Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1610,2,Mod(1609,1610)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1610, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("1610.1609");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 1610 = 2 \cdot 5 \cdot 7 \cdot 23 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 1610.g (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: | \(12.8559147254\) |
Analytic rank: | \(0\) |
Dimension: | \(96\) |
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.
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1609.1 | − | 1.00000i | 0.209068 | −1.00000 | 1.92511 | + | 1.13752i | − | 0.209068i | −0.855049 | − | 2.50378i | 1.00000i | −2.95629 | 1.13752 | − | 1.92511i | ||||||||||
1609.2 | 1.00000i | 0.209068 | −1.00000 | 1.92511 | − | 1.13752i | 0.209068i | −0.855049 | + | 2.50378i | − | 1.00000i | −2.95629 | 1.13752 | + | 1.92511i | |||||||||||
1609.3 | − | 1.00000i | 3.32094 | −1.00000 | −0.0164715 | + | 2.23601i | − | 3.32094i | 2.20809 | − | 1.45752i | 1.00000i | 8.02867 | 2.23601 | + | 0.0164715i | ||||||||||
1609.4 | 1.00000i | 3.32094 | −1.00000 | −0.0164715 | − | 2.23601i | 3.32094i | 2.20809 | + | 1.45752i | − | 1.00000i | 8.02867 | 2.23601 | − | 0.0164715i | |||||||||||
1609.5 | − | 1.00000i | 0.209068 | −1.00000 | −1.92511 | − | 1.13752i | − | 0.209068i | 0.855049 | + | 2.50378i | 1.00000i | −2.95629 | −1.13752 | + | 1.92511i | ||||||||||
1609.6 | 1.00000i | 0.209068 | −1.00000 | −1.92511 | + | 1.13752i | 0.209068i | 0.855049 | − | 2.50378i | − | 1.00000i | −2.95629 | −1.13752 | − | 1.92511i | |||||||||||
1609.7 | − | 1.00000i | 2.59834 | −1.00000 | −1.15059 | − | 1.91733i | − | 2.59834i | 2.54535 | − | 0.721949i | 1.00000i | 3.75138 | −1.91733 | + | 1.15059i | ||||||||||
1609.8 | 1.00000i | 2.59834 | −1.00000 | −1.15059 | + | 1.91733i | 2.59834i | 2.54535 | + | 0.721949i | − | 1.00000i | 3.75138 | −1.91733 | − | 1.15059i | |||||||||||
1609.9 | − | 1.00000i | −0.842190 | −1.00000 | 1.91245 | − | 1.15867i | 0.842190i | 1.90341 | + | 1.83767i | 1.00000i | −2.29072 | −1.15867 | − | 1.91245i | |||||||||||
1609.10 | 1.00000i | −0.842190 | −1.00000 | 1.91245 | + | 1.15867i | − | 0.842190i | 1.90341 | − | 1.83767i | − | 1.00000i | −2.29072 | −1.15867 | + | 1.91245i | ||||||||||
1609.11 | − | 1.00000i | 0.842190 | −1.00000 | −1.91245 | + | 1.15867i | − | 0.842190i | 1.90341 | − | 1.83767i | 1.00000i | −2.29072 | 1.15867 | + | 1.91245i | ||||||||||
1609.12 | 1.00000i | 0.842190 | −1.00000 | −1.91245 | − | 1.15867i | 0.842190i | 1.90341 | + | 1.83767i | − | 1.00000i | −2.29072 | 1.15867 | − | 1.91245i | |||||||||||
1609.13 | − | 1.00000i | −0.574023 | −1.00000 | −0.755769 | − | 2.10447i | 0.574023i | 2.38007 | − | 1.15553i | 1.00000i | −2.67050 | −2.10447 | + | 0.755769i | |||||||||||
1609.14 | 1.00000i | −0.574023 | −1.00000 | −0.755769 | + | 2.10447i | − | 0.574023i | 2.38007 | + | 1.15553i | − | 1.00000i | −2.67050 | −2.10447 | − | 0.755769i | ||||||||||
1609.15 | − | 1.00000i | 2.20972 | −1.00000 | −2.21000 | − | 0.340423i | − | 2.20972i | −2.40927 | − | 1.09336i | 1.00000i | 1.88286 | −0.340423 | + | 2.21000i | ||||||||||
1609.16 | 1.00000i | 2.20972 | −1.00000 | −2.21000 | + | 0.340423i | 2.20972i | −2.40927 | + | 1.09336i | − | 1.00000i | 1.88286 | −0.340423 | − | 2.21000i | |||||||||||
1609.17 | − | 1.00000i | −0.421830 | −1.00000 | −1.54042 | + | 1.62083i | 0.421830i | −2.63295 | − | 0.259941i | 1.00000i | −2.82206 | 1.62083 | + | 1.54042i | |||||||||||
1609.18 | 1.00000i | −0.421830 | −1.00000 | −1.54042 | − | 1.62083i | − | 0.421830i | −2.63295 | + | 0.259941i | − | 1.00000i | −2.82206 | 1.62083 | − | 1.54042i | ||||||||||
1609.19 | − | 1.00000i | −1.23117 | −1.00000 | 0.749972 | − | 2.10655i | 1.23117i | 0.117671 | − | 2.64313i | 1.00000i | −1.48421 | −2.10655 | − | 0.749972i | |||||||||||
1609.20 | 1.00000i | −1.23117 | −1.00000 | 0.749972 | + | 2.10655i | − | 1.23117i | 0.117671 | + | 2.64313i | − | 1.00000i | −1.48421 | −2.10655 | + | 0.749972i | ||||||||||
See all 96 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
7.b | odd | 2 | 1 | inner |
23.b | odd | 2 | 1 | inner |
35.c | odd | 2 | 1 | inner |
115.c | odd | 2 | 1 | inner |
161.c | even | 2 | 1 | inner |
805.d | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 1610.2.g.a | ✓ | 96 |
5.b | even | 2 | 1 | inner | 1610.2.g.a | ✓ | 96 |
7.b | odd | 2 | 1 | inner | 1610.2.g.a | ✓ | 96 |
23.b | odd | 2 | 1 | inner | 1610.2.g.a | ✓ | 96 |
35.c | odd | 2 | 1 | inner | 1610.2.g.a | ✓ | 96 |
115.c | odd | 2 | 1 | inner | 1610.2.g.a | ✓ | 96 |
161.c | even | 2 | 1 | inner | 1610.2.g.a | ✓ | 96 |
805.d | even | 2 | 1 | inner | 1610.2.g.a | ✓ | 96 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
1610.2.g.a | ✓ | 96 | 1.a | even | 1 | 1 | trivial |
1610.2.g.a | ✓ | 96 | 5.b | even | 2 | 1 | inner |
1610.2.g.a | ✓ | 96 | 7.b | odd | 2 | 1 | inner |
1610.2.g.a | ✓ | 96 | 23.b | odd | 2 | 1 | inner |
1610.2.g.a | ✓ | 96 | 35.c | odd | 2 | 1 | inner |
1610.2.g.a | ✓ | 96 | 115.c | odd | 2 | 1 | inner |
1610.2.g.a | ✓ | 96 | 161.c | even | 2 | 1 | inner |
1610.2.g.a | ✓ | 96 | 805.d | even | 2 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(1610, [\chi])\).