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.

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

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

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

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 \(S_{70}^{\mathrm{new}}(\Gamma_0(1))\).