LMFDB - Embedded newform 1050.3.e.a.701.3

Newspace parameters

Level:	$ N $	$=$	$ 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1050.e (of order $2$, degree $1$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$28.6104277578$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{7})$
Defining polynomial:	$ x^{4} + 8x^{2} + 9 $ x^4 + 8*x^2 + 9
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 42)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			701.3
Root			$-2.57794i$ of defining polynomial
Character	$\chi$	$=$	1050.701
Dual form			1050.3.e.a.701.1

$q$-expansion

$f(q)$	$=$	$q+1.41421i q^{2} +(-2.64575 + 1.41421i) q^{3} -2.00000 q^{4} +(-2.00000 - 3.74166i) q^{6} +2.64575 q^{7} -2.82843i q^{8} +(5.00000 - 7.48331i) q^{9} +O(q^{10})$	\(q+1.41421i q^{2} +(-2.64575 + 1.41421i) q^{3} -2.00000 q^{4} +(-2.00000 - 3.74166i) q^{6} +2.64575 q^{7} -2.82843i q^{8} +(5.00000 - 7.48331i) q^{9} +14.5544i q^{11} +(5.29150 - 2.82843i) q^{12} +0.583005 q^{13} +3.74166i q^{14} +4.00000 q^{16} -21.5367i q^{17} +(10.5830 + 7.07107i) q^{18} -16.0000 q^{19} +(-7.00000 + 3.74166i) q^{21} -20.5830 q^{22} +38.8308i q^{23} +(4.00000 + 7.48331i) q^{24} +0.824494i q^{26} +(-2.64575 + 26.8701i) q^{27} -5.29150 q^{28} -35.7676i q^{29} -58.4575 q^{31} +5.65685i q^{32} +(-20.5830 - 38.5073i) q^{33} +30.4575 q^{34} +(-10.0000 + 14.9666i) q^{36} -20.0000 q^{37} -22.6274i q^{38} +(-1.54249 + 0.824494i) q^{39} +8.75149i q^{41} +(-5.29150 - 9.89949i) q^{42} +11.7490 q^{43} -29.1088i q^{44} -54.9150 q^{46} -8.48528i q^{47} +(-10.5830 + 5.65685i) q^{48} +7.00000 q^{49} +(30.4575 + 56.9808i) q^{51} -1.16601 q^{52} -50.9117i q^{53} +(-38.0000 - 3.74166i) q^{54} -7.48331i q^{56} +(42.3320 - 22.6274i) q^{57} +50.5830 q^{58} -58.2175i q^{59} +38.9150 q^{61} -82.6714i q^{62} +(13.2288 - 19.7990i) q^{63} -8.00000 q^{64} +(54.4575 - 29.1088i) q^{66} -70.5830 q^{67} +43.0734i q^{68} +(-54.9150 - 102.737i) q^{69} +17.0279i q^{71} +(-21.1660 - 14.1421i) q^{72} -72.3320 q^{73} -28.2843i q^{74} +32.0000 q^{76} +38.5073i q^{77} +(-1.16601 - 2.18141i) q^{78} +20.9150 q^{79} +(-31.0000 - 74.8331i) q^{81} -12.3765 q^{82} -145.544i q^{83} +(14.0000 - 7.48331i) q^{84} +16.6156i q^{86} +(50.5830 + 94.6321i) q^{87} +41.1660 q^{88} -53.6514i q^{89} +1.54249 q^{91} -77.6616i q^{92} +(154.664 - 82.6714i) q^{93} +12.0000 q^{94} +(-8.00000 - 14.9666i) q^{96} +111.166 q^{97} +9.89949i q^{98} +(108.915 + 72.7719i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{4} - 8 q^{6} + 20 q^{9}+O(q^{10}) $ 4 * q - 8 * q^4 - 8 * q^6 + 20 * q^9	$ 4 q - 8 q^{4} - 8 q^{6} + 20 q^{9} - 40 q^{13} + 16 q^{16} - 64 q^{19} - 28 q^{21} - 40 q^{22} + 16 q^{24} - 128 q^{31} - 40 q^{33} + 16 q^{34} - 40 q^{36} - 80 q^{37} - 112 q^{39} - 80 q^{43} - 8 q^{46} + 28 q^{49} + 16 q^{51} + 80 q^{52} - 152 q^{54} + 160 q^{58} - 56 q^{61} - 32 q^{64} + 112 q^{66} - 240 q^{67} - 8 q^{69} - 120 q^{73} + 128 q^{76} + 80 q^{78} - 128 q^{79} - 124 q^{81} - 240 q^{82} + 56 q^{84} + 160 q^{87} + 80 q^{88} + 112 q^{91} + 280 q^{93} + 48 q^{94} - 32 q^{96} + 360 q^{97} + 224 q^{99}+O(q^{100}) $ 4 * q - 8 * q^4 - 8 * q^6 + 20 * q^9 - 40 * q^13 + 16 * q^16 - 64 * q^19 - 28 * q^21 - 40 * q^22 + 16 * q^24 - 128 * q^31 - 40 * q^33 + 16 * q^34 - 40 * q^36 - 80 * q^37 - 112 * q^39 - 80 * q^43 - 8 * q^46 + 28 * q^49 + 16 * q^51 + 80 * q^52 - 152 * q^54 + 160 * q^58 - 56 * q^61 - 32 * q^64 + 112 * q^66 - 240 * q^67 - 8 * q^69 - 120 * q^73 + 128 * q^76 + 80 * q^78 - 128 * q^79 - 124 * q^81 - 240 * q^82 + 56 * q^84 + 160 * q^87 + 80 * q^88 + 112 * q^91 + 280 * q^93 + 48 * q^94 - 32 * q^96 + 360 * q^97 + 224 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1050\mathbb{Z}\right)^\times$.

$n$	$127$	$451$	$701$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	−2.64575	+	1.41421i	−0.881917	+	0.471405i
$3$	−2.64575	+	1.41421i	−0.881917	+	0.471405i
$4$	−2.00000			−0.500000
$5$		0			0
$5$		0			0
$6$	−2.00000	−	3.74166i	−0.333333	−	0.623610i
$7$	2.64575			0.377964
$7$	2.64575			0.377964
$8$		−	2.82843i		−	0.353553i
$9$	5.00000	−	7.48331i	0.555556	−	0.831479i
$10$		0			0
$11$			14.5544i			1.32313i	0.749890	+	0.661563i	$0.230107\pi$
$11$			14.5544i			1.32313i	−0.749890	+	0.661563i	$0.769893\pi$
$12$	5.29150	−	2.82843i	0.440959	−	0.235702i
$13$	0.583005			0.0448466			0.0224233	−	0.999749i	$-0.492862\pi$
$13$	0.583005			0.0448466			0.0224233	+	0.999749i	$0.492862\pi$
$14$			3.74166i			0.267261i
$15$		0			0
$16$	4.00000			0.250000
$17$		−	21.5367i		−	1.26687i	−0.773798	−	0.633433i	$-0.781645\pi$
$17$		−	21.5367i		−	1.26687i	0.773798	−	0.633433i	$-0.218355\pi$
$18$	10.5830	+	7.07107i	0.587945	+	0.392837i
$19$	−16.0000			−0.842105			−0.421053	−	0.907036i	$-0.638339\pi$
$19$	−16.0000			−0.842105			−0.421053	+	0.907036i	$0.638339\pi$
$20$		0			0
$21$	−7.00000	+	3.74166i	−0.333333	+	0.178174i
$22$	−20.5830			−0.935591
$23$			38.8308i			1.68830i	0.536111	+	0.844148i	$0.319893\pi$
$23$			38.8308i			1.68830i	−0.536111	+	0.844148i	$0.680107\pi$
$24$	4.00000	+	7.48331i	0.166667	+	0.311805i
$25$		0			0
$26$			0.824494i			0.0317113i
$27$	−2.64575	+	26.8701i	−0.0979908	+	0.995187i
$28$	−5.29150			−0.188982
$29$		−	35.7676i		−	1.23337i	−0.787212	−	0.616683i	$-0.788476\pi$
$29$		−	35.7676i		−	1.23337i	0.787212	−	0.616683i	$-0.211524\pi$
$30$		0			0
$31$	−58.4575			−1.88573			−0.942863	−	0.333180i	$-0.891878\pi$
$31$	−58.4575			−1.88573			−0.942863	+	0.333180i	$0.891878\pi$
$32$			5.65685i			0.176777i
$33$	−20.5830	−	38.5073i	−0.623727	−	1.16689i
$34$	30.4575			0.895809
$35$		0			0
$36$	−10.0000	+	14.9666i	−0.277778	+	0.415740i
$37$	−20.0000			−0.540541			−0.270270	−	0.962784i	$-0.587113\pi$
$37$	−20.0000			−0.540541			−0.270270	+	0.962784i	$0.587113\pi$
$38$		−	22.6274i		−	0.595458i
$39$	−1.54249	+	0.824494i	−0.0395509	+	0.0211409i
$40$		0			0
$41$			8.75149i			0.213451i	0.994289	+	0.106725i	$0.0340366\pi$
$41$			8.75149i			0.213451i	−0.994289	+	0.106725i	$0.965963\pi$
$42$	−5.29150	−	9.89949i	−0.125988	−	0.235702i
$43$	11.7490			0.273233			0.136616	−	0.990624i	$-0.456377\pi$
$43$	11.7490			0.273233			0.136616	+	0.990624i	$0.456377\pi$
$44$		−	29.1088i		−	0.661563i
$45$		0			0
$46$	−54.9150			−1.19380
$47$		−	8.48528i		−	0.180538i	−0.995917	−	0.0902690i	$-0.971227\pi$
$47$		−	8.48528i		−	0.180538i	0.995917	−	0.0902690i	$-0.0287727\pi$
$48$	−10.5830	+	5.65685i	−0.220479	+	0.117851i
$49$	7.00000			0.142857
$50$		0			0
$51$	30.4575	+	56.9808i	0.597206	+	1.11727i
$52$	−1.16601			−0.0224233
$53$		−	50.9117i		−	0.960598i	−0.877105	−	0.480299i	$-0.840528\pi$
$53$		−	50.9117i		−	0.960598i	0.877105	−	0.480299i	$-0.159472\pi$
$54$	−38.0000	−	3.74166i	−0.703704	−	0.0692900i
$55$		0			0
$56$		−	7.48331i		−	0.133631i
$57$	42.3320	−	22.6274i	0.742667	−	0.396972i
$58$	50.5830			0.872121
$59$		−	58.2175i		−	0.986738i	−0.869820	−	0.493369i	$-0.835765\pi$
$59$		−	58.2175i		−	0.986738i	0.869820	−	0.493369i	$-0.164235\pi$
$60$		0			0
$61$	38.9150			0.637951			0.318976	−	0.947763i	$-0.396661\pi$
$61$	38.9150			0.637951			0.318976	+	0.947763i	$0.396661\pi$
$62$		−	82.6714i		−	1.33341i
$63$	13.2288	−	19.7990i	0.209980	−	0.314270i
$64$	−8.00000			−0.125000
$65$		0			0
$66$	54.4575	−	29.1088i	0.825114	−	0.441042i
$67$	−70.5830			−1.05348			−0.526739	−	0.850027i	$-0.676585\pi$
$67$	−70.5830			−1.05348			−0.526739	+	0.850027i	$0.676585\pi$
$68$			43.0734i			0.633433i
$69$	−54.9150	−	102.737i	−0.795870	−	1.48894i
$70$		0			0
$71$			17.0279i			0.239829i	0.992784	+	0.119915i	$0.0382621\pi$
$71$			17.0279i			0.239829i	−0.992784	+	0.119915i	$0.961738\pi$
$72$	−21.1660	−	14.1421i	−0.293972	−	0.196419i
$73$	−72.3320			−0.990850			−0.495425	−	0.868651i	$-0.664988\pi$
$73$	−72.3320			−0.990850			−0.495425	+	0.868651i	$0.664988\pi$
$74$		−	28.2843i		−	0.382220i
$75$		0			0
$76$	32.0000			0.421053
$77$			38.5073i			0.500095i
$78$	−1.16601	−	2.18141i	−0.0149489	−	0.0279667i
$79$	20.9150			0.264747			0.132374	−	0.991200i	$-0.457740\pi$
$79$	20.9150			0.264747			0.132374	+	0.991200i	$0.457740\pi$
$80$		0			0
$81$	−31.0000	−	74.8331i	−0.382716	−	0.923866i
$82$	−12.3765			−0.150933
$83$		−	145.544i		−	1.75354i	−0.480910	−	0.876770i	$-0.659694\pi$
$83$		−	145.544i		−	1.75354i	0.480910	−	0.876770i	$-0.340306\pi$
$84$	14.0000	−	7.48331i	0.166667	−	0.0890871i
$85$		0			0
$86$			16.6156i			0.193205i
$87$	50.5830	+	94.6321i	0.581414	+	1.08773i
$88$	41.1660			0.467796
$89$		−	53.6514i		−	0.602824i	−0.953494	−	0.301412i	$-0.902542\pi$
$89$		−	53.6514i		−	0.602824i	0.953494	−	0.301412i	$-0.0974580\pi$
$90$		0			0
$91$	1.54249			0.0169504
$92$		−	77.6616i		−	0.844148i
$93$	154.664	−	82.6714i	1.66305	−	0.888940i
$94$	12.0000			0.127660
$95$		0			0
$96$	−8.00000	−	14.9666i	−0.0833333	−	0.155902i
$97$	111.166			1.14604			0.573021	−	0.819541i	$-0.305771\pi$
$97$	111.166			1.14604			0.573021	+	0.819541i	$0.305771\pi$
$98$			9.89949i			0.101015i
$99$	108.915	+	72.7719i	1.10015	+	0.735070i
$100$		0			0
$101$			57.8367i			0.572641i	0.958134	+	0.286320i	$0.0924322\pi$
$101$			57.8367i			0.572641i	−0.958134	+	0.286320i	$0.907568\pi$
$102$	−80.5830	+	43.0734i	−0.790029	+	0.422289i
$103$	119.373			1.15896			0.579478	−	0.814988i	$-0.303256\pi$
$103$	119.373			1.15896			0.579478	+	0.814988i	$0.303256\pi$
$104$		−	1.64899i		−	0.0158557i
$105$		0			0
$106$	72.0000			0.679245
$107$		−	116.492i		−	1.08871i	−0.838854	−	0.544357i	$-0.816774\pi$
$107$		−	116.492i		−	1.08871i	0.838854	−	0.544357i	$-0.183226\pi$
$108$	5.29150	−	53.7401i	0.0489954	−	0.497594i
$109$	87.8301			0.805780			0.402890	−	0.915248i	$-0.368006\pi$
$109$	87.8301			0.805780			0.402890	+	0.915248i	$0.368006\pi$
$110$		0			0
$111$	52.9150	−	28.2843i	0.476712	−	0.254813i
$112$	10.5830			0.0944911
$113$			69.1763i			0.612180i	0.952003	+	0.306090i	$0.0990208\pi$
$113$			69.1763i			0.612180i	−0.952003	+	0.306090i	$0.900979\pi$
$114$	32.0000	+	59.8665i	0.280702	+	0.525145i
$115$		0			0
$116$			71.5352i			0.616683i
$117$	2.91503	−	4.36281i	0.0249148	−	0.0372890i
$118$	82.3320			0.697729
$119$		−	56.9808i		−	0.478830i
$120$		0			0
$121$	−90.8301			−0.750662
$122$			55.0342i			0.451100i
$123$	−12.3765	−	23.1543i	−0.100622	−	0.188246i
$124$	116.915			0.942863
$125$		0			0
$126$	28.0000	+	18.7083i	0.222222	+	0.148478i
$127$	27.0850			0.213268			0.106634	−	0.994298i	$-0.465993\pi$
$127$	27.0850			0.213268			0.106634	+	0.994298i	$0.465993\pi$
$128$		−	11.3137i		−	0.0883883i
$129$	−31.0850	+	16.6156i	−0.240969	+	0.128803i
$130$		0			0
$131$		−	145.544i		−	1.11102i	−0.831509	−	0.555511i	$-0.812523\pi$
$131$		−	145.544i		−	1.11102i	0.831509	−	0.555511i	$-0.187477\pi$
$132$	41.1660	+	77.0146i	0.311864	+	0.583444i
$133$	−42.3320			−0.318286
$134$		−	99.8194i		−	0.744921i
$135$		0			0
$136$	−60.9150			−0.447905
$137$			24.8088i			0.181086i	0.995893	+	0.0905431i	$0.0288603\pi$
$137$			24.8088i			0.181086i	−0.995893	+	0.0905431i	$0.971140\pi$
$138$	145.292	−	77.6616i	1.05284	−	0.562765i
$139$	146.458			1.05365			0.526826	−	0.849973i	$-0.323382\pi$
$139$	146.458			1.05365			0.526826	+	0.849973i	$0.323382\pi$
$140$		0			0
$141$	12.0000	+	22.4499i	0.0851064	+	0.159219i
$142$	−24.0810			−0.169585
$143$			8.48528i			0.0593376i
$144$	20.0000	−	29.9333i	0.138889	−	0.207870i
$145$		0			0
$146$		−	102.293i		−	0.700636i
$147$	−18.5203	+	9.89949i	−0.125988	+	0.0673435i
$148$	40.0000			0.270270
$149$		−	219.552i		−	1.47351i	−0.676162	−	0.736753i	$-0.736358\pi$
$149$		−	219.552i		−	1.47351i	0.676162	−	0.736753i	$-0.263642\pi$
$150$		0			0
$151$	−211.660			−1.40172			−0.700861	−	0.713298i	$-0.747201\pi$
$151$	−211.660			−1.40172			−0.700861	+	0.713298i	$0.747201\pi$
$152$			45.2548i			0.297729i
$153$	−161.166	−	107.684i	−1.05337	−	0.703814i
$154$	−54.4575			−0.353620
$155$		0			0
$156$	3.08497	−	1.64899i	0.0197755	−	0.0105704i
$157$	−208.745			−1.32959			−0.664793	−	0.747027i	$-0.731480\pi$
$157$	−208.745			−1.32959			−0.664793	+	0.747027i	$0.731480\pi$
$158$			29.5783i			0.187205i
$159$	72.0000	+	134.700i	0.452830	+	0.847168i
$160$		0			0
$161$			102.737i			0.638116i
$162$	105.830	−	43.8406i	0.653272	−	0.270621i
$163$	−266.996			−1.63801			−0.819006	−	0.573784i	$-0.805475\pi$
$163$	−266.996			−1.63801			−0.819006	+	0.573784i	$0.805475\pi$
$164$		−	17.5030i		−	0.106725i
$165$		0			0
$166$	205.830			1.23994
$167$		−	7.19124i		−	0.0430613i	−0.999768	−	0.0215307i	$-0.993146\pi$
$167$		−	7.19124i		−	0.0430613i	0.999768	−	0.0215307i	$-0.00685395\pi$
$168$	10.5830	+	19.7990i	0.0629941	+	0.117851i
$169$	−168.660			−0.997989
$170$		0			0
$171$	−80.0000	+	119.733i	−0.467836	+	0.700193i
$172$	−23.4980			−0.136616
$173$		−	192.536i		−	1.11293i	−0.830872	−	0.556464i	$-0.812158\pi$
$173$		−	192.536i		−	1.11293i	0.830872	−	0.556464i	$-0.187842\pi$
$174$	−133.830	+	71.5352i	−0.769138	+	0.411122i
$175$		0			0
$176$			58.2175i			0.330781i
$177$	82.3320	+	154.029i	0.465153	+	0.870221i
$178$	75.8745			0.426261
$179$			43.6631i			0.243928i	0.992535	+	0.121964i	$0.0389193\pi$
$179$			43.6631i			0.243928i	−0.992535	+	0.121964i	$0.961081\pi$
$180$		0			0
$181$	−81.0850			−0.447983			−0.223992	−	0.974591i	$-0.571909\pi$
$181$	−81.0850			−0.447983			−0.223992	+	0.974591i	$0.571909\pi$
$182$			2.18141i			0.0119857i
$183$	−102.959	+	55.0342i	−0.562620	+	0.300733i
$184$	109.830			0.596902
$185$		0			0
$186$	116.915	+	218.728i	0.628575	+	1.17596i
$187$	313.454			1.67622
$188$			16.9706i			0.0902690i
$189$	−7.00000	+	71.0915i	−0.0370370	+	0.376145i
$190$		0			0
$191$		−	53.5571i		−	0.280404i	−0.990123	−	0.140202i	$-0.955225\pi$
$191$		−	53.5571i		−	0.280404i	0.990123	−	0.140202i	$-0.0447751\pi$
$192$	21.1660	−	11.3137i	0.110240	−	0.0589256i
$193$	110.494			0.572508			0.286254	−	0.958154i	$-0.407590\pi$
$193$	110.494			0.572508			0.286254	+	0.958154i	$0.407590\pi$
$194$			157.212i			0.810374i
$195$		0			0
$196$	−14.0000			−0.0714286
$197$		−	86.1469i		−	0.437294i	−0.975804	−	0.218647i	$-0.929836\pi$
$197$		−	86.1469i		−	0.437294i	0.975804	−	0.218647i	$-0.0701643\pi$
$198$	−102.915	+	154.029i	−0.519773	+	0.777925i
$199$	−88.0000			−0.442211			−0.221106	−	0.975250i	$-0.570967\pi$
$199$	−88.0000			−0.442211			−0.221106	+	0.975250i	$0.570967\pi$
$200$		0			0
$201$	186.745	−	99.8194i	0.929080	−	0.496614i
$202$	−81.7935			−0.404918
$203$		−	94.6321i		−	0.466168i
$204$	−60.9150	−	113.962i	−0.298603	−	0.558635i
$205$		0			0
$206$			168.818i			0.819506i
$207$	290.583	+	194.154i	1.40378	+	0.937942i
$208$	2.33202			0.0112116
$209$		−	232.870i		−	1.11421i
$210$		0			0
$211$	−61.2549			−0.290308			−0.145154	−	0.989409i	$-0.546368\pi$
$211$	−61.2549			−0.290308			−0.145154	+	0.989409i	$0.546368\pi$
$212$			101.823i			0.480299i
$213$	−24.0810	−	45.0515i	−0.113057	−	0.211509i
$214$	164.745			0.769837
$215$		0			0
$216$	76.0000	+	7.48331i	0.351852	+	0.0346450i
$217$	−154.664			−0.712738
$218$			124.210i			0.569773i
$219$	191.373	−	102.293i	0.873847	−	0.467091i
$220$		0			0
$221$		−	12.5560i		−	0.0568146i
$222$	40.0000	+	74.8331i	0.180180	+	0.337086i
$223$	−150.494			−0.674861			−0.337431	−	0.941350i	$-0.609558\pi$
$223$	−150.494			−0.674861			−0.337431	+	0.941350i	$0.609558\pi$
$224$			14.9666i			0.0668153i
$225$		0			0
$226$	−97.8301			−0.432876
$227$		−	190.558i		−	0.839464i	−0.907648	−	0.419732i	$-0.862124\pi$
$227$		−	190.558i		−	0.839464i	0.907648	−	0.419732i	$-0.137876\pi$
$228$	−84.6640	+	45.2548i	−0.371334	+	0.198486i
$229$	−142.915			−0.624083			−0.312042	−	0.950068i	$-0.601013\pi$
$229$	−142.915			−0.624083			−0.312042	+	0.950068i	$0.601013\pi$
$230$		0			0
$231$	−54.4575	−	101.881i	−0.235747	−	0.441042i
$232$	−101.166			−0.436060
$233$		−	7.83826i		−	0.0336406i	−0.999859	−	0.0168203i	$-0.994646\pi$
$233$		−	7.83826i		−	0.0336406i	0.999859	−	0.0168203i	$-0.00535432\pi$
$234$	6.16995	+	4.12247i	0.0263673	+	0.0176174i
$235$		0			0
$236$			116.435i			0.493369i
$237$	−55.3360	+	29.5783i	−0.233485	+	0.124803i
$238$	80.5830			0.338584
$239$			213.369i			0.892756i	0.894844	+	0.446378i	$0.147286\pi$
$239$			213.369i			0.892756i	−0.894844	+	0.446378i	$0.852714\pi$
$240$		0			0
$241$	−130.000			−0.539419			−0.269710	−	0.962942i	$-0.586928\pi$
$241$	−130.000			−0.539419			−0.269710	+	0.962942i	$0.586928\pi$
$242$		−	128.453i		−	0.530798i
$243$	187.848	+	154.149i	0.773038	+	0.634359i
$244$	−77.8301			−0.318976
$245$		0			0
$246$	32.7451	−	17.5030i	0.133110	−	0.0711503i
$247$	−9.32808			−0.0377655
$248$			165.343i			0.666705i
$249$	205.830	+	385.073i	0.826627	+	1.54648i
$250$		0			0
$251$		−	389.258i		−	1.55083i	−0.631453	−	0.775415i	$-0.717541\pi$
$251$		−	389.258i		−	1.55083i	0.631453	−	0.775415i	$-0.282459\pi$
$252$	−26.4575	+	39.5980i	−0.104990	+	0.157135i
$253$	−565.158			−2.23383
$254$			38.3039i			0.150803i
$255$		0			0
$256$	16.0000			0.0625000
$257$		−	336.139i		−	1.30793i	−0.756523	−	0.653967i	$-0.773103\pi$
$257$		−	336.139i		−	1.30793i	0.756523	−	0.653967i	$-0.226897\pi$
$258$	−23.4980	−	43.9608i	−0.0910776	−	0.170391i
$259$	−52.9150			−0.204305
$260$		0			0
$261$	−267.660	−	178.838i	−1.02552	−	0.685203i
$262$	205.830			0.785611
$263$			438.933i			1.66895i	0.551048	+	0.834473i	$0.314228\pi$
$263$			438.933i			1.66895i	−0.551048	+	0.834473i	$0.685772\pi$
$264$	−108.915	+	58.2175i	−0.412557	+	0.220521i
$265$		0			0
$266$		−	59.8665i		−	0.225062i
$267$	75.8745	+	141.948i	0.284174	+	0.531641i
$268$	141.166			0.526739
$269$		−	14.4601i		−	0.0537549i	−0.999639	−	0.0268775i	$-0.991444\pi$
$269$		−	14.4601i		−	0.0537549i	0.999639	−	0.0268775i	$-0.00855639\pi$
$270$		0			0
$271$	61.5425			0.227094			0.113547	−	0.993533i	$-0.463779\pi$
$271$	61.5425			0.227094			0.113547	+	0.993533i	$0.463779\pi$
$272$		−	86.1469i		−	0.316716i
$273$	−4.08104	+	2.18141i	−0.0149489	+	0.00799050i
$274$	−35.0850			−0.128047
$275$		0			0
$276$	109.830	+	205.473i	0.397935	+	0.744468i
$277$	−150.494			−0.543300			−0.271650	−	0.962396i	$-0.587569\pi$
$277$	−150.494			−0.543300			−0.271650	+	0.962396i	$0.587569\pi$
$278$			207.122i			0.745044i
$279$	−292.288	+	437.456i	−1.04763	+	1.56794i
$280$		0			0
$281$			300.220i			1.06840i	0.845359	+	0.534199i	$0.179387\pi$
$281$			300.220i			1.06840i	−0.845359	+	0.534199i	$0.820613\pi$
$282$	−31.7490	+	16.9706i	−0.112585	+	0.0601793i
$283$	−98.8340			−0.349237			−0.174618	−	0.984636i	$-0.555869\pi$
$283$	−98.8340			−0.349237			−0.174618	+	0.984636i	$0.555869\pi$
$284$		−	34.0557i		−	0.119915i
$285$		0			0
$286$	−12.0000			−0.0419580
$287$			23.1543i			0.0806769i
$288$	42.3320	+	28.2843i	0.146986	+	0.0982093i
$289$	−174.830			−0.604948
$290$		0			0
$291$	−294.118	+	157.212i	−1.01071	+	0.540249i
$292$	144.664			0.495425
$293$		−	7.15424i		−	0.0244172i	−0.999925	−	0.0122086i	$-0.996114\pi$
$293$		−	7.15424i		−	0.0244172i	0.999925	−	0.0122086i	$-0.00388621\pi$
$294$	−14.0000	−	26.1916i	−0.0476190	−	0.0890871i
$295$		0			0
$296$			56.5685i			0.191110i
$297$	−391.077	−	38.5073i	−1.31676	−	0.129654i
$298$	310.494			1.04193
$299$			22.6386i			0.0757142i
$300$		0			0
$301$	31.0850			0.103272
$302$		−	299.333i		−	0.991168i
$303$	−81.7935	−	153.022i	−0.269945	−	0.505022i
$304$	−64.0000			−0.210526
$305$		0			0
$306$	152.288	−	227.923i	0.497672	−	0.744847i
$307$	−105.830			−0.344723			−0.172362	−	0.985034i	$-0.555140\pi$
$307$	−105.830			−0.344723			−0.172362	+	0.985034i	$0.555140\pi$
$308$		−	77.0146i		−	0.250047i
$309$	−315.830	+	168.818i	−1.02210	+	0.546337i
$310$		0			0
$311$		−	265.403i		−	0.853385i	−0.904397	−	0.426692i	$-0.859679\pi$
$311$		−	265.403i		−	0.853385i	0.904397	−	0.426692i	$-0.140321\pi$
$312$	2.33202	+	4.36281i	0.00747443	+	0.0139834i
$313$	−259.328			−0.828524			−0.414262	−	0.910158i	$-0.635960\pi$
$313$	−259.328			−0.828524			−0.414262	+	0.910158i	$0.635960\pi$
$314$		−	295.210i		−	0.940160i
$315$		0			0
$316$	−41.8301			−0.132374
$317$		−	37.8233i		−	0.119316i	−0.998219	−	0.0596581i	$-0.980999\pi$
$317$		−	37.8233i		−	0.119316i	0.998219	−	0.0596581i	$-0.0190011\pi$
$318$	−190.494	+	101.823i	−0.599038	+	0.320199i
$319$	520.575			1.63190
$320$		0			0
$321$	164.745	+	308.210i	0.513225	+	0.960155i
$322$	−145.292			−0.451216
$323$			344.587i			1.06683i
$324$	62.0000	+	149.666i	0.191358	+	0.461933i
$325$		0			0
$326$		−	377.589i		−	1.15825i
$327$	−232.376	+	124.210i	−0.710631	+	0.379848i
$328$	24.7530			0.0754663
$329$		−	22.4499i		−	0.0682369i
$330$		0			0
$331$	145.490			0.439547			0.219774	−	0.975551i	$-0.429468\pi$
$331$	145.490			0.439547			0.219774	+	0.975551i	$0.429468\pi$
$332$			291.088i			0.876770i
$333$	−100.000	+	149.666i	−0.300300	+	0.449448i
$334$	10.1699			0.0304489
$335$		0			0
$336$	−28.0000	+	14.9666i	−0.0833333	+	0.0445435i
$337$	600.316			1.78135			0.890677	−	0.454637i	$-0.150231\pi$
$337$	600.316			1.78135			0.890677	+	0.454637i	$0.150231\pi$
$338$		−	238.521i		−	0.705685i
$339$	−97.8301	−	183.023i	−0.288584	−	0.539892i
$340$		0			0
$341$		−	850.813i		−	2.49505i
$342$	−169.328	−	113.137i	−0.495111	−	0.330810i
$343$	18.5203			0.0539949
$344$		−	33.2312i		−	0.0966024i
$345$		0			0
$346$	272.288			0.786958
$347$		−	31.6395i		−	0.0911803i	−0.998960	−	0.0455901i	$-0.985483\pi$
$347$		−	31.6395i		−	0.0911803i	0.998960	−	0.0455901i	$-0.0145168\pi$
$348$	−101.166	−	189.264i	−0.290707	−	0.543863i
$349$	−592.405			−1.69744			−0.848718	−	0.528846i	$-0.822625\pi$
$349$	−592.405			−1.69744			−0.848718	+	0.528846i	$0.822625\pi$
$350$		0			0
$351$	−1.54249	+	15.6654i	−0.00439455	+	0.0446307i
$352$	−82.3320			−0.233898
$353$			621.977i			1.76197i	0.473141	+	0.880987i	$0.343120\pi$
$353$			621.977i			1.76197i	−0.473141	+	0.880987i	$0.656880\pi$
$354$	−217.830	+	116.435i	−0.615339	+	0.328913i
$355$		0			0
$356$			107.303i			0.301412i
$357$	80.5830	+	150.757i	0.225723	+	0.422289i
$358$	−61.7490			−0.172483
$359$			499.509i			1.39139i	0.718337	+	0.695696i	$0.244904\pi$
$359$			499.509i			1.39139i	−0.718337	+	0.695696i	$0.755096\pi$
$360$		0			0
$361$	−105.000			−0.290859
$362$		−	114.671i		−	0.316772i
$363$	240.314	−	128.453i	0.662021	−	0.353865i
$364$	−3.08497			−0.00847520
$365$		0			0
$366$	−77.8301	−	145.607i	−0.212650	−	0.397833i
$367$	−73.0039			−0.198921			−0.0994604	−	0.995042i	$-0.531712\pi$
$367$	−73.0039			−0.198921			−0.0994604	+	0.995042i	$0.531712\pi$
$368$			155.323i			0.422074i
$369$	65.4902	+	43.7575i	0.177480	+	0.118584i
$370$		0			0
$371$		−	134.700i		−	0.363072i
$372$	−309.328	+	165.343i	−0.831527	+	0.444470i
$373$	474.664			1.27256			0.636279	−	0.771459i	$-0.280473\pi$
$373$	474.664			1.27256			0.636279	+	0.771459i	$0.280473\pi$
$374$			443.290i			1.18527i
$375$		0			0
$376$	−24.0000			−0.0638298
$377$		−	20.8527i		−	0.0553122i
$378$	−100.539	−	9.89949i	−0.265975	−	0.0261891i
$379$	−223.660			−0.590132			−0.295066	−	0.955477i	$-0.595342\pi$
$379$	−223.660			−0.590132			−0.295066	+	0.955477i	$0.595342\pi$
$380$		0			0
$381$	−71.6601	+	38.3039i	−0.188084	+	0.100535i
$382$	75.7411			0.198275
$383$		−	518.175i		−	1.35294i	−0.736471	−	0.676469i	$-0.763509\pi$
$383$		−	518.175i		−	1.35294i	0.736471	−	0.676469i	$-0.236491\pi$
$384$	16.0000	+	29.9333i	0.0416667	+	0.0779512i
$385$		0			0
$386$			156.262i			0.404824i
$387$	58.7451	−	87.9216i	0.151796	−	0.227188i
$388$	−222.332			−0.573021
$389$			44.1383i			0.113466i	0.998389	+	0.0567330i	$0.0180684\pi$
$389$			44.1383i			0.113466i	−0.998389	+	0.0567330i	$0.981932\pi$
$390$		0			0
$391$	836.288			2.13884
$392$		−	19.7990i		−	0.0505076i
$393$	205.830	+	385.073i	0.523741	+	0.979829i
$394$	121.830			0.309213
$395$		0			0
$396$	−217.830	−	145.544i	−0.550076	−	0.367535i
$397$	−299.417			−0.754199			−0.377099	−	0.926173i	$-0.623079\pi$
$397$	−299.417			−0.754199			−0.377099	+	0.926173i	$0.623079\pi$
$398$		−	124.451i		−	0.312690i
$399$	112.000	−	59.8665i	0.280702	−	0.150041i
$400$		0			0
$401$			277.008i			0.690794i	0.938457	+	0.345397i	$0.112256\pi$
$401$			277.008i			0.690794i	−0.938457	+	0.345397i	$0.887744\pi$
$402$	141.166	+	264.097i	0.351159	+	0.656959i
$403$	−34.0810			−0.0845683
$404$		−	115.673i		−	0.286320i
$405$		0			0
$406$	133.830			0.329631
$407$		−	291.088i		−	0.715203i
$408$	161.166	−	86.1469i	0.395015	−	0.211144i
$409$	−740.980			−1.81169			−0.905844	−	0.423612i	$-0.860762\pi$
$409$	−740.980			−1.81169			−0.905844	+	0.423612i	$0.860762\pi$
$410$		0			0
$411$	−35.0850	−	65.6380i	−0.0853649	−	0.159703i
$412$	−238.745			−0.579478
$413$		−	154.029i		−	0.372952i
$414$	−274.575	+	410.946i	−0.663225	+	0.992624i
$415$		0			0
$416$			3.29798i			0.00792783i
$417$	−387.490	+	207.122i	−0.929233	+	0.496696i
$418$	329.328			0.787866
$419$			89.7998i			0.214319i	0.994242	+	0.107160i	$0.0341756\pi$
$419$			89.7998i			0.214319i	−0.994242	+	0.107160i	$0.965824\pi$
$420$		0			0
$421$	281.150			0.667815			0.333908	−	0.942606i	$-0.391633\pi$
$421$	281.150			0.667815			0.333908	+	0.942606i	$0.391633\pi$
$422$		−	86.6275i		−	0.205279i
$423$	−63.4980	−	42.4264i	−0.150114	−	0.100299i
$424$	−144.000			−0.339623
$425$		0			0
$426$	63.7124	−	34.0557i	0.149560	−	0.0799430i
$427$	102.959			0.241123
$428$			232.985i			0.544357i
$429$	−12.0000	−	22.4499i	−0.0279720	−	0.0523309i
$430$		0			0
$431$			560.200i			1.29977i	0.760033	+	0.649885i	$0.225183\pi$
$431$			560.200i			1.29977i	−0.760033	+	0.649885i	$0.774817\pi$
$432$	−10.5830	+	107.480i	−0.0244977	+	0.248797i
$433$	543.004			1.25405			0.627025	−	0.778999i	$-0.284272\pi$
$433$	543.004			1.25405			0.627025	+	0.778999i	$0.284272\pi$
$434$		−	218.728i		−	0.503982i
$435$		0			0
$436$	−175.660			−0.402890
$437$		−	621.293i		−	1.42172i
$438$	144.664	+	270.642i	0.330283	+	0.617903i
$439$	342.170			0.779430			0.389715	−	0.920935i	$-0.372573\pi$
$439$	342.170			0.779430			0.389715	+	0.920935i	$0.372573\pi$
$440$		0			0
$441$	35.0000	−	52.3832i	0.0793651	−	0.118783i
$442$	17.7569			0.0401740
$443$			399.816i			0.902519i	0.892393	+	0.451259i	$0.149025\pi$
$443$			399.816i			0.902519i	−0.892393	+	0.451259i	$0.850975\pi$
$444$	−105.830	+	56.5685i	−0.238356	+	0.127407i
$445$		0			0
$446$		−	212.831i		−	0.477199i
$447$	310.494	+	580.881i	0.694618	+	1.29951i
$448$	−21.1660			−0.0472456
$449$		−	737.040i		−	1.64151i	−0.571277	−	0.820757i	$-0.693552\pi$
$449$		−	737.040i		−	1.64151i	0.571277	−	0.820757i	$-0.306448\pi$
$450$		0			0
$451$	−127.373			−0.282422
$452$		−	138.353i		−	0.306090i
$453$	560.000	−	299.333i	1.23620	−	0.660778i
$454$	269.490			0.593591
$455$		0			0
$456$	−64.0000	−	119.733i	−0.140351	−	0.262572i
$457$	−665.336			−1.45588			−0.727939	−	0.685642i	$-0.759521\pi$
$457$	−665.336			−1.45588			−0.727939	+	0.685642i	$0.759521\pi$
$458$		−	202.112i		−	0.441293i
$459$	578.693	+	56.9808i	1.26077	+	0.124141i
$460$		0			0
$461$		−	318.865i		−	0.691682i	−0.938293	−	0.345841i	$-0.887594\pi$
$461$		−	318.865i		−	0.691682i	0.938293	−	0.345841i	$-0.112406\pi$
$462$	144.081	−	77.0146i	0.311864	−	0.166698i
$463$	−402.332			−0.868968			−0.434484	−	0.900680i	$-0.643069\pi$
$463$	−402.332			−0.868968			−0.434484	+	0.900680i	$0.643069\pi$
$464$		−	143.070i		−	0.308341i
$465$		0			0
$466$	11.0850			0.0237875
$467$			697.087i			1.49269i	0.665558	+	0.746346i	$0.268194\pi$
$467$			697.087i			1.49269i	−0.665558	+	0.746346i	$0.731806\pi$
$468$	−5.83005	+	8.72562i	−0.0124574	+	0.0186445i
$469$	−186.745			−0.398177
$470$		0			0
$471$	552.288	−	295.210i	1.17259	−	0.626773i
$472$	−164.664			−0.348864
$473$			171.000i			0.361522i
$474$	−41.8301	−	78.2569i	−0.0882491	−	0.165099i
$475$		0			0
$476$			113.962i			0.239415i
$477$	−380.988	−	254.558i	−0.798717	−	0.533665i
$478$	−301.749			−0.631274
$479$		−	163.808i		−	0.341980i	−0.985273	−	0.170990i	$-0.945303\pi$
$479$		−	163.808i		−	0.341980i	0.985273	−	0.170990i	$-0.0546966\pi$
$480$		0			0
$481$	−11.6601			−0.0242414
$482$		−	183.848i		−	0.381427i
$483$	−145.292	−	271.816i	−0.300811	−	0.562765i
$484$	181.660			0.375331
$485$		0			0
$486$	−218.000	+	265.658i	−0.448560	+	0.546621i
$487$	103.498			0.212522			0.106261	−	0.994338i	$-0.466112\pi$
$487$	103.498			0.212522			0.106261	+	0.994338i	$0.466112\pi$
$488$		−	110.068i		−	0.225550i
$489$	706.405	−	377.589i	1.44459	−	0.772167i
$490$		0			0
$491$			570.094i			1.16109i	0.814229	+	0.580544i	$0.197160\pi$
$491$			570.094i			1.16109i	−0.814229	+	0.580544i	$0.802840\pi$
$492$	24.7530	+	46.3085i	0.0503109	+	0.0941230i
$493$	−770.316			−1.56251
$494$		−	13.1919i		−	0.0267043i
$495$		0			0
$496$	−233.830			−0.471432
$497$			45.0515i			0.0906469i
$498$	−544.575	+	291.088i	−1.09352	+	0.584513i
$499$	−487.660			−0.977275			−0.488637	−	0.872487i	$-0.662506\pi$
$499$	−487.660			−0.977275			−0.488637	+	0.872487i	$0.662506\pi$
$500$		0			0
$501$	10.1699	+	19.0262i	0.0202993	+	0.0379765i
$502$	550.494			1.09660
$503$		−	97.9412i		−	0.194714i	−0.995250	−	0.0973571i	$-0.968961\pi$
$503$		−	97.9412i		−	0.194714i	0.995250	−	0.0973571i	$-0.0310389\pi$
$504$	−56.0000	−	37.4166i	−0.111111	−	0.0742392i
$505$		0			0
$506$		−	799.254i		−	1.57955i
$507$	446.233	−	238.521i	0.880143	−	0.470456i
$508$	−54.1699			−0.106634
$509$			318.104i			0.624958i	0.949925	+	0.312479i	$0.101159\pi$
$509$			318.104i			0.624958i	−0.949925	+	0.312479i	$0.898841\pi$
$510$		0			0
$511$	−191.373			−0.374506
$512$			22.6274i			0.0441942i
$513$	42.3320	−	429.921i	0.0825186	−	0.838052i
$514$	475.373			0.924849
$515$		0			0
$516$	62.1699	−	33.2312i	0.120484	−	0.0644016i
$517$	123.498			0.238874
$518$		−	74.8331i		−	0.144466i
$519$	272.288	+	509.403i	0.524639	+	0.981509i
$520$		0			0
$521$			727.150i			1.39568i	0.716253	+	0.697840i	$0.245856\pi$
$521$			727.150i			1.39568i	−0.716253	+	0.697840i	$0.754144\pi$
$522$	252.915	−	378.529i	0.484512	−	0.725150i
$523$	207.624			0.396986			0.198493	−	0.980102i	$-0.436395\pi$
$523$	207.624			0.396986			0.198493	+	0.980102i	$0.436395\pi$
$524$			291.088i			0.555511i
$525$		0			0
$526$	−620.745			−1.18012
$527$			1258.98i			2.38896i
$528$	−82.3320	−	154.029i	−0.155932	−	0.291722i
$529$	−978.830			−1.85034
$530$		0			0
$531$	−435.660	−	291.088i	−0.820452	−	0.548188i
$532$	84.6640			0.159143
$533$			5.10216i			0.00957254i
$534$	−200.745	+	107.303i	−0.375927	+	0.200941i
$535$		0			0
$536$			199.639i			0.372461i
$537$	−61.7490	−	115.522i	−0.114989	−	0.215124i
$538$	20.4496			0.0380105
$539$			101.881i			0.189018i
$540$		0			0
$541$	1025.15			1.89492			0.947459	−	0.319878i	$-0.103642\pi$
$541$	1025.15			1.89492			0.947459	+	0.319878i	$0.103642\pi$
$542$			87.0342i			0.160580i
$543$	214.531	−	114.671i	0.395084	−	0.211181i
$544$	121.830			0.223952
$545$		0			0
$546$	−3.08497	−	5.77146i	−0.00565014	−	0.0105704i
$547$	560.089			1.02393			0.511964	−	0.859007i	$-0.328918\pi$
$547$	560.089			1.02393			0.511964	+	0.859007i	$0.328918\pi$
$548$		−	49.6176i		−	0.0905431i
$549$	194.575	−	291.213i	0.354417	−	0.530443i
$550$		0			0
$551$			572.281i			1.03862i
$552$	−290.583	+	155.323i	−0.526418	+	0.281383i
$553$	55.3360			0.100065
$554$		−	212.831i		−	0.384171i
$555$		0			0
$556$	−292.915			−0.526826
$557$		−	575.705i		−	1.03358i	−0.856112	−	0.516791i	$-0.827126\pi$
$557$		−	575.705i		−	1.03358i	0.856112	−	0.516791i	$-0.172874\pi$
$558$	−618.656	−	413.357i	−1.10870	−	0.740783i
$559$	6.84974			0.0122536
$560$		0			0
$561$	−829.320	+	443.290i	−1.47829	+	0.790179i
$562$	−424.575			−0.755472
$563$			468.558i			0.832251i	0.909307	+	0.416126i	$0.136612\pi$
$563$			468.558i			0.832251i	−0.909307	+	0.416126i	$0.863388\pi$
$564$	−24.0000	−	44.8999i	−0.0425532	−	0.0796097i
$565$		0			0
$566$		−	139.772i		−	0.246948i
$567$	−82.0183	−	197.990i	−0.144653	−	0.349189i
$568$	48.1621			0.0847924
$569$		−	506.455i		−	0.890079i	−0.895511	−	0.445039i	$-0.853190\pi$
$569$		−	506.455i		−	0.890079i	0.895511	−	0.445039i	$-0.146810\pi$
$570$		0			0
$571$	32.9150			0.0576445			0.0288223	−	0.999585i	$-0.490824\pi$
$571$	32.9150			0.0576445			0.0288223	+	0.999585i	$0.490824\pi$
$572$		−	16.9706i		−	0.0296688i
$573$	75.7411	+	141.699i	0.132183	+	0.247293i
$574$	−32.7451			−0.0570472
$575$		0			0
$576$	−40.0000	+	59.8665i	−0.0694444	+	0.103935i
$577$	252.154			0.437009			0.218505	−	0.975836i	$-0.429882\pi$
$577$	252.154			0.437009			0.218505	+	0.975836i	$0.429882\pi$
$578$		−	247.247i		−	0.427763i
$579$	−292.340	+	156.262i	−0.504905	+	0.269883i
$580$		0			0
$581$		−	385.073i		−	0.662776i
$582$	−222.332	−	415.945i	−0.382014	−	0.714682i
$583$	740.988			1.27099
$584$			204.586i			0.350318i
$585$		0			0
$586$	10.1176			0.0172656
$587$		−	366.882i		−	0.625012i	−0.949916	−	0.312506i	$-0.898832\pi$
$587$		−	366.882i		−	0.625012i	0.949916	−	0.312506i	$-0.101168\pi$
$588$	37.0405	−	19.7990i	0.0629941	−	0.0336718i
$589$	935.320			1.58798
$590$		0			0
$591$	121.830	+	227.923i	0.206142	+	0.385657i
$592$	−80.0000			−0.135135
$593$			620.757i			1.04681i	0.852085	+	0.523404i	$0.175338\pi$
$593$			620.757i			1.04681i	−0.852085	+	0.523404i	$0.824662\pi$
$594$	54.4575	−	553.067i	0.0916793	−	0.931088i
$595$		0			0
$596$			439.105i			0.736753i
$597$	232.826	−	124.451i	0.389993	−	0.208460i
$598$	−32.0157			−0.0535380
$599$		−	26.3488i		−	0.0439879i	−0.999758	−	0.0219940i	$-0.992999\pi$
$599$		−	26.3488i		−	0.0439879i	0.999758	−	0.0219940i	$-0.00700146\pi$
$600$		0			0
$601$	930.470			1.54820			0.774102	−	0.633061i	$-0.218202\pi$
$601$	930.470			1.54820			0.774102	+	0.633061i	$0.218202\pi$
$602$			43.9608i			0.0730246i
$603$	−352.915	+	528.195i	−0.585265	+	0.875945i
$604$	423.320			0.700861
$605$		0			0
$606$	216.405	−	115.673i	0.357104	−	0.190880i
$607$	−216.146			−0.356089			−0.178045	−	0.984022i	$-0.556977\pi$
$607$	−216.146			−0.356089			−0.178045	+	0.984022i	$0.556977\pi$
$608$		−	90.5097i		−	0.148865i
$609$	133.830	+	250.373i	0.219754	+	0.411122i
$610$		0			0
$611$		−	4.94696i		−	0.00809650i
$612$	322.332	+	215.367i	0.526686	+	0.351907i
$613$	268.834			0.438555			0.219277	−	0.975663i	$-0.429630\pi$
$613$	268.834			0.438555			0.219277	+	0.975663i	$0.429630\pi$
$614$		−	149.666i		−	0.243756i
$615$		0			0
$616$	108.915			0.176810
$617$			531.338i			0.861163i	0.902552	+	0.430582i	$0.141692\pi$
$617$			531.338i			0.861163i	−0.902552	+	0.430582i	$0.858308\pi$
$618$	−238.745	−	446.651i	−0.386319	−	0.722736i
$619$	425.203			0.686919			0.343459	−	0.939168i	$-0.388401\pi$
$619$	425.203			0.686919			0.343459	+	0.939168i	$0.388401\pi$
$620$		0			0
$621$	−1043.39	−	102.737i	−1.68017	−	0.165437i
$622$	375.336			0.603434
$623$		−	141.948i		−	0.227846i
$624$	−6.16995	+	3.29798i	−0.00988774	+	0.00528522i
$625$		0			0
$626$		−	366.745i		−	0.585855i
$627$	329.328	+	616.116i	0.525244	+	0.982642i
$628$	417.490			0.664793
$629$			430.734i			0.684792i
$630$		0			0
$631$	−453.490			−0.718685			−0.359342	−	0.933206i	$-0.616999\pi$
$631$	−453.490			−0.718685			−0.359342	+	0.933206i	$0.616999\pi$
$632$		−	59.1566i		−	0.0936023i
$633$	162.065	−	86.6275i	0.256027	−	0.136852i
$634$	53.4902			0.0843693
$635$		0			0
$636$	−144.000	−	269.399i	−0.226415	−	0.423584i
$637$	4.08104			0.00640665
$638$			736.204i			1.15393i
$639$	127.425	+	85.1393i	0.199413	+	0.133238i
$640$		0			0
$641$		−	463.455i		−	0.723019i	−0.932368	−	0.361510i	$-0.882261\pi$
$641$		−	463.455i		−	0.723019i	0.932368	−	0.361510i	$-0.117739\pi$
$642$	−435.875	+	232.985i	−0.678932	+	0.362905i
$643$	−820.988			−1.27681			−0.638405	−	0.769701i	$-0.720405\pi$
$643$	−820.988			−1.27681			−0.638405	+	0.769701i	$0.720405\pi$
$644$		−	205.473i		−	0.319058i
$645$		0			0
$646$	−487.320			−0.754366
$647$		−	996.660i		−	1.54043i	−0.637783	−	0.770216i	$-0.720148\pi$
$647$		−	996.660i		−	1.54043i	0.637783	−	0.770216i	$-0.279852\pi$
$648$	−211.660	+	87.6812i	−0.326636	+	0.135311i
$649$	847.320			1.30558
$650$		0			0
$651$	409.203	−	218.728i	0.628575	−	0.335988i
$652$	533.992			0.819006
$653$		−	357.602i		−	0.547629i	−0.961782	−	0.273815i	$-0.911715\pi$
$653$		−	357.602i		−	0.547629i	0.961782	−	0.273815i	$-0.0882855\pi$
$654$	−175.660	−	328.630i	−0.268593	−	0.502492i
$655$		0			0
$656$			35.0060i			0.0533627i
$657$	−361.660	+	541.283i	−0.550472	+	0.823871i
$658$	31.7490			0.0482508
$659$			131.562i			0.199640i	0.995006	+	0.0998198i	$0.0318266\pi$
$659$			131.562i			0.199640i	−0.995006	+	0.0998198i	$0.968173\pi$
$660$		0			0
$661$	−250.915			−0.379599			−0.189800	−	0.981823i	$-0.560784\pi$
$661$	−250.915			−0.379599			−0.189800	+	0.981823i	$0.560784\pi$
$662$			205.754i			0.310807i
$663$	17.7569	+	33.2201i	0.0267826	+	0.0501057i
$664$	−411.660			−0.619970
$665$		0			0
$666$	−211.660	−	141.421i	−0.317808	−	0.212344i
$667$	1388.88			2.08228
$668$			14.3825i			0.0215307i
$669$	398.170	−	212.831i	0.595172	−	0.318133i
$670$		0			0
$671$			566.384i			0.844090i
$672$	−21.1660	−	39.5980i	−0.0314970	−	0.0589256i
$673$	−196.502			−0.291979			−0.145990	−	0.989286i	$-0.546637\pi$
$673$	−196.502			−0.291979			−0.145990	+	0.989286i	$0.546637\pi$
$674$			848.975i			1.25961i
$675$		0			0
$676$	337.320			0.498994
$677$			54.1838i			0.0800352i	0.999199	+	0.0400176i	$0.0127414\pi$
$677$			54.1838i			0.0800352i	−0.999199	+	0.0400176i	$0.987259\pi$
$678$	258.834	−	138.353i	0.381761	−	0.204060i
$679$	294.118			0.433163
$680$		0			0
$681$	269.490	+	504.170i	0.395727	+	0.740338i
$682$	1203.23			1.76427
$683$		−	749.006i		−	1.09664i	−0.836268	−	0.548321i	$-0.815267\pi$
$683$		−	749.006i		−	1.09664i	0.836268	−	0.548321i	$-0.184733\pi$
$684$	160.000	−	239.466i	0.233918	−	0.350097i
$685$		0			0
$686$			26.1916i			0.0381802i
$687$	378.118	−	202.112i	0.550390	−	0.294196i
$688$	46.9961			0.0683082
$689$		−	29.6818i		−	0.0430795i
$690$		0			0
$691$	475.137			0.687608			0.343804	−	0.939041i	$-0.388284\pi$
$691$	475.137			0.687608			0.343804	+	0.939041i	$0.388284\pi$
$692$			385.073i			0.556464i
$693$	288.162	+	192.536i	0.415818	+	0.277830i
$694$	44.7451			0.0644742
$695$		0			0
$696$	267.660	−	143.070i	0.384569	−	0.205561i
$697$	188.478			0.270414
$698$		−	837.787i		−	1.20027i
$699$	11.0850	+	20.7381i	0.0158583	+	0.0296682i
$700$		0			0
$701$			1251.49i			1.78529i	0.450760	+	0.892645i	$0.351153\pi$
$701$			1251.49i			1.78529i	−0.450760	+	0.892645i	$0.648847\pi$
$702$	−22.1542	−	2.18141i	−0.0315587	−	0.00310742i
$703$	320.000			0.455192
$704$		−	116.435i		−	0.165391i
$705$		0			0
$706$	−879.608			−1.24590
$707$			153.022i			0.216438i
$708$	−164.664	−	308.058i	−0.232576	−	0.435110i
$709$	−16.5098			−0.0232861			−0.0116430	−	0.999932i	$-0.503706\pi$
$709$	−16.5098			−0.0232861			−0.0116430	+	0.999932i	$0.503706\pi$
$710$		0			0
$711$	104.575	−	156.514i	0.147082	−	0.220132i
$712$	−151.749			−0.213131
$713$		−	2269.95i		−	3.18366i
$714$	−213.203	+	113.962i	−0.298603	+	0.159610i
$715$		0			0
$716$		−	87.3263i		−	0.121964i
$717$	−301.749	−	564.521i	−0.420849	−	0.787337i
$718$	−706.413			−0.983862
$719$		−	1327.59i		−	1.84643i	−0.384279	−	0.923217i	$-0.625550\pi$
$719$		−	1327.59i		−	1.84643i	0.384279	−	0.923217i	$-0.374450\pi$
$720$		0			0
$721$	315.830			0.438044
$722$		−	148.492i		−	0.205668i
$723$	343.948	−	183.848i	0.475723	−	0.254285i
$724$	162.170			0.223992
$725$		0			0
$726$	181.660	+	339.855i	0.250221	+	0.468120i
$727$	202.782			0.278929			0.139465	−	0.990227i	$-0.455462\pi$
$727$	202.782			0.278929			0.139465	+	0.990227i	$0.455462\pi$
$728$		−	4.36281i		−	0.00599287i
$729$	−715.000	−	142.183i	−0.980796	−	0.195038i
$730$		0			0
$731$		−	253.035i		−	0.346149i
$732$	205.919	−	110.068i	0.281310	−	0.150367i
$733$	582.559			0.794760			0.397380	−	0.917654i	$-0.369919\pi$
$733$	582.559			0.794760			0.397380	+	0.917654i	$0.369919\pi$
$734$		−	103.243i		−	0.140658i
$735$		0			0
$736$	−219.660			−0.298451
$737$		−	1027.29i		−	1.39388i
$738$	−61.8824	+	92.6171i	−0.0838515	+	0.125497i
$739$	−256.810			−0.347511			−0.173755	−	0.984789i	$-0.555590\pi$
$739$	−256.810			−0.347511			−0.173755	+	0.984789i	$0.555590\pi$
$740$		0			0
$741$	24.6798	−	13.1919i	0.0333061	−	0.0178028i
$742$	190.494			0.256731
$743$			573.256i			0.771542i	0.922595	+	0.385771i	$0.126064\pi$
$743$			573.256i			0.771542i	−0.922595	+	0.385771i	$0.873936\pi$
$744$	−233.830	−	437.456i	−0.314288	−	0.587978i
$745$		0			0
$746$			671.276i			0.899834i
$747$	−1089.15	−	727.719i	−1.45803	−	0.974189i
$748$	−626.907			−0.838111
$749$		−	308.210i		−	0.411495i
$750$		0			0
$751$	−579.085			−0.771085			−0.385543	−	0.922690i	$-0.625986\pi$
$751$	−579.085			−0.771085			−0.385543	+	0.922690i	$0.625986\pi$
$752$		−	33.9411i		−	0.0451345i
$753$	550.494	+	1029.88i	0.731068	+	1.36770i
$754$	29.4902			0.0391116
$755$		0			0
$756$	14.0000	−	142.183i	0.0185185	−	0.188073i
$757$	−1167.13			−1.54179			−0.770895	−	0.636963i	$-0.780191\pi$
$757$	−1167.13			−1.54179			−0.770895	+	0.636963i	$0.780191\pi$
$758$		−	316.303i		−	0.417286i
$759$	1495.27	−	799.254i	1.97005	−	1.05304i
$760$		0			0
$761$		−	800.970i		−	1.05252i	−0.850323	−	0.526261i	$-0.823593\pi$
$761$		−	800.970i		−	1.05252i	0.850323	−	0.526261i	$-0.176407\pi$
$762$	−54.1699	−	101.343i	−0.0710892	−	0.132996i
$763$	232.376			0.304556
$764$			107.114i			0.140202i
$765$		0			0
$766$	732.810			0.956671
$767$		−	33.9411i		−	0.0442518i
$768$	−42.3320	+	22.6274i	−0.0551198	+	0.0294628i
$769$	−242.680			−0.315578			−0.157789	−	0.987473i	$-0.550437\pi$
$769$	−242.680			−0.315578			−0.157789	+	0.987473i	$0.550437\pi$
$770$		0			0
$771$	475.373	+	889.341i	0.616566	+	1.15349i
$772$	−220.988			−0.286254
$773$			1262.83i			1.63367i	0.576870	+	0.816836i	$0.304274\pi$
$773$			1262.83i			1.63367i	−0.576870	+	0.816836i	$0.695726\pi$
$774$	124.340	+	83.0781i	0.160646	+	0.107336i
$775$		0			0
$776$		−	314.425i		−	0.405187i
$777$	140.000	−	74.8331i	0.180180	−	0.0963104i
$778$	−62.4209			−0.0802326
$779$		−	140.024i		−	0.179748i
$780$		0			0
$781$	−247.830			−0.317324
$782$			1182.69i			1.51239i
$783$	961.077	+	94.6321i	1.22743	+	0.120858i
$784$	28.0000			0.0357143
$785$		0			0
$786$	−544.575	+	291.088i	−0.692844	+	0.370341i
$787$	−1347.00			−1.71156			−0.855782	−

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:	\( N \)	\(=\)	\( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1050.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.41421i q^{2} +(-2.64575 + 1.41421i) q^{3} -2.00000 q^{4} +(-2.00000 - 3.74166i) q^{6} +2.64575 q^{7} -2.82843i q^{8} +(5.00000 - 7.48331i) q^{9} +O(q^{10})\)	\(q+1.41421i q^{2} +(-2.64575 + 1.41421i) q^{3} -2.00000 q^{4} +(-2.00000 - 3.74166i) q^{6} +2.64575 q^{7} -2.82843i q^{8} +(5.00000 - 7.48331i) q^{9} +14.5544i q^{11} +(5.29150 - 2.82843i) q^{12} +0.583005 q^{13} +3.74166i q^{14} +4.00000 q^{16} -21.5367i q^{17} +(10.5830 + 7.07107i) q^{18} -16.0000 q^{19} +(-7.00000 + 3.74166i) q^{21} -20.5830 q^{22} +38.8308i q^{23} +(4.00000 + 7.48331i) q^{24} +0.824494i q^{26} +(-2.64575 + 26.8701i) q^{27} -5.29150 q^{28} -35.7676i q^{29} -58.4575 q^{31} +5.65685i q^{32} +(-20.5830 - 38.5073i) q^{33} +30.4575 q^{34} +(-10.0000 + 14.9666i) q^{36} -20.0000 q^{37} -22.6274i q^{38} +(-1.54249 + 0.824494i) q^{39} +8.75149i q^{41} +(-5.29150 - 9.89949i) q^{42} +11.7490 q^{43} -29.1088i q^{44} -54.9150 q^{46} -8.48528i q^{47} +(-10.5830 + 5.65685i) q^{48} +7.00000 q^{49} +(30.4575 + 56.9808i) q^{51} -1.16601 q^{52} -50.9117i q^{53} +(-38.0000 - 3.74166i) q^{54} -7.48331i q^{56} +(42.3320 - 22.6274i) q^{57} +50.5830 q^{58} -58.2175i q^{59} +38.9150 q^{61} -82.6714i q^{62} +(13.2288 - 19.7990i) q^{63} -8.00000 q^{64} +(54.4575 - 29.1088i) q^{66} -70.5830 q^{67} +43.0734i q^{68} +(-54.9150 - 102.737i) q^{69} +17.0279i q^{71} +(-21.1660 - 14.1421i) q^{72} -72.3320 q^{73} -28.2843i q^{74} +32.0000 q^{76} +38.5073i q^{77} +(-1.16601 - 2.18141i) q^{78} +20.9150 q^{79} +(-31.0000 - 74.8331i) q^{81} -12.3765 q^{82} -145.544i q^{83} +(14.0000 - 7.48331i) q^{84} +16.6156i q^{86} +(50.5830 + 94.6321i) q^{87} +41.1660 q^{88} -53.6514i q^{89} +1.54249 q^{91} -77.6616i q^{92} +(154.664 - 82.6714i) q^{93} +12.0000 q^{94} +(-8.00000 - 14.9666i) q^{96} +111.166 q^{97} +9.89949i q^{98} +(108.915 + 72.7719i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{4} - 8 q^{6} + 20 q^{9}+O(q^{10}) \) 4 * q - 8 * q^4 - 8 * q^6 + 20 * q^9	\( 4 q - 8 q^{4} - 8 q^{6} + 20 q^{9} - 40 q^{13} + 16 q^{16} - 64 q^{19} - 28 q^{21} - 40 q^{22} + 16 q^{24} - 128 q^{31} - 40 q^{33} + 16 q^{34} - 40 q^{36} - 80 q^{37} - 112 q^{39} - 80 q^{43} - 8 q^{46} + 28 q^{49} + 16 q^{51} + 80 q^{52} - 152 q^{54} + 160 q^{58} - 56 q^{61} - 32 q^{64} + 112 q^{66} - 240 q^{67} - 8 q^{69} - 120 q^{73} + 128 q^{76} + 80 q^{78} - 128 q^{79} - 124 q^{81} - 240 q^{82} + 56 q^{84} + 160 q^{87} + 80 q^{88} + 112 q^{91} + 280 q^{93} + 48 q^{94} - 32 q^{96} + 360 q^{97} + 224 q^{99}+O(q^{100}) \) 4 * q - 8 * q^4 - 8 * q^6 + 20 * q^9 - 40 * q^13 + 16 * q^16 - 64 * q^19 - 28 * q^21 - 40 * q^22 + 16 * q^24 - 128 * q^31 - 40 * q^33 + 16 * q^34 - 40 * q^36 - 80 * q^37 - 112 * q^39 - 80 * q^43 - 8 * q^46 + 28 * q^49 + 16 * q^51 + 80 * q^52 - 152 * q^54 + 160 * q^58 - 56 * q^61 - 32 * q^64 + 112 * q^66 - 240 * q^67 - 8 * q^69 - 120 * q^73 + 128 * q^76 + 80 * q^78 - 128 * q^79 - 124 * q^81 - 240 * q^82 + 56 * q^84 + 160 * q^87 + 80 * q^88 + 112 * q^91 + 280 * q^93 + 48 * q^94 - 32 * q^96 + 360 * q^97 + 224 * q^99

Introduction

L-functions

Modular forms

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