Properties

Label 3600.2.a.x
Level $3600$
Weight $2$
Character orbit 3600.a
Self dual yes
Analytic conductor $28.746$
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] = [3600,2,Mod(1,3600)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3600, 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("3600.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3600 = 2^{4} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3600.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: \(28.7461447277\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 900)
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)
 
\(f(q)\) \(=\) \( q + q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{7} - 7 q^{13} + 7 q^{19} - 11 q^{31} - 10 q^{37} + 13 q^{43} - 6 q^{49} - q^{61} - 11 q^{67} - 10 q^{73} + 4 q^{79} - 7 q^{91} - 19 q^{97}+O(q^{100}) \) 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 1.00000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(5\) \(-1\)

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 3600.2.a.x 1
3.b odd 2 1 CM 3600.2.a.x 1
4.b odd 2 1 900.2.a.d 1
5.b even 2 1 3600.2.a.q 1
5.c odd 4 2 3600.2.f.h 2
12.b even 2 1 900.2.a.d 1
15.d odd 2 1 3600.2.a.q 1
15.e even 4 2 3600.2.f.h 2
20.d odd 2 1 900.2.a.f yes 1
20.e even 4 2 900.2.d.d 2
60.h even 2 1 900.2.a.f yes 1
60.l odd 4 2 900.2.d.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
900.2.a.d 1 4.b odd 2 1
900.2.a.d 1 12.b even 2 1
900.2.a.f yes 1 20.d odd 2 1
900.2.a.f yes 1 60.h even 2 1
900.2.d.d 2 20.e even 4 2
900.2.d.d 2 60.l odd 4 2
3600.2.a.q 1 5.b even 2 1
3600.2.a.q 1 15.d odd 2 1
3600.2.a.x 1 1.a even 1 1 trivial
3600.2.a.x 1 3.b odd 2 1 CM
3600.2.f.h 2 5.c odd 4 2
3600.2.f.h 2 15.e even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3600))\):

\( T_{7} - 1 \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display
\( T_{13} + 7 \) Copy content Toggle raw display
\( T_{17} \) 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
$7$ \( T - 1 \) Copy content Toggle raw display
$11$ \( T \) Copy content Toggle raw display
$13$ \( T + 7 \) Copy content Toggle raw display
$17$ \( T \) Copy content Toggle raw display
$19$ \( T - 7 \) Copy content Toggle raw display
$23$ \( T \) Copy content Toggle raw display
$29$ \( T \) Copy content Toggle raw display
$31$ \( T + 11 \) Copy content Toggle raw display
$37$ \( T + 10 \) Copy content Toggle raw display
$41$ \( T \) Copy content Toggle raw display
$43$ \( T - 13 \) 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
$61$ \( T + 1 \) Copy content Toggle raw display
$67$ \( T + 11 \) 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
$97$ \( T + 19 \) Copy content Toggle raw display
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