Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8021,2,Mod(1,8021)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8021, 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("8021.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 8021 = 13 \cdot 617 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 8021.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: | \(64.0480074613\) |
Analytic rank: | \(1\) |
Dimension: | \(134\) |
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 | −2.76381 | −0.591714 | 5.63866 | 2.62273 | 1.63539 | 1.07036 | −10.0566 | −2.64987 | −7.24873 | ||||||||||||||||||
1.2 | −2.68511 | 2.15795 | 5.20984 | −0.637263 | −5.79434 | 0.252834 | −8.61877 | 1.65675 | 1.71112 | ||||||||||||||||||
1.3 | −2.59685 | −2.33995 | 4.74363 | −3.06624 | 6.07651 | 0.887424 | −7.12481 | 2.47538 | 7.96257 | ||||||||||||||||||
1.4 | −2.56606 | 0.368471 | 4.58465 | 3.51803 | −0.945517 | 1.16169 | −6.63237 | −2.86423 | −9.02747 | ||||||||||||||||||
1.5 | −2.56319 | 1.05402 | 4.56992 | 0.0841128 | −2.70164 | 2.43328 | −6.58720 | −1.88905 | −0.215597 | ||||||||||||||||||
1.6 | −2.55271 | −2.07589 | 4.51632 | 1.36107 | 5.29914 | −3.73550 | −6.42344 | 1.30931 | −3.47441 | ||||||||||||||||||
1.7 | −2.51883 | −2.43350 | 4.34452 | −0.221192 | 6.12959 | −1.84924 | −5.90546 | 2.92194 | 0.557145 | ||||||||||||||||||
1.8 | −2.48492 | −2.97194 | 4.17482 | 3.21951 | 7.38503 | 2.90175 | −5.40425 | 5.83242 | −8.00022 | ||||||||||||||||||
1.9 | −2.48308 | 1.22996 | 4.16571 | 3.96003 | −3.05409 | −1.52214 | −5.37763 | −1.48721 | −9.83308 | ||||||||||||||||||
1.10 | −2.45973 | −2.31536 | 4.05029 | −2.20524 | 5.69516 | −0.890619 | −5.04316 | 2.36088 | 5.42429 | ||||||||||||||||||
1.11 | −2.38388 | 0.0725376 | 3.68286 | −3.05924 | −0.172921 | −4.27004 | −4.01173 | −2.99474 | 7.29285 | ||||||||||||||||||
1.12 | −2.37815 | 2.24175 | 3.65561 | 2.50193 | −5.33123 | −3.50735 | −3.93730 | 2.02545 | −5.94998 | ||||||||||||||||||
1.13 | −2.36088 | 0.842254 | 3.57377 | −1.49212 | −1.98846 | −1.11078 | −3.71549 | −2.29061 | 3.52272 | ||||||||||||||||||
1.14 | −2.34085 | 2.34194 | 3.47957 | −2.42598 | −5.48214 | 1.51832 | −3.46345 | 2.48470 | 5.67886 | ||||||||||||||||||
1.15 | −2.26323 | 2.87488 | 3.12222 | 0.727126 | −6.50652 | 0.231070 | −2.53984 | 5.26494 | −1.64565 | ||||||||||||||||||
1.16 | −2.25892 | 1.96923 | 3.10273 | −2.33615 | −4.44833 | −1.78149 | −2.49097 | 0.877859 | 5.27718 | ||||||||||||||||||
1.17 | −2.22845 | −2.09787 | 2.96601 | −1.11575 | 4.67502 | −1.94623 | −2.15271 | 1.40108 | 2.48640 | ||||||||||||||||||
1.18 | −2.20714 | −0.230793 | 2.87148 | −0.451913 | 0.509393 | 5.18852 | −1.92347 | −2.94673 | 0.997437 | ||||||||||||||||||
1.19 | −2.13613 | −0.639809 | 2.56307 | 0.116719 | 1.36672 | −4.16794 | −1.20279 | −2.59064 | −0.249327 | ||||||||||||||||||
1.20 | −2.09018 | −2.33308 | 2.36887 | 1.26900 | 4.87656 | 3.93400 | −0.771000 | 2.44324 | −2.65245 | ||||||||||||||||||
See next 80 embeddings (of 134 total) |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(13\) | \(-1\) |
\(617\) | \(-1\) |
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 | 8021.2.a.a | ✓ | 134 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
8021.2.a.a | ✓ | 134 | 1.a | even | 1 | 1 | trivial |