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.

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

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

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

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 \(S_{66}^{\mathrm{new}}(\Gamma_0(1))\).