Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8045,2,Mod(1,8045)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8045, 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("8045.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 8045 = 5 \cdot 1609 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 8045.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: | \(64.2396484261\) |
Analytic rank: | \(1\) |
Dimension: | \(126\) |
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 | −2.77015 | 2.48053 | 5.67370 | −1.00000 | −6.87144 | 2.38067 | −10.1767 | 3.15305 | 2.77015 | ||||||||||||||||||
1.2 | −2.68479 | −1.35137 | 5.20811 | −1.00000 | 3.62815 | −0.768058 | −8.61310 | −1.17380 | 2.68479 | ||||||||||||||||||
1.3 | −2.59235 | −0.476573 | 4.72028 | −1.00000 | 1.23544 | −4.74079 | −7.05191 | −2.77288 | 2.59235 | ||||||||||||||||||
1.4 | −2.58262 | 2.30051 | 4.66995 | −1.00000 | −5.94134 | −3.25366 | −6.89548 | 2.29232 | 2.58262 | ||||||||||||||||||
1.5 | −2.57758 | 3.01279 | 4.64389 | −1.00000 | −7.76569 | −3.99315 | −6.81483 | 6.07689 | 2.57758 | ||||||||||||||||||
1.6 | −2.55817 | −0.0316431 | 4.54425 | −1.00000 | 0.0809484 | 0.209260 | −6.50863 | −2.99900 | 2.55817 | ||||||||||||||||||
1.7 | −2.52674 | −0.799891 | 4.38441 | −1.00000 | 2.02112 | −0.269880 | −6.02477 | −2.36017 | 2.52674 | ||||||||||||||||||
1.8 | −2.51572 | −0.0214177 | 4.32884 | −1.00000 | 0.0538809 | 1.26771 | −5.85870 | −2.99954 | 2.51572 | ||||||||||||||||||
1.9 | −2.44573 | 0.659690 | 3.98158 | −1.00000 | −1.61342 | −0.125305 | −4.84641 | −2.56481 | 2.44573 | ||||||||||||||||||
1.10 | −2.44115 | 2.21626 | 3.95922 | −1.00000 | −5.41023 | 1.20932 | −4.78275 | 1.91181 | 2.44115 | ||||||||||||||||||
1.11 | −2.40759 | −3.16484 | 3.79648 | −1.00000 | 7.61962 | −1.46420 | −4.32518 | 7.01619 | 2.40759 | ||||||||||||||||||
1.12 | −2.37056 | −2.51302 | 3.61953 | −1.00000 | 5.95725 | 3.42592 | −3.83919 | 3.31526 | 2.37056 | ||||||||||||||||||
1.13 | −2.28169 | −1.33238 | 3.20613 | −1.00000 | 3.04008 | −1.04082 | −2.75203 | −1.22477 | 2.28169 | ||||||||||||||||||
1.14 | −2.25161 | −2.78948 | 3.06974 | −1.00000 | 6.28081 | 1.00140 | −2.40864 | 4.78119 | 2.25161 | ||||||||||||||||||
1.15 | −2.22019 | 1.33183 | 2.92925 | −1.00000 | −2.95691 | 2.14660 | −2.06312 | −1.22624 | 2.22019 | ||||||||||||||||||
1.16 | −2.20326 | −1.91505 | 2.85434 | −1.00000 | 4.21934 | 2.75796 | −1.88232 | 0.667415 | 2.20326 | ||||||||||||||||||
1.17 | −2.11951 | 0.398701 | 2.49230 | −1.00000 | −0.845049 | −4.30325 | −1.04344 | −2.84104 | 2.11951 | ||||||||||||||||||
1.18 | −2.05998 | 0.444613 | 2.24351 | −1.00000 | −0.915893 | 4.66376 | −0.501618 | −2.80232 | 2.05998 | ||||||||||||||||||
1.19 | −2.03743 | 2.30287 | 2.15111 | −1.00000 | −4.69193 | −3.80447 | −0.307877 | 2.30320 | 2.03743 | ||||||||||||||||||
1.20 | −2.02597 | −2.39891 | 2.10454 | −1.00000 | 4.86012 | −3.38637 | −0.211794 | 2.75479 | 2.02597 | ||||||||||||||||||
See next 80 embeddings (of 126 total) |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(5\) | \(1\) |
\(1609\) | \(1\) |
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 | 8045.2.a.b | ✓ | 126 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
8045.2.a.b | ✓ | 126 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{126} - 5 T_{2}^{125} - 168 T_{2}^{124} + 871 T_{2}^{123} + 13679 T_{2}^{122} - 73797 T_{2}^{121} + \cdots - 35354521 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8045))\).