Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8013,2,Mod(1,8013)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8013, 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("8013.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 8013 = 3 \cdot 2671 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 8013.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: | \(63.9841271397\) |
Analytic rank: | \(0\) |
Dimension: | \(129\) |
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 | −2.80990 | 1.00000 | 5.89553 | 3.04419 | −2.80990 | 3.35815 | −10.9460 | 1.00000 | −8.55386 | ||||||||||||||||||
1.2 | −2.75607 | 1.00000 | 5.59595 | 0.110089 | −2.75607 | 1.43985 | −9.91070 | 1.00000 | −0.303413 | ||||||||||||||||||
1.3 | −2.69445 | 1.00000 | 5.26005 | −1.49008 | −2.69445 | 1.50692 | −8.78405 | 1.00000 | 4.01496 | ||||||||||||||||||
1.4 | −2.69326 | 1.00000 | 5.25363 | −3.14797 | −2.69326 | 0.0430508 | −8.76286 | 1.00000 | 8.47829 | ||||||||||||||||||
1.5 | −2.62323 | 1.00000 | 4.88134 | −3.97507 | −2.62323 | 3.25996 | −7.55842 | 1.00000 | 10.4275 | ||||||||||||||||||
1.6 | −2.59970 | 1.00000 | 4.75846 | 1.02648 | −2.59970 | −4.07018 | −7.17119 | 1.00000 | −2.66854 | ||||||||||||||||||
1.7 | −2.59380 | 1.00000 | 4.72779 | 3.19035 | −2.59380 | −1.98667 | −7.07533 | 1.00000 | −8.27513 | ||||||||||||||||||
1.8 | −2.58900 | 1.00000 | 4.70290 | −3.13677 | −2.58900 | 5.18243 | −6.99780 | 1.00000 | 8.12109 | ||||||||||||||||||
1.9 | −2.52915 | 1.00000 | 4.39659 | 0.815749 | −2.52915 | 5.25378 | −6.06134 | 1.00000 | −2.06315 | ||||||||||||||||||
1.10 | −2.52474 | 1.00000 | 4.37429 | −3.85140 | −2.52474 | −2.38848 | −5.99445 | 1.00000 | 9.72375 | ||||||||||||||||||
1.11 | −2.51884 | 1.00000 | 4.34456 | 3.26876 | −2.51884 | 4.07622 | −5.90558 | 1.00000 | −8.23349 | ||||||||||||||||||
1.12 | −2.41705 | 1.00000 | 3.84214 | −2.77464 | −2.41705 | 0.900586 | −4.45255 | 1.00000 | 6.70646 | ||||||||||||||||||
1.13 | −2.41059 | 1.00000 | 3.81094 | −0.0291843 | −2.41059 | −4.28110 | −4.36543 | 1.00000 | 0.0703514 | ||||||||||||||||||
1.14 | −2.34305 | 1.00000 | 3.48991 | 4.17579 | −2.34305 | 2.00162 | −3.49093 | 1.00000 | −9.78411 | ||||||||||||||||||
1.15 | −2.27654 | 1.00000 | 3.18262 | −2.00184 | −2.27654 | 2.08885 | −2.69227 | 1.00000 | 4.55727 | ||||||||||||||||||
1.16 | −2.21092 | 1.00000 | 2.88816 | 1.51753 | −2.21092 | 0.0486600 | −1.96364 | 1.00000 | −3.35514 | ||||||||||||||||||
1.17 | −2.18651 | 1.00000 | 2.78083 | −0.652555 | −2.18651 | 2.68769 | −1.70730 | 1.00000 | 1.42682 | ||||||||||||||||||
1.18 | −2.17979 | 1.00000 | 2.75150 | 1.63083 | −2.17979 | 4.53329 | −1.63812 | 1.00000 | −3.55488 | ||||||||||||||||||
1.19 | −2.08790 | 1.00000 | 2.35934 | 0.999516 | −2.08790 | −2.01365 | −0.750272 | 1.00000 | −2.08689 | ||||||||||||||||||
1.20 | −2.06267 | 1.00000 | 2.25459 | −1.75345 | −2.06267 | −2.34820 | −0.525142 | 1.00000 | 3.61679 | ||||||||||||||||||
See next 80 embeddings (of 129 total) |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(3\) | \(-1\) |
\(2671\) | \(1\) |
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 | 8013.2.a.d | ✓ | 129 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
8013.2.a.d | ✓ | 129 | 1.a | even | 1 | 1 | trivial |