Properties

Label 3.68.a.a
Level 3
Weight 68
Character orbit 3.a
Self dual Yes
Analytic conductor 85.287
Analytic rank 1
Dimension 5
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 3 \)
Weight: \( k \) = \( 68 \)
Character orbit: \([\chi]\) = 3.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(85.287105579\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{40}\cdot 3^{20}\cdot 5^{3}\cdot 7^{2}\cdot 11\cdot 17 \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(f(q)\) \(=\) \( q +(-3251044618 - \beta_{1}) q^{2} +5559060566555523 q^{3} +(37545188503057400779 + 2560990459 \beta_{1} + \beta_{2}) q^{4} +(-\)\(15\!\cdots\!94\)\( + 4142290226862 \beta_{1} + 17 \beta_{2} + \beta_{3}) q^{5} +(-\)\(18\!\cdots\!14\)\( - 5559060566555523 \beta_{1}) q^{6} +(\)\(57\!\cdots\!11\)\( + 59871253268696882 \beta_{1} - 13937620 \beta_{2} - 590 \beta_{3} + 22013 \beta_{4}) q^{7} +(-\)\(89\!\cdots\!16\)\( - 52799222833224617488 \beta_{1} - 6592873520 \beta_{2} + 6721280 \beta_{3} - 397056 \beta_{4}) q^{8} +\)\(30\!\cdots\!29\)\( q^{9} +O(q^{10})\) \( q +(-3251044618 - \beta_{1}) q^{2} +5559060566555523 q^{3} +(37545188503057400779 + 2560990459 \beta_{1} + \beta_{2}) q^{4} +(-\)\(15\!\cdots\!94\)\( + 4142290226862 \beta_{1} + 17 \beta_{2} + \beta_{3}) q^{5} +(-\)\(18\!\cdots\!14\)\( - 5559060566555523 \beta_{1}) q^{6} +(\)\(57\!\cdots\!11\)\( + 59871253268696882 \beta_{1} - 13937620 \beta_{2} - 590 \beta_{3} + 22013 \beta_{4}) q^{7} +(-\)\(89\!\cdots\!16\)\( - 52799222833224617488 \beta_{1} - 6592873520 \beta_{2} + 6721280 \beta_{3} - 397056 \beta_{4}) q^{8} +\)\(30\!\cdots\!29\)\( q^{9} +(-\)\(67\!\cdots\!64\)\( + \)\(15\!\cdots\!22\)\( \beta_{1} + 4944925437552 \beta_{2} - 835396544 \beta_{3} + 315147200 \beta_{4}) q^{10} +(-\)\(21\!\cdots\!16\)\( + \)\(18\!\cdots\!52\)\( \beta_{1} + 47862157178902 \beta_{2} - 311482534330 \beta_{3} - 21603299784 \beta_{4}) q^{11} +(\)\(20\!\cdots\!17\)\( + \)\(14\!\cdots\!57\)\( \beta_{1} + 5559060566555523 \beta_{2}) q^{12} +(\)\(37\!\cdots\!51\)\( + \)\(28\!\cdots\!80\)\( \beta_{1} + 6757037482554021 \beta_{2} - 28968692525305 \beta_{3} - 13930476249209 \beta_{4}) q^{13} +(-\)\(12\!\cdots\!92\)\( + \)\(17\!\cdots\!40\)\( \beta_{1} - 407053386362401136 \beta_{2} + 103747362484160 \beta_{3} - 186448376467392 \beta_{4}) q^{14} +(-\)\(87\!\cdots\!62\)\( + \)\(23\!\cdots\!26\)\( \beta_{1} + 94504029631443891 \beta_{2} + 5559060566555523 \beta_{3}) q^{15} +(\)\(39\!\cdots\!96\)\( + \)\(75\!\cdots\!96\)\( \beta_{1} + 523788903724429568 \beta_{2} - 54246402364395520 \beta_{3} + 8240727220670464 \beta_{4}) q^{16} +(\)\(31\!\cdots\!32\)\( + \)\(36\!\cdots\!32\)\( \beta_{1} + 37944992246210411564 \beta_{2} - 562406648254731560 \beta_{3} + 65346583469962122 \beta_{4}) q^{17} +(-\)\(10\!\cdots\!22\)\( - \)\(30\!\cdots\!29\)\( \beta_{1}) q^{18} +(-\)\(12\!\cdots\!58\)\( - \)\(27\!\cdots\!44\)\( \beta_{1} - \)\(16\!\cdots\!56\)\( \beta_{2} - 12974117021434959560 \beta_{3} - 4172857839055283218 \beta_{4}) q^{19} +(\)\(17\!\cdots\!98\)\( - \)\(74\!\cdots\!54\)\( \beta_{1} - \)\(51\!\cdots\!14\)\( \beta_{2} - \)\(11\!\cdots\!92\)\( \beta_{3} - 4978104339150950400 \beta_{4}) q^{20} +(\)\(32\!\cdots\!53\)\( + \)\(33\!\cdots\!86\)\( \beta_{1} - \)\(77\!\cdots\!60\)\( \beta_{2} - 3279845734267758570 \beta_{3} + \)\(12\!\cdots\!99\)\( \beta_{4}) q^{21} +(-\)\(25\!\cdots\!36\)\( + \)\(14\!\cdots\!44\)\( \beta_{1} - \)\(45\!\cdots\!92\)\( \beta_{2} + \)\(42\!\cdots\!80\)\( \beta_{3} + 68953583922234546304 \beta_{4}) q^{22} +(-\)\(61\!\cdots\!14\)\( - \)\(28\!\cdots\!20\)\( \beta_{1} - \)\(20\!\cdots\!72\)\( \beta_{2} + \)\(11\!\cdots\!20\)\( \beta_{3} - \)\(14\!\cdots\!74\)\( \beta_{4}) q^{23} +(-\)\(49\!\cdots\!68\)\( - \)\(29\!\cdots\!24\)\( \beta_{1} - \)\(36\!\cdots\!60\)\( \beta_{2} + \)\(37\!\cdots\!40\)\( \beta_{3} - \)\(22\!\cdots\!88\)\( \beta_{4}) q^{24} +(-\)\(25\!\cdots\!35\)\( - \)\(72\!\cdots\!20\)\( \beta_{1} - \)\(21\!\cdots\!70\)\( \beta_{2} - \)\(19\!\cdots\!10\)\( \beta_{3} - \)\(59\!\cdots\!50\)\( \beta_{4}) q^{25} +(-\)\(61\!\cdots\!60\)\( - \)\(46\!\cdots\!38\)\( \beta_{1} - \)\(32\!\cdots\!04\)\( \beta_{2} - \)\(51\!\cdots\!00\)\( \beta_{3} + \)\(10\!\cdots\!40\)\( \beta_{4}) q^{26} +\)\(17\!\cdots\!67\)\( q^{27} +(-\)\(35\!\cdots\!64\)\( + \)\(71\!\cdots\!52\)\( \beta_{1} + \)\(62\!\cdots\!32\)\( \beta_{2} - \)\(44\!\cdots\!00\)\( \beta_{3} - \)\(14\!\cdots\!40\)\( \beta_{4}) q^{28} +(-\)\(34\!\cdots\!28\)\( + \)\(16\!\cdots\!46\)\( \beta_{1} + \)\(56\!\cdots\!17\)\( \beta_{2} + \)\(82\!\cdots\!65\)\( \beta_{3} + \)\(11\!\cdots\!82\)\( \beta_{4}) q^{29} +(-\)\(37\!\cdots\!72\)\( + \)\(87\!\cdots\!06\)\( \beta_{1} + \)\(27\!\cdots\!96\)\( \beta_{2} - \)\(46\!\cdots\!12\)\( \beta_{3} + \)\(17\!\cdots\!00\)\( \beta_{4}) q^{30} +(-\)\(33\!\cdots\!49\)\( - \)\(28\!\cdots\!54\)\( \beta_{1} - \)\(10\!\cdots\!48\)\( \beta_{2} + \)\(42\!\cdots\!50\)\( \beta_{3} + \)\(25\!\cdots\!05\)\( \beta_{4}) q^{31} +(-\)\(13\!\cdots\!40\)\( + \)\(42\!\cdots\!92\)\( \beta_{1} - \)\(43\!\cdots\!56\)\( \beta_{2} - \)\(86\!\cdots\!20\)\( \beta_{3} - \)\(30\!\cdots\!76\)\( \beta_{4}) q^{32} +(-\)\(11\!\cdots\!68\)\( + \)\(10\!\cdots\!96\)\( \beta_{1} + \)\(26\!\cdots\!46\)\( \beta_{2} - \)\(17\!\cdots\!90\)\( \beta_{3} - \)\(12\!\cdots\!32\)\( \beta_{4}) q^{33} +(-\)\(73\!\cdots\!48\)\( - \)\(34\!\cdots\!14\)\( \beta_{1} - \)\(10\!\cdots\!92\)\( \beta_{2} + \)\(13\!\cdots\!60\)\( \beta_{3} - \)\(76\!\cdots\!12\)\( \beta_{4}) q^{34} +(-\)\(11\!\cdots\!84\)\( - \)\(34\!\cdots\!68\)\( \beta_{1} - \)\(28\!\cdots\!38\)\( \beta_{2} + \)\(41\!\cdots\!86\)\( \beta_{3} + \)\(37\!\cdots\!00\)\( \beta_{4}) q^{35} +(\)\(11\!\cdots\!91\)\( + \)\(79\!\cdots\!11\)\( \beta_{1} + \)\(30\!\cdots\!29\)\( \beta_{2}) q^{36} +(\)\(60\!\cdots\!73\)\( + \)\(29\!\cdots\!68\)\( \beta_{1} + \)\(52\!\cdots\!81\)\( \beta_{2} + \)\(81\!\cdots\!75\)\( \beta_{3} - \)\(19\!\cdots\!35\)\( \beta_{4}) q^{37} +(\)\(52\!\cdots\!40\)\( + \)\(37\!\cdots\!12\)\( \beta_{1} + \)\(29\!\cdots\!24\)\( \beta_{2} - \)\(13\!\cdots\!40\)\( \beta_{3} + \)\(38\!\cdots\!68\)\( \beta_{4}) q^{38} +(\)\(20\!\cdots\!73\)\( + \)\(15\!\cdots\!40\)\( \beta_{1} + \)\(37\!\cdots\!83\)\( \beta_{2} - \)\(16\!\cdots\!15\)\( \beta_{3} - \)\(77\!\cdots\!07\)\( \beta_{4}) q^{39} +(\)\(22\!\cdots\!80\)\( + \)\(38\!\cdots\!60\)\( \beta_{1} - \)\(70\!\cdots\!40\)\( \beta_{2} - \)\(16\!\cdots\!20\)\( \beta_{3} - \)\(18\!\cdots\!00\)\( \beta_{4}) q^{40} +(-\)\(16\!\cdots\!36\)\( + \)\(10\!\cdots\!48\)\( \beta_{1} - \)\(37\!\cdots\!32\)\( \beta_{2} + \)\(77\!\cdots\!00\)\( \beta_{3} + \)\(83\!\cdots\!10\)\( \beta_{4}) q^{41} +(-\)\(68\!\cdots\!16\)\( + \)\(97\!\cdots\!20\)\( \beta_{1} - \)\(22\!\cdots\!28\)\( \beta_{2} + \)\(57\!\cdots\!80\)\( \beta_{3} - \)\(10\!\cdots\!16\)\( \beta_{4}) q^{42} +(-\)\(94\!\cdots\!74\)\( + \)\(34\!\cdots\!84\)\( \beta_{1} + \)\(47\!\cdots\!48\)\( \beta_{2} + \)\(59\!\cdots\!40\)\( \beta_{3} - \)\(28\!\cdots\!18\)\( \beta_{4}) q^{43} +(\)\(14\!\cdots\!72\)\( + \)\(73\!\cdots\!56\)\( \beta_{1} - \)\(10\!\cdots\!80\)\( \beta_{2} + \)\(15\!\cdots\!60\)\( \beta_{3} + \)\(45\!\cdots\!08\)\( \beta_{4}) q^{44} +(-\)\(48\!\cdots\!26\)\( + \)\(12\!\cdots\!98\)\( \beta_{1} + \)\(52\!\cdots\!93\)\( \beta_{2} + \)\(30\!\cdots\!29\)\( \beta_{3}) q^{45} +(\)\(25\!\cdots\!68\)\( + \)\(40\!\cdots\!36\)\( \beta_{1} + \)\(21\!\cdots\!64\)\( \beta_{2} - \)\(16\!\cdots\!40\)\( \beta_{3} + \)\(24\!\cdots\!08\)\( \beta_{4}) q^{46} +(\)\(21\!\cdots\!22\)\( + \)\(46\!\cdots\!28\)\( \beta_{1} + \)\(86\!\cdots\!72\)\( \beta_{2} - \)\(86\!\cdots\!80\)\( \beta_{3} - \)\(67\!\cdots\!54\)\( \beta_{4}) q^{47} +(\)\(22\!\cdots\!08\)\( + \)\(42\!\cdots\!08\)\( \beta_{1} + \)\(29\!\cdots\!64\)\( \beta_{2} - \)\(30\!\cdots\!60\)\( \beta_{3} + \)\(45\!\cdots\!72\)\( \beta_{4}) q^{48} +(\)\(17\!\cdots\!67\)\( + \)\(16\!\cdots\!00\)\( \beta_{1} - \)\(24\!\cdots\!34\)\( \beta_{2} + \)\(25\!\cdots\!70\)\( \beta_{3} - \)\(78\!\cdots\!94\)\( \beta_{4}) q^{49} +(\)\(21\!\cdots\!90\)\( + \)\(60\!\cdots\!05\)\( \beta_{1} - \)\(17\!\cdots\!20\)\( \beta_{2} - \)\(12\!\cdots\!60\)\( \beta_{3} + \)\(27\!\cdots\!00\)\( \beta_{4}) q^{50} +(\)\(17\!\cdots\!36\)\( + \)\(20\!\cdots\!36\)\( \beta_{1} + \)\(21\!\cdots\!72\)\( \beta_{2} - \)\(31\!\cdots\!80\)\( \beta_{3} + \)\(36\!\cdots\!06\)\( \beta_{4}) q^{51} +(\)\(46\!\cdots\!70\)\( + \)\(70\!\cdots\!90\)\( \beta_{1} + \)\(25\!\cdots\!74\)\( \beta_{2} + \)\(31\!\cdots\!20\)\( \beta_{3} + \)\(12\!\cdots\!16\)\( \beta_{4}) q^{52} +(\)\(64\!\cdots\!64\)\( - \)\(14\!\cdots\!62\)\( \beta_{1} + \)\(13\!\cdots\!73\)\( \beta_{2} - \)\(85\!\cdots\!75\)\( \beta_{3} - \)\(40\!\cdots\!70\)\( \beta_{4}) q^{53} +(-\)\(55\!\cdots\!06\)\( - \)\(17\!\cdots\!67\)\( \beta_{1}) q^{54} +(-\)\(10\!\cdots\!88\)\( + \)\(27\!\cdots\!24\)\( \beta_{1} + \)\(51\!\cdots\!84\)\( \beta_{2} + \)\(17\!\cdots\!52\)\( \beta_{3} - \)\(47\!\cdots\!00\)\( \beta_{4}) q^{55} +(-\)\(95\!\cdots\!80\)\( - \)\(88\!\cdots\!04\)\( \beta_{1} - \)\(51\!\cdots\!76\)\( \beta_{2} + \)\(18\!\cdots\!80\)\( \beta_{3} + \)\(36\!\cdots\!44\)\( \beta_{4}) q^{56} +(-\)\(70\!\cdots\!34\)\( - \)\(15\!\cdots\!12\)\( \beta_{1} - \)\(89\!\cdots\!88\)\( \beta_{2} - \)\(72\!\cdots\!80\)\( \beta_{3} - \)\(23\!\cdots\!14\)\( \beta_{4}) q^{57} +(-\)\(16\!\cdots\!68\)\( - \)\(57\!\cdots\!22\)\( \beta_{1} - \)\(33\!\cdots\!44\)\( \beta_{2} + \)\(38\!\cdots\!00\)\( \beta_{3} - \)\(29\!\cdots\!20\)\( \beta_{4}) q^{58} +(-\)\(35\!\cdots\!80\)\( - \)\(16\!\cdots\!92\)\( \beta_{1} + \)\(71\!\cdots\!84\)\( \beta_{2} - \)\(49\!\cdots\!60\)\( \beta_{3} - \)\(21\!\cdots\!48\)\( \beta_{4}) q^{59} +(\)\(97\!\cdots\!54\)\( - \)\(41\!\cdots\!42\)\( \beta_{1} - \)\(28\!\cdots\!22\)\( \beta_{2} - \)\(61\!\cdots\!16\)\( \beta_{3} - \)\(27\!\cdots\!00\)\( \beta_{4}) q^{60} +(\)\(11\!\cdots\!93\)\( - \)\(16\!\cdots\!00\)\( \beta_{1} + \)\(26\!\cdots\!69\)\( \beta_{2} + \)\(87\!\cdots\!55\)\( \beta_{3} + \)\(19\!\cdots\!49\)\( \beta_{4}) q^{61} +(\)\(60\!\cdots\!32\)\( + \)\(48\!\cdots\!32\)\( \beta_{1} + \)\(67\!\cdots\!92\)\( \beta_{2} - \)\(86\!\cdots\!40\)\( \beta_{3} - \)\(42\!\cdots\!32\)\( \beta_{4}) q^{62} +(\)\(17\!\cdots\!19\)\( + \)\(18\!\cdots\!78\)\( \beta_{1} - \)\(43\!\cdots\!80\)\( \beta_{2} - \)\(18\!\cdots\!10\)\( \beta_{3} + \)\(68\!\cdots\!77\)\( \beta_{4}) q^{63} +(-\)\(89\!\cdots\!32\)\( + \)\(94\!\cdots\!32\)\( \beta_{1} - \)\(99\!\cdots\!32\)\( \beta_{2} + \)\(56\!\cdots\!60\)\( \beta_{3} - \)\(10\!\cdots\!32\)\( \beta_{4}) q^{64} +(-\)\(89\!\cdots\!38\)\( + \)\(46\!\cdots\!24\)\( \beta_{1} + \)\(59\!\cdots\!84\)\( \beta_{2} + \)\(58\!\cdots\!52\)\( \beta_{3} - \)\(25\!\cdots\!50\)\( \beta_{4}) q^{65} +(-\)\(14\!\cdots\!28\)\( + \)\(82\!\cdots\!12\)\( \beta_{1} - \)\(25\!\cdots\!16\)\( \beta_{2} + \)\(23\!\cdots\!40\)\( \beta_{3} + \)\(38\!\cdots\!92\)\( \beta_{4}) q^{66} +(\)\(26\!\cdots\!80\)\( + \)\(36\!\cdots\!64\)\( \beta_{1} - \)\(34\!\cdots\!00\)\( \beta_{2} - \)\(66\!\cdots\!60\)\( \beta_{3} + \)\(94\!\cdots\!92\)\( \beta_{4}) q^{67} +(\)\(38\!\cdots\!14\)\( + \)\(18\!\cdots\!10\)\( \beta_{1} + \)\(96\!\cdots\!10\)\( \beta_{2} + \)\(69\!\cdots\!80\)\( \beta_{3} + \)\(14\!\cdots\!24\)\( \beta_{4}) q^{68} +(-\)\(34\!\cdots\!22\)\( - \)\(15\!\cdots\!60\)\( \beta_{1} - \)\(11\!\cdots\!56\)\( \beta_{2} + \)\(61\!\cdots\!60\)\( \beta_{3} - \)\(80\!\cdots\!02\)\( \beta_{4}) q^{69} +(\)\(63\!\cdots\!96\)\( + \)\(57\!\cdots\!92\)\( \beta_{1} + \)\(14\!\cdots\!72\)\( \beta_{2} + \)\(10\!\cdots\!16\)\( \beta_{3} - \)\(30\!\cdots\!00\)\( \beta_{4}) q^{70} +(\)\(41\!\cdots\!10\)\( + \)\(25\!\cdots\!04\)\( \beta_{1} + \)\(71\!\cdots\!64\)\( \beta_{2} + \)\(17\!\cdots\!60\)\( \beta_{3} + \)\(36\!\cdots\!58\)\( \beta_{4}) q^{71} +(-\)\(27\!\cdots\!64\)\( - \)\(16\!\cdots\!52\)\( \beta_{1} - \)\(20\!\cdots\!80\)\( \beta_{2} + \)\(20\!\cdots\!20\)\( \beta_{3} - \)\(12\!\cdots\!24\)\( \beta_{4}) q^{72} +(-\)\(50\!\cdots\!20\)\( - \)\(11\!\cdots\!68\)\( \beta_{1} - \)\(88\!\cdots\!50\)\( \beta_{2} - \)\(92\!\cdots\!50\)\( \beta_{3} - \)\(34\!\cdots\!30\)\( \beta_{4}) q^{73} +(-\)\(71\!\cdots\!04\)\( - \)\(14\!\cdots\!54\)\( \beta_{1} + \)\(57\!\cdots\!96\)\( \beta_{2} + \)\(10\!\cdots\!80\)\( \beta_{3} + \)\(17\!\cdots\!04\)\( \beta_{4}) q^{74} +(-\)\(14\!\cdots\!05\)\( - \)\(40\!\cdots\!60\)\( \beta_{1} - \)\(11\!\cdots\!10\)\( \beta_{2} - \)\(10\!\cdots\!30\)\( \beta_{3} - \)\(33\!\cdots\!50\)\( \beta_{4}) q^{75} +(-\)\(63\!\cdots\!04\)\( - \)\(57\!\cdots\!24\)\( \beta_{1} - \)\(44\!\cdots\!72\)\( \beta_{2} + \)\(43\!\cdots\!60\)\( \beta_{3} + \)\(11\!\cdots\!28\)\( \beta_{4}) q^{76} +(-\)\(54\!\cdots\!64\)\( - \)\(10\!\cdots\!68\)\( \beta_{1} + \)\(53\!\cdots\!76\)\( \beta_{2} - \)\(42\!\cdots\!20\)\( \beta_{3} - \)\(14\!\cdots\!56\)\( \beta_{4}) q^{77} +(-\)\(34\!\cdots\!80\)\( - \)\(25\!\cdots\!74\)\( \beta_{1} - \)\(18\!\cdots\!92\)\( \beta_{2} - \)\(28\!\cdots\!00\)\( \beta_{3} + \)\(60\!\cdots\!20\)\( \beta_{4}) q^{78} +(-\)\(68\!\cdots\!25\)\( + \)\(10\!\cdots\!06\)\( \beta_{1} + \)\(14\!\cdots\!44\)\( \beta_{2} + \)\(14\!\cdots\!30\)\( \beta_{3} + \)\(15\!\cdots\!29\)\( \beta_{4}) q^{79} +(-\)\(16\!\cdots\!64\)\( + \)\(50\!\cdots\!72\)\( \beta_{1} + \)\(53\!\cdots\!52\)\( \beta_{2} + \)\(11\!\cdots\!56\)\( \beta_{3} + \)\(11\!\cdots\!00\)\( \beta_{4}) q^{80} +\)\(95\!\cdots\!41\)\( q^{81} +(-\)\(12\!\cdots\!64\)\( + \)\(78\!\cdots\!54\)\( \beta_{1} - \)\(97\!\cdots\!76\)\( \beta_{2} - \)\(17\!\cdots\!60\)\( \beta_{3} - \)\(58\!\cdots\!48\)\( \beta_{4}) q^{82} +(-\)\(55\!\cdots\!96\)\( + \)\(61\!\cdots\!68\)\( \beta_{1} - \)\(83\!\cdots\!10\)\( \beta_{2} - \)\(35\!\cdots\!50\)\( \beta_{3} + \)\(17\!\cdots\!80\)\( \beta_{4}) q^{83} +(-\)\(19\!\cdots\!72\)\( + \)\(39\!\cdots\!96\)\( \beta_{1} + \)\(34\!\cdots\!36\)\( \beta_{2} - \)\(24\!\cdots\!00\)\( \beta_{3} - \)\(79\!\cdots\!20\)\( \beta_{4}) q^{84} +(-\)\(20\!\cdots\!84\)\( + \)\(15\!\cdots\!32\)\( \beta_{1} + \)\(78\!\cdots\!62\)\( \beta_{2} + \)\(11\!\cdots\!86\)\( \beta_{3} + \)\(94\!\cdots\!00\)\( \beta_{4}) q^{85} +(-\)\(57\!\cdots\!00\)\( + \)\(40\!\cdots\!08\)\( \beta_{1} - \)\(30\!\cdots\!48\)\( \beta_{2} + \)\(60\!\cdots\!80\)\( \beta_{3} + \)\(22\!\cdots\!44\)\( \beta_{4}) q^{86} +(-\)\(19\!\cdots\!44\)\( + \)\(89\!\cdots\!58\)\( \beta_{1} + \)\(31\!\cdots\!91\)\( \beta_{2} + \)\(45\!\cdots\!95\)\( \beta_{3} + \)\(65\!\cdots\!86\)\( \beta_{4}) q^{87} +(-\)\(95\!\cdots\!84\)\( - \)\(29\!\cdots\!12\)\( \beta_{1} - \)\(12\!\cdots\!84\)\( \beta_{2} - \)\(44\!\cdots\!80\)\( \beta_{3} - \)\(44\!\cdots\!24\)\( \beta_{4}) q^{88} +(-\)\(19\!\cdots\!42\)\( - \)\(43\!\cdots\!76\)\( \beta_{1} + \)\(43\!\cdots\!52\)\( \beta_{2} + \)\(18\!\cdots\!60\)\( \beta_{3} - \)\(31\!\cdots\!12\)\( \beta_{4}) q^{89} +(-\)\(20\!\cdots\!56\)\( + \)\(48\!\cdots\!38\)\( \beta_{1} + \)\(15\!\cdots\!08\)\( \beta_{2} - \)\(25\!\cdots\!76\)\( \beta_{3} + \)\(97\!\cdots\!00\)\( \beta_{4}) q^{90} +(-\)\(36\!\cdots\!46\)\( - \)\(10\!\cdots\!80\)\( \beta_{1} + \)\(17\!\cdots\!36\)\( \beta_{2} - \)\(17\!\cdots\!40\)\( \beta_{3} + \)\(73\!\cdots\!98\)\( \beta_{4}) q^{91} +(-\)\(62\!\cdots\!12\)\( - \)\(34\!\cdots\!88\)\( \beta_{1} - \)\(37\!\cdots\!08\)\( \beta_{2} + \)\(19\!\cdots\!20\)\( \beta_{3} - \)\(13\!\cdots\!04\)\( \beta_{4}) q^{92} +(-\)\(18\!\cdots\!27\)\( - \)\(15\!\cdots\!42\)\( \beta_{1} - \)\(56\!\cdots\!04\)\( \beta_{2} + \)\(23\!\cdots\!50\)\( \beta_{3} + \)\(14\!\cdots\!15\)\( \beta_{4}) q^{93} +(-\)\(87\!\cdots\!28\)\( - \)\(32\!\cdots\!40\)\( \beta_{1} - \)\(45\!\cdots\!28\)\( \beta_{2} + \)\(60\!\cdots\!80\)\( \beta_{3} + \)\(52\!\cdots\!64\)\( \beta_{4}) q^{94} +(-\)\(66\!\cdots\!20\)\( + \)\(31\!\cdots\!60\)\( \beta_{1} + \)\(50\!\cdots\!60\)\( \beta_{2} + \)\(35\!\cdots\!80\)\( \beta_{3} - \)\(56\!\cdots\!00\)\( \beta_{4}) q^{95} +(-\)\(73\!\cdots\!20\)\( + \)\(23\!\cdots\!16\)\( \beta_{1} - \)\(24\!\cdots\!88\)\( \beta_{2} - \)\(47\!\cdots\!60\)\( \beta_{3} - \)\(17\!\cdots\!48\)\( \beta_{4}) q^{96} +(-\)\(23\!\cdots\!86\)\( + \)\(17\!\cdots\!64\)\( \beta_{1} + \)\(75\!\cdots\!44\)\( \beta_{2} + \)\(70\!\cdots\!60\)\( \beta_{3} - \)\(16\!\cdots\!52\)\( \beta_{4}) q^{97} +(-\)\(34\!\cdots\!18\)\( + \)\(23\!\cdots\!31\)\( \beta_{1} + \)\(18\!\cdots\!36\)\( \beta_{2} - \)\(19\!\cdots\!00\)\( \beta_{3} + \)\(24\!\cdots\!60\)\( \beta_{4}) q^{98} +(-\)\(66\!\cdots\!64\)\( + \)\(57\!\cdots\!08\)\( \beta_{1} + \)\(14\!\cdots\!58\)\( \beta_{2} - \)\(96\!\cdots\!70\)\( \beta_{3} - \)\(66\!\cdots\!36\)\( \beta_{4}) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q - 16255223088q^{2} + 27795302832777615q^{3} + \)\(18\!\cdots\!76\)\(q^{4} - \)\(78\!\cdots\!10\)\(q^{5} - \)\(90\!\cdots\!24\)\(q^{6} + \)\(28\!\cdots\!08\)\(q^{7} - \)\(44\!\cdots\!48\)\(q^{8} + \)\(15\!\cdots\!45\)\(q^{9} + O(q^{10}) \) \( 5q - 16255223088q^{2} + 27795302832777615q^{3} + \)\(18\!\cdots\!76\)\(q^{4} - \)\(78\!\cdots\!10\)\(q^{5} - \)\(90\!\cdots\!24\)\(q^{6} + \)\(28\!\cdots\!08\)\(q^{7} - \)\(44\!\cdots\!48\)\(q^{8} + \)\(15\!\cdots\!45\)\(q^{9} - \)\(33\!\cdots\!60\)\(q^{10} - \)\(10\!\cdots\!32\)\(q^{11} + \)\(10\!\cdots\!48\)\(q^{12} + \)\(18\!\cdots\!78\)\(q^{13} - \)\(61\!\cdots\!52\)\(q^{14} - \)\(43\!\cdots\!30\)\(q^{15} + \)\(19\!\cdots\!36\)\(q^{16} + \)\(15\!\cdots\!50\)\(q^{17} - \)\(50\!\cdots\!52\)\(q^{18} - \)\(63\!\cdots\!88\)\(q^{19} + \)\(88\!\cdots\!20\)\(q^{20} + \)\(16\!\cdots\!84\)\(q^{21} - \)\(12\!\cdots\!00\)\(q^{22} - \)\(30\!\cdots\!64\)\(q^{23} - \)\(24\!\cdots\!04\)\(q^{24} - \)\(12\!\cdots\!25\)\(q^{25} - \)\(30\!\cdots\!60\)\(q^{26} + \)\(85\!\cdots\!35\)\(q^{27} - \)\(17\!\cdots\!16\)\(q^{28} - \)\(17\!\cdots\!66\)\(q^{29} - \)\(18\!\cdots\!80\)\(q^{30} - \)\(16\!\cdots\!44\)\(q^{31} - \)\(66\!\cdots\!72\)\(q^{32} - \)\(59\!\cdots\!36\)\(q^{33} - \)\(36\!\cdots\!48\)\(q^{34} - \)\(59\!\cdots\!60\)\(q^{35} + \)\(58\!\cdots\!04\)\(q^{36} + \)\(30\!\cdots\!58\)\(q^{37} + \)\(26\!\cdots\!44\)\(q^{38} + \)\(10\!\cdots\!94\)\(q^{39} + \)\(11\!\cdots\!00\)\(q^{40} - \)\(80\!\cdots\!54\)\(q^{41} - \)\(34\!\cdots\!96\)\(q^{42} - \)\(47\!\cdots\!28\)\(q^{43} + \)\(70\!\cdots\!80\)\(q^{44} - \)\(24\!\cdots\!90\)\(q^{45} + \)\(12\!\cdots\!56\)\(q^{46} + \)\(10\!\cdots\!56\)\(q^{47} + \)\(11\!\cdots\!28\)\(q^{48} + \)\(88\!\cdots\!33\)\(q^{49} + \)\(10\!\cdots\!00\)\(q^{50} + \)\(86\!\cdots\!50\)\(q^{51} + \)\(23\!\cdots\!00\)\(q^{52} + \)\(32\!\cdots\!66\)\(q^{53} - \)\(27\!\cdots\!96\)\(q^{54} - \)\(51\!\cdots\!20\)\(q^{55} - \)\(47\!\cdots\!80\)\(q^{56} - \)\(35\!\cdots\!24\)\(q^{57} - \)\(84\!\cdots\!32\)\(q^{58} - \)\(17\!\cdots\!12\)\(q^{59} + \)\(48\!\cdots\!60\)\(q^{60} + \)\(58\!\cdots\!02\)\(q^{61} + \)\(30\!\cdots\!96\)\(q^{62} + \)\(89\!\cdots\!32\)\(q^{63} - \)\(44\!\cdots\!00\)\(q^{64} - \)\(44\!\cdots\!20\)\(q^{65} - \)\(71\!\cdots\!00\)\(q^{66} + \)\(13\!\cdots\!20\)\(q^{67} + \)\(19\!\cdots\!96\)\(q^{68} - \)\(17\!\cdots\!72\)\(q^{69} + \)\(31\!\cdots\!40\)\(q^{70} + \)\(20\!\cdots\!80\)\(q^{71} - \)\(13\!\cdots\!92\)\(q^{72} - \)\(25\!\cdots\!34\)\(q^{73} - \)\(35\!\cdots\!32\)\(q^{74} - \)\(71\!\cdots\!75\)\(q^{75} - \)\(31\!\cdots\!68\)\(q^{76} - \)\(27\!\cdots\!24\)\(q^{77} - \)\(17\!\cdots\!80\)\(q^{78} - \)\(34\!\cdots\!80\)\(q^{79} - \)\(83\!\cdots\!60\)\(q^{80} + \)\(47\!\cdots\!05\)\(q^{81} - \)\(63\!\cdots\!64\)\(q^{82} - \)\(27\!\cdots\!36\)\(q^{83} - \)\(97\!\cdots\!68\)\(q^{84} - \)\(10\!\cdots\!60\)\(q^{85} - \)\(28\!\cdots\!32\)\(q^{86} - \)\(95\!\cdots\!18\)\(q^{87} - \)\(47\!\cdots\!68\)\(q^{88} - \)\(95\!\cdots\!38\)\(q^{89} - \)\(10\!\cdots\!40\)\(q^{90} - \)\(18\!\cdots\!44\)\(q^{91} - \)\(31\!\cdots\!52\)\(q^{92} - \)\(93\!\cdots\!12\)\(q^{93} - \)\(43\!\cdots\!16\)\(q^{94} - \)\(33\!\cdots\!00\)\(q^{95} - \)\(36\!\cdots\!56\)\(q^{96} - \)\(11\!\cdots\!90\)\(q^{97} - \)\(17\!\cdots\!48\)\(q^{98} - \)\(33\!\cdots\!28\)\(q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{5} - x^{4} - 189398708752260728 x^{3} + 10367208713972597963236992 x^{2} + 3357934964760116559921767608965120 x - 178120650251916997056936485095832786534400\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 48 \nu - 10 \)
\(\beta_{2}\)\(=\)\( 2304 \nu^{2} + 189172740336 \nu - 174549850023918035453 \)
\(\beta_{3}\)\(=\)\((\)\(125631 \nu^{4} - 20285201178639 \nu^{3} - 24287555793022213615416 \nu^{2} + 4089272190920375772005891850240 \nu + 248673114239406649464560671009488947200\)\()/ 2017580003532800 \)
\(\beta_{4}\)\(=\)\((\)\(425331 \nu^{4} + 43714584729981 \nu^{3} - 74827217973954639594456 \nu^{2} - 2849467079540794985601298513920 \nu + 980409903289358872957287217059326126080\)\()/ 403516000706560 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1} + 10\)\()/48\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{2} - 3941098757 \beta_{1} + 174549849984507047883\)\()/2304\)
\(\nu^{3}\)\(=\)\((\)\(24816 \beta_{4} - 420080 \beta_{3} - 197516269 \beta_{2} + 22167332404598804945 \beta_{1} - 42994887184245900373960652975\)\()/6912\)
\(\nu^{4}\)\(=\)\((\)\(12020859957712 \beta_{4} + 129524695496240 \beta_{3} + 1644244098821488241 \beta_{2} - 10180895620766242117007354533 \beta_{1} + 241831534082920720028966478728607384667\)\()/20736\)

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
3.76085e8
1.48809e8
5.25899e7
−1.37573e8
−4.39911e8
−2.13031e10 5.55906e15 3.06250e20 −4.34254e22 −1.18425e26 7.31825e27 −3.38029e30 3.09032e31 9.25096e32
1.2 −1.03939e10 5.55906e15 −3.95417e19 1.67944e23 −5.77801e25 −3.85597e28 1.94485e30 3.09032e31 −1.74559e33
1.3 −5.77536e9 5.55906e15 −1.14219e20 1.87613e23 −3.21056e25 3.73645e28 1.51195e30 3.09032e31 −1.08353e33
1.4 3.35246e9 5.55906e15 −1.36335e20 −3.80674e23 1.86366e25 2.28291e27 −9.51794e29 3.09032e31 −1.27619e33
1.5 1.78647e10 5.55906e15 1.71572e20 −1.00461e22 9.93107e25 −5.50883e27 4.28720e29 3.09032e31 −1.79470e32
\(n\): 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 \( T_{2}^{5} + 16255223088 T_{2}^{4} - \)\(33\!\cdots\!36\)\( T_{2}^{3} - \)\(50\!\cdots\!00\)\( T_{2}^{2} - \)\(29\!\cdots\!24\)\( T_{2} + \)\(76\!\cdots\!96\)\( \) acting on \(S_{68}^{\mathrm{new}}(\Gamma_0(3))\).