Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8041,2,Mod(1,8041)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8041, 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("8041.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 8041 = 11 \cdot 17 \cdot 43 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 8041.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: | \(64.2077082653\) |
Analytic rank: | \(0\) |
Dimension: | \(78\) |
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 | −2.81193 | 3.07426 | 5.90697 | 1.69870 | −8.64463 | 0.0326130 | −10.9861 | 6.45110 | −4.77662 | ||||||||||||||||||
1.2 | −2.68205 | 1.14586 | 5.19341 | 0.981432 | −3.07326 | −5.03314 | −8.56488 | −1.68700 | −2.63225 | ||||||||||||||||||
1.3 | −2.66059 | −0.870455 | 5.07876 | −2.53728 | 2.31593 | 1.57807 | −8.19134 | −2.24231 | 6.75067 | ||||||||||||||||||
1.4 | −2.59406 | −1.64341 | 4.72913 | −1.77081 | 4.26309 | −2.35395 | −7.07951 | −0.299219 | 4.59358 | ||||||||||||||||||
1.5 | −2.55392 | 1.24868 | 4.52253 | 4.26646 | −3.18904 | −0.595052 | −6.44234 | −1.44079 | −10.8962 | ||||||||||||||||||
1.6 | −2.53120 | −2.76336 | 4.40698 | 3.58460 | 6.99462 | −0.727911 | −6.09254 | 4.63616 | −9.07334 | ||||||||||||||||||
1.7 | −2.36526 | −1.70691 | 3.59444 | −0.509718 | 4.03728 | 0.312769 | −3.77127 | −0.0864584 | 1.20561 | ||||||||||||||||||
1.8 | −2.35944 | 3.36081 | 3.56697 | −2.07702 | −7.92965 | −0.337523 | −3.69718 | 8.29507 | 4.90061 | ||||||||||||||||||
1.9 | −2.30463 | −0.694087 | 3.31131 | 3.08057 | 1.59961 | 3.70671 | −3.02207 | −2.51824 | −7.09957 | ||||||||||||||||||
1.10 | −2.28473 | −0.648502 | 3.22000 | −1.19117 | 1.48165 | 3.55276 | −2.78738 | −2.57944 | 2.72151 | ||||||||||||||||||
1.11 | −2.24742 | 2.50303 | 3.05092 | 1.29850 | −5.62537 | 3.89591 | −2.36186 | 3.26514 | −2.91829 | ||||||||||||||||||
1.12 | −2.05622 | −3.10600 | 2.22804 | −2.37865 | 6.38662 | −3.40012 | −0.468901 | 6.64724 | 4.89103 | ||||||||||||||||||
1.13 | −1.93837 | −3.28485 | 1.75726 | −1.73342 | 6.36724 | 2.63625 | 0.470510 | 7.79024 | 3.36000 | ||||||||||||||||||
1.14 | −1.93129 | −1.02169 | 1.72989 | 3.42042 | 1.97319 | −3.76946 | 0.521655 | −1.95614 | −6.60584 | ||||||||||||||||||
1.15 | −1.90003 | 1.99775 | 1.61013 | −3.13238 | −3.79579 | −0.279659 | 0.740768 | 0.990997 | 5.95163 | ||||||||||||||||||
1.16 | −1.80887 | 0.743622 | 1.27201 | 1.40249 | −1.34511 | −0.305829 | 1.31684 | −2.44703 | −2.53691 | ||||||||||||||||||
1.17 | −1.73352 | 2.13034 | 1.00511 | −4.15912 | −3.69300 | −1.81917 | 1.72467 | 1.53836 | 7.20994 | ||||||||||||||||||
1.18 | −1.70714 | 3.00412 | 0.914329 | 2.58179 | −5.12845 | −2.99512 | 1.85339 | 6.02473 | −4.40747 | ||||||||||||||||||
1.19 | −1.53655 | 0.0485562 | 0.360996 | −0.584069 | −0.0746091 | −1.23435 | 2.51842 | −2.99764 | 0.897453 | ||||||||||||||||||
1.20 | −1.52440 | −2.61179 | 0.323806 | 2.08851 | 3.98143 | 2.52784 | 2.55520 | 3.82147 | −3.18374 | ||||||||||||||||||
See all 78 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(11\) | \(-1\) |
\(17\) | \(1\) |
\(43\) | \(1\) |
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 | 8041.2.a.i | ✓ | 78 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
8041.2.a.i | ✓ | 78 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{78} - 7 T_{2}^{77} - 99 T_{2}^{76} + 787 T_{2}^{75} + 4520 T_{2}^{74} - 42288 T_{2}^{73} + \cdots + 2176 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8041))\).