Properties

Label 2.50.a.b
Level 2
Weight 50
Character orbit 2.a
Self dual yes
Analytic conductor 30.413
Analytic rank 0
Dimension 3
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 2 \)
Weight: \( k \) \(=\) \( 50 \)
Character orbit: \([\chi]\) \(=\) 2.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(30.4132410198\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
Defining polynomial: \(x^{3} - x^{2} - 104434803447206332 x + 4289992005756109702361620\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{22}\cdot 3^{5}\cdot 5^{2}\cdot 7^{2} \)
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.

\(f(q)\) \(=\) \( q + 16777216 q^{2} + ( -5401204796 - \beta_{1} ) q^{3} + 281474976710656 q^{4} + ( -33937672133946810 - 78841 \beta_{1} - \beta_{2} ) q^{5} + ( -90617179522727936 - 16777216 \beta_{1} ) q^{6} + ( -41826393076734599032 + 99835658 \beta_{1} - 3700 \beta_{2} ) q^{7} + 4722366482869645213696 q^{8} + ( 129810977173232307350733 - 175523252718 \beta_{1} + 1418850 \beta_{2} ) q^{9} +O(q^{10})\) \( q +16777216 q^{2} +(-5401204796 - \beta_{1}) q^{3} +281474976710656 q^{4} +(-33937672133946810 - 78841 \beta_{1} - \beta_{2}) q^{5} +(-90617179522727936 - 16777216 \beta_{1}) q^{6} +(-41826393076734599032 + 99835658 \beta_{1} - 3700 \beta_{2}) q^{7} +\)\(47\!\cdots\!96\)\( q^{8} +(\)\(12\!\cdots\!33\)\( - 175523252718 \beta_{1} + 1418850 \beta_{2}) q^{9} +(-\)\(56\!\cdots\!60\)\( - 1322732486656 \beta_{1} - 16777216 \beta_{2}) q^{10} +(-\)\(86\!\cdots\!68\)\( - 76202146518283 \beta_{1} + 52537400 \beta_{2}) q^{11} +(-\)\(15\!\cdots\!76\)\( - 281474976710656 \beta_{1}) q^{12} +(\)\(98\!\cdots\!14\)\( - 340286117859137 \beta_{1} + 4260419575 \beta_{2}) q^{13} +(-\)\(70\!\cdots\!12\)\( + 1674964398768128 \beta_{1} - 62075699200 \beta_{2}) q^{14} +(\)\(29\!\cdots\!60\)\( + 125268300209547246 \beta_{1} + 303590419956 \beta_{2}) q^{15} +\)\(79\!\cdots\!36\)\( q^{16} +(\)\(13\!\cdots\!18\)\( - 2043132990929384798 \beta_{1} - 6325836222350 \beta_{2}) q^{17} +(\)\(21\!\cdots\!28\)\( - 2944791523872473088 \beta_{1} + 23804352921600 \beta_{2}) q^{18} +(\)\(48\!\cdots\!60\)\( + 18528996595190482627 \beta_{1} + 8505702406600 \beta_{2}) q^{19} +(-\)\(95\!\cdots\!60\)\( - 22191768638844829696 \beta_{1} - 281474976710656 \beta_{2}) q^{20} +(-\)\(36\!\cdots\!28\)\( + \)\(45\!\cdots\!44\)\( \beta_{1} + 567737584938900 \beta_{2}) q^{21} +(-\)\(14\!\cdots\!88\)\( - \)\(12\!\cdots\!28\)\( \beta_{1} + 881431307878400 \beta_{2}) q^{22} +(-\)\(16\!\cdots\!36\)\( - \)\(67\!\cdots\!66\)\( \beta_{1} - 4043926350742700 \beta_{2}) q^{23} +(-\)\(25\!\cdots\!16\)\( - \)\(47\!\cdots\!96\)\( \beta_{1}) q^{24} +(\)\(13\!\cdots\!75\)\( + \)\(34\!\cdots\!80\)\( \beta_{1} + 480088492044180 \beta_{2}) q^{25} +(\)\(16\!\cdots\!24\)\( - \)\(57\!\cdots\!92\)\( \beta_{1} + 71477979460403200 \beta_{2}) q^{26} +(\)\(65\!\cdots\!00\)\( - \)\(72\!\cdots\!02\)\( \beta_{1} - 22990498274413800 \beta_{2}) q^{27} +(-\)\(11\!\cdots\!92\)\( + \)\(28\!\cdots\!48\)\( \beta_{1} - 1041457413829427200 \beta_{2}) q^{28} +(\)\(31\!\cdots\!10\)\( - \)\(97\!\cdots\!81\)\( \beta_{1} + 2068317919687048475 \beta_{2}) q^{29} +(\)\(49\!\cdots\!60\)\( + \)\(21\!\cdots\!36\)\( \beta_{1} + 5093402051132522496 \beta_{2}) q^{30} +(\)\(80\!\cdots\!52\)\( + \)\(21\!\cdots\!04\)\( \beta_{1} - 21160996591207173600 \beta_{2}) q^{31} +\)\(13\!\cdots\!76\)\( q^{32} +(\)\(28\!\cdots\!28\)\( - \)\(10\!\cdots\!94\)\( \beta_{1} + 98046584479571070150 \beta_{2}) q^{33} +(\)\(21\!\cdots\!88\)\( - \)\(34\!\cdots\!68\)\( \beta_{1} - \)\(10\!\cdots\!00\)\( \beta_{2}) q^{34} +(\)\(10\!\cdots\!20\)\( + \)\(72\!\cdots\!32\)\( \beta_{1} - \)\(19\!\cdots\!48\)\( \beta_{2}) q^{35} +(\)\(36\!\cdots\!48\)\( - \)\(49\!\cdots\!08\)\( \beta_{1} + \)\(39\!\cdots\!00\)\( \beta_{2}) q^{36} +(\)\(10\!\cdots\!58\)\( + \)\(22\!\cdots\!27\)\( \beta_{1} - 1016938682892827725 \beta_{2}) q^{37} +(\)\(81\!\cdots\!60\)\( + \)\(31\!\cdots\!32\)\( \beta_{1} + \)\(14\!\cdots\!00\)\( \beta_{2}) q^{38} +(\)\(12\!\cdots\!56\)\( - \)\(14\!\cdots\!82\)\( \beta_{1} - \)\(33\!\cdots\!00\)\( \beta_{2}) q^{39} +(-\)\(16\!\cdots\!60\)\( - \)\(37\!\cdots\!36\)\( \beta_{1} - \)\(47\!\cdots\!96\)\( \beta_{2}) q^{40} +(-\)\(38\!\cdots\!18\)\( - \)\(10\!\cdots\!04\)\( \beta_{1} + \)\(88\!\cdots\!00\)\( \beta_{2}) q^{41} +(-\)\(61\!\cdots\!48\)\( + \)\(75\!\cdots\!04\)\( \beta_{1} + \)\(95\!\cdots\!00\)\( \beta_{2}) q^{42} +(-\)\(58\!\cdots\!16\)\( + \)\(10\!\cdots\!57\)\( \beta_{1} - \)\(32\!\cdots\!00\)\( \beta_{2}) q^{43} +(-\)\(24\!\cdots\!08\)\( - \)\(21\!\cdots\!48\)\( \beta_{1} + \)\(14\!\cdots\!00\)\( \beta_{2}) q^{44} +(-\)\(38\!\cdots\!30\)\( - \)\(19\!\cdots\!73\)\( \beta_{1} + \)\(33\!\cdots\!47\)\( \beta_{2}) q^{45} +(-\)\(27\!\cdots\!76\)\( - \)\(11\!\cdots\!56\)\( \beta_{1} - \)\(67\!\cdots\!00\)\( \beta_{2}) q^{46} +(-\)\(72\!\cdots\!52\)\( + \)\(72\!\cdots\!88\)\( \beta_{1} + \)\(31\!\cdots\!00\)\( \beta_{2}) q^{47} +(-\)\(42\!\cdots\!56\)\( - \)\(79\!\cdots\!36\)\( \beta_{1}) q^{48} +(\)\(12\!\cdots\!17\)\( + \)\(10\!\cdots\!48\)\( \beta_{1} - \)\(12\!\cdots\!00\)\( \beta_{2}) q^{49} +(\)\(22\!\cdots\!00\)\( + \)\(58\!\cdots\!80\)\( \beta_{1} + \)\(80\!\cdots\!80\)\( \beta_{2}) q^{50} +(\)\(74\!\cdots\!72\)\( - \)\(10\!\cdots\!90\)\( \beta_{1} + \)\(41\!\cdots\!00\)\( \beta_{2}) q^{51} +(\)\(27\!\cdots\!84\)\( - \)\(95\!\cdots\!72\)\( \beta_{1} + \)\(11\!\cdots\!00\)\( \beta_{2}) q^{52} +(\)\(24\!\cdots\!34\)\( - \)\(10\!\cdots\!37\)\( \beta_{1} - \)\(14\!\cdots\!25\)\( \beta_{2}) q^{53} +(\)\(10\!\cdots\!00\)\( - \)\(12\!\cdots\!32\)\( \beta_{1} - \)\(38\!\cdots\!00\)\( \beta_{2}) q^{54} +(\)\(10\!\cdots\!80\)\( + \)\(90\!\cdots\!18\)\( \beta_{1} + \)\(34\!\cdots\!48\)\( \beta_{2}) q^{55} +(-\)\(19\!\cdots\!72\)\( + \)\(47\!\cdots\!68\)\( \beta_{1} - \)\(17\!\cdots\!00\)\( \beta_{2}) q^{56} +(-\)\(68\!\cdots\!60\)\( - \)\(24\!\cdots\!82\)\( \beta_{1} - \)\(27\!\cdots\!50\)\( \beta_{2}) q^{57} +(\)\(52\!\cdots\!60\)\( - \)\(16\!\cdots\!96\)\( \beta_{1} + \)\(34\!\cdots\!00\)\( \beta_{2}) q^{58} +(-\)\(17\!\cdots\!80\)\( - \)\(19\!\cdots\!47\)\( \beta_{1} - \)\(81\!\cdots\!00\)\( \beta_{2}) q^{59} +(\)\(82\!\cdots\!60\)\( + \)\(35\!\cdots\!76\)\( \beta_{1} + \)\(85\!\cdots\!36\)\( \beta_{2}) q^{60} +(-\)\(41\!\cdots\!98\)\( + \)\(48\!\cdots\!43\)\( \beta_{1} + \)\(18\!\cdots\!75\)\( \beta_{2}) q^{61} +(\)\(13\!\cdots\!32\)\( + \)\(35\!\cdots\!64\)\( \beta_{1} - \)\(35\!\cdots\!00\)\( \beta_{2}) q^{62} +(-\)\(15\!\cdots\!56\)\( + \)\(34\!\cdots\!30\)\( \beta_{1} + \)\(13\!\cdots\!00\)\( \beta_{2}) q^{63} +\)\(22\!\cdots\!16\)\( q^{64} +(-\)\(14\!\cdots\!40\)\( - \)\(12\!\cdots\!04\)\( \beta_{1} - \)\(63\!\cdots\!44\)\( \beta_{2}) q^{65} +(\)\(47\!\cdots\!48\)\( - \)\(17\!\cdots\!04\)\( \beta_{1} + \)\(16\!\cdots\!00\)\( \beta_{2}) q^{66} +(\)\(37\!\cdots\!08\)\( - \)\(51\!\cdots\!01\)\( \beta_{1} - \)\(27\!\cdots\!00\)\( \beta_{2}) q^{67} +(\)\(36\!\cdots\!08\)\( - \)\(57\!\cdots\!88\)\( \beta_{1} - \)\(17\!\cdots\!00\)\( \beta_{2}) q^{68} +(\)\(25\!\cdots\!56\)\( + \)\(19\!\cdots\!12\)\( \beta_{1} + \)\(17\!\cdots\!00\)\( \beta_{2}) q^{69} +(\)\(16\!\cdots\!20\)\( + \)\(12\!\cdots\!12\)\( \beta_{1} - \)\(33\!\cdots\!68\)\( \beta_{2}) q^{70} +(-\)\(13\!\cdots\!08\)\( - \)\(18\!\cdots\!62\)\( \beta_{1} + \)\(31\!\cdots\!00\)\( \beta_{2}) q^{71} +(\)\(61\!\cdots\!68\)\( - \)\(82\!\cdots\!28\)\( \beta_{1} + \)\(67\!\cdots\!00\)\( \beta_{2}) q^{72} +(-\)\(33\!\cdots\!06\)\( - \)\(40\!\cdots\!62\)\( \beta_{1} + \)\(60\!\cdots\!50\)\( \beta_{2}) q^{73} +(\)\(17\!\cdots\!28\)\( + \)\(36\!\cdots\!32\)\( \beta_{1} - \)\(17\!\cdots\!00\)\( \beta_{2}) q^{74} +(-\)\(12\!\cdots\!00\)\( - \)\(69\!\cdots\!55\)\( \beta_{1} - \)\(49\!\cdots\!80\)\( \beta_{2}) q^{75} +(\)\(13\!\cdots\!60\)\( + \)\(52\!\cdots\!12\)\( \beta_{1} + \)\(23\!\cdots\!00\)\( \beta_{2}) q^{76} +(-\)\(77\!\cdots\!24\)\( + \)\(31\!\cdots\!52\)\( \beta_{1} + \)\(92\!\cdots\!00\)\( \beta_{2}) q^{77} +(\)\(20\!\cdots\!96\)\( - \)\(25\!\cdots\!12\)\( \beta_{1} - \)\(56\!\cdots\!00\)\( \beta_{2}) q^{78} +(\)\(12\!\cdots\!20\)\( + \)\(32\!\cdots\!32\)\( \beta_{1} - \)\(91\!\cdots\!00\)\( \beta_{2}) q^{79} +(-\)\(26\!\cdots\!60\)\( - \)\(62\!\cdots\!76\)\( \beta_{1} - \)\(79\!\cdots\!36\)\( \beta_{2}) q^{80} +(-\)\(48\!\cdots\!39\)\( - \)\(33\!\cdots\!34\)\( \beta_{1} - \)\(23\!\cdots\!50\)\( \beta_{2}) q^{81} +(-\)\(63\!\cdots\!88\)\( - \)\(17\!\cdots\!64\)\( \beta_{1} + \)\(14\!\cdots\!00\)\( \beta_{2}) q^{82} +(\)\(50\!\cdots\!64\)\( - \)\(72\!\cdots\!65\)\( \beta_{1} + \)\(57\!\cdots\!00\)\( \beta_{2}) q^{83} +(-\)\(10\!\cdots\!68\)\( + \)\(12\!\cdots\!64\)\( \beta_{1} + \)\(15\!\cdots\!00\)\( \beta_{2}) q^{84} +(\)\(18\!\cdots\!20\)\( + \)\(29\!\cdots\!42\)\( \beta_{1} - \)\(10\!\cdots\!38\)\( \beta_{2}) q^{85} +(-\)\(98\!\cdots\!56\)\( + \)\(17\!\cdots\!12\)\( \beta_{1} - \)\(54\!\cdots\!00\)\( \beta_{2}) q^{86} +(\)\(35\!\cdots\!40\)\( - \)\(70\!\cdots\!94\)\( \beta_{1} + \)\(98\!\cdots\!00\)\( \beta_{2}) q^{87} +(-\)\(40\!\cdots\!28\)\( - \)\(35\!\cdots\!68\)\( \beta_{1} + \)\(24\!\cdots\!00\)\( \beta_{2}) q^{88} +(\)\(18\!\cdots\!90\)\( + \)\(60\!\cdots\!10\)\( \beta_{1} + \)\(75\!\cdots\!50\)\( \beta_{2}) q^{89} +(-\)\(64\!\cdots\!80\)\( - \)\(33\!\cdots\!68\)\( \beta_{1} + \)\(56\!\cdots\!52\)\( \beta_{2}) q^{90} +(-\)\(48\!\cdots\!48\)\( + \)\(77\!\cdots\!56\)\( \beta_{1} - \)\(18\!\cdots\!00\)\( \beta_{2}) q^{91} +(-\)\(46\!\cdots\!16\)\( - \)\(18\!\cdots\!96\)\( \beta_{1} - \)\(11\!\cdots\!00\)\( \beta_{2}) q^{92} +(-\)\(78\!\cdots\!92\)\( + \)\(18\!\cdots\!04\)\( \beta_{1} + \)\(10\!\cdots\!00\)\( \beta_{2}) q^{93} +(-\)\(12\!\cdots\!32\)\( + \)\(12\!\cdots\!08\)\( \beta_{1} + \)\(52\!\cdots\!00\)\( \beta_{2}) q^{94} +(-\)\(93\!\cdots\!00\)\( - \)\(28\!\cdots\!30\)\( \beta_{1} - \)\(99\!\cdots\!80\)\( \beta_{2}) q^{95} +(-\)\(71\!\cdots\!96\)\( - \)\(13\!\cdots\!76\)\( \beta_{1}) q^{96} +(\)\(14\!\cdots\!38\)\( + \)\(28\!\cdots\!30\)\( \beta_{1} + \)\(28\!\cdots\!50\)\( \beta_{2}) q^{97} +(\)\(20\!\cdots\!72\)\( + \)\(17\!\cdots\!68\)\( \beta_{1} - \)\(21\!\cdots\!00\)\( \beta_{2}) q^{98} +(\)\(58\!\cdots\!56\)\( - \)\(22\!\cdots\!55\)\( \beta_{1} - \)\(16\!\cdots\!00\)\( \beta_{2}) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 50331648q^{2} - 16203614388q^{3} + 844424930131968q^{4} - 101813016401840430q^{5} - 271851538568183808q^{6} - 125479179230203797096q^{7} + 14167099448608935641088q^{8} + 389432931519696922052199q^{9} + O(q^{10}) \) \( 3q + 50331648q^{2} - 16203614388q^{3} + 844424930131968q^{4} - 101813016401840430q^{5} - 271851538568183808q^{6} - \)\(12\!\cdots\!96\)\(q^{7} + \)\(14\!\cdots\!88\)\(q^{8} + \)\(38\!\cdots\!99\)\(q^{9} - \)\(17\!\cdots\!80\)\(q^{10} - \)\(25\!\cdots\!04\)\(q^{11} - \)\(45\!\cdots\!28\)\(q^{12} + \)\(29\!\cdots\!42\)\(q^{13} - \)\(21\!\cdots\!36\)\(q^{14} + \)\(87\!\cdots\!80\)\(q^{15} + \)\(23\!\cdots\!08\)\(q^{16} + \)\(39\!\cdots\!54\)\(q^{17} + \)\(65\!\cdots\!84\)\(q^{18} + \)\(14\!\cdots\!80\)\(q^{19} - \)\(28\!\cdots\!80\)\(q^{20} - \)\(10\!\cdots\!84\)\(q^{21} - \)\(43\!\cdots\!64\)\(q^{22} - \)\(49\!\cdots\!08\)\(q^{23} - \)\(76\!\cdots\!48\)\(q^{24} + \)\(39\!\cdots\!25\)\(q^{25} + \)\(49\!\cdots\!72\)\(q^{26} + \)\(19\!\cdots\!00\)\(q^{27} - \)\(35\!\cdots\!76\)\(q^{28} + \)\(93\!\cdots\!30\)\(q^{29} + \)\(14\!\cdots\!80\)\(q^{30} + \)\(24\!\cdots\!56\)\(q^{31} + \)\(39\!\cdots\!28\)\(q^{32} + \)\(84\!\cdots\!84\)\(q^{33} + \)\(65\!\cdots\!64\)\(q^{34} + \)\(30\!\cdots\!60\)\(q^{35} + \)\(10\!\cdots\!44\)\(q^{36} + \)\(30\!\cdots\!74\)\(q^{37} + \)\(24\!\cdots\!80\)\(q^{38} + \)\(36\!\cdots\!68\)\(q^{39} - \)\(48\!\cdots\!80\)\(q^{40} - \)\(11\!\cdots\!54\)\(q^{41} - \)\(18\!\cdots\!44\)\(q^{42} - \)\(17\!\cdots\!48\)\(q^{43} - \)\(72\!\cdots\!24\)\(q^{44} - \)\(11\!\cdots\!90\)\(q^{45} - \)\(83\!\cdots\!28\)\(q^{46} - \)\(21\!\cdots\!56\)\(q^{47} - \)\(12\!\cdots\!68\)\(q^{48} + \)\(37\!\cdots\!51\)\(q^{49} + \)\(66\!\cdots\!00\)\(q^{50} + \)\(22\!\cdots\!16\)\(q^{51} + \)\(82\!\cdots\!52\)\(q^{52} + \)\(74\!\cdots\!02\)\(q^{53} + \)\(32\!\cdots\!00\)\(q^{54} + \)\(32\!\cdots\!40\)\(q^{55} - \)\(59\!\cdots\!16\)\(q^{56} - \)\(20\!\cdots\!80\)\(q^{57} + \)\(15\!\cdots\!80\)\(q^{58} - \)\(53\!\cdots\!40\)\(q^{59} + \)\(24\!\cdots\!80\)\(q^{60} - \)\(12\!\cdots\!94\)\(q^{61} + \)\(40\!\cdots\!96\)\(q^{62} - \)\(46\!\cdots\!68\)\(q^{63} + \)\(66\!\cdots\!48\)\(q^{64} - \)\(42\!\cdots\!20\)\(q^{65} + \)\(14\!\cdots\!44\)\(q^{66} + \)\(11\!\cdots\!24\)\(q^{67} + \)\(11\!\cdots\!24\)\(q^{68} + \)\(77\!\cdots\!68\)\(q^{69} + \)\(50\!\cdots\!60\)\(q^{70} - \)\(40\!\cdots\!24\)\(q^{71} + \)\(18\!\cdots\!04\)\(q^{72} - \)\(10\!\cdots\!18\)\(q^{73} + \)\(51\!\cdots\!84\)\(q^{74} - \)\(38\!\cdots\!00\)\(q^{75} + \)\(41\!\cdots\!80\)\(q^{76} - \)\(23\!\cdots\!72\)\(q^{77} + \)\(60\!\cdots\!88\)\(q^{78} + \)\(36\!\cdots\!60\)\(q^{79} - \)\(80\!\cdots\!80\)\(q^{80} - \)\(14\!\cdots\!17\)\(q^{81} - \)\(19\!\cdots\!64\)\(q^{82} + \)\(15\!\cdots\!92\)\(q^{83} - \)\(30\!\cdots\!04\)\(q^{84} + \)\(56\!\cdots\!60\)\(q^{85} - \)\(29\!\cdots\!68\)\(q^{86} + \)\(10\!\cdots\!20\)\(q^{87} - \)\(12\!\cdots\!84\)\(q^{88} + \)\(54\!\cdots\!70\)\(q^{89} - \)\(19\!\cdots\!40\)\(q^{90} - \)\(14\!\cdots\!44\)\(q^{91} - \)\(13\!\cdots\!48\)\(q^{92} - \)\(23\!\cdots\!76\)\(q^{93} - \)\(36\!\cdots\!96\)\(q^{94} - \)\(28\!\cdots\!00\)\(q^{95} - \)\(21\!\cdots\!88\)\(q^{96} + \)\(43\!\cdots\!14\)\(q^{97} + \)\(62\!\cdots\!16\)\(q^{98} + \)\(17\!\cdots\!68\)\(q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 104434803447206332 x + 4289992005756109702361620\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 160 \nu^{2} + 17410920480 \nu - 11139712373505648960 \)\()/12670249\)
\(\beta_{2}\)\(=\)\((\)\( -11698720 \nu^{2} + 6966187410422880 \nu + 814502346867205307288640 \)\()/12670249\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} + 73117 \beta_{1} + 216760320\)\()/ 650280960 \)
\(\nu^{2}\)\(=\)\((\)\(-12090917 \beta_{2} + 4837630146127 \beta_{1} + 5030515869856343941939200\)\()/72253440\)

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
3.00244e8
−3.42020e8
4.17763e7
1.67772e7 −6.77152e11 2.81475e14 −2.33026e17 −1.13607e19 −5.15430e20 4.72237e21 2.19235e23 −3.90952e24
1.2 1.67772e7 −1.33407e11 2.81475e14 1.87739e17 −2.23819e18 8.28497e20 4.72237e21 −2.21502e23 3.14974e24
1.3 1.67772e7 7.94355e11 2.81475e14 −5.65262e16 1.33271e19 −4.38546e20 4.72237e21 3.91700e23 −9.48353e23
\(n\): 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 2.50.a.b 3
4.b odd 2 1 16.50.a.b 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2.50.a.b 3 1.a even 1 1 trivial
16.50.a.b 3 4.b odd 2 1

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{3} + 16203614388 T_{3}^{2} - \)\(55\!\cdots\!52\)\( T_{3} - \)\(71\!\cdots\!64\)\( \) acting on \(S_{50}^{\mathrm{new}}(\Gamma_0(2))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - 16777216 T )^{3} \)
$3$ \( 1 + 16203614388 T + \)\(16\!\cdots\!97\)\( T^{2} - \)\(64\!\cdots\!56\)\( T^{3} + \)\(39\!\cdots\!51\)\( T^{4} + \)\(92\!\cdots\!32\)\( T^{5} + \)\(13\!\cdots\!87\)\( T^{6} \)
$5$ \( 1 + 101813016401840430 T + \)\(12\!\cdots\!75\)\( T^{2} + \)\(11\!\cdots\!00\)\( T^{3} + \)\(21\!\cdots\!75\)\( T^{4} + \)\(32\!\cdots\!50\)\( T^{5} + \)\(56\!\cdots\!25\)\( T^{6} \)
$7$ \( 1 + \)\(12\!\cdots\!96\)\( T + \)\(20\!\cdots\!93\)\( T^{2} - \)\(12\!\cdots\!88\)\( T^{3} + \)\(53\!\cdots\!51\)\( T^{4} + \)\(82\!\cdots\!04\)\( T^{5} + \)\(16\!\cdots\!43\)\( T^{6} \)
$11$ \( 1 + \)\(25\!\cdots\!04\)\( T + \)\(96\!\cdots\!45\)\( T^{2} - \)\(35\!\cdots\!40\)\( T^{3} + \)\(10\!\cdots\!95\)\( T^{4} + \)\(29\!\cdots\!24\)\( T^{5} + \)\(12\!\cdots\!71\)\( T^{6} \)
$13$ \( 1 - \)\(29\!\cdots\!42\)\( T + \)\(13\!\cdots\!07\)\( T^{2} - \)\(22\!\cdots\!76\)\( T^{3} + \)\(51\!\cdots\!11\)\( T^{4} - \)\(43\!\cdots\!18\)\( T^{5} + \)\(56\!\cdots\!17\)\( T^{6} \)
$17$ \( 1 - \)\(39\!\cdots\!54\)\( T + \)\(70\!\cdots\!63\)\( T^{2} - \)\(93\!\cdots\!08\)\( T^{3} + \)\(13\!\cdots\!11\)\( T^{4} - \)\(15\!\cdots\!86\)\( T^{5} + \)\(75\!\cdots\!73\)\( T^{6} \)
$19$ \( 1 - \)\(14\!\cdots\!80\)\( T + \)\(12\!\cdots\!37\)\( T^{2} - \)\(12\!\cdots\!40\)\( T^{3} + \)\(56\!\cdots\!23\)\( T^{4} - \)\(30\!\cdots\!80\)\( T^{5} + \)\(94\!\cdots\!39\)\( T^{6} \)
$23$ \( 1 + \)\(49\!\cdots\!08\)\( T + \)\(23\!\cdots\!77\)\( T^{2} + \)\(55\!\cdots\!64\)\( T^{3} + \)\(12\!\cdots\!51\)\( T^{4} + \)\(13\!\cdots\!52\)\( T^{5} + \)\(14\!\cdots\!47\)\( T^{6} \)
$29$ \( 1 - \)\(93\!\cdots\!30\)\( T + \)\(95\!\cdots\!07\)\( T^{2} - \)\(86\!\cdots\!40\)\( T^{3} + \)\(43\!\cdots\!83\)\( T^{4} - \)\(19\!\cdots\!30\)\( T^{5} + \)\(93\!\cdots\!09\)\( T^{6} \)
$31$ \( 1 - \)\(24\!\cdots\!56\)\( T + \)\(16\!\cdots\!25\)\( T^{2} - \)\(71\!\cdots\!60\)\( T^{3} + \)\(20\!\cdots\!75\)\( T^{4} - \)\(34\!\cdots\!96\)\( T^{5} + \)\(16\!\cdots\!11\)\( T^{6} \)
$37$ \( 1 - \)\(30\!\cdots\!74\)\( T + \)\(21\!\cdots\!23\)\( T^{2} - \)\(40\!\cdots\!08\)\( T^{3} + \)\(14\!\cdots\!71\)\( T^{4} - \)\(14\!\cdots\!46\)\( T^{5} + \)\(33\!\cdots\!33\)\( T^{6} \)
$41$ \( 1 + \)\(11\!\cdots\!54\)\( T + \)\(28\!\cdots\!55\)\( T^{2} + \)\(24\!\cdots\!20\)\( T^{3} + \)\(30\!\cdots\!55\)\( T^{4} + \)\(12\!\cdots\!34\)\( T^{5} + \)\(12\!\cdots\!81\)\( T^{6} \)
$43$ \( 1 + \)\(17\!\cdots\!48\)\( T + \)\(32\!\cdots\!97\)\( T^{2} + \)\(36\!\cdots\!24\)\( T^{3} + \)\(35\!\cdots\!71\)\( T^{4} + \)\(21\!\cdots\!52\)\( T^{5} + \)\(13\!\cdots\!07\)\( T^{6} \)
$47$ \( 1 + \)\(21\!\cdots\!56\)\( T + \)\(34\!\cdots\!13\)\( T^{2} + \)\(34\!\cdots\!12\)\( T^{3} + \)\(29\!\cdots\!71\)\( T^{4} + \)\(16\!\cdots\!84\)\( T^{5} + \)\(62\!\cdots\!63\)\( T^{6} \)
$53$ \( 1 - \)\(74\!\cdots\!02\)\( T + \)\(61\!\cdots\!67\)\( T^{2} - \)\(55\!\cdots\!36\)\( T^{3} + \)\(19\!\cdots\!11\)\( T^{4} - \)\(70\!\cdots\!78\)\( T^{5} + \)\(29\!\cdots\!37\)\( T^{6} \)
$59$ \( 1 + \)\(53\!\cdots\!40\)\( T + \)\(22\!\cdots\!17\)\( T^{2} + \)\(63\!\cdots\!20\)\( T^{3} + \)\(13\!\cdots\!63\)\( T^{4} + \)\(18\!\cdots\!40\)\( T^{5} + \)\(20\!\cdots\!19\)\( T^{6} \)
$61$ \( 1 + \)\(12\!\cdots\!94\)\( T + \)\(12\!\cdots\!35\)\( T^{2} + \)\(78\!\cdots\!00\)\( T^{3} + \)\(38\!\cdots\!35\)\( T^{4} + \)\(11\!\cdots\!14\)\( T^{5} + \)\(27\!\cdots\!21\)\( T^{6} \)
$67$ \( 1 - \)\(11\!\cdots\!24\)\( T + \)\(13\!\cdots\!33\)\( T^{2} - \)\(71\!\cdots\!68\)\( T^{3} + \)\(39\!\cdots\!51\)\( T^{4} - \)\(10\!\cdots\!16\)\( T^{5} + \)\(27\!\cdots\!23\)\( T^{6} \)
$71$ \( 1 + \)\(40\!\cdots\!24\)\( T + \)\(18\!\cdots\!85\)\( T^{2} + \)\(40\!\cdots\!00\)\( T^{3} + \)\(96\!\cdots\!35\)\( T^{4} + \)\(10\!\cdots\!64\)\( T^{5} + \)\(13\!\cdots\!91\)\( T^{6} \)
$73$ \( 1 + \)\(10\!\cdots\!18\)\( T + \)\(83\!\cdots\!47\)\( T^{2} + \)\(39\!\cdots\!84\)\( T^{3} + \)\(16\!\cdots\!11\)\( T^{4} + \)\(40\!\cdots\!42\)\( T^{5} + \)\(80\!\cdots\!97\)\( T^{6} \)
$79$ \( 1 - \)\(36\!\cdots\!60\)\( T + \)\(33\!\cdots\!57\)\( T^{2} - \)\(71\!\cdots\!80\)\( T^{3} + \)\(31\!\cdots\!83\)\( T^{4} - \)\(33\!\cdots\!60\)\( T^{5} + \)\(89\!\cdots\!59\)\( T^{6} \)
$83$ \( 1 - \)\(15\!\cdots\!92\)\( T + \)\(23\!\cdots\!97\)\( T^{2} - \)\(20\!\cdots\!96\)\( T^{3} + \)\(25\!\cdots\!91\)\( T^{4} - \)\(17\!\cdots\!28\)\( T^{5} + \)\(12\!\cdots\!27\)\( T^{6} \)
$89$ \( 1 - \)\(54\!\cdots\!70\)\( T + \)\(86\!\cdots\!27\)\( T^{2} - \)\(34\!\cdots\!60\)\( T^{3} + \)\(28\!\cdots\!43\)\( T^{4} - \)\(59\!\cdots\!70\)\( T^{5} + \)\(36\!\cdots\!29\)\( T^{6} \)
$97$ \( 1 - \)\(43\!\cdots\!14\)\( T + \)\(35\!\cdots\!83\)\( T^{2} - \)\(13\!\cdots\!48\)\( T^{3} + \)\(80\!\cdots\!11\)\( T^{4} - \)\(21\!\cdots\!46\)\( T^{5} + \)\(11\!\cdots\!13\)\( T^{6} \)
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