Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8043,2,Mod(1,8043)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8043, 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("8043.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 8043 = 3 \cdot 7 \cdot 383 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 8043.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: | \(64.2236783457\) |
Analytic rank: | \(0\) |
Dimension: | \(50\) |
Twist minimal: | yes |
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 | −2.82109 | −1.00000 | 5.95855 | −4.23433 | 2.82109 | −1.00000 | −11.1674 | 1.00000 | 11.9454 | ||||||||||||||||||
1.2 | −2.63224 | −1.00000 | 4.92870 | 3.86633 | 2.63224 | −1.00000 | −7.70903 | 1.00000 | −10.1771 | ||||||||||||||||||
1.3 | −2.63191 | −1.00000 | 4.92693 | 0.271396 | 2.63191 | −1.00000 | −7.70340 | 1.00000 | −0.714288 | ||||||||||||||||||
1.4 | −2.62429 | −1.00000 | 4.88687 | −1.77661 | 2.62429 | −1.00000 | −7.57598 | 1.00000 | 4.66233 | ||||||||||||||||||
1.5 | −2.39697 | −1.00000 | 3.74548 | 1.84555 | 2.39697 | −1.00000 | −4.18388 | 1.00000 | −4.42374 | ||||||||||||||||||
1.6 | −2.33599 | −1.00000 | 3.45686 | 4.00725 | 2.33599 | −1.00000 | −3.40322 | 1.00000 | −9.36091 | ||||||||||||||||||
1.7 | −2.33165 | −1.00000 | 3.43661 | −1.59780 | 2.33165 | −1.00000 | −3.34969 | 1.00000 | 3.72551 | ||||||||||||||||||
1.8 | −2.30033 | −1.00000 | 3.29153 | −3.56175 | 2.30033 | −1.00000 | −2.97095 | 1.00000 | 8.19322 | ||||||||||||||||||
1.9 | −2.16078 | −1.00000 | 2.66897 | −0.263100 | 2.16078 | −1.00000 | −1.44551 | 1.00000 | 0.568501 | ||||||||||||||||||
1.10 | −1.95593 | −1.00000 | 1.82566 | 0.355087 | 1.95593 | −1.00000 | 0.340990 | 1.00000 | −0.694525 | ||||||||||||||||||
1.11 | −1.84012 | −1.00000 | 1.38605 | −2.07033 | 1.84012 | −1.00000 | 1.12975 | 1.00000 | 3.80966 | ||||||||||||||||||
1.12 | −1.76145 | −1.00000 | 1.10269 | 2.11729 | 1.76145 | −1.00000 | 1.58056 | 1.00000 | −3.72950 | ||||||||||||||||||
1.13 | −1.46962 | −1.00000 | 0.159775 | 3.40283 | 1.46962 | −1.00000 | 2.70443 | 1.00000 | −5.00085 | ||||||||||||||||||
1.14 | −1.46567 | −1.00000 | 0.148201 | 0.335135 | 1.46567 | −1.00000 | 2.71413 | 1.00000 | −0.491199 | ||||||||||||||||||
1.15 | −1.19941 | −1.00000 | −0.561407 | −0.150113 | 1.19941 | −1.00000 | 3.07219 | 1.00000 | 0.180048 | ||||||||||||||||||
1.16 | −1.18848 | −1.00000 | −0.587510 | −4.24260 | 1.18848 | −1.00000 | 3.07521 | 1.00000 | 5.04226 | ||||||||||||||||||
1.17 | −1.18460 | −1.00000 | −0.596731 | 3.79711 | 1.18460 | −1.00000 | 3.07608 | 1.00000 | −4.49805 | ||||||||||||||||||
1.18 | −1.15299 | −1.00000 | −0.670610 | 0.489993 | 1.15299 | −1.00000 | 3.07919 | 1.00000 | −0.564958 | ||||||||||||||||||
1.19 | −1.12728 | −1.00000 | −0.729231 | −0.145556 | 1.12728 | −1.00000 | 3.07662 | 1.00000 | 0.164083 | ||||||||||||||||||
1.20 | −0.719015 | −1.00000 | −1.48302 | −1.74243 | 0.719015 | −1.00000 | 2.50434 | 1.00000 | 1.25284 | ||||||||||||||||||
See all 50 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(3\) | \(1\) |
\(7\) | \(1\) |
\(383\) | \(-1\) |
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 | 8043.2.a.s | ✓ | 50 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
8043.2.a.s | ✓ | 50 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8043))\):
\( T_{2}^{50} + T_{2}^{49} - 76 T_{2}^{48} - 73 T_{2}^{47} + 2697 T_{2}^{46} + 2484 T_{2}^{45} + \cdots + 1024 \) |
\( T_{5}^{50} - 11 T_{5}^{49} - 100 T_{5}^{48} + 1484 T_{5}^{47} + 3497 T_{5}^{46} - 91590 T_{5}^{45} + \cdots - 1449124352 \) |
\( T_{11}^{50} + 31 T_{11}^{49} + 169 T_{11}^{48} - 4399 T_{11}^{47} - 57153 T_{11}^{46} + \cdots - 30\!\cdots\!68 \) |