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: | \(28\) |
Twist minimal: | no (minimal twist has level 4024) |
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 | −3.18633 | 0 | −1.57145 | 0 | 3.06067 | 0 | 7.15268 | 0 | ||||||||||||||||||
1.2 | 0 | −2.89723 | 0 | −1.96349 | 0 | −2.82850 | 0 | 5.39392 | 0 | ||||||||||||||||||
1.3 | 0 | −2.84528 | 0 | −2.72112 | 0 | 3.71681 | 0 | 5.09562 | 0 | ||||||||||||||||||
1.4 | 0 | −2.31121 | 0 | 0.927476 | 0 | 1.63809 | 0 | 2.34167 | 0 | ||||||||||||||||||
1.5 | 0 | −2.11208 | 0 | 2.26631 | 0 | 4.06526 | 0 | 1.46088 | 0 | ||||||||||||||||||
1.6 | 0 | −1.95835 | 0 | 0.337153 | 0 | −0.830729 | 0 | 0.835124 | 0 | ||||||||||||||||||
1.7 | 0 | −1.67525 | 0 | −2.92493 | 0 | 1.65914 | 0 | −0.193522 | 0 | ||||||||||||||||||
1.8 | 0 | −1.66428 | 0 | −3.44274 | 0 | −4.43077 | 0 | −0.230172 | 0 | ||||||||||||||||||
1.9 | 0 | −1.59308 | 0 | 2.75433 | 0 | −1.46360 | 0 | −0.462108 | 0 | ||||||||||||||||||
1.10 | 0 | −1.49100 | 0 | −1.09730 | 0 | 0.799854 | 0 | −0.776913 | 0 | ||||||||||||||||||
1.11 | 0 | −0.823496 | 0 | −2.12271 | 0 | −4.15456 | 0 | −2.32185 | 0 | ||||||||||||||||||
1.12 | 0 | −0.799349 | 0 | 0.626001 | 0 | 0.555593 | 0 | −2.36104 | 0 | ||||||||||||||||||
1.13 | 0 | −0.682343 | 0 | 3.69511 | 0 | 3.46637 | 0 | −2.53441 | 0 | ||||||||||||||||||
1.14 | 0 | −0.659121 | 0 | 0.997881 | 0 | −0.366743 | 0 | −2.56556 | 0 | ||||||||||||||||||
1.15 | 0 | 0.0730681 | 0 | −4.10981 | 0 | 1.52145 | 0 | −2.99466 | 0 | ||||||||||||||||||
1.16 | 0 | 0.447624 | 0 | 0.901345 | 0 | 1.57888 | 0 | −2.79963 | 0 | ||||||||||||||||||
1.17 | 0 | 0.596863 | 0 | 3.74289 | 0 | −1.10831 | 0 | −2.64375 | 0 | ||||||||||||||||||
1.18 | 0 | 0.667261 | 0 | 0.307121 | 0 | −4.25072 | 0 | −2.55476 | 0 | ||||||||||||||||||
1.19 | 0 | 0.893789 | 0 | −3.85908 | 0 | −3.77625 | 0 | −2.20114 | 0 | ||||||||||||||||||
1.20 | 0 | 1.32661 | 0 | −1.90684 | 0 | 4.79236 | 0 | −1.24010 | 0 | ||||||||||||||||||
See all 28 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.v | 28 | |
4.b | odd | 2 | 1 | 4024.2.a.d | ✓ | 28 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
4024.2.a.d | ✓ | 28 | 4.b | odd | 2 | 1 | |
8048.2.a.v | 28 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8048))\):
\( T_{3}^{28} + 2 T_{3}^{27} - 49 T_{3}^{26} - 98 T_{3}^{25} + 1045 T_{3}^{24} + 2099 T_{3}^{23} + \cdots + 5657 \) |
\( T_{5}^{28} + 12 T_{5}^{27} - 9 T_{5}^{26} - 625 T_{5}^{25} - 1416 T_{5}^{24} + 12843 T_{5}^{23} + \cdots - 212992 \) |
\( T_{7}^{28} - 108 T_{7}^{26} + 21 T_{7}^{25} + 5077 T_{7}^{24} - 2013 T_{7}^{23} - 136595 T_{7}^{22} + \cdots - 58309973 \) |
\( T_{13}^{28} + 31 T_{13}^{27} + 296 T_{13}^{26} - 378 T_{13}^{25} - 23819 T_{13}^{24} - 103148 T_{13}^{23} + \cdots + 8128815559 \) |