Properties

Label 2160.1.c.b
Level $2160$
Weight $1$
Character orbit 2160.c
Self dual yes
Analytic conductor $1.078$
Analytic rank $0$
Dimension $1$
Projective image $D_{3}$
CM discriminant -15
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2160,1,Mod(1889,2160)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2160, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 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("2160.1889");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2160 = 2^{4} \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2160.c (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: \(1.07798042729\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 135)
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.135.1
Artin image: $D_6$
Artin field: Galois closure of 6.0.1166400.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q + q^{5}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{5} - q^{17} + q^{19} + q^{23} + q^{25} + q^{31} - 2 q^{47} + q^{49} - q^{53} - q^{61} + q^{79} + q^{83} - q^{85} + q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2160\mathbb{Z}\right)^\times\).

\(n\) \(271\) \(1297\) \(1621\) \(2081\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1889.1
0
0 0 0 1.00000 0 0 0 0 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
15.d odd 2 1 CM by \(\Q(\sqrt{-15}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2160.1.c.b 1
3.b odd 2 1 2160.1.c.a 1
4.b odd 2 1 135.1.d.a 1
5.b even 2 1 2160.1.c.a 1
12.b even 2 1 135.1.d.b yes 1
15.d odd 2 1 CM 2160.1.c.b 1
20.d odd 2 1 135.1.d.b yes 1
20.e even 4 2 675.1.c.c 2
36.f odd 6 2 405.1.h.b 2
36.h even 6 2 405.1.h.a 2
60.h even 2 1 135.1.d.a 1
60.l odd 4 2 675.1.c.c 2
108.j odd 18 6 3645.1.n.d 6
108.l even 18 6 3645.1.n.e 6
180.n even 6 2 405.1.h.b 2
180.p odd 6 2 405.1.h.a 2
180.v odd 12 4 2025.1.j.c 4
180.x even 12 4 2025.1.j.c 4
540.bb even 18 6 3645.1.n.d 6
540.bf odd 18 6 3645.1.n.e 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
135.1.d.a 1 4.b odd 2 1
135.1.d.a 1 60.h even 2 1
135.1.d.b yes 1 12.b even 2 1
135.1.d.b yes 1 20.d odd 2 1
405.1.h.a 2 36.h even 6 2
405.1.h.a 2 180.p odd 6 2
405.1.h.b 2 36.f odd 6 2
405.1.h.b 2 180.n even 6 2
675.1.c.c 2 20.e even 4 2
675.1.c.c 2 60.l odd 4 2
2025.1.j.c 4 180.v odd 12 4
2025.1.j.c 4 180.x even 12 4
2160.1.c.a 1 3.b odd 2 1
2160.1.c.a 1 5.b even 2 1
2160.1.c.b 1 1.a even 1 1 trivial
2160.1.c.b 1 15.d odd 2 1 CM
3645.1.n.d 6 108.j odd 18 6
3645.1.n.d 6 540.bb even 18 6
3645.1.n.e 6 108.l even 18 6
3645.1.n.e 6 540.bf odd 18 6

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(2160, [\chi])\):

\( T_{7} \) Copy content Toggle raw display
\( T_{17} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

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