Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [135,2,Mod(16,135)]
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([4, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("135.16");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 135 = 3^{3} \cdot 5 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 135.k (of order \(9\), 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: | \(42\) |
Relative dimension: | \(7\) over \(\Q(\zeta_{9})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{9}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
16.1 | −2.09652 | − | 1.75919i | −1.01713 | + | 1.40194i | 0.953350 | + | 5.40672i | −0.939693 | − | 0.342020i | 4.59871 | − | 1.14987i | 0.438601 | − | 2.48743i | 4.77590 | − | 8.27211i | −0.930888 | − | 2.85192i | 1.36840 | + | 2.37015i |
16.2 | −1.13174 | − | 0.949646i | 1.73105 | + | 0.0588713i | 0.0317206 | + | 0.179896i | −0.939693 | − | 0.342020i | −1.90320 | − | 1.71051i | 0.642118 | − | 3.64163i | −1.34245 | + | 2.32519i | 2.99307 | + | 0.203818i | 0.738693 | + | 1.27945i |
16.3 | −0.844223 | − | 0.708387i | −1.70750 | + | 0.290600i | −0.136396 | − | 0.773542i | −0.939693 | − | 0.342020i | 1.64737 | + | 0.964239i | −0.518523 | + | 2.94069i | −1.53487 | + | 2.65848i | 2.83110 | − | 0.992398i | 0.551027 | + | 0.954407i |
16.4 | 0.601643 | + | 0.504838i | −0.953823 | − | 1.44576i | −0.240184 | − | 1.36215i | −0.939693 | − | 0.342020i | 0.156014 | − | 1.35136i | 0.247142 | − | 1.40161i | 1.32855 | − | 2.30112i | −1.18044 | + | 2.75800i | −0.392694 | − | 0.680167i |
16.5 | 0.786397 | + | 0.659865i | 1.53844 | − | 0.795743i | −0.164299 | − | 0.931783i | −0.939693 | − | 0.342020i | 1.73491 | + | 0.389393i | −0.712481 | + | 4.04068i | 1.51222 | − | 2.61923i | 1.73359 | − | 2.44840i | −0.513284 | − | 0.889034i |
16.6 | 1.42971 | + | 1.19967i | 0.697646 | + | 1.58534i | 0.257569 | + | 1.46075i | −0.939693 | − | 0.342020i | −0.904448 | + | 3.10352i | 0.603654 | − | 3.42349i | 0.482190 | − | 0.835177i | −2.02658 | + | 2.21201i | −0.933178 | − | 1.61631i |
16.7 | 2.02078 | + | 1.69563i | −1.72837 | + | 0.112798i | 0.861072 | + | 4.88338i | −0.939693 | − | 0.342020i | −3.68392 | − | 2.70275i | −0.00591834 | + | 0.0335646i | −3.90246 | + | 6.75926i | 2.97455 | − | 0.389915i | −1.31897 | − | 2.28452i |
31.1 | −2.42143 | − | 0.881327i | 0.345069 | + | 1.69733i | 3.55448 | + | 2.98256i | 0.173648 | − | 0.984808i | 0.660344 | − | 4.41408i | −2.32834 | + | 1.95371i | −3.40147 | − | 5.89151i | −2.76186 | + | 1.17139i | −1.28841 | + | 2.23160i |
31.2 | −2.20793 | − | 0.803619i | 0.127781 | − | 1.72733i | 2.69705 | + | 2.26309i | 0.173648 | − | 0.984808i | −1.67025 | + | 3.71113i | 3.56068 | − | 2.98776i | −1.78659 | − | 3.09446i | −2.96734 | − | 0.441440i | −1.17481 | + | 2.03484i |
31.3 | −0.867363 | − | 0.315694i | −1.06092 | + | 1.36910i | −0.879434 | − | 0.737932i | 0.173648 | − | 0.984808i | 1.35242 | − | 0.852584i | 2.85743 | − | 2.39767i | 1.45286 | + | 2.51642i | −0.748897 | − | 2.90502i | −0.461514 | + | 0.799366i |
31.4 | 0.157838 | + | 0.0574483i | −0.188780 | − | 1.72173i | −1.51048 | − | 1.26744i | 0.173648 | − | 0.984808i | 0.0691141 | − | 0.282600i | −1.41056 | + | 1.18360i | −0.333566 | − | 0.577753i | −2.92872 | + | 0.650056i | 0.0839839 | − | 0.145464i |
31.5 | 0.281020 | + | 0.102283i | 1.72885 | − | 0.105208i | −1.46358 | − | 1.22809i | 0.173648 | − | 0.984808i | 0.496603 | + | 0.147266i | 1.00072 | − | 0.839702i | −0.584737 | − | 1.01279i | 2.97786 | − | 0.363779i | 0.149528 | − | 0.258989i |
31.6 | 1.71599 | + | 0.624569i | 0.0386016 | + | 1.73162i | 1.02244 | + | 0.857933i | 0.173648 | − | 0.984808i | −1.01528 | + | 2.99555i | 0.570617 | − | 0.478805i | −0.607452 | − | 1.05214i | −2.99702 | + | 0.133687i | 0.913059 | − | 1.58146i |
31.7 | 2.40218 | + | 0.874320i | −1.31696 | − | 1.12500i | 3.47392 | + | 2.91497i | 0.173648 | − | 0.984808i | −2.17995 | − | 3.85389i | −1.18637 | + | 0.995479i | 3.24001 | + | 5.61186i | 0.468745 | + | 2.96315i | 1.27817 | − | 2.21386i |
61.1 | −2.42143 | + | 0.881327i | 0.345069 | − | 1.69733i | 3.55448 | − | 2.98256i | 0.173648 | + | 0.984808i | 0.660344 | + | 4.41408i | −2.32834 | − | 1.95371i | −3.40147 | + | 5.89151i | −2.76186 | − | 1.17139i | −1.28841 | − | 2.23160i |
61.2 | −2.20793 | + | 0.803619i | 0.127781 | + | 1.72733i | 2.69705 | − | 2.26309i | 0.173648 | + | 0.984808i | −1.67025 | − | 3.71113i | 3.56068 | + | 2.98776i | −1.78659 | + | 3.09446i | −2.96734 | + | 0.441440i | −1.17481 | − | 2.03484i |
61.3 | −0.867363 | + | 0.315694i | −1.06092 | − | 1.36910i | −0.879434 | + | 0.737932i | 0.173648 | + | 0.984808i | 1.35242 | + | 0.852584i | 2.85743 | + | 2.39767i | 1.45286 | − | 2.51642i | −0.748897 | + | 2.90502i | −0.461514 | − | 0.799366i |
61.4 | 0.157838 | − | 0.0574483i | −0.188780 | + | 1.72173i | −1.51048 | + | 1.26744i | 0.173648 | + | 0.984808i | 0.0691141 | + | 0.282600i | −1.41056 | − | 1.18360i | −0.333566 | + | 0.577753i | −2.92872 | − | 0.650056i | 0.0839839 | + | 0.145464i |
61.5 | 0.281020 | − | 0.102283i | 1.72885 | + | 0.105208i | −1.46358 | + | 1.22809i | 0.173648 | + | 0.984808i | 0.496603 | − | 0.147266i | 1.00072 | + | 0.839702i | −0.584737 | + | 1.01279i | 2.97786 | + | 0.363779i | 0.149528 | + | 0.258989i |
61.6 | 1.71599 | − | 0.624569i | 0.0386016 | − | 1.73162i | 1.02244 | − | 0.857933i | 0.173648 | + | 0.984808i | −1.01528 | − | 2.99555i | 0.570617 | + | 0.478805i | −0.607452 | + | 1.05214i | −2.99702 | − | 0.133687i | 0.913059 | + | 1.58146i |
See all 42 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
27.e | even | 9 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 135.2.k.b | ✓ | 42 |
3.b | odd | 2 | 1 | 405.2.k.b | 42 | ||
5.b | even | 2 | 1 | 675.2.l.e | 42 | ||
5.c | odd | 4 | 2 | 675.2.u.d | 84 | ||
27.e | even | 9 | 1 | inner | 135.2.k.b | ✓ | 42 |
27.e | even | 9 | 1 | 3645.2.a.l | 21 | ||
27.f | odd | 18 | 1 | 405.2.k.b | 42 | ||
27.f | odd | 18 | 1 | 3645.2.a.k | 21 | ||
135.p | even | 18 | 1 | 675.2.l.e | 42 | ||
135.r | odd | 36 | 2 | 675.2.u.d | 84 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
135.2.k.b | ✓ | 42 | 1.a | even | 1 | 1 | trivial |
135.2.k.b | ✓ | 42 | 27.e | even | 9 | 1 | inner |
405.2.k.b | 42 | 3.b | odd | 2 | 1 | ||
405.2.k.b | 42 | 27.f | odd | 18 | 1 | ||
675.2.l.e | 42 | 5.b | even | 2 | 1 | ||
675.2.l.e | 42 | 135.p | even | 18 | 1 | ||
675.2.u.d | 84 | 5.c | odd | 4 | 2 | ||
675.2.u.d | 84 | 135.r | odd | 36 | 2 | ||
3645.2.a.k | 21 | 27.f | odd | 18 | 1 | ||
3645.2.a.l | 21 | 27.e | even | 9 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{42} + 7 T_{2}^{39} + 3 T_{2}^{38} - 36 T_{2}^{37} + 402 T_{2}^{36} + 153 T_{2}^{35} + \cdots + 29241 \) acting on \(S_{2}^{\mathrm{new}}(135, [\chi])\).