Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8037,2,Mod(1,8037)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8037, 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("8037.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 8037 = 3^{2} \cdot 19 \cdot 47 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 8037.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: | \(64.1757681045\) |
Analytic rank: | \(1\) |
Dimension: | \(24\) |
Twist minimal: | no (minimal twist has level 2679) |
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 | −2.74619 | 0 | 5.54156 | 1.39198 | 0 | 3.03199 | −9.72581 | 0 | −3.82263 | ||||||||||||||||||
1.2 | −2.73887 | 0 | 5.50141 | 1.01615 | 0 | 0.640714 | −9.58990 | 0 | −2.78309 | ||||||||||||||||||
1.3 | −2.62778 | 0 | 4.90524 | −2.66749 | 0 | −4.09726 | −7.63435 | 0 | 7.00959 | ||||||||||||||||||
1.4 | −2.30982 | 0 | 3.33525 | −0.261175 | 0 | 1.60168 | −3.08417 | 0 | 0.603266 | ||||||||||||||||||
1.5 | −2.26800 | 0 | 3.14381 | −4.22801 | 0 | 5.05999 | −2.59416 | 0 | 9.58911 | ||||||||||||||||||
1.6 | −2.20954 | 0 | 2.88208 | −2.70483 | 0 | −0.952275 | −1.94899 | 0 | 5.97643 | ||||||||||||||||||
1.7 | −1.92175 | 0 | 1.69311 | −0.940482 | 0 | 3.86468 | 0.589769 | 0 | 1.80737 | ||||||||||||||||||
1.8 | −1.72798 | 0 | 0.985905 | 4.09118 | 0 | 3.75957 | 1.75233 | 0 | −7.06946 | ||||||||||||||||||
1.9 | −1.39021 | 0 | −0.0673298 | −3.64442 | 0 | −1.02491 | 2.87401 | 0 | 5.06649 | ||||||||||||||||||
1.10 | −0.992877 | 0 | −1.01419 | 4.04955 | 0 | −4.64503 | 2.99273 | 0 | −4.02071 | ||||||||||||||||||
1.11 | −0.543143 | 0 | −1.70500 | 0.446531 | 0 | −4.73017 | 2.01234 | 0 | −0.242530 | ||||||||||||||||||
1.12 | −0.456458 | 0 | −1.79165 | 1.12773 | 0 | −0.162029 | 1.73073 | 0 | −0.514761 | ||||||||||||||||||
1.13 | −0.165573 | 0 | −1.97259 | −4.22825 | 0 | 2.94294 | 0.657754 | 0 | 0.700086 | ||||||||||||||||||
1.14 | −0.0666738 | 0 | −1.99555 | −2.45418 | 0 | 3.25184 | 0.266399 | 0 | 0.163629 | ||||||||||||||||||
1.15 | 0.0589819 | 0 | −1.99652 | 2.29389 | 0 | 2.75379 | −0.235722 | 0 | 0.135298 | ||||||||||||||||||
1.16 | 0.743704 | 0 | −1.44690 | −0.245686 | 0 | 1.64092 | −2.56348 | 0 | −0.182717 | ||||||||||||||||||
1.17 | 1.10355 | 0 | −0.782179 | −3.33604 | 0 | −4.44205 | −3.07027 | 0 | −3.68148 | ||||||||||||||||||
1.18 | 1.17388 | 0 | −0.622012 | 3.35404 | 0 | −1.78420 | −3.07792 | 0 | 3.93723 | ||||||||||||||||||
1.19 | 1.62396 | 0 | 0.637238 | 2.60823 | 0 | −0.198208 | −2.21307 | 0 | 4.23566 | ||||||||||||||||||
1.20 | 1.96663 | 0 | 1.86764 | −1.40576 | 0 | 4.82031 | −0.260313 | 0 | −2.76460 | ||||||||||||||||||
See all 24 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(3\) | \(-1\) |
\(19\) | \(1\) |
\(47\) | \(-1\) |
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 | 8037.2.a.s | 24 | |
3.b | odd | 2 | 1 | 2679.2.a.p | ✓ | 24 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
2679.2.a.p | ✓ | 24 | 3.b | odd | 2 | 1 | |
8037.2.a.s | 24 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8037))\):
\( T_{2}^{24} + 6 T_{2}^{23} - 22 T_{2}^{22} - 191 T_{2}^{21} + 104 T_{2}^{20} + 2544 T_{2}^{19} + \cdots + 16 \) |
\( T_{5}^{24} + 10 T_{5}^{23} - 39 T_{5}^{22} - 670 T_{5}^{21} - 107 T_{5}^{20} + 18359 T_{5}^{19} + \cdots + 48574 \) |