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: | \(0\) |
Dimension: | \(29\) |
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 | −3.11496 | 0 | 0.544032 | 0 | 4.06617 | 0 | 6.70298 | 0 | ||||||||||||||||||
1.2 | 0 | −2.62135 | 0 | −1.76626 | 0 | −2.03701 | 0 | 3.87147 | 0 | ||||||||||||||||||
1.3 | 0 | −2.38925 | 0 | −0.218073 | 0 | 2.23687 | 0 | 2.70851 | 0 | ||||||||||||||||||
1.4 | 0 | −2.22567 | 0 | −3.22615 | 0 | −0.00440441 | 0 | 1.95359 | 0 | ||||||||||||||||||
1.5 | 0 | −2.04161 | 0 | −4.02753 | 0 | −1.43352 | 0 | 1.16816 | 0 | ||||||||||||||||||
1.6 | 0 | −1.71451 | 0 | −1.13020 | 0 | 1.00303 | 0 | −0.0604433 | 0 | ||||||||||||||||||
1.7 | 0 | −1.62590 | 0 | 3.78842 | 0 | 2.93814 | 0 | −0.356460 | 0 | ||||||||||||||||||
1.8 | 0 | −1.30647 | 0 | 0.939477 | 0 | −2.17911 | 0 | −1.29314 | 0 | ||||||||||||||||||
1.9 | 0 | −1.29900 | 0 | 2.09702 | 0 | 3.13973 | 0 | −1.31260 | 0 | ||||||||||||||||||
1.10 | 0 | −1.28849 | 0 | −1.09309 | 0 | −0.799248 | 0 | −1.33979 | 0 | ||||||||||||||||||
1.11 | 0 | −0.936010 | 0 | 1.25557 | 0 | −0.290471 | 0 | −2.12388 | 0 | ||||||||||||||||||
1.12 | 0 | −0.454325 | 0 | 0.622756 | 0 | 4.14589 | 0 | −2.79359 | 0 | ||||||||||||||||||
1.13 | 0 | 0.137144 | 0 | 1.57958 | 0 | −4.44075 | 0 | −2.98119 | 0 | ||||||||||||||||||
1.14 | 0 | 0.278789 | 0 | −0.533193 | 0 | −1.93823 | 0 | −2.92228 | 0 | ||||||||||||||||||
1.15 | 0 | 0.502218 | 0 | −3.38571 | 0 | −3.39706 | 0 | −2.74778 | 0 | ||||||||||||||||||
1.16 | 0 | 0.582422 | 0 | −3.18976 | 0 | 3.00819 | 0 | −2.66078 | 0 | ||||||||||||||||||
1.17 | 0 | 0.782310 | 0 | 2.74195 | 0 | 3.13445 | 0 | −2.38799 | 0 | ||||||||||||||||||
1.18 | 0 | 0.795174 | 0 | 3.83818 | 0 | −0.297725 | 0 | −2.36770 | 0 | ||||||||||||||||||
1.19 | 0 | 1.10456 | 0 | −2.70098 | 0 | −3.17910 | 0 | −1.77994 | 0 | ||||||||||||||||||
1.20 | 0 | 1.15550 | 0 | −3.26045 | 0 | 1.92982 | 0 | −1.66482 | 0 | ||||||||||||||||||
See all 29 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.w | 29 | |
4.b | odd | 2 | 1 | 4024.2.a.e | ✓ | 29 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
4024.2.a.e | ✓ | 29 | 4.b | odd | 2 | 1 | |
8048.2.a.w | 29 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8048))\):
\( T_{3}^{29} - 7 T_{3}^{28} - 29 T_{3}^{27} + 295 T_{3}^{26} + 203 T_{3}^{25} - 5370 T_{3}^{24} + \cdots - 6975 \) |
\( T_{5}^{29} + 4 T_{5}^{28} - 75 T_{5}^{27} - 297 T_{5}^{26} + 2432 T_{5}^{25} + 9505 T_{5}^{24} + \cdots + 16384 \) |
\( T_{7}^{29} - 13 T_{7}^{28} - 24 T_{7}^{27} + 969 T_{7}^{26} - 1832 T_{7}^{25} - 29380 T_{7}^{24} + \cdots + 64747 \) |
\( T_{13}^{29} - 16 T_{13}^{28} - 47 T_{13}^{27} + 2000 T_{13}^{26} - 4659 T_{13}^{25} - 94601 T_{13}^{24} + \cdots - 3458323023 \) |