Properties

Label 9.74.a.a
Level 9
Weight 74
Character orbit 9.a
Self dual yes
Analytic conductor 303.736
Analytic rank 0
Dimension 5
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 74 \)
Character orbit: \([\chi]\) \(=\) 9.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(303.735576363\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
Defining polynomial: \(x^{5} - x^{4} - 10073499617947743056 x^{3} + 1429272143092482488433869600 x^{2} + 7661214288514935343595600445215756800 x + 1722510836040319301450745177697157900206688000\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{39}\cdot 3^{22}\cdot 5^{6}\cdot 7^{2} \)
Twist minimal: no (minimal twist has level 1)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q +(18417866698 + \beta_{1}) q^{2} +(\)\(17\!\cdots\!62\)\( + 26620058826 \beta_{1} - 2 \beta_{2} + \beta_{3}) q^{4} +(-\)\(46\!\cdots\!41\)\( - 14159859196422 \beta_{1} - 5610 \beta_{2} - 132 \beta_{3} + \beta_{4}) q^{5} +(-\)\(87\!\cdots\!18\)\( - 3945223266437249664 \beta_{1} - 947129022 \beta_{2} - 73498544 \beta_{3} + 186732 \beta_{4}) q^{7} +(\)\(76\!\cdots\!32\)\( + \)\(12\!\cdots\!20\)\( \beta_{1} - 750253536 \beta_{2} + 41549502192 \beta_{3} - 15102976 \beta_{4}) q^{8} +O(q^{10})\) \( q +(18417866698 + \beta_{1}) q^{2} +(\)\(17\!\cdots\!62\)\( + 26620058826 \beta_{1} - 2 \beta_{2} + \beta_{3}) q^{4} +(-\)\(46\!\cdots\!41\)\( - 14159859196422 \beta_{1} - 5610 \beta_{2} - 132 \beta_{3} + \beta_{4}) q^{5} +(-\)\(87\!\cdots\!18\)\( - 3945223266437249664 \beta_{1} - 947129022 \beta_{2} - 73498544 \beta_{3} + 186732 \beta_{4}) q^{7} +(\)\(76\!\cdots\!32\)\( + \)\(12\!\cdots\!20\)\( \beta_{1} - 750253536 \beta_{2} + 41549502192 \beta_{3} - 15102976 \beta_{4}) q^{8} +(-\)\(21\!\cdots\!88\)\( - \)\(57\!\cdots\!46\)\( \beta_{1} + 1083077017827520 \beta_{2} - 184621370870176 \beta_{3} + 52532895168 \beta_{4}) q^{10} +(-\)\(10\!\cdots\!15\)\( - \)\(18\!\cdots\!36\)\( \beta_{1} - 88539754426927815 \beta_{2} - 7662892565307936 \beta_{3} + 649409704328 \beta_{4}) q^{11} +(\)\(95\!\cdots\!85\)\( - \)\(98\!\cdots\!90\)\( \beta_{1} + 67012460580933470330 \beta_{2} - 3198282957929973212 \beta_{3} + 483886195074711 \beta_{4}) q^{13} +(-\)\(52\!\cdots\!04\)\( - \)\(15\!\cdots\!88\)\( \beta_{1} + \)\(20\!\cdots\!96\)\( \beta_{2} - 36504584384330910288 \beta_{3} + 10548213915031520 \beta_{4}) q^{14} +(\)\(11\!\cdots\!92\)\( + \)\(25\!\cdots\!64\)\( \beta_{1} + \)\(25\!\cdots\!96\)\( \beta_{2} - \)\(50\!\cdots\!36\)\( \beta_{3} - 1392819877299142656 \beta_{4}) q^{16} +(-\)\(13\!\cdots\!20\)\( - \)\(67\!\cdots\!76\)\( \beta_{1} - \)\(84\!\cdots\!44\)\( \beta_{2} - \)\(41\!\cdots\!84\)\( \beta_{3} - 8696329022352154898 \beta_{4}) q^{17} +(\)\(62\!\cdots\!21\)\( + \)\(42\!\cdots\!16\)\( \beta_{1} - \)\(46\!\cdots\!83\)\( \beta_{2} - \)\(22\!\cdots\!84\)\( \beta_{3} - \)\(24\!\cdots\!56\)\( \beta_{4}) q^{19} +(-\)\(13\!\cdots\!12\)\( - \)\(20\!\cdots\!04\)\( \beta_{1} + \)\(10\!\cdots\!80\)\( \beta_{2} - \)\(15\!\cdots\!74\)\( \beta_{3} - \)\(36\!\cdots\!68\)\( \beta_{4}) q^{20} +(-\)\(18\!\cdots\!60\)\( - \)\(84\!\cdots\!20\)\( \beta_{1} + \)\(88\!\cdots\!80\)\( \beta_{2} - \)\(42\!\cdots\!08\)\( \beta_{3} + \)\(11\!\cdots\!24\)\( \beta_{4}) q^{22} +(\)\(82\!\cdots\!02\)\( + \)\(26\!\cdots\!72\)\( \beta_{1} + \)\(55\!\cdots\!62\)\( \beta_{2} - \)\(26\!\cdots\!40\)\( \beta_{3} + \)\(16\!\cdots\!20\)\( \beta_{4}) q^{23} +(\)\(64\!\cdots\!75\)\( + \)\(41\!\cdots\!00\)\( \beta_{1} + \)\(10\!\cdots\!00\)\( \beta_{2} - \)\(45\!\cdots\!00\)\( \beta_{3} + \)\(18\!\cdots\!00\)\( \beta_{4}) q^{25} +(-\)\(89\!\cdots\!60\)\( - \)\(36\!\cdots\!34\)\( \beta_{1} + \)\(40\!\cdots\!96\)\( \beta_{2} - \)\(21\!\cdots\!84\)\( \beta_{3} + \)\(92\!\cdots\!68\)\( \beta_{4}) q^{26} +(-\)\(75\!\cdots\!92\)\( - \)\(41\!\cdots\!16\)\( \beta_{1} + \)\(20\!\cdots\!52\)\( \beta_{2} - \)\(31\!\cdots\!88\)\( \beta_{3} - \)\(61\!\cdots\!36\)\( \beta_{4}) q^{28} +(\)\(43\!\cdots\!23\)\( + \)\(12\!\cdots\!50\)\( \beta_{1} - \)\(19\!\cdots\!98\)\( \beta_{2} - \)\(10\!\cdots\!00\)\( \beta_{3} - \)\(71\!\cdots\!63\)\( \beta_{4}) q^{29} +(-\)\(79\!\cdots\!64\)\( + \)\(11\!\cdots\!48\)\( \beta_{1} - \)\(23\!\cdots\!80\)\( \beta_{2} - \)\(95\!\cdots\!52\)\( \beta_{3} + \)\(27\!\cdots\!96\)\( \beta_{4}) q^{31} +(\)\(18\!\cdots\!76\)\( - \)\(49\!\cdots\!88\)\( \beta_{1} - \)\(22\!\cdots\!60\)\( \beta_{2} + \)\(23\!\cdots\!16\)\( \beta_{3} + \)\(14\!\cdots\!52\)\( \beta_{4}) q^{32} +(-\)\(64\!\cdots\!60\)\( - \)\(56\!\cdots\!46\)\( \beta_{1} + \)\(42\!\cdots\!36\)\( \beta_{2} - \)\(60\!\cdots\!96\)\( \beta_{3} - \)\(11\!\cdots\!36\)\( \beta_{4}) q^{34} +(\)\(21\!\cdots\!28\)\( + \)\(87\!\cdots\!76\)\( \beta_{1} + \)\(18\!\cdots\!80\)\( \beta_{2} - \)\(77\!\cdots\!44\)\( \beta_{3} + \)\(36\!\cdots\!92\)\( \beta_{4}) q^{35} +(-\)\(13\!\cdots\!11\)\( + \)\(10\!\cdots\!42\)\( \beta_{1} + \)\(15\!\cdots\!46\)\( \beta_{2} + \)\(78\!\cdots\!56\)\( \beta_{3} - \)\(29\!\cdots\!93\)\( \beta_{4}) q^{37} +(\)\(40\!\cdots\!56\)\( - \)\(10\!\cdots\!44\)\( \beta_{1} - \)\(10\!\cdots\!44\)\( \beta_{2} + \)\(41\!\cdots\!64\)\( \beta_{3} + \)\(48\!\cdots\!08\)\( \beta_{4}) q^{38} +(-\)\(17\!\cdots\!20\)\( - \)\(14\!\cdots\!40\)\( \beta_{1} - \)\(10\!\cdots\!00\)\( \beta_{2} + \)\(60\!\cdots\!60\)\( \beta_{3} - \)\(42\!\cdots\!80\)\( \beta_{4}) q^{40} +(\)\(17\!\cdots\!34\)\( + \)\(39\!\cdots\!52\)\( \beta_{1} - \)\(72\!\cdots\!20\)\( \beta_{2} + \)\(21\!\cdots\!52\)\( \beta_{3} - \)\(26\!\cdots\!96\)\( \beta_{4}) q^{41} +(\)\(23\!\cdots\!07\)\( + \)\(23\!\cdots\!00\)\( \beta_{1} - \)\(10\!\cdots\!21\)\( \beta_{2} + \)\(73\!\cdots\!00\)\( \beta_{3} + \)\(43\!\cdots\!00\)\( \beta_{4}) q^{43} +(-\)\(72\!\cdots\!60\)\( - \)\(52\!\cdots\!24\)\( \beta_{1} + \)\(11\!\cdots\!84\)\( \beta_{2} - \)\(38\!\cdots\!24\)\( \beta_{3} + \)\(64\!\cdots\!16\)\( \beta_{4}) q^{44} +(\)\(26\!\cdots\!56\)\( - \)\(19\!\cdots\!72\)\( \beta_{1} - \)\(60\!\cdots\!40\)\( \beta_{2} + \)\(20\!\cdots\!28\)\( \beta_{3} + \)\(64\!\cdots\!96\)\( \beta_{4}) q^{46} +(-\)\(52\!\cdots\!20\)\( + \)\(36\!\cdots\!32\)\( \beta_{1} - \)\(78\!\cdots\!32\)\( \beta_{2} - \)\(21\!\cdots\!00\)\( \beta_{3} + \)\(20\!\cdots\!00\)\( \beta_{4}) q^{47} +(-\)\(95\!\cdots\!67\)\( + \)\(18\!\cdots\!52\)\( \beta_{1} + \)\(33\!\cdots\!60\)\( \beta_{2} - \)\(12\!\cdots\!48\)\( \beta_{3} + \)\(70\!\cdots\!84\)\( \beta_{4}) q^{49} +(\)\(39\!\cdots\!50\)\( - \)\(42\!\cdots\!25\)\( \beta_{1} + \)\(64\!\cdots\!00\)\( \beta_{2} + \)\(58\!\cdots\!00\)\( \beta_{3} + \)\(19\!\cdots\!00\)\( \beta_{4}) q^{50} +(-\)\(36\!\cdots\!36\)\( - \)\(24\!\cdots\!60\)\( \beta_{1} - \)\(47\!\cdots\!32\)\( \beta_{2} - \)\(24\!\cdots\!50\)\( \beta_{3} + \)\(35\!\cdots\!00\)\( \beta_{4}) q^{52} +(\)\(45\!\cdots\!87\)\( + \)\(38\!\cdots\!26\)\( \beta_{1} + \)\(83\!\cdots\!38\)\( \beta_{2} + \)\(82\!\cdots\!72\)\( \beta_{3} + \)\(35\!\cdots\!09\)\( \beta_{4}) q^{53} +(\)\(10\!\cdots\!62\)\( + \)\(14\!\cdots\!04\)\( \beta_{1} + \)\(25\!\cdots\!70\)\( \beta_{2} + \)\(75\!\cdots\!24\)\( \beta_{3} + \)\(78\!\cdots\!68\)\( \beta_{4}) q^{55} +(-\)\(34\!\cdots\!48\)\( - \)\(29\!\cdots\!80\)\( \beta_{1} - \)\(19\!\cdots\!92\)\( \beta_{2} - \)\(90\!\cdots\!80\)\( \beta_{3} - \)\(79\!\cdots\!12\)\( \beta_{4}) q^{56} +(\)\(12\!\cdots\!12\)\( - \)\(37\!\cdots\!22\)\( \beta_{1} - \)\(33\!\cdots\!64\)\( \beta_{2} + \)\(11\!\cdots\!16\)\( \beta_{3} + \)\(48\!\cdots\!52\)\( \beta_{4}) q^{58} +(-\)\(99\!\cdots\!35\)\( + \)\(22\!\cdots\!08\)\( \beta_{1} + \)\(20\!\cdots\!29\)\( \beta_{2} - \)\(47\!\cdots\!92\)\( \beta_{3} - \)\(53\!\cdots\!80\)\( \beta_{4}) q^{59} +(-\)\(40\!\cdots\!23\)\( + \)\(13\!\cdots\!90\)\( \beta_{1} - \)\(70\!\cdots\!50\)\( \beta_{2} - \)\(36\!\cdots\!60\)\( \beta_{3} + \)\(14\!\cdots\!55\)\( \beta_{4}) q^{61} +(\)\(91\!\cdots\!48\)\( - \)\(15\!\cdots\!24\)\( \beta_{1} + \)\(62\!\cdots\!60\)\( \beta_{2} + \)\(49\!\cdots\!44\)\( \beta_{3} + \)\(20\!\cdots\!68\)\( \beta_{4}) q^{62} +(-\)\(53\!\cdots\!96\)\( - \)\(58\!\cdots\!44\)\( \beta_{1} + \)\(23\!\cdots\!16\)\( \beta_{2} - \)\(26\!\cdots\!44\)\( \beta_{3} + \)\(19\!\cdots\!68\)\( \beta_{4}) q^{64} +(\)\(44\!\cdots\!64\)\( + \)\(11\!\cdots\!88\)\( \beta_{1} - \)\(15\!\cdots\!60\)\( \beta_{2} + \)\(36\!\cdots\!28\)\( \beta_{3} + \)\(41\!\cdots\!96\)\( \beta_{4}) q^{65} +(\)\(34\!\cdots\!85\)\( - \)\(12\!\cdots\!24\)\( \beta_{1} - \)\(11\!\cdots\!51\)\( \beta_{2} + \)\(10\!\cdots\!24\)\( \beta_{3} + \)\(13\!\cdots\!28\)\( \beta_{4}) q^{67} +(-\)\(51\!\cdots\!76\)\( - \)\(68\!\cdots\!08\)\( \beta_{1} + \)\(86\!\cdots\!96\)\( \beta_{2} - \)\(24\!\cdots\!02\)\( \beta_{3} + \)\(16\!\cdots\!56\)\( \beta_{4}) q^{68} +(\)\(12\!\cdots\!04\)\( + \)\(12\!\cdots\!68\)\( \beta_{1} + \)\(12\!\cdots\!40\)\( \beta_{2} + \)\(19\!\cdots\!08\)\( \beta_{3} + \)\(35\!\cdots\!56\)\( \beta_{4}) q^{70} +(-\)\(59\!\cdots\!22\)\( + \)\(30\!\cdots\!80\)\( \beta_{1} - \)\(26\!\cdots\!50\)\( \beta_{2} + \)\(64\!\cdots\!80\)\( \beta_{3} - \)\(15\!\cdots\!40\)\( \beta_{4}) q^{71} +(-\)\(47\!\cdots\!24\)\( - \)\(72\!\cdots\!76\)\( \beta_{1} - \)\(79\!\cdots\!32\)\( \beta_{2} + \)\(23\!\cdots\!96\)\( \beta_{3} - \)\(66\!\cdots\!38\)\( \beta_{4}) q^{73} +(\)\(77\!\cdots\!72\)\( - \)\(54\!\cdots\!78\)\( \beta_{1} - \)\(52\!\cdots\!40\)\( \beta_{2} + \)\(17\!\cdots\!72\)\( \beta_{3} - \)\(21\!\cdots\!76\)\( \beta_{4}) q^{74} +(-\)\(80\!\cdots\!84\)\( + \)\(42\!\cdots\!00\)\( \beta_{1} + \)\(48\!\cdots\!84\)\( \beta_{2} + \)\(16\!\cdots\!00\)\( \beta_{3} - \)\(38\!\cdots\!96\)\( \beta_{4}) q^{76} +(\)\(23\!\cdots\!00\)\( + \)\(29\!\cdots\!36\)\( \beta_{1} + \)\(42\!\cdots\!24\)\( \beta_{2} + \)\(15\!\cdots\!16\)\( \beta_{3} + \)\(11\!\cdots\!52\)\( \beta_{4}) q^{77} +(\)\(24\!\cdots\!56\)\( - \)\(14\!\cdots\!72\)\( \beta_{1} - \)\(13\!\cdots\!52\)\( \beta_{2} - \)\(32\!\cdots\!72\)\( \beta_{3} + \)\(44\!\cdots\!24\)\( \beta_{4}) q^{79} +(-\)\(15\!\cdots\!56\)\( + \)\(22\!\cdots\!48\)\( \beta_{1} - \)\(11\!\cdots\!60\)\( \beta_{2} + \)\(74\!\cdots\!88\)\( \beta_{3} + \)\(81\!\cdots\!16\)\( \beta_{4}) q^{80} +(\)\(39\!\cdots\!12\)\( + \)\(47\!\cdots\!94\)\( \beta_{1} - \)\(80\!\cdots\!60\)\( \beta_{2} + \)\(45\!\cdots\!56\)\( \beta_{3} - \)\(67\!\cdots\!68\)\( \beta_{4}) q^{82} +(-\)\(21\!\cdots\!25\)\( - \)\(78\!\cdots\!04\)\( \beta_{1} - \)\(25\!\cdots\!01\)\( \beta_{2} + \)\(12\!\cdots\!00\)\( \beta_{3} - \)\(11\!\cdots\!00\)\( \beta_{4}) q^{83} +(-\)\(57\!\cdots\!58\)\( + \)\(45\!\cdots\!64\)\( \beta_{1} + \)\(66\!\cdots\!20\)\( \beta_{2} + \)\(20\!\cdots\!84\)\( \beta_{3} - \)\(62\!\cdots\!62\)\( \beta_{4}) q^{85} +(\)\(26\!\cdots\!56\)\( + \)\(33\!\cdots\!12\)\( \beta_{1} - \)\(36\!\cdots\!72\)\( \beta_{2} + \)\(23\!\cdots\!12\)\( \beta_{3} - \)\(11\!\cdots\!88\)\( \beta_{4}) q^{86} +(-\)\(44\!\cdots\!44\)\( - \)\(42\!\cdots\!80\)\( \beta_{1} + \)\(46\!\cdots\!52\)\( \beta_{2} - \)\(26\!\cdots\!84\)\( \beta_{3} + \)\(13\!\cdots\!52\)\( \beta_{4}) q^{88} +(\)\(88\!\cdots\!64\)\( - \)\(81\!\cdots\!80\)\( \beta_{1} + \)\(87\!\cdots\!96\)\( \beta_{2} + \)\(78\!\cdots\!20\)\( \beta_{3} + \)\(88\!\cdots\!66\)\( \beta_{4}) q^{89} +(\)\(10\!\cdots\!80\)\( + \)\(22\!\cdots\!60\)\( \beta_{1} - \)\(26\!\cdots\!28\)\( \beta_{2} + \)\(79\!\cdots\!60\)\( \beta_{3} + \)\(75\!\cdots\!52\)\( \beta_{4}) q^{91} +(-\)\(21\!\cdots\!16\)\( + \)\(21\!\cdots\!72\)\( \beta_{1} - \)\(40\!\cdots\!24\)\( \beta_{2} - \)\(25\!\cdots\!16\)\( \beta_{3} - \)\(14\!\cdots\!52\)\( \beta_{4}) q^{92} +(\)\(24\!\cdots\!80\)\( - \)\(67\!\cdots\!64\)\( \beta_{1} + \)\(14\!\cdots\!12\)\( \beta_{2} - \)\(26\!\cdots\!64\)\( \beta_{3} + \)\(11\!\cdots\!04\)\( \beta_{4}) q^{94} +(-\)\(22\!\cdots\!10\)\( - \)\(12\!\cdots\!20\)\( \beta_{1} + \)\(14\!\cdots\!50\)\( \beta_{2} - \)\(46\!\cdots\!20\)\( \beta_{3} + \)\(11\!\cdots\!60\)\( \beta_{4}) q^{95} +(-\)\(94\!\cdots\!76\)\( + \)\(11\!\cdots\!36\)\( \beta_{1} - \)\(38\!\cdots\!68\)\( \beta_{2} - \)\(25\!\cdots\!72\)\( \beta_{3} + \)\(14\!\cdots\!66\)\( \beta_{4}) q^{97} +(\)\(15\!\cdots\!34\)\( - \)\(23\!\cdots\!07\)\( \beta_{1} + \)\(23\!\cdots\!80\)\( \beta_{2} + \)\(52\!\cdots\!96\)\( \beta_{3} + \)\(66\!\cdots\!12\)\( \beta_{4}) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q + 92089333488q^{2} + \)\(89\!\cdots\!60\)\(q^{4} - \)\(23\!\cdots\!50\)\(q^{5} - \)\(43\!\cdots\!08\)\(q^{7} + \)\(38\!\cdots\!80\)\(q^{8} + O(q^{10}) \) \( 5q + 92089333488q^{2} + \)\(89\!\cdots\!60\)\(q^{4} - \)\(23\!\cdots\!50\)\(q^{5} - \)\(43\!\cdots\!08\)\(q^{7} + \)\(38\!\cdots\!80\)\(q^{8} - \)\(10\!\cdots\!00\)\(q^{10} - \)\(50\!\cdots\!60\)\(q^{11} + \)\(47\!\cdots\!86\)\(q^{13} - \)\(26\!\cdots\!20\)\(q^{14} + \)\(57\!\cdots\!80\)\(q^{16} - \)\(66\!\cdots\!02\)\(q^{17} + \)\(31\!\cdots\!00\)\(q^{19} - \)\(68\!\cdots\!00\)\(q^{20} - \)\(94\!\cdots\!16\)\(q^{22} + \)\(41\!\cdots\!24\)\(q^{23} + \)\(32\!\cdots\!75\)\(q^{25} - \)\(44\!\cdots\!60\)\(q^{26} - \)\(37\!\cdots\!16\)\(q^{28} + \)\(21\!\cdots\!50\)\(q^{29} - \)\(39\!\cdots\!40\)\(q^{31} + \)\(94\!\cdots\!68\)\(q^{32} - \)\(32\!\cdots\!80\)\(q^{34} + \)\(10\!\cdots\!00\)\(q^{35} - \)\(67\!\cdots\!78\)\(q^{37} + \)\(20\!\cdots\!20\)\(q^{38} - \)\(87\!\cdots\!00\)\(q^{40} + \)\(89\!\cdots\!90\)\(q^{41} + \)\(11\!\cdots\!56\)\(q^{43} - \)\(36\!\cdots\!20\)\(q^{44} + \)\(13\!\cdots\!60\)\(q^{46} - \)\(26\!\cdots\!32\)\(q^{47} - \)\(47\!\cdots\!15\)\(q^{49} + \)\(19\!\cdots\!00\)\(q^{50} - \)\(18\!\cdots\!28\)\(q^{52} + \)\(22\!\cdots\!54\)\(q^{53} + \)\(52\!\cdots\!00\)\(q^{55} - \)\(17\!\cdots\!00\)\(q^{56} + \)\(63\!\cdots\!20\)\(q^{58} - \)\(49\!\cdots\!00\)\(q^{59} - \)\(20\!\cdots\!90\)\(q^{61} + \)\(45\!\cdots\!96\)\(q^{62} - \)\(26\!\cdots\!40\)\(q^{64} + \)\(22\!\cdots\!00\)\(q^{65} + \)\(17\!\cdots\!52\)\(q^{67} - \)\(25\!\cdots\!04\)\(q^{68} + \)\(60\!\cdots\!00\)\(q^{70} - \)\(29\!\cdots\!60\)\(q^{71} - \)\(23\!\cdots\!74\)\(q^{73} + \)\(38\!\cdots\!80\)\(q^{74} - \)\(40\!\cdots\!00\)\(q^{76} + \)\(11\!\cdots\!56\)\(q^{77} + \)\(12\!\cdots\!00\)\(q^{79} - \)\(76\!\cdots\!00\)\(q^{80} + \)\(19\!\cdots\!64\)\(q^{82} - \)\(10\!\cdots\!16\)\(q^{83} - \)\(28\!\cdots\!00\)\(q^{85} + \)\(13\!\cdots\!40\)\(q^{86} - \)\(22\!\cdots\!60\)\(q^{88} + \)\(44\!\cdots\!50\)\(q^{89} + \)\(50\!\cdots\!60\)\(q^{91} - \)\(10\!\cdots\!52\)\(q^{92} + \)\(12\!\cdots\!20\)\(q^{94} - \)\(11\!\cdots\!00\)\(q^{95} - \)\(47\!\cdots\!18\)\(q^{97} + \)\(76\!\cdots\!16\)\(q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{5} - x^{4} - 10073499617947743056 x^{3} + 1429272143092482488433869600 x^{2} + 7661214288514935343595600445215756800 x + 1722510836040319301450745177697157900206688000\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 48 \nu - 10 \)
\(\beta_{2}\)\(=\)\((\)\(-24381 \nu^{4} + 9422283614913 \nu^{3} + 238627368549740496490812 \nu^{2} - 127337480880295830634715778703440 \nu - 113246692547331213757777007193627862661056\)\()/ 69381191317447245824 \)
\(\beta_{3}\)\(=\)\((\)\(-24381 \nu^{4} + 9422283614913 \nu^{3} + 318554500947439723680060 \nu^{2} - 110326864790927987431551844802640 \nu - 435305067619486073586821241060813881374656\)\()/ 34690595658723622912 \)
\(\beta_{4}\)\(=\)\((\)\(-66468519 \nu^{4} - 228335508986952717 \nu^{3} + 578032012637896887558361332 \nu^{2} + 1809927158935047530768074784796932112 \nu - 234350229463453962607595222326602287177792320\)\()/ 34690595658723622912 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1} + 10\)\()/48\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} - 2 \beta_{2} - 10215674550 \beta_{1} + 9283737247896553731050\)\()/2304\)
\(\nu^{3}\)\(=\)\((\)\(-943936 \beta_{4} - 856506117 \beta_{3} + 6859809162 \beta_{2} + 1231550858259697254542 \beta_{1} - 5927477389563705417174812949458\)\()/6912\)
\(\nu^{4}\)\(=\)\((\)\(-1094380793198784 \beta_{4} + 87093867613749185745 \beta_{3} - 227229228278097165602 \beta_{2} - 1728287729978020779753014917782 \beta_{1} + 714587160964971173397365654587009050828362\)\()/20736\)

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
−3.13124e9
−6.50494e8
−2.60629e8
1.10442e9
2.93794e9
−1.31882e11 0 7.94807e21 7.75470e24 0 1.05974e30 1.97383e32 0 −1.02270e36
1.2 −1.28059e10 0 −9.28074e21 −4.05981e25 0 −7.12492e30 2.39796e32 0 5.19894e35
1.3 5.90767e9 0 −9.40983e21 −1.69056e25 0 −2.69951e30 −1.11387e32 0 −9.98727e34
1.4 7.14302e10 0 −4.34246e21 5.37334e25 0 1.02465e31 −9.84822e32 0 3.83819e36
1.5 1.59439e11 0 1.59761e22 −2.70839e25 0 −5.83802e30 1.04135e33 0 −4.31824e36
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 9.74.a.a 5
3.b odd 2 1 1.74.a.a 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.74.a.a 5 3.b odd 2 1
9.74.a.a 5 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{5} - 92089333488 T_{2}^{4} - \)\(19\!\cdots\!88\)\( T_{2}^{3} + \)\(13\!\cdots\!44\)\( T_{2}^{2} + \)\(11\!\cdots\!36\)\( T_{2} - \)\(11\!\cdots\!68\)\( \) acting on \(S_{74}^{\mathrm{new}}(\Gamma_0(9))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 92089333488 T + \)\(27\!\cdots\!72\)\( T^{2} - \)\(21\!\cdots\!40\)\( T^{3} + \)\(34\!\cdots\!88\)\( T^{4} - \)\(23\!\cdots\!64\)\( T^{5} + \)\(32\!\cdots\!96\)\( T^{6} - \)\(18\!\cdots\!60\)\( T^{7} + \)\(23\!\cdots\!36\)\( T^{8} - \)\(73\!\cdots\!48\)\( T^{9} + \)\(75\!\cdots\!32\)\( T^{10} \)
$3$ 1
$5$ \( 1 + \)\(23\!\cdots\!50\)\( T + \)\(27\!\cdots\!25\)\( T^{2} + \)\(13\!\cdots\!00\)\( T^{3} + \)\(29\!\cdots\!50\)\( T^{4} - \)\(15\!\cdots\!00\)\( T^{5} + \)\(31\!\cdots\!50\)\( T^{6} + \)\(15\!\cdots\!00\)\( T^{7} + \)\(32\!\cdots\!25\)\( T^{8} + \)\(29\!\cdots\!50\)\( T^{9} + \)\(13\!\cdots\!25\)\( T^{10} \)
$7$ \( 1 + \)\(43\!\cdots\!08\)\( T + \)\(15\!\cdots\!07\)\( T^{2} + \)\(27\!\cdots\!00\)\( T^{3} + \)\(10\!\cdots\!98\)\( T^{4} + \)\(70\!\cdots\!84\)\( T^{5} + \)\(51\!\cdots\!86\)\( T^{6} + \)\(66\!\cdots\!00\)\( T^{7} + \)\(18\!\cdots\!01\)\( T^{8} + \)\(25\!\cdots\!08\)\( T^{9} + \)\(28\!\cdots\!07\)\( T^{10} \)
$11$ \( 1 + \)\(50\!\cdots\!60\)\( T + \)\(30\!\cdots\!95\)\( T^{2} + \)\(75\!\cdots\!20\)\( T^{3} + \)\(44\!\cdots\!10\)\( T^{4} + \)\(73\!\cdots\!52\)\( T^{5} + \)\(46\!\cdots\!10\)\( T^{6} + \)\(83\!\cdots\!20\)\( T^{7} + \)\(35\!\cdots\!45\)\( T^{8} + \)\(61\!\cdots\!60\)\( T^{9} + \)\(12\!\cdots\!51\)\( T^{10} \)
$13$ \( 1 - \)\(47\!\cdots\!86\)\( T + \)\(33\!\cdots\!73\)\( T^{2} - \)\(75\!\cdots\!20\)\( T^{3} + \)\(10\!\cdots\!78\)\( T^{4} - \)\(68\!\cdots\!48\)\( T^{5} + \)\(21\!\cdots\!34\)\( T^{6} - \)\(32\!\cdots\!80\)\( T^{7} + \)\(30\!\cdots\!21\)\( T^{8} - \)\(88\!\cdots\!66\)\( T^{9} + \)\(38\!\cdots\!93\)\( T^{10} \)
$17$ \( 1 + \)\(66\!\cdots\!02\)\( T + \)\(11\!\cdots\!77\)\( T^{2} + \)\(48\!\cdots\!80\)\( T^{3} + \)\(70\!\cdots\!18\)\( T^{4} + \)\(14\!\cdots\!76\)\( T^{5} + \)\(47\!\cdots\!66\)\( T^{6} + \)\(21\!\cdots\!20\)\( T^{7} + \)\(34\!\cdots\!81\)\( T^{8} + \)\(12\!\cdots\!22\)\( T^{9} + \)\(12\!\cdots\!57\)\( T^{10} \)
$19$ \( 1 - \)\(31\!\cdots\!00\)\( T + \)\(41\!\cdots\!95\)\( T^{2} - \)\(41\!\cdots\!00\)\( T^{3} + \)\(82\!\cdots\!10\)\( T^{4} - \)\(14\!\cdots\!00\)\( T^{5} + \)\(18\!\cdots\!90\)\( T^{6} - \)\(20\!\cdots\!00\)\( T^{7} + \)\(46\!\cdots\!05\)\( T^{8} - \)\(78\!\cdots\!00\)\( T^{9} + \)\(55\!\cdots\!99\)\( T^{10} \)
$23$ \( 1 - \)\(41\!\cdots\!24\)\( T + \)\(74\!\cdots\!63\)\( T^{2} - \)\(22\!\cdots\!20\)\( T^{3} + \)\(28\!\cdots\!18\)\( T^{4} - \)\(65\!\cdots\!72\)\( T^{5} + \)\(72\!\cdots\!94\)\( T^{6} - \)\(14\!\cdots\!80\)\( T^{7} + \)\(12\!\cdots\!81\)\( T^{8} - \)\(17\!\cdots\!04\)\( T^{9} + \)\(10\!\cdots\!43\)\( T^{10} \)
$29$ \( 1 - \)\(21\!\cdots\!50\)\( T + \)\(20\!\cdots\!45\)\( T^{2} - \)\(36\!\cdots\!00\)\( T^{3} + \)\(19\!\cdots\!10\)\( T^{4} - \)\(28\!\cdots\!00\)\( T^{5} + \)\(11\!\cdots\!90\)\( T^{6} - \)\(11\!\cdots\!00\)\( T^{7} + \)\(38\!\cdots\!05\)\( T^{8} - \)\(22\!\cdots\!50\)\( T^{9} + \)\(59\!\cdots\!49\)\( T^{10} \)
$31$ \( 1 + \)\(39\!\cdots\!40\)\( T + \)\(31\!\cdots\!95\)\( T^{2} + \)\(94\!\cdots\!80\)\( T^{3} + \)\(43\!\cdots\!10\)\( T^{4} + \)\(98\!\cdots\!48\)\( T^{5} + \)\(32\!\cdots\!10\)\( T^{6} + \)\(51\!\cdots\!80\)\( T^{7} + \)\(12\!\cdots\!45\)\( T^{8} + \)\(11\!\cdots\!40\)\( T^{9} + \)\(22\!\cdots\!51\)\( T^{10} \)
$37$ \( 1 + \)\(67\!\cdots\!78\)\( T + \)\(24\!\cdots\!17\)\( T^{2} + \)\(63\!\cdots\!60\)\( T^{3} + \)\(14\!\cdots\!58\)\( T^{4} + \)\(26\!\cdots\!04\)\( T^{5} + \)\(42\!\cdots\!26\)\( T^{6} + \)\(57\!\cdots\!40\)\( T^{7} + \)\(66\!\cdots\!41\)\( T^{8} + \)\(55\!\cdots\!18\)\( T^{9} + \)\(24\!\cdots\!57\)\( T^{10} \)
$41$ \( 1 - \)\(89\!\cdots\!90\)\( T + \)\(21\!\cdots\!45\)\( T^{2} - \)\(15\!\cdots\!80\)\( T^{3} + \)\(20\!\cdots\!10\)\( T^{4} - \)\(11\!\cdots\!48\)\( T^{5} + \)\(10\!\cdots\!10\)\( T^{6} - \)\(46\!\cdots\!80\)\( T^{7} + \)\(33\!\cdots\!45\)\( T^{8} - \)\(77\!\cdots\!90\)\( T^{9} + \)\(46\!\cdots\!01\)\( T^{10} \)
$43$ \( 1 - \)\(11\!\cdots\!56\)\( T + \)\(12\!\cdots\!43\)\( T^{2} - \)\(85\!\cdots\!00\)\( T^{3} + \)\(51\!\cdots\!98\)\( T^{4} - \)\(22\!\cdots\!88\)\( T^{5} + \)\(90\!\cdots\!14\)\( T^{6} - \)\(26\!\cdots\!00\)\( T^{7} + \)\(68\!\cdots\!01\)\( T^{8} - \)\(11\!\cdots\!56\)\( T^{9} + \)\(16\!\cdots\!43\)\( T^{10} \)
$47$ \( 1 + \)\(26\!\cdots\!32\)\( T + \)\(64\!\cdots\!87\)\( T^{2} + \)\(10\!\cdots\!20\)\( T^{3} + \)\(14\!\cdots\!78\)\( T^{4} + \)\(16\!\cdots\!56\)\( T^{5} + \)\(17\!\cdots\!06\)\( T^{6} + \)\(13\!\cdots\!80\)\( T^{7} + \)\(99\!\cdots\!21\)\( T^{8} + \)\(47\!\cdots\!12\)\( T^{9} + \)\(20\!\cdots\!07\)\( T^{10} \)
$53$ \( 1 - \)\(22\!\cdots\!54\)\( T + \)\(53\!\cdots\!33\)\( T^{2} - \)\(71\!\cdots\!40\)\( T^{3} + \)\(92\!\cdots\!38\)\( T^{4} - \)\(81\!\cdots\!32\)\( T^{5} + \)\(68\!\cdots\!74\)\( T^{6} - \)\(39\!\cdots\!60\)\( T^{7} + \)\(22\!\cdots\!61\)\( T^{8} - \)\(69\!\cdots\!14\)\( T^{9} + \)\(22\!\cdots\!93\)\( T^{10} \)
$59$ \( 1 + \)\(49\!\cdots\!00\)\( T + \)\(80\!\cdots\!95\)\( T^{2} + \)\(32\!\cdots\!00\)\( T^{3} + \)\(27\!\cdots\!10\)\( T^{4} + \)\(87\!\cdots\!00\)\( T^{5} + \)\(52\!\cdots\!90\)\( T^{6} + \)\(11\!\cdots\!00\)\( T^{7} + \)\(52\!\cdots\!05\)\( T^{8} + \)\(61\!\cdots\!00\)\( T^{9} + \)\(22\!\cdots\!99\)\( T^{10} \)
$61$ \( 1 + \)\(20\!\cdots\!90\)\( T + \)\(10\!\cdots\!45\)\( T^{2} + \)\(15\!\cdots\!80\)\( T^{3} + \)\(46\!\cdots\!10\)\( T^{4} + \)\(49\!\cdots\!48\)\( T^{5} + \)\(99\!\cdots\!10\)\( T^{6} + \)\(72\!\cdots\!80\)\( T^{7} + \)\(10\!\cdots\!45\)\( T^{8} + \)\(41\!\cdots\!90\)\( T^{9} + \)\(44\!\cdots\!01\)\( T^{10} \)
$67$ \( 1 - \)\(17\!\cdots\!52\)\( T + \)\(45\!\cdots\!27\)\( T^{2} - \)\(13\!\cdots\!80\)\( T^{3} + \)\(14\!\cdots\!18\)\( T^{4} - \)\(28\!\cdots\!76\)\( T^{5} + \)\(29\!\cdots\!66\)\( T^{6} - \)\(53\!\cdots\!20\)\( T^{7} + \)\(37\!\cdots\!81\)\( T^{8} - \)\(28\!\cdots\!72\)\( T^{9} + \)\(32\!\cdots\!07\)\( T^{10} \)
$71$ \( 1 + \)\(29\!\cdots\!60\)\( T + \)\(44\!\cdots\!95\)\( T^{2} + \)\(85\!\cdots\!20\)\( T^{3} + \)\(91\!\cdots\!10\)\( T^{4} + \)\(13\!\cdots\!52\)\( T^{5} + \)\(12\!\cdots\!10\)\( T^{6} + \)\(16\!\cdots\!20\)\( T^{7} + \)\(11\!\cdots\!45\)\( T^{8} + \)\(10\!\cdots\!60\)\( T^{9} + \)\(51\!\cdots\!51\)\( T^{10} \)
$73$ \( 1 + \)\(23\!\cdots\!74\)\( T + \)\(55\!\cdots\!13\)\( T^{2} + \)\(77\!\cdots\!20\)\( T^{3} + \)\(10\!\cdots\!18\)\( T^{4} + \)\(11\!\cdots\!72\)\( T^{5} + \)\(11\!\cdots\!94\)\( T^{6} + \)\(85\!\cdots\!80\)\( T^{7} + \)\(64\!\cdots\!81\)\( T^{8} + \)\(29\!\cdots\!54\)\( T^{9} + \)\(12\!\cdots\!93\)\( T^{10} \)
$79$ \( 1 - \)\(12\!\cdots\!00\)\( T + \)\(10\!\cdots\!95\)\( T^{2} - \)\(10\!\cdots\!00\)\( T^{3} + \)\(55\!\cdots\!10\)\( T^{4} - \)\(42\!\cdots\!00\)\( T^{5} + \)\(18\!\cdots\!90\)\( T^{6} - \)\(12\!\cdots\!00\)\( T^{7} + \)\(39\!\cdots\!05\)\( T^{8} - \)\(15\!\cdots\!00\)\( T^{9} + \)\(43\!\cdots\!99\)\( T^{10} \)
$83$ \( 1 + \)\(10\!\cdots\!16\)\( T + \)\(23\!\cdots\!03\)\( T^{2} + \)\(14\!\cdots\!40\)\( T^{3} + \)\(46\!\cdots\!58\)\( T^{4} + \)\(38\!\cdots\!08\)\( T^{5} + \)\(57\!\cdots\!54\)\( T^{6} + \)\(22\!\cdots\!60\)\( T^{7} + \)\(44\!\cdots\!41\)\( T^{8} + \)\(24\!\cdots\!76\)\( T^{9} + \)\(29\!\cdots\!43\)\( T^{10} \)
$89$ \( 1 - \)\(44\!\cdots\!50\)\( T + \)\(62\!\cdots\!45\)\( T^{2} - \)\(39\!\cdots\!00\)\( T^{3} + \)\(17\!\cdots\!10\)\( T^{4} - \)\(12\!\cdots\!00\)\( T^{5} + \)\(36\!\cdots\!90\)\( T^{6} - \)\(16\!\cdots\!00\)\( T^{7} + \)\(51\!\cdots\!05\)\( T^{8} - \)\(73\!\cdots\!50\)\( T^{9} + \)\(33\!\cdots\!49\)\( T^{10} \)
$97$ \( 1 + \)\(47\!\cdots\!18\)\( T + \)\(59\!\cdots\!37\)\( T^{2} + \)\(20\!\cdots\!80\)\( T^{3} + \)\(13\!\cdots\!78\)\( T^{4} + \)\(32\!\cdots\!44\)\( T^{5} + \)\(14\!\cdots\!06\)\( T^{6} + \)\(23\!\cdots\!20\)\( T^{7} + \)\(75\!\cdots\!21\)\( T^{8} + \)\(64\!\cdots\!38\)\( T^{9} + \)\(14\!\cdots\!57\)\( T^{10} \)
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