Properties

Label 1.88.a.a
Level 1
Weight 88
Character orbit 1.a
Self dual Yes
Analytic conductor 47.933
Analytic rank 0
Dimension 7
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 1 \)
Weight: \( k \) = \( 88 \)
Character orbit: \([\chi]\) = 1.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(47.9333631461\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: multiple of \( 2^{76}\cdot 3^{35}\cdot 5^{8}\cdot 7^{4}\cdot 11^{2}\cdot 13\cdot 17\cdot 29^{2} \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(f(q)\) \(=\) \( q\) \(+(2599574577562 + \beta_{1}) q^{2}\) \(+(-10773887085464872817 + 11867285 \beta_{1} - \beta_{2}) q^{3}\) \(+(\)\(53\!\cdots\!39\)\( + 3273747124418 \beta_{1} - 55029 \beta_{2} + \beta_{3}) q^{4}\) \(+(\)\(47\!\cdots\!46\)\( - 26691298401857906 \beta_{1} + 1159759239 \beta_{2} - 1339 \beta_{3} + \beta_{4}) q^{5}\) \(+(\)\(23\!\cdots\!85\)\( + \)\(10\!\cdots\!97\)\( \beta_{1} - 2258267238311 \beta_{2} + 16021198 \beta_{3} - 461 \beta_{4} + \beta_{5}) q^{6}\) \(+(\)\(65\!\cdots\!13\)\( + \)\(12\!\cdots\!66\)\( \beta_{1} - 2558719881289795 \beta_{2} + 8344108529 \beta_{3} + 351922 \beta_{4} + 75 \beta_{5} - \beta_{6}) q^{7}\) \(+(\)\(39\!\cdots\!56\)\( + \)\(25\!\cdots\!28\)\( \beta_{1} - 521991805891439256 \beta_{2} + 3239851131088 \beta_{3} - 16926616 \beta_{4} + 172800 \beta_{5} - 72 \beta_{6}) q^{8}\) \(+(\)\(96\!\cdots\!61\)\( + \)\(77\!\cdots\!36\)\( \beta_{1} + 18460244025839105382 \beta_{2} + 389830491911874 \beta_{3} + 13195088382 \beta_{4} + 37818588 \beta_{5} + 24300 \beta_{6}) q^{9}\) \(+O(q^{10})\) \( q\) \(+(2599574577562 + \beta_{1}) q^{2}\) \(+(-10773887085464872817 + 11867285 \beta_{1} - \beta_{2}) q^{3}\) \(+(\)\(53\!\cdots\!39\)\( + 3273747124418 \beta_{1} - 55029 \beta_{2} + \beta_{3}) q^{4}\) \(+(\)\(47\!\cdots\!46\)\( - 26691298401857906 \beta_{1} + 1159759239 \beta_{2} - 1339 \beta_{3} + \beta_{4}) q^{5}\) \(+(\)\(23\!\cdots\!85\)\( + \)\(10\!\cdots\!97\)\( \beta_{1} - 2258267238311 \beta_{2} + 16021198 \beta_{3} - 461 \beta_{4} + \beta_{5}) q^{6}\) \(+(\)\(65\!\cdots\!13\)\( + \)\(12\!\cdots\!66\)\( \beta_{1} - 2558719881289795 \beta_{2} + 8344108529 \beta_{3} + 351922 \beta_{4} + 75 \beta_{5} - \beta_{6}) q^{7}\) \(+(\)\(39\!\cdots\!56\)\( + \)\(25\!\cdots\!28\)\( \beta_{1} - 521991805891439256 \beta_{2} + 3239851131088 \beta_{3} - 16926616 \beta_{4} + 172800 \beta_{5} - 72 \beta_{6}) q^{8}\) \(+(\)\(96\!\cdots\!61\)\( + \)\(77\!\cdots\!36\)\( \beta_{1} + 18460244025839105382 \beta_{2} + 389830491911874 \beta_{3} + 13195088382 \beta_{4} + 37818588 \beta_{5} + 24300 \beta_{6}) q^{9}\) \(+(-\)\(52\!\cdots\!64\)\( - \)\(25\!\cdots\!06\)\( \beta_{1} + \)\(12\!\cdots\!84\)\( \beta_{2} - 28646245621872264 \beta_{3} + 12629829380716 \beta_{4} - 2486792220 \beta_{5} - 2222080 \beta_{6}) q^{10}\) \(+(\)\(21\!\cdots\!59\)\( - \)\(18\!\cdots\!85\)\( \beta_{1} + \)\(72\!\cdots\!95\)\( \beta_{2} + 1434832638222415282 \beta_{3} + 475273083529412 \beta_{4} + 1984757958 \beta_{5} + 121027950 \beta_{6}) q^{11}\) \(+(\)\(28\!\cdots\!32\)\( + \)\(27\!\cdots\!92\)\( \beta_{1} - \)\(64\!\cdots\!32\)\( \beta_{2} + \)\(49\!\cdots\!68\)\( \beta_{3} - 19424243142677376 \beta_{4} + 4595807692800 \beta_{5} - 4678756992 \beta_{6}) q^{12}\) \(+(\)\(43\!\cdots\!58\)\( + \)\(28\!\cdots\!06\)\( \beta_{1} - \)\(63\!\cdots\!73\)\( \beta_{2} + \)\(13\!\cdots\!93\)\( \beta_{3} - 257405958719583151 \beta_{4} - 220454764203000 \beta_{5} + 139030911208 \beta_{6}) q^{13}\) \(+(\)\(24\!\cdots\!54\)\( + \)\(13\!\cdots\!34\)\( \beta_{1} - \)\(25\!\cdots\!02\)\( \beta_{2} + \)\(29\!\cdots\!28\)\( \beta_{3} + 16966725185054952590 \beta_{4} + 5772719908804410 \beta_{5} - 3323663769600 \beta_{6}) q^{14}\) \(+(-\)\(51\!\cdots\!73\)\( - \)\(19\!\cdots\!42\)\( \beta_{1} - \)\(38\!\cdots\!37\)\( \beta_{2} - \)\(69\!\cdots\!73\)\( \beta_{3} - \)\(19\!\cdots\!38\)\( \beta_{4} - 98787267301290115 \beta_{5} + 65806264868265 \beta_{6}) q^{15}\) \(+(-\)\(21\!\cdots\!72\)\( + \)\(34\!\cdots\!84\)\( \beta_{1} - \)\(29\!\cdots\!72\)\( \beta_{2} - \)\(50\!\cdots\!28\)\( \beta_{3} - \)\(15\!\cdots\!96\)\( \beta_{4} + 1098837490576091136 \beta_{5} - 1100370908964800 \beta_{6}) q^{16}\) \(+(\)\(43\!\cdots\!22\)\( + \)\(56\!\cdots\!28\)\( \beta_{1} - \)\(47\!\cdots\!34\)\( \beta_{2} + \)\(75\!\cdots\!66\)\( \beta_{3} + \)\(66\!\cdots\!38\)\( \beta_{4} - 5701810188248615700 \beta_{5} + 15745721338436796 \beta_{6}) q^{17}\) \(+(\)\(26\!\cdots\!50\)\( + \)\(14\!\cdots\!17\)\( \beta_{1} - \)\(76\!\cdots\!80\)\( \beta_{2} + \)\(11\!\cdots\!84\)\( \beta_{3} - \)\(67\!\cdots\!88\)\( \beta_{4} - 55702151964572941800 \beta_{5} - 194434583822954496 \beta_{6}) q^{18}\) \(+(-\)\(34\!\cdots\!95\)\( - \)\(37\!\cdots\!23\)\( \beta_{1} - \)\(28\!\cdots\!11\)\( \beta_{2} - \)\(13\!\cdots\!50\)\( \beta_{3} + \)\(27\!\cdots\!16\)\( \beta_{4} + \)\(17\!\cdots\!94\)\( \beta_{5} + 2080251453206650850 \beta_{6}) q^{19}\) \(+(-\)\(71\!\cdots\!22\)\( - \)\(59\!\cdots\!08\)\( \beta_{1} + \)\(47\!\cdots\!02\)\( \beta_{2} - \)\(72\!\cdots\!02\)\( \beta_{3} + \)\(68\!\cdots\!68\)\( \beta_{4} - \)\(24\!\cdots\!00\)\( \beta_{5} - 19266914499823065600 \beta_{6}) q^{20}\) \(+(\)\(12\!\cdots\!40\)\( + \)\(46\!\cdots\!64\)\( \beta_{1} + \)\(22\!\cdots\!08\)\( \beta_{2} + \)\(15\!\cdots\!68\)\( \beta_{3} - \)\(82\!\cdots\!80\)\( \beta_{4} + \)\(22\!\cdots\!80\)\( \beta_{5} + \)\(15\!\cdots\!00\)\( \beta_{6}) q^{21}\) \(+(-\)\(30\!\cdots\!77\)\( + \)\(29\!\cdots\!71\)\( \beta_{1} + \)\(53\!\cdots\!07\)\( \beta_{2} - \)\(36\!\cdots\!82\)\( \beta_{3} + \)\(42\!\cdots\!49\)\( \beta_{4} - \)\(14\!\cdots\!25\)\( \beta_{5} - \)\(10\!\cdots\!92\)\( \beta_{6}) q^{22}\) \(+(-\)\(27\!\cdots\!45\)\( + \)\(59\!\cdots\!14\)\( \beta_{1} - \)\(15\!\cdots\!29\)\( \beta_{2} - \)\(51\!\cdots\!85\)\( \beta_{3} + \)\(13\!\cdots\!70\)\( \beta_{4} + \)\(55\!\cdots\!25\)\( \beta_{5} + \)\(54\!\cdots\!65\)\( \beta_{6}) q^{23}\) \(+(\)\(26\!\cdots\!08\)\( + \)\(80\!\cdots\!36\)\( \beta_{1} - \)\(11\!\cdots\!88\)\( \beta_{2} + \)\(27\!\cdots\!48\)\( \beta_{3} - \)\(17\!\cdots\!24\)\( \beta_{4} + \)\(32\!\cdots\!84\)\( \beta_{5} - \)\(19\!\cdots\!00\)\( \beta_{6}) q^{24}\) \(+(\)\(13\!\cdots\!75\)\( + \)\(38\!\cdots\!00\)\( \beta_{1} + \)\(54\!\cdots\!00\)\( \beta_{2} + \)\(81\!\cdots\!00\)\( \beta_{3} + \)\(12\!\cdots\!00\)\( \beta_{4} - \)\(24\!\cdots\!00\)\( \beta_{5} - \)\(91\!\cdots\!00\)\( \beta_{6}) q^{25}\) \(+(\)\(68\!\cdots\!16\)\( + \)\(21\!\cdots\!26\)\( \beta_{1} + \)\(36\!\cdots\!12\)\( \beta_{2} - \)\(78\!\cdots\!36\)\( \beta_{3} - \)\(24\!\cdots\!08\)\( \beta_{4} + \)\(21\!\cdots\!28\)\( \beta_{5} + \)\(85\!\cdots\!00\)\( \beta_{6}) q^{26}\) \(+(-\)\(45\!\cdots\!76\)\( + \)\(12\!\cdots\!42\)\( \beta_{1} - \)\(49\!\cdots\!84\)\( \beta_{2} - \)\(20\!\cdots\!58\)\( \beta_{3} - \)\(14\!\cdots\!44\)\( \beta_{4} - \)\(11\!\cdots\!50\)\( \beta_{5} - \)\(88\!\cdots\!98\)\( \beta_{6}) q^{27}\) \(+(\)\(33\!\cdots\!96\)\( + \)\(44\!\cdots\!92\)\( \beta_{1} - \)\(81\!\cdots\!28\)\( \beta_{2} + \)\(28\!\cdots\!92\)\( \beta_{3} + \)\(11\!\cdots\!56\)\( \beta_{4} + \)\(37\!\cdots\!00\)\( \beta_{5} + \)\(62\!\cdots\!52\)\( \beta_{6}) q^{28}\) \(+(-\)\(41\!\cdots\!82\)\( + \)\(18\!\cdots\!98\)\( \beta_{1} - \)\(19\!\cdots\!29\)\( \beta_{2} - \)\(11\!\cdots\!47\)\( \beta_{3} - \)\(12\!\cdots\!23\)\( \beta_{4} - \)\(33\!\cdots\!32\)\( \beta_{5} - \)\(34\!\cdots\!00\)\( \beta_{6}) q^{29}\) \(+(-\)\(52\!\cdots\!38\)\( - \)\(13\!\cdots\!82\)\( \beta_{1} + \)\(20\!\cdots\!58\)\( \beta_{2} - \)\(64\!\cdots\!08\)\( \beta_{3} - \)\(23\!\cdots\!78\)\( \beta_{4} - \)\(36\!\cdots\!50\)\( \beta_{5} + \)\(15\!\cdots\!00\)\( \beta_{6}) q^{30}\) \(+(\)\(91\!\cdots\!28\)\( - \)\(40\!\cdots\!80\)\( \beta_{1} + \)\(36\!\cdots\!60\)\( \beta_{2} + \)\(19\!\cdots\!76\)\( \beta_{3} + \)\(14\!\cdots\!16\)\( \beta_{4} + \)\(18\!\cdots\!44\)\( \beta_{5} - \)\(51\!\cdots\!00\)\( \beta_{6}) q^{31}\) \(+(\)\(28\!\cdots\!20\)\( - \)\(87\!\cdots\!76\)\( \beta_{1} - \)\(20\!\cdots\!56\)\( \beta_{2} + \)\(55\!\cdots\!56\)\( \beta_{3} - \)\(28\!\cdots\!92\)\( \beta_{4} + \)\(15\!\cdots\!00\)\( \beta_{5} + \)\(11\!\cdots\!36\)\( \beta_{6}) q^{32}\) \(+(-\)\(32\!\cdots\!56\)\( - \)\(26\!\cdots\!32\)\( \beta_{1} - \)\(75\!\cdots\!58\)\( \beta_{2} - \)\(38\!\cdots\!14\)\( \beta_{3} - \)\(68\!\cdots\!02\)\( \beta_{4} - \)\(65\!\cdots\!00\)\( \beta_{5} - \)\(11\!\cdots\!84\)\( \beta_{6}) q^{33}\) \(+(\)\(12\!\cdots\!64\)\( + \)\(18\!\cdots\!06\)\( \beta_{1} + \)\(44\!\cdots\!52\)\( \beta_{2} + \)\(32\!\cdots\!28\)\( \beta_{3} + \)\(43\!\cdots\!16\)\( \beta_{4} + \)\(44\!\cdots\!44\)\( \beta_{5} - \)\(15\!\cdots\!00\)\( \beta_{6}) q^{34}\) \(+(\)\(42\!\cdots\!56\)\( + \)\(46\!\cdots\!24\)\( \beta_{1} + \)\(17\!\cdots\!64\)\( \beta_{2} + \)\(76\!\cdots\!56\)\( \beta_{3} + \)\(39\!\cdots\!36\)\( \beta_{4} - \)\(17\!\cdots\!20\)\( \beta_{5} + \)\(90\!\cdots\!20\)\( \beta_{6}) q^{35}\) \(+(\)\(14\!\cdots\!03\)\( + \)\(25\!\cdots\!34\)\( \beta_{1} - \)\(20\!\cdots\!97\)\( \beta_{2} + \)\(11\!\cdots\!77\)\( \beta_{3} - \)\(10\!\cdots\!16\)\( \beta_{4} + \)\(44\!\cdots\!56\)\( \beta_{5} - \)\(30\!\cdots\!00\)\( \beta_{6}) q^{36}\) \(+(-\)\(36\!\cdots\!78\)\( - \)\(39\!\cdots\!26\)\( \beta_{1} - \)\(40\!\cdots\!41\)\( \beta_{2} - \)\(63\!\cdots\!11\)\( \beta_{3} + \)\(38\!\cdots\!77\)\( \beta_{4} - \)\(38\!\cdots\!00\)\( \beta_{5} + \)\(56\!\cdots\!84\)\( \beta_{6}) q^{37}\) \(+(-\)\(83\!\cdots\!19\)\( - \)\(19\!\cdots\!71\)\( \beta_{1} - \)\(30\!\cdots\!43\)\( \beta_{2} + \)\(79\!\cdots\!86\)\( \beta_{3} - \)\(52\!\cdots\!77\)\( \beta_{4} - \)\(19\!\cdots\!75\)\( \beta_{5} + \)\(44\!\cdots\!16\)\( \beta_{6}) q^{38}\) \(+(\)\(31\!\cdots\!79\)\( - \)\(49\!\cdots\!86\)\( \beta_{1} + \)\(26\!\cdots\!23\)\( \beta_{2} + \)\(63\!\cdots\!15\)\( \beta_{3} - \)\(21\!\cdots\!98\)\( \beta_{4} + \)\(97\!\cdots\!93\)\( \beta_{5} - \)\(79\!\cdots\!75\)\( \beta_{6}) q^{39}\) \(+(-\)\(57\!\cdots\!60\)\( - \)\(17\!\cdots\!40\)\( \beta_{1} + \)\(48\!\cdots\!60\)\( \beta_{2} - \)\(99\!\cdots\!60\)\( \beta_{3} - \)\(31\!\cdots\!60\)\( \beta_{4} - \)\(19\!\cdots\!00\)\( \beta_{5} + \)\(31\!\cdots\!00\)\( \beta_{6}) q^{40}\) \(+(\)\(29\!\cdots\!78\)\( + \)\(29\!\cdots\!20\)\( \beta_{1} + \)\(16\!\cdots\!60\)\( \beta_{2} - \)\(36\!\cdots\!44\)\( \beta_{3} + \)\(38\!\cdots\!96\)\( \beta_{4} + \)\(10\!\cdots\!64\)\( \beta_{5} - \)\(66\!\cdots\!00\)\( \beta_{6}) q^{41}\) \(+(\)\(12\!\cdots\!32\)\( + \)\(29\!\cdots\!76\)\( \beta_{1} - \)\(54\!\cdots\!68\)\( \beta_{2} + \)\(96\!\cdots\!04\)\( \beta_{3} - \)\(13\!\cdots\!28\)\( \beta_{4} + \)\(10\!\cdots\!00\)\( \beta_{5} + \)\(10\!\cdots\!24\)\( \beta_{6}) q^{42}\) \(+(\)\(44\!\cdots\!57\)\( + \)\(33\!\cdots\!03\)\( \beta_{1} + \)\(32\!\cdots\!73\)\( \beta_{2} + \)\(15\!\cdots\!80\)\( \beta_{3} + \)\(19\!\cdots\!40\)\( \beta_{4} + \)\(20\!\cdots\!00\)\( \beta_{5} + \)\(54\!\cdots\!80\)\( \beta_{6}) q^{43}\) \(+(\)\(18\!\cdots\!44\)\( - \)\(49\!\cdots\!84\)\( \beta_{1} + \)\(21\!\cdots\!72\)\( \beta_{2} - \)\(35\!\cdots\!32\)\( \beta_{3} + \)\(14\!\cdots\!36\)\( \beta_{4} - \)\(12\!\cdots\!76\)\( \beta_{5} - \)\(21\!\cdots\!00\)\( \beta_{6}) q^{44}\) \(+(\)\(94\!\cdots\!82\)\( - \)\(47\!\cdots\!02\)\( \beta_{1} + \)\(62\!\cdots\!63\)\( \beta_{2} - \)\(24\!\cdots\!63\)\( \beta_{3} - \)\(52\!\cdots\!83\)\( \beta_{4} + \)\(18\!\cdots\!00\)\( \beta_{5} + \)\(37\!\cdots\!00\)\( \beta_{6}) q^{45}\) \(+(\)\(11\!\cdots\!18\)\( - \)\(43\!\cdots\!70\)\( \beta_{1} - \)\(15\!\cdots\!70\)\( \beta_{2} + \)\(80\!\cdots\!36\)\( \beta_{3} - \)\(20\!\cdots\!94\)\( \beta_{4} + \)\(73\!\cdots\!54\)\( \beta_{5} + \)\(16\!\cdots\!00\)\( \beta_{6}) q^{46}\) \(+(\)\(16\!\cdots\!18\)\( - \)\(28\!\cdots\!12\)\( \beta_{1} - \)\(34\!\cdots\!94\)\( \beta_{2} - \)\(47\!\cdots\!90\)\( \beta_{3} + \)\(83\!\cdots\!80\)\( \beta_{4} - \)\(45\!\cdots\!50\)\( \beta_{5} - \)\(30\!\cdots\!90\)\( \beta_{6}) q^{47}\) \(+(\)\(12\!\cdots\!08\)\( + \)\(34\!\cdots\!12\)\( \beta_{1} + \)\(90\!\cdots\!60\)\( \beta_{2} + \)\(10\!\cdots\!16\)\( \beta_{3} + \)\(57\!\cdots\!88\)\( \beta_{4} + \)\(10\!\cdots\!00\)\( \beta_{5} + \)\(89\!\cdots\!96\)\( \beta_{6}) q^{48}\) \(+(\)\(20\!\cdots\!89\)\( + \)\(10\!\cdots\!00\)\( \beta_{1} + \)\(19\!\cdots\!20\)\( \beta_{2} - \)\(12\!\cdots\!44\)\( \beta_{3} - \)\(90\!\cdots\!64\)\( \beta_{4} - \)\(27\!\cdots\!76\)\( \beta_{5} - \)\(93\!\cdots\!00\)\( \beta_{6}) q^{49}\) \(+(\)\(80\!\cdots\!50\)\( + \)\(20\!\cdots\!75\)\( \beta_{1} + \)\(45\!\cdots\!00\)\( \beta_{2} + \)\(26\!\cdots\!00\)\( \beta_{3} + \)\(20\!\cdots\!00\)\( \beta_{4} - \)\(58\!\cdots\!00\)\( \beta_{5} - \)\(22\!\cdots\!00\)\( \beta_{6}) q^{50}\) \(+(\)\(19\!\cdots\!92\)\( - \)\(12\!\cdots\!22\)\( \beta_{1} - \)\(86\!\cdots\!04\)\( \beta_{2} - \)\(39\!\cdots\!30\)\( \beta_{3} - \)\(11\!\cdots\!56\)\( \beta_{4} + \)\(15\!\cdots\!46\)\( \beta_{5} + \)\(11\!\cdots\!50\)\( \beta_{6}) q^{51}\) \(+(\)\(37\!\cdots\!06\)\( - \)\(92\!\cdots\!76\)\( \beta_{1} - \)\(39\!\cdots\!06\)\( \beta_{2} + \)\(61\!\cdots\!10\)\( \beta_{3} - \)\(15\!\cdots\!20\)\( \beta_{4} - \)\(11\!\cdots\!00\)\( \beta_{5} - \)\(18\!\cdots\!40\)\( \beta_{6}) q^{52}\) \(+(\)\(48\!\cdots\!62\)\( - \)\(54\!\cdots\!66\)\( \beta_{1} + \)\(23\!\cdots\!31\)\( \beta_{2} - \)\(54\!\cdots\!27\)\( \beta_{3} - \)\(38\!\cdots\!11\)\( \beta_{4} - \)\(29\!\cdots\!00\)\( \beta_{5} - \)\(51\!\cdots\!12\)\( \beta_{6}) q^{53}\) \(+(\)\(25\!\cdots\!10\)\( - \)\(14\!\cdots\!78\)\( \beta_{1} + \)\(14\!\cdots\!54\)\( \beta_{2} + \)\(96\!\cdots\!80\)\( \beta_{3} + \)\(75\!\cdots\!06\)\( \beta_{4} + \)\(67\!\cdots\!54\)\( \beta_{5} + \)\(87\!\cdots\!00\)\( \beta_{6}) q^{54}\) \(+(\)\(37\!\cdots\!37\)\( + \)\(11\!\cdots\!18\)\( \beta_{1} + \)\(23\!\cdots\!33\)\( \beta_{2} + \)\(48\!\cdots\!17\)\( \beta_{3} + \)\(76\!\cdots\!22\)\( \beta_{4} + \)\(10\!\cdots\!75\)\( \beta_{5} - \)\(15\!\cdots\!25\)\( \beta_{6}) q^{55}\) \(+(\)\(60\!\cdots\!08\)\( + \)\(58\!\cdots\!68\)\( \beta_{1} - \)\(71\!\cdots\!64\)\( \beta_{2} + \)\(74\!\cdots\!68\)\( \beta_{3} - \)\(29\!\cdots\!48\)\( \beta_{4} - \)\(45\!\cdots\!32\)\( \beta_{5} - \)\(33\!\cdots\!00\)\( \beta_{6}) q^{56}\) \(+(\)\(24\!\cdots\!16\)\( + \)\(35\!\cdots\!52\)\( \beta_{1} - \)\(10\!\cdots\!94\)\( \beta_{2} - \)\(91\!\cdots\!70\)\( \beta_{3} + \)\(31\!\cdots\!90\)\( \beta_{4} - \)\(84\!\cdots\!00\)\( \beta_{5} + \)\(47\!\cdots\!80\)\( \beta_{6}) q^{57}\) \(+(\)\(25\!\cdots\!00\)\( - \)\(33\!\cdots\!78\)\( \beta_{1} + \)\(18\!\cdots\!36\)\( \beta_{2} + \)\(60\!\cdots\!56\)\( \beta_{3} + \)\(30\!\cdots\!08\)\( \beta_{4} + \)\(28\!\cdots\!00\)\( \beta_{5} - \)\(14\!\cdots\!64\)\( \beta_{6}) q^{58}\) \(+(-\)\(29\!\cdots\!43\)\( - \)\(27\!\cdots\!49\)\( \beta_{1} + \)\(13\!\cdots\!97\)\( \beta_{2} + \)\(39\!\cdots\!12\)\( \beta_{3} - \)\(52\!\cdots\!20\)\( \beta_{4} - \)\(10\!\cdots\!80\)\( \beta_{5} - \)\(29\!\cdots\!00\)\( \beta_{6}) q^{59}\) \(+(-\)\(19\!\cdots\!64\)\( - \)\(12\!\cdots\!56\)\( \beta_{1} + \)\(23\!\cdots\!84\)\( \beta_{2} - \)\(66\!\cdots\!64\)\( \beta_{3} - \)\(62\!\cdots\!84\)\( \beta_{4} - \)\(14\!\cdots\!20\)\( \beta_{5} + \)\(38\!\cdots\!20\)\( \beta_{6}) q^{60}\) \(+(-\)\(24\!\cdots\!78\)\( + \)\(17\!\cdots\!50\)\( \beta_{1} - \)\(33\!\cdots\!25\)\( \beta_{2} + \)\(38\!\cdots\!25\)\( \beta_{3} - \)\(41\!\cdots\!75\)\( \beta_{4} + \)\(35\!\cdots\!00\)\( \beta_{5} + \)\(15\!\cdots\!00\)\( \beta_{6}) q^{61}\) \(+(-\)\(80\!\cdots\!44\)\( + \)\(17\!\cdots\!44\)\( \beta_{1} - \)\(11\!\cdots\!24\)\( \beta_{2} - \)\(27\!\cdots\!76\)\( \beta_{3} + \)\(24\!\cdots\!32\)\( \beta_{4} - \)\(17\!\cdots\!00\)\( \beta_{5} - \)\(63\!\cdots\!56\)\( \beta_{6}) q^{62}\) \(+(-\)\(78\!\cdots\!31\)\( + \)\(11\!\cdots\!34\)\( \beta_{1} - \)\(19\!\cdots\!79\)\( \beta_{2} + \)\(72\!\cdots\!01\)\( \beta_{3} - \)\(31\!\cdots\!82\)\( \beta_{4} - \)\(65\!\cdots\!25\)\( \beta_{5} + \)\(59\!\cdots\!31\)\( \beta_{6}) q^{63}\) \(+(-\)\(14\!\cdots\!88\)\( + \)\(31\!\cdots\!16\)\( \beta_{1} + \)\(40\!\cdots\!92\)\( \beta_{2} - \)\(12\!\cdots\!36\)\( \beta_{3} - \)\(38\!\cdots\!88\)\( \beta_{4} + \)\(10\!\cdots\!08\)\( \beta_{5} + \)\(20\!\cdots\!00\)\( \beta_{6}) q^{64}\) \(+(-\)\(26\!\cdots\!88\)\( - \)\(14\!\cdots\!52\)\( \beta_{1} - \)\(12\!\cdots\!72\)\( \beta_{2} - \)\(37\!\cdots\!88\)\( \beta_{3} + \)\(12\!\cdots\!72\)\( \beta_{4} - \)\(39\!\cdots\!40\)\( \beta_{5} - \)\(76\!\cdots\!60\)\( \beta_{6}) q^{65}\) \(+(-\)\(13\!\cdots\!36\)\( - \)\(71\!\cdots\!36\)\( \beta_{1} + \)\(12\!\cdots\!48\)\( \beta_{2} - \)\(24\!\cdots\!60\)\( \beta_{3} - \)\(35\!\cdots\!48\)\( \beta_{4} + \)\(37\!\cdots\!68\)\( \beta_{5} + \)\(73\!\cdots\!00\)\( \beta_{6}) q^{66}\) \(+(\)\(16\!\cdots\!57\)\( - \)\(82\!\cdots\!47\)\( \beta_{1} - \)\(31\!\cdots\!59\)\( \beta_{2} - \)\(25\!\cdots\!94\)\( \beta_{3} + \)\(51\!\cdots\!08\)\( \beta_{4} - \)\(15\!\cdots\!50\)\( \beta_{5} + \)\(16\!\cdots\!86\)\( \beta_{6}) q^{67}\) \(+(\)\(34\!\cdots\!70\)\( + \)\(83\!\cdots\!80\)\( \beta_{1} - \)\(23\!\cdots\!34\)\( \beta_{2} + \)\(26\!\cdots\!02\)\( \beta_{3} - \)\(81\!\cdots\!64\)\( \beta_{4} + \)\(68\!\cdots\!00\)\( \beta_{5} - \)\(59\!\cdots\!88\)\( \beta_{6}) q^{68}\) \(+(\)\(72\!\cdots\!32\)\( + \)\(14\!\cdots\!92\)\( \beta_{1} - \)\(33\!\cdots\!96\)\( \beta_{2} - \)\(17\!\cdots\!52\)\( \beta_{3} + \)\(99\!\cdots\!24\)\( \beta_{4} + \)\(81\!\cdots\!16\)\( \beta_{5} + \)\(55\!\cdots\!00\)\( \beta_{6}) q^{69}\) \(+(\)\(94\!\cdots\!36\)\( - \)\(17\!\cdots\!96\)\( \beta_{1} + \)\(36\!\cdots\!24\)\( \beta_{2} - \)\(42\!\cdots\!24\)\( \beta_{3} + \)\(22\!\cdots\!16\)\( \beta_{4} - \)\(18\!\cdots\!00\)\( \beta_{5} + \)\(80\!\cdots\!00\)\( \beta_{6}) q^{70}\) \(+(\)\(16\!\cdots\!17\)\( + \)\(13\!\cdots\!50\)\( \beta_{1} - \)\(23\!\cdots\!75\)\( \beta_{2} - \)\(52\!\cdots\!75\)\( \beta_{3} - \)\(42\!\cdots\!50\)\( \beta_{4} - \)\(19\!\cdots\!25\)\( \beta_{5} - \)\(27\!\cdots\!25\)\( \beta_{6}) q^{71}\) \(+(\)\(50\!\cdots\!88\)\( + \)\(94\!\cdots\!12\)\( \beta_{1} + \)\(45\!\cdots\!96\)\( \beta_{2} + \)\(29\!\cdots\!88\)\( \beta_{3} - \)\(37\!\cdots\!16\)\( \beta_{4} + \)\(61\!\cdots\!00\)\( \beta_{5} + \)\(20\!\cdots\!28\)\( \beta_{6}) q^{72}\) \(+(-\)\(53\!\cdots\!42\)\( - \)\(18\!\cdots\!48\)\( \beta_{1} - \)\(13\!\cdots\!70\)\( \beta_{2} - \)\(13\!\cdots\!14\)\( \beta_{3} + \)\(24\!\cdots\!98\)\( \beta_{4} - \)\(70\!\cdots\!00\)\( \beta_{5} + \)\(18\!\cdots\!16\)\( \beta_{6}) q^{73}\) \(+(-\)\(89\!\cdots\!40\)\( - \)\(10\!\cdots\!50\)\( \beta_{1} + \)\(14\!\cdots\!80\)\( \beta_{2} - \)\(58\!\cdots\!36\)\( \beta_{3} + \)\(40\!\cdots\!84\)\( \beta_{4} - \)\(94\!\cdots\!44\)\( \beta_{5} - \)\(17\!\cdots\!00\)\( \beta_{6}) q^{74}\) \(+(-\)\(12\!\cdots\!75\)\( - \)\(52\!\cdots\!25\)\( \beta_{1} - \)\(48\!\cdots\!75\)\( \beta_{2} + \)\(19\!\cdots\!00\)\( \beta_{3} - \)\(58\!\cdots\!00\)\( \beta_{4} + \)\(24\!\cdots\!00\)\( \beta_{5} - \)\(47\!\cdots\!00\)\( \beta_{6}) q^{75}\) \(+(-\)\(36\!\cdots\!64\)\( + \)\(20\!\cdots\!76\)\( \beta_{1} + \)\(45\!\cdots\!52\)\( \beta_{2} - \)\(54\!\cdots\!44\)\( \beta_{3} - \)\(63\!\cdots\!56\)\( \beta_{4} + \)\(91\!\cdots\!96\)\( \beta_{5} - \)\(10\!\cdots\!00\)\( \beta_{6}) q^{76}\) \(+(-\)\(51\!\cdots\!20\)\( + \)\(28\!\cdots\!36\)\( \beta_{1} + \)\(35\!\cdots\!72\)\( \beta_{2} + \)\(28\!\cdots\!76\)\( \beta_{3} + \)\(14\!\cdots\!68\)\( \beta_{4} - \)\(81\!\cdots\!00\)\( \beta_{5} + \)\(77\!\cdots\!56\)\( \beta_{6}) q^{77}\) \(+(-\)\(92\!\cdots\!30\)\( + \)\(10\!\cdots\!62\)\( \beta_{1} - \)\(18\!\cdots\!86\)\( \beta_{2} - \)\(40\!\cdots\!88\)\( \beta_{3} - \)\(17\!\cdots\!34\)\( \beta_{4} + \)\(70\!\cdots\!50\)\( \beta_{5} - \)\(97\!\cdots\!28\)\( \beta_{6}) q^{78}\) \(+(-\)\(12\!\cdots\!98\)\( + \)\(62\!\cdots\!08\)\( \beta_{1} - \)\(78\!\cdots\!54\)\( \beta_{2} - \)\(42\!\cdots\!98\)\( \beta_{3} + \)\(40\!\cdots\!76\)\( \beta_{4} + \)\(79\!\cdots\!34\)\( \beta_{5} - \)\(14\!\cdots\!50\)\( \beta_{6}) q^{79}\) \(+(-\)\(25\!\cdots\!44\)\( - \)\(14\!\cdots\!16\)\( \beta_{1} - \)\(15\!\cdots\!96\)\( \beta_{2} - \)\(19\!\cdots\!04\)\( \beta_{3} - \)\(14\!\cdots\!64\)\( \beta_{4} - \)\(27\!\cdots\!00\)\( \beta_{5} + \)\(58\!\cdots\!00\)\( \beta_{6}) q^{80}\) \(+(\)\(18\!\cdots\!05\)\( - \)\(38\!\cdots\!48\)\( \beta_{1} + \)\(96\!\cdots\!94\)\( \beta_{2} + \)\(18\!\cdots\!34\)\( \beta_{3} - \)\(17\!\cdots\!30\)\( \beta_{4} + \)\(56\!\cdots\!80\)\( \beta_{5} - \)\(43\!\cdots\!00\)\( \beta_{6}) q^{81}\) \(+(\)\(59\!\cdots\!56\)\( - \)\(40\!\cdots\!26\)\( \beta_{1} + \)\(59\!\cdots\!56\)\( \beta_{2} + \)\(43\!\cdots\!44\)\( \beta_{3} + \)\(55\!\cdots\!92\)\( \beta_{4} - \)\(86\!\cdots\!00\)\( \beta_{5} - \)\(77\!\cdots\!36\)\( \beta_{6}) q^{82}\) \(+(\)\(11\!\cdots\!71\)\( - \)\(79\!\cdots\!19\)\( \beta_{1} + \)\(18\!\cdots\!27\)\( \beta_{2} - \)\(52\!\cdots\!00\)\( \beta_{3} - \)\(45\!\cdots\!00\)\( \beta_{4} - \)\(46\!\cdots\!00\)\( \beta_{5} + \)\(15\!\cdots\!00\)\( \beta_{6}) q^{83}\) \(+(\)\(42\!\cdots\!24\)\( + \)\(24\!\cdots\!88\)\( \beta_{1} - \)\(76\!\cdots\!24\)\( \beta_{2} + \)\(72\!\cdots\!08\)\( \beta_{3} - \)\(67\!\cdots\!48\)\( \beta_{4} + \)\(48\!\cdots\!68\)\( \beta_{5} + \)\(11\!\cdots\!00\)\( \beta_{6}) q^{84}\) \(+(\)\(20\!\cdots\!96\)\( + \)\(84\!\cdots\!84\)\( \beta_{1} + \)\(30\!\cdots\!74\)\( \beta_{2} - \)\(37\!\cdots\!54\)\( \beta_{3} + \)\(54\!\cdots\!26\)\( \beta_{4} - \)\(66\!\cdots\!20\)\( \beta_{5} - \)\(63\!\cdots\!80\)\( \beta_{6}) q^{85}\) \(+(\)\(78\!\cdots\!71\)\( + \)\(64\!\cdots\!83\)\( \beta_{1} - \)\(46\!\cdots\!89\)\( \beta_{2} + \)\(33\!\cdots\!94\)\( \beta_{3} + \)\(13\!\cdots\!53\)\( \beta_{4} - \)\(20\!\cdots\!73\)\( \beta_{5} - \)\(57\!\cdots\!00\)\( \beta_{6}) q^{86}\) \(+(\)\(81\!\cdots\!51\)\( - \)\(18\!\cdots\!70\)\( \beta_{1} + \)\(16\!\cdots\!95\)\( \beta_{2} + \)\(48\!\cdots\!39\)\( \beta_{3} - \)\(95\!\cdots\!98\)\( \beta_{4} + \)\(10\!\cdots\!25\)\( \beta_{5} + \)\(88\!\cdots\!09\)\( \beta_{6}) q^{87}\) \(+(-\)\(47\!\cdots\!68\)\( - \)\(94\!\cdots\!84\)\( \beta_{1} + \)\(24\!\cdots\!68\)\( \beta_{2} - \)\(19\!\cdots\!64\)\( \beta_{3} + \)\(73\!\cdots\!48\)\( \beta_{4} + \)\(53\!\cdots\!00\)\( \beta_{5} + \)\(17\!\cdots\!16\)\( \beta_{6}) q^{88}\) \(+(-\)\(30\!\cdots\!06\)\( - \)\(17\!\cdots\!76\)\( \beta_{1} - \)\(41\!\cdots\!02\)\( \beta_{2} + \)\(15\!\cdots\!74\)\( \beta_{3} - \)\(49\!\cdots\!14\)\( \beta_{4} + \)\(75\!\cdots\!24\)\( \beta_{5} - \)\(58\!\cdots\!00\)\( \beta_{6}) q^{89}\) \(+(-\)\(93\!\cdots\!88\)\( - \)\(28\!\cdots\!02\)\( \beta_{1} + \)\(49\!\cdots\!28\)\( \beta_{2} - \)\(47\!\cdots\!88\)\( \beta_{3} - \)\(11\!\cdots\!28\)\( \beta_{4} - \)\(94\!\cdots\!40\)\( \beta_{5} + \)\(26\!\cdots\!40\)\( \beta_{6}) q^{90}\) \(+(-\)\(42\!\cdots\!20\)\( - \)\(10\!\cdots\!52\)\( \beta_{1} - \)\(10\!\cdots\!04\)\( \beta_{2} + \)\(46\!\cdots\!08\)\( \beta_{3} + \)\(59\!\cdots\!32\)\( \beta_{4} + \)\(12\!\cdots\!88\)\( \beta_{5} + \)\(96\!\cdots\!00\)\( \beta_{6}) q^{91}\) \(+(-\)\(52\!\cdots\!24\)\( + \)\(13\!\cdots\!72\)\( \beta_{1} + \)\(45\!\cdots\!08\)\( \beta_{2} + \)\(65\!\cdots\!24\)\( \beta_{3} - \)\(45\!\cdots\!68\)\( \beta_{4} + \)\(18\!\cdots\!00\)\( \beta_{5} - \)\(12\!\cdots\!56\)\( \beta_{6}) q^{92}\) \(+(-\)\(24\!\cdots\!80\)\( + \)\(57\!\cdots\!84\)\( \beta_{1} - \)\(44\!\cdots\!60\)\( \beta_{2} - \)\(10\!\cdots\!52\)\( \beta_{3} - \)\(92\!\cdots\!36\)\( \beta_{4} - \)\(56\!\cdots\!00\)\( \beta_{5} - \)\(41\!\cdots\!12\)\( \beta_{6}) q^{93}\) \(+(-\)\(15\!\cdots\!64\)\( + \)\(13\!\cdots\!72\)\( \beta_{1} + \)\(96\!\cdots\!04\)\( \beta_{2} - \)\(14\!\cdots\!60\)\( \beta_{3} + \)\(18\!\cdots\!16\)\( \beta_{4} + \)\(27\!\cdots\!44\)\( \beta_{5} + \)\(34\!\cdots\!00\)\( \beta_{6}) q^{94}\) \(+(\)\(69\!\cdots\!35\)\( - \)\(21\!\cdots\!10\)\( \beta_{1} - \)\(60\!\cdots\!85\)\( \beta_{2} - \)\(11\!\cdots\!65\)\( \beta_{3} - \)\(15\!\cdots\!90\)\( \beta_{4} + \)\(45\!\cdots\!25\)\( \beta_{5} + \)\(46\!\cdots\!25\)\( \beta_{6}) q^{95}\) \(+(\)\(60\!\cdots\!08\)\( + \)\(13\!\cdots\!12\)\( \beta_{1} - \)\(43\!\cdots\!96\)\( \beta_{2} + \)\(17\!\cdots\!96\)\( \beta_{3} + \)\(91\!\cdots\!72\)\( \beta_{4} - \)\(32\!\cdots\!52\)\( \beta_{5} - \)\(23\!\cdots\!00\)\( \beta_{6}) q^{96}\) \(+(\)\(87\!\cdots\!54\)\( - \)\(12\!\cdots\!76\)\( \beta_{1} - \)\(20\!\cdots\!46\)\( \beta_{2} + \)\(16\!\cdots\!02\)\( \beta_{3} + \)\(30\!\cdots\!86\)\( \beta_{4} + \)\(11\!\cdots\!00\)\( \beta_{5} - \)\(17\!\cdots\!88\)\( \beta_{6}) q^{97}\) \(+(\)\(26\!\cdots\!78\)\( + \)\(46\!\cdots\!45\)\( \beta_{1} - \)\(63\!\cdots\!64\)\( \beta_{2} + \)\(10\!\cdots\!84\)\( \beta_{3} - \)\(87\!\cdots\!88\)\( \beta_{4} - \)\(21\!\cdots\!00\)\( \beta_{5} + \)\(28\!\cdots\!04\)\( \beta_{6}) q^{98}\) \(+(\)\(22\!\cdots\!61\)\( - \)\(30\!\cdots\!53\)\( \beta_{1} + \)\(51\!\cdots\!69\)\( \beta_{2} - \)\(16\!\cdots\!88\)\( \beta_{3} - \)\(19\!\cdots\!52\)\( \beta_{4} + \)\(34\!\cdots\!32\)\( \beta_{5} + \)\(41\!\cdots\!00\)\( \beta_{6}) q^{99}\) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(7q \) \(\mathstrut +\mathstrut 18197022042936q^{2} \) \(\mathstrut -\mathstrut 75417209598230375148q^{3} \) \(\mathstrut +\mathstrut \)\(37\!\cdots\!36\)\(q^{4} \) \(\mathstrut +\mathstrut \)\(33\!\cdots\!50\)\(q^{5} \) \(\mathstrut +\mathstrut \)\(16\!\cdots\!44\)\(q^{6} \) \(\mathstrut +\mathstrut \)\(45\!\cdots\!56\)\(q^{7} \) \(\mathstrut +\mathstrut \)\(27\!\cdots\!40\)\(q^{8} \) \(\mathstrut +\mathstrut \)\(67\!\cdots\!39\)\(q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(7q \) \(\mathstrut +\mathstrut 18197022042936q^{2} \) \(\mathstrut -\mathstrut 75417209598230375148q^{3} \) \(\mathstrut +\mathstrut \)\(37\!\cdots\!36\)\(q^{4} \) \(\mathstrut +\mathstrut \)\(33\!\cdots\!50\)\(q^{5} \) \(\mathstrut +\mathstrut \)\(16\!\cdots\!44\)\(q^{6} \) \(\mathstrut +\mathstrut \)\(45\!\cdots\!56\)\(q^{7} \) \(\mathstrut +\mathstrut \)\(27\!\cdots\!40\)\(q^{8} \) \(\mathstrut +\mathstrut \)\(67\!\cdots\!39\)\(q^{9} \) \(\mathstrut -\mathstrut \)\(36\!\cdots\!00\)\(q^{10} \) \(\mathstrut +\mathstrut \)\(14\!\cdots\!04\)\(q^{11} \) \(\mathstrut +\mathstrut \)\(20\!\cdots\!36\)\(q^{12} \) \(\mathstrut +\mathstrut \)\(30\!\cdots\!62\)\(q^{13} \) \(\mathstrut +\mathstrut \)\(17\!\cdots\!92\)\(q^{14} \) \(\mathstrut -\mathstrut \)\(36\!\cdots\!00\)\(q^{15} \) \(\mathstrut -\mathstrut \)\(15\!\cdots\!68\)\(q^{16} \) \(\mathstrut +\mathstrut \)\(30\!\cdots\!46\)\(q^{17} \) \(\mathstrut +\mathstrut \)\(18\!\cdots\!12\)\(q^{18} \) \(\mathstrut -\mathstrut \)\(23\!\cdots\!40\)\(q^{19} \) \(\mathstrut -\mathstrut \)\(50\!\cdots\!00\)\(q^{20} \) \(\mathstrut +\mathstrut \)\(90\!\cdots\!64\)\(q^{21} \) \(\mathstrut -\mathstrut \)\(21\!\cdots\!08\)\(q^{22} \) \(\mathstrut -\mathstrut \)\(19\!\cdots\!28\)\(q^{23} \) \(\mathstrut +\mathstrut \)\(18\!\cdots\!80\)\(q^{24} \) \(\mathstrut +\mathstrut \)\(97\!\cdots\!25\)\(q^{25} \) \(\mathstrut +\mathstrut \)\(47\!\cdots\!44\)\(q^{26} \) \(\mathstrut -\mathstrut \)\(32\!\cdots\!40\)\(q^{27} \) \(\mathstrut +\mathstrut \)\(23\!\cdots\!08\)\(q^{28} \) \(\mathstrut -\mathstrut \)\(29\!\cdots\!10\)\(q^{29} \) \(\mathstrut -\mathstrut \)\(36\!\cdots\!00\)\(q^{30} \) \(\mathstrut +\mathstrut \)\(64\!\cdots\!84\)\(q^{31} \) \(\mathstrut +\mathstrut \)\(20\!\cdots\!76\)\(q^{32} \) \(\mathstrut -\mathstrut \)\(23\!\cdots\!56\)\(q^{33} \) \(\mathstrut +\mathstrut \)\(88\!\cdots\!12\)\(q^{34} \) \(\mathstrut +\mathstrut \)\(29\!\cdots\!00\)\(q^{35} \) \(\mathstrut +\mathstrut \)\(98\!\cdots\!72\)\(q^{36} \) \(\mathstrut -\mathstrut \)\(25\!\cdots\!74\)\(q^{37} \) \(\mathstrut -\mathstrut \)\(58\!\cdots\!40\)\(q^{38} \) \(\mathstrut +\mathstrut \)\(21\!\cdots\!48\)\(q^{39} \) \(\mathstrut -\mathstrut \)\(40\!\cdots\!00\)\(q^{40} \) \(\mathstrut +\mathstrut \)\(20\!\cdots\!74\)\(q^{41} \) \(\mathstrut +\mathstrut \)\(89\!\cdots\!32\)\(q^{42} \) \(\mathstrut +\mathstrut \)\(30\!\cdots\!92\)\(q^{43} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!92\)\(q^{44} \) \(\mathstrut +\mathstrut \)\(65\!\cdots\!50\)\(q^{45} \) \(\mathstrut +\mathstrut \)\(82\!\cdots\!44\)\(q^{46} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!16\)\(q^{47} \) \(\mathstrut +\mathstrut \)\(87\!\cdots\!72\)\(q^{48} \) \(\mathstrut +\mathstrut \)\(14\!\cdots\!51\)\(q^{49} \) \(\mathstrut +\mathstrut \)\(56\!\cdots\!00\)\(q^{50} \) \(\mathstrut +\mathstrut \)\(13\!\cdots\!04\)\(q^{51} \) \(\mathstrut +\mathstrut \)\(26\!\cdots\!16\)\(q^{52} \) \(\mathstrut +\mathstrut \)\(34\!\cdots\!02\)\(q^{53} \) \(\mathstrut +\mathstrut \)\(17\!\cdots\!60\)\(q^{54} \) \(\mathstrut +\mathstrut \)\(26\!\cdots\!00\)\(q^{55} \) \(\mathstrut +\mathstrut \)\(42\!\cdots\!40\)\(q^{56} \) \(\mathstrut +\mathstrut \)\(17\!\cdots\!20\)\(q^{57} \) \(\mathstrut +\mathstrut \)\(18\!\cdots\!40\)\(q^{58} \) \(\mathstrut -\mathstrut \)\(20\!\cdots\!20\)\(q^{59} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!00\)\(q^{60} \) \(\mathstrut -\mathstrut \)\(17\!\cdots\!46\)\(q^{61} \) \(\mathstrut -\mathstrut \)\(56\!\cdots\!68\)\(q^{62} \) \(\mathstrut -\mathstrut \)\(55\!\cdots\!48\)\(q^{63} \) \(\mathstrut -\mathstrut \)\(99\!\cdots\!84\)\(q^{64} \) \(\mathstrut -\mathstrut \)\(18\!\cdots\!00\)\(q^{65} \) \(\mathstrut -\mathstrut \)\(96\!\cdots\!32\)\(q^{66} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!96\)\(q^{67} \) \(\mathstrut +\mathstrut \)\(24\!\cdots\!28\)\(q^{68} \) \(\mathstrut +\mathstrut \)\(51\!\cdots\!48\)\(q^{69} \) \(\mathstrut +\mathstrut \)\(65\!\cdots\!00\)\(q^{70} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!44\)\(q^{71} \) \(\mathstrut +\mathstrut \)\(35\!\cdots\!80\)\(q^{72} \) \(\mathstrut -\mathstrut \)\(37\!\cdots\!78\)\(q^{73} \) \(\mathstrut -\mathstrut \)\(62\!\cdots\!48\)\(q^{74} \) \(\mathstrut -\mathstrut \)\(84\!\cdots\!00\)\(q^{75} \) \(\mathstrut -\mathstrut \)\(25\!\cdots\!20\)\(q^{76} \) \(\mathstrut -\mathstrut \)\(36\!\cdots\!68\)\(q^{77} \) \(\mathstrut -\mathstrut \)\(64\!\cdots\!36\)\(q^{78} \) \(\mathstrut -\mathstrut \)\(85\!\cdots\!60\)\(q^{79} \) \(\mathstrut -\mathstrut \)\(17\!\cdots\!00\)\(q^{80} \) \(\mathstrut +\mathstrut \)\(13\!\cdots\!27\)\(q^{81} \) \(\mathstrut +\mathstrut \)\(41\!\cdots\!52\)\(q^{82} \) \(\mathstrut +\mathstrut \)\(79\!\cdots\!32\)\(q^{83} \) \(\mathstrut +\mathstrut \)\(29\!\cdots\!72\)\(q^{84} \) \(\mathstrut +\mathstrut \)\(14\!\cdots\!00\)\(q^{85} \) \(\mathstrut +\mathstrut \)\(54\!\cdots\!44\)\(q^{86} \) \(\mathstrut +\mathstrut \)\(57\!\cdots\!80\)\(q^{87} \) \(\mathstrut -\mathstrut \)\(33\!\cdots\!20\)\(q^{88} \) \(\mathstrut -\mathstrut \)\(21\!\cdots\!30\)\(q^{89} \) \(\mathstrut -\mathstrut \)\(65\!\cdots\!00\)\(q^{90} \) \(\mathstrut -\mathstrut \)\(29\!\cdots\!36\)\(q^{91} \) \(\mathstrut -\mathstrut \)\(37\!\cdots\!04\)\(q^{92} \) \(\mathstrut -\mathstrut \)\(16\!\cdots\!76\)\(q^{93} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!28\)\(q^{94} \) \(\mathstrut +\mathstrut \)\(48\!\cdots\!00\)\(q^{95} \) \(\mathstrut +\mathstrut \)\(42\!\cdots\!04\)\(q^{96} \) \(\mathstrut +\mathstrut \)\(60\!\cdots\!66\)\(q^{97} \) \(\mathstrut +\mathstrut \)\(18\!\cdots\!48\)\(q^{98} \) \(\mathstrut +\mathstrut \)\(15\!\cdots\!08\)\(q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{7}\mathstrut -\mathstrut \) \(x^{6}\mathstrut -\mathstrut \) \(136216962554919874275824\) \(x^{5}\mathstrut +\mathstrut \) \(2428448265581416426698169507309556\) \(x^{4}\mathstrut +\mathstrut \) \(4947042984116303057415302671147699637405829712\) \(x^{3}\mathstrut -\mathstrut \) \(104175135964406876651844593123327151114904741702672521328\) \(x^{2}\mathstrut -\mathstrut \) \(39178147057757411803827754607509282883236218299345928694741829535360\) \(x\mathstrut -\mathstrut \) \(799990081389404679479129456354816396939504825273753348222638778162517453608256\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 72 \nu - 10 \)
\(\beta_{2}\)\(=\)\((\)\(99\!\cdots\!67\) \(\nu^{6}\mathstrut -\mathstrut \) \(16\!\cdots\!65\) \(\nu^{5}\mathstrut -\mathstrut \) \(15\!\cdots\!14\) \(\nu^{4}\mathstrut +\mathstrut \) \(36\!\cdots\!36\) \(\nu^{3}\mathstrut +\mathstrut \) \(63\!\cdots\!08\) \(\nu^{2}\mathstrut -\mathstrut \) \(18\!\cdots\!48\) \(\nu\mathstrut -\mathstrut \) \(35\!\cdots\!04\)\()/\)\(35\!\cdots\!64\)
\(\beta_{3}\)\(=\)\((\)\(54\!\cdots\!43\) \(\nu^{6}\mathstrut -\mathstrut \) \(89\!\cdots\!85\) \(\nu^{5}\mathstrut -\mathstrut \) \(86\!\cdots\!06\) \(\nu^{4}\mathstrut +\mathstrut \) \(20\!\cdots\!44\) \(\nu^{3}\mathstrut +\mathstrut \) \(53\!\cdots\!08\) \(\nu^{2}\mathstrut -\mathstrut \) \(49\!\cdots\!04\) \(\nu\mathstrut -\mathstrut \) \(91\!\cdots\!28\)\()/\)\(35\!\cdots\!64\)
\(\beta_{4}\)\(=\)\((\)\(-\)\(82\!\cdots\!71\) \(\nu^{6}\mathstrut -\mathstrut \) \(62\!\cdots\!91\) \(\nu^{5}\mathstrut +\mathstrut \) \(96\!\cdots\!02\) \(\nu^{4}\mathstrut +\mathstrut \) \(40\!\cdots\!68\) \(\nu^{3}\mathstrut -\mathstrut \) \(26\!\cdots\!56\) \(\nu^{2}\mathstrut -\mathstrut \) \(17\!\cdots\!44\) \(\nu\mathstrut +\mathstrut \) \(61\!\cdots\!92\)\()/\)\(22\!\cdots\!00\)
\(\beta_{5}\)\(=\)\((\)\(-\)\(15\!\cdots\!49\) \(\nu^{6}\mathstrut +\mathstrut \) \(28\!\cdots\!71\) \(\nu^{5}\mathstrut +\mathstrut \) \(20\!\cdots\!38\) \(\nu^{4}\mathstrut -\mathstrut \) \(19\!\cdots\!08\) \(\nu^{3}\mathstrut -\mathstrut \) \(55\!\cdots\!64\) \(\nu^{2}\mathstrut -\mathstrut \) \(10\!\cdots\!36\) \(\nu\mathstrut +\mathstrut \) \(15\!\cdots\!48\)\()/\)\(89\!\cdots\!00\)
\(\beta_{6}\)\(=\)\((\)\(-\)\(58\!\cdots\!89\) \(\nu^{6}\mathstrut +\mathstrut \) \(10\!\cdots\!31\) \(\nu^{5}\mathstrut +\mathstrut \) \(83\!\cdots\!18\) \(\nu^{4}\mathstrut -\mathstrut \) \(14\!\cdots\!88\) \(\nu^{3}\mathstrut -\mathstrut \) \(30\!\cdots\!04\) \(\nu^{2}\mathstrut +\mathstrut \) \(40\!\cdots\!04\) \(\nu\mathstrut +\mathstrut \) \(20\!\cdots\!28\)\()/\)\(12\!\cdots\!00\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1}\mathstrut +\mathstrut \) \(10\)\()/72\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3}\mathstrut -\mathstrut \) \(55029\) \(\beta_{2}\mathstrut -\mathstrut \) \(1925402030686\) \(\beta_{1}\mathstrut +\mathstrut \) \(201756781109916158185107823\)\()/5184\)
\(\nu^{3}\)\(=\)\((\)\(-\)\(9\) \(\beta_{6}\mathstrut +\mathstrut \) \(21600\) \(\beta_{5}\mathstrut -\mathstrut \) \(2115827\) \(\beta_{4}\mathstrut -\mathstrut \) \(569859075196\) \(\beta_{3}\mathstrut -\mathstrut \) \(11604479700889029\) \(\beta_{2}\mathstrut +\mathstrut \) \(41201269223519908505415611\) \(\beta_{1}\mathstrut -\mathstrut \) \(48557863831130868733665858581841875761\)\()/46656\)
\(\nu^{4}\)\(=\)\((\)\(-\)\(5495209853591\) \(\beta_{6}\mathstrut -\mathstrut \) \(10906069647310176\) \(\beta_{5}\mathstrut -\mathstrut \) \(22081402975112166637\) \(\beta_{4}\mathstrut +\mathstrut \) \(6566939912992225816485134\) \(\beta_{3}\mathstrut -\mathstrut \) \(811789349457676121999804129429\) \(\beta_{2}\mathstrut -\mathstrut \) \(24256377110901262298962187328286809559\) \(\beta_{1}\mathstrut +\mathstrut \) \(1039079432164905321644074588506346907349865760132767\)\()/419904\)
\(\nu^{5}\)\(=\)\((\)\(-\)\(1693618901523055879205617\) \(\beta_{6}\mathstrut +\mathstrut \) \(7661477248220168779180580960\) \(\beta_{5}\mathstrut -\mathstrut \) \(1418419879072061912128810799211\) \(\beta_{4}\mathstrut -\mathstrut \) \(200150112624151988311672356174744342\) \(\beta_{3}\mathstrut +\mathstrut \) \(11552023988019332938057323747015634783093\) \(\beta_{2}\mathstrut +\mathstrut \) \(9023503836567117528067551497955724660619494673055\) \(\beta_{1}\mathstrut -\mathstrut \) \(22656891146173623120281233176405996477782885834418410203331775\)\()/139968\)
\(\nu^{6}\)\(=\)\((\)\(-\)\(655886405368274699914169708381662519\) \(\beta_{6}\mathstrut -\mathstrut \) \(2065485815593754705581563406491719383392\) \(\beta_{5}\mathstrut -\mathstrut \) \(3496952326302575529108073514738588079850445\) \(\beta_{4}\mathstrut +\mathstrut \) \(531888815215264583737785243606481188932433049126\) \(\beta_{3}\mathstrut -\mathstrut \) \(84066922019005526188620285886820856360398640074100781\) \(\beta_{2}\mathstrut -\mathstrut \) \(2705670342599986856163460418538644062074220066307974993372103\) \(\beta_{1}\mathstrut +\mathstrut \) \(75856378636015798556289757586372838307977210483366335285110628244962249447\)\()/419904\)

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
−2.99978e11
−2.13963e11
−7.04801e10
−2.34744e10
1.42312e11
1.97356e11
2.68228e11
−1.89989e13 2.95637e20 2.06214e26 −5.49540e29 −5.61676e33 −2.67695e35 −9.77905e38 −2.35857e41 1.04406e43
1.2 −1.28058e13 −1.04195e21 9.24499e24 2.09154e30 1.33429e34 −6.15897e36 1.86321e39 7.62399e41 −2.67838e43
1.3 −2.47499e12 −1.66110e20 −1.48617e26 −1.54336e30 4.11120e32 1.02283e37 7.50812e38 −2.95665e41 3.81980e42
1.4 9.09417e11 7.58553e20 −1.53915e26 1.79518e30 6.89842e32 −6.36454e36 −2.80699e38 2.52145e41 1.63257e42
1.5 1.28460e13 −4.56067e20 1.02776e25 −3.44534e30 −5.85865e33 −1.04838e37 −1.85580e39 −1.15260e41 −4.42589e43
1.6 1.68092e13 −3.69059e20 1.27807e26 4.92178e30 −6.20358e33 5.74193e36 −4.52765e38 −1.87054e41 8.27312e43
1.7 2.19120e13 9.03577e20 3.25393e26 −2.93989e30 1.97992e34 7.30934e36 3.73929e39 4.93193e41 −6.44188e43
\(n\): 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 \(S_{88}^{\mathrm{new}}(\Gamma_0(1))\).