Properties

Label 4356.2.a.g
Level $4356$
Weight $2$
Character orbit 4356.a
Self dual yes
Analytic conductor $34.783$
Analytic rank $1$
Dimension $1$
CM discriminant -3
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4356,2,Mod(1,4356)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4356, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("4356.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4356 = 2^{2} \cdot 3^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4356.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: \(34.7828351205\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 36)
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)
 
\(f(q)\) \(=\) \( q + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + 4 q^{7} - 2 q^{13} - 8 q^{19} - 5 q^{25} - 4 q^{31} - 10 q^{37} - 8 q^{43} + 9 q^{49} - 14 q^{61} - 16 q^{67} + 10 q^{73} + 4 q^{79} - 8 q^{91} + 14 q^{97}+O(q^{100}) \) 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 0 0 4.00000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(11\) \(-1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4356.2.a.g 1
3.b odd 2 1 CM 4356.2.a.g 1
11.b odd 2 1 36.2.a.a 1
33.d even 2 1 36.2.a.a 1
44.c even 2 1 144.2.a.a 1
55.d odd 2 1 900.2.a.g 1
55.e even 4 2 900.2.d.b 2
77.b even 2 1 1764.2.a.e 1
77.h odd 6 2 1764.2.k.h 2
77.i even 6 2 1764.2.k.g 2
88.b odd 2 1 576.2.a.e 1
88.g even 2 1 576.2.a.f 1
99.g even 6 2 324.2.e.c 2
99.h odd 6 2 324.2.e.c 2
132.d odd 2 1 144.2.a.a 1
143.d odd 2 1 6084.2.a.i 1
143.g even 4 2 6084.2.b.f 2
165.d even 2 1 900.2.a.g 1
165.l odd 4 2 900.2.d.b 2
176.i even 4 2 2304.2.d.a 2
176.l odd 4 2 2304.2.d.q 2
220.g even 2 1 3600.2.a.e 1
220.i odd 4 2 3600.2.f.m 2
231.h odd 2 1 1764.2.a.e 1
231.k odd 6 2 1764.2.k.g 2
231.l even 6 2 1764.2.k.h 2
264.m even 2 1 576.2.a.e 1
264.p odd 2 1 576.2.a.f 1
308.g odd 2 1 7056.2.a.bb 1
396.k even 6 2 1296.2.i.h 2
396.o odd 6 2 1296.2.i.h 2
429.e even 2 1 6084.2.a.i 1
429.l odd 4 2 6084.2.b.f 2
528.s odd 4 2 2304.2.d.a 2
528.x even 4 2 2304.2.d.q 2
660.g odd 2 1 3600.2.a.e 1
660.q even 4 2 3600.2.f.m 2
924.n even 2 1 7056.2.a.bb 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.2.a.a 1 11.b odd 2 1
36.2.a.a 1 33.d even 2 1
144.2.a.a 1 44.c even 2 1
144.2.a.a 1 132.d odd 2 1
324.2.e.c 2 99.g even 6 2
324.2.e.c 2 99.h odd 6 2
576.2.a.e 1 88.b odd 2 1
576.2.a.e 1 264.m even 2 1
576.2.a.f 1 88.g even 2 1
576.2.a.f 1 264.p odd 2 1
900.2.a.g 1 55.d odd 2 1
900.2.a.g 1 165.d even 2 1
900.2.d.b 2 55.e even 4 2
900.2.d.b 2 165.l odd 4 2
1296.2.i.h 2 396.k even 6 2
1296.2.i.h 2 396.o odd 6 2
1764.2.a.e 1 77.b even 2 1
1764.2.a.e 1 231.h odd 2 1
1764.2.k.g 2 77.i even 6 2
1764.2.k.g 2 231.k odd 6 2
1764.2.k.h 2 77.h odd 6 2
1764.2.k.h 2 231.l even 6 2
2304.2.d.a 2 176.i even 4 2
2304.2.d.a 2 528.s odd 4 2
2304.2.d.q 2 176.l odd 4 2
2304.2.d.q 2 528.x even 4 2
3600.2.a.e 1 220.g even 2 1
3600.2.a.e 1 660.g odd 2 1
3600.2.f.m 2 220.i odd 4 2
3600.2.f.m 2 660.q even 4 2
4356.2.a.g 1 1.a even 1 1 trivial
4356.2.a.g 1 3.b odd 2 1 CM
6084.2.a.i 1 143.d odd 2 1
6084.2.a.i 1 429.e even 2 1
6084.2.b.f 2 143.g even 4 2
6084.2.b.f 2 429.l odd 4 2
7056.2.a.bb 1 308.g odd 2 1
7056.2.a.bb 1 924.n even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4356))\):

\( T_{5} \) Copy content Toggle raw display
\( T_{7} - 4 \) 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
$7$ \( T - 4 \) Copy content Toggle raw display
$11$ \( T \) Copy content Toggle raw display
$13$ \( T + 2 \) Copy content Toggle raw display
$17$ \( T \) Copy content Toggle raw display
$19$ \( T + 8 \) Copy content Toggle raw display
$23$ \( T \) Copy content Toggle raw display
$29$ \( T \) Copy content Toggle raw display
$31$ \( T + 4 \) Copy content Toggle raw display
$37$ \( T + 10 \) Copy content Toggle raw display
$41$ \( T \) Copy content Toggle raw display
$43$ \( T + 8 \) Copy content Toggle raw display
$47$ \( T \) Copy content Toggle raw display
$53$ \( T \) Copy content Toggle raw display
$59$ \( T \) Copy content Toggle raw display
$61$ \( T + 14 \) Copy content Toggle raw display
$67$ \( T + 16 \) Copy content Toggle raw display
$71$ \( T \) Copy content Toggle raw display
$73$ \( T - 10 \) Copy content Toggle raw display
$79$ \( T - 4 \) Copy content Toggle raw display
$83$ \( T \) Copy content Toggle raw display
$89$ \( T \) Copy content Toggle raw display
$97$ \( T - 14 \) Copy content Toggle raw display
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