Properties

Label 4.33.b.b
Level 4
Weight 33
Character orbit 4.b
Analytic conductor 25.947
Analytic rank 0
Dimension 14
CM no
Inner twists 2

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 4 = 2^{2} \)
Weight: \( k \) \(=\) \( 33 \)
Character orbit: \([\chi]\) \(=\) 4.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(25.9466620569\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} + \cdots)\)
Defining polynomial: \(x^{14} + 72511313626452 x^{12} + 2025191977179903324811336518 x^{10} + 27922884728028663894750078705223437415644 x^{8} + 203010662886800095440071970440402438747266446160157745 x^{6} + 758734102549599282271818004575465783845093632382487984186969965640 x^{4} + 1269648449115368448095465842606476325720277486461580161887038301255933321354000 x^{2} + 624216522131873762678666934520680301449631616035103441585151846396220724278849706601312000\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{182}\cdot 3^{20}\cdot 5^{6} \)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1699 + \beta_{1} ) q^{2} + ( 9 - 21 \beta_{1} + \beta_{2} ) q^{3} + ( -208774120 - 1782 \beta_{1} + 10 \beta_{2} + \beta_{3} ) q^{4} + ( 9865592530 + 532714 \beta_{1} + 22 \beta_{2} + 8 \beta_{3} + \beta_{4} ) q^{5} + ( 90195338916 + 35809 \beta_{1} + 8414 \beta_{2} - 32 \beta_{3} - \beta_{6} ) q^{6} + ( 55143902 - 128668274 \beta_{1} + 52118 \beta_{2} - 1267 \beta_{3} + 23 \beta_{4} + 6 \beta_{5} + \beta_{6} - \beta_{7} ) q^{7} + ( -13668947443585 - 214315405 \beta_{1} + 683481 \beta_{2} - 2968 \beta_{3} + 201 \beta_{4} - 40 \beta_{5} - 7 \beta_{6} + \beta_{11} ) q^{8} + ( -798819458325515 - 6252715161 \beta_{1} - 210575 \beta_{2} + 3664 \beta_{3} + 3675 \beta_{4} - 559 \beta_{5} - 7 \beta_{6} - \beta_{8} + 2 \beta_{9} + 4 \beta_{10} - 2 \beta_{11} - \beta_{12} ) q^{9} +O(q^{10})\) \( q +(-1699 + \beta_{1}) q^{2} +(9 - 21 \beta_{1} + \beta_{2}) q^{3} +(-208774120 - 1782 \beta_{1} + 10 \beta_{2} + \beta_{3}) q^{4} +(9865592530 + 532714 \beta_{1} + 22 \beta_{2} + 8 \beta_{3} + \beta_{4}) q^{5} +(90195338916 + 35809 \beta_{1} + 8414 \beta_{2} - 32 \beta_{3} - \beta_{6}) q^{6} +(55143902 - 128668274 \beta_{1} + 52118 \beta_{2} - 1267 \beta_{3} + 23 \beta_{4} + 6 \beta_{5} + \beta_{6} - \beta_{7}) q^{7} +(-13668947443585 - 214315405 \beta_{1} + 683481 \beta_{2} - 2968 \beta_{3} + 201 \beta_{4} - 40 \beta_{5} - 7 \beta_{6} + \beta_{11}) q^{8} +(-798819458325515 - 6252715161 \beta_{1} - 210575 \beta_{2} + 3664 \beta_{3} + 3675 \beta_{4} - 559 \beta_{5} - 7 \beta_{6} - \beta_{8} + 2 \beta_{9} + 4 \beta_{10} - 2 \beta_{11} - \beta_{12}) q^{9} +(2269681504813090 + 10770808509 \beta_{1} + 2189546 \beta_{2} + 566655 \beta_{3} - 9606 \beta_{4} + 2791 \beta_{5} - 116 \beta_{6} - 34 \beta_{7} - 2 \beta_{8} + 14 \beta_{9} + 6 \beta_{10} + 8 \beta_{11} + 3 \beta_{12} - \beta_{13}) q^{10} +(38893807894 - 90753095548 \beta_{1} - 10582231 \beta_{2} + 1317891 \beta_{3} + 658 \beta_{4} + 5606 \beta_{5} + 544 \beta_{6} - 123 \beta_{7} - 2 \beta_{8} - 7 \beta_{9} + 17 \beta_{10} - 14 \beta_{11} + \beta_{12} - 8 \beta_{13}) q^{11} +(-25455452462838834 + 80178527454 \beta_{1} - 252625420 \beta_{2} + 61523 \beta_{3} + 158910 \beta_{4} - 30124 \beta_{5} - 8544 \beta_{6} + 801 \beta_{7} - 80 \beta_{8} + 351 \beta_{9} + 358 \beta_{10} - 228 \beta_{11} + 32 \beta_{12} + 20 \beta_{13}) q^{12} +(-117172011244619788 - 1718682611943 \beta_{1} - 76709331 \beta_{2} - 40342417 \beta_{3} + 890263 \beta_{4} - 161620 \beta_{5} + 6984 \beta_{6} + 737 \beta_{7} + 158 \beta_{8} - 1053 \beta_{9} - 305 \beta_{10} + 332 \beta_{11} + 166 \beta_{12}) q^{13} +(552256883982614193 + 218484602198 \beta_{1} - 3161994519 \beta_{2} - 119741775 \beta_{3} - 1123334 \beta_{4} + 360769 \beta_{5} - 60902 \beta_{6} - 20648 \beta_{7} + 928 \beta_{8} - 664 \beta_{9} - 844 \beta_{10} - 1248 \beta_{11} + 1968 \beta_{12} + 128 \beta_{13}) q^{14} +(-414273066780 + 966745998772 \beta_{1} - 14897720202 \beta_{2} - 164634915 \beta_{3} + 1103957 \beta_{4} + 100078 \beta_{5} - 563793 \beta_{6} + 6823 \beta_{7} - 956 \beta_{8} + 302 \beta_{9} + 1278 \beta_{10} - 23076 \beta_{11} + 4574 \beta_{12} + 272 \beta_{13}) q^{15} +(-2183625418779930696 - 14543285858324 \beta_{1} - 60445494232 \beta_{2} - 164573328 \beta_{3} + 24497516 \beta_{4} - 3878004 \beta_{5} - 781780 \beta_{6} + 297980 \beta_{7} + 3520 \beta_{8} + 2308 \beta_{9} + 31960 \beta_{10} - 25412 \beta_{11} + 4928 \beta_{12} - 1104 \beta_{13}) q^{16} +(2255047583152878170 + 50289316697353 \beta_{1} + 2368589867 \beta_{2} + 1436736086 \beta_{3} + 5832763 \beta_{4} + 1400973 \beta_{5} + 5292973 \beta_{6} + 5002 \beta_{7} - 23641 \beta_{8} + 42280 \beta_{9} - 9878 \beta_{10} + 66542 \beta_{11} + 33271 \beta_{12}) q^{17} +(-25479955921074513147 - 809445773268161 \beta_{1} - 119478521604 \beta_{2} - 6515882754 \beta_{3} + 59976148 \beta_{4} - 3500850 \beta_{5} + 832504 \beta_{6} - 2556532 \beta_{7} + 18892 \beta_{8} + 108652 \beta_{9} + 308732 \beta_{10} - 124464 \beta_{11} - 42418 \beta_{12} - 5530 \beta_{13}) q^{18} +(-168858876809842 + 394003028697352 \beta_{1} + 1227632771387 \beta_{2} + 1553223893 \beta_{3} - 53953642 \beta_{4} - 780118 \beta_{5} - 56251592 \beta_{6} + 401667 \beta_{7} - 121278 \beta_{8} - 194713 \beta_{9} - 600497 \beta_{10} + 363470 \beta_{11} + 19679 \beta_{12} - 1784 \beta_{13}) q^{19} +(\)\(14\!\cdots\!20\)\( + 2328360794686596 \beta_{1} - 1183802876572 \beta_{2} + 14051968042 \beta_{3} + 968243024 \beta_{4} + 33129680 \beta_{5} - 6869440 \beta_{6} + 5168040 \beta_{7} + 388480 \beta_{8} + 504920 \beta_{9} + 470960 \beta_{10} + 345120 \beta_{11} - 220160 \beta_{12} + 28960 \beta_{13}) q^{20} +(-\)\(15\!\cdots\!06\)\( + 1863316758861683 \beta_{1} + 96639987723 \beta_{2} + 73096683955 \beta_{3} - 618317578 \beta_{4} - 1036380 \beta_{5} + 421474608 \beta_{6} + 3168181 \beta_{7} - 376850 \beta_{8} - 2414481 \beta_{9} - 4453925 \beta_{10} - 1026772 \beta_{11} - 513386 \beta_{12}) q^{21} +(\)\(38\!\cdots\!74\)\( + 154084504020279 \beta_{1} - 3470224452432 \beta_{2} - 79368984442 \beta_{3} - 13041193124 \beta_{4} + 130210374 \beta_{5} - 2353911 \beta_{6} + 16808464 \beta_{7} + 1420736 \beta_{8} + 2646640 \beta_{9} + 1000248 \beta_{10} + 1792704 \beta_{11} - 1064416 \beta_{12} + 131840 \beta_{13}) q^{22} +(-2086976762006056 + 4869782823481512 \beta_{1} + 14018455376118 \beta_{2} - 257203389299 \beta_{3} + 1259495049 \beta_{4} - 160859394 \beta_{5} - 654455645 \beta_{6} + 2337863 \beta_{7} - 1554116 \beta_{8} - 683758 \beta_{9} - 10678462 \beta_{10} + 8601764 \beta_{11} - 423070 \beta_{12} - 76560 \beta_{13}) q^{23} +(-\)\(18\!\cdots\!32\)\( - 26198512293714800 \beta_{1} - 13482147808384 \beta_{2} + 58544812928 \beta_{3} + 74257249488 \beta_{4} - 700110480 \beta_{5} + 298470480 \beta_{6} - 198290064 \beta_{7} - 1026304 \beta_{8} + 11439760 \beta_{9} + 10057056 \beta_{10} - 7081072 \beta_{11} - 3124992 \beta_{12} - 480064 \beta_{13}) q^{24} +(\)\(73\!\cdots\!75\)\( - 8207028047156820 \beta_{1} + 589320200240 \beta_{2} + 1901762065310 \beta_{3} + 19796138170 \beta_{4} - 413611750 \beta_{5} - 1179821150 \beta_{6} - 36976750 \beta_{7} - 2694150 \beta_{8} + 42365050 \beta_{9} + 9080150 \beta_{10} + 17716500 \beta_{11} + 8858250 \beta_{12}) q^{25} +(-\)\(71\!\cdots\!62\)\( - 120092493848287247 \beta_{1} - 7295993957646 \beta_{2} - 1790912630957 \beta_{3} - 107715443918 \beta_{4} - 7313998693 \beta_{5} + 208633692 \beta_{6} + 327965862 \beta_{7} - 8676794 \beta_{8} + 10613270 \beta_{9} - 45323794 \beta_{10} - 27146008 \beta_{11} + 15555223 \beta_{12} - 2056157 \beta_{13}) q^{26} +(40240665250224857 - 93886347923181187 \beta_{1} - 566478226518172 \beta_{2} - 12904647410341 \beta_{3} + 100903463690 \beta_{4} - 5169888074 \beta_{5} + 11992657992 \beta_{6} - 130808243 \beta_{7} + 23609502 \beta_{8} + 127089321 \beta_{9} + 65384769 \beta_{10} - 199408558 \beta_{11} + 18284465 \beta_{12} + 2368120 \beta_{13}) q^{27} +(\)\(20\!\cdots\!64\)\( + 560745704676958948 \beta_{1} - 487067624033064 \beta_{2} + 1068285792106 \beta_{3} + 309627352868 \beta_{4} - 2298818024 \beta_{5} + 2038526144 \beta_{6} + 998919790 \beta_{7} - 17839712 \beta_{8} + 23003282 \beta_{9} - 450495660 \beta_{10} + 44770568 \beta_{11} + 89597376 \beta_{12} + 5645720 \beta_{13}) q^{28} +(-\)\(27\!\cdots\!58\)\( - 652038978433254478 \beta_{1} - 12059842385362 \beta_{2} + 22148287317528 \beta_{3} + 406400543273 \beta_{4} - 41274492440 \beta_{5} - 33409098520 \beta_{6} - 408181264 \beta_{7} + 122673016 \beta_{8} + 162835232 \beta_{9} - 75832912 \beta_{10} - 153448976 \beta_{11} - 76724488 \beta_{12}) q^{29} +(-\)\(41\!\cdots\!65\)\( - 1610260745602822 \beta_{1} + 2273377729957739 \beta_{2} - 2423308689869 \beta_{3} + 677576386862 \beta_{4} + 11403217635 \beta_{5} + 20873995350 \beta_{6} - 5609436920 \beta_{7} - 143286560 \beta_{8} + 570529080 \beta_{9} - 929724260 \beta_{10} + 243955040 \beta_{11} - 31163120 \beta_{12} + 22696320 \beta_{13}) q^{30} +(153878973567913446 - 358978946316719002 \beta_{1} - 1232182105136100 \beta_{2} - 108780115559118 \beta_{3} + 796821259848 \beta_{4} - 56887492524 \beta_{5} + 63282509308 \beta_{6} - 1574205234 \beta_{7} + 71392380 \beta_{8} + 821243506 \beta_{9} - 863642206 \beta_{10} + 1590216548 \beta_{11} - 324566590 \beta_{12} - 35806736 \beta_{13}) q^{31} +(-\)\(21\!\cdots\!64\)\( - 2271425382640378736 \beta_{1} - 9400203141223008 \beta_{2} - 3768664063168 \beta_{3} - 1577553071600 \beta_{4} - 169655874992 \beta_{5} + 44958998800 \beta_{6} + 5243666320 \beta_{7} - 203824896 \beta_{8} + 1868008560 \beta_{9} - 520246368 \beta_{10} + 444854992 \beta_{11} - 712086784 \beta_{12} - 50198720 \beta_{13}) q^{32} +(\)\(21\!\cdots\!24\)\( + 2130216347481539045 \beta_{1} + 214772466853835 \beta_{2} + 312421164986244 \beta_{3} + 435737349749 \beta_{4} + 97701631303 \beta_{5} + 251341882927 \beta_{6} - 1650685172 \beta_{7} - 726768799 \beta_{8} + 3104222770 \beta_{9} - 2893605552 \beta_{10} - 125014142 \beta_{11} - 62507071 \beta_{12}) q^{33} +(\)\(21\!\cdots\!54\)\( + 2340824433332515016 \beta_{1} + 23063382535892012 \beta_{2} + 18152951653542 \beta_{3} + 12329138173124 \beta_{4} - 113526192074 \beta_{5} + 45400925464 \beta_{6} + 14012666716 \beta_{7} + 1535019676 \beta_{8} + 1125169852 \beta_{9} - 867604340 \beta_{10} + 299011728 \beta_{11} - 391696106 \beta_{12} - 183886130 \beta_{13}) q^{34} +(-3672045315354543170 + 8568406364837470358 \beta_{1} - 205986482744968 \beta_{2} - 453784603495230 \beta_{3} + 2268343549508 \beta_{4} - 418917146588 \beta_{5} - 745963980872 \beta_{6} + 7398164622 \beta_{7} - 1514740124 \beta_{8} + 907794398 \beta_{9} - 8347130578 \beta_{10} - 5555172164 \beta_{11} + 2055204046 \beta_{12} + 358545808 \beta_{13}) q^{35} +(\)\(28\!\cdots\!52\)\( - 25384428488076065942 \beta_{1} - 112247413097120470 \beta_{2} - 660558933978223 \beta_{3} - 10520614332896 \beta_{4} + 444768972064 \beta_{5} - 80843976064 \beta_{6} - 100145904880 \beta_{7} + 3690848000 \beta_{8} + 2051923184 \beta_{9} + 4652765152 \beta_{10} - 6857344192 \beta_{11} + 2485856256 \beta_{12} + 351833408 \beta_{13}) q^{36} +(-\)\(76\!\cdots\!84\)\( - 16682671261925429763 \beta_{1} - 720009214389711 \beta_{2} - 336801871576493 \beta_{3} + 6535484610527 \beta_{4} - 1851842468132 \beta_{5} + 1015422450696 \beta_{6} + 13491340765 \beta_{7} + 176001094 \beta_{8} - 13843342953 \beta_{9} - 12028923757 \beta_{10} + 10044704860 \beta_{11} + 5022352430 \beta_{12}) q^{37} +(-\)\(16\!\cdots\!14\)\( - 665657089123082195 \beta_{1} + 247047477896189056 \beta_{2} + 34803429861650 \beta_{3} + 6500564687636 \beta_{4} - 1118221228206 \beta_{5} - 757363280493 \beta_{6} + 107899529776 \beta_{7} - 4102858432 \beta_{8} + 17540827984 \beta_{9} + 33205904296 \beta_{10} - 21438717888 \beta_{11} + 2581978720 \beta_{12} + 1099680000 \beta_{13}) q^{38} +(624962358635587570 - 1459931781425078654 \beta_{1} + 15273370493315610 \beta_{2} + 2547950457687211 \beta_{3} - 17657127266063 \beta_{4} + 954389178506 \beta_{5} + 922618479351 \beta_{6} - 17144895895 \beta_{7} + 1622984256 \beta_{8} - 13256383776 \beta_{9} + 29156860896 \beta_{10} - 4377515072 \beta_{11} - 1211179808 \beta_{12} - 2576328448 \beta_{13}) q^{39} +(\)\(33\!\cdots\!10\)\( + \)\(14\!\cdots\!86\)\( \beta_{1} - 569102013066978766 \beta_{2} + 3188273335688400 \beta_{3} + 38516283175826 \beta_{4} + 7536273517104 \beta_{5} - 382718138894 \beta_{6} + 293724320384 \beta_{7} - 13916592128 \beta_{8} - 18092724864 \beta_{9} + 6895096064 \beta_{10} + 30098442242 \beta_{11} - 534816768 \beta_{12} - 2004749824 \beta_{13}) q^{40} +(\)\(25\!\cdots\!78\)\( - 74827106903692191600 \beta_{1} - 4964295670054316 \beta_{2} - 5273891891389358 \beta_{3} - 73111290498270 \beta_{4} + 196083311930 \beta_{5} - 9289809279598 \beta_{6} - 24442247362 \beta_{7} + 3097643794 \beta_{8} + 18246959774 \beta_{9} + 149321799722 \beta_{10} - 61513837052 \beta_{11} - 30756918526 \beta_{12}) q^{41} +(\)\(82\!\cdots\!72\)\( - \)\(14\!\cdots\!08\)\( \beta_{1} + 1788766687205266600 \beta_{2} - 411617736225916 \beta_{3} - 395733516354344 \beta_{4} + 31194245120996 \beta_{5} + 2853798294928 \beta_{6} - 926921803480 \beta_{7} - 940932952 \beta_{8} - 225206119064 \beta_{9} - 155474321208 \beta_{10} + 165802221408 \beta_{11} - 9277443836 \beta_{12} - 4681090220 \beta_{13}) q^{42} +(-\)\(16\!\cdots\!07\)\( + \)\(39\!\cdots\!99\)\( \beta_{1} + 185931105718975179 \beta_{2} + 14727067779205746 \beta_{3} - 145666920334884 \beta_{4} - 20327768160028 \beta_{5} + 12845360883568 \beta_{6} + 18187219198 \beta_{7} + 36477506388 \beta_{8} - 60572188506 \beta_{9} + 188820726966 \beta_{10} + 148431505228 \beta_{11} - 44553734570 \beta_{12} + 13509561680 \beta_{13}) q^{43} +(-\)\(24\!\cdots\!54\)\( + \)\(38\!\cdots\!70\)\( \beta_{1} - 2119332926193208276 \beta_{2} + 2729947231032101 \beta_{3} + 382920945820146 \beta_{4} + 2438731095756 \beta_{5} - 1525711660448 \beta_{6} + 1559743664007 \beta_{7} - 17309991984 \beta_{8} - 340819877895 \beta_{9} - 38185115766 \beta_{10} - 28770842812 \beta_{11} - 48387549984 \beta_{12} + 9499504908 \beta_{13}) q^{44} +(\)\(57\!\cdots\!60\)\( - \)\(40\!\cdots\!17\)\( \beta_{1} - 23188619904275261 \beta_{2} - 20698495577215619 \beta_{3} - 450632006803413 \beta_{4} - 10718189960500 \beta_{5} - 10064479887520 \beta_{6} + 138292738275 \beta_{7} + 76432138130 \beta_{8} - 291157014535 \beta_{9} - 25957963955 \beta_{10} + 125284508500 \beta_{11} + 62642254250 \beta_{12}) q^{45} +(-\)\(20\!\cdots\!45\)\( - 8228871592676831758 \beta_{1} + 2956665091073776119 \beta_{2} + 541377870988799 \beta_{3} + 56586366863334 \beta_{4} + 64123118839663 \beta_{5} - 11875055668578 \beta_{6} + 2389963429160 \beta_{7} - 33755153824 \beta_{8} + 161584235032 \beta_{9} + 608384253708 \beta_{10} - 610100319520 \beta_{11} + 39571972432 \beta_{12} + 12086122368 \beta_{13}) q^{46} +(-\)\(93\!\cdots\!54\)\( + \)\(21\!\cdots\!94\)\( \beta_{1} + 141961080888472528 \beta_{2} + 21118765888113880 \beta_{3} - 379944138409882 \beta_{4} - 93758569692776 \beta_{5} - 51216339351066 \beta_{6} + 1343920080320 \beta_{7} - 116700670428 \beta_{8} - 234323086914 \beta_{9} - 82996271346 \beta_{10} - 718812914948 \beta_{11} + 181024392430 \beta_{12} - 49734621040 \beta_{13}) q^{47} +(\)\(15\!\cdots\!20\)\( - \)\(17\!\cdots\!48\)\( \beta_{1} - 3913426185087773824 \beta_{2} - 19198073473731840 \beta_{3} - 1139533156589632 \beta_{4} + 412960441536704 \beta_{5} - 6119884024896 \beta_{6} - 10669206590528 \beta_{7} + 178567715840 \beta_{8} + 1042622981184 \beta_{9} + 945128281472 \beta_{10} - 55225166656 \beta_{11} + 363495994368 \beta_{12} - 37763994880 \beta_{13}) q^{48} +(-\)\(34\!\cdots\!23\)\( - \)\(77\!\cdots\!32\)\( \beta_{1} - 242874842737173500 \beta_{2} + 49002156370390384 \beta_{3} + 1346671197422684 \beta_{4} - 626640583928492 \beta_{5} + 149898996221492 \beta_{6} + 817826781520 \beta_{7} - 560952635124 \beta_{8} + 304078488728 \beta_{9} - 184409958656 \beta_{10} + 34327828248 \beta_{11} + 17163914124 \beta_{12}) q^{49} +(-\)\(36\!\cdots\!25\)\( + \)\(71\!\cdots\!55\)\( \beta_{1} - 4553048576003183880 \beta_{2} - 4308278025197300 \beta_{3} + 6586425808422280 \beta_{4} + 287460107685420 \beta_{5} - 24515446621520 \beta_{6} + 7184793830520 \beta_{7} + 321462119160 \beta_{8} + 1536307642680 \beta_{9} + 909235826520 \beta_{10} + 1335753865760 \beta_{11} - 64684993140 \beta_{12} - 870521220 \beta_{13}) q^{50} +(-\)\(60\!\cdots\!67\)\( + \)\(14\!\cdots\!93\)\( \beta_{1} - 321916589828146636 \beta_{2} - 223294730242036597 \beta_{3} + 37612054018498 \beta_{4} - 879010069817866 \beta_{5} - 115915147526880 \beta_{6} - 10188969603139 \beta_{7} - 295978543538 \beta_{8} + 1000787145233 \beta_{9} - 3163724917143 \beta_{10} + 865242766370 \beta_{11} + 88353150169 \beta_{12} + 105723746616 \beta_{13}) q^{51} +(-\)\(69\!\cdots\!28\)\( - \)\(74\!\cdots\!12\)\( \beta_{1} + 17234550105038492708 \beta_{2} - 154649190232210038 \beta_{3} + 541843091419152 \beta_{4} + 421248452347792 \beta_{5} + 19155498315072 \beta_{6} + 12052194600968 \beta_{7} + 211941033856 \beta_{8} + 478708967416 \beta_{9} - 2150499198224 \beta_{10} - 917053575264 \beta_{11} - 1362401782784 \beta_{12} + 123064043680 \beta_{13}) q^{52} +(\)\(13\!\cdots\!16\)\( - \)\(16\!\cdots\!89\)\( \beta_{1} - 344571749460908809 \beta_{2} + 469036988676569137 \beta_{3} - 7578301469499405 \beta_{4} - 714937796486884 \beta_{5} + 6274432427136 \beta_{6} - 4942880102769 \beta_{7} + 1235813485050 \beta_{8} + 2471253132669 \beta_{9} - 7736064803103 \beta_{10} + 810255395556 \beta_{11} + 405127697778 \beta_{12}) q^{53} +(\)\(40\!\cdots\!58\)\( + \)\(15\!\cdots\!78\)\( \beta_{1} - 51590271180882502142 \beta_{2} - 21713648473983506 \beta_{3} + 6090390725613484 \beta_{4} + 3400285280861582 \beta_{5} + 354411807738606 \beta_{6} - 54529548475696 \beta_{7} + 133536502464 \beta_{8} - 1420326203472 \beta_{9} - 12474673124904 \beta_{10} - 4359237133376 \beta_{11} - 678104917088 \beta_{12} - 151970863360 \beta_{13}) q^{54} +(-\)\(28\!\cdots\!50\)\( + \)\(67\!\cdots\!10\)\( \beta_{1} - 21131239313864370090 \beta_{2} - 445580966982237315 \beta_{3} - 4142953323769545 \beta_{4} - 4344447675548570 \beta_{5} + 814484194121345 \beta_{6} + 54237796380815 \beta_{7} + 1638449017440 \beta_{8} + 6080404530000 \beta_{9} + 739322542800 \beta_{10} + 6980488437920 \beta_{11} - 2239246842160 \beta_{12} + 49401727360 \beta_{13}) q^{55} +(\)\(45\!\cdots\!44\)\( + \)\(20\!\cdots\!12\)\( \beta_{1} + \)\(11\!\cdots\!32\)\( \beta_{2} + 410654820304519424 \beta_{3} + 20259319975728480 \beta_{4} + 5240480250980896 \beta_{5} + 488642020134496 \beta_{6} + 69600831394336 \beta_{7} - 3012924333568 \beta_{8} + 3634108442080 \beta_{9} - 12661281044672 \beta_{10} + 6416045341664 \beta_{11} + 1095473761792 \beta_{12} - 295137678720 \beta_{13}) q^{56} +(-\)\(32\!\cdots\!48\)\( - \)\(15\!\cdots\!13\)\( \beta_{1} - 5252220562755886959 \beta_{2} - 29181990059931708 \beta_{3} + 5628401614014711 \beta_{4} - 10010116305324627 \beta_{5} - 1555290715649019 \beta_{6} - 9289669868628 \beta_{7} - 335373816549 \beta_{8} + 9960417501726 \beta_{9} + 25151859990216 \beta_{10} - 10699424953098 \beta_{11} - 5349712476549 \beta_{12}) q^{57} +(-\)\(27\!\cdots\!86\)\( - \)\(28\!\cdots\!43\)\( \beta_{1} - \)\(14\!\cdots\!98\)\( \beta_{2} - 496086557641979873 \beta_{3} - 7326862751395526 \beta_{4} + 6647529160320135 \beta_{5} - 568400580516084 \beta_{6} + 33165386588062 \beta_{7} - 6436612199490 \beta_{8} + 8273574999502 \beta_{9} + 6060403451846 \beta_{10} + 27529640204552 \beta_{11} + 3521107166819 \beta_{12} + 711721189855 \beta_{13}) q^{58} +(-\)\(69\!\cdots\!29\)\( + \)\(16\!\cdots\!17\)\( \beta_{1} - 20226046600352454005 \beta_{2} + 771617582721959296 \beta_{3} - 22695857826181944 \beta_{4} - 9351872614143840 \beta_{5} + 945088582129032 \beta_{6} - 302490864945184 \beta_{7} + 1305082967664 \beta_{8} - 2797156249848 \beta_{9} + 23667893264328 \beta_{10} - 32963897623280 \beta_{11} + 4173335119048 \beta_{12} - 1351777519168 \beta_{13}) q^{59} +(\)\(33\!\cdots\!80\)\( - \)\(28\!\cdots\!52\)\( \beta_{1} + \)\(24\!\cdots\!52\)\( \beta_{2} - 336921482911497050 \beta_{3} + 82530528739555708 \beta_{4} + 20421188629217192 \beta_{5} - 2513480383644352 \beta_{6} - 232850599561918 \beta_{7} + 3441918898016 \beta_{8} - 13802377112642 \beta_{9} + 14486617233612 \beta_{10} - 9393216304584 \beta_{11} + 11781757980736 \beta_{12} + 283998071848 \beta_{13}) q^{60} +(\)\(76\!\cdots\!88\)\( - \)\(29\!\cdots\!83\)\( \beta_{1} - 11067847597283288195 \beta_{2} - 2528398285158594985 \beta_{3} - 70285230903406193 \beta_{4} - 18079807754869804 \beta_{5} + 1296834843568592 \beta_{6} + 38434706705465 \beta_{7} + 4043911909606 \beta_{8} - 46522530524677 \beta_{9} - 18462652241385 \beta_{10} + 36381624797020 \beta_{11} + 18190812398510 \beta_{12}) q^{61} +(\)\(15\!\cdots\!96\)\( + \)\(60\!\cdots\!96\)\( \beta_{1} - \)\(24\!\cdots\!76\)\( \beta_{2} - 8168113746513720 \beta_{3} - 41122418271324592 \beta_{4} + 32304337012268616 \beta_{5} - 665377241340568 \beta_{6} + 427942324472000 \beta_{7} + 13357088163072 \beta_{8} - 36173438317248 \beta_{9} + 66188079575712 \beta_{10} - 123045662105344 \beta_{11} - 217904985728 \beta_{12} - 1414508354560 \beta_{13}) q^{62} +(-\)\(37\!\cdots\!20\)\( + \)\(86\!\cdots\!28\)\( \beta_{1} + \)\(16\!\cdots\!54\)\( \beta_{2} + 7820290135500124891 \beta_{3} - 148684927622835773 \beta_{4} - 42030769194211870 \beta_{5} - 9411420395278887 \beta_{6} + 1214963348152193 \beta_{7} - 13441934324772 \beta_{8} - 63598526376126 \beta_{9} - 27136050265614 \beta_{10} + 13587575280388 \beta_{11} + 8436802857490 \beta_{12} + 5220327563120 \beta_{13}) q^{63} +(-\)\(14\!\cdots\!84\)\( - \)\(22\!\cdots\!88\)\( \beta_{1} - \)\(15\!\cdots\!04\)\( \beta_{2} - 2275779049219172608 \beta_{3} + 27066986011249856 \beta_{4} + 36274213362838464 \beta_{5} + 7823901582172352 \beta_{6} - 153869919503680 \beta_{7} + 15502390889472 \beta_{8} - 52983887604416 \beta_{9} + 138080046730112 \beta_{10} - 36639414741568 \beta_{11} - 42215603829760 \beta_{12} + 1593986864896 \beta_{13}) q^{64} +(\)\(80\!\cdots\!00\)\( - \)\(67\!\cdots\!80\)\( \beta_{1} - 24051940133571688380 \beta_{2} - 1978151470716266250 \beta_{3} - 294075411870343490 \beta_{4} - 39603505644138250 \beta_{5} + 11172014576290110 \beta_{6} + 98290423381050 \beta_{7} - 34334651283090 \beta_{8} - 29621120814870 \beta_{9} - 18051611523810 \beta_{10} - 27333265114500 \beta_{11} - 13666632557250 \beta_{12}) q^{65} +(\)\(91\!\cdots\!72\)\( + \)\(25\!\cdots\!22\)\( \beta_{1} + \)\(11\!\cdots\!88\)\( \beta_{2} + 550698024849580274 \beta_{3} + 342807180083865036 \beta_{4} + 70975338966816674 \beta_{5} - 262128904638264 \beta_{6} - 777447640020396 \beta_{7} + 33342451073044 \beta_{8} - 80374344355212 \beta_{9} + 14205622087780 \beta_{10} + 332010599620272 \beta_{11} - 34058726934814 \beta_{12} - 1184955094390 \beta_{13}) q^{66} +(-\)\(37\!\cdots\!86\)\( + \)\(86\!\cdots\!84\)\( \beta_{1} + \)\(51\!\cdots\!75\)\( \beta_{2} + 5248875414843367129 \beta_{3} - 129361396347498458 \beta_{4} - 48912942651924078 \beta_{5} + 1524935099442032 \beta_{6} - 3259800641435393 \beta_{7} - 6736126801158 \beta_{8} - 69136639813269 \beta_{9} - 38573969995341 \beta_{10} + 270277537123542 \beta_{11} - 43447242999165 \beta_{12} - 8675214038040 \beta_{13}) q^{67} +(\)\(25\!\cdots\!48\)\( + \)\(22\!\cdots\!68\)\( \beta_{1} - \)\(15\!\cdots\!36\)\( \beta_{2} + 4961073695857378002 \beta_{3} - 513190107722785376 \beta_{4} + 103245480736061600 \beta_{5} - 31328552075589504 \beta_{6} + 1786747840407248 \beta_{7} - 34055751245056 \beta_{8} + 126890905270576 \beta_{9} - 30310347247520 \beta_{10} + 166853022191168 \beta_{11} + 2692442523648 \beta_{12} - 10133329864640 \beta_{13}) q^{68} +(-\)\(37\!\cdots\!90\)\( - \)\(18\!\cdots\!99\)\( \beta_{1} - 61151521625505300935 \beta_{2} + 3227338130654475489 \beta_{3} + 344289048146692162 \beta_{4} - 130144595899725028 \beta_{5} - 16776777999241120 \beta_{6} - 72456144680425 \beta_{7} + 29280909960010 \beta_{8} + 13894324760405 \beta_{9} + 166628112444249 \beta_{10} - 161253808134652 \beta_{11} - 80626904067326 \beta_{12}) q^{69} +(-\)\(36\!\cdots\!60\)\( - \)\(14\!\cdots\!08\)\( \beta_{1} + \)\(33\!\cdots\!56\)\( \beta_{2} + 3395934435103076364 \beta_{3} + 872383964203438648 \beta_{4} + 85059851299668300 \beta_{5} + 3527828831476740 \beta_{6} - 1844064238623200 \beta_{7} - 145325742943360 \beta_{8} + 320063305949920 \beta_{9} - 94827731575440 \beta_{10} - 490000821187200 \beta_{11} + 49178941966400 \beta_{12} + 15929242150400 \beta_{13}) q^{70} +(-\)\(88\!\cdots\!08\)\( + \)\(20\!\cdots\!56\)\( \beta_{1} - \)\(23\!\cdots\!54\)\( \beta_{2} - 25273432796659264729 \beta_{3} - 45125429146698925 \beta_{4} - 141512757649889510 \beta_{5} + 40522040487710689 \beta_{6} + 7415058012498229 \beta_{7} + 91144864175812 \beta_{8} + 407284682137838 \beta_{9} + 323621378202302 \beta_{10} - 619988034986148 \beta_{11} + 39754662839454 \beta_{12} - 8440300623088 \beta_{13}) q^{71} +(\)\(81\!\cdots\!35\)\( + \)\(29\!\cdots\!95\)\( \beta_{1} - \)\(58\!\cdots\!59\)\( \beta_{2} - 17682635354849384344 \beta_{3} + 619613887407488073 \beta_{4} + 33011901584721112 \beta_{5} + 101210618841951097 \beta_{6} - 84545509138176 \beta_{7} - 84147236864 \beta_{8} - 210605122527488 \beta_{9} - 1053761217250816 \beta_{10} - 198097274913919 \beta_{11} + 253632140996608 \beta_{12} + 29055291724800 \beta_{13}) q^{72} +(\)\(10\!\cdots\!02\)\( - \)\(12\!\cdots\!85\)\( \beta_{1} - 29641656762028116347 \beta_{2} + 27858820601303250208 \beta_{3} - 787761844431007801 \beta_{4} - 20429000512377899 \beta_{5} - 68454404575771171 \beta_{6} - 951418843555856 \beta_{7} + 275639067954075 \beta_{8} + 400140707647706 \beta_{9} - 364483045230812 \beta_{10} + 578785076590134 \beta_{11} + 289392538295067 \beta_{12}) q^{73} +(-\)\(70\!\cdots\!90\)\( - \)\(79\!\cdots\!95\)\( \beta_{1} + \)\(38\!\cdots\!62\)\( \beta_{2} - 22986237972462102973 \beta_{3} - 786678737397106222 \beta_{4} - 74648037725693429 \beta_{5} + 9553945292314908 \beta_{6} + 5985387886064646 \beta_{7} + 12163390223270 \beta_{8} - 109474722416522 \beta_{9} - 1025951692157234 \beta_{10} - 90217123513496 \beta_{11} + 175295565139079 \beta_{12} - 42149816127277 \beta_{13}) q^{74} +(\)\(12\!\cdots\!25\)\( - \)\(28\!\cdots\!85\)\( \beta_{1} - \)\(16\!\cdots\!65\)\( \beta_{2} - \)\(12\!\cdots\!50\)\( \beta_{3} + 1207894653224895940 \beta_{4} + 83237225974700660 \beta_{5} + 31362134130332040 \beta_{6} - 16118680836887290 \beta_{7} + 41571825084180 \beta_{8} + 764911866456390 \beta_{9} - 429855329701290 \beta_{10} - 851194303164020 \beta_{11} + 142203192487030 \beta_{12} + 84156076047440 \beta_{13}) q^{75} +(-\)\(55\!\cdots\!90\)\( - \)\(17\!\cdots\!22\)\( \beta_{1} - \)\(48\!\cdots\!32\)\( \beta_{2} - 1671574893790659449 \beta_{3} + 4027480834177552582 \beta_{4} - 237949829067514620 \beta_{5} - 240161311231626208 \beta_{6} - 12201703200742115 \beta_{7} - 113536457688848 \beta_{8} + 937535302471011 \beta_{9} + 423372445371086 \beta_{10} - 289166499460948 \beta_{11} - 339144106610784 \beta_{12} - 32320185802300 \beta_{13}) q^{76} +(-\)\(22\!\cdots\!90\)\( + \)\(62\!\cdots\!97\)\( \beta_{1} + \)\(27\!\cdots\!41\)\( \beta_{2} + \)\(13\!\cdots\!85\)\( \beta_{3} + 4507787298673958794 \beta_{4} + 231285417534631564 \beta_{5} + 126283475461176160 \beta_{6} - 409428641975597 \beta_{7} - 793297175768782 \beta_{8} + 1996022993513161 \beta_{9} + 444789097022557 \beta_{10} - 788158854183532 \beta_{11} - 394079427091766 \beta_{12}) q^{77} +(\)\(62\!\cdots\!51\)\( + \)\(24\!\cdots\!14\)\( \beta_{1} - \)\(44\!\cdots\!09\)\( \beta_{2} + 6319436261616837127 \beta_{3} - 4976852084973537610 \beta_{4} - 499161179319605513 \beta_{5} - 5742151112068874 \beta_{6} + 5347501008673768 \beta_{7} + 540664041834336 \beta_{8} + 409077206794584 \beta_{9} + 25852545479340 \beta_{10} + 2736748574356448 \beta_{11} - 403330422995248 \beta_{12} + 15558244113280 \beta_{13}) q^{78} +(\)\(39\!\cdots\!52\)\( - \)\(92\!\cdots\!32\)\( \beta_{1} + \)\(83\!\cdots\!08\)\( \beta_{2} + \)\(11\!\cdots\!02\)\( \beta_{3} + 150182401018667102 \beta_{4} + 676181250988624508 \beta_{5} - 228422380144449438 \beta_{6} + 24207055764257478 \beta_{7} - 521885119932432 \beta_{8} - 1455291862964792 \beta_{9} - 2683227314307448 \beta_{10} + 4980344211023760 \beta_{11} - 547973161782264 \beta_{12} - 197517619699776 \beta_{13}) q^{79} +(\)\(44\!\cdots\!80\)\( + \)\(34\!\cdots\!44\)\( \beta_{1} + \)\(11\!\cdots\!72\)\( \beta_{2} + \)\(13\!\cdots\!48\)\( \beta_{3} - 2280711815466442024 \beta_{4} - 756703150548625640 \beta_{5} + 632632887478807640 \beta_{6} + 6784440749221880 \beta_{7} + 314272990174080 \beta_{8} + 1746536488878600 \beta_{9} + 3316588794452400 \beta_{10} + 1405800292286840 \beta_{11} - 829761355176320 \beta_{12} - 90002708613280 \beta_{13}) q^{80} +(-\)\(30\!\cdots\!51\)\( + \)\(20\!\cdots\!57\)\( \beta_{1} + \)\(58\!\cdots\!71\)\( \beta_{2} - \)\(27\!\cdots\!76\)\( \beta_{3} - 9413626155467891319 \beta_{4} + 1548473886787942227 \beta_{5} + 399205015052281851 \beta_{6} + 6045983688717972 \beta_{7} + 433585638544485 \beta_{8} - 6913154965806942 \beta_{9} - 5424283114325256 \beta_{10} - 280044488091126 \beta_{11} - 140022244045563 \beta_{12}) q^{81} +(-\)\(32\!\cdots\!02\)\( + \)\(24\!\cdots\!50\)\( \beta_{1} - \)\(40\!\cdots\!12\)\( \beta_{2} - 22414886565116349076 \beta_{3} - 2140624605146099000 \beta_{4} - 1321031252366689396 \beta_{5} - 17732452319540944 \beta_{6} - 34519842107369032 \beta_{7} - 363861210246088 \beta_{8} + 2093998605412472 \beta_{9} + 6841770277317272 \beta_{10} - 8030568674853600 \beta_{11} - 745275624915028 \beta_{12} + 219797194446620 \beta_{13}) q^{82} +(\)\(16\!\cdots\!61\)\( - \)\(39\!\cdots\!37\)\( \beta_{1} + \)\(93\!\cdots\!85\)\( \beta_{2} + \)\(37\!\cdots\!44\)\( \beta_{3} + 1627952615159610368 \beta_{4} + 2515451777417154280 \beta_{5} - 184670659002141848 \beta_{6} - 10847408289322468 \beta_{7} - 228836875678568 \beta_{8} - 3292787614702444 \beta_{9} + 2414208214272884 \beta_{10} - 1861915933042648 \beta_{11} + 494948286921140 \beta_{12} + 96213142884960 \beta_{13}) q^{83} +(\)\(13\!\cdots\!84\)\( + \)\(87\!\cdots\!48\)\( \beta_{1} + \)\(66\!\cdots\!80\)\( \beta_{2} - \)\(22\!\cdots\!36\)\( \beta_{3} - 23475324795005244992 \beta_{4} - 3004968330087273536 \beta_{5} - 1566129294141320448 \beta_{6} + 63497389415261408 \beta_{7} + 1667072826577408 \beta_{8} - 9972113588233440 \beta_{9} - 2513318836458944 \beta_{10} - 2098834731858560 \beta_{11} + 2159673530068992 \beta_{12} + 479582986584448 \beta_{13}) q^{84} +(\)\(48\!\cdots\!50\)\( + \)\(64\!\cdots\!45\)\( \beta_{1} + \)\(19\!\cdots\!45\)\( \beta_{2} - \)\(51\!\cdots\!75\)\( \beta_{3} + 27830538412527216060 \beta_{4} + 3476224051145146500 \beta_{5} - 684620759635015640 \beta_{6} + 603515387264175 \beta_{7} + 885197272211410 \beta_{8} - 2373909931686995 \beta_{9} + 9551028178964065 \beta_{10} + 2827561368164500 \beta_{11} + 1413780684082250 \beta_{12}) q^{85} +(-\)\(16\!\cdots\!04\)\( - \)\(66\!\cdots\!73\)\( \beta_{1} - \)\(52\!\cdots\!54\)\( \beta_{2} + \)\(44\!\cdots\!56\)\( \beta_{3} + 13403491618522679496 \beta_{4} - 4356444091734808012 \beta_{5} - 131979024085987967 \beta_{6} - 10848490449961504 \beta_{7} - 524953939796864 \beta_{8} - 13606243563022048 \beta_{9} - 957419748321136 \beta_{10} + 13117387875967616 \beta_{11} + 2172523429650368 \beta_{12} - 644317393036800 \beta_{13}) q^{86} +(\)\(37\!\cdots\!92\)\( - \)\(87\!\cdots\!72\)\( \beta_{1} - \)\(16\!\cdots\!26\)\( \beta_{2} + \)\(25\!\cdots\!61\)\( \beta_{3} + 7681300295509162037 \beta_{4} + 4814613048867159550 \beta_{5} + 861563400784604319 \beta_{6} - 45900114855052097 \beta_{7} + 2279934851352436 \beta_{8} + 836805725094998 \beta_{9} + 17361741284315622 \beta_{10} - 20895074615364564 \beta_{11} + 2186264984980550 \beta_{12} + 671127349964240 \beta_{13}) q^{87} +(\)\(79\!\cdots\!68\)\( - \)\(24\!\cdots\!20\)\( \beta_{1} + \)\(61\!\cdots\!60\)\( \beta_{2} + \)\(31\!\cdots\!68\)\( \beta_{3} - 27495284836670413904 \beta_{4} - 4398454945988937712 \beta_{5} + 2482643482656356400 \beta_{6} - 65310702071272432 \beta_{7} - 5537537188183808 \beta_{8} - 318957259452432 \beta_{9} + 4415971333239200 \beta_{10} - 390216578624784 \beta_{11} + 1220236370892544 \beta_{12} - 852007213895360 \beta_{13}) q^{88} +(-\)\(13\!\cdots\!62\)\( + \)\(43\!\cdots\!31\)\( \beta_{1} + \)\(11\!\cdots\!69\)\( \beta_{2} - \)\(59\!\cdots\!44\)\( \beta_{3} - 42012598081522528833 \beta_{4} + 5171666767763604557 \beta_{5} - 2022329651552276683 \beta_{6} - 20647056811043032 \beta_{7} - 2599622065803821 \beta_{8} + 25846300942650674 \beta_{9} + 35078606389352396 \beta_{10} - 1632544022651354 \beta_{11} - 816272011325677 \beta_{12}) q^{89} +(-\)\(18\!\cdots\!70\)\( + \)\(56\!\cdots\!33\)\( \beta_{1} - \)\(49\!\cdots\!58\)\( \beta_{2} - \)\(35\!\cdots\!85\)\( \beta_{3} - 19942209351896653982 \beta_{4} - 4929830523930362973 \beta_{5} + 51868158403475068 \beta_{6} + 171757861021128182 \beta_{7} - 1815710307444074 \beta_{8} - 3431518879650842 \beta_{9} - 17527236132008578 \beta_{10} - 16659366692228184 \beta_{11} + 3256077583951391 \beta_{12} + 381531168117323 \beta_{13}) q^{90} +(\)\(36\!\cdots\!90\)\( - \)\(85\!\cdots\!54\)\( \beta_{1} - \)\(19\!\cdots\!64\)\( \beta_{2} - 28545566221577393282 \beta_{3} + 9295142519937460876 \beta_{4} + 4545317079349973404 \beta_{5} + 1068383044133003544 \beta_{6} + 209743356370459122 \beta_{7} + 1592624101055868 \beta_{8} + 4329009542275250 \beta_{9} - 764671147247390 \beta_{10} + 29110465726221924 \beta_{11} - 6569914197446334 \beta_{12} - 1969261527116304 \beta_{13}) q^{91} +(\)\(43\!\cdots\!00\)\( - \)\(19\!\cdots\!68\)\( \beta_{1} - \)\(30\!\cdots\!92\)\( \beta_{2} + 76714562302967180654 \beta_{3} + 37502311959472526668 \beta_{4} - 5387538462386880632 \beta_{5} - 2764867129809765312 \beta_{6} - 215171225575053478 \beta_{7} - 3105104533080096 \beta_{8} + 14177974503968166 \beta_{9} - 5117926053481380 \beta_{10} + 4358674313058520 \beta_{11} - 7966308969712832 \beta_{12} - 180682699949560 \beta_{13}) q^{92} +(\)\(31\!\cdots\!92\)\( + \)\(15\!\cdots\!60\)\( \beta_{1} + \)\(62\!\cdots\!40\)\( \beta_{2} + \)\(22\!\cdots\!52\)\( \beta_{3} + \)\(12\!\cdots\!52\)\( \beta_{4} + 4901616137787564384 \beta_{5} + 2511712507136828496 \beta_{6} + 19284053631717764 \beta_{7} + 11996685107751368 \beta_{8} - 43277423847220500 \beta_{9} - 79881846909560836 \beta_{10} - 5443378226367536 \beta_{11} - 2721689113183768 \beta_{12}) q^{93} +(-\)\(93\!\cdots\!78\)\( - \)\(36\!\cdots\!76\)\( \beta_{1} + \)\(22\!\cdots\!38\)\( \beta_{2} + \)\(18\!\cdots\!14\)\( \beta_{3} - 69015450750684073564 \beta_{4} - 3164267239570940870 \beta_{5} + 325044674220515820 \beta_{6} + 18680095975069168 \beta_{7} + 1405281431077440 \beta_{8} + 50584350502907280 \beta_{9} - 5691758817761592 \beta_{10} + 22926190677071168 \beta_{11} - 10753745247418912 \beta_{12} + 2171009657380096 \beta_{13}) q^{94} +(\)\(13\!\cdots\!50\)\( - \)\(32\!\cdots\!70\)\( \beta_{1} + \)\(96\!\cdots\!30\)\( \beta_{2} - \)\(46\!\cdots\!45\)\( \beta_{3} + 36143388364453978765 \beta_{4} + 329965892797801490 \beta_{5} - 2647885979793982165 \beta_{6} - 612568579175096755 \beta_{7} - 7540638424940080 \beta_{8} + 13116336462880600 \beta_{9} - 73517857096260200 \beta_{10} + 42776618273066160 \beta_{11} - 1436914581659880 \beta_{12} + 1996288541673280 \beta_{13}) q^{95} +(\)\(24\!\cdots\!48\)\( + \)\(16\!\cdots\!52\)\( \beta_{1} - \)\(55\!\cdots\!84\)\( \beta_{2} - \)\(76\!\cdots\!96\)\( \beta_{3} + \)\(10\!\cdots\!40\)\( \beta_{4} + 2795227621827678464 \beta_{5} + 2833890918548217088 \beta_{6} + 293012631174403328 \beta_{7} + 26663387855441920 \beta_{8} + 11915277230631680 \beta_{9} - 53657126881242624 \beta_{10} + 6667552894732544 \beta_{11} + 6011770147434496 \beta_{12} + 4073706919535616 \beta_{13}) q^{96} +(-\)\(14\!\cdots\!50\)\( - \)\(31\!\cdots\!23\)\( \beta_{1} - \)\(93\!\cdots\!61\)\( \beta_{2} + \)\(30\!\cdots\!26\)\( \beta_{3} - \)\(21\!\cdots\!29\)\( \beta_{4} - 15526288599173407079 \beta_{5} + 8249900626631150393 \beta_{6} + 18409218639247138 \beta_{7} - 23531773474402485 \beta_{8} + 28654328309557832 \beta_{9} - 52032789243556094 \beta_{10} - 24694998573169482 \beta_{11} - 12347499286584741 \beta_{12}) q^{97} +(-\)\(32\!\cdots\!03\)\( - \)\(35\!\cdots\!03\)\( \beta_{1} + \)\(53\!\cdots\!36\)\( \beta_{2} - \)\(90\!\cdots\!60\)\( \beta_{3} + 34946027252161249616 \beta_{4} + 8569125808648625976 \beta_{5} + 1163636772809683424 \beta_{6} - 686528522605145808 \beta_{7} + 17026584095310384 \beta_{8} + 8737230557615280 \beta_{9} + 34173546162573552 \beta_{10} + 20955926159420224 \beta_{11} - 10646422407958344 \beta_{12} - 5955445916868840 \beta_{13}) q^{98} +(-\)\(23\!\cdots\!65\)\( + \)\(53\!\cdots\!57\)\( \beta_{1} - \)\(24\!\cdots\!97\)\( \beta_{2} - \)\(29\!\cdots\!72\)\( \beta_{3} - 37398086086498408744 \beta_{4} - 29199515035701865904 \beta_{5} - 6426992365656309624 \beta_{6} + 1232516962406283976 \beta_{7} - 10965214374903072 \beta_{8} + 19327116166881936 \beta_{9} - 48206328642683376 \beta_{10} - 111404288773106656 \beta_{11} + 26454656341048976 \beta_{12} + 1731395886787456 \beta_{13}) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14q - 23780q^{2} - 2922848368q^{4} + 138121491740q^{5} + 1262734959552q^{6} - 191366550113600q^{8} - 11183509932817650q^{9} + O(q^{10}) \) \( 14q - 23780q^{2} - 2922848368q^{4} + 138121491740q^{5} + 1262734959552q^{6} - 191366550113600q^{8} - 11183509932817650q^{9} + 31775605694457400q^{10} - 356375853407619840q^{12} - 1640418469677858020q^{13} + 7731597686180285568q^{14} - 30570843123186593536q^{16} + 31570967905797256220q^{17} - \)\(35\!\cdots\!40\)\(q^{18} + \)\(20\!\cdots\!60\)\(q^{20} - \)\(21\!\cdots\!16\)\(q^{21} + \)\(54\!\cdots\!80\)\(q^{22} - \)\(25\!\cdots\!32\)\(q^{24} + \)\(10\!\cdots\!50\)\(q^{25} - \)\(10\!\cdots\!24\)\(q^{26} + \)\(28\!\cdots\!20\)\(q^{28} - \)\(38\!\cdots\!16\)\(q^{29} - \)\(58\!\cdots\!20\)\(q^{30} - \)\(30\!\cdots\!00\)\(q^{32} + \)\(30\!\cdots\!00\)\(q^{33} + \)\(29\!\cdots\!16\)\(q^{34} + \)\(40\!\cdots\!96\)\(q^{36} - \)\(10\!\cdots\!40\)\(q^{37} - \)\(23\!\cdots\!60\)\(q^{38} + \)\(46\!\cdots\!00\)\(q^{40} + \)\(36\!\cdots\!48\)\(q^{41} + \)\(11\!\cdots\!00\)\(q^{42} - \)\(34\!\cdots\!40\)\(q^{44} + \)\(80\!\cdots\!80\)\(q^{45} - \)\(29\!\cdots\!48\)\(q^{46} + \)\(22\!\cdots\!60\)\(q^{48} - \)\(48\!\cdots\!90\)\(q^{49} - \)\(51\!\cdots\!00\)\(q^{50} - \)\(96\!\cdots\!20\)\(q^{52} + \)\(18\!\cdots\!80\)\(q^{53} + \)\(56\!\cdots\!36\)\(q^{54} + \)\(63\!\cdots\!52\)\(q^{56} - \)\(45\!\cdots\!40\)\(q^{57} - \)\(38\!\cdots\!60\)\(q^{58} + \)\(47\!\cdots\!00\)\(q^{60} + \)\(10\!\cdots\!08\)\(q^{61} + \)\(21\!\cdots\!00\)\(q^{62} - \)\(19\!\cdots\!88\)\(q^{64} + \)\(11\!\cdots\!00\)\(q^{65} + \)\(12\!\cdots\!60\)\(q^{66} + \)\(35\!\cdots\!00\)\(q^{68} - \)\(51\!\cdots\!84\)\(q^{69} - \)\(51\!\cdots\!80\)\(q^{70} + \)\(11\!\cdots\!60\)\(q^{72} + \)\(14\!\cdots\!60\)\(q^{73} - \)\(98\!\cdots\!84\)\(q^{74} - \)\(78\!\cdots\!00\)\(q^{76} - \)\(30\!\cdots\!40\)\(q^{77} + \)\(87\!\cdots\!00\)\(q^{78} + \)\(62\!\cdots\!40\)\(q^{80} - \)\(42\!\cdots\!22\)\(q^{81} - \)\(45\!\cdots\!80\)\(q^{82} + \)\(18\!\cdots\!96\)\(q^{84} + \)\(67\!\cdots\!00\)\(q^{85} - \)\(23\!\cdots\!48\)\(q^{86} + \)\(11\!\cdots\!80\)\(q^{88} - \)\(18\!\cdots\!76\)\(q^{89} - \)\(25\!\cdots\!00\)\(q^{90} + \)\(60\!\cdots\!00\)\(q^{92} + \)\(43\!\cdots\!40\)\(q^{93} - \)\(13\!\cdots\!12\)\(q^{94} + \)\(34\!\cdots\!32\)\(q^{96} - \)\(20\!\cdots\!20\)\(q^{97} - \)\(46\!\cdots\!00\)\(q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{14} + 72511313626452 x^{12} + 2025191977179903324811336518 x^{10} + 27922884728028663894750078705223437415644 x^{8} + 203010662886800095440071970440402438747266446160157745 x^{6} + 758734102549599282271818004575465783845093632382487984186969965640 x^{4} + 1269648449115368448095465842606476325720277486461580161887038301255933321354000 x^{2} + 624216522131873762678666934520680301449631616035103441585151846396220724278849706601312000\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(\)\(39\!\cdots\!85\)\( \nu^{13} + \)\(12\!\cdots\!61\)\( \nu^{12} + \)\(24\!\cdots\!05\)\( \nu^{11} + \)\(74\!\cdots\!97\)\( \nu^{10} + \)\(51\!\cdots\!75\)\( \nu^{9} + \)\(16\!\cdots\!79\)\( \nu^{8} + \)\(41\!\cdots\!15\)\( \nu^{7} + \)\(16\!\cdots\!47\)\( \nu^{6} + \)\(82\!\cdots\!20\)\( \nu^{5} + \)\(76\!\cdots\!96\)\( \nu^{4} - \)\(35\!\cdots\!40\)\( \nu^{3} + \)\(16\!\cdots\!40\)\( \nu^{2} - \)\(80\!\cdots\!20\)\( \nu + \)\(11\!\cdots\!60\)\(\)\()/ \)\(75\!\cdots\!20\)\( \)
\(\beta_{2}\)\(=\)\((\)\(\)\(39\!\cdots\!85\)\( \nu^{13} + \)\(12\!\cdots\!61\)\( \nu^{12} + \)\(24\!\cdots\!05\)\( \nu^{11} + \)\(74\!\cdots\!97\)\( \nu^{10} + \)\(51\!\cdots\!75\)\( \nu^{9} + \)\(16\!\cdots\!79\)\( \nu^{8} + \)\(41\!\cdots\!15\)\( \nu^{7} + \)\(16\!\cdots\!47\)\( \nu^{6} + \)\(82\!\cdots\!20\)\( \nu^{5} + \)\(76\!\cdots\!96\)\( \nu^{4} - \)\(35\!\cdots\!40\)\( \nu^{3} + \)\(16\!\cdots\!40\)\( \nu^{2} + \)\(49\!\cdots\!00\)\( \nu + \)\(11\!\cdots\!80\)\(\)\()/ \)\(36\!\cdots\!20\)\( \)
\(\beta_{3}\)\(=\)\((\)\(-\)\(98\!\cdots\!11\)\( \nu^{13} + \)\(38\!\cdots\!41\)\( \nu^{12} - \)\(64\!\cdots\!39\)\( \nu^{11} + \)\(12\!\cdots\!97\)\( \nu^{10} - \)\(15\!\cdots\!25\)\( \nu^{9} + \)\(71\!\cdots\!99\)\( \nu^{8} - \)\(16\!\cdots\!49\)\( \nu^{7} + \)\(15\!\cdots\!47\)\( \nu^{6} - \)\(83\!\cdots\!96\)\( \nu^{5} + \)\(13\!\cdots\!56\)\( \nu^{4} - \)\(17\!\cdots\!40\)\( \nu^{3} + \)\(41\!\cdots\!60\)\( \nu^{2} - \)\(12\!\cdots\!00\)\( \nu + \)\(11\!\cdots\!20\)\(\)\()/ \)\(67\!\cdots\!60\)\( \)
\(\beta_{4}\)\(=\)\((\)\(\)\(15\!\cdots\!77\)\( \nu^{13} - \)\(32\!\cdots\!91\)\( \nu^{12} + \)\(10\!\cdots\!93\)\( \nu^{11} - \)\(21\!\cdots\!79\)\( \nu^{10} + \)\(25\!\cdots\!75\)\( \nu^{9} - \)\(52\!\cdots\!53\)\( \nu^{8} + \)\(28\!\cdots\!43\)\( \nu^{7} - \)\(57\!\cdots\!33\)\( \nu^{6} + \)\(16\!\cdots\!52\)\( \nu^{5} - \)\(29\!\cdots\!84\)\( \nu^{4} + \)\(40\!\cdots\!60\)\( \nu^{3} - \)\(62\!\cdots\!00\)\( \nu^{2} + \)\(35\!\cdots\!60\)\( \nu - \)\(35\!\cdots\!40\)\(\)\()/ \)\(17\!\cdots\!40\)\( \)
\(\beta_{5}\)\(=\)\((\)\(-\)\(93\!\cdots\!97\)\( \nu^{13} + \)\(49\!\cdots\!23\)\( \nu^{12} - \)\(58\!\cdots\!73\)\( \nu^{11} + \)\(31\!\cdots\!83\)\( \nu^{10} - \)\(12\!\cdots\!75\)\( \nu^{9} + \)\(75\!\cdots\!81\)\( \nu^{8} - \)\(11\!\cdots\!23\)\( \nu^{7} + \)\(80\!\cdots\!17\)\( \nu^{6} - \)\(41\!\cdots\!72\)\( \nu^{5} + \)\(39\!\cdots\!96\)\( \nu^{4} - \)\(84\!\cdots\!60\)\( \nu^{3} + \)\(84\!\cdots\!80\)\( \nu^{2} + \)\(75\!\cdots\!00\)\( \nu + \)\(51\!\cdots\!80\)\(\)\()/ \)\(75\!\cdots\!20\)\( \)
\(\beta_{6}\)\(=\)\((\)\(-\)\(15\!\cdots\!97\)\( \nu^{13} + \)\(73\!\cdots\!79\)\( \nu^{12} - \)\(96\!\cdots\!61\)\( \nu^{11} + \)\(47\!\cdots\!23\)\( \nu^{10} - \)\(21\!\cdots\!39\)\( \nu^{9} + \)\(11\!\cdots\!21\)\( \nu^{8} - \)\(20\!\cdots\!91\)\( \nu^{7} + \)\(11\!\cdots\!73\)\( \nu^{6} - \)\(92\!\cdots\!12\)\( \nu^{5} + \)\(53\!\cdots\!24\)\( \nu^{4} - \)\(19\!\cdots\!00\)\( \nu^{3} + \)\(92\!\cdots\!00\)\( \nu^{2} - \)\(12\!\cdots\!60\)\( \nu + \)\(40\!\cdots\!00\)\(\)\()/ \)\(75\!\cdots\!20\)\( \)
\(\beta_{7}\)\(=\)\((\)\(-\)\(40\!\cdots\!87\)\( \nu^{13} - \)\(10\!\cdots\!67\)\( \nu^{12} - \)\(24\!\cdots\!51\)\( \nu^{11} - \)\(74\!\cdots\!95\)\( \nu^{10} - \)\(52\!\cdots\!29\)\( \nu^{9} - \)\(18\!\cdots\!85\)\( \nu^{8} - \)\(46\!\cdots\!41\)\( \nu^{7} - \)\(22\!\cdots\!57\)\( \nu^{6} - \)\(13\!\cdots\!12\)\( \nu^{5} - \)\(12\!\cdots\!76\)\( \nu^{4} + \)\(11\!\cdots\!20\)\( \nu^{3} - \)\(30\!\cdots\!00\)\( \nu^{2} + \)\(69\!\cdots\!60\)\( \nu - \)\(16\!\cdots\!20\)\(\)\()/ \)\(75\!\cdots\!20\)\( \)
\(\beta_{8}\)\(=\)\((\)\(-\)\(41\!\cdots\!21\)\( \nu^{13} - \)\(98\!\cdots\!85\)\( \nu^{12} - \)\(25\!\cdots\!05\)\( \nu^{11} - \)\(63\!\cdots\!41\)\( \nu^{10} - \)\(52\!\cdots\!03\)\( \nu^{9} - \)\(15\!\cdots\!99\)\( \nu^{8} - \)\(47\!\cdots\!35\)\( \nu^{7} - \)\(15\!\cdots\!19\)\( \nu^{6} - \)\(20\!\cdots\!96\)\( \nu^{5} - \)\(76\!\cdots\!76\)\( \nu^{4} - \)\(44\!\cdots\!00\)\( \nu^{3} - \)\(14\!\cdots\!20\)\( \nu^{2} - \)\(21\!\cdots\!80\)\( \nu - \)\(86\!\cdots\!60\)\(\)\()/ \)\(10\!\cdots\!60\)\( \)
\(\beta_{9}\)\(=\)\((\)\(\)\(19\!\cdots\!09\)\( \nu^{13} + \)\(85\!\cdots\!17\)\( \nu^{12} + \)\(12\!\cdots\!53\)\( \nu^{11} + \)\(55\!\cdots\!77\)\( \nu^{10} + \)\(30\!\cdots\!91\)\( \nu^{9} + \)\(13\!\cdots\!75\)\( \nu^{8} + \)\(33\!\cdots\!03\)\( \nu^{7} + \)\(14\!\cdots\!31\)\( \nu^{6} + \)\(17\!\cdots\!24\)\( \nu^{5} + \)\(76\!\cdots\!60\)\( \nu^{4} + \)\(37\!\cdots\!20\)\( \nu^{3} + \)\(16\!\cdots\!80\)\( \nu^{2} + \)\(27\!\cdots\!40\)\( \nu + \)\(85\!\cdots\!60\)\(\)\()/ \)\(25\!\cdots\!40\)\( \)
\(\beta_{10}\)\(=\)\((\)\(-\)\(14\!\cdots\!83\)\( \nu^{13} + \)\(21\!\cdots\!05\)\( \nu^{12} - \)\(95\!\cdots\!15\)\( \nu^{11} + \)\(13\!\cdots\!01\)\( \nu^{10} - \)\(22\!\cdots\!89\)\( \nu^{9} + \)\(31\!\cdots\!19\)\( \nu^{8} - \)\(24\!\cdots\!65\)\( \nu^{7} + \)\(31\!\cdots\!59\)\( \nu^{6} - \)\(12\!\cdots\!68\)\( \nu^{5} + \)\(13\!\cdots\!76\)\( \nu^{4} - \)\(26\!\cdots\!00\)\( \nu^{3} + \)\(22\!\cdots\!80\)\( \nu^{2} - \)\(18\!\cdots\!80\)\( \nu + \)\(12\!\cdots\!20\)\(\)\()/ \)\(18\!\cdots\!80\)\( \)
\(\beta_{11}\)\(=\)\((\)\(\)\(76\!\cdots\!49\)\( \nu^{13} + \)\(10\!\cdots\!09\)\( \nu^{12} + \)\(50\!\cdots\!05\)\( \nu^{11} + \)\(69\!\cdots\!45\)\( \nu^{10} + \)\(12\!\cdots\!07\)\( \nu^{9} + \)\(15\!\cdots\!59\)\( \nu^{8} + \)\(13\!\cdots\!55\)\( \nu^{7} + \)\(16\!\cdots\!91\)\( \nu^{6} + \)\(71\!\cdots\!64\)\( \nu^{5} + \)\(73\!\cdots\!76\)\( \nu^{4} + \)\(15\!\cdots\!20\)\( \nu^{3} + \)\(12\!\cdots\!80\)\( \nu^{2} + \)\(81\!\cdots\!40\)\( \nu + \)\(40\!\cdots\!40\)\(\)\()/ \)\(75\!\cdots\!20\)\( \)
\(\beta_{12}\)\(=\)\((\)\(-\)\(19\!\cdots\!31\)\( \nu^{13} + \)\(45\!\cdots\!41\)\( \nu^{12} - \)\(12\!\cdots\!95\)\( \nu^{11} + \)\(29\!\cdots\!21\)\( \nu^{10} - \)\(31\!\cdots\!73\)\( \nu^{9} + \)\(67\!\cdots\!91\)\( \nu^{8} - \)\(34\!\cdots\!65\)\( \nu^{7} + \)\(68\!\cdots\!71\)\( \nu^{6} - \)\(17\!\cdots\!16\)\( \nu^{5} + \)\(30\!\cdots\!76\)\( \nu^{4} - \)\(37\!\cdots\!60\)\( \nu^{3} + \)\(49\!\cdots\!40\)\( \nu^{2} - \)\(21\!\cdots\!20\)\( \nu + \)\(11\!\cdots\!80\)\(\)\()/ \)\(75\!\cdots\!20\)\( \)
\(\beta_{13}\)\(=\)\((\)\(\)\(36\!\cdots\!69\)\( \nu^{13} + \)\(99\!\cdots\!77\)\( \nu^{12} + \)\(24\!\cdots\!65\)\( \nu^{11} + \)\(65\!\cdots\!81\)\( \nu^{10} + \)\(60\!\cdots\!51\)\( \nu^{9} + \)\(15\!\cdots\!23\)\( \nu^{8} + \)\(69\!\cdots\!27\)\( \nu^{7} + \)\(16\!\cdots\!03\)\( \nu^{6} + \)\(37\!\cdots\!08\)\( \nu^{5} + \)\(82\!\cdots\!16\)\( \nu^{4} + \)\(80\!\cdots\!00\)\( \nu^{3} + \)\(16\!\cdots\!00\)\( \nu^{2} + \)\(31\!\cdots\!00\)\( \nu + \)\(99\!\cdots\!00\)\(\)\()/ \)\(28\!\cdots\!60\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} - 21 \beta_{1} + 9\)\()/16\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{12} - 2 \beta_{11} + 4 \beta_{10} + 2 \beta_{9} - \beta_{8} - 7 \beta_{6} - 559 \beta_{5} + 3675 \beta_{4} + 3664 \beta_{3} - 210575 \beta_{2} - 6252715161 \beta_{1} - 2651839647177356\)\()/256\)
\(\nu^{3}\)\(=\)\((\)\(2368120 \beta_{13} + 18284465 \beta_{12} - 199408558 \beta_{11} + 65384769 \beta_{10} + 127089321 \beta_{9} + 23609502 \beta_{8} - 130808243 \beta_{7} + 11992657992 \beta_{6} - 5169888074 \beta_{5} + 100903463690 \beta_{4} - 12904647410341 \beta_{3} - 4272518604221854 \beta_{2} - 16059499991403865 \beta_{1} + 6886301850891719\)\()/4096\)
\(\nu^{4}\)\(=\)\((\)\(1354759580627490 \beta_{12} + 2709519161254980 \beta_{11} - 6915131345136837 \beta_{10} - 4507819024729497 \beta_{9} + 1498161551275002 \beta_{8} + 1511495922179493 \beta_{7} + 109529609754542628 \beta_{6} + 1163997185873119896 \beta_{5} - 7460793434389859586 \beta_{4} - 72785252401383733437 \beta_{3} + 438955347347959830999 \beta_{2} + 13902146294867827507313415 \beta_{1} + 2826249605528988014915217064314\)\()/16384\)
\(\nu^{5}\)\(=\)\((\)\(-2975167109350098246840 \beta_{13} - 17404205291256975417705 \beta_{12} + 194315497368482967929790 \beta_{11} - 72213540727807007045145 \beta_{10} - 119305799499242600218305 \beta_{9} - 23960149959910327979310 \beta_{8} + 489168175296076404338973 \beta_{7} - 12247995432504084031454058 \beta_{6} - 1603777799699930303590722 \beta_{5} - 109491779285819603534563752 \beta_{4} + 12310664824859269278849390771 \beta_{3} + 2778454104966252737953753477326 \beta_{2} + 164097374677488017835835062886665 \beta_{1} - 70330933975816167950361121451967\)\()/131072\)
\(\nu^{6}\)\(=\)\((\)\(-1915584590821861847038184020008 \beta_{12} - 3831169181643723694076368040016 \beta_{11} + 11593912913253761285775775153383 \beta_{10} + 7899738800157405089966570333535 \beta_{9} - 2059819488110587032817267810608 \beta_{8} - 3780099823936231024332034712319 \beta_{7} - 317397628050282163466382499403418 \beta_{6} - 1906200796020491159778460003974054 \beta_{5} + 12061539565956529893326299662658440 \beta_{4} + 129962494224013943623244480557413543 \beta_{3} - 745778809421064783314210388438396747 \beta_{2} - 23699358333037732207494831680478072490911 \beta_{1} - 3666593018755509394532493686370548004348912774\)\()/1048576\)
\(\nu^{7}\)\(=\)\((\)\(\)\(27\!\cdots\!80\)\( \beta_{13} + \)\(16\!\cdots\!40\)\( \beta_{12} - \)\(19\!\cdots\!48\)\( \beta_{11} + \)\(79\!\cdots\!94\)\( \beta_{10} + \)\(12\!\cdots\!46\)\( \beta_{9} + \)\(25\!\cdots\!92\)\( \beta_{8} - \)\(76\!\cdots\!66\)\( \beta_{7} + \)\(13\!\cdots\!20\)\( \beta_{6} + \)\(75\!\cdots\!49\)\( \beta_{5} + \)\(12\!\cdots\!94\)\( \beta_{4} - \)\(12\!\cdots\!77\)\( \beta_{3} - \)\(24\!\cdots\!05\)\( \beta_{2} - \)\(28\!\cdots\!42\)\( \beta_{1} + \)\(12\!\cdots\!45\)\(\)\()/524288\)
\(\nu^{8}\)\(=\)\((\)\(\)\(21\!\cdots\!12\)\( \beta_{12} + \)\(43\!\cdots\!24\)\( \beta_{11} - \)\(15\!\cdots\!83\)\( \beta_{10} - \)\(10\!\cdots\!91\)\( \beta_{9} + \)\(22\!\cdots\!28\)\( \beta_{8} + \)\(56\!\cdots\!35\)\( \beta_{7} + \)\(52\!\cdots\!46\)\( \beta_{6} + \)\(23\!\cdots\!39\)\( \beta_{5} - \)\(14\!\cdots\!78\)\( \beta_{4} - \)\(13\!\cdots\!18\)\( \beta_{3} + \)\(96\!\cdots\!20\)\( \beta_{2} + \)\(30\!\cdots\!59\)\( \beta_{1} + \)\(40\!\cdots\!23\)\(\)\()/524288\)
\(\nu^{9}\)\(=\)\((\)\(-\)\(55\!\cdots\!60\)\( \beta_{13} - \)\(36\!\cdots\!00\)\( \beta_{12} + \)\(47\!\cdots\!24\)\( \beta_{11} - \)\(20\!\cdots\!12\)\( \beta_{10} - \)\(30\!\cdots\!08\)\( \beta_{9} - \)\(65\!\cdots\!36\)\( \beta_{8} + \)\(24\!\cdots\!16\)\( \beta_{7} - \)\(34\!\cdots\!88\)\( \beta_{6} - \)\(30\!\cdots\!99\)\( \beta_{5} - \)\(33\!\cdots\!54\)\( \beta_{4} + \)\(31\!\cdots\!77\)\( \beta_{3} + \)\(57\!\cdots\!01\)\( \beta_{2} + \)\(93\!\cdots\!56\)\( \beta_{1} - \)\(40\!\cdots\!47\)\(\)\()/524288\)
\(\nu^{10}\)\(=\)\((\)\(-\)\(10\!\cdots\!64\)\( \beta_{12} - \)\(20\!\cdots\!28\)\( \beta_{11} + \)\(77\!\cdots\!65\)\( \beta_{10} + \)\(50\!\cdots\!61\)\( \beta_{9} - \)\(97\!\cdots\!40\)\( \beta_{8} - \)\(30\!\cdots\!81\)\( \beta_{7} - \)\(31\!\cdots\!98\)\( \beta_{6} - \)\(11\!\cdots\!40\)\( \beta_{5} + \)\(67\!\cdots\!88\)\( \beta_{4} + \)\(51\!\cdots\!15\)\( \beta_{3} - \)\(49\!\cdots\!63\)\( \beta_{2} - \)\(15\!\cdots\!13\)\( \beta_{1} - \)\(18\!\cdots\!64\)\(\)\()/1048576\)
\(\nu^{11}\)\(=\)\((\)\(\)\(51\!\cdots\!00\)\( \beta_{13} + \)\(40\!\cdots\!00\)\( \beta_{12} - \)\(56\!\cdots\!72\)\( \beta_{11} + \)\(26\!\cdots\!11\)\( \beta_{10} + \)\(37\!\cdots\!99\)\( \beta_{9} + \)\(83\!\cdots\!68\)\( \beta_{8} - \)\(34\!\cdots\!11\)\( \beta_{7} + \)\(44\!\cdots\!82\)\( \beta_{6} + \)\(48\!\cdots\!34\)\( \beta_{5} + \)\(43\!\cdots\!64\)\( \beta_{4} - \)\(39\!\cdots\!57\)\( \beta_{3} - \)\(67\!\cdots\!79\)\( \beta_{2} - \)\(13\!\cdots\!83\)\( \beta_{1} + \)\(59\!\cdots\!58\)\(\)\()/262144\)
\(\nu^{12}\)\(=\)\((\)\(\)\(60\!\cdots\!72\)\( \beta_{12} + \)\(12\!\cdots\!44\)\( \beta_{11} - \)\(49\!\cdots\!96\)\( \beta_{10} - \)\(31\!\cdots\!84\)\( \beta_{9} + \)\(54\!\cdots\!36\)\( \beta_{8} + \)\(20\!\cdots\!12\)\( \beta_{7} + \)\(21\!\cdots\!28\)\( \beta_{6} + \)\(71\!\cdots\!83\)\( \beta_{5} - \)\(39\!\cdots\!02\)\( \beta_{4} - \)\(22\!\cdots\!81\)\( \beta_{3} + \)\(32\!\cdots\!71\)\( \beta_{2} + \)\(97\!\cdots\!64\)\( \beta_{1} + \)\(11\!\cdots\!19\)\(\)\()/262144\)
\(\nu^{13}\)\(=\)\((\)\(-\)\(90\!\cdots\!80\)\( \beta_{13} - \)\(87\!\cdots\!70\)\( \beta_{12} + \)\(13\!\cdots\!44\)\( \beta_{11} - \)\(68\!\cdots\!22\)\( \beta_{10} - \)\(91\!\cdots\!98\)\( \beta_{9} - \)\(21\!\cdots\!96\)\( \beta_{8} + \)\(95\!\cdots\!50\)\( \beta_{7} - \)\(11\!\cdots\!32\)\( \beta_{6} - \)\(14\!\cdots\!65\)\( \beta_{5} - \)\(11\!\cdots\!10\)\( \beta_{4} + \)\(96\!\cdots\!25\)\( \beta_{3} + \)\(16\!\cdots\!99\)\( \beta_{2} + \)\(38\!\cdots\!26\)\( \beta_{1} - \)\(16\!\cdots\!43\)\(\)\()/262144\)

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
2.77300e6i
2.77300e6i
2.55660e6i
2.55660e6i
1.78456e6i
1.78456e6i
4.36932e6i
4.36932e6i
895767.i
895767.i
5.00408e6i
5.00408e6i
3.18854e6i
3.18854e6i
−61401.1 22910.1i 4.43679e7i 3.24522e9 + 2.81341e9i 1.39224e11 1.01647e12 2.72424e12i 2.23454e13i −1.34805e14 2.47095e14i −1.15494e14 −8.54848e15 3.18962e15i
3.2 −61401.1 + 22910.1i 4.43679e7i 3.24522e9 2.81341e9i 1.39224e11 1.01647e12 + 2.72424e12i 2.23454e13i −1.34805e14 + 2.47095e14i −1.15494e14 −8.54848e15 + 3.18962e15i
3.3 −56022.2 34007.1i 4.09055e7i 1.98200e9 + 3.81031e9i −2.17196e11 −1.39108e12 + 2.29162e12i 2.81808e12i 1.85418e13 2.80864e14i 1.79757e14 1.21678e16 + 7.38622e15i
3.4 −56022.2 + 34007.1i 4.09055e7i 1.98200e9 3.81031e9i −2.17196e11 −1.39108e12 2.29162e12i 2.81808e12i 1.85418e13 + 2.80864e14i 1.79757e14 1.21678e16 7.38622e15i
3.5 −18058.8 62998.8i 2.85529e7i −3.64273e9 + 2.27536e9i 1.21258e11 −1.79880e12 + 5.15631e11i 3.17263e10i 2.09128e14 + 1.88397e14i 1.03775e15 −2.18978e15 7.63913e15i
3.6 −18058.8 + 62998.8i 2.85529e7i −3.64273e9 2.27536e9i 1.21258e11 −1.79880e12 5.15631e11i 3.17263e10i 2.09128e14 1.88397e14i 1.03775e15 −2.18978e15 + 7.63913e15i
3.7 −15224.0 63743.2i 6.99091e7i −3.83143e9 + 1.94085e9i −1.46681e11 4.45623e12 1.06430e12i 5.71644e13i 1.82046e14 + 2.14680e14i −3.03426e15 2.23307e15 + 9.34993e15i
3.8 −15224.0 + 63743.2i 6.99091e7i −3.83143e9 1.94085e9i −1.46681e11 4.45623e12 + 1.06430e12i 5.71644e13i 1.82046e14 2.14680e14i −3.03426e15 2.23307e15 9.34993e15i
3.9 32011.4 57186.0i 1.43323e7i −2.24550e9 3.66121e9i −4.45746e10 8.19605e11 + 4.58796e11i 4.42477e13i −2.81252e14 + 1.12108e13i 1.64761e15 −1.42690e15 + 2.54904e15i
3.10 32011.4 + 57186.0i 1.43323e7i −2.24550e9 + 3.66121e9i −4.45746e10 8.19605e11 4.58796e11i 4.42477e13i −2.81252e14 1.12108e13i 1.64761e15 −1.42690e15 2.54904e15i
3.11 46092.7 46587.9i 8.00653e7i −4.59006e7 4.29472e9i −3.25403e10 −3.73007e12 3.69042e12i 4.74477e13i −2.02198e14 1.95817e14i −4.55743e15 −1.49987e15 + 1.51599e15i
3.12 46092.7 + 46587.9i 8.00653e7i −4.59006e7 + 4.29472e9i −3.25403e10 −3.73007e12 + 3.69042e12i 4.74477e13i −2.02198e14 + 1.95817e14i −4.55743e15 −1.49987e15 1.51599e15i
3.13 60712.0 24678.5i 5.10167e7i 3.07691e9 2.99655e9i 2.49571e11 1.25901e12 + 3.09732e12i 4.65014e13i 1.12855e14 2.57860e14i −7.49683e14 1.51520e16 6.15903e15i
3.14 60712.0 + 24678.5i 5.10167e7i 3.07691e9 + 2.99655e9i 2.49571e11 1.25901e12 3.09732e12i 4.65014e13i 1.12855e14 + 2.57860e14i −7.49683e14 1.51520e16 + 6.15903e15i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.14
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 4.33.b.b 14
3.b odd 2 1 36.33.d.b 14
4.b odd 2 1 inner 4.33.b.b 14
12.b even 2 1 36.33.d.b 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4.33.b.b 14 1.a even 1 1 trivial
4.33.b.b 14 4.b odd 2 1 inner
36.33.d.b 14 3.b odd 2 1
36.33.d.b 14 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{14} + \)\(18\!\cdots\!12\)\( T_{3}^{12} + \)\(13\!\cdots\!48\)\( T_{3}^{10} + \)\(46\!\cdots\!04\)\( T_{3}^{8} + \)\(87\!\cdots\!20\)\( T_{3}^{6} + \)\(83\!\cdots\!40\)\( T_{3}^{4} + \)\(35\!\cdots\!00\)\( T_{3}^{2} + \)\(44\!\cdots\!00\)\( \) acting on \(S_{33}^{\mathrm{new}}(4, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 23780 T + 1744168384 T^{2} + 100782736158720 T^{3} + 10654023216358490112 T^{4} + \)\(93\!\cdots\!80\)\( T^{5} + \)\(65\!\cdots\!88\)\( T^{6} - \)\(27\!\cdots\!20\)\( T^{7} + \)\(28\!\cdots\!48\)\( T^{8} + \)\(17\!\cdots\!80\)\( T^{9} + \)\(84\!\cdots\!32\)\( T^{10} + \)\(34\!\cdots\!20\)\( T^{11} + \)\(25\!\cdots\!84\)\( T^{12} + \)\(14\!\cdots\!80\)\( T^{13} + \)\(26\!\cdots\!16\)\( T^{14} \)
$3$ \( 1 - 7379386355554062 T^{2} + \)\(32\!\cdots\!15\)\( T^{4} - \)\(10\!\cdots\!68\)\( T^{6} + \)\(25\!\cdots\!89\)\( T^{8} - \)\(55\!\cdots\!30\)\( T^{10} + \)\(11\!\cdots\!35\)\( T^{12} - \)\(22\!\cdots\!60\)\( T^{14} + \)\(39\!\cdots\!35\)\( T^{16} - \)\(65\!\cdots\!30\)\( T^{18} + \)\(10\!\cdots\!49\)\( T^{20} - \)\(13\!\cdots\!28\)\( T^{22} + \)\(15\!\cdots\!15\)\( T^{24} - \)\(12\!\cdots\!22\)\( T^{26} + \)\(56\!\cdots\!61\)\( T^{28} \)
$5$ \( ( 1 - 69060745870 T + \)\(81\!\cdots\!75\)\( T^{2} - \)\(60\!\cdots\!00\)\( T^{3} + \)\(35\!\cdots\!25\)\( T^{4} - \)\(26\!\cdots\!50\)\( T^{5} + \)\(10\!\cdots\!75\)\( T^{6} - \)\(77\!\cdots\!00\)\( T^{7} + \)\(25\!\cdots\!75\)\( T^{8} - \)\(14\!\cdots\!50\)\( T^{9} + \)\(45\!\cdots\!25\)\( T^{10} - \)\(17\!\cdots\!00\)\( T^{11} + \)\(55\!\cdots\!75\)\( T^{12} - \)\(11\!\cdots\!50\)\( T^{13} + \)\(37\!\cdots\!25\)\( T^{14} )^{2} \)
$7$ \( 1 - \)\(53\!\cdots\!62\)\( T^{2} + \)\(15\!\cdots\!35\)\( T^{4} - \)\(35\!\cdots\!08\)\( T^{6} + \)\(65\!\cdots\!89\)\( T^{8} - \)\(10\!\cdots\!10\)\( T^{10} + \)\(13\!\cdots\!75\)\( T^{12} - \)\(15\!\cdots\!40\)\( T^{14} + \)\(16\!\cdots\!75\)\( T^{16} - \)\(14\!\cdots\!10\)\( T^{18} + \)\(11\!\cdots\!89\)\( T^{20} - \)\(78\!\cdots\!08\)\( T^{22} + \)\(42\!\cdots\!35\)\( T^{24} - \)\(17\!\cdots\!62\)\( T^{26} + \)\(40\!\cdots\!01\)\( T^{28} \)
$11$ \( 1 - \)\(13\!\cdots\!94\)\( T^{2} + \)\(94\!\cdots\!91\)\( T^{4} - \)\(47\!\cdots\!04\)\( T^{6} + \)\(18\!\cdots\!81\)\( T^{8} - \)\(59\!\cdots\!02\)\( T^{10} + \)\(16\!\cdots\!63\)\( T^{12} - \)\(36\!\cdots\!32\)\( T^{14} + \)\(71\!\cdots\!83\)\( T^{16} - \)\(11\!\cdots\!62\)\( T^{18} + \)\(16\!\cdots\!01\)\( T^{20} - \)\(18\!\cdots\!44\)\( T^{22} + \)\(16\!\cdots\!91\)\( T^{24} - \)\(10\!\cdots\!54\)\( T^{26} + \)\(34\!\cdots\!81\)\( T^{28} \)
$13$ \( ( 1 + 820209234838929010 T + \)\(24\!\cdots\!39\)\( T^{2} + \)\(16\!\cdots\!00\)\( T^{3} + \)\(27\!\cdots\!57\)\( T^{4} + \)\(16\!\cdots\!30\)\( T^{5} + \)\(19\!\cdots\!43\)\( T^{6} + \)\(91\!\cdots\!80\)\( T^{7} + \)\(85\!\cdots\!83\)\( T^{8} + \)\(31\!\cdots\!30\)\( T^{9} + \)\(24\!\cdots\!37\)\( T^{10} + \)\(65\!\cdots\!00\)\( T^{11} + \)\(41\!\cdots\!39\)\( T^{12} + \)\(61\!\cdots\!10\)\( T^{13} + \)\(33\!\cdots\!61\)\( T^{14} )^{2} \)
$17$ \( ( 1 - 15785483952898628110 T + \)\(81\!\cdots\!39\)\( T^{2} - \)\(11\!\cdots\!00\)\( T^{3} + \)\(40\!\cdots\!57\)\( T^{4} - \)\(52\!\cdots\!50\)\( T^{5} + \)\(13\!\cdots\!03\)\( T^{6} - \)\(14\!\cdots\!20\)\( T^{7} + \)\(31\!\cdots\!83\)\( T^{8} - \)\(29\!\cdots\!50\)\( T^{9} + \)\(53\!\cdots\!17\)\( T^{10} - \)\(36\!\cdots\!00\)\( T^{11} + \)\(60\!\cdots\!39\)\( T^{12} - \)\(27\!\cdots\!10\)\( T^{13} + \)\(41\!\cdots\!21\)\( T^{14} )^{2} \)
$19$ \( 1 - \)\(56\!\cdots\!14\)\( T^{2} + \)\(15\!\cdots\!91\)\( T^{4} - \)\(29\!\cdots\!44\)\( T^{6} + \)\(41\!\cdots\!41\)\( T^{8} - \)\(49\!\cdots\!02\)\( T^{10} + \)\(51\!\cdots\!03\)\( T^{12} - \)\(45\!\cdots\!32\)\( T^{14} + \)\(35\!\cdots\!63\)\( T^{16} - \)\(23\!\cdots\!82\)\( T^{18} + \)\(13\!\cdots\!01\)\( T^{20} - \)\(67\!\cdots\!64\)\( T^{22} + \)\(25\!\cdots\!91\)\( T^{24} - \)\(62\!\cdots\!94\)\( T^{26} + \)\(76\!\cdots\!41\)\( T^{28} \)
$23$ \( 1 - \)\(39\!\cdots\!22\)\( T^{2} + \)\(75\!\cdots\!95\)\( T^{4} - \)\(90\!\cdots\!48\)\( T^{6} + \)\(78\!\cdots\!29\)\( T^{8} - \)\(51\!\cdots\!90\)\( T^{10} + \)\(26\!\cdots\!95\)\( T^{12} - \)\(11\!\cdots\!20\)\( T^{14} + \)\(37\!\cdots\!95\)\( T^{16} - \)\(10\!\cdots\!90\)\( T^{18} + \)\(22\!\cdots\!09\)\( T^{20} - \)\(36\!\cdots\!28\)\( T^{22} + \)\(42\!\cdots\!95\)\( T^{24} - \)\(31\!\cdots\!02\)\( T^{26} + \)\(11\!\cdots\!81\)\( T^{28} \)
$29$ \( ( 1 + \)\(19\!\cdots\!58\)\( T + \)\(22\!\cdots\!75\)\( T^{2} + \)\(23\!\cdots\!32\)\( T^{3} + \)\(19\!\cdots\!89\)\( T^{4} + \)\(11\!\cdots\!70\)\( T^{5} + \)\(88\!\cdots\!55\)\( T^{6} - \)\(79\!\cdots\!60\)\( T^{7} + \)\(55\!\cdots\!55\)\( T^{8} + \)\(46\!\cdots\!70\)\( T^{9} + \)\(46\!\cdots\!69\)\( T^{10} + \)\(35\!\cdots\!52\)\( T^{11} + \)\(21\!\cdots\!75\)\( T^{12} + \)\(11\!\cdots\!78\)\( T^{13} + \)\(37\!\cdots\!81\)\( T^{14} )^{2} \)
$31$ \( 1 - \)\(24\!\cdots\!14\)\( T^{2} + \)\(34\!\cdots\!91\)\( T^{4} - \)\(33\!\cdots\!44\)\( T^{6} + \)\(25\!\cdots\!41\)\( T^{8} - \)\(16\!\cdots\!02\)\( T^{10} + \)\(93\!\cdots\!03\)\( T^{12} - \)\(50\!\cdots\!32\)\( T^{14} + \)\(26\!\cdots\!63\)\( T^{16} - \)\(12\!\cdots\!82\)\( T^{18} + \)\(56\!\cdots\!01\)\( T^{20} - \)\(20\!\cdots\!64\)\( T^{22} + \)\(59\!\cdots\!91\)\( T^{24} - \)\(12\!\cdots\!94\)\( T^{26} + \)\(13\!\cdots\!41\)\( T^{28} \)
$37$ \( ( 1 + \)\(53\!\cdots\!70\)\( T + \)\(85\!\cdots\!99\)\( T^{2} + \)\(40\!\cdots\!20\)\( T^{3} + \)\(33\!\cdots\!97\)\( T^{4} + \)\(13\!\cdots\!30\)\( T^{5} + \)\(78\!\cdots\!63\)\( T^{6} + \)\(26\!\cdots\!60\)\( T^{7} + \)\(11\!\cdots\!03\)\( T^{8} + \)\(31\!\cdots\!30\)\( T^{9} + \)\(11\!\cdots\!77\)\( T^{10} + \)\(21\!\cdots\!20\)\( T^{11} + \)\(69\!\cdots\!99\)\( T^{12} + \)\(66\!\cdots\!70\)\( T^{13} + \)\(18\!\cdots\!61\)\( T^{14} )^{2} \)
$41$ \( ( 1 - \)\(18\!\cdots\!74\)\( T + \)\(16\!\cdots\!11\)\( T^{2} - \)\(37\!\cdots\!44\)\( T^{3} + \)\(12\!\cdots\!61\)\( T^{4} - \)\(25\!\cdots\!02\)\( T^{5} + \)\(68\!\cdots\!23\)\( T^{6} - \)\(11\!\cdots\!72\)\( T^{7} + \)\(27\!\cdots\!63\)\( T^{8} - \)\(42\!\cdots\!22\)\( T^{9} + \)\(84\!\cdots\!01\)\( T^{10} - \)\(10\!\cdots\!24\)\( T^{11} + \)\(17\!\cdots\!11\)\( T^{12} - \)\(81\!\cdots\!94\)\( T^{13} + \)\(18\!\cdots\!61\)\( T^{14} )^{2} \)
$43$ \( 1 - \)\(10\!\cdots\!42\)\( T^{2} + \)\(66\!\cdots\!55\)\( T^{4} - \)\(29\!\cdots\!68\)\( T^{6} + \)\(10\!\cdots\!29\)\( T^{8} - \)\(29\!\cdots\!50\)\( T^{10} + \)\(71\!\cdots\!15\)\( T^{12} - \)\(14\!\cdots\!80\)\( T^{14} + \)\(24\!\cdots\!15\)\( T^{16} - \)\(36\!\cdots\!50\)\( T^{18} + \)\(43\!\cdots\!29\)\( T^{20} - \)\(43\!\cdots\!68\)\( T^{22} + \)\(33\!\cdots\!55\)\( T^{24} - \)\(18\!\cdots\!42\)\( T^{26} + \)\(62\!\cdots\!01\)\( T^{28} \)
$47$ \( 1 - \)\(28\!\cdots\!22\)\( T^{2} + \)\(37\!\cdots\!15\)\( T^{4} - \)\(33\!\cdots\!08\)\( T^{6} + \)\(21\!\cdots\!89\)\( T^{8} - \)\(11\!\cdots\!30\)\( T^{10} + \)\(47\!\cdots\!35\)\( T^{12} - \)\(16\!\cdots\!60\)\( T^{14} + \)\(49\!\cdots\!35\)\( T^{16} - \)\(12\!\cdots\!30\)\( T^{18} + \)\(24\!\cdots\!49\)\( T^{20} - \)\(38\!\cdots\!68\)\( T^{22} + \)\(44\!\cdots\!15\)\( T^{24} - \)\(34\!\cdots\!82\)\( T^{26} + \)\(12\!\cdots\!61\)\( T^{28} \)
$53$ \( ( 1 - \)\(92\!\cdots\!90\)\( T + \)\(83\!\cdots\!59\)\( T^{2} - \)\(46\!\cdots\!20\)\( T^{3} + \)\(26\!\cdots\!57\)\( T^{4} - \)\(11\!\cdots\!10\)\( T^{5} + \)\(54\!\cdots\!63\)\( T^{6} - \)\(20\!\cdots\!60\)\( T^{7} + \)\(82\!\cdots\!83\)\( T^{8} - \)\(26\!\cdots\!10\)\( T^{9} + \)\(89\!\cdots\!97\)\( T^{10} - \)\(23\!\cdots\!20\)\( T^{11} + \)\(63\!\cdots\!59\)\( T^{12} - \)\(10\!\cdots\!90\)\( T^{13} + \)\(17\!\cdots\!81\)\( T^{14} )^{2} \)
$59$ \( 1 - \)\(29\!\cdots\!14\)\( T^{2} + \)\(31\!\cdots\!71\)\( T^{4} - \)\(10\!\cdots\!04\)\( T^{6} - \)\(51\!\cdots\!79\)\( T^{8} + \)\(42\!\cdots\!98\)\( T^{10} + \)\(82\!\cdots\!03\)\( T^{12} - \)\(14\!\cdots\!92\)\( T^{14} + \)\(17\!\cdots\!83\)\( T^{16} + \)\(19\!\cdots\!58\)\( T^{18} - \)\(52\!\cdots\!99\)\( T^{20} - \)\(23\!\cdots\!64\)\( T^{22} + \)\(14\!\cdots\!71\)\( T^{24} - \)\(29\!\cdots\!54\)\( T^{26} + \)\(21\!\cdots\!21\)\( T^{28} \)
$61$ \( ( 1 - \)\(53\!\cdots\!54\)\( T + \)\(52\!\cdots\!51\)\( T^{2} - \)\(16\!\cdots\!84\)\( T^{3} + \)\(11\!\cdots\!41\)\( T^{4} - \)\(31\!\cdots\!02\)\( T^{5} + \)\(22\!\cdots\!83\)\( T^{6} - \)\(55\!\cdots\!52\)\( T^{7} + \)\(30\!\cdots\!43\)\( T^{8} - \)\(57\!\cdots\!82\)\( T^{9} + \)\(28\!\cdots\!01\)\( T^{10} - \)\(53\!\cdots\!04\)\( T^{11} + \)\(23\!\cdots\!51\)\( T^{12} - \)\(32\!\cdots\!34\)\( T^{13} + \)\(82\!\cdots\!41\)\( T^{14} )^{2} \)
$67$ \( 1 - \)\(17\!\cdots\!62\)\( T^{2} + \)\(16\!\cdots\!55\)\( T^{4} - \)\(10\!\cdots\!68\)\( T^{6} + \)\(48\!\cdots\!89\)\( T^{8} - \)\(18\!\cdots\!10\)\( T^{10} + \)\(62\!\cdots\!15\)\( T^{12} - \)\(18\!\cdots\!40\)\( T^{14} + \)\(46\!\cdots\!15\)\( T^{16} - \)\(10\!\cdots\!10\)\( T^{18} + \)\(19\!\cdots\!29\)\( T^{20} - \)\(30\!\cdots\!08\)\( T^{22} + \)\(35\!\cdots\!55\)\( T^{24} - \)\(29\!\cdots\!02\)\( T^{26} + \)\(12\!\cdots\!41\)\( T^{28} \)
$71$ \( 1 - \)\(96\!\cdots\!14\)\( T^{2} + \)\(51\!\cdots\!11\)\( T^{4} - \)\(19\!\cdots\!84\)\( T^{6} + \)\(55\!\cdots\!61\)\( T^{8} - \)\(13\!\cdots\!02\)\( T^{10} + \)\(27\!\cdots\!03\)\( T^{12} - \)\(49\!\cdots\!72\)\( T^{14} + \)\(81\!\cdots\!43\)\( T^{16} - \)\(12\!\cdots\!22\)\( T^{18} + \)\(15\!\cdots\!01\)\( T^{20} - \)\(16\!\cdots\!64\)\( T^{22} + \)\(13\!\cdots\!11\)\( T^{24} - \)\(73\!\cdots\!34\)\( T^{26} + \)\(23\!\cdots\!61\)\( T^{28} \)
$73$ \( ( 1 - \)\(71\!\cdots\!30\)\( T + \)\(19\!\cdots\!99\)\( T^{2} - \)\(11\!\cdots\!20\)\( T^{3} + \)\(17\!\cdots\!17\)\( T^{4} - \)\(89\!\cdots\!30\)\( T^{5} + \)\(10\!\cdots\!03\)\( T^{6} - \)\(46\!\cdots\!80\)\( T^{7} + \)\(44\!\cdots\!63\)\( T^{8} - \)\(15\!\cdots\!30\)\( T^{9} + \)\(13\!\cdots\!37\)\( T^{10} - \)\(35\!\cdots\!20\)\( T^{11} + \)\(26\!\cdots\!99\)\( T^{12} - \)\(41\!\cdots\!30\)\( T^{13} + \)\(24\!\cdots\!41\)\( T^{14} )^{2} \)
$79$ \( 1 - \)\(32\!\cdots\!34\)\( T^{2} + \)\(63\!\cdots\!51\)\( T^{4} - \)\(84\!\cdots\!04\)\( T^{6} + \)\(88\!\cdots\!61\)\( T^{8} - \)\(73\!\cdots\!02\)\( T^{10} + \)\(50\!\cdots\!43\)\( T^{12} - \)\(29\!\cdots\!52\)\( T^{14} + \)\(14\!\cdots\!83\)\( T^{16} - \)\(57\!\cdots\!22\)\( T^{18} + \)\(19\!\cdots\!01\)\( T^{20} - \)\(52\!\cdots\!84\)\( T^{22} + \)\(10\!\cdots\!51\)\( T^{24} - \)\(16\!\cdots\!54\)\( T^{26} + \)\(13\!\cdots\!61\)\( T^{28} \)
$83$ \( 1 - \)\(22\!\cdots\!02\)\( T^{2} + \)\(24\!\cdots\!95\)\( T^{4} - \)\(17\!\cdots\!08\)\( T^{6} + \)\(97\!\cdots\!69\)\( T^{8} - \)\(41\!\cdots\!90\)\( T^{10} + \)\(14\!\cdots\!95\)\( T^{12} - \)\(40\!\cdots\!20\)\( T^{14} + \)\(95\!\cdots\!95\)\( T^{16} - \)\(18\!\cdots\!90\)\( T^{18} + \)\(28\!\cdots\!09\)\( T^{20} - \)\(34\!\cdots\!48\)\( T^{22} + \)\(31\!\cdots\!95\)\( T^{24} - \)\(18\!\cdots\!42\)\( T^{26} + \)\(55\!\cdots\!41\)\( T^{28} \)
$89$ \( ( 1 + \)\(92\!\cdots\!38\)\( T + \)\(73\!\cdots\!55\)\( T^{2} + \)\(12\!\cdots\!92\)\( T^{3} + \)\(36\!\cdots\!89\)\( T^{4} + \)\(55\!\cdots\!50\)\( T^{5} + \)\(12\!\cdots\!15\)\( T^{6} + \)\(16\!\cdots\!20\)\( T^{7} + \)\(30\!\cdots\!15\)\( T^{8} + \)\(32\!\cdots\!50\)\( T^{9} + \)\(50\!\cdots\!29\)\( T^{10} + \)\(40\!\cdots\!52\)\( T^{11} + \)\(58\!\cdots\!55\)\( T^{12} + \)\(17\!\cdots\!98\)\( T^{13} + \)\(46\!\cdots\!41\)\( T^{14} )^{2} \)
$97$ \( ( 1 + \)\(10\!\cdots\!10\)\( T + \)\(16\!\cdots\!39\)\( T^{2} + \)\(11\!\cdots\!20\)\( T^{3} + \)\(10\!\cdots\!57\)\( T^{4} + \)\(42\!\cdots\!30\)\( T^{5} + \)\(33\!\cdots\!63\)\( T^{6} + \)\(10\!\cdots\!40\)\( T^{7} + \)\(12\!\cdots\!83\)\( T^{8} + \)\(60\!\cdots\!30\)\( T^{9} + \)\(55\!\cdots\!97\)\( T^{10} + \)\(22\!\cdots\!20\)\( T^{11} + \)\(12\!\cdots\!39\)\( T^{12} + \)\(29\!\cdots\!10\)\( T^{13} + \)\(10\!\cdots\!81\)\( T^{14} )^{2} \)
show more
show less