Properties

Label 2304.2.a.i
Level $2304$
Weight $2$
Character orbit 2304.a
Self dual yes
Analytic conductor $18.398$
Analytic rank $0$
Dimension $1$
CM discriminant -8
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2304,2,Mod(1,2304)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2304, 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("2304.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2304 = 2^{8} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2304.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: \(18.3975326257\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 64)
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 \( q + 6 q^{11} + 6 q^{17} - 2 q^{19} - 5 q^{25} - 6 q^{41} + 10 q^{43} - 7 q^{49} + 6 q^{59} + 14 q^{67} - 2 q^{73} + 18 q^{83} + 18 q^{89} + 10 q^{97}+O(q^{100}) \) 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 0 0 0 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2304.2.a.i 1
3.b odd 2 1 256.2.a.a 1
4.b odd 2 1 2304.2.a.h 1
8.b even 2 1 2304.2.a.h 1
8.d odd 2 1 CM 2304.2.a.i 1
12.b even 2 1 256.2.a.d 1
15.d odd 2 1 6400.2.a.x 1
16.e even 4 2 576.2.d.a 2
16.f odd 4 2 576.2.d.a 2
24.f even 2 1 256.2.a.a 1
24.h odd 2 1 256.2.a.d 1
48.i odd 4 2 64.2.b.a 2
48.k even 4 2 64.2.b.a 2
60.h even 2 1 6400.2.a.a 1
96.o even 8 4 1024.2.e.l 4
96.p odd 8 4 1024.2.e.l 4
120.i odd 2 1 6400.2.a.a 1
120.m even 2 1 6400.2.a.x 1
240.t even 4 2 1600.2.d.a 2
240.z odd 4 2 1600.2.f.a 2
240.bb even 4 2 1600.2.f.b 2
240.bd odd 4 2 1600.2.f.b 2
240.bf even 4 2 1600.2.f.a 2
240.bm odd 4 2 1600.2.d.a 2
336.v odd 4 2 3136.2.b.b 2
336.y even 4 2 3136.2.b.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
64.2.b.a 2 48.i odd 4 2
64.2.b.a 2 48.k even 4 2
256.2.a.a 1 3.b odd 2 1
256.2.a.a 1 24.f even 2 1
256.2.a.d 1 12.b even 2 1
256.2.a.d 1 24.h odd 2 1
576.2.d.a 2 16.e even 4 2
576.2.d.a 2 16.f odd 4 2
1024.2.e.l 4 96.o even 8 4
1024.2.e.l 4 96.p odd 8 4
1600.2.d.a 2 240.t even 4 2
1600.2.d.a 2 240.bm odd 4 2
1600.2.f.a 2 240.z odd 4 2
1600.2.f.a 2 240.bf even 4 2
1600.2.f.b 2 240.bb even 4 2
1600.2.f.b 2 240.bd odd 4 2
2304.2.a.h 1 4.b odd 2 1
2304.2.a.h 1 8.b even 2 1
2304.2.a.i 1 1.a even 1 1 trivial
2304.2.a.i 1 8.d odd 2 1 CM
3136.2.b.b 2 336.v odd 4 2
3136.2.b.b 2 336.y even 4 2
6400.2.a.a 1 60.h even 2 1
6400.2.a.a 1 120.i odd 2 1
6400.2.a.x 1 15.d odd 2 1
6400.2.a.x 1 120.m even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2304))\):

\( T_{5} \) Copy content Toggle raw display
\( T_{7} \) Copy content Toggle raw display
\( T_{11} - 6 \) Copy content Toggle raw display
\( T_{13} \) Copy content Toggle raw display
\( T_{19} + 2 \) 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
$11$ \( T - 6 \) Copy content Toggle raw display
$13$ \( T \) Copy content Toggle raw display
$17$ \( T - 6 \) Copy content Toggle raw display
$19$ \( T + 2 \) Copy content Toggle raw display
$23$ \( T \) Copy content Toggle raw display
$29$ \( T \) Copy content Toggle raw display
$31$ \( T \) Copy content Toggle raw display
$37$ \( T \) Copy content Toggle raw display
$41$ \( T + 6 \) Copy content Toggle raw display
$43$ \( T - 10 \) Copy content Toggle raw display
$47$ \( T \) Copy content Toggle raw display
$53$ \( T \) Copy content Toggle raw display
$59$ \( T - 6 \) Copy content Toggle raw display
$61$ \( T \) Copy content Toggle raw display
$67$ \( T - 14 \) Copy content Toggle raw display
$71$ \( T \) Copy content Toggle raw display
$73$ \( T + 2 \) Copy content Toggle raw display
$79$ \( T \) Copy content Toggle raw display
$83$ \( T - 18 \) Copy content Toggle raw display
$89$ \( T - 18 \) Copy content Toggle raw display
$97$ \( T - 10 \) Copy content Toggle raw display
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