Properties

Label 6013.2.a.f
Level $6013$
Weight $2$
Character orbit 6013.a
Self dual yes
Analytic conductor $48.014$
Analytic rank $0$
Dimension $110$
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: \(110\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 110q + 16q^{2} + 29q^{3} + 118q^{4} + 12q^{6} - 110q^{7} + 57q^{8} + 127q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 110q + 16q^{2} + 29q^{3} + 118q^{4} + 12q^{6} - 110q^{7} + 57q^{8} + 127q^{9} + 3q^{10} + 52q^{11} + 62q^{12} - 9q^{13} - 16q^{14} + 39q^{15} + 146q^{16} + 11q^{17} + 60q^{18} + 14q^{19} + 18q^{20} - 29q^{21} + 32q^{22} + 73q^{23} + 24q^{24} + 132q^{25} - 7q^{26} + 116q^{27} - 118q^{28} + 35q^{29} + 18q^{30} + 36q^{31} + 140q^{32} + 42q^{33} - 7q^{34} + 180q^{36} + 49q^{37} + 45q^{39} + 6q^{40} - 14q^{41} - 12q^{42} + 58q^{43} + 92q^{44} + 17q^{45} + 27q^{46} + 87q^{47} + 98q^{48} + 110q^{49} + 91q^{50} + 42q^{51} + 16q^{52} + 95q^{53} + 41q^{54} + 8q^{55} - 57q^{56} + 61q^{57} + 46q^{58} + 114q^{59} + 81q^{60} - 47q^{61} + 31q^{62} - 127q^{63} + 199q^{64} + 62q^{65} + 21q^{66} + 95q^{67} + 60q^{68} - 39q^{69} - 3q^{70} + 131q^{71} + 186q^{72} + 31q^{73} + 23q^{74} + 121q^{75} + 14q^{76} - 52q^{77} + 110q^{78} + 9q^{79} + 61q^{80} + 194q^{81} + 45q^{82} + 73q^{83} - 62q^{84} + 59q^{85} + 72q^{86} + 64q^{87} + 100q^{88} - 17q^{89} + 11q^{90} + 9q^{91} + 192q^{92} + 85q^{93} - 11q^{94} + 108q^{95} + 68q^{96} + 32q^{97} + 16q^{98} + 160q^{99} + O(q^{100}) \)

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 −2.71420 2.40545 5.36690 2.49981 −6.52887 −1.00000 −9.13847 2.78617 −6.78498
1.2 −2.67470 1.47568 5.15405 −1.35988 −3.94701 −1.00000 −8.43614 −0.822364 3.63728
1.3 −2.64365 1.43905 4.98889 0.402261 −3.80434 −1.00000 −7.90158 −0.929138 −1.06344
1.4 −2.63934 −1.02733 4.96612 −0.539298 2.71147 −1.00000 −7.82859 −1.94460 1.42339
1.5 −2.59895 −1.56127 4.75456 1.42920 4.05766 −1.00000 −7.15899 −0.562451 −3.71443
1.6 −2.57438 3.21457 4.62742 −1.73819 −8.27551 −1.00000 −6.76398 7.33344 4.47476
1.7 −2.55830 −1.85077 4.54491 3.83295 4.73481 −1.00000 −6.51064 0.425331 −9.80585
1.8 −2.38030 −0.0622570 3.66584 −0.894729 0.148190 −1.00000 −3.96519 −2.99612 2.12972
1.9 −2.32328 −2.34605 3.39764 −4.09939 5.45055 −1.00000 −3.24712 2.50397 9.52403
1.10 −2.31912 −2.74143 3.37834 −1.26450 6.35772 −1.00000 −3.19654 4.51544 2.93254
1.11 −2.30986 1.63849 3.33543 −0.463511 −3.78466 −1.00000 −3.08465 −0.315366 1.07064
1.12 −2.24570 −1.05382 3.04316 1.30809 2.36656 −1.00000 −2.34263 −1.88947 −2.93757
1.13 −2.23147 2.27145 2.97947 −3.38307 −5.06867 −1.00000 −2.18565 2.15947 7.54923
1.14 −2.10326 2.95040 2.42369 4.24716 −6.20544 −1.00000 −0.891125 5.70483 −8.93287
1.15 −2.09677 −1.04747 2.39646 −2.55395 2.19632 −1.00000 −0.831292 −1.90280 5.35506
1.16 −2.05159 3.13328 2.20904 2.94582 −6.42821 −1.00000 −0.428856 6.81742 −6.04363
1.17 −1.99923 0.235231 1.99692 2.61420 −0.470281 −1.00000 0.00615784 −2.94467 −5.22639
1.18 −1.94077 −1.00614 1.76657 −2.69594 1.95267 −1.00000 0.453027 −1.98769 5.23218
1.19 −1.93774 1.87483 1.75485 1.38259 −3.63294 −1.00000 0.475036 0.514992 −2.67910
1.20 −1.87802 0.475349 1.52695 −2.58742 −0.892713 −1.00000 0.888402 −2.77404 4.85921
See next 80 embeddings (of 110 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.110
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.f 110
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6013.2.a.f 110 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(71\!\cdots\!78\)\( T_{2}^{94} + \)\(14\!\cdots\!02\)\( T_{2}^{93} + \)\(92\!\cdots\!92\)\( T_{2}^{92} - \)\(25\!\cdots\!81\)\( T_{2}^{91} - \)\(96\!\cdots\!00\)\( T_{2}^{90} + \)\(35\!\cdots\!81\)\( T_{2}^{89} + \)\(83\!\cdots\!37\)\( T_{2}^{88} - \)\(41\!\cdots\!99\)\( T_{2}^{87} - \)\(56\!\cdots\!69\)\( T_{2}^{86} + \)\(40\!\cdots\!10\)\( T_{2}^{85} + \)\(28\!\cdots\!65\)\( T_{2}^{84} - \)\(33\!\cdots\!17\)\( T_{2}^{83} - \)\(61\!\cdots\!56\)\( T_{2}^{82} + \)\(24\!\cdots\!21\)\( T_{2}^{81} - \)\(59\!\cdots\!35\)\( T_{2}^{80} - \)\(15\!\cdots\!79\)\( T_{2}^{79} + \)\(94\!\cdots\!22\)\( T_{2}^{78} + \)\(86\!\cdots\!02\)\( T_{2}^{77} - \)\(81\!\cdots\!16\)\( T_{2}^{76} - \)\(42\!\cdots\!20\)\( T_{2}^{75} + \)\(54\!\cdots\!05\)\( T_{2}^{74} + \)\(18\!\cdots\!71\)\( T_{2}^{73} - \)\(29\!\cdots\!15\)\( T_{2}^{72} - \)\(68\!\cdots\!63\)\( T_{2}^{71} + \)\(13\!\cdots\!66\)\( T_{2}^{70} + \)\(22\!\cdots\!42\)\( T_{2}^{69} - \)\(56\!\cdots\!32\)\( T_{2}^{68} - \)\(65\!\cdots\!31\)\( T_{2}^{67} + \)\(20\!\cdots\!95\)\( T_{2}^{66} + \)\(16\!\cdots\!80\)\( T_{2}^{65} - \)\(65\!\cdots\!55\)\( T_{2}^{64} - \)\(32\!\cdots\!59\)\( T_{2}^{63} + \)\(18\!\cdots\!99\)\( T_{2}^{62} + \)\(47\!\cdots\!83\)\( T_{2}^{61} - \)\(46\!\cdots\!38\)\( T_{2}^{60} - \)\(25\!\cdots\!83\)\( T_{2}^{59} + \)\(10\!\cdots\!01\)\( T_{2}^{58} - \)\(12\!\cdots\!31\)\( T_{2}^{57} - \)\(21\!\cdots\!35\)\( T_{2}^{56} + \)\(58\!\cdots\!55\)\( T_{2}^{55} + \)\(38\!\cdots\!83\)\( T_{2}^{54} - \)\(15\!\cdots\!39\)\( T_{2}^{53} - \)\(61\!\cdots\!98\)\( T_{2}^{52} + \)\(33\!\cdots\!84\)\( T_{2}^{51} + \)\(87\!\cdots\!58\)\( T_{2}^{50} - \)\(58\!\cdots\!29\)\( T_{2}^{49} - \)\(11\!\cdots\!97\)\( T_{2}^{48} + \)\(88\!\cdots\!62\)\( T_{2}^{47} + \)\(12\!\cdots\!84\)\( T_{2}^{46} - \)\(11\!\cdots\!96\)\( T_{2}^{45} - \)\(12\!\cdots\!71\)\( T_{2}^{44} + \)\(13\!\cdots\!39\)\( T_{2}^{43} + \)\(10\!\cdots\!93\)\( T_{2}^{42} - \)\(13\!\cdots\!46\)\( T_{2}^{41} - \)\(82\!\cdots\!09\)\( T_{2}^{40} + \)\(11\!\cdots\!84\)\( T_{2}^{39} + \)\(54\!\cdots\!73\)\( T_{2}^{38} - \)\(86\!\cdots\!84\)\( T_{2}^{37} - \)\(30\!\cdots\!08\)\( T_{2}^{36} + \)\(57\!\cdots\!96\)\( T_{2}^{35} + \)\(14\!\cdots\!37\)\( T_{2}^{34} - \)\(32\!\cdots\!74\)\( T_{2}^{33} - \)\(58\!\cdots\!33\)\( T_{2}^{32} + \)\(16\!\cdots\!28\)\( T_{2}^{31} + \)\(18\!\cdots\!20\)\( T_{2}^{30} - \)\(70\!\cdots\!32\)\( T_{2}^{29} - \)\(42\!\cdots\!01\)\( T_{2}^{28} + \)\(25\!\cdots\!50\)\( T_{2}^{27} + \)\(47\!\cdots\!44\)\( T_{2}^{26} - \)\(81\!\cdots\!53\)\( T_{2}^{25} + \)\(11\!\cdots\!84\)\( T_{2}^{24} + \)\(21\!\cdots\!44\)\( T_{2}^{23} - \)\(85\!\cdots\!38\)\( T_{2}^{22} - \)\(47\!\cdots\!41\)\( T_{2}^{21} + \)\(27\!\cdots\!84\)\( T_{2}^{20} + \)\(85\!\cdots\!27\)\( T_{2}^{19} - \)\(60\!\cdots\!07\)\( T_{2}^{18} - \)\(12\!\cdots\!74\)\( T_{2}^{17} + \)\(98\!\cdots\!11\)\( T_{2}^{16} + \)\(14\!\cdots\!59\)\( T_{2}^{15} - \)\(12\!\cdots\!95\)\( T_{2}^{14} - \)\(13\!\cdots\!43\)\( T_{2}^{13} + \)\(11\!\cdots\!88\)\( T_{2}^{12} + \)\(96\!\cdots\!49\)\( T_{2}^{11} - \)\(75\!\cdots\!28\)\( T_{2}^{10} - \)\(49\!\cdots\!08\)\( T_{2}^{9} + \)\(36\!\cdots\!74\)\( T_{2}^{8} + \)\(17\!\cdots\!67\)\( T_{2}^{7} - \)\(11\!\cdots\!10\)\( T_{2}^{6} - \)\(41\!\cdots\!25\)\( T_{2}^{5} + \)\(23\!\cdots\!08\)\( T_{2}^{4} + \)\(53\!\cdots\!66\)\( T_{2}^{3} - 206512941142 T_{2}^{2} - 27295135220 T_{2} + 82552988 \)">\(T_{2}^{110} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6013))\).

Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database