Properties

Label 1980.2.y.c
Level $1980$
Weight $2$
Character orbit 1980.y
Analytic conductor $15.810$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1980,2,Mod(1297,1980)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1980, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1980.1297");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1980 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1980.y (of order \(4\), degree \(2\), 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: \(15.8103796002\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 660)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 8 q^{5}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 8 q^{5} - 8 q^{11} - 16 q^{23} + 24 q^{25} + 16 q^{31} - 40 q^{37} - 64 q^{47} - 24 q^{53} + 24 q^{55} + 48 q^{67} - 24 q^{77} - 48 q^{91} - 72 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1297.1 0 0 0 −2.05777 0.874978i 0 −2.32148 + 2.32148i 0 0 0
1297.2 0 0 0 −2.05777 0.874978i 0 2.32148 2.32148i 0 0 0
1297.3 0 0 0 −1.53657 + 1.62448i 0 −2.61809 + 2.61809i 0 0 0
1297.4 0 0 0 −1.53657 + 1.62448i 0 2.61809 2.61809i 0 0 0
1297.5 0 0 0 −1.32558 1.80079i 0 −2.73815 + 2.73815i 0 0 0
1297.6 0 0 0 −1.32558 1.80079i 0 2.73815 2.73815i 0 0 0
1297.7 0 0 0 −1.14747 + 1.91919i 0 −0.593897 + 0.593897i 0 0 0
1297.8 0 0 0 −1.14747 + 1.91919i 0 0.593897 0.593897i 0 0 0
1297.9 0 0 0 1.86215 1.23791i 0 −1.37506 + 1.37506i 0 0 0
1297.10 0 0 0 1.86215 1.23791i 0 1.37506 1.37506i 0 0 0
1297.11 0 0 0 2.20524 + 0.369998i 0 −1.41964 + 1.41964i 0 0 0
1297.12 0 0 0 2.20524 + 0.369998i 0 1.41964 1.41964i 0 0 0
1693.1 0 0 0 −2.05777 + 0.874978i 0 −2.32148 2.32148i 0 0 0
1693.2 0 0 0 −2.05777 + 0.874978i 0 2.32148 + 2.32148i 0 0 0
1693.3 0 0 0 −1.53657 1.62448i 0 −2.61809 2.61809i 0 0 0
1693.4 0 0 0 −1.53657 1.62448i 0 2.61809 + 2.61809i 0 0 0
1693.5 0 0 0 −1.32558 + 1.80079i 0 −2.73815 2.73815i 0 0 0
1693.6 0 0 0 −1.32558 + 1.80079i 0 2.73815 + 2.73815i 0 0 0
1693.7 0 0 0 −1.14747 1.91919i 0 −0.593897 0.593897i 0 0 0
1693.8 0 0 0 −1.14747 1.91919i 0 0.593897 + 0.593897i 0 0 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1297.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
11.b odd 2 1 inner
55.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1980.2.y.c 24
3.b odd 2 1 660.2.x.a 24
5.c odd 4 1 inner 1980.2.y.c 24
11.b odd 2 1 inner 1980.2.y.c 24
15.d odd 2 1 3300.2.x.c 24
15.e even 4 1 660.2.x.a 24
15.e even 4 1 3300.2.x.c 24
33.d even 2 1 660.2.x.a 24
55.e even 4 1 inner 1980.2.y.c 24
165.d even 2 1 3300.2.x.c 24
165.l odd 4 1 660.2.x.a 24
165.l odd 4 1 3300.2.x.c 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
660.2.x.a 24 3.b odd 2 1
660.2.x.a 24 15.e even 4 1
660.2.x.a 24 33.d even 2 1
660.2.x.a 24 165.l odd 4 1
1980.2.y.c 24 1.a even 1 1 trivial
1980.2.y.c 24 5.c odd 4 1 inner
1980.2.y.c 24 11.b odd 2 1 inner
1980.2.y.c 24 55.e even 4 1 inner
3300.2.x.c 24 15.d odd 2 1
3300.2.x.c 24 15.e even 4 1
3300.2.x.c 24 165.d even 2 1
3300.2.x.c 24 165.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{24} + 560T_{7}^{20} + 106880T_{7}^{16} + 7840800T_{7}^{12} + 174796032T_{7}^{8} + 1225632768T_{7}^{4} + 567582976 \) acting on \(S_{2}^{\mathrm{new}}(1980, [\chi])\). Copy content Toggle raw display