Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [128,14,Mod(33,128)]
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([0, 3]))
N = Newforms(chi, 14, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("128.33");
S:= CuspForms(chi, 14);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 128 = 2^{7} \) |
Weight: | \( k \) | \(=\) | \( 14 \) |
Character orbit: | \([\chi]\) | \(=\) | 128.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: | \(137.255589058\) |
Analytic rank: | \(0\) |
Dimension: | \(50\) |
Relative dimension: | \(25\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
33.1 | 0 | −1621.89 | − | 1621.89i | 0 | −8037.57 | + | 8037.57i | 0 | − | 516249.i | 0 | 3.66675e6i | 0 | |||||||||||||
33.2 | 0 | −1610.25 | − | 1610.25i | 0 | −39780.2 | + | 39780.2i | 0 | 439933.i | 0 | 3.59146e6i | 0 | ||||||||||||||
33.3 | 0 | −1377.13 | − | 1377.13i | 0 | 6703.22 | − | 6703.22i | 0 | 90853.8i | 0 | 2.19866e6i | 0 | ||||||||||||||
33.4 | 0 | −1317.13 | − | 1317.13i | 0 | 24836.8 | − | 24836.8i | 0 | 73212.9i | 0 | 1.87536e6i | 0 | ||||||||||||||
33.5 | 0 | −1174.50 | − | 1174.50i | 0 | 33114.8 | − | 33114.8i | 0 | 548741.i | 0 | 1.16456e6i | 0 | ||||||||||||||
33.6 | 0 | −983.557 | − | 983.557i | 0 | 35371.2 | − | 35371.2i | 0 | − | 334186.i | 0 | 340445.i | 0 | |||||||||||||
33.7 | 0 | −913.885 | − | 913.885i | 0 | −15715.7 | + | 15715.7i | 0 | − | 356100.i | 0 | 76047.5i | 0 | |||||||||||||
33.8 | 0 | −805.865 | − | 805.865i | 0 | −18822.0 | + | 18822.0i | 0 | 17326.7i | 0 | − | 295487.i | 0 | |||||||||||||
33.9 | 0 | −381.223 | − | 381.223i | 0 | −29472.6 | + | 29472.6i | 0 | 167372.i | 0 | − | 1.30366e6i | 0 | |||||||||||||
33.10 | 0 | −375.766 | − | 375.766i | 0 | 11477.0 | − | 11477.0i | 0 | − | 9991.15i | 0 | − | 1.31192e6i | 0 | ||||||||||||
33.11 | 0 | −293.446 | − | 293.446i | 0 | −22716.6 | + | 22716.6i | 0 | 343566.i | 0 | − | 1.42210e6i | 0 | |||||||||||||
33.12 | 0 | −242.259 | − | 242.259i | 0 | −42421.3 | + | 42421.3i | 0 | − | 421955.i | 0 | − | 1.47694e6i | 0 | ||||||||||||
33.13 | 0 | 26.4760 | + | 26.4760i | 0 | 38066.2 | − | 38066.2i | 0 | − | 23704.0i | 0 | − | 1.59292e6i | 0 | ||||||||||||
33.14 | 0 | 217.732 | + | 217.732i | 0 | 36424.0 | − | 36424.0i | 0 | − | 397405.i | 0 | − | 1.49951e6i | 0 | ||||||||||||
33.15 | 0 | 230.668 | + | 230.668i | 0 | 3615.07 | − | 3615.07i | 0 | 262248.i | 0 | − | 1.48791e6i | 0 | |||||||||||||
33.16 | 0 | 336.873 | + | 336.873i | 0 | 6822.00 | − | 6822.00i | 0 | 509047.i | 0 | − | 1.36736e6i | 0 | |||||||||||||
33.17 | 0 | 660.650 | + | 660.650i | 0 | 6982.48 | − | 6982.48i | 0 | − | 294350.i | 0 | − | 721406.i | 0 | ||||||||||||
33.18 | 0 | 787.355 | + | 787.355i | 0 | −13376.5 | + | 13376.5i | 0 | − | 547213.i | 0 | − | 354466.i | 0 | ||||||||||||
33.19 | 0 | 924.404 | + | 924.404i | 0 | −42023.8 | + | 42023.8i | 0 | 291398.i | 0 | 114722.i | 0 | ||||||||||||||
33.20 | 0 | 1009.54 | + | 1009.54i | 0 | 19923.8 | − | 19923.8i | 0 | 484985.i | 0 | 444038.i | 0 | ||||||||||||||
See all 50 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 | 128.14.e.b | 50 | |
4.b | odd | 2 | 1 | 128.14.e.a | 50 | ||
8.b | even | 2 | 1 | 16.14.e.a | ✓ | 50 | |
8.d | odd | 2 | 1 | 64.14.e.a | 50 | ||
16.e | even | 4 | 1 | 16.14.e.a | ✓ | 50 | |
16.e | even | 4 | 1 | inner | 128.14.e.b | 50 | |
16.f | odd | 4 | 1 | 64.14.e.a | 50 | ||
16.f | odd | 4 | 1 | 128.14.e.a | 50 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
16.14.e.a | ✓ | 50 | 8.b | even | 2 | 1 | |
16.14.e.a | ✓ | 50 | 16.e | even | 4 | 1 | |
64.14.e.a | 50 | 8.d | odd | 2 | 1 | ||
64.14.e.a | 50 | 16.f | odd | 4 | 1 | ||
128.14.e.a | 50 | 4.b | odd | 2 | 1 | ||
128.14.e.a | 50 | 16.f | odd | 4 | 1 | ||
128.14.e.b | 50 | 1.a | even | 1 | 1 | trivial | |
128.14.e.b | 50 | 16.e | even | 4 | 1 | inner |