Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [450,2,Mod(23,450)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(450, base_ring=CyclotomicField(60))
chi = DirichletCharacter(H, H._module([50, 33]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("450.23");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 450.w (of order \(60\), degree \(16\), 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: | \(3.59326809096\) |
Analytic rank: | \(0\) |
Dimension: | \(480\) |
Relative dimension: | \(30\) over \(\Q(\zeta_{60})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{60}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
23.1 | −0.358368 | − | 0.933580i | −1.72733 | − | 0.127856i | −0.743145 | + | 0.669131i | −2.14941 | + | 0.616468i | 0.499654 | + | 1.65842i | −1.07243 | − | 0.287356i | 0.891007 | + | 0.453990i | 2.96731 | + | 0.441697i | 1.34580 | + | 1.78573i |
23.2 | −0.358368 | − | 0.933580i | −1.72496 | − | 0.156526i | −0.743145 | + | 0.669131i | −0.578851 | − | 2.15985i | 0.472042 | + | 1.66649i | −0.339629 | − | 0.0910032i | 0.891007 | + | 0.453990i | 2.95100 | + | 0.540003i | −1.80895 | + | 1.31442i |
23.3 | −0.358368 | − | 0.933580i | −1.48144 | − | 0.897405i | −0.743145 | + | 0.669131i | 1.69030 | − | 1.46386i | −0.306899 | + | 1.70464i | 3.92816 | + | 1.05255i | 0.891007 | + | 0.453990i | 1.38933 | + | 2.65890i | −1.97238 | − | 1.05343i |
23.4 | −0.358368 | − | 0.933580i | −1.40196 | + | 1.01710i | −0.743145 | + | 0.669131i | 2.16221 | + | 0.569949i | 1.45196 | + | 0.944351i | −4.53737 | − | 1.21579i | 0.891007 | + | 0.453990i | 0.931011 | − | 2.85188i | −0.242774 | − | 2.22285i |
23.5 | −0.358368 | − | 0.933580i | −1.02833 | + | 1.39375i | −0.743145 | + | 0.669131i | 0.468469 | − | 2.18644i | 1.66970 | + | 0.460550i | −0.570319 | − | 0.152817i | 0.891007 | + | 0.453990i | −0.885087 | − | 2.86646i | −2.20911 | + | 0.346198i |
23.6 | −0.358368 | − | 0.933580i | −0.466852 | − | 1.66795i | −0.743145 | + | 0.669131i | −0.954188 | + | 2.02226i | −1.38986 | + | 1.03358i | 3.63027 | + | 0.972729i | 0.891007 | + | 0.453990i | −2.56410 | + | 1.55737i | 2.22989 | + | 0.166099i |
23.7 | −0.358368 | − | 0.933580i | −0.0491411 | + | 1.73135i | −0.743145 | + | 0.669131i | 1.96089 | + | 1.07466i | 1.63397 | − | 0.574584i | 3.32195 | + | 0.890113i | 0.891007 | + | 0.453990i | −2.99517 | − | 0.170161i | 0.300562 | − | 2.21578i |
23.8 | −0.358368 | − | 0.933580i | 0.0371786 | − | 1.73165i | −0.743145 | + | 0.669131i | 1.91306 | − | 1.15766i | −1.62996 | + | 0.585859i | −2.33480 | − | 0.625609i | 0.891007 | + | 0.453990i | −2.99724 | − | 0.128761i | −1.76635 | − | 1.37113i |
23.9 | −0.358368 | − | 0.933580i | 0.169956 | + | 1.72369i | −0.743145 | + | 0.669131i | −1.60127 | + | 1.56075i | 1.54830 | − | 0.776384i | −2.59992 | − | 0.696648i | 0.891007 | + | 0.453990i | −2.94223 | + | 0.585904i | 2.03093 | + | 0.935586i |
23.10 | −0.358368 | − | 0.933580i | 0.807668 | − | 1.53221i | −0.743145 | + | 0.669131i | −1.96952 | + | 1.05877i | −1.71989 | − | 0.204927i | −3.23303 | − | 0.866288i | 0.891007 | + | 0.453990i | −1.69535 | − | 2.47504i | 1.69426 | + | 1.45927i |
23.11 | −0.358368 | − | 0.933580i | 0.866349 | + | 1.49981i | −0.743145 | + | 0.669131i | −1.65651 | − | 1.50200i | 1.08972 | − | 1.34629i | −1.52847 | − | 0.409552i | 0.891007 | + | 0.453990i | −1.49888 | + | 2.59872i | −0.808595 | + | 2.08475i |
23.12 | −0.358368 | − | 0.933580i | 1.34709 | − | 1.08874i | −0.743145 | + | 0.669131i | 1.80264 | + | 1.32305i | −1.49918 | − | 0.867446i | 0.216796 | + | 0.0580902i | 0.891007 | + | 0.453990i | 0.629292 | − | 2.93326i | 0.589168 | − | 2.15705i |
23.13 | −0.358368 | − | 0.933580i | 1.39329 | − | 1.02895i | −0.743145 | + | 0.669131i | −1.12618 | − | 1.93176i | −1.45992 | − | 0.932005i | 3.27132 | + | 0.876546i | 0.891007 | + | 0.453990i | 0.882516 | − | 2.86726i | −1.39987 | + | 1.74366i |
23.14 | −0.358368 | − | 0.933580i | 1.56679 | + | 0.738354i | −0.743145 | + | 0.669131i | 1.74716 | − | 1.39550i | 0.127826 | − | 1.72733i | −0.469048 | − | 0.125681i | 0.891007 | + | 0.453990i | 1.90967 | + | 2.31369i | −1.92894 | − | 1.13101i |
23.15 | −0.358368 | − | 0.933580i | 1.67532 | + | 0.439663i | −0.743145 | + | 0.669131i | −0.842800 | + | 2.07116i | −0.189920 | − | 1.72161i | 2.31653 | + | 0.620713i | 0.891007 | + | 0.453990i | 2.61339 | + | 1.47315i | 2.23562 | + | 0.0445853i |
23.16 | 0.358368 | + | 0.933580i | −1.73185 | − | 0.0261705i | −0.743145 | + | 0.669131i | −1.81212 | + | 1.31005i | −0.596208 | − | 1.62620i | 4.79708 | + | 1.28537i | −0.891007 | − | 0.453990i | 2.99863 | + | 0.0906470i | −1.87244 | − | 1.22227i |
23.17 | 0.358368 | + | 0.933580i | −1.69592 | + | 0.351926i | −0.743145 | + | 0.669131i | 2.22290 | + | 0.242291i | −0.936315 | − | 1.45716i | −0.250623 | − | 0.0671543i | −0.891007 | − | 0.453990i | 2.75230 | − | 1.19368i | 0.570419 | + | 2.16209i |
23.18 | 0.358368 | + | 0.933580i | −1.63302 | − | 0.577277i | −0.743145 | + | 0.669131i | 0.309636 | + | 2.21453i | −0.0462871 | − | 1.73143i | −4.14621 | − | 1.11097i | −0.891007 | − | 0.453990i | 2.33350 | + | 1.88541i | −1.95647 | + | 1.08269i |
23.19 | 0.358368 | + | 0.933580i | −1.58003 | + | 0.709574i | −0.743145 | + | 0.669131i | −0.430645 | − | 2.19421i | −1.22868 | − | 1.22080i | 0.475783 | + | 0.127486i | −0.891007 | − | 0.453990i | 1.99301 | − | 2.24230i | 1.89414 | − | 1.18837i |
23.20 | 0.358368 | + | 0.933580i | −0.955668 | − | 1.44454i | −0.743145 | + | 0.669131i | 2.22864 | − | 0.182152i | 1.00611 | − | 1.40987i | 2.61210 | + | 0.699910i | −0.891007 | − | 0.453990i | −1.17340 | + | 2.76100i | 0.968725 | + | 2.01533i |
See next 80 embeddings (of 480 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
9.d | odd | 6 | 1 | inner |
25.f | odd | 20 | 1 | inner |
225.w | even | 60 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 450.2.w.a | ✓ | 480 |
9.d | odd | 6 | 1 | inner | 450.2.w.a | ✓ | 480 |
25.f | odd | 20 | 1 | inner | 450.2.w.a | ✓ | 480 |
225.w | even | 60 | 1 | inner | 450.2.w.a | ✓ | 480 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
450.2.w.a | ✓ | 480 | 1.a | even | 1 | 1 | trivial |
450.2.w.a | ✓ | 480 | 9.d | odd | 6 | 1 | inner |
450.2.w.a | ✓ | 480 | 25.f | odd | 20 | 1 | inner |
450.2.w.a | ✓ | 480 | 225.w | even | 60 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(450, [\chi])\).