LMFDB - Embedded newform 300.2.m.a.61.1

Newspace parameters

Level:	$ N $	$=$	$ 300 = 2^{2} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	300.m (of order $5$, degree $4$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$2.39551206064$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$2$ over $\Q(\zeta_{5})$
Coefficient field:	$\Q(\zeta_{15})$
Defining polynomial:	$x^{8} - x^{7} + x^{5} - x^{4} + x^{3} - x + 1$
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 5 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label			61.1
Root			$0.913545 + 0.406737i$ of defining polynomial
Character	$\chi$	$=$	300.61
Dual form			300.2.m.a.241.1

$q$-expansion

$f(q)$	$=$	$q+(0.309017 - 0.951057i) q^{3} +(-1.49622 - 1.66172i) q^{5} -4.78339 q^{7} +(-0.809017 - 0.587785i) q^{9} +O(q^{10})$	\(q+(0.309017 - 0.951057i) q^{3} +(-1.49622 - 1.66172i) q^{5} -4.78339 q^{7} +(-0.809017 - 0.587785i) q^{9} +(-1.58268 + 1.14988i) q^{11} +(-0.873619 - 0.634721i) q^{13} +(-2.04275 + 0.909491i) q^{15} +(-1.17603 - 3.61946i) q^{17} +(1.31359 + 4.04280i) q^{19} +(-1.47815 + 4.54927i) q^{21} +(4.74346 - 3.44633i) q^{23} +(-0.522642 + 4.97261i) q^{25} +(-0.809017 + 0.587785i) q^{27} +(3.26015 - 10.0337i) q^{29} +(-1.33369 - 4.10468i) q^{31} +(0.604528 + 1.86055i) q^{33} +(7.15701 + 7.94866i) q^{35} +(-4.57890 - 3.32676i) q^{37} +(-0.873619 + 0.634721i) q^{39} +(-0.694596 - 0.504654i) q^{41} -10.8764 q^{43} +(0.233733 + 2.22382i) q^{45} +(0.927539 - 2.85467i) q^{47} +15.8808 q^{49} -3.80573 q^{51} +(1.30524 - 4.01711i) q^{53} +(4.27882 + 0.909491i) q^{55} +4.25085 q^{57} +(3.85916 + 2.80384i) q^{59} +(-2.93822 + 2.13474i) q^{61} +(3.86984 + 2.81160i) q^{63} +(0.252397 + 2.40140i) q^{65} +(2.14163 + 6.59126i) q^{67} +(-1.81184 - 5.57627i) q^{69} +(-3.70731 + 11.4099i) q^{71} +(13.7177 - 9.96647i) q^{73} +(4.56773 + 2.03368i) q^{75} +(7.57055 - 5.50033i) q^{77} +(-2.04587 + 6.29656i) q^{79} +(0.309017 + 0.951057i) q^{81} +(-0.797847 - 2.45552i) q^{83} +(-4.25493 + 7.36976i) q^{85} +(-8.53519 - 6.20118i) q^{87} +(0.673699 - 0.489471i) q^{89} +(4.17886 + 3.03612i) q^{91} -4.31592 q^{93} +(4.75260 - 8.23174i) q^{95} +(-2.81730 + 8.67076i) q^{97} +1.95630 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8q - 2q^{3} + 5q^{5} - 8q^{7} - 2q^{9} + O(q^{10}) $	$ 8q - 2q^{3} + 5q^{5} - 8q^{7} - 2q^{9} - 2q^{11} - 5q^{15} + 7q^{17} + 5q^{19} - 3q^{21} + 7q^{23} + 5q^{25} - 2q^{27} + 27q^{29} - 3q^{31} + 3q^{33} + 20q^{35} - 9q^{37} + 20q^{41} - 68q^{43} - 5q^{45} - 7q^{47} - 8q^{49} - 8q^{51} - 11q^{53} + 5q^{55} - 10q^{57} + 2q^{59} - 14q^{61} + 7q^{63} - 35q^{65} + 28q^{67} + 2q^{69} - 15q^{71} + 6q^{73} + 5q^{75} + 17q^{77} + 24q^{79} - 2q^{81} + 2q^{83} + 10q^{85} - 23q^{87} + 5q^{91} - 18q^{93} + 5q^{95} + 34q^{97} - 2q^{99} + O(q^{100}) $

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/300\mathbb{Z}\right)^\times$.

$n$	$101$	$151$	$277$
$\chi(n)$	$1$	$1$	$e\left(\frac{4}{5}\right)$

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
$3$	0.309017	−	0.951057i	0.178411	−	0.549093i
$3$	0.309017	−	0.951057i	0.178411	−	0.549093i
$4$		0			0
$5$	−1.49622	−	1.66172i	−0.669131	−	0.743145i
$5$	−1.49622	−	1.66172i	−0.669131	−	0.743145i
$6$		0			0
$7$	−4.78339			−1.80795			−0.903975	−	0.427585i	$-0.859364\pi$
$7$	−4.78339			−1.80795			−0.903975	+	0.427585i	$0.859364\pi$
$8$		0			0
$9$	−0.809017	−	0.587785i	−0.269672	−	0.195928i
$10$		0			0
$11$	−1.58268	+	1.14988i	−0.477195	+	0.346702i	−0.800239	−	0.599682i	$-0.795294\pi$
$11$	−1.58268	+	1.14988i	−0.477195	+	0.346702i	0.323044	+	0.946384i	$0.395294\pi$
$12$		0			0
$13$	−0.873619	−	0.634721i	−0.242298	−	0.176040i	0.460008	−	0.887915i	$-0.347846\pi$
$13$	−0.873619	−	0.634721i	−0.242298	−	0.176040i	−0.702307	+	0.711875i	$0.747846\pi$
$14$		0			0
$15$	−2.04275	+	0.909491i	−0.527436	+	0.234830i
$16$		0			0
$17$	−1.17603	−	3.61946i	−0.285230	−	0.877848i	−0.986330	−	0.164785i	$-0.947307\pi$
$17$	−1.17603	−	3.61946i	−0.285230	−	0.877848i	0.701099	−	0.713064i	$-0.252693\pi$
$18$		0			0
$19$	1.31359	+	4.04280i	0.301357	+	0.927482i	0.981012	+	0.193949i	$0.0621298\pi$
$19$	1.31359	+	4.04280i	0.301357	+	0.927482i	−0.679654	+	0.733533i	$0.737870\pi$
$20$		0			0
$21$	−1.47815	+	4.54927i	−0.322558	+	0.992732i
$22$		0			0
$23$	4.74346	−	3.44633i	0.989080	−	0.718608i	0.0293604	−	0.999569i	$-0.490653\pi$
$23$	4.74346	−	3.44633i	0.989080	−	0.718608i	0.959719	+	0.280960i	$0.0906530\pi$
$24$		0			0
$25$	−0.522642	+	4.97261i	−0.104528	+	0.994522i
$26$		0			0
$27$	−0.809017	+	0.587785i	−0.155695	+	0.113119i
$28$		0			0
$29$	3.26015	−	10.0337i	0.605395	−	1.86322i	0.111345	−	0.993782i	$-0.464484\pi$
$29$	3.26015	−	10.0337i	0.605395	−	1.86322i	0.494050	−	0.869433i	$-0.335516\pi$
$30$		0			0
$31$	−1.33369	−	4.10468i	−0.239538	−	0.737223i	−0.996487	−	0.0837487i	$-0.973311\pi$
$31$	−1.33369	−	4.10468i	−0.239538	−	0.737223i	0.756949	−	0.653474i	$-0.226689\pi$
$32$		0			0
$33$	0.604528	+	1.86055i	0.105235	+	0.323880i
$34$		0			0
$35$	7.15701	+	7.94866i	1.20975	+	1.34357i
$36$		0			0
$37$	−4.57890	−	3.32676i	−0.752766	−	0.546917i	0.143917	−	0.989590i	$-0.454030\pi$
$37$	−4.57890	−	3.32676i	−0.752766	−	0.546917i	−0.896683	+	0.442673i	$0.854030\pi$
$38$		0			0
$39$	−0.873619	+	0.634721i	−0.139891	+	0.101637i
$40$		0			0
$41$	−0.694596	−	0.504654i	−0.108478	−	0.0788137i	0.532224	−	0.846604i	$-0.321356\pi$
$41$	−0.694596	−	0.504654i	−0.108478	−	0.0788137i	−0.640702	+	0.767790i	$0.721356\pi$
$42$		0			0
$43$	−10.8764			−1.65864			−0.829321	−	0.558773i	$-0.811272\pi$
$43$	−10.8764			−1.65864			−0.829321	+	0.558773i	$0.811272\pi$
$44$		0			0
$45$	0.233733	+	2.22382i	0.0348428	+	0.331507i
$46$		0			0
$47$	0.927539	−	2.85467i	0.135295	−	0.416397i	−0.860340	−	0.509720i	$-0.829749\pi$
$47$	0.927539	−	2.85467i	0.135295	−	0.416397i	0.995636	+	0.0933233i	$0.0297490\pi$
$48$		0			0
$49$	15.8808			2.26868
$50$		0			0
$51$	−3.80573			−0.532908
$52$		0			0
$53$	1.30524	−	4.01711i	0.179288	−	0.551793i	−0.820515	−	0.571625i	$-0.806313\pi$
$53$	1.30524	−	4.01711i	0.179288	−	0.551793i	0.999803	+	0.0198323i	$0.00631324\pi$
$54$		0			0
$55$	4.27882	+	0.909491i	0.576956	+	0.122636i
$56$		0			0
$57$	4.25085			0.563039
$58$		0			0
$59$	3.85916	+	2.80384i	0.502420	+	0.365029i	0.809940	−	0.586512i	$-0.199499\pi$
$59$	3.85916	+	2.80384i	0.502420	+	0.365029i	−0.307521	+	0.951541i	$0.599499\pi$
$60$		0			0
$61$	−2.93822	+	2.13474i	−0.376201	+	0.273326i	−0.759777	−	0.650183i	$-0.774692\pi$
$61$	−2.93822	+	2.13474i	−0.376201	+	0.273326i	0.383577	+	0.923509i	$0.374692\pi$
$62$		0			0
$63$	3.86984	+	2.81160i	0.487554	+	0.354229i
$64$		0			0
$65$	0.252397	+	2.40140i	0.0313060	+	0.297857i
$66$		0			0
$67$	2.14163	+	6.59126i	0.261642	+	0.805251i	0.992448	+	0.122666i	$0.0391444\pi$
$67$	2.14163	+	6.59126i	0.261642	+	0.805251i	−0.730806	+	0.682585i	$0.760856\pi$
$68$		0			0
$69$	−1.81184	−	5.57627i	−0.218120	−	0.671304i
$70$		0			0
$71$	−3.70731	+	11.4099i	−0.439977	+	1.35411i	0.447922	+	0.894072i	$0.352164\pi$
$71$	−3.70731	+	11.4099i	−0.439977	+	1.35411i	−0.887899	+	0.460038i	$0.847836\pi$
$72$		0			0
$73$	13.7177	−	9.96647i	1.60553	−	1.16649i	0.729836	−	0.683622i	$-0.239596\pi$
$73$	13.7177	−	9.96647i	1.60553	−	1.16649i	0.875695	−	0.482864i	$-0.160404\pi$
$74$		0			0
$75$	4.56773	+	2.03368i	0.527436	+	0.234830i
$76$		0			0
$77$	7.57055	−	5.50033i	0.862744	−	0.626820i
$78$		0			0
$79$	−2.04587	+	6.29656i	−0.230179	+	0.708418i	0.767546	+	0.640994i	$0.221478\pi$
$79$	−2.04587	+	6.29656i	−0.230179	+	0.708418i	−0.997724	+	0.0674234i	$0.978522\pi$
$80$		0			0
$81$	0.309017	+	0.951057i	0.0343352	+	0.105673i
$82$		0			0
$83$	−0.797847	−	2.45552i	−0.0875751	−	0.269528i	0.897673	−	0.440663i	$-0.145257\pi$
$83$	−0.797847	−	2.45552i	−0.0875751	−	0.269528i	−0.985248	+	0.171135i	$0.945257\pi$
$84$		0			0
$85$	−4.25493	+	7.36976i	−0.461512	+	0.799362i
$86$		0			0
$87$	−8.53519	−	6.20118i	−0.915069	−	0.664836i
$88$		0			0
$89$	0.673699	−	0.489471i	0.0714120	−	0.0518838i	−0.551507	−	0.834171i	$-0.685947\pi$
$89$	0.673699	−	0.489471i	0.0714120	−	0.0518838i	0.622918	+	0.782287i	$0.285947\pi$
$90$		0			0
$91$	4.17886	+	3.03612i	0.438063	+	0.318272i
$92$		0			0
$93$	−4.31592			−0.447540
$94$		0			0
$95$	4.75260	−	8.23174i	0.487606	−	0.844559i
$96$		0			0
$97$	−2.81730	+	8.67076i	−0.286054	+	0.880382i	0.700027	+	0.714116i	$0.253171\pi$
$97$	−2.81730	+	8.67076i	−0.286054	+	0.880382i	−0.986081	+	0.166266i	$0.946829\pi$
$98$		0			0
$99$	1.95630			0.196615
$100$		0			0
$101$	1.49541			0.148799			0.0743994	−	0.997229i	$-0.476296\pi$
$101$	1.49541			0.148799			0.0743994	+	0.997229i	$0.476296\pi$
$102$		0			0
$103$	3.35664	−	10.3307i	0.330739	−	1.01791i	−0.638044	−	0.770000i	$-0.720256\pi$
$103$	3.35664	−	10.3307i	0.330739	−	1.01791i	0.968783	−	0.247911i	$-0.0797439\pi$
$104$		0			0
$105$	9.77126	−	4.35045i	0.953577	−	0.424560i
$106$		0			0
$107$	−19.8390			−1.91791			−0.958954	−	0.283563i	$-0.908484\pi$
$107$	−19.8390			−1.91791			−0.958954	+	0.283563i	$0.908484\pi$
$108$		0			0
$109$	1.89626	+	1.37771i	0.181629	+	0.131961i	0.674885	−	0.737923i	$-0.264193\pi$
$109$	1.89626	+	1.37771i	0.181629	+	0.131961i	−0.493256	+	0.869884i	$0.664193\pi$
$110$		0			0
$111$	−4.57890	+	3.32676i	−0.434610	+	0.315762i
$112$		0			0
$113$	2.61192	+	1.89767i	0.245709	+	0.178518i	0.703823	−	0.710376i	$-0.251475\pi$
$113$	2.61192	+	1.89767i	0.245709	+	0.178518i	−0.458114	+	0.888893i	$0.651475\pi$
$114$		0			0
$115$	−12.8241	−	2.72585i	−1.19585	−	0.254187i
$116$		0			0
$117$	0.333693	+	1.02700i	0.0308499	+	0.0949463i
$118$		0			0
$119$	5.62543	+	17.3133i	0.515682	+	1.58711i
$120$		0			0
$121$	−2.21655	+	6.82184i	−0.201505	+	0.620167i
$122$		0			0
$123$	−0.694596	+	0.504654i	−0.0626296	+	0.0455031i
$124$		0			0
$125$	9.04508	−	6.57164i	0.809017	−	0.587785i
$126$		0			0
$127$	−5.97565	+	4.34156i	−0.530253	+	0.385251i	−0.820452	−	0.571715i	$-0.806278\pi$
$127$	−5.97565	+	4.34156i	−0.530253	+	0.385251i	0.290199	+	0.956966i	$0.406278\pi$
$128$		0			0
$129$	−3.36101	+	10.3441i	−0.295920	+	0.910748i
$130$		0			0
$131$	−5.21740	−	16.0575i	−0.455847	−	1.40295i	−0.870138	−	0.492808i	$-0.835971\pi$
$131$	−5.21740	−	16.0575i	−0.455847	−	1.40295i	0.414292	−	0.910144i	$-0.364029\pi$
$132$		0			0
$133$	−6.28339	−	19.3383i	−0.544839	−	1.67684i
$134$		0			0
$135$	2.18720	+	0.464905i	0.188245	+	0.0400126i
$136$		0			0
$137$	−0.679023	−	0.493339i	−0.0580128	−	0.0421488i	0.558401	−	0.829571i	$-0.311415\pi$
$137$	−0.679023	−	0.493339i	−0.0580128	−	0.0421488i	−0.616414	+	0.787422i	$0.711415\pi$
$138$		0			0
$139$	9.02701	−	6.55851i	0.765661	−	0.556285i	−0.134980	−	0.990848i	$-0.543097\pi$
$139$	9.02701	−	6.55851i	0.765661	−	0.556285i	0.900641	+	0.434563i	$0.143097\pi$
$140$		0			0
$141$	−2.42833	−	1.76428i	−0.204502	−	0.148580i
$142$		0			0
$143$	2.11251			0.176657
$144$		0			0
$145$	−21.5512	+	9.59520i	−1.78973	+	0.796838i
$146$		0			0
$147$	4.90743	−	15.1035i	0.404758	−	1.24572i
$148$		0			0
$149$	−22.0277			−1.80458			−0.902288	−	0.431134i	$-0.858114\pi$
$149$	−22.0277			−1.80458			−0.902288	+	0.431134i	$0.858114\pi$
$150$		0			0
$151$	11.7890			0.959378			0.479689	−	0.877439i	$-0.340750\pi$
$151$	11.7890			0.959378			0.479689	+	0.877439i	$0.340750\pi$
$152$		0			0
$153$	−1.17603	+	3.61946i	−0.0950767	+	0.292616i
$154$		0			0
$155$	−4.82535	+	8.35774i	−0.387581	+	0.671310i
$156$		0			0
$157$	11.2769			0.899994			0.449997	−	0.893030i	$-0.351425\pi$
$157$	11.2769			0.899994			0.449997	+	0.893030i	$0.351425\pi$
$158$		0			0
$159$	−3.41716	−	2.48271i	−0.270998	−	0.196892i
$160$		0			0
$161$	−22.6898	+	16.4851i	−1.78821	+	1.29921i
$162$		0			0
$163$	−8.00627	−	5.81690i	−0.627100	−	0.455615i	0.228295	−	0.973592i	$-0.426685\pi$
$163$	−8.00627	−	5.81690i	−0.627100	−	0.455615i	−0.855394	+	0.517978i	$0.826685\pi$
$164$		0			0
$165$	2.18720	−	3.78835i	0.170274	−	0.294923i
$166$		0			0
$167$	−2.32246	−	7.14780i	−0.179717	−	0.553113i	0.820100	−	0.572220i	$-0.193918\pi$
$167$	−2.32246	−	7.14780i	−0.179717	−	0.553113i	−0.999817	+	0.0191070i	$0.993918\pi$
$168$		0			0
$169$	−3.65688	−	11.2547i	−0.281299	−	0.865748i
$170$		0			0
$171$	1.31359	−	4.04280i	0.100452	−	0.309161i
$172$		0			0
$173$	−9.93332	+	7.21698i	−0.755217	+	0.548697i	−0.897439	−	0.441138i	$-0.854575\pi$
$173$	−9.93332	+	7.21698i	−0.755217	+	0.548697i	0.142223	+	0.989835i	$0.454575\pi$
$174$		0			0
$175$	2.50000	−	23.7859i	0.188982	−	1.79805i
$176$		0			0
$177$	3.85916	−	2.80384i	0.290072	−	0.210750i
$178$		0			0
$179$	4.69982	−	14.4646i	0.351281	−	1.08113i	−0.606854	−	0.794813i	$-0.707569\pi$
$179$	4.69982	−	14.4646i	0.351281	−	1.08113i	0.958135	−	0.286318i	$-0.0924314\pi$
$180$		0			0
$181$	−6.45087	−	19.8537i	−0.479490	−	1.47572i	−0.839806	−	0.542887i	$-0.817331\pi$
$181$	−6.45087	−	19.8537i	−0.479490	−	1.47572i	0.360316	−	0.932830i	$-0.382669\pi$
$182$		0			0
$183$	1.12230	+	3.45409i	0.0829629	+	0.255333i
$184$		0			0
$185$	1.32289	+	12.5864i	0.0972606	+	0.925373i
$186$		0			0
$187$	6.02323	+	4.37613i	0.440462	+	0.320015i
$188$		0			0
$189$	3.86984	−	2.81160i	0.281489	−	0.204514i
$190$		0			0
$191$	4.24803	+	3.08637i	0.307377	+	0.223322i	0.730770	−	0.682624i	$-0.239161\pi$
$191$	4.24803	+	3.08637i	0.307377	+	0.223322i	−0.423393	+	0.905946i	$0.639161\pi$
$192$		0			0
$193$	7.94565			0.571940			0.285970	−	0.958239i	$-0.407684\pi$
$193$	7.94565			0.571940			0.285970	+	0.958239i	$0.407684\pi$
$194$		0			0
$195$	2.36186	+	0.502029i	0.169136	+	0.0359510i
$196$		0			0
$197$	−4.01905	+	12.3694i	−0.286345	+	0.881281i	0.699647	+	0.714489i	$0.253341\pi$
$197$	−4.01905	+	12.3694i	−0.286345	+	0.881281i	−0.985992	+	0.166792i	$0.946659\pi$
$198$		0			0
$199$	1.06434			0.0754490			0.0377245	−	0.999288i	$-0.487989\pi$
$199$	1.06434			0.0754490			0.0377245	+	0.999288i	$0.487989\pi$
$200$		0			0
$201$	6.93046			0.488837
$202$		0			0
$203$	−15.5946	+	47.9952i	−1.09452	+	3.36860i
$204$		0			0
$205$	0.200676	+	1.90930i	0.0140158	+	0.133351i
$206$		0			0
$207$	−5.86324			−0.407523
$208$		0			0
$209$	−6.72772	−	4.88798i	−0.465366	−	0.338108i
$210$		0			0
$211$	16.5791	−	12.0454i	1.14135	−	0.829239i	0.154043	−	0.988064i	$-0.450771\pi$
$211$	16.5791	−	12.0454i	1.14135	−	0.829239i	0.987307	+	0.158825i	$0.0507707\pi$
$212$		0			0
$213$	9.70587	+	7.05173i	0.665035	+	0.483176i
$214$		0			0
$215$	16.2736	+	18.0736i	1.10985	+	1.23261i
$216$		0			0
$217$	6.37957	+	19.6343i	0.433073	+	1.33286i
$218$		0			0
$219$	−5.23968	−	16.1261i	−0.354065	−	1.08970i
$220$		0			0
$221$	−1.26994	+	3.90848i	−0.0854257	+	0.262913i
$222$		0			0
$223$	−8.34266	+	6.06130i	−0.558666	+	0.405895i	−0.830971	−	0.556316i	$-0.812214\pi$
$223$	−8.34266	+	6.06130i	−0.558666	+	0.405895i	0.272304	+	0.962211i	$0.412214\pi$
$224$		0			0
$225$	3.34565	−	3.71572i	0.223044	−	0.247715i
$226$		0			0
$227$	14.6763	−	10.6630i	0.974100	−	0.707725i	0.0177176	−	0.999843i	$-0.494360\pi$
$227$	14.6763	−	10.6630i	0.974100	−	0.707725i	0.956382	+	0.292118i	$0.0943600\pi$
$228$		0			0
$229$	−3.00311	+	9.24263i	−0.198451	+	0.610770i	0.801468	+	0.598038i	$0.204053\pi$
$229$	−3.00311	+	9.24263i	−0.198451	+	0.610770i	−0.999919	+	0.0127320i	$0.995947\pi$
$230$		0			0
$231$	−2.89169	−	8.89972i	−0.190259	−	0.585558i
$232$		0			0
$233$	−6.86586	−	21.1310i	−0.449798	−	1.38433i	−0.877136	−	0.480243i	$-0.840549\pi$
$233$	−6.86586	−	21.1310i	−0.449798	−	1.38433i	0.427338	−	0.904092i	$-0.359451\pi$
$234$		0			0
$235$	−6.13148	+	2.72991i	−0.399973	+	0.178080i
$236$		0			0
$237$	5.35617	+	3.89149i	0.347921	+	0.252779i
$238$		0			0
$239$	−12.8871	+	9.36300i	−0.833595	+	0.605642i	−0.920574	−	0.390568i	$-0.872279\pi$
$239$	−12.8871	+	9.36300i	−0.833595	+	0.605642i	0.0869793	+	0.996210i	$0.472279\pi$
$240$		0			0
$241$	3.98964	+	2.89864i	0.256995	+	0.186718i	0.708821	−	0.705388i	$-0.249227\pi$
$241$	3.98964	+	2.89864i	0.256995	+	0.186718i	−0.451826	+	0.892106i	$0.649227\pi$
$242$		0			0
$243$	1.00000			0.0641500
$244$		0			0
$245$	−23.7612	−	26.3895i	−1.51805	−	1.68596i
$246$		0			0
$247$	1.41848	−	4.36563i	0.0902556	−	0.277778i
$248$		0			0
$249$	−2.58189			−0.163620
$250$		0			0
$251$	15.4351			0.974252			0.487126	−	0.873332i	$-0.338045\pi$
$251$	15.4351			0.974252			0.487126	+	0.873332i	$0.338045\pi$
$252$		0			0
$253$	−3.54449	+	10.9088i	−0.222840	+	0.685832i
$254$		0			0
$255$	5.69421	+	6.32406i	0.356585	+	0.396028i
$256$		0			0
$257$	21.1828			1.32134			0.660672	−	0.750674i	$-0.270271\pi$
$257$	21.1828			1.32134			0.660672	+	0.750674i	$0.270271\pi$
$258$		0			0
$259$	21.9026	+	15.9132i	1.36096	+	0.988798i
$260$		0			0
$261$	−8.53519	+	6.20118i	−0.528315	+	0.383843i
$262$		0			0
$263$	16.9298	+	12.3002i	1.04394	+	0.758465i	0.971050	−	0.238876i	$-0.0767788\pi$
$263$	16.9298	+	12.3002i	1.04394	+	0.758465i	0.0728867	+	0.997340i	$0.476779\pi$
$264$		0			0
$265$	−8.62825	+	3.84154i	−0.530029	+	0.235984i
$266$		0			0
$267$	−0.257330	−	0.791981i	−0.0157483	−	0.0484684i
$268$		0			0
$269$	−4.25938	−	13.1090i	−0.259699	−	0.799272i	−0.992867	−	0.119225i	$-0.961959\pi$
$269$	−4.25938	−	13.1090i	−0.259699	−	0.799272i	0.733168	−	0.680048i	$-0.238041\pi$
$270$		0			0
$271$	3.25998	−	10.0332i	0.198029	−	0.609472i	−0.801898	−	0.597460i	$-0.796177\pi$
$271$	3.25998	−	10.0332i	0.198029	−	0.609472i	0.999928	−	0.0120115i	$-0.00382346\pi$
$272$		0			0
$273$	4.17886	−	3.03612i	0.252916	−	0.183754i
$274$		0			0
$275$	−4.89074	−	8.47101i	−0.294923	−	0.510821i
$276$		0			0
$277$	−14.5317	+	10.5579i	−0.873127	+	0.634364i	−0.931424	−	0.363935i	$-0.881433\pi$
$277$	−14.5317	+	10.5579i	−0.873127	+	0.634364i	0.0582969	+	0.998299i	$0.481433\pi$
$278$		0			0
$279$	−1.33369	+	4.10468i	−0.0798461	+	0.245741i
$280$		0			0
$281$	−0.286375	−	0.881371i	−0.0170837	−	0.0525782i	0.942151	−	0.335188i	$-0.108800\pi$
$281$	−0.286375	−	0.881371i	−0.0170837	−	0.0525782i	−0.959235	+	0.282610i	$0.908800\pi$
$282$		0			0
$283$	−2.03256	−	6.25556i	−0.120823	−	0.371855i	0.872294	−	0.488982i	$-0.162632\pi$
$283$	−2.03256	−	6.25556i	−0.120823	−	0.371855i	−0.993117	+	0.117127i	$0.962632\pi$
$284$		0			0
$285$	−6.36022	−	7.06374i	−0.376747	−	0.418420i
$286$		0			0
$287$	3.32252	+	2.41395i	0.196122	+	0.142491i
$288$		0			0
$289$	2.03585	−	1.47913i	0.119756	−	0.0870076i
$290$		0			0
$291$	7.37579	+	5.35882i	0.432376	+	0.314140i
$292$		0			0
$293$	25.6208			1.49678			0.748391	−	0.663257i	$-0.230827\pi$
$293$	25.6208			1.49678			0.748391	+	0.663257i	$0.230827\pi$
$294$		0			0
$295$	−1.11495	−	10.6080i	−0.0649148	−	0.617623i
$296$		0			0
$297$	0.604528	−	1.86055i	0.0350783	−	0.107960i
$298$		0			0
$299$	−6.33143			−0.366156
$300$		0			0
$301$	52.0262			2.99874
$302$		0			0
$303$	0.462107	−	1.42222i	0.0265474	−	0.0817044i
$304$		0			0
$305$	7.94358	+	1.68846i	0.454848	+	0.0966809i
$306$		0			0
$307$	−6.18420			−0.352951			−0.176476	−	0.984305i	$-0.556470\pi$
$307$	−6.18420			−0.352951			−0.176476	+	0.984305i	$0.556470\pi$
$308$		0			0
$309$	−8.78779	−	6.38470i	−0.499920	−	0.363213i
$310$		0			0
$311$	19.6091	−	14.2468i	1.11193	−	0.807864i	0.128963	−	0.991649i	$-0.458835\pi$
$311$	19.6091	−	14.2468i	1.11193	−	0.807864i	0.982966	+	0.183785i	$0.0588351\pi$
$312$		0			0
$313$	24.2392	+	17.6108i	1.37008	+	0.995420i	0.997731	+	0.0673262i	$0.0214468\pi$
$313$	24.2392	+	17.6108i	1.37008	+	0.995420i	0.372347	+	0.928094i	$0.378553\pi$
$314$		0			0
$315$	−1.11803	−	10.6374i	−0.0629941	−	0.599349i
$316$		0			0
$317$	0.761007	+	2.34214i	0.0427424	+	0.131548i	0.970151	−	0.242503i	$-0.0779685\pi$
$317$	0.761007	+	2.34214i	0.0427424	+	0.131548i	−0.927408	+	0.374051i	$0.877969\pi$
$318$		0			0
$319$	6.37782	+	19.6289i	0.357090	+	1.09901i
$320$		0			0
$321$	−6.13058	+	18.8680i	−0.342176	+	1.05311i
$322$		0			0
$323$	13.0879	−	9.50894i	0.728232	−	0.529092i
$324$		0			0
$325$	3.61281	−	4.01243i	0.200403	−	0.222570i
$326$		0			0
$327$	1.89626	−	1.37771i	0.104863	−	0.0761878i
$328$		0			0
$329$	−4.43678	+	13.6550i	−0.244607	+	0.752824i
$330$		0			0
$331$	−8.27755	−	25.4757i	−0.454975	−	1.40027i	−0.871164	−	0.490992i	$-0.836634\pi$
$331$	−8.27755	−	25.4757i	−0.454975	−	1.40027i	0.416189	−	0.909278i	$-0.363366\pi$
$332$		0			0
$333$	1.74898	+	5.38282i	0.0958437	+	0.294977i
$334$		0			0
$335$	7.74850	−	13.4208i	0.423346	−	0.733256i
$336$		0			0
$337$	−1.97124	−	1.43219i	−0.107381	−	0.0780165i	0.532799	−	0.846242i	$-0.321140\pi$
$337$	−1.97124	−	1.43219i	−0.107381	−	0.0780165i	−0.640180	+	0.768225i	$0.721140\pi$
$338$		0			0
$339$	2.61192	−	1.89767i	0.141860	−	0.103067i
$340$		0			0
$341$	6.83070	+	4.96280i	0.369903	+	0.268751i
$342$		0			0
$343$	−42.4802			−2.29372
$344$		0			0
$345$	−6.55530	+	11.3541i	−0.352925	+	0.611285i
$346$		0			0
$347$	−2.36011	+	7.26368i	−0.126698	+	0.389935i	−0.994207	−	0.107487i	$-0.965720\pi$
$347$	−2.36011	+	7.26368i	−0.126698	+	0.389935i	0.867509	+	0.497421i	$0.165720\pi$
$348$		0			0
$349$	−20.2283			−1.08280			−0.541398	−	0.840766i	$-0.682105\pi$
$349$	−20.2283			−1.08280			−0.541398	+	0.840766i	$0.682105\pi$
$350$		0			0
$351$	1.07985			0.0576383
$352$		0			0
$353$	−8.74279	+	26.9075i	−0.465331	+	1.43214i	0.393234	+	0.919439i	$0.371356\pi$
$353$	−8.74279	+	26.9075i	−0.465331	+	1.43214i	−0.858565	+	0.512705i	$0.828644\pi$
$354$		0			0
$355$	24.5071	−	10.9113i	1.30070	−	0.579110i
$356$		0			0
$357$	18.2043			0.963472
$358$		0			0
$359$	−18.4713	−	13.4202i	−0.974880	−	0.708292i	−0.0183215	−	0.999832i	$-0.505832\pi$
$359$	−18.4713	−	13.4202i	−0.974880	−	0.708292i	−0.956559	+	0.291540i	$0.905832\pi$
$360$		0			0
$361$	0.752597	−	0.546793i	0.0396103	−	0.0287786i
$362$		0			0
$363$	5.80301	+	4.21613i	0.304579	+	0.221289i
$364$		0			0
$365$	−37.0862	−	7.88291i	−1.94118	−	0.412610i
$366$		0			0
$367$	−0.778091	−	2.39472i	−0.0406160	−	0.125003i	0.928693	−	0.370851i	$-0.120934\pi$
$367$	−0.778091	−	2.39472i	−0.0406160	−	0.125003i	−0.969309	+	0.245847i	$0.920934\pi$
$368$		0			0
$369$	0.265312	+	0.816547i	0.0138116	+	0.0425077i
$370$		0			0
$371$	−6.24346	+	19.2154i	−0.324144	+	0.997614i
$372$		0			0
$373$	−4.57416	+	3.32332i	−0.236841	+	0.172075i	−0.699875	−	0.714265i	$-0.746761\pi$
$373$	−4.57416	+	3.32332i	−0.236841	+	0.172075i	0.463034	+	0.886341i	$0.346761\pi$
$374$		0			0
$375$	−3.45492	−	10.6331i	−0.178411	−	0.549093i
$376$		0			0
$377$	−9.21675	+	6.69636i	−0.474687	+	0.344880i
$378$		0			0
$379$	−4.44729	+	13.6874i	−0.228442	+	0.703073i	0.769482	+	0.638669i	$0.220515\pi$
$379$	−4.44729	+	13.6874i	−0.228442	+	0.703073i	−0.997924	+	0.0644039i	$0.979485\pi$
$380$		0			0
$381$	2.28249	+	7.02479i	0.116936	+	0.359891i
$382$		0			0
$383$	−2.53391	−	7.79859i	−0.129477	−	0.398489i	0.865213	−	0.501404i	$-0.167183\pi$
$383$	−2.53391	−	7.79859i	−0.129477	−	0.398489i	−0.994690	+	0.102915i	$0.967183\pi$
$384$		0			0
$385$	−20.4672	−	4.35045i	−1.04311	−	0.221719i
$386$		0			0
$387$	8.79923	+	6.39301i	0.447290	+	0.324975i
$388$		0			0
$389$	−3.82719	+	2.78062i	−0.194046	+	0.140983i	−0.680566	−	0.732687i	$-0.738266\pi$
$389$	−3.82719	+	2.78062i	−0.194046	+	0.140983i	0.486520	+	0.873670i	$0.338266\pi$
$390$		0			0
$391$	−18.0523	−	13.1158i	−0.912945	−	0.663293i
$392$		0			0
$393$	−16.8839			−0.851679
$394$		0			0
$395$	13.5242	−	6.02137i	0.680477	−	0.302968i
$396$		0			0
$397$	−2.34336	+	7.21211i	−0.117610	+	0.361965i	−0.992482	−	0.122387i	$-0.960945\pi$
$397$	−2.34336	+	7.21211i	−0.117610	+	0.361965i	0.874873	+	0.484353i	$0.160945\pi$
$398$		0			0
$399$	−20.3335			−1.01795
$400$		0			0
$401$	10.1128			0.505011			0.252506	−	0.967595i	$-0.418745\pi$
$401$	10.1128			0.505011			0.252506	+	0.967595i	$0.418745\pi$
$402$		0			0
$403$	−1.44019	+	4.43245i	−0.0717411	+	0.220796i
$404$		0			0
$405$	1.11803	−	1.93649i	0.0555556	−	0.0962250i
$406$		0			0
$407$	11.0723			0.548833
$408$		0			0
$409$	−3.21927	−	2.33894i	−0.159183	−	0.115653i	0.505342	−	0.862919i	$-0.331366\pi$
$409$	−3.21927	−	2.33894i	−0.159183	−	0.115653i	−0.664525	+	0.747266i	$0.731366\pi$
$410$		0			0
$411$	−0.679023	+	0.493339i	−0.0334937	+	0.0243346i
$412$		0			0
$413$	−18.4598	−	13.4119i	−0.908350	−	0.659955i
$414$		0			0
$415$	−2.88664	+	4.99980i	−0.141699	+	0.245431i
$416$		0			0
$417$	−3.44801	−	10.6119i	−0.168850	−	0.519666i
$418$		0			0
$419$	0.555164	+	1.70862i	0.0271215	+	0.0834714i	0.963701	−	0.266984i	$-0.0860269\pi$
$419$	0.555164	+	1.70862i	0.0271215	+	0.0834714i	−0.936580	+	0.350455i	$0.886027\pi$
$420$		0			0
$421$	4.84234	−	14.9032i	0.236001	−	0.726337i	−0.760986	−	0.648769i	$-0.775284\pi$
$421$	4.84234	−	14.9032i	0.236001	−	0.726337i	0.996987	−	0.0775686i	$-0.0247157\pi$
$422$		0			0
$423$	−2.42833	+	1.76428i	−0.118069	+	0.0857824i
$424$		0			0
$425$	18.6128	−	3.95628i	0.902854	−	0.191908i
$426$		0			0
$427$	14.0546	−	10.2113i	0.680152	−	0.494159i
$428$		0			0
$429$	0.652802	−	2.00912i	0.0315176	−	0.0970011i
$430$		0			0
$431$	−11.9206	−	36.6880i	−0.574197	−	1.76720i	−0.638896	−	0.769293i	$-0.720609\pi$
$431$	−11.9206	−	36.6880i	−0.574197	−	1.76720i	0.0646985	−	0.997905i	$-0.479391\pi$
$432$		0			0
$433$	2.83353	+	8.72070i	0.136171	+	0.419090i	0.995770	−	0.0918780i	$-0.0292870\pi$
$433$	2.83353	+	8.72070i	0.136171	+	0.419090i	−0.859600	+	0.510968i	$0.829287\pi$
$434$		0			0
$435$	2.46590	+	23.4615i	0.118231	+	1.12489i
$436$		0			0
$437$	20.1637	+	14.6498i	0.964563	+	0.700796i
$438$		0			0
$439$	−14.3101	+	10.3969i	−0.682982	+	0.496216i	−0.874346	−	0.485304i	$-0.838709\pi$
$439$	−14.3101	+	10.3969i	−0.682982	+	0.496216i	0.191363	+	0.981519i	$0.438709\pi$
$440$		0			0
$441$	−12.8478	−	9.33449i	−0.611801	−	0.444500i
$442$		0			0
$443$	−18.6673			−0.886912			−0.443456	−	0.896296i	$-0.646248\pi$
$443$	−18.6673			−0.886912			−0.443456	+	0.896296i	$0.646248\pi$
$444$		0			0
$445$	−1.82137	−	0.387144i	−0.0863411	−	0.0183524i
$446$		0			0
$447$	−6.80692	+	20.9495i	−0.321956	+	0.990880i
$448$		0			0
$449$	41.2111			1.94487			0.972436	−	0.233171i	$-0.0749102\pi$
$449$	41.2111			1.94487			0.972436	+	0.233171i	$0.0749102\pi$
$450$		0			0
$451$	1.67961			0.0790899
$452$		0			0
$453$	3.64301	−	11.2120i	0.171164	−	0.526787i
$454$		0			0
$455$	−1.20731	−	11.4868i	−0.0565997	−	0.538510i
$456$		0			0
$457$	−5.56092			−0.260129			−0.130064	−	0.991506i	$-0.541518\pi$
$457$	−5.56092			−0.260129			−0.130064	+	0.991506i	$0.541518\pi$
$458$		0			0
$459$	3.07890	+	2.23695i	0.143711	+	0.104412i
$460$		0			0
$461$	−17.1210	+	12.4391i	−0.797404	+	0.579348i	−0.910151	−	0.414276i	$-0.864035\pi$
$461$	−17.1210	+	12.4391i	−0.797404	+	0.579348i	0.112747	+	0.993624i	$0.464035\pi$
$462$		0			0
$463$	6.26637	+	4.55278i	0.291223	+	0.211586i	0.723798	−	0.690012i	$-0.242395\pi$
$463$	6.26637	+	4.55278i	0.291223	+	0.211586i	−0.432575	+	0.901598i	$0.642395\pi$
$464$		0			0
$465$	6.45757	+	7.17186i	0.299463	+	0.332587i
$466$		0			0
$467$	−0.751544	−	2.31301i	−0.0347773	−	0.107034i	0.932161	−	0.362044i	$-0.117921\pi$
$467$	−0.751544	−	2.31301i	−0.0347773	−	0.107034i	−0.966938	+	0.255011i	$0.917921\pi$
$468$		0			0
$469$	−10.2442	−	31.5286i	−0.473036	−	1.45585i
$470$		0			0
$471$	3.48475	−	10.7250i	0.160569	−	0.494180i
$472$		0			0
$473$	17.2139	−	12.5066i	0.791495	−	0.575055i
$474$		0			0
$475$	−20.7898	+	4.41901i	−0.953902	+	0.202758i
$476$		0			0
$477$	−3.41716	+	2.48271i	−0.156461	+	0.113676i
$478$		0			0
$479$	10.5465	−	32.4587i	0.481880	−	1.48308i	−0.354569	−	0.935030i	$-0.615372\pi$
$479$	10.5465	−	32.4587i	0.481880	−	1.48308i	0.836449	−	0.548045i	$-0.184628\pi$
$480$		0			0
$481$	1.88864	+	5.81265i	0.0861148	+	0.265034i
$482$		0			0
$483$	8.66673	+	26.6735i	0.394350	+	1.21368i
$484$		0			0
$485$	18.6237	−	8.29181i	0.845659	−	0.376512i
$486$		0			0
$487$	25.9175	+	18.8302i	1.17443	+	0.853277i	0.991533	−	0.129854i	$-0.0414509\pi$
$487$	25.9175	+	18.8302i	1.17443	+	0.853277i	0.182902	+	0.983131i	$0.441451\pi$
$488$		0			0
$489$	−8.00627	+	5.81690i	−0.362056	+	0.263049i
$490$		0			0
$491$	−23.7860	−	17.2815i	−1.07345	−	0.779904i	−0.0969178	−	0.995292i	$-0.530898\pi$
$491$	−23.7860	−	17.2815i	−1.07345	−	0.779904i	−0.976529	+	0.215388i	$0.930898\pi$
$492$		0			0
$493$	−40.1507			−1.80830
$494$		0			0
$495$	−2.92705	−	3.25082i	−0.131561	−	0.146113i
$496$		0			0
$497$	17.7335	−	54.5781i	0.795456	−	2.44816i
$498$		0			0
$499$	−9.04298			−0.404819			−0.202410	−	0.979301i	$-0.564877\pi$
$499$	−9.04298			−0.404819			−0.202410	+	0.979301i	$0.564877\pi$
$500$		0			0
$501$	−7.51564			−0.335774
$502$		0			0
$503$	0.950848	−	2.92641i	0.0423962	−	0.130482i	−0.927618	−	0.373530i	$-0.878147\pi$
$503$	0.950848	−	2.92641i	0.0423962	−	0.130482i	0.970014	+	0.243048i	$0.0781473\pi$
$504$		0			0
$505$	−2.23747	−	2.48496i	−0.0995659	−	0.110579i
$506$		0			0
$507$	−11.8339			−0.525563
$508$		0			0
$509$	−21.2410	−	15.4325i	−0.941491	−	0.684033i	0.00728833	−	0.999973i	$-0.497680\pi$
$509$	−21.2410	−	15.4325i	−0.941491	−	0.684033i	−0.948779	+	0.315940i	$0.897680\pi$
$510$		0			0
$511$	−65.6169	+	47.6735i	−2.90272	+	2.10895i
$512$		0			0
$513$	−3.43901	−	2.49859i	−0.151836	−	0.110315i
$514$		0			0
$515$	−22.1890	+	9.87917i	−0.977763	+	0.435328i
$516$		0			0
$517$	1.81454	+	5.58458i	0.0798034	+	0.245610i
$518$		0			0
$519$	3.79419	+	11.6773i	0.166547	+	0.512578i
$520$		0			0
$521$	6.41127	−	19.7319i	0.280883	−	0.864469i	−0.706720	−	0.707494i	$-0.749826\pi$
$521$	6.41127	−	19.7319i	0.280883	−	0.864469i	0.987603	−	0.156975i	$-0.0501742\pi$
$522$		0			0
$523$	18.1623	−	13.1957i	0.794182	−	0.577007i	−0.115020	−	0.993363i	$-0.536693\pi$
$523$	18.1623	−	13.1957i	0.794182	−	0.577007i	0.909202	+	0.416356i	$0.136693\pi$
$524$		0			0
$525$	−21.8492	−	9.72789i	−0.953577	−	0.424560i
$526$		0			0
$527$	−13.2883	+	9.65450i	−0.578846	+	0.420557i
$528$		0			0
$529$	3.51586	−	10.8207i	0.152864	−	0.470466i
$530$		0			0
$531$	−1.47407	−	4.53671i	−0.0639691	−	0.196877i
$532$		0			0
$533$	0.286498	+	0.881750i	0.0124096	+	0.0381928i
$534$		0			0
$535$	29.6835	+	32.9669i	1.28333	+	1.42528i
$536$		0			0
$537$	−12.3043	−	8.93959i	−0.530969	−	0.385772i
$538$		0			0
$539$	−25.1341	+	18.2610i	−1.08260	+	0.786558i
$540$		0			0
$541$	20.4954	+	14.8908i	0.881166	+	0.640205i	0.933560	−	0.358422i	$-0.116685\pi$
$541$	20.4954	+	14.8908i	0.881166	+	0.640205i	−0.0523936	+	0.998627i	$0.516685\pi$
$542$		0			0
$543$	−20.8755			−0.895852
$544$		0			0
$545$	−0.547848	−	5.21243i	−0.0234672	−	0.223276i
$546$		0			0
$547$	−3.64866	+	11.2294i	−0.156005	+	0.480135i	−0.998261	−	0.0589426i	$-0.981227\pi$
$547$	−3.64866	+	11.2294i	−0.156005	+	0.480135i	0.842256	+	0.539078i	$0.181227\pi$
$548$		0			0
$549$	3.63184			0.155003
$550$		0			0
$551$	44.8468			1.91054
$552$		0			0
$553$	9.78621	−	30.1189i	0.416152	−	1.28078i
$554$		0			0
$555$	12.3792	+	2.63128i	0.525468	+	0.111692i
$556$		0			0
$557$	5.93356			0.251413			0.125706	−	0.992067i	$-0.459880\pi$
$557$	5.93356			0.251413			0.125706	+	0.992067i	$0.459880\pi$
$558$		0			0
$559$	9.50187	+	6.90351i	0.401886	+	0.291987i
$560$		0			0
$561$	6.02323	−	4.37613i	0.254301	−	0.184761i
$562$		0			0
$563$	−11.1180	−	8.07772i	−0.468569	−	0.340435i	0.328314	−	0.944569i	$-0.393520\pi$
$563$	−11.1180	−	8.07772i	−0.468569	−	0.340435i	−0.796883	+	0.604133i	$0.793520\pi$
$564$		0			0
$565$	−0.754609	−	7.17962i	−0.0317466	−	0.302049i
$566$		0			0
$567$	−1.47815	−	4.54927i	−0.0620764	−	0.191051i
$568$		0			0
$569$	10.2309	+	31.4874i	0.428900	+	1.32002i	0.899210	+	0.437518i	$0.144142\pi$
$569$	10.2309	+	31.4874i	0.428900	+	1.32002i	−0.470309	+	0.882502i	$0.655858\pi$
$570$		0			0
$571$	−7.91367	+	24.3558i	−0.331177	+	1.01926i	0.637398	+	0.770535i	$0.280011\pi$
$571$	−7.91367	+	24.3558i	−0.331177	+	1.01926i	−0.968575	+	0.248722i	$0.919989\pi$
$572$		0			0
$573$	4.24803	−	3.08637i	0.177464	−	0.128935i
$574$		0			0
$575$	14.6581	+	25.3886i	0.611285	+	1.05878i
$576$		0			0
$577$	−2.82151	+	2.04995i	−0.117461	+	0.0853405i	−0.644965	−	0.764212i	$-0.723128\pi$
$577$	−2.82151	+	2.04995i	−0.117461	+	0.0853405i	0.527504	+	0.849553i	$0.323128\pi$
$578$		0			0
$579$	2.45534	−	7.55676i	0.102040	−	0.314048i
$580$		0			0
$581$	3.81641	+	11.7457i	0.158331	+	0.487294i
$582$		0			0
$583$	2.55343	+	7.85866i	0.105752	+	0.325472i
$584$		0			0
$585$	1.20731	−	2.09113i	0.0499162	−	0.0864574i
$586$		0			0
$587$	−27.8690	−	20.2480i	−1.15028	−	0.835724i	−0.161758	−	0.986831i	$-0.551716\pi$
$587$	−27.8690	−	20.2480i	−1.15028	−	0.835724i	−0.988517	+	0.151107i	$0.951716\pi$
$588$		0			0
$589$	14.8425	−	10.7837i	0.611575	−	0.444335i
$590$		0			0
$591$	10.5220	+	7.64469i	0.432818	+	0.314460i
$592$		0			0
$593$	35.7248			1.46704			0.733522	−	0.679666i	$-0.237875\pi$
$593$	35.7248			1.46704			0.733522	+	0.679666i	$0.237875\pi$
$594$		0			0
$595$	20.3530	−	35.2524i	0.834391	−	1.44521i
$596$		0			0
$597$	0.328899	−	1.01225i	0.0134609	−	0.0414285i
$598$		0			0
$599$	39.7697			1.62495			0.812473	−	0.582998i	$-0.198121\pi$
$599$	39.7697			1.62495			0.812473	+	0.582998i	$0.198121\pi$
$600$		0			0
$601$	−15.5134			−0.632805			−0.316402	−	0.948625i	$-0.602475\pi$
$601$	−15.5134			−0.632805			−0.316402	+	0.948625i	$0.602475\pi$
$602$		0			0
$603$	2.14163	−	6.59126i	0.0872140	−	0.268417i
$604$		0			0
$605$	14.6525	−	6.52369i	0.595707	−	0.265226i
$606$		0			0
$607$	15.1855			0.616358			0.308179	−	0.951328i	$-0.400280\pi$
$607$	15.1855			0.616358			0.308179	+	0.951328i	$0.400280\pi$
$608$		0			0
$609$	40.8271	+	29.6626i	1.65440	+	1.20199i
$610$		0			0
$611$	−2.62224	+	1.90517i	−0.106084	+	0.0770748i
$612$		0			0
$613$	25.9974	+	18.8882i	1.05002	+	0.762887i	0.972217	−	0.234081i	$-0.0752082\pi$
$613$	25.9974	+	18.8882i	1.05002	+	0.762887i	0.0778070	+	0.996968i	$0.475208\pi$
$614$		0			0
$615$	1.87786	+	0.399152i	0.0757228	+	0.0160954i
$616$		0			0
$617$	1.33569	+	4.11082i	0.0537727	+	0.165495i	0.974336	−	0.225097i	$-0.0722700\pi$
$617$	1.33569	+	4.11082i	0.0537727	+	0.165495i	−0.920564	+	0.390593i	$0.872270\pi$
$618$		0			0
$619$	10.3201	+	31.7621i	0.414801	+	1.27663i	0.912428	+	0.409236i	$0.134205\pi$
$619$	10.3201	+	31.7621i	0.414801	+	1.27663i	−0.497627	+	0.867391i	$0.665795\pi$
$620$		0			0
$621$	−1.81184	+	5.57627i	−0.0727067	+	0.223768i
$622$		0			0
$623$	−3.22256	+	2.34133i	−0.129109	+	0.0938034i
$624$		0			0
$625$	−24.4537	−	5.19779i	−0.978148	−	0.207912i
$626$		0			0
$627$	−6.72772	+	4.88798i	−0.268679	+	0.195207i
$628$		0			0
$629$	−6.65615	+	20.4855i	−0.265398	+	0.816812i
$630$		0			0
$631$	3.39726	+	10.4557i	0.135243	+	0.416234i	0.995628	−	0.0934106i	$-0.0297769\pi$
$631$	3.39726	+	10.4557i	0.135243	+	0.416234i	−0.860385	+	0.509645i	$0.829777\pi$
$632$		0			0
$633$	−6.33264	−	19.4898i	−0.251700	−	0.774652i
$634$		0			0
$635$	16.1554	+	3.43393i	0.641106	+	0.136271i
$636$		0			0
$637$	−13.8738	−	10.0799i	−0.549698	−	0.399379i
$638$		0			0
$639$	9.70587	−	7.05173i	0.383958	−	0.278962i
$640$		0			0
$641$	5.55031	+	4.03253i	0.219224	+	0.159276i	0.691977	−	0.721919i	$-0.256740\pi$
$641$	5.55031	+	4.03253i	0.219224	+	0.159276i	−0.472753	+	0.881195i	$0.656740\pi$
$642$		0			0
$643$	23.1923			0.914615			0.457308	−	0.889309i	$-0.348814\pi$
$643$	23.1923			0.914615			0.457308	+	0.889309i	$0.348814\pi$
$644$		0			0
$645$	22.2179	−	9.89202i	0.874827	−	0.389498i
$646$		0			0
$647$	−9.51298	+	29.2779i	−0.373994	+	1.15103i	0.570161	+	0.821533i	$0.306881\pi$
$647$	−9.51298	+	29.2779i	−0.373994	+	1.15103i	−0.944155	+	0.329502i	$0.893119\pi$
$648$		0			0
$649$	−9.33188			−0.366309
$650$		0			0
$651$	20.6447			0.809130
$652$		0			0
$653$	−2.41245	+	7.42477i	−0.0944066	+	0.290554i	−0.987099	−	0.160114i	$-0.948814\pi$
$653$	−2.41245	+	7.42477i	−0.0944066	+	0.290554i	0.892692	+	0.450667i	$0.148814\pi$
$654$		0			0
$655$	−18.8767	+	32.6955i	−0.737575	+	1.27752i
$656$		0			0
$657$	−16.9560			−0.661515
$658$		0			0
$659$	−4.26388	−	3.09789i	−0.166097	−	0.120677i	0.501632	−	0.865081i	$-0.332733\pi$
$659$	−4.26388	−	3.09789i	−0.166097	−	0.120677i	−0.667729	+	0.744405i	$0.732733\pi$
$660$		0			0
$661$	29.8490	−	21.6865i	1.16099	−	0.843509i	0.171088	−	0.985256i	$-0.445272\pi$
$661$	29.8490	−	21.6865i	1.16099	−	0.843509i	0.989903	+	0.141747i	$0.0452719\pi$
$662$		0			0
$663$	3.32476	+	2.41558i	0.129123	+	0.0938132i
$664$		0			0
$665$	−22.7335	+	39.3756i	−0.881567	+	1.52692i
$666$		0			0
$667$	−19.1151	−	58.8301i	−0.740138	−	2.27791i
$668$		0			0
$669$	3.18661	+	9.80739i	0.123202	+	0.379176i
$670$		0			0
$671$	2.19555	−	6.75721i	0.0847583	−	0.260859i
$672$		0			0
$673$	36.4227	−	26.4626i	1.40399	−	1.02006i	0.409829	−	0.912162i	$-0.365588\pi$
$673$	36.4227	−	26.4626i	1.40399	−	1.02006i	0.994162	−	0.107897i	$-0.0344116\pi$
$674$		0			0
$675$	−2.50000	−	4.33013i	−0.0962250	−	0.166667i
$676$		0			0
$677$	−25.5161	+	18.5385i	−0.980662	+	0.712493i	−0.957856	−	0.287247i	$-0.907260\pi$
$677$	−25.5161	+	18.5385i	−0.980662	+	0.712493i	−0.0228056	+	0.999740i	$0.507260\pi$
$678$		0			0
$679$	13.4762	−	41.4756i	0.517171	−	1.59169i
$680$		0			0
$681$	−5.60585	−	17.2530i	−0.214816	−	0.661137i
$682$		0			0
$683$	−6.39320	−	19.6762i	−0.244629	−	0.752891i	−0.995697	−	0.0926660i	$-0.970461\pi$
$683$	−6.39320	−	19.6762i	−0.244629	−	0.752891i	0.751068	−	0.660225i	$-0.229539\pi$
$684$		0			0
$685$	0.196176	+	1.86649i	0.00749551	+	0.0713150i
$686$		0			0
$687$	7.86225	+	5.71226i	0.299964	+	0.217936i
$688$		0			0
$689$	−3.69003	+	2.68096i	−0.140579	+	0.102137i
$690$		0			0
$691$	27.6523	+	20.0906i	1.05194	+	0.764282i	0.972581	−	0.232564i	$-0.0747116\pi$
$691$	27.6523	+	20.0906i	1.05194	+	0.764282i	0.0793624	+	0.996846i	$0.474712\pi$
$692$		0			0
$693$	−9.35772			−0.355470
$694$		0			0
$695$	−24.4048	−	5.18741i	−0.925728	−	0.196770i
$696$		0			0
$697$	−1.00971	+	3.10755i	−0.0382453	+	0.117707i
$698$		0			0
$699$	−22.2184			−0.840377
$700$		0			0
$701$	−16.6859			−0.630216			−0.315108	−	0.949056i	$-0.602041\pi$
$701$	−16.6859			−0.630216			−0.315108	+	0.949056i	$0.602041\pi$
$702$		0			0
$703$	7.43467	−	22.8816i	0.280404	−	0.862994i
$704$		0			0
$705$	0.701568	+	6.67497i	0.0264226	+	0.251394i
$706$		0			0
$707$	−7.15312			−0.269021
$708$		0			0
$709$	−38.2796	−	27.8118i	−1.43762	−	1.04449i	−0.988533	−	0.151007i	$-0.951749\pi$
$709$	−38.2796	−	27.8118i	−1.43762	−	1.04449i	−0.449089	−	0.893487i	$-0.648251\pi$
$710$		0			0
$711$	5.35617	−	3.89149i	0.200872	−	0.145942i
$712$		0			0
$713$	−20.4724	−	14.8741i	−0.766697	−	0.557038i
$714$		0			0
$715$	−3.16078	−	3.51041i	−0.118207	−	0.131282i
$716$		0			0
$717$	4.92242	+	15.1496i	0.183831	+	0.565774i
$718$		0			0
$719$	1.73431	+	5.33767i	0.0646790	+	0.199062i	0.978174	−	0.207789i	$-0.0666268\pi$
$719$	1.73431	+	5.33767i	0.0646790	+	0.199062i	−0.913495	+	0.406851i	$0.866627\pi$
$720$		0			0
$721$	−16.0561	+	49.4156i	−0.597960	+	1.84033i
$722$		0			0
$723$	3.98964	−	2.89864i	0.148376	−	0.107802i
$724$		0			0
$725$	48.1899	+	21.4555i	1.78973	+	0.796838i
$726$		0			0
$727$	11.8412	−	8.60317i	0.439167	−	0.319074i	−0.346137	−	0.938184i	$-0.612507\pi$
$727$	11.8412	−	8.60317i	0.439167	−	0.319074i	0.785304	+	0.619110i	$0.212507\pi$
$728$		0			0
$729$	0.309017	−	0.951057i	0.0114451	−	0.0352243i
$730$		0			0
$731$	12.7911	+	39.3669i	0.473095	+	1.45604i
$732$		0			0
$733$	9.96818	+	30.6789i	0.368183	+	1.13315i	0.947964	+	0.318378i	$0.103138\pi$
$733$	9.96818	+	30.6789i	0.368183	+	1.13315i	−0.579781	+	0.814773i	$0.696862\pi$
$734$		0			0
$735$	−32.4405	+	14.4434i	−1.19658	+	0.532754i
$736$		0			0
$737$	−10.9687	−	7.96921i	−0.404037	−	0.293550i
$738$		0			0
$739$	19.2956	−	14.0191i	0.709800	−	0.515700i	−0.173309	−	0.984867i	$-0.555446\pi$
$739$	19.2956	−	14.0191i	0.709800	−	0.515700i	0.883109	+	0.469168i	$0.155446\pi$
$740$		0			0
$741$	−3.71363	−	2.69811i	−0.136423	−	0.0991174i
$742$		0			0
$743$	7.48188			0.274483			0.137242	−	0.990538i	$-0.456176\pi$
$743$	7.48188			0.274483			0.137242	+	0.990538i	$0.456176\pi$
$744$		0			0
$745$	32.9583	+	36.6039i	1.20750	+	1.34106i
$746$		0			0
$747$	−0.797847	+	2.45552i	−0.0291917	+	0.0898428i
$748$		0			0
$749$	94.8975			3.46748
$750$		0			0
$751$	−32.3687			−1.18115			−0.590576	−	0.806982i	$-0.701099\pi$
$751$	−32.3687			−1.18115			−0.590576	+	0.806982i	$0.701099\pi$
$752$		0			0
$753$	4.76969	−	14.6796i	0.173817	−	0.534955i
$754$		0			0
$755$	−17.6390	−	19.5901i	−0.641949	−	0.712957i
$756$		0			0
$757$	−10.2847			−0.373804			−0.186902	−	0.982379i	$-0.559845\pi$
$757$	−10.2847			−0.373804			−0.186902	+	0.982379i	$0.559845\pi$
$758$		0			0
$759$	9.27961	+	6.74203i	0.336828	+	0.244720i
$760$		0			0
$761$	−23.3033	+	16.9308i	−0.844743	+	0.613742i	−0.923691	−	0.383137i	$-0.874844\pi$
$761$	−23.3033	+	16.9308i	−0.844743	+	0.613742i	0.0789487	+	0.996879i	$0.474844\pi$
$762$		0			0
$763$	−9.07055	−	6.59014i	−0.328376	−	0.238579i
$764$		0			0
$765$	7.77415	−	3.46127i	0.281075	−	0.125143i
$766$		0			0
$767$	−1.59178	−	4.89898i	−0.0574757	−	0.176892i
$768$		0			0
$769$	−12.1673	−	37.4472i	−0.438765	−	1.35038i	−0.889179	−	0.457560i	$-0.848724\pi$
$769$	−12.1673	−	37.4472i	−0.438765	−	1.35038i	0.450414	−	0.892820i	$-0.351276\pi$
$770$		0			0
$771$	6.54584	−	20.1460i	0.235743	−	0.725541i
$772$		0			0
$773$	−4.09692	+	2.97658i	−0.147356	+	0.107060i	−0.659021	−	0.752125i	$-0.729029\pi$
$773$	−4.09692	+	2.97658i	−0.147356	+	0.107060i	0.511665	+	0.859185i	$0.329029\pi$
$774$		0			0
$775$	21.1080	−	4.48665i	0.758223	−	0.161165i
$776$		0			0
$777$	21.9026	−	15.9132i	0.785753	−	0.570883i
$778$		0			0
$779$	1.12780	−	3.47102i	0.0404077	−	0.124362i
$780$		0			0
$781$	−7.25260	−	22.3212i	−0.259518	−	0.798715i
$782$		0			0
$783$	3.26015	+	10.0337i	0.116508	+	0.358576i
$784$		0			0
$785$	−16.8727	−	18.7391i	−0.602213	−	0.668826i
$786$		0			0
$787$	−4.31328	−	3.13378i	−0.153752	−	0.111707i	0.508250	−	0.861210i	$-0.330293\pi$
$787$	−4.31328	−	3.13378i	−0.153752	−	0.111707i	−0.662001	+	0.749503i	$0.730293\pi$
$788$		0			0
$789$	16.9298	−	12.3002i	0.602717	−	0.437900i
$790$		0			0
$791$	−12.4938	−	9.07729i	−0.444229	−	0.322751i
$792$		0			0
$793$	3.92185			0.139269
$794$		0			0
$795$	0.987250	+	9.39306i	0.0350141	+

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:	\( N \)	\(=\)	\( 300 = 2^{2} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	300.m (of order \(5\), degree \(4\), minimal)

\(f(q)\)	\(=\)	\(q+(0.309017 - 0.951057i) q^{3} +(-1.49622 - 1.66172i) q^{5} -4.78339 q^{7} +(-0.809017 - 0.587785i) q^{9} +O(q^{10})\)	\(q+(0.309017 - 0.951057i) q^{3} +(-1.49622 - 1.66172i) q^{5} -4.78339 q^{7} +(-0.809017 - 0.587785i) q^{9} +(-1.58268 + 1.14988i) q^{11} +(-0.873619 - 0.634721i) q^{13} +(-2.04275 + 0.909491i) q^{15} +(-1.17603 - 3.61946i) q^{17} +(1.31359 + 4.04280i) q^{19} +(-1.47815 + 4.54927i) q^{21} +(4.74346 - 3.44633i) q^{23} +(-0.522642 + 4.97261i) q^{25} +(-0.809017 + 0.587785i) q^{27} +(3.26015 - 10.0337i) q^{29} +(-1.33369 - 4.10468i) q^{31} +(0.604528 + 1.86055i) q^{33} +(7.15701 + 7.94866i) q^{35} +(-4.57890 - 3.32676i) q^{37} +(-0.873619 + 0.634721i) q^{39} +(-0.694596 - 0.504654i) q^{41} -10.8764 q^{43} +(0.233733 + 2.22382i) q^{45} +(0.927539 - 2.85467i) q^{47} +15.8808 q^{49} -3.80573 q^{51} +(1.30524 - 4.01711i) q^{53} +(4.27882 + 0.909491i) q^{55} +4.25085 q^{57} +(3.85916 + 2.80384i) q^{59} +(-2.93822 + 2.13474i) q^{61} +(3.86984 + 2.81160i) q^{63} +(0.252397 + 2.40140i) q^{65} +(2.14163 + 6.59126i) q^{67} +(-1.81184 - 5.57627i) q^{69} +(-3.70731 + 11.4099i) q^{71} +(13.7177 - 9.96647i) q^{73} +(4.56773 + 2.03368i) q^{75} +(7.57055 - 5.50033i) q^{77} +(-2.04587 + 6.29656i) q^{79} +(0.309017 + 0.951057i) q^{81} +(-0.797847 - 2.45552i) q^{83} +(-4.25493 + 7.36976i) q^{85} +(-8.53519 - 6.20118i) q^{87} +(0.673699 - 0.489471i) q^{89} +(4.17886 + 3.03612i) q^{91} -4.31592 q^{93} +(4.75260 - 8.23174i) q^{95} +(-2.81730 + 8.67076i) q^{97} +1.95630 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8q - 2q^{3} + 5q^{5} - 8q^{7} - 2q^{9} + O(q^{10}) \)	\( 8q - 2q^{3} + 5q^{5} - 8q^{7} - 2q^{9} - 2q^{11} - 5q^{15} + 7q^{17} + 5q^{19} - 3q^{21} + 7q^{23} + 5q^{25} - 2q^{27} + 27q^{29} - 3q^{31} + 3q^{33} + 20q^{35} - 9q^{37} + 20q^{41} - 68q^{43} - 5q^{45} - 7q^{47} - 8q^{49} - 8q^{51} - 11q^{53} + 5q^{55} - 10q^{57} + 2q^{59} - 14q^{61} + 7q^{63} - 35q^{65} + 28q^{67} + 2q^{69} - 15q^{71} + 6q^{73} + 5q^{75} + 17q^{77} + 24q^{79} - 2q^{81} + 2q^{83} + 10q^{85} - 23q^{87} + 5q^{91} - 18q^{93} + 5q^{95} + 34q^{97} - 2q^{99} + O(q^{100}) \)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Properties

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