Properties

Label 7056.2.a.co
Level $7056$
Weight $2$
Character orbit 7056.a
Self dual yes
Analytic conductor $56.342$
Analytic rank $1$
Dimension $2$
CM discriminant -7
Inner twists $4$

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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: \(2\)
Coefficient field: \(\Q(\sqrt{7}) \)
Defining polynomial: \(x^{2} - 7\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 441)
Fricke sign: \(1\)
Sato-Tate group: $N(\mathrm{U}(1))$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{7}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q +O(q^{10})\) \( q + \beta q^{11} -\beta q^{23} -5 q^{25} -2 \beta q^{29} + 6 q^{37} -12 q^{43} -2 \beta q^{53} -4 q^{67} + \beta q^{71} -8 q^{79} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + O(q^{10}) \) \( 2q - 10q^{25} + 12q^{37} - 24q^{43} - 8q^{67} - 16q^{79} + O(q^{100}) \)

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
−2.64575
2.64575
0 0 0 0 0 0 0 0 0
1.2 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
7.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)
3.b odd 2 1 inner
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7056.2.a.co 2
3.b odd 2 1 inner 7056.2.a.co 2
4.b odd 2 1 441.2.a.h 2
7.b odd 2 1 CM 7056.2.a.co 2
12.b even 2 1 441.2.a.h 2
21.c even 2 1 inner 7056.2.a.co 2
28.d even 2 1 441.2.a.h 2
28.f even 6 2 441.2.e.h 4
28.g odd 6 2 441.2.e.h 4
84.h odd 2 1 441.2.a.h 2
84.j odd 6 2 441.2.e.h 4
84.n even 6 2 441.2.e.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
441.2.a.h 2 4.b odd 2 1
441.2.a.h 2 12.b even 2 1
441.2.a.h 2 28.d even 2 1
441.2.a.h 2 84.h odd 2 1
441.2.e.h 4 28.f even 6 2
441.2.e.h 4 28.g odd 6 2
441.2.e.h 4 84.j odd 6 2
441.2.e.h 4 84.n even 6 2
7056.2.a.co 2 1.a even 1 1 trivial
7056.2.a.co 2 3.b odd 2 1 inner
7056.2.a.co 2 7.b odd 2 1 CM
7056.2.a.co 2 21.c even 2 1 inner

Hecke kernels

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

\( T_{5} \)
\( T_{11}^{2} - 28 \)
\( T_{13} \)
\( T_{17} \)
\( T_{23}^{2} - 28 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( T^{2} \)
$11$ \( -28 + T^{2} \)
$13$ \( T^{2} \)
$17$ \( T^{2} \)
$19$ \( T^{2} \)
$23$ \( -28 + T^{2} \)
$29$ \( -112 + T^{2} \)
$31$ \( T^{2} \)
$37$ \( ( -6 + T )^{2} \)
$41$ \( T^{2} \)
$43$ \( ( 12 + T )^{2} \)
$47$ \( T^{2} \)
$53$ \( -112 + T^{2} \)
$59$ \( T^{2} \)
$61$ \( T^{2} \)
$67$ \( ( 4 + T )^{2} \)
$71$ \( -28 + T^{2} \)
$73$ \( T^{2} \)
$79$ \( ( 8 + T )^{2} \)
$83$ \( T^{2} \)
$89$ \( T^{2} \)
$97$ \( T^{2} \)
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