Properties

Label 3267.1.c.a
Level $3267$
Weight $1$
Character orbit 3267.c
Analytic conductor $1.630$
Analytic rank $0$
Dimension $4$
Projective image $D_{12}$
CM discriminant -3
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3267,1,Mod(2782,3267)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3267, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3267.2782");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3267 = 3^{3} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3267.c (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: no
Analytic conductor: \(1.63044539627\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 4x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{12}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{12} + \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q + q^{4} - \beta_1 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{4} - \beta_1 q^{7} - \beta_{3} q^{13} + q^{16} + ( - \beta_{3} + \beta_1) q^{19} - q^{25} - \beta_1 q^{28} - \beta_{2} q^{31} + (\beta_{3} - \beta_1) q^{43} + (\beta_{2} - 1) q^{49} - \beta_{3} q^{52} + (\beta_{3} - \beta_1) q^{61} + q^{64} + \beta_{2} q^{67} - \beta_{3} q^{73} + ( - \beta_{3} + \beta_1) q^{76} + \beta_{3} q^{79} - q^{91} + q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{4} + 4 q^{16} - 4 q^{25} - 4 q^{49} + 4 q^{64} - 4 q^{91} + 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 4x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} + 4\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - 4\beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3267\mathbb{Z}\right)^\times\).

\(n\) \(244\) \(3026\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2782.1
1.93185i
0.517638i
0.517638i
1.93185i
0 0 1.00000 0 0 1.93185i 0 0 0
2782.2 0 0 1.00000 0 0 0.517638i 0 0 0
2782.3 0 0 1.00000 0 0 0.517638i 0 0 0
2782.4 0 0 1.00000 0 0 1.93185i 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
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
11.b odd 2 1 inner
33.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3267.1.c.a 4
3.b odd 2 1 CM 3267.1.c.a 4
11.b odd 2 1 inner 3267.1.c.a 4
11.c even 5 4 3267.1.l.a 16
11.d odd 10 4 3267.1.l.a 16
33.d even 2 1 inner 3267.1.c.a 4
33.f even 10 4 3267.1.l.a 16
33.h odd 10 4 3267.1.l.a 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3267.1.c.a 4 1.a even 1 1 trivial
3267.1.c.a 4 3.b odd 2 1 CM
3267.1.c.a 4 11.b odd 2 1 inner
3267.1.c.a 4 33.d even 2 1 inner
3267.1.l.a 16 11.c even 5 4
3267.1.l.a 16 11.d odd 10 4
3267.1.l.a 16 33.f even 10 4
3267.1.l.a 16 33.h odd 10 4

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(3267, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 4T^{2} + 1 \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} + 4T^{2} + 1 \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} + 2)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} - 3)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} \) Copy content Toggle raw display
$41$ \( T^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 2)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( T^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} + 2)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} - 3)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( T^{4} + 4T^{2} + 1 \) Copy content Toggle raw display
$79$ \( T^{4} + 4T^{2} + 1 \) Copy content Toggle raw display
$83$ \( T^{4} \) Copy content Toggle raw display
$89$ \( T^{4} \) Copy content Toggle raw display
$97$ \( (T - 1)^{4} \) Copy content Toggle raw display
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