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.
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 |
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 \(S_{2}^{\mathrm{new}}(1980, [\chi])\).