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: | \(28\) |
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 | −2.65673 | 2.27240 | 5.05820 | −0.843549 | −6.03714 | −1.37381 | −8.12480 | 2.16379 | 2.24108 | ||||||||||||||||||
1.2 | −2.64864 | −1.20239 | 5.01532 | −2.61852 | 3.18469 | −1.89093 | −7.98650 | −1.55427 | 6.93554 | ||||||||||||||||||
1.3 | −2.42689 | 0.360604 | 3.88978 | −1.70693 | −0.875146 | −4.05794 | −4.58629 | −2.86996 | 4.14252 | ||||||||||||||||||
1.4 | −2.09853 | 2.13847 | 2.40381 | 3.18057 | −4.48764 | −2.64231 | −0.847411 | 1.57306 | −6.67450 | ||||||||||||||||||
1.5 | −1.94401 | −2.54293 | 1.77917 | 1.58898 | 4.94348 | 1.40808 | 0.429301 | 3.46650 | −3.08899 | ||||||||||||||||||
1.6 | −1.85401 | 2.41177 | 1.43734 | −0.990716 | −4.47143 | 3.76939 | 1.04318 | 2.81662 | 1.83679 | ||||||||||||||||||
1.7 | −1.48343 | −3.27337 | 0.200579 | 3.06348 | 4.85584 | 0.954273 | 2.66932 | 7.71497 | −4.54447 | ||||||||||||||||||
1.8 | −1.38336 | 0.109617 | −0.0863204 | −3.26254 | −0.151640 | 1.23528 | 2.88613 | −2.98798 | 4.51327 | ||||||||||||||||||
1.9 | −1.37246 | 0.796777 | −0.116357 | 0.411530 | −1.09354 | 1.05670 | 2.90461 | −2.36515 | −0.564808 | ||||||||||||||||||
1.10 | −0.810364 | 3.21601 | −1.34331 | −0.787794 | −2.60614 | −3.79576 | 2.70930 | 7.34274 | 0.638400 | ||||||||||||||||||
1.11 | −0.620276 | −1.18323 | −1.61526 | −0.583301 | 0.733929 | −4.46344 | 2.24246 | −1.59997 | 0.361807 | ||||||||||||||||||
1.12 | −0.542595 | −0.841086 | −1.70559 | −3.21547 | 0.456370 | 3.42767 | 2.01064 | −2.29257 | 1.74470 | ||||||||||||||||||
1.13 | −0.402915 | −2.36640 | −1.83766 | 4.07633 | 0.953461 | −2.03587 | 1.54625 | 2.59987 | −1.64241 | ||||||||||||||||||
1.14 | −0.0816744 | −2.10247 | −1.99333 | 0.911276 | 0.171718 | 3.20978 | 0.326153 | 1.42039 | −0.0744280 | ||||||||||||||||||
1.15 | 0.0672808 | 2.07633 | −1.99547 | 0.879559 | 0.139697 | 0.226756 | −0.268819 | 1.31116 | 0.0591775 | ||||||||||||||||||
1.16 | 0.0774088 | 1.19355 | −1.99401 | 2.17459 | 0.0923914 | 1.63543 | −0.309172 | −1.57544 | 0.168332 | ||||||||||||||||||
1.17 | 0.188180 | 0.234978 | −1.96459 | −4.09856 | 0.0442181 | −4.01742 | −0.746056 | −2.94479 | −0.771267 | ||||||||||||||||||
1.18 | 0.718188 | 2.92369 | −1.48421 | 3.34576 | 2.09976 | −4.97196 | −2.50231 | 5.54799 | 2.40288 | ||||||||||||||||||
1.19 | 1.00765 | −2.79972 | −0.984651 | −0.761194 | −2.82113 | 1.83038 | −3.00747 | 4.83846 | −0.767014 | ||||||||||||||||||
1.20 | 1.25336 | 2.57537 | −0.429096 | −3.04683 | 3.22786 | 2.79376 | −3.04452 | 3.63253 | −3.81877 | ||||||||||||||||||
See all 28 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.o | ✓ | 28 |
23.b | odd | 2 | 1 | 8993.2.a.p | yes | 28 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
8993.2.a.o | ✓ | 28 | 1.a | even | 1 | 1 | trivial |
8993.2.a.p | yes | 28 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8993))\):
\( T_{2}^{28} - 40 T_{2}^{26} - 2 T_{2}^{25} + 700 T_{2}^{24} + 72 T_{2}^{23} - 7049 T_{2}^{22} - 1110 T_{2}^{21} + \cdots + 1 \) |
\( T_{5}^{28} + 4 T_{5}^{27} - 72 T_{5}^{26} - 296 T_{5}^{25} + 2212 T_{5}^{24} + 9468 T_{5}^{23} + \cdots - 3458704 \) |