Properties

Label 1.50.a.a
Level 1
Weight 50
Character orbit 1.a
Self dual yes
Analytic conductor 15.207
Analytic rank 1
Dimension 3
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 1 \)
Weight: \( k \) \(=\) \( 50 \)
Character orbit: \([\chi]\) \(=\) 1.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(15.2066205099\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{11}\cdot 3^{5}\cdot 5^{2}\cdot 7^{2} \)
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.

\(f(q)\) \(=\) \( q + ( -8075056 + \beta_{1} ) q^{2} + ( -108984897468 + 4110 \beta_{1} + \beta_{2} ) q^{3} + ( 10500922661632 - 399312 \beta_{1} - 96 \beta_{2} ) q^{4} + ( 21294678572359750 - 830946040 \beta_{1} - 192420 \beta_{2} ) q^{5} + ( 2968712083082256192 - 140296680252 \beta_{1} - 24220416 \beta_{2} ) q^{6} + ( 169797159166237064 - 6197674750596 \beta_{1} + 657909714 \beta_{2} ) q^{7} + ( 4258131001838821888000 - 549479574747904 \beta_{1} + 2325616128 \beta_{2} ) q^{8} + ( 113875230170760151789773 - 11457427166565264 \beta_{1} - 286203921912 \beta_{2} ) q^{9} +O(q^{10})\) \( q +(-8075056 + \beta_{1}) q^{2} +(-108984897468 + 4110 \beta_{1} + \beta_{2}) q^{3} +(10500922661632 - 399312 \beta_{1} - 96 \beta_{2}) q^{4} +(21294678572359750 - 830946040 \beta_{1} - 192420 \beta_{2}) q^{5} +(2968712083082256192 - 140296680252 \beta_{1} - 24220416 \beta_{2}) q^{6} +(169797159166237064 - 6197674750596 \beta_{1} + 657909714 \beta_{2}) q^{7} +(\)\(42\!\cdots\!00\)\( - 549479574747904 \beta_{1} + 2325616128 \beta_{2}) q^{8} +(\)\(11\!\cdots\!73\)\( - 11457427166565264 \beta_{1} - 286203921912 \beta_{2}) q^{9} +(-\)\(59\!\cdots\!00\)\( + 27011895709376070 \beta_{1} + 4664342031360 \beta_{2}) q^{10} +(-\)\(69\!\cdots\!08\)\( + 1063535619042334570 \beta_{1} - 33484749559965 \beta_{2}) q^{11} +(-\)\(33\!\cdots\!16\)\( + 1100579697151435968 \beta_{1} + 27590671758976 \beta_{2}) q^{12} +(-\)\(62\!\cdots\!38\)\( - 57281740925974331832 \beta_{1} + 1632545471760060 \beta_{2}) q^{13} +(-\)\(31\!\cdots\!96\)\( - 88757573956408427896 \beta_{1} - 15080285330707968 \beta_{2}) q^{14} +(-\)\(68\!\cdots\!00\)\( + \)\(22\!\cdots\!60\)\( \beta_{1} + 56042589608616630 \beta_{2}) q^{15} +(-\)\(31\!\cdots\!04\)\( + \)\(11\!\cdots\!96\)\( \beta_{1} + 51383439727239168 \beta_{2}) q^{16} +(-\)\(11\!\cdots\!46\)\( - \)\(46\!\cdots\!40\)\( \beta_{1} - 1609842257538727608 \beta_{2}) q^{17} +(-\)\(67\!\cdots\!48\)\( + \)\(43\!\cdots\!45\)\( \beta_{1} + 7918966438100822016 \beta_{2}) q^{18} +(-\)\(15\!\cdots\!20\)\( + \)\(51\!\cdots\!22\)\( \beta_{1} - 16168497468226428699 \beta_{2}) q^{19} +(\)\(65\!\cdots\!00\)\( - \)\(21\!\cdots\!80\)\( \beta_{1} - 5402253524702091840 \beta_{2}) q^{20} +(\)\(20\!\cdots\!72\)\( - \)\(62\!\cdots\!24\)\( \beta_{1} + 27296567013048957008 \beta_{2}) q^{21} +(\)\(59\!\cdots\!48\)\( + \)\(33\!\cdots\!32\)\( \beta_{1} + \)\(69\!\cdots\!20\)\( \beta_{2}) q^{22} +(\)\(15\!\cdots\!92\)\( + \)\(41\!\cdots\!36\)\( \beta_{1} - \)\(46\!\cdots\!26\)\( \beta_{2}) q^{23} +(-\)\(83\!\cdots\!60\)\( + \)\(51\!\cdots\!76\)\( \beta_{1} + \)\(12\!\cdots\!08\)\( \beta_{2}) q^{24} +(-\)\(46\!\cdots\!25\)\( - \)\(42\!\cdots\!00\)\( \beta_{1} - \)\(10\!\cdots\!00\)\( \beta_{2}) q^{25} +(-\)\(28\!\cdots\!88\)\( - \)\(60\!\cdots\!86\)\( \beta_{1} - \)\(33\!\cdots\!88\)\( \beta_{2}) q^{26} +(-\)\(10\!\cdots\!00\)\( + \)\(37\!\cdots\!56\)\( \beta_{1} + \)\(91\!\cdots\!58\)\( \beta_{2}) q^{27} +(-\)\(19\!\cdots\!32\)\( + \)\(60\!\cdots\!64\)\( \beta_{1} - \)\(25\!\cdots\!44\)\( \beta_{2}) q^{28} +(\)\(37\!\cdots\!70\)\( - \)\(49\!\cdots\!92\)\( \beta_{1} - \)\(22\!\cdots\!36\)\( \beta_{2}) q^{29} +(\)\(16\!\cdots\!00\)\( - \)\(54\!\cdots\!80\)\( \beta_{1} - \)\(15\!\cdots\!40\)\( \beta_{2}) q^{30} +(\)\(31\!\cdots\!72\)\( + \)\(12\!\cdots\!60\)\( \beta_{1} + \)\(52\!\cdots\!80\)\( \beta_{2}) q^{31} +(\)\(24\!\cdots\!64\)\( - \)\(12\!\cdots\!64\)\( \beta_{1} - \)\(25\!\cdots\!60\)\( \beta_{2}) q^{32} +(-\)\(81\!\cdots\!56\)\( + \)\(21\!\cdots\!40\)\( \beta_{1} - \)\(20\!\cdots\!48\)\( \beta_{2}) q^{33} +(-\)\(14\!\cdots\!76\)\( - \)\(13\!\cdots\!14\)\( \beta_{1} + \)\(42\!\cdots\!88\)\( \beta_{2}) q^{34} +(-\)\(39\!\cdots\!00\)\( + \)\(12\!\cdots\!20\)\( \beta_{1} - \)\(46\!\cdots\!40\)\( \beta_{2}) q^{35} +(\)\(12\!\cdots\!36\)\( - \)\(45\!\cdots\!84\)\( \beta_{1} - \)\(31\!\cdots\!72\)\( \beta_{2}) q^{36} +(\)\(80\!\cdots\!34\)\( + \)\(78\!\cdots\!12\)\( \beta_{1} - \)\(16\!\cdots\!32\)\( \beta_{2}) q^{37} +(\)\(38\!\cdots\!00\)\( - \)\(10\!\cdots\!76\)\( \beta_{1} + \)\(33\!\cdots\!32\)\( \beta_{2}) q^{38} +(\)\(43\!\cdots\!16\)\( - \)\(11\!\cdots\!76\)\( \beta_{1} + \)\(68\!\cdots\!42\)\( \beta_{2}) q^{39} +(\)\(17\!\cdots\!00\)\( - \)\(99\!\cdots\!00\)\( \beta_{1} - \)\(24\!\cdots\!00\)\( \beta_{2}) q^{40} +(-\)\(20\!\cdots\!38\)\( + \)\(44\!\cdots\!60\)\( \beta_{1} + \)\(16\!\cdots\!80\)\( \beta_{2}) q^{41} +(-\)\(48\!\cdots\!92\)\( + \)\(15\!\cdots\!24\)\( \beta_{1} - \)\(46\!\cdots\!44\)\( \beta_{2}) q^{42} +(-\)\(25\!\cdots\!48\)\( - \)\(26\!\cdots\!18\)\( \beta_{1} + \)\(53\!\cdots\!07\)\( \beta_{2}) q^{43} +(\)\(78\!\cdots\!44\)\( - \)\(20\!\cdots\!64\)\( \beta_{1} + \)\(19\!\cdots\!88\)\( \beta_{2}) q^{44} +(\)\(25\!\cdots\!50\)\( - \)\(91\!\cdots\!20\)\( \beta_{1} - \)\(63\!\cdots\!60\)\( \beta_{2}) q^{45} +(\)\(89\!\cdots\!32\)\( + \)\(21\!\cdots\!00\)\( \beta_{1} + \)\(10\!\cdots\!00\)\( \beta_{2}) q^{46} +(\)\(24\!\cdots\!24\)\( + \)\(97\!\cdots\!08\)\( \beta_{1} + \)\(65\!\cdots\!16\)\( \beta_{2}) q^{47} +(\)\(52\!\cdots\!92\)\( - \)\(18\!\cdots\!24\)\( \beta_{1} - \)\(32\!\cdots\!56\)\( \beta_{2}) q^{48} +(-\)\(93\!\cdots\!43\)\( - \)\(33\!\cdots\!80\)\( \beta_{1} + \)\(15\!\cdots\!60\)\( \beta_{2}) q^{49} +(-\)\(17\!\cdots\!00\)\( - \)\(72\!\cdots\!25\)\( \beta_{1} + \)\(30\!\cdots\!00\)\( \beta_{2}) q^{50} +(-\)\(51\!\cdots\!68\)\( + \)\(17\!\cdots\!32\)\( \beta_{1} - \)\(17\!\cdots\!94\)\( \beta_{2}) q^{51} +(-\)\(41\!\cdots\!56\)\( + \)\(10\!\cdots\!24\)\( \beta_{1} - \)\(65\!\cdots\!36\)\( \beta_{2}) q^{52} +(-\)\(51\!\cdots\!18\)\( + \)\(37\!\cdots\!60\)\( \beta_{1} + \)\(57\!\cdots\!76\)\( \beta_{2}) q^{53} +(\)\(27\!\cdots\!80\)\( - \)\(82\!\cdots\!28\)\( \beta_{1} - \)\(25\!\cdots\!24\)\( \beta_{2}) q^{54} +(\)\(15\!\cdots\!00\)\( - \)\(40\!\cdots\!80\)\( \beta_{1} + \)\(38\!\cdots\!10\)\( \beta_{2}) q^{55} +(\)\(22\!\cdots\!80\)\( + \)\(35\!\cdots\!92\)\( \beta_{1} + \)\(84\!\cdots\!36\)\( \beta_{2}) q^{56} +(-\)\(26\!\cdots\!00\)\( + \)\(52\!\cdots\!12\)\( \beta_{1} - \)\(21\!\cdots\!84\)\( \beta_{2}) q^{57} +(-\)\(55\!\cdots\!00\)\( + \)\(33\!\cdots\!86\)\( \beta_{1} + \)\(10\!\cdots\!48\)\( \beta_{2}) q^{58} +(-\)\(20\!\cdots\!60\)\( - \)\(21\!\cdots\!14\)\( \beta_{1} + \)\(21\!\cdots\!63\)\( \beta_{2}) q^{59} +(-\)\(29\!\cdots\!00\)\( + \)\(10\!\cdots\!20\)\( \beta_{1} + \)\(10\!\cdots\!60\)\( \beta_{2}) q^{60} +(-\)\(80\!\cdots\!58\)\( - \)\(20\!\cdots\!00\)\( \beta_{1} + \)\(35\!\cdots\!00\)\( \beta_{2}) q^{61} +(\)\(62\!\cdots\!68\)\( + \)\(96\!\cdots\!92\)\( \beta_{1} - \)\(13\!\cdots\!40\)\( \beta_{2}) q^{62} +(-\)\(26\!\cdots\!88\)\( + \)\(27\!\cdots\!84\)\( \beta_{1} + \)\(14\!\cdots\!98\)\( \beta_{2}) q^{63} +(\)\(17\!\cdots\!12\)\( + \)\(24\!\cdots\!36\)\( \beta_{1} + \)\(32\!\cdots\!88\)\( \beta_{2}) q^{64} +(-\)\(81\!\cdots\!00\)\( + \)\(22\!\cdots\!60\)\( \beta_{1} - \)\(12\!\cdots\!20\)\( \beta_{2}) q^{65} +(\)\(17\!\cdots\!64\)\( - \)\(52\!\cdots\!44\)\( \beta_{1} + \)\(46\!\cdots\!48\)\( \beta_{2}) q^{66} +(-\)\(33\!\cdots\!96\)\( - \)\(14\!\cdots\!10\)\( \beta_{1} - \)\(96\!\cdots\!43\)\( \beta_{2}) q^{67} +(\)\(49\!\cdots\!48\)\( - \)\(16\!\cdots\!24\)\( \beta_{1} + \)\(16\!\cdots\!12\)\( \beta_{2}) q^{68} +(-\)\(16\!\cdots\!24\)\( + \)\(50\!\cdots\!72\)\( \beta_{1} + \)\(13\!\cdots\!76\)\( \beta_{2}) q^{69} +(\)\(93\!\cdots\!00\)\( - \)\(29\!\cdots\!60\)\( \beta_{1} - \)\(57\!\cdots\!80\)\( \beta_{2}) q^{70} +(-\)\(10\!\cdots\!68\)\( + \)\(40\!\cdots\!00\)\( \beta_{1} - \)\(15\!\cdots\!50\)\( \beta_{2}) q^{71} +(\)\(34\!\cdots\!00\)\( - \)\(13\!\cdots\!72\)\( \beta_{1} - \)\(36\!\cdots\!96\)\( \beta_{2}) q^{72} +(\)\(18\!\cdots\!42\)\( - \)\(22\!\cdots\!04\)\( \beta_{1} + \)\(14\!\cdots\!04\)\( \beta_{2}) q^{73} +(\)\(33\!\cdots\!64\)\( + \)\(15\!\cdots\!30\)\( \beta_{1} + \)\(32\!\cdots\!40\)\( \beta_{2}) q^{74} +(-\)\(40\!\cdots\!00\)\( + \)\(14\!\cdots\!50\)\( \beta_{1} + \)\(34\!\cdots\!75\)\( \beta_{2}) q^{75} +(\)\(25\!\cdots\!60\)\( - \)\(50\!\cdots\!76\)\( \beta_{1} + \)\(20\!\cdots\!92\)\( \beta_{2}) q^{76} +(-\)\(10\!\cdots\!12\)\( + \)\(10\!\cdots\!08\)\( \beta_{1} - \)\(19\!\cdots\!92\)\( \beta_{2}) q^{77} +(-\)\(92\!\cdots\!36\)\( + \)\(30\!\cdots\!64\)\( \beta_{1} - \)\(15\!\cdots\!56\)\( \beta_{2}) q^{78} +(-\)\(26\!\cdots\!80\)\( - \)\(14\!\cdots\!52\)\( \beta_{1} + \)\(39\!\cdots\!84\)\( \beta_{2}) q^{79} +(-\)\(10\!\cdots\!00\)\( + \)\(37\!\cdots\!60\)\( \beta_{1} + \)\(63\!\cdots\!80\)\( \beta_{2}) q^{80} +(\)\(22\!\cdots\!81\)\( + \)\(94\!\cdots\!48\)\( \beta_{1} - \)\(10\!\cdots\!16\)\( \beta_{2}) q^{81} +(\)\(39\!\cdots\!28\)\( - \)\(18\!\cdots\!18\)\( \beta_{1} - \)\(44\!\cdots\!40\)\( \beta_{2}) q^{82} +(\)\(31\!\cdots\!72\)\( + \)\(45\!\cdots\!34\)\( \beta_{1} + \)\(10\!\cdots\!53\)\( \beta_{2}) q^{83} +(\)\(25\!\cdots\!04\)\( - \)\(11\!\cdots\!92\)\( \beta_{1} - \)\(29\!\cdots\!36\)\( \beta_{2}) q^{84} +(\)\(99\!\cdots\!00\)\( - \)\(33\!\cdots\!80\)\( \beta_{1} + \)\(31\!\cdots\!60\)\( \beta_{2}) q^{85} +(-\)\(11\!\cdots\!28\)\( - \)\(50\!\cdots\!08\)\( \beta_{1} - \)\(10\!\cdots\!64\)\( \beta_{2}) q^{86} +(-\)\(58\!\cdots\!00\)\( + \)\(23\!\cdots\!68\)\( \beta_{1} + \)\(45\!\cdots\!74\)\( \beta_{2}) q^{87} +(-\)\(35\!\cdots\!00\)\( - \)\(13\!\cdots\!68\)\( \beta_{1} - \)\(43\!\cdots\!24\)\( \beta_{2}) q^{88} +(\)\(12\!\cdots\!10\)\( + \)\(21\!\cdots\!64\)\( \beta_{1} - \)\(11\!\cdots\!88\)\( \beta_{2}) q^{89} +(-\)\(66\!\cdots\!00\)\( + \)\(22\!\cdots\!10\)\( \beta_{1} + \)\(16\!\cdots\!80\)\( \beta_{2}) q^{90} +(\)\(53\!\cdots\!92\)\( - \)\(21\!\cdots\!68\)\( \beta_{1} + \)\(74\!\cdots\!56\)\( \beta_{2}) q^{91} +(\)\(15\!\cdots\!04\)\( - \)\(48\!\cdots\!00\)\( \beta_{1} - \)\(13\!\cdots\!88\)\( \beta_{2}) q^{92} +(\)\(19\!\cdots\!04\)\( - \)\(68\!\cdots\!20\)\( \beta_{1} - \)\(23\!\cdots\!48\)\( \beta_{2}) q^{93} +(\)\(30\!\cdots\!84\)\( + \)\(27\!\cdots\!92\)\( \beta_{1} - \)\(16\!\cdots\!64\)\( \beta_{2}) q^{94} +(\)\(49\!\cdots\!00\)\( - \)\(97\!\cdots\!00\)\( \beta_{1} + \)\(41\!\cdots\!50\)\( \beta_{2}) q^{95} +(-\)\(89\!\cdots\!48\)\( + \)\(29\!\cdots\!68\)\( \beta_{1} + \)\(72\!\cdots\!44\)\( \beta_{2}) q^{96} +(-\)\(34\!\cdots\!26\)\( + \)\(19\!\cdots\!48\)\( \beta_{1} + \)\(59\!\cdots\!36\)\( \beta_{2}) q^{97} +(-\)\(94\!\cdots\!92\)\( - \)\(12\!\cdots\!03\)\( \beta_{1} - \)\(33\!\cdots\!80\)\( \beta_{2}) q^{98} +(-\)\(37\!\cdots\!84\)\( - \)\(95\!\cdots\!78\)\( \beta_{1} - \)\(13\!\cdots\!49\)\( \beta_{2}) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 24225168q^{2} - 326954692404q^{3} + 31502767984896q^{4} + 63884035717079250q^{5} + 8906136249246768576q^{6} + 509391477498711192q^{7} + 12774393005516465664000q^{8} + 341625690512280455369319q^{9} + O(q^{10}) \) \( 3q - 24225168q^{2} - 326954692404q^{3} + 31502767984896q^{4} + 63884035717079250q^{5} + 8906136249246768576q^{6} + 509391477498711192q^{7} + \)\(12\!\cdots\!00\)\(q^{8} + \)\(34\!\cdots\!19\)\(q^{9} - \)\(17\!\cdots\!00\)\(q^{10} - \)\(20\!\cdots\!24\)\(q^{11} - \)\(10\!\cdots\!48\)\(q^{12} - \)\(18\!\cdots\!14\)\(q^{13} - \)\(94\!\cdots\!88\)\(q^{14} - \)\(20\!\cdots\!00\)\(q^{15} - \)\(95\!\cdots\!12\)\(q^{16} - \)\(33\!\cdots\!38\)\(q^{17} - \)\(20\!\cdots\!44\)\(q^{18} - \)\(45\!\cdots\!60\)\(q^{19} + \)\(19\!\cdots\!00\)\(q^{20} + \)\(61\!\cdots\!16\)\(q^{21} + \)\(17\!\cdots\!44\)\(q^{22} + \)\(45\!\cdots\!76\)\(q^{23} - \)\(25\!\cdots\!80\)\(q^{24} - \)\(13\!\cdots\!75\)\(q^{25} - \)\(85\!\cdots\!64\)\(q^{26} - \)\(31\!\cdots\!00\)\(q^{27} - \)\(59\!\cdots\!96\)\(q^{28} + \)\(11\!\cdots\!10\)\(q^{29} + \)\(50\!\cdots\!00\)\(q^{30} + \)\(93\!\cdots\!16\)\(q^{31} + \)\(73\!\cdots\!92\)\(q^{32} - \)\(24\!\cdots\!68\)\(q^{33} - \)\(44\!\cdots\!28\)\(q^{34} - \)\(11\!\cdots\!00\)\(q^{35} + \)\(37\!\cdots\!08\)\(q^{36} + \)\(24\!\cdots\!02\)\(q^{37} + \)\(11\!\cdots\!00\)\(q^{38} + \)\(12\!\cdots\!48\)\(q^{39} + \)\(52\!\cdots\!00\)\(q^{40} - \)\(62\!\cdots\!14\)\(q^{41} - \)\(14\!\cdots\!76\)\(q^{42} - \)\(77\!\cdots\!44\)\(q^{43} + \)\(23\!\cdots\!32\)\(q^{44} + \)\(76\!\cdots\!50\)\(q^{45} + \)\(26\!\cdots\!96\)\(q^{46} + \)\(72\!\cdots\!72\)\(q^{47} + \)\(15\!\cdots\!76\)\(q^{48} - \)\(28\!\cdots\!29\)\(q^{49} - \)\(53\!\cdots\!00\)\(q^{50} - \)\(15\!\cdots\!04\)\(q^{51} - \)\(12\!\cdots\!68\)\(q^{52} - \)\(15\!\cdots\!54\)\(q^{53} + \)\(82\!\cdots\!40\)\(q^{54} + \)\(46\!\cdots\!00\)\(q^{55} + \)\(67\!\cdots\!40\)\(q^{56} - \)\(79\!\cdots\!00\)\(q^{57} - \)\(16\!\cdots\!00\)\(q^{58} - \)\(62\!\cdots\!80\)\(q^{59} - \)\(88\!\cdots\!00\)\(q^{60} - \)\(24\!\cdots\!74\)\(q^{61} + \)\(18\!\cdots\!04\)\(q^{62} - \)\(79\!\cdots\!64\)\(q^{63} + \)\(51\!\cdots\!36\)\(q^{64} - \)\(24\!\cdots\!00\)\(q^{65} + \)\(52\!\cdots\!92\)\(q^{66} - \)\(10\!\cdots\!88\)\(q^{67} + \)\(14\!\cdots\!44\)\(q^{68} - \)\(48\!\cdots\!72\)\(q^{69} + \)\(28\!\cdots\!00\)\(q^{70} - \)\(32\!\cdots\!04\)\(q^{71} + \)\(10\!\cdots\!00\)\(q^{72} + \)\(56\!\cdots\!26\)\(q^{73} + \)\(10\!\cdots\!92\)\(q^{74} - \)\(12\!\cdots\!00\)\(q^{75} + \)\(75\!\cdots\!80\)\(q^{76} - \)\(32\!\cdots\!36\)\(q^{77} - \)\(27\!\cdots\!08\)\(q^{78} - \)\(78\!\cdots\!40\)\(q^{79} - \)\(30\!\cdots\!00\)\(q^{80} + \)\(67\!\cdots\!43\)\(q^{81} + \)\(11\!\cdots\!84\)\(q^{82} + \)\(94\!\cdots\!16\)\(q^{83} + \)\(77\!\cdots\!12\)\(q^{84} + \)\(29\!\cdots\!00\)\(q^{85} - \)\(34\!\cdots\!84\)\(q^{86} - \)\(17\!\cdots\!00\)\(q^{87} - \)\(10\!\cdots\!00\)\(q^{88} + \)\(36\!\cdots\!30\)\(q^{89} - \)\(20\!\cdots\!00\)\(q^{90} + \)\(16\!\cdots\!76\)\(q^{91} + \)\(46\!\cdots\!12\)\(q^{92} + \)\(59\!\cdots\!12\)\(q^{93} + \)\(90\!\cdots\!52\)\(q^{94} + \)\(14\!\cdots\!00\)\(q^{95} - \)\(26\!\cdots\!44\)\(q^{96} - \)\(10\!\cdots\!78\)\(q^{97} - \)\(28\!\cdots\!76\)\(q^{98} - \)\(11\!\cdots\!52\)\(q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 27962089502 x + 71708842875120\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -24 \nu^{2} + 58056 \nu + 447393412688 \)\()/14051\)
\(\beta_{2}\)\(=\)\((\)\( 39888 \nu^{2} + 59389824528 \nu - 743587680658656 \)\()/14051\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} + 1662 \beta_{1} + 1411200\)\()/4233600\)
\(\nu^{2}\)\(=\)\((\)\(2419 \beta_{2} - 2474576022 \beta_{1} + 78920201411856000\)\()/4233600\)

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
−168486.
165922.
2565.11
−2.54182e7 −8.64745e11 8.31363e13 1.67413e17 2.19803e19 −3.42668e20 1.21960e22 5.08484e23 −4.25535e24
1.2 −2.25719e7 5.57972e11 −5.34581e13 −1.06460e17 −1.25945e19 5.68014e20 1.39135e22 7.20337e22 2.40300e24
1.3 2.37650e7 −2.01823e10 1.82457e12 2.93049e15 −4.79631e17 −2.24836e20 −1.33351e22 −2.38892e23 6.96431e22
\(n\): 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 1.50.a.a 3
3.b odd 2 1 9.50.a.a 3
4.b odd 2 1 16.50.a.c 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.50.a.a 3 1.a even 1 1 trivial
9.50.a.a 3 3.b odd 2 1
16.50.a.c 3 4.b odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{50}^{\mathrm{new}}(\Gamma_0(1))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 24225168 T + 1122102928453632 T^{2} + \)\(13\!\cdots\!60\)\( T^{3} + \)\(63\!\cdots\!84\)\( T^{4} + \)\(76\!\cdots\!92\)\( T^{5} + \)\(17\!\cdots\!28\)\( T^{6} \)
$3$ \( 1 + 326954692404 T + \)\(24\!\cdots\!73\)\( T^{2} + \)\(14\!\cdots\!60\)\( T^{3} + \)\(57\!\cdots\!59\)\( T^{4} + \)\(18\!\cdots\!56\)\( T^{5} + \)\(13\!\cdots\!87\)\( T^{6} \)
$5$ \( 1 - 63884035717079250 T + \)\(35\!\cdots\!75\)\( T^{2} - \)\(22\!\cdots\!00\)\( T^{3} + \)\(63\!\cdots\!75\)\( T^{4} - \)\(20\!\cdots\!50\)\( T^{5} + \)\(56\!\cdots\!25\)\( T^{6} \)
$7$ \( 1 - 509391477498711192 T + \)\(52\!\cdots\!57\)\( T^{2} - \)\(44\!\cdots\!00\)\( T^{3} + \)\(13\!\cdots\!99\)\( T^{4} - \)\(33\!\cdots\!08\)\( T^{5} + \)\(16\!\cdots\!43\)\( T^{6} \)
$11$ \( 1 + \)\(20\!\cdots\!24\)\( T + \)\(19\!\cdots\!65\)\( T^{2} + \)\(41\!\cdots\!80\)\( T^{3} + \)\(20\!\cdots\!15\)\( T^{4} + \)\(23\!\cdots\!44\)\( T^{5} + \)\(12\!\cdots\!71\)\( T^{6} \)
$13$ \( 1 + \)\(18\!\cdots\!14\)\( T + \)\(76\!\cdots\!03\)\( T^{2} + \)\(72\!\cdots\!20\)\( T^{3} + \)\(29\!\cdots\!19\)\( T^{4} + \)\(27\!\cdots\!06\)\( T^{5} + \)\(56\!\cdots\!17\)\( T^{6} \)
$17$ \( 1 + \)\(33\!\cdots\!38\)\( T + \)\(65\!\cdots\!07\)\( T^{2} + \)\(96\!\cdots\!80\)\( T^{3} + \)\(12\!\cdots\!79\)\( T^{4} + \)\(12\!\cdots\!42\)\( T^{5} + \)\(75\!\cdots\!73\)\( T^{6} \)
$19$ \( 1 + \)\(45\!\cdots\!60\)\( T + \)\(17\!\cdots\!37\)\( T^{2} + \)\(41\!\cdots\!80\)\( T^{3} + \)\(79\!\cdots\!23\)\( T^{4} + \)\(95\!\cdots\!60\)\( T^{5} + \)\(94\!\cdots\!39\)\( T^{6} \)
$23$ \( 1 - \)\(45\!\cdots\!76\)\( T + \)\(10\!\cdots\!33\)\( T^{2} - \)\(18\!\cdots\!20\)\( T^{3} + \)\(56\!\cdots\!79\)\( T^{4} - \)\(12\!\cdots\!44\)\( T^{5} + \)\(14\!\cdots\!47\)\( T^{6} \)
$29$ \( 1 - \)\(11\!\cdots\!10\)\( T + \)\(17\!\cdots\!07\)\( T^{2} - \)\(10\!\cdots\!80\)\( T^{3} + \)\(80\!\cdots\!83\)\( T^{4} - \)\(23\!\cdots\!10\)\( T^{5} + \)\(93\!\cdots\!09\)\( T^{6} \)
$31$ \( 1 - \)\(93\!\cdots\!16\)\( T + \)\(96\!\cdots\!65\)\( T^{2} + \)\(30\!\cdots\!80\)\( T^{3} + \)\(11\!\cdots\!15\)\( T^{4} - \)\(13\!\cdots\!56\)\( T^{5} + \)\(16\!\cdots\!11\)\( T^{6} \)
$37$ \( 1 - \)\(24\!\cdots\!02\)\( T + \)\(16\!\cdots\!07\)\( T^{2} - \)\(30\!\cdots\!60\)\( T^{3} + \)\(11\!\cdots\!39\)\( T^{4} - \)\(11\!\cdots\!58\)\( T^{5} + \)\(33\!\cdots\!33\)\( T^{6} \)
$41$ \( 1 + \)\(62\!\cdots\!14\)\( T + \)\(41\!\cdots\!15\)\( T^{2} + \)\(13\!\cdots\!80\)\( T^{3} + \)\(44\!\cdots\!15\)\( T^{4} + \)\(70\!\cdots\!94\)\( T^{5} + \)\(12\!\cdots\!81\)\( T^{6} \)
$43$ \( 1 + \)\(77\!\cdots\!44\)\( T + \)\(27\!\cdots\!93\)\( T^{2} + \)\(16\!\cdots\!00\)\( T^{3} + \)\(30\!\cdots\!99\)\( T^{4} + \)\(93\!\cdots\!56\)\( T^{5} + \)\(13\!\cdots\!07\)\( T^{6} \)
$47$ \( 1 - \)\(72\!\cdots\!72\)\( T + \)\(24\!\cdots\!57\)\( T^{2} - \)\(11\!\cdots\!80\)\( T^{3} + \)\(21\!\cdots\!19\)\( T^{4} - \)\(53\!\cdots\!08\)\( T^{5} + \)\(62\!\cdots\!63\)\( T^{6} \)
$53$ \( 1 + \)\(15\!\cdots\!54\)\( T + \)\(82\!\cdots\!23\)\( T^{2} + \)\(50\!\cdots\!60\)\( T^{3} + \)\(25\!\cdots\!59\)\( T^{4} + \)\(14\!\cdots\!06\)\( T^{5} + \)\(29\!\cdots\!37\)\( T^{6} \)
$59$ \( 1 + \)\(62\!\cdots\!80\)\( T + \)\(28\!\cdots\!17\)\( T^{2} + \)\(75\!\cdots\!40\)\( T^{3} + \)\(16\!\cdots\!63\)\( T^{4} + \)\(21\!\cdots\!80\)\( T^{5} + \)\(20\!\cdots\!19\)\( T^{6} \)
$61$ \( 1 + \)\(24\!\cdots\!74\)\( T + \)\(61\!\cdots\!15\)\( T^{2} + \)\(18\!\cdots\!80\)\( T^{3} + \)\(18\!\cdots\!15\)\( T^{4} + \)\(22\!\cdots\!94\)\( T^{5} + \)\(27\!\cdots\!21\)\( T^{6} \)
$67$ \( 1 + \)\(10\!\cdots\!88\)\( T + \)\(61\!\cdots\!57\)\( T^{2} + \)\(24\!\cdots\!80\)\( T^{3} + \)\(18\!\cdots\!79\)\( T^{4} + \)\(91\!\cdots\!92\)\( T^{5} + \)\(27\!\cdots\!23\)\( T^{6} \)
$71$ \( 1 + \)\(32\!\cdots\!04\)\( T + \)\(17\!\cdots\!65\)\( T^{2} + \)\(32\!\cdots\!80\)\( T^{3} + \)\(90\!\cdots\!15\)\( T^{4} + \)\(85\!\cdots\!44\)\( T^{5} + \)\(13\!\cdots\!91\)\( T^{6} \)
$73$ \( 1 - \)\(56\!\cdots\!26\)\( T + \)\(30\!\cdots\!83\)\( T^{2} - \)\(65\!\cdots\!20\)\( T^{3} + \)\(61\!\cdots\!79\)\( T^{4} - \)\(22\!\cdots\!94\)\( T^{5} + \)\(80\!\cdots\!97\)\( T^{6} \)
$79$ \( 1 + \)\(78\!\cdots\!40\)\( T + \)\(49\!\cdots\!57\)\( T^{2} + \)\(50\!\cdots\!20\)\( T^{3} + \)\(47\!\cdots\!83\)\( T^{4} + \)\(73\!\cdots\!40\)\( T^{5} + \)\(89\!\cdots\!59\)\( T^{6} \)
$83$ \( 1 - \)\(94\!\cdots\!16\)\( T + \)\(14\!\cdots\!13\)\( T^{2} - \)\(11\!\cdots\!60\)\( T^{3} + \)\(15\!\cdots\!39\)\( T^{4} - \)\(11\!\cdots\!44\)\( T^{5} + \)\(12\!\cdots\!27\)\( T^{6} \)
$89$ \( 1 - \)\(36\!\cdots\!30\)\( T + \)\(37\!\cdots\!27\)\( T^{2} - \)\(88\!\cdots\!40\)\( T^{3} + \)\(12\!\cdots\!43\)\( T^{4} - \)\(40\!\cdots\!30\)\( T^{5} + \)\(36\!\cdots\!29\)\( T^{6} \)
$97$ \( 1 + \)\(10\!\cdots\!78\)\( T + \)\(73\!\cdots\!07\)\( T^{2} + \)\(34\!\cdots\!20\)\( T^{3} + \)\(16\!\cdots\!19\)\( T^{4} + \)\(52\!\cdots\!42\)\( T^{5} + \)\(11\!\cdots\!13\)\( T^{6} \)
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