Properties

Label 3344.1.p.d
Level $3344$
Weight $1$
Character orbit 3344.p
Analytic conductor $1.669$
Analytic rank $0$
Dimension $2$
Projective image $D_{6}$
CM discriminant -19
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3344,1,Mod(3343,3344)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3344, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 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("3344.3343");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3344 = 2^{4} \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3344.p (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: \(1.66887340224\)
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: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{6}\)
Projective field: Galois closure of 6.2.123005696.3

$q$-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.

\(f(q)\) \(=\) \( q + q^{5} + q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{5} + q^{7} - q^{9} - \zeta_{6}^{2} q^{11} + (\zeta_{6}^{2} + \zeta_{6}) q^{17} + q^{19} + q^{35} + q^{43} - q^{45} + ( - \zeta_{6}^{2} - \zeta_{6}) q^{47} - \zeta_{6}^{2} q^{55} + (\zeta_{6}^{2} + \zeta_{6}) q^{61} - q^{63} + ( - \zeta_{6}^{2} - \zeta_{6}) q^{73} - \zeta_{6}^{2} q^{77} + q^{81} - 2 q^{83} + (\zeta_{6}^{2} + \zeta_{6}) q^{85} + q^{95} + \zeta_{6}^{2} q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{5} + 2 q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{5} + 2 q^{7} - 2 q^{9} + q^{11} + 2 q^{19} + 2 q^{35} + 2 q^{43} - 2 q^{45} + q^{55} - 2 q^{63} + q^{77} + 2 q^{81} - 4 q^{83} + 2 q^{95} - q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3344\mathbb{Z}\right)^\times\).

\(n\) \(705\) \(837\) \(2433\) \(2927\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3343.1
0.500000 + 0.866025i
0.500000 0.866025i
0 0 0 1.00000 0 1.00000 0 −1.00000 0
3343.2 0 0 0 1.00000 0 1.00000 0 −1.00000 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
19.b odd 2 1 CM by \(\Q(\sqrt{-19}) \)
44.c even 2 1 inner
836.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3344.1.p.d yes 2
4.b odd 2 1 3344.1.p.c 2
11.b odd 2 1 3344.1.p.c 2
19.b odd 2 1 CM 3344.1.p.d yes 2
44.c even 2 1 inner 3344.1.p.d yes 2
76.d even 2 1 3344.1.p.c 2
209.d even 2 1 3344.1.p.c 2
836.h odd 2 1 inner 3344.1.p.d yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3344.1.p.c 2 4.b odd 2 1
3344.1.p.c 2 11.b odd 2 1
3344.1.p.c 2 76.d even 2 1
3344.1.p.c 2 209.d even 2 1
3344.1.p.d yes 2 1.a even 1 1 trivial
3344.1.p.d yes 2 19.b odd 2 1 CM
3344.1.p.d yes 2 44.c even 2 1 inner
3344.1.p.d yes 2 836.h odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(3344, [\chi])\):

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