Properties

Label 6015.2.a.h
Level 6015
Weight 2
Character orbit 6015.a
Self dual Yes
Analytic conductor 48.030
Analytic rank 0
Dimension 39
CM No

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Newspace parameters

Level: \( N \) = \( 6015 = 3 \cdot 5 \cdot 401 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6015.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(48.0300168158\)
Analytic rank: \(0\)
Dimension: \(39\)
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.

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

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 −2.73278 1.00000 5.46807 −1.00000 −2.73278 −1.71200 −9.47746 1.00000 2.73278
1.2 −2.72880 1.00000 5.44634 −1.00000 −2.72880 4.68686 −9.40437 1.00000 2.72880
1.3 −2.61259 1.00000 4.82563 −1.00000 −2.61259 1.06107 −7.38220 1.00000 2.61259
1.4 −2.40618 1.00000 3.78972 −1.00000 −2.40618 −5.00600 −4.30638 1.00000 2.40618
1.5 −2.39683 1.00000 3.74481 −1.00000 −2.39683 3.91052 −4.18201 1.00000 2.39683
1.6 −2.30681 1.00000 3.32135 −1.00000 −2.30681 0.182032 −3.04810 1.00000 2.30681
1.7 −2.17202 1.00000 2.71767 −1.00000 −2.17202 −0.722026 −1.55879 1.00000 2.17202
1.8 −2.05828 1.00000 2.23653 −1.00000 −2.05828 4.53201 −0.486836 1.00000 2.05828
1.9 −1.76442 1.00000 1.11320 −1.00000 −1.76442 −1.91400 1.56470 1.00000 1.76442
1.10 −1.60401 1.00000 0.572848 −1.00000 −1.60401 0.763953 2.28917 1.00000 1.60401
1.11 −1.46868 1.00000 0.157024 −1.00000 −1.46868 −1.36125 2.70674 1.00000 1.46868
1.12 −1.45937 1.00000 0.129770 −1.00000 −1.45937 4.22534 2.72936 1.00000 1.45937
1.13 −1.39292 1.00000 −0.0597773 −1.00000 −1.39292 0.772686 2.86910 1.00000 1.39292
1.14 −1.06568 1.00000 −0.864332 −1.00000 −1.06568 −3.38476 3.05245 1.00000 1.06568
1.15 −0.832555 1.00000 −1.30685 −1.00000 −0.832555 2.56306 2.75314 1.00000 0.832555
1.16 −0.749262 1.00000 −1.43861 −1.00000 −0.749262 −2.59303 2.57642 1.00000 0.749262
1.17 −0.693654 1.00000 −1.51884 −1.00000 −0.693654 5.04357 2.44086 1.00000 0.693654
1.18 −0.407003 1.00000 −1.83435 −1.00000 −0.407003 −1.58531 1.56059 1.00000 0.407003
1.19 −0.369183 1.00000 −1.86370 −1.00000 −0.369183 −1.65507 1.42641 1.00000 0.369183
1.20 0.104331 1.00000 −1.98912 −1.00000 0.104331 −3.27326 −0.416187 1.00000 −0.104331
See all 39 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.39
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(1\)
\(401\) \(1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{39} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6015))\).