LMFDB - Embedded newform 162.8.c.e.109.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [162,8,Mod(55,162)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(162, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4])) N = Newforms(chi, 8, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("162.55"); S:= CuspForms(chi, 8); N := Newforms(S);

Level:	$ N $	$=$	$ 162 = 2 \cdot 3^{4} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	162.c (of order $3$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,-8,0,-64,165] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$50.6063741284$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			109.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	162.109
Dual form			162.8.c.e.55.1

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-4.00000 - 6.92820i) q^{2} +(-32.0000 + 55.4256i) q^{4} +(82.5000 - 142.894i) q^{5} +(254.000 + 439.941i) q^{7} +512.000 q^{8} -1320.00 q^{10} +(-1512.00 - 2618.86i) q^{11} +(-2519.50 + 4363.90i) q^{13} +(2032.00 - 3519.53i) q^{14} +(-2048.00 - 3547.24i) q^{16} -3189.00 q^{17} +1508.00 q^{19} +(5280.00 + 9145.23i) q^{20} +(-12096.0 + 20950.9i) q^{22} +(37800.0 - 65471.5i) q^{23} +(25450.0 + 44080.7i) q^{25} +40312.0 q^{26} -32512.0 q^{28} +(41332.5 + 71590.0i) q^{29} +(87446.0 - 151461. i) q^{31} +(-16384.0 + 28377.9i) q^{32} +(12756.0 + 22094.0i) q^{34} +83820.0 q^{35} -323569. q^{37} +(-6032.00 - 10447.7i) q^{38} +(42240.0 - 73161.8i) q^{40} +(154059. - 266838. i) q^{41} +(-168340. - 291573. i) q^{43} +193536. q^{44} -604800. q^{46} +(191598. + 331857. i) q^{47} +(282740. - 489719. i) q^{49} +(203600. - 352646. i) q^{50} +(-161248. - 279290. i) q^{52} +760206. q^{53} -498960. q^{55} +(130048. + 225250. i) q^{56} +(330660. - 572720. i) q^{58} +(1.11283e6 - 1.92748e6i) q^{59} +(-1.12241e6 - 1.94407e6i) q^{61} -1.39914e6 q^{62} +262144. q^{64} +(415718. + 720044. i) q^{65} +(-736594. + 1.27582e6i) q^{67} +(102048. - 176752. i) q^{68} +(-335280. - 580722. i) q^{70} -5.00689e6 q^{71} -5.89830e6 q^{73} +(1.29428e6 + 2.24175e6i) q^{74} +(-48256.0 + 83581.8i) q^{76} +(768096. - 1.33038e6i) q^{77} +(-3.51438e6 - 6.08709e6i) q^{79} -675840. q^{80} -2.46494e6 q^{82} +(1.32560e6 + 2.29600e6i) q^{83} +(-263092. + 455690. i) q^{85} +(-1.34672e6 + 2.33259e6i) q^{86} +(-774144. - 1.34086e6i) q^{88} -6.77090e6 q^{89} -2.55981e6 q^{91} +(2.41920e6 + 4.19018e6i) q^{92} +(1.53278e6 - 2.65486e6i) q^{94} +(124410. - 215484. i) q^{95} +(-8.08819e6 - 1.40092e7i) q^{97} -4.52383e6 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 8 q^{2} - 64 q^{4} + 165 q^{5} + 508 q^{7} + 1024 q^{8} - 2640 q^{10} - 3024 q^{11} - 5039 q^{13} + 4064 q^{14} - 4096 q^{16} - 6378 q^{17} + 3016 q^{19} + 10560 q^{20} - 24192 q^{22} + 75600 q^{23}+ \cdots - 9047664 q^{98}+O(q^{100}) $ 2 * q - 8 * q^2 - 64 * q^4 + 165 * q^5 + 508 * q^7 + 1024 * q^8 - 2640 * q^10 - 3024 * q^11 - 5039 * q^13 + 4064 * q^14 - 4096 * q^16 - 6378 * q^17 + 3016 * q^19 + 10560 * q^20 - 24192 * q^22 + 75600 * q^23 + 50900 * q^25 + 80624 * q^26 - 65024 * q^28 + 82665 * q^29 + 174892 * q^31 - 32768 * q^32 + 25512 * q^34 + 167640 * q^35 - 647138 * q^37 - 12064 * q^38 + 84480 * q^40 + 308118 * q^41 - 336680 * q^43 + 387072 * q^44 - 1209600 * q^46 + 383196 * q^47 + 565479 * q^49 + 407200 * q^50 - 322496 * q^52 + 1520412 * q^53 - 997920 * q^55 + 260096 * q^56 + 661320 * q^58 + 2225664 * q^59 - 2244815 * q^61 - 2798272 * q^62 + 524288 * q^64 + 831435 * q^65 - 1473188 * q^67 + 204096 * q^68 - 670560 * q^70 - 10013784 * q^71 - 11796602 * q^73 + 2588552 * q^74 - 96512 * q^76 + 1536192 * q^77 - 7028768 * q^79 - 1351680 * q^80 - 4929888 * q^82 + 2651196 * q^83 - 526185 * q^85 - 2693440 * q^86 - 1548288 * q^88 - 13541802 * q^89 - 5119624 * q^91 + 4838400 * q^92 + 3065568 * q^94 + 248820 * q^95 - 16176386 * q^97 - 9047664 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/162\mathbb{Z}\right)^\times$.

$n$	$83$
$\chi(n)$	$e\left(\frac{1}{3}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	−4.00000	−	6.92820i	−0.353553	−	0.612372i
$2$	−4.00000	−	6.92820i	−0.353553	−	0.612372i
$3$		0			0
$3$		0			0
$4$	−32.0000	+	55.4256i	−0.250000	+	0.433013i
$5$	82.5000	−	142.894i	0.295161	−	0.511234i	−0.679861	−	0.733341i	$-0.737960\pi$
$5$	82.5000	−	142.894i	0.295161	−	0.511234i	0.975022	+	0.222107i	$0.0712934\pi$
$6$		0			0
$7$	254.000	+	439.941i	0.279892	+	0.484787i	0.971358	−	0.237622i	$-0.0763680\pi$
$7$	254.000	+	439.941i	0.279892	+	0.484787i	−0.691466	+	0.722409i	$0.743035\pi$
$8$	512.000			0.353553
$9$		0			0
$10$	−1320.00			−0.417421
$11$	−1512.00	−	2618.86i	−0.342513	−	0.593250i	0.642385	−	0.766382i	$-0.277945\pi$
$11$	−1512.00	−	2618.86i	−0.342513	−	0.593250i	−0.984899	+	0.173131i	$0.944611\pi$
$12$		0			0
$13$	−2519.50	+	4363.90i	−0.318063	+	0.550901i	−0.980084	−	0.198585i	$-0.936366\pi$
$13$	−2519.50	+	4363.90i	−0.318063	+	0.550901i	0.662021	+	0.749485i	$0.269699\pi$
$14$	2032.00	−	3519.53i	0.197914	−	0.342796i
$15$		0			0
$16$	−2048.00	−	3547.24i	−0.125000	−	0.216506i
$17$	−3189.00			−0.157428			−0.0787142	−	0.996897i	$-0.525081\pi$
$17$	−3189.00			−0.157428			−0.0787142	+	0.996897i	$0.525081\pi$
$18$		0			0
$19$	1508.00			0.0504387			0.0252193	−	0.999682i	$-0.491972\pi$
$19$	1508.00			0.0504387			0.0252193	+	0.999682i	$0.491972\pi$
$20$	5280.00	+	9145.23i	0.147580	+	0.255617i
$21$		0			0
$22$	−12096.0	+	20950.9i	−0.242193	+	0.419491i
$23$	37800.0	−	65471.5i	0.647805	−	1.12203i	−0.335841	−	0.941919i	$-0.609020\pi$
$23$	37800.0	−	65471.5i	0.647805	−	1.12203i	0.983646	−	0.180113i	$-0.0576462\pi$
$24$		0			0
$25$	25450.0	+	44080.7i	0.325760	+	0.564233i
$26$	40312.0			0.449808
$27$		0			0
$28$	−32512.0			−0.279892
$29$	41332.5	+	71590.0i	0.314701	+	0.545079i	0.979374	−	0.202056i	$-0.0647623\pi$
$29$	41332.5	+	71590.0i	0.314701	+	0.545079i	−0.664673	+	0.747135i	$0.731429\pi$
$30$		0			0
$31$	87446.0	−	151461.i	0.527198	−	0.913134i	−0.472299	−	0.881438i	$-0.656576\pi$
$31$	87446.0	−	151461.i	0.527198	−	0.913134i	0.999498	−	0.0316960i	$-0.0100908\pi$
$32$	−16384.0	+	28377.9i	−0.0883883	+	0.153093i
$33$		0			0
$34$	12756.0	+	22094.0i	0.0556594	+	0.0964049i
$35$	83820.0			0.330453
$36$		0			0
$37$	−323569.			−1.05017			−0.525087	−	0.851049i	$-0.675967\pi$
$37$	−323569.			−1.05017			−0.525087	+	0.851049i	$0.675967\pi$
$38$	−6032.00	−	10447.7i	−0.0178328	−	0.0308873i
$39$		0			0
$40$	42240.0	−	73161.8i	0.104355	−	0.180748i
$41$	154059.	−	266838.i	0.349095	−	0.604650i	−0.636994	−	0.770869i	$-0.719822\pi$
$41$	154059.	−	266838.i	0.349095	−	0.604650i	0.986089	+	0.166219i	$0.0531557\pi$
$42$		0			0
$43$	−168340.	−	291573.i	−0.322885	−	0.559253i	0.658197	−	0.752846i	$-0.271319\pi$
$43$	−168340.	−	291573.i	−0.322885	−	0.559253i	−0.981082	+	0.193593i	$0.937986\pi$
$44$	193536.			0.342513
$45$		0			0
$46$	−604800.			−0.916135
$47$	191598.	+	331857.i	0.269184	+	0.466240i	0.968651	−	0.248425i	$-0.0799128\pi$
$47$	191598.	+	331857.i	0.269184	+	0.466240i	−0.699468	+	0.714664i	$0.746580\pi$
$48$		0			0
$49$	282740.	−	489719.i	0.343321	−	0.594649i
$50$	203600.	−	352646.i	0.230347	−	0.398973i
$51$		0			0
$52$	−161248.	−	279290.i	−0.159031	−	0.275450i
$53$	760206.			0.701400			0.350700	−	0.936488i	$-0.385944\pi$
$53$	760206.			0.701400			0.350700	+	0.936488i	$0.385944\pi$
$54$		0			0
$55$	−498960.			−0.404386
$56$	130048.	+	225250.i	0.0989568	+	0.171398i
$57$		0			0
$58$	330660.	−	572720.i	0.222527	−	0.385429i
$59$	1.11283e6	−	1.92748e6i	0.705420	−	1.22182i	−0.261120	−	0.965306i	$-0.584092\pi$
$59$	1.11283e6	−	1.92748e6i	0.705420	−	1.22182i	0.966540	−	0.256516i	$-0.0825748\pi$
$60$		0			0
$61$	−1.12241e6	−	1.94407e6i	−0.633135	−	1.09662i	−0.986907	−	0.161290i	$-0.948435\pi$
$61$	−1.12241e6	−	1.94407e6i	−0.633135	−	1.09662i	0.353772	−	0.935332i	$-0.384899\pi$
$62$	−1.39914e6			−0.745571
$63$		0			0
$64$	262144.			0.125000
$65$	415718.	+	720044.i	0.187759	+	0.325209i
$66$		0			0
$67$	−736594.	+	1.27582e6i	−0.299203	+	0.518235i	−0.975954	−	0.217977i	$-0.930054\pi$
$67$	−736594.	+	1.27582e6i	−0.299203	+	0.518235i	0.676751	+	0.736212i	$0.263388\pi$
$68$	102048.	−	176752.i	0.0393571	−	0.0681685i
$69$		0			0
$70$	−335280.	−	580722.i	−0.116833	−	0.202360i
$71$	−5.00689e6			−1.66021			−0.830107	−	0.557604i	$-0.811721\pi$
$71$	−5.00689e6			−1.66021			−0.830107	+	0.557604i	$0.811721\pi$
$72$		0			0
$73$	−5.89830e6			−1.77459			−0.887293	−	0.461207i	$-0.847417\pi$
$73$	−5.89830e6			−1.77459			−0.887293	+	0.461207i	$0.847417\pi$
$74$	1.29428e6	+	2.24175e6i	0.371292	+	0.643097i
$75$		0			0
$76$	−48256.0	+	83581.8i	−0.0126097	+	0.0218406i
$77$	768096.	−	1.33038e6i	0.191733	−	0.332092i
$78$		0			0
$79$	−3.51438e6	−	6.08709e6i	−0.801963	−	1.38904i	−0.918322	−	0.395834i	$-0.870456\pi$
$79$	−3.51438e6	−	6.08709e6i	−0.801963	−	1.38904i	0.116359	−	0.993207i	$-0.462878\pi$
$80$	−675840.			−0.147580
$81$		0			0
$82$	−2.46494e6			−0.493695
$83$	1.32560e6	+	2.29600e6i	0.254471	+	0.440757i	0.964752	−	0.263162i	$-0.0847653\pi$
$83$	1.32560e6	+	2.29600e6i	0.254471	+	0.440757i	−0.710281	+	0.703919i	$0.751432\pi$
$84$		0			0
$85$	−263092.	+	455690.i	−0.0464667	+	0.0804828i
$86$	−1.34672e6	+	2.33259e6i	−0.228314	+	0.395452i
$87$		0			0
$88$	−774144.	−	1.34086e6i	−0.121097	−	0.209746i
$89$	−6.77090e6			−1.01808			−0.509039	−	0.860743i	$-0.669999\pi$
$89$	−6.77090e6			−1.01808			−0.509039	+	0.860743i	$0.669999\pi$
$90$		0			0
$91$	−2.55981e6			−0.356093
$92$	2.41920e6	+	4.19018e6i	0.323903	+	0.561016i
$93$		0			0
$94$	1.53278e6	−	2.65486e6i	0.190341	−	0.329681i
$95$	124410.	−	215484.i	0.0148875	−	0.0257860i
$96$		0			0
$97$	−8.08819e6	−	1.40092e7i	−0.899809	−	1.55852i	−0.827737	−	0.561116i	$-0.810372\pi$
$97$	−8.08819e6	−	1.40092e7i	−0.899809	−	1.55852i	−0.0720719	−	0.997399i	$-0.522961\pi$
$98$	−4.52383e6			−0.485529
$99$		0			0
$100$	−3.25760e6			−0.325760
$101$	−1.35081e6	−	2.33967e6i	−0.130457	−	0.225959i	0.793396	−	0.608706i	$-0.208311\pi$
$101$	−1.35081e6	−	2.33967e6i	−0.130457	−	0.225959i	−0.923853	+	0.382748i	$0.874978\pi$
$102$		0			0
$103$	−1.03446e6	+	1.79173e6i	−0.0932787	+	0.161563i	−0.908889	−	0.417038i	$-0.863068\pi$
$103$	−1.03446e6	+	1.79173e6i	−0.0932787	+	0.161563i	0.815610	+	0.578602i	$0.196401\pi$
$104$	−1.28998e6	+	2.23432e6i	−0.112452	+	0.194773i
$105$		0			0
$106$	−3.04082e6	−	5.26686e6i	−0.247982	−	0.429518i
$107$	−9.15450e6			−0.722423			−0.361211	−	0.932484i	$-0.617637\pi$
$107$	−9.15450e6			−0.722423			−0.361211	+	0.932484i	$0.617637\pi$
$108$		0			0
$109$	8.68645e6			0.642465			0.321233	−	0.947000i	$-0.395903\pi$
$109$	8.68645e6			0.642465			0.321233	+	0.947000i	$0.395903\pi$
$110$	1.99584e6	+	3.45690e6i	0.142972	+	0.247635i
$111$		0			0
$112$	1.04038e6	−	1.80200e6i	0.0699730	−	0.121197i
$113$	1.24888e7	−	2.16312e7i	0.814227	−	1.41028i	−0.0956551	−	0.995415i	$-0.530495\pi$
$113$	1.24888e7	−	2.16312e7i	0.814227	−	1.41028i	0.909882	−	0.414867i	$-0.136172\pi$
$114$		0			0
$115$	−6.23700e6	−	1.08028e7i	−0.382414	−	0.662360i
$116$	−5.29056e6			−0.314701
$117$		0			0
$118$	−1.78053e7			−0.997614
$119$	−810006.	−	1.40297e6i	−0.0440630	−	0.0763193i
$120$		0			0
$121$	5.17130e6	−	8.95695e6i	0.265369	−	0.459633i
$122$	−8.97926e6	+	1.55525e7i	−0.447694	+	0.775429i
$123$		0			0
$124$	5.59654e6	+	9.69350e6i	0.263599	+	0.456567i
$125$	2.12891e7			0.974929
$126$		0			0
$127$	−4.98659e6			−0.216018			−0.108009	−	0.994150i	$-0.534448\pi$
$127$	−4.98659e6			−0.216018			−0.108009	+	0.994150i	$0.534448\pi$
$128$	−1.04858e6	−	1.81619e6i	−0.0441942	−	0.0765466i
$129$		0			0
$130$	3.32574e6	−	5.76035e6i	0.132766	−	0.229957i
$131$	1.11702e7	−	1.93474e7i	0.434123	−	0.751923i	−0.563101	−	0.826388i	$-0.690392\pi$
$131$	1.11702e7	−	1.93474e7i	0.434123	−	0.751923i	0.997224	+	0.0744655i	$0.0237251\pi$
$132$		0			0
$133$	383032.	+	663431.i	0.0141174	+	0.0244520i
$134$	1.17855e7			0.423137
$135$		0			0
$136$	−1.63277e6			−0.0556594
$137$	−6.43154e6	−	1.11398e7i	−0.213694	−	0.370129i	0.739174	−	0.673515i	$-0.235216\pi$
$137$	−6.43154e6	−	1.11398e7i	−0.213694	−	0.370129i	−0.952868	+	0.303386i	$0.901883\pi$
$138$		0			0
$139$	−2.20951e7	+	3.82699e7i	−0.697822	+	1.20866i	0.271398	+	0.962467i	$0.412514\pi$
$139$	−2.20951e7	+	3.82699e7i	−0.697822	+	1.20866i	−0.969220	+	0.246196i	$0.920819\pi$
$140$	−2.68224e6	+	4.64578e6i	−0.0826132	+	0.143090i
$141$		0			0
$142$	2.00276e7	+	3.46888e7i	0.586974	+	1.01667i
$143$	1.52379e7			0.435763
$144$		0			0
$145$	1.36397e7			0.371550
$146$	2.35932e7	+	4.08646e7i	0.627411	+	1.08671i
$147$		0			0
$148$	1.03542e7	−	1.79340e7i	0.262543	−	0.454738i
$149$	1.31444e6	−	2.27667e6i	0.0325527	−	0.0563830i	−0.849290	−	0.527927i	$-0.822970\pi$
$149$	1.31444e6	−	2.27667e6i	0.0325527	−	0.0563830i	0.881843	+	0.471544i	$0.156303\pi$
$150$		0			0
$151$	2.76509e7	+	4.78928e7i	0.653568	+	1.13201i	0.982251	+	0.187572i	$0.0600619\pi$
$151$	2.76509e7	+	4.78928e7i	0.653568	+	1.13201i	−0.328683	+	0.944440i	$0.606605\pi$
$152$	772096.			0.0178328
$153$		0			0
$154$	−1.22895e7			−0.271152
$155$	−1.44286e7	−	2.49911e7i	−0.311217	−	0.539043i
$156$		0			0
$157$	1.11855e7	−	1.93739e7i	0.230679	−	0.399548i	−0.727329	−	0.686289i	$-0.759239\pi$
$157$	1.11855e7	−	1.93739e7i	0.230679	−	0.399548i	0.958008	+	0.286741i	$0.0925720\pi$
$158$	−2.81151e7	+	4.86967e7i	−0.567074	+	0.982200i
$159$		0			0
$160$	2.70336e6	+	4.68236e6i	0.0521776	+	0.0903742i
$161$	3.84048e7			0.725262
$162$		0			0
$163$	5.23606e7			0.946995			0.473498	−	0.880795i	$-0.342991\pi$
$163$	5.23606e7			0.946995			0.473498	+	0.880795i	$0.342991\pi$
$164$	9.85978e6	+	1.70776e7i	0.174547	+	0.302325i
$165$		0			0
$166$	1.06048e7	−	1.83680e7i	0.179938	−	0.311662i
$167$	2.73979e7	−	4.74545e7i	0.455207	−	0.788442i	−0.543493	−	0.839414i	$-0.682899\pi$
$167$	2.73979e7	−	4.74545e7i	0.455207	−	0.788442i	0.998700	+	0.0509718i	$0.0162319\pi$
$168$		0			0
$169$	1.86785e7	+	3.23521e7i	0.297672	+	0.515584i
$170$	4.20948e6			0.0657139
$171$		0			0
$172$	2.15475e7			0.322885
$173$	−6.18226e7	−	1.07080e8i	−0.907791	−	1.57234i	−0.817127	−	0.576458i	$-0.804434\pi$
$173$	−6.18226e7	−	1.07080e8i	−0.907791	−	1.57234i	−0.0906639	−	0.995882i	$-0.528899\pi$
$174$		0			0
$175$	−1.29286e7	+	2.23930e7i	−0.182355	+	0.315849i
$176$	−6.19315e6	+	1.07269e7i	−0.0856283	+	0.148313i
$177$		0			0
$178$	2.70836e7	+	4.69102e7i	0.359945	+	0.623443i
$179$	9.87297e7			1.28666			0.643328	−	0.765591i	$-0.277553\pi$
$179$	9.87297e7			1.28666			0.643328	+	0.765591i	$0.277553\pi$
$180$		0			0
$181$	9.19855e7			1.15304			0.576520	−	0.817083i	$-0.304410\pi$
$181$	9.19855e7			1.15304			0.576520	+	0.817083i	$0.304410\pi$
$182$	1.02392e7	+	1.77349e7i	0.125898	+	0.218061i
$183$		0			0
$184$	1.93536e7	−	3.35214e7i	0.229034	−	0.396698i
$185$	−2.66944e7	+	4.62361e7i	−0.309970	+	0.536884i
$186$		0			0
$187$	4.82177e6	+	8.35155e6i	0.0539213	+	0.0933945i
$188$	−2.45245e7			−0.269184
$189$		0			0
$190$	−1.99056e6			−0.0210541
$191$	−6.59893e7	−	1.14297e8i	−0.685262	−	1.18691i	−0.973354	−	0.229306i	$-0.926354\pi$
$191$	−6.59893e7	−	1.14297e8i	−0.685262	−	1.18691i	0.288092	−	0.957603i	$-0.406979\pi$
$192$		0			0
$193$	−7.39537e7	+	1.28092e8i	−0.740473	+	1.28254i	0.211807	+	0.977311i	$0.432065\pi$
$193$	−7.39537e7	+	1.28092e8i	−0.740473	+	1.28254i	−0.952280	+	0.305225i	$0.901268\pi$
$194$	−6.47055e7	+	1.12073e8i	−0.636261	+	1.10204i
$195$		0			0
$196$	1.80953e7	+	3.13420e7i	0.171660	+	0.297325i
$197$	−2.69561e7			−0.251203			−0.125602	−	0.992081i	$-0.540086\pi$
$197$	−2.69561e7			−0.251203			−0.125602	+	0.992081i	$0.540086\pi$
$198$		0			0
$199$	1.35831e8			1.22183			0.610916	−	0.791695i	$-0.290801\pi$
$199$	1.35831e8			1.22183			0.610916	+	0.791695i	$0.290801\pi$
$200$	1.30304e7	+	2.25693e7i	0.115174	+	0.199486i
$201$		0			0
$202$	−1.08065e7	+	1.87173e7i	−0.0922473	+	0.159777i
$203$	−2.09969e7	+	3.63677e7i	−0.176165	+	0.305126i
$204$		0			0
$205$	−2.54197e7	−	4.40283e7i	−0.206078	−	0.356938i
$206$	1.65513e7			0.131916
$207$		0			0
$208$	2.06397e7			0.159031
$209$	−2.28010e6	−	3.94924e6i	−0.0172759	−	0.0299228i
$210$		0			0
$211$	1.09670e6	−	1.89954e6i	0.00803710	−	0.0139207i	−0.861979	−	0.506944i	$-0.830775\pi$
$211$	1.09670e6	−	1.89954e6i	0.00803710	−	0.0139207i	0.870016	+	0.493024i	$0.164108\pi$
$212$	−2.43266e7	+	4.21349e7i	−0.175350	+	0.303715i
$213$		0			0
$214$	3.66180e7	+	6.34242e7i	0.255415	+	0.442392i
$215$	−5.55522e7			−0.381212
$216$		0			0
$217$	8.88451e7			0.590234
$218$	−3.47458e7	−	6.01815e7i	−0.227146	−	0.393428i
$219$		0			0
$220$	1.59667e7	−	2.76552e7i	0.101097	−	0.175104i
$221$	8.03469e6	−	1.39165e7i	0.0500721	−	0.0867274i
$222$		0			0
$223$	−9.97215e6	−	1.72723e7i	−0.0602174	−	0.104300i	0.834345	−	0.551242i	$-0.185846\pi$
$223$	−9.97215e6	−	1.72723e7i	−0.0602174	−	0.104300i	−0.894562	+	0.446943i	$0.852513\pi$
$224$	−1.66461e7			−0.0989568
$225$		0			0
$226$	−1.99820e8			−1.15149
$227$	6.29925e7	+	1.09106e8i	0.357436	+	0.619098i	0.987532	−	0.157420i	$-0.0503176\pi$
$227$	6.29925e7	+	1.09106e8i	0.357436	+	0.619098i	−0.630096	+	0.776518i	$0.716984\pi$
$228$		0			0
$229$	−1.17523e7	+	2.03556e7i	−0.0646693	+	0.112010i	−0.896547	−	0.442948i	$-0.853933\pi$
$229$	−1.17523e7	+	2.03556e7i	−0.0646693	+	0.112010i	0.831878	+	0.554959i	$0.187266\pi$
$230$	−4.98960e7	+	8.64224e7i	−0.270407	+	0.468359i
$231$		0			0
$232$	2.11622e7	+	3.66541e7i	0.111264	+	0.192714i
$233$	−1.27757e8			−0.661665			−0.330833	−	0.943689i	$-0.607330\pi$
$233$	−1.27757e8			−0.661665			−0.330833	+	0.943689i	$0.607330\pi$
$234$		0			0
$235$	6.32273e7			0.317810
$236$	7.12212e7	+	1.23359e8i	0.352710	+	0.610911i
$237$		0			0
$238$	−6.48005e6	+	1.12238e7i	−0.0311572	+	0.0539659i
$239$	4.67439e7	−	8.09628e7i	0.221479	−	0.383613i	−0.733778	−	0.679389i	$-0.762245\pi$
$239$	4.67439e7	−	8.09628e7i	0.221479	−	0.383613i	0.955257	+	0.295776i	$0.0955783\pi$
$240$		0			0
$241$	6.73901e7	+	1.16723e8i	0.310125	+	0.537152i	0.978389	−	0.206772i	$-0.0662958\pi$
$241$	6.73901e7	+	1.16723e8i	0.310125	+	0.537152i	−0.668264	+	0.743924i	$0.732963\pi$
$242$	−8.27408e7			−0.375289
$243$		0			0
$244$	1.43668e8			0.633135
$245$	−4.66520e7	−	8.08037e7i	−0.202670	−	0.351034i
$246$		0			0
$247$	−3.79941e6	+	6.58076e6i	−0.0160427	+	0.0277867i
$248$	4.47724e7	−	7.75480e7i	0.186393	−	0.322842i
$249$		0			0
$250$	−8.51565e7	−	1.47495e8i	−0.344689	−	0.597019i
$251$	1.31470e7			0.0524771			0.0262385	−	0.999656i	$-0.491647\pi$
$251$	1.31470e7			0.0524771			0.0262385	+	0.999656i	$0.491647\pi$
$252$		0			0
$253$	−2.28614e8			−0.887527
$254$	1.99464e7	+	3.45481e7i	0.0763740	+	0.132284i
$255$		0			0
$256$	−8.38861e6	+	1.45295e7i	−0.0312500	+	0.0541266i
$257$	6.42355e7	−	1.11259e8i	0.236053	−	0.408855i	−0.723525	−	0.690298i	$-0.757480\pi$
$257$	6.42355e7	−	1.11259e8i	0.236053	−	0.408855i	0.959578	+	0.281443i	$0.0908129\pi$
$258$		0			0
$259$	−8.21865e7	−	1.42351e8i	−0.293935	−	0.509111i
$260$	−5.32118e7			−0.187759
$261$		0			0
$262$	−1.78724e8			−0.613942
$263$	4.88758e7	+	8.46554e7i	0.165672	+	0.286952i	0.936894	−	0.349614i	$-0.113687\pi$
$263$	4.88758e7	+	8.46554e7i	0.165672	+	0.286952i	−0.771222	+	0.636567i	$0.780354\pi$
$264$		0			0
$265$	6.27170e7	−	1.08629e8i	0.207026	−	0.358579i
$266$	3.06426e6	−	5.30745e6i	0.00998250	−	0.0172902i
$267$		0			0
$268$	−4.71420e7	−	8.16524e7i	−0.149602	−	0.259118i
$269$	3.96851e7			0.124307			0.0621534	−	0.998067i	$-0.480203\pi$
$269$	3.96851e7			0.124307			0.0621534	+	0.998067i	$0.480203\pi$
$270$		0			0
$271$	−3.50733e8			−1.07049			−0.535246	−	0.844696i	$-0.679781\pi$
$271$	−3.50733e8			−1.07049			−0.535246	+	0.844696i	$0.679781\pi$
$272$	6.53107e6	+	1.13121e7i	0.0196786	+	0.0340843i
$273$		0			0
$274$	−5.14523e7	+	8.91180e7i	−0.151105	+	0.261721i
$275$	7.69608e7	−	1.33300e8i	0.223154	−	0.386514i
$276$		0			0
$277$	−2.69801e8	−	4.67309e8i	−0.762719	−	1.32107i	−0.941444	−	0.337169i	$-0.890531\pi$
$277$	−2.69801e8	−	4.67309e8i	−0.762719	−	1.32107i	0.178726	−	0.983899i	$-0.442803\pi$
$278$	3.53522e8			0.986869
$279$		0			0
$280$	4.29158e7			0.116833
$281$	5.04259e7	+	8.73403e7i	0.135576	+	0.234824i	0.925817	−	0.377971i	$-0.123378\pi$
$281$	5.04259e7	+	8.73403e7i	0.135576	+	0.234824i	−0.790242	+	0.612796i	$0.790045\pi$
$282$		0			0
$283$	1.27056e8	−	2.20068e8i	0.333229	−	0.577170i	−0.649914	−	0.760008i	$-0.725195\pi$
$283$	1.27056e8	−	2.20068e8i	0.333229	−	0.577170i	0.983143	+	0.182838i	$0.0585284\pi$
$284$	1.60221e8	−	2.77510e8i	0.415053	−	0.718894i
$285$		0			0
$286$	−6.09517e7	−	1.05572e8i	−0.154065	−	0.266849i
$287$	1.56524e8			0.390836
$288$		0			0
$289$	−4.00169e8			−0.975216
$290$	−5.45589e7	−	9.44988e7i	−0.131363	−	0.227527i
$291$		0			0
$292$	1.88746e8	−	3.26917e8i	0.443646	−	0.768418i
$293$	2.71542e8	−	4.70325e8i	0.630668	−	1.09235i	−0.356748	−	0.934201i	$-0.616114\pi$
$293$	2.71542e8	−	4.70325e8i	0.630668	−	1.09235i	0.987415	−	0.158148i	$-0.0505522\pi$
$294$		0			0
$295$	−1.83617e8	−	3.18034e8i	−0.416425	−	0.721269i
$296$	−1.65667e8			−0.371292
$297$		0			0
$298$	−2.10310e7			−0.0460365
$299$	1.90474e8	+	3.29911e8i	0.412085	+	0.713752i
$300$		0			0
$301$	8.55167e7	−	1.48119e8i	0.180746	−	0.313061i
$302$	2.21208e8	−	3.83143e8i	0.462142	−	0.800454i
$303$		0			0
$304$	−3.08838e6	−	5.34924e6i	−0.00630484	−	0.0109203i
$305$	−3.70394e8			−0.747507
$306$		0			0
$307$	−7.29877e8			−1.43968			−0.719839	−	0.694141i	$-0.755784\pi$
$307$	−7.29877e8			−1.43968			−0.719839	+	0.694141i	$0.755784\pi$
$308$	4.91581e7	+	8.51444e7i	0.0958667	+	0.166046i
$309$		0			0
$310$	−1.15429e8	+	1.99928e8i	−0.220063	+	0.381161i
$311$	−3.19385e8	+	5.53191e8i	−0.602078	+	1.04283i	0.390427	+	0.920634i	$0.372327\pi$
$311$	−3.19385e8	+	5.53191e8i	−0.602078	+	1.04283i	−0.992506	+	0.122197i	$0.961006\pi$
$312$		0			0
$313$	3.18880e8	+	5.52317e8i	0.587790	+	1.01808i	0.994521	+	0.104535i	$0.0333354\pi$
$313$	3.18880e8	+	5.52317e8i	0.587790	+	1.01808i	−0.406731	+	0.913548i	$0.633331\pi$
$314$	−1.78968e8			−0.326229
$315$		0			0
$316$	4.49841e8			0.801963
$317$	3.27189e8	+	5.66707e8i	0.576887	+	0.999198i	0.995834	+	0.0911869i	$0.0290661\pi$
$317$	3.27189e8	+	5.66707e8i	0.576887	+	0.999198i	−0.418947	+	0.908011i	$0.637601\pi$
$318$		0			0
$319$	1.24989e8	−	2.16488e8i	0.215579	−	0.373393i
$320$	2.16269e7	−	3.74589e7i	0.0368951	−	0.0639042i
$321$		0			0
$322$	−1.53619e8	−	2.66076e8i	−0.256419	−	0.444130i
$323$	−4.80901e6			−0.00794049
$324$		0			0
$325$	−2.56485e8			−0.414448
$326$	−2.09442e8	−	3.62765e8i	−0.334813	−	0.579914i
$327$		0			0
$328$	7.88782e7	−	1.36621e8i	0.123424	−	0.213776i
$329$	−9.73318e7	+	1.68584e8i	−0.150685	+	0.260993i
$330$		0			0
$331$	−4.03844e8	−	6.99479e8i	−0.612091	−	1.06017i	−0.990887	−	0.134693i	$-0.956995\pi$
$331$	−4.03844e8	−	6.99479e8i	−0.612091	−	1.06017i	0.378796	−	0.925480i	$-0.376338\pi$
$332$	−1.69677e8			−0.254471
$333$		0			0
$334$	−4.38366e8			−0.643760
$335$	1.21538e8	+	2.10510e8i	0.176626	+	0.305926i
$336$		0			0
$337$	9.36402e7	−	1.62190e8i	0.133278	−	0.230844i	−0.791660	−	0.610961i	$-0.790783\pi$
$337$	9.36402e7	−	1.62190e8i	0.133278	−	0.230844i	0.924938	+	0.380117i	$0.124116\pi$
$338$	1.49428e8	−	2.58817e8i	0.210486	−	0.364573i
$339$		0			0
$340$	−1.68379e7	−	2.91641e7i	−0.0232334	−	0.0402414i
$341$	−5.28873e8			−0.722290
$342$		0			0
$343$	7.05623e8			0.944155
$344$	−8.61901e7	−	1.49286e8i	−0.114157	−	0.197726i
$345$		0			0
$346$	−4.94581e8	+	8.56639e8i	−0.641905	+	1.11181i
$347$	−1.41278e8	+	2.44700e8i	−0.181518	+	0.314399i	−0.942398	−	0.334494i	$-0.891435\pi$
$347$	−1.41278e8	+	2.44700e8i	−0.181518	+	0.314399i	0.760879	+	0.648893i	$0.224768\pi$
$348$		0			0
$349$	3.80306e8	+	6.58709e8i	0.478899	+	0.829478i	0.999707	−	0.0241959i	$-0.00770255\pi$
$349$	3.80306e8	+	6.58709e8i	0.478899	+	0.829478i	−0.520808	+	0.853674i	$0.674369\pi$
$350$	2.06858e8			0.257889
$351$		0			0
$352$	9.90904e7			0.121097
$353$	2.34447e8	+	4.06074e8i	0.283683	+	0.491353i	0.972289	−	0.233783i	$-0.0751105\pi$
$353$	2.34447e8	+	4.06074e8i	0.283683	+	0.491353i	−0.688606	+	0.725135i	$0.741777\pi$
$354$		0			0
$355$	−4.13069e8	+	7.15456e8i	−0.490030	+	0.848757i
$356$	2.16669e8	−	3.75281e8i	0.254520	−	0.440841i
$357$		0			0
$358$	−3.94919e8	−	6.84020e8i	−0.454901	−	0.787912i
$359$	1.53906e8			0.175560			0.0877802	−	0.996140i	$-0.472023\pi$
$359$	1.53906e8			0.175560			0.0877802	+	0.996140i	$0.472023\pi$
$360$		0			0
$361$	−8.91598e8			−0.997456
$362$	−3.67942e8	−	6.37294e8i	−0.407661	−	0.706090i
$363$		0			0
$364$	8.19140e7	−	1.41879e8i	0.0890232	−	0.154193i
$365$	−4.86610e8	+	8.42833e8i	−0.523788	+	0.907228i
$366$		0			0
$367$	3.74764e8	+	6.49110e8i	0.395755	+	0.685468i	0.993197	−	0.116444i	$-0.0371496\pi$
$367$	3.74764e8	+	6.49110e8i	0.395755	+	0.685468i	−0.597442	+	0.801912i	$0.703816\pi$
$368$	−3.09658e8			−0.323903
$369$		0			0
$370$	4.27111e8			0.438364
$371$	1.93092e8	+	3.34446e8i	0.196316	+	0.340030i
$372$		0			0
$373$	2.70864e8	−	4.69150e8i	0.270253	−	0.468092i	−0.698674	−	0.715441i	$-0.746226\pi$
$373$	2.70864e8	−	4.69150e8i	0.270253	−	0.468092i	0.968926	+	0.247349i	$0.0795594\pi$
$374$	3.85741e7	−	6.68124e7i	0.0381281	−	0.0660399i
$375$		0			0
$376$	9.80982e7	+	1.69911e8i	0.0951707	+	0.164841i
$377$	−4.16549e8			−0.400379
$378$		0			0
$379$	1.32974e9			1.25467			0.627334	−	0.778750i	$-0.284146\pi$
$379$	1.32974e9			1.25467			0.627334	+	0.778750i	$0.284146\pi$
$380$	7.96224e6	+	1.37910e7i	0.00744377	+	0.0128930i
$381$		0			0
$382$	−5.27914e8	+	9.14375e8i	−0.484554	+	0.839271i
$383$	−3.59974e8	+	6.23494e8i	−0.327398	+	0.567070i	−0.981995	−	0.188908i	$-0.939505\pi$
$383$	−3.59974e8	+	6.23494e8i	−0.327398	+	0.567070i	0.654597	+	0.755978i	$0.272838\pi$
$384$		0			0
$385$	−1.26736e8	−	2.19513e8i	−0.113184	−	0.196041i
$386$	1.18326e9			1.04719
$387$		0			0
$388$	1.03529e9			0.899809
$389$	−1.16034e9	−	2.00976e9i	−0.999447	−	1.73109i	−0.528517	−	0.848922i	$-0.677252\pi$
$389$	−1.16034e9	−	2.00976e9i	−0.999447	−	1.73109i	−0.470930	−	0.882171i	$-0.656081\pi$
$390$		0			0
$391$	−1.20544e8	+	2.08789e8i	−0.101983	+	0.176640i
$392$	1.44763e8	−	2.50736e8i	0.121382	−	0.210240i
$393$		0			0
$394$	1.07824e8	+	1.86757e8i	0.0888138	+	0.153830i
$395$	−1.15975e9			−0.946833
$396$		0			0
$397$	1.18373e9			0.949477			0.474739	−	0.880127i	$-0.342543\pi$
$397$	1.18373e9			0.949477			0.474739	+	0.880127i	$0.342543\pi$
$398$	−5.43322e8	−	9.41061e8i	−0.431983	−	0.748216i
$399$		0			0
$400$	1.04243e8	−	1.80555e8i	0.0814400	−	0.141058i
$401$	−6.33820e8	+	1.09781e9i	−0.490863	+	0.850200i	−0.999945	−	0.0105184i	$-0.996652\pi$
$401$	−6.33820e8	+	1.09781e9i	−0.490863	+	0.850200i	0.509082	+	0.860718i	$0.329985\pi$
$402$		0			0
$403$	4.40640e8	+	7.63212e8i	0.335364	+	0.580868i
$404$	1.72903e8			0.130457
$405$		0			0
$406$	3.35951e8			0.249135
$407$	4.89236e8	+	8.47382e8i	0.359698	+	0.623016i
$408$		0			0
$409$	5.09744e8	−	8.82903e8i	0.368401	−	0.638089i	−0.620915	−	0.783878i	$-0.713239\pi$
$409$	5.09744e8	−	8.82903e8i	0.368401	−	0.638089i	0.989316	+	0.145789i	$0.0465721\pi$
$410$	−2.03358e8	+	3.52226e8i	−0.145719	+	0.252393i
$411$		0			0
$412$	−6.62053e7	−	1.14671e8i	−0.0466394	−	0.0807817i
$413$	1.13064e9			0.789765
$414$		0			0
$415$	4.37447e8			0.300440
$416$	−8.25590e7	−	1.42996e8i	−0.0562261	−	0.0973864i
$417$		0			0
$418$	−1.82408e7	+	3.15939e7i	−0.0122159	+	0.0211586i
$419$	−5.20643e8	+	9.01780e8i	−0.345773	+	0.598896i	−0.985494	−	0.169711i	$-0.945717\pi$
$419$	−5.20643e8	+	9.01780e8i	−0.345773	+	0.598896i	0.639721	+	0.768607i	$0.279050\pi$
$420$		0			0
$421$	9.79224e8	+	1.69607e9i	0.639580	+	1.10779i	0.985525	+	0.169530i	$0.0542251\pi$
$421$	9.79224e8	+	1.69607e9i	0.639580	+	1.10779i	−0.345945	+	0.938255i	$0.612442\pi$
$422$	−1.75472e7			−0.0113662
$423$		0			0
$424$	3.89225e8			0.247982
$425$	−8.11600e7	−	1.40573e8i	−0.0512839	−	0.0888263i
$426$		0			0
$427$	5.70183e8	−	9.87586e8i	0.354419	−	0.613871i
$428$	2.92944e8	−	5.07394e8i	0.180606	−	0.312818i
$429$		0			0
$430$	2.22209e8	+	3.84877e8i	0.134779	+	0.233444i
$431$	2.84256e9			1.71017			0.855084	−	0.518489i	$-0.173505\pi$
$431$	2.84256e9			1.71017			0.855084	+	0.518489i	$0.173505\pi$
$432$		0			0
$433$	−8.41292e7			−0.0498011			−0.0249006	−	0.999690i	$-0.507927\pi$
$433$	−8.41292e7			−0.0498011			−0.0249006	+	0.999690i	$0.507927\pi$
$434$	−3.55381e8	−	6.15537e8i	−0.208679	−	0.361443i
$435$		0			0
$436$	−2.77966e8	+	4.81452e8i	−0.160616	+	0.278196i
$437$	5.70024e7	−	9.87311e7i	0.0326744	−	0.0565938i
$438$		0			0
$439$	9.35233e8	+	1.61987e9i	0.527587	+	0.913807i	0.999483	+	0.0321530i	$0.0102364\pi$
$439$	9.35233e8	+	1.61987e9i	0.527587	+	0.913807i	−0.471896	+	0.881654i	$0.656430\pi$
$440$	−2.55468e8			−0.142972
$441$		0			0
$442$	−1.28555e8			−0.0708127
$443$	5.06663e8	+	8.77566e8i	0.276889	+	0.479587i	0.970610	−	0.240658i	$-0.0773631\pi$
$443$	5.06663e8	+	8.77566e8i	0.276889	+	0.479587i	−0.693721	+	0.720244i	$0.744030\pi$
$444$		0			0
$445$	−5.58599e8	+	9.67522e8i	−0.300497	+	0.520476i
$446$	−7.97772e7	+	1.38178e8i	−0.0425801	+	0.0737509i
$447$		0			0
$448$	6.65846e7	+	1.15328e8i	0.0349865	+	0.0605984i
$449$	−1.65756e8			−0.0864183			−0.0432092	−	0.999066i	$-0.513758\pi$
$449$	−1.65756e8			−0.0864183			−0.0432092	+	0.999066i	$0.513758\pi$
$450$		0			0
$451$	−9.31749e8			−0.478279
$452$	7.99282e8	+	1.38440e9i	0.407113	+	0.705141i
$453$		0			0
$454$	5.03940e8	−	8.72850e8i	0.252746	−	0.437768i
$455$	−2.11184e8	+	3.65782e8i	−0.105105	+	0.182047i
$456$		0			0
$457$	6.90398e8	+	1.19581e9i	0.338371	+	0.586076i	0.984126	−	0.177469i	$-0.0567908\pi$
$457$	6.90398e8	+	1.19581e9i	0.338371	+	0.586076i	−0.645756	+	0.763544i	$0.723457\pi$
$458$	1.88037e8			0.0914562
$459$		0			0
$460$	7.98336e8			0.382414
$461$	−1.39246e9	−	2.41181e9i	−0.661955	−	1.14654i	−0.980101	−	0.198498i	$-0.936394\pi$
$461$	−1.39246e9	−	2.41181e9i	−0.661955	−	1.14654i	0.318146	−	0.948042i	$-0.396940\pi$
$462$		0			0
$463$	−1.39459e9	+	2.41551e9i	−0.653001	+	1.13103i	0.329390	+	0.944194i	$0.393157\pi$
$463$	−1.39459e9	+	2.41551e9i	−0.653001	+	1.13103i	−0.982391	+	0.186837i	$0.940176\pi$
$464$	1.69298e8	−	2.93233e8i	0.0786754	−	0.136270i
$465$		0			0
$466$	5.11027e8	+	8.85125e8i	0.233934	+	0.405186i
$467$	3.97258e9			1.80494			0.902471	−	0.430751i	$-0.141751\pi$
$467$	3.97258e9			1.80494			0.902471	+	0.430751i	$0.141751\pi$
$468$		0			0
$469$	−7.48380e8			−0.334979
$470$	−2.52909e8	−	4.38052e8i	−0.112363	−	0.194618i
$471$		0			0
$472$	5.69770e8	−	9.86871e8i	0.249404	−	0.431980i
$473$	−5.09060e8	+	8.81718e8i	−0.221185	+	0.383103i
$474$		0			0
$475$	3.83786e7	+	6.64737e7i	0.0164309	+	0.0284592i
$476$	1.03681e8			0.0440630
$477$		0			0
$478$	−7.47902e8			−0.313218
$479$	1.17233e9	+	2.03054e9i	0.487391	+	0.844186i	0.999895	−	0.0144991i	$-0.00461537\pi$
$479$	1.17233e9	+	2.03054e9i	0.487391	+	0.844186i	−0.512504	+	0.858685i	$0.671282\pi$
$480$		0			0
$481$	8.15232e8	−	1.41202e9i	0.334021	−	0.578541i
$482$	5.39121e8	−	9.33785e8i	0.219291	−	0.379824i
$483$		0			0
$484$	3.30963e8	+	5.73245e8i	0.132685	+	0.229817i
$485$	−2.66910e9			−1.06235
$486$		0			0
$487$	3.08381e9			1.20986			0.604931	−	0.796278i	$-0.293201\pi$
$487$	3.08381e9			1.20986			0.604931	+	0.796278i	$0.293201\pi$
$488$	−5.74673e8	−	9.95362e8i	−0.223847	−	0.387714i
$489$		0			0
$490$	−3.73216e8	+	6.46429e8i	−0.143309	+	0.248219i
$491$	1.85925e9	−	3.22032e9i	0.708848	−	1.22776i	−0.256436	−	0.966561i	$-0.582548\pi$
$491$	1.85925e9	−	3.22032e9i	0.708848	−	1.22776i	0.965285	−	0.261200i	$-0.0841183\pi$
$492$		0			0
$493$	−1.31809e8	−	2.28300e8i	−0.0495430	−	0.0858109i
$494$	6.07905e7			0.0226877
$495$		0			0
$496$	−7.16358e8			−0.263599
$497$	−1.27175e9	−	2.20274e9i	−0.464681	−	0.804850i
$498$		0			0
$499$	−2.18213e8	+	3.77956e8i	−0.0786192	+	0.136172i	−0.902654	−	0.430366i	$-0.858384\pi$
$499$	−2.18213e8	+	3.77956e8i	−0.0786192	+	0.136172i	0.824035	+	0.566539i	$0.191718\pi$
$500$	−6.81252e8	+	1.17996e9i	−0.243732	+	0.422156i
$501$		0			0
$502$	−5.25881e7	−	9.10853e7i	−0.0185535	−	0.0321355i
$503$	−5.04603e8			−0.176792			−0.0883958	−	0.996085i	$-0.528174\pi$
$503$	−5.04603e8			−0.176792			−0.0883958	+	0.996085i	$0.528174\pi$
$504$		0			0
$505$	−4.45766e8			−0.154024
$506$	9.14458e8	+	1.58389e9i	0.313788	+	0.543497i
$507$		0			0
$508$	1.59571e8	−	2.76385e8i	0.0540046	−	0.0935387i
$509$	2.30043e9	−	3.98446e9i	0.773208	−	1.33923i	−0.162589	−	0.986694i	$-0.551984\pi$
$509$	2.30043e9	−	3.98446e9i	0.773208	−	1.33923i	0.935796	−	0.352541i	$-0.114682\pi$
$510$		0			0
$511$	−1.49817e9	−	2.59490e9i	−0.496692	−	0.860296i
$512$	1.34218e8			0.0441942
$513$		0			0
$514$	−1.02777e9			−0.333829
$515$	1.70686e8	+	2.95636e8i	0.0550645	+	0.0953745i
$516$		0			0
$517$	5.79392e8	−	1.00354e9i	0.184398	−	0.319386i
$518$	−6.57492e8	+	1.13881e9i	−0.207844	+	0.359996i
$519$		0			0
$520$	2.12847e8	+	3.68662e8i	0.0663830	+	0.114979i
$521$	7.14937e7			0.0221481			0.0110740	−	0.999939i	$-0.496475\pi$
$521$	7.14937e7			0.0221481			0.0110740	+	0.999939i	$0.496475\pi$
$522$		0			0
$523$	−2.12182e9			−0.648563			−0.324281	−	0.945961i	$-0.605122\pi$
$523$	−2.12182e9			−0.648563			−0.324281	+	0.945961i	$0.605122\pi$
$524$	7.14895e8	+	1.23823e9i	0.217061	+	0.375961i
$525$		0			0
$526$	3.91007e8	−	6.77243e8i	0.117148	−	0.202906i
$527$	−2.78865e8	+	4.83009e8i	−0.0829960	+	0.143753i
$528$		0			0
$529$	−1.15527e9	−	2.00098e9i	−0.339303	−	0.587690i
$530$	−1.00347e9			−0.292779
$531$		0			0
$532$	−4.90281e7			−0.0141174
$533$	7.76303e8	+	1.34460e9i	0.222068	+	0.384633i
$534$		0			0
$535$	−7.55246e8	+	1.30812e9i	−0.213231	+	0.369327i
$536$	−3.77136e8	+	6.53219e8i	−0.105784	+	0.183224i
$537$		0			0
$538$	−1.58740e8	−	2.74946e8i	−0.0439491	−	0.0761220i
$539$	−1.71001e9			−0.470368
$540$		0			0
$541$	−2.11351e9			−0.573870			−0.286935	−	0.957950i	$-0.592636\pi$
$541$	−2.11351e9			−0.573870			−0.286935	+	0.957950i	$0.592636\pi$
$542$	1.40293e9	+	2.42995e9i	0.378476	+	0.655540i
$543$		0			0
$544$	5.22486e7	−	9.04972e7i	0.0139148	−	0.0241012i
$545$	7.16632e8	−	1.24124e9i	0.189631	−	0.328450i
$546$		0			0
$547$	−3.12777e9	−	5.41745e9i	−0.817107	−	1.41527i	−0.907805	−	0.419392i	$-0.862243\pi$
$547$	−3.12777e9	−	5.41745e9i	−0.817107	−	1.41527i	0.0906987	−	0.995878i	$-0.471090\pi$
$548$	8.23237e8			0.213694
$549$		0			0
$550$	−1.23137e9			−0.315588
$551$	6.23294e7	+	1.07958e8i	0.0158731	+	0.0274931i
$552$		0			0
$553$	1.78531e9	−	3.09224e9i	0.448926	−	0.777563i
$554$	−2.15841e9	+	3.73847e9i	−0.539324	+	0.934136i
$555$		0			0
$556$	−1.41409e9	−	2.44927e9i	−0.348911	−	0.604331i
$557$	−6.50846e9			−1.59583			−0.797913	−	0.602773i	$-0.794062\pi$
$557$	−6.50846e9			−1.59583			−0.797913	+	0.602773i	$0.794062\pi$
$558$		0			0
$559$	1.69653e9			0.410790
$560$	−1.71663e8	−	2.97330e8i	−0.0413066	−	0.0715451i
$561$		0			0
$562$	4.03408e8	−	6.98722e8i	0.0958665	−	0.166046i
$563$	8.66244e7	−	1.50038e8i	0.0204579	−	0.0354341i	−0.855615	−	0.517612i	$-0.826821\pi$
$563$	8.66244e7	−	1.50038e8i	0.0204579	−	0.0354341i	0.876073	+	0.482178i	$0.160154\pi$
$564$		0			0
$565$	−2.06065e9	−	3.56915e9i	−0.480656	−	0.832520i
$566$	−2.03290e9			−0.471257
$567$		0			0
$568$	−2.56353e9			−0.586974
$569$	−3.64441e9	−	6.31230e9i	−0.829343	−	1.43646i	−0.898555	−	0.438862i	$-0.855382\pi$
$569$	−3.64441e9	−	6.31230e9i	−0.829343	−	1.43646i	0.0692121	−	0.997602i	$-0.477951\pi$
$570$		0			0
$571$	−2.14518e9	+	3.71556e9i	−0.482212	+	0.835215i	−0.999791	−	0.0204199i	$-0.993500\pi$
$571$	−2.14518e9	+	3.71556e9i	−0.482212	+	0.835215i	0.517580	+	0.855635i	$0.326833\pi$
$572$	−4.87614e8	+	8.44572e8i	−0.108941	+	0.188691i
$573$		0			0
$574$	−6.26096e8	−	1.08443e9i	−0.138181	−	0.239337i
$575$	3.84804e9			0.844116
$576$		0			0
$577$	6.28378e8			0.136178			0.0680888	−	0.997679i	$-0.478310\pi$
$577$	6.28378e8			0.136178			0.0680888	+	0.997679i	$0.478310\pi$
$578$	1.60068e9	+	2.77245e9i	0.344791	+	0.597196i
$579$		0			0
$580$	−4.36471e8	+	7.55990e8i	−0.0928876	+	0.160886i
$581$	−6.73404e8	+	1.16637e9i	−0.142449	+	0.246729i
$582$		0			0
$583$	−1.14943e9	−	1.99087e9i	−0.240239	−	0.416106i
$584$	−3.01993e9			−0.627411
$585$		0			0
$586$	−4.34467e9			−0.891899
$587$	4.48401e9	+	7.76654e9i	0.915026	+	1.58487i	0.806863	+	0.590738i	$0.201163\pi$
$587$	4.48401e9	+	7.76654e9i	0.915026	+	1.58487i	0.108163	+	0.994133i	$0.465503\pi$
$588$		0			0
$589$	1.31869e8	−	2.28403e8i	0.0265912	−	0.0460573i
$590$	−1.46894e9	+	2.54428e9i	−0.294457	+	0.510014i
$591$		0			0
$592$	6.62669e8	+	1.14778e9i	0.131272	+	0.227369i
$593$	−7.57484e9			−1.49170			−0.745851	−	0.666112i	$-0.767957\pi$
$593$	−7.57484e9			−1.49170			−0.745851	+	0.666112i	$0.767957\pi$
$594$		0			0
$595$	−2.67302e8			−0.0520227
$596$	8.41239e7	+	1.45707e8i	0.0162764	+	0.0281915i
$597$		0			0
$598$	1.52379e9	−	2.63929e9i	0.291388	−	0.504699i
$599$	3.02973e9	−	5.24765e9i	0.575984	−	0.997633i	−0.419950	−	0.907547i	$-0.637953\pi$
$599$	3.02973e9	−	5.24765e9i	0.575984	−	0.997633i	0.995934	−	0.0900859i	$-0.0287141\pi$
$600$		0			0
$601$	3.27899e8	+	5.67937e8i	0.0616139	+	0.106718i	0.895187	−	0.445691i	$-0.147042\pi$
$601$	3.27899e8	+	5.67937e8i	0.0616139	+	0.106718i	−0.833573	+	0.552409i	$0.813709\pi$
$602$	−1.36827e9			−0.255613
$603$		0			0
$604$	−3.53932e9			−0.653568
$605$	−8.53264e8	−	1.47790e9i	−0.156653	−	0.271332i
$606$		0			0
$607$	−1.92806e9	+	3.33949e9i	−0.349913	+	0.606066i	−0.986234	−	0.165358i	$-0.947122\pi$
$607$	−1.92806e9	+	3.33949e9i	−0.349913	+	0.606066i	0.636321	+	0.771424i	$0.280455\pi$
$608$	−2.47071e7	+	4.27939e7i	−0.00445819	+	0.00772181i
$609$		0			0
$610$	1.48158e9	+	2.56617e9i	0.264284	+	0.457752i
$611$	−1.93092e9			−0.342469
$612$		0			0
$613$	−4.89889e9			−0.858986			−0.429493	−	0.903070i	$-0.641308\pi$
$613$	−4.89889e9			−0.858986			−0.429493	+	0.903070i	$0.641308\pi$
$614$	2.91951e9	+	5.05674e9i	0.509003	+	0.881619i
$615$		0			0
$616$	3.93265e8	−	6.81155e8i	0.0677880	−	0.117412i
$617$	3.32728e9	−	5.76302e9i	0.570284	−	0.987761i	−0.426253	−	0.904604i	$-0.640167\pi$
$617$	3.32728e9	−	5.76302e9i	0.570284	−	0.987761i	0.996537	−	0.0831565i	$-0.0265001\pi$
$618$		0			0
$619$	−5.27657e9	−	9.13929e9i	−0.894199	−	1.54880i	−0.834792	−	0.550565i	$-0.814412\pi$
$619$	−5.27657e9	−	9.13929e9i	−0.894199	−	1.54880i	−0.0594070	−	0.998234i	$-0.518921\pi$
$620$	1.84686e9			0.311217
$621$		0			0
$622$	5.11016e9			0.851468
$623$	−1.71981e9	−	2.97880e9i	−0.284952	−	0.493552i
$624$		0			0
$625$	−2.31928e8	+	4.01712e8i	−0.0379992	+	0.0658165i
$626$	2.55104e9	−	4.41853e9i	0.415631	−	0.719893i
$627$		0			0
$628$	7.15874e8	+	1.23993e9i	0.115339	+	0.199774i
$629$	1.03186e9			0.165327
$630$		0			0
$631$	6.37775e9			1.01057			0.505283	−	0.862954i	$-0.331388\pi$
$631$	6.37775e9			1.01057			0.505283	+	0.862954i	$0.331388\pi$
$632$	−1.79936e9	−	3.11659e9i	−0.283537	−	0.491100i
$633$		0			0
$634$	2.61751e9	−	4.53366e9i	0.407921	−	0.706540i
$635$	−4.11394e8	+	7.12555e8i	−0.0637602	+	0.110436i
$636$		0			0
$637$	1.42472e9	+	2.46769e9i	0.218395	+	0.378271i
$638$	−1.99983e9			−0.304874
$639$		0			0
$640$	−3.46030e8			−0.0521776
$641$	−3.34351e8	−	5.79113e8i	−0.0501418	−	0.0868481i	0.839865	−	0.542795i	$-0.182634\pi$
$641$	−3.34351e8	−	5.79113e8i	−0.0501418	−	0.0868481i	−0.890007	+	0.455947i	$0.849301\pi$
$642$		0			0
$643$	−1.62316e9	+	2.81140e9i	−0.240782	+	0.417046i	−0.960937	−	0.276767i	$-0.910737\pi$
$643$	−1.62316e9	+	2.81140e9i	−0.240782	+	0.417046i	0.720155	+	0.693813i	$0.244070\pi$
$644$	−1.22895e9	+	2.12861e9i	−0.181316	+	0.314048i
$645$		0			0
$646$	1.92360e7	+	3.33178e7i	0.00280739	+	0.00486253i
$647$	7.46321e9			1.08333			0.541665	−	0.840595i	$-0.317794\pi$
$647$	7.46321e9			1.08333			0.541665	+	0.840595i	$0.317794\pi$
$648$		0			0
$649$	−6.73041e9			−0.966462
$650$	1.02594e9	+	1.77698e9i	0.146530	+	0.253797i
$651$		0			0
$652$	−1.67554e9	+	2.90212e9i	−0.236749	+	0.410061i
$653$	5.08150e9	−	8.80141e9i	0.714160	−	1.23696i	−0.249122	−	0.968472i	$-0.580142\pi$
$653$	5.08150e9	−	8.80141e9i	0.714160	−	1.23696i	0.963283	−	0.268490i	$-0.0865245\pi$
$654$		0			0
$655$	−1.84309e9	−	3.19232e9i	−0.256272	−	0.443876i
$656$	−1.26205e9			−0.174547
$657$		0			0
$658$	1.55731e9			0.213100
$659$	1.41534e9	+	2.45144e9i	0.192647	+	0.333674i	0.946127	−	0.323797i	$-0.104959\pi$
$659$	1.41534e9	+	2.45144e9i	0.192647	+	0.333674i	−0.753480	+	0.657471i	$0.771626\pi$
$660$		0			0
$661$	−1.98137e9	+	3.43184e9i	−0.266846	+	0.462192i	−0.968046	−	0.250774i	$-0.919315\pi$
$661$	−1.98137e9	+	3.43184e9i	−0.266846	+	0.462192i	0.701199	+	0.712965i	$0.252648\pi$
$662$	−3.23075e9	+	5.59583e9i	−0.432814	+	0.749655i
$663$		0			0
$664$	6.78706e8	+	1.17555e9i	0.0899691	+	0.155831i
$665$	1.26401e8			0.0166676
$666$		0			0
$667$	6.24947e9			0.815461
$668$	1.75346e9	+	3.03709e9i	0.227604	+	0.394221i
$669$		0			0
$670$	9.72304e8	−	1.68408e9i	0.124894	−	0.216322i
$671$	−3.39416e9	+	5.87886e9i	−0.433714	+	0.751215i
$672$		0			0
$673$	5.93077e9	+	1.02724e10i	0.749995	+	1.29903i	0.947824	+	0.318794i	$0.103278\pi$
$673$	5.93077e9	+	1.02724e10i	0.749995	+	1.29903i	−0.197829	+	0.980237i	$0.563389\pi$
$674$	−1.49824e9			−0.188483
$675$		0			0
$676$	−2.39085e9			−0.297672
$677$	−4.93418e9	−	8.54625e9i	−0.611159	−	1.05856i	−0.991045	−	0.133526i	$-0.957370\pi$
$677$	−4.93418e9	−	8.54625e9i	−0.611159	−	1.05856i	0.379886	−	0.925033i	$-0.375963\pi$
$678$		0			0
$679$	4.10880e9	−	7.11665e9i	0.503699	−	0.872432i
$680$	−1.34703e8	+	2.33313e8i	−0.0164285	+	0.0284550i
$681$		0			0
$682$	2.11549e9	+	3.66414e9i	0.255368	+	0.442310i
$683$	1.36287e10			1.63674			0.818372	−	0.574689i	$-0.194877\pi$
$683$	1.36287e10			1.63674			0.818372	+	0.574689i	$0.194877\pi$
$684$		0			0
$685$	−2.12241e9			−0.252297
$686$	−2.82249e9	−	4.88870e9i	−0.333809	−	0.578175i
$687$		0			0
$688$	−6.89521e8	+	1.19428e9i	−0.0807212	+	0.139813i
$689$	−1.91534e9	+	3.31746e9i	−0.223089	+	0.386402i
$690$		0			0
$691$	1.08135e9	+	1.87295e9i	0.124679	+	0.215950i	0.921607	−	0.388124i	$-0.126877\pi$
$691$	1.08135e9	+	1.87295e9i	0.124679	+	0.215950i	−0.796929	+	0.604074i	$0.793543\pi$
$692$	7.91329e9			0.907791
$693$		0			0
$694$	2.26045e9			0.256706
$695$	3.64569e9	+	6.31453e9i	0.411940	+	0.713500i
$696$		0			0
$697$	−4.91294e8	+	8.50946e8i	−0.0549575	+	0.0951892i
$698$	3.04245e9	−	5.26968e9i	0.338633	−	0.586530i
$699$		0			0
$700$	−8.27430e8	−	1.43315e9i	−0.0911776	−	0.157924i
$701$	2.93914e9			0.322260			0.161130	−	0.986933i	$-0.448486\pi$
$701$	2.93914e9			0.322260			0.161130	+	0.986933i	$0.448486\pi$
$702$		0			0
$703$	−4.87942e8			−0.0529693
$704$	−3.96362e8	−	6.86519e8i	−0.0428142	−	0.0741563i
$705$		0			0
$706$	1.87557e9	−	3.24859e9i	0.200594	−	0.347439i
$707$	6.86210e8	−	1.18855e9i	0.0730279	−	0.126488i
$708$		0			0
$709$	6.54572e9	+	1.13375e10i	0.689756	+	1.19469i	0.971917	+	0.235326i	$0.0756157\pi$
$709$	6.54572e9	+	1.13375e10i	0.689756	+	1.19469i	−0.282160	+	0.959367i	$0.591051\pi$
$710$	6.60910e9			0.693007
$711$		0			0
$712$	−3.46670e9			−0.359945
$713$	−6.61092e9	−	1.14504e10i	−0.683044	−	1.18307i
$714$		0			0
$715$	1.25713e9	−	2.17741e9i	0.128620	−	0.222777i
$716$	−3.15935e9	+	5.47216e9i	−0.321664	+	0.557138i
$717$		0			0
$718$	−6.15626e8	−	1.06630e9i	−0.0620699	−	0.107508i
$719$	−3.03015e9			−0.304027			−0.152014	−	0.988378i	$-0.548576\pi$
$719$	−3.03015e9			−0.304027			−0.152014	+	0.988378i	$0.548576\pi$
$720$		0			0
$721$	−1.05101e9			−0.104432
$722$	3.56639e9	+	6.17717e9i	0.352654	+	0.610815i
$723$		0			0
$724$	−2.94354e9	+	5.09836e9i	−0.288260	+	0.499281i
$725$	−2.10382e9	+	3.64393e9i	−0.205034	+	0.355130i
$726$		0			0
$727$	5.34238e9	+	9.25328e9i	0.515661	+	0.893151i	0.999835	+	0.0181794i	$0.00578701\pi$
$727$	5.34238e9	+	9.25328e9i	0.515661	+	0.893151i	−0.484174	+	0.874972i	$0.660880\pi$
$728$	−1.31062e9			−0.125898
$729$		0			0
$730$	7.78576e9			0.740749
$731$	5.36836e8	+	9.29828e8i	0.0508313	+	0.0880424i
$732$		0			0
$733$	8.38524e9	−	1.45237e10i	0.786414	−	1.36211i	−0.141736	−	0.989904i	$-0.545268\pi$
$733$	8.38524e9	−	1.45237e10i	0.786414	−	1.36211i	0.928150	−	0.372205i	$-0.121398\pi$
$734$	2.99811e9	−	5.19288e9i	0.279841	−	0.484699i
$735$		0			0
$736$	1.23863e9	+	2.14537e9i	0.114517	+	0.198349i
$737$	4.45492e9			0.409924
$738$		0			0
$739$	1.68052e10			1.53175			0.765877	−	0.642987i	$-0.222305\pi$
$739$	1.68052e10			1.53175			0.765877	+	0.642987i	$0.222305\pi$
$740$	−1.70844e9	−	2.95911e9i	−0.154985	−	0.268442i
$741$		0			0
$742$	1.54474e9	−	2.67557e9i	0.138817	−	0.240437i
$743$	−1.03917e10	+	1.79989e10i	−0.929448	+	1.60985i	−0.145203	+	0.989402i	$0.546383\pi$
$743$	−1.03917e10	+	1.79989e10i	−0.929448	+	1.60985i	−0.784246	+	0.620450i	$0.786950\pi$
$744$		0			0
$745$	−2.16882e8	−	3.75651e8i	−0.0192166	−	0.0332841i
$746$	−4.33382e9			−0.382195
$747$		0			0
$748$	−6.17186e8			−0.0539213
$749$	−2.32524e9	−	4.02744e9i	−0.202200	−	0.350221i
$750$		0			0
$751$	−5.03006e9	+	8.71232e9i	−0.433344	+	0.750575i	−0.997159	−	0.0753269i	$-0.976000\pi$
$751$	−5.03006e9	+	8.71232e9i	−0.433344	+	0.750575i	0.563814	+	0.825901i	$0.309333\pi$
$752$	7.84785e8	−	1.35929e9i	0.0672959	−	0.116560i
$753$		0			0
$754$	1.66620e9	+	2.88594e9i	0.141555	+	0.245181i
$755$	9.12481e9			0.771631
$756$		0			0
$757$	6.59893e9			0.552889			0.276445	−	0.961030i	$-0.410844\pi$
$757$	6.59893e9			0.552889			0.276445	+	0.961030i	$0.410844\pi$
$758$	−5.31896e9	−	9.21270e9i	−0.443592	−	0.768325i
$759$		0			0
$760$	6.36979e7	−	1.10328e8i	0.00526354	−	0.00911671i
$761$	3.03962e9	−	5.26478e9i	0.250019	−	0.433046i	−0.713511	−	0.700644i	$-0.752896\pi$
$761$	3.03962e9	−	5.26478e9i	0.250019	−	0.433046i	0.963531	+	0.267597i	$0.0862297\pi$
$762$		0			0
$763$	2.20636e9	+	3.82153e9i	0.179821	+	0.311459i
$764$	8.44663e9			0.685262
$765$		0			0
$766$	5.75959e9			0.463011
$767$	5.60756e9	+	9.71258e9i	0.448735	+	0.777232i
$768$		0			0
$769$	−2.42525e9	+	4.20065e9i	−0.192315	+	0.333100i	−0.946017	−	0.324117i	$-0.894933\pi$
$769$	−2.42525e9	+	4.20065e9i	−0.192315	+	0.333100i	0.753702	+	0.657216i	$0.228266\pi$
$770$	−1.01389e9	+	1.75610e9i	−0.0800335	+	0.138622i
$771$		0			0
$772$	−4.73304e9	−	8.19786e9i	−0.370236	−	0.641268i
$773$	−2.33466e10			−1.81800			−0.909002	−	0.416792i	$-0.863154\pi$
$773$	−2.33466e10			−1.81800			−0.909002	+	0.416792i	$0.863154\pi$
$774$		0			0
$775$	8.90200e9			0.686961
$776$	−4.14115e9	−	7.17269e9i	−0.318131	−	0.551018i
$777$		0			0
$778$	−9.28268e9	+	1.60781e10i	−0.706716	+	1.22407i
$779$	2.32321e8	−	4.02392e8i	0.0176079	−	0.0304978i
$780$		0			0
$781$	7.57042e9	+	1.31124e10i	0.568645	+	0.984922i
$782$	1.92871e9			0.144226
$783$		0			0
$784$	−2.31620e9			−0.171660
$785$	−1.84561e9	−	3.19669e9i	−0.136175	−	0.235862i
$786$		0			0
$787$	−4.68028e9	+	8.10648e9i	−0.342263	+	0.592817i	−0.984853	−	0.173394i	$-0.944527\pi$
$787$	−4.68028e9	+	8.10648e9i	−0.342263	+	0.592817i	0.642590	+	0.766211i	$0.277860\pi$
$788$	8.62595e8	−	1.49406e9i	0.0628008	−	0.108774i
$789$		0			0
$790$	4.63899e9	+	8.03496e9i	0.334756	+	0.579814i
$791$	1.26886e10			0.911582
$792$		0			0
$793$	1.13116e10			0.805506
$794$	−4.73491e9	−	8.20110e9i	−0.335691	−	0.581434i
$795$		0			0
$796$	−4.34658e9	+	7.52849e9i	−0.305458	+	0.529069i
$797$	−1.31186e10	+	2.27222e10i	−0.917878	+	1.58981i	−0.115245	+	0.993337i	$0.536765\pi$
$797$	−1.31186e10	+	2.27222e10i	−0.917878	+	1.58981i	−0.802633	+	0.596474i	$0.796568\pi$
$798$		0			0
$799$	−6.11006e8	−	1.05829e9i	−0.0423772	−	0.0733994i
$800$	−1.66789e9			−0.115174
$801$		0			0
$802$	1.01411e10			0.694185
$803$	8.91823e9	+	1.54468e10i	0.607819	+	1.05277i
$804$		0			0
$805$	3.16840e9	−	5.48782e9i	0.214069	−	0.370778i
$806$	3.52512e9	−	6.10569e9i	0.237138	−	0.410736i
$807$		0			0
$808$	−6.91613e8	−	1.19791e9i	−0.0461236	−	0.0798885i
$809$	−1.59304e10			−1.05780			−0.528902	−	0.848683i	$-0.677396\pi$
$809$	−1.59304e10			−1.05780			−0.528902	+	0.848683i	$0.677396\pi$
$810$		0			0
$811$	−1.69281e10			−1.11438			−0.557191	−	0.830384i	$-0.688121\pi$
$811$	−1.69281e10			−1.11438			−0.557191	+	0.830384i	$0.688121\pi$
$812$	−1.34380e9	−	2.32753e9i	−0.0880824	−	0.152563i
$813$		0			0
$814$	3.91389e9	−	6.77906e9i	0.254345	−	0.440539i
$815$	4.31975e9	−	7.48202e9i	0.279516	−	0.484136i
$816$		0			0
$817$	−2.53857e8	−	4.39693e8i	−0.0162859	−	0.0282080i
$818$	−8.15591e9			−0.520997
$819$		0			0
$820$	3.25373e9			0.206078
$821$	−4.36796e9	−	7.56553e9i	−0.275472	−	0.477131i	0.694782	−	0.719220i	$-0.255501\pi$
$821$	−4.36796e9	−	7.56553e9i	−0.275472	−	0.477131i	−0.970254	+	0.242089i	$0.922167\pi$
$822$		0			0
$823$	−7.21847e9	+	1.25028e10i	−0.451384	+	0.781820i	−0.998472	−	0.0552552i	$-0.982403\pi$
$823$	−7.21847e9	+	1.25028e10i	−0.451384	+	0.781820i	0.547089	+	0.837075i	$0.315736\pi$
$824$	−5.29642e8	+	9.17368e8i	−0.0329790	+	0.0571213i
$825$		0			0
$826$	−4.52255e9	−	7.83329e9i	−0.279224	−	0.483631i
$827$	5.56666e9			0.342236			0.171118	−	0.985251i	$-0.445262\pi$
$827$	5.56666e9			0.342236			0.171118	+	0.985251i	$0.445262\pi$
$828$		0			0
$829$	−1.50766e10			−0.919101			−0.459550	−	0.888152i	$-0.651989\pi$
$829$	−1.50766e10			−0.919101			−0.459550	+	0.888152i	$0.651989\pi$
$830$	−1.74979e9	−	3.03072e9i	−0.106222	−	0.183981i
$831$		0			0
$832$	−6.60472e8	+	1.14397e9i	−0.0397578	+	0.0688626i
$833$	−9.01656e8	+	1.56171e9i	−0.0540485	+	0.0936147i
$834$		0			0
$835$	−4.52065e9	−	7.82999e9i	−0.268719	−	0.465435i
$836$	2.91852e8			0.0172759
$837$		0			0
$838$	8.33029e9			0.488997
$839$	−6.00020e9	−	1.03927e10i	−0.350751	−	0.607519i	0.635630	−	0.771994i	$-0.280740\pi$
$839$	−6.00020e9	−	1.03927e10i	−0.350751	−	0.607519i	−0.986381	+	0.164475i	$0.947407\pi$
$840$		0			0
$841$	5.20819e9	−	9.02084e9i	0.301926	−	0.522951i
$842$	7.83380e9	−	1.35685e10i	0.452251	−	0.783322i
$843$		0			0
$844$	7.01888e7	+	1.21571e8i	0.00401855	+	0.00696033i
$845$	6.16390e9			0.351445
$846$		0			0
$847$	5.25404e9			0.297099
$848$	−1.55690e9	−	2.69663e9i	−0.0876750	−	0.151858i
$849$		0			0
$850$	−6.49280e8	+	1.12459e9i	−0.0362632	+	0.0628097i
$851$	−1.22309e10	+	2.11846e10i	−0.680307	+	1.17833i
$852$		0			0
$853$	4.50830e8	+	7.80861e8i	0.0248709	+	0.0430777i	0.878193	−	0.478307i	$-0.158749\pi$
$853$	4.50830e8	+	7.80861e8i	0.0248709	+	0.0430777i	−0.853322	+	0.521384i	$0.825416\pi$
$854$	−9.12293e9			−0.501224
$855$		0			0
$856$	−4.68710e9			−0.255415
$857$	1.40718e10	+	2.43731e10i	0.763690	+	1.32275i	0.940936	+	0.338583i	$0.109948\pi$
$857$	1.40718e10	+	2.43731e10i	0.763690	+	1.32275i	−0.177247	+	0.984166i	$0.556719\pi$
$858$		0			0
$859$	4.75771e9	−	8.24060e9i	0.256107	−	0.443591i	−0.709088	−	0.705120i	$-0.750893\pi$
$859$	4.75771e9	−	8.24060e9i	0.256107	−	0.443591i	0.965196	+	0.261529i	$0.0842266\pi$
$860$	1.77767e9	−	3.07902e9i	0.0953030	−	0.165070i
$861$		0			0
$862$	−1.13702e10	−	1.96938e10i	−0.604636	−	1.04726i
$863$	−1.24682e10			−0.660336			−0.330168	−	0.943922i	$-0.607105\pi$
$863$	−1.24682e10			−0.660336			−0.330168	+	0.943922i	$0.607105\pi$
$864$		0			0
$865$	−2.04014e10			−1.07178
$866$	3.36517e8	+	5.82864e8i	0.0176074	+	0.0304968i
$867$		0			0
$868$	−2.84304e9	+	4.92430e9i	−0.147559	+	0.255579i
$869$	−1.06275e10	+	1.84074e10i	−0.549366	+	0.951530i
$870$		0			0
$871$	−3.71170e9	−	6.42885e9i	−0.190331	−	0.329663i
$872$	4.44746e9			0.227146
$873$		0			0
$874$	−9.12038e8			−0.0462086
$875$	5.40744e9	+	9.36596e9i	0.272875	+	0.472633i
$876$		0			0
$877$	3.35537e9	−	5.81167e9i	0.167974	−	0.290939i	−0.769734	−	0.638365i	$-0.779611\pi$
$877$	3.35537e9	−	5.81167e9i	0.167974	−	0.290939i	0.937707	+	0.347426i	$0.112944\pi$
$878$	7.48186e9	−	1.29590e10i	0.373060	−	0.646159i
$879$		0			0
$880$	1.02187e9	+	1.76993e9i	0.0505483	+	0.0875522i
$881$	−3.59350e10			−1.77053			−0.885263	−	0.465091i	$-0.846022\pi$
$881$	−3.59350e10			−1.77053			−0.885263	+	0.465091i	$0.846022\pi$
$882$		0			0
$883$	−1.40241e9			−0.0685509			−0.0342754	−	0.999412i	$-0.510912\pi$
$883$	−1.40241e9			−0.0685509			−0.0342754	+	0.999412i	$0.510912\pi$
$884$	5.14220e8	+	8.90655e8i	0.0250361	+	0.0433637i
$885$		0			0
$886$	4.05331e9	−	7.02053e9i	0.195790	−	0.339119i
$887$	4.09845e8	−	7.09873e8i	0.0197191	−	0.0341545i	−0.855997	−	0.516980i	$-0.827056\pi$
$887$	4.09845e8	−	7.09873e8i	0.0197191	−	0.0341545i	0.875717	+	0.482826i	$0.160389\pi$
$888$		0			0
$889$	−1.26659e9	−	2.19381e9i	−0.0604618	−	0.104723i
$890$	8.93759e9			0.424967
$891$		0			0
$892$	1.27643e9			0.0602174
$893$	2.88930e8	+	5.00441e8i	0.0135773	+	0.0235165i
$894$		0			0
$895$	8.14520e9	−	1.41079e10i	0.379771	−	0.657782i
$896$	5.32677e8	−	9.22623e8i	0.0247392	−	0.0428495i
$897$		0			0
$898$	6.63022e8	+	1.14839e9i	0.0305535	+	0.0529202i
$899$	1.44574e10			0.663640
$900$		0			0
$901$	−2.42430e9			−0.110420
$902$	3.72700e9	+	6.45535e9i	0.169097	+	0.292885i
$903$		0			0
$904$	6.39425e9	−	1.10752e10i	0.287873	−	0.498610i
$905$	7.58881e9	−	1.31442e10i	0.340333	−	0.589473i
$906$		0			0
$907$	−1.42253e9	−	2.46390e9i	−0.0633047	−	0.109647i	0.832636	−	0.553821i	$-0.186831\pi$
$907$	−1.42253e9	−	2.46390e9i	−0.0633047	−	0.109647i	−0.895941	+	0.444174i	$0.853497\pi$
$908$	−8.06304e9			−0.357436
$909$		0			0
$910$	3.37895e9			0.148640
$911$	2.16498e10	+	3.74985e10i	0.948723	+	1.64324i	0.748119	+	0.663564i	$0.230957\pi$
$911$	2.16498e10	+	3.74985e10i	0.948723	+	1.64324i	0.200604	+	0.979672i	$0.435710\pi$
$912$		0			0
$913$	4.00861e9	−	6.94311e9i	0.174319	−	0.301930i
$914$	5.52319e9	−	9.56644e9i	0.239264	−	0.414418i
$915$		0			0
$916$	−7.52146e8	−	1.30276e9i	−0.0323346	−	0.0560052i
$917$	1.13490e10			0.486030
$918$		0			0
$919$	−3.26317e10			−1.38687			−0.693435	−	0.720520i	$-0.743903\pi$
$919$	−3.26317e10			−1.38687			−0.693435	+	0.720520i	$0.743903\pi$
$920$	−3.19334e9	−	5.53103e9i	−0.135204	−	0.234180i
$921$		0			0
$922$	−1.11397e10	+	1.92944e10i	−0.468073	+	0.810726i
$923$	1.26149e10	−	2.18496e10i	0.528052	−	0.914613i
$924$		0			0
$925$	−8.23483e9	−	1.42631e10i	−0.342104	−	0.592542i
$926$	2.23135e10			0.923483
$927$		0			0
$928$	−2.70877e9			−0.111264
$929$	−2.34773e10	−	4.06639e10i	−0.960712	−	1.66400i	−0.720718	−	0.693228i	$-0.756188\pi$
$929$	−2.34773e10	−	4.06639e10i	−0.960712	−	1.66400i	−0.239994	−	0.970774i	$-0.577146\pi$
$930$		0			0
$931$	4.26371e8	−	7.38497e8i	0.0173167	−	0.0299933i
$932$	4.08822e9	−	7.08100e9i	0.165416	−	0.286509i
$933$		0			0
$934$	−1.58903e10	−	2.75228e10i	−0.638143	−	1.10530i
$935$	1.59118e9			0.0636619
$936$		0			0
$937$	−2.80951e10			−1.11569			−0.557843	−	0.829947i	$-0.688371\pi$
$937$	−2.80951e10			−1.11569			−0.557843	+	0.829947i	$0.688371\pi$
$938$	2.99352e9	+	5.18493e9i	0.118433	+	0.205132i
$939$		0			0
$940$	−2.02327e9	+	3.50441e9i	−0.0794525	+	0.137616i
$941$	1.36099e10	−	2.35730e10i	0.532464	−	0.922254i	−0.466818	−	0.884354i	$-0.654600\pi$
$941$	1.36099e10	−	2.35730e10i	0.532464	−	0.922254i	0.999282	−	0.0379007i	$-0.0120671\pi$
$942$		0			0
$943$	−1.16469e10	−	2.01730e10i	−0.452291	−	0.783391i
$944$	−9.11632e9			−0.352710
$945$		0			0
$946$	8.14496e9			0.312802
$947$	1.15221e10	+	1.99568e10i	0.440864	+	0.763599i	0.997754	−	0.0669872i	$-0.0213387\pi$
$947$	1.15221e10	+	1.99568e10i	0.440864	+	0.763599i	−0.556890	+	0.830587i	$0.688005\pi$
$948$		0			0
$949$	1.48608e10	−	2.57396e10i	0.564429	−	0.977620i
$950$	3.07029e8	−	5.31789e8i	0.0116184	−	0.0201237i
$951$		0			0
$952$	−4.14723e8	−	7.18321e8i	−0.0155786	−	0.0269830i
$953$	2.22131e10			0.831352			0.415676	−	0.909513i	$-0.363545\pi$
$953$	2.22131e10			0.831352			0.415676	+	0.909513i	$0.363545\pi$
$954$		0			0
$955$	−2.17765e10			−0.809051
$956$	2.99161e9	+	5.18162e9i	0.110739	+	0.191806i
$957$		0			0
$958$	9.37868e9	−	1.62443e10i	0.344637	−	0.596929i
$959$	3.26722e9	−	5.65899e9i	0.119623	−	0.207193i
$960$		0			0
$961$	−1.53730e9	−	2.66268e9i	−0.0558761	−	0.0967803i
$962$	−1.30437e10			−0.472377
$963$		0			0
$964$	−8.62593e9			−0.310125
$965$	1.22024e10	+	2.11351e10i	0.437117	+	0.757110i
$966$		0			0
$967$	−1.35385e10	+	2.34494e10i	−0.481481	+	0.833950i	−0.999774	−	0.0212534i	$-0.993234\pi$
$967$	−1.35385e10	+	2.34494e10i	−0.481481	+	0.833950i	0.518293	+	0.855203i	$0.326568\pi$
$968$	2.64770e9	−	4.58596e9i	0.0938222	−	0.162505i
$969$		0			0
$970$	1.06764e10	+	1.84921e10i	0.375599	+	0.650556i
$971$	−2.57664e10			−0.903204			−0.451602	−	0.892220i	$-0.649147\pi$
$971$	−2.57664e10			−0.903204			−0.451602	+	0.892220i	$0.649147\pi$
$972$		0			0
$973$	−2.24486e10			−0.781259
$974$	−1.23352e10	−	2.13652e10i	−0.427751	−	0.740886i
$975$		0			0
$976$	−4.59738e9	+	7.96290e9i	−0.158284	+	0.274155i
$977$	4.75936e9	−	8.24346e9i	0.163274	−	0.282799i	−0.772767	−	0.634690i	$-0.781128\pi$
$977$	4.75936e9	−	8.24346e9i	0.163274	−	0.282799i	0.936041	+	0.351891i	$0.114461\pi$
$978$		0			0
$979$	1.02376e10	+	1.77320e10i	0.348706	+	0.603976i
$980$	5.97146e9			0.202670
$981$		0			0
$982$	−2.97481e10			−1.00246
$983$	2.75959e9	+	4.77974e9i	0.0926631	+	0.160497i	0.908631	−	0.417600i	$-0.137129\pi$
$983$	2.75959e9	+	4.77974e9i	0.0926631	+	0.160497i	−0.815968	+	0.578097i	$0.803795\pi$
$984$		0			0
$985$	−2.22388e9	+	3.85187e9i	−0.0741454	+	0.128424i
$986$	−1.05447e9	+	1.82640e9i	−0.0350322	+	0.0606775i
$987$		0			0
$988$	−2.43162e8	−	4.21169e8i	−0.00802133	−	0.0138934i
$989$	−2.54530e10			−0.836666
$990$		0			0
$991$	1.65440e10			0.539987			0.269994	−	0.962862i	$-0.412978\pi$
$991$	1.65440e10			0.539987			0.269994	+	0.962862i	$0.412978\pi$
$992$	2.86543e9	+	4.96307e9i	0.0931964	+	0.161421i
$993$		0			0
$994$	−1.01740e10	+	1.76219e10i	−0.328579	+	0.569115i
$995$	1.12060e10	−	1.94094e10i	0.360637	−	0.624642i
$996$		0			0
$997$	−2.05028e10	−	3.55118e10i	−0.655208	−	1.13485i	−0.981842	−	0.189703i	$-0.939248\pi$
$997$	−2.05028e10	−	3.55118e10i	−0.655208	−	1.13485i	0.326633	−	0.945151i	$-0.394086\pi$
$998$	3.49141e9			0.111184
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	162.8.c.e.109.1		2
3.2	odd	2		162.8.c.h.109.1		2
9.2	odd	6		162.8.c.h.55.1		2
9.4	even	3		162.8.a.b.1.1	yes	1
9.5	odd	6		162.8.a.a.1.1	✓	1
9.7	even	3	inner	162.8.c.e.55.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
162.8.a.a.1.1	✓	1	9.5	odd	6
162.8.a.b.1.1	yes	1	9.4	even	3
162.8.c.e.55.1		2	9.7	even	3	inner
162.8.c.e.109.1		2	1.1	even	1	trivial
162.8.c.h.55.1		2	9.2	odd	6
162.8.c.h.109.1		2	3.2	odd	2

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 162 = 2 \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	162.c (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-4.00000 - 6.92820i) q^{2} +(-32.0000 + 55.4256i) q^{4} +(82.5000 - 142.894i) q^{5} +(254.000 + 439.941i) q^{7} +512.000 q^{8} -1320.00 q^{10} +(-1512.00 - 2618.86i) q^{11} +(-2519.50 + 4363.90i) q^{13} +(2032.00 - 3519.53i) q^{14} +(-2048.00 - 3547.24i) q^{16} -3189.00 q^{17} +1508.00 q^{19} +(5280.00 + 9145.23i) q^{20} +(-12096.0 + 20950.9i) q^{22} +(37800.0 - 65471.5i) q^{23} +(25450.0 + 44080.7i) q^{25} +40312.0 q^{26} -32512.0 q^{28} +(41332.5 + 71590.0i) q^{29} +(87446.0 - 151461. i) q^{31} +(-16384.0 + 28377.9i) q^{32} +(12756.0 + 22094.0i) q^{34} +83820.0 q^{35} -323569. q^{37} +(-6032.00 - 10447.7i) q^{38} +(42240.0 - 73161.8i) q^{40} +(154059. - 266838. i) q^{41} +(-168340. - 291573. i) q^{43} +193536. q^{44} -604800. q^{46} +(191598. + 331857. i) q^{47} +(282740. - 489719. i) q^{49} +(203600. - 352646. i) q^{50} +(-161248. - 279290. i) q^{52} +760206. q^{53} -498960. q^{55} +(130048. + 225250. i) q^{56} +(330660. - 572720. i) q^{58} +(1.11283e6 - 1.92748e6i) q^{59} +(-1.12241e6 - 1.94407e6i) q^{61} -1.39914e6 q^{62} +262144. q^{64} +(415718. + 720044. i) q^{65} +(-736594. + 1.27582e6i) q^{67} +(102048. - 176752. i) q^{68} +(-335280. - 580722. i) q^{70} -5.00689e6 q^{71} -5.89830e6 q^{73} +(1.29428e6 + 2.24175e6i) q^{74} +(-48256.0 + 83581.8i) q^{76} +(768096. - 1.33038e6i) q^{77} +(-3.51438e6 - 6.08709e6i) q^{79} -675840. q^{80} -2.46494e6 q^{82} +(1.32560e6 + 2.29600e6i) q^{83} +(-263092. + 455690. i) q^{85} +(-1.34672e6 + 2.33259e6i) q^{86} +(-774144. - 1.34086e6i) q^{88} -6.77090e6 q^{89} -2.55981e6 q^{91} +(2.41920e6 + 4.19018e6i) q^{92} +(1.53278e6 - 2.65486e6i) q^{94} +(124410. - 215484. i) q^{95} +(-8.08819e6 - 1.40092e7i) q^{97} -4.52383e6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 8 q^{2} - 64 q^{4} + 165 q^{5} + 508 q^{7} + 1024 q^{8} - 2640 q^{10} - 3024 q^{11} - 5039 q^{13} + 4064 q^{14} - 4096 q^{16} - 6378 q^{17} + 3016 q^{19} + 10560 q^{20} - 24192 q^{22} + 75600 q^{23}+ \cdots - 9047664 q^{98}+O(q^{100}) \) 2 * q - 8 * q^2 - 64 * q^4 + 165 * q^5 + 508 * q^7 + 1024 * q^8 - 2640 * q^10 - 3024 * q^11 - 5039 * q^13 + 4064 * q^14 - 4096 * q^16 - 6378 * q^17 + 3016 * q^19 + 10560 * q^20 - 24192 * q^22 + 75600 * q^23 + 50900 * q^25 + 80624 * q^26 - 65024 * q^28 + 82665 * q^29 + 174892 * q^31 - 32768 * q^32 + 25512 * q^34 + 167640 * q^35 - 647138 * q^37 - 12064 * q^38 + 84480 * q^40 + 308118 * q^41 - 336680 * q^43 + 387072 * q^44 - 1209600 * q^46 + 383196 * q^47 + 565479 * q^49 + 407200 * q^50 - 322496 * q^52 + 1520412 * q^53 - 997920 * q^55 + 260096 * q^56 + 661320 * q^58 + 2225664 * q^59 - 2244815 * q^61 - 2798272 * q^62 + 524288 * q^64 + 831435 * q^65 - 1473188 * q^67 + 204096 * q^68 - 670560 * q^70 - 10013784 * q^71 - 11796602 * q^73 + 2588552 * q^74 - 96512 * q^76 + 1536192 * q^77 - 7028768 * q^79 - 1351680 * q^80 - 4929888 * q^82 + 2651196 * q^83 - 526185 * q^85 - 2693440 * q^86 - 1548288 * q^88 - 13541802 * q^89 - 5119624 * q^91 + 4838400 * q^92 + 3065568 * q^94 + 248820 * q^95 - 16176386 * q^97 - 9047664 * q^98

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