Properties

Label 1.52.a.a
Level 1
Weight 52
Character orbit 1.a
Self dual Yes
Analytic conductor 16.473
Analytic rank 0
Dimension 4
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 1 \)
Weight: \( k \) = \( 52 \)
Character orbit: \([\chi]\) = 1.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(16.4731353414\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{23}\cdot 3^{10}\cdot 5^{3}\cdot 7^{2}\cdot 13\cdot 17 \)
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.

\(f(q)\) \(=\) \( q\) \(+(8189010 + \beta_{1}) q^{2}\) \(+(100965943260 - 6677 \beta_{1} - \beta_{2}) q^{3}\) \(+(1994525867718928 - 7825327 \beta_{1} + 511 \beta_{2} - \beta_{3}) q^{4}\) \(+(303528278241889470 - 2170290676 \beta_{1} - 86148 \beta_{2} - 144 \beta_{3}) q^{5}\) \(+(-27078920080376717448 - 186291114948 \beta_{1} - 61334496 \beta_{2} + 16416 \beta_{3}) q^{6}\) \(+(\)\(16\!\cdots\!00\)\( + 3775025640302 \beta_{1} - 224454986 \beta_{2} - 736960 \beta_{3}) q^{7}\) \(+(-\)\(34\!\cdots\!80\)\( + 1959718379828168 \beta_{1} + 49429856952 \beta_{2} + 20189880 \beta_{3}) q^{8}\) \(+(\)\(12\!\cdots\!57\)\( + 41221787348461896 \beta_{1} - 666471696408 \beta_{2} - 386651232 \beta_{3}) q^{9}\) \(+O(q^{10})\) \( q\) \(+(8189010 + \beta_{1}) q^{2}\) \(+(100965943260 - 6677 \beta_{1} - \beta_{2}) q^{3}\) \(+(1994525867718928 - 7825327 \beta_{1} + 511 \beta_{2} - \beta_{3}) q^{4}\) \(+(303528278241889470 - 2170290676 \beta_{1} - 86148 \beta_{2} - 144 \beta_{3}) q^{5}\) \(+(-27078920080376717448 - 186291114948 \beta_{1} - 61334496 \beta_{2} + 16416 \beta_{3}) q^{6}\) \(+(\)\(16\!\cdots\!00\)\( + 3775025640302 \beta_{1} - 224454986 \beta_{2} - 736960 \beta_{3}) q^{7}\) \(+(-\)\(34\!\cdots\!80\)\( + 1959718379828168 \beta_{1} + 49429856952 \beta_{2} + 20189880 \beta_{3}) q^{8}\) \(+(\)\(12\!\cdots\!57\)\( + 41221787348461896 \beta_{1} - 666471696408 \beta_{2} - 386651232 \beta_{3}) q^{9}\) \(+(-\)\(65\!\cdots\!80\)\( + 576521306120640574 \beta_{1} - 2667384869248 \beta_{2} + 5506416256 \beta_{3}) q^{10}\) \(+(\)\(88\!\cdots\!12\)\( - 2258744929721205575 \beta_{1} + 155930790349845 \beta_{2} - 59943070080 \beta_{3}) q^{11}\) \(+(-\)\(12\!\cdots\!80\)\( - 64267735385758560740 \beta_{1} - 1787241473591644 \beta_{2} + 498954927780 \beta_{3}) q^{12}\) \(+(\)\(76\!\cdots\!70\)\( + 75259059392487711692 \beta_{1} + 8796829725936700 \beta_{2} - 3042585439120 \beta_{3}) q^{13}\) \(+(\)\(29\!\cdots\!56\)\( + \)\(28\!\cdots\!44\)\( \beta_{1} + 6489920058124608 \beta_{2} + 11190698955072 \beta_{3}) q^{14}\) \(+(\)\(36\!\cdots\!40\)\( - \)\(26\!\cdots\!02\)\( \beta_{1} - 365235659009187246 \beta_{2} + 10467930808512 \beta_{3}) q^{15}\) \(+(\)\(34\!\cdots\!36\)\( - \)\(67\!\cdots\!96\)\( \beta_{1} + 2232721294548523968 \beta_{2} - 539432326636608 \beta_{3}) q^{16}\) \(+(\)\(12\!\cdots\!30\)\( + \)\(12\!\cdots\!04\)\( \beta_{1} - 5095661844231778968 \beta_{2} + 4845742691513760 \beta_{3}) q^{17}\) \(+(\)\(18\!\cdots\!10\)\( + \)\(10\!\cdots\!33\)\( \beta_{1} - 8325439050872514816 \beta_{2} - 28026030112508160 \beta_{3}) q^{18}\) \(+(\)\(20\!\cdots\!20\)\( + \)\(21\!\cdots\!67\)\( \beta_{1} + 56753234257291972779 \beta_{2} + 115731936015363456 \beta_{3}) q^{19}\) \(+(\)\(16\!\cdots\!60\)\( - \)\(22\!\cdots\!78\)\( \beta_{1} + \)\(20\!\cdots\!06\)\( \beta_{2} - 321772238171413182 \beta_{3}) q^{20}\) \(+(\)\(78\!\cdots\!12\)\( - \)\(25\!\cdots\!96\)\( \beta_{1} - \)\(25\!\cdots\!72\)\( \beta_{2} + 357837433134495552 \beta_{3}) q^{21}\) \(+(-\)\(87\!\cdots\!80\)\( + \)\(29\!\cdots\!12\)\( \beta_{1} + \)\(93\!\cdots\!80\)\( \beta_{2} + 1779618650400099680 \beta_{3}) q^{22}\) \(+(-\)\(13\!\cdots\!20\)\( - \)\(14\!\cdots\!14\)\( \beta_{1} - \)\(12\!\cdots\!34\)\( \beta_{2} - 12037007883164961600 \beta_{3}) q^{23}\) \(+(-\)\(21\!\cdots\!40\)\( - \)\(14\!\cdots\!64\)\( \beta_{1} - \)\(10\!\cdots\!88\)\( \beta_{2} + 36055666143180714528 \beta_{3}) q^{24}\) \(+(-\)\(14\!\cdots\!25\)\( - \)\(36\!\cdots\!80\)\( \beta_{1} - \)\(29\!\cdots\!40\)\( \beta_{2} - 44536012526878198720 \beta_{3}) q^{25}\) \(+(\)\(37\!\cdots\!12\)\( + \)\(15\!\cdots\!06\)\( \beta_{1} + \)\(62\!\cdots\!12\)\( \beta_{2} - \)\(10\!\cdots\!52\)\( \beta_{3}) q^{26}\) \(+(\)\(91\!\cdots\!80\)\( + \)\(15\!\cdots\!62\)\( \beta_{1} - \)\(21\!\cdots\!82\)\( \beta_{2} + \)\(67\!\cdots\!20\)\( \beta_{3}) q^{27}\) \(+(\)\(86\!\cdots\!20\)\( - \)\(44\!\cdots\!76\)\( \beta_{1} + \)\(20\!\cdots\!84\)\( \beta_{2} - \)\(14\!\cdots\!80\)\( \beta_{3}) q^{28}\) \(+(\)\(61\!\cdots\!30\)\( - \)\(20\!\cdots\!12\)\( \beta_{1} + \)\(68\!\cdots\!36\)\( \beta_{2} + \)\(50\!\cdots\!64\)\( \beta_{3}) q^{29}\) \(+(-\)\(81\!\cdots\!60\)\( + \)\(24\!\cdots\!48\)\( \beta_{1} - \)\(22\!\cdots\!96\)\( \beta_{2} + \)\(60\!\cdots\!12\)\( \beta_{3}) q^{30}\) \(+(-\)\(18\!\cdots\!68\)\( + \)\(61\!\cdots\!00\)\( \beta_{1} + \)\(18\!\cdots\!60\)\( \beta_{2} - \)\(16\!\cdots\!40\)\( \beta_{3}) q^{31}\) \(+(-\)\(17\!\cdots\!40\)\( + \)\(19\!\cdots\!96\)\( \beta_{1} - \)\(34\!\cdots\!40\)\( \beta_{2} + \)\(93\!\cdots\!60\)\( \beta_{3}) q^{32}\) \(+(-\)\(43\!\cdots\!80\)\( - \)\(85\!\cdots\!24\)\( \beta_{1} + \)\(13\!\cdots\!48\)\( \beta_{2} + \)\(35\!\cdots\!60\)\( \beta_{3}) q^{33}\) \(+(\)\(14\!\cdots\!16\)\( + \)\(67\!\cdots\!86\)\( \beta_{1} - \)\(40\!\cdots\!88\)\( \beta_{2} - \)\(46\!\cdots\!72\)\( \beta_{3}) q^{34}\) \(+(\)\(13\!\cdots\!20\)\( - \)\(10\!\cdots\!56\)\( \beta_{1} - \)\(10\!\cdots\!88\)\( \beta_{2} - \)\(15\!\cdots\!64\)\( \beta_{3}) q^{35}\) \(+(\)\(31\!\cdots\!96\)\( + \)\(12\!\cdots\!89\)\( \beta_{1} + \)\(22\!\cdots\!63\)\( \beta_{2} + \)\(36\!\cdots\!47\)\( \beta_{3}) q^{36}\) \(+(\)\(23\!\cdots\!90\)\( - \)\(15\!\cdots\!92\)\( \beta_{1} - \)\(34\!\cdots\!92\)\( \beta_{2} + \)\(87\!\cdots\!40\)\( \beta_{3}) q^{37}\) \(+(\)\(10\!\cdots\!80\)\( - \)\(26\!\cdots\!28\)\( \beta_{1} + \)\(16\!\cdots\!48\)\( \beta_{2} - \)\(46\!\cdots\!40\)\( \beta_{3}) q^{38}\) \(+(-\)\(29\!\cdots\!76\)\( - \)\(51\!\cdots\!66\)\( \beta_{1} - \)\(56\!\cdots\!42\)\( \beta_{2} + \)\(51\!\cdots\!12\)\( \beta_{3}) q^{39}\) \(+(-\)\(65\!\cdots\!00\)\( + \)\(14\!\cdots\!20\)\( \beta_{1} + \)\(13\!\cdots\!60\)\( \beta_{2} + \)\(13\!\cdots\!80\)\( \beta_{3}) q^{40}\) \(+(\)\(36\!\cdots\!42\)\( + \)\(88\!\cdots\!00\)\( \beta_{1} + \)\(36\!\cdots\!60\)\( \beta_{2} - \)\(51\!\cdots\!40\)\( \beta_{3}) q^{41}\) \(+(-\)\(98\!\cdots\!20\)\( - \)\(49\!\cdots\!64\)\( \beta_{1} - \)\(16\!\cdots\!04\)\( \beta_{2} + \)\(43\!\cdots\!40\)\( \beta_{3}) q^{42}\) \(+(-\)\(95\!\cdots\!00\)\( - \)\(24\!\cdots\!51\)\( \beta_{1} + \)\(12\!\cdots\!33\)\( \beta_{2} + \)\(99\!\cdots\!00\)\( \beta_{3}) q^{43}\) \(+(\)\(97\!\cdots\!36\)\( - \)\(81\!\cdots\!24\)\( \beta_{1} + \)\(29\!\cdots\!92\)\( \beta_{2} - \)\(28\!\cdots\!52\)\( \beta_{3}) q^{44}\) \(+(\)\(63\!\cdots\!90\)\( + \)\(17\!\cdots\!68\)\( \beta_{1} - \)\(37\!\cdots\!36\)\( \beta_{2} + \)\(15\!\cdots\!92\)\( \beta_{3}) q^{45}\) \(+(-\)\(17\!\cdots\!28\)\( + \)\(44\!\cdots\!60\)\( \beta_{1} - \)\(43\!\cdots\!20\)\( \beta_{2} + \)\(34\!\cdots\!40\)\( \beta_{3}) q^{46}\) \(+(\)\(17\!\cdots\!20\)\( + \)\(37\!\cdots\!60\)\( \beta_{1} + \)\(13\!\cdots\!16\)\( \beta_{2} - \)\(42\!\cdots\!00\)\( \beta_{3}) q^{47}\) \(+(-\)\(49\!\cdots\!20\)\( - \)\(12\!\cdots\!04\)\( \beta_{1} + \)\(18\!\cdots\!16\)\( \beta_{2} - \)\(22\!\cdots\!40\)\( \beta_{3}) q^{48}\) \(+(-\)\(53\!\cdots\!07\)\( - \)\(84\!\cdots\!20\)\( \beta_{1} + \)\(95\!\cdots\!20\)\( \beta_{2} - \)\(72\!\cdots\!00\)\( \beta_{3}) q^{49}\) \(+(-\)\(16\!\cdots\!50\)\( - \)\(13\!\cdots\!05\)\( \beta_{1} - \)\(24\!\cdots\!40\)\( \beta_{2} + \)\(46\!\cdots\!80\)\( \beta_{3}) q^{50}\) \(+(\)\(17\!\cdots\!32\)\( + \)\(27\!\cdots\!58\)\( \beta_{1} - \)\(14\!\cdots\!54\)\( \beta_{2} - \)\(35\!\cdots\!56\)\( \beta_{3}) q^{51}\) \(+(\)\(51\!\cdots\!00\)\( + \)\(40\!\cdots\!82\)\( \beta_{1} + \)\(26\!\cdots\!74\)\( \beta_{2} - \)\(13\!\cdots\!50\)\( \beta_{3}) q^{52}\) \(+(-\)\(11\!\cdots\!90\)\( - \)\(26\!\cdots\!32\)\( \beta_{1} + \)\(39\!\cdots\!84\)\( \beta_{2} + \)\(28\!\cdots\!80\)\( \beta_{3}) q^{53}\) \(+(\)\(73\!\cdots\!20\)\( - \)\(14\!\cdots\!28\)\( \beta_{1} - \)\(13\!\cdots\!36\)\( \beta_{2} - \)\(63\!\cdots\!04\)\( \beta_{3}) q^{54}\) \(+(\)\(72\!\cdots\!40\)\( - \)\(25\!\cdots\!62\)\( \beta_{1} + \)\(64\!\cdots\!74\)\( \beta_{2} - \)\(35\!\cdots\!28\)\( \beta_{3}) q^{55}\) \(+(-\)\(17\!\cdots\!20\)\( + \)\(64\!\cdots\!68\)\( \beta_{1} + \)\(11\!\cdots\!96\)\( \beta_{2} + \)\(24\!\cdots\!04\)\( \beta_{3}) q^{56}\) \(+(-\)\(22\!\cdots\!40\)\( - \)\(38\!\cdots\!76\)\( \beta_{1} - \)\(18\!\cdots\!04\)\( \beta_{2} - \)\(56\!\cdots\!00\)\( \beta_{3}) q^{57}\) \(+(-\)\(81\!\cdots\!80\)\( + \)\(11\!\cdots\!58\)\( \beta_{1} + \)\(27\!\cdots\!92\)\( \beta_{2} + \)\(13\!\cdots\!60\)\( \beta_{3}) q^{58}\) \(+(\)\(27\!\cdots\!60\)\( - \)\(19\!\cdots\!99\)\( \beta_{1} - \)\(25\!\cdots\!43\)\( \beta_{2} - \)\(22\!\cdots\!12\)\( \beta_{3}) q^{59}\) \(+(\)\(13\!\cdots\!20\)\( - \)\(26\!\cdots\!56\)\( \beta_{1} - \)\(51\!\cdots\!88\)\( \beta_{2} - \)\(15\!\cdots\!64\)\( \beta_{3}) q^{60}\) \(+(\)\(85\!\cdots\!62\)\( + \)\(22\!\cdots\!00\)\( \beta_{1} + \)\(50\!\cdots\!00\)\( \beta_{2} + \)\(69\!\cdots\!00\)\( \beta_{3}) q^{61}\) \(+(\)\(24\!\cdots\!20\)\( + \)\(95\!\cdots\!32\)\( \beta_{1} + \)\(17\!\cdots\!40\)\( \beta_{2} - \)\(50\!\cdots\!60\)\( \beta_{3}) q^{62}\) \(+(\)\(54\!\cdots\!40\)\( + \)\(10\!\cdots\!50\)\( \beta_{1} - \)\(10\!\cdots\!38\)\( \beta_{2} + \)\(29\!\cdots\!60\)\( \beta_{3}) q^{63}\) \(+(-\)\(85\!\cdots\!32\)\( - \)\(74\!\cdots\!84\)\( \beta_{1} - \)\(44\!\cdots\!68\)\( \beta_{2} - \)\(89\!\cdots\!72\)\( \beta_{3}) q^{64}\) \(+(\)\(26\!\cdots\!40\)\( - \)\(23\!\cdots\!12\)\( \beta_{1} + \)\(29\!\cdots\!24\)\( \beta_{2} - \)\(10\!\cdots\!28\)\( \beta_{3}) q^{65}\) \(+(-\)\(39\!\cdots\!76\)\( - \)\(30\!\cdots\!76\)\( \beta_{1} + \)\(25\!\cdots\!88\)\( \beta_{2} + \)\(66\!\cdots\!32\)\( \beta_{3}) q^{66}\) \(+(\)\(77\!\cdots\!80\)\( + \)\(20\!\cdots\!39\)\( \beta_{1} + \)\(69\!\cdots\!47\)\( \beta_{2} - \)\(47\!\cdots\!40\)\( \beta_{3}) q^{67}\) \(+(-\)\(23\!\cdots\!40\)\( + \)\(40\!\cdots\!90\)\( \beta_{1} - \)\(10\!\cdots\!82\)\( \beta_{2} - \)\(68\!\cdots\!30\)\( \beta_{3}) q^{68}\) \(+(\)\(39\!\cdots\!44\)\( + \)\(22\!\cdots\!72\)\( \beta_{1} - \)\(51\!\cdots\!56\)\( \beta_{2} + \)\(21\!\cdots\!76\)\( \beta_{3}) q^{69}\) \(+(-\)\(31\!\cdots\!80\)\( + \)\(17\!\cdots\!44\)\( \beta_{1} - \)\(76\!\cdots\!88\)\( \beta_{2} + \)\(14\!\cdots\!36\)\( \beta_{3}) q^{70}\) \(+(\)\(99\!\cdots\!72\)\( - \)\(24\!\cdots\!50\)\( \beta_{1} - \)\(30\!\cdots\!50\)\( \beta_{2} + \)\(25\!\cdots\!00\)\( \beta_{3}) q^{71}\) \(+(\)\(12\!\cdots\!40\)\( - \)\(10\!\cdots\!04\)\( \beta_{1} + \)\(20\!\cdots\!64\)\( \beta_{2} - \)\(87\!\cdots\!20\)\( \beta_{3}) q^{72}\) \(+(\)\(27\!\cdots\!30\)\( - \)\(13\!\cdots\!44\)\( \beta_{1} - \)\(13\!\cdots\!44\)\( \beta_{2} - \)\(37\!\cdots\!40\)\( \beta_{3}) q^{73}\) \(+(-\)\(61\!\cdots\!64\)\( + \)\(16\!\cdots\!70\)\( \beta_{1} - \)\(30\!\cdots\!20\)\( \beta_{2} + \)\(16\!\cdots\!00\)\( \beta_{3}) q^{74}\) \(+(\)\(18\!\cdots\!00\)\( + \)\(12\!\cdots\!65\)\( \beta_{1} + \)\(28\!\cdots\!45\)\( \beta_{2} - \)\(46\!\cdots\!40\)\( \beta_{3}) q^{75}\) \(+(-\)\(48\!\cdots\!40\)\( + \)\(15\!\cdots\!16\)\( \beta_{1} + \)\(64\!\cdots\!52\)\( \beta_{2} - \)\(16\!\cdots\!52\)\( \beta_{3}) q^{76}\) \(+(\)\(34\!\cdots\!00\)\( - \)\(70\!\cdots\!76\)\( \beta_{1} + \)\(42\!\cdots\!48\)\( \beta_{2} - \)\(16\!\cdots\!40\)\( \beta_{3}) q^{77}\) \(+(-\)\(24\!\cdots\!00\)\( - \)\(33\!\cdots\!72\)\( \beta_{1} - \)\(71\!\cdots\!04\)\( \beta_{2} + \)\(48\!\cdots\!20\)\( \beta_{3}) q^{78}\) \(+(\)\(10\!\cdots\!80\)\( + \)\(34\!\cdots\!68\)\( \beta_{1} - \)\(25\!\cdots\!64\)\( \beta_{2} + \)\(32\!\cdots\!44\)\( \beta_{3}) q^{79}\) \(+(\)\(16\!\cdots\!20\)\( - \)\(57\!\cdots\!36\)\( \beta_{1} + \)\(75\!\cdots\!72\)\( \beta_{2} - \)\(10\!\cdots\!84\)\( \beta_{3}) q^{80}\) \(+(\)\(39\!\cdots\!61\)\( + \)\(13\!\cdots\!12\)\( \beta_{1} - \)\(18\!\cdots\!16\)\( \beta_{2} + \)\(15\!\cdots\!56\)\( \beta_{3}) q^{81}\) \(+(\)\(66\!\cdots\!20\)\( + \)\(14\!\cdots\!42\)\( \beta_{1} + \)\(34\!\cdots\!40\)\( \beta_{2} + \)\(44\!\cdots\!40\)\( \beta_{3}) q^{82}\) \(+(\)\(26\!\cdots\!40\)\( + \)\(51\!\cdots\!75\)\( \beta_{1} + \)\(14\!\cdots\!67\)\( \beta_{2} + \)\(30\!\cdots\!00\)\( \beta_{3}) q^{83}\) \(+(-\)\(46\!\cdots\!64\)\( - \)\(18\!\cdots\!52\)\( \beta_{1} - \)\(53\!\cdots\!44\)\( \beta_{2} + \)\(57\!\cdots\!44\)\( \beta_{3}) q^{84}\) \(+(-\)\(55\!\cdots\!80\)\( + \)\(50\!\cdots\!84\)\( \beta_{1} - \)\(37\!\cdots\!68\)\( \beta_{2} - \)\(15\!\cdots\!04\)\( \beta_{3}) q^{85}\) \(+(-\)\(10\!\cdots\!08\)\( - \)\(19\!\cdots\!32\)\( \beta_{1} + \)\(34\!\cdots\!56\)\( \beta_{2} - \)\(47\!\cdots\!36\)\( \beta_{3}) q^{86}\) \(+(-\)\(15\!\cdots\!60\)\( - \)\(27\!\cdots\!14\)\( \beta_{1} + \)\(11\!\cdots\!34\)\( \beta_{2} - \)\(17\!\cdots\!60\)\( \beta_{3}) q^{87}\) \(+(-\)\(62\!\cdots\!60\)\( + \)\(10\!\cdots\!16\)\( \beta_{1} - \)\(10\!\cdots\!76\)\( \beta_{2} + \)\(61\!\cdots\!60\)\( \beta_{3}) q^{88}\) \(+(-\)\(22\!\cdots\!10\)\( - \)\(25\!\cdots\!36\)\( \beta_{1} - \)\(11\!\cdots\!92\)\( \beta_{2} + \)\(60\!\cdots\!92\)\( \beta_{3}) q^{89}\) \(+(\)\(79\!\cdots\!40\)\( - \)\(98\!\cdots\!82\)\( \beta_{1} - \)\(16\!\cdots\!36\)\( \beta_{2} - \)\(17\!\cdots\!08\)\( \beta_{3}) q^{90}\) \(+(\)\(25\!\cdots\!72\)\( + \)\(20\!\cdots\!08\)\( \beta_{1} + \)\(24\!\cdots\!76\)\( \beta_{2} - \)\(79\!\cdots\!76\)\( \beta_{3}) q^{91}\) \(+(\)\(48\!\cdots\!80\)\( - \)\(10\!\cdots\!96\)\( \beta_{1} - \)\(33\!\cdots\!88\)\( \beta_{2} + \)\(20\!\cdots\!40\)\( \beta_{3}) q^{92}\) \(+(-\)\(77\!\cdots\!80\)\( - \)\(12\!\cdots\!64\)\( \beta_{1} - \)\(83\!\cdots\!52\)\( \beta_{2} + \)\(23\!\cdots\!80\)\( \beta_{3}) q^{93}\) \(+(\)\(15\!\cdots\!96\)\( + \)\(43\!\cdots\!12\)\( \beta_{1} + \)\(37\!\cdots\!44\)\( \beta_{2} - \)\(31\!\cdots\!84\)\( \beta_{3}) q^{94}\) \(+(-\)\(10\!\cdots\!00\)\( + \)\(21\!\cdots\!30\)\( \beta_{1} - \)\(20\!\cdots\!10\)\( \beta_{2} - \)\(25\!\cdots\!80\)\( \beta_{3}) q^{95}\) \(+(-\)\(61\!\cdots\!08\)\( + \)\(13\!\cdots\!92\)\( \beta_{1} + \)\(72\!\cdots\!64\)\( \beta_{2} + \)\(26\!\cdots\!16\)\( \beta_{3}) q^{96}\) \(+(-\)\(33\!\cdots\!30\)\( + \)\(19\!\cdots\!20\)\( \beta_{1} - \)\(31\!\cdots\!24\)\( \beta_{2} + \)\(34\!\cdots\!20\)\( \beta_{3}) q^{97}\) \(+(-\)\(79\!\cdots\!70\)\( - \)\(38\!\cdots\!27\)\( \beta_{1} + \)\(18\!\cdots\!60\)\( \beta_{2} + \)\(20\!\cdots\!40\)\( \beta_{3}) q^{98}\) \(+(-\)\(42\!\cdots\!16\)\( + \)\(44\!\cdots\!77\)\( \beta_{1} + \)\(69\!\cdots\!69\)\( \beta_{2} + \)\(90\!\cdots\!56\)\( \beta_{3}) q^{99}\) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(4q \) \(\mathstrut +\mathstrut 32756040q^{2} \) \(\mathstrut +\mathstrut 403863773040q^{3} \) \(\mathstrut +\mathstrut 7978103470875712q^{4} \) \(\mathstrut +\mathstrut 1214113112967557880q^{5} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!92\)\(q^{6} \) \(\mathstrut +\mathstrut \)\(65\!\cdots\!00\)\(q^{7} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!20\)\(q^{8} \) \(\mathstrut +\mathstrut \)\(50\!\cdots\!28\)\(q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(4q \) \(\mathstrut +\mathstrut 32756040q^{2} \) \(\mathstrut +\mathstrut 403863773040q^{3} \) \(\mathstrut +\mathstrut 7978103470875712q^{4} \) \(\mathstrut +\mathstrut 1214113112967557880q^{5} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!92\)\(q^{6} \) \(\mathstrut +\mathstrut \)\(65\!\cdots\!00\)\(q^{7} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!20\)\(q^{8} \) \(\mathstrut +\mathstrut \)\(50\!\cdots\!28\)\(q^{9} \) \(\mathstrut -\mathstrut \)\(26\!\cdots\!20\)\(q^{10} \) \(\mathstrut +\mathstrut \)\(35\!\cdots\!48\)\(q^{11} \) \(\mathstrut -\mathstrut \)\(49\!\cdots\!20\)\(q^{12} \) \(\mathstrut +\mathstrut \)\(30\!\cdots\!80\)\(q^{13} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!24\)\(q^{14} \) \(\mathstrut +\mathstrut \)\(14\!\cdots\!60\)\(q^{15} \) \(\mathstrut +\mathstrut \)\(13\!\cdots\!44\)\(q^{16} \) \(\mathstrut +\mathstrut \)\(48\!\cdots\!20\)\(q^{17} \) \(\mathstrut +\mathstrut \)\(73\!\cdots\!40\)\(q^{18} \) \(\mathstrut +\mathstrut \)\(81\!\cdots\!80\)\(q^{19} \) \(\mathstrut +\mathstrut \)\(66\!\cdots\!40\)\(q^{20} \) \(\mathstrut +\mathstrut \)\(31\!\cdots\!48\)\(q^{21} \) \(\mathstrut -\mathstrut \)\(34\!\cdots\!20\)\(q^{22} \) \(\mathstrut -\mathstrut \)\(54\!\cdots\!80\)\(q^{23} \) \(\mathstrut -\mathstrut \)\(87\!\cdots\!60\)\(q^{24} \) \(\mathstrut -\mathstrut \)\(57\!\cdots\!00\)\(q^{25} \) \(\mathstrut +\mathstrut \)\(15\!\cdots\!48\)\(q^{26} \) \(\mathstrut +\mathstrut \)\(36\!\cdots\!20\)\(q^{27} \) \(\mathstrut +\mathstrut \)\(34\!\cdots\!80\)\(q^{28} \) \(\mathstrut +\mathstrut \)\(24\!\cdots\!20\)\(q^{29} \) \(\mathstrut -\mathstrut \)\(32\!\cdots\!40\)\(q^{30} \) \(\mathstrut -\mathstrut \)\(74\!\cdots\!72\)\(q^{31} \) \(\mathstrut -\mathstrut \)\(69\!\cdots\!60\)\(q^{32} \) \(\mathstrut -\mathstrut \)\(17\!\cdots\!20\)\(q^{33} \) \(\mathstrut +\mathstrut \)\(59\!\cdots\!64\)\(q^{34} \) \(\mathstrut +\mathstrut \)\(54\!\cdots\!80\)\(q^{35} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!84\)\(q^{36} \) \(\mathstrut +\mathstrut \)\(92\!\cdots\!60\)\(q^{37} \) \(\mathstrut +\mathstrut \)\(42\!\cdots\!20\)\(q^{38} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!04\)\(q^{39} \) \(\mathstrut -\mathstrut \)\(26\!\cdots\!00\)\(q^{40} \) \(\mathstrut +\mathstrut \)\(14\!\cdots\!68\)\(q^{41} \) \(\mathstrut -\mathstrut \)\(39\!\cdots\!80\)\(q^{42} \) \(\mathstrut -\mathstrut \)\(38\!\cdots\!00\)\(q^{43} \) \(\mathstrut +\mathstrut \)\(39\!\cdots\!44\)\(q^{44} \) \(\mathstrut +\mathstrut \)\(25\!\cdots\!60\)\(q^{45} \) \(\mathstrut -\mathstrut \)\(68\!\cdots\!12\)\(q^{46} \) \(\mathstrut +\mathstrut \)\(70\!\cdots\!80\)\(q^{47} \) \(\mathstrut -\mathstrut \)\(19\!\cdots\!80\)\(q^{48} \) \(\mathstrut -\mathstrut \)\(21\!\cdots\!28\)\(q^{49} \) \(\mathstrut -\mathstrut \)\(65\!\cdots\!00\)\(q^{50} \) \(\mathstrut +\mathstrut \)\(69\!\cdots\!28\)\(q^{51} \) \(\mathstrut +\mathstrut \)\(20\!\cdots\!00\)\(q^{52} \) \(\mathstrut -\mathstrut \)\(46\!\cdots\!60\)\(q^{53} \) \(\mathstrut +\mathstrut \)\(29\!\cdots\!80\)\(q^{54} \) \(\mathstrut +\mathstrut \)\(28\!\cdots\!60\)\(q^{55} \) \(\mathstrut -\mathstrut \)\(71\!\cdots\!80\)\(q^{56} \) \(\mathstrut -\mathstrut \)\(88\!\cdots\!60\)\(q^{57} \) \(\mathstrut -\mathstrut \)\(32\!\cdots\!20\)\(q^{58} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!40\)\(q^{59} \) \(\mathstrut +\mathstrut \)\(55\!\cdots\!80\)\(q^{60} \) \(\mathstrut +\mathstrut \)\(34\!\cdots\!48\)\(q^{61} \) \(\mathstrut +\mathstrut \)\(96\!\cdots\!80\)\(q^{62} \) \(\mathstrut +\mathstrut \)\(21\!\cdots\!60\)\(q^{63} \) \(\mathstrut -\mathstrut \)\(34\!\cdots\!28\)\(q^{64} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!60\)\(q^{65} \) \(\mathstrut -\mathstrut \)\(15\!\cdots\!04\)\(q^{66} \) \(\mathstrut +\mathstrut \)\(30\!\cdots\!20\)\(q^{67} \) \(\mathstrut -\mathstrut \)\(92\!\cdots\!60\)\(q^{68} \) \(\mathstrut +\mathstrut \)\(15\!\cdots\!76\)\(q^{69} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!20\)\(q^{70} \) \(\mathstrut +\mathstrut \)\(39\!\cdots\!88\)\(q^{71} \) \(\mathstrut +\mathstrut \)\(50\!\cdots\!60\)\(q^{72} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!20\)\(q^{73} \) \(\mathstrut -\mathstrut \)\(24\!\cdots\!56\)\(q^{74} \) \(\mathstrut +\mathstrut \)\(72\!\cdots\!00\)\(q^{75} \) \(\mathstrut -\mathstrut \)\(19\!\cdots\!60\)\(q^{76} \) \(\mathstrut +\mathstrut \)\(13\!\cdots\!00\)\(q^{77} \) \(\mathstrut -\mathstrut \)\(96\!\cdots\!00\)\(q^{78} \) \(\mathstrut +\mathstrut \)\(40\!\cdots\!20\)\(q^{79} \) \(\mathstrut +\mathstrut \)\(65\!\cdots\!80\)\(q^{80} \) \(\mathstrut +\mathstrut \)\(15\!\cdots\!44\)\(q^{81} \) \(\mathstrut +\mathstrut \)\(26\!\cdots\!80\)\(q^{82} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!60\)\(q^{83} \) \(\mathstrut -\mathstrut \)\(18\!\cdots\!56\)\(q^{84} \) \(\mathstrut -\mathstrut \)\(22\!\cdots\!20\)\(q^{85} \) \(\mathstrut -\mathstrut \)\(43\!\cdots\!32\)\(q^{86} \) \(\mathstrut -\mathstrut \)\(62\!\cdots\!40\)\(q^{87} \) \(\mathstrut -\mathstrut \)\(25\!\cdots\!40\)\(q^{88} \) \(\mathstrut -\mathstrut \)\(90\!\cdots\!40\)\(q^{89} \) \(\mathstrut +\mathstrut \)\(31\!\cdots\!60\)\(q^{90} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!88\)\(q^{91} \) \(\mathstrut +\mathstrut \)\(19\!\cdots\!20\)\(q^{92} \) \(\mathstrut -\mathstrut \)\(31\!\cdots\!20\)\(q^{93} \) \(\mathstrut +\mathstrut \)\(63\!\cdots\!84\)\(q^{94} \) \(\mathstrut -\mathstrut \)\(41\!\cdots\!00\)\(q^{95} \) \(\mathstrut -\mathstrut \)\(24\!\cdots\!32\)\(q^{96} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!20\)\(q^{97} \) \(\mathstrut -\mathstrut \)\(31\!\cdots\!80\)\(q^{98} \) \(\mathstrut -\mathstrut \)\(17\!\cdots\!64\)\(q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4}\mathstrut -\mathstrut \) \(2\) \(x^{3}\mathstrut -\mathstrut \) \(495735060514\) \(x^{2}\mathstrut -\mathstrut \) \(23954614981416598\) \(x\mathstrut +\mathstrut \) \(48979992255622025570313\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -252006 \nu^{3} + 93063836718 \nu^{2} + 70872137420108946 \nu - 18539821380123502730388 \)\()/\)\(77619756584545\)
\(\beta_{2}\)\(=\)\((\)\( 3755577054 \nu^{3} - 2376866094373062 \nu^{2} - 745977194049162568314 \nu + 521672957126789442477814692 \)\()/\)\(77619756584545\)
\(\beta_{3}\)\(=\)\((\)\(-\)\(334830962412\) \(\nu^{3}\mathstrut +\mathstrut \) \(368681755057133436\) \(\nu^{2}\mathstrut +\mathstrut \) \(113767409889319937410692\) \(\nu\mathstrut -\mathstrut \) \(85368483888402915793673001576\)\()/\)\(11088536654935\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(31\) \(\beta_{3}\mathstrut +\mathstrut \) \(53711\) \(\beta_{2}\mathstrut +\mathstrut \) \(512120665\) \(\beta_{1}\mathstrut +\mathstrut \) \(134730086400\)\()/\)\(269460172800\)
\(\nu^{2}\)\(=\)\((\)\(4857031\) \(\beta_{3}\mathstrut -\mathstrut \) \(2148411769\) \(\beta_{2}\mathstrut -\mathstrut \) \(77190687886655\) \(\beta_{1}\mathstrut +\mathstrut \) \(33395213767414954291200\)\()/\)\(134730086400\)
\(\nu^{3}\)\(=\)\((\)\(251133001843\) \(\beta_{3}\mathstrut +\mathstrut \) \(275887056853763\) \(\beta_{2}\mathstrut +\mathstrut \) \(81981363281812165\) \(\beta_{1}\mathstrut +\mathstrut \) \(98802273274899860911564800\)\()/\)\(5499187200\)

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
−457245.
644100.
−511801.
324949.
−8.71140e7 6.49653e11 5.33705e15 8.70295e17 −5.65939e19 3.14283e21 −2.68769e23 −1.73165e24 −7.58149e25
1.2 −1.27049e7 −5.15198e11 −2.09038e15 −3.83602e17 6.54557e18 −2.02239e21 5.51672e22 −1.88826e24 4.87365e24
1.3 5.13381e7 2.68083e12 3.83803e14 4.84667e17 1.37629e20 2.59004e21 −9.58994e22 5.03313e24 2.48819e25
1.4 8.12369e7 −2.41142e12 4.34763e15 2.42754e17 −1.95896e20 2.85586e21 1.70259e23 3.66124e24 1.97206e25
\(n\): 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 \(S_{52}^{\mathrm{new}}(\Gamma_0(1))\).