Properties

Label 3.66.a.a
Level 3
Weight 66
Character orbit 3.a
Self dual Yes
Analytic conductor 80.272
Analytic rank 1
Dimension 5
CM No
Inner twists 1

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 3 \)
Weight: \( k \) = \( 66 \)
Character orbit: \([\chi]\) = 3.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(80.2717069417\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{37}\cdot 3^{20}\cdot 5^{2}\cdot 7^{2}\cdot 11^{2}\cdot 13^{2} \)
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.

\(f(q)\) \(=\) \( q +(-517306193 - \beta_{1}) q^{2} -1853020188851841 q^{3} +(18191329493565380240 + 801514303 \beta_{1} + \beta_{2}) q^{4} +(-\)\(17\!\cdots\!00\)\( + 2570555488797 \beta_{1} - 104 \beta_{2} - \beta_{4}) q^{5} +(\)\(95\!\cdots\!13\)\( + 1853020188851841 \beta_{1}) q^{6} +(\)\(11\!\cdots\!34\)\( + 171549858376018073 \beta_{1} + 16972753 \beta_{2} - 19117 \beta_{3} + 49714 \beta_{4}) q^{7} +(-\)\(34\!\cdots\!76\)\( - 30138530585785123900 \beta_{1} - 178094044 \beta_{2} - 4464024 \beta_{3} - 828392 \beta_{4}) q^{8} +\)\(34\!\cdots\!81\)\( q^{9} +O(q^{10})\) \( q +(-517306193 - \beta_{1}) q^{2} -1853020188851841 q^{3} +(18191329493565380240 + 801514303 \beta_{1} + \beta_{2}) q^{4} +(-\)\(17\!\cdots\!00\)\( + 2570555488797 \beta_{1} - 104 \beta_{2} - \beta_{4}) q^{5} +(\)\(95\!\cdots\!13\)\( + 1853020188851841 \beta_{1}) q^{6} +(\)\(11\!\cdots\!34\)\( + 171549858376018073 \beta_{1} + 16972753 \beta_{2} - 19117 \beta_{3} + 49714 \beta_{4}) q^{7} +(-\)\(34\!\cdots\!76\)\( - 30138530585785123900 \beta_{1} - 178094044 \beta_{2} - 4464024 \beta_{3} - 828392 \beta_{4}) q^{8} +\)\(34\!\cdots\!81\)\( q^{9} +(-\)\(13\!\cdots\!10\)\( + \)\(21\!\cdots\!58\)\( \beta_{1} - 778414271116 \beta_{2} + 3418011260 \beta_{3} + 415348036 \beta_{4}) q^{10} +(\)\(48\!\cdots\!48\)\( + \)\(49\!\cdots\!22\)\( \beta_{1} + 45908880202496 \beta_{2} + 143420360208 \beta_{3} - 49582296686 \beta_{4}) q^{11} +(-\)\(33\!\cdots\!40\)\( - \)\(14\!\cdots\!23\)\( \beta_{1} - 1853020188851841 \beta_{2}) q^{12} +(\)\(25\!\cdots\!78\)\( - \)\(16\!\cdots\!12\)\( \beta_{1} + 15746692115738911 \beta_{2} + 10334183212789 \beta_{3} - 28930566358363 \beta_{4}) q^{13} +(-\)\(99\!\cdots\!60\)\( - \)\(20\!\cdots\!08\)\( \beta_{1} - 164429895551028100 \beta_{2} - 182860778986284 \beta_{3} + 30479103186028 \beta_{4}) q^{14} +(\)\(31\!\cdots\!00\)\( - \)\(47\!\cdots\!77\)\( \beta_{1} + 192714099640591464 \beta_{2} + 1853020188851841 \beta_{4}) q^{15} +(\)\(99\!\cdots\!00\)\( + \)\(22\!\cdots\!32\)\( \beta_{1} + 15421276084687931184 \beta_{2} + 12523665882676192 \beta_{3} + 14642158952047136 \beta_{4}) q^{16} +(-\)\(28\!\cdots\!74\)\( - \)\(35\!\cdots\!14\)\( \beta_{1} - \)\(11\!\cdots\!06\)\( \beta_{2} - 78607431137893698 \beta_{3} + 146782101420019016 \beta_{4}) q^{17} +(-\)\(17\!\cdots\!33\)\( - \)\(34\!\cdots\!81\)\( \beta_{1}) q^{18} +(\)\(10\!\cdots\!88\)\( + \)\(39\!\cdots\!10\)\( \beta_{1} - \)\(11\!\cdots\!02\)\( \beta_{2} + 1587779776403887154 \beta_{3} + 1764160533840295432 \beta_{4}) q^{19} +(-\)\(47\!\cdots\!60\)\( + \)\(68\!\cdots\!78\)\( \beta_{1} - \)\(34\!\cdots\!06\)\( \beta_{2} - 4858846799949642240 \beta_{3} + 26512154908239141376 \beta_{4}) q^{20} +(-\)\(21\!\cdots\!94\)\( - \)\(31\!\cdots\!93\)\( \beta_{1} - \)\(31\!\cdots\!73\)\( \beta_{2} + 35424186950280644397 \beta_{3} - 92121045668580423474 \beta_{4}) q^{21} +(-\)\(27\!\cdots\!48\)\( - \)\(28\!\cdots\!28\)\( \beta_{1} - \)\(10\!\cdots\!12\)\( \beta_{2} - \)\(35\!\cdots\!16\)\( \beta_{3} - \)\(47\!\cdots\!28\)\( \beta_{4}) q^{22} +(-\)\(46\!\cdots\!60\)\( + \)\(44\!\cdots\!78\)\( \beta_{1} - \)\(34\!\cdots\!70\)\( \beta_{2} + \)\(47\!\cdots\!74\)\( \beta_{3} - \)\(13\!\cdots\!08\)\( \beta_{4}) q^{23} +(\)\(63\!\cdots\!16\)\( + \)\(55\!\cdots\!00\)\( \beta_{1} + \)\(33\!\cdots\!04\)\( \beta_{2} + \)\(82\!\cdots\!84\)\( \beta_{3} + \)\(15\!\cdots\!72\)\( \beta_{4}) q^{24} +(\)\(26\!\cdots\!95\)\( - \)\(65\!\cdots\!48\)\( \beta_{1} - \)\(26\!\cdots\!94\)\( \beta_{2} - \)\(33\!\cdots\!70\)\( \beta_{3} + \)\(52\!\cdots\!34\)\( \beta_{4}) q^{25} +(\)\(89\!\cdots\!14\)\( - \)\(97\!\cdots\!54\)\( \beta_{1} + \)\(18\!\cdots\!04\)\( \beta_{2} - \)\(63\!\cdots\!60\)\( \beta_{3} - \)\(36\!\cdots\!80\)\( \beta_{4}) q^{26} -\)\(63\!\cdots\!21\)\( q^{27} +(\)\(73\!\cdots\!16\)\( + \)\(12\!\cdots\!88\)\( \beta_{1} + \)\(20\!\cdots\!04\)\( \beta_{2} + \)\(17\!\cdots\!20\)\( \beta_{3} - \)\(11\!\cdots\!40\)\( \beta_{4}) q^{28} +(\)\(13\!\cdots\!92\)\( + \)\(13\!\cdots\!15\)\( \beta_{1} - \)\(62\!\cdots\!14\)\( \beta_{2} - \)\(51\!\cdots\!46\)\( \beta_{3} + \)\(19\!\cdots\!07\)\( \beta_{4}) q^{29} +(\)\(24\!\cdots\!10\)\( - \)\(40\!\cdots\!78\)\( \beta_{1} + \)\(14\!\cdots\!56\)\( \beta_{2} - \)\(63\!\cdots\!60\)\( \beta_{3} - \)\(76\!\cdots\!76\)\( \beta_{4}) q^{30} +(\)\(71\!\cdots\!26\)\( + \)\(11\!\cdots\!33\)\( \beta_{1} - \)\(38\!\cdots\!67\)\( \beta_{2} + \)\(38\!\cdots\!55\)\( \beta_{3} + \)\(22\!\cdots\!90\)\( \beta_{4}) q^{31} +(-\)\(45\!\cdots\!16\)\( - \)\(64\!\cdots\!20\)\( \beta_{1} - \)\(91\!\cdots\!64\)\( \beta_{2} + \)\(26\!\cdots\!64\)\( \beta_{3} - \)\(26\!\cdots\!88\)\( \beta_{4}) q^{32} +(-\)\(89\!\cdots\!68\)\( - \)\(92\!\cdots\!02\)\( \beta_{1} - \)\(85\!\cdots\!36\)\( \beta_{2} - \)\(26\!\cdots\!28\)\( \beta_{3} + \)\(91\!\cdots\!26\)\( \beta_{4}) q^{33} +(\)\(21\!\cdots\!62\)\( + \)\(84\!\cdots\!34\)\( \beta_{1} + \)\(38\!\cdots\!84\)\( \beta_{2} + \)\(23\!\cdots\!16\)\( \beta_{3} + \)\(29\!\cdots\!28\)\( \beta_{4}) q^{34} +(-\)\(88\!\cdots\!40\)\( + \)\(74\!\cdots\!30\)\( \beta_{1} + \)\(93\!\cdots\!00\)\( \beta_{2} + \)\(84\!\cdots\!40\)\( \beta_{3} - \)\(24\!\cdots\!90\)\( \beta_{4}) q^{35} +(\)\(62\!\cdots\!40\)\( + \)\(27\!\cdots\!43\)\( \beta_{1} + \)\(34\!\cdots\!81\)\( \beta_{2}) q^{36} +(-\)\(23\!\cdots\!34\)\( + \)\(35\!\cdots\!98\)\( \beta_{1} + \)\(55\!\cdots\!33\)\( \beta_{2} + \)\(29\!\cdots\!75\)\( \beta_{3} + \)\(75\!\cdots\!25\)\( \beta_{4}) q^{37} +(-\)\(26\!\cdots\!28\)\( - \)\(47\!\cdots\!68\)\( \beta_{1} - \)\(15\!\cdots\!80\)\( \beta_{2} - \)\(34\!\cdots\!08\)\( \beta_{3} - \)\(46\!\cdots\!64\)\( \beta_{4}) q^{38} +(-\)\(47\!\cdots\!98\)\( + \)\(30\!\cdots\!92\)\( \beta_{1} - \)\(29\!\cdots\!51\)\( \beta_{2} - \)\(19\!\cdots\!49\)\( \beta_{3} + \)\(53\!\cdots\!83\)\( \beta_{4}) q^{39} +(\)\(13\!\cdots\!60\)\( + \)\(13\!\cdots\!56\)\( \beta_{1} - \)\(88\!\cdots\!32\)\( \beta_{2} - \)\(38\!\cdots\!60\)\( \beta_{3} + \)\(20\!\cdots\!52\)\( \beta_{4}) q^{40} +(-\)\(63\!\cdots\!02\)\( + \)\(25\!\cdots\!74\)\( \beta_{1} - \)\(19\!\cdots\!58\)\( \beta_{2} + \)\(24\!\cdots\!30\)\( \beta_{3} - \)\(50\!\cdots\!60\)\( \beta_{4}) q^{41} +(\)\(18\!\cdots\!60\)\( + \)\(37\!\cdots\!28\)\( \beta_{1} + \)\(30\!\cdots\!00\)\( \beta_{2} + \)\(33\!\cdots\!44\)\( \beta_{3} - \)\(56\!\cdots\!48\)\( \beta_{4}) q^{42} +(\)\(66\!\cdots\!36\)\( - \)\(11\!\cdots\!78\)\( \beta_{1} + \)\(11\!\cdots\!54\)\( \beta_{2} - \)\(22\!\cdots\!62\)\( \beta_{3} + \)\(56\!\cdots\!04\)\( \beta_{4}) q^{43} +(\)\(15\!\cdots\!88\)\( + \)\(60\!\cdots\!84\)\( \beta_{1} + \)\(30\!\cdots\!56\)\( \beta_{2} + \)\(15\!\cdots\!56\)\( \beta_{3} + \)\(39\!\cdots\!48\)\( \beta_{4}) q^{44} +(-\)\(59\!\cdots\!00\)\( + \)\(88\!\cdots\!57\)\( \beta_{1} - \)\(35\!\cdots\!24\)\( \beta_{2} - \)\(34\!\cdots\!81\)\( \beta_{4}) q^{45} +(-\)\(22\!\cdots\!44\)\( + \)\(21\!\cdots\!96\)\( \beta_{1} - \)\(63\!\cdots\!20\)\( \beta_{2} + \)\(18\!\cdots\!44\)\( \beta_{3} + \)\(17\!\cdots\!52\)\( \beta_{4}) q^{46} +(-\)\(94\!\cdots\!76\)\( + \)\(28\!\cdots\!10\)\( \beta_{1} - \)\(22\!\cdots\!78\)\( \beta_{2} - \)\(37\!\cdots\!54\)\( \beta_{3} - \)\(19\!\cdots\!32\)\( \beta_{4}) q^{47} +(-\)\(18\!\cdots\!00\)\( - \)\(40\!\cdots\!12\)\( \beta_{1} - \)\(28\!\cdots\!44\)\( \beta_{2} - \)\(23\!\cdots\!72\)\( \beta_{3} - \)\(27\!\cdots\!76\)\( \beta_{4}) q^{48} +(-\)\(20\!\cdots\!67\)\( + \)\(51\!\cdots\!44\)\( \beta_{1} - \)\(15\!\cdots\!82\)\( \beta_{2} + \)\(84\!\cdots\!02\)\( \beta_{3} + \)\(68\!\cdots\!66\)\( \beta_{4}) q^{49} +(\)\(35\!\cdots\!05\)\( + \)\(12\!\cdots\!13\)\( \beta_{1} + \)\(10\!\cdots\!64\)\( \beta_{2} + \)\(31\!\cdots\!20\)\( \beta_{3} + \)\(11\!\cdots\!96\)\( \beta_{4}) q^{50} +(\)\(53\!\cdots\!34\)\( + \)\(66\!\cdots\!74\)\( \beta_{1} + \)\(20\!\cdots\!46\)\( \beta_{2} + \)\(14\!\cdots\!18\)\( \beta_{3} - \)\(27\!\cdots\!56\)\( \beta_{4}) q^{51} +(\)\(39\!\cdots\!92\)\( - \)\(11\!\cdots\!78\)\( \beta_{1} + \)\(60\!\cdots\!18\)\( \beta_{2} - \)\(10\!\cdots\!04\)\( \beta_{3} + \)\(94\!\cdots\!68\)\( \beta_{4}) q^{52} +(-\)\(34\!\cdots\!16\)\( - \)\(51\!\cdots\!53\)\( \beta_{1} - \)\(70\!\cdots\!14\)\( \beta_{2} + \)\(12\!\cdots\!70\)\( \beta_{3} - \)\(19\!\cdots\!65\)\( \beta_{4}) q^{53} +(\)\(32\!\cdots\!53\)\( + \)\(63\!\cdots\!21\)\( \beta_{1}) q^{54} +(\)\(45\!\cdots\!80\)\( - \)\(29\!\cdots\!28\)\( \beta_{1} - \)\(14\!\cdots\!24\)\( \beta_{2} + \)\(87\!\cdots\!20\)\( \beta_{3} + \)\(47\!\cdots\!24\)\( \beta_{4}) q^{55} +(-\)\(34\!\cdots\!44\)\( - \)\(10\!\cdots\!08\)\( \beta_{1} - \)\(10\!\cdots\!12\)\( \beta_{2} - \)\(28\!\cdots\!88\)\( \beta_{3} - \)\(79\!\cdots\!04\)\( \beta_{4}) q^{56} +(-\)\(18\!\cdots\!08\)\( - \)\(73\!\cdots\!10\)\( \beta_{1} + \)\(21\!\cdots\!82\)\( \beta_{2} - \)\(29\!\cdots\!14\)\( \beta_{3} - \)\(32\!\cdots\!12\)\( \beta_{4}) q^{57} +(-\)\(13\!\cdots\!34\)\( - \)\(10\!\cdots\!78\)\( \beta_{1} + \)\(21\!\cdots\!32\)\( \beta_{2} + \)\(12\!\cdots\!80\)\( \beta_{3} + \)\(16\!\cdots\!40\)\( \beta_{4}) q^{58} +(-\)\(94\!\cdots\!12\)\( + \)\(15\!\cdots\!84\)\( \beta_{1} + \)\(41\!\cdots\!48\)\( \beta_{2} - \)\(13\!\cdots\!56\)\( \beta_{3} + \)\(39\!\cdots\!52\)\( \beta_{4}) q^{59} +(\)\(88\!\cdots\!60\)\( - \)\(12\!\cdots\!98\)\( \beta_{1} + \)\(64\!\cdots\!46\)\( \beta_{2} + \)\(90\!\cdots\!40\)\( \beta_{3} - \)\(49\!\cdots\!16\)\( \beta_{4}) q^{60} +(\)\(33\!\cdots\!38\)\( + \)\(10\!\cdots\!14\)\( \beta_{1} + \)\(79\!\cdots\!13\)\( \beta_{2} - \)\(51\!\cdots\!73\)\( \beta_{3} - \)\(55\!\cdots\!59\)\( \beta_{4}) q^{61} +(-\)\(64\!\cdots\!56\)\( + \)\(11\!\cdots\!68\)\( \beta_{1} - \)\(35\!\cdots\!76\)\( \beta_{2} + \)\(22\!\cdots\!88\)\( \beta_{3} - \)\(98\!\cdots\!96\)\( \beta_{4}) q^{62} +(\)\(39\!\cdots\!54\)\( + \)\(58\!\cdots\!13\)\( \beta_{1} + \)\(58\!\cdots\!93\)\( \beta_{2} - \)\(65\!\cdots\!77\)\( \beta_{3} + \)\(17\!\cdots\!34\)\( \beta_{4}) q^{63} +(-\)\(13\!\cdots\!60\)\( + \)\(42\!\cdots\!96\)\( \beta_{1} - \)\(56\!\cdots\!76\)\( \beta_{2} - \)\(30\!\cdots\!04\)\( \beta_{3} - \)\(54\!\cdots\!32\)\( \beta_{4}) q^{64} +(\)\(19\!\cdots\!40\)\( + \)\(45\!\cdots\!26\)\( \beta_{1} - \)\(15\!\cdots\!42\)\( \beta_{2} + \)\(98\!\cdots\!10\)\( \beta_{3} + \)\(16\!\cdots\!92\)\( \beta_{4}) q^{65} +(\)\(51\!\cdots\!68\)\( + \)\(53\!\cdots\!48\)\( \beta_{1} + \)\(19\!\cdots\!92\)\( \beta_{2} + \)\(66\!\cdots\!56\)\( \beta_{3} + \)\(88\!\cdots\!48\)\( \beta_{4}) q^{66} +(-\)\(31\!\cdots\!00\)\( - \)\(24\!\cdots\!92\)\( \beta_{1} + \)\(42\!\cdots\!52\)\( \beta_{2} - \)\(11\!\cdots\!88\)\( \beta_{3} - \)\(15\!\cdots\!04\)\( \beta_{4}) q^{67} +(-\)\(36\!\cdots\!32\)\( - \)\(28\!\cdots\!78\)\( \beta_{1} - \)\(56\!\cdots\!94\)\( \beta_{2} - \)\(16\!\cdots\!04\)\( \beta_{3} - \)\(65\!\cdots\!32\)\( \beta_{4}) q^{68} +(\)\(85\!\cdots\!60\)\( - \)\(83\!\cdots\!98\)\( \beta_{1} + \)\(63\!\cdots\!70\)\( \beta_{2} - \)\(88\!\cdots\!34\)\( \beta_{3} + \)\(24\!\cdots\!28\)\( \beta_{4}) q^{69} +(\)\(46\!\cdots\!20\)\( + \)\(42\!\cdots\!00\)\( \beta_{1} + \)\(40\!\cdots\!20\)\( \beta_{2} + \)\(30\!\cdots\!80\)\( \beta_{3} - \)\(22\!\cdots\!00\)\( \beta_{4}) q^{70} +(-\)\(40\!\cdots\!24\)\( - \)\(16\!\cdots\!10\)\( \beta_{1} - \)\(21\!\cdots\!30\)\( \beta_{2} + \)\(72\!\cdots\!74\)\( \beta_{3} + \)\(14\!\cdots\!92\)\( \beta_{4}) q^{71} +(-\)\(11\!\cdots\!56\)\( - \)\(10\!\cdots\!00\)\( \beta_{1} - \)\(61\!\cdots\!64\)\( \beta_{2} - \)\(15\!\cdots\!44\)\( \beta_{3} - \)\(28\!\cdots\!52\)\( \beta_{4}) q^{72} +(-\)\(33\!\cdots\!34\)\( - \)\(55\!\cdots\!92\)\( \beta_{1} - \)\(49\!\cdots\!50\)\( \beta_{2} + \)\(12\!\cdots\!30\)\( \beta_{3} + \)\(27\!\cdots\!90\)\( \beta_{4}) q^{73} +(-\)\(18\!\cdots\!46\)\( - \)\(44\!\cdots\!42\)\( \beta_{1} - \)\(60\!\cdots\!76\)\( \beta_{2} - \)\(52\!\cdots\!92\)\( \beta_{3} - \)\(16\!\cdots\!36\)\( \beta_{4}) q^{74} +(-\)\(48\!\cdots\!95\)\( + \)\(12\!\cdots\!68\)\( \beta_{1} + \)\(48\!\cdots\!54\)\( \beta_{2} + \)\(61\!\cdots\!70\)\( \beta_{3} - \)\(97\!\cdots\!94\)\( \beta_{4}) q^{75} +(-\)\(10\!\cdots\!28\)\( + \)\(88\!\cdots\!56\)\( \beta_{1} + \)\(10\!\cdots\!16\)\( \beta_{2} + \)\(30\!\cdots\!04\)\( \beta_{3} - \)\(39\!\cdots\!68\)\( \beta_{4}) q^{76} +(\)\(92\!\cdots\!84\)\( + \)\(25\!\cdots\!20\)\( \beta_{1} + \)\(40\!\cdots\!92\)\( \beta_{2} + \)\(48\!\cdots\!64\)\( \beta_{3} + \)\(43\!\cdots\!12\)\( \beta_{4}) q^{77} +(-\)\(16\!\cdots\!74\)\( + \)\(18\!\cdots\!14\)\( \beta_{1} - \)\(33\!\cdots\!64\)\( \beta_{2} + \)\(11\!\cdots\!60\)\( \beta_{3} + \)\(67\!\cdots\!80\)\( \beta_{4}) q^{78} +(-\)\(32\!\cdots\!66\)\( - \)\(12\!\cdots\!27\)\( \beta_{1} - \)\(31\!\cdots\!03\)\( \beta_{2} - \)\(71\!\cdots\!77\)\( \beta_{3} + \)\(71\!\cdots\!34\)\( \beta_{4}) q^{79} +(-\)\(57\!\cdots\!40\)\( + \)\(31\!\cdots\!68\)\( \beta_{1} + \)\(14\!\cdots\!84\)\( \beta_{2} + \)\(59\!\cdots\!40\)\( \beta_{3} - \)\(78\!\cdots\!44\)\( \beta_{4}) q^{80} +\)\(11\!\cdots\!61\)\( q^{81} +(-\)\(13\!\cdots\!62\)\( + \)\(15\!\cdots\!78\)\( \beta_{1} - \)\(28\!\cdots\!36\)\( \beta_{2} + \)\(18\!\cdots\!72\)\( \beta_{3} - \)\(46\!\cdots\!24\)\( \beta_{4}) q^{82} +(-\)\(11\!\cdots\!84\)\( + \)\(13\!\cdots\!02\)\( \beta_{1} + \)\(31\!\cdots\!80\)\( \beta_{2} - \)\(16\!\cdots\!60\)\( \beta_{3} + \)\(66\!\cdots\!70\)\( \beta_{4}) q^{83} +(-\)\(13\!\cdots\!56\)\( - \)\(22\!\cdots\!08\)\( \beta_{1} - \)\(38\!\cdots\!64\)\( \beta_{2} - \)\(32\!\cdots\!20\)\( \beta_{3} + \)\(20\!\cdots\!40\)\( \beta_{4}) q^{84} +(-\)\(21\!\cdots\!00\)\( + \)\(30\!\cdots\!54\)\( \beta_{1} + \)\(79\!\cdots\!72\)\( \beta_{2} + \)\(29\!\cdots\!00\)\( \beta_{3} - \)\(86\!\cdots\!82\)\( \beta_{4}) q^{85} +(\)\(27\!\cdots\!64\)\( - \)\(12\!\cdots\!92\)\( \beta_{1} + \)\(11\!\cdots\!20\)\( \beta_{2} - \)\(22\!\cdots\!28\)\( \beta_{3} + \)\(60\!\cdots\!76\)\( \beta_{4}) q^{86} +(-\)\(24\!\cdots\!72\)\( - \)\(24\!\cdots\!15\)\( \beta_{1} + \)\(11\!\cdots\!74\)\( \beta_{2} + \)\(95\!\cdots\!86\)\( \beta_{3} - \)\(36\!\cdots\!87\)\( \beta_{4}) q^{87} +(-\)\(23\!\cdots\!68\)\( - \)\(21\!\cdots\!60\)\( \beta_{1} - \)\(34\!\cdots\!24\)\( \beta_{2} - \)\(15\!\cdots\!16\)\( \beta_{3} + \)\(89\!\cdots\!72\)\( \beta_{4}) q^{88} +(-\)\(21\!\cdots\!90\)\( + \)\(10\!\cdots\!40\)\( \beta_{1} + \)\(49\!\cdots\!48\)\( \beta_{2} + \)\(16\!\cdots\!36\)\( \beta_{3} + \)\(72\!\cdots\!88\)\( \beta_{4}) q^{89} +(-\)\(45\!\cdots\!10\)\( + \)\(74\!\cdots\!98\)\( \beta_{1} - \)\(26\!\cdots\!96\)\( \beta_{2} + \)\(11\!\cdots\!60\)\( \beta_{3} + \)\(14\!\cdots\!16\)\( \beta_{4}) q^{90} +(-\)\(32\!\cdots\!40\)\( - \)\(80\!\cdots\!02\)\( \beta_{1} + \)\(47\!\cdots\!98\)\( \beta_{2} - \)\(29\!\cdots\!66\)\( \beta_{3} - \)\(42\!\cdots\!28\)\( \beta_{4}) q^{91} +(-\)\(97\!\cdots\!76\)\( + \)\(30\!\cdots\!84\)\( \beta_{1} - \)\(17\!\cdots\!08\)\( \beta_{2} - \)\(32\!\cdots\!04\)\( \beta_{3} - \)\(37\!\cdots\!32\)\( \beta_{4}) q^{92} +(-\)\(13\!\cdots\!66\)\( - \)\(20\!\cdots\!53\)\( \beta_{1} + \)\(70\!\cdots\!47\)\( \beta_{2} - \)\(71\!\cdots\!55\)\( \beta_{3} - \)\(42\!\cdots\!90\)\( \beta_{4}) q^{93} +(-\)\(10\!\cdots\!08\)\( + \)\(20\!\cdots\!16\)\( \beta_{1} + \)\(16\!\cdots\!40\)\( \beta_{2} + \)\(23\!\cdots\!28\)\( \beta_{3} + \)\(14\!\cdots\!24\)\( \beta_{4}) q^{94} +(-\)\(56\!\cdots\!20\)\( - \)\(20\!\cdots\!52\)\( \beta_{1} + \)\(10\!\cdots\!44\)\( \beta_{2} + \)\(75\!\cdots\!20\)\( \beta_{3} - \)\(20\!\cdots\!84\)\( \beta_{4}) q^{95} +(\)\(85\!\cdots\!56\)\( + \)\(11\!\cdots\!20\)\( \beta_{1} + \)\(16\!\cdots\!24\)\( \beta_{2} - \)\(49\!\cdots\!24\)\( \beta_{3} + \)\(49\!\cdots\!08\)\( \beta_{4}) q^{96} +(\)\(64\!\cdots\!70\)\( + \)\(11\!\cdots\!80\)\( \beta_{1} + \)\(16\!\cdots\!60\)\( \beta_{2} - \)\(40\!\cdots\!72\)\( \beta_{3} + \)\(76\!\cdots\!24\)\( \beta_{4}) q^{97} +(-\)\(28\!\cdots\!85\)\( + \)\(13\!\cdots\!59\)\( \beta_{1} - \)\(10\!\cdots\!08\)\( \beta_{2} - \)\(37\!\cdots\!60\)\( \beta_{3} - \)\(28\!\cdots\!80\)\( \beta_{4}) q^{98} +(\)\(16\!\cdots\!88\)\( + \)\(17\!\cdots\!82\)\( \beta_{1} + \)\(15\!\cdots\!76\)\( \beta_{2} + \)\(49\!\cdots\!48\)\( \beta_{3} - \)\(17\!\cdots\!66\)\( \beta_{4}) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q - 2586530964q^{2} - 9265100944259205q^{3} + 90956647467025386896q^{4} - \)\(86\!\cdots\!94\)\(q^{5} + \)\(47\!\cdots\!24\)\(q^{6} + \)\(57\!\cdots\!24\)\(q^{7} - \)\(17\!\cdots\!76\)\(q^{8} + \)\(17\!\cdots\!05\)\(q^{9} + O(q^{10}) \) \( 5q - 2586530964q^{2} - 9265100944259205q^{3} + 90956647467025386896q^{4} - \)\(86\!\cdots\!94\)\(q^{5} + \)\(47\!\cdots\!24\)\(q^{6} + \)\(57\!\cdots\!24\)\(q^{7} - \)\(17\!\cdots\!76\)\(q^{8} + \)\(17\!\cdots\!05\)\(q^{9} - \)\(65\!\cdots\!36\)\(q^{10} + \)\(24\!\cdots\!52\)\(q^{11} - \)\(16\!\cdots\!36\)\(q^{12} + \)\(12\!\cdots\!06\)\(q^{13} - \)\(49\!\cdots\!32\)\(q^{14} + \)\(15\!\cdots\!54\)\(q^{15} + \)\(49\!\cdots\!04\)\(q^{16} - \)\(14\!\cdots\!30\)\(q^{17} - \)\(88\!\cdots\!84\)\(q^{18} + \)\(50\!\cdots\!72\)\(q^{19} - \)\(23\!\cdots\!76\)\(q^{20} - \)\(10\!\cdots\!84\)\(q^{21} - \)\(13\!\cdots\!60\)\(q^{22} - \)\(23\!\cdots\!68\)\(q^{23} + \)\(31\!\cdots\!16\)\(q^{24} + \)\(13\!\cdots\!11\)\(q^{25} + \)\(44\!\cdots\!20\)\(q^{26} - \)\(31\!\cdots\!05\)\(q^{27} + \)\(36\!\cdots\!88\)\(q^{28} + \)\(65\!\cdots\!74\)\(q^{29} + \)\(12\!\cdots\!76\)\(q^{30} + \)\(35\!\cdots\!64\)\(q^{31} - \)\(22\!\cdots\!56\)\(q^{32} - \)\(44\!\cdots\!32\)\(q^{33} + \)\(10\!\cdots\!52\)\(q^{34} - \)\(44\!\cdots\!40\)\(q^{35} + \)\(31\!\cdots\!76\)\(q^{36} - \)\(11\!\cdots\!26\)\(q^{37} - \)\(13\!\cdots\!72\)\(q^{38} - \)\(23\!\cdots\!46\)\(q^{39} + \)\(67\!\cdots\!08\)\(q^{40} - \)\(31\!\cdots\!26\)\(q^{41} + \)\(92\!\cdots\!12\)\(q^{42} + \)\(33\!\cdots\!84\)\(q^{43} + \)\(76\!\cdots\!60\)\(q^{44} - \)\(29\!\cdots\!14\)\(q^{45} - \)\(11\!\cdots\!56\)\(q^{46} - \)\(47\!\cdots\!52\)\(q^{47} - \)\(92\!\cdots\!64\)\(q^{48} - \)\(10\!\cdots\!27\)\(q^{49} + \)\(17\!\cdots\!84\)\(q^{50} + \)\(26\!\cdots\!30\)\(q^{51} + \)\(19\!\cdots\!80\)\(q^{52} - \)\(17\!\cdots\!38\)\(q^{53} + \)\(16\!\cdots\!44\)\(q^{54} + \)\(22\!\cdots\!16\)\(q^{55} - \)\(17\!\cdots\!80\)\(q^{56} - \)\(93\!\cdots\!52\)\(q^{57} - \)\(69\!\cdots\!24\)\(q^{58} - \)\(47\!\cdots\!52\)\(q^{59} + \)\(44\!\cdots\!16\)\(q^{60} + \)\(16\!\cdots\!58\)\(q^{61} - \)\(32\!\cdots\!92\)\(q^{62} + \)\(19\!\cdots\!44\)\(q^{63} - \)\(67\!\cdots\!60\)\(q^{64} + \)\(98\!\cdots\!28\)\(q^{65} + \)\(25\!\cdots\!60\)\(q^{66} - \)\(15\!\cdots\!40\)\(q^{67} - \)\(18\!\cdots\!28\)\(q^{68} + \)\(42\!\cdots\!88\)\(q^{69} + \)\(23\!\cdots\!40\)\(q^{70} - \)\(20\!\cdots\!40\)\(q^{71} - \)\(58\!\cdots\!56\)\(q^{72} - \)\(16\!\cdots\!78\)\(q^{73} - \)\(92\!\cdots\!32\)\(q^{74} - \)\(24\!\cdots\!51\)\(q^{75} - \)\(50\!\cdots\!72\)\(q^{76} + \)\(46\!\cdots\!48\)\(q^{77} - \)\(82\!\cdots\!20\)\(q^{78} - \)\(16\!\cdots\!20\)\(q^{79} - \)\(28\!\cdots\!16\)\(q^{80} + \)\(58\!\cdots\!05\)\(q^{81} - \)\(67\!\cdots\!32\)\(q^{82} - \)\(58\!\cdots\!52\)\(q^{83} - \)\(68\!\cdots\!08\)\(q^{84} - \)\(10\!\cdots\!08\)\(q^{85} + \)\(13\!\cdots\!12\)\(q^{86} - \)\(12\!\cdots\!34\)\(q^{87} - \)\(11\!\cdots\!16\)\(q^{88} - \)\(10\!\cdots\!78\)\(q^{89} - \)\(22\!\cdots\!16\)\(q^{90} - \)\(16\!\cdots\!56\)\(q^{91} - \)\(48\!\cdots\!96\)\(q^{92} - \)\(66\!\cdots\!24\)\(q^{93} - \)\(52\!\cdots\!16\)\(q^{94} - \)\(28\!\cdots\!36\)\(q^{95} + \)\(42\!\cdots\!96\)\(q^{96} + \)\(32\!\cdots\!90\)\(q^{97} - \)\(14\!\cdots\!76\)\(q^{98} + \)\(83\!\cdots\!12\)\(q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{5} - 2 x^{4} - 951687707356162322 x^{3} + 12324255588111120031125216 x^{2} + 110893397087607196931833643857303785 x + 11092688481415693693368751083964140538880250\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 12 \nu - 5 \)
\(\beta_{2}\)\(=\)\( 144 \nu^{2} + 2797176876 \nu - 54817211944833820613 \)
\(\beta_{3}\)\(=\)\((\)\(-8387469 \nu^{4} + 10908025077337047 \nu^{3} + 8665828280543129292267651 \nu^{2} - 8173462006937471544690536283502515 \nu - 923652763728045686755980799505889606615250\)\()/ 26849826340757504000 \)
\(\beta_{4}\)\(=\)\((\)\(45198243 \nu^{4} - 2773066158298809 \nu^{3} - 40286174486511372932888397 \nu^{2} + 3919930759364497102449458703314205 \nu + 2950595509101559396982085653800913632948750\)\()/ 26849826340757504000 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1} + 5\)\()/12\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{2} - 233098073 \beta_{1} + 54817211943668330248\)\()/144\)
\(\nu^{3}\)\(=\)\((\)\(103549 \beta_{4} + 558003 \beta_{3} - 171728065 \beta_{2} + 12935554878864422938 \beta_{1} - 1597223274954636433578258722\)\()/216\)
\(\nu^{4}\)\(=\)\((\)\(808001857580522 \beta_{4} + 205412308022134 \beta_{3} + 7958678629735950741 \beta_{2} - 6474616545229819910245523119 \beta_{1} + 354545526698056634174706274784570519312\)\()/1296\)

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
8.82783e8
4.30385e8
−1.14024e8
−2.77962e8
−9.21181e8
−1.11107e10 −1.85302e15 8.65541e19 −1.70324e22 2.05883e25 4.28205e27 −5.51764e29 3.43368e30 1.89242e32
1.2 −5.68192e9 −1.85302e15 −4.60926e18 5.44876e22 1.05287e25 4.11535e26 2.35815e29 3.43368e30 −3.09594e32
1.3 8.50981e8 −1.85302e15 −3.61693e19 −8.94186e22 −1.57689e24 3.62069e27 −6.21751e28 3.43368e30 −7.60936e31
1.4 2.81824e9 −1.85302e15 −2.89510e19 1.31338e22 −5.22225e24 −3.12329e27 −1.85565e29 3.43368e30 3.70141e31
1.5 1.05369e10 −1.85302e15 7.41322e19 −4.74965e22 −1.95250e25 5.36551e26 3.92379e29 3.43368e30 −5.00464e32
\(n\): 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 \( T_{2}^{5} + 2586530964 T_{2}^{4} - \)\(13\!\cdots\!80\)\( T_{2}^{3} - \)\(23\!\cdots\!44\)\( T_{2}^{2} + \)\(21\!\cdots\!28\)\( T_{2} - \)\(15\!\cdots\!96\)\( \) acting on \(S_{66}^{\mathrm{new}}(\Gamma_0(3))\).