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: | \(0\) |
Dimension: | \(168\) |
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 | −2.81579 | 1.75700 | 5.92868 | −0.878024 | −4.94733 | −1.13202 | −11.0623 | 0.0870368 | 2.47233 | ||||||||||||||||||
1.2 | −2.76928 | −1.56618 | 5.66891 | −1.99910 | 4.33718 | −3.75052 | −10.1602 | −0.547092 | 5.53606 | ||||||||||||||||||
1.3 | −2.76470 | 3.06859 | 5.64359 | 2.61743 | −8.48374 | 4.52897 | −10.0734 | 6.41624 | −7.23642 | ||||||||||||||||||
1.4 | −2.69715 | 2.56324 | 5.27461 | 3.51333 | −6.91344 | −4.43417 | −8.83210 | 3.57021 | −9.47597 | ||||||||||||||||||
1.5 | −2.59593 | −2.11181 | 4.73887 | −0.212258 | 5.48213 | 0.982044 | −7.10991 | 1.45976 | 0.551008 | ||||||||||||||||||
1.6 | −2.59413 | −3.39586 | 4.72950 | 2.81515 | 8.80929 | −0.0603743 | −7.08068 | 8.53185 | −7.30287 | ||||||||||||||||||
1.7 | −2.58760 | −0.932881 | 4.69570 | 1.35951 | 2.41393 | 0.312733 | −6.97540 | −2.12973 | −3.51787 | ||||||||||||||||||
1.8 | −2.56793 | 0.140851 | 4.59428 | −0.0428559 | −0.361697 | −5.13541 | −6.66194 | −2.98016 | 0.110051 | ||||||||||||||||||
1.9 | −2.56640 | 1.62937 | 4.58642 | 4.01357 | −4.18162 | 3.25555 | −6.63781 | −0.345155 | −10.3004 | ||||||||||||||||||
1.10 | −2.55763 | 0.266085 | 4.54145 | −4.02410 | −0.680547 | −2.52095 | −6.50007 | −2.92920 | 10.2921 | ||||||||||||||||||
1.11 | −2.52416 | 3.25012 | 4.37136 | −2.90640 | −8.20382 | 1.38708 | −5.98570 | 7.56330 | 7.33621 | ||||||||||||||||||
1.12 | −2.51705 | 1.87666 | 4.33553 | 0.537447 | −4.72364 | 0.939568 | −5.87865 | 0.521853 | −1.35278 | ||||||||||||||||||
1.13 | −2.51491 | 3.41977 | 4.32478 | −1.23152 | −8.60042 | −3.67991 | −5.84661 | 8.69483 | 3.09716 | ||||||||||||||||||
1.14 | −2.50460 | −2.26782 | 4.27304 | 0.233077 | 5.68000 | −0.679959 | −5.69306 | 2.14302 | −0.583765 | ||||||||||||||||||
1.15 | −2.43925 | −2.47662 | 3.94994 | 4.18680 | 6.04110 | 3.85857 | −4.75640 | 3.13366 | −10.2127 | ||||||||||||||||||
1.16 | −2.43547 | −1.20585 | 3.93153 | 2.83456 | 2.93681 | −2.65946 | −4.70420 | −1.54594 | −6.90350 | ||||||||||||||||||
1.17 | −2.39466 | −2.35552 | 3.73441 | 0.950970 | 5.64068 | −0.373318 | −4.15331 | 2.54848 | −2.27725 | ||||||||||||||||||
1.18 | −2.38728 | 1.74228 | 3.69910 | −3.20056 | −4.15931 | −0.986874 | −4.05622 | 0.0355448 | 7.64064 | ||||||||||||||||||
1.19 | −2.35312 | −0.357062 | 3.53719 | 1.67497 | 0.840212 | 4.28147 | −3.61721 | −2.87251 | −3.94141 | ||||||||||||||||||
1.20 | −2.33973 | −0.0830321 | 3.47431 | −2.47504 | 0.194272 | 1.67237 | −3.44949 | −2.99311 | 5.79091 | ||||||||||||||||||
See next 80 embeddings (of 168 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.e | ✓ | 168 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
8047.2.a.e | ✓ | 168 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{168} - 11 T_{2}^{167} - 198 T_{2}^{166} + 2598 T_{2}^{165} + 17984 T_{2}^{164} + \cdots - 21530657600 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8047))\).