Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8047,2,Mod(1,8047)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8047, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("8047.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 8047 = 13 \cdot 619 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 8047.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: | \(64.2556185065\) |
Analytic rank: | \(1\) |
Dimension: | \(151\) |
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 | −2.80247 | −1.33128 | 5.85385 | −2.29720 | 3.73087 | −2.08383 | −10.8003 | −1.22769 | 6.43783 | ||||||||||||||||||
1.2 | −2.76685 | 2.61084 | 5.65545 | 1.02194 | −7.22381 | 1.46189 | −10.1141 | 3.81650 | −2.82754 | ||||||||||||||||||
1.3 | −2.75813 | −3.14715 | 5.60727 | −3.12480 | 8.68025 | −2.02299 | −9.94932 | 6.90456 | 8.61861 | ||||||||||||||||||
1.4 | −2.73460 | 0.234090 | 5.47804 | −4.22814 | −0.640142 | 4.15879 | −9.51105 | −2.94520 | 11.5623 | ||||||||||||||||||
1.5 | −2.70117 | −0.798295 | 5.29633 | 1.50406 | 2.15633 | −4.14726 | −8.90395 | −2.36273 | −4.06273 | ||||||||||||||||||
1.6 | −2.69703 | 2.83475 | 5.27397 | −3.14903 | −7.64541 | 1.25916 | −8.82999 | 5.03583 | 8.49303 | ||||||||||||||||||
1.7 | −2.66878 | 1.69053 | 5.12237 | −3.40805 | −4.51165 | −4.86127 | −8.33291 | −0.142111 | 9.09532 | ||||||||||||||||||
1.8 | −2.66015 | −2.71534 | 5.07642 | 1.80330 | 7.22323 | −0.0430519 | −8.18376 | 4.37308 | −4.79706 | ||||||||||||||||||
1.9 | −2.65702 | 1.31176 | 5.05973 | 3.40250 | −3.48538 | 2.98608 | −8.12975 | −1.27927 | −9.04051 | ||||||||||||||||||
1.10 | −2.62707 | −2.08646 | 4.90149 | 1.65471 | 5.48127 | −3.37709 | −7.62241 | 1.35330 | −4.34704 | ||||||||||||||||||
1.11 | −2.59872 | 0.688709 | 4.75335 | −1.04737 | −1.78976 | 2.76372 | −7.15519 | −2.52568 | 2.72181 | ||||||||||||||||||
1.12 | −2.58760 | −0.469727 | 4.69569 | −2.90773 | 1.21547 | 0.483013 | −6.97536 | −2.77936 | 7.52406 | ||||||||||||||||||
1.13 | −2.53026 | −1.57653 | 4.40222 | −1.20027 | 3.98902 | 4.10699 | −6.07823 | −0.514564 | 3.03700 | ||||||||||||||||||
1.14 | −2.42822 | 2.05739 | 3.89625 | 1.53452 | −4.99579 | 1.53522 | −4.60451 | 1.23285 | −3.72615 | ||||||||||||||||||
1.15 | −2.37031 | 2.63830 | 3.61835 | 3.06059 | −6.25357 | −2.14204 | −3.83599 | 3.96060 | −7.25455 | ||||||||||||||||||
1.16 | −2.36725 | 0.254210 | 3.60387 | 1.17067 | −0.601779 | −2.57756 | −3.79676 | −2.93538 | −2.77126 | ||||||||||||||||||
1.17 | −2.34671 | 2.23237 | 3.50704 | −1.50112 | −5.23873 | −4.09520 | −3.53658 | 1.98350 | 3.52269 | ||||||||||||||||||
1.18 | −2.30742 | −2.51530 | 3.32419 | −4.19325 | 5.80386 | 2.00418 | −3.05546 | 3.32673 | 9.67559 | ||||||||||||||||||
1.19 | −2.28288 | −3.31283 | 3.21152 | 2.87679 | 7.56278 | −0.777401 | −2.76575 | 7.97486 | −6.56735 | ||||||||||||||||||
1.20 | −2.25853 | −0.662024 | 3.10095 | 2.15399 | 1.49520 | 0.474761 | −2.48652 | −2.56172 | −4.86485 | ||||||||||||||||||
See next 80 embeddings (of 151 total) |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(13\) | \(1\) |
\(619\) | \(1\) |
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 | 8047.2.a.c | ✓ | 151 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
8047.2.a.c | ✓ | 151 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{151} + 13 T_{2}^{150} - 142 T_{2}^{149} - 2548 T_{2}^{148} + 7532 T_{2}^{147} + 242111 T_{2}^{146} + \cdots - 89773312 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8047))\).