Properties

Label 1.82.a.a
Level 1
Weight 82
Character orbit 1.a
Self dual yes
Analytic conductor 41.550
Analytic rank 1
Dimension 6
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 1 \)
Weight: \( k \) \(=\) \( 82 \)
Character orbit: \([\chi]\) \(=\) 1.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(41.5501285538\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{58}\cdot 3^{26}\cdot 5^{7}\cdot 7^{3}\cdot 13 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q +(-76812004440 - \beta_{1}) q^{2} +(-2608048654315521660 - 4130706 \beta_{1} + \beta_{2}) q^{3} +(\)\(74\!\cdots\!92\)\( + 330453870896 \beta_{1} - 4668 \beta_{2} + \beta_{3}) q^{4} +(-\)\(30\!\cdots\!50\)\( + 3579253777224384 \beta_{1} + 21942656 \beta_{2} + 3247 \beta_{3} + \beta_{4}) q^{5} +(\)\(13\!\cdots\!72\)\( + 4513221122897092521 \beta_{1} + 247691680988 \beta_{2} + 2224483 \beta_{3} - 108 \beta_{4} - \beta_{5}) q^{6} +(-\)\(52\!\cdots\!00\)\( - \)\(20\!\cdots\!44\)\( \beta_{1} - 102284077821438 \beta_{2} - 1096811324 \beta_{3} - 12780 \beta_{4} + 408 \beta_{5}) q^{7} +(-\)\(91\!\cdots\!20\)\( - \)\(69\!\cdots\!92\)\( \beta_{1} - 23354748483639488 \beta_{2} - 483724211632 \beta_{3} - 73855360 \beta_{4} + 87264 \beta_{5}) q^{8} +(\)\(19\!\cdots\!33\)\( - \)\(31\!\cdots\!88\)\( \beta_{1} + 2385861390791192256 \beta_{2} - 63041239844574 \beta_{3} - 22295404386 \beta_{4} - 11158752 \beta_{5}) q^{9} +O(q^{10})\) \( q +(-76812004440 - \beta_{1}) q^{2} +(-2608048654315521660 - 4130706 \beta_{1} + \beta_{2}) q^{3} +(\)\(74\!\cdots\!92\)\( + 330453870896 \beta_{1} - 4668 \beta_{2} + \beta_{3}) q^{4} +(-\)\(30\!\cdots\!50\)\( + 3579253777224384 \beta_{1} + 21942656 \beta_{2} + 3247 \beta_{3} + \beta_{4}) q^{5} +(\)\(13\!\cdots\!72\)\( + 4513221122897092521 \beta_{1} + 247691680988 \beta_{2} + 2224483 \beta_{3} - 108 \beta_{4} - \beta_{5}) q^{6} +(-\)\(52\!\cdots\!00\)\( - \)\(20\!\cdots\!44\)\( \beta_{1} - 102284077821438 \beta_{2} - 1096811324 \beta_{3} - 12780 \beta_{4} + 408 \beta_{5}) q^{7} +(-\)\(91\!\cdots\!20\)\( - \)\(69\!\cdots\!92\)\( \beta_{1} - 23354748483639488 \beta_{2} - 483724211632 \beta_{3} - 73855360 \beta_{4} + 87264 \beta_{5}) q^{8} +(\)\(19\!\cdots\!33\)\( - \)\(31\!\cdots\!88\)\( \beta_{1} + 2385861390791192256 \beta_{2} - 63041239844574 \beta_{3} - 22295404386 \beta_{4} - 11158752 \beta_{5}) q^{9} +(-\)\(11\!\cdots\!00\)\( - \)\(52\!\cdots\!42\)\( \beta_{1} - \)\(12\!\cdots\!28\)\( \beta_{2} - 7823121005523436 \beta_{3} - 162776084688 \beta_{4} + 556990500 \beta_{5}) q^{10} +(-\)\(38\!\cdots\!88\)\( + \)\(75\!\cdots\!70\)\( \beta_{1} + \)\(60\!\cdots\!83\)\( \beta_{2} - 367894936402422424 \beta_{3} + 51221298912776 \beta_{4} - 15361340688 \beta_{5}) q^{11} +(-\)\(89\!\cdots\!80\)\( - \)\(92\!\cdots\!76\)\( \beta_{1} - \)\(27\!\cdots\!28\)\( \beta_{2} - 18961616389226821308 \beta_{3} - 1267640420797440 \beta_{4} + 218743785216 \beta_{5}) q^{12} +(\)\(21\!\cdots\!30\)\( + \)\(16\!\cdots\!44\)\( \beta_{1} - \)\(21\!\cdots\!76\)\( \beta_{2} + \)\(23\!\cdots\!63\)\( \beta_{3} + 768189821556105 \beta_{4} + 679320417984 \beta_{5}) q^{13} +(\)\(68\!\cdots\!04\)\( + \)\(81\!\cdots\!98\)\( \beta_{1} - \)\(86\!\cdots\!24\)\( \beta_{2} + \)\(81\!\cdots\!38\)\( \beta_{3} + 526793576543375720 \beta_{4} - 121314522929490 \beta_{5}) q^{14} +(-\)\(26\!\cdots\!00\)\( - \)\(80\!\cdots\!96\)\( \beta_{1} - \)\(20\!\cdots\!14\)\( \beta_{2} - \)\(13\!\cdots\!68\)\( \beta_{3} - 11969550314767844244 \beta_{4} + 3542243323729000 \beta_{5}) q^{15} +(\)\(46\!\cdots\!76\)\( + \)\(13\!\cdots\!12\)\( \beta_{1} - \)\(16\!\cdots\!20\)\( \beta_{2} - \)\(10\!\cdots\!96\)\( \beta_{3} + \)\(12\!\cdots\!32\)\( \beta_{4} - 67147365672685056 \beta_{5}) q^{16} +(\)\(39\!\cdots\!30\)\( - \)\(63\!\cdots\!16\)\( \beta_{1} + \)\(19\!\cdots\!44\)\( \beta_{2} + \)\(13\!\cdots\!94\)\( \beta_{3} - \)\(34\!\cdots\!70\)\( \beta_{4} + 963043109681077152 \beta_{5}) q^{17} +(\)\(84\!\cdots\!40\)\( - \)\(44\!\cdots\!25\)\( \beta_{1} + \)\(26\!\cdots\!08\)\( \beta_{2} - \)\(37\!\cdots\!96\)\( \beta_{3} - \)\(92\!\cdots\!20\)\( \beta_{4} - 10991883681013116168 \beta_{5}) q^{18} +(-\)\(29\!\cdots\!40\)\( + \)\(20\!\cdots\!74\)\( \beta_{1} + \)\(78\!\cdots\!49\)\( \beta_{2} - \)\(10\!\cdots\!84\)\( \beta_{3} + \)\(15\!\cdots\!12\)\( \beta_{4} + \)\(10\!\cdots\!44\)\( \beta_{5}) q^{19} +(\)\(24\!\cdots\!00\)\( + \)\(21\!\cdots\!28\)\( \beta_{1} - \)\(10\!\cdots\!48\)\( \beta_{2} + \)\(72\!\cdots\!74\)\( \beta_{3} - \)\(12\!\cdots\!08\)\( \beta_{4} - \)\(75\!\cdots\!00\)\( \beta_{5}) q^{20} +(-\)\(19\!\cdots\!48\)\( + \)\(19\!\cdots\!72\)\( \beta_{1} - \)\(98\!\cdots\!36\)\( \beta_{2} + \)\(23\!\cdots\!52\)\( \beta_{3} + \)\(44\!\cdots\!80\)\( \beta_{4} + \)\(42\!\cdots\!20\)\( \beta_{5}) q^{21} +(-\)\(20\!\cdots\!80\)\( + \)\(12\!\cdots\!71\)\( \beta_{1} + \)\(39\!\cdots\!16\)\( \beta_{2} - \)\(43\!\cdots\!43\)\( \beta_{3} + \)\(14\!\cdots\!20\)\( \beta_{4} - \)\(15\!\cdots\!99\)\( \beta_{5}) q^{22} +(\)\(17\!\cdots\!20\)\( + \)\(64\!\cdots\!60\)\( \beta_{1} + \)\(37\!\cdots\!54\)\( \beta_{2} + \)\(55\!\cdots\!40\)\( \beta_{3} - \)\(29\!\cdots\!00\)\( \beta_{4} - \)\(59\!\cdots\!80\)\( \beta_{5}) q^{23} +(-\)\(20\!\cdots\!20\)\( + \)\(43\!\cdots\!52\)\( \beta_{1} - \)\(64\!\cdots\!20\)\( \beta_{2} + \)\(15\!\cdots\!64\)\( \beta_{3} + \)\(18\!\cdots\!72\)\( \beta_{4} + \)\(66\!\cdots\!44\)\( \beta_{5}) q^{24} +(\)\(68\!\cdots\!75\)\( + \)\(73\!\cdots\!00\)\( \beta_{1} - \)\(96\!\cdots\!00\)\( \beta_{2} - \)\(80\!\cdots\!00\)\( \beta_{3} - \)\(46\!\cdots\!00\)\( \beta_{4} - \)\(64\!\cdots\!00\)\( \beta_{5}) q^{25} +(-\)\(53\!\cdots\!28\)\( - \)\(81\!\cdots\!22\)\( \beta_{1} - \)\(12\!\cdots\!56\)\( \beta_{2} - \)\(15\!\cdots\!16\)\( \beta_{3} - \)\(14\!\cdots\!24\)\( \beta_{4} + \)\(41\!\cdots\!52\)\( \beta_{5}) q^{26} +(\)\(24\!\cdots\!80\)\( - \)\(32\!\cdots\!52\)\( \beta_{1} + \)\(29\!\cdots\!82\)\( \beta_{2} + \)\(28\!\cdots\!68\)\( \beta_{3} + \)\(17\!\cdots\!40\)\( \beta_{4} - \)\(20\!\cdots\!36\)\( \beta_{5}) q^{27} +(-\)\(13\!\cdots\!20\)\( - \)\(23\!\cdots\!12\)\( \beta_{1} + \)\(80\!\cdots\!72\)\( \beta_{2} - \)\(88\!\cdots\!28\)\( \beta_{3} - \)\(57\!\cdots\!40\)\( \beta_{4} + \)\(76\!\cdots\!56\)\( \beta_{5}) q^{28} +(\)\(25\!\cdots\!90\)\( + \)\(15\!\cdots\!36\)\( \beta_{1} - \)\(34\!\cdots\!16\)\( \beta_{2} - \)\(16\!\cdots\!45\)\( \beta_{3} - \)\(73\!\cdots\!71\)\( \beta_{4} - \)\(21\!\cdots\!92\)\( \beta_{5}) q^{29} +(\)\(27\!\cdots\!00\)\( + \)\(32\!\cdots\!98\)\( \beta_{1} - \)\(88\!\cdots\!68\)\( \beta_{2} + \)\(16\!\cdots\!34\)\( \beta_{3} + \)\(15\!\cdots\!72\)\( \beta_{4} + \)\(41\!\cdots\!50\)\( \beta_{5}) q^{30} +(\)\(20\!\cdots\!32\)\( + \)\(61\!\cdots\!80\)\( \beta_{1} + \)\(52\!\cdots\!84\)\( \beta_{2} - \)\(19\!\cdots\!52\)\( \beta_{3} - \)\(69\!\cdots\!32\)\( \beta_{4} - \)\(40\!\cdots\!44\)\( \beta_{5}) q^{31} +(-\)\(21\!\cdots\!40\)\( + \)\(96\!\cdots\!60\)\( \beta_{1} + \)\(15\!\cdots\!32\)\( \beta_{2} - \)\(12\!\cdots\!36\)\( \beta_{3} + \)\(14\!\cdots\!80\)\( \beta_{4} + \)\(63\!\cdots\!12\)\( \beta_{5}) q^{32} +(\)\(35\!\cdots\!80\)\( - \)\(25\!\cdots\!88\)\( \beta_{1} - \)\(51\!\cdots\!80\)\( \beta_{2} + \)\(30\!\cdots\!66\)\( \beta_{3} - \)\(12\!\cdots\!10\)\( \beta_{4} - \)\(89\!\cdots\!92\)\( \beta_{5}) q^{33} +(\)\(19\!\cdots\!24\)\( - \)\(31\!\cdots\!38\)\( \beta_{1} - \)\(22\!\cdots\!60\)\( \beta_{2} + \)\(12\!\cdots\!84\)\( \beta_{3} - \)\(47\!\cdots\!48\)\( \beta_{4} + \)\(53\!\cdots\!24\)\( \beta_{5}) q^{34} +(-\)\(38\!\cdots\!00\)\( - \)\(13\!\cdots\!72\)\( \beta_{1} + \)\(41\!\cdots\!52\)\( \beta_{2} - \)\(62\!\cdots\!76\)\( \beta_{3} - \)\(60\!\cdots\!08\)\( \beta_{4} - \)\(13\!\cdots\!00\)\( \beta_{5}) q^{35} +(-\)\(34\!\cdots\!64\)\( + \)\(11\!\cdots\!72\)\( \beta_{1} + \)\(25\!\cdots\!20\)\( \beta_{2} + \)\(31\!\cdots\!69\)\( \beta_{3} + \)\(41\!\cdots\!72\)\( \beta_{4} - \)\(24\!\cdots\!16\)\( \beta_{5}) q^{36} +(\)\(69\!\cdots\!90\)\( + \)\(78\!\cdots\!44\)\( \beta_{1} - \)\(28\!\cdots\!24\)\( \beta_{2} + \)\(16\!\cdots\!91\)\( \beta_{3} - \)\(13\!\cdots\!95\)\( \beta_{4} + \)\(34\!\cdots\!68\)\( \beta_{5}) q^{37} +(-\)\(40\!\cdots\!80\)\( + \)\(52\!\cdots\!53\)\( \beta_{1} - \)\(17\!\cdots\!60\)\( \beta_{2} + \)\(65\!\cdots\!31\)\( \beta_{3} + \)\(18\!\cdots\!00\)\( \beta_{4} - \)\(14\!\cdots\!57\)\( \beta_{5}) q^{38} +(-\)\(15\!\cdots\!64\)\( - \)\(12\!\cdots\!52\)\( \beta_{1} - \)\(14\!\cdots\!62\)\( \beta_{2} - \)\(32\!\cdots\!88\)\( \beta_{3} + \)\(15\!\cdots\!04\)\( \beta_{4} + \)\(29\!\cdots\!48\)\( \beta_{5}) q^{39} +(-\)\(43\!\cdots\!00\)\( - \)\(35\!\cdots\!80\)\( \beta_{1} + \)\(15\!\cdots\!80\)\( \beta_{2} - \)\(76\!\cdots\!40\)\( \beta_{3} - \)\(93\!\cdots\!20\)\( \beta_{4} + \)\(19\!\cdots\!00\)\( \beta_{5}) q^{40} +(-\)\(41\!\cdots\!58\)\( - \)\(43\!\cdots\!00\)\( \beta_{1} + \)\(28\!\cdots\!64\)\( \beta_{2} + \)\(67\!\cdots\!08\)\( \beta_{3} - \)\(46\!\cdots\!52\)\( \beta_{4} - \)\(34\!\cdots\!44\)\( \beta_{5}) q^{41} +(-\)\(62\!\cdots\!20\)\( - \)\(92\!\cdots\!16\)\( \beta_{1} - \)\(11\!\cdots\!76\)\( \beta_{2} - \)\(14\!\cdots\!64\)\( \beta_{3} + \)\(42\!\cdots\!20\)\( \beta_{4} + \)\(11\!\cdots\!88\)\( \beta_{5}) q^{42} +(-\)\(89\!\cdots\!00\)\( + \)\(56\!\cdots\!46\)\( \beta_{1} - \)\(20\!\cdots\!13\)\( \beta_{2} + \)\(11\!\cdots\!40\)\( \beta_{3} + \)\(10\!\cdots\!80\)\( \beta_{4} - \)\(15\!\cdots\!60\)\( \beta_{5}) q^{43} +(-\)\(28\!\cdots\!96\)\( + \)\(73\!\cdots\!72\)\( \beta_{1} + \)\(43\!\cdots\!20\)\( \beta_{2} - \)\(59\!\cdots\!96\)\( \beta_{3} - \)\(11\!\cdots\!28\)\( \beta_{4} - \)\(28\!\cdots\!76\)\( \beta_{5}) q^{44} +(-\)\(10\!\cdots\!50\)\( + \)\(17\!\cdots\!72\)\( \beta_{1} + \)\(12\!\cdots\!48\)\( \beta_{2} + \)\(33\!\cdots\!51\)\( \beta_{3} + \)\(29\!\cdots\!33\)\( \beta_{4} + \)\(19\!\cdots\!00\)\( \beta_{5}) q^{45} +(-\)\(20\!\cdots\!28\)\( - \)\(43\!\cdots\!70\)\( \beta_{1} + \)\(15\!\cdots\!04\)\( \beta_{2} - \)\(62\!\cdots\!02\)\( \beta_{3} - \)\(11\!\cdots\!92\)\( \beta_{4} - \)\(42\!\cdots\!74\)\( \beta_{5}) q^{46} +(-\)\(38\!\cdots\!80\)\( - \)\(16\!\cdots\!04\)\( \beta_{1} - \)\(12\!\cdots\!24\)\( \beta_{2} - \)\(70\!\cdots\!80\)\( \beta_{3} - \)\(12\!\cdots\!60\)\( \beta_{4} + \)\(13\!\cdots\!20\)\( \beta_{5}) q^{47} +(-\)\(11\!\cdots\!80\)\( - \)\(21\!\cdots\!16\)\( \beta_{1} - \)\(11\!\cdots\!48\)\( \beta_{2} + \)\(37\!\cdots\!16\)\( \beta_{3} + \)\(28\!\cdots\!20\)\( \beta_{4} + \)\(17\!\cdots\!28\)\( \beta_{5}) q^{48} +(-\)\(24\!\cdots\!43\)\( + \)\(45\!\cdots\!60\)\( \beta_{1} + \)\(44\!\cdots\!16\)\( \beta_{2} + \)\(61\!\cdots\!72\)\( \beta_{3} - \)\(13\!\cdots\!28\)\( \beta_{4} - \)\(47\!\cdots\!16\)\( \beta_{5}) q^{49} +(-\)\(23\!\cdots\!00\)\( + \)\(88\!\cdots\!25\)\( \beta_{1} + \)\(13\!\cdots\!00\)\( \beta_{2} - \)\(11\!\cdots\!00\)\( \beta_{3} - \)\(19\!\cdots\!00\)\( \beta_{4} + \)\(35\!\cdots\!00\)\( \beta_{5}) q^{50} +(\)\(12\!\cdots\!12\)\( + \)\(28\!\cdots\!24\)\( \beta_{1} - \)\(20\!\cdots\!46\)\( \beta_{2} - \)\(26\!\cdots\!24\)\( \beta_{3} - \)\(19\!\cdots\!28\)\( \beta_{4} + \)\(86\!\cdots\!64\)\( \beta_{5}) q^{51} +(\)\(20\!\cdots\!00\)\( + \)\(68\!\cdots\!92\)\( \beta_{1} - \)\(36\!\cdots\!16\)\( \beta_{2} + \)\(92\!\cdots\!70\)\( \beta_{3} + \)\(53\!\cdots\!40\)\( \beta_{4} - \)\(19\!\cdots\!80\)\( \beta_{5}) q^{52} +(\)\(43\!\cdots\!90\)\( - \)\(12\!\cdots\!64\)\( \beta_{1} - \)\(91\!\cdots\!20\)\( \beta_{2} - \)\(71\!\cdots\!57\)\( \beta_{3} + \)\(44\!\cdots\!65\)\( \beta_{4} - \)\(12\!\cdots\!36\)\( \beta_{5}) q^{53} +(\)\(10\!\cdots\!60\)\( - \)\(79\!\cdots\!86\)\( \beta_{1} + \)\(40\!\cdots\!84\)\( \beta_{2} - \)\(12\!\cdots\!94\)\( \beta_{3} - \)\(44\!\cdots\!28\)\( \beta_{4} + \)\(62\!\cdots\!74\)\( \beta_{5}) q^{54} +(\)\(11\!\cdots\!00\)\( - \)\(34\!\cdots\!92\)\( \beta_{1} + \)\(11\!\cdots\!22\)\( \beta_{2} - \)\(57\!\cdots\!36\)\( \beta_{3} + \)\(72\!\cdots\!12\)\( \beta_{4} + \)\(14\!\cdots\!00\)\( \beta_{5}) q^{55} +(\)\(58\!\cdots\!60\)\( + \)\(20\!\cdots\!24\)\( \beta_{1} + \)\(65\!\cdots\!36\)\( \beta_{2} + \)\(17\!\cdots\!40\)\( \beta_{3} + \)\(13\!\cdots\!96\)\( \beta_{4} - \)\(93\!\cdots\!68\)\( \beta_{5}) q^{56} +(\)\(50\!\cdots\!60\)\( + \)\(52\!\cdots\!52\)\( \beta_{1} - \)\(53\!\cdots\!24\)\( \beta_{2} - \)\(92\!\cdots\!90\)\( \beta_{3} - \)\(12\!\cdots\!30\)\( \beta_{4} + \)\(12\!\cdots\!60\)\( \beta_{5}) q^{57} +(-\)\(52\!\cdots\!20\)\( + \)\(36\!\cdots\!30\)\( \beta_{1} + \)\(40\!\cdots\!56\)\( \beta_{2} - \)\(34\!\cdots\!24\)\( \beta_{3} - \)\(78\!\cdots\!40\)\( \beta_{4} + \)\(26\!\cdots\!68\)\( \beta_{5}) q^{58} +(-\)\(64\!\cdots\!20\)\( + \)\(10\!\cdots\!82\)\( \beta_{1} + \)\(75\!\cdots\!59\)\( \beta_{2} - \)\(40\!\cdots\!88\)\( \beta_{3} + \)\(35\!\cdots\!80\)\( \beta_{4} - \)\(12\!\cdots\!80\)\( \beta_{5}) q^{59} +(-\)\(97\!\cdots\!00\)\( - \)\(47\!\cdots\!32\)\( \beta_{1} + \)\(35\!\cdots\!12\)\( \beta_{2} + \)\(23\!\cdots\!44\)\( \beta_{3} + \)\(23\!\cdots\!52\)\( \beta_{4} + \)\(17\!\cdots\!00\)\( \beta_{5}) q^{60} +(-\)\(76\!\cdots\!38\)\( - \)\(33\!\cdots\!60\)\( \beta_{1} - \)\(68\!\cdots\!40\)\( \beta_{2} + \)\(34\!\cdots\!95\)\( \beta_{3} - \)\(57\!\cdots\!15\)\( \beta_{4} + \)\(12\!\cdots\!00\)\( \beta_{5}) q^{61} +(-\)\(19\!\cdots\!80\)\( + \)\(11\!\cdots\!12\)\( \beta_{1} + \)\(34\!\cdots\!68\)\( \beta_{2} - \)\(48\!\cdots\!64\)\( \beta_{3} - \)\(12\!\cdots\!80\)\( \beta_{4} - \)\(88\!\cdots\!12\)\( \beta_{5}) q^{62} +(-\)\(59\!\cdots\!40\)\( + \)\(23\!\cdots\!92\)\( \beta_{1} - \)\(21\!\cdots\!74\)\( \beta_{2} - \)\(93\!\cdots\!44\)\( \beta_{3} + \)\(18\!\cdots\!80\)\( \beta_{4} + \)\(14\!\cdots\!88\)\( \beta_{5}) q^{63} +(-\)\(40\!\cdots\!28\)\( + \)\(16\!\cdots\!32\)\( \beta_{1} + \)\(45\!\cdots\!16\)\( \beta_{2} - \)\(12\!\cdots\!24\)\( \beta_{3} - \)\(20\!\cdots\!96\)\( \beta_{4} - \)\(55\!\cdots\!12\)\( \beta_{5}) q^{64} +(\)\(66\!\cdots\!00\)\( + \)\(50\!\cdots\!04\)\( \beta_{1} + \)\(16\!\cdots\!36\)\( \beta_{2} + \)\(44\!\cdots\!32\)\( \beta_{3} - \)\(59\!\cdots\!44\)\( \beta_{4} - \)\(25\!\cdots\!00\)\( \beta_{5}) q^{65} +(\)\(77\!\cdots\!64\)\( - \)\(10\!\cdots\!28\)\( \beta_{1} - \)\(33\!\cdots\!68\)\( \beta_{2} + \)\(18\!\cdots\!68\)\( \beta_{3} - \)\(26\!\cdots\!44\)\( \beta_{4} + \)\(79\!\cdots\!72\)\( \beta_{5}) q^{66} +(\)\(46\!\cdots\!80\)\( - \)\(15\!\cdots\!66\)\( \beta_{1} - \)\(13\!\cdots\!91\)\( \beta_{2} - \)\(14\!\cdots\!56\)\( \beta_{3} + \)\(12\!\cdots\!00\)\( \beta_{4} - \)\(13\!\cdots\!68\)\( \beta_{5}) q^{67} +(\)\(98\!\cdots\!40\)\( - \)\(23\!\cdots\!40\)\( \beta_{1} - \)\(26\!\cdots\!72\)\( \beta_{2} + \)\(80\!\cdots\!02\)\( \beta_{3} + \)\(69\!\cdots\!60\)\( \beta_{4} + \)\(82\!\cdots\!96\)\( \beta_{5}) q^{68} +(\)\(13\!\cdots\!36\)\( - \)\(65\!\cdots\!76\)\( \beta_{1} + \)\(45\!\cdots\!72\)\( \beta_{2} - \)\(31\!\cdots\!08\)\( \beta_{3} - \)\(61\!\cdots\!52\)\( \beta_{4} + \)\(34\!\cdots\!16\)\( \beta_{5}) q^{69} +(\)\(43\!\cdots\!00\)\( + \)\(21\!\cdots\!36\)\( \beta_{1} + \)\(50\!\cdots\!24\)\( \beta_{2} + \)\(14\!\cdots\!88\)\( \beta_{3} + \)\(22\!\cdots\!04\)\( \beta_{4} - \)\(10\!\cdots\!00\)\( \beta_{5}) q^{70} +(-\)\(92\!\cdots\!28\)\( + \)\(12\!\cdots\!20\)\( \beta_{1} + \)\(18\!\cdots\!30\)\( \beta_{2} - \)\(84\!\cdots\!40\)\( \beta_{3} + \)\(11\!\cdots\!80\)\( \beta_{4} + \)\(82\!\cdots\!00\)\( \beta_{5}) q^{71} +(-\)\(52\!\cdots\!60\)\( + \)\(37\!\cdots\!76\)\( \beta_{1} - \)\(56\!\cdots\!40\)\( \beta_{2} - \)\(26\!\cdots\!48\)\( \beta_{3} + \)\(18\!\cdots\!40\)\( \beta_{4} + \)\(16\!\cdots\!16\)\( \beta_{5}) q^{72} +(-\)\(19\!\cdots\!30\)\( - \)\(55\!\cdots\!96\)\( \beta_{1} + \)\(89\!\cdots\!52\)\( \beta_{2} + \)\(54\!\cdots\!46\)\( \beta_{3} - \)\(94\!\cdots\!50\)\( \beta_{4} - \)\(36\!\cdots\!12\)\( \beta_{5}) q^{73} +(-\)\(24\!\cdots\!36\)\( - \)\(66\!\cdots\!70\)\( \beta_{1} - \)\(71\!\cdots\!36\)\( \beta_{2} + \)\(21\!\cdots\!88\)\( \beta_{3} + \)\(16\!\cdots\!28\)\( \beta_{4} - \)\(10\!\cdots\!04\)\( \beta_{5}) q^{74} +(-\)\(67\!\cdots\!00\)\( - \)\(21\!\cdots\!50\)\( \beta_{1} + \)\(23\!\cdots\!75\)\( \beta_{2} + \)\(25\!\cdots\!00\)\( \beta_{3} + \)\(35\!\cdots\!00\)\( \beta_{4} + \)\(63\!\cdots\!00\)\( \beta_{5}) q^{75} +(-\)\(93\!\cdots\!80\)\( - \)\(30\!\cdots\!12\)\( \beta_{1} + \)\(63\!\cdots\!52\)\( \beta_{2} - \)\(52\!\cdots\!80\)\( \beta_{3} - \)\(54\!\cdots\!08\)\( \beta_{4} + \)\(91\!\cdots\!64\)\( \beta_{5}) q^{76} +(-\)\(38\!\cdots\!00\)\( + \)\(43\!\cdots\!20\)\( \beta_{1} + \)\(12\!\cdots\!00\)\( \beta_{2} - \)\(41\!\cdots\!76\)\( \beta_{3} + \)\(68\!\cdots\!80\)\( \beta_{4} - \)\(33\!\cdots\!08\)\( \beta_{5}) q^{77} +(\)\(49\!\cdots\!00\)\( + \)\(22\!\cdots\!30\)\( \beta_{1} - \)\(59\!\cdots\!40\)\( \beta_{2} + \)\(55\!\cdots\!22\)\( \beta_{3} + \)\(58\!\cdots\!00\)\( \beta_{4} - \)\(16\!\cdots\!34\)\( \beta_{5}) q^{78} +(\)\(12\!\cdots\!40\)\( - \)\(14\!\cdots\!44\)\( \beta_{1} - \)\(17\!\cdots\!32\)\( \beta_{2} - \)\(13\!\cdots\!52\)\( \beta_{3} - \)\(13\!\cdots\!88\)\( \beta_{4} + \)\(14\!\cdots\!04\)\( \beta_{5}) q^{79} +(\)\(54\!\cdots\!00\)\( + \)\(18\!\cdots\!84\)\( \beta_{1} + \)\(27\!\cdots\!56\)\( \beta_{2} + \)\(21\!\cdots\!72\)\( \beta_{3} + \)\(35\!\cdots\!76\)\( \beta_{4} - \)\(75\!\cdots\!00\)\( \beta_{5}) q^{80} +(\)\(11\!\cdots\!81\)\( - \)\(90\!\cdots\!04\)\( \beta_{1} + \)\(15\!\cdots\!52\)\( \beta_{2} - \)\(46\!\cdots\!74\)\( \beta_{3} - \)\(26\!\cdots\!10\)\( \beta_{4} - \)\(33\!\cdots\!80\)\( \beta_{5}) q^{81} +(\)\(13\!\cdots\!20\)\( - \)\(99\!\cdots\!58\)\( \beta_{1} + \)\(62\!\cdots\!28\)\( \beta_{2} - \)\(21\!\cdots\!44\)\( \beta_{3} - \)\(90\!\cdots\!20\)\( \beta_{4} + \)\(38\!\cdots\!88\)\( \beta_{5}) q^{82} +(\)\(73\!\cdots\!60\)\( - \)\(87\!\cdots\!82\)\( \beta_{1} - \)\(15\!\cdots\!67\)\( \beta_{2} + \)\(21\!\cdots\!00\)\( \beta_{3} + \)\(15\!\cdots\!00\)\( \beta_{4} + \)\(69\!\cdots\!00\)\( \beta_{5}) q^{83} +(\)\(38\!\cdots\!84\)\( + \)\(48\!\cdots\!96\)\( \beta_{1} - \)\(14\!\cdots\!36\)\( \beta_{2} + \)\(23\!\cdots\!80\)\( \beta_{3} + \)\(13\!\cdots\!24\)\( \beta_{4} - \)\(12\!\cdots\!72\)\( \beta_{5}) q^{84} +(\)\(19\!\cdots\!00\)\( + \)\(29\!\cdots\!68\)\( \beta_{1} - \)\(16\!\cdots\!88\)\( \beta_{2} - \)\(53\!\cdots\!06\)\( \beta_{3} - \)\(40\!\cdots\!98\)\( \beta_{4} - \)\(13\!\cdots\!00\)\( \beta_{5}) q^{85} +(-\)\(17\!\cdots\!28\)\( + \)\(47\!\cdots\!79\)\( \beta_{1} + \)\(27\!\cdots\!80\)\( \beta_{2} - \)\(80\!\cdots\!67\)\( \beta_{3} - \)\(22\!\cdots\!16\)\( \beta_{4} + \)\(38\!\cdots\!53\)\( \beta_{5}) q^{86} +(-\)\(22\!\cdots\!60\)\( - \)\(72\!\cdots\!48\)\( \beta_{1} + \)\(64\!\cdots\!62\)\( \beta_{2} + \)\(61\!\cdots\!36\)\( \beta_{3} + \)\(10\!\cdots\!40\)\( \beta_{4} + \)\(13\!\cdots\!68\)\( \beta_{5}) q^{87} +(-\)\(16\!\cdots\!40\)\( + \)\(11\!\cdots\!56\)\( \beta_{1} + \)\(10\!\cdots\!84\)\( \beta_{2} + \)\(32\!\cdots\!96\)\( \beta_{3} - \)\(34\!\cdots\!80\)\( \beta_{4} - \)\(80\!\cdots\!32\)\( \beta_{5}) q^{88} +(-\)\(31\!\cdots\!30\)\( - \)\(47\!\cdots\!32\)\( \beta_{1} - \)\(12\!\cdots\!48\)\( \beta_{2} + \)\(86\!\cdots\!90\)\( \beta_{3} - \)\(32\!\cdots\!78\)\( \beta_{4} - \)\(26\!\cdots\!36\)\( \beta_{5}) q^{89} +(-\)\(45\!\cdots\!00\)\( + \)\(23\!\cdots\!14\)\( \beta_{1} - \)\(47\!\cdots\!24\)\( \beta_{2} - \)\(81\!\cdots\!88\)\( \beta_{3} - \)\(32\!\cdots\!04\)\( \beta_{4} + \)\(18\!\cdots\!00\)\( \beta_{5}) q^{90} +(-\)\(24\!\cdots\!48\)\( - \)\(96\!\cdots\!56\)\( \beta_{1} + \)\(23\!\cdots\!56\)\( \beta_{2} - \)\(25\!\cdots\!80\)\( \beta_{3} - \)\(23\!\cdots\!44\)\( \beta_{4} + \)\(81\!\cdots\!52\)\( \beta_{5}) q^{91} +(\)\(11\!\cdots\!80\)\( + \)\(20\!\cdots\!12\)\( \beta_{1} + \)\(15\!\cdots\!40\)\( \beta_{2} - \)\(17\!\cdots\!64\)\( \beta_{3} + \)\(70\!\cdots\!80\)\( \beta_{4} - \)\(55\!\cdots\!72\)\( \beta_{5}) q^{92} +(\)\(21\!\cdots\!80\)\( - \)\(79\!\cdots\!80\)\( \beta_{1} + \)\(13\!\cdots\!16\)\( \beta_{2} + \)\(33\!\cdots\!68\)\( \beta_{3} - \)\(30\!\cdots\!00\)\( \beta_{4} + \)\(18\!\cdots\!04\)\( \beta_{5}) q^{93} +(\)\(55\!\cdots\!84\)\( + \)\(43\!\cdots\!44\)\( \beta_{1} - \)\(59\!\cdots\!76\)\( \beta_{2} + \)\(25\!\cdots\!96\)\( \beta_{3} + \)\(23\!\cdots\!32\)\( \beta_{4} + \)\(78\!\cdots\!44\)\( \beta_{5}) q^{94} +(\)\(44\!\cdots\!00\)\( - \)\(45\!\cdots\!60\)\( \beta_{1} - \)\(96\!\cdots\!90\)\( \beta_{2} - \)\(43\!\cdots\!80\)\( \beta_{3} - \)\(16\!\cdots\!40\)\( \beta_{4} - \)\(60\!\cdots\!00\)\( \beta_{5}) q^{95} +(\)\(82\!\cdots\!12\)\( + \)\(64\!\cdots\!16\)\( \beta_{1} - \)\(27\!\cdots\!00\)\( \beta_{2} + \)\(38\!\cdots\!12\)\( \beta_{3} - \)\(24\!\cdots\!04\)\( \beta_{4} + \)\(23\!\cdots\!12\)\( \beta_{5}) q^{96} +(\)\(18\!\cdots\!70\)\( - \)\(51\!\cdots\!92\)\( \beta_{1} - \)\(26\!\cdots\!60\)\( \beta_{2} + \)\(55\!\cdots\!18\)\( \beta_{3} + \)\(14\!\cdots\!90\)\( \beta_{4} - \)\(21\!\cdots\!36\)\( \beta_{5}) q^{97} +(-\)\(14\!\cdots\!80\)\( - \)\(12\!\cdots\!01\)\( \beta_{1} + \)\(10\!\cdots\!12\)\( \beta_{2} - \)\(12\!\cdots\!36\)\( \beta_{3} - \)\(10\!\cdots\!80\)\( \beta_{4} + \)\(16\!\cdots\!72\)\( \beta_{5}) q^{98} +(-\)\(10\!\cdots\!04\)\( - \)\(56\!\cdots\!86\)\( \beta_{1} - \)\(14\!\cdots\!89\)\( \beta_{2} + \)\(15\!\cdots\!20\)\( \beta_{3} - \)\(16\!\cdots\!64\)\( \beta_{4} + \)\(62\!\cdots\!12\)\( \beta_{5}) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 460872026640q^{2} - 15648291925893129960q^{3} + \)\(44\!\cdots\!52\)\(q^{4} - \)\(18\!\cdots\!00\)\(q^{5} + \)\(79\!\cdots\!32\)\(q^{6} - \)\(31\!\cdots\!00\)\(q^{7} - \)\(54\!\cdots\!20\)\(q^{8} + \)\(11\!\cdots\!98\)\(q^{9} + O(q^{10}) \) \( 6q - 460872026640q^{2} - 15648291925893129960q^{3} + \)\(44\!\cdots\!52\)\(q^{4} - \)\(18\!\cdots\!00\)\(q^{5} + \)\(79\!\cdots\!32\)\(q^{6} - \)\(31\!\cdots\!00\)\(q^{7} - \)\(54\!\cdots\!20\)\(q^{8} + \)\(11\!\cdots\!98\)\(q^{9} - \)\(66\!\cdots\!00\)\(q^{10} - \)\(23\!\cdots\!28\)\(q^{11} - \)\(53\!\cdots\!80\)\(q^{12} + \)\(13\!\cdots\!80\)\(q^{13} + \)\(41\!\cdots\!24\)\(q^{14} - \)\(15\!\cdots\!00\)\(q^{15} + \)\(27\!\cdots\!56\)\(q^{16} + \)\(23\!\cdots\!80\)\(q^{17} + \)\(50\!\cdots\!40\)\(q^{18} - \)\(17\!\cdots\!40\)\(q^{19} + \)\(14\!\cdots\!00\)\(q^{20} - \)\(11\!\cdots\!88\)\(q^{21} - \)\(12\!\cdots\!80\)\(q^{22} + \)\(10\!\cdots\!20\)\(q^{23} - \)\(12\!\cdots\!20\)\(q^{24} + \)\(41\!\cdots\!50\)\(q^{25} - \)\(31\!\cdots\!68\)\(q^{26} + \)\(14\!\cdots\!80\)\(q^{27} - \)\(82\!\cdots\!20\)\(q^{28} + \)\(15\!\cdots\!40\)\(q^{29} + \)\(16\!\cdots\!00\)\(q^{30} + \)\(12\!\cdots\!92\)\(q^{31} - \)\(13\!\cdots\!40\)\(q^{32} + \)\(21\!\cdots\!80\)\(q^{33} + \)\(11\!\cdots\!44\)\(q^{34} - \)\(23\!\cdots\!00\)\(q^{35} - \)\(20\!\cdots\!84\)\(q^{36} + \)\(41\!\cdots\!40\)\(q^{37} - \)\(24\!\cdots\!80\)\(q^{38} - \)\(92\!\cdots\!84\)\(q^{39} - \)\(25\!\cdots\!00\)\(q^{40} - \)\(25\!\cdots\!48\)\(q^{41} - \)\(37\!\cdots\!20\)\(q^{42} - \)\(53\!\cdots\!00\)\(q^{43} - \)\(17\!\cdots\!76\)\(q^{44} - \)\(62\!\cdots\!00\)\(q^{45} - \)\(12\!\cdots\!68\)\(q^{46} - \)\(23\!\cdots\!80\)\(q^{47} - \)\(69\!\cdots\!80\)\(q^{48} - \)\(14\!\cdots\!58\)\(q^{49} - \)\(14\!\cdots\!00\)\(q^{50} + \)\(72\!\cdots\!72\)\(q^{51} + \)\(12\!\cdots\!00\)\(q^{52} + \)\(25\!\cdots\!40\)\(q^{53} + \)\(60\!\cdots\!60\)\(q^{54} + \)\(70\!\cdots\!00\)\(q^{55} + \)\(35\!\cdots\!60\)\(q^{56} + \)\(30\!\cdots\!60\)\(q^{57} - \)\(31\!\cdots\!20\)\(q^{58} - \)\(38\!\cdots\!20\)\(q^{59} - \)\(58\!\cdots\!00\)\(q^{60} - \)\(45\!\cdots\!28\)\(q^{61} - \)\(11\!\cdots\!80\)\(q^{62} - \)\(35\!\cdots\!40\)\(q^{63} - \)\(24\!\cdots\!68\)\(q^{64} + \)\(40\!\cdots\!00\)\(q^{65} + \)\(46\!\cdots\!84\)\(q^{66} + \)\(27\!\cdots\!80\)\(q^{67} + \)\(59\!\cdots\!40\)\(q^{68} + \)\(78\!\cdots\!16\)\(q^{69} + \)\(26\!\cdots\!00\)\(q^{70} - \)\(55\!\cdots\!68\)\(q^{71} - \)\(31\!\cdots\!60\)\(q^{72} - \)\(11\!\cdots\!80\)\(q^{73} - \)\(14\!\cdots\!16\)\(q^{74} - \)\(40\!\cdots\!00\)\(q^{75} - \)\(56\!\cdots\!80\)\(q^{76} - \)\(22\!\cdots\!00\)\(q^{77} + \)\(29\!\cdots\!00\)\(q^{78} + \)\(76\!\cdots\!40\)\(q^{79} + \)\(32\!\cdots\!00\)\(q^{80} + \)\(71\!\cdots\!86\)\(q^{81} + \)\(83\!\cdots\!20\)\(q^{82} + \)\(44\!\cdots\!60\)\(q^{83} + \)\(23\!\cdots\!04\)\(q^{84} + \)\(11\!\cdots\!00\)\(q^{85} - \)\(10\!\cdots\!68\)\(q^{86} - \)\(13\!\cdots\!60\)\(q^{87} - \)\(96\!\cdots\!40\)\(q^{88} - \)\(18\!\cdots\!80\)\(q^{89} - \)\(27\!\cdots\!00\)\(q^{90} - \)\(14\!\cdots\!88\)\(q^{91} + \)\(66\!\cdots\!80\)\(q^{92} + \)\(13\!\cdots\!80\)\(q^{93} + \)\(33\!\cdots\!04\)\(q^{94} + \)\(26\!\cdots\!00\)\(q^{95} + \)\(49\!\cdots\!72\)\(q^{96} + \)\(11\!\cdots\!20\)\(q^{97} - \)\(85\!\cdots\!80\)\(q^{98} - \)\(63\!\cdots\!24\)\(q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 3 x^{5} - 457312410974129060102 x^{4} - 374381904318715551718938507366 x^{3} + 44188779498690494552956207183337235151045 x^{2} + 43570916530513802078515169165814719895908350666425 x - 274514886475906972833021199940770859283985201013623777350000\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 144 \nu - 72 \)
\(\beta_{2}\)\(=\)\((\)\(-78099962708907 \nu^{5} - 1375691159918800801887660 \nu^{4} + 40226354593628598292993469800322382 \nu^{3} + 480737543221988187416410765264622639114508612 \nu^{2} - 4095995735119213827140200446492535473237363091214415291 \nu - 17130467806636669478709481975515934948439625934282286274145384368\)\()/ \)\(69\!\cdots\!88\)\( \)
\(\beta_{3}\)\(=\)\((\)\(-91142656481294469 \nu^{5} - 1605431583625240535802899220 \nu^{4} + 46944155810764574207923379256976219794 \nu^{3} + 4156478093089465029823565998569977702280963615996 \nu^{2} - 9195195307147830686701062611508965700721778119431024111477 \nu - 568073683618420740619803295994124603298936885106074825044960922999632\)\()/ \)\(17\!\cdots\!72\)\( \)
\(\beta_{4}\)\(=\)\((\)\(-606518390599808941384479 \nu^{5} + 1785096453037002977854674621991044 \nu^{4} + 272371810583872799198993532386144773328210262 \nu^{3} - 428159409320707302070751030867493812269669563678419532 \nu^{2} - 24830646082099940342148642768635140888295145794949529244263117551 \nu + 6940271488442406108725437805273034213041663446224288202723835834817750288\)\()/ \)\(19\!\cdots\!28\)\( \)
\(\beta_{5}\)\(=\)\((\)\(-289613468206311996503683251 \nu^{5} + 174972226505692037633509898090229300 \nu^{4} + 98117079324462934033155129479068624936612447902 \nu^{3} + 81654344557241561486631709673962807456824605977160938404 \nu^{2} - 3301073357707396020037591585583591704402760770904974187155182617667 \nu - 7051981094494430331918639309417795660138715723520929441699594261998477705264\)\()/ \)\(99\!\cdots\!64\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1} + 72\)\()/144\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} - 4668 \beta_{2} + 176829862160 \beta_{1} + 3160943384653180063456128\)\()/20736\)
\(\nu^{3}\)\(=\)\((\)\(-2727 \beta_{5} + 2307980 \beta_{4} + 7915256204 \beta_{3} + 763450743525280 \beta_{2} + 171109700008821821257987 \beta_{1} + 17467162191903096320225691284444736\)\()/93312\)
\(\nu^{4}\)\(=\)\((\)\(-52520409734386 \beta_{5} + 133986338060164584 \beta_{4} + 9163979507037693318775 \beta_{3} - 269216097411919964130756292 \beta_{2} + 2712908864519921503572803358879354 \beta_{1} + 22536169715180661308714439099709591398970306176\)\()/559872\)
\(\nu^{5}\)\(=\)\((\)\(-1875582683239936145189383 \beta_{5} + 1193105276567082329063209740 \beta_{4} + 7309604918011009905854873771614 \beta_{3} + 338424019486123722914639804333875304 \beta_{2} + 76621665860635844813220188031530036790903207 \beta_{1} + 14887763417556726787053963959092799432916570517581881184\)\()/139968\)

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
1.85696e10
1.12136e10
2.10202e9
−3.17952e9
−1.13941e10
−1.73117e10
−2.75084e12 −2.21125e19 5.14927e24 2.52982e28 6.08279e31 −2.62798e34 −7.51371e36 4.55346e37 −6.95912e40
1.2 −1.69157e12 1.74109e19 4.43572e23 −9.34342e27 −2.94518e31 1.25454e34 3.33964e36 −1.40288e38 1.58051e40
1.3 −3.79502e11 −3.74133e19 −2.27383e24 −3.57322e28 1.41984e31 −3.75122e33 1.78050e36 9.56330e38 1.35605e40
1.4 3.81039e11 −1.66183e18 −2.27266e24 2.26484e28 −6.33221e29 −1.01850e33 −1.78727e36 −4.40665e38 8.62992e39
1.5 1.56393e12 3.91464e19 2.80371e22 −1.92800e28 6.12224e31 −2.38829e34 −3.73751e36 1.08902e39 −3.01526e40
1.6 2.41607e12 −1.10180e19 3.41956e24 −1.95525e27 −2.66202e31 1.09563e34 2.42020e36 −3.22031e38 −4.72402e39
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1.82.a.a 6
3.b odd 2 1 9.82.a.b 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.82.a.a 6 1.a even 1 1 trivial
9.82.a.b 6 3.b odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{82}^{\mathrm{new}}(\Gamma_0(1))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 460872026640 T + \)\(51\!\cdots\!80\)\( T^{2} + \)\(37\!\cdots\!80\)\( T^{3} + \)\(15\!\cdots\!12\)\( T^{4} + \)\(14\!\cdots\!20\)\( T^{5} + \)\(42\!\cdots\!40\)\( T^{6} + \)\(34\!\cdots\!40\)\( T^{7} + \)\(92\!\cdots\!48\)\( T^{8} + \)\(53\!\cdots\!40\)\( T^{9} + \)\(17\!\cdots\!80\)\( T^{10} + \)\(38\!\cdots\!80\)\( T^{11} + \)\(19\!\cdots\!64\)\( T^{12} \)
$3$ \( 1 + 15648291925893129960 T + \)\(85\!\cdots\!10\)\( T^{2} + \)\(49\!\cdots\!80\)\( T^{3} + \)\(20\!\cdots\!27\)\( T^{4} - \)\(17\!\cdots\!20\)\( T^{5} + \)\(28\!\cdots\!80\)\( T^{6} - \)\(76\!\cdots\!60\)\( T^{7} + \)\(40\!\cdots\!43\)\( T^{8} + \)\(43\!\cdots\!60\)\( T^{9} + \)\(33\!\cdots\!10\)\( T^{10} + \)\(26\!\cdots\!80\)\( T^{11} + \)\(76\!\cdots\!29\)\( T^{12} \)
$5$ \( 1 + \)\(18\!\cdots\!00\)\( T + \)\(12\!\cdots\!50\)\( T^{2} + \)\(21\!\cdots\!00\)\( T^{3} + \)\(80\!\cdots\!75\)\( T^{4} + \)\(14\!\cdots\!00\)\( T^{5} + \)\(40\!\cdots\!00\)\( T^{6} + \)\(61\!\cdots\!00\)\( T^{7} + \)\(13\!\cdots\!75\)\( T^{8} + \)\(14\!\cdots\!00\)\( T^{9} + \)\(35\!\cdots\!50\)\( T^{10} + \)\(22\!\cdots\!00\)\( T^{11} + \)\(50\!\cdots\!25\)\( T^{12} \)
$7$ \( 1 + \)\(31\!\cdots\!00\)\( T + \)\(14\!\cdots\!50\)\( T^{2} + \)\(34\!\cdots\!00\)\( T^{3} + \)\(93\!\cdots\!47\)\( T^{4} + \)\(17\!\cdots\!00\)\( T^{5} + \)\(34\!\cdots\!00\)\( T^{6} + \)\(49\!\cdots\!00\)\( T^{7} + \)\(75\!\cdots\!03\)\( T^{8} + \)\(79\!\cdots\!00\)\( T^{9} + \)\(92\!\cdots\!50\)\( T^{10} + \)\(57\!\cdots\!00\)\( T^{11} + \)\(52\!\cdots\!49\)\( T^{12} \)
$11$ \( 1 + \)\(23\!\cdots\!28\)\( T + \)\(90\!\cdots\!26\)\( T^{2} + \)\(11\!\cdots\!80\)\( T^{3} + \)\(25\!\cdots\!95\)\( T^{4} + \)\(19\!\cdots\!08\)\( T^{5} + \)\(44\!\cdots\!44\)\( T^{6} + \)\(44\!\cdots\!88\)\( T^{7} + \)\(12\!\cdots\!95\)\( T^{8} + \)\(13\!\cdots\!80\)\( T^{9} + \)\(23\!\cdots\!66\)\( T^{10} + \)\(13\!\cdots\!28\)\( T^{11} + \)\(13\!\cdots\!61\)\( T^{12} \)
$13$ \( 1 - \)\(13\!\cdots\!80\)\( T + \)\(86\!\cdots\!70\)\( T^{2} - \)\(99\!\cdots\!40\)\( T^{3} + \)\(33\!\cdots\!07\)\( T^{4} - \)\(31\!\cdots\!40\)\( T^{5} + \)\(73\!\cdots\!60\)\( T^{6} - \)\(53\!\cdots\!20\)\( T^{7} + \)\(95\!\cdots\!83\)\( T^{8} - \)\(48\!\cdots\!80\)\( T^{9} + \)\(71\!\cdots\!70\)\( T^{10} - \)\(18\!\cdots\!40\)\( T^{11} + \)\(23\!\cdots\!09\)\( T^{12} \)
$17$ \( 1 - \)\(23\!\cdots\!80\)\( T + \)\(20\!\cdots\!90\)\( T^{2} - \)\(11\!\cdots\!60\)\( T^{3} + \)\(19\!\cdots\!67\)\( T^{4} - \)\(12\!\cdots\!40\)\( T^{5} + \)\(11\!\cdots\!20\)\( T^{6} - \)\(55\!\cdots\!80\)\( T^{7} + \)\(41\!\cdots\!63\)\( T^{8} - \)\(11\!\cdots\!80\)\( T^{9} + \)\(93\!\cdots\!90\)\( T^{10} - \)\(51\!\cdots\!60\)\( T^{11} + \)\(99\!\cdots\!69\)\( T^{12} \)
$19$ \( 1 + \)\(17\!\cdots\!40\)\( T + \)\(25\!\cdots\!14\)\( T^{2} + \)\(24\!\cdots\!00\)\( T^{3} + \)\(22\!\cdots\!15\)\( T^{4} + \)\(16\!\cdots\!00\)\( T^{5} + \)\(11\!\cdots\!80\)\( T^{6} + \)\(61\!\cdots\!00\)\( T^{7} + \)\(32\!\cdots\!15\)\( T^{8} + \)\(13\!\cdots\!00\)\( T^{9} + \)\(53\!\cdots\!94\)\( T^{10} + \)\(13\!\cdots\!60\)\( T^{11} + \)\(29\!\cdots\!81\)\( T^{12} \)
$23$ \( 1 - \)\(10\!\cdots\!20\)\( T + \)\(49\!\cdots\!30\)\( T^{2} - \)\(57\!\cdots\!60\)\( T^{3} + \)\(15\!\cdots\!87\)\( T^{4} - \)\(17\!\cdots\!60\)\( T^{5} + \)\(36\!\cdots\!40\)\( T^{6} - \)\(35\!\cdots\!80\)\( T^{7} + \)\(62\!\cdots\!23\)\( T^{8} - \)\(45\!\cdots\!20\)\( T^{9} + \)\(78\!\cdots\!30\)\( T^{10} - \)\(33\!\cdots\!60\)\( T^{11} + \)\(63\!\cdots\!89\)\( T^{12} \)
$29$ \( 1 - \)\(15\!\cdots\!40\)\( T + \)\(13\!\cdots\!74\)\( T^{2} - \)\(19\!\cdots\!00\)\( T^{3} + \)\(82\!\cdots\!15\)\( T^{4} - \)\(10\!\cdots\!00\)\( T^{5} + \)\(29\!\cdots\!80\)\( T^{6} - \)\(29\!\cdots\!00\)\( T^{7} + \)\(66\!\cdots\!15\)\( T^{8} - \)\(44\!\cdots\!00\)\( T^{9} + \)\(88\!\cdots\!94\)\( T^{10} - \)\(29\!\cdots\!60\)\( T^{11} + \)\(53\!\cdots\!21\)\( T^{12} \)
$31$ \( 1 - \)\(12\!\cdots\!92\)\( T + \)\(24\!\cdots\!46\)\( T^{2} - \)\(23\!\cdots\!20\)\( T^{3} + \)\(29\!\cdots\!95\)\( T^{4} - \)\(24\!\cdots\!92\)\( T^{5} + \)\(22\!\cdots\!84\)\( T^{6} - \)\(15\!\cdots\!52\)\( T^{7} + \)\(11\!\cdots\!95\)\( T^{8} - \)\(58\!\cdots\!20\)\( T^{9} + \)\(38\!\cdots\!66\)\( T^{10} - \)\(12\!\cdots\!92\)\( T^{11} + \)\(63\!\cdots\!81\)\( T^{12} \)
$37$ \( 1 - \)\(41\!\cdots\!40\)\( T + \)\(32\!\cdots\!70\)\( T^{2} + \)\(22\!\cdots\!20\)\( T^{3} + \)\(51\!\cdots\!07\)\( T^{4} + \)\(75\!\cdots\!80\)\( T^{5} + \)\(59\!\cdots\!60\)\( T^{6} + \)\(79\!\cdots\!60\)\( T^{7} + \)\(57\!\cdots\!83\)\( T^{8} + \)\(26\!\cdots\!60\)\( T^{9} + \)\(40\!\cdots\!70\)\( T^{10} - \)\(54\!\cdots\!80\)\( T^{11} + \)\(13\!\cdots\!09\)\( T^{12} \)
$41$ \( 1 + \)\(25\!\cdots\!48\)\( T + \)\(11\!\cdots\!06\)\( T^{2} + \)\(24\!\cdots\!80\)\( T^{3} + \)\(93\!\cdots\!95\)\( T^{4} + \)\(15\!\cdots\!08\)\( T^{5} + \)\(42\!\cdots\!04\)\( T^{6} + \)\(68\!\cdots\!28\)\( T^{7} + \)\(17\!\cdots\!95\)\( T^{8} + \)\(19\!\cdots\!80\)\( T^{9} + \)\(38\!\cdots\!66\)\( T^{10} + \)\(37\!\cdots\!48\)\( T^{11} + \)\(65\!\cdots\!41\)\( T^{12} \)
$43$ \( 1 + \)\(53\!\cdots\!00\)\( T + \)\(19\!\cdots\!50\)\( T^{2} + \)\(45\!\cdots\!00\)\( T^{3} + \)\(89\!\cdots\!47\)\( T^{4} + \)\(14\!\cdots\!00\)\( T^{5} + \)\(21\!\cdots\!00\)\( T^{6} + \)\(29\!\cdots\!00\)\( T^{7} + \)\(37\!\cdots\!03\)\( T^{8} + \)\(39\!\cdots\!00\)\( T^{9} + \)\(33\!\cdots\!50\)\( T^{10} + \)\(19\!\cdots\!00\)\( T^{11} + \)\(73\!\cdots\!49\)\( T^{12} \)
$47$ \( 1 + \)\(23\!\cdots\!80\)\( T + \)\(31\!\cdots\!10\)\( T^{2} + \)\(31\!\cdots\!60\)\( T^{3} + \)\(25\!\cdots\!27\)\( T^{4} + \)\(16\!\cdots\!40\)\( T^{5} + \)\(95\!\cdots\!80\)\( T^{6} + \)\(46\!\cdots\!80\)\( T^{7} + \)\(19\!\cdots\!43\)\( T^{8} + \)\(65\!\cdots\!80\)\( T^{9} + \)\(18\!\cdots\!10\)\( T^{10} + \)\(36\!\cdots\!60\)\( T^{11} + \)\(43\!\cdots\!29\)\( T^{12} \)
$53$ \( 1 - \)\(25\!\cdots\!40\)\( T + \)\(52\!\cdots\!10\)\( T^{2} - \)\(70\!\cdots\!20\)\( T^{3} + \)\(79\!\cdots\!27\)\( T^{4} - \)\(69\!\cdots\!20\)\( T^{5} + \)\(53\!\cdots\!80\)\( T^{6} - \)\(32\!\cdots\!60\)\( T^{7} + \)\(17\!\cdots\!43\)\( T^{8} - \)\(70\!\cdots\!40\)\( T^{9} + \)\(24\!\cdots\!10\)\( T^{10} - \)\(55\!\cdots\!20\)\( T^{11} + \)\(99\!\cdots\!29\)\( T^{12} \)
$59$ \( 1 + \)\(38\!\cdots\!20\)\( T + \)\(10\!\cdots\!54\)\( T^{2} + \)\(83\!\cdots\!00\)\( T^{3} + \)\(55\!\cdots\!15\)\( T^{4} + \)\(44\!\cdots\!00\)\( T^{5} + \)\(18\!\cdots\!80\)\( T^{6} + \)\(12\!\cdots\!00\)\( T^{7} + \)\(42\!\cdots\!15\)\( T^{8} + \)\(17\!\cdots\!00\)\( T^{9} + \)\(60\!\cdots\!94\)\( T^{10} + \)\(60\!\cdots\!80\)\( T^{11} + \)\(43\!\cdots\!41\)\( T^{12} \)
$61$ \( 1 + \)\(45\!\cdots\!28\)\( T + \)\(17\!\cdots\!26\)\( T^{2} + \)\(43\!\cdots\!80\)\( T^{3} + \)\(11\!\cdots\!95\)\( T^{4} + \)\(24\!\cdots\!08\)\( T^{5} + \)\(53\!\cdots\!44\)\( T^{6} + \)\(98\!\cdots\!88\)\( T^{7} + \)\(18\!\cdots\!95\)\( T^{8} + \)\(29\!\cdots\!80\)\( T^{9} + \)\(48\!\cdots\!66\)\( T^{10} + \)\(52\!\cdots\!28\)\( T^{11} + \)\(46\!\cdots\!61\)\( T^{12} \)
$67$ \( 1 - \)\(27\!\cdots\!80\)\( T + \)\(62\!\cdots\!90\)\( T^{2} - \)\(94\!\cdots\!60\)\( T^{3} + \)\(12\!\cdots\!67\)\( T^{4} - \)\(13\!\cdots\!40\)\( T^{5} + \)\(13\!\cdots\!20\)\( T^{6} - \)\(10\!\cdots\!80\)\( T^{7} + \)\(83\!\cdots\!63\)\( T^{8} - \)\(51\!\cdots\!80\)\( T^{9} + \)\(27\!\cdots\!90\)\( T^{10} - \)\(10\!\cdots\!60\)\( T^{11} + \)\(29\!\cdots\!69\)\( T^{12} \)
$71$ \( 1 + \)\(55\!\cdots\!68\)\( T + \)\(42\!\cdots\!86\)\( T^{2} + \)\(17\!\cdots\!80\)\( T^{3} + \)\(81\!\cdots\!95\)\( T^{4} + \)\(25\!\cdots\!08\)\( T^{5} + \)\(92\!\cdots\!64\)\( T^{6} + \)\(23\!\cdots\!68\)\( T^{7} + \)\(65\!\cdots\!95\)\( T^{8} + \)\(12\!\cdots\!80\)\( T^{9} + \)\(27\!\cdots\!66\)\( T^{10} + \)\(31\!\cdots\!68\)\( T^{11} + \)\(51\!\cdots\!21\)\( T^{12} \)
$73$ \( 1 + \)\(11\!\cdots\!80\)\( T + \)\(33\!\cdots\!30\)\( T^{2} + \)\(28\!\cdots\!40\)\( T^{3} + \)\(55\!\cdots\!87\)\( T^{4} + \)\(36\!\cdots\!40\)\( T^{5} + \)\(56\!\cdots\!40\)\( T^{6} + \)\(31\!\cdots\!20\)\( T^{7} + \)\(39\!\cdots\!23\)\( T^{8} + \)\(17\!\cdots\!80\)\( T^{9} + \)\(17\!\cdots\!30\)\( T^{10} + \)\(51\!\cdots\!40\)\( T^{11} + \)\(37\!\cdots\!89\)\( T^{12} \)
$79$ \( 1 - \)\(76\!\cdots\!40\)\( T + \)\(18\!\cdots\!74\)\( T^{2} - \)\(10\!\cdots\!00\)\( T^{3} + \)\(17\!\cdots\!15\)\( T^{4} - \)\(83\!\cdots\!00\)\( T^{5} + \)\(10\!\cdots\!80\)\( T^{6} - \)\(42\!\cdots\!00\)\( T^{7} + \)\(45\!\cdots\!15\)\( T^{8} - \)\(14\!\cdots\!00\)\( T^{9} + \)\(12\!\cdots\!94\)\( T^{10} - \)\(26\!\cdots\!60\)\( T^{11} + \)\(17\!\cdots\!21\)\( T^{12} \)
$83$ \( 1 - \)\(44\!\cdots\!60\)\( T + \)\(84\!\cdots\!90\)\( T^{2} - \)\(34\!\cdots\!80\)\( T^{3} + \)\(35\!\cdots\!67\)\( T^{4} - \)\(11\!\cdots\!80\)\( T^{5} + \)\(10\!\cdots\!20\)\( T^{6} - \)\(31\!\cdots\!40\)\( T^{7} + \)\(27\!\cdots\!63\)\( T^{8} - \)\(74\!\cdots\!60\)\( T^{9} + \)\(50\!\cdots\!90\)\( T^{10} - \)\(74\!\cdots\!80\)\( T^{11} + \)\(46\!\cdots\!69\)\( T^{12} \)
$89$ \( 1 + \)\(18\!\cdots\!80\)\( T + \)\(39\!\cdots\!34\)\( T^{2} + \)\(45\!\cdots\!00\)\( T^{3} + \)\(61\!\cdots\!15\)\( T^{4} + \)\(57\!\cdots\!00\)\( T^{5} + \)\(60\!\cdots\!80\)\( T^{6} + \)\(45\!\cdots\!00\)\( T^{7} + \)\(38\!\cdots\!15\)\( T^{8} + \)\(23\!\cdots\!00\)\( T^{9} + \)\(15\!\cdots\!94\)\( T^{10} + \)\(60\!\cdots\!20\)\( T^{11} + \)\(25\!\cdots\!61\)\( T^{12} \)
$97$ \( 1 - \)\(11\!\cdots\!20\)\( T + \)\(18\!\cdots\!10\)\( T^{2} - \)\(23\!\cdots\!40\)\( T^{3} + \)\(25\!\cdots\!27\)\( T^{4} - \)\(79\!\cdots\!60\)\( T^{5} + \)\(26\!\cdots\!80\)\( T^{6} - \)\(67\!\cdots\!20\)\( T^{7} + \)\(18\!\cdots\!43\)\( T^{8} - \)\(14\!\cdots\!20\)\( T^{9} + \)\(95\!\cdots\!10\)\( T^{10} - \)\(49\!\cdots\!40\)\( T^{11} + \)\(37\!\cdots\!29\)\( T^{12} \)
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