Properties

Label 8032.2.a.g
Level $8032$
Weight $2$
Character orbit 8032.a
Self dual yes
Analytic conductor $64.136$
Analytic rank $1$
Dimension $30$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8032 = 2^{5} \cdot 251 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8032.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(64.1358429035\)
Analytic rank: \(1\)
Dimension: \(30\)
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 30q - 9q^{3} - 5q^{7} + 35q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 30q - 9q^{3} - 5q^{7} + 35q^{9} - 31q^{11} + 5q^{13} - 8q^{15} - q^{17} - 29q^{19} + 6q^{21} - 27q^{23} + 34q^{25} - 36q^{27} + q^{29} + 9q^{31} - 8q^{33} - 51q^{35} + 5q^{37} + 8q^{39} - 3q^{41} - 43q^{43} - 42q^{47} + 41q^{49} - 54q^{51} - 11q^{53} + 10q^{55} + 2q^{57} - 49q^{59} + 21q^{61} - 15q^{63} - 14q^{65} - 68q^{67} - 10q^{69} - 44q^{71} + 25q^{73} - 51q^{75} - 20q^{77} + 49q^{79} + 6q^{81} - 88q^{83} - 5q^{85} - 16q^{87} - 16q^{89} - 46q^{91} - 4q^{93} - 28q^{95} - 4q^{97} - 71q^{99} + O(q^{100}) \)

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 −3.43537 0 2.52824 0 −2.67250 0 8.80174 0
1.2 0 −3.13996 0 −2.26457 0 2.38671 0 6.85933 0
1.3 0 −3.00046 0 2.74997 0 3.93356 0 6.00279 0
1.4 0 −2.99531 0 −2.48297 0 −3.43053 0 5.97187 0
1.5 0 −2.71488 0 −2.26191 0 1.04506 0 4.37055 0
1.6 0 −2.62556 0 1.77921 0 −2.01732 0 3.89356 0
1.7 0 −2.20649 0 2.47534 0 −0.802937 0 1.86861 0
1.8 0 −2.04450 0 0.222963 0 −5.15673 0 1.17996 0
1.9 0 −1.89707 0 −3.37686 0 3.27287 0 0.598876 0
1.10 0 −1.77983 0 3.93815 0 −3.71151 0 0.167788 0
1.11 0 −1.73507 0 −3.61768 0 3.69185 0 0.0104735 0
1.12 0 −1.21978 0 2.91757 0 −2.62203 0 −1.51215 0
1.13 0 −0.933880 0 0.0426876 0 2.12724 0 −2.12787 0
1.14 0 −0.824754 0 −2.00155 0 −1.90493 0 −2.31978 0
1.15 0 −0.692263 0 3.18782 0 3.79509 0 −2.52077 0
1.16 0 −0.369540 0 −3.94367 0 −3.05773 0 −2.86344 0
1.17 0 −0.193782 0 1.98943 0 −0.152201 0 −2.96245 0
1.18 0 0.222021 0 0.0345121 0 5.16899 0 −2.95071 0
1.19 0 0.685335 0 −1.94665 0 −1.34986 0 −2.53032 0
1.20 0 1.13597 0 −1.24613 0 −4.57789 0 −1.70957 0
See all 30 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.30
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(251\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8032.2.a.g 30
4.b odd 2 1 8032.2.a.j yes 30
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8032.2.a.g 30 1.a even 1 1 trivial
8032.2.a.j yes 30 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8032))\):

\(T_{3}^{30} + \cdots\)
\(T_{7}^{30} + \cdots\)
\(T_{11}^{30} + \cdots\)

Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database