Properties

Label 7056.2.a.cy
Level 7056
Weight 2
Character orbit 7056.a
Self dual yes
Analytic conductor 56.342
Analytic rank 0
Dimension 4
CM no
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: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{7})\)
Defining polynomial: \(x^{4} - 8 x^{2} + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 1764)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{5} +O(q^{10})\) \( q + \beta_{2} q^{5} + \beta_{3} q^{11} -3 \beta_{1} q^{13} + \beta_{2} q^{17} -2 \beta_{1} q^{19} -\beta_{3} q^{23} + 9 q^{25} + \beta_{3} q^{29} -6 \beta_{1} q^{31} + 4 q^{37} + \beta_{2} q^{41} -8 q^{43} + 2 \beta_{2} q^{47} + 2 \beta_{3} q^{53} + 14 \beta_{1} q^{55} + 2 \beta_{2} q^{59} + 7 \beta_{1} q^{61} -3 \beta_{3} q^{65} -12 q^{67} + 3 \beta_{3} q^{71} + \beta_{1} q^{73} + 4 q^{79} + 4 \beta_{2} q^{83} + 14 q^{85} -\beta_{2} q^{89} -2 \beta_{3} q^{95} -7 \beta_{1} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q + 36q^{25} + 16q^{37} - 32q^{43} - 48q^{67} + 16q^{79} + 56q^{85} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 8 x^{2} + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} - 5 \nu \)\()/3\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{3} + 11 \nu \)\()/3\)
\(\beta_{3}\)\(=\)\( 2 \nu^{2} - 8 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} + \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} + 8\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(5 \beta_{2} + 11 \beta_{1}\)\()/2\)

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
−1.16372
−2.57794
1.16372
2.57794
0 0 0 −3.74166 0 0 0 0 0
1.2 0 0 0 −3.74166 0 0 0 0 0
1.3 0 0 0 3.74166 0 0 0 0 0
1.4 0 0 0 3.74166 0 0 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.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.cy 4
3.b odd 2 1 inner 7056.2.a.cy 4
4.b odd 2 1 1764.2.a.m 4
7.b odd 2 1 inner 7056.2.a.cy 4
12.b even 2 1 1764.2.a.m 4
21.c even 2 1 inner 7056.2.a.cy 4
28.d even 2 1 1764.2.a.m 4
28.f even 6 2 1764.2.k.m 8
28.g odd 6 2 1764.2.k.m 8
84.h odd 2 1 1764.2.a.m 4
84.j odd 6 2 1764.2.k.m 8
84.n even 6 2 1764.2.k.m 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1764.2.a.m 4 4.b odd 2 1
1764.2.a.m 4 12.b even 2 1
1764.2.a.m 4 28.d even 2 1
1764.2.a.m 4 84.h odd 2 1
1764.2.k.m 8 28.f even 6 2
1764.2.k.m 8 28.g odd 6 2
1764.2.k.m 8 84.j odd 6 2
1764.2.k.m 8 84.n even 6 2
7056.2.a.cy 4 1.a even 1 1 trivial
7056.2.a.cy 4 3.b odd 2 1 inner
7056.2.a.cy 4 7.b odd 2 1 inner
7056.2.a.cy 4 21.c even 2 1 inner

Atkin-Lehner signs

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

Hecke kernels

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

\( T_{5}^{2} - 14 \)
\( T_{11}^{2} - 28 \)
\( T_{13}^{2} - 18 \)
\( T_{17}^{2} - 14 \)
\( T_{23}^{2} - 28 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( ( 1 - 4 T^{2} + 25 T^{4} )^{2} \)
$7$ 1
$11$ \( ( 1 - 6 T^{2} + 121 T^{4} )^{2} \)
$13$ \( ( 1 + 8 T^{2} + 169 T^{4} )^{2} \)
$17$ \( ( 1 + 20 T^{2} + 289 T^{4} )^{2} \)
$19$ \( ( 1 + 30 T^{2} + 361 T^{4} )^{2} \)
$23$ \( ( 1 + 18 T^{2} + 529 T^{4} )^{2} \)
$29$ \( ( 1 + 30 T^{2} + 841 T^{4} )^{2} \)
$31$ \( ( 1 - 10 T^{2} + 961 T^{4} )^{2} \)
$37$ \( ( 1 - 4 T + 37 T^{2} )^{4} \)
$41$ \( ( 1 + 68 T^{2} + 1681 T^{4} )^{2} \)
$43$ \( ( 1 + 8 T + 43 T^{2} )^{4} \)
$47$ \( ( 1 + 38 T^{2} + 2209 T^{4} )^{2} \)
$53$ \( ( 1 - 6 T^{2} + 2809 T^{4} )^{2} \)
$59$ \( ( 1 + 62 T^{2} + 3481 T^{4} )^{2} \)
$61$ \( ( 1 + 24 T^{2} + 3721 T^{4} )^{2} \)
$67$ \( ( 1 + 12 T + 67 T^{2} )^{4} \)
$71$ \( ( 1 - 110 T^{2} + 5041 T^{4} )^{2} \)
$73$ \( ( 1 + 144 T^{2} + 5329 T^{4} )^{2} \)
$79$ \( ( 1 - 4 T + 79 T^{2} )^{4} \)
$83$ \( ( 1 - 58 T^{2} + 6889 T^{4} )^{2} \)
$89$ \( ( 1 + 164 T^{2} + 7921 T^{4} )^{2} \)
$97$ \( ( 1 + 96 T^{2} + 9409 T^{4} )^{2} \)
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