Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [77,5,Mod(34,77)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(77, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0]))
N = Newforms(chi, 5, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("77.34");
S:= CuspForms(chi, 5);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 77 = 7 \cdot 11 \) |
Weight: | \( k \) | \(=\) | \( 5 \) |
Character orbit: | \([\chi]\) | \(=\) | 77.d (of order \(2\), degree \(1\), 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: | \(7.95948715746\) |
Analytic rank: | \(0\) |
Dimension: | \(28\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
34.1 | −7.57354 | − | 0.449581i | 41.3586 | 37.8002i | 3.40492i | 6.17570 | − | 48.6093i | −192.054 | 80.7979 | − | 286.281i | ||||||||||||||
34.2 | −7.57354 | 0.449581i | 41.3586 | − | 37.8002i | − | 3.40492i | 6.17570 | + | 48.6093i | −192.054 | 80.7979 | 286.281i | ||||||||||||||
34.3 | −6.56155 | − | 14.6451i | 27.0540 | 11.3204i | 96.0944i | −43.4696 | + | 22.6140i | −72.5314 | −133.478 | − | 74.2796i | ||||||||||||||
34.4 | −6.56155 | 14.6451i | 27.0540 | − | 11.3204i | − | 96.0944i | −43.4696 | − | 22.6140i | −72.5314 | −133.478 | 74.2796i | ||||||||||||||
34.5 | −5.70412 | − | 7.64626i | 16.5370 | − | 9.31094i | 43.6152i | 34.0002 | − | 35.2843i | −3.06288 | 22.5346 | 53.1107i | ||||||||||||||
34.6 | −5.70412 | 7.64626i | 16.5370 | 9.31094i | − | 43.6152i | 34.0002 | + | 35.2843i | −3.06288 | 22.5346 | − | 53.1107i | ||||||||||||||
34.7 | −4.40197 | − | 6.88637i | 3.37736 | − | 40.5957i | 30.3136i | −42.1123 | − | 25.0510i | 55.5645 | 33.5780 | 178.701i | ||||||||||||||
34.8 | −4.40197 | 6.88637i | 3.37736 | 40.5957i | − | 30.3136i | −42.1123 | + | 25.0510i | 55.5645 | 33.5780 | − | 178.701i | ||||||||||||||
34.9 | −3.85007 | − | 10.1487i | −1.17693 | 19.5148i | 39.0733i | 30.1430 | + | 38.6316i | 66.1325 | −21.9963 | − | 75.1335i | ||||||||||||||
34.10 | −3.85007 | 10.1487i | −1.17693 | − | 19.5148i | − | 39.0733i | 30.1430 | − | 38.6316i | 66.1325 | −21.9963 | 75.1335i | ||||||||||||||
34.11 | −1.51012 | − | 0.525884i | −13.7195 | − | 19.2027i | 0.794146i | −29.0302 | + | 39.4747i | 44.8800 | 80.7234 | 28.9984i | ||||||||||||||
34.12 | −1.51012 | 0.525884i | −13.7195 | 19.2027i | − | 0.794146i | −29.0302 | − | 39.4747i | 44.8800 | 80.7234 | − | 28.9984i | ||||||||||||||
34.13 | −1.01583 | − | 14.6362i | −14.9681 | 22.7511i | 14.8679i | −15.2165 | − | 46.5774i | 31.4583 | −133.218 | − | 23.1113i | ||||||||||||||
34.14 | −1.01583 | 14.6362i | −14.9681 | − | 22.7511i | − | 14.8679i | −15.2165 | + | 46.5774i | 31.4583 | −133.218 | 23.1113i | ||||||||||||||
34.15 | −0.518092 | − | 16.0828i | −15.7316 | − | 43.3519i | 8.33238i | 46.5272 | + | 15.3694i | 16.4399 | −177.657 | 22.4603i | ||||||||||||||
34.16 | −0.518092 | 16.0828i | −15.7316 | 43.3519i | − | 8.33238i | 46.5272 | − | 15.3694i | 16.4399 | −177.657 | − | 22.4603i | ||||||||||||||
34.17 | 2.59965 | − | 8.89988i | −9.24183 | 44.8437i | − | 23.1366i | −4.23946 | + | 48.8163i | −65.6199 | 1.79220 | 116.578i | ||||||||||||||
34.18 | 2.59965 | 8.89988i | −9.24183 | − | 44.8437i | 23.1366i | −4.23946 | − | 48.8163i | −65.6199 | 1.79220 | − | 116.578i | ||||||||||||||
34.19 | 3.32783 | − | 9.63594i | −4.92552 | − | 18.6477i | − | 32.0668i | −44.5022 | + | 20.5073i | −69.6367 | −11.8513 | − | 62.0566i | ||||||||||||
34.20 | 3.32783 | 9.63594i | −4.92552 | 18.6477i | 32.0668i | −44.5022 | − | 20.5073i | −69.6367 | −11.8513 | 62.0566i | ||||||||||||||||
See all 28 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.b | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 77.5.d.a | ✓ | 28 |
7.b | odd | 2 | 1 | inner | 77.5.d.a | ✓ | 28 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
77.5.d.a | ✓ | 28 | 1.a | even | 1 | 1 | trivial |
77.5.d.a | ✓ | 28 | 7.b | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(77, [\chi])\).