Properties

Label 6480.2.a.t
Level $6480$
Weight $2$
Character orbit 6480.a
Self dual yes
Analytic conductor $51.743$
Analytic rank $1$
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] = [6480,2,Mod(1,6480)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6480, 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("6480.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6480 = 2^{4} \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6480.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: \(51.7430605098\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 180)
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)
 
\(f(q)\) \(=\) \( q + q^{5} + q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{5} + q^{7} - 4 q^{13} - 6 q^{17} - 2 q^{19} + 3 q^{23} + q^{25} + 3 q^{29} + 10 q^{31} + q^{35} - 10 q^{37} + 9 q^{41} + 4 q^{43} - 9 q^{47} - 6 q^{49} - 6 q^{53} + 6 q^{59} - q^{61} - 4 q^{65} - 11 q^{67} - 12 q^{71} - 4 q^{73} + 10 q^{79} + 9 q^{83} - 6 q^{85} + 9 q^{89} - 4 q^{91} - 2 q^{95} - 10 q^{97}+O(q^{100}) \) 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 1.00000 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

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 6480.2.a.t 1
3.b odd 2 1 6480.2.a.h 1
4.b odd 2 1 1620.2.a.e 1
9.c even 3 2 720.2.q.a 2
9.d odd 6 2 2160.2.q.e 2
12.b even 2 1 1620.2.a.b 1
20.d odd 2 1 8100.2.a.i 1
20.e even 4 2 8100.2.d.e 2
36.f odd 6 2 180.2.i.a 2
36.h even 6 2 540.2.i.a 2
60.h even 2 1 8100.2.a.h 1
60.l odd 4 2 8100.2.d.f 2
180.n even 6 2 2700.2.i.a 2
180.p odd 6 2 900.2.i.a 2
180.v odd 12 4 2700.2.s.a 4
180.x even 12 4 900.2.s.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.2.i.a 2 36.f odd 6 2
540.2.i.a 2 36.h even 6 2
720.2.q.a 2 9.c even 3 2
900.2.i.a 2 180.p odd 6 2
900.2.s.a 4 180.x even 12 4
1620.2.a.b 1 12.b even 2 1
1620.2.a.e 1 4.b odd 2 1
2160.2.q.e 2 9.d odd 6 2
2700.2.i.a 2 180.n even 6 2
2700.2.s.a 4 180.v odd 12 4
6480.2.a.h 1 3.b odd 2 1
6480.2.a.t 1 1.a even 1 1 trivial
8100.2.a.h 1 60.h even 2 1
8100.2.a.i 1 20.d odd 2 1
8100.2.d.e 2 20.e even 4 2
8100.2.d.f 2 60.l odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6480))\):

\( T_{7} - 1 \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display
\( T_{13} + 4 \) Copy content Toggle raw display
\( T_{17} + 6 \) Copy content Toggle raw display
\( T_{19} + 2 \) Copy content Toggle raw display
\( T_{23} - 3 \) 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
$5$ \( T - 1 \) Copy content Toggle raw display
$7$ \( T - 1 \) Copy content Toggle raw display
$11$ \( T \) Copy content Toggle raw display
$13$ \( T + 4 \) Copy content Toggle raw display
$17$ \( T + 6 \) Copy content Toggle raw display
$19$ \( T + 2 \) Copy content Toggle raw display
$23$ \( T - 3 \) Copy content Toggle raw display
$29$ \( T - 3 \) Copy content Toggle raw display
$31$ \( T - 10 \) Copy content Toggle raw display
$37$ \( T + 10 \) Copy content Toggle raw display
$41$ \( T - 9 \) Copy content Toggle raw display
$43$ \( T - 4 \) Copy content Toggle raw display
$47$ \( T + 9 \) Copy content Toggle raw display
$53$ \( T + 6 \) Copy content Toggle raw display
$59$ \( T - 6 \) Copy content Toggle raw display
$61$ \( T + 1 \) Copy content Toggle raw display
$67$ \( T + 11 \) Copy content Toggle raw display
$71$ \( T + 12 \) Copy content Toggle raw display
$73$ \( T + 4 \) Copy content Toggle raw display
$79$ \( T - 10 \) Copy content Toggle raw display
$83$ \( T - 9 \) Copy content Toggle raw display
$89$ \( T - 9 \) Copy content Toggle raw display
$97$ \( T + 10 \) Copy content Toggle raw display
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