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

Related objects

Downloads

Learn more about

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.

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

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

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

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 \(S_{84}^{\mathrm{new}}(\Gamma_0(1))\).