Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [64,24,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, 24, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("64.17");
S:= CuspForms(chi, 24);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 64 = 2^{6} \) |
Weight: | \( k \) | \(=\) | \( 24 \) |
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: | \(214.530583901\) |
Analytic rank: | \(0\) |
Dimension: | \(90\) |
Relative dimension: | \(45\) 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 | −417071. | − | 417071.i | 0 | −1.17592e8 | + | 1.17592e8i | 0 | 1.67834e9i | 0 | 2.53754e11i | 0 | ||||||||||||||
17.2 | 0 | −391132. | − | 391132.i | 0 | 8.92131e7 | − | 8.92131e7i | 0 | 2.94140e9i | 0 | 2.11825e11i | 0 | ||||||||||||||
17.3 | 0 | −387560. | − | 387560.i | 0 | 1.05971e8 | − | 1.05971e8i | 0 | − | 7.59178e9i | 0 | 2.06262e11i | 0 | |||||||||||||
17.4 | 0 | −367312. | − | 367312.i | 0 | 3.42346e7 | − | 3.42346e7i | 0 | 1.00506e10i | 0 | 1.75693e11i | 0 | ||||||||||||||
17.5 | 0 | −358807. | − | 358807.i | 0 | −1.40867e7 | + | 1.40867e7i | 0 | − | 6.29989e9i | 0 | 1.63341e11i | 0 | |||||||||||||
17.6 | 0 | −332866. | − | 332866.i | 0 | −6.54212e7 | + | 6.54212e7i | 0 | 3.71667e9i | 0 | 1.27456e11i | 0 | ||||||||||||||
17.7 | 0 | −303564. | − | 303564.i | 0 | −5.98803e7 | + | 5.98803e7i | 0 | − | 4.78358e9i | 0 | 9.01589e10i | 0 | |||||||||||||
17.8 | 0 | −286130. | − | 286130.i | 0 | −1.30247e8 | + | 1.30247e8i | 0 | − | 1.92081e9i | 0 | 6.95980e10i | 0 | |||||||||||||
17.9 | 0 | −277696. | − | 277696.i | 0 | 4.72704e7 | − | 4.72704e7i | 0 | − | 2.08272e9i | 0 | 6.00869e10i | 0 | |||||||||||||
17.10 | 0 | −267941. | − | 267941.i | 0 | 6.26290e7 | − | 6.26290e7i | 0 | 1.19784e9i | 0 | 4.94417e10i | 0 | ||||||||||||||
17.11 | 0 | −220155. | − | 220155.i | 0 | 1.17841e7 | − | 1.17841e7i | 0 | 6.12920e9i | 0 | 2.79345e9i | 0 | ||||||||||||||
17.12 | 0 | −198285. | − | 198285.i | 0 | −1.18292e8 | + | 1.18292e8i | 0 | 6.05722e9i | 0 | − | 1.55096e10i | 0 | |||||||||||||
17.13 | 0 | −186750. | − | 186750.i | 0 | −4.02332e7 | + | 4.02332e7i | 0 | − | 5.66827e9i | 0 | − | 2.43918e10i | 0 | ||||||||||||
17.14 | 0 | −177640. | − | 177640.i | 0 | 1.42580e8 | − | 1.42580e8i | 0 | 6.56907e8i | 0 | − | 3.10312e10i | 0 | |||||||||||||
17.15 | 0 | −158154. | − | 158154.i | 0 | −2.60550e7 | + | 2.60550e7i | 0 | 6.36490e9i | 0 | − | 4.41178e10i | 0 | |||||||||||||
17.16 | 0 | −120876. | − | 120876.i | 0 | −7.90014e6 | + | 7.90014e6i | 0 | − | 8.37748e9i | 0 | − | 6.49214e10i | 0 | ||||||||||||
17.17 | 0 | −107278. | − | 107278.i | 0 | 1.32256e8 | − | 1.32256e8i | 0 | 2.70816e9i | 0 | − | 7.11262e10i | 0 | |||||||||||||
17.18 | 0 | −94609.7 | − | 94609.7i | 0 | 9.23513e7 | − | 9.23513e7i | 0 | − | 5.80111e9i | 0 | − | 7.62412e10i | 0 | ||||||||||||
17.19 | 0 | −85271.2 | − | 85271.2i | 0 | −7.95847e7 | + | 7.95847e7i | 0 | − | 8.89876e9i | 0 | − | 7.96008e10i | 0 | ||||||||||||
17.20 | 0 | −68436.9 | − | 68436.9i | 0 | −2.70386e6 | + | 2.70386e6i | 0 | 2.28175e9i | 0 | − | 8.47760e10i | 0 | |||||||||||||
See all 90 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 | 64.24.e.a | 90 | |
4.b | odd | 2 | 1 | 16.24.e.a | ✓ | 90 | |
16.e | even | 4 | 1 | inner | 64.24.e.a | 90 | |
16.f | odd | 4 | 1 | 16.24.e.a | ✓ | 90 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
16.24.e.a | ✓ | 90 | 4.b | odd | 2 | 1 | |
16.24.e.a | ✓ | 90 | 16.f | odd | 4 | 1 | |
64.24.e.a | 90 | 1.a | even | 1 | 1 | trivial | |
64.24.e.a | 90 | 16.e | even | 4 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{24}^{\mathrm{new}}(64, [\chi])\).