Properties

Label 3960.1.x.b
Level $3960$
Weight $1$
Character orbit 3960.x
Self dual yes
Analytic conductor $1.976$
Analytic rank $0$
Dimension $1$
Projective image $D_{3}$
CM discriminant -440
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3960,1,Mod(109,3960)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3960, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 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("3960.109");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3960 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3960.x (of order \(2\), degree \(1\), 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: \(1.97629745003\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 440)
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.440.1
Artin image: $D_6$
Artin field: Galois closure of 6.0.41817600.2
Stark unit: Root of $x^{6} - 4407x^{5} - 575652x^{4} - 20581769x^{3} - 575652x^{2} - 4407x + 1$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q - q^{2} + q^{4} + q^{5} - q^{7} - q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{4} + q^{5} - q^{7} - q^{8} - q^{10} + q^{11} + q^{14} + q^{16} + q^{17} + q^{19} + q^{20} - q^{22} + q^{25} - q^{28} - q^{29} - q^{31} - q^{32} - q^{34} - q^{35} + q^{37} - q^{38} - q^{40} + q^{44} - q^{50} - q^{53} + q^{55} + q^{56} + q^{58} + q^{61} + q^{62} + q^{64} - 2 q^{67} + q^{68} + q^{70} + q^{71} + 2 q^{73} - q^{74} + q^{76} - q^{77} + q^{80} + q^{85} - q^{88} + q^{89} + q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3960\mathbb{Z}\right)^\times\).

\(n\) \(991\) \(1981\) \(2377\) \(2521\) \(3521\)
\(\chi(n)\) \(0\) \(1\) \(1\) \(1\) \(0\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
109.1
0
−1.00000 0 1.00000 1.00000 0 −1.00000 −1.00000 0 −1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
440.o odd 2 1 CM by \(\Q(\sqrt{-110}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3960.1.x.b 1
3.b odd 2 1 440.1.o.d yes 1
5.b even 2 1 3960.1.x.c 1
8.b even 2 1 3960.1.x.a 1
11.b odd 2 1 3960.1.x.d 1
12.b even 2 1 1760.1.o.b 1
15.d odd 2 1 440.1.o.a 1
15.e even 4 2 2200.1.d.f 2
24.f even 2 1 1760.1.o.d 1
24.h odd 2 1 440.1.o.c yes 1
33.d even 2 1 440.1.o.b yes 1
40.f even 2 1 3960.1.x.d 1
55.d odd 2 1 3960.1.x.a 1
60.h even 2 1 1760.1.o.c 1
88.b odd 2 1 3960.1.x.c 1
120.i odd 2 1 440.1.o.b yes 1
120.m even 2 1 1760.1.o.a 1
120.w even 4 2 2200.1.d.e 2
132.d odd 2 1 1760.1.o.a 1
165.d even 2 1 440.1.o.c yes 1
165.l odd 4 2 2200.1.d.e 2
264.m even 2 1 440.1.o.a 1
264.p odd 2 1 1760.1.o.c 1
440.o odd 2 1 CM 3960.1.x.b 1
660.g odd 2 1 1760.1.o.d 1
1320.b odd 2 1 1760.1.o.b 1
1320.u even 2 1 440.1.o.d yes 1
1320.bn odd 4 2 2200.1.d.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
440.1.o.a 1 15.d odd 2 1
440.1.o.a 1 264.m even 2 1
440.1.o.b yes 1 33.d even 2 1
440.1.o.b yes 1 120.i odd 2 1
440.1.o.c yes 1 24.h odd 2 1
440.1.o.c yes 1 165.d even 2 1
440.1.o.d yes 1 3.b odd 2 1
440.1.o.d yes 1 1320.u even 2 1
1760.1.o.a 1 120.m even 2 1
1760.1.o.a 1 132.d odd 2 1
1760.1.o.b 1 12.b even 2 1
1760.1.o.b 1 1320.b odd 2 1
1760.1.o.c 1 60.h even 2 1
1760.1.o.c 1 264.p odd 2 1
1760.1.o.d 1 24.f even 2 1
1760.1.o.d 1 660.g odd 2 1
2200.1.d.e 2 120.w even 4 2
2200.1.d.e 2 165.l odd 4 2
2200.1.d.f 2 15.e even 4 2
2200.1.d.f 2 1320.bn odd 4 2
3960.1.x.a 1 8.b even 2 1
3960.1.x.a 1 55.d odd 2 1
3960.1.x.b 1 1.a even 1 1 trivial
3960.1.x.b 1 440.o odd 2 1 CM
3960.1.x.c 1 5.b even 2 1
3960.1.x.c 1 88.b odd 2 1
3960.1.x.d 1 11.b odd 2 1
3960.1.x.d 1 40.f even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(3960, [\chi])\):

\( T_{7} + 1 \) Copy content Toggle raw display
\( T_{17} - 1 \) Copy content Toggle raw display
\( T_{19} - 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 1 \) Copy content Toggle raw display
$3$ \( T \) 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
$13$ \( T \) Copy content Toggle raw display
$17$ \( T - 1 \) Copy content Toggle raw display
$19$ \( T - 1 \) Copy content Toggle raw display
$23$ \( T \) Copy content Toggle raw display
$29$ \( T + 1 \) Copy content Toggle raw display
$31$ \( T + 1 \) Copy content Toggle raw display
$37$ \( T - 1 \) Copy content Toggle raw display
$41$ \( T \) 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
$59$ \( T \) 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
$79$ \( T \) Copy content Toggle raw display
$83$ \( T \) Copy content Toggle raw display
$89$ \( T - 1 \) Copy content Toggle raw display
$97$ \( T \) Copy content Toggle raw display
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