Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [387,3,Mod(82,387)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(387, base_ring=CyclotomicField(14))
chi = DirichletCharacter(H, H._module([0, 11]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("387.82");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 387 = 3^{2} \cdot 43 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 387.w (of order \(14\), degree \(6\), 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: | \(42\) |
Relative dimension: | \(7\) over \(\Q(\zeta_{14})\) |
Twist minimal: | no (minimal twist has level 43) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{14}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
82.1 | −2.31044 | − | 1.84251i | 0 | 1.05318 | + | 4.61430i | −3.37568 | + | 7.00967i | 0 | 7.33610i | 0.939799 | − | 1.95151i | 0 | 20.7147 | − | 9.97566i | ||||||||
82.2 | −1.96310 | − | 1.56552i | 0 | 0.512821 | + | 2.24681i | 2.07465 | − | 4.30805i | 0 | − | 12.3653i | −1.84704 | + | 3.83541i | 0 | −10.8171 | + | 5.20922i | |||||||
82.3 | −0.405381 | − | 0.323281i | 0 | −0.830260 | − | 3.63761i | 0.149470 | − | 0.310378i | 0 | 9.37466i | −1.73928 | + | 3.61164i | 0 | −0.160932 | + | 0.0775006i | ||||||||
82.4 | 0.290487 | + | 0.231656i | 0 | −0.859365 | − | 3.76513i | 3.26645 | − | 6.78286i | 0 | 1.66643i | 1.26741 | − | 2.63181i | 0 | 2.52015 | − | 1.21364i | ||||||||
82.5 | 1.12245 | + | 0.895127i | 0 | −0.431434 | − | 1.89023i | −2.85407 | + | 5.92653i | 0 | − | 7.32927i | 3.69939 | − | 7.68187i | 0 | −8.50856 | + | 4.09751i | |||||||
82.6 | 2.20426 | + | 1.75784i | 0 | 0.878675 | + | 3.84973i | 1.34756 | − | 2.79824i | 0 | − | 7.15415i | 0.0627165 | − | 0.130232i | 0 | 7.88921 | − | 3.79924i | |||||||
82.7 | 2.96269 | + | 2.36266i | 0 | 2.30525 | + | 10.0999i | −0.910321 | + | 1.89030i | 0 | 8.82371i | −10.4564 | + | 21.7129i | 0 | −7.16314 | + | 3.44959i | ||||||||
118.1 | −2.31044 | + | 1.84251i | 0 | 1.05318 | − | 4.61430i | −3.37568 | − | 7.00967i | 0 | − | 7.33610i | 0.939799 | + | 1.95151i | 0 | 20.7147 | + | 9.97566i | |||||||
118.2 | −1.96310 | + | 1.56552i | 0 | 0.512821 | − | 2.24681i | 2.07465 | + | 4.30805i | 0 | 12.3653i | −1.84704 | − | 3.83541i | 0 | −10.8171 | − | 5.20922i | ||||||||
118.3 | −0.405381 | + | 0.323281i | 0 | −0.830260 | + | 3.63761i | 0.149470 | + | 0.310378i | 0 | − | 9.37466i | −1.73928 | − | 3.61164i | 0 | −0.160932 | − | 0.0775006i | |||||||
118.4 | 0.290487 | − | 0.231656i | 0 | −0.859365 | + | 3.76513i | 3.26645 | + | 6.78286i | 0 | − | 1.66643i | 1.26741 | + | 2.63181i | 0 | 2.52015 | + | 1.21364i | |||||||
118.5 | 1.12245 | − | 0.895127i | 0 | −0.431434 | + | 1.89023i | −2.85407 | − | 5.92653i | 0 | 7.32927i | 3.69939 | + | 7.68187i | 0 | −8.50856 | − | 4.09751i | ||||||||
118.6 | 2.20426 | − | 1.75784i | 0 | 0.878675 | − | 3.84973i | 1.34756 | + | 2.79824i | 0 | 7.15415i | 0.0627165 | + | 0.130232i | 0 | 7.88921 | + | 3.79924i | ||||||||
118.7 | 2.96269 | − | 2.36266i | 0 | 2.30525 | − | 10.0999i | −0.910321 | − | 1.89030i | 0 | − | 8.82371i | −10.4564 | − | 21.7129i | 0 | −7.16314 | − | 3.44959i | |||||||
199.1 | −3.62198 | + | 0.826694i | 0 | 8.83146 | − | 4.25301i | 2.82431 | − | 2.25231i | 0 | 9.33217i | −16.8530 | + | 13.4399i | 0 | −8.36764 | + | 10.4927i | ||||||||
199.2 | −2.11328 | + | 0.482341i | 0 | 0.629407 | − | 0.303106i | 3.78878 | − | 3.02145i | 0 | − | 10.9446i | 5.59495 | − | 4.46183i | 0 | −6.54936 | + | 8.21264i | |||||||
199.3 | −1.71853 | + | 0.392244i | 0 | −0.804371 | + | 0.387365i | −5.40310 | + | 4.30882i | 0 | 6.65286i | 6.74303 | − | 5.37738i | 0 | 7.59529 | − | 9.52420i | ||||||||
199.4 | 0.811544 | − | 0.185230i | 0 | −2.97958 | + | 1.43489i | 3.71827 | − | 2.96522i | 0 | 0.589128i | −4.75551 | + | 3.79239i | 0 | 2.46829 | − | 3.09514i | ||||||||
199.5 | 0.844666 | − | 0.192789i | 0 | −2.92758 | + | 1.40985i | −0.831553 | + | 0.663142i | 0 | 0.302383i | −4.91050 | + | 3.91600i | 0 | −0.574538 | + | 0.720448i | ||||||||
199.6 | 2.63550 | − | 0.601536i | 0 | 2.98015 | − | 1.43516i | −7.00305 | + | 5.58475i | 0 | − | 3.35098i | −1.46315 | + | 1.16682i | 0 | −15.0971 | + | 18.9312i | |||||||
See all 42 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
43.f | odd | 14 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 387.3.w.b | 42 | |
3.b | odd | 2 | 1 | 43.3.f.a | ✓ | 42 | |
43.f | odd | 14 | 1 | inner | 387.3.w.b | 42 | |
129.j | even | 14 | 1 | 43.3.f.a | ✓ | 42 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
43.3.f.a | ✓ | 42 | 3.b | odd | 2 | 1 | |
43.3.f.a | ✓ | 42 | 129.j | even | 14 | 1 | |
387.3.w.b | 42 | 1.a | even | 1 | 1 | trivial | |
387.3.w.b | 42 | 43.f | odd | 14 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{42} - 7 T_{2}^{41} + 8 T_{2}^{40} + 84 T_{2}^{39} - 246 T_{2}^{38} - 476 T_{2}^{37} + \cdots + 103657489047 \) acting on \(S_{3}^{\mathrm{new}}(387, [\chi])\).