Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8993,2,Mod(1,8993)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8993, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("8993.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 8993 = 17 \cdot 23^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 8993.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: | \(71.8094665377\) |
Analytic rank: | \(0\) |
Dimension: | \(90\) |
Twist minimal: | no (minimal twist has level 391) |
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 | −2.78943 | 0.152810 | 5.78091 | 0.458306 | −0.426253 | −4.47274 | −10.5466 | −2.97665 | −1.27841 | ||||||||||||||||||
1.2 | −2.75381 | 1.00406 | 5.58347 | 3.41558 | −2.76500 | −2.29121 | −9.86820 | −1.99185 | −9.40585 | ||||||||||||||||||
1.3 | −2.71401 | 3.16264 | 5.36584 | −3.30716 | −8.58342 | −1.44850 | −9.13490 | 7.00227 | 8.97566 | ||||||||||||||||||
1.4 | −2.57651 | −2.99955 | 4.63838 | 0.110603 | 7.72836 | 1.22146 | −6.79781 | 5.99729 | −0.284970 | ||||||||||||||||||
1.5 | −2.55642 | −1.52801 | 4.53529 | −3.04971 | 3.90623 | 3.12865 | −6.48128 | −0.665191 | 7.79636 | ||||||||||||||||||
1.6 | −2.49625 | 1.18938 | 4.23125 | 0.892664 | −2.96899 | −1.82513 | −5.56974 | −1.58537 | −2.22831 | ||||||||||||||||||
1.7 | −2.47925 | −1.56756 | 4.14668 | −4.32472 | 3.88638 | −2.44570 | −5.32215 | −0.542751 | 10.7221 | ||||||||||||||||||
1.8 | −2.45766 | 1.94047 | 4.04010 | −3.93953 | −4.76903 | 4.55543 | −5.01388 | 0.765442 | 9.68204 | ||||||||||||||||||
1.9 | −2.41552 | −0.509829 | 3.83473 | 3.12056 | 1.23150 | −2.26311 | −4.43182 | −2.74007 | −7.53778 | ||||||||||||||||||
1.10 | −2.37716 | −2.02408 | 3.65089 | −2.33294 | 4.81157 | −4.16483 | −3.92443 | 1.09690 | 5.54577 | ||||||||||||||||||
1.11 | −2.24566 | 2.48698 | 3.04301 | 3.24575 | −5.58493 | 3.64668 | −2.34224 | 3.18509 | −7.28886 | ||||||||||||||||||
1.12 | −2.22165 | −2.39230 | 2.93574 | −0.626912 | 5.31486 | 1.71533 | −2.07889 | 2.72311 | 1.39278 | ||||||||||||||||||
1.13 | −2.08596 | 2.77361 | 2.35123 | 2.18691 | −5.78563 | 4.72020 | −0.732662 | 4.69289 | −4.56182 | ||||||||||||||||||
1.14 | −2.02693 | 1.67419 | 2.10845 | −0.958934 | −3.39346 | −0.144869 | −0.219810 | −0.197092 | 1.94369 | ||||||||||||||||||
1.15 | −2.00839 | −0.268714 | 2.03362 | −2.53833 | 0.539682 | 0.944934 | −0.0675119 | −2.92779 | 5.09794 | ||||||||||||||||||
1.16 | −2.00548 | −1.48196 | 2.02196 | 2.45276 | 2.97204 | −1.23566 | −0.0440356 | −0.803797 | −4.91896 | ||||||||||||||||||
1.17 | −1.88676 | 3.01129 | 1.55987 | −3.29272 | −5.68159 | −2.53571 | 0.830415 | 6.06785 | 6.21259 | ||||||||||||||||||
1.18 | −1.80981 | −3.24553 | 1.27540 | −0.171831 | 5.87377 | 2.25177 | 1.31139 | 7.53345 | 0.310981 | ||||||||||||||||||
1.19 | −1.80678 | 1.07261 | 1.26445 | 1.84980 | −1.93797 | 0.873536 | 1.32898 | −1.84950 | −3.34218 | ||||||||||||||||||
1.20 | −1.71858 | 0.352612 | 0.953522 | 0.359463 | −0.605992 | −3.68964 | 1.79846 | −2.87567 | −0.617767 | ||||||||||||||||||
See all 90 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(17\) | \(-1\) |
\(23\) | \(1\) |
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 | 8993.2.a.x | 90 | |
23.b | odd | 2 | 1 | 8993.2.a.w | 90 | ||
23.c | even | 11 | 2 | 391.2.i.b | ✓ | 180 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
391.2.i.b | ✓ | 180 | 23.c | even | 11 | 2 | |
8993.2.a.w | 90 | 23.b | odd | 2 | 1 | ||
8993.2.a.x | 90 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8993))\):
\( T_{2}^{90} - 9 T_{2}^{89} - 104 T_{2}^{88} + 1153 T_{2}^{87} + 4718 T_{2}^{86} - 70918 T_{2}^{85} - 110317 T_{2}^{84} + 2788637 T_{2}^{83} + 678497 T_{2}^{82} - 78753075 T_{2}^{81} + 46234714 T_{2}^{80} + \cdots - 4643539 \) |
\( T_{5}^{90} - 8 T_{5}^{89} - 262 T_{5}^{88} + 2231 T_{5}^{87} + 32569 T_{5}^{86} - 298317 T_{5}^{85} - 2550719 T_{5}^{84} + 25472398 T_{5}^{83} + 140806923 T_{5}^{82} - 1560650598 T_{5}^{81} + \cdots + 48\!\cdots\!48 \) |