Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8001,2,Mod(1,8001)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8001, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("8001.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 8001 = 3^{2} \cdot 7 \cdot 127 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 8001.a (trivial) |
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: | yes |
Analytic conductor: | \(63.8883066572\) |
Analytic rank: | \(0\) |
Dimension: | \(28\) |
Twist minimal: | yes |
Fricke sign: | \(-1\) |
Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1.1 | −2.72816 | 0 | 5.44285 | 0.297532 | 0 | −1.00000 | −9.39264 | 0 | −0.811714 | ||||||||||||||||||
1.2 | −2.71687 | 0 | 5.38141 | −1.23671 | 0 | −1.00000 | −9.18686 | 0 | 3.35997 | ||||||||||||||||||
1.3 | −2.47474 | 0 | 4.12433 | −3.91787 | 0 | −1.00000 | −5.25716 | 0 | 9.69570 | ||||||||||||||||||
1.4 | −2.27499 | 0 | 3.17556 | 3.81133 | 0 | −1.00000 | −2.67439 | 0 | −8.67073 | ||||||||||||||||||
1.5 | −2.02489 | 0 | 2.10018 | 2.65037 | 0 | −1.00000 | −0.202857 | 0 | −5.36671 | ||||||||||||||||||
1.6 | −1.99802 | 0 | 1.99208 | −1.55092 | 0 | −1.00000 | 0.0158280 | 0 | 3.09877 | ||||||||||||||||||
1.7 | −1.55816 | 0 | 0.427866 | −3.44623 | 0 | −1.00000 | 2.44964 | 0 | 5.36979 | ||||||||||||||||||
1.8 | −1.42730 | 0 | 0.0371861 | 2.54057 | 0 | −1.00000 | 2.80152 | 0 | −3.62615 | ||||||||||||||||||
1.9 | −1.34710 | 0 | −0.185322 | 0.124848 | 0 | −1.00000 | 2.94385 | 0 | −0.168183 | ||||||||||||||||||
1.10 | −1.24740 | 0 | −0.443996 | 1.34151 | 0 | −1.00000 | 3.04864 | 0 | −1.67339 | ||||||||||||||||||
1.11 | −0.662421 | 0 | −1.56120 | −3.26980 | 0 | −1.00000 | 2.35901 | 0 | 2.16599 | ||||||||||||||||||
1.12 | −0.512239 | 0 | −1.73761 | 1.76990 | 0 | −1.00000 | 1.91455 | 0 | −0.906610 | ||||||||||||||||||
1.13 | −0.487315 | 0 | −1.76252 | 0.708639 | 0 | −1.00000 | 1.83353 | 0 | −0.345330 | ||||||||||||||||||
1.14 | −0.0958522 | 0 | −1.99081 | 1.26637 | 0 | −1.00000 | 0.382528 | 0 | −0.121384 | ||||||||||||||||||
1.15 | 0.0958522 | 0 | −1.99081 | −1.26637 | 0 | −1.00000 | −0.382528 | 0 | −0.121384 | ||||||||||||||||||
1.16 | 0.487315 | 0 | −1.76252 | −0.708639 | 0 | −1.00000 | −1.83353 | 0 | −0.345330 | ||||||||||||||||||
1.17 | 0.512239 | 0 | −1.73761 | −1.76990 | 0 | −1.00000 | −1.91455 | 0 | −0.906610 | ||||||||||||||||||
1.18 | 0.662421 | 0 | −1.56120 | 3.26980 | 0 | −1.00000 | −2.35901 | 0 | 2.16599 | ||||||||||||||||||
1.19 | 1.24740 | 0 | −0.443996 | −1.34151 | 0 | −1.00000 | −3.04864 | 0 | −1.67339 | ||||||||||||||||||
1.20 | 1.34710 | 0 | −0.185322 | −0.124848 | 0 | −1.00000 | −2.94385 | 0 | −0.168183 | ||||||||||||||||||
See all 28 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(3\) | \(1\) |
\(7\) | \(1\) |
\(127\) | \(-1\) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 8001.2.a.y | ✓ | 28 |
3.b | odd | 2 | 1 | inner | 8001.2.a.y | ✓ | 28 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
8001.2.a.y | ✓ | 28 | 1.a | even | 1 | 1 | trivial |
8001.2.a.y | ✓ | 28 | 3.b | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8001))\):
\( T_{2}^{28} - 43 T_{2}^{26} + 813 T_{2}^{24} - 8906 T_{2}^{22} + 62714 T_{2}^{20} - 297802 T_{2}^{18} + 973243 T_{2}^{16} - 2194191 T_{2}^{14} + 3368255 T_{2}^{12} - 3416593 T_{2}^{10} + 2175070 T_{2}^{8} + \cdots + 100 \) |
\( T_{5}^{28} - 77 T_{5}^{26} + 2556 T_{5}^{24} - 48052 T_{5}^{22} + 565977 T_{5}^{20} - 4371605 T_{5}^{18} + 22582659 T_{5}^{16} - 78348404 T_{5}^{14} + 181103072 T_{5}^{12} - 272571384 T_{5}^{10} + \cdots + 29584 \) |