Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [675,2,Mod(32,675)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(675, base_ring=CyclotomicField(36))
chi = DirichletCharacter(H, H._module([10, 9]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("675.32");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 675 = 3^{3} \cdot 5^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 675.ba (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: | \(5.38990213644\) |
Analytic rank: | \(0\) |
Dimension: | \(192\) |
Relative dimension: | \(16\) over \(\Q(\zeta_{36})\) |
Twist minimal: | no (minimal twist has level 135) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
32.1 | −0.237511 | + | 2.71476i | −0.914466 | + | 1.47097i | −5.34392 | − | 0.942278i | 0 | −3.77614 | − | 2.83193i | −0.907744 | + | 1.29639i | 2.41667 | − | 9.01913i | −1.32750 | − | 2.69030i | 0 | ||||
32.2 | −0.214747 | + | 2.45457i | 1.63667 | − | 0.566833i | −4.00918 | − | 0.706926i | 0 | 1.03986 | + | 4.13905i | 0.719449 | − | 1.02748i | 1.32073 | − | 4.92901i | 2.35740 | − | 1.85544i | 0 | ||||
32.3 | −0.191628 | + | 2.19031i | −0.699511 | − | 1.58451i | −2.79113 | − | 0.492152i | 0 | 3.60463 | − | 1.22851i | 0.000371431 | 0 | 0.000530458i | 0.474704 | − | 1.77162i | −2.02137 | + | 2.21677i | 0 | ||||
32.4 | −0.153697 | + | 1.75676i | −1.70488 | + | 0.305576i | −1.09297 | − | 0.192720i | 0 | −0.274789 | − | 3.04204i | 1.93217 | − | 2.75943i | −0.406292 | + | 1.51630i | 2.81325 | − | 1.04194i | 0 | ||||
32.5 | −0.119567 | + | 1.36666i | 1.50279 | − | 0.861169i | 0.116159 | + | 0.0204819i | 0 | 0.997239 | + | 2.15677i | −1.87969 | + | 2.68447i | −0.752017 | + | 2.80657i | 1.51678 | − | 2.58832i | 0 | ||||
32.6 | −0.0904240 | + | 1.03355i | 0.538569 | + | 1.64619i | 0.909563 | + | 0.160381i | 0 | −1.75012 | + | 0.407783i | −1.66856 | + | 2.38295i | −0.785057 | + | 2.92987i | −2.41989 | + | 1.77317i | 0 | ||||
32.7 | −0.0559313 | + | 0.639298i | −0.432570 | + | 1.67717i | 1.56404 | + | 0.275783i | 0 | −1.04801 | − | 0.370347i | 2.47086 | − | 3.52875i | −0.595975 | + | 2.22421i | −2.62577 | − | 1.45098i | 0 | ||||
32.8 | −0.0504771 | + | 0.576956i | 0.0483220 | − | 1.73138i | 1.63929 | + | 0.289050i | 0 | 0.996489 | + | 0.115275i | 1.94671 | − | 2.78018i | −0.549311 | + | 2.05006i | −2.99533 | − | 0.167327i | 0 | ||||
32.9 | −0.00133058 | + | 0.0152086i | 1.71244 | + | 0.259886i | 1.96939 | + | 0.347256i | 0 | −0.00623106 | + | 0.0256981i | 0.148030 | − | 0.211408i | −0.0158044 | + | 0.0589827i | 2.86492 | + | 0.890079i | 0 | ||||
32.10 | 0.0140240 | − | 0.160295i | −0.850222 | − | 1.50901i | 1.94412 | + | 0.342800i | 0 | −0.253811 | + | 0.115124i | −1.66092 | + | 2.37204i | 0.165506 | − | 0.617675i | −1.55424 | + | 2.56599i | 0 | ||||
32.11 | 0.0909691 | − | 1.03978i | −1.49497 | + | 0.874685i | 0.896746 | + | 0.158121i | 0 | 0.773485 | + | 1.63401i | −0.397456 | + | 0.567627i | 0.786273 | − | 2.93441i | 1.46985 | − | 2.61525i | 0 | ||||
32.12 | 0.118646 | − | 1.35613i | 1.53542 | + | 0.801546i | 0.144600 | + | 0.0254969i | 0 | 1.26917 | − | 1.98714i | 0.346787 | − | 0.495263i | 0.756400 | − | 2.82292i | 1.71505 | + | 2.46142i | 0 | ||||
32.13 | 0.162710 | − | 1.85978i | −0.207283 | + | 1.71960i | −1.46269 | − | 0.257912i | 0 | 3.16436 | + | 0.665296i | −1.79546 | + | 2.56418i | 0.248717 | − | 0.928224i | −2.91407 | − | 0.712888i | 0 | ||||
32.14 | 0.174019 | − | 1.98905i | 1.19471 | − | 1.25406i | −1.95641 | − | 0.344968i | 0 | −2.28648 | − | 2.59457i | 0.808730 | − | 1.15499i | 0.00692938 | − | 0.0258608i | −0.145327 | − | 2.99648i | 0 | ||||
32.15 | 0.182563 | − | 2.08671i | −1.65611 | − | 0.507262i | −2.35140 | − | 0.414615i | 0 | −1.36085 | + | 3.36320i | 2.36759 | − | 3.38127i | −0.210175 | + | 0.784385i | 2.48537 | + | 1.68016i | 0 | ||||
32.16 | 0.213925 | − | 2.44518i | −0.617760 | − | 1.61814i | −3.96351 | − | 0.698873i | 0 | −4.08879 | + | 1.16437i | −2.71258 | + | 3.87397i | −1.28621 | + | 4.80021i | −2.23674 | + | 1.99924i | 0 | ||||
68.1 | −1.43892 | + | 2.05499i | −1.64201 | − | 0.551185i | −1.46845 | − | 4.03452i | 0 | 3.49539 | − | 2.58120i | 2.44940 | − | 1.14217i | 5.55747 | + | 1.48912i | 2.39239 | + | 1.81010i | 0 | ||||
68.2 | −1.36466 | + | 1.94894i | 1.63096 | + | 0.583078i | −1.25203 | − | 3.43992i | 0 | −3.36209 | + | 2.38294i | 2.19758 | − | 1.02475i | 3.81650 | + | 1.02263i | 2.32004 | + | 1.90195i | 0 | ||||
68.3 | −1.15565 | + | 1.65044i | 0.352907 | + | 1.69572i | −0.704386 | − | 1.93528i | 0 | −3.20652 | − | 1.37721i | −1.32708 | + | 0.618828i | 0.115769 | + | 0.0310202i | −2.75091 | + | 1.19686i | 0 | ||||
68.4 | −0.866263 | + | 1.23715i | 0.938839 | − | 1.45553i | −0.0960923 | − | 0.264011i | 0 | 0.987435 | + | 2.42236i | −1.45360 | + | 0.677826i | −2.50778 | − | 0.671958i | −1.23716 | − | 2.73303i | 0 | ||||
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 | 675.2.ba.b | 192 | |
5.b | even | 2 | 1 | 135.2.q.a | ✓ | 192 | |
5.c | odd | 4 | 1 | 135.2.q.a | ✓ | 192 | |
5.c | odd | 4 | 1 | inner | 675.2.ba.b | 192 | |
15.d | odd | 2 | 1 | 405.2.r.a | 192 | ||
15.e | even | 4 | 1 | 405.2.r.a | 192 | ||
27.f | odd | 18 | 1 | inner | 675.2.ba.b | 192 | |
135.n | odd | 18 | 1 | 135.2.q.a | ✓ | 192 | |
135.p | even | 18 | 1 | 405.2.r.a | 192 | ||
135.q | even | 36 | 1 | 135.2.q.a | ✓ | 192 | |
135.q | even | 36 | 1 | inner | 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 | 5.b | even | 2 | 1 | |
135.2.q.a | ✓ | 192 | 5.c | odd | 4 | 1 | |
135.2.q.a | ✓ | 192 | 135.n | odd | 18 | 1 | |
135.2.q.a | ✓ | 192 | 135.q | even | 36 | 1 | |
405.2.r.a | 192 | 15.d | odd | 2 | 1 | ||
405.2.r.a | 192 | 15.e | even | 4 | 1 | ||
405.2.r.a | 192 | 135.p | even | 18 | 1 | ||
405.2.r.a | 192 | 135.r | odd | 36 | 1 | ||
675.2.ba.b | 192 | 1.a | even | 1 | 1 | trivial | |
675.2.ba.b | 192 | 5.c | odd | 4 | 1 | inner | |
675.2.ba.b | 192 | 27.f | odd | 18 | 1 | inner | |
675.2.ba.b | 192 | 135.q | even | 36 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{192} - 12 T_{2}^{191} + 72 T_{2}^{190} - 294 T_{2}^{189} + 942 T_{2}^{188} - 2544 T_{2}^{187} + \cdots + 130321 \) acting on \(S_{2}^{\mathrm{new}}(675, [\chi])\).