Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [135,2,Mod(2,135)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(135, base_ring=CyclotomicField(36))
chi = DirichletCharacter(H, H._module([2, 9]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("135.2");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 135 = 3^{3} \cdot 5 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 135.q (of order \(36\), degree \(12\), 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: | \(1.07798042729\) |
Analytic rank: | \(0\) |
Dimension: | \(192\) |
Relative dimension: | \(16\) over \(\Q(\zeta_{36})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{36}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
2.1 | −1.52658 | − | 2.18018i | 1.25829 | + | 1.19025i | −1.73870 | + | 4.77703i | −1.77382 | + | 1.36146i | 0.674088 | − | 4.56031i | 2.78918 | + | 1.30062i | 7.92739 | − | 2.12414i | 0.166591 | + | 2.99537i | 5.67609 | + | 1.78885i |
2.2 | −1.48477 | − | 2.12047i | 0.685393 | − | 1.59067i | −1.60780 | + | 4.41741i | 1.13563 | − | 1.92622i | −4.39061 | + | 0.908422i | −2.68394 | − | 1.25154i | 6.75336 | − | 1.80956i | −2.06047 | − | 2.18047i | −5.77064 | + | 0.451915i |
2.3 | −1.16608 | − | 1.66533i | −0.967587 | + | 1.43658i | −0.729546 | + | 2.00441i | 1.90549 | − | 1.17008i | 3.52066 | − | 0.0638164i | 2.43427 | + | 1.13512i | 0.261271 | − | 0.0700073i | −1.12755 | − | 2.78004i | −4.17052 | − | 1.80887i |
2.4 | −0.996232 | − | 1.42277i | −1.44830 | − | 0.949962i | −0.347747 | + | 0.955427i | −1.86340 | − | 1.23602i | 0.0912678 | + | 3.00698i | 0.920112 | + | 0.429055i | −1.64960 | + | 0.442010i | 1.19514 | + | 2.75166i | 0.0978025 | + | 3.88255i |
2.5 | −0.856769 | − | 1.22359i | −0.960742 | + | 1.44117i | −0.0790861 | + | 0.217287i | −1.10572 | + | 1.94355i | 2.58654 | − | 0.0591935i | −4.36642 | − | 2.03610i | −2.55204 | + | 0.683816i | −1.15395 | − | 2.76919i | 3.32546 | − | 0.312216i |
2.6 | −0.761943 | − | 1.08817i | 1.60100 | − | 0.660916i | 0.0804885 | − | 0.221140i | 1.62191 | + | 1.53928i | −1.93906 | − | 1.23857i | 1.77434 | + | 0.827388i | −2.86825 | + | 0.768546i | 2.12638 | − | 2.11625i | 0.439192 | − | 2.93776i |
2.7 | −0.565301 | − | 0.807333i | 1.55815 | + | 0.756415i | 0.351818 | − | 0.966613i | −0.542991 | − | 2.16914i | −0.270145 | − | 1.68555i | −0.590505 | − | 0.275357i | −2.88324 | + | 0.772562i | 1.85567 | + | 2.35722i | −1.44426 | + | 1.66459i |
2.8 | −0.264318 | − | 0.377486i | −0.938886 | − | 1.45550i | 0.611409 | − | 1.67983i | 2.04357 | + | 0.907638i | −0.301267 | + | 0.739132i | −3.26325 | − | 1.52168i | −1.68596 | + | 0.451753i | −1.23699 | + | 2.73311i | −0.197534 | − | 1.01132i |
2.9 | 0.0649826 | + | 0.0928047i | 0.728671 | − | 1.57132i | 0.679650 | − | 1.86732i | −2.22708 | + | 0.200340i | 0.193177 | − | 0.0344841i | 0.739528 | + | 0.344848i | 0.436329 | − | 0.116914i | −1.93808 | − | 2.28995i | −0.163314 | − | 0.193665i |
2.10 | 0.184358 | + | 0.263290i | −1.70366 | + | 0.312334i | 0.648706 | − | 1.78231i | −0.0853346 | + | 2.23444i | −0.396317 | − | 0.390975i | 4.46037 | + | 2.07991i | 1.20979 | − | 0.324162i | 2.80490 | − | 1.06422i | −0.604038 | + | 0.389469i |
2.11 | 0.343150 | + | 0.490069i | 0.150104 | + | 1.72553i | 0.561625 | − | 1.54305i | 2.04155 | − | 0.912186i | −0.794122 | + | 0.665679i | −0.136472 | − | 0.0636379i | 2.10468 | − | 0.563947i | −2.95494 | + | 0.518020i | 1.14759 | + | 0.687482i |
2.12 | 0.606324 | + | 0.865920i | 1.47671 | + | 0.905161i | 0.301851 | − | 0.829330i | −1.02123 | + | 1.98924i | 0.111569 | + | 1.82754i | −2.61975 | − | 1.22161i | 2.94330 | − | 0.788655i | 1.36137 | + | 2.67333i | −2.34172 | + | 0.321816i |
2.13 | 0.866263 | + | 1.23715i | −0.938839 | − | 1.45553i | −0.0960923 | + | 0.264011i | 0.525423 | − | 2.17346i | 0.987435 | − | 2.42236i | 1.45360 | + | 0.677826i | 2.50778 | − | 0.671958i | −1.23716 | + | 2.73303i | 3.14405 | − | 1.23276i |
2.14 | 1.15565 | + | 1.65044i | −0.352907 | + | 1.69572i | −0.704386 | + | 1.93528i | −2.13942 | − | 0.650308i | −3.20652 | + | 1.37721i | 1.32708 | + | 0.618828i | −0.115769 | + | 0.0310202i | −2.75091 | − | 1.19686i | −1.39912 | − | 4.28251i |
2.15 | 1.36466 | + | 1.94894i | −1.63096 | + | 0.583078i | −1.25203 | + | 3.43992i | 2.13659 | + | 0.659531i | −3.36209 | − | 2.38294i | −2.19758 | − | 1.02475i | −3.81650 | + | 1.02263i | 2.32004 | − | 1.90195i | 1.63034 | + | 5.06413i |
2.16 | 1.43892 | + | 2.05499i | 1.64201 | − | 0.551185i | −1.46845 | + | 4.03452i | −0.953539 | − | 2.02256i | 3.49539 | + | 2.58120i | −2.44940 | − | 1.14217i | −5.55747 | + | 1.48912i | 2.39239 | − | 1.81010i | 2.78428 | − | 4.86981i |
23.1 | −1.15365 | − | 2.47401i | −1.57848 | − | 0.713012i | −3.50423 | + | 4.17618i | −1.08960 | − | 1.95263i | 0.0570200 | + | 4.72775i | −0.00587402 | + | 0.0671403i | 9.10104 | + | 2.43862i | 1.98323 | + | 2.25096i | −3.57380 | + | 4.94833i |
23.2 | −0.849158 | − | 1.82102i | 1.18657 | − | 1.26176i | −1.30949 | + | 1.56058i | −1.94472 | + | 1.10365i | −3.30529 | − | 1.08934i | 0.278616 | − | 3.18459i | 0.0721951 | + | 0.0193446i | −0.184096 | − | 2.99435i | 3.66116 | + | 2.60421i |
23.3 | −0.846935 | − | 1.81626i | 1.73205 | − | 0.00445475i | −1.29592 | + | 1.54442i | 1.48888 | − | 1.66830i | −1.47502 | − | 3.14207i | −0.193523 | + | 2.21198i | 0.0311508 | + | 0.00834683i | 2.99996 | − | 0.0154316i | −4.29105 | − | 1.29125i |
23.4 | −0.768168 | − | 1.64734i | −1.46825 | + | 0.918833i | −0.838077 | + | 0.998782i | 0.104070 | + | 2.23364i | 2.64149 | + | 1.71288i | −0.393694 | + | 4.49995i | −1.22229 | − | 0.327512i | 1.31149 | − | 2.69815i | 3.59963 | − | 1.88725i |
See next 80 embeddings (of 192 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.c | odd | 4 | 1 | inner |
27.f | odd | 18 | 1 | inner |
135.q | even | 36 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 135.2.q.a | ✓ | 192 |
3.b | odd | 2 | 1 | 405.2.r.a | 192 | ||
5.b | even | 2 | 1 | 675.2.ba.b | 192 | ||
5.c | odd | 4 | 1 | inner | 135.2.q.a | ✓ | 192 |
5.c | odd | 4 | 1 | 675.2.ba.b | 192 | ||
15.e | even | 4 | 1 | 405.2.r.a | 192 | ||
27.e | even | 9 | 1 | 405.2.r.a | 192 | ||
27.f | odd | 18 | 1 | inner | 135.2.q.a | ✓ | 192 |
135.n | odd | 18 | 1 | 675.2.ba.b | 192 | ||
135.q | even | 36 | 1 | inner | 135.2.q.a | ✓ | 192 |
135.q | even | 36 | 1 | 675.2.ba.b | 192 | ||
135.r | odd | 36 | 1 | 405.2.r.a | 192 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
135.2.q.a | ✓ | 192 | 1.a | even | 1 | 1 | trivial |
135.2.q.a | ✓ | 192 | 5.c | odd | 4 | 1 | inner |
135.2.q.a | ✓ | 192 | 27.f | odd | 18 | 1 | inner |
135.2.q.a | ✓ | 192 | 135.q | even | 36 | 1 | inner |
405.2.r.a | 192 | 3.b | odd | 2 | 1 | ||
405.2.r.a | 192 | 15.e | even | 4 | 1 | ||
405.2.r.a | 192 | 27.e | even | 9 | 1 | ||
405.2.r.a | 192 | 135.r | odd | 36 | 1 | ||
675.2.ba.b | 192 | 5.b | even | 2 | 1 | ||
675.2.ba.b | 192 | 5.c | odd | 4 | 1 | ||
675.2.ba.b | 192 | 135.n | odd | 18 | 1 | ||
675.2.ba.b | 192 | 135.q | even | 36 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(135, [\chi])\).