Properties

Label 6013.2.a.e
Level $6013$
Weight $2$
Character orbit 6013.a
Self dual yes
Analytic conductor $48.014$
Analytic rank $0$
Dimension $109$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6013 = 7 \cdot 859 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6013.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(48.0140467354\)
Analytic rank: \(0\)
Dimension: \(109\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 109q + 19q^{2} + 38q^{3} + 111q^{4} + 43q^{5} + 14q^{6} + 109q^{7} + 48q^{8} + 119q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 109q + 19q^{2} + 38q^{3} + 111q^{4} + 43q^{5} + 14q^{6} + 109q^{7} + 48q^{8} + 119q^{9} + 15q^{10} + 48q^{11} + 72q^{12} + 29q^{13} + 19q^{14} + 29q^{15} + 115q^{16} + 72q^{17} + 33q^{18} + 58q^{19} + 88q^{20} + 38q^{21} + 4q^{22} + 65q^{23} + 46q^{24} + 124q^{25} + 49q^{26} + 131q^{27} + 111q^{28} + 25q^{29} + 2q^{30} + 41q^{31} + 75q^{32} + 54q^{33} + 23q^{34} + 43q^{35} + 111q^{36} + 25q^{37} + 54q^{38} + 27q^{39} + 30q^{40} + 109q^{41} + 14q^{42} + 38q^{43} + 68q^{44} + 84q^{45} - 9q^{46} + 121q^{47} + 106q^{48} + 109q^{49} + 14q^{50} + 36q^{51} + 38q^{52} + 61q^{53} + 31q^{54} + 50q^{55} + 48q^{56} + 5q^{57} - 20q^{58} + 181q^{59} + 25q^{60} + 34q^{61} + 75q^{62} + 119q^{63} + 96q^{64} + 12q^{65} + 19q^{66} + 87q^{67} + 150q^{68} + 89q^{69} + 15q^{70} + 83q^{71} + 65q^{72} + 32q^{73} - 19q^{74} + 112q^{75} + 84q^{76} + 48q^{77} - 34q^{78} - 9q^{79} + 137q^{80} + 109q^{81} - 19q^{82} + 136q^{83} + 72q^{84} - 32q^{85} - 24q^{86} + 28q^{87} - 24q^{88} + 142q^{89} + 19q^{90} + 29q^{91} + 96q^{92} + 29q^{93} + 9q^{94} + 52q^{95} + 88q^{96} + 75q^{97} + 19q^{98} + 84q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −2.71096 3.20670 5.34933 3.06653 −8.69325 1.00000 −9.07991 7.28293 −8.31326
1.2 −2.68013 −1.39322 5.18311 −1.95866 3.73403 1.00000 −8.53117 −1.05892 5.24948
1.3 −2.65451 −2.36774 5.04644 −0.160322 6.28521 1.00000 −8.08683 2.60621 0.425577
1.4 −2.64232 0.320634 4.98185 3.17576 −0.847217 1.00000 −7.87900 −2.89719 −8.39136
1.5 −2.53130 1.72523 4.40746 −1.30306 −4.36707 1.00000 −6.09399 −0.0235740 3.29843
1.6 −2.53104 0.289125 4.40615 −2.52674 −0.731786 1.00000 −6.09005 −2.91641 6.39527
1.7 −2.44055 1.60583 3.95629 4.05418 −3.91911 1.00000 −4.77442 −0.421314 −9.89444
1.8 −2.42826 1.22129 3.89645 0.497994 −2.96562 1.00000 −4.60506 −1.50844 −1.20926
1.9 −2.40830 −1.24102 3.79989 3.14441 2.98874 1.00000 −4.33466 −1.45987 −7.57267
1.10 −2.31924 3.03113 3.37885 1.64684 −7.02991 1.00000 −3.19788 6.18776 −3.81941
1.11 −2.28251 −0.277049 3.20984 0.585497 0.632366 1.00000 −2.76147 −2.92324 −1.33640
1.12 −2.23377 −0.900406 2.98971 4.46215 2.01130 1.00000 −2.21077 −2.18927 −9.96739
1.13 −2.18834 3.06167 2.78881 −2.53450 −6.69995 1.00000 −1.72618 6.37380 5.54634
1.14 −2.11721 0.0595888 2.48258 −0.217642 −0.126162 1.00000 −1.02172 −2.99645 0.460795
1.15 −2.09226 2.05769 2.37754 −3.21058 −4.30522 1.00000 −0.789909 1.23410 6.71736
1.16 −2.06498 −2.80647 2.26415 −1.02377 5.79530 1.00000 −0.545455 4.87626 2.11406
1.17 −2.02275 2.43668 2.09152 0.154664 −4.92881 1.00000 −0.185129 2.93743 −0.312846
1.18 −1.88015 −2.98311 1.53497 2.43013 5.60869 1.00000 0.874328 5.89893 −4.56900
1.19 −1.82060 2.13458 1.31457 −4.26535 −3.88622 1.00000 1.24789 1.55645 7.76548
1.20 −1.74589 −1.96009 1.04812 −3.28540 3.42209 1.00000 1.66188 0.841938 5.73592
See next 80 embeddings (of 109 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.109
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(7\) \(-1\)
\(859\) \(1\)

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 6013.2.a.e 109
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6013.2.a.e 109 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(14\!\cdots\!98\)\( T_{2}^{94} - \)\(26\!\cdots\!12\)\( T_{2}^{93} + \)\(24\!\cdots\!16\)\( T_{2}^{92} + \)\(13\!\cdots\!30\)\( T_{2}^{91} - \)\(33\!\cdots\!33\)\( T_{2}^{90} + \)\(12\!\cdots\!63\)\( T_{2}^{89} + \)\(36\!\cdots\!76\)\( T_{2}^{88} - \)\(40\!\cdots\!33\)\( T_{2}^{87} - \)\(33\!\cdots\!58\)\( T_{2}^{86} + \)\(60\!\cdots\!47\)\( T_{2}^{85} + \)\(24\!\cdots\!51\)\( T_{2}^{84} - \)\(65\!\cdots\!10\)\( T_{2}^{83} - \)\(15\!\cdots\!17\)\( T_{2}^{82} + \)\(57\!\cdots\!49\)\( T_{2}^{81} + \)\(75\!\cdots\!86\)\( T_{2}^{80} - \)\(41\!\cdots\!03\)\( T_{2}^{79} - \)\(26\!\cdots\!26\)\( T_{2}^{78} + \)\(25\!\cdots\!86\)\( T_{2}^{77} + \)\(24\!\cdots\!06\)\( T_{2}^{76} - \)\(13\!\cdots\!28\)\( T_{2}^{75} + \)\(47\!\cdots\!60\)\( T_{2}^{74} + \)\(64\!\cdots\!33\)\( T_{2}^{73} - \)\(47\!\cdots\!76\)\( T_{2}^{72} - \)\(25\!\cdots\!75\)\( T_{2}^{71} + \)\(29\!\cdots\!08\)\( T_{2}^{70} + \)\(90\!\cdots\!90\)\( T_{2}^{69} - \)\(13\!\cdots\!00\)\( T_{2}^{68} - \)\(27\!\cdots\!58\)\( T_{2}^{67} + \)\(55\!\cdots\!73\)\( T_{2}^{66} + \)\(69\!\cdots\!02\)\( T_{2}^{65} - \)\(19\!\cdots\!28\)\( T_{2}^{64} - \)\(14\!\cdots\!43\)\( T_{2}^{63} + \)\(57\!\cdots\!48\)\( T_{2}^{62} + \)\(22\!\cdots\!51\)\( T_{2}^{61} - \)\(14\!\cdots\!16\)\( T_{2}^{60} - \)\(15\!\cdots\!92\)\( T_{2}^{59} + \)\(34\!\cdots\!53\)\( T_{2}^{58} - \)\(49\!\cdots\!74\)\( T_{2}^{57} - \)\(69\!\cdots\!91\)\( T_{2}^{56} + \)\(25\!\cdots\!12\)\( T_{2}^{55} + \)\(12\!\cdots\!25\)\( T_{2}^{54} - \)\(71\!\cdots\!44\)\( T_{2}^{53} - \)\(19\!\cdots\!73\)\( T_{2}^{52} + \)\(15\!\cdots\!27\)\( T_{2}^{51} + \)\(25\!\cdots\!61\)\( T_{2}^{50} - \)\(26\!\cdots\!75\)\( T_{2}^{49} - \)\(30\!\cdots\!16\)\( T_{2}^{48} + \)\(39\!\cdots\!95\)\( T_{2}^{47} + \)\(30\!\cdots\!91\)\( T_{2}^{46} - \)\(50\!\cdots\!39\)\( T_{2}^{45} - \)\(24\!\cdots\!41\)\( T_{2}^{44} + \)\(56\!\cdots\!92\)\( T_{2}^{43} + \)\(15\!\cdots\!13\)\( T_{2}^{42} - \)\(54\!\cdots\!92\)\( T_{2}^{41} - \)\(56\!\cdots\!82\)\( T_{2}^{40} + \)\(44\!\cdots\!95\)\( T_{2}^{39} - \)\(17\!\cdots\!45\)\( T_{2}^{38} - \)\(32\!\cdots\!58\)\( T_{2}^{37} + \)\(52\!\cdots\!70\)\( T_{2}^{36} + \)\(19\!\cdots\!66\)\( T_{2}^{35} - \)\(55\!\cdots\!38\)\( T_{2}^{34} - \)\(10\!\cdots\!91\)\( T_{2}^{33} + \)\(40\!\cdots\!07\)\( T_{2}^{32} + \)\(45\!\cdots\!26\)\( T_{2}^{31} - \)\(23\!\cdots\!22\)\( T_{2}^{30} - \)\(16\!\cdots\!48\)\( T_{2}^{29} + \)\(10\!\cdots\!52\)\( T_{2}^{28} + \)\(47\!\cdots\!75\)\( T_{2}^{27} - \)\(39\!\cdots\!19\)\( T_{2}^{26} - \)\(10\!\cdots\!31\)\( T_{2}^{25} + \)\(11\!\cdots\!14\)\( T_{2}^{24} + \)\(14\!\cdots\!06\)\( T_{2}^{23} - \)\(28\!\cdots\!74\)\( T_{2}^{22} - \)\(35\!\cdots\!76\)\( T_{2}^{21} + \)\(53\!\cdots\!51\)\( T_{2}^{20} - \)\(43\!\cdots\!57\)\( T_{2}^{19} - \)\(76\!\cdots\!46\)\( T_{2}^{18} + \)\(12\!\cdots\!11\)\( T_{2}^{17} + \)\(79\!\cdots\!81\)\( T_{2}^{16} - \)\(20\!\cdots\!56\)\( T_{2}^{15} - \)\(53\!\cdots\!49\)\( T_{2}^{14} + \)\(20\!\cdots\!00\)\( T_{2}^{13} + \)\(17\!\cdots\!51\)\( T_{2}^{12} - \)\(13\!\cdots\!67\)\( T_{2}^{11} + \)\(39\!\cdots\!78\)\( T_{2}^{10} + \)\(50\!\cdots\!22\)\( T_{2}^{9} - \)\(58\!\cdots\!10\)\( T_{2}^{8} - \)\(77\!\cdots\!76\)\( T_{2}^{7} + \)\(18\!\cdots\!01\)\( T_{2}^{6} - \)\(50\!\cdots\!67\)\( T_{2}^{5} - \)\(16\!\cdots\!48\)\( T_{2}^{4} + 177695871490 T_{2}^{3} - 6965216658 T_{2}^{2} + 103028472 T_{2} - 451224 \)">\(T_{2}^{109} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6013))\).

Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database