Properties

Label 6004.2.a.h
Level 6004
Weight 2
Character orbit 6004.a
Self dual Yes
Analytic conductor 47.942
Analytic rank 0
Dimension 31
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: \(31\)
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.

\(\operatorname{Tr}(f)(q) = \) \(31q \) \(\mathstrut -\mathstrut 4q^{3} \) \(\mathstrut +\mathstrut 10q^{5} \) \(\mathstrut +\mathstrut 11q^{7} \) \(\mathstrut +\mathstrut 47q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \(31q \) \(\mathstrut -\mathstrut 4q^{3} \) \(\mathstrut +\mathstrut 10q^{5} \) \(\mathstrut +\mathstrut 11q^{7} \) \(\mathstrut +\mathstrut 47q^{9} \) \(\mathstrut -\mathstrut 4q^{11} \) \(\mathstrut +\mathstrut 11q^{13} \) \(\mathstrut +\mathstrut 5q^{15} \) \(\mathstrut +\mathstrut 14q^{17} \) \(\mathstrut -\mathstrut 31q^{19} \) \(\mathstrut +\mathstrut 22q^{21} \) \(\mathstrut +\mathstrut 15q^{23} \) \(\mathstrut +\mathstrut 59q^{25} \) \(\mathstrut +\mathstrut 5q^{27} \) \(\mathstrut +\mathstrut 34q^{29} \) \(\mathstrut -\mathstrut 12q^{31} \) \(\mathstrut +\mathstrut 10q^{33} \) \(\mathstrut +\mathstrut 8q^{35} \) \(\mathstrut +\mathstrut 16q^{37} \) \(\mathstrut +\mathstrut 18q^{39} \) \(\mathstrut +\mathstrut 27q^{41} \) \(\mathstrut +\mathstrut 2q^{43} \) \(\mathstrut +\mathstrut 22q^{45} \) \(\mathstrut +\mathstrut 30q^{47} \) \(\mathstrut +\mathstrut 62q^{49} \) \(\mathstrut -\mathstrut 14q^{51} \) \(\mathstrut +\mathstrut 35q^{53} \) \(\mathstrut +\mathstrut 8q^{55} \) \(\mathstrut +\mathstrut 4q^{57} \) \(\mathstrut -\mathstrut 16q^{59} \) \(\mathstrut +\mathstrut 37q^{61} \) \(\mathstrut +\mathstrut 31q^{63} \) \(\mathstrut +\mathstrut 80q^{65} \) \(\mathstrut +\mathstrut 16q^{67} \) \(\mathstrut +\mathstrut q^{69} \) \(\mathstrut +\mathstrut 19q^{71} \) \(\mathstrut +\mathstrut 38q^{73} \) \(\mathstrut +\mathstrut 21q^{75} \) \(\mathstrut +\mathstrut 44q^{77} \) \(\mathstrut -\mathstrut 31q^{79} \) \(\mathstrut +\mathstrut 55q^{81} \) \(\mathstrut -\mathstrut 12q^{83} \) \(\mathstrut +\mathstrut 66q^{85} \) \(\mathstrut +\mathstrut 58q^{87} \) \(\mathstrut +\mathstrut 16q^{89} \) \(\mathstrut -\mathstrut 42q^{91} \) \(\mathstrut +\mathstrut 10q^{93} \) \(\mathstrut -\mathstrut 10q^{95} \) \(\mathstrut -\mathstrut 12q^{97} \) \(\mathstrut +\mathstrut 12q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 −3.22742 0 2.43630 0 −2.42654 0 7.41621 0
1.2 0 −3.18617 0 0.345750 0 −2.57023 0 7.15170 0
1.3 0 −2.95660 0 −3.08337 0 1.14715 0 5.74148 0
1.4 0 −2.85328 0 0.175152 0 −4.13028 0 5.14119 0
1.5 0 −2.76211 0 3.21863 0 5.11880 0 4.62924 0
1.6 0 −2.75235 0 −3.07293 0 4.09507 0 4.57545 0
1.7 0 −2.21543 0 0.305779 0 4.25499 0 1.90813 0
1.8 0 −1.83450 0 −2.99399 0 −1.09223 0 0.365373 0
1.9 0 −1.79377 0 4.30057 0 0.947969 0 0.217622 0
1.10 0 −1.75073 0 2.38866 0 −2.17287 0 0.0650385 0
1.11 0 −1.68597 0 0.297828 0 −1.50278 0 −0.157509 0
1.12 0 −1.30458 0 4.26479 0 −3.65251 0 −1.29806 0
1.13 0 −1.12838 0 −3.66649 0 0.118556 0 −1.72677 0
1.14 0 −0.959654 0 −1.46666 0 −0.668604 0 −2.07906 0
1.15 0 −0.689110 0 −0.991060 0 0.742104 0 −2.52513 0
1.16 0 −0.199808 0 0.368036 0 1.63577 0 −2.96008 0
1.17 0 0.246226 0 2.74327 0 2.47378 0 −2.93937 0
1.18 0 0.413811 0 −0.213960 0 2.17396 0 −2.82876 0
1.19 0 0.654848 0 −2.95483 0 −0.614354 0 −2.57117 0
1.20 0 0.806301 0 1.48458 0 2.70067 0 −2.34988 0
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
\(2\) \(-1\)
\(19\) \(1\)
\(79\) \(1\)

Hecke kernels

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

\(T_{3}^{31} + \cdots\)
\(T_{5}^{31} - \cdots\)