Properties

Label 8020.2.a.f
Level 8020
Weight 2
Character orbit 8020.a
Self dual Yes
Analytic conductor 64.040
Analytic rank 0
Dimension 37
CM No

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Newspace parameters

Level: \( N \) = \( 8020 = 2^{2} \cdot 5 \cdot 401 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8020.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(64.0400224211\)
Analytic rank: \(0\)
Dimension: \(37\)
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.

\(\operatorname{Tr}(f)(q) = \) \(37q \) \(\mathstrut +\mathstrut 3q^{3} \) \(\mathstrut +\mathstrut 37q^{5} \) \(\mathstrut +\mathstrut 4q^{7} \) \(\mathstrut +\mathstrut 50q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \(37q \) \(\mathstrut +\mathstrut 3q^{3} \) \(\mathstrut +\mathstrut 37q^{5} \) \(\mathstrut +\mathstrut 4q^{7} \) \(\mathstrut +\mathstrut 50q^{9} \) \(\mathstrut +\mathstrut 2q^{11} \) \(\mathstrut +\mathstrut 27q^{13} \) \(\mathstrut +\mathstrut 3q^{15} \) \(\mathstrut +\mathstrut 36q^{17} \) \(\mathstrut -\mathstrut 6q^{19} \) \(\mathstrut +\mathstrut 20q^{21} \) \(\mathstrut +\mathstrut 17q^{23} \) \(\mathstrut +\mathstrut 37q^{25} \) \(\mathstrut +\mathstrut 9q^{27} \) \(\mathstrut +\mathstrut 29q^{29} \) \(\mathstrut +\mathstrut 5q^{31} \) \(\mathstrut +\mathstrut 36q^{33} \) \(\mathstrut +\mathstrut 4q^{35} \) \(\mathstrut +\mathstrut 35q^{37} \) \(\mathstrut +\mathstrut 21q^{39} \) \(\mathstrut +\mathstrut 24q^{41} \) \(\mathstrut +\mathstrut 11q^{43} \) \(\mathstrut +\mathstrut 50q^{45} \) \(\mathstrut +\mathstrut 19q^{47} \) \(\mathstrut +\mathstrut 57q^{49} \) \(\mathstrut +\mathstrut 8q^{51} \) \(\mathstrut +\mathstrut 65q^{53} \) \(\mathstrut +\mathstrut 2q^{55} \) \(\mathstrut +\mathstrut 62q^{57} \) \(\mathstrut -\mathstrut 9q^{59} \) \(\mathstrut +\mathstrut 13q^{61} \) \(\mathstrut +\mathstrut 26q^{63} \) \(\mathstrut +\mathstrut 27q^{65} \) \(\mathstrut +\mathstrut 13q^{67} \) \(\mathstrut +\mathstrut 20q^{69} \) \(\mathstrut +\mathstrut 33q^{71} \) \(\mathstrut +\mathstrut 67q^{73} \) \(\mathstrut +\mathstrut 3q^{75} \) \(\mathstrut +\mathstrut 62q^{77} \) \(\mathstrut +\mathstrut 23q^{79} \) \(\mathstrut +\mathstrut 97q^{81} \) \(\mathstrut +\mathstrut 2q^{83} \) \(\mathstrut +\mathstrut 36q^{85} \) \(\mathstrut +\mathstrut 32q^{87} \) \(\mathstrut +\mathstrut 34q^{89} \) \(\mathstrut +\mathstrut q^{91} \) \(\mathstrut +\mathstrut 41q^{93} \) \(\mathstrut -\mathstrut 6q^{95} \) \(\mathstrut +\mathstrut 66q^{97} \) \(\mathstrut -\mathstrut 7q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 −3.33477 0 1.00000 0 2.42678 0 8.12069 0
1.2 0 −3.32921 0 1.00000 0 −3.52319 0 8.08363 0
1.3 0 −3.05263 0 1.00000 0 2.86058 0 6.31852 0
1.4 0 −2.74673 0 1.00000 0 −2.35693 0 4.54455 0
1.5 0 −2.47047 0 1.00000 0 −0.726513 0 3.10322 0
1.6 0 −2.35696 0 1.00000 0 −1.59035 0 2.55528 0
1.7 0 −2.32976 0 1.00000 0 −1.33564 0 2.42776 0
1.8 0 −2.32562 0 1.00000 0 −4.92659 0 2.40852 0
1.9 0 −2.16998 0 1.00000 0 4.79835 0 1.70880 0
1.10 0 −1.70544 0 1.00000 0 −1.03894 0 −0.0914601 0
1.11 0 −1.65793 0 1.00000 0 4.10503 0 −0.251260 0
1.12 0 −1.55757 0 1.00000 0 3.26992 0 −0.573979 0
1.13 0 −0.881355 0 1.00000 0 3.19359 0 −2.22321 0
1.14 0 −0.869740 0 1.00000 0 −1.85510 0 −2.24355 0
1.15 0 −0.446086 0 1.00000 0 3.25733 0 −2.80101 0
1.16 0 −0.435929 0 1.00000 0 −3.81252 0 −2.80997 0
1.17 0 −0.210422 0 1.00000 0 −4.28610 0 −2.95572 0
1.18 0 −0.161183 0 1.00000 0 −0.359157 0 −2.97402 0
1.19 0 0.332188 0 1.00000 0 0.507569 0 −2.88965 0
1.20 0 0.353843 0 1.00000 0 1.60118 0 −2.87480 0
See all 37 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.37
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(-1\)
\(401\) \(-1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{3}^{37} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8020))\).