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Properties

Label 2808.1.b.d
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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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2808,1,Mod(701,2808)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://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);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2808, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1, 1])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
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

Copy content comment:select newform
 
Copy content sage:traces = [1,1,0,1,0,0,0,1,0,0,0,0,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content 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: $S_3$
Artin field: Galois closure of 3.1.2808.1
Stark unit: Root of $x^{3} - 13982913x^{2} + 7437x - 1$

$q$-expansion

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

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

\( T_{5} \) 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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