Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8015,2,Mod(1,8015)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8015, 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("8015.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 8015 = 5 \cdot 7 \cdot 229 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 8015.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: | \(64.0000972201\) |
Analytic rank: | \(1\) |
Dimension: | \(49\) |
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 | −2.67776 | −3.22300 | 5.17039 | −1.00000 | 8.63041 | −1.00000 | −8.48955 | 7.38771 | 2.67776 | ||||||||||||||||||
1.2 | −2.66490 | 2.35676 | 5.10167 | −1.00000 | −6.28053 | −1.00000 | −8.26563 | 2.55434 | 2.66490 | ||||||||||||||||||
1.3 | −2.62540 | −0.161903 | 4.89272 | −1.00000 | 0.425059 | −1.00000 | −7.59456 | −2.97379 | 2.62540 | ||||||||||||||||||
1.4 | −2.56687 | −1.38440 | 4.58882 | −1.00000 | 3.55358 | −1.00000 | −6.64517 | −1.08344 | 2.56687 | ||||||||||||||||||
1.5 | −2.52338 | −1.42453 | 4.36744 | −1.00000 | 3.59462 | −1.00000 | −5.97394 | −0.970719 | 2.52338 | ||||||||||||||||||
1.6 | −2.33828 | 2.05266 | 3.46754 | −1.00000 | −4.79969 | −1.00000 | −3.43150 | 1.21342 | 2.33828 | ||||||||||||||||||
1.7 | −2.29229 | −2.84815 | 3.25461 | −1.00000 | 6.52878 | −1.00000 | −2.87593 | 5.11193 | 2.29229 | ||||||||||||||||||
1.8 | −2.08886 | −0.324198 | 2.36335 | −1.00000 | 0.677204 | −1.00000 | −0.758978 | −2.89490 | 2.08886 | ||||||||||||||||||
1.9 | −2.05694 | 0.523578 | 2.23099 | −1.00000 | −1.07697 | −1.00000 | −0.475131 | −2.72587 | 2.05694 | ||||||||||||||||||
1.10 | −1.92336 | −1.33966 | 1.69931 | −1.00000 | 2.57665 | −1.00000 | 0.578330 | −1.20531 | 1.92336 | ||||||||||||||||||
1.11 | −1.91964 | 0.294978 | 1.68500 | −1.00000 | −0.566250 | −1.00000 | 0.604679 | −2.91299 | 1.91964 | ||||||||||||||||||
1.12 | −1.80610 | 2.35882 | 1.26200 | −1.00000 | −4.26027 | −1.00000 | 1.33289 | 2.56402 | 1.80610 | ||||||||||||||||||
1.13 | −1.73153 | −2.59444 | 0.998212 | −1.00000 | 4.49236 | −1.00000 | 1.73463 | 3.73110 | 1.73153 | ||||||||||||||||||
1.14 | −1.56783 | 2.04149 | 0.458083 | −1.00000 | −3.20071 | −1.00000 | 2.41746 | 1.16770 | 1.56783 | ||||||||||||||||||
1.15 | −1.36146 | −0.153831 | −0.146418 | −1.00000 | 0.209435 | −1.00000 | 2.92227 | −2.97634 | 1.36146 | ||||||||||||||||||
1.16 | −1.23019 | 2.26268 | −0.486637 | −1.00000 | −2.78352 | −1.00000 | 3.05903 | 2.11970 | 1.23019 | ||||||||||||||||||
1.17 | −1.12638 | −2.96921 | −0.731259 | −1.00000 | 3.34447 | −1.00000 | 3.07645 | 5.81619 | 1.12638 | ||||||||||||||||||
1.18 | −0.978041 | −3.27758 | −1.04343 | −1.00000 | 3.20561 | −1.00000 | 2.97661 | 7.74254 | 0.978041 | ||||||||||||||||||
1.19 | −0.788296 | −1.23031 | −1.37859 | −1.00000 | 0.969852 | −1.00000 | 2.66333 | −1.48633 | 0.788296 | ||||||||||||||||||
1.20 | −0.736107 | −1.41103 | −1.45815 | −1.00000 | 1.03867 | −1.00000 | 2.54557 | −1.00899 | 0.736107 | ||||||||||||||||||
See all 49 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(5\) | \(1\) |
\(7\) | \(1\) |
\(229\) | \(1\) |
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 | 8015.2.a.k | ✓ | 49 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
8015.2.a.k | ✓ | 49 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8015))\):
\( T_{2}^{49} + 3 T_{2}^{48} - 69 T_{2}^{47} - 210 T_{2}^{46} + 2208 T_{2}^{45} + 6832 T_{2}^{44} + \cdots + 3536 \) |
\( T_{3}^{49} + 10 T_{3}^{48} - 43 T_{3}^{47} - 724 T_{3}^{46} + 161 T_{3}^{45} + 23919 T_{3}^{44} + \cdots + 1178624 \) |