Properties

Label 2304.3.g.j
Level $2304$
Weight $3$
Character orbit 2304.g
Analytic conductor $62.779$
Analytic rank $0$
Dimension $2$
CM discriminant -8
Inner twists $4$

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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: no
Analytic conductor: \(62.7794529086\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 8)
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 \(\beta = 2i\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q+O(q^{10}) \) Copy content Toggle raw display \( q - 7 \beta q^{11} - 2 q^{17} + 17 \beta q^{19} - 25 q^{25} - 46 q^{41} + 7 \beta q^{43} + 49 q^{49} + 41 \beta q^{59} - 31 \beta q^{67} + 142 q^{73} + 79 \beta q^{83} + 146 q^{89} - 94 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{17} - 50 q^{25} - 92 q^{41} + 98 q^{49} + 284 q^{73} + 292 q^{89} - 188 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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
1.00000i
1.00000i
0 0 0 0 0 0 0 0 0
1279.2 0 0 0 0 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
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
4.b odd 2 1 inner
8.b 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.j 2
3.b odd 2 1 256.3.c.e 2
4.b odd 2 1 inner 2304.3.g.j 2
8.b even 2 1 inner 2304.3.g.j 2
8.d odd 2 1 CM 2304.3.g.j 2
12.b even 2 1 256.3.c.e 2
16.e even 4 1 72.3.b.a 1
16.e even 4 1 288.3.b.a 1
16.f odd 4 1 72.3.b.a 1
16.f odd 4 1 288.3.b.a 1
24.f even 2 1 256.3.c.e 2
24.h odd 2 1 256.3.c.e 2
48.i odd 4 1 8.3.d.a 1
48.i odd 4 1 32.3.d.a 1
48.k even 4 1 8.3.d.a 1
48.k even 4 1 32.3.d.a 1
240.t even 4 1 200.3.g.a 1
240.t even 4 1 800.3.g.a 1
240.z odd 4 1 200.3.e.a 2
240.z odd 4 1 800.3.e.a 2
240.bb even 4 1 200.3.e.a 2
240.bb even 4 1 800.3.e.a 2
240.bd odd 4 1 200.3.e.a 2
240.bd odd 4 1 800.3.e.a 2
240.bf even 4 1 200.3.e.a 2
240.bf even 4 1 800.3.e.a 2
240.bm odd 4 1 200.3.g.a 1
240.bm odd 4 1 800.3.g.a 1
336.v odd 4 1 392.3.g.a 1
336.v odd 4 1 1568.3.g.a 1
336.y even 4 1 392.3.g.a 1
336.y even 4 1 1568.3.g.a 1
336.bo even 12 2 392.3.k.b 2
336.br odd 12 2 392.3.k.b 2
336.bt odd 12 2 392.3.k.d 2
336.bu even 12 2 392.3.k.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.3.d.a 1 48.i odd 4 1
8.3.d.a 1 48.k even 4 1
32.3.d.a 1 48.i odd 4 1
32.3.d.a 1 48.k even 4 1
72.3.b.a 1 16.e even 4 1
72.3.b.a 1 16.f odd 4 1
200.3.e.a 2 240.z odd 4 1
200.3.e.a 2 240.bb even 4 1
200.3.e.a 2 240.bd odd 4 1
200.3.e.a 2 240.bf even 4 1
200.3.g.a 1 240.t even 4 1
200.3.g.a 1 240.bm odd 4 1
256.3.c.e 2 3.b odd 2 1
256.3.c.e 2 12.b even 2 1
256.3.c.e 2 24.f even 2 1
256.3.c.e 2 24.h odd 2 1
288.3.b.a 1 16.e even 4 1
288.3.b.a 1 16.f odd 4 1
392.3.g.a 1 336.v odd 4 1
392.3.g.a 1 336.y even 4 1
392.3.k.b 2 336.bo even 12 2
392.3.k.b 2 336.br odd 12 2
392.3.k.d 2 336.bt odd 12 2
392.3.k.d 2 336.bu even 12 2
800.3.e.a 2 240.z odd 4 1
800.3.e.a 2 240.bb even 4 1
800.3.e.a 2 240.bd odd 4 1
800.3.e.a 2 240.bf even 4 1
800.3.g.a 1 240.t even 4 1
800.3.g.a 1 240.bm odd 4 1
1568.3.g.a 1 336.v odd 4 1
1568.3.g.a 1 336.y even 4 1
2304.3.g.j 2 1.a even 1 1 trivial
2304.3.g.j 2 4.b odd 2 1 inner
2304.3.g.j 2 8.b even 2 1 inner
2304.3.g.j 2 8.d 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 \(S_{3}^{\mathrm{new}}(2304, [\chi])\):

\( T_{5} \) Copy content Toggle raw display
\( T_{7} \) Copy content Toggle raw display
\( T_{11}^{2} + 196 \) Copy content Toggle raw display
\( T_{13} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 196 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( (T + 2)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 1156 \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} \) Copy content Toggle raw display
$41$ \( (T + 46)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 196 \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 6724 \) Copy content Toggle raw display
$61$ \( T^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 3844 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( (T - 142)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 24964 \) Copy content Toggle raw display
$89$ \( (T - 146)^{2} \) Copy content Toggle raw display
$97$ \( (T + 94)^{2} \) Copy content Toggle raw display
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