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: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Defining polynomial: \(x^{4} - x^{3} - 886516819907 x^{2} - 42308083143723387 x + 94580276745082867224894\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{14}\cdot 3^{10}\cdot 5^{2}\cdot 7\cdot 11 \)
Twist minimal: yes
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.

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

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

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

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:

Atkin-Lehner signs

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

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3.44.a.b 4
3.b odd 2 1 9.44.a.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.44.a.b 4 1.a even 1 1 trivial
9.44.a.c 4 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - 1660014 T_{2}^{3} - \)30881238086592

'>\(30\!\cdots\!92\)\( T_{2}^{2} + \)17064803544699174912
'>\(17\!\cdots\!12\)\( T_{2} + \)120901670049507055201419264'>\(12\!\cdots\!64\)\( \) acting on \(S_{44}^{\mathrm{new}}(\Gamma_0(3))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 1660014 T + 4303134002240 T^{2} - 26740109141803597824 T^{3} + \)\(41\!\cdots\!76\)\( T^{4} - \)\(23\!\cdots\!92\)\( T^{5} + \)\(33\!\cdots\!60\)\( T^{6} - \)\(11\!\cdots\!68\)\( T^{7} + \)\(59\!\cdots\!96\)\( T^{8} \)
$3$ \( ( 1 + 10460353203 T )^{4} \)
$5$ \( 1 - 1650650807536920 T + \)\(25\!\cdots\!00\)\( T^{2} - \)\(38\!\cdots\!00\)\( T^{3} + \)\(45\!\cdots\!50\)\( T^{4} - \)\(44\!\cdots\!00\)\( T^{5} + \)\(33\!\cdots\!00\)\( T^{6} - \)\(24\!\cdots\!00\)\( T^{7} + \)\(16\!\cdots\!25\)\( T^{8} \)
$7$ \( 1 - 114481425784209728 T + \)\(65\!\cdots\!88\)\( T^{2} - \)\(12\!\cdots\!92\)\( T^{3} + \)\(19\!\cdots\!70\)\( T^{4} - \)\(27\!\cdots\!56\)\( T^{5} + \)\(31\!\cdots\!12\)\( T^{6} - \)\(11\!\cdots\!96\)\( T^{7} + \)\(22\!\cdots\!01\)\( T^{8} \)
$11$ \( 1 - \)\(87\!\cdots\!96\)\( T + \)\(13\!\cdots\!12\)\( T^{2} - \)\(69\!\cdots\!60\)\( T^{3} + \)\(11\!\cdots\!86\)\( T^{4} - \)\(41\!\cdots\!60\)\( T^{5} + \)\(50\!\cdots\!32\)\( T^{6} - \)\(19\!\cdots\!36\)\( T^{7} + \)\(13\!\cdots\!21\)\( T^{8} \)
$13$ \( 1 + \)\(16\!\cdots\!96\)\( T + \)\(27\!\cdots\!36\)\( T^{2} + \)\(32\!\cdots\!48\)\( T^{3} + \)\(32\!\cdots\!50\)\( T^{4} + \)\(25\!\cdots\!56\)\( T^{5} + \)\(17\!\cdots\!24\)\( T^{6} + \)\(81\!\cdots\!08\)\( T^{7} + \)\(39\!\cdots\!81\)\( T^{8} \)
$17$ \( 1 - \)\(29\!\cdots\!76\)\( T + \)\(29\!\cdots\!68\)\( T^{2} - \)\(64\!\cdots\!00\)\( T^{3} + \)\(35\!\cdots\!26\)\( T^{4} - \)\(52\!\cdots\!00\)\( T^{5} + \)\(19\!\cdots\!92\)\( T^{6} - \)\(15\!\cdots\!72\)\( T^{7} + \)\(43\!\cdots\!61\)\( T^{8} \)
$19$ \( 1 + \)\(25\!\cdots\!88\)\( T + \)\(20\!\cdots\!32\)\( T^{2} + \)\(42\!\cdots\!16\)\( T^{3} + \)\(20\!\cdots\!14\)\( T^{4} + \)\(40\!\cdots\!44\)\( T^{5} + \)\(19\!\cdots\!92\)\( T^{6} + \)\(23\!\cdots\!52\)\( T^{7} + \)\(88\!\cdots\!61\)\( T^{8} \)
$23$ \( 1 + \)\(16\!\cdots\!44\)\( T + \)\(11\!\cdots\!12\)\( T^{2} + \)\(15\!\cdots\!64\)\( T^{3} + \)\(56\!\cdots\!30\)\( T^{4} + \)\(55\!\cdots\!88\)\( T^{5} + \)\(14\!\cdots\!68\)\( T^{6} + \)\(76\!\cdots\!72\)\( T^{7} + \)\(16\!\cdots\!21\)\( T^{8} \)
$29$ \( 1 - \)\(50\!\cdots\!64\)\( T + \)\(33\!\cdots\!00\)\( T^{2} - \)\(10\!\cdots\!28\)\( T^{3} + \)\(38\!\cdots\!58\)\( T^{4} - \)\(83\!\cdots\!92\)\( T^{5} + \)\(19\!\cdots\!00\)\( T^{6} - \)\(22\!\cdots\!16\)\( T^{7} + \)\(34\!\cdots\!41\)\( T^{8} \)
$31$ \( 1 - \)\(15\!\cdots\!24\)\( T + \)\(45\!\cdots\!08\)\( T^{2} - \)\(44\!\cdots\!52\)\( T^{3} + \)\(83\!\cdots\!94\)\( T^{4} - \)\(60\!\cdots\!32\)\( T^{5} + \)\(81\!\cdots\!48\)\( T^{6} - \)\(37\!\cdots\!04\)\( T^{7} + \)\(32\!\cdots\!61\)\( T^{8} \)
$37$ \( 1 - \)\(19\!\cdots\!28\)\( T + \)\(22\!\cdots\!48\)\( T^{2} - \)\(18\!\cdots\!12\)\( T^{3} + \)\(10\!\cdots\!70\)\( T^{4} - \)\(49\!\cdots\!36\)\( T^{5} + \)\(16\!\cdots\!32\)\( T^{6} - \)\(38\!\cdots\!56\)\( T^{7} + \)\(53\!\cdots\!81\)\( T^{8} \)
$41$ \( 1 + \)\(90\!\cdots\!56\)\( T - \)\(37\!\cdots\!32\)\( T^{2} + \)\(41\!\cdots\!28\)\( T^{3} + \)\(71\!\cdots\!74\)\( T^{4} + \)\(92\!\cdots\!88\)\( T^{5} - \)\(18\!\cdots\!12\)\( T^{6} + \)\(10\!\cdots\!16\)\( T^{7} + \)\(25\!\cdots\!81\)\( T^{8} \)
$43$ \( 1 - \)\(57\!\cdots\!00\)\( T + \)\(28\!\cdots\!40\)\( T^{2} - \)\(91\!\cdots\!00\)\( T^{3} + \)\(40\!\cdots\!98\)\( T^{4} - \)\(15\!\cdots\!00\)\( T^{5} + \)\(85\!\cdots\!60\)\( T^{6} - \)\(30\!\cdots\!00\)\( T^{7} + \)\(90\!\cdots\!01\)\( T^{8} \)
$47$ \( 1 - \)\(48\!\cdots\!40\)\( T + \)\(19\!\cdots\!80\)\( T^{2} - \)\(82\!\cdots\!20\)\( T^{3} + \)\(20\!\cdots\!58\)\( T^{4} - \)\(65\!\cdots\!60\)\( T^{5} + \)\(12\!\cdots\!20\)\( T^{6} - \)\(24\!\cdots\!80\)\( T^{7} + \)\(39\!\cdots\!41\)\( T^{8} \)
$53$ \( 1 + \)\(40\!\cdots\!64\)\( T + \)\(11\!\cdots\!12\)\( T^{2} + \)\(19\!\cdots\!44\)\( T^{3} + \)\(27\!\cdots\!90\)\( T^{4} + \)\(27\!\cdots\!88\)\( T^{5} + \)\(21\!\cdots\!48\)\( T^{6} + \)\(10\!\cdots\!12\)\( T^{7} + \)\(37\!\cdots\!41\)\( T^{8} \)
$59$ \( 1 + \)\(31\!\cdots\!32\)\( T + \)\(32\!\cdots\!72\)\( T^{2} + \)\(69\!\cdots\!84\)\( T^{3} + \)\(57\!\cdots\!94\)\( T^{4} + \)\(97\!\cdots\!36\)\( T^{5} + \)\(63\!\cdots\!52\)\( T^{6} + \)\(86\!\cdots\!48\)\( T^{7} + \)\(38\!\cdots\!81\)\( T^{8} \)
$61$ \( 1 + \)\(24\!\cdots\!60\)\( T + \)\(18\!\cdots\!76\)\( T^{2} - \)\(57\!\cdots\!60\)\( T^{3} - \)\(20\!\cdots\!34\)\( T^{4} - \)\(33\!\cdots\!60\)\( T^{5} + \)\(65\!\cdots\!36\)\( T^{6} + \)\(50\!\cdots\!60\)\( T^{7} + \)\(11\!\cdots\!21\)\( T^{8} \)
$67$ \( 1 - \)\(10\!\cdots\!76\)\( T + \)\(13\!\cdots\!68\)\( T^{2} - \)\(10\!\cdots\!00\)\( T^{3} + \)\(65\!\cdots\!26\)\( T^{4} - \)\(33\!\cdots\!00\)\( T^{5} + \)\(14\!\cdots\!92\)\( T^{6} - \)\(37\!\cdots\!72\)\( T^{7} + \)\(12\!\cdots\!61\)\( T^{8} \)
$71$ \( 1 + \)\(71\!\cdots\!72\)\( T + \)\(94\!\cdots\!88\)\( T^{2} + \)\(62\!\cdots\!04\)\( T^{3} + \)\(46\!\cdots\!70\)\( T^{4} + \)\(25\!\cdots\!44\)\( T^{5} + \)\(15\!\cdots\!48\)\( T^{6} + \)\(46\!\cdots\!32\)\( T^{7} + \)\(26\!\cdots\!41\)\( T^{8} \)
$73$ \( 1 - \)\(54\!\cdots\!96\)\( T + \)\(34\!\cdots\!32\)\( T^{2} - \)\(18\!\cdots\!76\)\( T^{3} + \)\(60\!\cdots\!10\)\( T^{4} - \)\(24\!\cdots\!92\)\( T^{5} + \)\(59\!\cdots\!48\)\( T^{6} - \)\(12\!\cdots\!48\)\( T^{7} + \)\(31\!\cdots\!21\)\( T^{8} \)
$79$ \( 1 + \)\(18\!\cdots\!60\)\( T - \)\(12\!\cdots\!44\)\( T^{2} + \)\(59\!\cdots\!20\)\( T^{3} + \)\(25\!\cdots\!26\)\( T^{4} + \)\(23\!\cdots\!80\)\( T^{5} - \)\(19\!\cdots\!24\)\( T^{6} + \)\(11\!\cdots\!40\)\( T^{7} + \)\(24\!\cdots\!41\)\( T^{8} \)
$83$ \( 1 - \)\(20\!\cdots\!08\)\( T + \)\(10\!\cdots\!80\)\( T^{2} - \)\(13\!\cdots\!52\)\( T^{3} + \)\(43\!\cdots\!46\)\( T^{4} - \)\(44\!\cdots\!24\)\( T^{5} + \)\(11\!\cdots\!20\)\( T^{6} - \)\(75\!\cdots\!24\)\( T^{7} + \)\(12\!\cdots\!61\)\( T^{8} \)
$89$ \( 1 + \)\(44\!\cdots\!68\)\( T + \)\(24\!\cdots\!92\)\( T^{2} + \)\(88\!\cdots\!16\)\( T^{3} + \)\(23\!\cdots\!74\)\( T^{4} + \)\(58\!\cdots\!04\)\( T^{5} + \)\(10\!\cdots\!12\)\( T^{6} + \)\(13\!\cdots\!12\)\( T^{7} + \)\(19\!\cdots\!21\)\( T^{8} \)
$97$ \( 1 + \)\(50\!\cdots\!64\)\( T + \)\(60\!\cdots\!28\)\( T^{2} + \)\(93\!\cdots\!00\)\( T^{3} + \)\(13\!\cdots\!66\)\( T^{4} + \)\(25\!\cdots\!00\)\( T^{5} + \)\(44\!\cdots\!12\)\( T^{6} + \)\(99\!\cdots\!88\)\( T^{7} + \)\(53\!\cdots\!41\)\( T^{8} \)
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