Properties

Label 3960.1.x.i
Level $3960$
Weight $1$
Character orbit 3960.x
Analytic conductor $1.976$
Analytic rank $0$
Dimension $4$
Projective image $D_{4}$
CM discriminant -55
Inner twists $8$

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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: no
Analytic conductor: \(1.97629745003\)
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: no (minimal twist has level 440)
Projective image: \(D_{4}\)
Projective field: Galois closure of 4.0.38720.1

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{8} q^{2} + \zeta_{8}^{2} q^{4} + \zeta_{8}^{2} q^{5} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{7} - \zeta_{8}^{3} q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{8} q^{2} + \zeta_{8}^{2} q^{4} + \zeta_{8}^{2} q^{5} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{7} - \zeta_{8}^{3} q^{8} - \zeta_{8}^{3} q^{10} + \zeta_{8}^{2} q^{11} + (\zeta_{8}^{3} + \zeta_{8}) q^{13} + ( - \zeta_{8}^{2} - 1) q^{14} - q^{16} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{17} - q^{20} - \zeta_{8}^{3} q^{22} - q^{25} + ( - \zeta_{8}^{2} + 1) q^{26} + (\zeta_{8}^{3} + \zeta_{8}) q^{28} + \zeta_{8} q^{32} + ( - \zeta_{8}^{2} - 1) q^{34} + (\zeta_{8}^{3} + \zeta_{8}) q^{35} + \zeta_{8} q^{40} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{43} - q^{44} + q^{49} + \zeta_{8} q^{50} + (\zeta_{8}^{3} - \zeta_{8}) q^{52} - q^{55} + ( - \zeta_{8}^{2} + 1) q^{56} - \zeta_{8}^{2} q^{64} + (\zeta_{8}^{3} - \zeta_{8}) q^{65} + (\zeta_{8}^{3} + \zeta_{8}) q^{68} + ( - \zeta_{8}^{2} + 1) q^{70} - 2 q^{71} + (\zeta_{8}^{3} - \zeta_{8}) q^{73} + (\zeta_{8}^{3} + \zeta_{8}) q^{77} - \zeta_{8}^{2} q^{80} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{83} + (\zeta_{8}^{3} + \zeta_{8}) q^{85} + (\zeta_{8}^{2} - 1) q^{86} + \zeta_{8} q^{88} + 2 \zeta_{8}^{2} q^{91} - \zeta_{8} q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{14} - 4 q^{16} - 4 q^{20} - 4 q^{25} + 4 q^{26} - 4 q^{34} - 4 q^{44} + 4 q^{49} - 4 q^{55} + 4 q^{56} + 4 q^{70} - 8 q^{71} - 4 q^{86}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
109.1
0.707107 + 0.707107i
0.707107 0.707107i
−0.707107 + 0.707107i
−0.707107 0.707107i
−0.707107 0.707107i 0 1.00000i 1.00000i 0 1.41421 0.707107 0.707107i 0 0.707107 0.707107i
109.2 −0.707107 + 0.707107i 0 1.00000i 1.00000i 0 1.41421 0.707107 + 0.707107i 0 0.707107 + 0.707107i
109.3 0.707107 0.707107i 0 1.00000i 1.00000i 0 −1.41421 −0.707107 0.707107i 0 −0.707107 0.707107i
109.4 0.707107 + 0.707107i 0 1.00000i 1.00000i 0 −1.41421 −0.707107 + 0.707107i 0 −0.707107 + 0.707107i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
55.d odd 2 1 CM by \(\Q(\sqrt{-55}) \)
5.b even 2 1 inner
8.b even 2 1 inner
11.b odd 2 1 inner
40.f even 2 1 inner
88.b odd 2 1 inner
440.o odd 2 1 inner

Twists

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

Hecke kernels

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

\( T_{7}^{2} - 2 \) Copy content Toggle raw display
\( T_{17}^{2} - 2 \) Copy content Toggle raw display
\( T_{19} \) 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
$5$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} - 2)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 2)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 2)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( T^{4} \) 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
$43$ \( (T^{2} + 2)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} \) 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
$71$ \( (T + 2)^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} - 2)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} + 2)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} \) Copy content Toggle raw display
$97$ \( T^{4} \) Copy content Toggle raw display
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