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: \( 2^{25}\cdot 3^{10}\cdot 5^{2}\cdot 7\cdot 19 \)
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.

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

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

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

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.

Hecke kernels

There are no other newforms in \(S_{58}^{\mathrm{new}}(\Gamma_0(1))\).