LMFDB - Embedded newform 8712.2.a.ci.1.3

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:	$1$
Dimension:	$6$
Coefficient field:	6.6.62158000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 14x^{4} + 22x^{3} + 38x^{2} - 60x + 11 $ x^6 - 2x^5 - 14x^4 + 22x^3 + 38x^2 - 60*x + 11
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 792)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.216081$ 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)

$f(q)$	$=$	$q-1.13355 q^{5} +4.44328 q^{7} +O(q^{10})$	$q-1.13355 q^{5} +4.44328 q^{7} +3.39515 q^{13} -2.66356 q^{17} -3.12934 q^{19} +8.64153 q^{23} -3.71507 q^{25} -7.96650 q^{29} -4.17826 q^{31} -5.03666 q^{35} -5.49140 q^{37} -6.38540 q^{41} -4.96299 q^{43} -11.4591 q^{47} +12.7427 q^{49} -13.2039 q^{53} -2.19061 q^{59} -2.78824 q^{61} -3.84856 q^{65} +11.8576 q^{67} -5.13138 q^{71} -9.36764 q^{73} -4.30425 q^{79} -4.75768 q^{83} +3.01927 q^{85} -0.951061 q^{89} +15.0856 q^{91} +3.54725 q^{95} +5.77712 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 5 q^{5} + q^{7}+O(q^{10}) $ 6 * q - 5 * q^5 + q^7	$ 6 q - 5 q^{5} + q^{7} + 4 q^{13} + 6 q^{17} - 14 q^{19} - 6 q^{23} + 7 q^{25} + 7 q^{29} - 7 q^{31} - 2 q^{35} + 2 q^{37} - 6 q^{41} - 12 q^{43} - 26 q^{47} + 9 q^{49} - 9 q^{53} - 17 q^{59} + 20 q^{61} - 48 q^{65} + 14 q^{67} - 18 q^{71} + 7 q^{73} + 9 q^{79} + 11 q^{83} + 10 q^{85} - 14 q^{89} + 8 q^{91} + 30 q^{95} + 25 q^{97}+O(q^{100}) $ 6 * q - 5 * q^5 + q^7 + 4 * q^13 + 6 * q^17 - 14 * q^19 - 6 * q^23 + 7 * q^25 + 7 * q^29 - 7 * q^31 - 2 * q^35 + 2 * q^37 - 6 * q^41 - 12 * q^43 - 26 * q^47 + 9 * q^49 - 9 * q^53 - 17 * q^59 + 20 * q^61 - 48 * q^65 + 14 * q^67 - 18 * q^71 + 7 * q^73 + 9 * q^79 + 11 * q^83 + 10 * q^85 - 14 * q^89 + 8 * q^91 + 30 * q^95 + 25 * 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
$5$	−1.13355	−0.506937	−0.253468	−	0.967344i	$-0.581571\pi$
$5$	−1.13355	−0.506937	−0.253468	+	0.967344i	$0.581571\pi$
$6$	0	0
$7$	4.44328	1.67940	0.839701	−	0.543049i	$-0.182730\pi$
$7$	4.44328	1.67940	0.839701	+	0.543049i	$0.182730\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	3.39515	0.941646	0.470823	−	0.882228i	$-0.343957\pi$
$13$	3.39515	0.941646	0.470823	+	0.882228i	$0.343957\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−2.66356	−0.646009	−0.323004	−	0.946397i	$-0.604693\pi$
$17$	−2.66356	−0.646009	−0.323004	+	0.946397i	$0.604693\pi$
$18$	0	0
$19$	−3.12934	−0.717921	−0.358960	−	0.933353i	$-0.616869\pi$
$19$	−3.12934	−0.717921	−0.358960	+	0.933353i	$0.616869\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	8.64153	1.80188	0.900941	−	0.433941i	$-0.142877\pi$
$23$	8.64153	1.80188	0.900941	+	0.433941i	$0.142877\pi$
$24$	0	0
$25$	−3.71507	−0.743015
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−7.96650	−1.47934	−0.739671	−	0.672969i	$-0.765019\pi$
$29$	−7.96650	−1.47934	−0.739671	+	0.672969i	$0.765019\pi$
$30$	0	0
$31$	−4.17826	−0.750438	−0.375219	−	0.926936i	$-0.622432\pi$
$31$	−4.17826	−0.750438	−0.375219	+	0.926936i	$0.622432\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−5.03666	−0.851351
$36$	0	0
$37$	−5.49140	−0.902781	−0.451391	−	0.892326i	$-0.649072\pi$
$37$	−5.49140	−0.902781	−0.451391	+	0.892326i	$0.649072\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−6.38540	−0.997232	−0.498616	−	0.866823i	$-0.666158\pi$
$41$	−6.38540	−0.997232	−0.498616	+	0.866823i	$0.666158\pi$
$42$	0	0
$43$	−4.96299	−0.756849	−0.378425	−	0.925632i	$-0.623534\pi$
$43$	−4.96299	−0.756849	−0.378425	+	0.925632i	$0.623534\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−11.4591	−1.67148	−0.835738	−	0.549128i	$-0.814960\pi$
$47$	−11.4591	−1.67148	−0.835738	+	0.549128i	$0.814960\pi$
$48$	0	0
$49$	12.7427	1.82039
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−13.2039	−1.81369	−0.906846	−	0.421461i	$-0.861517\pi$
$53$	−13.2039	−1.81369	−0.906846	+	0.421461i	$0.861517\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−2.19061	−0.285193	−0.142597	−	0.989781i	$-0.545545\pi$
$59$	−2.19061	−0.285193	−0.142597	+	0.989781i	$0.545545\pi$
$60$	0	0
$61$	−2.78824	−0.356997	−0.178499	−	0.983940i	$-0.557124\pi$
$61$	−2.78824	−0.356997	−0.178499	+	0.983940i	$0.557124\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−3.84856	−0.477355
$66$	0	0
$67$	11.8576	1.44864	0.724319	−	0.689465i	$-0.242154\pi$
$67$	11.8576	1.44864	0.724319	+	0.689465i	$0.242154\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−5.13138	−0.608983	−0.304491	−	0.952515i	$-0.598486\pi$
$71$	−5.13138	−0.608983	−0.304491	+	0.952515i	$0.598486\pi$
$72$	0	0
$73$	−9.36764	−1.09640	−0.548199	−	0.836348i	$-0.684687\pi$
$73$	−9.36764	−1.09640	−0.548199	+	0.836348i	$0.684687\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−4.30425	−0.484266	−0.242133	−	0.970243i	$-0.577847\pi$
$79$	−4.30425	−0.484266	−0.242133	+	0.970243i	$0.577847\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−4.75768	−0.522223	−0.261112	−	0.965309i	$-0.584089\pi$
$83$	−4.75768	−0.522223	−0.261112	+	0.965309i	$0.584089\pi$
$84$	0	0
$85$	3.01927	0.327486
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−0.951061	−0.100812	−0.0504061	−	0.998729i	$-0.516052\pi$
$89$	−0.951061	−0.100812	−0.0504061	+	0.998729i	$0.516052\pi$
$90$	0	0
$91$	15.0856	1.58140
$92$	0	0
$93$	0	0
$94$	0	0
$95$	3.54725	0.363940
$96$	0	0
$97$	5.77712	0.586578	0.293289	−	0.956024i	$-0.405250\pi$
$97$	5.77712	0.586578	0.293289	+	0.956024i	$0.405250\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−6.22087	−0.618999	−0.309500	−	0.950900i	$-0.600162\pi$
$101$	−6.22087	−0.618999	−0.309500	+	0.950900i	$0.600162\pi$
$102$	0	0
$103$	14.8390	1.46213	0.731064	−	0.682309i	$-0.239024\pi$
$103$	14.8390	1.46213	0.731064	+	0.682309i	$0.239024\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1.77640	0.171731	0.0858656	−	0.996307i	$-0.472634\pi$
$107$	1.77640	0.171731	0.0858656	+	0.996307i	$0.472634\pi$
$108$	0	0
$109$	11.9643	1.14598	0.572988	−	0.819564i	$-0.305784\pi$
$109$	11.9643	1.14598	0.572988	+	0.819564i	$0.305784\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−6.68560	−0.628928	−0.314464	−	0.949269i	$-0.601825\pi$
$113$	−6.68560	−0.628928	−0.314464	+	0.949269i	$0.601825\pi$
$114$	0	0
$115$	−9.79556	−0.913441
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−11.8350	−1.08491
$120$	0	0
$121$	0	0
$122$	0	0
$123$	0	0
$124$	0	0
$125$	9.87893	0.883599
$126$	0	0
$127$	6.98695	0.619992	0.309996	−	0.950738i	$-0.399672\pi$
$127$	6.98695	0.619992	0.309996	+	0.950738i	$0.399672\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−8.64069	−0.754941	−0.377470	−	0.926022i	$-0.623206\pi$
$131$	−8.64069	−0.754941	−0.377470	+	0.926022i	$0.623206\pi$
$132$	0	0
$133$	−13.9045	−1.20568
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−4.39900	−0.375832	−0.187916	−	0.982185i	$-0.560173\pi$
$137$	−4.39900	−0.375832	−0.187916	+	0.982185i	$0.560173\pi$
$138$	0	0
$139$	−2.28197	−0.193554	−0.0967770	−	0.995306i	$-0.530853\pi$
$139$	−2.28197	−0.193554	−0.0967770	+	0.995306i	$0.530853\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	9.03039	0.749933
$146$	0	0
$147$	0	0
$148$	0	0
$149$	7.28634	0.596920	0.298460	−	0.954422i	$-0.403527\pi$
$149$	7.28634	0.596920	0.298460	+	0.954422i	$0.403527\pi$
$150$	0	0
$151$	−2.38069	−0.193737	−0.0968687	−	0.995297i	$-0.530883\pi$
$151$	−2.38069	−0.193737	−0.0968687	+	0.995297i	$0.530883\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	4.73625	0.380424
$156$	0	0
$157$	−21.6174	−1.72525	−0.862627	−	0.505840i	$-0.831183\pi$
$157$	−21.6174	−1.72525	−0.862627	+	0.505840i	$0.831183\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	38.3967	3.02608
$162$	0	0
$163$	−1.04334	−0.0817207	−0.0408604	−	0.999165i	$-0.513010\pi$
$163$	−1.04334	−0.0817207	−0.0408604	+	0.999165i	$0.513010\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	19.0753	1.47609	0.738047	−	0.674749i	$-0.235748\pi$
$167$	19.0753	1.47609	0.738047	+	0.674749i	$0.235748\pi$
$168$	0	0
$169$	−1.47293	−0.113302
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−20.2459	−1.53926	−0.769632	−	0.638488i	$-0.779560\pi$
$173$	−20.2459	−1.53926	−0.769632	+	0.638488i	$0.779560\pi$
$174$	0	0
$175$	−16.5071	−1.24782
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−8.31169	−0.621245	−0.310622	−	0.950533i	$-0.600537\pi$
$179$	−8.31169	−0.621245	−0.310622	+	0.950533i	$0.600537\pi$
$180$	0	0
$181$	−15.5605	−1.15660	−0.578300	−	0.815824i	$-0.696284\pi$
$181$	−15.5605	−1.15660	−0.578300	+	0.815824i	$0.696284\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	6.22476	0.457653
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	10.1940	0.737612	0.368806	−	0.929506i	$-0.379767\pi$
$191$	10.1940	0.737612	0.368806	+	0.929506i	$0.379767\pi$
$192$	0	0
$193$	25.4697	1.83335	0.916675	−	0.399633i	$-0.130863\pi$
$193$	25.4697	1.83335	0.916675	+	0.399633i	$0.130863\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	6.02962	0.429592	0.214796	−	0.976659i	$-0.431091\pi$
$197$	6.02962	0.429592	0.214796	+	0.976659i	$0.431091\pi$
$198$	0	0
$199$	−9.15227	−0.648788	−0.324394	−	0.945922i	$-0.605160\pi$
$199$	−9.15227	−0.648788	−0.324394	+	0.945922i	$0.605160\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−35.3974	−2.48441
$204$	0	0
$205$	7.23814	0.505534
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−11.8763	−0.817600	−0.408800	−	0.912624i	$-0.634053\pi$
$211$	−11.8763	−0.817600	−0.408800	+	0.912624i	$0.634053\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	5.62578	0.383675
$216$	0	0
$217$	−18.5652	−1.26029
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−9.04320	−0.608312
$222$	0	0
$223$	11.1259	0.745045	0.372523	−	0.928023i	$-0.378493\pi$
$223$	11.1259	0.745045	0.372523	+	0.928023i	$0.378493\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	24.7179	1.64058	0.820291	−	0.571946i	$-0.193811\pi$
$227$	24.7179	1.64058	0.820291	+	0.571946i	$0.193811\pi$
$228$	0	0
$229$	4.76137	0.314640	0.157320	−	0.987548i	$-0.449715\pi$
$229$	4.76137	0.314640	0.157320	+	0.987548i	$0.449715\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	19.4303	1.27292	0.636460	−	0.771310i	$-0.280398\pi$
$233$	19.4303	1.27292	0.636460	+	0.771310i	$0.280398\pi$
$234$	0	0
$235$	12.9894	0.847333
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−20.0002	−1.29370	−0.646852	−	0.762615i	$-0.723915\pi$
$239$	−20.0002	−1.29370	−0.646852	+	0.762615i	$0.723915\pi$
$240$	0	0
$241$	18.4246	1.18683	0.593415	−	0.804897i	$-0.297779\pi$
$241$	18.4246	1.18683	0.593415	+	0.804897i	$0.297779\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−14.4445	−0.922823
$246$	0	0
$247$	−10.6246	−0.676027
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−4.15611	−0.262331	−0.131166	−	0.991360i	$-0.541872\pi$
$251$	−4.15611	−0.262331	−0.131166	+	0.991360i	$0.541872\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−26.6349	−1.66144	−0.830720	−	0.556690i	$-0.812071\pi$
$257$	−26.6349	−1.66144	−0.830720	+	0.556690i	$0.812071\pi$
$258$	0	0
$259$	−24.3998	−1.51613
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−2.83968	−0.175102	−0.0875512	−	0.996160i	$-0.527904\pi$
$263$	−2.83968	−0.175102	−0.0875512	+	0.996160i	$0.527904\pi$
$264$	0	0
$265$	14.9672	0.919428
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−30.0375	−1.83142	−0.915708	−	0.401844i	$-0.868369\pi$
$269$	−30.0375	−1.83142	−0.915708	+	0.401844i	$0.868369\pi$
$270$	0	0
$271$	−14.1758	−0.861117	−0.430559	−	0.902563i	$-0.641683\pi$
$271$	−14.1758	−0.861117	−0.430559	+	0.902563i	$0.641683\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−17.2860	−1.03861	−0.519306	−	0.854588i	$-0.673810\pi$
$277$	−17.2860	−1.03861	−0.519306	+	0.854588i	$0.673810\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	8.36597	0.499072	0.249536	−	0.968365i	$-0.419722\pi$
$281$	8.36597	0.499072	0.249536	+	0.968365i	$0.419722\pi$
$282$	0	0
$283$	19.6394	1.16744	0.583721	−	0.811954i	$-0.301596\pi$
$283$	19.6394	1.16744	0.583721	+	0.811954i	$0.301596\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−28.3721	−1.67475
$288$	0	0
$289$	−9.90544	−0.582673
$290$	0	0
$291$	0	0
$292$	0	0
$293$	11.0786	0.647218	0.323609	−	0.946191i	$-0.395104\pi$
$293$	11.0786	0.647218	0.323609	+	0.946191i	$0.395104\pi$
$294$	0	0
$295$	2.48316	0.144575
$296$	0	0
$297$	0	0
$298$	0	0
$299$	29.3393	1.69674
$300$	0	0
$301$	−22.0520	−1.27105
$302$	0	0
$303$	0	0
$304$	0	0
$305$	3.16059	0.180975
$306$	0	0
$307$	7.33541	0.418654	0.209327	−	0.977846i	$-0.432873\pi$
$307$	7.33541	0.418654	0.209327	+	0.977846i	$0.432873\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−1.64074	−0.0930381	−0.0465190	−	0.998917i	$-0.514813\pi$
$311$	−1.64074	−0.0930381	−0.0465190	+	0.998917i	$0.514813\pi$
$312$	0	0
$313$	31.8255	1.79889	0.899443	−	0.437038i	$-0.143972\pi$
$313$	31.8255	1.79889	0.899443	+	0.437038i	$0.143972\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	12.6665	0.711420	0.355710	−	0.934596i	$-0.384239\pi$
$317$	12.6665	0.711420	0.355710	+	0.934596i	$0.384239\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	8.33520	0.463783
$324$	0	0
$325$	−12.6133	−0.699657
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−50.9158	−2.80708
$330$	0	0
$331$	3.53028	0.194042	0.0970210	−	0.995282i	$-0.469069\pi$
$331$	3.53028	0.194042	0.0970210	+	0.995282i	$0.469069\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−13.4411	−0.734368
$336$	0	0
$337$	18.9249	1.03091	0.515453	−	0.856918i	$-0.327624\pi$
$337$	18.9249	1.03091	0.515453	+	0.856918i	$0.327624\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	25.5166	1.37776
$344$	0	0
$345$	0	0
$346$	0	0
$347$	30.0421	1.61274	0.806372	−	0.591409i	$-0.201428\pi$
$347$	30.0421	1.61274	0.806372	+	0.591409i	$0.201428\pi$
$348$	0	0
$349$	27.4575	1.46977	0.734883	−	0.678194i	$-0.237237\pi$
$349$	27.4575	1.46977	0.734883	+	0.678194i	$0.237237\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	20.6756	1.10045	0.550227	−	0.835015i	$-0.314541\pi$
$353$	20.6756	1.10045	0.550227	+	0.835015i	$0.314541\pi$
$354$	0	0
$355$	5.81665	0.308716
$356$	0	0
$357$	0	0
$358$	0	0
$359$	32.9334	1.73816	0.869079	−	0.494673i	$-0.164712\pi$
$359$	32.9334	1.73816	0.869079	+	0.494673i	$0.164712\pi$
$360$	0	0
$361$	−9.20721	−0.484590
$362$	0	0
$363$	0	0
$364$	0	0
$365$	10.6186	0.555805
$366$	0	0
$367$	−28.3058	−1.47755	−0.738774	−	0.673953i	$-0.764595\pi$
$367$	−28.3058	−1.47755	−0.738774	+	0.673953i	$0.764595\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−58.6685	−3.04592
$372$	0	0
$373$	−5.00148	−0.258967	−0.129483	−	0.991582i	$-0.541332\pi$
$373$	−5.00148	−0.258967	−0.129483	+	0.991582i	$0.541332\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−27.0475	−1.39302
$378$	0	0
$379$	11.9875	0.615757	0.307878	−	0.951426i	$-0.400381\pi$
$379$	11.9875	0.615757	0.307878	+	0.951426i	$0.400381\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−0.0150482	−0.000768927 0	−0.000384464	−	1.00000i	$-0.500122\pi$
$383$	−0.0150482	−0.000768927 0	−0.000384464		1.00000i	$0.500122\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	12.8839	0.653238	0.326619	−	0.945156i	$-0.394091\pi$
$389$	12.8839	0.653238	0.326619	+	0.945156i	$0.394091\pi$
$390$	0	0
$391$	−23.0172	−1.16403
$392$	0	0
$393$	0	0
$394$	0	0
$395$	4.87907	0.245492
$396$	0	0
$397$	−23.9747	−1.20326	−0.601628	−	0.798777i	$-0.705481\pi$
$397$	−23.9747	−1.20326	−0.601628	+	0.798777i	$0.705481\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−21.5474	−1.07603	−0.538013	−	0.842936i	$-0.680825\pi$
$401$	−21.5474	−1.07603	−0.538013	+	0.842936i	$0.680825\pi$
$402$	0	0
$403$	−14.1858	−0.706647
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−2.90639	−0.143712	−0.0718560	−	0.997415i	$-0.522892\pi$
$409$	−2.90639	−0.143712	−0.0718560	+	0.997415i	$0.522892\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−9.73350	−0.478954
$414$	0	0
$415$	5.39305	0.264734
$416$	0	0
$417$	0	0
$418$	0	0
$419$	7.32085	0.357647	0.178824	−	0.983881i	$-0.442771\pi$
$419$	7.32085	0.357647	0.178824	+	0.983881i	$0.442771\pi$
$420$	0	0
$421$	15.5398	0.757364	0.378682	−	0.925527i	$-0.376377\pi$
$421$	15.5398	0.757364	0.378682	+	0.925527i	$0.376377\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	9.89533	0.479994
$426$	0	0
$427$	−12.3889	−0.599542
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−19.8088	−0.954154	−0.477077	−	0.878861i	$-0.658304\pi$
$431$	−19.8088	−0.954154	−0.477077	+	0.878861i	$0.658304\pi$
$432$	0	0
$433$	22.6895	1.09039	0.545193	−	0.838310i	$-0.316456\pi$
$433$	22.6895	1.09039	0.545193	+	0.838310i	$0.316456\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−27.0423	−1.29361
$438$	0	0
$439$	−35.2142	−1.68068	−0.840342	−	0.542057i	$-0.817646\pi$
$439$	−35.2142	−1.68068	−0.840342	+	0.542057i	$0.817646\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−8.83034	−0.419542	−0.209771	−	0.977751i	$-0.567272\pi$
$443$	−8.83034	−0.419542	−0.209771	+	0.977751i	$0.567272\pi$
$444$	0	0
$445$	1.07807	0.0511055
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−34.8511	−1.64473	−0.822363	−	0.568963i	$-0.807345\pi$
$449$	−34.8511	−1.64473	−0.822363	+	0.568963i	$0.807345\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−17.1002	−0.801671
$456$	0	0
$457$	21.0484	0.984601	0.492301	−	0.870425i	$-0.336156\pi$
$457$	21.0484	0.984601	0.492301	+	0.870425i	$0.336156\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−10.4969	−0.488888	−0.244444	−	0.969663i	$-0.578605\pi$
$461$	−10.4969	−0.488888	−0.244444	+	0.969663i	$0.578605\pi$
$462$	0	0
$463$	−25.3098	−1.17624	−0.588122	−	0.808772i	$-0.700133\pi$
$463$	−25.3098	−1.17624	−0.588122	+	0.808772i	$0.700133\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	2.41806	0.111894	0.0559472	−	0.998434i	$-0.482182\pi$
$467$	2.41806	0.111894	0.0559472	+	0.998434i	$0.482182\pi$
$468$	0	0
$469$	52.6867	2.43284
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	11.6257	0.533426
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−16.4930	−0.753582	−0.376791	−	0.926298i	$-0.622973\pi$
$479$	−16.4930	−0.753582	−0.376791	+	0.926298i	$0.622973\pi$
$480$	0	0
$481$	−18.6442	−0.850101
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−6.54863	−0.297358
$486$	0	0
$487$	17.1671	0.777915	0.388958	−	0.921256i	$-0.372835\pi$
$487$	17.1671	0.777915	0.388958	+	0.921256i	$0.372835\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−21.2365	−0.958390	−0.479195	−	0.877708i	$-0.659071\pi$
$491$	−21.2365	−0.958390	−0.479195	+	0.877708i	$0.659071\pi$
$492$	0	0
$493$	21.2193	0.955667
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−22.8002	−1.02273
$498$	0	0
$499$	−31.2113	−1.39721	−0.698606	−	0.715507i	$-0.746196\pi$
$499$	−31.2113	−1.39721	−0.698606	+	0.715507i	$0.746196\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	30.9158	1.37847	0.689233	−	0.724540i	$-0.257948\pi$
$503$	30.9158	1.37847	0.689233	+	0.724540i	$0.257948\pi$
$504$	0	0
$505$	7.05163	0.313794
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−1.60002	−0.0709198	−0.0354599	−	0.999371i	$-0.511290\pi$
$509$	−1.60002	−0.0709198	−0.0354599	+	0.999371i	$0.511290\pi$
$510$	0	0
$511$	−41.6230	−1.84129
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−16.8207	−0.741207
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	13.3146	0.583324	0.291662	−	0.956521i	$-0.405792\pi$
$521$	13.3146	0.583324	0.291662	+	0.956521i	$0.405792\pi$
$522$	0	0
$523$	−40.5247	−1.77202	−0.886010	−	0.463666i	$-0.846534\pi$
$523$	−40.5247	−1.77202	−0.886010	+	0.463666i	$0.846534\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	11.1291	0.484789
$528$	0	0
$529$	51.6760	2.24678
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−21.6794	−0.939040
$534$	0	0
$535$	−2.01363	−0.0870569
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−18.8778	−0.811618	−0.405809	−	0.913958i	$-0.633010\pi$
$541$	−18.8778	−0.811618	−0.405809	+	0.913958i	$0.633010\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−13.5621	−0.580937
$546$	0	0
$547$	−26.1131	−1.11652	−0.558258	−	0.829668i	$-0.688530\pi$
$547$	−26.1131	−1.11652	−0.558258	+	0.829668i	$0.688530\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	24.9299	1.06205
$552$	0	0
$553$	−19.1250	−0.813278
$554$	0	0
$555$	0	0
$556$	0	0
$557$	25.7585	1.09142	0.545712	−	0.837973i	$-0.316259\pi$
$557$	25.7585	1.09142	0.545712	+	0.837973i	$0.316259\pi$
$558$	0	0
$559$	−16.8501	−0.712684
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−9.57058	−0.403352	−0.201676	−	0.979452i	$-0.564639\pi$
$563$	−9.57058	−0.403352	−0.201676	+	0.979452i	$0.564639\pi$
$564$	0	0
$565$	7.57843	0.318827
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−2.37160	−0.0994226	−0.0497113	−	0.998764i	$-0.515830\pi$
$569$	−2.37160	−0.0994226	−0.0497113	+	0.998764i	$0.515830\pi$
$570$	0	0
$571$	−23.0358	−0.964017	−0.482009	−	0.876167i	$-0.660093\pi$
$571$	−23.0358	−0.964017	−0.482009	+	0.876167i	$0.660093\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−32.1039	−1.33883
$576$	0	0
$577$	8.54018	0.355532	0.177766	−	0.984073i	$-0.443113\pi$
$577$	8.54018	0.355532	0.177766	+	0.984073i	$0.443113\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−21.1397	−0.877023
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−2.85442	−0.117814	−0.0589072	−	0.998263i	$-0.518762\pi$
$587$	−2.85442	−0.117814	−0.0589072	+	0.998263i	$0.518762\pi$
$588$	0	0
$589$	13.0752	0.538755
$590$	0	0
$591$	0	0
$592$	0	0
$593$	30.5513	1.25459	0.627296	−	0.778781i	$-0.284162\pi$
$593$	30.5513	1.25459	0.627296	+	0.778781i	$0.284162\pi$
$594$	0	0
$595$	13.4155	0.549980
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−14.3866	−0.587820	−0.293910	−	0.955833i	$-0.594957\pi$
$599$	−14.3866	−0.587820	−0.293910	+	0.955833i	$0.594957\pi$
$600$	0	0
$601$	−2.95232	−0.120428	−0.0602139	−	0.998185i	$-0.519178\pi$
$601$	−2.95232	−0.120428	−0.0602139	+	0.998185i	$0.519178\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	21.8567	0.887135	0.443567	−	0.896241i	$-0.353713\pi$
$607$	21.8567	0.887135	0.443567	+	0.896241i	$0.353713\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−38.9053	−1.57394
$612$	0	0
$613$	1.67093	0.0674884	0.0337442	−	0.999431i	$-0.489257\pi$
$613$	1.67093	0.0674884	0.0337442	+	0.999431i	$0.489257\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	36.7149	1.47809	0.739043	−	0.673658i	$-0.235278\pi$
$617$	36.7149	1.47809	0.739043	+	0.673658i	$0.235278\pi$
$618$	0	0
$619$	−41.7055	−1.67628	−0.838142	−	0.545452i	$-0.816358\pi$
$619$	−41.7055	−1.67628	−0.838142	+	0.545452i	$0.816358\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−4.22583	−0.169304
$624$	0	0
$625$	7.37716	0.295086
$626$	0	0
$627$	0	0
$628$	0	0
$629$	14.6267	0.583205
$630$	0	0
$631$	18.3448	0.730293	0.365147	−	0.930950i	$-0.381019\pi$
$631$	18.3448	0.730293	0.365147	+	0.930950i	$0.381019\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−7.92002	−0.314297
$636$	0	0
$637$	43.2635	1.71416
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−35.3399	−1.39584	−0.697921	−	0.716174i	$-0.745891\pi$
$641$	−35.3399	−1.39584	−0.697921	+	0.716174i	$0.745891\pi$
$642$	0	0
$643$	−37.8176	−1.49138	−0.745691	−	0.666292i	$-0.767880\pi$
$643$	−37.8176	−1.49138	−0.745691	+	0.666292i	$0.767880\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−46.6512	−1.83405	−0.917025	−	0.398830i	$-0.869416\pi$
$647$	−46.6512	−1.83405	−0.917025	+	0.398830i	$0.869416\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	4.87070	0.190605	0.0953026	−	0.995448i	$-0.469618\pi$
$653$	4.87070	0.190605	0.0953026	+	0.995448i	$0.469618\pi$
$654$	0	0
$655$	9.79461	0.382707
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−43.3600	−1.68906	−0.844532	−	0.535505i	$-0.820121\pi$
$659$	−43.3600	−1.68906	−0.844532	+	0.535505i	$0.820121\pi$
$660$	0	0
$661$	−33.4417	−1.30073	−0.650366	−	0.759621i	$-0.725384\pi$
$661$	−33.4417	−1.30073	−0.650366	+	0.759621i	$0.725384\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	15.7614	0.611202
$666$	0	0
$667$	−68.8427	−2.66560
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−5.62711	−0.216909	−0.108455	−	0.994101i	$-0.534590\pi$
$673$	−5.62711	−0.216909	−0.108455	+	0.994101i	$0.534590\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	43.4243	1.66893	0.834466	−	0.551059i	$-0.185776\pi$
$677$	43.4243	1.66893	0.834466	+	0.551059i	$0.185776\pi$
$678$	0	0
$679$	25.6694	0.985100
$680$	0	0
$681$	0	0
$682$	0	0
$683$	14.5983	0.558589	0.279295	−	0.960205i	$-0.409899\pi$
$683$	14.5983	0.558589	0.279295	+	0.960205i	$0.409899\pi$
$684$	0	0
$685$	4.98646	0.190523
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−44.8292	−1.70786
$690$	0	0
$691$	−31.1590	−1.18534	−0.592672	−	0.805444i	$-0.701927\pi$
$691$	−31.1590	−1.18534	−0.592672	+	0.805444i	$0.701927\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	2.58671	0.0981197
$696$	0	0
$697$	17.0079	0.644220
$698$	0	0
$699$	0	0
$700$	0	0
$701$	28.1628	1.06369	0.531846	−	0.846841i	$-0.321498\pi$
$701$	28.1628	1.06369	0.531846	+	0.846841i	$0.321498\pi$
$702$	0	0
$703$	17.1845	0.648125
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−27.6410	−1.03955
$708$	0	0
$709$	29.0604	1.09138	0.545692	−	0.837986i	$-0.316267\pi$
$709$	29.0604	1.09138	0.545692	+	0.837986i	$0.316267\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−36.1065	−1.35220
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	46.4048	1.73061	0.865303	−	0.501249i	$-0.167126\pi$
$719$	46.4048	1.73061	0.865303	+	0.501249i	$0.167126\pi$
$720$	0	0
$721$	65.9337	2.45550
$722$	0	0
$723$	0	0
$724$	0	0
$725$	29.5961	1.09917
$726$	0	0
$727$	−5.65491	−0.209729	−0.104865	−	0.994487i	$-0.533441\pi$
$727$	−5.65491	−0.209729	−0.104865	+	0.994487i	$0.533441\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	13.2192	0.488931
$732$	0	0
$733$	−20.5106	−0.757578	−0.378789	−	0.925483i	$-0.623659\pi$
$733$	−20.5106	−0.757578	−0.378789	+	0.925483i	$0.623659\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−29.3675	−1.08030	−0.540150	−	0.841569i	$-0.681632\pi$
$739$	−29.3675	−1.08030	−0.540150	+	0.841569i	$0.681632\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−13.3639	−0.490276	−0.245138	−	0.969488i	$-0.578833\pi$
$743$	−13.3639	−0.490276	−0.245138	+	0.969488i	$0.578833\pi$
$744$	0	0
$745$	−8.25939	−0.302601
$746$	0	0
$747$	0	0
$748$	0	0
$749$	7.89305	0.288406
$750$	0	0
$751$	14.6699	0.535311	0.267656	−	0.963515i	$-0.413751\pi$
$751$	14.6699	0.535311	0.267656	+	0.963515i	$0.413751\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	2.69862	0.0982127
$756$	0	0
$757$	21.5814	0.784390	0.392195	−	0.919882i	$-0.371716\pi$
$757$	21.5814	0.784390	0.392195	+	0.919882i	$0.371716\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−11.2362	−0.407311	−0.203655	−	0.979043i	$-0.565282\pi$
$761$	−11.2362	−0.407311	−0.203655	+	0.979043i	$0.565282\pi$
$762$	0	0
$763$	53.1609	1.92455
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−7.43747	−0.268551
$768$	0	0
$769$	10.7163	0.386439	0.193219	−	0.981156i	$-0.438107\pi$
$769$	10.7163	0.386439	0.193219	+	0.981156i	$0.438107\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	8.33559	0.299810	0.149905	−	0.988700i	$-0.452103\pi$
$773$	8.33559	0.299810	0.149905	+	0.988700i	$0.452103\pi$
$774$	0	0
$775$	15.5225	0.557586
$776$	0	0
$777$	0	0
$778$	0	0
$779$	19.9821	0.715933
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	24.5043	0.874595
$786$	0	0
$787$	−5.75910	−0.205290	−0.102645	−	0.994718i	$-0.532731\pi$
$787$	−5.75910	−0.205290	−0.102645	+	0.994718i	$0.532731\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−29.7060	−1.05622
$792$	0	0
$793$	−9.46650	−0.336165
$794$	0	0
$795$	0	0
$796$	0	0
$797$	4.62882	0.163961	0.0819806	−	0.996634i	$-0.473875\pi$
$797$	4.62882	0.163961	0.0819806	+	0.996634i	$0.473875\pi$
$798$	0	0
$799$	30.5219	1.07979
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−43.5244	−1.53403
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−40.9378	−1.43930	−0.719648	−	0.694339i	$-0.755697\pi$
$809$	−40.9378	−1.43930	−0.719648	+	0.694339i	$0.755697\pi$
$810$	0	0
$811$	5.59302	0.196398	0.0981988	−	0.995167i	$-0.468692\pi$
$811$	5.59302	0.196398	0.0981988	+	0.995167i	$0.468692\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	1.18267	0.0414272
$816$	0	0
$817$	15.5309	0.543358
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−8.35997	−0.291765	−0.145882	−	0.989302i	$-0.546602\pi$
$821$	−8.35997	−0.291765	−0.145882	+	0.989302i	$0.546602\pi$
$822$	0	0
$823$	47.6717	1.66173	0.830866	−	0.556473i	$-0.187846\pi$
$823$	47.6717	1.66173	0.830866	+	0.556473i	$0.187846\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−44.7503	−1.55612	−0.778060	−	0.628189i	$-0.783796\pi$
$827$	−44.7503	−1.55612	−0.778060	+	0.628189i	$0.783796\pi$
$828$	0	0
$829$	−31.9827	−1.11080	−0.555402	−	0.831582i	$-0.687436\pi$
$829$	−31.9827	−1.11080	−0.555402	+	0.831582i	$0.687436\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−33.9411	−1.17599
$834$	0	0
$835$	−21.6228	−0.748286
$836$	0	0
$837$	0	0
$838$	0	0
$839$	24.4146	0.842886	0.421443	−	0.906855i	$-0.361524\pi$
$839$	24.4146	0.842886	0.421443	+	0.906855i	$0.361524\pi$
$840$	0	0
$841$	34.4651	1.18845
$842$	0	0
$843$	0	0
$844$	0	0
$845$	1.66963	0.0574370
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−47.4541	−1.62671
$852$	0	0
$853$	23.1476	0.792559	0.396280	−	0.918130i	$-0.370301\pi$
$853$	23.1476	0.792559	0.396280	+	0.918130i	$0.370301\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	34.1077	1.16510	0.582549	−	0.812795i	$-0.302055\pi$
$857$	34.1077	1.16510	0.582549	+	0.812795i	$0.302055\pi$
$858$	0	0
$859$	5.22073	0.178129	0.0890644	−	0.996026i	$-0.471612\pi$
$859$	5.22073	0.178129	0.0890644	+	0.996026i	$0.471612\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−35.2007	−1.19825	−0.599123	−	0.800657i	$-0.704484\pi$
$863$	−35.2007	−1.19825	−0.599123	+	0.800657i	$0.704484\pi$
$864$	0	0
$865$	22.9496	0.780310
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	40.2584	1.36410
$872$	0	0
$873$	0	0
$874$	0	0
$875$	43.8949	1.48392
$876$	0	0
$877$	12.7334	0.429976	0.214988	−	0.976617i	$-0.431029\pi$
$877$	12.7334	0.429976	0.214988	+	0.976617i	$0.431029\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−46.8110	−1.57710	−0.788552	−	0.614968i	$-0.789169\pi$
$881$	−46.8110	−1.57710	−0.788552	+	0.614968i	$0.789169\pi$
$882$	0	0
$883$	36.7634	1.23719	0.618594	−	0.785711i	$-0.287703\pi$
$883$	36.7634	1.23719	0.618594	+	0.785711i	$0.287703\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	41.7810	1.40287	0.701435	−	0.712733i	$-0.252543\pi$
$887$	41.7810	1.40287	0.701435	+	0.712733i	$0.252543\pi$
$888$	0	0
$889$	31.0450	1.04121
$890$	0	0
$891$	0	0
$892$	0	0
$893$	35.8593	1.19999
$894$	0	0
$895$	9.42167	0.314932
$896$	0	0
$897$	0	0
$898$	0	0
$899$	33.2861	1.11015
$900$	0	0
$901$	35.1694	1.17166
$902$	0	0
$903$	0	0
$904$	0	0
$905$	17.6385	0.586323
$906$	0	0
$907$	−5.63490	−0.187104	−0.0935518	−	0.995614i	$-0.529822\pi$
$907$	−5.63490	−0.187104	−0.0935518	+	0.995614i	$0.529822\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	10.8736	0.360259	0.180130	−	0.983643i	$-0.442348\pi$
$911$	10.8736	0.360259	0.180130	+	0.983643i	$0.442348\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−38.3930	−1.26785
$918$	0	0
$919$	0.528089	0.0174200	0.00871002	−	0.999962i	$-0.497227\pi$
$919$	0.528089	0.0174200	0.00871002	+	0.999962i	$0.497227\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−17.4218	−0.573446
$924$	0	0
$925$	20.4010	0.670780
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−13.7972	−0.452671	−0.226335	−	0.974049i	$-0.572675\pi$
$929$	−13.7972	−0.452671	−0.226335	+	0.974049i	$0.572675\pi$
$930$	0	0
$931$	−39.8764	−1.30690
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	21.0407	0.687371	0.343686	−	0.939085i	$-0.388325\pi$
$937$	21.0407	0.687371	0.343686	+	0.939085i	$0.388325\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−22.5853	−0.736259	−0.368130	−	0.929774i	$-0.620002\pi$
$941$	−22.5853	−0.736259	−0.368130	+	0.929774i	$0.620002\pi$
$942$	0	0
$943$	−55.1796	−1.79689
$944$	0	0
$945$	0	0
$946$	0	0
$947$	5.57107	0.181036	0.0905178	−	0.995895i	$-0.471148\pi$
$947$	5.57107	0.181036	0.0905178	+	0.995895i	$0.471148\pi$
$948$	0	0
$949$	−31.8046	−1.03242
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−4.68659	−0.151813	−0.0759067	−	0.997115i	$-0.524185\pi$
$953$	−4.68659	−0.151813	−0.0759067	+	0.997115i	$0.524185\pi$
$954$	0	0
$955$	−11.5554	−0.373922
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−19.5460	−0.631172
$960$	0	0
$961$	−13.5421	−0.436843
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−28.8711	−0.929393
$966$	0	0
$967$	−25.1290	−0.808095	−0.404047	−	0.914738i	$-0.632397\pi$
$967$	−25.1290	−0.808095	−0.404047	+	0.914738i	$0.632397\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	10.8550	0.348352	0.174176	−	0.984715i	$-0.444274\pi$
$971$	10.8550	0.348352	0.174176	+	0.984715i	$0.444274\pi$
$972$	0	0
$973$	−10.1394	−0.325055
$974$	0	0
$975$	0	0
$976$	0	0
$977$	52.4397	1.67769	0.838847	−	0.544367i	$-0.183230\pi$
$977$	52.4397	1.67769	0.838847	+	0.544367i	$0.183230\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−25.2016	−0.803807	−0.401903	−	0.915682i	$-0.631651\pi$
$983$	−25.2016	−0.803807	−0.401903	+	0.915682i	$0.631651\pi$
$984$	0	0
$985$	−6.83484	−0.217776
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−42.8878	−1.36375
$990$	0	0
$991$	9.95674	0.316286	0.158143	−	0.987416i	$-0.449449\pi$
$991$	9.95674	0.316286	0.158143	+	0.987416i	$0.449449\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	10.3745	0.328894
$996$	0	0
$997$	−18.9868	−0.601317	−0.300659	−	0.953732i	$-0.597206\pi$
$997$	−18.9868	−0.601317	−0.300659	+	0.953732i	$0.597206\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8712.2.a.ci.1.3		6
3.2	odd	2		8712.2.a.ck.1.4		6
11.7	odd	10		792.2.r.i.577.2	yes	12
11.8	odd	10		792.2.r.i.361.2	yes	12
11.10	odd	2		8712.2.a.ch.1.3		6
33.8	even	10		792.2.r.h.361.2	✓	12
33.29	even	10		792.2.r.h.577.2	yes	12
33.32	even	2		8712.2.a.cj.1.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
792.2.r.h.361.2	✓	12	33.8	even	10
792.2.r.h.577.2	yes	12	33.29	even	10
792.2.r.i.361.2	yes	12	11.8	odd	10
792.2.r.i.577.2	yes	12	11.7	odd	10
8712.2.a.ch.1.3		6	11.10	odd	2
8712.2.a.ci.1.3		6	1.1	even	1	trivial
8712.2.a.cj.1.4		6	33.32	even	2
8712.2.a.ck.1.4		6	3.2	odd	2

Overview	Random
Universe	Knowledge

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)

\(f(q)\)	\(=\)	\(q-1.13355 q^{5} +4.44328 q^{7} +O(q^{10})\)	\(q-1.13355 q^{5} +4.44328 q^{7} +3.39515 q^{13} -2.66356 q^{17} -3.12934 q^{19} +8.64153 q^{23} -3.71507 q^{25} -7.96650 q^{29} -4.17826 q^{31} -5.03666 q^{35} -5.49140 q^{37} -6.38540 q^{41} -4.96299 q^{43} -11.4591 q^{47} +12.7427 q^{49} -13.2039 q^{53} -2.19061 q^{59} -2.78824 q^{61} -3.84856 q^{65} +11.8576 q^{67} -5.13138 q^{71} -9.36764 q^{73} -4.30425 q^{79} -4.75768 q^{83} +3.01927 q^{85} -0.951061 q^{89} +15.0856 q^{91} +3.54725 q^{95} +5.77712 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 5 q^{5} + q^{7}+O(q^{10}) \) 6 * q - 5 * q^5 + q^7	\( 6 q - 5 q^{5} + q^{7} + 4 q^{13} + 6 q^{17} - 14 q^{19} - 6 q^{23} + 7 q^{25} + 7 q^{29} - 7 q^{31} - 2 q^{35} + 2 q^{37} - 6 q^{41} - 12 q^{43} - 26 q^{47} + 9 q^{49} - 9 q^{53} - 17 q^{59} + 20 q^{61} - 48 q^{65} + 14 q^{67} - 18 q^{71} + 7 q^{73} + 9 q^{79} + 11 q^{83} + 10 q^{85} - 14 q^{89} + 8 q^{91} + 30 q^{95} + 25 q^{97}+O(q^{100}) \) 6 * q - 5 * q^5 + q^7 + 4 * q^13 + 6 * q^17 - 14 * q^19 - 6 * q^23 + 7 * q^25 + 7 * q^29 - 7 * q^31 - 2 * q^35 + 2 * q^37 - 6 * q^41 - 12 * q^43 - 26 * q^47 + 9 * q^49 - 9 * q^53 - 17 * q^59 + 20 * q^61 - 48 * q^65 + 14 * q^67 - 18 * q^71 + 7 * q^73 + 9 * q^79 + 11 * q^83 + 10 * q^85 - 14 * q^89 + 8 * q^91 + 30 * q^95 + 25 * q^97

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