Properties

Label 6008.2.a.e
Level 6008
Weight 2
Character orbit 6008.a
Self dual Yes
Analytic conductor 47.974
Analytic rank 0
Dimension 50
CM No

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Newspace parameters

Level: \( N \) = \( 6008 = 2^{3} \cdot 751 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6008.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(47.9741215344\)
Analytic rank: \(0\)
Dimension: \(50\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 50q + 6q^{3} + 23q^{5} + 12q^{7} + 56q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 50q + 6q^{3} + 23q^{5} + 12q^{7} + 56q^{9} - 5q^{11} + 36q^{13} + 5q^{15} + 14q^{17} + 9q^{19} + 30q^{21} + 3q^{23} + 71q^{25} + 24q^{27} + 61q^{29} + 27q^{31} + 24q^{33} - 7q^{35} + 56q^{37} - 2q^{39} + 10q^{41} + 19q^{43} + 76q^{45} + 3q^{47} + 82q^{49} - q^{51} + 56q^{53} + 7q^{55} + 35q^{57} - q^{59} + 67q^{61} + 25q^{63} + 27q^{65} + 46q^{67} + 68q^{69} + 4q^{71} + 62q^{73} + 27q^{75} + 71q^{77} + 7q^{79} + 74q^{81} - q^{83} + 72q^{85} + 25q^{87} + 19q^{89} + 45q^{91} + 72q^{93} - 24q^{95} + 81q^{97} + 16q^{99} + O(q^{100}) \)

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 −3.22705 0 2.92731 0 0.525757 0 7.41384 0
1.2 0 −2.97826 0 −0.794149 0 −3.63896 0 5.87005 0
1.3 0 −2.97134 0 0.343630 0 −1.43593 0 5.82889 0
1.4 0 −2.96484 0 2.30894 0 −3.32968 0 5.79029 0
1.5 0 −2.83142 0 4.26484 0 3.89972 0 5.01693 0
1.6 0 −2.58697 0 0.0890087 0 4.21791 0 3.69244 0
1.7 0 −2.56656 0 −2.44840 0 −1.80922 0 3.58725 0
1.8 0 −2.49253 0 −3.92718 0 2.18622 0 3.21269 0
1.9 0 −2.41806 0 0.877422 0 2.89114 0 2.84699 0
1.10 0 −2.26403 0 −1.66165 0 −0.979480 0 2.12585 0
1.11 0 −2.24563 0 3.86835 0 −4.12735 0 2.04285 0
1.12 0 −1.56448 0 2.55356 0 −2.27627 0 −0.552416 0
1.13 0 −1.42091 0 −3.43075 0 0.534513 0 −0.981020 0
1.14 0 −1.38456 0 −1.32143 0 2.39545 0 −1.08299 0
1.15 0 −1.34723 0 0.0749039 0 −4.25247 0 −1.18497 0
1.16 0 −1.33859 0 3.94917 0 0.220759 0 −1.20816 0
1.17 0 −1.03653 0 2.53205 0 −0.333645 0 −1.92561 0
1.18 0 −0.691575 0 −1.88995 0 4.98805 0 −2.52172 0
1.19 0 −0.685000 0 2.14563 0 4.94928 0 −2.53078 0
1.20 0 −0.684026 0 −2.03205 0 −0.0739397 0 −2.53211 0
See all 50 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.50
Significant digits:
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Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(751\) \(1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{3}^{50} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6008))\).