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: | \(62\) |
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 | −2.78055 | 0.298033 | 5.73146 | −1.00000 | −0.828697 | −1.00000 | −10.3755 | −2.91118 | 2.78055 | ||||||||||||||||||
1.2 | −2.65622 | −1.51381 | 5.05553 | −1.00000 | 4.02102 | −1.00000 | −8.11618 | −0.708384 | 2.65622 | ||||||||||||||||||
1.3 | −2.61699 | 0.399813 | 4.84863 | −1.00000 | −1.04630 | −1.00000 | −7.45482 | −2.84015 | 2.61699 | ||||||||||||||||||
1.4 | −2.48664 | 3.28438 | 4.18340 | −1.00000 | −8.16709 | −1.00000 | −5.42933 | 7.78717 | 2.48664 | ||||||||||||||||||
1.5 | −2.41590 | 1.84179 | 3.83658 | −1.00000 | −4.44959 | −1.00000 | −4.43701 | 0.392199 | 2.41590 | ||||||||||||||||||
1.6 | −2.35302 | 1.42181 | 3.53669 | −1.00000 | −3.34553 | −1.00000 | −3.61585 | −0.978466 | 2.35302 | ||||||||||||||||||
1.7 | −2.28892 | −1.23476 | 3.23914 | −1.00000 | 2.82625 | −1.00000 | −2.83628 | −1.47538 | 2.28892 | ||||||||||||||||||
1.8 | −2.23328 | 1.54114 | 2.98754 | −1.00000 | −3.44181 | −1.00000 | −2.20546 | −0.624877 | 2.23328 | ||||||||||||||||||
1.9 | −2.20228 | −3.01220 | 2.85005 | −1.00000 | 6.63371 | −1.00000 | −1.87206 | 6.07333 | 2.20228 | ||||||||||||||||||
1.10 | −2.17211 | 3.11306 | 2.71804 | −1.00000 | −6.76191 | −1.00000 | −1.55967 | 6.69117 | 2.17211 | ||||||||||||||||||
1.11 | −2.12264 | −2.04208 | 2.50559 | −1.00000 | 4.33460 | −1.00000 | −1.07319 | 1.17010 | 2.12264 | ||||||||||||||||||
1.12 | −1.90469 | −0.920285 | 1.62783 | −1.00000 | 1.75286 | −1.00000 | 0.708859 | −2.15308 | 1.90469 | ||||||||||||||||||
1.13 | −1.87703 | −2.63781 | 1.52324 | −1.00000 | 4.95125 | −1.00000 | 0.894900 | 3.95806 | 1.87703 | ||||||||||||||||||
1.14 | −1.76681 | 2.93332 | 1.12163 | −1.00000 | −5.18264 | −1.00000 | 1.55191 | 5.60439 | 1.76681 | ||||||||||||||||||
1.15 | −1.54368 | 0.772565 | 0.382940 | −1.00000 | −1.19259 | −1.00000 | 2.49622 | −2.40314 | 1.54368 | ||||||||||||||||||
1.16 | −1.46418 | −1.88953 | 0.143813 | −1.00000 | 2.76661 | −1.00000 | 2.71779 | 0.570340 | 1.46418 | ||||||||||||||||||
1.17 | −1.44897 | −0.334420 | 0.0995284 | −1.00000 | 0.484566 | −1.00000 | 2.75374 | −2.88816 | 1.44897 | ||||||||||||||||||
1.18 | −1.34348 | −2.34976 | −0.195065 | −1.00000 | 3.15685 | −1.00000 | 2.94902 | 2.52136 | 1.34348 | ||||||||||||||||||
1.19 | −1.28523 | −0.829686 | −0.348175 | −1.00000 | 1.06634 | −1.00000 | 3.01795 | −2.31162 | 1.28523 | ||||||||||||||||||
1.20 | −1.21148 | 1.67937 | −0.532324 | −1.00000 | −2.03452 | −1.00000 | 3.06785 | −0.179714 | 1.21148 | ||||||||||||||||||
See all 62 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.l | ✓ | 62 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
8015.2.a.l | ✓ | 62 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8015))\):
\( T_{2}^{62} - 2 T_{2}^{61} - 92 T_{2}^{60} + 179 T_{2}^{59} + 4010 T_{2}^{58} - 7570 T_{2}^{57} - 110189 T_{2}^{56} + 201232 T_{2}^{55} + 2142834 T_{2}^{54} - 3773350 T_{2}^{53} - 31384856 T_{2}^{52} + 53093437 T_{2}^{51} + \cdots - 42758 \) |
\( T_{3}^{62} - 11 T_{3}^{61} - 67 T_{3}^{60} + 1139 T_{3}^{59} + 1017 T_{3}^{58} - 54828 T_{3}^{57} + 54202 T_{3}^{56} + 1627229 T_{3}^{55} - 3404316 T_{3}^{54} - 33266881 T_{3}^{53} + 98204795 T_{3}^{52} + \cdots - 82512 \) |