Properties

Label 1.74.a.a
Level 1
Weight 74
Character orbit 1.a
Self dual Yes
Analytic conductor 33.748
Analytic rank 1
Dimension 5
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 1 \)
Weight: \( k \) = \( 74 \)
Character orbit: \([\chi]\) = 1.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(33.7483973737\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{39}\cdot 3^{15}\cdot 5^{6}\cdot 7^{2} \)
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.

\(f(q)\) \(=\) \( q\) \(+(-18417866698 - \beta_{1}) q^{2}\) \(+(-25839159645211912 + 123123 \beta_{1} - \beta_{2}) q^{3}\) \(+(\)\(17\!\cdots\!30\)\( + 26620058935 \beta_{1} - 3441 \beta_{2} + \beta_{3}) q^{4}\) \(+(\)\(46\!\cdots\!55\)\( + 14159858805524 \beta_{1} + 12391606 \beta_{2} + 132 \beta_{3} + \beta_{4}) q^{5}\) \(+(-\)\(66\!\cdots\!40\)\( - 2320831478698956 \beta_{1} - 1747442760 \beta_{2} - 29592 \beta_{3} - 144 \beta_{4}) q^{6}\) \(+(-\)\(87\!\cdots\!00\)\( - 3945223198957624010 \beta_{1} - 2139829538826 \beta_{2} - 73498544 \beta_{3} - 186732 \beta_{4}) q^{7}\) \(+(-\)\(76\!\cdots\!04\)\( - \)\(12\!\cdots\!56\)\( \beta_{1} - 37115517216784 \beta_{2} - 41549502192 \beta_{3} - 15102976 \beta_{4}) q^{8}\) \(+(\)\(64\!\cdots\!19\)\( - \)\(16\!\cdots\!92\)\( \beta_{1} + 53110578546819540 \beta_{2} - 6698316560904 \beta_{3} + 1614263742 \beta_{4}) q^{9}\) \(+O(q^{10})\) \( q\) \(+(-18417866698 - \beta_{1}) q^{2}\) \(+(-25839159645211912 + 123123 \beta_{1} - \beta_{2}) q^{3}\) \(+(\)\(17\!\cdots\!30\)\( + 26620058935 \beta_{1} - 3441 \beta_{2} + \beta_{3}) q^{4}\) \(+(\)\(46\!\cdots\!55\)\( + 14159858805524 \beta_{1} + 12391606 \beta_{2} + 132 \beta_{3} + \beta_{4}) q^{5}\) \(+(-\)\(66\!\cdots\!40\)\( - 2320831478698956 \beta_{1} - 1747442760 \beta_{2} - 29592 \beta_{3} - 144 \beta_{4}) q^{6}\) \(+(-\)\(87\!\cdots\!00\)\( - 3945223198957624010 \beta_{1} - 2139829538826 \beta_{2} - 73498544 \beta_{3} - 186732 \beta_{4}) q^{7}\) \(+(-\)\(76\!\cdots\!04\)\( - \)\(12\!\cdots\!56\)\( \beta_{1} - 37115517216784 \beta_{2} - 41549502192 \beta_{3} - 15102976 \beta_{4}) q^{8}\) \(+(\)\(64\!\cdots\!19\)\( - \)\(16\!\cdots\!92\)\( \beta_{1} + 53110578546819540 \beta_{2} - 6698316560904 \beta_{3} + 1614263742 \beta_{4}) q^{9}\) \(+(-\)\(21\!\cdots\!40\)\( - \)\(57\!\cdots\!82\)\( \beta_{1} + 2196470269361916192 \beta_{2} - 184621370870176 \beta_{3} - 52532895168 \beta_{4}) q^{10}\) \(+(\)\(10\!\cdots\!32\)\( + \)\(18\!\cdots\!17\)\( \beta_{1} + \)\(20\!\cdots\!33\)\( \beta_{2} + 7662892565307936 \beta_{3} + 649409704328 \beta_{4}) q^{11}\) \(+(\)\(27\!\cdots\!44\)\( - \)\(22\!\cdots\!36\)\( \beta_{1} + \)\(91\!\cdots\!20\)\( \beta_{2} + 27025252969283172 \beta_{3} + 7724662972416 \beta_{4}) q^{12}\) \(+(\)\(95\!\cdots\!91\)\( - \)\(98\!\cdots\!32\)\( \beta_{1} + \)\(14\!\cdots\!34\)\( \beta_{2} - 3198282957929973212 \beta_{3} - 483886195074711 \beta_{4}) q^{13}\) \(+(\)\(52\!\cdots\!48\)\( + \)\(15\!\cdots\!60\)\( \beta_{1} - \)\(40\!\cdots\!48\)\( \beta_{2} + 36504584384330910288 \beta_{3} + 10548213915031520 \beta_{4}) q^{14}\) \(+(-\)\(10\!\cdots\!20\)\( + \)\(19\!\cdots\!74\)\( \beta_{1} - \)\(27\!\cdots\!94\)\( \beta_{2} + 71581385179388966832 \beta_{3} - 145876262636005524 \beta_{4}) q^{15}\) \(+(\)\(11\!\cdots\!32\)\( + \)\(25\!\cdots\!04\)\( \beta_{1} + \)\(81\!\cdots\!44\)\( \beta_{2} - \)\(50\!\cdots\!36\)\( \beta_{3} + 1392819877299142656 \beta_{4}) q^{16}\) \(+(\)\(13\!\cdots\!16\)\( + \)\(67\!\cdots\!44\)\( \beta_{1} + \)\(18\!\cdots\!60\)\( \beta_{2} + \)\(41\!\cdots\!84\)\( \beta_{3} - 8696329022352154898 \beta_{4}) q^{17}\) \(+(\)\(13\!\cdots\!78\)\( + \)\(64\!\cdots\!91\)\( \beta_{1} + \)\(38\!\cdots\!36\)\( \beta_{2} - \)\(10\!\cdots\!76\)\( \beta_{3} + 20923570108645529472 \beta_{4}) q^{18}\) \(+(\)\(62\!\cdots\!72\)\( + \)\(42\!\cdots\!99\)\( \beta_{1} - \)\(10\!\cdots\!13\)\( \beta_{2} - \)\(22\!\cdots\!84\)\( \beta_{3} + \)\(24\!\cdots\!56\)\( \beta_{4}) q^{19}\) \(+(\)\(13\!\cdots\!60\)\( + \)\(20\!\cdots\!18\)\( \beta_{1} - \)\(22\!\cdots\!58\)\( \beta_{2} + \)\(15\!\cdots\!74\)\( \beta_{3} - \)\(36\!\cdots\!68\)\( \beta_{4}) q^{20}\) \(+(\)\(17\!\cdots\!88\)\( - \)\(36\!\cdots\!00\)\( \beta_{1} + \)\(46\!\cdots\!88\)\( \beta_{2} - \)\(14\!\cdots\!08\)\( \beta_{3} + \)\(26\!\cdots\!20\)\( \beta_{4}) q^{21}\) \(+(-\)\(18\!\cdots\!36\)\( - \)\(84\!\cdots\!88\)\( \beta_{1} + \)\(15\!\cdots\!76\)\( \beta_{2} - \)\(42\!\cdots\!08\)\( \beta_{3} - \)\(11\!\cdots\!24\)\( \beta_{4}) q^{22}\) \(+(-\)\(82\!\cdots\!36\)\( - \)\(26\!\cdots\!54\)\( \beta_{1} - \)\(11\!\cdots\!74\)\( \beta_{2} + \)\(26\!\cdots\!40\)\( \beta_{3} + \)\(16\!\cdots\!20\)\( \beta_{4}) q^{23}\) \(+(\)\(32\!\cdots\!96\)\( - \)\(28\!\cdots\!36\)\( \beta_{1} + \)\(48\!\cdots\!84\)\( \beta_{2} - \)\(20\!\cdots\!56\)\( \beta_{3} + \)\(18\!\cdots\!96\)\( \beta_{4}) q^{24}\) \(+(\)\(64\!\cdots\!75\)\( + \)\(41\!\cdots\!00\)\( \beta_{1} + \)\(21\!\cdots\!00\)\( \beta_{2} - \)\(45\!\cdots\!00\)\( \beta_{3} - \)\(18\!\cdots\!00\)\( \beta_{4}) q^{25}\) \(+(\)\(89\!\cdots\!36\)\( + \)\(36\!\cdots\!82\)\( \beta_{1} - \)\(67\!\cdots\!40\)\( \beta_{2} + \)\(21\!\cdots\!84\)\( \beta_{3} + \)\(92\!\cdots\!68\)\( \beta_{4}) q^{26}\) \(+(-\)\(24\!\cdots\!84\)\( + \)\(26\!\cdots\!34\)\( \beta_{1} - \)\(39\!\cdots\!74\)\( \beta_{2} - \)\(87\!\cdots\!52\)\( \beta_{3} - \)\(30\!\cdots\!56\)\( \beta_{4}) q^{27}\) \(+(-\)\(75\!\cdots\!44\)\( - \)\(41\!\cdots\!32\)\( \beta_{1} + \)\(42\!\cdots\!00\)\( \beta_{2} - \)\(31\!\cdots\!88\)\( \beta_{3} + \)\(61\!\cdots\!36\)\( \beta_{4}) q^{28}\) \(+(-\)\(43\!\cdots\!97\)\( - \)\(12\!\cdots\!72\)\( \beta_{1} + \)\(42\!\cdots\!86\)\( \beta_{2} + \)\(10\!\cdots\!00\)\( \beta_{3} - \)\(71\!\cdots\!63\)\( \beta_{4}) q^{29}\) \(+(-\)\(16\!\cdots\!40\)\( - \)\(61\!\cdots\!32\)\( \beta_{1} - \)\(31\!\cdots\!08\)\( \beta_{2} + \)\(64\!\cdots\!24\)\( \beta_{3} + \)\(26\!\cdots\!32\)\( \beta_{4}) q^{30}\) \(+(-\)\(79\!\cdots\!08\)\( + \)\(11\!\cdots\!56\)\( \beta_{1} - \)\(52\!\cdots\!56\)\( \beta_{2} - \)\(95\!\cdots\!52\)\( \beta_{3} - \)\(27\!\cdots\!96\)\( \beta_{4}) q^{31}\) \(+(-\)\(18\!\cdots\!28\)\( + \)\(49\!\cdots\!52\)\( \beta_{1} + \)\(48\!\cdots\!52\)\( \beta_{2} - \)\(23\!\cdots\!16\)\( \beta_{3} + \)\(14\!\cdots\!52\)\( \beta_{4}) q^{32}\) \(+(-\)\(14\!\cdots\!34\)\( + \)\(46\!\cdots\!48\)\( \beta_{1} + \)\(39\!\cdots\!16\)\( \beta_{2} + \)\(17\!\cdots\!16\)\( \beta_{3} - \)\(38\!\cdots\!02\)\( \beta_{4}) q^{33}\) \(+(-\)\(64\!\cdots\!60\)\( - \)\(56\!\cdots\!26\)\( \beta_{1} + \)\(36\!\cdots\!44\)\( \beta_{2} - \)\(60\!\cdots\!96\)\( \beta_{3} + \)\(11\!\cdots\!36\)\( \beta_{4}) q^{34}\) \(+(-\)\(21\!\cdots\!40\)\( - \)\(87\!\cdots\!92\)\( \beta_{1} - \)\(39\!\cdots\!48\)\( \beta_{2} + \)\(77\!\cdots\!44\)\( \beta_{3} + \)\(36\!\cdots\!92\)\( \beta_{4}) q^{35}\) \(+(-\)\(68\!\cdots\!78\)\( - \)\(17\!\cdots\!41\)\( \beta_{1} - \)\(22\!\cdots\!81\)\( \beta_{2} - \)\(52\!\cdots\!51\)\( \beta_{3} - \)\(15\!\cdots\!24\)\( \beta_{4}) q^{36}\) \(+(-\)\(13\!\cdots\!97\)\( + \)\(10\!\cdots\!04\)\( \beta_{1} + \)\(34\!\cdots\!90\)\( \beta_{2} + \)\(78\!\cdots\!56\)\( \beta_{3} + \)\(29\!\cdots\!93\)\( \beta_{4}) q^{37}\) \(+(-\)\(40\!\cdots\!76\)\( + \)\(10\!\cdots\!24\)\( \beta_{1} + \)\(20\!\cdots\!56\)\( \beta_{2} - \)\(41\!\cdots\!64\)\( \beta_{3} + \)\(48\!\cdots\!08\)\( \beta_{4}) q^{38}\) \(+(-\)\(10\!\cdots\!60\)\( + \)\(13\!\cdots\!98\)\( \beta_{1} - \)\(22\!\cdots\!66\)\( \beta_{2} + \)\(81\!\cdots\!72\)\( \beta_{3} - \)\(17\!\cdots\!88\)\( \beta_{4}) q^{39}\) \(+(-\)\(17\!\cdots\!00\)\( - \)\(14\!\cdots\!80\)\( \beta_{1} - \)\(23\!\cdots\!20\)\( \beta_{2} + \)\(60\!\cdots\!60\)\( \beta_{3} + \)\(42\!\cdots\!80\)\( \beta_{4}) q^{40}\) \(+(-\)\(17\!\cdots\!78\)\( - \)\(39\!\cdots\!44\)\( \beta_{1} + \)\(15\!\cdots\!44\)\( \beta_{2} - \)\(21\!\cdots\!52\)\( \beta_{3} - \)\(26\!\cdots\!96\)\( \beta_{4}) q^{41}\) \(+(\)\(30\!\cdots\!16\)\( + \)\(13\!\cdots\!36\)\( \beta_{1} + \)\(56\!\cdots\!72\)\( \beta_{2} - \)\(10\!\cdots\!04\)\( \beta_{3} - \)\(52\!\cdots\!12\)\( \beta_{4}) q^{42}\) \(+(\)\(23\!\cdots\!20\)\( + \)\(23\!\cdots\!49\)\( \beta_{1} - \)\(21\!\cdots\!27\)\( \beta_{2} + \)\(73\!\cdots\!00\)\( \beta_{3} - \)\(43\!\cdots\!00\)\( \beta_{4}) q^{43}\) \(+(\)\(72\!\cdots\!60\)\( + \)\(52\!\cdots\!44\)\( \beta_{1} - \)\(23\!\cdots\!36\)\( \beta_{2} + \)\(38\!\cdots\!24\)\( \beta_{3} + \)\(64\!\cdots\!16\)\( \beta_{4}) q^{44}\) \(+(\)\(17\!\cdots\!15\)\( - \)\(78\!\cdots\!28\)\( \beta_{1} + \)\(53\!\cdots\!18\)\( \beta_{2} - \)\(18\!\cdots\!04\)\( \beta_{3} - \)\(28\!\cdots\!47\)\( \beta_{4}) q^{45}\) \(+(\)\(26\!\cdots\!52\)\( - \)\(19\!\cdots\!44\)\( \beta_{1} - \)\(11\!\cdots\!36\)\( \beta_{2} + \)\(20\!\cdots\!28\)\( \beta_{3} - \)\(64\!\cdots\!96\)\( \beta_{4}) q^{46}\) \(+(\)\(52\!\cdots\!24\)\( - \)\(36\!\cdots\!40\)\( \beta_{1} + \)\(17\!\cdots\!84\)\( \beta_{2} + \)\(21\!\cdots\!00\)\( \beta_{3} + \)\(20\!\cdots\!00\)\( \beta_{4}) q^{47}\) \(+(-\)\(61\!\cdots\!16\)\( + \)\(23\!\cdots\!68\)\( \beta_{1} - \)\(82\!\cdots\!36\)\( \beta_{2} - \)\(25\!\cdots\!44\)\( \beta_{3} - \)\(13\!\cdots\!32\)\( \beta_{4}) q^{48}\) \(+(-\)\(95\!\cdots\!63\)\( + \)\(18\!\cdots\!24\)\( \beta_{1} + \)\(72\!\cdots\!16\)\( \beta_{2} - \)\(12\!\cdots\!48\)\( \beta_{3} - \)\(70\!\cdots\!84\)\( \beta_{4}) q^{49}\) \(+(-\)\(39\!\cdots\!50\)\( + \)\(42\!\cdots\!25\)\( \beta_{1} - \)\(14\!\cdots\!00\)\( \beta_{2} - \)\(58\!\cdots\!00\)\( \beta_{3} + \)\(19\!\cdots\!00\)\( \beta_{4}) q^{50}\) \(+(-\)\(13\!\cdots\!36\)\( + \)\(18\!\cdots\!74\)\( \beta_{1} + \)\(52\!\cdots\!82\)\( \beta_{2} + \)\(15\!\cdots\!96\)\( \beta_{3} - \)\(98\!\cdots\!44\)\( \beta_{4}) q^{51}\) \(+(-\)\(36\!\cdots\!40\)\( - \)\(24\!\cdots\!02\)\( \beta_{1} - \)\(10\!\cdots\!34\)\( \beta_{2} - \)\(24\!\cdots\!50\)\( \beta_{3} - \)\(35\!\cdots\!00\)\( \beta_{4}) q^{52}\) \(+(-\)\(45\!\cdots\!97\)\( - \)\(38\!\cdots\!36\)\( \beta_{1} - \)\(19\!\cdots\!62\)\( \beta_{2} - \)\(82\!\cdots\!72\)\( \beta_{3} + \)\(35\!\cdots\!09\)\( \beta_{4}) q^{53}\) \(+(-\)\(19\!\cdots\!68\)\( + \)\(12\!\cdots\!24\)\( \beta_{1} - \)\(64\!\cdots\!08\)\( \beta_{2} + \)\(28\!\cdots\!36\)\( \beta_{3} + \)\(10\!\cdots\!56\)\( \beta_{4}) q^{54}\) \(+(\)\(10\!\cdots\!60\)\( + \)\(14\!\cdots\!18\)\( \beta_{1} + \)\(56\!\cdots\!42\)\( \beta_{2} + \)\(75\!\cdots\!24\)\( \beta_{3} - \)\(78\!\cdots\!68\)\( \beta_{4}) q^{55}\) \(+(\)\(34\!\cdots\!12\)\( + \)\(29\!\cdots\!72\)\( \beta_{1} + \)\(42\!\cdots\!84\)\( \beta_{2} + \)\(90\!\cdots\!80\)\( \beta_{3} - \)\(79\!\cdots\!12\)\( \beta_{4}) q^{56}\) \(+(\)\(79\!\cdots\!62\)\( - \)\(12\!\cdots\!64\)\( \beta_{1} - \)\(20\!\cdots\!76\)\( \beta_{2} - \)\(79\!\cdots\!00\)\( \beta_{3} + \)\(19\!\cdots\!50\)\( \beta_{4}) q^{57}\) \(+(\)\(12\!\cdots\!76\)\( - \)\(37\!\cdots\!10\)\( \beta_{1} - \)\(61\!\cdots\!00\)\( \beta_{2} + \)\(11\!\cdots\!16\)\( \beta_{3} - \)\(48\!\cdots\!52\)\( \beta_{4}) q^{58}\) \(+(\)\(99\!\cdots\!76\)\( - \)\(22\!\cdots\!75\)\( \beta_{1} - \)\(43\!\cdots\!87\)\( \beta_{2} + \)\(47\!\cdots\!92\)\( \beta_{3} - \)\(53\!\cdots\!80\)\( \beta_{4}) q^{59}\) \(+(\)\(18\!\cdots\!60\)\( - \)\(90\!\cdots\!32\)\( \beta_{1} + \)\(25\!\cdots\!92\)\( \beta_{2} - \)\(57\!\cdots\!76\)\( \beta_{3} + \)\(10\!\cdots\!32\)\( \beta_{4}) q^{60}\) \(+(-\)\(40\!\cdots\!93\)\( + \)\(13\!\cdots\!80\)\( \beta_{1} - \)\(15\!\cdots\!30\)\( \beta_{2} - \)\(36\!\cdots\!60\)\( \beta_{3} - \)\(14\!\cdots\!55\)\( \beta_{4}) q^{61}\) \(+(-\)\(91\!\cdots\!16\)\( + \)\(15\!\cdots\!00\)\( \beta_{1} - \)\(18\!\cdots\!32\)\( \beta_{2} - \)\(49\!\cdots\!44\)\( \beta_{3} + \)\(20\!\cdots\!68\)\( \beta_{4}) q^{62}\) \(+(-\)\(28\!\cdots\!48\)\( + \)\(14\!\cdots\!82\)\( \beta_{1} - \)\(98\!\cdots\!70\)\( \beta_{2} + \)\(34\!\cdots\!96\)\( \beta_{3} + \)\(12\!\cdots\!88\)\( \beta_{4}) q^{63}\) \(+(-\)\(53\!\cdots\!52\)\( - \)\(58\!\cdots\!32\)\( \beta_{1} + \)\(48\!\cdots\!00\)\( \beta_{2} - \)\(26\!\cdots\!44\)\( \beta_{3} - \)\(19\!\cdots\!68\)\( \beta_{4}) q^{64}\) \(+(-\)\(44\!\cdots\!20\)\( - \)\(11\!\cdots\!96\)\( \beta_{1} + \)\(33\!\cdots\!76\)\( \beta_{2} - \)\(36\!\cdots\!28\)\( \beta_{3} + \)\(41\!\cdots\!96\)\( \beta_{4}) q^{65}\) \(+(-\)\(15\!\cdots\!80\)\( - \)\(20\!\cdots\!68\)\( \beta_{1} - \)\(79\!\cdots\!44\)\( \beta_{2} - \)\(11\!\cdots\!52\)\( \beta_{3} + \)\(10\!\cdots\!08\)\( \beta_{4}) q^{66}\) \(+(\)\(34\!\cdots\!96\)\( - \)\(12\!\cdots\!61\)\( \beta_{1} - \)\(24\!\cdots\!85\)\( \beta_{2} + \)\(10\!\cdots\!24\)\( \beta_{3} - \)\(13\!\cdots\!28\)\( \beta_{4}) q^{67}\) \(+(\)\(51\!\cdots\!48\)\( + \)\(68\!\cdots\!94\)\( \beta_{1} - \)\(18\!\cdots\!06\)\( \beta_{2} + \)\(24\!\cdots\!02\)\( \beta_{3} + \)\(16\!\cdots\!56\)\( \beta_{4}) q^{68}\) \(+(\)\(86\!\cdots\!08\)\( - \)\(16\!\cdots\!44\)\( \beta_{1} + \)\(24\!\cdots\!20\)\( \beta_{2} - \)\(68\!\cdots\!68\)\( \beta_{3} + \)\(12\!\cdots\!44\)\( \beta_{4}) q^{69}\) \(+(\)\(12\!\cdots\!20\)\( + \)\(12\!\cdots\!56\)\( \beta_{1} + \)\(28\!\cdots\!64\)\( \beta_{2} + \)\(19\!\cdots\!08\)\( \beta_{3} - \)\(35\!\cdots\!56\)\( \beta_{4}) q^{70}\) \(+(\)\(59\!\cdots\!12\)\( - \)\(30\!\cdots\!10\)\( \beta_{1} + \)\(57\!\cdots\!10\)\( \beta_{2} - \)\(64\!\cdots\!80\)\( \beta_{3} - \)\(15\!\cdots\!40\)\( \beta_{4}) q^{71}\) \(+(\)\(11\!\cdots\!08\)\( + \)\(12\!\cdots\!88\)\( \beta_{1} - \)\(69\!\cdots\!28\)\( \beta_{2} + \)\(29\!\cdots\!52\)\( \beta_{3} + \)\(66\!\cdots\!56\)\( \beta_{4}) q^{72}\) \(+(-\)\(47\!\cdots\!96\)\( - \)\(72\!\cdots\!92\)\( \beta_{1} - \)\(17\!\cdots\!76\)\( \beta_{2} + \)\(23\!\cdots\!96\)\( \beta_{3} + \)\(66\!\cdots\!38\)\( \beta_{4}) q^{73}\) \(+(-\)\(77\!\cdots\!16\)\( + \)\(54\!\cdots\!86\)\( \beta_{1} + \)\(98\!\cdots\!24\)\( \beta_{2} - \)\(17\!\cdots\!72\)\( \beta_{3} - \)\(21\!\cdots\!76\)\( \beta_{4}) q^{74}\) \(+(-\)\(15\!\cdots\!00\)\( + \)\(32\!\cdots\!25\)\( \beta_{1} + \)\(29\!\cdots\!25\)\( \beta_{2} + \)\(12\!\cdots\!00\)\( \beta_{3} - \)\(47\!\cdots\!00\)\( \beta_{4}) q^{75}\) \(+(-\)\(80\!\cdots\!76\)\( + \)\(42\!\cdots\!24\)\( \beta_{1} + \)\(10\!\cdots\!88\)\( \beta_{2} + \)\(16\!\cdots\!00\)\( \beta_{3} + \)\(38\!\cdots\!96\)\( \beta_{4}) q^{76}\) \(+(-\)\(23\!\cdots\!00\)\( - \)\(29\!\cdots\!76\)\( \beta_{1} - \)\(94\!\cdots\!56\)\( \beta_{2} - \)\(15\!\cdots\!16\)\( \beta_{3} + \)\(11\!\cdots\!52\)\( \beta_{4}) q^{77}\) \(+(\)\(67\!\cdots\!40\)\( + \)\(16\!\cdots\!32\)\( \beta_{1} - \)\(47\!\cdots\!48\)\( \beta_{2} + \)\(54\!\cdots\!92\)\( \beta_{3} - \)\(58\!\cdots\!24\)\( \beta_{4}) q^{78}\) \(+(\)\(24\!\cdots\!08\)\( - \)\(14\!\cdots\!76\)\( \beta_{1} - \)\(29\!\cdots\!20\)\( \beta_{2} - \)\(32\!\cdots\!72\)\( \beta_{3} - \)\(44\!\cdots\!24\)\( \beta_{4}) q^{79}\) \(+(\)\(15\!\cdots\!80\)\( - \)\(22\!\cdots\!16\)\( \beta_{1} + \)\(24\!\cdots\!96\)\( \beta_{2} - \)\(74\!\cdots\!88\)\( \beta_{3} + \)\(81\!\cdots\!16\)\( \beta_{4}) q^{80}\) \(+(\)\(25\!\cdots\!79\)\( - \)\(81\!\cdots\!80\)\( \beta_{1} + \)\(75\!\cdots\!84\)\( \beta_{2} + \)\(19\!\cdots\!96\)\( \beta_{3} - \)\(27\!\cdots\!10\)\( \beta_{4}) q^{81}\) \(+(\)\(39\!\cdots\!44\)\( + \)\(47\!\cdots\!70\)\( \beta_{1} - \)\(13\!\cdots\!32\)\( \beta_{2} + \)\(45\!\cdots\!56\)\( \beta_{3} + \)\(67\!\cdots\!68\)\( \beta_{4}) q^{82}\) \(+(\)\(21\!\cdots\!72\)\( + \)\(78\!\cdots\!35\)\( \beta_{1} + \)\(43\!\cdots\!87\)\( \beta_{2} - \)\(12\!\cdots\!00\)\( \beta_{3} - \)\(11\!\cdots\!00\)\( \beta_{4}) q^{83}\) \(+(-\)\(34\!\cdots\!48\)\( + \)\(15\!\cdots\!88\)\( \beta_{1} - \)\(44\!\cdots\!64\)\( \beta_{2} + \)\(10\!\cdots\!20\)\( \beta_{3} - \)\(20\!\cdots\!48\)\( \beta_{4}) q^{84}\) \(+(-\)\(57\!\cdots\!90\)\( + \)\(45\!\cdots\!88\)\( \beta_{1} + \)\(16\!\cdots\!72\)\( \beta_{2} + \)\(20\!\cdots\!84\)\( \beta_{3} + \)\(62\!\cdots\!62\)\( \beta_{4}) q^{85}\) \(+(-\)\(26\!\cdots\!96\)\( - \)\(33\!\cdots\!92\)\( \beta_{1} + \)\(58\!\cdots\!28\)\( \beta_{2} - \)\(23\!\cdots\!12\)\( \beta_{3} - \)\(11\!\cdots\!88\)\( \beta_{4}) q^{86}\) \(+(-\)\(31\!\cdots\!12\)\( + \)\(53\!\cdots\!86\)\( \beta_{1} + \)\(11\!\cdots\!14\)\( \beta_{2} + \)\(33\!\cdots\!96\)\( \beta_{3} - \)\(79\!\cdots\!12\)\( \beta_{4}) q^{87}\) \(+(-\)\(44\!\cdots\!28\)\( - \)\(42\!\cdots\!72\)\( \beta_{1} + \)\(77\!\cdots\!92\)\( \beta_{2} - \)\(26\!\cdots\!84\)\( \beta_{3} - \)\(13\!\cdots\!52\)\( \beta_{4}) q^{88}\) \(+(-\)\(88\!\cdots\!56\)\( + \)\(81\!\cdots\!04\)\( \beta_{1} - \)\(19\!\cdots\!32\)\( \beta_{2} - \)\(78\!\cdots\!20\)\( \beta_{3} + \)\(88\!\cdots\!66\)\( \beta_{4}) q^{89}\) \(+(\)\(41\!\cdots\!80\)\( + \)\(36\!\cdots\!54\)\( \beta_{1} + \)\(85\!\cdots\!76\)\( \beta_{2} + \)\(10\!\cdots\!72\)\( \beta_{3} + \)\(36\!\cdots\!96\)\( \beta_{4}) q^{90}\) \(+(\)\(10\!\cdots\!64\)\( + \)\(22\!\cdots\!12\)\( \beta_{1} - \)\(58\!\cdots\!16\)\( \beta_{2} + \)\(79\!\cdots\!60\)\( \beta_{3} - \)\(75\!\cdots\!52\)\( \beta_{4}) q^{91}\) \(+(\)\(21\!\cdots\!16\)\( - \)\(21\!\cdots\!32\)\( \beta_{1} + \)\(89\!\cdots\!56\)\( \beta_{2} + \)\(25\!\cdots\!16\)\( \beta_{3} - \)\(14\!\cdots\!52\)\( \beta_{4}) q^{92}\) \(+(\)\(41\!\cdots\!96\)\( - \)\(14\!\cdots\!68\)\( \beta_{1} + \)\(96\!\cdots\!72\)\( \beta_{2} - \)\(38\!\cdots\!12\)\( \beta_{3} + \)\(13\!\cdots\!64\)\( \beta_{4}) q^{93}\) \(+(\)\(24\!\cdots\!36\)\( - \)\(67\!\cdots\!16\)\( \beta_{1} + \)\(32\!\cdots\!12\)\( \beta_{2} - \)\(26\!\cdots\!64\)\( \beta_{3} - \)\(11\!\cdots\!04\)\( \beta_{4}) q^{94}\) \(+(\)\(22\!\cdots\!00\)\( + \)\(12\!\cdots\!90\)\( \beta_{1} - \)\(31\!\cdots\!90\)\( \beta_{2} + \)\(46\!\cdots\!20\)\( \beta_{3} + \)\(11\!\cdots\!60\)\( \beta_{4}) q^{95}\) \(+(-\)\(24\!\cdots\!40\)\( + \)\(29\!\cdots\!32\)\( \beta_{1} - \)\(81\!\cdots\!08\)\( \beta_{2} + \)\(30\!\cdots\!72\)\( \beta_{3} - \)\(18\!\cdots\!52\)\( \beta_{4}) q^{96}\) \(+(-\)\(94\!\cdots\!96\)\( + \)\(11\!\cdots\!96\)\( \beta_{1} - \)\(84\!\cdots\!72\)\( \beta_{2} - \)\(25\!\cdots\!72\)\( \beta_{3} - \)\(14\!\cdots\!66\)\( \beta_{4}) q^{97}\) \(+(-\)\(15\!\cdots\!26\)\( + \)\(23\!\cdots\!51\)\( \beta_{1} - \)\(55\!\cdots\!68\)\( \beta_{2} - \)\(52\!\cdots\!96\)\( \beta_{3} + \)\(66\!\cdots\!12\)\( \beta_{4}) q^{98}\) \(+(-\)\(31\!\cdots\!92\)\( - \)\(13\!\cdots\!83\)\( \beta_{1} + \)\(25\!\cdots\!69\)\( \beta_{2} - \)\(15\!\cdots\!40\)\( \beta_{3} - \)\(53\!\cdots\!32\)\( \beta_{4}) q^{99}\) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(5q \) \(\mathstrut -\mathstrut 92089333488q^{2} \) \(\mathstrut -\mathstrut 129195798226305804q^{3} \) \(\mathstrut +\mathstrut \)\(89\!\cdots\!60\)\(q^{4} \) \(\mathstrut +\mathstrut \)\(23\!\cdots\!50\)\(q^{5} \) \(\mathstrut -\mathstrut \)\(33\!\cdots\!40\)\(q^{6} \) \(\mathstrut -\mathstrut \)\(43\!\cdots\!08\)\(q^{7} \) \(\mathstrut -\mathstrut \)\(38\!\cdots\!80\)\(q^{8} \) \(\mathstrut +\mathstrut \)\(32\!\cdots\!65\)\(q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(5q \) \(\mathstrut -\mathstrut 92089333488q^{2} \) \(\mathstrut -\mathstrut 129195798226305804q^{3} \) \(\mathstrut +\mathstrut \)\(89\!\cdots\!60\)\(q^{4} \) \(\mathstrut +\mathstrut \)\(23\!\cdots\!50\)\(q^{5} \) \(\mathstrut -\mathstrut \)\(33\!\cdots\!40\)\(q^{6} \) \(\mathstrut -\mathstrut \)\(43\!\cdots\!08\)\(q^{7} \) \(\mathstrut -\mathstrut \)\(38\!\cdots\!80\)\(q^{8} \) \(\mathstrut +\mathstrut \)\(32\!\cdots\!65\)\(q^{9} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!00\)\(q^{10} \) \(\mathstrut +\mathstrut \)\(50\!\cdots\!60\)\(q^{11} \) \(\mathstrut +\mathstrut \)\(13\!\cdots\!92\)\(q^{12} \) \(\mathstrut +\mathstrut \)\(47\!\cdots\!86\)\(q^{13} \) \(\mathstrut +\mathstrut \)\(26\!\cdots\!20\)\(q^{14} \) \(\mathstrut -\mathstrut \)\(50\!\cdots\!00\)\(q^{15} \) \(\mathstrut +\mathstrut \)\(57\!\cdots\!80\)\(q^{16} \) \(\mathstrut +\mathstrut \)\(66\!\cdots\!02\)\(q^{17} \) \(\mathstrut +\mathstrut \)\(69\!\cdots\!16\)\(q^{18} \) \(\mathstrut +\mathstrut \)\(31\!\cdots\!00\)\(q^{19} \) \(\mathstrut +\mathstrut \)\(68\!\cdots\!00\)\(q^{20} \) \(\mathstrut +\mathstrut \)\(87\!\cdots\!60\)\(q^{21} \) \(\mathstrut -\mathstrut \)\(94\!\cdots\!16\)\(q^{22} \) \(\mathstrut -\mathstrut \)\(41\!\cdots\!24\)\(q^{23} \) \(\mathstrut +\mathstrut \)\(16\!\cdots\!00\)\(q^{24} \) \(\mathstrut +\mathstrut \)\(32\!\cdots\!75\)\(q^{25} \) \(\mathstrut +\mathstrut \)\(44\!\cdots\!60\)\(q^{26} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!80\)\(q^{27} \) \(\mathstrut -\mathstrut \)\(37\!\cdots\!16\)\(q^{28} \) \(\mathstrut -\mathstrut \)\(21\!\cdots\!50\)\(q^{29} \) \(\mathstrut -\mathstrut \)\(80\!\cdots\!00\)\(q^{30} \) \(\mathstrut -\mathstrut \)\(39\!\cdots\!40\)\(q^{31} \) \(\mathstrut -\mathstrut \)\(94\!\cdots\!68\)\(q^{32} \) \(\mathstrut -\mathstrut \)\(73\!\cdots\!28\)\(q^{33} \) \(\mathstrut -\mathstrut \)\(32\!\cdots\!80\)\(q^{34} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!00\)\(q^{35} \) \(\mathstrut -\mathstrut \)\(34\!\cdots\!20\)\(q^{36} \) \(\mathstrut -\mathstrut \)\(67\!\cdots\!78\)\(q^{37} \) \(\mathstrut -\mathstrut \)\(20\!\cdots\!20\)\(q^{38} \) \(\mathstrut -\mathstrut \)\(53\!\cdots\!20\)\(q^{39} \) \(\mathstrut -\mathstrut \)\(87\!\cdots\!00\)\(q^{40} \) \(\mathstrut -\mathstrut \)\(89\!\cdots\!90\)\(q^{41} \) \(\mathstrut +\mathstrut \)\(15\!\cdots\!84\)\(q^{42} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!56\)\(q^{43} \) \(\mathstrut +\mathstrut \)\(36\!\cdots\!20\)\(q^{44} \) \(\mathstrut +\mathstrut \)\(86\!\cdots\!50\)\(q^{45} \) \(\mathstrut +\mathstrut \)\(13\!\cdots\!60\)\(q^{46} \) \(\mathstrut +\mathstrut \)\(26\!\cdots\!32\)\(q^{47} \) \(\mathstrut -\mathstrut \)\(30\!\cdots\!24\)\(q^{48} \) \(\mathstrut -\mathstrut \)\(47\!\cdots\!15\)\(q^{49} \) \(\mathstrut -\mathstrut \)\(19\!\cdots\!00\)\(q^{50} \) \(\mathstrut -\mathstrut \)\(66\!\cdots\!40\)\(q^{51} \) \(\mathstrut -\mathstrut \)\(18\!\cdots\!28\)\(q^{52} \) \(\mathstrut -\mathstrut \)\(22\!\cdots\!54\)\(q^{53} \) \(\mathstrut -\mathstrut \)\(98\!\cdots\!00\)\(q^{54} \) \(\mathstrut +\mathstrut \)\(52\!\cdots\!00\)\(q^{55} \) \(\mathstrut +\mathstrut \)\(17\!\cdots\!00\)\(q^{56} \) \(\mathstrut +\mathstrut \)\(39\!\cdots\!40\)\(q^{57} \) \(\mathstrut +\mathstrut \)\(63\!\cdots\!20\)\(q^{58} \) \(\mathstrut +\mathstrut \)\(49\!\cdots\!00\)\(q^{59} \) \(\mathstrut +\mathstrut \)\(91\!\cdots\!00\)\(q^{60} \) \(\mathstrut -\mathstrut \)\(20\!\cdots\!90\)\(q^{61} \) \(\mathstrut -\mathstrut \)\(45\!\cdots\!96\)\(q^{62} \) \(\mathstrut -\mathstrut \)\(14\!\cdots\!44\)\(q^{63} \) \(\mathstrut -\mathstrut \)\(26\!\cdots\!40\)\(q^{64} \) \(\mathstrut -\mathstrut \)\(22\!\cdots\!00\)\(q^{65} \) \(\mathstrut -\mathstrut \)\(79\!\cdots\!80\)\(q^{66} \) \(\mathstrut +\mathstrut \)\(17\!\cdots\!52\)\(q^{67} \) \(\mathstrut +\mathstrut \)\(25\!\cdots\!04\)\(q^{68} \) \(\mathstrut +\mathstrut \)\(43\!\cdots\!80\)\(q^{69} \) \(\mathstrut +\mathstrut \)\(60\!\cdots\!00\)\(q^{70} \) \(\mathstrut +\mathstrut \)\(29\!\cdots\!60\)\(q^{71} \) \(\mathstrut +\mathstrut \)\(59\!\cdots\!60\)\(q^{72} \) \(\mathstrut -\mathstrut \)\(23\!\cdots\!74\)\(q^{73} \) \(\mathstrut -\mathstrut \)\(38\!\cdots\!80\)\(q^{74} \) \(\mathstrut -\mathstrut \)\(78\!\cdots\!00\)\(q^{75} \) \(\mathstrut -\mathstrut \)\(40\!\cdots\!00\)\(q^{76} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!56\)\(q^{77} \) \(\mathstrut +\mathstrut \)\(33\!\cdots\!72\)\(q^{78} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!00\)\(q^{79} \) \(\mathstrut +\mathstrut \)\(76\!\cdots\!00\)\(q^{80} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!05\)\(q^{81} \) \(\mathstrut +\mathstrut \)\(19\!\cdots\!64\)\(q^{82} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!16\)\(q^{83} \) \(\mathstrut -\mathstrut \)\(17\!\cdots\!80\)\(q^{84} \) \(\mathstrut -\mathstrut \)\(28\!\cdots\!00\)\(q^{85} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!40\)\(q^{86} \) \(\mathstrut -\mathstrut \)\(15\!\cdots\!40\)\(q^{87} \) \(\mathstrut -\mathstrut \)\(22\!\cdots\!60\)\(q^{88} \) \(\mathstrut -\mathstrut \)\(44\!\cdots\!50\)\(q^{89} \) \(\mathstrut +\mathstrut \)\(20\!\cdots\!00\)\(q^{90} \) \(\mathstrut +\mathstrut \)\(50\!\cdots\!60\)\(q^{91} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!52\)\(q^{92} \) \(\mathstrut +\mathstrut \)\(20\!\cdots\!32\)\(q^{93} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!20\)\(q^{94} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!00\)\(q^{95} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!40\)\(q^{96} \) \(\mathstrut -\mathstrut \)\(47\!\cdots\!18\)\(q^{97} \) \(\mathstrut -\mathstrut \)\(76\!\cdots\!16\)\(q^{98} \) \(\mathstrut -\mathstrut \)\(15\!\cdots\!20\)\(q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{5}\mathstrut -\mathstrut \) \(x^{4}\mathstrut -\mathstrut \) \(10073499617947743056\) \(x^{3}\mathstrut +\mathstrut \) \(1429272143092482488433869600\) \(x^{2}\mathstrut +\mathstrut \) \(7661214288514935343595600445215756800\) \(x\mathstrut +\mathstrut \) \(1722510836040319301450745177697157900206688000\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 48 \nu - 10 \)
\(\beta_{2}\)\(=\)\((\)\(-\)\(301\) \(\nu^{4}\mathstrut +\mathstrut \) \(116324489073\) \(\nu^{3}\mathstrut +\mathstrut \) \(2946016895675808598652\) \(\nu^{2}\mathstrut -\mathstrut \) \(1572067662351917592485561734992\) \(\nu\mathstrut -\mathstrut \) \(1398107315399150788449578990722768882624\)\()/\)\(18\!\cdots\!48\)
\(\beta_{3}\)\(=\)\((\)\(-\)\(11137\) \(\nu^{4}\mathstrut +\mathstrut \) \(4304006095701\) \(\nu^{3}\mathstrut +\mathstrut \) \(155411927822539953292268\) \(\nu^{2}\mathstrut -\mathstrut \) \(48289371689549994763869174554256\) \(\nu\mathstrut -\mathstrut \) \(238731607808439142895551358181633844540096\)\()/\)\(20142926511516942336\)
\(\beta_{4}\)\(=\)\((\)\(1794556703\) \(\nu^{4}\mathstrut +\mathstrut \) \(6165094803239335989\) \(\nu^{3}\mathstrut -\mathstrut \) \(15605951075985556463410173844\) \(\nu^{2}\mathstrut -\mathstrut \) \(48868520632222511605431163829960893584\) \(\nu\mathstrut +\mathstrut \) \(6327022782245483253661063757607563332086973760\)\()/\)\(93\!\cdots\!24\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1}\mathstrut +\mathstrut \) \(10\)\()/48\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3}\mathstrut -\mathstrut \) \(3441\) \(\beta_{2}\mathstrut -\mathstrut \) \(10215674441\) \(\beta_{1}\mathstrut +\mathstrut \) \(9283737247896553729718\)\()/2304\)
\(\nu^{3}\)\(=\)\((\)\(943936\) \(\beta_{4}\mathstrut -\mathstrut \) \(856506117\) \(\beta_{3}\mathstrut +\mathstrut \) \(14202697189813\) \(\beta_{2}\mathstrut +\mathstrut \) \(1231550857811227978477\) \(\beta_{1}\mathstrut -\mathstrut \) \(5927477389563705411674836348238\)\()/6912\)
\(\nu^{4}\)\(=\)\((\)\(121597865910976\) \(\beta_{4}\mathstrut +\mathstrut \) \(9677096401527687305\) \(\beta_{3}\mathstrut -\mathstrut \) \(46188046919629143790841\) \(\beta_{2}\mathstrut -\mathstrut \) \(192031968536100477971908100241\) \(\beta_{1}\mathstrut +\mathstrut \) \(79398573440552352581825579517805287677222\)\()/2304\)

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
2.93794e9
1.10442e9
−2.60629e8
−6.50494e8
−3.13124e9
−1.59439e11 2.55219e16 1.59761e22 2.70839e25 −4.06919e27 −5.83802e30 −1.04135e33 −6.69338e34 −4.31824e36
1.2 −7.14302e10 −1.08968e17 −4.34246e21 −5.37334e25 7.78363e27 1.02465e31 9.84822e32 −5.57111e34 3.83819e36
1.3 −5.90767e9 3.95253e17 −9.40983e21 1.69056e25 −2.33502e27 −2.69951e30 1.11387e32 8.86395e34 −9.98727e34
1.4 1.28059e10 −4.48831e17 −9.28074e21 4.05981e25 −5.74767e27 −7.12492e30 −2.39796e32 1.33864e35 5.19894e35
1.5 1.31882e11 7.82937e15 7.94807e21 −7.75470e24 1.03255e27 1.05974e30 −1.97383e32 −6.75239e34 −1.02270e36
\(n\): 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.

Hecke kernels

There are no other newforms in \(S_{74}^{\mathrm{new}}(\Gamma_0(1))\).