LMFDB - Embedded newform 8712.2.a.bz.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8712,2,Mod(1,8712)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8712, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8712.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8712 = 2^{3} \cdot 3^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8712.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:	$69.5656702409$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.13625.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 11x^{2} + 12x + 31 $ x^4 - 2x^3 - 11x^2 + 12*x + 31
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 264)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-2.41309$ of defining polynomial
Character	$\chi$	$=$	8712.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.41309 q^{5} -4.72744 q^{7} +O(q^{10})$	$q+2.41309 q^{5} -4.72744 q^{7} -6.01386 q^{13} -0.890597 q^{17} -3.31435 q^{19} -0.204948 q^{23} +0.822982 q^{25} +9.57087 q^{29} -9.12129 q^{31} -11.4077 q^{35} +1.44102 q^{37} -3.54172 q^{41} -7.79505 q^{43} +9.17164 q^{47} +15.3487 q^{49} +4.64915 q^{53} +0.989331 q^{59} +6.43232 q^{61} -14.5120 q^{65} +6.17505 q^{67} -6.80375 q^{71} +3.86267 q^{73} -4.53975 q^{79} +9.99462 q^{83} -2.14909 q^{85} +0.930414 q^{89} +28.4301 q^{91} -7.99781 q^{95} +2.92369 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{5} - 5 q^{7}+O(q^{10}) $ 4 * q - 2 * q^5 - 5 * q^7	$ 4 q - 2 q^{5} - 5 q^{7} + 2 q^{13} - 13 q^{17} - 11 q^{19} - 8 q^{23} + 6 q^{25} + 11 q^{29} + 2 q^{31} + 5 q^{35} + 4 q^{37} - 6 q^{41} - 24 q^{43} - 5 q^{47} + 17 q^{49} - 2 q^{53} + 4 q^{59} + 27 q^{61} - 21 q^{65} + 19 q^{67} - 17 q^{71} + 15 q^{73} - 7 q^{79} + q^{83} + 4 q^{85} - 6 q^{89} + 50 q^{91} + 33 q^{95} + 8 q^{97}+O(q^{100}) $ 4 * q - 2 * q^5 - 5 * q^7 + 2 * q^13 - 13 * q^17 - 11 * q^19 - 8 * q^23 + 6 * q^25 + 11 * q^29 + 2 * q^31 + 5 * q^35 + 4 * q^37 - 6 * q^41 - 24 * q^43 - 5 * q^47 + 17 * q^49 - 2 * q^53 + 4 * q^59 + 27 * q^61 - 21 * q^65 + 19 * q^67 - 17 * q^71 + 15 * q^73 - 7 * q^79 + q^83 + 4 * q^85 - 6 * q^89 + 50 * q^91 + 33 * q^95 + 8 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	2.41309	1.07916	0.539582	−	0.841933i	$-0.318582\pi$
$5$	2.41309	1.07916	0.539582	+	0.841933i	$0.318582\pi$
$6$	0	0
$7$	−4.72744	−1.78680	−0.893402	−	0.449259i	$-0.851688\pi$
$7$	−4.72744	−1.78680	−0.893402	+	0.449259i	$0.851688\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−6.01386	−1.66794	−0.833972	−	0.551807i	$-0.813939\pi$
$13$	−6.01386	−1.66794	−0.833972	+	0.551807i	$0.813939\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−0.890597	−0.216002	−0.108001	−	0.994151i	$-0.534445\pi$
$17$	−0.890597	−0.216002	−0.108001	+	0.994151i	$0.534445\pi$
$18$	0	0
$19$	−3.31435	−0.760364	−0.380182	−	0.924912i	$-0.624139\pi$
$19$	−3.31435	−0.760364	−0.380182	+	0.924912i	$0.624139\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−0.204948	−0.0427347	−0.0213673	−	0.999772i	$-0.506802\pi$
$23$	−0.204948	−0.0427347	−0.0213673	+	0.999772i	$0.506802\pi$
$24$	0	0
$25$	0.822982	0.164596
$26$	0	0
$27$	0	0
$28$	0	0
$29$	9.57087	1.77727	0.888633	−	0.458619i	$-0.151656\pi$
$29$	9.57087	1.77727	0.888633	+	0.458619i	$0.151656\pi$
$30$	0	0
$31$	−9.12129	−1.63823	−0.819116	−	0.573628i	$-0.805536\pi$
$31$	−9.12129	−1.63823	−0.819116	+	0.573628i	$0.805536\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−11.4077	−1.92825
$36$	0	0
$37$	1.44102	0.236902	0.118451	−	0.992960i	$-0.462207\pi$
$37$	1.44102	0.236902	0.118451	+	0.992960i	$0.462207\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−3.54172	−0.553124	−0.276562	−	0.960996i	$-0.589195\pi$
$41$	−3.54172	−0.553124	−0.276562	+	0.960996i	$0.589195\pi$
$42$	0	0
$43$	−7.79505	−1.18873	−0.594367	−	0.804194i	$-0.702597\pi$
$43$	−7.79505	−1.18873	−0.594367	+	0.804194i	$0.702597\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	9.17164	1.33782	0.668911	−	0.743343i	$-0.266761\pi$
$47$	9.17164	1.33782	0.668911	+	0.743343i	$0.266761\pi$
$48$	0	0
$49$	15.3487	2.19267
$50$	0	0
$51$	0	0
$52$	0	0
$53$	4.64915	0.638610	0.319305	−	0.947652i	$-0.396550\pi$
$53$	4.64915	0.638610	0.319305	+	0.947652i	$0.396550\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	0.989331	0.128800	0.0644000	−	0.997924i	$-0.479487\pi$
$59$	0.989331	0.128800	0.0644000	+	0.997924i	$0.479487\pi$
$60$	0	0
$61$	6.43232	0.823574	0.411787	−	0.911280i	$-0.364905\pi$
$61$	6.43232	0.823574	0.411787	+	0.911280i	$0.364905\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−14.5120	−1.79999
$66$	0	0
$67$	6.17505	0.754402	0.377201	−	0.926131i	$-0.376887\pi$
$67$	6.17505	0.754402	0.377201	+	0.926131i	$0.376887\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−6.80375	−0.807456	−0.403728	−	0.914879i	$-0.632286\pi$
$71$	−6.80375	−0.807456	−0.403728	+	0.914879i	$0.632286\pi$
$72$	0	0
$73$	3.86267	0.452091	0.226045	−	0.974117i	$-0.427420\pi$
$73$	3.86267	0.452091	0.226045	+	0.974117i	$0.427420\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−4.53975	−0.510762	−0.255381	−	0.966841i	$-0.582201\pi$
$79$	−4.53975	−0.510762	−0.255381	+	0.966841i	$0.582201\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	9.99462	1.09705	0.548526	−	0.836133i	$-0.315189\pi$
$83$	9.99462	1.09705	0.548526	+	0.836133i	$0.315189\pi$
$84$	0	0
$85$	−2.14909	−0.233101
$86$	0	0
$87$	0	0
$88$	0	0
$89$	0.930414	0.0986237	0.0493119	−	0.998783i	$-0.484297\pi$
$89$	0.930414	0.0986237	0.0493119	+	0.998783i	$0.484297\pi$
$90$	0	0
$91$	28.4301	2.98029
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−7.99781	−0.820558
$96$	0	0
$97$	2.92369	0.296855	0.148428	−	0.988923i	$-0.452579\pi$
$97$	2.92369	0.296855	0.148428	+	0.988923i	$0.452579\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−13.5225	−1.34554	−0.672769	−	0.739853i	$-0.734895\pi$
$101$	−13.5225	−1.34554	−0.672769	+	0.739853i	$0.734895\pi$
$102$	0	0
$103$	12.8381	1.26497	0.632486	−	0.774572i	$-0.282035\pi$
$103$	12.8381	1.26497	0.632486	+	0.774572i	$0.282035\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	9.80694	0.948073	0.474036	−	0.880505i	$-0.342797\pi$
$107$	9.80694	0.948073	0.474036	+	0.880505i	$0.342797\pi$
$108$	0	0
$109$	10.5344	1.00901	0.504505	−	0.863409i	$-0.331675\pi$
$109$	10.5344	1.00901	0.504505	+	0.863409i	$0.331675\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	0.715550	0.0673133	0.0336567	−	0.999433i	$-0.489285\pi$
$113$	0.715550	0.0673133	0.0336567	+	0.999433i	$0.489285\pi$
$114$	0	0
$115$	−0.494558	−0.0461178
$116$	0	0
$117$	0	0
$118$	0	0
$119$	4.21024	0.385952
$120$	0	0
$121$	0	0
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−10.0795	−0.901538
$126$	0	0
$127$	0.663227	0.0588519	0.0294259	−	0.999567i	$-0.490632\pi$
$127$	0.663227	0.0588519	0.0294259	+	0.999567i	$0.490632\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−1.41309	−0.123462	−0.0617309	−	0.998093i	$-0.519662\pi$
$131$	−1.41309	−0.123462	−0.0617309	+	0.998093i	$0.519662\pi$
$132$	0	0
$133$	15.6684	1.35862
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−4.42375	−0.377947	−0.188973	−	0.981982i	$-0.560516\pi$
$137$	−4.42375	−0.377947	−0.188973	+	0.981982i	$0.560516\pi$
$138$	0	0
$139$	−11.7168	−0.993803	−0.496902	−	0.867807i	$-0.665529\pi$
$139$	−11.7168	−0.993803	−0.496902	+	0.867807i	$0.665529\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	23.0953	1.91796
$146$	0	0
$147$	0	0
$148$	0	0
$149$	17.3970	1.42522	0.712611	−	0.701560i	$-0.247513\pi$
$149$	17.3970	1.42522	0.712611	+	0.701560i	$0.247513\pi$
$150$	0	0
$151$	−2.11600	−0.172197	−0.0860987	−	0.996287i	$-0.527440\pi$
$151$	−2.11600	−0.172197	−0.0860987	+	0.996287i	$0.527440\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−22.0105	−1.76792
$156$	0	0
$157$	−3.19760	−0.255196	−0.127598	−	0.991826i	$-0.540727\pi$
$157$	−3.19760	−0.255196	−0.127598	+	0.991826i	$0.540727\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0.968880	0.0763585
$162$	0	0
$163$	4.16513	0.326238	0.163119	−	0.986606i	$-0.447845\pi$
$163$	4.16513	0.326238	0.163119	+	0.986606i	$0.447845\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	16.9667	1.31292	0.656461	−	0.754360i	$-0.272053\pi$
$167$	16.9667	1.31292	0.656461	+	0.754360i	$0.272053\pi$
$168$	0	0
$169$	23.1665	1.78204
$170$	0	0
$171$	0	0
$172$	0	0
$173$	25.1006	1.90837	0.954183	−	0.299225i	$-0.0967282\pi$
$173$	25.1006	1.90837	0.954183	+	0.299225i	$0.0967282\pi$
$174$	0	0
$175$	−3.89060	−0.294101
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−1.87015	−0.139781	−0.0698906	−	0.997555i	$-0.522265\pi$
$179$	−1.87015	−0.139781	−0.0698906	+	0.997555i	$0.522265\pi$
$180$	0	0
$181$	−13.2776	−0.986919	−0.493460	−	0.869769i	$-0.664268\pi$
$181$	−13.2776	−0.986919	−0.493460	+	0.869769i	$0.664268\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	3.47730	0.255656
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	7.65982	0.554245	0.277123	−	0.960835i	$-0.410619\pi$
$191$	7.65982	0.554245	0.277123	+	0.960835i	$0.410619\pi$
$192$	0	0
$193$	15.1039	1.08720	0.543601	−	0.839344i	$-0.317060\pi$
$193$	15.1039	1.08720	0.543601	+	0.839344i	$0.317060\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−6.48280	−0.461881	−0.230940	−	0.972968i	$-0.574180\pi$
$197$	−6.48280	−0.461881	−0.230940	+	0.972968i	$0.574180\pi$
$198$	0	0
$199$	7.03322	0.498572	0.249286	−	0.968430i	$-0.419804\pi$
$199$	7.03322	0.498572	0.249286	+	0.968430i	$0.419804\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−45.2457	−3.17562
$204$	0	0
$205$	−8.54648	−0.596912
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	5.52568	0.380403	0.190202	−	0.981745i	$-0.439086\pi$
$211$	5.52568	0.380403	0.190202	+	0.981745i	$0.439086\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−18.8101	−1.28284
$216$	0	0
$217$	43.1203	2.92720
$218$	0	0
$219$	0	0
$220$	0	0
$221$	5.35592	0.360278
$222$	0	0
$223$	−8.41090	−0.563235	−0.281618	−	0.959527i	$-0.590871\pi$
$223$	−8.41090	−0.563235	−0.281618	+	0.959527i	$0.590871\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	10.8593	0.720755	0.360377	−	0.932807i	$-0.382648\pi$
$227$	10.8593	0.720755	0.360377	+	0.932807i	$0.382648\pi$
$228$	0	0
$229$	22.6311	1.49551	0.747754	−	0.663976i	$-0.231132\pi$
$229$	22.6311	1.49551	0.747754	+	0.663976i	$0.231132\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−1.23077	−0.0806307	−0.0403154	−	0.999187i	$-0.512836\pi$
$233$	−1.23077	−0.0806307	−0.0403154	+	0.999187i	$0.512836\pi$
$234$	0	0
$235$	22.1320	1.44373
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−10.0708	−0.651426	−0.325713	−	0.945469i	$-0.605604\pi$
$239$	−10.0708	−0.651426	−0.325713	+	0.945469i	$0.605604\pi$
$240$	0	0
$241$	−4.56095	−0.293797	−0.146898	−	0.989152i	$-0.546929\pi$
$241$	−4.56095	−0.293797	−0.146898	+	0.989152i	$0.546929\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	37.0376	2.36625
$246$	0	0
$247$	19.9320	1.26825
$248$	0	0
$249$	0	0
$250$	0	0
$251$	6.74932	0.426013	0.213007	−	0.977051i	$-0.431674\pi$
$251$	6.74932	0.426013	0.213007	+	0.977051i	$0.431674\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	5.41111	0.337536	0.168768	−	0.985656i	$-0.446021\pi$
$257$	5.41111	0.337536	0.168768	+	0.985656i	$0.446021\pi$
$258$	0	0
$259$	−6.81231	−0.423297
$260$	0	0
$261$	0	0
$262$	0	0
$263$	22.3831	1.38020	0.690101	−	0.723713i	$-0.257566\pi$
$263$	22.3831	1.38020	0.690101	+	0.723713i	$0.257566\pi$
$264$	0	0
$265$	11.2188	0.689166
$266$	0	0
$267$	0	0
$268$	0	0
$269$	23.3819	1.42562	0.712809	−	0.701358i	$-0.247422\pi$
$269$	23.3819	1.42562	0.712809	+	0.701358i	$0.247422\pi$
$270$	0	0
$271$	−17.3830	−1.05594	−0.527970	−	0.849263i	$-0.677047\pi$
$271$	−17.3830	−1.05594	−0.527970	+	0.849263i	$0.677047\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−9.27235	−0.557121	−0.278561	−	0.960419i	$-0.589857\pi$
$277$	−9.27235	−0.557121	−0.278561	+	0.960419i	$0.589857\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−5.06211	−0.301980	−0.150990	−	0.988535i	$-0.548246\pi$
$281$	−5.06211	−0.301980	−0.150990	+	0.988535i	$0.548246\pi$
$282$	0	0
$283$	−9.33677	−0.555014	−0.277507	−	0.960724i	$-0.589508\pi$
$283$	−9.33677	−0.555014	−0.277507	+	0.960724i	$0.589508\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	16.7433	0.988324
$288$	0	0
$289$	−16.2068	−0.953343
$290$	0	0
$291$	0	0
$292$	0	0
$293$	5.51271	0.322056	0.161028	−	0.986950i	$-0.448519\pi$
$293$	5.51271	0.322056	0.161028	+	0.986950i	$0.448519\pi$
$294$	0	0
$295$	2.38734	0.138996
$296$	0	0
$297$	0	0
$298$	0	0
$299$	1.23253	0.0712791
$300$	0	0
$301$	36.8506	2.12403
$302$	0	0
$303$	0	0
$304$	0	0
$305$	15.5217	0.888772
$306$	0	0
$307$	−18.5988	−1.06149	−0.530745	−	0.847532i	$-0.678088\pi$
$307$	−18.5988	−1.06149	−0.530745	+	0.847532i	$0.678088\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−31.6103	−1.79246	−0.896229	−	0.443592i	$-0.853704\pi$
$311$	−31.6103	−1.79246	−0.896229	+	0.443592i	$0.853704\pi$
$312$	0	0
$313$	−16.4454	−0.929550	−0.464775	−	0.885429i	$-0.653865\pi$
$313$	−16.4454	−0.929550	−0.464775	+	0.885429i	$0.653865\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−26.4752	−1.48700	−0.743499	−	0.668737i	$-0.766835\pi$
$317$	−26.4752	−1.48700	−0.743499	+	0.668737i	$0.766835\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	2.95175	0.164240
$324$	0	0
$325$	−4.94930	−0.274538
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−43.3584	−2.39042
$330$	0	0
$331$	−18.4301	−1.01301	−0.506506	−	0.862237i	$-0.669063\pi$
$331$	−18.4301	−1.01301	−0.506506	+	0.862237i	$0.669063\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	14.9009	0.814124
$336$	0	0
$337$	−0.343284	−0.0186999	−0.00934995	−	0.999956i	$-0.502976\pi$
$337$	−0.343284	−0.0186999	−0.00934995	+	0.999956i	$0.502976\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−39.4678	−2.13106
$344$	0	0
$345$	0	0
$346$	0	0
$347$	8.87258	0.476305	0.238153	−	0.971228i	$-0.423458\pi$
$347$	8.87258	0.476305	0.238153	+	0.971228i	$0.423458\pi$
$348$	0	0
$349$	34.4368	1.84336	0.921679	−	0.387953i	$-0.126818\pi$
$349$	34.4368	1.84336	0.921679	+	0.387953i	$0.126818\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	35.5171	1.89039	0.945193	−	0.326513i	$-0.105874\pi$
$353$	35.5171	1.89039	0.945193	+	0.326513i	$0.105874\pi$
$354$	0	0
$355$	−16.4180	−0.871379
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−19.4377	−1.02588	−0.512942	−	0.858423i	$-0.671445\pi$
$359$	−19.4377	−1.02588	−0.512942	+	0.858423i	$0.671445\pi$
$360$	0	0
$361$	−8.01508	−0.421846
$362$	0	0
$363$	0	0
$364$	0	0
$365$	9.32094	0.487881
$366$	0	0
$367$	−13.3400	−0.696340	−0.348170	−	0.937431i	$-0.613197\pi$
$367$	−13.3400	−0.696340	−0.348170	+	0.937431i	$0.613197\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−21.9786	−1.14107
$372$	0	0
$373$	19.4513	1.00715	0.503576	−	0.863951i	$-0.332017\pi$
$373$	19.4513	1.00715	0.503576	+	0.863951i	$0.332017\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−57.5579	−2.96438
$378$	0	0
$379$	−24.6450	−1.26593	−0.632964	−	0.774181i	$-0.718162\pi$
$379$	−24.6450	−1.26593	−0.632964	+	0.774181i	$0.718162\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	27.1492	1.38726	0.693630	−	0.720331i	$-0.256010\pi$
$383$	27.1492	1.38726	0.693630	+	0.720331i	$0.256010\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	5.53643	0.280708	0.140354	−	0.990101i	$-0.455176\pi$
$389$	5.53643	0.280708	0.140354	+	0.990101i	$0.455176\pi$
$390$	0	0
$391$	0.182526	0.00923076
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−10.9548	−0.551196
$396$	0	0
$397$	26.4130	1.32563	0.662815	−	0.748783i	$-0.269361\pi$
$397$	26.4130	1.32563	0.662815	+	0.748783i	$0.269361\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−26.8625	−1.34145	−0.670723	−	0.741708i	$-0.734016\pi$
$401$	−26.8625	−1.34145	−0.670723	+	0.741708i	$0.734016\pi$
$402$	0	0
$403$	54.8541	2.73248
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	6.11994	0.302611	0.151306	−	0.988487i	$-0.451652\pi$
$409$	6.11994	0.302611	0.151306	+	0.988487i	$0.451652\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−4.67700	−0.230140
$414$	0	0
$415$	24.1179	1.18390
$416$	0	0
$417$	0	0
$418$	0	0
$419$	11.1984	0.547075	0.273538	−	0.961861i	$-0.411806\pi$
$419$	11.1984	0.547075	0.273538	+	0.961861i	$0.411806\pi$
$420$	0	0
$421$	20.6511	1.00647	0.503237	−	0.864148i	$-0.332142\pi$
$421$	20.6511	1.00647	0.503237	+	0.864148i	$0.332142\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−0.732946	−0.0355531
$426$	0	0
$427$	−30.4084	−1.47156
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−31.2231	−1.50397	−0.751983	−	0.659183i	$-0.770902\pi$
$431$	−31.2231	−1.50397	−0.751983	+	0.659183i	$0.770902\pi$
$432$	0	0
$433$	−38.6576	−1.85776	−0.928882	−	0.370376i	$-0.879229\pi$
$433$	−38.6576	−1.85776	−0.928882	+	0.370376i	$0.879229\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0.679271	0.0324939
$438$	0	0
$439$	−10.5363	−0.502872	−0.251436	−	0.967874i	$-0.580903\pi$
$439$	−10.5363	−0.502872	−0.251436	+	0.967874i	$0.580903\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−7.62971	−0.362498	−0.181249	−	0.983437i	$-0.558014\pi$
$443$	−7.62971	−0.362498	−0.181249	+	0.983437i	$0.558014\pi$
$444$	0	0
$445$	2.24517	0.106431
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−40.0130	−1.88833	−0.944166	−	0.329471i	$-0.893130\pi$
$449$	−40.0130	−1.88833	−0.944166	+	0.329471i	$0.893130\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	68.6043	3.21622
$456$	0	0
$457$	24.1910	1.13161	0.565804	−	0.824540i	$-0.308566\pi$
$457$	24.1910	1.13161	0.565804	+	0.824540i	$0.308566\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	31.9494	1.48803	0.744017	−	0.668161i	$-0.232918\pi$
$461$	31.9494	1.48803	0.744017	+	0.668161i	$0.232918\pi$
$462$	0	0
$463$	22.5612	1.04851	0.524254	−	0.851562i	$-0.324344\pi$
$463$	22.5612	1.04851	0.524254	+	0.851562i	$0.324344\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	22.4570	1.03919	0.519593	−	0.854414i	$-0.326084\pi$
$467$	22.4570	1.03919	0.519593	+	0.854414i	$0.326084\pi$
$468$	0	0
$469$	−29.1921	−1.34797
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−2.72765	−0.125153
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−2.02990	−0.0927486	−0.0463743	−	0.998924i	$-0.514767\pi$
$479$	−2.02990	−0.0927486	−0.0463743	+	0.998924i	$0.514767\pi$
$480$	0	0
$481$	−8.66607	−0.395139
$482$	0	0
$483$	0	0
$484$	0	0
$485$	7.05511	0.320356
$486$	0	0
$487$	22.4634	1.01791	0.508956	−	0.860793i	$-0.330032\pi$
$487$	22.4634	1.01791	0.508956	+	0.860793i	$0.330032\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	16.8414	0.760041	0.380020	−	0.924978i	$-0.375917\pi$
$491$	16.8414	0.760041	0.380020	+	0.924978i	$0.375917\pi$
$492$	0	0
$493$	−8.52379	−0.383892
$494$	0	0
$495$	0	0
$496$	0	0
$497$	32.1643	1.44277
$498$	0	0
$499$	39.8828	1.78540	0.892700	−	0.450651i	$-0.148808\pi$
$499$	39.8828	1.78540	0.892700	+	0.450651i	$0.148808\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	19.6975	0.878266	0.439133	−	0.898422i	$-0.355286\pi$
$503$	19.6975	0.878266	0.439133	+	0.898422i	$0.355286\pi$
$504$	0	0
$505$	−32.6309	−1.45206
$506$	0	0
$507$	0	0
$508$	0	0
$509$	41.4369	1.83666	0.918328	−	0.395820i	$-0.129539\pi$
$509$	41.4369	1.83666	0.918328	+	0.395820i	$0.129539\pi$
$510$	0	0
$511$	−18.2605	−0.807797
$512$	0	0
$513$	0	0
$514$	0	0
$515$	30.9793	1.36511
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−14.9788	−0.656233	−0.328116	−	0.944637i	$-0.606414\pi$
$521$	−14.9788	−0.656233	−0.328116	+	0.944637i	$0.606414\pi$
$522$	0	0
$523$	17.0215	0.744296	0.372148	−	0.928173i	$-0.378621\pi$
$523$	17.0215	0.744296	0.372148	+	0.928173i	$0.378621\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	8.12339	0.353861
$528$	0	0
$529$	−22.9580	−0.998174
$530$	0	0
$531$	0	0
$532$	0	0
$533$	21.2994	0.922580
$534$	0	0
$535$	23.6650	1.02313
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−8.74151	−0.375827	−0.187913	−	0.982186i	$-0.560172\pi$
$541$	−8.74151	−0.375827	−0.187913	+	0.982186i	$0.560172\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	25.4203	1.08889
$546$	0	0
$547$	−11.2101	−0.479310	−0.239655	−	0.970858i	$-0.577034\pi$
$547$	−11.2101	−0.479310	−0.239655	+	0.970858i	$0.577034\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−31.7212	−1.35137
$552$	0	0
$553$	21.4614	0.912631
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−5.33018	−0.225847	−0.112923	−	0.993604i	$-0.536021\pi$
$557$	−5.33018	−0.225847	−0.112923	+	0.993604i	$0.536021\pi$
$558$	0	0
$559$	46.8783	1.98274
$560$	0	0
$561$	0	0
$562$	0	0
$563$	7.75659	0.326901	0.163451	−	0.986552i	$-0.447738\pi$
$563$	7.75659	0.326901	0.163451	+	0.986552i	$0.447738\pi$
$564$	0	0
$565$	1.72668	0.0726421
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−7.78119	−0.326205	−0.163102	−	0.986609i	$-0.552150\pi$
$569$	−7.78119	−0.326205	−0.163102	+	0.986609i	$0.552150\pi$
$570$	0	0
$571$	−46.6962	−1.95418	−0.977088	−	0.212834i	$-0.931731\pi$
$571$	−46.6962	−1.95418	−0.977088	+	0.212834i	$0.931731\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−0.168669	−0.00703398
$576$	0	0
$577$	−16.7281	−0.696400	−0.348200	−	0.937420i	$-0.613207\pi$
$577$	−16.7281	−0.696400	−0.348200	+	0.937420i	$0.613207\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−47.2490	−1.96022
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	14.9987	0.619063	0.309531	−	0.950889i	$-0.399828\pi$
$587$	14.9987	0.619063	0.309531	+	0.950889i	$0.399828\pi$
$588$	0	0
$589$	30.2312	1.24565
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−9.84235	−0.404177	−0.202088	−	0.979367i	$-0.564773\pi$
$593$	−9.84235	−0.404177	−0.202088	+	0.979367i	$0.564773\pi$
$594$	0	0
$595$	10.1597	0.416506
$596$	0	0
$597$	0	0
$598$	0	0
$599$	1.87796	0.0767313	0.0383656	−	0.999264i	$-0.487785\pi$
$599$	1.87796	0.0767313	0.0383656	+	0.999264i	$0.487785\pi$
$600$	0	0
$601$	27.5536	1.12394	0.561968	−	0.827159i	$-0.310045\pi$
$601$	27.5536	1.12394	0.561968	+	0.827159i	$0.310045\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	13.5666	0.550650	0.275325	−	0.961351i	$-0.411214\pi$
$607$	13.5666	0.550650	0.275325	+	0.961351i	$0.411214\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−55.1569	−2.23141
$612$	0	0
$613$	−42.5753	−1.71960	−0.859800	−	0.510632i	$-0.829412\pi$
$613$	−42.5753	−1.71960	−0.859800	+	0.510632i	$0.829412\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	5.95843	0.239877	0.119939	−	0.992781i	$-0.461730\pi$
$617$	5.95843	0.239877	0.119939	+	0.992781i	$0.461730\pi$
$618$	0	0
$619$	22.4578	0.902657	0.451329	−	0.892358i	$-0.350950\pi$
$619$	22.4578	0.902657	0.451329	+	0.892358i	$0.350950\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−4.39848	−0.176221
$624$	0	0
$625$	−28.4376	−1.13750
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−1.28336	−0.0511711
$630$	0	0
$631$	1.06245	0.0422957	0.0211478	−	0.999776i	$-0.493268\pi$
$631$	1.06245	0.0422957	0.0211478	+	0.999776i	$0.493268\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	1.60042	0.0635108
$636$	0	0
$637$	−92.3047	−3.65724
$638$	0	0
$639$	0	0
$640$	0	0
$641$	31.2649	1.23489	0.617444	−	0.786615i	$-0.288168\pi$
$641$	31.2649	1.23489	0.617444	+	0.786615i	$0.288168\pi$
$642$	0	0
$643$	−23.8626	−0.941048	−0.470524	−	0.882387i	$-0.655935\pi$
$643$	−23.8626	−0.941048	−0.470524	+	0.882387i	$0.655935\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−7.50855	−0.295192	−0.147596	−	0.989048i	$-0.547153\pi$
$647$	−7.50855	−0.295192	−0.147596	+	0.989048i	$0.547153\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	18.4828	0.723288	0.361644	−	0.932316i	$-0.382216\pi$
$653$	18.4828	0.723288	0.361644	+	0.932316i	$0.382216\pi$
$654$	0	0
$655$	−3.40990	−0.133236
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−24.0641	−0.937403	−0.468702	−	0.883357i	$-0.655278\pi$
$659$	−24.0641	−0.937403	−0.468702	+	0.883357i	$0.655278\pi$
$660$	0	0
$661$	10.9489	0.425863	0.212931	−	0.977067i	$-0.431699\pi$
$661$	10.9489	0.425863	0.212931	+	0.977067i	$0.431699\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	37.8092	1.46618
$666$	0	0
$667$	−1.96153	−0.0759509
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−18.3141	−0.705958	−0.352979	−	0.935631i	$-0.614831\pi$
$673$	−18.3141	−0.705958	−0.352979	+	0.935631i	$0.614831\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	32.6791	1.25596	0.627979	−	0.778230i	$-0.283882\pi$
$677$	32.6791	1.25596	0.627979	+	0.778230i	$0.283882\pi$
$678$	0	0
$679$	−13.8215	−0.530422
$680$	0	0
$681$	0	0
$682$	0	0
$683$	42.7468	1.63566	0.817831	−	0.575458i	$-0.195176\pi$
$683$	42.7468	1.63566	0.817831	+	0.575458i	$0.195176\pi$
$684$	0	0
$685$	−10.6749	−0.407867
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−27.9593	−1.06517
$690$	0	0
$691$	16.7916	0.638784	0.319392	−	0.947623i	$-0.396521\pi$
$691$	16.7916	0.638784	0.319392	+	0.947623i	$0.396521\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−28.2736	−1.07248
$696$	0	0
$697$	3.15425	0.119476
$698$	0	0
$699$	0	0
$700$	0	0
$701$	12.4882	0.471672	0.235836	−	0.971793i	$-0.424217\pi$
$701$	12.4882	0.471672	0.235836	+	0.971793i	$0.424217\pi$
$702$	0	0
$703$	−4.77603	−0.180132
$704$	0	0
$705$	0	0
$706$	0	0
$707$	63.9267	2.40421
$708$	0	0
$709$	−38.9421	−1.46250	−0.731250	−	0.682109i	$-0.761063\pi$
$709$	−38.9421	−1.46250	−0.731250	+	0.682109i	$0.761063\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	1.86939	0.0700093
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−37.1659	−1.38605	−0.693027	−	0.720911i	$-0.743724\pi$
$719$	−37.1659	−1.38605	−0.693027	+	0.720911i	$0.743724\pi$
$720$	0	0
$721$	−60.6911	−2.26026
$722$	0	0
$723$	0	0
$724$	0	0
$725$	7.87666	0.292532
$726$	0	0
$727$	−1.69543	−0.0628801	−0.0314400	−	0.999506i	$-0.510009\pi$
$727$	−1.69543	−0.0628801	−0.0314400	+	0.999506i	$0.510009\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	6.94225	0.256768
$732$	0	0
$733$	−15.0276	−0.555056	−0.277528	−	0.960717i	$-0.589515\pi$
$733$	−15.0276	−0.555056	−0.277528	+	0.960717i	$0.589515\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−30.9197	−1.13740	−0.568699	−	0.822546i	$-0.692553\pi$
$739$	−30.9197	−1.13740	−0.568699	+	0.822546i	$0.692553\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	5.26705	0.193229	0.0966147	−	0.995322i	$-0.469199\pi$
$743$	5.26705	0.193229	0.0966147	+	0.995322i	$0.469199\pi$
$744$	0	0
$745$	41.9806	1.53805
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−46.3617	−1.69402
$750$	0	0
$751$	32.5183	1.18661	0.593305	−	0.804977i	$-0.297823\pi$
$751$	32.5183	1.18661	0.593305	+	0.804977i	$0.297823\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−5.10608	−0.185829
$756$	0	0
$757$	−37.9472	−1.37922	−0.689608	−	0.724183i	$-0.742217\pi$
$757$	−37.9472	−1.37922	−0.689608	+	0.724183i	$0.742217\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	49.9924	1.81222	0.906111	−	0.423041i	$-0.139037\pi$
$761$	49.9924	1.81222	0.906111	+	0.423041i	$0.139037\pi$
$762$	0	0
$763$	−49.8006	−1.80290
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−5.94970	−0.214831
$768$	0	0
$769$	21.4620	0.773940	0.386970	−	0.922092i	$-0.373522\pi$
$769$	21.4620	0.773940	0.386970	+	0.922092i	$0.373522\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	8.34622	0.300193	0.150096	−	0.988671i	$-0.452042\pi$
$773$	8.34622	0.300193	0.150096	+	0.988671i	$0.452042\pi$
$774$	0	0
$775$	−7.50666	−0.269647
$776$	0	0
$777$	0	0
$778$	0	0
$779$	11.7385	0.420576
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−7.71609	−0.275399
$786$	0	0
$787$	20.2703	0.722558	0.361279	−	0.932458i	$-0.382340\pi$
$787$	20.2703	0.722558	0.361279	+	0.932458i	$0.382340\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−3.38272	−0.120276
$792$	0	0
$793$	−38.6830	−1.37368
$794$	0	0
$795$	0	0
$796$	0	0
$797$	36.7714	1.30251	0.651256	−	0.758858i	$-0.274243\pi$
$797$	36.7714	1.30251	0.651256	+	0.758858i	$0.274243\pi$
$798$	0	0
$799$	−8.16824	−0.288971
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	2.33799	0.0824034
$806$	0	0
$807$	0	0
$808$	0	0
$809$	0.507745	0.0178514	0.00892568	−	0.999960i	$-0.497159\pi$
$809$	0.507745	0.0178514	0.00892568	+	0.999960i	$0.497159\pi$
$810$	0	0
$811$	42.6567	1.49788	0.748939	−	0.662639i	$-0.230564\pi$
$811$	42.6567	1.49788	0.748939	+	0.662639i	$0.230564\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	10.0508	0.352065
$816$	0	0
$817$	25.8355	0.903871
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−31.9937	−1.11659	−0.558293	−	0.829644i	$-0.688544\pi$
$821$	−31.9937	−1.11659	−0.558293	+	0.829644i	$0.688544\pi$
$822$	0	0
$823$	27.1129	0.945095	0.472547	−	0.881305i	$-0.343335\pi$
$823$	27.1129	0.945095	0.472547	+	0.881305i	$0.343335\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−13.5398	−0.470823	−0.235412	−	0.971896i	$-0.575644\pi$
$827$	−13.5398	−0.470823	−0.235412	+	0.971896i	$0.575644\pi$
$828$	0	0
$829$	−12.4462	−0.432274	−0.216137	−	0.976363i	$-0.569346\pi$
$829$	−12.4462	−0.432274	−0.216137	+	0.976363i	$0.569346\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−13.6695	−0.473619
$834$	0	0
$835$	40.9421	1.41686
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−1.10411	−0.0381181	−0.0190591	−	0.999818i	$-0.506067\pi$
$839$	−1.10411	−0.0381181	−0.0190591	+	0.999818i	$0.506067\pi$
$840$	0	0
$841$	62.6016	2.15867
$842$	0	0
$843$	0	0
$844$	0	0
$845$	55.9027	1.92311
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−0.295334	−0.0101239
$852$	0	0
$853$	20.1251	0.689072	0.344536	−	0.938773i	$-0.388036\pi$
$853$	20.1251	0.689072	0.344536	+	0.938773i	$0.388036\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	26.5205	0.905924	0.452962	−	0.891530i	$-0.350367\pi$
$857$	26.5205	0.905924	0.452962	+	0.891530i	$0.350367\pi$
$858$	0	0
$859$	−40.3123	−1.37544	−0.687719	−	0.725977i	$-0.741388\pi$
$859$	−40.3123	−1.37544	−0.687719	+	0.725977i	$0.741388\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−51.4679	−1.75199	−0.875994	−	0.482322i	$-0.839793\pi$
$863$	−51.4679	−1.75199	−0.875994	+	0.482322i	$0.839793\pi$
$864$	0	0
$865$	60.5700	2.05944
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−37.1359	−1.25830
$872$	0	0
$873$	0	0
$874$	0	0
$875$	47.6502	1.61087
$876$	0	0
$877$	41.9088	1.41516	0.707579	−	0.706634i	$-0.249787\pi$
$877$	41.9088	1.41516	0.707579	+	0.706634i	$0.249787\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	30.6635	1.03308	0.516539	−	0.856263i	$-0.327220\pi$
$881$	30.6635	1.03308	0.516539	+	0.856263i	$0.327220\pi$
$882$	0	0
$883$	17.9547	0.604225	0.302112	−	0.953272i	$-0.402308\pi$
$883$	17.9547	0.604225	0.302112	+	0.953272i	$0.402308\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−24.2327	−0.813653	−0.406827	−	0.913505i	$-0.633365\pi$
$887$	−24.2327	−0.813653	−0.406827	+	0.913505i	$0.633365\pi$
$888$	0	0
$889$	−3.13536	−0.105157
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−30.3980	−1.01723
$894$	0	0
$895$	−4.51282	−0.150847
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−87.2987	−2.91157
$900$	0	0
$901$	−4.14052	−0.137941
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−32.0401	−1.06505
$906$	0	0
$907$	−15.0716	−0.500445	−0.250222	−	0.968188i	$-0.580504\pi$
$907$	−15.0716	−0.500445	−0.250222	+	0.968188i	$0.580504\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	48.7523	1.61524	0.807619	−	0.589705i	$-0.200756\pi$
$911$	48.7523	1.61524	0.807619	+	0.589705i	$0.200756\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	6.68027	0.220602
$918$	0	0
$919$	26.2027	0.864347	0.432173	−	0.901791i	$-0.357747\pi$
$919$	26.2027	0.864347	0.432173	+	0.901791i	$0.357747\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	40.9168	1.34679
$924$	0	0
$925$	1.18593	0.0389932
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−12.1106	−0.397337	−0.198668	−	0.980067i	$-0.563662\pi$
$929$	−12.1106	−0.397337	−0.198668	+	0.980067i	$0.563662\pi$
$930$	0	0
$931$	−50.8709	−1.66722
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	9.88414	0.322901	0.161450	−	0.986881i	$-0.448383\pi$
$937$	9.88414	0.322901	0.161450	+	0.986881i	$0.448383\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−7.98706	−0.260371	−0.130185	−	0.991490i	$-0.541557\pi$
$941$	−7.98706	−0.260371	−0.130185	+	0.991490i	$0.541557\pi$
$942$	0	0
$943$	0.725870	0.0236376
$944$	0	0
$945$	0	0
$946$	0	0
$947$	29.3817	0.954776	0.477388	−	0.878693i	$-0.341584\pi$
$947$	29.3817	0.954776	0.477388	+	0.878693i	$0.341584\pi$
$948$	0	0
$949$	−23.2295	−0.754062
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−39.7401	−1.28731	−0.643654	−	0.765316i	$-0.722583\pi$
$953$	−39.7401	−1.28731	−0.643654	+	0.765316i	$0.722583\pi$
$954$	0	0
$955$	18.4838	0.598122
$956$	0	0
$957$	0	0
$958$	0	0
$959$	20.9130	0.675317
$960$	0	0
$961$	52.1979	1.68380
$962$	0	0
$963$	0	0
$964$	0	0
$965$	36.4470	1.17327
$966$	0	0
$967$	35.5630	1.14363	0.571814	−	0.820383i	$-0.306240\pi$
$967$	35.5630	1.14363	0.571814	+	0.820383i	$0.306240\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	17.0965	0.548652	0.274326	−	0.961637i	$-0.411545\pi$
$971$	17.0965	0.548652	0.274326	+	0.961637i	$0.411545\pi$
$972$	0	0
$973$	55.3903	1.77573
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−30.2430	−0.967559	−0.483779	−	0.875190i	$-0.660736\pi$
$977$	−30.2430	−0.967559	−0.483779	+	0.875190i	$0.660736\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	18.1698	0.579525	0.289763	−	0.957099i	$-0.406424\pi$
$983$	18.1698	0.579525	0.289763	+	0.957099i	$0.406424\pi$
$984$	0	0
$985$	−15.6436	−0.498445
$986$	0	0
$987$	0	0
$988$	0	0
$989$	1.59758	0.0508002
$990$	0	0
$991$	19.2500	0.611497	0.305748	−	0.952112i	$-0.401093\pi$
$991$	19.2500	0.611497	0.305748	+	0.952112i	$0.401093\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	16.9718	0.538041
$996$	0	0
$997$	−6.68565	−0.211737	−0.105868	−	0.994380i	$-0.533762\pi$
$997$	−6.68565	−0.211737	−0.105868	+	0.994380i	$0.533762\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8712.2.a.bz.1.4		4
3.2	odd	2		2904.2.a.bb.1.1		4
11.5	even	5		792.2.r.f.289.1		8
11.9	even	5		792.2.r.f.433.1		8
11.10	odd	2		8712.2.a.cc.1.4		4
12.11	even	2		5808.2.a.co.1.1		4
33.5	odd	10		264.2.q.e.25.2	✓	8
33.20	odd	10		264.2.q.e.169.2	yes	8
33.32	even	2		2904.2.a.be.1.1		4
132.71	even	10		528.2.y.k.289.2		8
132.119	even	10		528.2.y.k.433.2		8
132.131	odd	2		5808.2.a.cl.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
264.2.q.e.25.2	✓	8	33.5	odd	10
264.2.q.e.169.2	yes	8	33.20	odd	10
528.2.y.k.289.2		8	132.71	even	10
528.2.y.k.433.2		8	132.119	even	10
792.2.r.f.289.1		8	11.5	even	5
792.2.r.f.433.1		8	11.9	even	5
2904.2.a.bb.1.1		4	3.2	odd	2
2904.2.a.be.1.1		4	33.32	even	2
5808.2.a.cl.1.1		4	132.131	odd	2
5808.2.a.co.1.1		4	12.11	even	2
8712.2.a.bz.1.4		4	1.1	even	1	trivial
8712.2.a.cc.1.4		4	11.10	odd	2

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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 8712 = 2^{3} \cdot 3^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8712.a (trivial)

\(f(q)\)	\(=\)	\(q+2.41309 q^{5} -4.72744 q^{7} +O(q^{10})\)	\(q+2.41309 q^{5} -4.72744 q^{7} -6.01386 q^{13} -0.890597 q^{17} -3.31435 q^{19} -0.204948 q^{23} +0.822982 q^{25} +9.57087 q^{29} -9.12129 q^{31} -11.4077 q^{35} +1.44102 q^{37} -3.54172 q^{41} -7.79505 q^{43} +9.17164 q^{47} +15.3487 q^{49} +4.64915 q^{53} +0.989331 q^{59} +6.43232 q^{61} -14.5120 q^{65} +6.17505 q^{67} -6.80375 q^{71} +3.86267 q^{73} -4.53975 q^{79} +9.99462 q^{83} -2.14909 q^{85} +0.930414 q^{89} +28.4301 q^{91} -7.99781 q^{95} +2.92369 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{5} - 5 q^{7}+O(q^{10}) \) 4 * q - 2 * q^5 - 5 * q^7	\( 4 q - 2 q^{5} - 5 q^{7} + 2 q^{13} - 13 q^{17} - 11 q^{19} - 8 q^{23} + 6 q^{25} + 11 q^{29} + 2 q^{31} + 5 q^{35} + 4 q^{37} - 6 q^{41} - 24 q^{43} - 5 q^{47} + 17 q^{49} - 2 q^{53} + 4 q^{59} + 27 q^{61} - 21 q^{65} + 19 q^{67} - 17 q^{71} + 15 q^{73} - 7 q^{79} + q^{83} + 4 q^{85} - 6 q^{89} + 50 q^{91} + 33 q^{95} + 8 q^{97}+O(q^{100}) \) 4 * q - 2 * q^5 - 5 * q^7 + 2 * q^13 - 13 * q^17 - 11 * q^19 - 8 * q^23 + 6 * q^25 + 11 * q^29 + 2 * q^31 + 5 * q^35 + 4 * q^37 - 6 * q^41 - 24 * q^43 - 5 * q^47 + 17 * q^49 - 2 * q^53 + 4 * q^59 + 27 * q^61 - 21 * q^65 + 19 * q^67 - 17 * q^71 + 15 * q^73 - 7 * q^79 + q^83 + 4 * q^85 - 6 * q^89 + 50 * q^91 + 33 * q^95 + 8 * q^97

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