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: | \(53\) |
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 | −2.66147 | 1.00000 | 5.08344 | 2.81521 | −2.66147 | 1.00000 | −8.20648 | 1.00000 | −7.49261 | ||||||||||||||||||
1.2 | −2.65959 | 1.00000 | 5.07341 | 1.50911 | −2.65959 | 1.00000 | −8.17400 | 1.00000 | −4.01361 | ||||||||||||||||||
1.3 | −2.63741 | 1.00000 | 4.95592 | 0.460886 | −2.63741 | 1.00000 | −7.79596 | 1.00000 | −1.21554 | ||||||||||||||||||
1.4 | −2.39422 | 1.00000 | 3.73230 | −2.33465 | −2.39422 | 1.00000 | −4.14751 | 1.00000 | 5.58967 | ||||||||||||||||||
1.5 | −2.29113 | 1.00000 | 3.24926 | −2.33697 | −2.29113 | 1.00000 | −2.86221 | 1.00000 | 5.35430 | ||||||||||||||||||
1.6 | −2.26724 | 1.00000 | 3.14038 | 4.07533 | −2.26724 | 1.00000 | −2.58552 | 1.00000 | −9.23976 | ||||||||||||||||||
1.7 | −2.15559 | 1.00000 | 2.64657 | 3.65015 | −2.15559 | 1.00000 | −1.39373 | 1.00000 | −7.86823 | ||||||||||||||||||
1.8 | −2.10439 | 1.00000 | 2.42847 | 1.35273 | −2.10439 | 1.00000 | −0.901667 | 1.00000 | −2.84668 | ||||||||||||||||||
1.9 | −2.03619 | 1.00000 | 2.14608 | −0.995010 | −2.03619 | 1.00000 | −0.297449 | 1.00000 | 2.02603 | ||||||||||||||||||
1.10 | −1.92293 | 1.00000 | 1.69768 | 1.88319 | −1.92293 | 1.00000 | 0.581347 | 1.00000 | −3.62124 | ||||||||||||||||||
1.11 | −1.78404 | 1.00000 | 1.18281 | −2.51827 | −1.78404 | 1.00000 | 1.45789 | 1.00000 | 4.49270 | ||||||||||||||||||
1.12 | −1.46877 | 1.00000 | 0.157298 | −1.01808 | −1.46877 | 1.00000 | 2.70651 | 1.00000 | 1.49533 | ||||||||||||||||||
1.13 | −1.45637 | 1.00000 | 0.121016 | −1.02950 | −1.45637 | 1.00000 | 2.73650 | 1.00000 | 1.49933 | ||||||||||||||||||
1.14 | −1.32463 | 1.00000 | −0.245346 | 4.05110 | −1.32463 | 1.00000 | 2.97426 | 1.00000 | −5.36623 | ||||||||||||||||||
1.15 | −1.31380 | 1.00000 | −0.273938 | −2.47121 | −1.31380 | 1.00000 | 2.98749 | 1.00000 | 3.24666 | ||||||||||||||||||
1.16 | −1.27919 | 1.00000 | −0.363673 | 3.49939 | −1.27919 | 1.00000 | 3.02359 | 1.00000 | −4.47639 | ||||||||||||||||||
1.17 | −0.969084 | 1.00000 | −1.06088 | −0.461084 | −0.969084 | 1.00000 | 2.96625 | 1.00000 | 0.446829 | ||||||||||||||||||
1.18 | −0.817713 | 1.00000 | −1.33134 | 0.352247 | −0.817713 | 1.00000 | 2.72409 | 1.00000 | −0.288037 | ||||||||||||||||||
1.19 | −0.805347 | 1.00000 | −1.35142 | 3.29338 | −0.805347 | 1.00000 | 2.69905 | 1.00000 | −2.65231 | ||||||||||||||||||
1.20 | −0.701767 | 1.00000 | −1.50752 | −0.804883 | −0.701767 | 1.00000 | 2.46146 | 1.00000 | 0.564840 | ||||||||||||||||||
See all 53 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.u | ✓ | 53 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
8043.2.a.u | ✓ | 53 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8043))\):
\( T_{2}^{53} - 11 T_{2}^{52} - 24 T_{2}^{51} + 683 T_{2}^{50} - 878 T_{2}^{49} - 18989 T_{2}^{48} + 54676 T_{2}^{47} + 306335 T_{2}^{46} - 1332519 T_{2}^{45} - 3033531 T_{2}^{44} + 20098274 T_{2}^{43} + 16130675 T_{2}^{42} + \cdots - 172032 \) |
\( T_{5}^{53} - 24 T_{5}^{52} + 120 T_{5}^{51} + 1622 T_{5}^{50} - 18843 T_{5}^{49} - 9119 T_{5}^{48} + 941340 T_{5}^{47} - 2721436 T_{5}^{46} - 23788076 T_{5}^{45} + 137432557 T_{5}^{44} + \cdots + 28597059876864 \) |
\( T_{11}^{53} - 46 T_{11}^{52} + 725 T_{11}^{51} - 1278 T_{11}^{50} - 96835 T_{11}^{49} + 1028345 T_{11}^{48} + 1499498 T_{11}^{47} - 90444216 T_{11}^{46} + 404561498 T_{11}^{45} + 3125575358 T_{11}^{44} + \cdots - 11\!\cdots\!64 \) |