Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [320,10,Mod(161,320)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(320, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1, 0]))
N = Newforms(chi, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("320.161");
S:= CuspForms(chi, 10);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 320 = 2^{6} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| Character orbit: | \([\chi]\) | \(=\) | 320.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: | \(164.811467572\) |
| Analytic rank: | \(0\) |
| Dimension: | \(24\) |
| 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 161.1 | 0 | − | 264.920i | 0 | 625.000i | 0 | 1597.10 | 0 | −50499.5 | 0 | |||||||||||||||||
| 161.2 | 0 | − | 256.872i | 0 | − | 625.000i | 0 | −9756.19 | 0 | −46300.2 | 0 | ||||||||||||||||
| 161.3 | 0 | − | 212.383i | 0 | 625.000i | 0 | 6724.65 | 0 | −25423.5 | 0 | |||||||||||||||||
| 161.4 | 0 | − | 188.097i | 0 | − | 625.000i | 0 | 7734.95 | 0 | −15697.4 | 0 | ||||||||||||||||
| 161.5 | 0 | − | 161.013i | 0 | 625.000i | 0 | −5877.77 | 0 | −6242.04 | 0 | |||||||||||||||||
| 161.6 | 0 | − | 152.677i | 0 | − | 625.000i | 0 | −460.805 | 0 | −3627.17 | 0 | ||||||||||||||||
| 161.7 | 0 | − | 148.894i | 0 | − | 625.000i | 0 | 4203.72 | 0 | −2486.48 | 0 | ||||||||||||||||
| 161.8 | 0 | − | 91.6627i | 0 | 625.000i | 0 | 9986.35 | 0 | 11280.9 | 0 | |||||||||||||||||
| 161.9 | 0 | − | 89.5662i | 0 | 625.000i | 0 | −11082.3 | 0 | 11660.9 | 0 | |||||||||||||||||
| 161.10 | 0 | − | 86.0235i | 0 | − | 625.000i | 0 | −8680.12 | 0 | 12283.0 | 0 | ||||||||||||||||
| 161.11 | 0 | − | 39.0196i | 0 | 625.000i | 0 | −5789.92 | 0 | 18160.5 | 0 | |||||||||||||||||
| 161.12 | 0 | − | 26.0006i | 0 | − | 625.000i | 0 | 5544.30 | 0 | 19007.0 | 0 | ||||||||||||||||
| 161.13 | 0 | 26.0006i | 0 | 625.000i | 0 | 5544.30 | 0 | 19007.0 | 0 | ||||||||||||||||||
| 161.14 | 0 | 39.0196i | 0 | − | 625.000i | 0 | −5789.92 | 0 | 18160.5 | 0 | |||||||||||||||||
| 161.15 | 0 | 86.0235i | 0 | 625.000i | 0 | −8680.12 | 0 | 12283.0 | 0 | ||||||||||||||||||
| 161.16 | 0 | 89.5662i | 0 | − | 625.000i | 0 | −11082.3 | 0 | 11660.9 | 0 | |||||||||||||||||
| 161.17 | 0 | 91.6627i | 0 | − | 625.000i | 0 | 9986.35 | 0 | 11280.9 | 0 | |||||||||||||||||
| 161.18 | 0 | 148.894i | 0 | 625.000i | 0 | 4203.72 | 0 | −2486.48 | 0 | ||||||||||||||||||
| 161.19 | 0 | 152.677i | 0 | 625.000i | 0 | −460.805 | 0 | −3627.17 | 0 | ||||||||||||||||||
| 161.20 | 0 | 161.013i | 0 | − | 625.000i | 0 | −5877.77 | 0 | −6242.04 | 0 | |||||||||||||||||
| See all 24 embeddings | |||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 8.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 320.10.d.c | ✓ | 24 |
| 4.b | odd | 2 | 1 | 320.10.d.d | yes | 24 | |
| 8.b | even | 2 | 1 | inner | 320.10.d.c | ✓ | 24 |
| 8.d | odd | 2 | 1 | 320.10.d.d | yes | 24 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 320.10.d.c | ✓ | 24 | 1.a | even | 1 | 1 | trivial |
| 320.10.d.c | ✓ | 24 | 8.b | even | 2 | 1 | inner |
| 320.10.d.d | yes | 24 | 4.b | odd | 2 | 1 | |
| 320.10.d.d | yes | 24 | 8.d | odd | 2 | 1 | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{10}^{\mathrm{new}}(320, [\chi])\):
|
\( T_{3}^{24} + 314080 T_{3}^{22} + 42090536568 T_{3}^{20} + \cdots + 50\!\cdots\!44 \)
|
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\( T_{7}^{12} + 5856 T_{7}^{11} - 291537104 T_{7}^{10} - 1331046496560 T_{7}^{9} + \cdots + 28\!\cdots\!92 \)
|