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: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Defining polynomial: \(x^{4} - x^{3} - 196497525461 x^{2} + 10360343667016365 x + 6095744045744274504000\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{13}\cdot 3^{10}\cdot 5^{2} \)
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.

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

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

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

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:

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 3.42.a.b 4
3.b odd 2 1 9.42.a.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.42.a.b 4 1.a even 1 1 trivial
9.42.a.c 4 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 69822 T_{2}^{3} - \)7072082749728

'>\(70\!\cdots\!28\)\( T_{2}^{2} - \)2484749039974539264
'>\(24\!\cdots\!64\)\( T_{2} + \)7858870206246628249042944'>\(78\!\cdots\!44\)\( \) acting on \(S_{42}^{\mathrm{new}}(\Gamma_0(3))\).