Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3100,3,Mod(1549,3100)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3100, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1, 1]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3100.1549");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 3100 = 2^{2} \cdot 5^{2} \cdot 31 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3100.f (of order \(2\), degree \(1\), not 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: | \(84.4688819517\) |
| Analytic rank: | \(0\) |
| Dimension: | \(44\) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
The algebraic \(q\)-expansion of this newform has not been computed, 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1549.1 | 0 | −5.52468 | 0 | 0 | 0 | − | 8.23275i | 0 | 21.5221 | 0 | |||||||||||||||||
| 1549.2 | 0 | −5.52468 | 0 | 0 | 0 | 8.23275i | 0 | 21.5221 | 0 | ||||||||||||||||||
| 1549.3 | 0 | −5.49101 | 0 | 0 | 0 | − | 8.96078i | 0 | 21.1511 | 0 | |||||||||||||||||
| 1549.4 | 0 | −5.49101 | 0 | 0 | 0 | 8.96078i | 0 | 21.1511 | 0 | ||||||||||||||||||
| 1549.5 | 0 | −4.50171 | 0 | 0 | 0 | − | 3.34932i | 0 | 11.2654 | 0 | |||||||||||||||||
| 1549.6 | 0 | −4.50171 | 0 | 0 | 0 | 3.34932i | 0 | 11.2654 | 0 | ||||||||||||||||||
| 1549.7 | 0 | −3.71258 | 0 | 0 | 0 | 1.89499i | 0 | 4.78325 | 0 | ||||||||||||||||||
| 1549.8 | 0 | −3.71258 | 0 | 0 | 0 | − | 1.89499i | 0 | 4.78325 | 0 | |||||||||||||||||
| 1549.9 | 0 | −3.53436 | 0 | 0 | 0 | 7.77177i | 0 | 3.49167 | 0 | ||||||||||||||||||
| 1549.10 | 0 | −3.53436 | 0 | 0 | 0 | − | 7.77177i | 0 | 3.49167 | 0 | |||||||||||||||||
| 1549.11 | 0 | −3.39902 | 0 | 0 | 0 | − | 1.77923i | 0 | 2.55337 | 0 | |||||||||||||||||
| 1549.12 | 0 | −3.39902 | 0 | 0 | 0 | 1.77923i | 0 | 2.55337 | 0 | ||||||||||||||||||
| 1549.13 | 0 | −3.37243 | 0 | 0 | 0 | 12.8888i | 0 | 2.37326 | 0 | ||||||||||||||||||
| 1549.14 | 0 | −3.37243 | 0 | 0 | 0 | − | 12.8888i | 0 | 2.37326 | 0 | |||||||||||||||||
| 1549.15 | 0 | −1.74704 | 0 | 0 | 0 | 6.55009i | 0 | −5.94786 | 0 | ||||||||||||||||||
| 1549.16 | 0 | −1.74704 | 0 | 0 | 0 | − | 6.55009i | 0 | −5.94786 | 0 | |||||||||||||||||
| 1549.17 | 0 | −0.957337 | 0 | 0 | 0 | 12.5381i | 0 | −8.08351 | 0 | ||||||||||||||||||
| 1549.18 | 0 | −0.957337 | 0 | 0 | 0 | − | 12.5381i | 0 | −8.08351 | 0 | |||||||||||||||||
| 1549.19 | 0 | −0.880615 | 0 | 0 | 0 | − | 4.35842i | 0 | −8.22452 | 0 | |||||||||||||||||
| 1549.20 | 0 | −0.880615 | 0 | 0 | 0 | 4.35842i | 0 | −8.22452 | 0 | ||||||||||||||||||
| See all 44 embeddings | |||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
| 31.b | odd | 2 | 1 | inner |
| 155.c | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 3100.3.f.d | 44 | |
| 5.b | even | 2 | 1 | inner | 3100.3.f.d | 44 | |
| 5.c | odd | 4 | 1 | 3100.3.d.f | ✓ | 22 | |
| 5.c | odd | 4 | 1 | 3100.3.d.g | yes | 22 | |
| 31.b | odd | 2 | 1 | inner | 3100.3.f.d | 44 | |
| 155.c | odd | 2 | 1 | inner | 3100.3.f.d | 44 | |
| 155.f | even | 4 | 1 | 3100.3.d.f | ✓ | 22 | |
| 155.f | even | 4 | 1 | 3100.3.d.g | yes | 22 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3100.3.d.f | ✓ | 22 | 5.c | odd | 4 | 1 | |
| 3100.3.d.f | ✓ | 22 | 155.f | even | 4 | 1 | |
| 3100.3.d.g | yes | 22 | 5.c | odd | 4 | 1 | |
| 3100.3.d.g | yes | 22 | 155.f | even | 4 | 1 | |
| 3100.3.f.d | 44 | 1.a | even | 1 | 1 | trivial | |
| 3100.3.f.d | 44 | 5.b | even | 2 | 1 | inner | |
| 3100.3.f.d | 44 | 31.b | odd | 2 | 1 | inner | |
| 3100.3.f.d | 44 | 155.c | odd | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{22} - 135 T_{3}^{20} + 7677 T_{3}^{18} - 240196 T_{3}^{16} + 4529439 T_{3}^{14} + \cdots - 105850800 \)
acting on \(S_{3}^{\mathrm{new}}(3100, [\chi])\).