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: | \(1\) |
Dimension: | \(142\) |
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 | −2.78532 | −0.218444 | 5.75799 | 3.30077 | 0.608437 | −0.196953 | −10.4672 | −2.95228 | −9.19368 | ||||||||||||||||||
1.2 | −2.78511 | −1.73890 | 5.75683 | −2.18885 | 4.84303 | 2.36535 | −10.4632 | 0.0237834 | 6.09618 | ||||||||||||||||||
1.3 | −2.74745 | −2.94528 | 5.54847 | 1.95253 | 8.09201 | 3.66595 | −9.74923 | 5.67469 | −5.36448 | ||||||||||||||||||
1.4 | −2.73051 | 1.50429 | 5.45570 | −3.60561 | −4.10747 | 2.19221 | −9.43585 | −0.737125 | 9.84515 | ||||||||||||||||||
1.5 | −2.66381 | 1.33027 | 5.09589 | −0.440221 | −3.54360 | 2.49453 | −8.24687 | −1.23037 | 1.17267 | ||||||||||||||||||
1.6 | −2.65986 | −2.58294 | 5.07483 | 0.599360 | 6.87024 | −3.90786 | −8.17861 | 3.67156 | −1.59421 | ||||||||||||||||||
1.7 | −2.65232 | 2.65488 | 5.03481 | −0.675897 | −7.04159 | −2.02772 | −8.04928 | 4.04839 | 1.79269 | ||||||||||||||||||
1.8 | −2.56842 | 0.368799 | 4.59680 | −0.479572 | −0.947232 | 4.01594 | −6.66967 | −2.86399 | 1.23174 | ||||||||||||||||||
1.9 | −2.55403 | 1.62979 | 4.52309 | 1.02400 | −4.16254 | −2.90866 | −6.44404 | −0.343777 | −2.61533 | ||||||||||||||||||
1.10 | −2.52587 | −3.25300 | 4.38004 | −2.46689 | 8.21667 | −3.38874 | −6.01169 | 7.58202 | 6.23106 | ||||||||||||||||||
1.11 | −2.52091 | −0.134089 | 4.35498 | 2.66858 | 0.338025 | −2.31114 | −5.93668 | −2.98202 | −6.72724 | ||||||||||||||||||
1.12 | −2.51095 | −2.93259 | 4.30485 | −3.06715 | 7.36357 | 3.82085 | −5.78735 | 5.60007 | 7.70146 | ||||||||||||||||||
1.13 | −2.50524 | −0.199696 | 4.27622 | −1.81098 | 0.500286 | −1.67033 | −5.70247 | −2.96012 | 4.53694 | ||||||||||||||||||
1.14 | −2.45771 | 2.98686 | 4.04034 | −1.52748 | −7.34083 | 5.18217 | −5.01456 | 5.92132 | 3.75410 | ||||||||||||||||||
1.15 | −2.43267 | −0.839960 | 3.91786 | −3.64522 | 2.04334 | −2.93716 | −4.66552 | −2.29447 | 8.86761 | ||||||||||||||||||
1.16 | −2.41564 | 2.60884 | 3.83533 | 1.90553 | −6.30203 | −0.0638029 | −4.43349 | 3.80607 | −4.60307 | ||||||||||||||||||
1.17 | −2.27884 | −1.20174 | 3.19313 | −1.58295 | 2.73859 | 1.25281 | −2.71895 | −1.55581 | 3.60730 | ||||||||||||||||||
1.18 | −2.23848 | 1.32587 | 3.01078 | 3.90699 | −2.96792 | −1.03210 | −2.26261 | −1.24208 | −8.74571 | ||||||||||||||||||
1.19 | −2.23735 | −0.938044 | 3.00574 | 3.28954 | 2.09873 | 1.35040 | −2.25019 | −2.12007 | −7.35985 | ||||||||||||||||||
1.20 | −2.15511 | 1.98414 | 2.64449 | −3.20612 | −4.27605 | 1.44494 | −1.38895 | 0.936828 | 6.90953 | ||||||||||||||||||
See next 80 embeddings (of 142 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.b | ✓ | 142 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
8047.2.a.b | ✓ | 142 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{142} + 13 T_{2}^{141} - 122 T_{2}^{140} - 2288 T_{2}^{139} + 4951 T_{2}^{138} + \cdots + 33720492213 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8047))\).