Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [387,3,Mod(251,387)]
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([3, 2]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("387.251");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 387 = 3^{2} \cdot 43 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 387.i (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: | \(60\) |
Relative dimension: | \(30\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
251.1 | − | 3.82582i | 0 | −10.6369 | −4.07908 | + | 2.35506i | 0 | −0.828985 | + | 1.43584i | 25.3915i | 0 | 9.01003 | + | 15.6058i | |||||||||||
251.2 | − | 3.67873i | 0 | −9.53307 | 5.52313 | − | 3.18878i | 0 | 4.16366 | − | 7.21167i | 20.3547i | 0 | −11.7307 | − | 20.3181i | |||||||||||
251.3 | − | 3.60468i | 0 | −8.99370 | −6.10136 | + | 3.52262i | 0 | −1.11308 | + | 1.92790i | 18.0007i | 0 | 12.6979 | + | 21.9934i | |||||||||||
251.4 | − | 3.08493i | 0 | −5.51676 | 0.670722 | − | 0.387242i | 0 | 0.962002 | − | 1.66624i | 4.67910i | 0 | −1.19461 | − | 2.06913i | |||||||||||
251.5 | − | 2.92209i | 0 | −4.53862 | 1.25945 | − | 0.727146i | 0 | −4.31156 | + | 7.46785i | 1.57390i | 0 | −2.12479 | − | 3.68024i | |||||||||||
251.6 | − | 2.91777i | 0 | −4.51337 | 8.38570 | − | 4.84149i | 0 | −4.59727 | + | 7.96270i | 1.49789i | 0 | −14.1263 | − | 24.4675i | |||||||||||
251.7 | − | 2.74538i | 0 | −3.53712 | −2.82066 | + | 1.62851i | 0 | 0.221075 | − | 0.382913i | − | 1.27079i | 0 | 4.47088 | + | 7.74379i | ||||||||||
251.8 | − | 2.33960i | 0 | −1.47375 | −3.85397 | + | 2.22509i | 0 | 1.93315 | − | 3.34832i | − | 5.91043i | 0 | 5.20583 | + | 9.01676i | ||||||||||
251.9 | − | 2.00047i | 0 | −0.00186107 | 4.64054 | − | 2.67922i | 0 | 6.65424 | − | 11.5255i | − | 7.99814i | 0 | −5.35968 | − | 9.28324i | ||||||||||
251.10 | − | 1.57119i | 0 | 1.53137 | 4.22007 | − | 2.43646i | 0 | 3.93613 | − | 6.81758i | − | 8.69082i | 0 | −3.82813 | − | 6.63051i | ||||||||||
251.11 | − | 1.38607i | 0 | 2.07881 | −1.93550 | + | 1.11746i | 0 | −5.77497 | + | 10.0025i | − | 8.42566i | 0 | 1.54888 | + | 2.68274i | ||||||||||
251.12 | − | 1.15908i | 0 | 2.65654 | 3.71486 | − | 2.14478i | 0 | −4.06824 | + | 7.04639i | − | 7.71545i | 0 | −2.48596 | − | 4.30582i | ||||||||||
251.13 | − | 1.07818i | 0 | 2.83753 | −7.10171 | + | 4.10018i | 0 | 2.44369 | − | 4.23259i | − | 7.37208i | 0 | 4.42072 | + | 7.65691i | ||||||||||
251.14 | − | 0.560654i | 0 | 3.68567 | −5.94504 | + | 3.43237i | 0 | −2.88161 | + | 4.99110i | − | 4.30900i | 0 | 1.92437 | + | 3.33311i | ||||||||||
251.15 | − | 0.211632i | 0 | 3.95521 | 3.50663 | − | 2.02455i | 0 | 1.76176 | − | 3.05147i | − | 1.68358i | 0 | −0.428460 | − | 0.742115i | ||||||||||
251.16 | 0.211632i | 0 | 3.95521 | −3.50663 | + | 2.02455i | 0 | 1.76176 | − | 3.05147i | 1.68358i | 0 | −0.428460 | − | 0.742115i | ||||||||||||
251.17 | 0.560654i | 0 | 3.68567 | 5.94504 | − | 3.43237i | 0 | −2.88161 | + | 4.99110i | 4.30900i | 0 | 1.92437 | + | 3.33311i | ||||||||||||
251.18 | 1.07818i | 0 | 2.83753 | 7.10171 | − | 4.10018i | 0 | 2.44369 | − | 4.23259i | 7.37208i | 0 | 4.42072 | + | 7.65691i | ||||||||||||
251.19 | 1.15908i | 0 | 2.65654 | −3.71486 | + | 2.14478i | 0 | −4.06824 | + | 7.04639i | 7.71545i | 0 | −2.48596 | − | 4.30582i | ||||||||||||
251.20 | 1.38607i | 0 | 2.07881 | 1.93550 | − | 1.11746i | 0 | −5.77497 | + | 10.0025i | 8.42566i | 0 | 1.54888 | + | 2.68274i | ||||||||||||
See all 60 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
43.c | even | 3 | 1 | inner |
129.f | 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.i.a | ✓ | 60 |
3.b | odd | 2 | 1 | inner | 387.3.i.a | ✓ | 60 |
43.c | even | 3 | 1 | inner | 387.3.i.a | ✓ | 60 |
129.f | odd | 6 | 1 | inner | 387.3.i.a | ✓ | 60 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
387.3.i.a | ✓ | 60 | 1.a | even | 1 | 1 | trivial |
387.3.i.a | ✓ | 60 | 3.b | odd | 2 | 1 | inner |
387.3.i.a | ✓ | 60 | 43.c | even | 3 | 1 | inner |
387.3.i.a | ✓ | 60 | 129.f | odd | 6 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(387, [\chi])\).