Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [387,3,Mod(7,387)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(387, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([4, 5]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("387.7");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 387 = 3^{2} \cdot 43 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 387.r (of order \(6\), degree \(2\), 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: | \(10.5449862307\) |
Analytic rank: | \(0\) |
Dimension: | \(172\) |
Relative dimension: | \(86\) over \(\Q(\zeta_{6})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
7.1 | −3.33195 | − | 1.92370i | −2.87889 | − | 0.843786i | 5.40128 | + | 9.35529i | −5.56892 | − | 3.21522i | 7.96914 | + | 8.34960i | 11.0230 | − | 6.36414i | − | 26.1722i | 7.57605 | + | 4.85834i | 12.3703 | + | 21.4259i | |
7.2 | −3.31385 | − | 1.91325i | 2.97388 | + | 0.395039i | 5.32107 | + | 9.21636i | −3.68685 | − | 2.12860i | −9.09917 | − | 6.99888i | −2.02569 | + | 1.16953i | − | 25.4162i | 8.68789 | + | 2.34959i | 8.14511 | + | 14.1077i | |
7.3 | −3.26338 | − | 1.88411i | −0.988151 | − | 2.83259i | 5.09976 | + | 8.83304i | 3.54051 | + | 2.04412i | −2.11220 | + | 11.1056i | −2.60622 | + | 1.50470i | − | 23.3612i | −7.04711 | + | 5.59805i | −7.70268 | − | 13.3414i | |
7.4 | −3.21802 | − | 1.85793i | −0.00307485 | + | 3.00000i | 4.90377 | + | 8.49358i | 3.25716 | + | 1.88052i | 5.58367 | − | 9.64834i | −4.57150 | + | 2.63936i | − | 21.5800i | −8.99998 | − | 0.0184491i | −6.98774 | − | 12.1031i | |
7.5 | −3.15952 | − | 1.82415i | −2.09171 | + | 2.15053i | 4.65504 | + | 8.06277i | −4.21054 | − | 2.43095i | 10.5317 | − | 2.97905i | −6.38869 | + | 3.68851i | − | 19.3728i | −0.249528 | − | 8.99654i | 8.86885 | + | 15.3613i | |
7.6 | −3.12567 | − | 1.80461i | 1.46355 | − | 2.61878i | 4.51320 | + | 7.81709i | −0.264425 | − | 0.152666i | −9.30044 | + | 5.54431i | 7.65018 | − | 4.41683i | − | 18.1413i | −4.71604 | − | 7.66544i | 0.551003 | + | 0.954365i | |
7.7 | −3.11003 | − | 1.79558i | −2.85359 | + | 0.925757i | 4.44820 | + | 7.70451i | 6.98116 | + | 4.03058i | 10.5370 | + | 2.24471i | 4.86998 | − | 2.81168i | − | 17.5838i | 7.28595 | − | 5.28346i | −14.4744 | − | 25.0705i | |
7.8 | −2.88886 | − | 1.66788i | 0.527498 | − | 2.95326i | 3.56367 | + | 6.17245i | −7.21156 | − | 4.16359i | −6.44956 | + | 7.65174i | −8.45660 | + | 4.88242i | − | 10.4320i | −8.44349 | − | 3.11568i | 13.8888 | + | 24.0561i | |
7.9 | −2.86728 | − | 1.65542i | 2.96172 | − | 0.477749i | 3.48086 | + | 6.02902i | 7.89462 | + | 4.55796i | −9.28294 | − | 3.53306i | 8.80554 | − | 5.08388i | − | 9.80579i | 8.54351 | − | 2.82991i | −15.0907 | − | 26.1379i | |
7.10 | −2.82415 | − | 1.63052i | 2.33397 | + | 1.88483i | 3.31720 | + | 5.74556i | 0.737130 | + | 0.425582i | −3.51821 | − | 9.12863i | 2.72591 | − | 1.57381i | − | 8.59090i | 1.89481 | + | 8.79828i | −1.38784 | − | 2.40381i | |
7.11 | −2.79671 | − | 1.61468i | −2.76326 | − | 1.16807i | 3.21438 | + | 5.56746i | 2.89880 | + | 1.67363i | 5.84196 | + | 7.72854i | −8.94913 | + | 5.16678i | − | 7.84331i | 6.27121 | + | 6.45538i | −5.40474 | − | 9.36128i | |
7.12 | −2.79056 | − | 1.61113i | 0.481819 | + | 2.96106i | 3.19150 | + | 5.52784i | −6.91867 | − | 3.99450i | 3.42611 | − | 9.03929i | 5.55234 | − | 3.20565i | − | 7.67867i | −8.53570 | + | 2.85339i | 12.8713 | + | 22.2938i | |
7.13 | −2.57467 | − | 1.48649i | 2.09092 | + | 2.15129i | 2.41930 | + | 4.19035i | 6.34282 | + | 3.66203i | −2.18557 | − | 8.64700i | −5.59057 | + | 3.22772i | − | 2.49313i | −0.256102 | + | 8.99636i | −10.8871 | − | 18.8571i | |
7.14 | −2.55908 | − | 1.47749i | 2.41765 | − | 1.77622i | 2.36594 | + | 4.09793i | 2.13001 | + | 1.22976i | −8.81132 | + | 0.973459i | −5.45608 | + | 3.15007i | − | 2.16269i | 2.69006 | − | 8.58857i | −3.63391 | − | 6.29412i | |
7.15 | −2.48598 | − | 1.43528i | −2.97921 | + | 0.352563i | 2.12006 | + | 3.67206i | −1.19193 | − | 0.688159i | 7.91229 | + | 3.39954i | −2.78187 | + | 1.60611i | − | 0.689300i | 8.75140 | − | 2.10072i | 1.97540 | + | 3.42150i | |
7.16 | −2.39965 | − | 1.38544i | −1.96256 | − | 2.26900i | 1.83889 | + | 3.18505i | −5.76682 | − | 3.32948i | 1.56589 | + | 8.16382i | −0.255369 | + | 0.147437i | 0.892832i | −1.29674 | + | 8.90609i | 9.22558 | + | 15.9792i | ||
7.17 | −2.39615 | − | 1.38342i | −1.28201 | + | 2.71228i | 1.82768 | + | 3.16563i | 1.72160 | + | 0.993964i | 6.82409 | − | 4.72547i | 8.68573 | − | 5.01471i | 0.953570i | −5.71292 | − | 6.95432i | −2.75013 | − | 4.76336i | ||
7.18 | −2.30517 | − | 1.33089i | −1.40078 | − | 2.65289i | 1.54253 | + | 2.67174i | 1.93397 | + | 1.11658i | −0.301653 | + | 7.97963i | 5.35565 | − | 3.09209i | 2.43536i | −5.07560 | + | 7.43224i | −2.97209 | − | 5.14781i | ||
7.19 | −2.22575 | − | 1.28503i | 2.41565 | − | 1.77894i | 1.30263 | + | 2.25622i | −5.35135 | − | 3.08960i | −7.66262 | + | 0.855267i | 6.79901 | − | 3.92541i | 3.58459i | 2.67075 | − | 8.59460i | 7.94049 | + | 13.7533i | ||
7.20 | −2.02913 | − | 1.17152i | 2.99843 | − | 0.0971885i | 0.744905 | + | 1.29021i | −1.17017 | − | 0.675599i | −6.19804 | − | 3.31550i | −8.94397 | + | 5.16380i | 5.88146i | 8.98111 | − | 0.582825i | 1.58295 | + | 2.74175i | ||
See next 80 embeddings (of 172 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
387.r | odd | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 387.3.r.a | yes | 172 |
9.c | even | 3 | 1 | 387.3.n.a | ✓ | 172 | |
43.d | odd | 6 | 1 | 387.3.n.a | ✓ | 172 | |
387.r | odd | 6 | 1 | inner | 387.3.r.a | yes | 172 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
387.3.n.a | ✓ | 172 | 9.c | even | 3 | 1 | |
387.3.n.a | ✓ | 172 | 43.d | odd | 6 | 1 | |
387.3.r.a | yes | 172 | 1.a | even | 1 | 1 | trivial |
387.3.r.a | yes | 172 | 387.r | odd | 6 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(387, [\chi])\).