Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [800,2,Mod(63,800)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(800, base_ring=CyclotomicField(20))
chi = DirichletCharacter(H, H._module([10, 0, 19]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("800.63");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 800 = 2^{5} \cdot 5^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 800.bq (of order \(20\), degree \(8\), 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: | \(64\) |
Relative dimension: | \(8\) over \(\Q(\zeta_{20})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{20}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
63.1 | 0 | −2.49551 | − | 1.27153i | 0 | −1.46592 | + | 1.68851i | 0 | 1.17365 | + | 1.17365i | 0 | 2.84744 | + | 3.91917i | 0 | ||||||||||
63.2 | 0 | −2.31654 | − | 1.18033i | 0 | 2.18236 | − | 0.487149i | 0 | −2.41634 | − | 2.41634i | 0 | 2.20979 | + | 3.04152i | 0 | ||||||||||
63.3 | 0 | −0.696591 | − | 0.354931i | 0 | −2.17155 | − | 0.533278i | 0 | −1.62475 | − | 1.62475i | 0 | −1.40409 | − | 1.93257i | 0 | ||||||||||
63.4 | 0 | −0.685472 | − | 0.349265i | 0 | 1.93228 | + | 1.12530i | 0 | 3.37195 | + | 3.37195i | 0 | −1.41547 | − | 1.94823i | 0 | ||||||||||
63.5 | 0 | 0.306774 | + | 0.156309i | 0 | 0.187686 | − | 2.22818i | 0 | 1.04191 | + | 1.04191i | 0 | −1.69368 | − | 2.33115i | 0 | ||||||||||
63.6 | 0 | 0.675433 | + | 0.344150i | 0 | −0.294150 | + | 2.21664i | 0 | −2.26261 | − | 2.26261i | 0 | −1.42559 | − | 1.96215i | 0 | ||||||||||
63.7 | 0 | 2.21160 | + | 1.12686i | 0 | 2.10230 | + | 0.761808i | 0 | −0.390880 | − | 0.390880i | 0 | 1.85798 | + | 2.55729i | 0 | ||||||||||
63.8 | 0 | 3.00031 | + | 1.52873i | 0 | −1.83096 | − | 1.28358i | 0 | −0.177009 | − | 0.177009i | 0 | 4.90147 | + | 6.74629i | 0 | ||||||||||
127.1 | 0 | −2.49551 | + | 1.27153i | 0 | −1.46592 | − | 1.68851i | 0 | 1.17365 | − | 1.17365i | 0 | 2.84744 | − | 3.91917i | 0 | ||||||||||
127.2 | 0 | −2.31654 | + | 1.18033i | 0 | 2.18236 | + | 0.487149i | 0 | −2.41634 | + | 2.41634i | 0 | 2.20979 | − | 3.04152i | 0 | ||||||||||
127.3 | 0 | −0.696591 | + | 0.354931i | 0 | −2.17155 | + | 0.533278i | 0 | −1.62475 | + | 1.62475i | 0 | −1.40409 | + | 1.93257i | 0 | ||||||||||
127.4 | 0 | −0.685472 | + | 0.349265i | 0 | 1.93228 | − | 1.12530i | 0 | 3.37195 | − | 3.37195i | 0 | −1.41547 | + | 1.94823i | 0 | ||||||||||
127.5 | 0 | 0.306774 | − | 0.156309i | 0 | 0.187686 | + | 2.22818i | 0 | 1.04191 | − | 1.04191i | 0 | −1.69368 | + | 2.33115i | 0 | ||||||||||
127.6 | 0 | 0.675433 | − | 0.344150i | 0 | −0.294150 | − | 2.21664i | 0 | −2.26261 | + | 2.26261i | 0 | −1.42559 | + | 1.96215i | 0 | ||||||||||
127.7 | 0 | 2.21160 | − | 1.12686i | 0 | 2.10230 | − | 0.761808i | 0 | −0.390880 | + | 0.390880i | 0 | 1.85798 | − | 2.55729i | 0 | ||||||||||
127.8 | 0 | 3.00031 | − | 1.52873i | 0 | −1.83096 | + | 1.28358i | 0 | −0.177009 | + | 0.177009i | 0 | 4.90147 | − | 6.74629i | 0 | ||||||||||
223.1 | 0 | −1.39978 | − | 2.74723i | 0 | −1.70274 | − | 1.44937i | 0 | 2.85341 | + | 2.85341i | 0 | −3.82451 | + | 5.26398i | 0 | ||||||||||
223.2 | 0 | −1.07219 | − | 2.10429i | 0 | 1.72608 | − | 1.42149i | 0 | −3.37202 | − | 3.37202i | 0 | −1.51510 | + | 2.08536i | 0 | ||||||||||
223.3 | 0 | −0.751816 | − | 1.47552i | 0 | −0.850652 | + | 2.06794i | 0 | −0.0884893 | − | 0.0884893i | 0 | 0.151419 | − | 0.208410i | 0 | ||||||||||
223.4 | 0 | 0.0251148 | + | 0.0492906i | 0 | 1.96222 | + | 1.07225i | 0 | 0.693303 | + | 0.693303i | 0 | 1.76156 | − | 2.42458i | 0 | ||||||||||
See all 64 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
100.l | even | 20 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 800.2.bq.c | ✓ | 64 |
4.b | odd | 2 | 1 | 800.2.bq.d | yes | 64 | |
25.f | odd | 20 | 1 | 800.2.bq.d | yes | 64 | |
100.l | even | 20 | 1 | inner | 800.2.bq.c | ✓ | 64 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
800.2.bq.c | ✓ | 64 | 1.a | even | 1 | 1 | trivial |
800.2.bq.c | ✓ | 64 | 100.l | even | 20 | 1 | inner |
800.2.bq.d | yes | 64 | 4.b | odd | 2 | 1 | |
800.2.bq.d | yes | 64 | 25.f | odd | 20 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{64} + 6 T_{3}^{61} - 111 T_{3}^{60} + 148 T_{3}^{59} - 42 T_{3}^{58} + 358 T_{3}^{57} + \cdots + 110166016 \) acting on \(S_{2}^{\mathrm{new}}(800, [\chi])\).