Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [135,2,Mod(4,135)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(135, base_ring=CyclotomicField(18))
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("135.4");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 135 = 3^{3} \cdot 5 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 135.p (of order \(18\), degree \(6\), 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: | \(96\) |
Relative dimension: | \(16\) over \(\Q(\zeta_{18})\) |
Twist minimal: | yes |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
4.1 | −0.911584 | + | 2.50456i | −1.32331 | − | 1.11752i | −3.90973 | − | 3.28066i | 1.78103 | − | 1.35201i | 4.00520 | − | 2.29560i | −1.98995 | − | 2.37153i | 7.16422 | − | 4.13626i | 0.502310 | + | 2.95765i | 1.76262 | + | 5.69317i |
4.2 | −0.846914 | + | 2.32688i | 0.666374 | + | 1.59873i | −3.16501 | − | 2.65576i | −2.21912 | − | 0.274758i | −4.28442 | + | 0.196582i | −1.41496 | − | 1.68629i | 4.57119 | − | 2.63918i | −2.11189 | + | 2.13071i | 2.51874 | − | 4.93093i |
4.3 | −0.691324 | + | 1.89940i | 1.47769 | − | 0.903565i | −1.59769 | − | 1.34062i | 0.122356 | − | 2.23272i | 0.694666 | + | 3.43138i | 2.82126 | + | 3.36225i | 0.149906 | − | 0.0865484i | 1.36714 | − | 2.67038i | 4.15623 | + | 1.77593i |
4.4 | −0.613751 | + | 1.68627i | −1.05122 | + | 1.37657i | −0.934715 | − | 0.784319i | 1.43461 | + | 1.71520i | −1.67607 | − | 2.61750i | 0.337711 | + | 0.402468i | −1.21189 | + | 0.699685i | −0.789881 | − | 2.89415i | −3.77277 | + | 1.36642i |
4.5 | −0.557159 | + | 1.53078i | −0.844941 | − | 1.51198i | −0.500777 | − | 0.420202i | −1.06369 | + | 1.96687i | 2.78527 | − | 0.451009i | 1.77805 | + | 2.11900i | −1.89930 | + | 1.09656i | −1.57215 | + | 2.55506i | −2.41819 | − | 2.72414i |
4.6 | −0.289634 | + | 0.795763i | 1.07411 | − | 1.35878i | 0.982738 | + | 0.824615i | 2.08094 | + | 0.818343i | 0.770166 | + | 1.24829i | −3.00798 | − | 3.58478i | −2.40759 | + | 1.39002i | −0.692560 | − | 2.91897i | −1.25392 | + | 1.41892i |
4.7 | −0.228876 | + | 0.628833i | 1.71312 | + | 0.255373i | 1.18904 | + | 0.997725i | −2.05600 | + | 0.879131i | −0.552680 | + | 1.01882i | −0.158590 | − | 0.189000i | −2.05862 | + | 1.18854i | 2.86957 | + | 0.874971i | −0.0822570 | − | 1.49409i |
4.8 | −0.141924 | + | 0.389932i | 0.332960 | + | 1.69975i | 1.40018 | + | 1.17489i | 1.17994 | − | 1.89940i | −0.710041 | − | 0.111402i | −0.408725 | − | 0.487099i | −1.37557 | + | 0.794189i | −2.77828 | + | 1.13189i | 0.573177 | + | 0.729668i |
4.9 | 0.141924 | − | 0.389932i | −0.332960 | − | 1.69975i | 1.40018 | + | 1.17489i | −0.459147 | − | 2.18842i | −0.710041 | − | 0.111402i | 0.408725 | + | 0.487099i | 1.37557 | − | 0.794189i | −2.77828 | + | 1.13189i | −0.918499 | − | 0.131552i |
4.10 | 0.228876 | − | 0.628833i | −1.71312 | − | 0.255373i | 1.18904 | + | 0.997725i | 1.63133 | + | 1.52931i | −0.552680 | + | 1.01882i | 0.158590 | + | 0.189000i | 2.05862 | − | 1.18854i | 2.86957 | + | 0.874971i | 1.33505 | − | 0.675809i |
4.11 | 0.289634 | − | 0.795763i | −1.07411 | + | 1.35878i | 0.982738 | + | 0.824615i | −2.23533 | + | 0.0572668i | 0.770166 | + | 1.24829i | 3.00798 | + | 3.58478i | 2.40759 | − | 1.39002i | −0.692560 | − | 2.91897i | −0.601858 | + | 1.79538i |
4.12 | 0.557159 | − | 1.53078i | 0.844941 | + | 1.51198i | −0.500777 | − | 0.420202i | 0.326837 | + | 2.21205i | 2.78527 | − | 0.451009i | −1.77805 | − | 2.11900i | 1.89930 | − | 1.09656i | −1.57215 | + | 2.55506i | 3.56827 | + | 0.732149i |
4.13 | 0.613751 | − | 1.68627i | 1.05122 | − | 1.37657i | −0.934715 | − | 0.784319i | −1.93472 | + | 1.12109i | −1.67607 | − | 2.61750i | −0.337711 | − | 0.402468i | 1.21189 | − | 0.699685i | −0.789881 | − | 2.89415i | 0.703028 | + | 3.95053i |
4.14 | 0.691324 | − | 1.89940i | −1.47769 | + | 0.903565i | −1.59769 | − | 1.34062i | 0.648657 | − | 2.13992i | 0.694666 | + | 3.43138i | −2.82126 | − | 3.36225i | −0.149906 | + | 0.0865484i | 1.36714 | − | 2.67038i | −3.61612 | − | 2.71143i |
4.15 | 0.846914 | − | 2.32688i | −0.666374 | − | 1.59873i | −3.16501 | − | 2.65576i | 2.17927 | + | 0.500797i | −4.28442 | + | 0.196582i | 1.41496 | + | 1.68629i | −4.57119 | + | 2.63918i | −2.11189 | + | 2.13071i | 3.01095 | − | 4.64676i |
4.16 | 0.911584 | − | 2.50456i | 1.32331 | + | 1.11752i | −3.90973 | − | 3.28066i | −1.21121 | − | 1.87962i | 4.00520 | − | 2.29560i | 1.98995 | + | 2.37153i | −7.16422 | + | 4.13626i | 0.502310 | + | 2.95765i | −5.81174 | + | 1.32011i |
34.1 | −0.911584 | − | 2.50456i | −1.32331 | + | 1.11752i | −3.90973 | + | 3.28066i | 1.78103 | + | 1.35201i | 4.00520 | + | 2.29560i | −1.98995 | + | 2.37153i | 7.16422 | + | 4.13626i | 0.502310 | − | 2.95765i | 1.76262 | − | 5.69317i |
34.2 | −0.846914 | − | 2.32688i | 0.666374 | − | 1.59873i | −3.16501 | + | 2.65576i | −2.21912 | + | 0.274758i | −4.28442 | − | 0.196582i | −1.41496 | + | 1.68629i | 4.57119 | + | 2.63918i | −2.11189 | − | 2.13071i | 2.51874 | + | 4.93093i |
34.3 | −0.691324 | − | 1.89940i | 1.47769 | + | 0.903565i | −1.59769 | + | 1.34062i | 0.122356 | + | 2.23272i | 0.694666 | − | 3.43138i | 2.82126 | − | 3.36225i | 0.149906 | + | 0.0865484i | 1.36714 | + | 2.67038i | 4.15623 | − | 1.77593i |
34.4 | −0.613751 | − | 1.68627i | −1.05122 | − | 1.37657i | −0.934715 | + | 0.784319i | 1.43461 | − | 1.71520i | −1.67607 | + | 2.61750i | 0.337711 | − | 0.402468i | −1.21189 | − | 0.699685i | −0.789881 | + | 2.89415i | −3.77277 | − | 1.36642i |
See all 96 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
27.e | even | 9 | 1 | inner |
135.p | even | 18 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 135.2.p.a | ✓ | 96 |
3.b | odd | 2 | 1 | 405.2.p.a | 96 | ||
5.b | even | 2 | 1 | inner | 135.2.p.a | ✓ | 96 |
5.c | odd | 4 | 2 | 675.2.l.h | 96 | ||
15.d | odd | 2 | 1 | 405.2.p.a | 96 | ||
27.e | even | 9 | 1 | inner | 135.2.p.a | ✓ | 96 |
27.f | odd | 18 | 1 | 405.2.p.a | 96 | ||
135.n | odd | 18 | 1 | 405.2.p.a | 96 | ||
135.p | even | 18 | 1 | inner | 135.2.p.a | ✓ | 96 |
135.r | odd | 36 | 2 | 675.2.l.h | 96 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
135.2.p.a | ✓ | 96 | 1.a | even | 1 | 1 | trivial |
135.2.p.a | ✓ | 96 | 5.b | even | 2 | 1 | inner |
135.2.p.a | ✓ | 96 | 27.e | even | 9 | 1 | inner |
135.2.p.a | ✓ | 96 | 135.p | even | 18 | 1 | inner |
405.2.p.a | 96 | 3.b | odd | 2 | 1 | ||
405.2.p.a | 96 | 15.d | odd | 2 | 1 | ||
405.2.p.a | 96 | 27.f | odd | 18 | 1 | ||
405.2.p.a | 96 | 135.n | odd | 18 | 1 | ||
675.2.l.h | 96 | 5.c | odd | 4 | 2 | ||
675.2.l.h | 96 | 135.r | odd | 36 | 2 |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(135, [\chi])\).