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Properties

Label 1980.2.n.a
Level $1980$
Weight $2$
Character orbit 1980.n
Analytic conductor $15.810$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1980,2,Mod(989,1980)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1980, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1980.989");
 
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.n (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: \(15.8103796002\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 16 q^{25} + 8 q^{49} - 16 q^{55} - 64 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
989.1 0 0 0 −2.22447 − 0.227432i 0 −3.40219 0 0 0
989.2 0 0 0 −2.22447 − 0.227432i 0 3.40219 0 0 0
989.3 0 0 0 −2.22447 + 0.227432i 0 −3.40219 0 0 0
989.4 0 0 0 −2.22447 + 0.227432i 0 3.40219 0 0 0
989.5 0 0 0 −1.62156 − 1.53966i 0 −3.07186 0 0 0
989.6 0 0 0 −1.62156 − 1.53966i 0 3.07186 0 0 0
989.7 0 0 0 −1.62156 + 1.53966i 0 −3.07186 0 0 0
989.8 0 0 0 −1.62156 + 1.53966i 0 3.07186 0 0 0
989.9 0 0 0 −0.960355 − 2.01934i 0 −0.994380 0 0 0
989.10 0 0 0 −0.960355 − 2.01934i 0 0.994380 0 0 0
989.11 0 0 0 −0.960355 + 2.01934i 0 −0.994380 0 0 0
989.12 0 0 0 −0.960355 + 2.01934i 0 0.994380 0 0 0
989.13 0 0 0 0.960355 − 2.01934i 0 −0.994380 0 0 0
989.14 0 0 0 0.960355 − 2.01934i 0 0.994380 0 0 0
989.15 0 0 0 0.960355 + 2.01934i 0 −0.994380 0 0 0
989.16 0 0 0 0.960355 + 2.01934i 0 0.994380 0 0 0
989.17 0 0 0 1.62156 − 1.53966i 0 −3.07186 0 0 0
989.18 0 0 0 1.62156 − 1.53966i 0 3.07186 0 0 0
989.19 0 0 0 1.62156 + 1.53966i 0 −3.07186 0 0 0
989.20 0 0 0 1.62156 + 1.53966i 0 3.07186 0 0 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 989.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
11.b odd 2 1 inner
15.d odd 2 1 inner
33.d even 2 1 inner
55.d odd 2 1 inner
165.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1980.2.n.a ✓ 24
3.b odd 2 1 inner 1980.2.n.a ✓ 24
5.b even 2 1 inner 1980.2.n.a ✓ 24
5.c odd 4 2 9900.2.d.f 24
11.b odd 2 1 inner 1980.2.n.a ✓ 24
15.d odd 2 1 inner 1980.2.n.a ✓ 24
15.e even 4 2 9900.2.d.f 24
33.d even 2 1 inner 1980.2.n.a ✓ 24
55.d odd 2 1 inner 1980.2.n.a ✓ 24
55.e even 4 2 9900.2.d.f 24
165.d even 2 1 inner 1980.2.n.a ✓ 24
165.l odd 4 2 9900.2.d.f 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1980.2.n.a ✓ 24 1.a even 1 1 trivial
1980.2.n.a ✓ 24 3.b odd 2 1 inner
1980.2.n.a ✓ 24 5.b even 2 1 inner
1980.2.n.a ✓ 24 11.b odd 2 1 inner
1980.2.n.a ✓ 24 15.d odd 2 1 inner
1980.2.n.a ✓ 24 33.d even 2 1 inner
1980.2.n.a ✓ 24 55.d odd 2 1 inner
1980.2.n.a ✓ 24 165.d even 2 1 inner
9900.2.d.f 24 5.c odd 4 2
9900.2.d.f 24 15.e even 4 2
9900.2.d.f 24 55.e even 4 2
9900.2.d.f 24 165.l odd 4 2

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(1980, [\chi])\).