Properties

Label 8032.2.a.j
Level $8032$
Weight $2$
Character orbit 8032.a
Self dual yes
Analytic conductor $64.136$
Analytic rank $0$
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: \(0\)
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 −3.01325 0 0.418153 0 −0.933114 0 6.07967 0
1.2 0 −2.74123 0 −4.14653 0 −3.66203 0 4.51434 0
1.3 0 −2.60599 0 0.822843 0 3.53536 0 3.79120 0
1.4 0 −2.50495 0 0.621089 0 1.81379 0 3.27480 0
1.5 0 −2.26361 0 −0.762700 0 −1.25553 0 2.12395 0
1.6 0 −1.83124 0 4.45088 0 3.13811 0 0.353447 0
1.7 0 −1.73095 0 0.367690 0 0.998919 0 −0.00381480 0
1.8 0 −1.62030 0 −2.98882 0 −1.58362 0 −0.374619 0
1.9 0 −1.26148 0 2.94335 0 −0.461818 0 −1.40867 0
1.10 0 −1.19215 0 −0.449856 0 −2.62487 0 −1.57879 0
1.11 0 −1.13597 0 −1.24613 0 4.57789 0 −1.70957 0
1.12 0 −0.685335 0 −1.94665 0 1.34986 0 −2.53032 0
1.13 0 −0.222021 0 0.0345121 0 −5.16899 0 −2.95071 0
1.14 0 0.193782 0 1.98943 0 0.152201 0 −2.96245 0
1.15 0 0.369540 0 −3.94367 0 3.05773 0 −2.86344 0
1.16 0 0.692263 0 3.18782 0 −3.79509 0 −2.52077 0
1.17 0 0.824754 0 −2.00155 0 1.90493 0 −2.31978 0
1.18 0 0.933880 0 0.0426876 0 −2.12724 0 −2.12787 0
1.19 0 1.21978 0 2.91757 0 2.62203 0 −1.51215 0
1.20 0 1.73507 0 −3.61768 0 −3.69185 0 0.0104735 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.j yes 30
4.b odd 2 1 8032.2.a.g 30
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8032.2.a.g 30 4.b odd 2 1
8032.2.a.j yes 30 1.a even 1 1 trivial

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