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: \( 2^{11}\cdot 3^{5}\cdot 5^{2}\cdot 7^{2} \)
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.

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

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

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

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.

Hecke kernels

There are no other newforms in \(S_{50}^{\mathrm{new}}(\Gamma_0(1))\).