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: \( 2^{45}\cdot 3^{14}\cdot 5^{5}\cdot 7^{2}\cdot 11 \)
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.

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

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

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

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.

Atkin-Lehner signs

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

Hecke kernels

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