Properties

Label 3.40.a.b
Level 3
Weight 40
Character orbit 3.a
Self dual Yes
Analytic conductor 28.902
Analytic rank 0
Dimension 3
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 3 \)
Weight: \( k \) = \( 40 \)
Character orbit: \([\chi]\) = 3.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(28.9018654169\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{8}\cdot 3^{6}\cdot 5\cdot 7\cdot 13 \)
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.

\(f(q)\) \(=\) \( q + ( 177858 - \beta_{1} ) q^{2} -1162261467 q^{3} + ( 319147551244 - 227081 \beta_{1} + \beta_{2} ) q^{4} + ( -17793991314810 + 12092638 \beta_{1} + 54 \beta_{2} ) q^{5} + ( -206717499997686 + 1162261467 \beta_{1} ) q^{6} + ( -525770892074176 - 12596523938 \beta_{1} + 56758 \beta_{2} ) q^{7} + ( 149112778455452232 - 182657330678 \beta_{1} + 533574 \beta_{2} ) q^{8} + 1350851717672992089 q^{9} +O(q^{10})\) \( q +(177858 - \beta_{1}) q^{2} -1162261467 q^{3} +(319147551244 - 227081 \beta_{1} + \beta_{2}) q^{4} +(-17793991314810 + 12092638 \beta_{1} + 54 \beta_{2}) q^{5} +(-206717499997686 + 1162261467 \beta_{1}) q^{6} +(-525770892074176 - 12596523938 \beta_{1} + 56758 \beta_{2}) q^{7} +(149112778455452232 - 182657330678 \beta_{1} + 533574 \beta_{2}) q^{8} +1350851717672992089 q^{9} +(-13289593742193286260 - 3323523968902 \beta_{1} + 4457984 \beta_{2}) q^{10} +(-\)\(17\!\cdots\!80\)\( + 2617390166972 \beta_{1} - 42093972 \beta_{2}) q^{11} +(-\)\(37\!\cdots\!48\)\( + 263927496187827 \beta_{1} - 1162261467 \beta_{2}) q^{12} +(\)\(18\!\cdots\!30\)\( - 3296990601565580 \beta_{1} + 2302171012 \beta_{2}) q^{13} +(\)\(10\!\cdots\!84\)\( - 22915977673056320 \beta_{1} + 29992453632 \beta_{2}) q^{14} +(\)\(20\!\cdots\!70\)\( - 14054807181779946 \beta_{1} - 62762119218 \beta_{2}) q^{15} +(\)\(40\!\cdots\!12\)\( - 247808314916500164 \beta_{1} - 203561787228 \beta_{2}) q^{16} +(\)\(24\!\cdots\!86\)\( - 1432579830459091908 \beta_{1} + 73008839916 \beta_{2}) q^{17} +(\)\(24\!\cdots\!62\)\( - 1350851717672992089 \beta_{1}) q^{18} +(-\)\(35\!\cdots\!08\)\( - 3709293048070314324 \beta_{1} + 14019280007196 \beta_{2}) q^{19} +(\)\(10\!\cdots\!20\)\( + 4685500071252208714 \beta_{1} - 24996949090938 \beta_{2}) q^{20} +(\)\(61\!\cdots\!92\)\( + 14640454391280497046 \beta_{1} - 65967636343986 \beta_{2}) q^{21} +(-\)\(33\!\cdots\!08\)\( + \)\(19\!\cdots\!84\)\( \beta_{1} - 15518897927168 \beta_{2}) q^{22} +(\)\(13\!\cdots\!88\)\( + 71070379474158455740 \beta_{1} + 864912659360364 \beta_{2}) q^{23} +(-\)\(17\!\cdots\!44\)\( + \)\(21\!\cdots\!26\)\( \beta_{1} - 620152499993058 \beta_{2}) q^{24} +(-\)\(39\!\cdots\!25\)\( - 37236871224450578840 \beta_{1} - 3116005674041720 \beta_{2}) q^{25} +(\)\(30\!\cdots\!92\)\( - \)\(29\!\cdots\!78\)\( \beta_{1} + 4002589901546496 \beta_{2}) q^{26} -\)\(15\!\cdots\!63\)\( q^{27} +(\)\(21\!\cdots\!52\)\( - \)\(16\!\cdots\!88\)\( \beta_{1} + 905414279433792 \beta_{2}) q^{28} +(\)\(24\!\cdots\!54\)\( - \)\(60\!\cdots\!54\)\( \beta_{1} + 10883586973384674 \beta_{2}) q^{29} +(\)\(15\!\cdots\!20\)\( + \)\(38\!\cdots\!34\)\( \beta_{1} - 5181343023702528 \beta_{2}) q^{30} +(-\)\(12\!\cdots\!52\)\( + \)\(13\!\cdots\!26\)\( \beta_{1} - 51025473408711938 \beta_{2}) q^{31} +(\)\(12\!\cdots\!04\)\( + \)\(16\!\cdots\!32\)\( \beta_{1} - 107917356575846952 \beta_{2}) q^{32} +(\)\(20\!\cdots\!60\)\( - \)\(30\!\cdots\!24\)\( \beta_{1} + 48924201648576924 \beta_{2}) q^{33} +(\)\(12\!\cdots\!48\)\( - \)\(34\!\cdots\!14\)\( \beta_{1} + 1454956528831466496 \beta_{2}) q^{34} +(\)\(91\!\cdots\!40\)\( + \)\(42\!\cdots\!08\)\( \beta_{1} - 2423830555879147836 \beta_{2}) q^{35} +(\)\(43\!\cdots\!16\)\( - \)\(30\!\cdots\!09\)\( \beta_{1} + 1350851717672992089 \beta_{2}) q^{36} +(\)\(96\!\cdots\!74\)\( - \)\(33\!\cdots\!92\)\( \beta_{1} - 3950416425949998144 \beta_{2}) q^{37} +(\)\(24\!\cdots\!64\)\( - \)\(22\!\cdots\!08\)\( \beta_{1} + 8006104235315837952 \beta_{2}) q^{38} +(-\)\(21\!\cdots\!10\)\( + \)\(38\!\cdots\!60\)\( \beta_{1} - 2675724657691994604 \beta_{2}) q^{39} +(\)\(51\!\cdots\!00\)\( + \)\(19\!\cdots\!20\)\( \beta_{1} - 14797692611200750940 \beta_{2}) q^{40} +(-\)\(46\!\cdots\!42\)\( + \)\(18\!\cdots\!28\)\( \beta_{1} + 41049028078685916228 \beta_{2}) q^{41} +(-\)\(12\!\cdots\!28\)\( + \)\(26\!\cdots\!40\)\( \beta_{1} - 34859073157257798144 \beta_{2}) q^{42} +(-\)\(80\!\cdots\!76\)\( - \)\(20\!\cdots\!56\)\( \beta_{1} + 42956960985072135700 \beta_{2}) q^{43} +(-\)\(71\!\cdots\!04\)\( + \)\(48\!\cdots\!16\)\( \beta_{1} - \)\(17\!\cdots\!72\)\( \beta_{2}) q^{44} +(-\)\(24\!\cdots\!90\)\( + \)\(16\!\cdots\!82\)\( \beta_{1} + 72945992754341572806 \beta_{2}) q^{45} +(-\)\(34\!\cdots\!52\)\( - \)\(48\!\cdots\!44\)\( \beta_{1} + \)\(19\!\cdots\!12\)\( \beta_{2}) q^{46} +(-\)\(36\!\cdots\!68\)\( - \)\(27\!\cdots\!32\)\( \beta_{1} + \)\(15\!\cdots\!28\)\( \beta_{2}) q^{47} +(-\)\(46\!\cdots\!04\)\( + \)\(28\!\cdots\!88\)\( \beta_{1} + \)\(23\!\cdots\!76\)\( \beta_{2}) q^{48} +(\)\(30\!\cdots\!81\)\( - \)\(70\!\cdots\!80\)\( \beta_{1} - \)\(10\!\cdots\!68\)\( \beta_{2}) q^{49} +(-\)\(39\!\cdots\!50\)\( + \)\(16\!\cdots\!85\)\( \beta_{1} - \)\(91\!\cdots\!20\)\( \beta_{2}) q^{50} +(-\)\(28\!\cdots\!62\)\( + \)\(16\!\cdots\!36\)\( \beta_{1} - 84855361384738316772 \beta_{2}) q^{51} +(\)\(19\!\cdots\!96\)\( - \)\(30\!\cdots\!10\)\( \beta_{1} + \)\(28\!\cdots\!50\)\( \beta_{2}) q^{52} +(\)\(20\!\cdots\!78\)\( + \)\(33\!\cdots\!18\)\( \beta_{1} + \)\(29\!\cdots\!98\)\( \beta_{2}) q^{53} +(-\)\(27\!\cdots\!54\)\( + \)\(15\!\cdots\!63\)\( \beta_{1}) q^{54} +(\)\(24\!\cdots\!80\)\( - \)\(22\!\cdots\!24\)\( \beta_{1} - \)\(78\!\cdots\!92\)\( \beta_{2}) q^{55} +(\)\(12\!\cdots\!00\)\( - \)\(99\!\cdots\!44\)\( \beta_{1} + \)\(50\!\cdots\!28\)\( \beta_{2}) q^{56} +(\)\(41\!\cdots\!36\)\( + \)\(43\!\cdots\!08\)\( \beta_{1} - \)\(16\!\cdots\!32\)\( \beta_{2}) q^{57} +(\)\(93\!\cdots\!28\)\( - \)\(28\!\cdots\!62\)\( \beta_{1} + \)\(93\!\cdots\!36\)\( \beta_{2}) q^{58} +(-\)\(25\!\cdots\!52\)\( + \)\(10\!\cdots\!28\)\( \beta_{1} + \)\(27\!\cdots\!56\)\( \beta_{2}) q^{59} +(-\)\(11\!\cdots\!40\)\( - \)\(54\!\cdots\!38\)\( \beta_{1} + \)\(29\!\cdots\!46\)\( \beta_{2}) q^{60} +(-\)\(23\!\cdots\!66\)\( + \)\(74\!\cdots\!00\)\( \beta_{1} + \)\(14\!\cdots\!48\)\( \beta_{2}) q^{61} +(-\)\(11\!\cdots\!72\)\( + \)\(28\!\cdots\!92\)\( \beta_{1} - \)\(14\!\cdots\!60\)\( \beta_{2}) q^{62} +(-\)\(71\!\cdots\!64\)\( - \)\(17\!\cdots\!82\)\( \beta_{1} + \)\(76\!\cdots\!62\)\( \beta_{2}) q^{63} +(-\)\(11\!\cdots\!52\)\( + \)\(61\!\cdots\!32\)\( \beta_{1} - \)\(87\!\cdots\!04\)\( \beta_{2}) q^{64} +(-\)\(23\!\cdots\!20\)\( + \)\(87\!\cdots\!96\)\( \beta_{1} - \)\(12\!\cdots\!32\)\( \beta_{2}) q^{65} +(\)\(39\!\cdots\!36\)\( - \)\(22\!\cdots\!28\)\( \beta_{1} + \)\(18\!\cdots\!56\)\( \beta_{2}) q^{66} +(-\)\(17\!\cdots\!24\)\( + \)\(39\!\cdots\!84\)\( \beta_{1} + \)\(35\!\cdots\!32\)\( \beta_{2}) q^{67} +(\)\(37\!\cdots\!64\)\( - \)\(10\!\cdots\!30\)\( \beta_{1} + \)\(74\!\cdots\!34\)\( \beta_{2}) q^{68} +(-\)\(16\!\cdots\!96\)\( - \)\(82\!\cdots\!80\)\( \beta_{1} - \)\(10\!\cdots\!88\)\( \beta_{2}) q^{69} +(\)\(12\!\cdots\!40\)\( + \)\(63\!\cdots\!68\)\( \beta_{1} - \)\(78\!\cdots\!56\)\( \beta_{2}) q^{70} +(\)\(28\!\cdots\!92\)\( + \)\(73\!\cdots\!04\)\( \beta_{1} - \)\(74\!\cdots\!88\)\( \beta_{2}) q^{71} +(\)\(20\!\cdots\!48\)\( - \)\(24\!\cdots\!42\)\( \beta_{1} + \)\(72\!\cdots\!86\)\( \beta_{2}) q^{72} +(\)\(21\!\cdots\!38\)\( + \)\(25\!\cdots\!92\)\( \beta_{1} + \)\(69\!\cdots\!00\)\( \beta_{2}) q^{73} +(\)\(29\!\cdots\!04\)\( + \)\(45\!\cdots\!06\)\( \beta_{1} + \)\(21\!\cdots\!00\)\( \beta_{2}) q^{74} +(\)\(46\!\cdots\!75\)\( + \)\(43\!\cdots\!80\)\( \beta_{1} + \)\(36\!\cdots\!40\)\( \beta_{2}) q^{75} +(\)\(42\!\cdots\!12\)\( - \)\(37\!\cdots\!04\)\( \beta_{1} - \)\(29\!\cdots\!04\)\( \beta_{2}) q^{76} +(-\)\(73\!\cdots\!12\)\( + \)\(26\!\cdots\!32\)\( \beta_{1} - \)\(91\!\cdots\!40\)\( \beta_{2}) q^{77} +(-\)\(35\!\cdots\!64\)\( + \)\(33\!\cdots\!26\)\( \beta_{1} - \)\(46\!\cdots\!32\)\( \beta_{2}) q^{78} +(-\)\(54\!\cdots\!00\)\( - \)\(11\!\cdots\!74\)\( \beta_{1} + \)\(11\!\cdots\!38\)\( \beta_{2}) q^{79} +(-\)\(62\!\cdots\!60\)\( - \)\(17\!\cdots\!12\)\( \beta_{1} + \)\(72\!\cdots\!04\)\( \beta_{2}) q^{80} +\)\(18\!\cdots\!21\)\( q^{81} +(-\)\(16\!\cdots\!12\)\( - \)\(10\!\cdots\!66\)\( \beta_{1} - \)\(59\!\cdots\!24\)\( \beta_{2}) q^{82} +(\)\(19\!\cdots\!16\)\( + \)\(12\!\cdots\!88\)\( \beta_{1} - \)\(40\!\cdots\!00\)\( \beta_{2}) q^{83} +(-\)\(24\!\cdots\!84\)\( + \)\(19\!\cdots\!96\)\( \beta_{1} - \)\(10\!\cdots\!64\)\( \beta_{2}) q^{84} +(-\)\(17\!\cdots\!60\)\( - \)\(47\!\cdots\!32\)\( \beta_{1} + \)\(26\!\cdots\!44\)\( \beta_{2}) q^{85} +(\)\(28\!\cdots\!00\)\( + \)\(62\!\cdots\!88\)\( \beta_{1} + \)\(33\!\cdots\!56\)\( \beta_{2}) q^{86} +(-\)\(28\!\cdots\!18\)\( + \)\(70\!\cdots\!18\)\( \beta_{1} - \)\(12\!\cdots\!58\)\( \beta_{2}) q^{87} +(-\)\(34\!\cdots\!88\)\( + \)\(37\!\cdots\!28\)\( \beta_{1} - \)\(93\!\cdots\!28\)\( \beta_{2}) q^{88} +(\)\(60\!\cdots\!62\)\( - \)\(73\!\cdots\!36\)\( \beta_{1} + \)\(13\!\cdots\!40\)\( \beta_{2}) q^{89} +(-\)\(17\!\cdots\!40\)\( - \)\(44\!\cdots\!78\)\( \beta_{1} + \)\(60\!\cdots\!76\)\( \beta_{2}) q^{90} +(\)\(77\!\cdots\!08\)\( - \)\(10\!\cdots\!60\)\( \beta_{1} + \)\(11\!\cdots\!64\)\( \beta_{2}) q^{91} +(\)\(32\!\cdots\!44\)\( - \)\(10\!\cdots\!88\)\( \beta_{1} + \)\(67\!\cdots\!28\)\( \beta_{2}) q^{92} +(\)\(14\!\cdots\!84\)\( - \)\(15\!\cdots\!42\)\( \beta_{1} + \)\(59\!\cdots\!46\)\( \beta_{2}) q^{93} +(-\)\(41\!\cdots\!40\)\( + \)\(29\!\cdots\!80\)\( \beta_{1} + \)\(75\!\cdots\!36\)\( \beta_{2}) q^{94} +(\)\(28\!\cdots\!00\)\( - \)\(34\!\cdots\!60\)\( \beta_{1} - \)\(78\!\cdots\!80\)\( \beta_{2}) q^{95} +(-\)\(14\!\cdots\!68\)\( - \)\(19\!\cdots\!44\)\( \beta_{1} + \)\(12\!\cdots\!84\)\( \beta_{2}) q^{96} +(-\)\(33\!\cdots\!14\)\( + \)\(51\!\cdots\!24\)\( \beta_{1} + \)\(22\!\cdots\!12\)\( \beta_{2}) q^{97} +(\)\(64\!\cdots\!70\)\( + \)\(64\!\cdots\!91\)\( \beta_{1} + \)\(39\!\cdots\!56\)\( \beta_{2}) q^{98} +(-\)\(23\!\cdots\!20\)\( + \)\(35\!\cdots\!08\)\( \beta_{1} - \)\(56\!\cdots\!08\)\( \beta_{2}) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 533574q^{2} - 3486784401q^{3} + 957442653732q^{4} - 53381973944430q^{5} - 620152499993058q^{6} - 1577312676222528q^{7} + 447338335366356696q^{8} + 4052555153018976267q^{9} + O(q^{10}) \) \( 3q + 533574q^{2} - 3486784401q^{3} + 957442653732q^{4} - 53381973944430q^{5} - 620152499993058q^{6} - 1577312676222528q^{7} + 447338335366356696q^{8} + 4052555153018976267q^{9} - 39868781226579858780q^{10} - \)\(53\!\cdots\!40\)\(q^{11} - \)\(11\!\cdots\!44\)\(q^{12} + \)\(54\!\cdots\!90\)\(q^{13} + \)\(31\!\cdots\!52\)\(q^{14} + \)\(62\!\cdots\!10\)\(q^{15} + \)\(12\!\cdots\!36\)\(q^{16} + \)\(72\!\cdots\!58\)\(q^{17} + \)\(72\!\cdots\!86\)\(q^{18} - \)\(10\!\cdots\!24\)\(q^{19} + \)\(30\!\cdots\!60\)\(q^{20} + \)\(18\!\cdots\!76\)\(q^{21} - \)\(10\!\cdots\!24\)\(q^{22} + \)\(41\!\cdots\!64\)\(q^{23} - \)\(51\!\cdots\!32\)\(q^{24} - \)\(11\!\cdots\!75\)\(q^{25} + \)\(92\!\cdots\!76\)\(q^{26} - \)\(47\!\cdots\!89\)\(q^{27} + \)\(64\!\cdots\!56\)\(q^{28} + \)\(72\!\cdots\!62\)\(q^{29} + \)\(46\!\cdots\!60\)\(q^{30} - \)\(38\!\cdots\!56\)\(q^{31} + \)\(37\!\cdots\!12\)\(q^{32} + \)\(61\!\cdots\!80\)\(q^{33} + \)\(37\!\cdots\!44\)\(q^{34} + \)\(27\!\cdots\!20\)\(q^{35} + \)\(12\!\cdots\!48\)\(q^{36} + \)\(29\!\cdots\!22\)\(q^{37} + \)\(74\!\cdots\!92\)\(q^{38} - \)\(63\!\cdots\!30\)\(q^{39} + \)\(15\!\cdots\!00\)\(q^{40} - \)\(13\!\cdots\!26\)\(q^{41} - \)\(36\!\cdots\!84\)\(q^{42} - \)\(24\!\cdots\!28\)\(q^{43} - \)\(21\!\cdots\!12\)\(q^{44} - \)\(72\!\cdots\!70\)\(q^{45} - \)\(10\!\cdots\!56\)\(q^{46} - \)\(10\!\cdots\!04\)\(q^{47} - \)\(13\!\cdots\!12\)\(q^{48} + \)\(92\!\cdots\!43\)\(q^{49} - \)\(11\!\cdots\!50\)\(q^{50} - \)\(84\!\cdots\!86\)\(q^{51} + \)\(59\!\cdots\!88\)\(q^{52} + \)\(62\!\cdots\!34\)\(q^{53} - \)\(83\!\cdots\!62\)\(q^{54} + \)\(72\!\cdots\!40\)\(q^{55} + \)\(36\!\cdots\!00\)\(q^{56} + \)\(12\!\cdots\!08\)\(q^{57} + \)\(28\!\cdots\!84\)\(q^{58} - \)\(75\!\cdots\!56\)\(q^{59} - \)\(35\!\cdots\!20\)\(q^{60} - \)\(71\!\cdots\!98\)\(q^{61} - \)\(33\!\cdots\!16\)\(q^{62} - \)\(21\!\cdots\!92\)\(q^{63} - \)\(35\!\cdots\!56\)\(q^{64} - \)\(71\!\cdots\!60\)\(q^{65} + \)\(11\!\cdots\!08\)\(q^{66} - \)\(51\!\cdots\!72\)\(q^{67} + \)\(11\!\cdots\!92\)\(q^{68} - \)\(48\!\cdots\!88\)\(q^{69} + \)\(38\!\cdots\!20\)\(q^{70} + \)\(84\!\cdots\!76\)\(q^{71} + \)\(60\!\cdots\!44\)\(q^{72} + \)\(63\!\cdots\!14\)\(q^{73} + \)\(89\!\cdots\!12\)\(q^{74} + \)\(13\!\cdots\!25\)\(q^{75} + \)\(12\!\cdots\!36\)\(q^{76} - \)\(22\!\cdots\!36\)\(q^{77} - \)\(10\!\cdots\!92\)\(q^{78} - \)\(16\!\cdots\!00\)\(q^{79} - \)\(18\!\cdots\!80\)\(q^{80} + \)\(54\!\cdots\!63\)\(q^{81} - \)\(48\!\cdots\!36\)\(q^{82} + \)\(59\!\cdots\!48\)\(q^{83} - \)\(74\!\cdots\!52\)\(q^{84} - \)\(52\!\cdots\!80\)\(q^{85} + \)\(85\!\cdots\!00\)\(q^{86} - \)\(84\!\cdots\!54\)\(q^{87} - \)\(10\!\cdots\!64\)\(q^{88} + \)\(18\!\cdots\!86\)\(q^{89} - \)\(53\!\cdots\!20\)\(q^{90} + \)\(23\!\cdots\!24\)\(q^{91} + \)\(96\!\cdots\!32\)\(q^{92} + \)\(44\!\cdots\!52\)\(q^{93} - \)\(12\!\cdots\!20\)\(q^{94} + \)\(84\!\cdots\!00\)\(q^{95} - \)\(44\!\cdots\!04\)\(q^{96} - \)\(99\!\cdots\!42\)\(q^{97} + \)\(19\!\cdots\!10\)\(q^{98} - \)\(71\!\cdots\!60\)\(q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - 3876249523 x - 18467420411022\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 18 \nu \)
\(\beta_{2}\)\(=\)\( 324 \nu^{2} - 2315430 \nu - 837269896968 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)\(/18\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{2} + 128635 \beta_{1} + 837269896968\)\()/324\)

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
64517.4
−4792.65
−59724.7
−983454. −1.16226e9 4.17427e11 1.57969e13 1.14303e15 5.39162e15 1.30140e17 1.35085e18 −1.55355e19
1.2 264126. −1.16226e9 −4.79993e11 −6.30487e13 −3.06983e14 −4.59086e16 −2.71983e17 1.35085e18 −1.66528e19
1.3 1.25290e6 −1.16226e9 1.02001e12 −6.13018e12 −1.45620e15 3.89397e16 5.89182e17 1.35085e18 −7.68052e18
\(n\): 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
\(3\) \(1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \( T_{2}^{3} - 533574 T_{2}^{2} - \)\(11\!\cdots\!60\)\( T_{2} + \)\(32\!\cdots\!08\)\( \) acting on \(S_{40}^{\mathrm{new}}(\Gamma_0(3))\).