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: \( 2^{27}\cdot 3^{8}\cdot 5^{3}\cdot 7\cdot 13 \)
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.

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

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

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

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.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1.54.a.a 4
3.b odd 2 1 9.54.a.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.54.a.a 4 1.a even 1 1 trivial
9.54.a.b 4 3.b odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{54}^{\mathrm{new}}(\Gamma_0(1))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 68476320 T + 16444081999953920 T^{2} + \)\(76\!\cdots\!40\)\( T^{3} + \)\(17\!\cdots\!28\)\( T^{4} + \)\(69\!\cdots\!80\)\( T^{5} + \)\(13\!\cdots\!80\)\( T^{6} + \)\(50\!\cdots\!60\)\( T^{7} + \)\(65\!\cdots\!96\)\( T^{8} \)
$3$ \( 1 + 1048411007280 T + \)\(33\!\cdots\!40\)\( T^{2} - \)\(33\!\cdots\!60\)\( T^{3} + \)\(70\!\cdots\!58\)\( T^{4} - \)\(64\!\cdots\!80\)\( T^{5} + \)\(12\!\cdots\!60\)\( T^{6} + \)\(76\!\cdots\!60\)\( T^{7} + \)\(14\!\cdots\!41\)\( T^{8} \)
$5$ \( 1 + 4563895793294313000 T + \)\(14\!\cdots\!00\)\( T^{2} + \)\(19\!\cdots\!00\)\( T^{3} + \)\(10\!\cdots\!50\)\( T^{4} + \)\(21\!\cdots\!00\)\( T^{5} + \)\(17\!\cdots\!00\)\( T^{6} + \)\(62\!\cdots\!00\)\( T^{7} + \)\(15\!\cdots\!25\)\( T^{8} \)
$7$ \( 1 + \)\(22\!\cdots\!00\)\( T + \)\(12\!\cdots\!00\)\( T^{2} + \)\(33\!\cdots\!00\)\( T^{3} + \)\(10\!\cdots\!98\)\( T^{4} + \)\(20\!\cdots\!00\)\( T^{5} + \)\(45\!\cdots\!00\)\( T^{6} + \)\(52\!\cdots\!00\)\( T^{7} + \)\(14\!\cdots\!01\)\( T^{8} \)
$11$ \( 1 - \)\(24\!\cdots\!28\)\( T + \)\(45\!\cdots\!68\)\( T^{2} - \)\(92\!\cdots\!76\)\( T^{3} + \)\(96\!\cdots\!70\)\( T^{4} - \)\(14\!\cdots\!56\)\( T^{5} + \)\(11\!\cdots\!48\)\( T^{6} - \)\(94\!\cdots\!48\)\( T^{7} + \)\(59\!\cdots\!21\)\( T^{8} \)
$13$ \( 1 - \)\(38\!\cdots\!40\)\( T + \)\(15\!\cdots\!80\)\( T^{2} + \)\(15\!\cdots\!80\)\( T^{3} + \)\(12\!\cdots\!18\)\( T^{4} + \)\(16\!\cdots\!40\)\( T^{5} + \)\(19\!\cdots\!20\)\( T^{6} - \)\(50\!\cdots\!80\)\( T^{7} + \)\(14\!\cdots\!81\)\( T^{8} \)
$17$ \( 1 + \)\(86\!\cdots\!60\)\( T + \)\(53\!\cdots\!60\)\( T^{2} + \)\(20\!\cdots\!20\)\( T^{3} + \)\(89\!\cdots\!38\)\( T^{4} + \)\(33\!\cdots\!40\)\( T^{5} + \)\(14\!\cdots\!40\)\( T^{6} + \)\(38\!\cdots\!80\)\( T^{7} + \)\(71\!\cdots\!61\)\( T^{8} \)
$19$ \( 1 + \)\(25\!\cdots\!60\)\( T + \)\(37\!\cdots\!36\)\( T^{2} + \)\(37\!\cdots\!20\)\( T^{3} + \)\(31\!\cdots\!86\)\( T^{4} + \)\(22\!\cdots\!80\)\( T^{5} + \)\(13\!\cdots\!16\)\( T^{6} + \)\(53\!\cdots\!40\)\( T^{7} + \)\(12\!\cdots\!61\)\( T^{8} \)
$23$ \( 1 - \)\(69\!\cdots\!60\)\( T + \)\(50\!\cdots\!20\)\( T^{2} - \)\(27\!\cdots\!80\)\( T^{3} + \)\(10\!\cdots\!78\)\( T^{4} - \)\(40\!\cdots\!40\)\( T^{5} + \)\(11\!\cdots\!80\)\( T^{6} - \)\(22\!\cdots\!20\)\( T^{7} + \)\(48\!\cdots\!21\)\( T^{8} \)
$29$ \( 1 + \)\(14\!\cdots\!40\)\( T + \)\(18\!\cdots\!56\)\( T^{2} + \)\(14\!\cdots\!80\)\( T^{3} + \)\(10\!\cdots\!26\)\( T^{4} + \)\(46\!\cdots\!20\)\( T^{5} + \)\(19\!\cdots\!76\)\( T^{6} + \)\(46\!\cdots\!60\)\( T^{7} + \)\(10\!\cdots\!41\)\( T^{8} \)
$31$ \( 1 + \)\(39\!\cdots\!32\)\( T + \)\(33\!\cdots\!48\)\( T^{2} + \)\(99\!\cdots\!84\)\( T^{3} + \)\(52\!\cdots\!70\)\( T^{4} + \)\(10\!\cdots\!44\)\( T^{5} + \)\(40\!\cdots\!88\)\( T^{6} + \)\(53\!\cdots\!72\)\( T^{7} + \)\(14\!\cdots\!61\)\( T^{8} \)
$37$ \( 1 + \)\(28\!\cdots\!80\)\( T + \)\(36\!\cdots\!80\)\( T^{2} + \)\(12\!\cdots\!60\)\( T^{3} + \)\(60\!\cdots\!18\)\( T^{4} + \)\(15\!\cdots\!20\)\( T^{5} + \)\(61\!\cdots\!20\)\( T^{6} + \)\(62\!\cdots\!40\)\( T^{7} + \)\(28\!\cdots\!81\)\( T^{8} \)
$41$ \( 1 - \)\(81\!\cdots\!88\)\( T + \)\(68\!\cdots\!88\)\( T^{2} + \)\(84\!\cdots\!64\)\( T^{3} + \)\(22\!\cdots\!70\)\( T^{4} + \)\(25\!\cdots\!44\)\( T^{5} + \)\(61\!\cdots\!08\)\( T^{6} - \)\(22\!\cdots\!68\)\( T^{7} + \)\(81\!\cdots\!81\)\( T^{8} \)
$43$ \( 1 - \)\(45\!\cdots\!00\)\( T + \)\(22\!\cdots\!00\)\( T^{2} - \)\(55\!\cdots\!00\)\( T^{3} + \)\(13\!\cdots\!98\)\( T^{4} - \)\(20\!\cdots\!00\)\( T^{5} + \)\(31\!\cdots\!00\)\( T^{6} - \)\(24\!\cdots\!00\)\( T^{7} + \)\(19\!\cdots\!01\)\( T^{8} \)
$47$ \( 1 + \)\(45\!\cdots\!40\)\( T + \)\(21\!\cdots\!40\)\( T^{2} + \)\(57\!\cdots\!80\)\( T^{3} + \)\(14\!\cdots\!58\)\( T^{4} + \)\(24\!\cdots\!60\)\( T^{5} + \)\(38\!\cdots\!60\)\( T^{6} + \)\(32\!\cdots\!20\)\( T^{7} + \)\(30\!\cdots\!41\)\( T^{8} \)
$53$ \( 1 - \)\(38\!\cdots\!20\)\( T + \)\(69\!\cdots\!40\)\( T^{2} - \)\(16\!\cdots\!60\)\( T^{3} + \)\(21\!\cdots\!58\)\( T^{4} - \)\(40\!\cdots\!80\)\( T^{5} + \)\(41\!\cdots\!60\)\( T^{6} - \)\(56\!\cdots\!40\)\( T^{7} + \)\(35\!\cdots\!41\)\( T^{8} \)
$59$ \( 1 - \)\(16\!\cdots\!20\)\( T + \)\(23\!\cdots\!16\)\( T^{2} - \)\(19\!\cdots\!40\)\( T^{3} + \)\(20\!\cdots\!46\)\( T^{4} - \)\(14\!\cdots\!60\)\( T^{5} + \)\(12\!\cdots\!56\)\( T^{6} - \)\(59\!\cdots\!80\)\( T^{7} + \)\(26\!\cdots\!81\)\( T^{8} \)
$61$ \( 1 - \)\(65\!\cdots\!28\)\( T + \)\(79\!\cdots\!68\)\( T^{2} - \)\(10\!\cdots\!76\)\( T^{3} + \)\(33\!\cdots\!70\)\( T^{4} - \)\(42\!\cdots\!56\)\( T^{5} + \)\(14\!\cdots\!48\)\( T^{6} - \)\(48\!\cdots\!48\)\( T^{7} + \)\(30\!\cdots\!21\)\( T^{8} \)
$67$ \( 1 + \)\(11\!\cdots\!60\)\( T + \)\(16\!\cdots\!60\)\( T^{2} + \)\(27\!\cdots\!20\)\( T^{3} + \)\(12\!\cdots\!38\)\( T^{4} + \)\(16\!\cdots\!40\)\( T^{5} + \)\(60\!\cdots\!40\)\( T^{6} + \)\(24\!\cdots\!80\)\( T^{7} + \)\(13\!\cdots\!61\)\( T^{8} \)
$71$ \( 1 - \)\(35\!\cdots\!48\)\( T + \)\(37\!\cdots\!08\)\( T^{2} - \)\(14\!\cdots\!96\)\( T^{3} + \)\(65\!\cdots\!70\)\( T^{4} - \)\(19\!\cdots\!56\)\( T^{5} + \)\(64\!\cdots\!68\)\( T^{6} - \)\(79\!\cdots\!88\)\( T^{7} + \)\(29\!\cdots\!41\)\( T^{8} \)
$73$ \( 1 + \)\(70\!\cdots\!40\)\( T + \)\(40\!\cdots\!20\)\( T^{2} + \)\(14\!\cdots\!20\)\( T^{3} + \)\(41\!\cdots\!78\)\( T^{4} + \)\(80\!\cdots\!60\)\( T^{5} + \)\(13\!\cdots\!80\)\( T^{6} + \)\(13\!\cdots\!80\)\( T^{7} + \)\(10\!\cdots\!21\)\( T^{8} \)
$79$ \( 1 + \)\(15\!\cdots\!40\)\( T + \)\(10\!\cdots\!56\)\( T^{2} + \)\(11\!\cdots\!80\)\( T^{3} + \)\(51\!\cdots\!26\)\( T^{4} + \)\(44\!\cdots\!20\)\( T^{5} + \)\(14\!\cdots\!76\)\( T^{6} + \)\(82\!\cdots\!60\)\( T^{7} + \)\(19\!\cdots\!41\)\( T^{8} \)
$83$ \( 1 + \)\(26\!\cdots\!20\)\( T + \)\(41\!\cdots\!60\)\( T^{2} + \)\(45\!\cdots\!60\)\( T^{3} + \)\(37\!\cdots\!38\)\( T^{4} + \)\(23\!\cdots\!80\)\( T^{5} + \)\(11\!\cdots\!40\)\( T^{6} + \)\(35\!\cdots\!40\)\( T^{7} + \)\(69\!\cdots\!61\)\( T^{8} \)
$89$ \( 1 + \)\(37\!\cdots\!20\)\( T + \)\(51\!\cdots\!76\)\( T^{2} + \)\(97\!\cdots\!40\)\( T^{3} + \)\(12\!\cdots\!66\)\( T^{4} + \)\(20\!\cdots\!60\)\( T^{5} + \)\(22\!\cdots\!36\)\( T^{6} + \)\(33\!\cdots\!80\)\( T^{7} + \)\(18\!\cdots\!21\)\( T^{8} \)
$97$ \( 1 - \)\(10\!\cdots\!60\)\( T + \)\(42\!\cdots\!40\)\( T^{2} - \)\(14\!\cdots\!20\)\( T^{3} + \)\(11\!\cdots\!58\)\( T^{4} - \)\(28\!\cdots\!40\)\( T^{5} + \)\(16\!\cdots\!60\)\( T^{6} - \)\(81\!\cdots\!80\)\( T^{7} + \)\(15\!\cdots\!41\)\( T^{8} \)
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