Properties

Label 3600.2.a.g
Level $3600$
Weight $2$
Character orbit 3600.a
Self dual yes
Analytic conductor $28.746$
Analytic rank $0$
Dimension $1$
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] = [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: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 360)
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)
 
\(f(q)\) \(=\) \( q - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - 4 q^{7} + 4 q^{11} - 4 q^{13} - 6 q^{17} + 4 q^{19} + 4 q^{23} - 4 q^{29} - 4 q^{37} + 8 q^{41} + 12 q^{47} + 9 q^{49} - 2 q^{53} - 12 q^{59} + 2 q^{61} + 8 q^{67} - 8 q^{71} + 16 q^{73} - 16 q^{77} + 8 q^{79} + 8 q^{83} + 16 q^{91} + 8 q^{97}+O(q^{100}) \) 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
\(2\) \(1\)
\(3\) \(1\)
\(5\) \(-1\)

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 3600.2.a.g 1
3.b odd 2 1 3600.2.a.c 1
4.b odd 2 1 1800.2.a.u 1
5.b even 2 1 3600.2.a.bp 1
5.c odd 4 2 720.2.f.g 2
12.b even 2 1 1800.2.a.w 1
15.d odd 2 1 3600.2.a.bn 1
15.e even 4 2 720.2.f.a 2
20.d odd 2 1 1800.2.a.b 1
20.e even 4 2 360.2.f.d yes 2
40.i odd 4 2 2880.2.f.b 2
40.k even 4 2 2880.2.f.f 2
60.h even 2 1 1800.2.a.d 1
60.l odd 4 2 360.2.f.b 2
120.q odd 4 2 2880.2.f.q 2
120.w even 4 2 2880.2.f.u 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.2.f.b 2 60.l odd 4 2
360.2.f.d yes 2 20.e even 4 2
720.2.f.a 2 15.e even 4 2
720.2.f.g 2 5.c odd 4 2
1800.2.a.b 1 20.d odd 2 1
1800.2.a.d 1 60.h even 2 1
1800.2.a.u 1 4.b odd 2 1
1800.2.a.w 1 12.b even 2 1
2880.2.f.b 2 40.i odd 4 2
2880.2.f.f 2 40.k even 4 2
2880.2.f.q 2 120.q odd 4 2
2880.2.f.u 2 120.w even 4 2
3600.2.a.c 1 3.b odd 2 1
3600.2.a.g 1 1.a even 1 1 trivial
3600.2.a.bn 1 15.d odd 2 1
3600.2.a.bp 1 5.b even 2 1

Hecke kernels

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

\( T_{7} + 4 \) Copy content Toggle raw display
\( T_{11} - 4 \) Copy content Toggle raw display
\( T_{13} + 4 \) Copy content Toggle raw display
\( T_{17} + 6 \) 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
$11$ \( T - 4 \) Copy content Toggle raw display
$13$ \( T + 4 \) Copy content Toggle raw display
$17$ \( T + 6 \) Copy content Toggle raw display
$19$ \( T - 4 \) Copy content Toggle raw display
$23$ \( T - 4 \) Copy content Toggle raw display
$29$ \( T + 4 \) Copy content Toggle raw display
$31$ \( T \) Copy content Toggle raw display
$37$ \( T + 4 \) Copy content Toggle raw display
$41$ \( T - 8 \) Copy content Toggle raw display
$43$ \( T \) Copy content Toggle raw display
$47$ \( T - 12 \) Copy content Toggle raw display
$53$ \( T + 2 \) Copy content Toggle raw display
$59$ \( T + 12 \) Copy content Toggle raw display
$61$ \( T - 2 \) Copy content Toggle raw display
$67$ \( T - 8 \) Copy content Toggle raw display
$71$ \( T + 8 \) Copy content Toggle raw display
$73$ \( T - 16 \) Copy content Toggle raw display
$79$ \( T - 8 \) Copy content Toggle raw display
$83$ \( T - 8 \) Copy content Toggle raw display
$89$ \( T \) Copy content Toggle raw display
$97$ \( T - 8 \) Copy content Toggle raw display
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