LMFDB - Embedded newform 164.5.d.f.163.7

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [164,5,Mod(163,164)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("164.163"); S:= CuspForms(chi, 5); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(164, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 5, names="a")

Level:	$ N $	$=$	$ 164 = 2^{2} \cdot 41 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	164.d (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [72,6,0,-162,-8] 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:	$16.9526739458$
Analytic rank:	$0$
Dimension:	$72$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			163.7
Character	$\chi$	$=$	164.163
Dual form			164.5.d.f.163.5

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-3.64057 + 1.65719i) q^{2} -12.7376 q^{3} +(10.5074 - 12.0662i) q^{4} -20.4399 q^{5} +(46.3719 - 21.1085i) q^{6} +9.19140 q^{7} +(-18.2571 + 61.3407i) q^{8} +81.2453 q^{9} +(74.4128 - 33.8728i) q^{10} +123.546 q^{11} +(-133.839 + 153.694i) q^{12} +161.922i q^{13} +(-33.4619 + 15.2319i) q^{14} +260.354 q^{15} +(-35.1871 - 253.570i) q^{16} +238.118i q^{17} +(-295.779 + 134.639i) q^{18} +236.342 q^{19} +(-214.771 + 246.632i) q^{20} -117.076 q^{21} +(-449.776 + 204.738i) q^{22} +267.309i q^{23} +(232.550 - 781.330i) q^{24} -207.211 q^{25} +(-268.336 - 589.488i) q^{26} -3.12443 q^{27} +(96.5781 - 110.905i) q^{28} -419.632i q^{29} +(-947.837 + 431.456i) q^{30} -735.805i q^{31} +(548.315 + 864.828i) q^{32} -1573.67 q^{33} +(-394.606 - 866.884i) q^{34} -187.871 q^{35} +(853.681 - 980.323i) q^{36} -394.368 q^{37} +(-860.420 + 391.664i) q^{38} -2062.49i q^{39} +(373.172 - 1253.80i) q^{40} +(-1675.83 - 131.679i) q^{41} +(426.223 - 194.017i) q^{42} -3179.64i q^{43} +(1298.15 - 1490.73i) q^{44} -1660.65 q^{45} +(-442.981 - 973.156i) q^{46} +321.858 q^{47} +(448.198 + 3229.86i) q^{48} -2316.52 q^{49} +(754.364 - 343.387i) q^{50} -3033.04i q^{51} +(1953.79 + 1701.39i) q^{52} +3568.03i q^{53} +(11.3747 - 5.17777i) q^{54} -2525.26 q^{55} +(-167.808 + 563.807i) q^{56} -3010.42 q^{57} +(695.410 + 1527.70i) q^{58} +2646.01i q^{59} +(2735.66 - 3141.49i) q^{60} +2735.55 q^{61} +(1219.37 + 2678.75i) q^{62} +746.758 q^{63} +(-3429.36 - 2239.80i) q^{64} -3309.67i q^{65} +(5729.04 - 2607.87i) q^{66} -3660.04 q^{67} +(2873.18 + 2502.01i) q^{68} -3404.86i q^{69} +(683.958 - 311.338i) q^{70} -2837.05 q^{71} +(-1483.30 + 4983.64i) q^{72} -561.944 q^{73} +(1435.72 - 653.542i) q^{74} +2639.36 q^{75} +(2483.35 - 2851.76i) q^{76} +1135.56 q^{77} +(3417.94 + 7508.64i) q^{78} -2409.86 q^{79} +(719.221 + 5182.95i) q^{80} -6541.07 q^{81} +(6319.20 - 2297.79i) q^{82} +8044.10i q^{83} +(-1230.17 + 1412.66i) q^{84} -4867.10i q^{85} +(5269.27 + 11575.7i) q^{86} +5345.09i q^{87} +(-2255.58 + 7578.37i) q^{88} +8336.90i q^{89} +(6045.69 - 2752.00i) q^{90} +1488.29i q^{91} +(3225.41 + 2808.73i) q^{92} +9372.36i q^{93} +(-1171.74 + 533.379i) q^{94} -4830.81 q^{95} +(-6984.19 - 11015.8i) q^{96} -7959.56i q^{97} +(8433.44 - 3838.91i) q^{98} +10037.5 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 72 q + 6 q^{2} - 162 q^{4} - 8 q^{5} + 162 q^{8} + 1728 q^{9} - 272 q^{10} - 2842 q^{16} - 298 q^{18} - 584 q^{20} + 280 q^{21} + 4264 q^{25} + 66 q^{32} + 3512 q^{33} - 11338 q^{36} + 5720 q^{37} - 4648 q^{40}+ \cdots + 2902 q^{98}+O(q^{100}) $ 72 * q + 6 * q^2 - 162 * q^4 - 8 * q^5 + 162 * q^8 + 1728 * q^9 - 272 * q^10 - 2842 * q^16 - 298 * q^18 - 584 * q^20 + 280 * q^21 + 4264 * q^25 + 66 * q^32 + 3512 * q^33 - 11338 * q^36 + 5720 * q^37 - 4648 * q^40 - 7928 * q^41 - 10140 * q^42 - 9528 * q^45 - 5680 * q^46 + 6624 * q^49 + 554 * q^50 + 29864 * q^57 + 11712 * q^61 - 3936 * q^62 - 34554 * q^64 + 24852 * q^66 - 12166 * q^72 + 16632 * q^73 - 11120 * q^74 + 84456 * q^77 - 20496 * q^78 - 12104 * q^80 + 13408 * q^81 - 51114 * q^82 - 948 * q^84 - 20400 * q^86 + 64976 * q^90 - 2784 * q^92 + 2902 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/164\mathbb{Z}\right)^\times$.

$n$	$83$	$129$
$\chi(n)$	$-1$	$-1$

Coefficient data

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

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

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

$2$	−3.64057	+	1.65719i	−0.910142	+	0.414297i
$2$	−3.64057	+	1.65719i	−0.910142	+	0.414297i
$3$	−12.7376			−1.41528			−0.707642	−	0.706571i	$-0.750241\pi$
$3$	−12.7376			−1.41528			−0.707642	+	0.706571i	$0.750241\pi$
$4$	10.5074	−	12.0662i	0.656715	−	0.754138i
$5$	−20.4399			−0.817596			−0.408798	−	0.912625i	$-0.634052\pi$
$5$	−20.4399			−0.817596			−0.408798	+	0.912625i	$0.634052\pi$
$6$	46.3719	−	21.1085i	1.28811	−	0.586348i
$7$	9.19140			0.187580			0.0937898	−	0.995592i	$-0.470102\pi$
$7$	9.19140			0.187580			0.0937898	+	0.995592i	$0.470102\pi$
$8$	−18.2571	+	61.3407i	−0.285266	+	0.958448i
$9$	81.2453			1.00303
$10$	74.4128	−	33.8728i	0.744128	−	0.338728i
$11$	123.546			1.02104			0.510519	−	0.859867i	$-0.329453\pi$
$11$	123.546			1.02104			0.510519	+	0.859867i	$0.329453\pi$
$12$	−133.839	+	153.694i	−0.929439	+	1.06732i
$13$			161.922i			0.958119i	0.877782	+	0.479060i	$0.159022\pi$
$13$			161.922i			0.958119i	−0.877782	+	0.479060i	$0.840978\pi$
$14$	−33.4619	+	15.2319i	−0.170724	+	0.0777137i
$15$	260.354			1.15713
$16$	−35.1871	−	253.570i	−0.137450	−	0.990509i
$17$			238.118i			0.823937i	0.911198	+	0.411969i	$0.135159\pi$
$17$			238.118i			0.823937i	−0.911198	+	0.411969i	$0.864841\pi$
$18$	−295.779	+	134.639i	−0.912898	+	0.415552i
$19$	236.342			0.654688			0.327344	−	0.944905i	$-0.393846\pi$
$19$	236.342			0.654688			0.327344	+	0.944905i	$0.393846\pi$
$20$	−214.771	+	246.632i	−0.536928	+	0.616581i
$21$	−117.076			−0.265478
$22$	−449.776	+	204.738i	−0.929289	+	0.423013i
$23$			267.309i			0.505310i	0.967556	+	0.252655i	$0.0813037\pi$
$23$			267.309i			0.505310i	−0.967556	+	0.252655i	$0.918696\pi$
$24$	232.550	−	781.330i	0.403733	−	1.35648i
$25$	−207.211			−0.331537
$26$	−268.336	−	589.488i	−0.396946	−	0.872024i
$27$	−3.12443			−0.00428591
$28$	96.5781	−	110.905i	0.123186	−	0.141461i
$29$		−	419.632i		−	0.498968i	−0.968379	−	0.249484i	$-0.919739\pi$
$29$		−	419.632i		−	0.498968i	0.968379	−	0.249484i	$-0.0802610\pi$
$30$	−947.837	+	431.456i	−1.05315	+	0.479396i
$31$		−	735.805i		−	0.765666i	−0.923817	−	0.382833i	$-0.874948\pi$
$31$		−	735.805i		−	0.765666i	0.923817	−	0.382833i	$-0.125052\pi$
$32$	548.315	+	864.828i	0.535464	+	0.844558i
$33$	−1573.67			−1.44506
$34$	−394.606	−	866.884i	−0.341355	−	0.749900i
$35$	−187.871			−0.153364
$36$	853.681	−	980.323i	0.658704	−	0.756422i
$37$	−394.368			−0.288070			−0.144035	−	0.989573i	$-0.546008\pi$
$37$	−394.368			−0.288070			−0.144035	+	0.989573i	$0.546008\pi$
$38$	−860.420	+	391.664i	−0.595859	+	0.271235i
$39$		−	2062.49i		−	1.35601i
$40$	373.172	−	1253.80i	0.233233	−	0.783623i
$41$	−1675.83	−	131.679i	−0.996927	−	0.0783334i
$41$	−1675.83	−	131.679i	−0.996927	−	0.0783334i
$42$	426.223	−	194.017i	0.241623	−	0.109987i
$43$		−	3179.64i		−	1.71966i	−0.510584	−	0.859828i	$-0.670571\pi$
$43$		−	3179.64i		−	1.71966i	0.510584	−	0.859828i	$-0.329429\pi$
$44$	1298.15	−	1490.73i	0.670531	−	0.770004i
$45$	−1660.65			−0.820072
$46$	−442.981	−	973.156i	−0.209349	−	0.459903i
$47$	321.858			0.145703			0.0728514	−	0.997343i	$-0.476790\pi$
$47$	321.858			0.145703			0.0728514	+	0.997343i	$0.476790\pi$
$48$	448.198	+	3229.86i	0.194530	+	1.40185i
$49$	−2316.52			−0.964814
$50$	754.364	−	343.387i	0.301746	−	0.137355i
$51$		−	3033.04i		−	1.16610i
$52$	1953.79	+	1701.39i	0.722555	+	0.629212i
$53$			3568.03i			1.27021i	0.772425	+	0.635106i	$0.219044\pi$
$53$			3568.03i			1.27021i	−0.772425	+	0.635106i	$0.780956\pi$
$54$	11.3747	−	5.17777i	0.00390079	−	0.00177564i
$55$	−2525.26			−0.834796
$56$	−167.808	+	563.807i	−0.0535102	+	0.179785i
$57$	−3010.42			−0.926569
$58$	695.410	+	1527.70i	0.206721	+	0.454132i
$59$			2646.01i			0.760130i	0.924960	+	0.380065i	$0.124098\pi$
$59$			2646.01i			0.760130i	−0.924960	+	0.380065i	$0.875902\pi$
$60$	2735.66	−	3141.49i	0.759905	−	0.872636i
$61$	2735.55			0.735166			0.367583	−	0.929991i	$-0.380185\pi$
$61$	2735.55			0.735166			0.367583	+	0.929991i	$0.380185\pi$
$62$	1219.37	+	2678.75i	0.317214	+	0.696865i
$63$	746.758			0.188148
$64$	−3429.36	−	2239.80i	−0.837246	−	0.546826i
$65$		−	3309.67i		−	0.783354i
$66$	5729.04	−	2607.87i	1.31521	−	0.598684i
$67$	−3660.04			−0.815334			−0.407667	−	0.913131i	$-0.633658\pi$
$67$	−3660.04			−0.815334			−0.407667	+	0.913131i	$0.633658\pi$
$68$	2873.18	+	2502.01i	0.621363	+	0.541092i
$69$		−	3404.86i		−	0.715157i
$70$	683.958	−	311.338i	0.139583	−	0.0635384i
$71$	−2837.05			−0.562795			−0.281398	−	0.959591i	$-0.590798\pi$
$71$	−2837.05			−0.562795			−0.281398	+	0.959591i	$0.590798\pi$
$72$	−1483.30	+	4983.64i	−0.286130	+	0.961351i
$73$	−561.944			−0.105450			−0.0527251	−	0.998609i	$-0.516791\pi$
$73$	−561.944			−0.105450			−0.0527251	+	0.998609i	$0.516791\pi$
$74$	1435.72	−	653.542i	0.262184	−	0.119347i
$75$	2639.36			0.469219
$76$	2483.35	−	2851.76i	0.429944	−	0.493725i
$77$	1135.56			0.191526
$78$	3417.94	+	7508.64i	0.561792	+	1.23416i
$79$	−2409.86			−0.386134			−0.193067	−	0.981186i	$-0.561843\pi$
$79$	−2409.86			−0.386134			−0.193067	+	0.981186i	$0.561843\pi$
$80$	719.221	+	5182.95i	0.112378	+	0.809836i
$81$	−6541.07			−0.996963
$82$	6319.20	−	2297.79i	0.939798	−	0.341730i
$83$			8044.10i			1.16767i	0.811871	+	0.583836i	$0.198449\pi$
$83$			8044.10i			1.16767i	−0.811871	+	0.583836i	$0.801551\pi$
$84$	−1230.17	+	1412.66i	−0.174344	+	0.200207i
$85$		−	4867.10i		−	0.673648i
$86$	5269.27	+	11575.7i	0.712449	+	1.56513i
$87$			5345.09i			0.706181i
$88$	−2255.58	+	7578.37i	−0.291268	+	0.978612i
$89$			8336.90i			1.05251i	0.850328	+	0.526253i	$0.176403\pi$
$89$			8336.90i			1.05251i	−0.850328	+	0.526253i	$0.823597\pi$
$90$	6045.69	−	2752.00i	0.746381	−	0.339754i
$91$			1488.29i			0.179724i
$92$	3225.41	+	2808.73i	0.381074	+	0.331845i
$93$			9372.36i			1.08364i
$94$	−1171.74	+	533.379i	−0.132610	+	0.0603643i
$95$	−4830.81			−0.535270
$96$	−6984.19	−	11015.8i	−0.757833	−	1.19529i
$97$		−	7959.56i		−	0.845952i	−0.906141	−	0.422976i	$-0.860985\pi$
$97$		−	7959.56i		−	0.845952i	0.906141	−	0.422976i	$-0.139015\pi$
$98$	8433.44	−	3838.91i	0.878117	−	0.399720i
$99$	10037.5			1.02413
$100$	−2177.25	+	2500.25i	−0.217725	+	0.250025i
$101$		−	8429.91i		−	0.826381i	−0.910645	−	0.413190i	$-0.864414\pi$
$101$		−	8429.91i		−	0.826381i	0.910645	−	0.413190i	$-0.135586\pi$
$102$	5026.32	+	11042.0i	0.483114	+	1.06132i
$103$		−	11551.8i		−	1.08887i	−0.838803	−	0.544435i	$-0.816744\pi$
$103$		−	11551.8i		−	1.08887i	0.838803	−	0.544435i	$-0.183256\pi$
$104$	−9932.42	−	2956.22i	−0.918308	−	0.273319i
$105$	2393.02			0.217054
$106$	−5912.89	−	12989.6i	−0.526246	−	1.15607i
$107$		−	1843.67i		−	0.161033i	−0.996753	−	0.0805165i	$-0.974343\pi$
$107$		−	1843.67i		−	0.161033i	0.996753	−	0.0805165i	$-0.0256570\pi$
$108$	−32.8298	+	37.7001i	−0.00281463	+	0.00323217i
$109$		−	3489.38i		−	0.293694i	−0.989159	−	0.146847i	$-0.953087\pi$
$109$		−	3489.38i		−	0.293694i	0.989159	−	0.146847i	$-0.0469125\pi$
$110$	9193.37	−	4184.83i	0.759783	−	0.345854i
$111$	5023.28			0.407700
$112$	−323.419	−	2330.67i	−0.0257828	−	0.185799i
$113$	−14946.4			−1.17052			−0.585260	−	0.810846i	$-0.699008\pi$
$113$	−14946.4			−1.17052			−0.585260	+	0.810846i	$0.699008\pi$
$114$	10959.6	−	4988.84i	0.843309	−	0.383875i
$115$		−	5463.77i		−	0.413139i
$116$	−5063.37	−	4409.26i	−0.376291	−	0.327680i
$117$			13155.4i			0.961021i
$118$	−4384.94	−	9632.98i	−0.314920	−	0.691826i
$119$			2188.64i			0.154554i
$120$	−4753.30	+	15970.3i	−0.330090	+	1.10905i
$121$	622.508			0.0425182
$122$	−9958.96	+	4533.33i	−0.669105	+	0.304577i
$123$	21346.0	+	1677.26i	1.41093	+	0.110864i
$124$	−8878.39	−	7731.44i	−0.577419	−	0.502825i
$125$	17010.3			1.08866
$126$	−2718.62	+	1237.52i	−0.171241	+	0.0779491i
$127$			17115.5i			1.06116i	0.847633	+	0.530582i	$0.178027\pi$
$127$			17115.5i			1.06116i	−0.847633	+	0.530582i	$0.821973\pi$
$128$	16196.6	+	2471.04i	0.988561	+	0.150820i
$129$			40500.9i			2.43380i
$130$	5484.75	+	12049.1i	0.324542	+	0.712963i
$131$		−	10956.7i		−	0.638465i	−0.947676	−	0.319232i	$-0.896575\pi$
$131$		−	10956.7i		−	0.638465i	0.947676	−	0.319232i	$-0.103425\pi$
$132$	−16535.2	+	18988.2i	−0.948992	+	1.08977i
$133$	2172.32			0.122806
$134$	13324.6	−	6065.37i	0.742070	−	0.337791i
$135$	63.8630			0.00350415
$136$	−14606.3	−	4347.33i	−0.789701	−	0.235042i
$137$		−	34818.3i		−	1.85509i	−0.373707	−	0.927547i	$-0.621913\pi$
$137$		−	34818.3i		−	1.85509i	0.373707	−	0.927547i	$-0.378087\pi$
$138$	5642.50	+	12395.6i	0.296288	+	0.650894i
$139$		−	9639.17i		−	0.498896i	−0.968388	−	0.249448i	$-0.919751\pi$
$139$		−	9639.17i		−	0.498896i	0.968388	−	0.249448i	$-0.0802492\pi$
$140$	−1974.05	+	2266.90i	−0.100717	+	0.115658i
$141$	−4099.68			−0.206211
$142$	10328.5	−	4701.53i	0.512223	−	0.233165i
$143$			20004.8i			0.978276i
$144$	−2858.79	−	20601.4i	−0.137866	−	0.993508i
$145$			8577.24i			0.407954i
$146$	2045.79	−	931.248i	0.0959746	−	0.0436877i
$147$	29506.8			1.36549
$148$	−4143.80	+	4758.52i	−0.189180	+	0.217244i
$149$		−	27184.1i		−	1.22446i	−0.790682	−	0.612228i	$-0.790274\pi$
$149$		−	27184.1i		−	1.22446i	0.790682	−	0.612228i	$-0.209726\pi$
$150$	−9608.75	+	4373.91i	−0.427056	+	0.194396i
$151$	−19700.8			−0.864031			−0.432015	−	0.901866i	$-0.642197\pi$
$151$	−19700.8			−0.864031			−0.432015	+	0.901866i	$0.642197\pi$
$152$	−4314.91	+	14497.4i	−0.186760	+	0.627484i
$153$			19346.0i			0.826432i
$154$	−4134.07	+	1881.83i	−0.174316	+	0.0793487i
$155$			15039.8i			0.626006i
$156$	−24886.5	−	21671.5i	−1.02262	−	0.890513i
$157$			20847.8i			0.845788i	0.906179	+	0.422894i	$0.138986\pi$
$157$			20847.8i			0.845788i	−0.906179	+	0.422894i	$0.861014\pi$
$158$	8773.26	−	3993.60i	0.351437	−	0.159974i
$159$		−	45447.9i		−	1.79771i
$160$	−11207.5	−	17677.0i	−0.437793	−	0.690507i
$161$			2456.94i			0.0947858i
$162$	23813.2	−	10839.8i	0.907377	−	0.413039i
$163$		−	25362.6i		−	0.954593i	−0.878742	−	0.477297i	$-0.841617\pi$
$163$		−	25362.6i		−	0.954593i	0.878742	−	0.477297i	$-0.158383\pi$
$164$	−19197.6	+	18837.4i	−0.713772	+	0.700378i
$165$	32165.6			1.18147
$166$	−13330.6	−	29285.1i	−0.483764	−	1.06275i
$167$	−52169.4			−1.87061			−0.935304	−	0.353846i	$-0.884874\pi$
$167$	−52169.4			−1.87061			−0.935304	+	0.353846i	$0.884874\pi$
$168$	2137.46	−	7181.52i	0.0757321	−	0.254447i
$169$	2342.22			0.0820077
$170$	8065.71	+	17719.0i	0.279090	+	0.613115i
$171$	19201.7			0.656670
$172$	−38366.3	−	33409.9i	−1.29686	−	1.12932i
$173$	−5159.14			−0.172379			−0.0861896	−	0.996279i	$-0.527469\pi$
$173$	−5159.14			−0.172379			−0.0861896	+	0.996279i	$0.527469\pi$
$174$	−8857.82	−	19459.1i	−0.292569	−	0.642725i
$175$	−1904.56			−0.0621896
$176$	−4347.21	−	31327.5i	−0.140341	−	1.01135i
$177$		−	33703.7i		−	1.07580i
$178$	−13815.8	−	30351.0i	−0.436050	−	0.957929i
$179$	−11400.9			−0.355824			−0.177912	−	0.984046i	$-0.556934\pi$
$179$	−11400.9			−0.355824			−0.177912	+	0.984046i	$0.556934\pi$
$180$	−17449.1	+	20037.7i	−0.538554	+	0.618448i
$181$		−	41571.3i		−	1.26893i	−0.772953	−	0.634463i	$-0.781221\pi$
$181$		−	41571.3i		−	1.26893i	0.772953	−	0.634463i	$-0.218779\pi$
$182$	−2466.38	−	5418.22i	−0.0744590	−	0.163574i
$183$	−34844.3			−1.04047
$184$	−16396.9	−	4880.27i	−0.484313	−	0.144148i
$185$	8060.83			0.235525
$186$	−15531.8	−	34120.7i	−0.448947	−	0.986262i
$187$			29418.4i			0.841271i
$188$	3381.90	−	3883.60i	0.0956853	−	0.109880i
$189$	−28.7179			−0.000803950
$190$	17586.9	−	8005.57i	0.487172	−	0.221761i
$191$	−30122.3			−0.825698			−0.412849	−	0.910799i	$-0.635466\pi$
$191$	−30122.3			−0.825698			−0.412849	+	0.910799i	$0.635466\pi$
$192$	43681.7	+	28529.6i	1.18494	+	0.773914i
$193$		−	37272.9i		−	1.00064i	−0.865840	−	0.500321i	$-0.833215\pi$
$193$		−	37272.9i		−	1.00064i	0.865840	−	0.500321i	$-0.166785\pi$
$194$	13190.5	+	28977.3i	0.350476	+	0.769936i
$195$			42157.1i			1.10867i
$196$	−24340.7	+	27951.6i	−0.633608	+	0.727603i
$197$	−39523.9			−1.01842			−0.509211	−	0.860642i	$-0.670063\pi$
$197$	−39523.9			−1.01842			−0.509211	+	0.860642i	$0.670063\pi$
$198$	−36542.2	+	16634.0i	−0.932103	+	0.424294i
$199$	−24377.7			−0.615582			−0.307791	−	0.951454i	$-0.599590\pi$
$199$	−24377.7			−0.615582			−0.307791	+	0.951454i	$0.599590\pi$
$200$	3783.05	−	12710.4i	0.0945764	−	0.317761i
$201$	46619.9			1.15393
$202$	13970.0	+	30689.6i	0.342367	+	0.752123i
$203$		−	3857.01i		−	0.0935962i
$204$	−36597.3	−	31869.5i	−0.879405	−	0.765799i
$205$	34253.9	+	2691.50i	0.815084	+	0.0640451i
$206$	19143.6	+	42055.2i	0.451116	+	0.991027i
$207$			21717.6i			0.506840i
$208$	41058.6	−	5697.58i	0.949025	−	0.131693i
$209$	29199.0			0.668461
$210$	−8711.95	+	3965.69i	−0.197550	+	0.0899249i
$211$	11049.3			0.248181			0.124090	−	0.992271i	$-0.460399\pi$
$211$	11049.3			0.248181			0.124090	+	0.992271i	$0.460399\pi$
$212$	43052.6	+	37490.8i	0.957916	+	0.834168i
$213$	36137.1			0.796515
$214$	3055.30	+	6711.99i	0.0667155	+	0.146563i
$215$			64991.6i			1.40598i
$216$	57.0429	−	191.655i	0.00122263	−	0.00410783i
$217$		−	6763.08i		−	0.143623i
$218$	5782.57	+	12703.3i	0.121677	+	0.267303i
$219$	7157.79			0.149242
$220$	−26534.0	+	30470.3i	−0.548224	+	0.629552i
$221$	−38556.6			−0.789430
$222$	−18287.6	+	8324.52i	−0.371065	+	0.168909i
$223$		−	56199.0i		−	1.13011i	−0.825055	−	0.565053i	$-0.808856\pi$
$223$		−	56199.0i		−	1.13011i	0.825055	−	0.565053i	$-0.191144\pi$
$224$	5039.78	+	7948.98i	0.100442	+	0.158422i
$225$	−16834.9			−0.332541
$226$	54413.3	−	24769.0i	1.06534	−	0.484943i
$227$	46646.1			0.905239			0.452620	−	0.891704i	$-0.350490\pi$
$227$	46646.1			0.905239			0.452620	+	0.891704i	$0.350490\pi$
$228$	−31631.9	+	36324.4i	−0.608492	+	0.698761i
$229$			92417.7i			1.76232i	0.472820	+	0.881159i	$0.343236\pi$
$229$			92417.7i			1.76232i	−0.472820	+	0.881159i	$0.656764\pi$
$230$	9054.50	+	19891.2i	0.171162	+	0.376015i
$231$	−14464.2			−0.271063
$232$	25740.5	+	7661.25i	0.478235	+	0.142339i
$233$			89487.7i			1.64836i	0.566330	+	0.824179i	$0.308363\pi$
$233$			89487.7i			1.64836i	−0.566330	+	0.824179i	$0.691637\pi$
$234$	−21801.0	−	47893.1i	−0.398148	−	0.874665i
$235$	−6578.74			−0.119126
$236$	31927.4	+	27802.8i	0.573243	+	0.499189i
$237$	30695.7			0.546489
$238$	−3626.99	−	7967.88i	−0.0640312	−	0.140666i
$239$	86747.6			1.51866			0.759332	−	0.650703i	$-0.225526\pi$
$239$	86747.6			1.51866			0.759332	+	0.650703i	$0.225526\pi$
$240$	−9161.12	−	66018.1i	−0.159047	−	1.14615i
$241$	−55558.3			−0.956566			−0.478283	−	0.878206i	$-0.658741\pi$
$241$	−55558.3			−0.956566			−0.478283	+	0.878206i	$0.658741\pi$
$242$	−2266.28	+	1031.61i	−0.0386976	+	0.0176152i
$243$	83570.3			1.41527
$244$	28743.7	−	33007.8i	0.482795	−	0.554417i
$245$	47349.4			0.788828
$246$	−80491.2	+	29268.2i	−1.33008	+	0.483645i
$247$			38269.0i			0.627269i
$248$	45134.8	+	13433.6i	0.733852	+	0.218419i
$249$		−	102462.i		−	1.65259i
$250$	−61927.1	+	28189.3i	−0.990834	+	0.451029i
$251$		−	40524.1i		−	0.643229i	−0.946871	−	0.321615i	$-0.895774\pi$
$251$		−	40524.1i		−	0.643229i	0.946871	−	0.321615i	$-0.104226\pi$
$252$	7846.52	−	9010.54i	0.123559	−	0.141889i
$253$			33024.8i			0.515940i
$254$	−28363.7	−	62310.2i	−0.439638	−	0.965810i
$255$			61995.0i			0.953403i
$256$	−63059.7	+	17844.8i	−0.962215	+	0.272290i
$257$			113202.i			1.71392i	0.515386	+	0.856958i	$0.327649\pi$
$257$			113202.i			1.71392i	−0.515386	+	0.856958i	$0.672351\pi$
$258$	−67117.6	−	147446.i	−1.00832	−	2.21510i
$259$	−3624.79			−0.0540360
$260$	−39935.2	−	34776.2i	−0.590758	−	0.514441i
$261$		−	34093.1i		−	0.500479i
$262$	18157.3	+	39888.6i	0.264514	+	0.581093i
$263$	11567.7			0.167238			0.0836192	−	0.996498i	$-0.473352\pi$
$263$	11567.7			0.167238			0.0836192	+	0.996498i	$0.473352\pi$
$264$	28730.5	−	96529.9i	0.412227	−	1.38501i
$265$		−	72930.1i		−	1.03852i
$266$	−7908.46	+	3599.94i	−0.111771	+	0.0508782i
$267$		−	106192.i		−	1.48959i
$268$	−38457.6	+	44162.8i	−0.535443	+	0.614875i
$269$	101322.			1.40023			0.700117	−	0.714029i	$-0.253131\pi$
$269$	101322.			1.40023			0.700117	+	0.714029i	$0.253131\pi$
$270$	−232.498	+	105.833i	−0.00318927	+	0.00145176i
$271$		−	36359.5i		−	0.495085i	−0.968877	−	0.247543i	$-0.920377\pi$
$271$		−	36359.5i		−	0.495085i	0.968877	−	0.247543i	$-0.0796230\pi$
$272$	60379.6	−	8378.68i	0.816117	−	0.113250i
$273$		−	18957.2i		−	0.254360i
$274$	57700.4	+	126758.i	0.768560	+	1.68840i
$275$	−25600.0			−0.338512
$276$	−41083.8	−	35776.4i	−0.539327	−	0.469654i
$277$	−55294.7			−0.720649			−0.360325	−	0.932827i	$-0.617334\pi$
$277$	−55294.7			−0.720649			−0.360325	+	0.932827i	$0.617334\pi$
$278$	15973.9	+	35092.1i	0.206691	+	0.454066i
$279$		−	59780.7i		−	0.767985i
$280$	3429.98	−	11524.2i	0.0437497	−	0.146992i
$281$		−	18528.5i		−	0.234654i	−0.993093	−	0.117327i	$-0.962567\pi$
$281$		−	18528.5i		−	0.234654i	0.993093	−	0.117327i	$-0.0374326\pi$
$282$	14925.1	−	6793.94i	0.187681	−	0.0854326i
$283$			84117.3i			1.05030i	0.851010	+	0.525149i	$0.175990\pi$
$283$			84117.3i			1.05030i	−0.851010	+	0.525149i	$0.824010\pi$
$284$	−29810.2	+	34232.5i	−0.369596	+	0.424425i
$285$	61532.7			0.757559
$286$	−33151.7	−	72828.7i	−0.405297	−	0.890370i
$287$	−15403.3	−	1210.31i	−0.187003	−	0.0146938i
$288$	44548.0	+	70263.2i	0.537085	+	0.847116i
$289$	26820.9			0.321127
$290$	−14214.1	−	31226.0i	−0.169014	−	0.371296i
$291$			101385.i			1.19726i
$292$	−5904.60	+	6780.54i	−0.0692508	+	0.0795240i
$293$		−	99573.4i		−	1.15987i	−0.814664	−	0.579933i	$-0.803079\pi$
$293$		−	99573.4i		−	1.15987i	0.814664	−	0.579933i	$-0.196921\pi$
$294$	−107421.	+	48898.3i	−1.24279	+	0.565717i
$295$		−	54084.2i		−	0.621479i
$296$	7199.99	−	24190.8i	0.0821766	−	0.276100i
$297$	−386.010			−0.00437608
$298$	45049.2	+	98965.6i	0.507288	+	1.11443i
$299$	−43283.2			−0.484147
$300$	27732.9	−	31847.0i	0.308143	−	0.353856i
$301$		−	29225.4i		−	0.322572i
$302$	71721.9	−	32647.9i	0.786390	−	0.357966i
$303$			107376.i			1.16956i
$304$	−8316.21	−	59929.4i	−0.0899867	−	0.648474i
$305$	−55914.4			−0.601069
$306$	−32059.9	−	70430.2i	−0.342389	−	0.752170i
$307$		−	49363.4i		−	0.523755i	−0.965101	−	0.261877i	$-0.915658\pi$
$307$		−	49363.4i		−	0.523755i	0.965101	−	0.261877i	$-0.0843416\pi$
$308$	11931.8	−	13701.9i	0.125778	−	0.144437i
$309$			147142.i			1.54106i
$310$	−24923.8	−	54753.3i	−0.259353	−	0.569754i
$311$	39370.2			0.407049			0.203525	−	0.979070i	$-0.434760\pi$
$311$	39370.2			0.407049			0.203525	+	0.979070i	$0.434760\pi$
$312$	126515.	+	37655.0i	1.29967	+	0.386824i
$313$			34387.2i			0.351000i	0.984479	+	0.175500i	$0.0561543\pi$
$313$			34387.2i			0.351000i	−0.984479	+	0.175500i	$0.943846\pi$
$314$	−34548.8	−	75897.9i	−0.350408	−	0.769787i
$315$	−15263.7			−0.153829
$316$	−25321.5	+	29077.9i	−0.253580	+	0.291198i
$317$			12073.7i			0.120150i	0.998194	+	0.0600750i	$0.0191340\pi$
$317$			12073.7i			0.120150i	−0.998194	+	0.0600750i	$0.980866\pi$
$318$	75315.8	+	165456.i	0.744787	+	1.63617i
$319$		−	51843.7i		−	0.509465i
$320$	70095.8	+	45781.3i	0.684529	+	0.447083i
$321$			23483.8i			0.227907i
$322$	−4071.62	−	8944.66i	−0.0392695	−	0.0862685i
$323$			56277.3i			0.539422i
$324$	−68730.0	+	78926.0i	−0.654721	+	0.751848i
$325$		−	33552.0i		−	0.317652i
$326$	42030.6	+	92334.2i	0.395485	+	0.868815i
$327$			44446.2i			0.415661i
$328$	38673.0	−	100393.i	0.359468	−	0.933157i
$329$	2958.32			0.0273309
$330$	−117101.	+	53304.5i	−1.07531	+	0.489481i
$331$	−142970.			−1.30494			−0.652468	−	0.757816i	$-0.726266\pi$
$331$	−142970.			−1.30494			−0.652468	+	0.757816i	$0.726266\pi$
$332$	97061.8	+	84522.9i	0.880587	+	0.766828i
$333$	−32040.5			−0.288942
$334$	189926.	−	86454.5i	1.70252	−	0.774988i
$335$	74810.8			0.666614
$336$	4119.57	+	29687.0i	0.0364899	+	0.262959i
$337$	−163863.			−1.44285			−0.721425	−	0.692493i	$-0.756513\pi$
$337$	−163863.			−1.44285			−0.721425	+	0.692493i	$0.756513\pi$
$338$	−8527.01	+	3881.50i	−0.0746386	+	0.0339756i
$339$	190380.			1.65662
$340$	−58727.5	−	51140.8i	−0.508024	−	0.442395i
$341$		−	90905.5i		−	0.781774i
$342$	−69905.1	+	31820.9i	−0.597663	+	0.272057i
$343$	−43360.6			−0.368559
$344$	195042.	+	58050.9i	1.64820	+	0.490560i
$345$			69595.0i			0.584709i
$346$	18782.2	−	8549.67i	0.156889	−	0.0714162i
$347$	−229696.			−1.90764			−0.953818	−	0.300386i	$-0.902884\pi$
$347$	−229696.			−1.90764			−0.953818	+	0.300386i	$0.902884\pi$
$348$	64495.0	+	56163.2i	0.532559	+	0.463760i
$349$	−161790.			−1.32831			−0.664157	−	0.747593i	$-0.731209\pi$
$349$	−161790.			−1.32831			−0.664157	+	0.747593i	$0.731209\pi$
$350$	6933.66	−	3156.21i	0.0566013	−	0.0257650i
$351$		−	505.915i		−	0.00410642i
$352$	67741.9	+	106846.i	0.546729	+	0.862326i
$353$	−198778.			−1.59521			−0.797606	−	0.603179i	$-0.793900\pi$
$353$	−198778.			−1.59521			−0.797606	+	0.603179i	$0.793900\pi$
$354$	55853.5	+	122701.i	0.445701	+	0.979130i
$355$	57989.0			0.460139
$356$	100595.	+	87599.5i	0.793735	+	0.691197i
$357$		−	27877.9i		−	0.218738i
$358$	41505.9	−	18893.5i	0.323850	−	0.147417i
$359$		−	70546.3i		−	0.547376i	−0.961819	−	0.273688i	$-0.911756\pi$
$359$		−	70546.3i		−	0.547376i	0.961819	−	0.273688i	$-0.0882435\pi$
$360$	30318.5	−	101865.i	0.233939	−	0.785996i
$361$	−74463.3			−0.571384
$362$	68891.5	+	151343.i	0.525713	+	1.15490i
$363$	−7929.24			−0.0601753
$364$	17958.0	+	15638.1i	0.135536	+	0.118027i
$365$	11486.1			0.0862156
$366$	126853.	−	57743.5i	0.946974	−	0.431064i
$367$		−	63570.6i		−	0.471981i	−0.971755	−	0.235990i	$-0.924167\pi$
$367$		−	63570.6i		−	0.471981i	0.971755	−	0.235990i	$-0.0758334\pi$
$368$	67781.6	−	9405.83i	0.500514	−	0.0694547i
$369$	−136154.	−	10698.3i	−0.999946	−	0.0785707i
$370$	−29346.0	+	13358.3i	−0.214361	+	0.0975772i
$371$			32795.2i			0.238266i
$372$	113089.	+	98479.6i	0.817211	+	0.711640i
$373$	51703.3			0.371622			0.185811	−	0.982586i	$-0.440509\pi$
$373$	51703.3			0.371622			0.185811	+	0.982586i	$0.440509\pi$
$374$	−48751.9	−	107100.i	−0.348536	−	0.765676i
$375$	−216670.			−1.54076
$376$	−5876.17	+	19743.0i	−0.0415641	+	0.139649i
$377$	67947.7			0.478071
$378$	104.549	−	47.5910i	0.000731708	−	0.000333074i
$379$		−	267423.i		−	1.86174i	−0.365346	−	0.930872i	$-0.619049\pi$
$379$		−	267423.i		−	1.86174i	0.365346	−	0.930872i	$-0.380951\pi$
$380$	−50759.5	+	58289.6i	−0.351520	+	0.403668i
$381$		−	218010.i		−	1.50185i
$382$	109662.	−	49918.4i	0.751502	−	0.342085i
$383$	4628.99			0.0315565			0.0157782	−	0.999876i	$-0.494977\pi$
$383$	4628.99			0.0315565			0.0157782	+	0.999876i	$0.494977\pi$
$384$	−206305.	−	31475.0i	−1.39909	−	0.213454i
$385$	−23210.7			−0.156591
$386$	61768.3	+	135695.i	0.414564	+	0.910726i
$387$		−	258331.i		−	1.72486i
$388$	−96041.8	−	83634.7i	−0.637965	−	0.555550i
$389$	−90834.3			−0.600276			−0.300138	−	0.953896i	$-0.597033\pi$
$389$	−90834.3			−0.600276			−0.300138	+	0.953896i	$0.597033\pi$
$390$	−69862.3	−	153476.i	−0.459318	−	1.00905i
$391$	−63651.0			−0.416344
$392$	42292.8	−	142097.i	0.275229	−	0.924724i
$393$			139562.i			0.903609i
$394$	143889.	−	65498.6i	0.926908	−	0.421929i
$395$	49257.3			0.315702
$396$	105468.	−	121115.i	0.672562	−	0.772336i
$397$			73959.6i			0.469260i	0.972085	+	0.234630i	$0.0753879\pi$
$397$			73959.6i			0.469260i	−0.972085	+	0.234630i	$0.924612\pi$
$398$	88748.5	−	40398.4i	0.560267	−	0.255034i
$399$	−27670.0			−0.173805
$400$	7291.15	+	52542.4i	0.0455697	+	0.328390i
$401$	−114072.			−0.709397			−0.354699	−	0.934981i	$-0.615417\pi$
$401$	−114072.			−0.709397			−0.354699	+	0.934981i	$0.615417\pi$
$402$	−169723.	+	77258.0i	−1.05024	+	0.478070i
$403$	119143.			0.733600
$404$	−101717.	−	88576.8i	−0.623205	−	0.542697i
$405$	133699.			0.815112
$406$	6391.79	+	14041.7i	0.0387767	+	0.0851858i
$407$	−48722.4			−0.294130
$408$	186049.	+	55374.3i	1.11765	+	0.332651i
$409$	−46424.2			−0.277522			−0.138761	−	0.990326i	$-0.544312\pi$
$409$	−46424.2			−0.277522			−0.138761	+	0.990326i	$0.544312\pi$
$410$	−129164.	+	46966.6i	−0.768375	+	0.279397i
$411$			443499.i			2.62548i
$412$	−139387.	−	121380.i	−0.821159	−	0.715078i
$413$			24320.6i			0.142585i
$414$	−35990.2	−	79064.3i	−0.209982	−	0.461296i
$415$		−	164420.i		−	0.954684i
$416$	−140035.	+	88784.3i	−0.809187	+	0.513038i
$417$			122780.i			0.706080i
$418$	−106301.	+	48388.3i	−0.608394	+	0.276942i
$419$		−	161644.i		−	0.920727i	−0.887731	−	0.460363i	$-0.847719\pi$
$419$		−	161644.i		−	0.920727i	0.887731	−	0.460363i	$-0.152281\pi$
$420$	25144.5	−	28874.7i	0.142543	−	0.163689i
$421$			294394.i			1.66098i	0.557034	+	0.830490i	$0.311939\pi$
$421$			294394.i			1.66098i	−0.557034	+	0.830490i	$0.688061\pi$
$422$	−40225.6	+	18310.7i	−0.225880	+	0.102821i
$423$	26149.4			0.146144
$424$	−218865.	−	65141.6i	−1.21743	−	0.362349i
$425$		−	49340.5i		−	0.273166i
$426$	−131559.	+	59886.0i	−0.724941	+	0.329994i
$427$	25143.6			0.137902
$428$	−22246.1	−	19372.2i	−0.121441	−	0.105753i
$429$		−	254812.i		−	1.38454i
$430$	−107703.	−	236606.i	−0.582495	−	1.27964i
$431$		−	240746.i		−	1.29600i	−0.761640	−	0.648001i	$-0.775605\pi$
$431$		−	240746.i		−	1.29600i	0.761640	−	0.648001i	$-0.224395\pi$
$432$	109.940	+	792.263i	0.000589098	+	0.00424523i
$433$	302542.			1.61365			0.806826	−	0.590790i	$-0.201184\pi$
$433$	302542.			1.61365			0.806826	+	0.590790i	$0.201184\pi$
$434$	11207.7	+	24621.5i	0.0595028	+	0.130718i
$435$		−	109253.i		−	0.577371i
$436$	−42103.6	−	36664.5i	−0.221486	−	0.192874i
$437$			63176.4i			0.330820i
$438$	−26058.4	+	11861.8i	−0.135831	+	0.0618305i
$439$	264484.			1.37237			0.686183	−	0.727429i	$-0.259285\pi$
$439$	264484.			1.37237			0.686183	+	0.727429i	$0.259285\pi$
$440$	46103.8	−	154901.i	0.238139	−	0.800109i
$441$	−188206.			−0.967736
$442$	140368.	−	63895.5i	0.718493	−	0.327059i
$443$			117956.i			0.601054i	0.953773	+	0.300527i	$0.0971625\pi$
$443$			117956.i			0.601054i	−0.953773	+	0.300527i	$0.902837\pi$
$444$	52781.8	−	60612.0i	0.267743	−	0.307463i
$445$		−	170405.i		−	0.860524i
$446$	93132.5	+	204596.i	0.468200	+	1.02856i
$447$			346259.i			1.73295i
$448$	−31520.6	−	20586.9i	−0.157050	−	0.102573i
$449$	−329452.			−1.63418			−0.817090	−	0.576510i	$-0.804414\pi$
$449$	−329452.			−1.63418			−0.817090	+	0.576510i	$0.804414\pi$
$450$	61288.5	−	27898.6i	0.302659	−	0.137771i
$451$	−207042.	−	16268.3i	−1.01790	−	0.0799814i
$452$	−157048.	+	180346.i	−0.768699	+	0.882734i
$453$	250940.			1.22285
$454$	−169818.	+	77301.4i	−0.823896	+	0.375038i
$455$		−	30420.5i		−	0.146941i
$456$	54961.4	−	184661.i	0.264319	−	0.888068i
$457$		−	139636.i		−	0.668598i	−0.942467	−	0.334299i	$-0.891501\pi$
$457$		−	139636.i		−	0.668598i	0.942467	−	0.334299i	$-0.108499\pi$
$458$	−153154.	−	336453.i	−0.730124	−	1.60396i
$459$		−	743.983i		−	0.00353132i
$460$	−65927.0	−	57410.2i	−0.311564	−	0.271315i
$461$	−97954.6			−0.460917			−0.230459	−	0.973082i	$-0.574023\pi$
$461$	−97954.6			−0.460917			−0.230459	+	0.973082i	$0.574023\pi$
$462$	52657.9	−	23969.9i	0.246706	−	0.112301i
$463$	81767.7			0.381434			0.190717	−	0.981645i	$-0.438919\pi$
$463$	81767.7			0.381434			0.190717	+	0.981645i	$0.438919\pi$
$464$	−106406.	+	14765.7i	−0.494232	+	0.0685830i
$465$		−	191570.i		−	0.885976i
$466$	−148298.	−	325786.i	−0.682910	−	1.50024i
$467$		−	30540.8i		−	0.140038i	−0.997546	−	0.0700192i	$-0.977694\pi$
$467$		−	30540.8i		−	0.140038i	0.997546	−	0.0700192i	$-0.0223060\pi$
$468$	158736.	+	138230.i	0.724743	+	0.631117i
$469$	−33640.9			−0.152940
$470$	23950.3	−	10902.2i	0.108422	−	0.0493536i
$471$		−	265550.i		−	1.19703i
$472$	−162308.	−	48308.4i	−0.728545	−	0.216840i
$473$		−	392831.i		−	1.75583i
$474$	−111750.	+	50868.7i	−0.497382	+	0.226409i
$475$	−48972.6			−0.217053
$476$	26408.6	+	22997.0i	0.116555	+	0.101498i
$477$			289885.i			1.27406i
$478$	−315811.	+	143757.i	−1.38220	+	0.629179i
$479$	134802.			0.587525			0.293763	−	0.955878i	$-0.405092\pi$
$479$	134802.			0.587525			0.293763	+	0.955878i	$0.405092\pi$
$480$	142756.	+	225162.i	0.619601	+	0.977264i
$481$		−	63856.8i		−	0.276005i
$482$	202264.	−	92070.6i	0.870610	−	0.396303i
$483$		−	31295.4i		−	0.134149i
$484$	6540.97	−	7511.32i	0.0279223	−	0.0320646i
$485$			162693.i			0.691647i
$486$	−304243.	+	138492.i	−1.28810	+	0.586343i
$487$			384607.i			1.62166i	0.585284	+	0.810828i	$0.300983\pi$
$487$			384607.i			1.62166i	−0.585284	+	0.810828i	$0.699017\pi$
$488$	−49943.1	+	167801.i	−0.209718	+	0.704619i
$489$			323057.i			1.35102i
$490$	−172379.	+	78466.9i	−0.717945	+	0.326809i
$491$		−	300163.i		−	1.24507i	−0.782591	−	0.622536i	$-0.786102\pi$
$491$		−	300163.i		−	1.24507i	0.782591	−	0.622536i	$-0.213898\pi$
$492$	244531.	−	239942.i	1.01019	−	0.991234i
$493$	99921.9			0.411118
$494$	−63419.1	−	139321.i	−0.259876	−	0.570904i
$495$	−205165.			−0.837324
$496$	−186578.	+	25890.9i	−0.758399	+	0.105241i
$497$	−26076.5			−0.105569
$498$	169799.	+	373020.i	0.684663	+	1.50409i
$499$	348668.			1.40027			0.700134	−	0.714012i	$-0.253124\pi$
$499$	348668.			1.40027			0.700134	+	0.714012i	$0.253124\pi$
$500$	178735.	−	205250.i	0.714939	−	0.821000i
$501$	664510.			2.64744
$502$	67156.1	+	147531.i	0.266488	+	0.585430i
$503$	485393.			1.91848			0.959240	−	0.282592i	$-0.0911942\pi$
$503$	485393.			1.91848			0.959240	+	0.282592i	$0.0911942\pi$
$504$	−13633.6	+	45806.7i	−0.0536722	+	0.180330i
$505$			172306.i			0.675645i
$506$	−54728.4	−	120229.i	−0.213753	−	0.469579i
$507$	−29834.2			−0.116064
$508$	206520.	+	179840.i	0.800265	+	0.696883i
$509$			412610.i			1.59259i	0.604909	+	0.796295i	$0.293210\pi$
$509$			412610.i			1.59259i	−0.604909	+	0.796295i	$0.706790\pi$
$510$	−102737.	−	225697.i	−0.394992	−	0.867731i
$511$	−5165.05			−0.0197803
$512$	200001.	−	169467.i	0.762943	−	0.646466i
$513$	−738.435			−0.00280594
$514$	−187598.	−	412121.i	−0.710071	−	1.55991i
$515$			236118.i			0.890256i
$516$	488693.	+	425561.i	1.83542	+	1.59831i
$517$	39764.1			0.148768
$518$	13196.3	−	6006.96i	0.0491804	−	0.0223870i
$519$	65714.8			0.243965
$520$	203018.	+	60424.8i	0.750805	+	0.223465i
$521$			275507.i			1.01498i	0.861658	+	0.507489i	$0.169426\pi$
$521$			275507.i			1.01498i	−0.861658	+	0.507489i	$0.830574\pi$
$522$	56498.8	+	124118.i	0.207347	+	0.455507i
$523$			429200.i			1.56912i	0.620053	+	0.784560i	$0.287111\pi$
$523$			429200.i			1.56912i	−0.620053	+	0.784560i	$0.712889\pi$
$524$	−132206.	−	115127.i	−0.481491	−	0.419290i
$525$	24259.4			0.0880159
$526$	−42113.0	+	19169.9i	−0.152211	+	0.0692864i
$527$	175208.			0.630861
$528$	55372.9	+	399035.i	0.198623	+	1.43134i
$529$	208387.			0.744662
$530$	120859.	+	265507.i	0.430256	+	0.945200i
$531$			214976.i			0.762432i
$532$	22825.5	−	26211.6i	0.0806486	−	0.0926128i
$533$	21321.7	−	271355.i	0.0750528	−	0.955175i
$534$	175980.	+	386598.i	0.617135	+	1.35574i
$535$			37684.3i			0.131660i
$536$	66821.5	−	224509.i	0.232588	−	0.781456i
$537$	145220.			0.503592
$538$	−368870.	+	167910.i	−1.27441	+	0.580113i
$539$	−286196.			−0.985111
$540$	671.038	−	770.585i	0.00230123	−	0.00264261i
$541$	251897.			0.860654			0.430327	−	0.902673i	$-0.358398\pi$
$541$	251897.			0.860654			0.430327	+	0.902673i	$0.358398\pi$
$542$	60254.7	+	132369.i	0.205112	+	0.450598i
$543$			529517.i			1.79589i
$544$	−205931.	+	130564.i	−0.695863	+	0.441189i
$545$			71322.6i			0.240123i
$546$	31415.7	+	69014.9i	0.105381	+	0.231504i
$547$	−281478.			−0.940740			−0.470370	−	0.882469i	$-0.655879\pi$
$547$	−281478.			−0.940740			−0.470370	+	0.882469i	$0.655879\pi$
$548$	−420125.	−	365851.i	−1.39900	−	1.21827i
$549$	222251.			0.737393
$550$	93198.3	−	42424.0i	0.308094	−	0.140245i
$551$		−	99176.8i		−	0.326668i
$552$	208857.	+	62162.7i	0.685441	+	0.204010i
$553$	−22150.0			−0.0724308
$554$	201304.	−	91633.8i	0.655893	−	0.298563i
$555$	−102675.			−0.333334
$556$	−116308.	−	101283.i	−0.376237	−	0.327633i
$557$		−	372163.i		−	1.19956i	−0.800164	−	0.599781i	$-0.795254\pi$
$557$		−	372163.i		−	1.19956i	0.800164	−	0.599781i	$-0.204746\pi$
$558$	99068.0	+	217636.i	0.318174	+	0.698975i
$559$	514855.			1.64764
$560$	6610.65	+	47638.6i	0.0210799	+	0.151909i
$561$		−	374719.i		−	1.19064i
$562$	30705.3	+	67454.4i	0.0972167	+	0.213569i
$563$	−441661.			−1.39339			−0.696694	−	0.717368i	$-0.745346\pi$
$563$	−441661.			−1.39339			−0.696694	+	0.717368i	$0.745346\pi$
$564$	−43077.1	+	49467.6i	−0.135422	+	0.155512i
$565$	305502.			0.957013
$566$	−139398.	−	306235.i	−0.435136	−	0.955920i
$567$	−60121.6			−0.187010
$568$	51796.2	−	174027.i	0.160547	−	0.539410i
$569$	−155484.			−0.480244			−0.240122	−	0.970743i	$-0.577187\pi$
$569$	−155484.			−0.480244			−0.240122	+	0.970743i	$0.577187\pi$
$570$	−224014.	+	101971.i	−0.689486	+	0.313855i
$571$	−397304.			−1.21857			−0.609286	−	0.792951i	$-0.708544\pi$
$571$	−397304.			−1.21857			−0.609286	+	0.792951i	$0.708544\pi$
$572$	241382.	+	210199.i	0.737756	+	0.642449i
$573$	383685.			1.16860
$574$	58082.3	−	21119.9i	0.176287	−	0.0641015i
$575$		−	55389.2i		−	0.167529i
$576$	−278619.	−	181973.i	−0.839782	−	0.548482i
$577$		−	501200.i		−	1.50543i	−0.658348	−	0.752714i	$-0.728745\pi$
$577$		−	501200.i		−	1.50543i	0.658348	−	0.752714i	$-0.271255\pi$
$578$	−97643.2	+	44447.3i	−0.292271	+	0.133042i
$579$			474766.i			1.41619i
$580$	103495.	+	90124.9i	0.307654	+	0.267910i
$581$			73936.5i			0.219032i
$582$	−168015.	−	369100.i	−0.496023	−	1.08968i
$583$			440814.i			1.29693i
$584$	10259.4	−	34470.0i	0.0300814	−	0.101069i
$585$		−	268895.i		−	0.785727i
$586$	165012.	+	362503.i	0.480530	+	1.05564i
$587$	−385632.			−1.11917			−0.559586	−	0.828772i	$-0.689040\pi$
$587$	−385632.			−1.11917			−0.559586	+	0.828772i	$0.689040\pi$
$588$	310041.	−	356035.i	0.896735	−	1.02977i
$589$		−	173902.i		−	0.501272i
$590$	89627.8	+	196897.i	0.257477	+	0.565634i
$591$	503438.			1.44136
$592$	13876.7	+	99999.9i	0.0395951	+	0.285336i
$593$			236772.i			0.673318i	0.941627	+	0.336659i	$0.109297\pi$
$593$			236772.i			0.673318i	−0.941627	+	0.336659i	$0.890703\pi$
$594$	1405.29	−	639.691i	0.00398285	−	0.00181300i
$595$		−	44735.5i		−	0.126363i
$596$	−328010.	−	285636.i	−0.923409	−	0.804118i
$597$	310512.			0.871223
$598$	157575.	−	71728.5i	0.440642	−	0.200581i
$599$		−	285158.i		−	0.794752i	−0.917656	−	0.397376i	$-0.869921\pi$
$599$		−	285158.i		−	0.794752i	0.917656	−	0.397376i	$-0.130079\pi$
$600$	−48186.9	+	161900.i	−0.133852	+	0.449722i
$601$			145665.i			0.403280i	0.979460	+	0.201640i	$0.0646271\pi$
$601$			145665.i			0.403280i	−0.979460	+	0.201640i	$0.935373\pi$
$602$	48432.0	+	106397.i	0.133641	+	0.293587i
$603$	−297361.			−0.817803
$604$	−207005.	+	237714.i	−0.567422	+	0.651599i
$605$	−12724.0			−0.0347627
$606$	−177943.	−	390911.i	−0.484547	−	1.06447i
$607$		−	219232.i		−	0.595012i	−0.954720	−	0.297506i	$-0.903845\pi$
$607$		−	219232.i		−	0.595012i	0.954720	−	0.297506i	$-0.0961548\pi$
$608$	129590.	+	204395.i	0.350562	+	0.552922i
$609$			49128.8i			0.132465i
$610$	203560.	−	92660.8i	0.547058	−	0.249021i
$611$			52115.9i			0.139601i
$612$	233432.	+	203277.i	0.623244	+	0.542731i
$613$	281375.			0.748797			0.374398	−	0.927268i	$-0.377849\pi$
$613$	281375.			0.748797			0.374398	+	0.927268i	$0.377849\pi$
$614$	81804.5	+	179711.i	0.216990	+	0.476691i
$615$	−436311.	−	34283.1i	−1.15357	−	0.0906420i
$616$	−20731.9	+	69655.8i	−0.0546359	+	0.183568i
$617$	−623247.			−1.63715			−0.818577	−	0.574397i	$-0.805237\pi$
$617$	−623247.			−1.63715			−0.818577	+	0.574397i	$0.805237\pi$
$618$	−243842.	−	535681.i	−0.638458	−	1.40258i
$619$			22048.4i			0.0575433i	0.999586	+	0.0287717i	$0.00915957\pi$
$619$			22048.4i			0.0575433i	−0.999586	+	0.0287717i	$0.990840\pi$
$620$	181473.	+	158030.i	0.472095	+	0.411108i
$621$		−	835.188i		−	0.00216571i
$622$	−143330.	+	65243.9i	−0.370473	+	0.168639i
$623$			76627.7i			0.197429i
$624$	−522987.	+	72573.2i	−1.34314	+	0.186383i
$625$	−218182.			−0.558546
$626$	−56986.0	−	125189.i	−0.145419	−	0.319460i
$627$	−371924.			−0.946062
$628$	251554.	+	219057.i	0.637841	+	0.555442i
$629$		−	93905.9i		−	0.237351i
$630$	55568.4	−	25294.8i	0.140006	−	0.0637308i
$631$			351702.i			0.883316i	0.897183	+	0.441658i	$0.145610\pi$
$631$			351702.i			0.883316i	−0.897183	+	0.441658i	$0.854390\pi$
$632$	43997.0	−	147823.i	0.110151	−	0.370089i
$633$	−140741.			−0.351246
$634$	−20008.5	−	43955.3i	−0.0497778	−	0.109353i
$635$		−	349840.i		−	0.867604i
$636$	−548384.	−	477542.i	−1.35572	−	1.18058i
$637$		−	375096.i		−	0.924407i
$638$	85914.8	+	188740.i	0.211070	+	0.463686i
$639$	−230497.			−0.564499
$640$	−331057.	−	50507.8i	−0.808244	−	0.123310i
$641$			605323.i			1.47323i	0.676311	+	0.736616i	$0.263577\pi$
$641$			605323.i			1.47323i	−0.676311	+	0.736616i	$0.736423\pi$
$642$	−38917.1	−	85494.3i	−0.0944214	−	0.207428i
$643$	−11372.6			−0.0275067			−0.0137534	−	0.999905i	$-0.504378\pi$
$643$	−11372.6			−0.0275067			−0.0137534	+	0.999905i	$0.504378\pi$
$644$	29646.0	+	25816.2i	0.0714816	+	0.0622473i
$645$		−	827834.i		−	1.98987i
$646$	−93262.2	−	204881.i	−0.223481	−	0.490950i
$647$		−	775276.i		−	1.85203i	−0.377487	−	0.926015i	$-0.623212\pi$
$647$		−	775276.i		−	1.85203i	0.377487	−	0.926015i	$-0.376788\pi$
$648$	119421.	−	401234.i	0.284400	−	0.955537i
$649$			326903.i			0.776121i
$650$	55602.0	+	122148.i	0.131602	+	0.289108i
$651$			86145.1i			0.203268i
$652$	−306030.	−	266496.i	−0.719895	−	0.626896i
$653$		−	199228.i		−	0.467222i	−0.972330	−	0.233611i	$-0.924946\pi$
$653$		−	199228.i		−	0.467222i	0.972330	−	0.233611i	$-0.0750542\pi$
$654$	−73655.7	−	161809.i	−0.172207	−	0.378310i
$655$			223954.i			0.522006i
$656$	25578.1	+	429575.i	0.0594374	+	0.998232i
$657$	−45655.3			−0.105770
$658$	−10770.0	+	4902.50i	−0.0248750	+	0.0113231i
$659$	−58349.5			−0.134359			−0.0671794	−	0.997741i	$-0.521400\pi$
$659$	−58349.5			−0.134359			−0.0671794	+	0.997741i	$0.521400\pi$
$660$	337979.	−	388117.i	0.775892	−	0.890995i
$661$	788610.			1.80493			0.902463	−	0.430767i	$-0.141757\pi$
$661$	788610.			1.80493			0.902463	+	0.430767i	$0.141757\pi$
$662$	520492.	−	236929.i	1.18768	−	0.540632i
$663$	491116.			1.11727
$664$	−493430.	−	146861.i	−1.11915	−	0.333098i
$665$	−44401.9			−0.100406
$666$	116646.	−	53097.2i	0.262978	−	0.119708i
$667$	112171.			0.252133
$668$	−548167.	+	629487.i	−1.22846	+	1.41070i
$669$			715838.i			1.59942i
$670$	−272354.	+	123976.i	−0.606713	+	0.276176i
$671$	337966.			0.750633
$672$	−64194.5	−	101251.i	−0.142154	−	0.224212i
$673$			336101.i			0.742061i	0.928621	+	0.371031i	$0.120996\pi$
$673$			336101.i			0.742061i	−0.928621	+	0.371031i	$0.879004\pi$
$674$	596554.	−	271552.i	1.31320	−	0.597769i
$675$	647.415			0.00142094
$676$	24610.8	−	28261.7i	0.0538557	−	0.0618451i
$677$	−457.310			−0.000997777			−0.000498889	−	1.00000i	$-0.500159\pi$
$677$	−457.310			−0.000997777			−0.000498889		1.00000i	$0.500159\pi$
$678$	−693092.	+	315496.i	−1.50776	+	0.686333i
$679$		−	73159.5i		−	0.158683i
$680$	298552.	+	88859.0i	0.645656	+	0.192169i
$681$	−594157.			−1.28117
$682$	150648.	+	330948.i	0.323887	+	0.711525i
$683$	405013.			0.868215			0.434107	−	0.900861i	$-0.357064\pi$
$683$	405013.			0.868215			0.434107	+	0.900861i	$0.357064\pi$
$684$	201761.	−	231692.i	0.431246	−	0.495220i
$685$			711681.i			1.51672i
$686$	157857.	−	71856.7i	0.335441	−	0.152693i
$687$		−	1.17718e6i		−	2.49418i
$688$	−806263.	+	111883.i	−1.70333	+	0.236366i
$689$	−577742.			−1.21701
$690$	−115332.	−	253365.i	−0.242243	−	0.532168i
$691$	640269.			1.34093			0.670465	−	0.741941i	$-0.266095\pi$
$691$	640269.			1.34093			0.670465	+	0.741941i	$0.266095\pi$
$692$	−54209.3	+	62251.3i	−0.113204	+	0.129998i
$693$	92258.6			0.192106
$694$	836225.	−	380651.i	1.73622	−	0.790328i
$695$			197024.i			0.407896i
$696$	−327871.	−	97585.5i	−0.676838	−	0.201450i
$697$	31355.0	−	399046.i	0.0645418	−	0.821405i
$698$	589007.	−	268117.i	1.20895	−	0.550317i
$699$		−	1.13985e6i		−	2.33289i
$700$	−20012.0	+	22980.8i	−0.0408408	+	0.0468995i
$701$	70939.4			0.144361			0.0721807	−	0.997392i	$-0.477004\pi$
$701$	70939.4			0.144361			0.0721807	+	0.997392i	$0.477004\pi$
$702$	838.396	+	1841.82i	0.00170128	+	0.00373742i
$703$	−93205.7			−0.188596
$704$	−423682.	−	276717.i	−0.854860	−	0.558330i
$705$	83797.0			0.168597
$706$	723663.	−	329412.i	1.45187	−	0.660892i
$707$		−	77482.7i		−	0.155012i
$708$	−406676.	−	354140.i	−0.811302	−	0.706494i
$709$			279667.i			0.556351i	0.960530	+	0.278175i	$0.0897297\pi$
$709$			279667.i			0.556351i	−0.960530	+	0.278175i	$0.910270\pi$
$710$	−211113.	+	96098.8i	−0.418792	+	0.190634i
$711$	−195790.			−0.387303
$712$	−511391.	−	152207.i	−1.00877	−	0.300244i
$713$	196687.			0.386899
$714$	46198.9	+	101491.i	0.0906224	+	0.199082i
$715$		−	408895.i		−	0.799834i
$716$	−119795.	+	137566.i	−0.233675	+	0.268340i
$717$	−1.10495e6			−2.14934
$718$	116909.	+	256829.i	0.226776	+	0.498189i
$719$	818203.			1.58272			0.791359	−	0.611352i	$-0.209374\pi$
$719$	818203.			1.58272			0.791359	+	0.611352i	$0.209374\pi$
$720$	58433.4	+	421090.i	0.112719	+	0.812288i
$721$		−	106178.i		−	0.204250i
$722$	271089.	−	123400.i	0.520040	−	0.236723i
$723$	707677.			1.35381
$724$	−501608.	−	436808.i	−0.956946	−	0.833324i
$725$			86952.2i			0.165426i
$726$	28866.9	−	13140.2i	0.0547680	−	0.0249305i
$727$	−980994.			−1.85608			−0.928042	−	0.372476i	$-0.878509\pi$
$727$	−980994.			−1.85608			−0.928042	+	0.372476i	$0.878509\pi$
$728$	−91292.8	−	27171.8i	−0.172256	−	0.0512691i
$729$	−534655.			−1.00605
$730$	−41815.8	+	19034.6i	−0.0784684	+	0.0357189i
$731$	757130.			1.41689
$732$	−366124.	+	420438.i	−0.683292	+	0.784658i
$733$	206372.			0.384098			0.192049	−	0.981385i	$-0.438487\pi$
$733$	206372.			0.384098			0.192049	+	0.981385i	$0.438487\pi$
$734$	105349.	+	231433.i	0.195540	+	0.429570i
$735$	−603115.			−1.11642
$736$	−231176.	+	146569.i	−0.426763	+	0.270575i
$737$	−452181.			−0.832487
$738$	513406.	−	186685.i	0.942644	−	0.342765i
$739$			511327.i			0.936290i	0.883652	+	0.468145i	$0.155077\pi$
$739$			511327.i			0.936290i	−0.883652	+	0.468145i	$0.844923\pi$
$740$	84698.8	−	97263.7i	0.154673	−	0.177618i
$741$		−	487454.i		−	0.887764i
$742$	−54347.8	−	119393.i	−0.0987129	−	0.216856i
$743$			928747.i			1.68236i	0.540753	+	0.841182i	$0.318140\pi$
$743$			928747.i			1.68236i	−0.540753	+	0.841182i	$0.681860\pi$
$744$	−574907.	−	171112.i	−1.03861	−	0.309125i
$745$			555641.i			1.00111i
$746$	−188229.	+	85682.2i	−0.338228	+	0.153962i
$747$			653545.i			1.17121i
$748$	354969.	+	309112.i	0.634435	+	0.552476i
$749$		−	16945.9i		−	0.0302065i
$750$	788800.	−	359063.i	1.40231	−	0.638333i
$751$	570479.			1.01148			0.505742	−	0.862685i	$-0.331219\pi$
$751$	570479.			1.01148			0.505742	+	0.862685i	$0.331219\pi$
$752$	−11325.2	−	81613.5i	−0.0200268	−	0.144320i
$753$			516178.i			0.910352i
$754$	−247368.	+	112602.i	−0.435112	+	0.198064i
$755$	402682.			0.706428
$756$	−301.752	+	346.516i	−0.000527966	+	0.000606290i
$757$		−	607716.i		−	1.06050i	−0.847842	−	0.530248i	$-0.822099\pi$
$757$		−	607716.i		−	1.06050i	0.847842	−	0.530248i	$-0.177901\pi$
$758$	443170.	+	973570.i	0.771315	+	1.69445i
$759$		−	420656.i		−	0.730202i
$760$	88196.4	−	296325.i	0.152695	−	0.513029i
$761$	−1.07393e6			−1.85442			−0.927211	−	0.374540i	$-0.877801\pi$
$761$	−1.07393e6			−1.85442			−0.927211	+	0.374540i	$0.877801\pi$
$762$	361284.	+	793680.i	0.622212	+	1.36690i
$763$		−	32072.3i		−	0.0550910i
$764$	−316509.	+	363462.i	−0.542249	+	0.622691i
$765$		−	395429.i		−	0.675688i
$766$	−16852.1	+	7671.11i	−0.0287209	+	0.0130738i
$767$	−428448.			−0.728295
$768$	803227.	−	227299.i	1.36181	−	0.385368i
$769$	−228327.			−0.386104			−0.193052	−	0.981188i	$-0.561839\pi$
$769$	−228327.			−0.386104			−0.193052	+	0.981188i	$0.561839\pi$
$770$	84500.0	−	38464.5i	0.142520	−	0.0648751i
$771$		−	1.44192e6i		−	2.42568i
$772$	−449743.	−	391643.i	−0.754623	−	0.657137i
$773$			967760.i			1.61960i	0.586704	+	0.809802i	$0.300425\pi$
$773$			967760.i			1.61960i	−0.586704	+	0.809802i	$0.699575\pi$
$774$	428104.	+	940472.i	0.714607	+	1.56987i
$775$			152467.i			0.253847i
$776$	488245.	+	145318.i	0.810801	+	0.241322i
$777$	46171.0			0.0764763
$778$	330688.	−	150530.i	0.546336	−	0.248693i
$779$	−396071.	−	31121.2i	−0.652676	−	0.0512839i
$780$	508677.	+	442964.i	0.836090	+	0.728080i
$781$	−350505.			−0.574635
$782$	231726.	−	105482.i	0.378932	−	0.172490i
$783$			1311.11i			0.00213853i
$784$	81511.6	+	587400.i	0.132613	+	0.955657i
$785$		−	426127.i		−	0.691513i
$786$	−231280.	−	508083.i	−0.374363	−	0.822412i
$787$		−	461085.i		−	0.744443i	−0.928144	−	0.372222i	$-0.878596\pi$
$787$		−	461085.i		−	0.744443i	0.928144	−	0.372222i	$-0.121404\pi$
$788$	−415295.	+	476904.i	−0.668813	+	0.768031i
$789$	−147344.			−0.236690
$790$	−179325.	+	81628.7i	−0.287333	+	0.130794i
$791$	−137378.			−0.219566
$792$	−183255.	+	615707.i	−0.292150	+	0.981575i
$793$			442947.i			0.704377i
$794$	−122565.	−	269255.i	−0.194413	−	0.427093i
$795$			928951.i			1.46980i
$796$	−256147.	+	294146.i	−0.404262	+	0.464234i
$797$	−490245.			−0.771786			−0.385893	−	0.922544i	$-0.626107\pi$
$797$	−490245.			−0.771786			−0.385893	+	0.922544i	$0.626107\pi$
$798$	100734.	−	45854.4i	0.158188	−	0.0720071i
$799$			76640.0i			0.120050i
$800$	−113617.	−	179201.i	−0.177526	−	0.280002i
$801$			677334.i			1.05569i
$802$	415286.	−	189039.i	0.645652	−	0.293901i
$803$	−69425.7			−0.107669
$804$	489856.	−	562526.i	0.757803	−	0.870223i
$805$		−	50219.7i		−	0.0774965i
$806$	−433749.	+	197443.i	−0.667680	+	0.303928i
$807$	−1.29060e6			−1.98173
$808$	517096.	+	153905.i	0.792043	+	0.235739i
$809$		−	1.11500e6i		−	1.70363i	−0.523840	−	0.851817i	$-0.675501\pi$
$809$		−	1.11500e6i		−	1.70363i	0.523840	−	0.851817i	$-0.324499\pi$
$810$	−486739.	+	221564.i	−0.741868	+	0.337699i
$811$			476106.i			0.723873i	0.932203	+	0.361937i	$0.117884\pi$
$811$			476106.i			0.723873i	−0.932203	+	0.361937i	$0.882116\pi$
$812$	−46539.5	−	40527.3i	−0.0705845	−	0.0614661i
$813$			463132.i			0.700686i
$814$	177377.	−	80742.2i	0.267700	−	0.121857i
$815$			518409.i			0.780471i
$816$	−769088.	+	106724.i	−1.15504	+	0.160281i
$817$		−	751484.i		−	1.12584i
$818$	169010.	−	76933.7i	0.252584	−	0.114977i
$819$			120917.i			0.180268i
$820$	392397.	−	385034.i	0.583577	−	0.572627i
$821$	125583.			0.186313			0.0931565	−	0.995651i	$-0.470304\pi$
$821$	125583.			0.186313			0.0931565	+	0.995651i	$0.470304\pi$
$822$	−734962.	−	1.61459e6i	−1.08773	−	2.38956i
$823$	−1.00562e6			−1.48469			−0.742343	−	0.670020i	$-0.766285\pi$
$823$	−1.00562e6			−1.48469			−0.742343	+	0.670020i	$0.766285\pi$
$824$	708597.	+	210902.i	1.04363	+	0.310618i
$825$	326081.			0.479090
$826$	−40303.8	−	88540.6i	−0.0590725	−	0.129772i
$827$	−129029.			−0.188658			−0.0943291	−	0.995541i	$-0.530071\pi$
$827$	−129029.			−0.188658			−0.0943291	+	0.995541i	$0.530071\pi$
$828$	262049.	+	228196.i	0.382228	+	0.332850i
$829$	1.06280e6			1.54648			0.773240	−	0.634114i	$-0.218635\pi$
$829$	1.06280e6			1.54648			0.773240	+	0.634114i	$0.218635\pi$
$830$	272476.	+	598584.i	0.395523	+	0.868898i
$831$	704319.			1.01992
$832$	362673.	−	555289.i	0.523925	−	0.802182i
$833$		−	551604.i		−	0.794946i
$834$	−203469.	−	446987.i	−0.292527	−	0.642633i
$835$	1.06634e6			1.52940
$836$	306807.	−	352322.i	0.438989	−	0.504112i
$837$			2298.97i			0.00328158i
$838$	267874.	+	588475.i	0.381455	+	0.837992i
$839$	−606080.			−0.861006			−0.430503	−	0.902589i	$-0.641664\pi$
$839$	−606080.			−0.861006			−0.430503	+	0.902589i	$0.641664\pi$
$840$	−43689.5	+	146790.i	−0.0619182	+	0.208035i
$841$	531190.			0.751031
$842$	−487866.	−	1.07176e6i	−0.688139	−	1.51173i
$843$			236008.i			0.332103i
$844$	116099.	−	133323.i	0.162984	−	0.187163i
$845$	−47874.8			−0.0670491
$846$	−95198.7	+	43334.5i	−0.133012	+	0.0605471i
$847$	5721.72			0.00797554
$848$	904745.	−	125549.i	1.25816	−	0.174590i
$849$		−	1.07145e6i		−	1.48647i
$850$	81766.6	+	179628.i	0.113172	+	0.248619i
$851$		−	105418.i		−	0.145564i
$852$	379708.	−	436038.i	0.523084	−	0.600683i
$853$	375927.			0.516661			0.258331	−	0.966057i	$-0.416828\pi$
$853$	375927.			0.516661			0.258331	+	0.966057i	$0.416828\pi$
$854$	−91536.8	+	41667.7i	−0.125511	+	0.0571325i
$855$	−392481.			−0.536891
$856$	113092.	+	33659.9i	0.154342	+	0.0459373i
$857$	−143692.			−0.195646			−0.0978228	−	0.995204i	$-0.531188\pi$
$857$	−143692.			−0.195646			−0.0978228	+	0.995204i	$0.531188\pi$
$858$	422271.	+	927659.i	0.573610	+	1.26013i
$859$			101369.i			0.137379i	0.997638	+	0.0686894i	$0.0218817\pi$
$859$			101369.i			0.137379i	−0.997638	+	0.0686894i	$0.978118\pi$
$860$	784203.	+	682896.i	1.06031	+	0.923331i
$861$	196200.	+	15416.4i	0.264663	+	0.0207958i
$862$	398962.	+	876453.i	0.536930	+	1.17954i
$863$		−	1.25765e6i		−	1.68864i	−0.535837	−	0.844322i	$-0.680004\pi$
$863$		−	1.25765e6i		−	1.68864i	0.535837	−	0.844322i	$-0.319996\pi$
$864$	−1713.17	−	2702.09i	−0.00229495	−	0.00361970i
$865$	105452.			0.140937
$866$	−1.10142e6	+	501369.i	−1.46865	+	0.668531i
$867$	−341633.			−0.454487
$868$	−81604.8	−	71062.7i	−0.108312	−	0.0943197i
$869$	−297728.			−0.394257
$870$	181053.	+	397743.i	0.239203	+	0.525489i
$871$		−	592641.i		−	0.781187i
$872$	214041.	+	63705.8i	0.281491	+	0.0837811i
$873$		−	646677.i		−	0.848514i
$874$	−104695.	−	229998.i	−0.137058	−	0.301093i
$875$	156348.			0.204210
$876$	75210.1	−	86367.5i	0.0980095	−	0.112549i
$877$	−67546.7			−0.0878224			−0.0439112	−	0.999035i	$-0.513982\pi$
$877$	−67546.7			−0.0878224			−0.0439112	+	0.999035i	$0.513982\pi$
$878$	−962871.	+	438300.i	−1.24905	+	0.568568i
$879$			1.26832e6i			1.64154i
$880$	88856.6	+	640330.i	0.114743	+	0.826873i
$881$	−30852.7			−0.0397504			−0.0198752	−	0.999802i	$-0.506327\pi$
$881$	−30852.7			−0.0397504			−0.0198752	+	0.999802i	$0.506327\pi$
$882$	685177.	−	311893.i	0.880776	−	0.400930i
$883$	784152.			1.00572			0.502862	−	0.864367i	$-0.332280\pi$
$883$	784152.			1.00572			0.502862	+	0.864367i	$0.332280\pi$
$884$	−405131.	+	465232.i	−0.518431	+	0.595340i
$885$			688901.i			0.879569i
$886$	−195476.	−	429427.i	−0.249015	−	0.547044i
$887$	−254996.			−0.324105			−0.162052	−	0.986782i	$-0.551811\pi$
$887$	−254996.			−0.324105			−0.162052	+	0.986782i	$0.551811\pi$
$888$	−91710.2	+	308131.i	−0.116303	+	0.390760i
$889$			157316.i			0.199053i
$890$	282394.	+	620372.i	0.356513	+	0.783199i
$891$	−808120.			−1.01794
$892$	−678110.	−	590508.i	−0.852256	−	0.742158i
$893$	76068.5			0.0953899
$894$	−573817.	−	1.26058e6i	−0.717957	−	1.57723i
$895$	233034.			0.290920
$896$	148869.	+	22712.3i	0.185434	+	0.0282908i
$897$	551322.			0.685205
$898$	1.19939e6	−	545965.i	1.48734	−	0.677037i
$899$	−308768.			−0.382043
$900$	−176892.	+	203133.i	−0.218385	+	0.250782i
$901$	−849611.			−1.04657
$902$	780710.	−	283882.i	0.959570	−	0.348919i
$903$			372260.i			0.456532i
$904$	272877.	−	916821.i	0.333910	−	1.12188i
$905$			849713.i			1.03747i
$906$	−913562.	+	415854.i	−1.11297	+	0.506623i
$907$			434736.i			0.528458i	0.964460	+	0.264229i	$0.0851175\pi$
$907$			434736.i			0.528458i	−0.964460	+	0.264229i	$0.914882\pi$
$908$	490131.	−	562842.i	0.594485	−	0.682676i
$909$		−	684890.i		−	0.828883i
$910$	50412.6	+	110748.i	0.0608774	+	0.133737i
$911$			509973.i			0.614483i	0.951632	+	0.307242i	$0.0994060\pi$
$911$			509973.i			0.614483i	−0.951632	+	0.307242i	$0.900594\pi$
$912$	105928.	+	763354.i	0.127357	+	0.917775i
$913$			993812.i			1.19224i
$914$	231403.	+	508354.i	0.276998	+	0.608519i
$915$	712213.			0.850683
$916$	1.11513e6	+	971074.i	1.32903	+	1.15734i
$917$		−	100707.i		−	0.119763i
$918$	1232.92	+	2708.52i	0.00146302	+	0.00321400i
$919$	84574.6			0.100140			0.0500701	−	0.998746i	$-0.484056\pi$
$919$	84574.6			0.100140			0.0500701	+	0.998746i	$0.484056\pi$
$920$	335151.	+	99752.3i	0.395973	+	0.117855i
$921$			628769.i			0.741262i
$922$	356610.	−	162329.i	0.419500	−	0.190957i
$923$		−	459381.i		−	0.539225i
$924$	−151982.	+	174528.i	−0.178012	+	0.204419i
$925$	81717.1			0.0955058
$926$	−297681.	+	135504.i	−0.347159	+	0.158027i
$927$		−	938532.i		−	1.09217i
$928$	362909.	−	230091.i	0.421408	−	0.267179i
$929$			892909.i			1.03461i	0.855802	+	0.517304i	$0.173064\pi$
$929$			892909.i			1.03461i	−0.855802	+	0.517304i	$0.826936\pi$
$930$	317468.	+	697424.i	0.367057	+	0.806363i
$931$	−547491.			−0.631652
$932$	1.07978e6	+	940287.i	1.24309	+	1.08250i
$933$	−501480.			−0.576090
$934$	50611.9	+	111186.i	0.0580175	+	0.127455i
$935$		−	601309.i		−	0.687820i
$936$	−806962.	−	240179.i	−0.921089	−	0.274147i
$937$		−	878668.i		−	1.00080i	−0.865795	−	0.500398i	$-0.833187\pi$
$937$		−	878668.i		−	1.00080i	0.865795	−	0.500398i	$-0.166813\pi$
$938$	122472.	−	55749.3i	0.139197	−	0.0633627i
$939$		−	438008.i		−	0.496765i
$940$	−69125.7	+	79380.4i	−0.0782319	+	0.0898375i
$941$	−370968.			−0.418945			−0.209472	−	0.977815i	$-0.567175\pi$
$941$	−370968.			−0.418945			−0.209472	+	0.977815i	$0.567175\pi$
$942$	440067.	+	966754.i	0.495926	+	1.08947i
$943$	35198.8	−	447965.i	0.0395827	−	0.503757i
$944$	670950.	−	93105.6i	0.752915	−	0.104480i
$945$	586.991			0.000657306
$946$	650995.	+	1.43013e6i	0.727437	+	1.59806i
$947$			123684.i			0.137916i	0.997620	+	0.0689579i	$0.0219674\pi$
$947$			123684.i			0.137916i	−0.997620	+	0.0689579i	$0.978033\pi$
$948$	322534.	−	370381.i	0.358888	−	0.412128i
$949$		−	90991.2i		−	0.101034i
$950$	178288.	−	81156.9i	0.197549	−	0.0899246i
$951$		−	153790.i		−	0.170046i
$952$	−134252.	−	39958.0i	−0.148132	−	0.0440890i
$953$	−1.28041e6			−1.40982			−0.704908	−	0.709299i	$-0.749012\pi$
$953$	−1.28041e6			−1.40982			−0.704908	+	0.709299i	$0.749012\pi$
$954$	−480395.	−	1.05535e6i	−0.527839	−	1.15957i
$955$	615697.			0.675088
$956$	911496.	−	1.04672e6i	0.997331	−	1.14528i
$957$			660362.i			0.721038i
$958$	−490757.	+	223393.i	−0.534731	+	0.243410i
$959$		−	320028.i		−	0.347978i
$960$	−892849.	−	583142.i	−0.968803	−	0.632749i
$961$	382111.			0.413755
$962$	105823.	+	232475.i	0.114348	+	0.251204i
$963$		−	149789.i		−	0.161521i
$964$	−583776.	+	670379.i	−0.628192	+	0.721383i
$965$			761855.i			0.818121i
$966$	51862.5	+	113933.i	0.0555775	+	0.122094i
$967$	−1.15025e6			−1.23009			−0.615047	−	0.788490i	$-0.710863\pi$
$967$	−1.15025e6			−1.23009			−0.615047	+	0.788490i	$0.710863\pi$
$968$	−11365.2	+	38185.1i	−0.0121290	+	0.0407515i
$969$		−	716835.i		−	0.763435i
$970$	−269613.	−	592293.i	−0.286547	−	0.629497i
$971$	−821667.			−0.871480			−0.435740	−	0.900072i	$-0.643513\pi$
$971$	−821667.			−0.871480			−0.435740	+	0.900072i	$0.643513\pi$
$972$	878111.	−	1.00838e6i	0.929430	−	1.06731i
$973$		−	88597.5i		−	0.0935828i
$974$	−637366.	−	1.40019e6i	−0.671848	−	1.47594i
$975$			427370.i			0.449568i
$976$	−96256.3	−	693655.i	−0.101048	−	0.728189i
$977$		−	206218.i		−	0.216042i	−0.994149	−	0.108021i	$-0.965549\pi$
$977$		−	206218.i		−	0.216042i	0.994149	−	0.108021i	$-0.0344514\pi$
$978$	−535367.	−	1.17611e6i	−0.559724	−	1.22962i
$979$			1.02999e6i			1.07465i
$980$	497521.	−	571328.i	0.518035	−	0.594885i
$981$		−	283496.i		−	0.294584i
$982$	497427.	+	1.09276e6i	0.515830	+	1.13319i
$983$			1.23734e6i			1.28051i	0.768163	+	0.640254i	$0.221171\pi$
$983$			1.23734e6i			1.28051i	−0.768163	+	0.640254i	$0.778829\pi$
$984$	−492600.	+	1.27876e6i	−0.508750	+	1.32068i
$985$	807865.			0.832657
$986$	−363772.	+	165590.i	−0.374176	+	0.170325i
$987$	−37681.8			−0.0386809
$988$	461763.	+	402110.i	0.473048	+	0.411937i
$989$	849947.			0.868959
$990$	746918.	−	339998.i	0.762084	−	0.346901i
$991$	1.35832e6			1.38310			0.691550	−	0.722329i	$-0.256928\pi$
$991$	1.35832e6			1.38310			0.691550	+	0.722329i	$0.256928\pi$
$992$	636345.	−	403453.i	0.646650	−	0.409987i
$993$	1.82109e6			1.84686
$994$	94933.1	−	43213.6i	0.0960826	−	0.0437369i
$995$	498277.			0.503297
$996$	−1.23633e6	−	1.07662e6i	−1.24628	−	1.08528i
$997$		−	1.02180e6i		−	1.02796i	−0.857802	−	0.513980i	$-0.828171\pi$
$997$		−	1.02180e6i		−	1.02796i	0.857802	−	0.513980i	$-0.171829\pi$
$998$	−1.26935e6	+	577809.i	−1.27444	+	0.580127i
$999$	1232.17			0.00123464

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	164.5.d.f.163.7	yes	72
4.3	odd	2	inner	164.5.d.f.163.6	yes	72
41.40	even	2	inner	164.5.d.f.163.8	yes	72
164.163	odd	2	inner	164.5.d.f.163.5	✓	72

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
164.5.d.f.163.5	✓	72	164.163	odd	2	inner
164.5.d.f.163.6	yes	72	4.3	odd	2	inner
164.5.d.f.163.7	yes	72	1.1	even	1	trivial
164.5.d.f.163.8	yes	72	41.40	even	2	inner

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:	\( N \)	\(=\)	\( 164 = 2^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	164.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-3.64057 + 1.65719i) q^{2} -12.7376 q^{3} +(10.5074 - 12.0662i) q^{4} -20.4399 q^{5} +(46.3719 - 21.1085i) q^{6} +9.19140 q^{7} +(-18.2571 + 61.3407i) q^{8} +81.2453 q^{9} +(74.4128 - 33.8728i) q^{10} +123.546 q^{11} +(-133.839 + 153.694i) q^{12} +161.922i q^{13} +(-33.4619 + 15.2319i) q^{14} +260.354 q^{15} +(-35.1871 - 253.570i) q^{16} +238.118i q^{17} +(-295.779 + 134.639i) q^{18} +236.342 q^{19} +(-214.771 + 246.632i) q^{20} -117.076 q^{21} +(-449.776 + 204.738i) q^{22} +267.309i q^{23} +(232.550 - 781.330i) q^{24} -207.211 q^{25} +(-268.336 - 589.488i) q^{26} -3.12443 q^{27} +(96.5781 - 110.905i) q^{28} -419.632i q^{29} +(-947.837 + 431.456i) q^{30} -735.805i q^{31} +(548.315 + 864.828i) q^{32} -1573.67 q^{33} +(-394.606 - 866.884i) q^{34} -187.871 q^{35} +(853.681 - 980.323i) q^{36} -394.368 q^{37} +(-860.420 + 391.664i) q^{38} -2062.49i q^{39} +(373.172 - 1253.80i) q^{40} +(-1675.83 - 131.679i) q^{41} +(426.223 - 194.017i) q^{42} -3179.64i q^{43} +(1298.15 - 1490.73i) q^{44} -1660.65 q^{45} +(-442.981 - 973.156i) q^{46} +321.858 q^{47} +(448.198 + 3229.86i) q^{48} -2316.52 q^{49} +(754.364 - 343.387i) q^{50} -3033.04i q^{51} +(1953.79 + 1701.39i) q^{52} +3568.03i q^{53} +(11.3747 - 5.17777i) q^{54} -2525.26 q^{55} +(-167.808 + 563.807i) q^{56} -3010.42 q^{57} +(695.410 + 1527.70i) q^{58} +2646.01i q^{59} +(2735.66 - 3141.49i) q^{60} +2735.55 q^{61} +(1219.37 + 2678.75i) q^{62} +746.758 q^{63} +(-3429.36 - 2239.80i) q^{64} -3309.67i q^{65} +(5729.04 - 2607.87i) q^{66} -3660.04 q^{67} +(2873.18 + 2502.01i) q^{68} -3404.86i q^{69} +(683.958 - 311.338i) q^{70} -2837.05 q^{71} +(-1483.30 + 4983.64i) q^{72} -561.944 q^{73} +(1435.72 - 653.542i) q^{74} +2639.36 q^{75} +(2483.35 - 2851.76i) q^{76} +1135.56 q^{77} +(3417.94 + 7508.64i) q^{78} -2409.86 q^{79} +(719.221 + 5182.95i) q^{80} -6541.07 q^{81} +(6319.20 - 2297.79i) q^{82} +8044.10i q^{83} +(-1230.17 + 1412.66i) q^{84} -4867.10i q^{85} +(5269.27 + 11575.7i) q^{86} +5345.09i q^{87} +(-2255.58 + 7578.37i) q^{88} +8336.90i q^{89} +(6045.69 - 2752.00i) q^{90} +1488.29i q^{91} +(3225.41 + 2808.73i) q^{92} +9372.36i q^{93} +(-1171.74 + 533.379i) q^{94} -4830.81 q^{95} +(-6984.19 - 11015.8i) q^{96} -7959.56i q^{97} +(8433.44 - 3838.91i) q^{98} +10037.5 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q + 6 q^{2} - 162 q^{4} - 8 q^{5} + 162 q^{8} + 1728 q^{9} - 272 q^{10} - 2842 q^{16} - 298 q^{18} - 584 q^{20} + 280 q^{21} + 4264 q^{25} + 66 q^{32} + 3512 q^{33} - 11338 q^{36} + 5720 q^{37} - 4648 q^{40}+ \cdots + 2902 q^{98}+O(q^{100}) \) 72 * q + 6 * q^2 - 162 * q^4 - 8 * q^5 + 162 * q^8 + 1728 * q^9 - 272 * q^10 - 2842 * q^16 - 298 * q^18 - 584 * q^20 + 280 * q^21 + 4264 * q^25 + 66 * q^32 + 3512 * q^33 - 11338 * q^36 + 5720 * q^37 - 4648 * q^40 - 7928 * q^41 - 10140 * q^42 - 9528 * q^45 - 5680 * q^46 + 6624 * q^49 + 554 * q^50 + 29864 * q^57 + 11712 * q^61 - 3936 * q^62 - 34554 * q^64 + 24852 * q^66 - 12166 * q^72 + 16632 * q^73 - 11120 * q^74 + 84456 * q^77 - 20496 * q^78 - 12104 * q^80 + 13408 * q^81 - 51114 * q^82 - 948 * q^84 - 20400 * q^86 + 64976 * q^90 - 2784 * q^92 + 2902 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more