Properties

Label 7056.2.b.v
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: \(\Q(\sqrt{-3}, \sqrt{5})\)
Defining polynomial: \(x^{4} - x^{3} + 2 x^{2} + x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{4}\cdot 3 \)
Twist minimal: no (minimal twist has level 1008)
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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{5} +O(q^{10})\) \( q + \beta_{2} q^{5} + \beta_{2} q^{11} -2 \beta_{1} q^{13} + 4 q^{19} + 2 \beta_{2} q^{23} -10 q^{25} -\beta_{3} q^{29} - q^{31} + 4 q^{37} + 2 \beta_{2} q^{41} -4 \beta_{1} q^{43} -2 \beta_{3} q^{47} + \beta_{3} q^{53} -15 q^{55} -\beta_{3} q^{59} + 6 \beta_{1} q^{61} + 2 \beta_{3} q^{65} -4 \beta_{1} q^{67} -2 \beta_{2} q^{71} + 4 \beta_{1} q^{73} -7 \beta_{1} q^{79} -\beta_{3} q^{83} + 2 \beta_{2} q^{89} + 4 \beta_{2} q^{95} + 3 \beta_{1} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q + 16q^{19} - 40q^{25} - 4q^{31} + 16q^{37} - 60q^{55} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} + 2 x^{2} + x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( -\nu^{3} + 2 \nu^{2} - 2 \nu \)
\(\beta_{2}\)\(=\)\( 2 \nu^{3} - 2 \nu^{2} + 6 \nu + 1 \)
\(\beta_{3}\)\(=\)\( -3 \nu^{3} - 6 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + 3 \beta_{2} + 3 \beta_{1} + 3\)\()/12\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{3} + 3 \beta_{2} + 9 \beta_{1} - 9\)\()/12\)
\(\nu^{3}\)\(=\)\((\)\(-\beta_{3} - 6\)\()/3\)

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
−0.309017 0.535233i
0.809017 1.40126i
0.809017 + 1.40126i
−0.309017 + 0.535233i
0 0 0 3.87298i 0 0 0 0 0
1567.2 0 0 0 3.87298i 0 0 0 0 0
1567.3 0 0 0 3.87298i 0 0 0 0 0
1567.4 0 0 0 3.87298i 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
28.d even 2 1 inner
84.h odd 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.v 4
3.b odd 2 1 inner 7056.2.b.v 4
4.b odd 2 1 7056.2.b.o 4
7.b odd 2 1 7056.2.b.o 4
7.c even 3 1 1008.2.cs.o 4
7.d odd 6 1 1008.2.cs.p yes 4
12.b even 2 1 7056.2.b.o 4
21.c even 2 1 7056.2.b.o 4
21.g even 6 1 1008.2.cs.p yes 4
21.h odd 6 1 1008.2.cs.o 4
28.d even 2 1 inner 7056.2.b.v 4
28.f even 6 1 1008.2.cs.o 4
28.g odd 6 1 1008.2.cs.p yes 4
84.h odd 2 1 inner 7056.2.b.v 4
84.j odd 6 1 1008.2.cs.o 4
84.n even 6 1 1008.2.cs.p yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1008.2.cs.o 4 7.c even 3 1
1008.2.cs.o 4 21.h odd 6 1
1008.2.cs.o 4 28.f even 6 1
1008.2.cs.o 4 84.j odd 6 1
1008.2.cs.p yes 4 7.d odd 6 1
1008.2.cs.p yes 4 21.g even 6 1
1008.2.cs.p yes 4 28.g odd 6 1
1008.2.cs.p yes 4 84.n even 6 1
7056.2.b.o 4 4.b odd 2 1
7056.2.b.o 4 7.b odd 2 1
7056.2.b.o 4 12.b even 2 1
7056.2.b.o 4 21.c even 2 1
7056.2.b.v 4 1.a even 1 1 trivial
7056.2.b.v 4 3.b odd 2 1 inner
7056.2.b.v 4 28.d even 2 1 inner
7056.2.b.v 4 84.h odd 2 1 inner

Hecke kernels

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

\( T_{5}^{2} + 15 \)
\( T_{11}^{2} + 15 \)
\( T_{13}^{2} + 12 \)
\( T_{17} \)
\( T_{19} - 4 \)
\( T_{31} + 1 \)
\( T_{53}^{2} - 45 \)