Properties

Label 1.100.a.a
Level 1
Weight 100
Character orbit 1.a
Self dual Yes
Analytic conductor 62.068
Analytic rank 0
Dimension 8
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 1 \)
Weight: \( k \) = \( 100 \)
Character orbit: \([\chi]\) = 1.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(62.0676682981\)
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 \( 2^{109}\cdot 3^{44}\cdot 5^{13}\cdot 7^{9}\cdot 11^{3}\cdot 13\cdot 17 \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(f(q)\) \(=\) \( q\) \(+(-26005077112815 + \beta_{1}) q^{2}\) \(+(-\)\(35\!\cdots\!40\)\( - 10743446 \beta_{1} + \beta_{2}) q^{3}\) \(+(\)\(35\!\cdots\!28\)\( + 67444696288016 \beta_{1} - 111186 \beta_{2} + \beta_{3}) q^{4}\) \(+(-\)\(61\!\cdots\!70\)\( - 2800027452536314589 \beta_{1} + 27291602405 \beta_{2} - 11317 \beta_{3} - \beta_{4}) q^{5}\) \(+(-\)\(97\!\cdots\!68\)\( - \)\(55\!\cdots\!35\)\( \beta_{1} + 44505423613750 \beta_{2} - 136753408 \beta_{3} - 1016 \beta_{4} + \beta_{5}) q^{6}\) \(+(-\)\(71\!\cdots\!00\)\( - \)\(34\!\cdots\!87\)\( \beta_{1} - 3095970665067780 \beta_{2} - 194675541824 \beta_{3} + 2381438 \beta_{4} + 446 \beta_{5} - \beta_{6}) q^{7}\) \(+(\)\(74\!\cdots\!20\)\( + \)\(44\!\cdots\!22\)\( \beta_{1} - \)\(52\!\cdots\!58\)\( \beta_{2} - 92874580903099 \beta_{3} + 3005209639 \beta_{4} - 237529 \beta_{5} - 117 \beta_{6} + \beta_{7}) q^{8}\) \(+(\)\(19\!\cdots\!97\)\( + \)\(13\!\cdots\!98\)\( \beta_{1} - \)\(23\!\cdots\!02\)\( \beta_{2} - 42699188886526062 \beta_{3} + 501920451426 \beta_{4} + 102829320 \beta_{5} - 174420 \beta_{6} + 48 \beta_{7}) q^{9}\) \(+O(q^{10})\) \( q\) \(+(-26005077112815 + \beta_{1}) q^{2}\) \(+(-\)\(35\!\cdots\!40\)\( - 10743446 \beta_{1} + \beta_{2}) q^{3}\) \(+(\)\(35\!\cdots\!28\)\( + 67444696288016 \beta_{1} - 111186 \beta_{2} + \beta_{3}) q^{4}\) \(+(-\)\(61\!\cdots\!70\)\( - 2800027452536314589 \beta_{1} + 27291602405 \beta_{2} - 11317 \beta_{3} - \beta_{4}) q^{5}\) \(+(-\)\(97\!\cdots\!68\)\( - \)\(55\!\cdots\!35\)\( \beta_{1} + 44505423613750 \beta_{2} - 136753408 \beta_{3} - 1016 \beta_{4} + \beta_{5}) q^{6}\) \(+(-\)\(71\!\cdots\!00\)\( - \)\(34\!\cdots\!87\)\( \beta_{1} - 3095970665067780 \beta_{2} - 194675541824 \beta_{3} + 2381438 \beta_{4} + 446 \beta_{5} - \beta_{6}) q^{7}\) \(+(\)\(74\!\cdots\!20\)\( + \)\(44\!\cdots\!22\)\( \beta_{1} - \)\(52\!\cdots\!58\)\( \beta_{2} - 92874580903099 \beta_{3} + 3005209639 \beta_{4} - 237529 \beta_{5} - 117 \beta_{6} + \beta_{7}) q^{8}\) \(+(\)\(19\!\cdots\!97\)\( + \)\(13\!\cdots\!98\)\( \beta_{1} - \)\(23\!\cdots\!02\)\( \beta_{2} - 42699188886526062 \beta_{3} + 501920451426 \beta_{4} + 102829320 \beta_{5} - 174420 \beta_{6} + 48 \beta_{7}) q^{9}\) \(+(-\)\(26\!\cdots\!70\)\( - \)\(14\!\cdots\!10\)\( \beta_{1} + \)\(15\!\cdots\!04\)\( \beta_{2} - 10027815968442194416 \beta_{3} - 46716768015664 \beta_{4} + 61764223076 \beta_{5} - 31987600 \beta_{6} - 25776 \beta_{7}) q^{10}\) \(+(\)\(83\!\cdots\!92\)\( + \)\(81\!\cdots\!80\)\( \beta_{1} - \)\(79\!\cdots\!21\)\( \beta_{2} - \)\(99\!\cdots\!48\)\( \beta_{3} - 19100277639848132 \beta_{4} + 7359257331196 \beta_{5} + 3164749470 \beta_{6} + 2821952 \beta_{7}) q^{11}\) \(+(-\)\(32\!\cdots\!80\)\( - \)\(10\!\cdots\!20\)\( \beta_{1} + \)\(18\!\cdots\!48\)\( \beta_{2} - \)\(65\!\cdots\!84\)\( \beta_{3} - 3384274933769603616 \beta_{4} + 358620194605536 \beta_{5} - 59025596832 \beta_{6} - 180582624 \beta_{7}) q^{12}\) \(+(-\)\(66\!\cdots\!30\)\( + \)\(43\!\cdots\!31\)\( \beta_{1} + \)\(23\!\cdots\!97\)\( \beta_{2} - \)\(45\!\cdots\!37\)\( \beta_{3} - \)\(23\!\cdots\!17\)\( \beta_{4} - 29370066793870352 \beta_{5} - 4406514269912 \beta_{6} + 8179503264 \beta_{7}) q^{13}\) \(+(-\)\(34\!\cdots\!24\)\( - \)\(14\!\cdots\!98\)\( \beta_{1} + \)\(40\!\cdots\!76\)\( \beta_{2} + \)\(69\!\cdots\!04\)\( \beta_{3} - \)\(11\!\cdots\!28\)\( \beta_{4} + 191014078234987786 \beta_{5} + 345235597957440 \beta_{6} - 285208845376 \beta_{7}) q^{14}\) \(+(\)\(49\!\cdots\!60\)\( + \)\(30\!\cdots\!35\)\( \beta_{1} - \)\(79\!\cdots\!52\)\( \beta_{2} + \)\(63\!\cdots\!48\)\( \beta_{3} + \)\(15\!\cdots\!02\)\( \beta_{4} + 23770787889389594162 \beta_{5} - 12736493350453575 \beta_{6} + 8028484034688 \beta_{7}) q^{15}\) \(+(\)\(21\!\cdots\!76\)\( + \)\(17\!\cdots\!16\)\( \beta_{1} - \)\(15\!\cdots\!92\)\( \beta_{2} + \)\(73\!\cdots\!04\)\( \beta_{3} + \)\(57\!\cdots\!60\)\( \beta_{4} - \)\(79\!\cdots\!76\)\( \beta_{5} + 312042417392207320 \beta_{6} - 188094706545528 \beta_{7}) q^{16}\) \(+(\)\(37\!\cdots\!30\)\( + \)\(15\!\cdots\!02\)\( \beta_{1} - \)\(37\!\cdots\!66\)\( \beta_{2} + \)\(78\!\cdots\!14\)\( \beta_{3} - \)\(59\!\cdots\!58\)\( \beta_{4} + \)\(10\!\cdots\!44\)\( \beta_{5} - 5518382420607379524 \beta_{6} + 3745229142683760 \beta_{7}) q^{17}\) \(+(\)\(13\!\cdots\!85\)\( - \)\(26\!\cdots\!51\)\( \beta_{1} + \)\(90\!\cdots\!20\)\( \beta_{2} + \)\(74\!\cdots\!56\)\( \beta_{3} - \)\(17\!\cdots\!32\)\( \beta_{4} + \)\(83\!\cdots\!76\)\( \beta_{5} + 70804221690681945504 \beta_{6} - 64327834425641760 \beta_{7}) q^{18}\) \(+(-\)\(32\!\cdots\!40\)\( + \)\(50\!\cdots\!04\)\( \beta_{1} - \)\(23\!\cdots\!27\)\( \beta_{2} - \)\(10\!\cdots\!20\)\( \beta_{3} + \)\(33\!\cdots\!24\)\( \beta_{4} - \)\(24\!\cdots\!52\)\( \beta_{5} - \)\(59\!\cdots\!90\)\( \beta_{6} + 963297526631656896 \beta_{7}) q^{19}\) \(+(-\)\(10\!\cdots\!60\)\( - \)\(96\!\cdots\!12\)\( \beta_{1} + \)\(35\!\cdots\!00\)\( \beta_{2} - \)\(28\!\cdots\!26\)\( \beta_{3} - \)\(48\!\cdots\!68\)\( \beta_{4} + \)\(42\!\cdots\!40\)\( \beta_{5} + \)\(12\!\cdots\!00\)\( \beta_{6} - 12669558319686675840 \beta_{7}) q^{20}\) \(+(\)\(90\!\cdots\!52\)\( + \)\(73\!\cdots\!36\)\( \beta_{1} + \)\(47\!\cdots\!00\)\( \beta_{2} + \)\(70\!\cdots\!60\)\( \beta_{3} - \)\(31\!\cdots\!36\)\( \beta_{4} - \)\(36\!\cdots\!28\)\( \beta_{5} + \)\(56\!\cdots\!80\)\( \beta_{6} + \)\(14\!\cdots\!08\)\( \beta_{7}) q^{21}\) \(+(\)\(80\!\cdots\!20\)\( - \)\(53\!\cdots\!09\)\( \beta_{1} + \)\(59\!\cdots\!62\)\( \beta_{2} + \)\(28\!\cdots\!68\)\( \beta_{3} + \)\(23\!\cdots\!28\)\( \beta_{4} + \)\(11\!\cdots\!03\)\( \beta_{5} - \)\(11\!\cdots\!72\)\( \beta_{6} - \)\(15\!\cdots\!56\)\( \beta_{7}) q^{22}\) \(+(-\)\(73\!\cdots\!20\)\( + \)\(51\!\cdots\!35\)\( \beta_{1} + \)\(39\!\cdots\!24\)\( \beta_{2} - \)\(10\!\cdots\!00\)\( \beta_{3} + \)\(12\!\cdots\!10\)\( \beta_{4} + \)\(13\!\cdots\!50\)\( \beta_{5} + \)\(14\!\cdots\!65\)\( \beta_{6} + \)\(13\!\cdots\!60\)\( \beta_{7}) q^{23}\) \(+(-\)\(98\!\cdots\!20\)\( - \)\(57\!\cdots\!52\)\( \beta_{1} + \)\(31\!\cdots\!76\)\( \beta_{2} - \)\(20\!\cdots\!28\)\( \beta_{3} - \)\(23\!\cdots\!48\)\( \beta_{4} - \)\(20\!\cdots\!68\)\( \beta_{5} - \)\(13\!\cdots\!80\)\( \beta_{6} - \)\(10\!\cdots\!68\)\( \beta_{7}) q^{24}\) \(+(\)\(89\!\cdots\!75\)\( + \)\(32\!\cdots\!20\)\( \beta_{1} - \)\(16\!\cdots\!20\)\( \beta_{2} - \)\(13\!\cdots\!60\)\( \beta_{3} + \)\(56\!\cdots\!00\)\( \beta_{4} + \)\(10\!\cdots\!20\)\( \beta_{5} + \)\(89\!\cdots\!00\)\( \beta_{6} + \)\(73\!\cdots\!80\)\( \beta_{7}) q^{25}\) \(+(\)\(43\!\cdots\!82\)\( - \)\(33\!\cdots\!86\)\( \beta_{1} - \)\(19\!\cdots\!36\)\( \beta_{2} + \)\(10\!\cdots\!72\)\( \beta_{3} + \)\(94\!\cdots\!04\)\( \beta_{4} + \)\(18\!\cdots\!04\)\( \beta_{5} - \)\(40\!\cdots\!60\)\( \beta_{6} - \)\(41\!\cdots\!16\)\( \beta_{7}) q^{26}\) \(+(\)\(17\!\cdots\!80\)\( + \)\(17\!\cdots\!18\)\( \beta_{1} - \)\(72\!\cdots\!82\)\( \beta_{2} + \)\(51\!\cdots\!84\)\( \beta_{3} - \)\(71\!\cdots\!84\)\( \beta_{4} - \)\(71\!\cdots\!36\)\( \beta_{5} + \)\(58\!\cdots\!82\)\( \beta_{6} + \)\(17\!\cdots\!24\)\( \beta_{7}) q^{27}\) \(+(-\)\(14\!\cdots\!80\)\( + \)\(14\!\cdots\!84\)\( \beta_{1} + \)\(23\!\cdots\!92\)\( \beta_{2} - \)\(41\!\cdots\!76\)\( \beta_{3} - \)\(54\!\cdots\!24\)\( \beta_{4} + \)\(53\!\cdots\!04\)\( \beta_{5} + \)\(10\!\cdots\!52\)\( \beta_{6} - \)\(24\!\cdots\!36\)\( \beta_{7}) q^{28}\) \(+(-\)\(54\!\cdots\!10\)\( + \)\(71\!\cdots\!23\)\( \beta_{1} + \)\(30\!\cdots\!85\)\( \beta_{2} + \)\(25\!\cdots\!11\)\( \beta_{3} + \)\(29\!\cdots\!19\)\( \beta_{4} - \)\(15\!\cdots\!76\)\( \beta_{5} - \)\(12\!\cdots\!60\)\( \beta_{6} - \)\(41\!\cdots\!36\)\( \beta_{7}) q^{29}\) \(+(\)\(29\!\cdots\!60\)\( + \)\(99\!\cdots\!22\)\( \beta_{1} + \)\(12\!\cdots\!00\)\( \beta_{2} + \)\(10\!\cdots\!56\)\( \beta_{3} - \)\(11\!\cdots\!92\)\( \beta_{4} - \)\(70\!\cdots\!90\)\( \beta_{5} + \)\(81\!\cdots\!00\)\( \beta_{6} + \)\(54\!\cdots\!40\)\( \beta_{7}) q^{30}\) \(+(-\)\(53\!\cdots\!28\)\( + \)\(25\!\cdots\!12\)\( \beta_{1} - \)\(40\!\cdots\!36\)\( \beta_{2} - \)\(15\!\cdots\!76\)\( \beta_{3} - \)\(33\!\cdots\!60\)\( \beta_{4} + \)\(99\!\cdots\!36\)\( \beta_{5} - \)\(36\!\cdots\!20\)\( \beta_{6} - \)\(43\!\cdots\!92\)\( \beta_{7}) q^{31}\) \(+(-\)\(35\!\cdots\!40\)\( + \)\(45\!\cdots\!36\)\( \beta_{1} - \)\(68\!\cdots\!24\)\( \beta_{2} - \)\(15\!\cdots\!76\)\( \beta_{3} + \)\(44\!\cdots\!52\)\( \beta_{4} - \)\(50\!\cdots\!96\)\( \beta_{5} + \)\(99\!\cdots\!36\)\( \beta_{6} + \)\(26\!\cdots\!40\)\( \beta_{7}) q^{32}\) \(+(-\)\(14\!\cdots\!80\)\( + \)\(10\!\cdots\!02\)\( \beta_{1} + \)\(15\!\cdots\!22\)\( \beta_{2} - \)\(35\!\cdots\!50\)\( \beta_{3} - \)\(11\!\cdots\!66\)\( \beta_{4} + \)\(10\!\cdots\!00\)\( \beta_{5} - \)\(16\!\cdots\!44\)\( \beta_{6} - \)\(13\!\cdots\!16\)\( \beta_{7}) q^{33}\) \(+(\)\(14\!\cdots\!46\)\( + \)\(94\!\cdots\!62\)\( \beta_{1} + \)\(36\!\cdots\!20\)\( \beta_{2} + \)\(30\!\cdots\!08\)\( \beta_{3} - \)\(36\!\cdots\!56\)\( \beta_{4} + \)\(34\!\cdots\!68\)\( \beta_{5} - \)\(12\!\cdots\!40\)\( \beta_{6} + \)\(56\!\cdots\!16\)\( \beta_{7}) q^{34}\) \(+(-\)\(16\!\cdots\!20\)\( + \)\(18\!\cdots\!20\)\( \beta_{1} + \)\(35\!\cdots\!84\)\( \beta_{2} + \)\(85\!\cdots\!04\)\( \beta_{3} + \)\(24\!\cdots\!76\)\( \beta_{4} - \)\(31\!\cdots\!04\)\( \beta_{5} + \)\(48\!\cdots\!00\)\( \beta_{6} - \)\(17\!\cdots\!96\)\( \beta_{7}) q^{35}\) \(+(-\)\(28\!\cdots\!84\)\( + \)\(53\!\cdots\!64\)\( \beta_{1} + \)\(43\!\cdots\!26\)\( \beta_{2} - \)\(91\!\cdots\!79\)\( \beta_{3} + \)\(97\!\cdots\!44\)\( \beta_{4} + \)\(75\!\cdots\!96\)\( \beta_{5} + \)\(13\!\cdots\!00\)\( \beta_{6} + \)\(34\!\cdots\!40\)\( \beta_{7}) q^{36}\) \(+(\)\(87\!\cdots\!90\)\( - \)\(50\!\cdots\!17\)\( \beta_{1} - \)\(38\!\cdots\!91\)\( \beta_{2} - \)\(87\!\cdots\!57\)\( \beta_{3} - \)\(38\!\cdots\!93\)\( \beta_{4} + \)\(19\!\cdots\!28\)\( \beta_{5} - \)\(25\!\cdots\!36\)\( \beta_{6} + \)\(36\!\cdots\!48\)\( \beta_{7}) q^{37}\) \(+(\)\(51\!\cdots\!80\)\( - \)\(10\!\cdots\!35\)\( \beta_{1} - \)\(15\!\cdots\!66\)\( \beta_{2} + \)\(17\!\cdots\!08\)\( \beta_{3} + \)\(38\!\cdots\!60\)\( \beta_{4} - \)\(18\!\cdots\!07\)\( \beta_{5} + \)\(16\!\cdots\!96\)\( \beta_{6} - \)\(70\!\cdots\!44\)\( \beta_{7}) q^{38}\) \(+(\)\(38\!\cdots\!64\)\( - \)\(35\!\cdots\!97\)\( \beta_{1} - \)\(60\!\cdots\!44\)\( \beta_{2} - \)\(17\!\cdots\!60\)\( \beta_{3} + \)\(64\!\cdots\!98\)\( \beta_{4} + \)\(41\!\cdots\!26\)\( \beta_{5} - \)\(67\!\cdots\!55\)\( \beta_{6} + \)\(37\!\cdots\!32\)\( \beta_{7}) q^{39}\) \(+(-\)\(76\!\cdots\!00\)\( - \)\(23\!\cdots\!60\)\( \beta_{1} + \)\(38\!\cdots\!80\)\( \beta_{2} - \)\(14\!\cdots\!50\)\( \beta_{3} - \)\(27\!\cdots\!70\)\( \beta_{4} + \)\(14\!\cdots\!70\)\( \beta_{5} + \)\(17\!\cdots\!50\)\( \beta_{6} - \)\(12\!\cdots\!70\)\( \beta_{7}) q^{40}\) \(+(\)\(21\!\cdots\!62\)\( - \)\(86\!\cdots\!56\)\( \beta_{1} + \)\(20\!\cdots\!56\)\( \beta_{2} + \)\(13\!\cdots\!72\)\( \beta_{3} + \)\(70\!\cdots\!16\)\( \beta_{4} - \)\(13\!\cdots\!56\)\( \beta_{5} - \)\(18\!\cdots\!00\)\( \beta_{6} + \)\(22\!\cdots\!60\)\( \beta_{7}) q^{41}\) \(+(\)\(73\!\cdots\!80\)\( + \)\(47\!\cdots\!40\)\( \beta_{1} - \)\(26\!\cdots\!32\)\( \beta_{2} + \)\(10\!\cdots\!36\)\( \beta_{3} + \)\(22\!\cdots\!88\)\( \beta_{4} + \)\(44\!\cdots\!56\)\( \beta_{5} - \)\(89\!\cdots\!56\)\( \beta_{6} + \)\(21\!\cdots\!20\)\( \beta_{7}) q^{42}\) \(+(-\)\(14\!\cdots\!00\)\( + \)\(36\!\cdots\!02\)\( \beta_{1} - \)\(55\!\cdots\!17\)\( \beta_{2} + \)\(19\!\cdots\!44\)\( \beta_{3} - \)\(43\!\cdots\!32\)\( \beta_{4} - \)\(76\!\cdots\!76\)\( \beta_{5} + \)\(55\!\cdots\!20\)\( \beta_{6} - \)\(35\!\cdots\!04\)\( \beta_{7}) q^{43}\) \(+(-\)\(60\!\cdots\!24\)\( + \)\(21\!\cdots\!24\)\( \beta_{1} - \)\(55\!\cdots\!68\)\( \beta_{2} - \)\(21\!\cdots\!04\)\( \beta_{3} - \)\(93\!\cdots\!60\)\( \beta_{4} + \)\(76\!\cdots\!56\)\( \beta_{5} - \)\(15\!\cdots\!20\)\( \beta_{6} + \)\(14\!\cdots\!68\)\( \beta_{7}) q^{44}\) \(+(-\)\(52\!\cdots\!90\)\( + \)\(23\!\cdots\!27\)\( \beta_{1} + \)\(82\!\cdots\!05\)\( \beta_{2} + \)\(11\!\cdots\!51\)\( \beta_{3} + \)\(22\!\cdots\!23\)\( \beta_{4} - \)\(49\!\cdots\!20\)\( \beta_{5} + \)\(19\!\cdots\!00\)\( \beta_{6} - \)\(33\!\cdots\!80\)\( \beta_{7}) q^{45}\) \(+(\)\(51\!\cdots\!32\)\( - \)\(36\!\cdots\!06\)\( \beta_{1} + \)\(10\!\cdots\!96\)\( \beta_{2} + \)\(10\!\cdots\!52\)\( \beta_{3} + \)\(13\!\cdots\!16\)\( \beta_{4} + \)\(35\!\cdots\!34\)\( \beta_{5} + \)\(19\!\cdots\!00\)\( \beta_{6} + \)\(12\!\cdots\!80\)\( \beta_{7}) q^{46}\) \(+(\)\(36\!\cdots\!20\)\( - \)\(13\!\cdots\!54\)\( \beta_{1} - \)\(29\!\cdots\!24\)\( \beta_{2} + \)\(12\!\cdots\!48\)\( \beta_{3} - \)\(53\!\cdots\!44\)\( \beta_{4} - \)\(12\!\cdots\!92\)\( \beta_{5} - \)\(11\!\cdots\!10\)\( \beta_{6} + \)\(25\!\cdots\!32\)\( \beta_{7}) q^{47}\) \(+(-\)\(33\!\cdots\!20\)\( - \)\(18\!\cdots\!20\)\( \beta_{1} - \)\(12\!\cdots\!40\)\( \beta_{2} - \)\(10\!\cdots\!76\)\( \beta_{3} - \)\(19\!\cdots\!68\)\( \beta_{4} + \)\(16\!\cdots\!04\)\( \beta_{5} + \)\(79\!\cdots\!56\)\( \beta_{6} - \)\(11\!\cdots\!80\)\( \beta_{7}) q^{48}\) \(+(-\)\(17\!\cdots\!07\)\( - \)\(15\!\cdots\!56\)\( \beta_{1} - \)\(30\!\cdots\!64\)\( \beta_{2} - \)\(16\!\cdots\!08\)\( \beta_{3} + \)\(15\!\cdots\!36\)\( \beta_{4} + \)\(28\!\cdots\!04\)\( \beta_{5} + \)\(44\!\cdots\!00\)\( \beta_{6} + \)\(25\!\cdots\!00\)\( \beta_{7}) q^{49}\) \(+(\)\(31\!\cdots\!75\)\( + \)\(54\!\cdots\!95\)\( \beta_{1} + \)\(69\!\cdots\!80\)\( \beta_{2} + \)\(11\!\cdots\!40\)\( \beta_{3} + \)\(10\!\cdots\!00\)\( \beta_{4} - \)\(17\!\cdots\!80\)\( \beta_{5} + \)\(42\!\cdots\!00\)\( \beta_{6} - \)\(83\!\cdots\!20\)\( \beta_{7}) q^{50}\) \(+(-\)\(21\!\cdots\!08\)\( + \)\(47\!\cdots\!66\)\( \beta_{1} + \)\(58\!\cdots\!42\)\( \beta_{2} - \)\(30\!\cdots\!00\)\( \beta_{3} - \)\(66\!\cdots\!44\)\( \beta_{4} + \)\(32\!\cdots\!32\)\( \beta_{5} - \)\(12\!\cdots\!10\)\( \beta_{6} - \)\(16\!\cdots\!16\)\( \beta_{7}) q^{51}\) \(+(-\)\(24\!\cdots\!00\)\( + \)\(92\!\cdots\!60\)\( \beta_{1} - \)\(70\!\cdots\!52\)\( \beta_{2} - \)\(22\!\cdots\!22\)\( \beta_{3} + \)\(12\!\cdots\!16\)\( \beta_{4} - \)\(40\!\cdots\!12\)\( \beta_{5} + \)\(50\!\cdots\!40\)\( \beta_{6} + \)\(65\!\cdots\!52\)\( \beta_{7}) q^{52}\) \(+(-\)\(37\!\cdots\!90\)\( + \)\(52\!\cdots\!35\)\( \beta_{1} - \)\(13\!\cdots\!75\)\( \beta_{2} - \)\(13\!\cdots\!09\)\( \beta_{3} - \)\(93\!\cdots\!61\)\( \beta_{4} + \)\(17\!\cdots\!36\)\( \beta_{5} - \)\(66\!\cdots\!12\)\( \beta_{6} - \)\(11\!\cdots\!44\)\( \beta_{7}) q^{53}\) \(+(\)\(17\!\cdots\!60\)\( + \)\(83\!\cdots\!66\)\( \beta_{1} - \)\(54\!\cdots\!00\)\( \beta_{2} + \)\(36\!\cdots\!88\)\( \beta_{3} + \)\(46\!\cdots\!00\)\( \beta_{4} - \)\(71\!\cdots\!54\)\( \beta_{5} - \)\(20\!\cdots\!20\)\( \beta_{6} - \)\(48\!\cdots\!32\)\( \beta_{7}) q^{54}\) \(+(\)\(28\!\cdots\!60\)\( - \)\(30\!\cdots\!63\)\( \beta_{1} + \)\(86\!\cdots\!60\)\( \beta_{2} + \)\(30\!\cdots\!36\)\( \beta_{3} - \)\(27\!\cdots\!42\)\( \beta_{4} + \)\(94\!\cdots\!50\)\( \beta_{5} + \)\(11\!\cdots\!75\)\( \beta_{6} + \)\(87\!\cdots\!00\)\( \beta_{7}) q^{55}\) \(+(\)\(40\!\cdots\!40\)\( - \)\(34\!\cdots\!80\)\( \beta_{1} + \)\(35\!\cdots\!52\)\( \beta_{2} - \)\(28\!\cdots\!76\)\( \beta_{3} + \)\(50\!\cdots\!60\)\( \beta_{4} + \)\(31\!\cdots\!32\)\( \beta_{5} - \)\(24\!\cdots\!40\)\( \beta_{6} - \)\(24\!\cdots\!64\)\( \beta_{7}) q^{56}\) \(+(-\)\(34\!\cdots\!40\)\( - \)\(27\!\cdots\!86\)\( \beta_{1} + \)\(22\!\cdots\!50\)\( \beta_{2} - \)\(45\!\cdots\!46\)\( \beta_{3} + \)\(18\!\cdots\!38\)\( \beta_{4} - \)\(14\!\cdots\!16\)\( \beta_{5} - \)\(52\!\cdots\!80\)\( \beta_{6} + \)\(22\!\cdots\!36\)\( \beta_{7}) q^{57}\) \(+(\)\(71\!\cdots\!70\)\( + \)\(13\!\cdots\!74\)\( \beta_{1} - \)\(14\!\cdots\!48\)\( \beta_{2} + \)\(24\!\cdots\!96\)\( \beta_{3} - \)\(12\!\cdots\!04\)\( \beta_{4} + \)\(18\!\cdots\!16\)\( \beta_{5} + \)\(12\!\cdots\!36\)\( \beta_{6} + \)\(64\!\cdots\!48\)\( \beta_{7}) q^{58}\) \(+(\)\(18\!\cdots\!80\)\( + \)\(69\!\cdots\!94\)\( \beta_{1} - \)\(25\!\cdots\!85\)\( \beta_{2} + \)\(99\!\cdots\!00\)\( \beta_{3} - \)\(20\!\cdots\!84\)\( \beta_{4} + \)\(18\!\cdots\!68\)\( \beta_{5} - \)\(31\!\cdots\!80\)\( \beta_{6} - \)\(28\!\cdots\!48\)\( \beta_{7}) q^{59}\) \(+(\)\(66\!\cdots\!80\)\( + \)\(92\!\cdots\!60\)\( \beta_{1} - \)\(52\!\cdots\!16\)\( \beta_{2} + \)\(58\!\cdots\!24\)\( \beta_{3} + \)\(51\!\cdots\!36\)\( \beta_{4} - \)\(40\!\cdots\!04\)\( \beta_{5} + \)\(18\!\cdots\!00\)\( \beta_{6} + \)\(42\!\cdots\!04\)\( \beta_{7}) q^{60}\) \(+(\)\(47\!\cdots\!42\)\( + \)\(11\!\cdots\!99\)\( \beta_{1} + \)\(14\!\cdots\!09\)\( \beta_{2} - \)\(16\!\cdots\!69\)\( \beta_{3} + \)\(26\!\cdots\!87\)\( \beta_{4} - \)\(17\!\cdots\!84\)\( \beta_{5} + \)\(59\!\cdots\!40\)\( \beta_{6} + \)\(22\!\cdots\!84\)\( \beta_{7}) q^{61}\) \(+(\)\(25\!\cdots\!20\)\( - \)\(13\!\cdots\!08\)\( \beta_{1} + \)\(80\!\cdots\!64\)\( \beta_{2} - \)\(86\!\cdots\!64\)\( \beta_{3} - \)\(93\!\cdots\!72\)\( \beta_{4} + \)\(45\!\cdots\!56\)\( \beta_{5} - \)\(10\!\cdots\!96\)\( \beta_{6} - \)\(21\!\cdots\!40\)\( \beta_{7}) q^{62}\) \(+(\)\(82\!\cdots\!40\)\( - \)\(41\!\cdots\!75\)\( \beta_{1} - \)\(43\!\cdots\!04\)\( \beta_{2} - \)\(24\!\cdots\!68\)\( \beta_{3} + \)\(59\!\cdots\!78\)\( \beta_{4} + \)\(80\!\cdots\!22\)\( \beta_{5} - \)\(76\!\cdots\!49\)\( \beta_{6} + \)\(32\!\cdots\!12\)\( \beta_{7}) q^{63}\) \(+(\)\(32\!\cdots\!08\)\( - \)\(10\!\cdots\!40\)\( \beta_{1} - \)\(31\!\cdots\!20\)\( \beta_{2} + \)\(35\!\cdots\!36\)\( \beta_{3} + \)\(58\!\cdots\!12\)\( \beta_{4} - \)\(48\!\cdots\!12\)\( \beta_{5} + \)\(84\!\cdots\!00\)\( \beta_{6} + \)\(18\!\cdots\!60\)\( \beta_{7}) q^{64}\) \(+(\)\(33\!\cdots\!60\)\( - \)\(32\!\cdots\!20\)\( \beta_{1} - \)\(37\!\cdots\!92\)\( \beta_{2} - \)\(46\!\cdots\!32\)\( \beta_{3} + \)\(36\!\cdots\!72\)\( \beta_{4} + \)\(56\!\cdots\!52\)\( \beta_{5} + \)\(19\!\cdots\!00\)\( \beta_{6} - \)\(11\!\cdots\!52\)\( \beta_{7}) q^{65}\) \(+(\)\(99\!\cdots\!44\)\( - \)\(80\!\cdots\!92\)\( \beta_{1} + \)\(74\!\cdots\!76\)\( \beta_{2} - \)\(18\!\cdots\!00\)\( \beta_{3} - \)\(11\!\cdots\!92\)\( \beta_{4} + \)\(84\!\cdots\!16\)\( \beta_{5} - \)\(62\!\cdots\!80\)\( \beta_{6} - \)\(48\!\cdots\!68\)\( \beta_{7}) q^{66}\) \(+(\)\(14\!\cdots\!80\)\( + \)\(13\!\cdots\!08\)\( \beta_{1} + \)\(25\!\cdots\!05\)\( \beta_{2} - \)\(17\!\cdots\!52\)\( \beta_{3} - \)\(92\!\cdots\!20\)\( \beta_{4} - \)\(21\!\cdots\!92\)\( \beta_{5} + \)\(22\!\cdots\!06\)\( \beta_{6} + \)\(78\!\cdots\!56\)\( \beta_{7}) q^{67}\) \(+(\)\(70\!\cdots\!60\)\( + \)\(34\!\cdots\!56\)\( \beta_{1} + \)\(61\!\cdots\!68\)\( \beta_{2} + \)\(78\!\cdots\!54\)\( \beta_{3} + \)\(54\!\cdots\!16\)\( \beta_{4} - \)\(16\!\cdots\!16\)\( \beta_{5} + \)\(30\!\cdots\!72\)\( \beta_{6} - \)\(83\!\cdots\!36\)\( \beta_{7}) q^{68}\) \(+(\)\(68\!\cdots\!64\)\( + \)\(17\!\cdots\!72\)\( \beta_{1} - \)\(13\!\cdots\!48\)\( \beta_{2} - \)\(51\!\cdots\!60\)\( \beta_{3} - \)\(21\!\cdots\!80\)\( \beta_{4} + \)\(50\!\cdots\!84\)\( \beta_{5} - \)\(78\!\cdots\!80\)\( \beta_{6} - \)\(41\!\cdots\!68\)\( \beta_{7}) q^{69}\) \(+(\)\(17\!\cdots\!80\)\( + \)\(54\!\cdots\!76\)\( \beta_{1} - \)\(30\!\cdots\!40\)\( \beta_{2} - \)\(62\!\cdots\!92\)\( \beta_{3} + \)\(19\!\cdots\!04\)\( \beta_{4} + \)\(19\!\cdots\!20\)\( \beta_{5} + \)\(42\!\cdots\!00\)\( \beta_{6} + \)\(15\!\cdots\!80\)\( \beta_{7}) q^{70}\) \(+(\)\(65\!\cdots\!32\)\( - \)\(25\!\cdots\!03\)\( \beta_{1} + \)\(83\!\cdots\!52\)\( \beta_{2} - \)\(33\!\cdots\!32\)\( \beta_{3} - \)\(54\!\cdots\!14\)\( \beta_{4} - \)\(66\!\cdots\!02\)\( \beta_{5} + \)\(15\!\cdots\!95\)\( \beta_{6} - \)\(13\!\cdots\!48\)\( \beta_{7}) q^{71}\) \(+(\)\(45\!\cdots\!40\)\( - \)\(71\!\cdots\!18\)\( \beta_{1} + \)\(38\!\cdots\!54\)\( \beta_{2} + \)\(47\!\cdots\!37\)\( \beta_{3} + \)\(96\!\cdots\!31\)\( \beta_{4} + \)\(20\!\cdots\!27\)\( \beta_{5} - \)\(31\!\cdots\!37\)\( \beta_{6} - \)\(54\!\cdots\!75\)\( \beta_{7}) q^{72}\) \(+(\)\(63\!\cdots\!30\)\( + \)\(14\!\cdots\!66\)\( \beta_{1} + \)\(65\!\cdots\!66\)\( \beta_{2} + \)\(45\!\cdots\!98\)\( \beta_{3} + \)\(89\!\cdots\!70\)\( \beta_{4} + \)\(16\!\cdots\!08\)\( \beta_{5} + \)\(16\!\cdots\!16\)\( \beta_{6} + \)\(19\!\cdots\!96\)\( \beta_{7}) q^{73}\) \(+(-\)\(49\!\cdots\!14\)\( - \)\(58\!\cdots\!18\)\( \beta_{1} + \)\(20\!\cdots\!20\)\( \beta_{2} + \)\(33\!\cdots\!72\)\( \beta_{3} - \)\(25\!\cdots\!68\)\( \beta_{4} - \)\(16\!\cdots\!60\)\( \beta_{5} - \)\(52\!\cdots\!40\)\( \beta_{6} - \)\(18\!\cdots\!64\)\( \beta_{7}) q^{74}\) \(+(-\)\(29\!\cdots\!00\)\( + \)\(10\!\cdots\!90\)\( \beta_{1} - \)\(16\!\cdots\!65\)\( \beta_{2} - \)\(86\!\cdots\!20\)\( \beta_{3} - \)\(30\!\cdots\!00\)\( \beta_{4} - \)\(79\!\cdots\!60\)\( \beta_{5} + \)\(26\!\cdots\!00\)\( \beta_{6} - \)\(42\!\cdots\!40\)\( \beta_{7}) q^{75}\) \(+(-\)\(82\!\cdots\!20\)\( + \)\(13\!\cdots\!00\)\( \beta_{1} - \)\(21\!\cdots\!36\)\( \beta_{2} - \)\(10\!\cdots\!52\)\( \beta_{3} + \)\(25\!\cdots\!20\)\( \beta_{4} - \)\(99\!\cdots\!16\)\( \beta_{5} - \)\(16\!\cdots\!80\)\( \beta_{6} + \)\(15\!\cdots\!32\)\( \beta_{7}) q^{76}\) \(+(-\)\(80\!\cdots\!00\)\( + \)\(21\!\cdots\!76\)\( \beta_{1} - \)\(16\!\cdots\!92\)\( \beta_{2} + \)\(16\!\cdots\!24\)\( \beta_{3} + \)\(32\!\cdots\!72\)\( \beta_{4} + \)\(38\!\cdots\!04\)\( \beta_{5} - \)\(16\!\cdots\!84\)\( \beta_{6} - \)\(15\!\cdots\!40\)\( \beta_{7}) q^{77}\) \(+(-\)\(35\!\cdots\!00\)\( - \)\(11\!\cdots\!06\)\( \beta_{1} + \)\(37\!\cdots\!08\)\( \beta_{2} + \)\(54\!\cdots\!56\)\( \beta_{3} - \)\(59\!\cdots\!00\)\( \beta_{4} - \)\(10\!\cdots\!74\)\( \beta_{5} + \)\(45\!\cdots\!92\)\( \beta_{6} - \)\(21\!\cdots\!28\)\( \beta_{7}) q^{78}\) \(+(-\)\(38\!\cdots\!60\)\( + \)\(76\!\cdots\!78\)\( \beta_{1} + \)\(69\!\cdots\!48\)\( \beta_{2} + \)\(59\!\cdots\!80\)\( \beta_{3} + \)\(23\!\cdots\!20\)\( \beta_{4} - \)\(15\!\cdots\!24\)\( \beta_{5} - \)\(11\!\cdots\!70\)\( \beta_{6} + \)\(80\!\cdots\!68\)\( \beta_{7}) q^{79}\) \(+(-\)\(15\!\cdots\!20\)\( - \)\(14\!\cdots\!24\)\( \beta_{1} + \)\(18\!\cdots\!60\)\( \beta_{2} - \)\(20\!\cdots\!92\)\( \beta_{3} + \)\(81\!\cdots\!04\)\( \beta_{4} + \)\(25\!\cdots\!20\)\( \beta_{5} - \)\(15\!\cdots\!00\)\( \beta_{6} - \)\(72\!\cdots\!20\)\( \beta_{7}) q^{80}\) \(+(-\)\(13\!\cdots\!79\)\( - \)\(12\!\cdots\!74\)\( \beta_{1} - \)\(31\!\cdots\!62\)\( \beta_{2} + \)\(59\!\cdots\!02\)\( \beta_{3} + \)\(72\!\cdots\!86\)\( \beta_{4} + \)\(87\!\cdots\!68\)\( \beta_{5} + \)\(26\!\cdots\!20\)\( \beta_{6} - \)\(55\!\cdots\!88\)\( \beta_{7}) q^{81}\) \(+(-\)\(86\!\cdots\!30\)\( + \)\(10\!\cdots\!50\)\( \beta_{1} - \)\(10\!\cdots\!28\)\( \beta_{2} + \)\(93\!\cdots\!88\)\( \beta_{3} + \)\(37\!\cdots\!52\)\( \beta_{4} - \)\(19\!\cdots\!52\)\( \beta_{5} + \)\(14\!\cdots\!84\)\( \beta_{6} + \)\(13\!\cdots\!08\)\( \beta_{7}) q^{82}\) \(+(\)\(28\!\cdots\!40\)\( + \)\(41\!\cdots\!58\)\( \beta_{1} + \)\(78\!\cdots\!53\)\( \beta_{2} + \)\(36\!\cdots\!00\)\( \beta_{3} - \)\(11\!\cdots\!00\)\( \beta_{4} - \)\(42\!\cdots\!00\)\( \beta_{5} - \)\(89\!\cdots\!00\)\( \beta_{6} + \)\(45\!\cdots\!00\)\( \beta_{7}) q^{83}\) \(+(\)\(45\!\cdots\!56\)\( + \)\(11\!\cdots\!24\)\( \beta_{1} - \)\(66\!\cdots\!44\)\( \beta_{2} + \)\(44\!\cdots\!20\)\( \beta_{3} + \)\(51\!\cdots\!92\)\( \beta_{4} + \)\(56\!\cdots\!28\)\( \beta_{5} + \)\(35\!\cdots\!00\)\( \beta_{6} + \)\(21\!\cdots\!20\)\( \beta_{7}) q^{84}\) \(+(-\)\(82\!\cdots\!20\)\( + \)\(85\!\cdots\!30\)\( \beta_{1} + \)\(70\!\cdots\!94\)\( \beta_{2} - \)\(24\!\cdots\!06\)\( \beta_{3} - \)\(91\!\cdots\!94\)\( \beta_{4} + \)\(39\!\cdots\!36\)\( \beta_{5} + \)\(23\!\cdots\!00\)\( \beta_{6} - \)\(17\!\cdots\!36\)\( \beta_{7}) q^{85}\) \(+(\)\(36\!\cdots\!32\)\( + \)\(16\!\cdots\!47\)\( \beta_{1} - \)\(68\!\cdots\!34\)\( \beta_{2} + \)\(73\!\cdots\!48\)\( \beta_{3} + \)\(66\!\cdots\!60\)\( \beta_{4} - \)\(66\!\cdots\!77\)\( \beta_{5} - \)\(33\!\cdots\!60\)\( \beta_{6} + \)\(26\!\cdots\!64\)\( \beta_{7}) q^{86}\) \(+(\)\(51\!\cdots\!40\)\( - \)\(26\!\cdots\!17\)\( \beta_{1} + \)\(40\!\cdots\!36\)\( \beta_{2} - \)\(41\!\cdots\!80\)\( \beta_{3} - \)\(17\!\cdots\!34\)\( \beta_{4} + \)\(10\!\cdots\!70\)\( \beta_{5} - \)\(32\!\cdots\!91\)\( \beta_{6} + \)\(36\!\cdots\!56\)\( \beta_{7}) q^{87}\) \(+(\)\(16\!\cdots\!40\)\( - \)\(16\!\cdots\!68\)\( \beta_{1} - \)\(22\!\cdots\!48\)\( \beta_{2} + \)\(19\!\cdots\!64\)\( \beta_{3} - \)\(29\!\cdots\!88\)\( \beta_{4} + \)\(33\!\cdots\!44\)\( \beta_{5} + \)\(13\!\cdots\!56\)\( \beta_{6} - \)\(17\!\cdots\!20\)\( \beta_{7}) q^{88}\) \(+(\)\(16\!\cdots\!70\)\( - \)\(56\!\cdots\!42\)\( \beta_{1} + \)\(72\!\cdots\!34\)\( \beta_{2} - \)\(93\!\cdots\!46\)\( \beta_{3} + \)\(33\!\cdots\!54\)\( \beta_{4} + \)\(22\!\cdots\!88\)\( \beta_{5} - \)\(11\!\cdots\!40\)\( \beta_{6} + \)\(17\!\cdots\!56\)\( \beta_{7}) q^{89}\) \(+(\)\(23\!\cdots\!10\)\( + \)\(36\!\cdots\!10\)\( \beta_{1} - \)\(42\!\cdots\!72\)\( \beta_{2} - \)\(15\!\cdots\!72\)\( \beta_{3} + \)\(80\!\cdots\!72\)\( \beta_{4} + \)\(62\!\cdots\!32\)\( \beta_{5} - \)\(23\!\cdots\!00\)\( \beta_{6} + \)\(23\!\cdots\!68\)\( \beta_{7}) q^{90}\) \(+(-\)\(80\!\cdots\!48\)\( + \)\(61\!\cdots\!20\)\( \beta_{1} + \)\(20\!\cdots\!12\)\( \beta_{2} - \)\(16\!\cdots\!36\)\( \beta_{3} + \)\(52\!\cdots\!00\)\( \beta_{4} - \)\(18\!\cdots\!48\)\( \beta_{5} + \)\(22\!\cdots\!60\)\( \beta_{6} - \)\(70\!\cdots\!84\)\( \beta_{7}) q^{91}\) \(+(-\)\(45\!\cdots\!20\)\( + \)\(92\!\cdots\!80\)\( \beta_{1} - \)\(54\!\cdots\!40\)\( \beta_{2} - \)\(26\!\cdots\!52\)\( \beta_{3} - \)\(14\!\cdots\!28\)\( \beta_{4} + \)\(44\!\cdots\!08\)\( \beta_{5} - \)\(78\!\cdots\!16\)\( \beta_{6} + \)\(30\!\cdots\!48\)\( \beta_{7}) q^{92}\) \(+(-\)\(77\!\cdots\!80\)\( + \)\(13\!\cdots\!76\)\( \beta_{1} + \)\(13\!\cdots\!08\)\( \beta_{2} + \)\(13\!\cdots\!24\)\( \beta_{3} - \)\(19\!\cdots\!40\)\( \beta_{4} + \)\(46\!\cdots\!04\)\( \beta_{5} + \)\(16\!\cdots\!08\)\( \beta_{6} + \)\(61\!\cdots\!48\)\( \beta_{7}) q^{93}\) \(+(-\)\(13\!\cdots\!44\)\( + \)\(11\!\cdots\!56\)\( \beta_{1} - \)\(91\!\cdots\!00\)\( \beta_{2} + \)\(85\!\cdots\!48\)\( \beta_{3} + \)\(47\!\cdots\!80\)\( \beta_{4} - \)\(47\!\cdots\!44\)\( \beta_{5} - \)\(70\!\cdots\!20\)\( \beta_{6} + \)\(38\!\cdots\!88\)\( \beta_{7}) q^{94}\) \(+(-\)\(38\!\cdots\!00\)\( - \)\(28\!\cdots\!45\)\( \beta_{1} + \)\(78\!\cdots\!40\)\( \beta_{2} + \)\(17\!\cdots\!80\)\( \beta_{3} + \)\(20\!\cdots\!30\)\( \beta_{4} - \)\(54\!\cdots\!90\)\( \beta_{5} - \)\(38\!\cdots\!75\)\( \beta_{6} - \)\(26\!\cdots\!60\)\( \beta_{7}) q^{95}\) \(+(-\)\(12\!\cdots\!08\)\( - \)\(89\!\cdots\!88\)\( \beta_{1} - \)\(49\!\cdots\!92\)\( \beta_{2} - \)\(12\!\cdots\!72\)\( \beta_{3} + \)\(20\!\cdots\!72\)\( \beta_{4} + \)\(81\!\cdots\!48\)\( \beta_{5} + \)\(68\!\cdots\!00\)\( \beta_{6} - \)\(28\!\cdots\!80\)\( \beta_{7}) q^{96}\) \(+(-\)\(10\!\cdots\!30\)\( - \)\(35\!\cdots\!82\)\( \beta_{1} + \)\(22\!\cdots\!70\)\( \beta_{2} - \)\(54\!\cdots\!46\)\( \beta_{3} - \)\(27\!\cdots\!74\)\( \beta_{4} + \)\(92\!\cdots\!84\)\( \beta_{5} - \)\(39\!\cdots\!88\)\( \beta_{6} + \)\(19\!\cdots\!24\)\( \beta_{7}) q^{97}\) \(+(-\)\(15\!\cdots\!95\)\( - \)\(12\!\cdots\!39\)\( \beta_{1} + \)\(28\!\cdots\!52\)\( \beta_{2} + \)\(11\!\cdots\!88\)\( \beta_{3} + \)\(90\!\cdots\!72\)\( \beta_{4} - \)\(29\!\cdots\!52\)\( \beta_{5} + \)\(90\!\cdots\!64\)\( \beta_{6} - \)\(17\!\cdots\!72\)\( \beta_{7}) q^{98}\) \(+(-\)\(17\!\cdots\!76\)\( - \)\(45\!\cdots\!50\)\( \beta_{1} - \)\(10\!\cdots\!57\)\( \beta_{2} + \)\(35\!\cdots\!36\)\( \beta_{3} + \)\(36\!\cdots\!60\)\( \beta_{4} + \)\(68\!\cdots\!88\)\( \beta_{5} - \)\(23\!\cdots\!60\)\( \beta_{6} - \)\(34\!\cdots\!16\)\( \beta_{7}) q^{99}\) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(8q \) \(\mathstrut -\mathstrut 208040616902520q^{2} \) \(\mathstrut -\mathstrut \)\(28\!\cdots\!20\)\(q^{3} \) \(\mathstrut +\mathstrut \)\(28\!\cdots\!24\)\(q^{4} \) \(\mathstrut -\mathstrut \)\(48\!\cdots\!60\)\(q^{5} \) \(\mathstrut -\mathstrut \)\(77\!\cdots\!44\)\(q^{6} \) \(\mathstrut -\mathstrut \)\(56\!\cdots\!00\)\(q^{7} \) \(\mathstrut +\mathstrut \)\(59\!\cdots\!60\)\(q^{8} \) \(\mathstrut +\mathstrut \)\(15\!\cdots\!76\)\(q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(8q \) \(\mathstrut -\mathstrut 208040616902520q^{2} \) \(\mathstrut -\mathstrut \)\(28\!\cdots\!20\)\(q^{3} \) \(\mathstrut +\mathstrut \)\(28\!\cdots\!24\)\(q^{4} \) \(\mathstrut -\mathstrut \)\(48\!\cdots\!60\)\(q^{5} \) \(\mathstrut -\mathstrut \)\(77\!\cdots\!44\)\(q^{6} \) \(\mathstrut -\mathstrut \)\(56\!\cdots\!00\)\(q^{7} \) \(\mathstrut +\mathstrut \)\(59\!\cdots\!60\)\(q^{8} \) \(\mathstrut +\mathstrut \)\(15\!\cdots\!76\)\(q^{9} \) \(\mathstrut -\mathstrut \)\(20\!\cdots\!60\)\(q^{10} \) \(\mathstrut +\mathstrut \)\(66\!\cdots\!36\)\(q^{11} \) \(\mathstrut -\mathstrut \)\(26\!\cdots\!40\)\(q^{12} \) \(\mathstrut -\mathstrut \)\(53\!\cdots\!40\)\(q^{13} \) \(\mathstrut -\mathstrut \)\(27\!\cdots\!92\)\(q^{14} \) \(\mathstrut +\mathstrut \)\(39\!\cdots\!80\)\(q^{15} \) \(\mathstrut +\mathstrut \)\(16\!\cdots\!08\)\(q^{16} \) \(\mathstrut +\mathstrut \)\(29\!\cdots\!40\)\(q^{17} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!80\)\(q^{18} \) \(\mathstrut -\mathstrut \)\(25\!\cdots\!20\)\(q^{19} \) \(\mathstrut -\mathstrut \)\(87\!\cdots\!80\)\(q^{20} \) \(\mathstrut +\mathstrut \)\(72\!\cdots\!16\)\(q^{21} \) \(\mathstrut +\mathstrut \)\(64\!\cdots\!60\)\(q^{22} \) \(\mathstrut -\mathstrut \)\(59\!\cdots\!60\)\(q^{23} \) \(\mathstrut -\mathstrut \)\(78\!\cdots\!60\)\(q^{24} \) \(\mathstrut +\mathstrut \)\(71\!\cdots\!00\)\(q^{25} \) \(\mathstrut +\mathstrut \)\(34\!\cdots\!56\)\(q^{26} \) \(\mathstrut +\mathstrut \)\(13\!\cdots\!40\)\(q^{27} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!40\)\(q^{28} \) \(\mathstrut -\mathstrut \)\(43\!\cdots\!80\)\(q^{29} \) \(\mathstrut +\mathstrut \)\(23\!\cdots\!80\)\(q^{30} \) \(\mathstrut -\mathstrut \)\(42\!\cdots\!24\)\(q^{31} \) \(\mathstrut -\mathstrut \)\(28\!\cdots\!20\)\(q^{32} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!40\)\(q^{33} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!68\)\(q^{34} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!60\)\(q^{35} \) \(\mathstrut -\mathstrut \)\(22\!\cdots\!72\)\(q^{36} \) \(\mathstrut +\mathstrut \)\(70\!\cdots\!20\)\(q^{37} \) \(\mathstrut +\mathstrut \)\(40\!\cdots\!40\)\(q^{38} \) \(\mathstrut +\mathstrut \)\(30\!\cdots\!12\)\(q^{39} \) \(\mathstrut -\mathstrut \)\(61\!\cdots\!00\)\(q^{40} \) \(\mathstrut +\mathstrut \)\(16\!\cdots\!96\)\(q^{41} \) \(\mathstrut +\mathstrut \)\(58\!\cdots\!40\)\(q^{42} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!00\)\(q^{43} \) \(\mathstrut -\mathstrut \)\(48\!\cdots\!92\)\(q^{44} \) \(\mathstrut -\mathstrut \)\(42\!\cdots\!20\)\(q^{45} \) \(\mathstrut +\mathstrut \)\(41\!\cdots\!56\)\(q^{46} \) \(\mathstrut +\mathstrut \)\(28\!\cdots\!60\)\(q^{47} \) \(\mathstrut -\mathstrut \)\(27\!\cdots\!60\)\(q^{48} \) \(\mathstrut -\mathstrut \)\(14\!\cdots\!56\)\(q^{49} \) \(\mathstrut +\mathstrut \)\(25\!\cdots\!00\)\(q^{50} \) \(\mathstrut -\mathstrut \)\(17\!\cdots\!64\)\(q^{51} \) \(\mathstrut -\mathstrut \)\(19\!\cdots\!00\)\(q^{52} \) \(\mathstrut -\mathstrut \)\(30\!\cdots\!20\)\(q^{53} \) \(\mathstrut +\mathstrut \)\(13\!\cdots\!80\)\(q^{54} \) \(\mathstrut +\mathstrut \)\(22\!\cdots\!80\)\(q^{55} \) \(\mathstrut +\mathstrut \)\(32\!\cdots\!20\)\(q^{56} \) \(\mathstrut -\mathstrut \)\(27\!\cdots\!20\)\(q^{57} \) \(\mathstrut +\mathstrut \)\(57\!\cdots\!60\)\(q^{58} \) \(\mathstrut +\mathstrut \)\(14\!\cdots\!40\)\(q^{59} \) \(\mathstrut +\mathstrut \)\(53\!\cdots\!40\)\(q^{60} \) \(\mathstrut +\mathstrut \)\(38\!\cdots\!36\)\(q^{61} \) \(\mathstrut +\mathstrut \)\(20\!\cdots\!60\)\(q^{62} \) \(\mathstrut +\mathstrut \)\(65\!\cdots\!20\)\(q^{63} \) \(\mathstrut +\mathstrut \)\(25\!\cdots\!64\)\(q^{64} \) \(\mathstrut +\mathstrut \)\(26\!\cdots\!80\)\(q^{65} \) \(\mathstrut +\mathstrut \)\(79\!\cdots\!52\)\(q^{66} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!40\)\(q^{67} \) \(\mathstrut +\mathstrut \)\(56\!\cdots\!80\)\(q^{68} \) \(\mathstrut +\mathstrut \)\(54\!\cdots\!12\)\(q^{69} \) \(\mathstrut +\mathstrut \)\(14\!\cdots\!40\)\(q^{70} \) \(\mathstrut +\mathstrut \)\(52\!\cdots\!56\)\(q^{71} \) \(\mathstrut +\mathstrut \)\(36\!\cdots\!20\)\(q^{72} \) \(\mathstrut +\mathstrut \)\(50\!\cdots\!40\)\(q^{73} \) \(\mathstrut -\mathstrut \)\(39\!\cdots\!12\)\(q^{74} \) \(\mathstrut -\mathstrut \)\(23\!\cdots\!00\)\(q^{75} \) \(\mathstrut -\mathstrut \)\(66\!\cdots\!60\)\(q^{76} \) \(\mathstrut -\mathstrut \)\(64\!\cdots\!00\)\(q^{77} \) \(\mathstrut -\mathstrut \)\(28\!\cdots\!00\)\(q^{78} \) \(\mathstrut -\mathstrut \)\(30\!\cdots\!80\)\(q^{79} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!60\)\(q^{80} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!32\)\(q^{81} \) \(\mathstrut -\mathstrut \)\(68\!\cdots\!40\)\(q^{82} \) \(\mathstrut +\mathstrut \)\(22\!\cdots\!20\)\(q^{83} \) \(\mathstrut +\mathstrut \)\(36\!\cdots\!48\)\(q^{84} \) \(\mathstrut -\mathstrut \)\(65\!\cdots\!60\)\(q^{85} \) \(\mathstrut +\mathstrut \)\(29\!\cdots\!56\)\(q^{86} \) \(\mathstrut +\mathstrut \)\(41\!\cdots\!20\)\(q^{87} \) \(\mathstrut +\mathstrut \)\(13\!\cdots\!20\)\(q^{88} \) \(\mathstrut +\mathstrut \)\(13\!\cdots\!60\)\(q^{89} \) \(\mathstrut +\mathstrut \)\(18\!\cdots\!80\)\(q^{90} \) \(\mathstrut -\mathstrut \)\(64\!\cdots\!84\)\(q^{91} \) \(\mathstrut -\mathstrut \)\(36\!\cdots\!60\)\(q^{92} \) \(\mathstrut -\mathstrut \)\(62\!\cdots\!40\)\(q^{93} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!52\)\(q^{94} \) \(\mathstrut -\mathstrut \)\(31\!\cdots\!00\)\(q^{95} \) \(\mathstrut -\mathstrut \)\(96\!\cdots\!64\)\(q^{96} \) \(\mathstrut -\mathstrut \)\(87\!\cdots\!40\)\(q^{97} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!60\)\(q^{98} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!08\)\(q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8}\mathstrut -\mathstrut \) \(x^{7}\mathstrut -\mathstrut \) \(76\!\cdots\!19\) \(x^{6}\mathstrut -\mathstrut \) \(84\!\cdots\!41\) \(x^{5}\mathstrut +\mathstrut \) \(16\!\cdots\!91\) \(x^{4}\mathstrut +\mathstrut \) \(38\!\cdots\!65\) \(x^{3}\mathstrut -\mathstrut \) \(11\!\cdots\!25\) \(x^{2}\mathstrut -\mathstrut \) \(23\!\cdots\!75\) \(x\mathstrut +\mathstrut \) \(23\!\cdots\!00\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 72 \nu - 9 \)
\(\beta_{2}\)\(=\)\((\)\(25\!\cdots\!03\) \(\nu^{7}\mathstrut -\mathstrut \) \(25\!\cdots\!22\) \(\nu^{6}\mathstrut -\mathstrut \) \(19\!\cdots\!03\) \(\nu^{5}\mathstrut +\mathstrut \) \(17\!\cdots\!40\) \(\nu^{4}\mathstrut +\mathstrut \) \(42\!\cdots\!13\) \(\nu^{3}\mathstrut -\mathstrut \) \(29\!\cdots\!14\) \(\nu^{2}\mathstrut -\mathstrut \) \(24\!\cdots\!17\) \(\nu\mathstrut +\mathstrut \) \(13\!\cdots\!16\)\()/\)\(59\!\cdots\!92\)
\(\beta_{3}\)\(=\)\((\)\(14\!\cdots\!79\) \(\nu^{7}\mathstrut -\mathstrut \) \(14\!\cdots\!46\) \(\nu^{6}\mathstrut -\mathstrut \) \(11\!\cdots\!79\) \(\nu^{5}\mathstrut +\mathstrut \) \(94\!\cdots\!20\) \(\nu^{4}\mathstrut +\mathstrut \) \(23\!\cdots\!09\) \(\nu^{3}\mathstrut -\mathstrut \) \(10\!\cdots\!38\) \(\nu^{2}\mathstrut -\mathstrut \) \(16\!\cdots\!49\) \(\nu\mathstrut -\mathstrut \) \(21\!\cdots\!28\)\()/\)\(29\!\cdots\!96\)
\(\beta_{4}\)\(=\)\((\)\(35\!\cdots\!17\) \(\nu^{7}\mathstrut -\mathstrut \) \(38\!\cdots\!62\) \(\nu^{6}\mathstrut -\mathstrut \) \(23\!\cdots\!53\) \(\nu^{5}\mathstrut +\mathstrut \) \(22\!\cdots\!08\) \(\nu^{4}\mathstrut +\mathstrut \) \(37\!\cdots\!67\) \(\nu^{3}\mathstrut -\mathstrut \) \(28\!\cdots\!90\) \(\nu^{2}\mathstrut -\mathstrut \) \(16\!\cdots\!75\) \(\nu\mathstrut +\mathstrut \) \(10\!\cdots\!00\)\()/\)\(14\!\cdots\!00\)
\(\beta_{5}\)\(=\)\((\)\(-\)\(65\!\cdots\!89\) \(\nu^{7}\mathstrut -\mathstrut \) \(17\!\cdots\!46\) \(\nu^{6}\mathstrut +\mathstrut \) \(67\!\cdots\!01\) \(\nu^{5}\mathstrut +\mathstrut \) \(95\!\cdots\!64\) \(\nu^{4}\mathstrut -\mathstrut \) \(19\!\cdots\!39\) \(\nu^{3}\mathstrut -\mathstrut \) \(67\!\cdots\!70\) \(\nu^{2}\mathstrut +\mathstrut \) \(11\!\cdots\!75\) \(\nu\mathstrut -\mathstrut \) \(23\!\cdots\!00\)\()/\)\(74\!\cdots\!00\)
\(\beta_{6}\)\(=\)\((\)\(-\)\(24\!\cdots\!81\) \(\nu^{7}\mathstrut +\mathstrut \) \(21\!\cdots\!66\) \(\nu^{6}\mathstrut +\mathstrut \) \(16\!\cdots\!29\) \(\nu^{5}\mathstrut -\mathstrut \) \(12\!\cdots\!44\) \(\nu^{4}\mathstrut -\mathstrut \) \(27\!\cdots\!31\) \(\nu^{3}\mathstrut +\mathstrut \) \(14\!\cdots\!70\) \(\nu^{2}\mathstrut +\mathstrut \) \(12\!\cdots\!75\) \(\nu\mathstrut -\mathstrut \) \(51\!\cdots\!00\)\()/\)\(46\!\cdots\!00\)
\(\beta_{7}\)\(=\)\((\)\(-\)\(16\!\cdots\!89\) \(\nu^{7}\mathstrut +\mathstrut \) \(15\!\cdots\!54\) \(\nu^{6}\mathstrut +\mathstrut \) \(10\!\cdots\!01\) \(\nu^{5}\mathstrut -\mathstrut \) \(87\!\cdots\!36\) \(\nu^{4}\mathstrut -\mathstrut \) \(16\!\cdots\!39\) \(\nu^{3}\mathstrut +\mathstrut \) \(98\!\cdots\!30\) \(\nu^{2}\mathstrut +\mathstrut \) \(50\!\cdots\!75\) \(\nu\mathstrut -\mathstrut \) \(33\!\cdots\!00\)\()/\)\(14\!\cdots\!00\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1}\mathstrut +\mathstrut \) \(9\)\()/72\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3}\mathstrut -\mathstrut \) \(111186\) \(\beta_{2}\mathstrut +\mathstrut \) \(119454850513664\) \(\beta_{1}\mathstrut +\mathstrut \) \(992470782456187687053696265272\)\()/5184\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{7}\mathstrut -\mathstrut \) \(117\) \(\beta_{6}\mathstrut -\mathstrut \) \(237529\) \(\beta_{5}\mathstrut +\mathstrut \) \(3005209639\) \(\beta_{4}\mathstrut -\mathstrut \) \(14859349564627\) \(\beta_{3}\mathstrut -\mathstrut \) \(536064134954729281250\) \(\beta_{2}\mathstrut +\mathstrut \) \(1718829034221357594388531604578\) \(\beta_{1}\mathstrut +\mathstrut \) \(118555449007540404242398324722104382594371136\)\()/373248\)
\(\nu^{4}\)\(=\)\((\)\(-\)\(10509299761779\) \(\beta_{7}\mathstrut +\mathstrut \) \(37484005162925711\) \(\beta_{6}\mathstrut -\mathstrut \) \(102496235456722490245\) \(\beta_{5}\mathstrut +\mathstrut \) \(755301710738314363120763\) \(\beta_{4}\mathstrut +\mathstrut \) \(329081051286660577619057878065\) \(\beta_{3}\mathstrut -\mathstrut \) \(52333243149079756807364967660708810\) \(\beta_{2}\mathstrut +\mathstrut \) \(40460083325689924875803817842550525443865530\) \(\beta_{1}\mathstrut +\mathstrut \) \(213235949592922960711599612398604382838846469344248075010496\)\()/3359232\)
\(\nu^{5}\)\(=\)\((\)\(21769050311546271723143347339\) \(\beta_{7}\mathstrut -\mathstrut \) \(1227317391603745625026045668231\) \(\beta_{6}\mathstrut -\mathstrut \) \(9501291080923850876765723280318163\) \(\beta_{5}\mathstrut +\mathstrut \) \(100659460460658936633248002238881393965\) \(\beta_{4}\mathstrut -\mathstrut \) \(382582021842333704931897562608480612321713\) \(\beta_{3}\mathstrut -\mathstrut \) \(16161904747699125397534221040955087606691609992310\) \(\beta_{2}\mathstrut +\mathstrut \) \(28294254267772952734338185809341074272681475994691507756950\) \(\beta_{1}\mathstrut +\mathstrut \) \(2509715661253050086008505148011002099685486413050029970677160948789450528\)\()/15116544\)
\(\nu^{6}\)\(=\)\((\)\(-\)\(51\!\cdots\!99\) \(\beta_{7}\mathstrut +\mathstrut \) \(35\!\cdots\!11\) \(\beta_{6}\mathstrut -\mathstrut \) \(11\!\cdots\!81\) \(\beta_{5}\mathstrut +\mathstrut \) \(74\!\cdots\!99\) \(\beta_{4}\mathstrut +\mathstrut \) \(22\!\cdots\!37\) \(\beta_{3}\mathstrut -\mathstrut \) \(54\!\cdots\!82\) \(\beta_{2}\mathstrut +\mathstrut \) \(27\!\cdots\!70\) \(\beta_{1}\mathstrut +\mathstrut \) \(13\!\cdots\!04\)\()/5038848\)
\(\nu^{7}\)\(=\)\((\)\(20\!\cdots\!51\) \(\beta_{7}\mathstrut -\mathstrut \) \(49\!\cdots\!31\) \(\beta_{6}\mathstrut -\mathstrut \) \(12\!\cdots\!11\) \(\beta_{5}\mathstrut +\mathstrut \) \(11\!\cdots\!37\) \(\beta_{4}\mathstrut +\mathstrut \) \(96\!\cdots\!95\) \(\beta_{3}\mathstrut -\mathstrut \) \(17\!\cdots\!22\) \(\beta_{2}\mathstrut +\mathstrut \) \(24\!\cdots\!82\) \(\beta_{1}\mathstrut +\mathstrut \) \(22\!\cdots\!96\)\()/30233088\)

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
−2.10181e13
−1.08597e13
−9.79932e12
−8.46093e12
4.33987e12
8.50040e12
1.56804e13
2.16174e13
−1.53931e15 1.09471e23 1.73565e30 −1.42424e34 −1.68510e38 −4.25157e41 −1.69606e45 −1.59809e47 2.19235e49
1.2 −8.07907e14 −6.25114e23 1.88877e28 −5.34820e34 5.05034e38 1.04094e42 4.96812e44 2.18975e47 4.32085e49
1.3 −7.31556e14 −2.11461e23 −9.86512e28 5.46163e34 1.54696e38 −9.42405e41 5.35848e44 −1.27077e47 −3.99548e49
1.4 −6.35192e14 5.52801e23 −2.30357e29 4.78100e33 −3.51135e38 7.17401e41 5.48921e44 1.33796e47 −3.03685e48
1.5 2.86466e14 2.01536e23 −5.51763e29 −4.79915e34 5.77333e37 −6.92478e41 −3.39630e44 −1.31176e47 −1.37479e49
1.6 5.86024e14 −4.40976e23 −2.90401e29 2.74639e34 −2.58422e38 3.47138e41 −5.41619e44 2.26671e46 1.60945e49
1.7 1.10299e15 5.08811e23 5.82752e29 3.48045e34 5.61211e38 4.01523e41 −5.63335e43 8.70965e46 3.83889e49
1.8 1.53045e15 −3.78025e23 1.70845e30 −5.47796e34 −5.78549e38 −5.03791e41 1.64466e45 −2.88893e46 −8.38375e49
\(n\): 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 \(S_{100}^{\mathrm{new}}(\Gamma_0(1))\).