Properties

Label 1.104.a.a
Level 1
Weight 104
Character orbit 1.a
Self dual Yes
Analytic conductor 67.184
Analytic rank 0
Dimension 8
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 1 \)
Weight: \( k \) = \( 104 \)
Character orbit: \([\chi]\) = 1.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(67.1843880807\)
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 \( 2^{104}\cdot 3^{40}\cdot 5^{12}\cdot 7^{8}\cdot 11\cdot 13^{3}\cdot 17^{2} \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(f(q)\) \(=\) \( q +(548616194585055 - \beta_{1}) q^{2} +(\)\(63\!\cdots\!80\)\( + 1292206 \beta_{1} - \beta_{2}) q^{3} +(\)\(60\!\cdots\!08\)\( - 1006563777941965 \beta_{1} + 34249 \beta_{2} + \beta_{3}) q^{4} +(\)\(68\!\cdots\!90\)\( - 13577557385999983067 \beta_{1} + 23394938356 \beta_{2} - 2101 \beta_{3} + \beta_{4}) q^{5} +(\)\(32\!\cdots\!92\)\( - \)\(59\!\cdots\!29\)\( \beta_{1} - 2269182927810110 \beta_{2} + 289750017 \beta_{3} - 3331 \beta_{4} - \beta_{5}) q^{6} +(\)\(52\!\cdots\!00\)\( - \)\(14\!\cdots\!08\)\( \beta_{1} + 3436742111413277589 \beta_{2} + 49285015561 \beta_{3} - 4418774 \beta_{4} + 505 \beta_{5} + \beta_{6}) q^{7} +(\)\(13\!\cdots\!60\)\( - \)\(62\!\cdots\!67\)\( \beta_{1} - \)\(24\!\cdots\!96\)\( \beta_{2} + 913937500520892 \beta_{3} - 783459941 \beta_{4} + 75477 \beta_{5} - 134 \beta_{6} - \beta_{7}) q^{8} +(\)\(44\!\cdots\!77\)\( - \)\(19\!\cdots\!70\)\( \beta_{1} - \)\(37\!\cdots\!24\)\( \beta_{2} + 272254459805646246 \beta_{3} - 720712994562 \beta_{4} + 59282388 \beta_{5} + 63252 \beta_{6} - 288 \beta_{7}) q^{9} +O(q^{10})\) \( q +(548616194585055 - \beta_{1}) q^{2} +(\)\(63\!\cdots\!80\)\( + 1292206 \beta_{1} - \beta_{2}) q^{3} +(\)\(60\!\cdots\!08\)\( - 1006563777941965 \beta_{1} + 34249 \beta_{2} + \beta_{3}) q^{4} +(\)\(68\!\cdots\!90\)\( - 13577557385999983067 \beta_{1} + 23394938356 \beta_{2} - 2101 \beta_{3} + \beta_{4}) q^{5} +(\)\(32\!\cdots\!92\)\( - \)\(59\!\cdots\!29\)\( \beta_{1} - 2269182927810110 \beta_{2} + 289750017 \beta_{3} - 3331 \beta_{4} - \beta_{5}) q^{6} +(\)\(52\!\cdots\!00\)\( - \)\(14\!\cdots\!08\)\( \beta_{1} + 3436742111413277589 \beta_{2} + 49285015561 \beta_{3} - 4418774 \beta_{4} + 505 \beta_{5} + \beta_{6}) q^{7} +(\)\(13\!\cdots\!60\)\( - \)\(62\!\cdots\!67\)\( \beta_{1} - \)\(24\!\cdots\!96\)\( \beta_{2} + 913937500520892 \beta_{3} - 783459941 \beta_{4} + 75477 \beta_{5} - 134 \beta_{6} - \beta_{7}) q^{8} +(\)\(44\!\cdots\!77\)\( - \)\(19\!\cdots\!70\)\( \beta_{1} - \)\(37\!\cdots\!24\)\( \beta_{2} + 272254459805646246 \beta_{3} - 720712994562 \beta_{4} + 59282388 \beta_{5} + 63252 \beta_{6} - 288 \beta_{7}) q^{9} +(\)\(25\!\cdots\!90\)\( - \)\(55\!\cdots\!26\)\( \beta_{1} + \)\(22\!\cdots\!04\)\( \beta_{2} + 1659923428331358884 \beta_{3} - 433358967391916 \beta_{4} + 8111585308 \beta_{5} + 80046976 \beta_{6} + 12096 \beta_{7}) q^{10} +(-\)\(66\!\cdots\!68\)\( + \)\(85\!\cdots\!26\)\( \beta_{1} - \)\(11\!\cdots\!69\)\( \beta_{2} + \)\(24\!\cdots\!14\)\( \beta_{3} - 75497546757957324 \beta_{4} - 15023926325694 \beta_{5} + 6487484498 \beta_{6} + 3253888 \beta_{7}) q^{11} +(\)\(32\!\cdots\!60\)\( - \)\(33\!\cdots\!60\)\( \beta_{1} - \)\(10\!\cdots\!92\)\( \beta_{2} + \)\(11\!\cdots\!72\)\( \beta_{3} - 2004032933334384336 \beta_{4} - 3220545977664048 \beta_{5} - 846949617504 \beta_{6} - 441349776 \beta_{7}) q^{12} +(-\)\(19\!\cdots\!90\)\( + \)\(10\!\cdots\!29\)\( \beta_{1} + \)\(48\!\cdots\!48\)\( \beta_{2} - \)\(10\!\cdots\!93\)\( \beta_{3} + \)\(70\!\cdots\!85\)\( \beta_{4} - 44796893837044072 \beta_{5} + 26310306328088 \beta_{6} + 29237253696 \beta_{7}) q^{13} +(\)\(25\!\cdots\!76\)\( - \)\(64\!\cdots\!34\)\( \beta_{1} + \)\(13\!\cdots\!60\)\( \beta_{2} - \)\(44\!\cdots\!62\)\( \beta_{3} + \)\(10\!\cdots\!58\)\( \beta_{4} + 6383741660729125638 \beta_{5} + 177715194016256 \beta_{6} - 1306326904064 \beta_{7}) q^{14} +(-\)\(37\!\cdots\!20\)\( + \)\(23\!\cdots\!28\)\( \beta_{1} - \)\(56\!\cdots\!57\)\( \beta_{2} + \)\(34\!\cdots\!83\)\( \beta_{3} + \)\(29\!\cdots\!78\)\( \beta_{4} - 59490093613531007009 \beta_{5} - 38695683278765673 \beta_{6} + 43918762920192 \beta_{7}) q^{15} +(\)\(45\!\cdots\!36\)\( - \)\(15\!\cdots\!16\)\( \beta_{1} - \)\(70\!\cdots\!00\)\( \beta_{2} + \)\(57\!\cdots\!36\)\( \beta_{3} - \)\(36\!\cdots\!52\)\( \beta_{4} - \)\(32\!\cdots\!32\)\( \beta_{5} + 1452842140237683248 \beta_{6} - 1174576591128312 \beta_{7}) q^{16} +(\)\(25\!\cdots\!90\)\( + \)\(14\!\cdots\!18\)\( \beta_{1} + \)\(15\!\cdots\!72\)\( \beta_{2} + \)\(10\!\cdots\!54\)\( \beta_{3} - \)\(10\!\cdots\!86\)\( \beta_{4} + \)\(88\!\cdots\!20\)\( \beta_{5} - 31417385238378658236 \beta_{6} + 25834565725096800 \beta_{7}) q^{17} +(\)\(33\!\cdots\!55\)\( - \)\(80\!\cdots\!69\)\( \beta_{1} - \)\(78\!\cdots\!96\)\( \beta_{2} + \)\(45\!\cdots\!56\)\( \beta_{3} + \)\(20\!\cdots\!16\)\( \beta_{4} - \)\(33\!\cdots\!00\)\( \beta_{5} + \)\(42\!\cdots\!36\)\( \beta_{6} - 477433803334239360 \beta_{7}) q^{18} +(\)\(18\!\cdots\!00\)\( - \)\(91\!\cdots\!02\)\( \beta_{1} - \)\(32\!\cdots\!51\)\( \beta_{2} + \)\(41\!\cdots\!66\)\( \beta_{3} + \)\(38\!\cdots\!28\)\( \beta_{4} - \)\(20\!\cdots\!82\)\( \beta_{5} - \)\(26\!\cdots\!86\)\( \beta_{6} + 7517169711519398784 \beta_{7}) q^{19} +(\)\(31\!\cdots\!20\)\( - \)\(16\!\cdots\!06\)\( \beta_{1} + \)\(32\!\cdots\!38\)\( \beta_{2} - \)\(16\!\cdots\!58\)\( \beta_{3} - \)\(14\!\cdots\!52\)\( \beta_{4} + \)\(44\!\cdots\!40\)\( \beta_{5} - \)\(29\!\cdots\!20\)\( \beta_{6} - \)\(10\!\cdots\!20\)\( \beta_{7}) q^{20} +(-\)\(58\!\cdots\!88\)\( + \)\(12\!\cdots\!92\)\( \beta_{1} - \)\(13\!\cdots\!52\)\( \beta_{2} - \)\(81\!\cdots\!92\)\( \beta_{3} - \)\(68\!\cdots\!04\)\( \beta_{4} - \)\(35\!\cdots\!44\)\( \beta_{5} + \)\(11\!\cdots\!92\)\( \beta_{6} + \)\(11\!\cdots\!52\)\( \beta_{7}) q^{21} +(-\)\(17\!\cdots\!40\)\( - \)\(16\!\cdots\!71\)\( \beta_{1} - \)\(25\!\cdots\!22\)\( \beta_{2} - \)\(33\!\cdots\!13\)\( \beta_{3} + \)\(71\!\cdots\!15\)\( \beta_{4} - \)\(80\!\cdots\!47\)\( \beta_{5} - \)\(18\!\cdots\!32\)\( \beta_{6} - \)\(11\!\cdots\!04\)\( \beta_{7}) q^{22} +(\)\(50\!\cdots\!40\)\( - \)\(40\!\cdots\!60\)\( \beta_{1} - \)\(83\!\cdots\!29\)\( \beta_{2} + \)\(42\!\cdots\!35\)\( \beta_{3} + \)\(21\!\cdots\!50\)\( \beta_{4} + \)\(50\!\cdots\!15\)\( \beta_{5} + \)\(19\!\cdots\!15\)\( \beta_{6} + \)\(10\!\cdots\!80\)\( \beta_{7}) q^{23} +(\)\(52\!\cdots\!00\)\( - \)\(10\!\cdots\!48\)\( \beta_{1} - \)\(47\!\cdots\!12\)\( \beta_{2} + \)\(78\!\cdots\!44\)\( \beta_{3} - \)\(73\!\cdots\!60\)\( \beta_{4} - \)\(54\!\cdots\!00\)\( \beta_{5} - \)\(15\!\cdots\!72\)\( \beta_{6} - \)\(69\!\cdots\!32\)\( \beta_{7}) q^{24} +(-\)\(45\!\cdots\!25\)\( + \)\(49\!\cdots\!20\)\( \beta_{1} - \)\(11\!\cdots\!00\)\( \beta_{2} - \)\(37\!\cdots\!20\)\( \beta_{3} + \)\(26\!\cdots\!00\)\( \beta_{4} + \)\(25\!\cdots\!80\)\( \beta_{5} + \)\(79\!\cdots\!60\)\( \beta_{6} + \)\(36\!\cdots\!60\)\( \beta_{7}) q^{25} +(-\)\(17\!\cdots\!98\)\( + \)\(34\!\cdots\!74\)\( \beta_{1} - \)\(36\!\cdots\!92\)\( \beta_{2} - \)\(40\!\cdots\!84\)\( \beta_{3} + \)\(29\!\cdots\!60\)\( \beta_{4} + \)\(73\!\cdots\!80\)\( \beta_{5} - \)\(72\!\cdots\!64\)\( \beta_{6} - \)\(81\!\cdots\!84\)\( \beta_{7}) q^{26} +(\)\(78\!\cdots\!40\)\( - \)\(19\!\cdots\!68\)\( \beta_{1} - \)\(12\!\cdots\!84\)\( \beta_{2} + \)\(14\!\cdots\!98\)\( \beta_{3} - \)\(28\!\cdots\!84\)\( \beta_{4} - \)\(19\!\cdots\!42\)\( \beta_{5} - \)\(33\!\cdots\!06\)\( \beta_{6} - \)\(87\!\cdots\!04\)\( \beta_{7}) q^{27} +(\)\(64\!\cdots\!60\)\( - \)\(10\!\cdots\!44\)\( \beta_{1} + \)\(10\!\cdots\!32\)\( \beta_{2} + \)\(13\!\cdots\!88\)\( \beta_{3} + \)\(82\!\cdots\!76\)\( \beta_{4} + \)\(13\!\cdots\!28\)\( \beta_{5} + \)\(39\!\cdots\!24\)\( \beta_{6} + \)\(14\!\cdots\!36\)\( \beta_{7}) q^{28} +(-\)\(13\!\cdots\!50\)\( - \)\(89\!\cdots\!71\)\( \beta_{1} + \)\(13\!\cdots\!24\)\( \beta_{2} - \)\(39\!\cdots\!25\)\( \beta_{3} + \)\(10\!\cdots\!25\)\( \beta_{4} - \)\(29\!\cdots\!80\)\( \beta_{5} - \)\(27\!\cdots\!44\)\( \beta_{6} - \)\(13\!\cdots\!64\)\( \beta_{7}) q^{29} +(-\)\(39\!\cdots\!20\)\( + \)\(45\!\cdots\!06\)\( \beta_{1} - \)\(46\!\cdots\!88\)\( \beta_{2} - \)\(59\!\cdots\!42\)\( \beta_{3} - \)\(44\!\cdots\!98\)\( \beta_{4} - \)\(22\!\cdots\!90\)\( \beta_{5} + \)\(11\!\cdots\!20\)\( \beta_{6} + \)\(87\!\cdots\!20\)\( \beta_{7}) q^{30} +(\)\(13\!\cdots\!92\)\( - \)\(51\!\cdots\!24\)\( \beta_{1} - \)\(66\!\cdots\!08\)\( \beta_{2} - \)\(29\!\cdots\!48\)\( \beta_{3} - \)\(99\!\cdots\!48\)\( \beta_{4} + \)\(22\!\cdots\!32\)\( \beta_{5} - \)\(18\!\cdots\!28\)\( \beta_{6} - \)\(42\!\cdots\!68\)\( \beta_{7}) q^{31} +(\)\(12\!\cdots\!80\)\( - \)\(45\!\cdots\!16\)\( \beta_{1} - \)\(57\!\cdots\!00\)\( \beta_{2} + \)\(16\!\cdots\!04\)\( \beta_{3} + \)\(13\!\cdots\!84\)\( \beta_{4} - \)\(76\!\cdots\!60\)\( \beta_{5} - \)\(18\!\cdots\!96\)\( \beta_{6} + \)\(12\!\cdots\!40\)\( \beta_{7}) q^{32} +(\)\(20\!\cdots\!60\)\( - \)\(27\!\cdots\!42\)\( \beta_{1} + \)\(62\!\cdots\!56\)\( \beta_{2} + \)\(63\!\cdots\!14\)\( \beta_{3} - \)\(14\!\cdots\!58\)\( \beta_{4} - \)\(10\!\cdots\!92\)\( \beta_{5} + \)\(19\!\cdots\!40\)\( \beta_{6} + \)\(12\!\cdots\!36\)\( \beta_{7}) q^{33} +(-\)\(21\!\cdots\!54\)\( - \)\(11\!\cdots\!82\)\( \beta_{1} + \)\(11\!\cdots\!36\)\( \beta_{2} - \)\(38\!\cdots\!96\)\( \beta_{3} - \)\(19\!\cdots\!32\)\( \beta_{4} + \)\(20\!\cdots\!08\)\( \beta_{5} - \)\(10\!\cdots\!16\)\( \beta_{6} - \)\(50\!\cdots\!96\)\( \beta_{7}) q^{34} +(-\)\(19\!\cdots\!60\)\( - \)\(67\!\cdots\!96\)\( \beta_{1} + \)\(39\!\cdots\!64\)\( \beta_{2} - \)\(28\!\cdots\!76\)\( \beta_{3} + \)\(46\!\cdots\!44\)\( \beta_{4} - \)\(71\!\cdots\!92\)\( \beta_{5} + \)\(33\!\cdots\!76\)\( \beta_{6} + \)\(42\!\cdots\!96\)\( \beta_{7}) q^{35} +(\)\(10\!\cdots\!16\)\( - \)\(62\!\cdots\!49\)\( \beta_{1} - \)\(13\!\cdots\!19\)\( \beta_{2} + \)\(89\!\cdots\!01\)\( \beta_{3} + \)\(34\!\cdots\!84\)\( \beta_{4} - \)\(85\!\cdots\!36\)\( \beta_{5} - \)\(52\!\cdots\!80\)\( \beta_{6} - \)\(24\!\cdots\!80\)\( \beta_{7}) q^{36} +(-\)\(11\!\cdots\!30\)\( - \)\(74\!\cdots\!63\)\( \beta_{1} - \)\(28\!\cdots\!76\)\( \beta_{2} - \)\(10\!\cdots\!69\)\( \beta_{3} - \)\(15\!\cdots\!83\)\( \beta_{4} + \)\(16\!\cdots\!16\)\( \beta_{5} - \)\(61\!\cdots\!52\)\( \beta_{6} + \)\(10\!\cdots\!92\)\( \beta_{7}) q^{37} +(\)\(24\!\cdots\!40\)\( - \)\(59\!\cdots\!05\)\( \beta_{1} - \)\(39\!\cdots\!10\)\( \beta_{2} - \)\(52\!\cdots\!51\)\( \beta_{3} - \)\(39\!\cdots\!39\)\( \beta_{4} - \)\(58\!\cdots\!73\)\( \beta_{5} + \)\(10\!\cdots\!08\)\( \beta_{6} - \)\(33\!\cdots\!96\)\( \beta_{7}) q^{38} +(-\)\(98\!\cdots\!76\)\( - \)\(38\!\cdots\!04\)\( \beta_{1} + \)\(10\!\cdots\!83\)\( \beta_{2} - \)\(16\!\cdots\!53\)\( \beta_{3} + \)\(34\!\cdots\!06\)\( \beta_{4} + \)\(19\!\cdots\!11\)\( \beta_{5} + \)\(55\!\cdots\!03\)\( \beta_{6} + \)\(63\!\cdots\!68\)\( \beta_{7}) q^{39} +(\)\(24\!\cdots\!00\)\( + \)\(18\!\cdots\!70\)\( \beta_{1} + \)\(47\!\cdots\!60\)\( \beta_{2} + \)\(10\!\cdots\!00\)\( \beta_{3} + \)\(27\!\cdots\!10\)\( \beta_{4} + \)\(56\!\cdots\!10\)\( \beta_{5} - \)\(50\!\cdots\!80\)\( \beta_{6} + \)\(77\!\cdots\!70\)\( \beta_{7}) q^{40} +(\)\(27\!\cdots\!22\)\( + \)\(36\!\cdots\!84\)\( \beta_{1} + \)\(58\!\cdots\!76\)\( \beta_{2} + \)\(38\!\cdots\!52\)\( \beta_{3} - \)\(58\!\cdots\!64\)\( \beta_{4} - \)\(46\!\cdots\!44\)\( \beta_{5} + \)\(25\!\cdots\!80\)\( \beta_{6} - \)\(12\!\cdots\!20\)\( \beta_{7}) q^{41} +(-\)\(23\!\cdots\!60\)\( + \)\(14\!\cdots\!40\)\( \beta_{1} - \)\(70\!\cdots\!04\)\( \beta_{2} - \)\(19\!\cdots\!24\)\( \beta_{3} + \)\(13\!\cdots\!76\)\( \beta_{4} - \)\(12\!\cdots\!60\)\( \beta_{5} - \)\(79\!\cdots\!64\)\( \beta_{6} + \)\(61\!\cdots\!20\)\( \beta_{7}) q^{42} +(\)\(12\!\cdots\!00\)\( + \)\(18\!\cdots\!98\)\( \beta_{1} - \)\(57\!\cdots\!35\)\( \beta_{2} - \)\(30\!\cdots\!60\)\( \beta_{3} + \)\(40\!\cdots\!72\)\( \beta_{4} + \)\(71\!\cdots\!12\)\( \beta_{5} + \)\(13\!\cdots\!24\)\( \beta_{6} - \)\(18\!\cdots\!36\)\( \beta_{7}) q^{43} +(\)\(32\!\cdots\!56\)\( + \)\(33\!\cdots\!36\)\( \beta_{1} - \)\(36\!\cdots\!40\)\( \beta_{2} + \)\(18\!\cdots\!24\)\( \beta_{3} - \)\(22\!\cdots\!48\)\( \beta_{4} - \)\(17\!\cdots\!68\)\( \beta_{5} + \)\(62\!\cdots\!32\)\( \beta_{6} + \)\(24\!\cdots\!92\)\( \beta_{7}) q^{44} +(-\)\(18\!\cdots\!70\)\( - \)\(15\!\cdots\!99\)\( \beta_{1} + \)\(17\!\cdots\!92\)\( \beta_{2} + \)\(26\!\cdots\!23\)\( \beta_{3} + \)\(49\!\cdots\!57\)\( \beta_{4} + \)\(86\!\cdots\!80\)\( \beta_{5} - \)\(11\!\cdots\!40\)\( \beta_{6} + \)\(66\!\cdots\!60\)\( \beta_{7}) q^{45} +(\)\(34\!\cdots\!12\)\( - \)\(92\!\cdots\!66\)\( \beta_{1} + \)\(43\!\cdots\!56\)\( \beta_{2} - \)\(81\!\cdots\!78\)\( \beta_{3} + \)\(13\!\cdots\!86\)\( \beta_{4} + \)\(19\!\cdots\!06\)\( \beta_{5} + \)\(27\!\cdots\!80\)\( \beta_{6} - \)\(54\!\cdots\!20\)\( \beta_{7}) q^{46} +(-\)\(36\!\cdots\!40\)\( - \)\(69\!\cdots\!16\)\( \beta_{1} + \)\(59\!\cdots\!30\)\( \beta_{2} - \)\(68\!\cdots\!70\)\( \beta_{3} + \)\(95\!\cdots\!24\)\( \beta_{4} + \)\(31\!\cdots\!54\)\( \beta_{5} - \)\(10\!\cdots\!42\)\( \beta_{6} + \)\(18\!\cdots\!88\)\( \beta_{7}) q^{47} +(\)\(15\!\cdots\!40\)\( - \)\(97\!\cdots\!00\)\( \beta_{1} - \)\(11\!\cdots\!84\)\( \beta_{2} + \)\(19\!\cdots\!64\)\( \beta_{3} - \)\(24\!\cdots\!36\)\( \beta_{4} - \)\(23\!\cdots\!40\)\( \beta_{5} - \)\(55\!\cdots\!96\)\( \beta_{6} - \)\(30\!\cdots\!20\)\( \beta_{7}) q^{48} +(\)\(33\!\cdots\!93\)\( - \)\(57\!\cdots\!56\)\( \beta_{1} + \)\(74\!\cdots\!76\)\( \beta_{2} + \)\(22\!\cdots\!52\)\( \beta_{3} + \)\(54\!\cdots\!96\)\( \beta_{4} + \)\(68\!\cdots\!16\)\( \beta_{5} - \)\(39\!\cdots\!40\)\( \beta_{6} - \)\(27\!\cdots\!40\)\( \beta_{7}) q^{49} +(-\)\(81\!\cdots\!75\)\( + \)\(44\!\cdots\!45\)\( \beta_{1} + \)\(23\!\cdots\!00\)\( \beta_{2} - \)\(53\!\cdots\!20\)\( \beta_{3} + \)\(22\!\cdots\!00\)\( \beta_{4} - \)\(69\!\cdots\!20\)\( \beta_{5} + \)\(36\!\cdots\!60\)\( \beta_{6} + \)\(34\!\cdots\!60\)\( \beta_{7}) q^{50} +(-\)\(26\!\cdots\!48\)\( + \)\(12\!\cdots\!52\)\( \beta_{1} + \)\(29\!\cdots\!56\)\( \beta_{2} - \)\(23\!\cdots\!46\)\( \beta_{3} - \)\(76\!\cdots\!28\)\( \beta_{4} - \)\(99\!\cdots\!18\)\( \beta_{5} - \)\(11\!\cdots\!14\)\( \beta_{6} - \)\(11\!\cdots\!84\)\( \beta_{7}) q^{51} +(-\)\(45\!\cdots\!00\)\( + \)\(49\!\cdots\!90\)\( \beta_{1} + \)\(12\!\cdots\!18\)\( \beta_{2} - \)\(27\!\cdots\!70\)\( \beta_{3} - \)\(10\!\cdots\!56\)\( \beta_{4} + \)\(77\!\cdots\!24\)\( \beta_{5} + \)\(71\!\cdots\!48\)\( \beta_{6} + \)\(15\!\cdots\!28\)\( \beta_{7}) q^{52} +(\)\(20\!\cdots\!30\)\( + \)\(26\!\cdots\!25\)\( \beta_{1} + \)\(86\!\cdots\!76\)\( \beta_{2} + \)\(19\!\cdots\!47\)\( \beta_{3} + \)\(36\!\cdots\!49\)\( \beta_{4} + \)\(15\!\cdots\!12\)\( \beta_{5} + \)\(79\!\cdots\!16\)\( \beta_{6} + \)\(19\!\cdots\!44\)\( \beta_{7}) q^{53} +(\)\(30\!\cdots\!00\)\( - \)\(16\!\cdots\!94\)\( \beta_{1} - \)\(38\!\cdots\!12\)\( \beta_{2} + \)\(32\!\cdots\!86\)\( \beta_{3} - \)\(15\!\cdots\!38\)\( \beta_{4} - \)\(35\!\cdots\!58\)\( \beta_{5} - \)\(34\!\cdots\!28\)\( \beta_{6} - \)\(16\!\cdots\!68\)\( \beta_{7}) q^{54} +(-\)\(75\!\cdots\!20\)\( - \)\(38\!\cdots\!44\)\( \beta_{1} - \)\(73\!\cdots\!83\)\( \beta_{2} - \)\(16\!\cdots\!07\)\( \beta_{3} - \)\(49\!\cdots\!18\)\( \beta_{4} - \)\(99\!\cdots\!75\)\( \beta_{5} + \)\(58\!\cdots\!25\)\( \beta_{6} + \)\(38\!\cdots\!00\)\( \beta_{7}) q^{55} +(-\)\(62\!\cdots\!00\)\( - \)\(13\!\cdots\!28\)\( \beta_{1} + \)\(24\!\cdots\!68\)\( \beta_{2} - \)\(32\!\cdots\!80\)\( \beta_{3} - \)\(39\!\cdots\!36\)\( \beta_{4} + \)\(64\!\cdots\!24\)\( \beta_{5} + \)\(29\!\cdots\!24\)\( \beta_{6} - \)\(28\!\cdots\!56\)\( \beta_{7}) q^{56} +(\)\(69\!\cdots\!80\)\( - \)\(43\!\cdots\!54\)\( \beta_{1} - \)\(83\!\cdots\!32\)\( \beta_{2} + \)\(11\!\cdots\!90\)\( \beta_{3} + \)\(19\!\cdots\!22\)\( \beta_{4} - \)\(11\!\cdots\!88\)\( \beta_{5} - \)\(38\!\cdots\!76\)\( \beta_{6} - \)\(11\!\cdots\!36\)\( \beta_{7}) q^{57} +(\)\(13\!\cdots\!10\)\( + \)\(13\!\cdots\!06\)\( \beta_{1} + \)\(75\!\cdots\!68\)\( \beta_{2} + \)\(18\!\cdots\!24\)\( \beta_{3} - \)\(27\!\cdots\!60\)\( \beta_{4} + \)\(12\!\cdots\!16\)\( \beta_{5} + \)\(78\!\cdots\!56\)\( \beta_{6} + \)\(39\!\cdots\!12\)\( \beta_{7}) q^{58} +(-\)\(47\!\cdots\!00\)\( + \)\(23\!\cdots\!98\)\( \beta_{1} - \)\(15\!\cdots\!03\)\( \beta_{2} - \)\(40\!\cdots\!08\)\( \beta_{3} + \)\(18\!\cdots\!24\)\( \beta_{4} + \)\(19\!\cdots\!64\)\( \beta_{5} - \)\(31\!\cdots\!52\)\( \beta_{6} - \)\(46\!\cdots\!12\)\( \beta_{7}) q^{59} +(-\)\(55\!\cdots\!60\)\( + \)\(76\!\cdots\!64\)\( \beta_{1} - \)\(28\!\cdots\!36\)\( \beta_{2} - \)\(11\!\cdots\!36\)\( \beta_{3} - \)\(15\!\cdots\!56\)\( \beta_{4} + \)\(28\!\cdots\!48\)\( \beta_{5} - \)\(11\!\cdots\!44\)\( \beta_{6} - \)\(34\!\cdots\!24\)\( \beta_{7}) q^{60} +(\)\(70\!\cdots\!82\)\( + \)\(88\!\cdots\!21\)\( \beta_{1} + \)\(32\!\cdots\!28\)\( \beta_{2} - \)\(10\!\cdots\!65\)\( \beta_{3} + \)\(55\!\cdots\!53\)\( \beta_{4} - \)\(10\!\cdots\!92\)\( \beta_{5} - \)\(10\!\cdots\!24\)\( \beta_{6} + \)\(11\!\cdots\!56\)\( \beta_{7}) q^{61} +(\)\(88\!\cdots\!60\)\( - \)\(14\!\cdots\!52\)\( \beta_{1} + \)\(14\!\cdots\!80\)\( \beta_{2} + \)\(11\!\cdots\!36\)\( \beta_{3} - \)\(50\!\cdots\!64\)\( \beta_{4} - \)\(48\!\cdots\!60\)\( \beta_{5} + \)\(10\!\cdots\!96\)\( \beta_{6} + \)\(25\!\cdots\!20\)\( \beta_{7}) q^{62} +(\)\(12\!\cdots\!20\)\( - \)\(43\!\cdots\!80\)\( \beta_{1} + \)\(78\!\cdots\!01\)\( \beta_{2} - \)\(74\!\cdots\!71\)\( \beta_{3} - \)\(56\!\cdots\!02\)\( \beta_{4} + \)\(25\!\cdots\!89\)\( \beta_{5} + \)\(32\!\cdots\!97\)\( \beta_{6} - \)\(10\!\cdots\!32\)\( \beta_{7}) q^{63} +(\)\(33\!\cdots\!88\)\( - \)\(15\!\cdots\!52\)\( \beta_{1} - \)\(18\!\cdots\!80\)\( \beta_{2} - \)\(83\!\cdots\!44\)\( \beta_{3} - \)\(20\!\cdots\!88\)\( \beta_{4} - \)\(36\!\cdots\!48\)\( \beta_{5} - \)\(13\!\cdots\!40\)\( \beta_{6} - \)\(15\!\cdots\!40\)\( \beta_{7}) q^{64} +(\)\(64\!\cdots\!80\)\( - \)\(13\!\cdots\!72\)\( \beta_{1} - \)\(18\!\cdots\!12\)\( \beta_{2} - \)\(10\!\cdots\!52\)\( \beta_{3} + \)\(10\!\cdots\!48\)\( \beta_{4} - \)\(19\!\cdots\!24\)\( \beta_{5} + \)\(36\!\cdots\!72\)\( \beta_{6} + \)\(15\!\cdots\!12\)\( \beta_{7}) q^{65} +(\)\(45\!\cdots\!44\)\( - \)\(39\!\cdots\!44\)\( \beta_{1} - \)\(30\!\cdots\!12\)\( \beta_{2} + \)\(23\!\cdots\!52\)\( \beta_{3} + \)\(21\!\cdots\!56\)\( \beta_{4} + \)\(10\!\cdots\!36\)\( \beta_{5} - \)\(15\!\cdots\!32\)\( \beta_{6} - \)\(32\!\cdots\!92\)\( \beta_{7}) q^{66} +(\)\(45\!\cdots\!40\)\( + \)\(13\!\cdots\!62\)\( \beta_{1} + \)\(20\!\cdots\!29\)\( \beta_{2} + \)\(68\!\cdots\!34\)\( \beta_{3} - \)\(67\!\cdots\!24\)\( \beta_{4} - \)\(29\!\cdots\!18\)\( \beta_{5} - \)\(14\!\cdots\!22\)\( \beta_{6} - \)\(13\!\cdots\!36\)\( \beta_{7}) q^{67} +(\)\(15\!\cdots\!80\)\( + \)\(38\!\cdots\!94\)\( \beta_{1} + \)\(45\!\cdots\!46\)\( \beta_{2} + \)\(43\!\cdots\!98\)\( \beta_{3} + \)\(48\!\cdots\!76\)\( \beta_{4} + \)\(18\!\cdots\!68\)\( \beta_{5} + \)\(38\!\cdots\!64\)\( \beta_{6} + \)\(23\!\cdots\!16\)\( \beta_{7}) q^{68} +(\)\(18\!\cdots\!44\)\( + \)\(54\!\cdots\!24\)\( \beta_{1} - \)\(47\!\cdots\!56\)\( \beta_{2} - \)\(27\!\cdots\!68\)\( \beta_{3} - \)\(40\!\cdots\!12\)\( \beta_{4} - \)\(19\!\cdots\!92\)\( \beta_{5} - \)\(23\!\cdots\!72\)\( \beta_{6} - \)\(50\!\cdots\!32\)\( \beta_{7}) q^{69} +(\)\(10\!\cdots\!40\)\( + \)\(15\!\cdots\!48\)\( \beta_{1} + \)\(72\!\cdots\!76\)\( \beta_{2} + \)\(15\!\cdots\!24\)\( \beta_{3} + \)\(51\!\cdots\!96\)\( \beta_{4} + \)\(41\!\cdots\!20\)\( \beta_{5} - \)\(58\!\cdots\!60\)\( \beta_{6} + \)\(65\!\cdots\!40\)\( \beta_{7}) q^{70} +(\)\(16\!\cdots\!12\)\( - \)\(32\!\cdots\!92\)\( \beta_{1} - \)\(16\!\cdots\!31\)\( \beta_{2} - \)\(48\!\cdots\!95\)\( \beta_{3} - \)\(86\!\cdots\!06\)\( \beta_{4} + \)\(50\!\cdots\!09\)\( \beta_{5} + \)\(86\!\cdots\!73\)\( \beta_{6} + \)\(22\!\cdots\!88\)\( \beta_{7}) q^{71} +(\)\(70\!\cdots\!20\)\( - \)\(13\!\cdots\!27\)\( \beta_{1} - \)\(50\!\cdots\!16\)\( \beta_{2} + \)\(41\!\cdots\!12\)\( \beta_{3} - \)\(15\!\cdots\!73\)\( \beta_{4} - \)\(14\!\cdots\!55\)\( \beta_{5} + \)\(58\!\cdots\!62\)\( \beta_{6} - \)\(42\!\cdots\!05\)\( \beta_{7}) q^{72} +(\)\(55\!\cdots\!90\)\( - \)\(62\!\cdots\!26\)\( \beta_{1} + \)\(11\!\cdots\!88\)\( \beta_{2} - \)\(56\!\cdots\!86\)\( \beta_{3} + \)\(46\!\cdots\!86\)\( \beta_{4} + \)\(14\!\cdots\!12\)\( \beta_{5} + \)\(18\!\cdots\!68\)\( \beta_{6} - \)\(10\!\cdots\!76\)\( \beta_{7}) q^{73} +(\)\(11\!\cdots\!86\)\( + \)\(78\!\cdots\!30\)\( \beta_{1} + \)\(17\!\cdots\!36\)\( \beta_{2} - \)\(68\!\cdots\!72\)\( \beta_{3} + \)\(21\!\cdots\!16\)\( \beta_{4} - \)\(40\!\cdots\!84\)\( \beta_{5} - \)\(18\!\cdots\!96\)\( \beta_{6} + \)\(15\!\cdots\!24\)\( \beta_{7}) q^{74} +(\)\(19\!\cdots\!00\)\( + \)\(36\!\cdots\!90\)\( \beta_{1} + \)\(28\!\cdots\!25\)\( \beta_{2} - \)\(16\!\cdots\!40\)\( \beta_{3} + \)\(31\!\cdots\!00\)\( \beta_{4} + \)\(89\!\cdots\!60\)\( \beta_{5} + \)\(32\!\cdots\!20\)\( \beta_{6} - \)\(13\!\cdots\!80\)\( \beta_{7}) q^{75} +(\)\(77\!\cdots\!00\)\( + \)\(10\!\cdots\!44\)\( \beta_{1} - \)\(13\!\cdots\!44\)\( \beta_{2} + \)\(59\!\cdots\!00\)\( \beta_{3} - \)\(51\!\cdots\!52\)\( \beta_{4} - \)\(26\!\cdots\!32\)\( \beta_{5} + \)\(25\!\cdots\!28\)\( \beta_{6} - \)\(50\!\cdots\!32\)\( \beta_{7}) q^{76} +(\)\(47\!\cdots\!00\)\( + \)\(26\!\cdots\!24\)\( \beta_{1} - \)\(15\!\cdots\!92\)\( \beta_{2} - \)\(11\!\cdots\!96\)\( \beta_{3} + \)\(70\!\cdots\!84\)\( \beta_{4} - \)\(24\!\cdots\!60\)\( \beta_{5} - \)\(17\!\cdots\!96\)\( \beta_{6} + \)\(10\!\cdots\!40\)\( \beta_{7}) q^{77} +(\)\(55\!\cdots\!00\)\( + \)\(23\!\cdots\!86\)\( \beta_{1} + \)\(33\!\cdots\!40\)\( \beta_{2} - \)\(52\!\cdots\!22\)\( \beta_{3} + \)\(78\!\cdots\!82\)\( \beta_{4} + \)\(11\!\cdots\!34\)\( \beta_{5} + \)\(20\!\cdots\!56\)\( \beta_{6} + \)\(15\!\cdots\!68\)\( \beta_{7}) q^{78} +(\)\(30\!\cdots\!00\)\( - \)\(77\!\cdots\!24\)\( \beta_{1} - \)\(15\!\cdots\!34\)\( \beta_{2} - \)\(81\!\cdots\!22\)\( \beta_{3} - \)\(12\!\cdots\!08\)\( \beta_{4} - \)\(13\!\cdots\!78\)\( \beta_{5} - \)\(33\!\cdots\!58\)\( \beta_{6} - \)\(84\!\cdots\!48\)\( \beta_{7}) q^{79} +(-\)\(32\!\cdots\!60\)\( - \)\(81\!\cdots\!72\)\( \beta_{1} + \)\(12\!\cdots\!36\)\( \beta_{2} - \)\(17\!\cdots\!36\)\( \beta_{3} - \)\(19\!\cdots\!44\)\( \beta_{4} - \)\(31\!\cdots\!80\)\( \beta_{5} + \)\(22\!\cdots\!40\)\( \beta_{6} + \)\(74\!\cdots\!40\)\( \beta_{7}) q^{80} +(-\)\(39\!\cdots\!79\)\( - \)\(16\!\cdots\!62\)\( \beta_{1} - \)\(21\!\cdots\!40\)\( \beta_{2} + \)\(73\!\cdots\!54\)\( \beta_{3} - \)\(41\!\cdots\!06\)\( \beta_{4} - \)\(26\!\cdots\!16\)\( \beta_{5} - \)\(72\!\cdots\!92\)\( \beta_{6} + \)\(26\!\cdots\!48\)\( \beta_{7}) q^{81} +(-\)\(43\!\cdots\!90\)\( - \)\(64\!\cdots\!90\)\( \beta_{1} - \)\(12\!\cdots\!44\)\( \beta_{2} + \)\(35\!\cdots\!56\)\( \beta_{3} + \)\(26\!\cdots\!52\)\( \beta_{4} + \)\(11\!\cdots\!76\)\( \beta_{5} - \)\(26\!\cdots\!32\)\( \beta_{6} - \)\(82\!\cdots\!88\)\( \beta_{7}) q^{82} +(-\)\(32\!\cdots\!80\)\( + \)\(34\!\cdots\!62\)\( \beta_{1} - \)\(36\!\cdots\!53\)\( \beta_{2} + \)\(79\!\cdots\!00\)\( \beta_{3} - \)\(19\!\cdots\!40\)\( \beta_{4} + \)\(48\!\cdots\!60\)\( \beta_{5} + \)\(14\!\cdots\!20\)\( \beta_{6} + \)\(46\!\cdots\!20\)\( \beta_{7}) q^{83} +(-\)\(18\!\cdots\!04\)\( + \)\(37\!\cdots\!48\)\( \beta_{1} - \)\(13\!\cdots\!44\)\( \beta_{2} - \)\(89\!\cdots\!80\)\( \beta_{3} - \)\(19\!\cdots\!28\)\( \beta_{4} - \)\(64\!\cdots\!88\)\( \beta_{5} - \)\(16\!\cdots\!60\)\( \beta_{6} + \)\(20\!\cdots\!40\)\( \beta_{7}) q^{84} +(-\)\(96\!\cdots\!60\)\( + \)\(35\!\cdots\!14\)\( \beta_{1} + \)\(19\!\cdots\!84\)\( \beta_{2} + \)\(81\!\cdots\!54\)\( \beta_{3} - \)\(33\!\cdots\!86\)\( \beta_{4} + \)\(62\!\cdots\!08\)\( \beta_{5} - \)\(28\!\cdots\!24\)\( \beta_{6} - \)\(47\!\cdots\!04\)\( \beta_{7}) q^{85} +(-\)\(28\!\cdots\!68\)\( + \)\(26\!\cdots\!13\)\( \beta_{1} + \)\(11\!\cdots\!70\)\( \beta_{2} - \)\(29\!\cdots\!13\)\( \beta_{3} + \)\(51\!\cdots\!31\)\( \beta_{4} + \)\(80\!\cdots\!21\)\( \beta_{5} + \)\(10\!\cdots\!16\)\( \beta_{6} + \)\(17\!\cdots\!96\)\( \beta_{7}) q^{86} +(-\)\(24\!\cdots\!80\)\( - \)\(53\!\cdots\!28\)\( \beta_{1} - \)\(17\!\cdots\!29\)\( \beta_{2} + \)\(31\!\cdots\!91\)\( \beta_{3} + \)\(36\!\cdots\!98\)\( \beta_{4} - \)\(40\!\cdots\!73\)\( \beta_{5} - \)\(80\!\cdots\!65\)\( \beta_{6} + \)\(79\!\cdots\!84\)\( \beta_{7}) q^{87} +(-\)\(50\!\cdots\!80\)\( - \)\(18\!\cdots\!12\)\( \beta_{1} - \)\(15\!\cdots\!76\)\( \beta_{2} - \)\(13\!\cdots\!96\)\( \beta_{3} - \)\(58\!\cdots\!36\)\( \beta_{4} - \)\(38\!\cdots\!80\)\( \beta_{5} - \)\(15\!\cdots\!36\)\( \beta_{6} - \)\(10\!\cdots\!00\)\( \beta_{7}) q^{88} +(-\)\(30\!\cdots\!50\)\( - \)\(32\!\cdots\!82\)\( \beta_{1} - \)\(34\!\cdots\!40\)\( \beta_{2} - \)\(63\!\cdots\!70\)\( \beta_{3} - \)\(27\!\cdots\!42\)\( \beta_{4} + \)\(81\!\cdots\!48\)\( \beta_{5} + \)\(33\!\cdots\!04\)\( \beta_{6} - \)\(18\!\cdots\!76\)\( \beta_{7}) q^{89} +(\)\(24\!\cdots\!30\)\( - \)\(25\!\cdots\!62\)\( \beta_{1} + \)\(45\!\cdots\!28\)\( \beta_{2} - \)\(86\!\cdots\!32\)\( \beta_{3} - \)\(44\!\cdots\!12\)\( \beta_{4} + \)\(16\!\cdots\!36\)\( \beta_{5} - \)\(87\!\cdots\!08\)\( \beta_{6} - \)\(76\!\cdots\!68\)\( \beta_{7}) q^{90} +(\)\(19\!\cdots\!72\)\( + \)\(40\!\cdots\!72\)\( \beta_{1} - \)\(54\!\cdots\!12\)\( \beta_{2} + \)\(12\!\cdots\!40\)\( \beta_{3} + \)\(35\!\cdots\!24\)\( \beta_{4} - \)\(17\!\cdots\!16\)\( \beta_{5} - \)\(27\!\cdots\!36\)\( \beta_{6} + \)\(76\!\cdots\!84\)\( \beta_{7}) q^{91} +(\)\(96\!\cdots\!40\)\( + \)\(60\!\cdots\!60\)\( \beta_{1} + \)\(25\!\cdots\!68\)\( \beta_{2} + \)\(83\!\cdots\!16\)\( \beta_{3} + \)\(30\!\cdots\!32\)\( \beta_{4} - \)\(46\!\cdots\!04\)\( \beta_{5} + \)\(27\!\cdots\!68\)\( \beta_{6} - \)\(63\!\cdots\!48\)\( \beta_{7}) q^{92} +(\)\(12\!\cdots\!60\)\( + \)\(17\!\cdots\!04\)\( \beta_{1} - \)\(31\!\cdots\!16\)\( \beta_{2} - \)\(19\!\cdots\!08\)\( \beta_{3} - \)\(96\!\cdots\!72\)\( \beta_{4} - \)\(42\!\cdots\!44\)\( \beta_{5} - \)\(65\!\cdots\!56\)\( \beta_{6} - \)\(30\!\cdots\!88\)\( \beta_{7}) q^{93} +(\)\(90\!\cdots\!56\)\( + \)\(10\!\cdots\!16\)\( \beta_{1} + \)\(76\!\cdots\!08\)\( \beta_{2} - \)\(10\!\cdots\!64\)\( \beta_{3} + \)\(25\!\cdots\!52\)\( \beta_{4} + \)\(96\!\cdots\!32\)\( \beta_{5} + \)\(36\!\cdots\!72\)\( \beta_{6} + \)\(82\!\cdots\!32\)\( \beta_{7}) q^{94} +(\)\(34\!\cdots\!00\)\( + \)\(35\!\cdots\!20\)\( \beta_{1} - \)\(91\!\cdots\!65\)\( \beta_{2} - \)\(41\!\cdots\!25\)\( \beta_{3} - \)\(29\!\cdots\!90\)\( \beta_{4} + \)\(24\!\cdots\!35\)\( \beta_{5} + \)\(65\!\cdots\!95\)\( \beta_{6} - \)\(31\!\cdots\!80\)\( \beta_{7}) q^{95} +(\)\(11\!\cdots\!12\)\( - \)\(28\!\cdots\!72\)\( \beta_{1} - \)\(18\!\cdots\!52\)\( \beta_{2} + \)\(77\!\cdots\!28\)\( \beta_{3} + \)\(47\!\cdots\!72\)\( \beta_{4} - \)\(80\!\cdots\!88\)\( \beta_{5} - \)\(14\!\cdots\!60\)\( \beta_{6} - \)\(17\!\cdots\!60\)\( \beta_{7}) q^{96} +(\)\(62\!\cdots\!10\)\( - \)\(59\!\cdots\!18\)\( \beta_{1} - \)\(14\!\cdots\!96\)\( \beta_{2} - \)\(37\!\cdots\!22\)\( \beta_{3} - \)\(23\!\cdots\!54\)\( \beta_{4} + \)\(28\!\cdots\!08\)\( \beta_{5} - \)\(13\!\cdots\!76\)\( \beta_{6} + \)\(30\!\cdots\!96\)\( \beta_{7}) q^{97} +(\)\(11\!\cdots\!15\)\( - \)\(58\!\cdots\!21\)\( \beta_{1} + \)\(11\!\cdots\!76\)\( \beta_{2} + \)\(64\!\cdots\!96\)\( \beta_{3} - \)\(74\!\cdots\!08\)\( \beta_{4} + \)\(12\!\cdots\!76\)\( \beta_{5} + \)\(81\!\cdots\!08\)\( \beta_{6} - \)\(46\!\cdots\!88\)\( \beta_{7}) q^{98} +(-\)\(90\!\cdots\!36\)\( - \)\(26\!\cdots\!62\)\( \beta_{1} - \)\(59\!\cdots\!63\)\( \beta_{2} - \)\(47\!\cdots\!40\)\( \beta_{3} - \)\(93\!\cdots\!64\)\( \beta_{4} - \)\(11\!\cdots\!24\)\( \beta_{5} - \)\(62\!\cdots\!84\)\( \beta_{6} - \)\(20\!\cdots\!04\)\( \beta_{7}) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 4388929556680440q^{2} + \)\(50\!\cdots\!40\)\(q^{3} + \)\(48\!\cdots\!64\)\(q^{4} + \)\(55\!\cdots\!20\)\(q^{5} + \)\(26\!\cdots\!36\)\(q^{6} + \)\(41\!\cdots\!00\)\(q^{7} + \)\(10\!\cdots\!80\)\(q^{8} + \)\(35\!\cdots\!16\)\(q^{9} + O(q^{10}) \) \( 8q + 4388929556680440q^{2} + \)\(50\!\cdots\!40\)\(q^{3} + \)\(48\!\cdots\!64\)\(q^{4} + \)\(55\!\cdots\!20\)\(q^{5} + \)\(26\!\cdots\!36\)\(q^{6} + \)\(41\!\cdots\!00\)\(q^{7} + \)\(10\!\cdots\!80\)\(q^{8} + \)\(35\!\cdots\!16\)\(q^{9} + \)\(20\!\cdots\!20\)\(q^{10} - \)\(53\!\cdots\!44\)\(q^{11} + \)\(25\!\cdots\!80\)\(q^{12} - \)\(15\!\cdots\!20\)\(q^{13} + \)\(20\!\cdots\!08\)\(q^{14} - \)\(30\!\cdots\!60\)\(q^{15} + \)\(36\!\cdots\!88\)\(q^{16} + \)\(20\!\cdots\!20\)\(q^{17} + \)\(26\!\cdots\!40\)\(q^{18} + \)\(14\!\cdots\!00\)\(q^{19} + \)\(25\!\cdots\!60\)\(q^{20} - \)\(46\!\cdots\!04\)\(q^{21} - \)\(13\!\cdots\!20\)\(q^{22} + \)\(40\!\cdots\!20\)\(q^{23} + \)\(41\!\cdots\!00\)\(q^{24} - \)\(36\!\cdots\!00\)\(q^{25} - \)\(14\!\cdots\!84\)\(q^{26} + \)\(63\!\cdots\!20\)\(q^{27} + \)\(51\!\cdots\!80\)\(q^{28} - \)\(11\!\cdots\!00\)\(q^{29} - \)\(31\!\cdots\!60\)\(q^{30} + \)\(10\!\cdots\!36\)\(q^{31} + \)\(10\!\cdots\!40\)\(q^{32} + \)\(16\!\cdots\!80\)\(q^{33} - \)\(16\!\cdots\!32\)\(q^{34} - \)\(15\!\cdots\!80\)\(q^{35} + \)\(80\!\cdots\!28\)\(q^{36} - \)\(92\!\cdots\!40\)\(q^{37} + \)\(19\!\cdots\!20\)\(q^{38} - \)\(78\!\cdots\!08\)\(q^{39} + \)\(19\!\cdots\!00\)\(q^{40} + \)\(22\!\cdots\!76\)\(q^{41} - \)\(18\!\cdots\!80\)\(q^{42} + \)\(96\!\cdots\!00\)\(q^{43} + \)\(26\!\cdots\!48\)\(q^{44} - \)\(14\!\cdots\!60\)\(q^{45} + \)\(27\!\cdots\!96\)\(q^{46} - \)\(29\!\cdots\!20\)\(q^{47} + \)\(12\!\cdots\!20\)\(q^{48} + \)\(26\!\cdots\!44\)\(q^{49} - \)\(65\!\cdots\!00\)\(q^{50} - \)\(20\!\cdots\!84\)\(q^{51} - \)\(36\!\cdots\!00\)\(q^{52} + \)\(16\!\cdots\!40\)\(q^{53} + \)\(24\!\cdots\!00\)\(q^{54} - \)\(60\!\cdots\!60\)\(q^{55} - \)\(50\!\cdots\!00\)\(q^{56} + \)\(55\!\cdots\!40\)\(q^{57} + \)\(10\!\cdots\!80\)\(q^{58} - \)\(37\!\cdots\!00\)\(q^{59} - \)\(44\!\cdots\!80\)\(q^{60} + \)\(56\!\cdots\!56\)\(q^{61} + \)\(70\!\cdots\!80\)\(q^{62} + \)\(10\!\cdots\!60\)\(q^{63} + \)\(26\!\cdots\!04\)\(q^{64} + \)\(51\!\cdots\!40\)\(q^{65} + \)\(36\!\cdots\!52\)\(q^{66} + \)\(36\!\cdots\!20\)\(q^{67} + \)\(12\!\cdots\!40\)\(q^{68} + \)\(14\!\cdots\!52\)\(q^{69} + \)\(85\!\cdots\!20\)\(q^{70} + \)\(13\!\cdots\!96\)\(q^{71} + \)\(56\!\cdots\!60\)\(q^{72} + \)\(44\!\cdots\!20\)\(q^{73} + \)\(94\!\cdots\!88\)\(q^{74} + \)\(15\!\cdots\!00\)\(q^{75} + \)\(62\!\cdots\!00\)\(q^{76} + \)\(37\!\cdots\!00\)\(q^{77} + \)\(44\!\cdots\!00\)\(q^{78} + \)\(24\!\cdots\!00\)\(q^{79} - \)\(25\!\cdots\!80\)\(q^{80} - \)\(31\!\cdots\!32\)\(q^{81} - \)\(34\!\cdots\!20\)\(q^{82} - \)\(25\!\cdots\!40\)\(q^{83} - \)\(14\!\cdots\!32\)\(q^{84} - \)\(77\!\cdots\!80\)\(q^{85} - \)\(22\!\cdots\!44\)\(q^{86} - \)\(19\!\cdots\!40\)\(q^{87} - \)\(40\!\cdots\!40\)\(q^{88} - \)\(24\!\cdots\!00\)\(q^{89} + \)\(19\!\cdots\!40\)\(q^{90} + \)\(15\!\cdots\!76\)\(q^{91} + \)\(77\!\cdots\!20\)\(q^{92} + \)\(10\!\cdots\!80\)\(q^{93} + \)\(72\!\cdots\!48\)\(q^{94} + \)\(27\!\cdots\!00\)\(q^{95} + \)\(88\!\cdots\!96\)\(q^{96} + \)\(50\!\cdots\!80\)\(q^{97} + \)\(88\!\cdots\!20\)\(q^{98} - \)\(72\!\cdots\!88\)\(q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 3 x^{7} - \)\(11\!\cdots\!21\)\( x^{6} - \)\(27\!\cdots\!07\)\( x^{5} + \)\(36\!\cdots\!21\)\( x^{4} - \)\(47\!\cdots\!05\)\( x^{3} - \)\(36\!\cdots\!75\)\( x^{2} + \)\(17\!\cdots\!75\)\( x + \)\(10\!\cdots\!50\)\(\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 24 \nu - 9 \)
\(\beta_{2}\)\(=\)\((\)\(\)\(16\!\cdots\!41\)\( \nu^{7} + \)\(13\!\cdots\!88\)\( \nu^{6} - \)\(16\!\cdots\!69\)\( \nu^{5} - \)\(15\!\cdots\!70\)\( \nu^{4} + \)\(45\!\cdots\!31\)\( \nu^{3} + \)\(41\!\cdots\!16\)\( \nu^{2} - \)\(25\!\cdots\!71\)\( \nu - \)\(21\!\cdots\!46\)\(\)\()/ \)\(59\!\cdots\!28\)\( \)
\(\beta_{3}\)\(=\)\((\)\(-\)\(57\!\cdots\!09\)\( \nu^{7} - \)\(46\!\cdots\!12\)\( \nu^{6} + \)\(57\!\cdots\!81\)\( \nu^{5} + \)\(51\!\cdots\!30\)\( \nu^{4} - \)\(15\!\cdots\!19\)\( \nu^{3} + \)\(19\!\cdots\!44\)\( \nu^{2} + \)\(73\!\cdots\!43\)\( \nu - \)\(86\!\cdots\!86\)\(\)\()/ \)\(59\!\cdots\!28\)\( \)
\(\beta_{4}\)\(=\)\((\)\(\)\(44\!\cdots\!29\)\( \nu^{7} + \)\(15\!\cdots\!16\)\( \nu^{6} - \)\(46\!\cdots\!97\)\( \nu^{5} - \)\(18\!\cdots\!82\)\( \nu^{4} + \)\(13\!\cdots\!35\)\( \nu^{3} + \)\(52\!\cdots\!00\)\( \nu^{2} - \)\(80\!\cdots\!75\)\( \nu - \)\(27\!\cdots\!50\)\(\)\()/ \)\(82\!\cdots\!00\)\( \)
\(\beta_{5}\)\(=\)\((\)\(-\)\(17\!\cdots\!41\)\( \nu^{7} - \)\(16\!\cdots\!64\)\( \nu^{6} + \)\(18\!\cdots\!13\)\( \nu^{5} + \)\(16\!\cdots\!78\)\( \nu^{4} - \)\(50\!\cdots\!15\)\( \nu^{3} - \)\(40\!\cdots\!00\)\( \nu^{2} + \)\(28\!\cdots\!75\)\( \nu + \)\(19\!\cdots\!50\)\(\)\()/ \)\(14\!\cdots\!00\)\( \)
\(\beta_{6}\)\(=\)\((\)\(-\)\(55\!\cdots\!79\)\( \nu^{7} - \)\(13\!\cdots\!16\)\( \nu^{6} + \)\(60\!\cdots\!47\)\( \nu^{5} + \)\(13\!\cdots\!82\)\( \nu^{4} - \)\(18\!\cdots\!85\)\( \nu^{3} - \)\(37\!\cdots\!00\)\( \nu^{2} + \)\(11\!\cdots\!25\)\( \nu + \)\(19\!\cdots\!50\)\(\)\()/ \)\(94\!\cdots\!00\)\( \)
\(\beta_{7}\)\(=\)\((\)\(-\)\(36\!\cdots\!03\)\( \nu^{7} - \)\(23\!\cdots\!12\)\( \nu^{6} + \)\(36\!\cdots\!79\)\( \nu^{5} + \)\(25\!\cdots\!74\)\( \nu^{4} - \)\(90\!\cdots\!45\)\( \nu^{3} - \)\(72\!\cdots\!00\)\( \nu^{2} + \)\(24\!\cdots\!25\)\( \nu + \)\(40\!\cdots\!50\)\(\)\()/ \)\(48\!\cdots\!00\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1} + 9\)\()/24\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} + 34249 \beta_{2} + 90668611228163 \beta_{1} + 15880774323327381508482134600472\)\()/576\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{7} + 134 \beta_{6} - 75477 \beta_{5} + 783459941 \beta_{4} + 731911083234300 \beta_{3} + 2478085321776355741004 \beta_{2} + 25759876756015228360331701993191 \beta_{1} + 1439887753041709047949578036681177213775225536\)\()/13824\)
\(\nu^{4}\)\(=\)\((\)\(127486023401493 \beta_{7} + 218362552566909694 \beta_{6} - 430172487255977596793 \beta_{5} - 4356650832545394280174007 \beta_{4} + 4497838928225868883025303812548 \beta_{3} - 76823225147800960931291530835816428 \beta_{2} + 1386918490615083743415227142251113877567731451 \beta_{1} + 51135848669130346006836710771052948394485158826710775385653696\)\()/41472\)
\(\nu^{5}\)\(=\)\((\)\(152656288142514471733425820127 \beta_{7} + 45438221749047255954687266134714 \beta_{6} - 17993559956597123255555491648916619 \beta_{5} - 768246208699183609569551992420873901317 \beta_{4} + 173385253974466203701312559740646907720956512 \beta_{3} + 571363283579140025011367950090086947987690066381552 \beta_{2} + 2989070443962723331825630931119123352252371254739780813839645 \beta_{1} + 688291861077914391455873698555277860820710991874720231796873324604834841264\)\()/31104\)
\(\nu^{6}\)\(=\)\((\)\(\)\(23\!\cdots\!71\)\( \beta_{7} + \)\(33\!\cdots\!62\)\( \beta_{6} - \)\(73\!\cdots\!23\)\( \beta_{5} - \)\(85\!\cdots\!17\)\( \beta_{4} + \)\(43\!\cdots\!40\)\( \beta_{3} - \)\(56\!\cdots\!36\)\( \beta_{2} + \)\(24\!\cdots\!09\)\( \beta_{1} + \)\(43\!\cdots\!04\)\(\)\()/6912\)
\(\nu^{7}\)\(=\)\((\)\(\)\(12\!\cdots\!93\)\( \beta_{7} + \)\(53\!\cdots\!70\)\( \beta_{6} - \)\(21\!\cdots\!33\)\( \beta_{5} - \)\(10\!\cdots\!51\)\( \beta_{4} + \)\(19\!\cdots\!12\)\( \beta_{3} + \)\(56\!\cdots\!08\)\( \beta_{2} + \)\(21\!\cdots\!27\)\( \beta_{1} + \)\(98\!\cdots\!88\)\(\)\()/41472\)

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
2.50366e14
1.83024e14
8.88202e13
8.06342e13
−6.19123e13
−1.05899e14
−2.04878e14
−2.30156e14
−5.46018e15 −1.30099e24 1.96724e31 −5.30727e35 7.10366e39 2.61912e43 −5.20418e46 −1.22226e49 2.89786e51
1.2 −3.84396e15 3.76369e24 4.63479e30 1.31095e36 −1.44675e40 −5.54417e43 2.11664e46 2.50204e47 −5.03922e51
1.3 −1.58307e15 −4.68333e24 −7.63510e30 1.63346e35 7.41404e39 −1.85939e42 2.81411e46 8.01842e48 −2.58588e50
1.4 −1.38660e15 4.94875e24 −8.21853e30 −1.47562e36 −6.86195e39 3.27966e43 2.54577e46 1.05749e49 2.04610e51
1.5 2.03451e15 2.78778e24 −6.00197e30 1.50905e36 5.67177e39 3.17629e43 −3.28435e46 −6.14348e48 3.07017e51
1.6 3.09018e15 −9.30646e23 −5.91967e29 −7.61227e35 −2.87587e39 −3.39343e43 −3.31675e46 −1.30491e49 −2.35233e51
1.7 5.46569e15 −5.92176e24 1.97326e31 6.19706e35 −3.23665e40 6.38506e43 5.24235e46 2.11521e49 3.38712e51
1.8 6.07235e15 6.42408e24 2.67322e31 −2.83827e35 3.90093e40 −2.16284e43 1.00747e47 2.73537e49 −1.72350e51
\(n\): 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 \(S_{104}^{\mathrm{new}}(\Gamma_0(1))\).