Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [287,3,Mod(5,287)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(287, base_ring=CyclotomicField(60))
chi = DirichletCharacter(H, H._module([50, 33]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("287.5");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 287 = 7 \cdot 41 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 287.bd (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: | \(7.82018358714\) |
Analytic rank: | \(0\) |
Dimension: | \(864\) |
Relative dimension: | \(54\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
5.1 | −3.86332 | + | 0.406051i | 0.385359 | + | 0.103257i | 10.8478 | − | 2.30576i | 5.36436 | + | 5.95772i | −1.53069 | − | 0.242438i | 0.114923 | − | 6.99906i | −26.1942 | + | 8.51102i | −7.65639 | − | 4.42042i | −23.1434 | − | 20.8384i |
5.2 | −3.78322 | + | 0.397633i | −2.36963 | − | 0.634941i | 10.2421 | − | 2.17702i | −4.89261 | − | 5.43380i | 9.21731 | + | 1.45988i | −4.50728 | + | 5.35578i | −23.4109 | + | 7.60666i | −2.58223 | − | 1.49085i | 20.6705 | + | 18.6118i |
5.3 | −3.67223 | + | 0.385966i | 3.60494 | + | 0.965940i | 9.42368 | − | 2.00307i | −3.30438 | − | 3.66989i | −13.6110 | − | 2.15577i | 6.98845 | − | 0.401949i | −19.7858 | + | 6.42881i | 4.26831 | + | 2.46431i | 13.5509 | + | 12.2013i |
5.4 | −3.52452 | + | 0.370441i | −5.59949 | − | 1.50038i | 8.37239 | − | 1.77961i | 5.79187 | + | 6.43253i | 20.2913 | + | 3.21382i | −3.57911 | + | 6.01581i | −15.3675 | + | 4.99319i | 21.3089 | + | 12.3027i | −22.7964 | − | 20.5260i |
5.5 | −3.51461 | + | 0.369401i | 3.58086 | + | 0.959489i | 8.30344 | − | 1.76495i | −1.51949 | − | 1.68757i | −12.9398 | − | 2.04946i | −6.56906 | − | 2.41814i | −15.0874 | + | 4.90219i | 4.10772 | + | 2.37160i | 5.96381 | + | 5.36983i |
5.6 | −3.25516 | + | 0.342131i | −2.98484 | − | 0.799785i | 6.56644 | − | 1.39574i | −1.83918 | − | 2.04262i | 9.98976 | + | 1.58222i | −2.67170 | − | 6.47009i | −8.44572 | + | 2.74418i | 0.475370 | + | 0.274455i | 6.68568 | + | 6.01981i |
5.7 | −3.25016 | + | 0.341606i | −3.96280 | − | 1.06183i | 6.53427 | − | 1.38890i | −1.12109 | − | 1.24510i | 13.2425 | + | 2.09740i | 6.77770 | − | 1.75009i | −8.33052 | + | 2.70675i | 6.78206 | + | 3.91562i | 4.06906 | + | 3.66380i |
5.8 | −3.15003 | + | 0.331082i | 5.01583 | + | 1.34399i | 5.90050 | − | 1.25419i | 3.23952 | + | 3.59786i | −16.2450 | − | 2.57296i | 4.02106 | + | 5.72984i | −6.12207 | + | 1.98918i | 15.5580 | + | 8.98244i | −11.3958 | − | 10.2608i |
5.9 | −3.11963 | + | 0.327886i | 1.13607 | + | 0.304409i | 5.71200 | − | 1.21412i | 1.95582 | + | 2.17215i | −3.64393 | − | 0.577142i | −4.33634 | + | 5.49510i | −5.48808 | + | 1.78318i | −6.59624 | − | 3.80834i | −6.81364 | − | 6.13503i |
5.10 | −2.89312 | + | 0.304080i | −0.510002 | − | 0.136655i | 4.36511 | − | 0.927833i | 3.45934 | + | 3.84199i | 1.51705 | + | 0.240278i | 6.80668 | + | 1.63372i | −1.27995 | + | 0.415880i | −7.55280 | − | 4.36061i | −11.1766 | − | 10.0634i |
5.11 | −2.46927 | + | 0.259531i | 0.688721 | + | 0.184542i | 2.11735 | − | 0.450057i | −5.13570 | − | 5.70377i | −1.74853 | − | 0.276940i | 2.59590 | + | 6.50087i | 4.33389 | − | 1.40817i | −7.35395 | − | 4.24580i | 14.1617 | + | 12.7513i |
5.12 | −2.38273 | + | 0.250435i | 1.40604 | + | 0.376749i | 1.70209 | − | 0.361791i | −4.10188 | − | 4.55560i | −3.44457 | − | 0.545567i | 3.31786 | − | 6.16375i | 5.14935 | − | 1.67313i | −5.95921 | − | 3.44055i | 10.9146 | + | 9.82752i |
5.13 | −2.24269 | + | 0.235717i | 4.18226 | + | 1.12063i | 1.06153 | − | 0.225634i | 5.21001 | + | 5.78631i | −9.64368 | − | 1.52741i | −2.17446 | − | 6.65370i | 6.25122 | − | 2.03114i | 8.44125 | + | 4.87356i | −13.0484 | − | 11.7488i |
5.14 | −2.22184 | + | 0.233525i | −2.69284 | − | 0.721545i | 0.969471 | − | 0.206067i | 2.30180 | + | 2.55641i | 6.15158 | + | 0.974314i | −6.38644 | − | 2.86590i | 6.39307 | − | 2.07723i | −1.06346 | − | 0.613988i | −5.71124 | − | 5.14242i |
5.15 | −2.19003 | + | 0.230182i | 5.08333 | + | 1.36208i | 0.830673 | − | 0.176565i | −3.40707 | − | 3.78394i | −11.4462 | − | 1.81290i | −6.83271 | − | 1.52119i | 6.59871 | − | 2.14405i | 16.1908 | + | 9.34777i | 8.33260 | + | 7.50271i |
5.16 | −1.90739 | + | 0.200474i | 1.60329 | + | 0.429599i | −0.314656 | + | 0.0668823i | 1.19878 | + | 1.33139i | −3.14421 | − | 0.497994i | 4.18213 | − | 5.61335i | 7.88286 | − | 2.56130i | −5.40825 | − | 3.12246i | −2.55345 | − | 2.29914i |
5.17 | −1.84676 | + | 0.194102i | −4.62713 | − | 1.23984i | −0.539743 | + | 0.114726i | −4.15770 | − | 4.61759i | 8.78585 | + | 1.39154i | 5.25002 | + | 4.63004i | 8.03870 | − | 2.61193i | 12.0789 | + | 6.97376i | 8.57456 | + | 7.72057i |
5.18 | −1.80705 | + | 0.189928i | −2.98183 | − | 0.798979i | −0.683242 | + | 0.145228i | 0.710415 | + | 0.788996i | 5.54006 | + | 0.877459i | −2.34483 | + | 6.59559i | 8.11935 | − | 2.63814i | 0.458720 | + | 0.264842i | −1.43361 | − | 1.29082i |
5.19 | −1.32950 | + | 0.139736i | −1.23979 | − | 0.332200i | −2.16455 | + | 0.460089i | −5.09510 | − | 5.65868i | 1.69472 | + | 0.268417i | −6.54965 | − | 2.47023i | 7.89905 | − | 2.56656i | −6.36751 | − | 3.67628i | 7.56465 | + | 6.81124i |
5.20 | −1.22238 | + | 0.128477i | −2.41703 | − | 0.647641i | −2.43489 | + | 0.517552i | 6.07573 | + | 6.74778i | 3.03773 | + | 0.481129i | 6.66096 | + | 2.15212i | 7.58568 | − | 2.46474i | −2.37164 | − | 1.36927i | −8.29377 | − | 7.46774i |
See next 80 embeddings (of 864 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.d | odd | 6 | 1 | inner |
41.g | even | 20 | 1 | inner |
287.bd | odd | 60 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 287.3.bd.a | ✓ | 864 |
7.d | odd | 6 | 1 | inner | 287.3.bd.a | ✓ | 864 |
41.g | even | 20 | 1 | inner | 287.3.bd.a | ✓ | 864 |
287.bd | odd | 60 | 1 | inner | 287.3.bd.a | ✓ | 864 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
287.3.bd.a | ✓ | 864 | 1.a | even | 1 | 1 | trivial |
287.3.bd.a | ✓ | 864 | 7.d | odd | 6 | 1 | inner |
287.3.bd.a | ✓ | 864 | 41.g | even | 20 | 1 | inner |
287.3.bd.a | ✓ | 864 | 287.bd | odd | 60 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(287, [\chi])\).