Properties

Label 2304.3.g.d
Level $2304$
Weight $3$
Character orbit 2304.g
Self dual yes
Analytic conductor $62.779$
Analytic rank $0$
Dimension $1$
CM discriminant -4
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2304,3,Mod(1279,2304)]
 
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([1, 0, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2304.1279");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2304 = 2^{8} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2304.g (of order \(2\), degree \(1\), not minimal)

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: \(62.7794529086\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1152)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q + 6 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( q + 6 q^{5} - 24 q^{13} - 16 q^{17} + 11 q^{25} + 42 q^{29} + 24 q^{37} + 80 q^{41} + 49 q^{49} + 90 q^{53} + 120 q^{61} - 144 q^{65} + 110 q^{73} - 96 q^{85} + 160 q^{89} + 130 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2304\mathbb{Z}\right)^\times\).

\(n\) \(1279\) \(1793\) \(2053\)
\(\chi(n)\) \(1\) \(0\) \(0\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1279.1
0
0 0 0 6.00000 0 0 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2304.3.g.d 1
3.b odd 2 1 2304.3.g.b 1
4.b odd 2 1 CM 2304.3.g.d 1
8.b even 2 1 2304.3.g.c 1
8.d odd 2 1 2304.3.g.c 1
12.b even 2 1 2304.3.g.b 1
16.e even 4 2 1152.3.b.b 2
16.f odd 4 2 1152.3.b.b 2
24.f even 2 1 2304.3.g.e 1
24.h odd 2 1 2304.3.g.e 1
48.i odd 4 2 1152.3.b.d yes 2
48.k even 4 2 1152.3.b.d yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1152.3.b.b 2 16.e even 4 2
1152.3.b.b 2 16.f odd 4 2
1152.3.b.d yes 2 48.i odd 4 2
1152.3.b.d yes 2 48.k even 4 2
2304.3.g.b 1 3.b odd 2 1
2304.3.g.b 1 12.b even 2 1
2304.3.g.c 1 8.b even 2 1
2304.3.g.c 1 8.d odd 2 1
2304.3.g.d 1 1.a even 1 1 trivial
2304.3.g.d 1 4.b odd 2 1 CM
2304.3.g.e 1 24.f even 2 1
2304.3.g.e 1 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(2304, [\chi])\):

\( T_{5} - 6 \) Copy content Toggle raw display
\( T_{7} \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display
\( T_{13} + 24 \) 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
$5$ \( T - 6 \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T \) Copy content Toggle raw display
$13$ \( T + 24 \) Copy content Toggle raw display
$17$ \( T + 16 \) Copy content Toggle raw display
$19$ \( T \) Copy content Toggle raw display
$23$ \( T \) Copy content Toggle raw display
$29$ \( T - 42 \) Copy content Toggle raw display
$31$ \( T \) Copy content Toggle raw display
$37$ \( T - 24 \) Copy content Toggle raw display
$41$ \( T - 80 \) Copy content Toggle raw display
$43$ \( T \) Copy content Toggle raw display
$47$ \( T \) Copy content Toggle raw display
$53$ \( T - 90 \) Copy content Toggle raw display
$59$ \( T \) Copy content Toggle raw display
$61$ \( T - 120 \) Copy content Toggle raw display
$67$ \( T \) Copy content Toggle raw display
$71$ \( T \) Copy content Toggle raw display
$73$ \( T - 110 \) Copy content Toggle raw display
$79$ \( T \) Copy content Toggle raw display
$83$ \( T \) Copy content Toggle raw display
$89$ \( T - 160 \) Copy content Toggle raw display
$97$ \( T - 130 \) Copy content Toggle raw display
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