LMFDB - Embedded newform 8820.2.a.bg.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8820,2,Mod(1,8820)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("8820.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8820, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 8820 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8820.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,-2,0,0,0,0,0,2,0,-10,0,0,0,10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$70.4280545828$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 980)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	8820.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{5} -1.82843 q^{11} -6.41421 q^{13} +3.58579 q^{17} +7.65685 q^{19} -3.41421 q^{23} +1.00000 q^{25} +4.65685 q^{29} -7.41421 q^{31} -0.585786 q^{37} +3.41421 q^{41} +0.343146 q^{43} +10.8995 q^{47} +12.2426 q^{53} +1.82843 q^{55} +0.585786 q^{59} -10.8284 q^{61} +6.41421 q^{65} +3.07107 q^{67} +10.4853 q^{71} -10.8284 q^{73} -15.1421 q^{79} -8.00000 q^{83} -3.58579 q^{85} -16.9706 q^{89} -7.65685 q^{95} +9.72792 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{5} + 2 q^{11} - 10 q^{13} + 10 q^{17} + 4 q^{19} - 4 q^{23} + 2 q^{25} - 2 q^{29} - 12 q^{31} - 4 q^{37} + 4 q^{41} + 12 q^{43} + 2 q^{47} + 16 q^{53} - 2 q^{55} + 4 q^{59} - 16 q^{61} + 10 q^{65}+ \cdots - 6 q^{97}+O(q^{100}) $ 2 * q - 2 * q^5 + 2 * q^11 - 10 * q^13 + 10 * q^17 + 4 * q^19 - 4 * q^23 + 2 * q^25 - 2 * q^29 - 12 * q^31 - 4 * q^37 + 4 * q^41 + 12 * q^43 + 2 * q^47 + 16 * q^53 - 2 * q^55 + 4 * q^59 - 16 * q^61 + 10 * q^65 - 8 * q^67 + 4 * q^71 - 16 * q^73 - 2 * q^79 - 16 * q^83 - 10 * q^85 - 4 * q^95 - 6 * q^97

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
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−1.82843	−0.551292	−0.275646	−	0.961259i	$-0.588892\pi$
$11$	−1.82843	−0.551292	−0.275646	+	0.961259i	$0.588892\pi$
$12$	0	0
$13$	−6.41421	−1.77898	−0.889491	−	0.456952i	$-0.848941\pi$
$13$	−6.41421	−1.77898	−0.889491	+	0.456952i	$0.848941\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.58579	0.869681	0.434840	−	0.900508i	$-0.356805\pi$
$17$	3.58579	0.869681	0.434840	+	0.900508i	$0.356805\pi$
$18$	0	0
$19$	7.65685	1.75660	0.878301	−	0.478107i	$-0.158677\pi$
$19$	7.65685	1.75660	0.878301	+	0.478107i	$0.158677\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−3.41421	−0.711913	−0.355956	−	0.934503i	$-0.615845\pi$
$23$	−3.41421	−0.711913	−0.355956	+	0.934503i	$0.615845\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	4.65685	0.864756	0.432378	−	0.901692i	$-0.357675\pi$
$29$	4.65685	0.864756	0.432378	+	0.901692i	$0.357675\pi$
$30$	0	0
$31$	−7.41421	−1.33163	−0.665816	−	0.746116i	$-0.731916\pi$
$31$	−7.41421	−1.33163	−0.665816	+	0.746116i	$0.731916\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−0.585786	−0.0963027	−0.0481513	−	0.998840i	$-0.515333\pi$
$37$	−0.585786	−0.0963027	−0.0481513	+	0.998840i	$0.515333\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	3.41421	0.533211	0.266605	−	0.963806i	$-0.414098\pi$
$41$	3.41421	0.533211	0.266605	+	0.963806i	$0.414098\pi$
$42$	0	0
$43$	0.343146	0.0523292	0.0261646	−	0.999658i	$-0.491671\pi$
$43$	0.343146	0.0523292	0.0261646	+	0.999658i	$0.491671\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	10.8995	1.58985	0.794927	−	0.606705i	$-0.207509\pi$
$47$	10.8995	1.58985	0.794927	+	0.606705i	$0.207509\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0	0
$52$	0	0
$53$	12.2426	1.68166	0.840828	−	0.541302i	$-0.182069\pi$
$53$	12.2426	1.68166	0.840828	+	0.541302i	$0.182069\pi$
$54$	0	0
$55$	1.82843	0.246545
$56$	0	0
$57$	0	0
$58$	0	0
$59$	0.585786	0.0762629	0.0381314	−	0.999273i	$-0.487859\pi$
$59$	0.585786	0.0762629	0.0381314	+	0.999273i	$0.487859\pi$
$60$	0	0
$61$	−10.8284	−1.38644	−0.693219	−	0.720727i	$-0.743808\pi$
$61$	−10.8284	−1.38644	−0.693219	+	0.720727i	$0.743808\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	6.41421	0.795585
$66$	0	0
$67$	3.07107	0.375191	0.187595	−	0.982246i	$-0.439931\pi$
$67$	3.07107	0.375191	0.187595	+	0.982246i	$0.439931\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	10.4853	1.24437	0.622187	−	0.782869i	$-0.286244\pi$
$71$	10.4853	1.24437	0.622187	+	0.782869i	$0.286244\pi$
$72$	0	0
$73$	−10.8284	−1.26737	−0.633686	−	0.773591i	$-0.718459\pi$
$73$	−10.8284	−1.26737	−0.633686	+	0.773591i	$0.718459\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−15.1421	−1.70362	−0.851812	−	0.523848i	$-0.824496\pi$
$79$	−15.1421	−1.70362	−0.851812	+	0.523848i	$0.824496\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−8.00000	−0.878114	−0.439057	−	0.898459i	$-0.644687\pi$
$83$	−8.00000	−0.878114	−0.439057	+	0.898459i	$0.644687\pi$
$84$	0	0
$85$	−3.58579	−0.388933
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−16.9706	−1.79888	−0.899438	−	0.437048i	$-0.856024\pi$
$89$	−16.9706	−1.79888	−0.899438	+	0.437048i	$0.856024\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−7.65685	−0.785577
$96$	0	0
$97$	9.72792	0.987721	0.493860	−	0.869541i	$-0.335585\pi$
$97$	9.72792	0.987721	0.493860	+	0.869541i	$0.335585\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	0.828427	0.0824316	0.0412158	−	0.999150i	$-0.486877\pi$
$101$	0.828427	0.0824316	0.0412158	+	0.999150i	$0.486877\pi$
$102$	0	0
$103$	−4.41421	−0.434945	−0.217473	−	0.976066i	$-0.569781\pi$
$103$	−4.41421	−0.434945	−0.217473	+	0.976066i	$0.569781\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	11.1716	1.08000	0.539998	−	0.841666i	$-0.318425\pi$
$107$	11.1716	1.08000	0.539998	+	0.841666i	$0.318425\pi$
$108$	0	0
$109$	9.00000	0.862044	0.431022	−	0.902342i	$-0.358153\pi$
$109$	9.00000	0.862044	0.431022	+	0.902342i	$0.358153\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−9.07107	−0.853334	−0.426667	−	0.904409i	$-0.640312\pi$
$113$	−9.07107	−0.853334	−0.426667	+	0.904409i	$0.640312\pi$
$114$	0	0
$115$	3.41421	0.318377
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−7.65685	−0.696078
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−2.10051	−0.186390	−0.0931948	−	0.995648i	$-0.529708\pi$
$127$	−2.10051	−0.186390	−0.0931948	+	0.995648i	$0.529708\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−20.2426	−1.76861	−0.884304	−	0.466912i	$-0.845367\pi$
$131$	−20.2426	−1.76861	−0.884304	+	0.466912i	$0.845367\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	20.9706	1.79164	0.895818	−	0.444421i	$-0.146591\pi$
$137$	20.9706	1.79164	0.895818	+	0.444421i	$0.146591\pi$
$138$	0	0
$139$	−5.89949	−0.500389	−0.250194	−	0.968196i	$-0.580495\pi$
$139$	−5.89949	−0.500389	−0.250194	+	0.968196i	$0.580495\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	11.7279	0.980738
$144$	0	0
$145$	−4.65685	−0.386731
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−20.4853	−1.67822	−0.839110	−	0.543962i	$-0.816924\pi$
$149$	−20.4853	−1.67822	−0.839110	+	0.543962i	$0.816924\pi$
$150$	0	0
$151$	−6.17157	−0.502235	−0.251118	−	0.967957i	$-0.580798\pi$
$151$	−6.17157	−0.502235	−0.251118	+	0.967957i	$0.580798\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	7.41421	0.595524
$156$	0	0
$157$	6.48528	0.517582	0.258791	−	0.965933i	$-0.416676\pi$
$157$	6.48528	0.517582	0.258791	+	0.965933i	$0.416676\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−9.55635	−0.748511	−0.374256	−	0.927326i	$-0.622102\pi$
$163$	−9.55635	−0.748511	−0.374256	+	0.927326i	$0.622102\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−0.414214	−0.0320528	−0.0160264	−	0.999872i	$-0.505102\pi$
$167$	−0.414214	−0.0320528	−0.0160264	+	0.999872i	$0.505102\pi$
$168$	0	0
$169$	28.1421	2.16478
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−12.5563	−0.954642	−0.477321	−	0.878729i	$-0.658392\pi$
$173$	−12.5563	−0.954642	−0.477321	+	0.878729i	$0.658392\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	10.4853	0.783707	0.391853	−	0.920028i	$-0.371834\pi$
$179$	10.4853	0.783707	0.391853	+	0.920028i	$0.371834\pi$
$180$	0	0
$181$	−10.2426	−0.761329	−0.380665	−	0.924713i	$-0.624305\pi$
$181$	−10.2426	−0.761329	−0.380665	+	0.924713i	$0.624305\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0.585786	0.0430679
$186$	0	0
$187$	−6.55635	−0.479448
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−5.34315	−0.386616	−0.193308	−	0.981138i	$-0.561922\pi$
$191$	−5.34315	−0.386616	−0.193308	+	0.981138i	$0.561922\pi$
$192$	0	0
$193$	−5.65685	−0.407189	−0.203595	−	0.979055i	$-0.565262\pi$
$193$	−5.65685	−0.407189	−0.203595	+	0.979055i	$0.565262\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−3.55635	−0.253379	−0.126690	−	0.991942i	$-0.540435\pi$
$197$	−3.55635	−0.253379	−0.126690	+	0.991942i	$0.540435\pi$
$198$	0	0
$199$	1.27208	0.0901752	0.0450876	−	0.998983i	$-0.485643\pi$
$199$	1.27208	0.0901752	0.0450876	+	0.998983i	$0.485643\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−3.41421	−0.238459
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−14.0000	−0.968400
$210$	0	0
$211$	−1.68629	−0.116089	−0.0580445	−	0.998314i	$-0.518487\pi$
$211$	−1.68629	−0.116089	−0.0580445	+	0.998314i	$0.518487\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−0.343146	−0.0234023
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−23.0000	−1.54715
$222$	0	0
$223$	−9.92893	−0.664890	−0.332445	−	0.943123i	$-0.607874\pi$
$223$	−9.92893	−0.664890	−0.332445	+	0.943123i	$0.607874\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−17.7279	−1.17664	−0.588322	−	0.808627i	$-0.700211\pi$
$227$	−17.7279	−1.17664	−0.588322	+	0.808627i	$0.700211\pi$
$228$	0	0
$229$	−4.10051	−0.270969	−0.135485	−	0.990779i	$-0.543259\pi$
$229$	−4.10051	−0.270969	−0.135485	+	0.990779i	$0.543259\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	14.8284	0.971443	0.485721	−	0.874114i	$-0.338557\pi$
$233$	14.8284	0.971443	0.485721	+	0.874114i	$0.338557\pi$
$234$	0	0
$235$	−10.8995	−0.711005
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−19.8284	−1.28259	−0.641297	−	0.767293i	$-0.721603\pi$
$239$	−19.8284	−1.28259	−0.641297	+	0.767293i	$0.721603\pi$
$240$	0	0
$241$	−27.5563	−1.77506	−0.887530	−	0.460749i	$-0.847581\pi$
$241$	−27.5563	−1.77506	−0.887530	+	0.460749i	$0.847581\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−49.1127	−3.12497
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−12.9289	−0.816067	−0.408033	−	0.912967i	$-0.633785\pi$
$251$	−12.9289	−0.816067	−0.408033	+	0.912967i	$0.633785\pi$
$252$	0	0
$253$	6.24264	0.392471
$254$	0	0
$255$	0	0
$256$	0	0
$257$	13.7990	0.860757	0.430379	−	0.902648i	$-0.358380\pi$
$257$	13.7990	0.860757	0.430379	+	0.902648i	$0.358380\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−10.0000	−0.616626	−0.308313	−	0.951285i	$-0.599764\pi$
$263$	−10.0000	−0.616626	−0.308313	+	0.951285i	$0.599764\pi$
$264$	0	0
$265$	−12.2426	−0.752059
$266$	0	0
$267$	0	0
$268$	0	0
$269$	8.72792	0.532151	0.266075	−	0.963952i	$-0.414273\pi$
$269$	8.72792	0.532151	0.266075	+	0.963952i	$0.414273\pi$
$270$	0	0
$271$	4.97056	0.301940	0.150970	−	0.988538i	$-0.451760\pi$
$271$	4.97056	0.301940	0.150970	+	0.988538i	$0.451760\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−1.82843	−0.110258
$276$	0	0
$277$	−4.58579	−0.275533	−0.137767	−	0.990465i	$-0.543992\pi$
$277$	−4.58579	−0.275533	−0.137767	+	0.990465i	$0.543992\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	1.00000	0.0596550	0.0298275	−	0.999555i	$-0.490504\pi$
$281$	1.00000	0.0596550	0.0298275	+	0.999555i	$0.490504\pi$
$282$	0	0
$283$	27.2426	1.61941	0.809703	−	0.586839i	$-0.199628\pi$
$283$	27.2426	1.61941	0.809703	+	0.586839i	$0.199628\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−4.14214	−0.243655
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−11.5858	−0.676849	−0.338424	−	0.940994i	$-0.609894\pi$
$293$	−11.5858	−0.676849	−0.338424	+	0.940994i	$0.609894\pi$
$294$	0	0
$295$	−0.585786	−0.0341058
$296$	0	0
$297$	0	0
$298$	0	0
$299$	21.8995	1.26648
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	10.8284	0.620034
$306$	0	0
$307$	−6.41421	−0.366079	−0.183039	−	0.983106i	$-0.558594\pi$
$307$	−6.41421	−0.366079	−0.183039	+	0.983106i	$0.558594\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−26.9706	−1.52936	−0.764680	−	0.644410i	$-0.777103\pi$
$311$	−26.9706	−1.52936	−0.764680	+	0.644410i	$0.777103\pi$
$312$	0	0
$313$	14.5563	0.822774	0.411387	−	0.911461i	$-0.365045\pi$
$313$	14.5563	0.822774	0.411387	+	0.911461i	$0.365045\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−18.9706	−1.06549	−0.532746	−	0.846275i	$-0.678840\pi$
$317$	−18.9706	−1.06549	−0.532746	+	0.846275i	$0.678840\pi$
$318$	0	0
$319$	−8.51472	−0.476733
$320$	0	0
$321$	0	0
$322$	0	0
$323$	27.4558	1.52768
$324$	0	0
$325$	−6.41421	−0.355797
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−0.828427	−0.0455345	−0.0227672	−	0.999741i	$-0.507248\pi$
$331$	−0.828427	−0.0455345	−0.0227672	+	0.999741i	$0.507248\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−3.07107	−0.167790
$336$	0	0
$337$	6.72792	0.366493	0.183247	−	0.983067i	$-0.441339\pi$
$337$	6.72792	0.366493	0.183247	+	0.983067i	$0.441339\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	13.5563	0.734117
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	4.58579	0.246178	0.123089	−	0.992396i	$-0.460720\pi$
$347$	4.58579	0.246178	0.123089	+	0.992396i	$0.460720\pi$
$348$	0	0
$349$	−18.0000	−0.963518	−0.481759	−	0.876304i	$-0.660002\pi$
$349$	−18.0000	−0.963518	−0.481759	+	0.876304i	$0.660002\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	12.0711	0.642478	0.321239	−	0.946998i	$-0.395901\pi$
$353$	12.0711	0.642478	0.321239	+	0.946998i	$0.395901\pi$
$354$	0	0
$355$	−10.4853	−0.556501
$356$	0	0
$357$	0	0
$358$	0	0
$359$	11.3137	0.597115	0.298557	−	0.954392i	$-0.403495\pi$
$359$	11.3137	0.597115	0.298557	+	0.954392i	$0.403495\pi$
$360$	0	0
$361$	39.6274	2.08565
$362$	0	0
$363$	0	0
$364$	0	0
$365$	10.8284	0.566786
$366$	0	0
$367$	5.72792	0.298995	0.149498	−	0.988762i	$-0.452234\pi$
$367$	5.72792	0.298995	0.149498	+	0.988762i	$0.452234\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−0.485281	−0.0251269	−0.0125635	−	0.999921i	$-0.503999\pi$
$373$	−0.485281	−0.0251269	−0.0125635	+	0.999921i	$0.503999\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−29.8701	−1.53839
$378$	0	0
$379$	14.0000	0.719132	0.359566	−	0.933120i	$-0.382925\pi$
$379$	14.0000	0.719132	0.359566	+	0.933120i	$0.382925\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−20.4853	−1.04675	−0.523374	−	0.852103i	$-0.675327\pi$
$383$	−20.4853	−1.04675	−0.523374	+	0.852103i	$0.675327\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	15.1421	0.767737	0.383868	−	0.923388i	$-0.374592\pi$
$389$	15.1421	0.767737	0.383868	+	0.923388i	$0.374592\pi$
$390$	0	0
$391$	−12.2426	−0.619137
$392$	0	0
$393$	0	0
$394$	0	0
$395$	15.1421	0.761884
$396$	0	0
$397$	−11.5858	−0.581474	−0.290737	−	0.956803i	$-0.593900\pi$
$397$	−11.5858	−0.581474	−0.290737	+	0.956803i	$0.593900\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−17.4853	−0.873173	−0.436587	−	0.899662i	$-0.643813\pi$
$401$	−17.4853	−0.873173	−0.436587	+	0.899662i	$0.643813\pi$
$402$	0	0
$403$	47.5563	2.36895
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1.07107	0.0530909
$408$	0	0
$409$	−27.1716	−1.34355	−0.671774	−	0.740756i	$-0.734467\pi$
$409$	−27.1716	−1.34355	−0.671774	+	0.740756i	$0.734467\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	8.00000	0.392705
$416$	0	0
$417$	0	0
$418$	0	0
$419$	21.5563	1.05310	0.526548	−	0.850145i	$-0.323486\pi$
$419$	21.5563	1.05310	0.526548	+	0.850145i	$0.323486\pi$
$420$	0	0
$421$	−2.31371	−0.112763	−0.0563816	−	0.998409i	$-0.517956\pi$
$421$	−2.31371	−0.112763	−0.0563816	+	0.998409i	$0.517956\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	3.58579	0.173936
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	4.17157	0.200938	0.100469	−	0.994940i	$-0.467966\pi$
$431$	4.17157	0.200938	0.100469	+	0.994940i	$0.467966\pi$
$432$	0	0
$433$	−34.2843	−1.64760	−0.823798	−	0.566883i	$-0.808149\pi$
$433$	−34.2843	−1.64760	−0.823798	+	0.566883i	$0.808149\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−26.1421	−1.25055
$438$	0	0
$439$	−15.7574	−0.752058	−0.376029	−	0.926608i	$-0.622711\pi$
$439$	−15.7574	−0.752058	−0.376029	+	0.926608i	$0.622711\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−15.4558	−0.734329	−0.367165	−	0.930156i	$-0.619671\pi$
$443$	−15.4558	−0.734329	−0.367165	+	0.930156i	$0.619671\pi$
$444$	0	0
$445$	16.9706	0.804482
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−18.1716	−0.857570	−0.428785	−	0.903407i	$-0.641058\pi$
$449$	−18.1716	−0.857570	−0.428785	+	0.903407i	$0.641058\pi$
$450$	0	0
$451$	−6.24264	−0.293954
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−0.928932	−0.0434536	−0.0217268	−	0.999764i	$-0.506916\pi$
$457$	−0.928932	−0.0434536	−0.0217268	+	0.999764i	$0.506916\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	33.6569	1.56756	0.783778	−	0.621041i	$-0.213290\pi$
$461$	33.6569	1.56756	0.783778	+	0.621041i	$0.213290\pi$
$462$	0	0
$463$	37.4558	1.74072	0.870360	−	0.492415i	$-0.163886\pi$
$463$	37.4558	1.74072	0.870360	+	0.492415i	$0.163886\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−18.8995	−0.874564	−0.437282	−	0.899324i	$-0.644059\pi$
$467$	−18.8995	−0.874564	−0.437282	+	0.899324i	$0.644059\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−0.627417	−0.0288487
$474$	0	0
$475$	7.65685	0.351321
$476$	0	0
$477$	0	0
$478$	0	0
$479$	6.58579	0.300912	0.150456	−	0.988617i	$-0.451926\pi$
$479$	6.58579	0.300912	0.150456	+	0.988617i	$0.451926\pi$
$480$	0	0
$481$	3.75736	0.171321
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−9.72792	−0.441722
$486$	0	0
$487$	−4.58579	−0.207802	−0.103901	−	0.994588i	$-0.533132\pi$
$487$	−4.58579	−0.207802	−0.103901	+	0.994588i	$0.533132\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	39.2843	1.77287	0.886437	−	0.462849i	$-0.153173\pi$
$491$	39.2843	1.77287	0.886437	+	0.462849i	$0.153173\pi$
$492$	0	0
$493$	16.6985	0.752062
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	31.6274	1.41584	0.707919	−	0.706294i	$-0.249634\pi$
$499$	31.6274	1.41584	0.707919	+	0.706294i	$0.249634\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−9.04163	−0.403146	−0.201573	−	0.979473i	$-0.564605\pi$
$503$	−9.04163	−0.403146	−0.201573	+	0.979473i	$0.564605\pi$
$504$	0	0
$505$	−0.828427	−0.0368645
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−20.9289	−0.927659	−0.463829	−	0.885925i	$-0.653525\pi$
$509$	−20.9289	−0.927659	−0.463829	+	0.885925i	$0.653525\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	4.41421	0.194513
$516$	0	0
$517$	−19.9289	−0.876473
$518$	0	0
$519$	0	0
$520$	0	0
$521$	29.3137	1.28426	0.642128	−	0.766597i	$-0.278052\pi$
$521$	29.3137	1.28426	0.642128	+	0.766597i	$0.278052\pi$
$522$	0	0
$523$	−30.1421	−1.31802	−0.659012	−	0.752133i	$-0.729025\pi$
$523$	−30.1421	−1.31802	−0.659012	+	0.752133i	$0.729025\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−26.5858	−1.15810
$528$	0	0
$529$	−11.3431	−0.493180
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−21.8995	−0.948572
$534$	0	0
$535$	−11.1716	−0.482989
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−17.9706	−0.772615	−0.386307	−	0.922370i	$-0.626250\pi$
$541$	−17.9706	−0.772615	−0.386307	+	0.922370i	$0.626250\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−9.00000	−0.385518
$546$	0	0
$547$	24.4853	1.04692	0.523458	−	0.852052i	$-0.324642\pi$
$547$	24.4853	1.04692	0.523458	+	0.852052i	$0.324642\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	35.6569	1.51903
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−31.7990	−1.34737	−0.673683	−	0.739020i	$-0.735289\pi$
$557$	−31.7990	−1.34737	−0.673683	+	0.739020i	$0.735289\pi$
$558$	0	0
$559$	−2.20101	−0.0930928
$560$	0	0
$561$	0	0
$562$	0	0
$563$	24.6274	1.03792	0.518961	−	0.854798i	$-0.326319\pi$
$563$	24.6274	1.03792	0.518961	+	0.854798i	$0.326319\pi$
$564$	0	0
$565$	9.07107	0.381623
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−7.79899	−0.326951	−0.163475	−	0.986547i	$-0.552270\pi$
$569$	−7.79899	−0.326951	−0.163475	+	0.986547i	$0.552270\pi$
$570$	0	0
$571$	−0.828427	−0.0346686	−0.0173343	−	0.999850i	$-0.505518\pi$
$571$	−0.828427	−0.0346686	−0.0173343	+	0.999850i	$0.505518\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−3.41421	−0.142383
$576$	0	0
$577$	−0.0710678	−0.00295859	−0.00147930	−	0.999999i	$-0.500471\pi$
$577$	−0.0710678	−0.00295859	−0.00147930	+	0.999999i	$0.500471\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−22.3848	−0.927083
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−25.1716	−1.03894	−0.519471	−	0.854488i	$-0.673871\pi$
$587$	−25.1716	−1.03894	−0.519471	+	0.854488i	$0.673871\pi$
$588$	0	0
$589$	−56.7696	−2.33915
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−7.72792	−0.317348	−0.158674	−	0.987331i	$-0.550722\pi$
$593$	−7.72792	−0.317348	−0.158674	+	0.987331i	$0.550722\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−26.7990	−1.09498	−0.547489	−	0.836813i	$-0.684416\pi$
$599$	−26.7990	−1.09498	−0.547489	+	0.836813i	$0.684416\pi$
$600$	0	0
$601$	−28.3431	−1.15614	−0.578071	−	0.815987i	$-0.696194\pi$
$601$	−28.3431	−1.15614	−0.578071	+	0.815987i	$0.696194\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	7.65685	0.311295
$606$	0	0
$607$	−28.0711	−1.13937	−0.569685	−	0.821863i	$-0.692935\pi$
$607$	−28.0711	−1.13937	−0.569685	+	0.821863i	$0.692935\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−69.9117	−2.82832
$612$	0	0
$613$	−18.6863	−0.754732	−0.377366	−	0.926064i	$-0.623170\pi$
$613$	−18.6863	−0.754732	−0.377366	+	0.926064i	$0.623170\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	12.8701	0.518129	0.259065	−	0.965860i	$-0.416586\pi$
$617$	12.8701	0.518129	0.259065	+	0.965860i	$0.416586\pi$
$618$	0	0
$619$	42.8701	1.72309	0.861547	−	0.507679i	$-0.169496\pi$
$619$	42.8701	1.72309	0.861547	+	0.507679i	$0.169496\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−2.10051	−0.0837526
$630$	0	0
$631$	−22.5147	−0.896297	−0.448148	−	0.893959i	$-0.647916\pi$
$631$	−22.5147	−0.896297	−0.448148	+	0.893959i	$0.647916\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	2.10051	0.0833560
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−16.6863	−0.659069	−0.329534	−	0.944144i	$-0.606892\pi$
$641$	−16.6863	−0.659069	−0.329534	+	0.944144i	$0.606892\pi$
$642$	0	0
$643$	38.2132	1.50698	0.753491	−	0.657458i	$-0.228368\pi$
$643$	38.2132	1.50698	0.753491	+	0.657458i	$0.228368\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−27.1127	−1.06591	−0.532955	−	0.846144i	$-0.678919\pi$
$647$	−27.1127	−1.06591	−0.532955	+	0.846144i	$0.678919\pi$
$648$	0	0
$649$	−1.07107	−0.0420431
$650$	0	0
$651$	0	0
$652$	0	0
$653$	49.3137	1.92979	0.964897	−	0.262628i	$-0.0845891\pi$
$653$	49.3137	1.92979	0.964897	+	0.262628i	$0.0845891\pi$
$654$	0	0
$655$	20.2426	0.790945
$656$	0	0
$657$	0	0
$658$	0	0
$659$	9.34315	0.363957	0.181979	−	0.983302i	$-0.441750\pi$
$659$	9.34315	0.363957	0.181979	+	0.983302i	$0.441750\pi$
$660$	0	0
$661$	5.85786	0.227845	0.113922	−	0.993490i	$-0.463659\pi$
$661$	5.85786	0.227845	0.113922	+	0.993490i	$0.463659\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−15.8995	−0.615631
$668$	0	0
$669$	0	0
$670$	0	0
$671$	19.7990	0.764332
$672$	0	0
$673$	−44.4853	−1.71478	−0.857391	−	0.514666i	$-0.827916\pi$
$673$	−44.4853	−1.71478	−0.857391	+	0.514666i	$0.827916\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−35.7279	−1.37314	−0.686568	−	0.727066i	$-0.740884\pi$
$677$	−35.7279	−1.37314	−0.686568	+	0.727066i	$0.740884\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	27.1127	1.03744	0.518719	−	0.854945i	$-0.326409\pi$
$683$	27.1127	1.03744	0.518719	+	0.854945i	$0.326409\pi$
$684$	0	0
$685$	−20.9706	−0.801244
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−78.5269	−2.99164
$690$	0	0
$691$	−20.4853	−0.779297	−0.389648	−	0.920964i	$-0.627403\pi$
$691$	−20.4853	−0.779297	−0.389648	+	0.920964i	$0.627403\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	5.89949	0.223781
$696$	0	0
$697$	12.2426	0.463723
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−17.4853	−0.660410	−0.330205	−	0.943909i	$-0.607118\pi$
$701$	−17.4853	−0.660410	−0.330205	+	0.943909i	$0.607118\pi$
$702$	0	0
$703$	−4.48528	−0.169166
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−41.6274	−1.56335	−0.781675	−	0.623686i	$-0.785635\pi$
$709$	−41.6274	−1.56335	−0.781675	+	0.623686i	$0.785635\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	25.3137	0.948006
$714$	0	0
$715$	−11.7279	−0.438599
$716$	0	0
$717$	0	0
$718$	0	0
$719$	24.2426	0.904098	0.452049	−	0.891993i	$-0.350693\pi$
$719$	24.2426	0.904098	0.452049	+	0.891993i	$0.350693\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	4.65685	0.172951
$726$	0	0
$727$	18.0000	0.667583	0.333792	−	0.942647i	$-0.391672\pi$
$727$	18.0000	0.667583	0.333792	+	0.942647i	$0.391672\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	1.23045	0.0455097
$732$	0	0
$733$	−28.6985	−1.06000	−0.530001	−	0.847997i	$-0.677809\pi$
$733$	−28.6985	−1.06000	−0.530001	+	0.847997i	$0.677809\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−5.61522	−0.206839
$738$	0	0
$739$	9.34315	0.343693	0.171847	−	0.985124i	$-0.445027\pi$
$739$	9.34315	0.343693	0.171847	+	0.985124i	$0.445027\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	23.0711	0.846395	0.423198	−	0.906037i	$-0.360908\pi$
$743$	23.0711	0.846395	0.423198	+	0.906037i	$0.360908\pi$
$744$	0	0
$745$	20.4853	0.750523
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−27.1421	−0.990431	−0.495215	−	0.868770i	$-0.664911\pi$
$751$	−27.1421	−0.990431	−0.495215	+	0.868770i	$0.664911\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	6.17157	0.224607
$756$	0	0
$757$	42.8284	1.55663	0.778313	−	0.627877i	$-0.216076\pi$
$757$	42.8284	1.55663	0.778313	+	0.627877i	$0.216076\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	34.8701	1.26404	0.632019	−	0.774953i	$-0.282226\pi$
$761$	34.8701	1.26404	0.632019	+	0.774953i	$0.282226\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−3.75736	−0.135670
$768$	0	0
$769$	0.142136	0.00512554	0.00256277	−	0.999997i	$-0.499184\pi$
$769$	0.142136	0.00512554	0.00256277	+	0.999997i	$0.499184\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−18.2721	−0.657201	−0.328600	−	0.944469i	$-0.606577\pi$
$773$	−18.2721	−0.657201	−0.328600	+	0.944469i	$0.606577\pi$
$774$	0	0
$775$	−7.41421	−0.266326
$776$	0	0
$777$	0	0
$778$	0	0
$779$	26.1421	0.936639
$780$	0	0
$781$	−19.1716	−0.686013
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−6.48528	−0.231470
$786$	0	0
$787$	18.2721	0.651329	0.325665	−	0.945485i	$-0.394412\pi$
$787$	18.2721	0.651329	0.325665	+	0.945485i	$0.394412\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	69.4558	2.46645
$794$	0	0
$795$	0	0
$796$	0	0
$797$	1.38478	0.0490513	0.0245256	−	0.999699i	$-0.492192\pi$
$797$	1.38478	0.0490513	0.0245256	+	0.999699i	$0.492192\pi$
$798$	0	0
$799$	39.0833	1.38267
$800$	0	0
$801$	0	0
$802$	0	0
$803$	19.7990	0.698691
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−39.9706	−1.40529	−0.702645	−	0.711541i	$-0.747998\pi$
$809$	−39.9706	−1.40529	−0.702645	+	0.711541i	$0.747998\pi$
$810$	0	0
$811$	−11.5563	−0.405798	−0.202899	−	0.979200i	$-0.565036\pi$
$811$	−11.5563	−0.405798	−0.202899	+	0.979200i	$0.565036\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	9.55635	0.334744
$816$	0	0
$817$	2.62742	0.0919217
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−42.1716	−1.47180	−0.735899	−	0.677091i	$-0.763240\pi$
$821$	−42.1716	−1.47180	−0.735899	+	0.677091i	$0.763240\pi$
$822$	0	0
$823$	−28.5269	−0.994386	−0.497193	−	0.867640i	$-0.665636\pi$
$823$	−28.5269	−0.994386	−0.497193	+	0.867640i	$0.665636\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−38.0416	−1.32284	−0.661419	−	0.750017i	$-0.730045\pi$
$827$	−38.0416	−1.32284	−0.661419	+	0.750017i	$0.730045\pi$
$828$	0	0
$829$	2.72792	0.0947446	0.0473723	−	0.998877i	$-0.484915\pi$
$829$	2.72792	0.0947446	0.0473723	+	0.998877i	$0.484915\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0.414214	0.0143345
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−14.3848	−0.496618	−0.248309	−	0.968681i	$-0.579875\pi$
$839$	−14.3848	−0.496618	−0.248309	+	0.968681i	$0.579875\pi$
$840$	0	0
$841$	−7.31371	−0.252197
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−28.1421	−0.968119
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	2.00000	0.0685591
$852$	0	0
$853$	−26.2843	−0.899956	−0.449978	−	0.893040i	$-0.648568\pi$
$853$	−26.2843	−0.899956	−0.449978	+	0.893040i	$0.648568\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−15.1716	−0.518251	−0.259126	−	0.965844i	$-0.583434\pi$
$857$	−15.1716	−0.518251	−0.259126	+	0.965844i	$0.583434\pi$
$858$	0	0
$859$	−5.45584	−0.186151	−0.0930755	−	0.995659i	$-0.529670\pi$
$859$	−5.45584	−0.186151	−0.0930755	+	0.995659i	$0.529670\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	28.2426	0.961391	0.480695	−	0.876888i	$-0.340384\pi$
$863$	28.2426	0.961391	0.480695	+	0.876888i	$0.340384\pi$
$864$	0	0
$865$	12.5563	0.426929
$866$	0	0
$867$	0	0
$868$	0	0
$869$	27.6863	0.939193
$870$	0	0
$871$	−19.6985	−0.667458
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	33.3137	1.12492	0.562462	−	0.826823i	$-0.309854\pi$
$877$	33.3137	1.12492	0.562462	+	0.826823i	$0.309854\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	0.284271	0.00957734	0.00478867	−	0.999989i	$-0.498476\pi$
$881$	0.284271	0.00957734	0.00478867	+	0.999989i	$0.498476\pi$
$882$	0	0
$883$	55.1127	1.85469	0.927345	−	0.374208i	$-0.122085\pi$
$883$	55.1127	1.85469	0.927345	+	0.374208i	$0.122085\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−21.5980	−0.725189	−0.362595	−	0.931947i	$-0.618109\pi$
$887$	−21.5980	−0.725189	−0.362595	+	0.931947i	$0.618109\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	83.4558	2.79274
$894$	0	0
$895$	−10.4853	−0.350484
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−34.5269	−1.15154
$900$	0	0
$901$	43.8995	1.46250
$902$	0	0
$903$	0	0
$904$	0	0
$905$	10.2426	0.340477
$906$	0	0
$907$	−42.5269	−1.41208	−0.706041	−	0.708170i	$-0.749521\pi$
$907$	−42.5269	−1.41208	−0.706041	+	0.708170i	$0.749521\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−23.5980	−0.781836	−0.390918	−	0.920426i	$-0.627842\pi$
$911$	−23.5980	−0.781836	−0.390918	+	0.920426i	$0.627842\pi$
$912$	0	0
$913$	14.6274	0.484097
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−35.4853	−1.17055	−0.585276	−	0.810834i	$-0.699014\pi$
$919$	−35.4853	−1.17055	−0.585276	+	0.810834i	$0.699014\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−67.2548	−2.21372
$924$	0	0
$925$	−0.585786	−0.0192605
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−46.7279	−1.53309	−0.766547	−	0.642189i	$-0.778027\pi$
$929$	−46.7279	−1.53309	−0.766547	+	0.642189i	$0.778027\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	6.55635	0.214416
$936$	0	0
$937$	−17.7279	−0.579146	−0.289573	−	0.957156i	$-0.593513\pi$
$937$	−17.7279	−0.579146	−0.289573	+	0.957156i	$0.593513\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−4.00000	−0.130396	−0.0651981	−	0.997872i	$-0.520768\pi$
$941$	−4.00000	−0.130396	−0.0651981	+	0.997872i	$0.520768\pi$
$942$	0	0
$943$	−11.6569	−0.379599
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−16.0416	−0.521283	−0.260641	−	0.965436i	$-0.583934\pi$
$947$	−16.0416	−0.521283	−0.260641	+	0.965436i	$0.583934\pi$
$948$	0	0
$949$	69.4558	2.25463
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−14.8701	−0.481688	−0.240844	−	0.970564i	$-0.577424\pi$
$953$	−14.8701	−0.481688	−0.240844	+	0.970564i	$0.577424\pi$
$954$	0	0
$955$	5.34315	0.172900
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	23.9706	0.773244
$962$	0	0
$963$	0	0
$964$	0	0
$965$	5.65685	0.182101
$966$	0	0
$967$	43.4558	1.39745	0.698723	−	0.715392i	$-0.253752\pi$
$967$	43.4558	1.39745	0.698723	+	0.715392i	$0.253752\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	42.5269	1.36475	0.682377	−	0.731001i	$-0.260946\pi$
$971$	42.5269	1.36475	0.682377	+	0.731001i	$0.260946\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−20.1421	−0.644404	−0.322202	−	0.946671i	$-0.604423\pi$
$977$	−20.1421	−0.644404	−0.322202	+	0.946671i	$0.604423\pi$
$978$	0	0
$979$	31.0294	0.991705
$980$	0	0
$981$	0	0
$982$	0	0
$983$	40.0122	1.27619	0.638095	−	0.769957i	$-0.279723\pi$
$983$	40.0122	1.27619	0.638095	+	0.769957i	$0.279723\pi$
$984$	0	0
$985$	3.55635	0.113315
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−1.17157	−0.0372539
$990$	0	0
$991$	2.68629	0.0853329	0.0426664	−	0.999089i	$-0.486415\pi$
$991$	2.68629	0.0853329	0.0426664	+	0.999089i	$0.486415\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−1.27208	−0.0403276
$996$	0	0
$997$	−36.8406	−1.16675	−0.583377	−	0.812201i	$-0.698269\pi$
$997$	−36.8406	−1.16675	−0.583377	+	0.812201i	$0.698269\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8820.2.a.bg.1.1		2
3.2	odd	2		980.2.a.j.1.1	✓	2
7.6	odd	2		8820.2.a.bl.1.1		2
12.11	even	2		3920.2.a.bx.1.2		2
15.2	even	4		4900.2.e.q.2549.4		4
15.8	even	4		4900.2.e.q.2549.1		4
15.14	odd	2		4900.2.a.z.1.2		2
21.2	odd	6		980.2.i.l.361.2		4
21.5	even	6		980.2.i.k.361.1		4
21.11	odd	6		980.2.i.l.961.2		4
21.17	even	6		980.2.i.k.961.1		4
21.20	even	2		980.2.a.k.1.2	yes	2
84.83	odd	2		3920.2.a.bo.1.1		2
105.62	odd	4		4900.2.e.r.2549.1		4
105.83	odd	4		4900.2.e.r.2549.4		4
105.104	even	2		4900.2.a.x.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
980.2.a.j.1.1	✓	2	3.2	odd	2
980.2.a.k.1.2	yes	2	21.20	even	2
980.2.i.k.361.1		4	21.5	even	6
980.2.i.k.961.1		4	21.17	even	6
980.2.i.l.361.2		4	21.2	odd	6
980.2.i.l.961.2		4	21.11	odd	6
3920.2.a.bo.1.1		2	84.83	odd	2
3920.2.a.bx.1.2		2	12.11	even	2
4900.2.a.x.1.1		2	105.104	even	2
4900.2.a.z.1.2		2	15.14	odd	2
4900.2.e.q.2549.1		4	15.8	even	4
4900.2.e.q.2549.4		4	15.2	even	4
4900.2.e.r.2549.1		4	105.62	odd	4
4900.2.e.r.2549.4		4	105.83	odd	4
8820.2.a.bg.1.1		2	1.1	even	1	trivial
8820.2.a.bl.1.1		2	7.6	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 8820 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8820.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} -1.82843 q^{11} -6.41421 q^{13} +3.58579 q^{17} +7.65685 q^{19} -3.41421 q^{23} +1.00000 q^{25} +4.65685 q^{29} -7.41421 q^{31} -0.585786 q^{37} +3.41421 q^{41} +0.343146 q^{43} +10.8995 q^{47} +12.2426 q^{53} +1.82843 q^{55} +0.585786 q^{59} -10.8284 q^{61} +6.41421 q^{65} +3.07107 q^{67} +10.4853 q^{71} -10.8284 q^{73} -15.1421 q^{79} -8.00000 q^{83} -3.58579 q^{85} -16.9706 q^{89} -7.65685 q^{95} +9.72792 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5} + 2 q^{11} - 10 q^{13} + 10 q^{17} + 4 q^{19} - 4 q^{23} + 2 q^{25} - 2 q^{29} - 12 q^{31} - 4 q^{37} + 4 q^{41} + 12 q^{43} + 2 q^{47} + 16 q^{53} - 2 q^{55} + 4 q^{59} - 16 q^{61} + 10 q^{65}+ \cdots - 6 q^{97}+O(q^{100}) \) 2 * q - 2 * q^5 + 2 * q^11 - 10 * q^13 + 10 * q^17 + 4 * q^19 - 4 * q^23 + 2 * q^25 - 2 * q^29 - 12 * q^31 - 4 * q^37 + 4 * q^41 + 12 * q^43 + 2 * q^47 + 16 * q^53 - 2 * q^55 + 4 * q^59 - 16 * q^61 + 10 * q^65 - 8 * q^67 + 4 * q^71 - 16 * q^73 - 2 * q^79 - 16 * q^83 - 10 * q^85 - 4 * q^95 - 6 * q^97

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