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: | \(33\) |
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 | −3.42195 | 0 | −3.41581 | 0 | −3.07051 | 0 | 8.70973 | 0 | ||||||||||||||||||
1.2 | 0 | −3.41276 | 0 | 2.47839 | 0 | 2.17339 | 0 | 8.64692 | 0 | ||||||||||||||||||
1.3 | 0 | −3.14063 | 0 | 2.58012 | 0 | −1.35399 | 0 | 6.86353 | 0 | ||||||||||||||||||
1.4 | 0 | −2.91492 | 0 | 1.91993 | 0 | −3.81516 | 0 | 5.49674 | 0 | ||||||||||||||||||
1.5 | 0 | −2.90120 | 0 | −2.20357 | 0 | 1.74877 | 0 | 5.41693 | 0 | ||||||||||||||||||
1.6 | 0 | −2.74246 | 0 | 2.09147 | 0 | −3.15984 | 0 | 4.52108 | 0 | ||||||||||||||||||
1.7 | 0 | −2.60958 | 0 | −1.79981 | 0 | 2.96082 | 0 | 3.80991 | 0 | ||||||||||||||||||
1.8 | 0 | −2.40618 | 0 | 1.57414 | 0 | −2.69643 | 0 | 2.78968 | 0 | ||||||||||||||||||
1.9 | 0 | −2.03324 | 0 | 4.46117 | 0 | −0.828843 | 0 | 1.13405 | 0 | ||||||||||||||||||
1.10 | 0 | −1.94894 | 0 | 2.86334 | 0 | 4.75146 | 0 | 0.798369 | 0 | ||||||||||||||||||
1.11 | 0 | −1.78801 | 0 | −0.988126 | 0 | −3.73717 | 0 | 0.196987 | 0 | ||||||||||||||||||
1.12 | 0 | −1.65470 | 0 | −3.99833 | 0 | 4.68473 | 0 | −0.261967 | 0 | ||||||||||||||||||
1.13 | 0 | −1.05277 | 0 | −1.97996 | 0 | −4.35506 | 0 | −1.89168 | 0 | ||||||||||||||||||
1.14 | 0 | −0.940347 | 0 | 2.53897 | 0 | −0.182218 | 0 | −2.11575 | 0 | ||||||||||||||||||
1.15 | 0 | −0.885832 | 0 | −0.282055 | 0 | 2.95093 | 0 | −2.21530 | 0 | ||||||||||||||||||
1.16 | 0 | −0.649284 | 0 | −2.92377 | 0 | 1.90965 | 0 | −2.57843 | 0 | ||||||||||||||||||
1.17 | 0 | −0.410265 | 0 | −4.21207 | 0 | −3.36092 | 0 | −2.83168 | 0 | ||||||||||||||||||
1.18 | 0 | −0.281703 | 0 | −0.580455 | 0 | −2.82149 | 0 | −2.92064 | 0 | ||||||||||||||||||
1.19 | 0 | −0.221515 | 0 | 4.40155 | 0 | −5.16895 | 0 | −2.95093 | 0 | ||||||||||||||||||
1.20 | 0 | 0.105343 | 0 | −1.39521 | 0 | 1.93153 | 0 | −2.98890 | 0 | ||||||||||||||||||
See all 33 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.x | 33 | |
4.b | odd | 2 | 1 | 4024.2.a.g | ✓ | 33 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
4024.2.a.g | ✓ | 33 | 4.b | odd | 2 | 1 | |
8048.2.a.x | 33 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8048))\):
\( T_{3}^{33} + 10 T_{3}^{32} - 23 T_{3}^{31} - 532 T_{3}^{30} - 529 T_{3}^{29} + 12061 T_{3}^{28} + 28952 T_{3}^{27} - 149528 T_{3}^{26} - 539075 T_{3}^{25} + 1051227 T_{3}^{24} + 5729902 T_{3}^{23} - 3372238 T_{3}^{22} + \cdots + 116224 \) |
\( T_{5}^{33} - 106 T_{5}^{31} - 5 T_{5}^{30} + 5010 T_{5}^{29} + 446 T_{5}^{28} - 139884 T_{5}^{27} - 18352 T_{5}^{26} + 2576528 T_{5}^{25} + 461698 T_{5}^{24} - 33096020 T_{5}^{23} - 7865910 T_{5}^{22} + \cdots - 9535488 \) |
\( T_{7}^{33} + 12 T_{7}^{32} - 71 T_{7}^{31} - 1361 T_{7}^{30} + 486 T_{7}^{29} + 66228 T_{7}^{28} + 109528 T_{7}^{27} - 1810796 T_{7}^{26} - 5217651 T_{7}^{25} + 30521071 T_{7}^{24} + 121352211 T_{7}^{23} + \cdots + 1527027904 \) |
\( T_{13}^{33} + 17 T_{13}^{32} - 96 T_{13}^{31} - 3138 T_{13}^{30} - 3085 T_{13}^{29} + 246398 T_{13}^{28} + 911939 T_{13}^{27} - 10543390 T_{13}^{26} - 62183960 T_{13}^{25} + 253831251 T_{13}^{24} + \cdots - 116262986803488 \) |