Properties

Label 2304.3.g.t
Level $2304$
Weight $3$
Character orbit 2304.g
Analytic conductor $62.779$
Analytic rank $0$
Dimension $4$
CM discriminant -24
Inner twists $8$

Related objects

Downloads

Learn more

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: no
Analytic conductor: \(62.7794529086\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{7} \)
Twist minimal: no (minimal twist has level 576)
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)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{2} q^{5} - 5 \beta_1 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{5} - 5 \beta_1 q^{7} + \beta_{3} q^{11} + 71 q^{25} - 3 \beta_{2} q^{29} + 19 \beta_1 q^{31} + 5 \beta_{3} q^{35} - 51 q^{49} + 5 \beta_{2} q^{53} - 96 \beta_1 q^{55} + 6 \beta_{3} q^{59} + 50 q^{73} + 20 \beta_{2} q^{77} - 29 \beta_1 q^{79} - 5 \beta_{3} q^{83} - 190 q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 284 q^{25} - 204 q^{49} + 200 q^{73} - 760 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2\nu^{2} ) / 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -4\nu^{3} + 12\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 8\nu^{3} + 24\nu ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + 2\beta_{2} ) / 16 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3\beta_{3} - 6\beta_{2} ) / 16 \) Copy content Toggle raw display

Character values

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

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

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
1.22474 + 1.22474i
1.22474 1.22474i
−1.22474 1.22474i
−1.22474 + 1.22474i
0 0 0 −9.79796 0 10.0000i 0 0 0
1279.2 0 0 0 −9.79796 0 10.0000i 0 0 0
1279.3 0 0 0 9.79796 0 10.0000i 0 0 0
1279.4 0 0 0 9.79796 0 10.0000i 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
24.h odd 2 1 CM by \(\Q(\sqrt{-6}) \)
3.b odd 2 1 inner
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner
12.b even 2 1 inner
24.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2304.3.g.t 4
3.b odd 2 1 inner 2304.3.g.t 4
4.b odd 2 1 inner 2304.3.g.t 4
8.b even 2 1 inner 2304.3.g.t 4
8.d odd 2 1 inner 2304.3.g.t 4
12.b even 2 1 inner 2304.3.g.t 4
16.e even 4 2 576.3.b.b 4
16.f odd 4 2 576.3.b.b 4
24.f even 2 1 inner 2304.3.g.t 4
24.h odd 2 1 CM 2304.3.g.t 4
48.i odd 4 2 576.3.b.b 4
48.k even 4 2 576.3.b.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
576.3.b.b 4 16.e even 4 2
576.3.b.b 4 16.f odd 4 2
576.3.b.b 4 48.i odd 4 2
576.3.b.b 4 48.k even 4 2
2304.3.g.t 4 1.a even 1 1 trivial
2304.3.g.t 4 3.b odd 2 1 inner
2304.3.g.t 4 4.b odd 2 1 inner
2304.3.g.t 4 8.b even 2 1 inner
2304.3.g.t 4 8.d odd 2 1 inner
2304.3.g.t 4 12.b even 2 1 inner
2304.3.g.t 4 24.f even 2 1 inner
2304.3.g.t 4 24.h odd 2 1 CM

Hecke kernels

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

\( T_{5}^{2} - 96 \) Copy content Toggle raw display
\( T_{7}^{2} + 100 \) Copy content Toggle raw display
\( T_{11}^{2} + 384 \) Copy content Toggle raw display
\( T_{13} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} - 96)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 100)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 384)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( (T^{2} - 864)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 1444)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} \) Copy content Toggle raw display
$41$ \( T^{4} \) Copy content Toggle raw display
$43$ \( T^{4} \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( (T^{2} - 2400)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 13824)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} \) Copy content Toggle raw display
$67$ \( T^{4} \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( (T - 50)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} + 3364)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 9600)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} \) Copy content Toggle raw display
$97$ \( (T + 190)^{4} \) Copy content Toggle raw display
show more
show less