Properties

Label 7056.2.b.s
Level 7056
Weight 2
Character orbit 7056.b
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.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(56.3424436662\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{7})\)
Defining polynomial: \(x^{4} + 7 x^{2} + 49\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 112)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{5} +O(q^{10})\) \( q + \beta_{2} q^{5} + \beta_{3} q^{11} + 2 \beta_{2} q^{13} + 3 \beta_{2} q^{17} -\beta_{1} q^{19} -\beta_{3} q^{23} + 2 q^{25} + \beta_{1} q^{31} + 7 q^{37} + 2 \beta_{2} q^{41} + 2 \beta_{3} q^{43} -3 \beta_{1} q^{47} -3 q^{53} + 3 \beta_{1} q^{55} -3 \beta_{1} q^{59} + \beta_{2} q^{61} -6 q^{65} + \beta_{3} q^{67} -2 \beta_{3} q^{71} + 3 \beta_{2} q^{73} + \beta_{3} q^{79} -9 q^{85} -\beta_{2} q^{89} + \beta_{3} q^{95} -2 \beta_{2} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q + 8q^{25} + 28q^{37} - 12q^{53} - 24q^{65} - 36q^{85} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 7 x^{2} + 49\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{3} \)\(/7\)
\(\beta_{2}\)\(=\)\((\)\( 2 \nu^{2} + 7 \)\()/7\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{3} + 14 \nu \)\()/7\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} - \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(7 \beta_{2} - 7\)\()/2\)
\(\nu^{3}\)\(=\)\(7 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/7056\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1765\) \(4609\) \(6175\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1567.1
1.32288 2.29129i
−1.32288 + 2.29129i
−1.32288 2.29129i
1.32288 + 2.29129i
0 0 0 1.73205i 0 0 0 0 0
1567.2 0 0 0 1.73205i 0 0 0 0 0
1567.3 0 0 0 1.73205i 0 0 0 0 0
1567.4 0 0 0 1.73205i 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
4.b odd 2 1 inner
7.b odd 2 1 inner
28.d 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.b.s 4
3.b odd 2 1 784.2.f.d 4
4.b odd 2 1 inner 7056.2.b.s 4
7.b odd 2 1 inner 7056.2.b.s 4
7.c even 3 1 1008.2.cs.q 4
7.d odd 6 1 1008.2.cs.q 4
12.b even 2 1 784.2.f.d 4
21.c even 2 1 784.2.f.d 4
21.g even 6 1 112.2.p.c 4
21.g even 6 1 784.2.p.g 4
21.h odd 6 1 112.2.p.c 4
21.h odd 6 1 784.2.p.g 4
24.f even 2 1 3136.2.f.f 4
24.h odd 2 1 3136.2.f.f 4
28.d even 2 1 inner 7056.2.b.s 4
28.f even 6 1 1008.2.cs.q 4
28.g odd 6 1 1008.2.cs.q 4
84.h odd 2 1 784.2.f.d 4
84.j odd 6 1 112.2.p.c 4
84.j odd 6 1 784.2.p.g 4
84.n even 6 1 112.2.p.c 4
84.n even 6 1 784.2.p.g 4
168.e odd 2 1 3136.2.f.f 4
168.i even 2 1 3136.2.f.f 4
168.s odd 6 1 448.2.p.c 4
168.v even 6 1 448.2.p.c 4
168.ba even 6 1 448.2.p.c 4
168.be odd 6 1 448.2.p.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
112.2.p.c 4 21.g even 6 1
112.2.p.c 4 21.h odd 6 1
112.2.p.c 4 84.j odd 6 1
112.2.p.c 4 84.n even 6 1
448.2.p.c 4 168.s odd 6 1
448.2.p.c 4 168.v even 6 1
448.2.p.c 4 168.ba even 6 1
448.2.p.c 4 168.be odd 6 1
784.2.f.d 4 3.b odd 2 1
784.2.f.d 4 12.b even 2 1
784.2.f.d 4 21.c even 2 1
784.2.f.d 4 84.h odd 2 1
784.2.p.g 4 21.g even 6 1
784.2.p.g 4 21.h odd 6 1
784.2.p.g 4 84.j odd 6 1
784.2.p.g 4 84.n even 6 1
1008.2.cs.q 4 7.c even 3 1
1008.2.cs.q 4 7.d odd 6 1
1008.2.cs.q 4 28.f even 6 1
1008.2.cs.q 4 28.g odd 6 1
3136.2.f.f 4 24.f even 2 1
3136.2.f.f 4 24.h odd 2 1
3136.2.f.f 4 168.e odd 2 1
3136.2.f.f 4 168.i even 2 1
7056.2.b.s 4 1.a even 1 1 trivial
7056.2.b.s 4 4.b odd 2 1 inner
7056.2.b.s 4 7.b odd 2 1 inner
7056.2.b.s 4 28.d 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 \(S_{2}^{\mathrm{new}}(7056, [\chi])\):

\( T_{5}^{2} + 3 \)
\( T_{11}^{2} + 21 \)
\( T_{13}^{2} + 12 \)
\( T_{17}^{2} + 27 \)
\( T_{19}^{2} - 7 \)
\( T_{31}^{2} - 7 \)
\( T_{53} + 3 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( ( 1 - 7 T^{2} + 25 T^{4} )^{2} \)
$7$ 1
$11$ \( ( 1 - T^{2} + 121 T^{4} )^{2} \)
$13$ \( ( 1 - 14 T^{2} + 169 T^{4} )^{2} \)
$17$ \( ( 1 - 7 T^{2} + 289 T^{4} )^{2} \)
$19$ \( ( 1 + 31 T^{2} + 361 T^{4} )^{2} \)
$23$ \( ( 1 - 25 T^{2} + 529 T^{4} )^{2} \)
$29$ \( ( 1 + 29 T^{2} )^{4} \)
$31$ \( ( 1 + 55 T^{2} + 961 T^{4} )^{2} \)
$37$ \( ( 1 - 7 T + 37 T^{2} )^{4} \)
$41$ \( ( 1 - 70 T^{2} + 1681 T^{4} )^{2} \)
$43$ \( ( 1 - 2 T^{2} + 1849 T^{4} )^{2} \)
$47$ \( ( 1 + 31 T^{2} + 2209 T^{4} )^{2} \)
$53$ \( ( 1 + 3 T + 53 T^{2} )^{4} \)
$59$ \( ( 1 + 55 T^{2} + 3481 T^{4} )^{2} \)
$61$ \( ( 1 - 119 T^{2} + 3721 T^{4} )^{2} \)
$67$ \( ( 1 - 113 T^{2} + 4489 T^{4} )^{2} \)
$71$ \( ( 1 - 58 T^{2} + 5041 T^{4} )^{2} \)
$73$ \( ( 1 - 119 T^{2} + 5329 T^{4} )^{2} \)
$79$ \( ( 1 - 137 T^{2} + 6241 T^{4} )^{2} \)
$83$ \( ( 1 + 83 T^{2} )^{4} \)
$89$ \( ( 1 - 175 T^{2} + 7921 T^{4} )^{2} \)
$97$ \( ( 1 - 182 T^{2} + 9409 T^{4} )^{2} \)
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