Properties

Label 8020.2.a.e
Level 8020
Weight 2
Character orbit 8020.a
Self dual Yes
Analytic conductor 64.040
Analytic rank 0
Dimension 35
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: \(35\)
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.

\(\operatorname{Tr}(f)(q) = \) \(35q \) \(\mathstrut -\mathstrut q^{3} \) \(\mathstrut -\mathstrut 35q^{5} \) \(\mathstrut +\mathstrut 6q^{7} \) \(\mathstrut +\mathstrut 52q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \(35q \) \(\mathstrut -\mathstrut q^{3} \) \(\mathstrut -\mathstrut 35q^{5} \) \(\mathstrut +\mathstrut 6q^{7} \) \(\mathstrut +\mathstrut 52q^{9} \) \(\mathstrut -\mathstrut 2q^{11} \) \(\mathstrut -\mathstrut q^{13} \) \(\mathstrut +\mathstrut q^{15} \) \(\mathstrut +\mathstrut 18q^{17} \) \(\mathstrut +\mathstrut 2q^{19} \) \(\mathstrut +\mathstrut 12q^{21} \) \(\mathstrut +\mathstrut 13q^{23} \) \(\mathstrut +\mathstrut 35q^{25} \) \(\mathstrut -\mathstrut 7q^{27} \) \(\mathstrut +\mathstrut 25q^{29} \) \(\mathstrut +\mathstrut 13q^{31} \) \(\mathstrut +\mathstrut 14q^{33} \) \(\mathstrut -\mathstrut 6q^{35} \) \(\mathstrut -\mathstrut 19q^{37} \) \(\mathstrut -\mathstrut 3q^{39} \) \(\mathstrut +\mathstrut 24q^{41} \) \(\mathstrut -\mathstrut 5q^{43} \) \(\mathstrut -\mathstrut 52q^{45} \) \(\mathstrut +\mathstrut 19q^{47} \) \(\mathstrut +\mathstrut 55q^{49} \) \(\mathstrut +\mathstrut 41q^{53} \) \(\mathstrut +\mathstrut 2q^{55} \) \(\mathstrut +\mathstrut 14q^{57} \) \(\mathstrut +\mathstrut 3q^{59} \) \(\mathstrut +\mathstrut 13q^{61} \) \(\mathstrut +\mathstrut 70q^{63} \) \(\mathstrut +\mathstrut q^{65} \) \(\mathstrut -\mathstrut 17q^{67} \) \(\mathstrut +\mathstrut 64q^{69} \) \(\mathstrut +\mathstrut 17q^{71} \) \(\mathstrut -\mathstrut 63q^{73} \) \(\mathstrut -\mathstrut q^{75} \) \(\mathstrut +\mathstrut 54q^{77} \) \(\mathstrut +\mathstrut 11q^{79} \) \(\mathstrut +\mathstrut 107q^{81} \) \(\mathstrut -\mathstrut 8q^{83} \) \(\mathstrut -\mathstrut 18q^{85} \) \(\mathstrut +\mathstrut 36q^{87} \) \(\mathstrut +\mathstrut 38q^{89} \) \(\mathstrut -\mathstrut 27q^{91} \) \(\mathstrut +\mathstrut q^{93} \) \(\mathstrut -\mathstrut 2q^{95} \) \(\mathstrut -\mathstrut 54q^{97} \) \(\mathstrut -\mathstrut 51q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 −3.41155 0 −1.00000 0 0.311908 0 8.63864 0
1.2 0 −3.20562 0 −1.00000 0 −2.96453 0 7.27598 0
1.3 0 −3.08956 0 −1.00000 0 4.75064 0 6.54541 0
1.4 0 −2.93370 0 −1.00000 0 4.42731 0 5.60662 0
1.5 0 −2.91827 0 −1.00000 0 −4.31238 0 5.51629 0
1.6 0 −2.89929 0 −1.00000 0 2.68176 0 5.40586 0
1.7 0 −2.36832 0 −1.00000 0 −4.12967 0 2.60896 0
1.8 0 −2.25358 0 −1.00000 0 −0.859031 0 2.07862 0
1.9 0 −1.64682 0 −1.00000 0 1.19631 0 −0.287991 0
1.10 0 −1.51742 0 −1.00000 0 −1.44284 0 −0.697439 0
1.11 0 −1.37216 0 −1.00000 0 −1.82698 0 −1.11719 0
1.12 0 −1.36773 0 −1.00000 0 2.86190 0 −1.12932 0
1.13 0 −1.28894 0 −1.00000 0 0.0476003 0 −1.33863 0
1.14 0 −0.839936 0 −1.00000 0 3.45440 0 −2.29451 0
1.15 0 −0.713769 0 −1.00000 0 −1.26919 0 −2.49053 0
1.16 0 −0.505617 0 −1.00000 0 −0.615205 0 −2.74435 0
1.17 0 −0.478839 0 −1.00000 0 1.59155 0 −2.77071 0
1.18 0 −0.167962 0 −1.00000 0 2.31084 0 −2.97179 0
1.19 0 0.0559400 0 −1.00000 0 −0.736794 0 −2.99687 0
1.20 0 0.309116 0 −1.00000 0 −4.23374 0 −2.90445 0
See all 35 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.35
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 \(T_{3}^{35} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8020))\).