Properties

Label 1.122.a.a
Level 1
Weight 122
Character orbit 1.a
Self dual Yes
Analytic conductor 92.717
Analytic rank 1
Dimension 9
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 1 \)
Weight: \( k \) = \( 122 \)
Character orbit: \([\chi]\) = 1.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(92.7173263878\)
Analytic rank: \(1\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: multiple of \( 2^{145}\cdot 3^{53}\cdot 5^{20}\cdot 7^{8}\cdot 11^{6}\cdot 13^{2}\cdot 17^{2} \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \(+(-260556173583835525 + \beta_{1}) q^{2}\) \(+(-\)\(50\!\cdots\!47\)\( - 5694448773 \beta_{1} + \beta_{2}) q^{3}\) \(+(\)\(94\!\cdots\!52\)\( - 331283849028211141 \beta_{1} + 3414351 \beta_{2} + \beta_{3}) q^{4}\) \(+(-\)\(21\!\cdots\!17\)\( - \)\(69\!\cdots\!36\)\( \beta_{1} + 1315393166598 \beta_{2} + 60136 \beta_{3} + \beta_{4}) q^{5}\) \(+(-\)\(18\!\cdots\!28\)\( + \)\(13\!\cdots\!86\)\( \beta_{1} - 286415241388034362 \beta_{2} - 10945374801 \beta_{3} + 10106 \beta_{4} - \beta_{5}) q^{6}\) \(+(\)\(24\!\cdots\!97\)\( - \)\(18\!\cdots\!82\)\( \beta_{1} + \)\(22\!\cdots\!26\)\( \beta_{2} + 117761204996058 \beta_{3} + 27640033 \beta_{4} - 884 \beta_{5} + \beta_{6}) q^{7}\) \(+(-\)\(72\!\cdots\!85\)\( + \)\(11\!\cdots\!62\)\( \beta_{1} - \)\(10\!\cdots\!81\)\( \beta_{2} - 520035242032518678 \beta_{3} + 611104015289 \beta_{4} - 8716729 \beta_{5} - 100 \beta_{6} + \beta_{7}) q^{8}\) \(+(\)\(88\!\cdots\!41\)\( - \)\(15\!\cdots\!83\)\( \beta_{1} - \)\(14\!\cdots\!54\)\( \beta_{2} - \)\(29\!\cdots\!47\)\( \beta_{3} + 281352546038066 \beta_{4} + 136881610 \beta_{5} + 89884 \beta_{6} - 111 \beta_{7} + \beta_{8}) q^{9}\) \(+O(q^{10})\) \( q\) \(+(-260556173583835525 + \beta_{1}) q^{2}\) \(+(-\)\(50\!\cdots\!47\)\( - 5694448773 \beta_{1} + \beta_{2}) q^{3}\) \(+(\)\(94\!\cdots\!52\)\( - 331283849028211141 \beta_{1} + 3414351 \beta_{2} + \beta_{3}) q^{4}\) \(+(-\)\(21\!\cdots\!17\)\( - \)\(69\!\cdots\!36\)\( \beta_{1} + 1315393166598 \beta_{2} + 60136 \beta_{3} + \beta_{4}) q^{5}\) \(+(-\)\(18\!\cdots\!28\)\( + \)\(13\!\cdots\!86\)\( \beta_{1} - 286415241388034362 \beta_{2} - 10945374801 \beta_{3} + 10106 \beta_{4} - \beta_{5}) q^{6}\) \(+(\)\(24\!\cdots\!97\)\( - \)\(18\!\cdots\!82\)\( \beta_{1} + \)\(22\!\cdots\!26\)\( \beta_{2} + 117761204996058 \beta_{3} + 27640033 \beta_{4} - 884 \beta_{5} + \beta_{6}) q^{7}\) \(+(-\)\(72\!\cdots\!85\)\( + \)\(11\!\cdots\!62\)\( \beta_{1} - \)\(10\!\cdots\!81\)\( \beta_{2} - 520035242032518678 \beta_{3} + 611104015289 \beta_{4} - 8716729 \beta_{5} - 100 \beta_{6} + \beta_{7}) q^{8}\) \(+(\)\(88\!\cdots\!41\)\( - \)\(15\!\cdots\!83\)\( \beta_{1} - \)\(14\!\cdots\!54\)\( \beta_{2} - \)\(29\!\cdots\!47\)\( \beta_{3} + 281352546038066 \beta_{4} + 136881610 \beta_{5} + 89884 \beta_{6} - 111 \beta_{7} + \beta_{8}) q^{9}\) \(+(-\)\(19\!\cdots\!34\)\( - \)\(28\!\cdots\!02\)\( \beta_{1} + \)\(38\!\cdots\!76\)\( \beta_{2} + \)\(86\!\cdots\!52\)\( \beta_{3} - 85411994472303248 \beta_{4} + 223590377740 \beta_{5} - 21046880 \beta_{6} - 2520 \beta_{7} - 240 \beta_{8}) q^{10}\) \(+(-\)\(10\!\cdots\!35\)\( - \)\(67\!\cdots\!59\)\( \beta_{1} + \)\(31\!\cdots\!51\)\( \beta_{2} + \)\(82\!\cdots\!84\)\( \beta_{3} - 86913908567829979798 \beta_{4} + 547339293703264 \beta_{5} - 20065513526 \beta_{6} - 45440956 \beta_{7} - 168444 \beta_{8}) q^{11}\) \(+(\)\(22\!\cdots\!92\)\( - \)\(36\!\cdots\!40\)\( \beta_{1} + \)\(33\!\cdots\!80\)\( \beta_{2} + \)\(15\!\cdots\!88\)\( \beta_{3} - \)\(21\!\cdots\!04\)\( \beta_{4} + 321442233029511384 \beta_{5} - 13036671918240 \beta_{6} - 34941366936 \beta_{7} + 2046720 \beta_{8}) q^{12}\) \(+(\)\(22\!\cdots\!79\)\( - \)\(19\!\cdots\!46\)\( \beta_{1} - \)\(72\!\cdots\!74\)\( \beta_{2} + \)\(13\!\cdots\!46\)\( \beta_{3} - \)\(15\!\cdots\!51\)\( \beta_{4} + 39150026746055710940 \beta_{5} - 7810320313768024 \beta_{6} - 3364521724746 \beta_{7} + 2252737110 \beta_{8}) q^{13}\) \(+(-\)\(72\!\cdots\!52\)\( + \)\(58\!\cdots\!32\)\( \beta_{1} + \)\(88\!\cdots\!68\)\( \beta_{2} - \)\(21\!\cdots\!02\)\( \beta_{3} + \)\(47\!\cdots\!12\)\( \beta_{4} + \)\(74\!\cdots\!86\)\( \beta_{5} - 1424711924174411648 \beta_{6} + 319490020035872 \beta_{7} - 250410186432 \beta_{8}) q^{14}\) \(+(\)\(10\!\cdots\!83\)\( + \)\(78\!\cdots\!74\)\( \beta_{1} - \)\(91\!\cdots\!62\)\( \beta_{2} - \)\(46\!\cdots\!74\)\( \beta_{3} - \)\(21\!\cdots\!49\)\( \beta_{4} + \)\(49\!\cdots\!20\)\( \beta_{5} - 42708662191974162465 \beta_{6} - 1283199651234360 \beta_{7} + 14385279460680 \beta_{8}) q^{15}\) \(+(\)\(16\!\cdots\!84\)\( - \)\(14\!\cdots\!20\)\( \beta_{1} + \)\(29\!\cdots\!44\)\( \beta_{2} + \)\(13\!\cdots\!40\)\( \beta_{3} - \)\(11\!\cdots\!60\)\( \beta_{4} + \)\(49\!\cdots\!76\)\( \beta_{5} + \)\(30\!\cdots\!64\)\( \beta_{6} - 674307616144702896 \beta_{7} - 539902754586624 \beta_{8}) q^{16}\) \(+(\)\(32\!\cdots\!46\)\( + \)\(31\!\cdots\!33\)\( \beta_{1} - \)\(10\!\cdots\!90\)\( \beta_{2} - \)\(67\!\cdots\!23\)\( \beta_{3} - \)\(11\!\cdots\!58\)\( \beta_{4} + \)\(13\!\cdots\!74\)\( \beta_{5} + \)\(34\!\cdots\!84\)\( \beta_{6} + 33600631746098946165 \beta_{7} + 13988525269125285 \beta_{8}) q^{17}\) \(+(-\)\(28\!\cdots\!37\)\( - \)\(40\!\cdots\!99\)\( \beta_{1} + \)\(73\!\cdots\!64\)\( \beta_{2} - \)\(12\!\cdots\!88\)\( \beta_{3} - \)\(23\!\cdots\!88\)\( \beta_{4} + \)\(61\!\cdots\!84\)\( \beta_{5} - \)\(37\!\cdots\!96\)\( \beta_{6} - \)\(76\!\cdots\!60\)\( \beta_{7} - 233921939232398880 \beta_{8}) q^{18}\) \(+(\)\(33\!\cdots\!51\)\( + \)\(42\!\cdots\!03\)\( \beta_{1} - \)\(54\!\cdots\!59\)\( \beta_{2} - \)\(16\!\cdots\!88\)\( \beta_{3} - \)\(67\!\cdots\!06\)\( \beta_{4} - \)\(13\!\cdots\!72\)\( \beta_{5} + \)\(53\!\cdots\!58\)\( \beta_{6} + \)\(56\!\cdots\!48\)\( \beta_{7} + 1194095236891187052 \beta_{8}) q^{19}\) \(+(\)\(50\!\cdots\!36\)\( + \)\(26\!\cdots\!38\)\( \beta_{1} - \)\(63\!\cdots\!34\)\( \beta_{2} - \)\(24\!\cdots\!38\)\( \beta_{3} - \)\(20\!\cdots\!08\)\( \beta_{4} - \)\(29\!\cdots\!00\)\( \beta_{5} + \)\(81\!\cdots\!00\)\( \beta_{6} + \)\(20\!\cdots\!00\)\( \beta_{7} + 72483187728374092800 \beta_{8}) q^{20}\) \(+(\)\(39\!\cdots\!04\)\( - \)\(49\!\cdots\!22\)\( \beta_{1} + \)\(22\!\cdots\!92\)\( \beta_{2} + \)\(77\!\cdots\!42\)\( \beta_{3} - \)\(33\!\cdots\!84\)\( \beta_{4} + \)\(59\!\cdots\!08\)\( \beta_{5} - \)\(36\!\cdots\!24\)\( \beta_{6} - \)\(83\!\cdots\!14\)\( \beta_{7} - \)\(30\!\cdots\!66\)\( \beta_{8}) q^{21}\) \(+(-\)\(21\!\cdots\!84\)\( + \)\(14\!\cdots\!26\)\( \beta_{1} - \)\(37\!\cdots\!06\)\( \beta_{2} - \)\(23\!\cdots\!51\)\( \beta_{3} + \)\(47\!\cdots\!66\)\( \beta_{4} - \)\(42\!\cdots\!95\)\( \beta_{5} + \)\(45\!\cdots\!64\)\( \beta_{6} + \)\(16\!\cdots\!96\)\( \beta_{7} + \)\(76\!\cdots\!60\)\( \beta_{8}) q^{22}\) \(+(-\)\(73\!\cdots\!73\)\( - \)\(15\!\cdots\!06\)\( \beta_{1} + \)\(12\!\cdots\!14\)\( \beta_{2} - \)\(35\!\cdots\!70\)\( \beta_{3} + \)\(70\!\cdots\!35\)\( \beta_{4} - \)\(74\!\cdots\!80\)\( \beta_{5} + \)\(48\!\cdots\!95\)\( \beta_{6} - \)\(21\!\cdots\!40\)\( \beta_{7} - \)\(14\!\cdots\!40\)\( \beta_{8}) q^{23}\) \(+(-\)\(83\!\cdots\!68\)\( + \)\(67\!\cdots\!80\)\( \beta_{1} - \)\(57\!\cdots\!16\)\( \beta_{2} - \)\(10\!\cdots\!60\)\( \beta_{3} + \)\(64\!\cdots\!32\)\( \beta_{4} + \)\(95\!\cdots\!32\)\( \beta_{5} - \)\(90\!\cdots\!84\)\( \beta_{6} + \)\(16\!\cdots\!36\)\( \beta_{7} + \)\(22\!\cdots\!24\)\( \beta_{8}) q^{24}\) \(+(-\)\(67\!\cdots\!25\)\( + \)\(19\!\cdots\!50\)\( \beta_{1} + \)\(85\!\cdots\!00\)\( \beta_{2} - \)\(92\!\cdots\!50\)\( \beta_{3} - \)\(12\!\cdots\!00\)\( \beta_{4} + \)\(57\!\cdots\!00\)\( \beta_{5} + \)\(12\!\cdots\!00\)\( \beta_{6} - \)\(14\!\cdots\!50\)\( \beta_{7} - \)\(29\!\cdots\!50\)\( \beta_{8}) q^{25}\) \(+(-\)\(76\!\cdots\!82\)\( + \)\(24\!\cdots\!38\)\( \beta_{1} - \)\(12\!\cdots\!56\)\( \beta_{2} - \)\(11\!\cdots\!08\)\( \beta_{3} - \)\(21\!\cdots\!64\)\( \beta_{4} - \)\(14\!\cdots\!04\)\( \beta_{5} - \)\(61\!\cdots\!92\)\( \beta_{6} - \)\(21\!\cdots\!32\)\( \beta_{7} + \)\(32\!\cdots\!12\)\( \beta_{8}) q^{26}\) \(+(-\)\(66\!\cdots\!80\)\( + \)\(37\!\cdots\!22\)\( \beta_{1} + \)\(10\!\cdots\!14\)\( \beta_{2} + \)\(60\!\cdots\!92\)\( \beta_{3} + \)\(11\!\cdots\!54\)\( \beta_{4} + \)\(19\!\cdots\!36\)\( \beta_{5} - \)\(52\!\cdots\!30\)\( \beta_{6} + \)\(36\!\cdots\!56\)\( \beta_{7} - \)\(30\!\cdots\!40\)\( \beta_{8}) q^{27}\) \(+(\)\(16\!\cdots\!72\)\( - \)\(86\!\cdots\!04\)\( \beta_{1} + \)\(11\!\cdots\!00\)\( \beta_{2} + \)\(11\!\cdots\!88\)\( \beta_{3} + \)\(17\!\cdots\!76\)\( \beta_{4} + \)\(13\!\cdots\!44\)\( \beta_{5} + \)\(11\!\cdots\!20\)\( \beta_{6} - \)\(37\!\cdots\!76\)\( \beta_{7} + \)\(24\!\cdots\!80\)\( \beta_{8}) q^{28}\) \(+(\)\(10\!\cdots\!39\)\( + \)\(33\!\cdots\!08\)\( \beta_{1} - \)\(11\!\cdots\!30\)\( \beta_{2} + \)\(30\!\cdots\!52\)\( \beta_{3} - \)\(55\!\cdots\!71\)\( \beta_{4} - \)\(11\!\cdots\!96\)\( \beta_{5} - \)\(88\!\cdots\!48\)\( \beta_{6} + \)\(25\!\cdots\!12\)\( \beta_{7} - \)\(15\!\cdots\!12\)\( \beta_{8}) q^{29}\) \(+(\)\(24\!\cdots\!76\)\( - \)\(35\!\cdots\!92\)\( \beta_{1} - \)\(13\!\cdots\!44\)\( \beta_{2} + \)\(17\!\cdots\!42\)\( \beta_{3} - \)\(26\!\cdots\!28\)\( \beta_{4} - \)\(76\!\cdots\!50\)\( \beta_{5} + \)\(15\!\cdots\!00\)\( \beta_{6} - \)\(91\!\cdots\!00\)\( \beta_{7} + \)\(66\!\cdots\!00\)\( \beta_{8}) q^{30}\) \(+(-\)\(11\!\cdots\!36\)\( + \)\(52\!\cdots\!48\)\( \beta_{1} - \)\(11\!\cdots\!92\)\( \beta_{2} - \)\(18\!\cdots\!48\)\( \beta_{3} + \)\(32\!\cdots\!80\)\( \beta_{4} + \)\(62\!\cdots\!24\)\( \beta_{5} + \)\(34\!\cdots\!16\)\( \beta_{6} - \)\(32\!\cdots\!04\)\( \beta_{7} - \)\(33\!\cdots\!96\)\( \beta_{8}) q^{31}\) \(+(-\)\(36\!\cdots\!20\)\( + \)\(28\!\cdots\!00\)\( \beta_{1} - \)\(13\!\cdots\!72\)\( \beta_{2} - \)\(28\!\cdots\!68\)\( \beta_{3} + \)\(15\!\cdots\!32\)\( \beta_{4} - \)\(28\!\cdots\!96\)\( \beta_{5} - \)\(36\!\cdots\!36\)\( \beta_{6} + \)\(81\!\cdots\!60\)\( \beta_{7} - \)\(29\!\cdots\!20\)\( \beta_{8}) q^{32}\) \(+(\)\(21\!\cdots\!60\)\( - \)\(54\!\cdots\!57\)\( \beta_{1} - \)\(89\!\cdots\!86\)\( \beta_{2} + \)\(59\!\cdots\!95\)\( \beta_{3} - \)\(18\!\cdots\!22\)\( \beta_{4} - \)\(10\!\cdots\!22\)\( \beta_{5} + \)\(15\!\cdots\!84\)\( \beta_{6} - \)\(70\!\cdots\!41\)\( \beta_{7} + \)\(37\!\cdots\!55\)\( \beta_{8}) q^{33}\) \(+(\)\(10\!\cdots\!70\)\( + \)\(27\!\cdots\!30\)\( \beta_{1} - \)\(37\!\cdots\!16\)\( \beta_{2} + \)\(10\!\cdots\!40\)\( \beta_{3} - \)\(32\!\cdots\!76\)\( \beta_{4} + \)\(12\!\cdots\!28\)\( \beta_{5} + \)\(26\!\cdots\!28\)\( \beta_{6} + \)\(37\!\cdots\!88\)\( \beta_{7} - \)\(29\!\cdots\!08\)\( \beta_{8}) q^{34}\) \(+(\)\(14\!\cdots\!56\)\( + \)\(18\!\cdots\!68\)\( \beta_{1} - \)\(15\!\cdots\!84\)\( \beta_{2} + \)\(13\!\cdots\!32\)\( \beta_{3} + \)\(61\!\cdots\!32\)\( \beta_{4} - \)\(45\!\cdots\!60\)\( \beta_{5} - \)\(69\!\cdots\!80\)\( \beta_{6} - \)\(93\!\cdots\!20\)\( \beta_{7} + \)\(17\!\cdots\!60\)\( \beta_{8}) q^{35}\) \(+(-\)\(24\!\cdots\!64\)\( - \)\(55\!\cdots\!65\)\( \beta_{1} + \)\(90\!\cdots\!99\)\( \beta_{2} - \)\(16\!\cdots\!15\)\( \beta_{3} + \)\(56\!\cdots\!56\)\( \beta_{4} - \)\(42\!\cdots\!76\)\( \beta_{5} + \)\(36\!\cdots\!00\)\( \beta_{6} - \)\(36\!\cdots\!20\)\( \beta_{7} - \)\(78\!\cdots\!60\)\( \beta_{8}) q^{36}\) \(+(-\)\(98\!\cdots\!89\)\( + \)\(90\!\cdots\!38\)\( \beta_{1} + \)\(39\!\cdots\!10\)\( \beta_{2} - \)\(15\!\cdots\!06\)\( \beta_{3} - \)\(15\!\cdots\!07\)\( \beta_{4} + \)\(16\!\cdots\!32\)\( \beta_{5} - \)\(27\!\cdots\!80\)\( \beta_{6} + \)\(55\!\cdots\!22\)\( \beta_{7} + \)\(18\!\cdots\!50\)\( \beta_{8}) q^{37}\) \(+(\)\(14\!\cdots\!12\)\( - \)\(47\!\cdots\!14\)\( \beta_{1} + \)\(37\!\cdots\!54\)\( \beta_{2} + \)\(24\!\cdots\!31\)\( \beta_{3} - \)\(26\!\cdots\!02\)\( \beta_{4} + \)\(72\!\cdots\!99\)\( \beta_{5} - \)\(78\!\cdots\!12\)\( \beta_{6} - \)\(32\!\cdots\!84\)\( \beta_{7} + \)\(61\!\cdots\!20\)\( \beta_{8}) q^{38}\) \(+(\)\(24\!\cdots\!63\)\( - \)\(21\!\cdots\!54\)\( \beta_{1} + \)\(12\!\cdots\!62\)\( \beta_{2} + \)\(34\!\cdots\!34\)\( \beta_{3} + \)\(51\!\cdots\!23\)\( \beta_{4} - \)\(74\!\cdots\!84\)\( \beta_{5} + \)\(48\!\cdots\!71\)\( \beta_{6} + \)\(10\!\cdots\!96\)\( \beta_{7} - \)\(11\!\cdots\!16\)\( \beta_{8}) q^{39}\) \(+(\)\(12\!\cdots\!90\)\( - \)\(16\!\cdots\!80\)\( \beta_{1} - \)\(74\!\cdots\!10\)\( \beta_{2} + \)\(28\!\cdots\!80\)\( \beta_{3} - \)\(36\!\cdots\!70\)\( \beta_{4} + \)\(11\!\cdots\!50\)\( \beta_{5} - \)\(70\!\cdots\!00\)\( \beta_{6} - \)\(68\!\cdots\!50\)\( \beta_{7} + \)\(83\!\cdots\!00\)\( \beta_{8}) q^{40}\) \(+(-\)\(40\!\cdots\!38\)\( + \)\(71\!\cdots\!98\)\( \beta_{1} - \)\(13\!\cdots\!72\)\( \beta_{2} - \)\(32\!\cdots\!98\)\( \beta_{3} - \)\(12\!\cdots\!96\)\( \beta_{4} + \)\(11\!\cdots\!16\)\( \beta_{5} - \)\(67\!\cdots\!00\)\( \beta_{6} - \)\(17\!\cdots\!30\)\( \beta_{7} - \)\(44\!\cdots\!90\)\( \beta_{8}) q^{41}\) \(+(-\)\(27\!\cdots\!92\)\( + \)\(74\!\cdots\!52\)\( \beta_{1} - \)\(18\!\cdots\!72\)\( \beta_{2} - \)\(33\!\cdots\!52\)\( \beta_{3} + \)\(24\!\cdots\!48\)\( \beta_{4} - \)\(59\!\cdots\!84\)\( \beta_{5} + \)\(45\!\cdots\!36\)\( \beta_{6} + \)\(88\!\cdots\!80\)\( \beta_{7} + \)\(18\!\cdots\!40\)\( \beta_{8}) q^{42}\) \(+(-\)\(25\!\cdots\!53\)\( - \)\(83\!\cdots\!87\)\( \beta_{1} - \)\(16\!\cdots\!73\)\( \beta_{2} + \)\(65\!\cdots\!68\)\( \beta_{3} + \)\(11\!\cdots\!40\)\( \beta_{4} - \)\(25\!\cdots\!32\)\( \beta_{5} - \)\(84\!\cdots\!48\)\( \beta_{6} - \)\(88\!\cdots\!84\)\( \beta_{7} - \)\(62\!\cdots\!00\)\( \beta_{8}) q^{43}\) \(+(\)\(86\!\cdots\!92\)\( - \)\(72\!\cdots\!80\)\( \beta_{1} + \)\(91\!\cdots\!44\)\( \beta_{2} + \)\(53\!\cdots\!00\)\( \beta_{3} - \)\(30\!\cdots\!40\)\( \beta_{4} + \)\(10\!\cdots\!96\)\( \beta_{5} - \)\(37\!\cdots\!76\)\( \beta_{6} - \)\(15\!\cdots\!16\)\( \beta_{7} + \)\(14\!\cdots\!76\)\( \beta_{8}) q^{44}\) \(+(\)\(36\!\cdots\!39\)\( - \)\(19\!\cdots\!38\)\( \beta_{1} + \)\(58\!\cdots\!34\)\( \beta_{2} - \)\(24\!\cdots\!62\)\( \beta_{3} - \)\(12\!\cdots\!67\)\( \beta_{4} - \)\(25\!\cdots\!00\)\( \beta_{5} + \)\(25\!\cdots\!00\)\( \beta_{6} + \)\(12\!\cdots\!50\)\( \beta_{7} - \)\(19\!\cdots\!50\)\( \beta_{8}) q^{45}\) \(+(-\)\(35\!\cdots\!68\)\( - \)\(17\!\cdots\!92\)\( \beta_{1} + \)\(26\!\cdots\!68\)\( \beta_{2} - \)\(47\!\cdots\!58\)\( \beta_{3} + \)\(50\!\cdots\!84\)\( \beta_{4} - \)\(50\!\cdots\!94\)\( \beta_{5} - \)\(31\!\cdots\!20\)\( \beta_{6} - \)\(52\!\cdots\!40\)\( \beta_{7} - \)\(20\!\cdots\!40\)\( \beta_{8}) q^{46}\) \(+(-\)\(14\!\cdots\!54\)\( - \)\(34\!\cdots\!64\)\( \beta_{1} + \)\(26\!\cdots\!96\)\( \beta_{2} - \)\(20\!\cdots\!36\)\( \beta_{3} + \)\(32\!\cdots\!90\)\( \beta_{4} + \)\(27\!\cdots\!24\)\( \beta_{5} - \)\(29\!\cdots\!34\)\( \beta_{6} + \)\(13\!\cdots\!88\)\( \beta_{7} + \)\(14\!\cdots\!80\)\( \beta_{8}) q^{47}\) \(+(\)\(19\!\cdots\!52\)\( - \)\(29\!\cdots\!00\)\( \beta_{1} - \)\(21\!\cdots\!24\)\( \beta_{2} + \)\(85\!\cdots\!88\)\( \beta_{3} - \)\(57\!\cdots\!72\)\( \beta_{4} + \)\(63\!\cdots\!16\)\( \beta_{5} + \)\(16\!\cdots\!56\)\( \beta_{6} - \)\(22\!\cdots\!80\)\( \beta_{7} - \)\(82\!\cdots\!00\)\( \beta_{8}) q^{48}\) \(+(\)\(44\!\cdots\!17\)\( - \)\(83\!\cdots\!68\)\( \beta_{1} - \)\(13\!\cdots\!48\)\( \beta_{2} + \)\(15\!\cdots\!68\)\( \beta_{3} - \)\(27\!\cdots\!84\)\( \beta_{4} - \)\(57\!\cdots\!16\)\( \beta_{5} - \)\(25\!\cdots\!20\)\( \beta_{6} + \)\(35\!\cdots\!00\)\( \beta_{7} - \)\(13\!\cdots\!20\)\( \beta_{8}) q^{49}\) \(+(\)\(85\!\cdots\!25\)\( - \)\(32\!\cdots\!25\)\( \beta_{1} - \)\(13\!\cdots\!00\)\( \beta_{2} + \)\(50\!\cdots\!00\)\( \beta_{3} + \)\(29\!\cdots\!00\)\( \beta_{4} + \)\(59\!\cdots\!00\)\( \beta_{5} - \)\(60\!\cdots\!00\)\( \beta_{6} - \)\(29\!\cdots\!00\)\( \beta_{7} + \)\(22\!\cdots\!00\)\( \beta_{8}) q^{50}\) \(+(-\)\(14\!\cdots\!12\)\( - \)\(69\!\cdots\!02\)\( \beta_{1} + \)\(53\!\cdots\!26\)\( \beta_{2} - \)\(33\!\cdots\!08\)\( \beta_{3} + \)\(46\!\cdots\!14\)\( \beta_{4} + \)\(36\!\cdots\!08\)\( \beta_{5} + \)\(21\!\cdots\!58\)\( \beta_{6} + \)\(21\!\cdots\!68\)\( \beta_{7} + \)\(35\!\cdots\!12\)\( \beta_{8}) q^{51}\) \(+(\)\(46\!\cdots\!00\)\( - \)\(34\!\cdots\!02\)\( \beta_{1} + \)\(53\!\cdots\!54\)\( \beta_{2} + \)\(55\!\cdots\!74\)\( \beta_{3} - \)\(10\!\cdots\!80\)\( \beta_{4} - \)\(38\!\cdots\!76\)\( \beta_{5} + \)\(11\!\cdots\!36\)\( \beta_{6} - \)\(86\!\cdots\!12\)\( \beta_{7} - \)\(29\!\cdots\!00\)\( \beta_{8}) q^{52}\) \(+(\)\(12\!\cdots\!19\)\( - \)\(51\!\cdots\!58\)\( \beta_{1} + \)\(39\!\cdots\!62\)\( \beta_{2} - \)\(70\!\cdots\!58\)\( \beta_{3} - \)\(21\!\cdots\!31\)\( \beta_{4} - \)\(55\!\cdots\!44\)\( \beta_{5} - \)\(93\!\cdots\!40\)\( \beta_{6} + \)\(17\!\cdots\!46\)\( \beta_{7} + \)\(12\!\cdots\!70\)\( \beta_{8}) q^{53}\) \(+(\)\(14\!\cdots\!48\)\( - \)\(51\!\cdots\!32\)\( \beta_{1} + \)\(44\!\cdots\!96\)\( \beta_{2} + \)\(14\!\cdots\!22\)\( \beta_{3} + \)\(35\!\cdots\!80\)\( \beta_{4} + \)\(14\!\cdots\!26\)\( \beta_{5} + \)\(28\!\cdots\!24\)\( \beta_{6} + \)\(98\!\cdots\!44\)\( \beta_{7} - \)\(25\!\cdots\!44\)\( \beta_{8}) q^{54}\) \(+(-\)\(14\!\cdots\!79\)\( + \)\(34\!\cdots\!18\)\( \beta_{1} - \)\(92\!\cdots\!74\)\( \beta_{2} + \)\(21\!\cdots\!82\)\( \beta_{3} + \)\(20\!\cdots\!37\)\( \beta_{4} + \)\(36\!\cdots\!00\)\( \beta_{5} - \)\(38\!\cdots\!75\)\( \beta_{6} - \)\(20\!\cdots\!00\)\( \beta_{7} - \)\(22\!\cdots\!00\)\( \beta_{8}) q^{55}\) \(+(-\)\(15\!\cdots\!56\)\( + \)\(33\!\cdots\!12\)\( \beta_{1} - \)\(51\!\cdots\!40\)\( \beta_{2} - \)\(15\!\cdots\!72\)\( \beta_{3} + \)\(36\!\cdots\!80\)\( \beta_{4} - \)\(21\!\cdots\!12\)\( \beta_{5} + \)\(31\!\cdots\!72\)\( \beta_{6} + \)\(72\!\cdots\!32\)\( \beta_{7} + \)\(39\!\cdots\!68\)\( \beta_{8}) q^{56}\) \(+(-\)\(34\!\cdots\!88\)\( + \)\(65\!\cdots\!37\)\( \beta_{1} + \)\(32\!\cdots\!86\)\( \beta_{2} - \)\(35\!\cdots\!43\)\( \beta_{3} - \)\(44\!\cdots\!30\)\( \beta_{4} + \)\(25\!\cdots\!62\)\( \beta_{5} - \)\(36\!\cdots\!92\)\( \beta_{6} - \)\(11\!\cdots\!31\)\( \beta_{7} - \)\(16\!\cdots\!35\)\( \beta_{8}) q^{57}\) \(+(\)\(11\!\cdots\!02\)\( + \)\(87\!\cdots\!54\)\( \beta_{1} + \)\(30\!\cdots\!60\)\( \beta_{2} + \)\(10\!\cdots\!72\)\( \beta_{3} + \)\(20\!\cdots\!28\)\( \beta_{4} + \)\(16\!\cdots\!00\)\( \beta_{5} + \)\(23\!\cdots\!52\)\( \beta_{6} - \)\(24\!\cdots\!52\)\( \beta_{7} + \)\(31\!\cdots\!40\)\( \beta_{8}) q^{58}\) \(+(-\)\(23\!\cdots\!33\)\( + \)\(32\!\cdots\!73\)\( \beta_{1} + \)\(72\!\cdots\!27\)\( \beta_{2} + \)\(48\!\cdots\!72\)\( \beta_{3} + \)\(29\!\cdots\!76\)\( \beta_{4} - \)\(27\!\cdots\!92\)\( \beta_{5} - \)\(62\!\cdots\!84\)\( \beta_{6} + \)\(34\!\cdots\!36\)\( \beta_{7} + \)\(13\!\cdots\!24\)\( \beta_{8}) q^{59}\) \(+(-\)\(46\!\cdots\!64\)\( + \)\(14\!\cdots\!08\)\( \beta_{1} + \)\(64\!\cdots\!96\)\( \beta_{2} - \)\(12\!\cdots\!08\)\( \beta_{3} - \)\(12\!\cdots\!08\)\( \beta_{4} - \)\(43\!\cdots\!60\)\( \beta_{5} + \)\(41\!\cdots\!20\)\( \beta_{6} + \)\(43\!\cdots\!80\)\( \beta_{7} - \)\(32\!\cdots\!40\)\( \beta_{8}) q^{60}\) \(+(\)\(13\!\cdots\!03\)\( - \)\(17\!\cdots\!30\)\( \beta_{1} - \)\(69\!\cdots\!10\)\( \beta_{2} - \)\(16\!\cdots\!70\)\( \beta_{3} - \)\(17\!\cdots\!87\)\( \beta_{4} - \)\(30\!\cdots\!16\)\( \beta_{5} + \)\(20\!\cdots\!28\)\( \beta_{6} - \)\(42\!\cdots\!02\)\( \beta_{7} + \)\(11\!\cdots\!22\)\( \beta_{8}) q^{61}\) \(+(\)\(21\!\cdots\!60\)\( - \)\(71\!\cdots\!52\)\( \beta_{1} - \)\(10\!\cdots\!28\)\( \beta_{2} + \)\(57\!\cdots\!48\)\( \beta_{3} - \)\(18\!\cdots\!92\)\( \beta_{4} + \)\(13\!\cdots\!96\)\( \beta_{5} - \)\(46\!\cdots\!04\)\( \beta_{6} + \)\(10\!\cdots\!40\)\( \beta_{7} - \)\(17\!\cdots\!20\)\( \beta_{8}) q^{62}\) \(+(\)\(50\!\cdots\!81\)\( - \)\(21\!\cdots\!50\)\( \beta_{1} - \)\(24\!\cdots\!30\)\( \beta_{2} + \)\(35\!\cdots\!34\)\( \beta_{3} + \)\(19\!\cdots\!33\)\( \beta_{4} - \)\(41\!\cdots\!48\)\( \beta_{5} - \)\(28\!\cdots\!15\)\( \beta_{6} + \)\(60\!\cdots\!32\)\( \beta_{7} - \)\(18\!\cdots\!20\)\( \beta_{8}) q^{63}\) \(+(\)\(65\!\cdots\!72\)\( - \)\(89\!\cdots\!28\)\( \beta_{1} + \)\(80\!\cdots\!36\)\( \beta_{2} + \)\(30\!\cdots\!48\)\( \beta_{3} - \)\(39\!\cdots\!08\)\( \beta_{4} + \)\(49\!\cdots\!28\)\( \beta_{5} + \)\(30\!\cdots\!40\)\( \beta_{6} - \)\(24\!\cdots\!20\)\( \beta_{7} + \)\(17\!\cdots\!80\)\( \beta_{8}) q^{64}\) \(+(-\)\(45\!\cdots\!92\)\( - \)\(40\!\cdots\!26\)\( \beta_{1} + \)\(29\!\cdots\!88\)\( \beta_{2} - \)\(25\!\cdots\!74\)\( \beta_{3} - \)\(78\!\cdots\!24\)\( \beta_{4} - \)\(88\!\cdots\!80\)\( \beta_{5} + \)\(13\!\cdots\!60\)\( \beta_{6} + \)\(36\!\cdots\!90\)\( \beta_{7} - \)\(45\!\cdots\!70\)\( \beta_{8}) q^{65}\) \(+(-\)\(24\!\cdots\!68\)\( + \)\(32\!\cdots\!84\)\( \beta_{1} + \)\(35\!\cdots\!68\)\( \beta_{2} - \)\(22\!\cdots\!64\)\( \beta_{3} - \)\(82\!\cdots\!68\)\( \beta_{4} + \)\(57\!\cdots\!64\)\( \beta_{5} - \)\(56\!\cdots\!56\)\( \beta_{6} + \)\(56\!\cdots\!24\)\( \beta_{7} + \)\(37\!\cdots\!16\)\( \beta_{8}) q^{66}\) \(+(-\)\(15\!\cdots\!45\)\( + \)\(32\!\cdots\!31\)\( \beta_{1} - \)\(10\!\cdots\!55\)\( \beta_{2} - \)\(46\!\cdots\!56\)\( \beta_{3} + \)\(42\!\cdots\!62\)\( \beta_{4} - \)\(15\!\cdots\!44\)\( \beta_{5} + \)\(73\!\cdots\!22\)\( \beta_{6} - \)\(37\!\cdots\!56\)\( \beta_{7} + \)\(13\!\cdots\!20\)\( \beta_{8}) q^{67}\) \(+(-\)\(15\!\cdots\!40\)\( + \)\(28\!\cdots\!10\)\( \beta_{1} - \)\(32\!\cdots\!74\)\( \beta_{2} + \)\(10\!\cdots\!78\)\( \beta_{3} + \)\(11\!\cdots\!36\)\( \beta_{4} + \)\(10\!\cdots\!84\)\( \beta_{5} + \)\(92\!\cdots\!20\)\( \beta_{6} + \)\(70\!\cdots\!44\)\( \beta_{7} - \)\(52\!\cdots\!40\)\( \beta_{8}) q^{68}\) \(+(\)\(14\!\cdots\!28\)\( + \)\(57\!\cdots\!94\)\( \beta_{1} - \)\(81\!\cdots\!48\)\( \beta_{2} + \)\(12\!\cdots\!46\)\( \beta_{3} - \)\(33\!\cdots\!80\)\( \beta_{4} + \)\(72\!\cdots\!44\)\( \beta_{5} - \)\(35\!\cdots\!44\)\( \beta_{6} - \)\(11\!\cdots\!34\)\( \beta_{7} + \)\(72\!\cdots\!54\)\( \beta_{8}) q^{69}\) \(+(\)\(27\!\cdots\!32\)\( + \)\(42\!\cdots\!56\)\( \beta_{1} + \)\(74\!\cdots\!92\)\( \beta_{2} - \)\(38\!\cdots\!56\)\( \beta_{3} - \)\(48\!\cdots\!96\)\( \beta_{4} + \)\(10\!\cdots\!00\)\( \beta_{5} + \)\(59\!\cdots\!00\)\( \beta_{6} - \)\(16\!\cdots\!00\)\( \beta_{7} + \)\(32\!\cdots\!00\)\( \beta_{8}) q^{70}\) \(+(-\)\(23\!\cdots\!95\)\( - \)\(41\!\cdots\!50\)\( \beta_{1} + \)\(41\!\cdots\!70\)\( \beta_{2} - \)\(17\!\cdots\!50\)\( \beta_{3} + \)\(54\!\cdots\!09\)\( \beta_{4} - \)\(36\!\cdots\!28\)\( \beta_{5} + \)\(10\!\cdots\!69\)\( \beta_{6} + \)\(92\!\cdots\!84\)\( \beta_{7} - \)\(24\!\cdots\!04\)\( \beta_{8}) q^{71}\) \(+(-\)\(57\!\cdots\!93\)\( - \)\(68\!\cdots\!82\)\( \beta_{1} + \)\(11\!\cdots\!63\)\( \beta_{2} - \)\(50\!\cdots\!74\)\( \beta_{3} + \)\(67\!\cdots\!21\)\( \beta_{4} - \)\(19\!\cdots\!93\)\( \beta_{5} + \)\(60\!\cdots\!72\)\( \beta_{6} + \)\(50\!\cdots\!25\)\( \beta_{7} + \)\(87\!\cdots\!20\)\( \beta_{8}) q^{72}\) \(+(\)\(10\!\cdots\!98\)\( - \)\(13\!\cdots\!63\)\( \beta_{1} + \)\(54\!\cdots\!26\)\( \beta_{2} + \)\(41\!\cdots\!81\)\( \beta_{3} - \)\(11\!\cdots\!82\)\( \beta_{4} + \)\(22\!\cdots\!94\)\( \beta_{5} - \)\(22\!\cdots\!52\)\( \beta_{6} + \)\(10\!\cdots\!01\)\( \beta_{7} + \)\(75\!\cdots\!45\)\( \beta_{8}) q^{73}\) \(+(\)\(34\!\cdots\!10\)\( - \)\(51\!\cdots\!82\)\( \beta_{1} - \)\(47\!\cdots\!72\)\( \beta_{2} + \)\(19\!\cdots\!32\)\( \beta_{3} - \)\(78\!\cdots\!48\)\( \beta_{4} - \)\(16\!\cdots\!80\)\( \beta_{5} + \)\(18\!\cdots\!48\)\( \beta_{6} - \)\(11\!\cdots\!72\)\( \beta_{7} + \)\(82\!\cdots\!32\)\( \beta_{8}) q^{74}\) \(+(\)\(47\!\cdots\!75\)\( - \)\(61\!\cdots\!75\)\( \beta_{1} - \)\(16\!\cdots\!25\)\( \beta_{2} - \)\(94\!\cdots\!00\)\( \beta_{3} - \)\(32\!\cdots\!00\)\( \beta_{4} - \)\(31\!\cdots\!00\)\( \beta_{5} + \)\(59\!\cdots\!00\)\( \beta_{6} + \)\(26\!\cdots\!00\)\( \beta_{7} - \)\(13\!\cdots\!00\)\( \beta_{8}) q^{75}\) \(+(-\)\(18\!\cdots\!52\)\( + \)\(96\!\cdots\!64\)\( \beta_{1} - \)\(20\!\cdots\!40\)\( \beta_{2} - \)\(12\!\cdots\!84\)\( \beta_{3} + \)\(24\!\cdots\!20\)\( \beta_{4} - \)\(15\!\cdots\!24\)\( \beta_{5} - \)\(14\!\cdots\!16\)\( \beta_{6} + \)\(38\!\cdots\!64\)\( \beta_{7} + \)\(33\!\cdots\!76\)\( \beta_{8}) q^{76}\) \(+(\)\(65\!\cdots\!04\)\( + \)\(15\!\cdots\!38\)\( \beta_{1} + \)\(76\!\cdots\!36\)\( \beta_{2} + \)\(39\!\cdots\!62\)\( \beta_{3} - \)\(35\!\cdots\!08\)\( \beta_{4} + \)\(83\!\cdots\!04\)\( \beta_{5} + \)\(30\!\cdots\!04\)\( \beta_{6} - \)\(14\!\cdots\!10\)\( \beta_{7} + \)\(57\!\cdots\!50\)\( \beta_{8}) q^{77}\) \(+(-\)\(76\!\cdots\!16\)\( + \)\(10\!\cdots\!08\)\( \beta_{1} + \)\(16\!\cdots\!28\)\( \beta_{2} + \)\(75\!\cdots\!82\)\( \beta_{3} + \)\(34\!\cdots\!16\)\( \beta_{4} - \)\(83\!\cdots\!82\)\( \beta_{5} - \)\(14\!\cdots\!64\)\( \beta_{6} + \)\(30\!\cdots\!52\)\( \beta_{7} - \)\(23\!\cdots\!00\)\( \beta_{8}) q^{78}\) \(+(\)\(13\!\cdots\!38\)\( + \)\(10\!\cdots\!96\)\( \beta_{1} + \)\(82\!\cdots\!88\)\( \beta_{2} + \)\(71\!\cdots\!64\)\( \beta_{3} - \)\(18\!\cdots\!10\)\( \beta_{4} - \)\(11\!\cdots\!44\)\( \beta_{5} + \)\(19\!\cdots\!34\)\( \beta_{6} - \)\(98\!\cdots\!56\)\( \beta_{7} + \)\(58\!\cdots\!16\)\( \beta_{8}) q^{79}\) \(+(-\)\(22\!\cdots\!92\)\( + \)\(13\!\cdots\!64\)\( \beta_{1} - \)\(14\!\cdots\!52\)\( \beta_{2} + \)\(37\!\cdots\!36\)\( \beta_{3} + \)\(39\!\cdots\!76\)\( \beta_{4} + \)\(71\!\cdots\!00\)\( \beta_{5} - \)\(38\!\cdots\!00\)\( \beta_{6} - \)\(53\!\cdots\!00\)\( \beta_{7} - \)\(22\!\cdots\!00\)\( \beta_{8}) q^{80}\) \(+(-\)\(45\!\cdots\!51\)\( + \)\(25\!\cdots\!79\)\( \beta_{1} - \)\(12\!\cdots\!14\)\( \beta_{2} - \)\(23\!\cdots\!69\)\( \beta_{3} - \)\(25\!\cdots\!46\)\( \beta_{4} + \)\(49\!\cdots\!22\)\( \beta_{5} - \)\(30\!\cdots\!76\)\( \beta_{6} + \)\(38\!\cdots\!19\)\( \beta_{7} - \)\(22\!\cdots\!69\)\( \beta_{8}) q^{81}\) \(+(\)\(35\!\cdots\!02\)\( - \)\(13\!\cdots\!38\)\( \beta_{1} - \)\(32\!\cdots\!28\)\( \beta_{2} - \)\(56\!\cdots\!36\)\( \beta_{3} + \)\(99\!\cdots\!48\)\( \beta_{4} + \)\(96\!\cdots\!52\)\( \beta_{5} + \)\(21\!\cdots\!20\)\( \beta_{6} + \)\(68\!\cdots\!32\)\( \beta_{7} + \)\(57\!\cdots\!40\)\( \beta_{8}) q^{82}\) \(+(-\)\(10\!\cdots\!31\)\( - \)\(27\!\cdots\!65\)\( \beta_{1} - \)\(15\!\cdots\!67\)\( \beta_{2} - \)\(73\!\cdots\!60\)\( \beta_{3} - \)\(33\!\cdots\!80\)\( \beta_{4} - \)\(65\!\cdots\!00\)\( \beta_{5} - \)\(20\!\cdots\!20\)\( \beta_{6} + \)\(46\!\cdots\!00\)\( \beta_{7} - \)\(28\!\cdots\!20\)\( \beta_{8}) q^{83}\) \(+(\)\(16\!\cdots\!64\)\( - \)\(10\!\cdots\!12\)\( \beta_{1} + \)\(19\!\cdots\!40\)\( \beta_{2} + \)\(52\!\cdots\!72\)\( \beta_{3} - \)\(41\!\cdots\!88\)\( \beta_{4} + \)\(75\!\cdots\!08\)\( \beta_{5} - \)\(73\!\cdots\!60\)\( \beta_{6} - \)\(24\!\cdots\!40\)\( \beta_{7} - \)\(14\!\cdots\!80\)\( \beta_{8}) q^{84}\) \(+(-\)\(50\!\cdots\!14\)\( - \)\(30\!\cdots\!42\)\( \beta_{1} + \)\(22\!\cdots\!96\)\( \beta_{2} + \)\(27\!\cdots\!42\)\( \beta_{3} + \)\(77\!\cdots\!42\)\( \beta_{4} + \)\(22\!\cdots\!40\)\( \beta_{5} - \)\(66\!\cdots\!80\)\( \beta_{6} + \)\(29\!\cdots\!30\)\( \beta_{7} + \)\(27\!\cdots\!10\)\( \beta_{8}) q^{85}\) \(+(-\)\(28\!\cdots\!12\)\( + \)\(15\!\cdots\!50\)\( \beta_{1} + \)\(88\!\cdots\!38\)\( \beta_{2} - \)\(84\!\cdots\!35\)\( \beta_{3} + \)\(13\!\cdots\!90\)\( \beta_{4} + \)\(15\!\cdots\!17\)\( \beta_{5} + \)\(28\!\cdots\!28\)\( \beta_{6} + \)\(60\!\cdots\!08\)\( \beta_{7} + \)\(98\!\cdots\!52\)\( \beta_{8}) q^{86}\) \(+(-\)\(76\!\cdots\!49\)\( + \)\(40\!\cdots\!10\)\( \beta_{1} - \)\(67\!\cdots\!14\)\( \beta_{2} - \)\(12\!\cdots\!30\)\( \beta_{3} - \)\(25\!\cdots\!17\)\( \beta_{4} - \)\(40\!\cdots\!52\)\( \beta_{5} + \)\(21\!\cdots\!59\)\( \beta_{6} - \)\(22\!\cdots\!16\)\( \beta_{7} - \)\(82\!\cdots\!40\)\( \beta_{8}) q^{87}\) \(+(-\)\(22\!\cdots\!32\)\( + \)\(20\!\cdots\!20\)\( \beta_{1} - \)\(29\!\cdots\!72\)\( \beta_{2} - \)\(78\!\cdots\!32\)\( \beta_{3} - \)\(92\!\cdots\!52\)\( \beta_{4} - \)\(80\!\cdots\!24\)\( \beta_{5} - \)\(15\!\cdots\!24\)\( \beta_{6} + \)\(17\!\cdots\!80\)\( \beta_{7} - \)\(41\!\cdots\!80\)\( \beta_{8}) q^{88}\) \(+(-\)\(67\!\cdots\!14\)\( + \)\(11\!\cdots\!49\)\( \beta_{1} - \)\(18\!\cdots\!10\)\( \beta_{2} + \)\(18\!\cdots\!81\)\( \beta_{3} + \)\(15\!\cdots\!14\)\( \beta_{4} + \)\(28\!\cdots\!58\)\( \beta_{5} + \)\(22\!\cdots\!08\)\( \beta_{6} + \)\(18\!\cdots\!93\)\( \beta_{7} + \)\(57\!\cdots\!37\)\( \beta_{8}) q^{89}\) \(+(-\)\(70\!\cdots\!22\)\( + \)\(16\!\cdots\!34\)\( \beta_{1} + \)\(45\!\cdots\!08\)\( \beta_{2} + \)\(96\!\cdots\!16\)\( \beta_{3} + \)\(48\!\cdots\!16\)\( \beta_{4} - \)\(11\!\cdots\!80\)\( \beta_{5} + \)\(44\!\cdots\!60\)\( \beta_{6} + \)\(35\!\cdots\!40\)\( \beta_{7} - \)\(76\!\cdots\!20\)\( \beta_{8}) q^{90}\) \(+(-\)\(13\!\cdots\!84\)\( - \)\(84\!\cdots\!88\)\( \beta_{1} + \)\(22\!\cdots\!40\)\( \beta_{2} + \)\(45\!\cdots\!28\)\( \beta_{3} + \)\(55\!\cdots\!60\)\( \beta_{4} + \)\(24\!\cdots\!48\)\( \beta_{5} + \)\(30\!\cdots\!32\)\( \beta_{6} - \)\(12\!\cdots\!88\)\( \beta_{7} - \)\(16\!\cdots\!32\)\( \beta_{8}) q^{91}\) \(+(-\)\(40\!\cdots\!72\)\( - \)\(99\!\cdots\!36\)\( \beta_{1} + \)\(17\!\cdots\!04\)\( \beta_{2} - \)\(21\!\cdots\!56\)\( \beta_{3} - \)\(16\!\cdots\!72\)\( \beta_{4} - \)\(22\!\cdots\!08\)\( \beta_{5} - \)\(12\!\cdots\!00\)\( \beta_{6} - \)\(13\!\cdots\!48\)\( \beta_{7} + \)\(71\!\cdots\!00\)\( \beta_{8}) q^{92}\) \(+(-\)\(73\!\cdots\!76\)\( - \)\(20\!\cdots\!72\)\( \beta_{1} + \)\(77\!\cdots\!08\)\( \beta_{2} + \)\(14\!\cdots\!28\)\( \beta_{3} + \)\(54\!\cdots\!84\)\( \beta_{4} + \)\(22\!\cdots\!52\)\( \beta_{5} + \)\(16\!\cdots\!44\)\( \beta_{6} + \)\(14\!\cdots\!08\)\( \beta_{7} - \)\(86\!\cdots\!80\)\( \beta_{8}) q^{93}\) \(+(-\)\(11\!\cdots\!68\)\( - \)\(70\!\cdots\!72\)\( \beta_{1} - \)\(55\!\cdots\!64\)\( \beta_{2} + \)\(83\!\cdots\!12\)\( \beta_{3} - \)\(23\!\cdots\!80\)\( \beta_{4} + \)\(59\!\cdots\!96\)\( \beta_{5} + \)\(83\!\cdots\!64\)\( \beta_{6} - \)\(22\!\cdots\!56\)\( \beta_{7} - \)\(85\!\cdots\!04\)\( \beta_{8}) q^{94}\) \(+(-\)\(21\!\cdots\!45\)\( - \)\(23\!\cdots\!10\)\( \beta_{1} - \)\(25\!\cdots\!70\)\( \beta_{2} + \)\(90\!\cdots\!10\)\( \beta_{3} + \)\(10\!\cdots\!35\)\( \beta_{4} - \)\(57\!\cdots\!00\)\( \beta_{5} - \)\(12\!\cdots\!25\)\( \beta_{6} - \)\(15\!\cdots\!00\)\( \beta_{7} + \)\(47\!\cdots\!00\)\( \beta_{8}) q^{95}\) \(+(-\)\(88\!\cdots\!48\)\( + \)\(26\!\cdots\!80\)\( \beta_{1} - \)\(16\!\cdots\!48\)\( \beta_{2} - \)\(90\!\cdots\!20\)\( \beta_{3} + \)\(36\!\cdots\!48\)\( \beta_{4} - \)\(13\!\cdots\!68\)\( \beta_{5} - \)\(46\!\cdots\!40\)\( \beta_{6} + \)\(93\!\cdots\!40\)\( \beta_{7} - \)\(78\!\cdots\!20\)\( \beta_{8}) q^{96}\) \(+(-\)\(11\!\cdots\!26\)\( + \)\(24\!\cdots\!85\)\( \beta_{1} + \)\(39\!\cdots\!22\)\( \beta_{2} + \)\(14\!\cdots\!37\)\( \beta_{3} - \)\(34\!\cdots\!26\)\( \beta_{4} - \)\(20\!\cdots\!34\)\( \beta_{5} + \)\(10\!\cdots\!20\)\( \beta_{6} - \)\(98\!\cdots\!59\)\( \beta_{7} + \)\(40\!\cdots\!65\)\( \beta_{8}) q^{97}\) \(+(-\)\(30\!\cdots\!57\)\( + \)\(86\!\cdots\!17\)\( \beta_{1} + \)\(14\!\cdots\!48\)\( \beta_{2} - \)\(69\!\cdots\!44\)\( \beta_{3} - \)\(61\!\cdots\!08\)\( \beta_{4} + \)\(13\!\cdots\!68\)\( \beta_{5} + \)\(41\!\cdots\!20\)\( \beta_{6} - \)\(58\!\cdots\!32\)\( \beta_{7} + \)\(14\!\cdots\!80\)\( \beta_{8}) q^{98}\) \(+(-\)\(48\!\cdots\!05\)\( + \)\(43\!\cdots\!57\)\( \beta_{1} + \)\(27\!\cdots\!15\)\( \beta_{2} + \)\(69\!\cdots\!08\)\( \beta_{3} - \)\(27\!\cdots\!60\)\( \beta_{4} - \)\(66\!\cdots\!32\)\( \beta_{5} + \)\(50\!\cdots\!32\)\( \beta_{6} - \)\(33\!\cdots\!68\)\( \beta_{7} - \)\(53\!\cdots\!72\)\( \beta_{8}) q^{99}\) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(9q \) \(\mathstrut -\mathstrut 2345005562254519728q^{2} \) \(\mathstrut -\mathstrut \)\(45\!\cdots\!04\)\(q^{3} \) \(\mathstrut +\mathstrut \)\(84\!\cdots\!88\)\(q^{4} \) \(\mathstrut -\mathstrut \)\(18\!\cdots\!50\)\(q^{5} \) \(\mathstrut -\mathstrut \)\(16\!\cdots\!92\)\(q^{6} \) \(\mathstrut +\mathstrut \)\(21\!\cdots\!92\)\(q^{7} \) \(\mathstrut -\mathstrut \)\(64\!\cdots\!40\)\(q^{8} \) \(\mathstrut +\mathstrut \)\(79\!\cdots\!17\)\(q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(9q \) \(\mathstrut -\mathstrut 2345005562254519728q^{2} \) \(\mathstrut -\mathstrut \)\(45\!\cdots\!04\)\(q^{3} \) \(\mathstrut +\mathstrut \)\(84\!\cdots\!88\)\(q^{4} \) \(\mathstrut -\mathstrut \)\(18\!\cdots\!50\)\(q^{5} \) \(\mathstrut -\mathstrut \)\(16\!\cdots\!92\)\(q^{6} \) \(\mathstrut +\mathstrut \)\(21\!\cdots\!92\)\(q^{7} \) \(\mathstrut -\mathstrut \)\(64\!\cdots\!40\)\(q^{8} \) \(\mathstrut +\mathstrut \)\(79\!\cdots\!17\)\(q^{9} \) \(\mathstrut -\mathstrut \)\(17\!\cdots\!00\)\(q^{10} \) \(\mathstrut -\mathstrut \)\(94\!\cdots\!92\)\(q^{11} \) \(\mathstrut +\mathstrut \)\(20\!\cdots\!72\)\(q^{12} \) \(\mathstrut +\mathstrut \)\(20\!\cdots\!86\)\(q^{13} \) \(\mathstrut -\mathstrut \)\(65\!\cdots\!84\)\(q^{14} \) \(\mathstrut +\mathstrut \)\(94\!\cdots\!00\)\(q^{15} \) \(\mathstrut +\mathstrut \)\(15\!\cdots\!04\)\(q^{16} \) \(\mathstrut +\mathstrut \)\(29\!\cdots\!82\)\(q^{17} \) \(\mathstrut -\mathstrut \)\(25\!\cdots\!64\)\(q^{18} \) \(\mathstrut +\mathstrut \)\(30\!\cdots\!80\)\(q^{19} \) \(\mathstrut +\mathstrut \)\(45\!\cdots\!00\)\(q^{20} \) \(\mathstrut +\mathstrut \)\(35\!\cdots\!88\)\(q^{21} \) \(\mathstrut -\mathstrut \)\(19\!\cdots\!36\)\(q^{22} \) \(\mathstrut -\mathstrut \)\(66\!\cdots\!24\)\(q^{23} \) \(\mathstrut -\mathstrut \)\(75\!\cdots\!60\)\(q^{24} \) \(\mathstrut -\mathstrut \)\(60\!\cdots\!25\)\(q^{25} \) \(\mathstrut -\mathstrut \)\(68\!\cdots\!72\)\(q^{26} \) \(\mathstrut -\mathstrut \)\(59\!\cdots\!40\)\(q^{27} \) \(\mathstrut +\mathstrut \)\(14\!\cdots\!44\)\(q^{28} \) \(\mathstrut +\mathstrut \)\(90\!\cdots\!70\)\(q^{29} \) \(\mathstrut +\mathstrut \)\(22\!\cdots\!00\)\(q^{30} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!12\)\(q^{31} \) \(\mathstrut -\mathstrut \)\(32\!\cdots\!88\)\(q^{32} \) \(\mathstrut +\mathstrut \)\(18\!\cdots\!52\)\(q^{33} \) \(\mathstrut +\mathstrut \)\(93\!\cdots\!36\)\(q^{34} \) \(\mathstrut +\mathstrut \)\(13\!\cdots\!00\)\(q^{35} \) \(\mathstrut -\mathstrut \)\(21\!\cdots\!56\)\(q^{36} \) \(\mathstrut -\mathstrut \)\(88\!\cdots\!38\)\(q^{37} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!40\)\(q^{38} \) \(\mathstrut +\mathstrut \)\(21\!\cdots\!04\)\(q^{39} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!00\)\(q^{40} \) \(\mathstrut -\mathstrut \)\(36\!\cdots\!22\)\(q^{41} \) \(\mathstrut -\mathstrut \)\(24\!\cdots\!96\)\(q^{42} \) \(\mathstrut -\mathstrut \)\(23\!\cdots\!44\)\(q^{43} \) \(\mathstrut +\mathstrut \)\(77\!\cdots\!56\)\(q^{44} \) \(\mathstrut +\mathstrut \)\(32\!\cdots\!50\)\(q^{45} \) \(\mathstrut -\mathstrut \)\(32\!\cdots\!52\)\(q^{46} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!48\)\(q^{47} \) \(\mathstrut +\mathstrut \)\(17\!\cdots\!76\)\(q^{48} \) \(\mathstrut +\mathstrut \)\(39\!\cdots\!13\)\(q^{49} \) \(\mathstrut +\mathstrut \)\(76\!\cdots\!00\)\(q^{50} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!52\)\(q^{51} \) \(\mathstrut +\mathstrut \)\(41\!\cdots\!52\)\(q^{52} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!46\)\(q^{53} \) \(\mathstrut +\mathstrut \)\(13\!\cdots\!80\)\(q^{54} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!00\)\(q^{55} \) \(\mathstrut -\mathstrut \)\(14\!\cdots\!20\)\(q^{56} \) \(\mathstrut -\mathstrut \)\(31\!\cdots\!80\)\(q^{57} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!60\)\(q^{58} \) \(\mathstrut -\mathstrut \)\(21\!\cdots\!60\)\(q^{59} \) \(\mathstrut -\mathstrut \)\(42\!\cdots\!00\)\(q^{60} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!58\)\(q^{61} \) \(\mathstrut +\mathstrut \)\(19\!\cdots\!04\)\(q^{62} \) \(\mathstrut +\mathstrut \)\(45\!\cdots\!96\)\(q^{63} \) \(\mathstrut +\mathstrut \)\(58\!\cdots\!68\)\(q^{64} \) \(\mathstrut -\mathstrut \)\(41\!\cdots\!00\)\(q^{65} \) \(\mathstrut -\mathstrut \)\(22\!\cdots\!04\)\(q^{66} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!68\)\(q^{67} \) \(\mathstrut -\mathstrut \)\(14\!\cdots\!76\)\(q^{68} \) \(\mathstrut +\mathstrut \)\(13\!\cdots\!64\)\(q^{69} \) \(\mathstrut +\mathstrut \)\(25\!\cdots\!00\)\(q^{70} \) \(\mathstrut -\mathstrut \)\(21\!\cdots\!52\)\(q^{71} \) \(\mathstrut -\mathstrut \)\(51\!\cdots\!20\)\(q^{72} \) \(\mathstrut +\mathstrut \)\(91\!\cdots\!26\)\(q^{73} \) \(\mathstrut +\mathstrut \)\(31\!\cdots\!76\)\(q^{74} \) \(\mathstrut +\mathstrut \)\(43\!\cdots\!00\)\(q^{75} \) \(\mathstrut -\mathstrut \)\(16\!\cdots\!40\)\(q^{76} \) \(\mathstrut +\mathstrut \)\(59\!\cdots\!04\)\(q^{77} \) \(\mathstrut -\mathstrut \)\(68\!\cdots\!68\)\(q^{78} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!20\)\(q^{79} \) \(\mathstrut -\mathstrut \)\(20\!\cdots\!00\)\(q^{80} \) \(\mathstrut -\mathstrut \)\(40\!\cdots\!11\)\(q^{81} \) \(\mathstrut +\mathstrut \)\(32\!\cdots\!24\)\(q^{82} \) \(\mathstrut -\mathstrut \)\(92\!\cdots\!84\)\(q^{83} \) \(\mathstrut +\mathstrut \)\(14\!\cdots\!16\)\(q^{84} \) \(\mathstrut -\mathstrut \)\(45\!\cdots\!00\)\(q^{85} \) \(\mathstrut -\mathstrut \)\(25\!\cdots\!12\)\(q^{86} \) \(\mathstrut -\mathstrut \)\(68\!\cdots\!20\)\(q^{87} \) \(\mathstrut -\mathstrut \)\(20\!\cdots\!80\)\(q^{88} \) \(\mathstrut -\mathstrut \)\(60\!\cdots\!90\)\(q^{89} \) \(\mathstrut -\mathstrut \)\(63\!\cdots\!00\)\(q^{90} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!92\)\(q^{91} \) \(\mathstrut -\mathstrut \)\(36\!\cdots\!68\)\(q^{92} \) \(\mathstrut -\mathstrut \)\(66\!\cdots\!28\)\(q^{93} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!04\)\(q^{94} \) \(\mathstrut -\mathstrut \)\(19\!\cdots\!00\)\(q^{95} \) \(\mathstrut -\mathstrut \)\(79\!\cdots\!32\)\(q^{96} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!98\)\(q^{97} \) \(\mathstrut -\mathstrut \)\(27\!\cdots\!96\)\(q^{98} \) \(\mathstrut -\mathstrut \)\(44\!\cdots\!96\)\(q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{9}\mathstrut -\mathstrut \) \(2\) \(x^{8}\mathstrut -\mathstrut \) \(68\!\cdots\!36\) \(x^{7}\mathstrut -\mathstrut \) \(18\!\cdots\!28\) \(x^{6}\mathstrut +\mathstrut \) \(14\!\cdots\!06\) \(x^{5}\mathstrut +\mathstrut \) \(61\!\cdots\!64\) \(x^{4}\mathstrut -\mathstrut \) \(98\!\cdots\!68\) \(x^{3}\mathstrut -\mathstrut \) \(28\!\cdots\!44\) \(x^{2}\mathstrut +\mathstrut \) \(18\!\cdots\!33\) \(x\mathstrut +\mathstrut \) \(32\!\cdots\!74\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 48 \nu - 11 \)
\(\beta_{2}\)\(=\)\((\)\(-\)\(69\!\cdots\!11\) \(\nu^{8}\mathstrut +\mathstrut \) \(15\!\cdots\!53\) \(\nu^{7}\mathstrut +\mathstrut \) \(40\!\cdots\!19\) \(\nu^{6}\mathstrut -\mathstrut \) \(80\!\cdots\!47\) \(\nu^{5}\mathstrut -\mathstrut \) \(59\!\cdots\!71\) \(\nu^{4}\mathstrut +\mathstrut \) \(10\!\cdots\!27\) \(\nu^{3}\mathstrut +\mathstrut \) \(16\!\cdots\!17\) \(\nu^{2}\mathstrut -\mathstrut \) \(44\!\cdots\!81\) \(\nu\mathstrut +\mathstrut \) \(15\!\cdots\!02\)\()/\)\(44\!\cdots\!52\)
\(\beta_{3}\)\(=\)\((\)\(79\!\cdots\!87\) \(\nu^{8}\mathstrut -\mathstrut \) \(17\!\cdots\!01\) \(\nu^{7}\mathstrut -\mathstrut \) \(46\!\cdots\!23\) \(\nu^{6}\mathstrut +\mathstrut \) \(91\!\cdots\!99\) \(\nu^{5}\mathstrut +\mathstrut \) \(67\!\cdots\!07\) \(\nu^{4}\mathstrut -\mathstrut \) \(11\!\cdots\!59\) \(\nu^{3}\mathstrut +\mathstrut \) \(14\!\cdots\!47\) \(\nu^{2}\mathstrut +\mathstrut \) \(36\!\cdots\!85\) \(\nu\mathstrut -\mathstrut \) \(53\!\cdots\!90\)\()/\)\(14\!\cdots\!84\)
\(\beta_{4}\)\(=\)\((\)\(-\)\(14\!\cdots\!31\) \(\nu^{8}\mathstrut -\mathstrut \) \(35\!\cdots\!19\) \(\nu^{7}\mathstrut +\mathstrut \) \(10\!\cdots\!47\) \(\nu^{6}\mathstrut +\mathstrut \) \(23\!\cdots\!65\) \(\nu^{5}\mathstrut -\mathstrut \) \(21\!\cdots\!71\) \(\nu^{4}\mathstrut -\mathstrut \) \(38\!\cdots\!05\) \(\nu^{3}\mathstrut +\mathstrut \) \(13\!\cdots\!53\) \(\nu^{2}\mathstrut +\mathstrut \) \(12\!\cdots\!67\) \(\nu\mathstrut -\mathstrut \) \(15\!\cdots\!06\)\()/\)\(46\!\cdots\!00\)
\(\beta_{5}\)\(=\)\((\)\(-\)\(55\!\cdots\!91\) \(\nu^{8}\mathstrut +\mathstrut \) \(19\!\cdots\!41\) \(\nu^{7}\mathstrut +\mathstrut \) \(34\!\cdots\!67\) \(\nu^{6}\mathstrut -\mathstrut \) \(10\!\cdots\!35\) \(\nu^{5}\mathstrut -\mathstrut \) \(57\!\cdots\!31\) \(\nu^{4}\mathstrut +\mathstrut \) \(13\!\cdots\!95\) \(\nu^{3}\mathstrut +\mathstrut \) \(27\!\cdots\!33\) \(\nu^{2}\mathstrut -\mathstrut \) \(34\!\cdots\!13\) \(\nu\mathstrut -\mathstrut \) \(47\!\cdots\!66\)\()/\)\(27\!\cdots\!00\)
\(\beta_{6}\)\(=\)\((\)\(33\!\cdots\!11\) \(\nu^{8}\mathstrut -\mathstrut \) \(11\!\cdots\!61\) \(\nu^{7}\mathstrut -\mathstrut \) \(18\!\cdots\!07\) \(\nu^{6}\mathstrut +\mathstrut \) \(60\!\cdots\!35\) \(\nu^{5}\mathstrut +\mathstrut \) \(27\!\cdots\!51\) \(\nu^{4}\mathstrut -\mathstrut \) \(70\!\cdots\!95\) \(\nu^{3}\mathstrut -\mathstrut \) \(11\!\cdots\!93\) \(\nu^{2}\mathstrut +\mathstrut \) \(17\!\cdots\!73\) \(\nu\mathstrut +\mathstrut \) \(11\!\cdots\!86\)\()/\)\(69\!\cdots\!00\)
\(\beta_{7}\)\(=\)\((\)\(-\)\(95\!\cdots\!97\) \(\nu^{8}\mathstrut +\mathstrut \) \(14\!\cdots\!47\) \(\nu^{7}\mathstrut +\mathstrut \) \(31\!\cdots\!89\) \(\nu^{6}\mathstrut -\mathstrut \) \(85\!\cdots\!45\) \(\nu^{5}\mathstrut +\mathstrut \) \(32\!\cdots\!23\) \(\nu^{4}\mathstrut +\mathstrut \) \(13\!\cdots\!65\) \(\nu^{3}\mathstrut -\mathstrut \) \(10\!\cdots\!89\) \(\nu^{2}\mathstrut -\mathstrut \) \(76\!\cdots\!71\) \(\nu\mathstrut +\mathstrut \) \(31\!\cdots\!78\)\()/\)\(12\!\cdots\!00\)
\(\beta_{8}\)\(=\)\((\)\(50\!\cdots\!01\) \(\nu^{8}\mathstrut -\mathstrut \) \(85\!\cdots\!51\) \(\nu^{7}\mathstrut -\mathstrut \) \(27\!\cdots\!37\) \(\nu^{6}\mathstrut +\mathstrut \) \(28\!\cdots\!85\) \(\nu^{5}\mathstrut +\mathstrut \) \(33\!\cdots\!41\) \(\nu^{4}\mathstrut +\mathstrut \) \(82\!\cdots\!55\) \(\nu^{3}\mathstrut -\mathstrut \) \(48\!\cdots\!63\) \(\nu^{2}\mathstrut -\mathstrut \) \(24\!\cdots\!57\) \(\nu\mathstrut -\mathstrut \) \(26\!\cdots\!74\)\()/\)\(55\!\cdots\!00\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1}\mathstrut +\mathstrut \) \(11\)\()/48\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3}\mathstrut +\mathstrut \) \(3414351\) \(\beta_{2}\mathstrut +\mathstrut \) \(189828498139459931\) \(\beta_{1}\mathstrut +\mathstrut \) \(3530888858815588277459233691606826100\)\()/2304\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{7}\mathstrut -\mathstrut \) \(100\) \(\beta_{6}\mathstrut -\mathstrut \) \(8716729\) \(\beta_{5}\mathstrut +\mathstrut \) \(611104015289\) \(\beta_{4}\mathstrut +\mathstrut \) \(261633278718987930\) \(\beta_{3}\mathstrut -\mathstrut \) \(7852379881357415610550873\) \(\beta_{2}\mathstrut +\mathstrut \) \(6392297614796539847227904912944140426\) \(\beta_{1}\mathstrut +\mathstrut \) \(670263329160507439760026724614221985787578575603698103\)\()/110592\)
\(\nu^{4}\)\(=\)\((\)\(-\)\(33743922161664\) \(\beta_{8}\mathstrut +\mathstrut \) \(22994817386914953\) \(\beta_{7}\mathstrut +\mathstrut \) \(185098793423871777404\) \(\beta_{6}\mathstrut +\mathstrut \) \(2504204149487045150693375\) \(\beta_{5}\mathstrut -\mathstrut \) \(30109893022878964364258792959\) \(\beta_{4}\mathstrut +\mathstrut \) \(576961848958890314932551097193358490\) \(\beta_{3}\mathstrut +\mathstrut \) \(2924045494290224782521215472492163016111407\) \(\beta_{2}\mathstrut +\mathstrut \) \(160690031202447687101840435956556935214181300547721706\) \(\beta_{1}\mathstrut +\mathstrut \) \(1410655776895744825634505644465995221491836208865564342313094813099015599\)\()/331776\)
\(\nu^{5}\)\(=\)\((\)\(-\)\(977710028302938496794410972160\) \(\beta_{8}\mathstrut +\mathstrut \) \(10982027054486582087150658654706277\) \(\beta_{7}\mathstrut -\mathstrut \) \(804239144813557581512932253680820084\) \(\beta_{6}\mathstrut -\mathstrut \) \(61749678315799720816719286887561336919037\) \(\beta_{5}\mathstrut +\mathstrut \) \(6885741500484871576889697797771796121426834941\) \(\beta_{4}\mathstrut +\mathstrut \) \(3515690829166565982410567204916235296549117532215482\) \(\beta_{3}\mathstrut -\mathstrut \) \(178763837174342394896597444434978400630110164948157560128405\) \(\beta_{2}\mathstrut +\mathstrut \) \(48077367563605355957548557806232488176704933143418647772251942064726234\) \(\beta_{1}\mathstrut +\mathstrut \) \(8865291263913725629783995190494740789504015754670068598075343284757366059693509637005139\)\()/248832\)
\(\nu^{6}\)\(=\)\((\)\(-\)\(86\!\cdots\!80\) \(\beta_{8}\mathstrut +\mathstrut \) \(83\!\cdots\!21\) \(\beta_{7}\mathstrut +\mathstrut \) \(49\!\cdots\!88\) \(\beta_{6}\mathstrut +\mathstrut \) \(75\!\cdots\!75\) \(\beta_{5}\mathstrut -\mathstrut \) \(47\!\cdots\!83\) \(\beta_{4}\mathstrut +\mathstrut \) \(11\!\cdots\!94\) \(\beta_{3}\mathstrut +\mathstrut \) \(47\!\cdots\!79\) \(\beta_{2}\mathstrut +\mathstrut \) \(37\!\cdots\!66\) \(\beta_{1}\mathstrut +\mathstrut \) \(23\!\cdots\!35\)\()/165888\)
\(\nu^{7}\)\(=\)\((\)\(-\)\(78\!\cdots\!60\) \(\beta_{8}\mathstrut +\mathstrut \) \(61\!\cdots\!01\) \(\beta_{7}\mathstrut -\mathstrut \) \(13\!\cdots\!28\) \(\beta_{6}\mathstrut -\mathstrut \) \(16\!\cdots\!49\) \(\beta_{5}\mathstrut +\mathstrut \) \(37\!\cdots\!97\) \(\beta_{4}\mathstrut +\mathstrut \) \(23\!\cdots\!10\) \(\beta_{3}\mathstrut -\mathstrut \) \(12\!\cdots\!21\) \(\beta_{2}\mathstrut +\mathstrut \) \(23\!\cdots\!82\) \(\beta_{1}\mathstrut +\mathstrut \) \(54\!\cdots\!07\)\()/331776\)
\(\nu^{8}\)\(=\)\((\)\(-\)\(75\!\cdots\!56\) \(\beta_{8}\mathstrut +\mathstrut \) \(90\!\cdots\!07\) \(\beta_{7}\mathstrut +\mathstrut \) \(42\!\cdots\!40\) \(\beta_{6}\mathstrut +\mathstrut \) \(69\!\cdots\!21\) \(\beta_{5}\mathstrut -\mathstrut \) \(25\!\cdots\!41\) \(\beta_{4}\mathstrut +\mathstrut \) \(84\!\cdots\!38\) \(\beta_{3}\mathstrut +\mathstrut \) \(25\!\cdots\!05\) \(\beta_{2}\mathstrut +\mathstrut \) \(33\!\cdots\!26\) \(\beta_{1}\mathstrut +\mathstrut \) \(17\!\cdots\!21\)\()/331776\)

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
−6.03020e16
−3.77468e16
−3.52210e16
−1.87667e16
−1.78332e14
1.54621e16
2.68569e16
4.65892e16
6.33066e16
−3.15505e18 6.48457e28 7.29589e36 −2.23954e41 −2.04592e47 1.98659e51 −1.46313e55 −1.18606e57 7.06586e59
1.2 −2.07240e18 −2.64580e28 1.63640e36 −5.16214e41 5.48316e46 −2.39892e51 2.11812e54 −4.69101e57 1.06980e60
1.3 −1.95116e18 −1.11766e29 1.14858e36 9.20370e41 2.18074e47 2.38466e51 2.94601e54 7.10063e57 −1.79579e60
1.4 −1.16136e18 1.09371e29 −1.30970e36 1.47325e42 −1.27019e47 −1.67990e50 4.60846e54 6.57089e57 −1.71098e60
1.5 −2.69116e17 3.14043e28 −2.58603e36 −3.76295e42 −8.45141e45 1.39208e51 1.41138e54 −4.40480e57 1.01267e60
1.6 4.81626e17 −2.96062e28 −2.42649e36 2.39673e42 −1.42591e46 2.39069e50 −2.44905e54 −4.51451e57 1.15433e60
1.7 1.02857e18 −1.22395e29 −1.60049e36 −2.11004e42 −1.25893e47 −1.42065e51 −4.38064e54 9.58953e57 −2.17033e60
1.8 1.97572e18 8.85385e28 1.24503e36 −2.55522e41 1.74928e47 −5.36128e50 −2.79253e54 2.44803e57 −5.04842e59
1.9 2.77816e18 −4.92324e28 5.05972e36 1.85701e41 −1.36775e47 7.17028e50 6.67109e54 −2.96721e57 5.15907e59
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
Significant digits:
Format:

Inner twists

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

Hecke kernels

There are no other newforms in \(S_{122}^{\mathrm{new}}(\Gamma_0(1))\).