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: | \(1\) |
Dimension: | \(69\) |
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 | −2.81130 | −0.519316 | 5.90341 | −3.87319 | 1.45995 | 3.50752 | −10.9737 | −2.73031 | 10.8887 | ||||||||||||||||||
1.2 | −2.75821 | 0.843025 | 5.60771 | 3.43649 | −2.32524 | −0.344850 | −9.95082 | −2.28931 | −9.47854 | ||||||||||||||||||
1.3 | −2.75592 | 2.83072 | 5.59510 | −1.76173 | −7.80125 | 2.47832 | −9.90779 | 5.01300 | 4.85519 | ||||||||||||||||||
1.4 | −2.67122 | 2.62074 | 5.13539 | 0.949400 | −7.00056 | −4.03564 | −8.37532 | 3.86827 | −2.53605 | ||||||||||||||||||
1.5 | −2.56568 | −3.40788 | 4.58273 | 3.86072 | 8.74353 | 2.70727 | −6.62646 | 8.61362 | −9.90537 | ||||||||||||||||||
1.6 | −2.53420 | −1.82986 | 4.42217 | −1.56039 | 4.63723 | −0.999486 | −6.13827 | 0.348383 | 3.95435 | ||||||||||||||||||
1.7 | −2.47238 | −0.879695 | 4.11267 | −2.32585 | 2.17494 | 1.92542 | −5.22332 | −2.22614 | 5.75038 | ||||||||||||||||||
1.8 | −2.45687 | −0.886506 | 4.03623 | −0.806805 | 2.17803 | −3.77933 | −5.00275 | −2.21411 | 1.98222 | ||||||||||||||||||
1.9 | −2.42698 | 0.265461 | 3.89025 | 3.56663 | −0.644270 | −2.27175 | −4.58762 | −2.92953 | −8.65616 | ||||||||||||||||||
1.10 | −2.20984 | 0.750857 | 2.88340 | 0.114086 | −1.65928 | 4.65666 | −1.95217 | −2.43621 | −0.252113 | ||||||||||||||||||
1.11 | −2.20191 | −3.02493 | 2.84839 | −2.51341 | 6.66061 | 0.0943373 | −1.86807 | 6.15021 | 5.53430 | ||||||||||||||||||
1.12 | −2.12129 | 1.35005 | 2.49986 | −1.60395 | −2.86383 | −4.53308 | −1.06034 | −1.17738 | 3.40244 | ||||||||||||||||||
1.13 | −2.10431 | −0.209589 | 2.42810 | 1.85985 | 0.441039 | 3.06120 | −0.900860 | −2.95607 | −3.91370 | ||||||||||||||||||
1.14 | −2.02215 | 2.02595 | 2.08909 | −3.90857 | −4.09678 | −0.616656 | −0.180153 | 1.10448 | 7.90371 | ||||||||||||||||||
1.15 | −1.87945 | −2.61165 | 1.53233 | −1.23543 | 4.90847 | −3.71692 | 0.878969 | 3.82074 | 2.32194 | ||||||||||||||||||
1.16 | −1.85886 | 3.31260 | 1.45535 | 0.943187 | −6.15766 | −1.28029 | 1.01243 | 7.97335 | −1.75325 | ||||||||||||||||||
1.17 | −1.71913 | 3.18493 | 0.955421 | −4.00741 | −5.47533 | −0.822562 | 1.79577 | 7.14381 | 6.88927 | ||||||||||||||||||
1.18 | −1.69256 | −2.67363 | 0.864767 | 3.80511 | 4.52529 | −4.13916 | 1.92145 | 4.14830 | −6.44038 | ||||||||||||||||||
1.19 | −1.67732 | −1.15146 | 0.813414 | 0.864739 | 1.93138 | 4.27290 | 1.99029 | −1.67413 | −1.45045 | ||||||||||||||||||
1.20 | −1.66399 | 1.84747 | 0.768874 | 1.24336 | −3.07417 | 0.0311850 | 2.04859 | 0.413132 | −2.06894 | ||||||||||||||||||
See all 69 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.g | ✓ | 69 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
8041.2.a.g | ✓ | 69 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{69} + 11 T_{2}^{68} - 41 T_{2}^{67} - 869 T_{2}^{66} - 346 T_{2}^{65} + 31814 T_{2}^{64} + \cdots + 5248 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8041))\).