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

Related objects

Downloads

Learn more about

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.

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

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

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

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 \(T_{2}^{4} \) \(\mathstrut -\mathstrut 437562 T_{2}^{3} \) \(\mathstrut -\mathstrut 352197132768 T_{2}^{2} \) \(\mathstrut +\mathstrut \)\(95\!\cdots\!24\)\( T_{2} \) \(\mathstrut +\mathstrut \)\(97\!\cdots\!04\)\( \) acting on \(S_{38}^{\mathrm{new}}(\Gamma_0(3))\).