Properties

Label 1.94.a.a
Level 1
Weight 94
Character orbit 1.a
Self dual Yes
Analytic conductor 54.773
Analytic rank 1
Dimension 7
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 1 \)
Weight: \( k \) = \( 94 \)
Character orbit: \([\chi]\) = 1.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(54.7725430605\)
Analytic rank: \(1\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: multiple of \( 2^{88}\cdot 3^{34}\cdot 5^{10}\cdot 7^{6}\cdot 13^{2}\cdot 19\cdot 23\cdot 31^{2} \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \(+(6247918101970 + \beta_{1}) q^{2}\) \(+(-\)\(52\!\cdots\!50\)\( - 31923367 \beta_{1} + \beta_{2}) q^{3}\) \(+(\)\(53\!\cdots\!54\)\( + 7774038780732 \beta_{1} - 61939 \beta_{2} + \beta_{3}) q^{4}\) \(+(-\)\(34\!\cdots\!44\)\( + 546350848183256180 \beta_{1} - 7877047084 \beta_{2} - 7000 \beta_{3} + \beta_{4}) q^{5}\) \(+(-\)\(48\!\cdots\!57\)\( - \)\(16\!\cdots\!28\)\( \beta_{1} + 43260183106140 \beta_{2} - 45704118 \beta_{3} - 418 \beta_{4} + \beta_{5}) q^{6}\) \(+(-\)\(13\!\cdots\!28\)\( - \)\(77\!\cdots\!61\)\( \beta_{1} + 14582367553629345 \beta_{2} + 17728695287 \beta_{3} - 1347406 \beta_{4} - 123 \beta_{5} - \beta_{6}) q^{7}\) \(+(\)\(89\!\cdots\!96\)\( + \)\(34\!\cdots\!24\)\( \beta_{1} - 9147820446924023504 \beta_{2} - 4555802756144 \beta_{3} + 603919232 \beta_{4} - 13824 \beta_{5} + 432 \beta_{6}) q^{8}\) \(+(\)\(51\!\cdots\!05\)\( + \)\(72\!\cdots\!24\)\( \beta_{1} - \)\(45\!\cdots\!40\)\( \beta_{2} + 2604286330686324 \beta_{3} + 117986092854 \beta_{4} - 74272308 \beta_{5} + 77220 \beta_{6}) q^{9}\) \(+O(q^{10})\) \( q\) \(+(6247918101970 + \beta_{1}) q^{2}\) \(+(-\)\(52\!\cdots\!50\)\( - 31923367 \beta_{1} + \beta_{2}) q^{3}\) \(+(\)\(53\!\cdots\!54\)\( + 7774038780732 \beta_{1} - 61939 \beta_{2} + \beta_{3}) q^{4}\) \(+(-\)\(34\!\cdots\!44\)\( + 546350848183256180 \beta_{1} - 7877047084 \beta_{2} - 7000 \beta_{3} + \beta_{4}) q^{5}\) \(+(-\)\(48\!\cdots\!57\)\( - \)\(16\!\cdots\!28\)\( \beta_{1} + 43260183106140 \beta_{2} - 45704118 \beta_{3} - 418 \beta_{4} + \beta_{5}) q^{6}\) \(+(-\)\(13\!\cdots\!28\)\( - \)\(77\!\cdots\!61\)\( \beta_{1} + 14582367553629345 \beta_{2} + 17728695287 \beta_{3} - 1347406 \beta_{4} - 123 \beta_{5} - \beta_{6}) q^{7}\) \(+(\)\(89\!\cdots\!96\)\( + \)\(34\!\cdots\!24\)\( \beta_{1} - 9147820446924023504 \beta_{2} - 4555802756144 \beta_{3} + 603919232 \beta_{4} - 13824 \beta_{5} + 432 \beta_{6}) q^{8}\) \(+(\)\(51\!\cdots\!05\)\( + \)\(72\!\cdots\!24\)\( \beta_{1} - \)\(45\!\cdots\!40\)\( \beta_{2} + 2604286330686324 \beta_{3} + 117986092854 \beta_{4} - 74272308 \beta_{5} + 77220 \beta_{6}) q^{9}\) \(+(\)\(80\!\cdots\!92\)\( - \)\(80\!\cdots\!90\)\( \beta_{1} - \)\(46\!\cdots\!88\)\( \beta_{2} + 1221375546245830200 \beta_{3} - 29240772708568 \beta_{4} - 20407593300 \beta_{5} - 13148800 \beta_{6}) q^{10}\) \(+(\)\(15\!\cdots\!74\)\( + \)\(31\!\cdots\!25\)\( \beta_{1} + \)\(40\!\cdots\!37\)\( \beta_{2} + 6948722001898227958 \beta_{3} - 4354860374068204 \beta_{4} + 346104683298 \beta_{5} + 846185670 \beta_{6}) q^{11}\) \(+(-\)\(23\!\cdots\!56\)\( - \)\(58\!\cdots\!08\)\( \beta_{1} + \)\(63\!\cdots\!88\)\( \beta_{2} - \)\(35\!\cdots\!96\)\( \beta_{3} - 501852475502963712 \beta_{4} + 49903145385984 \beta_{5} - 31667594112 \beta_{6}) q^{12}\) \(+(\)\(27\!\cdots\!56\)\( + \)\(73\!\cdots\!32\)\( \beta_{1} - \)\(94\!\cdots\!92\)\( \beta_{2} - \)\(98\!\cdots\!64\)\( \beta_{3} + 69203956420168297 \beta_{4} - 2512210575556344 \beta_{5} + 744622284952 \beta_{6}) q^{13}\) \(+(-\)\(11\!\cdots\!10\)\( - \)\(19\!\cdots\!44\)\( \beta_{1} - \)\(54\!\cdots\!72\)\( \beta_{2} - \)\(19\!\cdots\!52\)\( \beta_{3} + \)\(41\!\cdots\!80\)\( \beta_{4} + 44315536909975650 \beta_{5} - 9315909757440 \beta_{6}) q^{14}\) \(+(-\)\(23\!\cdots\!44\)\( - \)\(50\!\cdots\!95\)\( \beta_{1} + \)\(42\!\cdots\!91\)\( \beta_{2} - \)\(43\!\cdots\!75\)\( \beta_{3} - \)\(23\!\cdots\!74\)\( \beta_{4} + 113377891508026475 \beta_{5} - 57126559913775 \beta_{6}) q^{15}\) \(+(-\)\(26\!\cdots\!96\)\( - \)\(78\!\cdots\!96\)\( \beta_{1} - \)\(23\!\cdots\!04\)\( \beta_{2} - \)\(13\!\cdots\!52\)\( \beta_{3} - \)\(14\!\cdots\!88\)\( \beta_{4} - 22360325777271128064 \beta_{5} + 6225619981419520 \beta_{6}) q^{16}\) \(+(\)\(11\!\cdots\!38\)\( + \)\(71\!\cdots\!68\)\( \beta_{1} - \)\(27\!\cdots\!44\)\( \beta_{2} + \)\(50\!\cdots\!68\)\( \beta_{3} + \)\(21\!\cdots\!66\)\( \beta_{4} + \)\(53\!\cdots\!28\)\( \beta_{5} - 182593265909684364 \beta_{6}) q^{17}\) \(+(\)\(11\!\cdots\!18\)\( + \)\(78\!\cdots\!85\)\( \beta_{1} - \)\(13\!\cdots\!60\)\( \beta_{2} + \)\(15\!\cdots\!48\)\( \beta_{3} + \)\(10\!\cdots\!76\)\( \beta_{4} - \)\(71\!\cdots\!92\)\( \beta_{5} + 3648063859426135296 \beta_{6}) q^{18}\) \(+(-\)\(70\!\cdots\!38\)\( - \)\(37\!\cdots\!17\)\( \beta_{1} - \)\(80\!\cdots\!17\)\( \beta_{2} - \)\(93\!\cdots\!90\)\( \beta_{3} - \)\(56\!\cdots\!08\)\( \beta_{4} + \)\(51\!\cdots\!46\)\( \beta_{5} - 56835316641090318310 \beta_{6}) q^{19}\) \(+(-\)\(83\!\cdots\!08\)\( + \)\(12\!\cdots\!60\)\( \beta_{1} - \)\(22\!\cdots\!38\)\( \beta_{2} - \)\(20\!\cdots\!50\)\( \beta_{3} + \)\(55\!\cdots\!32\)\( \beta_{4} + \)\(40\!\cdots\!00\)\( \beta_{5} + \)\(72\!\cdots\!00\)\( \beta_{6}) q^{20}\) \(+(\)\(77\!\cdots\!96\)\( - \)\(82\!\cdots\!76\)\( \beta_{1} - \)\(12\!\cdots\!88\)\( \beta_{2} + \)\(14\!\cdots\!12\)\( \beta_{3} - \)\(63\!\cdots\!40\)\( \beta_{4} - \)\(61\!\cdots\!40\)\( \beta_{5} - \)\(77\!\cdots\!20\)\( \beta_{6}) q^{21}\) \(+(\)\(48\!\cdots\!29\)\( + \)\(16\!\cdots\!68\)\( \beta_{1} + \)\(69\!\cdots\!52\)\( \beta_{2} + \)\(10\!\cdots\!74\)\( \beta_{3} - \)\(31\!\cdots\!02\)\( \beta_{4} + \)\(84\!\cdots\!79\)\( \beta_{5} + \)\(69\!\cdots\!68\)\( \beta_{6}) q^{22}\) \(+(-\)\(37\!\cdots\!00\)\( - \)\(58\!\cdots\!07\)\( \beta_{1} + \)\(54\!\cdots\!59\)\( \beta_{2} - \)\(67\!\cdots\!95\)\( \beta_{3} + \)\(24\!\cdots\!10\)\( \beta_{4} - \)\(67\!\cdots\!45\)\( \beta_{5} - \)\(51\!\cdots\!15\)\( \beta_{6}) q^{23}\) \(+(-\)\(42\!\cdots\!12\)\( - \)\(17\!\cdots\!76\)\( \beta_{1} + \)\(64\!\cdots\!76\)\( \beta_{2} - \)\(72\!\cdots\!32\)\( \beta_{3} + \)\(20\!\cdots\!32\)\( \beta_{4} + \)\(31\!\cdots\!56\)\( \beta_{5} + \)\(30\!\cdots\!80\)\( \beta_{6}) q^{24}\) \(+(\)\(26\!\cdots\!75\)\( - \)\(30\!\cdots\!00\)\( \beta_{1} - \)\(59\!\cdots\!00\)\( \beta_{2} + \)\(57\!\cdots\!00\)\( \beta_{3} - \)\(14\!\cdots\!00\)\( \beta_{4} - \)\(70\!\cdots\!00\)\( \beta_{5} - \)\(13\!\cdots\!00\)\( \beta_{6}) q^{25}\) \(+(\)\(11\!\cdots\!40\)\( - \)\(38\!\cdots\!34\)\( \beta_{1} - \)\(38\!\cdots\!40\)\( \beta_{2} + \)\(14\!\cdots\!76\)\( \beta_{3} + \)\(92\!\cdots\!56\)\( \beta_{4} - \)\(11\!\cdots\!52\)\( \beta_{5} + \)\(21\!\cdots\!40\)\( \beta_{6}) q^{26}\) \(+(-\)\(15\!\cdots\!24\)\( - \)\(18\!\cdots\!56\)\( \beta_{1} - \)\(27\!\cdots\!04\)\( \beta_{2} - \)\(23\!\cdots\!14\)\( \beta_{3} - \)\(36\!\cdots\!08\)\( \beta_{4} + \)\(10\!\cdots\!06\)\( \beta_{5} + \)\(29\!\cdots\!42\)\( \beta_{6}) q^{27}\) \(+(\)\(27\!\cdots\!44\)\( - \)\(19\!\cdots\!72\)\( \beta_{1} + \)\(51\!\cdots\!68\)\( \beta_{2} + \)\(23\!\cdots\!84\)\( \beta_{3} - \)\(27\!\cdots\!52\)\( \beta_{4} - \)\(50\!\cdots\!36\)\( \beta_{5} - \)\(38\!\cdots\!52\)\( \beta_{6}) q^{28}\) \(+(\)\(16\!\cdots\!20\)\( - \)\(69\!\cdots\!48\)\( \beta_{1} + \)\(22\!\cdots\!24\)\( \beta_{2} + \)\(34\!\cdots\!28\)\( \beta_{3} + \)\(16\!\cdots\!29\)\( \beta_{4} + \)\(90\!\cdots\!32\)\( \beta_{5} + \)\(29\!\cdots\!60\)\( \beta_{6}) q^{29}\) \(+(-\)\(92\!\cdots\!58\)\( - \)\(59\!\cdots\!40\)\( \beta_{1} + \)\(67\!\cdots\!12\)\( \beta_{2} - \)\(14\!\cdots\!00\)\( \beta_{3} - \)\(23\!\cdots\!68\)\( \beta_{4} + \)\(52\!\cdots\!50\)\( \beta_{5} - \)\(16\!\cdots\!00\)\( \beta_{6}) q^{30}\) \(+(-\)\(16\!\cdots\!32\)\( - \)\(93\!\cdots\!60\)\( \beta_{1} - \)\(83\!\cdots\!44\)\( \beta_{2} - \)\(76\!\cdots\!96\)\( \beta_{3} - \)\(15\!\cdots\!92\)\( \beta_{4} - \)\(47\!\cdots\!76\)\( \beta_{5} + \)\(77\!\cdots\!80\)\( \beta_{6}) q^{31}\) \(+(-\)\(10\!\cdots\!92\)\( - \)\(44\!\cdots\!92\)\( \beta_{1} - \)\(20\!\cdots\!36\)\( \beta_{2} + \)\(23\!\cdots\!68\)\( \beta_{3} + \)\(60\!\cdots\!16\)\( \beta_{4} + \)\(14\!\cdots\!28\)\( \beta_{5} - \)\(28\!\cdots\!64\)\( \beta_{6}) q^{32}\) \(+(\)\(95\!\cdots\!92\)\( + \)\(11\!\cdots\!84\)\( \beta_{1} + \)\(51\!\cdots\!48\)\( \beta_{2} + \)\(15\!\cdots\!92\)\( \beta_{3} + \)\(23\!\cdots\!94\)\( \beta_{4} + \)\(25\!\cdots\!32\)\( \beta_{5} + \)\(78\!\cdots\!64\)\( \beta_{6}) q^{33}\) \(+(\)\(11\!\cdots\!72\)\( + \)\(55\!\cdots\!54\)\( \beta_{1} + \)\(57\!\cdots\!36\)\( \beta_{2} + \)\(29\!\cdots\!68\)\( \beta_{3} - \)\(21\!\cdots\!48\)\( \beta_{4} - \)\(47\!\cdots\!24\)\( \beta_{5} - \)\(14\!\cdots\!60\)\( \beta_{6}) q^{34}\) \(+(-\)\(23\!\cdots\!48\)\( + \)\(42\!\cdots\!60\)\( \beta_{1} + \)\(92\!\cdots\!72\)\( \beta_{2} - \)\(39\!\cdots\!00\)\( \beta_{3} + \)\(35\!\cdots\!92\)\( \beta_{4} + \)\(24\!\cdots\!00\)\( \beta_{5} + \)\(96\!\cdots\!00\)\( \beta_{6}) q^{35}\) \(+(\)\(75\!\cdots\!34\)\( + \)\(17\!\cdots\!04\)\( \beta_{1} - \)\(12\!\cdots\!19\)\( \beta_{2} + \)\(84\!\cdots\!53\)\( \beta_{3} + \)\(21\!\cdots\!72\)\( \beta_{4} - \)\(70\!\cdots\!04\)\( \beta_{5} + \)\(24\!\cdots\!00\)\( \beta_{6}) q^{36}\) \(+(\)\(15\!\cdots\!72\)\( + \)\(86\!\cdots\!04\)\( \beta_{1} - \)\(51\!\cdots\!56\)\( \beta_{2} + \)\(31\!\cdots\!92\)\( \beta_{3} - \)\(11\!\cdots\!51\)\( \beta_{4} + \)\(10\!\cdots\!32\)\( \beta_{5} + \)\(60\!\cdots\!24\)\( \beta_{6}) q^{37}\) \(+(-\)\(60\!\cdots\!73\)\( - \)\(13\!\cdots\!72\)\( \beta_{1} + \)\(54\!\cdots\!28\)\( \beta_{2} - \)\(94\!\cdots\!38\)\( \beta_{3} + \)\(75\!\cdots\!54\)\( \beta_{4} - \)\(76\!\cdots\!23\)\( \beta_{5} - \)\(10\!\cdots\!56\)\( \beta_{6}) q^{38}\) \(+(-\)\(29\!\cdots\!20\)\( - \)\(67\!\cdots\!79\)\( \beta_{1} + \)\(29\!\cdots\!31\)\( \beta_{2} - \)\(41\!\cdots\!15\)\( \beta_{3} + \)\(90\!\cdots\!34\)\( \beta_{4} + \)\(12\!\cdots\!67\)\( \beta_{5} + \)\(32\!\cdots\!05\)\( \beta_{6}) q^{39}\) \(+(\)\(10\!\cdots\!80\)\( - \)\(14\!\cdots\!00\)\( \beta_{1} - \)\(44\!\cdots\!20\)\( \beta_{2} + \)\(17\!\cdots\!00\)\( \beta_{3} - \)\(13\!\cdots\!20\)\( \beta_{4} - \)\(10\!\cdots\!00\)\( \beta_{5} + \)\(71\!\cdots\!00\)\( \beta_{6}) q^{40}\) \(+(-\)\(72\!\cdots\!82\)\( - \)\(21\!\cdots\!20\)\( \beta_{1} - \)\(17\!\cdots\!04\)\( \beta_{2} + \)\(12\!\cdots\!64\)\( \beta_{3} - \)\(13\!\cdots\!12\)\( \beta_{4} + \)\(47\!\cdots\!84\)\( \beta_{5} - \)\(11\!\cdots\!00\)\( \beta_{6}) q^{41}\) \(+(-\)\(77\!\cdots\!48\)\( + \)\(20\!\cdots\!00\)\( \beta_{1} - \)\(14\!\cdots\!68\)\( \beta_{2} - \)\(10\!\cdots\!08\)\( \beta_{3} + \)\(68\!\cdots\!04\)\( \beta_{4} - \)\(10\!\cdots\!68\)\( \beta_{5} + \)\(53\!\cdots\!84\)\( \beta_{6}) q^{42}\) \(+(-\)\(10\!\cdots\!90\)\( + \)\(30\!\cdots\!63\)\( \beta_{1} + \)\(29\!\cdots\!47\)\( \beta_{2} - \)\(14\!\cdots\!60\)\( \beta_{3} - \)\(10\!\cdots\!60\)\( \beta_{4} + \)\(15\!\cdots\!40\)\( \beta_{5} - \)\(13\!\cdots\!00\)\( \beta_{6}) q^{43}\) \(+(\)\(13\!\cdots\!32\)\( + \)\(90\!\cdots\!84\)\( \beta_{1} + \)\(10\!\cdots\!76\)\( \beta_{2} + \)\(60\!\cdots\!48\)\( \beta_{3} - \)\(62\!\cdots\!68\)\( \beta_{4} + \)\(72\!\cdots\!96\)\( \beta_{5} + \)\(82\!\cdots\!20\)\( \beta_{6}) q^{44}\) \(+(\)\(23\!\cdots\!68\)\( + \)\(17\!\cdots\!40\)\( \beta_{1} - \)\(12\!\cdots\!52\)\( \beta_{2} + \)\(30\!\cdots\!00\)\( \beta_{3} - \)\(47\!\cdots\!47\)\( \beta_{4} - \)\(23\!\cdots\!00\)\( \beta_{5} + \)\(13\!\cdots\!00\)\( \beta_{6}) q^{45}\) \(+(-\)\(91\!\cdots\!82\)\( - \)\(97\!\cdots\!60\)\( \beta_{1} - \)\(68\!\cdots\!24\)\( \beta_{2} - \)\(11\!\cdots\!56\)\( \beta_{3} + \)\(34\!\cdots\!88\)\( \beta_{4} + \)\(25\!\cdots\!34\)\( \beta_{5} - \)\(59\!\cdots\!00\)\( \beta_{6}) q^{46}\) \(+(-\)\(53\!\cdots\!80\)\( - \)\(28\!\cdots\!14\)\( \beta_{1} + \)\(98\!\cdots\!26\)\( \beta_{2} - \)\(43\!\cdots\!30\)\( \beta_{3} - \)\(12\!\cdots\!80\)\( \beta_{4} + \)\(16\!\cdots\!70\)\( \beta_{5} + \)\(12\!\cdots\!50\)\( \beta_{6}) q^{47}\) \(+(-\)\(59\!\cdots\!88\)\( - \)\(48\!\cdots\!04\)\( \beta_{1} + \)\(12\!\cdots\!80\)\( \beta_{2} + \)\(26\!\cdots\!12\)\( \beta_{3} + \)\(18\!\cdots\!44\)\( \beta_{4} + \)\(63\!\cdots\!52\)\( \beta_{5} + \)\(31\!\cdots\!24\)\( \beta_{6}) q^{48}\) \(+(\)\(36\!\cdots\!61\)\( - \)\(66\!\cdots\!60\)\( \beta_{1} + \)\(88\!\cdots\!44\)\( \beta_{2} + \)\(19\!\cdots\!16\)\( \beta_{3} - \)\(15\!\cdots\!48\)\( \beta_{4} - \)\(88\!\cdots\!64\)\( \beta_{5} - \)\(10\!\cdots\!00\)\( \beta_{6}) q^{49}\) \(+(-\)\(44\!\cdots\!50\)\( + \)\(71\!\cdots\!75\)\( \beta_{1} - \)\(34\!\cdots\!00\)\( \beta_{2} - \)\(59\!\cdots\!00\)\( \beta_{3} + \)\(12\!\cdots\!00\)\( \beta_{4} + \)\(22\!\cdots\!00\)\( \beta_{5} + \)\(35\!\cdots\!00\)\( \beta_{6}) q^{50}\) \(+(-\)\(79\!\cdots\!76\)\( + \)\(87\!\cdots\!68\)\( \beta_{1} - \)\(14\!\cdots\!12\)\( \beta_{2} - \)\(14\!\cdots\!10\)\( \beta_{3} - \)\(50\!\cdots\!08\)\( \beta_{4} + \)\(18\!\cdots\!46\)\( \beta_{5} - \)\(45\!\cdots\!10\)\( \beta_{6}) q^{51}\) \(+(-\)\(78\!\cdots\!60\)\( + \)\(19\!\cdots\!56\)\( \beta_{1} - \)\(25\!\cdots\!06\)\( \beta_{2} + \)\(14\!\cdots\!70\)\( \beta_{3} + \)\(46\!\cdots\!20\)\( \beta_{4} - \)\(25\!\cdots\!80\)\( \beta_{5} - \)\(79\!\cdots\!00\)\( \beta_{6}) q^{52}\) \(+(-\)\(51\!\cdots\!84\)\( + \)\(98\!\cdots\!32\)\( \beta_{1} + \)\(17\!\cdots\!04\)\( \beta_{2} + \)\(35\!\cdots\!96\)\( \beta_{3} + \)\(22\!\cdots\!37\)\( \beta_{4} + \)\(63\!\cdots\!16\)\( \beta_{5} + \)\(51\!\cdots\!12\)\( \beta_{6}) q^{53}\) \(+(-\)\(28\!\cdots\!18\)\( - \)\(33\!\cdots\!92\)\( \beta_{1} + \)\(16\!\cdots\!88\)\( \beta_{2} - \)\(10\!\cdots\!80\)\( \beta_{3} - \)\(70\!\cdots\!88\)\( \beta_{4} - \)\(33\!\cdots\!14\)\( \beta_{5} - \)\(10\!\cdots\!80\)\( \beta_{6}) q^{54}\) \(+(-\)\(50\!\cdots\!08\)\( - \)\(33\!\cdots\!65\)\( \beta_{1} - \)\(51\!\cdots\!63\)\( \beta_{2} + \)\(29\!\cdots\!75\)\( \beta_{3} + \)\(41\!\cdots\!82\)\( \beta_{4} - \)\(19\!\cdots\!75\)\( \beta_{5} + \)\(19\!\cdots\!75\)\( \beta_{6}) q^{55}\) \(+(-\)\(18\!\cdots\!40\)\( + \)\(14\!\cdots\!48\)\( \beta_{1} - \)\(28\!\cdots\!04\)\( \beta_{2} - \)\(74\!\cdots\!88\)\( \beta_{3} + \)\(18\!\cdots\!76\)\( \beta_{4} + \)\(45\!\cdots\!28\)\( \beta_{5} + \)\(42\!\cdots\!60\)\( \beta_{6}) q^{56}\) \(+(-\)\(19\!\cdots\!80\)\( + \)\(15\!\cdots\!20\)\( \beta_{1} - \)\(16\!\cdots\!44\)\( \beta_{2} + \)\(33\!\cdots\!60\)\( \beta_{3} + \)\(50\!\cdots\!10\)\( \beta_{4} - \)\(19\!\cdots\!40\)\( \beta_{5} - \)\(10\!\cdots\!00\)\( \beta_{6}) q^{57}\) \(+(-\)\(10\!\cdots\!08\)\( + \)\(40\!\cdots\!38\)\( \beta_{1} + \)\(21\!\cdots\!44\)\( \beta_{2} - \)\(36\!\cdots\!48\)\( \beta_{3} - \)\(17\!\cdots\!96\)\( \beta_{4} + \)\(52\!\cdots\!92\)\( \beta_{5} + \)\(64\!\cdots\!64\)\( \beta_{6}) q^{58}\) \(+(-\)\(17\!\cdots\!66\)\( + \)\(86\!\cdots\!99\)\( \beta_{1} + \)\(36\!\cdots\!87\)\( \beta_{2} + \)\(32\!\cdots\!12\)\( \beta_{3} + \)\(52\!\cdots\!60\)\( \beta_{4} - \)\(38\!\cdots\!40\)\( \beta_{5} + \)\(21\!\cdots\!80\)\( \beta_{6}) q^{59}\) \(+(-\)\(66\!\cdots\!08\)\( - \)\(70\!\cdots\!40\)\( \beta_{1} + \)\(99\!\cdots\!12\)\( \beta_{2} - \)\(35\!\cdots\!00\)\( \beta_{3} + \)\(54\!\cdots\!32\)\( \beta_{4} + \)\(10\!\cdots\!00\)\( \beta_{5} - \)\(40\!\cdots\!00\)\( \beta_{6}) q^{60}\) \(+(-\)\(46\!\cdots\!48\)\( - \)\(35\!\cdots\!20\)\( \beta_{1} - \)\(14\!\cdots\!40\)\( \beta_{2} + \)\(84\!\cdots\!40\)\( \beta_{3} - \)\(69\!\cdots\!75\)\( \beta_{4} + \)\(45\!\cdots\!40\)\( \beta_{5} - \)\(44\!\cdots\!60\)\( \beta_{6}) q^{61}\) \(+(-\)\(14\!\cdots\!68\)\( - \)\(69\!\cdots\!60\)\( \beta_{1} - \)\(87\!\cdots\!24\)\( \beta_{2} - \)\(21\!\cdots\!88\)\( \beta_{3} + \)\(16\!\cdots\!44\)\( \beta_{4} - \)\(79\!\cdots\!48\)\( \beta_{5} + \)\(15\!\cdots\!24\)\( \beta_{6}) q^{62}\) \(+(-\)\(32\!\cdots\!08\)\( - \)\(13\!\cdots\!69\)\( \beta_{1} + \)\(51\!\cdots\!89\)\( \beta_{2} - \)\(49\!\cdots\!53\)\( \beta_{3} - \)\(46\!\cdots\!66\)\( \beta_{4} + \)\(15\!\cdots\!37\)\( \beta_{5} + \)\(38\!\cdots\!59\)\( \beta_{6}) q^{63}\) \(+(-\)\(67\!\cdots\!56\)\( - \)\(59\!\cdots\!56\)\( \beta_{1} + \)\(15\!\cdots\!60\)\( \beta_{2} - \)\(54\!\cdots\!16\)\( \beta_{3} + \)\(15\!\cdots\!04\)\( \beta_{4} - \)\(10\!\cdots\!28\)\( \beta_{5} - \)\(22\!\cdots\!00\)\( \beta_{6}) q^{64}\) \(+(\)\(34\!\cdots\!76\)\( + \)\(10\!\cdots\!80\)\( \beta_{1} + \)\(24\!\cdots\!36\)\( \beta_{2} + \)\(67\!\cdots\!00\)\( \beta_{3} - \)\(54\!\cdots\!04\)\( \beta_{4} - \)\(32\!\cdots\!00\)\( \beta_{5} + \)\(30\!\cdots\!00\)\( \beta_{6}) q^{65}\) \(+(\)\(17\!\cdots\!80\)\( + \)\(21\!\cdots\!64\)\( \beta_{1} + \)\(37\!\cdots\!04\)\( \beta_{2} + \)\(18\!\cdots\!40\)\( \beta_{3} - \)\(36\!\cdots\!44\)\( \beta_{4} + \)\(21\!\cdots\!28\)\( \beta_{5} + \)\(63\!\cdots\!20\)\( \beta_{6}) q^{66}\) \(+(\)\(13\!\cdots\!98\)\( - \)\(23\!\cdots\!37\)\( \beta_{1} - \)\(21\!\cdots\!09\)\( \beta_{2} - \)\(19\!\cdots\!42\)\( \beta_{3} + \)\(31\!\cdots\!36\)\( \beta_{4} + \)\(74\!\cdots\!18\)\( \beta_{5} - \)\(31\!\cdots\!54\)\( \beta_{6}) q^{67}\) \(+(\)\(73\!\cdots\!64\)\( + \)\(13\!\cdots\!08\)\( \beta_{1} - \)\(15\!\cdots\!66\)\( \beta_{2} - \)\(37\!\cdots\!86\)\( \beta_{3} + \)\(92\!\cdots\!08\)\( \beta_{4} + \)\(30\!\cdots\!44\)\( \beta_{5} + \)\(44\!\cdots\!08\)\( \beta_{6}) q^{68}\) \(+(\)\(17\!\cdots\!40\)\( - \)\(95\!\cdots\!32\)\( \beta_{1} - \)\(72\!\cdots\!40\)\( \beta_{2} - \)\(11\!\cdots\!52\)\( \beta_{3} + \)\(36\!\cdots\!88\)\( \beta_{4} - \)\(10\!\cdots\!36\)\( \beta_{5} + \)\(33\!\cdots\!80\)\( \beta_{6}) q^{69}\) \(+(\)\(62\!\cdots\!64\)\( - \)\(55\!\cdots\!80\)\( \beta_{1} + \)\(37\!\cdots\!04\)\( \beta_{2} + \)\(52\!\cdots\!00\)\( \beta_{3} - \)\(15\!\cdots\!56\)\( \beta_{4} + \)\(37\!\cdots\!00\)\( \beta_{5} - \)\(24\!\cdots\!00\)\( \beta_{6}) q^{70}\) \(+(\)\(60\!\cdots\!72\)\( + \)\(18\!\cdots\!15\)\( \beta_{1} + \)\(45\!\cdots\!05\)\( \beta_{2} - \)\(39\!\cdots\!05\)\( \beta_{3} + \)\(10\!\cdots\!50\)\( \beta_{4} + \)\(40\!\cdots\!45\)\( \beta_{5} + \)\(40\!\cdots\!95\)\( \beta_{6}) q^{71}\) \(+(\)\(16\!\cdots\!24\)\( + \)\(81\!\cdots\!96\)\( \beta_{1} - \)\(40\!\cdots\!24\)\( \beta_{2} + \)\(35\!\cdots\!04\)\( \beta_{3} + \)\(25\!\cdots\!48\)\( \beta_{4} - \)\(79\!\cdots\!16\)\( \beta_{5} - \)\(95\!\cdots\!92\)\( \beta_{6}) q^{72}\) \(+(\)\(35\!\cdots\!62\)\( + \)\(16\!\cdots\!16\)\( \beta_{1} - \)\(15\!\cdots\!00\)\( \beta_{2} - \)\(20\!\cdots\!68\)\( \beta_{3} - \)\(28\!\cdots\!06\)\( \beta_{4} + \)\(82\!\cdots\!72\)\( \beta_{5} - \)\(96\!\cdots\!16\)\( \beta_{6}) q^{73}\) \(+(\)\(14\!\cdots\!60\)\( + \)\(41\!\cdots\!90\)\( \beta_{1} + \)\(11\!\cdots\!96\)\( \beta_{2} + \)\(17\!\cdots\!44\)\( \beta_{3} + \)\(42\!\cdots\!88\)\( \beta_{4} + \)\(10\!\cdots\!24\)\( \beta_{5} + \)\(24\!\cdots\!40\)\( \beta_{6}) q^{74}\) \(+(\)\(13\!\cdots\!50\)\( - \)\(22\!\cdots\!25\)\( \beta_{1} + \)\(11\!\cdots\!75\)\( \beta_{2} + \)\(82\!\cdots\!00\)\( \beta_{3} - \)\(21\!\cdots\!00\)\( \beta_{4} + \)\(25\!\cdots\!00\)\( \beta_{5} - \)\(31\!\cdots\!00\)\( \beta_{6}) q^{75}\) \(+(-\)\(14\!\cdots\!00\)\( - \)\(10\!\cdots\!04\)\( \beta_{1} + \)\(10\!\cdots\!72\)\( \beta_{2} - \)\(47\!\cdots\!56\)\( \beta_{3} + \)\(52\!\cdots\!92\)\( \beta_{4} - \)\(86\!\cdots\!24\)\( \beta_{5} + \)\(56\!\cdots\!20\)\( \beta_{6}) q^{76}\) \(+(-\)\(23\!\cdots\!52\)\( - \)\(11\!\cdots\!16\)\( \beta_{1} - \)\(16\!\cdots\!28\)\( \beta_{2} + \)\(24\!\cdots\!68\)\( \beta_{3} + \)\(11\!\cdots\!16\)\( \beta_{4} - \)\(80\!\cdots\!72\)\( \beta_{5} + \)\(10\!\cdots\!36\)\( \beta_{6}) q^{77}\) \(+(-\)\(10\!\cdots\!86\)\( - \)\(66\!\cdots\!08\)\( \beta_{1} + \)\(34\!\cdots\!96\)\( \beta_{2} - \)\(56\!\cdots\!36\)\( \beta_{3} - \)\(12\!\cdots\!12\)\( \beta_{4} + \)\(16\!\cdots\!94\)\( \beta_{5} - \)\(27\!\cdots\!32\)\( \beta_{6}) q^{78}\) \(+(-\)\(61\!\cdots\!64\)\( + \)\(59\!\cdots\!22\)\( \beta_{1} - \)\(96\!\cdots\!10\)\( \beta_{2} + \)\(69\!\cdots\!42\)\( \beta_{3} - \)\(37\!\cdots\!48\)\( \beta_{4} + \)\(11\!\cdots\!06\)\( \beta_{5} + \)\(26\!\cdots\!70\)\( \beta_{6}) q^{79}\) \(+(-\)\(12\!\cdots\!04\)\( + \)\(12\!\cdots\!80\)\( \beta_{1} + \)\(52\!\cdots\!56\)\( \beta_{2} + \)\(33\!\cdots\!00\)\( \beta_{3} + \)\(10\!\cdots\!16\)\( \beta_{4} - \)\(28\!\cdots\!00\)\( \beta_{5} + \)\(36\!\cdots\!00\)\( \beta_{6}) q^{80}\) \(+(-\)\(10\!\cdots\!07\)\( + \)\(23\!\cdots\!52\)\( \beta_{1} - \)\(15\!\cdots\!24\)\( \beta_{2} + \)\(14\!\cdots\!16\)\( \beta_{3} + \)\(75\!\cdots\!10\)\( \beta_{4} - \)\(10\!\cdots\!00\)\( \beta_{5} - \)\(12\!\cdots\!80\)\( \beta_{6}) q^{81}\) \(+(-\)\(33\!\cdots\!28\)\( + \)\(28\!\cdots\!70\)\( \beta_{1} + \)\(68\!\cdots\!16\)\( \beta_{2} - \)\(34\!\cdots\!08\)\( \beta_{3} - \)\(14\!\cdots\!76\)\( \beta_{4} + \)\(18\!\cdots\!32\)\( \beta_{5} + \)\(78\!\cdots\!24\)\( \beta_{6}) q^{82}\) \(+(-\)\(29\!\cdots\!70\)\( - \)\(73\!\cdots\!07\)\( \beta_{1} - \)\(14\!\cdots\!87\)\( \beta_{2} - \)\(79\!\cdots\!00\)\( \beta_{3} - \)\(68\!\cdots\!00\)\( \beta_{4} + \)\(56\!\cdots\!00\)\( \beta_{5} + \)\(13\!\cdots\!00\)\( \beta_{6}) q^{83}\) \(+(\)\(23\!\cdots\!04\)\( - \)\(64\!\cdots\!88\)\( \beta_{1} - \)\(98\!\cdots\!96\)\( \beta_{2} + \)\(17\!\cdots\!28\)\( \beta_{3} + \)\(20\!\cdots\!44\)\( \beta_{4} - \)\(18\!\cdots\!08\)\( \beta_{5} - \)\(37\!\cdots\!00\)\( \beta_{6}) q^{84}\) \(+(\)\(23\!\cdots\!72\)\( - \)\(20\!\cdots\!40\)\( \beta_{1} - \)\(16\!\cdots\!08\)\( \beta_{2} + \)\(34\!\cdots\!00\)\( \beta_{3} + \)\(11\!\cdots\!62\)\( \beta_{4} + \)\(72\!\cdots\!00\)\( \beta_{5} - \)\(60\!\cdots\!00\)\( \beta_{6}) q^{85}\) \(+(\)\(46\!\cdots\!13\)\( - \)\(24\!\cdots\!72\)\( \beta_{1} + \)\(15\!\cdots\!52\)\( \beta_{2} + \)\(95\!\cdots\!06\)\( \beta_{3} + \)\(43\!\cdots\!74\)\( \beta_{4} + \)\(43\!\cdots\!47\)\( \beta_{5} + \)\(21\!\cdots\!40\)\( \beta_{6}) q^{86}\) \(+(\)\(93\!\cdots\!12\)\( + \)\(25\!\cdots\!73\)\( \beta_{1} + \)\(12\!\cdots\!95\)\( \beta_{2} - \)\(40\!\cdots\!23\)\( \beta_{3} - \)\(17\!\cdots\!86\)\( \beta_{4} - \)\(77\!\cdots\!33\)\( \beta_{5} + \)\(36\!\cdots\!09\)\( \beta_{6}) q^{87}\) \(+(\)\(90\!\cdots\!52\)\( + \)\(34\!\cdots\!48\)\( \beta_{1} + \)\(67\!\cdots\!92\)\( \beta_{2} - \)\(23\!\cdots\!68\)\( \beta_{3} - \)\(23\!\cdots\!16\)\( \beta_{4} + \)\(33\!\cdots\!72\)\( \beta_{5} - \)\(47\!\cdots\!36\)\( \beta_{6}) q^{88}\) \(+(\)\(79\!\cdots\!30\)\( + \)\(12\!\cdots\!16\)\( \beta_{1} + \)\(73\!\cdots\!32\)\( \beta_{2} - \)\(39\!\cdots\!36\)\( \beta_{3} + \)\(18\!\cdots\!62\)\( \beta_{4} + \)\(34\!\cdots\!56\)\( \beta_{5} - \)\(74\!\cdots\!60\)\( \beta_{6}) q^{89}\) \(+(\)\(28\!\cdots\!76\)\( + \)\(49\!\cdots\!30\)\( \beta_{1} - \)\(40\!\cdots\!64\)\( \beta_{2} + \)\(26\!\cdots\!00\)\( \beta_{3} + \)\(90\!\cdots\!96\)\( \beta_{4} - \)\(13\!\cdots\!00\)\( \beta_{5} + \)\(21\!\cdots\!00\)\( \beta_{6}) q^{90}\) \(+(-\)\(25\!\cdots\!08\)\( - \)\(13\!\cdots\!32\)\( \beta_{1} + \)\(18\!\cdots\!96\)\( \beta_{2} + \)\(16\!\cdots\!32\)\( \beta_{3} - \)\(11\!\cdots\!04\)\( \beta_{4} + \)\(40\!\cdots\!88\)\( \beta_{5} + \)\(23\!\cdots\!60\)\( \beta_{6}) q^{91}\) \(+(-\)\(11\!\cdots\!48\)\( - \)\(11\!\cdots\!76\)\( \beta_{1} - \)\(39\!\cdots\!32\)\( \beta_{2} - \)\(51\!\cdots\!88\)\( \beta_{3} - \)\(31\!\cdots\!36\)\( \beta_{4} - \)\(40\!\cdots\!48\)\( \beta_{5} - \)\(56\!\cdots\!36\)\( \beta_{6}) q^{92}\) \(+(-\)\(17\!\cdots\!24\)\( - \)\(12\!\cdots\!60\)\( \beta_{1} + \)\(57\!\cdots\!00\)\( \beta_{2} - \)\(15\!\cdots\!04\)\( \beta_{3} + \)\(58\!\cdots\!32\)\( \beta_{4} - \)\(75\!\cdots\!84\)\( \beta_{5} + \)\(15\!\cdots\!52\)\( \beta_{6}) q^{93}\) \(+(-\)\(43\!\cdots\!24\)\( - \)\(92\!\cdots\!52\)\( \beta_{1} + \)\(67\!\cdots\!68\)\( \beta_{2} - \)\(20\!\cdots\!20\)\( \beta_{3} - \)\(12\!\cdots\!08\)\( \beta_{4} + \)\(14\!\cdots\!76\)\( \beta_{5} + \)\(14\!\cdots\!20\)\( \beta_{6}) q^{94}\) \(+(-\)\(30\!\cdots\!40\)\( - \)\(23\!\cdots\!75\)\( \beta_{1} + \)\(27\!\cdots\!35\)\( \beta_{2} + \)\(42\!\cdots\!25\)\( \beta_{3} + \)\(52\!\cdots\!10\)\( \beta_{4} + \)\(15\!\cdots\!75\)\( \beta_{5} - \)\(10\!\cdots\!75\)\( \beta_{6}) q^{95}\) \(+(-\)\(32\!\cdots\!84\)\( + \)\(29\!\cdots\!32\)\( \beta_{1} - \)\(51\!\cdots\!92\)\( \beta_{2} + \)\(30\!\cdots\!64\)\( \beta_{3} - \)\(10\!\cdots\!44\)\( \beta_{4} - \)\(38\!\cdots\!92\)\( \beta_{5} - \)\(37\!\cdots\!00\)\( \beta_{6}) q^{96}\) \(+(\)\(62\!\cdots\!26\)\( + \)\(13\!\cdots\!60\)\( \beta_{1} - \)\(15\!\cdots\!36\)\( \beta_{2} + \)\(26\!\cdots\!96\)\( \beta_{3} - \)\(38\!\cdots\!38\)\( \beta_{4} - \)\(44\!\cdots\!84\)\( \beta_{5} + \)\(44\!\cdots\!12\)\( \beta_{6}) q^{97}\) \(+(-\)\(98\!\cdots\!22\)\( + \)\(17\!\cdots\!89\)\( \beta_{1} - \)\(92\!\cdots\!76\)\( \beta_{2} - \)\(24\!\cdots\!72\)\( \beta_{3} + \)\(86\!\cdots\!16\)\( \beta_{4} + \)\(12\!\cdots\!88\)\( \beta_{5} + \)\(76\!\cdots\!16\)\( \beta_{6}) q^{98}\) \(+(\)\(43\!\cdots\!06\)\( - \)\(83\!\cdots\!37\)\( \beta_{1} + \)\(95\!\cdots\!11\)\( \beta_{2} - \)\(17\!\cdots\!88\)\( \beta_{3} + \)\(21\!\cdots\!36\)\( \beta_{4} + \)\(48\!\cdots\!08\)\( \beta_{5} - \)\(13\!\cdots\!40\)\( \beta_{6}) q^{99}\) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(7q \) \(\mathstrut +\mathstrut 43735426713792q^{2} \) \(\mathstrut -\mathstrut \)\(36\!\cdots\!84\)\(q^{3} \) \(\mathstrut +\mathstrut \)\(37\!\cdots\!44\)\(q^{4} \) \(\mathstrut -\mathstrut \)\(24\!\cdots\!50\)\(q^{5} \) \(\mathstrut -\mathstrut \)\(34\!\cdots\!56\)\(q^{6} \) \(\mathstrut -\mathstrut \)\(92\!\cdots\!08\)\(q^{7} \) \(\mathstrut +\mathstrut \)\(62\!\cdots\!60\)\(q^{8} \) \(\mathstrut +\mathstrut \)\(36\!\cdots\!11\)\(q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(7q \) \(\mathstrut +\mathstrut 43735426713792q^{2} \) \(\mathstrut -\mathstrut \)\(36\!\cdots\!84\)\(q^{3} \) \(\mathstrut +\mathstrut \)\(37\!\cdots\!44\)\(q^{4} \) \(\mathstrut -\mathstrut \)\(24\!\cdots\!50\)\(q^{5} \) \(\mathstrut -\mathstrut \)\(34\!\cdots\!56\)\(q^{6} \) \(\mathstrut -\mathstrut \)\(92\!\cdots\!08\)\(q^{7} \) \(\mathstrut +\mathstrut \)\(62\!\cdots\!60\)\(q^{8} \) \(\mathstrut +\mathstrut \)\(36\!\cdots\!11\)\(q^{9} \) \(\mathstrut +\mathstrut \)\(56\!\cdots\!00\)\(q^{10} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!24\)\(q^{11} \) \(\mathstrut -\mathstrut \)\(16\!\cdots\!48\)\(q^{12} \) \(\mathstrut +\mathstrut \)\(19\!\cdots\!26\)\(q^{13} \) \(\mathstrut -\mathstrut \)\(82\!\cdots\!32\)\(q^{14} \) \(\mathstrut -\mathstrut \)\(16\!\cdots\!00\)\(q^{15} \) \(\mathstrut -\mathstrut \)\(18\!\cdots\!48\)\(q^{16} \) \(\mathstrut +\mathstrut \)\(80\!\cdots\!42\)\(q^{17} \) \(\mathstrut +\mathstrut \)\(79\!\cdots\!36\)\(q^{18} \) \(\mathstrut -\mathstrut \)\(49\!\cdots\!00\)\(q^{19} \) \(\mathstrut -\mathstrut \)\(58\!\cdots\!00\)\(q^{20} \) \(\mathstrut +\mathstrut \)\(54\!\cdots\!84\)\(q^{21} \) \(\mathstrut +\mathstrut \)\(34\!\cdots\!44\)\(q^{22} \) \(\mathstrut -\mathstrut \)\(25\!\cdots\!64\)\(q^{23} \) \(\mathstrut -\mathstrut \)\(29\!\cdots\!00\)\(q^{24} \) \(\mathstrut +\mathstrut \)\(18\!\cdots\!25\)\(q^{25} \) \(\mathstrut +\mathstrut \)\(79\!\cdots\!64\)\(q^{26} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!40\)\(q^{27} \) \(\mathstrut +\mathstrut \)\(19\!\cdots\!24\)\(q^{28} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!50\)\(q^{29} \) \(\mathstrut -\mathstrut \)\(64\!\cdots\!00\)\(q^{30} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!56\)\(q^{31} \) \(\mathstrut -\mathstrut \)\(70\!\cdots\!88\)\(q^{32} \) \(\mathstrut +\mathstrut \)\(66\!\cdots\!12\)\(q^{33} \) \(\mathstrut +\mathstrut \)\(80\!\cdots\!28\)\(q^{34} \) \(\mathstrut -\mathstrut \)\(16\!\cdots\!00\)\(q^{35} \) \(\mathstrut +\mathstrut \)\(52\!\cdots\!12\)\(q^{36} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!42\)\(q^{37} \) \(\mathstrut -\mathstrut \)\(42\!\cdots\!60\)\(q^{38} \) \(\mathstrut -\mathstrut \)\(20\!\cdots\!68\)\(q^{39} \) \(\mathstrut +\mathstrut \)\(76\!\cdots\!00\)\(q^{40} \) \(\mathstrut -\mathstrut \)\(50\!\cdots\!46\)\(q^{41} \) \(\mathstrut -\mathstrut \)\(54\!\cdots\!76\)\(q^{42} \) \(\mathstrut -\mathstrut \)\(72\!\cdots\!44\)\(q^{43} \) \(\mathstrut +\mathstrut \)\(94\!\cdots\!08\)\(q^{44} \) \(\mathstrut +\mathstrut \)\(16\!\cdots\!50\)\(q^{45} \) \(\mathstrut -\mathstrut \)\(63\!\cdots\!16\)\(q^{46} \) \(\mathstrut -\mathstrut \)\(37\!\cdots\!08\)\(q^{47} \) \(\mathstrut -\mathstrut \)\(41\!\cdots\!64\)\(q^{48} \) \(\mathstrut +\mathstrut \)\(25\!\cdots\!99\)\(q^{49} \) \(\mathstrut -\mathstrut \)\(31\!\cdots\!00\)\(q^{50} \) \(\mathstrut -\mathstrut \)\(55\!\cdots\!36\)\(q^{51} \) \(\mathstrut -\mathstrut \)\(55\!\cdots\!28\)\(q^{52} \) \(\mathstrut -\mathstrut \)\(36\!\cdots\!34\)\(q^{53} \) \(\mathstrut -\mathstrut \)\(19\!\cdots\!00\)\(q^{54} \) \(\mathstrut -\mathstrut \)\(35\!\cdots\!00\)\(q^{55} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!00\)\(q^{56} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!80\)\(q^{57} \) \(\mathstrut -\mathstrut \)\(73\!\cdots\!40\)\(q^{58} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!00\)\(q^{59} \) \(\mathstrut -\mathstrut \)\(46\!\cdots\!00\)\(q^{60} \) \(\mathstrut -\mathstrut \)\(32\!\cdots\!26\)\(q^{61} \) \(\mathstrut -\mathstrut \)\(99\!\cdots\!36\)\(q^{62} \) \(\mathstrut -\mathstrut \)\(22\!\cdots\!64\)\(q^{63} \) \(\mathstrut -\mathstrut \)\(47\!\cdots\!16\)\(q^{64} \) \(\mathstrut +\mathstrut \)\(24\!\cdots\!00\)\(q^{65} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!08\)\(q^{66} \) \(\mathstrut +\mathstrut \)\(97\!\cdots\!92\)\(q^{67} \) \(\mathstrut +\mathstrut \)\(51\!\cdots\!24\)\(q^{68} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!92\)\(q^{69} \) \(\mathstrut +\mathstrut \)\(43\!\cdots\!00\)\(q^{70} \) \(\mathstrut +\mathstrut \)\(42\!\cdots\!84\)\(q^{71} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!80\)\(q^{72} \) \(\mathstrut +\mathstrut \)\(24\!\cdots\!86\)\(q^{73} \) \(\mathstrut +\mathstrut \)\(98\!\cdots\!48\)\(q^{74} \) \(\mathstrut +\mathstrut \)\(94\!\cdots\!00\)\(q^{75} \) \(\mathstrut -\mathstrut \)\(98\!\cdots\!00\)\(q^{76} \) \(\mathstrut -\mathstrut \)\(16\!\cdots\!56\)\(q^{77} \) \(\mathstrut -\mathstrut \)\(73\!\cdots\!28\)\(q^{78} \) \(\mathstrut -\mathstrut \)\(43\!\cdots\!00\)\(q^{79} \) \(\mathstrut -\mathstrut \)\(88\!\cdots\!00\)\(q^{80} \) \(\mathstrut -\mathstrut \)\(70\!\cdots\!53\)\(q^{81} \) \(\mathstrut -\mathstrut \)\(23\!\cdots\!76\)\(q^{82} \) \(\mathstrut -\mathstrut \)\(20\!\cdots\!04\)\(q^{83} \) \(\mathstrut +\mathstrut \)\(16\!\cdots\!28\)\(q^{84} \) \(\mathstrut +\mathstrut \)\(16\!\cdots\!00\)\(q^{85} \) \(\mathstrut +\mathstrut \)\(32\!\cdots\!24\)\(q^{86} \) \(\mathstrut +\mathstrut \)\(65\!\cdots\!80\)\(q^{87} \) \(\mathstrut +\mathstrut \)\(63\!\cdots\!20\)\(q^{88} \) \(\mathstrut +\mathstrut \)\(55\!\cdots\!50\)\(q^{89} \) \(\mathstrut +\mathstrut \)\(19\!\cdots\!00\)\(q^{90} \) \(\mathstrut -\mathstrut \)\(18\!\cdots\!96\)\(q^{91} \) \(\mathstrut -\mathstrut \)\(81\!\cdots\!08\)\(q^{92} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!28\)\(q^{93} \) \(\mathstrut -\mathstrut \)\(30\!\cdots\!92\)\(q^{94} \) \(\mathstrut -\mathstrut \)\(21\!\cdots\!00\)\(q^{95} \) \(\mathstrut -\mathstrut \)\(22\!\cdots\!16\)\(q^{96} \) \(\mathstrut +\mathstrut \)\(43\!\cdots\!42\)\(q^{97} \) \(\mathstrut -\mathstrut \)\(69\!\cdots\!56\)\(q^{98} \) \(\mathstrut +\mathstrut \)\(30\!\cdots\!52\)\(q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{7}\mathstrut -\mathstrut \) \(x^{6}\mathstrut -\mathstrut \) \(160477500301516091326739\) \(x^{5}\mathstrut +\mathstrut \) \(877016488484326647371325741724874\) \(x^{4}\mathstrut +\mathstrut \) \(7260529465737129707868752892581169765229378456\) \(x^{3}\mathstrut -\mathstrut \) \(20781038399188480098606854392326662967337072615105929280\) \(x^{2}\mathstrut -\mathstrut \) \(71309214652872234197294752847774640455181142633761719353245451878000\) \(x\mathstrut -\mathstrut \) \(1353216958878139720025204995487184336935523797943751976847532373756765247900000\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 576 \nu - 82 \)
\(\beta_{2}\)\(=\)\((\)\(72\!\cdots\!11\) \(\nu^{6}\mathstrut -\mathstrut \) \(34\!\cdots\!61\) \(\nu^{5}\mathstrut -\mathstrut \) \(67\!\cdots\!07\) \(\nu^{4}\mathstrut +\mathstrut \) \(33\!\cdots\!40\) \(\nu^{3}\mathstrut +\mathstrut \) \(47\!\cdots\!76\) \(\nu^{2}\mathstrut -\mathstrut \) \(58\!\cdots\!24\) \(\nu\mathstrut -\mathstrut \) \(37\!\cdots\!40\)\()/\)\(15\!\cdots\!56\)
\(\beta_{3}\)\(=\)\((\)\(44\!\cdots\!29\) \(\nu^{6}\mathstrut -\mathstrut \) \(21\!\cdots\!79\) \(\nu^{5}\mathstrut -\mathstrut \) \(41\!\cdots\!73\) \(\nu^{4}\mathstrut +\mathstrut \) \(20\!\cdots\!60\) \(\nu^{3}\mathstrut +\mathstrut \) \(80\!\cdots\!20\) \(\nu^{2}\mathstrut -\mathstrut \) \(32\!\cdots\!72\) \(\nu\mathstrut -\mathstrut \) \(25\!\cdots\!28\)\()/\)\(15\!\cdots\!56\)
\(\beta_{4}\)\(=\)\((\)\(43\!\cdots\!13\) \(\nu^{6}\mathstrut +\mathstrut \) \(48\!\cdots\!69\) \(\nu^{5}\mathstrut -\mathstrut \) \(61\!\cdots\!61\) \(\nu^{4}\mathstrut -\mathstrut \) \(36\!\cdots\!32\) \(\nu^{3}\mathstrut +\mathstrut \) \(23\!\cdots\!20\) \(\nu^{2}\mathstrut +\mathstrut \) \(68\!\cdots\!60\) \(\nu\mathstrut -\mathstrut \) \(13\!\cdots\!40\)\()/\)\(23\!\cdots\!40\)
\(\beta_{5}\)\(=\)\((\)\(-\)\(47\!\cdots\!31\) \(\nu^{6}\mathstrut -\mathstrut \) \(46\!\cdots\!23\) \(\nu^{5}\mathstrut +\mathstrut \) \(50\!\cdots\!47\) \(\nu^{4}\mathstrut +\mathstrut \) \(29\!\cdots\!84\) \(\nu^{3}\mathstrut -\mathstrut \) \(10\!\cdots\!60\) \(\nu^{2}\mathstrut +\mathstrut \) \(65\!\cdots\!00\) \(\nu\mathstrut +\mathstrut \) \(16\!\cdots\!40\)\()/\)\(38\!\cdots\!40\)
\(\beta_{6}\)\(=\)\((\)\(-\)\(21\!\cdots\!85\) \(\nu^{6}\mathstrut -\mathstrut \) \(62\!\cdots\!73\) \(\nu^{5}\mathstrut +\mathstrut \) \(23\!\cdots\!61\) \(\nu^{4}\mathstrut +\mathstrut \) \(89\!\cdots\!24\) \(\nu^{3}\mathstrut -\mathstrut \) \(47\!\cdots\!96\) \(\nu^{2}\mathstrut -\mathstrut \) \(30\!\cdots\!92\) \(\nu\mathstrut -\mathstrut \) \(29\!\cdots\!36\)\()/\)\(76\!\cdots\!28\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1}\mathstrut +\mathstrut \) \(82\)\()/576\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3}\mathstrut -\mathstrut \) \(61939\) \(\beta_{2}\mathstrut -\mathstrut \) \(4721797423044\) \(\beta_{1}\mathstrut +\mathstrut \) \(15212166611438802125090784970\)\()/331776\)
\(\nu^{3}\)\(=\)\((\)\(27\) \(\beta_{6}\mathstrut -\mathstrut \) \(864\) \(\beta_{5}\mathstrut +\mathstrut \) \(37744952\) \(\beta_{4}\mathstrut -\mathstrut \) \(1456222316363\) \(\beta_{3}\mathstrut -\mathstrut \) \(499178190561593813\) \(\beta_{2}\mathstrut +\mathstrut \) \(1451331883235195183810578437\) \(\beta_{1}\mathstrut -\mathstrut \) \(4489298060351043874647950339581169424175\)\()/11943936\)
\(\nu^{4}\)\(=\)\((\)\(-\)\(4463654783831\) \(\beta_{6}\mathstrut -\mathstrut \) \(21498868064364384\) \(\beta_{5}\mathstrut -\mathstrut \) \(155388708286642932248\) \(\beta_{4}\mathstrut +\mathstrut \) \(28081700571652480371245099\) \(\beta_{3}\mathstrut -\mathstrut \) \(1815877728278393136954118531219\) \(\beta_{2}\mathstrut -\mathstrut \) \(348763612805097247584437129526130994905\) \(\beta_{1}\mathstrut +\mathstrut \) \(344967225271537957345817879516443292173032621371678371\)\()/\)\(107495424\)
\(\nu^{5}\)\(=\)\((\)\(4031902847950727791358793\) \(\beta_{6}\mathstrut +\mathstrut \) \(446030637567255082180432384\) \(\beta_{5}\mathstrut +\mathstrut \) \(9799409161359655276620666800168\) \(\beta_{4}\mathstrut -\mathstrut \) \(296422306245977895528775566747803031\) \(\beta_{3}\mathstrut -\mathstrut \) \(142017271835412325593196941825801689218797\) \(\beta_{2}\mathstrut +\mathstrut \) \(165215070082703970546710102086700872073524127175695\) \(\beta_{1}\mathstrut -\mathstrut \) \(1535141835048048933707568034851741984779886813917883504203651633\)\()/17915904\)
\(\nu^{6}\)\(=\)\((\)\(-\)\(1335541450894285404984266582091313909\) \(\beta_{6}\mathstrut -\mathstrut \) \(3426487876318248168893117888497548692064\) \(\beta_{5}\mathstrut -\mathstrut \) \(19968275914226553615587787533690160626665288\) \(\beta_{4}\mathstrut +\mathstrut \) \(2314388201240813675181945981146024360029919939773\) \(\beta_{3}\mathstrut -\mathstrut \) \(89910593683805545602419486337132984127228260542427725\) \(\beta_{2}\mathstrut -\mathstrut \) \(46930504248390006308237368207327007259269959952589989701125323\) \(\beta_{1}\mathstrut +\mathstrut \) \(26179991386820870474566739391248959942327523521524187102176189183833960599121\)\()/\)\(107495424\)

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
−3.01170e11
−2.45391e11
−9.86534e10
−1.98834e10
1.30998e11
2.59428e11
2.74671e11
−1.67226e14 1.22637e22 1.80609e28 −5.69707e32 −2.05080e36 3.48367e39 −1.36413e42 −8.52575e43 9.52696e46
1.2 −1.35097e14 −1.21702e22 8.34770e27 3.52234e32 1.64416e36 −7.46408e38 2.10186e41 −8.75419e43 −4.75858e46
1.3 −5.05764e13 2.16277e22 −7.34554e27 1.20105e32 −1.09385e36 −1.85159e39 8.72396e41 2.32102e44 −6.07451e45
1.4 −5.20495e12 −1.59625e22 −9.87643e27 −3.34454e32 8.30840e34 −2.46426e38 1.02954e41 1.91468e43 1.74082e45
1.5 8.17029e13 3.94920e21 −3.22816e27 2.56302e32 3.22661e35 2.42990e39 −1.07290e42 −2.20059e44 2.09406e46
1.6 1.55678e14 1.43559e22 1.43322e28 −3.85879e32 2.23491e36 −1.93687e39 6.89447e41 −2.95620e43 −6.00730e46
1.7 1.64459e14 −2.77426e22 1.71431e28 3.18897e32 −4.56250e36 −2.05341e39 1.19061e42 5.33996e44 5.24454e46
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
Significant digits:
Format:

Inner twists

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

Hecke kernels

There are no other newforms in \(S_{94}^{\mathrm{new}}(\Gamma_0(1))\).