Properties

Label 1.110.a.a
Level 1
Weight 110
Character orbit 1.a
Self dual Yes
Analytic conductor 75.239
Analytic rank 1
Dimension 8
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 1 \)
Weight: \( k \) = \( 110 \)
Character orbit: \([\chi]\) = 1.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(75.2394221917\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: multiple of \( 2^{118}\cdot 3^{40}\cdot 5^{14}\cdot 7^{6}\cdot 11^{3}\cdot 13 \)
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.

\(f(q)\) \(=\) \( q\) \(+(286049075864400 - \beta_{1}) q^{2}\) \(+(-\)\(99\!\cdots\!00\)\( - 852079581 \beta_{1} + \beta_{2}) q^{3}\) \(+(\)\(32\!\cdots\!32\)\( + 1401168417189145 \beta_{1} - 733523 \beta_{2} + \beta_{3}) q^{4}\) \(+(-\)\(26\!\cdots\!50\)\( - \)\(24\!\cdots\!63\)\( \beta_{1} + 256532420749 \beta_{2} - 43853 \beta_{3} - \beta_{4}) q^{5}\) \(+(\)\(82\!\cdots\!92\)\( + \)\(18\!\cdots\!44\)\( \beta_{1} - 4952444410002432 \beta_{2} + 1320331506 \beta_{3} + 1134 \beta_{4} + \beta_{5}) q^{6}\) \(+(\)\(29\!\cdots\!00\)\( + \)\(65\!\cdots\!95\)\( \beta_{1} - 29921845411816752196 \beta_{2} + 5353680223264 \beta_{3} + 7148829 \beta_{4} + 67 \beta_{5} - \beta_{6}) q^{7}\) \(+(-\)\(14\!\cdots\!00\)\( - \)\(36\!\cdots\!36\)\( \beta_{1} + \)\(36\!\cdots\!21\)\( \beta_{2} - 526031820577024 \beta_{3} + 23678718391 \beta_{4} - 2011782 \beta_{5} - 79 \beta_{6} - \beta_{7}) q^{8}\) \(+(\)\(30\!\cdots\!73\)\( + \)\(76\!\cdots\!54\)\( \beta_{1} - \)\(71\!\cdots\!02\)\( \beta_{2} - 15308494200942822 \beta_{3} - 5285573276346 \beta_{4} + 431210340 \beta_{5} + 40716 \beta_{6} + 216 \beta_{7}) q^{9}\) \(+O(q^{10})\) \( q\) \(+(286049075864400 - \beta_{1}) q^{2}\) \(+(-\)\(99\!\cdots\!00\)\( - 852079581 \beta_{1} + \beta_{2}) q^{3}\) \(+(\)\(32\!\cdots\!32\)\( + 1401168417189145 \beta_{1} - 733523 \beta_{2} + \beta_{3}) q^{4}\) \(+(-\)\(26\!\cdots\!50\)\( - \)\(24\!\cdots\!63\)\( \beta_{1} + 256532420749 \beta_{2} - 43853 \beta_{3} - \beta_{4}) q^{5}\) \(+(\)\(82\!\cdots\!92\)\( + \)\(18\!\cdots\!44\)\( \beta_{1} - 4952444410002432 \beta_{2} + 1320331506 \beta_{3} + 1134 \beta_{4} + \beta_{5}) q^{6}\) \(+(\)\(29\!\cdots\!00\)\( + \)\(65\!\cdots\!95\)\( \beta_{1} - 29921845411816752196 \beta_{2} + 5353680223264 \beta_{3} + 7148829 \beta_{4} + 67 \beta_{5} - \beta_{6}) q^{7}\) \(+(-\)\(14\!\cdots\!00\)\( - \)\(36\!\cdots\!36\)\( \beta_{1} + \)\(36\!\cdots\!21\)\( \beta_{2} - 526031820577024 \beta_{3} + 23678718391 \beta_{4} - 2011782 \beta_{5} - 79 \beta_{6} - \beta_{7}) q^{8}\) \(+(\)\(30\!\cdots\!73\)\( + \)\(76\!\cdots\!54\)\( \beta_{1} - \)\(71\!\cdots\!02\)\( \beta_{2} - 15308494200942822 \beta_{3} - 5285573276346 \beta_{4} + 431210340 \beta_{5} + 40716 \beta_{6} + 216 \beta_{7}) q^{9}\) \(+(\)\(22\!\cdots\!00\)\( + \)\(58\!\cdots\!34\)\( \beta_{1} - \)\(44\!\cdots\!32\)\( \beta_{2} - \)\(24\!\cdots\!96\)\( \beta_{3} - 4561081552650912 \beta_{4} + 335860266940 \beta_{5} + 74398280 \beta_{6} + 173880 \beta_{7}) q^{10}\) \(+(-\)\(20\!\cdots\!08\)\( - \)\(19\!\cdots\!97\)\( \beta_{1} + \)\(10\!\cdots\!35\)\( \beta_{2} - \)\(21\!\cdots\!64\)\( \beta_{3} - 322715467190875022 \beta_{4} - 14356077251154 \beta_{5} + 36376798246 \beta_{6} + 2070496 \beta_{7}) q^{11}\) \(+(-\)\(11\!\cdots\!00\)\( - \)\(12\!\cdots\!52\)\( \beta_{1} + \)\(24\!\cdots\!88\)\( \beta_{2} - \)\(33\!\cdots\!76\)\( \beta_{3} + 18868717616355676584 \beta_{4} - 13421783109153168 \beta_{5} + 3844922465304 \beta_{6} - 2246540184 \beta_{7}) q^{12}\) \(+(-\)\(89\!\cdots\!50\)\( + \)\(55\!\cdots\!57\)\( \beta_{1} - \)\(61\!\cdots\!83\)\( \beta_{2} - \)\(76\!\cdots\!77\)\( \beta_{3} - \)\(27\!\cdots\!77\)\( \beta_{4} - 1174394252859531496 \beta_{5} - 194701425963512 \beta_{6} + 196226359056 \beta_{7}) q^{13}\) \(+(-\)\(63\!\cdots\!96\)\( - \)\(42\!\cdots\!72\)\( \beta_{1} + \)\(41\!\cdots\!12\)\( \beta_{2} - \)\(56\!\cdots\!64\)\( \beta_{3} + \)\(67\!\cdots\!28\)\( \beta_{4} - 97742305884835341726 \beta_{5} - 6693323875281632 \beta_{6} - 9107509497632 \beta_{7}) q^{14}\) \(+(\)\(29\!\cdots\!00\)\( + \)\(97\!\cdots\!97\)\( \beta_{1} - \)\(12\!\cdots\!56\)\( \beta_{2} + \)\(11\!\cdots\!32\)\( \beta_{3} + \)\(48\!\cdots\!79\)\( \beta_{4} + \)\(20\!\cdots\!45\)\( \beta_{5} + 657840936324634065 \beta_{6} + 268557631842240 \beta_{7}) q^{15}\) \(+(\)\(14\!\cdots\!96\)\( + \)\(17\!\cdots\!20\)\( \beta_{1} - \)\(15\!\cdots\!88\)\( \beta_{2} + \)\(32\!\cdots\!68\)\( \beta_{3} + \)\(33\!\cdots\!60\)\( \beta_{4} + \)\(81\!\cdots\!44\)\( \beta_{5} - 18752777077785451904 \beta_{6} - 4948338306443904 \beta_{7}) q^{16}\) \(+(-\)\(10\!\cdots\!50\)\( - \)\(30\!\cdots\!30\)\( \beta_{1} + \)\(26\!\cdots\!62\)\( \beta_{2} - \)\(37\!\cdots\!34\)\( \beta_{3} + \)\(48\!\cdots\!26\)\( \beta_{4} - \)\(20\!\cdots\!52\)\( \beta_{5} + \)\(17\!\cdots\!56\)\( \beta_{6} + 33117702287146680 \beta_{7}) q^{17}\) \(+(-\)\(74\!\cdots\!00\)\( - \)\(47\!\cdots\!09\)\( \beta_{1} + \)\(42\!\cdots\!76\)\( \beta_{2} - \)\(20\!\cdots\!44\)\( \beta_{3} - \)\(73\!\cdots\!84\)\( \beta_{4} - \)\(15\!\cdots\!32\)\( \beta_{5} + \)\(42\!\cdots\!96\)\( \beta_{6} + 1288165981990582800 \beta_{7}) q^{18}\) \(+(-\)\(14\!\cdots\!20\)\( - \)\(19\!\cdots\!99\)\( \beta_{1} + \)\(13\!\cdots\!33\)\( \beta_{2} - \)\(36\!\cdots\!64\)\( \beta_{3} - \)\(21\!\cdots\!54\)\( \beta_{4} + \)\(96\!\cdots\!02\)\( \beta_{5} - \)\(18\!\cdots\!58\)\( \beta_{6} - 58545559046077715808 \beta_{7}) q^{19}\) \(+(-\)\(39\!\cdots\!00\)\( - \)\(33\!\cdots\!26\)\( \beta_{1} + \)\(15\!\cdots\!98\)\( \beta_{2} - \)\(13\!\cdots\!06\)\( \beta_{3} + \)\(30\!\cdots\!48\)\( \beta_{4} - \)\(80\!\cdots\!00\)\( \beta_{5} + \)\(36\!\cdots\!00\)\( \beta_{6} + \)\(14\!\cdots\!00\)\( \beta_{7}) q^{20}\) \(+(-\)\(34\!\cdots\!28\)\( - \)\(85\!\cdots\!76\)\( \beta_{1} - \)\(58\!\cdots\!68\)\( \beta_{2} - \)\(92\!\cdots\!24\)\( \beta_{3} + \)\(61\!\cdots\!64\)\( \beta_{4} - \)\(13\!\cdots\!48\)\( \beta_{5} - \)\(41\!\cdots\!56\)\( \beta_{6} - \)\(25\!\cdots\!56\)\( \beta_{7}) q^{21}\) \(+(\)\(18\!\cdots\!00\)\( + \)\(17\!\cdots\!88\)\( \beta_{1} - \)\(25\!\cdots\!72\)\( \beta_{2} + \)\(58\!\cdots\!62\)\( \beta_{3} - \)\(31\!\cdots\!38\)\( \beta_{4} + \)\(35\!\cdots\!51\)\( \beta_{5} + \)\(21\!\cdots\!72\)\( \beta_{6} + \)\(37\!\cdots\!64\)\( \beta_{7}) q^{22}\) \(+(\)\(67\!\cdots\!00\)\( - \)\(18\!\cdots\!35\)\( \beta_{1} + \)\(11\!\cdots\!44\)\( \beta_{2} + \)\(76\!\cdots\!20\)\( \beta_{3} - \)\(58\!\cdots\!05\)\( \beta_{4} - \)\(32\!\cdots\!15\)\( \beta_{5} + \)\(15\!\cdots\!45\)\( \beta_{6} - \)\(44\!\cdots\!40\)\( \beta_{7}) q^{23}\) \(+(\)\(66\!\cdots\!40\)\( + \)\(26\!\cdots\!68\)\( \beta_{1} - \)\(10\!\cdots\!12\)\( \beta_{2} + \)\(18\!\cdots\!36\)\( \beta_{3} + \)\(38\!\cdots\!68\)\( \beta_{4} + \)\(41\!\cdots\!08\)\( \beta_{5} - \)\(48\!\cdots\!56\)\( \beta_{6} + \)\(44\!\cdots\!44\)\( \beta_{7}) q^{24}\) \(+(\)\(22\!\cdots\!75\)\( - \)\(40\!\cdots\!00\)\( \beta_{1} - \)\(77\!\cdots\!00\)\( \beta_{2} - \)\(18\!\cdots\!00\)\( \beta_{3} + \)\(49\!\cdots\!00\)\( \beta_{4} + \)\(18\!\cdots\!00\)\( \beta_{5} + \)\(57\!\cdots\!00\)\( \beta_{6} - \)\(35\!\cdots\!00\)\( \beta_{7}) q^{25}\) \(+(-\)\(54\!\cdots\!88\)\( + \)\(12\!\cdots\!54\)\( \beta_{1} - \)\(91\!\cdots\!32\)\( \beta_{2} - \)\(14\!\cdots\!80\)\( \beta_{3} - \)\(52\!\cdots\!36\)\( \beta_{4} - \)\(15\!\cdots\!76\)\( \beta_{5} - \)\(39\!\cdots\!28\)\( \beta_{6} + \)\(22\!\cdots\!72\)\( \beta_{7}) q^{26}\) \(+(-\)\(34\!\cdots\!00\)\( + \)\(31\!\cdots\!24\)\( \beta_{1} - \)\(78\!\cdots\!34\)\( \beta_{2} - \)\(37\!\cdots\!64\)\( \beta_{3} - \)\(97\!\cdots\!74\)\( \beta_{4} - \)\(57\!\cdots\!02\)\( \beta_{5} + \)\(11\!\cdots\!06\)\( \beta_{6} - \)\(89\!\cdots\!56\)\( \beta_{7}) q^{27}\) \(+(\)\(39\!\cdots\!00\)\( + \)\(69\!\cdots\!12\)\( \beta_{1} - \)\(65\!\cdots\!72\)\( \beta_{2} + \)\(78\!\cdots\!84\)\( \beta_{3} + \)\(30\!\cdots\!44\)\( \beta_{4} + \)\(10\!\cdots\!12\)\( \beta_{5} + \)\(93\!\cdots\!64\)\( \beta_{6} - \)\(84\!\cdots\!04\)\( \beta_{7}) q^{28}\) \(+(\)\(37\!\cdots\!70\)\( + \)\(29\!\cdots\!01\)\( \beta_{1} - \)\(16\!\cdots\!55\)\( \beta_{2} + \)\(24\!\cdots\!99\)\( \beta_{3} - \)\(50\!\cdots\!89\)\( \beta_{4} - \)\(71\!\cdots\!24\)\( \beta_{5} - \)\(15\!\cdots\!72\)\( \beta_{6} + \)\(55\!\cdots\!28\)\( \beta_{7}) q^{29}\) \(+(-\)\(93\!\cdots\!00\)\( - \)\(39\!\cdots\!56\)\( \beta_{1} + \)\(18\!\cdots\!88\)\( \beta_{2} - \)\(26\!\cdots\!36\)\( \beta_{3} - \)\(10\!\cdots\!12\)\( \beta_{4} + \)\(95\!\cdots\!50\)\( \beta_{5} + \)\(10\!\cdots\!00\)\( \beta_{6} - \)\(61\!\cdots\!00\)\( \beta_{7}) q^{30}\) \(+(-\)\(18\!\cdots\!28\)\( + \)\(13\!\cdots\!60\)\( \beta_{1} + \)\(82\!\cdots\!12\)\( \beta_{2} - \)\(62\!\cdots\!76\)\( \beta_{3} + \)\(40\!\cdots\!80\)\( \beta_{4} + \)\(31\!\cdots\!36\)\( \beta_{5} - \)\(39\!\cdots\!16\)\( \beta_{6} + \)\(44\!\cdots\!84\)\( \beta_{7}) q^{31}\) \(+(-\)\(74\!\cdots\!00\)\( - \)\(14\!\cdots\!24\)\( \beta_{1} + \)\(66\!\cdots\!00\)\( \beta_{2} + \)\(17\!\cdots\!96\)\( \beta_{3} + \)\(28\!\cdots\!56\)\( \beta_{4} - \)\(22\!\cdots\!12\)\( \beta_{5} - \)\(37\!\cdots\!64\)\( \beta_{6} - \)\(23\!\cdots\!00\)\( \beta_{7}) q^{32}\) \(+(\)\(12\!\cdots\!00\)\( + \)\(42\!\cdots\!22\)\( \beta_{1} + \)\(36\!\cdots\!94\)\( \beta_{2} + \)\(32\!\cdots\!70\)\( \beta_{3} - \)\(22\!\cdots\!10\)\( \beta_{4} + \)\(48\!\cdots\!20\)\( \beta_{5} + \)\(15\!\cdots\!40\)\( \beta_{6} + \)\(82\!\cdots\!76\)\( \beta_{7}) q^{33}\) \(+(\)\(28\!\cdots\!24\)\( + \)\(36\!\cdots\!06\)\( \beta_{1} - \)\(17\!\cdots\!16\)\( \beta_{2} + \)\(64\!\cdots\!44\)\( \beta_{3} - \)\(80\!\cdots\!84\)\( \beta_{4} + \)\(29\!\cdots\!52\)\( \beta_{5} - \)\(10\!\cdots\!28\)\( \beta_{6} - \)\(39\!\cdots\!28\)\( \beta_{7}) q^{34}\) \(+(-\)\(36\!\cdots\!00\)\( + \)\(48\!\cdots\!84\)\( \beta_{1} - \)\(57\!\cdots\!32\)\( \beta_{2} - \)\(13\!\cdots\!96\)\( \beta_{3} + \)\(50\!\cdots\!88\)\( \beta_{4} - \)\(26\!\cdots\!60\)\( \beta_{5} + \)\(33\!\cdots\!80\)\( \beta_{6} - \)\(20\!\cdots\!20\)\( \beta_{7}) q^{35}\) \(+(\)\(24\!\cdots\!36\)\( + \)\(17\!\cdots\!09\)\( \beta_{1} - \)\(17\!\cdots\!15\)\( \beta_{2} - \)\(50\!\cdots\!03\)\( \beta_{3} - \)\(10\!\cdots\!96\)\( \beta_{4} + \)\(74\!\cdots\!76\)\( \beta_{5} + \)\(17\!\cdots\!80\)\( \beta_{6} + \)\(19\!\cdots\!80\)\( \beta_{7}) q^{36}\) \(+(\)\(41\!\cdots\!50\)\( + \)\(17\!\cdots\!41\)\( \beta_{1} + \)\(12\!\cdots\!29\)\( \beta_{2} + \)\(24\!\cdots\!07\)\( \beta_{3} - \)\(56\!\cdots\!13\)\( \beta_{4} + \)\(89\!\cdots\!76\)\( \beta_{5} - \)\(78\!\cdots\!28\)\( \beta_{6} - \)\(11\!\cdots\!52\)\( \beta_{7}) q^{37}\) \(+(\)\(14\!\cdots\!00\)\( + \)\(39\!\cdots\!96\)\( \beta_{1} + \)\(62\!\cdots\!40\)\( \beta_{2} + \)\(61\!\cdots\!18\)\( \beta_{3} + \)\(10\!\cdots\!78\)\( \beta_{4} - \)\(93\!\cdots\!31\)\( \beta_{5} + \)\(40\!\cdots\!68\)\( \beta_{6} + \)\(49\!\cdots\!84\)\( \beta_{7}) q^{38}\) \(+(-\)\(96\!\cdots\!84\)\( + \)\(96\!\cdots\!17\)\( \beta_{1} + \)\(13\!\cdots\!56\)\( \beta_{2} - \)\(22\!\cdots\!48\)\( \beta_{3} + \)\(16\!\cdots\!07\)\( \beta_{4} - \)\(15\!\cdots\!31\)\( \beta_{5} - \)\(71\!\cdots\!71\)\( \beta_{6} - \)\(16\!\cdots\!96\)\( \beta_{7}) q^{39}\) \(+(-\)\(12\!\cdots\!00\)\( + \)\(95\!\cdots\!60\)\( \beta_{1} - \)\(86\!\cdots\!30\)\( \beta_{2} - \)\(59\!\cdots\!40\)\( \beta_{3} - \)\(65\!\cdots\!30\)\( \beta_{4} + \)\(33\!\cdots\!00\)\( \beta_{5} - \)\(38\!\cdots\!50\)\( \beta_{6} + \)\(41\!\cdots\!50\)\( \beta_{7}) q^{40}\) \(+(\)\(32\!\cdots\!62\)\( + \)\(23\!\cdots\!16\)\( \beta_{1} - \)\(24\!\cdots\!16\)\( \beta_{2} + \)\(30\!\cdots\!80\)\( \beta_{3} - \)\(10\!\cdots\!44\)\( \beta_{4} - \)\(11\!\cdots\!36\)\( \beta_{5} + \)\(34\!\cdots\!20\)\( \beta_{6} - \)\(60\!\cdots\!80\)\( \beta_{7}) q^{41}\) \(+(\)\(81\!\cdots\!00\)\( + \)\(98\!\cdots\!08\)\( \beta_{1} - \)\(12\!\cdots\!04\)\( \beta_{2} + \)\(27\!\cdots\!84\)\( \beta_{3} + \)\(10\!\cdots\!24\)\( \beta_{4} - \)\(22\!\cdots\!48\)\( \beta_{5} - \)\(13\!\cdots\!56\)\( \beta_{6} + \)\(84\!\cdots\!00\)\( \beta_{7}) q^{42}\) \(+(-\)\(39\!\cdots\!00\)\( + \)\(15\!\cdots\!65\)\( \beta_{1} + \)\(39\!\cdots\!15\)\( \beta_{2} - \)\(32\!\cdots\!76\)\( \beta_{3} - \)\(33\!\cdots\!56\)\( \beta_{4} + \)\(15\!\cdots\!12\)\( \beta_{5} + \)\(29\!\cdots\!64\)\( \beta_{6} + \)\(20\!\cdots\!04\)\( \beta_{7}) q^{43}\) \(+(-\)\(15\!\cdots\!56\)\( - \)\(14\!\cdots\!40\)\( \beta_{1} + \)\(24\!\cdots\!92\)\( \beta_{2} - \)\(33\!\cdots\!32\)\( \beta_{3} - \)\(12\!\cdots\!00\)\( \beta_{4} - \)\(40\!\cdots\!16\)\( \beta_{5} - \)\(24\!\cdots\!84\)\( \beta_{6} + \)\(14\!\cdots\!16\)\( \beta_{7}) q^{44}\) \(+(\)\(35\!\cdots\!50\)\( - \)\(20\!\cdots\!19\)\( \beta_{1} + \)\(20\!\cdots\!37\)\( \beta_{2} + \)\(16\!\cdots\!11\)\( \beta_{3} + \)\(12\!\cdots\!87\)\( \beta_{4} - \)\(65\!\cdots\!00\)\( \beta_{5} + \)\(74\!\cdots\!00\)\( \beta_{6} - \)\(84\!\cdots\!00\)\( \beta_{7}) q^{45}\) \(+(\)\(18\!\cdots\!32\)\( - \)\(55\!\cdots\!64\)\( \beta_{1} - \)\(26\!\cdots\!36\)\( \beta_{2} + \)\(46\!\cdots\!40\)\( \beta_{3} + \)\(13\!\cdots\!76\)\( \beta_{4} + \)\(50\!\cdots\!14\)\( \beta_{5} - \)\(12\!\cdots\!00\)\( \beta_{6} + \)\(12\!\cdots\!00\)\( \beta_{7}) q^{46}\) \(+(\)\(20\!\cdots\!00\)\( - \)\(31\!\cdots\!82\)\( \beta_{1} - \)\(38\!\cdots\!00\)\( \beta_{2} - \)\(47\!\cdots\!48\)\( \beta_{3} - \)\(30\!\cdots\!38\)\( \beta_{4} + \)\(21\!\cdots\!26\)\( \beta_{5} + \)\(78\!\cdots\!22\)\( \beta_{6} + \)\(11\!\cdots\!92\)\( \beta_{7}) q^{47}\) \(+(-\)\(17\!\cdots\!00\)\( - \)\(12\!\cdots\!12\)\( \beta_{1} + \)\(69\!\cdots\!44\)\( \beta_{2} - \)\(29\!\cdots\!36\)\( \beta_{3} - \)\(76\!\cdots\!96\)\( \beta_{4} - \)\(92\!\cdots\!08\)\( \beta_{5} - \)\(26\!\cdots\!76\)\( \beta_{6} - \)\(88\!\cdots\!20\)\( \beta_{7}) q^{48}\) \(+(\)\(19\!\cdots\!57\)\( - \)\(15\!\cdots\!56\)\( \beta_{1} + \)\(31\!\cdots\!56\)\( \beta_{2} - \)\(59\!\cdots\!60\)\( \beta_{3} + \)\(21\!\cdots\!04\)\( \beta_{4} + \)\(33\!\cdots\!16\)\( \beta_{5} + \)\(43\!\cdots\!40\)\( \beta_{6} + \)\(33\!\cdots\!40\)\( \beta_{7}) q^{49}\) \(+(\)\(40\!\cdots\!00\)\( - \)\(15\!\cdots\!75\)\( \beta_{1} + \)\(18\!\cdots\!00\)\( \beta_{2} + \)\(18\!\cdots\!00\)\( \beta_{3} + \)\(83\!\cdots\!00\)\( \beta_{4} - \)\(41\!\cdots\!00\)\( \beta_{5} + \)\(44\!\cdots\!00\)\( \beta_{6} - \)\(58\!\cdots\!00\)\( \beta_{7}) q^{50}\) \(+(\)\(61\!\cdots\!32\)\( + \)\(21\!\cdots\!96\)\( \beta_{1} - \)\(33\!\cdots\!82\)\( \beta_{2} + \)\(56\!\cdots\!36\)\( \beta_{3} - \)\(22\!\cdots\!34\)\( \beta_{4} - \)\(23\!\cdots\!98\)\( \beta_{5} - \)\(44\!\cdots\!78\)\( \beta_{6} - \)\(88\!\cdots\!28\)\( \beta_{7}) q^{51}\) \(+(-\)\(67\!\cdots\!00\)\( + \)\(11\!\cdots\!54\)\( \beta_{1} - \)\(97\!\cdots\!82\)\( \beta_{2} - \)\(26\!\cdots\!38\)\( \beta_{3} - \)\(94\!\cdots\!28\)\( \beta_{4} - \)\(16\!\cdots\!44\)\( \beta_{5} + \)\(10\!\cdots\!32\)\( \beta_{6} + \)\(98\!\cdots\!52\)\( \beta_{7}) q^{52}\) \(+(\)\(11\!\cdots\!50\)\( + \)\(14\!\cdots\!17\)\( \beta_{1} - \)\(43\!\cdots\!11\)\( \beta_{2} - \)\(22\!\cdots\!69\)\( \beta_{3} + \)\(40\!\cdots\!71\)\( \beta_{4} + \)\(19\!\cdots\!08\)\( \beta_{5} - \)\(93\!\cdots\!24\)\( \beta_{6} - \)\(33\!\cdots\!16\)\( \beta_{7}) q^{53}\) \(+(-\)\(30\!\cdots\!20\)\( + \)\(22\!\cdots\!60\)\( \beta_{1} + \)\(17\!\cdots\!36\)\( \beta_{2} - \)\(30\!\cdots\!40\)\( \beta_{3} - \)\(82\!\cdots\!40\)\( \beta_{4} - \)\(62\!\cdots\!46\)\( \beta_{5} + \)\(68\!\cdots\!36\)\( \beta_{6} + \)\(50\!\cdots\!36\)\( \beta_{7}) q^{54}\) \(+(\)\(14\!\cdots\!00\)\( - \)\(31\!\cdots\!21\)\( \beta_{1} + \)\(19\!\cdots\!08\)\( \beta_{2} + \)\(54\!\cdots\!24\)\( \beta_{3} - \)\(15\!\cdots\!67\)\( \beta_{4} + \)\(90\!\cdots\!75\)\( \beta_{5} - \)\(12\!\cdots\!25\)\( \beta_{6} + \)\(75\!\cdots\!00\)\( \beta_{7}) q^{55}\) \(+(-\)\(25\!\cdots\!20\)\( - \)\(71\!\cdots\!20\)\( \beta_{1} + \)\(11\!\cdots\!12\)\( \beta_{2} - \)\(49\!\cdots\!48\)\( \beta_{3} - \)\(24\!\cdots\!60\)\( \beta_{4} - \)\(13\!\cdots\!68\)\( \beta_{5} + \)\(61\!\cdots\!28\)\( \beta_{6} - \)\(64\!\cdots\!72\)\( \beta_{7}) q^{56}\) \(+(\)\(28\!\cdots\!00\)\( - \)\(28\!\cdots\!22\)\( \beta_{1} - \)\(18\!\cdots\!02\)\( \beta_{2} - \)\(90\!\cdots\!54\)\( \beta_{3} + \)\(17\!\cdots\!26\)\( \beta_{4} + \)\(79\!\cdots\!48\)\( \beta_{5} - \)\(81\!\cdots\!44\)\( \beta_{6} + \)\(17\!\cdots\!16\)\( \beta_{7}) q^{57}\) \(+(-\)\(28\!\cdots\!00\)\( - \)\(22\!\cdots\!74\)\( \beta_{1} - \)\(41\!\cdots\!48\)\( \beta_{2} - \)\(53\!\cdots\!44\)\( \beta_{3} - \)\(57\!\cdots\!44\)\( \beta_{4} - \)\(25\!\cdots\!12\)\( \beta_{5} - \)\(31\!\cdots\!64\)\( \beta_{6} - \)\(14\!\cdots\!68\)\( \beta_{7}) q^{58}\) \(+(\)\(52\!\cdots\!40\)\( - \)\(18\!\cdots\!11\)\( \beta_{1} - \)\(44\!\cdots\!73\)\( \beta_{2} + \)\(69\!\cdots\!36\)\( \beta_{3} - \)\(29\!\cdots\!96\)\( \beta_{4} + \)\(64\!\cdots\!72\)\( \beta_{5} + \)\(17\!\cdots\!84\)\( \beta_{6} - \)\(53\!\cdots\!16\)\( \beta_{7}) q^{59}\) \(+(\)\(19\!\cdots\!00\)\( + \)\(23\!\cdots\!44\)\( \beta_{1} - \)\(12\!\cdots\!12\)\( \beta_{2} + \)\(64\!\cdots\!64\)\( \beta_{3} - \)\(26\!\cdots\!92\)\( \beta_{4} + \)\(18\!\cdots\!40\)\( \beta_{5} - \)\(31\!\cdots\!20\)\( \beta_{6} + \)\(22\!\cdots\!80\)\( \beta_{7}) q^{60}\) \(+(-\)\(37\!\cdots\!58\)\( + \)\(15\!\cdots\!77\)\( \beta_{1} + \)\(93\!\cdots\!89\)\( \beta_{2} - \)\(57\!\cdots\!93\)\( \beta_{3} + \)\(11\!\cdots\!67\)\( \beta_{4} - \)\(48\!\cdots\!04\)\( \beta_{5} - \)\(21\!\cdots\!08\)\( \beta_{6} - \)\(35\!\cdots\!08\)\( \beta_{7}) q^{61}\) \(+(-\)\(12\!\cdots\!00\)\( + \)\(64\!\cdots\!96\)\( \beta_{1} + \)\(21\!\cdots\!40\)\( \beta_{2} - \)\(36\!\cdots\!36\)\( \beta_{3} - \)\(11\!\cdots\!96\)\( \beta_{4} + \)\(28\!\cdots\!92\)\( \beta_{5} + \)\(11\!\cdots\!24\)\( \beta_{6} - \)\(92\!\cdots\!80\)\( \beta_{7}) q^{62}\) \(+(\)\(18\!\cdots\!00\)\( + \)\(15\!\cdots\!59\)\( \beta_{1} - \)\(26\!\cdots\!16\)\( \beta_{2} + \)\(10\!\cdots\!92\)\( \beta_{3} + \)\(47\!\cdots\!97\)\( \beta_{4} + \)\(21\!\cdots\!31\)\( \beta_{5} - \)\(14\!\cdots\!93\)\( \beta_{6} + \)\(13\!\cdots\!48\)\( \beta_{7}) q^{63}\) \(+(\)\(43\!\cdots\!12\)\( - \)\(91\!\cdots\!12\)\( \beta_{1} - \)\(10\!\cdots\!04\)\( \beta_{2} + \)\(14\!\cdots\!92\)\( \beta_{3} - \)\(11\!\cdots\!32\)\( \beta_{4} + \)\(32\!\cdots\!52\)\( \beta_{5} - \)\(12\!\cdots\!00\)\( \beta_{6} - \)\(14\!\cdots\!00\)\( \beta_{7}) q^{64}\) \(+(\)\(59\!\cdots\!00\)\( - \)\(11\!\cdots\!68\)\( \beta_{1} - \)\(23\!\cdots\!36\)\( \beta_{2} + \)\(25\!\cdots\!92\)\( \beta_{3} + \)\(38\!\cdots\!24\)\( \beta_{4} - \)\(12\!\cdots\!80\)\( \beta_{5} - \)\(41\!\cdots\!60\)\( \beta_{6} - \)\(19\!\cdots\!60\)\( \beta_{7}) q^{65}\) \(+(-\)\(40\!\cdots\!36\)\( - \)\(35\!\cdots\!72\)\( \beta_{1} + \)\(48\!\cdots\!24\)\( \beta_{2} - \)\(10\!\cdots\!32\)\( \beta_{3} - \)\(34\!\cdots\!12\)\( \beta_{4} + \)\(28\!\cdots\!76\)\( \beta_{5} + \)\(48\!\cdots\!56\)\( \beta_{6} - \)\(76\!\cdots\!44\)\( \beta_{7}) q^{66}\) \(+(-\)\(30\!\cdots\!00\)\( - \)\(18\!\cdots\!79\)\( \beta_{1} + \)\(65\!\cdots\!09\)\( \beta_{2} - \)\(14\!\cdots\!28\)\( \beta_{3} + \)\(17\!\cdots\!62\)\( \beta_{4} + \)\(67\!\cdots\!26\)\( \beta_{5} - \)\(10\!\cdots\!78\)\( \beta_{6} + \)\(44\!\cdots\!96\)\( \beta_{7}) q^{67}\) \(+(-\)\(28\!\cdots\!00\)\( - \)\(72\!\cdots\!06\)\( \beta_{1} + \)\(34\!\cdots\!54\)\( \beta_{2} - \)\(21\!\cdots\!26\)\( \beta_{3} - \)\(25\!\cdots\!16\)\( \beta_{4} - \)\(12\!\cdots\!68\)\( \beta_{5} - \)\(59\!\cdots\!96\)\( \beta_{6} - \)\(12\!\cdots\!44\)\( \beta_{7}) q^{68}\) \(+(\)\(26\!\cdots\!76\)\( + \)\(23\!\cdots\!40\)\( \beta_{1} - \)\(27\!\cdots\!20\)\( \beta_{2} + \)\(10\!\cdots\!88\)\( \beta_{3} + \)\(58\!\cdots\!80\)\( \beta_{4} - \)\(73\!\cdots\!04\)\( \beta_{5} + \)\(74\!\cdots\!24\)\( \beta_{6} - \)\(13\!\cdots\!76\)\( \beta_{7}) q^{69}\) \(+(-\)\(48\!\cdots\!00\)\( + \)\(12\!\cdots\!68\)\( \beta_{1} - \)\(22\!\cdots\!64\)\( \beta_{2} + \)\(83\!\cdots\!08\)\( \beta_{3} + \)\(28\!\cdots\!36\)\( \beta_{4} + \)\(38\!\cdots\!00\)\( \beta_{5} - \)\(13\!\cdots\!00\)\( \beta_{6} + \)\(92\!\cdots\!00\)\( \beta_{7}) q^{70}\) \(+(-\)\(22\!\cdots\!68\)\( + \)\(65\!\cdots\!91\)\( \beta_{1} - \)\(12\!\cdots\!88\)\( \beta_{2} - \)\(10\!\cdots\!44\)\( \beta_{3} - \)\(10\!\cdots\!39\)\( \beta_{4} + \)\(93\!\cdots\!43\)\( \beta_{5} - \)\(13\!\cdots\!89\)\( \beta_{6} - \)\(24\!\cdots\!64\)\( \beta_{7}) q^{71}\) \(+(-\)\(11\!\cdots\!00\)\( - \)\(68\!\cdots\!56\)\( \beta_{1} + \)\(49\!\cdots\!89\)\( \beta_{2} - \)\(16\!\cdots\!92\)\( \beta_{3} - \)\(21\!\cdots\!37\)\( \beta_{4} - \)\(18\!\cdots\!26\)\( \beta_{5} + \)\(27\!\cdots\!53\)\( \beta_{6} + \)\(11\!\cdots\!15\)\( \beta_{7}) q^{72}\) \(+(-\)\(12\!\cdots\!50\)\( - \)\(21\!\cdots\!18\)\( \beta_{1} + \)\(12\!\cdots\!82\)\( \beta_{2} + \)\(12\!\cdots\!58\)\( \beta_{3} - \)\(42\!\cdots\!82\)\( \beta_{4} - \)\(20\!\cdots\!36\)\( \beta_{5} + \)\(10\!\cdots\!08\)\( \beta_{6} + \)\(89\!\cdots\!44\)\( \beta_{7}) q^{73}\) \(+(-\)\(16\!\cdots\!36\)\( - \)\(20\!\cdots\!62\)\( \beta_{1} + \)\(99\!\cdots\!28\)\( \beta_{2} + \)\(49\!\cdots\!92\)\( \beta_{3} + \)\(44\!\cdots\!68\)\( \beta_{4} + \)\(80\!\cdots\!00\)\( \beta_{5} - \)\(86\!\cdots\!48\)\( \beta_{6} - \)\(23\!\cdots\!48\)\( \beta_{7}) q^{74}\) \(+(-\)\(73\!\cdots\!00\)\( - \)\(19\!\cdots\!75\)\( \beta_{1} - \)\(20\!\cdots\!25\)\( \beta_{2} + \)\(74\!\cdots\!00\)\( \beta_{3} - \)\(59\!\cdots\!00\)\( \beta_{4} - \)\(20\!\cdots\!00\)\( \beta_{5} - \)\(27\!\cdots\!00\)\( \beta_{6} + \)\(12\!\cdots\!00\)\( \beta_{7}) q^{75}\) \(+(-\)\(28\!\cdots\!40\)\( - \)\(65\!\cdots\!20\)\( \beta_{1} - \)\(78\!\cdots\!36\)\( \beta_{2} - \)\(38\!\cdots\!16\)\( \beta_{3} - \)\(24\!\cdots\!00\)\( \beta_{4} + \)\(11\!\cdots\!44\)\( \beta_{5} + \)\(14\!\cdots\!56\)\( \beta_{6} + \)\(54\!\cdots\!56\)\( \beta_{7}) q^{76}\) \(+(-\)\(55\!\cdots\!00\)\( + \)\(45\!\cdots\!96\)\( \beta_{1} - \)\(11\!\cdots\!12\)\( \beta_{2} - \)\(31\!\cdots\!64\)\( \beta_{3} - \)\(90\!\cdots\!04\)\( \beta_{4} - \)\(11\!\cdots\!92\)\( \beta_{5} - \)\(14\!\cdots\!24\)\( \beta_{6} - \)\(10\!\cdots\!40\)\( \beta_{7}) q^{77}\) \(+(-\)\(93\!\cdots\!00\)\( + \)\(24\!\cdots\!56\)\( \beta_{1} - \)\(16\!\cdots\!20\)\( \beta_{2} + \)\(10\!\cdots\!96\)\( \beta_{3} + \)\(58\!\cdots\!16\)\( \beta_{4} + \)\(18\!\cdots\!18\)\( \beta_{5} - \)\(34\!\cdots\!04\)\( \beta_{6} - \)\(50\!\cdots\!72\)\( \beta_{7}) q^{78}\) \(+(-\)\(11\!\cdots\!80\)\( + \)\(52\!\cdots\!10\)\( \beta_{1} + \)\(81\!\cdots\!20\)\( \beta_{2} + \)\(29\!\cdots\!92\)\( \beta_{3} - \)\(98\!\cdots\!30\)\( \beta_{4} + \)\(25\!\cdots\!14\)\( \beta_{5} + \)\(10\!\cdots\!66\)\( \beta_{6} + \)\(30\!\cdots\!16\)\( \beta_{7}) q^{79}\) \(+(-\)\(67\!\cdots\!00\)\( + \)\(33\!\cdots\!32\)\( \beta_{1} + \)\(24\!\cdots\!64\)\( \beta_{2} - \)\(42\!\cdots\!08\)\( \beta_{3} - \)\(18\!\cdots\!36\)\( \beta_{4} - \)\(85\!\cdots\!00\)\( \beta_{5} - \)\(39\!\cdots\!00\)\( \beta_{6} + \)\(30\!\cdots\!00\)\( \beta_{7}) q^{80}\) \(+(-\)\(80\!\cdots\!19\)\( - \)\(81\!\cdots\!54\)\( \beta_{1} - \)\(27\!\cdots\!86\)\( \beta_{2} + \)\(35\!\cdots\!42\)\( \beta_{3} + \)\(98\!\cdots\!46\)\( \beta_{4} - \)\(54\!\cdots\!32\)\( \beta_{5} - \)\(97\!\cdots\!24\)\( \beta_{6} - \)\(24\!\cdots\!24\)\( \beta_{7}) q^{81}\) \(+(-\)\(23\!\cdots\!00\)\( - \)\(11\!\cdots\!06\)\( \beta_{1} - \)\(70\!\cdots\!24\)\( \beta_{2} - \)\(20\!\cdots\!08\)\( \beta_{3} - \)\(49\!\cdots\!28\)\( \beta_{4} + \)\(68\!\cdots\!56\)\( \beta_{5} - \)\(24\!\cdots\!68\)\( \beta_{6} + \)\(27\!\cdots\!88\)\( \beta_{7}) q^{82}\) \(+(-\)\(10\!\cdots\!00\)\( - \)\(20\!\cdots\!37\)\( \beta_{1} - \)\(22\!\cdots\!27\)\( \beta_{2} + \)\(37\!\cdots\!60\)\( \beta_{3} + \)\(15\!\cdots\!60\)\( \beta_{4} + \)\(67\!\cdots\!80\)\( \beta_{5} + \)\(93\!\cdots\!60\)\( \beta_{6} + \)\(71\!\cdots\!60\)\( \beta_{7}) q^{83}\) \(+(-\)\(73\!\cdots\!96\)\( - \)\(23\!\cdots\!52\)\( \beta_{1} + \)\(25\!\cdots\!44\)\( \beta_{2} - \)\(33\!\cdots\!84\)\( \beta_{3} - \)\(23\!\cdots\!52\)\( \beta_{4} - \)\(16\!\cdots\!08\)\( \beta_{5} + \)\(79\!\cdots\!80\)\( \beta_{6} - \)\(27\!\cdots\!20\)\( \beta_{7}) q^{84}\) \(+(-\)\(58\!\cdots\!00\)\( + \)\(67\!\cdots\!34\)\( \beta_{1} + \)\(10\!\cdots\!18\)\( \beta_{2} + \)\(12\!\cdots\!54\)\( \beta_{3} - \)\(11\!\cdots\!62\)\( \beta_{4} - \)\(46\!\cdots\!60\)\( \beta_{5} - \)\(67\!\cdots\!20\)\( \beta_{6} + \)\(31\!\cdots\!80\)\( \beta_{7}) q^{85}\) \(+(-\)\(14\!\cdots\!28\)\( + \)\(26\!\cdots\!80\)\( \beta_{1} + \)\(10\!\cdots\!64\)\( \beta_{2} - \)\(16\!\cdots\!54\)\( \beta_{3} - \)\(22\!\cdots\!90\)\( \beta_{4} + \)\(50\!\cdots\!63\)\( \beta_{5} + \)\(83\!\cdots\!52\)\( \beta_{6} + \)\(28\!\cdots\!52\)\( \beta_{7}) q^{86}\) \(+(-\)\(18\!\cdots\!00\)\( + \)\(19\!\cdots\!37\)\( \beta_{1} + \)\(88\!\cdots\!16\)\( \beta_{2} - \)\(16\!\cdots\!40\)\( \beta_{3} + \)\(16\!\cdots\!55\)\( \beta_{4} - \)\(46\!\cdots\!35\)\( \beta_{5} + \)\(57\!\cdots\!05\)\( \beta_{6} - \)\(15\!\cdots\!24\)\( \beta_{7}) q^{87}\) \(+(\)\(19\!\cdots\!00\)\( + \)\(28\!\cdots\!40\)\( \beta_{1} - \)\(40\!\cdots\!64\)\( \beta_{2} + \)\(10\!\cdots\!44\)\( \beta_{3} + \)\(78\!\cdots\!84\)\( \beta_{4} + \)\(61\!\cdots\!32\)\( \beta_{5} - \)\(19\!\cdots\!96\)\( \beta_{6} + \)\(28\!\cdots\!60\)\( \beta_{7}) q^{88}\) \(+(\)\(67\!\cdots\!10\)\( + \)\(39\!\cdots\!66\)\( \beta_{1} - \)\(34\!\cdots\!54\)\( \beta_{2} - \)\(75\!\cdots\!70\)\( \beta_{3} - \)\(68\!\cdots\!94\)\( \beta_{4} - \)\(31\!\cdots\!68\)\( \beta_{5} + \)\(80\!\cdots\!52\)\( \beta_{6} - \)\(25\!\cdots\!48\)\( \beta_{7}) q^{89}\) \(+(\)\(20\!\cdots\!00\)\( - \)\(96\!\cdots\!58\)\( \beta_{1} - \)\(69\!\cdots\!16\)\( \beta_{2} + \)\(28\!\cdots\!52\)\( \beta_{3} + \)\(34\!\cdots\!44\)\( \beta_{4} + \)\(29\!\cdots\!20\)\( \beta_{5} - \)\(39\!\cdots\!60\)\( \beta_{6} - \)\(28\!\cdots\!60\)\( \beta_{7}) q^{90}\) \(+(\)\(24\!\cdots\!92\)\( - \)\(20\!\cdots\!40\)\( \beta_{1} + \)\(48\!\cdots\!32\)\( \beta_{2} - \)\(30\!\cdots\!88\)\( \beta_{3} - \)\(84\!\cdots\!80\)\( \beta_{4} + \)\(10\!\cdots\!72\)\( \beta_{5} + \)\(17\!\cdots\!08\)\( \beta_{6} + \)\(19\!\cdots\!08\)\( \beta_{7}) q^{91}\) \(+(\)\(49\!\cdots\!00\)\( - \)\(38\!\cdots\!56\)\( \beta_{1} + \)\(58\!\cdots\!48\)\( \beta_{2} - \)\(65\!\cdots\!48\)\( \beta_{3} + \)\(53\!\cdots\!32\)\( \beta_{4} - \)\(23\!\cdots\!64\)\( \beta_{5} - \)\(92\!\cdots\!08\)\( \beta_{6} - \)\(41\!\cdots\!52\)\( \beta_{7}) q^{92}\) \(+(\)\(71\!\cdots\!00\)\( - \)\(16\!\cdots\!96\)\( \beta_{1} + \)\(18\!\cdots\!56\)\( \beta_{2} - \)\(12\!\cdots\!76\)\( \beta_{3} - \)\(37\!\cdots\!96\)\( \beta_{4} + \)\(23\!\cdots\!92\)\( \beta_{5} - \)\(64\!\cdots\!76\)\( \beta_{6} + \)\(17\!\cdots\!92\)\( \beta_{7}) q^{93}\) \(+(\)\(30\!\cdots\!84\)\( + \)\(32\!\cdots\!00\)\( \beta_{1} - \)\(98\!\cdots\!44\)\( \beta_{2} + \)\(20\!\cdots\!60\)\( \beta_{3} - \)\(10\!\cdots\!00\)\( \beta_{4} + \)\(10\!\cdots\!04\)\( \beta_{5} + \)\(90\!\cdots\!96\)\( \beta_{6} + \)\(11\!\cdots\!96\)\( \beta_{7}) q^{94}\) \(+(\)\(46\!\cdots\!00\)\( + \)\(13\!\cdots\!45\)\( \beta_{1} - \)\(22\!\cdots\!60\)\( \beta_{2} + \)\(58\!\cdots\!20\)\( \beta_{3} + \)\(55\!\cdots\!15\)\( \beta_{4} + \)\(57\!\cdots\!25\)\( \beta_{5} + \)\(12\!\cdots\!25\)\( \beta_{6} - \)\(24\!\cdots\!00\)\( \beta_{7}) q^{95}\) \(+(\)\(76\!\cdots\!52\)\( + \)\(22\!\cdots\!72\)\( \beta_{1} - \)\(25\!\cdots\!20\)\( \beta_{2} + \)\(63\!\cdots\!36\)\( \beta_{3} + \)\(87\!\cdots\!32\)\( \beta_{4} - \)\(47\!\cdots\!72\)\( \beta_{5} - \)\(21\!\cdots\!80\)\( \beta_{6} + \)\(68\!\cdots\!20\)\( \beta_{7}) q^{96}\) \(+(\)\(38\!\cdots\!50\)\( + \)\(43\!\cdots\!14\)\( \beta_{1} - \)\(63\!\cdots\!86\)\( \beta_{2} - \)\(10\!\cdots\!74\)\( \beta_{3} + \)\(62\!\cdots\!66\)\( \beta_{4} - \)\(19\!\cdots\!32\)\( \beta_{5} - \)\(67\!\cdots\!04\)\( \beta_{6} + \)\(38\!\cdots\!04\)\( \beta_{7}) q^{97}\) \(+(\)\(15\!\cdots\!00\)\( - \)\(16\!\cdots\!93\)\( \beta_{1} + \)\(26\!\cdots\!44\)\( \beta_{2} - \)\(19\!\cdots\!52\)\( \beta_{3} - \)\(12\!\cdots\!32\)\( \beta_{4} + \)\(22\!\cdots\!64\)\( \beta_{5} + \)\(17\!\cdots\!08\)\( \beta_{6} - \)\(29\!\cdots\!68\)\( \beta_{7}) q^{98}\) \(+(\)\(89\!\cdots\!16\)\( - \)\(62\!\cdots\!35\)\( \beta_{1} + \)\(91\!\cdots\!07\)\( \beta_{2} - \)\(20\!\cdots\!68\)\( \beta_{3} - \)\(14\!\cdots\!00\)\( \beta_{4} - \)\(10\!\cdots\!48\)\( \beta_{5} + \)\(44\!\cdots\!48\)\( \beta_{6} - \)\(58\!\cdots\!52\)\( \beta_{7}) q^{99}\) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(8q \) \(\mathstrut +\mathstrut 2288392606915200q^{2} \) \(\mathstrut -\mathstrut \)\(79\!\cdots\!00\)\(q^{3} \) \(\mathstrut +\mathstrut \)\(25\!\cdots\!56\)\(q^{4} \) \(\mathstrut -\mathstrut \)\(21\!\cdots\!00\)\(q^{5} \) \(\mathstrut +\mathstrut \)\(65\!\cdots\!36\)\(q^{6} \) \(\mathstrut +\mathstrut \)\(23\!\cdots\!00\)\(q^{7} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!00\)\(q^{8} \) \(\mathstrut +\mathstrut \)\(24\!\cdots\!84\)\(q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(8q \) \(\mathstrut +\mathstrut 2288392606915200q^{2} \) \(\mathstrut -\mathstrut \)\(79\!\cdots\!00\)\(q^{3} \) \(\mathstrut +\mathstrut \)\(25\!\cdots\!56\)\(q^{4} \) \(\mathstrut -\mathstrut \)\(21\!\cdots\!00\)\(q^{5} \) \(\mathstrut +\mathstrut \)\(65\!\cdots\!36\)\(q^{6} \) \(\mathstrut +\mathstrut \)\(23\!\cdots\!00\)\(q^{7} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!00\)\(q^{8} \) \(\mathstrut +\mathstrut \)\(24\!\cdots\!84\)\(q^{9} \) \(\mathstrut +\mathstrut \)\(18\!\cdots\!00\)\(q^{10} \) \(\mathstrut -\mathstrut \)\(16\!\cdots\!64\)\(q^{11} \) \(\mathstrut -\mathstrut \)\(91\!\cdots\!00\)\(q^{12} \) \(\mathstrut -\mathstrut \)\(71\!\cdots\!00\)\(q^{13} \) \(\mathstrut -\mathstrut \)\(50\!\cdots\!68\)\(q^{14} \) \(\mathstrut +\mathstrut \)\(23\!\cdots\!00\)\(q^{15} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!68\)\(q^{16} \) \(\mathstrut -\mathstrut \)\(86\!\cdots\!00\)\(q^{17} \) \(\mathstrut -\mathstrut \)\(59\!\cdots\!00\)\(q^{18} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!60\)\(q^{19} \) \(\mathstrut -\mathstrut \)\(31\!\cdots\!00\)\(q^{20} \) \(\mathstrut -\mathstrut \)\(27\!\cdots\!24\)\(q^{21} \) \(\mathstrut +\mathstrut \)\(14\!\cdots\!00\)\(q^{22} \) \(\mathstrut +\mathstrut \)\(53\!\cdots\!00\)\(q^{23} \) \(\mathstrut +\mathstrut \)\(53\!\cdots\!20\)\(q^{24} \) \(\mathstrut +\mathstrut \)\(18\!\cdots\!00\)\(q^{25} \) \(\mathstrut -\mathstrut \)\(43\!\cdots\!04\)\(q^{26} \) \(\mathstrut -\mathstrut \)\(27\!\cdots\!00\)\(q^{27} \) \(\mathstrut +\mathstrut \)\(31\!\cdots\!00\)\(q^{28} \) \(\mathstrut +\mathstrut \)\(29\!\cdots\!60\)\(q^{29} \) \(\mathstrut -\mathstrut \)\(75\!\cdots\!00\)\(q^{30} \) \(\mathstrut -\mathstrut \)\(14\!\cdots\!24\)\(q^{31} \) \(\mathstrut -\mathstrut \)\(59\!\cdots\!00\)\(q^{32} \) \(\mathstrut +\mathstrut \)\(98\!\cdots\!00\)\(q^{33} \) \(\mathstrut +\mathstrut \)\(23\!\cdots\!92\)\(q^{34} \) \(\mathstrut -\mathstrut \)\(28\!\cdots\!00\)\(q^{35} \) \(\mathstrut +\mathstrut \)\(19\!\cdots\!88\)\(q^{36} \) \(\mathstrut +\mathstrut \)\(33\!\cdots\!00\)\(q^{37} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!00\)\(q^{38} \) \(\mathstrut -\mathstrut \)\(77\!\cdots\!72\)\(q^{39} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!00\)\(q^{40} \) \(\mathstrut +\mathstrut \)\(26\!\cdots\!96\)\(q^{41} \) \(\mathstrut +\mathstrut \)\(65\!\cdots\!00\)\(q^{42} \) \(\mathstrut -\mathstrut \)\(31\!\cdots\!00\)\(q^{43} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!48\)\(q^{44} \) \(\mathstrut +\mathstrut \)\(28\!\cdots\!00\)\(q^{45} \) \(\mathstrut +\mathstrut \)\(14\!\cdots\!56\)\(q^{46} \) \(\mathstrut +\mathstrut \)\(16\!\cdots\!00\)\(q^{47} \) \(\mathstrut -\mathstrut \)\(14\!\cdots\!00\)\(q^{48} \) \(\mathstrut +\mathstrut \)\(15\!\cdots\!56\)\(q^{49} \) \(\mathstrut +\mathstrut \)\(32\!\cdots\!00\)\(q^{50} \) \(\mathstrut +\mathstrut \)\(48\!\cdots\!56\)\(q^{51} \) \(\mathstrut -\mathstrut \)\(54\!\cdots\!00\)\(q^{52} \) \(\mathstrut +\mathstrut \)\(90\!\cdots\!00\)\(q^{53} \) \(\mathstrut -\mathstrut \)\(24\!\cdots\!60\)\(q^{54} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!00\)\(q^{55} \) \(\mathstrut -\mathstrut \)\(20\!\cdots\!60\)\(q^{56} \) \(\mathstrut +\mathstrut \)\(22\!\cdots\!00\)\(q^{57} \) \(\mathstrut -\mathstrut \)\(22\!\cdots\!00\)\(q^{58} \) \(\mathstrut +\mathstrut \)\(42\!\cdots\!20\)\(q^{59} \) \(\mathstrut +\mathstrut \)\(15\!\cdots\!00\)\(q^{60} \) \(\mathstrut -\mathstrut \)\(30\!\cdots\!64\)\(q^{61} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!00\)\(q^{62} \) \(\mathstrut +\mathstrut \)\(14\!\cdots\!00\)\(q^{63} \) \(\mathstrut +\mathstrut \)\(35\!\cdots\!96\)\(q^{64} \) \(\mathstrut +\mathstrut \)\(47\!\cdots\!00\)\(q^{65} \) \(\mathstrut -\mathstrut \)\(32\!\cdots\!88\)\(q^{66} \) \(\mathstrut -\mathstrut \)\(24\!\cdots\!00\)\(q^{67} \) \(\mathstrut -\mathstrut \)\(22\!\cdots\!00\)\(q^{68} \) \(\mathstrut +\mathstrut \)\(20\!\cdots\!08\)\(q^{69} \) \(\mathstrut -\mathstrut \)\(38\!\cdots\!00\)\(q^{70} \) \(\mathstrut -\mathstrut \)\(18\!\cdots\!44\)\(q^{71} \) \(\mathstrut -\mathstrut \)\(93\!\cdots\!00\)\(q^{72} \) \(\mathstrut -\mathstrut \)\(99\!\cdots\!00\)\(q^{73} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!88\)\(q^{74} \) \(\mathstrut -\mathstrut \)\(58\!\cdots\!00\)\(q^{75} \) \(\mathstrut -\mathstrut \)\(23\!\cdots\!20\)\(q^{76} \) \(\mathstrut -\mathstrut \)\(44\!\cdots\!00\)\(q^{77} \) \(\mathstrut -\mathstrut \)\(74\!\cdots\!00\)\(q^{78} \) \(\mathstrut -\mathstrut \)\(90\!\cdots\!40\)\(q^{79} \) \(\mathstrut -\mathstrut \)\(53\!\cdots\!00\)\(q^{80} \) \(\mathstrut -\mathstrut \)\(64\!\cdots\!52\)\(q^{81} \) \(\mathstrut -\mathstrut \)\(18\!\cdots\!00\)\(q^{82} \) \(\mathstrut -\mathstrut \)\(86\!\cdots\!00\)\(q^{83} \) \(\mathstrut -\mathstrut \)\(58\!\cdots\!68\)\(q^{84} \) \(\mathstrut -\mathstrut \)\(46\!\cdots\!00\)\(q^{85} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!24\)\(q^{86} \) \(\mathstrut -\mathstrut \)\(14\!\cdots\!00\)\(q^{87} \) \(\mathstrut +\mathstrut \)\(15\!\cdots\!00\)\(q^{88} \) \(\mathstrut +\mathstrut \)\(53\!\cdots\!80\)\(q^{89} \) \(\mathstrut +\mathstrut \)\(16\!\cdots\!00\)\(q^{90} \) \(\mathstrut +\mathstrut \)\(19\!\cdots\!36\)\(q^{91} \) \(\mathstrut +\mathstrut \)\(39\!\cdots\!00\)\(q^{92} \) \(\mathstrut +\mathstrut \)\(56\!\cdots\!00\)\(q^{93} \) \(\mathstrut +\mathstrut \)\(24\!\cdots\!72\)\(q^{94} \) \(\mathstrut +\mathstrut \)\(37\!\cdots\!00\)\(q^{95} \) \(\mathstrut +\mathstrut \)\(61\!\cdots\!16\)\(q^{96} \) \(\mathstrut +\mathstrut \)\(31\!\cdots\!00\)\(q^{97} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!00\)\(q^{98} \) \(\mathstrut +\mathstrut \)\(71\!\cdots\!28\)\(q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8}\mathstrut -\mathstrut \) \(2\) \(x^{7}\mathstrut -\mathstrut \) \(10\!\cdots\!01\) \(x^{6}\mathstrut -\mathstrut \) \(72\!\cdots\!28\) \(x^{5}\mathstrut +\mathstrut \) \(31\!\cdots\!56\) \(x^{4}\mathstrut +\mathstrut \) \(37\!\cdots\!80\) \(x^{3}\mathstrut -\mathstrut \) \(26\!\cdots\!00\) \(x^{2}\mathstrut -\mathstrut \) \(71\!\cdots\!00\) \(x\mathstrut +\mathstrut \) \(46\!\cdots\!00\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 192 \nu - 48 \)
\(\beta_{2}\)\(=\)\((\)\(-\)\(18\!\cdots\!09\) \(\nu^{7}\mathstrut +\mathstrut \) \(28\!\cdots\!72\) \(\nu^{6}\mathstrut +\mathstrut \) \(15\!\cdots\!61\) \(\nu^{5}\mathstrut -\mathstrut \) \(22\!\cdots\!90\) \(\nu^{4}\mathstrut -\mathstrut \) \(26\!\cdots\!64\) \(\nu^{3}\mathstrut +\mathstrut \) \(33\!\cdots\!04\) \(\nu^{2}\mathstrut +\mathstrut \) \(52\!\cdots\!64\) \(\nu\mathstrut -\mathstrut \) \(59\!\cdots\!64\)\()/\)\(10\!\cdots\!76\)
\(\beta_{3}\)\(=\)\((\)\(-\)\(19\!\cdots\!01\) \(\nu^{7}\mathstrut +\mathstrut \) \(29\!\cdots\!08\) \(\nu^{6}\mathstrut +\mathstrut \) \(16\!\cdots\!29\) \(\nu^{5}\mathstrut -\mathstrut \) \(23\!\cdots\!10\) \(\nu^{4}\mathstrut -\mathstrut \) \(27\!\cdots\!96\) \(\nu^{3}\mathstrut +\mathstrut \) \(35\!\cdots\!08\) \(\nu^{2}\mathstrut +\mathstrut \) \(49\!\cdots\!00\) \(\nu\mathstrut -\mathstrut \) \(20\!\cdots\!36\)\()/\)\(15\!\cdots\!68\)
\(\beta_{4}\)\(=\)\((\)\(-\)\(25\!\cdots\!07\) \(\nu^{7}\mathstrut +\mathstrut \) \(86\!\cdots\!88\) \(\nu^{6}\mathstrut +\mathstrut \) \(24\!\cdots\!91\) \(\nu^{5}\mathstrut -\mathstrut \) \(34\!\cdots\!66\) \(\nu^{4}\mathstrut -\mathstrut \) \(58\!\cdots\!80\) \(\nu^{3}\mathstrut -\mathstrut \) \(91\!\cdots\!80\) \(\nu^{2}\mathstrut +\mathstrut \) \(31\!\cdots\!00\) \(\nu\mathstrut +\mathstrut \) \(99\!\cdots\!60\)\()/\)\(64\!\cdots\!40\)
\(\beta_{5}\)\(=\)\((\)\(-\)\(37\!\cdots\!77\) \(\nu^{7}\mathstrut +\mathstrut \) \(56\!\cdots\!08\) \(\nu^{6}\mathstrut +\mathstrut \) \(31\!\cdots\!21\) \(\nu^{5}\mathstrut -\mathstrut \) \(44\!\cdots\!06\) \(\nu^{4}\mathstrut -\mathstrut \) \(54\!\cdots\!20\) \(\nu^{3}\mathstrut +\mathstrut \) \(66\!\cdots\!20\) \(\nu^{2}\mathstrut +\mathstrut \) \(23\!\cdots\!20\) \(\nu\mathstrut -\mathstrut \) \(11\!\cdots\!00\)\()/\)\(64\!\cdots\!40\)
\(\beta_{6}\)\(=\)\((\)\(-\)\(29\!\cdots\!71\) \(\nu^{7}\mathstrut +\mathstrut \) \(42\!\cdots\!44\) \(\nu^{6}\mathstrut +\mathstrut \) \(24\!\cdots\!63\) \(\nu^{5}\mathstrut -\mathstrut \) \(33\!\cdots\!98\) \(\nu^{4}\mathstrut -\mathstrut \) \(44\!\cdots\!60\) \(\nu^{3}\mathstrut +\mathstrut \) \(49\!\cdots\!80\) \(\nu^{2}\mathstrut +\mathstrut \) \(45\!\cdots\!40\) \(\nu\mathstrut -\mathstrut \) \(92\!\cdots\!20\)\()/\)\(19\!\cdots\!20\)
\(\beta_{7}\)\(=\)\((\)\(21\!\cdots\!61\) \(\nu^{7}\mathstrut -\mathstrut \) \(32\!\cdots\!28\) \(\nu^{6}\mathstrut -\mathstrut \) \(18\!\cdots\!81\) \(\nu^{5}\mathstrut +\mathstrut \) \(25\!\cdots\!70\) \(\nu^{4}\mathstrut +\mathstrut \) \(32\!\cdots\!48\) \(\nu^{3}\mathstrut -\mathstrut \) \(38\!\cdots\!80\) \(\nu^{2}\mathstrut -\mathstrut \) \(31\!\cdots\!16\) \(\nu\mathstrut +\mathstrut \) \(76\!\cdots\!00\)\()/\)\(33\!\cdots\!76\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1}\mathstrut +\mathstrut \) \(48\)\()/192\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3}\mathstrut -\mathstrut \) \(733523\) \(\beta_{2}\mathstrut +\mathstrut \) \(1973266568918041\) \(\beta_{1}\mathstrut +\mathstrut \) \(969131934460330611034670049512448\)\()/36864\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{7}\mathstrut +\mathstrut \) \(79\) \(\beta_{6}\mathstrut +\mathstrut \) \(2011782\) \(\beta_{5}\mathstrut -\mathstrut \) \(23678718391\) \(\beta_{4}\mathstrut +\mathstrut \) \(1384179048170368\) \(\beta_{3}\mathstrut -\mathstrut \) \(37016620395754701897533\) \(\beta_{2}\mathstrut +\mathstrut \) \(1662515232343756217142566943276552\) \(\beta_{1}\mathstrut +\mathstrut \) \(1912355646808829026711061052601232496903554531328\)\()/7077888\)
\(\nu^{4}\)\(=\)\((\)\(-\)\(29719859398329\) \(\beta_{7}\mathstrut -\mathstrut \) \(145799887264158487\) \(\beta_{6}\mathstrut +\mathstrut \) \(652495846394054466346\) \(\beta_{5}\mathstrut +\mathstrut \) \(25721593923614443863307071\) \(\beta_{4}\mathstrut +\mathstrut \) \(17723063282997046737413637201312\) \(\beta_{3}\mathstrut -\mathstrut \) \(23663368907972630582218452884002380875\) \(\beta_{2}\mathstrut +\mathstrut \) \(50079391174913080622776913061670979014869966936\) \(\beta_{1}\mathstrut +\mathstrut \) \(12587473462299087857831108687417268263598920316167838249896411136\)\()/10616832\)
\(\nu^{5}\)\(=\)\((\)\(690081898468688194372948118093\) \(\beta_{7}\mathstrut +\mathstrut \) \(52712705502696217851643644636611\) \(\beta_{6}\mathstrut +\mathstrut \) \(1851396046364780165157952552942886478\) \(\beta_{5}\mathstrut -\mathstrut \) \(20736003028845115535796221761437035534859\) \(\beta_{4}\mathstrut +\mathstrut \) \(686874930443596549204656381482858041942526976\) \(\beta_{3}\mathstrut -\mathstrut \) \(40384008968747526252706688812928724407833709582813913\) \(\beta_{2}\mathstrut +\mathstrut \) \(780750341310787037771810044164864831223409858253746164631892712\) \(\beta_{1}\mathstrut +\mathstrut \) \(1516673038600256798883004165784241181727374958343599126469274830406359768956928\)\()/63700992\)
\(\nu^{6}\)\(=\)\((\)\(-\)\(33\!\cdots\!81\) \(\beta_{7}\mathstrut -\mathstrut \) \(16\!\cdots\!47\) \(\beta_{6}\mathstrut +\mathstrut \) \(81\!\cdots\!26\) \(\beta_{5}\mathstrut +\mathstrut \) \(25\!\cdots\!31\) \(\beta_{4}\mathstrut +\mathstrut \) \(13\!\cdots\!76\) \(\beta_{3}\mathstrut -\mathstrut \) \(24\!\cdots\!31\) \(\beta_{2}\mathstrut +\mathstrut \) \(41\!\cdots\!32\) \(\beta_{1}\mathstrut +\mathstrut \) \(82\!\cdots\!88\)\()/1327104\)
\(\nu^{7}\)\(=\)\((\)\(14\!\cdots\!23\) \(\beta_{7}\mathstrut +\mathstrut \) \(62\!\cdots\!45\) \(\beta_{6}\mathstrut +\mathstrut \) \(46\!\cdots\!46\) \(\beta_{5}\mathstrut -\mathstrut \) \(47\!\cdots\!81\) \(\beta_{4}\mathstrut +\mathstrut \) \(15\!\cdots\!28\) \(\beta_{3}\mathstrut -\mathstrut \) \(10\!\cdots\!07\) \(\beta_{2}\mathstrut +\mathstrut \) \(14\!\cdots\!44\) \(\beta_{1}\mathstrut +\mathstrut \) \(33\!\cdots\!64\)\()/21233664\)

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
2.47185e14
1.86979e14
1.06740e14
1.20500e13
−1.45972e13
−1.48022e14
−1.48210e14
−2.42126e14
−4.71735e16 −1.38575e26 1.57631e33 −1.39304e38 6.53705e42 1.59904e46 −4.37425e49 9.05875e51 6.57147e54
1.2 −3.56139e16 1.00458e26 6.19316e32 −8.07110e36 −3.57771e42 −7.18940e45 1.05848e48 −5.23267e49 2.87444e53
1.3 −2.02081e16 −7.88466e25 −2.40671e32 1.60701e38 1.59334e42 1.73591e45 1.79793e49 −3.92739e51 −3.24746e54
1.4 −2.02754e15 −9.96254e25 −6.44926e32 −2.14431e38 2.01995e41 −1.47165e46 2.62357e48 −2.18949e50 4.34769e53
1.5 3.08871e15 1.03355e26 −6.39497e32 −4.34627e37 3.19235e41 1.35697e46 −3.97991e48 5.38156e50 −1.34244e53
1.6 2.87062e16 1.16403e26 1.75010e32 1.23460e38 3.34150e42 −1.99654e46 −1.36075e49 3.40559e51 3.54406e54
1.7 2.87423e16 −1.13913e26 1.77081e32 7.39260e37 −3.27412e42 4.38791e45 −1.35651e49 2.83200e51 2.12480e54
1.8 4.67743e16 3.08304e25 1.53880e33 −1.66073e38 1.44207e42 8.52831e45 4.16178e49 −9.19366e51 −7.76792e54
\(n\): 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 \(S_{110}^{\mathrm{new}}(\Gamma_0(1))\).