Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [354,5,Mod(235,354)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(354, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 5, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("354.235");
S:= CuspForms(chi, 5);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 354 = 2 \cdot 3 \cdot 59 \) |
Weight: | \( k \) | \(=\) | \( 5 \) |
Character orbit: | \([\chi]\) | \(=\) | 354.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: | \(36.5929669317\) |
Analytic rank: | \(0\) |
Dimension: | \(40\) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
235.1 | − | 2.82843i | 5.19615 | −8.00000 | −10.0054 | − | 14.6969i | 10.1747 | 22.6274i | 27.0000 | 28.2996i | ||||||||||||||||
235.2 | 2.82843i | 5.19615 | −8.00000 | −10.0054 | 14.6969i | 10.1747 | − | 22.6274i | 27.0000 | − | 28.2996i | ||||||||||||||||
235.3 | − | 2.82843i | −5.19615 | −8.00000 | 21.2806 | 14.6969i | −81.9829 | 22.6274i | 27.0000 | − | 60.1907i | ||||||||||||||||
235.4 | 2.82843i | −5.19615 | −8.00000 | 21.2806 | − | 14.6969i | −81.9829 | − | 22.6274i | 27.0000 | 60.1907i | ||||||||||||||||
235.5 | − | 2.82843i | −5.19615 | −8.00000 | 0.141718 | 14.6969i | −89.1589 | 22.6274i | 27.0000 | − | 0.400839i | ||||||||||||||||
235.6 | 2.82843i | −5.19615 | −8.00000 | 0.141718 | − | 14.6969i | −89.1589 | − | 22.6274i | 27.0000 | 0.400839i | ||||||||||||||||
235.7 | − | 2.82843i | −5.19615 | −8.00000 | −8.58312 | 14.6969i | 95.1629 | 22.6274i | 27.0000 | 24.2767i | |||||||||||||||||
235.8 | 2.82843i | −5.19615 | −8.00000 | −8.58312 | − | 14.6969i | 95.1629 | − | 22.6274i | 27.0000 | − | 24.2767i | |||||||||||||||
235.9 | − | 2.82843i | −5.19615 | −8.00000 | −44.0140 | 14.6969i | 46.8163 | 22.6274i | 27.0000 | 124.490i | |||||||||||||||||
235.10 | 2.82843i | −5.19615 | −8.00000 | −44.0140 | − | 14.6969i | 46.8163 | − | 22.6274i | 27.0000 | − | 124.490i | |||||||||||||||
235.11 | − | 2.82843i | −5.19615 | −8.00000 | 0.345303 | 14.6969i | 29.1133 | 22.6274i | 27.0000 | − | 0.976665i | ||||||||||||||||
235.12 | 2.82843i | −5.19615 | −8.00000 | 0.345303 | − | 14.6969i | 29.1133 | − | 22.6274i | 27.0000 | 0.976665i | ||||||||||||||||
235.13 | − | 2.82843i | 5.19615 | −8.00000 | −17.0264 | − | 14.6969i | 47.1245 | 22.6274i | 27.0000 | 48.1580i | ||||||||||||||||
235.14 | 2.82843i | 5.19615 | −8.00000 | −17.0264 | 14.6969i | 47.1245 | − | 22.6274i | 27.0000 | − | 48.1580i | ||||||||||||||||
235.15 | − | 2.82843i | −5.19615 | −8.00000 | −27.9908 | 14.6969i | −34.1747 | 22.6274i | 27.0000 | 79.1699i | |||||||||||||||||
235.16 | 2.82843i | −5.19615 | −8.00000 | −27.9908 | − | 14.6969i | −34.1747 | − | 22.6274i | 27.0000 | − | 79.1699i | |||||||||||||||
235.17 | − | 2.82843i | 5.19615 | −8.00000 | −4.63644 | − | 14.6969i | −33.3282 | 22.6274i | 27.0000 | 13.1138i | ||||||||||||||||
235.18 | 2.82843i | 5.19615 | −8.00000 | −4.63644 | 14.6969i | −33.3282 | − | 22.6274i | 27.0000 | − | 13.1138i | ||||||||||||||||
235.19 | − | 2.82843i | 5.19615 | −8.00000 | 26.1412 | − | 14.6969i | −51.7193 | 22.6274i | 27.0000 | − | 73.9384i | |||||||||||||||
235.20 | 2.82843i | 5.19615 | −8.00000 | 26.1412 | 14.6969i | −51.7193 | − | 22.6274i | 27.0000 | 73.9384i | |||||||||||||||||
See all 40 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
59.b | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 354.5.d.a | ✓ | 40 |
3.b | odd | 2 | 1 | 1062.5.d.b | 40 | ||
59.b | odd | 2 | 1 | inner | 354.5.d.a | ✓ | 40 |
177.d | even | 2 | 1 | 1062.5.d.b | 40 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
354.5.d.a | ✓ | 40 | 1.a | even | 1 | 1 | trivial |
354.5.d.a | ✓ | 40 | 59.b | odd | 2 | 1 | inner |
1062.5.d.b | 40 | 3.b | odd | 2 | 1 | ||
1062.5.d.b | 40 | 177.d | even | 2 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(354, [\chi])\).