Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [324,3,Mod(19,324)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(324, base_ring=CyclotomicField(18))
chi = DirichletCharacter(H, H._module([9, 16]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("324.19");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 324 = 2^{2} \cdot 3^{4} \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 324.j (of order \(18\), degree \(6\), 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: | \(8.82836056527\) |
Analytic rank: | \(0\) |
Dimension: | \(204\) |
Relative dimension: | \(34\) over \(\Q(\zeta_{18})\) |
Twist minimal: | no (minimal twist has level 108) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{18}]$ |
$q$-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 | −1.99738 | − | 0.102316i | 0 | 3.97906 | + | 0.408727i | −1.35104 | − | 7.66211i | 0 | −0.787463 | + | 0.938462i | −7.90589 | − | 1.22351i | 0 | 1.91458 | + | 15.4424i | ||||||
19.2 | −1.99036 | − | 0.196104i | 0 | 3.92309 | + | 0.780636i | 0.479555 | + | 2.71969i | 0 | −6.88504 | + | 8.20527i | −7.65528 | − | 2.32308i | 0 | −0.421146 | − | 5.50722i | ||||||
19.3 | −1.97794 | + | 0.296237i | 0 | 3.82449 | − | 1.17188i | 0.00321173 | + | 0.0182146i | 0 | 0.446686 | − | 0.532339i | −7.21745 | + | 3.45086i | 0 | −0.0117484 | − | 0.0350759i | ||||||
19.4 | −1.91833 | − | 0.565700i | 0 | 3.35997 | + | 2.17040i | 1.66005 | + | 9.41462i | 0 | 4.14643 | − | 4.94152i | −5.21773 | − | 6.06427i | 0 | 2.14133 | − | 18.9994i | ||||||
19.5 | −1.70561 | + | 1.04446i | 0 | 1.81819 | − | 3.56289i | 0.00321173 | + | 0.0182146i | 0 | −0.446686 | + | 0.532339i | 0.620191 | + | 7.97592i | 0 | −0.0245024 | − | 0.0277124i | ||||||
19.6 | −1.63904 | − | 1.14610i | 0 | 1.37291 | + | 3.75701i | −1.09499 | − | 6.21002i | 0 | 5.65814 | − | 6.74311i | 2.05564 | − | 7.73139i | 0 | −5.32256 | + | 11.4335i | ||||||
19.7 | −1.48574 | − | 1.33887i | 0 | 0.414837 | + | 3.97843i | 0.542224 | + | 3.07510i | 0 | 2.98812 | − | 3.56110i | 4.71027 | − | 6.46632i | 0 | 3.31157 | − | 5.29477i | ||||||
19.8 | −1.46432 | + | 1.36227i | 0 | 0.288439 | − | 3.98959i | −1.35104 | − | 7.66211i | 0 | 0.787463 | − | 0.938462i | 5.01253 | + | 6.23495i | 0 | 12.4162 | + | 9.37927i | ||||||
19.9 | −1.43960 | − | 1.38836i | 0 | 0.144916 | + | 3.99737i | 0.133306 | + | 0.756014i | 0 | −3.08121 | + | 3.67204i | 5.34117 | − | 5.95583i | 0 | 0.857712 | − | 1.27344i | ||||||
19.10 | −1.39865 | + | 1.42960i | 0 | −0.0875399 | − | 3.99904i | 0.479555 | + | 2.71969i | 0 | 6.88504 | − | 8.20527i | 5.83949 | + | 5.46812i | 0 | −4.55882 | − | 3.11833i | ||||||
19.11 | −1.10590 | + | 1.66643i | 0 | −1.55397 | − | 3.68581i | 1.66005 | + | 9.41462i | 0 | −4.14643 | + | 4.94152i | 7.86067 | + | 1.48655i | 0 | −17.5246 | − | 7.64527i | ||||||
19.12 | −0.747520 | − | 1.85505i | 0 | −2.88243 | + | 2.77337i | −0.463207 | − | 2.62698i | 0 | −4.34885 | + | 5.18275i | 7.29942 | + | 3.27390i | 0 | −4.52692 | + | 2.82299i | ||||||
19.13 | −0.518880 | + | 1.93152i | 0 | −3.46153 | − | 2.00445i | −1.09499 | − | 6.21002i | 0 | −5.65814 | + | 6.74311i | 5.66776 | − | 5.64593i | 0 | 12.5629 | + | 1.10726i | ||||||
19.14 | −0.376263 | − | 1.96429i | 0 | −3.71685 | + | 1.47818i | 1.26933 | + | 7.19876i | 0 | 2.67622 | − | 3.18940i | 4.30208 | + | 6.74478i | 0 | 13.6628 | − | 5.20197i | ||||||
19.15 | −0.343802 | − | 1.97023i | 0 | −3.76360 | + | 1.35474i | −0.608002 | − | 3.44815i | 0 | −5.71739 | + | 6.81372i | 3.96308 | + | 6.94939i | 0 | −6.58462 | + | 2.38339i | ||||||
19.16 | −0.277531 | + | 1.98065i | 0 | −3.84595 | − | 1.09938i | 0.542224 | + | 3.07510i | 0 | −2.98812 | + | 3.56110i | 3.24486 | − | 7.31238i | 0 | −6.24119 | + | 0.220520i | ||||||
19.17 | −0.210380 | + | 1.98890i | 0 | −3.91148 | − | 0.836851i | 0.133306 | + | 0.756014i | 0 | 3.08121 | − | 3.67204i | 2.48731 | − | 7.60350i | 0 | −1.53168 | + | 0.106082i | ||||||
19.18 | −0.103865 | − | 1.99730i | 0 | −3.97842 | + | 0.414901i | −1.50670 | − | 8.54492i | 0 | 7.53806 | − | 8.98351i | 1.24190 | + | 7.90302i | 0 | −16.9103 | + | 3.89685i | ||||||
19.19 | 0.443343 | − | 1.95024i | 0 | −3.60689 | − | 1.72925i | 0.890579 | + | 5.05073i | 0 | −0.0265129 | + | 0.0315969i | −4.97155 | + | 6.26767i | 0 | 10.2450 | + | 0.502357i | ||||||
19.20 | 0.619770 | + | 1.90155i | 0 | −3.23177 | + | 2.35705i | −0.463207 | − | 2.62698i | 0 | 4.34885 | − | 5.18275i | −6.48499 | − | 4.68454i | 0 | 4.70825 | − | 2.50894i | ||||||
See next 80 embeddings (of 204 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
27.e | even | 9 | 1 | inner |
108.j | odd | 18 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 324.3.j.a | 204 | |
3.b | odd | 2 | 1 | 108.3.j.a | ✓ | 204 | |
4.b | odd | 2 | 1 | inner | 324.3.j.a | 204 | |
12.b | even | 2 | 1 | 108.3.j.a | ✓ | 204 | |
27.e | even | 9 | 1 | inner | 324.3.j.a | 204 | |
27.f | odd | 18 | 1 | 108.3.j.a | ✓ | 204 | |
108.j | odd | 18 | 1 | inner | 324.3.j.a | 204 | |
108.l | even | 18 | 1 | 108.3.j.a | ✓ | 204 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
108.3.j.a | ✓ | 204 | 3.b | odd | 2 | 1 | |
108.3.j.a | ✓ | 204 | 12.b | even | 2 | 1 | |
108.3.j.a | ✓ | 204 | 27.f | odd | 18 | 1 | |
108.3.j.a | ✓ | 204 | 108.l | even | 18 | 1 | |
324.3.j.a | 204 | 1.a | even | 1 | 1 | trivial | |
324.3.j.a | 204 | 4.b | odd | 2 | 1 | inner | |
324.3.j.a | 204 | 27.e | even | 9 | 1 | inner | |
324.3.j.a | 204 | 108.j | odd | 18 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(324, [\chi])\).