Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8048,2,Mod(1,8048)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8048, 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("8048.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 8048 = 2^{4} \cdot 503 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 8048.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: | \(64.2636035467\) |
Analytic rank: | \(1\) |
Dimension: | \(21\) |
Twist minimal: | no (minimal twist has level 2012) |
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 | 0 | −3.31593 | 0 | −2.21364 | 0 | 1.66896 | 0 | 7.99538 | 0 | ||||||||||||||||||
1.2 | 0 | −3.16551 | 0 | −0.445451 | 0 | −3.10818 | 0 | 7.02047 | 0 | ||||||||||||||||||
1.3 | 0 | −3.07762 | 0 | 3.88947 | 0 | 0.0828849 | 0 | 6.47174 | 0 | ||||||||||||||||||
1.4 | 0 | −2.69810 | 0 | 1.28348 | 0 | −4.83591 | 0 | 4.27974 | 0 | ||||||||||||||||||
1.5 | 0 | −2.44429 | 0 | 3.11768 | 0 | 1.05875 | 0 | 2.97454 | 0 | ||||||||||||||||||
1.6 | 0 | −2.30121 | 0 | 1.38461 | 0 | 2.51996 | 0 | 2.29558 | 0 | ||||||||||||||||||
1.7 | 0 | −1.93158 | 0 | −3.25684 | 0 | −2.49607 | 0 | 0.730999 | 0 | ||||||||||||||||||
1.8 | 0 | −1.38206 | 0 | −2.65136 | 0 | −3.15617 | 0 | −1.08990 | 0 | ||||||||||||||||||
1.9 | 0 | −1.32334 | 0 | 2.33991 | 0 | −2.12211 | 0 | −1.24878 | 0 | ||||||||||||||||||
1.10 | 0 | −1.10653 | 0 | 2.64866 | 0 | −3.45308 | 0 | −1.77559 | 0 | ||||||||||||||||||
1.11 | 0 | −0.207528 | 0 | −1.91018 | 0 | 4.59256 | 0 | −2.95693 | 0 | ||||||||||||||||||
1.12 | 0 | 0.112957 | 0 | 0.539075 | 0 | 1.01688 | 0 | −2.98724 | 0 | ||||||||||||||||||
1.13 | 0 | 0.175511 | 0 | 3.29961 | 0 | −3.77232 | 0 | −2.96920 | 0 | ||||||||||||||||||
1.14 | 0 | 0.273229 | 0 | −1.56489 | 0 | 3.17063 | 0 | −2.92535 | 0 | ||||||||||||||||||
1.15 | 0 | 0.388717 | 0 | −4.11083 | 0 | 1.27214 | 0 | −2.84890 | 0 | ||||||||||||||||||
1.16 | 0 | 1.27408 | 0 | 3.64848 | 0 | 0.903884 | 0 | −1.37671 | 0 | ||||||||||||||||||
1.17 | 0 | 1.30141 | 0 | −2.99143 | 0 | −4.27894 | 0 | −1.30633 | 0 | ||||||||||||||||||
1.18 | 0 | 1.91007 | 0 | 1.16052 | 0 | −1.88100 | 0 | 0.648366 | 0 | ||||||||||||||||||
1.19 | 0 | 2.22301 | 0 | −1.09600 | 0 | 1.95566 | 0 | 1.94176 | 0 | ||||||||||||||||||
1.20 | 0 | 2.41348 | 0 | −0.469286 | 0 | 1.19654 | 0 | 2.82487 | 0 | ||||||||||||||||||
See all 21 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(2\) | \(-1\) |
\(503\) | \(-1\) |
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 | 8048.2.a.s | 21 | |
4.b | odd | 2 | 1 | 2012.2.a.b | ✓ | 21 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
2012.2.a.b | ✓ | 21 | 4.b | odd | 2 | 1 | |
8048.2.a.s | 21 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8048))\):
\( T_{3}^{21} + 10 T_{3}^{20} + 8 T_{3}^{19} - 221 T_{3}^{18} - 627 T_{3}^{17} + 1614 T_{3}^{16} + \cdots + 41 \) |
\( T_{5}^{21} - 3 T_{5}^{20} - 57 T_{5}^{19} + 167 T_{5}^{18} + 1337 T_{5}^{17} - 3785 T_{5}^{16} + \cdots - 64512 \) |
\( T_{7}^{21} + 13 T_{7}^{20} + 3 T_{7}^{19} - 609 T_{7}^{18} - 1837 T_{7}^{17} + 10228 T_{7}^{16} + \cdots - 1292009 \) |
\( T_{13}^{21} - 12 T_{13}^{20} - 64 T_{13}^{19} + 1204 T_{13}^{18} + 184 T_{13}^{17} - 46562 T_{13}^{16} + \cdots + 2465001 \) |