Properties

Label 6015.2.a.e
Level 6015
Weight 2
Character orbit 6015.a
Self dual Yes
Analytic conductor 48.030
Analytic rank 0
Dimension 31
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: \(31\)
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.

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

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 −2.46359 −1.00000 4.06929 −1.00000 2.46359 −1.56442 −5.09790 1.00000 2.46359
1.2 −2.32925 −1.00000 3.42539 −1.00000 2.32925 −3.29161 −3.32009 1.00000 2.32925
1.3 −2.16689 −1.00000 2.69543 −1.00000 2.16689 0.00504296 −1.50693 1.00000 2.16689
1.4 −2.16128 −1.00000 2.67112 −1.00000 2.16128 1.73409 −1.45047 1.00000 2.16128
1.5 −2.02086 −1.00000 2.08386 −1.00000 2.02086 0.941480 −0.169465 1.00000 2.02086
1.6 −1.61585 −1.00000 0.610967 −1.00000 1.61585 5.09249 2.24447 1.00000 1.61585
1.7 −1.60901 −1.00000 0.588914 −1.00000 1.60901 −4.04417 2.27045 1.00000 1.60901
1.8 −1.52097 −1.00000 0.313357 −1.00000 1.52097 −2.49520 2.56534 1.00000 1.52097
1.9 −1.22966 −1.00000 −0.487935 −1.00000 1.22966 −1.61954 3.05932 1.00000 1.22966
1.10 −0.900767 −1.00000 −1.18862 −1.00000 0.900767 3.65606 2.87220 1.00000 0.900767
1.11 −0.707695 −1.00000 −1.49917 −1.00000 0.707695 −2.27723 2.47634 1.00000 0.707695
1.12 −0.349021 −1.00000 −1.87818 −1.00000 0.349021 0.242299 1.35357 1.00000 0.349021
1.13 −0.325483 −1.00000 −1.89406 −1.00000 0.325483 −4.65914 1.26745 1.00000 0.325483
1.14 −0.0812850 −1.00000 −1.99339 −1.00000 0.0812850 2.03254 0.324603 1.00000 0.0812850
1.15 0.0495237 −1.00000 −1.99755 −1.00000 −0.0495237 −1.50762 −0.197974 1.00000 −0.0495237
1.16 0.0951324 −1.00000 −1.99095 −1.00000 −0.0951324 0.303287 −0.379669 1.00000 −0.0951324
1.17 0.363417 −1.00000 −1.86793 −1.00000 −0.363417 3.29378 −1.40567 1.00000 −0.363417
1.18 0.698893 −1.00000 −1.51155 −1.00000 −0.698893 4.36738 −2.45420 1.00000 −0.698893
1.19 0.917882 −1.00000 −1.15749 −1.00000 −0.917882 −0.377330 −2.89821 1.00000 −0.917882
1.20 1.04920 −1.00000 −0.899176 −1.00000 −1.04920 2.71724 −3.04182 1.00000 −1.04920
See all 31 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.31
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 \(T_{2}^{31} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6015))\).