Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [64,8,Mod(17,64)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(64, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 3]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("64.17");
S:= CuspForms(chi, 8);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 64 = 2^{6} \) |
Weight: | \( k \) | \(=\) | \( 8 \) |
Character orbit: | \([\chi]\) | \(=\) | 64.e (of order \(4\), degree \(2\), 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: | \(19.9926416310\) |
Analytic rank: | \(0\) |
Dimension: | \(26\) |
Relative dimension: | \(13\) over \(\Q(i)\) |
Twist minimal: | no (minimal twist has level 16) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
17.1 | 0 | −61.7547 | − | 61.7547i | 0 | 160.824 | − | 160.824i | 0 | 1202.46i | 0 | 5440.30i | 0 | ||||||||||||||
17.2 | 0 | −50.5872 | − | 50.5872i | 0 | −364.370 | + | 364.370i | 0 | − | 182.346i | 0 | 2931.14i | 0 | |||||||||||||
17.3 | 0 | −42.2145 | − | 42.2145i | 0 | 46.9348 | − | 46.9348i | 0 | − | 1384.27i | 0 | 1377.13i | 0 | |||||||||||||
17.4 | 0 | −26.1364 | − | 26.1364i | 0 | −31.6175 | + | 31.6175i | 0 | − | 444.381i | 0 | − | 820.775i | 0 | ||||||||||||
17.5 | 0 | −20.1662 | − | 20.1662i | 0 | 269.345 | − | 269.345i | 0 | 147.771i | 0 | − | 1373.65i | 0 | |||||||||||||
17.6 | 0 | −8.25573 | − | 8.25573i | 0 | 63.0500 | − | 63.0500i | 0 | 847.676i | 0 | − | 2050.69i | 0 | |||||||||||||
17.7 | 0 | −2.27414 | − | 2.27414i | 0 | −242.456 | + | 242.456i | 0 | 1642.72i | 0 | − | 2176.66i | 0 | |||||||||||||
17.8 | 0 | 12.2643 | + | 12.2643i | 0 | −210.432 | + | 210.432i | 0 | − | 920.183i | 0 | − | 1886.17i | 0 | ||||||||||||
17.9 | 0 | 21.1050 | + | 21.1050i | 0 | 328.807 | − | 328.807i | 0 | − | 874.718i | 0 | − | 1296.15i | 0 | ||||||||||||
17.10 | 0 | 35.6625 | + | 35.6625i | 0 | 50.2015 | − | 50.2015i | 0 | − | 699.226i | 0 | 356.628i | 0 | |||||||||||||
17.11 | 0 | 37.3394 | + | 37.3394i | 0 | −233.214 | + | 233.214i | 0 | 241.506i | 0 | 601.466i | 0 | ||||||||||||||
17.12 | 0 | 49.4488 | + | 49.4488i | 0 | 252.094 | − | 252.094i | 0 | 1482.88i | 0 | 2703.37i | 0 | ||||||||||||||
17.13 | 0 | 56.5688 | + | 56.5688i | 0 | −90.1684 | + | 90.1684i | 0 | − | 373.897i | 0 | 4213.06i | 0 | |||||||||||||
49.1 | 0 | −61.7547 | + | 61.7547i | 0 | 160.824 | + | 160.824i | 0 | − | 1202.46i | 0 | − | 5440.30i | 0 | ||||||||||||
49.2 | 0 | −50.5872 | + | 50.5872i | 0 | −364.370 | − | 364.370i | 0 | 182.346i | 0 | − | 2931.14i | 0 | |||||||||||||
49.3 | 0 | −42.2145 | + | 42.2145i | 0 | 46.9348 | + | 46.9348i | 0 | 1384.27i | 0 | − | 1377.13i | 0 | |||||||||||||
49.4 | 0 | −26.1364 | + | 26.1364i | 0 | −31.6175 | − | 31.6175i | 0 | 444.381i | 0 | 820.775i | 0 | ||||||||||||||
49.5 | 0 | −20.1662 | + | 20.1662i | 0 | 269.345 | + | 269.345i | 0 | − | 147.771i | 0 | 1373.65i | 0 | |||||||||||||
49.6 | 0 | −8.25573 | + | 8.25573i | 0 | 63.0500 | + | 63.0500i | 0 | − | 847.676i | 0 | 2050.69i | 0 | |||||||||||||
49.7 | 0 | −2.27414 | + | 2.27414i | 0 | −242.456 | − | 242.456i | 0 | − | 1642.72i | 0 | 2176.66i | 0 | |||||||||||||
See all 26 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
16.e | even | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 64.8.e.a | 26 | |
4.b | odd | 2 | 1 | 16.8.e.a | ✓ | 26 | |
8.b | even | 2 | 1 | 128.8.e.a | 26 | ||
8.d | odd | 2 | 1 | 128.8.e.b | 26 | ||
16.e | even | 4 | 1 | inner | 64.8.e.a | 26 | |
16.e | even | 4 | 1 | 128.8.e.a | 26 | ||
16.f | odd | 4 | 1 | 16.8.e.a | ✓ | 26 | |
16.f | odd | 4 | 1 | 128.8.e.b | 26 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
16.8.e.a | ✓ | 26 | 4.b | odd | 2 | 1 | |
16.8.e.a | ✓ | 26 | 16.f | odd | 4 | 1 | |
64.8.e.a | 26 | 1.a | even | 1 | 1 | trivial | |
64.8.e.a | 26 | 16.e | even | 4 | 1 | inner | |
128.8.e.a | 26 | 8.b | even | 2 | 1 | ||
128.8.e.a | 26 | 16.e | even | 4 | 1 | ||
128.8.e.b | 26 | 8.d | odd | 2 | 1 | ||
128.8.e.b | 26 | 16.f | odd | 4 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{8}^{\mathrm{new}}(64, [\chi])\).