Properties

Label 1.44.a.a
Level 1
Weight 44
Character orbit 1.a
Self dual yes
Analytic conductor 11.711
Analytic rank 0
Dimension 3
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 1 \)
Weight: \( k \) \(=\) \( 44 \)
Character orbit: \([\chi]\) \(=\) 1.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(11.7110395346\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{11}\cdot 3^{4}\cdot 5\cdot 7 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(f(q)\) \(=\) \( q + ( -736648 - \beta_{1} ) q^{2} + ( 8133812604 + 2135 \beta_{1} - \beta_{2} ) q^{3} + ( 3274206140608 + 810816 \beta_{1} - 408 \beta_{2} ) q^{4} + ( 178401793591390 - 23658932 \beta_{1} - 29844 \beta_{2} ) q^{5} + ( -30597991586147808 - 21343496316 \beta_{1} + 2270208 \beta_{2} ) q^{6} + ( 100657141888492952 - 411863382714 \beta_{1} - 60797226 \beta_{2} ) q^{7} + ( -5276954637929116160 + 136805587456 \beta_{1} + 901657152 \beta_{2} ) q^{8} + ( 159160708390355581077 + 108619653530952 \beta_{1} - 7904835576 \beta_{2} ) q^{9} +O(q^{10})\) \( q +(-736648 - \beta_{1}) q^{2} +(8133812604 + 2135 \beta_{1} - \beta_{2}) q^{3} +(3274206140608 + 810816 \beta_{1} - 408 \beta_{2}) q^{4} +(178401793591390 - 23658932 \beta_{1} - 29844 \beta_{2}) q^{5} +(-30597991586147808 - 21343496316 \beta_{1} + 2270208 \beta_{2}) q^{6} +(100657141888492952 - 411863382714 \beta_{1} - 60797226 \beta_{2}) q^{7} +(-5276954637929116160 + 136805587456 \beta_{1} + 901657152 \beta_{2}) q^{8} +(\)\(15\!\cdots\!77\)\( + 108619653530952 \beta_{1} - 7904835576 \beta_{2}) q^{9} +(\)\(14\!\cdots\!40\)\( - 566151100617822 \beta_{1} + 32102731776 \beta_{2}) q^{10} +(\)\(88\!\cdots\!32\)\( - 4605494939944195 \beta_{1} + 137241701685 \beta_{2}) q^{11} +(\)\(19\!\cdots\!12\)\( + 43030582618825472 \beta_{1} - 3088365053344 \beta_{2}) q^{12} +(\)\(89\!\cdots\!94\)\( - 6729074743426932 \beta_{1} + 22807920190380 \beta_{2}) q^{13} +(\)\(46\!\cdots\!76\)\( - 863595024061582232 \beta_{1} - 82977158928384 \beta_{2}) q^{14} +(\)\(11\!\cdots\!80\)\( + 1376784350493064386 \beta_{1} - 37572060903438 \beta_{2}) q^{15} +(-\)\(26\!\cdots\!64\)\( + 9902624655975026688 \beta_{1} + 2383088864979456 \beta_{2}) q^{16} +(-\)\(13\!\cdots\!98\)\( - 4605229780369521720 \beta_{1} - 15383154249178488 \beta_{2}) q^{17} +(-\)\(13\!\cdots\!16\)\( - \)\(27\!\cdots\!45\)\( \beta_{1} + 55376695430406144 \beta_{2}) q^{18} +(\)\(52\!\cdots\!00\)\( + \)\(85\!\cdots\!19\)\( \beta_{1} - 107308048565285397 \beta_{2}) q^{19} +(\)\(48\!\cdots\!20\)\( + \)\(52\!\cdots\!44\)\( \beta_{1} - 13394879801587152 \beta_{2}) q^{20} +(\)\(13\!\cdots\!12\)\( - \)\(49\!\cdots\!32\)\( \beta_{1} + 743940509412748816 \beta_{2}) q^{21} +(\)\(46\!\cdots\!64\)\( - \)\(67\!\cdots\!52\)\( \beta_{1} - 2071060643092362240 \beta_{2}) q^{22} +(-\)\(40\!\cdots\!16\)\( + \)\(54\!\cdots\!86\)\( \beta_{1} + 1876113512966847906 \beta_{2}) q^{23} +(-\)\(37\!\cdots\!00\)\( - \)\(52\!\cdots\!88\)\( \beta_{1} + 1908534981075097344 \beta_{2}) q^{24} +(-\)\(77\!\cdots\!25\)\( - \)\(83\!\cdots\!40\)\( \beta_{1} + 1683060411116573520 \beta_{2}) q^{25} +(-\)\(57\!\cdots\!48\)\( - \)\(59\!\cdots\!58\)\( \beta_{1} - 34656662255444176896 \beta_{2}) q^{26} +(\)\(42\!\cdots\!60\)\( + \)\(21\!\cdots\!74\)\( \beta_{1} + 30213151036337641158 \beta_{2}) q^{27} +(\)\(56\!\cdots\!56\)\( - \)\(20\!\cdots\!76\)\( \beta_{1} + \)\(29\!\cdots\!04\)\( \beta_{2}) q^{28} +(\)\(19\!\cdots\!50\)\( + \)\(17\!\cdots\!76\)\( \beta_{1} - \)\(10\!\cdots\!88\)\( \beta_{2}) q^{29} +(-\)\(24\!\cdots\!20\)\( - \)\(12\!\cdots\!44\)\( \beta_{1} + \)\(61\!\cdots\!52\)\( \beta_{2}) q^{30} +(-\)\(85\!\cdots\!08\)\( + \)\(11\!\cdots\!40\)\( \beta_{1} + \)\(33\!\cdots\!80\)\( \beta_{2}) q^{31} +(-\)\(48\!\cdots\!28\)\( + \)\(55\!\cdots\!24\)\( \beta_{1} - \)\(72\!\cdots\!80\)\( \beta_{2}) q^{32} +(-\)\(91\!\cdots\!72\)\( - \)\(78\!\cdots\!20\)\( \beta_{1} - \)\(23\!\cdots\!12\)\( \beta_{2}) q^{33} +(\)\(15\!\cdots\!96\)\( - \)\(64\!\cdots\!58\)\( \beta_{1} + \)\(19\!\cdots\!04\)\( \beta_{2}) q^{34} +(\)\(79\!\cdots\!40\)\( - \)\(22\!\cdots\!92\)\( \beta_{1} + \)\(14\!\cdots\!36\)\( \beta_{2}) q^{35} +(\)\(27\!\cdots\!16\)\( + \)\(11\!\cdots\!68\)\( \beta_{1} - \)\(11\!\cdots\!84\)\( \beta_{2}) q^{36} +(-\)\(77\!\cdots\!98\)\( - \)\(43\!\cdots\!72\)\( \beta_{1} + \)\(46\!\cdots\!08\)\( \beta_{2}) q^{37} +(-\)\(10\!\cdots\!40\)\( - \)\(19\!\cdots\!96\)\( \beta_{1} + \)\(49\!\cdots\!68\)\( \beta_{2}) q^{38} +(-\)\(13\!\cdots\!76\)\( + \)\(65\!\cdots\!78\)\( \beta_{1} - \)\(96\!\cdots\!14\)\( \beta_{2}) q^{39} +(-\)\(10\!\cdots\!00\)\( - \)\(86\!\cdots\!00\)\( \beta_{1} - \)\(48\!\cdots\!00\)\( \beta_{2}) q^{40} +(\)\(85\!\cdots\!22\)\( + \)\(87\!\cdots\!40\)\( \beta_{1} + \)\(23\!\cdots\!80\)\( \beta_{2}) q^{41} +(\)\(47\!\cdots\!44\)\( - \)\(30\!\cdots\!24\)\( \beta_{1} - \)\(30\!\cdots\!04\)\( \beta_{2}) q^{42} +(\)\(80\!\cdots\!64\)\( - \)\(12\!\cdots\!83\)\( \beta_{1} + \)\(17\!\cdots\!73\)\( \beta_{2}) q^{43} +(-\)\(34\!\cdots\!44\)\( - \)\(32\!\cdots\!48\)\( \beta_{1} - \)\(10\!\cdots\!76\)\( \beta_{2}) q^{44} +(\)\(85\!\cdots\!30\)\( + \)\(62\!\cdots\!36\)\( \beta_{1} - \)\(40\!\cdots\!88\)\( \beta_{2}) q^{45} +(-\)\(63\!\cdots\!88\)\( + \)\(20\!\cdots\!60\)\( \beta_{1} + \)\(19\!\cdots\!20\)\( \beta_{2}) q^{46} +(\)\(10\!\cdots\!52\)\( + \)\(31\!\cdots\!32\)\( \beta_{1} - \)\(25\!\cdots\!84\)\( \beta_{2}) q^{47} +(-\)\(85\!\cdots\!16\)\( + \)\(22\!\cdots\!16\)\( \beta_{1} + \)\(29\!\cdots\!16\)\( \beta_{2}) q^{48} +(\)\(11\!\cdots\!93\)\( - \)\(77\!\cdots\!00\)\( \beta_{1} - \)\(30\!\cdots\!00\)\( \beta_{2}) q^{49} +(\)\(66\!\cdots\!00\)\( + \)\(79\!\cdots\!85\)\( \beta_{1} - \)\(57\!\cdots\!80\)\( \beta_{2}) q^{50} +(\)\(44\!\cdots\!52\)\( + \)\(37\!\cdots\!26\)\( \beta_{1} + \)\(19\!\cdots\!62\)\( \beta_{2}) q^{51} +(-\)\(57\!\cdots\!68\)\( + \)\(23\!\cdots\!76\)\( \beta_{1} - \)\(39\!\cdots\!16\)\( \beta_{2}) q^{52} +(\)\(50\!\cdots\!54\)\( + \)\(47\!\cdots\!80\)\( \beta_{1} - \)\(90\!\cdots\!36\)\( \beta_{2}) q^{53} +(-\)\(28\!\cdots\!00\)\( - \)\(39\!\cdots\!36\)\( \beta_{1} + \)\(84\!\cdots\!68\)\( \beta_{2}) q^{54} +(\)\(13\!\cdots\!80\)\( - \)\(29\!\cdots\!74\)\( \beta_{1} - \)\(24\!\cdots\!58\)\( \beta_{2}) q^{55} +(-\)\(21\!\cdots\!00\)\( + \)\(60\!\cdots\!96\)\( \beta_{1} - \)\(53\!\cdots\!48\)\( \beta_{2}) q^{56} +(\)\(64\!\cdots\!20\)\( + \)\(23\!\cdots\!28\)\( \beta_{1} - \)\(14\!\cdots\!24\)\( \beta_{2}) q^{57} +(-\)\(34\!\cdots\!60\)\( - \)\(32\!\cdots\!34\)\( \beta_{1} + \)\(21\!\cdots\!72\)\( \beta_{2}) q^{58} +(\)\(75\!\cdots\!00\)\( - \)\(62\!\cdots\!43\)\( \beta_{1} + \)\(37\!\cdots\!09\)\( \beta_{2}) q^{59} +(\)\(57\!\cdots\!40\)\( + \)\(21\!\cdots\!88\)\( \beta_{1} - \)\(56\!\cdots\!04\)\( \beta_{2}) q^{60} +(-\)\(31\!\cdots\!18\)\( - \)\(68\!\cdots\!00\)\( \beta_{1} - \)\(30\!\cdots\!00\)\( \beta_{2}) q^{61} +(-\)\(69\!\cdots\!16\)\( + \)\(12\!\cdots\!48\)\( \beta_{1} - \)\(20\!\cdots\!20\)\( \beta_{2}) q^{62} +(-\)\(32\!\cdots\!16\)\( + \)\(28\!\cdots\!14\)\( \beta_{1} + \)\(11\!\cdots\!62\)\( \beta_{2}) q^{63} +(-\)\(37\!\cdots\!12\)\( - \)\(13\!\cdots\!68\)\( \beta_{1} + \)\(11\!\cdots\!84\)\( \beta_{2}) q^{64} +(-\)\(90\!\cdots\!20\)\( - \)\(32\!\cdots\!84\)\( \beta_{1} - \)\(31\!\cdots\!28\)\( \beta_{2}) q^{65} +(\)\(97\!\cdots\!44\)\( + \)\(67\!\cdots\!48\)\( \beta_{1} - \)\(28\!\cdots\!24\)\( \beta_{2}) q^{66} +(-\)\(24\!\cdots\!48\)\( - \)\(23\!\cdots\!25\)\( \beta_{1} + \)\(46\!\cdots\!27\)\( \beta_{2}) q^{67} +(\)\(18\!\cdots\!56\)\( + \)\(14\!\cdots\!36\)\( \beta_{1} + \)\(81\!\cdots\!28\)\( \beta_{2}) q^{68} +(\)\(65\!\cdots\!44\)\( + \)\(99\!\cdots\!24\)\( \beta_{1} - \)\(90\!\cdots\!12\)\( \beta_{2}) q^{69} +(\)\(19\!\cdots\!40\)\( - \)\(58\!\cdots\!32\)\( \beta_{1} - \)\(11\!\cdots\!44\)\( \beta_{2}) q^{70} +(-\)\(61\!\cdots\!88\)\( - \)\(47\!\cdots\!50\)\( \beta_{1} + \)\(97\!\cdots\!50\)\( \beta_{2}) q^{71} +(-\)\(32\!\cdots\!20\)\( - \)\(19\!\cdots\!68\)\( \beta_{1} + \)\(15\!\cdots\!44\)\( \beta_{2}) q^{72} +(-\)\(67\!\cdots\!66\)\( + \)\(18\!\cdots\!36\)\( \beta_{1} - \)\(13\!\cdots\!44\)\( \beta_{2}) q^{73} +(\)\(55\!\cdots\!36\)\( + \)\(14\!\cdots\!50\)\( \beta_{1} - \)\(24\!\cdots\!00\)\( \beta_{2}) q^{74} +(-\)\(70\!\cdots\!00\)\( - \)\(18\!\cdots\!55\)\( \beta_{1} + \)\(77\!\cdots\!65\)\( \beta_{2}) q^{75} +(\)\(25\!\cdots\!00\)\( + \)\(93\!\cdots\!92\)\( \beta_{1} - \)\(56\!\cdots\!96\)\( \beta_{2}) q^{76} +(\)\(19\!\cdots\!64\)\( - \)\(84\!\cdots\!68\)\( \beta_{1} - \)\(12\!\cdots\!72\)\( \beta_{2}) q^{77} +(-\)\(65\!\cdots\!32\)\( - \)\(11\!\cdots\!76\)\( \beta_{1} + \)\(16\!\cdots\!16\)\( \beta_{2}) q^{78} +(\)\(50\!\cdots\!00\)\( + \)\(26\!\cdots\!96\)\( \beta_{1} + \)\(11\!\cdots\!52\)\( \beta_{2}) q^{79} +(-\)\(33\!\cdots\!60\)\( + \)\(56\!\cdots\!48\)\( \beta_{1} + \)\(15\!\cdots\!16\)\( \beta_{2}) q^{80} +(\)\(24\!\cdots\!21\)\( + \)\(14\!\cdots\!44\)\( \beta_{1} - \)\(50\!\cdots\!72\)\( \beta_{2}) q^{81} +(-\)\(10\!\cdots\!56\)\( + \)\(20\!\cdots\!18\)\( \beta_{1} + \)\(32\!\cdots\!80\)\( \beta_{2}) q^{82} +(-\)\(29\!\cdots\!76\)\( - \)\(22\!\cdots\!81\)\( \beta_{1} + \)\(66\!\cdots\!87\)\( \beta_{2}) q^{83} +(-\)\(11\!\cdots\!04\)\( - \)\(43\!\cdots\!84\)\( \beta_{1} - \)\(34\!\cdots\!08\)\( \beta_{2}) q^{84} +(\)\(14\!\cdots\!40\)\( + \)\(55\!\cdots\!68\)\( \beta_{1} + \)\(76\!\cdots\!56\)\( \beta_{2}) q^{85} +(\)\(83\!\cdots\!32\)\( - \)\(55\!\cdots\!84\)\( \beta_{1} - \)\(75\!\cdots\!08\)\( \beta_{2}) q^{86} +(\)\(57\!\cdots\!80\)\( + \)\(12\!\cdots\!62\)\( \beta_{1} - \)\(18\!\cdots\!46\)\( \beta_{2}) q^{87} +(-\)\(85\!\cdots\!20\)\( + \)\(82\!\cdots\!92\)\( \beta_{1} + \)\(64\!\cdots\!64\)\( \beta_{2}) q^{88} +(\)\(67\!\cdots\!50\)\( - \)\(45\!\cdots\!52\)\( \beta_{1} + \)\(31\!\cdots\!76\)\( \beta_{2}) q^{89} +(-\)\(77\!\cdots\!20\)\( - \)\(14\!\cdots\!94\)\( \beta_{1} + \)\(31\!\cdots\!52\)\( \beta_{2}) q^{90} +(-\)\(38\!\cdots\!28\)\( - \)\(28\!\cdots\!64\)\( \beta_{1} - \)\(77\!\cdots\!68\)\( \beta_{2}) q^{91} +(\)\(22\!\cdots\!52\)\( + \)\(40\!\cdots\!60\)\( \beta_{1} - \)\(35\!\cdots\!28\)\( \beta_{2}) q^{92} +(-\)\(16\!\cdots\!32\)\( - \)\(11\!\cdots\!00\)\( \beta_{1} + \)\(55\!\cdots\!68\)\( \beta_{2}) q^{93} +(-\)\(44\!\cdots\!44\)\( - \)\(43\!\cdots\!16\)\( \beta_{1} + \)\(48\!\cdots\!08\)\( \beta_{2}) q^{94} +(\)\(10\!\cdots\!00\)\( + \)\(52\!\cdots\!50\)\( \beta_{1} + \)\(12\!\cdots\!50\)\( \beta_{2}) q^{95} +(\)\(36\!\cdots\!12\)\( + \)\(13\!\cdots\!44\)\( \beta_{1} - \)\(11\!\cdots\!72\)\( \beta_{2}) q^{96} +(-\)\(12\!\cdots\!98\)\( - \)\(11\!\cdots\!28\)\( \beta_{1} + \)\(21\!\cdots\!96\)\( \beta_{2}) q^{97} +(\)\(80\!\cdots\!36\)\( - \)\(10\!\cdots\!93\)\( \beta_{1} - \)\(31\!\cdots\!00\)\( \beta_{2}) q^{98} +(-\)\(47\!\cdots\!36\)\( - \)\(12\!\cdots\!51\)\( \beta_{1} + \)\(17\!\cdots\!13\)\( \beta_{2}) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 2209944q^{2} + 24401437812q^{3} + 9822618421824q^{4} + 535205380774170q^{5} - 91793974758443424q^{6} + 301971425665478856q^{7} - 15830863913787348480q^{8} + 477482125171066743231q^{9} + O(q^{10}) \) \( 3q - 2209944q^{2} + 24401437812q^{3} + 9822618421824q^{4} + 535205380774170q^{5} - 91793974758443424q^{6} + 301971425665478856q^{7} - 15830863913787348480q^{8} + \)\(47\!\cdots\!31\)\(q^{9} + \)\(42\!\cdots\!20\)\(q^{10} + \)\(26\!\cdots\!96\)\(q^{11} + \)\(59\!\cdots\!36\)\(q^{12} + \)\(26\!\cdots\!82\)\(q^{13} + \)\(14\!\cdots\!28\)\(q^{14} + \)\(35\!\cdots\!40\)\(q^{15} - \)\(79\!\cdots\!92\)\(q^{16} - \)\(40\!\cdots\!94\)\(q^{17} - \)\(41\!\cdots\!48\)\(q^{18} + \)\(15\!\cdots\!00\)\(q^{19} + \)\(14\!\cdots\!60\)\(q^{20} + \)\(39\!\cdots\!36\)\(q^{21} + \)\(13\!\cdots\!92\)\(q^{22} - \)\(12\!\cdots\!48\)\(q^{23} - \)\(11\!\cdots\!00\)\(q^{24} - \)\(23\!\cdots\!75\)\(q^{25} - \)\(17\!\cdots\!44\)\(q^{26} + \)\(12\!\cdots\!80\)\(q^{27} + \)\(16\!\cdots\!68\)\(q^{28} + \)\(57\!\cdots\!50\)\(q^{29} - \)\(73\!\cdots\!60\)\(q^{30} - \)\(25\!\cdots\!24\)\(q^{31} - \)\(14\!\cdots\!84\)\(q^{32} - \)\(27\!\cdots\!16\)\(q^{33} + \)\(46\!\cdots\!88\)\(q^{34} + \)\(23\!\cdots\!20\)\(q^{35} + \)\(81\!\cdots\!48\)\(q^{36} - \)\(23\!\cdots\!94\)\(q^{37} - \)\(30\!\cdots\!20\)\(q^{38} - \)\(39\!\cdots\!28\)\(q^{39} - \)\(32\!\cdots\!00\)\(q^{40} + \)\(25\!\cdots\!66\)\(q^{41} + \)\(14\!\cdots\!32\)\(q^{42} + \)\(24\!\cdots\!92\)\(q^{43} - \)\(10\!\cdots\!32\)\(q^{44} + \)\(25\!\cdots\!90\)\(q^{45} - \)\(18\!\cdots\!64\)\(q^{46} + \)\(30\!\cdots\!56\)\(q^{47} - \)\(25\!\cdots\!48\)\(q^{48} + \)\(34\!\cdots\!79\)\(q^{49} + \)\(19\!\cdots\!00\)\(q^{50} + \)\(13\!\cdots\!56\)\(q^{51} - \)\(17\!\cdots\!04\)\(q^{52} + \)\(15\!\cdots\!62\)\(q^{53} - \)\(84\!\cdots\!00\)\(q^{54} + \)\(39\!\cdots\!40\)\(q^{55} - \)\(64\!\cdots\!00\)\(q^{56} + \)\(19\!\cdots\!60\)\(q^{57} - \)\(10\!\cdots\!80\)\(q^{58} + \)\(22\!\cdots\!00\)\(q^{59} + \)\(17\!\cdots\!20\)\(q^{60} - \)\(93\!\cdots\!54\)\(q^{61} - \)\(20\!\cdots\!48\)\(q^{62} - \)\(96\!\cdots\!48\)\(q^{63} - \)\(11\!\cdots\!36\)\(q^{64} - \)\(27\!\cdots\!60\)\(q^{65} + \)\(29\!\cdots\!32\)\(q^{66} - \)\(73\!\cdots\!44\)\(q^{67} + \)\(54\!\cdots\!68\)\(q^{68} + \)\(19\!\cdots\!32\)\(q^{69} + \)\(59\!\cdots\!20\)\(q^{70} - \)\(18\!\cdots\!64\)\(q^{71} - \)\(98\!\cdots\!60\)\(q^{72} - \)\(20\!\cdots\!98\)\(q^{73} + \)\(16\!\cdots\!08\)\(q^{74} - \)\(21\!\cdots\!00\)\(q^{75} + \)\(77\!\cdots\!00\)\(q^{76} + \)\(59\!\cdots\!92\)\(q^{77} - \)\(19\!\cdots\!96\)\(q^{78} + \)\(15\!\cdots\!00\)\(q^{79} - \)\(10\!\cdots\!80\)\(q^{80} + \)\(73\!\cdots\!63\)\(q^{81} - \)\(32\!\cdots\!68\)\(q^{82} - \)\(89\!\cdots\!28\)\(q^{83} - \)\(34\!\cdots\!12\)\(q^{84} + \)\(43\!\cdots\!20\)\(q^{85} + \)\(25\!\cdots\!96\)\(q^{86} + \)\(17\!\cdots\!40\)\(q^{87} - \)\(25\!\cdots\!60\)\(q^{88} + \)\(20\!\cdots\!50\)\(q^{89} - \)\(23\!\cdots\!60\)\(q^{90} - \)\(11\!\cdots\!84\)\(q^{91} + \)\(68\!\cdots\!56\)\(q^{92} - \)\(49\!\cdots\!96\)\(q^{93} - \)\(13\!\cdots\!32\)\(q^{94} + \)\(31\!\cdots\!00\)\(q^{95} + \)\(10\!\cdots\!36\)\(q^{96} - \)\(38\!\cdots\!94\)\(q^{97} + \)\(24\!\cdots\!08\)\(q^{98} - \)\(14\!\cdots\!08\)\(q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - 11258260111 x - 264759545317170\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -6 \nu^{2} + 343074 \nu + 45033040444 \)\()/8369\)
\(\beta_{2}\)\(=\)\((\)\( 14838 \nu^{2} + 1176206478 \nu - 111366709018012 \)\()/8369\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} + 2473 \beta_{1}\)\()/241920\)
\(\nu^{2}\)\(=\)\((\)\(57179 \beta_{2} - 196034413 \beta_{1} + 1815732190702080\)\()/241920\)

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
−24885.9
116336.
−91450.2
−4.65343e6 3.22027e10 1.28583e13 5.54482e14 −1.49853e17 −5.57604e17 −1.89031e19 7.08758e20 −2.58024e21
1.2 −1.18359e6 −1.79507e10 −7.39521e12 −6.39116e14 2.12463e16 −1.72730e18 1.91639e19 −6.02939e18 7.56451e20
1.3 3.62707e6 1.01494e10 4.35955e12 6.19839e14 3.68127e16 2.58688e18 −1.60917e19 −2.25246e20 2.24820e21
\(n\): e.g. 2-40 or 990-1000
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 1.44.a.a 3
3.b odd 2 1 9.44.a.b 3
4.b odd 2 1 16.44.a.c 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.44.a.a 3 1.a even 1 1 trivial
9.44.a.b 3 3.b odd 2 1
16.44.a.c 3 4.b odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{44}^{\mathrm{new}}(\Gamma_0(1))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 2209944 T + 10724756563968 T^{2} + 18900761561310167040 T^{3} + \)\(94\!\cdots\!44\)\( T^{4} + \)\(17\!\cdots\!16\)\( T^{5} + \)\(68\!\cdots\!12\)\( T^{6} \)
$3$ \( 1 - 24401437812 T + \)\(55\!\cdots\!97\)\( T^{2} - \)\(10\!\cdots\!40\)\( T^{3} + \)\(18\!\cdots\!19\)\( T^{4} - \)\(26\!\cdots\!48\)\( T^{5} + \)\(35\!\cdots\!83\)\( T^{6} \)
$5$ \( 1 - 535205380774170 T + \)\(30\!\cdots\!75\)\( T^{2} - \)\(99\!\cdots\!00\)\( T^{3} + \)\(34\!\cdots\!75\)\( T^{4} - \)\(69\!\cdots\!50\)\( T^{5} + \)\(14\!\cdots\!25\)\( T^{6} \)
$7$ \( 1 - 301971425665478856 T + \)\(16\!\cdots\!93\)\( T^{2} - \)\(38\!\cdots\!00\)\( T^{3} + \)\(35\!\cdots\!99\)\( T^{4} - \)\(14\!\cdots\!44\)\( T^{5} + \)\(10\!\cdots\!07\)\( T^{6} \)
$11$ \( 1 - \)\(26\!\cdots\!96\)\( T + \)\(16\!\cdots\!65\)\( T^{2} - \)\(28\!\cdots\!20\)\( T^{3} + \)\(10\!\cdots\!15\)\( T^{4} - \)\(96\!\cdots\!56\)\( T^{5} + \)\(21\!\cdots\!91\)\( T^{6} \)
$13$ \( 1 - \)\(26\!\cdots\!82\)\( T + \)\(44\!\cdots\!87\)\( T^{2} - \)\(47\!\cdots\!80\)\( T^{3} + \)\(35\!\cdots\!39\)\( T^{4} - \)\(16\!\cdots\!38\)\( T^{5} + \)\(49\!\cdots\!73\)\( T^{6} \)
$17$ \( 1 + \)\(40\!\cdots\!94\)\( T + \)\(16\!\cdots\!43\)\( T^{2} + \)\(69\!\cdots\!20\)\( T^{3} + \)\(13\!\cdots\!59\)\( T^{4} + \)\(26\!\cdots\!86\)\( T^{5} + \)\(53\!\cdots\!97\)\( T^{6} \)
$19$ \( 1 - \)\(15\!\cdots\!00\)\( T + \)\(10\!\cdots\!77\)\( T^{2} - \)\(52\!\cdots\!00\)\( T^{3} + \)\(10\!\cdots\!43\)\( T^{4} - \)\(14\!\cdots\!00\)\( T^{5} + \)\(91\!\cdots\!79\)\( T^{6} \)
$23$ \( 1 + \)\(12\!\cdots\!48\)\( T + \)\(53\!\cdots\!77\)\( T^{2} + \)\(37\!\cdots\!80\)\( T^{3} + \)\(19\!\cdots\!59\)\( T^{4} + \)\(15\!\cdots\!72\)\( T^{5} + \)\(46\!\cdots\!63\)\( T^{6} \)
$29$ \( 1 - \)\(57\!\cdots\!50\)\( T + \)\(27\!\cdots\!67\)\( T^{2} - \)\(79\!\cdots\!00\)\( T^{3} + \)\(21\!\cdots\!63\)\( T^{4} - \)\(33\!\cdots\!50\)\( T^{5} + \)\(44\!\cdots\!69\)\( T^{6} \)
$31$ \( 1 + \)\(25\!\cdots\!24\)\( T + \)\(53\!\cdots\!65\)\( T^{2} + \)\(66\!\cdots\!80\)\( T^{3} + \)\(72\!\cdots\!15\)\( T^{4} + \)\(46\!\cdots\!44\)\( T^{5} + \)\(24\!\cdots\!71\)\( T^{6} \)
$37$ \( 1 + \)\(23\!\cdots\!94\)\( T + \)\(78\!\cdots\!43\)\( T^{2} + \)\(12\!\cdots\!60\)\( T^{3} + \)\(21\!\cdots\!79\)\( T^{4} + \)\(17\!\cdots\!46\)\( T^{5} + \)\(19\!\cdots\!77\)\( T^{6} \)
$41$ \( 1 - \)\(25\!\cdots\!66\)\( T + \)\(26\!\cdots\!15\)\( T^{2} - \)\(88\!\cdots\!20\)\( T^{3} + \)\(59\!\cdots\!15\)\( T^{4} - \)\(12\!\cdots\!06\)\( T^{5} + \)\(11\!\cdots\!61\)\( T^{6} \)
$43$ \( 1 - \)\(24\!\cdots\!92\)\( T + \)\(66\!\cdots\!57\)\( T^{2} - \)\(83\!\cdots\!00\)\( T^{3} + \)\(11\!\cdots\!99\)\( T^{4} - \)\(72\!\cdots\!08\)\( T^{5} + \)\(52\!\cdots\!43\)\( T^{6} \)
$47$ \( 1 - \)\(30\!\cdots\!56\)\( T + \)\(20\!\cdots\!93\)\( T^{2} - \)\(40\!\cdots\!20\)\( T^{3} + \)\(16\!\cdots\!39\)\( T^{4} - \)\(19\!\cdots\!24\)\( T^{5} + \)\(50\!\cdots\!67\)\( T^{6} \)
$53$ \( 1 - \)\(15\!\cdots\!62\)\( T + \)\(48\!\cdots\!47\)\( T^{2} - \)\(42\!\cdots\!40\)\( T^{3} + \)\(67\!\cdots\!19\)\( T^{4} - \)\(29\!\cdots\!98\)\( T^{5} + \)\(27\!\cdots\!33\)\( T^{6} \)
$59$ \( 1 - \)\(22\!\cdots\!00\)\( T + \)\(50\!\cdots\!37\)\( T^{2} - \)\(62\!\cdots\!00\)\( T^{3} + \)\(71\!\cdots\!23\)\( T^{4} - \)\(44\!\cdots\!00\)\( T^{5} + \)\(27\!\cdots\!39\)\( T^{6} \)
$61$ \( 1 + \)\(93\!\cdots\!54\)\( T + \)\(89\!\cdots\!15\)\( T^{2} - \)\(75\!\cdots\!20\)\( T^{3} + \)\(52\!\cdots\!15\)\( T^{4} + \)\(32\!\cdots\!94\)\( T^{5} + \)\(20\!\cdots\!41\)\( T^{6} \)
$67$ \( 1 + \)\(73\!\cdots\!44\)\( T + \)\(79\!\cdots\!93\)\( T^{2} + \)\(52\!\cdots\!20\)\( T^{3} + \)\(26\!\cdots\!59\)\( T^{4} + \)\(80\!\cdots\!36\)\( T^{5} + \)\(36\!\cdots\!47\)\( T^{6} \)
$71$ \( 1 + \)\(18\!\cdots\!64\)\( T + \)\(22\!\cdots\!65\)\( T^{2} + \)\(16\!\cdots\!80\)\( T^{3} + \)\(90\!\cdots\!15\)\( T^{4} + \)\(29\!\cdots\!44\)\( T^{5} + \)\(64\!\cdots\!31\)\( T^{6} \)
$73$ \( 1 + \)\(20\!\cdots\!98\)\( T + \)\(46\!\cdots\!27\)\( T^{2} + \)\(50\!\cdots\!80\)\( T^{3} + \)\(61\!\cdots\!59\)\( T^{4} + \)\(35\!\cdots\!22\)\( T^{5} + \)\(23\!\cdots\!13\)\( T^{6} \)
$79$ \( 1 - \)\(15\!\cdots\!00\)\( T + \)\(11\!\cdots\!17\)\( T^{2} - \)\(12\!\cdots\!00\)\( T^{3} + \)\(43\!\cdots\!63\)\( T^{4} - \)\(23\!\cdots\!00\)\( T^{5} + \)\(62\!\cdots\!19\)\( T^{6} \)
$83$ \( 1 + \)\(89\!\cdots\!28\)\( T + \)\(68\!\cdots\!17\)\( T^{2} + \)\(57\!\cdots\!40\)\( T^{3} + \)\(22\!\cdots\!79\)\( T^{4} + \)\(97\!\cdots\!32\)\( T^{5} + \)\(36\!\cdots\!03\)\( T^{6} \)
$89$ \( 1 - \)\(20\!\cdots\!50\)\( T + \)\(14\!\cdots\!07\)\( T^{2} - \)\(37\!\cdots\!00\)\( T^{3} + \)\(98\!\cdots\!83\)\( T^{4} - \)\(90\!\cdots\!50\)\( T^{5} + \)\(29\!\cdots\!09\)\( T^{6} \)
$97$ \( 1 + \)\(38\!\cdots\!94\)\( T + \)\(63\!\cdots\!43\)\( T^{2} + \)\(18\!\cdots\!80\)\( T^{3} + \)\(17\!\cdots\!39\)\( T^{4} + \)\(28\!\cdots\!26\)\( T^{5} + \)\(19\!\cdots\!17\)\( T^{6} \)
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