Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [124,5,Mod(37,124)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(124, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 5]))
N = Newforms(chi, 5, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("124.37");
S:= CuspForms(chi, 5);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 124 = 2^{2} \cdot 31 \) |
Weight: | \( k \) | \(=\) | \( 5 \) |
Character orbit: | \([\chi]\) | \(=\) | 124.h (of order \(6\), degree \(2\), 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: | \(12.8178754224\) |
Analytic rank: | \(0\) |
Dimension: | \(22\) |
Relative dimension: | \(11\) over \(\Q(\zeta_{6})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
37.1 | 0 | −13.5307 | + | 7.81198i | 0 | 9.80731 | − | 16.9868i | 0 | −5.82661 | − | 10.0920i | 0 | 81.5539 | − | 141.256i | 0 | ||||||||||
37.2 | 0 | −10.0456 | + | 5.79983i | 0 | −15.8301 | + | 27.4186i | 0 | 43.3790 | + | 75.1347i | 0 | 26.7761 | − | 46.3775i | 0 | ||||||||||
37.3 | 0 | −7.72554 | + | 4.46034i | 0 | −21.4163 | + | 37.0942i | 0 | −45.4534 | − | 78.7276i | 0 | −0.710729 | + | 1.23102i | 0 | ||||||||||
37.4 | 0 | −6.25901 | + | 3.61364i | 0 | 0.373689 | − | 0.647249i | 0 | 0.470241 | + | 0.814482i | 0 | −14.3832 | + | 24.9124i | 0 | ||||||||||
37.5 | 0 | −2.65144 | + | 1.53081i | 0 | 22.6791 | − | 39.2814i | 0 | −12.5671 | − | 21.7669i | 0 | −35.8132 | + | 62.0304i | 0 | ||||||||||
37.6 | 0 | 0.0892917 | − | 0.0515526i | 0 | −0.578296 | + | 1.00164i | 0 | −11.2982 | − | 19.5690i | 0 | −40.4947 | + | 70.1389i | 0 | ||||||||||
37.7 | 0 | 2.97484 | − | 1.71753i | 0 | 7.94217 | − | 13.7562i | 0 | 43.7017 | + | 75.6935i | 0 | −34.6002 | + | 59.9293i | 0 | ||||||||||
37.8 | 0 | 6.25440 | − | 3.61098i | 0 | −16.2328 | + | 28.1161i | 0 | 5.12079 | + | 8.86947i | 0 | −14.4216 | + | 24.9790i | 0 | ||||||||||
37.9 | 0 | 10.3346 | − | 5.96666i | 0 | 3.25406 | − | 5.63620i | 0 | −33.7901 | − | 58.5261i | 0 | 30.7021 | − | 53.1776i | 0 | ||||||||||
37.10 | 0 | 12.3007 | − | 7.10184i | 0 | 21.1797 | − | 36.6844i | 0 | 11.7026 | + | 20.2694i | 0 | 60.3722 | − | 104.568i | 0 | ||||||||||
37.11 | 0 | 12.7585 | − | 7.36612i | 0 | −12.6785 | + | 21.9598i | 0 | 12.0611 | + | 20.8905i | 0 | 68.0193 | − | 117.813i | 0 | ||||||||||
57.1 | 0 | −13.5307 | − | 7.81198i | 0 | 9.80731 | + | 16.9868i | 0 | −5.82661 | + | 10.0920i | 0 | 81.5539 | + | 141.256i | 0 | ||||||||||
57.2 | 0 | −10.0456 | − | 5.79983i | 0 | −15.8301 | − | 27.4186i | 0 | 43.3790 | − | 75.1347i | 0 | 26.7761 | + | 46.3775i | 0 | ||||||||||
57.3 | 0 | −7.72554 | − | 4.46034i | 0 | −21.4163 | − | 37.0942i | 0 | −45.4534 | + | 78.7276i | 0 | −0.710729 | − | 1.23102i | 0 | ||||||||||
57.4 | 0 | −6.25901 | − | 3.61364i | 0 | 0.373689 | + | 0.647249i | 0 | 0.470241 | − | 0.814482i | 0 | −14.3832 | − | 24.9124i | 0 | ||||||||||
57.5 | 0 | −2.65144 | − | 1.53081i | 0 | 22.6791 | + | 39.2814i | 0 | −12.5671 | + | 21.7669i | 0 | −35.8132 | − | 62.0304i | 0 | ||||||||||
57.6 | 0 | 0.0892917 | + | 0.0515526i | 0 | −0.578296 | − | 1.00164i | 0 | −11.2982 | + | 19.5690i | 0 | −40.4947 | − | 70.1389i | 0 | ||||||||||
57.7 | 0 | 2.97484 | + | 1.71753i | 0 | 7.94217 | + | 13.7562i | 0 | 43.7017 | − | 75.6935i | 0 | −34.6002 | − | 59.9293i | 0 | ||||||||||
57.8 | 0 | 6.25440 | + | 3.61098i | 0 | −16.2328 | − | 28.1161i | 0 | 5.12079 | − | 8.86947i | 0 | −14.4216 | − | 24.9790i | 0 | ||||||||||
57.9 | 0 | 10.3346 | + | 5.96666i | 0 | 3.25406 | + | 5.63620i | 0 | −33.7901 | + | 58.5261i | 0 | 30.7021 | + | 53.1776i | 0 | ||||||||||
See all 22 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
31.e | odd | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 124.5.h.a | ✓ | 22 |
31.e | odd | 6 | 1 | inner | 124.5.h.a | ✓ | 22 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
124.5.h.a | ✓ | 22 | 1.a | even | 1 | 1 | trivial |
124.5.h.a | ✓ | 22 | 31.e | odd | 6 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(124, [\chi])\).