Properties

Label 1.54.a.a
Level 1
Weight 54
Character orbit 1.a
Self dual Yes
Analytic conductor 17.790
Analytic rank 1
Dimension 4
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 1 \)
Weight: \( k \) = \( 54 \)
Character orbit: \([\chi]\) = 1.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(17.7903107608\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{27}\cdot 3^{8}\cdot 5^{3}\cdot 7\cdot 13 \)
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.

\(f(q)\) \(=\) \( q\) \(+(-17119080 + \beta_{1}) q^{2}\) \(+(-262102751820 + 8136 \beta_{1} - \beta_{2}) q^{3}\) \(+(1957409854949632 - 8049420 \beta_{1} + 374 \beta_{2} + \beta_{3}) q^{4}\) \(+(-1140973948323578250 + 3784860800 \beta_{1} - 97500 \beta_{2} - 600 \beta_{3}) q^{5}\) \(+(91310821379248619232 - 940459617900 \beta_{1} + 89361744 \beta_{2} + 36216 \beta_{3}) q^{6}\) \(+(-56210442687111357400 - 54402495474672 \beta_{1} + 5099906366 \beta_{2} - 909920 \beta_{3}) q^{7}\) \(+(\)\(34\!\cdots\!60\)\( - 2009331196034176 \beta_{1} - 102810683328 \beta_{2} + 7340640 \beta_{3}) q^{8}\) \(+(\)\(28\!\cdots\!53\)\( - 153884364943507200 \beta_{1} - 1414745890632 \beta_{2} + 216501552 \beta_{3}) q^{9}\) \(+O(q^{10})\) \( q\) \(+(-17119080 + \beta_{1}) q^{2}\) \(+(-262102751820 + 8136 \beta_{1} - \beta_{2}) q^{3}\) \(+(1957409854949632 - 8049420 \beta_{1} + 374 \beta_{2} + \beta_{3}) q^{4}\) \(+(-1140973948323578250 + 3784860800 \beta_{1} - 97500 \beta_{2} - 600 \beta_{3}) q^{5}\) \(+(91310821379248619232 - 940459617900 \beta_{1} + 89361744 \beta_{2} + 36216 \beta_{3}) q^{6}\) \(+(-56210442687111357400 - 54402495474672 \beta_{1} + 5099906366 \beta_{2} - 909920 \beta_{3}) q^{7}\) \(+(\)\(34\!\cdots\!60\)\( - 2009331196034176 \beta_{1} - 102810683328 \beta_{2} + 7340640 \beta_{3}) q^{8}\) \(+(\)\(28\!\cdots\!53\)\( - 153884364943507200 \beta_{1} - 1414745890632 \beta_{2} + 216501552 \beta_{3}) q^{9}\) \(+(\)\(59\!\cdots\!00\)\( - 4079277238575184650 \beta_{1} + 50341792216000 \beta_{2} - 9012527200 \beta_{3}) q^{10}\) \(+(\)\(62\!\cdots\!32\)\( - 20797035850880243240 \beta_{1} - 400485890554035 \beta_{2} + 180852555840 \beta_{3}) q^{11}\) \(+(-\)\(92\!\cdots\!60\)\( + \)\(25\!\cdots\!64\)\( \beta_{1} - 1503331766739976 \beta_{2} - 2512033441740 \beta_{3}) q^{12}\) \(+(\)\(97\!\cdots\!10\)\( + \)\(20\!\cdots\!08\)\( \beta_{1} + 51373303912919300 \beta_{2} + 26522640225640 \beta_{3}) q^{13}\) \(+(-\)\(57\!\cdots\!56\)\( - \)\(11\!\cdots\!60\)\( \beta_{1} - 399129725674167648 \beta_{2} - 221167496673552 \beta_{3}) q^{14}\) \(+(\)\(27\!\cdots\!00\)\( - \)\(13\!\cdots\!00\)\( \beta_{1} + 1080428233798364250 \beta_{2} + 1479299373410400 \beta_{3}) q^{15}\) \(+(-\)\(39\!\cdots\!84\)\( + \)\(47\!\cdots\!80\)\( \beta_{1} + 4258703104277157888 \beta_{2} - 7939542758857728 \beta_{3}) q^{16}\) \(+(-\)\(21\!\cdots\!90\)\( + \)\(31\!\cdots\!92\)\( \beta_{1} - 40747405116440959752 \beta_{2} + 33675934369779120 \beta_{3}) q^{17}\) \(+(-\)\(16\!\cdots\!20\)\( + \)\(14\!\cdots\!21\)\( \beta_{1} + 49949006167583512704 \beta_{2} - 108552646480207680 \beta_{3}) q^{18}\) \(+(-\)\(63\!\cdots\!40\)\( - \)\(60\!\cdots\!60\)\( \beta_{1} + \)\(75\!\cdots\!11\)\( \beta_{2} + 243892004379685824 \beta_{3}) q^{19}\) \(+(-\)\(34\!\cdots\!00\)\( - \)\(16\!\cdots\!00\)\( \beta_{1} - \)\(43\!\cdots\!00\)\( \beta_{2} - 321907385167725450 \beta_{3}) q^{20}\) \(+(-\)\(11\!\cdots\!08\)\( + \)\(62\!\cdots\!60\)\( \beta_{1} + \)\(75\!\cdots\!48\)\( \beta_{2} + 475413659635414752 \beta_{3}) q^{21}\) \(+(-\)\(23\!\cdots\!60\)\( + \)\(10\!\cdots\!92\)\( \beta_{1} + \)\(14\!\cdots\!20\)\( \beta_{2} - 4868761285364777240 \beta_{3}) q^{22}\) \(+(\)\(17\!\cdots\!40\)\( - \)\(64\!\cdots\!20\)\( \beta_{1} - \)\(37\!\cdots\!94\)\( \beta_{2} + 31701177037073551200 \beta_{3}) q^{23}\) \(+(\)\(20\!\cdots\!80\)\( - \)\(11\!\cdots\!20\)\( \beta_{1} - \)\(41\!\cdots\!12\)\( \beta_{2} - 97319344971076633728 \beta_{3}) q^{24}\) \(+(\)\(91\!\cdots\!75\)\( + \)\(26\!\cdots\!00\)\( \beta_{1} + \)\(24\!\cdots\!00\)\( \beta_{2} + 5532490760056300000 \beta_{3}) q^{25}\) \(+(\)\(22\!\cdots\!72\)\( + \)\(19\!\cdots\!50\)\( \beta_{1} - \)\(54\!\cdots\!28\)\( \beta_{2} + \)\(13\!\cdots\!08\)\( \beta_{3}) q^{26}\) \(+(\)\(21\!\cdots\!60\)\( - \)\(17\!\cdots\!96\)\( \beta_{1} + \)\(22\!\cdots\!62\)\( \beta_{2} - \)\(61\!\cdots\!60\)\( \beta_{3}) q^{27}\) \(+(-\)\(14\!\cdots\!40\)\( - \)\(14\!\cdots\!00\)\( \beta_{1} + \)\(30\!\cdots\!84\)\( \beta_{2} + \)\(12\!\cdots\!60\)\( \beta_{3}) q^{28}\) \(+(-\)\(35\!\cdots\!10\)\( - \)\(20\!\cdots\!80\)\( \beta_{1} + \)\(57\!\cdots\!04\)\( \beta_{2} + \)\(63\!\cdots\!16\)\( \beta_{3}) q^{29}\) \(+(-\)\(14\!\cdots\!00\)\( + \)\(94\!\cdots\!00\)\( \beta_{1} - \)\(24\!\cdots\!00\)\( \beta_{2} - \)\(12\!\cdots\!00\)\( \beta_{3}) q^{30}\) \(+(-\)\(99\!\cdots\!08\)\( + \)\(77\!\cdots\!80\)\( \beta_{1} + \)\(15\!\cdots\!20\)\( \beta_{2} + \)\(36\!\cdots\!20\)\( \beta_{3}) q^{31}\) \(+(\)\(86\!\cdots\!20\)\( - \)\(56\!\cdots\!44\)\( \beta_{1} + \)\(11\!\cdots\!40\)\( \beta_{2} - \)\(34\!\cdots\!80\)\( \beta_{3}) q^{32}\) \(+(\)\(66\!\cdots\!60\)\( - \)\(47\!\cdots\!68\)\( \beta_{1} - \)\(29\!\cdots\!72\)\( \beta_{2} - \)\(11\!\cdots\!20\)\( \beta_{3}) q^{33}\) \(+(\)\(37\!\cdots\!84\)\( - \)\(56\!\cdots\!70\)\( \beta_{1} + \)\(24\!\cdots\!08\)\( \beta_{2} + \)\(51\!\cdots\!52\)\( \beta_{3}) q^{34}\) \(+(\)\(15\!\cdots\!00\)\( + \)\(80\!\cdots\!00\)\( \beta_{1} - \)\(27\!\cdots\!00\)\( \beta_{2} - \)\(75\!\cdots\!00\)\( \beta_{3}) q^{35}\) \(+(\)\(18\!\cdots\!96\)\( - \)\(77\!\cdots\!20\)\( \beta_{1} + \)\(16\!\cdots\!18\)\( \beta_{2} - \)\(47\!\cdots\!83\)\( \beta_{3}) q^{36}\) \(+(-\)\(70\!\cdots\!70\)\( - \)\(17\!\cdots\!80\)\( \beta_{1} - \)\(13\!\cdots\!68\)\( \beta_{2} + \)\(42\!\cdots\!80\)\( \beta_{3}) q^{37}\) \(+(-\)\(53\!\cdots\!60\)\( - \)\(51\!\cdots\!24\)\( \beta_{1} - \)\(10\!\cdots\!92\)\( \beta_{2} - \)\(75\!\cdots\!20\)\( \beta_{3}) q^{38}\) \(+(-\)\(92\!\cdots\!04\)\( + \)\(10\!\cdots\!80\)\( \beta_{1} + \)\(26\!\cdots\!82\)\( \beta_{2} + \)\(19\!\cdots\!88\)\( \beta_{3}) q^{39}\) \(+(-\)\(13\!\cdots\!00\)\( - \)\(25\!\cdots\!00\)\( \beta_{1} - \)\(59\!\cdots\!00\)\( \beta_{2} + \)\(17\!\cdots\!00\)\( \beta_{3}) q^{40}\) \(+(\)\(20\!\cdots\!22\)\( + \)\(37\!\cdots\!80\)\( \beta_{1} - \)\(31\!\cdots\!80\)\( \beta_{2} - \)\(40\!\cdots\!80\)\( \beta_{3}) q^{41}\) \(+(\)\(86\!\cdots\!60\)\( - \)\(10\!\cdots\!40\)\( \beta_{1} - \)\(44\!\cdots\!56\)\( \beta_{2} + \)\(42\!\cdots\!80\)\( \beta_{3}) q^{42}\) \(+(\)\(11\!\cdots\!00\)\( + \)\(24\!\cdots\!44\)\( \beta_{1} + \)\(11\!\cdots\!73\)\( \beta_{2} - \)\(40\!\cdots\!00\)\( \beta_{3}) q^{43}\) \(+(\)\(91\!\cdots\!24\)\( - \)\(48\!\cdots\!20\)\( \beta_{1} + \)\(30\!\cdots\!48\)\( \beta_{2} - \)\(11\!\cdots\!88\)\( \beta_{3}) q^{44}\) \(+(-\)\(13\!\cdots\!50\)\( + \)\(46\!\cdots\!00\)\( \beta_{1} - \)\(76\!\cdots\!00\)\( \beta_{2} + \)\(36\!\cdots\!00\)\( \beta_{3}) q^{45}\) \(+(-\)\(71\!\cdots\!88\)\( + \)\(23\!\cdots\!80\)\( \beta_{1} - \)\(13\!\cdots\!60\)\( \beta_{2} - \)\(45\!\cdots\!00\)\( \beta_{3}) q^{46}\) \(+(-\)\(11\!\cdots\!60\)\( - \)\(10\!\cdots\!16\)\( \beta_{1} - \)\(97\!\cdots\!16\)\( \beta_{2} - \)\(39\!\cdots\!00\)\( \beta_{3}) q^{47}\) \(+(-\)\(76\!\cdots\!60\)\( - \)\(11\!\cdots\!60\)\( \beta_{1} + \)\(51\!\cdots\!56\)\( \beta_{2} + \)\(19\!\cdots\!80\)\( \beta_{3}) q^{48}\) \(+(\)\(15\!\cdots\!57\)\( - \)\(20\!\cdots\!80\)\( \beta_{1} - \)\(36\!\cdots\!80\)\( \beta_{2} - \)\(10\!\cdots\!60\)\( \beta_{3}) q^{49}\) \(+(\)\(13\!\cdots\!00\)\( + \)\(11\!\cdots\!75\)\( \beta_{1} - \)\(20\!\cdots\!00\)\( \beta_{2} - \)\(43\!\cdots\!00\)\( \beta_{3}) q^{50}\) \(+(\)\(12\!\cdots\!12\)\( - \)\(43\!\cdots\!60\)\( \beta_{1} + \)\(36\!\cdots\!06\)\( \beta_{2} + \)\(35\!\cdots\!04\)\( \beta_{3}) q^{51}\) \(+(\)\(16\!\cdots\!00\)\( + \)\(73\!\cdots\!08\)\( \beta_{1} - \)\(90\!\cdots\!84\)\( \beta_{2} + \)\(14\!\cdots\!50\)\( \beta_{3}) q^{52}\) \(+(\)\(97\!\cdots\!30\)\( - \)\(35\!\cdots\!64\)\( \beta_{1} - \)\(22\!\cdots\!16\)\( \beta_{2} - \)\(19\!\cdots\!60\)\( \beta_{3}) q^{53}\) \(+(-\)\(22\!\cdots\!40\)\( - \)\(84\!\cdots\!60\)\( \beta_{1} + \)\(15\!\cdots\!16\)\( \beta_{2} - \)\(40\!\cdots\!56\)\( \beta_{3}) q^{54}\) \(+(-\)\(63\!\cdots\!00\)\( + \)\(29\!\cdots\!00\)\( \beta_{1} - \)\(14\!\cdots\!50\)\( \beta_{2} + \)\(84\!\cdots\!00\)\( \beta_{3}) q^{55}\) \(+(-\)\(10\!\cdots\!40\)\( + \)\(58\!\cdots\!80\)\( \beta_{1} + \)\(19\!\cdots\!16\)\( \beta_{2} + \)\(76\!\cdots\!64\)\( \beta_{3}) q^{56}\) \(+(-\)\(19\!\cdots\!80\)\( + \)\(14\!\cdots\!48\)\( \beta_{1} + \)\(37\!\cdots\!24\)\( \beta_{2} - \)\(26\!\cdots\!00\)\( \beta_{3}) q^{57}\) \(+(\)\(57\!\cdots\!60\)\( - \)\(27\!\cdots\!66\)\( \beta_{1} - \)\(53\!\cdots\!88\)\( \beta_{2} - \)\(14\!\cdots\!20\)\( \beta_{3}) q^{58}\) \(+(\)\(40\!\cdots\!80\)\( + \)\(34\!\cdots\!60\)\( \beta_{1} - \)\(12\!\cdots\!27\)\( \beta_{2} + \)\(87\!\cdots\!52\)\( \beta_{3}) q^{59}\) \(+(\)\(10\!\cdots\!00\)\( - \)\(95\!\cdots\!00\)\( \beta_{1} + \)\(23\!\cdots\!00\)\( \beta_{2} - \)\(22\!\cdots\!00\)\( \beta_{3}) q^{60}\) \(+(\)\(16\!\cdots\!82\)\( + \)\(12\!\cdots\!00\)\( \beta_{1} + \)\(35\!\cdots\!00\)\( \beta_{2} - \)\(23\!\cdots\!00\)\( \beta_{3}) q^{61}\) \(+(\)\(99\!\cdots\!40\)\( + \)\(97\!\cdots\!72\)\( \beta_{1} - \)\(35\!\cdots\!40\)\( \beta_{2} + \)\(12\!\cdots\!80\)\( \beta_{3}) q^{62}\) \(+(-\)\(75\!\cdots\!80\)\( + \)\(79\!\cdots\!92\)\( \beta_{1} + \)\(77\!\cdots\!02\)\( \beta_{2} + \)\(39\!\cdots\!80\)\( \beta_{3}) q^{63}\) \(+(-\)\(25\!\cdots\!08\)\( - \)\(89\!\cdots\!00\)\( \beta_{1} - \)\(13\!\cdots\!12\)\( \beta_{2} - \)\(24\!\cdots\!68\)\( \beta_{3}) q^{64}\) \(+(-\)\(85\!\cdots\!00\)\( - \)\(57\!\cdots\!00\)\( \beta_{1} - \)\(27\!\cdots\!00\)\( \beta_{2} - \)\(88\!\cdots\!00\)\( \beta_{3}) q^{65}\) \(+(-\)\(16\!\cdots\!76\)\( - \)\(14\!\cdots\!80\)\( \beta_{1} + \)\(33\!\cdots\!88\)\( \beta_{2} + \)\(48\!\cdots\!92\)\( \beta_{3}) q^{66}\) \(+(-\)\(27\!\cdots\!40\)\( - \)\(59\!\cdots\!88\)\( \beta_{1} - \)\(10\!\cdots\!17\)\( \beta_{2} + \)\(25\!\cdots\!20\)\( \beta_{3}) q^{67}\) \(+(\)\(71\!\cdots\!80\)\( + \)\(35\!\cdots\!88\)\( \beta_{1} - \)\(21\!\cdots\!92\)\( \beta_{2} - \)\(29\!\cdots\!90\)\( \beta_{3}) q^{68}\) \(+(\)\(19\!\cdots\!16\)\( + \)\(67\!\cdots\!00\)\( \beta_{1} - \)\(71\!\cdots\!84\)\( \beta_{2} - \)\(30\!\cdots\!76\)\( \beta_{3}) q^{69}\) \(+(\)\(82\!\cdots\!00\)\( - \)\(16\!\cdots\!00\)\( \beta_{1} + \)\(10\!\cdots\!00\)\( \beta_{2} + \)\(68\!\cdots\!00\)\( \beta_{3}) q^{70}\) \(+(\)\(89\!\cdots\!12\)\( - \)\(66\!\cdots\!00\)\( \beta_{1} + \)\(11\!\cdots\!50\)\( \beta_{2} - \)\(60\!\cdots\!00\)\( \beta_{3}) q^{71}\) \(+(\)\(66\!\cdots\!80\)\( - \)\(11\!\cdots\!08\)\( \beta_{1} - \)\(18\!\cdots\!64\)\( \beta_{2} - \)\(38\!\cdots\!40\)\( \beta_{3}) q^{72}\) \(+(-\)\(17\!\cdots\!10\)\( - \)\(19\!\cdots\!80\)\( \beta_{1} + \)\(54\!\cdots\!96\)\( \beta_{2} - \)\(58\!\cdots\!20\)\( \beta_{3}) q^{73}\) \(+(-\)\(61\!\cdots\!36\)\( + \)\(12\!\cdots\!30\)\( \beta_{1} - \)\(17\!\cdots\!20\)\( \beta_{2} + \)\(12\!\cdots\!60\)\( \beta_{3}) q^{74}\) \(+(-\)\(53\!\cdots\!00\)\( + \)\(41\!\cdots\!00\)\( \beta_{1} - \)\(15\!\cdots\!75\)\( \beta_{2} + \)\(11\!\cdots\!00\)\( \beta_{3}) q^{75}\) \(+(\)\(11\!\cdots\!20\)\( - \)\(48\!\cdots\!40\)\( \beta_{1} + \)\(53\!\cdots\!32\)\( \beta_{2} - \)\(64\!\cdots\!72\)\( \beta_{3}) q^{76}\) \(+(-\)\(40\!\cdots\!00\)\( + \)\(52\!\cdots\!56\)\( \beta_{1} + \)\(13\!\cdots\!32\)\( \beta_{2} + \)\(76\!\cdots\!20\)\( \beta_{3}) q^{77}\) \(+(\)\(12\!\cdots\!00\)\( - \)\(53\!\cdots\!12\)\( \beta_{1} - \)\(20\!\cdots\!04\)\( \beta_{2} + \)\(36\!\cdots\!60\)\( \beta_{3}) q^{78}\) \(+(-\)\(39\!\cdots\!60\)\( - \)\(55\!\cdots\!00\)\( \beta_{1} - \)\(23\!\cdots\!36\)\( \beta_{2} - \)\(16\!\cdots\!04\)\( \beta_{3}) q^{79}\) \(+(\)\(28\!\cdots\!00\)\( + \)\(81\!\cdots\!00\)\( \beta_{1} + \)\(31\!\cdots\!00\)\( \beta_{2} + \)\(66\!\cdots\!00\)\( \beta_{3}) q^{80}\) \(+(-\)\(12\!\cdots\!39\)\( + \)\(25\!\cdots\!80\)\( \beta_{1} - \)\(28\!\cdots\!16\)\( \beta_{2} + \)\(33\!\cdots\!16\)\( \beta_{3}) q^{81}\) \(+(\)\(39\!\cdots\!40\)\( - \)\(16\!\cdots\!98\)\( \beta_{1} + \)\(68\!\cdots\!60\)\( \beta_{2} + \)\(35\!\cdots\!80\)\( \beta_{3}) q^{82}\) \(+(-\)\(65\!\cdots\!80\)\( + \)\(62\!\cdots\!32\)\( \beta_{1} - \)\(71\!\cdots\!13\)\( \beta_{2} - \)\(47\!\cdots\!00\)\( \beta_{3}) q^{83}\) \(+(-\)\(19\!\cdots\!56\)\( + \)\(38\!\cdots\!20\)\( \beta_{1} - \)\(95\!\cdots\!36\)\( \beta_{2} - \)\(81\!\cdots\!44\)\( \beta_{3}) q^{84}\) \(+(-\)\(58\!\cdots\!00\)\( - \)\(20\!\cdots\!00\)\( \beta_{1} + \)\(82\!\cdots\!00\)\( \beta_{2} + \)\(17\!\cdots\!00\)\( \beta_{3}) q^{85}\) \(+(-\)\(19\!\cdots\!08\)\( + \)\(12\!\cdots\!60\)\( \beta_{1} - \)\(97\!\cdots\!84\)\( \beta_{2} - \)\(33\!\cdots\!96\)\( \beta_{3}) q^{86}\) \(+(-\)\(11\!\cdots\!20\)\( + \)\(65\!\cdots\!32\)\( \beta_{1} + \)\(48\!\cdots\!86\)\( \beta_{2} - \)\(10\!\cdots\!20\)\( \beta_{3}) q^{87}\) \(+(\)\(14\!\cdots\!20\)\( - \)\(36\!\cdots\!32\)\( \beta_{1} - \)\(33\!\cdots\!96\)\( \beta_{2} - \)\(12\!\cdots\!20\)\( \beta_{3}) q^{88}\) \(+(-\)\(94\!\cdots\!30\)\( + \)\(31\!\cdots\!60\)\( \beta_{1} - \)\(31\!\cdots\!08\)\( \beta_{2} - \)\(24\!\cdots\!32\)\( \beta_{3}) q^{89}\) \(+(\)\(52\!\cdots\!00\)\( + \)\(28\!\cdots\!50\)\( \beta_{1} + \)\(58\!\cdots\!00\)\( \beta_{2} + \)\(77\!\cdots\!00\)\( \beta_{3}) q^{90}\) \(+(\)\(31\!\cdots\!32\)\( - \)\(67\!\cdots\!20\)\( \beta_{1} - \)\(13\!\cdots\!44\)\( \beta_{2} - \)\(26\!\cdots\!76\)\( \beta_{3}) q^{91}\) \(+(\)\(22\!\cdots\!60\)\( - \)\(34\!\cdots\!48\)\( \beta_{1} + \)\(84\!\cdots\!68\)\( \beta_{2} - \)\(12\!\cdots\!20\)\( \beta_{3}) q^{92}\) \(+(-\)\(24\!\cdots\!40\)\( + \)\(73\!\cdots\!52\)\( \beta_{1} + \)\(19\!\cdots\!88\)\( \beta_{2} - \)\(53\!\cdots\!60\)\( \beta_{3}) q^{93}\) \(+(-\)\(92\!\cdots\!96\)\( - \)\(14\!\cdots\!60\)\( \beta_{1} - \)\(11\!\cdots\!04\)\( \beta_{2} - \)\(11\!\cdots\!36\)\( \beta_{3}) q^{94}\) \(+(-\)\(42\!\cdots\!00\)\( + \)\(26\!\cdots\!00\)\( \beta_{1} - \)\(36\!\cdots\!50\)\( \beta_{2} + \)\(41\!\cdots\!00\)\( \beta_{3}) q^{95}\) \(+(-\)\(28\!\cdots\!48\)\( + \)\(15\!\cdots\!40\)\( \beta_{1} - \)\(24\!\cdots\!56\)\( \beta_{2} - \)\(11\!\cdots\!64\)\( \beta_{3}) q^{96}\) \(+(\)\(25\!\cdots\!90\)\( - \)\(24\!\cdots\!56\)\( \beta_{1} + \)\(59\!\cdots\!24\)\( \beta_{2} - \)\(29\!\cdots\!60\)\( \beta_{3}) q^{97}\) \(+(-\)\(22\!\cdots\!60\)\( - \)\(79\!\cdots\!83\)\( \beta_{1} + \)\(30\!\cdots\!60\)\( \beta_{2} - \)\(12\!\cdots\!80\)\( \beta_{3}) q^{98}\) \(+(\)\(50\!\cdots\!96\)\( - \)\(18\!\cdots\!20\)\( \beta_{1} - \)\(62\!\cdots\!79\)\( \beta_{2} - \)\(12\!\cdots\!16\)\( \beta_{3}) q^{99}\) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(4q \) \(\mathstrut -\mathstrut 68476320q^{2} \) \(\mathstrut -\mathstrut 1048411007280q^{3} \) \(\mathstrut +\mathstrut 7829639419798528q^{4} \) \(\mathstrut -\mathstrut 4563895793294313000q^{5} \) \(\mathstrut +\mathstrut \)\(36\!\cdots\!28\)\(q^{6} \) \(\mathstrut -\mathstrut \)\(22\!\cdots\!00\)\(q^{7} \) \(\mathstrut +\mathstrut \)\(13\!\cdots\!40\)\(q^{8} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!12\)\(q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(4q \) \(\mathstrut -\mathstrut 68476320q^{2} \) \(\mathstrut -\mathstrut 1048411007280q^{3} \) \(\mathstrut +\mathstrut 7829639419798528q^{4} \) \(\mathstrut -\mathstrut 4563895793294313000q^{5} \) \(\mathstrut +\mathstrut \)\(36\!\cdots\!28\)\(q^{6} \) \(\mathstrut -\mathstrut \)\(22\!\cdots\!00\)\(q^{7} \) \(\mathstrut +\mathstrut \)\(13\!\cdots\!40\)\(q^{8} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!12\)\(q^{9} \) \(\mathstrut +\mathstrut \)\(23\!\cdots\!00\)\(q^{10} \) \(\mathstrut +\mathstrut \)\(24\!\cdots\!28\)\(q^{11} \) \(\mathstrut -\mathstrut \)\(36\!\cdots\!40\)\(q^{12} \) \(\mathstrut +\mathstrut \)\(38\!\cdots\!40\)\(q^{13} \) \(\mathstrut -\mathstrut \)\(23\!\cdots\!24\)\(q^{14} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!00\)\(q^{15} \) \(\mathstrut -\mathstrut \)\(15\!\cdots\!36\)\(q^{16} \) \(\mathstrut -\mathstrut \)\(86\!\cdots\!60\)\(q^{17} \) \(\mathstrut -\mathstrut \)\(67\!\cdots\!80\)\(q^{18} \) \(\mathstrut -\mathstrut \)\(25\!\cdots\!60\)\(q^{19} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!00\)\(q^{20} \) \(\mathstrut -\mathstrut \)\(45\!\cdots\!32\)\(q^{21} \) \(\mathstrut -\mathstrut \)\(93\!\cdots\!40\)\(q^{22} \) \(\mathstrut +\mathstrut \)\(69\!\cdots\!60\)\(q^{23} \) \(\mathstrut +\mathstrut \)\(80\!\cdots\!20\)\(q^{24} \) \(\mathstrut +\mathstrut \)\(36\!\cdots\!00\)\(q^{25} \) \(\mathstrut +\mathstrut \)\(88\!\cdots\!88\)\(q^{26} \) \(\mathstrut +\mathstrut \)\(85\!\cdots\!40\)\(q^{27} \) \(\mathstrut -\mathstrut \)\(56\!\cdots\!60\)\(q^{28} \) \(\mathstrut -\mathstrut \)\(14\!\cdots\!40\)\(q^{29} \) \(\mathstrut -\mathstrut \)\(57\!\cdots\!00\)\(q^{30} \) \(\mathstrut -\mathstrut \)\(39\!\cdots\!32\)\(q^{31} \) \(\mathstrut +\mathstrut \)\(34\!\cdots\!80\)\(q^{32} \) \(\mathstrut +\mathstrut \)\(26\!\cdots\!40\)\(q^{33} \) \(\mathstrut +\mathstrut \)\(14\!\cdots\!36\)\(q^{34} \) \(\mathstrut +\mathstrut \)\(61\!\cdots\!00\)\(q^{35} \) \(\mathstrut +\mathstrut \)\(74\!\cdots\!84\)\(q^{36} \) \(\mathstrut -\mathstrut \)\(28\!\cdots\!80\)\(q^{37} \) \(\mathstrut -\mathstrut \)\(21\!\cdots\!40\)\(q^{38} \) \(\mathstrut -\mathstrut \)\(36\!\cdots\!16\)\(q^{39} \) \(\mathstrut -\mathstrut \)\(53\!\cdots\!00\)\(q^{40} \) \(\mathstrut +\mathstrut \)\(81\!\cdots\!88\)\(q^{41} \) \(\mathstrut +\mathstrut \)\(34\!\cdots\!40\)\(q^{42} \) \(\mathstrut +\mathstrut \)\(45\!\cdots\!00\)\(q^{43} \) \(\mathstrut +\mathstrut \)\(36\!\cdots\!96\)\(q^{44} \) \(\mathstrut -\mathstrut \)\(52\!\cdots\!00\)\(q^{45} \) \(\mathstrut -\mathstrut \)\(28\!\cdots\!52\)\(q^{46} \) \(\mathstrut -\mathstrut \)\(45\!\cdots\!40\)\(q^{47} \) \(\mathstrut -\mathstrut \)\(30\!\cdots\!40\)\(q^{48} \) \(\mathstrut +\mathstrut \)\(61\!\cdots\!28\)\(q^{49} \) \(\mathstrut +\mathstrut \)\(52\!\cdots\!00\)\(q^{50} \) \(\mathstrut +\mathstrut \)\(48\!\cdots\!48\)\(q^{51} \) \(\mathstrut +\mathstrut \)\(64\!\cdots\!00\)\(q^{52} \) \(\mathstrut +\mathstrut \)\(38\!\cdots\!20\)\(q^{53} \) \(\mathstrut -\mathstrut \)\(90\!\cdots\!60\)\(q^{54} \) \(\mathstrut -\mathstrut \)\(25\!\cdots\!00\)\(q^{55} \) \(\mathstrut -\mathstrut \)\(41\!\cdots\!60\)\(q^{56} \) \(\mathstrut -\mathstrut \)\(78\!\cdots\!20\)\(q^{57} \) \(\mathstrut +\mathstrut \)\(23\!\cdots\!40\)\(q^{58} \) \(\mathstrut +\mathstrut \)\(16\!\cdots\!20\)\(q^{59} \) \(\mathstrut +\mathstrut \)\(40\!\cdots\!00\)\(q^{60} \) \(\mathstrut +\mathstrut \)\(65\!\cdots\!28\)\(q^{61} \) \(\mathstrut +\mathstrut \)\(39\!\cdots\!60\)\(q^{62} \) \(\mathstrut -\mathstrut \)\(30\!\cdots\!20\)\(q^{63} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!32\)\(q^{64} \) \(\mathstrut -\mathstrut \)\(34\!\cdots\!00\)\(q^{65} \) \(\mathstrut -\mathstrut \)\(65\!\cdots\!04\)\(q^{66} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!60\)\(q^{67} \) \(\mathstrut +\mathstrut \)\(28\!\cdots\!20\)\(q^{68} \) \(\mathstrut +\mathstrut \)\(77\!\cdots\!64\)\(q^{69} \) \(\mathstrut +\mathstrut \)\(33\!\cdots\!00\)\(q^{70} \) \(\mathstrut +\mathstrut \)\(35\!\cdots\!48\)\(q^{71} \) \(\mathstrut +\mathstrut \)\(26\!\cdots\!20\)\(q^{72} \) \(\mathstrut -\mathstrut \)\(70\!\cdots\!40\)\(q^{73} \) \(\mathstrut -\mathstrut \)\(24\!\cdots\!44\)\(q^{74} \) \(\mathstrut -\mathstrut \)\(21\!\cdots\!00\)\(q^{75} \) \(\mathstrut +\mathstrut \)\(45\!\cdots\!80\)\(q^{76} \) \(\mathstrut -\mathstrut \)\(16\!\cdots\!00\)\(q^{77} \) \(\mathstrut +\mathstrut \)\(51\!\cdots\!00\)\(q^{78} \) \(\mathstrut -\mathstrut \)\(15\!\cdots\!40\)\(q^{79} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!00\)\(q^{80} \) \(\mathstrut -\mathstrut \)\(49\!\cdots\!56\)\(q^{81} \) \(\mathstrut +\mathstrut \)\(15\!\cdots\!60\)\(q^{82} \) \(\mathstrut -\mathstrut \)\(26\!\cdots\!20\)\(q^{83} \) \(\mathstrut -\mathstrut \)\(76\!\cdots\!24\)\(q^{84} \) \(\mathstrut -\mathstrut \)\(23\!\cdots\!00\)\(q^{85} \) \(\mathstrut -\mathstrut \)\(77\!\cdots\!32\)\(q^{86} \) \(\mathstrut -\mathstrut \)\(45\!\cdots\!80\)\(q^{87} \) \(\mathstrut +\mathstrut \)\(56\!\cdots\!80\)\(q^{88} \) \(\mathstrut -\mathstrut \)\(37\!\cdots\!20\)\(q^{89} \) \(\mathstrut +\mathstrut \)\(20\!\cdots\!00\)\(q^{90} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!28\)\(q^{91} \) \(\mathstrut +\mathstrut \)\(89\!\cdots\!40\)\(q^{92} \) \(\mathstrut -\mathstrut \)\(99\!\cdots\!60\)\(q^{93} \) \(\mathstrut -\mathstrut \)\(37\!\cdots\!84\)\(q^{94} \) \(\mathstrut -\mathstrut \)\(17\!\cdots\!00\)\(q^{95} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!92\)\(q^{96} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!60\)\(q^{97} \) \(\mathstrut -\mathstrut \)\(88\!\cdots\!40\)\(q^{98} \) \(\mathstrut +\mathstrut \)\(20\!\cdots\!84\)\(q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4}\mathstrut -\mathstrut \) \(x^{3}\mathstrut -\mathstrut \) \(2315873743412\) \(x^{2}\mathstrut -\mathstrut \) \(421178019174503472\) \(x\mathstrut +\mathstrut \) \(612167648493870378955584\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 96 \nu - 24 \)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{3} + 611437 \nu^{2} + 1620402960440 \nu - 392120600840009328 \)\()/119308\)
\(\beta_{3}\)\(=\)\((\)\( 187 \nu^{3} + 435432545 \nu^{2} - 452992885700072 \nu - 563273827738686130416 \)\()/59654\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1}\mathstrut +\mathstrut \) \(24\)\()/96\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3}\mathstrut +\mathstrut \) \(374\) \(\beta_{2}\mathstrut +\mathstrut \) \(26188788\) \(\beta_{1}\mathstrut +\mathstrut \) \(10671546209644800\)\()/9216\)
\(\nu^{3}\)\(=\)\((\)\(611437\) \(\beta_{3}\mathstrut -\mathstrut \) \(870865090\) \(\beta_{2}\mathstrut +\mathstrut \) \(171571478170596\) \(\beta_{1}\mathstrut +\mathstrut \) \(2911198475853482464512\)\()/9216\)

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
−1.26509e6
−708531.
447512.
1.52611e6
−1.38568e8 −5.95414e12 1.01938e16 −5.35900e18 8.25051e20 2.55363e22 −1.64427e23 1.60685e25 7.42585e26
1.2 −8.51381e7 6.54009e12 −1.75871e15 2.26338e17 −5.56810e20 −3.24923e22 9.16589e23 2.33895e25 −1.92700e25
1.3 2.58420e7 −2.97907e12 −8.33939e15 5.38136e18 −7.69850e19 2.33436e22 −4.48271e23 −1.05084e25 1.39065e26
1.4 1.29387e8 1.34470e12 7.73392e15 −4.81259e18 1.73988e20 −1.66125e22 −1.64746e23 −1.75750e25 −6.22689e26
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Hecke kernels

There are no other newforms in \(S_{54}^{\mathrm{new}}(\Gamma_0(1))\).