Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [828,2,Mod(19,828)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(828, base_ring=CyclotomicField(22))
chi = DirichletCharacter(H, H._module([11, 0, 15]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("828.19");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 828 = 2^{2} \cdot 3^{2} \cdot 23 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 828.u (of order \(22\), degree \(10\), 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: | \(6.61161328736\) |
Analytic rank: | \(0\) |
Dimension: | \(240\) |
Relative dimension: | \(24\) over \(\Q(\zeta_{22})\) |
Twist minimal: | no (minimal twist has level 276) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{22}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
19.1 | −1.41418 | + | 0.0101703i | 0 | 1.99979 | − | 0.0287651i | 1.32696 | − | 2.06479i | 0 | 0.0202392 | + | 0.140767i | −2.82777 | + | 0.0610173i | 0 | −1.85556 | + | 2.93347i | ||||||
19.2 | −1.37328 | + | 0.337778i | 0 | 1.77181 | − | 0.927729i | −0.802068 | + | 1.24804i | 0 | 0.333353 | + | 2.31852i | −2.11983 | + | 1.87251i | 0 | 0.679905 | − | 1.98484i | ||||||
19.3 | −1.36812 | − | 0.358110i | 0 | 1.74351 | + | 0.979876i | −0.606007 | + | 0.942966i | 0 | −0.442225 | − | 3.07574i | −2.03444 | − | 1.96496i | 0 | 1.16678 | − | 1.07308i | ||||||
19.4 | −1.31756 | + | 0.513838i | 0 | 1.47194 | − | 1.35403i | −0.897443 | + | 1.39645i | 0 | −0.549207 | − | 3.81982i | −1.24362 | + | 2.54035i | 0 | 0.464889 | − | 2.30105i | ||||||
19.5 | −1.12363 | − | 0.858753i | 0 | 0.525087 | + | 1.92984i | −0.628133 | + | 0.977395i | 0 | 0.140631 | + | 0.978108i | 1.06725 | − | 2.61935i | 0 | 1.54513 | − | 0.558818i | ||||||
19.6 | −1.04496 | + | 0.952920i | 0 | 0.183889 | − | 1.99153i | −1.44457 | + | 2.24780i | 0 | 0.629892 | + | 4.38099i | 1.70561 | + | 2.25630i | 0 | −0.632448 | − | 3.72542i | ||||||
19.7 | −0.881142 | + | 1.10616i | 0 | −0.447177 | − | 1.94937i | 0.505840 | − | 0.787103i | 0 | −0.334710 | − | 2.32796i | 2.55034 | + | 1.22302i | 0 | 0.424944 | + | 1.25309i | ||||||
19.8 | −0.631897 | − | 1.26519i | 0 | −1.20141 | + | 1.59894i | 1.27480 | − | 1.98363i | 0 | 0.0657938 | + | 0.457606i | 2.78213 | + | 0.509647i | 0 | −3.31522 | − | 0.359416i | ||||||
19.9 | −0.569834 | + | 1.29433i | 0 | −1.35058 | − | 1.47511i | 2.32731 | − | 3.62137i | 0 | 0.331623 | + | 2.30649i | 2.67888 | − | 0.907528i | 0 | 3.36106 | + | 5.07589i | ||||||
19.10 | −0.542362 | − | 1.30608i | 0 | −1.41169 | + | 1.41674i | 1.27480 | − | 1.98363i | 0 | −0.0657938 | − | 0.457606i | 2.61601 | + | 1.07539i | 0 | −3.28219 | − | 0.589147i | ||||||
19.11 | −0.298132 | + | 1.38243i | 0 | −1.82223 | − | 0.824294i | −0.197236 | + | 0.306906i | 0 | −0.608715 | − | 4.23371i | 1.68280 | − | 2.27337i | 0 | −0.365474 | − | 0.364164i | ||||||
19.12 | 0.000698111 | 1.41421i | 0 | −2.00000 | + | 0.00197456i | −2.33778 | + | 3.63766i | 0 | 0.00906423 | + | 0.0630431i | −0.00418866 | − | 2.82842i | 0 | −5.14605 | − | 3.30358i | |||||||
19.13 | 0.0868188 | − | 1.41155i | 0 | −1.98492 | − | 0.245097i | −0.628133 | + | 0.977395i | 0 | −0.140631 | − | 0.978108i | −0.518295 | + | 2.78053i | 0 | 1.32510 | + | 0.971495i | ||||||
19.14 | 0.186283 | + | 1.40189i | 0 | −1.93060 | + | 0.522296i | 1.47832 | − | 2.30032i | 0 | 0.464774 | + | 3.23258i | −1.09184 | − | 2.60919i | 0 | 3.50018 | + | 1.64394i | ||||||
19.15 | 0.625288 | − | 1.26847i | 0 | −1.21803 | − | 1.58632i | −0.606007 | + | 0.942966i | 0 | 0.442225 | + | 3.07574i | −2.77381 | + | 0.553129i | 0 | 0.817195 | + | 1.35833i | ||||||
19.16 | 0.933775 | − | 1.06210i | 0 | −0.256128 | − | 1.98353i | 1.32696 | − | 2.06479i | 0 | −0.0202392 | − | 0.140767i | −2.34588 | − | 1.58014i | 0 | −0.953939 | − | 3.33742i | ||||||
19.17 | 0.937490 | + | 1.05883i | 0 | −0.242227 | + | 1.98528i | 1.47832 | − | 2.30032i | 0 | −0.464774 | − | 3.23258i | −2.32915 | + | 1.60470i | 0 | 3.82155 | − | 0.591235i | ||||||
19.18 | 1.06833 | + | 0.926640i | 0 | 0.282675 | + | 1.97992i | −2.33778 | + | 3.63766i | 0 | −0.00906423 | − | 0.0630431i | −1.53269 | + | 2.37716i | 0 | −5.86833 | + | 1.71995i | ||||||
19.19 | 1.15458 | − | 0.816661i | 0 | 0.666131 | − | 1.88581i | −0.802068 | + | 1.24804i | 0 | −0.333353 | − | 2.31852i | −0.770961 | − | 2.72133i | 0 | 0.0931716 | + | 2.09599i | ||||||
19.20 | 1.24001 | + | 0.679987i | 0 | 1.07524 | + | 1.68638i | −0.197236 | + | 0.306906i | 0 | 0.608715 | + | 4.23371i | 0.186585 | + | 2.82227i | 0 | −0.453266 | + | 0.246447i | ||||||
See next 80 embeddings (of 240 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
23.d | odd | 22 | 1 | inner |
92.h | even | 22 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 828.2.u.c | 240 | |
3.b | odd | 2 | 1 | 276.2.m.a | ✓ | 240 | |
4.b | odd | 2 | 1 | inner | 828.2.u.c | 240 | |
12.b | even | 2 | 1 | 276.2.m.a | ✓ | 240 | |
23.d | odd | 22 | 1 | inner | 828.2.u.c | 240 | |
69.g | even | 22 | 1 | 276.2.m.a | ✓ | 240 | |
92.h | even | 22 | 1 | inner | 828.2.u.c | 240 | |
276.j | odd | 22 | 1 | 276.2.m.a | ✓ | 240 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
276.2.m.a | ✓ | 240 | 3.b | odd | 2 | 1 | |
276.2.m.a | ✓ | 240 | 12.b | even | 2 | 1 | |
276.2.m.a | ✓ | 240 | 69.g | even | 22 | 1 | |
276.2.m.a | ✓ | 240 | 276.j | odd | 22 | 1 | |
828.2.u.c | 240 | 1.a | even | 1 | 1 | trivial | |
828.2.u.c | 240 | 4.b | odd | 2 | 1 | inner | |
828.2.u.c | 240 | 23.d | odd | 22 | 1 | inner | |
828.2.u.c | 240 | 92.h | even | 22 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{120} - 36 T_{5}^{118} + 924 T_{5}^{116} + 44 T_{5}^{115} - 20280 T_{5}^{114} + \cdots + 37\!\cdots\!29 \) acting on \(S_{2}^{\mathrm{new}}(828, [\chi])\).