Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8039,2,Mod(1,8039)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8039, 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("8039.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 8039 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 8039.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: | \(64.1917381849\) |
Analytic rank: | \(1\) |
Dimension: | \(279\) |
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 | −2.75826 | 0.543669 | 5.60799 | 4.07368 | −1.49958 | −1.45954 | −9.95176 | −2.70442 | −11.2363 | ||||||||||||||||||
1.2 | −2.72307 | −0.226737 | 5.41513 | 0.679891 | 0.617421 | −1.82848 | −9.29967 | −2.94859 | −1.85139 | ||||||||||||||||||
1.3 | −2.72088 | 2.51741 | 5.40317 | −2.91261 | −6.84957 | −4.81699 | −9.25961 | 3.33737 | 7.92486 | ||||||||||||||||||
1.4 | −2.69202 | −2.07215 | 5.24695 | 2.43355 | 5.57825 | −0.413821 | −8.74083 | 1.29379 | −6.55116 | ||||||||||||||||||
1.5 | −2.68146 | 1.09614 | 5.19020 | 0.161648 | −2.93925 | 3.85315 | −8.55439 | −1.79847 | −0.433452 | ||||||||||||||||||
1.6 | −2.66452 | 2.42009 | 5.09965 | 1.98258 | −6.44836 | 1.42101 | −8.25906 | 2.85681 | −5.28261 | ||||||||||||||||||
1.7 | −2.65513 | −1.36605 | 5.04972 | 2.66534 | 3.62703 | −0.858733 | −8.09741 | −1.13392 | −7.07683 | ||||||||||||||||||
1.8 | −2.65123 | 3.28882 | 5.02904 | −0.364681 | −8.71943 | −0.0824305 | −8.03069 | 7.81633 | 0.966855 | ||||||||||||||||||
1.9 | −2.64944 | 0.544800 | 5.01955 | −2.56078 | −1.44342 | −0.780813 | −8.00012 | −2.70319 | 6.78465 | ||||||||||||||||||
1.10 | −2.64457 | −1.20149 | 4.99374 | −2.42138 | 3.17741 | 3.07885 | −7.91715 | −1.55643 | 6.40351 | ||||||||||||||||||
1.11 | −2.62410 | 1.81984 | 4.88588 | −0.731341 | −4.77543 | −1.89048 | −7.57282 | 0.311808 | 1.91911 | ||||||||||||||||||
1.12 | −2.61158 | −1.07466 | 4.82033 | −1.60450 | 2.80656 | −4.85582 | −7.36550 | −1.84510 | 4.19028 | ||||||||||||||||||
1.13 | −2.60916 | −2.67838 | 4.80773 | −2.56224 | 6.98832 | −1.44552 | −7.32584 | 4.17369 | 6.68531 | ||||||||||||||||||
1.14 | −2.59800 | −2.63260 | 4.74962 | 0.476137 | 6.83949 | 1.24122 | −7.14351 | 3.93057 | −1.23700 | ||||||||||||||||||
1.15 | −2.55932 | −0.558527 | 4.55013 | 0.514137 | 1.42945 | 2.79738 | −6.52662 | −2.68805 | −1.31584 | ||||||||||||||||||
1.16 | −2.55424 | 2.06857 | 4.52414 | −3.35787 | −5.28363 | 0.600266 | −6.44725 | 1.27899 | 8.57679 | ||||||||||||||||||
1.17 | −2.54048 | −0.00862555 | 4.45405 | 1.93048 | 0.0219131 | 2.12668 | −6.23447 | −2.99993 | −4.90434 | ||||||||||||||||||
1.18 | −2.53338 | 3.09967 | 4.41800 | 3.04096 | −7.85264 | −3.65810 | −6.12571 | 6.60796 | −7.70391 | ||||||||||||||||||
1.19 | −2.51220 | −0.224247 | 4.31117 | 3.23846 | 0.563355 | −3.25585 | −5.80612 | −2.94971 | −8.13568 | ||||||||||||||||||
1.20 | −2.47449 | 2.58655 | 4.12308 | 0.927265 | −6.40039 | −0.828816 | −5.25355 | 3.69026 | −2.29451 | ||||||||||||||||||
See next 80 embeddings (of 279 total) |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(8039\) | \(1\) |
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 | 8039.2.a.a | ✓ | 279 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
8039.2.a.a | ✓ | 279 | 1.a | even | 1 | 1 | trivial |