Properties

Label 2808.1.b.a
Level $2808$
Weight $1$
Character orbit 2808.b
Self dual yes
Analytic conductor $1.401$
Analytic rank $0$
Dimension $1$
Projective image $D_{3}$
CM discriminant -312
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2808,1,Mod(701,2808)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2808, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2808.701");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2808 = 2^{3} \cdot 3^{3} \cdot 13 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2808.b (of order \(2\), degree \(1\), 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: \(1.40137455547\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.2808.1
Artin image: $D_6$
Artin field: Galois closure of 6.2.102503232.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q - q^{2} + q^{4} - q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{4} - q^{8} - q^{13} + q^{16} + q^{19} + q^{25} + q^{26} - q^{29} - q^{32} + q^{37} - q^{38} + q^{41} + q^{47} + q^{49} - q^{50} - q^{52} + 2 q^{53} + q^{58} + q^{64} - 2 q^{67} + q^{71} - q^{74} + q^{76} - q^{79} - q^{82} + q^{89} - q^{94} - q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2808\mathbb{Z}\right)^\times\).

\(n\) \(703\) \(1081\) \(1405\) \(2081\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
701.1
0
−1.00000 0 1.00000 0 0 0 −1.00000 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
312.b odd 2 1 CM by \(\Q(\sqrt{-78}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2808.1.b.a 1
3.b odd 2 1 2808.1.b.c yes 1
8.b even 2 1 2808.1.b.b yes 1
13.b even 2 1 2808.1.b.d yes 1
24.h odd 2 1 2808.1.b.d yes 1
39.d odd 2 1 2808.1.b.b yes 1
104.e even 2 1 2808.1.b.c yes 1
312.b odd 2 1 CM 2808.1.b.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2808.1.b.a 1 1.a even 1 1 trivial
2808.1.b.a 1 312.b odd 2 1 CM
2808.1.b.b yes 1 8.b even 2 1
2808.1.b.b yes 1 39.d odd 2 1
2808.1.b.c yes 1 3.b odd 2 1
2808.1.b.c yes 1 104.e even 2 1
2808.1.b.d yes 1 13.b even 2 1
2808.1.b.d yes 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 \(S_{1}^{\mathrm{new}}(2808, [\chi])\):

\( T_{5} \) Copy content Toggle raw display
\( T_{19} - 1 \) Copy content Toggle raw display
\( T_{29} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 1 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T \) Copy content Toggle raw display
$13$ \( T + 1 \) Copy content Toggle raw display
$17$ \( T \) Copy content Toggle raw display
$19$ \( T - 1 \) Copy content Toggle raw display
$23$ \( T \) Copy content Toggle raw display
$29$ \( T + 1 \) Copy content Toggle raw display
$31$ \( T \) Copy content Toggle raw display
$37$ \( T - 1 \) Copy content Toggle raw display
$41$ \( T - 1 \) Copy content Toggle raw display
$43$ \( T \) Copy content Toggle raw display
$47$ \( T - 1 \) Copy content Toggle raw display
$53$ \( T - 2 \) Copy content Toggle raw display
$59$ \( T \) Copy content Toggle raw display
$61$ \( T \) Copy content Toggle raw display
$67$ \( T + 2 \) Copy content Toggle raw display
$71$ \( T - 1 \) Copy content Toggle raw display
$73$ \( T \) Copy content Toggle raw display
$79$ \( T + 1 \) Copy content Toggle raw display
$83$ \( T \) Copy content Toggle raw display
$89$ \( T - 1 \) Copy content Toggle raw display
$97$ \( T \) Copy content Toggle raw display
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