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: | \(38\) |
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 | −2.60567 | −1.96600 | 4.78952 | 1.00000 | 5.12274 | 1.00000 | −7.26856 | 0.865151 | −2.60567 | ||||||||||||||||||
1.2 | −2.59504 | 0.990389 | 4.73422 | 1.00000 | −2.57010 | 1.00000 | −7.09539 | −2.01913 | −2.59504 | ||||||||||||||||||
1.3 | −2.51011 | −2.35176 | 4.30066 | 1.00000 | 5.90318 | 1.00000 | −5.77493 | 2.53077 | −2.51011 | ||||||||||||||||||
1.4 | −2.46620 | 0.936734 | 4.08216 | 1.00000 | −2.31018 | 1.00000 | −5.13504 | −2.12253 | −2.46620 | ||||||||||||||||||
1.5 | −2.38102 | 2.43622 | 3.66928 | 1.00000 | −5.80069 | 1.00000 | −3.97459 | 2.93515 | −2.38102 | ||||||||||||||||||
1.6 | −2.02615 | 0.680619 | 2.10529 | 1.00000 | −1.37904 | 1.00000 | −0.213342 | −2.53676 | −2.02615 | ||||||||||||||||||
1.7 | −1.89678 | −2.50257 | 1.59778 | 1.00000 | 4.74683 | 1.00000 | 0.762920 | 3.26285 | −1.89678 | ||||||||||||||||||
1.8 | −1.84626 | −2.48565 | 1.40867 | 1.00000 | 4.58916 | 1.00000 | 1.09175 | 3.17847 | −1.84626 | ||||||||||||||||||
1.9 | −1.83849 | 1.40018 | 1.38005 | 1.00000 | −2.57423 | 1.00000 | 1.13977 | −1.03949 | −1.83849 | ||||||||||||||||||
1.10 | −1.74109 | −0.00460406 | 1.03141 | 1.00000 | 0.00801611 | 1.00000 | 1.68640 | −2.99998 | −1.74109 | ||||||||||||||||||
1.11 | −1.66719 | 2.43876 | 0.779527 | 1.00000 | −4.06588 | 1.00000 | 2.03476 | 2.94756 | −1.66719 | ||||||||||||||||||
1.12 | −1.65687 | −0.430057 | 0.745215 | 1.00000 | 0.712549 | 1.00000 | 2.07901 | −2.81505 | −1.65687 | ||||||||||||||||||
1.13 | −1.23429 | −1.22031 | −0.476531 | 1.00000 | 1.50621 | 1.00000 | 3.05675 | −1.51085 | −1.23429 | ||||||||||||||||||
1.14 | −1.08292 | −0.496145 | −0.827289 | 1.00000 | 0.537284 | 1.00000 | 3.06172 | −2.75384 | −1.08292 | ||||||||||||||||||
1.15 | −0.697175 | 1.52901 | −1.51395 | 1.00000 | −1.06599 | 1.00000 | 2.44984 | −0.662121 | −0.697175 | ||||||||||||||||||
1.16 | −0.656564 | 2.76581 | −1.56892 | 1.00000 | −1.81593 | 1.00000 | 2.34323 | 4.64973 | −0.656564 | ||||||||||||||||||
1.17 | −0.520713 | −3.35310 | −1.72886 | 1.00000 | 1.74600 | 1.00000 | 1.94166 | 8.24329 | −0.520713 | ||||||||||||||||||
1.18 | −0.217856 | −2.35998 | −1.95254 | 1.00000 | 0.514135 | 1.00000 | 0.861085 | 2.56949 | −0.217856 | ||||||||||||||||||
1.19 | −0.0912753 | 1.39308 | −1.99167 | 1.00000 | −0.127154 | 1.00000 | 0.364341 | −1.05934 | −0.0912753 | ||||||||||||||||||
1.20 | −0.0830831 | 2.10839 | −1.99310 | 1.00000 | −0.175171 | 1.00000 | 0.331759 | 1.44530 | −0.0830831 | ||||||||||||||||||
See all 38 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.h | ✓ | 38 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
8015.2.a.h | ✓ | 38 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8015))\):
\( T_{2}^{38} + 6 T_{2}^{37} - 32 T_{2}^{36} - 249 T_{2}^{35} + 382 T_{2}^{34} + 4672 T_{2}^{33} - 1265 T_{2}^{32} + \cdots + 3 \) |
\( T_{3}^{38} + 9 T_{3}^{37} - 23 T_{3}^{36} - 427 T_{3}^{35} - 266 T_{3}^{34} + 8797 T_{3}^{33} + \cdots - 80 \) |