Properties

Label 6004.2.a.e
Level 6004
Weight 2
Character orbit 6004.a
Self dual Yes
Analytic conductor 47.942
Analytic rank 0
Dimension 24
CM No

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Newspace parameters

Level: \( N \) = \( 6004 = 2^{2} \cdot 19 \cdot 79 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6004.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(47.9421813736\)
Analytic rank: \(0\)
Dimension: \(24\)
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.

\(\operatorname{Tr}(f)(q) = \) \(24q \) \(\mathstrut +\mathstrut q^{3} \) \(\mathstrut +\mathstrut 9q^{5} \) \(\mathstrut +\mathstrut 2q^{7} \) \(\mathstrut +\mathstrut 75q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \(24q \) \(\mathstrut +\mathstrut q^{3} \) \(\mathstrut +\mathstrut 9q^{5} \) \(\mathstrut +\mathstrut 2q^{7} \) \(\mathstrut +\mathstrut 75q^{9} \) \(\mathstrut +\mathstrut 10q^{11} \) \(\mathstrut +\mathstrut 18q^{13} \) \(\mathstrut +\mathstrut 16q^{15} \) \(\mathstrut +\mathstrut 18q^{17} \) \(\mathstrut +\mathstrut 24q^{19} \) \(\mathstrut +\mathstrut 25q^{21} \) \(\mathstrut +\mathstrut 9q^{23} \) \(\mathstrut +\mathstrut 25q^{25} \) \(\mathstrut +\mathstrut 4q^{27} \) \(\mathstrut +\mathstrut 32q^{29} \) \(\mathstrut +\mathstrut 20q^{31} \) \(\mathstrut -\mathstrut 4q^{33} \) \(\mathstrut +\mathstrut 3q^{35} \) \(\mathstrut +\mathstrut 20q^{37} \) \(\mathstrut +\mathstrut 13q^{39} \) \(\mathstrut +\mathstrut 41q^{41} \) \(\mathstrut -\mathstrut 8q^{43} \) \(\mathstrut +\mathstrut 48q^{45} \) \(\mathstrut -\mathstrut 5q^{47} \) \(\mathstrut +\mathstrut 12q^{49} \) \(\mathstrut +\mathstrut 24q^{51} \) \(\mathstrut +\mathstrut 15q^{53} \) \(\mathstrut +\mathstrut 14q^{55} \) \(\mathstrut +\mathstrut q^{57} \) \(\mathstrut +\mathstrut 5q^{59} \) \(\mathstrut -\mathstrut 13q^{61} \) \(\mathstrut +\mathstrut 9q^{63} \) \(\mathstrut +\mathstrut 59q^{65} \) \(\mathstrut -\mathstrut 30q^{67} \) \(\mathstrut +\mathstrut 51q^{69} \) \(\mathstrut +\mathstrut 20q^{73} \) \(\mathstrut -\mathstrut 31q^{75} \) \(\mathstrut +\mathstrut 6q^{77} \) \(\mathstrut +\mathstrut 24q^{79} \) \(\mathstrut +\mathstrut 32q^{81} \) \(\mathstrut +\mathstrut 8q^{83} \) \(\mathstrut +\mathstrut 4q^{85} \) \(\mathstrut -\mathstrut 32q^{87} \) \(\mathstrut +\mathstrut 47q^{89} \) \(\mathstrut -\mathstrut 27q^{91} \) \(\mathstrut +\mathstrut 34q^{93} \) \(\mathstrut +\mathstrut 9q^{95} \) \(\mathstrut +\mathstrut 69q^{97} \) \(\mathstrut -\mathstrut 34q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 0 −3.35141 0 −3.29178 0 −3.21633 0 8.23192 0
1.2 0 −3.19734 0 3.86202 0 0.0940904 0 7.22301 0
1.3 0 −3.17665 0 0.661665 0 4.43999 0 7.09108 0
1.4 0 −2.94455 0 3.51674 0 −3.98269 0 5.67040 0
1.5 0 −2.58747 0 −3.96262 0 2.03599 0 3.69501 0
1.6 0 −2.48790 0 −1.86011 0 −0.813337 0 3.18963 0
1.7 0 −2.24876 0 −0.730072 0 −4.54869 0 2.05693 0
1.8 0 −1.95303 0 2.03258 0 1.63097 0 0.814318 0
1.9 0 −1.93761 0 0.795901 0 1.65858 0 0.754317 0
1.10 0 −1.53751 0 2.83612 0 −2.00746 0 −0.636048 0
1.11 0 −1.34737 0 −0.232989 0 3.14804 0 −1.18461 0
1.12 0 −1.33751 0 −1.75116 0 −1.54762 0 −1.21107 0
1.13 0 1.34688 0 3.43943 0 0.297446 0 −1.18593 0
1.14 0 1.70311 0 −2.13957 0 −3.96784 0 −0.0994128 0
1.15 0 1.84886 0 0.871764 0 −3.24231 0 0.418286 0
1.16 0 1.98454 0 1.21429 0 4.02832 0 0.938380 0
1.17 0 2.05400 0 −2.46181 0 3.02591 0 1.21890 0
1.18 0 2.22506 0 −3.09603 0 −0.873879 0 1.95088 0
1.19 0 2.59829 0 0.131830 0 4.07797 0 3.75113 0
1.20 0 2.68661 0 2.87584 0 2.53518 0 4.21786 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.24
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(19\) \(-1\)
\(79\) \(-1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6004))\):

\(T_{3}^{24} - \cdots\)
\(T_{5}^{24} - \cdots\)