Properties

Label 3344.1.p.f
Level $3344$
Weight $1$
Character orbit 3344.p
Self dual yes
Analytic conductor $1.669$
Analytic rank $0$
Dimension $4$
Projective image $D_{10}$
CM discriminant -836
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

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: yes
Analytic conductor: \(1.66887340224\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{20})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 5x^{2} + 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{10}\)
Projective field: Galois closure of 10.2.21492047227904.1

$q$-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.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} - \beta_{2} q^{5} + (\beta_{2} + 1) q^{7} + (\beta_{2} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} - \beta_{2} q^{5} + (\beta_{2} + 1) q^{7} + (\beta_{2} + 2) q^{9} - q^{11} + \beta_{3} q^{13} + \beta_{3} q^{15} + q^{19} + ( - \beta_{3} - \beta_1) q^{21} - \beta_{2} q^{25} + ( - \beta_{3} - \beta_1) q^{27} + \beta_1 q^{29} - \beta_{3} q^{31} + \beta_1 q^{33} - q^{35} + ( - 2 \beta_{2} - 1) q^{39} - \beta_{3} q^{41} - \beta_{2} q^{43} + ( - \beta_{2} - 1) q^{45} + (\beta_{2} + 1) q^{49} + \beta_{2} q^{55} - \beta_1 q^{57} + (2 \beta_{2} + 3) q^{63} + (\beta_{3} - \beta_1) q^{65} + \beta_{3} q^{67} + \beta_1 q^{71} + \beta_{3} q^{75} + ( - \beta_{2} - 1) q^{77} + (2 \beta_{2} + 2) q^{81} - \beta_{2} q^{83} + ( - \beta_{2} - 3) q^{87} + \beta_1 q^{91} + (2 \beta_{2} + 1) q^{93} - \beta_{2} q^{95} + ( - \beta_{2} - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{5} + 2 q^{7} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{5} + 2 q^{7} + 6 q^{9} - 4 q^{11} + 4 q^{19} + 2 q^{25} - 4 q^{35} + 2 q^{43} - 2 q^{45} + 2 q^{49} - 2 q^{55} + 8 q^{63} - 2 q^{77} + 4 q^{81} + 2 q^{83} - 10 q^{87} + 2 q^{95} - 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of \(\nu = \zeta_{20} + \zeta_{20}^{-1}\):

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 3\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 3\beta_1 \) 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
1.90211
1.17557
−1.17557
−1.90211
0 −1.90211 0 −0.618034 0 1.61803 0 2.61803 0
3343.2 0 −1.17557 0 1.61803 0 −0.618034 0 0.381966 0
3343.3 0 1.17557 0 1.61803 0 −0.618034 0 0.381966 0
3343.4 0 1.90211 0 −0.618034 0 1.61803 0 2.61803 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
836.h odd 2 1 CM by \(\Q(\sqrt{-209}) \)
19.b odd 2 1 inner
44.c even 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.f yes 4
4.b odd 2 1 3344.1.p.e 4
11.b odd 2 1 3344.1.p.e 4
19.b odd 2 1 inner 3344.1.p.f yes 4
44.c even 2 1 inner 3344.1.p.f yes 4
76.d even 2 1 3344.1.p.e 4
209.d even 2 1 3344.1.p.e 4
836.h odd 2 1 CM 3344.1.p.f yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3344.1.p.e 4 4.b odd 2 1
3344.1.p.e 4 11.b odd 2 1
3344.1.p.e 4 76.d even 2 1
3344.1.p.e 4 209.d even 2 1
3344.1.p.f yes 4 1.a even 1 1 trivial
3344.1.p.f yes 4 19.b odd 2 1 inner
3344.1.p.f yes 4 44.c even 2 1 inner
3344.1.p.f yes 4 836.h 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 \(S_{1}^{\mathrm{new}}(3344, [\chi])\):

\( T_{3}^{4} - 5T_{3}^{2} + 5 \) Copy content Toggle raw display
\( T_{5}^{2} - T_{5} - 1 \) Copy content Toggle raw display
\( T_{7}^{2} - T_{7} - 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

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