Properties

Label 7056.2.a.bg
Level $7056$
Weight $2$
Character orbit 7056.a
Self dual yes
Analytic conductor $56.342$
Analytic rank $0$
Dimension $1$
CM discriminant -7
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [7056,2,Mod(1,7056)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7056, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("7056.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7056 = 2^{4} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7056.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(56.3424436662\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 49)
Fricke sign: \(-1\)
Sato-Tate group: $N(\mathrm{U}(1))$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q+O(q^{10}) \) Copy content Toggle raw display \( q + 4 q^{11} + 8 q^{23} - 5 q^{25} - 2 q^{29} - 6 q^{37} + 12 q^{43} + 10 q^{53} - 4 q^{67} + 16 q^{71} - 8 q^{79}+O(q^{100}) \) Copy content Toggle raw display

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
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
7.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7056.2.a.bg 1
3.b odd 2 1 784.2.a.f 1
4.b odd 2 1 441.2.a.c 1
7.b odd 2 1 CM 7056.2.a.bg 1
12.b even 2 1 49.2.a.a 1
21.c even 2 1 784.2.a.f 1
21.g even 6 2 784.2.i.f 2
21.h odd 6 2 784.2.i.f 2
24.f even 2 1 3136.2.a.n 1
24.h odd 2 1 3136.2.a.o 1
28.d even 2 1 441.2.a.c 1
28.f even 6 2 441.2.e.d 2
28.g odd 6 2 441.2.e.d 2
60.h even 2 1 1225.2.a.c 1
60.l odd 4 2 1225.2.b.c 2
84.h odd 2 1 49.2.a.a 1
84.j odd 6 2 49.2.c.a 2
84.n even 6 2 49.2.c.a 2
132.d odd 2 1 5929.2.a.c 1
156.h even 2 1 8281.2.a.d 1
168.e odd 2 1 3136.2.a.n 1
168.i even 2 1 3136.2.a.o 1
420.o odd 2 1 1225.2.a.c 1
420.w even 4 2 1225.2.b.c 2
924.n even 2 1 5929.2.a.c 1
1092.d odd 2 1 8281.2.a.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
49.2.a.a 1 12.b even 2 1
49.2.a.a 1 84.h odd 2 1
49.2.c.a 2 84.j odd 6 2
49.2.c.a 2 84.n even 6 2
441.2.a.c 1 4.b odd 2 1
441.2.a.c 1 28.d even 2 1
441.2.e.d 2 28.f even 6 2
441.2.e.d 2 28.g odd 6 2
784.2.a.f 1 3.b odd 2 1
784.2.a.f 1 21.c even 2 1
784.2.i.f 2 21.g even 6 2
784.2.i.f 2 21.h odd 6 2
1225.2.a.c 1 60.h even 2 1
1225.2.a.c 1 420.o odd 2 1
1225.2.b.c 2 60.l odd 4 2
1225.2.b.c 2 420.w even 4 2
3136.2.a.n 1 24.f even 2 1
3136.2.a.n 1 168.e odd 2 1
3136.2.a.o 1 24.h odd 2 1
3136.2.a.o 1 168.i even 2 1
5929.2.a.c 1 132.d odd 2 1
5929.2.a.c 1 924.n even 2 1
7056.2.a.bg 1 1.a even 1 1 trivial
7056.2.a.bg 1 7.b odd 2 1 CM
8281.2.a.d 1 156.h even 2 1
8281.2.a.d 1 1092.d odd 2 1

Hecke kernels

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

\( T_{5} \) Copy content Toggle raw display
\( T_{11} - 4 \) Copy content Toggle raw display
\( T_{13} \) Copy content Toggle raw display
\( T_{17} \) Copy content Toggle raw display
\( T_{23} - 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T - 4 \) Copy content Toggle raw display
$13$ \( T \) Copy content Toggle raw display
$17$ \( T \) Copy content Toggle raw display
$19$ \( T \) Copy content Toggle raw display
$23$ \( T - 8 \) Copy content Toggle raw display
$29$ \( T + 2 \) Copy content Toggle raw display
$31$ \( T \) Copy content Toggle raw display
$37$ \( T + 6 \) Copy content Toggle raw display
$41$ \( T \) Copy content Toggle raw display
$43$ \( T - 12 \) Copy content Toggle raw display
$47$ \( T \) Copy content Toggle raw display
$53$ \( T - 10 \) Copy content Toggle raw display
$59$ \( T \) Copy content Toggle raw display
$61$ \( T \) Copy content Toggle raw display
$67$ \( T + 4 \) Copy content Toggle raw display
$71$ \( T - 16 \) Copy content Toggle raw display
$73$ \( T \) Copy content Toggle raw display
$79$ \( T + 8 \) Copy content Toggle raw display
$83$ \( T \) Copy content Toggle raw display
$89$ \( T \) Copy content Toggle raw display
$97$ \( T \) Copy content Toggle raw display
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