Properties

Label 7056.2.a.bf
Level $7056$
Weight $2$
Character orbit 7056.a
Self dual yes
Analytic conductor $56.342$
Analytic rank $1$
Dimension $1$
CM discriminant -3
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 7056 = 2^{4} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7056.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(56.3424436662\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 63)
Fricke sign: \(1\)
Sato-Tate group: $N(\mathrm{U}(1))$

$q$-expansion

\(f(q)\) \(=\) \( q + O(q^{10}) \) \( q + 7q^{13} - 7q^{19} - 5q^{25} - 7q^{31} - q^{37} - 5q^{43} - 14q^{61} - 11q^{67} + 7q^{73} + 13q^{79} - 14q^{97} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
0 0 0 0 0 0 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(7\) \(-1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7056.2.a.bf 1
3.b odd 2 1 CM 7056.2.a.bf 1
4.b odd 2 1 441.2.a.e 1
7.b odd 2 1 7056.2.a.y 1
7.d odd 6 2 1008.2.s.j 2
12.b even 2 1 441.2.a.e 1
21.c even 2 1 7056.2.a.y 1
21.g even 6 2 1008.2.s.j 2
28.d even 2 1 441.2.a.d 1
28.f even 6 2 63.2.e.a 2
28.g odd 6 2 441.2.e.c 2
84.h odd 2 1 441.2.a.d 1
84.j odd 6 2 63.2.e.a 2
84.n even 6 2 441.2.e.c 2
252.n even 6 2 567.2.g.d 2
252.r odd 6 2 567.2.h.c 2
252.bj even 6 2 567.2.h.c 2
252.bn odd 6 2 567.2.g.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.e.a 2 28.f even 6 2
63.2.e.a 2 84.j odd 6 2
441.2.a.d 1 28.d even 2 1
441.2.a.d 1 84.h odd 2 1
441.2.a.e 1 4.b odd 2 1
441.2.a.e 1 12.b even 2 1
441.2.e.c 2 28.g odd 6 2
441.2.e.c 2 84.n even 6 2
567.2.g.d 2 252.n even 6 2
567.2.g.d 2 252.bn odd 6 2
567.2.h.c 2 252.r odd 6 2
567.2.h.c 2 252.bj even 6 2
1008.2.s.j 2 7.d odd 6 2
1008.2.s.j 2 21.g even 6 2
7056.2.a.y 1 7.b odd 2 1
7056.2.a.y 1 21.c even 2 1
7056.2.a.bf 1 1.a even 1 1 trivial
7056.2.a.bf 1 3.b odd 2 1 CM

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7056))\):

\( T_{5} \)
\( T_{11} \)
\( T_{13} - 7 \)
\( T_{17} \)
\( T_{23} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( T \)
$5$ \( T \)
$7$ \( T \)
$11$ \( T \)
$13$ \( -7 + T \)
$17$ \( T \)
$19$ \( 7 + T \)
$23$ \( T \)
$29$ \( T \)
$31$ \( 7 + T \)
$37$ \( 1 + T \)
$41$ \( T \)
$43$ \( 5 + T \)
$47$ \( T \)
$53$ \( T \)
$59$ \( T \)
$61$ \( 14 + T \)
$67$ \( 11 + T \)
$71$ \( T \)
$73$ \( -7 + T \)
$79$ \( -13 + T \)
$83$ \( T \)
$89$ \( T \)
$97$ \( 14 + T \)
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