Properties

Label 2304.4.a.d
Level $2304$
Weight $4$
Character orbit 2304.a
Self dual yes
Analytic conductor $135.940$
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] = [2304,4,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, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2304.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2304 = 2^{8} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 4 \)
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: \(135.940400653\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 128)
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)
 
\(f(q)\) \(=\) \( q - 12 q^{5} + 32 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - 12 q^{5} + 32 q^{7} + 8 q^{11} - 20 q^{13} + 98 q^{17} - 88 q^{19} + 32 q^{23} + 19 q^{25} - 172 q^{29} - 256 q^{31} - 384 q^{35} + 92 q^{37} - 102 q^{41} - 296 q^{43} + 320 q^{47} + 681 q^{49} - 76 q^{53} - 96 q^{55} - 408 q^{59} + 636 q^{61} + 240 q^{65} + 552 q^{67} - 416 q^{71} + 138 q^{73} + 256 q^{77} - 64 q^{79} - 392 q^{83} - 1176 q^{85} + 582 q^{89} - 640 q^{91} + 1056 q^{95} + 238 q^{97}+O(q^{100}) \) 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 −12.0000 0 32.0000 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 2304.4.a.d 1
3.b odd 2 1 256.4.a.g 1
4.b odd 2 1 2304.4.a.c 1
8.b even 2 1 2304.4.a.n 1
8.d odd 2 1 2304.4.a.m 1
12.b even 2 1 256.4.a.c 1
16.e even 4 2 1152.4.d.a 2
16.f odd 4 2 1152.4.d.h 2
24.f even 2 1 256.4.a.f 1
24.h odd 2 1 256.4.a.b 1
48.i odd 4 2 128.4.b.a 2
48.k even 4 2 128.4.b.d yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
128.4.b.a 2 48.i odd 4 2
128.4.b.d yes 2 48.k even 4 2
256.4.a.b 1 24.h odd 2 1
256.4.a.c 1 12.b even 2 1
256.4.a.f 1 24.f even 2 1
256.4.a.g 1 3.b odd 2 1
1152.4.d.a 2 16.e even 4 2
1152.4.d.h 2 16.f odd 4 2
2304.4.a.c 1 4.b odd 2 1
2304.4.a.d 1 1.a even 1 1 trivial
2304.4.a.m 1 8.d odd 2 1
2304.4.a.n 1 8.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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2304))\):

\( T_{5} + 12 \) Copy content Toggle raw display
\( T_{7} - 32 \) Copy content Toggle raw display
\( T_{11} - 8 \) Copy content Toggle raw display
\( T_{13} + 20 \) Copy content Toggle raw display
\( T_{17} - 98 \) Copy content Toggle raw display
\( T_{19} + 88 \) 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
$5$ \( T + 12 \) Copy content Toggle raw display
$7$ \( T - 32 \) Copy content Toggle raw display
$11$ \( T - 8 \) Copy content Toggle raw display
$13$ \( T + 20 \) Copy content Toggle raw display
$17$ \( T - 98 \) Copy content Toggle raw display
$19$ \( T + 88 \) Copy content Toggle raw display
$23$ \( T - 32 \) Copy content Toggle raw display
$29$ \( T + 172 \) Copy content Toggle raw display
$31$ \( T + 256 \) Copy content Toggle raw display
$37$ \( T - 92 \) Copy content Toggle raw display
$41$ \( T + 102 \) Copy content Toggle raw display
$43$ \( T + 296 \) Copy content Toggle raw display
$47$ \( T - 320 \) Copy content Toggle raw display
$53$ \( T + 76 \) Copy content Toggle raw display
$59$ \( T + 408 \) Copy content Toggle raw display
$61$ \( T - 636 \) Copy content Toggle raw display
$67$ \( T - 552 \) Copy content Toggle raw display
$71$ \( T + 416 \) Copy content Toggle raw display
$73$ \( T - 138 \) Copy content Toggle raw display
$79$ \( T + 64 \) Copy content Toggle raw display
$83$ \( T + 392 \) Copy content Toggle raw display
$89$ \( T - 582 \) Copy content Toggle raw display
$97$ \( T - 238 \) Copy content Toggle raw display
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