Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [216,4,Mod(37,216)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(216, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 3, 2]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("216.37");
S:= CuspForms(chi, 4);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 216 = 2^{3} \cdot 3^{3} \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 216.n (of order \(6\), degree \(2\), not 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: | \(12.7444125612\) |
Analytic rank: | \(0\) |
Dimension: | \(68\) |
Relative dimension: | \(34\) over \(\Q(\zeta_{6})\) |
Twist minimal: | no (minimal twist has level 72) |
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 | −2.82826 | − | 0.0306697i | 0 | 7.99812 | + | 0.173484i | 4.46589 | + | 2.57838i | 0 | −4.47414 | − | 7.74943i | −22.6154 | − | 0.735958i | 0 | −12.5516 | − | 7.42931i | ||||||
37.2 | −2.81886 | − | 0.232479i | 0 | 7.89191 | + | 1.31065i | −2.06876 | − | 1.19440i | 0 | 17.9335 | + | 31.0618i | −21.9415 | − | 5.52925i | 0 | 5.55387 | + | 3.84779i | ||||||
37.3 | −2.80631 | + | 0.353019i | 0 | 7.75076 | − | 1.98136i | −13.3655 | − | 7.71659i | 0 | −14.8241 | − | 25.6761i | −21.0516 | + | 8.29647i | 0 | 40.2319 | + | 16.9369i | ||||||
37.4 | −2.67089 | + | 0.930772i | 0 | 6.26733 | − | 4.97198i | 14.6790 | + | 8.47493i | 0 | 9.54171 | + | 16.5267i | −12.1116 | + | 19.1131i | 0 | −47.0943 | − | 8.97282i | ||||||
37.5 | −2.48873 | − | 1.34396i | 0 | 4.38752 | + | 6.68952i | −6.59200 | − | 3.80589i | 0 | −7.67810 | − | 13.2989i | −1.92888 | − | 22.5451i | 0 | 11.2907 | + | 18.3312i | ||||||
37.6 | −2.34837 | + | 1.57644i | 0 | 3.02970 | − | 7.40412i | −6.66588 | − | 3.84855i | 0 | 4.44653 | + | 7.70162i | 4.55727 | + | 22.1637i | 0 | 21.7209 | − | 1.47052i | ||||||
37.7 | −2.30783 | − | 1.63521i | 0 | 2.65219 | + | 7.54757i | −3.38868 | − | 1.95645i | 0 | 1.48218 | + | 2.56721i | 6.22103 | − | 21.7554i | 0 | 4.62129 | + | 10.0564i | ||||||
37.8 | −2.17295 | + | 1.81060i | 0 | 1.44346 | − | 7.86870i | 7.42736 | + | 4.28819i | 0 | −10.3525 | − | 17.9310i | 11.1105 | + | 19.7119i | 0 | −23.9035 | + | 4.12993i | ||||||
37.9 | −2.16881 | − | 1.81556i | 0 | 1.40750 | + | 7.87521i | 17.5797 | + | 10.1497i | 0 | −10.6896 | − | 18.5150i | 11.2453 | − | 19.6353i | 0 | −19.6999 | − | 53.9298i | ||||||
37.10 | −1.77410 | + | 2.20286i | 0 | −1.70517 | − | 7.81616i | −13.2040 | − | 7.62335i | 0 | 6.18711 | + | 10.7164i | 20.2430 | + | 10.1104i | 0 | 40.2184 | − | 15.5620i | ||||||
37.11 | −1.59019 | − | 2.33908i | 0 | −2.94259 | + | 7.43916i | 11.0897 | + | 6.40265i | 0 | 11.0213 | + | 19.0894i | 22.0801 | − | 4.94672i | 0 | −2.65844 | − | 36.1212i | ||||||
37.12 | −1.23061 | − | 2.54668i | 0 | −4.97121 | + | 6.26794i | −11.0897 | − | 6.40265i | 0 | 11.0213 | + | 19.0894i | 22.0801 | + | 4.94672i | 0 | −2.65844 | + | 36.1212i | ||||||
37.13 | −1.08796 | + | 2.61081i | 0 | −5.63270 | − | 5.68091i | 10.5646 | + | 6.09948i | 0 | −9.53003 | − | 16.5065i | 20.9599 | − | 8.52535i | 0 | −27.4185 | + | 20.9463i | ||||||
37.14 | −0.833385 | + | 2.70286i | 0 | −6.61094 | − | 4.50505i | 2.76156 | + | 1.59439i | 0 | 1.73091 | + | 2.99802i | 17.6860 | − | 14.1140i | 0 | −6.61085 | + | 6.13538i | ||||||
37.15 | −0.487912 | − | 2.78603i | 0 | −7.52388 | + | 2.71867i | −17.5797 | − | 10.1497i | 0 | −10.6896 | − | 18.5150i | 11.2453 | + | 19.6353i | 0 | −19.6999 | + | 53.9298i | ||||||
37.16 | −0.262215 | − | 2.81625i | 0 | −7.86249 | + | 1.47692i | 3.38868 | + | 1.95645i | 0 | 1.48218 | + | 2.56721i | 6.22103 | + | 21.7554i | 0 | 4.62129 | − | 10.0564i | ||||||
37.17 | −0.181886 | + | 2.82257i | 0 | −7.93383 | − | 1.02677i | 13.2324 | + | 7.63974i | 0 | 11.2586 | + | 19.5004i | 4.34120 | − | 22.2071i | 0 | −23.9705 | + | 35.9599i | ||||||
37.18 | −0.176838 | + | 2.82289i | 0 | −7.93746 | − | 0.998392i | −17.3643 | − | 10.0253i | 0 | 1.69820 | + | 2.94137i | 4.22200 | − | 22.2300i | 0 | 31.3710 | − | 47.2447i | ||||||
37.19 | 0.0804569 | − | 2.82728i | 0 | −7.98705 | − | 0.454949i | 6.59200 | + | 3.80589i | 0 | −7.67810 | − | 13.2989i | −1.92888 | + | 22.5451i | 0 | 11.2907 | − | 18.3312i | ||||||
37.20 | 0.980526 | + | 2.65303i | 0 | −6.07714 | + | 5.20273i | −0.772898 | − | 0.446233i | 0 | −14.6308 | − | 25.3412i | −19.7618 | − | 11.0214i | 0 | 0.426023 | − | 2.48806i | ||||||
See all 68 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
8.b | even | 2 | 1 | inner |
9.c | even | 3 | 1 | inner |
72.n | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 216.4.n.a | 68 | |
3.b | odd | 2 | 1 | 72.4.n.a | ✓ | 68 | |
4.b | odd | 2 | 1 | 864.4.r.a | 68 | ||
8.b | even | 2 | 1 | inner | 216.4.n.a | 68 | |
8.d | odd | 2 | 1 | 864.4.r.a | 68 | ||
9.c | even | 3 | 1 | inner | 216.4.n.a | 68 | |
9.d | odd | 6 | 1 | 72.4.n.a | ✓ | 68 | |
12.b | even | 2 | 1 | 288.4.r.a | 68 | ||
24.f | even | 2 | 1 | 288.4.r.a | 68 | ||
24.h | odd | 2 | 1 | 72.4.n.a | ✓ | 68 | |
36.f | odd | 6 | 1 | 864.4.r.a | 68 | ||
36.h | even | 6 | 1 | 288.4.r.a | 68 | ||
72.j | odd | 6 | 1 | 72.4.n.a | ✓ | 68 | |
72.l | even | 6 | 1 | 288.4.r.a | 68 | ||
72.n | even | 6 | 1 | inner | 216.4.n.a | 68 | |
72.p | odd | 6 | 1 | 864.4.r.a | 68 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
72.4.n.a | ✓ | 68 | 3.b | odd | 2 | 1 | |
72.4.n.a | ✓ | 68 | 9.d | odd | 6 | 1 | |
72.4.n.a | ✓ | 68 | 24.h | odd | 2 | 1 | |
72.4.n.a | ✓ | 68 | 72.j | odd | 6 | 1 | |
216.4.n.a | 68 | 1.a | even | 1 | 1 | trivial | |
216.4.n.a | 68 | 8.b | even | 2 | 1 | inner | |
216.4.n.a | 68 | 9.c | even | 3 | 1 | inner | |
216.4.n.a | 68 | 72.n | even | 6 | 1 | inner | |
288.4.r.a | 68 | 12.b | even | 2 | 1 | ||
288.4.r.a | 68 | 24.f | even | 2 | 1 | ||
288.4.r.a | 68 | 36.h | even | 6 | 1 | ||
288.4.r.a | 68 | 72.l | even | 6 | 1 | ||
864.4.r.a | 68 | 4.b | odd | 2 | 1 | ||
864.4.r.a | 68 | 8.d | odd | 2 | 1 | ||
864.4.r.a | 68 | 36.f | odd | 6 | 1 | ||
864.4.r.a | 68 | 72.p | odd | 6 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(216, [\chi])\).