Properties

Label 3332.1.w.c
Level $3332$
Weight $1$
Character orbit 3332.w
Analytic conductor $1.663$
Analytic rank $0$
Dimension $4$
Projective image $D_{8}$
CM discriminant -4
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3332,1,Mod(491,3332)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3332, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([4, 0, 3]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3332.491");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3332 = 2^{2} \cdot 7^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3332.w (of order \(8\), degree \(4\), 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: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{8}\)
Projective field: Galois closure of 8.2.3089659810545728.4

$q$-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.

\(f(q)\) \(=\) \( q + \zeta_{8} q^{2} + \zeta_{8}^{2} q^{4} + ( - \zeta_{8}^{2} - \zeta_{8}) q^{5} + \zeta_{8}^{3} q^{8} - \zeta_{8}^{3} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{8} q^{2} + \zeta_{8}^{2} q^{4} + ( - \zeta_{8}^{2} - \zeta_{8}) q^{5} + \zeta_{8}^{3} q^{8} - \zeta_{8}^{3} q^{9} + ( - \zeta_{8}^{3} - \zeta_{8}^{2}) q^{10} - q^{16} - \zeta_{8} q^{17} + q^{18} + ( - \zeta_{8}^{3} + 1) q^{20} + (\zeta_{8}^{3} + \zeta_{8}^{2} - 1) q^{25} + ( - \zeta_{8}^{2} - \zeta_{8}) q^{29} - \zeta_{8} q^{32} - \zeta_{8}^{2} q^{34} + \zeta_{8} q^{36} + ( - \zeta_{8}^{3} + 1) q^{37} + (\zeta_{8} + 1) q^{40} + ( - \zeta_{8} + 1) q^{41} + ( - \zeta_{8} - 1) q^{45} + (\zeta_{8}^{3} - \zeta_{8} - 1) q^{50} + ( - \zeta_{8}^{2} - 1) q^{53} + ( - \zeta_{8}^{3} - \zeta_{8}^{2}) q^{58} + ( - \zeta_{8} + 1) q^{61} - \zeta_{8}^{2} q^{64} - \zeta_{8}^{3} q^{68} + \zeta_{8}^{2} q^{72} + (\zeta_{8}^{2} + \zeta_{8}) q^{73} + (\zeta_{8} + 1) q^{74} + (\zeta_{8}^{2} + \zeta_{8}) q^{80} - \zeta_{8}^{2} q^{81} + ( - \zeta_{8}^{2} + \zeta_{8}) q^{82} + (\zeta_{8}^{3} + \zeta_{8}^{2}) q^{85} + \zeta_{8}^{2} q^{89} + ( - \zeta_{8}^{2} - \zeta_{8}) q^{90} + ( - \zeta_{8}^{2} - \zeta_{8}) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{16} + 4 q^{18} + 4 q^{20} - 4 q^{25} + 4 q^{37} + 4 q^{40} + 4 q^{41} - 4 q^{45} - 4 q^{50} - 4 q^{53} + 4 q^{61} + 4 q^{74}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(785\) \(885\) \(1667\)
\(\chi(n)\) \(-\zeta_{8}^{3}\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
491.1
−0.707107 0.707107i
−0.707107 + 0.707107i
0.707107 0.707107i
0.707107 + 0.707107i
−0.707107 0.707107i 0 1.00000i 0.707107 0.292893i 0 0 0.707107 0.707107i −0.707107 + 0.707107i −0.707107 0.292893i
1079.1 −0.707107 + 0.707107i 0 1.00000i 0.707107 + 0.292893i 0 0 0.707107 + 0.707107i −0.707107 0.707107i −0.707107 + 0.292893i
1471.1 0.707107 0.707107i 0 1.00000i −0.707107 + 1.70711i 0 0 −0.707107 0.707107i 0.707107 + 0.707107i 0.707107 + 1.70711i
2059.1 0.707107 + 0.707107i 0 1.00000i −0.707107 1.70711i 0 0 −0.707107 + 0.707107i 0.707107 0.707107i 0.707107 1.70711i
\(n\): 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 \(\Q(\sqrt{-1}) \)
17.d even 8 1 inner
68.g odd 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3332.1.w.c yes 4
4.b odd 2 1 CM 3332.1.w.c yes 4
7.b odd 2 1 3332.1.w.b 4
7.c even 3 2 3332.1.bp.c 8
7.d odd 6 2 3332.1.bp.b 8
17.d even 8 1 inner 3332.1.w.c yes 4
28.d even 2 1 3332.1.w.b 4
28.f even 6 2 3332.1.bp.b 8
28.g odd 6 2 3332.1.bp.c 8
68.g odd 8 1 inner 3332.1.w.c yes 4
119.l odd 8 1 3332.1.w.b 4
119.q even 24 2 3332.1.bp.c 8
119.r odd 24 2 3332.1.bp.b 8
476.w even 8 1 3332.1.w.b 4
476.bg odd 24 2 3332.1.bp.c 8
476.bj even 24 2 3332.1.bp.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3332.1.w.b 4 7.b odd 2 1
3332.1.w.b 4 28.d even 2 1
3332.1.w.b 4 119.l odd 8 1
3332.1.w.b 4 476.w even 8 1
3332.1.w.c yes 4 1.a even 1 1 trivial
3332.1.w.c yes 4 4.b odd 2 1 CM
3332.1.w.c yes 4 17.d even 8 1 inner
3332.1.w.c yes 4 68.g odd 8 1 inner
3332.1.bp.b 8 7.d odd 6 2
3332.1.bp.b 8 28.f even 6 2
3332.1.bp.b 8 119.r odd 24 2
3332.1.bp.b 8 476.bj even 24 2
3332.1.bp.c 8 7.c even 3 2
3332.1.bp.c 8 28.g odd 6 2
3332.1.bp.c 8 119.q even 24 2
3332.1.bp.c 8 476.bg odd 24 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 2T_{5}^{2} - 4T_{5} + 2 \) acting on \(S_{1}^{\mathrm{new}}(3332, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 1 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 2 T^{2} - 4 T + 2 \) 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
$17$ \( T^{4} + 1 \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( T^{4} + 2 T^{2} - 4 T + 2 \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( T^{4} - 4 T^{3} + 6 T^{2} - 4 T + 2 \) Copy content Toggle raw display
$41$ \( T^{4} - 4 T^{3} + 6 T^{2} - 4 T + 2 \) Copy content Toggle raw display
$43$ \( T^{4} \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( (T^{2} + 2 T + 2)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} \) Copy content Toggle raw display
$61$ \( T^{4} - 4 T^{3} + 6 T^{2} - 4 T + 2 \) Copy content Toggle raw display
$67$ \( T^{4} \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( T^{4} + 2 T^{2} + 4 T + 2 \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( T^{4} \) Copy content Toggle raw display
$89$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 2 T^{2} - 4 T + 2 \) Copy content Toggle raw display
show more
show less