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Properties

Label 3332.1.o.e
Level $3332$
Weight $1$
Character orbit 3332.o
Analytic conductor $1.663$
Analytic rank $0$
Dimension $4$
Projective image $D_{2}$
CM/RM discs -4, -119, 476
Inner twists $16$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3332,1,Mod(67,3332)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3332, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 4, 3]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3332.67");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3332 = 2^{2} \cdot 7^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3332.o (of order \(6\), degree \(2\), not 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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{2}\)
Projective field: Galois closure of \(\Q(i, \sqrt{119})\)
Artin image: $D_4:C_6$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{24} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{12}^{2} q^{2} + \zeta_{12}^{4} q^{4} + \zeta_{12}^{5} q^{5} + q^{8} + \zeta_{12}^{2} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{12}^{2} q^{2} + \zeta_{12}^{4} q^{4} + \zeta_{12}^{5} q^{5} + q^{8} + \zeta_{12}^{2} q^{9} + 2 \zeta_{12} q^{10} - \zeta_{12}^{2} q^{16} + \zeta_{12} q^{17} - \zeta_{12}^{4} q^{18} - 2 \zeta_{12}^{3} q^{20} - 3 \zeta_{12}^{4} q^{25} + \zeta_{12}^{4} q^{32} - \zeta_{12}^{3} q^{34} - q^{36} + 2 \zeta_{12}^{5} q^{40} + \zeta_{12}^{3} q^{41} - 2 \zeta_{12} q^{45} - 3 q^{50} + \zeta_{12}^{4} q^{53} + \zeta_{12}^{5} q^{61} + q^{64} + \zeta_{12}^{5} q^{68} + \zeta_{12}^{2} q^{72} - \zeta_{12} q^{73} + 2 \zeta_{12} q^{80} + \zeta_{12}^{4} q^{81} - 2 \zeta_{12}^{5} q^{82} - 2 q^{85} + 2 \zeta_{12}^{3} q^{90} - \zeta_{12}^{3} q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{2} - 2 q^{4} + 4 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{2} - 2 q^{4} + 4 q^{8} + 2 q^{9} - 2 q^{16} + 2 q^{18} + 6 q^{25} - 2 q^{32} - 4 q^{36} - 12 q^{50} - 4 q^{53} + 4 q^{64} + 2 q^{72} - 2 q^{81} - 8 q^{85}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

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

\(n\) \(785\) \(885\) \(1667\)
\(\chi(n)\) \(-1\) \(-\zeta_{12}^{2}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
67.1
0.866025 + 0.500000i
−0.866025 − 0.500000i
0.866025 − 0.500000i
−0.866025 + 0.500000i
−0.500000 − 0.866025i 0 −0.500000 + 0.866025i −1.73205 + 1.00000i 0 0 1.00000 0.500000 + 0.866025i 1.73205 + 1.00000i
67.2 −0.500000 − 0.866025i 0 −0.500000 + 0.866025i 1.73205 − 1.00000i 0 0 1.00000 0.500000 + 0.866025i −1.73205 − 1.00000i
2039.1 −0.500000 + 0.866025i 0 −0.500000 − 0.866025i −1.73205 − 1.00000i 0 0 1.00000 0.500000 − 0.866025i 1.73205 − 1.00000i
2039.2 −0.500000 + 0.866025i 0 −0.500000 − 0.866025i 1.73205 + 1.00000i 0 0 1.00000 0.500000 − 0.866025i −1.73205 + 1.00000i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
119.d odd 2 1 CM by \(\Q(\sqrt{-119}) \)
476.e even 2 1 RM by \(\Q(\sqrt{119}) \)
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner
17.b even 2 1 inner
28.d even 2 1 inner
28.f even 6 1 inner
28.g odd 6 1 inner
68.d odd 2 1 inner
119.h odd 6 1 inner
119.j even 6 1 inner
476.o odd 6 1 inner
476.q even 6 1 inner

Twists

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

Hecke kernels

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

\( T_{3} \) Copy content Toggle raw display
\( T_{5}^{4} - 4T_{5}^{2} + 16 \) Copy content Toggle raw display
\( T_{13} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( T^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( (T^{2} + 2 T + 4)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} \) Copy content Toggle raw display
$61$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
$67$ \( T^{4} \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( T^{4} \) Copy content Toggle raw display
$89$ \( T^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
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