Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6008,2,Mod(1,6008)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6008, 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("6008.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 6008 = 2^{3} \cdot 751 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 6008.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: | \(47.9741215344\) |
Analytic rank: | \(0\) |
Dimension: | \(50\) |
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 | 0 | −3.22705 | 0 | 2.92731 | 0 | 0.525757 | 0 | 7.41384 | 0 | ||||||||||||||||||
1.2 | 0 | −2.97826 | 0 | −0.794149 | 0 | −3.63896 | 0 | 5.87005 | 0 | ||||||||||||||||||
1.3 | 0 | −2.97134 | 0 | 0.343630 | 0 | −1.43593 | 0 | 5.82889 | 0 | ||||||||||||||||||
1.4 | 0 | −2.96484 | 0 | 2.30894 | 0 | −3.32968 | 0 | 5.79029 | 0 | ||||||||||||||||||
1.5 | 0 | −2.83142 | 0 | 4.26484 | 0 | 3.89972 | 0 | 5.01693 | 0 | ||||||||||||||||||
1.6 | 0 | −2.58697 | 0 | 0.0890087 | 0 | 4.21791 | 0 | 3.69244 | 0 | ||||||||||||||||||
1.7 | 0 | −2.56656 | 0 | −2.44840 | 0 | −1.80922 | 0 | 3.58725 | 0 | ||||||||||||||||||
1.8 | 0 | −2.49253 | 0 | −3.92718 | 0 | 2.18622 | 0 | 3.21269 | 0 | ||||||||||||||||||
1.9 | 0 | −2.41806 | 0 | 0.877422 | 0 | 2.89114 | 0 | 2.84699 | 0 | ||||||||||||||||||
1.10 | 0 | −2.26403 | 0 | −1.66165 | 0 | −0.979480 | 0 | 2.12585 | 0 | ||||||||||||||||||
1.11 | 0 | −2.24563 | 0 | 3.86835 | 0 | −4.12735 | 0 | 2.04285 | 0 | ||||||||||||||||||
1.12 | 0 | −1.56448 | 0 | 2.55356 | 0 | −2.27627 | 0 | −0.552416 | 0 | ||||||||||||||||||
1.13 | 0 | −1.42091 | 0 | −3.43075 | 0 | 0.534513 | 0 | −0.981020 | 0 | ||||||||||||||||||
1.14 | 0 | −1.38456 | 0 | −1.32143 | 0 | 2.39545 | 0 | −1.08299 | 0 | ||||||||||||||||||
1.15 | 0 | −1.34723 | 0 | 0.0749039 | 0 | −4.25247 | 0 | −1.18497 | 0 | ||||||||||||||||||
1.16 | 0 | −1.33859 | 0 | 3.94917 | 0 | 0.220759 | 0 | −1.20816 | 0 | ||||||||||||||||||
1.17 | 0 | −1.03653 | 0 | 2.53205 | 0 | −0.333645 | 0 | −1.92561 | 0 | ||||||||||||||||||
1.18 | 0 | −0.691575 | 0 | −1.88995 | 0 | 4.98805 | 0 | −2.52172 | 0 | ||||||||||||||||||
1.19 | 0 | −0.685000 | 0 | 2.14563 | 0 | 4.94928 | 0 | −2.53078 | 0 | ||||||||||||||||||
1.20 | 0 | −0.684026 | 0 | −2.03205 | 0 | −0.0739397 | 0 | −2.53211 | 0 | ||||||||||||||||||
See all 50 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(2\) | \(-1\) |
\(751\) | \(1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 6008.2.a.e | ✓ | 50 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
6008.2.a.e | ✓ | 50 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{50} - 6 T_{3}^{49} - 85 T_{3}^{48} + 562 T_{3}^{47} + 3255 T_{3}^{46} - 24398 T_{3}^{45} + \cdots - 769024 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6008))\).