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

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Newspace parameters

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

Newform invariants

Self dual: No
Analytic conductor: \(25.9466620569\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} + \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{182}\cdot 3^{20}\cdot 5^{6} \)
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.

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

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

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

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

Hecke kernels

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