Properties

Label 3120.1.be.a
Level $3120$
Weight $1$
Character orbit 3120.be
Self dual yes
Analytic conductor $1.557$
Analytic rank $0$
Dimension $1$
Projective image $D_{3}$
CM discriminant -195
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3120,1,Mod(1169,3120)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3120, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 1, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3120.1169");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3120 = 2^{4} \cdot 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3120.be (of order \(2\), degree \(1\), 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: yes
Analytic conductor: \(1.55708283941\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 780)
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.780.1
Artin image: $D_6$
Artin field: Galois closure of 6.0.126547200.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q - q^{3} - q^{5} - q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{3} - q^{5} - q^{7} + q^{9} - q^{11} - q^{13} + q^{15} - q^{17} + q^{21} + q^{23} + q^{25} - q^{27} + q^{33} + q^{35} + q^{37} + q^{39} + q^{41} - q^{45} + q^{51} - q^{53} + q^{55} + 2 q^{59} - q^{61} - q^{63} + q^{65} + 2 q^{67} - q^{69} - q^{71} - 2 q^{73} - q^{75} + q^{77} + q^{79} + q^{81} + q^{85} + q^{89} + q^{91} + q^{97} - q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3120\mathbb{Z}\right)^\times\).

\(n\) \(1951\) \(2081\) \(2341\) \(2497\) \(2641\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1169.1
0
0 −1.00000 0 −1.00000 0 −1.00000 0 1.00000 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
195.e odd 2 1 CM by \(\Q(\sqrt{-195}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3120.1.be.a 1
3.b odd 2 1 3120.1.be.d 1
4.b odd 2 1 780.1.o.c yes 1
5.b even 2 1 3120.1.be.c 1
12.b even 2 1 780.1.o.b yes 1
13.b even 2 1 3120.1.be.b 1
15.d odd 2 1 3120.1.be.b 1
20.d odd 2 1 780.1.o.a 1
20.e even 4 2 3900.1.f.d 2
39.d odd 2 1 3120.1.be.c 1
52.b odd 2 1 780.1.o.d yes 1
60.h even 2 1 780.1.o.d yes 1
60.l odd 4 2 3900.1.f.c 2
65.d even 2 1 3120.1.be.d 1
156.h even 2 1 780.1.o.a 1
195.e odd 2 1 CM 3120.1.be.a 1
260.g odd 2 1 780.1.o.b yes 1
260.p even 4 2 3900.1.f.c 2
780.d even 2 1 780.1.o.c yes 1
780.w odd 4 2 3900.1.f.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
780.1.o.a 1 20.d odd 2 1
780.1.o.a 1 156.h even 2 1
780.1.o.b yes 1 12.b even 2 1
780.1.o.b yes 1 260.g odd 2 1
780.1.o.c yes 1 4.b odd 2 1
780.1.o.c yes 1 780.d even 2 1
780.1.o.d yes 1 52.b odd 2 1
780.1.o.d yes 1 60.h even 2 1
3120.1.be.a 1 1.a even 1 1 trivial
3120.1.be.a 1 195.e odd 2 1 CM
3120.1.be.b 1 13.b even 2 1
3120.1.be.b 1 15.d odd 2 1
3120.1.be.c 1 5.b even 2 1
3120.1.be.c 1 39.d odd 2 1
3120.1.be.d 1 3.b odd 2 1
3120.1.be.d 1 65.d even 2 1
3900.1.f.c 2 60.l odd 4 2
3900.1.f.c 2 260.p even 4 2
3900.1.f.d 2 20.e even 4 2
3900.1.f.d 2 780.w odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(3120, [\chi])\):

\( T_{7} + 1 \) Copy content Toggle raw display
\( T_{11} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T + 1 \) Copy content Toggle raw display
$5$ \( T + 1 \) Copy content Toggle raw display
$7$ \( T + 1 \) 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
$23$ \( T - 1 \) Copy content Toggle raw display
$29$ \( T \) Copy content Toggle raw display
$31$ \( T \) Copy content Toggle raw display
$37$ \( T - 1 \) Copy content Toggle raw display
$41$ \( T - 1 \) 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
$59$ \( T - 2 \) Copy content Toggle raw display
$61$ \( T + 1 \) Copy content Toggle raw display
$67$ \( T - 2 \) Copy content Toggle raw display
$71$ \( T + 1 \) Copy content Toggle raw display
$73$ \( T + 2 \) 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
$97$ \( T - 1 \) Copy content Toggle raw display
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