Properties

Label 1296.2.a.l
Level $1296$
Weight $2$
Character orbit 1296.a
Self dual yes
Analytic conductor $10.349$
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] = [1296,2,Mod(1,1296)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1296, 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("1296.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1296 = 2^{4} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1296.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: \(10.3486121020\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 162)
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)
 
\(f(q)\) \(=\) \( q + 3 q^{5} + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + 3 q^{5} + 4 q^{7} - q^{13} + 3 q^{17} + 4 q^{19} + 4 q^{25} - 9 q^{29} + 4 q^{31} + 12 q^{35} - q^{37} - 6 q^{41} - 8 q^{43} - 12 q^{47} + 9 q^{49} + 6 q^{53} - q^{61} - 3 q^{65} + 4 q^{67} - 12 q^{71} + 11 q^{73} + 16 q^{79} - 12 q^{83} + 9 q^{85} + 3 q^{89} - 4 q^{91} + 12 q^{95} + 2 q^{97}+O(q^{100}) \) 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 3.00000 0 4.00000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 1296.2.a.l 1
3.b odd 2 1 1296.2.a.c 1
4.b odd 2 1 162.2.a.d yes 1
8.b even 2 1 5184.2.a.h 1
8.d odd 2 1 5184.2.a.c 1
9.c even 3 2 1296.2.i.b 2
9.d odd 6 2 1296.2.i.n 2
12.b even 2 1 162.2.a.a 1
20.d odd 2 1 4050.2.a.r 1
20.e even 4 2 4050.2.c.n 2
24.f even 2 1 5184.2.a.y 1
24.h odd 2 1 5184.2.a.bd 1
28.d even 2 1 7938.2.a.s 1
36.f odd 6 2 162.2.c.a 2
36.h even 6 2 162.2.c.d 2
60.h even 2 1 4050.2.a.bh 1
60.l odd 4 2 4050.2.c.g 2
84.h odd 2 1 7938.2.a.n 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
162.2.a.a 1 12.b even 2 1
162.2.a.d yes 1 4.b odd 2 1
162.2.c.a 2 36.f odd 6 2
162.2.c.d 2 36.h even 6 2
1296.2.a.c 1 3.b odd 2 1
1296.2.a.l 1 1.a even 1 1 trivial
1296.2.i.b 2 9.c even 3 2
1296.2.i.n 2 9.d odd 6 2
4050.2.a.r 1 20.d odd 2 1
4050.2.a.bh 1 60.h even 2 1
4050.2.c.g 2 60.l odd 4 2
4050.2.c.n 2 20.e even 4 2
5184.2.a.c 1 8.d odd 2 1
5184.2.a.h 1 8.b even 2 1
5184.2.a.y 1 24.f even 2 1
5184.2.a.bd 1 24.h odd 2 1
7938.2.a.n 1 84.h odd 2 1
7938.2.a.s 1 28.d even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1296))\):

\( T_{5} - 3 \) Copy content Toggle raw display
\( T_{7} - 4 \) Copy content Toggle raw display
\( T_{11} \) 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
$5$ \( T - 3 \) Copy content Toggle raw display
$7$ \( T - 4 \) Copy content Toggle raw display
$11$ \( T \) Copy content Toggle raw display
$13$ \( T + 1 \) Copy content Toggle raw display
$17$ \( T - 3 \) Copy content Toggle raw display
$19$ \( T - 4 \) Copy content Toggle raw display
$23$ \( T \) Copy content Toggle raw display
$29$ \( T + 9 \) Copy content Toggle raw display
$31$ \( T - 4 \) Copy content Toggle raw display
$37$ \( T + 1 \) Copy content Toggle raw display
$41$ \( T + 6 \) Copy content Toggle raw display
$43$ \( T + 8 \) Copy content Toggle raw display
$47$ \( T + 12 \) Copy content Toggle raw display
$53$ \( T - 6 \) Copy content Toggle raw display
$59$ \( T \) Copy content Toggle raw display
$61$ \( T + 1 \) Copy content Toggle raw display
$67$ \( T - 4 \) Copy content Toggle raw display
$71$ \( T + 12 \) Copy content Toggle raw display
$73$ \( T - 11 \) Copy content Toggle raw display
$79$ \( T - 16 \) Copy content Toggle raw display
$83$ \( T + 12 \) Copy content Toggle raw display
$89$ \( T - 3 \) Copy content Toggle raw display
$97$ \( T - 2 \) Copy content Toggle raw display
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