Properties

Label 3332.1.g.e
Level $3332$
Weight $1$
Character orbit 3332.g
Self dual yes
Analytic conductor $1.663$
Analytic rank $0$
Dimension $1$
Projective image $D_{3}$
CM discriminant -68
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3332,1,Mod(883,3332)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3332, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 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("3332.883");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3332 = 2^{2} \cdot 7^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3332.g (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: \(1.66288462209\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 476)
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.3332.1
Artin image: $D_6$
Artin field: Galois closure of 6.0.44408896.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q + q^{2} + q^{3} + q^{4} + q^{6} + q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + q^{3} + q^{4} + q^{6} + q^{8} + q^{11} + q^{12} - q^{13} + q^{16} + q^{17} + q^{22} - 2 q^{23} + q^{24} + q^{25} - q^{26} - q^{27} - 2 q^{31} + q^{32} + q^{33} + q^{34} - q^{39} + q^{44} - 2 q^{46} + q^{48} + q^{50} + q^{51} - q^{52} - q^{53} - q^{54} - 2 q^{62} + q^{64} + q^{66} + q^{68} - 2 q^{69} + q^{71} + q^{75} - q^{78} + q^{79} - q^{81} + q^{88} - q^{89} - 2 q^{92} - 2 q^{93} + q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3332\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(885\) \(1667\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
883.1
0
1.00000 1.00000 1.00000 0 1.00000 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
68.d odd 2 1 CM by \(\Q(\sqrt{-17}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3332.1.g.e 1
4.b odd 2 1 3332.1.g.b 1
7.b odd 2 1 3332.1.g.c 1
7.c even 3 2 476.1.o.a 2
7.d odd 6 2 3332.1.o.b 2
17.b even 2 1 3332.1.g.b 1
28.d even 2 1 3332.1.g.d 1
28.f even 6 2 3332.1.o.a 2
28.g odd 6 2 476.1.o.b yes 2
68.d odd 2 1 CM 3332.1.g.e 1
119.d odd 2 1 3332.1.g.d 1
119.h odd 6 2 3332.1.o.a 2
119.j even 6 2 476.1.o.b yes 2
476.e even 2 1 3332.1.g.c 1
476.o odd 6 2 476.1.o.a 2
476.q even 6 2 3332.1.o.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
476.1.o.a 2 7.c even 3 2
476.1.o.a 2 476.o odd 6 2
476.1.o.b yes 2 28.g odd 6 2
476.1.o.b yes 2 119.j even 6 2
3332.1.g.b 1 4.b odd 2 1
3332.1.g.b 1 17.b even 2 1
3332.1.g.c 1 7.b odd 2 1
3332.1.g.c 1 476.e even 2 1
3332.1.g.d 1 28.d even 2 1
3332.1.g.d 1 119.d odd 2 1
3332.1.g.e 1 1.a even 1 1 trivial
3332.1.g.e 1 68.d odd 2 1 CM
3332.1.o.a 2 28.f even 6 2
3332.1.o.a 2 119.h odd 6 2
3332.1.o.b 2 7.d odd 6 2
3332.1.o.b 2 476.q even 6 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(3332, [\chi])\):

\( T_{3} - 1 \) Copy content Toggle raw display
\( T_{5} \) Copy content Toggle raw display
\( T_{11} - 1 \) Copy content Toggle raw display
\( T_{13} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

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