Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1640,2,Mod(329,1640)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1640, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("1640.329");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 1640 = 2^{3} \cdot 5 \cdot 41 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 1640.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: | \(13.0954659315\) |
Analytic rank: | \(0\) |
Dimension: | \(32\) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
329.1 | 0 | − | 3.33506i | 0 | −2.22964 | + | 0.169488i | 0 | − | 3.43583i | 0 | −8.12259 | 0 | ||||||||||||||
329.2 | 0 | − | 3.25653i | 0 | 1.77169 | + | 1.36422i | 0 | 3.46019i | 0 | −7.60502 | 0 | |||||||||||||||
329.3 | 0 | − | 3.18640i | 0 | 1.45061 | − | 1.70169i | 0 | − | 3.18951i | 0 | −7.15316 | 0 | ||||||||||||||
329.4 | 0 | − | 2.96258i | 0 | −2.20941 | + | 0.344263i | 0 | 2.24753i | 0 | −5.77689 | 0 | |||||||||||||||
329.5 | 0 | − | 2.48752i | 0 | −0.503071 | − | 2.17874i | 0 | 4.84804i | 0 | −3.18777 | 0 | |||||||||||||||
329.6 | 0 | − | 2.47088i | 0 | 2.23106 | − | 0.149608i | 0 | 0.197424i | 0 | −3.10526 | 0 | |||||||||||||||
329.7 | 0 | − | 2.15759i | 0 | 0.185392 | − | 2.22837i | 0 | 4.34072i | 0 | −1.65520 | 0 | |||||||||||||||
329.8 | 0 | − | 2.09348i | 0 | 0.953958 | + | 2.02237i | 0 | − | 0.161846i | 0 | −1.38265 | 0 | ||||||||||||||
329.9 | 0 | − | 1.80651i | 0 | −0.893955 | − | 2.04960i | 0 | − | 2.88368i | 0 | −0.263493 | 0 | ||||||||||||||
329.10 | 0 | − | 1.54952i | 0 | 2.14609 | + | 0.627939i | 0 | − | 5.11156i | 0 | 0.598994 | 0 | ||||||||||||||
329.11 | 0 | − | 1.54459i | 0 | −1.97568 | + | 1.04723i | 0 | 0.909473i | 0 | 0.614235 | 0 | |||||||||||||||
329.12 | 0 | − | 0.862924i | 0 | 1.29968 | + | 1.81957i | 0 | 2.20025i | 0 | 2.25536 | 0 | |||||||||||||||
329.13 | 0 | − | 0.843007i | 0 | 1.49227 | + | 1.66527i | 0 | − | 2.89984i | 0 | 2.28934 | 0 | ||||||||||||||
329.14 | 0 | − | 0.483326i | 0 | −1.55986 | + | 1.60214i | 0 | − | 1.45691i | 0 | 2.76640 | 0 | ||||||||||||||
329.15 | 0 | − | 0.405841i | 0 | 1.06357 | − | 1.96693i | 0 | 0.855869i | 0 | 2.83529 | 0 | |||||||||||||||
329.16 | 0 | − | 0.328008i | 0 | −2.22271 | − | 0.244068i | 0 | 2.28281i | 0 | 2.89241 | 0 | |||||||||||||||
329.17 | 0 | 0.328008i | 0 | −2.22271 | + | 0.244068i | 0 | − | 2.28281i | 0 | 2.89241 | 0 | |||||||||||||||
329.18 | 0 | 0.405841i | 0 | 1.06357 | + | 1.96693i | 0 | − | 0.855869i | 0 | 2.83529 | 0 | |||||||||||||||
329.19 | 0 | 0.483326i | 0 | −1.55986 | − | 1.60214i | 0 | 1.45691i | 0 | 2.76640 | 0 | ||||||||||||||||
329.20 | 0 | 0.843007i | 0 | 1.49227 | − | 1.66527i | 0 | 2.89984i | 0 | 2.28934 | 0 | ||||||||||||||||
See all 32 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 1640.2.d.c | ✓ | 32 |
5.b | even | 2 | 1 | inner | 1640.2.d.c | ✓ | 32 |
5.c | odd | 4 | 1 | 8200.2.a.bn | 16 | ||
5.c | odd | 4 | 1 | 8200.2.a.bo | 16 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
1640.2.d.c | ✓ | 32 | 1.a | even | 1 | 1 | trivial |
1640.2.d.c | ✓ | 32 | 5.b | even | 2 | 1 | inner |
8200.2.a.bn | 16 | 5.c | odd | 4 | 1 | ||
8200.2.a.bo | 16 | 5.c | odd | 4 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{32} + 72 T_{3}^{30} + 2314 T_{3}^{28} + 43860 T_{3}^{26} + 545809 T_{3}^{24} + 4698728 T_{3}^{22} + \cdots + 331776 \) acting on \(S_{2}^{\mathrm{new}}(1640, [\chi])\).