LMFDB - Embedded newform 177.3.b.a.119.3

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	$4.82290067918$
Analytic rank:	$0$
Dimension:	$38$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			119.3
Character	$\chi$	$=$	177.119
Dual form			177.3.b.a.119.36

$q$-expansion

$f(q)$	$=$	$q-3.55815i q^{2} +(-1.69290 - 2.47671i) q^{3} -8.66046 q^{4} +6.82324i q^{5} +(-8.81252 + 6.02359i) q^{6} +0.148532 q^{7} +16.5826i q^{8} +(-3.26820 + 8.38564i) q^{9} +O(q^{10})$	\(q-3.55815i q^{2} +(-1.69290 - 2.47671i) q^{3} -8.66046 q^{4} +6.82324i q^{5} +(-8.81252 + 6.02359i) q^{6} +0.148532 q^{7} +16.5826i q^{8} +(-3.26820 + 8.38564i) q^{9} +24.2782 q^{10} -1.80659i q^{11} +(14.6613 + 21.4495i) q^{12} -17.1364 q^{13} -0.528501i q^{14} +(16.8992 - 11.5511i) q^{15} +24.3617 q^{16} +18.3201i q^{17} +(29.8374 + 11.6287i) q^{18} -19.6285 q^{19} -59.0924i q^{20} +(-0.251450 - 0.367871i) q^{21} -6.42813 q^{22} -40.3036i q^{23} +(41.0704 - 28.0727i) q^{24} -21.5567 q^{25} +60.9738i q^{26} +(26.3015 - 6.10165i) q^{27} -1.28636 q^{28} +33.8473i q^{29} +(-41.1004 - 60.1300i) q^{30} -46.0774 q^{31} -20.3523i q^{32} +(-4.47440 + 3.05837i) q^{33} +65.1857 q^{34} +1.01347i q^{35} +(28.3041 - 72.6235i) q^{36} +40.6329 q^{37} +69.8412i q^{38} +(29.0101 + 42.4418i) q^{39} -113.147 q^{40} -21.3927i q^{41} +(-1.30894 + 0.894697i) q^{42} +4.07233 q^{43} +15.6459i q^{44} +(-57.2172 - 22.2997i) q^{45} -143.406 q^{46} +45.7277i q^{47} +(-41.2419 - 60.3370i) q^{48} -48.9779 q^{49} +76.7019i q^{50} +(45.3736 - 31.0140i) q^{51} +148.409 q^{52} +29.4395i q^{53} +(-21.7106 - 93.5849i) q^{54} +12.3268 q^{55} +2.46306i q^{56} +(33.2290 + 48.6141i) q^{57} +120.434 q^{58} -7.68115i q^{59} +(-146.355 + 100.037i) q^{60} -28.5354 q^{61} +163.950i q^{62} +(-0.485432 + 1.24554i) q^{63} +25.0305 q^{64} -116.926i q^{65} +(10.8822 + 15.9206i) q^{66} -6.55405 q^{67} -158.660i q^{68} +(-99.8203 + 68.2298i) q^{69} +3.60609 q^{70} +11.4894i q^{71} +(-139.056 - 54.1953i) q^{72} -113.523 q^{73} -144.578i q^{74} +(36.4932 + 53.3896i) q^{75} +169.992 q^{76} -0.268337i q^{77} +(151.014 - 103.222i) q^{78} -32.0057 q^{79} +166.226i q^{80} +(-59.6378 - 54.8118i) q^{81} -76.1184 q^{82} -145.911i q^{83} +(2.17767 + 3.18594i) q^{84} -125.002 q^{85} -14.4900i q^{86} +(83.8300 - 57.3000i) q^{87} +29.9580 q^{88} -148.358i q^{89} +(-79.3458 + 203.588i) q^{90} -2.54530 q^{91} +349.048i q^{92} +(78.0043 + 114.120i) q^{93} +162.706 q^{94} -133.930i q^{95} +(-50.4066 + 34.4543i) q^{96} -133.709 q^{97} +174.271i q^{98} +(15.1494 + 5.90429i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 38q - 76q^{4} - 8q^{6} - 12q^{7} + 20q^{9} + O(q^{10}) $	$ 38q - 76q^{4} - 8q^{6} - 12q^{7} + 20q^{9} + 36q^{10} - 4q^{13} - 17q^{15} + 100q^{16} - 2q^{18} - 28q^{19} - 11q^{21} + 84q^{22} - 6q^{24} - 166q^{25} + 3q^{27} + 12q^{28} + 102q^{30} - 40q^{31} - 46q^{33} - 148q^{34} - 96q^{36} + 112q^{37} + 62q^{39} - 56q^{40} + 14q^{42} + 164q^{43} + 55q^{45} - 4q^{46} - 124q^{48} + 242q^{49} + 52q^{51} + 8q^{52} + 18q^{54} - 228q^{55} - 147q^{57} - 80q^{58} + 128q^{60} + 12q^{61} + 86q^{63} + 48q^{64} - 24q^{66} + 124q^{67} - 240q^{69} + 148q^{70} + 166q^{72} - 192q^{73} - 78q^{75} - 304q^{76} + 244q^{78} + 64q^{79} - 156q^{81} - 180q^{82} + 300q^{84} - 52q^{85} - 83q^{87} - 96q^{88} - 376q^{90} - 332q^{91} + 454q^{93} + 768q^{94} - 722q^{96} + 416q^{97} + 494q^{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 $

$2$		−	3.55815i		−	1.77908i	−0.456860	−	0.889539i	$-0.651026\pi$
$2$		−	3.55815i		−	1.77908i	0.456860	−	0.889539i	$-0.348974\pi$
$3$	−1.69290	−	2.47671i	−0.564299	−	0.825570i
$3$	−1.69290	−	2.47671i	−0.564299	−	0.825570i
$4$	−8.66046			−2.16512
$5$			6.82324i			1.36465i	0.731049	+	0.682324i	$0.239031\pi$
$5$			6.82324i			1.36465i	−0.731049	+	0.682324i	$0.760969\pi$
$6$	−8.81252	+	6.02359i	−1.46875	+	1.00393i
$7$	0.148532			0.0212189			0.0106094	−	0.999944i	$-0.496623\pi$
$7$	0.148532			0.0212189			0.0106094	+	0.999944i	$0.496623\pi$
$8$			16.5826i			2.07283i
$9$	−3.26820	+	8.38564i	−0.363133	+	0.931737i
$10$	24.2782			2.42782
$11$		−	1.80659i		−	0.164235i	−0.996623	−	0.0821177i	$-0.973832\pi$
$11$		−	1.80659i		−	0.164235i	0.996623	−	0.0821177i	$-0.0261684\pi$
$12$	14.6613	+	21.4495i	1.22177	+	1.78745i
$13$	−17.1364			−1.31818			−0.659091	−	0.752064i	$-0.729059\pi$
$13$	−17.1364			−1.31818			−0.659091	+	0.752064i	$0.729059\pi$
$14$		−	0.528501i		−	0.0377500i
$15$	16.8992	−	11.5511i	1.12661	−	0.770070i
$16$	24.3617			1.52261
$17$			18.3201i			1.07765i	0.842417	+	0.538826i	$0.181132\pi$
$17$			18.3201i			1.07765i	−0.842417	+	0.538826i	$0.818868\pi$
$18$	29.8374	+	11.6287i	1.65763	+	0.646041i
$19$	−19.6285			−1.03308			−0.516539	−	0.856264i	$-0.672780\pi$
$19$	−19.6285			−1.03308			−0.516539	+	0.856264i	$0.672780\pi$
$20$		−	59.0924i		−	2.95462i
$21$	−0.251450	−	0.367871i	−0.0119738	−	0.0175177i
$22$	−6.42813			−0.292188
$23$		−	40.3036i		−	1.75233i	−0.482012	−	0.876165i	$-0.660094\pi$
$23$		−	40.3036i		−	1.75233i	0.482012	−	0.876165i	$-0.339906\pi$
$24$	41.0704	−	28.0727i	1.71127	−	1.16970i
$25$	−21.5567			−0.862267
$26$			60.9738i			2.34515i
$27$	26.3015	−	6.10165i	0.974130	−	0.225987i
$28$	−1.28636			−0.0459413
$29$			33.8473i			1.16715i	0.812060	+	0.583574i	$0.198346\pi$
$29$			33.8473i			1.16715i	−0.812060	+	0.583574i	$0.801654\pi$
$30$	−41.1004	−	60.1300i	−1.37001	−	2.00433i
$31$	−46.0774			−1.48637			−0.743184	−	0.669087i	$-0.766685\pi$
$31$	−46.0774			−1.48637			−0.743184	+	0.669087i	$0.766685\pi$
$32$		−	20.3523i		−	0.636008i
$33$	−4.47440	+	3.05837i	−0.135588	+	0.0926779i
$34$	65.1857			1.91723
$35$			1.01347i			0.0289563i
$36$	28.3041	−	72.6235i	0.786224	−	2.01732i
$37$	40.6329			1.09819			0.549093	−	0.835761i	$-0.314973\pi$
$37$	40.6329			1.09819			0.549093	+	0.835761i	$0.314973\pi$
$38$			69.8412i			1.83793i
$39$	29.0101	+	42.4418i	0.743849	+	1.08825i
$40$	−113.147			−2.82868
$41$		−	21.3927i		−	0.521772i	−0.965370	−	0.260886i	$-0.915985\pi$
$41$		−	21.3927i		−	0.521772i	0.965370	−	0.260886i	$-0.0840147\pi$
$42$	−1.30894	+	0.894697i	−0.0311653	+	0.0213023i
$43$	4.07233			0.0947053			0.0473527	−	0.998878i	$-0.484922\pi$
$43$	4.07233			0.0947053			0.0473527	+	0.998878i	$0.484922\pi$
$44$			15.6459i			0.355589i
$45$	−57.2172	−	22.2997i	−1.27149	−	0.495549i
$46$	−143.406			−3.11753
$47$			45.7277i			0.972930i	0.873700	+	0.486465i	$0.161714\pi$
$47$			45.7277i			0.972930i	−0.873700	+	0.486465i	$0.838286\pi$
$48$	−41.2419	−	60.3370i	−0.859207	−	1.25702i
$49$	−48.9779			−0.999550
$50$			76.7019i			1.53404i
$51$	45.3736	−	31.0140i	0.889678	−	0.608118i
$52$	148.409			2.85401
$53$			29.4395i			0.555462i	0.960659	+	0.277731i	$0.0895823\pi$
$53$			29.4395i			0.555462i	−0.960659	+	0.277731i	$0.910418\pi$
$54$	−21.7106	−	93.5849i	−0.402048	−	1.73305i
$55$	12.3268			0.224124
$56$			2.46306i			0.0439831i
$57$	33.2290	+	48.6141i	0.582965	+	0.852879i
$58$	120.434			2.07645
$59$		−	7.68115i		−	0.130189i
$59$		−	7.68115i		−	0.130189i
$60$	−146.355	+	100.037i	−2.43925	+	1.66729i
$61$	−28.5354			−0.467793			−0.233897	−	0.972261i	$-0.575148\pi$
$61$	−28.5354			−0.467793			−0.233897	+	0.972261i	$0.575148\pi$
$62$			163.950i			2.64436i
$63$	−0.485432	+	1.24554i	−0.00770528	+	0.0197704i
$64$	25.0305			0.391101
$65$		−	116.926i		−	1.79885i
$66$	10.8822	+	15.9206i	0.164881	+	0.241221i
$67$	−6.55405			−0.0978217			−0.0489108	−	0.998803i	$-0.515575\pi$
$67$	−6.55405			−0.0978217			−0.0489108	+	0.998803i	$0.515575\pi$
$68$		−	158.660i		−	2.33324i
$69$	−99.8203	+	68.2298i	−1.44667	+	0.988838i
$70$	3.60609			0.0515155
$71$			11.4894i			0.161823i	0.996721	+	0.0809115i	$0.0257831\pi$
$71$			11.4894i			0.161823i	−0.996721	+	0.0809115i	$0.974217\pi$
$72$	−139.056	−	54.1953i	−1.93133	−	0.752712i
$73$	−113.523			−1.55510			−0.777552	−	0.628818i	$-0.783539\pi$
$73$	−113.523			−1.55510			−0.777552	+	0.628818i	$0.783539\pi$
$74$		−	144.578i		−	1.95376i
$75$	36.4932	+	53.3896i	0.486576	+	0.711862i
$76$	169.992			2.23673
$77$		−	0.268337i		−	0.00348489i
$78$	151.014	−	103.222i	1.93608	−	1.32336i
$79$	−32.0057			−0.405135			−0.202568	−	0.979268i	$-0.564929\pi$
$79$	−32.0057			−0.405135			−0.202568	+	0.979268i	$0.564929\pi$
$80$			166.226i			2.07783i
$81$	−59.6378	−	54.8118i	−0.736269	−	0.676689i
$82$	−76.1184			−0.928273
$83$		−	145.911i		−	1.75797i	−0.476854	−	0.878983i	$-0.658223\pi$
$83$		−	145.911i		−	1.75797i	0.476854	−	0.878983i	$-0.341777\pi$
$84$	2.17767	+	3.18594i	0.0259247	+	0.0379278i
$85$	−125.002			−1.47062
$86$		−	14.4900i		−	0.168488i
$87$	83.8300	−	57.3000i	0.963563	−	0.658621i
$88$	29.9580			0.340432
$89$		−	148.358i		−	1.66695i	−0.552560	−	0.833473i	$-0.686349\pi$
$89$		−	148.358i		−	1.66695i	0.552560	−	0.833473i	$-0.313651\pi$
$90$	−79.3458	+	203.588i	−0.881620	+	2.26209i
$91$	−2.54530			−0.0279703
$92$			349.048i			3.79399i
$93$	78.0043	+	114.120i	0.838756	+	1.22710i
$94$	162.706			1.73092
$95$		−	133.930i		−	1.40979i
$96$	−50.4066	+	34.4543i	−0.525069	+	0.358899i
$97$	−133.709			−1.37844			−0.689222	−	0.724550i	$-0.742048\pi$
$97$	−133.709			−1.37844			−0.689222	+	0.724550i	$0.742048\pi$
$98$			174.271i			1.77828i
$99$	15.1494	+	5.90429i	0.153024	+	0.0596393i
$100$	186.691			1.86691
$101$		−	98.8253i		−	0.978468i	−0.872152	−	0.489234i	$-0.837276\pi$
$101$		−	98.8253i		−	0.978468i	0.872152	−	0.489234i	$-0.162724\pi$
$102$	−110.353	−	161.446i	−1.08189	−	1.58281i
$103$	9.65442			0.0937322			0.0468661	−	0.998901i	$-0.485077\pi$
$103$	9.65442			0.0937322			0.0468661	+	0.998901i	$0.485077\pi$
$104$		−	284.166i		−	2.73236i
$105$	2.51008	−	1.71570i	0.0239055	−	0.0163400i
$106$	104.750			0.988209
$107$			179.075i			1.67360i	0.547510	+	0.836799i	$0.315576\pi$
$107$			179.075i			1.67360i	−0.547510	+	0.836799i	$0.684424\pi$
$108$	−227.783	+	52.8431i	−2.10910	+	0.489288i
$109$	173.922			1.59561			0.797805	−	0.602915i	$-0.205994\pi$
$109$	173.922			1.59561			0.797805	+	0.602915i	$0.205994\pi$
$110$		−	43.8607i		−	0.398733i
$111$	−68.7873	−	100.636i	−0.619705	−	0.906629i
$112$	3.61850			0.0323081
$113$			119.474i			1.05729i	0.848843	+	0.528646i	$0.177300\pi$
$113$			119.474i			1.05729i	−0.848843	+	0.528646i	$0.822700\pi$
$114$	172.976	−	118.234i	1.51734	−	1.03714i
$115$	275.001			2.39131
$116$		−	293.133i		−	2.52701i
$117$	56.0050	−	143.699i	0.478675	−	1.22820i
$118$	−27.3307			−0.231616
$119$			2.72112i			0.0228666i
$120$	191.547	+	280.233i	1.59622	+	2.33528i
$121$	117.736			0.973027
$122$			101.533i			0.832241i
$123$	−52.9835	+	36.2156i	−0.430760	+	0.294436i
$124$	399.051			3.21816
$125$			23.4947i			0.187958i
$126$	4.43181	+	1.72724i	0.0351731	+	0.0137083i
$127$	4.26470			0.0335803			0.0167902	−	0.999859i	$-0.494655\pi$
$127$	4.26470			0.0335803			0.0167902	+	0.999859i	$0.494655\pi$
$128$		−	170.471i		−	1.33181i
$129$	−6.89404	−	10.0860i	−0.0534421	−	0.0781859i
$130$	−416.039			−3.20030
$131$			177.455i			1.35462i	0.735697	+	0.677311i	$0.236855\pi$
$131$			177.455i			1.35462i	−0.735697	+	0.677311i	$0.763145\pi$
$132$	38.7504	−	26.4869i	0.293563	−	0.200658i
$133$	−2.91546			−0.0219208
$134$			23.3203i			0.174032i
$135$	41.6330	+	179.462i	0.308393	+	1.32935i
$136$	−303.795			−2.23379
$137$			75.8430i			0.553599i	0.960928	+	0.276799i	$0.0892737\pi$
$137$			75.8430i			0.553599i	−0.960928	+	0.276799i	$0.910726\pi$
$138$	242.772	+	355.176i	1.75922	+	2.57374i
$139$	−75.6188			−0.544020			−0.272010	−	0.962294i	$-0.587688\pi$
$139$	−75.6188			−0.544020			−0.272010	+	0.962294i	$0.587688\pi$
$140$		−	8.77713i		−	0.0626938i
$141$	113.254	−	77.4123i	0.803222	−	0.549024i
$142$	40.8812			0.287896
$143$			30.9584i			0.216492i
$144$	−79.6189	+	204.289i	−0.552909	+	1.41867i
$145$	−230.948			−1.59275
$146$			403.931i			2.76665i
$147$	82.9146	+	121.304i	0.564045	+	0.825199i
$148$	−351.899			−2.37770
$149$		−	186.834i		−	1.25392i	−0.779051	−	0.626960i	$-0.784299\pi$
$149$		−	186.834i		−	1.25392i	0.779051	−	0.626960i	$-0.215701\pi$
$150$	189.969	−	129.849i	1.26646	−	0.865657i
$151$	−59.5580			−0.394424			−0.197212	−	0.980361i	$-0.563189\pi$
$151$	−59.5580			−0.394424			−0.197212	+	0.980361i	$0.563189\pi$
$152$		−	325.492i		−	2.14140i
$153$	−153.626	−	59.8736i	−1.00409	−	0.391331i
$154$	−0.954784			−0.00619989
$155$		−	314.397i		−	2.02837i
$156$	−251.241	−	367.566i	−1.61052	−	2.35619i
$157$	163.179			1.03936			0.519679	−	0.854362i	$-0.326052\pi$
$157$	163.179			1.03936			0.519679	+	0.854362i	$0.326052\pi$
$158$			113.881i			0.720767i
$159$	72.9131	−	49.8380i	0.458573	−	0.313447i
$160$	138.868			0.867927
$161$		−	5.98638i		−	0.0371825i
$162$	−195.029	+	212.200i	−1.20388	+	1.30988i
$163$	121.738			0.746856			0.373428	−	0.927659i	$-0.378182\pi$
$163$	121.738			0.746856			0.373428	+	0.927659i	$0.378182\pi$
$164$			185.270i			1.12970i
$165$	−20.8680	−	30.5299i	−0.126473	−	0.185030i
$166$	−519.174			−3.12756
$167$			53.8804i			0.322637i	0.986902	+	0.161319i	$0.0515747\pi$
$167$			53.8804i			0.322637i	−0.986902	+	0.161319i	$0.948425\pi$
$168$	6.10028	−	4.16970i	0.0363112	−	0.0248196i
$169$	124.655			0.737601
$170$			444.778i			2.61634i
$171$	64.1497	−	164.597i	0.375145	−	0.962558i
$172$	−35.2682			−0.205048
$173$		−	116.857i		−	0.675472i	−0.941241	−	0.337736i	$-0.890339\pi$
$173$		−	116.857i		−	0.675472i	0.941241	−	0.337736i	$-0.109661\pi$
$174$	−203.882	−	298.280i	−1.17174	−	1.71425i
$175$	−3.20186			−0.0182963
$176$		−	44.0117i		−	0.250066i
$177$	−19.0240	+	13.0034i	−0.107480	+	0.0734655i
$178$	−527.881			−2.96563
$179$			200.408i			1.11960i	0.828628	+	0.559800i	$0.189122\pi$
$179$			200.408i			1.11960i	−0.828628	+	0.559800i	$0.810878\pi$
$180$	495.528	+	193.126i	2.75293	+	1.07292i
$181$	−188.879			−1.04353			−0.521765	−	0.853089i	$-0.674726\pi$
$181$	−188.879			−1.04353			−0.521765	+	0.853089i	$0.674726\pi$
$182$			9.05657i			0.0497614i
$183$	48.3075	+	70.6739i	0.263975	+	0.386196i
$184$	668.340			3.63228
$185$			277.248i			1.49864i
$186$	406.058	−	277.551i	2.18311	−	1.49221i
$187$	33.0969			0.176989
$188$		−	396.023i		−	2.10651i
$189$	3.90662	−	0.906291i	0.0206700	−	0.00479519i
$190$	−476.543			−2.50812
$191$		−	334.562i		−	1.75163i	−0.482646	−	0.875816i	$-0.660324\pi$
$191$		−	334.562i		−	1.75163i	0.482646	−	0.875816i	$-0.339676\pi$
$192$	−42.3740	−	61.9933i	−0.220698	−	0.322882i
$193$	258.231			1.33798			0.668991	−	0.743270i	$-0.266726\pi$
$193$	258.231			1.33798			0.668991	+	0.743270i	$0.266726\pi$
$194$			475.757i			2.45236i
$195$	−289.591	+	197.943i	−1.48508	+	1.01509i
$196$	424.171			2.16414
$197$			88.8263i			0.450895i	0.974255	+	0.225447i	$0.0723844\pi$
$197$			88.8263i			0.450895i	−0.974255	+	0.225447i	$0.927616\pi$
$198$	21.0084	−	53.9039i	0.106103	−	0.272242i
$199$	−49.8385			−0.250445			−0.125222	−	0.992129i	$-0.539964\pi$
$199$	−49.8385			−0.250445			−0.125222	+	0.992129i	$0.539964\pi$
$200$		−	357.466i		−	1.78733i
$201$	11.0953	+	16.2325i	0.0552007	+	0.0807587i
$202$	−351.636			−1.74077
$203$			5.02742i			0.0247656i
$204$	−392.956	+	268.596i	−1.92625	+	1.31665i
$205$	145.967			0.712036
$206$		−	34.3519i		−	0.166757i
$207$	337.971	+	131.720i	1.63271	+	0.636328i
$208$	−417.471			−2.00707
$209$			35.4606i			0.169668i
$210$	−6.10474	−	8.93124i	−0.0290702	−	0.0425297i
$211$	−162.127			−0.768375			−0.384187	−	0.923255i	$-0.625518\pi$
$211$	−162.127			−0.768375			−0.384187	+	0.923255i	$0.625518\pi$
$212$		−	254.959i		−	1.20264i
$213$	28.4560	−	19.4504i	0.133596	−	0.0913166i
$214$	637.176			2.97746
$215$			27.7865i			0.129240i
$216$	101.181	+	436.149i	0.468432	+	2.01921i
$217$	−6.84398			−0.0315391
$218$		−	618.840i		−	2.83871i
$219$	192.182	+	281.163i	0.877544	+	1.28385i
$220$	−106.756			−0.485254
$221$		−	313.939i		−	1.42054i
$222$	−358.078	+	244.756i	−1.61296	+	1.10250i
$223$	−132.740			−0.595248			−0.297624	−	0.954683i	$-0.596194\pi$
$223$	−132.740			−0.595248			−0.297624	+	0.954683i	$0.596194\pi$
$224$		−	3.02297i		−	0.0134954i
$225$	70.4514	−	180.766i	0.313117	−	0.803406i
$226$	425.107			1.88100
$227$			104.312i			0.459525i	0.973247	+	0.229762i	$0.0737949\pi$
$227$			104.312i			0.459525i	−0.973247	+	0.229762i	$0.926205\pi$
$228$	−287.779	−	421.020i	−1.26219	−	1.84658i
$229$	62.2509			0.271838			0.135919	−	0.990720i	$-0.456601\pi$
$229$	62.2509			0.271838			0.135919	+	0.990720i	$0.456601\pi$
$230$		−	978.496i		−	4.25433i
$231$	−0.664593	+	0.454267i	−0.00287703	+	0.00196652i
$232$	−561.277			−2.41930
$233$		−	18.0515i		−	0.0774742i	−0.999249	−	0.0387371i	$-0.987667\pi$
$233$		−	18.0515i		−	0.0774742i	0.999249	−	0.0387371i	$-0.0123335\pi$
$234$	−511.304	−	199.274i	−2.18506	−	0.851599i
$235$	−312.011			−1.32771
$236$			66.5223i			0.281874i
$237$	54.1824	+	79.2689i	0.228618	+	0.334468i
$238$	9.68217			0.0406814
$239$			17.4492i			0.0730091i	0.999333	+	0.0365045i	$0.0116223\pi$
$239$			17.4492i			0.0730091i	−0.999333	+	0.0365045i	$0.988378\pi$
$240$	411.694	−	281.404i	1.71539	−	1.17252i
$241$	−211.523			−0.877689			−0.438845	−	0.898563i	$-0.644612\pi$
$241$	−211.523			−0.877689			−0.438845	+	0.898563i	$0.644612\pi$
$242$		−	418.924i		−	1.73109i
$243$	−34.7923	+	240.496i	−0.143178	+	0.989697i
$244$	247.130			1.01283
$245$		−	334.188i		−	1.36403i
$246$	128.861	+	188.523i	0.523824	+	0.766355i
$247$	336.361			1.36178
$248$		−	764.085i		−	3.08099i
$249$	−361.380	+	247.013i	−1.45132	+	0.992018i
$250$	83.5978			0.334391
$251$			127.331i			0.507295i	0.967297	+	0.253647i	$0.0816303\pi$
$251$			127.331i			0.507295i	−0.967297	+	0.253647i	$0.918370\pi$
$252$	4.20407	−	10.7869i	0.0166828	−	0.0428053i
$253$	−72.8120			−0.287795
$254$		−	15.1745i		−	0.0597419i
$255$	211.616	+	309.595i	0.829868	+	1.21410i
$256$	−506.441			−1.97829
$257$			136.584i			0.531457i	0.964048	+	0.265729i	$0.0856125\pi$
$257$			136.584i			0.531457i	−0.964048	+	0.265729i	$0.914388\pi$
$258$	−35.8875	+	24.5300i	−0.139099	+	0.0950777i
$259$	6.03529			0.0233023
$260$			1012.63i			3.89473i
$261$	−283.831	−	110.620i	−1.08748	−	0.423830i
$262$	631.414			2.40998
$263$			232.681i			0.884718i	0.896838	+	0.442359i	$0.145858\pi$
$263$			232.681i			0.884718i	−0.896838	+	0.442359i	$0.854142\pi$
$264$	−50.7159	−	74.1974i	−0.192106	−	0.281051i
$265$	−200.873			−0.758010
$266$			10.3737i			0.0389987i
$267$	−367.440	+	251.155i	−1.37618	+	0.940656i
$268$	56.7611			0.211795
$269$		−	139.297i		−	0.517832i	−0.965900	−	0.258916i	$-0.916635\pi$
$269$		−	139.297i		−	0.517832i	0.965900	−	0.258916i	$-0.0833652\pi$
$270$	638.552	−	148.137i	2.36501	−	0.548655i
$271$	340.939			1.25808			0.629038	−	0.777375i	$-0.283449\pi$
$271$	340.939			1.25808			0.629038	+	0.777375i	$0.283449\pi$
$272$			446.309i			1.64084i
$273$	4.30893	+	6.30397i	0.0157836	+	0.0230915i
$274$	269.861			0.984895
$275$			38.9441i			0.141615i
$276$	864.490	−	590.902i	3.13221	−	2.14095i
$277$	−181.171			−0.654047			−0.327024	−	0.945016i	$-0.606046\pi$
$277$	−181.171			−0.654047			−0.327024	+	0.945016i	$0.606046\pi$
$278$			269.063i			0.967853i
$279$	150.590	−	386.388i	0.539749	−	1.38490i
$280$	−16.8060			−0.0600215
$281$			382.735i			1.36205i	0.732262	+	0.681023i	$0.238465\pi$
$281$			382.735i			1.36205i	−0.732262	+	0.681023i	$0.761535\pi$
$282$	−275.445	−	402.976i	−0.976755	−	1.42899i
$283$	−35.7887			−0.126462			−0.0632309	−	0.997999i	$-0.520140\pi$
$283$	−35.7887			−0.126462			−0.0632309	+	0.997999i	$0.520140\pi$
$284$		−	99.5038i		−	0.350365i
$285$	−331.706	+	226.730i	−1.16388	+	0.795543i
$286$	110.155			0.385156
$287$		−	3.17750i		−	0.0110714i
$288$	170.667	+	66.5151i	0.592592	+	0.230955i
$289$	−46.6255			−0.161334
$290$			821.750i			2.83362i
$291$	226.356	+	331.159i	0.777855	+	1.13800i
$292$	983.158			3.36698
$293$			243.685i			0.831688i	0.909436	+	0.415844i	$0.136514\pi$
$293$			243.685i			0.831688i	−0.909436	+	0.415844i	$0.863486\pi$
$294$	431.619	−	295.023i	1.46809	−	1.00348i
$295$	52.4103			0.177662
$296$			673.800i			2.27635i
$297$	−11.0232	−	47.5161i	−0.0371151	−	0.159987i
$298$	−664.785			−2.23082
$299$			690.656i			2.30989i
$300$	−316.048	−	462.379i	−1.05349	−	1.54126i
$301$	0.604872			0.00200954
$302$			211.917i			0.701711i
$303$	−244.762	+	167.301i	−0.807794	+	0.552149i
$304$	−478.184			−1.57297
$305$		−	194.704i		−	0.638374i
$306$	−213.040	+	546.623i	−0.696208	+	1.78635i
$307$	153.107			0.498718			0.249359	−	0.968411i	$-0.419780\pi$
$307$	153.107			0.498718			0.249359	+	0.968411i	$0.419780\pi$
$308$			2.32392i			0.00754520i
$309$	−16.3439	−	23.9112i	−0.0528930	−	0.0773825i
$310$	−1118.67			−3.60863
$311$		−	173.317i		−	0.557290i	−0.960394	−	0.278645i	$-0.910115\pi$
$311$		−	173.317i		−	0.557290i	0.960394	−	0.278645i	$-0.0898853\pi$
$312$	−703.797	+	481.064i	−2.25576	+	1.54187i
$313$	440.611			1.40770			0.703851	−	0.710347i	$-0.251462\pi$
$313$	440.611			1.40770			0.703851	+	0.710347i	$0.251462\pi$
$314$		−	580.617i		−	1.84910i
$315$	−8.49860	−	3.31222i	−0.0269797	−	0.0105150i
$316$	277.184			0.877165
$317$			122.299i			0.385800i	0.981218	+	0.192900i	$0.0617894\pi$
$317$			122.299i			0.385800i	−0.981218	+	0.192900i	$0.938211\pi$
$318$	−177.331	−	259.436i	−0.557646	−	0.815836i
$319$	61.1482			0.191687
$320$			170.789i			0.533716i
$321$	443.517	−	303.156i	1.38167	−	0.944410i
$322$	−21.3005			−0.0661505
$323$		−	359.596i		−	1.11330i
$324$	516.491	+	474.695i	1.59411	+	1.46511i
$325$	369.403			1.13662
$326$		−	433.161i		−	1.32872i
$327$	−294.431	−	430.753i	−0.900402	−	1.31729i
$328$	354.747			1.08155
$329$			6.79204i			0.0206445i
$330$	−108.630	+	74.2516i	−0.329182	+	0.225005i
$331$	198.661			0.600183			0.300092	−	0.953910i	$-0.402983\pi$
$331$	198.661			0.600183			0.300092	+	0.953910i	$0.402983\pi$
$332$			1263.66i			3.80620i
$333$	−132.796	+	340.732i	−0.398787	+	1.02322i
$334$	191.715			0.573996
$335$		−	44.7199i		−	0.133492i
$336$	−6.12575	−	8.96198i	−0.0182314	−	0.0266726i
$337$	−31.0112			−0.0920215			−0.0460108	−	0.998941i	$-0.514651\pi$
$337$	−31.0112			−0.0920215			−0.0460108	+	0.998941i	$0.514651\pi$
$338$		−	443.540i		−	1.31225i
$339$	295.902	−	202.257i	0.872868	−	0.596629i
$340$	1082.58			3.18405
$341$			83.2430i			0.244114i
$342$	−585.663	−	228.255i	−1.71246	−	0.667411i
$343$	−14.5529			−0.0424282
$344$			67.5300i			0.196308i
$345$	−465.549	−	681.098i	−1.34942	−	1.97420i
$346$	−415.794			−1.20172
$347$		−	384.717i		−	1.10869i	−0.832286	−	0.554347i	$-0.812968\pi$
$347$		−	384.717i		−	1.10869i	0.832286	−	0.554347i	$-0.187032\pi$
$348$	−726.006	+	496.245i	−2.08622	+	1.42599i
$349$	122.340			0.350545			0.175273	−	0.984520i	$-0.443919\pi$
$349$	122.340			0.350545			0.175273	+	0.984520i	$0.443919\pi$
$350$			11.3927i			0.0325506i
$351$	−450.712	+	104.560i	−1.28408	+	0.297892i
$352$	−36.7682			−0.104455
$353$			659.650i			1.86870i	0.356360	+	0.934349i	$0.384018\pi$
$353$			659.650i			1.86870i	−0.356360	+	0.934349i	$0.615982\pi$
$354$	46.2681	+	67.6902i	0.130701	+	0.191215i
$355$	−78.3952			−0.220832
$356$			1284.85i			3.60913i
$357$	6.73944	−	4.60658i	0.0188780	−	0.0129036i
$358$	713.084			1.99186
$359$		−	107.767i		−	0.300186i	−0.988672	−	0.150093i	$-0.952043\pi$
$359$		−	107.767i		−	0.300186i	0.988672	−	0.150093i	$-0.0479573\pi$
$360$	369.788	−	948.813i	1.02719	−	2.63559i
$361$	24.2775			0.0672507
$362$			672.061i			1.85652i
$363$	−199.315	−	291.599i	−0.549078	−	0.803302i
$364$	22.0435			0.0605590
$365$		−	774.593i		−	2.12217i
$366$	251.469	−	171.886i	0.687073	−	0.469633i
$367$	−606.039			−1.65133			−0.825666	−	0.564159i	$-0.809201\pi$
$367$	−606.039			−1.65133			−0.825666	+	0.564159i	$0.809201\pi$
$368$		−	981.865i		−	2.66811i
$369$	179.391	+	69.9154i	0.486155	+	0.189473i
$370$	986.491			2.66619
$371$			4.37271i			0.0117863i
$372$	−675.553	−	988.335i	−1.81600	−	2.65681i
$373$	−42.6708			−0.114399			−0.0571995	−	0.998363i	$-0.518217\pi$
$373$	−42.6708			−0.114399			−0.0571995	+	0.998363i	$0.518217\pi$
$374$		−	117.764i		−	0.314877i
$375$	58.1896	−	39.7742i	0.155172	−	0.106064i
$376$	−758.286			−2.01672
$377$		−	580.019i		−	1.53851i
$378$	−3.22472	−	13.9004i	−0.00853102	−	0.0367735i
$379$	−548.047			−1.44604			−0.723018	−	0.690830i	$-0.757245\pi$
$379$	−548.047			−1.44604			−0.723018	+	0.690830i	$0.757245\pi$
$380$			1159.90i			3.05236i
$381$	−7.21970	−	10.5624i	−0.0189493	−	0.0277229i
$382$	−1190.42			−3.11629
$383$		−	435.197i		−	1.13628i	−0.822931	−	0.568142i	$-0.807663\pi$
$383$		−	435.197i		−	1.13628i	0.822931	−	0.568142i	$-0.192337\pi$
$384$	−422.208	+	288.590i	−1.09950	+	0.751538i
$385$	1.83093			0.00475566
$386$		−	918.824i		−	2.38037i
$387$	−13.3092	+	34.1491i	−0.0343906	+	0.0882405i
$388$	1157.98			2.98449
$389$			227.077i			0.583746i	0.956457	+	0.291873i	$0.0942784\pi$
$389$			227.077i			0.583746i	−0.956457	+	0.291873i	$0.905722\pi$
$390$	704.312	+	1030.41i	1.80593	+	2.64207i
$391$	738.365			1.88840
$392$		−	812.183i		−	2.07190i
$393$	439.506	−	300.414i	1.11834	−	0.764412i
$394$	316.058			0.802177
$395$		−	218.383i		−	0.552868i
$396$	−131.201	−	51.1339i	−0.331315	−	0.129126i
$397$	740.164			1.86439			0.932197	−	0.361952i	$-0.117890\pi$
$397$	740.164			1.86439			0.932197	+	0.361952i	$0.117890\pi$
$398$			177.333i			0.445560i
$399$	4.93558	+	7.22076i	0.0123699	+	0.0180971i
$400$	−525.158			−1.31289
$401$			306.471i			0.764268i	0.924107	+	0.382134i	$0.124811\pi$
$401$			306.471i			0.764268i	−0.924107	+	0.382134i	$0.875189\pi$
$402$	57.7577	−	39.4789i	0.143676	−	0.0982063i
$403$	789.599			1.95930
$404$			855.873i			2.11850i
$405$	373.994	−	406.923i	0.923443	−	1.00475i
$406$	17.8883			0.0440599
$407$		−	73.4069i		−	0.180361i
$408$	514.294	+	752.413i	1.26053	+	1.84415i
$409$	−572.519			−1.39980			−0.699901	−	0.714240i	$-0.746772\pi$
$409$	−572.519			−1.39980			−0.699901	+	0.714240i	$0.746772\pi$
$410$		−	519.374i		−	1.26677i
$411$	187.841	−	128.394i	0.457035	−	0.312395i
$412$	−83.6117			−0.202941
$413$		−	1.14090i		−	0.00276246i
$414$	468.680	−	1202.55i	1.13208	−	2.90472i
$415$	995.587			2.39901
$416$			348.763i			0.838374i
$417$	128.015	+	187.286i	0.306990	+	0.449127i
$418$	126.174			0.301853
$419$			50.8652i			0.121397i	0.998156	+	0.0606984i	$0.0193328\pi$
$419$			50.8652i			0.121397i	−0.998156	+	0.0606984i	$0.980667\pi$
$420$	−21.7384	+	14.8588i	−0.0517581	+	0.0353781i
$421$	−320.698			−0.761753			−0.380876	−	0.924626i	$-0.624378\pi$
$421$	−320.698			−0.761753			−0.380876	+	0.924626i	$0.624378\pi$
$422$			576.873i			1.36700i
$423$	−383.456	−	149.447i	−0.906515	−	0.353303i
$424$	−488.184			−1.15138
$425$		−	394.920i		−	0.929223i
$426$	−69.2076	−	101.251i	−0.162459	−	0.237678i
$427$	−4.23843			−0.00992606
$428$		−	1550.87i		−	3.62353i
$429$	76.6749	−	52.4093i	0.178729	−	0.122166i
$430$	98.8686			0.229927
$431$		−	137.310i		−	0.318584i	−0.987232	−	0.159292i	$-0.949079\pi$
$431$		−	137.310i		−	0.318584i	0.987232	−	0.159292i	$-0.0509211\pi$
$432$	640.751	−	148.647i	1.48322	−	0.344090i
$433$	−721.580			−1.66647			−0.833233	−	0.552921i	$-0.813513\pi$
$433$	−721.580			−1.66647			−0.833233	+	0.552921i	$0.813513\pi$
$434$			24.3519i			0.0561104i
$435$	390.972	+	571.993i	0.898786	+	1.31493i
$436$	−1506.24			−3.45468
$437$			791.098i			1.81029i
$438$	1000.42	−	683.814i	2.28407	−	1.56122i
$439$	−288.042			−0.656132			−0.328066	−	0.944655i	$-0.606397\pi$
$439$	−288.042			−0.656132			−0.328066	+	0.944655i	$0.606397\pi$
$440$			204.411i			0.464570i
$441$	160.069	−	410.711i	0.362969	−	0.931318i
$442$	−1117.05			−2.52725
$443$		−	656.684i		−	1.48236i	−0.671308	−	0.741178i	$-0.734267\pi$
$443$		−	656.684i		−	1.48236i	0.671308	−	0.741178i	$-0.265733\pi$
$444$	595.730	+	871.553i	1.34173	+	1.96296i
$445$	1012.28			2.27480
$446$			472.311i			1.05899i
$447$	−462.734	+	316.291i	−1.03520	+	0.707586i
$448$	3.71783			0.00829873
$449$		−	265.029i		−	0.590266i	−0.955456	−	0.295133i	$-0.904636\pi$
$449$		−	265.029i		−	0.590266i	0.955456	−	0.295133i	$-0.0953640\pi$
$450$	−643.195	−	250.677i	−1.42932	−	0.557060i
$451$	−38.6478			−0.0856935
$452$		−	1034.70i		−	2.28916i
$453$	100.826	+	147.508i	0.222573	+	0.325625i
$454$	371.159			0.817530
$455$		−	17.3672i		−	0.0381697i
$456$	−806.150	+	551.025i	−1.76787	+	1.20839i
$457$	−751.980			−1.64547			−0.822736	−	0.568424i	$-0.807553\pi$
$457$	−751.980			−1.64547			−0.822736	+	0.568424i	$0.807553\pi$
$458$		−	221.498i		−	0.483621i
$459$	111.783	+	481.846i	0.243535	+	1.04977i
$460$	−2381.64			−5.17747
$461$			342.840i			0.743689i	0.928295	+	0.371844i	$0.121274\pi$
$461$			342.840i			0.743689i	−0.928295	+	0.371844i	$0.878726\pi$
$462$	1.61635	+	2.36472i	0.00349860	+	0.00511845i
$463$	−610.900			−1.31944			−0.659719	−	0.751512i	$-0.729325\pi$
$463$	−610.900			−1.31944			−0.659719	+	0.751512i	$0.729325\pi$
$464$			824.579i			1.77711i
$465$	−778.671	+	532.242i	−1.67456	+	1.14461i
$466$	−64.2300			−0.137833
$467$		−	70.7483i		−	0.151495i	−0.997127	−	0.0757477i	$-0.975866\pi$
$467$		−	70.7483i		−	0.151495i	0.997127	−	0.0757477i	$-0.0241344\pi$
$468$	−485.029	+	1244.50i	−1.03639	+	2.65919i
$469$	−0.973488			−0.00207567
$470$			1110.18i			2.36209i
$471$	−276.246	−	404.148i	−0.586509	−	0.858063i
$472$	127.374			0.269859
$473$		−	7.35703i		−	0.0155540i
$474$	282.051	−	192.789i	0.595044	−	0.406728i
$475$	423.125			0.890789
$476$		−	23.5662i		−	0.0495088i
$477$	−246.869	−	96.2140i	−0.517544	−	0.201706i
$478$	62.0868			0.129889
$479$			86.5788i			0.180749i	0.995908	+	0.0903746i	$0.0288064\pi$
$479$			86.5788i			0.180749i	−0.995908	+	0.0903746i	$0.971194\pi$
$480$	−235.090	−	343.937i	−0.489771	−	0.716535i
$481$	−696.299			−1.44761
$482$			752.632i			1.56148i
$483$	−14.8265	+	10.1343i	−0.0306968	+	0.0209820i
$484$	−1019.65			−2.10671
$485$		−	912.330i		−	1.88109i
$486$	855.723	+	123.796i	1.76075	+	0.254725i
$487$	167.140			0.343203			0.171601	−	0.985166i	$-0.445106\pi$
$487$	167.140			0.343203			0.171601	+	0.985166i	$0.445106\pi$
$488$		−	473.192i		−	0.969656i
$489$	−206.089	−	301.509i	−0.421450	−	0.616583i
$490$	−1189.09			−2.42672
$491$			732.046i			1.49093i	0.666546	+	0.745464i	$0.267772\pi$
$491$			732.046i			1.49093i	−0.666546	+	0.745464i	$0.732228\pi$
$492$	458.861	−	313.644i	0.932645	−	0.637487i
$493$	−620.085			−1.25778
$494$		−	1196.82i		−	2.42272i
$495$	−40.2864	+	103.368i	−0.0813867	+	0.208824i
$496$	−1122.53			−2.26316
$497$			1.70655i			0.00343370i
$498$	878.909	+	1285.84i	1.76488	+	2.58202i
$499$	756.899			1.51683			0.758416	−	0.651771i	$-0.225974\pi$
$499$	756.899			1.51683			0.758416	+	0.651771i	$0.225974\pi$
$500$		−	203.475i		−	0.406950i
$501$	133.446	−	91.2140i	0.266360	−	0.182064i
$502$	453.063			0.902516
$503$			424.962i			0.844856i	0.906397	+	0.422428i	$0.138822\pi$
$503$			424.962i			0.844856i	−0.906397	+	0.422428i	$0.861178\pi$
$504$	−20.6543	−	8.04975i	−0.0409807	−	0.0159717i
$505$	674.309			1.33527
$506$			259.076i			0.512009i
$507$	−211.028	−	308.734i	−0.416228	−	0.608942i
$508$	−36.9343			−0.0727052
$509$		−	163.430i		−	0.321081i	−0.987029	−	0.160540i	$-0.948676\pi$
$509$		−	163.430i		−	0.321081i	0.987029	−	0.160540i	$-0.0513237\pi$
$510$	1101.59	−	752.963i	2.15997	−	1.47640i
$511$	−16.8618			−0.0329976
$512$			1120.11i			2.18772i
$513$	−516.259	+	119.766i	−1.00635	+	0.233462i
$514$	485.989			0.945503
$515$			65.8744i			0.127912i
$516$	59.7055	+	87.3493i	0.115708	+	0.169282i
$517$	82.6112			0.159790
$518$		−	21.4745i		−	0.0414565i
$519$	−289.420	+	197.826i	−0.557650	+	0.381168i
$520$	1938.93			3.72872
$521$		−	643.967i		−	1.23602i	−0.786169	−	0.618011i	$-0.787939\pi$
$521$		−	643.967i		−	1.23602i	0.786169	−	0.618011i	$-0.212061\pi$
$522$	−393.602	+	1009.92i	−0.754026	+	1.93470i
$523$	371.188			0.709729			0.354864	−	0.934918i	$-0.384527\pi$
$523$	371.188			0.709729			0.354864	+	0.934918i	$0.384527\pi$
$524$		−	1536.85i		−	2.93291i
$525$	5.42042	+	7.93008i	0.0103246	+	0.0151049i
$526$	827.914			1.57398
$527$		−	844.142i		−	1.60179i
$528$	−109.004	+	74.5072i	−0.206447	+	0.141112i
$529$	−1095.38			−2.07066
$530$			714.736i			1.34856i
$531$	64.4113	+	25.1035i	0.121302	+	0.0472759i
$532$	25.2493			0.0474610
$533$			366.592i			0.687790i
$534$	893.649	+	1307.41i	1.67350	+	2.44833i
$535$	−1221.87			−2.28387
$536$		−	108.683i		−	0.202768i
$537$	496.354	−	339.271i	0.924309	−	0.631790i
$538$	−495.639			−0.921262
$539$			88.4831i			0.164162i
$540$	−360.561	−	1554.22i	−0.667706	−	2.87819i
$541$	175.446			0.324300			0.162150	−	0.986766i	$-0.448157\pi$
$541$	175.446			0.324300			0.162150	+	0.986766i	$0.448157\pi$
$542$		−	1213.11i		−	2.23821i
$543$	319.753	+	467.799i	0.588864	+	0.861508i
$544$	372.855			0.685395
$545$			1186.71i			2.17745i
$546$	22.4305	−	15.3318i	0.0410815	−	0.0280803i
$547$	209.300			0.382632			0.191316	−	0.981528i	$-0.438724\pi$
$547$	209.300			0.382632			0.191316	+	0.981528i	$0.438724\pi$
$548$		−	656.835i		−	1.19860i
$549$	93.2593	−	239.288i	0.169871	−	0.435861i
$550$	138.569			0.251944
$551$		−	664.371i		−	1.20576i
$552$	−1131.43	−	1655.28i	−2.04969	−	2.99870i
$553$	−4.75388			−0.00859652
$554$			644.635i			1.16360i
$555$	686.663	−	469.352i	1.23723	−	0.845680i
$556$	654.893			1.17787
$557$			46.3436i			0.0832022i	0.999134	+	0.0416011i	$0.0132459\pi$
$557$			46.3436i			0.0832022i	−0.999134	+	0.0416011i	$0.986754\pi$
$558$	−1374.83	−	535.822i	−2.46385	−	0.960255i
$559$	−69.7849			−0.124839
$560$			24.6899i			0.0440892i
$561$	−56.0296	−	81.9714i	−0.0998746	−	0.146117i
$562$	1361.83			2.42318
$563$			866.791i			1.53959i	0.638290	+	0.769796i	$0.279642\pi$
$563$			866.791i			1.53959i	−0.638290	+	0.769796i	$0.720358\pi$
$564$	−980.835	+	670.426i	−1.73907	+	1.18870i
$565$	−815.200			−1.44283
$566$			127.342i			0.224985i
$567$	−8.85813	−	8.14132i	−0.0156228	−	0.0143586i
$568$	−190.525			−0.335431
$569$			116.360i			0.204500i	0.994759	+	0.102250i	$0.0326041\pi$
$569$			116.360i			0.204500i	−0.994759	+	0.102250i	$0.967396\pi$
$570$	806.739	+	1180.26i	1.41533	+	2.07063i
$571$	487.416			0.853618			0.426809	−	0.904342i	$-0.359638\pi$
$571$	487.416			0.853618			0.426809	+	0.904342i	$0.359638\pi$
$572$		−	268.114i		−	0.468730i
$573$	−828.612	+	566.378i	−1.44609	+	0.988444i
$574$	−11.3060			−0.0196969
$575$			868.811i			1.51098i
$576$	−81.8045	+	209.897i	−0.142022	+	0.364404i
$577$	672.310			1.16518			0.582591	−	0.812766i	$-0.302039\pi$
$577$	672.310			1.16518			0.582591	+	0.812766i	$0.302039\pi$
$578$			165.901i			0.287026i
$579$	−437.158	−	639.563i	−0.755023	−	1.10460i
$580$	2000.12			3.44848
$581$		−	21.6725i		−	0.0373021i
$582$	1178.31	−	805.409i	2.02459	−	1.38386i
$583$	53.1851			0.0912265
$584$		−	1882.50i		−	3.22347i
$585$	980.495	+	382.136i	1.67606	+	0.653223i
$586$	867.067			1.47964
$587$			320.616i			0.546194i	0.961986	+	0.273097i	$0.0880480\pi$
$587$			320.616i			0.546194i	−0.961986	+	0.273097i	$0.911952\pi$
$588$	−718.079	−	1050.55i	−1.22122	−	1.78665i
$589$	904.429			1.53553
$590$		−	186.484i		−	0.316075i
$591$	219.997	−	150.374i	0.372245	−	0.254440i
$592$	989.887			1.67211
$593$			779.831i			1.31506i	0.753428	+	0.657530i	$0.228399\pi$
$593$			779.831i			1.31506i	−0.753428	+	0.657530i	$0.771601\pi$
$594$	−169.069	+	39.2222i	−0.284629	+	0.0660306i
$595$	−18.5669			−0.0312049
$596$			1618.07i			2.71488i
$597$	84.3714	+	123.436i	0.141326	+	0.206760i
$598$	2457.46			4.10947
$599$		−	315.093i		−	0.526032i	−0.964791	−	0.263016i	$-0.915283\pi$
$599$		−	315.093i		−	0.526032i	0.964791	−	0.263016i	$-0.0847172\pi$
$600$	−885.341	+	605.154i	−1.47557	+	1.00859i
$601$	−474.472			−0.789470			−0.394735	−	0.918795i	$-0.629164\pi$
$601$	−474.472			−0.789470			−0.394735	+	0.918795i	$0.629164\pi$
$602$		−	2.15223i		−	0.00357513i
$603$	21.4199	−	54.9599i	0.0355223	−	0.0911441i
$604$	515.800			0.853973
$605$			803.343i			1.32784i
$606$	595.283	+	870.900i	0.982315	+	1.43713i
$607$	368.145			0.606500			0.303250	−	0.952911i	$-0.401928\pi$
$607$	368.145			0.606500			0.303250	+	0.952911i	$0.401928\pi$
$608$			399.484i			0.657046i
$609$	12.4515	−	8.51090i	0.0204457	−	0.0139752i
$610$	−692.787			−1.13572
$611$		−	783.606i		−	1.28250i
$612$	1330.47	+	518.533i	2.17397	+	0.847276i
$613$	25.3403			0.0413382			0.0206691	−	0.999786i	$-0.493420\pi$
$613$	25.3403			0.0413382			0.0206691	+	0.999786i	$0.493420\pi$
$614$		−	544.777i		−	0.887259i
$615$	−247.108	−	361.519i	−0.401801	−	0.587836i
$616$	4.44973			0.00722359
$617$		−	410.846i		−	0.665876i	−0.942949	−	0.332938i	$-0.891960\pi$
$617$		−	410.846i		−	0.665876i	0.942949	−	0.332938i	$-0.108040\pi$
$618$	−85.0797	+	58.1543i	−0.137669	+	0.0941007i
$619$	−652.761			−1.05454			−0.527271	−	0.849697i	$-0.676785\pi$
$619$	−652.761			−1.05454			−0.527271	+	0.849697i	$0.676785\pi$
$620$			2722.83i			4.39165i
$621$	−245.918	−	1060.05i	−0.396004	−	1.70700i
$622$	−616.689			−0.991462
$623$		−	22.0360i		−	0.0353707i
$624$	706.736	+	1033.96i	1.13259	+	1.65698i
$625$	−699.227			−1.11876
$626$		−	1567.76i		−	2.50441i
$627$	87.8257	−	60.0312i	0.140073	−	0.0957436i
$628$	−1413.21			−2.25033
$629$			744.398i			1.18346i
$630$	−11.7854	+	30.2393i	−0.0187070	+	0.0479990i
$631$	−869.264			−1.37760			−0.688799	−	0.724952i	$-0.741862\pi$
$631$	−869.264			−1.37760			−0.688799	+	0.724952i	$0.741862\pi$
$632$		−	530.739i		−	0.839777i
$633$	274.465	+	401.542i	0.433593	+	0.634347i
$634$	435.158			0.686369
$635$			29.0991i			0.0458253i
$636$	−631.461	+	431.620i	−0.992863	+	0.678648i
$637$	839.303			1.31759
$638$		−	217.575i		−	0.341026i
$639$	−96.3462	−	37.5497i	−0.150777	−	0.0587632i
$640$	1163.17			1.81745
$641$			252.973i			0.394653i	0.980338	+	0.197327i	$0.0632259\pi$
$641$			252.973i			0.394653i	−0.980338	+	0.197327i	$0.936774\pi$
$642$	−1078.67	−	1578.10i	−1.68018	−	2.45810i
$643$	−1098.13			−1.70782			−0.853910	−	0.520421i	$-0.825775\pi$
$643$	−1098.13			−1.70782			−0.853910	+	0.520421i	$0.825775\pi$
$644$			51.8448i			0.0805044i
$645$	68.8191	−	47.0397i	0.106696	−	0.0729298i
$646$	−1279.50			−1.98064
$647$		−	634.640i		−	0.980896i	−0.871470	−	0.490448i	$-0.836833\pi$
$647$		−	634.640i		−	0.980896i	0.871470	−	0.490448i	$-0.163167\pi$
$648$	908.924	−	988.952i	1.40266	−	1.52616i
$649$	−13.8767			−0.0213816
$650$		−	1314.39i		−	2.02214i
$651$	11.5862	+	16.9506i	0.0177975	+	0.0260377i
$652$	−1054.30			−1.61703
$653$		−	743.777i		−	1.13901i	−0.821986	−	0.569507i	$-0.807134\pi$
$653$		−	743.777i		−	1.13901i	0.821986	−	0.569507i	$-0.192866\pi$
$654$	−1532.69	+	1047.63i	−2.34356	+	1.60188i
$655$	−1210.82			−1.84858
$656$		−	521.162i		−	0.794455i
$657$	371.014	−	951.960i	0.564710	−	1.44895i
$658$	24.1671			0.0367281
$659$		−	9.11578i		−	0.0138327i	−0.999976	−	0.00691637i	$-0.997798\pi$
$659$		−	9.11578i		−	0.0138327i	0.999976	−	0.00691637i	$-0.00220157\pi$
$660$	180.727	+	264.403i	0.273828	+	0.400611i
$661$	−350.843			−0.530776			−0.265388	−	0.964142i	$-0.585500\pi$
$661$	−350.843			−0.530776			−0.265388	+	0.964142i	$0.585500\pi$
$662$		−	706.865i		−	1.06777i
$663$	−777.537	+	531.467i	−1.17276	+	0.801610i
$664$	2419.59			3.64396
$665$		−	19.8929i		−	0.0299142i
$666$	1212.38	+	472.509i	1.82039	+	0.709473i
$667$	1364.17			2.04523
$668$		−	466.629i		−	0.698547i
$669$	224.716	+	328.759i	0.335898	+	0.491419i
$670$	−159.120			−0.237493
$671$			51.5518i			0.0768283i
$672$	−7.48701	+	5.11757i	−0.0111414	+	0.00761543i
$673$	−1036.60			−1.54027			−0.770134	−	0.637882i	$-0.779811\pi$
$673$	−1036.60			−1.54027			−0.770134	+	0.637882i	$0.779811\pi$
$674$			110.343i			0.163713i
$675$	−566.973	+	131.531i	−0.839960	+	0.194861i
$676$	−1079.57			−1.59699
$677$			446.603i			0.659680i	0.944037	+	0.329840i	$0.106995\pi$
$677$			446.603i			0.659680i	−0.944037	+	0.329840i	$0.893005\pi$
$678$	−719.662	−	1052.87i	−1.06145	−	1.55290i
$679$	−19.8601			−0.0292491
$680$		−	2072.87i		−	3.04834i
$681$	258.351	−	176.590i	0.379370	−	0.259309i
$682$	296.191			0.434298
$683$			751.761i			1.10067i	0.834942	+	0.550337i	$0.185501\pi$
$683$			751.761i			1.10067i	−0.834942	+	0.550337i	$0.814499\pi$
$684$	−555.566	+	1425.49i	−0.812231	+	2.08405i
$685$	−517.495			−0.755468
$686$			51.7814i			0.0754831i
$687$	−105.384	−	154.178i	−0.153398	−	0.224422i
$688$	99.2090			0.144199
$689$		−	504.485i		−	0.732199i
$690$	−2423.45	+	1656.49i	−3.51225	+	2.40072i
$691$	296.414			0.428964			0.214482	−	0.976728i	$-0.431194\pi$
$691$	296.414			0.428964			0.214482	+	0.976728i	$0.431194\pi$
$692$			1012.03i			1.46247i
$693$	2.25018	+	0.876977i	0.00324701	+	0.00126548i
$694$	−1368.88			−1.97245
$695$		−	515.965i		−	0.742396i
$696$	950.185	+	1390.12i	1.36521	+	1.99730i
$697$	391.915			0.562289
$698$		−	435.305i		−	0.623647i
$699$	−44.7083	+	30.5593i	−0.0639604	+	0.0437186i
$700$	27.7296			0.0396137
$701$			317.412i			0.452799i	0.974035	+	0.226400i	$0.0726955\pi$
$701$			317.412i			0.452799i	−0.974035	+	0.226400i	$0.927304\pi$
$702$	372.041	+	1603.70i	0.529972	+	2.28448i
$703$	−797.562			−1.13451
$704$		−	45.2198i		−	0.0642327i
$705$	528.203	+	772.762i	0.749224	+	1.09612i
$706$	2347.14			3.32456
$707$		−	14.6787i		−	0.0207620i
$708$	164.756	−	112.615i	0.232707	−	0.159061i
$709$	−981.184			−1.38390			−0.691950	−	0.721946i	$-0.743248\pi$
$709$	−981.184			−1.38390			−0.691950	+	0.721946i	$0.743248\pi$
$710$			278.942i			0.392876i
$711$	104.601	−	268.388i	0.147118	−	0.377480i
$712$	2460.17			3.45530
$713$			1857.08i			2.60461i
$714$	−16.3909	−	23.9800i	−0.0229565	−	0.0335854i
$715$	−211.237			−0.295436
$716$		−	1735.63i		−	2.42406i
$717$	43.2165	−	29.5397i	0.0602741	−	0.0411990i
$718$	−383.450			−0.534054
$719$		−	73.9197i		−	0.102809i	−0.998678	−	0.0514045i	$-0.983630\pi$
$719$		−	73.9197i		−	0.102809i	0.998678	−	0.0514045i	$-0.0163698\pi$
$720$	−1393.91	−	543.259i	−1.93599	−	0.754527i
$721$	1.43399			0.00198889
$722$		−	86.3831i		−	0.119644i
$723$	358.087	+	523.882i	0.495279	+	0.724594i
$724$	1635.78			2.25936
$725$		−	729.635i		−	1.00639i
$726$	−1037.55	+	709.195i	−1.42914	+	0.976852i
$727$	1232.61			1.69547			0.847735	−	0.530420i	$-0.177966\pi$
$727$	1232.61			1.69547			0.847735	+	0.530420i	$0.177966\pi$
$728$		−	42.2078i		−	0.0579777i
$729$	654.540	−	320.965i	0.897860	−	0.440281i
$730$	−2756.12			−3.77551
$731$			74.6054i			0.102059i
$732$	−418.365	−	612.069i	−0.571537	−	0.836160i
$733$	325.363			0.443878			0.221939	−	0.975061i	$-0.428761\pi$
$733$	325.363			0.443878			0.221939	+	0.975061i	$0.428761\pi$
$734$			2156.38i			2.93785i
$735$	−827.688	+	565.747i	−1.12611	+	0.769724i
$736$	−820.269			−1.11450
$737$			11.8405i			0.0160658i
$738$	248.770	−	638.301i	0.337086	−	0.864907i
$739$	411.718			0.557129			0.278564	−	0.960418i	$-0.410141\pi$
$739$	411.718			0.557129			0.278564	+	0.960418i	$0.410141\pi$
$740$		−	2401.10i		−	3.24472i
$741$	−569.424	−	833.068i	−0.768454	−	1.12425i
$742$	15.5588			0.0209687
$743$		−	31.6240i		−	0.0425626i	−0.999774	−	0.0212813i	$-0.993225\pi$
$743$		−	31.6240i		−	0.0425626i	0.999774	−	0.0212813i	$-0.00677456\pi$
$744$	−1892.42	+	1293.52i	−2.54357	+	1.73860i
$745$	1274.81			1.71116
$746$			151.829i			0.203525i
$747$	1223.56	+	476.866i	1.63796	+	0.638375i
$748$	−286.634			−0.383201
$749$			26.5984i			0.0355119i
$750$	−141.523	−	207.048i	−0.188697	−	0.276064i
$751$	−827.119			−1.10136			−0.550678	−	0.834718i	$-0.685631\pi$
$751$	−827.119			−1.10136			−0.550678	+	0.834718i	$0.685631\pi$
$752$			1114.01i			1.48139i
$753$	315.362	−	215.558i	0.418807	−	0.286266i
$754$	−2063.80			−2.73713
$755$		−	406.379i		−	0.538250i
$756$	−33.8332	+	7.84890i	−0.0447528	+	0.0103821i
$757$	422.114			0.557615			0.278807	−	0.960347i	$-0.410061\pi$
$757$	422.114			0.557615			0.278807	+	0.960347i	$0.410061\pi$
$758$			1950.04i			2.57261i
$759$	123.263	+	180.334i	0.162402	+	0.237595i
$760$	2220.91			2.92225
$761$			37.4749i			0.0492443i	0.999697	+	0.0246222i	$0.00783827\pi$
$761$			37.4749i			0.0492443i	−0.999697	+	0.0246222i	$0.992162\pi$
$762$	−37.5827	+	25.6888i	−0.0493212	+	0.0337123i
$763$	25.8330			0.0338571
$764$			2897.46i			3.79248i
$765$	408.532	−	1048.22i	0.534029	−	1.37023i
$766$	−1548.50			−2.02154
$767$			131.627i			0.171613i
$768$	857.353	+	1254.31i	1.11635	+	1.63321i
$769$	162.942			0.211888			0.105944	−	0.994372i	$-0.466214\pi$
$769$	162.942			0.211888			0.105944	+	0.994372i	$0.466214\pi$
$770$		−	6.51472i		−	0.00846068i
$771$	338.280	−	231.224i	0.438755	−	0.299901i
$772$	−2236.40			−2.89689
$773$			817.034i			1.05697i	0.848944	+	0.528483i	$0.177239\pi$
$773$			817.034i			1.05697i	−0.848944	+	0.528483i	$0.822761\pi$
$774$	121.508	+	47.3561i	0.156987	+	0.0611836i
$775$	993.275			1.28164
$776$		−	2217.25i		−	2.85728i
$777$	−10.2171	−	14.9477i	−0.0131495	−	0.0192377i
$778$	807.975			1.03853
$779$			419.906i			0.539032i
$780$	2507.99	−	1714.28i	3.21537	−	2.19779i
$781$	20.7567			0.0265771
$782$		−	2627.22i		−	3.35961i
$783$	206.524	+	890.235i	0.263760	+	1.13695i
$784$	−1193.19			−1.52192
$785$			1113.41i			1.41836i
$786$	−1068.92	−	1563.83i	−1.35995	−	1.98960i

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

\(f(q)\)	\(=\)	\(q-3.55815i q^{2} +(-1.69290 - 2.47671i) q^{3} -8.66046 q^{4} +6.82324i q^{5} +(-8.81252 + 6.02359i) q^{6} +0.148532 q^{7} +16.5826i q^{8} +(-3.26820 + 8.38564i) q^{9} +O(q^{10})\)	\(q-3.55815i q^{2} +(-1.69290 - 2.47671i) q^{3} -8.66046 q^{4} +6.82324i q^{5} +(-8.81252 + 6.02359i) q^{6} +0.148532 q^{7} +16.5826i q^{8} +(-3.26820 + 8.38564i) q^{9} +24.2782 q^{10} -1.80659i q^{11} +(14.6613 + 21.4495i) q^{12} -17.1364 q^{13} -0.528501i q^{14} +(16.8992 - 11.5511i) q^{15} +24.3617 q^{16} +18.3201i q^{17} +(29.8374 + 11.6287i) q^{18} -19.6285 q^{19} -59.0924i q^{20} +(-0.251450 - 0.367871i) q^{21} -6.42813 q^{22} -40.3036i q^{23} +(41.0704 - 28.0727i) q^{24} -21.5567 q^{25} +60.9738i q^{26} +(26.3015 - 6.10165i) q^{27} -1.28636 q^{28} +33.8473i q^{29} +(-41.1004 - 60.1300i) q^{30} -46.0774 q^{31} -20.3523i q^{32} +(-4.47440 + 3.05837i) q^{33} +65.1857 q^{34} +1.01347i q^{35} +(28.3041 - 72.6235i) q^{36} +40.6329 q^{37} +69.8412i q^{38} +(29.0101 + 42.4418i) q^{39} -113.147 q^{40} -21.3927i q^{41} +(-1.30894 + 0.894697i) q^{42} +4.07233 q^{43} +15.6459i q^{44} +(-57.2172 - 22.2997i) q^{45} -143.406 q^{46} +45.7277i q^{47} +(-41.2419 - 60.3370i) q^{48} -48.9779 q^{49} +76.7019i q^{50} +(45.3736 - 31.0140i) q^{51} +148.409 q^{52} +29.4395i q^{53} +(-21.7106 - 93.5849i) q^{54} +12.3268 q^{55} +2.46306i q^{56} +(33.2290 + 48.6141i) q^{57} +120.434 q^{58} -7.68115i q^{59} +(-146.355 + 100.037i) q^{60} -28.5354 q^{61} +163.950i q^{62} +(-0.485432 + 1.24554i) q^{63} +25.0305 q^{64} -116.926i q^{65} +(10.8822 + 15.9206i) q^{66} -6.55405 q^{67} -158.660i q^{68} +(-99.8203 + 68.2298i) q^{69} +3.60609 q^{70} +11.4894i q^{71} +(-139.056 - 54.1953i) q^{72} -113.523 q^{73} -144.578i q^{74} +(36.4932 + 53.3896i) q^{75} +169.992 q^{76} -0.268337i q^{77} +(151.014 - 103.222i) q^{78} -32.0057 q^{79} +166.226i q^{80} +(-59.6378 - 54.8118i) q^{81} -76.1184 q^{82} -145.911i q^{83} +(2.17767 + 3.18594i) q^{84} -125.002 q^{85} -14.4900i q^{86} +(83.8300 - 57.3000i) q^{87} +29.9580 q^{88} -148.358i q^{89} +(-79.3458 + 203.588i) q^{90} -2.54530 q^{91} +349.048i q^{92} +(78.0043 + 114.120i) q^{93} +162.706 q^{94} -133.930i q^{95} +(-50.4066 + 34.4543i) q^{96} -133.709 q^{97} +174.271i q^{98} +(15.1494 + 5.90429i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 38q - 76q^{4} - 8q^{6} - 12q^{7} + 20q^{9} + O(q^{10}) \)	\( 38q - 76q^{4} - 8q^{6} - 12q^{7} + 20q^{9} + 36q^{10} - 4q^{13} - 17q^{15} + 100q^{16} - 2q^{18} - 28q^{19} - 11q^{21} + 84q^{22} - 6q^{24} - 166q^{25} + 3q^{27} + 12q^{28} + 102q^{30} - 40q^{31} - 46q^{33} - 148q^{34} - 96q^{36} + 112q^{37} + 62q^{39} - 56q^{40} + 14q^{42} + 164q^{43} + 55q^{45} - 4q^{46} - 124q^{48} + 242q^{49} + 52q^{51} + 8q^{52} + 18q^{54} - 228q^{55} - 147q^{57} - 80q^{58} + 128q^{60} + 12q^{61} + 86q^{63} + 48q^{64} - 24q^{66} + 124q^{67} - 240q^{69} + 148q^{70} + 166q^{72} - 192q^{73} - 78q^{75} - 304q^{76} + 244q^{78} + 64q^{79} - 156q^{81} - 180q^{82} + 300q^{84} - 52q^{85} - 83q^{87} - 96q^{88} - 376q^{90} - 332q^{91} + 454q^{93} + 768q^{94} - 722q^{96} + 416q^{97} + 494q^{99} + O(q^{100}) \)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Properties

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