Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [252,9,Mod(29,252)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(252, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 1, 0]))
N = Newforms(chi, 9, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("252.29");
S:= CuspForms(chi, 9);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 9 \) |
Character orbit: | \([\chi]\) | \(=\) | 252.bg (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: | \(102.659409735\) |
Analytic rank: | \(0\) |
Dimension: | \(96\) |
Relative dimension: | \(48\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
29.1 | 0 | −80.9976 | − | 0.623037i | 0 | 25.8029 | − | 14.8973i | 0 | 453.746 | − | 785.912i | 0 | 6560.22 | + | 100.929i | 0 | ||||||||||
29.2 | 0 | −80.2339 | − | 11.1142i | 0 | −207.070 | + | 119.552i | 0 | −453.746 | + | 785.912i | 0 | 6313.95 | + | 1783.47i | 0 | ||||||||||
29.3 | 0 | −79.9711 | − | 12.8694i | 0 | −908.444 | + | 524.491i | 0 | 453.746 | − | 785.912i | 0 | 6229.76 | + | 2058.36i | 0 | ||||||||||
29.4 | 0 | −79.7801 | − | 14.0049i | 0 | 225.541 | − | 130.216i | 0 | −453.746 | + | 785.912i | 0 | 6168.73 | + | 2234.62i | 0 | ||||||||||
29.5 | 0 | −75.7614 | + | 28.6568i | 0 | 425.975 | − | 245.937i | 0 | 453.746 | − | 785.912i | 0 | 4918.57 | − | 4342.16i | 0 | ||||||||||
29.6 | 0 | −75.7417 | + | 28.7087i | 0 | 820.829 | − | 473.906i | 0 | 453.746 | − | 785.912i | 0 | 4912.62 | − | 4348.90i | 0 | ||||||||||
29.7 | 0 | −73.8531 | + | 33.2674i | 0 | −290.393 | + | 167.658i | 0 | −453.746 | + | 785.912i | 0 | 4347.56 | − | 4913.80i | 0 | ||||||||||
29.8 | 0 | −71.6703 | + | 37.7408i | 0 | 900.834 | − | 520.096i | 0 | −453.746 | + | 785.912i | 0 | 3712.26 | − | 5409.79i | 0 | ||||||||||
29.9 | 0 | −71.3437 | − | 38.3547i | 0 | 330.062 | − | 190.561i | 0 | 453.746 | − | 785.912i | 0 | 3618.84 | + | 5472.73i | 0 | ||||||||||
29.10 | 0 | −68.4056 | − | 43.3783i | 0 | 645.917 | − | 372.921i | 0 | −453.746 | + | 785.912i | 0 | 2797.64 | + | 5934.64i | 0 | ||||||||||
29.11 | 0 | −63.2760 | + | 50.5682i | 0 | −405.178 | + | 233.930i | 0 | −453.746 | + | 785.912i | 0 | 1446.71 | − | 6399.51i | 0 | ||||||||||
29.12 | 0 | −62.9498 | − | 50.9738i | 0 | −401.865 | + | 232.017i | 0 | 453.746 | − | 785.912i | 0 | 1364.35 | + | 6417.57i | 0 | ||||||||||
29.13 | 0 | −55.9608 | − | 58.5610i | 0 | −883.740 | + | 510.228i | 0 | −453.746 | + | 785.912i | 0 | −297.778 | + | 6554.24i | 0 | ||||||||||
29.14 | 0 | −55.8369 | + | 58.6791i | 0 | −400.624 | + | 231.301i | 0 | 453.746 | − | 785.912i | 0 | −325.483 | − | 6552.92i | 0 | ||||||||||
29.15 | 0 | −45.2321 | − | 67.1942i | 0 | −4.45748 | + | 2.57353i | 0 | −453.746 | + | 785.912i | 0 | −2469.12 | + | 6078.67i | 0 | ||||||||||
29.16 | 0 | −41.3930 | + | 69.6248i | 0 | −865.452 | + | 499.669i | 0 | 453.746 | − | 785.912i | 0 | −3134.24 | − | 5763.96i | 0 | ||||||||||
29.17 | 0 | −31.9190 | − | 74.4458i | 0 | 867.572 | − | 500.893i | 0 | 453.746 | − | 785.912i | 0 | −4523.36 | + | 4752.47i | 0 | ||||||||||
29.18 | 0 | −26.4497 | + | 76.5599i | 0 | 151.995 | − | 87.7545i | 0 | −453.746 | + | 785.912i | 0 | −5161.83 | − | 4049.97i | 0 | ||||||||||
29.19 | 0 | −24.5606 | − | 77.1866i | 0 | −248.615 | + | 143.538i | 0 | 453.746 | − | 785.912i | 0 | −5354.55 | + | 3791.50i | 0 | ||||||||||
29.20 | 0 | −21.6670 | + | 78.0483i | 0 | 514.619 | − | 297.115i | 0 | 453.746 | − | 785.912i | 0 | −5622.08 | − | 3382.15i | 0 | ||||||||||
See all 96 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
9.d | odd | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 252.9.bg.a | ✓ | 96 |
9.d | odd | 6 | 1 | inner | 252.9.bg.a | ✓ | 96 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
252.9.bg.a | ✓ | 96 | 1.a | even | 1 | 1 | trivial |
252.9.bg.a | ✓ | 96 | 9.d | odd | 6 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{9}^{\mathrm{new}}(252, [\chi])\).