Properties

Label 1.58.a.a
Level 1
Weight 58
Character orbit 1.a
Self dual yes
Analytic conductor 20.577
Analytic rank 1
Dimension 4
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 1 \)
Weight: \( k \) \(=\) \( 58 \)
Character orbit: \([\chi]\) \(=\) 1.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(20.5766433651\)
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: \( 2^{25}\cdot 3^{10}\cdot 5^{2}\cdot 7\cdot 19 \)
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.

\(f(q)\) \(=\) \( q +(-54436140 + \beta_{1}) q^{2} +(9368965543140 - 37249 \beta_{1} + \beta_{2}) q^{3} +(73281224077844032 - 121220263 \beta_{1} + 610 \beta_{2} + \beta_{3}) q^{4} +(-26740319051960674650 + 45065074460 \beta_{1} - 257940 \beta_{2} - 360 \beta_{3}) q^{5} +(-\)\(84\!\cdots\!28\)\( + 15802420234572 \beta_{1} - 145347120 \beta_{2} - 105624 \beta_{3}) q^{6} +(\)\(23\!\cdots\!00\)\( + 142544382259118 \beta_{1} - 21508096526 \beta_{2} + 6492640 \beta_{3}) q^{7} +(-\)\(22\!\cdots\!20\)\( + 59447165194026704 \beta_{1} - 1415339625312 \beta_{2} - 136828080 \beta_{3}) q^{8} +(\)\(20\!\cdots\!33\)\( - 1450085833261199736 \beta_{1} + 14657512147560 \beta_{2} + 245717712 \beta_{3}) q^{9} +O(q^{10})\) \( q +(-54436140 + \beta_{1}) q^{2} +(9368965543140 - 37249 \beta_{1} + \beta_{2}) q^{3} +(73281224077844032 - 121220263 \beta_{1} + 610 \beta_{2} + \beta_{3}) q^{4} +(-26740319051960674650 + 45065074460 \beta_{1} - 257940 \beta_{2} - 360 \beta_{3}) q^{5} +(-\)\(84\!\cdots\!28\)\( + 15802420234572 \beta_{1} - 145347120 \beta_{2} - 105624 \beta_{3}) q^{6} +(\)\(23\!\cdots\!00\)\( + 142544382259118 \beta_{1} - 21508096526 \beta_{2} + 6492640 \beta_{3}) q^{7} +(-\)\(22\!\cdots\!20\)\( + 59447165194026704 \beta_{1} - 1415339625312 \beta_{2} - 136828080 \beta_{3}) q^{8} +(\)\(20\!\cdots\!33\)\( - 1450085833261199736 \beta_{1} + 14657512147560 \beta_{2} + 245717712 \beta_{3}) q^{9} +(\)\(11\!\cdots\!00\)\( - 73888027479641467130 \beta_{1} + 515093442096320 \beta_{2} + 53305386080 \beta_{3}) q^{10} +(-\)\(11\!\cdots\!28\)\( - \)\(80\!\cdots\!75\)\( \beta_{1} - 9128019995338365 \beta_{2} - 1480064351040 \beta_{3}) q^{11} +(\)\(25\!\cdots\!20\)\( - \)\(17\!\cdots\!36\)\( \beta_{1} + 17130223695919816 \beta_{2} + 22983644617380 \beta_{3}) q^{12} +(-\)\(89\!\cdots\!70\)\( - \)\(22\!\cdots\!52\)\( \beta_{1} + 683468672788458700 \beta_{2} - 242325085843880 \beta_{3}) q^{13} +(\)\(17\!\cdots\!84\)\( + \)\(92\!\cdots\!56\)\( \beta_{1} - 5499150697325063520 \beta_{2} + 1782624443857488 \beta_{3}) q^{14} +(-\)\(96\!\cdots\!00\)\( + \)\(63\!\cdots\!10\)\( \beta_{1} - 6728508434120677290 \beta_{2} - 8283284992481760 \beta_{3}) q^{15} +(\)\(33\!\cdots\!36\)\( - \)\(30\!\cdots\!36\)\( \beta_{1} + \)\(29\!\cdots\!40\)\( \beta_{2} + 8534483450280192 \beta_{3}) q^{16} +(\)\(34\!\cdots\!30\)\( - \)\(19\!\cdots\!08\)\( \beta_{1} - \)\(12\!\cdots\!08\)\( \beta_{2} + 236164889075956560 \beta_{3}) q^{17} +(-\)\(32\!\cdots\!60\)\( + \)\(38\!\cdots\!01\)\( \beta_{1} - \)\(29\!\cdots\!44\)\( \beta_{2} - 2445879764812789440 \beta_{3}) q^{18} +(\)\(84\!\cdots\!40\)\( + \)\(18\!\cdots\!63\)\( \beta_{1} + \)\(47\!\cdots\!85\)\( \beta_{2} + 14227278615873217344 \beta_{3}) q^{19} +(-\)\(12\!\cdots\!00\)\( + \)\(17\!\cdots\!70\)\( \beta_{1} - \)\(13\!\cdots\!80\)\( \beta_{2} - 55834753565056020570 \beta_{3}) q^{20} +(-\)\(28\!\cdots\!88\)\( - \)\(76\!\cdots\!76\)\( \beta_{1} - \)\(25\!\cdots\!80\)\( \beta_{2} + \)\(14\!\cdots\!52\)\( \beta_{3}) q^{21} +(-\)\(16\!\cdots\!80\)\( - \)\(27\!\cdots\!28\)\( \beta_{1} + \)\(25\!\cdots\!80\)\( \beta_{2} - \)\(21\!\cdots\!20\)\( \beta_{3}) q^{22} +(-\)\(74\!\cdots\!80\)\( + \)\(72\!\cdots\!10\)\( \beta_{1} - \)\(26\!\cdots\!86\)\( \beta_{2} + \)\(27\!\cdots\!00\)\( \beta_{3}) q^{23} +(-\)\(26\!\cdots\!80\)\( + \)\(42\!\cdots\!44\)\( \beta_{1} - \)\(20\!\cdots\!60\)\( \beta_{2} - \)\(27\!\cdots\!68\)\( \beta_{3}) q^{24} +(-\)\(23\!\cdots\!25\)\( - \)\(67\!\cdots\!00\)\( \beta_{1} + \)\(50\!\cdots\!00\)\( \beta_{2} + \)\(20\!\cdots\!00\)\( \beta_{3}) q^{25} +(-\)\(43\!\cdots\!68\)\( - \)\(33\!\cdots\!74\)\( \beta_{1} + \)\(20\!\cdots\!40\)\( \beta_{2} - \)\(75\!\cdots\!92\)\( \beta_{3}) q^{26} +(\)\(19\!\cdots\!80\)\( + \)\(19\!\cdots\!74\)\( \beta_{1} - \)\(12\!\cdots\!22\)\( \beta_{2} + \)\(10\!\cdots\!20\)\( \beta_{3}) q^{27} +(\)\(16\!\cdots\!80\)\( + \)\(12\!\cdots\!60\)\( \beta_{1} + \)\(20\!\cdots\!16\)\( \beta_{2} + \)\(40\!\cdots\!80\)\( \beta_{3}) q^{28} +(\)\(17\!\cdots\!10\)\( + \)\(76\!\cdots\!72\)\( \beta_{1} - \)\(17\!\cdots\!00\)\( \beta_{2} - \)\(26\!\cdots\!04\)\( \beta_{3}) q^{29} +(\)\(14\!\cdots\!00\)\( - \)\(24\!\cdots\!80\)\( \beta_{1} + \)\(15\!\cdots\!20\)\( \beta_{2} + \)\(65\!\cdots\!80\)\( \beta_{3}) q^{30} +(-\)\(22\!\cdots\!88\)\( - \)\(93\!\cdots\!00\)\( \beta_{1} - \)\(78\!\cdots\!20\)\( \beta_{2} - \)\(30\!\cdots\!20\)\( \beta_{3}) q^{31} +(-\)\(35\!\cdots\!40\)\( - \)\(94\!\cdots\!84\)\( \beta_{1} + \)\(13\!\cdots\!60\)\( \beta_{2} - \)\(30\!\cdots\!40\)\( \beta_{3}) q^{32} +(-\)\(73\!\cdots\!20\)\( + \)\(17\!\cdots\!72\)\( \beta_{1} + \)\(41\!\cdots\!12\)\( \beta_{2} + \)\(99\!\cdots\!40\)\( \beta_{3}) q^{33} +(-\)\(42\!\cdots\!56\)\( + \)\(39\!\cdots\!74\)\( \beta_{1} - \)\(26\!\cdots\!60\)\( \beta_{2} - \)\(10\!\cdots\!28\)\( \beta_{3}) q^{34} +(-\)\(57\!\cdots\!00\)\( - \)\(42\!\cdots\!80\)\( \beta_{1} - \)\(73\!\cdots\!80\)\( \beta_{2} - \)\(15\!\cdots\!20\)\( \beta_{3}) q^{35} +(\)\(71\!\cdots\!56\)\( - \)\(44\!\cdots\!11\)\( \beta_{1} + \)\(15\!\cdots\!90\)\( \beta_{2} + \)\(49\!\cdots\!17\)\( \beta_{3}) q^{36} +(\)\(27\!\cdots\!90\)\( + \)\(78\!\cdots\!00\)\( \beta_{1} + \)\(65\!\cdots\!88\)\( \beta_{2} + \)\(23\!\cdots\!40\)\( \beta_{3}) q^{37} +(\)\(38\!\cdots\!20\)\( + \)\(18\!\cdots\!96\)\( \beta_{1} - \)\(22\!\cdots\!28\)\( \beta_{2} - \)\(10\!\cdots\!60\)\( \beta_{3}) q^{38} +(\)\(10\!\cdots\!76\)\( + \)\(26\!\cdots\!66\)\( \beta_{1} + \)\(10\!\cdots\!70\)\( \beta_{2} - \)\(31\!\cdots\!92\)\( \beta_{3}) q^{39} +(\)\(29\!\cdots\!00\)\( - \)\(10\!\cdots\!00\)\( \beta_{1} + \)\(24\!\cdots\!00\)\( \beta_{2} + \)\(18\!\cdots\!00\)\( \beta_{3}) q^{40} +(-\)\(27\!\cdots\!18\)\( - \)\(29\!\cdots\!00\)\( \beta_{1} + \)\(59\!\cdots\!80\)\( \beta_{2} - \)\(21\!\cdots\!20\)\( \beta_{3}) q^{41} +(-\)\(14\!\cdots\!20\)\( - \)\(72\!\cdots\!00\)\( \beta_{1} - \)\(19\!\cdots\!24\)\( \beta_{2} - \)\(54\!\cdots\!60\)\( \beta_{3}) q^{42} +(-\)\(15\!\cdots\!00\)\( + \)\(75\!\cdots\!29\)\( \beta_{1} - \)\(24\!\cdots\!53\)\( \beta_{2} + \)\(21\!\cdots\!00\)\( \beta_{3}) q^{43} +(-\)\(33\!\cdots\!96\)\( - \)\(47\!\cdots\!36\)\( \beta_{1} + \)\(11\!\cdots\!40\)\( \beta_{2} - \)\(23\!\cdots\!08\)\( \beta_{3}) q^{44} +(-\)\(26\!\cdots\!50\)\( + \)\(16\!\cdots\!80\)\( \beta_{1} - \)\(58\!\cdots\!20\)\( \beta_{2} - \)\(18\!\cdots\!80\)\( \beta_{3}) q^{45} +(\)\(19\!\cdots\!92\)\( - \)\(77\!\cdots\!40\)\( \beta_{1} + \)\(42\!\cdots\!80\)\( \beta_{2} + \)\(91\!\cdots\!60\)\( \beta_{3}) q^{46} +(\)\(90\!\cdots\!20\)\( + \)\(76\!\cdots\!84\)\( \beta_{1} - \)\(61\!\cdots\!84\)\( \beta_{2} - \)\(15\!\cdots\!00\)\( \beta_{3}) q^{47} +(\)\(68\!\cdots\!20\)\( - \)\(82\!\cdots\!60\)\( \beta_{1} + \)\(61\!\cdots\!64\)\( \beta_{2} + \)\(22\!\cdots\!40\)\( \beta_{3}) q^{48} +(\)\(30\!\cdots\!57\)\( + \)\(27\!\cdots\!20\)\( \beta_{1} + \)\(20\!\cdots\!20\)\( \beta_{2} - \)\(10\!\cdots\!20\)\( \beta_{3}) q^{49} +(-\)\(13\!\cdots\!00\)\( + \)\(72\!\cdots\!75\)\( \beta_{1} - \)\(36\!\cdots\!00\)\( \beta_{2} - \)\(96\!\cdots\!00\)\( \beta_{3}) q^{50} +(-\)\(19\!\cdots\!08\)\( - \)\(50\!\cdots\!82\)\( \beta_{1} + \)\(26\!\cdots\!10\)\( \beta_{2} + \)\(28\!\cdots\!84\)\( \beta_{3}) q^{51} +(-\)\(57\!\cdots\!00\)\( - \)\(70\!\cdots\!62\)\( \beta_{1} - \)\(48\!\cdots\!96\)\( \beta_{2} - \)\(15\!\cdots\!50\)\( \beta_{3}) q^{52} +(-\)\(88\!\cdots\!10\)\( - \)\(25\!\cdots\!64\)\( \beta_{1} + \)\(18\!\cdots\!16\)\( \beta_{2} - \)\(66\!\cdots\!80\)\( \beta_{3}) q^{53} +(\)\(30\!\cdots\!40\)\( + \)\(25\!\cdots\!68\)\( \beta_{1} + \)\(32\!\cdots\!60\)\( \beta_{2} + \)\(10\!\cdots\!84\)\( \beta_{3}) q^{54} +(\)\(12\!\cdots\!00\)\( + \)\(16\!\cdots\!70\)\( \beta_{1} - \)\(40\!\cdots\!30\)\( \beta_{2} + \)\(82\!\cdots\!80\)\( \beta_{3}) q^{55} +(\)\(15\!\cdots\!40\)\( + \)\(78\!\cdots\!48\)\( \beta_{1} + \)\(98\!\cdots\!00\)\( \beta_{2} - \)\(26\!\cdots\!36\)\( \beta_{3}) q^{56} +(\)\(51\!\cdots\!60\)\( - \)\(18\!\cdots\!92\)\( \beta_{1} - \)\(96\!\cdots\!64\)\( \beta_{2} - \)\(19\!\cdots\!00\)\( \beta_{3}) q^{57} +(\)\(15\!\cdots\!80\)\( - \)\(20\!\cdots\!26\)\( \beta_{1} + \)\(40\!\cdots\!48\)\( \beta_{2} + \)\(81\!\cdots\!40\)\( \beta_{3}) q^{58} +(\)\(16\!\cdots\!20\)\( - \)\(23\!\cdots\!91\)\( \beta_{1} - \)\(65\!\cdots\!05\)\( \beta_{2} + \)\(22\!\cdots\!32\)\( \beta_{3}) q^{59} +(-\)\(45\!\cdots\!00\)\( + \)\(15\!\cdots\!20\)\( \beta_{1} - \)\(10\!\cdots\!80\)\( \beta_{2} - \)\(20\!\cdots\!20\)\( \beta_{3}) q^{60} +(-\)\(38\!\cdots\!78\)\( + \)\(90\!\cdots\!00\)\( \beta_{1} + \)\(55\!\cdots\!00\)\( \beta_{2} + \)\(24\!\cdots\!00\)\( \beta_{3}) q^{61} +(-\)\(18\!\cdots\!80\)\( - \)\(84\!\cdots\!88\)\( \beta_{1} + \)\(12\!\cdots\!40\)\( \beta_{2} + \)\(43\!\cdots\!40\)\( \beta_{3}) q^{62} +(-\)\(39\!\cdots\!40\)\( - \)\(18\!\cdots\!58\)\( \beta_{1} + \)\(15\!\cdots\!18\)\( \beta_{2} + \)\(63\!\cdots\!40\)\( \beta_{3}) q^{63} +(-\)\(49\!\cdots\!28\)\( - \)\(22\!\cdots\!16\)\( \beta_{1} - \)\(20\!\cdots\!40\)\( \beta_{2} - \)\(12\!\cdots\!28\)\( \beta_{3}) q^{64} +(\)\(20\!\cdots\!00\)\( + \)\(24\!\cdots\!60\)\( \beta_{1} + \)\(17\!\cdots\!60\)\( \beta_{2} + \)\(55\!\cdots\!40\)\( \beta_{3}) q^{65} +(\)\(41\!\cdots\!84\)\( + \)\(36\!\cdots\!84\)\( \beta_{1} - \)\(12\!\cdots\!20\)\( \beta_{2} + \)\(17\!\cdots\!92\)\( \beta_{3}) q^{66} +(\)\(38\!\cdots\!80\)\( + \)\(17\!\cdots\!87\)\( \beta_{1} + \)\(15\!\cdots\!57\)\( \beta_{2} - \)\(17\!\cdots\!40\)\( \beta_{3}) q^{67} +(\)\(10\!\cdots\!40\)\( - \)\(29\!\cdots\!42\)\( \beta_{1} + \)\(37\!\cdots\!12\)\( \beta_{2} + \)\(20\!\cdots\!30\)\( \beta_{3}) q^{68} +(-\)\(16\!\cdots\!44\)\( + \)\(35\!\cdots\!08\)\( \beta_{1} - \)\(83\!\cdots\!80\)\( \beta_{2} - \)\(63\!\cdots\!36\)\( \beta_{3}) q^{69} +(-\)\(60\!\cdots\!00\)\( - \)\(75\!\cdots\!60\)\( \beta_{1} + \)\(25\!\cdots\!40\)\( \beta_{2} + \)\(35\!\cdots\!60\)\( \beta_{3}) q^{70} +(-\)\(18\!\cdots\!08\)\( - \)\(31\!\cdots\!50\)\( \beta_{1} + \)\(31\!\cdots\!50\)\( \beta_{2} + \)\(21\!\cdots\!00\)\( \beta_{3}) q^{71} +(-\)\(52\!\cdots\!60\)\( + \)\(11\!\cdots\!52\)\( \beta_{1} - \)\(65\!\cdots\!36\)\( \beta_{2} - \)\(18\!\cdots\!20\)\( \beta_{3}) q^{72} +(-\)\(32\!\cdots\!30\)\( + \)\(14\!\cdots\!20\)\( \beta_{1} + \)\(20\!\cdots\!24\)\( \beta_{2} - \)\(59\!\cdots\!60\)\( \beta_{3}) q^{73} +(\)\(19\!\cdots\!64\)\( + \)\(29\!\cdots\!30\)\( \beta_{1} - \)\(75\!\cdots\!20\)\( \beta_{2} - \)\(36\!\cdots\!80\)\( \beta_{3}) q^{74} +(\)\(10\!\cdots\!00\)\( - \)\(29\!\cdots\!75\)\( \beta_{1} - \)\(30\!\cdots\!25\)\( \beta_{2} + \)\(77\!\cdots\!00\)\( \beta_{3}) q^{75} +(\)\(36\!\cdots\!80\)\( - \)\(21\!\cdots\!64\)\( \beta_{1} - \)\(16\!\cdots\!00\)\( \beta_{2} + \)\(13\!\cdots\!48\)\( \beta_{3}) q^{76} +(-\)\(26\!\cdots\!00\)\( - \)\(74\!\cdots\!04\)\( \beta_{1} + \)\(61\!\cdots\!08\)\( \beta_{2} - \)\(40\!\cdots\!40\)\( \beta_{3}) q^{77} +(\)\(51\!\cdots\!00\)\( + \)\(53\!\cdots\!68\)\( \beta_{1} + \)\(42\!\cdots\!04\)\( \beta_{2} + \)\(18\!\cdots\!80\)\( \beta_{3}) q^{78} +(-\)\(20\!\cdots\!40\)\( - \)\(16\!\cdots\!08\)\( \beta_{1} - \)\(56\!\cdots\!20\)\( \beta_{2} + \)\(30\!\cdots\!36\)\( \beta_{3}) q^{79} +(-\)\(57\!\cdots\!00\)\( + \)\(33\!\cdots\!60\)\( \beta_{1} - \)\(12\!\cdots\!40\)\( \beta_{2} - \)\(35\!\cdots\!60\)\( \beta_{3}) q^{80} +(-\)\(19\!\cdots\!59\)\( + \)\(12\!\cdots\!32\)\( \beta_{1} - \)\(14\!\cdots\!40\)\( \beta_{2} + \)\(33\!\cdots\!36\)\( \beta_{3}) q^{81} +(-\)\(47\!\cdots\!80\)\( - \)\(48\!\cdots\!18\)\( \beta_{1} + \)\(17\!\cdots\!40\)\( \beta_{2} - \)\(74\!\cdots\!60\)\( \beta_{3}) q^{82} +(-\)\(24\!\cdots\!40\)\( - \)\(24\!\cdots\!53\)\( \beta_{1} - \)\(10\!\cdots\!27\)\( \beta_{2} + \)\(11\!\cdots\!00\)\( \beta_{3}) q^{83} +(\)\(33\!\cdots\!84\)\( - \)\(10\!\cdots\!68\)\( \beta_{1} + \)\(12\!\cdots\!00\)\( \beta_{2} - \)\(16\!\cdots\!24\)\( \beta_{3}) q^{84} +(-\)\(37\!\cdots\!00\)\( + \)\(10\!\cdots\!20\)\( \beta_{1} - \)\(13\!\cdots\!80\)\( \beta_{2} - \)\(86\!\cdots\!20\)\( \beta_{3}) q^{85} +(\)\(16\!\cdots\!12\)\( + \)\(44\!\cdots\!68\)\( \beta_{1} - \)\(19\!\cdots\!20\)\( \beta_{2} + \)\(97\!\cdots\!04\)\( \beta_{3}) q^{86} +(-\)\(68\!\cdots\!60\)\( + \)\(38\!\cdots\!02\)\( \beta_{1} + \)\(28\!\cdots\!74\)\( \beta_{2} - \)\(10\!\cdots\!60\)\( \beta_{3}) q^{87} +(\)\(15\!\cdots\!60\)\( - \)\(14\!\cdots\!12\)\( \beta_{1} - \)\(22\!\cdots\!64\)\( \beta_{2} - \)\(98\!\cdots\!60\)\( \beta_{3}) q^{88} +(-\)\(26\!\cdots\!70\)\( - \)\(24\!\cdots\!64\)\( \beta_{1} + \)\(35\!\cdots\!00\)\( \beta_{2} + \)\(14\!\cdots\!48\)\( \beta_{3}) q^{89} +(\)\(36\!\cdots\!00\)\( - \)\(61\!\cdots\!90\)\( \beta_{1} + \)\(39\!\cdots\!60\)\( \beta_{2} + \)\(19\!\cdots\!40\)\( \beta_{3}) q^{90} +(-\)\(63\!\cdots\!28\)\( - \)\(11\!\cdots\!72\)\( \beta_{1} - \)\(60\!\cdots\!00\)\( \beta_{2} - \)\(16\!\cdots\!96\)\( \beta_{3}) q^{91} +(-\)\(68\!\cdots\!20\)\( + \)\(25\!\cdots\!92\)\( \beta_{1} - \)\(12\!\cdots\!28\)\( \beta_{2} - \)\(81\!\cdots\!60\)\( \beta_{3}) q^{92} +(-\)\(10\!\cdots\!20\)\( + \)\(80\!\cdots\!12\)\( \beta_{1} - \)\(78\!\cdots\!68\)\( \beta_{2} + \)\(34\!\cdots\!20\)\( \beta_{3}) q^{93} +(\)\(15\!\cdots\!24\)\( - \)\(17\!\cdots\!32\)\( \beta_{1} + \)\(31\!\cdots\!60\)\( \beta_{2} + \)\(11\!\cdots\!84\)\( \beta_{3}) q^{94} +(-\)\(13\!\cdots\!00\)\( + \)\(83\!\cdots\!50\)\( \beta_{1} + \)\(61\!\cdots\!50\)\( \beta_{2} - \)\(47\!\cdots\!00\)\( \beta_{3}) q^{95} +(\)\(16\!\cdots\!52\)\( + \)\(42\!\cdots\!52\)\( \beta_{1} - \)\(11\!\cdots\!80\)\( \beta_{2} - \)\(79\!\cdots\!44\)\( \beta_{3}) q^{96} +(-\)\(38\!\cdots\!30\)\( - \)\(10\!\cdots\!96\)\( \beta_{1} - \)\(52\!\cdots\!24\)\( \beta_{2} - \)\(97\!\cdots\!80\)\( \beta_{3}) q^{97} +(\)\(57\!\cdots\!20\)\( + \)\(70\!\cdots\!97\)\( \beta_{1} + \)\(56\!\cdots\!40\)\( \beta_{2} + \)\(13\!\cdots\!60\)\( \beta_{3}) q^{98} +(\)\(30\!\cdots\!76\)\( + \)\(48\!\cdots\!33\)\( \beta_{1} - \)\(13\!\cdots\!25\)\( \beta_{2} + \)\(15\!\cdots\!44\)\( \beta_{3}) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 217744560q^{2} + 37475862172560q^{3} + 293124896311376128q^{4} - \)\(10\!\cdots\!00\)\(q^{5} - \)\(33\!\cdots\!12\)\(q^{6} + \)\(95\!\cdots\!00\)\(q^{7} - \)\(88\!\cdots\!80\)\(q^{8} + \)\(80\!\cdots\!32\)\(q^{9} + O(q^{10}) \) \( 4q - 217744560q^{2} + 37475862172560q^{3} + 293124896311376128q^{4} - \)\(10\!\cdots\!00\)\(q^{5} - \)\(33\!\cdots\!12\)\(q^{6} + \)\(95\!\cdots\!00\)\(q^{7} - \)\(88\!\cdots\!80\)\(q^{8} + \)\(80\!\cdots\!32\)\(q^{9} + \)\(44\!\cdots\!00\)\(q^{10} - \)\(46\!\cdots\!12\)\(q^{11} + \)\(10\!\cdots\!80\)\(q^{12} - \)\(35\!\cdots\!80\)\(q^{13} + \)\(70\!\cdots\!36\)\(q^{14} - \)\(38\!\cdots\!00\)\(q^{15} + \)\(13\!\cdots\!44\)\(q^{16} + \)\(13\!\cdots\!20\)\(q^{17} - \)\(12\!\cdots\!40\)\(q^{18} + \)\(33\!\cdots\!60\)\(q^{19} - \)\(50\!\cdots\!00\)\(q^{20} - \)\(11\!\cdots\!52\)\(q^{21} - \)\(66\!\cdots\!20\)\(q^{22} - \)\(29\!\cdots\!20\)\(q^{23} - \)\(10\!\cdots\!20\)\(q^{24} - \)\(94\!\cdots\!00\)\(q^{25} - \)\(17\!\cdots\!72\)\(q^{26} + \)\(76\!\cdots\!20\)\(q^{27} + \)\(64\!\cdots\!20\)\(q^{28} + \)\(69\!\cdots\!40\)\(q^{29} + \)\(56\!\cdots\!00\)\(q^{30} - \)\(89\!\cdots\!52\)\(q^{31} - \)\(14\!\cdots\!60\)\(q^{32} - \)\(29\!\cdots\!80\)\(q^{33} - \)\(16\!\cdots\!24\)\(q^{34} - \)\(22\!\cdots\!00\)\(q^{35} + \)\(28\!\cdots\!24\)\(q^{36} + \)\(10\!\cdots\!60\)\(q^{37} + \)\(15\!\cdots\!80\)\(q^{38} + \)\(41\!\cdots\!04\)\(q^{39} + \)\(11\!\cdots\!00\)\(q^{40} - \)\(11\!\cdots\!72\)\(q^{41} - \)\(59\!\cdots\!80\)\(q^{42} - \)\(61\!\cdots\!00\)\(q^{43} - \)\(13\!\cdots\!84\)\(q^{44} - \)\(10\!\cdots\!00\)\(q^{45} + \)\(78\!\cdots\!68\)\(q^{46} + \)\(36\!\cdots\!80\)\(q^{47} + \)\(27\!\cdots\!80\)\(q^{48} + \)\(12\!\cdots\!28\)\(q^{49} - \)\(52\!\cdots\!00\)\(q^{50} - \)\(76\!\cdots\!32\)\(q^{51} - \)\(22\!\cdots\!00\)\(q^{52} - \)\(35\!\cdots\!40\)\(q^{53} + \)\(12\!\cdots\!60\)\(q^{54} + \)\(49\!\cdots\!00\)\(q^{55} + \)\(63\!\cdots\!60\)\(q^{56} + \)\(20\!\cdots\!40\)\(q^{57} + \)\(61\!\cdots\!20\)\(q^{58} + \)\(67\!\cdots\!80\)\(q^{59} - \)\(18\!\cdots\!00\)\(q^{60} - \)\(15\!\cdots\!12\)\(q^{61} - \)\(74\!\cdots\!20\)\(q^{62} - \)\(15\!\cdots\!60\)\(q^{63} - \)\(19\!\cdots\!12\)\(q^{64} + \)\(80\!\cdots\!00\)\(q^{65} + \)\(16\!\cdots\!36\)\(q^{66} + \)\(15\!\cdots\!20\)\(q^{67} + \)\(41\!\cdots\!60\)\(q^{68} - \)\(65\!\cdots\!76\)\(q^{69} - \)\(24\!\cdots\!00\)\(q^{70} - \)\(73\!\cdots\!32\)\(q^{71} - \)\(21\!\cdots\!40\)\(q^{72} - \)\(13\!\cdots\!20\)\(q^{73} + \)\(76\!\cdots\!56\)\(q^{74} + \)\(40\!\cdots\!00\)\(q^{75} + \)\(14\!\cdots\!20\)\(q^{76} - \)\(10\!\cdots\!00\)\(q^{77} + \)\(20\!\cdots\!00\)\(q^{78} - \)\(82\!\cdots\!60\)\(q^{79} - \)\(22\!\cdots\!00\)\(q^{80} - \)\(79\!\cdots\!36\)\(q^{81} - \)\(18\!\cdots\!20\)\(q^{82} - \)\(98\!\cdots\!60\)\(q^{83} + \)\(13\!\cdots\!36\)\(q^{84} - \)\(14\!\cdots\!00\)\(q^{85} + \)\(67\!\cdots\!48\)\(q^{86} - \)\(27\!\cdots\!40\)\(q^{87} + \)\(61\!\cdots\!40\)\(q^{88} - \)\(10\!\cdots\!80\)\(q^{89} + \)\(14\!\cdots\!00\)\(q^{90} - \)\(25\!\cdots\!12\)\(q^{91} - \)\(27\!\cdots\!80\)\(q^{92} - \)\(41\!\cdots\!80\)\(q^{93} + \)\(63\!\cdots\!96\)\(q^{94} - \)\(53\!\cdots\!00\)\(q^{95} + \)\(66\!\cdots\!08\)\(q^{96} - \)\(15\!\cdots\!20\)\(q^{97} + \)\(22\!\cdots\!80\)\(q^{98} + \)\(12\!\cdots\!04\)\(q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 20682206675887 x^{2} + 1182366456513663853 x + 45927816189452762789055234\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 144 \nu - 36 \)
\(\beta_{2}\)\(=\)\((\)\( -27 \nu^{3} + 4965084 \nu^{2} + 438929828931105 \nu - 75287058387744845106 \)\()/12043136\)
\(\beta_{3}\)\(=\)\((\)\( 8235 \nu^{3} + 123348883428 \nu^{2} - 123166652676786513 \nu - 1268261056268698604351598 \)\()/6021568\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1} + 36\)\()/144\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} + 610 \beta_{2} - 12347911 \beta_{1} + 214433118815601600\)\()/20736\)
\(\nu^{3}\)\(=\)\((\)\(45973 \beta_{3} - 2284238582 \beta_{2} + 584672101395737 \beta_{1} - 4596960370505569210512\)\()/5184\)

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
−4.29777e6
−1.55353e6
1.62926e6
4.22205e6
−6.73315e8 5.51192e13 3.09237e17 −1.13437e20 −3.71126e22 6.16816e23 −1.11179e26 1.46809e27 7.63789e28
1.2 −2.78145e8 −3.57694e13 −6.67506e16 2.54023e19 9.94908e21 4.82921e23 5.86512e25 −2.90592e26 −7.06551e27
1.3 1.80177e8 4.51574e13 −1.11652e17 3.84628e19 8.13632e21 −1.87845e24 −4.60832e25 4.69152e26 6.93010e27
1.4 5.53538e8 −2.70314e13 1.62289e17 −5.73891e19 −1.49629e22 1.73157e24 1.00602e25 −8.39347e26 −3.17671e28
\(n\): 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 1.58.a.a 4
3.b odd 2 1 9.58.a.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.58.a.a 4 1.a even 1 1 trivial
9.58.a.b 4 3.b odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{58}^{\mathrm{new}}(\Gamma_0(1))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 217744560 T + 165374274700820480 T^{2} + \)\(51\!\cdots\!20\)\( T^{3} + \)\(24\!\cdots\!68\)\( T^{4} + \)\(74\!\cdots\!40\)\( T^{5} + \)\(34\!\cdots\!20\)\( T^{6} + \)\(65\!\cdots\!80\)\( T^{7} + \)\(43\!\cdots\!56\)\( T^{8} \)
$3$ \( 1 - 37475862172560 T + \)\(34\!\cdots\!60\)\( T^{2} - \)\(11\!\cdots\!80\)\( T^{3} + \)\(82\!\cdots\!38\)\( T^{4} - \)\(18\!\cdots\!40\)\( T^{5} + \)\(84\!\cdots\!40\)\( T^{6} - \)\(14\!\cdots\!20\)\( T^{7} + \)\(60\!\cdots\!61\)\( T^{8} \)
$5$ \( 1 + \)\(10\!\cdots\!00\)\( T + \)\(24\!\cdots\!00\)\( T^{2} + \)\(19\!\cdots\!00\)\( T^{3} + \)\(24\!\cdots\!50\)\( T^{4} + \)\(13\!\cdots\!00\)\( T^{5} + \)\(11\!\cdots\!00\)\( T^{6} + \)\(35\!\cdots\!00\)\( T^{7} + \)\(23\!\cdots\!25\)\( T^{8} \)
$7$ \( 1 - \)\(95\!\cdots\!00\)\( T + \)\(28\!\cdots\!00\)\( T^{2} - \)\(61\!\cdots\!00\)\( T^{3} + \)\(29\!\cdots\!98\)\( T^{4} - \)\(90\!\cdots\!00\)\( T^{5} + \)\(61\!\cdots\!00\)\( T^{6} - \)\(30\!\cdots\!00\)\( T^{7} + \)\(48\!\cdots\!01\)\( T^{8} \)
$11$ \( 1 + \)\(46\!\cdots\!12\)\( T + \)\(37\!\cdots\!88\)\( T^{2} + \)\(52\!\cdots\!64\)\( T^{3} + \)\(56\!\cdots\!70\)\( T^{4} + \)\(11\!\cdots\!44\)\( T^{5} + \)\(19\!\cdots\!08\)\( T^{6} + \)\(55\!\cdots\!32\)\( T^{7} + \)\(27\!\cdots\!81\)\( T^{8} \)
$13$ \( 1 + \)\(35\!\cdots\!80\)\( T + \)\(84\!\cdots\!20\)\( T^{2} + \)\(23\!\cdots\!40\)\( T^{3} + \)\(32\!\cdots\!78\)\( T^{4} + \)\(73\!\cdots\!20\)\( T^{5} + \)\(82\!\cdots\!80\)\( T^{6} + \)\(10\!\cdots\!60\)\( T^{7} + \)\(95\!\cdots\!21\)\( T^{8} \)
$17$ \( 1 - \)\(13\!\cdots\!20\)\( T + \)\(30\!\cdots\!40\)\( T^{2} - \)\(12\!\cdots\!40\)\( T^{3} + \)\(55\!\cdots\!58\)\( T^{4} - \)\(17\!\cdots\!80\)\( T^{5} + \)\(57\!\cdots\!60\)\( T^{6} - \)\(35\!\cdots\!60\)\( T^{7} + \)\(34\!\cdots\!41\)\( T^{8} \)
$19$ \( 1 - \)\(33\!\cdots\!60\)\( T + \)\(12\!\cdots\!56\)\( T^{2} + \)\(22\!\cdots\!80\)\( T^{3} + \)\(79\!\cdots\!26\)\( T^{4} + \)\(17\!\cdots\!20\)\( T^{5} + \)\(77\!\cdots\!76\)\( T^{6} - \)\(15\!\cdots\!40\)\( T^{7} + \)\(35\!\cdots\!41\)\( T^{8} \)
$23$ \( 1 + \)\(29\!\cdots\!20\)\( T + \)\(47\!\cdots\!80\)\( T^{2} + \)\(49\!\cdots\!60\)\( T^{3} + \)\(37\!\cdots\!18\)\( T^{4} + \)\(20\!\cdots\!80\)\( T^{5} + \)\(81\!\cdots\!20\)\( T^{6} + \)\(21\!\cdots\!40\)\( T^{7} + \)\(29\!\cdots\!81\)\( T^{8} \)
$29$ \( 1 - \)\(69\!\cdots\!40\)\( T + \)\(46\!\cdots\!36\)\( T^{2} - \)\(10\!\cdots\!80\)\( T^{3} + \)\(55\!\cdots\!86\)\( T^{4} - \)\(24\!\cdots\!20\)\( T^{5} + \)\(23\!\cdots\!16\)\( T^{6} - \)\(81\!\cdots\!60\)\( T^{7} + \)\(26\!\cdots\!61\)\( T^{8} \)
$31$ \( 1 + \)\(89\!\cdots\!52\)\( T + \)\(23\!\cdots\!08\)\( T^{2} - \)\(59\!\cdots\!96\)\( T^{3} + \)\(25\!\cdots\!70\)\( T^{4} - \)\(61\!\cdots\!56\)\( T^{5} + \)\(23\!\cdots\!68\)\( T^{6} + \)\(94\!\cdots\!12\)\( T^{7} + \)\(10\!\cdots\!41\)\( T^{8} \)
$37$ \( 1 - \)\(10\!\cdots\!60\)\( T + \)\(12\!\cdots\!20\)\( T^{2} - \)\(80\!\cdots\!20\)\( T^{3} + \)\(51\!\cdots\!78\)\( T^{4} - \)\(19\!\cdots\!40\)\( T^{5} + \)\(77\!\cdots\!80\)\( T^{6} - \)\(15\!\cdots\!80\)\( T^{7} + \)\(35\!\cdots\!21\)\( T^{8} \)
$41$ \( 1 + \)\(11\!\cdots\!72\)\( T + \)\(34\!\cdots\!68\)\( T^{2} + \)\(27\!\cdots\!24\)\( T^{3} + \)\(44\!\cdots\!70\)\( T^{4} + \)\(23\!\cdots\!44\)\( T^{5} + \)\(25\!\cdots\!48\)\( T^{6} + \)\(67\!\cdots\!52\)\( T^{7} + \)\(51\!\cdots\!21\)\( T^{8} \)
$43$ \( 1 + \)\(61\!\cdots\!00\)\( T + \)\(15\!\cdots\!00\)\( T^{2} - \)\(88\!\cdots\!00\)\( T^{3} - \)\(51\!\cdots\!02\)\( T^{4} - \)\(11\!\cdots\!00\)\( T^{5} + \)\(25\!\cdots\!00\)\( T^{6} + \)\(12\!\cdots\!00\)\( T^{7} + \)\(26\!\cdots\!01\)\( T^{8} \)
$47$ \( 1 - \)\(36\!\cdots\!80\)\( T + \)\(38\!\cdots\!60\)\( T^{2} + \)\(33\!\cdots\!40\)\( T^{3} + \)\(38\!\cdots\!38\)\( T^{4} + \)\(67\!\cdots\!80\)\( T^{5} + \)\(16\!\cdots\!40\)\( T^{6} - \)\(30\!\cdots\!40\)\( T^{7} + \)\(17\!\cdots\!61\)\( T^{8} \)
$53$ \( 1 + \)\(35\!\cdots\!40\)\( T + \)\(91\!\cdots\!60\)\( T^{2} + \)\(16\!\cdots\!20\)\( T^{3} + \)\(24\!\cdots\!38\)\( T^{4} + \)\(30\!\cdots\!60\)\( T^{5} + \)\(33\!\cdots\!40\)\( T^{6} + \)\(25\!\cdots\!80\)\( T^{7} + \)\(13\!\cdots\!61\)\( T^{8} \)
$59$ \( 1 - \)\(67\!\cdots\!80\)\( T + \)\(32\!\cdots\!76\)\( T^{2} - \)\(17\!\cdots\!60\)\( T^{3} + \)\(40\!\cdots\!66\)\( T^{4} - \)\(15\!\cdots\!40\)\( T^{5} + \)\(24\!\cdots\!36\)\( T^{6} - \)\(44\!\cdots\!20\)\( T^{7} + \)\(56\!\cdots\!21\)\( T^{8} \)
$61$ \( 1 + \)\(15\!\cdots\!12\)\( T + \)\(31\!\cdots\!88\)\( T^{2} + \)\(28\!\cdots\!64\)\( T^{3} + \)\(29\!\cdots\!70\)\( T^{4} + \)\(16\!\cdots\!44\)\( T^{5} + \)\(10\!\cdots\!08\)\( T^{6} + \)\(30\!\cdots\!32\)\( T^{7} + \)\(11\!\cdots\!81\)\( T^{8} \)
$67$ \( 1 - \)\(15\!\cdots\!20\)\( T + \)\(36\!\cdots\!40\)\( T^{2} - \)\(45\!\cdots\!40\)\( T^{3} + \)\(64\!\cdots\!58\)\( T^{4} - \)\(55\!\cdots\!80\)\( T^{5} + \)\(53\!\cdots\!60\)\( T^{6} - \)\(27\!\cdots\!60\)\( T^{7} + \)\(22\!\cdots\!41\)\( T^{8} \)
$71$ \( 1 + \)\(73\!\cdots\!32\)\( T + \)\(12\!\cdots\!48\)\( T^{2} + \)\(61\!\cdots\!84\)\( T^{3} + \)\(59\!\cdots\!70\)\( T^{4} + \)\(20\!\cdots\!44\)\( T^{5} + \)\(13\!\cdots\!88\)\( T^{6} + \)\(27\!\cdots\!72\)\( T^{7} + \)\(12\!\cdots\!61\)\( T^{8} \)
$73$ \( 1 + \)\(13\!\cdots\!20\)\( T + \)\(59\!\cdots\!80\)\( T^{2} + \)\(56\!\cdots\!60\)\( T^{3} + \)\(13\!\cdots\!18\)\( T^{4} + \)\(91\!\cdots\!80\)\( T^{5} + \)\(15\!\cdots\!20\)\( T^{6} + \)\(55\!\cdots\!40\)\( T^{7} + \)\(68\!\cdots\!81\)\( T^{8} \)
$79$ \( 1 + \)\(82\!\cdots\!60\)\( T + \)\(56\!\cdots\!36\)\( T^{2} + \)\(34\!\cdots\!20\)\( T^{3} + \)\(12\!\cdots\!86\)\( T^{4} + \)\(50\!\cdots\!80\)\( T^{5} + \)\(11\!\cdots\!16\)\( T^{6} + \)\(25\!\cdots\!40\)\( T^{7} + \)\(45\!\cdots\!61\)\( T^{8} \)
$83$ \( 1 + \)\(98\!\cdots\!60\)\( T + \)\(12\!\cdots\!40\)\( T^{2} + \)\(72\!\cdots\!80\)\( T^{3} + \)\(48\!\cdots\!58\)\( T^{4} + \)\(17\!\cdots\!40\)\( T^{5} + \)\(74\!\cdots\!60\)\( T^{6} + \)\(14\!\cdots\!20\)\( T^{7} + \)\(35\!\cdots\!41\)\( T^{8} \)
$89$ \( 1 + \)\(10\!\cdots\!80\)\( T + \)\(35\!\cdots\!16\)\( T^{2} + \)\(17\!\cdots\!60\)\( T^{3} + \)\(60\!\cdots\!46\)\( T^{4} + \)\(22\!\cdots\!40\)\( T^{5} + \)\(60\!\cdots\!56\)\( T^{6} + \)\(23\!\cdots\!20\)\( T^{7} + \)\(28\!\cdots\!81\)\( T^{8} \)
$97$ \( 1 + \)\(15\!\cdots\!20\)\( T + \)\(10\!\cdots\!60\)\( T^{2} + \)\(68\!\cdots\!40\)\( T^{3} + \)\(32\!\cdots\!38\)\( T^{4} + \)\(11\!\cdots\!80\)\( T^{5} + \)\(31\!\cdots\!40\)\( T^{6} + \)\(83\!\cdots\!60\)\( T^{7} + \)\(96\!\cdots\!61\)\( T^{8} \)
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