Properties

Label 1.104.a.a
Level 1
Weight 104
Character orbit 1.a
Self dual Yes
Analytic conductor 67.184
Analytic rank 0
Dimension 8
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 1 \)
Weight: \( k \) = \( 104 \)
Character orbit: \([\chi]\) = 1.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(67.1843880807\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: multiple of \( 2^{104}\cdot 3^{40}\cdot 5^{12}\cdot 7^{8}\cdot 11\cdot 13^{3}\cdot 17^{2} \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \(+(548616194585055 - \beta_{1}) q^{2}\) \(+(\)\(63\!\cdots\!80\)\( + 1292206 \beta_{1} - \beta_{2}) q^{3}\) \(+(\)\(60\!\cdots\!08\)\( - 1006563777941965 \beta_{1} + 34249 \beta_{2} + \beta_{3}) q^{4}\) \(+(\)\(68\!\cdots\!90\)\( - 13577557385999983067 \beta_{1} + 23394938356 \beta_{2} - 2101 \beta_{3} + \beta_{4}) q^{5}\) \(+(\)\(32\!\cdots\!92\)\( - \)\(59\!\cdots\!29\)\( \beta_{1} - 2269182927810110 \beta_{2} + 289750017 \beta_{3} - 3331 \beta_{4} - \beta_{5}) q^{6}\) \(+(\)\(52\!\cdots\!00\)\( - \)\(14\!\cdots\!08\)\( \beta_{1} + 3436742111413277589 \beta_{2} + 49285015561 \beta_{3} - 4418774 \beta_{4} + 505 \beta_{5} + \beta_{6}) q^{7}\) \(+(\)\(13\!\cdots\!60\)\( - \)\(62\!\cdots\!67\)\( \beta_{1} - \)\(24\!\cdots\!96\)\( \beta_{2} + 913937500520892 \beta_{3} - 783459941 \beta_{4} + 75477 \beta_{5} - 134 \beta_{6} - \beta_{7}) q^{8}\) \(+(\)\(44\!\cdots\!77\)\( - \)\(19\!\cdots\!70\)\( \beta_{1} - \)\(37\!\cdots\!24\)\( \beta_{2} + 272254459805646246 \beta_{3} - 720712994562 \beta_{4} + 59282388 \beta_{5} + 63252 \beta_{6} - 288 \beta_{7}) q^{9}\) \(+O(q^{10})\) \( q\) \(+(548616194585055 - \beta_{1}) q^{2}\) \(+(\)\(63\!\cdots\!80\)\( + 1292206 \beta_{1} - \beta_{2}) q^{3}\) \(+(\)\(60\!\cdots\!08\)\( - 1006563777941965 \beta_{1} + 34249 \beta_{2} + \beta_{3}) q^{4}\) \(+(\)\(68\!\cdots\!90\)\( - 13577557385999983067 \beta_{1} + 23394938356 \beta_{2} - 2101 \beta_{3} + \beta_{4}) q^{5}\) \(+(\)\(32\!\cdots\!92\)\( - \)\(59\!\cdots\!29\)\( \beta_{1} - 2269182927810110 \beta_{2} + 289750017 \beta_{3} - 3331 \beta_{4} - \beta_{5}) q^{6}\) \(+(\)\(52\!\cdots\!00\)\( - \)\(14\!\cdots\!08\)\( \beta_{1} + 3436742111413277589 \beta_{2} + 49285015561 \beta_{3} - 4418774 \beta_{4} + 505 \beta_{5} + \beta_{6}) q^{7}\) \(+(\)\(13\!\cdots\!60\)\( - \)\(62\!\cdots\!67\)\( \beta_{1} - \)\(24\!\cdots\!96\)\( \beta_{2} + 913937500520892 \beta_{3} - 783459941 \beta_{4} + 75477 \beta_{5} - 134 \beta_{6} - \beta_{7}) q^{8}\) \(+(\)\(44\!\cdots\!77\)\( - \)\(19\!\cdots\!70\)\( \beta_{1} - \)\(37\!\cdots\!24\)\( \beta_{2} + 272254459805646246 \beta_{3} - 720712994562 \beta_{4} + 59282388 \beta_{5} + 63252 \beta_{6} - 288 \beta_{7}) q^{9}\) \(+(\)\(25\!\cdots\!90\)\( - \)\(55\!\cdots\!26\)\( \beta_{1} + \)\(22\!\cdots\!04\)\( \beta_{2} + 1659923428331358884 \beta_{3} - 433358967391916 \beta_{4} + 8111585308 \beta_{5} + 80046976 \beta_{6} + 12096 \beta_{7}) q^{10}\) \(+(-\)\(66\!\cdots\!68\)\( + \)\(85\!\cdots\!26\)\( \beta_{1} - \)\(11\!\cdots\!69\)\( \beta_{2} + \)\(24\!\cdots\!14\)\( \beta_{3} - 75497546757957324 \beta_{4} - 15023926325694 \beta_{5} + 6487484498 \beta_{6} + 3253888 \beta_{7}) q^{11}\) \(+(\)\(32\!\cdots\!60\)\( - \)\(33\!\cdots\!60\)\( \beta_{1} - \)\(10\!\cdots\!92\)\( \beta_{2} + \)\(11\!\cdots\!72\)\( \beta_{3} - 2004032933334384336 \beta_{4} - 3220545977664048 \beta_{5} - 846949617504 \beta_{6} - 441349776 \beta_{7}) q^{12}\) \(+(-\)\(19\!\cdots\!90\)\( + \)\(10\!\cdots\!29\)\( \beta_{1} + \)\(48\!\cdots\!48\)\( \beta_{2} - \)\(10\!\cdots\!93\)\( \beta_{3} + \)\(70\!\cdots\!85\)\( \beta_{4} - 44796893837044072 \beta_{5} + 26310306328088 \beta_{6} + 29237253696 \beta_{7}) q^{13}\) \(+(\)\(25\!\cdots\!76\)\( - \)\(64\!\cdots\!34\)\( \beta_{1} + \)\(13\!\cdots\!60\)\( \beta_{2} - \)\(44\!\cdots\!62\)\( \beta_{3} + \)\(10\!\cdots\!58\)\( \beta_{4} + 6383741660729125638 \beta_{5} + 177715194016256 \beta_{6} - 1306326904064 \beta_{7}) q^{14}\) \(+(-\)\(37\!\cdots\!20\)\( + \)\(23\!\cdots\!28\)\( \beta_{1} - \)\(56\!\cdots\!57\)\( \beta_{2} + \)\(34\!\cdots\!83\)\( \beta_{3} + \)\(29\!\cdots\!78\)\( \beta_{4} - 59490093613531007009 \beta_{5} - 38695683278765673 \beta_{6} + 43918762920192 \beta_{7}) q^{15}\) \(+(\)\(45\!\cdots\!36\)\( - \)\(15\!\cdots\!16\)\( \beta_{1} - \)\(70\!\cdots\!00\)\( \beta_{2} + \)\(57\!\cdots\!36\)\( \beta_{3} - \)\(36\!\cdots\!52\)\( \beta_{4} - \)\(32\!\cdots\!32\)\( \beta_{5} + 1452842140237683248 \beta_{6} - 1174576591128312 \beta_{7}) q^{16}\) \(+(\)\(25\!\cdots\!90\)\( + \)\(14\!\cdots\!18\)\( \beta_{1} + \)\(15\!\cdots\!72\)\( \beta_{2} + \)\(10\!\cdots\!54\)\( \beta_{3} - \)\(10\!\cdots\!86\)\( \beta_{4} + \)\(88\!\cdots\!20\)\( \beta_{5} - 31417385238378658236 \beta_{6} + 25834565725096800 \beta_{7}) q^{17}\) \(+(\)\(33\!\cdots\!55\)\( - \)\(80\!\cdots\!69\)\( \beta_{1} - \)\(78\!\cdots\!96\)\( \beta_{2} + \)\(45\!\cdots\!56\)\( \beta_{3} + \)\(20\!\cdots\!16\)\( \beta_{4} - \)\(33\!\cdots\!00\)\( \beta_{5} + \)\(42\!\cdots\!36\)\( \beta_{6} - 477433803334239360 \beta_{7}) q^{18}\) \(+(\)\(18\!\cdots\!00\)\( - \)\(91\!\cdots\!02\)\( \beta_{1} - \)\(32\!\cdots\!51\)\( \beta_{2} + \)\(41\!\cdots\!66\)\( \beta_{3} + \)\(38\!\cdots\!28\)\( \beta_{4} - \)\(20\!\cdots\!82\)\( \beta_{5} - \)\(26\!\cdots\!86\)\( \beta_{6} + 7517169711519398784 \beta_{7}) q^{19}\) \(+(\)\(31\!\cdots\!20\)\( - \)\(16\!\cdots\!06\)\( \beta_{1} + \)\(32\!\cdots\!38\)\( \beta_{2} - \)\(16\!\cdots\!58\)\( \beta_{3} - \)\(14\!\cdots\!52\)\( \beta_{4} + \)\(44\!\cdots\!40\)\( \beta_{5} - \)\(29\!\cdots\!20\)\( \beta_{6} - \)\(10\!\cdots\!20\)\( \beta_{7}) q^{20}\) \(+(-\)\(58\!\cdots\!88\)\( + \)\(12\!\cdots\!92\)\( \beta_{1} - \)\(13\!\cdots\!52\)\( \beta_{2} - \)\(81\!\cdots\!92\)\( \beta_{3} - \)\(68\!\cdots\!04\)\( \beta_{4} - \)\(35\!\cdots\!44\)\( \beta_{5} + \)\(11\!\cdots\!92\)\( \beta_{6} + \)\(11\!\cdots\!52\)\( \beta_{7}) q^{21}\) \(+(-\)\(17\!\cdots\!40\)\( - \)\(16\!\cdots\!71\)\( \beta_{1} - \)\(25\!\cdots\!22\)\( \beta_{2} - \)\(33\!\cdots\!13\)\( \beta_{3} + \)\(71\!\cdots\!15\)\( \beta_{4} - \)\(80\!\cdots\!47\)\( \beta_{5} - \)\(18\!\cdots\!32\)\( \beta_{6} - \)\(11\!\cdots\!04\)\( \beta_{7}) q^{22}\) \(+(\)\(50\!\cdots\!40\)\( - \)\(40\!\cdots\!60\)\( \beta_{1} - \)\(83\!\cdots\!29\)\( \beta_{2} + \)\(42\!\cdots\!35\)\( \beta_{3} + \)\(21\!\cdots\!50\)\( \beta_{4} + \)\(50\!\cdots\!15\)\( \beta_{5} + \)\(19\!\cdots\!15\)\( \beta_{6} + \)\(10\!\cdots\!80\)\( \beta_{7}) q^{23}\) \(+(\)\(52\!\cdots\!00\)\( - \)\(10\!\cdots\!48\)\( \beta_{1} - \)\(47\!\cdots\!12\)\( \beta_{2} + \)\(78\!\cdots\!44\)\( \beta_{3} - \)\(73\!\cdots\!60\)\( \beta_{4} - \)\(54\!\cdots\!00\)\( \beta_{5} - \)\(15\!\cdots\!72\)\( \beta_{6} - \)\(69\!\cdots\!32\)\( \beta_{7}) q^{24}\) \(+(-\)\(45\!\cdots\!25\)\( + \)\(49\!\cdots\!20\)\( \beta_{1} - \)\(11\!\cdots\!00\)\( \beta_{2} - \)\(37\!\cdots\!20\)\( \beta_{3} + \)\(26\!\cdots\!00\)\( \beta_{4} + \)\(25\!\cdots\!80\)\( \beta_{5} + \)\(79\!\cdots\!60\)\( \beta_{6} + \)\(36\!\cdots\!60\)\( \beta_{7}) q^{25}\) \(+(-\)\(17\!\cdots\!98\)\( + \)\(34\!\cdots\!74\)\( \beta_{1} - \)\(36\!\cdots\!92\)\( \beta_{2} - \)\(40\!\cdots\!84\)\( \beta_{3} + \)\(29\!\cdots\!60\)\( \beta_{4} + \)\(73\!\cdots\!80\)\( \beta_{5} - \)\(72\!\cdots\!64\)\( \beta_{6} - \)\(81\!\cdots\!84\)\( \beta_{7}) q^{26}\) \(+(\)\(78\!\cdots\!40\)\( - \)\(19\!\cdots\!68\)\( \beta_{1} - \)\(12\!\cdots\!84\)\( \beta_{2} + \)\(14\!\cdots\!98\)\( \beta_{3} - \)\(28\!\cdots\!84\)\( \beta_{4} - \)\(19\!\cdots\!42\)\( \beta_{5} - \)\(33\!\cdots\!06\)\( \beta_{6} - \)\(87\!\cdots\!04\)\( \beta_{7}) q^{27}\) \(+(\)\(64\!\cdots\!60\)\( - \)\(10\!\cdots\!44\)\( \beta_{1} + \)\(10\!\cdots\!32\)\( \beta_{2} + \)\(13\!\cdots\!88\)\( \beta_{3} + \)\(82\!\cdots\!76\)\( \beta_{4} + \)\(13\!\cdots\!28\)\( \beta_{5} + \)\(39\!\cdots\!24\)\( \beta_{6} + \)\(14\!\cdots\!36\)\( \beta_{7}) q^{28}\) \(+(-\)\(13\!\cdots\!50\)\( - \)\(89\!\cdots\!71\)\( \beta_{1} + \)\(13\!\cdots\!24\)\( \beta_{2} - \)\(39\!\cdots\!25\)\( \beta_{3} + \)\(10\!\cdots\!25\)\( \beta_{4} - \)\(29\!\cdots\!80\)\( \beta_{5} - \)\(27\!\cdots\!44\)\( \beta_{6} - \)\(13\!\cdots\!64\)\( \beta_{7}) q^{29}\) \(+(-\)\(39\!\cdots\!20\)\( + \)\(45\!\cdots\!06\)\( \beta_{1} - \)\(46\!\cdots\!88\)\( \beta_{2} - \)\(59\!\cdots\!42\)\( \beta_{3} - \)\(44\!\cdots\!98\)\( \beta_{4} - \)\(22\!\cdots\!90\)\( \beta_{5} + \)\(11\!\cdots\!20\)\( \beta_{6} + \)\(87\!\cdots\!20\)\( \beta_{7}) q^{30}\) \(+(\)\(13\!\cdots\!92\)\( - \)\(51\!\cdots\!24\)\( \beta_{1} - \)\(66\!\cdots\!08\)\( \beta_{2} - \)\(29\!\cdots\!48\)\( \beta_{3} - \)\(99\!\cdots\!48\)\( \beta_{4} + \)\(22\!\cdots\!32\)\( \beta_{5} - \)\(18\!\cdots\!28\)\( \beta_{6} - \)\(42\!\cdots\!68\)\( \beta_{7}) q^{31}\) \(+(\)\(12\!\cdots\!80\)\( - \)\(45\!\cdots\!16\)\( \beta_{1} - \)\(57\!\cdots\!00\)\( \beta_{2} + \)\(16\!\cdots\!04\)\( \beta_{3} + \)\(13\!\cdots\!84\)\( \beta_{4} - \)\(76\!\cdots\!60\)\( \beta_{5} - \)\(18\!\cdots\!96\)\( \beta_{6} + \)\(12\!\cdots\!40\)\( \beta_{7}) q^{32}\) \(+(\)\(20\!\cdots\!60\)\( - \)\(27\!\cdots\!42\)\( \beta_{1} + \)\(62\!\cdots\!56\)\( \beta_{2} + \)\(63\!\cdots\!14\)\( \beta_{3} - \)\(14\!\cdots\!58\)\( \beta_{4} - \)\(10\!\cdots\!92\)\( \beta_{5} + \)\(19\!\cdots\!40\)\( \beta_{6} + \)\(12\!\cdots\!36\)\( \beta_{7}) q^{33}\) \(+(-\)\(21\!\cdots\!54\)\( - \)\(11\!\cdots\!82\)\( \beta_{1} + \)\(11\!\cdots\!36\)\( \beta_{2} - \)\(38\!\cdots\!96\)\( \beta_{3} - \)\(19\!\cdots\!32\)\( \beta_{4} + \)\(20\!\cdots\!08\)\( \beta_{5} - \)\(10\!\cdots\!16\)\( \beta_{6} - \)\(50\!\cdots\!96\)\( \beta_{7}) q^{34}\) \(+(-\)\(19\!\cdots\!60\)\( - \)\(67\!\cdots\!96\)\( \beta_{1} + \)\(39\!\cdots\!64\)\( \beta_{2} - \)\(28\!\cdots\!76\)\( \beta_{3} + \)\(46\!\cdots\!44\)\( \beta_{4} - \)\(71\!\cdots\!92\)\( \beta_{5} + \)\(33\!\cdots\!76\)\( \beta_{6} + \)\(42\!\cdots\!96\)\( \beta_{7}) q^{35}\) \(+(\)\(10\!\cdots\!16\)\( - \)\(62\!\cdots\!49\)\( \beta_{1} - \)\(13\!\cdots\!19\)\( \beta_{2} + \)\(89\!\cdots\!01\)\( \beta_{3} + \)\(34\!\cdots\!84\)\( \beta_{4} - \)\(85\!\cdots\!36\)\( \beta_{5} - \)\(52\!\cdots\!80\)\( \beta_{6} - \)\(24\!\cdots\!80\)\( \beta_{7}) q^{36}\) \(+(-\)\(11\!\cdots\!30\)\( - \)\(74\!\cdots\!63\)\( \beta_{1} - \)\(28\!\cdots\!76\)\( \beta_{2} - \)\(10\!\cdots\!69\)\( \beta_{3} - \)\(15\!\cdots\!83\)\( \beta_{4} + \)\(16\!\cdots\!16\)\( \beta_{5} - \)\(61\!\cdots\!52\)\( \beta_{6} + \)\(10\!\cdots\!92\)\( \beta_{7}) q^{37}\) \(+(\)\(24\!\cdots\!40\)\( - \)\(59\!\cdots\!05\)\( \beta_{1} - \)\(39\!\cdots\!10\)\( \beta_{2} - \)\(52\!\cdots\!51\)\( \beta_{3} - \)\(39\!\cdots\!39\)\( \beta_{4} - \)\(58\!\cdots\!73\)\( \beta_{5} + \)\(10\!\cdots\!08\)\( \beta_{6} - \)\(33\!\cdots\!96\)\( \beta_{7}) q^{38}\) \(+(-\)\(98\!\cdots\!76\)\( - \)\(38\!\cdots\!04\)\( \beta_{1} + \)\(10\!\cdots\!83\)\( \beta_{2} - \)\(16\!\cdots\!53\)\( \beta_{3} + \)\(34\!\cdots\!06\)\( \beta_{4} + \)\(19\!\cdots\!11\)\( \beta_{5} + \)\(55\!\cdots\!03\)\( \beta_{6} + \)\(63\!\cdots\!68\)\( \beta_{7}) q^{39}\) \(+(\)\(24\!\cdots\!00\)\( + \)\(18\!\cdots\!70\)\( \beta_{1} + \)\(47\!\cdots\!60\)\( \beta_{2} + \)\(10\!\cdots\!00\)\( \beta_{3} + \)\(27\!\cdots\!10\)\( \beta_{4} + \)\(56\!\cdots\!10\)\( \beta_{5} - \)\(50\!\cdots\!80\)\( \beta_{6} + \)\(77\!\cdots\!70\)\( \beta_{7}) q^{40}\) \(+(\)\(27\!\cdots\!22\)\( + \)\(36\!\cdots\!84\)\( \beta_{1} + \)\(58\!\cdots\!76\)\( \beta_{2} + \)\(38\!\cdots\!52\)\( \beta_{3} - \)\(58\!\cdots\!64\)\( \beta_{4} - \)\(46\!\cdots\!44\)\( \beta_{5} + \)\(25\!\cdots\!80\)\( \beta_{6} - \)\(12\!\cdots\!20\)\( \beta_{7}) q^{41}\) \(+(-\)\(23\!\cdots\!60\)\( + \)\(14\!\cdots\!40\)\( \beta_{1} - \)\(70\!\cdots\!04\)\( \beta_{2} - \)\(19\!\cdots\!24\)\( \beta_{3} + \)\(13\!\cdots\!76\)\( \beta_{4} - \)\(12\!\cdots\!60\)\( \beta_{5} - \)\(79\!\cdots\!64\)\( \beta_{6} + \)\(61\!\cdots\!20\)\( \beta_{7}) q^{42}\) \(+(\)\(12\!\cdots\!00\)\( + \)\(18\!\cdots\!98\)\( \beta_{1} - \)\(57\!\cdots\!35\)\( \beta_{2} - \)\(30\!\cdots\!60\)\( \beta_{3} + \)\(40\!\cdots\!72\)\( \beta_{4} + \)\(71\!\cdots\!12\)\( \beta_{5} + \)\(13\!\cdots\!24\)\( \beta_{6} - \)\(18\!\cdots\!36\)\( \beta_{7}) q^{43}\) \(+(\)\(32\!\cdots\!56\)\( + \)\(33\!\cdots\!36\)\( \beta_{1} - \)\(36\!\cdots\!40\)\( \beta_{2} + \)\(18\!\cdots\!24\)\( \beta_{3} - \)\(22\!\cdots\!48\)\( \beta_{4} - \)\(17\!\cdots\!68\)\( \beta_{5} + \)\(62\!\cdots\!32\)\( \beta_{6} + \)\(24\!\cdots\!92\)\( \beta_{7}) q^{44}\) \(+(-\)\(18\!\cdots\!70\)\( - \)\(15\!\cdots\!99\)\( \beta_{1} + \)\(17\!\cdots\!92\)\( \beta_{2} + \)\(26\!\cdots\!23\)\( \beta_{3} + \)\(49\!\cdots\!57\)\( \beta_{4} + \)\(86\!\cdots\!80\)\( \beta_{5} - \)\(11\!\cdots\!40\)\( \beta_{6} + \)\(66\!\cdots\!60\)\( \beta_{7}) q^{45}\) \(+(\)\(34\!\cdots\!12\)\( - \)\(92\!\cdots\!66\)\( \beta_{1} + \)\(43\!\cdots\!56\)\( \beta_{2} - \)\(81\!\cdots\!78\)\( \beta_{3} + \)\(13\!\cdots\!86\)\( \beta_{4} + \)\(19\!\cdots\!06\)\( \beta_{5} + \)\(27\!\cdots\!80\)\( \beta_{6} - \)\(54\!\cdots\!20\)\( \beta_{7}) q^{46}\) \(+(-\)\(36\!\cdots\!40\)\( - \)\(69\!\cdots\!16\)\( \beta_{1} + \)\(59\!\cdots\!30\)\( \beta_{2} - \)\(68\!\cdots\!70\)\( \beta_{3} + \)\(95\!\cdots\!24\)\( \beta_{4} + \)\(31\!\cdots\!54\)\( \beta_{5} - \)\(10\!\cdots\!42\)\( \beta_{6} + \)\(18\!\cdots\!88\)\( \beta_{7}) q^{47}\) \(+(\)\(15\!\cdots\!40\)\( - \)\(97\!\cdots\!00\)\( \beta_{1} - \)\(11\!\cdots\!84\)\( \beta_{2} + \)\(19\!\cdots\!64\)\( \beta_{3} - \)\(24\!\cdots\!36\)\( \beta_{4} - \)\(23\!\cdots\!40\)\( \beta_{5} - \)\(55\!\cdots\!96\)\( \beta_{6} - \)\(30\!\cdots\!20\)\( \beta_{7}) q^{48}\) \(+(\)\(33\!\cdots\!93\)\( - \)\(57\!\cdots\!56\)\( \beta_{1} + \)\(74\!\cdots\!76\)\( \beta_{2} + \)\(22\!\cdots\!52\)\( \beta_{3} + \)\(54\!\cdots\!96\)\( \beta_{4} + \)\(68\!\cdots\!16\)\( \beta_{5} - \)\(39\!\cdots\!40\)\( \beta_{6} - \)\(27\!\cdots\!40\)\( \beta_{7}) q^{49}\) \(+(-\)\(81\!\cdots\!75\)\( + \)\(44\!\cdots\!45\)\( \beta_{1} + \)\(23\!\cdots\!00\)\( \beta_{2} - \)\(53\!\cdots\!20\)\( \beta_{3} + \)\(22\!\cdots\!00\)\( \beta_{4} - \)\(69\!\cdots\!20\)\( \beta_{5} + \)\(36\!\cdots\!60\)\( \beta_{6} + \)\(34\!\cdots\!60\)\( \beta_{7}) q^{50}\) \(+(-\)\(26\!\cdots\!48\)\( + \)\(12\!\cdots\!52\)\( \beta_{1} + \)\(29\!\cdots\!56\)\( \beta_{2} - \)\(23\!\cdots\!46\)\( \beta_{3} - \)\(76\!\cdots\!28\)\( \beta_{4} - \)\(99\!\cdots\!18\)\( \beta_{5} - \)\(11\!\cdots\!14\)\( \beta_{6} - \)\(11\!\cdots\!84\)\( \beta_{7}) q^{51}\) \(+(-\)\(45\!\cdots\!00\)\( + \)\(49\!\cdots\!90\)\( \beta_{1} + \)\(12\!\cdots\!18\)\( \beta_{2} - \)\(27\!\cdots\!70\)\( \beta_{3} - \)\(10\!\cdots\!56\)\( \beta_{4} + \)\(77\!\cdots\!24\)\( \beta_{5} + \)\(71\!\cdots\!48\)\( \beta_{6} + \)\(15\!\cdots\!28\)\( \beta_{7}) q^{52}\) \(+(\)\(20\!\cdots\!30\)\( + \)\(26\!\cdots\!25\)\( \beta_{1} + \)\(86\!\cdots\!76\)\( \beta_{2} + \)\(19\!\cdots\!47\)\( \beta_{3} + \)\(36\!\cdots\!49\)\( \beta_{4} + \)\(15\!\cdots\!12\)\( \beta_{5} + \)\(79\!\cdots\!16\)\( \beta_{6} + \)\(19\!\cdots\!44\)\( \beta_{7}) q^{53}\) \(+(\)\(30\!\cdots\!00\)\( - \)\(16\!\cdots\!94\)\( \beta_{1} - \)\(38\!\cdots\!12\)\( \beta_{2} + \)\(32\!\cdots\!86\)\( \beta_{3} - \)\(15\!\cdots\!38\)\( \beta_{4} - \)\(35\!\cdots\!58\)\( \beta_{5} - \)\(34\!\cdots\!28\)\( \beta_{6} - \)\(16\!\cdots\!68\)\( \beta_{7}) q^{54}\) \(+(-\)\(75\!\cdots\!20\)\( - \)\(38\!\cdots\!44\)\( \beta_{1} - \)\(73\!\cdots\!83\)\( \beta_{2} - \)\(16\!\cdots\!07\)\( \beta_{3} - \)\(49\!\cdots\!18\)\( \beta_{4} - \)\(99\!\cdots\!75\)\( \beta_{5} + \)\(58\!\cdots\!25\)\( \beta_{6} + \)\(38\!\cdots\!00\)\( \beta_{7}) q^{55}\) \(+(-\)\(62\!\cdots\!00\)\( - \)\(13\!\cdots\!28\)\( \beta_{1} + \)\(24\!\cdots\!68\)\( \beta_{2} - \)\(32\!\cdots\!80\)\( \beta_{3} - \)\(39\!\cdots\!36\)\( \beta_{4} + \)\(64\!\cdots\!24\)\( \beta_{5} + \)\(29\!\cdots\!24\)\( \beta_{6} - \)\(28\!\cdots\!56\)\( \beta_{7}) q^{56}\) \(+(\)\(69\!\cdots\!80\)\( - \)\(43\!\cdots\!54\)\( \beta_{1} - \)\(83\!\cdots\!32\)\( \beta_{2} + \)\(11\!\cdots\!90\)\( \beta_{3} + \)\(19\!\cdots\!22\)\( \beta_{4} - \)\(11\!\cdots\!88\)\( \beta_{5} - \)\(38\!\cdots\!76\)\( \beta_{6} - \)\(11\!\cdots\!36\)\( \beta_{7}) q^{57}\) \(+(\)\(13\!\cdots\!10\)\( + \)\(13\!\cdots\!06\)\( \beta_{1} + \)\(75\!\cdots\!68\)\( \beta_{2} + \)\(18\!\cdots\!24\)\( \beta_{3} - \)\(27\!\cdots\!60\)\( \beta_{4} + \)\(12\!\cdots\!16\)\( \beta_{5} + \)\(78\!\cdots\!56\)\( \beta_{6} + \)\(39\!\cdots\!12\)\( \beta_{7}) q^{58}\) \(+(-\)\(47\!\cdots\!00\)\( + \)\(23\!\cdots\!98\)\( \beta_{1} - \)\(15\!\cdots\!03\)\( \beta_{2} - \)\(40\!\cdots\!08\)\( \beta_{3} + \)\(18\!\cdots\!24\)\( \beta_{4} + \)\(19\!\cdots\!64\)\( \beta_{5} - \)\(31\!\cdots\!52\)\( \beta_{6} - \)\(46\!\cdots\!12\)\( \beta_{7}) q^{59}\) \(+(-\)\(55\!\cdots\!60\)\( + \)\(76\!\cdots\!64\)\( \beta_{1} - \)\(28\!\cdots\!36\)\( \beta_{2} - \)\(11\!\cdots\!36\)\( \beta_{3} - \)\(15\!\cdots\!56\)\( \beta_{4} + \)\(28\!\cdots\!48\)\( \beta_{5} - \)\(11\!\cdots\!44\)\( \beta_{6} - \)\(34\!\cdots\!24\)\( \beta_{7}) q^{60}\) \(+(\)\(70\!\cdots\!82\)\( + \)\(88\!\cdots\!21\)\( \beta_{1} + \)\(32\!\cdots\!28\)\( \beta_{2} - \)\(10\!\cdots\!65\)\( \beta_{3} + \)\(55\!\cdots\!53\)\( \beta_{4} - \)\(10\!\cdots\!92\)\( \beta_{5} - \)\(10\!\cdots\!24\)\( \beta_{6} + \)\(11\!\cdots\!56\)\( \beta_{7}) q^{61}\) \(+(\)\(88\!\cdots\!60\)\( - \)\(14\!\cdots\!52\)\( \beta_{1} + \)\(14\!\cdots\!80\)\( \beta_{2} + \)\(11\!\cdots\!36\)\( \beta_{3} - \)\(50\!\cdots\!64\)\( \beta_{4} - \)\(48\!\cdots\!60\)\( \beta_{5} + \)\(10\!\cdots\!96\)\( \beta_{6} + \)\(25\!\cdots\!20\)\( \beta_{7}) q^{62}\) \(+(\)\(12\!\cdots\!20\)\( - \)\(43\!\cdots\!80\)\( \beta_{1} + \)\(78\!\cdots\!01\)\( \beta_{2} - \)\(74\!\cdots\!71\)\( \beta_{3} - \)\(56\!\cdots\!02\)\( \beta_{4} + \)\(25\!\cdots\!89\)\( \beta_{5} + \)\(32\!\cdots\!97\)\( \beta_{6} - \)\(10\!\cdots\!32\)\( \beta_{7}) q^{63}\) \(+(\)\(33\!\cdots\!88\)\( - \)\(15\!\cdots\!52\)\( \beta_{1} - \)\(18\!\cdots\!80\)\( \beta_{2} - \)\(83\!\cdots\!44\)\( \beta_{3} - \)\(20\!\cdots\!88\)\( \beta_{4} - \)\(36\!\cdots\!48\)\( \beta_{5} - \)\(13\!\cdots\!40\)\( \beta_{6} - \)\(15\!\cdots\!40\)\( \beta_{7}) q^{64}\) \(+(\)\(64\!\cdots\!80\)\( - \)\(13\!\cdots\!72\)\( \beta_{1} - \)\(18\!\cdots\!12\)\( \beta_{2} - \)\(10\!\cdots\!52\)\( \beta_{3} + \)\(10\!\cdots\!48\)\( \beta_{4} - \)\(19\!\cdots\!24\)\( \beta_{5} + \)\(36\!\cdots\!72\)\( \beta_{6} + \)\(15\!\cdots\!12\)\( \beta_{7}) q^{65}\) \(+(\)\(45\!\cdots\!44\)\( - \)\(39\!\cdots\!44\)\( \beta_{1} - \)\(30\!\cdots\!12\)\( \beta_{2} + \)\(23\!\cdots\!52\)\( \beta_{3} + \)\(21\!\cdots\!56\)\( \beta_{4} + \)\(10\!\cdots\!36\)\( \beta_{5} - \)\(15\!\cdots\!32\)\( \beta_{6} - \)\(32\!\cdots\!92\)\( \beta_{7}) q^{66}\) \(+(\)\(45\!\cdots\!40\)\( + \)\(13\!\cdots\!62\)\( \beta_{1} + \)\(20\!\cdots\!29\)\( \beta_{2} + \)\(68\!\cdots\!34\)\( \beta_{3} - \)\(67\!\cdots\!24\)\( \beta_{4} - \)\(29\!\cdots\!18\)\( \beta_{5} - \)\(14\!\cdots\!22\)\( \beta_{6} - \)\(13\!\cdots\!36\)\( \beta_{7}) q^{67}\) \(+(\)\(15\!\cdots\!80\)\( + \)\(38\!\cdots\!94\)\( \beta_{1} + \)\(45\!\cdots\!46\)\( \beta_{2} + \)\(43\!\cdots\!98\)\( \beta_{3} + \)\(48\!\cdots\!76\)\( \beta_{4} + \)\(18\!\cdots\!68\)\( \beta_{5} + \)\(38\!\cdots\!64\)\( \beta_{6} + \)\(23\!\cdots\!16\)\( \beta_{7}) q^{68}\) \(+(\)\(18\!\cdots\!44\)\( + \)\(54\!\cdots\!24\)\( \beta_{1} - \)\(47\!\cdots\!56\)\( \beta_{2} - \)\(27\!\cdots\!68\)\( \beta_{3} - \)\(40\!\cdots\!12\)\( \beta_{4} - \)\(19\!\cdots\!92\)\( \beta_{5} - \)\(23\!\cdots\!72\)\( \beta_{6} - \)\(50\!\cdots\!32\)\( \beta_{7}) q^{69}\) \(+(\)\(10\!\cdots\!40\)\( + \)\(15\!\cdots\!48\)\( \beta_{1} + \)\(72\!\cdots\!76\)\( \beta_{2} + \)\(15\!\cdots\!24\)\( \beta_{3} + \)\(51\!\cdots\!96\)\( \beta_{4} + \)\(41\!\cdots\!20\)\( \beta_{5} - \)\(58\!\cdots\!60\)\( \beta_{6} + \)\(65\!\cdots\!40\)\( \beta_{7}) q^{70}\) \(+(\)\(16\!\cdots\!12\)\( - \)\(32\!\cdots\!92\)\( \beta_{1} - \)\(16\!\cdots\!31\)\( \beta_{2} - \)\(48\!\cdots\!95\)\( \beta_{3} - \)\(86\!\cdots\!06\)\( \beta_{4} + \)\(50\!\cdots\!09\)\( \beta_{5} + \)\(86\!\cdots\!73\)\( \beta_{6} + \)\(22\!\cdots\!88\)\( \beta_{7}) q^{71}\) \(+(\)\(70\!\cdots\!20\)\( - \)\(13\!\cdots\!27\)\( \beta_{1} - \)\(50\!\cdots\!16\)\( \beta_{2} + \)\(41\!\cdots\!12\)\( \beta_{3} - \)\(15\!\cdots\!73\)\( \beta_{4} - \)\(14\!\cdots\!55\)\( \beta_{5} + \)\(58\!\cdots\!62\)\( \beta_{6} - \)\(42\!\cdots\!05\)\( \beta_{7}) q^{72}\) \(+(\)\(55\!\cdots\!90\)\( - \)\(62\!\cdots\!26\)\( \beta_{1} + \)\(11\!\cdots\!88\)\( \beta_{2} - \)\(56\!\cdots\!86\)\( \beta_{3} + \)\(46\!\cdots\!86\)\( \beta_{4} + \)\(14\!\cdots\!12\)\( \beta_{5} + \)\(18\!\cdots\!68\)\( \beta_{6} - \)\(10\!\cdots\!76\)\( \beta_{7}) q^{73}\) \(+(\)\(11\!\cdots\!86\)\( + \)\(78\!\cdots\!30\)\( \beta_{1} + \)\(17\!\cdots\!36\)\( \beta_{2} - \)\(68\!\cdots\!72\)\( \beta_{3} + \)\(21\!\cdots\!16\)\( \beta_{4} - \)\(40\!\cdots\!84\)\( \beta_{5} - \)\(18\!\cdots\!96\)\( \beta_{6} + \)\(15\!\cdots\!24\)\( \beta_{7}) q^{74}\) \(+(\)\(19\!\cdots\!00\)\( + \)\(36\!\cdots\!90\)\( \beta_{1} + \)\(28\!\cdots\!25\)\( \beta_{2} - \)\(16\!\cdots\!40\)\( \beta_{3} + \)\(31\!\cdots\!00\)\( \beta_{4} + \)\(89\!\cdots\!60\)\( \beta_{5} + \)\(32\!\cdots\!20\)\( \beta_{6} - \)\(13\!\cdots\!80\)\( \beta_{7}) q^{75}\) \(+(\)\(77\!\cdots\!00\)\( + \)\(10\!\cdots\!44\)\( \beta_{1} - \)\(13\!\cdots\!44\)\( \beta_{2} + \)\(59\!\cdots\!00\)\( \beta_{3} - \)\(51\!\cdots\!52\)\( \beta_{4} - \)\(26\!\cdots\!32\)\( \beta_{5} + \)\(25\!\cdots\!28\)\( \beta_{6} - \)\(50\!\cdots\!32\)\( \beta_{7}) q^{76}\) \(+(\)\(47\!\cdots\!00\)\( + \)\(26\!\cdots\!24\)\( \beta_{1} - \)\(15\!\cdots\!92\)\( \beta_{2} - \)\(11\!\cdots\!96\)\( \beta_{3} + \)\(70\!\cdots\!84\)\( \beta_{4} - \)\(24\!\cdots\!60\)\( \beta_{5} - \)\(17\!\cdots\!96\)\( \beta_{6} + \)\(10\!\cdots\!40\)\( \beta_{7}) q^{77}\) \(+(\)\(55\!\cdots\!00\)\( + \)\(23\!\cdots\!86\)\( \beta_{1} + \)\(33\!\cdots\!40\)\( \beta_{2} - \)\(52\!\cdots\!22\)\( \beta_{3} + \)\(78\!\cdots\!82\)\( \beta_{4} + \)\(11\!\cdots\!34\)\( \beta_{5} + \)\(20\!\cdots\!56\)\( \beta_{6} + \)\(15\!\cdots\!68\)\( \beta_{7}) q^{78}\) \(+(\)\(30\!\cdots\!00\)\( - \)\(77\!\cdots\!24\)\( \beta_{1} - \)\(15\!\cdots\!34\)\( \beta_{2} - \)\(81\!\cdots\!22\)\( \beta_{3} - \)\(12\!\cdots\!08\)\( \beta_{4} - \)\(13\!\cdots\!78\)\( \beta_{5} - \)\(33\!\cdots\!58\)\( \beta_{6} - \)\(84\!\cdots\!48\)\( \beta_{7}) q^{79}\) \(+(-\)\(32\!\cdots\!60\)\( - \)\(81\!\cdots\!72\)\( \beta_{1} + \)\(12\!\cdots\!36\)\( \beta_{2} - \)\(17\!\cdots\!36\)\( \beta_{3} - \)\(19\!\cdots\!44\)\( \beta_{4} - \)\(31\!\cdots\!80\)\( \beta_{5} + \)\(22\!\cdots\!40\)\( \beta_{6} + \)\(74\!\cdots\!40\)\( \beta_{7}) q^{80}\) \(+(-\)\(39\!\cdots\!79\)\( - \)\(16\!\cdots\!62\)\( \beta_{1} - \)\(21\!\cdots\!40\)\( \beta_{2} + \)\(73\!\cdots\!54\)\( \beta_{3} - \)\(41\!\cdots\!06\)\( \beta_{4} - \)\(26\!\cdots\!16\)\( \beta_{5} - \)\(72\!\cdots\!92\)\( \beta_{6} + \)\(26\!\cdots\!48\)\( \beta_{7}) q^{81}\) \(+(-\)\(43\!\cdots\!90\)\( - \)\(64\!\cdots\!90\)\( \beta_{1} - \)\(12\!\cdots\!44\)\( \beta_{2} + \)\(35\!\cdots\!56\)\( \beta_{3} + \)\(26\!\cdots\!52\)\( \beta_{4} + \)\(11\!\cdots\!76\)\( \beta_{5} - \)\(26\!\cdots\!32\)\( \beta_{6} - \)\(82\!\cdots\!88\)\( \beta_{7}) q^{82}\) \(+(-\)\(32\!\cdots\!80\)\( + \)\(34\!\cdots\!62\)\( \beta_{1} - \)\(36\!\cdots\!53\)\( \beta_{2} + \)\(79\!\cdots\!00\)\( \beta_{3} - \)\(19\!\cdots\!40\)\( \beta_{4} + \)\(48\!\cdots\!60\)\( \beta_{5} + \)\(14\!\cdots\!20\)\( \beta_{6} + \)\(46\!\cdots\!20\)\( \beta_{7}) q^{83}\) \(+(-\)\(18\!\cdots\!04\)\( + \)\(37\!\cdots\!48\)\( \beta_{1} - \)\(13\!\cdots\!44\)\( \beta_{2} - \)\(89\!\cdots\!80\)\( \beta_{3} - \)\(19\!\cdots\!28\)\( \beta_{4} - \)\(64\!\cdots\!88\)\( \beta_{5} - \)\(16\!\cdots\!60\)\( \beta_{6} + \)\(20\!\cdots\!40\)\( \beta_{7}) q^{84}\) \(+(-\)\(96\!\cdots\!60\)\( + \)\(35\!\cdots\!14\)\( \beta_{1} + \)\(19\!\cdots\!84\)\( \beta_{2} + \)\(81\!\cdots\!54\)\( \beta_{3} - \)\(33\!\cdots\!86\)\( \beta_{4} + \)\(62\!\cdots\!08\)\( \beta_{5} - \)\(28\!\cdots\!24\)\( \beta_{6} - \)\(47\!\cdots\!04\)\( \beta_{7}) q^{85}\) \(+(-\)\(28\!\cdots\!68\)\( + \)\(26\!\cdots\!13\)\( \beta_{1} + \)\(11\!\cdots\!70\)\( \beta_{2} - \)\(29\!\cdots\!13\)\( \beta_{3} + \)\(51\!\cdots\!31\)\( \beta_{4} + \)\(80\!\cdots\!21\)\( \beta_{5} + \)\(10\!\cdots\!16\)\( \beta_{6} + \)\(17\!\cdots\!96\)\( \beta_{7}) q^{86}\) \(+(-\)\(24\!\cdots\!80\)\( - \)\(53\!\cdots\!28\)\( \beta_{1} - \)\(17\!\cdots\!29\)\( \beta_{2} + \)\(31\!\cdots\!91\)\( \beta_{3} + \)\(36\!\cdots\!98\)\( \beta_{4} - \)\(40\!\cdots\!73\)\( \beta_{5} - \)\(80\!\cdots\!65\)\( \beta_{6} + \)\(79\!\cdots\!84\)\( \beta_{7}) q^{87}\) \(+(-\)\(50\!\cdots\!80\)\( - \)\(18\!\cdots\!12\)\( \beta_{1} - \)\(15\!\cdots\!76\)\( \beta_{2} - \)\(13\!\cdots\!96\)\( \beta_{3} - \)\(58\!\cdots\!36\)\( \beta_{4} - \)\(38\!\cdots\!80\)\( \beta_{5} - \)\(15\!\cdots\!36\)\( \beta_{6} - \)\(10\!\cdots\!00\)\( \beta_{7}) q^{88}\) \(+(-\)\(30\!\cdots\!50\)\( - \)\(32\!\cdots\!82\)\( \beta_{1} - \)\(34\!\cdots\!40\)\( \beta_{2} - \)\(63\!\cdots\!70\)\( \beta_{3} - \)\(27\!\cdots\!42\)\( \beta_{4} + \)\(81\!\cdots\!48\)\( \beta_{5} + \)\(33\!\cdots\!04\)\( \beta_{6} - \)\(18\!\cdots\!76\)\( \beta_{7}) q^{89}\) \(+(\)\(24\!\cdots\!30\)\( - \)\(25\!\cdots\!62\)\( \beta_{1} + \)\(45\!\cdots\!28\)\( \beta_{2} - \)\(86\!\cdots\!32\)\( \beta_{3} - \)\(44\!\cdots\!12\)\( \beta_{4} + \)\(16\!\cdots\!36\)\( \beta_{5} - \)\(87\!\cdots\!08\)\( \beta_{6} - \)\(76\!\cdots\!68\)\( \beta_{7}) q^{90}\) \(+(\)\(19\!\cdots\!72\)\( + \)\(40\!\cdots\!72\)\( \beta_{1} - \)\(54\!\cdots\!12\)\( \beta_{2} + \)\(12\!\cdots\!40\)\( \beta_{3} + \)\(35\!\cdots\!24\)\( \beta_{4} - \)\(17\!\cdots\!16\)\( \beta_{5} - \)\(27\!\cdots\!36\)\( \beta_{6} + \)\(76\!\cdots\!84\)\( \beta_{7}) q^{91}\) \(+(\)\(96\!\cdots\!40\)\( + \)\(60\!\cdots\!60\)\( \beta_{1} + \)\(25\!\cdots\!68\)\( \beta_{2} + \)\(83\!\cdots\!16\)\( \beta_{3} + \)\(30\!\cdots\!32\)\( \beta_{4} - \)\(46\!\cdots\!04\)\( \beta_{5} + \)\(27\!\cdots\!68\)\( \beta_{6} - \)\(63\!\cdots\!48\)\( \beta_{7}) q^{92}\) \(+(\)\(12\!\cdots\!60\)\( + \)\(17\!\cdots\!04\)\( \beta_{1} - \)\(31\!\cdots\!16\)\( \beta_{2} - \)\(19\!\cdots\!08\)\( \beta_{3} - \)\(96\!\cdots\!72\)\( \beta_{4} - \)\(42\!\cdots\!44\)\( \beta_{5} - \)\(65\!\cdots\!56\)\( \beta_{6} - \)\(30\!\cdots\!88\)\( \beta_{7}) q^{93}\) \(+(\)\(90\!\cdots\!56\)\( + \)\(10\!\cdots\!16\)\( \beta_{1} + \)\(76\!\cdots\!08\)\( \beta_{2} - \)\(10\!\cdots\!64\)\( \beta_{3} + \)\(25\!\cdots\!52\)\( \beta_{4} + \)\(96\!\cdots\!32\)\( \beta_{5} + \)\(36\!\cdots\!72\)\( \beta_{6} + \)\(82\!\cdots\!32\)\( \beta_{7}) q^{94}\) \(+(\)\(34\!\cdots\!00\)\( + \)\(35\!\cdots\!20\)\( \beta_{1} - \)\(91\!\cdots\!65\)\( \beta_{2} - \)\(41\!\cdots\!25\)\( \beta_{3} - \)\(29\!\cdots\!90\)\( \beta_{4} + \)\(24\!\cdots\!35\)\( \beta_{5} + \)\(65\!\cdots\!95\)\( \beta_{6} - \)\(31\!\cdots\!80\)\( \beta_{7}) q^{95}\) \(+(\)\(11\!\cdots\!12\)\( - \)\(28\!\cdots\!72\)\( \beta_{1} - \)\(18\!\cdots\!52\)\( \beta_{2} + \)\(77\!\cdots\!28\)\( \beta_{3} + \)\(47\!\cdots\!72\)\( \beta_{4} - \)\(80\!\cdots\!88\)\( \beta_{5} - \)\(14\!\cdots\!60\)\( \beta_{6} - \)\(17\!\cdots\!60\)\( \beta_{7}) q^{96}\) \(+(\)\(62\!\cdots\!10\)\( - \)\(59\!\cdots\!18\)\( \beta_{1} - \)\(14\!\cdots\!96\)\( \beta_{2} - \)\(37\!\cdots\!22\)\( \beta_{3} - \)\(23\!\cdots\!54\)\( \beta_{4} + \)\(28\!\cdots\!08\)\( \beta_{5} - \)\(13\!\cdots\!76\)\( \beta_{6} + \)\(30\!\cdots\!96\)\( \beta_{7}) q^{97}\) \(+(\)\(11\!\cdots\!15\)\( - \)\(58\!\cdots\!21\)\( \beta_{1} + \)\(11\!\cdots\!76\)\( \beta_{2} + \)\(64\!\cdots\!96\)\( \beta_{3} - \)\(74\!\cdots\!08\)\( \beta_{4} + \)\(12\!\cdots\!76\)\( \beta_{5} + \)\(81\!\cdots\!08\)\( \beta_{6} - \)\(46\!\cdots\!88\)\( \beta_{7}) q^{98}\) \(+(-\)\(90\!\cdots\!36\)\( - \)\(26\!\cdots\!62\)\( \beta_{1} - \)\(59\!\cdots\!63\)\( \beta_{2} - \)\(47\!\cdots\!40\)\( \beta_{3} - \)\(93\!\cdots\!64\)\( \beta_{4} - \)\(11\!\cdots\!24\)\( \beta_{5} - \)\(62\!\cdots\!84\)\( \beta_{6} - \)\(20\!\cdots\!04\)\( \beta_{7}) q^{99}\) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(8q \) \(\mathstrut +\mathstrut 4388929556680440q^{2} \) \(\mathstrut +\mathstrut \)\(50\!\cdots\!40\)\(q^{3} \) \(\mathstrut +\mathstrut \)\(48\!\cdots\!64\)\(q^{4} \) \(\mathstrut +\mathstrut \)\(55\!\cdots\!20\)\(q^{5} \) \(\mathstrut +\mathstrut \)\(26\!\cdots\!36\)\(q^{6} \) \(\mathstrut +\mathstrut \)\(41\!\cdots\!00\)\(q^{7} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!80\)\(q^{8} \) \(\mathstrut +\mathstrut \)\(35\!\cdots\!16\)\(q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(8q \) \(\mathstrut +\mathstrut 4388929556680440q^{2} \) \(\mathstrut +\mathstrut \)\(50\!\cdots\!40\)\(q^{3} \) \(\mathstrut +\mathstrut \)\(48\!\cdots\!64\)\(q^{4} \) \(\mathstrut +\mathstrut \)\(55\!\cdots\!20\)\(q^{5} \) \(\mathstrut +\mathstrut \)\(26\!\cdots\!36\)\(q^{6} \) \(\mathstrut +\mathstrut \)\(41\!\cdots\!00\)\(q^{7} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!80\)\(q^{8} \) \(\mathstrut +\mathstrut \)\(35\!\cdots\!16\)\(q^{9} \) \(\mathstrut +\mathstrut \)\(20\!\cdots\!20\)\(q^{10} \) \(\mathstrut -\mathstrut \)\(53\!\cdots\!44\)\(q^{11} \) \(\mathstrut +\mathstrut \)\(25\!\cdots\!80\)\(q^{12} \) \(\mathstrut -\mathstrut \)\(15\!\cdots\!20\)\(q^{13} \) \(\mathstrut +\mathstrut \)\(20\!\cdots\!08\)\(q^{14} \) \(\mathstrut -\mathstrut \)\(30\!\cdots\!60\)\(q^{15} \) \(\mathstrut +\mathstrut \)\(36\!\cdots\!88\)\(q^{16} \) \(\mathstrut +\mathstrut \)\(20\!\cdots\!20\)\(q^{17} \) \(\mathstrut +\mathstrut \)\(26\!\cdots\!40\)\(q^{18} \) \(\mathstrut +\mathstrut \)\(14\!\cdots\!00\)\(q^{19} \) \(\mathstrut +\mathstrut \)\(25\!\cdots\!60\)\(q^{20} \) \(\mathstrut -\mathstrut \)\(46\!\cdots\!04\)\(q^{21} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!20\)\(q^{22} \) \(\mathstrut +\mathstrut \)\(40\!\cdots\!20\)\(q^{23} \) \(\mathstrut +\mathstrut \)\(41\!\cdots\!00\)\(q^{24} \) \(\mathstrut -\mathstrut \)\(36\!\cdots\!00\)\(q^{25} \) \(\mathstrut -\mathstrut \)\(14\!\cdots\!84\)\(q^{26} \) \(\mathstrut +\mathstrut \)\(63\!\cdots\!20\)\(q^{27} \) \(\mathstrut +\mathstrut \)\(51\!\cdots\!80\)\(q^{28} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!00\)\(q^{29} \) \(\mathstrut -\mathstrut \)\(31\!\cdots\!60\)\(q^{30} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!36\)\(q^{31} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!40\)\(q^{32} \) \(\mathstrut +\mathstrut \)\(16\!\cdots\!80\)\(q^{33} \) \(\mathstrut -\mathstrut \)\(16\!\cdots\!32\)\(q^{34} \) \(\mathstrut -\mathstrut \)\(15\!\cdots\!80\)\(q^{35} \) \(\mathstrut +\mathstrut \)\(80\!\cdots\!28\)\(q^{36} \) \(\mathstrut -\mathstrut \)\(92\!\cdots\!40\)\(q^{37} \) \(\mathstrut +\mathstrut \)\(19\!\cdots\!20\)\(q^{38} \) \(\mathstrut -\mathstrut \)\(78\!\cdots\!08\)\(q^{39} \) \(\mathstrut +\mathstrut \)\(19\!\cdots\!00\)\(q^{40} \) \(\mathstrut +\mathstrut \)\(22\!\cdots\!76\)\(q^{41} \) \(\mathstrut -\mathstrut \)\(18\!\cdots\!80\)\(q^{42} \) \(\mathstrut +\mathstrut \)\(96\!\cdots\!00\)\(q^{43} \) \(\mathstrut +\mathstrut \)\(26\!\cdots\!48\)\(q^{44} \) \(\mathstrut -\mathstrut \)\(14\!\cdots\!60\)\(q^{45} \) \(\mathstrut +\mathstrut \)\(27\!\cdots\!96\)\(q^{46} \) \(\mathstrut -\mathstrut \)\(29\!\cdots\!20\)\(q^{47} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!20\)\(q^{48} \) \(\mathstrut +\mathstrut \)\(26\!\cdots\!44\)\(q^{49} \) \(\mathstrut -\mathstrut \)\(65\!\cdots\!00\)\(q^{50} \) \(\mathstrut -\mathstrut \)\(20\!\cdots\!84\)\(q^{51} \) \(\mathstrut -\mathstrut \)\(36\!\cdots\!00\)\(q^{52} \) \(\mathstrut +\mathstrut \)\(16\!\cdots\!40\)\(q^{53} \) \(\mathstrut +\mathstrut \)\(24\!\cdots\!00\)\(q^{54} \) \(\mathstrut -\mathstrut \)\(60\!\cdots\!60\)\(q^{55} \) \(\mathstrut -\mathstrut \)\(50\!\cdots\!00\)\(q^{56} \) \(\mathstrut +\mathstrut \)\(55\!\cdots\!40\)\(q^{57} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!80\)\(q^{58} \) \(\mathstrut -\mathstrut \)\(37\!\cdots\!00\)\(q^{59} \) \(\mathstrut -\mathstrut \)\(44\!\cdots\!80\)\(q^{60} \) \(\mathstrut +\mathstrut \)\(56\!\cdots\!56\)\(q^{61} \) \(\mathstrut +\mathstrut \)\(70\!\cdots\!80\)\(q^{62} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!60\)\(q^{63} \) \(\mathstrut +\mathstrut \)\(26\!\cdots\!04\)\(q^{64} \) \(\mathstrut +\mathstrut \)\(51\!\cdots\!40\)\(q^{65} \) \(\mathstrut +\mathstrut \)\(36\!\cdots\!52\)\(q^{66} \) \(\mathstrut +\mathstrut \)\(36\!\cdots\!20\)\(q^{67} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!40\)\(q^{68} \) \(\mathstrut +\mathstrut \)\(14\!\cdots\!52\)\(q^{69} \) \(\mathstrut +\mathstrut \)\(85\!\cdots\!20\)\(q^{70} \) \(\mathstrut +\mathstrut \)\(13\!\cdots\!96\)\(q^{71} \) \(\mathstrut +\mathstrut \)\(56\!\cdots\!60\)\(q^{72} \) \(\mathstrut +\mathstrut \)\(44\!\cdots\!20\)\(q^{73} \) \(\mathstrut +\mathstrut \)\(94\!\cdots\!88\)\(q^{74} \) \(\mathstrut +\mathstrut \)\(15\!\cdots\!00\)\(q^{75} \) \(\mathstrut +\mathstrut \)\(62\!\cdots\!00\)\(q^{76} \) \(\mathstrut +\mathstrut \)\(37\!\cdots\!00\)\(q^{77} \) \(\mathstrut +\mathstrut \)\(44\!\cdots\!00\)\(q^{78} \) \(\mathstrut +\mathstrut \)\(24\!\cdots\!00\)\(q^{79} \) \(\mathstrut -\mathstrut \)\(25\!\cdots\!80\)\(q^{80} \) \(\mathstrut -\mathstrut \)\(31\!\cdots\!32\)\(q^{81} \) \(\mathstrut -\mathstrut \)\(34\!\cdots\!20\)\(q^{82} \) \(\mathstrut -\mathstrut \)\(25\!\cdots\!40\)\(q^{83} \) \(\mathstrut -\mathstrut \)\(14\!\cdots\!32\)\(q^{84} \) \(\mathstrut -\mathstrut \)\(77\!\cdots\!80\)\(q^{85} \) \(\mathstrut -\mathstrut \)\(22\!\cdots\!44\)\(q^{86} \) \(\mathstrut -\mathstrut \)\(19\!\cdots\!40\)\(q^{87} \) \(\mathstrut -\mathstrut \)\(40\!\cdots\!40\)\(q^{88} \) \(\mathstrut -\mathstrut \)\(24\!\cdots\!00\)\(q^{89} \) \(\mathstrut +\mathstrut \)\(19\!\cdots\!40\)\(q^{90} \) \(\mathstrut +\mathstrut \)\(15\!\cdots\!76\)\(q^{91} \) \(\mathstrut +\mathstrut \)\(77\!\cdots\!20\)\(q^{92} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!80\)\(q^{93} \) \(\mathstrut +\mathstrut \)\(72\!\cdots\!48\)\(q^{94} \) \(\mathstrut +\mathstrut \)\(27\!\cdots\!00\)\(q^{95} \) \(\mathstrut +\mathstrut \)\(88\!\cdots\!96\)\(q^{96} \) \(\mathstrut +\mathstrut \)\(50\!\cdots\!80\)\(q^{97} \) \(\mathstrut +\mathstrut \)\(88\!\cdots\!20\)\(q^{98} \) \(\mathstrut -\mathstrut \)\(72\!\cdots\!88\)\(q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8}\mathstrut -\mathstrut \) \(3\) \(x^{7}\mathstrut -\mathstrut \) \(11\!\cdots\!21\) \(x^{6}\mathstrut -\mathstrut \) \(27\!\cdots\!07\) \(x^{5}\mathstrut +\mathstrut \) \(36\!\cdots\!21\) \(x^{4}\mathstrut -\mathstrut \) \(47\!\cdots\!05\) \(x^{3}\mathstrut -\mathstrut \) \(36\!\cdots\!75\) \(x^{2}\mathstrut +\mathstrut \) \(17\!\cdots\!75\) \(x\mathstrut +\mathstrut \) \(10\!\cdots\!50\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 24 \nu - 9 \)
\(\beta_{2}\)\(=\)\((\)\(16\!\cdots\!41\) \(\nu^{7}\mathstrut +\mathstrut \) \(13\!\cdots\!88\) \(\nu^{6}\mathstrut -\mathstrut \) \(16\!\cdots\!69\) \(\nu^{5}\mathstrut -\mathstrut \) \(15\!\cdots\!70\) \(\nu^{4}\mathstrut +\mathstrut \) \(45\!\cdots\!31\) \(\nu^{3}\mathstrut +\mathstrut \) \(41\!\cdots\!16\) \(\nu^{2}\mathstrut -\mathstrut \) \(25\!\cdots\!71\) \(\nu\mathstrut -\mathstrut \) \(21\!\cdots\!46\)\()/\)\(59\!\cdots\!28\)
\(\beta_{3}\)\(=\)\((\)\(-\)\(57\!\cdots\!09\) \(\nu^{7}\mathstrut -\mathstrut \) \(46\!\cdots\!12\) \(\nu^{6}\mathstrut +\mathstrut \) \(57\!\cdots\!81\) \(\nu^{5}\mathstrut +\mathstrut \) \(51\!\cdots\!30\) \(\nu^{4}\mathstrut -\mathstrut \) \(15\!\cdots\!19\) \(\nu^{3}\mathstrut +\mathstrut \) \(19\!\cdots\!44\) \(\nu^{2}\mathstrut +\mathstrut \) \(73\!\cdots\!43\) \(\nu\mathstrut -\mathstrut \) \(86\!\cdots\!86\)\()/\)\(59\!\cdots\!28\)
\(\beta_{4}\)\(=\)\((\)\(44\!\cdots\!29\) \(\nu^{7}\mathstrut +\mathstrut \) \(15\!\cdots\!16\) \(\nu^{6}\mathstrut -\mathstrut \) \(46\!\cdots\!97\) \(\nu^{5}\mathstrut -\mathstrut \) \(18\!\cdots\!82\) \(\nu^{4}\mathstrut +\mathstrut \) \(13\!\cdots\!35\) \(\nu^{3}\mathstrut +\mathstrut \) \(52\!\cdots\!00\) \(\nu^{2}\mathstrut -\mathstrut \) \(80\!\cdots\!75\) \(\nu\mathstrut -\mathstrut \) \(27\!\cdots\!50\)\()/\)\(82\!\cdots\!00\)
\(\beta_{5}\)\(=\)\((\)\(-\)\(17\!\cdots\!41\) \(\nu^{7}\mathstrut -\mathstrut \) \(16\!\cdots\!64\) \(\nu^{6}\mathstrut +\mathstrut \) \(18\!\cdots\!13\) \(\nu^{5}\mathstrut +\mathstrut \) \(16\!\cdots\!78\) \(\nu^{4}\mathstrut -\mathstrut \) \(50\!\cdots\!15\) \(\nu^{3}\mathstrut -\mathstrut \) \(40\!\cdots\!00\) \(\nu^{2}\mathstrut +\mathstrut \) \(28\!\cdots\!75\) \(\nu\mathstrut +\mathstrut \) \(19\!\cdots\!50\)\()/\)\(14\!\cdots\!00\)
\(\beta_{6}\)\(=\)\((\)\(-\)\(55\!\cdots\!79\) \(\nu^{7}\mathstrut -\mathstrut \) \(13\!\cdots\!16\) \(\nu^{6}\mathstrut +\mathstrut \) \(60\!\cdots\!47\) \(\nu^{5}\mathstrut +\mathstrut \) \(13\!\cdots\!82\) \(\nu^{4}\mathstrut -\mathstrut \) \(18\!\cdots\!85\) \(\nu^{3}\mathstrut -\mathstrut \) \(37\!\cdots\!00\) \(\nu^{2}\mathstrut +\mathstrut \) \(11\!\cdots\!25\) \(\nu\mathstrut +\mathstrut \) \(19\!\cdots\!50\)\()/\)\(94\!\cdots\!00\)
\(\beta_{7}\)\(=\)\((\)\(-\)\(36\!\cdots\!03\) \(\nu^{7}\mathstrut -\mathstrut \) \(23\!\cdots\!12\) \(\nu^{6}\mathstrut +\mathstrut \) \(36\!\cdots\!79\) \(\nu^{5}\mathstrut +\mathstrut \) \(25\!\cdots\!74\) \(\nu^{4}\mathstrut -\mathstrut \) \(90\!\cdots\!45\) \(\nu^{3}\mathstrut -\mathstrut \) \(72\!\cdots\!00\) \(\nu^{2}\mathstrut +\mathstrut \) \(24\!\cdots\!25\) \(\nu\mathstrut +\mathstrut \) \(40\!\cdots\!50\)\()/\)\(48\!\cdots\!00\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1}\mathstrut +\mathstrut \) \(9\)\()/24\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3}\mathstrut +\mathstrut \) \(34249\) \(\beta_{2}\mathstrut +\mathstrut \) \(90668611228163\) \(\beta_{1}\mathstrut +\mathstrut \) \(15880774323327381508482134600472\)\()/576\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{7}\mathstrut +\mathstrut \) \(134\) \(\beta_{6}\mathstrut -\mathstrut \) \(75477\) \(\beta_{5}\mathstrut +\mathstrut \) \(783459941\) \(\beta_{4}\mathstrut +\mathstrut \) \(731911083234300\) \(\beta_{3}\mathstrut +\mathstrut \) \(2478085321776355741004\) \(\beta_{2}\mathstrut +\mathstrut \) \(25759876756015228360331701993191\) \(\beta_{1}\mathstrut +\mathstrut \) \(1439887753041709047949578036681177213775225536\)\()/13824\)
\(\nu^{4}\)\(=\)\((\)\(127486023401493\) \(\beta_{7}\mathstrut +\mathstrut \) \(218362552566909694\) \(\beta_{6}\mathstrut -\mathstrut \) \(430172487255977596793\) \(\beta_{5}\mathstrut -\mathstrut \) \(4356650832545394280174007\) \(\beta_{4}\mathstrut +\mathstrut \) \(4497838928225868883025303812548\) \(\beta_{3}\mathstrut -\mathstrut \) \(76823225147800960931291530835816428\) \(\beta_{2}\mathstrut +\mathstrut \) \(1386918490615083743415227142251113877567731451\) \(\beta_{1}\mathstrut +\mathstrut \) \(51135848669130346006836710771052948394485158826710775385653696\)\()/41472\)
\(\nu^{5}\)\(=\)\((\)\(152656288142514471733425820127\) \(\beta_{7}\mathstrut +\mathstrut \) \(45438221749047255954687266134714\) \(\beta_{6}\mathstrut -\mathstrut \) \(17993559956597123255555491648916619\) \(\beta_{5}\mathstrut -\mathstrut \) \(768246208699183609569551992420873901317\) \(\beta_{4}\mathstrut +\mathstrut \) \(173385253974466203701312559740646907720956512\) \(\beta_{3}\mathstrut +\mathstrut \) \(571363283579140025011367950090086947987690066381552\) \(\beta_{2}\mathstrut +\mathstrut \) \(2989070443962723331825630931119123352252371254739780813839645\) \(\beta_{1}\mathstrut +\mathstrut \) \(688291861077914391455873698555277860820710991874720231796873324604834841264\)\()/31104\)
\(\nu^{6}\)\(=\)\((\)\(23\!\cdots\!71\) \(\beta_{7}\mathstrut +\mathstrut \) \(33\!\cdots\!62\) \(\beta_{6}\mathstrut -\mathstrut \) \(73\!\cdots\!23\) \(\beta_{5}\mathstrut -\mathstrut \) \(85\!\cdots\!17\) \(\beta_{4}\mathstrut +\mathstrut \) \(43\!\cdots\!40\) \(\beta_{3}\mathstrut -\mathstrut \) \(56\!\cdots\!36\) \(\beta_{2}\mathstrut +\mathstrut \) \(24\!\cdots\!09\) \(\beta_{1}\mathstrut +\mathstrut \) \(43\!\cdots\!04\)\()/6912\)
\(\nu^{7}\)\(=\)\((\)\(12\!\cdots\!93\) \(\beta_{7}\mathstrut +\mathstrut \) \(53\!\cdots\!70\) \(\beta_{6}\mathstrut -\mathstrut \) \(21\!\cdots\!33\) \(\beta_{5}\mathstrut -\mathstrut \) \(10\!\cdots\!51\) \(\beta_{4}\mathstrut +\mathstrut \) \(19\!\cdots\!12\) \(\beta_{3}\mathstrut +\mathstrut \) \(56\!\cdots\!08\) \(\beta_{2}\mathstrut +\mathstrut \) \(21\!\cdots\!27\) \(\beta_{1}\mathstrut +\mathstrut \) \(98\!\cdots\!88\)\()/41472\)

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
2.50366e14
1.83024e14
8.88202e13
8.06342e13
−6.19123e13
−1.05899e14
−2.04878e14
−2.30156e14
−5.46018e15 −1.30099e24 1.96724e31 −5.30727e35 7.10366e39 2.61912e43 −5.20418e46 −1.22226e49 2.89786e51
1.2 −3.84396e15 3.76369e24 4.63479e30 1.31095e36 −1.44675e40 −5.54417e43 2.11664e46 2.50204e47 −5.03922e51
1.3 −1.58307e15 −4.68333e24 −7.63510e30 1.63346e35 7.41404e39 −1.85939e42 2.81411e46 8.01842e48 −2.58588e50
1.4 −1.38660e15 4.94875e24 −8.21853e30 −1.47562e36 −6.86195e39 3.27966e43 2.54577e46 1.05749e49 2.04610e51
1.5 2.03451e15 2.78778e24 −6.00197e30 1.50905e36 5.67177e39 3.17629e43 −3.28435e46 −6.14348e48 3.07017e51
1.6 3.09018e15 −9.30646e23 −5.91967e29 −7.61227e35 −2.87587e39 −3.39343e43 −3.31675e46 −1.30491e49 −2.35233e51
1.7 5.46569e15 −5.92176e24 1.97326e31 6.19706e35 −3.23665e40 6.38506e43 5.24235e46 2.11521e49 3.38712e51
1.8 6.07235e15 6.42408e24 2.67322e31 −2.83827e35 3.90093e40 −2.16284e43 1.00747e47 2.73537e49 −1.72350e51
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Inner twists

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

Hecke kernels

There are no other newforms in \(S_{104}^{\mathrm{new}}(\Gamma_0(1))\).