Properties

Label 1.92.a.a
Level 1
Weight 92
Character orbit 1.a
Self dual Yes
Analytic conductor 52.442
Analytic rank 0
Dimension 7
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 1 \)
Weight: \( k \) = \( 92 \)
Character orbit: \([\chi]\) = 1.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(52.442155831\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: multiple of \( 2^{83}\cdot 3^{31}\cdot 5^{8}\cdot 7^{6}\cdot 11\cdot 13^{3}\cdot 23 \)
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.

\(f(q)\) \(=\) \( q\) \(+(548816691151 + \beta_{1}) q^{2}\) \(+(\)\(88\!\cdots\!63\)\( + 19467112 \beta_{1} + \beta_{2}) q^{3}\) \(+(\)\(81\!\cdots\!85\)\( - 173326932865 \beta_{1} + 49571 \beta_{2} + \beta_{3}) q^{4}\) \(+(\)\(33\!\cdots\!67\)\( + 203684509503647605 \beta_{1} + 3408688857 \beta_{2} + 3464 \beta_{3} - \beta_{4}) q^{5}\) \(+(\)\(64\!\cdots\!93\)\( + \)\(13\!\cdots\!16\)\( \beta_{1} + 7916444742738 \beta_{2} - 24868 \beta_{3} - 1212 \beta_{4} + \beta_{5}) q^{6}\) \(+(-\)\(24\!\cdots\!13\)\( + \)\(77\!\cdots\!83\)\( \beta_{1} - 15418806458979859 \beta_{2} - 20904077593 \beta_{3} + 79668 \beta_{4} + 29 \beta_{5} + \beta_{6}) q^{7}\) \(+(-\)\(14\!\cdots\!32\)\( + \)\(36\!\cdots\!44\)\( \beta_{1} - 4021839698115592272 \beta_{2} - 6483639273776 \beta_{3} - 101411744 \beta_{4} + 103728 \beta_{5} + 312 \beta_{6}) q^{8}\) \(+(\)\(54\!\cdots\!55\)\( + \)\(10\!\cdots\!58\)\( \beta_{1} + \)\(81\!\cdots\!14\)\( \beta_{2} + 204820844789196 \beta_{3} - 127519054326 \beta_{4} + 80849268 \beta_{5} - 4860 \beta_{6}) q^{9}\) \(+O(q^{10})\) \( q\) \(+(548816691151 + \beta_{1}) q^{2}\) \(+(\)\(88\!\cdots\!63\)\( + 19467112 \beta_{1} + \beta_{2}) q^{3}\) \(+(\)\(81\!\cdots\!85\)\( - 173326932865 \beta_{1} + 49571 \beta_{2} + \beta_{3}) q^{4}\) \(+(\)\(33\!\cdots\!67\)\( + 203684509503647605 \beta_{1} + 3408688857 \beta_{2} + 3464 \beta_{3} - \beta_{4}) q^{5}\) \(+(\)\(64\!\cdots\!93\)\( + \)\(13\!\cdots\!16\)\( \beta_{1} + 7916444742738 \beta_{2} - 24868 \beta_{3} - 1212 \beta_{4} + \beta_{5}) q^{6}\) \(+(-\)\(24\!\cdots\!13\)\( + \)\(77\!\cdots\!83\)\( \beta_{1} - 15418806458979859 \beta_{2} - 20904077593 \beta_{3} + 79668 \beta_{4} + 29 \beta_{5} + \beta_{6}) q^{7}\) \(+(-\)\(14\!\cdots\!32\)\( + \)\(36\!\cdots\!44\)\( \beta_{1} - 4021839698115592272 \beta_{2} - 6483639273776 \beta_{3} - 101411744 \beta_{4} + 103728 \beta_{5} + 312 \beta_{6}) q^{8}\) \(+(\)\(54\!\cdots\!55\)\( + \)\(10\!\cdots\!58\)\( \beta_{1} + \)\(81\!\cdots\!14\)\( \beta_{2} + 204820844789196 \beta_{3} - 127519054326 \beta_{4} + 80849268 \beta_{5} - 4860 \beta_{6}) q^{9}\) \(+(\)\(67\!\cdots\!02\)\( + \)\(11\!\cdots\!90\)\( \beta_{1} + \)\(17\!\cdots\!12\)\( \beta_{2} + 143330903119276464 \beta_{3} - 28411298625456 \beta_{4} + 7949474900 \beta_{5} - 3447680 \beta_{6}) q^{10}\) \(+(-\)\(34\!\cdots\!69\)\( + \)\(12\!\cdots\!98\)\( \beta_{1} + \)\(99\!\cdots\!77\)\( \beta_{2} + 8437080972595758094 \beta_{3} - 2522095344612056 \beta_{4} - 615036062982 \beta_{5} + 360284610 \beta_{6}) q^{11}\) \(+(\)\(21\!\cdots\!00\)\( + \)\(18\!\cdots\!00\)\( \beta_{1} + \)\(84\!\cdots\!24\)\( \beta_{2} + \)\(19\!\cdots\!44\)\( \beta_{3} - 137230898165028864 \beta_{4} + 5600704186368 \beta_{5} - 19730262528 \beta_{6}) q^{12}\) \(+(\)\(10\!\cdots\!67\)\( + \)\(61\!\cdots\!01\)\( \beta_{1} - \)\(13\!\cdots\!35\)\( \beta_{2} + \)\(83\!\cdots\!84\)\( \beta_{3} + 3130478306245636911 \beta_{4} + 660012374861848 \beta_{5} + 736959274232 \beta_{6}) q^{13}\) \(+(\)\(25\!\cdots\!22\)\( - \)\(73\!\cdots\!40\)\( \beta_{1} + \)\(95\!\cdots\!48\)\( \beta_{2} + \)\(53\!\cdots\!88\)\( \beta_{3} + \)\(12\!\cdots\!80\)\( \beta_{4} - 31314710120571030 \beta_{5} - 20712747194880 \beta_{6}) q^{14}\) \(+(\)\(11\!\cdots\!29\)\( + \)\(20\!\cdots\!05\)\( \beta_{1} + \)\(18\!\cdots\!99\)\( \beta_{2} + \)\(86\!\cdots\!53\)\( \beta_{3} - \)\(40\!\cdots\!12\)\( \beta_{4} + 709496409196973675 \beta_{5} + 460952841048615 \beta_{6}) q^{15}\) \(+(-\)\(79\!\cdots\!20\)\( - \)\(16\!\cdots\!04\)\( \beta_{1} + \)\(22\!\cdots\!72\)\( \beta_{2} - \)\(61\!\cdots\!64\)\( \beta_{3} + \)\(24\!\cdots\!28\)\( \beta_{4} - 9148275738935924864 \beta_{5} - 8369862307627840 \beta_{6}) q^{16}\) \(+(-\)\(16\!\cdots\!96\)\( + \)\(19\!\cdots\!94\)\( \beta_{1} + \)\(36\!\cdots\!38\)\( \beta_{2} - \)\(31\!\cdots\!92\)\( \beta_{3} + \)\(61\!\cdots\!42\)\( \beta_{4} + 43604599257025439076 \beta_{5} + 126235309562590644 \beta_{6}) q^{17}\) \(+(\)\(36\!\cdots\!43\)\( + \)\(61\!\cdots\!49\)\( \beta_{1} + \)\(27\!\cdots\!96\)\( \beta_{2} + \)\(13\!\cdots\!72\)\( \beta_{3} - \)\(12\!\cdots\!72\)\( \beta_{4} + \)\(83\!\cdots\!84\)\( \beta_{5} - 1595655892723739904 \beta_{6}) q^{18}\) \(+(\)\(35\!\cdots\!45\)\( + \)\(27\!\cdots\!14\)\( \beta_{1} + \)\(23\!\cdots\!35\)\( \beta_{2} + \)\(33\!\cdots\!26\)\( \beta_{3} + \)\(71\!\cdots\!72\)\( \beta_{4} - \)\(22\!\cdots\!66\)\( \beta_{5} + 16907326006883839630 \beta_{6}) q^{19}\) \(+(\)\(30\!\cdots\!26\)\( + \)\(52\!\cdots\!90\)\( \beta_{1} + \)\(21\!\cdots\!46\)\( \beta_{2} - \)\(90\!\cdots\!58\)\( \beta_{3} + \)\(43\!\cdots\!72\)\( \beta_{4} + \)\(29\!\cdots\!00\)\( \beta_{5} - \)\(14\!\cdots\!00\)\( \beta_{6}) q^{20}\) \(+(-\)\(42\!\cdots\!32\)\( + \)\(17\!\cdots\!00\)\( \beta_{1} - \)\(26\!\cdots\!32\)\( \beta_{2} - \)\(30\!\cdots\!52\)\( \beta_{3} - \)\(88\!\cdots\!60\)\( \beta_{4} - \)\(25\!\cdots\!00\)\( \beta_{5} + \)\(10\!\cdots\!40\)\( \beta_{6}) q^{21}\) \(+(\)\(39\!\cdots\!87\)\( - \)\(17\!\cdots\!84\)\( \beta_{1} - \)\(14\!\cdots\!90\)\( \beta_{2} + \)\(11\!\cdots\!24\)\( \beta_{3} + \)\(34\!\cdots\!96\)\( \beta_{4} + \)\(11\!\cdots\!03\)\( \beta_{5} - \)\(51\!\cdots\!48\)\( \beta_{6}) q^{22}\) \(+(\)\(40\!\cdots\!21\)\( - \)\(60\!\cdots\!43\)\( \beta_{1} - \)\(10\!\cdots\!41\)\( \beta_{2} + \)\(15\!\cdots\!45\)\( \beta_{3} + \)\(29\!\cdots\!80\)\( \beta_{4} + \)\(60\!\cdots\!15\)\( \beta_{5} + \)\(75\!\cdots\!35\)\( \beta_{6}) q^{23}\) \(+(-\)\(96\!\cdots\!84\)\( + \)\(31\!\cdots\!24\)\( \beta_{1} + \)\(17\!\cdots\!28\)\( \beta_{2} - \)\(82\!\cdots\!76\)\( \beta_{3} - \)\(39\!\cdots\!28\)\( \beta_{4} - \)\(69\!\cdots\!76\)\( \beta_{5} + \)\(17\!\cdots\!60\)\( \beta_{6}) q^{24}\) \(+(\)\(46\!\cdots\!35\)\( + \)\(42\!\cdots\!00\)\( \beta_{1} + \)\(49\!\cdots\!60\)\( \beta_{2} - \)\(55\!\cdots\!80\)\( \beta_{3} + \)\(11\!\cdots\!20\)\( \beta_{4} + \)\(68\!\cdots\!00\)\( \beta_{5} - \)\(24\!\cdots\!00\)\( \beta_{6}) q^{25}\) \(+(\)\(20\!\cdots\!10\)\( + \)\(26\!\cdots\!38\)\( \beta_{1} + \)\(10\!\cdots\!64\)\( \beta_{2} + \)\(39\!\cdots\!36\)\( \beta_{3} + \)\(98\!\cdots\!24\)\( \beta_{4} - \)\(38\!\cdots\!92\)\( \beta_{5} + \)\(20\!\cdots\!20\)\( \beta_{6}) q^{26}\) \(+(\)\(12\!\cdots\!12\)\( + \)\(21\!\cdots\!06\)\( \beta_{1} - \)\(33\!\cdots\!48\)\( \beta_{2} + \)\(14\!\cdots\!26\)\( \beta_{3} - \)\(10\!\cdots\!56\)\( \beta_{4} + \)\(10\!\cdots\!22\)\( \beta_{5} - \)\(12\!\cdots\!62\)\( \beta_{6}) q^{27}\) \(+(-\)\(18\!\cdots\!16\)\( + \)\(11\!\cdots\!64\)\( \beta_{1} - \)\(10\!\cdots\!64\)\( \beta_{2} - \)\(16\!\cdots\!04\)\( \beta_{3} + \)\(33\!\cdots\!24\)\( \beta_{4} + \)\(32\!\cdots\!12\)\( \beta_{5} + \)\(49\!\cdots\!48\)\( \beta_{6}) q^{28}\) \(+(-\)\(28\!\cdots\!29\)\( + \)\(53\!\cdots\!13\)\( \beta_{1} - \)\(35\!\cdots\!07\)\( \beta_{2} - \)\(86\!\cdots\!00\)\( \beta_{3} + \)\(12\!\cdots\!79\)\( \beta_{4} - \)\(50\!\cdots\!32\)\( \beta_{5} - \)\(74\!\cdots\!80\)\( \beta_{6}) q^{29}\) \(+(\)\(66\!\cdots\!34\)\( + \)\(29\!\cdots\!60\)\( \beta_{1} + \)\(26\!\cdots\!64\)\( \beta_{2} + \)\(37\!\cdots\!28\)\( \beta_{3} - \)\(14\!\cdots\!52\)\( \beta_{4} + \)\(28\!\cdots\!50\)\( \beta_{5} - \)\(81\!\cdots\!00\)\( \beta_{6}) q^{30}\) \(+(-\)\(85\!\cdots\!76\)\( + \)\(25\!\cdots\!84\)\( \beta_{1} + \)\(54\!\cdots\!36\)\( \beta_{2} - \)\(72\!\cdots\!88\)\( \beta_{3} + \)\(45\!\cdots\!72\)\( \beta_{4} - \)\(83\!\cdots\!36\)\( \beta_{5} + \)\(94\!\cdots\!40\)\( \beta_{6}) q^{31}\) \(+(-\)\(49\!\cdots\!24\)\( - \)\(28\!\cdots\!16\)\( \beta_{1} - \)\(18\!\cdots\!40\)\( \beta_{2} - \)\(43\!\cdots\!52\)\( \beta_{3} + \)\(63\!\cdots\!52\)\( \beta_{4} + \)\(15\!\cdots\!56\)\( \beta_{5} - \)\(60\!\cdots\!36\)\( \beta_{6}) q^{32}\) \(+(\)\(69\!\cdots\!46\)\( - \)\(13\!\cdots\!18\)\( \beta_{1} - \)\(11\!\cdots\!98\)\( \beta_{2} + \)\(98\!\cdots\!88\)\( \beta_{3} - \)\(93\!\cdots\!18\)\( \beta_{4} + \)\(94\!\cdots\!36\)\( \beta_{5} + \)\(28\!\cdots\!04\)\( \beta_{6}) q^{33}\) \(+(\)\(63\!\cdots\!02\)\( - \)\(79\!\cdots\!26\)\( \beta_{1} + \)\(20\!\cdots\!68\)\( \beta_{2} + \)\(83\!\cdots\!44\)\( \beta_{3} + \)\(20\!\cdots\!12\)\( \beta_{4} - \)\(28\!\cdots\!36\)\( \beta_{5} - \)\(95\!\cdots\!20\)\( \beta_{6}) q^{34}\) \(+(-\)\(19\!\cdots\!88\)\( - \)\(11\!\cdots\!60\)\( \beta_{1} + \)\(11\!\cdots\!72\)\( \beta_{2} - \)\(30\!\cdots\!16\)\( \beta_{3} + \)\(78\!\cdots\!64\)\( \beta_{4} - \)\(12\!\cdots\!00\)\( \beta_{5} + \)\(20\!\cdots\!20\)\( \beta_{6}) q^{35}\) \(+(\)\(68\!\cdots\!65\)\( + \)\(47\!\cdots\!71\)\( \beta_{1} + \)\(41\!\cdots\!47\)\( \beta_{2} - \)\(61\!\cdots\!79\)\( \beta_{3} - \)\(56\!\cdots\!92\)\( \beta_{4} + \)\(15\!\cdots\!16\)\( \beta_{5} + \)\(76\!\cdots\!00\)\( \beta_{6}) q^{36}\) \(+(-\)\(27\!\cdots\!97\)\( + \)\(83\!\cdots\!73\)\( \beta_{1} - \)\(18\!\cdots\!43\)\( \beta_{2} + \)\(40\!\cdots\!12\)\( \beta_{3} + \)\(97\!\cdots\!03\)\( \beta_{4} - \)\(74\!\cdots\!36\)\( \beta_{5} - \)\(29\!\cdots\!44\)\( \beta_{6}) q^{37}\) \(+(\)\(92\!\cdots\!97\)\( + \)\(11\!\cdots\!28\)\( \beta_{1} - \)\(77\!\cdots\!58\)\( \beta_{2} + \)\(15\!\cdots\!08\)\( \beta_{3} + \)\(26\!\cdots\!72\)\( \beta_{4} + \)\(21\!\cdots\!01\)\( \beta_{5} + \)\(14\!\cdots\!24\)\( \beta_{6}) q^{38}\) \(+(-\)\(26\!\cdots\!71\)\( + \)\(15\!\cdots\!13\)\( \beta_{1} + \)\(18\!\cdots\!35\)\( \beta_{2} - \)\(25\!\cdots\!43\)\( \beta_{3} - \)\(14\!\cdots\!76\)\( \beta_{4} - \)\(33\!\cdots\!97\)\( \beta_{5} - \)\(42\!\cdots\!65\)\( \beta_{6}) q^{39}\) \(+(\)\(87\!\cdots\!60\)\( - \)\(76\!\cdots\!00\)\( \beta_{1} + \)\(59\!\cdots\!60\)\( \beta_{2} - \)\(60\!\cdots\!80\)\( \beta_{3} + \)\(10\!\cdots\!20\)\( \beta_{4} - \)\(99\!\cdots\!00\)\( \beta_{5} + \)\(54\!\cdots\!00\)\( \beta_{6}) q^{40}\) \(+(\)\(39\!\cdots\!54\)\( - \)\(23\!\cdots\!56\)\( \beta_{1} + \)\(17\!\cdots\!96\)\( \beta_{2} + \)\(34\!\cdots\!52\)\( \beta_{3} + \)\(43\!\cdots\!72\)\( \beta_{4} + \)\(13\!\cdots\!44\)\( \beta_{5} + \)\(15\!\cdots\!00\)\( \beta_{6}) q^{41}\) \(+(\)\(35\!\cdots\!16\)\( - \)\(10\!\cdots\!36\)\( \beta_{1} - \)\(49\!\cdots\!16\)\( \beta_{2} + \)\(50\!\cdots\!52\)\( \beta_{3} + \)\(18\!\cdots\!48\)\( \beta_{4} - \)\(90\!\cdots\!56\)\( \beta_{5} - \)\(11\!\cdots\!64\)\( \beta_{6}) q^{42}\) \(+(-\)\(43\!\cdots\!31\)\( - \)\(59\!\cdots\!28\)\( \beta_{1} - \)\(13\!\cdots\!13\)\( \beta_{2} - \)\(36\!\cdots\!80\)\( \beta_{3} - \)\(21\!\cdots\!40\)\( \beta_{4} - \)\(10\!\cdots\!60\)\( \beta_{5} + \)\(34\!\cdots\!40\)\( \beta_{6}) q^{43}\) \(+(\)\(29\!\cdots\!52\)\( + \)\(18\!\cdots\!84\)\( \beta_{1} + \)\(11\!\cdots\!08\)\( \beta_{2} - \)\(29\!\cdots\!36\)\( \beta_{3} + \)\(71\!\cdots\!12\)\( \beta_{4} - \)\(47\!\cdots\!56\)\( \beta_{5} - \)\(46\!\cdots\!60\)\( \beta_{6}) q^{44}\) \(+(\)\(68\!\cdots\!19\)\( + \)\(25\!\cdots\!85\)\( \beta_{1} + \)\(22\!\cdots\!49\)\( \beta_{2} + \)\(30\!\cdots\!48\)\( \beta_{3} - \)\(11\!\cdots\!57\)\( \beta_{4} + \)\(25\!\cdots\!00\)\( \beta_{5} - \)\(82\!\cdots\!00\)\( \beta_{6}) q^{45}\) \(+(-\)\(19\!\cdots\!86\)\( + \)\(31\!\cdots\!24\)\( \beta_{1} - \)\(17\!\cdots\!64\)\( \beta_{2} + \)\(88\!\cdots\!12\)\( \beta_{3} + \)\(19\!\cdots\!12\)\( \beta_{4} - \)\(11\!\cdots\!26\)\( \beta_{5} + \)\(61\!\cdots\!00\)\( \beta_{6}) q^{46}\) \(+(\)\(85\!\cdots\!50\)\( + \)\(22\!\cdots\!06\)\( \beta_{1} - \)\(51\!\cdots\!26\)\( \beta_{2} - \)\(17\!\cdots\!10\)\( \beta_{3} - \)\(92\!\cdots\!80\)\( \beta_{4} + \)\(19\!\cdots\!30\)\( \beta_{5} - \)\(15\!\cdots\!70\)\( \beta_{6}) q^{47}\) \(+(\)\(48\!\cdots\!36\)\( - \)\(30\!\cdots\!60\)\( \beta_{1} - \)\(28\!\cdots\!96\)\( \beta_{2} - \)\(11\!\cdots\!12\)\( \beta_{3} + \)\(31\!\cdots\!12\)\( \beta_{4} + \)\(19\!\cdots\!36\)\( \beta_{5} + \)\(12\!\cdots\!84\)\( \beta_{6}) q^{48}\) \(+(\)\(21\!\cdots\!45\)\( - \)\(90\!\cdots\!16\)\( \beta_{1} + \)\(39\!\cdots\!16\)\( \beta_{2} + \)\(91\!\cdots\!32\)\( \beta_{3} - \)\(28\!\cdots\!08\)\( \beta_{4} - \)\(18\!\cdots\!16\)\( \beta_{5} + \)\(45\!\cdots\!00\)\( \beta_{6}) q^{49}\) \(+(\)\(14\!\cdots\!85\)\( - \)\(45\!\cdots\!25\)\( \beta_{1} + \)\(26\!\cdots\!60\)\( \beta_{2} + \)\(30\!\cdots\!20\)\( \beta_{3} - \)\(12\!\cdots\!80\)\( \beta_{4} + \)\(39\!\cdots\!00\)\( \beta_{5} - \)\(18\!\cdots\!00\)\( \beta_{6}) q^{50}\) \(+(\)\(10\!\cdots\!72\)\( + \)\(46\!\cdots\!66\)\( \beta_{1} - \)\(19\!\cdots\!40\)\( \beta_{2} - \)\(14\!\cdots\!86\)\( \beta_{3} + \)\(36\!\cdots\!68\)\( \beta_{4} - \)\(18\!\cdots\!54\)\( \beta_{5} + \)\(25\!\cdots\!70\)\( \beta_{6}) q^{51}\) \(+(\)\(61\!\cdots\!02\)\( + \)\(87\!\cdots\!66\)\( \beta_{1} - \)\(93\!\cdots\!34\)\( \beta_{2} + \)\(70\!\cdots\!90\)\( \beta_{3} - \)\(95\!\cdots\!80\)\( \beta_{4} - \)\(80\!\cdots\!20\)\( \beta_{5} + \)\(96\!\cdots\!80\)\( \beta_{6}) q^{52}\) \(+(\)\(19\!\cdots\!63\)\( + \)\(25\!\cdots\!21\)\( \beta_{1} - \)\(14\!\cdots\!79\)\( \beta_{2} + \)\(36\!\cdots\!24\)\( \beta_{3} - \)\(18\!\cdots\!69\)\( \beta_{4} + \)\(68\!\cdots\!28\)\( \beta_{5} - \)\(77\!\cdots\!88\)\( \beta_{6}) q^{53}\) \(+(\)\(71\!\cdots\!50\)\( + \)\(39\!\cdots\!04\)\( \beta_{1} + \)\(49\!\cdots\!00\)\( \beta_{2} + \)\(17\!\cdots\!16\)\( \beta_{3} - \)\(45\!\cdots\!88\)\( \beta_{4} + \)\(31\!\cdots\!94\)\( \beta_{5} - \)\(23\!\cdots\!60\)\( \beta_{6}) q^{54}\) \(+(\)\(10\!\cdots\!79\)\( - \)\(83\!\cdots\!65\)\( \beta_{1} + \)\(48\!\cdots\!09\)\( \beta_{2} - \)\(53\!\cdots\!57\)\( \beta_{3} + \)\(16\!\cdots\!88\)\( \beta_{4} + \)\(19\!\cdots\!25\)\( \beta_{5} + \)\(33\!\cdots\!25\)\( \beta_{6}) q^{55}\) \(+(\)\(31\!\cdots\!16\)\( - \)\(38\!\cdots\!92\)\( \beta_{1} + \)\(77\!\cdots\!28\)\( \beta_{2} + \)\(32\!\cdots\!80\)\( \beta_{3} - \)\(69\!\cdots\!16\)\( \beta_{4} - \)\(57\!\cdots\!92\)\( \beta_{5} + \)\(33\!\cdots\!80\)\( \beta_{6}) q^{56}\) \(+(\)\(75\!\cdots\!74\)\( - \)\(46\!\cdots\!10\)\( \beta_{1} - \)\(20\!\cdots\!26\)\( \beta_{2} - \)\(39\!\cdots\!80\)\( \beta_{3} - \)\(57\!\cdots\!90\)\( \beta_{4} + \)\(15\!\cdots\!40\)\( \beta_{5} - \)\(54\!\cdots\!60\)\( \beta_{6}) q^{57}\) \(+(\)\(17\!\cdots\!78\)\( - \)\(35\!\cdots\!62\)\( \beta_{1} - \)\(14\!\cdots\!12\)\( \beta_{2} + \)\(47\!\cdots\!68\)\( \beta_{3} + \)\(75\!\cdots\!72\)\( \beta_{4} - \)\(39\!\cdots\!04\)\( \beta_{5} + \)\(13\!\cdots\!64\)\( \beta_{6}) q^{58}\) \(+(\)\(18\!\cdots\!49\)\( + \)\(20\!\cdots\!00\)\( \beta_{1} - \)\(36\!\cdots\!33\)\( \beta_{2} - \)\(90\!\cdots\!88\)\( \beta_{3} + \)\(12\!\cdots\!60\)\( \beta_{4} - \)\(70\!\cdots\!00\)\( \beta_{5} + \)\(22\!\cdots\!60\)\( \beta_{6}) q^{59}\) \(+(\)\(68\!\cdots\!12\)\( + \)\(10\!\cdots\!40\)\( \beta_{1} + \)\(80\!\cdots\!72\)\( \beta_{2} + \)\(27\!\cdots\!84\)\( \beta_{3} - \)\(47\!\cdots\!36\)\( \beta_{4} + \)\(16\!\cdots\!00\)\( \beta_{5} - \)\(92\!\cdots\!80\)\( \beta_{6}) q^{60}\) \(+(\)\(72\!\cdots\!27\)\( + \)\(14\!\cdots\!45\)\( \beta_{1} + \)\(10\!\cdots\!45\)\( \beta_{2} - \)\(80\!\cdots\!20\)\( \beta_{3} - \)\(14\!\cdots\!25\)\( \beta_{4} - \)\(57\!\cdots\!40\)\( \beta_{5} + \)\(21\!\cdots\!20\)\( \beta_{6}) q^{61}\) \(+(\)\(84\!\cdots\!52\)\( - \)\(23\!\cdots\!56\)\( \beta_{1} - \)\(27\!\cdots\!60\)\( \beta_{2} + \)\(60\!\cdots\!52\)\( \beta_{3} + \)\(28\!\cdots\!48\)\( \beta_{4} - \)\(25\!\cdots\!56\)\( \beta_{5} - \)\(36\!\cdots\!64\)\( \beta_{6}) q^{62}\) \(+(-\)\(51\!\cdots\!45\)\( - \)\(85\!\cdots\!25\)\( \beta_{1} - \)\(38\!\cdots\!27\)\( \beta_{2} + \)\(27\!\cdots\!83\)\( \beta_{3} + \)\(18\!\cdots\!52\)\( \beta_{4} + \)\(46\!\cdots\!01\)\( \beta_{5} - \)\(89\!\cdots\!71\)\( \beta_{6}) q^{63}\) \(+(-\)\(73\!\cdots\!64\)\( - \)\(10\!\cdots\!92\)\( \beta_{1} - \)\(68\!\cdots\!76\)\( \beta_{2} - \)\(32\!\cdots\!04\)\( \beta_{3} - \)\(84\!\cdots\!56\)\( \beta_{4} + \)\(79\!\cdots\!88\)\( \beta_{5} + \)\(18\!\cdots\!00\)\( \beta_{6}) q^{64}\) \(+(-\)\(11\!\cdots\!76\)\( - \)\(98\!\cdots\!20\)\( \beta_{1} + \)\(29\!\cdots\!44\)\( \beta_{2} + \)\(12\!\cdots\!68\)\( \beta_{3} - \)\(33\!\cdots\!72\)\( \beta_{4} + \)\(16\!\cdots\!00\)\( \beta_{5} + \)\(27\!\cdots\!40\)\( \beta_{6}) q^{65}\) \(+(-\)\(44\!\cdots\!96\)\( + \)\(16\!\cdots\!68\)\( \beta_{1} + \)\(24\!\cdots\!60\)\( \beta_{2} + \)\(10\!\cdots\!52\)\( \beta_{3} + \)\(10\!\cdots\!64\)\( \beta_{4} - \)\(11\!\cdots\!92\)\( \beta_{5} - \)\(64\!\cdots\!40\)\( \beta_{6}) q^{66}\) \(+(-\)\(53\!\cdots\!71\)\( + \)\(12\!\cdots\!34\)\( \beta_{1} + \)\(22\!\cdots\!43\)\( \beta_{2} + \)\(25\!\cdots\!58\)\( \beta_{3} + \)\(81\!\cdots\!72\)\( \beta_{4} + \)\(15\!\cdots\!26\)\( \beta_{5} + \)\(52\!\cdots\!74\)\( \beta_{6}) q^{67}\) \(+(-\)\(21\!\cdots\!10\)\( + \)\(19\!\cdots\!94\)\( \beta_{1} - \)\(27\!\cdots\!98\)\( \beta_{2} - \)\(51\!\cdots\!14\)\( \beta_{3} - \)\(20\!\cdots\!16\)\( \beta_{4} + \)\(21\!\cdots\!92\)\( \beta_{5} + \)\(21\!\cdots\!68\)\( \beta_{6}) q^{68}\) \(+(-\)\(31\!\cdots\!20\)\( - \)\(96\!\cdots\!84\)\( \beta_{1} + \)\(34\!\cdots\!48\)\( \beta_{2} + \)\(29\!\cdots\!32\)\( \beta_{3} - \)\(17\!\cdots\!92\)\( \beta_{4} - \)\(77\!\cdots\!04\)\( \beta_{5} - \)\(42\!\cdots\!40\)\( \beta_{6}) q^{69}\) \(+(-\)\(38\!\cdots\!48\)\( - \)\(74\!\cdots\!20\)\( \beta_{1} - \)\(33\!\cdots\!08\)\( \beta_{2} - \)\(47\!\cdots\!16\)\( \beta_{3} + \)\(39\!\cdots\!44\)\( \beta_{4} + \)\(40\!\cdots\!00\)\( \beta_{5} - \)\(71\!\cdots\!00\)\( \beta_{6}) q^{70}\) \(+(\)\(13\!\cdots\!67\)\( - \)\(80\!\cdots\!65\)\( \beta_{1} + \)\(20\!\cdots\!85\)\( \beta_{2} + \)\(22\!\cdots\!15\)\( \beta_{3} + \)\(97\!\cdots\!00\)\( \beta_{4} + \)\(33\!\cdots\!05\)\( \beta_{5} + \)\(35\!\cdots\!85\)\( \beta_{6}) q^{71}\) \(+(\)\(67\!\cdots\!76\)\( - \)\(19\!\cdots\!96\)\( \beta_{1} - \)\(51\!\cdots\!04\)\( \beta_{2} + \)\(25\!\cdots\!64\)\( \beta_{3} - \)\(16\!\cdots\!64\)\( \beta_{4} + \)\(29\!\cdots\!08\)\( \beta_{5} - \)\(28\!\cdots\!48\)\( \beta_{6}) q^{72}\) \(+(\)\(13\!\cdots\!36\)\( + \)\(63\!\cdots\!70\)\( \beta_{1} + \)\(47\!\cdots\!34\)\( \beta_{2} - \)\(78\!\cdots\!72\)\( \beta_{3} - \)\(42\!\cdots\!98\)\( \beta_{4} - \)\(77\!\cdots\!84\)\( \beta_{5} - \)\(11\!\cdots\!16\)\( \beta_{6}) q^{73}\) \(+(\)\(27\!\cdots\!34\)\( + \)\(45\!\cdots\!26\)\( \beta_{1} - \)\(26\!\cdots\!16\)\( \beta_{2} - \)\(80\!\cdots\!72\)\( \beta_{3} + \)\(87\!\cdots\!48\)\( \beta_{4} - \)\(10\!\cdots\!64\)\( \beta_{5} + \)\(31\!\cdots\!80\)\( \beta_{6}) q^{74}\) \(+(\)\(17\!\cdots\!45\)\( + \)\(26\!\cdots\!00\)\( \beta_{1} + \)\(17\!\cdots\!95\)\( \beta_{2} - \)\(98\!\cdots\!60\)\( \beta_{3} - \)\(87\!\cdots\!60\)\( \beta_{4} + \)\(64\!\cdots\!00\)\( \beta_{5} - \)\(14\!\cdots\!00\)\( \beta_{6}) q^{75}\) \(+(\)\(28\!\cdots\!52\)\( + \)\(12\!\cdots\!16\)\( \beta_{1} + \)\(19\!\cdots\!16\)\( \beta_{2} + \)\(12\!\cdots\!20\)\( \beta_{3} - \)\(85\!\cdots\!32\)\( \beta_{4} - \)\(68\!\cdots\!84\)\( \beta_{5} - \)\(76\!\cdots\!40\)\( \beta_{6}) q^{76}\) \(+(\)\(29\!\cdots\!84\)\( + \)\(68\!\cdots\!00\)\( \beta_{1} + \)\(76\!\cdots\!72\)\( \beta_{2} - \)\(10\!\cdots\!32\)\( \beta_{3} + \)\(57\!\cdots\!32\)\( \beta_{4} - \)\(10\!\cdots\!04\)\( \beta_{5} + \)\(15\!\cdots\!24\)\( \beta_{6}) q^{77}\) \(+(\)\(49\!\cdots\!78\)\( - \)\(71\!\cdots\!80\)\( \beta_{1} - \)\(46\!\cdots\!16\)\( \beta_{2} - \)\(68\!\cdots\!04\)\( \beta_{3} - \)\(65\!\cdots\!36\)\( \beta_{4} + \)\(31\!\cdots\!62\)\( \beta_{5} - \)\(37\!\cdots\!12\)\( \beta_{6}) q^{78}\) \(+(-\)\(83\!\cdots\!86\)\( - \)\(24\!\cdots\!46\)\( \beta_{1} - \)\(12\!\cdots\!38\)\( \beta_{2} - \)\(21\!\cdots\!42\)\( \beta_{3} + \)\(20\!\cdots\!52\)\( \beta_{4} - \)\(15\!\cdots\!26\)\( \beta_{5} - \)\(22\!\cdots\!10\)\( \beta_{6}) q^{79}\) \(+(-\)\(10\!\cdots\!88\)\( - \)\(21\!\cdots\!20\)\( \beta_{1} - \)\(50\!\cdots\!48\)\( \beta_{2} + \)\(28\!\cdots\!04\)\( \beta_{3} - \)\(85\!\cdots\!36\)\( \beta_{4} - \)\(29\!\cdots\!00\)\( \beta_{5} + \)\(16\!\cdots\!00\)\( \beta_{6}) q^{80}\) \(+(-\)\(92\!\cdots\!21\)\( + \)\(43\!\cdots\!30\)\( \beta_{1} + \)\(44\!\cdots\!54\)\( \beta_{2} + \)\(14\!\cdots\!24\)\( \beta_{3} - \)\(33\!\cdots\!10\)\( \beta_{4} + \)\(33\!\cdots\!60\)\( \beta_{5} + \)\(14\!\cdots\!60\)\( \beta_{6}) q^{81}\) \(+(-\)\(76\!\cdots\!18\)\( + \)\(11\!\cdots\!14\)\( \beta_{1} + \)\(92\!\cdots\!00\)\( \beta_{2} - \)\(18\!\cdots\!08\)\( \beta_{3} + \)\(89\!\cdots\!48\)\( \beta_{4} - \)\(15\!\cdots\!76\)\( \beta_{5} + \)\(86\!\cdots\!96\)\( \beta_{6}) q^{82}\) \(+(-\)\(10\!\cdots\!05\)\( + \)\(15\!\cdots\!12\)\( \beta_{1} + \)\(20\!\cdots\!33\)\( \beta_{2} + \)\(19\!\cdots\!00\)\( \beta_{3} + \)\(37\!\cdots\!00\)\( \beta_{4} + \)\(25\!\cdots\!00\)\( \beta_{5} - \)\(34\!\cdots\!00\)\( \beta_{6}) q^{83}\) \(+(-\)\(24\!\cdots\!88\)\( + \)\(87\!\cdots\!88\)\( \beta_{1} - \)\(75\!\cdots\!52\)\( \beta_{2} - \)\(71\!\cdots\!00\)\( \beta_{3} + \)\(20\!\cdots\!64\)\( \beta_{4} + \)\(23\!\cdots\!28\)\( \beta_{5} + \)\(15\!\cdots\!00\)\( \beta_{6}) q^{84}\) \(+(-\)\(27\!\cdots\!18\)\( - \)\(27\!\cdots\!10\)\( \beta_{1} - \)\(49\!\cdots\!58\)\( \beta_{2} + \)\(11\!\cdots\!24\)\( \beta_{3} - \)\(51\!\cdots\!46\)\( \beta_{4} - \)\(66\!\cdots\!00\)\( \beta_{5} + \)\(11\!\cdots\!20\)\( \beta_{6}) q^{85}\) \(+(-\)\(19\!\cdots\!25\)\( - \)\(12\!\cdots\!48\)\( \beta_{1} + \)\(64\!\cdots\!54\)\( \beta_{2} + \)\(12\!\cdots\!72\)\( \beta_{3} + \)\(11\!\cdots\!36\)\( \beta_{4} - \)\(99\!\cdots\!93\)\( \beta_{5} - \)\(24\!\cdots\!80\)\( \beta_{6}) q^{86}\) \(+(\)\(21\!\cdots\!61\)\( - \)\(92\!\cdots\!03\)\( \beta_{1} - \)\(97\!\cdots\!09\)\( \beta_{2} - \)\(96\!\cdots\!23\)\( \beta_{3} + \)\(17\!\cdots\!28\)\( \beta_{4} + \)\(56\!\cdots\!19\)\( \beta_{5} + \)\(45\!\cdots\!91\)\( \beta_{6}) q^{87}\) \(+(\)\(50\!\cdots\!36\)\( + \)\(16\!\cdots\!88\)\( \beta_{1} + \)\(26\!\cdots\!96\)\( \beta_{2} - \)\(33\!\cdots\!72\)\( \beta_{3} + \)\(29\!\cdots\!72\)\( \beta_{4} - \)\(43\!\cdots\!84\)\( \beta_{5} + \)\(42\!\cdots\!04\)\( \beta_{6}) q^{88}\) \(+(\)\(84\!\cdots\!48\)\( - \)\(37\!\cdots\!26\)\( \beta_{1} - \)\(12\!\cdots\!66\)\( \beta_{2} - \)\(68\!\cdots\!00\)\( \beta_{3} + \)\(30\!\cdots\!82\)\( \beta_{4} - \)\(13\!\cdots\!96\)\( \beta_{5} - \)\(52\!\cdots\!20\)\( \beta_{6}) q^{89}\) \(+(\)\(85\!\cdots\!14\)\( + \)\(13\!\cdots\!30\)\( \beta_{1} + \)\(11\!\cdots\!84\)\( \beta_{2} + \)\(18\!\cdots\!48\)\( \beta_{3} - \)\(90\!\cdots\!92\)\( \beta_{4} + \)\(28\!\cdots\!00\)\( \beta_{5} + \)\(15\!\cdots\!40\)\( \beta_{6}) q^{90}\) \(+(\)\(61\!\cdots\!08\)\( + \)\(40\!\cdots\!88\)\( \beta_{1} - \)\(15\!\cdots\!72\)\( \beta_{2} + \)\(30\!\cdots\!00\)\( \beta_{3} - \)\(17\!\cdots\!76\)\( \beta_{4} - \)\(53\!\cdots\!12\)\( \beta_{5} - \)\(62\!\cdots\!20\)\( \beta_{6}) q^{91}\) \(+(\)\(92\!\cdots\!64\)\( + \)\(30\!\cdots\!28\)\( \beta_{1} - \)\(10\!\cdots\!72\)\( \beta_{2} + \)\(11\!\cdots\!52\)\( \beta_{3} + \)\(16\!\cdots\!88\)\( \beta_{4} - \)\(27\!\cdots\!56\)\( \beta_{5} + \)\(16\!\cdots\!76\)\( \beta_{6}) q^{92}\) \(+(\)\(25\!\cdots\!76\)\( - \)\(26\!\cdots\!52\)\( \beta_{1} - \)\(30\!\cdots\!28\)\( \beta_{2} - \)\(77\!\cdots\!76\)\( \beta_{3} + \)\(13\!\cdots\!16\)\( \beta_{4} + \)\(44\!\cdots\!28\)\( \beta_{5} + \)\(13\!\cdots\!72\)\( \beta_{6}) q^{93}\) \(+(\)\(75\!\cdots\!96\)\( - \)\(29\!\cdots\!76\)\( \beta_{1} + \)\(77\!\cdots\!60\)\( \beta_{2} - \)\(50\!\cdots\!44\)\( \beta_{3} - \)\(49\!\cdots\!28\)\( \beta_{4} - \)\(13\!\cdots\!36\)\( \beta_{5} - \)\(10\!\cdots\!60\)\( \beta_{6}) q^{94}\) \(+(\)\(65\!\cdots\!65\)\( - \)\(30\!\cdots\!75\)\( \beta_{1} + \)\(68\!\cdots\!15\)\( \beta_{2} + \)\(88\!\cdots\!05\)\( \beta_{3} - \)\(53\!\cdots\!20\)\( \beta_{4} + \)\(19\!\cdots\!75\)\( \beta_{5} + \)\(13\!\cdots\!75\)\( \beta_{6}) q^{95}\) \(+(-\)\(75\!\cdots\!56\)\( - \)\(63\!\cdots\!72\)\( \beta_{1} - \)\(65\!\cdots\!24\)\( \beta_{2} + \)\(21\!\cdots\!08\)\( \beta_{3} + \)\(20\!\cdots\!44\)\( \beta_{4} - \)\(29\!\cdots\!12\)\( \beta_{5} + \)\(15\!\cdots\!00\)\( \beta_{6}) q^{96}\) \(+(-\)\(10\!\cdots\!00\)\( + \)\(42\!\cdots\!02\)\( \beta_{1} + \)\(55\!\cdots\!94\)\( \beta_{2} + \)\(42\!\cdots\!76\)\( \beta_{3} - \)\(15\!\cdots\!06\)\( \beta_{4} - \)\(11\!\cdots\!28\)\( \beta_{5} - \)\(22\!\cdots\!12\)\( \beta_{6}) q^{97}\) \(+(-\)\(29\!\cdots\!37\)\( + \)\(41\!\cdots\!25\)\( \beta_{1} - \)\(56\!\cdots\!20\)\( \beta_{2} - \)\(96\!\cdots\!48\)\( \beta_{3} + \)\(10\!\cdots\!88\)\( \beta_{4} + \)\(64\!\cdots\!44\)\( \beta_{5} + \)\(35\!\cdots\!76\)\( \beta_{6}) q^{98}\) \(+(-\)\(34\!\cdots\!07\)\( + \)\(10\!\cdots\!72\)\( \beta_{1} + \)\(17\!\cdots\!07\)\( \beta_{2} - \)\(20\!\cdots\!00\)\( \beta_{3} + \)\(12\!\cdots\!56\)\( \beta_{4} - \)\(90\!\cdots\!28\)\( \beta_{5} - \)\(59\!\cdots\!80\)\( \beta_{6}) q^{99}\) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(7q \) \(\mathstrut +\mathstrut 3841716838056q^{2} \) \(\mathstrut +\mathstrut \)\(62\!\cdots\!32\)\(q^{3} \) \(\mathstrut +\mathstrut \)\(56\!\cdots\!76\)\(q^{4} \) \(\mathstrut +\mathstrut \)\(23\!\cdots\!30\)\(q^{5} \) \(\mathstrut +\mathstrut \)\(45\!\cdots\!84\)\(q^{6} \) \(\mathstrut -\mathstrut \)\(17\!\cdots\!44\)\(q^{7} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!60\)\(q^{8} \) \(\mathstrut +\mathstrut \)\(38\!\cdots\!59\)\(q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(7q \) \(\mathstrut +\mathstrut 3841716838056q^{2} \) \(\mathstrut +\mathstrut \)\(62\!\cdots\!32\)\(q^{3} \) \(\mathstrut +\mathstrut \)\(56\!\cdots\!76\)\(q^{4} \) \(\mathstrut +\mathstrut \)\(23\!\cdots\!30\)\(q^{5} \) \(\mathstrut +\mathstrut \)\(45\!\cdots\!84\)\(q^{6} \) \(\mathstrut -\mathstrut \)\(17\!\cdots\!44\)\(q^{7} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!60\)\(q^{8} \) \(\mathstrut +\mathstrut \)\(38\!\cdots\!59\)\(q^{9} \) \(\mathstrut +\mathstrut \)\(46\!\cdots\!80\)\(q^{10} \) \(\mathstrut -\mathstrut \)\(24\!\cdots\!16\)\(q^{11} \) \(\mathstrut +\mathstrut \)\(15\!\cdots\!16\)\(q^{12} \) \(\mathstrut +\mathstrut \)\(71\!\cdots\!22\)\(q^{13} \) \(\mathstrut +\mathstrut \)\(17\!\cdots\!32\)\(q^{14} \) \(\mathstrut +\mathstrut \)\(81\!\cdots\!60\)\(q^{15} \) \(\mathstrut -\mathstrut \)\(55\!\cdots\!08\)\(q^{16} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!94\)\(q^{17} \) \(\mathstrut +\mathstrut \)\(25\!\cdots\!12\)\(q^{18} \) \(\mathstrut +\mathstrut \)\(24\!\cdots\!60\)\(q^{19} \) \(\mathstrut +\mathstrut \)\(21\!\cdots\!40\)\(q^{20} \) \(\mathstrut -\mathstrut \)\(29\!\cdots\!76\)\(q^{21} \) \(\mathstrut +\mathstrut \)\(27\!\cdots\!72\)\(q^{22} \) \(\mathstrut +\mathstrut \)\(28\!\cdots\!12\)\(q^{23} \) \(\mathstrut -\mathstrut \)\(67\!\cdots\!20\)\(q^{24} \) \(\mathstrut +\mathstrut \)\(32\!\cdots\!25\)\(q^{25} \) \(\mathstrut +\mathstrut \)\(14\!\cdots\!44\)\(q^{26} \) \(\mathstrut +\mathstrut \)\(88\!\cdots\!60\)\(q^{27} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!72\)\(q^{28} \) \(\mathstrut -\mathstrut \)\(19\!\cdots\!10\)\(q^{29} \) \(\mathstrut +\mathstrut \)\(46\!\cdots\!60\)\(q^{30} \) \(\mathstrut -\mathstrut \)\(59\!\cdots\!76\)\(q^{31} \) \(\mathstrut -\mathstrut \)\(34\!\cdots\!24\)\(q^{32} \) \(\mathstrut +\mathstrut \)\(48\!\cdots\!84\)\(q^{33} \) \(\mathstrut +\mathstrut \)\(44\!\cdots\!72\)\(q^{34} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!20\)\(q^{35} \) \(\mathstrut +\mathstrut \)\(48\!\cdots\!12\)\(q^{36} \) \(\mathstrut -\mathstrut \)\(19\!\cdots\!94\)\(q^{37} \) \(\mathstrut +\mathstrut \)\(64\!\cdots\!60\)\(q^{38} \) \(\mathstrut -\mathstrut \)\(18\!\cdots\!92\)\(q^{39} \) \(\mathstrut +\mathstrut \)\(61\!\cdots\!00\)\(q^{40} \) \(\mathstrut +\mathstrut \)\(27\!\cdots\!94\)\(q^{41} \) \(\mathstrut +\mathstrut \)\(24\!\cdots\!52\)\(q^{42} \) \(\mathstrut -\mathstrut \)\(30\!\cdots\!08\)\(q^{43} \) \(\mathstrut +\mathstrut \)\(20\!\cdots\!12\)\(q^{44} \) \(\mathstrut +\mathstrut \)\(47\!\cdots\!10\)\(q^{45} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!96\)\(q^{46} \) \(\mathstrut +\mathstrut \)\(60\!\cdots\!56\)\(q^{47} \) \(\mathstrut +\mathstrut \)\(34\!\cdots\!12\)\(q^{48} \) \(\mathstrut +\mathstrut \)\(14\!\cdots\!51\)\(q^{49} \) \(\mathstrut +\mathstrut \)\(98\!\cdots\!00\)\(q^{50} \) \(\mathstrut +\mathstrut \)\(74\!\cdots\!04\)\(q^{51} \) \(\mathstrut +\mathstrut \)\(43\!\cdots\!36\)\(q^{52} \) \(\mathstrut +\mathstrut \)\(13\!\cdots\!82\)\(q^{53} \) \(\mathstrut +\mathstrut \)\(49\!\cdots\!60\)\(q^{54} \) \(\mathstrut +\mathstrut \)\(71\!\cdots\!60\)\(q^{55} \) \(\mathstrut +\mathstrut \)\(22\!\cdots\!40\)\(q^{56} \) \(\mathstrut +\mathstrut \)\(52\!\cdots\!20\)\(q^{57} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!40\)\(q^{58} \) \(\mathstrut +\mathstrut \)\(13\!\cdots\!80\)\(q^{59} \) \(\mathstrut +\mathstrut \)\(48\!\cdots\!80\)\(q^{60} \) \(\mathstrut +\mathstrut \)\(50\!\cdots\!34\)\(q^{61} \) \(\mathstrut +\mathstrut \)\(59\!\cdots\!92\)\(q^{62} \) \(\mathstrut -\mathstrut \)\(36\!\cdots\!88\)\(q^{63} \) \(\mathstrut -\mathstrut \)\(51\!\cdots\!64\)\(q^{64} \) \(\mathstrut -\mathstrut \)\(77\!\cdots\!40\)\(q^{65} \) \(\mathstrut -\mathstrut \)\(30\!\cdots\!92\)\(q^{66} \) \(\mathstrut -\mathstrut \)\(37\!\cdots\!44\)\(q^{67} \) \(\mathstrut -\mathstrut \)\(15\!\cdots\!72\)\(q^{68} \) \(\mathstrut -\mathstrut \)\(21\!\cdots\!72\)\(q^{69} \) \(\mathstrut -\mathstrut \)\(26\!\cdots\!20\)\(q^{70} \) \(\mathstrut +\mathstrut \)\(96\!\cdots\!04\)\(q^{71} \) \(\mathstrut +\mathstrut \)\(47\!\cdots\!80\)\(q^{72} \) \(\mathstrut +\mathstrut \)\(95\!\cdots\!62\)\(q^{73} \) \(\mathstrut +\mathstrut \)\(19\!\cdots\!52\)\(q^{74} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!00\)\(q^{75} \) \(\mathstrut +\mathstrut \)\(20\!\cdots\!80\)\(q^{76} \) \(\mathstrut +\mathstrut \)\(20\!\cdots\!72\)\(q^{77} \) \(\mathstrut +\mathstrut \)\(34\!\cdots\!24\)\(q^{78} \) \(\mathstrut -\mathstrut \)\(58\!\cdots\!60\)\(q^{79} \) \(\mathstrut -\mathstrut \)\(70\!\cdots\!20\)\(q^{80} \) \(\mathstrut -\mathstrut \)\(64\!\cdots\!53\)\(q^{81} \) \(\mathstrut -\mathstrut \)\(53\!\cdots\!48\)\(q^{82} \) \(\mathstrut -\mathstrut \)\(71\!\cdots\!48\)\(q^{83} \) \(\mathstrut -\mathstrut \)\(17\!\cdots\!68\)\(q^{84} \) \(\mathstrut -\mathstrut \)\(18\!\cdots\!20\)\(q^{85} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!76\)\(q^{86} \) \(\mathstrut +\mathstrut \)\(14\!\cdots\!80\)\(q^{87} \) \(\mathstrut +\mathstrut \)\(35\!\cdots\!80\)\(q^{88} \) \(\mathstrut +\mathstrut \)\(58\!\cdots\!70\)\(q^{89} \) \(\mathstrut +\mathstrut \)\(59\!\cdots\!60\)\(q^{90} \) \(\mathstrut +\mathstrut \)\(43\!\cdots\!84\)\(q^{91} \) \(\mathstrut +\mathstrut \)\(64\!\cdots\!56\)\(q^{92} \) \(\mathstrut +\mathstrut \)\(17\!\cdots\!24\)\(q^{93} \) \(\mathstrut +\mathstrut \)\(52\!\cdots\!92\)\(q^{94} \) \(\mathstrut +\mathstrut \)\(46\!\cdots\!00\)\(q^{95} \) \(\mathstrut -\mathstrut \)\(53\!\cdots\!36\)\(q^{96} \) \(\mathstrut -\mathstrut \)\(70\!\cdots\!94\)\(q^{97} \) \(\mathstrut -\mathstrut \)\(20\!\cdots\!92\)\(q^{98} \) \(\mathstrut -\mathstrut \)\(23\!\cdots\!92\)\(q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{7}\mathstrut -\mathstrut \) \(2\) \(x^{6}\mathstrut -\mathstrut \) \(19965898505076685758763041\) \(x^{5}\mathstrut +\mathstrut \) \(704885126882155619610489099544084432\) \(x^{4}\mathstrut +\mathstrut \) \(107067825809672902244360085998174063583690648295951\) \(x^{3}\mathstrut +\mathstrut \) \(2760974505633618159712492393750116169743299337348578140163910\) \(x^{2}\mathstrut -\mathstrut \) \(117290449585568076678331590341364312042793082454528443905095325102023447375\) \(x\mathstrut +\mathstrut \) \(37970201148992037926462243830462339624462239310525820322458780504727338931040585255500\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 24 \nu - 7 \)
\(\beta_{2}\)\(=\)\((\)\(-\)\(34\!\cdots\!59\) \(\nu^{6}\mathstrut -\mathstrut \) \(93\!\cdots\!77\) \(\nu^{5}\mathstrut +\mathstrut \) \(59\!\cdots\!42\) \(\nu^{4}\mathstrut +\mathstrut \) \(12\!\cdots\!50\) \(\nu^{3}\mathstrut -\mathstrut \) \(27\!\cdots\!39\) \(\nu^{2}\mathstrut -\mathstrut \) \(35\!\cdots\!53\) \(\nu\mathstrut +\mathstrut \) \(26\!\cdots\!08\)\()/\)\(24\!\cdots\!72\)
\(\beta_{3}\)\(=\)\((\)\(90\!\cdots\!31\) \(\nu^{6}\mathstrut +\mathstrut \) \(24\!\cdots\!93\) \(\nu^{5}\mathstrut -\mathstrut \) \(15\!\cdots\!78\) \(\nu^{4}\mathstrut -\mathstrut \) \(33\!\cdots\!50\) \(\nu^{3}\mathstrut +\mathstrut \) \(14\!\cdots\!39\) \(\nu^{2}\mathstrut +\mathstrut \) \(96\!\cdots\!13\) \(\nu\mathstrut -\mathstrut \) \(49\!\cdots\!48\)\()/\)\(12\!\cdots\!88\)
\(\beta_{4}\)\(=\)\((\)\(-\)\(65\!\cdots\!79\) \(\nu^{6}\mathstrut +\mathstrut \) \(30\!\cdots\!11\) \(\nu^{5}\mathstrut +\mathstrut \) \(14\!\cdots\!62\) \(\nu^{4}\mathstrut -\mathstrut \) \(11\!\cdots\!62\) \(\nu^{3}\mathstrut -\mathstrut \) \(79\!\cdots\!95\) \(\nu^{2}\mathstrut -\mathstrut \) \(26\!\cdots\!25\) \(\nu\mathstrut +\mathstrut \) \(55\!\cdots\!00\)\()/\)\(30\!\cdots\!00\)
\(\beta_{5}\)\(=\)\((\)\(-\)\(16\!\cdots\!49\) \(\nu^{6}\mathstrut -\mathstrut \) \(89\!\cdots\!59\) \(\nu^{5}\mathstrut +\mathstrut \) \(25\!\cdots\!22\) \(\nu^{4}\mathstrut +\mathstrut \) \(76\!\cdots\!78\) \(\nu^{3}\mathstrut -\mathstrut \) \(85\!\cdots\!45\) \(\nu^{2}\mathstrut -\mathstrut \) \(22\!\cdots\!75\) \(\nu\mathstrut +\mathstrut \) \(31\!\cdots\!00\)\()/\)\(15\!\cdots\!00\)
\(\beta_{6}\)\(=\)\((\)\(15\!\cdots\!61\) \(\nu^{6}\mathstrut -\mathstrut \) \(28\!\cdots\!49\) \(\nu^{5}\mathstrut -\mathstrut \) \(14\!\cdots\!58\) \(\nu^{4}\mathstrut +\mathstrut \) \(11\!\cdots\!58\) \(\nu^{3}\mathstrut -\mathstrut \) \(22\!\cdots\!95\) \(\nu^{2}\mathstrut -\mathstrut \) \(71\!\cdots\!25\) \(\nu\mathstrut +\mathstrut \) \(61\!\cdots\!00\)\()/\)\(12\!\cdots\!00\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1}\mathstrut +\mathstrut \) \(7\)\()/24\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3}\mathstrut +\mathstrut \) \(49571\) \(\beta_{2}\mathstrut -\mathstrut \) \(1270960315153\) \(\beta_{1}\mathstrut +\mathstrut \) \(3285816439692620103304937381\)\()/576\)
\(\nu^{3}\)\(=\)\((\)\(39\) \(\beta_{6}\mathstrut +\mathstrut \) \(12966\) \(\beta_{5}\mathstrut -\mathstrut \) \(12676468\) \(\beta_{4}\mathstrut -\mathstrut \) \(1016261168401\) \(\beta_{3}\mathstrut -\mathstrut \) \(512931984338211243\) \(\beta_{2}\mathstrut +\mathstrut \) \(665339679315744541705428191\) \(\beta_{1}\mathstrut -\mathstrut \) \(522017785364203279525612267220887567696\)\()/1728\)
\(\nu^{4}\)\(=\)\((\)\(-\)\(141481024033993\) \(\beta_{6}\mathstrut -\mathstrut \) \(146499787029560378\) \(\beta_{5}\mathstrut +\mathstrut \) \(383626286810111142348\) \(\beta_{4}\mathstrut +\mathstrut \) \(106762309451665755675362048\) \(\beta_{3}\mathstrut +\mathstrut \) \(9384661690135860719271389741736\) \(\beta_{2}\mathstrut -\mathstrut \) \(455311605135943378511136613941824967370\) \(\beta_{1}\mathstrut +\mathstrut \) \(273273007015307998096220902015072945718157332183285693\)\()/5184\)
\(\nu^{5}\)\(=\)\((\)\(272294119230702643338278143\) \(\beta_{6}\mathstrut +\mathstrut \) \(115955420708801071468255024662\) \(\beta_{5}\mathstrut -\mathstrut \) \(46965447327553276381417610283156\) \(\beta_{4}\mathstrut -\mathstrut \) \(13713050651072167295090703870908934459\) \(\beta_{3}\mathstrut -\mathstrut \) \(6454898015889767005127124500408305904301609\) \(\beta_{2}\mathstrut +\mathstrut \) \(3423404559609344185123436271357033475350202101321557\) \(\beta_{1}\mathstrut -\mathstrut \) \(10389377462060720364413531410820663002534052824647214875962006902\)\()/864\)
\(\nu^{6}\)\(=\)\((\)\(-\)\(845081535489787599813217876804493212586\) \(\beta_{6}\mathstrut -\mathstrut \) \(983551911903246004064930950812150880066660\) \(\beta_{5}\mathstrut +\mathstrut \) \(1971838849993081965086934612786280371118306744\) \(\beta_{4}\mathstrut +\mathstrut \) \(404479468114429999630179233938341116865447981180449\) \(\beta_{3}\mathstrut +\mathstrut \) \(45311208431768090051123574350547561815149654442811609363\) \(\beta_{2}\mathstrut -\mathstrut \) \(3769018111755297590653218515154387604191873885598880668897146997\) \(\beta_{1}\mathstrut +\mathstrut \) \(937389914979775814832843332260231511597800305402118168526359478476802655371255\)\()/1728\)

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
−3.52994e12
−2.36516e12
−1.48073e12
3.73440e11
8.78141e11
2.95949e12
3.16475e12
−8.41697e13 2.94143e21 4.60866e27 2.26211e30 −2.47580e35 −5.00335e38 −1.79515e41 −1.75319e43 −1.90401e44
1.2 −5.62150e13 −7.69269e21 6.84242e26 −3.78165e31 4.32444e35 1.92974e38 1.00717e41 3.29936e43 2.12585e45
1.3 −3.49886e13 3.81854e21 −1.25168e27 4.98385e31 −1.33605e35 4.56727e38 1.30422e41 −1.16026e43 −1.74378e45
1.4 9.51137e12 5.28795e21 −2.38541e27 −1.05575e32 5.02957e34 −1.59557e38 −4.62376e40 1.77856e42 −1.00417e45
1.5 2.16242e13 −4.57928e21 −2.00827e27 4.97855e31 −9.90233e34 −2.82571e38 −9.69662e40 −5.21406e42 1.07657e45
1.6 7.15766e13 9.10839e21 2.64733e27 1.09575e32 6.51948e35 −1.62560e38 1.22718e40 5.67789e43 7.84298e45
1.7 7.65028e13 −2.65748e21 3.37680e27 −4.45639e31 −2.03305e35 2.84237e38 6.89228e40 −1.91217e43 −3.40927e45
\(n\): 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 \(S_{92}^{\mathrm{new}}(\Gamma_0(1))\).