Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [287,3,Mod(73,287)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(287, base_ring=CyclotomicField(12))
chi = DirichletCharacter(H, H._module([2, 3]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("287.73");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 287 = 7 \cdot 41 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 287.q (of order \(12\), degree \(4\), 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: | \(216\) |
Relative dimension: | \(54\) over \(\Q(\zeta_{12})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{12}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
73.1 | −3.32140 | + | 1.91761i | 1.15997 | − | 0.310813i | 5.35446 | − | 9.27420i | −2.42895 | − | 4.20707i | −3.25670 | + | 3.25670i | 6.43721 | − | 2.74996i | 25.7302i | −6.54530 | + | 3.77893i | 16.1351 | + | 9.31558i | ||
73.2 | −3.29726 | + | 1.90368i | 2.58181 | − | 0.691795i | 5.24797 | − | 9.08975i | 3.40762 | + | 5.90217i | −7.19597 | + | 7.19597i | 0.200258 | + | 6.99713i | 24.7323i | −1.60705 | + | 0.927831i | −22.4717 | − | 12.9740i | ||
73.3 | −3.26748 | + | 1.88648i | −4.81897 | + | 1.29124i | 5.11761 | − | 8.86396i | −2.61661 | − | 4.53210i | 13.3100 | − | 13.3100i | 2.60313 | + | 6.49798i | 23.5252i | 13.7610 | − | 7.94489i | 17.0994 | + | 9.87236i | ||
73.4 | −3.02057 | + | 1.74393i | 4.33367 | − | 1.16120i | 4.08257 | − | 7.07121i | −3.12206 | − | 5.40756i | −11.0651 | + | 11.0651i | −6.98293 | + | 0.488488i | 14.5274i | 9.63808 | − | 5.56455i | 18.8608 | + | 10.8893i | ||
73.5 | −2.99250 | + | 1.72772i | −3.76632 | + | 1.00918i | 3.97002 | − | 6.87628i | 0.907514 | + | 1.57186i | 9.52711 | − | 9.52711i | −0.422895 | − | 6.98721i | 13.6146i | 5.37248 | − | 3.10180i | −5.43146 | − | 3.13586i | ||
73.6 | −2.86475 | + | 1.65396i | −1.20386 | + | 0.322573i | 3.47118 | − | 6.01226i | −0.352767 | − | 0.611010i | 2.91523 | − | 2.91523i | −6.65179 | + | 2.18029i | 9.73311i | −6.44901 | + | 3.72334i | 2.02117 | + | 1.16693i | ||
73.7 | −2.81671 | + | 1.62623i | 1.49616 | − | 0.400895i | 3.28926 | − | 5.69716i | 4.01378 | + | 6.95206i | −3.56231 | + | 3.56231i | −0.176070 | − | 6.99779i | 8.38651i | −5.71645 | + | 3.30039i | −22.6113 | − | 13.0547i | ||
73.8 | −2.67372 | + | 1.54367i | −3.71258 | + | 0.994784i | 2.76586 | − | 4.79061i | 3.73945 | + | 6.47692i | 8.39080 | − | 8.39080i | 6.61930 | + | 2.27703i | 4.72895i | 4.99946 | − | 2.88644i | −19.9965 | − | 11.5450i | ||
73.9 | −2.60389 | + | 1.50336i | 5.11320 | − | 1.37008i | 2.52016 | − | 4.36505i | 1.80353 | + | 3.12380i | −11.2545 | + | 11.2545i | 1.87307 | − | 6.74475i | 3.12796i | 16.4735 | − | 9.51099i | −9.39238 | − | 5.42269i | ||
73.10 | −2.27032 | + | 1.31077i | 4.22026 | − | 1.13082i | 1.43624 | − | 2.48764i | −0.688501 | − | 1.19252i | −8.09911 | + | 8.09911i | 4.55185 | + | 5.31796i | − | 2.95585i | 8.73764 | − | 5.04468i | 3.12624 | + | 1.80493i | |
73.11 | −2.27028 | + | 1.31075i | −0.713090 | + | 0.191072i | 1.43611 | − | 2.48742i | −3.48098 | − | 6.02923i | 1.36847 | − | 1.36847i | 3.27191 | + | 6.18826i | − | 2.95646i | −7.32224 | + | 4.22750i | 15.8056 | + | 9.12535i | |
73.12 | −2.20801 | + | 1.27480i | 0.503298 | − | 0.134858i | 1.25022 | − | 2.16544i | −0.660961 | − | 1.14482i | −0.939373 | + | 0.939373i | 6.75058 | − | 1.85194i | − | 3.82329i | −7.55911 | + | 4.36425i | 2.91882 | + | 1.68518i | |
73.13 | −2.14335 | + | 1.23746i | −3.28848 | + | 0.881145i | 1.06263 | − | 1.84052i | −4.85635 | − | 8.41145i | 5.95797 | − | 5.95797i | −2.38873 | − | 6.57982i | − | 4.63986i | 2.24345 | − | 1.29525i | 20.8177 | + | 12.0191i | |
73.14 | −2.04653 | + | 1.18156i | −2.47138 | + | 0.662204i | 0.792187 | − | 1.37211i | 3.06597 | + | 5.31041i | 4.27531 | − | 4.27531i | −1.68831 | + | 6.79335i | − | 5.70843i | −2.12504 | + | 1.22689i | −12.5492 | − | 7.24528i | |
73.15 | −1.69267 | + | 0.977266i | 1.83287 | − | 0.491117i | −0.0899034 | + | 0.155717i | −2.23127 | − | 3.86467i | −2.62250 | + | 2.62250i | −4.95227 | − | 4.94722i | − | 8.16956i | −4.67600 | + | 2.69969i | 7.55362 | + | 4.36108i | |
73.16 | −1.65656 | + | 0.956418i | 1.46388 | − | 0.392246i | −0.170530 | + | 0.295367i | 2.30350 | + | 3.98978i | −2.04986 | + | 2.04986i | −6.96672 | − | 0.681733i | − | 8.30373i | −5.80513 | + | 3.35159i | −7.63179 | − | 4.40622i | |
73.17 | −1.63115 | + | 0.941746i | −5.53126 | + | 1.48210i | −0.226228 | + | 0.391838i | 1.72421 | + | 2.98642i | 7.62657 | − | 7.62657i | −6.86836 | − | 1.35117i | − | 8.38617i | 20.6040 | − | 11.8957i | −5.62490 | − | 3.24754i | |
73.18 | −1.21135 | + | 0.699374i | −5.22513 | + | 1.40007i | −1.02175 | + | 1.76973i | −1.85825 | − | 3.21859i | 5.35029 | − | 5.35029i | 6.57733 | + | 2.39557i | − | 8.45334i | 17.5475 | − | 10.1311i | 4.50199 | + | 2.59923i | |
73.19 | −1.04789 | + | 0.604999i | 1.82152 | − | 0.488075i | −1.26795 | + | 2.19616i | 3.38546 | + | 5.86378i | −1.61347 | + | 1.61347i | 6.99425 | − | 0.283563i | − | 7.90844i | −4.71451 | + | 2.72192i | −7.09517 | − | 4.09640i | |
73.20 | −1.04052 | + | 0.600745i | −2.35139 | + | 0.630054i | −1.27821 | + | 2.21393i | 0.621112 | + | 1.07580i | 2.06817 | − | 2.06817i | 4.10040 | − | 5.67333i | − | 7.87747i | −2.66215 | + | 1.53699i | −1.29256 | − | 0.746260i | |
See next 80 embeddings (of 216 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.d | odd | 6 | 1 | inner |
41.c | even | 4 | 1 | inner |
287.q | odd | 12 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 287.3.q.a | ✓ | 216 |
7.d | odd | 6 | 1 | inner | 287.3.q.a | ✓ | 216 |
41.c | even | 4 | 1 | inner | 287.3.q.a | ✓ | 216 |
287.q | odd | 12 | 1 | inner | 287.3.q.a | ✓ | 216 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
287.3.q.a | ✓ | 216 | 1.a | even | 1 | 1 | trivial |
287.3.q.a | ✓ | 216 | 7.d | odd | 6 | 1 | inner |
287.3.q.a | ✓ | 216 | 41.c | even | 4 | 1 | inner |
287.3.q.a | ✓ | 216 | 287.q | odd | 12 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(287, [\chi])\).