Properties

Label 7056.2.a.bi
Level $7056$
Weight $2$
Character orbit 7056.a
Self dual yes
Analytic conductor $56.342$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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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 56)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q + q^{5}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{5} - q^{11} + 2 q^{13} - 3 q^{17} - 5 q^{19} - 3 q^{23} - 4 q^{25} + 6 q^{29} + q^{31} - 5 q^{37} + 10 q^{41} + 4 q^{43} + q^{47} + 9 q^{53} - q^{55} + 3 q^{59} + 3 q^{61} + 2 q^{65} - 11 q^{67} + 16 q^{71} + 7 q^{73} + 11 q^{79} - 4 q^{83} - 3 q^{85} + 9 q^{89} - 5 q^{95} + 6 q^{97}+O(q^{100}) \) 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 1.00000 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

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7056.2.a.bi 1
3.b odd 2 1 784.2.a.a 1
4.b odd 2 1 3528.2.a.r 1
7.b odd 2 1 7056.2.a.s 1
7.c even 3 2 1008.2.s.e 2
12.b even 2 1 392.2.a.f 1
21.c even 2 1 784.2.a.j 1
21.g even 6 2 784.2.i.a 2
21.h odd 6 2 112.2.i.c 2
24.f even 2 1 3136.2.a.b 1
24.h odd 2 1 3136.2.a.bc 1
28.d even 2 1 3528.2.a.k 1
28.f even 6 2 3528.2.s.o 2
28.g odd 6 2 504.2.s.e 2
60.h even 2 1 9800.2.a.b 1
84.h odd 2 1 392.2.a.a 1
84.j odd 6 2 392.2.i.f 2
84.n even 6 2 56.2.i.a 2
168.e odd 2 1 3136.2.a.bb 1
168.i even 2 1 3136.2.a.a 1
168.s odd 6 2 448.2.i.a 2
168.v even 6 2 448.2.i.f 2
420.o odd 2 1 9800.2.a.bp 1
420.ba even 6 2 1400.2.q.g 2
420.bp odd 12 4 1400.2.bh.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.2.i.a 2 84.n even 6 2
112.2.i.c 2 21.h odd 6 2
392.2.a.a 1 84.h odd 2 1
392.2.a.f 1 12.b even 2 1
392.2.i.f 2 84.j odd 6 2
448.2.i.a 2 168.s odd 6 2
448.2.i.f 2 168.v even 6 2
504.2.s.e 2 28.g odd 6 2
784.2.a.a 1 3.b odd 2 1
784.2.a.j 1 21.c even 2 1
784.2.i.a 2 21.g even 6 2
1008.2.s.e 2 7.c even 3 2
1400.2.q.g 2 420.ba even 6 2
1400.2.bh.f 4 420.bp odd 12 4
3136.2.a.a 1 168.i even 2 1
3136.2.a.b 1 24.f even 2 1
3136.2.a.bb 1 168.e odd 2 1
3136.2.a.bc 1 24.h odd 2 1
3528.2.a.k 1 28.d even 2 1
3528.2.a.r 1 4.b odd 2 1
3528.2.s.o 2 28.f even 6 2
7056.2.a.s 1 7.b odd 2 1
7056.2.a.bi 1 1.a even 1 1 trivial
9800.2.a.b 1 60.h even 2 1
9800.2.a.bp 1 420.o odd 2 1

Hecke kernels

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

\( T_{5} - 1 \) Copy content Toggle raw display
\( T_{11} + 1 \) Copy content Toggle raw display
\( T_{13} - 2 \) Copy content Toggle raw display
\( T_{17} + 3 \) Copy content Toggle raw display
\( T_{23} + 3 \) 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
$5$ \( T - 1 \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T + 1 \) Copy content Toggle raw display
$13$ \( T - 2 \) Copy content Toggle raw display
$17$ \( T + 3 \) Copy content Toggle raw display
$19$ \( T + 5 \) Copy content Toggle raw display
$23$ \( T + 3 \) Copy content Toggle raw display
$29$ \( T - 6 \) Copy content Toggle raw display
$31$ \( T - 1 \) Copy content Toggle raw display
$37$ \( T + 5 \) Copy content Toggle raw display
$41$ \( T - 10 \) Copy content Toggle raw display
$43$ \( T - 4 \) Copy content Toggle raw display
$47$ \( T - 1 \) Copy content Toggle raw display
$53$ \( T - 9 \) Copy content Toggle raw display
$59$ \( T - 3 \) Copy content Toggle raw display
$61$ \( T - 3 \) Copy content Toggle raw display
$67$ \( T + 11 \) Copy content Toggle raw display
$71$ \( T - 16 \) Copy content Toggle raw display
$73$ \( T - 7 \) Copy content Toggle raw display
$79$ \( T - 11 \) Copy content Toggle raw display
$83$ \( T + 4 \) Copy content Toggle raw display
$89$ \( T - 9 \) Copy content Toggle raw display
$97$ \( T - 6 \) Copy content Toggle raw display
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