LMFDB - Embedded newform 108.5.k.a.29.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [108,5,Mod(5,108)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(108, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([0, 5]))

N = Newforms(chi, 5, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("108.5");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 108 = 2^{2} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	108.k (of order $18$, degree $6$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$11.1639560131$
Analytic rank:	$0$
Dimension:	$72$
Relative dimension:	$12$ over $\Q(\zeta_{18})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			29.4
Character	$\chi$	$=$	108.29
Dual form			108.5.k.a.41.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-3.84126 + 8.13909i) q^{3} +(0.0201223 + 0.0239808i) q^{5} +(-60.3934 + 21.9814i) q^{7} +(-51.4895 - 62.5287i) q^{9} +O(q^{10})$	\(q+(-3.84126 + 8.13909i) q^{3} +(0.0201223 + 0.0239808i) q^{5} +(-60.3934 + 21.9814i) q^{7} +(-51.4895 - 62.5287i) q^{9} +(122.535 - 146.032i) q^{11} +(8.21590 + 46.5947i) q^{13} +(-0.272476 + 0.0716604i) q^{15} +(-383.840 + 221.610i) q^{17} +(299.637 - 518.987i) q^{19} +(53.0781 - 575.983i) q^{21} +(158.524 - 435.540i) q^{23} +(108.530 - 615.504i) q^{25} +(706.711 - 178.888i) q^{27} +(-396.668 - 69.9433i) q^{29} +(-1472.40 - 535.909i) q^{31} +(717.877 + 1558.27i) q^{33} +(-1.74238 - 1.00596i) q^{35} +(-532.781 - 922.804i) q^{37} +(-410.798 - 112.112i) q^{39} +(-2327.41 + 410.385i) q^{41} +(1343.05 + 1126.95i) q^{43} +(0.463402 - 2.49298i) q^{45} +(864.494 + 2375.18i) q^{47} +(1324.91 - 1111.73i) q^{49} +(-329.275 - 3975.37i) q^{51} +2410.75i q^{53} +5.96765 q^{55} +(3073.09 + 4432.33i) q^{57} +(165.882 + 197.690i) q^{59} +(-3689.65 + 1342.92i) q^{61} +(4484.09 + 2644.51i) q^{63} +(-0.952054 + 1.13461i) q^{65} +(-702.984 - 3986.82i) q^{67} +(2935.97 + 2963.26i) q^{69} +(-2144.37 + 1238.05i) q^{71} +(761.304 - 1318.62i) q^{73} +(4592.75 + 3247.64i) q^{75} +(-4190.34 + 11512.9i) q^{77} +(1351.07 - 7662.27i) q^{79} +(-1258.67 + 6439.14i) q^{81} +(-12237.2 - 2157.76i) q^{83} +(-13.0381 - 4.74548i) q^{85} +(2092.98 - 2959.85i) q^{87} +(4114.03 + 2375.24i) q^{89} +(-1520.40 - 2633.41i) q^{91} +(10017.7 - 9925.42i) q^{93} +(18.4751 - 3.25765i) q^{95} +(-4423.75 - 3711.97i) q^{97} +(-15440.5 - 142.867i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 72 q + 9 q^{5} - 102 q^{9}+O(q^{10}) $ 72 * q + 9 * q^5 - 102 * q^9	$ 72 q + 9 q^{5} - 102 q^{9} + 18 q^{11} - 225 q^{15} - 282 q^{21} - 1278 q^{23} + 441 q^{25} + 54 q^{27} + 1854 q^{29} - 1665 q^{31} - 45 q^{33} - 2673 q^{35} + 6951 q^{39} - 5472 q^{41} + 1260 q^{43} + 5553 q^{45} + 5103 q^{47} - 5904 q^{49} + 1899 q^{51} + 1107 q^{57} - 10944 q^{59} + 8352 q^{61} - 11985 q^{63} + 8757 q^{65} + 378 q^{67} + 5607 q^{69} - 19764 q^{71} + 6111 q^{73} - 3453 q^{75} - 5679 q^{77} - 5652 q^{79} - 20466 q^{81} - 20061 q^{83} + 26100 q^{85} + 40545 q^{87} + 15633 q^{89} - 6039 q^{91} + 40179 q^{93} + 48024 q^{95} - 37530 q^{97} + 12177 q^{99}+O(q^{100}) $ 72 * q + 9 * q^5 - 102 * q^9 + 18 * q^11 - 225 * q^15 - 282 * q^21 - 1278 * q^23 + 441 * q^25 + 54 * q^27 + 1854 * q^29 - 1665 * q^31 - 45 * q^33 - 2673 * q^35 + 6951 * q^39 - 5472 * q^41 + 1260 * q^43 + 5553 * q^45 + 5103 * q^47 - 5904 * q^49 + 1899 * q^51 + 1107 * q^57 - 10944 * q^59 + 8352 * q^61 - 11985 * q^63 + 8757 * q^65 + 378 * q^67 + 5607 * q^69 - 19764 * q^71 + 6111 * q^73 - 3453 * q^75 - 5679 * q^77 - 5652 * q^79 - 20466 * q^81 - 20061 * q^83 + 26100 * q^85 + 40545 * q^87 + 15633 * q^89 - 6039 * q^91 + 40179 * q^93 + 48024 * q^95 - 37530 * q^97 + 12177 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/108\mathbb{Z}\right)^\times$.

$n$	$29$	$55$
$\chi(n)$	$e\left(\frac{1}{18}\right)$	$1$

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 $

$2$		0			0
$2$		0			0
$3$	−3.84126	+	8.13909i	−0.426807	+	0.904343i
$3$	−3.84126	+	8.13909i	−0.426807	+	0.904343i
$4$		0			0
$5$	0.0201223	+	0.0239808i	0.000804890	+	0.000959231i	0.766447	−	0.642308i	$-0.222023\pi$
$5$	0.0201223	+	0.0239808i	0.000804890	+	0.000959231i	−0.765642	+	0.643267i	$0.777578\pi$
$6$		0			0
$7$	−60.3934	+	21.9814i	−1.23252	+	0.448600i	−0.874459	−	0.485099i	$-0.838783\pi$
$7$	−60.3934	+	21.9814i	−1.23252	+	0.448600i	−0.358059	+	0.933699i	$0.616561\pi$
$8$		0			0
$9$	−51.4895	−	62.5287i	−0.635672	−	0.771959i
$10$		0			0
$11$	122.535	−	146.032i	1.01269	−	1.20688i	0.0344472	−	0.999407i	$-0.489033\pi$
$11$	122.535	−	146.032i	1.01269	−	1.20688i	0.978242	−	0.207469i	$-0.0665226\pi$
$12$		0			0
$13$	8.21590	+	46.5947i	0.0486148	+	0.275708i	0.999419	−	0.0340844i	$-0.0108515\pi$
$13$	8.21590	+	46.5947i	0.0486148	+	0.275708i	−0.950804	+	0.309793i	$0.899740\pi$
$14$		0			0
$15$	−0.272476	+	0.0716604i	−0.00121101	+	0.000318491i
$16$		0			0
$17$	−383.840	+	221.610i	−1.32817	+	0.766817i	−0.985016	−	0.172464i	$-0.944827\pi$
$17$	−383.840	+	221.610i	−1.32817	+	0.766817i	−0.343150	+	0.939281i	$0.611494\pi$
$18$		0			0
$19$	299.637	−	518.987i	0.830020	−	1.43764i	−0.0680020	−	0.997685i	$-0.521662\pi$
$19$	299.637	−	518.987i	0.830020	−	1.43764i	0.898022	−	0.439951i	$-0.145004\pi$
$20$		0			0
$21$	53.0781	−	575.983i	0.120359	−	1.30608i
$22$		0			0
$23$	158.524	−	435.540i	0.299666	−	0.823327i	−0.694889	−	0.719117i	$-0.744546\pi$
$23$	158.524	−	435.540i	0.299666	−	0.823327i	0.994555	−	0.104210i	$-0.0332313\pi$
$24$		0			0
$25$	108.530	−	615.504i	0.173648	−	0.984806i
$26$		0			0
$27$	706.711	−	178.888i	0.969425	−	0.245389i
$28$		0			0
$29$	−396.668	−	69.9433i	−0.471663	−	0.0831668i	−0.0672356	−	0.997737i	$-0.521418\pi$
$29$	−396.668	−	69.9433i	−0.471663	−	0.0831668i	−0.404427	+	0.914570i	$0.632529\pi$
$30$		0			0
$31$	−1472.40	−	535.909i	−1.53215	−	0.557658i	−0.568005	−	0.823025i	$-0.692285\pi$
$31$	−1472.40	−	535.909i	−1.53215	−	0.557658i	−0.964148	+	0.265367i	$0.914507\pi$
$32$		0			0
$33$	717.877	+	1558.27i	0.659207	+	1.43092i
$34$		0			0
$35$	−1.74238	−	1.00596i	−0.00142235	−	0.000821196i
$36$		0			0
$37$	−532.781	−	922.804i	−0.389175	−	0.674071i	0.603163	−	0.797618i	$-0.293907\pi$
$37$	−532.781	−	922.804i	−0.389175	−	0.674071i	−0.992339	+	0.123546i	$0.960573\pi$
$38$		0			0
$39$	−410.798	−	112.112i	−0.270084	−	0.0737096i
$40$		0			0
$41$	−2327.41	+	410.385i	−1.38454	+	0.244132i	−0.815775	−	0.578369i	$-0.803689\pi$
$41$	−2327.41	+	410.385i	−1.38454	+	0.244132i	−0.568764	+	0.822501i	$0.692578\pi$
$42$		0			0
$43$	1343.05	+	1126.95i	0.726364	+	0.609491i	0.929138	−	0.369734i	$-0.120551\pi$
$43$	1343.05	+	1126.95i	0.726364	+	0.609491i	−0.202774	+	0.979226i	$0.564996\pi$
$44$		0			0
$45$	0.463402	−	2.49298i	0.000228840	−	0.00123110i
$46$		0			0
$47$	864.494	+	2375.18i	0.391351	+	1.07523i	0.966385	+	0.257099i	$0.0827665\pi$
$47$	864.494	+	2375.18i	0.391351	+	1.07523i	−0.575034	+	0.818129i	$0.695011\pi$
$48$		0			0
$49$	1324.91	−	1111.73i	0.551814	−	0.463027i
$50$		0			0
$51$	−329.275	−	3975.37i	−0.126596	−	1.52840i
$52$		0			0
$53$			2410.75i			0.858225i	0.903251	+	0.429112i	$0.141174\pi$
$53$			2410.75i			0.858225i	−0.903251	+	0.429112i	$0.858826\pi$
$54$		0			0
$55$	5.96765			0.00197278
$56$		0			0
$57$	3073.09	+	4432.33i	0.945858	+	1.36422i
$58$		0			0
$59$	165.882	+	197.690i	0.0476535	+	0.0567912i	0.789345	−	0.613950i	$-0.210420\pi$
$59$	165.882	+	197.690i	0.0476535	+	0.0567912i	−0.741691	+	0.670741i	$0.765976\pi$
$60$		0			0
$61$	−3689.65	+	1342.92i	−0.991574	+	0.360903i	−0.786329	−	0.617807i	$-0.788021\pi$
$61$	−3689.65	+	1342.92i	−0.991574	+	0.360903i	−0.205244	+	0.978711i	$0.565799\pi$
$62$		0			0
$63$	4484.09	+	2644.51i	1.12978	+	0.666291i
$64$		0			0
$65$	−0.952054	+	1.13461i	−0.000225338	+	0.000268548i
$66$		0			0
$67$	−702.984	−	3986.82i	−0.156601	−	0.888131i	−0.957307	−	0.289073i	$-0.906653\pi$
$67$	−702.984	−	3986.82i	−0.156601	−	0.888131i	0.800706	−	0.599058i	$-0.204458\pi$
$68$		0			0
$69$	2935.97	+	2963.26i	0.616670	+	0.622403i
$70$		0			0
$71$	−2144.37	+	1238.05i	−0.425385	+	0.245596i	−0.697379	−	0.716703i	$-0.745650\pi$
$71$	−2144.37	+	1238.05i	−0.425385	+	0.245596i	0.271994	+	0.962299i	$0.412317\pi$
$72$		0			0
$73$	761.304	−	1318.62i	0.142861	−	0.247442i	−0.785712	−	0.618592i	$-0.787703\pi$
$73$	761.304	−	1318.62i	0.142861	−	0.247442i	0.928573	+	0.371150i	$0.121037\pi$
$74$		0			0
$75$	4592.75	+	3247.64i	0.816489	+	0.577359i
$76$		0			0
$77$	−4190.34	+	11512.9i	−0.706753	+	1.94179i
$78$		0			0
$79$	1351.07	−	7662.27i	0.216482	−	1.22773i	−0.661834	−	0.749651i	$-0.730221\pi$
$79$	1351.07	−	7662.27i	0.216482	−	1.22773i	0.878316	−	0.478081i	$-0.158667\pi$
$80$		0			0
$81$	−1258.67	+	6439.14i	−0.191841	+	0.981426i
$82$		0			0
$83$	−12237.2	−	2157.76i	−1.77634	−	0.313218i	−0.813156	−	0.582045i	$-0.802253\pi$
$83$	−12237.2	−	2157.76i	−1.77634	−	0.313218i	−0.963188	+	0.268828i	$0.913364\pi$
$84$		0			0
$85$	−13.0381	−	4.74548i	−0.00180458	−	0.000656814i
$86$		0			0
$87$	2092.98	−	2959.85i	0.276520	−	0.391049i
$88$		0			0
$89$	4114.03	+	2375.24i	0.519383	+	0.299866i	0.736682	−	0.676239i	$-0.236391\pi$
$89$	4114.03	+	2375.24i	0.519383	+	0.299866i	−0.217299	+	0.976105i	$0.569725\pi$
$90$		0			0
$91$	−1520.40	−	2633.41i	−0.183601	−	0.318007i
$92$		0			0
$93$	10017.7	−	9925.42i	1.15825	−	1.14758i
$94$		0			0
$95$	18.4751	−	3.25765i	0.00204710	−	0.000360959i
$96$		0			0
$97$	−4423.75	−	3711.97i	−0.470162	−	0.394512i	0.376692	−	0.926339i	$-0.377062\pi$
$97$	−4423.75	−	3711.97i	−0.470162	−	0.394512i	−0.846854	+	0.531826i	$0.821506\pi$
$98$		0			0
$99$	−15440.5	−	142.867i	−1.57540	−	0.0145767i
$100$		0			0
$101$	191.499	+	526.138i	0.0187725	+	0.0515771i	0.948725	−	0.316103i	$-0.102374\pi$
$101$	191.499	+	526.138i	0.0187725	+	0.0515771i	−0.929952	+	0.367680i	$0.880152\pi$
$102$		0			0
$103$	−5058.09	+	4244.24i	−0.476773	+	0.400060i	−0.849258	−	0.527978i	$-0.822950\pi$
$103$	−5058.09	+	4244.24i	−0.476773	+	0.400060i	0.372485	+	0.928038i	$0.378506\pi$
$104$		0			0
$105$	14.8806	−	10.3172i	0.00134971	−	0.000935803i
$106$		0			0
$107$			4467.37i			0.390198i	0.980784	+	0.195099i	$0.0625027\pi$
$107$			4467.37i			0.390198i	−0.980784	+	0.195099i	$0.937497\pi$
$108$		0			0
$109$	9782.65			0.823386			0.411693	−	0.911323i	$-0.364938\pi$
$109$	9782.65			0.823386			0.411693	+	0.911323i	$0.364938\pi$
$110$		0			0
$111$	9557.33	−	791.623i	0.775694	−	0.0642499i
$112$		0			0
$113$	−8140.66	−	9701.66i	−0.637533	−	0.759782i	0.346446	−	0.938070i	$-0.387389\pi$
$113$	−8140.66	−	9701.66i	−0.637533	−	0.759782i	−0.983978	+	0.178288i	$0.942944\pi$
$114$		0			0
$115$	13.6344	−	4.96253i	0.00103096	−	0.000375238i
$116$		0			0
$117$	2490.47	−	2912.86i	0.181932	−	0.212789i
$118$		0			0
$119$	18310.1	−	21821.1i	1.29299	−	1.54093i
$120$		0			0
$121$	−3768.03	−	21369.6i	−0.257362	−	1.45957i
$122$		0			0
$123$	5600.02	−	20519.4i	0.370152	−	1.35630i
$124$		0			0
$125$	33.8883	−	19.5654i	0.00216885	−	0.00125219i
$126$		0			0
$127$	11131.3	−	19280.0i	0.690142	−	1.19536i	−0.281649	−	0.959518i	$-0.590881\pi$
$127$	11131.3	−	19280.0i	0.690142	−	1.19536i	0.971791	−	0.235844i	$-0.0757853\pi$
$128$		0			0
$129$	−14331.3	+	6602.26i	−0.861206	+	0.396747i
$130$		0			0
$131$	2326.48	−	6391.94i	0.135568	−	0.372469i	−0.853269	−	0.521470i	$-0.825384\pi$
$131$	2326.48	−	6391.94i	0.135568	−	0.372469i	0.988837	+	0.149001i	$0.0476059\pi$
$132$		0			0
$133$	−6688.05	+	37929.8i	−0.378091	+	2.14426i
$134$		0			0
$135$	18.5105	+	13.3478i	0.00101567	+	0.000732391i
$136$		0			0
$137$	6309.25	+	1112.49i	0.336153	+	0.0592728i	0.339177	−	0.940723i	$-0.389852\pi$
$137$	6309.25	+	1112.49i	0.336153	+	0.0592728i	−0.00302422	+	0.999995i	$0.500963\pi$
$138$		0			0
$139$	−21971.8	−	7997.08i	−1.13720	−	0.413906i	−0.296297	−	0.955096i	$-0.595752\pi$
$139$	−21971.8	−	7997.08i	−1.13720	−	0.413906i	−0.840901	+	0.541190i	$0.817974\pi$
$140$		0			0
$141$	−22652.5	−	2087.48i	−1.13941	−	0.104999i
$142$		0			0
$143$	7811.05	+	4509.71i	0.381977	+	0.220535i
$144$		0			0
$145$	−6.30457	−	10.9198i	−0.000299860	−	0.000519373i
$146$		0			0
$147$	3959.14	+	15054.0i	0.183217	+	0.696652i
$148$		0			0
$149$	−33711.6	+	5944.26i	−1.51847	+	0.267747i	−0.869832	−	0.493348i	$-0.835773\pi$
$149$	−33711.6	+	5944.26i	−1.51847	+	0.267747i	−0.648639	+	0.761096i	$0.724662\pi$
$150$		0			0
$151$	22011.9	+	18470.2i	0.965390	+	0.810059i	0.981822	−	0.189806i	$-0.0607860\pi$
$151$	22011.9	+	18470.2i	0.965390	+	0.810059i	−0.0164311	+	0.999865i	$0.505230\pi$
$152$		0			0
$153$	33620.7	+	12590.4i	1.43623	+	0.537845i
$154$		0			0
$155$	−16.7765	−	46.0930i	−0.000698292	−	0.00191854i
$156$		0			0
$157$	−5881.23	+	4934.94i	−0.238599	+	0.200208i	−0.754244	−	0.656594i	$-0.771997\pi$
$157$	−5881.23	+	4934.94i	−0.238599	+	0.200208i	0.515645	+	0.856802i	$0.327552\pi$
$158$		0			0
$159$	−19621.3	−	9260.33i	−0.776130	−	0.366296i
$160$		0			0
$161$			29788.3i			1.14920i
$162$		0			0
$163$	9213.77			0.346786			0.173393	−	0.984853i	$-0.444527\pi$
$163$	9213.77			0.346786			0.173393	+	0.984853i	$0.444527\pi$
$164$		0			0
$165$	−22.9233	+	48.5712i	−0.000841994	+	0.00178407i
$166$		0			0
$167$	−9340.18	−	11131.2i	−0.334905	−	0.399125i	0.572141	−	0.820155i	$-0.306113\pi$
$167$	−9340.18	−	11131.2i	−0.334905	−	0.399125i	−0.907047	+	0.421030i	$0.861669\pi$
$168$		0			0
$169$	24735.0	−	9002.80i	0.866041	−	0.315213i
$170$		0			0
$171$	−47879.7	+	7986.44i	−1.63742	+	0.273125i
$172$		0			0
$173$	14480.8	−	17257.6i	0.483839	−	0.576617i	−0.467800	−	0.883834i	$-0.654953\pi$
$173$	14480.8	−	17257.6i	0.483839	−	0.576617i	0.951639	+	0.307217i	$0.0993979\pi$
$174$		0			0
$175$	6975.14	+	39558.0i	0.227760	+	1.29169i
$176$		0			0
$177$	−2246.21	+	590.747i	−0.0716975	+	0.0188562i
$178$		0			0
$179$	35962.9	−	20763.2i	1.12240	−	0.648020i	0.180390	−	0.983595i	$-0.442264\pi$
$179$	35962.9	−	20763.2i	1.12240	−	0.648020i	0.942013	+	0.335576i	$0.108931\pi$
$180$		0			0
$181$	896.024	−	1551.96i	0.0273503	−	0.0473721i	−0.852026	−	0.523499i	$-0.824626\pi$
$181$	896.024	−	1551.96i	0.0273503	−	0.0473721i	0.879377	+	0.476127i	$0.157960\pi$
$182$		0			0
$183$	3242.73	−	35188.9i	0.0968298	−	1.05076i
$184$		0			0
$185$	11.4088	−	31.3454i	0.000333347	−	0.000915863i
$186$		0			0
$187$	−14671.8	+	83208.0i	−0.419566	+	2.37948i
$188$		0			0
$189$	−38748.4	+	26338.2i	−1.08475	+	0.737330i
$190$		0			0
$191$	55200.3	+	9733.31i	1.51313	+	0.266805i	0.867727	−	0.497041i	$-0.165580\pi$
$191$	55200.3	+	9733.31i	1.51313	+	0.266805i	0.645399	+	0.763846i	$0.276691\pi$
$192$		0			0
$193$	−39783.2	−	14479.9i	−1.06803	−	0.388733i	−0.252593	−	0.967573i	$-0.581283\pi$
$193$	−39783.2	−	14479.9i	−1.06803	−	0.388733i	−0.815441	+	0.578840i	$0.803506\pi$
$194$		0			0
$195$	−5.57764	−	12.1072i	−0.000146683	−	0.000318401i
$196$		0			0
$197$	51989.7	+	30016.3i	1.33963	+	0.773436i	0.986752	−	0.162233i	$-0.0518696\pi$
$197$	51989.7	+	30016.3i	1.33963	+	0.773436i	0.352878	+	0.935669i	$0.385203\pi$
$198$		0			0
$199$	29053.9	+	50322.8i	0.733665	+	1.27074i	0.955307	+	0.295616i	$0.0955250\pi$
$199$	29053.9	+	50322.8i	0.733665	+	1.27074i	−0.221642	+	0.975128i	$0.571142\pi$
$200$		0			0
$201$	35149.4	+	9592.76i	0.870013	+	0.237439i
$202$		0			0
$203$	25493.6	−	4495.21i	0.618641	−	0.109083i
$204$		0			0
$205$	−56.6741	−	47.5552i	−0.00134858	−	0.00113159i
$206$		0			0
$207$	−35396.0	+	12513.4i	−0.826064	+	0.292036i
$208$		0			0
$209$	−39072.5	−	107351.i	−0.894496	−	2.45761i
$210$		0			0
$211$	−7858.30	+	6593.90i	−0.176508	+	0.148108i	−0.726761	−	0.686890i	$-0.758975\pi$
$211$	−7858.30	+	6593.90i	−0.176508	+	0.148108i	0.550254	+	0.834998i	$0.314531\pi$
$212$		0			0
$213$	−1839.53	−	22208.8i	−0.0405460	−	0.489516i
$214$		0			0
$215$			54.8841i			0.00118732i
$216$		0			0
$217$	100703.			2.13857
$218$		0			0
$219$	7807.98	+	11261.5i	0.162798	+	0.234805i
$220$		0			0
$221$	−13479.4	−	16064.2i	−0.275986	−	0.328908i
$222$		0			0
$223$	−74762.5	+	27211.3i	−1.50340	+	0.547192i	−0.956938	−	0.290293i	$-0.906247\pi$
$223$	−74762.5	+	27211.3i	−1.50340	+	0.547192i	−0.546460	+	0.837485i	$0.684025\pi$
$224$		0			0
$225$	−44074.8	+	24905.7i	−0.870613	+	0.491965i
$226$		0			0
$227$	14091.0	−	16793.0i	0.273458	−	0.325895i	−0.611784	−	0.791025i	$-0.709548\pi$
$227$	14091.0	−	16793.0i	0.273458	−	0.325895i	0.885242	+	0.465130i	$0.153992\pi$
$228$		0			0
$229$	−4457.80	−	25281.5i	−0.0850061	−	0.482093i	−0.997355	−	0.0726804i	$-0.976845\pi$
$229$	−4457.80	−	25281.5i	−0.0850061	−	0.482093i	0.912349	−	0.409413i	$-0.134266\pi$
$230$		0			0
$231$	−77608.0	−	78329.4i	−1.45440	−	1.46791i
$232$		0			0
$233$	−42121.6	+	24318.9i	−0.775877	+	0.447953i	−0.834967	−	0.550300i	$-0.814513\pi$
$233$	−42121.6	+	24318.9i	−0.775877	+	0.447953i	0.0590903	+	0.998253i	$0.481180\pi$
$234$		0			0
$235$	−39.5630	+	68.5252i	−0.000716397	+	0.00124084i
$236$		0			0
$237$	57174.1	+	40429.2i	1.01789	+	0.719778i
$238$		0			0
$239$	−34021.3	+	93472.8i	−0.595601	+	1.63640i	0.164337	+	0.986404i	$0.447452\pi$
$239$	−34021.3	+	93472.8i	−0.595601	+	1.63640i	−0.759938	+	0.649996i	$0.774771\pi$
$240$		0			0
$241$	8476.64	−	48073.4i	0.145945	−	0.827696i	−0.820658	−	0.571419i	$-0.806393\pi$
$241$	8476.64	−	48073.4i	0.145945	−	0.827696i	0.966603	−	0.256277i	$-0.0824958\pi$
$242$		0			0
$243$	−47573.8	−	34978.8i	−0.805667	−	0.592369i
$244$		0			0
$245$	53.3202	+	9.40178i	0.000888299	+	0.000156631i
$246$		0			0
$247$	26643.8	+	9697.56i	0.436719	+	0.158953i
$248$		0			0
$249$	64568.6	−	91311.4i	1.04141	−	1.47274i
$250$		0			0
$251$	35448.9	+	20466.5i	0.562673	+	0.324859i	0.754217	−	0.656625i	$-0.228016\pi$
$251$	35448.9	+	20466.5i	0.562673	+	0.324859i	−0.191545	+	0.981484i	$0.561350\pi$
$252$		0			0
$253$	−44178.0	−	76518.5i	−0.690184	−	1.19543i
$254$		0			0
$255$	88.7066	−	87.8896i	0.00136419	−	0.00135163i
$256$		0			0
$257$	−432.210	+	76.2102i	−0.00654377	+	0.00115384i	−0.176919	−	0.984225i	$-0.556613\pi$
$257$	−432.210	+	76.2102i	−0.00654377	+	0.00115384i	0.170375	+	0.985379i	$0.445502\pi$
$258$		0			0
$259$	52461.0	+	44020.0i	0.782054	+	0.656221i
$260$		0			0
$261$	16050.8	+	28404.5i	0.235621	+	0.416971i
$262$		0			0
$263$	−26508.2	−	72830.8i	−0.383239	−	1.05294i	−0.969984	−	0.243168i	$-0.921813\pi$
$263$	−26508.2	−	72830.8i	−0.383239	−	1.05294i	0.586745	−	0.809772i	$-0.300409\pi$
$264$		0			0
$265$	−57.8117	+	48.5098i	−0.000823236	+	0.000690777i
$266$		0			0
$267$	−35135.3	+	24360.5i	−0.492857	+	0.341715i
$268$		0			0
$269$		−	46665.4i		−	0.644897i	−0.946587	−	0.322449i	$-0.895494\pi$
$269$		−	46665.4i		−	0.644897i	0.946587	−	0.322449i	$-0.104506\pi$
$270$		0			0
$271$	−64614.8			−0.879819			−0.439910	−	0.898042i	$-0.644989\pi$
$271$	−64614.8			−0.879819			−0.439910	+	0.898042i	$0.644989\pi$
$272$		0			0
$273$	27273.8	−	2259.06i	0.365949	−	0.0303112i
$274$		0			0
$275$	−76584.5	−	91269.8i	−1.01269	−	1.20687i
$276$		0			0
$277$	−23345.1	+	8496.92i	−0.304254	+	0.110739i	−0.489635	−	0.871927i	$-0.662870\pi$
$277$	−23345.1	+	8496.92i	−0.304254	+	0.110739i	0.185382	+	0.982667i	$0.440648\pi$
$278$		0			0
$279$	42303.3	+	119661.i	0.543458	+	1.53725i
$280$		0			0
$281$	−10338.9	+	12321.4i	−0.130937	+	0.156044i	−0.827529	−	0.561423i	$-0.810254\pi$
$281$	−10338.9	+	12321.4i	−0.130937	+	0.156044i	0.696592	+	0.717467i	$0.254699\pi$
$282$		0			0
$283$	−23776.7	−	134845.i	−0.296879	−	1.68368i	−0.659467	−	0.751734i	$-0.729218\pi$
$283$	−23776.7	−	134845.i	−0.296879	−	1.68368i	0.362588	−	0.931950i	$-0.381893\pi$
$284$		0			0
$285$	−44.4532	+	162.884i	−0.000547285	+	0.00200534i
$286$		0			0
$287$	131539.	−	75944.3i	1.59695	−	0.922001i
$288$		0			0
$289$	56461.5	−	97794.3i	0.676016	−	1.17089i
$290$		0			0
$291$	47204.8	−	21746.7i	0.557443	−	0.256807i
$292$		0			0
$293$	−20530.8	+	56407.8i	−0.239150	+	0.657058i	0.760818	+	0.648966i	$0.224798\pi$
$293$	−20530.8	+	56407.8i	−0.239150	+	0.657058i	−0.999967	+	0.00809254i	$0.997424\pi$
$294$		0			0
$295$	−1.40285	+	7.95594i	−1.61200e−5	+	9.14213e-5i
$296$		0			0
$297$	60473.6	−	125122.i	0.685572	−	1.41848i
$298$		0			0
$299$	21596.3	+	3808.00i	0.241566	+	0.0425946i
$300$		0			0
$301$	−105883.	−	38538.3i	−1.16867	−	0.425362i
$302$		0			0
$303$	−5017.88	−	462.409i	−0.0546556	−	0.00503664i
$304$		0			0
$305$	−106.448	−	61.4580i	−0.00114430	−	0.000660661i
$306$		0			0
$307$	57552.3	+	99683.5i	0.610641	+	1.05766i	0.991133	+	0.132877i	$0.0424215\pi$
$307$	57552.3	+	99683.5i	0.610641	+	1.05766i	−0.380492	+	0.924784i	$0.624245\pi$
$308$		0			0
$309$	−15114.8	−	57471.4i	−0.158302	−	0.601915i
$310$		0			0
$311$	−34145.6	+	6020.79i	−0.353032	+	0.0622491i	−0.347352	−	0.937735i	$-0.612919\pi$
$311$	−34145.6	+	6020.79i	−0.353032	+	0.0622491i	−0.00567992	+	0.999984i	$0.501808\pi$
$312$		0			0
$313$	89576.8	+	75163.9i	0.914339	+	0.767221i	0.972939	−	0.231060i	$-0.0742195\pi$
$313$	89576.8	+	75163.9i	0.914339	+	0.767221i	−0.0586009	+	0.998281i	$0.518664\pi$
$314$		0			0
$315$	26.8127	+	160.745i	0.000270221	+	0.00162001i
$316$		0			0
$317$	53640.5	+	147376.i	0.533795	+	1.46659i	0.854521	+	0.519418i	$0.173851\pi$
$317$	53640.5	+	147376.i	0.533795	+	1.46659i	−0.320726	+	0.947172i	$0.603927\pi$
$318$		0			0
$319$	−58819.8	+	49355.7i	−0.578019	+	0.485016i
$320$		0			0
$321$	−36360.3	−	17160.3i	−0.352872	−	0.166539i
$322$		0			0
$323$			265610.i			2.54589i
$324$		0			0
$325$	29570.9			0.279961
$326$		0			0
$327$	−37577.7	+	79621.9i	−0.351427	+	0.744624i
$328$		0			0
$329$	−104419.	−	124442.i	−0.964694	−	1.14968i
$330$		0			0
$331$	−2040.65	+	742.736i	−0.0186257	+	0.00677920i	−0.351316	−	0.936257i	$-0.614266\pi$
$331$	−2040.65	+	742.736i	−0.0186257	+	0.00677920i	0.332690	+	0.943036i	$0.392044\pi$
$332$		0			0
$333$	−30269.1	+	80828.8i	−0.272967	+	0.728916i
$334$		0			0
$335$	81.4614	−	97.0819i	0.000725876	−	0.000865065i
$336$		0			0
$337$	−10362.9	−	58770.8i	−0.0912474	−	0.517490i	−0.995833	−	0.0911974i	$-0.970931\pi$
$337$	−10362.9	−	58770.8i	−0.0912474	−	0.517490i	0.904585	−	0.426292i	$-0.140181\pi$
$338$		0			0
$339$	110233.	−	28990.9i	0.959207	−	0.252268i
$340$		0			0
$341$	−258681.	+	149349.i	−2.22462	+	1.28438i
$342$		0			0
$343$	21577.1	−	37372.6i	0.183402	−	0.317662i
$344$		0			0
$345$	−11.9829	+	130.034i	−0.000100676	+	0.00109249i
$346$		0			0
$347$	−31580.8	+	86767.6i	−0.262280	+	0.720607i	0.736733	+	0.676184i	$0.236367\pi$
$347$	−31580.8	+	86767.6i	−0.262280	+	0.720607i	−0.999013	+	0.0444237i	$0.985855\pi$
$348$		0			0
$349$	−4641.37	+	26322.5i	−0.0381062	+	0.216111i	−0.997915	−	0.0645444i	$-0.979441\pi$
$349$	−4641.37	+	26322.5i	−0.0381062	+	0.216111i	0.959809	+	0.280655i	$0.0905517\pi$
$350$		0			0
$351$	14141.5	+	31459.2i	0.114784	+	0.255349i
$352$		0			0
$353$	51899.9	+	9151.35i	0.416502	+	0.0734405i	0.377972	−	0.925817i	$-0.376621\pi$
$353$	51899.9	+	9151.35i	0.416502	+	0.0734405i	0.0385297	+	0.999257i	$0.487733\pi$
$354$		0			0
$355$	−72.8389	−	26.5112i	−0.000577972	−	0.000210364i
$356$		0			0
$357$	107270.	+	232848.i	0.841671	+	1.82699i
$358$		0			0
$359$	−31426.2	−	18143.9i	−0.243839	−	0.140781i	0.373101	−	0.927791i	$-0.378295\pi$
$359$	−31426.2	−	18143.9i	−0.243839	−	0.140781i	−0.616940	+	0.787010i	$0.711628\pi$
$360$		0			0
$361$	−114404.	−	198154.i	−0.877865	−	1.52051i
$362$		0			0
$363$	188403.	+	51417.7i	1.42980	+	0.390211i
$364$		0			0
$365$	46.9406	−	8.27690i	0.000352341	−	6.21272e-5i
$366$		0			0
$367$	−160886.	−	135000.i	−1.19450	−	1.00231i	−0.999770	−	0.0214507i	$-0.993172\pi$
$367$	−160886.	−	135000.i	−1.19450	−	1.00231i	−0.194733	−	0.980856i	$-0.562384\pi$
$368$		0			0
$369$	145498.	+	124399.i	1.06857	+	0.913620i
$370$		0			0
$371$	−52991.7	−	145594.i	−0.385000	−	1.05778i
$372$		0			0
$373$	10634.3	−	8923.25i	0.0764350	−	0.0641366i	−0.603770	−	0.797159i	$-0.706335\pi$
$373$	10634.3	−	8923.25i	0.0764350	−	0.0641366i	0.680205	+	0.733022i	$0.261891\pi$
$374$		0			0
$375$	29.0709	+	350.975i	0.000206726	+	0.00249583i
$376$		0			0
$377$		−	19057.3i		−	0.134084i
$378$		0			0
$379$	123341.			0.858677			0.429338	−	0.903144i	$-0.358747\pi$
$379$	123341.			0.858677			0.429338	+	0.903144i	$0.358747\pi$
$380$		0			0
$381$	114163.	+	164658.i	0.786459	+	1.13431i
$382$		0			0
$383$	158981.	+	189467.i	1.08380	+	1.29162i	0.953910	+	0.300093i	$0.0970176\pi$
$383$	158981.	+	189467.i	1.08380	+	1.29162i	0.129889	+	0.991529i	$0.458538\pi$
$384$		0			0
$385$	−360.406	+	131.177i	−0.00243148	+	0.000884987i
$386$		0			0
$387$	1313.94	−	142005.i	0.00877308	−	0.948160i
$388$		0			0
$389$	−45832.1	+	54620.6i	−0.302880	+	0.360958i	−0.895921	−	0.444214i	$-0.853483\pi$
$389$	−45832.1	+	54620.6i	−0.302880	+	0.360958i	0.593041	+	0.805173i	$0.297927\pi$
$390$		0			0
$391$	35672.4	+	202308.i	0.233334	+	1.32330i
$392$		0			0
$393$	43088.0	+	43488.5i	0.278979	+	0.281572i
$394$		0			0
$395$	210.934	−	121.783i	0.00135192	−	0.000780533i
$396$		0			0
$397$	35045.3	−	60700.3i	0.222356	−	0.385132i	−0.733167	−	0.680049i	$-0.761959\pi$
$397$	35045.3	−	60700.3i	0.222356	−	0.385132i	0.955523	+	0.294917i	$0.0952919\pi$
$398$		0			0
$399$	−283023.	−	200133.i	−1.77777	−	1.25711i
$400$		0			0
$401$	−51680.5	+	141991.i	−0.321394	+	0.883023i	0.668815	+	0.743429i	$0.266802\pi$
$401$	−51680.5	+	141991.i	−0.321394	+	0.883023i	−0.990209	+	0.139594i	$0.955420\pi$
$402$		0			0
$403$	12873.4	−	73009.0i	0.0792656	−	0.449538i
$404$		0			0
$405$	−179.743	+	99.3861i	−0.00109583	+	0.000605920i
$406$		0			0
$407$	−200043.	−	35273.0i	−1.20763	−	0.212938i
$408$		0			0
$409$	−35042.7	−	12754.5i	−0.209484	−	0.0762460i	0.235147	−	0.971960i	$-0.424443\pi$
$409$	−35042.7	−	12754.5i	−0.209484	−	0.0762460i	−0.444631	+	0.895714i	$0.646665\pi$
$410$		0			0
$411$	−33290.1	+	47078.1i	−0.197075	+	0.278699i
$412$		0			0
$413$	−14363.7	−	8292.86i	−0.0842102	−	0.0486188i
$414$		0			0
$415$	−194.496	−	336.877i	−0.00112931	−	0.00195603i
$416$		0			0
$417$	149488.	−	148112.i	0.859677	−	0.851759i
$418$		0			0
$419$	242548.	−	42767.8i	1.38156	−	0.243607i	0.567017	−	0.823706i	$-0.308098\pi$
$419$	242548.	−	42767.8i	1.38156	−	0.243607i	0.814545	+	0.580100i	$0.196986\pi$
$420$		0			0
$421$	−13777.8	−	11560.9i	−0.0777348	−	0.0652272i	0.603092	−	0.797671i	$-0.293935\pi$
$421$	−13777.8	−	11560.9i	−0.0777348	−	0.0652272i	−0.680827	+	0.732444i	$0.738379\pi$
$422$		0			0
$423$	104004.	−	176352.i	0.581261	−	0.985600i
$424$		0			0
$425$	94743.7	+	260306.i	0.524533	+	1.44114i
$426$		0			0
$427$	193311.	−	162207.i	1.06023	−	0.889640i
$428$		0			0
$429$	−66709.2	+	46251.8i	−0.362469	+	0.251313i
$430$		0			0
$431$		−	314703.i		−	1.69413i	−0.531490	−	0.847064i	$-0.678368\pi$
$431$		−	314703.i		−	1.69413i	0.531490	−	0.847064i	$-0.321632\pi$
$432$		0			0
$433$	−208556.			−1.11236			−0.556182	−	0.831060i	$-0.687734\pi$
$433$	−208556.			−1.11236			−0.556182	+	0.831060i	$0.687734\pi$
$434$		0			0
$435$	113.095	−	9.36752i	0.000597674	−	4.95047e-5i
$436$		0			0
$437$	−178540.	−	212776.i	−0.934916	−	1.11419i
$438$		0			0
$439$	175311.	−	63807.9i	0.909661	−	0.331090i	0.155544	−	0.987829i	$-0.450287\pi$
$439$	175311.	−	63807.9i	0.909661	−	0.331090i	0.754118	+	0.656739i	$0.228065\pi$
$440$		0			0
$441$	−137734.	−	25602.3i	−0.708211	−	0.131644i
$442$		0			0
$443$	44298.3	−	52792.7i	0.225725	−	0.269009i	−0.641281	−	0.767306i	$-0.721597\pi$
$443$	44298.3	−	52792.7i	0.225725	−	0.269009i	0.867006	+	0.498297i	$0.166041\pi$
$444$		0			0
$445$	25.8236	+	146.453i	0.000130406	+	0.000739567i
$446$		0			0
$447$	81114.0	−	297215.i	0.405958	−	1.48750i
$448$		0			0
$449$	174409.	−	100695.i	0.865117	−	0.499476i	−0.000605245	−	1.00000i	$-0.500193\pi$
$449$	174409.	−	100695.i	0.865117	−	0.499476i	0.865723	+	0.500524i	$0.166859\pi$
$450$		0			0
$451$	−225261.	+	390163.i	−1.10747	+	1.91820i
$452$		0			0
$453$	−234883.	+	108208.i	−1.14461	+	0.527306i
$454$		0			0
$455$	32.5574	−	89.4506i	0.000157263	−	0.000432077i
$456$		0			0
$457$	12388.7	−	70259.5i	0.0593187	−	0.336413i	−0.940677	−	0.339303i	$-0.889809\pi$
$457$	12388.7	−	70259.5i	0.0593187	−	0.336413i	0.999996	+	0.00288993i	$0.000919895\pi$
$458$		0			0
$459$	−231620.	+	225279.i	−1.09939	+	1.06929i
$460$		0			0
$461$	−284665.	−	50194.1i	−1.33947	−	0.236184i	−0.542423	−	0.840105i	$-0.682493\pi$
$461$	−284665.	−	50194.1i	−1.33947	−	0.236184i	−0.797044	+	0.603921i	$0.793604\pi$
$462$		0			0
$463$	68653.9	+	24988.0i	0.320260	+	0.116565i	0.497148	−	0.867666i	$-0.334381\pi$
$463$	68653.9	+	24988.0i	0.320260	+	0.116565i	−0.176888	+	0.984231i	$0.556603\pi$
$464$		0			0
$465$	439.597	+	40.5099i	0.00203306	+	0.000187351i
$466$		0			0
$467$	−79976.9	−	46174.7i	−0.366717	−	0.211724i	0.305306	−	0.952254i	$-0.401241\pi$
$467$	−79976.9	−	46174.7i	−0.366717	−	0.211724i	−0.672023	+	0.740530i	$0.734574\pi$
$468$		0			0
$469$	130091.	+	225325.i	0.591429	+	1.02439i
$470$		0			0
$471$	−17574.6	−	66824.2i	−0.0792214	−	0.301226i
$472$		0			0
$473$	329141.	−	58036.5i	1.47116	−	0.259405i
$474$		0			0
$475$	−286919.	−	240753.i	−1.27166	−	1.06705i
$476$		0			0
$477$	150741.	−	124128.i	0.662514	−	0.545550i
$478$		0			0
$479$	94771.9	+	260384.i	0.413056	+	1.13486i	0.955557	+	0.294806i	$0.0952551\pi$
$479$	94771.9	+	260384.i	0.413056	+	1.13486i	−0.542501	+	0.840055i	$0.682523\pi$
$480$		0			0
$481$	38620.5	−	32406.4i	0.166927	−	0.140069i
$482$		0			0
$483$	−242450.	−	114425.i	−1.03927	−	0.490484i
$484$		0			0
$485$		−	180.778i		−	0.000768533i
$486$		0			0
$487$	−231576.			−0.976416			−0.488208	−	0.872727i	$-0.662349\pi$
$487$	−231576.			−0.976416			−0.488208	+	0.872727i	$0.662349\pi$
$488$		0			0
$489$	−35392.5	+	74991.7i	−0.148011	+	0.313614i
$490$		0			0
$491$	−271099.	−	323083.i	−1.12451	−	1.34014i	−0.933509	−	0.358553i	$-0.883270\pi$
$491$	−271099.	−	323083.i	−1.12451	−	1.34014i	−0.191003	−	0.981589i	$-0.561174\pi$
$492$		0			0
$493$	167757.	−	61058.6i	0.690220	−	0.251219i
$494$		0			0
$495$	−307.271	−	373.149i	−0.00125404	−	0.00152290i
$496$		0			0
$497$	102291.	−	121906.i	0.414120	−	0.493529i
$498$		0			0
$499$	−12436.4	−	70530.2i	−0.0499451	−	0.283253i	0.949598	−	0.313470i	$-0.101491\pi$
$499$	−12436.4	−	70530.2i	−0.0499451	−	0.283253i	−0.999543	+	0.0302167i	$0.990380\pi$
$500$		0			0
$501$	126476.	−	33262.7i	0.503886	−	0.132520i
$502$		0			0
$503$	259012.	−	149541.i	1.02373	−	0.591048i	0.108545	−	0.994092i	$-0.465381\pi$
$503$	259012.	−	149541.i	1.02373	−	0.591048i	0.915181	+	0.403043i	$0.132048\pi$
$504$		0			0
$505$	−8.76382	+	15.1794i	−3.43645e−5	+	5.95211e-5i
$506$		0			0
$507$	−21738.9	+	235902.i	−0.0845712	+	0.917733i
$508$		0			0
$509$	−108249.	+	297413.i	−0.417820	+	1.14795i	0.535115	+	0.844779i	$0.320268\pi$
$509$	−108249.	+	297413.i	−0.417820	+	1.14795i	−0.952936	+	0.303173i	$0.901954\pi$
$510$		0			0
$511$	−16992.7	+	96370.3i	−0.0650759	+	0.369064i
$512$		0			0
$513$	118916.	−	420375.i	0.451862	−	1.59736i
$514$		0			0
$515$	−203.560	−	35.8932i	−0.000767500	−	0.000135331i
$516$		0			0
$517$	452783.	+	164800.i	1.69398	+	0.616559i
$518$		0			0
$519$	84836.2	+	184151.i	0.314954	+	0.683660i
$520$		0			0
$521$	−175042.	−	101061.i	−0.644862	−	0.372311i	0.141623	−	0.989921i	$-0.454768\pi$
$521$	−175042.	−	101061.i	−0.644862	−	0.372311i	−0.786485	+	0.617609i	$0.788101\pi$
$522$		0			0
$523$	−53298.1	−	92314.9i	−0.194853	−	0.337496i	0.751999	−	0.659164i	$-0.229090\pi$
$523$	−53298.1	−	92314.9i	−0.194853	−	0.337496i	−0.946852	+	0.321668i	$0.895756\pi$
$524$		0			0
$525$	−348759.	−	95181.2i	−1.26534	−	0.345329i
$526$		0			0
$527$	683928.	−	120595.i	2.46257	−	0.434218i
$528$		0			0
$529$	49805.3	+	41791.6i	0.177977	+	0.149341i
$530$		0			0
$531$	3820.14	−	20551.3i	0.0135485	−	0.0728871i
$532$		0			0
$533$	−38243.5	−	105073.i	−0.134618	−	0.369860i
$534$		0			0
$535$	−107.131	+	89.8936i	−0.000374290	+	0.000314066i
$536$		0			0
$537$	30850.6	+	372462.i	0.106983	+	1.29162i
$538$		0			0
$539$		−	329704.i		−	1.13487i
$540$		0			0
$541$	−53821.8			−0.183892			−0.0919462	−	0.995764i	$-0.529309\pi$
$541$	−53821.8			−0.183892			−0.0919462	+	0.995764i	$0.529309\pi$
$542$		0			0
$543$	9189.67	+	13254.3i	0.0311674	+	0.0449528i
$544$		0			0
$545$	196.849	+	234.596i	0.000662736	+	0.000789818i
$546$		0			0
$547$	−35772.0	+	13019.9i	−0.119555	+	0.0435145i	−0.401105	−	0.916032i	$-0.631374\pi$
$547$	−35772.0	+	13019.9i	−0.119555	+	0.0435145i	0.281550	+	0.959547i	$0.409152\pi$
$548$		0			0
$549$	273949.	+	161562.i	0.908919	+	0.536038i
$550$		0			0
$551$	−155156.	+	184908.i	−0.511053	+	0.609049i
$552$		0			0
$553$	86832.0	+	492449.i	0.283942	+	1.61032i
$554$		0			0
$555$	211.299	+	213.263i	0.000685979	+	0.000692356i
$556$		0			0
$557$	−369490.	+	213325.i	−1.19095	+	0.687594i	−0.958522	−	0.285020i	$-0.908000\pi$
$557$	−369490.	+	213325.i	−1.19095	+	0.687594i	−0.232426	+	0.972614i	$0.574666\pi$
$558$		0			0
$559$	−41475.5	+	71837.7i	−0.132730	+	0.229895i
$560$		0			0
$561$	−620879.	−	439038.i	−1.97279	−	1.39501i
$562$		0			0
$563$	−77677.9	+	213418.i	−0.245065	+	0.673310i	0.754785	+	0.655972i	$0.227741\pi$
$563$	−77677.9	+	213418.i	−0.245065	+	0.673310i	−0.999850	+	0.0173374i	$0.994481\pi$
$564$		0			0
$565$	68.8448	−	390.438i	0.000215662	−	0.00122308i
$566$		0			0
$567$	−65525.8	−	416548.i	−0.203820	−	1.29568i
$568$		0			0
$569$	508866.	+	89726.9i	1.57173	+	0.277139i	0.890520	−	0.454945i	$-0.150341\pi$
$569$	508866.	+	89726.9i	1.57173	+	0.277139i	0.681215	+	0.732084i	$0.261452\pi$
$570$		0			0
$571$	470667.	+	171309.i	1.44358	+	0.525421i	0.940791	−	0.338987i	$-0.110084\pi$
$571$	470667.	+	171309.i	1.44358	+	0.525421i	0.502791	+	0.864408i	$0.332306\pi$
$572$		0			0
$573$	−291259.	+	411892.i	−0.887095	+	1.25451i
$574$		0			0
$575$	−250872.	−	144841.i	−0.758781	−	0.438082i
$576$		0			0
$577$	139405.	+	241457.i	0.418723	+	0.725250i	0.995811	−	0.0914318i	$-0.0291444\pi$
$577$	139405.	+	241457.i	0.418723	+	0.725250i	−0.577088	+	0.816682i	$0.695811\pi$
$578$		0			0
$579$	270671.	−	268178.i	0.807392	−	0.799956i
$580$		0			0
$581$	786479.	−	138677.i	2.32989	−	0.410822i
$582$		0			0
$583$	352047.	+	295403.i	1.03577	+	0.869115i
$584$		0			0
$585$	119.967	+	1.11002i	0.000350549	+	3.24354e-6i
$586$		0			0
$587$	107700.	+	295904.i	0.312565	+	0.858764i	0.992137	+	0.125156i	$0.0399431\pi$
$587$	107700.	+	295904.i	0.312565	+	0.858764i	−0.679572	+	0.733608i	$0.737835\pi$
$588$		0			0
$589$	−719315.	+	603577.i	−2.07343	+	1.73981i
$590$		0			0
$591$	−444011.	+	307849.i	−1.27121	+	0.881378i
$592$		0			0
$593$			36185.2i			0.102902i	0.998676	+	0.0514508i	$0.0163845\pi$
$593$			36185.2i			0.102902i	−0.998676	+	0.0514508i	$0.983615\pi$
$594$		0			0
$595$	891.727			0.00251883
$596$		0			0
$597$	−521185.	+	43169.1i	−1.46232	+	0.121122i
$598$		0			0
$599$	65094.9	+	77577.1i	0.181424	+	0.216212i	0.849090	−	0.528249i	$-0.177151\pi$
$599$	65094.9	+	77577.1i	0.181424	+	0.216212i	−0.667666	+	0.744461i	$0.732707\pi$
$600$		0			0
$601$	−267323.	+	97297.5i	−0.740094	+	0.269372i	−0.684431	−	0.729077i	$-0.739949\pi$
$601$	−267323.	+	97297.5i	−0.740094	+	0.269372i	−0.0556627	+	0.998450i	$0.517727\pi$
$602$		0			0
$603$	−213094.	+	249236.i	−0.586053	+	0.685450i
$604$		0			0
$605$	436.638	−	520.364i	0.00119292	−	0.00142166i
$606$		0			0
$607$	71270.1	+	404193.i	0.193433	+	1.09701i	0.914633	+	0.404285i	$0.132480\pi$
$607$	71270.1	+	404193.i	0.193433	+	1.09701i	−0.721201	+	0.692726i	$0.756409\pi$
$608$		0			0
$609$	−61340.6	+	224762.i	−0.165392	+	0.606021i
$610$		0			0
$611$	−103568.	+	59795.1i	−0.277424	+	0.160171i
$612$		0			0
$613$	−163074.	+	282453.i	−0.433974	+	0.751666i	−0.997211	−	0.0746297i	$-0.976223\pi$
$613$	−163074.	+	282453.i	−0.433974	+	0.751666i	0.563237	+	0.826295i	$0.309556\pi$
$614$		0			0
$615$	604.756	−	278.604i	0.00159893	−	0.000736608i
$616$		0			0
$617$	249596.	−	685760.i	0.655643	−	1.80137i	0.0598795	−	0.998206i	$-0.480928\pi$
$617$	249596.	−	685760.i	0.655643	−	1.80137i	0.595764	−	0.803160i	$-0.296849\pi$
$618$		0			0
$619$	119433.	−	677338.i	0.311705	−	1.76776i	−0.278427	−	0.960457i	$-0.589813\pi$
$619$	119433.	−	677338.i	0.311705	−	1.76776i	0.590131	−	0.807307i	$-0.299076\pi$
$620$		0			0
$621$	34117.3	−	336159.i	0.0884690	−	0.871688i
$622$		0			0
$623$	−300671.	−	53016.4i	−0.774668	−	0.136595i
$624$		0			0
$625$	−367066.	−	133601.i	−0.939688	−	0.342019i
$626$		0			0
$627$	1.02383e6	+	94347.8i	2.60430	+	0.239992i
$628$		0			0
$629$	409005.	+	236139.i	1.03378	+	0.596852i
$630$		0			0
$631$	−43866.1	−	75978.2i	−0.110172	−	0.190823i	0.805668	−	0.592368i	$-0.201807\pi$
$631$	−43866.1	−	75978.2i	−0.110172	−	0.190823i	−0.915839	+	0.401545i	$0.868473\pi$
$632$		0			0
$633$	−23482.5	−	89288.3i	−0.0586054	−	0.222837i
$634$		0			0
$635$	686.336	−	121.020i	0.00170212	−	0.000300129i
$636$		0			0
$637$	62685.9	+	52599.7i	0.154487	+	0.129630i
$638$		0			0
$639$	187826.	+	70337.8i	0.459996	+	0.172261i
$640$		0			0
$641$	−95751.6	−	263075.i	−0.233040	−	0.640272i	0.766959	−	0.641696i	$-0.221769\pi$
$641$	−95751.6	−	263075.i	−0.233040	−	0.640272i	−0.999999	+	0.00142423i	$0.999547\pi$
$642$		0			0
$643$	395316.	−	331709.i	0.956142	−	0.802298i	−0.0241793	−	0.999708i	$-0.507697\pi$
$643$	395316.	−	331709.i	0.956142	−	0.802298i	0.980321	+	0.197409i	$0.0632528\pi$
$644$		0			0
$645$	−446.706	−	210.824i	−0.00107375	−	0.000506758i
$646$		0			0
$647$		−	230598.i		−	0.550868i	−0.961320	−	0.275434i	$-0.911178\pi$
$647$		−	230598.i		−	0.550868i	0.961320	−	0.275434i	$-0.0888215\pi$
$648$		0			0
$649$	49195.4			0.116798
$650$		0			0
$651$	−386827.	+	819632.i	−0.912756	+	1.93400i
$652$		0			0
$653$	−344862.	−	410991.i	−0.808760	−	0.963842i	0.191083	−	0.981574i	$-0.438800\pi$
$653$	−344862.	−	410991.i	−0.808760	−	0.963842i	−0.999843	+	0.0177317i	$0.994356\pi$
$654$		0			0
$655$	200.098	−	72.8296i	0.000466401	−	0.000169756i
$656$		0			0
$657$	−121651.	+	20291.6i	−0.281828	+	0.0470094i
$658$		0			0
$659$	−432211.	+	515089.i	−0.995234	+	1.18607i	−0.0127133	+	0.999919i	$0.504047\pi$
$659$	−432211.	+	515089.i	−0.995234	+	1.18607i	−0.982521	+	0.186154i	$0.940398\pi$
$660$		0			0
$661$	−90978.6	−	515965.i	−0.208227	−	1.18091i	−0.892281	−	0.451481i	$-0.850896\pi$
$661$	−90978.6	−	515965.i	−0.208227	−	1.18091i	0.684054	−	0.729431i	$-0.260215\pi$
$662$		0			0
$663$	182526.	−	48003.7i	0.415238	−	0.109206i
$664$		0			0
$665$	−1044.16	+	602.849i	−0.00236116	+	0.00136322i
$666$		0			0
$667$	−93344.4	+	161677.i	−0.209815	+	0.363410i
$668$		0			0
$669$	65706.8	−	713024.i	0.146811	−	1.59313i
$670$		0			0
$671$	−256003.	+	703362.i	−0.568590	+	1.56219i
$672$		0			0
$673$	2915.44	−	16534.3i	0.00643685	−	0.0365052i	−0.981420	−	0.191871i	$-0.938544\pi$
$673$	2915.44	−	16534.3i	0.00643685	−	0.0365052i	0.987857	+	0.155366i	$0.0496556\pi$
$674$		0			0
$675$	−33407.2	−	454398.i	−0.0733218	−	0.997307i
$676$		0			0
$677$	886585.	+	156329.i	1.93438	+	0.341084i	0.999889	−	0.0148909i	$-0.00474009\pi$
$677$	886585.	+	156329.i	1.93438	+	0.341084i	0.934495	+	0.355975i	$0.115851\pi$
$678$		0			0
$679$	348759.	+	126938.i	0.756461	+	0.275329i
$680$		0			0
$681$	82552.6	+	179194.i	0.178007	+	0.386394i
$682$		0			0
$683$	−515809.	−	297802.i	−1.10573	−	0.638391i	−0.168007	−	0.985786i	$-0.553733\pi$
$683$	−515809.	−	297802.i	−1.10573	−	0.638391i	−0.937719	+	0.347395i	$0.887066\pi$
$684$		0			0
$685$	100.278	+	173.686i	0.000213710	+	0.000370156i
$686$		0			0
$687$	222892.	+	60830.2i	0.472259	+	0.128886i
$688$		0			0
$689$	−112328.	+	19806.5i	−0.236620	+	0.0417224i
$690$		0			0
$691$	−285792.	−	239808.i	−0.598541	−	0.502236i	0.292435	−	0.956285i	$-0.405534\pi$
$691$	−285792.	−	239808.i	−0.598541	−	0.502236i	−0.890976	+	0.454050i	$0.849979\pi$
$692$		0			0
$693$	935642.	−	330775.i	1.94824	−	0.688757i
$694$		0			0
$695$	−250.346	−	687.820i	−0.000518288	−	0.00142398i
$696$		0			0
$697$	802407.	−	673300.i	1.65169	−	1.38594i
$698$		0			0
$699$	−36133.8	−	436246.i	−0.0739536	−	0.892848i
$700$		0			0
$701$		−	182277.i		−	0.370934i	−0.982650	−	0.185467i	$-0.940620\pi$
$701$		−	182277.i		−	0.370934i	0.982650	−	0.185467i	$-0.0593798\pi$
$702$		0			0
$703$	−638564.			−1.29209
$704$		0			0
$705$	−405.761	−	585.230i	−0.000816379	−	0.00117747i
$706$		0			0
$707$	−23130.5	−	27565.9i	−0.0462750	−	0.0551484i
$708$		0			0
$709$	219638.	−	79941.7i	0.436933	−	0.159031i	−0.114182	−	0.993460i	$-0.536425\pi$
$709$	219638.	−	79941.7i	0.436933	−	0.159031i	0.551115	+	0.834429i	$0.314202\pi$
$710$		0			0
$711$	−548677.	+	310046.i	−1.08537	+	0.613320i
$712$		0			0
$713$	−466820.	+	556334.i	−0.918270	+	1.09435i
$714$		0			0
$715$	49.0296	+	278.061i	9.59061e−5	+	0.000543910i
$716$		0			0
$717$	−630099.	−	635956.i	−1.22566	−	1.23705i
$718$		0			0
$719$	355326.	−	205148.i	0.687336	−	0.396834i	−0.115277	−	0.993333i	$-0.536776\pi$
$719$	355326.	−	205148.i	0.687336	−	0.396834i	0.802613	+	0.596500i	$0.203442\pi$
$720$		0			0
$721$	212181.	−	367508.i	0.408164	−	0.706962i
$722$		0			0
$723$	358713.	+	253654.i	0.686231	+	0.485250i
$724$		0			0
$725$	−86100.8	+	236560.i	−0.163806	+	0.450054i
$726$		0			0
$727$	167973.	−	952624.i	0.317813	−	1.80241i	−0.238183	−	0.971220i	$-0.576552\pi$
$727$	167973.	−	952624.i	0.317813	−	1.80241i	0.555996	−	0.831185i	$-0.312337\pi$
$728$		0			0
$729$	467439.	−	252845.i	0.879569	−	0.475772i
$730$		0			0
$731$	−765258.	−	134936.i	−1.43210	−	0.252518i
$732$		0			0
$733$	−165250.	−	60146.2i	−0.307563	−	0.111944i	0.183628	−	0.982996i	$-0.441216\pi$
$733$	−165250.	−	60146.2i	−0.307563	−	0.111944i	−0.491191	+	0.871052i	$0.663438\pi$
$734$		0			0
$735$	−281.339	+	397.863i	−0.000520780	+	0.000736476i
$736$		0			0
$737$	−668343.	−	385868.i	−1.23045	−	0.710402i
$738$		0			0
$739$	−366610.	−	634988.i	−0.671299	−	1.16272i	−0.977536	−	0.210769i	$-0.932403\pi$
$739$	−366610.	−	634988.i	−0.671299	−	1.16272i	0.306237	−	0.951955i	$-0.400930\pi$
$740$		0			0
$741$	−181275.	+	179605.i	−0.330143	+	0.327102i
$742$		0			0
$743$	730405.	−	128790.i	1.32308	−	0.233295i	0.532904	−	0.846176i	$-0.321101\pi$
$743$	730405.	−	128790.i	1.32308	−	0.233295i	0.790175	+	0.612881i	$0.209989\pi$
$744$		0			0
$745$	−820.901	−	688.818i	−0.00147903	−	0.00124106i
$746$		0			0
$747$	495167.	+	876280.i	0.887382	+	1.57037i
$748$		0			0
$749$	−98199.0	−	269800.i	−0.175043	−	0.480925i
$750$		0			0
$751$	−72745.4	+	61040.7i	−0.128981	+	0.108228i	−0.704996	−	0.709211i	$-0.749051\pi$
$751$	−72745.4	+	61040.7i	−0.128981	+	0.108228i	0.576015	+	0.817439i	$0.304607\pi$
$752$		0			0
$753$	−302747.	+	209905.i	−0.533936	+	0.370197i
$754$		0			0
$755$			899.523i			0.00157804i
$756$		0			0
$757$	−499006.			−0.870791			−0.435396	−	0.900239i	$-0.643391\pi$
$757$	−499006.			−0.870791			−0.435396	+	0.900239i	$0.643391\pi$
$758$		0			0
$759$	792490.	−	65641.1i	1.37566	−	0.113944i
$760$		0			0
$761$	−314122.	−	374356.i	−0.542412	−	0.646421i	0.423315	−	0.905983i	$-0.360866\pi$
$761$	−314122.	−	374356.i	−0.542412	−	0.646421i	−0.965727	+	0.259561i	$0.916422\pi$
$762$		0			0
$763$	−590807.	+	215036.i	−1.01484	+	0.369371i
$764$		0			0
$765$	374.596	+	1059.60i	0.000640089	+	0.00181058i
$766$		0			0
$767$	−7848.44	+	9353.41i	−0.0133411	+	0.0158993i
$768$		0			0
$769$	96572.4	+	547689.i	0.163305	+	0.926150i	0.950795	+	0.309822i	$0.100269\pi$
$769$	96572.4	+	547689.i	0.163305	+	0.926150i	−0.787489	+	0.616328i	$0.788619\pi$
$770$		0			0
$771$	1039.95	−	3810.53i	0.00174945	−	0.00641028i
$772$		0			0
$773$	297119.	−	171542.i	0.497247	−	0.287086i	−0.230329	−	0.973113i	$-0.573980\pi$
$773$	297119.	−	171542.i	0.497247	−	0.287086i	0.727576	+	0.686027i	$0.240647\pi$
$774$		0			0
$775$	−489654.	+	848105.i	−0.815240	+	1.41204i
$776$		0			0
$777$	−559798.	+	257892.i	−0.927235	+	0.427165i
$778$		0			0
$779$	−484394.	+	1.33086e6i	−0.798222	+	2.19310i
$780$		0			0
$781$	−81965.7	+	464851.i	−0.134379	+	0.762099i
$782$		0			0
$783$	−292842.	+	21529.7i	−0.477650	+	0.0351167i
$784$		0			0
$785$	−236.687	−	41.7344i	−0.000384092	−	6.77258e-5i
$786$		0			0
$787$	−191036.	−	69531.5i	−0.308437	−	0.112262i	0.183164	−	0.983082i	$-0.441366\pi$
$787$	−191036.	−	69531.5i	−0.308437	−	0.112262i	−0.491601	+	0.870820i	$0.663588\pi$
$788$		0			0
$789$	694601.	+	64009.1i	1.11579	+	0.102822i
$790$		0			0
$791$	704898.	+	406973.i	1.12661	+	0.650448i
$792$		0			0
$793$	−92886.8	−	160885.i	−0.147709	−	0.255840i
$794$		0			0
$795$	−172.756	−	656.873i	−0.000273337	−	0.00103932i
$796$		0			0
$797$	750642.	−	132358.i	1.18172	−	0.208370i	0.451940	−	0.892048i	$-0.350732\pi$
$797$	750642.	−	132358.i	1.18172	−	0.208370i	0.729783	+	0.683678i	$0.239621\pi$
$798$		0			0
$799$	−858191.	−	720108.i	−1.34428	−	1.12799i
$800$		0			0
$801$	−63308.8	−	379544.i	−0.0986732	−	0.591558i
$802$		0			0
$803$	−99273.6	−	272752.i	−0.153958	−	0.422997i
$804$		0			0
$805$	−714.346	+	599.408i	−0.00110234	+	0.000924976i
$806$		0			0
$807$	379814.	+	179254.i	0.583208	+	0.275246i
$808$		0			0
$809$			524519.i			0.801428i	0.916203	+	0.400714i	$0.131238\pi$
$809$			524519.i			0.801428i	−0.916203	+	0.400714i	$0.868762\pi$
$810$		0			0
$811$	1.30005e6			1.97660			0.988298	−	0.152532i	$-0.0487428\pi$
$811$	1.30005e6			1.97660			0.988298	+	0.152532i	$0.0487428\pi$
$812$		0			0
$813$	248202.	−	525905.i	0.375513	−	0.795658i
$814$		0			0
$815$	185.402	+	220.953i	0.000279125	+	0.000332648i
$816$		0			0
$817$	987298.	−	359347.i	1.47912	−	0.538357i
$818$		0			0
$819$	−86379.2	+	230662.i	−0.128778	+	0.343881i
$820$		0			0
$821$	38691.2	−	46110.3i	0.0574018	−	0.0684088i	−0.736580	−	0.676351i	$-0.763560\pi$
$821$	38691.2	−	46110.3i	0.0574018	−	0.0684088i	0.793981	+	0.607942i	$0.208005\pi$
$822$		0			0
$823$	−144022.	−	816790.i	−0.212632	−	1.20590i	−0.884968	−	0.465652i	$-0.845820\pi$
$823$	−144022.	−	816790.i	−0.212632	−	1.20590i	0.672336	−	0.740246i	$-0.265291\pi$
$824$		0			0
$825$	1.03703e6	−	272737.i	1.52365	−	0.400715i
$826$		0			0
$827$	−1.00315e6	+	579170.i	−1.46675	+	0.846827i	−0.999308	−	0.0371972i	$-0.988157\pi$
$827$	−1.00315e6	+	579170.i	−1.46675	+	0.846827i	−0.467440	+	0.884025i	$0.654824\pi$
$828$		0			0
$829$	565538.	−	979540.i	0.822910	−	1.42532i	−0.0805960	−	0.996747i	$-0.525682\pi$
$829$	565538.	−	979540.i	0.822910	−	1.42532i	0.903506	−	0.428575i	$-0.140984\pi$
$830$		0			0
$831$	20517.4	−	222647.i	0.0297112	−	0.322414i
$832$		0			0
$833$	−262181.	+	720338.i	−0.377844	+	1.03812i
$834$		0			0
$835$	78.9891	−	447.969i	0.000113291	−	0.000642503i
$836$		0			0
$837$	−1.13643e6	−	115338.i	−1.62215	−	0.164634i
$838$		0			0
$839$	−736507.	−	129866.i	−1.04629	−	0.184490i	−0.376025	−	0.926609i	$-0.622709\pi$
$839$	−736507.	−	129866.i	−1.04629	−	0.184490i	−0.670267	+	0.742120i	$0.733820\pi$
$840$		0			0
$841$	−512173.	−	186416.i	−0.724144	−	0.263567i
$842$		0			0
$843$	−60570.7	−	131479.i	−0.0852330	−	0.185013i
$844$		0			0
$845$	713.618	+	412.008i	0.000999430	+	0.000577021i
$846$		0			0
$847$	697297.	+	1.20775e6i	0.971966	+	1.68349i
$848$		0			0
$849$	1.18884e6	+	324452.i	1.64934	+	0.450127i
$850$		0			0
$851$	−486376.	+	85761.3i	−0.671604	+	0.118422i
$852$		0			0
$853$	20927.0	+	17559.8i	0.0287613	+	0.0241336i	0.657055	−	0.753843i	$-0.271802\pi$
$853$	20927.0	+	17559.8i	0.0287613	+	0.0241336i	−0.628294	+	0.777976i	$0.716246\pi$
$854$		0			0
$855$	−1154.97	−	987.487i	−0.00157993	−	0.00135083i
$856$		0			0
$857$	274182.	+	753309.i	0.373316	+	1.02568i	0.974071	+	0.226245i	$0.0726449\pi$
$857$	274182.	+	753309.i	0.373316	+	1.02568i	−0.600754	+	0.799434i	$0.705133\pi$
$858$		0			0
$859$	68325.2	−	57331.6i	0.0925965	−	0.0776977i	−0.595314	−	0.803493i	$-0.702972\pi$
$859$	68325.2	−	57331.6i	0.0925965	−	0.0776977i	0.687911	+	0.725795i	$0.258528\pi$
$860$		0			0
$861$	112840.	+	1.36233e6i	0.152215	+	1.83771i
$862$		0			0
$863$			822294.i			1.10409i	0.833813	+	0.552047i	$0.186153\pi$
$863$			822294.i			1.10409i	−0.833813	+	0.552047i	$0.813847\pi$
$864$		0			0
$865$	705.236			0.000942546
$866$		0			0
$867$	579073.	+	835198.i	0.770362	+	1.11110i
$868$		0			0
$869$	−953383.	−	1.13620e6i	−1.26249	−	1.50458i
$870$		0			0
$871$	179989.	−	65510.6i	0.237252	−	0.0863526i
$872$		0			0
$873$	−4327.87	+	467738.i	−0.00567865	+	0.613726i
$874$		0			0
$875$	−1616.55	+	1926.53i	−0.00211142	+	0.00251629i
$876$		0			0
$877$	−1480.55	−	8396.64i	−0.00192498	−	0.0109171i	0.983830	−	0.179104i	$-0.0573199\pi$
$877$	−1480.55	−	8396.64i	−0.00192498	−	0.0109171i	−0.985755	+	0.168187i	$0.946209\pi$
$878$		0			0
$879$	−380244.	−	383779.i	−0.492135	−	0.496710i
$880$		0			0
$881$	−442659.	+	255570.i	−0.570319	+	0.329274i	−0.757277	−	0.653094i	$-0.773471\pi$
$881$	−442659.	+	255570.i	−0.570319	+	0.329274i	0.186958	+	0.982368i	$0.440137\pi$
$882$		0			0
$883$	590245.	−	1.02233e6i	0.757026	−	1.31121i	−0.187335	−	0.982296i	$-0.559985\pi$
$883$	590245.	−	1.02233e6i	0.757026	−	1.31121i	0.944361	−	0.328912i	$-0.106682\pi$
$884$		0			0
$885$	−59.3654	−	41.9787i	−7.57961e−5	−	5.35973e-5i
$886$		0			0
$887$	16612.2	−	45641.5i	0.0211144	−	0.0580113i	−0.928688	−	0.370863i	$-0.879062\pi$
$887$	16612.2	−	45641.5i	0.0211144	−	0.0580113i	0.949802	+	0.312852i	$0.101284\pi$
$888$		0			0
$889$	−248456.	+	1.40906e6i	−0.314374	+	1.78290i
$890$		0			0
$891$	786088.	+	972828.i	0.990183	+	1.22541i
$892$		0			0
$893$	1.49172e6	+	263031.i	1.87062	+	0.329840i
$894$		0			0
$895$	1221.57	+	444.616i	0.00152501	+	0.000555059i
$896$		0			0
$897$	−113951.	+	161146.i	−0.141622	+	0.200279i
$898$		0			0
$899$	546571.	+	315563.i	0.676281	+	0.390451i
$900$		0			0
$901$	−534247.	−	925343.i	−0.658101	−	1.13986i
$902$		0			0
$903$	720390.	−	713756.i	0.883471	−	0.875334i
$904$		0			0
$905$	55.2472	−	9.74157i	6.74548e−5	−	1.18941e-5i
$906$		0			0
$907$	171541.	+	143940.i	0.208523	+	0.174972i	0.741068	−	0.671430i	$-0.234320\pi$
$907$	171541.	+	143940.i	0.208523	+	0.174972i	−0.532545	+	0.846402i	$0.678764\pi$
$908$		0			0
$909$	23038.6	−	39064.7i	0.0278822	−	0.0472778i
$910$		0			0
$911$	−117743.	−	323495.i	−0.141872	−	0.389790i	0.848324	−	0.529478i	$-0.177612\pi$
$911$	−117743.	−	323495.i	−0.141872	−	0.389790i	−0.990196	+	0.139688i	$0.955390\pi$
$912$		0			0
$913$	−1.81460e6	+	1.52263e6i	−2.17690	+	1.82664i
$914$		0			0
$915$	909.107	−	630.316i	0.00108586	−	0.000752863i
$916$		0			0
$917$			437170.i			0.519890i
$918$		0			0
$919$	−744201.			−0.881169			−0.440585	−	0.897711i	$-0.645229\pi$
$919$	−744201.			−0.881169			−0.440585	+	0.897711i	$0.645229\pi$
$920$		0			0
$921$	−1.03241e6	+	85513.0i	−1.21711	+	0.100812i
$922$		0			0
$923$	−75304.4	−	89744.3i	−0.0883929	−	0.105343i
$924$		0			0
$925$	−625812.	+	227777.i	−0.731409	+	0.266211i
$926$		0			0
$927$	525825.	+	97741.9i	0.611902	+	0.113742i
$928$		0			0
$929$	861768.	−	1.02701e6i	0.998525	−	1.19000i	0.0167668	−	0.999859i	$-0.494663\pi$
$929$	861768.	−	1.02701e6i	0.998525	−	1.19000i	0.981758	−	0.190136i	$-0.0608929\pi$
$930$		0			0
$931$	−179981.	−	1.02072e6i	−0.207648	−	1.17763i
$932$		0			0
$933$	82158.4	−	301041.i	0.0943818	−	0.345830i
$934$		0			0
$935$	−2290.62	+	1322.49i	−0.00262017	+	0.00151276i
$936$		0			0
$937$	41933.4	−	72630.8i	0.0477618	−	0.0827259i	−0.841156	−	0.540792i	$-0.818124\pi$
$937$	41933.4	−	72630.8i	0.0477618	−	0.0827259i	0.888918	+	0.458066i	$0.151458\pi$
$938$		0			0
$939$	−955853.	+	440350.i	−1.08408	+	0.499421i
$940$		0			0
$941$	−60997.9	+	167590.i	−0.0688867	+	0.189265i	−0.969359	−	0.245649i	$-0.920999\pi$
$941$	−60997.9	+	167590.i	−0.0688867	+	0.189265i	0.900472	+	0.434914i	$0.143221\pi$
$942$		0			0
$943$	−190210.	+	1.07874e6i	−0.213900	+	1.21309i
$944$		0			0
$945$	−1411.32	−	399.234i	−0.00158038	−	0.000447058i
$946$		0			0
$947$	−319363.	−	56312.3i	−0.356110	−	0.0627918i	−0.00726867	−	0.999974i	$-0.502314\pi$
$947$	−319363.	−	56312.3i	−0.356110	−	0.0627918i	−0.348841	+	0.937182i	$0.613425\pi$
$948$		0			0
$949$	67695.4	+	24639.1i	0.0751669	+	0.0273585i
$950$		0			0
$951$	−1.40555e6	−	129525.i	−1.55413	−	0.143216i
$952$		0			0
$953$	111927.	+	64620.9i	0.123239	+	0.0711520i	0.560352	−	0.828254i	$-0.310666\pi$
$953$	111927.	+	64620.9i	0.123239	+	0.0711520i	−0.437113	+	0.899406i	$0.643999\pi$
$954$		0			0
$955$	877.343	+	1519.60i	0.000961973	+	0.00166619i
$956$		0			0
$957$	−175768.	−	668328.i	−0.191918	−	0.729736i
$958$		0			0
$959$	−405491.	+	71499.0i	−0.440904	+	0.0777432i
$960$		0			0
$961$	1.17330e6	+	984517.i	1.27047	+	1.06605i
$962$		0			0
$963$	279339.	−	230023.i	0.301216	−	0.248038i
$964$		0			0
$965$	−453.289	−	1245.40i	−0.000486766	−	0.00133738i
$966$		0			0
$967$	866669.	−	727222.i	0.926831	−	0.777703i	−0.0484150	−	0.998827i	$-0.515417\pi$
$967$	866669.	−	727222.i	0.926831	−	0.777703i	0.975246	+	0.221124i	$0.0709725\pi$
$968$		0			0
$969$	−2.16183e6	−	1.02028e6i	−2.30236	−	1.08660i
$970$		0			0
$971$		−	1.53192e6i		−	1.62479i	−0.583105	−	0.812397i	$-0.698162\pi$
$971$		−	1.53192e6i		−	1.62479i	0.583105	−	0.812397i	$-0.301838\pi$
$972$		0			0
$973$	1.50274e6			1.58729
$974$		0			0
$975$	−113589.	+	240680.i	−0.119489	+	0.253181i
$976$		0			0
$977$	15911.4	+	18962.5i	0.0166694	+	0.0198658i	0.774315	−	0.632801i	$-0.218095\pi$
$977$	15911.4	+	18962.5i	0.0166694	+	0.0198658i	−0.757645	+	0.652666i	$0.773650\pi$
$978$		0			0
$979$	850974.	−	309729.i	0.887873	−	0.323159i
$980$		0			0
$981$	−503704.	−	611696.i	−0.523404	−	0.635620i
$982$		0			0
$983$	632658.	−	753973.i	0.654730	−	0.780277i	−0.331889	−	0.943318i	$-0.607686\pi$
$983$	632658.	−	753973.i	0.654730	−	0.780277i	0.986619	+	0.163042i	$0.0521305\pi$
$984$		0			0
$985$	326.337	+	1850.75i	0.000336352	+	0.00190755i
$986$		0			0
$987$	1.41395e6	−	371864.i	1.45144	−	0.381724i
$988$		0			0
$989$	703736.	−	406302.i	0.719477	−	0.415391i
$990$		0			0
$991$	−351721.	+	609199.i	−0.358138	+	0.620314i	−0.987650	−	0.156677i	$-0.949922\pi$
$991$	−351721.	+	609199.i	−0.358138	+	0.620314i	0.629511	+	0.776991i	$0.283255\pi$
$992$		0			0
$993$	1793.47	−	19462.1i	0.00181885	−	0.0197374i
$994$		0			0
$995$	−622.149	+	1709.34i	−0.000628418	+	0.00172656i
$996$		0			0
$997$	−139091.	+	788824.i	−0.139929	+	0.793579i	0.831370	+	0.555719i	$0.187557\pi$
$997$	−139091.	+	788824.i	−0.139929	+	0.793579i	−0.971300	+	0.237860i	$0.923554\pi$
$998$		0			0
$999$	−541601.	−	556847.i	−0.542686	−	0.557962i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	108.5.k.a.29.4	✓	72
3.2	odd	2		324.5.k.a.89.7		72
27.13	even	9		324.5.k.a.233.7		72
27.14	odd	18	inner	108.5.k.a.41.4	yes	72

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.5.k.a.29.4	✓	72	1.1	even	1	trivial
108.5.k.a.41.4	yes	72	27.14	odd	18	inner
324.5.k.a.89.7		72	3.2	odd	2
324.5.k.a.233.7		72	27.13	even	9

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 108 = 2^{2} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	108.k (of order \(18\), degree \(6\), minimal)

\(f(q)\)	\(=\)	\(q+(-3.84126 + 8.13909i) q^{3} +(0.0201223 + 0.0239808i) q^{5} +(-60.3934 + 21.9814i) q^{7} +(-51.4895 - 62.5287i) q^{9} +O(q^{10})\)	\(q+(-3.84126 + 8.13909i) q^{3} +(0.0201223 + 0.0239808i) q^{5} +(-60.3934 + 21.9814i) q^{7} +(-51.4895 - 62.5287i) q^{9} +(122.535 - 146.032i) q^{11} +(8.21590 + 46.5947i) q^{13} +(-0.272476 + 0.0716604i) q^{15} +(-383.840 + 221.610i) q^{17} +(299.637 - 518.987i) q^{19} +(53.0781 - 575.983i) q^{21} +(158.524 - 435.540i) q^{23} +(108.530 - 615.504i) q^{25} +(706.711 - 178.888i) q^{27} +(-396.668 - 69.9433i) q^{29} +(-1472.40 - 535.909i) q^{31} +(717.877 + 1558.27i) q^{33} +(-1.74238 - 1.00596i) q^{35} +(-532.781 - 922.804i) q^{37} +(-410.798 - 112.112i) q^{39} +(-2327.41 + 410.385i) q^{41} +(1343.05 + 1126.95i) q^{43} +(0.463402 - 2.49298i) q^{45} +(864.494 + 2375.18i) q^{47} +(1324.91 - 1111.73i) q^{49} +(-329.275 - 3975.37i) q^{51} +2410.75i q^{53} +5.96765 q^{55} +(3073.09 + 4432.33i) q^{57} +(165.882 + 197.690i) q^{59} +(-3689.65 + 1342.92i) q^{61} +(4484.09 + 2644.51i) q^{63} +(-0.952054 + 1.13461i) q^{65} +(-702.984 - 3986.82i) q^{67} +(2935.97 + 2963.26i) q^{69} +(-2144.37 + 1238.05i) q^{71} +(761.304 - 1318.62i) q^{73} +(4592.75 + 3247.64i) q^{75} +(-4190.34 + 11512.9i) q^{77} +(1351.07 - 7662.27i) q^{79} +(-1258.67 + 6439.14i) q^{81} +(-12237.2 - 2157.76i) q^{83} +(-13.0381 - 4.74548i) q^{85} +(2092.98 - 2959.85i) q^{87} +(4114.03 + 2375.24i) q^{89} +(-1520.40 - 2633.41i) q^{91} +(10017.7 - 9925.42i) q^{93} +(18.4751 - 3.25765i) q^{95} +(-4423.75 - 3711.97i) q^{97} +(-15440.5 - 142.867i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q + 9 q^{5} - 102 q^{9}+O(q^{10}) \) 72 * q + 9 * q^5 - 102 * q^9	\( 72 q + 9 q^{5} - 102 q^{9} + 18 q^{11} - 225 q^{15} - 282 q^{21} - 1278 q^{23} + 441 q^{25} + 54 q^{27} + 1854 q^{29} - 1665 q^{31} - 45 q^{33} - 2673 q^{35} + 6951 q^{39} - 5472 q^{41} + 1260 q^{43} + 5553 q^{45} + 5103 q^{47} - 5904 q^{49} + 1899 q^{51} + 1107 q^{57} - 10944 q^{59} + 8352 q^{61} - 11985 q^{63} + 8757 q^{65} + 378 q^{67} + 5607 q^{69} - 19764 q^{71} + 6111 q^{73} - 3453 q^{75} - 5679 q^{77} - 5652 q^{79} - 20466 q^{81} - 20061 q^{83} + 26100 q^{85} + 40545 q^{87} + 15633 q^{89} - 6039 q^{91} + 40179 q^{93} + 48024 q^{95} - 37530 q^{97} + 12177 q^{99}+O(q^{100}) \) 72 * q + 9 * q^5 - 102 * q^9 + 18 * q^11 - 225 * q^15 - 282 * q^21 - 1278 * q^23 + 441 * q^25 + 54 * q^27 + 1854 * q^29 - 1665 * q^31 - 45 * q^33 - 2673 * q^35 + 6951 * q^39 - 5472 * q^41 + 1260 * q^43 + 5553 * q^45 + 5103 * q^47 - 5904 * q^49 + 1899 * q^51 + 1107 * q^57 - 10944 * q^59 + 8352 * q^61 - 11985 * q^63 + 8757 * q^65 + 378 * q^67 + 5607 * q^69 - 19764 * q^71 + 6111 * q^73 - 3453 * q^75 - 5679 * q^77 - 5652 * q^79 - 20466 * q^81 - 20061 * q^83 + 26100 * q^85 + 40545 * q^87 + 15633 * q^89 - 6039 * q^91 + 40179 * q^93 + 48024 * q^95 - 37530 * q^97 + 12177 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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