Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [128,13,Mod(31,128)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(128, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([2, 1]))
N = Newforms(chi, 13, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("128.31");
S:= CuspForms(chi, 13);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 128 = 2^{7} \) |
Weight: | \( k \) | \(=\) | \( 13 \) |
Character orbit: | \([\chi]\) | \(=\) | 128.f (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: | \(116.991208611\) |
Analytic rank: | \(0\) |
Dimension: | \(46\) |
Relative dimension: | \(23\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
31.1 | 0 | −949.435 | − | 949.435i | 0 | −10755.1 | − | 10755.1i | 0 | 35819.9 | 0 | 1.27141e6i | 0 | ||||||||||||||
31.2 | 0 | −837.232 | − | 837.232i | 0 | −5671.68 | − | 5671.68i | 0 | −4585.41 | 0 | 870475.i | 0 | ||||||||||||||
31.3 | 0 | −769.939 | − | 769.939i | 0 | 17644.5 | + | 17644.5i | 0 | 15384.5 | 0 | 654171.i | 0 | ||||||||||||||
31.4 | 0 | −739.301 | − | 739.301i | 0 | 805.082 | + | 805.082i | 0 | −97582.7 | 0 | 561690.i | 0 | ||||||||||||||
31.5 | 0 | −652.763 | − | 652.763i | 0 | 7919.92 | + | 7919.92i | 0 | 182451. | 0 | 320757.i | 0 | ||||||||||||||
31.6 | 0 | −543.347 | − | 543.347i | 0 | 12801.7 | + | 12801.7i | 0 | −188146. | 0 | 59011.2i | 0 | ||||||||||||||
31.7 | 0 | −398.082 | − | 398.082i | 0 | −14857.0 | − | 14857.0i | 0 | 113515. | 0 | − | 214502.i | 0 | |||||||||||||
31.8 | 0 | −386.327 | − | 386.327i | 0 | −20822.4 | − | 20822.4i | 0 | −171983. | 0 | − | 232944.i | 0 | |||||||||||||
31.9 | 0 | −252.451 | − | 252.451i | 0 | 9569.73 | + | 9569.73i | 0 | 168743. | 0 | − | 403978.i | 0 | |||||||||||||
31.10 | 0 | −193.300 | − | 193.300i | 0 | 10355.8 | + | 10355.8i | 0 | −108466. | 0 | − | 456712.i | 0 | |||||||||||||
31.11 | 0 | −87.0633 | − | 87.0633i | 0 | −8119.76 | − | 8119.76i | 0 | 149756. | 0 | − | 516281.i | 0 | |||||||||||||
31.12 | 0 | −78.2777 | − | 78.2777i | 0 | −5406.83 | − | 5406.83i | 0 | −30061.3 | 0 | − | 519186.i | 0 | |||||||||||||
31.13 | 0 | −37.8347 | − | 37.8347i | 0 | −5270.38 | − | 5270.38i | 0 | −115235. | 0 | − | 528578.i | 0 | |||||||||||||
31.14 | 0 | 234.930 | + | 234.930i | 0 | 18167.9 | + | 18167.9i | 0 | 112043. | 0 | − | 421057.i | 0 | |||||||||||||
31.15 | 0 | 322.288 | + | 322.288i | 0 | 16463.9 | + | 16463.9i | 0 | 21051.9 | 0 | − | 323701.i | 0 | |||||||||||||
31.16 | 0 | 429.462 | + | 429.462i | 0 | 2473.08 | + | 2473.08i | 0 | −198740. | 0 | − | 162567.i | 0 | |||||||||||||
31.17 | 0 | 435.597 | + | 435.597i | 0 | −17337.6 | − | 17337.6i | 0 | 102408. | 0 | − | 151952.i | 0 | |||||||||||||
31.18 | 0 | 513.579 | + | 513.579i | 0 | −208.882 | − | 208.882i | 0 | 42912.4 | 0 | − | 3914.51i | 0 | |||||||||||||
31.19 | 0 | 623.967 | + | 623.967i | 0 | 4768.39 | + | 4768.39i | 0 | −21612.8 | 0 | 247229.i | 0 | ||||||||||||||
31.20 | 0 | 640.742 | + | 640.742i | 0 | −10737.8 | − | 10737.8i | 0 | 25233.9 | 0 | 289658.i | 0 | ||||||||||||||
See all 46 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
16.f | odd | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 128.13.f.b | 46 | |
4.b | odd | 2 | 1 | 128.13.f.a | 46 | ||
8.b | even | 2 | 1 | 16.13.f.a | ✓ | 46 | |
8.d | odd | 2 | 1 | 64.13.f.a | 46 | ||
16.e | even | 4 | 1 | 64.13.f.a | 46 | ||
16.e | even | 4 | 1 | 128.13.f.a | 46 | ||
16.f | odd | 4 | 1 | 16.13.f.a | ✓ | 46 | |
16.f | odd | 4 | 1 | inner | 128.13.f.b | 46 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
16.13.f.a | ✓ | 46 | 8.b | even | 2 | 1 | |
16.13.f.a | ✓ | 46 | 16.f | odd | 4 | 1 | |
64.13.f.a | 46 | 8.d | odd | 2 | 1 | ||
64.13.f.a | 46 | 16.e | even | 4 | 1 | ||
128.13.f.a | 46 | 4.b | odd | 2 | 1 | ||
128.13.f.a | 46 | 16.e | even | 4 | 1 | ||
128.13.f.b | 46 | 1.a | even | 1 | 1 | trivial | |
128.13.f.b | 46 | 16.f | odd | 4 | 1 | inner |