Properties

Label 3.72.a.a
Level 3
Weight 72
Character orbit 3.a
Self dual Yes
Analytic conductor 95.774
Analytic rank 1
Dimension 5
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 3 \)
Weight: \( k \) = \( 72 \)
Character orbit: \([\chi]\) = 3.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(95.7738481683\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{36}\cdot 3^{20}\cdot 5^{4}\cdot 7^{2} \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \(+(-5010255538 - \beta_{1}) q^{2}\) \(+50031545098999707 q^{3}\) \(+(\)\(38\!\cdots\!24\)\( + 14833524748 \beta_{1} + \beta_{2}) q^{4}\) \(+(\)\(28\!\cdots\!47\)\( + 22205806960703 \beta_{1} - 1156 \beta_{2} - \beta_{3}) q^{5}\) \(+(-\)\(25\!\cdots\!66\)\( - 50031545098999707 \beta_{1}) q^{6}\) \(+(-\)\(25\!\cdots\!77\)\( - 12693346119954501029 \beta_{1} - 105331541 \beta_{2} + 63091 \beta_{3} - 3983 \beta_{4}) q^{7}\) \(+(-\)\(30\!\cdots\!56\)\( - 72150789618238048416 \beta_{1} - 41145611768 \beta_{2} + 1969408 \beta_{3} - 991104 \beta_{4}) q^{8}\) \(+\)\(25\!\cdots\!49\)\( q^{9}\) \(+O(q^{10})\) \( q\) \(+(-5010255538 - \beta_{1}) q^{2}\) \(+50031545098999707 q^{3}\) \(+(\)\(38\!\cdots\!24\)\( + 14833524748 \beta_{1} + \beta_{2}) q^{4}\) \(+(\)\(28\!\cdots\!47\)\( + 22205806960703 \beta_{1} - 1156 \beta_{2} - \beta_{3}) q^{5}\) \(+(-\)\(25\!\cdots\!66\)\( - 50031545098999707 \beta_{1}) q^{6}\) \(+(-\)\(25\!\cdots\!77\)\( - 12693346119954501029 \beta_{1} - 105331541 \beta_{2} + 63091 \beta_{3} - 3983 \beta_{4}) q^{7}\) \(+(-\)\(30\!\cdots\!56\)\( - 72150789618238048416 \beta_{1} - 41145611768 \beta_{2} + 1969408 \beta_{3} - 991104 \beta_{4}) q^{8}\) \(+\)\(25\!\cdots\!49\)\( q^{9}\) \(+(-\)\(61\!\cdots\!84\)\( + \)\(17\!\cdots\!34\)\( \beta_{1} - 7653174060568 \beta_{2} + 13194541072 \beta_{3} + 3346421800 \beta_{4}) q^{10}\) \(+(\)\(27\!\cdots\!26\)\( - \)\(72\!\cdots\!66\)\( \beta_{1} + 941677448344864 \beta_{2} - 653169989582 \beta_{3} - 113546592984 \beta_{4}) q^{11}\) \(+(\)\(19\!\cdots\!68\)\( + \)\(74\!\cdots\!36\)\( \beta_{1} + 50031545098999707 \beta_{2}) q^{12}\) \(+(-\)\(74\!\cdots\!36\)\( + \)\(13\!\cdots\!36\)\( \beta_{1} + 131383178781509317 \beta_{2} - 59886625166368 \beta_{3} + 72687561798059 \beta_{4}) q^{13}\) \(+(\)\(35\!\cdots\!92\)\( + \)\(57\!\cdots\!64\)\( \beta_{1} + 19345982584020187480 \beta_{2} - 350761150967696 \beta_{3} - 160175633400552 \beta_{4}) q^{14}\) \(+(\)\(14\!\cdots\!29\)\( + \)\(11\!\cdots\!21\)\( \beta_{1} - 57836466134443661292 \beta_{2} - 50031545098999707 \beta_{3}) q^{15}\) \(+(-\)\(55\!\cdots\!84\)\( + \)\(74\!\cdots\!48\)\( \beta_{1} - \)\(45\!\cdots\!44\)\( \beta_{2} + 95719960882767872 \beta_{3} + 5160789222769664 \beta_{4}) q^{16}\) \(+(-\)\(55\!\cdots\!08\)\( + \)\(45\!\cdots\!10\)\( \beta_{1} + \)\(11\!\cdots\!82\)\( \beta_{2} + 2400820664738926274 \beta_{3} - 21924450278509662 \beta_{4}) q^{17}\) \(+(-\)\(12\!\cdots\!62\)\( - \)\(25\!\cdots\!49\)\( \beta_{1}) q^{18}\) \(+(\)\(58\!\cdots\!94\)\( + \)\(56\!\cdots\!86\)\( \beta_{1} + \)\(58\!\cdots\!78\)\( \beta_{2} + \)\(24\!\cdots\!26\)\( \beta_{3} - 7744471202131557338 \beta_{4}) q^{19}\) \(+(-\)\(49\!\cdots\!08\)\( + \)\(72\!\cdots\!08\)\( \beta_{1} - \)\(97\!\cdots\!66\)\( \beta_{2} + \)\(14\!\cdots\!64\)\( \beta_{3} + 84179304783469209600 \beta_{4}) q^{20}\) \(+(-\)\(12\!\cdots\!39\)\( - \)\(63\!\cdots\!03\)\( \beta_{1} - \)\(52\!\cdots\!87\)\( \beta_{2} + \)\(31\!\cdots\!37\)\( \beta_{3} - \)\(19\!\cdots\!81\)\( \beta_{4}) q^{21}\) \(+(\)\(19\!\cdots\!84\)\( - \)\(13\!\cdots\!80\)\( \beta_{1} + \)\(11\!\cdots\!24\)\( \beta_{2} + \)\(35\!\cdots\!48\)\( \beta_{3} - \)\(30\!\cdots\!24\)\( \beta_{4}) q^{22}\) \(+(-\)\(84\!\cdots\!74\)\( - \)\(14\!\cdots\!98\)\( \beta_{1} - \)\(35\!\cdots\!10\)\( \beta_{2} + \)\(12\!\cdots\!82\)\( \beta_{3} + \)\(29\!\cdots\!34\)\( \beta_{4}) q^{23}\) \(+(-\)\(15\!\cdots\!92\)\( - \)\(36\!\cdots\!12\)\( \beta_{1} - \)\(20\!\cdots\!76\)\( \beta_{2} + \)\(98\!\cdots\!56\)\( \beta_{3} - \)\(49\!\cdots\!28\)\( \beta_{4}) q^{24}\) \(+(\)\(39\!\cdots\!35\)\( + \)\(41\!\cdots\!40\)\( \beta_{1} - \)\(13\!\cdots\!30\)\( \beta_{2} - \)\(36\!\cdots\!80\)\( \beta_{3} - \)\(48\!\cdots\!50\)\( \beta_{4}) q^{25}\) \(+(-\)\(32\!\cdots\!08\)\( + \)\(36\!\cdots\!30\)\( \beta_{1} - \)\(70\!\cdots\!64\)\( \beta_{2} - \)\(14\!\cdots\!00\)\( \beta_{3} + \)\(22\!\cdots\!00\)\( \beta_{4}) q^{26}\) \(+\)\(12\!\cdots\!43\)\( q^{27}\) \(+(-\)\(11\!\cdots\!32\)\( - \)\(48\!\cdots\!12\)\( \beta_{1} - \)\(72\!\cdots\!32\)\( \beta_{2} - \)\(71\!\cdots\!20\)\( \beta_{3} - \)\(14\!\cdots\!40\)\( \beta_{4}) q^{28}\) \(+(-\)\(73\!\cdots\!19\)\( + \)\(30\!\cdots\!77\)\( \beta_{1} - \)\(15\!\cdots\!94\)\( \beta_{2} + \)\(11\!\cdots\!41\)\( \beta_{3} - \)\(87\!\cdots\!58\)\( \beta_{4}) q^{29}\) \(+(-\)\(30\!\cdots\!88\)\( + \)\(85\!\cdots\!38\)\( \beta_{1} - \)\(38\!\cdots\!76\)\( \beta_{2} + \)\(66\!\cdots\!04\)\( \beta_{3} + \)\(16\!\cdots\!00\)\( \beta_{4}) q^{30}\) \(+(\)\(14\!\cdots\!63\)\( + \)\(70\!\cdots\!03\)\( \beta_{1} + \)\(34\!\cdots\!27\)\( \beta_{2} - \)\(23\!\cdots\!25\)\( \beta_{3} - \)\(63\!\cdots\!75\)\( \beta_{4}) q^{31}\) \(+(-\)\(12\!\cdots\!68\)\( + \)\(86\!\cdots\!16\)\( \beta_{1} + \)\(32\!\cdots\!48\)\( \beta_{2} - \)\(81\!\cdots\!92\)\( \beta_{3} + \)\(27\!\cdots\!96\)\( \beta_{4}) q^{32}\) \(+(\)\(13\!\cdots\!82\)\( - \)\(36\!\cdots\!62\)\( \beta_{1} + \)\(47\!\cdots\!48\)\( \beta_{2} - \)\(32\!\cdots\!74\)\( \beta_{3} - \)\(56\!\cdots\!88\)\( \beta_{4}) q^{33}\) \(+(-\)\(12\!\cdots\!64\)\( - \)\(67\!\cdots\!94\)\( \beta_{1} - \)\(43\!\cdots\!16\)\( \beta_{2} - \)\(30\!\cdots\!96\)\( \beta_{3} - \)\(71\!\cdots\!52\)\( \beta_{4}) q^{34}\) \(+(-\)\(26\!\cdots\!70\)\( + \)\(26\!\cdots\!70\)\( \beta_{1} + \)\(70\!\cdots\!60\)\( \beta_{2} + \)\(10\!\cdots\!10\)\( \beta_{3} + \)\(33\!\cdots\!00\)\( \beta_{4}) q^{35}\) \(+(\)\(95\!\cdots\!76\)\( + \)\(37\!\cdots\!52\)\( \beta_{1} + \)\(25\!\cdots\!49\)\( \beta_{2}) q^{36}\) \(+(-\)\(32\!\cdots\!14\)\( + \)\(12\!\cdots\!66\)\( \beta_{1} - \)\(18\!\cdots\!61\)\( \beta_{2} + \)\(39\!\cdots\!10\)\( \beta_{3} + \)\(14\!\cdots\!45\)\( \beta_{4}) q^{37}\) \(+(-\)\(18\!\cdots\!32\)\( - \)\(17\!\cdots\!52\)\( \beta_{1} - \)\(11\!\cdots\!80\)\( \beta_{2} - \)\(10\!\cdots\!24\)\( \beta_{3} - \)\(13\!\cdots\!88\)\( \beta_{4}) q^{38}\) \(+(-\)\(37\!\cdots\!52\)\( + \)\(67\!\cdots\!52\)\( \beta_{1} + \)\(65\!\cdots\!19\)\( \beta_{2} - \)\(29\!\cdots\!76\)\( \beta_{3} + \)\(36\!\cdots\!13\)\( \beta_{4}) q^{39}\) \(+(\)\(15\!\cdots\!08\)\( + \)\(27\!\cdots\!92\)\( \beta_{1} - \)\(13\!\cdots\!84\)\( \beta_{2} - \)\(72\!\cdots\!64\)\( \beta_{3} - \)\(74\!\cdots\!00\)\( \beta_{4}) q^{40}\) \(+(\)\(45\!\cdots\!88\)\( + \)\(14\!\cdots\!98\)\( \beta_{1} + \)\(42\!\cdots\!78\)\( \beta_{2} + \)\(18\!\cdots\!30\)\( \beta_{3} + \)\(15\!\cdots\!10\)\( \beta_{4}) q^{41}\) \(+(\)\(17\!\cdots\!44\)\( + \)\(28\!\cdots\!48\)\( \beta_{1} + \)\(96\!\cdots\!60\)\( \beta_{2} - \)\(17\!\cdots\!72\)\( \beta_{3} - \)\(80\!\cdots\!64\)\( \beta_{4}) q^{42}\) \(+(\)\(13\!\cdots\!34\)\( + \)\(41\!\cdots\!14\)\( \beta_{1} - \)\(12\!\cdots\!42\)\( \beta_{2} + \)\(61\!\cdots\!74\)\( \beta_{3} - \)\(27\!\cdots\!62\)\( \beta_{4}) q^{43}\) \(+(\)\(20\!\cdots\!84\)\( - \)\(23\!\cdots\!20\)\( \beta_{1} - \)\(12\!\cdots\!04\)\( \beta_{2} + \)\(18\!\cdots\!24\)\( \beta_{3} - \)\(26\!\cdots\!12\)\( \beta_{4}) q^{44}\) \(+(\)\(71\!\cdots\!03\)\( + \)\(55\!\cdots\!47\)\( \beta_{1} - \)\(28\!\cdots\!44\)\( \beta_{2} - \)\(25\!\cdots\!49\)\( \beta_{3}) q^{45}\) \(+(\)\(43\!\cdots\!36\)\( + \)\(16\!\cdots\!28\)\( \beta_{1} + \)\(43\!\cdots\!60\)\( \beta_{2} - \)\(86\!\cdots\!16\)\( \beta_{3} + \)\(10\!\cdots\!08\)\( \beta_{4}) q^{46}\) \(+(-\)\(93\!\cdots\!58\)\( + \)\(11\!\cdots\!82\)\( \beta_{1} - \)\(83\!\cdots\!14\)\( \beta_{2} - \)\(15\!\cdots\!58\)\( \beta_{3} - \)\(20\!\cdots\!46\)\( \beta_{4}) q^{47}\) \(+(-\)\(27\!\cdots\!88\)\( + \)\(37\!\cdots\!36\)\( \beta_{1} - \)\(22\!\cdots\!08\)\( \beta_{2} + \)\(47\!\cdots\!04\)\( \beta_{3} + \)\(25\!\cdots\!48\)\( \beta_{4}) q^{48}\) \(+(-\)\(23\!\cdots\!23\)\( + \)\(12\!\cdots\!44\)\( \beta_{1} + \)\(30\!\cdots\!18\)\( \beta_{2} + \)\(61\!\cdots\!88\)\( \beta_{3} - \)\(99\!\cdots\!94\)\( \beta_{4}) q^{49}\) \(+(-\)\(13\!\cdots\!70\)\( + \)\(20\!\cdots\!45\)\( \beta_{1} + \)\(60\!\cdots\!60\)\( \beta_{2} + \)\(13\!\cdots\!60\)\( \beta_{3} + \)\(58\!\cdots\!00\)\( \beta_{4}) q^{50}\) \(+(-\)\(27\!\cdots\!56\)\( + \)\(22\!\cdots\!70\)\( \beta_{1} + \)\(59\!\cdots\!74\)\( \beta_{2} + \)\(12\!\cdots\!18\)\( \beta_{3} - \)\(10\!\cdots\!34\)\( \beta_{4}) q^{51}\) \(+(\)\(93\!\cdots\!28\)\( + \)\(13\!\cdots\!20\)\( \beta_{1} - \)\(70\!\cdots\!66\)\( \beta_{2} - \)\(26\!\cdots\!48\)\( \beta_{3} + \)\(14\!\cdots\!24\)\( \beta_{4}) q^{52}\) \(+(\)\(33\!\cdots\!57\)\( + \)\(36\!\cdots\!37\)\( \beta_{1} - \)\(32\!\cdots\!18\)\( \beta_{2} - \)\(87\!\cdots\!15\)\( \beta_{3} - \)\(34\!\cdots\!30\)\( \beta_{4}) q^{53}\) \(+(-\)\(62\!\cdots\!34\)\( - \)\(12\!\cdots\!43\)\( \beta_{1}) q^{54}\) \(+(\)\(13\!\cdots\!92\)\( + \)\(75\!\cdots\!08\)\( \beta_{1} - \)\(21\!\cdots\!16\)\( \beta_{2} - \)\(14\!\cdots\!36\)\( \beta_{3} - \)\(10\!\cdots\!00\)\( \beta_{4}) q^{55}\) \(+(\)\(52\!\cdots\!04\)\( + \)\(16\!\cdots\!60\)\( \beta_{1} + \)\(30\!\cdots\!92\)\( \beta_{2} + \)\(34\!\cdots\!12\)\( \beta_{3} + \)\(81\!\cdots\!44\)\( \beta_{4}) q^{56}\) \(+(\)\(29\!\cdots\!58\)\( + \)\(28\!\cdots\!02\)\( \beta_{1} + \)\(29\!\cdots\!46\)\( \beta_{2} + \)\(12\!\cdots\!82\)\( \beta_{3} - \)\(38\!\cdots\!66\)\( \beta_{4}) q^{57}\) \(+(-\)\(78\!\cdots\!08\)\( + \)\(74\!\cdots\!26\)\( \beta_{1} - \)\(20\!\cdots\!16\)\( \beta_{2} - \)\(15\!\cdots\!20\)\( \beta_{3} - \)\(25\!\cdots\!40\)\( \beta_{4}) q^{58}\) \(+(-\)\(78\!\cdots\!00\)\( + \)\(34\!\cdots\!04\)\( \beta_{1} + \)\(12\!\cdots\!68\)\( \beta_{2} + \)\(22\!\cdots\!96\)\( \beta_{3} + \)\(98\!\cdots\!52\)\( \beta_{4}) q^{59}\) \(+(-\)\(24\!\cdots\!56\)\( + \)\(36\!\cdots\!56\)\( \beta_{1} - \)\(49\!\cdots\!62\)\( \beta_{2} + \)\(72\!\cdots\!48\)\( \beta_{3} + \)\(42\!\cdots\!00\)\( \beta_{4}) q^{60}\) \(+(\)\(14\!\cdots\!78\)\( + \)\(13\!\cdots\!06\)\( \beta_{1} + \)\(78\!\cdots\!87\)\( \beta_{2} - \)\(41\!\cdots\!38\)\( \beta_{3} + \)\(17\!\cdots\!69\)\( \beta_{4}) q^{61}\) \(+(-\)\(19\!\cdots\!76\)\( - \)\(86\!\cdots\!88\)\( \beta_{1} - \)\(11\!\cdots\!68\)\( \beta_{2} + \)\(23\!\cdots\!16\)\( \beta_{3} - \)\(48\!\cdots\!08\)\( \beta_{4}) q^{62}\) \(+(-\)\(63\!\cdots\!73\)\( - \)\(31\!\cdots\!21\)\( \beta_{1} - \)\(26\!\cdots\!09\)\( \beta_{2} + \)\(15\!\cdots\!59\)\( \beta_{3} - \)\(99\!\cdots\!67\)\( \beta_{4}) q^{63}\) \(+(-\)\(40\!\cdots\!24\)\( - \)\(11\!\cdots\!20\)\( \beta_{1} - \)\(27\!\cdots\!36\)\( \beta_{2} - \)\(61\!\cdots\!56\)\( \beta_{3} + \)\(60\!\cdots\!28\)\( \beta_{4}) q^{64}\) \(+(\)\(26\!\cdots\!58\)\( - \)\(36\!\cdots\!58\)\( \beta_{1} + \)\(69\!\cdots\!66\)\( \beta_{2} + \)\(27\!\cdots\!86\)\( \beta_{3} + \)\(15\!\cdots\!50\)\( \beta_{4}) q^{65}\) \(+(\)\(98\!\cdots\!88\)\( - \)\(67\!\cdots\!60\)\( \beta_{1} + \)\(59\!\cdots\!68\)\( \beta_{2} + \)\(17\!\cdots\!36\)\( \beta_{3} - \)\(15\!\cdots\!68\)\( \beta_{4}) q^{66}\) \(+(\)\(13\!\cdots\!72\)\( - \)\(52\!\cdots\!40\)\( \beta_{1} - \)\(69\!\cdots\!24\)\( \beta_{2} - \)\(76\!\cdots\!16\)\( \beta_{3} - \)\(58\!\cdots\!92\)\( \beta_{4}) q^{67}\) \(+(\)\(24\!\cdots\!12\)\( + \)\(10\!\cdots\!12\)\( \beta_{1} + \)\(20\!\cdots\!42\)\( \beta_{2} - \)\(45\!\cdots\!72\)\( \beta_{3} + \)\(32\!\cdots\!36\)\( \beta_{4}) q^{68}\) \(+(-\)\(42\!\cdots\!18\)\( - \)\(71\!\cdots\!86\)\( \beta_{1} - \)\(17\!\cdots\!70\)\( \beta_{2} + \)\(60\!\cdots\!74\)\( \beta_{3} + \)\(14\!\cdots\!38\)\( \beta_{4}) q^{69}\) \(+(-\)\(58\!\cdots\!60\)\( + \)\(10\!\cdots\!60\)\( \beta_{1} - \)\(67\!\cdots\!20\)\( \beta_{2} - \)\(73\!\cdots\!20\)\( \beta_{3} + \)\(12\!\cdots\!00\)\( \beta_{4}) q^{70}\) \(+(\)\(36\!\cdots\!50\)\( + \)\(31\!\cdots\!50\)\( \beta_{1} + \)\(10\!\cdots\!50\)\( \beta_{2} + \)\(69\!\cdots\!54\)\( \beta_{3} - \)\(34\!\cdots\!02\)\( \beta_{4}) q^{71}\) \(+(-\)\(76\!\cdots\!44\)\( - \)\(18\!\cdots\!84\)\( \beta_{1} - \)\(10\!\cdots\!32\)\( \beta_{2} + \)\(49\!\cdots\!92\)\( \beta_{3} - \)\(24\!\cdots\!96\)\( \beta_{4}) q^{72}\) \(+(-\)\(62\!\cdots\!38\)\( + \)\(60\!\cdots\!92\)\( \beta_{1} - \)\(20\!\cdots\!30\)\( \beta_{2} + \)\(17\!\cdots\!40\)\( \beta_{3} + \)\(65\!\cdots\!30\)\( \beta_{4}) q^{73}\) \(+(-\)\(18\!\cdots\!52\)\( + \)\(65\!\cdots\!34\)\( \beta_{1} + \)\(30\!\cdots\!64\)\( \beta_{2} - \)\(12\!\cdots\!68\)\( \beta_{3} + \)\(13\!\cdots\!84\)\( \beta_{4}) q^{74}\) \(+(\)\(19\!\cdots\!45\)\( + \)\(20\!\cdots\!80\)\( \beta_{1} - \)\(66\!\cdots\!10\)\( \beta_{2} - \)\(18\!\cdots\!60\)\( \beta_{3} - \)\(24\!\cdots\!50\)\( \beta_{4}) q^{75}\) \(+(\)\(34\!\cdots\!00\)\( + \)\(44\!\cdots\!56\)\( \beta_{1} + \)\(16\!\cdots\!24\)\( \beta_{2} - \)\(29\!\cdots\!36\)\( \beta_{3} - \)\(10\!\cdots\!32\)\( \beta_{4}) q^{76}\) \(+(\)\(20\!\cdots\!88\)\( - \)\(28\!\cdots\!68\)\( \beta_{1} + \)\(26\!\cdots\!96\)\( \beta_{2} + \)\(64\!\cdots\!68\)\( \beta_{3} - \)\(74\!\cdots\!84\)\( \beta_{4}) q^{77}\) \(+(-\)\(16\!\cdots\!56\)\( + \)\(18\!\cdots\!10\)\( \beta_{1} - \)\(35\!\cdots\!48\)\( \beta_{2} - \)\(70\!\cdots\!00\)\( \beta_{3} + \)\(11\!\cdots\!00\)\( \beta_{4}) q^{78}\) \(+(-\)\(54\!\cdots\!21\)\( - \)\(14\!\cdots\!65\)\( \beta_{1} - \)\(37\!\cdots\!33\)\( \beta_{2} + \)\(35\!\cdots\!87\)\( \beta_{3} - \)\(15\!\cdots\!31\)\( \beta_{4}) q^{79}\) \(+(\)\(34\!\cdots\!08\)\( - \)\(19\!\cdots\!08\)\( \beta_{1} + \)\(38\!\cdots\!16\)\( \beta_{2} - \)\(72\!\cdots\!64\)\( \beta_{3} - \)\(27\!\cdots\!00\)\( \beta_{4}) q^{80}\) \(+\)\(62\!\cdots\!01\)\( q^{81}\) \(+(-\)\(42\!\cdots\!60\)\( - \)\(14\!\cdots\!38\)\( \beta_{1} - \)\(36\!\cdots\!28\)\( \beta_{2} - \)\(26\!\cdots\!16\)\( \beta_{3} + \)\(29\!\cdots\!08\)\( \beta_{4}) q^{82}\) \(+(-\)\(23\!\cdots\!46\)\( - \)\(15\!\cdots\!02\)\( \beta_{1} + \)\(60\!\cdots\!40\)\( \beta_{2} - \)\(27\!\cdots\!50\)\( \beta_{3} + \)\(11\!\cdots\!00\)\( \beta_{4}) q^{83}\) \(+(-\)\(57\!\cdots\!24\)\( - \)\(24\!\cdots\!84\)\( \beta_{1} - \)\(36\!\cdots\!24\)\( \beta_{2} - \)\(35\!\cdots\!40\)\( \beta_{3} - \)\(70\!\cdots\!80\)\( \beta_{4}) q^{84}\) \(+(-\)\(70\!\cdots\!62\)\( - \)\(95\!\cdots\!38\)\( \beta_{1} + \)\(37\!\cdots\!76\)\( \beta_{2} + \)\(37\!\cdots\!46\)\( \beta_{3} - \)\(80\!\cdots\!00\)\( \beta_{4}) q^{85}\) \(+(-\)\(11\!\cdots\!96\)\( + \)\(70\!\cdots\!64\)\( \beta_{1} + \)\(22\!\cdots\!60\)\( \beta_{2} - \)\(62\!\cdots\!88\)\( \beta_{3} - \)\(95\!\cdots\!56\)\( \beta_{4}) q^{86}\) \(+(-\)\(36\!\cdots\!33\)\( + \)\(15\!\cdots\!39\)\( \beta_{1} - \)\(79\!\cdots\!58\)\( \beta_{2} + \)\(56\!\cdots\!87\)\( \beta_{3} - \)\(43\!\cdots\!06\)\( \beta_{4}) q^{87}\) \(+(\)\(17\!\cdots\!12\)\( + \)\(58\!\cdots\!64\)\( \beta_{1} + \)\(37\!\cdots\!32\)\( \beta_{2} - \)\(11\!\cdots\!88\)\( \beta_{3} + \)\(35\!\cdots\!44\)\( \beta_{4}) q^{88}\) \(+(\)\(18\!\cdots\!50\)\( + \)\(18\!\cdots\!76\)\( \beta_{1} + \)\(15\!\cdots\!48\)\( \beta_{2} + \)\(19\!\cdots\!04\)\( \beta_{3} + \)\(30\!\cdots\!48\)\( \beta_{4}) q^{89}\) \(+(-\)\(15\!\cdots\!16\)\( + \)\(42\!\cdots\!66\)\( \beta_{1} - \)\(19\!\cdots\!32\)\( \beta_{2} + \)\(33\!\cdots\!28\)\( \beta_{3} + \)\(83\!\cdots\!00\)\( \beta_{4}) q^{90}\) \(+(-\)\(16\!\cdots\!34\)\( + \)\(15\!\cdots\!78\)\( \beta_{1} - \)\(97\!\cdots\!78\)\( \beta_{2} - \)\(15\!\cdots\!86\)\( \beta_{3} - \)\(36\!\cdots\!82\)\( \beta_{4}) q^{91}\) \(+(-\)\(27\!\cdots\!60\)\( - \)\(34\!\cdots\!24\)\( \beta_{1} - \)\(18\!\cdots\!04\)\( \beta_{2} - \)\(35\!\cdots\!88\)\( \beta_{3} - \)\(22\!\cdots\!56\)\( \beta_{4}) q^{92}\) \(+(\)\(74\!\cdots\!41\)\( + \)\(35\!\cdots\!21\)\( \beta_{1} + \)\(17\!\cdots\!89\)\( \beta_{2} - \)\(11\!\cdots\!75\)\( \beta_{3} - \)\(31\!\cdots\!25\)\( \beta_{4}) q^{93}\) \(+(-\)\(26\!\cdots\!16\)\( + \)\(24\!\cdots\!72\)\( \beta_{1} + \)\(25\!\cdots\!20\)\( \beta_{2} + \)\(29\!\cdots\!72\)\( \beta_{3} + \)\(20\!\cdots\!64\)\( \beta_{4}) q^{94}\) \(+(-\)\(12\!\cdots\!88\)\( + \)\(27\!\cdots\!88\)\( \beta_{1} + \)\(16\!\cdots\!24\)\( \beta_{2} + \)\(53\!\cdots\!04\)\( \beta_{3} + \)\(10\!\cdots\!00\)\( \beta_{4}) q^{95}\) \(+(-\)\(64\!\cdots\!76\)\( + \)\(43\!\cdots\!12\)\( \beta_{1} + \)\(16\!\cdots\!36\)\( \beta_{2} - \)\(40\!\cdots\!44\)\( \beta_{3} + \)\(13\!\cdots\!72\)\( \beta_{4}) q^{96}\) \(+(-\)\(29\!\cdots\!94\)\( - \)\(63\!\cdots\!80\)\( \beta_{1} + \)\(78\!\cdots\!20\)\( \beta_{2} + \)\(15\!\cdots\!36\)\( \beta_{3} - \)\(18\!\cdots\!68\)\( \beta_{4}) q^{97}\) \(+(-\)\(32\!\cdots\!30\)\( - \)\(48\!\cdots\!81\)\( \beta_{1} - \)\(20\!\cdots\!96\)\( \beta_{2} + \)\(61\!\cdots\!20\)\( \beta_{3} - \)\(31\!\cdots\!60\)\( \beta_{4}) q^{98}\) \(+(\)\(67\!\cdots\!74\)\( - \)\(18\!\cdots\!34\)\( \beta_{1} + \)\(23\!\cdots\!36\)\( \beta_{2} - \)\(16\!\cdots\!18\)\( \beta_{3} - \)\(28\!\cdots\!16\)\( \beta_{4}) q^{99}\) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(5q \) \(\mathstrut -\mathstrut 25051277688q^{2} \) \(\mathstrut +\mathstrut 250157725494998535q^{3} \) \(\mathstrut +\mathstrut \)\(19\!\cdots\!24\)\(q^{4} \) \(\mathstrut +\mathstrut \)\(14\!\cdots\!30\)\(q^{5} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!16\)\(q^{6} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!52\)\(q^{7} \) \(\mathstrut -\mathstrut \)\(15\!\cdots\!48\)\(q^{8} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!45\)\(q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(5q \) \(\mathstrut -\mathstrut 25051277688q^{2} \) \(\mathstrut +\mathstrut 250157725494998535q^{3} \) \(\mathstrut +\mathstrut \)\(19\!\cdots\!24\)\(q^{4} \) \(\mathstrut +\mathstrut \)\(14\!\cdots\!30\)\(q^{5} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!16\)\(q^{6} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!52\)\(q^{7} \) \(\mathstrut -\mathstrut \)\(15\!\cdots\!48\)\(q^{8} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!45\)\(q^{9} \) \(\mathstrut -\mathstrut \)\(30\!\cdots\!60\)\(q^{10} \) \(\mathstrut +\mathstrut \)\(13\!\cdots\!12\)\(q^{11} \) \(\mathstrut +\mathstrut \)\(95\!\cdots\!68\)\(q^{12} \) \(\mathstrut -\mathstrut \)\(37\!\cdots\!02\)\(q^{13} \) \(\mathstrut +\mathstrut \)\(17\!\cdots\!32\)\(q^{14} \) \(\mathstrut +\mathstrut \)\(71\!\cdots\!10\)\(q^{15} \) \(\mathstrut -\mathstrut \)\(27\!\cdots\!16\)\(q^{16} \) \(\mathstrut -\mathstrut \)\(27\!\cdots\!10\)\(q^{17} \) \(\mathstrut -\mathstrut \)\(62\!\cdots\!12\)\(q^{18} \) \(\mathstrut +\mathstrut \)\(29\!\cdots\!48\)\(q^{19} \) \(\mathstrut -\mathstrut \)\(24\!\cdots\!20\)\(q^{20} \) \(\mathstrut -\mathstrut \)\(63\!\cdots\!64\)\(q^{21} \) \(\mathstrut +\mathstrut \)\(98\!\cdots\!80\)\(q^{22} \) \(\mathstrut -\mathstrut \)\(42\!\cdots\!24\)\(q^{23} \) \(\mathstrut -\mathstrut \)\(76\!\cdots\!36\)\(q^{24} \) \(\mathstrut +\mathstrut \)\(19\!\cdots\!75\)\(q^{25} \) \(\mathstrut -\mathstrut \)\(16\!\cdots\!00\)\(q^{26} \) \(\mathstrut +\mathstrut \)\(62\!\cdots\!15\)\(q^{27} \) \(\mathstrut -\mathstrut \)\(57\!\cdots\!36\)\(q^{28} \) \(\mathstrut -\mathstrut \)\(36\!\cdots\!74\)\(q^{29} \) \(\mathstrut -\mathstrut \)\(15\!\cdots\!20\)\(q^{30} \) \(\mathstrut +\mathstrut \)\(74\!\cdots\!84\)\(q^{31} \) \(\mathstrut -\mathstrut \)\(63\!\cdots\!72\)\(q^{32} \) \(\mathstrut +\mathstrut \)\(67\!\cdots\!84\)\(q^{33} \) \(\mathstrut -\mathstrut \)\(62\!\cdots\!32\)\(q^{34} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!00\)\(q^{35} \) \(\mathstrut +\mathstrut \)\(47\!\cdots\!76\)\(q^{36} \) \(\mathstrut -\mathstrut \)\(16\!\cdots\!02\)\(q^{37} \) \(\mathstrut -\mathstrut \)\(91\!\cdots\!56\)\(q^{38} \) \(\mathstrut -\mathstrut \)\(18\!\cdots\!14\)\(q^{39} \) \(\mathstrut +\mathstrut \)\(75\!\cdots\!20\)\(q^{40} \) \(\mathstrut +\mathstrut \)\(22\!\cdots\!94\)\(q^{41} \) \(\mathstrut +\mathstrut \)\(89\!\cdots\!24\)\(q^{42} \) \(\mathstrut +\mathstrut \)\(67\!\cdots\!92\)\(q^{43} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!60\)\(q^{44} \) \(\mathstrut +\mathstrut \)\(35\!\cdots\!70\)\(q^{45} \) \(\mathstrut +\mathstrut \)\(21\!\cdots\!24\)\(q^{46} \) \(\mathstrut -\mathstrut \)\(46\!\cdots\!04\)\(q^{47} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!12\)\(q^{48} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!03\)\(q^{49} \) \(\mathstrut -\mathstrut \)\(66\!\cdots\!00\)\(q^{50} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!70\)\(q^{51} \) \(\mathstrut +\mathstrut \)\(46\!\cdots\!00\)\(q^{52} \) \(\mathstrut +\mathstrut \)\(16\!\cdots\!86\)\(q^{53} \) \(\mathstrut -\mathstrut \)\(31\!\cdots\!84\)\(q^{54} \) \(\mathstrut +\mathstrut \)\(69\!\cdots\!80\)\(q^{55} \) \(\mathstrut +\mathstrut \)\(26\!\cdots\!00\)\(q^{56} \) \(\mathstrut +\mathstrut \)\(14\!\cdots\!36\)\(q^{57} \) \(\mathstrut -\mathstrut \)\(39\!\cdots\!92\)\(q^{58} \) \(\mathstrut -\mathstrut \)\(39\!\cdots\!08\)\(q^{59} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!40\)\(q^{60} \) \(\mathstrut +\mathstrut \)\(72\!\cdots\!78\)\(q^{61} \) \(\mathstrut -\mathstrut \)\(99\!\cdots\!04\)\(q^{62} \) \(\mathstrut -\mathstrut \)\(31\!\cdots\!48\)\(q^{63} \) \(\mathstrut -\mathstrut \)\(20\!\cdots\!80\)\(q^{64} \) \(\mathstrut +\mathstrut \)\(13\!\cdots\!20\)\(q^{65} \) \(\mathstrut +\mathstrut \)\(49\!\cdots\!60\)\(q^{66} \) \(\mathstrut +\mathstrut \)\(66\!\cdots\!40\)\(q^{67} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!36\)\(q^{68} \) \(\mathstrut -\mathstrut \)\(21\!\cdots\!68\)\(q^{69} \) \(\mathstrut -\mathstrut \)\(29\!\cdots\!00\)\(q^{70} \) \(\mathstrut +\mathstrut \)\(18\!\cdots\!00\)\(q^{71} \) \(\mathstrut -\mathstrut \)\(38\!\cdots\!52\)\(q^{72} \) \(\mathstrut -\mathstrut \)\(31\!\cdots\!74\)\(q^{73} \) \(\mathstrut -\mathstrut \)\(90\!\cdots\!28\)\(q^{74} \) \(\mathstrut +\mathstrut \)\(98\!\cdots\!25\)\(q^{75} \) \(\mathstrut +\mathstrut \)\(17\!\cdots\!88\)\(q^{76} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!76\)\(q^{77} \) \(\mathstrut -\mathstrut \)\(81\!\cdots\!00\)\(q^{78} \) \(\mathstrut -\mathstrut \)\(27\!\cdots\!00\)\(q^{79} \) \(\mathstrut +\mathstrut \)\(17\!\cdots\!20\)\(q^{80} \) \(\mathstrut +\mathstrut \)\(31\!\cdots\!05\)\(q^{81} \) \(\mathstrut -\mathstrut \)\(21\!\cdots\!24\)\(q^{82} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!76\)\(q^{83} \) \(\mathstrut -\mathstrut \)\(28\!\cdots\!52\)\(q^{84} \) \(\mathstrut -\mathstrut \)\(35\!\cdots\!80\)\(q^{85} \) \(\mathstrut -\mathstrut \)\(59\!\cdots\!08\)\(q^{86} \) \(\mathstrut -\mathstrut \)\(18\!\cdots\!18\)\(q^{87} \) \(\mathstrut +\mathstrut \)\(88\!\cdots\!32\)\(q^{88} \) \(\mathstrut +\mathstrut \)\(92\!\cdots\!98\)\(q^{89} \) \(\mathstrut -\mathstrut \)\(77\!\cdots\!40\)\(q^{90} \) \(\mathstrut -\mathstrut \)\(80\!\cdots\!76\)\(q^{91} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!52\)\(q^{92} \) \(\mathstrut +\mathstrut \)\(37\!\cdots\!88\)\(q^{93} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!24\)\(q^{94} \) \(\mathstrut -\mathstrut \)\(61\!\cdots\!20\)\(q^{95} \) \(\mathstrut -\mathstrut \)\(32\!\cdots\!04\)\(q^{96} \) \(\mathstrut -\mathstrut \)\(14\!\cdots\!10\)\(q^{97} \) \(\mathstrut -\mathstrut \)\(16\!\cdots\!88\)\(q^{98} \) \(\mathstrut +\mathstrut \)\(33\!\cdots\!88\)\(q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{5}\mathstrut -\mathstrut \) \(2\) \(x^{4}\mathstrut -\mathstrut \) \(11794880527205841026\) \(x^{3}\mathstrut -\mathstrut \) \(1576914464895109132006308600\) \(x^{2}\mathstrut +\mathstrut \) \(21982147283297468753929167380086357881\) \(x\mathstrut -\mathstrut \) \(10487125185557323126488817758602332290924256254\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 24 \nu - 10 \)
\(\beta_{2}\)\(=\)\( 576 \nu^{2} - 115512328608 \nu - 2717540473422020841408 \)
\(\beta_{3}\)\(=\)\((\)\(627183\) \(\nu^{4}\mathstrut +\mathstrut \) \(387649018384857\) \(\nu^{3}\mathstrut -\mathstrut \) \(6901727976782003164396941\) \(\nu^{2}\mathstrut -\mathstrut \) \(5609906443063199591123859798720621\) \(\nu\mathstrut +\mathstrut \) \(8323431865157550665308972831447685029379122\)\()/\)\(200332900414259200\)
\(\beta_{4}\)\(=\)\((\)\(89019\) \(\nu^{4}\mathstrut +\mathstrut \) \(254610810147501\) \(\nu^{3}\mathstrut -\mathstrut \) \(1196771911116752539811313\) \(\nu^{2}\mathstrut -\mathstrut \) \(2362874168895880655055146438671953\) \(\nu\mathstrut +\mathstrut \) \(2017174951889050070721069247792617717110746\)\()/\)\(14309492886732800\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1}\mathstrut +\mathstrut \) \(10\)\()/24\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{2}\mathstrut +\mathstrut \) \(4813013692\) \(\beta_{1}\mathstrut +\mathstrut \) \(2717540473470150978328\)\()/576\)
\(\nu^{3}\)\(=\)\((\)\(15486\) \(\beta_{4}\mathstrut -\mathstrut \) \(30772\) \(\beta_{3}\mathstrut +\mathstrut \) \(408044456\) \(\beta_{2}\mathstrut +\mathstrut \) \(72607281339847992463\) \(\beta_{1}\mathstrut +\mathstrut \) \(204368117736682088695710785870\)\()/216\)
\(\nu^{4}\)\(=\)\((\)\(-\)\(229717936820656\) \(\beta_{4}\mathstrut +\mathstrut \) \(2112326721223712\) \(\beta_{3}\mathstrut +\mathstrut \) \(92986064678441917467\) \(\beta_{2}\mathstrut +\mathstrut \) \(1331661414327886771669041456900\) \(\beta_{1}\mathstrut +\mathstrut \) \(197313225711958020400085616512940566052648\)\()/5184\)

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
3.23970e9
9.01834e8
6.47992e8
−1.95252e9
−2.83700e9
−8.27630e10 5.00315e16 4.48853e21 −1.31164e24 −4.14076e27 −1.62984e30 −1.76066e32 2.50316e33 1.08555e35
1.2 −2.66543e10 5.00315e16 −1.65073e21 1.17208e25 −1.33355e27 −5.64464e29 1.06935e32 2.50316e33 −3.12409e35
1.3 −2.05621e10 5.00315e16 −1.93838e21 −6.43665e24 −1.02875e27 4.31786e29 8.84080e31 2.50316e33 1.32351e35
1.4 4.18503e10 5.00315e16 −6.09737e20 3.60424e24 2.09383e27 −2.88017e29 −1.24334e32 2.50316e33 1.50838e35
1.5 6.30778e10 5.00315e16 1.61762e21 −6.15414e24 3.15588e27 7.89316e29 −4.69021e31 2.50316e33 −3.88189e35
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{5} \) \(\mathstrut +\mathstrut 25051277688 T_{2}^{4} \) \(\mathstrut -\mathstrut \)\(65\!\cdots\!60\)\( T_{2}^{3} \) \(\mathstrut -\mathstrut \)\(79\!\cdots\!72\)\( T_{2}^{2} \) \(\mathstrut +\mathstrut \)\(70\!\cdots\!68\)\( T_{2} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!68\)\( \) acting on \(S_{72}^{\mathrm{new}}(\Gamma_0(3))\).