Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3712,2,Mod(1857,3712)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3712, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3712.1857");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 3712 = 2^{7} \cdot 29 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 3712.c (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: | \(29.6404692303\) |
Analytic rank: | \(0\) |
Dimension: | \(28\) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1857.1 | 0 | − | 3.22320i | 0 | − | 3.16181i | 0 | 1.12719 | 0 | −7.38904 | 0 | ||||||||||||||||
1857.2 | 0 | − | 3.22320i | 0 | 3.16181i | 0 | −1.12719 | 0 | −7.38904 | 0 | |||||||||||||||||
1857.3 | 0 | − | 2.65617i | 0 | − | 0.843315i | 0 | 3.39137 | 0 | −4.05525 | 0 | ||||||||||||||||
1857.4 | 0 | − | 2.65617i | 0 | 0.843315i | 0 | −3.39137 | 0 | −4.05525 | 0 | |||||||||||||||||
1857.5 | 0 | − | 2.24189i | 0 | − | 2.96670i | 0 | −1.63911 | 0 | −2.02606 | 0 | ||||||||||||||||
1857.6 | 0 | − | 2.24189i | 0 | 2.96670i | 0 | 1.63911 | 0 | −2.02606 | 0 | |||||||||||||||||
1857.7 | 0 | − | 1.58779i | 0 | − | 1.36523i | 0 | −4.86306 | 0 | 0.478909 | 0 | ||||||||||||||||
1857.8 | 0 | − | 1.58779i | 0 | 1.36523i | 0 | 4.86306 | 0 | 0.478909 | 0 | |||||||||||||||||
1857.9 | 0 | − | 0.892520i | 0 | − | 1.46740i | 0 | 0.506033 | 0 | 2.20341 | 0 | ||||||||||||||||
1857.10 | 0 | − | 0.892520i | 0 | 1.46740i | 0 | −0.506033 | 0 | 2.20341 | 0 | |||||||||||||||||
1857.11 | 0 | − | 0.445350i | 0 | − | 1.09301i | 0 | −0.731427 | 0 | 2.80166 | 0 | ||||||||||||||||
1857.12 | 0 | − | 0.445350i | 0 | 1.09301i | 0 | 0.731427 | 0 | 2.80166 | 0 | |||||||||||||||||
1857.13 | 0 | − | 0.116746i | 0 | − | 4.15676i | 0 | −4.01260 | 0 | 2.98637 | 0 | ||||||||||||||||
1857.14 | 0 | − | 0.116746i | 0 | 4.15676i | 0 | 4.01260 | 0 | 2.98637 | 0 | |||||||||||||||||
1857.15 | 0 | 0.116746i | 0 | − | 4.15676i | 0 | 4.01260 | 0 | 2.98637 | 0 | |||||||||||||||||
1857.16 | 0 | 0.116746i | 0 | 4.15676i | 0 | −4.01260 | 0 | 2.98637 | 0 | ||||||||||||||||||
1857.17 | 0 | 0.445350i | 0 | − | 1.09301i | 0 | 0.731427 | 0 | 2.80166 | 0 | |||||||||||||||||
1857.18 | 0 | 0.445350i | 0 | 1.09301i | 0 | −0.731427 | 0 | 2.80166 | 0 | ||||||||||||||||||
1857.19 | 0 | 0.892520i | 0 | − | 1.46740i | 0 | −0.506033 | 0 | 2.20341 | 0 | |||||||||||||||||
1857.20 | 0 | 0.892520i | 0 | 1.46740i | 0 | 0.506033 | 0 | 2.20341 | 0 | ||||||||||||||||||
See all 28 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
8.b | even | 2 | 1 | inner |
8.d | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 3712.2.c.g | ✓ | 28 |
4.b | odd | 2 | 1 | inner | 3712.2.c.g | ✓ | 28 |
8.b | even | 2 | 1 | inner | 3712.2.c.g | ✓ | 28 |
8.d | odd | 2 | 1 | inner | 3712.2.c.g | ✓ | 28 |
16.e | even | 4 | 1 | 7424.2.a.w | 14 | ||
16.e | even | 4 | 1 | 7424.2.a.x | 14 | ||
16.f | odd | 4 | 1 | 7424.2.a.w | 14 | ||
16.f | odd | 4 | 1 | 7424.2.a.x | 14 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
3712.2.c.g | ✓ | 28 | 1.a | even | 1 | 1 | trivial |
3712.2.c.g | ✓ | 28 | 4.b | odd | 2 | 1 | inner |
3712.2.c.g | ✓ | 28 | 8.b | even | 2 | 1 | inner |
3712.2.c.g | ✓ | 28 | 8.d | odd | 2 | 1 | inner |
7424.2.a.w | 14 | 16.e | even | 4 | 1 | ||
7424.2.a.w | 14 | 16.f | odd | 4 | 1 | ||
7424.2.a.x | 14 | 16.e | even | 4 | 1 | ||
7424.2.a.x | 14 | 16.f | odd | 4 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{14} + 26T_{3}^{12} + 243T_{3}^{10} + 998T_{3}^{8} + 1747T_{3}^{6} + 1070T_{3}^{4} + 161T_{3}^{2} + 2 \) acting on \(S_{2}^{\mathrm{new}}(3712, [\chi])\).