Properties

Label 7056.2.h.g
Level $7056$
Weight $2$
Character orbit 7056.h
Analytic conductor $56.342$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [7056,2,Mod(4607,7056)]
 
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([1, 0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("7056.4607");
 
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.h (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(56.3424436662\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 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.

\(f(q)\) \(=\) \( q - 2 \beta_1 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 \beta_1 q^{5} - \beta_{2} q^{11} - 2 \beta_1 q^{17} - 2 \beta_{3} q^{19} + \beta_{2} q^{23} - 3 q^{25} - 3 \beta_1 q^{29} - 2 \beta_{3} q^{31} + 4 q^{37} + 2 \beta_1 q^{41} + 2 \beta_{3} q^{43} - 4 \beta_{2} q^{47} - 3 \beta_1 q^{53} + 2 \beta_{3} q^{55} - 4 \beta_{2} q^{59} + 12 q^{61} + 3 \beta_{3} q^{67} - 5 \beta_{2} q^{71} + 12 q^{73} - \beta_{3} q^{79} + 4 \beta_{2} q^{83} - 8 q^{85} + 2 \beta_1 q^{89} - 8 \beta_{2} q^{95} - 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 12 q^{25} + 16 q^{37} + 48 q^{61} + 48 q^{73} - 32 q^{85} - 48 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} + 4\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\nu^{2} - 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_1 \) Copy content Toggle raw display

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4607.1
1.22474 + 0.707107i
−1.22474 + 0.707107i
1.22474 0.707107i
−1.22474 0.707107i
0 0 0 2.82843i 0 0 0 0 0
4607.2 0 0 0 2.82843i 0 0 0 0 0
4607.3 0 0 0 2.82843i 0 0 0 0 0
4607.4 0 0 0 2.82843i 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
4.b odd 2 1 inner
12.b 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.h.g yes 4
3.b odd 2 1 inner 7056.2.h.g yes 4
4.b odd 2 1 inner 7056.2.h.g yes 4
7.b odd 2 1 7056.2.h.f 4
12.b even 2 1 inner 7056.2.h.g yes 4
21.c even 2 1 7056.2.h.f 4
28.d even 2 1 7056.2.h.f 4
84.h odd 2 1 7056.2.h.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7056.2.h.f 4 7.b odd 2 1
7056.2.h.f 4 21.c even 2 1
7056.2.h.f 4 28.d even 2 1
7056.2.h.f 4 84.h odd 2 1
7056.2.h.g yes 4 1.a even 1 1 trivial
7056.2.h.g yes 4 3.b odd 2 1 inner
7056.2.h.g yes 4 4.b odd 2 1 inner
7056.2.h.g yes 4 12.b 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 \(S_{2}^{\mathrm{new}}(7056, [\chi])\):

\( T_{5}^{2} + 8 \) Copy content Toggle raw display
\( T_{11}^{2} - 6 \) Copy content Toggle raw display
\( T_{13} \) Copy content Toggle raw display
\( T_{61} - 12 \) Copy content Toggle raw display
\( T_{83}^{2} - 96 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} + 8)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} - 6)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} + 8)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 48)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} - 6)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 18)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 48)^{2} \) Copy content Toggle raw display
$37$ \( (T - 4)^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} + 8)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 48)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 96)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 18)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 96)^{2} \) Copy content Toggle raw display
$61$ \( (T - 12)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} + 108)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 150)^{2} \) Copy content Toggle raw display
$73$ \( (T - 12)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 96)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 8)^{2} \) Copy content Toggle raw display
$97$ \( (T + 12)^{4} \) Copy content Toggle raw display
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