LMFDB - Embedded newform 210.3.h.a.139.4

Newspace parameters

Level:	$ N $	$=$	$ 210 = 2 \cdot 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	210.h (of order $2$, degree $1$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$5.72208555157$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
Defining polynomial:	$ x^{16} - 8 x^{15} + 96 x^{14} - 532 x^{13} + 3236 x^{12} - 12864 x^{11} + 49526 x^{10} - 141436 x^{9} + 362298 x^{8} - 722060 x^{7} + 1208164 x^{6} - 1570812 x^{5} + \cdots + 33750 $ x^16 - 8x^15 + 96x^14 - 532x^13 + 3236x^12 - 12864x^11 + 49526x^10 - 141436x^9 + 362298x^8 - 722060x^7 + 1208164x^6 - 1570812x^5 + 1591101x^4 - 1183860x^3 + 619650x^2 - 202500*x + 33750
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			139.4
Root			$0.500000 - 4.10071i$ of defining polynomial
Character	$\chi$	$=$	210.139
Dual form			210.3.h.a.139.12

$q$-expansion

$f(q)$	$=$	$q-1.41421i q^{2} -1.73205 q^{3} -2.00000 q^{4} +(4.59769 - 1.96501i) q^{5} +2.44949i q^{6} +(6.81963 + 1.57881i) q^{7} +2.82843i q^{8} +3.00000 q^{9} +O(q^{10})$	\(q-1.41421i q^{2} -1.73205 q^{3} -2.00000 q^{4} +(4.59769 - 1.96501i) q^{5} +2.44949i q^{6} +(6.81963 + 1.57881i) q^{7} +2.82843i q^{8} +3.00000 q^{9} +(-2.77894 - 6.50212i) q^{10} -8.15965 q^{11} +3.46410 q^{12} +14.6064 q^{13} +(2.23278 - 9.64441i) q^{14} +(-7.96343 + 3.40349i) q^{15} +4.00000 q^{16} +5.81421 q^{17} -4.24264i q^{18} -33.7736i q^{19} +(-9.19538 + 3.93001i) q^{20} +(-11.8119 - 2.73459i) q^{21} +11.5395i q^{22} -37.2576i q^{23} -4.89898i q^{24} +(17.2775 - 18.0690i) q^{25} -20.6566i q^{26} -5.19615 q^{27} +(-13.6393 - 3.15763i) q^{28} -9.25305 q^{29} +(4.81326 + 11.2620i) q^{30} +19.2558i q^{31} -5.65685i q^{32} +14.1329 q^{33} -8.22253i q^{34} +(34.4569 - 6.14171i) q^{35} -6.00000 q^{36} +63.4350i q^{37} -47.7631 q^{38} -25.2990 q^{39} +(5.55788 + 13.0042i) q^{40} -8.25880i q^{41} +(-3.86729 + 16.7046i) q^{42} +42.0893i q^{43} +16.3193 q^{44} +(13.7931 - 5.89502i) q^{45} -52.6901 q^{46} +23.3380 q^{47} -6.92820 q^{48} +(44.0147 + 21.5339i) q^{49} +(-25.5534 - 24.4341i) q^{50} -10.0705 q^{51} -29.2128 q^{52} -71.3497i q^{53} +7.34847i q^{54} +(-37.5155 + 16.0337i) q^{55} +(-4.46556 + 19.2888i) q^{56} +58.4976i q^{57} +13.0858i q^{58} +42.9350i q^{59} +(15.9269 - 6.80698i) q^{60} +34.2864i q^{61} +27.2318 q^{62} +(20.4589 + 4.73644i) q^{63} -8.00000 q^{64} +(67.1557 - 28.7017i) q^{65} -19.9870i q^{66} +4.99889i q^{67} -11.6284 q^{68} +64.5320i q^{69} +(-8.68569 - 48.7294i) q^{70} -38.8120 q^{71} +8.48528i q^{72} +124.629 q^{73} +89.7107 q^{74} +(-29.9255 + 31.2964i) q^{75} +67.5472i q^{76} +(-55.6458 - 12.8826i) q^{77} +35.7782i q^{78} -56.1842 q^{79} +(18.3908 - 7.86002i) q^{80} +9.00000 q^{81} -11.6797 q^{82} -90.3980 q^{83} +(23.6239 + 5.46917i) q^{84} +(26.7319 - 11.4249i) q^{85} +59.5233 q^{86} +16.0267 q^{87} -23.0790i q^{88} -16.2289i q^{89} +(-8.33681 - 19.5063i) q^{90} +(99.6102 + 23.0608i) q^{91} +74.5151i q^{92} -33.3520i q^{93} -33.0048i q^{94} +(-66.3653 - 155.280i) q^{95} +9.79796i q^{96} +82.7605 q^{97} +(30.4535 - 62.2462i) q^{98} -24.4789 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 32 q^{4} + 48 q^{9}+O(q^{10}) $ 16 * q - 32 * q^4 + 48 * q^9	$ 16 q - 32 q^{4} + 48 q^{9} + 96 q^{11} + 16 q^{14} - 24 q^{15} + 64 q^{16} + 24 q^{21} + 24 q^{25} + 64 q^{29} + 24 q^{30} - 8 q^{35} - 96 q^{36} - 144 q^{39} - 192 q^{44} - 176 q^{46} + 224 q^{49} - 96 q^{50} - 48 q^{51} - 32 q^{56} + 48 q^{60} - 128 q^{64} + 368 q^{65} - 56 q^{70} - 384 q^{71} + 224 q^{74} - 608 q^{79} + 144 q^{81} - 48 q^{84} - 440 q^{85} + 416 q^{86} + 224 q^{91} - 560 q^{95} + 288 q^{99}+O(q^{100}) $ 16 * q - 32 * q^4 + 48 * q^9 + 96 * q^11 + 16 * q^14 - 24 * q^15 + 64 * q^16 + 24 * q^21 + 24 * q^25 + 64 * q^29 + 24 * q^30 - 8 * q^35 - 96 * q^36 - 144 * q^39 - 192 * q^44 - 176 * q^46 + 224 * q^49 - 96 * q^50 - 48 * q^51 - 32 * q^56 + 48 * q^60 - 128 * q^64 + 368 * q^65 - 56 * q^70 - 384 * q^71 + 224 * q^74 - 608 * q^79 + 144 * q^81 - 48 * q^84 - 440 * q^85 + 416 * q^86 + 224 * q^91 - 560 * q^95 + 288 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/210\mathbb{Z}\right)^\times$.

$n$	$31$	$71$	$127$
$\chi(n)$	$-1$	$1$	$-1$

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 $

$2$		−	1.41421i		−	0.707107i
$2$		−	1.41421i		−	0.707107i
$3$	−1.73205			−0.577350
$3$	−1.73205			−0.577350
$4$	−2.00000			−0.500000
$5$	4.59769	−	1.96501i	0.919538	−	0.393001i
$5$	4.59769	−	1.96501i	0.919538	−	0.393001i
$6$			2.44949i			0.408248i
$7$	6.81963	+	1.57881i	0.974233	+	0.225545i
$7$	6.81963	+	1.57881i	0.974233	+	0.225545i
$8$			2.82843i			0.353553i
$9$	3.00000			0.333333
$10$	−2.77894	−	6.50212i	−0.277894	−	0.650212i
$11$	−8.15965			−0.741786			−0.370893	−	0.928676i	$-0.620948\pi$
$11$	−8.15965			−0.741786			−0.370893	+	0.928676i	$0.620948\pi$
$12$	3.46410			0.288675
$13$	14.6064			1.12357			0.561785	−	0.827284i	$-0.310115\pi$
$13$	14.6064			1.12357			0.561785	+	0.827284i	$0.310115\pi$
$14$	2.23278	−	9.64441i	0.159484	−	0.688887i
$15$	−7.96343	+	3.40349i	−0.530896	+	0.226899i
$16$	4.00000			0.250000
$17$	5.81421			0.342012			0.171006	−	0.985270i	$-0.445298\pi$
$17$	5.81421			0.342012			0.171006	+	0.985270i	$0.445298\pi$
$18$		−	4.24264i		−	0.235702i
$19$		−	33.7736i		−	1.77756i	−0.458337	−	0.888778i	$-0.651555\pi$
$19$		−	33.7736i		−	1.77756i	0.458337	−	0.888778i	$-0.348445\pi$
$20$	−9.19538	+	3.93001i	−0.459769	+	0.196501i
$21$	−11.8119	−	2.73459i	−0.562474	−	0.130218i
$22$			11.5395i			0.524522i
$23$		−	37.2576i		−	1.61989i	−0.586503	−	0.809947i	$-0.699496\pi$
$23$		−	37.2576i		−	1.61989i	0.586503	−	0.809947i	$-0.300504\pi$
$24$		−	4.89898i		−	0.204124i
$25$	17.2775	−	18.0690i	0.691100	−	0.722759i
$26$		−	20.6566i		−	0.794483i
$27$	−5.19615			−0.192450
$28$	−13.6393	−	3.15763i	−0.487116	−	0.112772i
$29$	−9.25305			−0.319071			−0.159535	−	0.987192i	$-0.551000\pi$
$29$	−9.25305			−0.319071			−0.159535	+	0.987192i	$0.551000\pi$
$30$	4.81326	+	11.2620i	0.160442	+	0.375400i
$31$			19.2558i			0.621154i	0.950548	+	0.310577i	$0.100522\pi$
$31$			19.2558i			0.621154i	−0.950548	+	0.310577i	$0.899478\pi$
$32$		−	5.65685i		−	0.176777i
$33$	14.1329			0.428270
$34$		−	8.22253i		−	0.241839i
$35$	34.4569	−	6.14171i	0.984483	−	0.175478i
$36$	−6.00000			−0.166667
$37$			63.4350i			1.71446i	0.514933	+	0.857230i	$0.327817\pi$
$37$			63.4350i			1.71446i	−0.514933	+	0.857230i	$0.672183\pi$
$38$	−47.7631			−1.25692
$39$	−25.2990			−0.648693
$40$	5.55788	+	13.0042i	0.138947	+	0.325106i
$41$		−	8.25880i		−	0.201434i	−0.994915	−	0.100717i	$-0.967886\pi$
$41$		−	8.25880i		−	0.201434i	0.994915	−	0.100717i	$-0.0321137\pi$
$42$	−3.86729	+	16.7046i	−0.0920783	+	0.397729i
$43$			42.0893i			0.978822i	0.872053	+	0.489411i	$0.162788\pi$
$43$			42.0893i			0.978822i	−0.872053	+	0.489411i	$0.837212\pi$
$44$	16.3193			0.370893
$45$	13.7931	−	5.89502i	0.306513	−	0.131000i
$46$	−52.6901			−1.14544
$47$	23.3380			0.496552			0.248276	−	0.968689i	$-0.420136\pi$
$47$	23.3380			0.496552			0.248276	+	0.968689i	$0.420136\pi$
$48$	−6.92820			−0.144338
$49$	44.0147	+	21.5339i	0.898259	+	0.439466i
$50$	−25.5534	−	24.4341i	−0.511068	−	0.488682i
$51$	−10.0705			−0.197461
$52$	−29.2128			−0.561785
$53$		−	71.3497i		−	1.34622i	−0.739542	−	0.673110i	$-0.764958\pi$
$53$		−	71.3497i		−	1.34622i	0.739542	−	0.673110i	$-0.235042\pi$
$54$			7.34847i			0.136083i
$55$	−37.5155	+	16.0337i	−0.682100	+	0.291523i
$56$	−4.46556	+	19.2888i	−0.0797422	+	0.344443i
$57$			58.4976i			1.02627i
$58$			13.0858i			0.225617i
$59$			42.9350i			0.727712i	0.931455	+	0.363856i	$0.118540\pi$
$59$			42.9350i			0.727712i	−0.931455	+	0.363856i	$0.881460\pi$
$60$	15.9269	−	6.80698i	0.265448	−	0.113450i
$61$			34.2864i			0.562072i	0.959697	+	0.281036i	$0.0906781\pi$
$61$			34.2864i			0.562072i	−0.959697	+	0.281036i	$0.909322\pi$
$62$	27.2318			0.439222
$63$	20.4589	+	4.73644i	0.324744	+	0.0751816i
$64$	−8.00000			−0.125000
$65$	67.1557	−	28.7017i	1.03316	−	0.441564i
$66$		−	19.9870i		−	0.302833i
$67$			4.99889i			0.0746102i	0.999304	+	0.0373051i	$0.0118773\pi$
$67$			4.99889i			0.0746102i	−0.999304	+	0.0373051i	$0.988123\pi$
$68$	−11.6284			−0.171006
$69$			64.5320i			0.935246i
$70$	−8.68569	−	48.7294i	−0.124081	−	0.696135i
$71$	−38.8120			−0.546648			−0.273324	−	0.961922i	$-0.588123\pi$
$71$	−38.8120			−0.546648			−0.273324	+	0.961922i	$0.588123\pi$
$72$			8.48528i			0.117851i
$73$	124.629			1.70725			0.853626	−	0.520886i	$-0.174398\pi$
$73$	124.629			1.70725			0.853626	+	0.520886i	$0.174398\pi$
$74$	89.7107			1.21231
$75$	−29.9255	+	31.2964i	−0.399007	+	0.417285i
$76$			67.5472i			0.888778i
$77$	−55.6458	−	12.8826i	−0.722672	−	0.167306i
$78$			35.7782i			0.458695i
$79$	−56.1842			−0.711192			−0.355596	−	0.934640i	$-0.615722\pi$
$79$	−56.1842			−0.711192			−0.355596	+	0.934640i	$0.615722\pi$
$80$	18.3908	−	7.86002i	0.229884	−	0.0982503i
$81$	9.00000			0.111111
$82$	−11.6797			−0.142435
$83$	−90.3980			−1.08913			−0.544566	−	0.838718i	$-0.683306\pi$
$83$	−90.3980			−1.08913			−0.544566	+	0.838718i	$0.683306\pi$
$84$	23.6239	+	5.46917i	0.281237	+	0.0651092i
$85$	26.7319	−	11.4249i	0.314493	−	0.134411i
$86$	59.5233			0.692131
$87$	16.0267			0.184215
$88$		−	23.0790i		−	0.262261i
$89$		−	16.2289i		−	0.182347i	−0.995835	−	0.0911736i	$-0.970938\pi$
$89$		−	16.2289i		−	0.182347i	0.995835	−	0.0911736i	$-0.0290618\pi$
$90$	−8.33681	−	19.5063i	−0.0926313	−	0.216737i
$91$	99.6102	+	23.0608i	1.09462	+	0.253415i
$92$			74.5151i			0.809947i
$93$		−	33.3520i		−	0.358624i
$94$		−	33.0048i		−	0.351115i
$95$	−66.3653	−	155.280i	−0.698582	−	1.63453i
$96$			9.79796i			0.102062i
$97$	82.7605			0.853201			0.426601	−	0.904440i	$-0.359711\pi$
$97$	82.7605			0.853201			0.426601	+	0.904440i	$0.359711\pi$
$98$	30.4535	−	62.2462i	0.310750	−	0.635165i
$99$	−24.4789			−0.247262
$100$	−34.5550	+	36.1379i	−0.345550	+	0.361379i
$101$			87.2055i			0.863420i	0.902012	+	0.431710i	$0.142090\pi$
$101$			87.2055i			0.863420i	−0.902012	+	0.431710i	$0.857910\pi$
$102$			14.2418i			0.139626i
$103$	−187.567			−1.82104			−0.910521	−	0.413462i	$-0.864319\pi$
$103$	−187.567			−1.82104			−0.910521	+	0.413462i	$0.864319\pi$
$104$			41.3131i			0.397242i
$105$	−59.6811	+	10.6378i	−0.568392	+	0.101312i
$106$	−100.904			−0.951922
$107$			157.858i			1.47531i	0.675180	+	0.737653i	$0.264066\pi$
$107$			157.858i			1.47531i	−0.675180	+	0.737653i	$0.735934\pi$
$108$	10.3923			0.0962250
$109$	−112.073			−1.02819			−0.514097	−	0.857732i	$-0.671873\pi$
$109$	−112.073			−1.02819			−0.514097	+	0.857732i	$0.671873\pi$
$110$	22.6751	+	53.0550i	0.206138	+	0.482318i
$111$		−	109.873i		−	0.989844i
$112$	27.2785	+	6.31526i	0.243558	+	0.0563862i
$113$			141.987i			1.25653i	0.778001	+	0.628263i	$0.216234\pi$
$113$			141.987i			1.25653i	−0.778001	+	0.628263i	$0.783766\pi$
$114$	82.7280			0.725684
$115$	−73.2113	−	171.299i	−0.636620	−	1.48955i
$116$	18.5061			0.159535
$117$	43.8192			0.374523
$118$	60.7192			0.514570
$119$	39.6507	+	9.17955i	0.333199	+	0.0771391i
$120$	−9.62652	−	22.5240i	−0.0802210	−	0.187700i
$121$	−54.4202			−0.449754
$122$	48.4883			0.397445
$123$			14.3047i			0.116298i
$124$		−	38.5116i		−	0.310577i
$125$	43.9310	−	117.026i	0.351448	−	0.936207i
$126$	6.69834	−	28.9332i	0.0531614	−	0.229629i
$127$			202.414i			1.59381i	0.604104	+	0.796905i	$0.293531\pi$
$127$			202.414i			1.59381i	−0.604104	+	0.796905i	$0.706469\pi$
$128$			11.3137i			0.0883883i
$129$		−	72.9008i		−	0.565123i
$130$	−40.5903	−	94.9725i	−0.312233	−	0.730558i
$131$		−	141.260i		−	1.07832i	−0.842203	−	0.539160i	$-0.818742\pi$
$131$		−	141.260i		−	1.07832i	0.842203	−	0.539160i	$-0.181258\pi$
$132$	−28.2658			−0.214135
$133$	53.3222	−	230.323i	0.400919	−	1.73175i
$134$	7.06949			0.0527574
$135$	−23.8903	+	10.2105i	−0.176965	+	0.0756331i
$136$			16.4451i			0.120920i
$137$		−	50.6430i		−	0.369657i	−0.982771	−	0.184829i	$-0.940827\pi$
$137$		−	50.6430i		−	0.369657i	0.982771	−	0.184829i	$-0.0591730\pi$
$138$	91.2620			0.661319
$139$		−	74.0256i		−	0.532559i	−0.963896	−	0.266279i	$-0.914206\pi$
$139$		−	74.0256i		−	0.532559i	0.963896	−	0.266279i	$-0.0857943\pi$
$140$	−68.9138	+	12.2834i	−0.492242	+	0.0877388i
$141$	−40.4225			−0.286685
$142$			54.8885i			0.386538i
$143$	−119.183			−0.833448
$144$	12.0000			0.0833333
$145$	−42.5426	+	18.1823i	−0.293397	+	0.125395i
$146$		−	176.253i		−	1.20721i
$147$	−76.2357	−	37.2977i	−0.518610	−	0.253726i
$148$		−	126.870i		−	0.857230i
$149$	−226.979			−1.52335			−0.761676	−	0.647958i	$-0.775623\pi$
$149$	−226.979			−1.52335			−0.761676	+	0.647958i	$0.775623\pi$
$150$	44.2598	+	42.3211i	0.295065	+	0.282140i
$151$	294.426			1.94984			0.974920	−	0.222558i	$-0.0714406\pi$
$151$	294.426			1.94984			0.974920	+	0.222558i	$0.0714406\pi$
$152$	95.5261			0.628461
$153$	17.4426			0.114004
$154$	−18.2187	+	78.6950i	−0.118303	+	0.511006i
$155$	37.8377	+	88.5321i	0.244114	+	0.571175i
$156$	50.5981			0.324346
$157$	−65.8596			−0.419488			−0.209744	−	0.977756i	$-0.567263\pi$
$157$	−65.8596			−0.419488			−0.209744	+	0.977756i	$0.567263\pi$
$158$			79.4564i			0.502889i
$159$			123.581i			0.777241i
$160$	−11.1158	−	26.0085i	−0.0694734	−	0.162553i
$161$	58.8227	−	254.083i	0.365359	−	1.57815i
$162$		−	12.7279i		−	0.0785674i
$163$			28.2075i			0.173052i	0.996250	+	0.0865260i	$0.0275766\pi$
$163$			28.2075i			0.173052i	−0.996250	+	0.0865260i	$0.972423\pi$
$164$			16.5176i			0.100717i
$165$	64.9788	−	27.7713i	0.393811	−	0.168311i
$166$			127.842i			0.770133i
$167$	128.461			0.769227			0.384614	−	0.923078i	$-0.374335\pi$
$167$	128.461			0.769227			0.384614	+	0.923078i	$0.374335\pi$
$168$	7.73458	−	33.4092i	0.0460392	−	0.198864i
$169$	44.3469			0.262408
$170$	−16.1573	−	37.8046i	−0.0950431	−	0.222380i
$171$		−	101.321i		−	0.592519i
$172$		−	84.1786i		−	0.489411i
$173$	112.446			0.649976			0.324988	−	0.945718i	$-0.394640\pi$
$173$	112.446			0.649976			0.324988	+	0.945718i	$0.394640\pi$
$174$		−	22.6652i		−	0.130260i
$175$	146.354	−	95.9457i	0.836307	−	0.548261i
$176$	−32.6386			−0.185446
$177$		−	74.3656i		−	0.420144i
$178$	−22.9511			−0.128939
$179$	−68.9019			−0.384927			−0.192464	−	0.981304i	$-0.561648\pi$
$179$	−68.9019			−0.384927			−0.192464	+	0.981304i	$0.561648\pi$
$180$	−27.5861	+	11.7900i	−0.153256	+	0.0655002i
$181$			166.537i			0.920094i	0.887895	+	0.460047i	$0.152167\pi$
$181$			166.537i			0.920094i	−0.887895	+	0.460047i	$0.847833\pi$
$182$	32.6129	−	140.870i	0.179192	−	0.774012i
$183$		−	59.3858i		−	0.324513i
$184$	105.380			0.572719
$185$	124.650	+	291.655i	0.673785	+	1.57651i
$186$	−47.1668			−0.253585
$187$	−47.4419			−0.253700
$188$	−46.6759			−0.248276
$189$	−35.4358	−	8.20376i	−0.187491	−	0.0434061i
$190$	−219.600	+	93.8547i	−1.15579	+	0.493972i
$191$	−148.289			−0.776382			−0.388191	−	0.921579i	$-0.626900\pi$
$191$	−148.289			−0.776382			−0.388191	+	0.921579i	$0.626900\pi$
$192$	13.8564			0.0721688
$193$		−	35.1886i		−	0.182324i	−0.995836	−	0.0911622i	$-0.970942\pi$
$193$		−	35.1886i		−	0.182324i	0.995836	−	0.0911622i	$-0.0290582\pi$
$194$		−	117.041i		−	0.603304i
$195$	−116.317	+	49.7127i	−0.596498	+	0.254937i
$196$	−88.0294	−	43.0677i	−0.449130	−	0.219733i
$197$			193.634i			0.982915i	0.870902	+	0.491457i	$0.163536\pi$
$197$			193.634i			0.982915i	−0.870902	+	0.491457i	$0.836464\pi$
$198$			34.6184i			0.174841i
$199$		−	181.838i		−	0.913759i	−0.889529	−	0.456880i	$-0.848967\pi$
$199$		−	181.838i		−	0.913759i	0.889529	−	0.456880i	$-0.151033\pi$
$200$	51.1068	+	48.8682i	0.255534	+	0.244341i
$201$		−	8.65832i		−	0.0430762i
$202$	123.327			0.610530
$203$	−63.1023	−	14.6088i	−0.310849	−	0.0719647i
$204$	20.1410			0.0987304
$205$	−16.2286	−	37.9714i	−0.0791638	−	0.185226i
$206$			265.260i			1.28767i
$207$		−	111.773i		−	0.539965i
$208$	58.4256			0.280892
$209$			275.580i			1.31857i
$210$	15.0441	+	84.4019i	0.0716384	+	0.401914i
$211$	175.914			0.833717			0.416859	−	0.908971i	$-0.363131\pi$
$211$	175.914			0.833717			0.416859	+	0.908971i	$0.363131\pi$
$212$			142.699i			0.673110i
$213$	67.2244			0.315607
$214$	223.244			1.04320
$215$	82.7058	+	193.514i	0.384678	+	0.900064i
$216$		−	14.6969i		−	0.0680414i
$217$	−30.4013	+	131.317i	−0.140098	+	0.605149i
$218$			158.495i			0.727042i
$219$	−215.864			−0.985683
$220$	75.0310	−	32.0675i	0.341050	−	0.145761i
$221$	84.9246			0.384274
$222$	−155.384			−0.699926
$223$	20.5675			0.0922308			0.0461154	−	0.998936i	$-0.485316\pi$
$223$	20.5675			0.0922308			0.0461154	+	0.998936i	$0.485316\pi$
$224$	8.93112	−	38.5776i	0.0398711	−	0.172222i
$225$	51.8325	−	54.2069i	0.230367	−	0.240920i
$226$	200.801			0.888498
$227$	−414.087			−1.82417			−0.912085	−	0.410001i	$-0.865528\pi$
$227$	−414.087			−1.82417			−0.912085	+	0.410001i	$0.865528\pi$
$228$		−	116.995i		−	0.513136i
$229$		−	195.520i		−	0.853799i	−0.904299	−	0.426899i	$-0.859606\pi$
$229$		−	195.520i		−	0.853799i	0.904299	−	0.426899i	$-0.140394\pi$
$230$	−242.253	+	103.536i	−1.05327	+	0.450158i
$231$	96.3813	+	22.3133i	0.417235	+	0.0965942i
$232$		−	26.1716i		−	0.112808i
$233$		−	81.3006i		−	0.348930i	−0.984663	−	0.174465i	$-0.944180\pi$
$233$		−	81.3006i		−	0.348930i	0.984663	−	0.174465i	$-0.0558195\pi$
$234$		−	61.9697i		−	0.264828i
$235$	107.301	−	45.8592i	0.456599	−	0.195146i
$236$		−	85.8700i		−	0.363856i
$237$	97.3138			0.410607
$238$	12.9818	−	56.0746i	0.0545456	−	0.235608i
$239$	−249.265			−1.04295			−0.521475	−	0.853267i	$-0.674618\pi$
$239$	−249.265			−1.04295			−0.521475	+	0.853267i	$0.674618\pi$
$240$	−31.8537	+	13.6140i	−0.132724	+	0.0567248i
$241$			262.343i			1.08856i	0.838904	+	0.544280i	$0.183197\pi$
$241$			262.343i			1.08856i	−0.838904	+	0.544280i	$0.816803\pi$
$242$			76.9618i			0.318024i
$243$	−15.5885			−0.0641500
$244$		−	68.5728i		−	0.281036i
$245$	244.680	+	12.5169i	0.998694	+	0.0510892i
$246$	20.2298			0.0822351
$247$		−	493.310i		−	1.99721i
$248$	−54.4636			−0.219611
$249$	156.574			0.628811
$250$	−165.500	−	62.1278i	−0.661999	−	0.248511i
$251$			141.917i			0.565406i	0.959208	+	0.282703i	$0.0912310\pi$
$251$			141.917i			0.565406i	−0.959208	+	0.282703i	$0.908769\pi$
$252$	−40.9178	−	9.47288i	−0.162372	−	0.0375908i
$253$			304.008i			1.20161i
$254$	286.257			1.12699
$255$	−46.3010	+	19.7886i	−0.181573	+	0.0776023i
$256$	16.0000			0.0625000
$257$	170.843			0.664757			0.332379	−	0.943146i	$-0.392149\pi$
$257$	170.843			0.664757			0.332379	+	0.943146i	$0.392149\pi$
$258$	−103.097			−0.399602
$259$	−100.152	+	432.604i	−0.386688	+	1.67028i
$260$	−134.311	+	57.4033i	−0.516582	+	0.220782i
$261$	−27.7591			−0.106357
$262$	−199.772			−0.762488
$263$			383.417i			1.45786i	0.684588	+	0.728930i	$0.259982\pi$
$263$			383.417i			1.45786i	−0.684588	+	0.728930i	$0.740018\pi$
$264$			39.9739i			0.151416i
$265$	−140.203	−	328.044i	−0.529066	−	1.23790i
$266$	−325.726	−	75.4090i	−1.22454	−	0.283492i
$267$			28.1093i			0.105278i
$268$		−	9.99777i		−	0.0373051i
$269$			154.512i			0.574393i	0.957872	+	0.287196i	$0.0927232\pi$
$269$			154.512i			0.574393i	−0.957872	+	0.287196i	$0.907277\pi$
$270$	14.4398	+	33.7860i	0.0534807	+	0.125133i
$271$			320.328i			1.18202i	0.806664	+	0.591010i	$0.201271\pi$
$271$			320.328i			1.18202i	−0.806664	+	0.591010i	$0.798729\pi$
$272$	23.2568			0.0855030
$273$	−172.530	−	39.9425i	−0.631978	−	0.146309i
$274$	−71.6200			−0.261387
$275$	−140.978	+	147.436i	−0.512648	+	0.536132i
$276$		−	129.064i		−	0.467623i
$277$		−	222.854i		−	0.804526i	−0.915524	−	0.402263i	$-0.868224\pi$
$277$		−	222.854i		−	0.804526i	0.915524	−	0.402263i	$-0.131776\pi$
$278$	−104.688			−0.376576
$279$			57.7673i			0.207051i
$280$	17.3714	+	97.4589i	0.0620407	+	0.348067i
$281$	218.973			0.779263			0.389631	−	0.920971i	$-0.372602\pi$
$281$	218.973			0.779263			0.389631	+	0.920971i	$0.372602\pi$
$282$			57.1661i			0.202717i
$283$	36.3957			0.128607			0.0643034	−	0.997930i	$-0.479517\pi$
$283$	36.3957			0.128607			0.0643034	+	0.997930i	$0.479517\pi$
$284$	77.6240			0.273324
$285$	114.948	+	268.954i	0.403326	+	0.943697i
$286$			168.550i			0.589337i
$287$	13.0391	−	56.3219i	0.0454324	−	0.196244i
$288$		−	16.9706i		−	0.0589256i
$289$	−255.195			−0.883028
$290$	25.7136	+	60.1644i	0.0886677	+	0.207463i
$291$	−143.345			−0.492596
$292$	−249.259			−0.853626
$293$	−168.440			−0.574882			−0.287441	−	0.957798i	$-0.592805\pi$
$293$	−168.440			−0.574882			−0.287441	+	0.957798i	$0.592805\pi$
$294$	−52.7470	+	107.814i	−0.179411	+	0.366713i
$295$	84.3675	+	197.402i	0.285991	+	0.669158i
$296$	−179.421			−0.606153
$297$	42.3988			0.142757
$298$			320.997i			1.07717i
$299$		−	544.199i		−	1.82006i
$300$	59.8510	−	62.5928i	0.199503	−	0.208643i
$301$	−66.4512	+	287.034i	−0.220768	+	0.953600i
$302$		−	416.381i		−	1.37874i
$303$		−	151.044i		−	0.498496i
$304$		−	135.094i		−	0.444389i
$305$	67.3730	+	157.638i	0.220895	+	0.516847i
$306$		−	24.6676i		−	0.0806130i
$307$	223.400			0.727688			0.363844	−	0.931460i	$-0.381464\pi$
$307$	223.400			0.727688			0.363844	+	0.931460i	$0.381464\pi$
$308$	111.292	+	25.7651i	0.361336	+	0.0836530i
$309$	324.876			1.05138
$310$	125.203	−	53.5106i	0.403882	−	0.172615i
$311$		−	46.0396i		−	0.148037i	−0.997257	−	0.0740186i	$-0.976418\pi$
$311$		−	46.0396i		−	0.148037i	0.997257	−	0.0740186i	$-0.0235824\pi$
$312$		−	71.5565i		−	0.229348i
$313$	−80.0375			−0.255711			−0.127855	−	0.991793i	$-0.540809\pi$
$313$	−80.0375			−0.255711			−0.127855	+	0.991793i	$0.540809\pi$
$314$			93.1395i			0.296623i
$315$	103.371	−	18.4251i	0.328161	−	0.0584925i
$316$	112.368			0.355596
$317$		−	185.237i		−	0.584343i	−0.956366	−	0.292171i	$-0.905622\pi$
$317$		−	185.237i		−	0.584343i	0.956366	−	0.292171i	$-0.0943777\pi$
$318$	174.770			0.549592
$319$	75.5016			0.236682
$320$	−36.7815	+	15.7200i	−0.114942	+	0.0491251i
$321$		−	273.418i		−	0.851768i
$322$	−359.327	−	83.1879i	−1.11592	−	0.258348i
$323$		−	196.367i		−	0.607946i
$324$	−18.0000			−0.0555556
$325$	252.362	−	263.923i	0.776499	−	0.812070i
$326$	39.8914			0.122366
$327$	194.116			0.593628
$328$	23.3594			0.0712177
$329$	159.156	+	36.8463i	0.483757	+	0.111995i
$330$	−39.2745	−	91.8939i	−0.119014	−	0.278466i
$331$	61.4686			0.185706			0.0928529	−	0.995680i	$-0.470401\pi$
$331$	61.4686			0.185706			0.0928529	+	0.995680i	$0.470401\pi$
$332$	180.796			0.544566
$333$			190.305i			0.571487i
$334$		−	181.671i		−	0.543926i
$335$	9.82284	+	22.9833i	0.0293219	+	0.0686069i
$336$	−47.2478	−	10.9383i	−0.140618	−	0.0325546i
$337$		−	656.501i		−	1.94807i	−0.226387	−	0.974037i	$-0.572691\pi$
$337$		−	656.501i		−	1.94807i	0.226387	−	0.974037i	$-0.427309\pi$
$338$		−	62.7160i		−	0.185550i
$339$		−	245.929i		−	0.725456i
$340$	−53.4638	+	22.8499i	−0.157247	+	0.0672056i
$341$		−	157.120i		−	0.460764i
$342$	−143.289			−0.418974
$343$	266.166	+	216.344i	0.775994	+	0.630740i
$344$	−119.047			−0.346066
$345$	126.806	+	296.698i	0.367553	+	0.859994i
$346$		−	159.022i		−	0.459603i
$347$		−	321.323i		−	0.926002i	−0.886358	−	0.463001i	$-0.846773\pi$
$347$		−	321.323i		−	0.926002i	0.886358	−	0.463001i	$-0.153227\pi$
$348$	−32.0535			−0.0921077
$349$			185.135i			0.530474i	0.964183	+	0.265237i	$0.0854501\pi$
$349$			185.135i			0.530474i	−0.964183	+	0.265237i	$0.914550\pi$
$350$	−135.688	−	206.975i	−0.387679	−	0.591358i
$351$	−75.8971			−0.216231
$352$			46.1579i			0.131130i
$353$	597.338			1.69218			0.846088	−	0.533043i	$-0.178952\pi$
$353$	597.338			1.69218			0.846088	+	0.533043i	$0.178952\pi$
$354$	−105.169			−0.297087
$355$	−178.446	+	76.2658i	−0.502664	+	0.214833i
$356$			32.4578i			0.0911736i
$357$	−68.6771	−	15.8995i	−0.192373	−	0.0445363i
$358$			97.4421i			0.272185i
$359$	239.446			0.666982			0.333491	−	0.942753i	$-0.391773\pi$
$359$	239.446			0.666982			0.333491	+	0.942753i	$0.391773\pi$
$360$	16.6736	+	39.0127i	0.0463156	+	0.108369i
$361$	−779.655			−2.15971
$362$	235.519			0.650604
$363$	94.2585			0.259665
$364$	−199.220	−	46.1216i	−0.547309	−	0.126708i
$365$	573.007	−	244.898i	1.56988	−	0.670952i
$366$	−83.9842			−0.229465
$367$	398.743			1.08649			0.543246	−	0.839573i	$-0.317195\pi$
$367$	398.743			1.08649			0.543246	+	0.839573i	$0.317195\pi$
$368$		−	149.030i		−	0.404973i
$369$		−	24.7764i		−	0.0671447i
$370$	412.462	−	176.282i	1.11476	−	0.476438i
$371$	112.648	−	486.578i	0.303633	−	1.31153i
$372$			66.7040i			0.179312i
$373$		−	34.7064i		−	0.0930466i	−0.998917	−	0.0465233i	$-0.985186\pi$
$373$		−	34.7064i		−	0.0930466i	0.998917	−	0.0465233i	$-0.0148142\pi$
$374$			67.0929i			0.179393i
$375$	−76.0907	+	202.695i	−0.202908	+	0.540520i
$376$			66.0097i			0.175558i
$377$	−135.154			−0.358498
$378$	−11.6019	+	50.1138i	−0.0306928	+	0.132576i
$379$	−109.217			−0.288172			−0.144086	−	0.989565i	$-0.546024\pi$
$379$	−109.217			−0.288172			−0.144086	+	0.989565i	$0.546024\pi$
$380$	132.731	+	310.561i	0.349291	+	0.817265i
$381$		−	350.591i		−	0.920187i
$382$			209.712i			0.548985i
$383$	336.420			0.878381			0.439191	−	0.898394i	$-0.355265\pi$
$383$	336.420			0.878381			0.439191	+	0.898394i	$0.355265\pi$
$384$		−	19.5959i		−	0.0510310i
$385$	−281.156	+	50.1142i	−0.730276	+	0.130167i
$386$	−49.7642			−0.128923
$387$			126.268i			0.326274i
$388$	−165.521			−0.426601
$389$	−311.173			−0.799930			−0.399965	−	0.916530i	$-0.630978\pi$
$389$	−311.173			−0.799930			−0.399965	+	0.916530i	$0.630978\pi$
$390$	70.3044	+	164.497i	0.180268	+	0.421788i
$391$		−	216.623i		−	0.554023i
$392$	−60.9069	+	124.492i	−0.155375	+	0.317583i
$393$			244.669i			0.622568i
$394$	273.840			0.695026
$395$	−258.317	+	110.402i	−0.653968	+	0.279499i
$396$	48.9579			0.123631
$397$	−41.2679			−0.103949			−0.0519747	−	0.998648i	$-0.516552\pi$
$397$	−41.2679			−0.103949			−0.0519747	+	0.998648i	$0.516552\pi$
$398$	−257.158			−0.646125
$399$	−92.3568	+	398.932i	−0.231471	+	0.999829i
$400$	69.1100	−	72.2759i	0.172775	−	0.180690i
$401$	−495.642			−1.23601			−0.618007	−	0.786173i	$-0.712060\pi$
$401$	−495.642			−1.23601			−0.618007	+	0.786173i	$0.712060\pi$
$402$	−12.2447			−0.0304595
$403$			281.258i			0.697910i
$404$		−	174.411i		−	0.431710i
$405$	41.3792	−	17.6851i	0.102171	−	0.0436668i
$406$	−20.6600	+	89.2402i	−0.0508867	+	0.219803i
$407$		−	517.608i		−	1.27176i
$408$		−	28.4837i		−	0.0698129i
$409$		−	359.920i		−	0.880001i	−0.897998	−	0.440000i	$-0.854978\pi$
$409$		−	359.920i		−	0.880001i	0.897998	−	0.440000i	$-0.145022\pi$
$410$	−53.6997	+	22.9507i	−0.130975	+	0.0559773i
$411$			87.7163i			0.213422i
$412$	375.135			0.910521
$413$	−67.7863	+	292.801i	−0.164132	+	0.708960i
$414$	−158.070			−0.381813
$415$	−415.622	+	177.633i	−1.00150	+	0.428030i
$416$		−	82.6263i		−	0.198621i
$417$			128.216i			0.307473i
$418$	389.730			0.932367
$419$			482.910i			1.15253i	0.817263	+	0.576265i	$0.195490\pi$
$419$			482.910i			1.15253i	−0.817263	+	0.576265i	$0.804510\pi$
$420$	119.362	−	21.2755i	0.284196	−	0.0506560i
$421$	−523.520			−1.24351			−0.621757	−	0.783210i	$-0.713581\pi$
$421$	−523.520			−1.24351			−0.621757	+	0.783210i	$0.713581\pi$
$422$		−	248.780i		−	0.589527i
$423$	70.0139			0.165517
$424$	201.807			0.475961
$425$	100.455	−	105.057i	0.236365	−	0.247192i
$426$		−	95.0696i		−	0.223168i
$427$	−54.1319	+	233.821i	−0.126773	+	0.547589i
$428$		−	315.715i		−	0.737653i
$429$	206.431			0.481191
$430$	273.670	−	116.964i	0.636441	−	0.272008i
$431$	517.027			1.19960			0.599799	−	0.800150i	$-0.295247\pi$
$431$	517.027			1.19960			0.599799	+	0.800150i	$0.295247\pi$
$432$	−20.7846			−0.0481125
$433$	−567.712			−1.31111			−0.655557	−	0.755146i	$-0.727566\pi$
$433$	−567.712			−1.31111			−0.655557	+	0.755146i	$0.727566\pi$
$434$	185.711	+	42.9939i	0.427905	+	0.0990644i
$435$	73.6860	−	31.4926i	0.169393	−	0.0723969i
$436$	224.146			0.514097
$437$	−1258.32			−2.87945
$438$			305.279i			0.696983i
$439$		−	622.875i		−	1.41885i	−0.704781	−	0.709425i	$-0.748955\pi$
$439$		−	622.875i		−	1.41885i	0.704781	−	0.709425i	$-0.251045\pi$
$440$	−45.3503	−	106.110i	−0.103069	−	0.241159i
$441$	132.044	+	64.6016i	0.299420	+	0.146489i
$442$		−	120.102i		−	0.271723i
$443$		−	476.699i		−	1.07607i	−0.842923	−	0.538035i	$-0.819167\pi$
$443$		−	476.699i		−	1.07607i	0.842923	−	0.538035i	$-0.180833\pi$
$444$			219.745i			0.494922i
$445$	−31.8899	−	74.6154i	−0.0716626	−	0.167675i
$446$		−	29.0868i		−	0.0652170i
$447$	393.140			0.879507
$448$	−54.5570	−	12.6305i	−0.121779	−	0.0281931i
$449$	861.931			1.91967			0.959834	−	0.280570i	$-0.0905233\pi$
$449$	861.931			1.91967			0.959834	+	0.280570i	$0.0905233\pi$
$450$	−76.6602	−	73.3022i	−0.170356	−	0.162894i
$451$			67.3889i			0.149421i
$452$		−	283.975i		−	0.628263i
$453$	−509.960			−1.12574
$454$			585.607i			1.28988i
$455$	503.292	−	89.7083i	1.10614	−	0.197161i
$456$	−165.456			−0.362842
$457$			27.5401i			0.0602629i	0.999546	+	0.0301314i	$0.00959258\pi$
$457$			27.5401i			0.0602629i	−0.999546	+	0.0301314i	$0.990407\pi$
$458$	−276.507			−0.603727
$459$	−30.2115			−0.0658203
$460$	146.423	+	342.597i	0.318310	+	0.744777i
$461$		−	378.183i		−	0.820353i	−0.912006	−	0.410176i	$-0.865467\pi$
$461$		−	378.183i		−	0.820353i	0.912006	−	0.410176i	$-0.134533\pi$
$462$	31.5557	−	136.304i	0.0683024	−	0.295030i
$463$		−	724.994i		−	1.56586i	−0.622108	−	0.782931i	$-0.713724\pi$
$463$		−	724.994i		−	1.56586i	0.622108	−	0.782931i	$-0.286276\pi$
$464$	−37.0122			−0.0797676
$465$	−65.5369	−	153.342i	−0.140939	−	0.329768i
$466$	−114.976			−0.246731
$467$	−371.451			−0.795398			−0.397699	−	0.917516i	$-0.630191\pi$
$467$	−371.451			−0.795398			−0.397699	+	0.917516i	$0.630191\pi$
$468$	−87.6384			−0.187262
$469$	−7.89231	+	34.0905i	−0.0168280	+	0.0726877i
$470$	−64.8547	−	151.746i	−0.137989	−	0.322864i
$471$	114.072			0.242191
$472$	−121.438			−0.257285
$473$		−	343.434i		−	0.726076i
$474$		−	137.623i		−	0.290343i
$475$	−610.254	−	583.523i	−1.28475	−	1.22847i
$476$	−79.3015	−	18.3591i	−0.166600	−	0.0385695i
$477$		−	214.049i		−	0.448740i
$478$			352.514i			0.737477i
$479$			860.650i			1.79677i	0.439214	+	0.898383i	$0.355257\pi$
$479$			860.650i			1.79677i	−0.439214	+	0.898383i	$0.644743\pi$
$480$	19.2530	+	45.0480i	0.0401105	+	0.0938500i
$481$			926.558i			1.92632i
$482$	371.009			0.769728
$483$	−101.884	+	440.084i	−0.210940	+	0.911147i
$484$	108.840			0.224877
$485$	380.507	−	162.625i	0.784551	−	0.335309i
$486$			22.0454i			0.0453609i
$487$			887.173i			1.82171i	0.412726	+	0.910855i	$0.364577\pi$
$487$			887.173i			1.82171i	−0.412726	+	0.910855i	$0.635423\pi$
$488$	−96.9766			−0.198723
$489$		−	48.8568i		−	0.0999116i
$490$	17.7015	−	346.030i	0.0361255	−	0.706183i
$491$	−50.1823			−0.102204			−0.0511021	−	0.998693i	$-0.516273\pi$
$491$	−50.1823			−0.102204			−0.0511021	+	0.998693i	$0.516273\pi$
$492$		−	28.6093i		−	0.0581490i
$493$	−53.7991			−0.109126
$494$	−697.646			−1.41224
$495$	−112.547	+	48.1012i	−0.227367	+	0.0971742i
$496$			77.0231i			0.155289i
$497$	−264.683	−	61.2769i	−0.532562	−	0.123294i
$498$		−	221.429i		−	0.444636i
$499$	160.443			0.321529			0.160765	−	0.986993i	$-0.448604\pi$
$499$	160.443			0.321529			0.160765	+	0.986993i	$0.448604\pi$
$500$	−87.8620	+	234.052i	−0.175724	+	0.468104i
$501$	−222.501			−0.444113
$502$	200.701			0.399802
$503$	742.042			1.47523			0.737616	−	0.675220i	$-0.235951\pi$
$503$	742.042			1.47523			0.737616	+	0.675220i	$0.235951\pi$
$504$	−13.3967	+	57.8665i	−0.0265807	+	0.114814i
$505$	171.359	+	400.944i	0.339325	+	0.793948i
$506$	429.933			0.849670
$507$	−76.8111			−0.151501
$508$		−	404.828i		−	0.796905i
$509$		−	978.447i		−	1.92229i	−0.276038	−	0.961147i	$-0.589021\pi$
$509$		−	978.447i		−	1.92229i	0.276038	−	0.961147i	$-0.410979\pi$
$510$	27.9853	+	65.4796i	0.0548731	+	0.128391i
$511$	849.927	+	196.767i	1.66326	+	0.385062i
$512$		−	22.6274i		−	0.0441942i
$513$			175.493i			0.342091i
$514$		−	241.608i		−	0.470054i
$515$	−862.377	+	368.571i	−1.67452	+	0.715672i
$516$			145.802i			0.282561i
$517$	−190.429			−0.368335
$518$	611.794	+	141.637i	1.18107	+	0.273430i
$519$	−194.762			−0.375264
$520$	81.1805	+	189.945i	0.156116	+	0.365279i
$521$			575.650i			1.10489i	0.833548	+	0.552447i	$0.186306\pi$
$521$			575.650i			1.10489i	−0.833548	+	0.552447i	$0.813694\pi$
$522$			39.2573i			0.0752056i
$523$	−339.090			−0.648355			−0.324178	−	0.945996i	$-0.605088\pi$
$523$	−339.090			−0.648355			−0.324178	+	0.945996i	$0.605088\pi$
$524$			282.520i			0.539160i
$525$	−253.492	+	166.183i	−0.482842	+	0.316539i
$526$	542.234			1.03086
$527$			111.957i			0.212442i
$528$	56.5317			0.107068
$529$	−859.125			−1.62406
$530$	−463.924	+	198.276i	−0.875328	+	0.374106i
$531$			128.805i			0.242571i
$532$	−106.644	+	460.647i	−0.200459	+	0.865877i
$533$		−	120.631i		−	0.226325i
$534$	39.7525			0.0744429
$535$	310.191	+	725.781i	0.579797	+	1.35660i
$536$	−14.1390			−0.0263787
$537$	119.342			0.222238
$538$	218.512			0.406157
$539$	−359.144	−	175.709i	−0.666316	−	0.325990i
$540$	47.7806	−	20.4209i	0.0884826	−	0.0378166i
$541$	35.8638			0.0662916			0.0331458	−	0.999451i	$-0.489447\pi$
$541$	35.8638			0.0662916			0.0331458	+	0.999451i	$0.489447\pi$
$542$	453.012			0.835815
$543$		−	288.450i		−	0.531216i
$544$		−	32.8901i		−	0.0604598i
$545$	−515.277	+	220.224i	−0.945463	+	0.404081i
$546$	−56.4872	+	243.994i	−0.103456	+	0.446876i
$547$		−	700.845i		−	1.28125i	−0.767853	−	0.640626i	$-0.778675\pi$
$547$		−	700.845i		−	1.28125i	0.767853	−	0.640626i	$-0.221325\pi$
$548$			101.286i			0.184829i
$549$			102.859i			0.187357i
$550$	208.507	+	199.373i	0.379103	+	0.362497i
$551$			312.508i			0.567166i
$552$	−182.524			−0.330659
$553$	−383.155	−	88.7044i	−0.692867	−	0.160406i
$554$	−315.163			−0.568886
$555$	−215.901	−	505.161i	−0.389010	−	0.910200i
$556$			148.051i			0.266279i
$557$			639.349i			1.14784i	0.818910	+	0.573922i	$0.194579\pi$
$557$			639.349i			1.14784i	−0.818910	+	0.573922i	$0.805421\pi$
$558$	81.6954			0.146407
$559$			614.773i			1.09977i
$560$	137.828	−	24.5669i	0.246121	−	0.0438694i
$561$	82.1717			0.146474
$562$		−	309.674i		−	0.551022i
$563$	−227.519			−0.404120			−0.202060	−	0.979373i	$-0.564764\pi$
$563$	−227.519			−0.404120			−0.202060	+	0.979373i	$0.564764\pi$
$564$	80.8450			0.143342
$565$	279.006	+	652.814i	0.493816	+	1.15542i
$566$		−	51.4713i		−	0.0909387i
$567$	61.3767	+	14.2093i	0.108248	+	0.0250605i
$568$		−	109.777i		−	0.193269i
$569$	613.901			1.07891			0.539456	−	0.842014i	$-0.318630\pi$
$569$	613.901			1.07891			0.539456	+	0.842014i	$0.318630\pi$
$570$	380.358	−	162.561i	0.667294	−	0.285195i
$571$	366.674			0.642161			0.321081	−	0.947052i	$-0.395954\pi$
$571$	366.674			0.642161			0.321081	+	0.947052i	$0.395954\pi$
$572$	238.366			0.416724
$573$	256.844			0.448244
$574$	−79.6513	−	18.4401i	−0.138765	−	0.0321256i
$575$	−673.206	−	643.718i	−1.17079	−	1.11951i
$576$	−24.0000			−0.0416667
$577$	−62.1951			−0.107790			−0.0538952	−	0.998547i	$-0.517164\pi$
$577$	−62.1951			−0.107790			−0.0538952	+	0.998547i	$0.517164\pi$
$578$			360.900i			0.624395i
$579$			60.9485i			0.105265i
$580$	85.0853	−	36.3646i	0.146699	−	0.0626975i
$581$	−616.481	−	142.722i	−1.06107	−	0.245648i
$582$			202.721i			0.348318i
$583$			582.188i			0.998607i
$584$			352.505i			0.603605i
$585$	201.467	−	86.1050i	0.344388	−	0.147188i
$586$			238.211i			0.406503i
$587$	−176.872			−0.301315			−0.150658	−	0.988586i	$-0.548139\pi$
$587$	−176.872			−0.301315			−0.150658	+	0.988586i	$0.548139\pi$
$588$	152.471	+	74.5955i	0.259305	+	0.126863i
$589$	650.337			1.10414
$590$	279.168	−	119.314i	0.473166	−	0.202226i
$591$		−	335.384i		−	0.567486i
$592$			253.740i			0.428615i
$593$	154.878			0.261176			0.130588	−	0.991437i	$-0.458313\pi$
$593$	154.878			0.261176			0.130588	+	0.991437i	$0.458313\pi$
$594$		−	59.9609i		−	0.100944i
$595$	200.340	−	35.7092i	0.336705	−	0.0600154i
$596$	453.959			0.761676
$597$			314.953i			0.527559i
$598$	−769.613			−1.28698
$599$	900.825			1.50388			0.751940	−	0.659231i	$-0.229118\pi$
$599$	900.825			1.50388			0.751940	+	0.659231i	$0.229118\pi$
$600$	−88.5195	−	84.6421i	−0.147533	−	0.141070i
$601$		−	682.387i		−	1.13542i	−0.823229	−	0.567710i	$-0.807830\pi$
$601$		−	682.387i		−	1.13542i	0.823229	−	0.567710i	$-0.192170\pi$
$602$	405.927	+	93.9762i	0.674297	+	0.156107i
$603$			14.9967i			0.0248701i
$604$	−588.851			−0.974920
$605$	−250.207	+	106.936i	−0.413566	+	0.176754i
$606$	−213.609			−0.352490
$607$	367.814			0.605954			0.302977	−	0.952998i	$-0.402019\pi$
$607$	367.814			0.605954			0.302977	+	0.952998i	$0.402019\pi$
$608$	−191.052			−0.314231
$609$	109.296	+	25.3032i	0.179469	+	0.0415488i
$610$	222.934	−	95.2798i	0.365466	−	0.156196i
$611$	340.883			0.557911
$612$	−34.8852			−0.0570020
$613$			117.271i			0.191306i	0.995415	+	0.0956532i	$0.0304940\pi$
$613$			117.271i			0.191306i	−0.995415	+	0.0956532i	$0.969506\pi$
$614$		−	315.936i		−	0.514553i
$615$	28.1087	+	65.7684i	0.0457053	+	0.106940i
$616$	36.4374	−	157.390i	0.0591516	−	0.255503i
$617$			1070.18i			1.73449i	0.497881	+	0.867245i	$0.334112\pi$
$617$			1070.18i			1.73449i	−0.497881	+	0.867245i	$0.665888\pi$
$618$		−	459.444i		−	0.743438i
$619$		−	532.192i		−	0.859761i	−0.902886	−	0.429880i	$-0.858556\pi$
$619$		−	532.192i		−	0.859761i	0.902886	−	0.429880i	$-0.141444\pi$
$620$	−75.6754	−	177.064i	−0.122057	−	0.285587i
$621$			193.596i			0.311749i
$622$	−65.1098			−0.104678
$623$	25.6224	−	110.675i	0.0411275	−	0.177649i
$624$	−101.196			−0.162173
$625$	−27.9756	−	624.374i	−0.0447610	−	0.998998i
$626$			113.190i			0.180815i
$627$		−	477.319i		−	0.761275i
$628$	131.719			0.209744
$629$			368.825i			0.586366i
$630$	−26.0571	−	146.188i	−0.0413604	−	0.232045i
$631$	472.933			0.749498			0.374749	−	0.927126i	$-0.377729\pi$
$631$	472.933			0.749498			0.374749	+	0.927126i	$0.377729\pi$
$632$		−	158.913i		−	0.251444i
$633$	−304.693			−0.481347
$634$	−261.964			−0.413193
$635$	397.745	+	930.636i	0.626369	+	1.46557i
$636$		−	247.163i		−	0.388620i
$637$	642.896	+	314.532i	1.00926	+	0.493771i
$638$		−	106.775i		−	0.167359i
$639$	−116.436			−0.182216
$640$	22.2315	+	52.0169i	0.0347367	+	0.0812764i
$641$	−486.990			−0.759734			−0.379867	−	0.925041i	$-0.624030\pi$
$641$	−486.990			−0.759734			−0.379867	+	0.925041i	$0.624030\pi$
$642$	−386.671			−0.602291
$643$	−111.425			−0.173289			−0.0866447	−	0.996239i	$-0.527614\pi$
$643$	−111.425			−0.173289			−0.0866447	+	0.996239i	$0.527614\pi$
$644$	−117.645	+	508.165i	−0.182679	+	0.789077i
$645$	−143.251	−	335.176i	−0.222094	−	0.519652i
$646$	−277.704			−0.429883
$647$	737.696			1.14018			0.570090	−	0.821583i	$-0.306908\pi$
$647$	737.696			1.14018			0.570090	+	0.821583i	$0.306908\pi$
$648$			25.4558i			0.0392837i
$649$		−	350.334i		−	0.539806i
$650$	−373.243	−	356.894i	−0.574220	−	0.549068i
$651$	52.6566	−	227.448i	0.0808857	−	0.349383i
$652$		−	56.4149i		−	0.0865260i
$653$			193.734i			0.296682i	0.988936	+	0.148341i	$0.0473934\pi$
$653$			193.734i			0.296682i	−0.988936	+	0.148341i	$0.952607\pi$
$654$		−	274.522i		−	0.419758i
$655$	−277.577	−	649.469i	−0.423781	−	0.991556i
$656$		−	33.0352i		−	0.0503585i
$657$	373.888			0.569084
$658$	52.1085	−	225.081i	0.0791923	−	0.342068i
$659$	−823.339			−1.24938			−0.624688	−	0.780874i	$-0.714774\pi$
$659$	−823.339			−1.24938			−0.624688	+	0.780874i	$0.714774\pi$
$660$	−129.958	+	55.5425i	−0.196905	+	0.0841554i
$661$		−	657.833i		−	0.995208i	−0.867404	−	0.497604i	$-0.834213\pi$
$661$		−	657.833i		−	0.995208i	0.867404	−	0.497604i	$-0.165787\pi$
$662$		−	86.9298i		−	0.131314i
$663$	−147.094			−0.221861
$664$		−	255.684i		−	0.385066i
$665$	−207.428	−	1163.73i	−0.311921	−	1.74998i
$666$	269.132			0.404102
$667$			344.746i			0.516860i
$668$	−256.922			−0.384614
$669$	−35.6239			−0.0532495
$670$	32.5033	−	13.8916i	0.0485124	−	0.0207337i
$671$		−	279.765i		−	0.416937i
$672$	−15.4692	+	66.8184i	−0.0230196	+	0.0994322i
$673$		−	727.300i		−	1.08068i	−0.841446	−	0.540342i	$-0.818295\pi$
$673$		−	727.300i		−	1.08068i	0.841446	−	0.540342i	$-0.181705\pi$
$674$	−928.433			−1.37750
$675$	−89.7766	+	93.8891i	−0.133002	+	0.139095i
$676$	−88.6938			−0.131204
$677$	−385.964			−0.570110			−0.285055	−	0.958511i	$-0.592012\pi$
$677$	−385.964			−0.570110			−0.285055	+	0.958511i	$0.592012\pi$
$678$	−347.797			−0.512975
$679$	564.396	+	130.663i	0.831217	+	0.192435i
$680$	32.3146	+	75.6093i	0.0475215	+	0.111190i
$681$	717.219			1.05319
$682$	−222.202			−0.325809
$683$			236.488i			0.346249i	0.984900	+	0.173124i	$0.0553863\pi$
$683$			236.488i			0.346249i	−0.984900	+	0.173124i	$0.944614\pi$
$684$			202.641i			0.296259i
$685$	−99.5138	−	232.841i	−0.145276	−	0.339914i
$686$	305.957	−	376.415i	0.446001	−	0.548711i
$687$			338.650i			0.492941i
$688$			168.357i			0.244705i
$689$		−	1042.16i		−	1.51257i
$690$	419.594	−	179.330i	0.608108	−	0.259899i
$691$		−	337.426i		−	0.488316i	−0.969735	−	0.244158i	$-0.921488\pi$
$691$		−	337.426i		−	0.488316i	0.969735	−	0.244158i	$-0.0785116\pi$
$692$	−224.892			−0.324988
$693$	−166.937	−	38.6477i	−0.240891	−	0.0557687i
$694$	−454.419			−0.654782
$695$	−145.461	−	340.347i	−0.209296	−	0.489708i
$696$			45.3305i			0.0651300i
$697$		−	48.0184i		−	0.0688929i
$698$	261.821			0.375101
$699$			140.817i			0.201455i
$700$	−292.707	+	191.891i	−0.418154	+	0.274131i
$701$	89.2192			0.127274			0.0636371	−	0.997973i	$-0.479730\pi$
$701$	89.2192			0.127274			0.0636371	+	0.997973i	$0.479730\pi$
$702$			107.335i			0.152898i
$703$	2142.43			3.04755
$704$	65.2772			0.0927232
$705$	−185.850	+	79.4305i	−0.263617	+	0.112667i
$706$		−	844.764i		−	1.19655i
$707$	−137.681	+	594.709i	−0.194740	+	0.841172i
$708$			148.731i			0.210072i
$709$	569.646			0.803450			0.401725	−	0.915760i	$-0.368411\pi$
$709$	569.646			0.803450			0.401725	+	0.915760i	$0.368411\pi$
$710$	107.856	+	252.360i	0.151910	+	0.355437i
$711$	−168.552			−0.237064
$712$	45.9023			0.0644695
$713$	717.423			1.00620
$714$	−22.4852	+	97.1241i	−0.0314919	+	0.136028i
$715$	−547.967	+	234.195i	−0.766387	+	0.327546i
$716$	137.804			0.192464
$717$	431.740			0.602147
$718$		−	338.628i		−	0.471627i
$719$		−	1261.33i		−	1.75428i	−0.480233	−	0.877141i	$-0.659448\pi$
$719$		−	1261.33i		−	1.75428i	0.480233	−	0.877141i	$-0.340552\pi$
$720$	55.1723	−	23.5801i	0.0766282	−	0.0327501i
$721$	−1279.14	−	296.134i	−1.77412	−	0.410727i
$722$			1102.60i			1.52714i
$723$		−	454.391i		−	0.628481i
$724$		−	333.074i		−	0.460047i
$725$	−159.870	+	167.193i	−0.220510	+	0.230611i
$726$		−	133.302i		−	0.183611i
$727$	−307.746			−0.423310			−0.211655	−	0.977344i	$-0.567885\pi$
$727$	−307.746			−0.423310			−0.211655	+	0.977344i	$0.567885\pi$
$728$	−65.2258	+	281.740i	−0.0895958	+	0.387006i
$729$	27.0000			0.0370370
$730$	−346.337	−	810.355i	−0.474435	−	1.11008i
$731$			244.716i			0.334769i
$732$			118.772i			0.162256i
$733$	−633.490			−0.864243			−0.432121	−	0.901815i	$-0.642235\pi$
$733$	−633.490			−0.864243			−0.432121	+	0.901815i	$0.642235\pi$
$734$		−	563.908i		−	0.768266i
$735$	−423.798	−	21.6798i	−0.576596	−	0.0294964i
$736$	−210.761			−0.286359
$737$		−	40.7891i		−	0.0553448i
$738$	−35.0391			−0.0474785
$739$	749.557			1.01428			0.507142	−	0.861862i	$-0.330702\pi$
$739$	749.557			1.01428			0.507142	+	0.861862i	$0.330702\pi$
$740$	−249.300	−	583.309i	−0.336893	−	0.788256i
$741$			854.439i			1.15309i
$742$	−688.126	−	159.308i	−0.927393	−	0.214701i
$743$		−	145.699i		−	0.196095i	−0.995182	−	0.0980476i	$-0.968740\pi$
$743$		−	145.699i		−	0.196095i	0.995182	−	0.0980476i	$-0.0312597\pi$
$744$	94.3337			0.126793
$745$	−1043.58	+	446.016i	−1.40078	+	0.598679i
$746$	−49.0822			−0.0657939
$747$	−271.194			−0.363044
$748$	94.8837			0.126850
$749$	−249.228	+	1076.53i	−0.332748	+	1.43729i
$750$	286.654	+	107.608i	0.382205	+	0.143478i
$751$	156.811			0.208803			0.104402	−	0.994535i	$-0.466707\pi$
$751$	156.811			0.208803			0.104402	+	0.994535i	$0.466707\pi$
$752$	93.3518			0.124138
$753$		−	245.807i		−	0.326437i
$754$			191.136i			0.253496i
$755$	1353.68	−	578.548i	1.79295	−	0.766289i
$756$	70.8717	+	16.4075i	0.0937456	+	0.0217031i
$757$			296.377i			0.391515i	0.980652	+	0.195758i	$0.0627166\pi$
$757$			296.377i			0.391515i	−0.980652	+	0.195758i	$0.937283\pi$
$758$			154.457i			0.203769i
$759$		−	526.558i		−	0.693752i
$760$	439.199	−	187.709i	0.577894	−	0.246986i
$761$			312.747i			0.410968i	0.978660	+	0.205484i	$0.0658769\pi$
$761$			312.747i			0.410968i	−0.978660	+	0.205484i	$0.934123\pi$
$762$	−495.811			−0.650670
$763$	−764.297	−	176.942i	−1.00170	−	0.231904i
$764$	296.578			0.388191
$765$	80.1958	−	34.2748i	0.104831	−	0.0448037i
$766$		−	475.770i		−	0.621109i
$767$			627.125i			0.817634i
$768$	−27.7128			−0.0360844
$769$			91.1460i			0.118525i	0.998242	+	0.0592627i	$0.0188750\pi$
$769$			91.1460i			0.118525i	−0.998242	+	0.0592627i	$0.981125\pi$
$770$	70.8722	+	397.615i	0.0920418	+	0.516383i
$771$	−295.908			−0.383798
$772$			70.3772i			0.0911622i
$773$	417.539			0.540154			0.270077	−	0.962839i	$-0.412951\pi$
$773$	417.539			0.540154			0.270077	+	0.962839i	$0.412951\pi$
$774$	178.570			0.230710
$775$	347.932	+	332.692i	0.448945	+	0.429280i
$776$			234.082i			0.301652i
$777$	173.469	−	749.291i	0.223254	−	0.964339i
$778$			440.065i			0.565636i
$779$	−278.929			−0.358061
$780$	232.634	−	99.4255i	0.298249	−	0.127469i
$781$	316.692			0.405496

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:	\( N \)	\(=\)	\( 210 = 2 \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	210.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.41421i q^{2} -1.73205 q^{3} -2.00000 q^{4} +(4.59769 - 1.96501i) q^{5} +2.44949i q^{6} +(6.81963 + 1.57881i) q^{7} +2.82843i q^{8} +3.00000 q^{9} +O(q^{10})\)	\(q-1.41421i q^{2} -1.73205 q^{3} -2.00000 q^{4} +(4.59769 - 1.96501i) q^{5} +2.44949i q^{6} +(6.81963 + 1.57881i) q^{7} +2.82843i q^{8} +3.00000 q^{9} +(-2.77894 - 6.50212i) q^{10} -8.15965 q^{11} +3.46410 q^{12} +14.6064 q^{13} +(2.23278 - 9.64441i) q^{14} +(-7.96343 + 3.40349i) q^{15} +4.00000 q^{16} +5.81421 q^{17} -4.24264i q^{18} -33.7736i q^{19} +(-9.19538 + 3.93001i) q^{20} +(-11.8119 - 2.73459i) q^{21} +11.5395i q^{22} -37.2576i q^{23} -4.89898i q^{24} +(17.2775 - 18.0690i) q^{25} -20.6566i q^{26} -5.19615 q^{27} +(-13.6393 - 3.15763i) q^{28} -9.25305 q^{29} +(4.81326 + 11.2620i) q^{30} +19.2558i q^{31} -5.65685i q^{32} +14.1329 q^{33} -8.22253i q^{34} +(34.4569 - 6.14171i) q^{35} -6.00000 q^{36} +63.4350i q^{37} -47.7631 q^{38} -25.2990 q^{39} +(5.55788 + 13.0042i) q^{40} -8.25880i q^{41} +(-3.86729 + 16.7046i) q^{42} +42.0893i q^{43} +16.3193 q^{44} +(13.7931 - 5.89502i) q^{45} -52.6901 q^{46} +23.3380 q^{47} -6.92820 q^{48} +(44.0147 + 21.5339i) q^{49} +(-25.5534 - 24.4341i) q^{50} -10.0705 q^{51} -29.2128 q^{52} -71.3497i q^{53} +7.34847i q^{54} +(-37.5155 + 16.0337i) q^{55} +(-4.46556 + 19.2888i) q^{56} +58.4976i q^{57} +13.0858i q^{58} +42.9350i q^{59} +(15.9269 - 6.80698i) q^{60} +34.2864i q^{61} +27.2318 q^{62} +(20.4589 + 4.73644i) q^{63} -8.00000 q^{64} +(67.1557 - 28.7017i) q^{65} -19.9870i q^{66} +4.99889i q^{67} -11.6284 q^{68} +64.5320i q^{69} +(-8.68569 - 48.7294i) q^{70} -38.8120 q^{71} +8.48528i q^{72} +124.629 q^{73} +89.7107 q^{74} +(-29.9255 + 31.2964i) q^{75} +67.5472i q^{76} +(-55.6458 - 12.8826i) q^{77} +35.7782i q^{78} -56.1842 q^{79} +(18.3908 - 7.86002i) q^{80} +9.00000 q^{81} -11.6797 q^{82} -90.3980 q^{83} +(23.6239 + 5.46917i) q^{84} +(26.7319 - 11.4249i) q^{85} +59.5233 q^{86} +16.0267 q^{87} -23.0790i q^{88} -16.2289i q^{89} +(-8.33681 - 19.5063i) q^{90} +(99.6102 + 23.0608i) q^{91} +74.5151i q^{92} -33.3520i q^{93} -33.0048i q^{94} +(-66.3653 - 155.280i) q^{95} +9.79796i q^{96} +82.7605 q^{97} +(30.4535 - 62.2462i) q^{98} -24.4789 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 32 q^{4} + 48 q^{9}+O(q^{10}) \) 16 * q - 32 * q^4 + 48 * q^9	\( 16 q - 32 q^{4} + 48 q^{9} + 96 q^{11} + 16 q^{14} - 24 q^{15} + 64 q^{16} + 24 q^{21} + 24 q^{25} + 64 q^{29} + 24 q^{30} - 8 q^{35} - 96 q^{36} - 144 q^{39} - 192 q^{44} - 176 q^{46} + 224 q^{49} - 96 q^{50} - 48 q^{51} - 32 q^{56} + 48 q^{60} - 128 q^{64} + 368 q^{65} - 56 q^{70} - 384 q^{71} + 224 q^{74} - 608 q^{79} + 144 q^{81} - 48 q^{84} - 440 q^{85} + 416 q^{86} + 224 q^{91} - 560 q^{95} + 288 q^{99}+O(q^{100}) \) 16 * q - 32 * q^4 + 48 * q^9 + 96 * q^11 + 16 * q^14 - 24 * q^15 + 64 * q^16 + 24 * q^21 + 24 * q^25 + 64 * q^29 + 24 * q^30 - 8 * q^35 - 96 * q^36 - 144 * q^39 - 192 * q^44 - 176 * q^46 + 224 * q^49 - 96 * q^50 - 48 * q^51 - 32 * q^56 + 48 * q^60 - 128 * q^64 + 368 * q^65 - 56 * q^70 - 384 * q^71 + 224 * q^74 - 608 * q^79 + 144 * q^81 - 48 * q^84 - 440 * q^85 + 416 * q^86 + 224 * q^91 - 560 * q^95 + 288 * q^99

Introduction

L-functions

Modular forms

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