LMFDB - Embedded newform 177.5.b.a.119.1

Newspace parameters

Level:	$ N $	$=$	$ 177 = 3 \cdot 59 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	177.b (of order $2$, degree $1$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$18.2964834658$
Analytic rank:	$0$
Dimension:	$78$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			119.1
Character	$\chi$	$=$	177.119
Dual form			177.5.b.a.119.78

$q$-expansion

$f(q)$	$=$	$q-7.85862i q^{2} +(1.36439 + 8.89598i) q^{3} -45.7580 q^{4} +33.8616i q^{5} +(69.9102 - 10.7222i) q^{6} +13.8137 q^{7} +233.857i q^{8} +(-77.2769 + 24.2751i) q^{9} +O(q^{10})$	\(q-7.85862i q^{2} +(1.36439 + 8.89598i) q^{3} -45.7580 q^{4} +33.8616i q^{5} +(69.9102 - 10.7222i) q^{6} +13.8137 q^{7} +233.857i q^{8} +(-77.2769 + 24.2751i) q^{9} +266.105 q^{10} -192.320i q^{11} +(-62.4316 - 407.062i) q^{12} -170.965 q^{13} -108.557i q^{14} +(-301.232 + 46.2003i) q^{15} +1105.66 q^{16} -492.040i q^{17} +(190.769 + 607.290i) q^{18} +210.495 q^{19} -1549.44i q^{20} +(18.8472 + 122.886i) q^{21} -1511.37 q^{22} -115.548i q^{23} +(-2080.38 + 319.071i) q^{24} -521.606 q^{25} +1343.55i q^{26} +(-321.387 - 654.333i) q^{27} -632.087 q^{28} -439.260i q^{29} +(363.071 + 2367.27i) q^{30} +175.455 q^{31} -4947.30i q^{32} +(1710.88 - 262.400i) q^{33} -3866.76 q^{34} +467.753i q^{35} +(3536.03 - 1110.78i) q^{36} +1951.00 q^{37} -1654.20i q^{38} +(-233.263 - 1520.90i) q^{39} -7918.76 q^{40} -1818.50i q^{41} +(965.717 - 148.113i) q^{42} +1528.80 q^{43} +8800.20i q^{44} +(-821.994 - 2616.72i) q^{45} -908.046 q^{46} +1691.29i q^{47} +(1508.56 + 9835.97i) q^{48} -2210.18 q^{49} +4099.10i q^{50} +(4377.18 - 671.334i) q^{51} +7823.03 q^{52} -873.418i q^{53} +(-5142.16 + 2525.66i) q^{54} +6512.27 q^{55} +3230.43i q^{56} +(287.197 + 1872.56i) q^{57} -3451.98 q^{58} +453.188i q^{59} +(13783.8 - 2114.03i) q^{60} -4977.77 q^{61} -1378.83i q^{62} +(-1067.48 + 335.329i) q^{63} -21188.3 q^{64} -5789.15i q^{65} +(-2062.10 - 13445.2i) q^{66} -4017.43 q^{67} +22514.8i q^{68} +(1027.91 - 157.652i) q^{69} +3675.90 q^{70} -233.784i q^{71} +(-5676.91 - 18071.7i) q^{72} -2072.57 q^{73} -15332.1i q^{74} +(-711.672 - 4640.19i) q^{75} -9631.84 q^{76} -2656.66i q^{77} +(-11952.2 + 1833.13i) q^{78} +336.550 q^{79} +37439.5i q^{80} +(5382.44 - 3751.81i) q^{81} -14290.9 q^{82} +687.683i q^{83} +(-862.412 - 5623.03i) q^{84} +16661.3 q^{85} -12014.3i q^{86} +(3907.65 - 599.322i) q^{87} +44975.4 q^{88} -5666.49i q^{89} +(-20563.8 + 6459.74i) q^{90} -2361.66 q^{91} +5287.23i q^{92} +(239.388 + 1560.84i) q^{93} +13291.2 q^{94} +7127.70i q^{95} +(44011.0 - 6750.03i) q^{96} -3420.98 q^{97} +17369.0i q^{98} +(4668.61 + 14861.9i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 78q - 612q^{4} + 64q^{6} + 76q^{7} - 100q^{9} + O(q^{10}) $	$ 78q - 612q^{4} + 64q^{6} + 76q^{7} - 100q^{9} - 156q^{10} - 4q^{13} + 13q^{15} + 4948q^{16} + 22q^{18} + 812q^{19} - 173q^{21} - 1644q^{22} - 678q^{24} - 8238q^{25} + 777q^{27} - 3764q^{28} + 1374q^{30} + 4664q^{31} - 1042q^{33} + 3244q^{34} + 3648q^{36} - 3960q^{37} - 7078q^{39} - 1576q^{40} + 4934q^{42} - 1492q^{43} - 2063q^{45} - 2036q^{46} - 2620q^{48} + 24274q^{49} + 7300q^{51} + 8408q^{52} - 14766q^{54} + 9780q^{55} + 6939q^{57} - 3856q^{58} + 4712q^{60} - 212q^{61} - 7438q^{63} - 45760q^{64} + 3048q^{66} - 12972q^{67} + 21672q^{69} + 5828q^{70} - 866q^{72} - 5240q^{73} - 20922q^{75} + 12368q^{76} - 16508q^{78} - 14976q^{79} + 25524q^{81} - 14484q^{82} + 9540q^{84} + 11572q^{85} + 5695q^{87} + 62160q^{88} - 31672q^{90} + 8284q^{91} - 9590q^{93} - 10992q^{94} + 34102q^{96} - 55000q^{97} - 14254q^{99} + O(q^{100}) $

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/177\mathbb{Z}\right)^\times$.

$n$	$61$	$119$
$\chi(n)$	$1$	$-1$

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 $

$2$		−	7.85862i		−	1.96466i	−0.187167	−	0.982328i	$-0.559931\pi$
$2$		−	7.85862i		−	1.96466i	0.187167	−	0.982328i	$-0.440069\pi$
$3$	1.36439	+	8.89598i	0.151599	+	0.988442i
$3$	1.36439	+	8.89598i	0.151599	+	0.988442i
$4$	−45.7580			−2.85987
$5$			33.8616i			1.35446i	0.735770	+	0.677231i	$0.236820\pi$
$5$			33.8616i			1.35446i	−0.735770	+	0.677231i	$0.763180\pi$
$6$	69.9102	−	10.7222i	1.94195	−	0.297839i
$7$	13.8137			0.281912			0.140956	−	0.990016i	$-0.454982\pi$
$7$	13.8137			0.281912			0.140956	+	0.990016i	$0.454982\pi$
$8$			233.857i			3.65401i
$9$	−77.2769	+	24.2751i	−0.954036	+	0.299693i
$10$	266.105			2.66105
$11$		−	192.320i		−	1.58943i	−0.606986	−	0.794713i	$-0.707621\pi$
$11$		−	192.320i		−	1.58943i	0.606986	−	0.794713i	$-0.292379\pi$
$12$	−62.4316	−	407.062i	−0.433553	−	2.82682i
$13$	−170.965			−1.01163			−0.505815	−	0.862642i	$-0.668808\pi$
$13$	−170.965			−1.01163			−0.505815	+	0.862642i	$0.668808\pi$
$14$		−	108.557i		−	0.553860i
$15$	−301.232	+	46.2003i	−1.33881	+	0.205335i
$16$	1105.66			4.31900
$17$		−	492.040i		−	1.70256i	−0.524711	−	0.851281i	$-0.675826\pi$
$17$		−	492.040i		−	1.70256i	0.524711	−	0.851281i	$-0.324174\pi$
$18$	190.769	+	607.290i	0.588794	+	1.87435i
$19$	210.495			0.583089			0.291545	−	0.956557i	$-0.405831\pi$
$19$	210.495			0.583089			0.291545	+	0.956557i	$0.405831\pi$
$20$		−	1549.44i		−	3.87359i
$21$	18.8472	+	122.886i	0.0427375	+	0.278654i
$22$	−1511.37			−3.12267
$23$		−	115.548i		−	0.218427i	−0.994018	−	0.109213i	$-0.965167\pi$
$23$		−	115.548i		−	0.218427i	0.994018	−	0.109213i	$-0.0348332\pi$
$24$	−2080.38	+	319.071i	−3.61178	+	0.553943i
$25$	−521.606			−0.834569
$26$			1343.55i			1.98750i
$27$	−321.387	−	654.333i	−0.440860	−	0.897576i
$28$	−632.087			−0.806233
$29$		−	439.260i		−	0.522307i	−0.965297	−	0.261154i	$-0.915897\pi$
$29$		−	439.260i		−	0.522307i	0.965297	−	0.261154i	$-0.0841029\pi$
$30$	363.071	+	2367.27i	0.403412	+	2.63030i
$31$	175.455			0.182575			0.0912876	−	0.995825i	$-0.470902\pi$
$31$	175.455			0.182575			0.0912876	+	0.995825i	$0.470902\pi$
$32$		−	4947.30i		−	4.83134i
$33$	1710.88	−	262.400i	1.57106	−	0.240955i
$34$	−3866.76			−3.34495
$35$			467.753i			0.381839i
$36$	3536.03	−	1110.78i	2.72842	−	0.857084i
$37$	1951.00			1.42512			0.712562	−	0.701609i	$-0.247534\pi$
$37$	1951.00			1.42512			0.712562	+	0.701609i	$0.247534\pi$
$38$		−	1654.20i		−	1.14557i
$39$	−233.263	−	1520.90i	−0.153362	−	0.999937i
$40$	−7918.76			−4.94922
$41$		−	1818.50i		−	1.08179i	−0.841089	−	0.540897i	$-0.818085\pi$
$41$		−	1818.50i		−	1.08179i	0.841089	−	0.540897i	$-0.181915\pi$
$42$	965.717	−	148.113i	0.547459	−	0.0839645i
$43$	1528.80			0.826825			0.413413	−	0.910544i	$-0.364337\pi$
$43$	1528.80			0.826825			0.413413	+	0.910544i	$0.364337\pi$
$44$			8800.20i			4.54556i
$45$	−821.994	−	2616.72i	−0.405923	−	1.29221i
$46$	−908.046			−0.429133
$47$			1691.29i			0.765637i	0.923824	+	0.382819i	$0.125047\pi$
$47$			1691.29i			0.765637i	−0.923824	+	0.382819i	$0.874953\pi$
$48$	1508.56	+	9835.97i	0.654755	+	4.26908i
$49$	−2210.18			−0.920526
$50$			4099.10i			1.63964i
$51$	4377.18	−	671.334i	1.68288	−	0.258106i
$52$	7823.03			2.89313
$53$		−	873.418i		−	0.310936i	−0.987841	−	0.155468i	$-0.950312\pi$
$53$		−	873.418i		−	0.310936i	0.987841	−	0.155468i	$-0.0496885\pi$
$54$	−5142.16	+	2525.66i	−1.76343	+	0.866138i
$55$	6512.27			2.15282
$56$			3230.43i			1.03011i
$57$	287.197	+	1872.56i	0.0883956	+	0.576350i
$58$	−3451.98			−1.02615
$59$			453.188i			0.130189i
$59$			453.188i			0.130189i
$60$	13783.8	−	2114.03i	3.82882	−	0.587231i
$61$	−4977.77			−1.33775			−0.668876	−	0.743374i	$-0.733224\pi$
$61$	−4977.77			−1.33775			−0.668876	+	0.743374i	$0.733224\pi$
$62$		−	1378.83i		−	0.358697i
$63$	−1067.48	+	335.329i	−0.268954	+	0.0844871i
$64$	−21188.3			−5.17293
$65$		−	5789.15i		−	1.37021i
$66$	−2062.10	−	13445.2i	−0.473393	−	3.08658i
$67$	−4017.43			−0.894951			−0.447475	−	0.894296i	$-0.647677\pi$
$67$	−4017.43			−0.894951			−0.447475	+	0.894296i	$0.647677\pi$
$68$			22514.8i			4.86911i
$69$	1027.91	−	157.652i	0.215902	−	0.0331132i
$70$	3675.90			0.750183
$71$		−	233.784i		−	0.0463765i	−0.999731	−	0.0231883i	$-0.992618\pi$
$71$		−	233.784i		−	0.0463765i	0.999731	−	0.0231883i	$-0.00738172\pi$
$72$	−5676.91	−	18071.7i	−1.09508	−	3.48606i
$73$	−2072.57			−0.388924			−0.194462	−	0.980910i	$-0.562296\pi$
$73$	−2072.57			−0.388924			−0.194462	+	0.980910i	$0.562296\pi$
$74$		−	15332.1i		−	2.79988i
$75$	−711.672	−	4640.19i	−0.126520	−	0.824923i
$76$	−9631.84			−1.66756
$77$		−	2656.66i		−	0.448078i
$78$	−11952.2	+	1833.13i	−1.96453	+	0.301303i
$79$	336.550			0.0539257			0.0269628	−	0.999636i	$-0.491416\pi$
$79$	336.550			0.0539257			0.0269628	+	0.999636i	$0.491416\pi$
$80$			37439.5i			5.84993i
$81$	5382.44	−	3751.81i	0.820368	−	0.571836i
$82$	−14290.9			−2.12535
$83$			687.683i			0.0998234i	0.998754	+	0.0499117i	$0.0158940\pi$
$83$			687.683i			0.0998234i	−0.998754	+	0.0499117i	$0.984106\pi$
$84$	−862.412	−	5623.03i	−0.122224	−	0.796915i
$85$	16661.3			2.30606
$86$		−	12014.3i		−	1.62443i
$87$	3907.65	−	599.322i	0.516270	−	0.0791811i
$88$	44975.4			5.80778
$89$		−	5666.49i		−	0.715376i	−0.933841	−	0.357688i	$-0.883565\pi$
$89$		−	5666.49i		−	0.715376i	0.933841	−	0.357688i	$-0.116435\pi$
$90$	−20563.8	+	6459.74i	−2.53874	+	0.797499i
$91$	−2361.66			−0.285191
$92$			5287.23i			0.624673i
$93$	239.388	+	1560.84i	0.0276782	+	0.180465i
$94$	13291.2			1.50421
$95$			7127.70i			0.789773i
$96$	44011.0	−	6750.03i	4.77550	−	0.732425i
$97$	−3420.98			−0.363586			−0.181793	−	0.983337i	$-0.558190\pi$
$97$	−3420.98			−0.363586			−0.181793	+	0.983337i	$0.558190\pi$
$98$			17369.0i			1.80852i
$99$	4668.61	+	14861.9i	0.476340	+	1.51637i
$100$	23867.6			2.38676
$101$		−	584.413i		−	0.0572898i	−0.999590	−	0.0286449i	$-0.990881\pi$
$101$		−	584.413i		−	0.0572898i	0.999590	−	0.0286449i	$-0.00911920\pi$
$102$	−5275.76	−	34398.6i	−0.507090	−	3.30629i
$103$	−19980.2			−1.88332			−0.941662	−	0.336560i	$-0.890737\pi$
$103$	−19980.2			−1.88332			−0.941662	+	0.336560i	$0.890737\pi$
$104$		−	39981.4i		−	3.69651i
$105$	−4161.12	+	638.197i	−0.377426	+	0.0578864i
$106$	−6863.87			−0.610882
$107$		−	10860.0i		−	0.948556i	−0.880375	−	0.474278i	$-0.842709\pi$
$107$		−	10860.0i		−	0.948556i	0.880375	−	0.474278i	$-0.157291\pi$
$108$	14706.0	+	29940.9i	1.26080	+	2.56695i
$109$	−5033.51			−0.423660			−0.211830	−	0.977306i	$-0.567942\pi$
$109$	−5033.51			−0.423660			−0.211830	+	0.977306i	$0.567942\pi$
$110$		−	51177.5i		−	4.22955i
$111$	2661.92	+	17356.0i	0.216047	+	1.40865i
$112$	15273.3			1.21758
$113$		−	16750.8i		−	1.31183i	−0.754833	−	0.655917i	$-0.772282\pi$
$113$		−	16750.8i		−	1.31183i	0.754833	−	0.655917i	$-0.227718\pi$
$114$	14715.8	−	2256.98i	1.13233	−	0.173667i
$115$	3912.63			0.295851
$116$			20099.7i			1.49373i
$117$	13211.7	−	4150.21i	0.965130	−	0.303178i
$118$	3561.43			0.255776
$119$		−	6796.89i		−	0.479973i
$120$	−10804.3	−	70445.1i	−0.750296	−	4.89202i
$121$	−22346.2			−1.52627
$122$			39118.5i			2.62822i
$123$	16177.3	−	2481.13i	1.06929	−	0.163999i
$124$	−8028.45			−0.522142
$125$			3501.10i			0.224070i
$126$	2635.23	+	8388.92i	0.165988	+	0.528402i
$127$	8026.58			0.497649			0.248825	−	0.968549i	$-0.419956\pi$
$127$	8026.58			0.497649			0.248825	+	0.968549i	$0.419956\pi$
$128$			87354.2i			5.33168i
$129$	2085.88	+	13600.2i	0.125346	+	0.817269i
$130$	−45494.8			−2.69200
$131$		−	21267.7i		−	1.23931i	−0.784876	−	0.619653i	$-0.787273\pi$
$131$		−	21267.7i		−	1.23931i	0.784876	−	0.619653i	$-0.212727\pi$
$132$	−78286.4	+	12006.9i	−4.49302	+	0.689100i
$133$	2907.72			0.164380
$134$			31571.5i			1.75827i
$135$	22156.7	−	10882.7i	1.21573	−	0.597128i
$136$	115067.			6.22118
$137$		−	32442.8i		−	1.72853i	−0.503034	−	0.864266i	$-0.667783\pi$
$137$		−	32442.8i		−	1.72853i	0.503034	−	0.864266i	$-0.332217\pi$
$138$	−1238.93	−	8077.96i	−0.0650561	−	0.424173i
$139$	18614.4			0.963428			0.481714	−	0.876329i	$-0.340014\pi$
$139$	18614.4			0.963428			0.481714	+	0.876329i	$0.340014\pi$
$140$		−	21403.4i		−	1.09201i
$141$	−15045.7	+	2307.58i	−0.756788	+	0.116070i
$142$	−1837.22			−0.0911139
$143$			32880.1i			1.60791i
$144$	−85442.3	+	26840.2i	−4.12048	+	1.29438i
$145$	14874.0			0.707446
$146$			16287.6i			0.764101i
$147$	−3015.55	−	19661.7i	−0.139550	−	0.909886i
$148$	−89273.6			−4.07568
$149$			4095.47i			0.184472i	0.995737	+	0.0922362i	$0.0294015\pi$
$149$			4095.47i			0.184472i	−0.995737	+	0.0922362i	$0.970599\pi$
$150$	−36465.5	+	5592.77i	−1.62069	+	0.248567i
$151$	−25391.9			−1.11363			−0.556816	−	0.830636i	$-0.687977\pi$
$151$	−25391.9			−1.11363			−0.556816	+	0.830636i	$0.687977\pi$
$152$			49225.7i			2.13062i
$153$	11944.3	+	38023.3i	0.510246	+	1.62430i
$154$	−20877.7			−0.880320
$155$			5941.17i			0.247291i
$156$	10673.6	+	69593.5i	0.438595	+	2.85969i
$157$	−11978.8			−0.485975			−0.242988	−	0.970029i	$-0.578127\pi$
$157$	−11978.8			−0.485975			−0.242988	+	0.970029i	$0.578127\pi$
$158$		−	2644.82i		−	0.105945i
$159$	7769.91	−	1191.68i	0.307342	−	0.0471374i
$160$	167523.			6.54387
$161$		−	1596.14i		−	0.0615771i
$162$	−29484.1	−	42298.5i	−1.12346	−	1.61174i
$163$	−5885.26			−0.221509			−0.110754	−	0.993848i	$-0.535327\pi$
$163$	−5885.26			−0.221509			−0.110754	+	0.993848i	$0.535327\pi$
$164$			83210.7i			3.09379i
$165$	8885.27	+	57933.0i	0.326364	+	2.12794i
$166$	5404.24			0.196119
$167$			16035.8i			0.574987i	0.957783	+	0.287493i	$0.0928219\pi$
$167$			16035.8i			0.574987i	−0.957783	+	0.287493i	$0.907178\pi$
$168$	−28737.8	+	4407.55i	−1.01820	+	0.156163i
$169$	668.154			0.0233939
$170$		−	130935.i		−	4.53061i
$171$	−16266.4	+	5109.80i	−0.556288	+	0.174748i
$172$	−69954.8			−2.36462
$173$			26851.9i			0.897187i	0.893736	+	0.448593i	$0.148075\pi$
$173$			26851.9i			0.897187i	−0.893736	+	0.448593i	$0.851925\pi$
$174$	−4709.84	−	30708.8i	−0.155564	−	1.01429i
$175$	−7205.30			−0.235275
$176$		−	212642.i		−	6.86473i
$177$	−4031.55	+	618.324i	−0.128684	+	0.0197365i
$178$	−44530.8			−1.40547
$179$		−	36640.6i		−	1.14355i	−0.820410	−	0.571776i	$-0.806255\pi$
$179$		−	36640.6i		−	1.14355i	0.820410	−	0.571776i	$-0.193745\pi$
$180$	37612.8	+	119736.i	1.16089	+	3.69554i
$181$	40626.2			1.24008			0.620039	−	0.784571i	$-0.287117\pi$
$181$	40626.2			1.24008			0.620039	+	0.784571i	$0.287117\pi$
$182$			18559.4i			0.560301i
$183$	−6791.62	−	44282.2i	−0.202801	−	1.32229i
$184$	27021.6			0.798134
$185$			66063.8i			1.93028i
$186$	12266.1	−	1881.26i	0.354552	−	0.0543781i
$187$	−94629.4			−2.70609
$188$		−	77390.1i		−	2.18963i
$189$	−4439.54	−	9038.75i	−0.124284	−	0.253038i
$190$	56013.9			1.55163
$191$			14680.3i			0.402409i	0.979549	+	0.201204i	$0.0644855\pi$
$191$			14680.3i			0.402409i	−0.979549	+	0.201204i	$0.935514\pi$
$192$	−28909.1	−	188491.i	−0.784209	−	5.11314i
$193$	70347.1			1.88856			0.944282	−	0.329137i	$-0.106758\pi$
$193$	70347.1			1.88856			0.944282	+	0.329137i	$0.106758\pi$
$194$			26884.2i			0.714321i
$195$	51500.2	−	7898.65i	1.35438	−	0.207723i
$196$	101133.			2.63259
$197$			63492.6i			1.63603i	0.575198	+	0.818014i	$0.304925\pi$
$197$			63492.6i			1.63603i	−0.575198	+	0.818014i	$0.695075\pi$
$198$	116794.	−	36688.8i	2.97914	−	0.935844i
$199$	20815.5			0.525630			0.262815	−	0.964846i	$-0.415349\pi$
$199$	20815.5			0.525630			0.262815	+	0.964846i	$0.415349\pi$
$200$		−	121981.i		−	3.04952i
$201$	−5481.34	−	35739.0i	−0.135673	−	0.884607i
$202$	−4592.68			−0.112555
$203$		−	6067.81i		−	0.147245i
$204$	−200291.	+	30718.9i	−4.81283	+	0.738151i
$205$	61577.1			1.46525
$206$			157017.i			3.70008i
$207$	2804.94	+	8929.17i	0.0654610	+	0.208387i
$208$	−189030.			−4.36923
$209$		−	40482.5i		−	0.926777i
$210$	5015.35	+	32700.7i	0.113727	+	0.741513i
$211$	49115.2			1.10319			0.551596	−	0.834112i	$-0.314019\pi$
$211$	49115.2			1.10319			0.551596	+	0.834112i	$0.314019\pi$
$212$			39965.8i			0.889237i
$213$	2079.74	−	318.972i	0.0458405	−	0.00703062i
$214$	−85344.8			−1.86359
$215$			51767.6i			1.11990i
$216$	153020.	−	75158.5i	3.27975	−	1.61091i
$217$	2423.68			0.0514701
$218$			39556.4i			0.832347i
$219$	−2827.80	−	18437.6i	−0.0589603	−	0.384429i
$220$	−297988.			−6.15679
$221$			84121.8i			1.72236i
$222$	136394.	−	20919.0i	2.76752	−	0.424458i
$223$	6583.44			0.132386			0.0661932	−	0.997807i	$-0.478915\pi$
$223$	6583.44			0.132386			0.0661932	+	0.997807i	$0.478915\pi$
$224$		−	68340.4i		−	1.36201i
$225$	40308.1	−	12662.0i	0.796208	−	0.250115i
$226$	−131638.			−2.57730
$227$		−	627.937i		−	0.0121861i	−0.999981	−	0.00609305i	$-0.998061\pi$
$227$		−	627.937i		−	0.0121861i	0.999981	−	0.00609305i	$-0.00193949\pi$
$228$	−13141.6	−	85684.6i	−0.252800	−	1.64829i
$229$	28854.0			0.550219			0.275110	−	0.961413i	$-0.411286\pi$
$229$	28854.0			0.550219			0.275110	+	0.961413i	$0.411286\pi$
$230$		−	30747.9i		−	0.581245i
$231$	23633.6	−	3624.71i	0.442899	−	0.0679281i
$232$	102724.			1.90852
$233$			22896.0i			0.421742i	0.977514	+	0.210871i	$0.0676300\pi$
$233$			22896.0i			0.421742i	−0.977514	+	0.210871i	$0.932370\pi$
$234$	−32614.9	−	103826.i	−0.595641	−	1.89615i
$235$	−57269.8			−1.03703
$236$		−	20736.9i		−	0.372324i
$237$	459.185	+	2993.94i	0.00817506	+	0.0533024i
$238$	−53414.2			−0.942981
$239$		−	106275.i		−	1.86053i	−0.366887	−	0.930266i	$-0.619576\pi$
$239$		−	106275.i		−	1.86053i	0.366887	−	0.930266i	$-0.380424\pi$
$240$	−333061.	+	51082.1i	−5.78231	+	0.886841i
$241$	−88624.7			−1.52588			−0.762941	−	0.646468i	$-0.776245\pi$
$241$	−88624.7			−1.52588			−0.762941	+	0.646468i	$0.776245\pi$
$242$			175610.i			2.99860i
$243$	40719.8	+	42763.1i	0.689593	+	0.724197i
$244$	227773.			3.82580
$245$		−	74840.2i		−	1.24682i
$246$	−19498.3	−	127131.i	−0.322201	−	2.10079i
$247$	−35987.4			−0.589870
$248$			41031.3i			0.667132i
$249$	−6117.61	+	938.267i	−0.0986696	+	0.0151331i
$250$	27513.8			0.440221
$251$		−	39392.8i		−	0.625273i	−0.949873	−	0.312636i	$-0.898788\pi$
$251$		−	39392.8i		−	0.625273i	0.949873	−	0.312636i	$-0.101212\pi$
$252$	48845.7	−	15344.0i	0.769175	−	0.241622i
$253$	−22222.2			−0.347173
$254$		−	63077.9i		−	0.977709i
$255$	22732.4	+	148218.i	0.349595	+	2.27940i
$256$	347471.			5.30198
$257$			47148.3i			0.713838i	0.934135	+	0.356919i	$0.116173\pi$
$257$			47148.3i			0.713838i	−0.934135	+	0.356919i	$0.883827\pi$
$258$	106879.	−	16392.1i	1.60565	−	0.246261i
$259$	26950.5			0.401760
$260$			264900.i			3.91864i
$261$	10663.1	+	33944.7i	0.156532	+	0.498300i
$262$	−167135.			−2.43481
$263$		−	11954.9i		−	0.172836i	−0.996259	−	0.0864181i	$-0.972458\pi$
$263$		−	11954.9i		−	0.172836i	0.996259	−	0.0864181i	$-0.0275421\pi$
$264$	61364.0	+	400101.i	0.880452	+	5.74065i
$265$	29575.3			0.421151
$266$		−	22850.7i		−	0.322950i
$267$	50409.0	−	7731.29i	0.707107	−	0.108450i
$268$	183830.			2.55945
$269$			31888.6i			0.440688i	0.975422	+	0.220344i	$0.0707180\pi$
$269$			31888.6i			0.440688i	−0.975422	+	0.220344i	$0.929282\pi$
$270$	−85522.7	−	174121.i	−1.17315	−	2.38850i
$271$	−76590.4			−1.04288			−0.521441	−	0.853287i	$-0.674606\pi$
$271$	−76590.4			−1.04288			−0.521441	+	0.853287i	$0.674606\pi$
$272$		−	544032.i		−	7.35337i
$273$	−3222.23	−	21009.3i	−0.0432345	−	0.281894i
$274$	−254956.			−3.39597
$275$			100315.i			1.32649i
$276$	−47035.1	+	7213.83i	−0.617453	+	0.0946996i
$277$	50328.2			0.655922			0.327961	−	0.944691i	$-0.393639\pi$
$277$	50328.2			0.655922			0.327961	+	0.944691i	$0.393639\pi$
$278$		−	146284.i		−	1.89280i
$279$	−13558.6	+	4259.19i	−0.174183	+	0.0547165i
$280$	−109387.			−1.39525
$281$			22681.3i			0.287247i	0.989632	+	0.143624i	$0.0458755\pi$
$281$			22681.3i			0.287247i	−0.989632	+	0.143624i	$0.954124\pi$
$282$	18134.4	+	118239.i	0.228037	+	1.48683i
$283$	20552.3			0.256619			0.128309	−	0.991734i	$-0.459045\pi$
$283$	20552.3			0.256619			0.128309	+	0.991734i	$0.459045\pi$
$284$			10697.5i			0.132631i
$285$	−63407.9	+	9724.95i	−0.780645	+	0.119729i
$286$	258393.			3.15899
$287$		−	25120.1i		−	0.304971i
$288$	120096.	+	382312.i	1.44792	+	4.60927i
$289$	−158583.			−1.89872
$290$		−	116890.i		−	1.38989i
$291$	−4667.54	−	30433.0i	−0.0551191	−	0.359384i
$292$	94836.8			1.11227
$293$			78542.5i			0.914891i	0.889238	+	0.457446i	$0.151236\pi$
$293$			78542.5i			0.914891i	−0.889238	+	0.457446i	$0.848764\pi$
$294$	−154514.	+	23698.0i	−1.78761	+	0.274169i
$295$	−15345.6			−0.176336
$296$			456254.i			5.20742i
$297$	−125842.	+	61809.3i	−1.42663	+	0.700714i
$298$	32184.8			0.362425
$299$			19754.7i			0.220967i
$300$	32564.7	+	212326.i	0.361830	+	2.35918i
$301$	21118.4			0.233092
$302$			199546.i			2.18790i
$303$	5198.93	−	797.366i	0.0566276	−	0.00868505i
$304$	232737.			2.51836
$305$		−	168555.i		−	1.81193i
$306$	298811.	−	93866.1i	3.19120	−	1.00246i
$307$	−143095.			−1.51827			−0.759135	−	0.650934i	$-0.774378\pi$
$307$	−143095.			−1.51827			−0.759135	+	0.650934i	$0.774378\pi$
$308$			121563.i			1.28145i
$309$	−27260.7	−	177743.i	−0.285510	−	1.86156i
$310$	46689.4			0.485842
$311$		−	63503.8i		−	0.656567i	−0.944579	−	0.328283i	$-0.893530\pi$
$311$		−	63503.8i		−	0.656567i	0.944579	−	0.328283i	$-0.106470\pi$
$312$	355674.	−	54550.2i	3.65378	−	0.560385i
$313$	119628.			1.22108			0.610542	−	0.791984i	$-0.290952\pi$
$313$	119628.			1.22108			0.610542	+	0.791984i	$0.290952\pi$
$314$			94136.9i			0.954774i
$315$	−11354.8	−	36146.5i	−0.114435	−	0.364288i
$316$	−15399.9			−0.154221
$317$		−	137237.i		−	1.36569i	−0.730561	−	0.682847i	$-0.760741\pi$
$317$		−	137237.i		−	1.36569i	0.730561	−	0.682847i	$-0.239259\pi$
$318$	−9364.98	−	61060.8i	−0.0926088	−	0.603821i
$319$	−84478.8			−0.830168
$320$		−	717469.i		−	7.00653i
$321$	96610.5	−	14817.3i	0.937593	−	0.143800i
$322$	−12543.5			−0.120978
$323$		−	103572.i		−	0.992745i
$324$	−246289.	+	171675.i	−2.34615	+	1.63538i
$325$	89176.5			0.844274
$326$			46250.1i			0.435188i
$327$	−6867.66	−	44778.0i	−0.0642263	−	0.418764i
$328$	425267.			3.95289
$329$			23363.0i			0.215842i
$330$	455274.	−	69826.0i	4.18066	−	0.641194i
$331$	−118240.			−1.07921			−0.539607	−	0.841917i	$-0.681427\pi$
$331$	−118240.			−1.07921			−0.539607	+	0.841917i	$0.681427\pi$
$332$		−	31467.0i		−	0.285482i
$333$	−150767.	+	47360.7i	−1.35962	+	0.427100i
$334$	126019.			1.12965
$335$		−	136037.i		−	1.21218i
$336$	20838.7	+	135871.i	0.184583	+	1.20351i
$337$	47426.4			0.417600			0.208800	−	0.977958i	$-0.433044\pi$
$337$	47426.4			0.417600			0.208800	+	0.977958i	$0.433044\pi$
$338$		−	5250.77i		−	0.0459610i
$339$	149015.	−	22854.6i	1.29667	−	0.198872i
$340$	−762385.			−6.59503
$341$		−	33743.5i		−	0.290190i
$342$	40156.0	+	127832.i	0.343319	+	1.09291i
$343$	−63697.5			−0.541419
$344$			357520.i			3.02123i
$345$	5338.34	+	34806.6i	0.0448506	+	0.292431i
$346$	211019.			1.76266
$347$		−	38766.0i		−	0.321953i	−0.986958	−	0.160976i	$-0.948536\pi$
$347$		−	38766.0i		−	0.321953i	0.986958	−	0.160976i	$-0.0514643\pi$
$348$	−178806.	+	27423.7i	−1.47647	+	0.226448i
$349$	−174398.			−1.43183			−0.715914	−	0.698189i	$-0.753990\pi$
$349$	−174398.			−1.43183			−0.715914	+	0.698189i	$0.753990\pi$
$350$			56623.7i			0.462235i
$351$	54946.0	+	111868.i	0.445987	+	0.908014i
$352$	−951466.			−7.67906
$353$		−	122807.i		−	0.985536i	−0.870161	−	0.492768i	$-0.835985\pi$
$353$		−	122807.i		−	0.985536i	0.870161	−	0.492768i	$-0.164015\pi$
$354$	4859.17	+	31682.4i	0.0387754	+	0.252820i
$355$	7916.29			0.0628153
$356$			259287.i			2.04588i
$357$	60465.0	−	9273.60i	0.474425	−	0.0727632i
$358$	−287944.			−2.24669
$359$			19827.2i			0.153841i	0.997037	+	0.0769204i	$0.0245087\pi$
$359$			19827.2i			0.153841i	−0.997037	+	0.0769204i	$0.975491\pi$
$360$	611937.	−	192229.i	4.72173	−	1.48325i
$361$	−86012.8			−0.660007
$362$		−	319266.i		−	2.43633i
$363$	−30488.9	−	198791.i	−0.231381	−	1.50863i
$364$	108065.			0.815609
$365$		−	70180.6i		−	0.526783i
$366$	−347997.	+	53372.8i	−2.59784	+	0.398435i
$367$	3598.99			0.0267207			0.0133604	−	0.999911i	$-0.495747\pi$
$367$	3598.99			0.0267207			0.0133604	+	0.999911i	$0.495747\pi$
$368$		−	127757.i		−	0.943386i
$369$	44144.2	+	140528.i	0.324206	+	1.03207i
$370$	519170.			3.79233
$371$		−	12065.1i		−	0.0876565i
$372$	−10953.9	−	71421.0i	−0.0791560	−	0.516107i
$373$	−51310.8			−0.368800			−0.184400	−	0.982851i	$-0.559034\pi$
$373$	−51310.8			−0.368800			−0.184400	+	0.982851i	$0.559034\pi$
$374$			743657.i			5.31655i
$375$	−31145.7	+	4776.86i	−0.221481	+	0.0339688i
$376$	−395520.			−2.79765
$377$			75098.3i			0.528381i
$378$	−71032.2	+	34888.7i	−0.497132	+	0.244175i
$379$	134311.			0.935046			0.467523	−	0.883981i	$-0.345147\pi$
$379$	134311.			0.935046			0.467523	+	0.883981i	$0.345147\pi$
$380$		−	326149.i		−	2.25865i
$381$	10951.4	+	71404.3i	0.0754429	+	0.491897i
$382$	115367.			0.790595
$383$		−	19879.1i		−	0.135518i	−0.997702	−	0.0677592i	$-0.978415\pi$
$383$		−	19879.1i		−	0.135518i	0.997702	−	0.0677592i	$-0.0215850\pi$
$384$	−777101.	+	119185.i	−5.27005	+	0.808275i
$385$	89958.5			0.606905
$386$		−	552832.i		−	3.71038i
$387$	−118141.	+	37111.8i	−0.788821	+	0.247794i
$388$	156537.			1.03981
$389$			63588.9i			0.420225i	0.977677	+	0.210113i	$0.0673831\pi$
$389$			63588.9i			0.420225i	−0.977677	+	0.210113i	$0.932617\pi$
$390$	−62072.6	−	404721.i	−0.408104	−	2.66089i
$391$	−56854.1			−0.371885
$392$		−	516866.i		−	3.36361i
$393$	189197.	−	29017.4i	1.22498	−	0.187877i
$394$	498965.			3.21423
$395$			11396.1i			0.0730403i
$396$	−213626.	−	680052.i	−1.36227	−	4.33662i
$397$	88252.9			0.559948			0.279974	−	0.960008i	$-0.409674\pi$
$397$	88252.9			0.559948			0.279974	+	0.960008i	$0.409674\pi$
$398$		−	163581.i		−	1.03268i
$399$	3967.25	+	25867.0i	0.0249198	+	0.162480i
$400$	−576721.			−3.60451
$401$			129776.i			0.807062i	0.914966	+	0.403531i	$0.132217\pi$
$401$			129776.i			0.807062i	−0.914966	+	0.403531i	$0.867783\pi$
$402$	−280859.	+	43075.8i	−1.73795	+	0.266551i
$403$	−29996.7			−0.184698
$404$			26741.6i			0.163841i
$405$	127042.	+	182258.i	0.774530	+	1.11116i
$406$	−47684.6			−0.289285
$407$		−	375217.i		−	2.26513i
$408$	156996.	+	1.02363e6i	0.943123	+	6.14928i
$409$	226902.			1.35641			0.678207	−	0.734871i	$-0.262757\pi$
$409$	226902.			1.35641			0.678207	+	0.734871i	$0.262757\pi$
$410$		−	483911.i		−	2.87871i
$411$	288611.	−	44264.6i	1.70855	−	0.262043i
$412$	914253.			5.38607
$413$			6260.19i			0.0367018i
$414$	70171.0	−	22042.9i	0.409409	−	0.128608i
$415$	−23286.0			−0.135207
$416$			845816.i			4.88753i
$417$	25397.3	+	165593.i	0.146054	+	0.952293i
$418$	−318137.			−1.82080
$419$		−	333074.i		−	1.89720i	−0.316477	−	0.948600i	$-0.602500\pi$
$419$		−	333074.i		−	1.89720i	0.316477	−	0.948600i	$-0.397500\pi$
$420$	190405.	−	29202.6i	1.07939	−	0.165548i
$421$	−109937.			−0.620267			−0.310133	−	0.950693i	$-0.600374\pi$
$421$	−109937.			−0.620267			−0.310133	+	0.950693i	$0.600374\pi$
$422$		−	385978.i		−	2.16739i
$423$	−41056.4	−	130698.i	−0.229456	−	0.730445i
$424$	204255.			1.13616
$425$			256651.i			1.42090i
$426$	−2506.68	−	16343.9i	−0.0138128	−	0.0900608i
$427$	−68761.4			−0.377128
$428$			496933.i			2.71275i
$429$	−292501.	+	44861.3i	−1.58933	+	0.243757i
$430$	406822.			2.20023
$431$			216245.i			1.16410i	0.813153	+	0.582050i	$0.197749\pi$
$431$			216245.i			1.16410i	−0.813153	+	0.582050i	$0.802251\pi$
$432$	−355346.	−	723473.i	−1.90407	−	3.87663i
$433$	107632.			0.574071			0.287035	−	0.957920i	$-0.407330\pi$
$433$	107632.			0.574071			0.287035	+	0.957920i	$0.407330\pi$
$434$		−	19046.8i		−	0.101121i
$435$	20294.0	+	132319.i	0.107248	+	0.699269i
$436$	230323.			1.21161
$437$		−	24322.2i		−	0.127362i
$438$	−144894.	+	22222.6i	−0.755270	+	0.115837i
$439$	148945.			0.772853			0.386427	−	0.922320i	$-0.373709\pi$
$439$	148945.			0.772853			0.386427	+	0.922320i	$0.373709\pi$
$440$			1.52294e6i			7.86642i
$441$	170796.	−	53652.5i	0.878214	−	0.275875i
$442$	661082.			3.38385
$443$		−	9322.46i		−	0.0475032i	−0.999718	−	0.0237516i	$-0.992439\pi$
$443$		−	9322.46i		−	0.0475032i	0.999718	−	0.0237516i	$-0.00756108\pi$
$444$	−121804.	−	794176.i	−0.617867	−	4.02857i
$445$	191876.			0.968949
$446$		−	51736.8i		−	0.260094i
$447$	−36433.2	+	5587.81i	−0.182340	+	0.0279658i
$448$	−292689.			−1.45831
$449$			153270.i			0.760266i	0.924932	+	0.380133i	$0.124122\pi$
$449$			153270.i			0.760266i	−0.924932	+	0.380133i	$0.875878\pi$
$450$	−99506.3	−	316766.i	−0.491389	−	1.56428i
$451$	−349734.			−1.71943
$452$			766483.i			3.75168i
$453$	−34644.5	−	225886.i	−0.168825	−	1.10076i
$454$	−4934.72			−0.0239415
$455$		−	79969.6i		−	0.386280i
$456$	−437911.	+	67163.0i	−2.10599	+	0.322998i
$457$	27315.8			0.130792			0.0653961	−	0.997859i	$-0.479169\pi$
$457$	27315.8			0.130792			0.0653961	+	0.997859i	$0.479169\pi$
$458$		−	226753.i		−	1.08099i
$459$	−321958.	+	158135.i	−1.52818	+	0.750591i
$460$	−179034.			−0.846096
$461$		−	175848.i		−	0.827436i	−0.910405	−	0.413718i	$-0.864230\pi$
$461$		−	175848.i		−	0.827436i	0.910405	−	0.413718i	$-0.135770\pi$
$462$	−28485.2	−	185727.i	−0.133455	−	0.870145i
$463$	−189562.			−0.884278			−0.442139	−	0.896946i	$-0.645780\pi$
$463$	−189562.			−0.884278			−0.442139	+	0.896946i	$0.645780\pi$
$464$		−	485675.i		−	2.25585i
$465$	−52852.5	+	8106.07i	−0.244433	+	0.0374890i
$466$	179931.			0.828578
$467$		−	121958.i		−	0.559213i	−0.960115	−	0.279607i	$-0.909796\pi$
$467$		−	121958.i		−	0.559213i	0.960115	−	0.279607i	$-0.0902041\pi$
$468$	−604539.	+	189905.i	−2.76015	+	0.867052i
$469$	−55495.6			−0.252297
$470$			450062.i			2.03740i
$471$	−16343.7	−	106563.i	−0.0736732	−	0.480358i
$472$	−105981.			−0.475712
$473$		−	294020.i		−	1.31418i
$474$	23528.3	−	3608.56i	0.104721	−	0.0160612i
$475$	−109795.			−0.486628
$476$			311012.i			1.37266i
$477$	21202.3	+	67495.0i	0.0931853	+	0.296644i
$478$	−835178.			−3.65530
$479$			304764.i			1.32829i	0.747604	+	0.664145i	$0.231204\pi$
$479$			304764.i			1.32829i	−0.747604	+	0.664145i	$0.768796\pi$
$480$	228567.	+	1.49028e6i	0.992043	+	6.46824i
$481$	−333553.			−1.44170
$482$			696469.i			2.99783i
$483$	14199.2	−	2177.76i	0.0608654	−	0.00933501i
$484$	1.02252e6			4.36495
$485$		−	115840.i		−	0.492463i
$486$	336059.	−	320002.i	1.42280	−	1.35481i
$487$	−187435.			−0.790302			−0.395151	−	0.918616i	$-0.629308\pi$
$487$	−187435.			−0.790302			−0.395151	+	0.918616i	$0.629308\pi$
$488$		−	1.16409e6i		−	4.88816i
$489$	−8029.78	−	52355.2i	−0.0335804	−	0.218948i
$490$	−588141.			−2.44957
$491$			463302.i			1.92177i	0.276950	+	0.960884i	$0.410676\pi$
$491$			463302.i			1.92177i	−0.276950	+	0.960884i	$0.589324\pi$
$492$	−740240.	+	113532.i	−3.05804	+	0.469015i
$493$	−216134.			−0.889260
$494$			282811.i			1.15889i
$495$	−503248.	+	158086.i	−2.05386	+	0.645184i
$496$	193994.			0.788543
$497$		−	3229.42i		−	0.0130741i
$498$	7373.49	+	48076.0i	0.0297313	+	0.193852i
$499$	92062.3			0.369727			0.184863	−	0.982764i	$-0.440816\pi$
$499$	92062.3			0.369727			0.184863	+	0.982764i	$0.440816\pi$
$500$		−	160203.i		−	0.640813i
$501$	−142654.	+	21879.1i	−0.568341	+	0.0871672i
$502$	−309573.			−1.22845
$503$			325821.i			1.28778i	0.765116	+	0.643892i	$0.222681\pi$
$503$			325821.i			1.28778i	−0.765116	+	0.643892i	$0.777319\pi$
$504$	−78419.0	−	249637.i	−0.308717	−	0.982762i
$505$	19789.1			0.0775968
$506$			174636.i			0.682076i
$507$	911.621	+	5943.88i	0.00354649	+	0.0231235i
$508$	−367280.			−1.42321
$509$		−	392471.i		−	1.51486i	−0.652917	−	0.757429i	$-0.726455\pi$
$509$		−	392471.i		−	1.51486i	0.652917	−	0.757429i	$-0.273545\pi$
$510$	1.16479e6	−	178646.i	4.47824	−	0.686834i
$511$	−28629.9			−0.109642
$512$		−	1.33298e6i		−	5.08490i
$513$	−67650.4	−	137734.i	−0.257061	−	0.523367i
$514$	370521.			1.40245
$515$		−	676560.i		−	2.55089i
$516$	−95445.5	−	622316.i	−0.358473	−	2.33729i
$517$	325270.			1.21692
$518$		−	211794.i		−	0.789320i
$519$	−238874.	+	36636.4i	−0.886817	+	0.136012i
$520$	1.35383e6			5.00678
$521$			513011.i			1.88995i	0.327137	+	0.944977i	$0.393916\pi$
$521$			513011.i			1.88995i	−0.327137	+	0.944977i	$0.606084\pi$
$522$	266758.	−	83797.3i	0.978988	−	0.307531i
$523$	−290306.			−1.06133			−0.530667	−	0.847580i	$-0.678059\pi$
$523$	−290306.			−1.06133			−0.530667	+	0.847580i	$0.678059\pi$
$524$			973168.i			3.54426i
$525$	−9830.83	−	64098.2i	−0.0356674	−	0.232556i
$526$	−93949.1			−0.339564
$527$		−	86330.8i		−	0.310845i
$528$	1.89166e6	−	290126.i	6.78539	−	1.04068i
$529$	266490.			0.952290
$530$		−	232421.i		−	0.827416i
$531$	−11001.2	−	35020.9i	−0.0390167	−	0.124205i
$532$	−133051.			−0.470106
$533$			310900.i			1.09437i
$534$	−60757.3	−	396145.i	−0.213067	−	1.38922i
$535$	367737.			1.28478
$536$		−	939504.i		−	3.27016i
$537$	325954.	−	49992.0i	1.13034	−	0.173361i
$538$	250601.			0.865800
$539$			425063.i			1.46311i
$540$	−1.01385e6	+	497968.i	−3.47684	+	1.70771i
$541$	−42782.2			−0.146173			−0.0730867	−	0.997326i	$-0.523285\pi$
$541$	−42782.2			−0.146173			−0.0730867	+	0.997326i	$0.523285\pi$
$542$			601895.i			2.04891i
$543$	55429.9	+	361410.i	0.187994	+	1.22575i
$544$	−2.43427e6			−8.22566
$545$		−	170442.i		−	0.573832i
$546$	−165104.	+	25322.3i	−0.553825	+	0.0849410i
$547$	348295.			1.16405			0.582026	−	0.813170i	$-0.302260\pi$
$547$	348295.			1.16405			0.582026	+	0.813170i	$0.302260\pi$
$548$			1.48452e6i			4.94339i
$549$	384667.	−	120836.i	1.27626	−	0.400915i
$550$	788341.			2.60609
$551$		−	92462.2i		−	0.304552i
$552$	36868.0	+	240384.i	0.120996	+	0.788909i
$553$	4649.00			0.0152023
$554$		−	395511.i		−	1.28866i
$555$	−587702.	+	90136.6i	−1.90797	+	0.292628i
$556$	−851757.			−2.75528
$557$		−	570919.i		−	1.84020i	−0.391689	−	0.920098i	$-0.628109\pi$
$557$		−	570919.i		−	1.84020i	0.391689	−	0.920098i	$-0.371891\pi$
$558$	33471.4	+	106552.i	0.107499	+	0.342210i
$559$	−261372.			−0.836441
$560$			517178.i			1.64917i
$561$	−129111.	−	841821.i	−0.410240	−	2.67482i
$562$	178244.			0.564343
$563$			65361.5i			0.206208i	0.994671	+	0.103104i	$0.0328774\pi$
$563$			65361.5i			0.206208i	−0.994671	+	0.103104i	$0.967123\pi$
$564$	688461.	−	105590.i	2.16432	−	0.331944i
$565$	567208.			1.77683
$566$		−	161513.i		−	0.504168i
$567$	74351.3	−	51826.4i	0.231272	−	0.161207i
$568$	54672.0			0.169460
$569$			384477.i			1.18753i	0.804637	+	0.593767i	$0.202360\pi$
$569$			384477.i			1.18753i	−0.804637	+	0.593767i	$0.797640\pi$
$570$	76424.7	+	498298.i	0.235225	+	1.53370i
$571$	−54990.0			−0.168660			−0.0843298	−	0.996438i	$-0.526875\pi$
$571$	−54990.0			−0.168660			−0.0843298	+	0.996438i	$0.526875\pi$
$572$		−	1.50453e6i		−	4.59842i
$573$	−130595.	+	20029.6i	−0.397758	+	0.0610046i
$574$	−197410.			−0.599163
$575$			60270.3i			0.182292i
$576$	1.63737e6	−	514349.i	4.93516	−	1.55029i
$577$	421721.			1.26670			0.633350	−	0.773866i	$-0.281680\pi$
$577$	421721.			1.26670			0.633350	+	0.773866i	$0.281680\pi$
$578$			1.24624e6i			3.73032i
$579$	95980.8	+	625807.i	0.286304	+	1.86674i
$580$	−680606.			−2.02320
$581$			9499.44i			0.0281414i
$582$	−239161.	+	36680.5i	−0.706065	+	0.108290i
$583$	−167976.			−0.494209
$584$		−	484686.i		−	1.42113i
$585$	140533.	+	447368.i	0.410644	+	1.30723i
$586$	617236.			1.79745
$587$		−	228158.i		−	0.662156i	−0.943603	−	0.331078i	$-0.892588\pi$
$587$		−	228158.i		−	0.662156i	0.943603	−	0.331078i	$-0.107412\pi$
$588$	137985.	+	899681.i	0.399097	+	2.60216i
$589$	36932.4			0.106458
$590$			120596.i			0.346440i
$591$	−564829.	+	86628.6i	−1.61712	+	0.248020i
$592$	2.15715e6			6.15512
$593$			205389.i			0.584073i	0.956407	+	0.292037i	$0.0943329\pi$
$593$			205389.i			0.584073i	−0.956407	+	0.292037i	$0.905667\pi$
$594$	485736.	+	988942.i	1.37666	+	2.80284i
$595$	230153.			0.650105
$596$		−	187401.i		−	0.527568i
$597$	28400.4	+	185174.i	0.0796848	+	0.519554i
$598$	155244.			0.434124
$599$			88620.3i			0.246990i	0.992345	+	0.123495i	$0.0394103\pi$
$599$			88620.3i			0.246990i	−0.992345	+	0.123495i	$0.960590\pi$
$600$	1.08514e6	−	166429.i	3.01428	−	0.462304i
$601$	−363771.			−1.00711			−0.503557	−	0.863962i	$-0.667976\pi$
$601$	−363771.			−1.00711			−0.503557	+	0.863962i	$0.667976\pi$
$602$		−	165961.i		−	0.457946i
$603$	310455.	−	97523.7i	0.853815	−	0.268210i
$604$	1.16188e6			3.18485
$605$		−	756676.i		−	2.06728i
$606$	−6266.20	−	40856.4i	−0.0170631	−	0.111254i
$607$	−150200.			−0.407653			−0.203827	−	0.979007i	$-0.565338\pi$
$607$	−150200.			−0.407653			−0.203827	+	0.979007i	$0.565338\pi$
$608$		−	1.04138e6i		−	2.81710i
$609$	53979.1	−	8278.85i	0.145543	−	0.0223221i
$610$	−1.32461e6			−3.55983
$611$		−	289152.i		−	0.774541i
$612$	−546549.	−	1.73987e6i	−1.45924	−	4.64530i
$613$	90472.5			0.240766			0.120383	−	0.992728i	$-0.461588\pi$
$613$	90472.5			0.240766			0.120383	+	0.992728i	$0.461588\pi$
$614$			1.12453e6i			2.98288i
$615$	84015.1	+	547789.i	0.222130	+	1.44831i
$616$	621277.			1.63728
$617$		−	42407.5i		−	0.111397i	−0.998448	−	0.0556984i	$-0.982261\pi$
$617$		−	42407.5i		−	0.111397i	0.998448	−	0.0556984i	$-0.0177385\pi$
$618$	−1.39682e6	+	214232.i	−3.65732	+	0.560928i
$619$	57960.9			0.151270			0.0756352	−	0.997136i	$-0.475902\pi$
$619$	57960.9			0.151270			0.0756352	+	0.997136i	$0.475902\pi$
$620$		−	271856.i		−	0.707222i
$621$	−75606.7	+	37135.5i	−0.196055	+	0.0962956i
$622$	−499052.			−1.28993
$623$		−	78275.2i		−	0.201673i
$624$	−257911.	−	1.68161e6i	−0.662370	−	4.31873i
$625$	−444556.			−1.13806
$626$		−	940114.i		−	2.39901i
$627$	360132.	−	55233.9i	0.916065	−	0.140498i
$628$	548126.			1.38983
$629$		−	959969.i		−	2.42636i
$630$	−284062.	+	89232.9i	−0.715701	+	0.224825i
$631$	−214265.			−0.538136			−0.269068	−	0.963121i	$-0.586716\pi$
$631$	−214265.			−0.538136			−0.269068	+	0.963121i	$0.586716\pi$
$632$			78704.5i			0.197045i
$633$	67012.2	+	436928.i	0.167242	+	1.09044i
$634$	−1.07850e6			−2.68312
$635$			271793.i			0.674047i
$636$	−355535.	+	54528.9i	−0.878959	+	0.134807i
$637$	377865.			0.931231
$638$			663887.i			1.63100i
$639$	5675.14	+	18066.1i	0.0138987	+	0.0442449i
$640$	−2.95795e6			−7.22156
$641$		−	55617.2i		−	0.135361i	−0.997707	−	0.0676804i	$-0.978440\pi$
$641$		−	55617.2i		−	0.135361i	0.997707	−	0.0676804i	$-0.0215598\pi$
$642$	−116443.	−	759226.i	−0.282517	−	1.84205i
$643$	173926.			0.420672			0.210336	−	0.977629i	$-0.432544\pi$
$643$	173926.			0.420672			0.210336	+	0.977629i	$0.432544\pi$
$644$			73036.2i			0.176103i
$645$	−460523.	+	70631.0i	−1.10696	+	0.169776i
$646$	−813934.			−1.95040
$647$			698038.i			1.66752i	0.552128	+	0.833759i	$0.313816\pi$
$647$			698038.i			1.66752i	−0.552128	+	0.833759i	$0.686184\pi$
$648$	877387.	+	1.25872e6i	2.08949	+	2.99763i
$649$	87157.3			0.206926
$650$		−	700804.i		−	1.65871i
$651$	3306.84	+	21561.0i	0.00780281	+	0.0508753i
$652$	269298.			0.633487
$653$			13267.9i			0.0311154i	0.999879	+	0.0155577i	$0.00495237\pi$
$653$			13267.9i			0.0311154i	−0.999879	+	0.0155577i	$0.995048\pi$
$654$	−351893.	+	53970.4i	−0.822727	+	0.126183i
$655$	720159.			1.67859
$656$		−	2.01065e6i		−	4.67227i
$657$	160162.	−	50312.0i	0.371047	−	0.116558i
$658$	183601.			0.424056
$659$		−	481694.i		−	1.10917i	−0.832125	−	0.554587i	$-0.812876\pi$
$659$		−	481694.i		−	1.10917i	0.832125	−	0.554587i	$-0.187124\pi$
$660$	−406572.	−	2.65090e6i	−0.933361	−	6.08563i
$661$	586120.			1.34148			0.670739	−	0.741693i	$-0.265977\pi$
$661$	586120.			1.34148			0.670739	+	0.741693i	$0.265977\pi$
$662$			929201.i			2.12028i
$663$	−748346.	+	114775.i	−1.70245	+	0.261108i
$664$	−160819.			−0.364756
$665$			98459.8i			0.222646i
$666$	372190.	+	1.18482e6i	0.839105	+	2.67119i
$667$	−50755.5			−0.114086
$668$		−	733766.i		−	1.64439i
$669$	8982.37	+	58566.2i	0.0200696	+	0.130856i
$670$	−1.06906e6			−2.38151
$671$			957328.i			2.12626i
$672$	607955.	−	93242.9i	1.34627	−	0.206480i
$673$	148675.			0.328251			0.164126	−	0.986439i	$-0.447520\pi$
$673$	148675.			0.328251			0.164126	+	0.986439i	$0.447520\pi$
$674$		−	372706.i		−	0.820440i
$675$	167637.	+	341304.i	0.367928	+	0.749089i
$676$	−30573.4			−0.0669037
$677$			315103.i			0.687504i	0.939060	+	0.343752i	$0.111698\pi$
$677$			315103.i			0.687504i	−0.939060	+	0.343752i	$0.888302\pi$
$678$	−179606.	−	1.17105e6i	−0.390716	−	2.54751i
$679$	−47256.4			−0.102499
$680$			3.89635e6i			8.42635i
$681$	5586.12	−	856.750i	0.0120452	−	0.00184740i
$682$	−265178.			−0.570123
$683$		−	71563.2i		−	0.153408i	−0.997054	−	0.0767040i	$-0.975560\pi$
$683$		−	71563.2i		−	0.153408i	0.997054	−	0.0767040i	$-0.0244397\pi$
$684$	744318.	−	233814.i	1.59091	−	0.499757i
$685$	1.09857e6			2.34123
$686$			500574.i			1.06370i
$687$	39368.1	+	256685.i	0.0834125	+	0.543860i
$688$	1.69034e6			3.57106
$689$			149324.i			0.314552i
$690$	273532.	−	41952.0i	0.574527	−	0.0881160i
$691$	−600494.			−1.25763			−0.628814	−	0.777556i	$-0.716459\pi$
$691$	−600494.			−1.25763			−0.628814	+	0.777556i	$0.716459\pi$
$692$		−	1.22869e6i		−	2.56584i
$693$	64490.7	+	205298.i	0.134286	+	0.427483i
$694$	−304648.			−0.632527
$695$			630312.i			1.30493i
$696$	140155.	+	913831.i	0.289329	+	1.88646i
$697$	−894773.			−1.84182
$698$			1.37053e6i			2.81305i
$699$	−203682.	+	31239.0i	−0.416868	+	0.0639355i
$700$	329700.			0.672857
$701$		−	743521.i		−	1.51306i	−0.653957	−	0.756531i	$-0.726892\pi$
$701$		−	743521.i		−	1.51306i	0.653957	−	0.756531i	$-0.273108\pi$
$702$	879131.	−	431800.i	1.78394	−	0.876211i
$703$	410675.			0.830975
$704$			4.07494e6i			8.22198i
$705$	−78138.3	−	509471.i	−0.157212	−	1.02504i
$706$	−965091.			−1.93624
$707$		−	8072.90i		−	0.0161507i
$708$	184475.	−	28293.2i	0.368021	−	0.0564438i
$709$	578585.			1.15100			0.575499	−	0.817802i	$-0.304808\pi$
$709$	578585.			1.15100			0.575499	+	0.817802i	$0.304808\pi$
$710$		−	62211.2i		−	0.123410i
$711$	−26007.5	+	8169.80i	−0.0514470	+	0.0161612i
$712$	1.32515e6			2.61399
$713$		−	20273.4i		−	0.0398793i
$714$	−72877.7	−	475172.i	−0.142955	−	0.932082i
$715$	−1.11337e6			−2.17785
$716$			1.67660e6i			3.27042i
$717$	945424.	−	145001.i	1.83903	−	0.282054i
$718$	155814.			0.302244
$719$			754589.i			1.45966i	0.683627	+	0.729832i	$0.260402\pi$
$719$			754589.i			1.45966i	−0.683627	+	0.729832i	$0.739598\pi$
$720$	−908850.	−	2.89321e6i	−1.75318	−	5.58104i
$721$	−276000.			−0.530932
$722$			675942.i			1.29669i
$723$	−120919.	−	788404.i	−0.231322	−	1.50825i
$724$	−1.85897e6			−3.54647
$725$			229121.i			0.435901i
$726$	−1.56222e6	+	239600.i	−2.96395	+	0.454584i
$727$	−239352.			−0.452864			−0.226432	−	0.974027i	$-0.572706\pi$
$727$	−239352.			−0.452864			−0.226432	+	0.974027i	$0.572706\pi$
$728$		−	552291.i		−	1.04209i
$729$	−324862.	+	420588.i	−0.611285	+	0.791410i
$730$	−551523.			−1.03495
$731$		−	752231.i		−	1.40772i
$732$	310771.	+	2.02626e6i	0.579986	+	3.78158i
$733$	231674.			0.431191			0.215595	−	0.976483i	$-0.430831\pi$
$733$	231674.			0.431191			0.215595	+	0.976483i	$0.430831\pi$
$734$		−	28283.1i		−	0.0524970i
$735$	665777.	−	102111.i	1.23241	−	0.189016i
$736$	−571649.			−1.05529
$737$			772635.i			1.42246i
$738$	1.10435e6	−	346913.i	2.02766	−	0.636954i
$739$	−731305.			−1.33909			−0.669545	−	0.742772i	$-0.733511\pi$
$739$	−731305.			−1.33909			−0.669545	+	0.742772i	$0.733511\pi$
$740$		−	3.02294e6i		−	5.52035i
$741$	−49100.8	−	320143.i	−0.0894236	−	0.583053i
$742$	−94815.3			−0.172215
$743$		−	797019.i		−	1.44375i	−0.692025	−	0.721873i	$-0.743281\pi$
$743$		−	797019.i		−	1.44375i	0.692025	−	0.721873i	$-0.256719\pi$
$744$	−365013.	+	55982.6i	−0.659421	+	0.101136i
$745$	−138679.			−0.249861
$746$			403232.i			0.724565i
$747$	−16693.6	−	53142.0i	−0.0299164	−	0.0952350i
$748$	4.33005e6			7.73909
$749$		−	150017.i		−	0.267410i
$750$	37539.5	+	244762.i	0.0667369	+	0.435133i
$751$	1.02196e6			1.81198			0.905991	−	0.423296i	$-0.139127\pi$
$751$	1.02196e6			1.81198			0.905991	+	0.423296i	$0.139127\pi$
$752$			1.87000e6i			3.30679i
$753$	350438.	−	53747.1i	0.618046	−	0.0947906i
$754$	590169.			1.03809
$755$		−	859811.i		−	1.50837i
$756$	203144.	+	413595.i	0.355436	+	0.723655i
$757$	711548.			1.24169			0.620844	−	0.783934i	$-0.286790\pi$
$757$	711548.			1.24169			0.620844	+	0.783934i	$0.286790\pi$
$758$		−	1.05550e6i		−	1.83704i
$759$	−30319.7	−	197688.i	−0.0526310	−	0.343160i
$760$	−1.66686e6			−2.88584
$761$			1.11176e6i			1.91973i	0.280459	+	0.959866i	$0.409513\pi$
$761$			1.11176e6i			1.91973i	−0.280459	+	0.959866i	$0.590487\pi$
$762$	561140.	−	86062.7i	0.966409	−	0.148219i
$763$	−69531.3			−0.119435
$764$		−	671740.i		−	1.15084i
$765$	−1.28753e6	+	404454.i	−2.20006	+	0.691109i
$766$	−156222.			−0.266247
$767$		−	77479.4i		−	0.131703i
$768$	474085.	+	3.09109e6i	0.803774	+	5.24070i
$769$	864703.			1.46223			0.731113	−	0.682257i	$-0.239001\pi$
$769$	864703.			1.46223			0.731113	+	0.682257i	$0.239001\pi$
$770$		−	706950.i		−	1.19236i
$771$	−419430.	+	64328.6i	−0.705588	+	0.108217i
$772$	−3.21894e6			−5.40105
$773$		−	770578.i		−	1.28961i	−0.764348	−	0.644804i	$-0.776939\pi$
$773$		−	770578.i		−	1.28961i	0.764348	−	0.644804i	$-0.223061\pi$
$774$	291648.	+	928425.i	0.486830	+	1.54976i
$775$	−91518.2			−0.152372
$776$		−	800019.i		−	1.32855i
$777$	36770.9	+	239751.i	0.0609063	+	0.397116i
$778$	499721.			0.825598
$779$		−	382785.i		−	0.630782i
$780$	−2.35654e6	+	361426.i	−3.87335	+	0.594061i
$781$	−44961.5			−0.0737120
$782$			446795.i			0.730626i

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:	\( N \)	\(=\)	\( 177 = 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	177.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-7.85862i q^{2} +(1.36439 + 8.89598i) q^{3} -45.7580 q^{4} +33.8616i q^{5} +(69.9102 - 10.7222i) q^{6} +13.8137 q^{7} +233.857i q^{8} +(-77.2769 + 24.2751i) q^{9} +O(q^{10})\)	\(q-7.85862i q^{2} +(1.36439 + 8.89598i) q^{3} -45.7580 q^{4} +33.8616i q^{5} +(69.9102 - 10.7222i) q^{6} +13.8137 q^{7} +233.857i q^{8} +(-77.2769 + 24.2751i) q^{9} +266.105 q^{10} -192.320i q^{11} +(-62.4316 - 407.062i) q^{12} -170.965 q^{13} -108.557i q^{14} +(-301.232 + 46.2003i) q^{15} +1105.66 q^{16} -492.040i q^{17} +(190.769 + 607.290i) q^{18} +210.495 q^{19} -1549.44i q^{20} +(18.8472 + 122.886i) q^{21} -1511.37 q^{22} -115.548i q^{23} +(-2080.38 + 319.071i) q^{24} -521.606 q^{25} +1343.55i q^{26} +(-321.387 - 654.333i) q^{27} -632.087 q^{28} -439.260i q^{29} +(363.071 + 2367.27i) q^{30} +175.455 q^{31} -4947.30i q^{32} +(1710.88 - 262.400i) q^{33} -3866.76 q^{34} +467.753i q^{35} +(3536.03 - 1110.78i) q^{36} +1951.00 q^{37} -1654.20i q^{38} +(-233.263 - 1520.90i) q^{39} -7918.76 q^{40} -1818.50i q^{41} +(965.717 - 148.113i) q^{42} +1528.80 q^{43} +8800.20i q^{44} +(-821.994 - 2616.72i) q^{45} -908.046 q^{46} +1691.29i q^{47} +(1508.56 + 9835.97i) q^{48} -2210.18 q^{49} +4099.10i q^{50} +(4377.18 - 671.334i) q^{51} +7823.03 q^{52} -873.418i q^{53} +(-5142.16 + 2525.66i) q^{54} +6512.27 q^{55} +3230.43i q^{56} +(287.197 + 1872.56i) q^{57} -3451.98 q^{58} +453.188i q^{59} +(13783.8 - 2114.03i) q^{60} -4977.77 q^{61} -1378.83i q^{62} +(-1067.48 + 335.329i) q^{63} -21188.3 q^{64} -5789.15i q^{65} +(-2062.10 - 13445.2i) q^{66} -4017.43 q^{67} +22514.8i q^{68} +(1027.91 - 157.652i) q^{69} +3675.90 q^{70} -233.784i q^{71} +(-5676.91 - 18071.7i) q^{72} -2072.57 q^{73} -15332.1i q^{74} +(-711.672 - 4640.19i) q^{75} -9631.84 q^{76} -2656.66i q^{77} +(-11952.2 + 1833.13i) q^{78} +336.550 q^{79} +37439.5i q^{80} +(5382.44 - 3751.81i) q^{81} -14290.9 q^{82} +687.683i q^{83} +(-862.412 - 5623.03i) q^{84} +16661.3 q^{85} -12014.3i q^{86} +(3907.65 - 599.322i) q^{87} +44975.4 q^{88} -5666.49i q^{89} +(-20563.8 + 6459.74i) q^{90} -2361.66 q^{91} +5287.23i q^{92} +(239.388 + 1560.84i) q^{93} +13291.2 q^{94} +7127.70i q^{95} +(44011.0 - 6750.03i) q^{96} -3420.98 q^{97} +17369.0i q^{98} +(4668.61 + 14861.9i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 78q - 612q^{4} + 64q^{6} + 76q^{7} - 100q^{9} + O(q^{10}) \)	\( 78q - 612q^{4} + 64q^{6} + 76q^{7} - 100q^{9} - 156q^{10} - 4q^{13} + 13q^{15} + 4948q^{16} + 22q^{18} + 812q^{19} - 173q^{21} - 1644q^{22} - 678q^{24} - 8238q^{25} + 777q^{27} - 3764q^{28} + 1374q^{30} + 4664q^{31} - 1042q^{33} + 3244q^{34} + 3648q^{36} - 3960q^{37} - 7078q^{39} - 1576q^{40} + 4934q^{42} - 1492q^{43} - 2063q^{45} - 2036q^{46} - 2620q^{48} + 24274q^{49} + 7300q^{51} + 8408q^{52} - 14766q^{54} + 9780q^{55} + 6939q^{57} - 3856q^{58} + 4712q^{60} - 212q^{61} - 7438q^{63} - 45760q^{64} + 3048q^{66} - 12972q^{67} + 21672q^{69} + 5828q^{70} - 866q^{72} - 5240q^{73} - 20922q^{75} + 12368q^{76} - 16508q^{78} - 14976q^{79} + 25524q^{81} - 14484q^{82} + 9540q^{84} + 11572q^{85} + 5695q^{87} + 62160q^{88} - 31672q^{90} + 8284q^{91} - 9590q^{93} - 10992q^{94} + 34102q^{96} - 55000q^{97} - 14254q^{99} + O(q^{100}) \)

Introduction

L-functions

Modular forms

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Properties

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