Properties

Label 3.38.a.b
Level 3
Weight 38
Character orbit 3.a
Self dual Yes
Analytic conductor 26.014
Analytic rank 0
Dimension 4
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 3 \)
Weight: \( k \) = \( 38 \)
Character orbit: \([\chi]\) = 3.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(26.0142114374\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{15}\cdot 3^{10}\cdot 5\cdot 7 \)
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.

\(f(q)\) \(=\) \( q + ( 109391 - \beta_{1} ) q^{2} + 387420489 q^{3} + ( 86524834843 - 191939 \beta_{1} + \beta_{3} ) q^{4} + ( -1024957187485 - 503509 \beta_{1} + \beta_{2} - 28 \beta_{3} ) q^{5} + ( 42380314712199 - 387420489 \beta_{1} ) q^{6} + ( 1651450994333561 + 2985430279 \beta_{1} + 405 \beta_{2} + 20404 \beta_{3} ) q^{7} + ( 35121079860249342 - 103049124558 \beta_{1} - 5440 \beta_{2} + 299578 \beta_{3} ) q^{8} + 150094635296999121 q^{9} +O(q^{10})\) \( q +(109391 - \beta_{1}) q^{2} +387420489 q^{3} +(86524834843 - 191939 \beta_{1} + \beta_{3}) q^{4} +(-1024957187485 - 503509 \beta_{1} + \beta_{2} - 28 \beta_{3}) q^{5} +(42380314712199 - 387420489 \beta_{1}) q^{6} +(1651450994333561 + 2985430279 \beta_{1} + 405 \beta_{2} + 20404 \beta_{3}) q^{7} +(35121079860249342 - 103049124558 \beta_{1} - 5440 \beta_{2} + 299578 \beta_{3}) q^{8} +150094635296999121 q^{9} +(-5380189282634790 + 4867048785914 \beta_{1} + 290304 \beta_{2} - 15949312 \beta_{3}) q^{10} +(5238439959104483494 - 25492139539658 \beta_{1} - 1387870 \beta_{2} - 26831288 \beta_{3}) q^{11} +(33521493825519298227 - 74361101238171 \beta_{1} + 387420489 \beta_{3}) q^{12} +(12957972542954522468 - 498690534219638 \beta_{1} + 14789790 \beta_{2} + 865098424 \beta_{3}) q^{13} +(-\)\(45\!\cdots\!20\)\( - 4216580839272024 \beta_{1} - 55114240 \beta_{2} - 6231930368 \beta_{3}) q^{14} +(-\)\(39\!\cdots\!65\)\( - 195069702995901 \beta_{1} + 387420489 \beta_{2} - 10847773692 \beta_{3}) q^{15} +(\)\(13\!\cdots\!28\)\( - 58713905237284428 \beta_{1} - 2380337280 \beta_{2} + 70964221188 \beta_{3}) q^{16} +(\)\(20\!\cdots\!68\)\( - 26353299406777634 \beta_{1} + 6152361370 \beta_{2} - 41466582232 \beta_{3}) q^{17} +(\)\(16\!\cdots\!11\)\( - 150094635296999121 \beta_{1}) q^{18} +(-\)\(13\!\cdots\!30\)\( + 796715087148526350 \beta_{1} - 23805860310 \beta_{2} + 691505159016 \beta_{3}) q^{19} +(-\)\(89\!\cdots\!90\)\( + 2683989303798924654 \beta_{1} - 10617389056 \beta_{2} - 6636900927482 \beta_{3}) q^{20} +(\)\(63\!\cdots\!29\)\( + 1156616858565586431 \beta_{1} + 156905298045 \beta_{2} + 7904927657556 \beta_{3}) q^{21} +(\)\(59\!\cdots\!56\)\( - 3659504875238983628 \beta_{1} - 45541647360 \beta_{2} + 41255532921856 \beta_{3}) q^{22} +(-\)\(15\!\cdots\!30\)\( + 16703658020546342574 \beta_{1} - 178310885110 \beta_{2} - 114016465995032 \beta_{3}) q^{23} +(\)\(13\!\cdots\!38\)\( - 39923342227282268862 \beta_{1} - 2107567460160 \beta_{2} + 116062655253642 \beta_{3}) q^{24} +(\)\(23\!\cdots\!15\)\( - \)\(10\!\cdots\!64\)\( \beta_{1} + 6346460995596 \beta_{2} - 254265946315088 \beta_{3}) q^{25} +(\)\(10\!\cdots\!10\)\( - \)\(17\!\cdots\!62\)\( \beta_{1} - 2665381043200 \beta_{2} + 393049925745664 \beta_{3}) q^{26} +\)\(58\!\cdots\!69\)\( q^{27} +(\)\(61\!\cdots\!04\)\( + \)\(55\!\cdots\!60\)\( \beta_{1} - 29365958246400 \beta_{2} + 1482154037997144 \beta_{3}) q^{28} +(\)\(10\!\cdots\!69\)\( + \)\(14\!\cdots\!09\)\( \beta_{1} + 23642784642315 \beta_{2} - 2583712585520436 \beta_{3}) q^{29} +(-\)\(20\!\cdots\!10\)\( + \)\(18\!\cdots\!46\)\( \beta_{1} + 112469717638656 \beta_{2} - 6179090254253568 \beta_{3}) q^{30} +(\)\(22\!\cdots\!29\)\( + \)\(56\!\cdots\!67\)\( \beta_{1} - 144867143474775 \beta_{2} - 6432129197675132 \beta_{3}) q^{31} +(\)\(91\!\cdots\!96\)\( - \)\(14\!\cdots\!96\)\( \beta_{1} + 33174084381440 \beta_{2} + 57167919940477864 \beta_{3}) q^{32} +(\)\(20\!\cdots\!66\)\( - \)\(98\!\cdots\!62\)\( \beta_{1} - 537689274068430 \beta_{2} - 10394990717459832 \beta_{3}) q^{33} +(\)\(78\!\cdots\!14\)\( - \)\(16\!\cdots\!50\)\( \beta_{1} + 1074505638620160 \beta_{2} - 60791269671865344 \beta_{3}) q^{34} +(\)\(10\!\cdots\!50\)\( - \)\(42\!\cdots\!10\)\( \beta_{1} - 126089450324010 \beta_{2} - 175002471366658920 \beta_{3}) q^{35} +(\)\(12\!\cdots\!03\)\( - \)\(28\!\cdots\!19\)\( \beta_{1} + 150094635296999121 \beta_{3}) q^{36} +(\)\(13\!\cdots\!14\)\( + \)\(11\!\cdots\!12\)\( \beta_{1} + 556310835696600 \beta_{2} + 508236733739231328 \beta_{3}) q^{37} +(-\)\(18\!\cdots\!40\)\( + \)\(10\!\cdots\!80\)\( \beta_{1} - 7046615894062080 \beta_{2} - 402356896847195136 \beta_{3}) q^{38} +(\)\(50\!\cdots\!52\)\( - \)\(19\!\cdots\!82\)\( \beta_{1} + 5729867674007310 \beta_{2} + 335156854459209336 \beta_{3}) q^{39} +(-\)\(66\!\cdots\!80\)\( + \)\(13\!\cdots\!08\)\( \beta_{1} - 5259366714736512 \beta_{2} - 1063635595164787364 \beta_{3}) q^{40} +(-\)\(21\!\cdots\!68\)\( - \)\(50\!\cdots\!30\)\( \beta_{1} + 46408136639113950 \beta_{2} - 101394467803539528 \beta_{3}) q^{41} +(-\)\(17\!\cdots\!80\)\( - \)\(16\!\cdots\!36\)\( \beta_{1} - 21352385811663360 \beta_{2} - 2414377510584509952 \beta_{3}) q^{42} +(-\)\(12\!\cdots\!18\)\( + \)\(36\!\cdots\!66\)\( \beta_{1} - 106767768332859570 \beta_{2} + 3642447885337803640 \beta_{3}) q^{43} +(\)\(70\!\cdots\!20\)\( - \)\(84\!\cdots\!24\)\( \beta_{1} - 39966717409034240 \beta_{2} + 12399904291854550988 \beta_{3}) q^{44} +(-\)\(15\!\cdots\!85\)\( - \)\(75\!\cdots\!89\)\( \beta_{1} + 150094635296999121 \beta_{2} - 4202649788315975388 \beta_{3}) q^{45} +(-\)\(37\!\cdots\!96\)\( + \)\(18\!\cdots\!52\)\( \beta_{1} + 595645525841955840 \beta_{2} - 26579969078865144832 \beta_{3}) q^{46} +(\)\(10\!\cdots\!42\)\( + \)\(12\!\cdots\!70\)\( \beta_{1} - 580153156762703090 \beta_{2} - 13645452734161870216 \beta_{3}) q^{47} +(\)\(53\!\cdots\!92\)\( - \)\(22\!\cdots\!92\)\( \beta_{1} - 922191433002529920 \beta_{2} + 27492993274159120932 \beta_{3}) q^{48} +(\)\(10\!\cdots\!93\)\( + \)\(55\!\cdots\!36\)\( \beta_{1} - 335873647667195280 \beta_{2} + 40861585860796998592 \beta_{3}) q^{49} +(\)\(25\!\cdots\!85\)\( + \)\(23\!\cdots\!69\)\( \beta_{1} + 2258916821970397184 \beta_{2} - 3778457216840394752 \beta_{3}) q^{50} +(\)\(78\!\cdots\!52\)\( - \)\(10\!\cdots\!26\)\( \beta_{1} + 2383550850470109930 \beta_{2} - 16065003565480151448 \beta_{3}) q^{51} +(\)\(46\!\cdots\!42\)\( - \)\(10\!\cdots\!10\)\( \beta_{1} - 4538664793591971840 \beta_{2} + \)\(13\!\cdots\!30\)\( \beta_{3}) q^{52} +(-\)\(31\!\cdots\!59\)\( - \)\(45\!\cdots\!55\)\( \beta_{1} - 1922334687103569825 \beta_{2} - \)\(40\!\cdots\!04\)\( \beta_{3}) q^{53} +(\)\(63\!\cdots\!79\)\( - \)\(58\!\cdots\!69\)\( \beta_{1}) q^{54} +(-\)\(81\!\cdots\!60\)\( + \)\(16\!\cdots\!96\)\( \beta_{1} + 8408678824795228056 \beta_{2} + 22892580198991444832 \beta_{3}) q^{55} +(\)\(12\!\cdots\!84\)\( - \)\(19\!\cdots\!56\)\( \beta_{1} - 4540106882371079680 \beta_{2} + \)\(85\!\cdots\!52\)\( \beta_{3}) q^{56} +(-\)\(52\!\cdots\!70\)\( + \)\(30\!\cdots\!50\)\( \beta_{1} - 9222878042365891590 \beta_{2} + \)\(26\!\cdots\!24\)\( \beta_{3}) q^{57} +(-\)\(18\!\cdots\!22\)\( - \)\(56\!\cdots\!82\)\( \beta_{1} + 17317722461316364800 \beta_{2} - \)\(20\!\cdots\!48\)\( \beta_{3}) q^{58} +(-\)\(33\!\cdots\!36\)\( + \)\(15\!\cdots\!44\)\( \beta_{1} + 3830895234533430040 \beta_{2} + \)\(24\!\cdots\!76\)\( \beta_{3}) q^{59} +(-\)\(34\!\cdots\!10\)\( + \)\(10\!\cdots\!06\)\( \beta_{1} - 4113394059978768384 \beta_{2} - \)\(25\!\cdots\!98\)\( \beta_{3}) q^{60} +(-\)\(30\!\cdots\!62\)\( - \)\(70\!\cdots\!84\)\( \beta_{1} - 56929543616748524580 \beta_{2} + \)\(14\!\cdots\!52\)\( \beta_{3}) q^{61} +(-\)\(94\!\cdots\!24\)\( - \)\(89\!\cdots\!08\)\( \beta_{1} + 15001434910129364480 \beta_{2} - \)\(43\!\cdots\!40\)\( \beta_{3}) q^{62} +(\)\(24\!\cdots\!81\)\( + \)\(44\!\cdots\!59\)\( \beta_{1} + 60788327295284644005 \beta_{2} + \)\(30\!\cdots\!84\)\( \beta_{3}) q^{63} +(\)\(21\!\cdots\!64\)\( - \)\(10\!\cdots\!08\)\( \beta_{1} + 20735073056676072960 \beta_{2} + \)\(10\!\cdots\!96\)\( \beta_{3}) q^{64} +(\)\(39\!\cdots\!70\)\( + \)\(30\!\cdots\!78\)\( \beta_{1} + 42166817812091905458 \beta_{2} - \)\(13\!\cdots\!24\)\( \beta_{3}) q^{65} +(\)\(23\!\cdots\!84\)\( - \)\(14\!\cdots\!92\)\( \beta_{1} - 17643767290076759040 \beta_{2} + \)\(15\!\cdots\!84\)\( \beta_{3}) q^{66} +(\)\(40\!\cdots\!44\)\( + \)\(87\!\cdots\!32\)\( \beta_{1} - \)\(14\!\cdots\!80\)\( \beta_{2} - \)\(11\!\cdots\!24\)\( \beta_{3}) q^{67} +(\)\(15\!\cdots\!98\)\( + \)\(28\!\cdots\!54\)\( \beta_{1} - \)\(36\!\cdots\!40\)\( \beta_{2} + \)\(13\!\cdots\!98\)\( \beta_{3}) q^{68} +(-\)\(59\!\cdots\!70\)\( + \)\(64\!\cdots\!86\)\( \beta_{1} - 69081290303339018790 \beta_{2} - \)\(44\!\cdots\!48\)\( \beta_{3}) q^{69} +(\)\(20\!\cdots\!00\)\( + \)\(13\!\cdots\!60\)\( \beta_{1} + \)\(93\!\cdots\!60\)\( \beta_{2} - \)\(12\!\cdots\!80\)\( \beta_{3}) q^{70} +(\)\(27\!\cdots\!38\)\( + \)\(10\!\cdots\!26\)\( \beta_{1} + \)\(11\!\cdots\!90\)\( \beta_{2} + \)\(95\!\cdots\!68\)\( \beta_{3}) q^{71} +(\)\(52\!\cdots\!82\)\( - \)\(15\!\cdots\!18\)\( \beta_{1} - \)\(81\!\cdots\!40\)\( \beta_{2} + \)\(44\!\cdots\!38\)\( \beta_{3}) q^{72} +(-\)\(48\!\cdots\!86\)\( + \)\(22\!\cdots\!88\)\( \beta_{1} - \)\(19\!\cdots\!00\)\( \beta_{2} - \)\(24\!\cdots\!60\)\( \beta_{3}) q^{73} +(-\)\(22\!\cdots\!94\)\( - \)\(74\!\cdots\!58\)\( \beta_{1} - \)\(26\!\cdots\!20\)\( \beta_{2} - \)\(65\!\cdots\!20\)\( \beta_{3}) q^{74} +(\)\(92\!\cdots\!35\)\( - \)\(42\!\cdots\!96\)\( \beta_{1} + \)\(24\!\cdots\!44\)\( \beta_{2} - \)\(98\!\cdots\!32\)\( \beta_{3}) q^{75} +(-\)\(23\!\cdots\!00\)\( + \)\(13\!\cdots\!40\)\( \beta_{1} + \)\(44\!\cdots\!40\)\( \beta_{2} - \)\(14\!\cdots\!16\)\( \beta_{3}) q^{76} +(-\)\(64\!\cdots\!64\)\( - \)\(17\!\cdots\!28\)\( \beta_{1} + \)\(20\!\cdots\!40\)\( \beta_{2} + \)\(10\!\cdots\!00\)\( \beta_{3}) q^{77} +(\)\(41\!\cdots\!90\)\( - \)\(67\!\cdots\!18\)\( \beta_{1} - \)\(10\!\cdots\!00\)\( \beta_{2} + \)\(15\!\cdots\!96\)\( \beta_{3}) q^{78} +(\)\(10\!\cdots\!01\)\( - \)\(96\!\cdots\!09\)\( \beta_{1} - \)\(13\!\cdots\!95\)\( \beta_{2} + \)\(69\!\cdots\!28\)\( \beta_{3}) q^{79} +(-\)\(23\!\cdots\!40\)\( + \)\(55\!\cdots\!44\)\( \beta_{1} + \)\(65\!\cdots\!84\)\( \beta_{2} - \)\(49\!\cdots\!52\)\( \beta_{3}) q^{80} +\)\(22\!\cdots\!41\)\( q^{81} +(\)\(83\!\cdots\!42\)\( + \)\(19\!\cdots\!98\)\( \beta_{1} + \)\(69\!\cdots\!20\)\( \beta_{2} - \)\(12\!\cdots\!12\)\( \beta_{3}) q^{82} +(-\)\(11\!\cdots\!62\)\( - \)\(11\!\cdots\!98\)\( \beta_{1} - \)\(74\!\cdots\!50\)\( \beta_{2} - \)\(12\!\cdots\!60\)\( \beta_{3}) q^{83} +(\)\(23\!\cdots\!56\)\( + \)\(21\!\cdots\!40\)\( \beta_{1} - \)\(11\!\cdots\!00\)\( \beta_{2} + \)\(57\!\cdots\!16\)\( \beta_{3}) q^{84} +(\)\(46\!\cdots\!30\)\( - \)\(33\!\cdots\!98\)\( \beta_{1} + \)\(52\!\cdots\!22\)\( \beta_{2} - \)\(28\!\cdots\!16\)\( \beta_{3}) q^{85} +(-\)\(90\!\cdots\!12\)\( - \)\(34\!\cdots\!84\)\( \beta_{1} - \)\(34\!\cdots\!80\)\( \beta_{2} + \)\(14\!\cdots\!24\)\( \beta_{3}) q^{86} +(\)\(40\!\cdots\!41\)\( + \)\(55\!\cdots\!01\)\( \beta_{1} + \)\(91\!\cdots\!35\)\( \beta_{2} - \)\(10\!\cdots\!04\)\( \beta_{3}) q^{87} +(\)\(10\!\cdots\!84\)\( - \)\(26\!\cdots\!16\)\( \beta_{1} - \)\(66\!\cdots\!60\)\( \beta_{2} + \)\(46\!\cdots\!84\)\( \beta_{3}) q^{88} +(-\)\(78\!\cdots\!22\)\( - \)\(10\!\cdots\!72\)\( \beta_{1} + \)\(24\!\cdots\!60\)\( \beta_{2} + \)\(46\!\cdots\!00\)\( \beta_{3}) q^{89} +(-\)\(80\!\cdots\!90\)\( + \)\(73\!\cdots\!94\)\( \beta_{1} + \)\(43\!\cdots\!84\)\( \beta_{2} - \)\(23\!\cdots\!52\)\( \beta_{3}) q^{90} +(\)\(65\!\cdots\!10\)\( - \)\(71\!\cdots\!62\)\( \beta_{1} - \)\(97\!\cdots\!10\)\( \beta_{2} - \)\(44\!\cdots\!44\)\( \beta_{3}) q^{91} +(-\)\(41\!\cdots\!44\)\( + \)\(66\!\cdots\!84\)\( \beta_{1} + \)\(25\!\cdots\!60\)\( \beta_{2} - \)\(13\!\cdots\!56\)\( \beta_{3}) q^{92} +(\)\(87\!\cdots\!81\)\( + \)\(21\!\cdots\!63\)\( \beta_{1} - \)\(56\!\cdots\!75\)\( \beta_{2} - \)\(24\!\cdots\!48\)\( \beta_{3}) q^{93} +(-\)\(14\!\cdots\!48\)\( + \)\(90\!\cdots\!68\)\( \beta_{1} - \)\(58\!\cdots\!20\)\( \beta_{2} + \)\(50\!\cdots\!16\)\( \beta_{3}) q^{94} +(-\)\(22\!\cdots\!40\)\( - \)\(11\!\cdots\!76\)\( \beta_{1} - \)\(45\!\cdots\!36\)\( \beta_{2} + \)\(20\!\cdots\!08\)\( \beta_{3}) q^{95} +(\)\(35\!\cdots\!44\)\( - \)\(55\!\cdots\!44\)\( \beta_{1} + \)\(12\!\cdots\!60\)\( \beta_{2} + \)\(22\!\cdots\!96\)\( \beta_{3}) q^{96} +(\)\(11\!\cdots\!34\)\( + \)\(93\!\cdots\!72\)\( \beta_{1} + \)\(73\!\cdots\!80\)\( \beta_{2} - \)\(65\!\cdots\!64\)\( \beta_{3}) q^{97} +(-\)\(10\!\cdots\!21\)\( - \)\(11\!\cdots\!25\)\( \beta_{1} - \)\(26\!\cdots\!00\)\( \beta_{2} - \)\(47\!\cdots\!28\)\( \beta_{3}) q^{98} +(\)\(78\!\cdots\!74\)\( - \)\(38\!\cdots\!18\)\( \beta_{1} - \)\(20\!\cdots\!70\)\( \beta_{2} - \)\(40\!\cdots\!48\)\( \beta_{3}) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 437562q^{2} + 1549681956q^{3} + 346098955492q^{4} - 4099829756904q^{5} + 169520484007818q^{6} + 6605809948153184q^{7} + 140484113342159976q^{8} + 600378541187996484q^{9} + O(q^{10}) \) \( 4q + 437562q^{2} + 1549681956q^{3} + 346098955492q^{4} - 4099829756904q^{5} + 169520484007818q^{6} + 6605809948153184q^{7} + 140484113342159976q^{8} + 600378541187996484q^{9} - 21511023001649316q^{10} + 20953708852195292976q^{11} + \)\(13\!\cdots\!88\)\(q^{12} + 51830892788989874168q^{13} - \)\(18\!\cdots\!12\)\(q^{14} - \)\(15\!\cdots\!56\)\(q^{15} + \)\(55\!\cdots\!40\)\(q^{16} + \)\(81\!\cdots\!28\)\(q^{17} + \)\(65\!\cdots\!02\)\(q^{18} - \)\(54\!\cdots\!32\)\(q^{19} - \)\(35\!\cdots\!76\)\(q^{20} + \)\(25\!\cdots\!76\)\(q^{21} + \)\(23\!\cdots\!76\)\(q^{22} - \)\(61\!\cdots\!88\)\(q^{23} + \)\(54\!\cdots\!64\)\(q^{24} + \)\(95\!\cdots\!16\)\(q^{25} + \)\(42\!\cdots\!88\)\(q^{26} + \)\(23\!\cdots\!76\)\(q^{27} + \)\(24\!\cdots\!48\)\(q^{28} + \)\(41\!\cdots\!36\)\(q^{29} - \)\(83\!\cdots\!24\)\(q^{30} + \)\(89\!\cdots\!64\)\(q^{31} + \)\(36\!\cdots\!84\)\(q^{32} + \)\(81\!\cdots\!64\)\(q^{33} + \)\(31\!\cdots\!24\)\(q^{34} + \)\(42\!\cdots\!40\)\(q^{35} + \)\(51\!\cdots\!32\)\(q^{36} + \)\(55\!\cdots\!24\)\(q^{37} - \)\(73\!\cdots\!68\)\(q^{38} + \)\(20\!\cdots\!52\)\(q^{39} - \)\(26\!\cdots\!52\)\(q^{40} - \)\(86\!\cdots\!76\)\(q^{41} - \)\(70\!\cdots\!68\)\(q^{42} - \)\(50\!\cdots\!80\)\(q^{43} + \)\(28\!\cdots\!36\)\(q^{44} - \)\(61\!\cdots\!84\)\(q^{45} - \)\(14\!\cdots\!96\)\(q^{46} + \)\(42\!\cdots\!20\)\(q^{47} + \)\(21\!\cdots\!60\)\(q^{48} + \)\(40\!\cdots\!20\)\(q^{49} + \)\(10\!\cdots\!14\)\(q^{50} + \)\(31\!\cdots\!92\)\(q^{51} + \)\(18\!\cdots\!68\)\(q^{52} - \)\(12\!\cdots\!88\)\(q^{53} + \)\(25\!\cdots\!78\)\(q^{54} - \)\(32\!\cdots\!24\)\(q^{55} + \)\(49\!\cdots\!80\)\(q^{56} - \)\(21\!\cdots\!48\)\(q^{57} - \)\(75\!\cdots\!56\)\(q^{58} - \)\(13\!\cdots\!88\)\(q^{59} - \)\(13\!\cdots\!64\)\(q^{60} - \)\(12\!\cdots\!60\)\(q^{61} - \)\(37\!\cdots\!92\)\(q^{62} + \)\(99\!\cdots\!64\)\(q^{63} + \)\(85\!\cdots\!28\)\(q^{64} + \)\(15\!\cdots\!68\)\(q^{65} + \)\(92\!\cdots\!64\)\(q^{66} + \)\(16\!\cdots\!48\)\(q^{67} + \)\(63\!\cdots\!84\)\(q^{68} - \)\(23\!\cdots\!32\)\(q^{69} + \)\(82\!\cdots\!60\)\(q^{70} + \)\(10\!\cdots\!88\)\(q^{71} + \)\(21\!\cdots\!96\)\(q^{72} - \)\(19\!\cdots\!48\)\(q^{73} - \)\(89\!\cdots\!12\)\(q^{74} + \)\(36\!\cdots\!24\)\(q^{75} - \)\(95\!\cdots\!68\)\(q^{76} - \)\(25\!\cdots\!92\)\(q^{77} + \)\(16\!\cdots\!32\)\(q^{78} + \)\(42\!\cdots\!20\)\(q^{79} - \)\(95\!\cdots\!36\)\(q^{80} + \)\(90\!\cdots\!64\)\(q^{81} + \)\(33\!\cdots\!48\)\(q^{82} - \)\(46\!\cdots\!24\)\(q^{83} + \)\(95\!\cdots\!72\)\(q^{84} + \)\(18\!\cdots\!12\)\(q^{85} - \)\(36\!\cdots\!04\)\(q^{86} + \)\(16\!\cdots\!04\)\(q^{87} + \)\(42\!\cdots\!56\)\(q^{88} - \)\(31\!\cdots\!52\)\(q^{89} - \)\(32\!\cdots\!36\)\(q^{90} + \)\(26\!\cdots\!24\)\(q^{91} - \)\(16\!\cdots\!16\)\(q^{92} + \)\(34\!\cdots\!96\)\(q^{93} - \)\(57\!\cdots\!48\)\(q^{94} - \)\(89\!\cdots\!56\)\(q^{95} + \)\(14\!\cdots\!76\)\(q^{96} + \)\(44\!\cdots\!48\)\(q^{97} - \)\(43\!\cdots\!78\)\(q^{98} + \)\(31\!\cdots\!96\)\(q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 11777633936 x^{2} - 35120319927360 x + 11967042111800832000\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 6 \nu - 1 \)
\(\beta_{2}\)\(=\)\((\)\( 27 \nu^{3} - 128691 \nu^{2} - 262552030440 \nu + 46478406852080 \)\()/680\)
\(\beta_{3}\)\(=\)\( 36 \nu^{2} - 161070 \nu - 211997370590 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1} + 1\)\()/6\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} + 26845 \beta_{1} + 211997397435\)\()/36\)
\(\nu^{3}\)\(=\)\((\)\(14299 \beta_{3} + 2720 \beta_{2} + 175418543615 \beta_{1} + 2845612193201705\)\()/108\)

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
105009.
31743.2
−35434.6
−101317.
−520663. 3.87420e8 1.33651e11 −2.63441e12 −2.01716e14 8.34393e15 1.97212e15 1.50095e17 1.37164e18
1.2 −81067.1 3.87420e8 −1.30867e11 −7.16603e12 −3.14071e13 −5.96869e15 2.17508e16 1.50095e17 5.80929e17
1.3 322000. 3.87420e8 −3.37552e10 1.53382e13 1.24749e14 2.48687e15 −5.51245e16 1.50095e17 4.93889e18
1.4 717293. 3.87420e8 3.77070e11 −9.63758e12 2.77894e14 1.74370e15 1.71886e17 1.50095e17 −6.91297e18
\(n\): 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 \( T_{2}^{4} - 437562 T_{2}^{3} - 352197132768 T_{2}^{2} + \)\(95\!\cdots\!24\)\( T_{2} + \)\(97\!\cdots\!04\)\( \) acting on \(S_{38}^{\mathrm{new}}(\Gamma_0(3))\).