Properties

Label 1.68.a.a
Level 1
Weight 68
Character orbit 1.a
Self dual Yes
Analytic conductor 28.429
Analytic rank 0
Dimension 5
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 1 \)
Weight: \( k \) = \( 68 \)
Character orbit: \([\chi]\) = 1.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(28.429035193\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{40}\cdot 3^{15}\cdot 5^{4}\cdot 7^{2}\cdot 11\cdot 13\cdot 17 \)
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.

\(f(q)\) \(=\) \( q\) \(+(1110980251 - \beta_{1}) q^{2}\) \(+(688672053797479 - 131979 \beta_{1} + \beta_{2}) q^{3}\) \(+(70094400875776967549 - 3411750909 \beta_{1} - 5994 \beta_{2} + \beta_{3}) q^{4}\) \(+(\)\(66\!\cdots\!25\)\( + 5225164572589 \beta_{1} - 881784 \beta_{2} - 172 \beta_{3} - \beta_{4}) q^{5}\) \(+(\)\(29\!\cdots\!36\)\( + 1611846772269804 \beta_{1} - 781391136 \beta_{2} + 16704 \beta_{3} - 360 \beta_{4}) q^{6}\) \(+(\)\(67\!\cdots\!98\)\( - 234147910821976578 \beta_{1} - 494133901254 \beta_{2} + 83062000 \beta_{3} - 11340 \beta_{4}) q^{7}\) \(+(\)\(65\!\cdots\!36\)\( - 61324159169951174696 \beta_{1} - 51096613883472 \beta_{2} + 12774152200 \beta_{3} + 3277120 \beta_{4}) q^{8}\) \(+(\)\(55\!\cdots\!99\)\( + \)\(96\!\cdots\!02\)\( \beta_{1} + 4943811769157232 \beta_{2} - 181017634248 \beta_{3} - 196006230 \beta_{4}) q^{9}\) \(+O(q^{10})\) \( q\) \(+(1110980251 - \beta_{1}) q^{2}\) \(+(688672053797479 - 131979 \beta_{1} + \beta_{2}) q^{3}\) \(+(70094400875776967549 - 3411750909 \beta_{1} - 5994 \beta_{2} + \beta_{3}) q^{4}\) \(+(\)\(66\!\cdots\!25\)\( + 5225164572589 \beta_{1} - 881784 \beta_{2} - 172 \beta_{3} - \beta_{4}) q^{5}\) \(+(\)\(29\!\cdots\!36\)\( + 1611846772269804 \beta_{1} - 781391136 \beta_{2} + 16704 \beta_{3} - 360 \beta_{4}) q^{6}\) \(+(\)\(67\!\cdots\!98\)\( - 234147910821976578 \beta_{1} - 494133901254 \beta_{2} + 83062000 \beta_{3} - 11340 \beta_{4}) q^{7}\) \(+(\)\(65\!\cdots\!36\)\( - 61324159169951174696 \beta_{1} - 51096613883472 \beta_{2} + 12774152200 \beta_{3} + 3277120 \beta_{4}) q^{8}\) \(+(\)\(55\!\cdots\!99\)\( + \)\(96\!\cdots\!02\)\( \beta_{1} + 4943811769157232 \beta_{2} - 181017634248 \beta_{3} - 196006230 \beta_{4}) q^{9}\) \(+(-\)\(10\!\cdots\!50\)\( - \)\(36\!\cdots\!46\)\( \beta_{1} + 387747904434926976 \beta_{2} - 10724715572992 \beta_{3} + 6662909664 \beta_{4}) q^{10}\) \(+(\)\(40\!\cdots\!97\)\( - \)\(14\!\cdots\!25\)\( \beta_{1} + 4432914057561345675 \beta_{2} + 337416804371680 \beta_{3} - 154884117080 \beta_{4}) q^{11}\) \(+(-\)\(41\!\cdots\!16\)\( - \)\(10\!\cdots\!44\)\( \beta_{1} - 12007852213004039576 \beta_{2} - 2856419893635300 \beta_{3} + 2653671352320 \beta_{4}) q^{12}\) \(+(\)\(35\!\cdots\!77\)\( - \)\(19\!\cdots\!87\)\( \beta_{1} - \)\(15\!\cdots\!60\)\( \beta_{2} - 29875083592222700 \beta_{3} - 34622077787505 \beta_{4}) q^{13}\) \(+(\)\(58\!\cdots\!44\)\( - \)\(18\!\cdots\!92\)\( \beta_{1} + 33318747846508840128 \beta_{2} + 964772473179561088 \beta_{3} + 344998573654000 \beta_{4}) q^{14}\) \(+(-\)\(18\!\cdots\!50\)\( - \)\(17\!\cdots\!22\)\( \beta_{1} + \)\(22\!\cdots\!82\)\( \beta_{2} - 10229182088434956144 \beta_{3} - 2522507816770452 \beta_{4}) q^{15}\) \(+(\)\(36\!\cdots\!04\)\( - \)\(18\!\cdots\!52\)\( \beta_{1} - \)\(10\!\cdots\!32\)\( \beta_{2} + 43878715027197261888 \beta_{3} + 11266877309667840 \beta_{4}) q^{16}\) \(+(\)\(15\!\cdots\!32\)\( - \)\(71\!\cdots\!02\)\( \beta_{1} - \)\(95\!\cdots\!12\)\( \beta_{2} + \)\(18\!\cdots\!00\)\( \beta_{3} + 8396275243314890 \beta_{4}) q^{17}\) \(+(-\)\(20\!\cdots\!21\)\( + \)\(30\!\cdots\!91\)\( \beta_{1} + \)\(80\!\cdots\!76\)\( \beta_{2} - \)\(39\!\cdots\!00\)\( \beta_{3} - 700896140455976640 \beta_{4}) q^{18}\) \(+(\)\(79\!\cdots\!79\)\( + \)\(29\!\cdots\!09\)\( \beta_{1} + \)\(63\!\cdots\!69\)\( \beta_{2} + \)\(24\!\cdots\!64\)\( \beta_{3} + 7385036747071057560 \beta_{4}) q^{19}\) \(+(-\)\(30\!\cdots\!50\)\( + \)\(24\!\cdots\!02\)\( \beta_{1} - \)\(20\!\cdots\!12\)\( \beta_{2} - \)\(48\!\cdots\!46\)\( \beta_{3} - 47581246658342739968 \beta_{4}) q^{20}\) \(+(-\)\(35\!\cdots\!00\)\( - \)\(27\!\cdots\!32\)\( \beta_{1} + \)\(16\!\cdots\!88\)\( \beta_{2} - \)\(36\!\cdots\!52\)\( \beta_{3} + \)\(19\!\cdots\!00\)\( \beta_{4}) q^{21}\) \(+(\)\(36\!\cdots\!72\)\( - \)\(71\!\cdots\!32\)\( \beta_{1} + \)\(35\!\cdots\!60\)\( \beta_{2} + \)\(32\!\cdots\!00\)\( \beta_{3} - \)\(20\!\cdots\!80\)\( \beta_{4}) q^{22}\) \(+(-\)\(83\!\cdots\!90\)\( - \)\(73\!\cdots\!10\)\( \beta_{1} - \)\(14\!\cdots\!86\)\( \beta_{2} - \)\(10\!\cdots\!00\)\( \beta_{3} - \)\(38\!\cdots\!00\)\( \beta_{4}) q^{23}\) \(+(-\)\(26\!\cdots\!48\)\( + \)\(45\!\cdots\!92\)\( \beta_{1} - \)\(78\!\cdots\!28\)\( \beta_{2} - \)\(48\!\cdots\!48\)\( \beta_{3} + \)\(36\!\cdots\!60\)\( \beta_{4}) q^{24}\) \(+(\)\(35\!\cdots\!75\)\( + \)\(38\!\cdots\!00\)\( \beta_{1} + \)\(94\!\cdots\!00\)\( \beta_{2} + \)\(13\!\cdots\!00\)\( \beta_{3} - \)\(20\!\cdots\!00\)\( \beta_{4}) q^{25}\) \(+(\)\(42\!\cdots\!94\)\( - \)\(62\!\cdots\!18\)\( \beta_{1} + \)\(12\!\cdots\!12\)\( \beta_{2} - \)\(20\!\cdots\!68\)\( \beta_{3} + \)\(74\!\cdots\!20\)\( \beta_{4}) q^{26}\) \(+(\)\(37\!\cdots\!34\)\( - \)\(15\!\cdots\!74\)\( \beta_{1} - \)\(25\!\cdots\!18\)\( \beta_{2} - \)\(23\!\cdots\!00\)\( \beta_{3} - \)\(17\!\cdots\!20\)\( \beta_{4}) q^{27}\) \(+(\)\(29\!\cdots\!20\)\( - \)\(17\!\cdots\!60\)\( \beta_{1} - \)\(18\!\cdots\!24\)\( \beta_{2} + \)\(15\!\cdots\!00\)\( \beta_{3} + \)\(48\!\cdots\!80\)\( \beta_{4}) q^{28}\) \(+(\)\(37\!\cdots\!01\)\( + \)\(75\!\cdots\!21\)\( \beta_{1} + \)\(75\!\cdots\!36\)\( \beta_{2} - \)\(43\!\cdots\!64\)\( \beta_{3} + \)\(15\!\cdots\!95\)\( \beta_{4}) q^{29}\) \(+(\)\(36\!\cdots\!00\)\( + \)\(15\!\cdots\!08\)\( \beta_{1} + \)\(17\!\cdots\!52\)\( \beta_{2} + \)\(45\!\cdots\!16\)\( \beta_{3} - \)\(76\!\cdots\!72\)\( \beta_{4}) q^{30}\) \(+(\)\(73\!\cdots\!12\)\( + \)\(24\!\cdots\!00\)\( \beta_{1} - \)\(37\!\cdots\!00\)\( \beta_{2} - \)\(46\!\cdots\!60\)\( \beta_{3} + \)\(18\!\cdots\!60\)\( \beta_{4}) q^{31}\) \(+(\)\(31\!\cdots\!16\)\( - \)\(66\!\cdots\!36\)\( \beta_{1} - \)\(91\!\cdots\!80\)\( \beta_{2} + \)\(55\!\cdots\!00\)\( \beta_{3} - \)\(13\!\cdots\!60\)\( \beta_{4}) q^{32}\) \(+(\)\(48\!\cdots\!38\)\( - \)\(23\!\cdots\!58\)\( \beta_{1} + \)\(63\!\cdots\!92\)\( \beta_{2} - \)\(38\!\cdots\!00\)\( \beta_{3} - \)\(80\!\cdots\!10\)\( \beta_{4}) q^{33}\) \(+(\)\(15\!\cdots\!18\)\( - \)\(77\!\cdots\!18\)\( \beta_{1} - \)\(51\!\cdots\!88\)\( \beta_{2} + \)\(98\!\cdots\!92\)\( \beta_{3} + \)\(35\!\cdots\!60\)\( \beta_{4}) q^{34}\) \(+(\)\(90\!\cdots\!00\)\( + \)\(24\!\cdots\!84\)\( \beta_{1} - \)\(16\!\cdots\!04\)\( \beta_{2} - \)\(28\!\cdots\!32\)\( \beta_{3} - \)\(63\!\cdots\!56\)\( \beta_{4}) q^{35}\) \(+(-\)\(76\!\cdots\!51\)\( + \)\(78\!\cdots\!23\)\( \beta_{1} - \)\(18\!\cdots\!82\)\( \beta_{2} - \)\(49\!\cdots\!87\)\( \beta_{3} + \)\(26\!\cdots\!40\)\( \beta_{4}) q^{36}\) \(+(-\)\(11\!\cdots\!15\)\( - \)\(90\!\cdots\!15\)\( \beta_{1} + \)\(19\!\cdots\!32\)\( \beta_{2} + \)\(11\!\cdots\!00\)\( \beta_{3} + \)\(24\!\cdots\!35\)\( \beta_{4}) q^{37}\) \(+(-\)\(62\!\cdots\!36\)\( - \)\(27\!\cdots\!84\)\( \beta_{1} - \)\(15\!\cdots\!48\)\( \beta_{2} - \)\(56\!\cdots\!00\)\( \beta_{3} - \)\(43\!\cdots\!60\)\( \beta_{4}) q^{38}\) \(+(-\)\(14\!\cdots\!18\)\( - \)\(64\!\cdots\!62\)\( \beta_{1} - \)\(43\!\cdots\!42\)\( \beta_{2} + \)\(20\!\cdots\!48\)\( \beta_{3} - \)\(81\!\cdots\!80\)\( \beta_{4}) q^{39}\) \(+(-\)\(37\!\cdots\!00\)\( + \)\(14\!\cdots\!60\)\( \beta_{1} - \)\(24\!\cdots\!60\)\( \beta_{2} - \)\(12\!\cdots\!80\)\( \beta_{3} + \)\(28\!\cdots\!60\)\( \beta_{4}) q^{40}\) \(+(\)\(22\!\cdots\!82\)\( + \)\(16\!\cdots\!00\)\( \beta_{1} + \)\(78\!\cdots\!00\)\( \beta_{2} + \)\(38\!\cdots\!40\)\( \beta_{3} + \)\(62\!\cdots\!60\)\( \beta_{4}) q^{41}\) \(+(\)\(56\!\cdots\!20\)\( + \)\(75\!\cdots\!00\)\( \beta_{1} - \)\(72\!\cdots\!56\)\( \beta_{2} + \)\(21\!\cdots\!00\)\( \beta_{3} - \)\(22\!\cdots\!40\)\( \beta_{4}) q^{42}\) \(+(\)\(13\!\cdots\!81\)\( - \)\(39\!\cdots\!81\)\( \beta_{1} + \)\(21\!\cdots\!47\)\( \beta_{2} - \)\(23\!\cdots\!00\)\( \beta_{3} + \)\(67\!\cdots\!00\)\( \beta_{4}) q^{43}\) \(+(\)\(98\!\cdots\!28\)\( - \)\(59\!\cdots\!48\)\( \beta_{1} - \)\(11\!\cdots\!68\)\( \beta_{2} + \)\(45\!\cdots\!12\)\( \beta_{3} + \)\(15\!\cdots\!60\)\( \beta_{4}) q^{44}\) \(+(\)\(19\!\cdots\!25\)\( - \)\(33\!\cdots\!87\)\( \beta_{1} + \)\(14\!\cdots\!72\)\( \beta_{2} - \)\(91\!\cdots\!24\)\( \beta_{3} - \)\(43\!\cdots\!17\)\( \beta_{4}) q^{45}\) \(+(\)\(15\!\cdots\!12\)\( + \)\(10\!\cdots\!20\)\( \beta_{1} + \)\(12\!\cdots\!20\)\( \beta_{2} - \)\(28\!\cdots\!40\)\( \beta_{3} + \)\(18\!\cdots\!60\)\( \beta_{4}) q^{46}\) \(+(-\)\(25\!\cdots\!96\)\( + \)\(44\!\cdots\!96\)\( \beta_{1} - \)\(19\!\cdots\!16\)\( \beta_{2} - \)\(14\!\cdots\!00\)\( \beta_{3} + \)\(15\!\cdots\!00\)\( \beta_{4}) q^{47}\) \(+(-\)\(39\!\cdots\!20\)\( + \)\(36\!\cdots\!00\)\( \beta_{1} - \)\(83\!\cdots\!56\)\( \beta_{2} + \)\(16\!\cdots\!00\)\( \beta_{3} - \)\(35\!\cdots\!60\)\( \beta_{4}) q^{48}\) \(+(-\)\(15\!\cdots\!47\)\( - \)\(12\!\cdots\!40\)\( \beta_{1} - \)\(17\!\cdots\!40\)\( \beta_{2} + \)\(12\!\cdots\!00\)\( \beta_{3} - \)\(26\!\cdots\!40\)\( \beta_{4}) q^{49}\) \(+(-\)\(78\!\cdots\!75\)\( - \)\(39\!\cdots\!75\)\( \beta_{1} + \)\(89\!\cdots\!00\)\( \beta_{2} - \)\(35\!\cdots\!00\)\( \beta_{3} + \)\(11\!\cdots\!00\)\( \beta_{4}) q^{50}\) \(+(-\)\(67\!\cdots\!42\)\( + \)\(36\!\cdots\!26\)\( \beta_{1} - \)\(51\!\cdots\!34\)\( \beta_{2} + \)\(12\!\cdots\!96\)\( \beta_{3} - \)\(57\!\cdots\!60\)\( \beta_{4}) q^{51}\) \(+(\)\(13\!\cdots\!18\)\( + \)\(54\!\cdots\!82\)\( \beta_{1} - \)\(32\!\cdots\!04\)\( \beta_{2} + \)\(48\!\cdots\!50\)\( \beta_{3} - \)\(46\!\cdots\!00\)\( \beta_{4}) q^{52}\) \(+(\)\(20\!\cdots\!69\)\( + \)\(31\!\cdots\!21\)\( \beta_{1} - \)\(35\!\cdots\!44\)\( \beta_{2} + \)\(20\!\cdots\!00\)\( \beta_{3} + \)\(66\!\cdots\!95\)\( \beta_{4}) q^{53}\) \(+(\)\(38\!\cdots\!44\)\( - \)\(20\!\cdots\!76\)\( \beta_{1} + \)\(58\!\cdots\!84\)\( \beta_{2} - \)\(87\!\cdots\!96\)\( \beta_{3} + \)\(17\!\cdots\!60\)\( \beta_{4}) q^{54}\) \(+(\)\(13\!\cdots\!50\)\( + \)\(32\!\cdots\!58\)\( \beta_{1} + \)\(96\!\cdots\!02\)\( \beta_{2} - \)\(67\!\cdots\!84\)\( \beta_{3} - \)\(64\!\cdots\!72\)\( \beta_{4}) q^{55}\) \(+(\)\(32\!\cdots\!16\)\( - \)\(30\!\cdots\!64\)\( \beta_{1} - \)\(17\!\cdots\!24\)\( \beta_{2} + \)\(19\!\cdots\!76\)\( \beta_{3} + \)\(30\!\cdots\!20\)\( \beta_{4}) q^{56}\) \(+(-\)\(18\!\cdots\!22\)\( + \)\(51\!\cdots\!22\)\( \beta_{1} - \)\(62\!\cdots\!36\)\( \beta_{2} - \)\(33\!\cdots\!00\)\( \beta_{3} + \)\(19\!\cdots\!50\)\( \beta_{4}) q^{57}\) \(+(-\)\(12\!\cdots\!34\)\( + \)\(34\!\cdots\!14\)\( \beta_{1} - \)\(37\!\cdots\!72\)\( \beta_{2} - \)\(47\!\cdots\!00\)\( \beta_{3} - \)\(42\!\cdots\!60\)\( \beta_{4}) q^{58}\) \(+(-\)\(61\!\cdots\!03\)\( + \)\(11\!\cdots\!87\)\( \beta_{1} + \)\(12\!\cdots\!67\)\( \beta_{2} + \)\(49\!\cdots\!32\)\( \beta_{3} + \)\(78\!\cdots\!00\)\( \beta_{4}) q^{59}\) \(+(-\)\(26\!\cdots\!00\)\( - \)\(12\!\cdots\!96\)\( \beta_{1} + \)\(14\!\cdots\!76\)\( \beta_{2} + \)\(14\!\cdots\!08\)\( \beta_{3} + \)\(85\!\cdots\!64\)\( \beta_{4}) q^{60}\) \(+(-\)\(22\!\cdots\!03\)\( + \)\(14\!\cdots\!25\)\( \beta_{1} - \)\(12\!\cdots\!00\)\( \beta_{2} + \)\(81\!\cdots\!00\)\( \beta_{3} - \)\(11\!\cdots\!25\)\( \beta_{4}) q^{61}\) \(+(-\)\(43\!\cdots\!88\)\( - \)\(76\!\cdots\!92\)\( \beta_{1} - \)\(51\!\cdots\!20\)\( \beta_{2} - \)\(12\!\cdots\!00\)\( \beta_{3} + \)\(11\!\cdots\!60\)\( \beta_{4}) q^{62}\) \(+(-\)\(41\!\cdots\!02\)\( + \)\(62\!\cdots\!82\)\( \beta_{1} + \)\(27\!\cdots\!98\)\( \beta_{2} - \)\(43\!\cdots\!00\)\( \beta_{3} - \)\(50\!\cdots\!60\)\( \beta_{4}) q^{63}\) \(+(\)\(12\!\cdots\!20\)\( - \)\(14\!\cdots\!88\)\( \beta_{1} + \)\(14\!\cdots\!92\)\( \beta_{2} + \)\(55\!\cdots\!12\)\( \beta_{3} + \)\(31\!\cdots\!20\)\( \beta_{4}) q^{64}\) \(+(\)\(34\!\cdots\!00\)\( + \)\(33\!\cdots\!68\)\( \beta_{1} - \)\(94\!\cdots\!08\)\( \beta_{2} + \)\(17\!\cdots\!36\)\( \beta_{3} + \)\(16\!\cdots\!88\)\( \beta_{4}) q^{65}\) \(+(\)\(55\!\cdots\!92\)\( + \)\(68\!\cdots\!88\)\( \beta_{1} + \)\(26\!\cdots\!08\)\( \beta_{2} - \)\(20\!\cdots\!52\)\( \beta_{3} - \)\(21\!\cdots\!80\)\( \beta_{4}) q^{66}\) \(+(-\)\(23\!\cdots\!73\)\( + \)\(58\!\cdots\!53\)\( \beta_{1} - \)\(60\!\cdots\!27\)\( \beta_{2} - \)\(73\!\cdots\!00\)\( \beta_{3} + \)\(33\!\cdots\!40\)\( \beta_{4}) q^{67}\) \(+(\)\(16\!\cdots\!02\)\( - \)\(19\!\cdots\!42\)\( \beta_{1} - \)\(62\!\cdots\!28\)\( \beta_{2} + \)\(15\!\cdots\!50\)\( \beta_{3} + \)\(50\!\cdots\!80\)\( \beta_{4}) q^{68}\) \(+(\)\(71\!\cdots\!48\)\( - \)\(43\!\cdots\!56\)\( \beta_{1} + \)\(76\!\cdots\!04\)\( \beta_{2} + \)\(26\!\cdots\!44\)\( \beta_{3} - \)\(70\!\cdots\!60\)\( \beta_{4}) q^{69}\) \(+(-\)\(52\!\cdots\!00\)\( - \)\(43\!\cdots\!76\)\( \beta_{1} + \)\(38\!\cdots\!56\)\( \beta_{2} - \)\(27\!\cdots\!52\)\( \beta_{3} + \)\(96\!\cdots\!84\)\( \beta_{4}) q^{70}\) \(+(-\)\(22\!\cdots\!58\)\( + \)\(16\!\cdots\!50\)\( \beta_{1} - \)\(23\!\cdots\!50\)\( \beta_{2} + \)\(41\!\cdots\!00\)\( \beta_{3} - \)\(10\!\cdots\!00\)\( \beta_{4}) q^{71}\) \(+(-\)\(14\!\cdots\!68\)\( + \)\(10\!\cdots\!08\)\( \beta_{1} - \)\(56\!\cdots\!24\)\( \beta_{2} - \)\(62\!\cdots\!00\)\( \beta_{3} + \)\(11\!\cdots\!20\)\( \beta_{4}) q^{72}\) \(+(-\)\(60\!\cdots\!00\)\( - \)\(28\!\cdots\!70\)\( \beta_{1} + \)\(14\!\cdots\!04\)\( \beta_{2} + \)\(33\!\cdots\!00\)\( \beta_{3} + \)\(14\!\cdots\!90\)\( \beta_{4}) q^{73}\) \(+(\)\(18\!\cdots\!26\)\( + \)\(42\!\cdots\!90\)\( \beta_{1} - \)\(17\!\cdots\!60\)\( \beta_{2} + \)\(18\!\cdots\!00\)\( \beta_{3} - \)\(72\!\cdots\!60\)\( \beta_{4}) q^{74}\) \(+(\)\(43\!\cdots\!25\)\( - \)\(41\!\cdots\!25\)\( \beta_{1} + \)\(68\!\cdots\!75\)\( \beta_{2} - \)\(29\!\cdots\!00\)\( \beta_{3} + \)\(83\!\cdots\!00\)\( \beta_{4}) q^{75}\) \(+(\)\(41\!\cdots\!12\)\( + \)\(15\!\cdots\!52\)\( \beta_{1} - \)\(89\!\cdots\!68\)\( \beta_{2} - \)\(13\!\cdots\!68\)\( \beta_{3} - \)\(29\!\cdots\!60\)\( \beta_{4}) q^{76}\) \(+(\)\(10\!\cdots\!56\)\( - \)\(47\!\cdots\!76\)\( \beta_{1} - \)\(81\!\cdots\!28\)\( \beta_{2} + \)\(29\!\cdots\!00\)\( \beta_{3} + \)\(54\!\cdots\!40\)\( \beta_{4}) q^{77}\) \(+(\)\(12\!\cdots\!52\)\( + \)\(12\!\cdots\!08\)\( \beta_{1} - \)\(36\!\cdots\!36\)\( \beta_{2} + \)\(66\!\cdots\!00\)\( \beta_{3} + \)\(23\!\cdots\!80\)\( \beta_{4}) q^{78}\) \(+(\)\(59\!\cdots\!36\)\( + \)\(85\!\cdots\!56\)\( \beta_{1} - \)\(81\!\cdots\!04\)\( \beta_{2} - \)\(63\!\cdots\!44\)\( \beta_{3} - \)\(77\!\cdots\!40\)\( \beta_{4}) q^{79}\) \(+(-\)\(30\!\cdots\!00\)\( + \)\(12\!\cdots\!04\)\( \beta_{1} + \)\(42\!\cdots\!76\)\( \beta_{2} - \)\(14\!\cdots\!92\)\( \beta_{3} + \)\(14\!\cdots\!64\)\( \beta_{4}) q^{80}\) \(+(-\)\(22\!\cdots\!25\)\( - \)\(39\!\cdots\!26\)\( \beta_{1} + \)\(13\!\cdots\!84\)\( \beta_{2} + \)\(12\!\cdots\!64\)\( \beta_{3} + \)\(24\!\cdots\!50\)\( \beta_{4}) q^{81}\) \(+(-\)\(35\!\cdots\!18\)\( - \)\(22\!\cdots\!62\)\( \beta_{1} - \)\(23\!\cdots\!20\)\( \beta_{2} + \)\(10\!\cdots\!00\)\( \beta_{3} - \)\(28\!\cdots\!40\)\( \beta_{4}) q^{82}\) \(+(-\)\(94\!\cdots\!57\)\( - \)\(35\!\cdots\!43\)\( \beta_{1} - \)\(14\!\cdots\!27\)\( \beta_{2} + \)\(14\!\cdots\!00\)\( \beta_{3} + \)\(10\!\cdots\!00\)\( \beta_{4}) q^{83}\) \(+(-\)\(10\!\cdots\!68\)\( - \)\(19\!\cdots\!24\)\( \beta_{1} + \)\(99\!\cdots\!16\)\( \beta_{2} - \)\(20\!\cdots\!84\)\( \beta_{3} + \)\(12\!\cdots\!20\)\( \beta_{4}) q^{84}\) \(+(-\)\(78\!\cdots\!50\)\( + \)\(15\!\cdots\!94\)\( \beta_{1} + \)\(65\!\cdots\!36\)\( \beta_{2} - \)\(53\!\cdots\!12\)\( \beta_{3} + \)\(64\!\cdots\!54\)\( \beta_{4}) q^{85}\) \(+(\)\(99\!\cdots\!28\)\( + \)\(18\!\cdots\!76\)\( \beta_{1} + \)\(23\!\cdots\!16\)\( \beta_{2} - \)\(18\!\cdots\!44\)\( \beta_{3} - \)\(10\!\cdots\!20\)\( \beta_{4}) q^{86}\) \(+(\)\(71\!\cdots\!82\)\( + \)\(24\!\cdots\!38\)\( \beta_{1} + \)\(71\!\cdots\!46\)\( \beta_{2} + \)\(20\!\cdots\!00\)\( \beta_{3} - \)\(15\!\cdots\!40\)\( \beta_{4}) q^{87}\) \(+(\)\(86\!\cdots\!92\)\( - \)\(88\!\cdots\!12\)\( \beta_{1} - \)\(15\!\cdots\!84\)\( \beta_{2} + \)\(69\!\cdots\!00\)\( \beta_{3} + \)\(38\!\cdots\!40\)\( \beta_{4}) q^{88}\) \(+(-\)\(23\!\cdots\!12\)\( + \)\(45\!\cdots\!98\)\( \beta_{1} - \)\(86\!\cdots\!32\)\( \beta_{2} - \)\(12\!\cdots\!32\)\( \beta_{3} - \)\(74\!\cdots\!90\)\( \beta_{4}) q^{89}\) \(+(\)\(95\!\cdots\!50\)\( - \)\(15\!\cdots\!82\)\( \beta_{1} + \)\(13\!\cdots\!92\)\( \beta_{2} - \)\(84\!\cdots\!64\)\( \beta_{3} - \)\(25\!\cdots\!12\)\( \beta_{4}) q^{90}\) \(+(\)\(46\!\cdots\!08\)\( + \)\(82\!\cdots\!96\)\( \beta_{1} - \)\(38\!\cdots\!64\)\( \beta_{2} + \)\(60\!\cdots\!36\)\( \beta_{3} - \)\(37\!\cdots\!80\)\( \beta_{4}) q^{91}\) \(+(-\)\(19\!\cdots\!68\)\( + \)\(38\!\cdots\!88\)\( \beta_{1} + \)\(26\!\cdots\!48\)\( \beta_{2} + \)\(14\!\cdots\!00\)\( \beta_{3} - \)\(25\!\cdots\!40\)\( \beta_{4}) q^{92}\) \(+(-\)\(36\!\cdots\!52\)\( + \)\(11\!\cdots\!92\)\( \beta_{1} + \)\(34\!\cdots\!72\)\( \beta_{2} + \)\(27\!\cdots\!00\)\( \beta_{3} + \)\(21\!\cdots\!20\)\( \beta_{4}) q^{93}\) \(+(-\)\(98\!\cdots\!40\)\( + \)\(51\!\cdots\!24\)\( \beta_{1} - \)\(24\!\cdots\!16\)\( \beta_{2} - \)\(49\!\cdots\!96\)\( \beta_{3} - \)\(11\!\cdots\!40\)\( \beta_{4}) q^{94}\) \(+(-\)\(40\!\cdots\!50\)\( + \)\(11\!\cdots\!90\)\( \beta_{1} - \)\(18\!\cdots\!90\)\( \beta_{2} + \)\(52\!\cdots\!80\)\( \beta_{3} - \)\(30\!\cdots\!60\)\( \beta_{4}) q^{95}\) \(+(-\)\(44\!\cdots\!24\)\( - \)\(60\!\cdots\!36\)\( \beta_{1} + \)\(25\!\cdots\!24\)\( \beta_{2} - \)\(19\!\cdots\!16\)\( \beta_{3} + \)\(50\!\cdots\!20\)\( \beta_{4}) q^{96}\) \(+(\)\(57\!\cdots\!24\)\( - \)\(21\!\cdots\!14\)\( \beta_{1} - \)\(76\!\cdots\!16\)\( \beta_{2} + \)\(27\!\cdots\!00\)\( \beta_{3} + \)\(50\!\cdots\!30\)\( \beta_{4}) q^{97}\) \(+(\)\(24\!\cdots\!03\)\( - \)\(68\!\cdots\!33\)\( \beta_{1} - \)\(10\!\cdots\!40\)\( \beta_{2} + \)\(25\!\cdots\!00\)\( \beta_{3} + \)\(79\!\cdots\!60\)\( \beta_{4}) q^{98}\) \(+(\)\(32\!\cdots\!53\)\( + \)\(51\!\cdots\!19\)\( \beta_{1} + \)\(62\!\cdots\!79\)\( \beta_{2} - \)\(36\!\cdots\!96\)\( \beta_{3} - \)\(18\!\cdots\!20\)\( \beta_{4}) q^{99}\) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(5q \) \(\mathstrut +\mathstrut 5554901256q^{2} \) \(\mathstrut +\mathstrut 3443360269119372q^{3} \) \(\mathstrut +\mathstrut \)\(35\!\cdots\!40\)\(q^{4} \) \(\mathstrut +\mathstrut \)\(33\!\cdots\!50\)\(q^{5} \) \(\mathstrut +\mathstrut \)\(14\!\cdots\!60\)\(q^{6} \) \(\mathstrut +\mathstrut \)\(33\!\cdots\!56\)\(q^{7} \) \(\mathstrut +\mathstrut \)\(32\!\cdots\!80\)\(q^{8} \) \(\mathstrut +\mathstrut \)\(27\!\cdots\!85\)\(q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(5q \) \(\mathstrut +\mathstrut 5554901256q^{2} \) \(\mathstrut +\mathstrut 3443360269119372q^{3} \) \(\mathstrut +\mathstrut \)\(35\!\cdots\!40\)\(q^{4} \) \(\mathstrut +\mathstrut \)\(33\!\cdots\!50\)\(q^{5} \) \(\mathstrut +\mathstrut \)\(14\!\cdots\!60\)\(q^{6} \) \(\mathstrut +\mathstrut \)\(33\!\cdots\!56\)\(q^{7} \) \(\mathstrut +\mathstrut \)\(32\!\cdots\!80\)\(q^{8} \) \(\mathstrut +\mathstrut \)\(27\!\cdots\!85\)\(q^{9} \) \(\mathstrut -\mathstrut \)\(52\!\cdots\!00\)\(q^{10} \) \(\mathstrut +\mathstrut \)\(20\!\cdots\!60\)\(q^{11} \) \(\mathstrut -\mathstrut \)\(20\!\cdots\!24\)\(q^{12} \) \(\mathstrut +\mathstrut \)\(17\!\cdots\!02\)\(q^{13} \) \(\mathstrut +\mathstrut \)\(29\!\cdots\!80\)\(q^{14} \) \(\mathstrut -\mathstrut \)\(93\!\cdots\!00\)\(q^{15} \) \(\mathstrut +\mathstrut \)\(18\!\cdots\!80\)\(q^{16} \) \(\mathstrut +\mathstrut \)\(75\!\cdots\!06\)\(q^{17} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!68\)\(q^{18} \) \(\mathstrut +\mathstrut \)\(39\!\cdots\!00\)\(q^{19} \) \(\mathstrut -\mathstrut \)\(15\!\cdots\!00\)\(q^{20} \) \(\mathstrut -\mathstrut \)\(17\!\cdots\!40\)\(q^{21} \) \(\mathstrut +\mathstrut \)\(18\!\cdots\!32\)\(q^{22} \) \(\mathstrut -\mathstrut \)\(41\!\cdots\!68\)\(q^{23} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!00\)\(q^{24} \) \(\mathstrut +\mathstrut \)\(17\!\cdots\!75\)\(q^{25} \) \(\mathstrut +\mathstrut \)\(21\!\cdots\!60\)\(q^{26} \) \(\mathstrut +\mathstrut \)\(18\!\cdots\!20\)\(q^{27} \) \(\mathstrut +\mathstrut \)\(14\!\cdots\!48\)\(q^{28} \) \(\mathstrut +\mathstrut \)\(18\!\cdots\!50\)\(q^{29} \) \(\mathstrut +\mathstrut \)\(18\!\cdots\!00\)\(q^{30} \) \(\mathstrut +\mathstrut \)\(36\!\cdots\!60\)\(q^{31} \) \(\mathstrut +\mathstrut \)\(15\!\cdots\!96\)\(q^{32} \) \(\mathstrut +\mathstrut \)\(24\!\cdots\!84\)\(q^{33} \) \(\mathstrut +\mathstrut \)\(78\!\cdots\!80\)\(q^{34} \) \(\mathstrut +\mathstrut \)\(45\!\cdots\!00\)\(q^{35} \) \(\mathstrut -\mathstrut \)\(38\!\cdots\!20\)\(q^{36} \) \(\mathstrut -\mathstrut \)\(56\!\cdots\!94\)\(q^{37} \) \(\mathstrut -\mathstrut \)\(31\!\cdots\!80\)\(q^{38} \) \(\mathstrut -\mathstrut \)\(71\!\cdots\!80\)\(q^{39} \) \(\mathstrut -\mathstrut \)\(18\!\cdots\!00\)\(q^{40} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!10\)\(q^{41} \) \(\mathstrut +\mathstrut \)\(28\!\cdots\!92\)\(q^{42} \) \(\mathstrut +\mathstrut \)\(65\!\cdots\!92\)\(q^{43} \) \(\mathstrut +\mathstrut \)\(49\!\cdots\!80\)\(q^{44} \) \(\mathstrut +\mathstrut \)\(99\!\cdots\!50\)\(q^{45} \) \(\mathstrut +\mathstrut \)\(79\!\cdots\!60\)\(q^{46} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!44\)\(q^{47} \) \(\mathstrut -\mathstrut \)\(19\!\cdots\!68\)\(q^{48} \) \(\mathstrut -\mathstrut \)\(77\!\cdots\!35\)\(q^{49} \) \(\mathstrut -\mathstrut \)\(39\!\cdots\!00\)\(q^{50} \) \(\mathstrut -\mathstrut \)\(33\!\cdots\!40\)\(q^{51} \) \(\mathstrut +\mathstrut \)\(67\!\cdots\!16\)\(q^{52} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!22\)\(q^{53} \) \(\mathstrut +\mathstrut \)\(19\!\cdots\!00\)\(q^{54} \) \(\mathstrut +\mathstrut \)\(66\!\cdots\!00\)\(q^{55} \) \(\mathstrut +\mathstrut \)\(16\!\cdots\!00\)\(q^{56} \) \(\mathstrut -\mathstrut \)\(92\!\cdots\!60\)\(q^{57} \) \(\mathstrut -\mathstrut \)\(61\!\cdots\!20\)\(q^{58} \) \(\mathstrut -\mathstrut \)\(30\!\cdots\!00\)\(q^{59} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!00\)\(q^{60} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!90\)\(q^{61} \) \(\mathstrut -\mathstrut \)\(21\!\cdots\!28\)\(q^{62} \) \(\mathstrut -\mathstrut \)\(20\!\cdots\!68\)\(q^{63} \) \(\mathstrut +\mathstrut \)\(63\!\cdots\!40\)\(q^{64} \) \(\mathstrut +\mathstrut \)\(17\!\cdots\!00\)\(q^{65} \) \(\mathstrut +\mathstrut \)\(27\!\cdots\!20\)\(q^{66} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!44\)\(q^{67} \) \(\mathstrut +\mathstrut \)\(81\!\cdots\!48\)\(q^{68} \) \(\mathstrut +\mathstrut \)\(35\!\cdots\!20\)\(q^{69} \) \(\mathstrut -\mathstrut \)\(26\!\cdots\!00\)\(q^{70} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!40\)\(q^{71} \) \(\mathstrut -\mathstrut \)\(74\!\cdots\!40\)\(q^{72} \) \(\mathstrut -\mathstrut \)\(30\!\cdots\!18\)\(q^{73} \) \(\mathstrut +\mathstrut \)\(92\!\cdots\!80\)\(q^{74} \) \(\mathstrut +\mathstrut \)\(21\!\cdots\!00\)\(q^{75} \) \(\mathstrut +\mathstrut \)\(20\!\cdots\!00\)\(q^{76} \) \(\mathstrut +\mathstrut \)\(50\!\cdots\!32\)\(q^{77} \) \(\mathstrut +\mathstrut \)\(61\!\cdots\!64\)\(q^{78} \) \(\mathstrut +\mathstrut \)\(29\!\cdots\!00\)\(q^{79} \) \(\mathstrut -\mathstrut \)\(15\!\cdots\!00\)\(q^{80} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!95\)\(q^{81} \) \(\mathstrut -\mathstrut \)\(17\!\cdots\!08\)\(q^{82} \) \(\mathstrut -\mathstrut \)\(47\!\cdots\!88\)\(q^{83} \) \(\mathstrut -\mathstrut \)\(52\!\cdots\!20\)\(q^{84} \) \(\mathstrut -\mathstrut \)\(39\!\cdots\!00\)\(q^{85} \) \(\mathstrut +\mathstrut \)\(49\!\cdots\!60\)\(q^{86} \) \(\mathstrut +\mathstrut \)\(35\!\cdots\!60\)\(q^{87} \) \(\mathstrut +\mathstrut \)\(43\!\cdots\!60\)\(q^{88} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!50\)\(q^{89} \) \(\mathstrut +\mathstrut \)\(47\!\cdots\!00\)\(q^{90} \) \(\mathstrut +\mathstrut \)\(23\!\cdots\!60\)\(q^{91} \) \(\mathstrut -\mathstrut \)\(98\!\cdots\!44\)\(q^{92} \) \(\mathstrut -\mathstrut \)\(18\!\cdots\!36\)\(q^{93} \) \(\mathstrut -\mathstrut \)\(49\!\cdots\!20\)\(q^{94} \) \(\mathstrut -\mathstrut \)\(20\!\cdots\!00\)\(q^{95} \) \(\mathstrut -\mathstrut \)\(22\!\cdots\!40\)\(q^{96} \) \(\mathstrut +\mathstrut \)\(28\!\cdots\!06\)\(q^{97} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!08\)\(q^{98} \) \(\mathstrut +\mathstrut \)\(16\!\cdots\!20\)\(q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{5}\mathstrut -\mathstrut \) \(x^{4}\mathstrut -\mathstrut \) \(939384011925257456\) \(x^{3}\mathstrut +\mathstrut \) \(31046449413968483513911200\) \(x^{2}\mathstrut +\mathstrut \) \(156793504704482691874379743265203200\) \(x\mathstrut +\mathstrut \) \(20916736226052669578405116700517591609696000\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 24 \nu - 5 \)
\(\beta_{2}\)\(=\)\((\)\(-\)\(1463\) \(\nu^{4}\mathstrut +\mathstrut \) \(61316688179\) \(\nu^{3}\mathstrut +\mathstrut \) \(1134677465176013610916\) \(\nu^{2}\mathstrut -\mathstrut \) \(175871625402178878144147591024\) \(\nu\mathstrut -\mathstrut \) \(92322819705534579210406721528027067456\)\()/\)\(37\!\cdots\!28\)
\(\beta_{3}\)\(=\)\((\)\(-\)\(1461537\) \(\nu^{4}\mathstrut +\mathstrut \) \(61255371490821\) \(\nu^{3}\mathstrut +\mathstrut \) \(1493224099349187733125372\) \(\nu^{2}\mathstrut -\mathstrut \) \(157864696669790310577560760925712\) \(\nu\mathstrut -\mathstrut \) \(227382046305944113582206155994463766078912\)\()/\)\(62\!\cdots\!88\)
\(\beta_{4}\)\(=\)\((\)\(4738030953\) \(\nu^{4}\mathstrut -\mathstrut \) \(6783896933686506093\) \(\nu^{3}\mathstrut -\mathstrut \) \(6265292406150844870725632988\) \(\nu^{2}\mathstrut +\mathstrut \) \(4428965898625782506113044897501160080\) \(\nu\mathstrut +\mathstrut \) \(1149735030546567532337013572577831183558493120\)\()/\)\(15\!\cdots\!20\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1}\mathstrut +\mathstrut \) \(5\)\()/24\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3}\mathstrut -\mathstrut \) \(5994\) \(\beta_{2}\mathstrut -\mathstrut \) \(1189790397\) \(\beta_{1}\mathstrut +\mathstrut \) \(216434076347341357501\)\()/576\)
\(\nu^{3}\)\(=\)\((\)\(-\)\(409640\) \(\beta_{4}\mathstrut -\mathstrut \) \(1180151429\) \(\beta_{3}\mathstrut +\mathstrut \) \(3889870865010\) \(\beta_{2}\mathstrut +\mathstrut \) \(43600466505262789881\) \(\beta_{1}\mathstrut -\mathstrut \) \(32188957769729085286587521881\)\()/1728\)
\(\nu^{4}\)\(=\)\((\)\(-\)\(17168672690120\) \(\beta_{4}\mathstrut +\mathstrut \) \(2277286000240633339\) \(\beta_{3}\mathstrut -\mathstrut \) \(18208831024879688328462\) \(\beta_{2}\mathstrut -\mathstrut \) \(9596312969973284759959813095\) \(\beta_{1}\mathstrut +\mathstrut \) \(393192791921864844386864518484726790407\)\()/1728\)

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
7.63188e8
6.06038e8
−1.64845e8
−3.05046e8
−8.99335e8
−1.72055e10 −1.09955e16 1.48456e20 4.93311e23 1.89183e26 1.79351e28 −1.51842e28 2.81916e31 −8.48767e33
1.2 −1.34339e10 7.87958e15 3.28965e19 −3.01383e23 −1.05854e26 −4.20742e26 1.54057e30 −3.06216e31 4.04876e33
1.3 5.06726e9 −7.82458e15 −1.21897e20 −8.33551e22 −3.96491e25 −1.15070e28 −1.36548e30 −3.14854e31 −4.22381e32
1.4 8.43209e9 1.56682e16 −7.64738e19 3.83370e23 1.32115e26 −1.75533e27 −1.88919e30 1.52782e32 3.23261e33
1.5 2.26950e10 −1.28429e15 3.67490e20 −1.61198e23 −2.91471e25 2.93806e28 4.99099e30 −9.10601e31 −3.65838e33
\(n\): 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 \(S_{68}^{\mathrm{new}}(\Gamma_0(1))\).