Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3100,3,Mod(1301,3100)]
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, 0, 1]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3100.1301");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 3100 = 2^{2} \cdot 5^{2} \cdot 31 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3100.d (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: | \(84.4688819517\) |
| Analytic rank: | \(0\) |
| Dimension: | \(22\) |
| 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1301.1 | 0 | − | 5.52468i | 0 | 0 | 0 | 8.23275 | 0 | −21.5221 | 0 | |||||||||||||||||
| 1301.2 | 0 | − | 5.49101i | 0 | 0 | 0 | −8.96078 | 0 | −21.1511 | 0 | |||||||||||||||||
| 1301.3 | 0 | − | 4.50171i | 0 | 0 | 0 | −3.34932 | 0 | −11.2654 | 0 | |||||||||||||||||
| 1301.4 | 0 | − | 3.71258i | 0 | 0 | 0 | 1.89499 | 0 | −4.78325 | 0 | |||||||||||||||||
| 1301.5 | 0 | − | 3.53436i | 0 | 0 | 0 | −7.77177 | 0 | −3.49167 | 0 | |||||||||||||||||
| 1301.6 | 0 | − | 3.39902i | 0 | 0 | 0 | −1.77923 | 0 | −2.55337 | 0 | |||||||||||||||||
| 1301.7 | 0 | − | 3.37243i | 0 | 0 | 0 | 12.8888 | 0 | −2.37326 | 0 | |||||||||||||||||
| 1301.8 | 0 | − | 1.74704i | 0 | 0 | 0 | 6.55009 | 0 | 5.94786 | 0 | |||||||||||||||||
| 1301.9 | 0 | − | 0.957337i | 0 | 0 | 0 | −12.5381 | 0 | 8.08351 | 0 | |||||||||||||||||
| 1301.10 | 0 | − | 0.880615i | 0 | 0 | 0 | 4.35842 | 0 | 8.22452 | 0 | |||||||||||||||||
| 1301.11 | 0 | − | 0.340074i | 0 | 0 | 0 | −1.52585 | 0 | 8.88435 | 0 | |||||||||||||||||
| 1301.12 | 0 | 0.340074i | 0 | 0 | 0 | −1.52585 | 0 | 8.88435 | 0 | ||||||||||||||||||
| 1301.13 | 0 | 0.880615i | 0 | 0 | 0 | 4.35842 | 0 | 8.22452 | 0 | ||||||||||||||||||
| 1301.14 | 0 | 0.957337i | 0 | 0 | 0 | −12.5381 | 0 | 8.08351 | 0 | ||||||||||||||||||
| 1301.15 | 0 | 1.74704i | 0 | 0 | 0 | 6.55009 | 0 | 5.94786 | 0 | ||||||||||||||||||
| 1301.16 | 0 | 3.37243i | 0 | 0 | 0 | 12.8888 | 0 | −2.37326 | 0 | ||||||||||||||||||
| 1301.17 | 0 | 3.39902i | 0 | 0 | 0 | −1.77923 | 0 | −2.55337 | 0 | ||||||||||||||||||
| 1301.18 | 0 | 3.53436i | 0 | 0 | 0 | −7.77177 | 0 | −3.49167 | 0 | ||||||||||||||||||
| 1301.19 | 0 | 3.71258i | 0 | 0 | 0 | 1.89499 | 0 | −4.78325 | 0 | ||||||||||||||||||
| 1301.20 | 0 | 4.50171i | 0 | 0 | 0 | −3.34932 | 0 | −11.2654 | 0 | ||||||||||||||||||
| See all 22 embeddings | |||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 31.b | 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.d.f | ✓ | 22 |
| 5.b | even | 2 | 1 | 3100.3.d.g | yes | 22 | |
| 5.c | odd | 4 | 2 | 3100.3.f.d | 44 | ||
| 31.b | odd | 2 | 1 | inner | 3100.3.d.f | ✓ | 22 |
| 155.c | odd | 2 | 1 | 3100.3.d.g | yes | 22 | |
| 155.f | even | 4 | 2 | 3100.3.f.d | 44 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3100.3.d.f | ✓ | 22 | 1.a | even | 1 | 1 | trivial |
| 3100.3.d.f | ✓ | 22 | 31.b | odd | 2 | 1 | inner |
| 3100.3.d.g | yes | 22 | 5.b | even | 2 | 1 | |
| 3100.3.d.g | yes | 22 | 155.c | odd | 2 | 1 | |
| 3100.3.f.d | 44 | 5.c | odd | 4 | 2 | ||
| 3100.3.f.d | 44 | 155.f | even | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(3100, [\chi])\):
|
\( T_{3}^{22} + 135 T_{3}^{20} + 7677 T_{3}^{18} + 240196 T_{3}^{16} + 4529439 T_{3}^{14} + \cdots + 105850800 \)
|
|
\( T_{7}^{11} + 2 T_{7}^{10} - 305 T_{7}^{9} - 567 T_{7}^{8} + 29264 T_{7}^{7} + 43089 T_{7}^{6} + \cdots - 45576250 \)
|