$$$49\!\cdots\!92$$$$\beta_{5} -$$$$49\!\cdots\!50$$$$\beta_{6} +$$$$46\!\cdots\!86$$$$\beta_{7} - 1009085372086197372 \beta_{8} + 12675154800278245008 \beta_{9} - 138035369955975786 \beta_{10} + 325596710186470092 \beta_{11} - 32873487782049248 \beta_{12} - 525802504145485344 \beta_{13} + 19403007602812680 \beta_{14} - 105419674236138528 \beta_{15} + 1833061896789592 \beta_{16} - 1348581994745066 \beta_{17}) q^{99}$$ $$+O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$18q$$ $$\mathstrut +\mathstrut 364228q^{2}$$ $$\mathstrut +\mathstrut 200567335824q^{4}$$ $$\mathstrut -\mathstrut 8991287507020q^{5}$$ $$\mathstrut +\mathstrut 817599417526752q^{6}$$ $$\mathstrut +\mathstrut 90410803724813632q^{8}$$ $$\mathstrut -\mathstrut 7529771892957713214q^{9}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$18q$$ $$\mathstrut +\mathstrut 364228q^{2}$$ $$\mathstrut +\mathstrut 200567335824q^{4}$$ $$\mathstrut -\mathstrut 8991287507020q^{5}$$ $$\mathstrut +\mathstrut 817599417526752q^{6}$$ $$\mathstrut +\mathstrut 90410803724813632q^{8}$$ $$\mathstrut -\mathstrut 7529771892957713214q^{9}$$ $$\mathstrut -\mathstrut 3330064307292995160q^{10}$$ $$\mathstrut +\mathstrut$$$$21\!\cdots\!80$$$$q^{12}$$ $$\mathstrut -\mathstrut$$$$13\!\cdots\!56$$$$q^{13}$$ $$\mathstrut -\mathstrut$$$$75\!\cdots\!28$$$$q^{14}$$ $$\mathstrut +\mathstrut$$$$14\!\cdots\!88$$$$q^{16}$$ $$\mathstrut -\mathstrut$$$$10\!\cdots\!44$$$$q^{17}$$ $$\mathstrut -\mathstrut$$$$22\!\cdots\!28$$$$q^{18}$$ $$\mathstrut +\mathstrut$$$$14\!\cdots\!60$$$$q^{20}$$ $$\mathstrut -\mathstrut$$$$36\!\cdots\!76$$$$q^{21}$$ $$\mathstrut -\mathstrut$$$$48\!\cdots\!60$$$$q^{22}$$ $$\mathstrut -\mathstrut$$$$50\!\cdots\!48$$$$q^{24}$$ $$\mathstrut +\mathstrut$$$$44\!\cdots\!90$$$$q^{25}$$ $$\mathstrut +\mathstrut$$$$14\!\cdots\!84$$$$q^{26}$$ $$\mathstrut -\mathstrut$$$$24\!\cdots\!80$$$$q^{28}$$ $$\mathstrut +\mathstrut$$$$12\!\cdots\!48$$$$q^{29}$$ $$\mathstrut -\mathstrut$$$$36\!\cdots\!80$$$$q^{30}$$ $$\mathstrut +\mathstrut$$$$40\!\cdots\!28$$$$q^{32}$$ $$\mathstrut -\mathstrut$$$$25\!\cdots\!40$$$$q^{33}$$ $$\mathstrut +\mathstrut$$$$36\!\cdots\!76$$$$q^{34}$$ $$\mathstrut -\mathstrut$$$$58\!\cdots\!80$$$$q^{36}$$ $$\mathstrut -\mathstrut$$$$12\!\cdots\!44$$$$q^{37}$$ $$\mathstrut +\mathstrut$$$$10\!\cdots\!60$$$$q^{38}$$ $$\mathstrut +\mathstrut$$$$36\!\cdots\!20$$$$q^{40}$$ $$\mathstrut -\mathstrut$$$$42\!\cdots\!84$$$$q^{41}$$ $$\mathstrut -\mathstrut$$$$14\!\cdots\!40$$$$q^{42}$$ $$\mathstrut -\mathstrut$$$$10\!\cdots\!60$$$$q^{44}$$ $$\mathstrut +\mathstrut$$$$59\!\cdots\!40$$$$q^{45}$$ $$\mathstrut +\mathstrut$$$$22\!\cdots\!52$$$$q^{46}$$ $$\mathstrut -\mathstrut$$$$23\!\cdots\!00$$$$q^{48}$$ $$\mathstrut -\mathstrut$$$$31\!\cdots\!94$$$$q^{49}$$ $$\mathstrut +\mathstrut$$$$24\!\cdots\!80$$$$q^{50}$$ $$\mathstrut +\mathstrut$$$$10\!\cdots\!76$$$$q^{52}$$ $$\mathstrut +\mathstrut$$$$91\!\cdots\!64$$$$q^{53}$$ $$\mathstrut -\mathstrut$$$$78\!\cdots\!96$$$$q^{54}$$ $$\mathstrut +\mathstrut$$$$11\!\cdots\!12$$$$q^{56}$$ $$\mathstrut +\mathstrut$$$$16\!\cdots\!00$$$$q^{57}$$ $$\mathstrut +\mathstrut$$$$61\!\cdots\!24$$$$q^{58}$$ $$\mathstrut -\mathstrut$$$$11\!\cdots\!20$$$$q^{60}$$ $$\mathstrut -\mathstrut$$$$39\!\cdots\!44$$$$q^{61}$$ $$\mathstrut -\mathstrut$$$$89\!\cdots\!60$$$$q^{62}$$ $$\mathstrut +\mathstrut$$$$24\!\cdots\!64$$$$q^{64}$$ $$\mathstrut +\mathstrut$$$$95\!\cdots\!60$$$$q^{65}$$ $$\mathstrut -\mathstrut$$$$59\!\cdots\!40$$$$q^{66}$$ $$\mathstrut -\mathstrut$$$$37\!\cdots\!16$$$$q^{68}$$ $$\mathstrut -\mathstrut$$$$37\!\cdots\!56$$$$q^{69}$$ $$\mathstrut +\mathstrut$$$$18\!\cdots\!00$$$$q^{70}$$ $$\mathstrut -\mathstrut$$$$26\!\cdots\!52$$$$q^{72}$$ $$\mathstrut +\mathstrut$$$$76\!\cdots\!64$$$$q^{73}$$ $$\mathstrut -\mathstrut$$$$25\!\cdots\!24$$$$q^{74}$$ $$\mathstrut +\mathstrut$$$$34\!\cdots\!00$$$$q^{76}$$ $$\mathstrut -\mathstrut$$$$11\!\cdots\!20$$$$q^{77}$$ $$\mathstrut -\mathstrut$$$$14\!\cdots\!60$$$$q^{78}$$ $$\mathstrut +\mathstrut$$$$39\!\cdots\!20$$$$q^{80}$$ $$\mathstrut +\mathstrut$$$$88\!\cdots\!82$$$$q^{81}$$ $$\mathstrut +\mathstrut$$$$34\!\cdots\!76$$$$q^{82}$$ $$\mathstrut -\mathstrut$$$$67\!\cdots\!16$$$$q^{84}$$ $$\mathstrut -\mathstrut$$$$31\!\cdots\!00$$$$q^{85}$$ $$\mathstrut +\mathstrut$$$$86\!\cdots\!32$$$$q^{86}$$ $$\mathstrut -\mathstrut$$$$81\!\cdots\!80$$$$q^{88}$$ $$\mathstrut -\mathstrut$$$$62\!\cdots\!92$$$$q^{89}$$ $$\mathstrut -\mathstrut$$$$23\!\cdots\!00$$$$q^{90}$$ $$\mathstrut +\mathstrut$$$$11\!\cdots\!60$$$$q^{92}$$ $$\mathstrut +\mathstrut$$$$41\!\cdots\!00$$$$q^{93}$$ $$\mathstrut -\mathstrut$$$$33\!\cdots\!88$$$$q^{94}$$ $$\mathstrut -\mathstrut$$$$31\!\cdots\!28$$$$q^{96}$$ $$\mathstrut -\mathstrut$$$$62\!\cdots\!44$$$$q^{97}$$ $$\mathstrut +\mathstrut$$$$79\!\cdots\!72$$$$q^{98}$$ $$\mathstrut +\mathstrut O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{18}\mathstrut -\mathstrut$$ $$9$$ $$x^{17}\mathstrut +\mathstrut$$ $$62197466427874200$$ $$x^{16}\mathstrut -\mathstrut$$ $$497579731422993396$$ $$x^{15}\mathstrut +\mathstrut$$ $$15\!\cdots\!58$$ $$x^{14}\mathstrut -\mathstrut$$ $$10\!\cdots\!90$$ $$x^{13}\mathstrut +\mathstrut$$ $$19\!\cdots\!44$$ $$x^{12}\mathstrut -\mathstrut$$ $$11\!\cdots\!60$$ $$x^{11}\mathstrut +\mathstrut$$ $$13\!\cdots\!01$$ $$x^{10}\mathstrut -\mathstrut$$ $$69\!\cdots\!53$$ $$x^{9}\mathstrut +\mathstrut$$ $$56\!\cdots\!48$$ $$x^{8}\mathstrut -\mathstrut$$ $$22\!\cdots\!96$$ $$x^{7}\mathstrut +\mathstrut$$ $$12\!\cdots\!80$$ $$x^{6}\mathstrut -\mathstrut$$ $$38\!\cdots\!64$$ $$x^{5}\mathstrut +\mathstrut$$ $$14\!\cdots\!12$$ $$x^{4}\mathstrut -\mathstrut$$ $$28\!\cdots\!36$$ $$x^{3}\mathstrut +\mathstrut$$ $$59\!\cdots\!60$$ $$x^{2}\mathstrut -\mathstrut$$ $$59\!\cdots\!00$$ $$x\mathstrut +\mathstrut$$ $$98\!\cdots\!00$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$46\!\cdots\!75$$ $$\nu^{17}\mathstrut +\mathstrut$$ $$35\!\cdots\!68$$ $$\nu^{16}\mathstrut +\mathstrut$$ $$26\!\cdots\!56$$ $$\nu^{15}\mathstrut +\mathstrut$$ $$33\!\cdots\!32$$ $$\nu^{14}\mathstrut +\mathstrut$$ $$58\!\cdots\!46$$ $$\nu^{13}\mathstrut +\mathstrut$$ $$11\!\cdots\!76$$ $$\nu^{12}\mathstrut +\mathstrut$$ $$60\!\cdots\!44$$ $$\nu^{11}\mathstrut +\mathstrut$$ $$18\!\cdots\!72$$ $$\nu^{10}\mathstrut +\mathstrut$$ $$31\!\cdots\!43$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$15\!\cdots\!72$$ $$\nu^{8}\mathstrut +\mathstrut$$ $$68\!\cdots\!92$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$65\!\cdots\!36$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$33\!\cdots\!40$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$12\!\cdots\!48$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$17\!\cdots\!80$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$62\!\cdots\!80$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$17\!\cdots\!00$$ $$\nu\mathstrut -\mathstrut$$ $$16\!\cdots\!00$$$$)/$$$$37\!\cdots\!00$$ $$\beta_{2}$$ $$=$$ $$($$$$46\!\cdots\!75$$ $$\nu^{17}\mathstrut +\mathstrut$$ $$35\!\cdots\!68$$ $$\nu^{16}\mathstrut +\mathstrut$$ $$26\!\cdots\!56$$ $$\nu^{15}\mathstrut +\mathstrut$$ $$33\!\cdots\!32$$ $$\nu^{14}\mathstrut +\mathstrut$$ $$58\!\cdots\!46$$ $$\nu^{13}\mathstrut +\mathstrut$$ $$11\!\cdots\!76$$ $$\nu^{12}\mathstrut +\mathstrut$$ $$60\!\cdots\!44$$ $$\nu^{11}\mathstrut +\mathstrut$$ $$18\!\cdots\!72$$ $$\nu^{10}\mathstrut +\mathstrut$$ $$31\!\cdots\!43$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$15\!\cdots\!72$$ $$\nu^{8}\mathstrut +\mathstrut$$ $$68\!\cdots\!92$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$65\!\cdots\!36$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$33\!\cdots\!40$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$12\!\cdots\!48$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$17\!\cdots\!80$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$62\!\cdots\!80$$ $$\nu^{2}\mathstrut +\mathstrut$$ $$19\!\cdots\!00$$ $$\nu\mathstrut -\mathstrut$$ $$16\!\cdots\!00$$$$)/$$$$22\!\cdots\!00$$ $$\beta_{3}$$ $$=$$ $$($$$$14\!\cdots\!97$$ $$\nu^{17}\mathstrut -\mathstrut$$ $$63\!\cdots\!68$$ $$\nu^{16}\mathstrut +\mathstrut$$ $$82\!\cdots\!40$$ $$\nu^{15}\mathstrut -\mathstrut$$ $$39\!\cdots\!44$$ $$\nu^{14}\mathstrut +\mathstrut$$ $$17\!\cdots\!38$$ $$\nu^{13}\mathstrut -\mathstrut$$ $$94\!\cdots\!00$$ $$\nu^{12}\mathstrut +\mathstrut$$ $$18\!\cdots\!56$$ $$\nu^{11}\mathstrut -\mathstrut$$ $$11\!\cdots\!60$$ $$\nu^{10}\mathstrut +\mathstrut$$ $$93\!\cdots\!17$$ $$\nu^{9}\mathstrut -\mathstrut$$ $$74\!\cdots\!00$$ $$\nu^{8}\mathstrut +\mathstrut$$ $$21\!\cdots\!52$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$25\!\cdots\!96$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$37\!\cdots\!60$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$41\!\cdots\!80$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$57\!\cdots\!52$$ $$\nu^{3}\mathstrut -\mathstrut$$ $$23\!\cdots\!60$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$54\!\cdots\!00$$ $$\nu\mathstrut -\mathstrut$$ $$27\!\cdots\!00$$$$)/$$$$12\!\cdots\!00$$ $$\beta_{4}$$ $$=$$ $$($$$$-$$$$25\!\cdots\!26$$ $$\nu^{17}\mathstrut +\mathstrut$$ $$31\!\cdots\!11$$ $$\nu^{16}\mathstrut -\mathstrut$$ $$14\!\cdots\!56$$ $$\nu^{15}\mathstrut +\mathstrut$$ $$18\!\cdots\!72$$ $$\nu^{14}\mathstrut -\mathstrut$$ $$30\!\cdots\!64$$ $$\nu^{13}\mathstrut +\mathstrut$$ $$43\!\cdots\!38$$ $$\nu^{12}\mathstrut -\mathstrut$$ $$31\!\cdots\!84$$ $$\nu^{11}\mathstrut +\mathstrut$$ $$50\!\cdots\!40$$ $$\nu^{10}\mathstrut -\mathstrut$$ $$16\!\cdots\!46$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$31\!\cdots\!99$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$37\!\cdots\!40$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$10\!\cdots\!88$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$68\!\cdots\!56$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$16\!\cdots\!60$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$10\!\cdots\!84$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$96\!\cdots\!80$$ $$\nu^{2}\mathstrut +\mathstrut$$ $$96\!\cdots\!00$$ $$\nu\mathstrut +\mathstrut$$ $$64\!\cdots\!00$$$$)/$$$$23\!\cdots\!00$$ $$\beta_{5}$$ $$=$$ $$($$$$-$$$$41\!\cdots\!87$$ $$\nu^{17}\mathstrut +\mathstrut$$ $$16\!\cdots\!28$$ $$\nu^{16}\mathstrut -\mathstrut$$ $$82\!\cdots\!92$$ $$\nu^{15}\mathstrut +\mathstrut$$ $$99\!\cdots\!64$$ $$\nu^{14}\mathstrut -\mathstrut$$ $$37\!\cdots\!14$$ $$\nu^{13}\mathstrut +\mathstrut$$ $$22\!\cdots\!64$$ $$\nu^{12}\mathstrut -\mathstrut$$ $$70\!\cdots\!84$$ $$\nu^{11}\mathstrut +\mathstrut$$ $$25\!\cdots\!88$$ $$\nu^{10}\mathstrut -\mathstrut$$ $$64\!\cdots\!99$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$15\!\cdots\!00$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$28\!\cdots\!04$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$47\!\cdots\!60$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$55\!\cdots\!16$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$71\!\cdots\!76$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$33\!\cdots\!24$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$39\!\cdots\!40$$ $$\nu^{2}\mathstrut +\mathstrut$$ $$18\!\cdots\!00$$ $$\nu\mathstrut +\mathstrut$$ $$18\!\cdots\!00$$$$)/$$$$10\!\cdots\!00$$ $$\beta_{6}$$ $$=$$ $$($$$$-$$$$22\!\cdots\!25$$ $$\nu^{17}\mathstrut +\mathstrut$$ $$30\!\cdots\!04$$ $$\nu^{16}\mathstrut -\mathstrut$$ $$12\!\cdots\!32$$ $$\nu^{15}\mathstrut +\mathstrut$$ $$22\!\cdots\!36$$ $$\nu^{14}\mathstrut -\mathstrut$$ $$27\!\cdots\!42$$ $$\nu^{13}\mathstrut +\mathstrut$$ $$68\!\cdots\!08$$ $$\nu^{12}\mathstrut -\mathstrut$$ $$28\!\cdots\!08$$ $$\nu^{11}\mathstrut +\mathstrut$$ $$10\!\cdots\!56$$ $$\nu^{10}\mathstrut -\mathstrut$$ $$14\!\cdots\!61$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$80\!\cdots\!36$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$32\!\cdots\!64$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$32\!\cdots\!28$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$18\!\cdots\!00$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$58\!\cdots\!04$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$84\!\cdots\!20$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$30\!\cdots\!40$$ $$\nu^{2}\mathstrut +\mathstrut$$ $$81\!\cdots\!00$$ $$\nu\mathstrut -\mathstrut$$ $$63\!\cdots\!00$$$$)/$$$$37\!\cdots\!00$$ $$\beta_{7}$$ $$=$$ $$($$$$47\!\cdots\!09$$ $$\nu^{17}\mathstrut -\mathstrut$$ $$60\!\cdots\!12$$ $$\nu^{16}\mathstrut +\mathstrut$$ $$37\!\cdots\!04$$ $$\nu^{15}\mathstrut -\mathstrut$$ $$36\!\cdots\!72$$ $$\nu^{14}\mathstrut +\mathstrut$$ $$11\!\cdots\!02$$ $$\nu^{13}\mathstrut -\mathstrut$$ $$85\!\cdots\!84$$ $$\nu^{12}\mathstrut +\mathstrut$$ $$18\!\cdots\!88$$ $$\nu^{11}\mathstrut -\mathstrut$$ $$10\!\cdots\!28$$ $$\nu^{10}\mathstrut +\mathstrut$$ $$15\!\cdots\!49$$ $$\nu^{9}\mathstrut -\mathstrut$$ $$64\!\cdots\!64$$ $$\nu^{8}\mathstrut +\mathstrut$$ $$73\!\cdots\!16$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$21\!\cdots\!56$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$17\!\cdots\!28$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$33\!\cdots\!44$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$20\!\cdots\!84$$ $$\nu^{3}\mathstrut -\mathstrut$$ $$18\!\cdots\!20$$ $$\nu^{2}\mathstrut +\mathstrut$$ $$73\!\cdots\!00$$ $$\nu\mathstrut +\mathstrut$$ $$67\!\cdots\!00$$$$)/$$$$18\!\cdots\!00$$ $$\beta_{8}$$ $$=$$ $$($$$$-$$$$47\!\cdots\!53$$ $$\nu^{17}\mathstrut -\mathstrut$$ $$17\!\cdots\!16$$ $$\nu^{16}\mathstrut -\mathstrut$$ $$24\!\cdots\!24$$ $$\nu^{15}\mathstrut -\mathstrut$$ $$13\!\cdots\!96$$ $$\nu^{14}\mathstrut -\mathstrut$$ $$46\!\cdots\!82$$ $$\nu^{13}\mathstrut -\mathstrut$$ $$38\!\cdots\!20$$ $$\nu^{12}\mathstrut -\mathstrut$$ $$35\!\cdots\!60$$ $$\nu^{11}\mathstrut -\mathstrut$$ $$57\!\cdots\!40$$ $$\nu^{10}\mathstrut -\mathstrut$$ $$77\!\cdots\!29$$ $$\nu^{9}\mathstrut -\mathstrut$$ $$45\!\cdots\!72$$ $$\nu^{8}\mathstrut +\mathstrut$$ $$41\!\cdots\!12$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$18\!\cdots\!16$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$24\!\cdots\!56$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$32\!\cdots\!44$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$42\!\cdots\!04$$ $$\nu^{3}\mathstrut -\mathstrut$$ $$12\!\cdots\!20$$ $$\nu^{2}\mathstrut +\mathstrut$$ $$24\!\cdots\!00$$ $$\nu\mathstrut +\mathstrut$$ $$29\!\cdots\!00$$$$)/$$$$13\!\cdots\!00$$ $$\beta_{9}$$ $$=$$ $$($$$$21\!\cdots\!59$$ $$\nu^{17}\mathstrut -\mathstrut$$ $$74\!\cdots\!40$$ $$\nu^{16}\mathstrut +\mathstrut$$ $$12\!\cdots\!84$$ $$\nu^{15}\mathstrut -\mathstrut$$ $$42\!\cdots\!20$$ $$\nu^{14}\mathstrut +\mathstrut$$ $$29\!\cdots\!26$$ $$\nu^{13}\mathstrut -\mathstrut$$ $$90\!\cdots\!04$$ $$\nu^{12}\mathstrut +\mathstrut$$ $$33\!\cdots\!84$$ $$\nu^{11}\mathstrut -\mathstrut$$ $$92\!\cdots\!20$$ $$\nu^{10}\mathstrut +\mathstrut$$ $$21\!\cdots\!23$$ $$\nu^{9}\mathstrut -\mathstrut$$ $$48\!\cdots\!44$$ $$\nu^{8}\mathstrut +\mathstrut$$ $$74\!\cdots\!76$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$12\!\cdots\!92$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$13\!\cdots\!24$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$13\!\cdots\!44$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$11\!\cdots\!88$$ $$\nu^{3}\mathstrut -\mathstrut$$ $$23\!\cdots\!00$$ $$\nu^{2}\mathstrut +\mathstrut$$ $$40\!\cdots\!00$$ $$\nu\mathstrut +\mathstrut$$ $$59\!\cdots\!00$$$$)/$$$$12\!\cdots\!00$$ $$\beta_{10}$$ $$=$$ $$($$$$15\!\cdots\!77$$ $$\nu^{17}\mathstrut +\mathstrut$$ $$81\!\cdots\!44$$ $$\nu^{16}\mathstrut +\mathstrut$$ $$86\!\cdots\!32$$ $$\nu^{15}\mathstrut +\mathstrut$$ $$49\!\cdots\!56$$ $$\nu^{14}\mathstrut +\mathstrut$$ $$18\!\cdots\!82$$ $$\nu^{13}\mathstrut +\mathstrut$$ $$11\!\cdots\!32$$ $$\nu^{12}\mathstrut +\mathstrut$$ $$19\!\cdots\!12$$ $$\nu^{11}\mathstrut +\mathstrut$$ $$14\!\cdots\!28$$ $$\nu^{10}\mathstrut +\mathstrut$$ $$10\!\cdots\!45$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$91\!\cdots\!84$$ $$\nu^{8}\mathstrut +\mathstrut$$ $$23\!\cdots\!76$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$31\!\cdots\!88$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$46\!\cdots\!08$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$49\!\cdots\!36$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$63\!\cdots\!60$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$28\!\cdots\!60$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$61\!\cdots\!00$$ $$\nu\mathstrut +\mathstrut$$ $$32\!\cdots\!00$$$$)/$$$$18\!\cdots\!00$$ $$\beta_{11}$$ $$=$$ $$($$$$-$$$$47\!\cdots\!55$$ $$\nu^{17}\mathstrut +\mathstrut$$ $$10\!\cdots\!56$$ $$\nu^{16}\mathstrut -\mathstrut$$ $$28\!\cdots\!96$$ $$\nu^{15}\mathstrut +\mathstrut$$ $$62\!\cdots\!68$$ $$\nu^{14}\mathstrut -\mathstrut$$ $$67\!\cdots\!30$$ $$\nu^{13}\mathstrut +\mathstrut$$ $$14\!\cdots\!56$$ $$\nu^{12}\mathstrut -\mathstrut$$ $$79\!\cdots\!28$$ $$\nu^{11}\mathstrut +\mathstrut$$ $$16\!\cdots\!00$$ $$\nu^{10}\mathstrut -\mathstrut$$ $$51\!\cdots\!51$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$99\!\cdots\!96$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$17\!\cdots\!28$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$30\!\cdots\!60$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$31\!\cdots\!96$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$39\!\cdots\!36$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$23\!\cdots\!08$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$94\!\cdots\!20$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$48\!\cdots\!00$$ $$\nu\mathstrut -\mathstrut$$ $$37\!\cdots\!00$$$$)/$$$$37\!\cdots\!00$$ $$\beta_{12}$$ $$=$$ $$($$$$-$$$$20\!\cdots\!25$$ $$\nu^{17}\mathstrut +\mathstrut$$ $$15\!\cdots\!56$$ $$\nu^{16}\mathstrut -\mathstrut$$ $$63\!\cdots\!64$$ $$\nu^{15}\mathstrut +\mathstrut$$ $$92\!\cdots\!64$$ $$\nu^{14}\mathstrut +\mathstrut$$ $$35\!\cdots\!74$$ $$\nu^{13}\mathstrut +\mathstrut$$ $$20\!\cdots\!24$$ $$\nu^{12}\mathstrut +\mathstrut$$ $$30\!\cdots\!64$$ $$\nu^{11}\mathstrut +\mathstrut$$ $$22\!\cdots\!88$$ $$\nu^{10}\mathstrut +\mathstrut$$ $$36\!\cdots\!15$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$13\!\cdots\!64$$ $$\nu^{8}\mathstrut +\mathstrut$$ $$16\!\cdots\!28$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$38\!\cdots\!24$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$23\!\cdots\!12$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$55\!\cdots\!24$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$14\!\cdots\!88$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$29\!\cdots\!40$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$31\!\cdots\!00$$ $$\nu\mathstrut +\mathstrut$$ $$35\!\cdots\!00$$$$)/$$$$93\!\cdots\!00$$ $$\beta_{13}$$ $$=$$ $$($$$$-$$$$13\!\cdots\!91$$ $$\nu^{17}\mathstrut -\mathstrut$$ $$14\!\cdots\!16$$ $$\nu^{16}\mathstrut -\mathstrut$$ $$82\!\cdots\!72$$ $$\nu^{15}\mathstrut -\mathstrut$$ $$85\!\cdots\!24$$ $$\nu^{14}\mathstrut -\mathstrut$$ $$20\!\cdots\!18$$ $$\nu^{13}\mathstrut -\mathstrut$$ $$19\!\cdots\!88$$ $$\nu^{12}\mathstrut -\mathstrut$$ $$25\!\cdots\!40$$ $$\nu^{11}\mathstrut -\mathstrut$$ $$21\!\cdots\!48$$ $$\nu^{10}\mathstrut -\mathstrut$$ $$18\!\cdots\!67$$ $$\nu^{9}\mathstrut -\mathstrut$$ $$12\!\cdots\!88$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$83\!\cdots\!32$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$38\!\cdots\!56$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$21\!\cdots\!88$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$48\!\cdots\!08$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$27\!\cdots\!04$$ $$\nu^{3}\mathstrut -\mathstrut$$ $$11\!\cdots\!60$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$11\!\cdots\!00$$ $$\nu\mathstrut +\mathstrut$$ $$46\!\cdots\!00$$$$)/$$$$41\!\cdots\!00$$ $$\beta_{14}$$ $$=$$ $$($$$$-$$$$31\!\cdots\!09$$ $$\nu^{17}\mathstrut -\mathstrut$$ $$28\!\cdots\!04$$ $$\nu^{16}\mathstrut -\mathstrut$$ $$17\!\cdots\!36$$ $$\nu^{15}\mathstrut -\mathstrut$$ $$16\!\cdots\!32$$ $$\nu^{14}\mathstrut -\mathstrut$$ $$37\!\cdots\!34$$ $$\nu^{13}\mathstrut -\mathstrut$$ $$36\!\cdots\!48$$ $$\nu^{12}\mathstrut -\mathstrut$$ $$35\!\cdots\!36$$ $$\nu^{11}\mathstrut -\mathstrut$$ $$39\!\cdots\!56$$ $$\nu^{10}\mathstrut -\mathstrut$$ $$13\!\cdots\!45$$ $$\nu^{9}\mathstrut -\mathstrut$$ $$22\!\cdots\!00$$ $$\nu^{8}\mathstrut +\mathstrut$$ $$56\!\cdots\!20$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$67\!\cdots\!16$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$19\!\cdots\!92$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$99\!\cdots\!12$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$45\!\cdots\!96$$ $$\nu^{3}\mathstrut -\mathstrut$$ $$51\!\cdots\!80$$ $$\nu^{2}\mathstrut +\mathstrut$$ $$30\!\cdots\!00$$ $$\nu\mathstrut +\mathstrut$$ $$28\!\cdots\!00$$$$)/$$$$37\!\cdots\!00$$ $$\beta_{15}$$ $$=$$ $$($$$$-$$$$20\!\cdots\!41$$ $$\nu^{17}\mathstrut -\mathstrut$$ $$20\!\cdots\!60$$ $$\nu^{16}\mathstrut -\mathstrut$$ $$12\!\cdots\!20$$ $$\nu^{15}\mathstrut -\mathstrut$$ $$11\!\cdots\!04$$ $$\nu^{14}\mathstrut -\mathstrut$$ $$31\!\cdots\!98$$ $$\nu^{13}\mathstrut -\mathstrut$$ $$25\!\cdots\!84$$ $$\nu^{12}\mathstrut -\mathstrut$$ $$39\!\cdots\!48$$ $$\nu^{11}\mathstrut -\mathstrut$$ $$26\!\cdots\!48$$ $$\nu^{10}\mathstrut -\mathstrut$$ $$27\!\cdots\!37$$ $$\nu^{9}\mathstrut -\mathstrut$$ $$15\!\cdots\!76$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$10\!\cdots\!84$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$45\!\cdots\!16$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$21\!\cdots\!16$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$68\!\cdots\!68$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$21\!\cdots\!20$$ $$\nu^{3}\mathstrut -\mathstrut$$ $$37\!\cdots\!80$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$75\!\cdots\!00$$ $$\nu\mathstrut -\mathstrut$$ $$57\!\cdots\!00$$$$)/$$$$18\!\cdots\!00$$ $$\beta_{16}$$ $$=$$ $$($$$$-$$$$84\!\cdots\!57$$ $$\nu^{17}\mathstrut +\mathstrut$$ $$91\!\cdots\!28$$ $$\nu^{16}\mathstrut -\mathstrut$$ $$51\!\cdots\!72$$ $$\nu^{15}\mathstrut +\mathstrut$$ $$46\!\cdots\!48$$ $$\nu^{14}\mathstrut -\mathstrut$$ $$12\!\cdots\!42$$ $$\nu^{13}\mathstrut +\mathstrut$$ $$83\!\cdots\!32$$ $$\nu^{12}\mathstrut -\mathstrut$$ $$14\!\cdots\!68$$ $$\nu^{11}\mathstrut +\mathstrut$$ $$64\!\cdots\!72$$ $$\nu^{10}\mathstrut -\mathstrut$$ $$93\!\cdots\!93$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$22\!\cdots\!72$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$32\!\cdots\!68$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$45\!\cdots\!52$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$56\!\cdots\!68$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$17\!\cdots\!72$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$39\!\cdots\!88$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$36\!\cdots\!80$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$54\!\cdots\!00$$ $$\nu\mathstrut +\mathstrut$$ $$89\!\cdots\!00$$$$)/$$$$37\!\cdots\!00$$ $$\beta_{17}$$ $$=$$ $$($$$$-$$$$38\!\cdots\!89$$ $$\nu^{17}\mathstrut -\mathstrut$$ $$13\!\cdots\!40$$ $$\nu^{16}\mathstrut -\mathstrut$$ $$23\!\cdots\!08$$ $$\nu^{15}\mathstrut -\mathstrut$$ $$75\!\cdots\!84$$ $$\nu^{14}\mathstrut -\mathstrut$$ $$53\!\cdots\!30$$ $$\nu^{13}\mathstrut -\mathstrut$$ $$16\!\cdots\!96$$ $$\nu^{12}\mathstrut -\mathstrut$$ $$61\!\cdots\!16$$ $$\nu^{11}\mathstrut -\mathstrut$$ $$16\!\cdots\!48$$ $$\nu^{10}\mathstrut -\mathstrut$$ $$38\!\cdots\!33$$ $$\nu^{9}\mathstrut -\mathstrut$$ $$89\!\cdots\!88$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$13\!\cdots\!56$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$23\!\cdots\!92$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$23\!\cdots\!64$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$25\!\cdots\!00$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$19\!\cdots\!76$$ $$\nu^{3}\mathstrut -\mathstrut$$ $$53\!\cdots\!80$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$75\!\cdots\!00$$ $$\nu\mathstrut +\mathstrut$$ $$11\!\cdots\!00$$$$)/$$$$13\!\cdots\!00$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2}\mathstrut -\mathstrut$$ $$165$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$26$$$$)/16$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{8}\mathstrut -\mathstrut$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$93$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$184$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$787$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$338301$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$11133470$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$428473845010$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$1769172426004743437$$$$)/256$$ $$\nu^{3}$$ $$=$$ $$($$$$-$$$$229606$$ $$\beta_{17}\mathstrut +\mathstrut$$ $$238184$$ $$\beta_{16}\mathstrut +\mathstrut$$ $$1952544$$ $$\beta_{15}\mathstrut -\mathstrut$$ $$87240$$ $$\beta_{14}\mathstrut -\mathstrut$$ $$1207407$$ $$\beta_{13}\mathstrut +\mathstrut$$ $$1485152$$ $$\beta_{12}\mathstrut +\mathstrut$$ $$59625861$$ $$\beta_{11}\mathstrut +\mathstrut$$ $$4200570$$ $$\beta_{10}\mathstrut +\mathstrut$$ $$4878603483$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$103242324$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$2049249323$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$2841323328$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$1922669237196$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$529762565348$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$216021976603912$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$3298911496266147799$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$604795752649597289228$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$108560051681127728282$$$$)/4096$$ $$\nu^{4}$$ $$=$$ $$($$$$102656040740096$$ $$\beta_{17}\mathstrut -\mathstrut$$ $$70908194140864$$ $$\beta_{16}\mathstrut -\mathstrut$$ $$282262760903076$$ $$\beta_{15}\mathstrut -\mathstrut$$ $$212039595419538$$ $$\beta_{14}\mathstrut -\mathstrut$$ $$43708674916398$$ $$\beta_{13}\mathstrut -\mathstrut$$ $$381055508518804$$ $$\beta_{12}\mathstrut +\mathstrut$$ $$20554294076322450$$ $$\beta_{11}\mathstrut +\mathstrut$$ $$19092201540691185$$ $$\beta_{10}\mathstrut +\mathstrut$$ $$1978294453282779405$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$2416625490949913661$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$2820250906814885215$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$504603937807855245927$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$623320456436519879628$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$1500361087232372304643$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$2295860450123087405269603$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$60245941537853261446343631$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$2314033128747739400296082329252$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$2919496263477059026244597432835344842$$$$)/32768$$ $$\nu^{5}$$ $$=$$ $$($$$$278553342535061832543095$$ $$\beta_{17}\mathstrut -\mathstrut$$ $$295016010732912147948580$$ $$\beta_{16}\mathstrut -\mathstrut$$ $$3360868292987206089119520$$ $$\beta_{15}\mathstrut +\mathstrut$$ $$265128701001451638320940$$ $$\beta_{14}\mathstrut +\mathstrut$$ $$5799575232057650419817790$$ $$\beta_{13}\mathstrut -\mathstrut$$ $$2515231369396537832274880$$ $$\beta_{12}\mathstrut -\mathstrut$$ $$68250258209554812253744500$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$800336132640805343282325$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$6305737357526197984418924850$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$162085964526078635278791130$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$26974301495069992782807247661$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$31847568948782669445577484343$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$3311285057784038683736891227052$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$1937499328677437213953538535366$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$849759011473019786496716458458569$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$3406736189531140076372807541000431292$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$746839858717381813644982424478794432345$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$140236642021934914620826670017363817674$$$$)/262144$$ $$\nu^{6}$$ $$=$$ $$($$$$-$$$$11\!\cdots\!08$$ $$\beta_{17}\mathstrut +\mathstrut$$ $$77\!\cdots\!92$$ $$\beta_{16}\mathstrut +\mathstrut$$ $$36\!\cdots\!10$$ $$\beta_{15}\mathstrut +\mathstrut$$ $$26\!\cdots\!87$$ $$\beta_{14}\mathstrut +\mathstrut$$ $$24\!\cdots\!07$$ $$\beta_{13}\mathstrut +\mathstrut$$ $$24\!\cdots\!36$$ $$\beta_{12}\mathstrut -\mathstrut$$ $$24\!\cdots\!43$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$17\!\cdots\!60$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$22\!\cdots\!63$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$14\!\cdots\!22$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$96\!\cdots\!23$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$38\!\cdots\!28$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$31\!\cdots\!02$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$73\!\cdots\!03$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$19\!\cdots\!86$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$45\!\cdots\!13$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$17\!\cdots\!09$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$15\!\cdots\!90$$$$)/1048576$$ $$\nu^{7}$$ $$=$$ $$($$$$-$$$$31\!\cdots\!33$$ $$\beta_{17}\mathstrut +\mathstrut$$ $$31\!\cdots\!96$$ $$\beta_{16}\mathstrut +\mathstrut$$ $$49\!\cdots\!24$$ $$\beta_{15}\mathstrut -\mathstrut$$ $$51\!\cdots\!28$$ $$\beta_{14}\mathstrut -\mathstrut$$ $$10\!\cdots\!24$$ $$\beta_{13}\mathstrut +\mathstrut$$ $$35\!\cdots\!12$$ $$\beta_{12}\mathstrut +\mathstrut$$ $$65\!\cdots\!70$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$53\!\cdots\!05$$ $$\beta_{10}\mathstrut +\mathstrut$$ $$74\!\cdots\!96$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$21\!\cdots\!46$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$59\!\cdots\!93$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$35\!\cdots\!75$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$46\!\cdots\!44$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$35\!\cdots\!50$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$16\!\cdots\!95$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$38\!\cdots\!94$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$70\!\cdots\!61$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$16\!\cdots\!96$$$$)/16777216$$ $$\nu^{8}$$ $$=$$ $$($$$$90\!\cdots\!84$$ $$\beta_{17}\mathstrut -\mathstrut$$ $$60\!\cdots\!44$$ $$\beta_{16}\mathstrut -\mathstrut$$ $$31\!\cdots\!84$$ $$\beta_{15}\mathstrut -\mathstrut$$ $$21\!\cdots\!80$$ $$\beta_{14}\mathstrut -\mathstrut$$ $$29\!\cdots\!68$$ $$\beta_{13}\mathstrut -\mathstrut$$ $$10\!\cdots\!32$$ $$\beta_{12}\mathstrut +\mathstrut$$ $$19\!\cdots\!12$$ $$\beta_{11}\mathstrut +\mathstrut$$ $$12\!\cdots\!97$$ $$\beta_{10}\mathstrut +\mathstrut$$ $$18\!\cdots\!71$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$87\!\cdots\!37$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$96\!\cdots\!07$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$25\!\cdots\!35$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$12\!\cdots\!16$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$86\!\cdots\!37$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$14\!\cdots\!59$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$30\!\cdots\!09$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$11\!\cdots\!22$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$86\!\cdots\!54$$$$)/33554432$$ $$\nu^{9}$$ $$=$$ $$($$$$45\!\cdots\!40$$ $$\beta_{17}\mathstrut -\mathstrut$$ $$41\!\cdots\!20$$ $$\beta_{16}\mathstrut -\mathstrut$$ $$89\!\cdots\!00$$ $$\beta_{15}\mathstrut +\mathstrut$$ $$10\!\cdots\!80$$ $$\beta_{14}\mathstrut +\mathstrut$$ $$20\!\cdots\!72$$ $$\beta_{13}\mathstrut -\mathstrut$$ $$58\!\cdots\!52$$ $$\beta_{12}\mathstrut -\mathstrut$$ $$73\!\cdots\!28$$ $$\beta_{11}\mathstrut +\mathstrut$$ $$15\!\cdots\!16$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$10\!\cdots\!32$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$34\!\cdots\!16$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$12\!\cdots\!32$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$84\!\cdots\!41$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$78\!\cdots\!88$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$68\!\cdots\!42$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$32\!\cdots\!36$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$58\!\cdots\!42$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$66\!\cdots\!01$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$22\!\cdots\!97$$$$)/$$$$134217728$$ $$\nu^{10}$$ $$=$$ $$($$$$-$$$$82\!\cdots\!52$$ $$\beta_{17}\mathstrut +\mathstrut$$ $$53\!\cdots\!88$$ $$\beta_{16}\mathstrut +\mathstrut$$ $$29\!\cdots\!20$$ $$\beta_{15}\mathstrut +\mathstrut$$ $$20\!\cdots\!88$$ $$\beta_{14}\mathstrut +\mathstrut$$ $$32\!\cdots\!52$$ $$\beta_{13}\mathstrut +\mathstrut$$ $$42\!\cdots\!80$$ $$\beta_{12}\mathstrut -\mathstrut$$ $$17\!\cdots\!48$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$11\!\cdots\!36$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$17\!\cdots\!20$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$68\!\cdots\!60$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$44\!\cdots\!28$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$21\!\cdots\!63$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$39\!\cdots\!08$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$96\!\cdots\!38$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$12\!\cdots\!28$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$25\!\cdots\!06$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$96\!\cdots\!59$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$64\!\cdots\!19$$$$)/$$$$134217728$$ $$\nu^{11}$$ $$=$$ $$($$$$-$$$$82\!\cdots\!36$$ $$\beta_{17}\mathstrut +\mathstrut$$ $$68\!\cdots\!48$$ $$\beta_{16}\mathstrut +\mathstrut$$ $$19\!\cdots\!88$$ $$\beta_{15}\mathstrut -\mathstrut$$ $$25\!\cdots\!00$$ $$\beta_{14}\mathstrut -\mathstrut$$ $$45\!\cdots\!96$$ $$\beta_{13}\mathstrut +\mathstrut$$ $$11\!\cdots\!56$$ $$\beta_{12}\mathstrut +\mathstrut$$ $$97\!\cdots\!16$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$42\!\cdots\!28$$ $$\beta_{10}\mathstrut +\mathstrut$$ $$20\!\cdots\!08$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$69\!\cdots\!12$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$29\!\cdots\!04$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$34\!\cdots\!93$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$16\!\cdots\!56$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$15\!\cdots\!38$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$73\!\cdots\!00$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$11\!\cdots\!98$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$51\!\cdots\!61$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$40\!\cdots\!05$$$$)/$$$$134217728$$ $$\nu^{12}$$ $$=$$ $$($$$$17\!\cdots\!08$$ $$\beta_{17}\mathstrut -\mathstrut$$ $$11\!\cdots\!16$$ $$\beta_{16}\mathstrut -\mathstrut$$ $$66\!\cdots\!20$$ $$\beta_{15}\mathstrut -\mathstrut$$ $$44\!\cdots\!36$$ $$\beta_{14}\mathstrut -\mathstrut$$ $$77\!\cdots\!52$$ $$\beta_{13}\mathstrut -\mathstrut$$ $$79\!\cdots\!52$$ $$\beta_{12}\mathstrut +\mathstrut$$ $$38\!\cdots\!32$$ $$\beta_{11}\mathstrut +\mathstrut$$ $$25\!\cdots\!80$$ $$\beta_{10}\mathstrut +\mathstrut$$ $$37\!\cdots\!08$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$13\!\cdots\!20$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$15\!\cdots\!00$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$43\!\cdots\!35$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$18\!\cdots\!68$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$23\!\cdots\!42$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$28\!\cdots\!28$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$51\!\cdots\!54$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$19\!\cdots\!19$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$12\!\cdots\!55$$$$)/$$$$134217728$$ $$\nu^{13}$$ $$=$$ $$($$$$76\!\cdots\!60$$ $$\beta_{17}\mathstrut -\mathstrut$$ $$57\!\cdots\!12$$ $$\beta_{16}\mathstrut -\mathstrut$$ $$20\!\cdots\!32$$ $$\beta_{15}\mathstrut +\mathstrut$$ $$28\!\cdots\!60$$ $$\beta_{14}\mathstrut +\mathstrut$$ $$48\!\cdots\!44$$ $$\beta_{13}\mathstrut -\mathstrut$$ $$12\!\cdots\!44$$ $$\beta_{12}\mathstrut -\mathstrut$$ $$58\!\cdots\!88$$ $$\beta_{11}\mathstrut +\mathstrut$$ $$51\!\cdots\!96$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$19\!\cdots\!84$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$69\!\cdots\!36$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$32\!\cdots\!48$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$48\!\cdots\!41$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$16\!\cdots\!92$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$16\!\cdots\!02$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$79\!\cdots\!36$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$11\!\cdots\!22$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$14\!\cdots\!19$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$38\!\cdots\!57$$$$)/67108864$$ $$\nu^{14}$$ $$=$$ $$($$$$-$$$$18\!\cdots\!56$$ $$\beta_{17}\mathstrut +\mathstrut$$ $$11\!\cdots\!40$$ $$\beta_{16}\mathstrut +\mathstrut$$ $$72\!\cdots\!08$$ $$\beta_{15}\mathstrut +\mathstrut$$ $$47\!\cdots\!52$$ $$\beta_{14}\mathstrut +\mathstrut$$ $$88\!\cdots\!88$$ $$\beta_{13}\mathstrut -\mathstrut$$ $$54\!\cdots\!00$$ $$\beta_{12}\mathstrut -\mathstrut$$ $$40\!\cdots\!72$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$27\!\cdots\!48$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$40\!\cdots\!72$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$13\!\cdots\!16$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$21\!\cdots\!48$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$44\!\cdots\!95$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$95\!\cdots\!68$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$27\!\cdots\!58$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$30\!\cdots\!36$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$52\!\cdots\!66$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$19\!\cdots\!99$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$12\!\cdots\!71$$$$)/67108864$$ $$\nu^{15}$$ $$=$$ $$($$$$-$$$$14\!\cdots\!00$$ $$\beta_{17}\mathstrut +\mathstrut$$ $$98\!\cdots\!88$$ $$\beta_{16}\mathstrut +\mathstrut$$ $$44\!\cdots\!52$$ $$\beta_{15}\mathstrut -\mathstrut$$ $$63\!\cdots\!64$$ $$\beta_{14}\mathstrut -\mathstrut$$ $$10\!\cdots\!00$$ $$\beta_{13}\mathstrut +\mathstrut$$ $$24\!\cdots\!44$$ $$\beta_{12}\mathstrut +\mathstrut$$ $$56\!\cdots\!36$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$11\!\cdots\!72$$ $$\beta_{10}\mathstrut +\mathstrut$$ $$36\!\cdots\!76$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$13\!\cdots\!52$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$69\!\cdots\!00$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$12\!\cdots\!79$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$33\!\cdots\!88$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$34\!\cdots\!74$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$16\!\cdots\!60$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$21\!\cdots\!90$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$13\!\cdots\!93$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$75\!\cdots\!71$$$$)/67108864$$ $$\nu^{16}$$ $$=$$ $$($$$$39\!\cdots\!56$$ $$\beta_{17}\mathstrut -\mathstrut$$ $$24\!\cdots\!12$$ $$\beta_{16}\mathstrut -\mathstrut$$ $$15\!\cdots\!88$$ $$\beta_{15}\mathstrut -\mathstrut$$ $$10\!\cdots\!24$$ $$\beta_{14}\mathstrut -\mathstrut$$ $$19\!\cdots\!68$$ $$\beta_{13}\mathstrut +\mathstrut$$ $$20\!\cdots\!72$$ $$\beta_{12}\mathstrut +\mathstrut$$ $$85\!\cdots\!64$$ $$\beta_{11}\mathstrut +\mathstrut$$ $$57\!\cdots\!14$$ $$\beta_{10}\mathstrut +\mathstrut$$ $$85\!\cdots\!10$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$26\!\cdots\!54$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$51\!\cdots\!26$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$89\!\cdots\!79$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$30\!\cdots\!20$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$60\!\cdots\!20$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$63\!\cdots\!34$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$10\!\cdots\!40$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$40\!\cdots\!69$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$23\!\cdots\!37$$$$)/67108864$$ $$\nu^{17}$$ $$=$$ $$($$$$56\!\cdots\!56$$ $$\beta_{17}\mathstrut -\mathstrut$$ $$34\!\cdots\!52$$ $$\beta_{16}\mathstrut -\mathstrut$$ $$18\!\cdots\!60$$ $$\beta_{15}\mathstrut +\mathstrut$$ $$27\!\cdots\!28$$ $$\beta_{14}\mathstrut +\mathstrut$$ $$42\!\cdots\!56$$ $$\beta_{13}\mathstrut -\mathstrut$$ $$98\!\cdots\!64$$ $$\beta_{12}\mathstrut -\mathstrut$$ $$44\!\cdots\!56$$ $$\beta_{11}\mathstrut +\mathstrut$$ $$50\!\cdots\!00$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$14\!\cdots\!44$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$55\!\cdots\!68$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$29\!\cdots\!88$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$55\!\cdots\!19$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$13\!\cdots\!44$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$14\!\cdots\!98$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$70\!\cdots\!36$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$86\!\cdots\!02$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$85\!\cdots\!57$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$30\!\cdots\!31$$$$)/$$$$134217728$$

## 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
 0.5 + 9.26410e7i 0.5 − 9.26410e7i 0.5 − 4.81893e7i 0.5 + 4.81893e7i 0.5 + 7.90224e7i 0.5 − 7.90224e7i 0.5 − 5.46285e7i 0.5 + 5.46285e7i 0.5 − 1.22705e8i 0.5 + 1.22705e8i 0.5 + 7.59483e7i 0.5 − 7.59483e7i 0.5 − 2.95082e7i 0.5 + 2.95082e7i 0.5 − 4.14425e6i 0.5 + 4.14425e6i 0.5 + 1.42659e8i 0.5 − 1.42659e8i
−521279. 56087.1i 1.48226e9i 2.68586e11 + 5.84741e10i 1.90217e13 −8.31355e13 + 7.72670e14i 1.45250e16i −1.36729e17 4.55456e16i −8.46232e17 −9.91563e18 1.06687e18i
3.2 −521279. + 56087.1i 1.48226e9i 2.68586e11 5.84741e10i 1.90217e13 −8.31355e13 7.72670e14i 1.45250e16i −1.36729e17 + 4.55456e16i −8.46232e17 −9.91563e18 + 1.06687e18i
3.3 −453236. 263543.i 7.71029e8i 1.35968e11 + 2.38895e11i −2.10588e13 2.03199e14 3.49458e14i 5.85221e15i 1.33358e15 1.44109e17i 7.56367e17 9.54463e18 + 5.54992e18i
3.4 −453236. + 263543.i 7.71029e8i 1.35968e11 2.38895e11i −2.10588e13 2.03199e14 + 3.49458e14i 5.85221e15i 1.33358e15 + 1.44109e17i 7.56367e17 9.54463e18 5.54992e18i
3.5 −306058. 425684.i 1.26436e9i −8.75354e10 + 2.60567e11i −4.90299e12 −5.38217e14 + 3.86966e14i 1.75338e16i 1.37710e17 4.24863e16i −2.47749e17 1.50060e18 + 2.08712e18i
3.6 −306058. + 425684.i 1.26436e9i −8.75354e10 2.60567e11i −4.90299e12 −5.38217e14 3.86966e14i 1.75338e16i 1.37710e17 + 4.24863e16i −2.47749e17 1.50060e18 2.08712e18i
3.7 −173179. 494861.i 8.74057e8i −2.14896e11 + 1.71399e11i 2.48589e13 4.32536e14 1.51368e14i 8.29070e15i 1.22034e17 + 7.66608e16i 5.86877e17 −4.30504e18 1.23017e19i
3.8 −173179. + 494861.i 8.74057e8i −2.14896e11 1.71399e11i 2.48589e13 4.32536e14 + 1.51368e14i 8.29070e15i 1.22034e17 7.66608e16i 5.86877e17 −4.30504e18 + 1.23017e19i
3.9 112582. 512058.i 1.96328e9i −2.49528e11 1.15297e11i −3.09501e13 1.00531e15 + 2.21030e14i 9.14517e15i −8.71314e16 + 1.14792e17i −2.50360e18 −3.48443e18 + 1.58482e19i
3.10 112582. + 512058.i 1.96328e9i −2.49528e11 + 1.15297e11i −3.09501e13 1.00531e15 2.21030e14i 9.14517e15i −8.71314e16 1.14792e17i −2.50360e18 −3.48443e18 1.58482e19i
3.11 130787. 507713.i 1.21517e9i −2.40667e11 1.32805e11i −7.30752e12 −6.16959e14 1.58929e14i 7.26783e15i −9.89028e16 + 1.04821e17i −1.25791e17 −9.55728e17 + 3.71012e18i
3.12 130787. + 507713.i 1.21517e9i −2.40667e11 + 1.32805e11i −7.30752e12 −6.16959e14 + 1.58929e14i 7.26783e15i −9.89028e16 1.04821e17i −1.25791e17 −9.55728e17 3.71012e18i
3.13 383569. 357425.i 4.72132e8i 1.93720e10 2.74194e11i 2.09077e13 1.68752e14 + 1.81095e14i 1.08901e16i −9.05736e16 1.12096e17i 1.12794e18 8.01953e18 7.47293e18i
3.14 383569. + 357425.i 4.72132e8i 1.93720e10 + 2.74194e11i 2.09077e13 1.68752e14 1.81095e14i 1.08901e16i −9.05736e16 + 1.12096e17i 1.12794e18 8.01953e18 + 7.47293e18i
3.15 490276. 185761.i 6.63080e7i 2.05863e11 1.82149e11i −1.96572e13 1.23175e13 + 3.25092e13i 1.51470e16i 6.70938e16 1.27545e17i 1.34645e18 −9.63744e18 + 3.65154e18i
3.16 490276. + 185761.i 6.63080e7i 2.05863e11 + 1.82149e11i −1.96572e13 1.23175e13 3.25092e13i 1.51470e16i 6.70938e16 + 1.27545e17i 1.34645e18 −9.63744e18 3.65154e18i
3.17 518652. 76670.9i 2.28254e9i 2.63121e11 7.95309e10i 1.45926e13 −1.75005e14 1.18385e15i 8.90761e15i 1.30370e17 6.14226e16i −3.85916e18 7.56849e18 1.11883e18i
3.18 518652. + 76670.9i 2.28254e9i 2.63121e11 + 7.95309e10i 1.45926e13 −1.75005e14 + 1.18385e15i 8.90761e15i 1.30370e17 + 6.14226e16i −3.85916e18 7.56849e18 + 1.11883e18i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 3.18 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
4.b Odd 1 yes

## Hecke kernels

There are no other newforms in $$S_{39}^{\mathrm{new}}(4, [\chi])$$.