Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [567,2,Mod(64,567)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(567, base_ring=CyclotomicField(18))
chi = DirichletCharacter(H, H._module([2, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("567.64");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 567 = 3^{4} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 567.v (of order \(9\), 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: | \(4.52751779461\) |
Analytic rank: | \(0\) |
Dimension: | \(54\) |
Relative dimension: | \(9\) over \(\Q(\zeta_{9})\) |
Twist minimal: | no (minimal twist has level 189) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
64.1 | −2.16628 | − | 0.788461i | 0 | 2.53900 | + | 2.13048i | −0.276602 | + | 1.56869i | 0 | 0.766044 | − | 0.642788i | −1.51508 | − | 2.62420i | 0 | 1.83605 | − | 3.18012i | ||||||
64.2 | −1.66659 | − | 0.606588i | 0 | 0.877471 | + | 0.736286i | 0.343651 | − | 1.94894i | 0 | 0.766044 | − | 0.642788i | 0.757784 | + | 1.31252i | 0 | −1.75493 | + | 3.03962i | ||||||
64.3 | −0.754940 | − | 0.274776i | 0 | −1.03766 | − | 0.870696i | −0.713335 | + | 4.04552i | 0 | 0.766044 | − | 0.642788i | 1.34751 | + | 2.33396i | 0 | 1.65014 | − | 2.85812i | ||||||
64.4 | −0.435562 | − | 0.158531i | 0 | −1.36751 | − | 1.14747i | 0.0549550 | − | 0.311665i | 0 | 0.766044 | − | 0.642788i | 0.877238 | + | 1.51942i | 0 | −0.0733450 | + | 0.127037i | ||||||
64.5 | 0.252618 | + | 0.0919454i | 0 | −1.47673 | − | 1.23912i | −0.344603 | + | 1.95434i | 0 | 0.766044 | − | 0.642788i | −0.527947 | − | 0.914430i | 0 | −0.266746 | + | 0.462017i | ||||||
64.6 | 1.08295 | + | 0.394160i | 0 | −0.514680 | − | 0.431868i | 0.132560 | − | 0.751786i | 0 | 0.766044 | − | 0.642788i | −1.53959 | − | 2.66665i | 0 | 0.439879 | − | 0.761893i | ||||||
64.7 | 1.80206 | + | 0.655894i | 0 | 1.28512 | + | 1.07834i | 0.489098 | − | 2.77381i | 0 | 0.766044 | − | 0.642788i | −0.309134 | − | 0.535436i | 0 | 2.70071 | − | 4.67777i | ||||||
64.8 | 2.17921 | + | 0.793168i | 0 | 2.58775 | + | 2.17138i | −0.443161 | + | 2.51329i | 0 | 0.766044 | − | 0.642788i | 1.59792 | + | 2.76768i | 0 | −2.95920 | + | 5.12549i | ||||||
64.9 | 2.52562 | + | 0.919249i | 0 | 4.00163 | + | 3.35776i | −0.203200 | + | 1.15240i | 0 | 0.766044 | − | 0.642788i | 4.33224 | + | 7.50367i | 0 | −1.57255 | + | 2.72374i | ||||||
127.1 | −2.00215 | − | 1.68000i | 0 | 0.838893 | + | 4.75760i | 3.29720 | + | 1.20008i | 0 | 0.173648 | − | 0.984808i | 3.69957 | − | 6.40784i | 0 | −4.58534 | − | 7.94204i | ||||||
127.2 | −1.89179 | − | 1.58740i | 0 | 0.711733 | + | 4.03644i | −2.16637 | − | 0.788496i | 0 | 0.173648 | − | 0.984808i | 2.59144 | − | 4.48850i | 0 | 2.84667 | + | 4.93057i | ||||||
127.3 | −0.969537 | − | 0.813538i | 0 | −0.0691389 | − | 0.392106i | 0.855069 | + | 0.311220i | 0 | 0.173648 | − | 0.984808i | −1.51760 | + | 2.62856i | 0 | −0.575832 | − | 0.997370i | ||||||
127.4 | −0.786541 | − | 0.659987i | 0 | −0.164231 | − | 0.931402i | −3.56827 | − | 1.29874i | 0 | 0.173648 | − | 0.984808i | −1.51229 | + | 2.61937i | 0 | 1.94944 | + | 3.37653i | ||||||
127.5 | −0.731503 | − | 0.613803i | 0 | −0.188955 | − | 1.07162i | 3.78665 | + | 1.37823i | 0 | 0.173648 | − | 0.984808i | −1.47445 | + | 2.55382i | 0 | −1.92398 | − | 3.33244i | ||||||
127.6 | −0.0808704 | − | 0.0678583i | 0 | −0.345361 | − | 1.95864i | −0.0449304 | − | 0.0163533i | 0 | 0.173648 | − | 0.984808i | −0.210549 | + | 0.364682i | 0 | 0.00252383 | + | 0.00437140i | ||||||
127.7 | 1.09798 | + | 0.921318i | 0 | 0.00944557 | + | 0.0535685i | −1.35272 | − | 0.492351i | 0 | 0.173648 | − | 0.984808i | 1.39433 | − | 2.41506i | 0 | −1.03166 | − | 1.78688i | ||||||
127.8 | 1.17963 | + | 0.989828i | 0 | 0.0644735 | + | 0.365648i | 1.99174 | + | 0.724932i | 0 | 0.173648 | − | 0.984808i | 1.25403 | − | 2.17204i | 0 | 1.63195 | + | 2.82663i | ||||||
127.9 | 1.88664 | + | 1.58308i | 0 | 0.705975 | + | 4.00378i | 1.28676 | + | 0.468343i | 0 | 0.173648 | − | 0.984808i | −2.54355 | + | 4.40555i | 0 | 1.68623 | + | 2.92064i | ||||||
253.1 | −0.483276 | − | 2.74079i | 0 | −5.39901 | + | 1.96508i | 1.86046 | + | 1.56111i | 0 | −0.939693 | − | 0.342020i | 5.21200 | + | 9.02746i | 0 | 3.37956 | − | 5.85358i | ||||||
253.2 | −0.399059 | − | 2.26318i | 0 | −3.08333 | + | 1.12224i | −2.84760 | − | 2.38942i | 0 | −0.939693 | − | 0.342020i | 1.47217 | + | 2.54988i | 0 | −4.27132 | + | 7.39815i | ||||||
See all 54 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 | 567.2.v.b | 54 | |
3.b | odd | 2 | 1 | 189.2.v.a | ✓ | 54 | |
27.e | even | 9 | 1 | inner | 567.2.v.b | 54 | |
27.e | even | 9 | 1 | 5103.2.a.f | 27 | ||
27.f | odd | 18 | 1 | 189.2.v.a | ✓ | 54 | |
27.f | odd | 18 | 1 | 5103.2.a.i | 27 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
189.2.v.a | ✓ | 54 | 3.b | odd | 2 | 1 | |
189.2.v.a | ✓ | 54 | 27.f | odd | 18 | 1 | |
567.2.v.b | 54 | 1.a | even | 1 | 1 | trivial | |
567.2.v.b | 54 | 27.e | even | 9 | 1 | inner | |
5103.2.a.f | 27 | 27.e | even | 9 | 1 | ||
5103.2.a.i | 27 | 27.f | odd | 18 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{54} - 21 T_{2}^{51} + 27 T_{2}^{49} + 612 T_{2}^{48} - 117 T_{2}^{47} - 648 T_{2}^{46} + \cdots + 729 \) acting on \(S_{2}^{\mathrm{new}}(567, [\chi])\).