Properties

Label 6004.2.a.f
Level 6004
Weight 2
Character orbit 6004.a
Self dual Yes
Analytic conductor 47.942
Analytic rank 1
Dimension 25
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: \(1\)
Dimension: \(25\)
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.

\(\operatorname{Tr}(f)(q) = \) \(25q \) \(\mathstrut +\mathstrut 4q^{3} \) \(\mathstrut -\mathstrut 8q^{5} \) \(\mathstrut +\mathstrut 2q^{7} \) \(\mathstrut +\mathstrut 13q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \(25q \) \(\mathstrut +\mathstrut 4q^{3} \) \(\mathstrut -\mathstrut 8q^{5} \) \(\mathstrut +\mathstrut 2q^{7} \) \(\mathstrut +\mathstrut 13q^{9} \) \(\mathstrut -\mathstrut 3q^{11} \) \(\mathstrut +\mathstrut q^{13} \) \(\mathstrut -\mathstrut 5q^{15} \) \(\mathstrut -\mathstrut 13q^{17} \) \(\mathstrut -\mathstrut 25q^{19} \) \(\mathstrut -\mathstrut 24q^{21} \) \(\mathstrut -\mathstrut 31q^{23} \) \(\mathstrut +\mathstrut 21q^{25} \) \(\mathstrut +\mathstrut 7q^{27} \) \(\mathstrut -\mathstrut 19q^{29} \) \(\mathstrut -\mathstrut 7q^{31} \) \(\mathstrut -\mathstrut 30q^{33} \) \(\mathstrut -\mathstrut q^{35} \) \(\mathstrut -\mathstrut 29q^{37} \) \(\mathstrut -\mathstrut 26q^{39} \) \(\mathstrut -\mathstrut 40q^{41} \) \(\mathstrut -\mathstrut 40q^{45} \) \(\mathstrut -\mathstrut 8q^{47} \) \(\mathstrut -\mathstrut 9q^{49} \) \(\mathstrut +\mathstrut 12q^{51} \) \(\mathstrut -\mathstrut 38q^{53} \) \(\mathstrut -\mathstrut 29q^{55} \) \(\mathstrut -\mathstrut 4q^{57} \) \(\mathstrut +\mathstrut 18q^{59} \) \(\mathstrut -\mathstrut 26q^{61} \) \(\mathstrut -\mathstrut 40q^{63} \) \(\mathstrut -\mathstrut 70q^{65} \) \(\mathstrut -\mathstrut 13q^{67} \) \(\mathstrut +\mathstrut q^{69} \) \(\mathstrut -\mathstrut 47q^{71} \) \(\mathstrut -\mathstrut 8q^{73} \) \(\mathstrut +\mathstrut 7q^{75} \) \(\mathstrut -\mathstrut 19q^{77} \) \(\mathstrut +\mathstrut 25q^{79} \) \(\mathstrut -\mathstrut 19q^{81} \) \(\mathstrut -\mathstrut 8q^{83} \) \(\mathstrut -\mathstrut 33q^{85} \) \(\mathstrut -\mathstrut 50q^{87} \) \(\mathstrut -\mathstrut 54q^{89} \) \(\mathstrut -\mathstrut 12q^{91} \) \(\mathstrut -\mathstrut 24q^{93} \) \(\mathstrut +\mathstrut 8q^{95} \) \(\mathstrut -\mathstrut 4q^{97} \) \(\mathstrut -\mathstrut 39q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 −3.10476 0 −3.52338 0 0.220122 0 6.63952 0
1.2 0 −2.61174 0 0.965145 0 0.0126812 0 3.82116 0
1.3 0 −2.45149 0 1.53010 0 1.05276 0 3.00979 0
1.4 0 −2.38552 0 −2.90737 0 −2.69828 0 2.69068 0
1.5 0 −1.61532 0 1.18505 0 3.44544 0 −0.390754 0
1.6 0 −1.50951 0 −2.01073 0 3.94863 0 −0.721387 0
1.7 0 −1.45593 0 2.84743 0 −2.57267 0 −0.880256 0
1.8 0 −1.19435 0 3.68128 0 2.98419 0 −1.57353 0
1.9 0 −0.904784 0 −1.47158 0 0.136041 0 −2.18137 0
1.10 0 −0.891665 0 −1.39999 0 3.43195 0 −2.20493 0
1.11 0 −0.642577 0 −3.12522 0 −2.66487 0 −2.58709 0
1.12 0 −0.0249148 0 1.33267 0 −1.80071 0 −2.99938 0
1.13 0 0.565175 0 2.63421 0 0.231397 0 −2.68058 0
1.14 0 0.681472 0 −3.07187 0 −2.21919 0 −2.53560 0
1.15 0 0.749226 0 −1.84486 0 4.01817 0 −2.43866 0
1.16 0 1.01059 0 3.45227 0 −1.97653 0 −1.97870 0
1.17 0 1.32467 0 −0.559509 0 4.61440 0 −1.24526 0
1.18 0 1.53952 0 0.492772 0 −1.61008 0 −0.629866 0
1.19 0 1.71727 0 1.91187 0 −0.564608 0 −0.0509783 0
1.20 0 1.83994 0 −0.734016 0 1.69773 0 0.385379 0
See all 25 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.25
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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6004))\):

\(T_{3}^{25} - \cdots\)
\(T_{5}^{25} + \cdots\)