Properties

Label 7200.2.a.be
Level $7200$
Weight $2$
Character orbit 7200.a
Self dual yes
Analytic conductor $57.492$
Analytic rank $1$
Dimension $1$
CM discriminant -4
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [7200,2,Mod(1,7200)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7200, 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("7200.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7200 = 2^{5} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7200.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: \(57.4922894553\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 288)
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)
 
\(f(q)\) \(=\) \( q+O(q^{10}) \) Copy content Toggle raw display \( q + 6 q^{13} - 8 q^{17} - 4 q^{29} + 2 q^{37} - 8 q^{41} - 7 q^{49} + 4 q^{53} - 10 q^{61} - 6 q^{73} + 16 q^{89} + 18 q^{97}+O(q^{100}) \) 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 0 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
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7200.2.a.be 1
3.b odd 2 1 7200.2.a.bf 1
4.b odd 2 1 CM 7200.2.a.be 1
5.b even 2 1 288.2.a.e yes 1
5.c odd 4 2 7200.2.f.q 2
12.b even 2 1 7200.2.a.bf 1
15.d odd 2 1 288.2.a.a 1
15.e even 4 2 7200.2.f.n 2
20.d odd 2 1 288.2.a.e yes 1
20.e even 4 2 7200.2.f.q 2
40.e odd 2 1 576.2.a.a 1
40.f even 2 1 576.2.a.a 1
45.h odd 6 2 2592.2.i.x 2
45.j even 6 2 2592.2.i.a 2
60.h even 2 1 288.2.a.a 1
60.l odd 4 2 7200.2.f.n 2
80.k odd 4 2 2304.2.d.l 2
80.q even 4 2 2304.2.d.l 2
120.i odd 2 1 576.2.a.i 1
120.m even 2 1 576.2.a.i 1
180.n even 6 2 2592.2.i.x 2
180.p odd 6 2 2592.2.i.a 2
240.t even 4 2 2304.2.d.h 2
240.bm odd 4 2 2304.2.d.h 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
288.2.a.a 1 15.d odd 2 1
288.2.a.a 1 60.h even 2 1
288.2.a.e yes 1 5.b even 2 1
288.2.a.e yes 1 20.d odd 2 1
576.2.a.a 1 40.e odd 2 1
576.2.a.a 1 40.f even 2 1
576.2.a.i 1 120.i odd 2 1
576.2.a.i 1 120.m even 2 1
2304.2.d.h 2 240.t even 4 2
2304.2.d.h 2 240.bm odd 4 2
2304.2.d.l 2 80.k odd 4 2
2304.2.d.l 2 80.q even 4 2
2592.2.i.a 2 45.j even 6 2
2592.2.i.a 2 180.p odd 6 2
2592.2.i.x 2 45.h odd 6 2
2592.2.i.x 2 180.n even 6 2
7200.2.a.be 1 1.a even 1 1 trivial
7200.2.a.be 1 4.b odd 2 1 CM
7200.2.a.bf 1 3.b odd 2 1
7200.2.a.bf 1 12.b even 2 1
7200.2.f.n 2 15.e even 4 2
7200.2.f.n 2 60.l odd 4 2
7200.2.f.q 2 5.c odd 4 2
7200.2.f.q 2 20.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7200))\):

\( T_{7} \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display
\( T_{13} - 6 \) Copy content Toggle raw display
\( T_{17} + 8 \) Copy content Toggle raw display
\( T_{19} \) Copy content Toggle raw display
\( T_{23} \) 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
$7$ \( T \) Copy content Toggle raw display
$11$ \( T \) Copy content Toggle raw display
$13$ \( T - 6 \) Copy content Toggle raw display
$17$ \( T + 8 \) Copy content Toggle raw display
$19$ \( T \) Copy content Toggle raw display
$23$ \( T \) Copy content Toggle raw display
$29$ \( T + 4 \) Copy content Toggle raw display
$31$ \( T \) Copy content Toggle raw display
$37$ \( T - 2 \) Copy content Toggle raw display
$41$ \( T + 8 \) Copy content Toggle raw display
$43$ \( T \) Copy content Toggle raw display
$47$ \( T \) Copy content Toggle raw display
$53$ \( T - 4 \) Copy content Toggle raw display
$59$ \( T \) Copy content Toggle raw display
$61$ \( T + 10 \) Copy content Toggle raw display
$67$ \( T \) Copy content Toggle raw display
$71$ \( T \) Copy content Toggle raw display
$73$ \( T + 6 \) Copy content Toggle raw display
$79$ \( T \) Copy content Toggle raw display
$83$ \( T \) Copy content Toggle raw display
$89$ \( T - 16 \) Copy content Toggle raw display
$97$ \( T - 18 \) Copy content Toggle raw display
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