Properties

Label 1.66.a.a
Level 1
Weight 66
Character orbit 1.a
Self dual Yes
Analytic conductor 26.757
Analytic rank 1
Dimension 5
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 1 \)
Weight: \( k \) = \( 66 \)
Character orbit: \([\chi]\) = 1.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(26.7572356472\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{40}\cdot 3^{12}\cdot 5^{4}\cdot 7^{2}\cdot 11\cdot 13^{2} \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(f(q)\) \(=\) \( q\) \(+(-791941930 - \beta_{1}) q^{2}\) \(+(-446261486940055 + 117564 \beta_{1} + \beta_{2}) q^{3}\) \(+(23382438400147059430 + 1677780488 \beta_{1} + 2416 \beta_{2} + \beta_{3}) q^{4}\) \(+(\)\(53\!\cdots\!67\)\( + 84352482422 \beta_{1} + 1436250 \beta_{2} + 455 \beta_{3} - \beta_{4}) q^{5}\) \(+(-\)\(66\!\cdots\!08\)\( - 292080021098340 \beta_{1} - 1253959584 \beta_{2} - 425544 \beta_{3} + 624 \beta_{4}) q^{6}\) \(+(-\)\(13\!\cdots\!34\)\( - 119693477507882512 \beta_{1} - 133339578326 \beta_{2} - 48407060 \beta_{3} - 50868 \beta_{4}) q^{7}\) \(+(-\)\(89\!\cdots\!12\)\( - 8333849689741716096 \beta_{1} - 28497504421632 \beta_{2} - 4577904080 \beta_{3} + 1931776 \beta_{4}) q^{8}\) \(+(\)\(63\!\cdots\!03\)\( + 92186899734703151700 \beta_{1} - 1809402202914228 \beta_{2} + 235975728402 \beta_{3} - 39193182 \beta_{4}) q^{9}\) \(+O(q^{10})\) \( q\) \(+(-791941930 - \beta_{1}) q^{2}\) \(+(-446261486940055 + 117564 \beta_{1} + \beta_{2}) q^{3}\) \(+(23382438400147059430 + 1677780488 \beta_{1} + 2416 \beta_{2} + \beta_{3}) q^{4}\) \(+(\)\(53\!\cdots\!67\)\( + 84352482422 \beta_{1} + 1436250 \beta_{2} + 455 \beta_{3} - \beta_{4}) q^{5}\) \(+(-\)\(66\!\cdots\!08\)\( - 292080021098340 \beta_{1} - 1253959584 \beta_{2} - 425544 \beta_{3} + 624 \beta_{4}) q^{6}\) \(+(-\)\(13\!\cdots\!34\)\( - 119693477507882512 \beta_{1} - 133339578326 \beta_{2} - 48407060 \beta_{3} - 50868 \beta_{4}) q^{7}\) \(+(-\)\(89\!\cdots\!12\)\( - 8333849689741716096 \beta_{1} - 28497504421632 \beta_{2} - 4577904080 \beta_{3} + 1931776 \beta_{4}) q^{8}\) \(+(\)\(63\!\cdots\!03\)\( + 92186899734703151700 \beta_{1} - 1809402202914228 \beta_{2} + 235975728402 \beta_{3} - 39193182 \beta_{4}) q^{9}\) \(+(-\)\(92\!\cdots\!16\)\( - \)\(14\!\cdots\!06\)\( \beta_{1} - 107220246340720000 \beta_{2} - 2415018319840 \beta_{3} + 245187648 \beta_{4}) q^{10}\) \(+(-\)\(10\!\cdots\!17\)\( - \)\(61\!\cdots\!04\)\( \beta_{1} - 604982467023176685 \beta_{2} - 60559729888520 \beta_{3} + 10840785592 \beta_{4}) q^{11}\) \(+(\)\(39\!\cdots\!72\)\( + \)\(11\!\cdots\!48\)\( \beta_{1} + 34067405179750899136 \beta_{2} + 2161939703720580 \beta_{3} - 436302360576 \beta_{4}) q^{12}\) \(+(-\)\(57\!\cdots\!17\)\( + \)\(11\!\cdots\!70\)\( \beta_{1} - 2387059676413490390 \beta_{2} - 30367983471959105 \beta_{3} + 9402040635831 \beta_{4}) q^{13}\) \(+(\)\(82\!\cdots\!48\)\( + \)\(24\!\cdots\!04\)\( \beta_{1} - \)\(33\!\cdots\!12\)\( \beta_{2} + 218973106651069168 \beta_{3} - 146672396540960 \beta_{4}) q^{14}\) \(+(\)\(20\!\cdots\!82\)\( + \)\(29\!\cdots\!12\)\( \beta_{1} + \)\(42\!\cdots\!50\)\( \beta_{2} - 369543698628563820 \beta_{3} + 1802001986692404 \beta_{4}) q^{15}\) \(+(-\)\(29\!\cdots\!76\)\( + \)\(13\!\cdots\!08\)\( \beta_{1} + \)\(24\!\cdots\!92\)\( \beta_{2} - 8160057313016817408 \beta_{3} - 18093857387913216 \beta_{4}) q^{16}\) \(+(\)\(27\!\cdots\!44\)\( - \)\(25\!\cdots\!48\)\( \beta_{1} - \)\(10\!\cdots\!48\)\( \beta_{2} + 74625002521923602210 \beta_{3} + 151276694420287538 \beta_{4}) q^{17}\) \(+(-\)\(10\!\cdots\!06\)\( - \)\(97\!\cdots\!17\)\( \beta_{1} - \)\(74\!\cdots\!44\)\( \beta_{2} - 79262375341337202240 \beta_{3} - 1062048560483820672 \beta_{4}) q^{18}\) \(+(-\)\(10\!\cdots\!91\)\( - \)\(26\!\cdots\!76\)\( \beta_{1} + \)\(51\!\cdots\!49\)\( \beta_{2} - \)\(36\!\cdots\!56\)\( \beta_{3} + 6265373194870209864 \beta_{4}) q^{19}\) \(+(\)\(69\!\cdots\!24\)\( + \)\(13\!\cdots\!84\)\( \beta_{1} + \)\(16\!\cdots\!00\)\( \beta_{2} + \)\(36\!\cdots\!10\)\( \beta_{3} - 30829428908515868672 \beta_{4}) q^{20}\) \(+(-\)\(23\!\cdots\!72\)\( + \)\(68\!\cdots\!56\)\( \beta_{1} - \)\(21\!\cdots\!68\)\( \beta_{2} - \)\(18\!\cdots\!48\)\( \beta_{3} + \)\(12\!\cdots\!60\)\( \beta_{4}) q^{21}\) \(+(\)\(45\!\cdots\!32\)\( + \)\(26\!\cdots\!60\)\( \beta_{1} + \)\(31\!\cdots\!20\)\( \beta_{2} + \)\(38\!\cdots\!80\)\( \beta_{3} - \)\(39\!\cdots\!76\)\( \beta_{4}) q^{22}\) \(+(-\)\(18\!\cdots\!10\)\( - \)\(81\!\cdots\!00\)\( \beta_{1} + \)\(29\!\cdots\!54\)\( \beta_{2} + \)\(10\!\cdots\!00\)\( \beta_{3} + \)\(90\!\cdots\!80\)\( \beta_{4}) q^{23}\) \(+(-\)\(46\!\cdots\!12\)\( - \)\(10\!\cdots\!52\)\( \beta_{1} - \)\(10\!\cdots\!88\)\( \beta_{2} - \)\(11\!\cdots\!88\)\( \beta_{3} - \)\(12\!\cdots\!56\)\( \beta_{4}) q^{24}\) \(+(-\)\(17\!\cdots\!25\)\( + \)\(37\!\cdots\!00\)\( \beta_{1} - \)\(20\!\cdots\!00\)\( \beta_{2} + \)\(37\!\cdots\!00\)\( \beta_{3} + \)\(54\!\cdots\!00\)\( \beta_{4}) q^{25}\) \(+(-\)\(61\!\cdots\!28\)\( + \)\(10\!\cdots\!30\)\( \beta_{1} + \)\(13\!\cdots\!68\)\( \beta_{2} - \)\(35\!\cdots\!12\)\( \beta_{3} - \)\(64\!\cdots\!48\)\( \beta_{4}) q^{26}\) \(+(-\)\(25\!\cdots\!82\)\( + \)\(22\!\cdots\!84\)\( \beta_{1} + \)\(16\!\cdots\!18\)\( \beta_{2} - \)\(10\!\cdots\!80\)\( \beta_{3} + \)\(66\!\cdots\!36\)\( \beta_{4}) q^{27}\) \(+(-\)\(10\!\cdots\!08\)\( - \)\(80\!\cdots\!68\)\( \beta_{1} - \)\(16\!\cdots\!44\)\( \beta_{2} - \)\(26\!\cdots\!20\)\( \beta_{3} - \)\(25\!\cdots\!76\)\( \beta_{4}) q^{28}\) \(+(-\)\(22\!\cdots\!33\)\( - \)\(23\!\cdots\!58\)\( \beta_{1} + \)\(40\!\cdots\!06\)\( \beta_{2} + \)\(59\!\cdots\!51\)\( \beta_{3} + \)\(11\!\cdots\!03\)\( \beta_{4}) q^{29}\) \(+(-\)\(17\!\cdots\!36\)\( - \)\(42\!\cdots\!76\)\( \beta_{1} + \)\(10\!\cdots\!00\)\( \beta_{2} - \)\(28\!\cdots\!40\)\( \beta_{3} + \)\(39\!\cdots\!08\)\( \beta_{4}) q^{30}\) \(+(-\)\(94\!\cdots\!20\)\( + \)\(32\!\cdots\!48\)\( \beta_{1} + \)\(67\!\cdots\!20\)\( \beta_{2} + \)\(66\!\cdots\!40\)\( \beta_{3} - \)\(22\!\cdots\!04\)\( \beta_{4}) q^{31}\) \(+(-\)\(47\!\cdots\!96\)\( + \)\(47\!\cdots\!48\)\( \beta_{1} - \)\(10\!\cdots\!60\)\( \beta_{2} - \)\(44\!\cdots\!40\)\( \beta_{3} + \)\(62\!\cdots\!48\)\( \beta_{4}) q^{32}\) \(+(-\)\(94\!\cdots\!74\)\( - \)\(97\!\cdots\!76\)\( \beta_{1} - \)\(10\!\cdots\!88\)\( \beta_{2} - \)\(18\!\cdots\!10\)\( \beta_{3} - \)\(31\!\cdots\!58\)\( \beta_{4}) q^{33}\) \(+(\)\(13\!\cdots\!72\)\( - \)\(32\!\cdots\!82\)\( \beta_{1} + \)\(14\!\cdots\!72\)\( \beta_{2} + \)\(54\!\cdots\!72\)\( \beta_{3} - \)\(48\!\cdots\!76\)\( \beta_{4}) q^{34}\) \(+(\)\(86\!\cdots\!64\)\( - \)\(85\!\cdots\!76\)\( \beta_{1} - \)\(22\!\cdots\!00\)\( \beta_{2} - \)\(71\!\cdots\!40\)\( \beta_{3} + \)\(21\!\cdots\!08\)\( \beta_{4}) q^{35}\) \(+(\)\(35\!\cdots\!74\)\( + \)\(21\!\cdots\!08\)\( \beta_{1} - \)\(22\!\cdots\!48\)\( \beta_{2} + \)\(25\!\cdots\!77\)\( \beta_{3} - \)\(41\!\cdots\!76\)\( \beta_{4}) q^{36}\) \(+(\)\(50\!\cdots\!91\)\( + \)\(45\!\cdots\!66\)\( \beta_{1} - \)\(12\!\cdots\!02\)\( \beta_{2} - \)\(12\!\cdots\!85\)\( \beta_{3} - \)\(20\!\cdots\!73\)\( \beta_{4}) q^{37}\) \(+(\)\(16\!\cdots\!56\)\( + \)\(16\!\cdots\!40\)\( \beta_{1} + \)\(69\!\cdots\!52\)\( \beta_{2} + \)\(23\!\cdots\!40\)\( \beta_{3} + \)\(35\!\cdots\!92\)\( \beta_{4}) q^{38}\) \(+(\)\(99\!\cdots\!14\)\( - \)\(52\!\cdots\!92\)\( \beta_{1} - \)\(87\!\cdots\!22\)\( \beta_{2} + \)\(26\!\cdots\!68\)\( \beta_{3} - \)\(99\!\cdots\!32\)\( \beta_{4}) q^{39}\) \(+(-\)\(81\!\cdots\!40\)\( - \)\(10\!\cdots\!40\)\( \beta_{1} - \)\(27\!\cdots\!00\)\( \beta_{2} - \)\(20\!\cdots\!00\)\( \beta_{3} + \)\(82\!\cdots\!20\)\( \beta_{4}) q^{40}\) \(+(-\)\(64\!\cdots\!30\)\( + \)\(56\!\cdots\!88\)\( \beta_{1} - \)\(29\!\cdots\!80\)\( \beta_{2} + \)\(23\!\cdots\!40\)\( \beta_{3} + \)\(32\!\cdots\!76\)\( \beta_{4}) q^{41}\) \(+(-\)\(39\!\cdots\!24\)\( + \)\(64\!\cdots\!16\)\( \beta_{1} + \)\(16\!\cdots\!36\)\( \beta_{2} + \)\(48\!\cdots\!40\)\( \beta_{3} - \)\(13\!\cdots\!28\)\( \beta_{4}) q^{42}\) \(+(-\)\(80\!\cdots\!85\)\( + \)\(44\!\cdots\!76\)\( \beta_{1} - \)\(11\!\cdots\!13\)\( \beta_{2} - \)\(10\!\cdots\!00\)\( \beta_{3} + \)\(18\!\cdots\!20\)\( \beta_{4}) q^{43}\) \(+(-\)\(12\!\cdots\!28\)\( - \)\(13\!\cdots\!32\)\( \beta_{1} - \)\(33\!\cdots\!88\)\( \beta_{2} - \)\(25\!\cdots\!88\)\( \beta_{3} + \)\(13\!\cdots\!84\)\( \beta_{4}) q^{44}\) \(+(\)\(20\!\cdots\!91\)\( - \)\(40\!\cdots\!94\)\( \beta_{1} + \)\(18\!\cdots\!50\)\( \beta_{2} + \)\(10\!\cdots\!15\)\( \beta_{3} - \)\(11\!\cdots\!73\)\( \beta_{4}) q^{45}\) \(+(\)\(50\!\cdots\!00\)\( - \)\(12\!\cdots\!32\)\( \beta_{1} + \)\(54\!\cdots\!00\)\( \beta_{2} - \)\(24\!\cdots\!80\)\( \beta_{3} + \)\(19\!\cdots\!56\)\( \beta_{4}) q^{46}\) \(+(\)\(31\!\cdots\!60\)\( + \)\(38\!\cdots\!16\)\( \beta_{1} + \)\(27\!\cdots\!16\)\( \beta_{2} - \)\(42\!\cdots\!00\)\( \beta_{3} - \)\(43\!\cdots\!40\)\( \beta_{4}) q^{47}\) \(+(\)\(49\!\cdots\!16\)\( + \)\(41\!\cdots\!56\)\( \beta_{1} - \)\(78\!\cdots\!76\)\( \beta_{2} + \)\(76\!\cdots\!40\)\( \beta_{3} - \)\(54\!\cdots\!48\)\( \beta_{4}) q^{48}\) \(+(\)\(33\!\cdots\!29\)\( + \)\(11\!\cdots\!52\)\( \beta_{1} + \)\(13\!\cdots\!40\)\( \beta_{2} - \)\(10\!\cdots\!80\)\( \beta_{3} + \)\(12\!\cdots\!44\)\( \beta_{4}) q^{49}\) \(+(-\)\(22\!\cdots\!50\)\( - \)\(58\!\cdots\!75\)\( \beta_{1} - \)\(66\!\cdots\!00\)\( \beta_{2} - \)\(53\!\cdots\!00\)\( \beta_{3} - \)\(11\!\cdots\!00\)\( \beta_{4}) q^{50}\) \(+(-\)\(19\!\cdots\!94\)\( - \)\(30\!\cdots\!56\)\( \beta_{1} + \)\(58\!\cdots\!34\)\( \beta_{2} - \)\(54\!\cdots\!96\)\( \beta_{3} - \)\(72\!\cdots\!56\)\( \beta_{4}) q^{51}\) \(+(-\)\(36\!\cdots\!60\)\( + \)\(94\!\cdots\!12\)\( \beta_{1} - \)\(35\!\cdots\!56\)\( \beta_{2} - \)\(26\!\cdots\!50\)\( \beta_{3} + \)\(44\!\cdots\!60\)\( \beta_{4}) q^{52}\) \(+(-\)\(19\!\cdots\!57\)\( + \)\(63\!\cdots\!82\)\( \beta_{1} - \)\(12\!\cdots\!74\)\( \beta_{2} + \)\(18\!\cdots\!95\)\( \beta_{3} - \)\(94\!\cdots\!69\)\( \beta_{4}) q^{53}\) \(+(-\)\(11\!\cdots\!56\)\( + \)\(24\!\cdots\!84\)\( \beta_{1} + \)\(17\!\cdots\!04\)\( \beta_{2} - \)\(24\!\cdots\!76\)\( \beta_{3} + \)\(10\!\cdots\!04\)\( \beta_{4}) q^{54}\) \(+(-\)\(76\!\cdots\!66\)\( - \)\(19\!\cdots\!56\)\( \beta_{1} + \)\(46\!\cdots\!50\)\( \beta_{2} - \)\(20\!\cdots\!40\)\( \beta_{3} + \)\(86\!\cdots\!48\)\( \beta_{4}) q^{55}\) \(+(\)\(25\!\cdots\!88\)\( + \)\(34\!\cdots\!88\)\( \beta_{1} + \)\(14\!\cdots\!64\)\( \beta_{2} + \)\(54\!\cdots\!44\)\( \beta_{3} - \)\(53\!\cdots\!88\)\( \beta_{4}) q^{56}\) \(+(\)\(66\!\cdots\!50\)\( - \)\(19\!\cdots\!68\)\( \beta_{1} - \)\(20\!\cdots\!44\)\( \beta_{2} + \)\(18\!\cdots\!50\)\( \beta_{3} + \)\(63\!\cdots\!30\)\( \beta_{4}) q^{57}\) \(+(\)\(16\!\cdots\!56\)\( + \)\(13\!\cdots\!82\)\( \beta_{1} - \)\(66\!\cdots\!12\)\( \beta_{2} + \)\(99\!\cdots\!40\)\( \beta_{3} + \)\(51\!\cdots\!72\)\( \beta_{4}) q^{58}\) \(+(-\)\(51\!\cdots\!29\)\( - \)\(28\!\cdots\!24\)\( \beta_{1} - \)\(23\!\cdots\!53\)\( \beta_{2} - \)\(78\!\cdots\!08\)\( \beta_{3} - \)\(16\!\cdots\!40\)\( \beta_{4}) q^{59}\) \(+(\)\(31\!\cdots\!04\)\( + \)\(11\!\cdots\!64\)\( \beta_{1} + \)\(85\!\cdots\!00\)\( \beta_{2} + \)\(88\!\cdots\!60\)\( \beta_{3} - \)\(97\!\cdots\!12\)\( \beta_{4}) q^{60}\) \(+(\)\(78\!\cdots\!87\)\( - \)\(76\!\cdots\!90\)\( \beta_{1} + \)\(15\!\cdots\!50\)\( \beta_{2} + \)\(17\!\cdots\!75\)\( \beta_{3} + \)\(48\!\cdots\!95\)\( \beta_{4}) q^{61}\) \(+(-\)\(18\!\cdots\!64\)\( - \)\(64\!\cdots\!96\)\( \beta_{1} - \)\(45\!\cdots\!40\)\( \beta_{2} - \)\(50\!\cdots\!60\)\( \beta_{3} + \)\(50\!\cdots\!12\)\( \beta_{4}) q^{62}\) \(+(-\)\(13\!\cdots\!54\)\( - \)\(37\!\cdots\!76\)\( \beta_{1} + \)\(67\!\cdots\!78\)\( \beta_{2} + \)\(34\!\cdots\!40\)\( \beta_{3} - \)\(26\!\cdots\!68\)\( \beta_{4}) q^{63}\) \(+(-\)\(13\!\cdots\!48\)\( + \)\(72\!\cdots\!60\)\( \beta_{1} - \)\(21\!\cdots\!08\)\( \beta_{2} + \)\(30\!\cdots\!72\)\( \beta_{3} + \)\(33\!\cdots\!68\)\( \beta_{4}) q^{64}\) \(+(-\)\(40\!\cdots\!88\)\( + \)\(55\!\cdots\!92\)\( \beta_{1} + \)\(13\!\cdots\!00\)\( \beta_{2} - \)\(98\!\cdots\!20\)\( \beta_{3} + \)\(89\!\cdots\!64\)\( \beta_{4}) q^{65}\) \(+(\)\(65\!\cdots\!16\)\( + \)\(14\!\cdots\!32\)\( \beta_{1} + \)\(44\!\cdots\!12\)\( \beta_{2} + \)\(16\!\cdots\!72\)\( \beta_{3} - \)\(73\!\cdots\!28\)\( \beta_{4}) q^{66}\) \(+(\)\(95\!\cdots\!49\)\( + \)\(65\!\cdots\!12\)\( \beta_{1} - \)\(22\!\cdots\!83\)\( \beta_{2} - \)\(28\!\cdots\!40\)\( \beta_{3} - \)\(22\!\cdots\!52\)\( \beta_{4}) q^{67}\) \(+(\)\(84\!\cdots\!52\)\( - \)\(20\!\cdots\!16\)\( \beta_{1} - \)\(26\!\cdots\!08\)\( \beta_{2} - \)\(55\!\cdots\!70\)\( \beta_{3} + \)\(32\!\cdots\!24\)\( \beta_{4}) q^{68}\) \(+(\)\(41\!\cdots\!56\)\( - \)\(14\!\cdots\!00\)\( \beta_{1} - \)\(46\!\cdots\!76\)\( \beta_{2} + \)\(41\!\cdots\!84\)\( \beta_{3} + \)\(52\!\cdots\!56\)\( \beta_{4}) q^{69}\) \(+(\)\(44\!\cdots\!28\)\( - \)\(51\!\cdots\!52\)\( \beta_{1} + \)\(26\!\cdots\!00\)\( \beta_{2} + \)\(18\!\cdots\!20\)\( \beta_{3} - \)\(12\!\cdots\!84\)\( \beta_{4}) q^{70}\) \(+(-\)\(69\!\cdots\!18\)\( + \)\(25\!\cdots\!80\)\( \beta_{1} - \)\(75\!\cdots\!50\)\( \beta_{2} - \)\(27\!\cdots\!00\)\( \beta_{3} - \)\(79\!\cdots\!40\)\( \beta_{4}) q^{71}\) \(+(-\)\(12\!\cdots\!68\)\( - \)\(61\!\cdots\!00\)\( \beta_{1} - \)\(22\!\cdots\!96\)\( \beta_{2} - \)\(27\!\cdots\!20\)\( \beta_{3} + \)\(35\!\cdots\!24\)\( \beta_{4}) q^{72}\) \(+(\)\(14\!\cdots\!16\)\( + \)\(54\!\cdots\!76\)\( \beta_{1} - \)\(23\!\cdots\!16\)\( \beta_{2} + \)\(61\!\cdots\!90\)\( \beta_{3} + \)\(59\!\cdots\!02\)\( \beta_{4}) q^{73}\) \(+(-\)\(31\!\cdots\!08\)\( - \)\(23\!\cdots\!02\)\( \beta_{1} + \)\(93\!\cdots\!60\)\( \beta_{2} + \)\(17\!\cdots\!80\)\( \beta_{3} - \)\(83\!\cdots\!44\)\( \beta_{4}) q^{74}\) \(+(-\)\(29\!\cdots\!25\)\( + \)\(26\!\cdots\!00\)\( \beta_{1} + \)\(79\!\cdots\!75\)\( \beta_{2} + \)\(15\!\cdots\!00\)\( \beta_{3} - \)\(11\!\cdots\!00\)\( \beta_{4}) q^{75}\) \(+(-\)\(72\!\cdots\!64\)\( - \)\(17\!\cdots\!64\)\( \beta_{1} - \)\(77\!\cdots\!12\)\( \beta_{2} - \)\(26\!\cdots\!52\)\( \beta_{3} + \)\(24\!\cdots\!84\)\( \beta_{4}) q^{76}\) \(+(\)\(51\!\cdots\!44\)\( + \)\(11\!\cdots\!08\)\( \beta_{1} + \)\(24\!\cdots\!68\)\( \beta_{2} + \)\(86\!\cdots\!60\)\( \beta_{3} - \)\(52\!\cdots\!72\)\( \beta_{4}) q^{77}\) \(+(\)\(30\!\cdots\!12\)\( - \)\(47\!\cdots\!96\)\( \beta_{1} - \)\(79\!\cdots\!56\)\( \beta_{2} + \)\(64\!\cdots\!80\)\( \beta_{3} - \)\(62\!\cdots\!56\)\( \beta_{4}) q^{78}\) \(+(\)\(11\!\cdots\!80\)\( + \)\(21\!\cdots\!00\)\( \beta_{1} + \)\(11\!\cdots\!36\)\( \beta_{2} - \)\(67\!\cdots\!24\)\( \beta_{3} + \)\(49\!\cdots\!84\)\( \beta_{4}) q^{79}\) \(+(\)\(43\!\cdots\!52\)\( + \)\(81\!\cdots\!32\)\( \beta_{1} + \)\(91\!\cdots\!00\)\( \beta_{2} + \)\(30\!\cdots\!80\)\( \beta_{3} + \)\(94\!\cdots\!44\)\( \beta_{4}) q^{80}\) \(+(-\)\(13\!\cdots\!61\)\( - \)\(58\!\cdots\!72\)\( \beta_{1} - \)\(96\!\cdots\!84\)\( \beta_{2} - \)\(12\!\cdots\!74\)\( \beta_{3} - \)\(10\!\cdots\!70\)\( \beta_{4}) q^{81}\) \(+(-\)\(28\!\cdots\!84\)\( + \)\(34\!\cdots\!34\)\( \beta_{1} + \)\(27\!\cdots\!60\)\( \beta_{2} + \)\(15\!\cdots\!40\)\( \beta_{3} - \)\(14\!\cdots\!28\)\( \beta_{4}) q^{82}\) \(+(\)\(41\!\cdots\!65\)\( - \)\(20\!\cdots\!92\)\( \beta_{1} - \)\(61\!\cdots\!07\)\( \beta_{2} + \)\(57\!\cdots\!00\)\( \beta_{3} + \)\(14\!\cdots\!00\)\( \beta_{4}) q^{83}\) \(+(-\)\(27\!\cdots\!84\)\( - \)\(11\!\cdots\!48\)\( \beta_{1} - \)\(88\!\cdots\!84\)\( \beta_{2} - \)\(70\!\cdots\!64\)\( \beta_{3} + \)\(46\!\cdots\!88\)\( \beta_{4}) q^{84}\) \(+(-\)\(33\!\cdots\!26\)\( - \)\(25\!\cdots\!16\)\( \beta_{1} + \)\(12\!\cdots\!00\)\( \beta_{2} - \)\(80\!\cdots\!90\)\( \beta_{3} - \)\(64\!\cdots\!22\)\( \beta_{4}) q^{85}\) \(+(-\)\(20\!\cdots\!52\)\( + \)\(10\!\cdots\!16\)\( \beta_{1} + \)\(20\!\cdots\!24\)\( \beta_{2} + \)\(31\!\cdots\!24\)\( \beta_{3} - \)\(62\!\cdots\!72\)\( \beta_{4}) q^{86}\) \(+(-\)\(26\!\cdots\!34\)\( + \)\(13\!\cdots\!64\)\( \beta_{1} - \)\(16\!\cdots\!86\)\( \beta_{2} - \)\(33\!\cdots\!60\)\( \beta_{3} + \)\(18\!\cdots\!72\)\( \beta_{4}) q^{87}\) \(+(\)\(75\!\cdots\!16\)\( + \)\(10\!\cdots\!48\)\( \beta_{1} + \)\(13\!\cdots\!36\)\( \beta_{2} + \)\(16\!\cdots\!40\)\( \beta_{3} - \)\(62\!\cdots\!68\)\( \beta_{4}) q^{88}\) \(+(\)\(29\!\cdots\!16\)\( - \)\(11\!\cdots\!84\)\( \beta_{1} + \)\(10\!\cdots\!08\)\( \beta_{2} - \)\(14\!\cdots\!82\)\( \beta_{3} - \)\(13\!\cdots\!26\)\( \beta_{4}) q^{89}\) \(+(\)\(22\!\cdots\!32\)\( - \)\(37\!\cdots\!38\)\( \beta_{1} - \)\(12\!\cdots\!00\)\( \beta_{2} + \)\(23\!\cdots\!80\)\( \beta_{3} + \)\(63\!\cdots\!04\)\( \beta_{4}) q^{90}\) \(+(\)\(52\!\cdots\!20\)\( - \)\(22\!\cdots\!32\)\( \beta_{1} + \)\(97\!\cdots\!64\)\( \beta_{2} - \)\(31\!\cdots\!56\)\( \beta_{3} + \)\(16\!\cdots\!72\)\( \beta_{4}) q^{91}\) \(+(\)\(10\!\cdots\!56\)\( - \)\(17\!\cdots\!16\)\( \beta_{1} + \)\(82\!\cdots\!72\)\( \beta_{2} - \)\(19\!\cdots\!60\)\( \beta_{3} + \)\(16\!\cdots\!52\)\( \beta_{4}) q^{92}\) \(+(\)\(37\!\cdots\!08\)\( + \)\(10\!\cdots\!76\)\( \beta_{1} - \)\(11\!\cdots\!68\)\( \beta_{2} + \)\(19\!\cdots\!20\)\( \beta_{3} - \)\(12\!\cdots\!04\)\( \beta_{4}) q^{93}\) \(+(-\)\(25\!\cdots\!12\)\( + \)\(27\!\cdots\!64\)\( \beta_{1} + \)\(15\!\cdots\!64\)\( \beta_{2} - \)\(35\!\cdots\!16\)\( \beta_{3} + \)\(14\!\cdots\!04\)\( \beta_{4}) q^{94}\) \(+(-\)\(16\!\cdots\!30\)\( + \)\(10\!\cdots\!20\)\( \beta_{1} - \)\(19\!\cdots\!50\)\( \beta_{2} - \)\(36\!\cdots\!00\)\( \beta_{3} + \)\(14\!\cdots\!40\)\( \beta_{4}) q^{95}\) \(+(-\)\(11\!\cdots\!44\)\( - \)\(25\!\cdots\!16\)\( \beta_{1} - \)\(32\!\cdots\!24\)\( \beta_{2} + \)\(30\!\cdots\!76\)\( \beta_{3} - \)\(45\!\cdots\!28\)\( \beta_{4}) q^{96}\) \(+(-\)\(62\!\cdots\!72\)\( + \)\(74\!\cdots\!44\)\( \beta_{1} + \)\(20\!\cdots\!36\)\( \beta_{2} + \)\(54\!\cdots\!70\)\( \beta_{3} + \)\(19\!\cdots\!06\)\( \beta_{4}) q^{97}\) \(+(-\)\(93\!\cdots\!86\)\( - \)\(33\!\cdots\!33\)\( \beta_{1} + \)\(11\!\cdots\!20\)\( \beta_{2} - \)\(15\!\cdots\!40\)\( \beta_{3} + \)\(18\!\cdots\!28\)\( \beta_{4}) q^{98}\) \(+(-\)\(72\!\cdots\!21\)\( - \)\(15\!\cdots\!32\)\( \beta_{1} - \)\(48\!\cdots\!61\)\( \beta_{2} - \)\(19\!\cdots\!56\)\( \beta_{3} + \)\(41\!\cdots\!72\)\( \beta_{4}) q^{99}\) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(5q \) \(\mathstrut -\mathstrut 3959709648q^{2} \) \(\mathstrut -\mathstrut 2231307434935404q^{3} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!60\)\(q^{4} \) \(\mathstrut +\mathstrut \)\(26\!\cdots\!50\)\(q^{5} \) \(\mathstrut -\mathstrut \)\(33\!\cdots\!40\)\(q^{6} \) \(\mathstrut -\mathstrut \)\(69\!\cdots\!08\)\(q^{7} \) \(\mathstrut -\mathstrut \)\(44\!\cdots\!20\)\(q^{8} \) \(\mathstrut +\mathstrut \)\(31\!\cdots\!65\)\(q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(5q \) \(\mathstrut -\mathstrut 3959709648q^{2} \) \(\mathstrut -\mathstrut 2231307434935404q^{3} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!60\)\(q^{4} \) \(\mathstrut +\mathstrut \)\(26\!\cdots\!50\)\(q^{5} \) \(\mathstrut -\mathstrut \)\(33\!\cdots\!40\)\(q^{6} \) \(\mathstrut -\mathstrut \)\(69\!\cdots\!08\)\(q^{7} \) \(\mathstrut -\mathstrut \)\(44\!\cdots\!20\)\(q^{8} \) \(\mathstrut +\mathstrut \)\(31\!\cdots\!65\)\(q^{9} \) \(\mathstrut -\mathstrut \)\(46\!\cdots\!00\)\(q^{10} \) \(\mathstrut -\mathstrut \)\(54\!\cdots\!40\)\(q^{11} \) \(\mathstrut +\mathstrut \)\(19\!\cdots\!12\)\(q^{12} \) \(\mathstrut -\mathstrut \)\(28\!\cdots\!14\)\(q^{13} \) \(\mathstrut +\mathstrut \)\(41\!\cdots\!20\)\(q^{14} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!00\)\(q^{15} \) \(\mathstrut -\mathstrut \)\(14\!\cdots\!20\)\(q^{16} \) \(\mathstrut +\mathstrut \)\(13\!\cdots\!22\)\(q^{17} \) \(\mathstrut -\mathstrut \)\(52\!\cdots\!04\)\(q^{18} \) \(\mathstrut -\mathstrut \)\(53\!\cdots\!00\)\(q^{19} \) \(\mathstrut +\mathstrut \)\(34\!\cdots\!00\)\(q^{20} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!40\)\(q^{21} \) \(\mathstrut +\mathstrut \)\(22\!\cdots\!04\)\(q^{22} \) \(\mathstrut -\mathstrut \)\(94\!\cdots\!24\)\(q^{23} \) \(\mathstrut -\mathstrut \)\(23\!\cdots\!00\)\(q^{24} \) \(\mathstrut -\mathstrut \)\(89\!\cdots\!25\)\(q^{25} \) \(\mathstrut -\mathstrut \)\(30\!\cdots\!40\)\(q^{26} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!20\)\(q^{27} \) \(\mathstrut -\mathstrut \)\(51\!\cdots\!76\)\(q^{28} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!50\)\(q^{29} \) \(\mathstrut -\mathstrut \)\(88\!\cdots\!00\)\(q^{30} \) \(\mathstrut -\mathstrut \)\(47\!\cdots\!40\)\(q^{31} \) \(\mathstrut -\mathstrut \)\(23\!\cdots\!48\)\(q^{32} \) \(\mathstrut -\mathstrut \)\(47\!\cdots\!08\)\(q^{33} \) \(\mathstrut +\mathstrut \)\(65\!\cdots\!20\)\(q^{34} \) \(\mathstrut +\mathstrut \)\(43\!\cdots\!00\)\(q^{35} \) \(\mathstrut +\mathstrut \)\(17\!\cdots\!80\)\(q^{36} \) \(\mathstrut +\mathstrut \)\(25\!\cdots\!82\)\(q^{37} \) \(\mathstrut +\mathstrut \)\(82\!\cdots\!20\)\(q^{38} \) \(\mathstrut +\mathstrut \)\(49\!\cdots\!80\)\(q^{39} \) \(\mathstrut -\mathstrut \)\(40\!\cdots\!00\)\(q^{40} \) \(\mathstrut -\mathstrut \)\(32\!\cdots\!90\)\(q^{41} \) \(\mathstrut -\mathstrut \)\(19\!\cdots\!36\)\(q^{42} \) \(\mathstrut -\mathstrut \)\(40\!\cdots\!44\)\(q^{43} \) \(\mathstrut -\mathstrut \)\(61\!\cdots\!80\)\(q^{44} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!50\)\(q^{45} \) \(\mathstrut +\mathstrut \)\(25\!\cdots\!60\)\(q^{46} \) \(\mathstrut +\mathstrut \)\(15\!\cdots\!12\)\(q^{47} \) \(\mathstrut +\mathstrut \)\(24\!\cdots\!76\)\(q^{48} \) \(\mathstrut +\mathstrut \)\(16\!\cdots\!85\)\(q^{49} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!00\)\(q^{50} \) \(\mathstrut -\mathstrut \)\(95\!\cdots\!40\)\(q^{51} \) \(\mathstrut -\mathstrut \)\(18\!\cdots\!08\)\(q^{52} \) \(\mathstrut -\mathstrut \)\(95\!\cdots\!54\)\(q^{53} \) \(\mathstrut -\mathstrut \)\(55\!\cdots\!00\)\(q^{54} \) \(\mathstrut -\mathstrut \)\(38\!\cdots\!00\)\(q^{55} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!00\)\(q^{56} \) \(\mathstrut +\mathstrut \)\(33\!\cdots\!60\)\(q^{57} \) \(\mathstrut +\mathstrut \)\(80\!\cdots\!80\)\(q^{58} \) \(\mathstrut -\mathstrut \)\(25\!\cdots\!00\)\(q^{59} \) \(\mathstrut +\mathstrut \)\(15\!\cdots\!00\)\(q^{60} \) \(\mathstrut +\mathstrut \)\(39\!\cdots\!10\)\(q^{61} \) \(\mathstrut -\mathstrut \)\(92\!\cdots\!96\)\(q^{62} \) \(\mathstrut -\mathstrut \)\(68\!\cdots\!84\)\(q^{63} \) \(\mathstrut -\mathstrut \)\(69\!\cdots\!40\)\(q^{64} \) \(\mathstrut -\mathstrut \)\(20\!\cdots\!00\)\(q^{65} \) \(\mathstrut +\mathstrut \)\(32\!\cdots\!20\)\(q^{66} \) \(\mathstrut +\mathstrut \)\(47\!\cdots\!72\)\(q^{67} \) \(\mathstrut +\mathstrut \)\(42\!\cdots\!84\)\(q^{68} \) \(\mathstrut +\mathstrut \)\(20\!\cdots\!80\)\(q^{69} \) \(\mathstrut +\mathstrut \)\(22\!\cdots\!00\)\(q^{70} \) \(\mathstrut -\mathstrut \)\(34\!\cdots\!40\)\(q^{71} \) \(\mathstrut -\mathstrut \)\(60\!\cdots\!60\)\(q^{72} \) \(\mathstrut +\mathstrut \)\(73\!\cdots\!26\)\(q^{73} \) \(\mathstrut -\mathstrut \)\(15\!\cdots\!80\)\(q^{74} \) \(\mathstrut -\mathstrut \)\(14\!\cdots\!00\)\(q^{75} \) \(\mathstrut -\mathstrut \)\(36\!\cdots\!00\)\(q^{76} \) \(\mathstrut +\mathstrut \)\(25\!\cdots\!84\)\(q^{77} \) \(\mathstrut +\mathstrut \)\(15\!\cdots\!12\)\(q^{78} \) \(\mathstrut +\mathstrut \)\(59\!\cdots\!00\)\(q^{79} \) \(\mathstrut +\mathstrut \)\(21\!\cdots\!00\)\(q^{80} \) \(\mathstrut -\mathstrut \)\(65\!\cdots\!95\)\(q^{81} \) \(\mathstrut -\mathstrut \)\(14\!\cdots\!96\)\(q^{82} \) \(\mathstrut +\mathstrut \)\(20\!\cdots\!16\)\(q^{83} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!80\)\(q^{84} \) \(\mathstrut -\mathstrut \)\(16\!\cdots\!00\)\(q^{85} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!40\)\(q^{86} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!60\)\(q^{87} \) \(\mathstrut +\mathstrut \)\(37\!\cdots\!60\)\(q^{88} \) \(\mathstrut +\mathstrut \)\(14\!\cdots\!50\)\(q^{89} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!00\)\(q^{90} \) \(\mathstrut +\mathstrut \)\(26\!\cdots\!60\)\(q^{91} \) \(\mathstrut +\mathstrut \)\(52\!\cdots\!72\)\(q^{92} \) \(\mathstrut +\mathstrut \)\(18\!\cdots\!92\)\(q^{93} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!80\)\(q^{94} \) \(\mathstrut -\mathstrut \)\(82\!\cdots\!00\)\(q^{95} \) \(\mathstrut -\mathstrut \)\(56\!\cdots\!40\)\(q^{96} \) \(\mathstrut -\mathstrut \)\(31\!\cdots\!38\)\(q^{97} \) \(\mathstrut -\mathstrut \)\(46\!\cdots\!36\)\(q^{98} \) \(\mathstrut -\mathstrut \)\(36\!\cdots\!20\)\(q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{5}\mathstrut -\mathstrut \) \(x^{4}\mathstrut -\mathstrut \) \(64723040936454512\) \(x^{3}\mathstrut -\mathstrut \) \(84407031146177217128088\) \(x^{2}\mathstrut +\mathstrut \) \(970644106288906308951592802733936\) \(x\mathstrut -\mathstrut \) \(10693396258963037598644570162849209944336\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 48 \nu - 10 \)
\(\beta_{2}\)\(=\)\((\)\(539\) \(\nu^{4}\mathstrut -\mathstrut \) \(69605886173\) \(\nu^{3}\mathstrut -\mathstrut \) \(18278217129738643138\) \(\nu^{2}\mathstrut +\mathstrut \) \(2116747767049990541221596996\) \(\nu\mathstrut -\mathstrut \) \(7888334440378384983674043800884296\)\()/\)\(10236912155953201152\)
\(\beta_{3}\)\(=\)\((\)\(-\)\(81389\) \(\nu^{4}\mathstrut +\mathstrut \) \(10510488812123\) \(\nu^{3}\mathstrut +\mathstrut \) \(4234126137047796079726\) \(\nu^{2}\mathstrut -\mathstrut \) \(322512548036491948223588761884\) \(\nu\mathstrut -\mathstrut \) \(36972552768006653800706820158499260808\)\()/\)\(639807009747075072\)
\(\beta_{4}\)\(=\)\((\)\(2432657389\) \(\nu^{4}\mathstrut -\mathstrut \) \(607176630283444315\) \(\nu^{3}\mathstrut -\mathstrut \) \(69051655044660831141992558\) \(\nu^{2}\mathstrut +\mathstrut \) \(19703748660506529942426355399555356\) \(\nu\mathstrut -\mathstrut \) \(368791919050534846695324845193663120828024\)\()/\)\(5118456077976600576\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1}\mathstrut +\mathstrut \) \(10\)\()/48\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3}\mathstrut +\mathstrut \) \(2416\) \(\beta_{2}\mathstrut +\mathstrut \) \(93896648\) \(\beta_{1}\mathstrut +\mathstrut \) \(59648754527074037862\)\()/2304\)
\(\nu^{3}\)\(=\)\((\)\(-\)\(120736\) \(\beta_{4}\mathstrut +\mathstrut \) \(137629895\) \(\beta_{3}\mathstrut +\mathstrut \) \(1422344336592\) \(\beta_{2}\mathstrut +\mathstrut \) \(5001014243470250680\) \(\beta_{1}\mathstrut +\mathstrut \) \(350053109989227508339699882\)\()/6912\)
\(\nu^{4}\)\(=\)\((\)\(-\)\(15591718502752\) \(\beta_{4}\mathstrut +\mathstrut \) \(119507425221869891\) \(\beta_{3}\mathstrut +\mathstrut \) \(560745069703020709776\) \(\beta_{2}\mathstrut +\mathstrut \) \(89864823163338141981679192\) \(\beta_{1}\mathstrut +\mathstrut \) \(6214672693553272320265454552957776594\)\()/6912\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
2.10736e8
1.39707e8
1.11190e7
−1.76800e8
−1.84762e8
−1.09073e10 4.46080e15 8.20747e19 5.92338e22 −4.86551e25 −3.00901e27 −4.92803e29 9.59772e30 −6.46079e32
1.2 −7.49786e9 −4.87294e15 1.93244e19 −6.28165e22 3.65366e25 −4.41462e27 1.31731e29 1.34445e31 4.70989e32
1.3 −1.32566e9 9.15766e14 −3.51361e19 9.89719e21 −1.21399e24 2.98681e27 9.54865e28 −9.46242e30 −1.31203e31
1.4 7.69444e9 −5.56240e15 2.23108e19 6.36641e22 −4.27995e25 1.82854e27 −1.12205e29 2.06393e31 4.89860e32
1.5 8.07662e9 2.82746e15 2.83384e19 −4.30990e22 2.28363e25 −4.31488e27 −6.90965e28 −2.30651e30 −3.48095e32
\(n\): 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 \(S_{66}^{\mathrm{new}}(\Gamma_0(1))\).