Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [800,2,Mod(81,800)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(800, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([0, 5, 4]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("800.81");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 800 = 2^{5} \cdot 5^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 800.bj (of order \(10\), degree \(4\), not 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: | \(6.38803216170\) |
Analytic rank: | \(0\) |
Dimension: | \(112\) |
Relative dimension: | \(28\) over \(\Q(\zeta_{10})\) |
Twist minimal: | no (minimal twist has level 200) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{10}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
81.1 | 0 | −1.91233 | − | 2.63210i | 0 | 0.977596 | − | 2.01105i | 0 | 1.36581 | 0 | −2.34388 | + | 7.21373i | 0 | ||||||||||||
81.2 | 0 | −1.75086 | − | 2.40986i | 0 | −2.21918 | + | 0.274336i | 0 | 3.41885 | 0 | −1.81484 | + | 5.58549i | 0 | ||||||||||||
81.3 | 0 | −1.68097 | − | 2.31366i | 0 | −1.82826 | − | 1.28743i | 0 | −4.93157 | 0 | −1.60031 | + | 4.92524i | 0 | ||||||||||||
81.4 | 0 | −1.52910 | − | 2.10462i | 0 | 0.881108 | + | 2.05515i | 0 | 1.53499 | 0 | −1.16425 | + | 3.58319i | 0 | ||||||||||||
81.5 | 0 | −1.35783 | − | 1.86889i | 0 | 1.91517 | − | 1.15417i | 0 | −0.146324 | 0 | −0.722003 | + | 2.22210i | 0 | ||||||||||||
81.6 | 0 | −1.13822 | − | 1.56662i | 0 | 0.658250 | + | 2.13699i | 0 | −3.40673 | 0 | −0.231711 | + | 0.713133i | 0 | ||||||||||||
81.7 | 0 | −1.07287 | − | 1.47668i | 0 | −1.36252 | + | 1.77300i | 0 | 2.31401 | 0 | −0.102477 | + | 0.315391i | 0 | ||||||||||||
81.8 | 0 | −1.00642 | − | 1.38522i | 0 | 2.06230 | + | 0.864242i | 0 | −0.719899 | 0 | 0.0211043 | − | 0.0649523i | 0 | ||||||||||||
81.9 | 0 | −0.910283 | − | 1.25290i | 0 | −0.0584731 | − | 2.23530i | 0 | 1.01594 | 0 | 0.185915 | − | 0.572186i | 0 | ||||||||||||
81.10 | 0 | −0.801761 | − | 1.10353i | 0 | −2.06049 | + | 0.868546i | 0 | −1.98026 | 0 | 0.352095 | − | 1.08364i | 0 | ||||||||||||
81.11 | 0 | −0.289798 | − | 0.398873i | 0 | 0.143005 | − | 2.23149i | 0 | 4.76281 | 0 | 0.851934 | − | 2.62198i | 0 | ||||||||||||
81.12 | 0 | −0.260015 | − | 0.357881i | 0 | −1.65958 | − | 1.49860i | 0 | −1.14468 | 0 | 0.866581 | − | 2.66706i | 0 | ||||||||||||
81.13 | 0 | −0.239455 | − | 0.329581i | 0 | 2.14380 | + | 0.635713i | 0 | 2.90112 | 0 | 0.875766 | − | 2.69533i | 0 | ||||||||||||
81.14 | 0 | −0.0135478 | − | 0.0186469i | 0 | −1.58615 | + | 1.57611i | 0 | −2.98406 | 0 | 0.926887 | − | 2.85266i | 0 | ||||||||||||
81.15 | 0 | 0.0135478 | + | 0.0186469i | 0 | 1.58615 | − | 1.57611i | 0 | −2.98406 | 0 | 0.926887 | − | 2.85266i | 0 | ||||||||||||
81.16 | 0 | 0.239455 | + | 0.329581i | 0 | −2.14380 | − | 0.635713i | 0 | 2.90112 | 0 | 0.875766 | − | 2.69533i | 0 | ||||||||||||
81.17 | 0 | 0.260015 | + | 0.357881i | 0 | 1.65958 | + | 1.49860i | 0 | −1.14468 | 0 | 0.866581 | − | 2.66706i | 0 | ||||||||||||
81.18 | 0 | 0.289798 | + | 0.398873i | 0 | −0.143005 | + | 2.23149i | 0 | 4.76281 | 0 | 0.851934 | − | 2.62198i | 0 | ||||||||||||
81.19 | 0 | 0.801761 | + | 1.10353i | 0 | 2.06049 | − | 0.868546i | 0 | −1.98026 | 0 | 0.352095 | − | 1.08364i | 0 | ||||||||||||
81.20 | 0 | 0.910283 | + | 1.25290i | 0 | 0.0584731 | + | 2.23530i | 0 | 1.01594 | 0 | 0.185915 | − | 0.572186i | 0 | ||||||||||||
See next 80 embeddings (of 112 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
8.b | even | 2 | 1 | inner |
25.d | even | 5 | 1 | inner |
200.t | even | 10 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 800.2.bj.a | 112 | |
4.b | odd | 2 | 1 | 200.2.t.a | ✓ | 112 | |
8.b | even | 2 | 1 | inner | 800.2.bj.a | 112 | |
8.d | odd | 2 | 1 | 200.2.t.a | ✓ | 112 | |
20.d | odd | 2 | 1 | 1000.2.t.a | 112 | ||
20.e | even | 4 | 2 | 1000.2.o.b | 224 | ||
25.d | even | 5 | 1 | inner | 800.2.bj.a | 112 | |
40.e | odd | 2 | 1 | 1000.2.t.a | 112 | ||
40.k | even | 4 | 2 | 1000.2.o.b | 224 | ||
100.h | odd | 10 | 1 | 1000.2.t.a | 112 | ||
100.j | odd | 10 | 1 | 200.2.t.a | ✓ | 112 | |
100.l | even | 20 | 2 | 1000.2.o.b | 224 | ||
200.n | odd | 10 | 1 | 200.2.t.a | ✓ | 112 | |
200.s | odd | 10 | 1 | 1000.2.t.a | 112 | ||
200.t | even | 10 | 1 | inner | 800.2.bj.a | 112 | |
200.v | even | 20 | 2 | 1000.2.o.b | 224 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
200.2.t.a | ✓ | 112 | 4.b | odd | 2 | 1 | |
200.2.t.a | ✓ | 112 | 8.d | odd | 2 | 1 | |
200.2.t.a | ✓ | 112 | 100.j | odd | 10 | 1 | |
200.2.t.a | ✓ | 112 | 200.n | odd | 10 | 1 | |
800.2.bj.a | 112 | 1.a | even | 1 | 1 | trivial | |
800.2.bj.a | 112 | 8.b | even | 2 | 1 | inner | |
800.2.bj.a | 112 | 25.d | even | 5 | 1 | inner | |
800.2.bj.a | 112 | 200.t | even | 10 | 1 | inner | |
1000.2.o.b | 224 | 20.e | even | 4 | 2 | ||
1000.2.o.b | 224 | 40.k | even | 4 | 2 | ||
1000.2.o.b | 224 | 100.l | even | 20 | 2 | ||
1000.2.o.b | 224 | 200.v | even | 20 | 2 | ||
1000.2.t.a | 112 | 20.d | odd | 2 | 1 | ||
1000.2.t.a | 112 | 40.e | odd | 2 | 1 | ||
1000.2.t.a | 112 | 100.h | odd | 10 | 1 | ||
1000.2.t.a | 112 | 200.s | odd | 10 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(800, [\chi])\).