Properties

Label 1.70.a.a
Level 1
Weight 70
Character orbit 1.a
Self dual Yes
Analytic conductor 30.151
Analytic rank 1
Dimension 5
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 1 \)
Weight: \( k \) = \( 70 \)
Character orbit: \([\chi]\) = 1.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(30.1514953292\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{43}\cdot 3^{17}\cdot 5^{5}\cdot 7^{2}\cdot 17\cdot 23 \)
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.

\(f(q)\) \(=\) \( q\) \(+(-3601146874 - \beta_{1}) q^{2}\) \(+(-971616465424800 - 154097 \beta_{1} + \beta_{2}) q^{3}\) \(+(\)\(25\!\cdots\!59\)\( + 12546134069 \beta_{1} - 599 \beta_{2} + \beta_{3}) q^{4}\) \(+(-\)\(37\!\cdots\!81\)\( - 9075890696221 \beta_{1} - 7910543 \beta_{2} - 202 \beta_{3} + \beta_{4}) q^{5}\) \(+(\)\(13\!\cdots\!28\)\( + 2892181780399908 \beta_{1} + 3062596728 \beta_{2} + 30264 \beta_{3} - 384 \beta_{4}) q^{6}\) \(+(\)\(15\!\cdots\!36\)\( - 152192544280326790 \beta_{1} + 797210195830 \beta_{2} - 62779496 \beta_{3} - 96732 \beta_{4}) q^{7}\) \(+(-\)\(91\!\cdots\!88\)\( - \)\(37\!\cdots\!56\)\( \beta_{1} - 76003319375584 \beta_{2} - 25126773728 \beta_{3} + 8935424 \beta_{4}) q^{8}\) \(+(-\)\(63\!\cdots\!25\)\( - \)\(55\!\cdots\!94\)\( \beta_{1} - 3343842553844754 \beta_{2} - 684227811852 \beta_{3} - 318733938 \beta_{4}) q^{9}\) \(+O(q^{10})\) \( q\) \(+(-3601146874 - \beta_{1}) q^{2}\) \(+(-971616465424800 - 154097 \beta_{1} + \beta_{2}) q^{3}\) \(+(\)\(25\!\cdots\!59\)\( + 12546134069 \beta_{1} - 599 \beta_{2} + \beta_{3}) q^{4}\) \(+(-\)\(37\!\cdots\!81\)\( - 9075890696221 \beta_{1} - 7910543 \beta_{2} - 202 \beta_{3} + \beta_{4}) q^{5}\) \(+(\)\(13\!\cdots\!28\)\( + 2892181780399908 \beta_{1} + 3062596728 \beta_{2} + 30264 \beta_{3} - 384 \beta_{4}) q^{6}\) \(+(\)\(15\!\cdots\!36\)\( - 152192544280326790 \beta_{1} + 797210195830 \beta_{2} - 62779496 \beta_{3} - 96732 \beta_{4}) q^{7}\) \(+(-\)\(91\!\cdots\!88\)\( - \)\(37\!\cdots\!56\)\( \beta_{1} - 76003319375584 \beta_{2} - 25126773728 \beta_{3} + 8935424 \beta_{4}) q^{8}\) \(+(-\)\(63\!\cdots\!25\)\( - \)\(55\!\cdots\!94\)\( \beta_{1} - 3343842553844754 \beta_{2} - 684227811852 \beta_{3} - 318733938 \beta_{4}) q^{9}\) \(+(\)\(88\!\cdots\!48\)\( + \)\(57\!\cdots\!18\)\( \beta_{1} - 787945402321615456 \beta_{2} + 39226581408416 \beta_{3} + 5319470592 \beta_{4}) q^{10}\) \(+(-\)\(12\!\cdots\!16\)\( + \)\(28\!\cdots\!17\)\( \beta_{1} - 29934397995194428549 \beta_{2} - 164648498145296 \beta_{3} + 2862315368 \beta_{4}) q^{11}\) \(+(-\)\(22\!\cdots\!08\)\( - \)\(82\!\cdots\!76\)\( \beta_{1} - \)\(28\!\cdots\!72\)\( \beta_{2} - 13874513027942412 \beta_{3} - 2464057442304 \beta_{4}) q^{12}\) \(+(\)\(48\!\cdots\!59\)\( + \)\(37\!\cdots\!83\)\( \beta_{1} + \)\(44\!\cdots\!33\)\( \beta_{2} + 281874899377195462 \beta_{3} + 70878040791129 \beta_{4}) q^{13}\) \(+(\)\(70\!\cdots\!80\)\( + \)\(23\!\cdots\!12\)\( \beta_{1} + \)\(82\!\cdots\!88\)\( \beta_{2} - 1726571337424387472 \beta_{3} - 1243477543740160 \beta_{4}) q^{14}\) \(+(-\)\(44\!\cdots\!16\)\( + \)\(83\!\cdots\!94\)\( \beta_{1} - \)\(71\!\cdots\!98\)\( \beta_{2} - 14200285018930299672 \beta_{3} + 15827790179758236 \beta_{4}) q^{15}\) \(+(\)\(19\!\cdots\!04\)\( + \)\(20\!\cdots\!12\)\( \beta_{1} - \)\(49\!\cdots\!36\)\( \beta_{2} + \)\(34\!\cdots\!20\)\( \beta_{3} - 153451619270590464 \beta_{4}) q^{16}\) \(+(-\)\(68\!\cdots\!32\)\( + \)\(64\!\cdots\!54\)\( \beta_{1} + \)\(41\!\cdots\!26\)\( \beta_{2} - \)\(26\!\cdots\!04\)\( \beta_{3} + 1132362595620713182 \beta_{4}) q^{17}\) \(+(\)\(48\!\cdots\!98\)\( + \)\(51\!\cdots\!91\)\( \beta_{1} + \)\(28\!\cdots\!80\)\( \beta_{2} + \)\(61\!\cdots\!16\)\( \beta_{3} - 5960232076511075328 \beta_{4}) q^{18}\) \(+(\)\(10\!\cdots\!32\)\( - \)\(72\!\cdots\!85\)\( \beta_{1} - \)\(27\!\cdots\!39\)\( \beta_{2} + \)\(68\!\cdots\!52\)\( \beta_{3} + 15789470671886921496 \beta_{4}) q^{19}\) \(+(-\)\(28\!\cdots\!62\)\( - \)\(32\!\cdots\!42\)\( \beta_{1} - \)\(47\!\cdots\!86\)\( \beta_{2} - \)\(70\!\cdots\!54\)\( \beta_{3} + 75250953882173898752 \beta_{4}) q^{20}\) \(+(\)\(60\!\cdots\!96\)\( - \)\(73\!\cdots\!92\)\( \beta_{1} + \)\(92\!\cdots\!12\)\( \beta_{2} + \)\(25\!\cdots\!92\)\( \beta_{3} - \)\(13\!\cdots\!20\)\( \beta_{4}) q^{21}\) \(+(-\)\(23\!\cdots\!64\)\( + \)\(68\!\cdots\!52\)\( \beta_{1} - \)\(81\!\cdots\!68\)\( \beta_{2} + \)\(30\!\cdots\!28\)\( \beta_{3} + \)\(10\!\cdots\!76\)\( \beta_{4}) q^{22}\) \(+(\)\(99\!\cdots\!76\)\( + \)\(24\!\cdots\!06\)\( \beta_{1} - \)\(13\!\cdots\!86\)\( \beta_{2} - \)\(65\!\cdots\!40\)\( \beta_{3} - \)\(52\!\cdots\!80\)\( \beta_{4}) q^{23}\) \(+(-\)\(75\!\cdots\!72\)\( + \)\(94\!\cdots\!12\)\( \beta_{1} + \)\(28\!\cdots\!04\)\( \beta_{2} + \)\(20\!\cdots\!00\)\( \beta_{3} + \)\(20\!\cdots\!76\)\( \beta_{4}) q^{24}\) \(+(\)\(92\!\cdots\!75\)\( - \)\(92\!\cdots\!00\)\( \beta_{1} + \)\(34\!\cdots\!00\)\( \beta_{2} + \)\(45\!\cdots\!00\)\( \beta_{3} - \)\(57\!\cdots\!00\)\( \beta_{4}) q^{25}\) \(+(-\)\(32\!\cdots\!60\)\( - \)\(24\!\cdots\!26\)\( \beta_{1} - \)\(61\!\cdots\!56\)\( \beta_{2} - \)\(59\!\cdots\!88\)\( \beta_{3} + \)\(10\!\cdots\!08\)\( \beta_{4}) q^{26}\) \(+(-\)\(93\!\cdots\!08\)\( - \)\(39\!\cdots\!06\)\( \beta_{1} - \)\(37\!\cdots\!94\)\( \beta_{2} + \)\(19\!\cdots\!12\)\( \beta_{3} - \)\(11\!\cdots\!96\)\( \beta_{4}) q^{27}\) \(+(-\)\(29\!\cdots\!72\)\( + \)\(88\!\cdots\!08\)\( \beta_{1} + \)\(90\!\cdots\!28\)\( \beta_{2} - \)\(82\!\cdots\!32\)\( \beta_{3} + \)\(53\!\cdots\!56\)\( \beta_{4}) q^{28}\) \(+(-\)\(12\!\cdots\!45\)\( + \)\(56\!\cdots\!87\)\( \beta_{1} + \)\(41\!\cdots\!25\)\( \beta_{2} - \)\(92\!\cdots\!18\)\( \beta_{3} - \)\(46\!\cdots\!83\)\( \beta_{4}) q^{29}\) \(+(-\)\(67\!\cdots\!72\)\( + \)\(50\!\cdots\!48\)\( \beta_{1} - \)\(12\!\cdots\!16\)\( \beta_{2} - \)\(14\!\cdots\!24\)\( \beta_{3} + \)\(21\!\cdots\!12\)\( \beta_{4}) q^{30}\) \(+(-\)\(15\!\cdots\!52\)\( - \)\(40\!\cdots\!24\)\( \beta_{1} - \)\(32\!\cdots\!72\)\( \beta_{2} + \)\(32\!\cdots\!12\)\( \beta_{3} - \)\(42\!\cdots\!96\)\( \beta_{4}) q^{31}\) \(+(-\)\(11\!\cdots\!56\)\( - \)\(36\!\cdots\!68\)\( \beta_{1} + \)\(13\!\cdots\!64\)\( \beta_{2} - \)\(13\!\cdots\!44\)\( \beta_{3} - \)\(97\!\cdots\!48\)\( \beta_{4}) q^{32}\) \(+(-\)\(22\!\cdots\!18\)\( + \)\(18\!\cdots\!58\)\( \beta_{1} + \)\(14\!\cdots\!38\)\( \beta_{2} + \)\(22\!\cdots\!84\)\( \beta_{3} + \)\(10\!\cdots\!78\)\( \beta_{4}) q^{33}\) \(+(-\)\(51\!\cdots\!28\)\( + \)\(17\!\cdots\!62\)\( \beta_{1} - \)\(48\!\cdots\!16\)\( \beta_{2} - \)\(59\!\cdots\!40\)\( \beta_{3} - \)\(34\!\cdots\!44\)\( \beta_{4}) q^{34}\) \(+(-\)\(14\!\cdots\!52\)\( + \)\(35\!\cdots\!68\)\( \beta_{1} - \)\(28\!\cdots\!56\)\( \beta_{2} + \)\(14\!\cdots\!16\)\( \beta_{3} + \)\(48\!\cdots\!92\)\( \beta_{4}) q^{35}\) \(+(-\)\(41\!\cdots\!41\)\( - \)\(96\!\cdots\!43\)\( \beta_{1} + \)\(73\!\cdots\!09\)\( \beta_{2} - \)\(38\!\cdots\!95\)\( \beta_{3} + \)\(10\!\cdots\!76\)\( \beta_{4}) q^{36}\) \(+(\)\(23\!\cdots\!39\)\( - \)\(90\!\cdots\!61\)\( \beta_{1} + \)\(12\!\cdots\!89\)\( \beta_{2} + \)\(12\!\cdots\!74\)\( \beta_{3} - \)\(73\!\cdots\!67\)\( \beta_{4}) q^{37}\) \(+(\)\(23\!\cdots\!36\)\( - \)\(51\!\cdots\!16\)\( \beta_{1} - \)\(26\!\cdots\!72\)\( \beta_{2} - \)\(31\!\cdots\!76\)\( \beta_{3} + \)\(17\!\cdots\!08\)\( \beta_{4}) q^{38}\) \(+(\)\(28\!\cdots\!60\)\( + \)\(15\!\cdots\!90\)\( \beta_{1} - \)\(15\!\cdots\!58\)\( \beta_{2} - \)\(87\!\cdots\!96\)\( \beta_{3} - \)\(69\!\cdots\!28\)\( \beta_{4}) q^{39}\) \(+(\)\(22\!\cdots\!20\)\( + \)\(65\!\cdots\!20\)\( \beta_{1} + \)\(43\!\cdots\!60\)\( \beta_{2} + \)\(20\!\cdots\!40\)\( \beta_{3} - \)\(71\!\cdots\!20\)\( \beta_{4}) q^{40}\) \(+(\)\(25\!\cdots\!78\)\( + \)\(11\!\cdots\!16\)\( \beta_{1} - \)\(16\!\cdots\!52\)\( \beta_{2} - \)\(11\!\cdots\!08\)\( \beta_{3} + \)\(21\!\cdots\!64\)\( \beta_{4}) q^{41}\) \(+(\)\(58\!\cdots\!96\)\( - \)\(14\!\cdots\!04\)\( \beta_{1} + \)\(10\!\cdots\!52\)\( \beta_{2} - \)\(52\!\cdots\!76\)\( \beta_{3} - \)\(18\!\cdots\!92\)\( \beta_{4}) q^{42}\) \(+(\)\(36\!\cdots\!72\)\( - \)\(47\!\cdots\!31\)\( \beta_{1} - \)\(68\!\cdots\!13\)\( \beta_{2} - \)\(10\!\cdots\!80\)\( \beta_{3} - \)\(35\!\cdots\!60\)\( \beta_{4}) q^{43}\) \(+(-\)\(41\!\cdots\!68\)\( - \)\(18\!\cdots\!48\)\( \beta_{1} + \)\(94\!\cdots\!04\)\( \beta_{2} + \)\(26\!\cdots\!40\)\( \beta_{3} + \)\(96\!\cdots\!16\)\( \beta_{4}) q^{44}\) \(+(-\)\(32\!\cdots\!53\)\( + \)\(21\!\cdots\!27\)\( \beta_{1} - \)\(27\!\cdots\!59\)\( \beta_{2} + \)\(47\!\cdots\!74\)\( \beta_{3} + \)\(25\!\cdots\!13\)\( \beta_{4}) q^{45}\) \(+(-\)\(20\!\cdots\!52\)\( + \)\(69\!\cdots\!76\)\( \beta_{1} + \)\(43\!\cdots\!48\)\( \beta_{2} - \)\(28\!\cdots\!48\)\( \beta_{3} - \)\(25\!\cdots\!76\)\( \beta_{4}) q^{46}\) \(+(-\)\(20\!\cdots\!72\)\( + \)\(86\!\cdots\!80\)\( \beta_{1} - \)\(13\!\cdots\!44\)\( \beta_{2} + \)\(34\!\cdots\!60\)\( \beta_{3} - \)\(63\!\cdots\!80\)\( \beta_{4}) q^{47}\) \(+(-\)\(64\!\cdots\!32\)\( - \)\(16\!\cdots\!12\)\( \beta_{1} - \)\(15\!\cdots\!60\)\( \beta_{2} + \)\(15\!\cdots\!24\)\( \beta_{3} + \)\(40\!\cdots\!08\)\( \beta_{4}) q^{48}\) \(+(-\)\(36\!\cdots\!39\)\( - \)\(41\!\cdots\!76\)\( \beta_{1} + \)\(20\!\cdots\!12\)\( \beta_{2} - \)\(77\!\cdots\!32\)\( \beta_{3} - \)\(51\!\cdots\!64\)\( \beta_{4}) q^{49}\) \(+(\)\(73\!\cdots\!50\)\( - \)\(26\!\cdots\!75\)\( \beta_{1} + \)\(50\!\cdots\!00\)\( \beta_{2} - \)\(15\!\cdots\!00\)\( \beta_{3} - \)\(11\!\cdots\!00\)\( \beta_{4}) q^{50}\) \(+(\)\(23\!\cdots\!64\)\( + \)\(54\!\cdots\!10\)\( \beta_{1} - \)\(62\!\cdots\!74\)\( \beta_{2} - \)\(19\!\cdots\!48\)\( \beta_{3} + \)\(52\!\cdots\!56\)\( \beta_{4}) q^{51}\) \(+(\)\(18\!\cdots\!14\)\( + \)\(68\!\cdots\!38\)\( \beta_{1} - \)\(32\!\cdots\!46\)\( \beta_{2} + \)\(19\!\cdots\!10\)\( \beta_{3} - \)\(65\!\cdots\!80\)\( \beta_{4}) q^{52}\) \(+(\)\(12\!\cdots\!31\)\( - \)\(36\!\cdots\!61\)\( \beta_{1} + \)\(50\!\cdots\!69\)\( \beta_{2} - \)\(26\!\cdots\!58\)\( \beta_{3} - \)\(49\!\cdots\!11\)\( \beta_{4}) q^{53}\) \(+(\)\(36\!\cdots\!52\)\( - \)\(10\!\cdots\!40\)\( \beta_{1} - \)\(18\!\cdots\!04\)\( \beta_{2} - \)\(18\!\cdots\!08\)\( \beta_{3} + \)\(30\!\cdots\!76\)\( \beta_{4}) q^{54}\) \(+(\)\(18\!\cdots\!48\)\( - \)\(19\!\cdots\!82\)\( \beta_{1} + \)\(22\!\cdots\!94\)\( \beta_{2} + \)\(53\!\cdots\!16\)\( \beta_{3} - \)\(49\!\cdots\!08\)\( \beta_{4}) q^{55}\) \(+(-\)\(66\!\cdots\!60\)\( + \)\(12\!\cdots\!08\)\( \beta_{1} - \)\(48\!\cdots\!40\)\( \beta_{2} + \)\(27\!\cdots\!08\)\( \beta_{3} + \)\(33\!\cdots\!88\)\( \beta_{4}) q^{56}\) \(+(-\)\(20\!\cdots\!58\)\( + \)\(24\!\cdots\!86\)\( \beta_{1} - \)\(19\!\cdots\!14\)\( \beta_{2} + \)\(57\!\cdots\!80\)\( \beta_{3} + \)\(42\!\cdots\!10\)\( \beta_{4}) q^{57}\) \(+(-\)\(42\!\cdots\!92\)\( + \)\(13\!\cdots\!50\)\( \beta_{1} + \)\(60\!\cdots\!04\)\( \beta_{2} - \)\(62\!\cdots\!36\)\( \beta_{3} - \)\(25\!\cdots\!12\)\( \beta_{4}) q^{58}\) \(+(-\)\(67\!\cdots\!96\)\( - \)\(50\!\cdots\!87\)\( \beta_{1} + \)\(72\!\cdots\!07\)\( \beta_{2} + \)\(54\!\cdots\!12\)\( \beta_{3} + \)\(62\!\cdots\!80\)\( \beta_{4}) q^{59}\) \(+(\)\(82\!\cdots\!68\)\( + \)\(21\!\cdots\!88\)\( \beta_{1} + \)\(23\!\cdots\!04\)\( \beta_{2} + \)\(12\!\cdots\!56\)\( \beta_{3} - \)\(47\!\cdots\!28\)\( \beta_{4}) q^{60}\) \(+(\)\(52\!\cdots\!67\)\( - \)\(11\!\cdots\!25\)\( \beta_{1} - \)\(82\!\cdots\!75\)\( \beta_{2} - \)\(76\!\cdots\!50\)\( \beta_{3} - \)\(14\!\cdots\!75\)\( \beta_{4}) q^{61}\) \(+(\)\(89\!\cdots\!04\)\( - \)\(33\!\cdots\!40\)\( \beta_{1} - \)\(40\!\cdots\!04\)\( \beta_{2} - \)\(43\!\cdots\!16\)\( \beta_{3} + \)\(38\!\cdots\!28\)\( \beta_{4}) q^{62}\) \(+(\)\(65\!\cdots\!16\)\( - \)\(13\!\cdots\!38\)\( \beta_{1} - \)\(34\!\cdots\!66\)\( \beta_{2} - \)\(14\!\cdots\!16\)\( \beta_{3} - \)\(11\!\cdots\!72\)\( \beta_{4}) q^{63}\) \(+(\)\(22\!\cdots\!44\)\( + \)\(11\!\cdots\!36\)\( \beta_{1} + \)\(49\!\cdots\!36\)\( \beta_{2} + \)\(28\!\cdots\!08\)\( \beta_{3} - \)\(77\!\cdots\!68\)\( \beta_{4}) q^{64}\) \(+(\)\(64\!\cdots\!04\)\( - \)\(67\!\cdots\!36\)\( \beta_{1} - \)\(15\!\cdots\!88\)\( \beta_{2} - \)\(30\!\cdots\!32\)\( \beta_{3} + \)\(56\!\cdots\!16\)\( \beta_{4}) q^{65}\) \(+(-\)\(67\!\cdots\!00\)\( + \)\(76\!\cdots\!60\)\( \beta_{1} - \)\(91\!\cdots\!52\)\( \beta_{2} - \)\(21\!\cdots\!24\)\( \beta_{3} + \)\(18\!\cdots\!68\)\( \beta_{4}) q^{66}\) \(+(-\)\(25\!\cdots\!32\)\( - \)\(29\!\cdots\!61\)\( \beta_{1} - \)\(30\!\cdots\!99\)\( \beta_{2} + \)\(54\!\cdots\!36\)\( \beta_{3} - \)\(15\!\cdots\!88\)\( \beta_{4}) q^{67}\) \(+(-\)\(84\!\cdots\!30\)\( + \)\(83\!\cdots\!14\)\( \beta_{1} + \)\(23\!\cdots\!54\)\( \beta_{2} - \)\(93\!\cdots\!02\)\( \beta_{3} - \)\(67\!\cdots\!84\)\( \beta_{4}) q^{68}\) \(+(-\)\(13\!\cdots\!40\)\( - \)\(34\!\cdots\!08\)\( \beta_{1} + \)\(56\!\cdots\!32\)\( \beta_{2} + \)\(30\!\cdots\!56\)\( \beta_{3} + \)\(94\!\cdots\!44\)\( \beta_{4}) q^{69}\) \(+(-\)\(24\!\cdots\!84\)\( + \)\(10\!\cdots\!56\)\( \beta_{1} - \)\(45\!\cdots\!52\)\( \beta_{2} - \)\(20\!\cdots\!28\)\( \beta_{3} + \)\(14\!\cdots\!64\)\( \beta_{4}) q^{70}\) \(+(-\)\(22\!\cdots\!68\)\( - \)\(62\!\cdots\!50\)\( \beta_{1} + \)\(22\!\cdots\!50\)\( \beta_{2} - \)\(87\!\cdots\!00\)\( \beta_{3} - \)\(33\!\cdots\!00\)\( \beta_{4}) q^{71}\) \(+(\)\(66\!\cdots\!52\)\( + \)\(42\!\cdots\!08\)\( \beta_{1} - \)\(20\!\cdots\!04\)\( \beta_{2} + \)\(12\!\cdots\!48\)\( \beta_{3} - \)\(22\!\cdots\!84\)\( \beta_{4}) q^{72}\) \(+(\)\(67\!\cdots\!28\)\( - \)\(22\!\cdots\!62\)\( \beta_{1} + \)\(14\!\cdots\!70\)\( \beta_{2} + \)\(12\!\cdots\!04\)\( \beta_{3} + \)\(91\!\cdots\!18\)\( \beta_{4}) q^{73}\) \(+(\)\(66\!\cdots\!40\)\( - \)\(88\!\cdots\!94\)\( \beta_{1} + \)\(50\!\cdots\!28\)\( \beta_{2} - \)\(27\!\cdots\!08\)\( \beta_{3} + \)\(28\!\cdots\!84\)\( \beta_{4}) q^{74}\) \(+(\)\(26\!\cdots\!00\)\( - \)\(60\!\cdots\!75\)\( \beta_{1} + \)\(13\!\cdots\!75\)\( \beta_{2} - \)\(16\!\cdots\!00\)\( \beta_{3} - \)\(24\!\cdots\!00\)\( \beta_{4}) q^{75}\) \(+(\)\(36\!\cdots\!60\)\( + \)\(48\!\cdots\!56\)\( \beta_{1} + \)\(17\!\cdots\!60\)\( \beta_{2} + \)\(60\!\cdots\!36\)\( \beta_{3} + \)\(74\!\cdots\!56\)\( \beta_{4}) q^{76}\) \(+(-\)\(13\!\cdots\!24\)\( + \)\(20\!\cdots\!84\)\( \beta_{1} - \)\(29\!\cdots\!48\)\( \beta_{2} - \)\(73\!\cdots\!64\)\( \beta_{3} + \)\(39\!\cdots\!12\)\( \beta_{4}) q^{77}\) \(+(-\)\(11\!\cdots\!12\)\( + \)\(22\!\cdots\!92\)\( \beta_{1} + \)\(75\!\cdots\!36\)\( \beta_{2} + \)\(10\!\cdots\!48\)\( \beta_{3} - \)\(18\!\cdots\!84\)\( \beta_{4}) q^{78}\) \(+(-\)\(73\!\cdots\!24\)\( + \)\(71\!\cdots\!28\)\( \beta_{1} + \)\(17\!\cdots\!88\)\( \beta_{2} + \)\(12\!\cdots\!04\)\( \beta_{3} - \)\(39\!\cdots\!04\)\( \beta_{4}) q^{79}\) \(+(-\)\(45\!\cdots\!16\)\( - \)\(21\!\cdots\!56\)\( \beta_{1} + \)\(84\!\cdots\!52\)\( \beta_{2} - \)\(74\!\cdots\!72\)\( \beta_{3} - \)\(56\!\cdots\!64\)\( \beta_{4}) q^{80}\) \(+(-\)\(22\!\cdots\!57\)\( + \)\(59\!\cdots\!94\)\( \beta_{1} - \)\(47\!\cdots\!94\)\( \beta_{2} + \)\(58\!\cdots\!36\)\( \beta_{3} + \)\(18\!\cdots\!30\)\( \beta_{4}) q^{81}\) \(+(-\)\(18\!\cdots\!76\)\( - \)\(19\!\cdots\!50\)\( \beta_{1} - \)\(16\!\cdots\!64\)\( \beta_{2} + \)\(62\!\cdots\!44\)\( \beta_{3} + \)\(49\!\cdots\!48\)\( \beta_{4}) q^{82}\) \(+(\)\(23\!\cdots\!04\)\( + \)\(25\!\cdots\!55\)\( \beta_{1} + \)\(71\!\cdots\!53\)\( \beta_{2} - \)\(29\!\cdots\!00\)\( \beta_{3} - \)\(36\!\cdots\!00\)\( \beta_{4}) q^{83}\) \(+(\)\(59\!\cdots\!24\)\( + \)\(88\!\cdots\!72\)\( \beta_{1} - \)\(37\!\cdots\!20\)\( \beta_{2} - \)\(65\!\cdots\!48\)\( \beta_{3} + \)\(24\!\cdots\!32\)\( \beta_{4}) q^{84}\) \(+(\)\(14\!\cdots\!98\)\( + \)\(45\!\cdots\!18\)\( \beta_{1} + \)\(10\!\cdots\!94\)\( \beta_{2} - \)\(29\!\cdots\!84\)\( \beta_{3} - \)\(65\!\cdots\!58\)\( \beta_{4}) q^{85}\) \(+(\)\(37\!\cdots\!68\)\( + \)\(63\!\cdots\!04\)\( \beta_{1} + \)\(11\!\cdots\!68\)\( \beta_{2} + \)\(59\!\cdots\!00\)\( \beta_{3} + \)\(15\!\cdots\!92\)\( \beta_{4}) q^{86}\) \(+(\)\(25\!\cdots\!60\)\( - \)\(50\!\cdots\!14\)\( \beta_{1} - \)\(11\!\cdots\!30\)\( \beta_{2} - \)\(68\!\cdots\!76\)\( \beta_{3} + \)\(10\!\cdots\!08\)\( \beta_{4}) q^{87}\) \(+(\)\(30\!\cdots\!04\)\( - \)\(13\!\cdots\!52\)\( \beta_{1} - \)\(20\!\cdots\!28\)\( \beta_{2} - \)\(19\!\cdots\!76\)\( \beta_{3} - \)\(68\!\cdots\!92\)\( \beta_{4}) q^{88}\) \(+(-\)\(37\!\cdots\!20\)\( - \)\(30\!\cdots\!14\)\( \beta_{1} - \)\(98\!\cdots\!30\)\( \beta_{2} + \)\(49\!\cdots\!36\)\( \beta_{3} + \)\(45\!\cdots\!46\)\( \beta_{4}) q^{89}\) \(+(-\)\(17\!\cdots\!76\)\( - \)\(15\!\cdots\!66\)\( \beta_{1} - \)\(54\!\cdots\!28\)\( \beta_{2} - \)\(26\!\cdots\!92\)\( \beta_{3} + \)\(52\!\cdots\!96\)\( \beta_{4}) q^{90}\) \(+(-\)\(16\!\cdots\!88\)\( + \)\(96\!\cdots\!28\)\( \beta_{1} + \)\(42\!\cdots\!40\)\( \beta_{2} + \)\(14\!\cdots\!88\)\( \beta_{3} + \)\(17\!\cdots\!88\)\( \beta_{4}) q^{91}\) \(+(-\)\(42\!\cdots\!28\)\( + \)\(22\!\cdots\!08\)\( \beta_{1} + \)\(13\!\cdots\!88\)\( \beta_{2} + \)\(55\!\cdots\!44\)\( \beta_{3} - \)\(11\!\cdots\!52\)\( \beta_{4}) q^{92}\) \(+(-\)\(22\!\cdots\!04\)\( + \)\(92\!\cdots\!12\)\( \beta_{1} - \)\(26\!\cdots\!40\)\( \beta_{2} - \)\(12\!\cdots\!48\)\( \beta_{3} - \)\(23\!\cdots\!16\)\( \beta_{4}) q^{93}\) \(+(-\)\(64\!\cdots\!84\)\( - \)\(87\!\cdots\!00\)\( \beta_{1} - \)\(15\!\cdots\!24\)\( \beta_{2} - \)\(13\!\cdots\!08\)\( \beta_{3} + \)\(78\!\cdots\!96\)\( \beta_{4}) q^{94}\) \(+(\)\(30\!\cdots\!40\)\( - \)\(38\!\cdots\!10\)\( \beta_{1} + \)\(10\!\cdots\!70\)\( \beta_{2} + \)\(40\!\cdots\!80\)\( \beta_{3} - \)\(37\!\cdots\!40\)\( \beta_{4}) q^{95}\) \(+(\)\(15\!\cdots\!16\)\( + \)\(14\!\cdots\!96\)\( \beta_{1} - \)\(34\!\cdots\!88\)\( \beta_{2} + \)\(12\!\cdots\!60\)\( \beta_{3} - \)\(91\!\cdots\!12\)\( \beta_{4}) q^{96}\) \(+(\)\(73\!\cdots\!44\)\( - \)\(81\!\cdots\!94\)\( \beta_{1} + \)\(91\!\cdots\!54\)\( \beta_{2} - \)\(41\!\cdots\!08\)\( \beta_{3} + \)\(15\!\cdots\!14\)\( \beta_{4}) q^{97}\) \(+(\)\(35\!\cdots\!90\)\( + \)\(79\!\cdots\!71\)\( \beta_{1} + \)\(44\!\cdots\!24\)\( \beta_{2} + \)\(26\!\cdots\!76\)\( \beta_{3} - \)\(10\!\cdots\!08\)\( \beta_{4}) q^{98}\) \(+(\)\(11\!\cdots\!36\)\( + \)\(65\!\cdots\!13\)\( \beta_{1} - \)\(15\!\cdots\!85\)\( \beta_{2} - \)\(51\!\cdots\!52\)\( \beta_{3} + \)\(19\!\cdots\!48\)\( \beta_{4}) q^{99}\) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(5q \) \(\mathstrut -\mathstrut 18005734368q^{2} \) \(\mathstrut -\mathstrut 4858082326815804q^{3} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!60\)\(q^{4} \) \(\mathstrut -\mathstrut \)\(18\!\cdots\!50\)\(q^{5} \) \(\mathstrut +\mathstrut \)\(65\!\cdots\!60\)\(q^{6} \) \(\mathstrut +\mathstrut \)\(76\!\cdots\!92\)\(q^{7} \) \(\mathstrut -\mathstrut \)\(45\!\cdots\!00\)\(q^{8} \) \(\mathstrut -\mathstrut \)\(31\!\cdots\!35\)\(q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(5q \) \(\mathstrut -\mathstrut 18005734368q^{2} \) \(\mathstrut -\mathstrut 4858082326815804q^{3} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!60\)\(q^{4} \) \(\mathstrut -\mathstrut \)\(18\!\cdots\!50\)\(q^{5} \) \(\mathstrut +\mathstrut \)\(65\!\cdots\!60\)\(q^{6} \) \(\mathstrut +\mathstrut \)\(76\!\cdots\!92\)\(q^{7} \) \(\mathstrut -\mathstrut \)\(45\!\cdots\!00\)\(q^{8} \) \(\mathstrut -\mathstrut \)\(31\!\cdots\!35\)\(q^{9} \) \(\mathstrut +\mathstrut \)\(44\!\cdots\!00\)\(q^{10} \) \(\mathstrut -\mathstrut \)\(60\!\cdots\!40\)\(q^{11} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!48\)\(q^{12} \) \(\mathstrut +\mathstrut \)\(24\!\cdots\!86\)\(q^{13} \) \(\mathstrut +\mathstrut \)\(35\!\cdots\!20\)\(q^{14} \) \(\mathstrut -\mathstrut \)\(22\!\cdots\!00\)\(q^{15} \) \(\mathstrut +\mathstrut \)\(95\!\cdots\!80\)\(q^{16} \) \(\mathstrut -\mathstrut \)\(34\!\cdots\!38\)\(q^{17} \) \(\mathstrut +\mathstrut \)\(24\!\cdots\!56\)\(q^{18} \) \(\mathstrut +\mathstrut \)\(50\!\cdots\!00\)\(q^{19} \) \(\mathstrut -\mathstrut \)\(14\!\cdots\!00\)\(q^{20} \) \(\mathstrut +\mathstrut \)\(30\!\cdots\!60\)\(q^{21} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!56\)\(q^{22} \) \(\mathstrut +\mathstrut \)\(49\!\cdots\!76\)\(q^{23} \) \(\mathstrut -\mathstrut \)\(37\!\cdots\!00\)\(q^{24} \) \(\mathstrut +\mathstrut \)\(46\!\cdots\!75\)\(q^{25} \) \(\mathstrut -\mathstrut \)\(16\!\cdots\!40\)\(q^{26} \) \(\mathstrut -\mathstrut \)\(46\!\cdots\!00\)\(q^{27} \) \(\mathstrut -\mathstrut \)\(14\!\cdots\!96\)\(q^{28} \) \(\mathstrut -\mathstrut \)\(62\!\cdots\!50\)\(q^{29} \) \(\mathstrut -\mathstrut \)\(33\!\cdots\!00\)\(q^{30} \) \(\mathstrut -\mathstrut \)\(77\!\cdots\!40\)\(q^{31} \) \(\mathstrut -\mathstrut \)\(59\!\cdots\!08\)\(q^{32} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!68\)\(q^{33} \) \(\mathstrut -\mathstrut \)\(25\!\cdots\!80\)\(q^{34} \) \(\mathstrut -\mathstrut \)\(74\!\cdots\!00\)\(q^{35} \) \(\mathstrut -\mathstrut \)\(20\!\cdots\!20\)\(q^{36} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!02\)\(q^{37} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!00\)\(q^{38} \) \(\mathstrut +\mathstrut \)\(14\!\cdots\!80\)\(q^{39} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!00\)\(q^{40} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!10\)\(q^{41} \) \(\mathstrut +\mathstrut \)\(29\!\cdots\!24\)\(q^{42} \) \(\mathstrut +\mathstrut \)\(18\!\cdots\!56\)\(q^{43} \) \(\mathstrut -\mathstrut \)\(20\!\cdots\!80\)\(q^{44} \) \(\mathstrut -\mathstrut \)\(16\!\cdots\!50\)\(q^{45} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!40\)\(q^{46} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!28\)\(q^{47} \) \(\mathstrut -\mathstrut \)\(32\!\cdots\!24\)\(q^{48} \) \(\mathstrut -\mathstrut \)\(18\!\cdots\!15\)\(q^{49} \) \(\mathstrut +\mathstrut \)\(36\!\cdots\!00\)\(q^{50} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!60\)\(q^{51} \) \(\mathstrut +\mathstrut \)\(94\!\cdots\!32\)\(q^{52} \) \(\mathstrut +\mathstrut \)\(63\!\cdots\!46\)\(q^{53} \) \(\mathstrut +\mathstrut \)\(18\!\cdots\!00\)\(q^{54} \) \(\mathstrut +\mathstrut \)\(90\!\cdots\!00\)\(q^{55} \) \(\mathstrut -\mathstrut \)\(33\!\cdots\!00\)\(q^{56} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!00\)\(q^{57} \) \(\mathstrut -\mathstrut \)\(21\!\cdots\!00\)\(q^{58} \) \(\mathstrut -\mathstrut \)\(33\!\cdots\!00\)\(q^{59} \) \(\mathstrut +\mathstrut \)\(41\!\cdots\!00\)\(q^{60} \) \(\mathstrut +\mathstrut \)\(26\!\cdots\!10\)\(q^{61} \) \(\mathstrut +\mathstrut \)\(44\!\cdots\!04\)\(q^{62} \) \(\mathstrut +\mathstrut \)\(32\!\cdots\!36\)\(q^{63} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!60\)\(q^{64} \) \(\mathstrut +\mathstrut \)\(32\!\cdots\!00\)\(q^{65} \) \(\mathstrut -\mathstrut \)\(33\!\cdots\!80\)\(q^{66} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!88\)\(q^{67} \) \(\mathstrut -\mathstrut \)\(42\!\cdots\!56\)\(q^{68} \) \(\mathstrut -\mathstrut \)\(68\!\cdots\!20\)\(q^{69} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!00\)\(q^{70} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!40\)\(q^{71} \) \(\mathstrut +\mathstrut \)\(33\!\cdots\!00\)\(q^{72} \) \(\mathstrut +\mathstrut \)\(33\!\cdots\!26\)\(q^{73} \) \(\mathstrut +\mathstrut \)\(33\!\cdots\!20\)\(q^{74} \) \(\mathstrut +\mathstrut \)\(13\!\cdots\!00\)\(q^{75} \) \(\mathstrut +\mathstrut \)\(18\!\cdots\!00\)\(q^{76} \) \(\mathstrut -\mathstrut \)\(68\!\cdots\!36\)\(q^{77} \) \(\mathstrut -\mathstrut \)\(57\!\cdots\!08\)\(q^{78} \) \(\mathstrut -\mathstrut \)\(36\!\cdots\!00\)\(q^{79} \) \(\mathstrut -\mathstrut \)\(22\!\cdots\!00\)\(q^{80} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!95\)\(q^{81} \) \(\mathstrut -\mathstrut \)\(93\!\cdots\!16\)\(q^{82} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!16\)\(q^{83} \) \(\mathstrut +\mathstrut \)\(29\!\cdots\!20\)\(q^{84} \) \(\mathstrut +\mathstrut \)\(74\!\cdots\!00\)\(q^{85} \) \(\mathstrut +\mathstrut \)\(18\!\cdots\!60\)\(q^{86} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!00\)\(q^{87} \) \(\mathstrut +\mathstrut \)\(15\!\cdots\!00\)\(q^{88} \) \(\mathstrut -\mathstrut \)\(18\!\cdots\!50\)\(q^{89} \) \(\mathstrut -\mathstrut \)\(85\!\cdots\!00\)\(q^{90} \) \(\mathstrut -\mathstrut \)\(80\!\cdots\!40\)\(q^{91} \) \(\mathstrut -\mathstrut \)\(21\!\cdots\!88\)\(q^{92} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!88\)\(q^{93} \) \(\mathstrut -\mathstrut \)\(32\!\cdots\!80\)\(q^{94} \) \(\mathstrut +\mathstrut \)\(15\!\cdots\!00\)\(q^{95} \) \(\mathstrut +\mathstrut \)\(78\!\cdots\!60\)\(q^{96} \) \(\mathstrut +\mathstrut \)\(36\!\cdots\!22\)\(q^{97} \) \(\mathstrut +\mathstrut \)\(17\!\cdots\!24\)\(q^{98} \) \(\mathstrut +\mathstrut \)\(58\!\cdots\!80\)\(q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{5}\mathstrut -\mathstrut \) \(x^{4}\mathstrut -\mathstrut \) \(24985894640345560\) \(x^{3}\mathstrut -\mathstrut \) \(309075533549354721261224\) \(x^{2}\mathstrut +\mathstrut \) \(93046016582444120336711740360848\) \(x\mathstrut -\mathstrut \) \(941262570723617919770907675844106102736\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 288 \nu - 58 \)
\(\beta_{2}\)\(=\)\((\)\(-\)\(13089\) \(\nu^{4}\mathstrut +\mathstrut \) \(125836814991\) \(\nu^{3}\mathstrut +\mathstrut \) \(282900780149205261990\) \(\nu^{2}\mathstrut +\mathstrut \) \(3890285472254492554232513748\) \(\nu\mathstrut -\mathstrut \) \(556492390667412783765229916175949480\)\()/\)\(12043643989083004928\)
\(\beta_{3}\)\(=\)\((\)\(-\)\(7840311\) \(\nu^{4}\mathstrut +\mathstrut \) \(75376252179609\) \(\nu^{3}\mathstrut +\mathstrut \) \(1168405574339874712680042\) \(\nu^{2}\mathstrut -\mathstrut \) \(16205200788335358465725126243316\) \(\nu\mathstrut -\mathstrut \) \(10317182800241591202024916588626964847384\)\()/\)\(12043643989083004928\)
\(\beta_{4}\)\(=\)\((\)\(-\)\(133380259041\) \(\nu^{4}\mathstrut -\mathstrut \) \(30915046488318323505\) \(\nu^{3}\mathstrut +\mathstrut \) \(4484128064840743575244333350\) \(\nu^{2}\mathstrut +\mathstrut \) \(574608538311537764497110501713672148\) \(\nu\mathstrut -\mathstrut \) \(15703883069245078406372403354648420339590312\)\()/\)\(12043643989083004928\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1}\mathstrut +\mathstrut \) \(58\)\()/288\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3}\mathstrut -\mathstrut \) \(599\) \(\beta_{2}\mathstrut +\mathstrut \) \(5343840437\) \(\beta_{1}\mathstrut +\mathstrut \) \(828972018021666404459\)\()/82944\)
\(\nu^{3}\)\(=\)\((\)\(-\)\(279232\) \(\beta_{4}\mathstrut +\mathstrut \) \(447604165\) \(\beta_{3}\mathstrut +\mathstrut \) \(2577330631373\) \(\beta_{2}\mathstrut +\mathstrut \) \(45458525839157972873\) \(\beta_{1}\mathstrut +\mathstrut \) \(138434200904781113769852318231\)\()/746496\)
\(\nu^{4}\)\(=\)\((\)\(-\)\(2684518719808\) \(\beta_{4}\mathstrut +\mathstrut \) \(198825892263973825\) \(\beta_{3}\mathstrut -\mathstrut \) \(778617701216911222615\) \(\beta_{2}\mathstrut +\mathstrut \) \(2246922278818159470404999861\) \(\beta_{1}\mathstrut +\mathstrut \) \(130846684347970439133698906592804797931\)\()/746496\)

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
1.52577e8
5.13065e7
1.08482e7
−8.63045e7
−1.28427e8
−4.75433e10 −9.70523e15 1.67007e21 −1.40726e24 4.61419e26 −1.18429e27 −5.13361e31 −7.40194e32 6.69057e34
1.2 −1.83774e10 2.28315e16 −2.52566e20 1.12358e24 −4.19583e26 −8.82897e28 1.54896e31 −3.13110e32 −2.06484e34
1.3 −6.72544e9 −4.13926e16 −5.45064e20 −6.57446e23 2.78383e26 1.09182e29 7.63579e30 8.78961e32 4.42161e33
1.4 2.12545e10 3.67245e16 −1.38540e20 −2.01597e24 7.80563e26 2.05778e29 −1.54911e31 5.14307e32 −4.28486e34
1.5 3.33859e10 −1.33163e16 5.24322e20 1.09328e24 −4.44575e26 −1.48686e29 −2.20259e30 −6.57062e32 3.65000e34
\(n\): 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 \(S_{70}^{\mathrm{new}}(\Gamma_0(1))\).