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: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{22}\cdot 3^{5}\cdot 5^{2}\cdot 7^{2} \)
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.

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

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

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

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.

Atkin-Lehner signs

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

Hecke kernels

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