Properties

Label 8020.2.a.d
Level 8020
Weight 2
Character orbit 8020.a
Self dual Yes
Analytic conductor 64.040
Analytic rank 1
Dimension 29
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: \(1\)
Dimension: \(29\)
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.

\(\operatorname{Tr}(f)(q) = \) \(29q \) \(\mathstrut -\mathstrut 3q^{3} \) \(\mathstrut +\mathstrut 29q^{5} \) \(\mathstrut -\mathstrut 8q^{7} \) \(\mathstrut +\mathstrut 10q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \(29q \) \(\mathstrut -\mathstrut 3q^{3} \) \(\mathstrut +\mathstrut 29q^{5} \) \(\mathstrut -\mathstrut 8q^{7} \) \(\mathstrut +\mathstrut 10q^{9} \) \(\mathstrut +\mathstrut 2q^{11} \) \(\mathstrut -\mathstrut 23q^{13} \) \(\mathstrut -\mathstrut 3q^{15} \) \(\mathstrut -\mathstrut 30q^{17} \) \(\mathstrut -\mathstrut 6q^{19} \) \(\mathstrut -\mathstrut 16q^{21} \) \(\mathstrut -\mathstrut 21q^{23} \) \(\mathstrut +\mathstrut 29q^{25} \) \(\mathstrut -\mathstrut 15q^{27} \) \(\mathstrut -\mathstrut 35q^{29} \) \(\mathstrut -\mathstrut 7q^{31} \) \(\mathstrut -\mathstrut 36q^{33} \) \(\mathstrut -\mathstrut 8q^{35} \) \(\mathstrut -\mathstrut 31q^{37} \) \(\mathstrut -\mathstrut 11q^{39} \) \(\mathstrut -\mathstrut 24q^{41} \) \(\mathstrut -\mathstrut 17q^{43} \) \(\mathstrut +\mathstrut 10q^{45} \) \(\mathstrut -\mathstrut 17q^{47} \) \(\mathstrut +\mathstrut q^{49} \) \(\mathstrut +\mathstrut 8q^{51} \) \(\mathstrut -\mathstrut 57q^{53} \) \(\mathstrut +\mathstrut 2q^{55} \) \(\mathstrut -\mathstrut 46q^{57} \) \(\mathstrut -\mathstrut 9q^{59} \) \(\mathstrut -\mathstrut 27q^{61} \) \(\mathstrut -\mathstrut 34q^{63} \) \(\mathstrut -\mathstrut 23q^{65} \) \(\mathstrut -\mathstrut 21q^{67} \) \(\mathstrut -\mathstrut 28q^{69} \) \(\mathstrut -\mathstrut 19q^{71} \) \(\mathstrut -\mathstrut 81q^{73} \) \(\mathstrut -\mathstrut 3q^{75} \) \(\mathstrut -\mathstrut 66q^{77} \) \(\mathstrut -\mathstrut 17q^{79} \) \(\mathstrut -\mathstrut 39q^{81} \) \(\mathstrut -\mathstrut 30q^{83} \) \(\mathstrut -\mathstrut 30q^{85} \) \(\mathstrut -\mathstrut 20q^{87} \) \(\mathstrut -\mathstrut 38q^{89} \) \(\mathstrut +\mathstrut q^{91} \) \(\mathstrut -\mathstrut 75q^{93} \) \(\mathstrut -\mathstrut 6q^{95} \) \(\mathstrut -\mathstrut 48q^{97} \) \(\mathstrut -\mathstrut 7q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 −3.11347 0 1.00000 0 3.18726 0 6.69370 0
1.2 0 −2.75932 0 1.00000 0 −2.48714 0 4.61387 0
1.3 0 −2.73282 0 1.00000 0 1.81499 0 4.46832 0
1.4 0 −2.62437 0 1.00000 0 −4.07774 0 3.88732 0
1.5 0 −2.50668 0 1.00000 0 −0.362320 0 3.28346 0
1.6 0 −2.25625 0 1.00000 0 0.417578 0 2.09065 0
1.7 0 −1.73579 0 1.00000 0 2.72114 0 0.0129592 0
1.8 0 −1.59991 0 1.00000 0 −1.91437 0 −0.440296 0
1.9 0 −1.35123 0 1.00000 0 0.606242 0 −1.17417 0
1.10 0 −1.12115 0 1.00000 0 0.722262 0 −1.74302 0
1.11 0 −0.848679 0 1.00000 0 −3.71403 0 −2.27974 0
1.12 0 −0.697718 0 1.00000 0 −4.99015 0 −2.51319 0
1.13 0 −0.537747 0 1.00000 0 −1.46052 0 −2.71083 0
1.14 0 −0.202645 0 1.00000 0 1.27188 0 −2.95893 0
1.15 0 −0.107530 0 1.00000 0 2.68164 0 −2.98844 0
1.16 0 0.0219398 0 1.00000 0 2.04593 0 −2.99952 0
1.17 0 0.121745 0 1.00000 0 5.25775 0 −2.98518 0
1.18 0 0.358058 0 1.00000 0 1.78556 0 −2.87179 0
1.19 0 0.540664 0 1.00000 0 −1.71282 0 −2.70768 0
1.20 0 1.03867 0 1.00000 0 −0.0217287 0 −1.92117 0
See all 29 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.29
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 \(T_{3}^{29} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8020))\).