Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [128,3,Mod(15,128)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(128, base_ring=CyclotomicField(8))
chi = DirichletCharacter(H, H._module([4, 1]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("128.15");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 128 = 2^{7} \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 128.h (of order \(8\), degree \(4\), 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: | \(3.48774738381\) |
Analytic rank: | \(0\) |
Dimension: | \(28\) |
Relative dimension: | \(7\) over \(\Q(\zeta_{8})\) |
Twist minimal: | no (minimal twist has level 32) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{8}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
15.1 | 0 | −4.35131 | + | 1.80237i | 0 | −2.81639 | − | 1.16659i | 0 | 6.23443 | − | 6.23443i | 0 | 9.32143 | − | 9.32143i | 0 | ||||||||||
15.2 | 0 | −2.49683 | + | 1.03422i | 0 | −0.452310 | − | 0.187353i | 0 | −0.429965 | + | 0.429965i | 0 | −1.19943 | + | 1.19943i | 0 | ||||||||||
15.3 | 0 | −1.58190 | + | 0.655246i | 0 | 4.18866 | + | 1.73500i | 0 | −3.93197 | + | 3.93197i | 0 | −4.29089 | + | 4.29089i | 0 | ||||||||||
15.4 | 0 | 0.374985 | − | 0.155324i | 0 | −7.60625 | − | 3.15061i | 0 | −6.84161 | + | 6.84161i | 0 | −6.24747 | + | 6.24747i | 0 | ||||||||||
15.5 | 0 | 1.37292 | − | 0.568682i | 0 | 2.28872 | + | 0.948019i | 0 | 6.37744 | − | 6.37744i | 0 | −4.80245 | + | 4.80245i | 0 | ||||||||||
15.6 | 0 | 3.70255 | − | 1.53365i | 0 | 7.20074 | + | 2.98264i | 0 | −4.26150 | + | 4.26150i | 0 | 4.99283 | − | 4.99283i | 0 | ||||||||||
15.7 | 0 | 4.68670 | − | 1.94129i | 0 | −4.51028 | − | 1.86822i | 0 | 3.85317 | − | 3.85317i | 0 | 11.8326 | − | 11.8326i | 0 | ||||||||||
47.1 | 0 | −2.10187 | + | 5.07436i | 0 | 1.74699 | + | 4.21761i | 0 | 0.392379 | + | 0.392379i | 0 | −14.9674 | − | 14.9674i | 0 | ||||||||||
47.2 | 0 | −0.936461 | + | 2.26082i | 0 | −3.18221 | − | 7.68254i | 0 | −3.67370 | − | 3.67370i | 0 | 2.12963 | + | 2.12963i | 0 | ||||||||||
47.3 | 0 | −0.527719 | + | 1.27403i | 0 | −0.642823 | − | 1.55191i | 0 | 4.95044 | + | 4.95044i | 0 | 5.01930 | + | 5.01930i | 0 | ||||||||||
47.4 | 0 | −0.299792 | + | 0.723762i | 0 | 1.34740 | + | 3.25291i | 0 | −0.583225 | − | 0.583225i | 0 | 5.93000 | + | 5.93000i | 0 | ||||||||||
47.5 | 0 | 1.10785 | − | 2.67458i | 0 | 2.95565 | + | 7.13556i | 0 | 4.18452 | + | 4.18452i | 0 | 0.437918 | + | 0.437918i | 0 | ||||||||||
47.6 | 0 | 1.31872 | − | 3.18367i | 0 | −0.659338 | − | 1.59178i | 0 | −9.54718 | − | 9.54718i | 0 | −2.03276 | − | 2.03276i | 0 | ||||||||||
47.7 | 0 | 1.73217 | − | 4.18183i | 0 | −1.85856 | − | 4.48696i | 0 | 5.27676 | + | 5.27676i | 0 | −8.12333 | − | 8.12333i | 0 | ||||||||||
79.1 | 0 | −2.10187 | − | 5.07436i | 0 | 1.74699 | − | 4.21761i | 0 | 0.392379 | − | 0.392379i | 0 | −14.9674 | + | 14.9674i | 0 | ||||||||||
79.2 | 0 | −0.936461 | − | 2.26082i | 0 | −3.18221 | + | 7.68254i | 0 | −3.67370 | + | 3.67370i | 0 | 2.12963 | − | 2.12963i | 0 | ||||||||||
79.3 | 0 | −0.527719 | − | 1.27403i | 0 | −0.642823 | + | 1.55191i | 0 | 4.95044 | − | 4.95044i | 0 | 5.01930 | − | 5.01930i | 0 | ||||||||||
79.4 | 0 | −0.299792 | − | 0.723762i | 0 | 1.34740 | − | 3.25291i | 0 | −0.583225 | + | 0.583225i | 0 | 5.93000 | − | 5.93000i | 0 | ||||||||||
79.5 | 0 | 1.10785 | + | 2.67458i | 0 | 2.95565 | − | 7.13556i | 0 | 4.18452 | − | 4.18452i | 0 | 0.437918 | − | 0.437918i | 0 | ||||||||||
79.6 | 0 | 1.31872 | + | 3.18367i | 0 | −0.659338 | + | 1.59178i | 0 | −9.54718 | + | 9.54718i | 0 | −2.03276 | + | 2.03276i | 0 | ||||||||||
See all 28 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
32.h | odd | 8 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 128.3.h.a | 28 | |
4.b | odd | 2 | 1 | 32.3.h.a | ✓ | 28 | |
8.b | even | 2 | 1 | 256.3.h.a | 28 | ||
8.d | odd | 2 | 1 | 256.3.h.b | 28 | ||
12.b | even | 2 | 1 | 288.3.u.a | 28 | ||
32.g | even | 8 | 1 | 32.3.h.a | ✓ | 28 | |
32.g | even | 8 | 1 | 256.3.h.b | 28 | ||
32.h | odd | 8 | 1 | inner | 128.3.h.a | 28 | |
32.h | odd | 8 | 1 | 256.3.h.a | 28 | ||
96.p | odd | 8 | 1 | 288.3.u.a | 28 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
32.3.h.a | ✓ | 28 | 4.b | odd | 2 | 1 | |
32.3.h.a | ✓ | 28 | 32.g | even | 8 | 1 | |
128.3.h.a | 28 | 1.a | even | 1 | 1 | trivial | |
128.3.h.a | 28 | 32.h | odd | 8 | 1 | inner | |
256.3.h.a | 28 | 8.b | even | 2 | 1 | ||
256.3.h.a | 28 | 32.h | odd | 8 | 1 | ||
256.3.h.b | 28 | 8.d | odd | 2 | 1 | ||
256.3.h.b | 28 | 32.g | even | 8 | 1 | ||
288.3.u.a | 28 | 12.b | even | 2 | 1 | ||
288.3.u.a | 28 | 96.p | odd | 8 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(128, [\chi])\).