Properties

Label 3332.1.g.i
Level $3332$
Weight $1$
Character orbit 3332.g
Analytic conductor $1.663$
Analytic rank $0$
Dimension $8$
Projective image $D_{10}$
CM discriminant -119
Inner twists $8$

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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: no
Analytic conductor: \(1.66288462209\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{20})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{6} + x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{10}\)
Projective field: Galois closure of 10.0.205346735104.4

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{20}^{6} q^{2} + ( - \zeta_{20}^{9} + \zeta_{20}) q^{3} - \zeta_{20}^{2} q^{4} + ( - \zeta_{20}^{7} - \zeta_{20}^{3}) q^{5} + ( - \zeta_{20}^{7} - \zeta_{20}^{5}) q^{6} + \zeta_{20}^{8} q^{8} + ( - \zeta_{20}^{8} + \zeta_{20}^{2} + 1) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{20}^{6} q^{2} + ( - \zeta_{20}^{9} + \zeta_{20}) q^{3} - \zeta_{20}^{2} q^{4} + ( - \zeta_{20}^{7} - \zeta_{20}^{3}) q^{5} + ( - \zeta_{20}^{7} - \zeta_{20}^{5}) q^{6} + \zeta_{20}^{8} q^{8} + ( - \zeta_{20}^{8} + \zeta_{20}^{2} + 1) q^{9} + (\zeta_{20}^{9} - \zeta_{20}^{3}) q^{10} + ( - \zeta_{20}^{3} - \zeta_{20}) q^{12} + ( - \zeta_{20}^{8} + \cdots - \zeta_{20}^{2}) q^{15} + \cdots + ( - \zeta_{20}^{7} - \zeta_{20}^{3}) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{2} - 2 q^{4} - 2 q^{8} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 2 q^{2} - 2 q^{4} - 2 q^{8} + 12 q^{9} - 2 q^{16} + 2 q^{18} - 4 q^{25} - 10 q^{30} + 8 q^{32} - 8 q^{36} - 4 q^{50} - 4 q^{53} - 10 q^{60} - 2 q^{64} - 8 q^{72} + 8 q^{81} + 4 q^{85} + 10 q^{86}+O(q^{100}) \) 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
−0.587785 + 0.809017i
0.587785 0.809017i
−0.587785 0.809017i
0.587785 + 0.809017i
−0.951057 0.309017i
0.951057 + 0.309017i
−0.951057 + 0.309017i
0.951057 0.309017i
−0.809017 0.587785i −1.17557 0.309017 + 0.951057i 0.618034i 0.951057 + 0.690983i 0 0.309017 0.951057i 0.381966 −0.363271 + 0.500000i
883.2 −0.809017 0.587785i 1.17557 0.309017 + 0.951057i 0.618034i −0.951057 0.690983i 0 0.309017 0.951057i 0.381966 0.363271 0.500000i
883.3 −0.809017 + 0.587785i −1.17557 0.309017 0.951057i 0.618034i 0.951057 0.690983i 0 0.309017 + 0.951057i 0.381966 −0.363271 0.500000i
883.4 −0.809017 + 0.587785i 1.17557 0.309017 0.951057i 0.618034i −0.951057 + 0.690983i 0 0.309017 + 0.951057i 0.381966 0.363271 + 0.500000i
883.5 0.309017 0.951057i −1.90211 −0.809017 0.587785i 1.61803i −0.587785 + 1.80902i 0 −0.809017 + 0.587785i 2.61803 1.53884 + 0.500000i
883.6 0.309017 0.951057i 1.90211 −0.809017 0.587785i 1.61803i 0.587785 1.80902i 0 −0.809017 + 0.587785i 2.61803 −1.53884 0.500000i
883.7 0.309017 + 0.951057i −1.90211 −0.809017 + 0.587785i 1.61803i −0.587785 1.80902i 0 −0.809017 0.587785i 2.61803 1.53884 0.500000i
883.8 0.309017 + 0.951057i 1.90211 −0.809017 + 0.587785i 1.61803i 0.587785 + 1.80902i 0 −0.809017 0.587785i 2.61803 −1.53884 + 0.500000i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 883.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
119.d odd 2 1 CM by \(\Q(\sqrt{-119}) \)
4.b odd 2 1 inner
7.b odd 2 1 inner
17.b even 2 1 inner
28.d even 2 1 inner
68.d odd 2 1 inner
476.e even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3332.1.g.i 8
4.b odd 2 1 inner 3332.1.g.i 8
7.b odd 2 1 inner 3332.1.g.i 8
7.c even 3 2 3332.1.o.g 16
7.d odd 6 2 3332.1.o.g 16
17.b even 2 1 inner 3332.1.g.i 8
28.d even 2 1 inner 3332.1.g.i 8
28.f even 6 2 3332.1.o.g 16
28.g odd 6 2 3332.1.o.g 16
68.d odd 2 1 inner 3332.1.g.i 8
119.d odd 2 1 CM 3332.1.g.i 8
119.h odd 6 2 3332.1.o.g 16
119.j even 6 2 3332.1.o.g 16
476.e even 2 1 inner 3332.1.g.i 8
476.o odd 6 2 3332.1.o.g 16
476.q even 6 2 3332.1.o.g 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3332.1.g.i 8 1.a even 1 1 trivial
3332.1.g.i 8 4.b odd 2 1 inner
3332.1.g.i 8 7.b odd 2 1 inner
3332.1.g.i 8 17.b even 2 1 inner
3332.1.g.i 8 28.d even 2 1 inner
3332.1.g.i 8 68.d odd 2 1 inner
3332.1.g.i 8 119.d odd 2 1 CM
3332.1.g.i 8 476.e even 2 1 inner
3332.1.o.g 16 7.c even 3 2
3332.1.o.g 16 7.d odd 6 2
3332.1.o.g 16 28.f even 6 2
3332.1.o.g 16 28.g odd 6 2
3332.1.o.g 16 119.h odd 6 2
3332.1.o.g 16 119.j even 6 2
3332.1.o.g 16 476.o odd 6 2
3332.1.o.g 16 476.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 \(S_{1}^{\mathrm{new}}(3332, [\chi])\):

\( T_{3}^{4} - 5T_{3}^{2} + 5 \) Copy content Toggle raw display
\( T_{5}^{4} + 3T_{5}^{2} + 1 \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display
\( T_{13} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + T^{3} + T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$3$ \( (T^{4} - 5 T^{2} + 5)^{2} \) Copy content Toggle raw display
$5$ \( (T^{4} + 3 T^{2} + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$19$ \( T^{8} \) Copy content Toggle raw display
$23$ \( T^{8} \) Copy content Toggle raw display
$29$ \( T^{8} \) Copy content Toggle raw display
$31$ \( (T^{4} - 5 T^{2} + 5)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} \) Copy content Toggle raw display
$41$ \( (T^{4} + 3 T^{2} + 1)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 5 T^{2} + 5)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} \) Copy content Toggle raw display
$53$ \( (T^{2} + T - 1)^{4} \) Copy content Toggle raw display
$59$ \( T^{8} \) Copy content Toggle raw display
$61$ \( (T^{4} + 3 T^{2} + 1)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 5 T^{2} + 5)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} \) Copy content Toggle raw display
$73$ \( (T^{4} + 3 T^{2} + 1)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} \) Copy content Toggle raw display
$83$ \( T^{8} \) Copy content Toggle raw display
$89$ \( T^{8} \) Copy content Toggle raw display
$97$ \( (T^{4} + 3 T^{2} + 1)^{2} \) Copy content Toggle raw display
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