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

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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: \(110\)
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.

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

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:

Inner twists

This newform does not have CM; other inner twists have not been computed.

Atkin-Lehner signs

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

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{110} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6013))\).