Properties

Label 9.68.a.a
Level 9
Weight 68
Character orbit 9.a
Self dual yes
Analytic conductor 255.861
Analytic rank 1
Dimension 5
CM no
Inner twists 1

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 68 \)
Character orbit: \([\chi]\) \(=\) 9.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(255.861316737\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
Defining polynomial: \(x^{5} - x^{4} - 939384011925257456 x^{3} + 31046449413968483513911200 x^{2} + 156793504704482691874379743265203200 x + 20916736226052669578405116700517591609696000\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{40}\cdot 3^{20}\cdot 5^{4}\cdot 7^{2}\cdot 11\cdot 13\cdot 17 \)
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.

\(f(q)\) \(=\) \( q +(-1110980251 + \beta_{1}) q^{2} +(70094400875776969497 - 3411753104 \beta_{1} + 25 \beta_{2} + \beta_{3}) q^{4} +(-\)\(66\!\cdots\!11\)\( - 5225164249574 \beta_{1} - 3571 \beta_{2} + 172 \beta_{3} - \beta_{4}) q^{5} +(\)\(67\!\cdots\!00\)\( - 234148091773369480 \beta_{1} + 2061155558 \beta_{2} + 83062000 \beta_{3} + 11340 \beta_{4}) q^{7} +(-\)\(65\!\cdots\!32\)\( + 61324177880091651072 \beta_{1} - 214533517064 \beta_{2} - 12774152200 \beta_{3} + 3277120 \beta_{4}) q^{8} +O(q^{10})\) \( q +(-1110980251 + \beta_{1}) q^{2} +(70094400875776969497 - 3411753104 \beta_{1} + 25 \beta_{2} + \beta_{3}) q^{4} +(-\)\(66\!\cdots\!11\)\( - 5225164249574 \beta_{1} - 3571 \beta_{2} + 172 \beta_{3} - \beta_{4}) q^{5} +(\)\(67\!\cdots\!00\)\( - 234148091773369480 \beta_{1} + 2061155558 \beta_{2} + 83062000 \beta_{3} + 11340 \beta_{4}) q^{7} +(-\)\(65\!\cdots\!32\)\( + 61324177880091651072 \beta_{1} - 214533517064 \beta_{2} - 12774152200 \beta_{3} + 3277120 \beta_{4}) q^{8} +(-\)\(10\!\cdots\!54\)\( - \)\(36\!\cdots\!86\)\( \beta_{1} - 1599242579293344 \beta_{2} - 10724715572992 \beta_{3} - 6662909664 \beta_{4}) q^{10} +(-\)\(40\!\cdots\!82\)\( + \)\(14\!\cdots\!60\)\( \beta_{1} + 18130035514135105 \beta_{2} - 337416804371680 \beta_{3} - 154884117080 \beta_{4}) q^{11} +(\)\(35\!\cdots\!87\)\( - \)\(19\!\cdots\!62\)\( \beta_{1} + 6293359129886549955 \beta_{2} - 29875083592222700 \beta_{3} + 34622077787505 \beta_{4}) q^{13} +(-\)\(58\!\cdots\!68\)\( + \)\(18\!\cdots\!52\)\( \beta_{1} - 184617190698878000 \beta_{2} - 964772473179561088 \beta_{3} + 344998573654000 \beta_{4}) q^{14} +(\)\(36\!\cdots\!68\)\( - \)\(18\!\cdots\!32\)\( \beta_{1} + \)\(42\!\cdots\!40\)\( \beta_{2} + 43878715027197261888 \beta_{3} - 11266877309667840 \beta_{4}) q^{16} +(-\)\(15\!\cdots\!28\)\( + \)\(71\!\cdots\!68\)\( \beta_{1} - \)\(39\!\cdots\!54\)\( \beta_{2} - \)\(18\!\cdots\!00\)\( \beta_{3} + 8396275243314890 \beta_{4}) q^{17} +(\)\(79\!\cdots\!86\)\( + \)\(29\!\cdots\!24\)\( \beta_{1} - \)\(25\!\cdots\!15\)\( \beta_{2} + \)\(24\!\cdots\!64\)\( \beta_{3} - 7385036747071057560 \beta_{4}) q^{19} +(\)\(30\!\cdots\!02\)\( - \)\(24\!\cdots\!32\)\( \beta_{1} - \)\(86\!\cdots\!78\)\( \beta_{2} + \)\(48\!\cdots\!46\)\( \beta_{3} - 47581246658342739968 \beta_{4}) q^{20} +(\)\(36\!\cdots\!72\)\( - \)\(71\!\cdots\!72\)\( \beta_{1} - \)\(14\!\cdots\!60\)\( \beta_{2} + \)\(32\!\cdots\!00\)\( \beta_{3} + \)\(20\!\cdots\!80\)\( \beta_{4}) q^{22} +(\)\(83\!\cdots\!32\)\( + \)\(73\!\cdots\!88\)\( \beta_{1} - \)\(55\!\cdots\!02\)\( \beta_{2} + \)\(10\!\cdots\!00\)\( \beta_{3} - \)\(38\!\cdots\!00\)\( \beta_{4}) q^{23} +(\)\(35\!\cdots\!75\)\( + \)\(38\!\cdots\!00\)\( \beta_{1} - \)\(38\!\cdots\!00\)\( \beta_{2} + \)\(13\!\cdots\!00\)\( \beta_{3} + \)\(20\!\cdots\!00\)\( \beta_{4}) q^{25} +(-\)\(42\!\cdots\!50\)\( + \)\(62\!\cdots\!18\)\( \beta_{1} + \)\(53\!\cdots\!80\)\( \beta_{2} + \)\(20\!\cdots\!68\)\( \beta_{3} + \)\(74\!\cdots\!20\)\( \beta_{4}) q^{26} +(\)\(29\!\cdots\!12\)\( - \)\(17\!\cdots\!92\)\( \beta_{1} + \)\(77\!\cdots\!08\)\( \beta_{2} + \)\(15\!\cdots\!00\)\( \beta_{3} - \)\(48\!\cdots\!80\)\( \beta_{4}) q^{28} +(-\)\(37\!\cdots\!99\)\( - \)\(75\!\cdots\!66\)\( \beta_{1} + \)\(31\!\cdots\!05\)\( \beta_{2} + \)\(43\!\cdots\!64\)\( \beta_{3} + \)\(15\!\cdots\!95\)\( \beta_{4}) q^{29} +(\)\(73\!\cdots\!92\)\( + \)\(24\!\cdots\!20\)\( \beta_{1} + \)\(15\!\cdots\!60\)\( \beta_{2} - \)\(46\!\cdots\!60\)\( \beta_{3} - \)\(18\!\cdots\!60\)\( \beta_{4}) q^{31} +(-\)\(31\!\cdots\!16\)\( + \)\(66\!\cdots\!16\)\( \beta_{1} - \)\(37\!\cdots\!80\)\( \beta_{2} - \)\(55\!\cdots\!00\)\( \beta_{3} - \)\(13\!\cdots\!60\)\( \beta_{4}) q^{32} +(\)\(15\!\cdots\!94\)\( - \)\(77\!\cdots\!38\)\( \beta_{1} + \)\(21\!\cdots\!60\)\( \beta_{2} + \)\(98\!\cdots\!92\)\( \beta_{3} - \)\(35\!\cdots\!60\)\( \beta_{4}) q^{34} +(-\)\(90\!\cdots\!16\)\( - \)\(24\!\cdots\!44\)\( \beta_{1} - \)\(66\!\cdots\!76\)\( \beta_{2} + \)\(28\!\cdots\!32\)\( \beta_{3} - \)\(63\!\cdots\!56\)\( \beta_{4}) q^{35} +(-\)\(11\!\cdots\!21\)\( - \)\(90\!\cdots\!14\)\( \beta_{1} - \)\(79\!\cdots\!69\)\( \beta_{2} + \)\(11\!\cdots\!00\)\( \beta_{3} - \)\(24\!\cdots\!35\)\( \beta_{4}) q^{37} +(\)\(62\!\cdots\!32\)\( + \)\(27\!\cdots\!28\)\( \beta_{1} - \)\(65\!\cdots\!56\)\( \beta_{2} + \)\(56\!\cdots\!00\)\( \beta_{3} - \)\(43\!\cdots\!60\)\( \beta_{4}) q^{38} +(-\)\(37\!\cdots\!60\)\( + \)\(14\!\cdots\!60\)\( \beta_{1} + \)\(99\!\cdots\!40\)\( \beta_{2} - \)\(12\!\cdots\!80\)\( \beta_{3} - \)\(28\!\cdots\!60\)\( \beta_{4}) q^{40} +(-\)\(22\!\cdots\!62\)\( - \)\(16\!\cdots\!20\)\( \beta_{1} + \)\(32\!\cdots\!40\)\( \beta_{2} - \)\(38\!\cdots\!40\)\( \beta_{3} + \)\(62\!\cdots\!60\)\( \beta_{4}) q^{41} +(\)\(13\!\cdots\!90\)\( - \)\(39\!\cdots\!00\)\( \beta_{1} - \)\(89\!\cdots\!29\)\( \beta_{2} - \)\(23\!\cdots\!00\)\( \beta_{3} - \)\(67\!\cdots\!00\)\( \beta_{4}) q^{43} +(-\)\(98\!\cdots\!64\)\( + \)\(59\!\cdots\!68\)\( \beta_{1} - \)\(45\!\cdots\!60\)\( \beta_{2} - \)\(45\!\cdots\!12\)\( \beta_{3} + \)\(15\!\cdots\!60\)\( \beta_{4}) q^{44} +(\)\(15\!\cdots\!52\)\( + \)\(10\!\cdots\!40\)\( \beta_{1} - \)\(49\!\cdots\!40\)\( \beta_{2} - \)\(28\!\cdots\!40\)\( \beta_{3} - \)\(18\!\cdots\!60\)\( \beta_{4}) q^{46} +(\)\(25\!\cdots\!48\)\( - \)\(44\!\cdots\!28\)\( \beta_{1} - \)\(75\!\cdots\!12\)\( \beta_{2} + \)\(14\!\cdots\!00\)\( \beta_{3} + \)\(15\!\cdots\!00\)\( \beta_{4}) q^{47} +(-\)\(15\!\cdots\!87\)\( - \)\(12\!\cdots\!20\)\( \beta_{1} + \)\(73\!\cdots\!60\)\( \beta_{2} + \)\(12\!\cdots\!00\)\( \beta_{3} + \)\(26\!\cdots\!40\)\( \beta_{4}) q^{49} +(\)\(78\!\cdots\!75\)\( + \)\(39\!\cdots\!75\)\( \beta_{1} + \)\(36\!\cdots\!00\)\( \beta_{2} + \)\(35\!\cdots\!00\)\( \beta_{3} + \)\(11\!\cdots\!00\)\( \beta_{4}) q^{50} +(\)\(13\!\cdots\!30\)\( + \)\(54\!\cdots\!40\)\( \beta_{1} + \)\(13\!\cdots\!78\)\( \beta_{2} + \)\(48\!\cdots\!50\)\( \beta_{3} + \)\(46\!\cdots\!00\)\( \beta_{4}) q^{52} +(-\)\(20\!\cdots\!31\)\( - \)\(31\!\cdots\!14\)\( \beta_{1} - \)\(14\!\cdots\!43\)\( \beta_{2} - \)\(20\!\cdots\!00\)\( \beta_{3} + \)\(66\!\cdots\!95\)\( \beta_{4}) q^{53} +(\)\(13\!\cdots\!92\)\( + \)\(32\!\cdots\!28\)\( \beta_{1} - \)\(40\!\cdots\!38\)\( \beta_{2} - \)\(67\!\cdots\!84\)\( \beta_{3} + \)\(64\!\cdots\!72\)\( \beta_{4}) q^{55} +(-\)\(32\!\cdots\!84\)\( + \)\(30\!\cdots\!44\)\( \beta_{1} - \)\(71\!\cdots\!20\)\( \beta_{2} - \)\(19\!\cdots\!76\)\( \beta_{3} + \)\(30\!\cdots\!20\)\( \beta_{4}) q^{56} +(-\)\(12\!\cdots\!58\)\( + \)\(34\!\cdots\!18\)\( \beta_{1} + \)\(15\!\cdots\!24\)\( \beta_{2} - \)\(47\!\cdots\!00\)\( \beta_{3} + \)\(42\!\cdots\!60\)\( \beta_{4}) q^{58} +(\)\(61\!\cdots\!42\)\( - \)\(11\!\cdots\!72\)\( \beta_{1} + \)\(49\!\cdots\!25\)\( \beta_{2} - \)\(49\!\cdots\!32\)\( \beta_{3} + \)\(78\!\cdots\!00\)\( \beta_{4}) q^{59} +(-\)\(22\!\cdots\!53\)\( + \)\(14\!\cdots\!50\)\( \beta_{1} + \)\(52\!\cdots\!75\)\( \beta_{2} + \)\(81\!\cdots\!00\)\( \beta_{3} + \)\(11\!\cdots\!25\)\( \beta_{4}) q^{61} +(\)\(43\!\cdots\!88\)\( + \)\(76\!\cdots\!12\)\( \beta_{1} - \)\(20\!\cdots\!20\)\( \beta_{2} + \)\(12\!\cdots\!00\)\( \beta_{3} + \)\(11\!\cdots\!60\)\( \beta_{4}) q^{62} +(\)\(12\!\cdots\!16\)\( - \)\(14\!\cdots\!88\)\( \beta_{1} - \)\(58\!\cdots\!80\)\( \beta_{2} + \)\(55\!\cdots\!12\)\( \beta_{3} - \)\(31\!\cdots\!20\)\( \beta_{4}) q^{64} +(-\)\(34\!\cdots\!32\)\( - \)\(33\!\cdots\!88\)\( \beta_{1} - \)\(39\!\cdots\!52\)\( \beta_{2} - \)\(17\!\cdots\!36\)\( \beta_{3} + \)\(16\!\cdots\!88\)\( \beta_{4}) q^{65} +(-\)\(23\!\cdots\!82\)\( + \)\(58\!\cdots\!92\)\( \beta_{1} + \)\(24\!\cdots\!09\)\( \beta_{2} - \)\(73\!\cdots\!00\)\( \beta_{3} - \)\(33\!\cdots\!40\)\( \beta_{4}) q^{67} +(-\)\(16\!\cdots\!06\)\( + \)\(19\!\cdots\!16\)\( \beta_{1} - \)\(26\!\cdots\!86\)\( \beta_{2} - \)\(15\!\cdots\!50\)\( \beta_{3} + \)\(50\!\cdots\!80\)\( \beta_{4}) q^{68} +(-\)\(52\!\cdots\!24\)\( - \)\(43\!\cdots\!16\)\( \beta_{1} - \)\(15\!\cdots\!64\)\( \beta_{2} - \)\(27\!\cdots\!52\)\( \beta_{3} - \)\(96\!\cdots\!84\)\( \beta_{4}) q^{70} +(\)\(22\!\cdots\!08\)\( - \)\(16\!\cdots\!00\)\( \beta_{1} - \)\(96\!\cdots\!50\)\( \beta_{2} - \)\(41\!\cdots\!00\)\( \beta_{3} - \)\(10\!\cdots\!00\)\( \beta_{4}) q^{71} +(-\)\(60\!\cdots\!52\)\( - \)\(28\!\cdots\!68\)\( \beta_{1} - \)\(47\!\cdots\!58\)\( \beta_{2} + \)\(33\!\cdots\!00\)\( \beta_{3} - \)\(14\!\cdots\!90\)\( \beta_{4}) q^{73} +(-\)\(18\!\cdots\!66\)\( - \)\(42\!\cdots\!70\)\( \beta_{1} - \)\(73\!\cdots\!40\)\( \beta_{2} - \)\(18\!\cdots\!00\)\( \beta_{3} - \)\(72\!\cdots\!60\)\( \beta_{4}) q^{74} +(\)\(41\!\cdots\!88\)\( + \)\(15\!\cdots\!92\)\( \beta_{1} + \)\(36\!\cdots\!40\)\( \beta_{2} - \)\(13\!\cdots\!68\)\( \beta_{3} + \)\(29\!\cdots\!60\)\( \beta_{4}) q^{76} +(-\)\(10\!\cdots\!00\)\( + \)\(47\!\cdots\!60\)\( \beta_{1} - \)\(33\!\cdots\!16\)\( \beta_{2} - \)\(29\!\cdots\!00\)\( \beta_{3} + \)\(54\!\cdots\!40\)\( \beta_{4}) q^{77} +(\)\(59\!\cdots\!84\)\( + \)\(85\!\cdots\!56\)\( \beta_{1} + \)\(33\!\cdots\!60\)\( \beta_{2} - \)\(63\!\cdots\!44\)\( \beta_{3} + \)\(77\!\cdots\!40\)\( \beta_{4}) q^{79} +(\)\(30\!\cdots\!04\)\( - \)\(12\!\cdots\!64\)\( \beta_{1} + \)\(17\!\cdots\!44\)\( \beta_{2} + \)\(14\!\cdots\!92\)\( \beta_{3} + \)\(14\!\cdots\!64\)\( \beta_{4}) q^{80} +(-\)\(35\!\cdots\!18\)\( - \)\(22\!\cdots\!82\)\( \beta_{1} + \)\(97\!\cdots\!20\)\( \beta_{2} + \)\(10\!\cdots\!00\)\( \beta_{3} + \)\(28\!\cdots\!40\)\( \beta_{4}) q^{82} +(\)\(94\!\cdots\!26\)\( + \)\(35\!\cdots\!64\)\( \beta_{1} - \)\(60\!\cdots\!89\)\( \beta_{2} - \)\(14\!\cdots\!00\)\( \beta_{3} + \)\(10\!\cdots\!00\)\( \beta_{4}) q^{83} +(-\)\(78\!\cdots\!94\)\( + \)\(15\!\cdots\!04\)\( \beta_{1} - \)\(26\!\cdots\!34\)\( \beta_{2} - \)\(53\!\cdots\!12\)\( \beta_{3} - \)\(64\!\cdots\!54\)\( \beta_{4}) q^{85} +(-\)\(99\!\cdots\!96\)\( - \)\(18\!\cdots\!16\)\( \beta_{1} + \)\(10\!\cdots\!20\)\( \beta_{2} + \)\(18\!\cdots\!44\)\( \beta_{3} - \)\(10\!\cdots\!20\)\( \beta_{4}) q^{86} +(\)\(86\!\cdots\!04\)\( - \)\(88\!\cdots\!84\)\( \beta_{1} + \)\(64\!\cdots\!08\)\( \beta_{2} + \)\(69\!\cdots\!00\)\( \beta_{3} - \)\(38\!\cdots\!40\)\( \beta_{4}) q^{88} +(\)\(23\!\cdots\!88\)\( - \)\(45\!\cdots\!08\)\( \beta_{1} - \)\(31\!\cdots\!10\)\( \beta_{2} + \)\(12\!\cdots\!32\)\( \beta_{3} - \)\(74\!\cdots\!90\)\( \beta_{4}) q^{89} +(\)\(46\!\cdots\!56\)\( + \)\(82\!\cdots\!16\)\( \beta_{1} + \)\(16\!\cdots\!20\)\( \beta_{2} + \)\(60\!\cdots\!36\)\( \beta_{3} + \)\(37\!\cdots\!80\)\( \beta_{4}) q^{91} +(\)\(19\!\cdots\!72\)\( - \)\(38\!\cdots\!32\)\( \beta_{1} + \)\(11\!\cdots\!56\)\( \beta_{2} - \)\(14\!\cdots\!00\)\( \beta_{3} - \)\(25\!\cdots\!40\)\( \beta_{4}) q^{92} +(-\)\(98\!\cdots\!88\)\( + \)\(51\!\cdots\!64\)\( \beta_{1} + \)\(98\!\cdots\!60\)\( \beta_{2} - \)\(49\!\cdots\!96\)\( \beta_{3} + \)\(11\!\cdots\!40\)\( \beta_{4}) q^{94} +(\)\(40\!\cdots\!40\)\( - \)\(11\!\cdots\!40\)\( \beta_{1} - \)\(74\!\cdots\!10\)\( \beta_{2} - \)\(52\!\cdots\!80\)\( \beta_{3} - \)\(30\!\cdots\!60\)\( \beta_{4}) q^{95} +(\)\(57\!\cdots\!92\)\( - \)\(21\!\cdots\!12\)\( \beta_{1} + \)\(31\!\cdots\!02\)\( \beta_{2} + \)\(27\!\cdots\!00\)\( \beta_{3} - \)\(50\!\cdots\!30\)\( \beta_{4}) q^{97} +(-\)\(24\!\cdots\!63\)\( + \)\(68\!\cdots\!13\)\( \beta_{1} - \)\(43\!\cdots\!60\)\( \beta_{2} - \)\(25\!\cdots\!00\)\( \beta_{3} + \)\(79\!\cdots\!60\)\( \beta_{4}) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q - 5554901256q^{2} + \)\(35\!\cdots\!40\)\(q^{4} - \)\(33\!\cdots\!50\)\(q^{5} + \)\(33\!\cdots\!56\)\(q^{7} - \)\(32\!\cdots\!80\)\(q^{8} + O(q^{10}) \) \( 5q - 5554901256q^{2} + \)\(35\!\cdots\!40\)\(q^{4} - \)\(33\!\cdots\!50\)\(q^{5} + \)\(33\!\cdots\!56\)\(q^{7} - \)\(32\!\cdots\!80\)\(q^{8} - \)\(52\!\cdots\!00\)\(q^{10} - \)\(20\!\cdots\!60\)\(q^{11} + \)\(17\!\cdots\!02\)\(q^{13} - \)\(29\!\cdots\!80\)\(q^{14} + \)\(18\!\cdots\!80\)\(q^{16} - \)\(75\!\cdots\!06\)\(q^{17} + \)\(39\!\cdots\!00\)\(q^{19} + \)\(15\!\cdots\!00\)\(q^{20} + \)\(18\!\cdots\!32\)\(q^{22} + \)\(41\!\cdots\!68\)\(q^{23} + \)\(17\!\cdots\!75\)\(q^{25} - \)\(21\!\cdots\!60\)\(q^{26} + \)\(14\!\cdots\!48\)\(q^{28} - \)\(18\!\cdots\!50\)\(q^{29} + \)\(36\!\cdots\!60\)\(q^{31} - \)\(15\!\cdots\!96\)\(q^{32} + \)\(78\!\cdots\!80\)\(q^{34} - \)\(45\!\cdots\!00\)\(q^{35} - \)\(56\!\cdots\!94\)\(q^{37} + \)\(31\!\cdots\!80\)\(q^{38} - \)\(18\!\cdots\!00\)\(q^{40} - \)\(11\!\cdots\!10\)\(q^{41} + \)\(65\!\cdots\!92\)\(q^{43} - \)\(49\!\cdots\!80\)\(q^{44} + \)\(79\!\cdots\!60\)\(q^{46} + \)\(12\!\cdots\!44\)\(q^{47} - \)\(77\!\cdots\!35\)\(q^{49} + \)\(39\!\cdots\!00\)\(q^{50} + \)\(67\!\cdots\!16\)\(q^{52} - \)\(10\!\cdots\!22\)\(q^{53} + \)\(66\!\cdots\!00\)\(q^{55} - \)\(16\!\cdots\!00\)\(q^{56} - \)\(61\!\cdots\!20\)\(q^{58} + \)\(30\!\cdots\!00\)\(q^{59} - \)\(11\!\cdots\!90\)\(q^{61} + \)\(21\!\cdots\!28\)\(q^{62} + \)\(63\!\cdots\!40\)\(q^{64} - \)\(17\!\cdots\!00\)\(q^{65} - \)\(11\!\cdots\!44\)\(q^{67} - \)\(81\!\cdots\!48\)\(q^{68} - \)\(26\!\cdots\!00\)\(q^{70} + \)\(11\!\cdots\!40\)\(q^{71} - \)\(30\!\cdots\!18\)\(q^{73} - \)\(92\!\cdots\!80\)\(q^{74} + \)\(20\!\cdots\!00\)\(q^{76} - \)\(50\!\cdots\!32\)\(q^{77} + \)\(29\!\cdots\!00\)\(q^{79} + \)\(15\!\cdots\!00\)\(q^{80} - \)\(17\!\cdots\!08\)\(q^{82} + \)\(47\!\cdots\!88\)\(q^{83} - \)\(39\!\cdots\!00\)\(q^{85} - \)\(49\!\cdots\!60\)\(q^{86} + \)\(43\!\cdots\!60\)\(q^{88} + \)\(11\!\cdots\!50\)\(q^{89} + \)\(23\!\cdots\!60\)\(q^{91} + \)\(98\!\cdots\!44\)\(q^{92} - \)\(49\!\cdots\!20\)\(q^{94} + \)\(20\!\cdots\!00\)\(q^{95} + \)\(28\!\cdots\!06\)\(q^{97} - \)\(12\!\cdots\!08\)\(q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{5} - x^{4} - 939384011925257456 x^{3} + 31046449413968483513911200 x^{2} + 156793504704482691874379743265203200 x + 20916736226052669578405116700517591609696000\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 24 \nu - 5 \)
\(\beta_{2}\)\(=\)\((\)\(118503 \nu^{4} - 4966651742499 \nu^{3} - 91908874679257102484196 \nu^{2} + 14245604325212883780772795540080 \nu + 7478148396148300261622780212883139790912\)\()/ \)\(12\!\cdots\!76\)\( \)
\(\beta_{3}\)\(=\)\((\)\(-2962575 \nu^{4} + 124166293562475 \nu^{3} + 3017084490258127833745476 \nu^{2} - 320477928124642729960888846359792 \nu - 457256808743937660582039154025781575339584\)\()/ \)\(12\!\cdots\!76\)\( \)
\(\beta_{4}\)\(=\)\((\)\(-18951882417 \nu^{4} + 27135577617492474837 \nu^{3} + 25060982402821625440656730812 \nu^{2} - 17715834575679543445262710553614817040 \nu - 4598924888921018716143814644719268592040927680\)\()/ \)\(62\!\cdots\!80\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1} + 5\)\()/24\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} + 25 \beta_{2} - 1189792592 \beta_{1} + 216434076347341359449\)\()/576\)
\(\nu^{3}\)\(=\)\((\)\(409640 \beta_{4} - 1180151429 \beta_{3} - 16401249733 \beta_{2} + 43600467929554726208 \beta_{1} - 32188957769729086550408644085\)\()/1728\)
\(\nu^{4}\)\(=\)\((\)\(51506018070360 \beta_{4} + 6831858000721900017 \beta_{3} + 227077648039513374161 \beta_{2} - 28788958914876643991563272064 \beta_{1} + 1179578375765594550915269284254841951361\)\()/5184\)

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
−8.99335e8
−3.05046e8
−1.64845e8
6.06038e8
7.63188e8
−2.26950e10 0 3.67490e20 1.61198e23 0 2.93806e28 −4.99099e30 0 −3.65838e33
1.2 −8.43209e9 0 −7.64738e19 −3.83370e23 0 −1.75533e27 1.88919e30 0 3.23261e33
1.3 −5.06726e9 0 −1.21897e20 8.33551e22 0 −1.15070e28 1.36548e30 0 −4.22381e32
1.4 1.34339e10 0 3.28965e19 3.01383e23 0 −4.20742e26 −1.54057e30 0 4.04876e33
1.5 1.72055e10 0 1.48456e20 −4.93311e23 0 1.79351e28 1.51842e28 0 −8.48767e33
\(n\): 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 9.68.a.a 5
3.b odd 2 1 1.68.a.a 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.68.a.a 5 3.b odd 2 1
9.68.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 \( T_{2}^{5} + 5554901256 T_{2}^{4} - \)\(52\!\cdots\!72\)\( T_{2}^{3} - \)\(13\!\cdots\!32\)\( T_{2}^{2} + \)\(50\!\cdots\!96\)\( T_{2} + \)\(22\!\cdots\!76\)\( \) acting on \(S_{68}^{\mathrm{new}}(\Gamma_0(9))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 5554901256 T + \)\(20\!\cdots\!68\)\( T^{2} + \)\(19\!\cdots\!40\)\( T^{3} + \)\(34\!\cdots\!88\)\( T^{4} + \)\(54\!\cdots\!08\)\( T^{5} + \)\(51\!\cdots\!64\)\( T^{6} + \)\(41\!\cdots\!60\)\( T^{7} + \)\(67\!\cdots\!36\)\( T^{8} + \)\(26\!\cdots\!36\)\( T^{9} + \)\(69\!\cdots\!68\)\( T^{10} \)
$3$ 1
$5$ \( 1 + \)\(33\!\cdots\!50\)\( T + \)\(13\!\cdots\!25\)\( T^{2} + \)\(58\!\cdots\!00\)\( T^{3} + \)\(17\!\cdots\!50\)\( T^{4} + \)\(41\!\cdots\!00\)\( T^{5} + \)\(12\!\cdots\!50\)\( T^{6} + \)\(26\!\cdots\!00\)\( T^{7} + \)\(42\!\cdots\!25\)\( T^{8} + \)\(69\!\cdots\!50\)\( T^{9} + \)\(14\!\cdots\!25\)\( T^{10} \)
$7$ \( 1 - \)\(33\!\cdots\!56\)\( T + \)\(19\!\cdots\!43\)\( T^{2} - \)\(50\!\cdots\!00\)\( T^{3} + \)\(16\!\cdots\!98\)\( T^{4} - \)\(30\!\cdots\!88\)\( T^{5} + \)\(68\!\cdots\!14\)\( T^{6} - \)\(88\!\cdots\!00\)\( T^{7} + \)\(14\!\cdots\!01\)\( T^{8} - \)\(10\!\cdots\!56\)\( T^{9} + \)\(12\!\cdots\!43\)\( T^{10} \)
$11$ \( 1 + \)\(20\!\cdots\!60\)\( T + \)\(27\!\cdots\!95\)\( T^{2} + \)\(25\!\cdots\!20\)\( T^{3} + \)\(19\!\cdots\!10\)\( T^{4} + \)\(14\!\cdots\!52\)\( T^{5} + \)\(11\!\cdots\!10\)\( T^{6} + \)\(88\!\cdots\!20\)\( T^{7} + \)\(57\!\cdots\!45\)\( T^{8} + \)\(25\!\cdots\!60\)\( T^{9} + \)\(73\!\cdots\!51\)\( T^{10} \)
$13$ \( 1 - \)\(17\!\cdots\!02\)\( T + \)\(12\!\cdots\!77\)\( T^{2} + \)\(44\!\cdots\!80\)\( T^{3} + \)\(79\!\cdots\!98\)\( T^{4} + \)\(39\!\cdots\!44\)\( T^{5} + \)\(34\!\cdots\!66\)\( T^{6} + \)\(82\!\cdots\!20\)\( T^{7} + \)\(10\!\cdots\!01\)\( T^{8} - \)\(61\!\cdots\!42\)\( T^{9} + \)\(14\!\cdots\!57\)\( T^{10} \)
$17$ \( 1 + \)\(75\!\cdots\!06\)\( T + \)\(88\!\cdots\!93\)\( T^{2} + \)\(84\!\cdots\!20\)\( T^{3} + \)\(38\!\cdots\!18\)\( T^{4} + \)\(34\!\cdots\!48\)\( T^{5} + \)\(10\!\cdots\!14\)\( T^{6} + \)\(64\!\cdots\!80\)\( T^{7} + \)\(18\!\cdots\!81\)\( T^{8} + \)\(43\!\cdots\!46\)\( T^{9} + \)\(15\!\cdots\!93\)\( T^{10} \)
$19$ \( 1 - \)\(39\!\cdots\!00\)\( T + \)\(14\!\cdots\!95\)\( T^{2} - \)\(67\!\cdots\!00\)\( T^{3} + \)\(11\!\cdots\!10\)\( T^{4} - \)\(45\!\cdots\!00\)\( T^{5} + \)\(53\!\cdots\!90\)\( T^{6} - \)\(15\!\cdots\!00\)\( T^{7} + \)\(15\!\cdots\!05\)\( T^{8} - \)\(20\!\cdots\!00\)\( T^{9} + \)\(24\!\cdots\!99\)\( T^{10} \)
$23$ \( 1 - \)\(41\!\cdots\!68\)\( T + \)\(73\!\cdots\!87\)\( T^{2} - \)\(30\!\cdots\!20\)\( T^{3} + \)\(23\!\cdots\!98\)\( T^{4} - \)\(80\!\cdots\!24\)\( T^{5} + \)\(40\!\cdots\!06\)\( T^{6} - \)\(89\!\cdots\!80\)\( T^{7} + \)\(37\!\cdots\!01\)\( T^{8} - \)\(36\!\cdots\!08\)\( T^{9} + \)\(15\!\cdots\!07\)\( T^{10} \)
$29$ \( 1 + \)\(18\!\cdots\!50\)\( T + \)\(30\!\cdots\!45\)\( T^{2} + \)\(33\!\cdots\!00\)\( T^{3} + \)\(48\!\cdots\!10\)\( T^{4} + \)\(48\!\cdots\!00\)\( T^{5} + \)\(46\!\cdots\!90\)\( T^{6} + \)\(30\!\cdots\!00\)\( T^{7} + \)\(26\!\cdots\!05\)\( T^{8} + \)\(15\!\cdots\!50\)\( T^{9} + \)\(80\!\cdots\!49\)\( T^{10} \)
$31$ \( 1 - \)\(36\!\cdots\!60\)\( T + \)\(80\!\cdots\!95\)\( T^{2} - \)\(12\!\cdots\!20\)\( T^{3} + \)\(15\!\cdots\!10\)\( T^{4} - \)\(14\!\cdots\!52\)\( T^{5} + \)\(12\!\cdots\!10\)\( T^{6} - \)\(86\!\cdots\!20\)\( T^{7} + \)\(46\!\cdots\!45\)\( T^{8} - \)\(17\!\cdots\!60\)\( T^{9} + \)\(40\!\cdots\!51\)\( T^{10} \)
$37$ \( 1 + \)\(56\!\cdots\!94\)\( T + \)\(47\!\cdots\!93\)\( T^{2} + \)\(21\!\cdots\!40\)\( T^{3} + \)\(10\!\cdots\!58\)\( T^{4} + \)\(34\!\cdots\!32\)\( T^{5} + \)\(12\!\cdots\!14\)\( T^{6} + \)\(29\!\cdots\!60\)\( T^{7} + \)\(77\!\cdots\!41\)\( T^{8} + \)\(10\!\cdots\!74\)\( T^{9} + \)\(22\!\cdots\!93\)\( T^{10} \)
$41$ \( 1 + \)\(11\!\cdots\!10\)\( T + \)\(30\!\cdots\!45\)\( T^{2} - \)\(95\!\cdots\!80\)\( T^{3} + \)\(43\!\cdots\!10\)\( T^{4} - \)\(48\!\cdots\!48\)\( T^{5} + \)\(49\!\cdots\!10\)\( T^{6} - \)\(12\!\cdots\!80\)\( T^{7} + \)\(45\!\cdots\!45\)\( T^{8} + \)\(19\!\cdots\!10\)\( T^{9} + \)\(19\!\cdots\!01\)\( T^{10} \)
$43$ \( 1 - \)\(65\!\cdots\!92\)\( T + \)\(10\!\cdots\!07\)\( T^{2} - \)\(56\!\cdots\!00\)\( T^{3} + \)\(55\!\cdots\!98\)\( T^{4} - \)\(21\!\cdots\!16\)\( T^{5} + \)\(15\!\cdots\!86\)\( T^{6} - \)\(43\!\cdots\!00\)\( T^{7} + \)\(23\!\cdots\!01\)\( T^{8} - \)\(38\!\cdots\!92\)\( T^{9} + \)\(16\!\cdots\!07\)\( T^{10} \)
$47$ \( 1 - \)\(12\!\cdots\!44\)\( T + \)\(42\!\cdots\!43\)\( T^{2} - \)\(42\!\cdots\!20\)\( T^{3} + \)\(82\!\cdots\!78\)\( T^{4} - \)\(62\!\cdots\!72\)\( T^{5} + \)\(88\!\cdots\!14\)\( T^{6} - \)\(49\!\cdots\!80\)\( T^{7} + \)\(52\!\cdots\!21\)\( T^{8} - \)\(16\!\cdots\!84\)\( T^{9} + \)\(14\!\cdots\!43\)\( T^{10} \)
$53$ \( 1 + \)\(10\!\cdots\!22\)\( T + \)\(74\!\cdots\!17\)\( T^{2} - \)\(11\!\cdots\!40\)\( T^{3} + \)\(31\!\cdots\!98\)\( T^{4} - \)\(63\!\cdots\!64\)\( T^{5} + \)\(10\!\cdots\!26\)\( T^{6} - \)\(12\!\cdots\!60\)\( T^{7} + \)\(28\!\cdots\!01\)\( T^{8} + \)\(13\!\cdots\!42\)\( T^{9} + \)\(42\!\cdots\!57\)\( T^{10} \)
$59$ \( 1 - \)\(30\!\cdots\!00\)\( T + \)\(13\!\cdots\!95\)\( T^{2} - \)\(29\!\cdots\!00\)\( T^{3} + \)\(87\!\cdots\!10\)\( T^{4} - \)\(14\!\cdots\!00\)\( T^{5} + \)\(38\!\cdots\!90\)\( T^{6} - \)\(58\!\cdots\!00\)\( T^{7} + \)\(11\!\cdots\!05\)\( T^{8} - \)\(11\!\cdots\!00\)\( T^{9} + \)\(17\!\cdots\!99\)\( T^{10} \)
$61$ \( 1 + \)\(11\!\cdots\!90\)\( T + \)\(21\!\cdots\!45\)\( T^{2} + \)\(16\!\cdots\!80\)\( T^{3} + \)\(17\!\cdots\!10\)\( T^{4} + \)\(93\!\cdots\!48\)\( T^{5} + \)\(71\!\cdots\!10\)\( T^{6} + \)\(27\!\cdots\!80\)\( T^{7} + \)\(15\!\cdots\!45\)\( T^{8} + \)\(33\!\cdots\!90\)\( T^{9} + \)\(12\!\cdots\!01\)\( T^{10} \)
$67$ \( 1 + \)\(11\!\cdots\!44\)\( T + \)\(30\!\cdots\!43\)\( T^{2} + \)\(16\!\cdots\!80\)\( T^{3} + \)\(58\!\cdots\!18\)\( T^{4} - \)\(37\!\cdots\!48\)\( T^{5} + \)\(12\!\cdots\!14\)\( T^{6} + \)\(79\!\cdots\!20\)\( T^{7} + \)\(33\!\cdots\!81\)\( T^{8} + \)\(29\!\cdots\!04\)\( T^{9} + \)\(54\!\cdots\!43\)\( T^{10} \)
$71$ \( 1 - \)\(11\!\cdots\!40\)\( T + \)\(43\!\cdots\!95\)\( T^{2} - \)\(37\!\cdots\!80\)\( T^{3} + \)\(80\!\cdots\!10\)\( T^{4} - \)\(54\!\cdots\!48\)\( T^{5} + \)\(87\!\cdots\!10\)\( T^{6} - \)\(43\!\cdots\!80\)\( T^{7} + \)\(54\!\cdots\!45\)\( T^{8} - \)\(15\!\cdots\!40\)\( T^{9} + \)\(14\!\cdots\!51\)\( T^{10} \)
$73$ \( 1 + \)\(30\!\cdots\!18\)\( T + \)\(36\!\cdots\!37\)\( T^{2} + \)\(83\!\cdots\!20\)\( T^{3} + \)\(52\!\cdots\!98\)\( T^{4} + \)\(86\!\cdots\!24\)\( T^{5} + \)\(36\!\cdots\!06\)\( T^{6} + \)\(40\!\cdots\!80\)\( T^{7} + \)\(12\!\cdots\!01\)\( T^{8} + \)\(71\!\cdots\!58\)\( T^{9} + \)\(16\!\cdots\!57\)\( T^{10} \)
$79$ \( 1 - \)\(29\!\cdots\!00\)\( T + \)\(51\!\cdots\!95\)\( T^{2} - \)\(13\!\cdots\!00\)\( T^{3} + \)\(12\!\cdots\!10\)\( T^{4} - \)\(26\!\cdots\!00\)\( T^{5} + \)\(17\!\cdots\!90\)\( T^{6} - \)\(25\!\cdots\!00\)\( T^{7} + \)\(13\!\cdots\!05\)\( T^{8} - \)\(10\!\cdots\!00\)\( T^{9} + \)\(50\!\cdots\!99\)\( T^{10} \)
$83$ \( 1 - \)\(47\!\cdots\!88\)\( T + \)\(21\!\cdots\!47\)\( T^{2} - \)\(64\!\cdots\!60\)\( T^{3} + \)\(17\!\cdots\!98\)\( T^{4} - \)\(35\!\cdots\!04\)\( T^{5} + \)\(65\!\cdots\!46\)\( T^{6} - \)\(92\!\cdots\!40\)\( T^{7} + \)\(11\!\cdots\!01\)\( T^{8} - \)\(97\!\cdots\!08\)\( T^{9} + \)\(77\!\cdots\!07\)\( T^{10} \)
$89$ \( 1 - \)\(11\!\cdots\!50\)\( T + \)\(98\!\cdots\!45\)\( T^{2} - \)\(38\!\cdots\!00\)\( T^{3} + \)\(39\!\cdots\!10\)\( T^{4} - \)\(27\!\cdots\!00\)\( T^{5} + \)\(15\!\cdots\!90\)\( T^{6} - \)\(64\!\cdots\!00\)\( T^{7} + \)\(65\!\cdots\!05\)\( T^{8} - \)\(32\!\cdots\!50\)\( T^{9} + \)\(11\!\cdots\!49\)\( T^{10} \)
$97$ \( 1 - \)\(28\!\cdots\!06\)\( T + \)\(36\!\cdots\!93\)\( T^{2} - \)\(13\!\cdots\!80\)\( T^{3} + \)\(66\!\cdots\!78\)\( T^{4} - \)\(24\!\cdots\!28\)\( T^{5} + \)\(86\!\cdots\!14\)\( T^{6} - \)\(22\!\cdots\!20\)\( T^{7} + \)\(79\!\cdots\!21\)\( T^{8} - \)\(82\!\cdots\!66\)\( T^{9} + \)\(37\!\cdots\!93\)\( T^{10} \)
show more
show less