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.

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

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

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

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 \(S_{100}^{\mathrm{new}}(\Gamma_0(1))\).