Properties

Label 3.44.a.a
Level 3
Weight 44
Character orbit 3.a
Self dual yes
Analytic conductor 35.133
Analytic rank 1
Dimension 3
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 3 \)
Weight: \( k \) \(=\) \( 44 \)
Character orbit: \([\chi]\) \(=\) 3.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(35.1331186037\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
Defining polynomial: \(x^{3} - x^{2} - 908401710 x + 974756489742\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{13}\cdot 3^{5}\cdot 5\cdot 11 \)
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.

\(f(q)\) \(=\) \( q + ( 1619008 - \beta_{1} ) q^{2} + 10460353203 q^{3} + ( -593686009856 - 3392488 \beta_{1} + 4 \beta_{2} ) q^{4} + ( -169251107035090 + 87483970 \beta_{1} - 1317 \beta_{2} ) q^{5} + ( 16935395518482624 - 10460353203 \beta_{1} ) q^{6} + ( -544389767902363096 + 407863957886 \beta_{1} + 1184197 \beta_{2} ) q^{7} + ( 3732086898157617152 + 606520958464 \beta_{1} + 19428096 \beta_{2} ) q^{8} + 109418989131512359209 q^{9} +O(q^{10})\) \( q +(1619008 - \beta_{1}) q^{2} +10460353203 q^{3} +(-593686009856 - 3392488 \beta_{1} + 4 \beta_{2}) q^{4} +(-169251107035090 + 87483970 \beta_{1} - 1317 \beta_{2}) q^{5} +(16935395518482624 - 10460353203 \beta_{1}) q^{6} +(-544389767902363096 + 407863957886 \beta_{1} + 1184197 \beta_{2}) q^{7} +(3732086898157617152 + 606520958464 \beta_{1} + 19428096 \beta_{2}) q^{8} +\)\(10\!\cdots\!09\)\( q^{9} +(-\)\(76\!\cdots\!40\)\( + 1235354107077970 \beta_{1} - 2278729792 \beta_{2}) q^{10} +(-\)\(92\!\cdots\!40\)\( + 7617306526252796 \beta_{1} - 14933373558 \beta_{2}) q^{11} +(-\)\(62\!\cdots\!68\)\( - 35486622716939064 \beta_{1} + 41841412812 \beta_{2}) q^{12} +(-\)\(33\!\cdots\!50\)\( - 202202396925505444 \beta_{1} + 641923722778 \beta_{2}) q^{13} +(-\)\(31\!\cdots\!76\)\( + 448634534786039896 \beta_{1} + 102843306048 \beta_{2}) q^{14} +(-\)\(17\!\cdots\!70\)\( + 915113225800655910 \beta_{1} - 13776285168351 \beta_{2}) q^{15} +(\)\(78\!\cdots\!12\)\( + 13746042266337042432 \beta_{1} - 9157309919232 \beta_{2}) q^{16} +(-\)\(53\!\cdots\!94\)\( + 90683027194690706820 \beta_{1} + 201871787631414 \beta_{2}) q^{17} +(\)\(17\!\cdots\!72\)\( - \)\(10\!\cdots\!09\)\( \beta_{1}) q^{18} +(-\)\(10\!\cdots\!88\)\( + \)\(40\!\cdots\!04\)\( \beta_{1} - 2147995640621886 \beta_{2}) q^{19} +(-\)\(66\!\cdots\!20\)\( + \)\(37\!\cdots\!60\)\( \beta_{1} + 3305756267279544 \beta_{2}) q^{20} +(-\)\(56\!\cdots\!88\)\( + \)\(42\!\cdots\!58\)\( \beta_{1} + 12387118881932991 \beta_{2}) q^{21} +(-\)\(57\!\cdots\!08\)\( + \)\(33\!\cdots\!40\)\( \beta_{1} - 52339689282150272 \beta_{2}) q^{22} +(\)\(38\!\cdots\!68\)\( - \)\(60\!\cdots\!28\)\( \beta_{1} + 32453105180134026 \beta_{2}) q^{23} +(\)\(39\!\cdots\!56\)\( + \)\(63\!\cdots\!92\)\( \beta_{1} + 203224746221791488 \beta_{2}) q^{24} +(\)\(60\!\cdots\!75\)\( - \)\(14\!\cdots\!00\)\( \beta_{1} - 758897906364611660 \beta_{2}) q^{25} +(\)\(59\!\cdots\!12\)\( - \)\(47\!\cdots\!90\)\( \beta_{1} + 1748929988964422784 \beta_{2}) q^{26} +\)\(11\!\cdots\!27\)\( q^{27} +(-\)\(28\!\cdots\!48\)\( + \)\(29\!\cdots\!48\)\( \beta_{1} - 12060227383697492832 \beta_{2}) q^{28} +(-\)\(94\!\cdots\!86\)\( - \)\(12\!\cdots\!42\)\( \beta_{1} + 20892595961060032401 \beta_{2}) q^{29} +(-\)\(79\!\cdots\!20\)\( + \)\(12\!\cdots\!10\)\( \beta_{1} - 23836318478518723776 \beta_{2}) q^{30} +(-\)\(18\!\cdots\!12\)\( + \)\(35\!\cdots\!42\)\( \beta_{1} + \)\(13\!\cdots\!53\)\( \beta_{2}) q^{31} +(-\)\(96\!\cdots\!96\)\( + \)\(17\!\cdots\!16\)\( \beta_{1} - \)\(23\!\cdots\!48\)\( \beta_{2}) q^{32} +(-\)\(96\!\cdots\!20\)\( + \)\(79\!\cdots\!88\)\( \beta_{1} - \)\(15\!\cdots\!74\)\( \beta_{2}) q^{33} +(-\)\(59\!\cdots\!92\)\( + \)\(74\!\cdots\!34\)\( \beta_{1} - 67083608408202293376 \beta_{2}) q^{34} +(-\)\(12\!\cdots\!40\)\( - \)\(46\!\cdots\!80\)\( \beta_{1} + \)\(16\!\cdots\!58\)\( \beta_{2}) q^{35} +(-\)\(64\!\cdots\!04\)\( - \)\(37\!\cdots\!92\)\( \beta_{1} + \)\(43\!\cdots\!36\)\( \beta_{2}) q^{36} +(-\)\(25\!\cdots\!26\)\( - \)\(15\!\cdots\!64\)\( \beta_{1} - \)\(37\!\cdots\!36\)\( \beta_{2}) q^{37} +(-\)\(39\!\cdots\!36\)\( + \)\(32\!\cdots\!48\)\( \beta_{1} - \)\(47\!\cdots\!12\)\( \beta_{2}) q^{38} +(-\)\(34\!\cdots\!50\)\( - \)\(21\!\cdots\!32\)\( \beta_{1} + \)\(67\!\cdots\!34\)\( \beta_{2}) q^{39} +(-\)\(25\!\cdots\!00\)\( + \)\(15\!\cdots\!00\)\( \beta_{1} + \)\(98\!\cdots\!80\)\( \beta_{2}) q^{40} +(-\)\(50\!\cdots\!22\)\( - \)\(21\!\cdots\!68\)\( \beta_{1} - \)\(13\!\cdots\!78\)\( \beta_{2}) q^{41} +(-\)\(33\!\cdots\!28\)\( + \)\(46\!\cdots\!88\)\( \beta_{1} + \)\(10\!\cdots\!44\)\( \beta_{2}) q^{42} +(-\)\(16\!\cdots\!96\)\( + \)\(14\!\cdots\!44\)\( \beta_{1} + \)\(35\!\cdots\!70\)\( \beta_{2}) q^{43} +(-\)\(19\!\cdots\!24\)\( + \)\(85\!\cdots\!20\)\( \beta_{1} - \)\(77\!\cdots\!88\)\( \beta_{2}) q^{44} +(-\)\(18\!\cdots\!10\)\( + \)\(95\!\cdots\!30\)\( \beta_{1} - \)\(14\!\cdots\!53\)\( \beta_{2}) q^{45} +(\)\(40\!\cdots\!88\)\( - \)\(16\!\cdots\!48\)\( \beta_{1} + \)\(29\!\cdots\!48\)\( \beta_{2}) q^{46} +(\)\(17\!\cdots\!12\)\( - \)\(41\!\cdots\!68\)\( \beta_{1} + \)\(50\!\cdots\!22\)\( \beta_{2}) q^{47} +(\)\(82\!\cdots\!36\)\( + \)\(14\!\cdots\!96\)\( \beta_{1} - \)\(95\!\cdots\!96\)\( \beta_{2}) q^{48} +(\)\(39\!\cdots\!01\)\( + \)\(23\!\cdots\!36\)\( \beta_{1} - \)\(14\!\cdots\!52\)\( \beta_{2}) q^{49} +(\)\(17\!\cdots\!00\)\( - \)\(33\!\cdots\!75\)\( \beta_{1} - \)\(53\!\cdots\!60\)\( \beta_{2}) q^{50} +(-\)\(55\!\cdots\!82\)\( + \)\(94\!\cdots\!60\)\( \beta_{1} + \)\(21\!\cdots\!42\)\( \beta_{2}) q^{51} +(\)\(65\!\cdots\!36\)\( - \)\(85\!\cdots\!20\)\( \beta_{1} - \)\(12\!\cdots\!40\)\( \beta_{2}) q^{52} +(\)\(10\!\cdots\!78\)\( + \)\(57\!\cdots\!02\)\( \beta_{1} + \)\(38\!\cdots\!57\)\( \beta_{2}) q^{53} +(\)\(18\!\cdots\!16\)\( - \)\(11\!\cdots\!27\)\( \beta_{1}) q^{54} +(\)\(24\!\cdots\!20\)\( - \)\(95\!\cdots\!60\)\( \beta_{1} + \)\(20\!\cdots\!16\)\( \beta_{2}) q^{55} +(\)\(21\!\cdots\!40\)\( + \)\(77\!\cdots\!00\)\( \beta_{1} - \)\(19\!\cdots\!28\)\( \beta_{2}) q^{56} +(-\)\(11\!\cdots\!64\)\( + \)\(42\!\cdots\!12\)\( \beta_{1} - \)\(22\!\cdots\!58\)\( \beta_{2}) q^{57} +(\)\(55\!\cdots\!88\)\( - \)\(27\!\cdots\!14\)\( \beta_{1} + \)\(81\!\cdots\!04\)\( \beta_{2}) q^{58} +(-\)\(59\!\cdots\!92\)\( + \)\(17\!\cdots\!36\)\( \beta_{1} - \)\(22\!\cdots\!36\)\( \beta_{2}) q^{59} +(-\)\(69\!\cdots\!60\)\( + \)\(39\!\cdots\!80\)\( \beta_{1} + \)\(34\!\cdots\!32\)\( \beta_{2}) q^{60} +(-\)\(41\!\cdots\!06\)\( - \)\(87\!\cdots\!96\)\( \beta_{1} - \)\(17\!\cdots\!68\)\( \beta_{2}) q^{61} +(-\)\(22\!\cdots\!52\)\( - \)\(14\!\cdots\!48\)\( \beta_{1} + \)\(62\!\cdots\!40\)\( \beta_{2}) q^{62} +(-\)\(59\!\cdots\!64\)\( + \)\(44\!\cdots\!74\)\( \beta_{1} + \)\(12\!\cdots\!73\)\( \beta_{2}) q^{63} +(-\)\(32\!\cdots\!12\)\( + \)\(17\!\cdots\!40\)\( \beta_{1} - \)\(33\!\cdots\!36\)\( \beta_{2}) q^{64} +(-\)\(85\!\cdots\!80\)\( + \)\(20\!\cdots\!40\)\( \beta_{1} + \)\(89\!\cdots\!26\)\( \beta_{2}) q^{65} +(-\)\(60\!\cdots\!24\)\( + \)\(34\!\cdots\!20\)\( \beta_{1} - \)\(54\!\cdots\!16\)\( \beta_{2}) q^{66} +(\)\(43\!\cdots\!16\)\( - \)\(72\!\cdots\!40\)\( \beta_{1} + \)\(22\!\cdots\!08\)\( \beta_{2}) q^{67} +(-\)\(90\!\cdots\!56\)\( - \)\(26\!\cdots\!08\)\( \beta_{1} - \)\(21\!\cdots\!84\)\( \beta_{2}) q^{68} +(\)\(40\!\cdots\!04\)\( - \)\(63\!\cdots\!84\)\( \beta_{1} + \)\(33\!\cdots\!78\)\( \beta_{2}) q^{69} +(\)\(62\!\cdots\!60\)\( - \)\(76\!\cdots\!80\)\( \beta_{1} + \)\(43\!\cdots\!08\)\( \beta_{2}) q^{70} +(\)\(74\!\cdots\!72\)\( + \)\(47\!\cdots\!60\)\( \beta_{1} - \)\(13\!\cdots\!02\)\( \beta_{2}) q^{71} +(\)\(40\!\cdots\!68\)\( + \)\(66\!\cdots\!76\)\( \beta_{1} + \)\(21\!\cdots\!64\)\( \beta_{2}) q^{72} +(\)\(10\!\cdots\!18\)\( + \)\(89\!\cdots\!92\)\( \beta_{1} - \)\(57\!\cdots\!40\)\( \beta_{2}) q^{73} +(\)\(42\!\cdots\!44\)\( + \)\(25\!\cdots\!46\)\( \beta_{1} + \)\(52\!\cdots\!60\)\( \beta_{2}) q^{74} +(\)\(63\!\cdots\!25\)\( - \)\(15\!\cdots\!00\)\( \beta_{1} - \)\(79\!\cdots\!80\)\( \beta_{2}) q^{75} +(-\)\(15\!\cdots\!68\)\( + \)\(95\!\cdots\!24\)\( \beta_{1} - \)\(11\!\cdots\!36\)\( \beta_{2}) q^{76} +(\)\(53\!\cdots\!48\)\( - \)\(68\!\cdots\!28\)\( \beta_{1} + \)\(22\!\cdots\!80\)\( \beta_{2}) q^{77} +(\)\(61\!\cdots\!36\)\( - \)\(49\!\cdots\!70\)\( \beta_{1} + \)\(18\!\cdots\!52\)\( \beta_{2}) q^{78} +(-\)\(14\!\cdots\!60\)\( + \)\(10\!\cdots\!50\)\( \beta_{1} - \)\(24\!\cdots\!83\)\( \beta_{2}) q^{79} +(\)\(17\!\cdots\!60\)\( - \)\(14\!\cdots\!80\)\( \beta_{1} - \)\(15\!\cdots\!72\)\( \beta_{2}) q^{80} +\)\(11\!\cdots\!81\)\( q^{81} +(-\)\(69\!\cdots\!32\)\( + \)\(47\!\cdots\!02\)\( \beta_{1} + \)\(66\!\cdots\!64\)\( \beta_{2}) q^{82} +(\)\(29\!\cdots\!56\)\( - \)\(90\!\cdots\!92\)\( \beta_{1} - \)\(21\!\cdots\!90\)\( \beta_{2}) q^{83} +(-\)\(29\!\cdots\!44\)\( + \)\(30\!\cdots\!44\)\( \beta_{1} - \)\(12\!\cdots\!96\)\( \beta_{2}) q^{84} +(-\)\(20\!\cdots\!40\)\( - \)\(98\!\cdots\!80\)\( \beta_{1} + \)\(23\!\cdots\!38\)\( \beta_{2}) q^{85} +(-\)\(35\!\cdots\!40\)\( + \)\(19\!\cdots\!16\)\( \beta_{1} - \)\(51\!\cdots\!56\)\( \beta_{2}) q^{86} +(-\)\(98\!\cdots\!58\)\( - \)\(13\!\cdots\!26\)\( \beta_{1} + \)\(21\!\cdots\!03\)\( \beta_{2}) q^{87} +(-\)\(28\!\cdots\!28\)\( + \)\(11\!\cdots\!24\)\( \beta_{1} + \)\(51\!\cdots\!28\)\( \beta_{2}) q^{88} +(\)\(21\!\cdots\!02\)\( + \)\(24\!\cdots\!04\)\( \beta_{1} - \)\(14\!\cdots\!20\)\( \beta_{2}) q^{89} +(-\)\(83\!\cdots\!60\)\( + \)\(13\!\cdots\!30\)\( \beta_{1} - \)\(24\!\cdots\!28\)\( \beta_{2}) q^{90} +(\)\(45\!\cdots\!08\)\( + \)\(23\!\cdots\!72\)\( \beta_{1} - \)\(15\!\cdots\!34\)\( \beta_{2}) q^{91} +(\)\(12\!\cdots\!24\)\( - \)\(36\!\cdots\!24\)\( \beta_{1} + \)\(81\!\cdots\!12\)\( \beta_{2}) q^{92} +(-\)\(19\!\cdots\!36\)\( + \)\(37\!\cdots\!26\)\( \beta_{1} + \)\(14\!\cdots\!59\)\( \beta_{2}) q^{93} +(\)\(26\!\cdots\!40\)\( - \)\(12\!\cdots\!32\)\( \beta_{1} + \)\(24\!\cdots\!64\)\( \beta_{2}) q^{94} +(\)\(31\!\cdots\!00\)\( - \)\(59\!\cdots\!00\)\( \beta_{1} - \)\(14\!\cdots\!60\)\( \beta_{2}) q^{95} +(-\)\(10\!\cdots\!88\)\( + \)\(18\!\cdots\!48\)\( \beta_{1} - \)\(25\!\cdots\!44\)\( \beta_{2}) q^{96} +(\)\(20\!\cdots\!26\)\( + \)\(21\!\cdots\!80\)\( \beta_{1} - \)\(24\!\cdots\!92\)\( \beta_{2}) q^{97} +(-\)\(67\!\cdots\!20\)\( + \)\(10\!\cdots\!59\)\( \beta_{1} - \)\(30\!\cdots\!16\)\( \beta_{2}) q^{98} +(-\)\(10\!\cdots\!60\)\( + \)\(83\!\cdots\!64\)\( \beta_{1} - \)\(16\!\cdots\!22\)\( \beta_{2}) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 4857024q^{2} + 31381059609q^{3} - 1781058029568q^{4} - 507753321105270q^{5} + 50806186555447872q^{6} - 1633169303707089288q^{7} + 11196260694472851456q^{8} + 328256967394537077627q^{9} + O(q^{10}) \) \( 3q + 4857024q^{2} + 31381059609q^{3} - 1781058029568q^{4} - 507753321105270q^{5} + 50806186555447872q^{6} - 1633169303707089288q^{7} + 11196260694472851456q^{8} + \)\(32\!\cdots\!27\)\(q^{9} - \)\(22\!\cdots\!20\)\(q^{10} - \)\(27\!\cdots\!20\)\(q^{11} - \)\(18\!\cdots\!04\)\(q^{12} - \)\(99\!\cdots\!50\)\(q^{13} - \)\(94\!\cdots\!28\)\(q^{14} - \)\(53\!\cdots\!10\)\(q^{15} + \)\(23\!\cdots\!36\)\(q^{16} - \)\(16\!\cdots\!82\)\(q^{17} + \)\(53\!\cdots\!16\)\(q^{18} - \)\(32\!\cdots\!64\)\(q^{19} - \)\(19\!\cdots\!60\)\(q^{20} - \)\(17\!\cdots\!64\)\(q^{21} - \)\(17\!\cdots\!24\)\(q^{22} + \)\(11\!\cdots\!04\)\(q^{23} + \)\(11\!\cdots\!68\)\(q^{24} + \)\(18\!\cdots\!25\)\(q^{25} + \)\(17\!\cdots\!36\)\(q^{26} + \)\(34\!\cdots\!81\)\(q^{27} - \)\(84\!\cdots\!44\)\(q^{28} - \)\(28\!\cdots\!58\)\(q^{29} - \)\(23\!\cdots\!60\)\(q^{30} - \)\(56\!\cdots\!36\)\(q^{31} - \)\(29\!\cdots\!88\)\(q^{32} - \)\(29\!\cdots\!60\)\(q^{33} - \)\(17\!\cdots\!76\)\(q^{34} - \)\(36\!\cdots\!20\)\(q^{35} - \)\(19\!\cdots\!12\)\(q^{36} - \)\(77\!\cdots\!78\)\(q^{37} - \)\(11\!\cdots\!08\)\(q^{38} - \)\(10\!\cdots\!50\)\(q^{39} - \)\(75\!\cdots\!00\)\(q^{40} - \)\(15\!\cdots\!66\)\(q^{41} - \)\(99\!\cdots\!84\)\(q^{42} - \)\(50\!\cdots\!88\)\(q^{43} - \)\(58\!\cdots\!72\)\(q^{44} - \)\(55\!\cdots\!30\)\(q^{45} + \)\(12\!\cdots\!64\)\(q^{46} + \)\(51\!\cdots\!36\)\(q^{47} + \)\(24\!\cdots\!08\)\(q^{48} + \)\(11\!\cdots\!03\)\(q^{49} + \)\(53\!\cdots\!00\)\(q^{50} - \)\(16\!\cdots\!46\)\(q^{51} + \)\(19\!\cdots\!08\)\(q^{52} + \)\(31\!\cdots\!34\)\(q^{53} + \)\(55\!\cdots\!48\)\(q^{54} + \)\(72\!\cdots\!60\)\(q^{55} + \)\(64\!\cdots\!20\)\(q^{56} - \)\(33\!\cdots\!92\)\(q^{57} + \)\(16\!\cdots\!64\)\(q^{58} - \)\(17\!\cdots\!76\)\(q^{59} - \)\(20\!\cdots\!80\)\(q^{60} - \)\(12\!\cdots\!18\)\(q^{61} - \)\(68\!\cdots\!56\)\(q^{62} - \)\(17\!\cdots\!92\)\(q^{63} - \)\(97\!\cdots\!36\)\(q^{64} - \)\(25\!\cdots\!40\)\(q^{65} - \)\(18\!\cdots\!72\)\(q^{66} + \)\(12\!\cdots\!48\)\(q^{67} - \)\(27\!\cdots\!68\)\(q^{68} + \)\(12\!\cdots\!12\)\(q^{69} + \)\(18\!\cdots\!80\)\(q^{70} + \)\(22\!\cdots\!16\)\(q^{71} + \)\(12\!\cdots\!04\)\(q^{72} + \)\(31\!\cdots\!54\)\(q^{73} + \)\(12\!\cdots\!32\)\(q^{74} + \)\(19\!\cdots\!75\)\(q^{75} - \)\(45\!\cdots\!04\)\(q^{76} + \)\(15\!\cdots\!44\)\(q^{77} + \)\(18\!\cdots\!08\)\(q^{78} - \)\(43\!\cdots\!80\)\(q^{79} + \)\(51\!\cdots\!80\)\(q^{80} + \)\(35\!\cdots\!43\)\(q^{81} - \)\(20\!\cdots\!96\)\(q^{82} + \)\(89\!\cdots\!68\)\(q^{83} - \)\(88\!\cdots\!32\)\(q^{84} - \)\(60\!\cdots\!20\)\(q^{85} - \)\(10\!\cdots\!20\)\(q^{86} - \)\(29\!\cdots\!74\)\(q^{87} - \)\(86\!\cdots\!84\)\(q^{88} + \)\(65\!\cdots\!06\)\(q^{89} - \)\(25\!\cdots\!80\)\(q^{90} + \)\(13\!\cdots\!24\)\(q^{91} + \)\(37\!\cdots\!72\)\(q^{92} - \)\(59\!\cdots\!08\)\(q^{93} + \)\(78\!\cdots\!20\)\(q^{94} + \)\(93\!\cdots\!00\)\(q^{95} - \)\(30\!\cdots\!64\)\(q^{96} + \)\(61\!\cdots\!78\)\(q^{97} - \)\(20\!\cdots\!60\)\(q^{98} - \)\(30\!\cdots\!80\)\(q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 908401710 x + 974756489742\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 96 \nu - 32 \)
\(\beta_{2}\)\(=\)\( 2304 \nu^{2} + 3705792 \nu - 1395306262592 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1} + 32\)\()/96\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{2} - 38602 \beta_{1} + 1395305027328\)\()/2304\)

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
29588.6
1074.41
−30662.1
−1.22147e6 1.04604e10 −7.30411e12 −8.84097e14 −1.27770e16 1.48034e18 1.96659e19 1.09419e20 1.07990e21
1.2 1.51590e6 1.04604e10 −6.49815e12 1.66864e15 1.58568e16 −2.14679e18 −2.31845e19 1.09419e20 2.52949e21
1.3 4.56260e6 1.04604e10 1.20212e13 −1.29230e15 4.77264e16 −9.66721e17 1.47149e19 1.09419e20 −5.89624e21
\(n\): 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 3.44.a.a 3
3.b odd 2 1 9.44.a.a 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.44.a.a 3 1.a even 1 1 trivial
9.44.a.a 3 3.b odd 2 1

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{3} - 4857024 T_{2}^{2} - 508269450240 T_{2} + \)\(84\!\cdots\!88\)\( \) acting on \(S_{44}^{\mathrm{new}}(\Gamma_0(3))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 4857024 T + 25880009616384 T^{2} - 76997466648892932096 T^{3} + \)\(22\!\cdots\!72\)\( T^{4} - \)\(37\!\cdots\!36\)\( T^{5} + \)\(68\!\cdots\!12\)\( T^{6} \)
$3$ \( ( 1 - 10460353203 T )^{3} \)
$5$ \( 1 + 507753321105270 T + \)\(92\!\cdots\!75\)\( T^{2} - \)\(75\!\cdots\!00\)\( T^{3} + \)\(10\!\cdots\!75\)\( T^{4} + \)\(65\!\cdots\!50\)\( T^{5} + \)\(14\!\cdots\!25\)\( T^{6} \)
$7$ \( 1 + 1633169303707089288 T + \)\(40\!\cdots\!85\)\( T^{2} + \)\(40\!\cdots\!48\)\( T^{3} + \)\(87\!\cdots\!55\)\( T^{4} + \)\(77\!\cdots\!12\)\( T^{5} + \)\(10\!\cdots\!07\)\( T^{6} \)
$11$ \( 1 + \)\(27\!\cdots\!20\)\( T + \)\(12\!\cdots\!61\)\( T^{2} + \)\(34\!\cdots\!16\)\( T^{3} + \)\(75\!\cdots\!91\)\( T^{4} + \)\(10\!\cdots\!20\)\( T^{5} + \)\(21\!\cdots\!91\)\( T^{6} \)
$13$ \( 1 + \)\(99\!\cdots\!50\)\( T + \)\(17\!\cdots\!59\)\( T^{2} + \)\(12\!\cdots\!56\)\( T^{3} + \)\(14\!\cdots\!23\)\( T^{4} + \)\(62\!\cdots\!50\)\( T^{5} + \)\(49\!\cdots\!73\)\( T^{6} \)
$17$ \( 1 + \)\(16\!\cdots\!82\)\( T + \)\(12\!\cdots\!47\)\( T^{2} + \)\(46\!\cdots\!16\)\( T^{3} + \)\(10\!\cdots\!11\)\( T^{4} + \)\(10\!\cdots\!58\)\( T^{5} + \)\(53\!\cdots\!97\)\( T^{6} \)
$19$ \( 1 + \)\(32\!\cdots\!64\)\( T + \)\(24\!\cdots\!37\)\( T^{2} + \)\(51\!\cdots\!52\)\( T^{3} + \)\(23\!\cdots\!83\)\( T^{4} + \)\(30\!\cdots\!84\)\( T^{5} + \)\(91\!\cdots\!79\)\( T^{6} \)
$23$ \( 1 - \)\(11\!\cdots\!04\)\( T + \)\(79\!\cdots\!65\)\( T^{2} - \)\(85\!\cdots\!56\)\( T^{3} + \)\(28\!\cdots\!55\)\( T^{4} - \)\(14\!\cdots\!56\)\( T^{5} + \)\(46\!\cdots\!63\)\( T^{6} \)
$29$ \( 1 + \)\(28\!\cdots\!58\)\( T + \)\(58\!\cdots\!27\)\( T^{2} - \)\(63\!\cdots\!76\)\( T^{3} + \)\(45\!\cdots\!03\)\( T^{4} + \)\(16\!\cdots\!18\)\( T^{5} + \)\(44\!\cdots\!69\)\( T^{6} \)
$31$ \( 1 + \)\(56\!\cdots\!36\)\( T + \)\(27\!\cdots\!97\)\( T^{2} + \)\(31\!\cdots\!52\)\( T^{3} + \)\(36\!\cdots\!27\)\( T^{4} + \)\(10\!\cdots\!16\)\( T^{5} + \)\(24\!\cdots\!71\)\( T^{6} \)
$37$ \( 1 + \)\(77\!\cdots\!78\)\( T + \)\(61\!\cdots\!75\)\( T^{2} + \)\(40\!\cdots\!48\)\( T^{3} + \)\(16\!\cdots\!75\)\( T^{4} + \)\(56\!\cdots\!02\)\( T^{5} + \)\(19\!\cdots\!77\)\( T^{6} \)
$41$ \( 1 + \)\(15\!\cdots\!66\)\( T + \)\(14\!\cdots\!07\)\( T^{2} + \)\(79\!\cdots\!12\)\( T^{3} + \)\(31\!\cdots\!47\)\( T^{4} + \)\(75\!\cdots\!06\)\( T^{5} + \)\(11\!\cdots\!61\)\( T^{6} \)
$43$ \( 1 + \)\(50\!\cdots\!88\)\( T + \)\(13\!\cdots\!17\)\( T^{2} + \)\(21\!\cdots\!80\)\( T^{3} + \)\(23\!\cdots\!19\)\( T^{4} + \)\(15\!\cdots\!12\)\( T^{5} + \)\(52\!\cdots\!43\)\( T^{6} \)
$47$ \( 1 - \)\(51\!\cdots\!36\)\( T + \)\(63\!\cdots\!93\)\( T^{2} - \)\(14\!\cdots\!20\)\( T^{3} + \)\(50\!\cdots\!39\)\( T^{4} - \)\(32\!\cdots\!44\)\( T^{5} + \)\(50\!\cdots\!67\)\( T^{6} \)
$53$ \( 1 - \)\(31\!\cdots\!34\)\( T + \)\(11\!\cdots\!15\)\( T^{2} - \)\(18\!\cdots\!76\)\( T^{3} + \)\(16\!\cdots\!55\)\( T^{4} - \)\(61\!\cdots\!86\)\( T^{5} + \)\(27\!\cdots\!33\)\( T^{6} \)
$59$ \( 1 + \)\(17\!\cdots\!76\)\( T + \)\(51\!\cdots\!17\)\( T^{2} + \)\(51\!\cdots\!08\)\( T^{3} + \)\(72\!\cdots\!43\)\( T^{4} + \)\(34\!\cdots\!16\)\( T^{5} + \)\(27\!\cdots\!39\)\( T^{6} \)
$61$ \( 1 + \)\(12\!\cdots\!18\)\( T + \)\(74\!\cdots\!59\)\( T^{2} + \)\(22\!\cdots\!04\)\( T^{3} + \)\(43\!\cdots\!79\)\( T^{4} + \)\(42\!\cdots\!98\)\( T^{5} + \)\(20\!\cdots\!41\)\( T^{6} \)
$67$ \( 1 - \)\(12\!\cdots\!48\)\( T + \)\(60\!\cdots\!57\)\( T^{2} - \)\(84\!\cdots\!44\)\( T^{3} + \)\(20\!\cdots\!91\)\( T^{4} - \)\(14\!\cdots\!12\)\( T^{5} + \)\(36\!\cdots\!47\)\( T^{6} \)
$71$ \( 1 - \)\(22\!\cdots\!16\)\( T + \)\(28\!\cdots\!85\)\( T^{2} - \)\(21\!\cdots\!00\)\( T^{3} + \)\(11\!\cdots\!35\)\( T^{4} - \)\(36\!\cdots\!36\)\( T^{5} + \)\(64\!\cdots\!31\)\( T^{6} \)
$73$ \( 1 - \)\(31\!\cdots\!54\)\( T + \)\(67\!\cdots\!75\)\( T^{2} - \)\(89\!\cdots\!76\)\( T^{3} + \)\(88\!\cdots\!75\)\( T^{4} - \)\(55\!\cdots\!06\)\( T^{5} + \)\(23\!\cdots\!13\)\( T^{6} \)
$79$ \( 1 + \)\(43\!\cdots\!80\)\( T + \)\(10\!\cdots\!17\)\( T^{2} + \)\(34\!\cdots\!40\)\( T^{3} + \)\(42\!\cdots\!63\)\( T^{4} + \)\(68\!\cdots\!80\)\( T^{5} + \)\(62\!\cdots\!19\)\( T^{6} \)
$83$ \( 1 - \)\(89\!\cdots\!68\)\( T + \)\(32\!\cdots\!21\)\( T^{2} - \)\(24\!\cdots\!88\)\( T^{3} + \)\(10\!\cdots\!27\)\( T^{4} - \)\(98\!\cdots\!92\)\( T^{5} + \)\(36\!\cdots\!03\)\( T^{6} \)
$89$ \( 1 - \)\(65\!\cdots\!06\)\( T + \)\(14\!\cdots\!07\)\( T^{2} + \)\(34\!\cdots\!72\)\( T^{3} + \)\(97\!\cdots\!83\)\( T^{4} - \)\(28\!\cdots\!66\)\( T^{5} + \)\(29\!\cdots\!09\)\( T^{6} \)
$97$ \( 1 - \)\(61\!\cdots\!78\)\( T + \)\(84\!\cdots\!47\)\( T^{2} - \)\(32\!\cdots\!64\)\( T^{3} + \)\(22\!\cdots\!31\)\( T^{4} - \)\(44\!\cdots\!62\)\( T^{5} + \)\(19\!\cdots\!17\)\( T^{6} \)
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