Properties

Label 3920.1.bt.e
Level $3920$
Weight $1$
Character orbit 3920.bt
Analytic conductor $1.956$
Analytic rank $0$
Dimension $4$
Projective image $D_{6}$
CM discriminant -35
Inner twists $8$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3920,1,Mod(79,3920)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3920, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 3, 2]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3920.79");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3920 = 2^{4} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3920.bt (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: \(1.95633484952\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{6}\)
Projective field: Galois closure of 6.0.313600.1

$q$-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.

\(f(q)\) \(=\) \( q + ( - \zeta_{12}^{3} - \zeta_{12}) q^{3} + \zeta_{12} q^{5} + (\zeta_{12}^{4} + \zeta_{12}^{2} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{12}^{3} - \zeta_{12}) q^{3} + \zeta_{12} q^{5} + (\zeta_{12}^{4} + \zeta_{12}^{2} - 1) q^{9} + ( - \zeta_{12}^{4} + 1) q^{11} - \zeta_{12}^{3} q^{13} + ( - \zeta_{12}^{4} - \zeta_{12}^{2}) q^{15} + \zeta_{12}^{5} q^{17} + \zeta_{12}^{2} q^{25} + ( - \zeta_{12}^{5} - \zeta_{12}) q^{27} + q^{29} + (\zeta_{12}^{5} - \zeta_{12}^{3} - \zeta_{12}) q^{33} + (\zeta_{12}^{4} - 1) q^{39} + (\zeta_{12}^{5} + \zeta_{12}^{3} - \zeta_{12}) q^{45} + ( - \zeta_{12}^{5} - \zeta_{12}^{3}) q^{47} + (\zeta_{12}^{2} + 1) q^{51} + ( - \zeta_{12}^{5} + \zeta_{12}) q^{55} - \zeta_{12}^{4} q^{65} - \zeta_{12}^{5} q^{73} + ( - \zeta_{12}^{5} - \zeta_{12}^{3}) q^{75} + ( - \zeta_{12}^{2} - 1) q^{79} + (\zeta_{12}^{2} + 1) q^{81} - q^{85} + ( - \zeta_{12}^{3} - \zeta_{12}) q^{87} - \zeta_{12}^{3} q^{97} + (2 \zeta_{12}^{4} + \zeta_{12}^{2} + 1) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{9} + 6 q^{11} + 2 q^{25} + 4 q^{29} - 6 q^{39} + 6 q^{51} + 2 q^{65} - 6 q^{79} - 2 q^{81} - 4 q^{85}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3920\mathbb{Z}\right)^\times\).

\(n\) \(981\) \(1471\) \(3041\) \(3137\)
\(\chi(n)\) \(1\) \(-1\) \(\zeta_{12}^{4}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
79.1
0.866025 + 0.500000i
−0.866025 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
0 −0.866025 1.50000i 0 0.866025 + 0.500000i 0 0 0 −1.00000 + 1.73205i 0
79.2 0 0.866025 + 1.50000i 0 −0.866025 0.500000i 0 0 0 −1.00000 + 1.73205i 0
1439.1 0 −0.866025 + 1.50000i 0 0.866025 0.500000i 0 0 0 −1.00000 1.73205i 0
1439.2 0 0.866025 1.50000i 0 −0.866025 + 0.500000i 0 0 0 −1.00000 1.73205i 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
35.c odd 2 1 CM by \(\Q(\sqrt{-35}) \)
5.b even 2 1 inner
7.b odd 2 1 inner
28.f even 6 1 inner
28.g odd 6 1 inner
140.p odd 6 1 inner
140.s even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3920.1.bt.e 4
4.b odd 2 1 3920.1.bt.d 4
5.b even 2 1 inner 3920.1.bt.e 4
7.b odd 2 1 inner 3920.1.bt.e 4
7.c even 3 1 3920.1.j.e 4
7.c even 3 1 3920.1.bt.d 4
7.d odd 6 1 3920.1.j.e 4
7.d odd 6 1 3920.1.bt.d 4
20.d odd 2 1 3920.1.bt.d 4
28.d even 2 1 3920.1.bt.d 4
28.f even 6 1 3920.1.j.e 4
28.f even 6 1 inner 3920.1.bt.e 4
28.g odd 6 1 3920.1.j.e 4
28.g odd 6 1 inner 3920.1.bt.e 4
35.c odd 2 1 CM 3920.1.bt.e 4
35.i odd 6 1 3920.1.j.e 4
35.i odd 6 1 3920.1.bt.d 4
35.j even 6 1 3920.1.j.e 4
35.j even 6 1 3920.1.bt.d 4
140.c even 2 1 3920.1.bt.d 4
140.p odd 6 1 3920.1.j.e 4
140.p odd 6 1 inner 3920.1.bt.e 4
140.s even 6 1 3920.1.j.e 4
140.s even 6 1 inner 3920.1.bt.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3920.1.j.e 4 7.c even 3 1
3920.1.j.e 4 7.d odd 6 1
3920.1.j.e 4 28.f even 6 1
3920.1.j.e 4 28.g odd 6 1
3920.1.j.e 4 35.i odd 6 1
3920.1.j.e 4 35.j even 6 1
3920.1.j.e 4 140.p odd 6 1
3920.1.j.e 4 140.s even 6 1
3920.1.bt.d 4 4.b odd 2 1
3920.1.bt.d 4 7.c even 3 1
3920.1.bt.d 4 7.d odd 6 1
3920.1.bt.d 4 20.d odd 2 1
3920.1.bt.d 4 28.d even 2 1
3920.1.bt.d 4 35.i odd 6 1
3920.1.bt.d 4 35.j even 6 1
3920.1.bt.d 4 140.c even 2 1
3920.1.bt.e 4 1.a even 1 1 trivial
3920.1.bt.e 4 5.b even 2 1 inner
3920.1.bt.e 4 7.b odd 2 1 inner
3920.1.bt.e 4 28.f even 6 1 inner
3920.1.bt.e 4 28.g odd 6 1 inner
3920.1.bt.e 4 35.c odd 2 1 CM
3920.1.bt.e 4 140.p odd 6 1 inner
3920.1.bt.e 4 140.s 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 \(S_{1}^{\mathrm{new}}(3920, [\chi])\):

\( T_{3}^{4} + 3T_{3}^{2} + 9 \) Copy content Toggle raw display
\( T_{11}^{2} - 3T_{11} + 3 \) Copy content Toggle raw display
\( T_{13}^{2} + 1 \) Copy content Toggle raw display
\( T_{41} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$5$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} - 3 T + 3)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 1)^{2} \) 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
$29$ \( (T - 1)^{4} \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( T^{4} \) Copy content Toggle raw display
$41$ \( T^{4} \) Copy content Toggle raw display
$43$ \( T^{4} \) Copy content Toggle raw display
$47$ \( T^{4} + 3T^{2} + 9 \) 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
$67$ \( T^{4} \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
$79$ \( (T^{2} + 3 T + 3)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} \) Copy content Toggle raw display
$89$ \( T^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
show more
show less