Properties

Label 2304.1.q.a
Level $2304$
Weight $1$
Character orbit 2304.q
Analytic conductor $1.150$
Analytic rank $0$
Dimension $2$
Projective image $D_{6}$
CM discriminant -8
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2304,1,Mod(257,2304)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2304, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 5]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2304.257");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2304 = 2^{8} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2304.q (of order \(6\), degree \(2\), 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: \(1.14984578911\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 576)
Projective image: \(D_{6}\)
Projective field: Galois closure of 6.2.10077696.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q + \zeta_{6}^{2} q^{3} - \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{6}^{2} q^{3} - \zeta_{6} q^{9} + (\zeta_{6} + 1) q^{11} + ( - \zeta_{6}^{2} - \zeta_{6}) q^{17} + q^{19} + \zeta_{6}^{2} q^{25} + q^{27} + (\zeta_{6}^{2} - 1) q^{33} + ( - \zeta_{6}^{2} + 1) q^{41} + \zeta_{6}^{2} q^{43} + \zeta_{6} q^{49} + (\zeta_{6} + 1) q^{51} + \zeta_{6}^{2} q^{57} + (\zeta_{6}^{2} - 1) q^{59} + \zeta_{6} q^{67} + q^{73} - \zeta_{6} q^{75} + \zeta_{6}^{2} q^{81} + \zeta_{6}^{2} q^{97} + ( - \zeta_{6}^{2} - \zeta_{6}) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{3} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{3} - q^{9} + 3 q^{11} + 2 q^{19} - q^{25} + 2 q^{27} - 3 q^{33} + 3 q^{41} - q^{43} + q^{49} + 3 q^{51} - q^{57} - 3 q^{59} + q^{67} + 2 q^{73} - q^{75} - q^{81} - q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(1279\) \(1793\) \(2053\)
\(\chi(n)\) \(1\) \(\zeta_{6}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
257.1
0.500000 0.866025i
0.500000 + 0.866025i
0 −0.500000 0.866025i 0 0 0 0 0 −0.500000 + 0.866025i 0
1793.1 0 −0.500000 + 0.866025i 0 0 0 0 0 −0.500000 0.866025i 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 \(\Q(\sqrt{-2}) \)
9.d odd 6 1 inner
72.l even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2304.1.q.a 2
4.b odd 2 1 2304.1.q.b 2
8.b even 2 1 2304.1.q.b 2
8.d odd 2 1 CM 2304.1.q.a 2
9.d odd 6 1 inner 2304.1.q.a 2
16.e even 4 2 576.1.n.a 4
16.f odd 4 2 576.1.n.a 4
36.h even 6 1 2304.1.q.b 2
48.i odd 4 2 1728.1.n.a 4
48.k even 4 2 1728.1.n.a 4
72.j odd 6 1 2304.1.q.b 2
72.l even 6 1 inner 2304.1.q.a 2
144.u even 12 2 576.1.n.a 4
144.v odd 12 2 1728.1.n.a 4
144.w odd 12 2 576.1.n.a 4
144.x even 12 2 1728.1.n.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
576.1.n.a 4 16.e even 4 2
576.1.n.a 4 16.f odd 4 2
576.1.n.a 4 144.u even 12 2
576.1.n.a 4 144.w odd 12 2
1728.1.n.a 4 48.i odd 4 2
1728.1.n.a 4 48.k even 4 2
1728.1.n.a 4 144.v odd 12 2
1728.1.n.a 4 144.x even 12 2
2304.1.q.a 2 1.a even 1 1 trivial
2304.1.q.a 2 8.d odd 2 1 CM
2304.1.q.a 2 9.d odd 6 1 inner
2304.1.q.a 2 72.l even 6 1 inner
2304.1.q.b 2 4.b odd 2 1
2304.1.q.b 2 8.b even 2 1
2304.1.q.b 2 36.h even 6 1
2304.1.q.b 2 72.j odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{2} - 3T_{11} + 3 \) acting on \(S_{1}^{\mathrm{new}}(2304, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 3T + 3 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 3 \) Copy content Toggle raw display
$19$ \( (T - 1)^{2} \) 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
$41$ \( T^{2} - 3T + 3 \) Copy content Toggle raw display
$43$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 3T + 3 \) Copy content Toggle raw display
$61$ \( T^{2} \) Copy content Toggle raw display
$67$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( (T - 1)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + T + 1 \) Copy content Toggle raw display
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