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: | \(0\) |
Dimension: | \(67\) |
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 | −2.79998 | −2.08197 | 5.83989 | 1.00000 | 5.82947 | −1.00000 | −10.7516 | 1.33460 | −2.79998 | ||||||||||||||||||
1.2 | −2.72902 | 1.60790 | 5.44753 | 1.00000 | −4.38799 | −1.00000 | −9.40835 | −0.414653 | −2.72902 | ||||||||||||||||||
1.3 | −2.66598 | 2.60914 | 5.10744 | 1.00000 | −6.95591 | −1.00000 | −8.28438 | 3.80761 | −2.66598 | ||||||||||||||||||
1.4 | −2.55168 | −1.07267 | 4.51109 | 1.00000 | 2.73711 | −1.00000 | −6.40752 | −1.84938 | −2.55168 | ||||||||||||||||||
1.5 | −2.54658 | −2.85949 | 4.48508 | 1.00000 | 7.28192 | −1.00000 | −6.32845 | 5.17667 | −2.54658 | ||||||||||||||||||
1.6 | −2.53656 | −2.78475 | 4.43415 | 1.00000 | 7.06368 | −1.00000 | −6.17438 | 4.75481 | −2.53656 | ||||||||||||||||||
1.7 | −2.40339 | −1.71467 | 3.77630 | 1.00000 | 4.12102 | −1.00000 | −4.26915 | −0.0599106 | −2.40339 | ||||||||||||||||||
1.8 | −2.30662 | 1.34001 | 3.32049 | 1.00000 | −3.09089 | −1.00000 | −3.04587 | −1.20437 | −2.30662 | ||||||||||||||||||
1.9 | −2.25277 | 3.29965 | 3.07495 | 1.00000 | −7.43334 | −1.00000 | −2.42162 | 7.88769 | −2.25277 | ||||||||||||||||||
1.10 | −2.05677 | 1.58909 | 2.23031 | 1.00000 | −3.26839 | −1.00000 | −0.473688 | −0.474802 | −2.05677 | ||||||||||||||||||
1.11 | −1.94503 | 2.18285 | 1.78313 | 1.00000 | −4.24571 | −1.00000 | 0.421811 | 1.76484 | −1.94503 | ||||||||||||||||||
1.12 | −1.89706 | −1.32001 | 1.59885 | 1.00000 | 2.50414 | −1.00000 | 0.761013 | −1.25758 | −1.89706 | ||||||||||||||||||
1.13 | −1.83785 | 1.36680 | 1.37770 | 1.00000 | −2.51199 | −1.00000 | 1.14369 | −1.13184 | −1.83785 | ||||||||||||||||||
1.14 | −1.82956 | −1.24341 | 1.34731 | 1.00000 | 2.27489 | −1.00000 | 1.19414 | −1.45394 | −1.82956 | ||||||||||||||||||
1.15 | −1.80341 | 0.567965 | 1.25230 | 1.00000 | −1.02428 | −1.00000 | 1.34841 | −2.67742 | −1.80341 | ||||||||||||||||||
1.16 | −1.76232 | −3.28691 | 1.10578 | 1.00000 | 5.79259 | −1.00000 | 1.57590 | 7.80377 | −1.76232 | ||||||||||||||||||
1.17 | −1.76209 | −0.304110 | 1.10496 | 1.00000 | 0.535869 | −1.00000 | 1.57714 | −2.90752 | −1.76209 | ||||||||||||||||||
1.18 | −1.50021 | 0.266806 | 0.250644 | 1.00000 | −0.400267 | −1.00000 | 2.62441 | −2.92881 | −1.50021 | ||||||||||||||||||
1.19 | −1.44039 | 2.88342 | 0.0747197 | 1.00000 | −4.15325 | −1.00000 | 2.77315 | 5.31413 | −1.44039 | ||||||||||||||||||
1.20 | −1.42994 | −3.33526 | 0.0447232 | 1.00000 | 4.76921 | −1.00000 | 2.79592 | 8.12394 | −1.42994 | ||||||||||||||||||
See all 67 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.m | ✓ | 67 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
8015.2.a.m | ✓ | 67 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8015))\):
\( T_{2}^{67} - 3 T_{2}^{66} - 99 T_{2}^{65} + 298 T_{2}^{64} + 4651 T_{2}^{63} - 14049 T_{2}^{62} + \cdots - 2440 \) |
\( T_{3}^{67} - 149 T_{3}^{65} - 2 T_{3}^{64} + 10535 T_{3}^{63} + 281 T_{3}^{62} - 470399 T_{3}^{61} + \cdots + 4713433856 \) |