LMFDB - Embedded newform 63.4.e.d.46.3

Newspace parameters

Level:	$ N $	$=$	$ 63 = 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	63.e (of order $3$, degree $2$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$3.71712033036$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
Defining polynomial:	$x^{8} + 19 x^{6} + 319 x^{4} + 798 x^{2} + 1764$
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2}\cdot 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			46.3
Root			$-0.799027 + 1.38396i$ of defining polynomial
Character	$\chi$	$=$	63.46
Dual form			63.4.e.d.37.3

$q$-expansion

$f(q)$	$=$	$q+(0.799027 + 1.38396i) q^{2} +(2.72311 - 4.71657i) q^{4} +(-9.14584 - 15.8411i) q^{5} +(12.3924 - 13.7633i) q^{7} +21.4878 q^{8} +O(q^{10})$	\(q+(0.799027 + 1.38396i) q^{2} +(2.72311 - 4.71657i) q^{4} +(-9.14584 - 15.8411i) q^{5} +(12.3924 - 13.7633i) q^{7} +21.4878 q^{8} +(14.6156 - 25.3149i) q^{10} +(-30.6336 + 53.0590i) q^{11} +32.4462 q^{13} +(28.9496 + 6.15337i) q^{14} +(-4.61555 - 7.99438i) q^{16} +(40.6644 - 70.4329i) q^{17} +(10.4542 + 18.1072i) q^{19} -99.6206 q^{20} -97.9084 q^{22} +(16.8655 + 29.2119i) q^{23} +(-104.793 + 181.507i) q^{25} +(25.9254 + 44.9041i) q^{26} +(-31.1693 - 95.9287i) q^{28} +52.0227 q^{29} +(-96.9622 + 167.943i) q^{31} +(93.3271 - 161.647i) q^{32} +129.968 q^{34} +(-331.364 - 70.4329i) q^{35} +(133.578 + 231.363i) q^{37} +(-16.7064 + 28.9364i) q^{38} +(-196.524 - 340.390i) q^{40} +203.176 q^{41} -21.9520 q^{43} +(166.838 + 288.971i) q^{44} +(-26.9520 + 46.6822i) q^{46} +(123.961 + 214.706i) q^{47} +(-35.8547 - 341.121i) q^{49} -334.929 q^{50} +(88.3547 - 153.035i) q^{52} +(-70.4131 + 121.959i) q^{53} +1120.68 q^{55} +(266.286 - 295.742i) q^{56} +(41.5676 + 71.9971i) q^{58} +(110.734 - 191.797i) q^{59} +(-326.263 - 565.104i) q^{61} -309.902 q^{62} +224.435 q^{64} +(-296.748 - 513.983i) q^{65} +(-302.239 + 523.493i) q^{67} +(-221.468 - 383.593i) q^{68} +(-167.293 - 514.871i) q^{70} +716.031 q^{71} +(-194.438 + 336.777i) q^{73} +(-213.465 + 369.731i) q^{74} +113.872 q^{76} +(350.639 + 1079.15i) q^{77} +(144.871 + 250.923i) q^{79} +(-84.4263 + 146.231i) q^{80} +(162.343 + 281.186i) q^{82} -115.652 q^{83} -1487.64 q^{85} +(-17.5403 - 30.3806i) q^{86} +(-658.249 + 1140.12i) q^{88} +(-469.682 - 813.513i) q^{89} +(402.088 - 446.566i) q^{91} +183.707 q^{92} +(-198.096 + 343.112i) q^{94} +(191.225 - 331.212i) q^{95} +120.394 q^{97} +(443.447 - 322.186i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8q - 6q^{4} - 12q^{7} + O(q^{10}) $	$ 8q - 6q^{4} - 12q^{7} - 22q^{10} + 204q^{13} + 102q^{16} - 222q^{19} - 172q^{22} - 366q^{25} - 166q^{28} - 220q^{31} + 2040q^{34} + 374q^{37} - 822q^{40} - 1676q^{43} - 1716q^{46} + 380q^{49} + 40q^{52} + 5020q^{55} + 1694q^{58} - 1332q^{61} - 1372q^{64} - 1890q^{67} - 866q^{70} - 1750q^{73} + 4912q^{76} - 8q^{79} - 2480q^{82} - 2232q^{85} - 2682q^{88} + 466q^{91} + 1416q^{94} + 6020q^{97} + O(q^{100}) $

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/63\mathbb{Z}\right)^\times$.

$n$	$10$	$29$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$1$

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 $

$2$	0.799027	+	1.38396i	0.282499	+	0.489302i	0.972000	−	0.234983i	$-0.0755034\pi$
$2$	0.799027	+	1.38396i	0.282499	+	0.489302i	−0.689501	+	0.724285i	$0.742170\pi$
$3$		0			0
$3$		0			0
$4$	2.72311	−	4.71657i	0.340389	−	0.589571i
$5$	−9.14584	−	15.8411i	−0.818029	−	1.41687i	−0.907132	−	0.420846i	$-0.861733\pi$
$5$	−9.14584	−	15.8411i	−0.818029	−	1.41687i	0.0891033	−	0.996022i	$-0.471600\pi$
$6$		0			0
$7$	12.3924	−	13.7633i	0.669129	−	0.743146i
$7$	12.3924	−	13.7633i	0.669129	−	0.743146i
$8$	21.4878			0.949635
$9$		0			0
$10$	14.6156	−	25.3149i	0.462184	−	0.800527i
$11$	−30.6336	+	53.0590i	−0.839672	+	1.45435i	0.0504975	+	0.998724i	$0.483919\pi$
$11$	−30.6336	+	53.0590i	−0.839672	+	1.45435i	−0.890169	+	0.455630i	$0.849414\pi$
$12$		0			0
$13$	32.4462			0.692228			0.346114	−	0.938192i	$-0.387501\pi$
$13$	32.4462			0.692228			0.346114	+	0.938192i	$0.387501\pi$
$14$	28.9496	+	6.15337i	0.552651	+	0.117468i
$15$		0			0
$16$	−4.61555	−	7.99438i	−0.0721180	−	0.124912i
$17$	40.6644	−	70.4329i	0.580152	−	1.00485i	−0.415309	−	0.909680i	$-0.636327\pi$
$17$	40.6644	−	70.4329i	0.580152	−	1.00485i	0.995461	−	0.0951718i	$-0.0303400\pi$
$18$		0			0
$19$	10.4542	+	18.1072i	0.126230	+	0.218636i	0.922213	−	0.386682i	$-0.126379\pi$
$19$	10.4542	+	18.1072i	0.126230	+	0.218636i	−0.795983	+	0.605319i	$0.793046\pi$
$20$	−99.6206			−1.11379
$21$		0			0
$22$	−97.9084			−0.948825
$23$	16.8655	+	29.2119i	0.152900	+	0.264831i	0.932292	−	0.361705i	$-0.117805\pi$
$23$	16.8655	+	29.2119i	0.152900	+	0.264831i	−0.779392	+	0.626536i	$0.784472\pi$
$24$		0			0
$25$	−104.793	+	181.507i	−0.838343	+	1.45205i
$26$	25.9254	+	44.9041i	0.195554	+	0.338709i
$27$		0			0
$28$	−31.1693	−	95.9287i	−0.210373	−	0.647458i
$29$	52.0227			0.333116			0.166558	−	0.986032i	$-0.446735\pi$
$29$	52.0227			0.333116			0.166558	+	0.986032i	$0.446735\pi$
$30$		0			0
$31$	−96.9622	+	167.943i	−0.561772	+	0.973017i	0.435570	+	0.900155i	$0.356547\pi$
$31$	−96.9622	+	167.943i	−0.561772	+	0.973017i	−0.997342	+	0.0728626i	$0.976787\pi$
$32$	93.3271	−	161.647i	0.515564	−	0.892983i
$33$		0			0
$34$	129.968			0.655568
$35$	−331.364	−	70.4329i	−1.60031	−	0.340152i
$36$		0			0
$37$	133.578	+	231.363i	0.593515	+	1.02800i	0.993755	+	0.111587i	$0.0355935\pi$
$37$	133.578	+	231.363i	0.593515	+	1.02800i	−0.400240	+	0.916410i	$0.631073\pi$
$38$	−16.7064	+	28.9364i	−0.0713194	+	0.123529i
$39$		0			0
$40$	−196.524	−	340.390i	−0.776829	−	1.34551i
$41$	203.176			0.773921			0.386960	−	0.922096i	$-0.373525\pi$
$41$	203.176			0.773921			0.386960	+	0.922096i	$0.373525\pi$
$42$		0			0
$43$	−21.9520			−0.0778523			−0.0389262	−	0.999242i	$-0.512394\pi$
$43$	−21.9520			−0.0778523			−0.0389262	+	0.999242i	$0.512394\pi$
$44$	166.838	+	288.971i	0.571630	+	0.990092i
$45$		0			0
$46$	−26.9520	+	46.6822i	−0.0863882	+	0.149629i
$47$	123.961	+	214.706i	0.384713	+	0.666343i	0.991729	−	0.128346i	$-0.0409669\pi$
$47$	123.961	+	214.706i	0.384713	+	0.666343i	−0.607016	+	0.794690i	$0.707634\pi$
$48$		0			0
$49$	−35.8547	−	341.121i	−0.104533	−	0.994521i
$50$	−334.929			−0.947324
$51$		0			0
$52$	88.3547	−	153.035i	0.235627	−	0.408117i
$53$	−70.4131	+	121.959i	−0.182490	+	0.316082i	−0.942728	−	0.333563i	$-0.891749\pi$
$53$	−70.4131	+	121.959i	−0.182490	+	0.316082i	0.760238	+	0.649645i	$0.225082\pi$
$54$		0			0
$55$	1120.68			2.74750
$56$	266.286	−	295.742i	0.635429	−	0.705718i
$57$		0			0
$58$	41.5676	+	71.9971i	0.0941050	+	0.162995i
$59$	110.734	−	191.797i	0.244344	−	0.423217i	−0.717603	−	0.696453i	$-0.754761\pi$
$59$	110.734	−	191.797i	0.244344	−	0.423217i	0.961947	+	0.273236i	$0.0880940\pi$
$60$		0			0
$61$	−326.263	−	565.104i	−0.684815	−	1.18613i	−0.973495	−	0.228709i	$-0.926550\pi$
$61$	−326.263	−	565.104i	−0.684815	−	1.18613i	0.288680	−	0.957426i	$-0.406784\pi$
$62$	−309.902			−0.634800
$63$		0			0
$64$	224.435			0.438349
$65$	−296.748	−	513.983i	−0.566263	−	0.980796i
$66$		0			0
$67$	−302.239	+	523.493i	−0.551110	+	0.954551i	0.447085	+	0.894492i	$0.352462\pi$
$67$	−302.239	+	523.493i	−0.551110	+	0.954551i	−0.998195	+	0.0600592i	$0.980871\pi$
$68$	−221.468	−	383.593i	−0.394954	−	0.684081i
$69$		0			0
$70$	−167.293	−	514.871i	−0.285647	−	0.879126i
$71$	716.031			1.19686			0.598431	−	0.801174i	$-0.295791\pi$
$71$	716.031			1.19686			0.598431	+	0.801174i	$0.295791\pi$
$72$		0			0
$73$	−194.438	+	336.777i	−0.311743	+	0.539956i	−0.978740	−	0.205105i	$-0.934246\pi$
$73$	−194.438	+	336.777i	−0.311743	+	0.539956i	0.666996	+	0.745061i	$0.267580\pi$
$74$	−213.465	+	369.731i	−0.335334	+	0.580816i
$75$		0			0
$76$	113.872			0.171869
$77$	350.639	+	1079.15i	0.518949	+	1.59715i
$78$		0			0
$79$	144.871	+	250.923i	0.206319	+	0.357355i	0.950552	−	0.310565i	$-0.100518\pi$
$79$	144.871	+	250.923i	0.206319	+	0.357355i	−0.744233	+	0.667920i	$0.767185\pi$
$80$	−84.4263	+	146.231i	−0.117989	+	0.204363i
$81$		0			0
$82$	162.343	+	281.186i	0.218632	+	0.378681i
$83$	−115.652			−0.152946			−0.0764728	−	0.997072i	$-0.524366\pi$
$83$	−115.652			−0.152946			−0.0764728	+	0.997072i	$0.524366\pi$
$84$		0			0
$85$	−1487.64			−1.89832
$86$	−17.5403	−	30.3806i	−0.0219932	−	0.0380933i
$87$		0			0
$88$	−658.249	+	1140.12i	−0.797382	+	1.38111i
$89$	−469.682	−	813.513i	−0.559395	−	0.968901i	−0.997547	−	0.0699997i	$-0.977700\pi$
$89$	−469.682	−	813.513i	−0.559395	−	0.968901i	0.438152	−	0.898901i	$-0.355633\pi$
$90$		0			0
$91$	402.088	−	446.566i	0.463190	−	0.514427i
$92$	183.707			0.208182
$93$		0			0
$94$	−198.096	+	343.112i	−0.217362	+	0.376482i
$95$	191.225	−	331.212i	0.206519	−	0.357701i
$96$		0			0
$97$	120.394			0.126022			0.0630110	−	0.998013i	$-0.479930\pi$
$97$	120.394			0.126022			0.0630110	+	0.998013i	$0.479930\pi$
$98$	443.447	−	322.186i	0.457091	−	0.332099i
$99$		0			0
$100$	570.725	+	988.525i	0.570725	+	0.988525i
$101$	−640.502	+	1109.38i	−0.631013	+	1.09295i	0.356332	+	0.934359i	$0.384027\pi$
$101$	−640.502	+	1109.38i	−0.631013	+	1.09295i	−0.987345	+	0.158587i	$0.949306\pi$
$102$		0			0
$103$	−265.669	−	460.153i	−0.254147	−	0.440196i	0.710516	−	0.703681i	$-0.248461\pi$
$103$	−265.669	−	460.153i	−0.254147	−	0.440196i	−0.964664	+	0.263485i	$0.915128\pi$
$104$	697.198			0.657364
$105$		0			0
$106$	−225.048			−0.206213
$107$	−66.6758	−	115.486i	−0.0602411	−	0.104341i	0.834332	−	0.551262i	$-0.185854\pi$
$107$	−66.6758	−	115.486i	−0.0602411	−	0.104341i	−0.894573	+	0.446922i	$0.852520\pi$
$108$		0			0
$109$	108.884	−	188.593i	0.0956811	−	0.165725i	−0.814212	−	0.580568i	$-0.802830\pi$
$109$	108.884	−	188.593i	0.0956811	−	0.165725i	0.909893	+	0.414844i	$0.136164\pi$
$110$	895.455	+	1550.97i	0.776166	+	1.34436i
$111$		0			0
$112$	−167.227	−	35.5448i	−0.141084	−	0.0299881i
$113$	−2006.09			−1.67006			−0.835031	−	0.550204i	$-0.814550\pi$
$113$	−2006.09			−1.67006			−0.835031	+	0.550204i	$0.814550\pi$
$114$		0			0
$115$	308.499	−	534.335i	0.250153	−	0.433278i
$116$	141.664	−	245.369i	0.113389	−	0.196396i
$117$		0			0
$118$	353.917			0.276108
$119$	−465.454	−	1432.51i	−0.358556	−	1.10351i
$120$		0			0
$121$	−1211.34	−	2098.10i	−0.910097	−	1.57633i
$122$	521.386	−	903.067i	0.386919	−	0.670163i
$123$		0			0
$124$	528.078	+	914.658i	0.382442	+	0.662409i
$125$	1547.22			1.10710
$126$		0			0
$127$	1638.92			1.14512			0.572562	−	0.819861i	$-0.305950\pi$
$127$	1638.92			1.14512			0.572562	+	0.819861i	$0.305950\pi$
$128$	−567.287	−	982.570i	−0.391731	−	0.678498i
$129$		0			0
$130$	474.220	−	821.372i	0.319937	−	0.554147i
$131$	−45.8755	−	79.4587i	−0.0305967	−	0.0529950i	0.850322	−	0.526263i	$-0.176407\pi$
$131$	−45.8755	−	79.4587i	−0.0305967	−	0.0529950i	−0.880918	+	0.473268i	$0.843074\pi$
$132$		0			0
$133$	378.768	+	80.5088i	0.246943	+	0.0524887i
$134$	−965.989			−0.622752
$135$		0			0
$136$	873.789	−	1513.45i	0.550932	−	0.954243i
$137$	933.564	−	1616.98i	0.582188	−	1.00838i	−0.413032	−	0.910717i	$-0.635530\pi$
$137$	933.564	−	1616.98i	0.582188	−	1.00838i	0.995220	−	0.0976621i	$-0.0311364\pi$
$138$		0			0
$139$	639.778			0.390397			0.195199	−	0.980764i	$-0.437465\pi$
$139$	639.778			0.390397			0.195199	+	0.980764i	$0.437465\pi$
$140$	−1234.54	+	1371.10i	−0.745271	+	0.827710i
$141$		0			0
$142$	572.128	+	990.955i	0.338112	+	0.585627i
$143$	−993.946	+	1721.56i	−0.581244	+	1.00674i
$144$		0			0
$145$	−475.792	−	824.095i	−0.272499	−	0.471982i
$146$	−621.446			−0.352269
$147$		0			0
$148$	1454.99			0.808103
$149$	1568.27	+	2716.32i	0.862264	+	1.49348i	0.869739	+	0.493513i	$0.164287\pi$
$149$	1568.27	+	2716.32i	0.862264	+	1.49348i	−0.00747495	+	0.999972i	$0.502379\pi$
$150$		0			0
$151$	1360.68	−	2356.77i	0.733317	−	1.27014i	−0.222141	−	0.975015i	$-0.571304\pi$
$151$	1360.68	−	2356.77i	0.733317	−	1.27014i	0.955458	−	0.295128i	$-0.0953623\pi$
$152$	224.638	+	389.085i	0.119872	+	0.207625i
$153$		0			0
$154$	−1213.32	+	1347.54i	−0.634886	+	0.705116i
$155$	3547.20			1.83818
$156$		0			0
$157$	−1439.87	+	2493.93i	−0.731939	+	1.26776i	0.224114	+	0.974563i	$0.428051\pi$
$157$	−1439.87	+	2493.93i	−0.731939	+	1.26776i	−0.956053	+	0.293193i	$0.905282\pi$
$158$	−231.511	+	400.989i	−0.116570	+	0.201905i
$159$		0			0
$160$	−3414.22			−1.68699
$161$	611.056	+	129.883i	0.299118	+	0.0635788i
$162$		0			0
$163$	−323.071	−	559.576i	−0.155245	−	0.268892i	0.777903	−	0.628384i	$-0.216283\pi$
$163$	−323.071	−	559.576i	−0.155245	−	0.268892i	−0.933148	+	0.359492i	$0.882950\pi$
$164$	553.271	−	958.293i	0.263434	−	0.456281i
$165$		0			0
$166$	−92.4093	−	160.058i	−0.0432069	−	0.0748366i
$167$	−3765.03			−1.74459			−0.872296	−	0.488979i	$-0.837370\pi$
$167$	−3765.03			−1.74459			−0.872296	+	0.488979i	$0.837370\pi$
$168$		0			0
$169$	−1144.24			−0.520821
$170$	−1188.67	−	2058.83i	−0.536274	−	0.928854i
$171$		0			0
$172$	−59.7777	+	103.538i	−0.0265001	+	0.0458995i
$173$	−1154.49	−	1999.64i	−0.507366	−	0.878783i	−0.999964	−	0.00852600i	$-0.997286\pi$
$173$	−1154.49	−	1999.64i	−0.507366	−	0.878783i	0.492598	−	0.870257i	$-0.336047\pi$
$174$		0			0
$175$	1199.48	+	3691.60i	0.518128	+	1.59462i
$176$	565.565			0.242222
$177$		0			0
$178$	750.577	−	1300.04i	0.316057	−	0.547427i
$179$	−1516.88	+	2627.31i	−0.633390	+	1.09706i	0.353464	+	0.935448i	$0.385004\pi$
$179$	−1516.88	+	2627.31i	−0.633390	+	1.09706i	−0.986854	+	0.161616i	$0.948329\pi$
$180$		0			0
$181$	−4079.71			−1.67537			−0.837686	−	0.546152i	$-0.816092\pi$
$181$	−4079.71			−1.67537			−0.837686	+	0.546152i	$0.816092\pi$
$182$	939.307	+	199.654i	0.382561	+	0.0813149i
$183$		0			0
$184$	362.403	+	627.700i	0.145199	+	0.251493i
$185$	2443.36	−	4232.03i	0.971025	−	1.68186i
$186$		0			0
$187$	2491.40	+	4315.23i	0.974274	+	1.68749i
$188$	1350.24			0.523809
$189$		0			0
$190$	611.177			0.233365
$191$	438.554	+	759.599i	0.166140	+	0.287762i	0.937059	−	0.349170i	$-0.113536\pi$
$191$	438.554	+	759.599i	0.166140	+	0.287762i	−0.770920	+	0.636932i	$0.780203\pi$
$192$		0			0
$193$	729.356	−	1263.28i	0.272022	−	0.471155i	−0.697358	−	0.716723i	$-0.745641\pi$
$193$	729.356	−	1263.28i	0.272022	−	0.471155i	0.969379	+	0.245568i	$0.0789744\pi$
$194$	96.1979	+	166.620i	0.0356011	+	0.0616629i
$195$		0			0
$196$	−1706.56	−	759.799i	−0.621923	−	0.276895i
$197$	−952.250			−0.344391			−0.172195	−	0.985063i	$-0.555086\pi$
$197$	−952.250			−0.344391			−0.172195	+	0.985063i	$0.555086\pi$
$198$		0			0
$199$	−1671.11	+	2894.44i	−0.595285	+	1.03106i	0.398222	+	0.917289i	$0.369627\pi$
$199$	−1671.11	+	2894.44i	−0.595285	+	1.03106i	−0.993507	+	0.113774i	$0.963706\pi$
$200$	−2251.77	+	3900.18i	−0.796120	+	1.37892i
$201$		0			0
$202$	−2047.11			−0.713041
$203$	644.689	−	716.002i	0.222898	−	0.247554i
$204$		0			0
$205$	−1858.22	−	3218.52i	−0.633090	−	1.09654i
$206$	424.554	−	735.349i	0.143593	−	0.248710i
$207$		0			0
$208$	−149.757	−	259.387i	−0.0499221	−	0.0864677i
$209$	−1281.00			−0.423966
$210$		0			0
$211$	1439.27			0.469589			0.234794	−	0.972045i	$-0.424558\pi$
$211$	1439.27			0.469589			0.234794	+	0.972045i	$0.424558\pi$
$212$	383.485	+	664.216i	0.124235	+	0.215182i
$213$		0			0
$214$	106.552	−	184.553i	0.0340361	−	0.0589522i
$215$	200.770	+	347.743i	0.0636855	+	0.110306i
$216$		0			0
$217$	1109.85	+	3415.75i	0.347196	+	1.06855i
$218$	348.007			0.108119
$219$		0			0
$220$	3051.74	−	5285.77i	0.935220	−	1.61985i
$221$	1319.41	−	2285.28i	0.401597	−	0.695587i
$222$		0			0
$223$	1009.86			0.303253			0.151626	−	0.988438i	$-0.451549\pi$
$223$	1009.86			0.303253			0.151626	+	0.988438i	$0.451549\pi$
$224$	−1068.24	−	3287.69i	−0.318638	−	0.980661i
$225$		0			0
$226$	−1602.92	−	2776.34i	−0.471790	−	0.817165i
$227$	−1474.30	+	2553.57i	−0.431070	+	0.746635i	−0.996966	−	0.0778419i	$-0.975197\pi$
$227$	−1474.30	+	2553.57i	−0.431070	+	0.746635i	0.565896	+	0.824477i	$0.308530\pi$
$228$		0			0
$229$	2019.42	+	3497.74i	0.582739	+	1.00933i	0.995153	+	0.0983374i	$0.0313524\pi$
$229$	2019.42	+	3497.74i	0.582739	+	1.00933i	−0.412414	+	0.910997i	$0.635314\pi$
$230$	985.995			0.282672
$231$		0			0
$232$	1117.85			0.316339
$233$	1497.90	+	2594.44i	0.421162	+	0.729473i	0.996053	−	0.0887561i	$-0.0282892\pi$
$233$	1497.90	+	2594.44i	0.421162	+	0.729473i	−0.574892	+	0.818230i	$0.694956\pi$
$234$		0			0
$235$	2267.45	−	3927.34i	0.629413	−	1.09018i
$236$	−603.081	−	1044.57i	−0.166344	−	0.288117i
$237$		0			0
$238$	1610.62	−	1788.78i	0.438660	−	0.487183i
$239$	1810.28			0.489948			0.244974	−	0.969530i	$-0.421221\pi$
$239$	1810.28			0.489948			0.244974	+	0.969530i	$0.421221\pi$
$240$		0			0
$241$	1874.71	−	3247.10i	0.501083	−	0.867900i	−0.498917	−	0.866650i	$-0.666269\pi$
$241$	1874.71	−	3247.10i	0.501083	−	0.867900i	0.999999	−	0.00125048i	$-0.000398042\pi$
$242$	1935.79	−	3352.88i	0.514203	−	0.890625i
$243$		0			0
$244$	−3553.80			−0.932414
$245$	−5075.80	+	3687.81i	−1.32359	+	0.961656i
$246$		0			0
$247$	339.200	+	587.512i	0.0873797	+	0.151346i
$248$	−2083.50	+	3608.74i	−0.533478	+	0.924012i
$249$		0			0
$250$	1236.27	+	2141.28i	0.312754	+	0.541706i
$251$	2706.96			0.680724			0.340362	−	0.940295i	$-0.389450\pi$
$251$	2706.96			0.680724			0.340362	+	0.940295i	$0.389450\pi$
$252$		0			0
$253$	−2066.61			−0.513544
$254$	1309.54	+	2268.20i	0.323496	+	0.560312i
$255$		0			0
$256$	1804.29	−	3125.13i	0.440502	−	0.762971i
$257$	2687.64	+	4655.13i	0.652337	+	1.12988i	0.982554	+	0.185975i	$0.0595445\pi$
$257$	2687.64	+	4655.13i	0.652337	+	1.12988i	−0.330218	+	0.943905i	$0.607122\pi$
$258$		0			0
$259$	4839.67	+	1028.69i	1.16109	+	0.246795i
$260$	−3232.31			−0.770998
$261$		0			0
$262$	73.3116	−	126.979i	0.0172870	−	0.0299420i
$263$	2623.28	−	4543.66i	0.615051	−	1.06530i	−0.375324	−	0.926893i	$-0.622469\pi$
$263$	2623.28	−	4543.66i	0.615051	−	1.06530i	0.990376	−	0.138406i	$-0.0441979\pi$
$264$		0			0
$265$	2575.95			0.597129
$266$	191.225	+	588.527i	0.0440781	+	0.135658i
$267$		0			0
$268$	1646.06	+	2851.06i	0.375184	+	0.649837i
$269$	1506.66	−	2609.61i	0.341496	−	0.591489i	−0.643214	−	0.765686i	$-0.722400\pi$
$269$	1506.66	−	2609.61i	0.341496	−	0.591489i	0.984711	+	0.174197i	$0.0557329\pi$
$270$		0			0
$271$	−3448.62	−	5973.19i	−0.773022	−	1.33891i	−0.935899	−	0.352267i	$-0.885411\pi$
$271$	−3448.62	−	5973.19i	−0.773022	−	1.33891i	0.162877	−	0.986646i	$-0.447923\pi$
$272$	−750.756			−0.167358
$273$		0			0
$274$	2983.77			0.657869
$275$	−6420.37	−	11120.4i	−1.40787	−	2.43850i
$276$		0			0
$277$	−1659.30	+	2874.00i	−0.359920	+	0.623400i	−0.987947	−	0.154792i	$-0.950529\pi$
$277$	−1659.30	+	2874.00i	−0.359920	+	0.623400i	0.628027	+	0.778191i	$0.283863\pi$
$278$	511.200	+	885.424i	0.110287	+	0.191022i
$279$		0			0
$280$	−7120.28	−	1513.45i	−1.51971	−	0.323021i
$281$	−6274.14			−1.33197			−0.665986	−	0.745964i	$-0.731989\pi$
$281$	−6274.14			−1.33197			−0.665986	+	0.745964i	$0.731989\pi$
$282$		0			0
$283$	−3886.24	+	6731.16i	−0.816300	+	1.41387i	0.0920914	+	0.995751i	$0.470645\pi$
$283$	−3886.24	+	6731.16i	−0.816300	+	1.41387i	−0.908391	+	0.418122i	$0.862689\pi$
$284$	1949.83	−	3377.21i	0.407399	−	0.705635i
$285$		0			0
$286$	−3176.76			−0.656803
$287$	2517.85	−	2796.36i	0.517853	−	0.575136i
$288$		0			0
$289$	−850.695	−	1473.45i	−0.173152	−	0.299908i
$290$	760.341	−	1316.95i	0.153961	−	0.266669i
$291$		0			0
$292$	1058.95	+	1834.16i	0.212228	+	0.367590i
$293$	854.897			0.170456			0.0852280	−	0.996361i	$-0.472838\pi$
$293$	854.897			0.170456			0.0852280	+	0.996361i	$0.472838\pi$
$294$		0			0
$295$	−4051.02			−0.799523
$296$	2870.29	+	4971.49i	0.563623	+	0.976223i
$297$		0			0
$298$	−2506.17	+	4340.82i	−0.487177	+	0.843815i
$299$	547.222	+	947.817i	0.105842	+	0.183323i
$300$		0			0
$301$	−272.039	+	302.131i	−0.0520932	+	0.0578557i
$302$	4348.89			0.828645
$303$		0			0
$304$	96.5041	−	167.150i	0.0182069	−	0.0315352i
$305$	−5967.90	+	10336.7i	−1.12040	+	1.94058i
$306$		0			0
$307$	2550.68			0.474185			0.237092	−	0.971487i	$-0.423806\pi$
$307$	2550.68			0.474185			0.237092	+	0.971487i	$0.423806\pi$
$308$	6044.71	+	1284.83i	1.11828	+	0.237695i
$309$		0			0
$310$	2834.31	+	4909.17i	0.519284	+	0.899427i
$311$	−3740.25	+	6478.31i	−0.681962	+	1.18119i	0.292419	+	0.956290i	$0.405540\pi$
$311$	−3740.25	+	6478.31i	−0.681962	+	1.18119i	−0.974381	+	0.224903i	$0.927794\pi$
$312$		0			0
$313$	−3.12392	−	5.41079i	−0.000564135	−	0.000977111i	0.865743	−	0.500488i	$-0.166846\pi$
$313$	−3.12392	−	5.41079i	−0.000564135	−	0.000977111i	−0.866307	+	0.499511i	$0.833513\pi$
$314$	−4601.99			−0.827088
$315$		0			0
$316$	1578.00			0.280915
$317$	−482.902	−	836.410i	−0.0855598	−	0.148194i	0.820070	−	0.572263i	$-0.193935\pi$
$317$	−482.902	−	836.410i	−0.0855598	−	0.148194i	−0.905630	+	0.424070i	$0.860601\pi$
$318$		0			0
$319$	−1593.64	+	2760.27i	−0.279708	+	0.484469i
$320$	−2052.64	−	3555.28i	−0.358582	−	0.621083i
$321$		0			0
$322$	308.499	+	949.455i	0.0533912	+	0.164320i
$323$	1700.46			0.292929
$324$		0			0
$325$	−3400.13	+	5889.20i	−0.580324	+	1.00515i
$326$	516.285	−	894.232i	0.0877129	−	0.151923i
$327$		0			0
$328$	4365.80			0.734942
$329$	4491.23	+	954.632i	0.752613	+	0.159971i
$330$		0			0
$331$	−3355.10	−	5811.20i	−0.557139	−	0.964992i	−0.997734	−	0.0672865i	$-0.978566\pi$
$331$	−3355.10	−	5811.20i	−0.557139	−	0.964992i	0.440595	−	0.897706i	$-0.354767\pi$
$332$	−314.934	+	545.482i	−0.0520610	+	0.0901723i
$333$		0			0
$334$	−3008.36	−	5210.64i	−0.492845	−	0.853633i
$335$	11056.9			1.80330
$336$		0			0
$337$	−605.546			−0.0978819			−0.0489409	−	0.998802i	$-0.515585\pi$
$337$	−605.546			−0.0978819			−0.0489409	+	0.998802i	$0.515585\pi$
$338$	−914.281	−	1583.58i	−0.147131	−	0.254839i
$339$		0			0
$340$	−4051.02	+	7016.57i	−0.646168	+	1.11920i
$341$	−5940.61	−	10289.4i	−0.943408	−	1.63403i
$342$		0			0
$343$	−5139.26	−	3733.84i	−0.809021	−	0.587780i
$344$	−471.700			−0.0739313
$345$		0			0
$346$	1844.94	−	3195.53i	0.286660	−	0.496510i
$347$	−3469.08	+	6008.62i	−0.536686	+	0.929567i	0.462394	+	0.886675i	$0.346991\pi$
$347$	−3469.08	+	6008.62i	−0.536686	+	0.929567i	−0.999080	+	0.0428923i	$0.986343\pi$
$348$		0			0
$349$	10368.9			1.59035			0.795176	−	0.606378i	$-0.207378\pi$
$349$	10368.9			1.59035			0.795176	+	0.606378i	$0.207378\pi$
$350$	−4150.59	+	4609.72i	−0.633882	+	0.704000i
$351$		0			0
$352$	5717.90	+	9903.69i	0.865809	+	1.49963i
$353$	2940.88	−	5093.75i	0.443420	−	0.768026i	−0.554521	−	0.832170i	$-0.687098\pi$
$353$	2940.88	−	5093.75i	0.443420	−	0.768026i	0.997941	+	0.0641440i	$0.0204317\pi$
$354$		0			0
$355$	−6548.70	−	11342.7i	−0.979068	−	1.69580i
$356$	−5115.98			−0.761647
$357$		0			0
$358$	−4848.11			−0.715728
$359$	−294.634	−	510.322i	−0.0433153	−	0.0750244i	0.843555	−	0.537043i	$-0.180459\pi$
$359$	−294.634	−	510.322i	−0.0433153	−	0.0750244i	−0.886870	+	0.462019i	$0.847125\pi$
$360$		0			0
$361$	3210.92	−	5561.47i	0.468132	−	0.810829i
$362$	−3259.80	−	5646.14i	−0.473291	−	0.819763i
$363$		0			0
$364$	−1011.33	−	3112.52i	−0.145626	−	0.448188i
$365$	7113.21			1.02006
$366$		0			0
$367$	1774.36	−	3073.28i	0.252373	−	0.437122i	−0.711806	−	0.702376i	$-0.752122\pi$
$367$	1774.36	−	3073.28i	0.252373	−	0.437122i	0.964179	+	0.265254i	$0.0854558\pi$
$368$	155.687	−	269.658i	0.0220537	−	0.0381982i
$369$		0			0
$370$	7809.25			1.09725
$371$	805.964	+	2480.49i	0.112786	+	0.347117i
$372$		0			0
$373$	790.667	+	1369.47i	0.109756	+	0.190104i	0.915672	−	0.401927i	$-0.131660\pi$
$373$	790.667	+	1369.47i	0.109756	+	0.190104i	−0.805915	+	0.592031i	$0.798326\pi$
$374$	−3981.39	+	6895.97i	−0.550462	+	0.953429i
$375$		0			0
$376$	2663.64	+	4613.56i	0.365337	+	0.632783i
$377$	1687.94			0.230592
$378$		0			0
$379$	3057.01			0.414322			0.207161	−	0.978307i	$-0.433578\pi$
$379$	3057.01			0.414322			0.207161	+	0.978307i	$0.433578\pi$
$380$	−1041.46	−	1803.85i	−0.140594	−	0.243515i
$381$		0			0
$382$	−700.834	+	1213.88i	−0.0938685	+	0.162585i
$383$	5289.87	+	9162.33i	0.705744	+	1.22238i	0.966422	+	0.256959i	$0.0827204\pi$
$383$	5289.87	+	9162.33i	0.705744	+	1.22238i	−0.260678	+	0.965426i	$0.583946\pi$
$384$		0			0
$385$	13888.0	−	15424.2i	1.83843	−	2.04180i
$386$	2331.10			0.307383
$387$		0			0
$388$	327.846	−	567.845i	0.0428965	−	0.0742989i
$389$	3696.65	−	6402.78i	0.481819	−	0.834534i	−0.517964	−	0.855403i	$-0.673310\pi$
$389$	3696.65	−	6402.78i	0.481819	−	0.834534i	0.999782	+	0.0208684i	$0.00664310\pi$
$390$		0			0
$391$	2743.31			0.354821
$392$	−770.438	−	7329.94i	−0.0992678	−	0.944433i
$393$		0			0
$394$	−760.874	−	1317.87i	−0.0972900	−	0.168511i
$395$	2649.93	−	4589.81i	0.337550	−	0.584654i
$396$		0			0
$397$	−889.086	−	1539.94i	−0.112398	−	0.194679i	0.804339	−	0.594171i	$-0.202520\pi$
$397$	−889.086	−	1539.94i	−0.112398	−	0.194679i	−0.916737	+	0.399492i	$0.869186\pi$
$398$	−5341.04			−0.672669
$399$		0			0
$400$	1934.71			0.241839
$401$	73.8031	+	127.831i	0.00919090	+	0.0159191i	0.870584	−	0.492019i	$-0.163741\pi$
$401$	73.8031	+	127.831i	0.00919090	+	0.0159191i	−0.861393	+	0.507938i	$0.830408\pi$
$402$		0			0
$403$	−3146.06	+	5449.13i	−0.388874	+	0.673550i
$404$	3488.31	+	6041.94i	0.429579	+	0.744053i
$405$		0			0
$406$	1506.04	+	320.115i	0.184097	+	0.0391307i
$407$	−16367.9			−1.99343
$408$		0			0
$409$	1080.03	−	1870.66i	0.130572	−	0.226157i	−0.793325	−	0.608798i	$-0.791652\pi$
$409$	1080.03	−	1870.66i	0.130572	−	0.226157i	0.923897	+	0.382641i	$0.124985\pi$
$410$	2969.53	−	5143.37i	0.357694	−	0.619544i
$411$		0			0
$412$	−2893.79			−0.346036
$413$	−1267.48	−	3900.89i	−0.151014	−	0.464770i
$414$		0			0
$415$	1057.74	+	1832.05i	0.125114	+	0.216704i
$416$	3028.11	−	5244.84i	0.356888	−	0.618148i
$417$		0			0
$418$	−1023.56	−	1772.85i	−0.119770	−	0.207447i
$419$	−13491.0			−1.57298			−0.786488	−	0.617605i	$-0.788103\pi$
$419$	−13491.0			−1.57298			−0.786488	+	0.617605i	$0.788103\pi$
$420$		0			0
$421$	−14146.7			−1.63769			−0.818847	−	0.574012i	$-0.805386\pi$
$421$	−14146.7			−1.63769			−0.818847	+	0.574012i	$0.805386\pi$
$422$	1150.01	+	1991.88i	0.132658	+	0.229771i
$423$		0			0
$424$	−1513.02	+	2620.63i	−0.173299	+	0.300163i
$425$	8522.69	+	14761.7i	0.972732	+	1.68482i
$426$		0			0
$427$	−11820.9	−	2512.58i	−1.33970	−	0.284759i
$428$	−726.262			−0.0820215
$429$		0			0
$430$	−320.841	+	555.712i	−0.0359821	+	0.0623229i
$431$	4544.26	−	7870.89i	0.507864	−	0.879646i	−0.492095	−	0.870542i	$-0.663769\pi$
$431$	4544.26	−	7870.89i	0.507864	−	0.879646i	0.999959	−	0.00910411i	$-0.00289797\pi$
$432$		0			0
$433$	15461.2			1.71597			0.857986	−	0.513673i	$-0.171716\pi$
$433$	15461.2			1.71597			0.857986	+	0.513673i	$0.171716\pi$
$434$	−3840.44	+	4265.26i	−0.424763	+	0.471749i
$435$		0			0
$436$	−593.009	−	1027.12i	−0.0651376	−	0.112822i
$437$	−352.632	+	610.776i	−0.0386010	+	0.0668590i
$438$		0			0
$439$	891.091	+	1543.41i	0.0968780	+	0.167798i	0.910391	−	0.413749i	$-0.135781\pi$
$439$	891.091	+	1543.41i	0.0968780	+	0.167798i	−0.813513	+	0.581547i	$0.802448\pi$
$440$	24081.0			2.60913
$441$		0			0
$442$	4216.97			0.453803
$443$	6712.36	+	11626.1i	0.719896	+	1.24690i	0.961041	+	0.276407i	$0.0891437\pi$
$443$	6712.36	+	11626.1i	0.719896	+	1.24690i	−0.241145	+	0.970489i	$0.577523\pi$
$444$		0			0
$445$	−8591.27	+	14880.5i	−0.915203	+	1.58518i
$446$	806.907	+	1397.60i	0.0856685	+	0.148382i
$447$		0			0
$448$	2781.29	−	3088.95i	0.293312	−	0.325757i
$449$	−418.639			−0.0440018			−0.0220009	−	0.999758i	$-0.507004\pi$
$449$	−418.639			−0.0440018			−0.0220009	+	0.999758i	$0.507004\pi$
$450$		0			0
$451$	−6224.02	+	10780.3i	−0.649839	+	1.12555i
$452$	−5462.80	+	9461.85i	−0.568470	+	0.984619i
$453$		0			0
$454$	−4712.03			−0.487107
$455$	−10751.5	−	2285.28i	−1.10778	−	0.235463i
$456$		0			0
$457$	354.205	+	613.501i	0.0362560	+	0.0627973i	0.883584	−	0.468273i	$-0.155123\pi$
$457$	354.205	+	613.501i	0.0362560	+	0.0627973i	−0.847328	+	0.531070i	$0.821790\pi$
$458$	−3227.15	+	5589.59i	−0.329246	+	0.570271i
$459$		0			0
$460$	−1680.15	−	2910.11i	−0.170299	−	0.294966i
$461$	−8223.97			−0.830865			−0.415432	−	0.909624i	$-0.636370\pi$
$461$	−8223.97			−0.830865			−0.415432	+	0.909624i	$0.636370\pi$
$462$		0			0
$463$	−9414.17			−0.944954			−0.472477	−	0.881343i	$-0.656640\pi$
$463$	−9414.17			−0.944954			−0.472477	+	0.881343i	$0.656640\pi$
$464$	−240.114	−	415.889i	−0.0240237	−	0.0416103i
$465$		0			0
$466$	−2393.73	+	4146.05i	−0.237955	+	0.412151i
$467$	−5410.76	−	9371.72i	−0.536146	−	0.928632i	−0.999107	−	0.0422535i	$-0.986546\pi$
$467$	−5410.76	−	9371.72i	−0.536146	−	0.928632i	0.462961	−	0.886379i	$-0.346787\pi$
$468$		0			0
$469$	3459.50	+	10647.2i	0.340607	+	1.04827i
$470$	7247.02			0.711234
$471$		0			0
$472$	2379.43	−	4121.29i	0.232038	−	0.401902i
$473$	672.470	−	1164.75i	0.0653704	−	0.113225i
$474$		0			0
$475$	−4382.11			−0.423295
$476$	−8024.02	−	1705.54i	−0.772648	−	0.164230i
$477$		0			0
$478$	1446.47	+	2505.35i	0.138410	+	0.239733i
$479$	4092.75	−	7088.85i	0.390402	−	0.676196i	−0.602100	−	0.798420i	$-0.705669\pi$
$479$	4092.75	−	7088.85i	0.390402	−	0.676196i	0.992503	+	0.122224i	$0.0390026\pi$
$480$		0			0
$481$	4334.09	+	7506.87i	0.410848	+	0.711609i
$482$	5991.79			0.566221
$483$		0			0
$484$	−13194.4			−1.23915
$485$	−1101.10	−	1907.17i	−0.103090	−	0.178557i
$486$		0			0
$487$	2001.58	−	3466.83i	0.186242	−	0.322581i	−0.757752	−	0.652543i	$-0.773702\pi$
$487$	2001.58	−	3466.83i	0.186242	−	0.322581i	0.943994	+	0.329961i	$0.107036\pi$
$488$	−7010.67	−	12142.8i	−0.650324	−	1.12639i
$489$		0			0
$490$	−9159.47	−	4078.01i	−0.844455	−	0.375971i
$491$	11180.8			1.02766			0.513831	−	0.857891i	$-0.328226\pi$
$491$	11180.8			1.02766			0.513831	+	0.857891i	$0.328226\pi$
$492$		0			0
$493$	2115.47	−	3664.11i	0.193258	−	0.334733i
$494$	−542.060	+	938.876i	−0.0493693	+	0.0855101i
$495$		0			0
$496$	1790.14			0.162056
$497$	8873.37	−	9854.92i	0.800855	−	0.889443i
$498$		0			0
$499$	1885.54	+	3265.85i	0.169155	+	0.292985i	0.938123	−	0.346302i	$-0.112563\pi$
$499$	1885.54	+	3265.85i	0.169155	+	0.292985i	−0.768968	+	0.639287i	$0.779229\pi$
$500$	4213.24	−	7297.55i	0.376844	−	0.652713i
$501$		0			0
$502$	2162.93	+	3746.31i	0.192304	+	0.333080i
$503$	−13597.2			−1.20531			−0.602654	−	0.798003i	$-0.705890\pi$
$503$	−13597.2			−1.20531			−0.602654	+	0.798003i	$0.705890\pi$
$504$		0			0
$505$	23431.7			2.06475
$506$	−1651.28	−	2860.09i	−0.145075	−	0.251278i
$507$		0			0
$508$	4462.97	−	7730.08i	0.389788	−	0.675132i
$509$	−3680.38	−	6374.60i	−0.320491	−	0.555106i	0.660099	−	0.751179i	$-0.270515\pi$
$509$	−3680.38	−	6374.60i	−0.320491	−	0.555106i	−0.980589	+	0.196073i	$0.937181\pi$
$510$		0			0
$511$	2225.58	+	6849.59i	0.192669	+	0.592971i
$512$	−3309.87			−0.285698
$513$		0			0
$514$	−4295.00	+	7439.16i	−0.368569	+	0.638380i
$515$	−4859.54	+	8416.97i	−0.415800	+	0.720186i
$516$		0			0
$517$	−15189.5			−1.29213
$518$	2443.36	+	7519.84i	0.207249	+	0.637844i
$519$		0			0
$520$	−6376.46	−	11044.4i	−0.537743	−	0.931398i
$521$	−6899.40	+	11950.1i	−0.580169	+	1.00488i	0.415290	+	0.909689i	$0.363680\pi$
$521$	−6899.40	+	11950.1i	−0.580169	+	1.00488i	−0.995459	+	0.0951930i	$0.969653\pi$
$522$		0			0
$523$	−9423.58	−	16322.1i	−0.787886	−	1.36466i	−0.927260	−	0.374419i	$-0.877842\pi$
$523$	−9423.58	−	16322.1i	−0.787886	−	1.36466i	0.139373	−	0.990240i	$-0.455491\pi$
$524$	−499.697			−0.0416591
$525$		0			0
$526$	8384.29			0.695005
$527$	7885.83	+	13658.7i	0.651826	+	1.12900i
$528$		0			0
$529$	5514.61	−	9551.58i	0.453243	−	0.785040i
$530$	2058.25	+	3565.00i	0.168688	+	0.292177i
$531$		0			0
$532$	1411.15	−	1567.25i	0.115002	−	0.127724i
$533$	6592.29			0.535730
$534$		0			0
$535$	−1219.61	+	2112.43i	−0.0985579	+	0.170707i
$536$	−6494.45	+	11248.7i	−0.523354	+	0.906475i
$537$		0			0
$538$	4815.44			0.385889
$539$	19197.9	+	8547.36i	1.53416	+	0.683044i
$540$		0			0
$541$	7234.77	+	12531.0i	0.574948	+	0.995839i	0.996047	+	0.0888248i	$0.0283111\pi$
$541$	7234.77	+	12531.0i	0.574948	+	0.995839i	−0.421099	+	0.907015i	$0.638356\pi$
$542$	5511.09	−	9545.49i	0.436756	−	0.756483i
$543$		0			0
$544$	−7590.19	−	13146.6i	−0.598211	−	1.03613i
$545$	−3983.36			−0.313080
$546$		0			0
$547$	5749.63			0.449427			0.224713	−	0.974425i	$-0.427855\pi$
$547$	5749.63			0.449427			0.224713	+	0.974425i	$0.427855\pi$
$548$	−5084.39	−	8806.43i	−0.396340	−	0.686482i
$549$		0			0
$550$	10260.1	−	17771.0i	0.795441	−	1.37774i
$551$	543.857	+	941.988i	0.0420492	+	0.0728313i
$552$		0			0
$553$	5248.82	+	1115.66i	0.403622	+	0.0857916i
$554$	−5303.31			−0.406708
$555$		0			0
$556$	1742.19	−	3017.55i	0.132887	−	0.230167i
$557$	−2715.39	+	4703.19i	−0.206561	+	0.357775i	−0.950629	−	0.310330i	$-0.899561\pi$
$557$	−2715.39	+	4703.19i	−0.206561	+	0.357775i	0.744068	+	0.668104i	$0.232894\pi$
$558$		0			0
$559$	−712.260			−0.0538915
$560$	966.362	+	2974.14i	0.0729219	+	0.224429i
$561$		0			0
$562$	−5013.21	−	8683.14i	−0.376280	−	0.651737i
$563$	12065.3	−	20897.8i	0.903185	−	1.56436i	0.0798500	−	0.996807i	$-0.474556\pi$
$563$	12065.3	−	20897.8i	0.903185	−	1.56436i	0.823335	−	0.567556i	$-0.192111\pi$
$564$		0			0
$565$	18347.4	+	31778.6i	1.36616	+	2.36626i
$566$	−12420.8			−0.922415
$567$		0			0
$568$	15385.9			1.13658
$569$	−9024.27	−	15630.5i	−0.664880	−	1.15161i	−0.979318	−	0.202328i	$-0.935149\pi$
$569$	−9024.27	−	15630.5i	−0.664880	−	1.15161i	0.314437	−	0.949278i	$-0.398184\pi$
$570$		0			0
$571$	5637.42	−	9764.30i	0.413168	−	0.715628i	−0.582066	−	0.813141i	$-0.697756\pi$
$571$	5637.42	−	9764.30i	0.413168	−	0.715628i	0.995234	+	0.0975136i	$0.0310889\pi$
$572$	5413.25	+	9376.02i	0.395698	+	0.685369i
$573$		0			0
$574$	5881.87	+	1250.22i	0.427708	+	0.0909113i
$575$	−7069.54			−0.512731
$576$		0			0
$577$	12047.5	−	20866.8i	0.869225	−	1.50554i	0.00643457	−	0.999979i	$-0.497952\pi$
$577$	12047.5	−	20866.8i	0.869225	−	1.50554i	0.862790	−	0.505562i	$-0.168715\pi$
$578$	1359.46	−	2354.65i	0.0978303	−	0.169447i
$579$		0			0
$580$	−5182.53			−0.371022
$581$	−1433.21	+	1591.75i	−0.102340	+	0.113661i
$582$		0			0
$583$	−4314.02	−	7472.10i	−0.306464	−	0.530811i
$584$	−4178.05	+	7236.59i	−0.296043	+	0.512761i
$585$		0			0
$586$	683.086	+	1183.14i	0.0481536	+	0.0834045i
$587$	11438.9			0.804315			0.402157	−	0.915571i	$-0.368260\pi$
$587$	11438.9			0.804315			0.402157	+	0.915571i	$0.368260\pi$
$588$		0			0
$589$	−4054.66			−0.283649
$590$	−3236.87	−	5606.43i	−0.225864	−	0.391208i
$591$		0			0
$592$	1233.07	−	2135.74i	0.0856063	−	0.148274i
$593$	2087.22	+	3615.17i	0.144539	+	0.250349i	0.929201	−	0.369575i	$-0.120497\pi$
$593$	2087.22	+	3615.17i	0.144539	+	0.250349i	−0.784662	+	0.619924i	$0.787163\pi$
$594$		0			0
$595$	−18435.5	+	20474.8i	−1.27022	+	1.41073i
$596$	17082.2			1.17402
$597$		0			0
$598$	−874.491	+	1514.66i	−0.0598003	+	0.103577i
$599$	5727.77	−	9920.79i	0.390702	−	0.676715i	−0.601841	−	0.798616i	$-0.705566\pi$
$599$	5727.77	−	9920.79i	0.390702	−	0.676715i	0.992542	+	0.121901i	$0.0388991\pi$
$600$		0			0
$601$	−17539.2			−1.19042			−0.595208	−	0.803572i	$-0.702930\pi$
$601$	−17539.2			−1.19042			−0.595208	+	0.803572i	$0.702930\pi$
$602$	−635.503	−	135.079i	−0.0430252	−	0.00914519i
$603$		0			0
$604$	−7410.59	−	12835.5i	−0.499226	−	0.864685i
$605$	−22157.4	+	38377.8i	−1.48897	+	2.57898i
$606$		0			0
$607$	−1692.86	−	2932.12i	−0.113198	−	0.196064i	0.803860	−	0.594818i	$-0.202776\pi$
$607$	−1692.86	−	2932.12i	−0.113198	−	0.196064i	−0.917058	+	0.398754i	$0.869443\pi$
$608$	3902.65			0.260318
$609$		0			0
$610$	−19074.1			−1.26604
$611$	4022.06	+	6966.41i	0.266309	+	0.461261i
$612$		0			0
$613$	−2135.94	+	3699.56i	−0.140734	+	0.243758i	−0.927773	−	0.373145i	$-0.878279\pi$
$613$	−2135.94	+	3699.56i	−0.140734	+	0.243758i	0.787039	+	0.616903i	$0.211613\pi$
$614$	2038.06	+	3530.02i	0.133957	+	0.232020i
$615$		0			0
$616$	7534.47	+	23188.5i	0.492812	+	1.51671i
$617$	−13123.9			−0.856321			−0.428160	−	0.903703i	$-0.640838\pi$
$617$	−13123.9			−0.856321			−0.428160	+	0.903703i	$0.640838\pi$
$618$		0			0
$619$	946.969	−	1640.20i	0.0614893	−	0.106503i	−0.833642	−	0.552305i	$-0.813748\pi$
$619$	946.969	−	1640.20i	0.0614893	−	0.106503i	0.895131	+	0.445803i	$0.147082\pi$
$620$	9659.43	−	16730.6i	0.625697	−	1.08374i
$621$		0			0
$622$	−11954.3			−0.770614
$623$	−17017.1	−	3617.06i	−1.09434	−	0.232607i
$624$		0			0
$625$	−1051.49	−	1821.23i	−0.0672952	−	0.116559i
$626$	4.99219	−	8.64673i	0.000318735	−	0.000552066i
$627$		0			0
$628$	7841.87	+	13582.5i	0.498288	+	0.863060i
$629$	21727.5			1.37731
$630$		0			0
$631$	20443.8			1.28979			0.644894	−	0.764272i	$-0.276902\pi$
$631$	20443.8			1.28979			0.644894	+	0.764272i	$0.276902\pi$
$632$	3112.95	+	5391.79i	0.195928	+	0.339357i
$633$		0			0
$634$	771.703	−	1336.63i	0.0483411	−	0.0837292i
$635$	−14989.3	−	25962.3i	−0.936745	−	1.62249i
$636$		0			0
$637$	−1163.35	−	11068.1i	−0.0723603	−	0.688436i
$638$	−5093.46			−0.316069
$639$		0			0
$640$	−10376.6	+	17972.9i	−0.640895	+	1.11006i
$641$	9614.27	−	16652.4i	0.592419	−	1.02610i	−0.401486	−	0.915865i	$-0.631506\pi$
$641$	9614.27	−	16652.4i	0.592419	−	1.02610i	0.993906	−	0.110235i	$-0.0351604\pi$
$642$		0			0
$643$	−18525.1			−1.13617			−0.568087	−	0.822969i	$-0.692316\pi$
$643$	−18525.1			−1.13617			−0.568087	+	0.822969i	$0.692316\pi$
$644$	2276.57	−	2528.40i	0.139301	−	0.154710i
$645$		0			0
$646$	1358.71	+	2353.36i	0.0827522	+	0.143331i
$647$	4011.20	−	6947.60i	0.243735	−	0.422161i	−0.718040	−	0.696001i	$-0.754961\pi$
$647$	4011.20	−	6947.60i	0.243735	−	0.422161i	0.961775	+	0.273840i	$0.0882940\pi$
$648$		0			0
$649$	6784.36	+	11750.9i	0.410338	+	0.710726i
$650$	−10867.2			−0.655764
$651$		0			0
$652$	−3519.03			−0.211374
$653$	1025.85	+	1776.82i	0.0614769	+	0.106481i	0.895126	−	0.445814i	$-0.147086\pi$
$653$	1025.85	+	1776.82i	0.0614769	+	0.106481i	−0.833649	+	0.552295i	$0.813752\pi$
$654$		0			0
$655$	−839.141	+	1453.43i	−0.0500579	+	0.0867029i
$656$	−937.770	−	1624.26i	−0.0558136	−	0.0966721i
$657$		0			0
$658$	2267.45	+	6978.45i	0.134338	+	0.413447i
$659$	−14765.2			−0.872792			−0.436396	−	0.899755i	$-0.643745\pi$
$659$	−14765.2			−0.872792			−0.436396	+	0.899755i	$0.643745\pi$
$660$		0			0
$661$	−323.532	+	560.374i	−0.0190377	+	0.0329743i	−0.875387	−	0.483422i	$-0.839394\pi$
$661$	−323.532	+	560.374i	−0.0190377	+	0.0329743i	0.856350	+	0.516397i	$0.172727\pi$
$662$	5361.63	−	9286.62i	0.314782	−	0.545218i
$663$		0			0
$664$	−2485.11			−0.145243
$665$	−2188.81	−	6736.41i	−0.127637	−	0.392822i
$666$		0			0
$667$	877.390	+	1519.68i	0.0509335	+	0.0882195i
$668$	−10252.6	+	17758.0i	−0.593840	+	1.02856i
$669$		0			0
$670$	8834.78	+	15302.3i	0.509429	+	0.882357i
$671$	39978.5			2.30008
$672$		0			0
$673$	−22596.6			−1.29426			−0.647130	−	0.762380i	$-0.724031\pi$
$673$	−22596.6			−1.29426			−0.647130	+	0.762380i	$0.724031\pi$
$674$	−483.848	−	838.049i	−0.0276515	−	0.0478938i
$675$		0			0
$676$	−3115.90	+	5396.90i	−0.177282	+	0.307061i
$677$	12602.1	+	21827.5i	0.715420	+	1.23914i	0.962797	+	0.270225i	$0.0870980\pi$
$677$	12602.1	+	21827.5i	0.715420	+	1.23914i	−0.247377	+	0.968919i	$0.579569\pi$
$678$		0			0
$679$	1491.97	−	1657.01i	0.0843250	−	0.0936528i
$680$	−31966.2			−1.80272
$681$		0			0
$682$	9493.42	−	16443.1i	0.533023	−	0.923223i
$683$	−8510.27	+	14740.2i	−0.476774	+	0.825796i	−0.999646	−	0.0266151i	$-0.991527\pi$
$683$	−8510.27	+	14740.2i	−0.476774	+	0.825796i	0.522872	+	0.852411i	$0.324860\pi$
$684$		0			0
$685$	−34152.9			−1.90499
$686$	1061.06	−	10096.0i	0.0590549	−	0.561903i
$687$		0			0
$688$	101.321	+	175.493i	0.00561456	+	0.00972470i
$689$	−2284.64	+	3957.11i	−0.126325	+	0.218801i
$690$		0			0
$691$	9645.26	+	16706.1i	0.531003	+	0.919724i	0.999345	+	0.0361772i	$0.0115181\pi$
$691$	9645.26	+	16706.1i	0.531003	+	0.919724i	−0.468342	+	0.883547i	$0.655149\pi$
$692$	−12575.2			−0.690806
$693$		0			0
$694$	−11087.6			−0.606452
$695$	−5851.31	−	10134.8i	−0.319356	−	0.553142i
$696$		0			0
$697$	8262.04	−	14310.3i	0.448991	−	0.777676i
$698$	8285.01	+	14350.1i	0.449273	+	0.778163i
$699$		0			0
$700$	20678.0	+	4395.20i	1.11651	+	0.237319i
$701$	28511.4			1.53618			0.768088	−	0.640345i	$-0.221208\pi$
$701$	28511.4			1.53618			0.768088	+	0.640345i	$0.221208\pi$
$702$		0			0
$703$	−2792.90	+	4837.45i	−0.149838	+	0.259528i
$704$	−6875.25	+	11908.3i	−0.368069	+	0.637515i
$705$		0			0
$706$	9399.37			0.501062
$707$	7331.32	+	22563.3i	0.389990	+	1.20026i
$708$		0			0
$709$	−14213.3	−	24618.1i	−0.752877	−	1.30402i	−0.946423	−	0.322930i	$-0.895332\pi$
$709$	−14213.3	−	24618.1i	−0.752877	−	1.30402i	0.193546	−	0.981091i	$-0.438001\pi$
$710$	10465.2	−	18126.2i	0.553171	−	0.958120i
$711$		0			0
$712$	−10092.4	−	17480.6i	−0.531221	−	0.920102i
$713$	−6541.27			−0.343580
$714$		0			0
$715$	36361.9			1.90190
$716$	8261.26	+	14308.9i	0.431198	+	0.746857i
$717$		0			0
$718$	470.842	−	815.522i	0.0244731	−	0.0423886i
$719$	10881.7	+	18847.6i	0.564420	+	0.977603i	0.997103	+	0.0760577i	$0.0242333\pi$
$719$	10881.7	+	18847.6i	0.564420	+	0.977603i	−0.432684	+	0.901546i	$0.642433\pi$
$720$		0			0
$721$	−9625.49	−	2045.94i	−0.497187	−	0.105679i
$722$	10262.4			0.528987
$723$		0			0
$724$	−11109.5	+	19242.2i	−0.570278	+	0.987750i
$725$	−5451.61	+	9442.47i	−0.279266	+	0.483703i
$726$		0			0
$727$	−13422.8			−0.684763			−0.342382	−	0.939561i	$-0.611234\pi$
$727$	−13422.8			−0.684763			−0.342382	+	0.939561i	$0.611234\pi$
$728$	8639.98	−	9595.71i	0.439861	−	0.488518i
$729$		0			0
$730$	5683.65	+	9844.36i	0.288166	+	0.499118i
$731$	−892.666	+	1546.14i	−0.0451661	+	0.0782301i
$732$		0			0
$733$	−2279.76	−	3948.66i	−0.114877	−	0.198973i	0.802854	−	0.596176i	$-0.203314\pi$
$733$	−2279.76	−	3948.66i	−0.114877	−	0.198973i	−0.917731	+	0.397204i	$0.869981\pi$
$734$	5671.04			0.285180
$735$		0			0
$736$	6296.04			0.315319
$737$	−18517.4	−	32073.0i	−0.925503	−	1.60302i
$738$		0			0
$739$	18332.7	−	31753.2i	0.912557	−	1.58059i	0.102118	−	0.994772i	$-0.467438\pi$
$739$	18332.7	−	31753.2i	0.912557	−	1.58059i	0.810439	−	0.585823i	$-0.199228\pi$
$740$	−13307.1	−	23048.6i	−0.661052	−	1.14498i
$741$		0			0
$742$	−2788.89	+	3097.39i	−0.137983	+	0.153247i
$743$	−10321.3			−0.509625			−0.254813	−	0.966990i	$-0.582014\pi$
$743$	−10321.3			−0.509625			−0.254813	+	0.966990i	$0.582014\pi$
$744$		0			0
$745$	28686.2	−	49686.0i	1.41071	−	2.44343i
$746$	−1263.53	+	2188.50i	−0.0620121	+	0.107408i
$747$		0			0
$748$	27137.4			1.32653
$749$	−2415.74	−	513.476i	−0.117849	−	0.0250494i
$750$		0			0
$751$	13339.0	+	23103.9i	0.648134	+	1.12260i	0.983568	+	0.180537i	$0.0577836\pi$
$751$	13339.0	+	23103.9i	0.648134	+	1.12260i	−0.335434	+	0.942064i	$0.608883\pi$
$752$	1144.30	−	1981.98i	0.0554896	−	0.0961107i
$753$		0			0
$754$	1348.71	+	2336.03i	0.0651421	+	0.112829i
$755$	−49778.4			−2.39950
$756$		0			0
$757$	−11630.8			−0.558425			−0.279212	−	0.960229i	$-0.590073\pi$
$757$	−11630.8			−0.558425			−0.279212	+	0.960229i	$0.590073\pi$
$758$	2442.63	+	4230.77i	0.117045	+	0.202729i
$759$		0			0
$760$	4109.01	−	7117.02i	0.196118	−	0.339686i
$761$	−18045.8	−	31256.3i	−0.859607	−	1.48888i	−0.872304	−	0.488963i	$-0.837375\pi$
$761$	−18045.8	−	31256.3i	−0.859607	−	1.48888i	0.0126976	−	0.999919i	$-0.495958\pi$
$762$		0			0
$763$	−1246.32	−	3835.74i	−0.0591345	−	0.181996i
$764$	4776.93			0.226208
$765$		0			0
$766$	−8453.51	+	14641.9i	−0.398744	+	0.690644i
$767$	3592.89	−	6223.07i	0.169142	−	0.292962i
$768$		0			0
$769$	33089.3			1.55167			0.775833	−	0.630938i	$-0.217330\pi$
$769$	33089.3			1.55167			0.775833	+	0.630938i	$0.217330\pi$
$770$	32443.3	+	6895.97i	1.51841	+	0.322745i
$771$		0			0
$772$	−3972.23	−	6880.11i	−0.185186	−	0.320752i
$773$	15495.4	−	26838.8i	0.720997	−	1.24880i	−0.239603	−	0.970871i	$-0.577017\pi$
$773$	15495.4	−	26838.8i	0.720997	−	1.24880i	0.960601	−	0.277933i	$-0.0896493\pi$
$774$		0			0
$775$	−20321.9	−	35198.6i	−0.941915	−	1.63144i
$776$	2587.00			0.119675
$777$		0			0
$778$	11814.9			0.544453
$779$	2124.05	+	3678.96i	0.0976917	+	0.169207i
$780$		0			0
$781$	−21934.6	+	37991.9i	−1.00497	+	1.74066i
$782$	2191.98	+	3796.62i	0.100236	+	0.173615i
$783$		0			0
$784$	−2561.56	+	1861.10i	−0.116689	+	0.0847803i
$785$	52675.4			2.39499
$786$		0			0
$787$	12710.6	−	22015.4i	0.575711	−	0.997161i	−0.420253	−	0.907407i	$-0.638059\pi$
$787$	12710.6	−	22015.4i	0.575711	−	0.997161i	0.995964	−	0.0897537i	$-0.0286080\pi$

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:	\( N \)	\(=\)	\( 63 = 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	63.e (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(0.799027 + 1.38396i) q^{2} +(2.72311 - 4.71657i) q^{4} +(-9.14584 - 15.8411i) q^{5} +(12.3924 - 13.7633i) q^{7} +21.4878 q^{8} +O(q^{10})\)	\(q+(0.799027 + 1.38396i) q^{2} +(2.72311 - 4.71657i) q^{4} +(-9.14584 - 15.8411i) q^{5} +(12.3924 - 13.7633i) q^{7} +21.4878 q^{8} +(14.6156 - 25.3149i) q^{10} +(-30.6336 + 53.0590i) q^{11} +32.4462 q^{13} +(28.9496 + 6.15337i) q^{14} +(-4.61555 - 7.99438i) q^{16} +(40.6644 - 70.4329i) q^{17} +(10.4542 + 18.1072i) q^{19} -99.6206 q^{20} -97.9084 q^{22} +(16.8655 + 29.2119i) q^{23} +(-104.793 + 181.507i) q^{25} +(25.9254 + 44.9041i) q^{26} +(-31.1693 - 95.9287i) q^{28} +52.0227 q^{29} +(-96.9622 + 167.943i) q^{31} +(93.3271 - 161.647i) q^{32} +129.968 q^{34} +(-331.364 - 70.4329i) q^{35} +(133.578 + 231.363i) q^{37} +(-16.7064 + 28.9364i) q^{38} +(-196.524 - 340.390i) q^{40} +203.176 q^{41} -21.9520 q^{43} +(166.838 + 288.971i) q^{44} +(-26.9520 + 46.6822i) q^{46} +(123.961 + 214.706i) q^{47} +(-35.8547 - 341.121i) q^{49} -334.929 q^{50} +(88.3547 - 153.035i) q^{52} +(-70.4131 + 121.959i) q^{53} +1120.68 q^{55} +(266.286 - 295.742i) q^{56} +(41.5676 + 71.9971i) q^{58} +(110.734 - 191.797i) q^{59} +(-326.263 - 565.104i) q^{61} -309.902 q^{62} +224.435 q^{64} +(-296.748 - 513.983i) q^{65} +(-302.239 + 523.493i) q^{67} +(-221.468 - 383.593i) q^{68} +(-167.293 - 514.871i) q^{70} +716.031 q^{71} +(-194.438 + 336.777i) q^{73} +(-213.465 + 369.731i) q^{74} +113.872 q^{76} +(350.639 + 1079.15i) q^{77} +(144.871 + 250.923i) q^{79} +(-84.4263 + 146.231i) q^{80} +(162.343 + 281.186i) q^{82} -115.652 q^{83} -1487.64 q^{85} +(-17.5403 - 30.3806i) q^{86} +(-658.249 + 1140.12i) q^{88} +(-469.682 - 813.513i) q^{89} +(402.088 - 446.566i) q^{91} +183.707 q^{92} +(-198.096 + 343.112i) q^{94} +(191.225 - 331.212i) q^{95} +120.394 q^{97} +(443.447 - 322.186i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8q - 6q^{4} - 12q^{7} + O(q^{10}) \)	\( 8q - 6q^{4} - 12q^{7} - 22q^{10} + 204q^{13} + 102q^{16} - 222q^{19} - 172q^{22} - 366q^{25} - 166q^{28} - 220q^{31} + 2040q^{34} + 374q^{37} - 822q^{40} - 1676q^{43} - 1716q^{46} + 380q^{49} + 40q^{52} + 5020q^{55} + 1694q^{58} - 1332q^{61} - 1372q^{64} - 1890q^{67} - 866q^{70} - 1750q^{73} + 4912q^{76} - 8q^{79} - 2480q^{82} - 2232q^{85} - 2682q^{88} + 466q^{91} + 1416q^{94} + 6020q^{97} + O(q^{100}) \)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Properties

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