Properties

Label 1.84.a.a
Level 1
Weight 84
Character orbit 1.a
Self dual Yes
Analytic conductor 43.627
Analytic rank 0
Dimension 7
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 1 \)
Weight: \( k \) = \( 84 \)
Character orbit: \([\chi]\) = 1.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(43.6272128266\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: multiple of \( 2^{82}\cdot 3^{30}\cdot 5^{8}\cdot 7^{4}\cdot 17 \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q +(49635823059 + \beta_{1}) q^{2} +(13252713247479580889 - 2898083 \beta_{1} - \beta_{2}) q^{3} +(\)\(50\!\cdots\!92\)\( - 499409171338 \beta_{1} - 55717 \beta_{2} + \beta_{3}) q^{4} +(\)\(13\!\cdots\!60\)\( + 4793275464288047 \beta_{1} - 19973746 \beta_{2} - 302 \beta_{3} - \beta_{4}) q^{5} +(-\)\(42\!\cdots\!74\)\( + 32655979982060558238 \beta_{1} + 392089455434 \beta_{2} - 3661857 \beta_{3} - 141 \beta_{4} - \beta_{5}) q^{6} +(\)\(59\!\cdots\!49\)\( - \)\(64\!\cdots\!29\)\( \beta_{1} + 18218040499108 \beta_{2} - 3449182225 \beta_{3} + 9081 \beta_{4} - 138 \beta_{5} - \beta_{6}) q^{7} +(-\)\(75\!\cdots\!44\)\( + \)\(74\!\cdots\!00\)\( \beta_{1} + 66297220963752280 \beta_{2} - 99013320520 \beta_{3} - 102492128 \beta_{4} - 159336 \beta_{5} + 168 \beta_{6}) q^{8} +(\)\(10\!\cdots\!89\)\( - \)\(27\!\cdots\!86\)\( \beta_{1} - 5290078316587194228 \beta_{2} - 2920893642936 \beta_{3} - 29890996578 \beta_{4} + 8089752 \beta_{5} - 13860 \beta_{6}) q^{9} +O(q^{10})\) \( q +(49635823059 + \beta_{1}) q^{2} +(13252713247479580889 - 2898083 \beta_{1} - \beta_{2}) q^{3} +(\)\(50\!\cdots\!92\)\( - 499409171338 \beta_{1} - 55717 \beta_{2} + \beta_{3}) q^{4} +(\)\(13\!\cdots\!60\)\( + 4793275464288047 \beta_{1} - 19973746 \beta_{2} - 302 \beta_{3} - \beta_{4}) q^{5} +(-\)\(42\!\cdots\!74\)\( + 32655979982060558238 \beta_{1} + 392089455434 \beta_{2} - 3661857 \beta_{3} - 141 \beta_{4} - \beta_{5}) q^{6} +(\)\(59\!\cdots\!49\)\( - \)\(64\!\cdots\!29\)\( \beta_{1} + 18218040499108 \beta_{2} - 3449182225 \beta_{3} + 9081 \beta_{4} - 138 \beta_{5} - \beta_{6}) q^{7} +(-\)\(75\!\cdots\!44\)\( + \)\(74\!\cdots\!00\)\( \beta_{1} + 66297220963752280 \beta_{2} - 99013320520 \beta_{3} - 102492128 \beta_{4} - 159336 \beta_{5} + 168 \beta_{6}) q^{8} +(\)\(10\!\cdots\!89\)\( - \)\(27\!\cdots\!86\)\( \beta_{1} - 5290078316587194228 \beta_{2} - 2920893642936 \beta_{3} - 29890996578 \beta_{4} + 8089752 \beta_{5} - 13860 \beta_{6}) q^{9} +(\)\(71\!\cdots\!90\)\( + \)\(81\!\cdots\!62\)\( \beta_{1} - \)\(54\!\cdots\!76\)\( \beta_{2} + 10114105121331468 \beta_{3} - 2278279661796 \beta_{4} + 217072140 \beta_{5} + 748160 \beta_{6}) q^{10} +(\)\(36\!\cdots\!21\)\( - \)\(25\!\cdots\!15\)\( \beta_{1} - \)\(53\!\cdots\!79\)\( \beta_{2} + 997560625897275902 \beta_{3} + 88118652158482 \beta_{4} - 33811281108 \beta_{5} - 29704290 \beta_{6}) q^{11} +(\)\(35\!\cdots\!16\)\( - \)\(78\!\cdots\!24\)\( \beta_{1} - \)\(43\!\cdots\!96\)\( \beta_{2} + 69068083393168540380 \beta_{3} + 1345917072586752 \beta_{4} + 1580793249024 \beta_{5} + 924473088 \beta_{6}) q^{12} +(\)\(25\!\cdots\!96\)\( + \)\(13\!\cdots\!07\)\( \beta_{1} - \)\(52\!\cdots\!42\)\( \beta_{2} + \)\(15\!\cdots\!70\)\( \beta_{3} - 108455952770922393 \beta_{4} - 45372512759856 \beta_{5} - 23471191352 \beta_{6}) q^{13} +(-\)\(95\!\cdots\!32\)\( - \)\(27\!\cdots\!24\)\( \beta_{1} - \)\(13\!\cdots\!36\)\( \beta_{2} - \)\(21\!\cdots\!82\)\( \beta_{3} + 2033650722027610750 \beta_{4} + 918273332707830 \beta_{5} + 499474398720 \beta_{6}) q^{14} +(\)\(70\!\cdots\!55\)\( + \)\(21\!\cdots\!69\)\( \beta_{1} - \)\(70\!\cdots\!12\)\( \beta_{2} - \)\(20\!\cdots\!59\)\( \beta_{3} - 9709046302923571377 \beta_{4} - 13692004076022470 \beta_{5} - 9083657330055 \beta_{6}) q^{15} +(\)\(59\!\cdots\!76\)\( - \)\(90\!\cdots\!84\)\( \beta_{1} - \)\(18\!\cdots\!04\)\( \beta_{2} + \)\(72\!\cdots\!12\)\( \beta_{3} - \)\(26\!\cdots\!76\)\( \beta_{4} + 148577170751258304 \beta_{5} + 143225916285760 \beta_{6}) q^{16} +(-\)\(26\!\cdots\!02\)\( - \)\(35\!\cdots\!34\)\( \beta_{1} - \)\(83\!\cdots\!12\)\( \beta_{2} - \)\(16\!\cdots\!00\)\( \beta_{3} + \)\(61\!\cdots\!34\)\( \beta_{4} - 1032676104402524232 \beta_{5} - 1979270827255764 \beta_{6}) q^{17} +(-\)\(40\!\cdots\!97\)\( + \)\(12\!\cdots\!37\)\( \beta_{1} + \)\(10\!\cdots\!08\)\( \beta_{2} - \)\(66\!\cdots\!00\)\( \beta_{3} - \)\(58\!\cdots\!24\)\( \beta_{4} + 905385868195668552 \beta_{5} + 24170027844904704 \beta_{6}) q^{18} +(\)\(10\!\cdots\!43\)\( - \)\(10\!\cdots\!17\)\( \beta_{1} + \)\(10\!\cdots\!35\)\( \beta_{2} + \)\(42\!\cdots\!90\)\( \beta_{3} + \)\(19\!\cdots\!86\)\( \beta_{4} + 94412277239963646516 \beta_{5} - 262419578102711470 \beta_{6}) q^{19} +(-\)\(82\!\cdots\!20\)\( + \)\(13\!\cdots\!76\)\( \beta_{1} + \)\(20\!\cdots\!82\)\( \beta_{2} + \)\(33\!\cdots\!34\)\( \beta_{3} + \)\(19\!\cdots\!92\)\( \beta_{4} - \)\(16\!\cdots\!00\)\( \beta_{5} + 2544200774633548800 \beta_{6}) q^{20} +(\)\(27\!\cdots\!44\)\( + \)\(43\!\cdots\!76\)\( \beta_{1} - \)\(52\!\cdots\!36\)\( \beta_{2} - \)\(43\!\cdots\!72\)\( \beta_{3} - \)\(31\!\cdots\!60\)\( \beta_{4} + \)\(17\!\cdots\!00\)\( \beta_{5} - 22086255152920575960 \beta_{6}) q^{21} +(-\)\(36\!\cdots\!74\)\( + \)\(15\!\cdots\!42\)\( \beta_{1} - \)\(32\!\cdots\!42\)\( \beta_{2} + \)\(36\!\cdots\!45\)\( \beta_{3} + \)\(18\!\cdots\!37\)\( \beta_{4} - \)\(13\!\cdots\!71\)\( \beta_{5} + \)\(17\!\cdots\!68\)\( \beta_{6}) q^{22} +(\)\(11\!\cdots\!67\)\( + \)\(52\!\cdots\!77\)\( \beta_{1} + \)\(47\!\cdots\!96\)\( \beta_{2} + \)\(15\!\cdots\!25\)\( \beta_{3} - \)\(17\!\cdots\!65\)\( \beta_{4} + \)\(82\!\cdots\!70\)\( \beta_{5} - \)\(11\!\cdots\!35\)\( \beta_{6}) q^{23} +(-\)\(72\!\cdots\!40\)\( + \)\(89\!\cdots\!84\)\( \beta_{1} + \)\(22\!\cdots\!44\)\( \beta_{2} - \)\(64\!\cdots\!72\)\( \beta_{3} - \)\(60\!\cdots\!04\)\( \beta_{4} - \)\(37\!\cdots\!04\)\( \beta_{5} + \)\(74\!\cdots\!60\)\( \beta_{6}) q^{24} +(-\)\(65\!\cdots\!25\)\( + \)\(14\!\cdots\!40\)\( \beta_{1} - \)\(11\!\cdots\!20\)\( \beta_{2} - \)\(32\!\cdots\!40\)\( \beta_{3} + \)\(51\!\cdots\!80\)\( \beta_{4} + \)\(10\!\cdots\!00\)\( \beta_{5} - \)\(41\!\cdots\!00\)\( \beta_{6}) q^{25} +(\)\(20\!\cdots\!34\)\( + \)\(43\!\cdots\!94\)\( \beta_{1} - \)\(63\!\cdots\!08\)\( \beta_{2} + \)\(32\!\cdots\!44\)\( \beta_{3} - \)\(17\!\cdots\!68\)\( \beta_{4} - \)\(99\!\cdots\!68\)\( \beta_{5} + \)\(19\!\cdots\!20\)\( \beta_{6}) q^{26} +(-\)\(26\!\cdots\!56\)\( - \)\(29\!\cdots\!00\)\( \beta_{1} - \)\(70\!\cdots\!10\)\( \beta_{2} - \)\(46\!\cdots\!30\)\( \beta_{3} - \)\(13\!\cdots\!82\)\( \beta_{4} - \)\(17\!\cdots\!84\)\( \beta_{5} - \)\(81\!\cdots\!58\)\( \beta_{6}) q^{27} +(-\)\(47\!\cdots\!80\)\( - \)\(22\!\cdots\!92\)\( \beta_{1} + \)\(12\!\cdots\!96\)\( \beta_{2} - \)\(40\!\cdots\!80\)\( \beta_{3} + \)\(44\!\cdots\!08\)\( \beta_{4} + \)\(80\!\cdots\!96\)\( \beta_{5} + \)\(27\!\cdots\!52\)\( \beta_{6}) q^{28} +(\)\(91\!\cdots\!08\)\( - \)\(10\!\cdots\!93\)\( \beta_{1} + \)\(60\!\cdots\!74\)\( \beta_{2} + \)\(13\!\cdots\!98\)\( \beta_{3} - \)\(20\!\cdots\!13\)\( \beta_{4} + \)\(56\!\cdots\!12\)\( \beta_{5} - \)\(61\!\cdots\!80\)\( \beta_{6}) q^{29} +(\)\(31\!\cdots\!20\)\( - \)\(10\!\cdots\!76\)\( \beta_{1} - \)\(16\!\cdots\!32\)\( \beta_{2} + \)\(45\!\cdots\!66\)\( \beta_{3} + \)\(18\!\cdots\!58\)\( \beta_{4} - \)\(31\!\cdots\!50\)\( \beta_{5} + \)\(14\!\cdots\!00\)\( \beta_{6}) q^{30} +(-\)\(53\!\cdots\!96\)\( + \)\(19\!\cdots\!80\)\( \beta_{1} - \)\(29\!\cdots\!12\)\( \beta_{2} - \)\(23\!\cdots\!64\)\( \beta_{3} + \)\(21\!\cdots\!16\)\( \beta_{4} + \)\(24\!\cdots\!36\)\( \beta_{5} + \)\(86\!\cdots\!40\)\( \beta_{6}) q^{31} +(-\)\(57\!\cdots\!80\)\( + \)\(10\!\cdots\!56\)\( \beta_{1} + \)\(21\!\cdots\!96\)\( \beta_{2} - \)\(76\!\cdots\!00\)\( \beta_{3} - \)\(97\!\cdots\!16\)\( \beta_{4} - \)\(11\!\cdots\!32\)\( \beta_{5} - \)\(52\!\cdots\!64\)\( \beta_{6}) q^{32} +(\)\(31\!\cdots\!84\)\( + \)\(55\!\cdots\!14\)\( \beta_{1} + \)\(29\!\cdots\!24\)\( \beta_{2} + \)\(68\!\cdots\!20\)\( \beta_{3} - \)\(26\!\cdots\!46\)\( \beta_{4} + \)\(36\!\cdots\!88\)\( \beta_{5} + \)\(19\!\cdots\!36\)\( \beta_{6}) q^{33} +(-\)\(52\!\cdots\!74\)\( - \)\(35\!\cdots\!06\)\( \beta_{1} - \)\(72\!\cdots\!76\)\( \beta_{2} - \)\(80\!\cdots\!52\)\( \beta_{3} + \)\(18\!\cdots\!16\)\( \beta_{4} - \)\(62\!\cdots\!04\)\( \beta_{5} - \)\(44\!\cdots\!20\)\( \beta_{6}) q^{34} +(-\)\(29\!\cdots\!60\)\( - \)\(14\!\cdots\!68\)\( \beta_{1} - \)\(62\!\cdots\!36\)\( \beta_{2} - \)\(55\!\cdots\!52\)\( \beta_{3} - \)\(87\!\cdots\!56\)\( \beta_{4} - \)\(79\!\cdots\!60\)\( \beta_{5} + \)\(84\!\cdots\!60\)\( \beta_{6}) q^{35} +(\)\(81\!\cdots\!76\)\( - \)\(11\!\cdots\!74\)\( \beta_{1} - \)\(64\!\cdots\!09\)\( \beta_{2} + \)\(17\!\cdots\!97\)\( \beta_{3} + \)\(19\!\cdots\!64\)\( \beta_{4} + \)\(97\!\cdots\!04\)\( \beta_{5} + \)\(45\!\cdots\!00\)\( \beta_{6}) q^{36} +(\)\(16\!\cdots\!40\)\( + \)\(50\!\cdots\!11\)\( \beta_{1} + \)\(89\!\cdots\!62\)\( \beta_{2} + \)\(10\!\cdots\!90\)\( \beta_{3} - \)\(19\!\cdots\!29\)\( \beta_{4} - \)\(34\!\cdots\!48\)\( \beta_{5} - \)\(23\!\cdots\!76\)\( \beta_{6}) q^{37} +(-\)\(14\!\cdots\!62\)\( + \)\(36\!\cdots\!30\)\( \beta_{1} + \)\(11\!\cdots\!54\)\( \beta_{2} - \)\(77\!\cdots\!45\)\( \beta_{3} + \)\(48\!\cdots\!19\)\( \beta_{4} + \)\(63\!\cdots\!83\)\( \beta_{5} + \)\(64\!\cdots\!56\)\( \beta_{6}) q^{38} +(\)\(22\!\cdots\!95\)\( + \)\(20\!\cdots\!21\)\( \beta_{1} - \)\(60\!\cdots\!60\)\( \beta_{2} - \)\(30\!\cdots\!55\)\( \beta_{3} - \)\(49\!\cdots\!53\)\( \beta_{4} + \)\(59\!\cdots\!82\)\( \beta_{5} - \)\(65\!\cdots\!15\)\( \beta_{6}) q^{39} +(\)\(13\!\cdots\!00\)\( + \)\(15\!\cdots\!00\)\( \beta_{1} - \)\(20\!\cdots\!00\)\( \beta_{2} + \)\(15\!\cdots\!00\)\( \beta_{3} + \)\(21\!\cdots\!00\)\( \beta_{4} - \)\(37\!\cdots\!00\)\( \beta_{5} - \)\(31\!\cdots\!00\)\( \beta_{6}) q^{40} +(\)\(24\!\cdots\!94\)\( - \)\(31\!\cdots\!20\)\( \beta_{1} + \)\(44\!\cdots\!48\)\( \beta_{2} - \)\(11\!\cdots\!64\)\( \beta_{3} - \)\(40\!\cdots\!44\)\( \beta_{4} + \)\(93\!\cdots\!16\)\( \beta_{5} + \)\(19\!\cdots\!00\)\( \beta_{6}) q^{41} +(\)\(64\!\cdots\!36\)\( - \)\(36\!\cdots\!88\)\( \beta_{1} + \)\(17\!\cdots\!40\)\( \beta_{2} - \)\(66\!\cdots\!00\)\( \beta_{3} - \)\(28\!\cdots\!04\)\( \beta_{4} - \)\(45\!\cdots\!08\)\( \beta_{5} - \)\(52\!\cdots\!16\)\( \beta_{6}) q^{42} +(\)\(22\!\cdots\!15\)\( - \)\(82\!\cdots\!13\)\( \beta_{1} - \)\(41\!\cdots\!87\)\( \beta_{2} - \)\(15\!\cdots\!20\)\( \beta_{3} + \)\(30\!\cdots\!80\)\( \beta_{4} - \)\(17\!\cdots\!20\)\( \beta_{5} + \)\(46\!\cdots\!00\)\( \beta_{6}) q^{43} +(\)\(19\!\cdots\!96\)\( + \)\(11\!\cdots\!64\)\( \beta_{1} - \)\(18\!\cdots\!96\)\( \beta_{2} + \)\(12\!\cdots\!88\)\( \beta_{3} - \)\(58\!\cdots\!64\)\( \beta_{4} - \)\(46\!\cdots\!44\)\( \beta_{5} + \)\(23\!\cdots\!40\)\( \beta_{6}) q^{44} +(\)\(26\!\cdots\!20\)\( + \)\(39\!\cdots\!19\)\( \beta_{1} + \)\(47\!\cdots\!58\)\( \beta_{2} - \)\(21\!\cdots\!54\)\( \beta_{3} + \)\(25\!\cdots\!23\)\( \beta_{4} + \)\(51\!\cdots\!00\)\( \beta_{5} - \)\(12\!\cdots\!00\)\( \beta_{6}) q^{45} +(\)\(78\!\cdots\!64\)\( + \)\(24\!\cdots\!20\)\( \beta_{1} + \)\(71\!\cdots\!08\)\( \beta_{2} + \)\(87\!\cdots\!06\)\( \beta_{3} - \)\(10\!\cdots\!94\)\( \beta_{4} - \)\(12\!\cdots\!34\)\( \beta_{5} + \)\(26\!\cdots\!00\)\( \beta_{6}) q^{46} +(\)\(15\!\cdots\!18\)\( + \)\(13\!\cdots\!86\)\( \beta_{1} + \)\(15\!\cdots\!96\)\( \beta_{2} - \)\(95\!\cdots\!90\)\( \beta_{3} + \)\(10\!\cdots\!10\)\( \beta_{4} - \)\(10\!\cdots\!40\)\( \beta_{5} - \)\(54\!\cdots\!50\)\( \beta_{6}) q^{47} +(\)\(97\!\cdots\!48\)\( - \)\(15\!\cdots\!28\)\( \beta_{1} - \)\(61\!\cdots\!68\)\( \beta_{2} + \)\(52\!\cdots\!00\)\( \beta_{3} - \)\(24\!\cdots\!56\)\( \beta_{4} + \)\(14\!\cdots\!88\)\( \beta_{5} - \)\(16\!\cdots\!24\)\( \beta_{6}) q^{48} +(\)\(69\!\cdots\!45\)\( - \)\(20\!\cdots\!00\)\( \beta_{1} - \)\(32\!\cdots\!12\)\( \beta_{2} - \)\(80\!\cdots\!84\)\( \beta_{3} - \)\(17\!\cdots\!24\)\( \beta_{4} - \)\(39\!\cdots\!64\)\( \beta_{5} + \)\(55\!\cdots\!00\)\( \beta_{6}) q^{49} +(\)\(21\!\cdots\!25\)\( - \)\(47\!\cdots\!85\)\( \beta_{1} + \)\(55\!\cdots\!80\)\( \beta_{2} - \)\(12\!\cdots\!40\)\( \beta_{3} + \)\(19\!\cdots\!80\)\( \beta_{4} + \)\(21\!\cdots\!00\)\( \beta_{5} - \)\(74\!\cdots\!00\)\( \beta_{6}) q^{50} +(\)\(40\!\cdots\!64\)\( + \)\(18\!\cdots\!32\)\( \beta_{1} + \)\(40\!\cdots\!50\)\( \beta_{2} - \)\(72\!\cdots\!70\)\( \beta_{3} - \)\(33\!\cdots\!86\)\( \beta_{4} + \)\(17\!\cdots\!84\)\( \beta_{5} - \)\(10\!\cdots\!30\)\( \beta_{6}) q^{51} +(\)\(41\!\cdots\!00\)\( + \)\(50\!\cdots\!56\)\( \beta_{1} + \)\(43\!\cdots\!14\)\( \beta_{2} + \)\(84\!\cdots\!10\)\( \beta_{3} - \)\(57\!\cdots\!40\)\( \beta_{4} - \)\(57\!\cdots\!40\)\( \beta_{5} + \)\(74\!\cdots\!00\)\( \beta_{6}) q^{52} +(\)\(39\!\cdots\!12\)\( - \)\(19\!\cdots\!73\)\( \beta_{1} - \)\(26\!\cdots\!38\)\( \beta_{2} - \)\(53\!\cdots\!70\)\( \beta_{3} + \)\(51\!\cdots\!27\)\( \beta_{4} + \)\(61\!\cdots\!24\)\( \beta_{5} - \)\(15\!\cdots\!12\)\( \beta_{6}) q^{53} +(-\)\(43\!\cdots\!92\)\( - \)\(35\!\cdots\!52\)\( \beta_{1} - \)\(56\!\cdots\!20\)\( \beta_{2} - \)\(39\!\cdots\!90\)\( \beta_{3} + \)\(15\!\cdots\!26\)\( \beta_{4} + \)\(93\!\cdots\!46\)\( \beta_{5} + \)\(29\!\cdots\!40\)\( \beta_{6}) q^{54} +(-\)\(81\!\cdots\!55\)\( - \)\(42\!\cdots\!21\)\( \beta_{1} - \)\(34\!\cdots\!72\)\( \beta_{2} + \)\(66\!\cdots\!11\)\( \beta_{3} - \)\(44\!\cdots\!07\)\( \beta_{4} - \)\(41\!\cdots\!50\)\( \beta_{5} + \)\(67\!\cdots\!75\)\( \beta_{6}) q^{55} +(-\)\(23\!\cdots\!52\)\( - \)\(74\!\cdots\!28\)\( \beta_{1} + \)\(32\!\cdots\!44\)\( \beta_{2} - \)\(28\!\cdots\!32\)\( \beta_{3} - \)\(87\!\cdots\!48\)\( \beta_{4} + \)\(52\!\cdots\!92\)\( \beta_{5} - \)\(18\!\cdots\!20\)\( \beta_{6}) q^{56} +(-\)\(48\!\cdots\!96\)\( - \)\(45\!\cdots\!30\)\( \beta_{1} - \)\(25\!\cdots\!84\)\( \beta_{2} + \)\(31\!\cdots\!80\)\( \beta_{3} + \)\(47\!\cdots\!30\)\( \beta_{4} - \)\(11\!\cdots\!20\)\( \beta_{5} + \)\(15\!\cdots\!00\)\( \beta_{6}) q^{57} +(-\)\(15\!\cdots\!86\)\( + \)\(28\!\cdots\!10\)\( \beta_{1} + \)\(61\!\cdots\!68\)\( \beta_{2} - \)\(10\!\cdots\!60\)\( \beta_{3} - \)\(49\!\cdots\!76\)\( \beta_{4} - \)\(14\!\cdots\!92\)\( \beta_{5} + \)\(43\!\cdots\!36\)\( \beta_{6}) q^{58} +(-\)\(33\!\cdots\!57\)\( + \)\(19\!\cdots\!99\)\( \beta_{1} - \)\(39\!\cdots\!39\)\( \beta_{2} + \)\(12\!\cdots\!72\)\( \beta_{3} - \)\(60\!\cdots\!40\)\( \beta_{4} + \)\(66\!\cdots\!00\)\( \beta_{5} - \)\(16\!\cdots\!40\)\( \beta_{6}) q^{59} +(-\)\(15\!\cdots\!60\)\( + \)\(95\!\cdots\!52\)\( \beta_{1} + \)\(41\!\cdots\!04\)\( \beta_{2} + \)\(21\!\cdots\!28\)\( \beta_{3} + \)\(10\!\cdots\!84\)\( \beta_{4} - \)\(61\!\cdots\!60\)\( \beta_{5} + \)\(19\!\cdots\!60\)\( \beta_{6}) q^{60} +(\)\(24\!\cdots\!12\)\( - \)\(13\!\cdots\!25\)\( \beta_{1} - \)\(21\!\cdots\!90\)\( \beta_{2} + \)\(23\!\cdots\!30\)\( \beta_{3} + \)\(12\!\cdots\!35\)\( \beta_{4} + \)\(27\!\cdots\!40\)\( \beta_{5} + \)\(17\!\cdots\!20\)\( \beta_{6}) q^{61} +(\)\(28\!\cdots\!92\)\( - \)\(26\!\cdots\!28\)\( \beta_{1} + \)\(28\!\cdots\!64\)\( \beta_{2} - \)\(79\!\cdots\!00\)\( \beta_{3} + \)\(23\!\cdots\!56\)\( \beta_{4} - \)\(32\!\cdots\!88\)\( \beta_{5} - \)\(99\!\cdots\!76\)\( \beta_{6}) q^{62} +(\)\(20\!\cdots\!17\)\( - \)\(14\!\cdots\!93\)\( \beta_{1} - \)\(39\!\cdots\!52\)\( \beta_{2} - \)\(22\!\cdots\!65\)\( \beta_{3} - \)\(26\!\cdots\!11\)\( \beta_{4} - \)\(74\!\cdots\!82\)\( \beta_{5} + \)\(13\!\cdots\!91\)\( \beta_{6}) q^{63} +(\)\(94\!\cdots\!60\)\( - \)\(14\!\cdots\!76\)\( \beta_{1} - \)\(29\!\cdots\!08\)\( \beta_{2} + \)\(70\!\cdots\!84\)\( \beta_{3} + \)\(36\!\cdots\!52\)\( \beta_{4} + \)\(28\!\cdots\!72\)\( \beta_{5} + \)\(23\!\cdots\!00\)\( \beta_{6}) q^{64} +(\)\(11\!\cdots\!80\)\( + \)\(14\!\cdots\!64\)\( \beta_{1} - \)\(11\!\cdots\!72\)\( \beta_{2} - \)\(22\!\cdots\!04\)\( \beta_{3} + \)\(44\!\cdots\!88\)\( \beta_{4} - \)\(17\!\cdots\!20\)\( \beta_{5} - \)\(35\!\cdots\!80\)\( \beta_{6}) q^{65} +(\)\(83\!\cdots\!80\)\( + \)\(96\!\cdots\!36\)\( \beta_{1} + \)\(41\!\cdots\!40\)\( \beta_{2} - \)\(10\!\cdots\!80\)\( \beta_{3} - \)\(14\!\cdots\!48\)\( \beta_{4} - \)\(78\!\cdots\!88\)\( \beta_{5} + \)\(46\!\cdots\!60\)\( \beta_{6}) q^{66} +(\)\(60\!\cdots\!03\)\( + \)\(11\!\cdots\!91\)\( \beta_{1} + \)\(54\!\cdots\!03\)\( \beta_{2} - \)\(22\!\cdots\!70\)\( \beta_{3} + \)\(18\!\cdots\!54\)\( \beta_{4} + \)\(14\!\cdots\!28\)\( \beta_{5} - \)\(17\!\cdots\!54\)\( \beta_{6}) q^{67} +(-\)\(29\!\cdots\!44\)\( - \)\(89\!\cdots\!52\)\( \beta_{1} + \)\(12\!\cdots\!10\)\( \beta_{2} + \)\(13\!\cdots\!70\)\( \beta_{3} + \)\(53\!\cdots\!68\)\( \beta_{4} + \)\(14\!\cdots\!16\)\( \beta_{5} + \)\(34\!\cdots\!92\)\( \beta_{6}) q^{68} +(-\)\(29\!\cdots\!52\)\( - \)\(23\!\cdots\!32\)\( \beta_{1} - \)\(21\!\cdots\!36\)\( \beta_{2} + \)\(86\!\cdots\!08\)\( \beta_{3} + \)\(78\!\cdots\!24\)\( \beta_{4} - \)\(67\!\cdots\!96\)\( \beta_{5} - \)\(13\!\cdots\!40\)\( \beta_{6}) q^{69} +(-\)\(20\!\cdots\!40\)\( - \)\(43\!\cdots\!28\)\( \beta_{1} - \)\(39\!\cdots\!96\)\( \beta_{2} - \)\(91\!\cdots\!52\)\( \beta_{3} - \)\(14\!\cdots\!76\)\( \beta_{4} + \)\(18\!\cdots\!00\)\( \beta_{5} - \)\(12\!\cdots\!00\)\( \beta_{6}) q^{70} +(-\)\(36\!\cdots\!23\)\( - \)\(62\!\cdots\!25\)\( \beta_{1} + \)\(24\!\cdots\!80\)\( \beta_{2} + \)\(10\!\cdots\!15\)\( \beta_{3} - \)\(14\!\cdots\!95\)\( \beta_{4} + \)\(23\!\cdots\!70\)\( \beta_{5} + \)\(14\!\cdots\!35\)\( \beta_{6}) q^{71} +(-\)\(12\!\cdots\!16\)\( + \)\(22\!\cdots\!40\)\( \beta_{1} + \)\(87\!\cdots\!72\)\( \beta_{2} - \)\(76\!\cdots\!00\)\( \beta_{3} + \)\(64\!\cdots\!52\)\( \beta_{4} - \)\(39\!\cdots\!96\)\( \beta_{5} - \)\(26\!\cdots\!92\)\( \beta_{6}) q^{72} +(-\)\(43\!\cdots\!42\)\( + \)\(33\!\cdots\!22\)\( \beta_{1} + \)\(56\!\cdots\!52\)\( \beta_{2} + \)\(93\!\cdots\!80\)\( \beta_{3} - \)\(54\!\cdots\!66\)\( \beta_{4} - \)\(14\!\cdots\!12\)\( \beta_{5} - \)\(72\!\cdots\!84\)\( \beta_{6}) q^{73} +(\)\(74\!\cdots\!94\)\( - \)\(52\!\cdots\!50\)\( \beta_{1} - \)\(46\!\cdots\!88\)\( \beta_{2} + \)\(15\!\cdots\!24\)\( \beta_{3} + \)\(37\!\cdots\!84\)\( \beta_{4} + \)\(11\!\cdots\!44\)\( \beta_{5} + \)\(10\!\cdots\!80\)\( \beta_{6}) q^{74} +(-\)\(15\!\cdots\!25\)\( + \)\(15\!\cdots\!55\)\( \beta_{1} + \)\(15\!\cdots\!85\)\( \beta_{2} - \)\(23\!\cdots\!80\)\( \beta_{3} - \)\(12\!\cdots\!40\)\( \beta_{4} - \)\(13\!\cdots\!00\)\( \beta_{5} - \)\(13\!\cdots\!00\)\( \beta_{6}) q^{75} +(\)\(44\!\cdots\!56\)\( - \)\(12\!\cdots\!56\)\( \beta_{1} - \)\(51\!\cdots\!72\)\( \beta_{2} - \)\(20\!\cdots\!84\)\( \beta_{3} - \)\(16\!\cdots\!16\)\( \beta_{4} + \)\(11\!\cdots\!64\)\( \beta_{5} - \)\(62\!\cdots\!40\)\( \beta_{6}) q^{76} +(\)\(28\!\cdots\!16\)\( - \)\(24\!\cdots\!68\)\( \beta_{1} + \)\(24\!\cdots\!04\)\( \beta_{2} - \)\(41\!\cdots\!00\)\( \beta_{3} + \)\(11\!\cdots\!84\)\( \beta_{4} + \)\(39\!\cdots\!68\)\( \beta_{5} + \)\(31\!\cdots\!36\)\( \beta_{6}) q^{77} +(\)\(30\!\cdots\!32\)\( - \)\(11\!\cdots\!16\)\( \beta_{1} - \)\(11\!\cdots\!92\)\( \beta_{2} + \)\(20\!\cdots\!10\)\( \beta_{3} - \)\(84\!\cdots\!82\)\( \beta_{4} - \)\(16\!\cdots\!74\)\( \beta_{5} - \)\(72\!\cdots\!68\)\( \beta_{6}) q^{78} +(\)\(33\!\cdots\!66\)\( - \)\(41\!\cdots\!78\)\( \beta_{1} + \)\(62\!\cdots\!56\)\( \beta_{2} - \)\(23\!\cdots\!18\)\( \beta_{3} - \)\(25\!\cdots\!54\)\( \beta_{4} + \)\(10\!\cdots\!16\)\( \beta_{5} - \)\(37\!\cdots\!10\)\( \beta_{6}) q^{79} +(\)\(24\!\cdots\!60\)\( + \)\(19\!\cdots\!92\)\( \beta_{1} - \)\(58\!\cdots\!56\)\( \beta_{2} - \)\(15\!\cdots\!72\)\( \beta_{3} + \)\(23\!\cdots\!64\)\( \beta_{4} - \)\(21\!\cdots\!00\)\( \beta_{5} - \)\(83\!\cdots\!00\)\( \beta_{6}) q^{80} +(-\)\(25\!\cdots\!43\)\( + \)\(98\!\cdots\!58\)\( \beta_{1} - \)\(29\!\cdots\!88\)\( \beta_{2} - \)\(42\!\cdots\!56\)\( \beta_{3} + \)\(84\!\cdots\!50\)\( \beta_{4} + \)\(59\!\cdots\!40\)\( \beta_{5} + \)\(33\!\cdots\!60\)\( \beta_{6}) q^{81} +(-\)\(44\!\cdots\!58\)\( + \)\(17\!\cdots\!02\)\( \beta_{1} + \)\(10\!\cdots\!84\)\( \beta_{2} - \)\(48\!\cdots\!60\)\( \beta_{3} - \)\(12\!\cdots\!04\)\( \beta_{4} + \)\(38\!\cdots\!52\)\( \beta_{5} + \)\(11\!\cdots\!24\)\( \beta_{6}) q^{82} +(-\)\(65\!\cdots\!19\)\( - \)\(62\!\cdots\!43\)\( \beta_{1} - \)\(22\!\cdots\!13\)\( \beta_{2} + \)\(20\!\cdots\!00\)\( \beta_{3} - \)\(11\!\cdots\!00\)\( \beta_{4} - \)\(68\!\cdots\!00\)\( \beta_{5} - \)\(13\!\cdots\!00\)\( \beta_{6}) q^{83} +(-\)\(56\!\cdots\!76\)\( - \)\(59\!\cdots\!88\)\( \beta_{1} + \)\(59\!\cdots\!44\)\( \beta_{2} - \)\(29\!\cdots\!72\)\( \beta_{3} + \)\(41\!\cdots\!72\)\( \beta_{4} + \)\(10\!\cdots\!92\)\( \beta_{5} + \)\(17\!\cdots\!00\)\( \beta_{6}) q^{84} +(-\)\(57\!\cdots\!60\)\( - \)\(11\!\cdots\!78\)\( \beta_{1} - \)\(85\!\cdots\!56\)\( \beta_{2} - \)\(91\!\cdots\!92\)\( \beta_{3} - \)\(59\!\cdots\!26\)\( \beta_{4} - \)\(21\!\cdots\!60\)\( \beta_{5} + \)\(15\!\cdots\!60\)\( \beta_{6}) q^{85} +(-\)\(12\!\cdots\!78\)\( + \)\(11\!\cdots\!62\)\( \beta_{1} + \)\(30\!\cdots\!62\)\( \beta_{2} - \)\(91\!\cdots\!11\)\( \beta_{3} + \)\(80\!\cdots\!73\)\( \beta_{4} + \)\(44\!\cdots\!33\)\( \beta_{5} - \)\(51\!\cdots\!80\)\( \beta_{6}) q^{86} +(\)\(28\!\cdots\!15\)\( - \)\(41\!\cdots\!15\)\( \beta_{1} - \)\(29\!\cdots\!12\)\( \beta_{2} + \)\(86\!\cdots\!05\)\( \beta_{3} + \)\(51\!\cdots\!51\)\( \beta_{4} + \)\(97\!\cdots\!22\)\( \beta_{5} + \)\(59\!\cdots\!09\)\( \beta_{6}) q^{87} +(\)\(46\!\cdots\!32\)\( + \)\(21\!\cdots\!00\)\( \beta_{1} + \)\(32\!\cdots\!40\)\( \beta_{2} + \)\(10\!\cdots\!00\)\( \beta_{3} - \)\(49\!\cdots\!16\)\( \beta_{4} - \)\(18\!\cdots\!32\)\( \beta_{5} + \)\(82\!\cdots\!36\)\( \beta_{6}) q^{88} +(\)\(14\!\cdots\!34\)\( - \)\(52\!\cdots\!14\)\( \beta_{1} + \)\(25\!\cdots\!92\)\( \beta_{2} + \)\(11\!\cdots\!44\)\( \beta_{3} + \)\(60\!\cdots\!46\)\( \beta_{4} + \)\(13\!\cdots\!76\)\( \beta_{5} + \)\(64\!\cdots\!80\)\( \beta_{6}) q^{89} +(\)\(59\!\cdots\!30\)\( + \)\(26\!\cdots\!74\)\( \beta_{1} + \)\(33\!\cdots\!48\)\( \beta_{2} - \)\(13\!\cdots\!64\)\( \beta_{3} + \)\(39\!\cdots\!08\)\( \beta_{4} + \)\(28\!\cdots\!80\)\( \beta_{5} - \)\(35\!\cdots\!80\)\( \beta_{6}) q^{90} +(\)\(54\!\cdots\!60\)\( - \)\(37\!\cdots\!48\)\( \beta_{1} + \)\(31\!\cdots\!84\)\( \beta_{2} - \)\(14\!\cdots\!52\)\( \beta_{3} - \)\(10\!\cdots\!08\)\( \beta_{4} - \)\(83\!\cdots\!68\)\( \beta_{5} - \)\(63\!\cdots\!20\)\( \beta_{6}) q^{91} +(\)\(25\!\cdots\!36\)\( + \)\(10\!\cdots\!76\)\( \beta_{1} - \)\(35\!\cdots\!84\)\( \beta_{2} + \)\(20\!\cdots\!40\)\( \beta_{3} - \)\(30\!\cdots\!44\)\( \beta_{4} - \)\(16\!\cdots\!28\)\( \beta_{5} + \)\(13\!\cdots\!64\)\( \beta_{6}) q^{92} +(\)\(12\!\cdots\!96\)\( - \)\(82\!\cdots\!76\)\( \beta_{1} + \)\(97\!\cdots\!56\)\( \beta_{2} - \)\(49\!\cdots\!60\)\( \beta_{3} + \)\(48\!\cdots\!52\)\( \beta_{4} - \)\(10\!\cdots\!36\)\( \beta_{5} - \)\(10\!\cdots\!52\)\( \beta_{6}) q^{93} +(\)\(20\!\cdots\!24\)\( + \)\(21\!\cdots\!88\)\( \beta_{1} - \)\(23\!\cdots\!40\)\( \beta_{2} + \)\(76\!\cdots\!20\)\( \beta_{3} + \)\(37\!\cdots\!16\)\( \beta_{4} + \)\(24\!\cdots\!36\)\( \beta_{5} - \)\(22\!\cdots\!60\)\( \beta_{6}) q^{94} +(-\)\(20\!\cdots\!25\)\( - \)\(11\!\cdots\!75\)\( \beta_{1} + \)\(11\!\cdots\!00\)\( \beta_{2} + \)\(33\!\cdots\!25\)\( \beta_{3} - \)\(21\!\cdots\!25\)\( \beta_{4} + \)\(12\!\cdots\!50\)\( \beta_{5} + \)\(34\!\cdots\!25\)\( \beta_{6}) q^{95} +(-\)\(15\!\cdots\!88\)\( + \)\(87\!\cdots\!08\)\( \beta_{1} + \)\(67\!\cdots\!08\)\( \beta_{2} - \)\(14\!\cdots\!64\)\( \beta_{3} - \)\(49\!\cdots\!28\)\( \beta_{4} - \)\(79\!\cdots\!08\)\( \beta_{5} + \)\(28\!\cdots\!00\)\( \beta_{6}) q^{96} +(-\)\(10\!\cdots\!70\)\( - \)\(49\!\cdots\!74\)\( \beta_{1} - \)\(11\!\cdots\!52\)\( \beta_{2} + \)\(22\!\cdots\!20\)\( \beta_{3} + \)\(21\!\cdots\!98\)\( \beta_{4} + \)\(52\!\cdots\!76\)\( \beta_{5} - \)\(55\!\cdots\!88\)\( \beta_{6}) q^{97} +(-\)\(27\!\cdots\!29\)\( - \)\(96\!\cdots\!27\)\( \beta_{1} - \)\(48\!\cdots\!36\)\( \beta_{2} + \)\(20\!\cdots\!40\)\( \beta_{3} - \)\(16\!\cdots\!64\)\( \beta_{4} + \)\(61\!\cdots\!32\)\( \beta_{5} - \)\(11\!\cdots\!16\)\( \beta_{6}) q^{98} +(-\)\(26\!\cdots\!69\)\( - \)\(14\!\cdots\!57\)\( \beta_{1} - \)\(21\!\cdots\!19\)\( \beta_{2} + \)\(18\!\cdots\!32\)\( \beta_{3} + \)\(15\!\cdots\!28\)\( \beta_{4} - \)\(46\!\cdots\!12\)\( \beta_{5} + \)\(21\!\cdots\!20\)\( \beta_{6}) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7q + 347450761416q^{2} + 92768992732348371972q^{3} + \)\(35\!\cdots\!96\)\(q^{4} + \)\(95\!\cdots\!70\)\(q^{5} - \)\(29\!\cdots\!96\)\(q^{6} + \)\(41\!\cdots\!56\)\(q^{7} - \)\(53\!\cdots\!60\)\(q^{8} + \)\(71\!\cdots\!19\)\(q^{9} + O(q^{10}) \) \( 7q + 347450761416q^{2} + 92768992732348371972q^{3} + \)\(35\!\cdots\!96\)\(q^{4} + \)\(95\!\cdots\!70\)\(q^{5} - \)\(29\!\cdots\!96\)\(q^{6} + \)\(41\!\cdots\!56\)\(q^{7} - \)\(53\!\cdots\!60\)\(q^{8} + \)\(71\!\cdots\!19\)\(q^{9} + \)\(49\!\cdots\!20\)\(q^{10} + \)\(25\!\cdots\!24\)\(q^{11} + \)\(24\!\cdots\!56\)\(q^{12} + \)\(17\!\cdots\!02\)\(q^{13} - \)\(66\!\cdots\!48\)\(q^{14} + \)\(49\!\cdots\!40\)\(q^{15} + \)\(41\!\cdots\!72\)\(q^{16} - \)\(18\!\cdots\!14\)\(q^{17} - \)\(28\!\cdots\!88\)\(q^{18} + \)\(70\!\cdots\!60\)\(q^{19} - \)\(57\!\cdots\!40\)\(q^{20} + \)\(18\!\cdots\!04\)\(q^{21} - \)\(25\!\cdots\!88\)\(q^{22} + \)\(82\!\cdots\!32\)\(q^{23} - \)\(51\!\cdots\!20\)\(q^{24} - \)\(45\!\cdots\!75\)\(q^{25} + \)\(14\!\cdots\!44\)\(q^{26} - \)\(18\!\cdots\!40\)\(q^{27} - \)\(33\!\cdots\!12\)\(q^{28} + \)\(64\!\cdots\!90\)\(q^{29} + \)\(22\!\cdots\!40\)\(q^{30} - \)\(37\!\cdots\!56\)\(q^{31} - \)\(40\!\cdots\!24\)\(q^{32} + \)\(21\!\cdots\!04\)\(q^{33} - \)\(36\!\cdots\!48\)\(q^{34} - \)\(20\!\cdots\!80\)\(q^{35} + \)\(57\!\cdots\!32\)\(q^{36} + \)\(11\!\cdots\!46\)\(q^{37} - \)\(10\!\cdots\!40\)\(q^{38} + \)\(15\!\cdots\!88\)\(q^{39} + \)\(94\!\cdots\!00\)\(q^{40} + \)\(17\!\cdots\!54\)\(q^{41} + \)\(44\!\cdots\!12\)\(q^{42} + \)\(15\!\cdots\!92\)\(q^{43} + \)\(13\!\cdots\!72\)\(q^{44} + \)\(18\!\cdots\!90\)\(q^{45} + \)\(55\!\cdots\!84\)\(q^{46} + \)\(11\!\cdots\!76\)\(q^{47} + \)\(68\!\cdots\!32\)\(q^{48} + \)\(48\!\cdots\!51\)\(q^{49} + \)\(15\!\cdots\!00\)\(q^{50} + \)\(28\!\cdots\!04\)\(q^{51} + \)\(28\!\cdots\!96\)\(q^{52} + \)\(27\!\cdots\!22\)\(q^{53} - \)\(30\!\cdots\!40\)\(q^{54} - \)\(57\!\cdots\!60\)\(q^{55} - \)\(16\!\cdots\!60\)\(q^{56} - \)\(34\!\cdots\!80\)\(q^{57} - \)\(10\!\cdots\!60\)\(q^{58} - \)\(23\!\cdots\!20\)\(q^{59} - \)\(11\!\cdots\!80\)\(q^{60} + \)\(17\!\cdots\!74\)\(q^{61} + \)\(20\!\cdots\!72\)\(q^{62} + \)\(14\!\cdots\!92\)\(q^{63} + \)\(66\!\cdots\!96\)\(q^{64} + \)\(78\!\cdots\!40\)\(q^{65} + \)\(58\!\cdots\!28\)\(q^{66} + \)\(42\!\cdots\!36\)\(q^{67} - \)\(20\!\cdots\!72\)\(q^{68} - \)\(20\!\cdots\!32\)\(q^{69} - \)\(14\!\cdots\!80\)\(q^{70} - \)\(25\!\cdots\!16\)\(q^{71} - \)\(87\!\cdots\!20\)\(q^{72} - \)\(30\!\cdots\!18\)\(q^{73} + \)\(51\!\cdots\!52\)\(q^{74} - \)\(10\!\cdots\!00\)\(q^{75} + \)\(31\!\cdots\!80\)\(q^{76} + \)\(19\!\cdots\!92\)\(q^{77} + \)\(21\!\cdots\!04\)\(q^{78} + \)\(23\!\cdots\!40\)\(q^{79} + \)\(17\!\cdots\!20\)\(q^{80} - \)\(17\!\cdots\!93\)\(q^{81} - \)\(31\!\cdots\!48\)\(q^{82} - \)\(46\!\cdots\!88\)\(q^{83} - \)\(39\!\cdots\!88\)\(q^{84} - \)\(40\!\cdots\!80\)\(q^{85} - \)\(84\!\cdots\!36\)\(q^{86} + \)\(19\!\cdots\!80\)\(q^{87} + \)\(32\!\cdots\!80\)\(q^{88} + \)\(10\!\cdots\!70\)\(q^{89} + \)\(41\!\cdots\!40\)\(q^{90} + \)\(38\!\cdots\!44\)\(q^{91} + \)\(18\!\cdots\!36\)\(q^{92} + \)\(86\!\cdots\!24\)\(q^{93} + \)\(14\!\cdots\!52\)\(q^{94} - \)\(14\!\cdots\!00\)\(q^{95} - \)\(10\!\cdots\!56\)\(q^{96} - \)\(74\!\cdots\!74\)\(q^{97} - \)\(18\!\cdots\!12\)\(q^{98} - \)\(18\!\cdots\!92\)\(q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{7} - x^{6} - 89641237459195851367368 x^{5} + 1490735848862859486144964924638452 x^{4} + 1930211822410210561241520455818961338113682960 x^{3} - 58721021531954131531332854115146412227388949851414322800 x^{2} - 9636367192739742930950668323030502132711885071308786705499042816000 x + 225425399226476103878513452100286122913961367345316581915847454103589416760000\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 24 \nu - 3 \)
\(\beta_{2}\)\(=\)\((\)\(\)\(33\!\cdots\!59\)\( \nu^{6} + \)\(38\!\cdots\!59\)\( \nu^{5} - \)\(26\!\cdots\!62\)\( \nu^{4} - \)\(24\!\cdots\!44\)\( \nu^{3} + \)\(37\!\cdots\!96\)\( \nu^{2} + \)\(22\!\cdots\!28\)\( \nu - \)\(74\!\cdots\!48\)\(\)\()/ \)\(10\!\cdots\!52\)\( \)
\(\beta_{3}\)\(=\)\((\)\(\)\(18\!\cdots\!03\)\( \nu^{6} + \)\(21\!\cdots\!03\)\( \nu^{5} - \)\(14\!\cdots\!54\)\( \nu^{4} - \)\(13\!\cdots\!48\)\( \nu^{3} + \)\(21\!\cdots\!84\)\( \nu^{2} + \)\(12\!\cdots\!76\)\( \nu - \)\(56\!\cdots\!72\)\(\)\()/ \)\(10\!\cdots\!52\)\( \)
\(\beta_{4}\)\(=\)\((\)\(-\)\(66\!\cdots\!99\)\( \nu^{6} - \)\(78\!\cdots\!91\)\( \nu^{5} + \)\(50\!\cdots\!22\)\( \nu^{4} + \)\(49\!\cdots\!72\)\( \nu^{3} - \)\(70\!\cdots\!20\)\( \nu^{2} - \)\(45\!\cdots\!00\)\( \nu + \)\(12\!\cdots\!00\)\(\)\()/ \)\(70\!\cdots\!00\)\( \)
\(\beta_{5}\)\(=\)\((\)\(\)\(28\!\cdots\!99\)\( \nu^{6} + \)\(32\!\cdots\!91\)\( \nu^{5} - \)\(21\!\cdots\!22\)\( \nu^{4} - \)\(20\!\cdots\!72\)\( \nu^{3} + \)\(30\!\cdots\!20\)\( \nu^{2} + \)\(18\!\cdots\!00\)\( \nu - \)\(56\!\cdots\!00\)\(\)\()/ \)\(25\!\cdots\!00\)\( \)
\(\beta_{6}\)\(=\)\((\)\(\)\(31\!\cdots\!43\)\( \nu^{6} + \)\(32\!\cdots\!87\)\( \nu^{5} - \)\(24\!\cdots\!54\)\( \nu^{4} - \)\(19\!\cdots\!04\)\( \nu^{3} + \)\(34\!\cdots\!40\)\( \nu^{2} + \)\(16\!\cdots\!00\)\( \nu - \)\(61\!\cdots\!00\)\(\)\()/ \)\(85\!\cdots\!00\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1} + 3\)\()/24\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} - 55717 \beta_{2} - 598680817450 \beta_{1} + 14752386507570773831110328\)\()/576\)
\(\nu^{3}\)\(=\)\((\)\(21 \beta_{6} - 19917 \beta_{5} - 12811516 \beta_{4} - 30990098711 \beta_{3} + 9324237302923217 \beta_{2} + 3353470015785282503673651 \beta_{1} - 1103996377150060544464718742566708093\)\()/1728\)
\(\nu^{4}\)\(=\)\((\)\(1716728799877 \beta_{6} + 2815816636891587 \beta_{5} - 3880270734803784636 \beta_{4} + 567492910382550157904063 \beta_{3} - 54010293718434759060258961661 \beta_{2} - 451261134655122535170485467287287193 \beta_{1} + 6183960731059307947203775791742670090324795964163\)\()/5184\)
\(\nu^{5}\)\(=\)\((\)\(1289009423366260981283899 \beta_{6} - 1598635181494851381896254083 \beta_{5} - 1058770791342485363752163466564 \beta_{4} - 4453777458051671636672903816446135 \beta_{3} + 1214520439369559055842656024965072018517 \beta_{2} + 187531216987097574618989614645447376912689365897 \beta_{1} - 92460817660760799002781722708421287361656622355436362908803\)\()/1728\)
\(\nu^{6}\)\(=\)\((\)\(48693722572342833834731096897963399 \beta_{6} + 110727202095643716925987449749626134033 \beta_{5} - 71258167992269751133676344318933820842708 \beta_{4} + 11468225868854744137077312048346953991299666605 \beta_{3} - 1263383964865374527236870463461437955571946519984983 \beta_{2} - 11422068171564705047611248719271590459898086130320376664563 \beta_{1} + 115272208319394359104551408745791183512749492933635115387941267540913041\)\()/1728\)

Embeddings

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

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

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−2.54071e11
−1.58823e11
−7.80143e10
2.25782e10
1.15014e11
1.16386e11
2.36930e11
−6.04807e12 9.44466e19 2.69078e25 −3.42567e28 −5.71220e32 −1.33934e34 −1.04247e38 4.92932e39 2.07187e41
1.2 −3.76211e12 −6.68720e19 4.48205e24 1.95538e28 2.51580e32 1.66208e35 1.95229e37 4.81025e38 −7.35637e40
1.3 −1.82271e12 1.57108e19 −6.34915e24 −2.84572e28 −2.86362e31 −2.18145e35 2.92008e37 −3.74401e39 5.18692e40
1.4 5.91512e11 9.95960e19 −9.32152e24 1.29632e29 5.89123e31 1.88228e35 −1.12345e37 5.92853e39 7.66792e40
1.5 2.80997e12 −9.69898e19 −1.77546e24 1.10914e29 −2.72539e32 −5.50994e34 −3.21654e37 5.41619e39 3.11666e41
1.6 2.84290e12 5.71915e17 −1.58931e24 −1.75639e29 1.62590e30 1.13550e35 −3.20131e37 −3.99051e39 −4.99324e41
1.7 5.73595e12 4.63055e19 2.32297e25 7.41344e28 2.65606e32 −1.39412e35 7.77697e37 −1.84664e39 4.25231e41
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
Significant digits:
Format:

Inner twists

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

Hecke kernels

There are no other newforms in \(S_{84}^{\mathrm{new}}(\Gamma_0(1))\).