Properties

Label 2.74.a.b
Level 2
Weight 74
Character orbit 2.a
Self dual yes
Analytic conductor 67.497
Analytic rank 0
Dimension 4
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 2 \)
Weight: \( k \) \(=\) \( 74 \)
Character orbit: \([\chi]\) \(=\) 2.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(67.4967947474\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{45}\cdot 3^{14}\cdot 5^{5}\cdot 7^{2}\cdot 11 \)
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.

\(f(q)\) \(=\) \( q +68719476736 q^{2} +(76266581430811044 - \beta_{1}) q^{3} +\)\(47\!\cdots\!96\)\( q^{4} +(-\)\(19\!\cdots\!90\)\( + 19575108 \beta_{1} - \beta_{2}) q^{5} +(\)\(52\!\cdots\!84\)\( - 68719476736 \beta_{1}) q^{6} +(\)\(90\!\cdots\!28\)\( - 1043925364863 \beta_{1} - 59221 \beta_{2} - 3 \beta_{3}) q^{7} +\)\(32\!\cdots\!56\)\( q^{8} +(\)\(30\!\cdots\!13\)\( - 63252985212755460 \beta_{1} + 965244438 \beta_{2} - 22916 \beta_{3}) q^{9} +O(q^{10})\) \( q +68719476736 q^{2} +(76266581430811044 - \beta_{1}) q^{3} +\)\(47\!\cdots\!96\)\( q^{4} +(-\)\(19\!\cdots\!90\)\( + 19575108 \beta_{1} - \beta_{2}) q^{5} +(\)\(52\!\cdots\!84\)\( - 68719476736 \beta_{1}) q^{6} +(\)\(90\!\cdots\!28\)\( - 1043925364863 \beta_{1} - 59221 \beta_{2} - 3 \beta_{3}) q^{7} +\)\(32\!\cdots\!56\)\( q^{8} +(\)\(30\!\cdots\!13\)\( - 63252985212755460 \beta_{1} + 965244438 \beta_{2} - 22916 \beta_{3}) q^{9} +(-\)\(13\!\cdots\!40\)\( + 1345191178810687488 \beta_{1} - 68719476736 \beta_{2}) q^{10} +(\)\(13\!\cdots\!12\)\( - \)\(15\!\cdots\!61\)\( \beta_{1} + 107182329902 \beta_{2} - 39941214 \beta_{3}) q^{11} +(\)\(36\!\cdots\!24\)\( - \)\(47\!\cdots\!96\)\( \beta_{1}) q^{12} +(\)\(33\!\cdots\!94\)\( - \)\(30\!\cdots\!16\)\( \beta_{1} - 763803984604529 \beta_{2} + 18201073128 \beta_{3}) q^{13} +(\)\(62\!\cdots\!08\)\( - \)\(71\!\cdots\!68\)\( \beta_{1} - 4069636131782656 \beta_{2} - 206158430208 \beta_{3}) q^{14} +(-\)\(19\!\cdots\!60\)\( + \)\(21\!\cdots\!87\)\( \beta_{1} - 321977322694328139 \beta_{2} + 303823613475 \beta_{3}) q^{15} +\)\(22\!\cdots\!16\)\( q^{16} +(\)\(23\!\cdots\!38\)\( - \)\(10\!\cdots\!36\)\( \beta_{1} + 5025317190866718262 \beta_{2} + 133258150737516 \beta_{3}) q^{17} +(\)\(21\!\cdots\!68\)\( - \)\(43\!\cdots\!60\)\( \beta_{1} + 66331092701694394368 \beta_{2} - 1574775528882176 \beta_{3}) q^{18} +(\)\(23\!\cdots\!00\)\( + \)\(29\!\cdots\!25\)\( \beta_{1} + \)\(61\!\cdots\!18\)\( \beta_{2} + 5689296285428574 \beta_{3}) q^{19} +(-\)\(92\!\cdots\!40\)\( + \)\(92\!\cdots\!68\)\( \beta_{1} - \)\(47\!\cdots\!96\)\( \beta_{2}) q^{20} +(\)\(16\!\cdots\!32\)\( - \)\(41\!\cdots\!44\)\( \beta_{1} - \)\(18\!\cdots\!48\)\( \beta_{2} + 151312254540761736 \beta_{3}) q^{21} +(\)\(90\!\cdots\!32\)\( - \)\(10\!\cdots\!96\)\( \beta_{1} + \)\(73\!\cdots\!72\)\( \beta_{2} - 2744739326280597504 \beta_{3}) q^{22} +(-\)\(16\!\cdots\!76\)\( - \)\(11\!\cdots\!17\)\( \beta_{1} + \)\(91\!\cdots\!09\)\( \beta_{2} + 14198855229780916587 \beta_{3}) q^{23} +(\)\(24\!\cdots\!64\)\( - \)\(32\!\cdots\!56\)\( \beta_{1}) q^{24} +(\)\(86\!\cdots\!75\)\( - \)\(54\!\cdots\!80\)\( \beta_{1} - \)\(11\!\cdots\!40\)\( \beta_{2} - \)\(31\!\cdots\!00\)\( \beta_{3}) q^{25} +(\)\(23\!\cdots\!84\)\( - \)\(20\!\cdots\!76\)\( \beta_{1} - \)\(52\!\cdots\!44\)\( \beta_{2} + \)\(12\!\cdots\!08\)\( \beta_{3}) q^{26} +(\)\(30\!\cdots\!60\)\( - \)\(15\!\cdots\!76\)\( \beta_{1} + \)\(34\!\cdots\!06\)\( \beta_{2} + 94268353549554494058 \beta_{3}) q^{27} +(\)\(42\!\cdots\!88\)\( - \)\(49\!\cdots\!48\)\( \beta_{1} - \)\(27\!\cdots\!16\)\( \beta_{2} - \)\(14\!\cdots\!88\)\( \beta_{3}) q^{28} +(-\)\(10\!\cdots\!10\)\( + \)\(33\!\cdots\!52\)\( \beta_{1} + \)\(27\!\cdots\!43\)\( \beta_{2} + \)\(34\!\cdots\!24\)\( \beta_{3}) q^{29} +(-\)\(13\!\cdots\!60\)\( + \)\(14\!\cdots\!32\)\( \beta_{1} - \)\(22\!\cdots\!04\)\( \beta_{2} + \)\(20\!\cdots\!00\)\( \beta_{3}) q^{30} +(\)\(94\!\cdots\!52\)\( + \)\(80\!\cdots\!76\)\( \beta_{1} + \)\(36\!\cdots\!32\)\( \beta_{2} - \)\(14\!\cdots\!24\)\( \beta_{3}) q^{31} +\)\(15\!\cdots\!76\)\( q^{32} +(\)\(15\!\cdots\!28\)\( - \)\(72\!\cdots\!92\)\( \beta_{1} + \)\(16\!\cdots\!22\)\( \beta_{2} - \)\(11\!\cdots\!04\)\( \beta_{3}) q^{33} +(\)\(16\!\cdots\!68\)\( - \)\(72\!\cdots\!96\)\( \beta_{1} + \)\(34\!\cdots\!32\)\( \beta_{2} + \)\(91\!\cdots\!76\)\( \beta_{3}) q^{34} +(\)\(10\!\cdots\!80\)\( - \)\(19\!\cdots\!56\)\( \beta_{1} - \)\(52\!\cdots\!68\)\( \beta_{2} - \)\(37\!\cdots\!00\)\( \beta_{3}) q^{35} +(\)\(14\!\cdots\!48\)\( - \)\(29\!\cdots\!60\)\( \beta_{1} + \)\(45\!\cdots\!48\)\( \beta_{2} - \)\(10\!\cdots\!36\)\( \beta_{3}) q^{36} +(\)\(17\!\cdots\!38\)\( + \)\(81\!\cdots\!44\)\( \beta_{1} + \)\(17\!\cdots\!27\)\( \beta_{2} + \)\(24\!\cdots\!36\)\( \beta_{3}) q^{37} +(\)\(16\!\cdots\!00\)\( + \)\(20\!\cdots\!00\)\( \beta_{1} + \)\(42\!\cdots\!48\)\( \beta_{2} + \)\(39\!\cdots\!64\)\( \beta_{3}) q^{38} +(\)\(28\!\cdots\!36\)\( + \)\(42\!\cdots\!83\)\( \beta_{1} - \)\(19\!\cdots\!75\)\( \beta_{2} - \)\(19\!\cdots\!25\)\( \beta_{3}) q^{39} +(-\)\(63\!\cdots\!40\)\( + \)\(63\!\cdots\!48\)\( \beta_{1} - \)\(32\!\cdots\!56\)\( \beta_{2}) q^{40} +(\)\(14\!\cdots\!62\)\( - \)\(16\!\cdots\!80\)\( \beta_{1} + \)\(14\!\cdots\!64\)\( \beta_{2} + \)\(51\!\cdots\!52\)\( \beta_{3}) q^{41} +(\)\(11\!\cdots\!52\)\( - \)\(28\!\cdots\!84\)\( \beta_{1} - \)\(12\!\cdots\!28\)\( \beta_{2} + \)\(10\!\cdots\!96\)\( \beta_{3}) q^{42} +(-\)\(48\!\cdots\!76\)\( - \)\(26\!\cdots\!19\)\( \beta_{1} + \)\(33\!\cdots\!84\)\( \beta_{2} - \)\(18\!\cdots\!88\)\( \beta_{3}) q^{43} +(\)\(62\!\cdots\!52\)\( - \)\(74\!\cdots\!56\)\( \beta_{1} + \)\(50\!\cdots\!92\)\( \beta_{2} - \)\(18\!\cdots\!44\)\( \beta_{3}) q^{44} +(-\)\(19\!\cdots\!70\)\( + \)\(71\!\cdots\!84\)\( \beta_{1} - \)\(50\!\cdots\!73\)\( \beta_{2} + \)\(42\!\cdots\!00\)\( \beta_{3}) q^{45} +(-\)\(11\!\cdots\!36\)\( - \)\(76\!\cdots\!12\)\( \beta_{1} + \)\(62\!\cdots\!24\)\( \beta_{2} + \)\(97\!\cdots\!32\)\( \beta_{3}) q^{46} +(-\)\(67\!\cdots\!72\)\( + \)\(29\!\cdots\!06\)\( \beta_{1} + \)\(11\!\cdots\!78\)\( \beta_{2} - \)\(41\!\cdots\!46\)\( \beta_{3}) q^{47} +(\)\(17\!\cdots\!04\)\( - \)\(22\!\cdots\!16\)\( \beta_{1}) q^{48} +(\)\(12\!\cdots\!77\)\( + \)\(52\!\cdots\!84\)\( \beta_{1} - \)\(42\!\cdots\!24\)\( \beta_{2} + \)\(18\!\cdots\!68\)\( \beta_{3}) q^{49} +(\)\(59\!\cdots\!00\)\( - \)\(37\!\cdots\!80\)\( \beta_{1} - \)\(82\!\cdots\!40\)\( \beta_{2} - \)\(21\!\cdots\!00\)\( \beta_{3}) q^{50} +(\)\(11\!\cdots\!72\)\( - \)\(12\!\cdots\!88\)\( \beta_{1} + \)\(26\!\cdots\!42\)\( \beta_{2} - \)\(31\!\cdots\!94\)\( \beta_{3}) q^{51} +(\)\(16\!\cdots\!24\)\( - \)\(14\!\cdots\!36\)\( \beta_{1} - \)\(36\!\cdots\!84\)\( \beta_{2} + \)\(85\!\cdots\!88\)\( \beta_{3}) q^{52} +(\)\(11\!\cdots\!54\)\( + \)\(29\!\cdots\!20\)\( \beta_{1} + \)\(72\!\cdots\!47\)\( \beta_{2} - \)\(60\!\cdots\!04\)\( \beta_{3}) q^{53} +(\)\(20\!\cdots\!60\)\( - \)\(10\!\cdots\!36\)\( \beta_{1} + \)\(23\!\cdots\!16\)\( \beta_{2} + \)\(64\!\cdots\!88\)\( \beta_{3}) q^{54} +(-\)\(51\!\cdots\!80\)\( + \)\(47\!\cdots\!01\)\( \beta_{1} - \)\(89\!\cdots\!97\)\( \beta_{2} + \)\(26\!\cdots\!25\)\( \beta_{3}) q^{55} +(\)\(29\!\cdots\!68\)\( - \)\(33\!\cdots\!28\)\( \beta_{1} - \)\(19\!\cdots\!76\)\( \beta_{2} - \)\(97\!\cdots\!68\)\( \beta_{3}) q^{56} +(-\)\(89\!\cdots\!00\)\( - \)\(27\!\cdots\!36\)\( \beta_{1} + \)\(16\!\cdots\!86\)\( \beta_{2} + \)\(41\!\cdots\!48\)\( \beta_{3}) q^{57} +(-\)\(70\!\cdots\!60\)\( + \)\(22\!\cdots\!72\)\( \beta_{1} + \)\(18\!\cdots\!48\)\( \beta_{2} + \)\(23\!\cdots\!64\)\( \beta_{3}) q^{58} +(-\)\(90\!\cdots\!20\)\( - \)\(12\!\cdots\!51\)\( \beta_{1} + \)\(15\!\cdots\!96\)\( \beta_{2} - \)\(20\!\cdots\!72\)\( \beta_{3}) q^{59} +(-\)\(92\!\cdots\!60\)\( + \)\(10\!\cdots\!52\)\( \beta_{1} - \)\(15\!\cdots\!44\)\( \beta_{2} + \)\(14\!\cdots\!00\)\( \beta_{3}) q^{60} +(\)\(88\!\cdots\!02\)\( + \)\(31\!\cdots\!12\)\( \beta_{1} - \)\(48\!\cdots\!41\)\( \beta_{2} - \)\(12\!\cdots\!88\)\( \beta_{3}) q^{61} +(\)\(65\!\cdots\!72\)\( + \)\(55\!\cdots\!36\)\( \beta_{1} + \)\(24\!\cdots\!52\)\( \beta_{2} - \)\(10\!\cdots\!64\)\( \beta_{3}) q^{62} +(\)\(33\!\cdots\!64\)\( + \)\(54\!\cdots\!09\)\( \beta_{1} + \)\(25\!\cdots\!59\)\( \beta_{2} + \)\(94\!\cdots\!37\)\( \beta_{3}) q^{63} +\)\(10\!\cdots\!36\)\( q^{64} +(\)\(13\!\cdots\!40\)\( - \)\(35\!\cdots\!28\)\( \beta_{1} - \)\(42\!\cdots\!84\)\( \beta_{2} - \)\(31\!\cdots\!00\)\( \beta_{3}) q^{65} +(\)\(10\!\cdots\!08\)\( - \)\(49\!\cdots\!12\)\( \beta_{1} + \)\(11\!\cdots\!92\)\( \beta_{2} - \)\(78\!\cdots\!44\)\( \beta_{3}) q^{66} +(\)\(62\!\cdots\!68\)\( + \)\(61\!\cdots\!37\)\( \beta_{1} - \)\(90\!\cdots\!82\)\( \beta_{2} + \)\(95\!\cdots\!74\)\( \beta_{3}) q^{67} +(\)\(11\!\cdots\!48\)\( - \)\(49\!\cdots\!56\)\( \beta_{1} + \)\(23\!\cdots\!52\)\( \beta_{2} + \)\(62\!\cdots\!36\)\( \beta_{3}) q^{68} +(\)\(90\!\cdots\!56\)\( + \)\(21\!\cdots\!36\)\( \beta_{1} + \)\(39\!\cdots\!64\)\( \beta_{2} - \)\(32\!\cdots\!48\)\( \beta_{3}) q^{69} +(\)\(74\!\cdots\!80\)\( - \)\(13\!\cdots\!16\)\( \beta_{1} - \)\(35\!\cdots\!48\)\( \beta_{2} - \)\(25\!\cdots\!00\)\( \beta_{3}) q^{70} +(\)\(15\!\cdots\!32\)\( + \)\(74\!\cdots\!25\)\( \beta_{1} - \)\(19\!\cdots\!73\)\( \beta_{2} + \)\(61\!\cdots\!61\)\( \beta_{3}) q^{71} +(\)\(99\!\cdots\!28\)\( - \)\(20\!\cdots\!60\)\( \beta_{1} + \)\(31\!\cdots\!28\)\( \beta_{2} - \)\(74\!\cdots\!96\)\( \beta_{3}) q^{72} +(\)\(11\!\cdots\!34\)\( - \)\(44\!\cdots\!60\)\( \beta_{1} - \)\(12\!\cdots\!98\)\( \beta_{2} + \)\(12\!\cdots\!36\)\( \beta_{3}) q^{73} +(\)\(11\!\cdots\!68\)\( + \)\(56\!\cdots\!84\)\( \beta_{1} + \)\(11\!\cdots\!72\)\( \beta_{2} + \)\(17\!\cdots\!96\)\( \beta_{3}) q^{74} +(\)\(56\!\cdots\!00\)\( - \)\(10\!\cdots\!95\)\( \beta_{1} + \)\(14\!\cdots\!40\)\( \beta_{2} - \)\(10\!\cdots\!00\)\( \beta_{3}) q^{75} +(\)\(11\!\cdots\!00\)\( + \)\(13\!\cdots\!00\)\( \beta_{1} + \)\(29\!\cdots\!28\)\( \beta_{2} + \)\(26\!\cdots\!04\)\( \beta_{3}) q^{76} +(\)\(73\!\cdots\!36\)\( + \)\(35\!\cdots\!44\)\( \beta_{1} - \)\(26\!\cdots\!28\)\( \beta_{2} + \)\(29\!\cdots\!96\)\( \beta_{3}) q^{77} +(\)\(19\!\cdots\!96\)\( + \)\(28\!\cdots\!88\)\( \beta_{1} - \)\(13\!\cdots\!00\)\( \beta_{2} - \)\(13\!\cdots\!00\)\( \beta_{3}) q^{78} +(\)\(50\!\cdots\!60\)\( + \)\(20\!\cdots\!78\)\( \beta_{1} + \)\(61\!\cdots\!66\)\( \beta_{2} - \)\(56\!\cdots\!62\)\( \beta_{3}) q^{79} +(-\)\(43\!\cdots\!40\)\( + \)\(43\!\cdots\!28\)\( \beta_{1} - \)\(22\!\cdots\!16\)\( \beta_{2}) q^{80} +(-\)\(42\!\cdots\!59\)\( - \)\(51\!\cdots\!08\)\( \beta_{1} + \)\(53\!\cdots\!26\)\( \beta_{2} + \)\(12\!\cdots\!68\)\( \beta_{3}) q^{81} +(\)\(97\!\cdots\!32\)\( - \)\(11\!\cdots\!80\)\( \beta_{1} + \)\(99\!\cdots\!04\)\( \beta_{2} + \)\(35\!\cdots\!72\)\( \beta_{3}) q^{82} +(\)\(39\!\cdots\!04\)\( - \)\(37\!\cdots\!73\)\( \beta_{1} - \)\(12\!\cdots\!68\)\( \beta_{2} - \)\(26\!\cdots\!24\)\( \beta_{3}) q^{83} +(\)\(78\!\cdots\!72\)\( - \)\(19\!\cdots\!24\)\( \beta_{1} - \)\(86\!\cdots\!08\)\( \beta_{2} + \)\(71\!\cdots\!56\)\( \beta_{3}) q^{84} +(-\)\(11\!\cdots\!20\)\( + \)\(48\!\cdots\!84\)\( \beta_{1} - \)\(27\!\cdots\!98\)\( \beta_{2} + \)\(12\!\cdots\!00\)\( \beta_{3}) q^{85} +(-\)\(33\!\cdots\!36\)\( - \)\(18\!\cdots\!84\)\( \beta_{1} + \)\(22\!\cdots\!24\)\( \beta_{2} - \)\(12\!\cdots\!68\)\( \beta_{3}) q^{86} +(-\)\(38\!\cdots\!40\)\( + \)\(94\!\cdots\!31\)\( \beta_{1} + \)\(52\!\cdots\!85\)\( \beta_{2} + \)\(59\!\cdots\!55\)\( \beta_{3}) q^{87} +(\)\(42\!\cdots\!72\)\( - \)\(51\!\cdots\!16\)\( \beta_{1} + \)\(34\!\cdots\!12\)\( \beta_{2} - \)\(12\!\cdots\!84\)\( \beta_{3}) q^{88} +(-\)\(61\!\cdots\!10\)\( + \)\(47\!\cdots\!00\)\( \beta_{1} + \)\(18\!\cdots\!90\)\( \beta_{2} + \)\(12\!\cdots\!20\)\( \beta_{3}) q^{89} +(-\)\(13\!\cdots\!20\)\( + \)\(49\!\cdots\!24\)\( \beta_{1} - \)\(34\!\cdots\!28\)\( \beta_{2} + \)\(29\!\cdots\!00\)\( \beta_{3}) q^{90} +(-\)\(23\!\cdots\!68\)\( - \)\(79\!\cdots\!08\)\( \beta_{1} - \)\(45\!\cdots\!84\)\( \beta_{2} - \)\(11\!\cdots\!12\)\( \beta_{3}) q^{91} +(-\)\(79\!\cdots\!96\)\( - \)\(52\!\cdots\!32\)\( \beta_{1} + \)\(43\!\cdots\!64\)\( \beta_{2} + \)\(67\!\cdots\!52\)\( \beta_{3}) q^{92} +(-\)\(66\!\cdots\!12\)\( - \)\(19\!\cdots\!00\)\( \beta_{1} + \)\(31\!\cdots\!76\)\( \beta_{2} + \)\(19\!\cdots\!68\)\( \beta_{3}) q^{93} +(-\)\(46\!\cdots\!92\)\( + \)\(20\!\cdots\!16\)\( \beta_{1} + \)\(76\!\cdots\!08\)\( \beta_{2} - \)\(28\!\cdots\!56\)\( \beta_{3}) q^{94} +(-\)\(11\!\cdots\!00\)\( + \)\(28\!\cdots\!55\)\( \beta_{1} + \)\(50\!\cdots\!65\)\( \beta_{2} + \)\(15\!\cdots\!75\)\( \beta_{3}) q^{95} +(\)\(11\!\cdots\!44\)\( - \)\(15\!\cdots\!76\)\( \beta_{1}) q^{96} +(\)\(14\!\cdots\!58\)\( + \)\(72\!\cdots\!88\)\( \beta_{1} - \)\(40\!\cdots\!82\)\( \beta_{2} - \)\(89\!\cdots\!76\)\( \beta_{3}) q^{97} +(\)\(84\!\cdots\!72\)\( + \)\(35\!\cdots\!24\)\( \beta_{1} - \)\(28\!\cdots\!64\)\( \beta_{2} + \)\(12\!\cdots\!48\)\( \beta_{3}) q^{98} +(\)\(70\!\cdots\!56\)\( - \)\(89\!\cdots\!91\)\( \beta_{1} + \)\(11\!\cdots\!44\)\( \beta_{2} + \)\(11\!\cdots\!92\)\( \beta_{3}) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 274877906944q^{2} + 305066325723244176q^{3} + \)\(18\!\cdots\!84\)\(q^{4} - \)\(78\!\cdots\!60\)\(q^{5} + \)\(20\!\cdots\!36\)\(q^{6} + \)\(36\!\cdots\!12\)\(q^{7} + \)\(12\!\cdots\!24\)\(q^{8} + \)\(12\!\cdots\!52\)\(q^{9} + O(q^{10}) \) \( 4q + 274877906944q^{2} + 305066325723244176q^{3} + \)\(18\!\cdots\!84\)\(q^{4} - \)\(78\!\cdots\!60\)\(q^{5} + \)\(20\!\cdots\!36\)\(q^{6} + \)\(36\!\cdots\!12\)\(q^{7} + \)\(12\!\cdots\!24\)\(q^{8} + \)\(12\!\cdots\!52\)\(q^{9} - \)\(53\!\cdots\!60\)\(q^{10} + \)\(52\!\cdots\!48\)\(q^{11} + \)\(14\!\cdots\!96\)\(q^{12} + \)\(13\!\cdots\!76\)\(q^{13} + \)\(24\!\cdots\!32\)\(q^{14} - \)\(78\!\cdots\!40\)\(q^{15} + \)\(89\!\cdots\!64\)\(q^{16} + \)\(95\!\cdots\!52\)\(q^{17} + \)\(84\!\cdots\!72\)\(q^{18} + \)\(95\!\cdots\!00\)\(q^{19} - \)\(37\!\cdots\!60\)\(q^{20} + \)\(66\!\cdots\!28\)\(q^{21} + \)\(36\!\cdots\!28\)\(q^{22} - \)\(67\!\cdots\!04\)\(q^{23} + \)\(98\!\cdots\!56\)\(q^{24} + \)\(34\!\cdots\!00\)\(q^{25} + \)\(93\!\cdots\!36\)\(q^{26} + \)\(12\!\cdots\!40\)\(q^{27} + \)\(17\!\cdots\!52\)\(q^{28} - \)\(40\!\cdots\!40\)\(q^{29} - \)\(53\!\cdots\!40\)\(q^{30} + \)\(37\!\cdots\!08\)\(q^{31} + \)\(61\!\cdots\!04\)\(q^{32} + \)\(62\!\cdots\!12\)\(q^{33} + \)\(65\!\cdots\!72\)\(q^{34} + \)\(43\!\cdots\!20\)\(q^{35} + \)\(57\!\cdots\!92\)\(q^{36} + \)\(68\!\cdots\!52\)\(q^{37} + \)\(65\!\cdots\!00\)\(q^{38} + \)\(11\!\cdots\!44\)\(q^{39} - \)\(25\!\cdots\!60\)\(q^{40} + \)\(56\!\cdots\!48\)\(q^{41} + \)\(45\!\cdots\!08\)\(q^{42} - \)\(19\!\cdots\!04\)\(q^{43} + \)\(24\!\cdots\!08\)\(q^{44} - \)\(79\!\cdots\!80\)\(q^{45} - \)\(46\!\cdots\!44\)\(q^{46} - \)\(26\!\cdots\!88\)\(q^{47} + \)\(68\!\cdots\!16\)\(q^{48} + \)\(49\!\cdots\!08\)\(q^{49} + \)\(23\!\cdots\!00\)\(q^{50} + \)\(46\!\cdots\!88\)\(q^{51} + \)\(64\!\cdots\!96\)\(q^{52} + \)\(45\!\cdots\!16\)\(q^{53} + \)\(83\!\cdots\!40\)\(q^{54} - \)\(20\!\cdots\!20\)\(q^{55} + \)\(11\!\cdots\!72\)\(q^{56} - \)\(35\!\cdots\!00\)\(q^{57} - \)\(28\!\cdots\!40\)\(q^{58} - \)\(36\!\cdots\!80\)\(q^{59} - \)\(36\!\cdots\!40\)\(q^{60} + \)\(35\!\cdots\!08\)\(q^{61} + \)\(26\!\cdots\!88\)\(q^{62} + \)\(13\!\cdots\!56\)\(q^{63} + \)\(42\!\cdots\!44\)\(q^{64} + \)\(55\!\cdots\!60\)\(q^{65} + \)\(42\!\cdots\!32\)\(q^{66} + \)\(24\!\cdots\!72\)\(q^{67} + \)\(45\!\cdots\!92\)\(q^{68} + \)\(36\!\cdots\!24\)\(q^{69} + \)\(29\!\cdots\!20\)\(q^{70} + \)\(61\!\cdots\!28\)\(q^{71} + \)\(39\!\cdots\!12\)\(q^{72} + \)\(45\!\cdots\!36\)\(q^{73} + \)\(46\!\cdots\!72\)\(q^{74} + \)\(22\!\cdots\!00\)\(q^{75} + \)\(45\!\cdots\!00\)\(q^{76} + \)\(29\!\cdots\!44\)\(q^{77} + \)\(77\!\cdots\!84\)\(q^{78} + \)\(20\!\cdots\!40\)\(q^{79} - \)\(17\!\cdots\!60\)\(q^{80} - \)\(16\!\cdots\!36\)\(q^{81} + \)\(39\!\cdots\!28\)\(q^{82} + \)\(15\!\cdots\!16\)\(q^{83} + \)\(31\!\cdots\!88\)\(q^{84} - \)\(47\!\cdots\!80\)\(q^{85} - \)\(13\!\cdots\!44\)\(q^{86} - \)\(15\!\cdots\!60\)\(q^{87} + \)\(17\!\cdots\!88\)\(q^{88} - \)\(24\!\cdots\!40\)\(q^{89} - \)\(54\!\cdots\!80\)\(q^{90} - \)\(95\!\cdots\!72\)\(q^{91} - \)\(31\!\cdots\!84\)\(q^{92} - \)\(26\!\cdots\!48\)\(q^{93} - \)\(18\!\cdots\!68\)\(q^{94} - \)\(46\!\cdots\!00\)\(q^{95} + \)\(46\!\cdots\!76\)\(q^{96} + \)\(58\!\cdots\!32\)\(q^{97} + \)\(33\!\cdots\!88\)\(q^{98} + \)\(28\!\cdots\!24\)\(q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 2 x^{3} - 50110216549679282411939889123 x^{2} - 1553419798812705152849518883586567702011976 x + 212476488708962349290223416738491633360102623610323343644\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 1920 \nu - 960 \)
\(\beta_{2}\)\(=\)\((\)\(-12014583808 \nu^{3} + 1457477848748036954808320 \nu^{2} + 432392473222768464554335054948276568064 \nu - 22519496037180424106837040399750952105785206652432384\)\()/ \)\(58\!\cdots\!19\)\( \)
\(\beta_{3}\)\(=\)\((\)\(-6247729332224 \nu^{3} - 411946603233703090049392640 \nu^{2} + 279247593385092711098269803296947448456832 \nu + 17600376579197898873192899424418450842106870550214491328\)\()/ \)\(72\!\cdots\!99\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1} + 960\)\()/1920\)
\(\nu^{2}\)\(=\)\((\)\(-11458 \beta_{3} + 482622219 \beta_{2} + 44640088824434274 \beta_{1} + 46181575572184426670843801817600000\)\()/1843200\)
\(\nu^{3}\)\(=\)\((\)\(-35582955291700120967 \beta_{3} - 814640499558836677099629 \beta_{2} + 1023095206494676459264880467925559 \beta_{1} + 54974656764897846989462821323522143300270248427520\)\()/47185920\)

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
2.29885e14
5.25645e13
−9.26366e13
−1.89813e14
6.87195e10 −3.65112e17 4.72237e21 −6.80519e24 −2.50903e28 6.19950e30 3.24519e32 6.57217e34 −4.67649e35
1.2 6.87195e10 −2.46573e16 4.72237e21 −4.21515e24 −1.69444e27 −1.19188e31 3.24519e32 −6.69772e34 −2.89663e35
1.3 6.87195e10 2.54129e17 4.72237e21 6.33389e25 1.74636e28 7.98841e30 3.24519e32 −3.00372e33 4.35262e36
1.4 6.87195e10 4.40707e17 4.72237e21 −6.01656e25 3.02852e28 1.36725e30 3.24519e32 1.26637e35 −4.13455e36
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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 2.74.a.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2.74.a.b 4 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - \)\(30\!\cdots\!76\)\( T_{3}^{3} - \)\(14\!\cdots\!84\)\( T_{3}^{2} + \)\(37\!\cdots\!64\)\( T_{3} + \)\(10\!\cdots\!96\)\( \) acting on \(S_{74}^{\mathrm{new}}(\Gamma_0(2))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - 68719476736 T )^{4} \)
$3$ \( 1 - 305066325723244176 T + \)\(12\!\cdots\!08\)\( T^{2} - \)\(24\!\cdots\!80\)\( T^{3} + \)\(81\!\cdots\!06\)\( T^{4} - \)\(16\!\cdots\!40\)\( T^{5} + \)\(55\!\cdots\!32\)\( T^{6} - \)\(94\!\cdots\!92\)\( T^{7} + \)\(20\!\cdots\!41\)\( T^{8} \)
$5$ \( 1 + \)\(78\!\cdots\!60\)\( T + \)\(41\!\cdots\!00\)\( T^{2} - \)\(17\!\cdots\!00\)\( T^{3} - \)\(14\!\cdots\!50\)\( T^{4} - \)\(18\!\cdots\!00\)\( T^{5} + \)\(46\!\cdots\!00\)\( T^{6} + \)\(93\!\cdots\!00\)\( T^{7} + \)\(12\!\cdots\!25\)\( T^{8} \)
$7$ \( 1 - \)\(36\!\cdots\!12\)\( T + \)\(80\!\cdots\!32\)\( T^{2} + \)\(21\!\cdots\!40\)\( T^{3} + \)\(22\!\cdots\!06\)\( T^{4} + \)\(10\!\cdots\!80\)\( T^{5} + \)\(19\!\cdots\!68\)\( T^{6} - \)\(43\!\cdots\!16\)\( T^{7} + \)\(58\!\cdots\!01\)\( T^{8} \)
$11$ \( 1 - \)\(52\!\cdots\!48\)\( T + \)\(19\!\cdots\!88\)\( T^{2} + \)\(50\!\cdots\!24\)\( T^{3} + \)\(13\!\cdots\!70\)\( T^{4} + \)\(52\!\cdots\!44\)\( T^{5} + \)\(21\!\cdots\!68\)\( T^{6} - \)\(61\!\cdots\!68\)\( T^{7} + \)\(12\!\cdots\!21\)\( T^{8} \)
$13$ \( 1 - \)\(13\!\cdots\!76\)\( T + \)\(19\!\cdots\!28\)\( T^{2} - \)\(18\!\cdots\!20\)\( T^{3} + \)\(79\!\cdots\!46\)\( T^{4} - \)\(39\!\cdots\!60\)\( T^{5} + \)\(85\!\cdots\!52\)\( T^{6} - \)\(12\!\cdots\!52\)\( T^{7} + \)\(18\!\cdots\!81\)\( T^{8} \)
$17$ \( 1 - \)\(95\!\cdots\!52\)\( T + \)\(24\!\cdots\!12\)\( T^{2} - \)\(16\!\cdots\!60\)\( T^{3} + \)\(24\!\cdots\!86\)\( T^{4} - \)\(11\!\cdots\!20\)\( T^{5} + \)\(10\!\cdots\!28\)\( T^{6} - \)\(28\!\cdots\!56\)\( T^{7} + \)\(19\!\cdots\!61\)\( T^{8} \)
$19$ \( 1 - \)\(95\!\cdots\!00\)\( T + \)\(10\!\cdots\!36\)\( T^{2} - \)\(55\!\cdots\!00\)\( T^{3} + \)\(34\!\cdots\!86\)\( T^{4} - \)\(12\!\cdots\!00\)\( T^{5} + \)\(51\!\cdots\!16\)\( T^{6} - \)\(10\!\cdots\!00\)\( T^{7} + \)\(24\!\cdots\!61\)\( T^{8} \)
$23$ \( 1 + \)\(67\!\cdots\!04\)\( T + \)\(40\!\cdots\!88\)\( T^{2} + \)\(25\!\cdots\!00\)\( T^{3} + \)\(19\!\cdots\!06\)\( T^{4} + \)\(66\!\cdots\!00\)\( T^{5} + \)\(26\!\cdots\!32\)\( T^{6} + \)\(11\!\cdots\!48\)\( T^{7} + \)\(42\!\cdots\!21\)\( T^{8} \)
$29$ \( 1 + \)\(40\!\cdots\!40\)\( T + \)\(22\!\cdots\!56\)\( T^{2} + \)\(65\!\cdots\!80\)\( T^{3} + \)\(19\!\cdots\!26\)\( T^{4} + \)\(37\!\cdots\!20\)\( T^{5} + \)\(73\!\cdots\!76\)\( T^{6} + \)\(75\!\cdots\!60\)\( T^{7} + \)\(10\!\cdots\!41\)\( T^{8} \)
$31$ \( 1 - \)\(37\!\cdots\!08\)\( T + \)\(17\!\cdots\!88\)\( T^{2} - \)\(64\!\cdots\!16\)\( T^{3} + \)\(15\!\cdots\!70\)\( T^{4} - \)\(47\!\cdots\!56\)\( T^{5} + \)\(98\!\cdots\!28\)\( T^{6} - \)\(15\!\cdots\!68\)\( T^{7} + \)\(30\!\cdots\!61\)\( T^{8} \)
$37$ \( 1 - \)\(68\!\cdots\!52\)\( T + \)\(27\!\cdots\!52\)\( T^{2} - \)\(73\!\cdots\!20\)\( T^{3} + \)\(14\!\cdots\!06\)\( T^{4} - \)\(22\!\cdots\!40\)\( T^{5} + \)\(24\!\cdots\!68\)\( T^{6} - \)\(18\!\cdots\!96\)\( T^{7} + \)\(82\!\cdots\!81\)\( T^{8} \)
$41$ \( 1 - \)\(56\!\cdots\!48\)\( T + \)\(97\!\cdots\!48\)\( T^{2} - \)\(10\!\cdots\!36\)\( T^{3} + \)\(58\!\cdots\!70\)\( T^{4} - \)\(55\!\cdots\!56\)\( T^{5} + \)\(28\!\cdots\!68\)\( T^{6} - \)\(90\!\cdots\!28\)\( T^{7} + \)\(85\!\cdots\!81\)\( T^{8} \)
$43$ \( 1 + \)\(19\!\cdots\!04\)\( T + \)\(65\!\cdots\!28\)\( T^{2} + \)\(91\!\cdots\!20\)\( T^{3} + \)\(16\!\cdots\!86\)\( T^{4} + \)\(16\!\cdots\!60\)\( T^{5} + \)\(20\!\cdots\!72\)\( T^{6} + \)\(10\!\cdots\!28\)\( T^{7} + \)\(93\!\cdots\!01\)\( T^{8} \)
$47$ \( 1 + \)\(26\!\cdots\!88\)\( T + \)\(32\!\cdots\!12\)\( T^{2} + \)\(16\!\cdots\!20\)\( T^{3} + \)\(60\!\cdots\!46\)\( T^{4} + \)\(19\!\cdots\!40\)\( T^{5} + \)\(42\!\cdots\!48\)\( T^{6} + \)\(41\!\cdots\!04\)\( T^{7} + \)\(17\!\cdots\!41\)\( T^{8} \)
$53$ \( 1 - \)\(45\!\cdots\!16\)\( T + \)\(11\!\cdots\!88\)\( T^{2} - \)\(84\!\cdots\!60\)\( T^{3} + \)\(70\!\cdots\!46\)\( T^{4} - \)\(62\!\cdots\!80\)\( T^{5} + \)\(64\!\cdots\!52\)\( T^{6} - \)\(18\!\cdots\!72\)\( T^{7} + \)\(30\!\cdots\!41\)\( T^{8} \)
$59$ \( 1 + \)\(36\!\cdots\!80\)\( T + \)\(48\!\cdots\!16\)\( T^{2} + \)\(13\!\cdots\!60\)\( T^{3} + \)\(11\!\cdots\!46\)\( T^{4} + \)\(24\!\cdots\!40\)\( T^{5} + \)\(16\!\cdots\!56\)\( T^{6} + \)\(23\!\cdots\!20\)\( T^{7} + \)\(12\!\cdots\!81\)\( T^{8} \)
$61$ \( 1 - \)\(35\!\cdots\!08\)\( T + \)\(12\!\cdots\!48\)\( T^{2} - \)\(25\!\cdots\!76\)\( T^{3} + \)\(46\!\cdots\!70\)\( T^{4} - \)\(53\!\cdots\!56\)\( T^{5} + \)\(59\!\cdots\!28\)\( T^{6} - \)\(34\!\cdots\!28\)\( T^{7} + \)\(20\!\cdots\!21\)\( T^{8} \)
$67$ \( 1 - \)\(24\!\cdots\!72\)\( T + \)\(33\!\cdots\!92\)\( T^{2} + \)\(22\!\cdots\!80\)\( T^{3} + \)\(53\!\cdots\!46\)\( T^{4} + \)\(45\!\cdots\!60\)\( T^{5} + \)\(13\!\cdots\!48\)\( T^{6} - \)\(20\!\cdots\!16\)\( T^{7} + \)\(16\!\cdots\!61\)\( T^{8} \)
$71$ \( 1 - \)\(61\!\cdots\!28\)\( T + \)\(53\!\cdots\!88\)\( T^{2} - \)\(20\!\cdots\!96\)\( T^{3} + \)\(10\!\cdots\!70\)\( T^{4} - \)\(28\!\cdots\!56\)\( T^{5} + \)\(10\!\cdots\!48\)\( T^{6} - \)\(16\!\cdots\!68\)\( T^{7} + \)\(36\!\cdots\!41\)\( T^{8} \)
$73$ \( 1 - \)\(45\!\cdots\!36\)\( T + \)\(11\!\cdots\!68\)\( T^{2} - \)\(18\!\cdots\!80\)\( T^{3} + \)\(22\!\cdots\!46\)\( T^{4} - \)\(19\!\cdots\!40\)\( T^{5} + \)\(12\!\cdots\!52\)\( T^{6} - \)\(53\!\cdots\!32\)\( T^{7} + \)\(12\!\cdots\!21\)\( T^{8} \)
$79$ \( 1 - \)\(20\!\cdots\!40\)\( T + \)\(10\!\cdots\!56\)\( T^{2} - \)\(16\!\cdots\!80\)\( T^{3} + \)\(45\!\cdots\!26\)\( T^{4} - \)\(56\!\cdots\!20\)\( T^{5} + \)\(11\!\cdots\!76\)\( T^{6} - \)\(77\!\cdots\!60\)\( T^{7} + \)\(12\!\cdots\!41\)\( T^{8} \)
$83$ \( 1 - \)\(15\!\cdots\!16\)\( T + \)\(44\!\cdots\!48\)\( T^{2} - \)\(49\!\cdots\!80\)\( T^{3} + \)\(77\!\cdots\!66\)\( T^{4} - \)\(61\!\cdots\!40\)\( T^{5} + \)\(67\!\cdots\!12\)\( T^{6} - \)\(29\!\cdots\!52\)\( T^{7} + \)\(23\!\cdots\!61\)\( T^{8} \)
$89$ \( 1 + \)\(24\!\cdots\!40\)\( T + \)\(87\!\cdots\!76\)\( T^{2} + \)\(14\!\cdots\!80\)\( T^{3} + \)\(27\!\cdots\!66\)\( T^{4} + \)\(29\!\cdots\!20\)\( T^{5} + \)\(35\!\cdots\!36\)\( T^{6} + \)\(20\!\cdots\!60\)\( T^{7} + \)\(16\!\cdots\!21\)\( T^{8} \)
$97$ \( 1 - \)\(58\!\cdots\!32\)\( T + \)\(30\!\cdots\!92\)\( T^{2} - \)\(13\!\cdots\!40\)\( T^{3} + \)\(51\!\cdots\!06\)\( T^{4} - \)\(14\!\cdots\!80\)\( T^{5} + \)\(35\!\cdots\!68\)\( T^{6} - \)\(73\!\cdots\!56\)\( T^{7} + \)\(13\!\cdots\!41\)\( T^{8} \)
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