Properties

Label 7056.2.a.cx
Level $7056$
Weight $2$
Character orbit 7056.a
Self dual yes
Analytic conductor $56.342$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1176)
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)
 

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta + 2) q^{5}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta + 2) q^{5} + (2 \beta + 2) q^{11} - \beta q^{13} + ( - 3 \beta - 2) q^{17} + ( - 2 \beta + 4) q^{19} + ( - 2 \beta + 2) q^{23} + (4 \beta + 1) q^{25} + 6 \beta q^{29} + (2 \beta + 8) q^{31} + ( - 4 \beta - 4) q^{37} + ( - \beta - 2) q^{41} + 8 q^{43} + (2 \beta - 4) q^{47} + ( - 8 \beta + 2) q^{53} + (6 \beta + 8) q^{55} + ( - 2 \beta - 8) q^{59} + (7 \beta - 4) q^{61} + ( - 2 \beta - 2) q^{65} + 8 q^{67} + (2 \beta + 2) q^{71} + ( - 5 \beta + 4) q^{73} + (4 \beta + 8) q^{79} + (8 \beta - 4) q^{83} + ( - 8 \beta - 10) q^{85} + (9 \beta + 2) q^{89} + 4 q^{95} + (3 \beta + 12) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{5} + 4 q^{11} - 4 q^{17} + 8 q^{19} + 4 q^{23} + 2 q^{25} + 16 q^{31} - 8 q^{37} - 4 q^{41} + 16 q^{43} - 8 q^{47} + 4 q^{53} + 16 q^{55} - 16 q^{59} - 8 q^{61} - 4 q^{65} + 16 q^{67} + 4 q^{71} + 8 q^{73} + 16 q^{79} - 8 q^{83} - 20 q^{85} + 4 q^{89} + 8 q^{95} + 24 q^{97}+O(q^{100}) \) 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
−1.41421
1.41421
0 0 0 0.585786 0 0 0 0 0
1.2 0 0 0 3.41421 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.cx 2
3.b odd 2 1 2352.2.a.bd 2
4.b odd 2 1 3528.2.a.bl 2
7.b odd 2 1 7056.2.a.cg 2
12.b even 2 1 1176.2.a.j 2
21.c even 2 1 2352.2.a.bb 2
21.g even 6 2 2352.2.q.be 4
21.h odd 6 2 2352.2.q.bc 4
24.f even 2 1 9408.2.a.ee 2
24.h odd 2 1 9408.2.a.ds 2
28.d even 2 1 3528.2.a.bb 2
28.f even 6 2 3528.2.s.bm 4
28.g odd 6 2 3528.2.s.bd 4
84.h odd 2 1 1176.2.a.o yes 2
84.j odd 6 2 1176.2.q.k 4
84.n even 6 2 1176.2.q.o 4
168.e odd 2 1 9408.2.a.dg 2
168.i even 2 1 9408.2.a.du 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1176.2.a.j 2 12.b even 2 1
1176.2.a.o yes 2 84.h odd 2 1
1176.2.q.k 4 84.j odd 6 2
1176.2.q.o 4 84.n even 6 2
2352.2.a.bb 2 21.c even 2 1
2352.2.a.bd 2 3.b odd 2 1
2352.2.q.bc 4 21.h odd 6 2
2352.2.q.be 4 21.g even 6 2
3528.2.a.bb 2 28.d even 2 1
3528.2.a.bl 2 4.b odd 2 1
3528.2.s.bd 4 28.g odd 6 2
3528.2.s.bm 4 28.f even 6 2
7056.2.a.cg 2 7.b odd 2 1
7056.2.a.cx 2 1.a even 1 1 trivial
9408.2.a.dg 2 168.e odd 2 1
9408.2.a.ds 2 24.h odd 2 1
9408.2.a.du 2 168.i even 2 1
9408.2.a.ee 2 24.f even 2 1

Hecke kernels

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

\( T_{5}^{2} - 4T_{5} + 2 \) Copy content Toggle raw display
\( T_{11}^{2} - 4T_{11} - 4 \) Copy content Toggle raw display
\( T_{13}^{2} - 2 \) Copy content Toggle raw display
\( T_{17}^{2} + 4T_{17} - 14 \) Copy content Toggle raw display
\( T_{23}^{2} - 4T_{23} - 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 4T + 2 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 4T - 4 \) Copy content Toggle raw display
$13$ \( T^{2} - 2 \) Copy content Toggle raw display
$17$ \( T^{2} + 4T - 14 \) Copy content Toggle raw display
$19$ \( T^{2} - 8T + 8 \) Copy content Toggle raw display
$23$ \( T^{2} - 4T - 4 \) Copy content Toggle raw display
$29$ \( T^{2} - 72 \) Copy content Toggle raw display
$31$ \( T^{2} - 16T + 56 \) Copy content Toggle raw display
$37$ \( T^{2} + 8T - 16 \) Copy content Toggle raw display
$41$ \( T^{2} + 4T + 2 \) Copy content Toggle raw display
$43$ \( (T - 8)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 8T + 8 \) Copy content Toggle raw display
$53$ \( T^{2} - 4T - 124 \) Copy content Toggle raw display
$59$ \( T^{2} + 16T + 56 \) Copy content Toggle raw display
$61$ \( T^{2} + 8T - 82 \) Copy content Toggle raw display
$67$ \( (T - 8)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} - 4T - 4 \) Copy content Toggle raw display
$73$ \( T^{2} - 8T - 34 \) Copy content Toggle raw display
$79$ \( T^{2} - 16T + 32 \) Copy content Toggle raw display
$83$ \( T^{2} + 8T - 112 \) Copy content Toggle raw display
$89$ \( T^{2} - 4T - 158 \) Copy content Toggle raw display
$97$ \( T^{2} - 24T + 126 \) Copy content Toggle raw display
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