Properties

Label 1.72.a.a
Level 1
Weight 72
Character orbit 1.a
Self dual Yes
Analytic conductor 31.925
Analytic rank 0
Dimension 6
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 1 \)
Weight: \( k \) = \( 72 \)
Character orbit: \([\chi]\) = 1.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(31.9246160561\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{55}\cdot 3^{20}\cdot 5^{6}\cdot 7^{3} \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \(+(11026222740 - \beta_{1}) q^{2}\) \(+(14982825462961740 + 223315 \beta_{1} + \beta_{2}) q^{3}\) \(+(\)\(13\!\cdots\!28\)\( - 15119266233 \beta_{1} + 6859 \beta_{2} + \beta_{3}) q^{4}\) \(+(-\)\(71\!\cdots\!70\)\( + 9986066292712 \beta_{1} + 19552951 \beta_{2} - 40 \beta_{3} + \beta_{4}) q^{5}\) \(+(-\)\(64\!\cdots\!48\)\( - 36755474236375553 \beta_{1} + 23852632969 \beta_{2} - 37549 \beta_{3} + 569 \beta_{4} + \beta_{5}) q^{6}\) \(+(\)\(56\!\cdots\!00\)\( + 6085020597208821870 \beta_{1} + 704467142494 \beta_{2} + 68764912 \beta_{3} + 104364 \beta_{4} - 144 \beta_{5}) q^{7}\) \(+(\)\(43\!\cdots\!80\)\( - \)\(93\!\cdots\!76\)\( \beta_{1} + 373662647287184 \beta_{2} + 33223483504 \beta_{3} - 85912 \beta_{4} + 10152 \beta_{5}) q^{8}\) \(+(\)\(48\!\cdots\!57\)\( - \)\(30\!\cdots\!44\)\( \beta_{1} + 12565567900962762 \beta_{2} + 3101985626448 \beta_{3} - 549102138 \beta_{4} - 466752 \beta_{5}) q^{9}\) \(+O(q^{10})\) \( q\) \(+(11026222740 - \beta_{1}) q^{2}\) \(+(14982825462961740 + 223315 \beta_{1} + \beta_{2}) q^{3}\) \(+(\)\(13\!\cdots\!28\)\( - 15119266233 \beta_{1} + 6859 \beta_{2} + \beta_{3}) q^{4}\) \(+(-\)\(71\!\cdots\!70\)\( + 9986066292712 \beta_{1} + 19552951 \beta_{2} - 40 \beta_{3} + \beta_{4}) q^{5}\) \(+(-\)\(64\!\cdots\!48\)\( - 36755474236375553 \beta_{1} + 23852632969 \beta_{2} - 37549 \beta_{3} + 569 \beta_{4} + \beta_{5}) q^{6}\) \(+(\)\(56\!\cdots\!00\)\( + 6085020597208821870 \beta_{1} + 704467142494 \beta_{2} + 68764912 \beta_{3} + 104364 \beta_{4} - 144 \beta_{5}) q^{7}\) \(+(\)\(43\!\cdots\!80\)\( - \)\(93\!\cdots\!76\)\( \beta_{1} + 373662647287184 \beta_{2} + 33223483504 \beta_{3} - 85912 \beta_{4} + 10152 \beta_{5}) q^{8}\) \(+(\)\(48\!\cdots\!57\)\( - \)\(30\!\cdots\!44\)\( \beta_{1} + 12565567900962762 \beta_{2} + 3101985626448 \beta_{3} - 549102138 \beta_{4} - 466752 \beta_{5}) q^{9}\) \(+(-\)\(43\!\cdots\!20\)\( + \)\(37\!\cdots\!22\)\( \beta_{1} + 2030985545242499396 \beta_{2} - 28842510271060 \beta_{3} + 32895007236 \beta_{4} + 15727140 \beta_{5}) q^{10}\) \(+(-\)\(12\!\cdots\!88\)\( + \)\(64\!\cdots\!57\)\( \beta_{1} + 48683821575851961539 \beta_{2} - 2448158380737824 \beta_{3} - 1029826965928 \beta_{4} - 413754912 \beta_{5}) q^{11}\) \(+(\)\(90\!\cdots\!80\)\( - \)\(51\!\cdots\!80\)\( \beta_{1} + \)\(31\!\cdots\!60\)\( \beta_{2} + 52097436999376524 \beta_{3} + 21112675878528 \beta_{4} + 8840886912 \beta_{5}) q^{12}\) \(+(\)\(32\!\cdots\!30\)\( + \)\(21\!\cdots\!16\)\( \beta_{1} - \)\(13\!\cdots\!09\)\( \beta_{2} - 35883420918929576 \beta_{3} - 304264002590847 \beta_{4} - 157563209088 \beta_{5}) q^{13}\) \(+(-\)\(21\!\cdots\!44\)\( - \)\(16\!\cdots\!34\)\( \beta_{1} - \)\(39\!\cdots\!18\)\( \beta_{2} - 10523246779171378802 \beta_{3} + 3088202886856490 \beta_{4} + 2386586841210 \beta_{5}) q^{14}\) \(+(\)\(23\!\cdots\!60\)\( - \)\(59\!\cdots\!06\)\( \beta_{1} - \)\(20\!\cdots\!58\)\( \beta_{2} + \)\(14\!\cdots\!80\)\( \beta_{3} - 19837023582415628 \beta_{4} - 31143253089520 \beta_{5}) q^{15}\) \(+(\)\(61\!\cdots\!36\)\( - \)\(76\!\cdots\!00\)\( \beta_{1} + \)\(40\!\cdots\!00\)\( \beta_{2} - \)\(55\!\cdots\!40\)\( \beta_{3} + 28441592013392064 \beta_{4} + 353579854769856 \beta_{5}) q^{16}\) \(+(\)\(53\!\cdots\!70\)\( - \)\(66\!\cdots\!44\)\( \beta_{1} + \)\(11\!\cdots\!70\)\( \beta_{2} - \)\(57\!\cdots\!72\)\( \beta_{3} + 1061975335613560966 \beta_{4} - 3516764067625536 \beta_{5}) q^{17}\) \(+(\)\(16\!\cdots\!40\)\( - \)\(10\!\cdots\!41\)\( \beta_{1} - \)\(18\!\cdots\!16\)\( \beta_{2} + \)\(85\!\cdots\!52\)\( \beta_{3} - 14799504281961238056 \beta_{4} + 30776805316993176 \beta_{5}) q^{18}\) \(+(-\)\(35\!\cdots\!80\)\( - \)\(10\!\cdots\!41\)\( \beta_{1} - \)\(97\!\cdots\!07\)\( \beta_{2} - \)\(39\!\cdots\!08\)\( \beta_{3} + \)\(10\!\cdots\!36\)\( \beta_{4} - 237437426670073056 \beta_{5}) q^{19}\) \(+(-\)\(16\!\cdots\!60\)\( + \)\(25\!\cdots\!86\)\( \beta_{1} + \)\(11\!\cdots\!78\)\( \beta_{2} - \)\(73\!\cdots\!70\)\( \beta_{3} - \)\(45\!\cdots\!72\)\( \beta_{4} + 1613400289468070400 \beta_{5}) q^{20}\) \(+(\)\(14\!\cdots\!12\)\( + \)\(14\!\cdots\!36\)\( \beta_{1} + \)\(95\!\cdots\!72\)\( \beta_{2} + \)\(18\!\cdots\!08\)\( \beta_{3} - 75478716933227776980 \beta_{4} - 9612887800803073920 \beta_{5}) q^{21}\) \(+(-\)\(24\!\cdots\!20\)\( + \)\(45\!\cdots\!61\)\( \beta_{1} - \)\(26\!\cdots\!29\)\( \beta_{2} - \)\(84\!\cdots\!11\)\( \beta_{3} + \)\(17\!\cdots\!83\)\( \beta_{4} + 49716646820528064507 \beta_{5}) q^{22}\) \(+(\)\(16\!\cdots\!20\)\( + \)\(75\!\cdots\!46\)\( \beta_{1} - \)\(52\!\cdots\!86\)\( \beta_{2} - \)\(55\!\cdots\!80\)\( \beta_{3} - \)\(14\!\cdots\!60\)\( \beta_{4} - \)\(21\!\cdots\!40\)\( \beta_{5}) q^{23}\) \(+(\)\(43\!\cdots\!60\)\( - \)\(16\!\cdots\!40\)\( \beta_{1} + \)\(99\!\cdots\!20\)\( \beta_{2} + \)\(18\!\cdots\!20\)\( \beta_{3} + \)\(66\!\cdots\!36\)\( \beta_{4} + \)\(78\!\cdots\!44\)\( \beta_{5}) q^{24}\) \(+(-\)\(32\!\cdots\!25\)\( - \)\(33\!\cdots\!40\)\( \beta_{1} - \)\(15\!\cdots\!20\)\( \beta_{2} - \)\(76\!\cdots\!00\)\( \beta_{3} - \)\(12\!\cdots\!20\)\( \beta_{4} - \)\(20\!\cdots\!00\)\( \beta_{5}) q^{25}\) \(+(-\)\(42\!\cdots\!88\)\( - \)\(22\!\cdots\!54\)\( \beta_{1} - \)\(11\!\cdots\!08\)\( \beta_{2} - \)\(18\!\cdots\!32\)\( \beta_{3} - \)\(64\!\cdots\!88\)\( \beta_{4} + \)\(16\!\cdots\!48\)\( \beta_{5}) q^{26}\) \(+(\)\(85\!\cdots\!20\)\( + \)\(65\!\cdots\!86\)\( \beta_{1} + \)\(37\!\cdots\!46\)\( \beta_{2} + \)\(10\!\cdots\!96\)\( \beta_{3} + \)\(72\!\cdots\!12\)\( \beta_{4} + \)\(17\!\cdots\!48\)\( \beta_{5}) q^{27}\) \(+(\)\(22\!\cdots\!80\)\( + \)\(33\!\cdots\!56\)\( \beta_{1} + \)\(24\!\cdots\!60\)\( \beta_{2} - \)\(25\!\cdots\!84\)\( \beta_{3} - \)\(37\!\cdots\!48\)\( \beta_{4} - \)\(12\!\cdots\!92\)\( \beta_{5}) q^{28}\) \(+(-\)\(21\!\cdots\!70\)\( - \)\(21\!\cdots\!36\)\( \beta_{1} + \)\(22\!\cdots\!03\)\( \beta_{2} - \)\(97\!\cdots\!28\)\( \beta_{3} + \)\(11\!\cdots\!77\)\( \beta_{4} + \)\(49\!\cdots\!08\)\( \beta_{5}) q^{29}\) \(+(\)\(23\!\cdots\!60\)\( - \)\(45\!\cdots\!26\)\( \beta_{1} - \)\(14\!\cdots\!98\)\( \beta_{2} + \)\(76\!\cdots\!70\)\( \beta_{3} - \)\(20\!\cdots\!98\)\( \beta_{4} - \)\(11\!\cdots\!50\)\( \beta_{5}) q^{30}\) \(+(\)\(34\!\cdots\!32\)\( - \)\(42\!\cdots\!84\)\( \beta_{1} + \)\(91\!\cdots\!32\)\( \beta_{2} - \)\(17\!\cdots\!12\)\( \beta_{3} - \)\(18\!\cdots\!64\)\( \beta_{4} + \)\(34\!\cdots\!44\)\( \beta_{5}) q^{31}\) \(+(\)\(18\!\cdots\!40\)\( + \)\(12\!\cdots\!80\)\( \beta_{1} + \)\(20\!\cdots\!52\)\( \beta_{2} + \)\(45\!\cdots\!68\)\( \beta_{3} + \)\(24\!\cdots\!96\)\( \beta_{4} + \)\(10\!\cdots\!84\)\( \beta_{5}) q^{32}\) \(+(\)\(61\!\cdots\!80\)\( + \)\(10\!\cdots\!96\)\( \beta_{1} - \)\(43\!\cdots\!06\)\( \beta_{2} + \)\(31\!\cdots\!88\)\( \beta_{3} - \)\(80\!\cdots\!14\)\( \beta_{4} - \)\(52\!\cdots\!56\)\( \beta_{5}) q^{33}\) \(+(\)\(24\!\cdots\!16\)\( - \)\(72\!\cdots\!70\)\( \beta_{1} - \)\(46\!\cdots\!40\)\( \beta_{2} + \)\(48\!\cdots\!80\)\( \beta_{3} + \)\(12\!\cdots\!56\)\( \beta_{4} + \)\(13\!\cdots\!24\)\( \beta_{5}) q^{34}\) \(+(\)\(38\!\cdots\!80\)\( + \)\(48\!\cdots\!32\)\( \beta_{1} - \)\(30\!\cdots\!24\)\( \beta_{2} - \)\(25\!\cdots\!60\)\( \beta_{3} + \)\(24\!\cdots\!16\)\( \beta_{4} - \)\(95\!\cdots\!60\)\( \beta_{5}) q^{35}\) \(+(\)\(28\!\cdots\!96\)\( - \)\(23\!\cdots\!45\)\( \beta_{1} + \)\(96\!\cdots\!35\)\( \beta_{2} + \)\(53\!\cdots\!25\)\( \beta_{3} - \)\(54\!\cdots\!96\)\( \beta_{4} - \)\(76\!\cdots\!84\)\( \beta_{5}) q^{36}\) \(+(\)\(34\!\cdots\!10\)\( - \)\(31\!\cdots\!28\)\( \beta_{1} + \)\(20\!\cdots\!35\)\( \beta_{2} - \)\(22\!\cdots\!48\)\( \beta_{3} + \)\(94\!\cdots\!69\)\( \beta_{4} + \)\(38\!\cdots\!76\)\( \beta_{5}) q^{37}\) \(+(\)\(35\!\cdots\!20\)\( + \)\(12\!\cdots\!95\)\( \beta_{1} - \)\(92\!\cdots\!91\)\( \beta_{2} - \)\(54\!\cdots\!77\)\( \beta_{3} - \)\(43\!\cdots\!19\)\( \beta_{4} - \)\(83\!\cdots\!51\)\( \beta_{5}) q^{38}\) \(+(-\)\(99\!\cdots\!76\)\( + \)\(37\!\cdots\!58\)\( \beta_{1} + \)\(75\!\cdots\!66\)\( \beta_{2} - \)\(25\!\cdots\!96\)\( \beta_{3} + \)\(38\!\cdots\!12\)\( \beta_{4} + \)\(19\!\cdots\!48\)\( \beta_{5}) q^{39}\) \(+(\)\(11\!\cdots\!00\)\( - \)\(19\!\cdots\!40\)\( \beta_{1} + \)\(31\!\cdots\!80\)\( \beta_{2} + \)\(10\!\cdots\!00\)\( \beta_{3} - \)\(28\!\cdots\!20\)\( \beta_{4} + \)\(53\!\cdots\!00\)\( \beta_{5}) q^{40}\) \(+(-\)\(48\!\cdots\!58\)\( - \)\(75\!\cdots\!84\)\( \beta_{1} + \)\(49\!\cdots\!32\)\( \beta_{2} + \)\(15\!\cdots\!88\)\( \beta_{3} + \)\(70\!\cdots\!36\)\( \beta_{4} - \)\(19\!\cdots\!56\)\( \beta_{5}) q^{41}\) \(+(-\)\(51\!\cdots\!80\)\( - \)\(41\!\cdots\!40\)\( \beta_{1} - \)\(42\!\cdots\!08\)\( \beta_{2} - \)\(13\!\cdots\!68\)\( \beta_{3} + \)\(27\!\cdots\!04\)\( \beta_{4} + \)\(27\!\cdots\!16\)\( \beta_{5}) q^{42}\) \(+(-\)\(33\!\cdots\!00\)\( - \)\(59\!\cdots\!23\)\( \beta_{1} + \)\(38\!\cdots\!27\)\( \beta_{2} + \)\(24\!\cdots\!40\)\( \beta_{3} - \)\(59\!\cdots\!20\)\( \beta_{4} + \)\(33\!\cdots\!20\)\( \beta_{5}) q^{43}\) \(+(-\)\(16\!\cdots\!64\)\( + \)\(31\!\cdots\!40\)\( \beta_{1} - \)\(72\!\cdots\!20\)\( \beta_{2} + \)\(12\!\cdots\!60\)\( \beta_{3} + \)\(13\!\cdots\!56\)\( \beta_{4} - \)\(26\!\cdots\!76\)\( \beta_{5}) q^{44}\) \(+(-\)\(20\!\cdots\!90\)\( + \)\(33\!\cdots\!84\)\( \beta_{1} + \)\(44\!\cdots\!07\)\( \beta_{2} - \)\(24\!\cdots\!80\)\( \beta_{3} - \)\(25\!\cdots\!43\)\( \beta_{4} + \)\(58\!\cdots\!00\)\( \beta_{5}) q^{45}\) \(+(-\)\(25\!\cdots\!28\)\( - \)\(14\!\cdots\!74\)\( \beta_{1} - \)\(12\!\cdots\!98\)\( \beta_{2} - \)\(10\!\cdots\!82\)\( \beta_{3} - \)\(50\!\cdots\!94\)\( \beta_{4} - \)\(19\!\cdots\!26\)\( \beta_{5}) q^{46}\) \(+(\)\(13\!\cdots\!80\)\( - \)\(10\!\cdots\!08\)\( \beta_{1} + \)\(41\!\cdots\!04\)\( \beta_{2} + \)\(15\!\cdots\!80\)\( \beta_{3} + \)\(73\!\cdots\!60\)\( \beta_{4} - \)\(24\!\cdots\!60\)\( \beta_{5}) q^{47}\) \(+(\)\(43\!\cdots\!20\)\( - \)\(91\!\cdots\!68\)\( \beta_{1} + \)\(37\!\cdots\!56\)\( \beta_{2} + \)\(96\!\cdots\!08\)\( \beta_{3} + \)\(10\!\cdots\!76\)\( \beta_{4} + \)\(72\!\cdots\!04\)\( \beta_{5}) q^{48}\) \(+(\)\(46\!\cdots\!93\)\( + \)\(92\!\cdots\!96\)\( \beta_{1} + \)\(61\!\cdots\!92\)\( \beta_{2} - \)\(29\!\cdots\!72\)\( \beta_{3} - \)\(25\!\cdots\!04\)\( \beta_{4} - \)\(66\!\cdots\!16\)\( \beta_{5}) q^{49}\) \(+(\)\(11\!\cdots\!00\)\( + \)\(18\!\cdots\!85\)\( \beta_{1} - \)\(12\!\cdots\!20\)\( \beta_{2} + \)\(15\!\cdots\!00\)\( \beta_{3} - \)\(77\!\cdots\!20\)\( \beta_{4} - \)\(15\!\cdots\!00\)\( \beta_{5}) q^{50}\) \(+(\)\(86\!\cdots\!32\)\( + \)\(79\!\cdots\!86\)\( \beta_{1} + \)\(66\!\cdots\!22\)\( \beta_{2} + \)\(34\!\cdots\!68\)\( \beta_{3} + \)\(31\!\cdots\!84\)\( \beta_{4} + \)\(61\!\cdots\!36\)\( \beta_{5}) q^{51}\) \(+(-\)\(24\!\cdots\!00\)\( + \)\(24\!\cdots\!66\)\( \beta_{1} - \)\(28\!\cdots\!54\)\( \beta_{2} + \)\(27\!\cdots\!30\)\( \beta_{3} - \)\(73\!\cdots\!40\)\( \beta_{4} - \)\(76\!\cdots\!60\)\( \beta_{5}) q^{52}\) \(+(-\)\(42\!\cdots\!10\)\( - \)\(78\!\cdots\!92\)\( \beta_{1} + \)\(11\!\cdots\!27\)\( \beta_{2} - \)\(11\!\cdots\!96\)\( \beta_{3} - \)\(16\!\cdots\!87\)\( \beta_{4} - \)\(59\!\cdots\!48\)\( \beta_{5}) q^{53}\) \(+(-\)\(22\!\cdots\!80\)\( - \)\(32\!\cdots\!26\)\( \beta_{1} + \)\(24\!\cdots\!98\)\( \beta_{2} - \)\(53\!\cdots\!38\)\( \beta_{3} + \)\(32\!\cdots\!06\)\( \beta_{4} + \)\(35\!\cdots\!74\)\( \beta_{5}) q^{54}\) \(+(-\)\(19\!\cdots\!40\)\( - \)\(14\!\cdots\!06\)\( \beta_{1} - \)\(28\!\cdots\!38\)\( \beta_{2} + \)\(17\!\cdots\!20\)\( \beta_{3} + \)\(11\!\cdots\!12\)\( \beta_{4} - \)\(47\!\cdots\!00\)\( \beta_{5}) q^{55}\) \(+(-\)\(66\!\cdots\!20\)\( + \)\(68\!\cdots\!24\)\( \beta_{1} - \)\(34\!\cdots\!52\)\( \beta_{2} - \)\(70\!\cdots\!48\)\( \beta_{3} - \)\(12\!\cdots\!08\)\( \beta_{4} - \)\(10\!\cdots\!32\)\( \beta_{5}) q^{56}\) \(+(-\)\(13\!\cdots\!60\)\( + \)\(22\!\cdots\!08\)\( \beta_{1} - \)\(75\!\cdots\!86\)\( \beta_{2} - \)\(25\!\cdots\!60\)\( \beta_{3} + \)\(89\!\cdots\!30\)\( \beta_{4} + \)\(11\!\cdots\!20\)\( \beta_{5}) q^{57}\) \(+(\)\(74\!\cdots\!80\)\( + \)\(17\!\cdots\!62\)\( \beta_{1} + \)\(38\!\cdots\!40\)\( \beta_{2} - \)\(87\!\cdots\!52\)\( \beta_{3} + \)\(28\!\cdots\!56\)\( \beta_{4} - \)\(11\!\cdots\!76\)\( \beta_{5}) q^{58}\) \(+(\)\(42\!\cdots\!60\)\( - \)\(37\!\cdots\!91\)\( \beta_{1} - \)\(10\!\cdots\!57\)\( \beta_{2} + \)\(58\!\cdots\!52\)\( \beta_{3} - \)\(37\!\cdots\!20\)\( \beta_{4} + \)\(28\!\cdots\!20\)\( \beta_{5}) q^{59}\) \(+(\)\(13\!\cdots\!80\)\( - \)\(21\!\cdots\!68\)\( \beta_{1} - \)\(25\!\cdots\!24\)\( \beta_{2} + \)\(23\!\cdots\!40\)\( \beta_{3} - \)\(10\!\cdots\!84\)\( \beta_{4} - \)\(26\!\cdots\!60\)\( \beta_{5}) q^{60}\) \(+(\)\(79\!\cdots\!62\)\( + \)\(76\!\cdots\!00\)\( \beta_{1} - \)\(90\!\cdots\!25\)\( \beta_{2} - \)\(68\!\cdots\!00\)\( \beta_{3} + \)\(27\!\cdots\!25\)\( \beta_{4} + \)\(83\!\cdots\!00\)\( \beta_{5}) q^{61}\) \(+(\)\(19\!\cdots\!80\)\( - \)\(78\!\cdots\!08\)\( \beta_{1} + \)\(18\!\cdots\!48\)\( \beta_{2} + \)\(18\!\cdots\!32\)\( \beta_{3} + \)\(82\!\cdots\!04\)\( \beta_{4} + \)\(35\!\cdots\!16\)\( \beta_{5}) q^{62}\) \(+(\)\(21\!\cdots\!60\)\( + \)\(13\!\cdots\!50\)\( \beta_{1} + \)\(28\!\cdots\!50\)\( \beta_{2} + \)\(72\!\cdots\!68\)\( \beta_{3} - \)\(11\!\cdots\!04\)\( \beta_{4} - \)\(57\!\cdots\!16\)\( \beta_{5}) q^{63}\) \(+(-\)\(41\!\cdots\!32\)\( - \)\(47\!\cdots\!04\)\( \beta_{1} + \)\(24\!\cdots\!92\)\( \beta_{2} + \)\(17\!\cdots\!68\)\( \beta_{3} + \)\(15\!\cdots\!72\)\( \beta_{4} + \)\(95\!\cdots\!88\)\( \beta_{5}) q^{64}\) \(+(-\)\(10\!\cdots\!40\)\( + \)\(10\!\cdots\!64\)\( \beta_{1} + \)\(44\!\cdots\!52\)\( \beta_{2} + \)\(13\!\cdots\!80\)\( \beta_{3} + \)\(60\!\cdots\!32\)\( \beta_{4} + \)\(37\!\cdots\!80\)\( \beta_{5}) q^{65}\) \(+(-\)\(31\!\cdots\!76\)\( - \)\(47\!\cdots\!12\)\( \beta_{1} - \)\(50\!\cdots\!24\)\( \beta_{2} - \)\(10\!\cdots\!56\)\( \beta_{3} - \)\(39\!\cdots\!68\)\( \beta_{4} - \)\(33\!\cdots\!72\)\( \beta_{5}) q^{66}\) \(+(-\)\(24\!\cdots\!80\)\( - \)\(58\!\cdots\!09\)\( \beta_{1} + \)\(43\!\cdots\!85\)\( \beta_{2} + \)\(89\!\cdots\!48\)\( \beta_{3} + \)\(52\!\cdots\!56\)\( \beta_{4} + \)\(33\!\cdots\!24\)\( \beta_{5}) q^{67}\) \(+(\)\(17\!\cdots\!40\)\( - \)\(16\!\cdots\!22\)\( \beta_{1} + \)\(37\!\cdots\!86\)\( \beta_{2} + \)\(22\!\cdots\!06\)\( \beta_{3} - \)\(50\!\cdots\!68\)\( \beta_{4} + \)\(30\!\cdots\!28\)\( \beta_{5}) q^{68}\) \(+(-\)\(54\!\cdots\!56\)\( + \)\(40\!\cdots\!12\)\( \beta_{1} + \)\(42\!\cdots\!24\)\( \beta_{2} - \)\(20\!\cdots\!04\)\( \beta_{3} + \)\(34\!\cdots\!04\)\( \beta_{4} - \)\(58\!\cdots\!84\)\( \beta_{5}) q^{69}\) \(+(\)\(24\!\cdots\!80\)\( + \)\(56\!\cdots\!72\)\( \beta_{1} - \)\(60\!\cdots\!44\)\( \beta_{2} - \)\(50\!\cdots\!40\)\( \beta_{3} + \)\(11\!\cdots\!56\)\( \beta_{4} - \)\(76\!\cdots\!00\)\( \beta_{5}) q^{70}\) \(+(\)\(20\!\cdots\!72\)\( + \)\(50\!\cdots\!50\)\( \beta_{1} - \)\(41\!\cdots\!50\)\( \beta_{2} + \)\(31\!\cdots\!00\)\( \beta_{3} - \)\(42\!\cdots\!00\)\( \beta_{4} + \)\(95\!\cdots\!00\)\( \beta_{5}) q^{71}\) \(+(\)\(79\!\cdots\!60\)\( - \)\(15\!\cdots\!20\)\( \beta_{1} + \)\(51\!\cdots\!52\)\( \beta_{2} + \)\(13\!\cdots\!24\)\( \beta_{3} + \)\(60\!\cdots\!28\)\( \beta_{4} + \)\(53\!\cdots\!12\)\( \beta_{5}) q^{72}\) \(+(\)\(39\!\cdots\!70\)\( + \)\(77\!\cdots\!72\)\( \beta_{1} - \)\(38\!\cdots\!94\)\( \beta_{2} - \)\(19\!\cdots\!32\)\( \beta_{3} - \)\(46\!\cdots\!54\)\( \beta_{4} - \)\(11\!\cdots\!16\)\( \beta_{5}) q^{73}\) \(+(\)\(15\!\cdots\!36\)\( - \)\(31\!\cdots\!26\)\( \beta_{1} + \)\(17\!\cdots\!48\)\( \beta_{2} + \)\(34\!\cdots\!32\)\( \beta_{3} + \)\(11\!\cdots\!24\)\( \beta_{4} - \)\(57\!\cdots\!04\)\( \beta_{5}) q^{74}\) \(+(-\)\(22\!\cdots\!00\)\( + \)\(24\!\cdots\!45\)\( \beta_{1} - \)\(78\!\cdots\!65\)\( \beta_{2} - \)\(78\!\cdots\!00\)\( \beta_{3} + \)\(19\!\cdots\!60\)\( \beta_{4} + \)\(40\!\cdots\!00\)\( \beta_{5}) q^{75}\) \(+(-\)\(30\!\cdots\!40\)\( + \)\(25\!\cdots\!68\)\( \beta_{1} - \)\(41\!\cdots\!64\)\( \beta_{2} - \)\(73\!\cdots\!36\)\( \beta_{3} - \)\(73\!\cdots\!16\)\( \beta_{4} - \)\(27\!\cdots\!64\)\( \beta_{5}) q^{76}\) \(+(-\)\(22\!\cdots\!00\)\( + \)\(63\!\cdots\!48\)\( \beta_{1} + \)\(20\!\cdots\!44\)\( \beta_{2} + \)\(49\!\cdots\!88\)\( \beta_{3} + \)\(89\!\cdots\!36\)\( \beta_{4} - \)\(59\!\cdots\!56\)\( \beta_{5}) q^{77}\) \(+(-\)\(13\!\cdots\!00\)\( + \)\(52\!\cdots\!66\)\( \beta_{1} + \)\(40\!\cdots\!38\)\( \beta_{2} - \)\(41\!\cdots\!74\)\( \beta_{3} + \)\(60\!\cdots\!22\)\( \beta_{4} + \)\(62\!\cdots\!38\)\( \beta_{5}) q^{78}\) \(+(-\)\(10\!\cdots\!20\)\( - \)\(89\!\cdots\!72\)\( \beta_{1} + \)\(91\!\cdots\!56\)\( \beta_{2} - \)\(48\!\cdots\!76\)\( \beta_{3} - \)\(25\!\cdots\!24\)\( \beta_{4} + \)\(12\!\cdots\!04\)\( \beta_{5}) q^{79}\) \(+(\)\(74\!\cdots\!80\)\( - \)\(32\!\cdots\!68\)\( \beta_{1} - \)\(77\!\cdots\!64\)\( \beta_{2} + \)\(78\!\cdots\!60\)\( \beta_{3} + \)\(15\!\cdots\!36\)\( \beta_{4} - \)\(92\!\cdots\!00\)\( \beta_{5}) q^{80}\) \(+(\)\(15\!\cdots\!61\)\( - \)\(25\!\cdots\!32\)\( \beta_{1} - \)\(27\!\cdots\!14\)\( \beta_{2} - \)\(26\!\cdots\!96\)\( \beta_{3} - \)\(37\!\cdots\!30\)\( \beta_{4} - \)\(55\!\cdots\!20\)\( \beta_{5}) q^{81}\) \(+(\)\(21\!\cdots\!80\)\( + \)\(74\!\cdots\!82\)\( \beta_{1} - \)\(42\!\cdots\!52\)\( \beta_{2} + \)\(49\!\cdots\!32\)\( \beta_{3} + \)\(46\!\cdots\!04\)\( \beta_{4} + \)\(76\!\cdots\!16\)\( \beta_{5}) q^{82}\) \(+(\)\(79\!\cdots\!60\)\( - \)\(44\!\cdots\!21\)\( \beta_{1} + \)\(21\!\cdots\!33\)\( \beta_{2} + \)\(63\!\cdots\!00\)\( \beta_{3} - \)\(34\!\cdots\!00\)\( \beta_{4} + \)\(10\!\cdots\!00\)\( \beta_{5}) q^{83}\) \(+(\)\(59\!\cdots\!36\)\( + \)\(47\!\cdots\!84\)\( \beta_{1} - \)\(48\!\cdots\!32\)\( \beta_{2} - \)\(26\!\cdots\!68\)\( \beta_{3} - \)\(19\!\cdots\!68\)\( \beta_{4} - \)\(27\!\cdots\!72\)\( \beta_{5}) q^{84}\) \(+(\)\(35\!\cdots\!80\)\( - \)\(24\!\cdots\!48\)\( \beta_{1} + \)\(96\!\cdots\!86\)\( \beta_{2} - \)\(48\!\cdots\!60\)\( \beta_{3} + \)\(33\!\cdots\!26\)\( \beta_{4} - \)\(18\!\cdots\!60\)\( \beta_{5}) q^{85}\) \(+(\)\(17\!\cdots\!92\)\( - \)\(18\!\cdots\!95\)\( \beta_{1} + \)\(19\!\cdots\!35\)\( \beta_{2} + \)\(12\!\cdots\!85\)\( \beta_{3} - \)\(20\!\cdots\!57\)\( \beta_{4} + \)\(44\!\cdots\!47\)\( \beta_{5}) q^{86}\) \(+(\)\(62\!\cdots\!60\)\( - \)\(13\!\cdots\!94\)\( \beta_{1} - \)\(33\!\cdots\!66\)\( \beta_{2} + \)\(59\!\cdots\!52\)\( \beta_{3} - \)\(20\!\cdots\!56\)\( \beta_{4} + \)\(10\!\cdots\!76\)\( \beta_{5}) q^{87}\) \(+(-\)\(74\!\cdots\!40\)\( + \)\(67\!\cdots\!88\)\( \beta_{1} - \)\(38\!\cdots\!92\)\( \beta_{2} - \)\(21\!\cdots\!52\)\( \beta_{3} - \)\(23\!\cdots\!44\)\( \beta_{4} - \)\(95\!\cdots\!76\)\( \beta_{5}) q^{88}\) \(+(-\)\(79\!\cdots\!10\)\( - \)\(78\!\cdots\!48\)\( \beta_{1} - \)\(86\!\cdots\!46\)\( \beta_{2} - \)\(43\!\cdots\!04\)\( \beta_{3} - \)\(79\!\cdots\!34\)\( \beta_{4} - \)\(67\!\cdots\!36\)\( \beta_{5}) q^{89}\) \(+(-\)\(14\!\cdots\!40\)\( + \)\(16\!\cdots\!54\)\( \beta_{1} + \)\(30\!\cdots\!72\)\( \beta_{2} - \)\(14\!\cdots\!20\)\( \beta_{3} + \)\(22\!\cdots\!52\)\( \beta_{4} + \)\(35\!\cdots\!80\)\( \beta_{5}) q^{90}\) \(+(-\)\(13\!\cdots\!28\)\( + \)\(13\!\cdots\!04\)\( \beta_{1} + \)\(57\!\cdots\!08\)\( \beta_{2} + \)\(86\!\cdots\!92\)\( \beta_{3} + \)\(20\!\cdots\!12\)\( \beta_{4} - \)\(74\!\cdots\!52\)\( \beta_{5}) q^{91}\) \(+(-\)\(65\!\cdots\!80\)\( + \)\(54\!\cdots\!04\)\( \beta_{1} - \)\(35\!\cdots\!84\)\( \beta_{2} - \)\(33\!\cdots\!08\)\( \beta_{3} - \)\(41\!\cdots\!76\)\( \beta_{4} - \)\(71\!\cdots\!04\)\( \beta_{5}) q^{92}\) \(+(\)\(12\!\cdots\!80\)\( - \)\(84\!\cdots\!12\)\( \beta_{1} + \)\(13\!\cdots\!48\)\( \beta_{2} - \)\(45\!\cdots\!56\)\( \beta_{3} - \)\(70\!\cdots\!32\)\( \beta_{4} + \)\(51\!\cdots\!72\)\( \beta_{5}) q^{93}\) \(+(\)\(51\!\cdots\!96\)\( - \)\(17\!\cdots\!36\)\( \beta_{1} - \)\(60\!\cdots\!72\)\( \beta_{2} + \)\(61\!\cdots\!32\)\( \beta_{3} + \)\(45\!\cdots\!36\)\( \beta_{4} + \)\(75\!\cdots\!44\)\( \beta_{5}) q^{94}\) \(+(\)\(13\!\cdots\!00\)\( + \)\(62\!\cdots\!90\)\( \beta_{1} + \)\(19\!\cdots\!70\)\( \beta_{2} - \)\(50\!\cdots\!00\)\( \beta_{3} + \)\(13\!\cdots\!20\)\( \beta_{4} - \)\(23\!\cdots\!00\)\( \beta_{5}) q^{95}\) \(+(\)\(27\!\cdots\!92\)\( - \)\(25\!\cdots\!00\)\( \beta_{1} + \)\(13\!\cdots\!00\)\( \beta_{2} + \)\(81\!\cdots\!80\)\( \beta_{3} - \)\(18\!\cdots\!28\)\( \beta_{4} - \)\(21\!\cdots\!12\)\( \beta_{5}) q^{96}\) \(+(\)\(15\!\cdots\!30\)\( + \)\(24\!\cdots\!24\)\( \beta_{1} + \)\(64\!\cdots\!38\)\( \beta_{2} + \)\(46\!\cdots\!96\)\( \beta_{3} - \)\(31\!\cdots\!38\)\( \beta_{4} + \)\(31\!\cdots\!48\)\( \beta_{5}) q^{97}\) \(+(-\)\(28\!\cdots\!80\)\( - \)\(53\!\cdots\!89\)\( \beta_{1} - \)\(22\!\cdots\!32\)\( \beta_{2} - \)\(14\!\cdots\!08\)\( \beta_{3} + \)\(44\!\cdots\!24\)\( \beta_{4} + \)\(61\!\cdots\!96\)\( \beta_{5}) q^{98}\) \(+(-\)\(24\!\cdots\!16\)\( + \)\(13\!\cdots\!01\)\( \beta_{1} + \)\(25\!\cdots\!27\)\( \beta_{2} + \)\(45\!\cdots\!48\)\( \beta_{3} + \)\(51\!\cdots\!28\)\( \beta_{4} - \)\(56\!\cdots\!88\)\( \beta_{5}) q^{99}\) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(6q \) \(\mathstrut +\mathstrut 66157336440q^{2} \) \(\mathstrut +\mathstrut 89896952777770440q^{3} \) \(\mathstrut +\mathstrut \)\(82\!\cdots\!68\)\(q^{4} \) \(\mathstrut -\mathstrut \)\(42\!\cdots\!20\)\(q^{5} \) \(\mathstrut -\mathstrut \)\(38\!\cdots\!88\)\(q^{6} \) \(\mathstrut +\mathstrut \)\(33\!\cdots\!00\)\(q^{7} \) \(\mathstrut +\mathstrut \)\(26\!\cdots\!80\)\(q^{8} \) \(\mathstrut +\mathstrut \)\(28\!\cdots\!42\)\(q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(6q \) \(\mathstrut +\mathstrut 66157336440q^{2} \) \(\mathstrut +\mathstrut 89896952777770440q^{3} \) \(\mathstrut +\mathstrut \)\(82\!\cdots\!68\)\(q^{4} \) \(\mathstrut -\mathstrut \)\(42\!\cdots\!20\)\(q^{5} \) \(\mathstrut -\mathstrut \)\(38\!\cdots\!88\)\(q^{6} \) \(\mathstrut +\mathstrut \)\(33\!\cdots\!00\)\(q^{7} \) \(\mathstrut +\mathstrut \)\(26\!\cdots\!80\)\(q^{8} \) \(\mathstrut +\mathstrut \)\(28\!\cdots\!42\)\(q^{9} \) \(\mathstrut -\mathstrut \)\(26\!\cdots\!20\)\(q^{10} \) \(\mathstrut -\mathstrut \)\(72\!\cdots\!28\)\(q^{11} \) \(\mathstrut +\mathstrut \)\(54\!\cdots\!80\)\(q^{12} \) \(\mathstrut +\mathstrut \)\(19\!\cdots\!80\)\(q^{13} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!64\)\(q^{14} \) \(\mathstrut +\mathstrut \)\(13\!\cdots\!60\)\(q^{15} \) \(\mathstrut +\mathstrut \)\(36\!\cdots\!16\)\(q^{16} \) \(\mathstrut +\mathstrut \)\(31\!\cdots\!20\)\(q^{17} \) \(\mathstrut +\mathstrut \)\(97\!\cdots\!40\)\(q^{18} \) \(\mathstrut -\mathstrut \)\(21\!\cdots\!80\)\(q^{19} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!60\)\(q^{20} \) \(\mathstrut +\mathstrut \)\(84\!\cdots\!72\)\(q^{21} \) \(\mathstrut -\mathstrut \)\(14\!\cdots\!20\)\(q^{22} \) \(\mathstrut +\mathstrut \)\(96\!\cdots\!20\)\(q^{23} \) \(\mathstrut +\mathstrut \)\(26\!\cdots\!60\)\(q^{24} \) \(\mathstrut -\mathstrut \)\(19\!\cdots\!50\)\(q^{25} \) \(\mathstrut -\mathstrut \)\(25\!\cdots\!28\)\(q^{26} \) \(\mathstrut +\mathstrut \)\(51\!\cdots\!20\)\(q^{27} \) \(\mathstrut +\mathstrut \)\(13\!\cdots\!80\)\(q^{28} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!20\)\(q^{29} \) \(\mathstrut +\mathstrut \)\(14\!\cdots\!60\)\(q^{30} \) \(\mathstrut +\mathstrut \)\(20\!\cdots\!92\)\(q^{31} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!40\)\(q^{32} \) \(\mathstrut +\mathstrut \)\(36\!\cdots\!80\)\(q^{33} \) \(\mathstrut +\mathstrut \)\(14\!\cdots\!96\)\(q^{34} \) \(\mathstrut +\mathstrut \)\(23\!\cdots\!80\)\(q^{35} \) \(\mathstrut +\mathstrut \)\(17\!\cdots\!76\)\(q^{36} \) \(\mathstrut +\mathstrut \)\(20\!\cdots\!60\)\(q^{37} \) \(\mathstrut +\mathstrut \)\(21\!\cdots\!20\)\(q^{38} \) \(\mathstrut -\mathstrut \)\(59\!\cdots\!56\)\(q^{39} \) \(\mathstrut +\mathstrut \)\(68\!\cdots\!00\)\(q^{40} \) \(\mathstrut -\mathstrut \)\(29\!\cdots\!48\)\(q^{41} \) \(\mathstrut -\mathstrut \)\(30\!\cdots\!80\)\(q^{42} \) \(\mathstrut -\mathstrut \)\(20\!\cdots\!00\)\(q^{43} \) \(\mathstrut -\mathstrut \)\(97\!\cdots\!84\)\(q^{44} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!40\)\(q^{45} \) \(\mathstrut -\mathstrut \)\(15\!\cdots\!68\)\(q^{46} \) \(\mathstrut +\mathstrut \)\(80\!\cdots\!80\)\(q^{47} \) \(\mathstrut +\mathstrut \)\(26\!\cdots\!20\)\(q^{48} \) \(\mathstrut +\mathstrut \)\(27\!\cdots\!58\)\(q^{49} \) \(\mathstrut +\mathstrut \)\(70\!\cdots\!00\)\(q^{50} \) \(\mathstrut +\mathstrut \)\(52\!\cdots\!92\)\(q^{51} \) \(\mathstrut -\mathstrut \)\(14\!\cdots\!00\)\(q^{52} \) \(\mathstrut -\mathstrut \)\(25\!\cdots\!60\)\(q^{53} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!80\)\(q^{54} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!40\)\(q^{55} \) \(\mathstrut -\mathstrut \)\(39\!\cdots\!20\)\(q^{56} \) \(\mathstrut -\mathstrut \)\(78\!\cdots\!60\)\(q^{57} \) \(\mathstrut +\mathstrut \)\(44\!\cdots\!80\)\(q^{58} \) \(\mathstrut +\mathstrut \)\(25\!\cdots\!60\)\(q^{59} \) \(\mathstrut +\mathstrut \)\(81\!\cdots\!80\)\(q^{60} \) \(\mathstrut +\mathstrut \)\(47\!\cdots\!72\)\(q^{61} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!80\)\(q^{62} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!60\)\(q^{63} \) \(\mathstrut -\mathstrut \)\(24\!\cdots\!92\)\(q^{64} \) \(\mathstrut -\mathstrut \)\(64\!\cdots\!40\)\(q^{65} \) \(\mathstrut -\mathstrut \)\(19\!\cdots\!56\)\(q^{66} \) \(\mathstrut -\mathstrut \)\(14\!\cdots\!80\)\(q^{67} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!40\)\(q^{68} \) \(\mathstrut -\mathstrut \)\(32\!\cdots\!36\)\(q^{69} \) \(\mathstrut +\mathstrut \)\(14\!\cdots\!80\)\(q^{70} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!32\)\(q^{71} \) \(\mathstrut +\mathstrut \)\(47\!\cdots\!60\)\(q^{72} \) \(\mathstrut +\mathstrut \)\(23\!\cdots\!20\)\(q^{73} \) \(\mathstrut +\mathstrut \)\(90\!\cdots\!16\)\(q^{74} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!00\)\(q^{75} \) \(\mathstrut -\mathstrut \)\(18\!\cdots\!40\)\(q^{76} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!00\)\(q^{77} \) \(\mathstrut -\mathstrut \)\(82\!\cdots\!00\)\(q^{78} \) \(\mathstrut -\mathstrut \)\(60\!\cdots\!20\)\(q^{79} \) \(\mathstrut +\mathstrut \)\(44\!\cdots\!80\)\(q^{80} \) \(\mathstrut +\mathstrut \)\(90\!\cdots\!66\)\(q^{81} \) \(\mathstrut +\mathstrut \)\(13\!\cdots\!80\)\(q^{82} \) \(\mathstrut +\mathstrut \)\(47\!\cdots\!60\)\(q^{83} \) \(\mathstrut +\mathstrut \)\(35\!\cdots\!16\)\(q^{84} \) \(\mathstrut +\mathstrut \)\(21\!\cdots\!80\)\(q^{85} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!52\)\(q^{86} \) \(\mathstrut +\mathstrut \)\(37\!\cdots\!60\)\(q^{87} \) \(\mathstrut -\mathstrut \)\(44\!\cdots\!40\)\(q^{88} \) \(\mathstrut -\mathstrut \)\(47\!\cdots\!60\)\(q^{89} \) \(\mathstrut -\mathstrut \)\(86\!\cdots\!40\)\(q^{90} \) \(\mathstrut -\mathstrut \)\(78\!\cdots\!68\)\(q^{91} \) \(\mathstrut -\mathstrut \)\(39\!\cdots\!80\)\(q^{92} \) \(\mathstrut +\mathstrut \)\(76\!\cdots\!80\)\(q^{93} \) \(\mathstrut +\mathstrut \)\(31\!\cdots\!76\)\(q^{94} \) \(\mathstrut +\mathstrut \)\(83\!\cdots\!00\)\(q^{95} \) \(\mathstrut +\mathstrut \)\(16\!\cdots\!52\)\(q^{96} \) \(\mathstrut +\mathstrut \)\(90\!\cdots\!80\)\(q^{97} \) \(\mathstrut -\mathstrut \)\(16\!\cdots\!80\)\(q^{98} \) \(\mathstrut -\mathstrut \)\(14\!\cdots\!96\)\(q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6}\mathstrut -\mathstrut \) \(3\) \(x^{5}\mathstrut -\mathstrut \) \(18795872470415627868\) \(x^{4}\mathstrut -\mathstrut \) \(3619866001951210071877967584\) \(x^{3}\mathstrut +\mathstrut \) \(86589594972295987044342709087705454976\) \(x^{2}\mathstrut +\mathstrut \) \(35564509575256206574563332087691180650482000128\) \(x\mathstrut -\mathstrut \) \(11608057594368249332625196663730471558406093241461869568\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 24 \nu - 12 \)
\(\beta_{2}\)\(=\)\((\)\(1725241141\) \(\nu^{5}\mathstrut +\mathstrut \) \(52370219295084693165\) \(\nu^{4}\mathstrut -\mathstrut \) \(49498807386833879951613619104\) \(\nu^{3}\mathstrut -\mathstrut \) \(528775848438262569600754436589454711264\) \(\nu^{2}\mathstrut +\mathstrut \) \(234519987467691230702307692269422720705140245248\) \(\nu\mathstrut +\mathstrut \) \(211778903923627520139425348255827122068172212718485498112\)\()/\)\(63\!\cdots\!64\)
\(\beta_{3}\)\(=\)\((\)\(-\)\(622812051901\) \(\nu^{5}\mathstrut -\mathstrut \) \(18905649165525574232565\) \(\nu^{4}\mathstrut +\mathstrut \) \(17869069466647030662532516496544\) \(\nu^{3}\mathstrut +\mathstrut \) \(382695705267597042646168653945716402011360\) \(\nu^{2}\mathstrut -\mathstrut \) \(140071575584482192899887077227607649818311761885952\) \(\nu\mathstrut -\mathstrut \) \(1278182730691235506753379480697661790587354972675857065283840\)\()/\)\(33\!\cdots\!56\)
\(\beta_{4}\)\(=\)\((\)\(-\)\(687580198211778593\) \(\nu^{5}\mathstrut +\mathstrut \) \(261149215098436825652325447\) \(\nu^{4}\mathstrut +\mathstrut \) \(11559216341078523099923912171229656352\) \(\nu^{3}\mathstrut -\mathstrut \) \(1831957080661742895586860354230197619463579040\) \(\nu^{2}\mathstrut -\mathstrut \) \(45068668662605440818298141845697038686098198714223785728\) \(\nu\mathstrut -\mathstrut \) \(6514727813269645208229386763944768843836092419206003637663237376\)\()/\)\(20\!\cdots\!00\)
\(\beta_{5}\)\(=\)\((\)\(-\)\(1079038690906363417057\) \(\nu^{5}\mathstrut -\mathstrut \) \(18626326216530086775315818821497\) \(\nu^{4}\mathstrut -\mathstrut \) \(189888900813385683268026335502486713225952\) \(\nu^{3}\mathstrut +\mathstrut \) \(187306220558887538447517093208279813704122145555040\) \(\nu^{2}\mathstrut +\mathstrut \) \(1949929883211596391434605232574955365872129939317018400540928\) \(\nu\mathstrut +\mathstrut \) \(317594279206985688318277247102119634699065221984689566533231834053376\)\()/\)\(15\!\cdots\!00\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1}\mathstrut +\mathstrut \) \(12\)\()/24\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3}\mathstrut +\mathstrut \) \(6859\) \(\beta_{2}\mathstrut +\mathstrut \) \(6933179271\) \(\beta_{1}\mathstrut +\mathstrut \) \(3608807514319800551520\)\()/576\)
\(\nu^{3}\)\(=\)\((\)\(-\)\(1269\) \(\beta_{5}\mathstrut +\mathstrut \) \(10739\) \(\beta_{4}\mathstrut -\mathstrut \) \(18101906\) \(\beta_{3}\mathstrut -\mathstrut \) \(18347007714910\) \(\beta_{2}\mathstrut +\mathstrut \) \(689917724744602829892\) \(\beta_{1}\mathstrut +\mathstrut \) \(3127564274404746945419871434208\)\()/1728\)
\(\nu^{4}\)\(=\)\((\)\(-\)\(490484368455\) \(\beta_{5}\mathstrut +\mathstrut \) \(167868392758705\) \(\beta_{4}\mathstrut +\mathstrut \) \(30169391367716115638\) \(\beta_{3}\mathstrut +\mathstrut \) \(406073038002404360756170\) \(\beta_{2}\mathstrut +\mathstrut \) \(310989773189543403973035133392\) \(\beta_{1}\mathstrut +\mathstrut \) \(103740844032723052780031393952765939440928\)\()/1728\)
\(\nu^{5}\)\(=\)\((\)\(-\)\(7173337831791001913787\) \(\beta_{5}\mathstrut -\mathstrut \) \(1595861326796321577498403\) \(\beta_{4}\mathstrut +\mathstrut \) \(1053742672105498200047372126\) \(\beta_{3}\mathstrut -\mathstrut \) \(69677960216467034790725601629790\) \(\beta_{2}\mathstrut +\mathstrut \) \(2313948240110542399345406901819640590768\) \(\beta_{1}\mathstrut +\mathstrut \) \(15587534859598899444262181668865818402373281376928\)\()/576\)

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
3.39604e9
2.88052e9
2.15491e8
−6.56883e8
−2.55796e9
−3.27721e9
−7.04788e10 1.46397e17 2.60608e21 3.41813e23 −1.03179e28 1.73558e30 −1.72603e31 1.39227e34 −2.40906e34
1.2 −5.81063e10 −8.58335e16 1.01516e21 8.21398e23 4.98747e27 −4.94216e29 7.82125e31 −1.42084e32 −4.77284e34
1.3 5.85445e9 5.36564e16 −2.32691e21 −7.86170e24 3.14128e26 −1.67406e30 −2.74462e31 −4.63046e33 −4.60259e34
1.4 2.67914e10 −1.17507e16 −1.64340e21 8.36674e24 −3.14817e26 1.48756e30 −1.07289e32 −7.37139e33 2.24157e35
1.5 7.24174e10 −1.51457e17 2.88309e21 −9.47506e24 −1.09681e28 1.67109e28 3.77951e31 1.54297e34 −6.86158e35
1.6 8.96793e10 1.38884e17 5.68119e21 3.52866e24 1.24550e28 −7.32344e29 2.97736e32 1.17794e34 3.16448e35
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Inner twists

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

Hecke kernels

There are no other newforms in \(S_{72}^{\mathrm{new}}(\Gamma_0(1))\).