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: | \(1\) |
Dimension: | \(60\) |
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 | −2.75139 | −2.61726 | 5.57017 | 3.67428 | 7.20112 | −3.69830 | −9.82293 | 3.85006 | −10.1094 | ||||||||||||||||||
1.2 | −2.73909 | 2.01751 | 5.50261 | −2.00983 | −5.52615 | −4.50525 | −9.59398 | 1.07037 | 5.50510 | ||||||||||||||||||
1.3 | −2.72186 | 1.16873 | 5.40850 | 3.33668 | −3.18112 | 0.121576 | −9.27745 | −1.63406 | −9.08197 | ||||||||||||||||||
1.4 | −2.63857 | −2.69878 | 4.96206 | −3.68596 | 7.12092 | −1.95163 | −7.81561 | 4.28341 | 9.72566 | ||||||||||||||||||
1.5 | −2.45640 | −1.04637 | 4.03388 | 1.11649 | 2.57030 | −5.12832 | −4.99602 | −1.90511 | −2.74254 | ||||||||||||||||||
1.6 | −2.44956 | 1.54457 | 4.00036 | 1.84227 | −3.78353 | 3.24730 | −4.90002 | −0.614299 | −4.51276 | ||||||||||||||||||
1.7 | −2.33416 | 2.99604 | 3.44829 | −4.14609 | −6.99323 | −0.932097 | −3.38053 | 5.97627 | 9.67762 | ||||||||||||||||||
1.8 | −2.31836 | −1.91788 | 3.37477 | −1.08742 | 4.44632 | 3.16877 | −3.18721 | 0.678259 | 2.52103 | ||||||||||||||||||
1.9 | −2.24598 | 0.778915 | 3.04443 | −0.486460 | −1.74943 | 3.60332 | −2.34577 | −2.39329 | 1.09258 | ||||||||||||||||||
1.10 | −2.22533 | −2.62054 | 2.95208 | −3.22919 | 5.83156 | −0.0263326 | −2.11869 | 3.86723 | 7.18600 | ||||||||||||||||||
1.11 | −2.21189 | 1.34816 | 2.89245 | 0.0332713 | −2.98198 | −0.532643 | −1.97399 | −1.18247 | −0.0735923 | ||||||||||||||||||
1.12 | −2.08350 | 3.18490 | 2.34096 | −0.383847 | −6.63573 | −1.77595 | −0.710394 | 7.14359 | 0.799743 | ||||||||||||||||||
1.13 | −1.88961 | −1.56134 | 1.57061 | 0.136642 | 2.95031 | −4.65970 | 0.811376 | −0.562224 | −0.258199 | ||||||||||||||||||
1.14 | −1.85908 | −2.21837 | 1.45616 | 2.21766 | 4.12412 | 3.50843 | 1.01104 | 1.92117 | −4.12280 | ||||||||||||||||||
1.15 | −1.76421 | −0.616396 | 1.11244 | 0.592705 | 1.08745 | −0.184513 | 1.56585 | −2.62006 | −1.04566 | ||||||||||||||||||
1.16 | −1.56930 | 3.26822 | 0.462708 | −3.90153 | −5.12882 | 2.79245 | 2.41248 | 7.68125 | 6.12267 | ||||||||||||||||||
1.17 | −1.50369 | 0.405610 | 0.261082 | 4.23179 | −0.609911 | −3.07647 | 2.61479 | −2.83548 | −6.36330 | ||||||||||||||||||
1.18 | −1.40634 | 2.04244 | −0.0222050 | −3.82162 | −2.87236 | 0.994724 | 2.84391 | 1.17155 | 5.37450 | ||||||||||||||||||
1.19 | −1.20319 | 0.683964 | −0.552340 | −2.45715 | −0.822937 | −1.30297 | 3.07094 | −2.53219 | 2.95642 | ||||||||||||||||||
1.20 | −1.16307 | −0.290693 | −0.647264 | 2.95820 | 0.338097 | −1.06319 | 3.07896 | −2.91550 | −3.44060 | ||||||||||||||||||
See all 60 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.s | ✓ | 60 |
23.b | odd | 2 | 1 | 8993.2.a.t | yes | 60 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
8993.2.a.s | ✓ | 60 | 1.a | even | 1 | 1 | trivial |
8993.2.a.t | yes | 60 | 23.b | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8993))\):
\( T_{2}^{60} - 92 T_{2}^{58} + 8 T_{2}^{57} + 4002 T_{2}^{56} - 692 T_{2}^{55} - 109472 T_{2}^{54} + \cdots + 873 \) |
\( T_{5}^{60} + 8 T_{5}^{59} - 152 T_{5}^{58} - 1360 T_{5}^{57} + 10384 T_{5}^{56} + 107828 T_{5}^{55} + \cdots + 131647560768 \) |