Properties

Label 3.42.a.b
Level 3
Weight 42
Character orbit 3.a
Self dual Yes
Analytic conductor 31.942
Analytic rank 0
Dimension 4
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 3 \)
Weight: \( k \) = \( 42 \)
Character orbit: \([\chi]\) = 3.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(31.9415011369\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{13}\cdot 3^{10}\cdot 5^{2} \)
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.

\(f(q)\) \(=\) \( q\) \( + ( -17455 - \beta_{1} ) q^{2} \) \( + 3486784401 q^{3} \) \( + ( 1338237117038 - 439610 \beta_{1} + \beta_{2} ) q^{4} \) \( + ( 29634266666058 - 51443969 \beta_{1} + 18 \beta_{2} - 7 \beta_{3} ) q^{5} \) \( + ( -60861821719455 - 3486784401 \beta_{1} ) q^{6} \) \( + ( 37564083753994332 - 35063520317 \beta_{1} + 15946 \beta_{2} - 4779 \beta_{3} ) q^{7} \) \( + ( 1569906771202510460 - 418345277780 \beta_{1} + 1435842 \beta_{2} - 45248 \beta_{3} ) q^{8} \) \( + 12157665459056928801 q^{9} \) \(+O(q^{10})\) \( q\) \(+(-17455 - \beta_{1}) q^{2}\) \(+3486784401 q^{3}\) \(+(1338237117038 - 439610 \beta_{1} + \beta_{2}) q^{4}\) \(+(29634266666058 - 51443969 \beta_{1} + 18 \beta_{2} - 7 \beta_{3}) q^{5}\) \(+(-60861821719455 - 3486784401 \beta_{1}) q^{6}\) \(+(37564083753994332 - 35063520317 \beta_{1} + 15946 \beta_{2} - 4779 \beta_{3}) q^{7}\) \(+(1569906771202510460 - 418345277780 \beta_{1} + 1435842 \beta_{2} - 45248 \beta_{3}) q^{8}\) \(+12157665459056928801 q^{9}\) \(+(\)\(18\!\cdots\!14\)\( - 74836662113542 \beta_{1} + 181536704 \beta_{2} + 9725184 \beta_{3}) q^{10}\) \(+(\)\(18\!\cdots\!76\)\( + 445969526243582 \beta_{1} - 1110357468 \beta_{2} - 53582606 \beta_{3}) q^{11}\) \(+(\)\(46\!\cdots\!38\)\( - 1532825290523610 \beta_{1} + 3486784401 \beta_{2}) q^{12}\) \(+(-\)\(22\!\cdots\!02\)\( + 4459831718836882 \beta_{1} - 36533853764 \beta_{2} - 137466882 \beta_{3}) q^{13}\) \(+(\)\(12\!\cdots\!88\)\( - 72341024860883240 \beta_{1} + 127523051712 \beta_{2} + 6474043648 \beta_{3}) q^{14}\) \(+(\)\(10\!\cdots\!58\)\( - 179374028634727569 \beta_{1} + 62762119218 \beta_{2} - 24407490807 \beta_{3}) q^{15}\) \(+(-\)\(14\!\cdots\!60\)\( - 2357279389979362536 \beta_{1} + 374759622948 \beta_{2} + 3159305856 \beta_{3}) q^{16}\) \(+(-\)\(95\!\cdots\!14\)\( - 6485295754548475050 \beta_{1} - 5106626451468 \beta_{2} + 63847564634 \beta_{3}) q^{17}\) \(+(-\)\(21\!\cdots\!55\)\( - 12157665459056928801 \beta_{1}) q^{18}\) \(+(\)\(65\!\cdots\!40\)\( - 81391276309975697418 \beta_{1} + 18349548635316 \beta_{2} + 1793065426362 \beta_{3}) q^{19}\) \(+(\)\(19\!\cdots\!16\)\( - \)\(29\!\cdots\!48\)\( \beta_{1} + 60280906667526 \beta_{2} - 7463869435904 \beta_{3}) q^{20}\) \(+(\)\(13\!\cdots\!32\)\( - \)\(12\!\cdots\!17\)\( \beta_{1} + 55600264058346 \beta_{2} - 16663342652379 \beta_{3}) q^{21}\) \(+(-\)\(15\!\cdots\!88\)\( + \)\(12\!\cdots\!40\)\( \beta_{1} - 693592447070336 \beta_{2} + 130918855592448 \beta_{3}) q^{22}\) \(+(-\)\(38\!\cdots\!12\)\( + \)\(31\!\cdots\!10\)\( \beta_{1} - 1690662527110188 \beta_{2} - 164397073048502 \beta_{3}) q^{23}\) \(+(\)\(54\!\cdots\!60\)\( - \)\(14\!\cdots\!80\)\( \beta_{1} + 5006471487900642 \beta_{2} - 157770020576448 \beta_{3}) q^{24}\) \(+(\)\(29\!\cdots\!15\)\( + \)\(15\!\cdots\!80\)\( \beta_{1} + 14362136572628440 \beta_{2} - 153575990203860 \beta_{3}) q^{25}\) \(+(-\)\(15\!\cdots\!06\)\( + \)\(43\!\cdots\!66\)\( \beta_{1} - 38653402731981696 \beta_{2} + 1860062750533120 \beta_{3}) q^{26}\) \(+\)\(42\!\cdots\!01\)\( q^{27}\) \(+(\)\(17\!\cdots\!40\)\( - \)\(21\!\cdots\!84\)\( \beta_{1} + 60584751373846056 \beta_{2} - 5008765360803840 \beta_{3}) q^{28}\) \(+(-\)\(25\!\cdots\!70\)\( - \)\(27\!\cdots\!39\)\( \beta_{1} - 283149395406383706 \beta_{2} - 8521223156703453 \beta_{3}) q^{29}\) \(+(\)\(63\!\cdots\!14\)\( - \)\(26\!\cdots\!42\)\( \beta_{1} + 632979347716154304 \beta_{2} + 33909619868054784 \beta_{3}) q^{30}\) \(+(\)\(23\!\cdots\!88\)\( - \)\(10\!\cdots\!89\)\( \beta_{1} + 283781381122053778 \beta_{2} - 31160787502593735 \beta_{3}) q^{31}\) \(+(\)\(49\!\cdots\!48\)\( + \)\(93\!\cdots\!24\)\( \beta_{1} - 477444389324619384 \beta_{2} + 77787427755697408 \beta_{3}) q^{32}\) \(+(\)\(63\!\cdots\!76\)\( + \)\(15\!\cdots\!82\)\( \beta_{1} - 3871577098956256668 \beta_{2} - 186830994765729006 \beta_{3}) q^{33}\) \(+(\)\(23\!\cdots\!62\)\( + \)\(12\!\cdots\!90\)\( \beta_{1} + 374885565815596416 \beta_{2} + 134931654118937088 \beta_{3}) q^{34}\) \(+(\)\(51\!\cdots\!24\)\( + \)\(87\!\cdots\!58\)\( \beta_{1} + 9946665426813493644 \beta_{2} - 235208506978322586 \beta_{3}) q^{35}\) \(+(\)\(16\!\cdots\!38\)\( - \)\(53\!\cdots\!10\)\( \beta_{1} + 12157665459056928801 \beta_{2}) q^{36}\) \(+(\)\(50\!\cdots\!82\)\( + \)\(20\!\cdots\!04\)\( \beta_{1} - 45237321564510086448 \beta_{2} + 2776647317048496360 \beta_{3}) q^{37}\) \(+(\)\(28\!\cdots\!76\)\( - \)\(12\!\cdots\!52\)\( \beta_{1} + 70941509482557361536 \beta_{2} - 3530034438768692736 \beta_{3}) q^{38}\) \(+(-\)\(77\!\cdots\!02\)\( + \)\(15\!\cdots\!82\)\( \beta_{1} - \)\(12\!\cdots\!64\)\( \beta_{2} - 479317379811707682 \beta_{3}) q^{39}\) \(+(\)\(64\!\cdots\!80\)\( - \)\(23\!\cdots\!00\)\( \beta_{1} + 75523271032104212620 \beta_{2} - 12875416735073477760 \beta_{3}) q^{40}\) \(+(\)\(58\!\cdots\!30\)\( - \)\(46\!\cdots\!38\)\( \beta_{1} - \)\(17\!\cdots\!48\)\( \beta_{2} + 27650179365112504830 \beta_{3}) q^{41}\) \(+(\)\(43\!\cdots\!88\)\( - \)\(25\!\cdots\!40\)\( \beta_{1} + \)\(44\!\cdots\!12\)\( \beta_{2} + 22573594403239534848 \beta_{3}) q^{42}\) \(+(\)\(98\!\cdots\!60\)\( + \)\(90\!\cdots\!14\)\( \beta_{1} + \)\(65\!\cdots\!00\)\( \beta_{2} - 41372446718770080354 \beta_{3}) q^{43}\) \(+(-\)\(46\!\cdots\!88\)\( + \)\(19\!\cdots\!56\)\( \beta_{1} - \)\(15\!\cdots\!32\)\( \beta_{2} - 47906740054628933632 \beta_{3}) q^{44}\) \(+(\)\(36\!\cdots\!58\)\( - \)\(62\!\cdots\!69\)\( \beta_{1} + \)\(21\!\cdots\!18\)\( \beta_{2} - 85103658213398501607 \beta_{3}) q^{45}\) \(+(-\)\(11\!\cdots\!96\)\( + \)\(70\!\cdots\!32\)\( \beta_{1} - \)\(21\!\cdots\!52\)\( \beta_{2} + \)\(32\!\cdots\!52\)\( \beta_{3}) q^{46}\) \(+(-\)\(22\!\cdots\!72\)\( - \)\(69\!\cdots\!38\)\( \beta_{1} + \)\(26\!\cdots\!96\)\( \beta_{2} - 68913360574891567378 \beta_{3}) q^{47}\) \(+(-\)\(51\!\cdots\!60\)\( - \)\(82\!\cdots\!36\)\( \beta_{1} + \)\(13\!\cdots\!48\)\( \beta_{2} + 11015818376688752256 \beta_{3}) q^{48}\) \(+(-\)\(82\!\cdots\!35\)\( + \)\(48\!\cdots\!24\)\( \beta_{1} + \)\(69\!\cdots\!52\)\( \beta_{2} - \)\(24\!\cdots\!44\)\( \beta_{3}) q^{49}\) \(+(-\)\(54\!\cdots\!05\)\( - \)\(37\!\cdots\!35\)\( \beta_{1} + \)\(16\!\cdots\!20\)\( \beta_{2} - \)\(41\!\cdots\!80\)\( \beta_{3}) q^{50}\) \(+(-\)\(33\!\cdots\!14\)\( - \)\(22\!\cdots\!50\)\( \beta_{1} - \)\(17\!\cdots\!68\)\( \beta_{2} + \)\(22\!\cdots\!34\)\( \beta_{3}) q^{51}\) \(+(-\)\(14\!\cdots\!36\)\( + \)\(68\!\cdots\!76\)\( \beta_{1} - \)\(31\!\cdots\!50\)\( \beta_{2} - \)\(74\!\cdots\!08\)\( \beta_{3}) q^{52}\) \(+(\)\(23\!\cdots\!70\)\( + \)\(31\!\cdots\!69\)\( \beta_{1} + \)\(38\!\cdots\!54\)\( \beta_{2} + \)\(46\!\cdots\!55\)\( \beta_{3}) q^{53}\) \(+(-\)\(73\!\cdots\!55\)\( - \)\(42\!\cdots\!01\)\( \beta_{1}) q^{54}\) \(+(\)\(32\!\cdots\!16\)\( + \)\(42\!\cdots\!92\)\( \beta_{1} + \)\(11\!\cdots\!16\)\( \beta_{2} - \)\(60\!\cdots\!44\)\( \beta_{3}) q^{55}\) \(+(\)\(48\!\cdots\!04\)\( - \)\(17\!\cdots\!24\)\( \beta_{1} + \)\(74\!\cdots\!32\)\( \beta_{2} - \)\(94\!\cdots\!24\)\( \beta_{3}) q^{56}\) \(+(\)\(22\!\cdots\!40\)\( - \)\(28\!\cdots\!18\)\( \beta_{1} + \)\(63\!\cdots\!16\)\( \beta_{2} + \)\(62\!\cdots\!62\)\( \beta_{3}) q^{57}\) \(+(\)\(96\!\cdots\!66\)\( + \)\(43\!\cdots\!46\)\( \beta_{1} + \)\(12\!\cdots\!08\)\( \beta_{2} + \)\(25\!\cdots\!80\)\( \beta_{3}) q^{58}\) \(+(-\)\(45\!\cdots\!32\)\( - \)\(24\!\cdots\!32\)\( \beta_{1} - \)\(11\!\cdots\!04\)\( \beta_{2} - \)\(76\!\cdots\!08\)\( \beta_{3}) q^{59}\) \(+(\)\(68\!\cdots\!16\)\( - \)\(10\!\cdots\!48\)\( \beta_{1} + \)\(21\!\cdots\!26\)\( \beta_{2} - \)\(26\!\cdots\!04\)\( \beta_{3}) q^{60}\) \(+(\)\(13\!\cdots\!30\)\( + \)\(60\!\cdots\!84\)\( \beta_{1} + \)\(66\!\cdots\!52\)\( \beta_{2} + \)\(10\!\cdots\!76\)\( \beta_{3}) q^{61}\) \(+(\)\(35\!\cdots\!60\)\( - \)\(30\!\cdots\!40\)\( \beta_{1} + \)\(17\!\cdots\!80\)\( \beta_{2} + \)\(34\!\cdots\!96\)\( \beta_{3}) q^{62}\) \(+(\)\(45\!\cdots\!32\)\( - \)\(42\!\cdots\!17\)\( \beta_{1} + \)\(19\!\cdots\!46\)\( \beta_{2} - \)\(58\!\cdots\!79\)\( \beta_{3}) q^{63}\) \(+(-\)\(97\!\cdots\!96\)\( + \)\(12\!\cdots\!88\)\( \beta_{1} - \)\(34\!\cdots\!04\)\( \beta_{2} - \)\(10\!\cdots\!92\)\( \beta_{3}) q^{64}\) \(+(-\)\(24\!\cdots\!48\)\( + \)\(79\!\cdots\!74\)\( \beta_{1} + \)\(85\!\cdots\!52\)\( \beta_{2} + \)\(49\!\cdots\!82\)\( \beta_{3}) q^{65}\) \(+(-\)\(55\!\cdots\!88\)\( + \)\(41\!\cdots\!40\)\( \beta_{1} - \)\(24\!\cdots\!36\)\( \beta_{2} + \)\(45\!\cdots\!48\)\( \beta_{3}) q^{66}\) \(+(-\)\(18\!\cdots\!48\)\( + \)\(59\!\cdots\!68\)\( \beta_{1} + \)\(43\!\cdots\!64\)\( \beta_{2} - \)\(95\!\cdots\!36\)\( \beta_{3}) q^{67}\) \(+(-\)\(22\!\cdots\!08\)\( - \)\(36\!\cdots\!28\)\( \beta_{1} - \)\(26\!\cdots\!98\)\( \beta_{2} - \)\(36\!\cdots\!68\)\( \beta_{3}) q^{68}\) \(+(-\)\(13\!\cdots\!12\)\( + \)\(10\!\cdots\!10\)\( \beta_{1} - \)\(58\!\cdots\!88\)\( \beta_{2} - \)\(57\!\cdots\!02\)\( \beta_{3}) q^{69}\) \(+(-\)\(31\!\cdots\!28\)\( - \)\(58\!\cdots\!76\)\( \beta_{1} + \)\(49\!\cdots\!32\)\( \beta_{2} - \)\(95\!\cdots\!08\)\( \beta_{3}) q^{70}\) \(+(-\)\(21\!\cdots\!68\)\( + \)\(98\!\cdots\!38\)\( \beta_{1} + \)\(31\!\cdots\!48\)\( \beta_{2} + \)\(12\!\cdots\!22\)\( \beta_{3}) q^{71}\) \(+(\)\(19\!\cdots\!60\)\( - \)\(50\!\cdots\!80\)\( \beta_{1} + \)\(17\!\cdots\!42\)\( \beta_{2} - \)\(55\!\cdots\!48\)\( \beta_{3}) q^{72}\) \(+(-\)\(11\!\cdots\!22\)\( - \)\(12\!\cdots\!92\)\( \beta_{1} - \)\(61\!\cdots\!40\)\( \beta_{2} + \)\(26\!\cdots\!20\)\( \beta_{3}) q^{73}\) \(+(-\)\(74\!\cdots\!10\)\( + \)\(90\!\cdots\!50\)\( \beta_{1} - \)\(11\!\cdots\!60\)\( \beta_{2} - \)\(21\!\cdots\!36\)\( \beta_{3}) q^{74}\) \(+(\)\(10\!\cdots\!15\)\( + \)\(52\!\cdots\!80\)\( \beta_{1} + \)\(50\!\cdots\!40\)\( \beta_{2} - \)\(53\!\cdots\!60\)\( \beta_{3}) q^{75}\) \(+(\)\(28\!\cdots\!12\)\( - \)\(24\!\cdots\!40\)\( \beta_{1} + \)\(20\!\cdots\!04\)\( \beta_{2} - \)\(18\!\cdots\!48\)\( \beta_{3}) q^{76}\) \(+(\)\(20\!\cdots\!16\)\( + \)\(29\!\cdots\!16\)\( \beta_{1} - \)\(10\!\cdots\!80\)\( \beta_{2} - \)\(51\!\cdots\!72\)\( \beta_{3}) q^{77}\) \(+(-\)\(54\!\cdots\!06\)\( + \)\(15\!\cdots\!66\)\( \beta_{1} - \)\(13\!\cdots\!96\)\( \beta_{2} + \)\(64\!\cdots\!20\)\( \beta_{3}) q^{78}\) \(+(\)\(35\!\cdots\!36\)\( + \)\(33\!\cdots\!59\)\( \beta_{1} - \)\(14\!\cdots\!62\)\( \beta_{2} - \)\(51\!\cdots\!91\)\( \beta_{3}) q^{79}\) \(+(\)\(38\!\cdots\!88\)\( - \)\(18\!\cdots\!24\)\( \beta_{1} + \)\(38\!\cdots\!08\)\( \beta_{2} + \)\(32\!\cdots\!88\)\( \beta_{3}) q^{80}\) \(+\)\(14\!\cdots\!01\)\( q^{81}\) \(+(\)\(15\!\cdots\!90\)\( - \)\(41\!\cdots\!58\)\( \beta_{1} - \)\(56\!\cdots\!08\)\( \beta_{2} - \)\(33\!\cdots\!16\)\( \beta_{3}) q^{82}\) \(+(\)\(38\!\cdots\!40\)\( - \)\(56\!\cdots\!98\)\( \beta_{1} - \)\(28\!\cdots\!40\)\( \beta_{2} - \)\(53\!\cdots\!30\)\( \beta_{3}) q^{83}\) \(+(\)\(59\!\cdots\!40\)\( - \)\(74\!\cdots\!84\)\( \beta_{1} + \)\(21\!\cdots\!56\)\( \beta_{2} - \)\(17\!\cdots\!40\)\( \beta_{3}) q^{84}\) \(+(-\)\(71\!\cdots\!56\)\( + \)\(76\!\cdots\!58\)\( \beta_{1} + \)\(11\!\cdots\!24\)\( \beta_{2} + \)\(14\!\cdots\!74\)\( \beta_{3}) q^{85}\) \(+(-\)\(32\!\cdots\!32\)\( - \)\(12\!\cdots\!04\)\( \beta_{1} + \)\(40\!\cdots\!84\)\( \beta_{2} + \)\(32\!\cdots\!56\)\( \beta_{3}) q^{86}\) \(+(-\)\(90\!\cdots\!70\)\( - \)\(94\!\cdots\!39\)\( \beta_{1} - \)\(98\!\cdots\!06\)\( \beta_{2} - \)\(29\!\cdots\!53\)\( \beta_{3}) q^{87}\) \(+(-\)\(33\!\cdots\!80\)\( + \)\(45\!\cdots\!24\)\( \beta_{1} - \)\(11\!\cdots\!24\)\( \beta_{2} - \)\(14\!\cdots\!12\)\( \beta_{3}) q^{88}\) \(+(-\)\(98\!\cdots\!30\)\( - \)\(15\!\cdots\!24\)\( \beta_{1} + \)\(80\!\cdots\!20\)\( \beta_{2} - \)\(34\!\cdots\!52\)\( \beta_{3}) q^{89}\) \(+(\)\(22\!\cdots\!14\)\( - \)\(90\!\cdots\!42\)\( \beta_{1} + \)\(22\!\cdots\!04\)\( \beta_{2} + \)\(11\!\cdots\!84\)\( \beta_{3}) q^{90}\) \(+(-\)\(22\!\cdots\!52\)\( + \)\(56\!\cdots\!94\)\( \beta_{1} + \)\(76\!\cdots\!76\)\( \beta_{2} + \)\(26\!\cdots\!02\)\( \beta_{3}) q^{91}\) \(+(-\)\(16\!\cdots\!20\)\( + \)\(98\!\cdots\!96\)\( \beta_{1} - \)\(10\!\cdots\!44\)\( \beta_{2} - \)\(26\!\cdots\!28\)\( \beta_{3}) q^{92}\) \(+(\)\(80\!\cdots\!88\)\( - \)\(35\!\cdots\!89\)\( \beta_{1} + \)\(98\!\cdots\!78\)\( \beta_{2} - \)\(10\!\cdots\!35\)\( \beta_{3}) q^{93}\) \(+(\)\(24\!\cdots\!16\)\( - \)\(37\!\cdots\!00\)\( \beta_{1} + \)\(10\!\cdots\!96\)\( \beta_{2} - \)\(14\!\cdots\!16\)\( \beta_{3}) q^{94}\) \(+(\)\(19\!\cdots\!80\)\( - \)\(19\!\cdots\!60\)\( \beta_{1} + \)\(13\!\cdots\!60\)\( \beta_{2} + \)\(38\!\cdots\!00\)\( \beta_{3}) q^{95}\) \(+(\)\(17\!\cdots\!48\)\( + \)\(32\!\cdots\!24\)\( \beta_{1} - \)\(16\!\cdots\!84\)\( \beta_{2} + \)\(27\!\cdots\!08\)\( \beta_{3}) q^{96}\) \(+(\)\(91\!\cdots\!78\)\( - \)\(20\!\cdots\!16\)\( \beta_{1} - \)\(26\!\cdots\!96\)\( \beta_{2} - \)\(68\!\cdots\!64\)\( \beta_{3}) q^{97}\) \(+(-\)\(17\!\cdots\!47\)\( + \)\(28\!\cdots\!63\)\( \beta_{1} + \)\(60\!\cdots\!68\)\( \beta_{2} + \)\(58\!\cdots\!20\)\( \beta_{3}) q^{98}\) \(+(\)\(22\!\cdots\!76\)\( + \)\(54\!\cdots\!82\)\( \beta_{1} - \)\(13\!\cdots\!68\)\( \beta_{2} - \)\(65\!\cdots\!06\)\( \beta_{3}) q^{99}\) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(4q \) \(\mathstrut -\mathstrut 69822q^{2} \) \(\mathstrut +\mathstrut 13947137604q^{3} \) \(\mathstrut +\mathstrut 5352947588932q^{4} \) \(\mathstrut +\mathstrut 118536963776280q^{5} \) \(\mathstrut -\mathstrut 243454260446622q^{6} \) \(\mathstrut +\mathstrut 150256264888927136q^{7} \) \(\mathstrut +\mathstrut 6279626248119395784q^{8} \) \(\mathstrut +\mathstrut 48630661836227715204q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(4q \) \(\mathstrut -\mathstrut 69822q^{2} \) \(\mathstrut +\mathstrut 13947137604q^{3} \) \(\mathstrut +\mathstrut 5352947588932q^{4} \) \(\mathstrut +\mathstrut 118536963776280q^{5} \) \(\mathstrut -\mathstrut 243454260446622q^{6} \) \(\mathstrut +\mathstrut 150256264888927136q^{7} \) \(\mathstrut +\mathstrut 6279626248119395784q^{8} \) \(\mathstrut +\mathstrut 48630661836227715204q^{9} \) \(\mathstrut +\mathstrut \)\(72\!\cdots\!40\)\(q^{10} \) \(\mathstrut +\mathstrut \)\(72\!\cdots\!56\)\(q^{11} \) \(\mathstrut +\mathstrut \)\(18\!\cdots\!32\)\(q^{12} \) \(\mathstrut -\mathstrut \)\(88\!\cdots\!08\)\(q^{13} \) \(\mathstrut +\mathstrut \)\(49\!\cdots\!68\)\(q^{14} \) \(\mathstrut +\mathstrut \)\(41\!\cdots\!80\)\(q^{15} \) \(\mathstrut -\mathstrut \)\(59\!\cdots\!00\)\(q^{16} \) \(\mathstrut -\mathstrut \)\(38\!\cdots\!88\)\(q^{17} \) \(\mathstrut -\mathstrut \)\(84\!\cdots\!22\)\(q^{18} \) \(\mathstrut +\mathstrut \)\(26\!\cdots\!48\)\(q^{19} \) \(\mathstrut +\mathstrut \)\(78\!\cdots\!60\)\(q^{20} \) \(\mathstrut +\mathstrut \)\(52\!\cdots\!36\)\(q^{21} \) \(\mathstrut -\mathstrut \)\(63\!\cdots\!76\)\(q^{22} \) \(\mathstrut -\mathstrut \)\(15\!\cdots\!32\)\(q^{23} \) \(\mathstrut +\mathstrut \)\(21\!\cdots\!84\)\(q^{24} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!00\)\(q^{25} \) \(\mathstrut -\mathstrut \)\(62\!\cdots\!52\)\(q^{26} \) \(\mathstrut +\mathstrut \)\(16\!\cdots\!04\)\(q^{27} \) \(\mathstrut +\mathstrut \)\(68\!\cdots\!12\)\(q^{28} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!64\)\(q^{29} \) \(\mathstrut +\mathstrut \)\(25\!\cdots\!40\)\(q^{30} \) \(\mathstrut +\mathstrut \)\(92\!\cdots\!04\)\(q^{31} \) \(\mathstrut +\mathstrut \)\(19\!\cdots\!56\)\(q^{32} \) \(\mathstrut +\mathstrut \)\(25\!\cdots\!56\)\(q^{33} \) \(\mathstrut +\mathstrut \)\(92\!\cdots\!04\)\(q^{34} \) \(\mathstrut +\mathstrut \)\(20\!\cdots\!40\)\(q^{35} \) \(\mathstrut +\mathstrut \)\(65\!\cdots\!32\)\(q^{36} \) \(\mathstrut +\mathstrut \)\(20\!\cdots\!56\)\(q^{37} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!28\)\(q^{38} \) \(\mathstrut -\mathstrut \)\(30\!\cdots\!08\)\(q^{39} \) \(\mathstrut +\mathstrut \)\(25\!\cdots\!00\)\(q^{40} \) \(\mathstrut +\mathstrut \)\(23\!\cdots\!04\)\(q^{41} \) \(\mathstrut +\mathstrut \)\(17\!\cdots\!68\)\(q^{42} \) \(\mathstrut +\mathstrut \)\(39\!\cdots\!60\)\(q^{43} \) \(\mathstrut -\mathstrut \)\(18\!\cdots\!04\)\(q^{44} \) \(\mathstrut +\mathstrut \)\(14\!\cdots\!80\)\(q^{45} \) \(\mathstrut -\mathstrut \)\(44\!\cdots\!16\)\(q^{46} \) \(\mathstrut -\mathstrut \)\(88\!\cdots\!20\)\(q^{47} \) \(\mathstrut -\mathstrut \)\(20\!\cdots\!00\)\(q^{48} \) \(\mathstrut -\mathstrut \)\(33\!\cdots\!80\)\(q^{49} \) \(\mathstrut -\mathstrut \)\(21\!\cdots\!50\)\(q^{50} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!88\)\(q^{51} \) \(\mathstrut -\mathstrut \)\(59\!\cdots\!08\)\(q^{52} \) \(\mathstrut +\mathstrut \)\(95\!\cdots\!28\)\(q^{53} \) \(\mathstrut -\mathstrut \)\(29\!\cdots\!22\)\(q^{54} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!60\)\(q^{55} \) \(\mathstrut +\mathstrut \)\(19\!\cdots\!20\)\(q^{56} \) \(\mathstrut +\mathstrut \)\(91\!\cdots\!48\)\(q^{57} \) \(\mathstrut +\mathstrut \)\(38\!\cdots\!16\)\(q^{58} \) \(\mathstrut -\mathstrut \)\(18\!\cdots\!08\)\(q^{59} \) \(\mathstrut +\mathstrut \)\(27\!\cdots\!60\)\(q^{60} \) \(\mathstrut +\mathstrut \)\(53\!\cdots\!40\)\(q^{61} \) \(\mathstrut +\mathstrut \)\(14\!\cdots\!52\)\(q^{62} \) \(\mathstrut +\mathstrut \)\(18\!\cdots\!36\)\(q^{63} \) \(\mathstrut -\mathstrut \)\(38\!\cdots\!92\)\(q^{64} \) \(\mathstrut -\mathstrut \)\(97\!\cdots\!80\)\(q^{65} \) \(\mathstrut -\mathstrut \)\(22\!\cdots\!76\)\(q^{66} \) \(\mathstrut -\mathstrut \)\(73\!\cdots\!28\)\(q^{67} \) \(\mathstrut -\mathstrut \)\(89\!\cdots\!24\)\(q^{68} \) \(\mathstrut -\mathstrut \)\(53\!\cdots\!32\)\(q^{69} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!80\)\(q^{70} \) \(\mathstrut -\mathstrut \)\(84\!\cdots\!52\)\(q^{71} \) \(\mathstrut +\mathstrut \)\(76\!\cdots\!84\)\(q^{72} \) \(\mathstrut -\mathstrut \)\(44\!\cdots\!32\)\(q^{73} \) \(\mathstrut -\mathstrut \)\(29\!\cdots\!12\)\(q^{74} \) \(\mathstrut +\mathstrut \)\(40\!\cdots\!00\)\(q^{75} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!72\)\(q^{76} \) \(\mathstrut +\mathstrut \)\(83\!\cdots\!52\)\(q^{77} \) \(\mathstrut -\mathstrut \)\(21\!\cdots\!52\)\(q^{78} \) \(\mathstrut +\mathstrut \)\(14\!\cdots\!80\)\(q^{79} \) \(\mathstrut +\mathstrut \)\(15\!\cdots\!80\)\(q^{80} \) \(\mathstrut +\mathstrut \)\(59\!\cdots\!04\)\(q^{81} \) \(\mathstrut +\mathstrut \)\(61\!\cdots\!12\)\(q^{82} \) \(\mathstrut +\mathstrut \)\(15\!\cdots\!04\)\(q^{83} \) \(\mathstrut +\mathstrut \)\(23\!\cdots\!12\)\(q^{84} \) \(\mathstrut -\mathstrut \)\(28\!\cdots\!60\)\(q^{85} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!24\)\(q^{86} \) \(\mathstrut -\mathstrut \)\(36\!\cdots\!64\)\(q^{87} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!96\)\(q^{88} \) \(\mathstrut -\mathstrut \)\(39\!\cdots\!72\)\(q^{89} \) \(\mathstrut +\mathstrut \)\(88\!\cdots\!40\)\(q^{90} \) \(\mathstrut -\mathstrut \)\(88\!\cdots\!16\)\(q^{91} \) \(\mathstrut -\mathstrut \)\(65\!\cdots\!44\)\(q^{92} \) \(\mathstrut +\mathstrut \)\(32\!\cdots\!04\)\(q^{93} \) \(\mathstrut +\mathstrut \)\(98\!\cdots\!32\)\(q^{94} \) \(\mathstrut +\mathstrut \)\(78\!\cdots\!00\)\(q^{95} \) \(\mathstrut +\mathstrut \)\(68\!\cdots\!56\)\(q^{96} \) \(\mathstrut +\mathstrut \)\(36\!\cdots\!52\)\(q^{97} \) \(\mathstrut -\mathstrut \)\(68\!\cdots\!22\)\(q^{98} \) \(\mathstrut +\mathstrut \)\(88\!\cdots\!56\)\(q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4}\mathstrut -\mathstrut \) \(x^{3}\mathstrut -\mathstrut \) \(196497525461\) \(x^{2}\mathstrut +\mathstrut \) \(10360343667016365\) \(x\mathstrut +\mathstrut \) \(6095744045744274504000\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 6 \nu - 1 \)
\(\beta_{2}\)\(=\)\( 36 \nu^{2} + 2847108 \nu - 3536956170084 \)
\(\beta_{3}\)\(=\)\((\)\( 27 \nu^{3} + 6696918 \nu^{2} - 3100608741285 \nu - 448170152274909880 \)\()/5656\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1}\mathstrut +\mathstrut \) \(1\)\()/6\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{2}\mathstrut -\mathstrut \) \(474518\) \(\beta_{1}\mathstrut +\mathstrut \) \(3536955695566\)\()/36\)
\(\nu^{3}\)\(=\)\((\)\(11312\) \(\beta_{3}\mathstrut -\mathstrut \) \(372051\) \(\beta_{2}\mathstrut +\mathstrut \) \(1210081143513\) \(\beta_{1}\mathstrut -\mathstrut \) \(419586565404959011\)\()/54\)

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
330995.
263663.
−161109.
−433547.
−2.00342e6 3.48678e9 1.81468e12 −3.43221e14 −6.98551e15 −2.11939e17 7.69998e17 1.21577e19 6.87617e20
1.2 −1.59943e6 3.48678e9 3.59152e11 3.20915e14 −5.57687e15 2.35480e17 2.94274e18 1.21577e19 −5.13282e20
1.3 949201. 3.48678e9 −1.29804e12 −1.14706e14 3.30966e15 −7.22302e16 −3.31942e18 1.21577e19 −1.08879e20
1.4 2.58383e6 3.48678e9 4.47715e12 2.55548e14 9.00926e15 1.98945e17 5.88630e18 1.21577e19 6.60294e20
\(n\): 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 \(T_{2}^{4} \) \(\mathstrut +\mathstrut 69822 T_{2}^{3} \) \(\mathstrut -\mathstrut \)\(70\!\cdots\!28\)\( T_{2}^{2} \) \(\mathstrut -\mathstrut \)\(24\!\cdots\!64\)\( T_{2} \) \(\mathstrut +\mathstrut \)\(78\!\cdots\!44\)\( \) acting on \(S_{42}^{\mathrm{new}}(\Gamma_0(3))\).