Properties

Label 1.76.a.a
Level 1
Weight 76
Character orbit 1.a
Self dual yes
Analytic conductor 35.623
Analytic rank 0
Dimension 6
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 1 \)
Weight: \( k \) \(=\) \( 76 \)
Character orbit: \([\chi]\) \(=\) 1.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(35.6228392822\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: multiple of \( 2^{58}\cdot 3^{26}\cdot 5^{7}\cdot 7^{3}\cdot 11\cdot 13\cdot 19 \)
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.

\(f(q)\) \(=\) \( q +(-9513470340 + \beta_{1}) q^{2} +(-130848727303118340 + 60703 \beta_{1} - \beta_{2}) q^{3} +(\)\(28\!\cdots\!88\)\( + 9832402747 \beta_{1} + 16382 \beta_{2} + \beta_{3}) q^{4} +(-\)\(64\!\cdots\!90\)\( + 333300085276404 \beta_{1} + 56977362 \beta_{2} + 340 \beta_{3} + \beta_{4}) q^{5} +(\)\(52\!\cdots\!32\)\( - 362421303601527504 \beta_{1} + 81392735112 \beta_{2} - 1201700 \beta_{3} + 10 \beta_{4} - \beta_{5}) q^{6} +(\)\(32\!\cdots\!00\)\( + 20646145567510125430 \beta_{1} + 17742168701918 \beta_{2} - 28211984 \beta_{3} + 166380 \beta_{4} - 96 \beta_{5}) q^{7} +(\)\(73\!\cdots\!20\)\( + \)\(23\!\cdots\!24\)\( \beta_{1} + 1349226115346160 \beta_{2} + 81567629192 \beta_{3} + 43973360 \beta_{4} + 22248 \beta_{5}) q^{8} +(\)\(35\!\cdots\!97\)\( - \)\(10\!\cdots\!32\)\( \beta_{1} + 220475936644473996 \beta_{2} + 9061217513400 \beta_{3} - 1671114570 \beta_{4} - 1665408 \beta_{5}) q^{9} +O(q^{10})\) \( q +(-9513470340 + \beta_{1}) q^{2} +(-130848727303118340 + 60703 \beta_{1} - \beta_{2}) q^{3} +(\)\(28\!\cdots\!88\)\( + 9832402747 \beta_{1} + 16382 \beta_{2} + \beta_{3}) q^{4} +(-\)\(64\!\cdots\!90\)\( + 333300085276404 \beta_{1} + 56977362 \beta_{2} + 340 \beta_{3} + \beta_{4}) q^{5} +(\)\(52\!\cdots\!32\)\( - 362421303601527504 \beta_{1} + 81392735112 \beta_{2} - 1201700 \beta_{3} + 10 \beta_{4} - \beta_{5}) q^{6} +(\)\(32\!\cdots\!00\)\( + 20646145567510125430 \beta_{1} + 17742168701918 \beta_{2} - 28211984 \beta_{3} + 166380 \beta_{4} - 96 \beta_{5}) q^{7} +(\)\(73\!\cdots\!20\)\( + \)\(23\!\cdots\!24\)\( \beta_{1} + 1349226115346160 \beta_{2} + 81567629192 \beta_{3} + 43973360 \beta_{4} + 22248 \beta_{5}) q^{8} +(\)\(35\!\cdots\!97\)\( - \)\(10\!\cdots\!32\)\( \beta_{1} + 220475936644473996 \beta_{2} + 9061217513400 \beta_{3} - 1671114570 \beta_{4} - 1665408 \beta_{5}) q^{9} +(\)\(22\!\cdots\!60\)\( + \)\(22\!\cdots\!74\)\( \beta_{1} + 1529304067596928672 \beta_{2} + 869735375021360 \beta_{3} - 38186921784 \beta_{4} + 75315180 \beta_{5}) q^{10} +(-\)\(15\!\cdots\!48\)\( + \)\(36\!\cdots\!69\)\( \beta_{1} - 81125168161221157563 \beta_{2} - 4721555795882272 \beta_{3} + 3747958317080 \beta_{4} - 2418303168 \beta_{5}) q^{11} +(-\)\(19\!\cdots\!80\)\( - \)\(18\!\cdots\!84\)\( \beta_{1} - \)\(47\!\cdots\!36\)\( \beta_{2} - 523043049227823108 \beta_{3} - 111178174832640 \beta_{4} + 59452674048 \beta_{5}) q^{12} +(\)\(88\!\cdots\!70\)\( + \)\(30\!\cdots\!88\)\( \beta_{1} - \)\(44\!\cdots\!82\)\( \beta_{2} + 5942217982613612692 \beta_{3} + 1816281211946385 \beta_{4} - 1168346295552 \beta_{5}) q^{13} +(\)\(13\!\cdots\!96\)\( + \)\(40\!\cdots\!76\)\( \beta_{1} - \)\(78\!\cdots\!84\)\( \beta_{2} + 96176929298921698328 \beta_{3} - 15630807774027100 \beta_{4} + 18850673461590 \beta_{5}) q^{14} +(-\)\(51\!\cdots\!80\)\( - \)\(71\!\cdots\!22\)\( \beta_{1} + \)\(53\!\cdots\!34\)\( \beta_{2} - \)\(24\!\cdots\!80\)\( \beta_{3} - 13957315940273548 \beta_{4} - 253911008827040 \beta_{5}) q^{15} +(\)\(44\!\cdots\!36\)\( + \)\(34\!\cdots\!04\)\( \beta_{1} + \)\(14\!\cdots\!60\)\( \beta_{2} + \)\(15\!\cdots\!64\)\( \beta_{3} + 2536277958175820160 \beta_{4} + 2881303642780224 \beta_{5}) q^{16} +(\)\(30\!\cdots\!30\)\( + \)\(80\!\cdots\!56\)\( \beta_{1} - \)\(43\!\cdots\!52\)\( \beta_{2} + \)\(64\!\cdots\!44\)\( \beta_{3} - 41006711266916156330 \beta_{4} - 27593889782868864 \beta_{5}) q^{17} +(-\)\(73\!\cdots\!40\)\( + \)\(63\!\cdots\!01\)\( \beta_{1} - \)\(12\!\cdots\!32\)\( \beta_{2} - \)\(14\!\cdots\!64\)\( \beta_{3} + \)\(39\!\cdots\!80\)\( \beta_{4} + 221245792113362184 \beta_{5}) q^{18} +(\)\(17\!\cdots\!80\)\( + \)\(31\!\cdots\!75\)\( \beta_{1} + \)\(10\!\cdots\!59\)\( \beta_{2} + \)\(41\!\cdots\!08\)\( \beta_{3} - \)\(23\!\cdots\!60\)\( \beta_{4} - 1446527802339380544 \beta_{5}) q^{19} +(\)\(15\!\cdots\!80\)\( + \)\(34\!\cdots\!02\)\( \beta_{1} + \)\(55\!\cdots\!56\)\( \beta_{2} + \)\(54\!\cdots\!70\)\( \beta_{3} + \)\(68\!\cdots\!88\)\( \beta_{4} + 7140887499810816000 \beta_{5}) q^{20} +(-\)\(16\!\cdots\!88\)\( + \)\(10\!\cdots\!96\)\( \beta_{1} - \)\(50\!\cdots\!44\)\( \beta_{2} - \)\(55\!\cdots\!72\)\( \beta_{3} + \)\(31\!\cdots\!00\)\( \beta_{4} - 18929889334740199680 \beta_{5}) q^{21} +(\)\(25\!\cdots\!20\)\( - \)\(30\!\cdots\!52\)\( \beta_{1} - \)\(16\!\cdots\!92\)\( \beta_{2} + \)\(11\!\cdots\!32\)\( \beta_{3} - \)\(54\!\cdots\!90\)\( \beta_{4} - 83973431097663605067 \beta_{5}) q^{22} +(\)\(25\!\cdots\!80\)\( - \)\(20\!\cdots\!54\)\( \beta_{1} + \)\(20\!\cdots\!86\)\( \beta_{2} + \)\(11\!\cdots\!60\)\( \beta_{3} + \)\(36\!\cdots\!00\)\( \beta_{4} + \)\(16\!\cdots\!40\)\( \beta_{5}) q^{23} +(-\)\(12\!\cdots\!60\)\( - \)\(31\!\cdots\!24\)\( \beta_{1} + \)\(33\!\cdots\!00\)\( \beta_{2} - \)\(97\!\cdots\!64\)\( \beta_{3} - \)\(13\!\cdots\!60\)\( \beta_{4} - \)\(13\!\cdots\!04\)\( \beta_{5}) q^{24} +(\)\(32\!\cdots\!75\)\( + \)\(83\!\cdots\!40\)\( \beta_{1} - \)\(38\!\cdots\!80\)\( \beta_{2} + \)\(22\!\cdots\!00\)\( \beta_{3} + \)\(19\!\cdots\!60\)\( \beta_{4} + \)\(84\!\cdots\!00\)\( \beta_{5}) q^{25} +(\)\(19\!\cdots\!92\)\( + \)\(15\!\cdots\!18\)\( \beta_{1} - \)\(25\!\cdots\!44\)\( \beta_{2} + \)\(61\!\cdots\!20\)\( \beta_{3} + \)\(10\!\cdots\!80\)\( \beta_{4} - \)\(40\!\cdots\!68\)\( \beta_{5}) q^{26} +(-\)\(17\!\cdots\!20\)\( + \)\(19\!\cdots\!66\)\( \beta_{1} - \)\(24\!\cdots\!90\)\( \beta_{2} - \)\(42\!\cdots\!12\)\( \beta_{3} - \)\(63\!\cdots\!60\)\( \beta_{4} + \)\(14\!\cdots\!72\)\( \beta_{5}) q^{27} +(\)\(24\!\cdots\!20\)\( + \)\(15\!\cdots\!48\)\( \beta_{1} + \)\(15\!\cdots\!16\)\( \beta_{2} - \)\(30\!\cdots\!92\)\( \beta_{3} - \)\(68\!\cdots\!60\)\( \beta_{4} - \)\(36\!\cdots\!48\)\( \beta_{5}) q^{28} +(\)\(24\!\cdots\!70\)\( - \)\(40\!\cdots\!84\)\( \beta_{1} + \)\(21\!\cdots\!74\)\( \beta_{2} + \)\(83\!\cdots\!64\)\( \beta_{3} + \)\(18\!\cdots\!05\)\( \beta_{4} + \)\(22\!\cdots\!52\)\( \beta_{5}) q^{29} +(-\)\(42\!\cdots\!80\)\( - \)\(10\!\cdots\!72\)\( \beta_{1} - \)\(29\!\cdots\!16\)\( \beta_{2} - \)\(18\!\cdots\!20\)\( \beta_{3} - \)\(12\!\cdots\!68\)\( \beta_{4} + \)\(34\!\cdots\!50\)\( \beta_{5}) q^{30} +(-\)\(69\!\cdots\!48\)\( - \)\(73\!\cdots\!48\)\( \beta_{1} + \)\(30\!\cdots\!96\)\( \beta_{2} - \)\(70\!\cdots\!76\)\( \beta_{3} + \)\(45\!\cdots\!40\)\( \beta_{4} - \)\(21\!\cdots\!44\)\( \beta_{5}) q^{31} +(\)\(19\!\cdots\!60\)\( + \)\(40\!\cdots\!48\)\( \beta_{1} + \)\(13\!\cdots\!56\)\( \beta_{2} + \)\(46\!\cdots\!24\)\( \beta_{3} - \)\(95\!\cdots\!80\)\( \beta_{4} + \)\(73\!\cdots\!56\)\( \beta_{5}) q^{32} +(\)\(98\!\cdots\!20\)\( + \)\(23\!\cdots\!28\)\( \beta_{1} + \)\(13\!\cdots\!04\)\( \beta_{2} - \)\(69\!\cdots\!76\)\( \beta_{3} + \)\(50\!\cdots\!70\)\( \beta_{4} - \)\(13\!\cdots\!44\)\( \beta_{5}) q^{33} +(\)\(50\!\cdots\!76\)\( + \)\(49\!\cdots\!46\)\( \beta_{1} - \)\(15\!\cdots\!80\)\( \beta_{2} - \)\(13\!\cdots\!04\)\( \beta_{3} + \)\(34\!\cdots\!40\)\( \beta_{4} - \)\(11\!\cdots\!04\)\( \beta_{5}) q^{34} +(\)\(45\!\cdots\!60\)\( - \)\(80\!\cdots\!16\)\( \beta_{1} - \)\(32\!\cdots\!48\)\( \beta_{2} + \)\(21\!\cdots\!60\)\( \beta_{3} - \)\(83\!\cdots\!44\)\( \beta_{4} + \)\(17\!\cdots\!80\)\( \beta_{5}) q^{35} +(\)\(29\!\cdots\!36\)\( - \)\(90\!\cdots\!41\)\( \beta_{1} + \)\(24\!\cdots\!90\)\( \beta_{2} + \)\(28\!\cdots\!29\)\( \beta_{3} - \)\(54\!\cdots\!40\)\( \beta_{4} - \)\(63\!\cdots\!76\)\( \beta_{5}) q^{36} +(\)\(16\!\cdots\!90\)\( - \)\(58\!\cdots\!24\)\( \beta_{1} - \)\(14\!\cdots\!58\)\( \beta_{2} - \)\(12\!\cdots\!84\)\( \beta_{3} + \)\(29\!\cdots\!05\)\( \beta_{4} + \)\(11\!\cdots\!04\)\( \beta_{5}) q^{37} +(\)\(20\!\cdots\!80\)\( + \)\(38\!\cdots\!84\)\( \beta_{1} - \)\(46\!\cdots\!96\)\( \beta_{2} + \)\(21\!\cdots\!84\)\( \beta_{3} + \)\(22\!\cdots\!70\)\( \beta_{4} + \)\(71\!\cdots\!71\)\( \beta_{5}) q^{38} +(\)\(41\!\cdots\!24\)\( + \)\(39\!\cdots\!30\)\( \beta_{1} - \)\(20\!\cdots\!62\)\( \beta_{2} - \)\(27\!\cdots\!64\)\( \beta_{3} - \)\(15\!\cdots\!20\)\( \beta_{4} - \)\(10\!\cdots\!28\)\( \beta_{5}) q^{39} +(\)\(14\!\cdots\!00\)\( + \)\(41\!\cdots\!80\)\( \beta_{1} + \)\(70\!\cdots\!40\)\( \beta_{2} + \)\(16\!\cdots\!00\)\( \beta_{3} + \)\(38\!\cdots\!20\)\( \beta_{4} + \)\(32\!\cdots\!00\)\( \beta_{5}) q^{40} +(\)\(83\!\cdots\!02\)\( - \)\(62\!\cdots\!08\)\( \beta_{1} + \)\(49\!\cdots\!16\)\( \beta_{2} - \)\(66\!\cdots\!96\)\( \beta_{3} + \)\(12\!\cdots\!40\)\( \beta_{4} - \)\(37\!\cdots\!24\)\( \beta_{5}) q^{41} +(\)\(72\!\cdots\!80\)\( - \)\(31\!\cdots\!36\)\( \beta_{1} - \)\(55\!\cdots\!40\)\( \beta_{2} + \)\(83\!\cdots\!36\)\( \beta_{3} - \)\(27\!\cdots\!20\)\( \beta_{4} - \)\(80\!\cdots\!16\)\( \beta_{5}) q^{42} +(\)\(46\!\cdots\!00\)\( + \)\(18\!\cdots\!69\)\( \beta_{1} - \)\(10\!\cdots\!07\)\( \beta_{2} + \)\(16\!\cdots\!20\)\( \beta_{3} + \)\(77\!\cdots\!00\)\( \beta_{4} + \)\(44\!\cdots\!80\)\( \beta_{5}) q^{43} +(-\)\(14\!\cdots\!24\)\( + \)\(35\!\cdots\!36\)\( \beta_{1} + \)\(89\!\cdots\!80\)\( \beta_{2} - \)\(53\!\cdots\!44\)\( \beta_{3} - \)\(63\!\cdots\!60\)\( \beta_{4} - \)\(80\!\cdots\!24\)\( \beta_{5}) q^{44} +(-\)\(39\!\cdots\!30\)\( + \)\(20\!\cdots\!88\)\( \beta_{1} + \)\(22\!\cdots\!14\)\( \beta_{2} - \)\(84\!\cdots\!20\)\( \beta_{3} - \)\(15\!\cdots\!03\)\( \beta_{4} - \)\(41\!\cdots\!00\)\( \beta_{5}) q^{45} +(-\)\(13\!\cdots\!48\)\( + \)\(59\!\cdots\!92\)\( \beta_{1} + \)\(92\!\cdots\!36\)\( \beta_{2} + \)\(77\!\cdots\!44\)\( \beta_{3} + \)\(42\!\cdots\!40\)\( \beta_{4} + \)\(35\!\cdots\!06\)\( \beta_{5}) q^{46} +(-\)\(23\!\cdots\!80\)\( - \)\(77\!\cdots\!40\)\( \beta_{1} - \)\(29\!\cdots\!24\)\( \beta_{2} + \)\(61\!\cdots\!40\)\( \beta_{3} - \)\(95\!\cdots\!00\)\( \beta_{4} - \)\(72\!\cdots\!40\)\( \beta_{5}) q^{47} +(-\)\(13\!\cdots\!20\)\( - \)\(31\!\cdots\!52\)\( \beta_{1} - \)\(19\!\cdots\!08\)\( \beta_{2} - \)\(21\!\cdots\!76\)\( \beta_{3} - \)\(20\!\cdots\!80\)\( \beta_{4} + \)\(53\!\cdots\!56\)\( \beta_{5}) q^{48} +(-\)\(95\!\cdots\!07\)\( - \)\(43\!\cdots\!28\)\( \beta_{1} - \)\(54\!\cdots\!84\)\( \beta_{2} + \)\(11\!\cdots\!84\)\( \beta_{3} - \)\(39\!\cdots\!60\)\( \beta_{4} + \)\(20\!\cdots\!56\)\( \beta_{5}) q^{49} +(\)\(52\!\cdots\!00\)\( + \)\(89\!\cdots\!15\)\( \beta_{1} + \)\(69\!\cdots\!20\)\( \beta_{2} + \)\(41\!\cdots\!00\)\( \beta_{3} + \)\(13\!\cdots\!60\)\( \beta_{4} - \)\(92\!\cdots\!00\)\( \beta_{5}) q^{50} +(\)\(47\!\cdots\!72\)\( + \)\(21\!\cdots\!90\)\( \beta_{1} - \)\(80\!\cdots\!74\)\( \beta_{2} - \)\(13\!\cdots\!48\)\( \beta_{3} - \)\(43\!\cdots\!40\)\( \beta_{4} - \)\(10\!\cdots\!76\)\( \beta_{5}) q^{51} +(\)\(69\!\cdots\!00\)\( + \)\(22\!\cdots\!22\)\( \beta_{1} - \)\(49\!\cdots\!96\)\( \beta_{2} - \)\(10\!\cdots\!10\)\( \beta_{3} - \)\(69\!\cdots\!00\)\( \beta_{4} + \)\(17\!\cdots\!60\)\( \beta_{5}) q^{52} +(\)\(10\!\cdots\!10\)\( + \)\(47\!\cdots\!04\)\( \beta_{1} - \)\(34\!\cdots\!58\)\( \beta_{2} - \)\(32\!\cdots\!48\)\( \beta_{3} + \)\(16\!\cdots\!85\)\( \beta_{4} + \)\(56\!\cdots\!88\)\( \beta_{5}) q^{53} +(\)\(13\!\cdots\!80\)\( - \)\(31\!\cdots\!40\)\( \beta_{1} + \)\(14\!\cdots\!44\)\( \beta_{2} + \)\(14\!\cdots\!88\)\( \beta_{3} - \)\(65\!\cdots\!60\)\( \beta_{4} - \)\(25\!\cdots\!94\)\( \beta_{5}) q^{54} +(\)\(72\!\cdots\!20\)\( - \)\(37\!\cdots\!42\)\( \beta_{1} - \)\(78\!\cdots\!26\)\( \beta_{2} - \)\(84\!\cdots\!20\)\( \beta_{3} - \)\(25\!\cdots\!48\)\( \beta_{4} + \)\(26\!\cdots\!00\)\( \beta_{5}) q^{55} +(\)\(48\!\cdots\!20\)\( + \)\(40\!\cdots\!36\)\( \beta_{1} - \)\(69\!\cdots\!36\)\( \beta_{2} - \)\(10\!\cdots\!36\)\( \beta_{3} + \)\(23\!\cdots\!80\)\( \beta_{4} + \)\(80\!\cdots\!32\)\( \beta_{5}) q^{56} +(-\)\(11\!\cdots\!40\)\( + \)\(25\!\cdots\!84\)\( \beta_{1} - \)\(27\!\cdots\!64\)\( \beta_{2} - \)\(51\!\cdots\!80\)\( \beta_{3} - \)\(78\!\cdots\!50\)\( \beta_{4} - \)\(30\!\cdots\!20\)\( \beta_{5}) q^{57} +(-\)\(29\!\cdots\!80\)\( + \)\(52\!\cdots\!50\)\( \beta_{1} + \)\(12\!\cdots\!48\)\( \beta_{2} + \)\(11\!\cdots\!04\)\( \beta_{3} + \)\(28\!\cdots\!20\)\( \beta_{4} + \)\(29\!\cdots\!76\)\( \beta_{5}) q^{58} +(-\)\(41\!\cdots\!60\)\( + \)\(28\!\cdots\!09\)\( \beta_{1} + \)\(10\!\cdots\!49\)\( \beta_{2} + \)\(31\!\cdots\!12\)\( \beta_{3} - \)\(87\!\cdots\!00\)\( \beta_{4} + \)\(62\!\cdots\!80\)\( \beta_{5}) q^{59} +(-\)\(52\!\cdots\!40\)\( - \)\(15\!\cdots\!36\)\( \beta_{1} + \)\(52\!\cdots\!92\)\( \beta_{2} - \)\(11\!\cdots\!40\)\( \beta_{3} + \)\(91\!\cdots\!76\)\( \beta_{4} - \)\(22\!\cdots\!20\)\( \beta_{5}) q^{60} +(-\)\(42\!\cdots\!98\)\( - \)\(26\!\cdots\!00\)\( \beta_{1} + \)\(16\!\cdots\!50\)\( \beta_{2} + \)\(60\!\cdots\!00\)\( \beta_{3} + \)\(40\!\cdots\!25\)\( \beta_{4} + \)\(19\!\cdots\!00\)\( \beta_{5}) q^{61} +(-\)\(48\!\cdots\!80\)\( - \)\(21\!\cdots\!80\)\( \beta_{1} - \)\(20\!\cdots\!36\)\( \beta_{2} + \)\(85\!\cdots\!56\)\( \beta_{3} - \)\(67\!\cdots\!20\)\( \beta_{4} + \)\(30\!\cdots\!64\)\( \beta_{5}) q^{62} +(\)\(71\!\cdots\!40\)\( - \)\(19\!\cdots\!58\)\( \beta_{1} + \)\(23\!\cdots\!18\)\( \beta_{2} + \)\(61\!\cdots\!44\)\( \beta_{3} - \)\(31\!\cdots\!80\)\( \beta_{4} - \)\(85\!\cdots\!64\)\( \beta_{5}) q^{63} +(\)\(79\!\cdots\!08\)\( + \)\(23\!\cdots\!88\)\( \beta_{1} + \)\(63\!\cdots\!76\)\( \beta_{2} - \)\(57\!\cdots\!20\)\( \beta_{3} + \)\(20\!\cdots\!80\)\( \beta_{4} + \)\(45\!\cdots\!32\)\( \beta_{5}) q^{64} +(\)\(20\!\cdots\!20\)\( + \)\(68\!\cdots\!68\)\( \beta_{1} + \)\(36\!\cdots\!04\)\( \beta_{2} - \)\(12\!\cdots\!80\)\( \beta_{3} - \)\(72\!\cdots\!88\)\( \beta_{4} + \)\(49\!\cdots\!60\)\( \beta_{5}) q^{65} +(\)\(15\!\cdots\!64\)\( - \)\(20\!\cdots\!20\)\( \beta_{1} - \)\(82\!\cdots\!52\)\( \beta_{2} + \)\(21\!\cdots\!56\)\( \beta_{3} - \)\(40\!\cdots\!20\)\( \beta_{4} + \)\(78\!\cdots\!12\)\( \beta_{5}) q^{66} +(\)\(15\!\cdots\!80\)\( - \)\(88\!\cdots\!29\)\( \beta_{1} - \)\(70\!\cdots\!97\)\( \beta_{2} + \)\(11\!\cdots\!04\)\( \beta_{3} + \)\(36\!\cdots\!20\)\( \beta_{4} - \)\(31\!\cdots\!24\)\( \beta_{5}) q^{67} +(\)\(20\!\cdots\!60\)\( - \)\(44\!\cdots\!46\)\( \beta_{1} + \)\(11\!\cdots\!80\)\( \beta_{2} + \)\(10\!\cdots\!58\)\( \beta_{3} + \)\(73\!\cdots\!40\)\( \beta_{4} - \)\(58\!\cdots\!48\)\( \beta_{5}) q^{68} +(-\)\(23\!\cdots\!56\)\( - \)\(10\!\cdots\!04\)\( \beta_{1} - \)\(32\!\cdots\!48\)\( \beta_{2} - \)\(63\!\cdots\!20\)\( \beta_{3} - \)\(17\!\cdots\!40\)\( \beta_{4} + \)\(31\!\cdots\!84\)\( \beta_{5}) q^{69} +(-\)\(57\!\cdots\!40\)\( + \)\(42\!\cdots\!84\)\( \beta_{1} + \)\(14\!\cdots\!52\)\( \beta_{2} - \)\(11\!\cdots\!60\)\( \beta_{3} + \)\(24\!\cdots\!96\)\( \beta_{4} - \)\(30\!\cdots\!00\)\( \beta_{5}) q^{70} +(-\)\(42\!\cdots\!48\)\( - \)\(12\!\cdots\!50\)\( \beta_{1} - \)\(36\!\cdots\!50\)\( \beta_{2} + \)\(41\!\cdots\!00\)\( \beta_{3} - \)\(35\!\cdots\!00\)\( \beta_{4} - \)\(58\!\cdots\!00\)\( \beta_{5}) q^{71} +(-\)\(35\!\cdots\!60\)\( + \)\(17\!\cdots\!72\)\( \beta_{1} - \)\(23\!\cdots\!88\)\( \beta_{2} - \)\(79\!\cdots\!88\)\( \beta_{3} - \)\(58\!\cdots\!40\)\( \beta_{4} + \)\(16\!\cdots\!28\)\( \beta_{5}) q^{72} +(-\)\(51\!\cdots\!70\)\( - \)\(10\!\cdots\!52\)\( \beta_{1} - \)\(24\!\cdots\!48\)\( \beta_{2} - \)\(21\!\cdots\!96\)\( \beta_{3} + \)\(39\!\cdots\!70\)\( \beta_{4} - \)\(72\!\cdots\!24\)\( \beta_{5}) q^{73} +(-\)\(40\!\cdots\!64\)\( - \)\(24\!\cdots\!82\)\( \beta_{1} + \)\(47\!\cdots\!04\)\( \beta_{2} - \)\(10\!\cdots\!04\)\( \beta_{3} - \)\(51\!\cdots\!40\)\( \beta_{4} - \)\(19\!\cdots\!36\)\( \beta_{5}) q^{74} +(\)\(32\!\cdots\!00\)\( - \)\(98\!\cdots\!95\)\( \beta_{1} + \)\(36\!\cdots\!65\)\( \beta_{2} + \)\(43\!\cdots\!00\)\( \beta_{3} - \)\(47\!\cdots\!80\)\( \beta_{4} + \)\(13\!\cdots\!00\)\( \beta_{5}) q^{75} +(\)\(16\!\cdots\!40\)\( + \)\(14\!\cdots\!72\)\( \beta_{1} + \)\(16\!\cdots\!68\)\( \beta_{2} + \)\(39\!\cdots\!08\)\( \beta_{3} + \)\(17\!\cdots\!60\)\( \beta_{4} + \)\(25\!\cdots\!24\)\( \beta_{5}) q^{76} +(\)\(26\!\cdots\!00\)\( - \)\(63\!\cdots\!44\)\( \beta_{1} - \)\(20\!\cdots\!56\)\( \beta_{2} + \)\(45\!\cdots\!64\)\( \beta_{3} - \)\(82\!\cdots\!80\)\( \beta_{4} + \)\(48\!\cdots\!16\)\( \beta_{5}) q^{77} +(\)\(22\!\cdots\!00\)\( + \)\(29\!\cdots\!32\)\( \beta_{1} - \)\(58\!\cdots\!52\)\( \beta_{2} - \)\(93\!\cdots\!12\)\( \beta_{3} - \)\(96\!\cdots\!60\)\( \beta_{4} - \)\(26\!\cdots\!78\)\( \beta_{5}) q^{78} +(\)\(19\!\cdots\!20\)\( - \)\(22\!\cdots\!56\)\( \beta_{1} + \)\(68\!\cdots\!28\)\( \beta_{2} - \)\(19\!\cdots\!80\)\( \beta_{3} - \)\(21\!\cdots\!60\)\( \beta_{4} + \)\(15\!\cdots\!76\)\( \beta_{5}) q^{79} +(\)\(20\!\cdots\!60\)\( + \)\(85\!\cdots\!44\)\( \beta_{1} + \)\(69\!\cdots\!32\)\( \beta_{2} + \)\(62\!\cdots\!40\)\( \beta_{3} + \)\(43\!\cdots\!36\)\( \beta_{4} + \)\(64\!\cdots\!00\)\( \beta_{5}) q^{80} +(\)\(49\!\cdots\!81\)\( - \)\(16\!\cdots\!32\)\( \beta_{1} + \)\(42\!\cdots\!88\)\( \beta_{2} - \)\(94\!\cdots\!96\)\( \beta_{3} + \)\(11\!\cdots\!50\)\( \beta_{4} - \)\(11\!\cdots\!80\)\( \beta_{5}) q^{81} +(-\)\(42\!\cdots\!80\)\( - \)\(10\!\cdots\!70\)\( \beta_{1} - \)\(56\!\cdots\!56\)\( \beta_{2} - \)\(98\!\cdots\!24\)\( \beta_{3} - \)\(31\!\cdots\!20\)\( \beta_{4} + \)\(45\!\cdots\!44\)\( \beta_{5}) q^{82} +(-\)\(13\!\cdots\!60\)\( - \)\(11\!\cdots\!65\)\( \beta_{1} - \)\(11\!\cdots\!93\)\( \beta_{2} + \)\(13\!\cdots\!00\)\( \beta_{3} + \)\(12\!\cdots\!00\)\( \beta_{4} + \)\(11\!\cdots\!00\)\( \beta_{5}) q^{83} +(-\)\(15\!\cdots\!44\)\( + \)\(48\!\cdots\!56\)\( \beta_{1} - \)\(62\!\cdots\!96\)\( \beta_{2} - \)\(20\!\cdots\!36\)\( \beta_{3} + \)\(34\!\cdots\!80\)\( \beta_{4} + \)\(76\!\cdots\!12\)\( \beta_{5}) q^{84} +(-\)\(60\!\cdots\!40\)\( + \)\(10\!\cdots\!04\)\( \beta_{1} + \)\(12\!\cdots\!12\)\( \beta_{2} + \)\(12\!\cdots\!60\)\( \beta_{3} - \)\(46\!\cdots\!14\)\( \beta_{4} + \)\(20\!\cdots\!80\)\( \beta_{5}) q^{85} +(\)\(12\!\cdots\!72\)\( + \)\(73\!\cdots\!48\)\( \beta_{1} + \)\(46\!\cdots\!00\)\( \beta_{2} + \)\(58\!\cdots\!28\)\( \beta_{3} + \)\(58\!\cdots\!70\)\( \beta_{4} - \)\(73\!\cdots\!67\)\( \beta_{5}) q^{86} +(-\)\(23\!\cdots\!60\)\( - \)\(12\!\cdots\!86\)\( \beta_{1} - \)\(35\!\cdots\!42\)\( \beta_{2} - \)\(67\!\cdots\!84\)\( \beta_{3} - \)\(95\!\cdots\!20\)\( \beta_{4} + \)\(11\!\cdots\!04\)\( \beta_{5}) q^{87} +(-\)\(80\!\cdots\!60\)\( - \)\(16\!\cdots\!52\)\( \beta_{1} - \)\(13\!\cdots\!80\)\( \beta_{2} - \)\(10\!\cdots\!16\)\( \beta_{3} - \)\(45\!\cdots\!80\)\( \beta_{4} + \)\(67\!\cdots\!96\)\( \beta_{5}) q^{88} +(\)\(89\!\cdots\!10\)\( - \)\(17\!\cdots\!92\)\( \beta_{1} - \)\(37\!\cdots\!08\)\( \beta_{2} + \)\(12\!\cdots\!92\)\( \beta_{3} + \)\(25\!\cdots\!90\)\( \beta_{4} - \)\(28\!\cdots\!04\)\( \beta_{5}) q^{89} +(\)\(14\!\cdots\!20\)\( - \)\(32\!\cdots\!22\)\( \beta_{1} + \)\(11\!\cdots\!84\)\( \beta_{2} + \)\(16\!\cdots\!20\)\( \beta_{3} + \)\(40\!\cdots\!52\)\( \beta_{4} + \)\(19\!\cdots\!60\)\( \beta_{5}) q^{90} +(\)\(57\!\cdots\!72\)\( + \)\(40\!\cdots\!56\)\( \beta_{1} + \)\(21\!\cdots\!24\)\( \beta_{2} - \)\(30\!\cdots\!96\)\( \beta_{3} - \)\(23\!\cdots\!20\)\( \beta_{4} - \)\(16\!\cdots\!08\)\( \beta_{5}) q^{91} +(\)\(31\!\cdots\!80\)\( - \)\(56\!\cdots\!28\)\( \beta_{1} + \)\(21\!\cdots\!16\)\( \beta_{2} + \)\(56\!\cdots\!16\)\( \beta_{3} - \)\(10\!\cdots\!20\)\( \beta_{4} + \)\(48\!\cdots\!04\)\( \beta_{5}) q^{92} +(-\)\(27\!\cdots\!80\)\( + \)\(29\!\cdots\!32\)\( \beta_{1} + \)\(25\!\cdots\!96\)\( \beta_{2} - \)\(89\!\cdots\!08\)\( \beta_{3} + \)\(13\!\cdots\!60\)\( \beta_{4} + \)\(32\!\cdots\!48\)\( \beta_{5}) q^{93} +(-\)\(49\!\cdots\!84\)\( - \)\(14\!\cdots\!20\)\( \beta_{1} - \)\(23\!\cdots\!36\)\( \beta_{2} - \)\(10\!\cdots\!52\)\( \beta_{3} + \)\(24\!\cdots\!40\)\( \beta_{4} - \)\(27\!\cdots\!44\)\( \beta_{5}) q^{94} +(\)\(32\!\cdots\!00\)\( + \)\(15\!\cdots\!70\)\( \beta_{1} + \)\(10\!\cdots\!10\)\( \beta_{2} + \)\(34\!\cdots\!00\)\( \beta_{3} - \)\(34\!\cdots\!20\)\( \beta_{4} + \)\(23\!\cdots\!00\)\( \beta_{5}) q^{95} +(-\)\(14\!\cdots\!68\)\( - \)\(89\!\cdots\!88\)\( \beta_{1} - \)\(21\!\cdots\!20\)\( \beta_{2} - \)\(90\!\cdots\!08\)\( \beta_{3} - \)\(42\!\cdots\!20\)\( \beta_{4} + \)\(19\!\cdots\!72\)\( \beta_{5}) q^{96} +(-\)\(12\!\cdots\!30\)\( + \)\(94\!\cdots\!44\)\( \beta_{1} + \)\(25\!\cdots\!68\)\( \beta_{2} + \)\(16\!\cdots\!48\)\( \beta_{3} + \)\(53\!\cdots\!90\)\( \beta_{4} - \)\(41\!\cdots\!88\)\( \beta_{5}) q^{97} +(-\)\(27\!\cdots\!20\)\( - \)\(84\!\cdots\!99\)\( \beta_{1} + \)\(14\!\cdots\!64\)\( \beta_{2} - \)\(57\!\cdots\!24\)\( \beta_{3} + \)\(83\!\cdots\!80\)\( \beta_{4} - \)\(51\!\cdots\!56\)\( \beta_{5}) q^{98} +(-\)\(31\!\cdots\!56\)\( + \)\(62\!\cdots\!69\)\( \beta_{1} + \)\(10\!\cdots\!01\)\( \beta_{2} - \)\(13\!\cdots\!04\)\( \beta_{3} - \)\(98\!\cdots\!80\)\( \beta_{4} - \)\(12\!\cdots\!92\)\( \beta_{5}) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 57080822040q^{2} - 785092363818710040q^{3} + \)\(17\!\cdots\!28\)\(q^{4} - \)\(38\!\cdots\!40\)\(q^{5} + \)\(31\!\cdots\!92\)\(q^{6} + \)\(19\!\cdots\!00\)\(q^{7} + \)\(44\!\cdots\!20\)\(q^{8} + \)\(21\!\cdots\!82\)\(q^{9} + O(q^{10}) \) \( 6q - 57080822040q^{2} - 785092363818710040q^{3} + \)\(17\!\cdots\!28\)\(q^{4} - \)\(38\!\cdots\!40\)\(q^{5} + \)\(31\!\cdots\!92\)\(q^{6} + \)\(19\!\cdots\!00\)\(q^{7} + \)\(44\!\cdots\!20\)\(q^{8} + \)\(21\!\cdots\!82\)\(q^{9} + \)\(13\!\cdots\!60\)\(q^{10} - \)\(94\!\cdots\!88\)\(q^{11} - \)\(11\!\cdots\!80\)\(q^{12} + \)\(53\!\cdots\!20\)\(q^{13} + \)\(82\!\cdots\!76\)\(q^{14} - \)\(30\!\cdots\!80\)\(q^{15} + \)\(26\!\cdots\!16\)\(q^{16} + \)\(18\!\cdots\!80\)\(q^{17} - \)\(43\!\cdots\!40\)\(q^{18} + \)\(10\!\cdots\!80\)\(q^{19} + \)\(92\!\cdots\!80\)\(q^{20} - \)\(10\!\cdots\!28\)\(q^{21} + \)\(15\!\cdots\!20\)\(q^{22} + \)\(15\!\cdots\!80\)\(q^{23} - \)\(76\!\cdots\!60\)\(q^{24} + \)\(19\!\cdots\!50\)\(q^{25} + \)\(11\!\cdots\!52\)\(q^{26} - \)\(10\!\cdots\!20\)\(q^{27} + \)\(14\!\cdots\!20\)\(q^{28} + \)\(14\!\cdots\!20\)\(q^{29} - \)\(25\!\cdots\!80\)\(q^{30} - \)\(41\!\cdots\!88\)\(q^{31} + \)\(11\!\cdots\!60\)\(q^{32} + \)\(59\!\cdots\!20\)\(q^{33} + \)\(30\!\cdots\!56\)\(q^{34} + \)\(27\!\cdots\!60\)\(q^{35} + \)\(17\!\cdots\!16\)\(q^{36} + \)\(98\!\cdots\!40\)\(q^{37} + \)\(12\!\cdots\!80\)\(q^{38} + \)\(24\!\cdots\!44\)\(q^{39} + \)\(88\!\cdots\!00\)\(q^{40} + \)\(50\!\cdots\!12\)\(q^{41} + \)\(43\!\cdots\!80\)\(q^{42} + \)\(27\!\cdots\!00\)\(q^{43} - \)\(86\!\cdots\!44\)\(q^{44} - \)\(23\!\cdots\!80\)\(q^{45} - \)\(82\!\cdots\!88\)\(q^{46} - \)\(13\!\cdots\!80\)\(q^{47} - \)\(82\!\cdots\!20\)\(q^{48} - \)\(57\!\cdots\!42\)\(q^{49} + \)\(31\!\cdots\!00\)\(q^{50} + \)\(28\!\cdots\!32\)\(q^{51} + \)\(41\!\cdots\!00\)\(q^{52} + \)\(64\!\cdots\!60\)\(q^{53} + \)\(78\!\cdots\!80\)\(q^{54} + \)\(43\!\cdots\!20\)\(q^{55} + \)\(28\!\cdots\!20\)\(q^{56} - \)\(67\!\cdots\!40\)\(q^{57} - \)\(17\!\cdots\!80\)\(q^{58} - \)\(24\!\cdots\!60\)\(q^{59} - \)\(31\!\cdots\!40\)\(q^{60} - \)\(25\!\cdots\!88\)\(q^{61} - \)\(29\!\cdots\!80\)\(q^{62} + \)\(42\!\cdots\!40\)\(q^{63} + \)\(47\!\cdots\!48\)\(q^{64} + \)\(12\!\cdots\!20\)\(q^{65} + \)\(93\!\cdots\!84\)\(q^{66} + \)\(95\!\cdots\!80\)\(q^{67} + \)\(12\!\cdots\!60\)\(q^{68} - \)\(14\!\cdots\!36\)\(q^{69} - \)\(34\!\cdots\!40\)\(q^{70} - \)\(25\!\cdots\!88\)\(q^{71} - \)\(21\!\cdots\!60\)\(q^{72} - \)\(30\!\cdots\!20\)\(q^{73} - \)\(24\!\cdots\!84\)\(q^{74} + \)\(19\!\cdots\!00\)\(q^{75} + \)\(10\!\cdots\!40\)\(q^{76} + \)\(15\!\cdots\!00\)\(q^{77} + \)\(13\!\cdots\!00\)\(q^{78} + \)\(11\!\cdots\!20\)\(q^{79} + \)\(12\!\cdots\!60\)\(q^{80} + \)\(29\!\cdots\!86\)\(q^{81} - \)\(25\!\cdots\!80\)\(q^{82} - \)\(79\!\cdots\!60\)\(q^{83} - \)\(91\!\cdots\!64\)\(q^{84} - \)\(36\!\cdots\!40\)\(q^{85} + \)\(72\!\cdots\!32\)\(q^{86} - \)\(14\!\cdots\!60\)\(q^{87} - \)\(48\!\cdots\!60\)\(q^{88} + \)\(53\!\cdots\!60\)\(q^{89} + \)\(85\!\cdots\!20\)\(q^{90} + \)\(34\!\cdots\!32\)\(q^{91} + \)\(18\!\cdots\!80\)\(q^{92} - \)\(16\!\cdots\!80\)\(q^{93} - \)\(29\!\cdots\!04\)\(q^{94} + \)\(19\!\cdots\!00\)\(q^{95} - \)\(89\!\cdots\!08\)\(q^{96} - \)\(74\!\cdots\!80\)\(q^{97} - \)\(16\!\cdots\!20\)\(q^{98} - \)\(18\!\cdots\!36\)\(q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 3 x^{5} - 38457853073924058692 x^{4} - 10276556354621685339901678086 x^{3} + 371187556674475060057870954681799784505 x^{2} + 52686123927652036687598761277591247931691204025 x - 675344021115865838575279495800656435684060652010336995750\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 72 \nu - 36 \)
\(\beta_{2}\)\(=\)\((\)\(-251418152991 \nu^{5} + 2839090148245371662730 \nu^{4} + 2672662334648937824773191854526 \nu^{3} - 61047240176099464621737168584181023168544 \nu^{2} + 16219856538754443854395028975803863024780609343353 \nu + 143492946370941019722293327053082709869576915420152549389438\)\()/ \)\(10\!\cdots\!92\)\( \)
\(\beta_{3}\)\(=\)\((\)\(2059366091149281 \nu^{5} - 23254987404277839289421430 \nu^{4} - 21891777183109449722717214480422466 \nu^{3} + 776789357333974287671708684008658965330995168 \nu^{2} - 243785512868494979667958427387112398214451850405400711 \nu - 4723105784047957510086218814866656934311529249589537274536594306\)\()/ \)\(53\!\cdots\!96\)\( \)
\(\beta_{4}\)\(=\)\((\)\(-228096834986897734083 \nu^{5} + 304701294131890017647369088546 \nu^{4} + 7710030476352223652010557341900000847622 \nu^{3} - 5414352335550905547108516528571235876349656871840 \nu^{2} - 55981577684731823185470442962565582167839748553685625756555 \nu + 20082325532061210174094112435059165634057808851228748401625287634630\)\()/ \)\(26\!\cdots\!80\)\( \)
\(\beta_{5}\)\(=\)\((\)\(45120217709417106638349 \nu^{5} - 60638732387766545668302315952062 \nu^{4} - 1076466198217589466216190694618622730035354 \nu^{3} + 394220893411765307992869176482553788864527033340512 \nu^{2} + 2759324350025362153979169793754824502140431417815846490908613 \nu + 2487895388050538770512495457517733458287438811317899784587471321948966\)\()/ \)\(26\!\cdots\!48\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1} + 36\)\()/72\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} + 16382 \beta_{2} + 28859343499 \beta_{1} + 66455170111740773427552\)\()/5184\)
\(\nu^{3}\)\(=\)\((\)\(2781 \beta_{5} + 5496670 \beta_{4} + 13763505040 \beta_{3} + 227096891305632 \beta_{2} + 12411954401428958554888 \beta_{1} + 239731509332049065134727670101088\)\()/46656\)
\(\nu^{4}\)\(=\)\((\)\(19416283325423 \beta_{5} + 21925182234107050 \beta_{4} + 690181245893389417501 \beta_{3} + 17287986212334257062927110 \beta_{2} + 43286015777229834388292642264079 \beta_{1} + 34368272254786976926167523762875484141163424\)\()/139968\)
\(\nu^{5}\)\(=\)\((\)\(51323926323315844825348 \beta_{5} + 70480058977104454915417560 \beta_{4} + 279459332649115966158549368997 \beta_{3} + 5937244294005086129294402815328694 \beta_{2} + 136807307066496780727894546870876202630299 \beta_{1} + 6658747698433783854938116507636604591239766182012464\)\()/23328\)

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
−4.40594e9
−3.89081e9
−1.60397e9
1.46671e9
3.19848e9
5.23553e9
−3.26741e11 −1.51453e18 6.89808e22 −4.33451e25 4.94860e29 4.24638e31 −1.01949e34 1.68554e36 1.41626e37
1.2 −2.89652e11 1.03577e18 4.61192e22 −2.47463e26 −3.00013e29 −6.83079e31 −2.41577e33 4.64556e35 7.16781e37
1.3 −1.25000e11 1.35148e17 −2.21540e22 1.80927e26 −1.68935e28 2.76229e31 7.49160e33 −5.90002e35 −2.26158e37
1.4 9.60898e10 −6.47242e17 −2.85457e22 −1.49650e26 −6.21934e28 −3.16100e31 −6.37312e33 −1.89344e35 −1.43799e37
1.5 2.20777e11 1.08881e18 1.09637e22 −2.30711e25 2.40385e29 1.96027e31 −5.92020e33 5.77242e35 −5.09358e36
1.6 3.67444e11 −8.83049e17 9.72365e22 2.43619e26 −3.24471e29 1.21530e31 2.18473e34 1.71508e35 8.95165e37
\(n\): 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 1.76.a.a 6
3.b odd 2 1 9.76.a.c 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.76.a.a 6 1.a even 1 1 trivial
9.76.a.c 6 3.b odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{76}^{\mathrm{new}}(\Gamma_0(1))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 57080822040 T + \)\(28\!\cdots\!40\)\( T^{2} - \)\(62\!\cdots\!80\)\( T^{3} + \)\(12\!\cdots\!72\)\( T^{4} - \)\(18\!\cdots\!80\)\( T^{5} + \)\(27\!\cdots\!20\)\( T^{6} - \)\(70\!\cdots\!40\)\( T^{7} + \)\(17\!\cdots\!28\)\( T^{8} - \)\(33\!\cdots\!60\)\( T^{9} + \)\(58\!\cdots\!40\)\( T^{10} + \)\(43\!\cdots\!20\)\( T^{11} + \)\(29\!\cdots\!24\)\( T^{12} \)
$3$ \( 1 + 785092363818710040 T + \)\(10\!\cdots\!30\)\( T^{2} + \)\(88\!\cdots\!80\)\( T^{3} + \)\(94\!\cdots\!47\)\( T^{4} + \)\(93\!\cdots\!20\)\( T^{5} + \)\(67\!\cdots\!40\)\( T^{6} + \)\(57\!\cdots\!40\)\( T^{7} + \)\(35\!\cdots\!03\)\( T^{8} + \)\(19\!\cdots\!40\)\( T^{9} + \)\(14\!\cdots\!30\)\( T^{10} + \)\(65\!\cdots\!80\)\( T^{11} + \)\(50\!\cdots\!49\)\( T^{12} \)
$5$ \( 1 + \)\(38\!\cdots\!40\)\( T + \)\(70\!\cdots\!50\)\( T^{2} + \)\(11\!\cdots\!00\)\( T^{3} + \)\(28\!\cdots\!75\)\( T^{4} + \)\(61\!\cdots\!00\)\( T^{5} + \)\(89\!\cdots\!00\)\( T^{6} + \)\(16\!\cdots\!00\)\( T^{7} + \)\(19\!\cdots\!75\)\( T^{8} + \)\(20\!\cdots\!00\)\( T^{9} + \)\(34\!\cdots\!50\)\( T^{10} + \)\(50\!\cdots\!00\)\( T^{11} + \)\(34\!\cdots\!25\)\( T^{12} \)
$7$ \( 1 - \)\(19\!\cdots\!00\)\( T + \)\(10\!\cdots\!50\)\( T^{2} + \)\(66\!\cdots\!00\)\( T^{3} + \)\(47\!\cdots\!47\)\( T^{4} + \)\(44\!\cdots\!00\)\( T^{5} + \)\(14\!\cdots\!00\)\( T^{6} + \)\(10\!\cdots\!00\)\( T^{7} + \)\(27\!\cdots\!03\)\( T^{8} + \)\(93\!\cdots\!00\)\( T^{9} + \)\(34\!\cdots\!50\)\( T^{10} - \)\(15\!\cdots\!00\)\( T^{11} + \)\(19\!\cdots\!49\)\( T^{12} \)
$11$ \( 1 + \)\(94\!\cdots\!88\)\( T + \)\(58\!\cdots\!66\)\( T^{2} + \)\(47\!\cdots\!80\)\( T^{3} + \)\(16\!\cdots\!95\)\( T^{4} + \)\(10\!\cdots\!08\)\( T^{5} + \)\(26\!\cdots\!24\)\( T^{6} + \)\(13\!\cdots\!08\)\( T^{7} + \)\(26\!\cdots\!95\)\( T^{8} + \)\(97\!\cdots\!80\)\( T^{9} + \)\(15\!\cdots\!66\)\( T^{10} + \)\(31\!\cdots\!88\)\( T^{11} + \)\(42\!\cdots\!01\)\( T^{12} \)
$13$ \( 1 - \)\(53\!\cdots\!20\)\( T + \)\(98\!\cdots\!10\)\( T^{2} - \)\(43\!\cdots\!40\)\( T^{3} + \)\(27\!\cdots\!47\)\( T^{4} - \)\(14\!\cdots\!60\)\( T^{5} + \)\(38\!\cdots\!80\)\( T^{6} - \)\(51\!\cdots\!20\)\( T^{7} + \)\(33\!\cdots\!03\)\( T^{8} - \)\(18\!\cdots\!20\)\( T^{9} + \)\(14\!\cdots\!10\)\( T^{10} - \)\(28\!\cdots\!40\)\( T^{11} + \)\(18\!\cdots\!49\)\( T^{12} \)
$17$ \( 1 - \)\(18\!\cdots\!80\)\( T + \)\(10\!\cdots\!70\)\( T^{2} - \)\(14\!\cdots\!40\)\( T^{3} + \)\(43\!\cdots\!47\)\( T^{4} - \)\(48\!\cdots\!40\)\( T^{5} + \)\(10\!\cdots\!60\)\( T^{6} - \)\(92\!\cdots\!20\)\( T^{7} + \)\(15\!\cdots\!03\)\( T^{8} - \)\(10\!\cdots\!80\)\( T^{9} + \)\(13\!\cdots\!70\)\( T^{10} - \)\(47\!\cdots\!40\)\( T^{11} + \)\(50\!\cdots\!49\)\( T^{12} \)
$19$ \( 1 - \)\(10\!\cdots\!80\)\( T + \)\(24\!\cdots\!94\)\( T^{2} - \)\(22\!\cdots\!00\)\( T^{3} + \)\(35\!\cdots\!15\)\( T^{4} - \)\(27\!\cdots\!00\)\( T^{5} + \)\(34\!\cdots\!80\)\( T^{6} - \)\(22\!\cdots\!00\)\( T^{7} + \)\(22\!\cdots\!15\)\( T^{8} - \)\(11\!\cdots\!00\)\( T^{9} + \)\(10\!\cdots\!94\)\( T^{10} - \)\(36\!\cdots\!20\)\( T^{11} + \)\(27\!\cdots\!01\)\( T^{12} \)
$23$ \( 1 - \)\(15\!\cdots\!80\)\( T + \)\(60\!\cdots\!90\)\( T^{2} - \)\(61\!\cdots\!60\)\( T^{3} + \)\(16\!\cdots\!47\)\( T^{4} - \)\(13\!\cdots\!40\)\( T^{5} + \)\(28\!\cdots\!20\)\( T^{6} - \)\(18\!\cdots\!80\)\( T^{7} + \)\(29\!\cdots\!03\)\( T^{8} - \)\(15\!\cdots\!80\)\( T^{9} + \)\(19\!\cdots\!90\)\( T^{10} - \)\(67\!\cdots\!60\)\( T^{11} + \)\(59\!\cdots\!49\)\( T^{12} \)
$29$ \( 1 - \)\(14\!\cdots\!20\)\( T + \)\(30\!\cdots\!94\)\( T^{2} - \)\(29\!\cdots\!00\)\( T^{3} + \)\(35\!\cdots\!15\)\( T^{4} - \)\(26\!\cdots\!00\)\( T^{5} + \)\(22\!\cdots\!80\)\( T^{6} - \)\(12\!\cdots\!00\)\( T^{7} + \)\(81\!\cdots\!15\)\( T^{8} - \)\(32\!\cdots\!00\)\( T^{9} + \)\(15\!\cdots\!94\)\( T^{10} - \)\(36\!\cdots\!80\)\( T^{11} + \)\(11\!\cdots\!01\)\( T^{12} \)
$31$ \( 1 + \)\(41\!\cdots\!88\)\( T + \)\(16\!\cdots\!66\)\( T^{2} + \)\(74\!\cdots\!80\)\( T^{3} + \)\(16\!\cdots\!95\)\( T^{4} + \)\(99\!\cdots\!08\)\( T^{5} + \)\(13\!\cdots\!24\)\( T^{6} + \)\(70\!\cdots\!08\)\( T^{7} + \)\(83\!\cdots\!95\)\( T^{8} + \)\(26\!\cdots\!80\)\( T^{9} + \)\(41\!\cdots\!66\)\( T^{10} + \)\(76\!\cdots\!88\)\( T^{11} + \)\(12\!\cdots\!01\)\( T^{12} \)
$37$ \( 1 - \)\(98\!\cdots\!40\)\( T + \)\(15\!\cdots\!10\)\( T^{2} - \)\(14\!\cdots\!20\)\( T^{3} + \)\(14\!\cdots\!47\)\( T^{4} - \)\(96\!\cdots\!20\)\( T^{5} + \)\(74\!\cdots\!80\)\( T^{6} - \)\(39\!\cdots\!60\)\( T^{7} + \)\(23\!\cdots\!03\)\( T^{8} - \)\(10\!\cdots\!40\)\( T^{9} + \)\(45\!\cdots\!10\)\( T^{10} - \)\(11\!\cdots\!20\)\( T^{11} + \)\(49\!\cdots\!49\)\( T^{12} \)
$41$ \( 1 - \)\(50\!\cdots\!12\)\( T + \)\(29\!\cdots\!66\)\( T^{2} - \)\(66\!\cdots\!20\)\( T^{3} + \)\(24\!\cdots\!95\)\( T^{4} - \)\(19\!\cdots\!92\)\( T^{5} + \)\(12\!\cdots\!24\)\( T^{6} - \)\(17\!\cdots\!92\)\( T^{7} + \)\(20\!\cdots\!95\)\( T^{8} - \)\(50\!\cdots\!20\)\( T^{9} + \)\(19\!\cdots\!66\)\( T^{10} - \)\(31\!\cdots\!12\)\( T^{11} + \)\(56\!\cdots\!01\)\( T^{12} \)
$43$ \( 1 - \)\(27\!\cdots\!00\)\( T + \)\(10\!\cdots\!50\)\( T^{2} - \)\(17\!\cdots\!00\)\( T^{3} + \)\(53\!\cdots\!47\)\( T^{4} - \)\(84\!\cdots\!00\)\( T^{5} + \)\(20\!\cdots\!00\)\( T^{6} - \)\(27\!\cdots\!00\)\( T^{7} + \)\(55\!\cdots\!03\)\( T^{8} - \)\(59\!\cdots\!00\)\( T^{9} + \)\(11\!\cdots\!50\)\( T^{10} - \)\(99\!\cdots\!00\)\( T^{11} + \)\(11\!\cdots\!49\)\( T^{12} \)
$47$ \( 1 + \)\(13\!\cdots\!80\)\( T + \)\(16\!\cdots\!30\)\( T^{2} + \)\(14\!\cdots\!40\)\( T^{3} + \)\(10\!\cdots\!47\)\( T^{4} + \)\(65\!\cdots\!40\)\( T^{5} + \)\(35\!\cdots\!40\)\( T^{6} + \)\(16\!\cdots\!20\)\( T^{7} + \)\(68\!\cdots\!03\)\( T^{8} + \)\(23\!\cdots\!80\)\( T^{9} + \)\(70\!\cdots\!30\)\( T^{10} + \)\(15\!\cdots\!40\)\( T^{11} + \)\(27\!\cdots\!49\)\( T^{12} \)
$53$ \( 1 - \)\(64\!\cdots\!60\)\( T + \)\(76\!\cdots\!30\)\( T^{2} - \)\(28\!\cdots\!20\)\( T^{3} + \)\(18\!\cdots\!47\)\( T^{4} - \)\(37\!\cdots\!80\)\( T^{5} + \)\(26\!\cdots\!40\)\( T^{6} - \)\(78\!\cdots\!60\)\( T^{7} + \)\(79\!\cdots\!03\)\( T^{8} - \)\(26\!\cdots\!60\)\( T^{9} + \)\(14\!\cdots\!30\)\( T^{10} - \)\(25\!\cdots\!20\)\( T^{11} + \)\(83\!\cdots\!49\)\( T^{12} \)
$59$ \( 1 + \)\(24\!\cdots\!60\)\( T + \)\(20\!\cdots\!94\)\( T^{2} + \)\(59\!\cdots\!00\)\( T^{3} + \)\(26\!\cdots\!15\)\( T^{4} + \)\(59\!\cdots\!00\)\( T^{5} + \)\(21\!\cdots\!80\)\( T^{6} + \)\(39\!\cdots\!00\)\( T^{7} + \)\(11\!\cdots\!15\)\( T^{8} + \)\(16\!\cdots\!00\)\( T^{9} + \)\(37\!\cdots\!94\)\( T^{10} + \)\(29\!\cdots\!40\)\( T^{11} + \)\(76\!\cdots\!01\)\( T^{12} \)
$61$ \( 1 + \)\(25\!\cdots\!88\)\( T + \)\(55\!\cdots\!66\)\( T^{2} + \)\(85\!\cdots\!80\)\( T^{3} + \)\(11\!\cdots\!95\)\( T^{4} + \)\(12\!\cdots\!08\)\( T^{5} + \)\(12\!\cdots\!24\)\( T^{6} + \)\(99\!\cdots\!08\)\( T^{7} + \)\(71\!\cdots\!95\)\( T^{8} + \)\(42\!\cdots\!80\)\( T^{9} + \)\(22\!\cdots\!66\)\( T^{10} + \)\(80\!\cdots\!88\)\( T^{11} + \)\(25\!\cdots\!01\)\( T^{12} \)
$67$ \( 1 - \)\(95\!\cdots\!80\)\( T + \)\(72\!\cdots\!70\)\( T^{2} - \)\(36\!\cdots\!40\)\( T^{3} + \)\(16\!\cdots\!47\)\( T^{4} - \)\(60\!\cdots\!40\)\( T^{5} + \)\(20\!\cdots\!60\)\( T^{6} - \)\(54\!\cdots\!20\)\( T^{7} + \)\(13\!\cdots\!03\)\( T^{8} - \)\(27\!\cdots\!80\)\( T^{9} + \)\(48\!\cdots\!70\)\( T^{10} - \)\(57\!\cdots\!40\)\( T^{11} + \)\(54\!\cdots\!49\)\( T^{12} \)
$71$ \( 1 + \)\(25\!\cdots\!88\)\( T + \)\(24\!\cdots\!66\)\( T^{2} + \)\(16\!\cdots\!80\)\( T^{3} + \)\(31\!\cdots\!95\)\( T^{4} + \)\(19\!\cdots\!08\)\( T^{5} + \)\(27\!\cdots\!24\)\( T^{6} + \)\(13\!\cdots\!08\)\( T^{7} + \)\(15\!\cdots\!95\)\( T^{8} + \)\(54\!\cdots\!80\)\( T^{9} + \)\(57\!\cdots\!66\)\( T^{10} + \)\(42\!\cdots\!88\)\( T^{11} + \)\(11\!\cdots\!01\)\( T^{12} \)
$73$ \( 1 + \)\(30\!\cdots\!20\)\( T + \)\(58\!\cdots\!90\)\( T^{2} + \)\(81\!\cdots\!40\)\( T^{3} + \)\(89\!\cdots\!47\)\( T^{4} + \)\(83\!\cdots\!60\)\( T^{5} + \)\(66\!\cdots\!20\)\( T^{6} + \)\(46\!\cdots\!20\)\( T^{7} + \)\(28\!\cdots\!03\)\( T^{8} + \)\(14\!\cdots\!20\)\( T^{9} + \)\(58\!\cdots\!90\)\( T^{10} + \)\(17\!\cdots\!40\)\( T^{11} + \)\(31\!\cdots\!49\)\( T^{12} \)
$79$ \( 1 - \)\(11\!\cdots\!20\)\( T + \)\(92\!\cdots\!94\)\( T^{2} - \)\(90\!\cdots\!00\)\( T^{3} + \)\(40\!\cdots\!15\)\( T^{4} - \)\(33\!\cdots\!00\)\( T^{5} + \)\(10\!\cdots\!80\)\( T^{6} - \)\(70\!\cdots\!00\)\( T^{7} + \)\(17\!\cdots\!15\)\( T^{8} - \)\(84\!\cdots\!00\)\( T^{9} + \)\(17\!\cdots\!94\)\( T^{10} - \)\(47\!\cdots\!80\)\( T^{11} + \)\(85\!\cdots\!01\)\( T^{12} \)
$83$ \( 1 + \)\(79\!\cdots\!60\)\( T + \)\(37\!\cdots\!70\)\( T^{2} + \)\(25\!\cdots\!20\)\( T^{3} + \)\(64\!\cdots\!47\)\( T^{4} + \)\(36\!\cdots\!80\)\( T^{5} + \)\(67\!\cdots\!60\)\( T^{6} + \)\(31\!\cdots\!60\)\( T^{7} + \)\(46\!\cdots\!03\)\( T^{8} + \)\(15\!\cdots\!60\)\( T^{9} + \)\(19\!\cdots\!70\)\( T^{10} + \)\(36\!\cdots\!20\)\( T^{11} + \)\(38\!\cdots\!49\)\( T^{12} \)
$89$ \( 1 - \)\(53\!\cdots\!60\)\( T + \)\(17\!\cdots\!94\)\( T^{2} - \)\(40\!\cdots\!00\)\( T^{3} + \)\(76\!\cdots\!15\)\( T^{4} - \)\(11\!\cdots\!00\)\( T^{5} + \)\(15\!\cdots\!80\)\( T^{6} - \)\(18\!\cdots\!00\)\( T^{7} + \)\(19\!\cdots\!15\)\( T^{8} - \)\(16\!\cdots\!00\)\( T^{9} + \)\(11\!\cdots\!94\)\( T^{10} - \)\(56\!\cdots\!40\)\( T^{11} + \)\(16\!\cdots\!01\)\( T^{12} \)
$97$ \( 1 + \)\(74\!\cdots\!80\)\( T + \)\(63\!\cdots\!30\)\( T^{2} + \)\(32\!\cdots\!40\)\( T^{3} + \)\(16\!\cdots\!47\)\( T^{4} + \)\(61\!\cdots\!40\)\( T^{5} + \)\(22\!\cdots\!40\)\( T^{6} + \)\(62\!\cdots\!20\)\( T^{7} + \)\(16\!\cdots\!03\)\( T^{8} + \)\(34\!\cdots\!80\)\( T^{9} + \)\(68\!\cdots\!30\)\( T^{10} + \)\(81\!\cdots\!40\)\( T^{11} + \)\(11\!\cdots\!49\)\( T^{12} \)
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