Properties

Label 9.76.a.c
Level $9$
Weight $76$
Character orbit 9.a
Self dual yes
Analytic conductor $320.606$
Analytic rank $1$
Dimension $6$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 76 \)
Character orbit: \([\chi]\) \(=\) 9.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(320.605553540\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Defining polynomial: \(x^{6} - 3 x^{5} - 38457853073924058692 x^{4} - 10276556354621685339901678086 x^{3} + 371187556674475060057870954681799784505 x^{2} + 52686123927652036687598761277591247931691204025 x - 675344021115865838575279495800656435684060652010336995750\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: multiple of \( 2^{58}\cdot 3^{36}\cdot 5^{7}\cdot 7^{3}\cdot 11\cdot 13\cdot 19 \)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q +(9513470340 - \beta_{1}) q^{2} +(\)\(28\!\cdots\!88\)\( + 9832395084 \beta_{1} + \beta_{2} + \beta_{3}) q^{4} +(\)\(64\!\cdots\!90\)\( - 333300058621800 \beta_{1} - 2952 \beta_{2} - 340 \beta_{3} + \beta_{4}) q^{5} +(\)\(32\!\cdots\!00\)\( + 20646137267438940438 \beta_{1} + 896702338 \beta_{2} - 28211952 \beta_{3} - 166380 \beta_{4} - 32 \beta_{5}) q^{7} +(-\)\(73\!\cdots\!20\)\( - \)\(23\!\cdots\!76\)\( \beta_{1} - 82237429424 \beta_{2} - 81567621776 \beta_{3} + 43973360 \beta_{4} - 7416 \beta_{5}) q^{8} +O(q^{10})\) \( q +(9513470340 - \beta_{1}) q^{2} +(\)\(28\!\cdots\!88\)\( + 9832395084 \beta_{1} + \beta_{2} + \beta_{3}) q^{4} +(\)\(64\!\cdots\!90\)\( - 333300058621800 \beta_{1} - 2952 \beta_{2} - 340 \beta_{3} + \beta_{4}) q^{5} +(\)\(32\!\cdots\!00\)\( + 20646137267438940438 \beta_{1} + 896702338 \beta_{2} - 28211952 \beta_{3} - 166380 \beta_{4} - 32 \beta_{5}) q^{7} +(-\)\(73\!\cdots\!20\)\( - \)\(23\!\cdots\!76\)\( \beta_{1} - 82237429424 \beta_{2} - 81567621776 \beta_{3} + 43973360 \beta_{4} - 7416 \beta_{5}) q^{8} +(\)\(22\!\cdots\!60\)\( + \)\(22\!\cdots\!50\)\( \beta_{1} + 223549665063932 \beta_{2} + 869735349916300 \beta_{3} + 38186921784 \beta_{4} + 25105060 \beta_{5}) q^{10} +(\)\(15\!\cdots\!48\)\( - \)\(36\!\cdots\!85\)\( \beta_{1} + 4912569329548505 \beta_{2} + 4721554989781216 \beta_{3} + 3747958317080 \beta_{4} + 806101056 \beta_{5}) q^{11} +(\)\(88\!\cdots\!70\)\( + \)\(30\!\cdots\!32\)\( \beta_{1} - 21816932345480852600 \beta_{2} + 5942218372062377876 \beta_{3} - 1816281211946385 \beta_{4} - 389448765184 \beta_{5}) q^{13} +(-\)\(13\!\cdots\!96\)\( - \)\(40\!\cdots\!62\)\( \beta_{1} + \)\(38\!\cdots\!82\)\( \beta_{2} - 96176923015363877798 \beta_{3} - 15630807774027100 \beta_{4} - 6283557820530 \beta_{5}) q^{14} +(\)\(44\!\cdots\!36\)\( + \)\(34\!\cdots\!32\)\( \beta_{1} + \)\(73\!\cdots\!08\)\( \beta_{2} + \)\(15\!\cdots\!56\)\( \beta_{3} - 2536277958175820160 \beta_{4} + 960434547593408 \beta_{5}) q^{16} +(-\)\(30\!\cdots\!30\)\( - \)\(80\!\cdots\!92\)\( \beta_{1} + \)\(11\!\cdots\!16\)\( \beta_{2} - \)\(64\!\cdots\!32\)\( \beta_{3} - 41006711266916156330 \beta_{4} + 9197963260956288 \beta_{5}) q^{17} +(\)\(17\!\cdots\!80\)\( + \)\(31\!\cdots\!11\)\( \beta_{1} + \)\(61\!\cdots\!69\)\( \beta_{2} + \)\(41\!\cdots\!56\)\( \beta_{3} + \)\(23\!\cdots\!60\)\( \beta_{4} - 482175934113126848 \beta_{5}) q^{19} +(-\)\(15\!\cdots\!80\)\( - \)\(34\!\cdots\!00\)\( \beta_{1} - \)\(29\!\cdots\!26\)\( \beta_{2} - \)\(54\!\cdots\!70\)\( \beta_{3} + \)\(68\!\cdots\!88\)\( \beta_{4} - 2380295833270272000 \beta_{5}) q^{20} +(\)\(25\!\cdots\!20\)\( - \)\(30\!\cdots\!83\)\( \beta_{1} - \)\(83\!\cdots\!15\)\( \beta_{2} + \)\(11\!\cdots\!21\)\( \beta_{3} + \)\(54\!\cdots\!90\)\( \beta_{4} - 27991143699221201689 \beta_{5}) q^{22} +(-\)\(25\!\cdots\!80\)\( + \)\(20\!\cdots\!30\)\( \beta_{1} - \)\(12\!\cdots\!62\)\( \beta_{2} - \)\(11\!\cdots\!80\)\( \beta_{3} + \)\(36\!\cdots\!00\)\( \beta_{4} - \)\(54\!\cdots\!80\)\( \beta_{5}) q^{23} +(\)\(32\!\cdots\!75\)\( + \)\(83\!\cdots\!00\)\( \beta_{1} - \)\(15\!\cdots\!80\)\( \beta_{2} + \)\(22\!\cdots\!00\)\( \beta_{3} - \)\(19\!\cdots\!60\)\( \beta_{4} + \)\(28\!\cdots\!00\)\( \beta_{5}) q^{25} +(-\)\(19\!\cdots\!92\)\( - \)\(15\!\cdots\!38\)\( \beta_{1} + \)\(12\!\cdots\!88\)\( \beta_{2} - \)\(61\!\cdots\!76\)\( \beta_{3} + \)\(10\!\cdots\!80\)\( \beta_{4} + \)\(13\!\cdots\!56\)\( \beta_{5}) q^{26} +(\)\(24\!\cdots\!20\)\( + \)\(15\!\cdots\!20\)\( \beta_{1} + \)\(80\!\cdots\!48\)\( \beta_{2} - \)\(30\!\cdots\!76\)\( \beta_{3} + \)\(68\!\cdots\!60\)\( \beta_{4} - \)\(12\!\cdots\!16\)\( \beta_{5}) q^{28} +(-\)\(24\!\cdots\!70\)\( + \)\(40\!\cdots\!36\)\( \beta_{1} - \)\(12\!\cdots\!76\)\( \beta_{2} - \)\(83\!\cdots\!80\)\( \beta_{3} + \)\(18\!\cdots\!05\)\( \beta_{4} - \)\(74\!\cdots\!84\)\( \beta_{5}) q^{29} +(-\)\(69\!\cdots\!48\)\( - \)\(73\!\cdots\!20\)\( \beta_{1} + \)\(14\!\cdots\!60\)\( \beta_{2} - \)\(70\!\cdots\!28\)\( \beta_{3} - \)\(45\!\cdots\!40\)\( \beta_{4} - \)\(72\!\cdots\!48\)\( \beta_{5}) q^{31} +(-\)\(19\!\cdots\!60\)\( - \)\(40\!\cdots\!56\)\( \beta_{1} - \)\(74\!\cdots\!20\)\( \beta_{2} - \)\(46\!\cdots\!72\)\( \beta_{3} - \)\(95\!\cdots\!80\)\( \beta_{4} - \)\(24\!\cdots\!52\)\( \beta_{5}) q^{32} +(\)\(50\!\cdots\!76\)\( + \)\(49\!\cdots\!38\)\( \beta_{1} - \)\(83\!\cdots\!28\)\( \beta_{2} - \)\(13\!\cdots\!36\)\( \beta_{3} - \)\(34\!\cdots\!40\)\( \beta_{4} - \)\(38\!\cdots\!68\)\( \beta_{5}) q^{34} +(-\)\(45\!\cdots\!60\)\( + \)\(80\!\cdots\!00\)\( \beta_{1} + \)\(16\!\cdots\!88\)\( \beta_{2} - \)\(21\!\cdots\!00\)\( \beta_{3} - \)\(83\!\cdots\!44\)\( \beta_{4} - \)\(58\!\cdots\!60\)\( \beta_{5}) q^{35} +(\)\(16\!\cdots\!90\)\( - \)\(58\!\cdots\!40\)\( \beta_{1} - \)\(93\!\cdots\!84\)\( \beta_{2} - \)\(12\!\cdots\!52\)\( \beta_{3} - \)\(29\!\cdots\!05\)\( \beta_{4} + \)\(37\!\cdots\!68\)\( \beta_{5}) q^{37} +(-\)\(20\!\cdots\!80\)\( - \)\(38\!\cdots\!79\)\( \beta_{1} + \)\(20\!\cdots\!29\)\( \beta_{2} - \)\(21\!\cdots\!27\)\( \beta_{3} + \)\(22\!\cdots\!70\)\( \beta_{4} - \)\(23\!\cdots\!57\)\( \beta_{5}) q^{38} +(\)\(14\!\cdots\!00\)\( + \)\(41\!\cdots\!00\)\( \beta_{1} + \)\(38\!\cdots\!40\)\( \beta_{2} + \)\(16\!\cdots\!00\)\( \beta_{3} - \)\(38\!\cdots\!20\)\( \beta_{4} + \)\(10\!\cdots\!00\)\( \beta_{5}) q^{40} +(-\)\(83\!\cdots\!02\)\( + \)\(62\!\cdots\!20\)\( \beta_{1} - \)\(13\!\cdots\!60\)\( \beta_{2} + \)\(66\!\cdots\!88\)\( \beta_{3} + \)\(12\!\cdots\!40\)\( \beta_{4} + \)\(12\!\cdots\!08\)\( \beta_{5}) q^{41} +(\)\(46\!\cdots\!00\)\( + \)\(18\!\cdots\!21\)\( \beta_{1} - \)\(49\!\cdots\!89\)\( \beta_{2} + \)\(16\!\cdots\!60\)\( \beta_{3} - \)\(77\!\cdots\!00\)\( \beta_{4} + \)\(14\!\cdots\!60\)\( \beta_{5}) q^{43} +(\)\(14\!\cdots\!24\)\( - \)\(35\!\cdots\!48\)\( \beta_{1} - \)\(36\!\cdots\!12\)\( \beta_{2} + \)\(53\!\cdots\!36\)\( \beta_{3} - \)\(63\!\cdots\!60\)\( \beta_{4} + \)\(26\!\cdots\!08\)\( \beta_{5}) q^{44} +(-\)\(13\!\cdots\!48\)\( + \)\(59\!\cdots\!10\)\( \beta_{1} + \)\(48\!\cdots\!30\)\( \beta_{2} + \)\(77\!\cdots\!42\)\( \beta_{3} - \)\(42\!\cdots\!40\)\( \beta_{4} + \)\(11\!\cdots\!02\)\( \beta_{5}) q^{46} +(\)\(23\!\cdots\!80\)\( + \)\(77\!\cdots\!76\)\( \beta_{1} + \)\(13\!\cdots\!48\)\( \beta_{2} - \)\(61\!\cdots\!20\)\( \beta_{3} - \)\(95\!\cdots\!00\)\( \beta_{4} + \)\(24\!\cdots\!80\)\( \beta_{5}) q^{47} +(-\)\(95\!\cdots\!07\)\( - \)\(43\!\cdots\!80\)\( \beta_{1} - \)\(25\!\cdots\!80\)\( \beta_{2} + \)\(11\!\cdots\!32\)\( \beta_{3} + \)\(39\!\cdots\!60\)\( \beta_{4} + \)\(67\!\cdots\!52\)\( \beta_{5}) q^{49} +(-\)\(52\!\cdots\!00\)\( - \)\(89\!\cdots\!75\)\( \beta_{1} - \)\(35\!\cdots\!20\)\( \beta_{2} - \)\(41\!\cdots\!00\)\( \beta_{3} + \)\(13\!\cdots\!60\)\( \beta_{4} + \)\(30\!\cdots\!00\)\( \beta_{5}) q^{50} +(\)\(69\!\cdots\!00\)\( + \)\(22\!\cdots\!08\)\( \beta_{1} - \)\(26\!\cdots\!82\)\( \beta_{2} - \)\(10\!\cdots\!30\)\( \beta_{3} + \)\(69\!\cdots\!00\)\( \beta_{4} + \)\(58\!\cdots\!20\)\( \beta_{5}) q^{52} +(-\)\(10\!\cdots\!10\)\( - \)\(47\!\cdots\!84\)\( \beta_{1} + \)\(18\!\cdots\!52\)\( \beta_{2} + \)\(32\!\cdots\!44\)\( \beta_{3} + \)\(16\!\cdots\!85\)\( \beta_{4} - \)\(18\!\cdots\!96\)\( \beta_{5}) q^{53} +(\)\(72\!\cdots\!20\)\( - \)\(37\!\cdots\!50\)\( \beta_{1} - \)\(41\!\cdots\!46\)\( \beta_{2} - \)\(84\!\cdots\!20\)\( \beta_{3} + \)\(25\!\cdots\!48\)\( \beta_{4} + \)\(88\!\cdots\!00\)\( \beta_{5}) q^{55} +(-\)\(48\!\cdots\!20\)\( - \)\(40\!\cdots\!84\)\( \beta_{1} + \)\(36\!\cdots\!44\)\( \beta_{2} + \)\(10\!\cdots\!80\)\( \beta_{3} + \)\(23\!\cdots\!80\)\( \beta_{4} - \)\(26\!\cdots\!44\)\( \beta_{5}) q^{56} +(-\)\(29\!\cdots\!80\)\( + \)\(52\!\cdots\!46\)\( \beta_{1} + \)\(82\!\cdots\!04\)\( \beta_{2} + \)\(11\!\cdots\!12\)\( \beta_{3} - \)\(28\!\cdots\!20\)\( \beta_{4} + \)\(98\!\cdots\!92\)\( \beta_{5}) q^{58} +(\)\(41\!\cdots\!60\)\( - \)\(28\!\cdots\!53\)\( \beta_{1} - \)\(58\!\cdots\!67\)\( \beta_{2} - \)\(31\!\cdots\!52\)\( \beta_{3} - \)\(87\!\cdots\!00\)\( \beta_{4} - \)\(20\!\cdots\!60\)\( \beta_{5}) q^{59} +(-\)\(42\!\cdots\!98\)\( - \)\(26\!\cdots\!00\)\( \beta_{1} + \)\(94\!\cdots\!00\)\( \beta_{2} + \)\(60\!\cdots\!00\)\( \beta_{3} - \)\(40\!\cdots\!25\)\( \beta_{4} + \)\(64\!\cdots\!00\)\( \beta_{5}) q^{61} +(\)\(48\!\cdots\!80\)\( + \)\(21\!\cdots\!28\)\( \beta_{1} + \)\(10\!\cdots\!20\)\( \beta_{2} - \)\(85\!\cdots\!68\)\( \beta_{3} - \)\(67\!\cdots\!20\)\( \beta_{4} - \)\(10\!\cdots\!88\)\( \beta_{5}) q^{62} +(\)\(79\!\cdots\!08\)\( + \)\(23\!\cdots\!28\)\( \beta_{1} + \)\(31\!\cdots\!72\)\( \beta_{2} - \)\(57\!\cdots\!64\)\( \beta_{3} - \)\(20\!\cdots\!80\)\( \beta_{4} + \)\(15\!\cdots\!44\)\( \beta_{5}) q^{64} +(-\)\(20\!\cdots\!20\)\( - \)\(68\!\cdots\!00\)\( \beta_{1} - \)\(16\!\cdots\!24\)\( \beta_{2} + \)\(12\!\cdots\!00\)\( \beta_{3} - \)\(72\!\cdots\!88\)\( \beta_{4} - \)\(16\!\cdots\!20\)\( \beta_{5}) q^{65} +(\)\(15\!\cdots\!80\)\( - \)\(88\!\cdots\!13\)\( \beta_{1} - \)\(35\!\cdots\!11\)\( \beta_{2} + \)\(11\!\cdots\!12\)\( \beta_{3} - \)\(36\!\cdots\!20\)\( \beta_{4} - \)\(10\!\cdots\!08\)\( \beta_{5}) q^{67} +(-\)\(20\!\cdots\!60\)\( + \)\(44\!\cdots\!88\)\( \beta_{1} - \)\(58\!\cdots\!06\)\( \beta_{2} - \)\(10\!\cdots\!74\)\( \beta_{3} + \)\(73\!\cdots\!40\)\( \beta_{4} + \)\(19\!\cdots\!16\)\( \beta_{5}) q^{68} +(-\)\(57\!\cdots\!40\)\( + \)\(42\!\cdots\!00\)\( \beta_{1} + \)\(69\!\cdots\!92\)\( \beta_{2} - \)\(11\!\cdots\!60\)\( \beta_{3} - \)\(24\!\cdots\!96\)\( \beta_{4} - \)\(10\!\cdots\!00\)\( \beta_{5}) q^{70} +(\)\(42\!\cdots\!48\)\( + \)\(12\!\cdots\!50\)\( \beta_{1} + \)\(11\!\cdots\!50\)\( \beta_{2} - \)\(41\!\cdots\!00\)\( \beta_{3} - \)\(35\!\cdots\!00\)\( \beta_{4} + \)\(19\!\cdots\!00\)\( \beta_{5}) q^{71} +(-\)\(51\!\cdots\!70\)\( - \)\(10\!\cdots\!60\)\( \beta_{1} - \)\(12\!\cdots\!08\)\( \beta_{2} - \)\(21\!\cdots\!88\)\( \beta_{3} - \)\(39\!\cdots\!70\)\( \beta_{4} - \)\(24\!\cdots\!08\)\( \beta_{5}) q^{73} +(\)\(40\!\cdots\!64\)\( + \)\(24\!\cdots\!70\)\( \beta_{1} - \)\(22\!\cdots\!80\)\( \beta_{2} + \)\(10\!\cdots\!92\)\( \beta_{3} - \)\(51\!\cdots\!40\)\( \beta_{4} + \)\(65\!\cdots\!12\)\( \beta_{5}) q^{74} +(\)\(16\!\cdots\!40\)\( + \)\(14\!\cdots\!28\)\( \beta_{1} + \)\(89\!\cdots\!52\)\( \beta_{2} + \)\(39\!\cdots\!00\)\( \beta_{3} - \)\(17\!\cdots\!60\)\( \beta_{4} + \)\(84\!\cdots\!08\)\( \beta_{5}) q^{76} +(-\)\(26\!\cdots\!00\)\( + \)\(63\!\cdots\!84\)\( \beta_{1} + \)\(10\!\cdots\!64\)\( \beta_{2} - \)\(45\!\cdots\!92\)\( \beta_{3} - \)\(82\!\cdots\!80\)\( \beta_{4} - \)\(16\!\cdots\!72\)\( \beta_{5}) q^{77} +(\)\(19\!\cdots\!20\)\( - \)\(22\!\cdots\!76\)\( \beta_{1} - \)\(29\!\cdots\!24\)\( \beta_{2} - \)\(19\!\cdots\!72\)\( \beta_{3} + \)\(21\!\cdots\!60\)\( \beta_{4} + \)\(52\!\cdots\!92\)\( \beta_{5}) q^{79} +(-\)\(20\!\cdots\!60\)\( - \)\(85\!\cdots\!00\)\( \beta_{1} - \)\(45\!\cdots\!72\)\( \beta_{2} - \)\(62\!\cdots\!40\)\( \beta_{3} + \)\(43\!\cdots\!36\)\( \beta_{4} - \)\(21\!\cdots\!00\)\( \beta_{5}) q^{80} +(-\)\(42\!\cdots\!80\)\( - \)\(10\!\cdots\!78\)\( \beta_{1} - \)\(30\!\cdots\!20\)\( \beta_{2} - \)\(98\!\cdots\!72\)\( \beta_{3} + \)\(31\!\cdots\!20\)\( \beta_{4} + \)\(15\!\cdots\!48\)\( \beta_{5}) q^{82} +(\)\(13\!\cdots\!60\)\( + \)\(11\!\cdots\!97\)\( \beta_{1} + \)\(35\!\cdots\!71\)\( \beta_{2} - \)\(13\!\cdots\!00\)\( \beta_{3} + \)\(12\!\cdots\!00\)\( \beta_{4} - \)\(39\!\cdots\!00\)\( \beta_{5}) q^{83} +(-\)\(60\!\cdots\!40\)\( + \)\(10\!\cdots\!00\)\( \beta_{1} + \)\(86\!\cdots\!72\)\( \beta_{2} + \)\(12\!\cdots\!00\)\( \beta_{3} + \)\(46\!\cdots\!14\)\( \beta_{4} + \)\(69\!\cdots\!60\)\( \beta_{5}) q^{85} +(-\)\(12\!\cdots\!72\)\( - \)\(73\!\cdots\!29\)\( \beta_{1} - \)\(24\!\cdots\!01\)\( \beta_{2} - \)\(58\!\cdots\!17\)\( \beta_{3} + \)\(58\!\cdots\!70\)\( \beta_{4} + \)\(24\!\cdots\!89\)\( \beta_{5}) q^{86} +(-\)\(80\!\cdots\!60\)\( - \)\(16\!\cdots\!48\)\( \beta_{1} - \)\(84\!\cdots\!52\)\( \beta_{2} - \)\(10\!\cdots\!48\)\( \beta_{3} + \)\(45\!\cdots\!80\)\( \beta_{4} + \)\(22\!\cdots\!32\)\( \beta_{5}) q^{88} +(-\)\(89\!\cdots\!10\)\( + \)\(17\!\cdots\!48\)\( \beta_{1} + \)\(16\!\cdots\!32\)\( \beta_{2} - \)\(12\!\cdots\!60\)\( \beta_{3} + \)\(25\!\cdots\!90\)\( \beta_{4} + \)\(93\!\cdots\!68\)\( \beta_{5}) q^{89} +(\)\(57\!\cdots\!72\)\( + \)\(40\!\cdots\!84\)\( \beta_{1} - \)\(39\!\cdots\!44\)\( \beta_{2} - \)\(30\!\cdots\!60\)\( \beta_{3} + \)\(23\!\cdots\!20\)\( \beta_{4} - \)\(55\!\cdots\!36\)\( \beta_{5}) q^{91} +(-\)\(31\!\cdots\!80\)\( + \)\(56\!\cdots\!68\)\( \beta_{1} - \)\(11\!\cdots\!44\)\( \beta_{2} - \)\(56\!\cdots\!48\)\( \beta_{3} - \)\(10\!\cdots\!20\)\( \beta_{4} - \)\(16\!\cdots\!68\)\( \beta_{5}) q^{92} +(-\)\(49\!\cdots\!84\)\( - \)\(14\!\cdots\!04\)\( \beta_{1} - \)\(13\!\cdots\!16\)\( \beta_{2} - \)\(10\!\cdots\!04\)\( \beta_{3} - \)\(24\!\cdots\!40\)\( \beta_{4} - \)\(90\!\cdots\!48\)\( \beta_{5}) q^{94} +(-\)\(32\!\cdots\!00\)\( - \)\(15\!\cdots\!50\)\( \beta_{1} - \)\(10\!\cdots\!10\)\( \beta_{2} - \)\(34\!\cdots\!00\)\( \beta_{3} - \)\(34\!\cdots\!20\)\( \beta_{4} - \)\(78\!\cdots\!00\)\( \beta_{5}) q^{95} +(-\)\(12\!\cdots\!30\)\( + \)\(94\!\cdots\!04\)\( \beta_{1} + \)\(13\!\cdots\!72\)\( \beta_{2} + \)\(16\!\cdots\!44\)\( \beta_{3} - \)\(53\!\cdots\!90\)\( \beta_{4} - \)\(13\!\cdots\!96\)\( \beta_{5}) q^{97} +(\)\(27\!\cdots\!20\)\( + \)\(84\!\cdots\!27\)\( \beta_{1} - \)\(63\!\cdots\!20\)\( \beta_{2} + \)\(57\!\cdots\!72\)\( \beta_{3} + \)\(83\!\cdots\!80\)\( \beta_{4} + \)\(17\!\cdots\!52\)\( \beta_{5}) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 57080822040q^{2} + \)\(17\!\cdots\!28\)\(q^{4} + \)\(38\!\cdots\!40\)\(q^{5} + \)\(19\!\cdots\!00\)\(q^{7} - \)\(44\!\cdots\!20\)\(q^{8} + O(q^{10}) \) \( 6q + 57080822040q^{2} + \)\(17\!\cdots\!28\)\(q^{4} + \)\(38\!\cdots\!40\)\(q^{5} + \)\(19\!\cdots\!00\)\(q^{7} - \)\(44\!\cdots\!20\)\(q^{8} + \)\(13\!\cdots\!60\)\(q^{10} + \)\(94\!\cdots\!88\)\(q^{11} + \)\(53\!\cdots\!20\)\(q^{13} - \)\(82\!\cdots\!76\)\(q^{14} + \)\(26\!\cdots\!16\)\(q^{16} - \)\(18\!\cdots\!80\)\(q^{17} + \)\(10\!\cdots\!80\)\(q^{19} - \)\(92\!\cdots\!80\)\(q^{20} + \)\(15\!\cdots\!20\)\(q^{22} - \)\(15\!\cdots\!80\)\(q^{23} + \)\(19\!\cdots\!50\)\(q^{25} - \)\(11\!\cdots\!52\)\(q^{26} + \)\(14\!\cdots\!20\)\(q^{28} - \)\(14\!\cdots\!20\)\(q^{29} - \)\(41\!\cdots\!88\)\(q^{31} - \)\(11\!\cdots\!60\)\(q^{32} + \)\(30\!\cdots\!56\)\(q^{34} - \)\(27\!\cdots\!60\)\(q^{35} + \)\(98\!\cdots\!40\)\(q^{37} - \)\(12\!\cdots\!80\)\(q^{38} + \)\(88\!\cdots\!00\)\(q^{40} - \)\(50\!\cdots\!12\)\(q^{41} + \)\(27\!\cdots\!00\)\(q^{43} + \)\(86\!\cdots\!44\)\(q^{44} - \)\(82\!\cdots\!88\)\(q^{46} + \)\(13\!\cdots\!80\)\(q^{47} - \)\(57\!\cdots\!42\)\(q^{49} - \)\(31\!\cdots\!00\)\(q^{50} + \)\(41\!\cdots\!00\)\(q^{52} - \)\(64\!\cdots\!60\)\(q^{53} + \)\(43\!\cdots\!20\)\(q^{55} - \)\(28\!\cdots\!20\)\(q^{56} - \)\(17\!\cdots\!80\)\(q^{58} + \)\(24\!\cdots\!60\)\(q^{59} - \)\(25\!\cdots\!88\)\(q^{61} + \)\(29\!\cdots\!80\)\(q^{62} + \)\(47\!\cdots\!48\)\(q^{64} - \)\(12\!\cdots\!20\)\(q^{65} + \)\(95\!\cdots\!80\)\(q^{67} - \)\(12\!\cdots\!60\)\(q^{68} - \)\(34\!\cdots\!40\)\(q^{70} + \)\(25\!\cdots\!88\)\(q^{71} - \)\(30\!\cdots\!20\)\(q^{73} + \)\(24\!\cdots\!84\)\(q^{74} + \)\(10\!\cdots\!40\)\(q^{76} - \)\(15\!\cdots\!00\)\(q^{77} + \)\(11\!\cdots\!20\)\(q^{79} - \)\(12\!\cdots\!60\)\(q^{80} - \)\(25\!\cdots\!80\)\(q^{82} + \)\(79\!\cdots\!60\)\(q^{83} - \)\(36\!\cdots\!40\)\(q^{85} - \)\(72\!\cdots\!32\)\(q^{86} - \)\(48\!\cdots\!60\)\(q^{88} - \)\(53\!\cdots\!60\)\(q^{89} + \)\(34\!\cdots\!32\)\(q^{91} - \)\(18\!\cdots\!80\)\(q^{92} - \)\(29\!\cdots\!04\)\(q^{94} - \)\(19\!\cdots\!00\)\(q^{95} - \)\(74\!\cdots\!80\)\(q^{97} + \)\(16\!\cdots\!20\)\(q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 3 x^{5} - 38457853073924058692 x^{4} - 10276556354621685339901678086 x^{3} + 371187556674475060057870954681799784505 x^{2} + 52686123927652036687598761277591247931691204025 x - 675344021115865838575279495800656435684060652010336995750\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 72 \nu - 36 \)
\(\beta_{2}\)\(=\)\((\)\(-4948663505321853 \nu^{5} + 55881811387913650437514590 \nu^{4} + 52606012732895043205010735272635258 \nu^{3} - 1201592828386165762149652689242435079026451552 \nu^{2} + 319255507039165145569757904502642123052905054955529291 \nu + 2824371663419232055800468842793976703776935082646788469007152058\)\()/ \)\(10\!\cdots\!92\)\( \)
\(\beta_{3}\)\(=\)\((\)\(4948663505321853 \nu^{5} - 55881811387913650437514590 \nu^{4} - 52606012732895043205010735272635258 \nu^{3} + 1755095654489252908059771761513699488141354080 \nu^{2} - 541112784048666521961353184744658440761253271177159755 \nu - 9919881784066391320337103830604500724241879350737223725145748410\)\()/ \)\(10\!\cdots\!92\)\( \)
\(\beta_{4}\)\(=\)\((\)\(28511749716630153831 \nu^{5} - 38083656874945883578519397562 \nu^{4} - 963750039419722142343400597053740315214 \nu^{3} + 676707927180689783081282528077970424187899981600 \nu^{2} + 6997720090731681654577112441812622855800067824107978126735 \nu - 2510088276770176763088355182310893932211704380215188959388288448270\)\()/ \)\(33\!\cdots\!60\)\( \)
\(\beta_{5}\)\(=\)\((\)\(135360654096714045236379 \nu^{5} - 181916208099474888046078592692146 \nu^{4} - 3229398604947863711716280584882203213740214 \nu^{3} + 1182663053764971608085531729909002820674984623978656 \nu^{2} + 8277972932132865971256201289051763726161866888924340410394275 \nu + 7463683837539256728882869193632480747348831952586787915716598591246730\)\()/ \)\(26\!\cdots\!48\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1} + 36\)\()/72\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} + \beta_{2} + 28859335836 \beta_{1} + 66455170111740773427552\)\()/5184\)
\(\nu^{3}\)\(=\)\((\)\(927 \beta_{5} - 5496670 \beta_{4} + 13763504113 \beta_{3} + 13847230069 \beta_{2} + 12411954295199886478449 \beta_{1} + 239731509332049065134727670101088\)\()/46656\)
\(\nu^{4}\)\(=\)\((\)\(19416283325423 \beta_{5} - 65775546702321150 \beta_{4} + 2070543718263884927080 \beta_{3} + 2982224271853793103276 \beta_{2} + 129858023070497361753663350041341 \beta_{1} + 103104816764360930778502571288626452423490272\)\()/419904\)
\(\nu^{5}\)\(=\)\((\)\(1385746010729527810284396 \beta_{5} - 5708884777145460848148822360 \beta_{4} + 22636204558832382529314688604361 \beta_{3} + 28230689814524467376000755560217 \beta_{2} + 11081391647423040002963344026763817931453572 \beta_{1} + 539358563573136492249987437118564971890421060743009584\)\()/1889568\)

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
5.23553e9
3.19848e9
1.46671e9
−1.60397e9
−3.89081e9
−4.40594e9
−3.67444e11 0 9.72365e22 −2.43619e26 0 1.21530e31 −2.18473e34 0 8.95165e37
1.2 −2.20777e11 0 1.09637e22 2.30711e25 0 1.96027e31 5.92020e33 0 −5.09358e36
1.3 −9.60898e10 0 −2.85457e22 1.49650e26 0 −3.16100e31 6.37312e33 0 −1.43799e37
1.4 1.25000e11 0 −2.21540e22 −1.80927e26 0 2.76229e31 −7.49160e33 0 −2.26158e37
1.5 2.89652e11 0 4.61192e22 2.47463e26 0 −6.83079e31 2.41577e33 0 7.16781e37
1.6 3.26741e11 0 6.89808e22 4.33451e25 0 4.24638e31 1.01949e34 0 1.41626e37
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Atkin-Lehner signs

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

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 9.76.a.c 6
3.b odd 2 1 1.76.a.a 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.76.a.a 6 3.b odd 2 1
9.76.a.c 6 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} - 57080822040 T_{2}^{5} - \)\(19\!\cdots\!68\)\( T_{2}^{4} + \)\(11\!\cdots\!80\)\( T_{2}^{3} + \)\(97\!\cdots\!08\)\( T_{2}^{2} - \)\(29\!\cdots\!40\)\( T_{2} - \)\(92\!\cdots\!16\)\( \) acting on \(S_{76}^{\mathrm{new}}(\Gamma_0(9))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -\)\(92\!\cdots\!16\)\( - \)\(29\!\cdots\!40\)\( T + \)\(97\!\cdots\!08\)\( T^{2} + \)\(11\!\cdots\!80\)\( T^{3} - \)\(19\!\cdots\!68\)\( T^{4} - 57080822040 T^{5} + T^{6} \)
$3$ \( T^{6} \)
$5$ \( \)\(16\!\cdots\!00\)\( - \)\(11\!\cdots\!00\)\( T + \)\(16\!\cdots\!00\)\( T^{2} + \)\(40\!\cdots\!00\)\( T^{3} - \)\(88\!\cdots\!00\)\( T^{4} - \)\(38\!\cdots\!40\)\( T^{5} + T^{6} \)
$7$ \( \)\(60\!\cdots\!44\)\( - \)\(88\!\cdots\!00\)\( T + \)\(27\!\cdots\!88\)\( T^{2} + \)\(89\!\cdots\!00\)\( T^{3} - \)\(43\!\cdots\!08\)\( T^{4} - \)\(19\!\cdots\!00\)\( T^{5} + T^{6} \)
$11$ \( -\)\(76\!\cdots\!36\)\( - \)\(39\!\cdots\!08\)\( T + \)\(84\!\cdots\!40\)\( T^{2} + \)\(12\!\cdots\!60\)\( T^{3} - \)\(17\!\cdots\!40\)\( T^{4} - \)\(94\!\cdots\!88\)\( T^{5} + T^{6} \)
$13$ \( \)\(87\!\cdots\!16\)\( - \)\(18\!\cdots\!20\)\( T + \)\(79\!\cdots\!08\)\( T^{2} + \)\(50\!\cdots\!60\)\( T^{3} - \)\(11\!\cdots\!32\)\( T^{4} - \)\(53\!\cdots\!20\)\( T^{5} + T^{6} \)
$17$ \( -\)\(35\!\cdots\!36\)\( - \)\(12\!\cdots\!20\)\( T - \)\(11\!\cdots\!52\)\( T^{2} - \)\(33\!\cdots\!60\)\( T^{3} - \)\(14\!\cdots\!88\)\( T^{4} + \)\(18\!\cdots\!80\)\( T^{5} + T^{6} \)
$19$ \( -\)\(12\!\cdots\!00\)\( - \)\(71\!\cdots\!00\)\( T + \)\(15\!\cdots\!00\)\( T^{2} + \)\(19\!\cdots\!00\)\( T^{3} - \)\(24\!\cdots\!00\)\( T^{4} - \)\(10\!\cdots\!80\)\( T^{5} + T^{6} \)
$23$ \( \)\(10\!\cdots\!96\)\( + \)\(26\!\cdots\!80\)\( T + \)\(91\!\cdots\!68\)\( T^{2} - \)\(40\!\cdots\!40\)\( T^{3} - \)\(20\!\cdots\!52\)\( T^{4} + \)\(15\!\cdots\!80\)\( T^{5} + T^{6} \)
$29$ \( \)\(99\!\cdots\!00\)\( + \)\(26\!\cdots\!00\)\( T - \)\(16\!\cdots\!00\)\( T^{2} - \)\(54\!\cdots\!00\)\( T^{3} + \)\(14\!\cdots\!00\)\( T^{4} + \)\(14\!\cdots\!20\)\( T^{5} + T^{6} \)
$31$ \( -\)\(14\!\cdots\!36\)\( + \)\(46\!\cdots\!08\)\( T + \)\(16\!\cdots\!40\)\( T^{2} - \)\(74\!\cdots\!60\)\( T^{3} - \)\(26\!\cdots\!40\)\( T^{4} + \)\(41\!\cdots\!88\)\( T^{5} + T^{6} \)
$37$ \( -\)\(18\!\cdots\!96\)\( - \)\(32\!\cdots\!40\)\( T + \)\(33\!\cdots\!68\)\( T^{2} + \)\(59\!\cdots\!80\)\( T^{3} - \)\(89\!\cdots\!48\)\( T^{4} - \)\(98\!\cdots\!40\)\( T^{5} + T^{6} \)
$41$ \( \)\(16\!\cdots\!64\)\( + \)\(46\!\cdots\!92\)\( T - \)\(70\!\cdots\!60\)\( T^{2} - \)\(16\!\cdots\!40\)\( T^{3} - \)\(25\!\cdots\!40\)\( T^{4} + \)\(50\!\cdots\!12\)\( T^{5} + T^{6} \)
$43$ \( \)\(52\!\cdots\!56\)\( - \)\(59\!\cdots\!00\)\( T + \)\(16\!\cdots\!88\)\( T^{2} + \)\(27\!\cdots\!00\)\( T^{3} - \)\(92\!\cdots\!92\)\( T^{4} - \)\(27\!\cdots\!00\)\( T^{5} + T^{6} \)
$47$ \( \)\(11\!\cdots\!24\)\( - \)\(22\!\cdots\!80\)\( T - \)\(53\!\cdots\!72\)\( T^{2} + \)\(35\!\cdots\!60\)\( T^{3} + \)\(11\!\cdots\!72\)\( T^{4} - \)\(13\!\cdots\!80\)\( T^{5} + T^{6} \)
$53$ \( \)\(83\!\cdots\!36\)\( - \)\(12\!\cdots\!40\)\( T - \)\(66\!\cdots\!52\)\( T^{2} - \)\(38\!\cdots\!80\)\( T^{3} - \)\(48\!\cdots\!12\)\( T^{4} + \)\(64\!\cdots\!60\)\( T^{5} + T^{6} \)
$59$ \( -\)\(41\!\cdots\!00\)\( + \)\(42\!\cdots\!00\)\( T + \)\(10\!\cdots\!00\)\( T^{2} + \)\(20\!\cdots\!00\)\( T^{3} - \)\(18\!\cdots\!00\)\( T^{4} - \)\(24\!\cdots\!60\)\( T^{5} + T^{6} \)
$61$ \( \)\(57\!\cdots\!64\)\( + \)\(28\!\cdots\!08\)\( T - \)\(54\!\cdots\!60\)\( T^{2} - \)\(16\!\cdots\!60\)\( T^{3} + \)\(78\!\cdots\!60\)\( T^{4} + \)\(25\!\cdots\!88\)\( T^{5} + T^{6} \)
$67$ \( \)\(21\!\cdots\!64\)\( + \)\(86\!\cdots\!20\)\( T - \)\(20\!\cdots\!52\)\( T^{2} + \)\(63\!\cdots\!60\)\( T^{3} + \)\(18\!\cdots\!12\)\( T^{4} - \)\(95\!\cdots\!80\)\( T^{5} + T^{6} \)
$71$ \( \)\(14\!\cdots\!64\)\( + \)\(81\!\cdots\!92\)\( T + \)\(82\!\cdots\!40\)\( T^{2} - \)\(70\!\cdots\!40\)\( T^{3} - \)\(17\!\cdots\!40\)\( T^{4} - \)\(25\!\cdots\!88\)\( T^{5} + T^{6} \)
$73$ \( -\)\(66\!\cdots\!04\)\( - \)\(55\!\cdots\!80\)\( T - \)\(13\!\cdots\!32\)\( T^{2} - \)\(46\!\cdots\!60\)\( T^{3} + \)\(25\!\cdots\!48\)\( T^{4} + \)\(30\!\cdots\!20\)\( T^{5} + T^{6} \)
$79$ \( \)\(15\!\cdots\!00\)\( - \)\(18\!\cdots\!00\)\( T + \)\(19\!\cdots\!00\)\( T^{2} + \)\(31\!\cdots\!00\)\( T^{3} - \)\(33\!\cdots\!00\)\( T^{4} - \)\(11\!\cdots\!20\)\( T^{5} + T^{6} \)
$83$ \( \)\(13\!\cdots\!76\)\( - \)\(50\!\cdots\!60\)\( T + \)\(25\!\cdots\!28\)\( T^{2} + \)\(85\!\cdots\!80\)\( T^{3} - \)\(13\!\cdots\!72\)\( T^{4} - \)\(79\!\cdots\!60\)\( T^{5} + T^{6} \)
$89$ \( -\)\(16\!\cdots\!00\)\( - \)\(10\!\cdots\!00\)\( T - \)\(14\!\cdots\!00\)\( T^{2} - \)\(20\!\cdots\!00\)\( T^{3} + \)\(80\!\cdots\!00\)\( T^{4} + \)\(53\!\cdots\!60\)\( T^{5} + T^{6} \)
$97$ \( \)\(36\!\cdots\!24\)\( + \)\(64\!\cdots\!80\)\( T - \)\(19\!\cdots\!72\)\( T^{2} - \)\(54\!\cdots\!60\)\( T^{3} + \)\(25\!\cdots\!72\)\( T^{4} + \)\(74\!\cdots\!80\)\( T^{5} + T^{6} \)
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