Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [192,8,Mod(191,192)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(192, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0, 1]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("192.191");
S:= CuspForms(chi, 8);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 192 = 2^{6} \cdot 3 \) |
Weight: | \( k \) | \(=\) | \( 8 \) |
Character orbit: | \([\chi]\) | \(=\) | 192.c (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: | \(59.9779248930\) |
Analytic rank: | \(0\) |
Dimension: | \(28\) |
Twist minimal: | no (minimal twist has level 96) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
191.1 | 0 | −45.5869 | − | 10.4324i | 0 | 183.679i | 0 | 1238.80i | 0 | 1969.33 | + | 951.162i | 0 | ||||||||||||||
191.2 | 0 | −45.5869 | + | 10.4324i | 0 | − | 183.679i | 0 | − | 1238.80i | 0 | 1969.33 | − | 951.162i | 0 | ||||||||||||
191.3 | 0 | −43.8532 | − | 16.2450i | 0 | 235.987i | 0 | − | 481.583i | 0 | 1659.20 | + | 1424.79i | 0 | |||||||||||||
191.4 | 0 | −43.8532 | + | 16.2450i | 0 | − | 235.987i | 0 | 481.583i | 0 | 1659.20 | − | 1424.79i | 0 | |||||||||||||
191.5 | 0 | −39.5690 | − | 24.9257i | 0 | − | 508.011i | 0 | − | 1121.87i | 0 | 944.419 | + | 1972.57i | 0 | ||||||||||||
191.6 | 0 | −39.5690 | + | 24.9257i | 0 | 508.011i | 0 | 1121.87i | 0 | 944.419 | − | 1972.57i | 0 | ||||||||||||||
191.7 | 0 | −36.3965 | − | 29.3649i | 0 | 163.559i | 0 | − | 284.735i | 0 | 462.403 | + | 2137.56i | 0 | |||||||||||||
191.8 | 0 | −36.3965 | + | 29.3649i | 0 | − | 163.559i | 0 | 284.735i | 0 | 462.403 | − | 2137.56i | 0 | |||||||||||||
191.9 | 0 | −28.9719 | − | 36.7101i | 0 | − | 331.036i | 0 | 1721.03i | 0 | −508.262 | + | 2127.12i | 0 | |||||||||||||
191.10 | 0 | −28.9719 | + | 36.7101i | 0 | 331.036i | 0 | − | 1721.03i | 0 | −508.262 | − | 2127.12i | 0 | |||||||||||||
191.11 | 0 | −12.9879 | − | 44.9256i | 0 | 361.105i | 0 | − | 928.878i | 0 | −1849.63 | + | 1166.98i | 0 | |||||||||||||
191.12 | 0 | −12.9879 | + | 44.9256i | 0 | − | 361.105i | 0 | 928.878i | 0 | −1849.63 | − | 1166.98i | 0 | |||||||||||||
191.13 | 0 | −5.85395 | − | 46.3975i | 0 | − | 192.070i | 0 | 198.065i | 0 | −2118.46 | + | 543.217i | 0 | |||||||||||||
191.14 | 0 | −5.85395 | + | 46.3975i | 0 | 192.070i | 0 | − | 198.065i | 0 | −2118.46 | − | 543.217i | 0 | |||||||||||||
191.15 | 0 | 5.85395 | − | 46.3975i | 0 | 192.070i | 0 | 198.065i | 0 | −2118.46 | − | 543.217i | 0 | ||||||||||||||
191.16 | 0 | 5.85395 | + | 46.3975i | 0 | − | 192.070i | 0 | − | 198.065i | 0 | −2118.46 | + | 543.217i | 0 | ||||||||||||
191.17 | 0 | 12.9879 | − | 44.9256i | 0 | − | 361.105i | 0 | − | 928.878i | 0 | −1849.63 | − | 1166.98i | 0 | ||||||||||||
191.18 | 0 | 12.9879 | + | 44.9256i | 0 | 361.105i | 0 | 928.878i | 0 | −1849.63 | + | 1166.98i | 0 | ||||||||||||||
191.19 | 0 | 28.9719 | − | 36.7101i | 0 | 331.036i | 0 | 1721.03i | 0 | −508.262 | − | 2127.12i | 0 | ||||||||||||||
191.20 | 0 | 28.9719 | + | 36.7101i | 0 | − | 331.036i | 0 | − | 1721.03i | 0 | −508.262 | + | 2127.12i | 0 | ||||||||||||
See all 28 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
4.b | odd | 2 | 1 | inner |
12.b | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 192.8.c.f | 28 | |
3.b | odd | 2 | 1 | inner | 192.8.c.f | 28 | |
4.b | odd | 2 | 1 | inner | 192.8.c.f | 28 | |
8.b | even | 2 | 1 | 96.8.c.a | ✓ | 28 | |
8.d | odd | 2 | 1 | 96.8.c.a | ✓ | 28 | |
12.b | even | 2 | 1 | inner | 192.8.c.f | 28 | |
24.f | even | 2 | 1 | 96.8.c.a | ✓ | 28 | |
24.h | odd | 2 | 1 | 96.8.c.a | ✓ | 28 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
96.8.c.a | ✓ | 28 | 8.b | even | 2 | 1 | |
96.8.c.a | ✓ | 28 | 8.d | odd | 2 | 1 | |
96.8.c.a | ✓ | 28 | 24.f | even | 2 | 1 | |
96.8.c.a | ✓ | 28 | 24.h | odd | 2 | 1 | |
192.8.c.f | 28 | 1.a | even | 1 | 1 | trivial | |
192.8.c.f | 28 | 3.b | odd | 2 | 1 | inner | |
192.8.c.f | 28 | 4.b | odd | 2 | 1 | inner | |
192.8.c.f | 28 | 12.b | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{14} + 651128 T_{5}^{12} + 161018125632 T_{5}^{10} + \cdots + 68\!\cdots\!00 \)
acting on \(S_{8}^{\mathrm{new}}(192, [\chi])\).