Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [128,5,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, 5, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("128.15");
S:= CuspForms(chi, 5);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 128 = 2^{7} \) |
Weight: | \( k \) | \(=\) | \( 5 \) |
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: | \(13.2313552747\) |
Analytic rank: | \(0\) |
Dimension: | \(60\) |
Relative dimension: | \(15\) 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 | −15.8039 | + | 6.54620i | 0 | 21.3091 | + | 8.82651i | 0 | 18.2180 | − | 18.2180i | 0 | 149.636 | − | 149.636i | 0 | ||||||||||
15.2 | 0 | −12.3594 | + | 5.11942i | 0 | −32.9803 | − | 13.6609i | 0 | −59.2707 | + | 59.2707i | 0 | 69.2702 | − | 69.2702i | 0 | ||||||||||
15.3 | 0 | −10.5427 | + | 4.36694i | 0 | −27.2781 | − | 11.2990i | 0 | 51.6577 | − | 51.6577i | 0 | 34.8033 | − | 34.8033i | 0 | ||||||||||
15.4 | 0 | −8.76479 | + | 3.63049i | 0 | 8.51003 | + | 3.52497i | 0 | −21.5633 | + | 21.5633i | 0 | 6.36534 | − | 6.36534i | 0 | ||||||||||
15.5 | 0 | −7.43335 | + | 3.07899i | 0 | 8.82167 | + | 3.65406i | 0 | −10.5220 | + | 10.5220i | 0 | −11.5012 | + | 11.5012i | 0 | ||||||||||
15.6 | 0 | −4.81945 | + | 1.99628i | 0 | 42.1637 | + | 17.4648i | 0 | 27.8831 | − | 27.8831i | 0 | −38.0337 | + | 38.0337i | 0 | ||||||||||
15.7 | 0 | −0.421454 | + | 0.174572i | 0 | −33.4127 | − | 13.8400i | 0 | −1.19092 | + | 1.19092i | 0 | −57.1285 | + | 57.1285i | 0 | ||||||||||
15.8 | 0 | 0.239783 | − | 0.0993213i | 0 | −10.4583 | − | 4.33197i | 0 | 51.7271 | − | 51.7271i | 0 | −57.2280 | + | 57.2280i | 0 | ||||||||||
15.9 | 0 | 3.21611 | − | 1.33215i | 0 | 23.1494 | + | 9.58878i | 0 | −30.5337 | + | 30.5337i | 0 | −48.7069 | + | 48.7069i | 0 | ||||||||||
15.10 | 0 | 4.25478 | − | 1.76239i | 0 | −19.2280 | − | 7.96449i | 0 | −0.0963230 | + | 0.0963230i | 0 | −42.2785 | + | 42.2785i | 0 | ||||||||||
15.11 | 0 | 5.05753 | − | 2.09490i | 0 | 14.0253 | + | 5.80946i | 0 | −61.2653 | + | 61.2653i | 0 | −36.0856 | + | 36.0856i | 0 | ||||||||||
15.12 | 0 | 10.9321 | − | 4.52821i | 0 | −0.302610 | − | 0.125345i | 0 | 32.5244 | − | 32.5244i | 0 | 41.7299 | − | 41.7299i | 0 | ||||||||||
15.13 | 0 | 10.9680 | − | 4.54309i | 0 | 35.3175 | + | 14.6290i | 0 | 47.3057 | − | 47.3057i | 0 | 42.3816 | − | 42.3816i | 0 | ||||||||||
15.14 | 0 | 12.3801 | − | 5.12803i | 0 | −36.8127 | − | 15.2483i | 0 | −16.3058 | + | 16.3058i | 0 | 69.6958 | − | 69.6958i | 0 | ||||||||||
15.15 | 0 | 14.8038 | − | 6.13192i | 0 | 5.46900 | + | 2.26533i | 0 | −27.5680 | + | 27.5680i | 0 | 124.275 | − | 124.275i | 0 | ||||||||||
47.1 | 0 | −6.02892 | + | 14.5551i | 0 | 1.24804 | + | 3.01304i | 0 | −48.9953 | − | 48.9953i | 0 | −118.227 | − | 118.227i | 0 | ||||||||||
47.2 | 0 | −5.09563 | + | 12.3019i | 0 | −6.07195 | − | 14.6590i | 0 | −3.01401 | − | 3.01401i | 0 | −68.0967 | − | 68.0967i | 0 | ||||||||||
47.3 | 0 | −4.50708 | + | 10.8811i | 0 | −0.960017 | − | 2.31769i | 0 | 43.4088 | + | 43.4088i | 0 | −40.8081 | − | 40.8081i | 0 | ||||||||||
47.4 | 0 | −3.05748 | + | 7.38140i | 0 | −13.2656 | − | 32.0260i | 0 | 14.4988 | + | 14.4988i | 0 | 12.1387 | + | 12.1387i | 0 | ||||||||||
47.5 | 0 | −3.03024 | + | 7.31565i | 0 | 12.4171 | + | 29.9775i | 0 | 51.5981 | + | 51.5981i | 0 | 12.9392 | + | 12.9392i | 0 | ||||||||||
See all 60 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 | 128.5.h.a | 60 | |
4.b | odd | 2 | 1 | 32.5.h.a | ✓ | 60 | |
32.g | even | 8 | 1 | 32.5.h.a | ✓ | 60 | |
32.h | odd | 8 | 1 | inner | 128.5.h.a | 60 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
32.5.h.a | ✓ | 60 | 4.b | odd | 2 | 1 | |
32.5.h.a | ✓ | 60 | 32.g | even | 8 | 1 | |
128.5.h.a | 60 | 1.a | even | 1 | 1 | trivial | |
128.5.h.a | 60 | 32.h | odd | 8 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(128, [\chi])\).