Properties

Label 3.44.a.b
Level 3
Weight 44
Character orbit 3.a
Self dual Yes
Analytic conductor 35.133
Analytic rank 0
Dimension 4
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 3 \)
Weight: \( k \) = \( 44 \)
Character orbit: \([\chi]\) = 3.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(35.1331186037\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{14}\cdot 3^{10}\cdot 5^{2}\cdot 7\cdot 11 \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + ( 415003 + \beta_{1} ) q^{2} \) \( -10460353203 q^{3} \) \( + ( 7333437011374 + 1259527 \beta_{1} + \beta_{3} ) q^{4} \) \( + ( 412662665270741 + 73226956 \beta_{1} + 13 \beta_{2} - 22 \beta_{3} ) q^{5} \) \( + ( -4341077960304609 - 10460353203 \beta_{1} ) q^{6} \) \( + ( 28620462315178775 - 211738302932 \beta_{1} + 2997 \beta_{2} - 50246 \beta_{3} ) q^{7} \) \( + ( 19491643319063864324 + 7692359694018 \beta_{1} + 344576 \beta_{2} + 1836142 \beta_{3} ) q^{8} \) \( + 109418989131512359209 q^{9} \) \(+O(q^{10})\) \( q\) \(+(415003 + \beta_{1}) q^{2}\) \(-10460353203 q^{3}\) \(+(7333437011374 + 1259527 \beta_{1} + \beta_{3}) q^{4}\) \(+(412662665270741 + 73226956 \beta_{1} + 13 \beta_{2} - 22 \beta_{3}) q^{5}\) \(+(-4341077960304609 - 10460353203 \beta_{1}) q^{6}\) \(+(28620462315178775 - 211738302932 \beta_{1} + 2997 \beta_{2} - 50246 \beta_{3}) q^{7}\) \(+(19491643319063864324 + 7692359694018 \beta_{1} + 344576 \beta_{2} + 1836142 \beta_{3}) q^{8}\) \(+\)\(10\!\cdots\!09\)\( q^{9}\) \(+(\)\(13\!\cdots\!98\)\( + 326714929810118 \beta_{1} - 9870336 \beta_{2} + 346741184 \beta_{3}) q^{10}\) \(+(\)\(21\!\cdots\!02\)\( + 1103916130966808 \beta_{1} - 92037574 \beta_{2} + 1868332564 \beta_{3}) q^{11}\) \(+(-\)\(76\!\cdots\!22\)\( - 13175097288714981 \beta_{1} - 10460353203 \beta_{3}) q^{12}\) \(+(-\)\(40\!\cdots\!24\)\( + 106131612461552392 \beta_{1} + 7515407934 \beta_{2} - 21941631908 \beta_{3}) q^{13}\) \(+(-\)\(33\!\cdots\!48\)\( - 549787121670036528 \beta_{1} - 17841421312 \beta_{2} - 174730848320 \beta_{3}) q^{14}\) \(+(-\)\(43\!\cdots\!23\)\( - 765979823740540068 \beta_{1} - 135984591639 \beta_{2} + 230127770466 \beta_{3}) q^{15}\) \(+(\)\(66\!\cdots\!48\)\( + 30566899560284838300 \beta_{1} + 572000984064 \beta_{2} + 7540980445956 \beta_{3}) q^{16}\) \(+(\)\(73\!\cdots\!52\)\( - 962490835473988264 \beta_{1} - 89546779862 \beta_{2} - 15082051169548 \beta_{3}) q^{17}\) \(+(\)\(45\!\cdots\!27\)\( + \)\(10\!\cdots\!09\)\( \beta_{1}) q^{18}\) \(+(-\)\(64\!\cdots\!82\)\( + \)\(50\!\cdots\!68\)\( \beta_{1} - 9646988077278 \beta_{2} - 202970742410652 \beta_{3}) q^{19}\) \(+(\)\(21\!\cdots\!16\)\( + \)\(37\!\cdots\!06\)\( \beta_{1} + 6867923468288 \beta_{2} + 502866084527878 \beta_{3}) q^{20}\) \(+(-\)\(29\!\cdots\!25\)\( + \)\(22\!\cdots\!96\)\( \beta_{1} - 31349678549391 \beta_{2} + 525590907037938 \beta_{3}) q^{21}\) \(+(\)\(18\!\cdots\!04\)\( + \)\(18\!\cdots\!92\)\( \beta_{1} + 659992955406336 \beta_{2} + 154983834995584 \beta_{3}) q^{22}\) \(+(-\)\(41\!\cdots\!54\)\( + \)\(12\!\cdots\!56\)\( \beta_{1} - 1440843693152086 \beta_{2} - 124021677613580 \beta_{3}) q^{23}\) \(+(-\)\(20\!\cdots\!72\)\( - \)\(80\!\cdots\!54\)\( \beta_{1} - 3604386665276928 \beta_{2} - 19206693850862826 \beta_{3}) q^{24}\) \(+(\)\(54\!\cdots\!95\)\( - \)\(10\!\cdots\!80\)\( \beta_{1} + 15204023814482460 \beta_{2} - 46618766772767240 \beta_{3}) q^{25}\) \(+(\)\(15\!\cdots\!50\)\( - \)\(47\!\cdots\!70\)\( \beta_{1} - 8884233524930560 \beta_{2} + 258934194685234816 \beta_{3}) q^{26}\) \(-\)\(11\!\cdots\!27\)\( q^{27}\) \(+(-\)\(10\!\cdots\!16\)\( - \)\(34\!\cdots\!48\)\( \beta_{1} - 83427573725429760 \beta_{2} - 601356479924090352 \beta_{3}) q^{28}\) \(+(\)\(12\!\cdots\!83\)\( + \)\(22\!\cdots\!72\)\( \beta_{1} + 127100696144575287 \beta_{2} + 947731908380650206 \beta_{3}) q^{29}\) \(+(-\)\(14\!\cdots\!94\)\( - \)\(34\!\cdots\!54\)\( \beta_{1} + 103247200792286208 \beta_{2} - 3627035254666412352 \beta_{3}) q^{30}\) \(+(\)\(38\!\cdots\!51\)\( + \)\(21\!\cdots\!32\)\( \beta_{1} - 10998884783113695 \beta_{2} - 2991683904270408878 \beta_{3}) q^{31}\) \(+(\)\(34\!\cdots\!48\)\( + \)\(86\!\cdots\!16\)\( \beta_{1} - 533227060395833344 \beta_{2} + 31357077182675843128 \beta_{3}) q^{32}\) \(+(-\)\(22\!\cdots\!06\)\( - \)\(11\!\cdots\!24\)\( \beta_{1} + 962745531987249522 \beta_{2} - 19543418520106602492 \beta_{3}) q^{33}\) \(+(\)\(15\!\cdots\!66\)\( - \)\(49\!\cdots\!30\)\( \beta_{1} - 5181141168554637312 \beta_{2} - 11630432903386267776 \beta_{3}) q^{34}\) \(+(\)\(19\!\cdots\!70\)\( - \)\(22\!\cdots\!80\)\( \beta_{1} + 9258237867206759310 \beta_{2} - \)\(10\!\cdots\!40\)\( \beta_{3}) q^{35}\) \(+(\)\(80\!\cdots\!66\)\( + \)\(13\!\cdots\!43\)\( \beta_{1} + \)\(10\!\cdots\!09\)\( \beta_{3}) q^{36}\) \(+(\)\(48\!\cdots\!62\)\( - \)\(74\!\cdots\!56\)\( \beta_{1} + 20544301497367946160 \beta_{2} + 16927790591022379104 \beta_{3}) q^{37}\) \(+(\)\(77\!\cdots\!76\)\( - \)\(18\!\cdots\!84\)\( \beta_{1} - 68239741820818003968 \beta_{2} + \)\(17\!\cdots\!40\)\( \beta_{3}) q^{38}\) \(+(\)\(42\!\cdots\!72\)\( - \)\(11\!\cdots\!76\)\( \beta_{1} - 78613821454268512602 \beta_{2} + \)\(22\!\cdots\!24\)\( \beta_{3}) q^{39}\) \(+(\)\(49\!\cdots\!64\)\( + \)\(65\!\cdots\!24\)\( \beta_{1} + \)\(25\!\cdots\!52\)\( \beta_{2} + \)\(11\!\cdots\!12\)\( \beta_{3}) q^{40}\) \(+(-\)\(22\!\cdots\!16\)\( + \)\(23\!\cdots\!72\)\( \beta_{1} + 55181475299084350110 \beta_{2} - \)\(59\!\cdots\!32\)\( \beta_{3}) q^{41}\) \(+(\)\(35\!\cdots\!44\)\( + \)\(57\!\cdots\!84\)\( \beta_{1} + \)\(18\!\cdots\!36\)\( \beta_{2} + \)\(18\!\cdots\!60\)\( \beta_{3}) q^{42}\) \(+(\)\(14\!\cdots\!58\)\( + \)\(68\!\cdots\!92\)\( \beta_{1} - \)\(15\!\cdots\!42\)\( \beta_{2} + \)\(11\!\cdots\!08\)\( \beta_{3}) q^{43}\) \(+(\)\(27\!\cdots\!16\)\( + \)\(26\!\cdots\!96\)\( \beta_{1} + \)\(74\!\cdots\!68\)\( \beta_{2} + \)\(16\!\cdots\!16\)\( \beta_{3}) q^{44}\) \(+(\)\(45\!\cdots\!69\)\( + \)\(80\!\cdots\!04\)\( \beta_{1} + \)\(14\!\cdots\!17\)\( \beta_{2} - \)\(24\!\cdots\!98\)\( \beta_{3}) q^{45}\) \(+(\)\(17\!\cdots\!92\)\( - \)\(35\!\cdots\!92\)\( \beta_{1} + \)\(21\!\cdots\!28\)\( \beta_{2} - \)\(19\!\cdots\!20\)\( \beta_{3}) q^{46}\) \(+(\)\(12\!\cdots\!10\)\( - \)\(20\!\cdots\!64\)\( \beta_{1} + \)\(60\!\cdots\!14\)\( \beta_{2} + \)\(24\!\cdots\!36\)\( \beta_{3}) q^{47}\) \(+(-\)\(69\!\cdots\!44\)\( - \)\(31\!\cdots\!00\)\( \beta_{1} - \)\(59\!\cdots\!92\)\( \beta_{2} - \)\(78\!\cdots\!68\)\( \beta_{3}) q^{48}\) \(+(-\)\(10\!\cdots\!95\)\( + \)\(12\!\cdots\!28\)\( \beta_{1} + \)\(99\!\cdots\!36\)\( \beta_{2} + \)\(17\!\cdots\!48\)\( \beta_{3}) q^{49}\) \(+(-\)\(13\!\cdots\!15\)\( + \)\(11\!\cdots\!35\)\( \beta_{1} - \)\(18\!\cdots\!20\)\( \beta_{2} + \)\(20\!\cdots\!80\)\( \beta_{3}) q^{50}\) \(+(-\)\(76\!\cdots\!56\)\( + \)\(10\!\cdots\!92\)\( \beta_{1} + \)\(93\!\cdots\!86\)\( \beta_{2} + \)\(15\!\cdots\!44\)\( \beta_{3}) q^{51}\) \(+(-\)\(33\!\cdots\!28\)\( + \)\(22\!\cdots\!78\)\( \beta_{1} + \)\(24\!\cdots\!24\)\( \beta_{2} - \)\(32\!\cdots\!26\)\( \beta_{3}) q^{52}\) \(+(-\)\(10\!\cdots\!29\)\( + \)\(66\!\cdots\!68\)\( \beta_{1} + \)\(43\!\cdots\!95\)\( \beta_{2} - \)\(21\!\cdots\!58\)\( \beta_{3}) q^{53}\) \(+(-\)\(47\!\cdots\!81\)\( - \)\(11\!\cdots\!27\)\( \beta_{1}) q^{54}\) \(+(-\)\(11\!\cdots\!84\)\( + \)\(64\!\cdots\!56\)\( \beta_{1} - \)\(18\!\cdots\!12\)\( \beta_{2} + \)\(11\!\cdots\!28\)\( \beta_{3}) q^{55}\) \(+(-\)\(29\!\cdots\!68\)\( - \)\(13\!\cdots\!72\)\( \beta_{1} - \)\(35\!\cdots\!76\)\( \beta_{2} - \)\(40\!\cdots\!28\)\( \beta_{3}) q^{56}\) \(+(\)\(67\!\cdots\!46\)\( - \)\(52\!\cdots\!04\)\( \beta_{1} + \)\(10\!\cdots\!34\)\( \beta_{2} + \)\(21\!\cdots\!56\)\( \beta_{3}) q^{57}\) \(+(\)\(41\!\cdots\!38\)\( + \)\(22\!\cdots\!14\)\( \beta_{1} + \)\(30\!\cdots\!20\)\( \beta_{2} + \)\(56\!\cdots\!04\)\( \beta_{3}) q^{58}\) \(+(-\)\(78\!\cdots\!04\)\( - \)\(23\!\cdots\!40\)\( \beta_{1} + \)\(31\!\cdots\!52\)\( \beta_{2} + \)\(43\!\cdots\!68\)\( \beta_{3}) q^{59}\) \(+(-\)\(22\!\cdots\!48\)\( - \)\(39\!\cdots\!18\)\( \beta_{1} - \)\(71\!\cdots\!64\)\( \beta_{2} - \)\(52\!\cdots\!34\)\( \beta_{3}) q^{60}\) \(+(-\)\(61\!\cdots\!30\)\( - \)\(43\!\cdots\!88\)\( \beta_{1} - \)\(13\!\cdots\!36\)\( \beta_{2} - \)\(27\!\cdots\!68\)\( \beta_{3}) q^{61}\) \(+(\)\(36\!\cdots\!44\)\( + \)\(32\!\cdots\!92\)\( \beta_{1} - \)\(10\!\cdots\!68\)\( \beta_{2} + \)\(19\!\cdots\!92\)\( \beta_{3}) q^{62}\) \(+(\)\(31\!\cdots\!75\)\( - \)\(23\!\cdots\!88\)\( \beta_{1} + \)\(32\!\cdots\!73\)\( \beta_{2} - \)\(54\!\cdots\!14\)\( \beta_{3}) q^{63}\) \(+(\)\(94\!\cdots\!32\)\( + \)\(40\!\cdots\!96\)\( \beta_{1} + \)\(58\!\cdots\!48\)\( \beta_{2} + \)\(26\!\cdots\!84\)\( \beta_{3}) q^{64}\) \(+(\)\(80\!\cdots\!34\)\( - \)\(11\!\cdots\!56\)\( \beta_{1} + \)\(80\!\cdots\!62\)\( \beta_{2} + \)\(79\!\cdots\!72\)\( \beta_{3}) q^{65}\) \(+(-\)\(19\!\cdots\!12\)\( - \)\(18\!\cdots\!76\)\( \beta_{1} - \)\(69\!\cdots\!08\)\( \beta_{2} - \)\(16\!\cdots\!52\)\( \beta_{3}) q^{66}\) \(+(\)\(25\!\cdots\!84\)\( + \)\(12\!\cdots\!16\)\( \beta_{1} + \)\(42\!\cdots\!68\)\( \beta_{2} - \)\(96\!\cdots\!04\)\( \beta_{3}) q^{67}\) \(+(-\)\(14\!\cdots\!16\)\( - \)\(12\!\cdots\!50\)\( \beta_{1} - \)\(23\!\cdots\!44\)\( \beta_{2} - \)\(37\!\cdots\!78\)\( \beta_{3}) q^{68}\) \(+(\)\(43\!\cdots\!62\)\( - \)\(12\!\cdots\!68\)\( \beta_{1} + \)\(15\!\cdots\!58\)\( \beta_{2} + \)\(12\!\cdots\!40\)\( \beta_{3}) q^{69}\) \(+(-\)\(34\!\cdots\!40\)\( - \)\(78\!\cdots\!40\)\( \beta_{1} - \)\(36\!\cdots\!20\)\( \beta_{2} - \)\(78\!\cdots\!20\)\( \beta_{3}) q^{70}\) \(+(-\)\(17\!\cdots\!86\)\( - \)\(13\!\cdots\!84\)\( \beta_{1} + \)\(24\!\cdots\!78\)\( \beta_{2} - \)\(24\!\cdots\!20\)\( \beta_{3}) q^{71}\) \(+(\)\(21\!\cdots\!16\)\( + \)\(84\!\cdots\!62\)\( \beta_{1} + \)\(37\!\cdots\!84\)\( \beta_{2} + \)\(20\!\cdots\!78\)\( \beta_{3}) q^{72}\) \(+(\)\(13\!\cdots\!70\)\( - \)\(74\!\cdots\!32\)\( \beta_{1} - \)\(27\!\cdots\!20\)\( \beta_{2} + \)\(84\!\cdots\!60\)\( \beta_{3}) q^{73}\) \(+(-\)\(98\!\cdots\!42\)\( + \)\(44\!\cdots\!54\)\( \beta_{1} + \)\(22\!\cdots\!24\)\( \beta_{2} - \)\(28\!\cdots\!36\)\( \beta_{3}) q^{74}\) \(+(-\)\(56\!\cdots\!85\)\( + \)\(10\!\cdots\!40\)\( \beta_{1} - \)\(15\!\cdots\!80\)\( \beta_{2} + \)\(48\!\cdots\!20\)\( \beta_{3}) q^{75}\) \(+(-\)\(21\!\cdots\!40\)\( + \)\(29\!\cdots\!20\)\( \beta_{1} + \)\(15\!\cdots\!68\)\( \beta_{2} - \)\(15\!\cdots\!56\)\( \beta_{3}) q^{76}\) \(+(-\)\(17\!\cdots\!28\)\( - \)\(24\!\cdots\!00\)\( \beta_{1} - \)\(22\!\cdots\!84\)\( \beta_{2} + \)\(13\!\cdots\!56\)\( \beta_{3}) q^{77}\) \(+(-\)\(15\!\cdots\!50\)\( + \)\(49\!\cdots\!10\)\( \beta_{1} + \)\(92\!\cdots\!80\)\( \beta_{2} - \)\(27\!\cdots\!48\)\( \beta_{3}) q^{78}\) \(+(-\)\(46\!\cdots\!93\)\( - \)\(21\!\cdots\!92\)\( \beta_{1} + \)\(40\!\cdots\!89\)\( \beta_{2} - \)\(68\!\cdots\!98\)\( \beta_{3}) q^{79}\) \(+(\)\(10\!\cdots\!64\)\( + \)\(31\!\cdots\!24\)\( \beta_{1} + \)\(28\!\cdots\!52\)\( \beta_{2} + \)\(84\!\cdots\!12\)\( \beta_{3}) q^{80}\) \(+\)\(11\!\cdots\!81\)\( q^{81}\) \(+(\)\(36\!\cdots\!38\)\( - \)\(48\!\cdots\!46\)\( \beta_{1} - \)\(20\!\cdots\!12\)\( \beta_{2} + \)\(14\!\cdots\!52\)\( \beta_{3}) q^{82}\) \(+(\)\(51\!\cdots\!86\)\( - \)\(34\!\cdots\!68\)\( \beta_{1} + \)\(19\!\cdots\!70\)\( \beta_{2} - \)\(58\!\cdots\!00\)\( \beta_{3}) q^{83}\) \(+(\)\(10\!\cdots\!48\)\( + \)\(35\!\cdots\!44\)\( \beta_{1} + \)\(87\!\cdots\!80\)\( \beta_{2} + \)\(62\!\cdots\!56\)\( \beta_{3}) q^{84}\) \(+(\)\(55\!\cdots\!14\)\( - \)\(37\!\cdots\!76\)\( \beta_{1} + \)\(19\!\cdots\!02\)\( \beta_{2} - \)\(48\!\cdots\!88\)\( \beta_{3}) q^{85}\) \(+(\)\(11\!\cdots\!32\)\( + \)\(25\!\cdots\!84\)\( \beta_{1} + \)\(65\!\cdots\!84\)\( \beta_{2} - \)\(27\!\cdots\!60\)\( \beta_{3}) q^{86}\) \(+(-\)\(13\!\cdots\!49\)\( - \)\(23\!\cdots\!16\)\( \beta_{1} - \)\(13\!\cdots\!61\)\( \beta_{2} - \)\(99\!\cdots\!18\)\( \beta_{3}) q^{87}\) \(+(\)\(37\!\cdots\!96\)\( + \)\(27\!\cdots\!04\)\( \beta_{1} - \)\(35\!\cdots\!76\)\( \beta_{2} + \)\(51\!\cdots\!52\)\( \beta_{3}) q^{88}\) \(+(-\)\(11\!\cdots\!78\)\( + \)\(17\!\cdots\!60\)\( \beta_{1} - \)\(40\!\cdots\!32\)\( \beta_{2} + \)\(12\!\cdots\!88\)\( \beta_{3}) q^{89}\) \(+(\)\(14\!\cdots\!82\)\( + \)\(35\!\cdots\!62\)\( \beta_{1} - \)\(10\!\cdots\!24\)\( \beta_{2} + \)\(37\!\cdots\!56\)\( \beta_{3}) q^{90}\) \(+(-\)\(60\!\cdots\!82\)\( - \)\(10\!\cdots\!52\)\( \beta_{1} + \)\(29\!\cdots\!98\)\( \beta_{2} - \)\(35\!\cdots\!48\)\( \beta_{3}) q^{91}\) \(+(-\)\(12\!\cdots\!76\)\( - \)\(11\!\cdots\!68\)\( \beta_{1} + \)\(59\!\cdots\!84\)\( \beta_{2} - \)\(40\!\cdots\!04\)\( \beta_{3}) q^{92}\) \(+(-\)\(40\!\cdots\!53\)\( - \)\(22\!\cdots\!96\)\( \beta_{1} + \)\(11\!\cdots\!85\)\( \beta_{2} + \)\(31\!\cdots\!34\)\( \beta_{3}) q^{93}\) \(+(-\)\(31\!\cdots\!96\)\( - \)\(27\!\cdots\!44\)\( \beta_{1} + \)\(73\!\cdots\!44\)\( \beta_{2} - \)\(18\!\cdots\!00\)\( \beta_{3}) q^{94}\) \(+(-\)\(21\!\cdots\!84\)\( - \)\(40\!\cdots\!44\)\( \beta_{1} - \)\(90\!\cdots\!12\)\( \beta_{2} + \)\(19\!\cdots\!28\)\( \beta_{3}) q^{95}\) \(+(-\)\(35\!\cdots\!44\)\( - \)\(90\!\cdots\!48\)\( \beta_{1} + \)\(55\!\cdots\!32\)\( \beta_{2} - \)\(32\!\cdots\!84\)\( \beta_{3}) q^{96}\) \(+(-\)\(12\!\cdots\!78\)\( + \)\(11\!\cdots\!84\)\( \beta_{1} - \)\(10\!\cdots\!68\)\( \beta_{2} + \)\(24\!\cdots\!60\)\( \beta_{3}) q^{97}\) \(+(\)\(15\!\cdots\!83\)\( - \)\(81\!\cdots\!19\)\( \beta_{1} + \)\(41\!\cdots\!40\)\( \beta_{2} + \)\(35\!\cdots\!24\)\( \beta_{3}) q^{98}\) \(+(\)\(23\!\cdots\!18\)\( + \)\(12\!\cdots\!72\)\( \beta_{1} - \)\(10\!\cdots\!66\)\( \beta_{2} + \)\(20\!\cdots\!76\)\( \beta_{3}) q^{99}\) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(4q \) \(\mathstrut +\mathstrut 1660014q^{2} \) \(\mathstrut -\mathstrut 41841412812q^{3} \) \(\mathstrut +\mathstrut 29333750564548q^{4} \) \(\mathstrut +\mathstrut 1650650807536920q^{5} \) \(\mathstrut -\mathstrut 17364332761924842q^{6} \) \(\mathstrut +\mathstrut 114481425784209728q^{7} \) \(\mathstrut +\mathstrut 77966588660971173048q^{8} \) \(\mathstrut +\mathstrut 437675956526049436836q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(4q \) \(\mathstrut +\mathstrut 1660014q^{2} \) \(\mathstrut -\mathstrut 41841412812q^{3} \) \(\mathstrut +\mathstrut 29333750564548q^{4} \) \(\mathstrut +\mathstrut 1650650807536920q^{5} \) \(\mathstrut -\mathstrut 17364332761924842q^{6} \) \(\mathstrut +\mathstrut 114481425784209728q^{7} \) \(\mathstrut +\mathstrut 77966588660971173048q^{8} \) \(\mathstrut +\mathstrut \)\(43\!\cdots\!36\)\(q^{9} \) \(\mathstrut +\mathstrut \)\(53\!\cdots\!60\)\(q^{10} \) \(\mathstrut +\mathstrut \)\(87\!\cdots\!96\)\(q^{11} \) \(\mathstrut -\mathstrut \)\(30\!\cdots\!44\)\(q^{12} \) \(\mathstrut -\mathstrut \)\(16\!\cdots\!96\)\(q^{13} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!08\)\(q^{14} \) \(\mathstrut -\mathstrut \)\(17\!\cdots\!60\)\(q^{15} \) \(\mathstrut +\mathstrut \)\(26\!\cdots\!80\)\(q^{16} \) \(\mathstrut +\mathstrut \)\(29\!\cdots\!76\)\(q^{17} \) \(\mathstrut +\mathstrut \)\(18\!\cdots\!26\)\(q^{18} \) \(\mathstrut -\mathstrut \)\(25\!\cdots\!88\)\(q^{19} \) \(\mathstrut +\mathstrut \)\(85\!\cdots\!20\)\(q^{20} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!84\)\(q^{21} \) \(\mathstrut +\mathstrut \)\(74\!\cdots\!32\)\(q^{22} \) \(\mathstrut -\mathstrut \)\(16\!\cdots\!44\)\(q^{23} \) \(\mathstrut -\mathstrut \)\(81\!\cdots\!44\)\(q^{24} \) \(\mathstrut +\mathstrut \)\(21\!\cdots\!00\)\(q^{25} \) \(\mathstrut +\mathstrut \)\(61\!\cdots\!28\)\(q^{26} \) \(\mathstrut -\mathstrut \)\(45\!\cdots\!08\)\(q^{27} \) \(\mathstrut -\mathstrut \)\(41\!\cdots\!56\)\(q^{28} \) \(\mathstrut +\mathstrut \)\(50\!\cdots\!64\)\(q^{29} \) \(\mathstrut -\mathstrut \)\(56\!\cdots\!80\)\(q^{30} \) \(\mathstrut +\mathstrut \)\(15\!\cdots\!24\)\(q^{31} \) \(\mathstrut +\mathstrut \)\(13\!\cdots\!68\)\(q^{32} \) \(\mathstrut -\mathstrut \)\(91\!\cdots\!88\)\(q^{33} \) \(\mathstrut +\mathstrut \)\(60\!\cdots\!56\)\(q^{34} \) \(\mathstrut +\mathstrut \)\(79\!\cdots\!00\)\(q^{35} \) \(\mathstrut +\mathstrut \)\(32\!\cdots\!32\)\(q^{36} \) \(\mathstrut +\mathstrut \)\(19\!\cdots\!28\)\(q^{37} \) \(\mathstrut +\mathstrut \)\(30\!\cdots\!56\)\(q^{38} \) \(\mathstrut +\mathstrut \)\(16\!\cdots\!88\)\(q^{39} \) \(\mathstrut +\mathstrut \)\(19\!\cdots\!80\)\(q^{40} \) \(\mathstrut -\mathstrut \)\(90\!\cdots\!56\)\(q^{41} \) \(\mathstrut +\mathstrut \)\(14\!\cdots\!24\)\(q^{42} \) \(\mathstrut +\mathstrut \)\(57\!\cdots\!00\)\(q^{43} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!24\)\(q^{44} \) \(\mathstrut +\mathstrut \)\(18\!\cdots\!80\)\(q^{45} \) \(\mathstrut +\mathstrut \)\(71\!\cdots\!24\)\(q^{46} \) \(\mathstrut +\mathstrut \)\(48\!\cdots\!40\)\(q^{47} \) \(\mathstrut -\mathstrut \)\(27\!\cdots\!40\)\(q^{48} \) \(\mathstrut -\mathstrut \)\(43\!\cdots\!20\)\(q^{49} \) \(\mathstrut -\mathstrut \)\(55\!\cdots\!50\)\(q^{50} \) \(\mathstrut -\mathstrut \)\(30\!\cdots\!28\)\(q^{51} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!04\)\(q^{52} \) \(\mathstrut -\mathstrut \)\(40\!\cdots\!64\)\(q^{53} \) \(\mathstrut -\mathstrut \)\(18\!\cdots\!78\)\(q^{54} \) \(\mathstrut -\mathstrut \)\(47\!\cdots\!80\)\(q^{55} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!60\)\(q^{56} \) \(\mathstrut +\mathstrut \)\(27\!\cdots\!64\)\(q^{57} \) \(\mathstrut +\mathstrut \)\(16\!\cdots\!72\)\(q^{58} \) \(\mathstrut -\mathstrut \)\(31\!\cdots\!32\)\(q^{59} \) \(\mathstrut -\mathstrut \)\(89\!\cdots\!60\)\(q^{60} \) \(\mathstrut -\mathstrut \)\(24\!\cdots\!60\)\(q^{61} \) \(\mathstrut +\mathstrut \)\(14\!\cdots\!76\)\(q^{62} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!52\)\(q^{63} \) \(\mathstrut +\mathstrut \)\(37\!\cdots\!52\)\(q^{64} \) \(\mathstrut +\mathstrut \)\(32\!\cdots\!80\)\(q^{65} \) \(\mathstrut -\mathstrut \)\(77\!\cdots\!96\)\(q^{66} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!76\)\(q^{67} \) \(\mathstrut -\mathstrut \)\(57\!\cdots\!08\)\(q^{68} \) \(\mathstrut +\mathstrut \)\(17\!\cdots\!32\)\(q^{69} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!00\)\(q^{70} \) \(\mathstrut -\mathstrut \)\(71\!\cdots\!72\)\(q^{71} \) \(\mathstrut +\mathstrut \)\(85\!\cdots\!32\)\(q^{72} \) \(\mathstrut +\mathstrut \)\(54\!\cdots\!96\)\(q^{73} \) \(\mathstrut -\mathstrut \)\(39\!\cdots\!88\)\(q^{74} \) \(\mathstrut -\mathstrut \)\(22\!\cdots\!00\)\(q^{75} \) \(\mathstrut -\mathstrut \)\(84\!\cdots\!08\)\(q^{76} \) \(\mathstrut -\mathstrut \)\(70\!\cdots\!24\)\(q^{77} \) \(\mathstrut -\mathstrut \)\(63\!\cdots\!84\)\(q^{78} \) \(\mathstrut -\mathstrut \)\(18\!\cdots\!60\)\(q^{79} \) \(\mathstrut +\mathstrut \)\(42\!\cdots\!80\)\(q^{80} \) \(\mathstrut +\mathstrut \)\(47\!\cdots\!24\)\(q^{81} \) \(\mathstrut +\mathstrut \)\(14\!\cdots\!56\)\(q^{82} \) \(\mathstrut +\mathstrut \)\(20\!\cdots\!08\)\(q^{83} \) \(\mathstrut +\mathstrut \)\(43\!\cdots\!68\)\(q^{84} \) \(\mathstrut +\mathstrut \)\(22\!\cdots\!80\)\(q^{85} \) \(\mathstrut +\mathstrut \)\(45\!\cdots\!16\)\(q^{86} \) \(\mathstrut -\mathstrut \)\(52\!\cdots\!92\)\(q^{87} \) \(\mathstrut +\mathstrut \)\(15\!\cdots\!88\)\(q^{88} \) \(\mathstrut -\mathstrut \)\(44\!\cdots\!68\)\(q^{89} \) \(\mathstrut +\mathstrut \)\(58\!\cdots\!40\)\(q^{90} \) \(\mathstrut -\mathstrut \)\(24\!\cdots\!36\)\(q^{91} \) \(\mathstrut -\mathstrut \)\(50\!\cdots\!32\)\(q^{92} \) \(\mathstrut -\mathstrut \)\(16\!\cdots\!72\)\(q^{93} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!72\)\(q^{94} \) \(\mathstrut -\mathstrut \)\(87\!\cdots\!80\)\(q^{95} \) \(\mathstrut -\mathstrut \)\(14\!\cdots\!04\)\(q^{96} \) \(\mathstrut -\mathstrut \)\(50\!\cdots\!64\)\(q^{97} \) \(\mathstrut +\mathstrut \)\(62\!\cdots\!46\)\(q^{98} \) \(\mathstrut +\mathstrut \)\(95\!\cdots\!64\)\(q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4}\mathstrut -\mathstrut \) \(x^{3}\mathstrut -\mathstrut \) \(886516819907\) \(x^{2}\mathstrut -\mathstrut \) \(42308083143723387\) \(x\mathstrut +\mathstrut \) \(94580276745082867224894\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 6 \nu - 1 \)
\(\beta_{2}\)\(=\)\((\)\( 27 \nu^{3} - 2660112 \nu^{2} - 17984400403617 \nu + 322364875892989014 \)\()/43072\)
\(\beta_{3}\)\(=\)\( 36 \nu^{2} - 2577138 \nu - 15957302114051 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1}\mathstrut +\mathstrut \) \(1\)\()/6\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3}\mathstrut +\mathstrut \) \(429523\) \(\beta_{1}\mathstrut +\mathstrut \) \(15957302543574\)\()/36\)
\(\nu^{3}\)\(=\)\((\)\(147784\) \(\beta_{3}\mathstrut +\mathstrut \) \(86144\) \(\beta_{2}\mathstrut +\mathstrut \) \(6058276761571\) \(\beta_{1}\mathstrut +\mathstrut \) \(1713510242113696527\)\()/54\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−837306.
−388484.
321562.
904229.
−4.60883e6 −1.04604e10 1.24452e13 −9.11231e14 4.82100e16 3.55193e17 −1.68183e19 1.09419e20 4.19971e21
1.2 −1.91590e6 −1.04604e10 −5.12540e12 2.05854e15 2.00410e16 1.37114e18 2.66723e19 1.09419e20 −3.94396e21
1.3 2.34437e6 −1.04604e10 −3.30001e12 −6.18875e14 −2.45230e16 −6.01464e16 −2.83578e19 1.09419e20 −1.45087e21
1.4 5.84038e6 −1.04604e10 2.53139e13 1.12222e15 −6.10924e16 −1.55171e18 9.64704e19 1.09419e20 6.55417e21
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{4} \) \(\mathstrut -\mathstrut 1660014 T_{2}^{3} \) \(\mathstrut -\mathstrut \)\(30\!\cdots\!92\)\( T_{2}^{2} \) \(\mathstrut +\mathstrut \)\(17\!\cdots\!12\)\( T_{2} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!64\)\( \) acting on \(S_{44}^{\mathrm{new}}(\Gamma_0(3))\).