Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8011,2,Mod(1,8011)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8011, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("8011.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 8011 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 8011.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: | \(63.9681570592\) |
Analytic rank: | \(1\) |
Dimension: | \(309\) |
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 | −2.81833 | 1.84881 | 5.94296 | −3.66460 | −5.21056 | −0.909168 | −11.1126 | 0.418106 | 10.3280 | ||||||||||||||||||
1.2 | −2.78702 | −1.95051 | 5.76751 | −0.484177 | 5.43612 | 0.734493 | −10.5001 | 0.804484 | 1.34941 | ||||||||||||||||||
1.3 | −2.76917 | 0.105018 | 5.66833 | −1.01486 | −0.290815 | 4.45070 | −10.1582 | −2.98897 | 2.81033 | ||||||||||||||||||
1.4 | −2.75364 | −0.00102412 | 5.58256 | −3.57698 | 0.00282007 | 1.05818 | −9.86509 | −3.00000 | 9.84973 | ||||||||||||||||||
1.5 | −2.74707 | 2.55044 | 5.54642 | 3.00412 | −7.00626 | 1.74411 | −9.74228 | 3.50477 | −8.25255 | ||||||||||||||||||
1.6 | −2.74693 | 2.61520 | 5.54560 | −1.49229 | −7.18376 | 4.10131 | −9.73951 | 3.83927 | 4.09920 | ||||||||||||||||||
1.7 | −2.74292 | −0.194612 | 5.52364 | 4.36086 | 0.533805 | −1.38725 | −9.66507 | −2.96213 | −11.9615 | ||||||||||||||||||
1.8 | −2.69623 | −0.432527 | 5.26965 | 2.72420 | 1.16619 | −0.628855 | −8.81573 | −2.81292 | −7.34507 | ||||||||||||||||||
1.9 | −2.68837 | −2.21810 | 5.22734 | 1.47525 | 5.96309 | −1.22716 | −8.67630 | 1.91998 | −3.96602 | ||||||||||||||||||
1.10 | −2.67443 | −1.89818 | 5.15257 | −2.43152 | 5.07654 | 2.86566 | −8.43133 | 0.603070 | 6.50294 | ||||||||||||||||||
1.11 | −2.67381 | −2.54689 | 5.14927 | −3.71609 | 6.80992 | 3.32085 | −8.42057 | 3.48667 | 9.93612 | ||||||||||||||||||
1.12 | −2.67151 | −0.346621 | 5.13696 | 1.63571 | 0.926001 | −0.275244 | −8.38042 | −2.87985 | −4.36981 | ||||||||||||||||||
1.13 | −2.65734 | −0.429161 | 5.06145 | 2.18236 | 1.14043 | 3.77949 | −8.13530 | −2.81582 | −5.79927 | ||||||||||||||||||
1.14 | −2.64340 | 2.84192 | 4.98759 | 0.696294 | −7.51235 | −3.08264 | −7.89741 | 5.07651 | −1.84059 | ||||||||||||||||||
1.15 | −2.61747 | −3.23549 | 4.85115 | −2.81151 | 8.46879 | −1.74803 | −7.46281 | 7.46837 | 7.35905 | ||||||||||||||||||
1.16 | −2.61660 | −0.249789 | 4.84659 | −1.85842 | 0.653599 | −0.258344 | −7.44839 | −2.93761 | 4.86274 | ||||||||||||||||||
1.17 | −2.58476 | −1.81476 | 4.68100 | 2.17232 | 4.69073 | −4.47910 | −6.92974 | 0.293363 | −5.61492 | ||||||||||||||||||
1.18 | −2.57419 | 1.43469 | 4.62646 | 0.684490 | −3.69317 | 3.09372 | −6.76100 | −0.941663 | −1.76201 | ||||||||||||||||||
1.19 | −2.56976 | 2.00808 | 4.60368 | −0.878358 | −5.16030 | 1.06025 | −6.69084 | 1.03240 | 2.25717 | ||||||||||||||||||
1.20 | −2.56050 | 2.53687 | 4.55617 | −3.00342 | −6.49566 | 2.39644 | −6.54508 | 3.43571 | 7.69027 | ||||||||||||||||||
See next 80 embeddings (of 309 total) |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(8011\) | \(1\) |
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 | 8011.2.a.a | ✓ | 309 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
8011.2.a.a | ✓ | 309 | 1.a | even | 1 | 1 | trivial |