Properties

Label 6015.2.a.d
Level 6015
Weight 2
Character orbit 6015.a
Self dual Yes
Analytic conductor 48.030
Analytic rank 1
Dimension 29
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: \(1\)
Dimension: \(29\)
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.

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

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 −2.79568 −1.00000 5.81581 1.00000 2.79568 1.96975 −10.6678 1.00000 −2.79568
1.2 −2.50063 −1.00000 4.25317 1.00000 2.50063 4.30119 −5.63435 1.00000 −2.50063
1.3 −2.37855 −1.00000 3.65751 1.00000 2.37855 2.87479 −3.94247 1.00000 −2.37855
1.4 −2.21779 −1.00000 2.91860 1.00000 2.21779 3.34170 −2.03727 1.00000 −2.21779
1.5 −2.17430 −1.00000 2.72756 1.00000 2.17430 −3.13141 −1.58193 1.00000 −2.17430
1.6 −2.12396 −1.00000 2.51122 1.00000 2.12396 −2.76942 −1.08582 1.00000 −2.12396
1.7 −1.69527 −1.00000 0.873943 1.00000 1.69527 −0.214929 1.90897 1.00000 −1.69527
1.8 −1.64211 −1.00000 0.696514 1.00000 1.64211 −2.05425 2.14046 1.00000 −1.64211
1.9 −1.55350 −1.00000 0.413352 1.00000 1.55350 0.802812 2.46485 1.00000 −1.55350
1.10 −1.15935 −1.00000 −0.655911 1.00000 1.15935 2.44720 3.07913 1.00000 −1.15935
1.11 −0.705071 −1.00000 −1.50287 1.00000 0.705071 −2.52019 2.46978 1.00000 −0.705071
1.12 −0.462418 −1.00000 −1.78617 1.00000 0.462418 −1.50774 1.75079 1.00000 −0.462418
1.13 −0.319268 −1.00000 −1.89807 1.00000 0.319268 0.830562 1.24453 1.00000 −0.319268
1.14 −0.256270 −1.00000 −1.93433 1.00000 0.256270 0.958240 1.00825 1.00000 −0.256270
1.15 −0.0716495 −1.00000 −1.99487 1.00000 0.0716495 4.62140 0.286230 1.00000 −0.0716495
1.16 −0.0247562 −1.00000 −1.99939 1.00000 0.0247562 −4.38402 0.0990096 1.00000 −0.0247562
1.17 0.242491 −1.00000 −1.94120 1.00000 −0.242491 2.88846 −0.955706 1.00000 0.242491
1.18 0.806575 −1.00000 −1.34944 1.00000 −0.806575 2.45232 −2.70157 1.00000 0.806575
1.19 0.843993 −1.00000 −1.28768 1.00000 −0.843993 −1.98856 −2.77478 1.00000 0.843993
1.20 1.09145 −1.00000 −0.808735 1.00000 −1.09145 0.212319 −3.06560 1.00000 1.09145
See all 29 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.29
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 \(T_{2}^{29} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6015))\).