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: | \(1\) |
Dimension: | \(40\) |
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 | −2.75032 | 1.00000 | 5.56428 | 0.854023 | −2.75032 | 1.00000 | −9.80291 | 1.00000 | −2.34884 | ||||||||||||||||||
1.2 | −2.74138 | 1.00000 | 5.51515 | −1.10050 | −2.74138 | 1.00000 | −9.63635 | 1.00000 | 3.01688 | ||||||||||||||||||
1.3 | −2.64192 | 1.00000 | 4.97976 | −3.51496 | −2.64192 | 1.00000 | −7.87228 | 1.00000 | 9.28626 | ||||||||||||||||||
1.4 | −2.58104 | 1.00000 | 4.66178 | 2.25631 | −2.58104 | 1.00000 | −6.87016 | 1.00000 | −5.82363 | ||||||||||||||||||
1.5 | −2.32186 | 1.00000 | 3.39104 | 0.609544 | −2.32186 | 1.00000 | −3.22980 | 1.00000 | −1.41528 | ||||||||||||||||||
1.6 | −2.16260 | 1.00000 | 2.67684 | −4.31110 | −2.16260 | 1.00000 | −1.46374 | 1.00000 | 9.32319 | ||||||||||||||||||
1.7 | −2.08373 | 1.00000 | 2.34192 | −2.93682 | −2.08373 | 1.00000 | −0.712460 | 1.00000 | 6.11952 | ||||||||||||||||||
1.8 | −1.97104 | 1.00000 | 1.88501 | 1.28397 | −1.97104 | 1.00000 | 0.226654 | 1.00000 | −2.53076 | ||||||||||||||||||
1.9 | −1.85383 | 1.00000 | 1.43669 | −2.52734 | −1.85383 | 1.00000 | 1.04428 | 1.00000 | 4.68526 | ||||||||||||||||||
1.10 | −1.83203 | 1.00000 | 1.35635 | 1.36968 | −1.83203 | 1.00000 | 1.17919 | 1.00000 | −2.50930 | ||||||||||||||||||
1.11 | −1.65595 | 1.00000 | 0.742162 | −4.41540 | −1.65595 | 1.00000 | 2.08291 | 1.00000 | 7.31167 | ||||||||||||||||||
1.12 | −1.59287 | 1.00000 | 0.537230 | 2.23699 | −1.59287 | 1.00000 | 2.33000 | 1.00000 | −3.56323 | ||||||||||||||||||
1.13 | −1.43635 | 1.00000 | 0.0630943 | 3.31438 | −1.43635 | 1.00000 | 2.78207 | 1.00000 | −4.76060 | ||||||||||||||||||
1.14 | −1.36925 | 1.00000 | −0.125168 | 0.412992 | −1.36925 | 1.00000 | 2.90988 | 1.00000 | −0.565488 | ||||||||||||||||||
1.15 | −1.12599 | 1.00000 | −0.732150 | 2.85700 | −1.12599 | 1.00000 | 3.07637 | 1.00000 | −3.21695 | ||||||||||||||||||
1.16 | −1.09091 | 1.00000 | −0.809919 | −1.12988 | −1.09091 | 1.00000 | 3.06536 | 1.00000 | 1.23260 | ||||||||||||||||||
1.17 | −0.765993 | 1.00000 | −1.41325 | −3.34403 | −0.765993 | 1.00000 | 2.61453 | 1.00000 | 2.56151 | ||||||||||||||||||
1.18 | −0.717010 | 1.00000 | −1.48590 | 1.86418 | −0.717010 | 1.00000 | 2.49942 | 1.00000 | −1.33664 | ||||||||||||||||||
1.19 | −0.619181 | 1.00000 | −1.61661 | −0.937229 | −0.619181 | 1.00000 | 2.23934 | 1.00000 | 0.580315 | ||||||||||||||||||
1.20 | −0.506041 | 1.00000 | −1.74392 | −0.0764488 | −0.506041 | 1.00000 | 1.89458 | 1.00000 | 0.0386862 | ||||||||||||||||||
See all 40 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.n | ✓ | 40 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
8043.2.a.n | ✓ | 40 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8043))\):
\( T_{2}^{40} + 9 T_{2}^{39} - 16 T_{2}^{38} - 369 T_{2}^{37} - 392 T_{2}^{36} + 6561 T_{2}^{35} + 15388 T_{2}^{34} - 65043 T_{2}^{33} - 230012 T_{2}^{32} + 372678 T_{2}^{31} + 2050662 T_{2}^{30} - 967682 T_{2}^{29} + \cdots - 112 \) |
\( T_{5}^{40} + 27 T_{5}^{39} + 247 T_{5}^{38} + 245 T_{5}^{37} - 10748 T_{5}^{36} - 62279 T_{5}^{35} + 71897 T_{5}^{34} + 1754561 T_{5}^{33} + 3575503 T_{5}^{32} - 20685092 T_{5}^{31} - 95797377 T_{5}^{30} + \cdots + 507904 \) |
\( T_{11}^{40} + 29 T_{11}^{39} + 198 T_{11}^{38} - 2150 T_{11}^{37} - 34978 T_{11}^{36} - 36899 T_{11}^{35} + 1900314 T_{11}^{34} + 9394822 T_{11}^{33} - 42601700 T_{11}^{32} - 443917469 T_{11}^{31} + \cdots + 221970893321008 \) |