LMFDB - Embedded newform 76.8.i.a.73.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [76,8,Mod(5,76)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(76, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([0, 16]))

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("76.5");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 76 = 2^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	76.i (of order $9$, 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:	$23.7412619368$
Analytic rank:	$0$
Dimension:	$72$
Relative dimension:	$12$ over $\Q(\zeta_{9})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label			73.10
Character	$\chi$	$=$	76.73
Dual form			76.8.i.a.25.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(42.4972 - 15.4677i) q^{3} +(-0.750345 + 4.25542i) q^{5} +(0.108800 + 0.188448i) q^{7} +(-108.581 + 91.1103i) q^{9} +O(q^{10})$	\(q+(42.4972 - 15.4677i) q^{3} +(-0.750345 + 4.25542i) q^{5} +(0.108800 + 0.188448i) q^{7} +(-108.581 + 91.1103i) q^{9} +(3434.86 - 5949.36i) q^{11} +(10124.0 + 3684.82i) q^{13} +(33.9340 + 192.449i) q^{15} +(-15579.1 - 13072.4i) q^{17} +(29559.9 - 4481.44i) q^{19} +(7.53855 + 6.32560i) q^{21} +(-11844.6 - 67173.8i) q^{23} +(73395.9 + 26713.9i) q^{25} +(-52658.1 + 91206.6i) q^{27} +(1317.04 - 1105.13i) q^{29} +(-86596.8 - 149990. i) q^{31} +(53949.0 - 305960. i) q^{33} +(-0.883561 + 0.321590i) q^{35} +339566. q^{37} +487235. q^{39} +(-77647.7 + 28261.5i) q^{41} +(110847. - 628646. i) q^{43} +(-306.239 - 530.421i) q^{45} +(982238. - 824195. i) q^{47} +(411771. - 713209. i) q^{49} +(-864269. - 314568. i) q^{51} +(248605. + 1.40991e6i) q^{53} +(22739.7 + 19080.8i) q^{55} +(1.18689e6 - 647672. i) q^{57} +(-1.09768e6 - 921063. i) q^{59} +(301874. + 1.71201e6i) q^{61} +(-28.9831 - 10.5490i) q^{63} +(-23276.9 + 40316.8i) q^{65} +(-3.14124e6 + 2.63582e6i) q^{67} +(-1.54238e6 - 2.67149e6i) q^{69} +(-426449. + 2.41851e6i) q^{71} +(-4.61451e6 + 1.67954e6i) q^{73} +3.53232e6 q^{75} +1494.86 q^{77} +(250438. - 91151.9i) q^{79} +(-773236. + 4.38524e6i) q^{81} +(2.05684e6 + 3.56255e6i) q^{83} +(67318.4 - 56486.8i) q^{85} +(38876.7 - 67336.4i) q^{87} +(-3.89264e6 - 1.41680e6i) q^{89} +(407.094 + 2308.74i) q^{91} +(-6.00012e6 - 5.03470e6i) q^{93} +(-3109.75 + 129152. i) q^{95} +(1.02867e7 + 8.63156e6i) q^{97} +(169087. + 958938. i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 72 q - 39 q^{3} + 591 q^{7} - 1677 q^{9}+O(q^{10}) $ 72 * q - 39 * q^3 + 591 * q^7 - 1677 * q^9	$ 72 q - 39 q^{3} + 591 q^{7} - 1677 q^{9} - 5793 q^{11} + 22083 q^{13} - 70707 q^{15} + 29409 q^{17} + 12756 q^{19} - 40419 q^{21} - 116796 q^{23} + 26934 q^{25} + 439347 q^{27} - 148005 q^{29} + 141366 q^{31} + 1206654 q^{33} - 531651 q^{35} - 1508004 q^{37} - 2140644 q^{39} + 494157 q^{41} + 923511 q^{43} + 1790028 q^{45} - 662955 q^{47} - 4191369 q^{49} + 3753363 q^{51} - 246513 q^{53} + 2650185 q^{55} + 5831286 q^{57} - 2244432 q^{59} - 10302306 q^{61} - 12102117 q^{63} + 3985635 q^{65} + 11688288 q^{67} + 12632535 q^{69} + 10541736 q^{71} + 560730 q^{73} - 17718750 q^{75} - 22266192 q^{77} + 9424767 q^{79} + 16585437 q^{81} + 2359839 q^{83} - 20859762 q^{85} + 2348574 q^{87} - 4397130 q^{89} - 12653112 q^{91} + 52259766 q^{93} + 46248669 q^{95} - 34112766 q^{97} - 93757926 q^{99}+O(q^{100}) $ 72 * q - 39 * q^3 + 591 * q^7 - 1677 * q^9 - 5793 * q^11 + 22083 * q^13 - 70707 * q^15 + 29409 * q^17 + 12756 * q^19 - 40419 * q^21 - 116796 * q^23 + 26934 * q^25 + 439347 * q^27 - 148005 * q^29 + 141366 * q^31 + 1206654 * q^33 - 531651 * q^35 - 1508004 * q^37 - 2140644 * q^39 + 494157 * q^41 + 923511 * q^43 + 1790028 * q^45 - 662955 * q^47 - 4191369 * q^49 + 3753363 * q^51 - 246513 * q^53 + 2650185 * q^55 + 5831286 * q^57 - 2244432 * q^59 - 10302306 * q^61 - 12102117 * q^63 + 3985635 * q^65 + 11688288 * q^67 + 12632535 * q^69 + 10541736 * q^71 + 560730 * q^73 - 17718750 * q^75 - 22266192 * q^77 + 9424767 * q^79 + 16585437 * q^81 + 2359839 * q^83 - 20859762 * q^85 + 2348574 * q^87 - 4397130 * q^89 - 12653112 * q^91 + 52259766 * q^93 + 46248669 * q^95 - 34112766 * q^97 - 93757926 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/76\mathbb{Z}\right)^\times$.

$n$	$21$	$39$
$\chi(n)$	$e\left(\frac{2}{9}\right)$	$1$

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
$3$	42.4972	−	15.4677i	0.908731	−	0.330751i	0.154985	−	0.987917i	$-0.450467\pi$
$3$	42.4972	−	15.4677i	0.908731	−	0.330751i	0.753746	+	0.657166i	$0.228245\pi$
$4$		0			0
$5$	−0.750345	+	4.25542i	−0.00268451	+	0.0152246i	−0.986121	−	0.166031i	$-0.946905\pi$
$5$	−0.750345	+	4.25542i	−0.00268451	+	0.0152246i	0.983436	+	0.181255i	$0.0580160\pi$
$6$		0			0
$7$	0.108800	+	0.188448i	0.000119891	+	0.000207657i	0.866085	−	0.499896i	$-0.166629\pi$
$7$	0.108800	+	0.188448i	0.000119891	+	0.000207657i	−0.865965	+	0.500104i	$0.833295\pi$
$8$		0			0
$9$	−108.581	+	91.1103i	−0.0496484	+	0.0416599i
$10$		0			0
$11$	3434.86	−	5949.36i	0.778099	−	1.34771i	−0.154937	−	0.987924i	$-0.549517\pi$
$11$	3434.86	−	5949.36i	0.778099	−	1.34771i	0.933036	−	0.359783i	$-0.117149\pi$
$12$		0			0
$13$	10124.0	+	3684.82i	1.27805	+	0.465173i	0.889788	−	0.456375i	$-0.150852\pi$
$13$	10124.0	+	3684.82i	1.27805	+	0.465173i	0.388265	+	0.921548i	$0.373075\pi$
$14$		0			0
$15$	33.9340	+	192.449i	0.00259606	+	0.0147230i
$16$		0			0
$17$	−15579.1	−	13072.4i	−0.769080	−	0.645335i	0.171393	−	0.985203i	$-0.445173\pi$
$17$	−15579.1	−	13072.4i	−0.769080	−	0.645335i	−0.940473	+	0.339868i	$0.889618\pi$
$18$		0			0
$19$	29559.9	−	4481.44i	0.988702	−	0.149892i
$19$	29559.9	−	4481.44i	0.988702	−	0.149892i
$20$		0			0
$21$	7.53855	+	6.32560i	0.000177632	+	0.000149051i
$22$		0			0
$23$	−11844.6	−	67173.8i	−0.202988	−	1.15120i	−0.900574	−	0.434703i	$-0.856853\pi$
$23$	−11844.6	−	67173.8i	−0.202988	−	1.15120i	0.697585	−	0.716502i	$-0.254258\pi$
$24$		0			0
$25$	73395.9	+	26713.9i	0.939468	+	0.341938i
$26$		0			0
$27$	−52658.1	+	91206.6i	−0.514864	+	0.891770i
$28$		0			0
$29$	1317.04	−	1105.13i	0.0100278	−	0.00841434i	−0.637760	−	0.770235i	$-0.720139\pi$
$29$	1317.04	−	1105.13i	0.0100278	−	0.00841434i	0.647788	+	0.761821i	$0.275694\pi$
$30$		0			0
$31$	−86596.8	−	149990.i	−0.522078	−	0.904266i	−0.999670	−	0.0256845i	$-0.991823\pi$
$31$	−86596.8	−	149990.i	−0.522078	−	0.904266i	0.477592	−	0.878582i	$-0.341510\pi$
$32$		0			0
$33$	53949.0	−	305960.i	0.261327	−	1.48206i
$34$		0			0
$35$	−0.883561	+	0.321590i	−3.48336e−6	+	1.26784e-6i
$36$		0			0
$37$	339566.			1.10209			0.551046	−	0.834475i	$-0.314229\pi$
$37$	339566.			1.10209			0.551046	+	0.834475i	$0.314229\pi$
$38$		0			0
$39$	487235.			1.31526
$40$		0			0
$41$	−77647.7	+	28261.5i	−0.175948	+	0.0640400i	−0.428492	−	0.903545i	$-0.640955\pi$
$41$	−77647.7	+	28261.5i	−0.175948	+	0.0640400i	0.252544	+	0.967585i	$0.418733\pi$
$42$		0			0
$43$	110847.	−	628646.i	0.212611	−	1.20578i	−0.672394	−	0.740194i	$-0.734734\pi$
$43$	110847.	−	628646.i	0.212611	−	1.20578i	0.885005	−	0.465582i	$-0.154155\pi$
$44$		0			0
$45$	−306.239	−	530.421i	−0.000500976	−	0.000867715i
$46$		0			0
$47$	982238.	−	824195.i	1.37998	−	1.15794i	0.410766	−	0.911741i	$-0.365261\pi$
$47$	982238.	−	824195.i	1.37998	−	1.15794i	0.969218	−	0.246204i	$-0.0791833\pi$
$48$		0			0
$49$	411771.	−	713209.i	0.500000	−	0.866025i
$50$		0			0
$51$	−864269.	−	314568.i	−0.912333	−	0.332062i
$52$		0			0
$53$	248605.	+	1.40991e6i	0.229374	+	1.30085i	0.854144	+	0.520037i	$0.174082\pi$
$53$	248605.	+	1.40991e6i	0.229374	+	1.30085i	−0.624770	+	0.780809i	$0.714807\pi$
$54$		0			0
$55$	22739.7	+	19080.8i	0.0184295	+	0.0154642i
$56$		0			0
$57$	1.18689e6	−	647672.i	0.848888	−	0.463226i
$58$		0			0
$59$	−1.09768e6	−	921063.i	−0.695815	−	0.583858i	0.224764	−	0.974413i	$-0.427839\pi$
$59$	−1.09768e6	−	921063.i	−0.695815	−	0.583858i	−0.920580	+	0.390555i	$0.872283\pi$
$60$		0			0
$61$	301874.	+	1.71201e6i	0.170283	+	0.965724i	0.943448	+	0.331519i	$0.107561\pi$
$61$	301874.	+	1.71201e6i	0.170283	+	0.965724i	−0.773165	+	0.634205i	$0.781328\pi$
$62$		0			0
$63$	−28.9831	−	10.5490i	−1.46034e−5	−	5.31520e-6i
$64$		0			0
$65$	−23276.9	+	40316.8i	−0.0105130	+	0.0182091i
$66$		0			0
$67$	−3.14124e6	+	2.63582e6i	−1.27597	+	1.07066i	−0.282181	+	0.959361i	$0.591058\pi$
$67$	−3.14124e6	+	2.63582e6i	−1.27597	+	1.07066i	−0.993787	+	0.111303i	$0.964498\pi$
$68$		0			0
$69$	−1.54238e6	−	2.67149e6i	−0.565224	−	0.978997i
$70$		0			0
$71$	−426449.	+	2.41851e6i	−0.141404	+	0.801944i	0.828780	+	0.559575i	$0.189036\pi$
$71$	−426449.	+	2.41851e6i	−0.141404	+	0.801944i	−0.970184	+	0.242369i	$0.922075\pi$
$72$		0			0
$73$	−4.61451e6	+	1.67954e6i	−1.38834	+	0.505314i	−0.924695	−	0.380708i	$-0.875680\pi$
$73$	−4.61451e6	+	1.67954e6i	−1.38834	+	0.505314i	−0.463643	+	0.886022i	$0.653458\pi$
$74$		0			0
$75$	3.53232e6			0.966820
$76$		0			0
$77$	1494.86			0.000373149
$78$		0			0
$79$	250438.	−	91151.9i	0.0571485	−	0.0208004i	−0.313288	−	0.949658i	$-0.601430\pi$
$79$	250438.	−	91151.9i	0.0571485	−	0.0208004i	0.370436	+	0.928858i	$0.379208\pi$
$80$		0			0
$81$	−773236.	+	4.38524e6i	−0.161664	+	0.916844i
$82$		0			0
$83$	2.05684e6	+	3.56255e6i	0.394846	+	0.683893i	0.993081	−	0.117427i	$-0.0374647\pi$
$83$	2.05684e6	+	3.56255e6i	0.394846	+	0.683893i	−0.598236	+	0.801320i	$0.704131\pi$
$84$		0			0
$85$	67318.4	−	56486.8i	0.0118896	−	0.00997656i
$86$		0			0
$87$	38876.7	−	67336.4i	0.00632954	−	0.0109631i
$88$		0			0
$89$	−3.89264e6	−	1.41680e6i	−0.585301	−	0.213032i	0.0323606	−	0.999476i	$-0.489698\pi$
$89$	−3.89264e6	−	1.41680e6i	−0.585301	−	0.213032i	−0.617661	+	0.786444i	$0.711920\pi$
$90$		0			0
$91$	407.094	+	2308.74i	5.66304e−5	+	0.000321167i
$92$		0			0
$93$	−6.00012e6	−	5.03470e6i	−0.773516	−	0.649057i
$94$		0			0
$95$	−3109.75	+	129152.i	−0.000372128	+	0.0154550i
$96$		0			0
$97$	1.02867e7	+	8.63156e6i	1.14439	+	0.960258i	0.999574	−	0.0292003i	$-0.00929606\pi$
$97$	1.02867e7	+	8.63156e6i	1.14439	+	0.960258i	0.144817	+	0.989458i	$0.453741\pi$
$98$		0			0
$99$	169087.	+	958938.i	0.0175140	+	0.0993270i
$100$		0			0
$101$	−9.86294e6	−	3.58982e6i	−0.952536	−	0.346695i	−0.181432	−	0.983404i	$-0.558073\pi$
$101$	−9.86294e6	−	3.58982e6i	−0.952536	−	0.346695i	−0.771105	+	0.636709i	$0.780295\pi$
$102$		0			0
$103$	−4.63368e6	+	8.02577e6i	−0.417827	+	0.723697i	−0.995721	−	0.0924150i	$-0.970541\pi$
$103$	−4.63368e6	+	8.02577e6i	−0.417827	+	0.723697i	0.577894	+	0.816112i	$0.303875\pi$
$104$		0			0
$105$	−32.5746	+	27.3333i	−2.74610e−6	+	2.30425e-6i
$106$		0			0
$107$	5.17959e6	+	8.97131e6i	0.408745	+	0.707966i	0.994749	−	0.102341i	$-0.0326334\pi$
$107$	5.17959e6	+	8.97131e6i	0.408745	+	0.707966i	−0.586005	+	0.810308i	$0.699300\pi$
$108$		0			0
$109$	784407.	−	4.44859e6i	0.0580161	−	0.329026i	−0.941961	−	0.335721i	$-0.891020\pi$
$109$	784407.	−	4.44859e6i	0.0580161	−	0.329026i	0.999978	+	0.00669526i	$0.00213118\pi$
$110$		0			0
$111$	1.44306e7	−	5.25230e6i	1.00151	−	0.364518i
$112$		0			0
$113$	−1.69675e7			−1.10623			−0.553114	−	0.833106i	$-0.686561\pi$
$113$	−1.69675e7			−1.10623			−0.553114	+	0.833106i	$0.686561\pi$
$114$		0			0
$115$	294740.			0.0180716
$116$		0			0
$117$	−1.43499e6	+	522295.i	−0.0828323	+	0.0301485i
$118$		0			0
$119$	768.456	−	4358.13i	4.18027e−5	−	0.000237075i
$120$		0			0
$121$	−1.38530e7	−	2.39941e7i	−0.710877	−	1.23127i
$122$		0			0
$123$	−2.86267e6	+	2.40206e6i	−0.138709	+	0.116390i
$124$		0			0
$125$	−337543.	+	584641.i	−0.0154577	+	0.0267734i
$126$		0			0
$127$	1.70169e7	+	6.19365e6i	0.737170	+	0.268308i	0.683197	−	0.730235i	$-0.260589\pi$
$127$	1.70169e7	+	6.19365e6i	0.737170	+	0.268308i	0.0539732	+	0.998542i	$0.482811\pi$
$128$		0			0
$129$	−5.01301e6	−	2.84302e7i	−0.205606	−	1.16605i
$130$		0			0
$131$	2.97788e7	+	2.49874e7i	1.15733	+	0.971116i	0.999865	−	0.0164084i	$-0.00522319\pi$
$131$	2.97788e7	+	2.49874e7i	1.15733	+	0.971116i	0.157466	+	0.987524i	$0.449668\pi$
$132$		0			0
$133$	4060.64	+	5082.91i	0.000149663	+	0.000187341i
$134$		0			0
$135$	−348610.	−	292519.i	−0.0121947	−	0.0102326i
$136$		0			0
$137$	−2.27784e6	−	1.29183e7i	−0.0756835	−	0.429223i	−0.998981	−	0.0451366i	$-0.985628\pi$
$137$	−2.27784e6	−	1.29183e7i	−0.0756835	−	0.429223i	0.923297	−	0.384086i	$-0.125483\pi$
$138$		0			0
$139$	2.78055e7	+	1.01204e7i	0.878172	+	0.319628i	0.741472	−	0.670984i	$-0.234128\pi$
$139$	2.78055e7	+	1.01204e7i	0.878172	+	0.319628i	0.136700	+	0.990612i	$0.456350\pi$
$140$		0			0
$141$	2.89939e7	−	5.02189e7i	0.871044	−	1.50869i
$142$		0			0
$143$	5.66967e7	−	4.75742e7i	1.62137	−	1.36049i
$144$		0			0
$145$	3714.55	+	6433.79i	0.000101185	+	0.000175258i
$146$		0			0
$147$	6.46741e6	−	3.66785e7i	0.167927	−	0.952360i
$148$		0			0
$149$	−2.48270e7	+	9.03630e6i	−0.614855	+	0.223789i	−0.630626	−	0.776087i	$-0.717202\pi$
$149$	−2.48270e7	+	9.03630e6i	−0.614855	+	0.223789i	0.0157712	+	0.999876i	$0.494980\pi$
$150$		0			0
$151$	−4.04360e7			−0.955760			−0.477880	−	0.878425i	$-0.658595\pi$
$151$	−4.04360e7			−0.955760			−0.477880	+	0.878425i	$0.658595\pi$
$152$		0			0
$153$	2.88263e6			0.0650682
$154$		0			0
$155$	703247.	−	255961.i	0.0151687	−	0.00552094i
$156$		0			0
$157$	−5.27128e6	+	2.98949e7i	−0.108709	+	0.616522i	0.880964	+	0.473183i	$0.156895\pi$
$157$	−5.27128e6	+	2.98949e7i	−0.108709	+	0.616522i	−0.989674	+	0.143339i	$0.954216\pi$
$158$		0			0
$159$	3.23731e7	+	5.60718e7i	0.638696	+	1.10625i
$160$		0			0
$161$	11370.1	−	9540.60i	0.000214720	−	0.000180171i
$162$		0			0
$163$	1.18385e7	−	2.05048e7i	0.214111	−	0.370851i	−0.738886	−	0.673830i	$-0.764648\pi$
$163$	1.18385e7	−	2.05048e7i	0.214111	−	0.370851i	0.952997	+	0.302979i	$0.0979812\pi$
$164$		0			0
$165$	1.26151e6	+	459151.i	0.0218623	+	0.00795723i
$166$		0			0
$167$	−1.07280e7	−	6.08414e7i	−0.178242	−	1.01086i	−0.934335	−	0.356397i	$-0.884005\pi$
$167$	−1.07280e7	−	6.08414e7i	−0.178242	−	1.01086i	0.756093	−	0.654465i	$-0.227106\pi$
$168$		0			0
$169$	4.08485e7	+	3.42759e7i	0.650987	+	0.546243i
$170$		0			0
$171$	−2.80134e6	+	3.17981e6i	−0.0428429	+	0.0486312i
$172$		0			0
$173$	−3.91570e7	−	3.28567e7i	−0.574974	−	0.482461i	0.308318	−	0.951283i	$-0.400234\pi$
$173$	−3.91570e7	−	3.28567e7i	−0.574974	−	0.482461i	−0.883292	+	0.468823i	$0.844678\pi$
$174$		0			0
$175$	2951.32	+	16737.8i	4.16278e−5	+	0.000236083i
$176$		0			0
$177$	−6.08950e7	−	2.21640e7i	−0.825421	−	0.300429i
$178$		0			0
$179$	−6.48684e7	+	1.12355e8i	−0.845372	+	1.46423i	0.0399266	+	0.999203i	$0.487288\pi$
$179$	−6.48684e7	+	1.12355e8i	−0.845372	+	1.46423i	−0.885298	+	0.465024i	$0.846046\pi$
$180$		0			0
$181$	−7.27550e7	+	6.10487e7i	−0.911986	+	0.765247i	−0.972496	−	0.232921i	$-0.925172\pi$
$181$	−7.27550e7	+	6.10487e7i	−0.911986	+	0.765247i	0.0605101	+	0.998168i	$0.480727\pi$
$182$		0			0
$183$	3.93097e7	+	6.80865e7i	0.474156	+	0.821262i
$184$		0			0
$185$	−254791.	+	1.44499e6i	−0.00295858	+	0.0167790i
$186$		0			0
$187$	−1.31285e8	+	4.77837e7i	−1.46814	+	0.534361i
$188$		0			0
$189$	−22916.9			−0.000246910
$190$		0			0
$191$	4.51558e7			0.468917			0.234459	−	0.972126i	$-0.424668\pi$
$191$	4.51558e7			0.468917			0.234459	+	0.972126i	$0.424668\pi$
$192$		0			0
$193$	−7.12196e7	+	2.59218e7i	−0.713097	+	0.259546i	−0.672993	−	0.739649i	$-0.734991\pi$
$193$	−7.12196e7	+	2.59218e7i	−0.713097	+	0.259546i	−0.0401048	+	0.999195i	$0.512769\pi$
$194$		0			0
$195$	−365594.	+	2.07339e6i	−0.00353084	+	0.0200244i
$196$		0			0
$197$	−8.27090e6	−	1.43256e7i	−0.0770763	−	0.133500i	0.824911	−	0.565263i	$-0.191225\pi$
$197$	−8.27090e6	−	1.43256e7i	−0.0770763	−	0.133500i	−0.901987	+	0.431763i	$0.857892\pi$
$198$		0			0
$199$	9.38504e7	−	7.87498e7i	0.844209	−	0.708376i	−0.114297	−	0.993447i	$-0.536462\pi$
$199$	9.38504e7	−	7.87498e7i	0.844209	−	0.708376i	0.958506	+	0.285071i	$0.0920171\pi$
$200$		0			0
$201$	−9.27239e7	+	1.60602e8i	−0.805388	+	1.39497i
$202$		0			0
$203$	351.553	+	127.955i	2.94955e−6	+	1.07355e-6i
$204$		0			0
$205$	−62001.7	−	351629.i	−0.000502650	−	0.00285067i
$206$		0			0
$207$	7.40632e6	+	6.21464e6i	0.0580371	+	0.0486989i
$208$		0			0
$209$	7.48726e7	−	1.91256e8i	0.567297	−	1.44911i
$210$		0			0
$211$	1.46521e8	+	1.22945e8i	1.07377	+	0.900998i	0.995389	−	0.0959252i	$-0.0305810\pi$
$211$	1.46521e8	+	1.22945e8i	1.07377	+	0.900998i	0.0783795	+	0.996924i	$0.475025\pi$
$212$		0			0
$213$	1.92860e7	+	1.09376e8i	0.136745	+	0.775521i
$214$		0			0
$215$	2.59198e6	+	943402.i	0.0177867	+	0.00647385i
$216$		0			0
$217$	18843.5	−	32637.9i	0.000125185	−	0.000216827i
$218$		0			0
$219$	−1.70125e8	+	1.42752e8i	−1.09449	+	0.918389i
$220$		0			0
$221$	−1.09553e8	−	1.89751e8i	−0.682732	−	1.18253i
$222$		0			0
$223$	1.46018e7	−	8.28107e7i	0.0881736	−	0.500057i	−0.908453	−	0.417987i	$-0.862736\pi$
$223$	1.46018e7	−	8.28107e7i	0.0881736	−	0.500057i	0.996627	−	0.0820701i	$-0.0261531\pi$
$224$		0			0
$225$	−1.04033e7	+	3.78650e6i	−0.0608882	+	0.0221615i
$226$		0			0
$227$	−1.37858e8			−0.782242			−0.391121	−	0.920339i	$-0.627913\pi$
$227$	−1.37858e8			−0.782242			−0.391121	+	0.920339i	$0.627913\pi$
$228$		0			0
$229$	2.59725e8			1.42919			0.714595	−	0.699538i	$-0.246611\pi$
$229$	2.59725e8			1.42919			0.714595	+	0.699538i	$0.246611\pi$
$230$		0			0
$231$	63527.1	−	23122.0i	0.000339092	−	0.000123419i
$232$		0			0
$233$	−4.11844e7	+	2.33568e8i	−0.213298	+	1.20967i	0.670537	+	0.741876i	$0.266064\pi$
$233$	−4.11844e7	+	2.33568e8i	−0.213298	+	1.20967i	−0.883835	+	0.467798i	$0.845047\pi$
$234$		0			0
$235$	2.77028e6	+	4.79826e6i	0.0139247	+	0.0241183i
$236$		0			0
$237$	9.23298e6	−	7.74739e6i	0.0450529	−	0.0378039i
$238$		0			0
$239$	1.78835e8	−	3.09752e8i	0.847346	−	1.46765i	−0.0362213	−	0.999344i	$-0.511532\pi$
$239$	1.78835e8	−	3.09752e8i	0.847346	−	1.46765i	0.883568	−	0.468303i	$-0.155135\pi$
$240$		0			0
$241$	−1.81336e8	−	6.60010e7i	−0.834497	−	0.303732i	−0.110794	−	0.993843i	$-0.535339\pi$
$241$	−1.81336e8	−	6.60010e7i	−0.834497	−	0.303732i	−0.723703	+	0.690111i	$0.757562\pi$
$242$		0			0
$243$	−5.02659e6	−	2.85072e7i	−0.0224725	−	0.127448i
$244$		0			0
$245$	2.72603e6	+	2.28741e6i	0.0118427	+	0.00993718i
$246$		0			0
$247$	3.15777e8	+	6.35531e7i	1.33334	+	0.268347i
$248$		0			0
$249$	1.42514e8	+	1.19584e8i	0.585007	+	0.490879i
$250$		0			0
$251$	3.12292e6	+	1.77110e7i	0.0124653	+	0.0706944i	0.990406	−	0.138189i	$-0.0441282\pi$
$251$	3.12292e6	+	1.77110e7i	0.0124653	+	0.0706944i	−0.977941	+	0.208883i	$0.933017\pi$
$252$		0			0
$253$	−4.40325e8	−	1.60265e8i	−1.70943	−	0.622182i
$254$		0			0
$255$	1.98712e6	−	3.44179e6i	0.00750469	−	0.0129985i
$256$		0			0
$257$	2.18246e8	−	1.83130e8i	0.802012	−	0.672968i	−0.146675	−	0.989185i	$-0.546857\pi$
$257$	2.18246e8	−	1.83130e8i	0.802012	−	0.672968i	0.948687	+	0.316217i	$0.102413\pi$
$258$		0			0
$259$	36944.9	+	63990.4i	0.000132131	+	0.000228858i
$260$		0			0
$261$	−42317.1	+	239992.i	−0.000147324	+	0.000835516i
$262$		0			0
$263$	−3.52571e8	+	1.28325e8i	−1.19509	+	0.434978i	−0.861509	−	0.507743i	$-0.830480\pi$
$263$	−3.52571e8	+	1.28325e8i	−1.19509	+	0.434978i	−0.333583	+	0.942721i	$0.608258\pi$
$264$		0			0
$265$	−6.18629e6			−0.0204207
$266$		0			0
$267$	−1.87341e8			−0.602342
$268$		0			0
$269$	3.05352e8	−	1.11139e8i	0.956461	−	0.348123i	0.183815	−	0.982961i	$-0.441155\pi$
$269$	3.05352e8	−	1.11139e8i	0.956461	−	0.348123i	0.772646	+	0.634838i	$0.218933\pi$
$270$		0			0
$271$	−1.87895e7	+	1.06561e8i	−0.0573487	+	0.325241i	−0.999963	−	0.00864391i	$-0.997249\pi$
$271$	−1.87895e7	+	1.06561e8i	−0.0573487	+	0.325241i	0.942614	+	0.333885i	$0.108360\pi$
$272$		0			0
$273$	53011.3	+	91818.3i	0.000157688	+	0.000273124i
$274$		0			0
$275$	4.11036e8	−	3.44900e8i	1.19183	−	1.00007i
$276$		0			0
$277$	1.68367e8	−	2.91621e8i	0.475969	−	0.824402i	−0.523652	−	0.851932i	$-0.675431\pi$
$277$	1.68367e8	−	2.91621e8i	0.475969	−	0.824402i	0.999621	+	0.0275299i	$0.00876415\pi$
$278$		0			0
$279$	2.30684e7	+	8.39621e6i	0.0635920	+	0.0231456i
$280$		0			0
$281$	−9.55988e7	−	5.42168e8i	−0.257028	−	1.45768i	−0.790813	−	0.612058i	$-0.790342\pi$
$281$	−9.55988e7	−	5.42168e8i	−0.257028	−	1.45768i	0.533785	−	0.845620i	$-0.320769\pi$
$282$		0			0
$283$	2.64426e8	+	2.21880e8i	0.693508	+	0.581923i	0.919919	−	0.392109i	$-0.128255\pi$
$283$	2.64426e8	+	2.21880e8i	0.693508	+	0.581923i	−0.226410	+	0.974032i	$0.572699\pi$
$284$		0			0
$285$	1.86553e6	+	5.53671e6i	0.00477360	+	0.0141675i
$286$		0			0
$287$	−13773.9	−	11557.7i	−3.43930e−5	−	2.88592e-5i
$288$		0			0
$289$	565903.	+	3.20939e6i	0.00137911	+	0.00782133i
$290$		0			0
$291$	5.70665e8	+	2.07705e8i	1.35755	+	0.494108i
$292$		0			0
$293$	2.30531e8	−	3.99291e8i	0.535417	−	0.927370i	−0.463726	−	0.885979i	$-0.653488\pi$
$293$	2.30531e8	−	3.99291e8i	0.535417	−	0.927370i	0.999143	−	0.0413913i	$-0.0131790\pi$
$294$		0			0
$295$	4.74315e6	−	3.97997e6i	0.0107570	−	0.00902616i
$296$		0			0
$297$	3.61747e8	+	6.26564e8i	0.801230	+	1.38777i
$298$		0			0
$299$	1.27610e8	−	7.23710e8i	0.276079	−	1.56572i
$300$		0			0
$301$	130527.	−	47508.0i	0.000275878	−	0.000100412i
$302$		0			0
$303$	−4.74673e8			−0.980269
$304$		0			0
$305$	−7.51185e6			−0.0151599
$306$		0			0
$307$	−6.10069e8	+	2.22047e8i	−1.20336	+	0.437986i	−0.864395	−	0.502813i	$-0.832298\pi$
$307$	−6.10069e8	+	2.22047e8i	−1.20336	+	0.437986i	−0.338963	+	0.940800i	$0.610076\pi$
$308$		0			0
$309$	−7.27781e7	+	4.12745e8i	−0.140328	+	0.795842i
$310$		0			0
$311$	−4.00156e8	−	6.93090e8i	−0.754341	−	1.30656i	−0.945701	−	0.325038i	$-0.894623\pi$
$311$	−4.00156e8	−	6.93090e8i	−0.754341	−	1.30656i	0.191360	−	0.981520i	$-0.438710\pi$
$312$		0			0
$313$	−2.88443e8	+	2.42032e8i	−0.531686	+	0.446137i	−0.868683	−	0.495368i	$-0.835033\pi$
$313$	−2.88443e8	+	2.42032e8i	−0.531686	+	0.446137i	0.336997	+	0.941506i	$0.390589\pi$
$314$		0			0
$315$	66.6377	−	115.420i	1.20125e−7	−	2.08063e-7i
$316$		0			0
$317$	4.95087e8	+	1.80197e8i	0.872920	+	0.317717i	0.739349	−	0.673322i	$-0.235133\pi$
$317$	4.95087e8	+	1.80197e8i	0.872920	+	0.317717i	0.133571	+	0.991039i	$0.457356\pi$
$318$		0			0
$319$	−2.05095e6	−	1.16315e7i	−0.00353743	−	0.0200618i
$320$		0			0
$321$	3.58883e8	+	3.01139e8i	0.605600	+	0.508158i
$322$		0			0
$323$	−5.19101e8	−	3.16603e8i	−0.857122	−	0.522765i
$324$		0			0
$325$	6.44621e8	+	5.40902e8i	1.04163	+	0.874030i
$326$		0			0
$327$	−3.54744e7	−	2.01185e8i	−0.0561046	−	0.318185i
$328$		0			0
$329$	262185.	+	95427.7i	0.000405904	+	0.000147737i
$330$		0			0
$331$	−3.70049e8	+	6.40944e8i	−0.560869	+	0.971453i	0.436552	+	0.899679i	$0.356200\pi$
$331$	−3.70049e8	+	6.40944e8i	−0.560869	+	0.971453i	−0.997421	+	0.0717744i	$0.977134\pi$
$332$		0			0
$333$	−3.68704e7	+	3.09379e7i	−0.0547171	+	0.0459131i
$334$		0			0
$335$	−8.85948e6	−	1.53451e7i	−0.0128751	−	0.0223004i
$336$		0			0
$337$	−6.15631e7	+	3.49142e8i	−0.0876226	+	0.496932i	0.909137	+	0.416497i	$0.136742\pi$
$337$	−6.15631e7	+	3.49142e8i	−0.0876226	+	0.496932i	−0.996760	+	0.0804356i	$0.974369\pi$
$338$		0			0
$339$	−7.21072e8	+	2.62449e8i	−1.00526	+	0.365886i
$340$		0			0
$341$	−1.18979e9			−1.62492
$342$		0			0
$343$	358407.			0.000479564
$344$		0			0
$345$	1.25256e7	−	4.55895e6i	0.0164222	−	0.00597720i
$346$		0			0
$347$	−2.37356e6	+	1.34611e7i	−0.00304963	+	0.0172953i	−0.986295	−	0.164994i	$-0.947240\pi$
$347$	−2.37356e6	+	1.34611e7i	−0.00304963	+	0.0172953i	0.983245	+	0.182289i	$0.0583507\pi$
$348$		0			0
$349$	3.12565e8	+	5.41378e8i	0.393596	+	0.681729i	0.992921	−	0.118777i	$-0.0378975\pi$
$349$	3.12565e8	+	5.41378e8i	0.393596	+	0.681729i	−0.599325	+	0.800506i	$0.704564\pi$
$350$		0			0
$351$	−8.69188e8	+	7.29336e8i	−1.07285	+	0.900228i
$352$		0			0
$353$	−7.27457e8	+	1.25999e9i	−0.880229	+	1.52460i	−0.0291434	+	0.999575i	$0.509278\pi$
$353$	−7.27457e8	+	1.25999e9i	−0.880229	+	1.52460i	−0.851086	+	0.525027i	$0.824055\pi$
$354$		0			0
$355$	−9.97179e6	−	3.62944e6i	−0.0118297	−	0.00430566i
$356$		0			0
$357$	−34753.1	−	197094.i	−4.04254e−5	−	0.000229264i
$358$		0			0
$359$	1.25314e8	+	1.05151e8i	0.142945	+	0.119945i	0.711456	−	0.702730i	$-0.248036\pi$
$359$	1.25314e8	+	1.05151e8i	0.142945	+	0.119945i	−0.568511	+	0.822675i	$0.692480\pi$
$360$		0			0
$361$	8.53705e8	−	2.64942e8i	0.955065	−	0.296398i
$362$		0			0
$363$	−9.59845e8	−	8.05405e8i	−1.05324	−	0.883774i
$364$		0			0
$365$	−3.68469e6	−	2.08969e7i	−0.00396621	−	0.0224935i
$366$		0			0
$367$	−2.94344e8	−	1.07132e8i	−0.310831	−	0.113133i	0.181895	−	0.983318i	$-0.441777\pi$
$367$	−2.94344e8	−	1.07132e8i	−0.310831	−	0.113133i	−0.492725	+	0.870185i	$0.663999\pi$
$368$		0			0
$369$	5.85616e6	−	1.01432e7i	0.00606765	−	0.0105095i
$370$		0			0
$371$	−238646.	+	200248.i	−0.000242630	+	0.000203591i
$372$		0			0
$373$	−8.96522e7	−	1.55282e8i	−0.0894499	−	0.154932i	0.817829	−	0.575462i	$-0.195178\pi$
$373$	−8.96522e7	−	1.55282e8i	−0.0894499	−	0.154932i	−0.907279	+	0.420530i	$0.861844\pi$
$374$		0			0
$375$	−5.30155e6	+	3.00666e7i	−0.00519151	+	0.0294425i
$376$		0			0
$377$	1.74059e7	−	6.33522e6i	0.0167302	−	0.00608929i
$378$		0			0
$379$	3.42174e8			0.322857			0.161429	−	0.986884i	$-0.448390\pi$
$379$	3.42174e8			0.322857			0.161429	+	0.986884i	$0.448390\pi$
$380$		0			0
$381$	8.18971e8			0.758632
$382$		0			0
$383$	6.31353e8	−	2.29794e8i	0.574218	−	0.208998i	−0.0385560	−	0.999256i	$-0.512276\pi$
$383$	6.31353e8	−	2.29794e8i	0.574218	−	0.208998i	0.612774	+	0.790258i	$0.290054\pi$
$384$		0			0
$385$	−1121.66	+	6361.23i	−1.00172e−6	+	5.68105e-6i
$386$		0			0
$387$	4.52402e7	+	7.83583e7i	0.0396768	+	0.0687221i
$388$		0			0
$389$	9.24311e8	−	7.75589e8i	0.796149	−	0.668049i	−0.151110	−	0.988517i	$-0.548285\pi$
$389$	9.24311e8	−	7.75589e8i	0.796149	−	0.668049i	0.947259	+	0.320468i	$0.103840\pi$
$390$		0			0
$391$	−6.93597e8	+	1.20135e9i	−0.586798	+	1.01636i
$392$		0			0
$393$	1.65201e9	+	6.01283e8i	1.37290	+	0.499695i
$394$		0			0
$395$	199975.	+	1.13411e6i	0.000163262	+	0.000925905i
$396$		0			0
$397$	1.15022e8	+	9.65153e7i	0.0922605	+	0.0774158i	0.687751	−	0.725946i	$-0.258598\pi$
$397$	1.15022e8	+	9.65153e7i	0.0922605	+	0.0774158i	−0.595491	+	0.803362i	$0.703042\pi$
$398$		0			0
$399$	251187.	+	153201.i	0.000197966	+	0.000120741i
$400$		0			0
$401$	−4.98324e8	−	4.18143e8i	−0.385928	−	0.323832i	0.429096	−	0.903259i	$-0.358832\pi$
$401$	−4.98324e8	−	4.18143e8i	−0.385928	−	0.323832i	−0.815024	+	0.579427i	$0.803277\pi$
$402$		0			0
$403$	−3.24016e8	−	1.83759e9i	−0.246603	−	1.39856i
$404$		0			0
$405$	−1.80808e7	−	6.58088e6i	−0.0135246	−	0.00492256i
$406$		0			0
$407$	1.16636e9	−	2.02020e9i	0.857537	−	1.48530i
$408$		0			0
$409$	−1.80900e9	+	1.51793e9i	−1.30740	+	1.09704i	−0.318583	+	0.947895i	$0.603207\pi$
$409$	−1.80900e9	+	1.51793e9i	−1.30740	+	1.09704i	−0.988816	+	0.149143i	$0.952349\pi$
$410$		0			0
$411$	−2.96618e8	−	5.13757e8i	−0.210742	−	0.365016i
$412$		0			0
$413$	54144.2	−	307067.i	3.78205e−5	−	0.000214491i
$414$		0			0
$415$	−1.67035e7	+	6.07957e6i	−0.0114720	+	0.00417546i
$416$		0			0
$417$	1.33820e9			0.903740
$418$		0			0
$419$	1.12245e8			0.0745450			0.0372725	−	0.999305i	$-0.488133\pi$
$419$	1.12245e8			0.0745450			0.0372725	+	0.999305i	$0.488133\pi$
$420$		0			0
$421$	7.48102e8	−	2.72287e8i	0.488623	−	0.177844i	−0.0859471	−	0.996300i	$-0.527392\pi$
$421$	7.48102e8	−	2.72287e8i	0.488623	−	0.177844i	0.574570	+	0.818456i	$0.305169\pi$
$422$		0			0
$423$	−3.15597e7	+	1.78984e8i	−0.0202741	+	0.114980i
$424$		0			0
$425$	−7.94228e8	−	1.37564e9i	−0.501862	−	0.869250i
$426$		0			0
$427$	−289781.	+	243155.i	−0.000180124	+	0.000151142i
$428$		0			0
$429$	1.67359e9	−	2.89874e9i	1.02340	−	1.77259i
$430$		0			0
$431$	−7.63236e8	−	2.77795e8i	−0.459186	−	0.167130i	0.102061	−	0.994778i	$-0.467456\pi$
$431$	−7.63236e8	−	2.77795e8i	−0.459186	−	0.167130i	−0.561247	+	0.827648i	$0.689678\pi$
$432$		0			0
$433$	5.24608e8	+	2.97520e9i	0.310547	+	1.76120i	0.596172	+	0.802856i	$0.296687\pi$
$433$	5.24608e8	+	2.97520e9i	0.310547	+	1.76120i	−0.285626	+	0.958341i	$0.592201\pi$
$434$		0			0
$435$	257374.	+	215962.i	0.000149917	+	0.000125796i
$436$		0			0
$437$	−6.51159e8	−	1.93257e9i	−0.373252	−	1.10777i
$438$		0			0
$439$	1.34280e9	+	1.12674e9i	0.757506	+	0.635623i	0.937476	−	0.348049i	$-0.113156\pi$
$439$	1.34280e9	+	1.12674e9i	0.757506	+	0.635623i	−0.179971	+	0.983672i	$0.557600\pi$
$440$		0			0
$441$	2.02701e7	+	1.14958e8i	0.0112544	+	0.0638267i
$442$		0			0
$443$	1.77952e9	+	6.47692e8i	0.972500	+	0.353961i	0.778920	−	0.627123i	$-0.215768\pi$
$443$	1.77952e9	+	6.47692e8i	0.972500	+	0.353961i	0.193580	+	0.981084i	$0.437990\pi$
$444$		0			0
$445$	8.94991e6	−	1.55017e7i	0.00481458	−	0.00833910i
$446$		0			0
$447$	−9.15307e8	+	7.68034e8i	−0.484719	+	0.406728i
$448$		0			0
$449$	2.34721e8	+	4.06549e8i	0.122374	+	0.211958i	0.920704	−	0.390263i	$-0.127616\pi$
$449$	2.34721e8	+	4.06549e8i	0.122374	+	0.211958i	−0.798329	+	0.602221i	$0.794283\pi$
$450$		0			0
$451$	−9.85718e7	+	5.59028e8i	−0.0505982	+	0.286956i
$452$		0			0
$453$	−1.71842e9	+	6.25452e8i	−0.868529	+	0.316119i
$454$		0			0
$455$	−10130.1			−5.04168e−6
$456$		0			0
$457$	−2.94512e9			−1.44343			−0.721716	−	0.692189i	$-0.756646\pi$
$457$	−2.94512e9			−1.44343			−0.721716	+	0.692189i	$0.756646\pi$
$458$		0			0
$459$	2.01266e9	−	7.32548e8i	0.971462	−	0.353583i
$460$		0			0
$461$	5.47471e8	−	3.10486e9i	0.260260	−	1.47601i	−0.521939	−	0.852983i	$-0.674791\pi$
$461$	5.47471e8	−	3.10486e9i	0.260260	−	1.47601i	0.782200	−	0.623028i	$-0.214098\pi$
$462$		0			0
$463$	−1.09733e9	−	1.90063e9i	−0.513811	−	0.889947i	−0.999872	−	0.0160218i	$-0.994900\pi$
$463$	−1.09733e9	−	1.90063e9i	−0.513811	−	0.889947i	0.486061	−	0.873925i	$-0.338433\pi$
$464$		0			0
$465$	2.59269e7	−	2.17552e7i	0.0119582	−	0.0100341i
$466$		0			0
$467$	1.16964e9	−	2.02588e9i	0.531427	−	0.920459i	−0.467900	−	0.883781i	$-0.654989\pi$
$467$	1.16964e9	−	2.02588e9i	0.531427	−	0.920459i	0.999327	−	0.0366772i	$-0.0116773\pi$
$468$		0			0
$469$	−838481.	−	305182.i	−0.000375308	−	0.000136601i
$470$		0			0
$471$	2.38391e8	+	1.35198e9i	0.105128	+	0.596208i
$472$		0			0
$473$	−3.35929e9	−	2.81878e9i	−1.45960	−	1.22475i
$474$		0			0
$475$	2.28929e9	+	4.60742e8i	0.980108	+	0.197256i
$476$		0			0
$477$	−1.55451e8	−	1.30439e8i	−0.0655812	−	0.0550292i
$478$		0			0
$479$	6.42798e8	+	3.64549e9i	0.267239	+	1.51559i	0.762582	+	0.646892i	$0.223931\pi$
$479$	6.42798e8	+	3.64549e9i	0.267239	+	1.51559i	−0.495343	+	0.868698i	$0.664958\pi$
$480$		0			0
$481$	3.43775e9	+	1.25124e9i	1.40853	+	0.512664i
$482$		0			0
$483$	335624.	−	581317.i	0.000135531	−	0.000234746i
$484$		0			0
$485$	−4.44494e7	+	3.72975e7i	−0.0176917	+	0.0148451i
$486$		0			0
$487$	4.03875e8	+	6.99532e8i	0.158451	+	0.274446i	0.934310	−	0.356461i	$-0.116017\pi$
$487$	4.03875e8	+	6.99532e8i	0.158451	+	0.274446i	−0.775859	+	0.630906i	$0.782683\pi$
$488$		0			0
$489$	1.85939e8	−	1.05451e9i	0.0719099	−	0.407822i
$490$		0			0
$491$	1.51643e9	−	5.51934e8i	0.578144	−	0.210427i	−0.0363631	−	0.999339i	$-0.511577\pi$
$491$	1.51643e9	−	5.51934e8i	0.578144	−	0.210427i	0.614507	+	0.788912i	$0.289355\pi$
$492$		0			0
$493$	−3.49651e7			−0.0131423
$494$		0			0
$495$	−4.20755e6			−0.00155923
$496$		0			0
$497$	−502161.	+	182772.i	−0.000183483	+	6.67823e-5i
$498$		0			0
$499$	−2.35799e8	+	1.33728e9i	−0.0849552	+	0.481805i	0.912411	+	0.409275i	$0.134218\pi$
$499$	−2.35799e8	+	1.33728e9i	−0.0849552	+	0.481805i	−0.997366	+	0.0725299i	$0.976893\pi$
$500$		0			0
$501$	−1.39699e9	−	2.41965e9i	−0.496318	−	0.859648i
$502$		0			0
$503$	−1.79149e9	+	1.50324e9i	−0.627664	+	0.526673i	−0.900202	−	0.435472i	$-0.856581\pi$
$503$	−1.79149e9	+	1.50324e9i	−0.627664	+	0.526673i	0.272538	+	0.962145i	$0.412137\pi$
$504$		0			0
$505$	2.26768e7	−	3.92773e7i	0.00783540	−	0.0135713i
$506$		0			0
$507$	2.26611e9	+	8.24798e8i	0.772243	+	0.281073i
$508$		0			0
$509$	8.73515e8	+	4.95395e9i	0.293602	+	1.66510i	0.672832	+	0.739795i	$0.265077\pi$
$509$	8.73515e8	+	4.95395e9i	0.293602	+	1.66510i	−0.379231	+	0.925302i	$0.623811\pi$
$510$		0			0
$511$	−818566.	−	686858.i	−0.000271382	−	0.000227716i
$512$		0			0
$513$	−1.14783e9	+	2.93204e9i	−0.375377	+	0.958869i
$514$		0			0
$515$	−3.06761e7	−	2.57403e7i	−0.00989636	−	0.00830403i
$516$		0			0
$517$	−1.52958e9	−	8.67468e9i	−0.486805	−	2.76081i
$518$		0			0
$519$	−2.17228e9	−	7.90645e8i	−0.682071	−	0.248254i
$520$		0			0
$521$	−1.26997e8	+	2.19965e8i	−0.0393424	+	0.0681430i	−0.885026	−	0.465541i	$-0.845860\pi$
$521$	−1.26997e8	+	2.19965e8i	−0.0393424	+	0.0681430i	0.845684	+	0.533684i	$0.179193\pi$
$522$		0			0
$523$	5.82594e8	−	4.88854e8i	0.178078	−	0.149425i	−0.549392	−	0.835565i	$-0.685141\pi$
$523$	5.82594e8	−	4.88854e8i	0.178078	−	0.149425i	0.727470	+	0.686140i	$0.240696\pi$
$524$		0			0
$525$	384318.	+	665657.i	0.000115913	+	0.000200767i
$526$		0			0
$527$	−6.11633e8	+	3.46874e9i	−0.182035	+	1.03237i
$528$		0			0
$529$	−1.17254e9	+	4.26769e8i	−0.344375	+	0.125342i
$530$		0			0
$531$	2.03106e8			0.0588696
$532$		0			0
$533$	−8.90241e8			−0.254661
$534$		0			0
$535$	−4.20631e7	+	1.53097e7i	−0.0118758	+	0.00432244i
$536$		0			0
$537$	−1.01884e9	+	5.77815e9i	−0.283921	+	1.61020i
$538$		0			0
$539$	−2.82876e9	−	4.89955e9i	−0.778099	−	1.34771i
$540$		0			0
$541$	−5.30615e9	+	4.45239e9i	−1.44075	+	1.20893i	−0.501754	+	0.865011i	$0.667312\pi$
$541$	−5.30615e9	+	4.45239e9i	−1.44075	+	1.20893i	−0.938998	+	0.343923i	$0.888244\pi$
$542$		0			0
$543$	−2.14760e9	+	3.71975e9i	−0.575644	+	0.997044i
$544$		0			0
$545$	1.83420e7	+	6.67595e6i	0.00485355	+	0.00176655i
$546$		0			0
$547$	1.13368e9	+	6.42944e9i	0.296167	+	1.67965i	0.662422	+	0.749131i	$0.269529\pi$
$547$	1.13368e9	+	6.42944e9i	0.296167	+	1.67965i	−0.366256	+	0.930514i	$0.619360\pi$
$548$		0			0
$549$	−1.88760e8	−	1.58388e8i	−0.0486863	−	0.0408526i
$550$		0			0
$551$	3.39791e7	−	3.85698e7i	0.00865328	−	0.00982237i
$552$		0			0
$553$	44425.0	+	37277.0i	1.11709e−5	+	9.37354e-6i
$554$		0			0
$555$	1.15228e7	+	6.53492e7i	0.00286110	+	0.0162261i
$556$		0			0
$557$	5.60033e9	+	2.03835e9i	1.37316	+	0.499789i	0.920097	−	0.391691i	$-0.128110\pi$
$557$	5.60033e9	+	2.03835e9i	1.37316	+	0.499789i	0.453061	+	0.891480i	$0.350332\pi$
$558$		0			0
$559$	3.43866e9	−	5.95593e9i	0.832622	−	1.44214i
$560$		0			0
$561$	−4.84012e9	+	4.06134e9i	−1.15741	+	0.971180i
$562$		0			0
$563$	8.83352e8	+	1.53001e9i	0.208619	+	0.361339i	0.951280	−	0.308329i	$-0.0997697\pi$
$563$	8.83352e8	+	1.53001e9i	0.208619	+	0.361339i	−0.742661	+	0.669668i	$0.766436\pi$
$564$		0			0
$565$	1.27315e7	−	7.22040e7i	0.00296968	−	0.0168419i
$566$		0			0
$567$	−910516.	+	331401.i	−0.000209772	+	7.63506e-5i
$568$		0			0
$569$	−7.11561e9			−1.61927			−0.809635	−	0.586934i	$-0.800335\pi$
$569$	−7.11561e9			−1.61927			−0.809635	+	0.586934i	$0.800335\pi$
$570$		0			0
$571$	−6.04687e9			−1.35926			−0.679632	−	0.733553i	$-0.737861\pi$
$571$	−6.04687e9			−1.35926			−0.679632	+	0.733553i	$0.737861\pi$
$572$		0			0
$573$	1.91899e9	−	6.98456e8i	0.426120	−	0.155095i
$574$		0			0
$575$	9.25135e8	−	5.24670e9i	0.202940	−	1.15093i
$576$		0			0
$577$	−9.43767e8	−	1.63465e9i	−0.204526	−	0.354250i	0.745455	−	0.666556i	$-0.232232\pi$
$577$	−9.43767e8	−	1.63465e9i	−0.204526	−	0.354250i	−0.949982	+	0.312305i	$0.898899\pi$
$578$		0			0
$579$	−2.62568e9	+	2.20321e9i	−0.562169	+	0.471716i
$580$		0			0
$581$	−447570.	+	775214.i	−9.46770e−5	+	0.000163985i
$582$		0			0
$583$	9.24198e9	+	3.36381e9i	1.93164	+	0.703058i
$584$		0			0
$585$	−1.14584e6	−	6.49840e6i	−0.000236635	−	0.00134203i
$586$		0			0
$587$	1.39468e9	+	1.17027e9i	0.284604	+	0.238811i	0.773902	−	0.633306i	$-0.218302\pi$
$587$	1.39468e9	+	1.17027e9i	0.284604	+	0.238811i	−0.489298	+	0.872117i	$0.662747\pi$
$588$		0			0
$589$	−3.23196e9	−	4.04561e9i	−0.651723	−	0.815795i
$590$		0			0
$591$	−5.73074e8	−	4.80866e8i	−0.114197	−	0.0958227i
$592$		0			0
$593$	−1.65583e8	−	9.39069e8i	−0.0326081	−	0.184929i	0.964153	−	0.265346i	$-0.0854861\pi$
$593$	−1.65583e8	−	9.39069e8i	−0.0326081	−	0.184929i	−0.996761	+	0.0804163i	$0.974375\pi$
$594$		0			0
$595$	17969.1	+	6540.20i	3.49716e−6	+	1.27286e-6i
$596$		0			0
$597$	2.77029e9	−	4.79829e9i	0.532863	−	0.922946i
$598$		0			0
$599$	−7.10012e9	+	5.95771e9i	−1.34981	+	1.13262i	−0.370819	+	0.928705i	$0.620923\pi$
$599$	−7.10012e9	+	5.95771e9i	−1.34981	+	1.13262i	−0.978988	+	0.203917i	$0.934633\pi$
$600$		0			0
$601$	9.21639e8	+	1.59633e9i	0.173181	+	0.299958i	0.939530	−	0.342466i	$-0.111262\pi$
$601$	9.21639e8	+	1.59633e9i	0.173181	+	0.299958i	−0.766349	+	0.642424i	$0.777929\pi$
$602$		0			0
$603$	1.00929e8	−	5.72399e8i	0.0187459	−	0.106313i
$604$		0			0
$605$	1.12499e8	−	4.09464e7i	0.0206541	−	0.00751747i
$606$		0			0
$607$	4.81913e9			0.874598			0.437299	−	0.899316i	$-0.355935\pi$
$607$	4.81913e9			0.874598			0.437299	+	0.899316i	$0.355935\pi$
$608$		0			0
$609$	16919.2			3.03542e−6
$610$		0			0
$611$	1.29811e10	−	4.72475e9i	2.30234	−	0.837982i
$612$		0			0
$613$	1.34505e9	−	7.62816e9i	0.235845	−	1.33754i	−0.604982	−	0.796239i	$-0.706820\pi$
$613$	1.34505e9	−	7.62816e9i	0.235845	−	1.33754i	0.840827	−	0.541304i	$-0.182069\pi$
$614$		0			0
$615$	−8.07379e6	−	1.39842e7i	−0.00139963	−	0.00242424i
$616$		0			0
$617$	2.81628e9	−	2.36314e9i	0.482700	−	0.405033i	−0.368702	−	0.929548i	$-0.620198\pi$
$617$	2.81628e9	−	2.36314e9i	0.482700	−	0.405033i	0.851401	+	0.524515i	$0.175753\pi$
$618$		0			0
$619$	−3.14506e9	+	5.44740e9i	−0.532981	+	0.923149i	0.466278	+	0.884638i	$0.345595\pi$
$619$	−3.14506e9	+	5.44740e9i	−0.532981	+	0.923149i	−0.999258	+	0.0385110i	$0.987739\pi$
$620$		0			0
$621$	6.75040e9	+	2.45695e9i	1.13112	+	0.411694i
$622$		0			0
$623$	−156527.	−	887707.i	−2.59346e−5	−	0.000147083i
$624$		0			0
$625$	4.67221e9	+	3.92045e9i	0.765495	+	0.642327i
$626$		0			0
$627$	2.23588e8	−	9.28592e9i	0.0362252	−	1.50449i
$628$		0			0
$629$	−5.29014e9	−	4.43895e9i	−0.847598	−	0.711219i
$630$		0			0
$631$	−1.98515e9	−	1.12583e10i	−0.314550	−	1.78390i	−0.574731	−	0.818342i	$-0.694893\pi$
$631$	−1.98515e9	−	1.12583e10i	−0.314550	−	1.78390i	0.260181	−	0.965560i	$-0.416218\pi$
$632$		0			0
$633$	8.12840e9	+	2.95849e9i	1.27377	+	0.463615i
$634$		0			0
$635$	−3.91251e7	+	6.77666e7i	−0.00606383	+	0.0105029i
$636$		0			0
$637$	6.79680e9	−	5.70320e9i	1.04188	−	0.874239i
$638$		0			0
$639$	−1.74047e8	−	3.01458e8i	−0.0263884	−	0.0457061i
$640$		0			0
$641$	−2.90722e8	+	1.64877e9i	−0.0435988	+	0.247261i	−0.998816	−	0.0486437i	$-0.984510\pi$
$641$	−2.90722e8	+	1.64877e9i	−0.0435988	+	0.247261i	0.955217	+	0.295905i	$0.0956212\pi$
$642$		0			0
$643$	−5.20112e9	+	1.89305e9i	−0.771540	+	0.280818i	−0.697640	−	0.716448i	$-0.745767\pi$
$643$	−5.20112e9	+	1.89305e9i	−0.771540	+	0.280818i	−0.0738995	+	0.997266i	$0.523544\pi$
$644$		0			0
$645$	1.24744e8			0.0183046
$646$		0			0
$647$	6.89225e9			1.00045			0.500226	−	0.865895i	$-0.333250\pi$
$647$	6.89225e9			1.00045			0.500226	+	0.865895i	$0.333250\pi$
$648$		0			0
$649$	−9.25012e9	+	3.36677e9i	−1.32828	+	0.483456i
$650$		0			0
$651$	295962.	−	1.67848e6i	4.20438e−5	−	0.000238443i
$652$		0			0
$653$	−2.92489e9	−	5.06605e9i	−0.411067	−	0.711989i	0.583939	−	0.811797i	$-0.301511\pi$
$653$	−2.92489e9	−	5.06605e9i	−0.411067	−	0.711989i	−0.995007	+	0.0998078i	$0.968177\pi$
$654$		0			0
$655$	−1.28676e8	+	1.07972e8i	−0.0178918	+	0.0150130i
$656$		0			0
$657$	3.48024e8	−	6.02796e8i	0.0478774	−	0.0829261i
$658$		0			0
$659$	3.55126e9	+	1.29255e9i	0.483375	+	0.175934i	0.572201	−	0.820113i	$-0.306090\pi$
$659$	3.55126e9	+	1.29255e9i	0.483375	+	0.175934i	−0.0888268	+	0.996047i	$0.528312\pi$
$660$		0			0
$661$	3.86894e8	+	2.19419e9i	0.0521059	+	0.295507i	0.999714	−	0.0239290i	$-0.00761757\pi$
$661$	3.86894e8	+	2.19419e9i	0.0521059	+	0.295507i	−0.947608	+	0.319436i	$0.896506\pi$
$662$		0			0
$663$	−7.59069e9	−	6.36935e9i	−1.01154	−	0.848785i
$664$		0			0
$665$	−24676.8	+	13465.8i	−3.25397e−6	+	1.77564e-6i
$666$		0			0
$667$	−8.98355e7	−	7.53809e7i	−0.0117222	−	0.00983606i
$668$		0			0
$669$	−6.60358e8	−	3.74508e9i	−0.0852684	−	0.483581i
$670$		0			0
$671$	1.12223e10	+	4.08458e9i	1.43401	+	0.521937i
$672$		0			0
$673$	3.74106e9	−	6.47971e9i	0.473088	−	0.819413i	−0.526437	−	0.850214i	$-0.676473\pi$
$673$	3.74106e9	−	6.47971e9i	0.473088	−	0.819413i	0.999526	+	0.0308012i	$0.00980586\pi$
$674$		0			0
$675$	−6.30138e9	+	5.28749e9i	−0.788628	+	0.661738i
$676$		0			0
$677$	1.38267e9	+	2.39485e9i	0.171260	+	0.296632i	0.938861	−	0.344297i	$-0.111883\pi$
$677$	1.38267e9	+	2.39485e9i	0.171260	+	0.296632i	−0.767600	+	0.640929i	$0.778549\pi$
$678$		0			0
$679$	−507401.	+	2.87762e6i	−6.22025e−5	+	0.000352768i
$680$		0			0
$681$	−5.85857e9	+	2.13234e9i	−0.710847	+	0.258727i
$682$		0			0
$683$	−4.29187e9			−0.515435			−0.257718	−	0.966220i	$-0.582970\pi$
$683$	−4.29187e9			−0.515435			−0.257718	+	0.966220i	$0.582970\pi$
$684$		0			0
$685$	5.66818e7			0.00673793
$686$		0			0
$687$	1.10376e10	−	4.01735e9i	1.29875	−	0.472706i
$688$		0			0
$689$	−2.67840e9	+	1.51899e10i	−0.311966	+	1.76925i
$690$		0			0
$691$	−6.11839e9	−	1.05974e10i	−0.705447	−	1.22187i	−0.966530	−	0.256553i	$-0.917413\pi$
$691$	−6.11839e9	−	1.05974e10i	−0.705447	−	1.22187i	0.261083	−	0.965316i	$-0.415920\pi$
$692$		0			0
$693$	−162313.	+	136197.i	−1.85262e−5	+	1.55453e-5i
$694$		0			0
$695$	−6.39302e7	+	1.10730e8i	−0.00722369	+	0.0125118i
$696$		0			0
$697$	1.57913e9	+	5.74756e8i	0.176646	+	0.0642938i
$698$		0			0
$699$	1.86255e9	+	1.05630e10i	0.206270	+	1.16982i
$700$		0			0
$701$	−1.02544e9	−	8.60447e8i	−0.112434	−	0.0943433i	0.584837	−	0.811151i	$-0.301158\pi$
$701$	−1.02544e9	−	8.60447e8i	−0.112434	−	0.0943433i	−0.697271	+	0.716807i	$0.745603\pi$
$702$		0			0
$703$	1.00375e10	−	1.52174e9i	1.08964	−	0.165195i
$704$		0			0
$705$	1.91947e8	+	1.61063e8i	0.0206310	+	0.0173114i
$706$		0			0
$707$	−396598.	−	2.24922e6i	−4.22068e−5	−	0.000239367i
$708$		0			0
$709$	−6.49257e9	−	2.36310e9i	−0.684155	−	0.249012i	−0.0235245	−	0.999723i	$-0.507489\pi$
$709$	−6.49257e9	−	2.36310e9i	−0.684155	−	0.249012i	−0.660631	+	0.750711i	$0.729711\pi$
$710$		0			0
$711$	−1.88879e7	+	3.27148e7i	−0.00197079	+	0.00341351i
$712$		0			0
$713$	−9.04970e9	+	7.59360e9i	−0.935020	+	0.784575i
$714$		0			0
$715$	1.59906e8	+	2.76965e8i	0.0163604	+	0.0283370i
$716$		0			0
$717$	2.80885e9	−	1.59298e10i	0.284584	−	1.61396i
$718$		0			0
$719$	−1.47834e10	+	5.38071e9i	−1.48328	+	0.539870i	−0.951671	−	0.307120i	$-0.900635\pi$
$719$	−1.47834e10	+	5.38071e9i	−1.48328	+	0.539870i	−0.531609	+	0.846990i	$0.678412\pi$
$720$		0			0
$721$	−2.01658e6			−0.000200375
$722$		0			0
$723$	−8.72715e9			−0.858793
$724$		0			0
$725$	1.26188e8	−	4.59286e7i	0.0122980	−	0.00447611i
$726$		0			0
$727$	8.87023e8	−	5.03056e9i	0.0856179	−	0.485563i	−0.911604	−	0.411070i	$-0.865155\pi$
$727$	8.87023e8	−	5.03056e9i	0.0856179	−	0.485563i	0.997222	−	0.0744930i	$-0.0237338\pi$
$728$		0			0
$729$	−5.52379e9	−	9.56748e9i	−0.528069	−	0.914642i
$730$		0			0
$731$	−9.94484e9	+	8.34471e9i	−0.941644	+	0.790133i
$732$		0			0
$733$	7.68262e9	−	1.33067e10i	0.720519	−	1.24797i	−0.240274	−	0.970705i	$-0.577237\pi$
$733$	7.68262e9	−	1.33067e10i	0.720519	−	1.24797i	0.960792	−	0.277270i	$-0.0894295\pi$
$734$		0			0
$735$	1.51230e8	+	5.50431e7i	0.0140485	+	0.00511325i
$736$		0			0
$737$	4.89167e9	+	2.77420e10i	0.450112	+	2.55271i
$738$		0			0
$739$	1.21336e10	+	1.01813e10i	1.10594	+	0.927997i	0.997810	−	0.0661394i	$-0.0210682\pi$
$739$	1.21336e10	+	1.01813e10i	1.10594	+	0.927997i	0.108133	+	0.994136i	$0.465513\pi$
$740$		0			0
$741$	1.44026e10	−	2.18351e9i	1.30040	−	0.197148i
$742$		0			0
$743$	1.26618e10	+	1.06245e10i	1.13249	+	0.950272i	0.999167	−	0.0408007i	$-0.0129909\pi$
$743$	1.26618e10	+	1.06245e10i	1.13249	+	0.950272i	0.133323	+	0.991073i	$0.457435\pi$
$744$		0			0
$745$	−1.98244e7	−	1.12430e8i	−0.00175652	−	0.00996171i
$746$		0			0
$747$	−5.47919e8	−	1.99426e8i	−0.0480944	−	0.0175049i
$748$		0			0
$749$	−1.12708e6	+	1.95216e6i	−9.80096e−5	+	0.000169758i
$750$		0			0
$751$	−1.13424e10	+	9.51741e9i	−0.977160	+	0.819934i	−0.983658	−	0.180045i	$-0.942376\pi$
$751$	−1.13424e10	+	9.51741e9i	−0.977160	+	0.819934i	0.00649872	+	0.999979i	$0.497931\pi$
$752$		0			0
$753$	4.06664e8	+	7.04362e8i	0.0347099	+	0.0601192i
$754$		0			0
$755$	3.03409e7	−	1.72072e8i	0.00256575	−	0.0145511i
$756$		0			0
$757$	−8.30615e9	+	3.02319e9i	−0.695928	+	0.253297i	−0.665671	−	0.746245i	$-0.731855\pi$
$757$	−8.30615e9	+	3.02319e9i	−0.695928	+	0.253297i	−0.0302567	+	0.999542i	$0.509632\pi$
$758$		0			0
$759$	−2.11915e10			−1.75920
$760$		0			0
$761$	8.90724e9			0.732651			0.366325	−	0.930487i	$-0.380616\pi$
$761$	8.90724e9			0.732651			0.366325	+	0.930487i	$0.380616\pi$
$762$		0			0
$763$	923670.	−	336188.i	7.52802e−5	−	2.73998e-5i
$764$		0			0
$765$	−2.16297e6	+	1.22668e7i	−0.000174677	+	0.000990640i
$766$		0			0
$767$	−7.71892e9	−	1.33696e10i	−0.617693	−	1.06988i
$768$		0			0
$769$	1.66632e9	−	1.39821e9i	0.132135	−	0.110874i	−0.574325	−	0.818627i	$-0.694736\pi$
$769$	1.66632e9	−	1.39821e9i	0.132135	−	0.110874i	0.706460	+	0.707753i	$0.250291\pi$
$770$		0			0
$771$	6.44224e9	−	1.11583e10i	0.506229	−	0.876814i
$772$		0			0
$773$	1.14264e10	+	4.15887e9i	0.889778	+	0.323853i	0.746149	−	0.665779i	$-0.231901\pi$
$773$	1.14264e10	+	4.15887e9i	0.889778	+	0.323853i	0.143629	+	0.989632i	$0.454123\pi$
$774$		0			0
$775$	−2.34903e9	−	1.33220e10i	−0.181273	−	1.02805i
$776$		0			0
$777$	2.55984e6	+	2.14796e6i	0.000195767	+	0.000164268i
$778$		0			0
$779$	−2.16861e9	+	1.18338e9i	−0.164361	+	0.0896898i
$780$		0			0
$781$	1.29238e10	+	1.08444e10i	0.970759	+	0.814564i
$782$		0			0
$783$	3.14421e7	+	1.78317e8i	0.00234070	+	0.0132747i
$784$		0			0
$785$	−1.23260e8	−	4.48630e7i	−0.00909449	−	0.00331012i
$786$		0			0
$787$	−1.32038e9	+	2.28697e9i	−0.0965578	+	0.167243i	−0.910258	−	0.414042i	$-0.864117\pi$
$787$	−1.32038e9	+	2.28697e9i	−0.0965578	+	0.167243i	0.813700	+	0.581285i	$0.197450\pi$
$788$		0			0
$789$	−1.29984e10	+	1.09069e10i	−0.942148	+	0.790556i
$790$		0			0
$791$	−1.84607e6	−	3.19749e6i	−0.000132627	−	0.000229716i
$792$		0			0
$793$	−3.25230e9	+	1.84447e10i	−0.231598	+	1.31346i
$794$		0			0
$795$	−2.62900e8	+	9.56877e7i	−0.0185569	+	0.00675416i
$796$		0			0
$797$	−2.10719e10			−1.47434			−0.737172	−	0.675706i	$-0.763839\pi$
$797$	−2.10719e10			−1.47434			−0.737172	+	0.675706i	$0.763839\pi$
$798$		0			0
$799$	−2.60766e10			−1.80858
$800$		0			0
$801$	5.51752e8	−	2.00821e8i	0.0379341	−	0.0138069i
$802$		0			0
$803$	−5.85800e9	+	3.32224e10i	−0.399250	+	2.26426i
$804$		0			0
$805$	32067.8	+	55543.0i	2.16662e−6	+	3.75270e-6i
$806$		0			0
$807$	1.12575e10	−	9.44617e9i	0.754023	−	0.632701i
$808$		0			0
$809$	−2.62513e9	+	4.54687e9i	−0.174314	+	0.301920i	−0.939924	−	0.341385i	$-0.889104\pi$
$809$	−2.62513e9	+	4.54687e9i	−0.174314	+	0.301920i	0.765610	+	0.643305i	$0.222437\pi$
$810$		0			0
$811$	−1.34809e10	−	4.90666e9i	−0.887457	−	0.323008i	−0.142242	−	0.989832i	$-0.545431\pi$
$811$	−1.34809e10	−	4.90666e9i	−0.887457	−	0.323008i	−0.745215	+	0.666824i	$0.767653\pi$
$812$		0			0
$813$	8.49748e8	+	4.81916e9i	0.0554592	+	0.314525i
$814$		0			0
$815$	7.83737e7	+	6.57633e7i	0.00507129	+	0.00425532i
$816$		0			0
$817$	4.59398e8	−	1.90795e10i	0.0294721	−	1.22402i
$818$		0			0
$819$	−254553.	−	213595.i	−1.61914e−5	−	1.35862e-5i
$820$		0			0
$821$	9.38665e8	+	5.32344e9i	0.0591984	+	0.335731i	0.999995	−	0.00324158i	$-0.00103183\pi$
$821$	9.38665e8	+	5.32344e9i	0.0591984	+	0.335731i	−0.940796	+	0.338972i	$0.889921\pi$
$822$		0			0
$823$	1.07659e10	+	3.91848e9i	0.673212	+	0.245029i	0.655930	−	0.754821i	$-0.272276\pi$
$823$	1.07659e10	+	3.91848e9i	0.673212	+	0.245029i	0.0172820	+	0.999851i	$0.494499\pi$
$824$		0			0
$825$	1.21330e10	−	2.10150e10i	0.752282	−	1.30299i
$826$		0			0
$827$	3.70179e9	−	3.10617e9i	0.227584	−	0.190966i	−0.521864	−	0.853029i	$-0.674763\pi$
$827$	3.70179e9	−	3.10617e9i	0.227584	−	0.190966i	0.749448	+	0.662063i	$0.230319\pi$
$828$		0			0
$829$	1.11838e10	+	1.93710e10i	0.681789	+	1.18089i	0.974435	+	0.224672i	$0.0721309\pi$
$829$	1.11838e10	+	1.93710e10i	0.681789	+	1.18089i	−0.292646	+	0.956221i	$0.594536\pi$
$830$		0			0
$831$	2.64443e9	−	1.49973e10i	0.159856	−	0.906587i
$832$		0			0
$833$	−1.57384e10	+	5.72832e9i	−0.943417	+	0.343376i
$834$		0			0
$835$	2.66955e8			0.0158685
$836$		0			0
$837$	1.82401e10			1.07520
$838$		0			0
$839$	6.98853e9	−	2.54362e9i	0.408525	−	0.148691i	−0.129579	−	0.991569i	$-0.541363\pi$
$839$	6.98853e9	−	2.54362e9i	0.408525	−	0.148691i	0.538105	+	0.842878i	$0.319141\pi$
$840$		0			0
$841$	−2.99490e9	+	1.69849e10i	−0.173618	+	0.984639i
$842$		0			0
$843$	−1.24488e10	−	2.15619e10i	−0.715698	−	1.23963i
$844$		0			0
$845$	−1.76509e8	+	1.48109e8i	−0.0100639	+	0.00844465i
$846$		0			0
$847$	3.01441e6	−	5.22112e6i	0.000170456	−	0.000295238i
$848$		0			0
$849$	1.46693e10	+	5.33920e9i	0.822684	+	0.299433i
$850$		0			0
$851$	−4.02201e9	−	2.28099e10i	−0.223712	−	1.26873i
$852$		0			0
$853$	−1.06665e10	−	8.95023e9i	−0.588436	−	0.493756i	0.299269	−	0.954169i	$-0.403257\pi$
$853$	−1.06665e10	−	8.95023e9i	−0.588436	−	0.493756i	−0.887705	+	0.460412i	$0.847702\pi$
$854$		0			0
$855$	−1.14294e7	−	1.43068e7i	−0.000625380	−	0.000782820i
$856$		0			0
$857$	1.44211e10	+	1.21008e10i	0.782648	+	0.656719i	0.943914	−	0.330192i	$-0.107113\pi$
$857$	1.44211e10	+	1.21008e10i	0.782648	+	0.656719i	−0.161266	+	0.986911i	$0.551558\pi$
$858$		0			0
$859$	−3.01326e9	−	1.70890e10i	−0.162203	−	0.919901i	−0.951901	−	0.306405i	$-0.900874\pi$
$859$	−3.01326e9	−	1.70890e10i	−0.162203	−	0.919901i	0.789698	−	0.613496i	$-0.210237\pi$
$860$		0			0
$861$	−764122.	−	278118.i	−4.07992e−5	−	1.48497e-5i
$862$		0			0
$863$	−3.17119e8	+	5.49266e8i	−0.0167952	+	0.0290901i	−0.874301	−	0.485384i	$-0.838680\pi$
$863$	−3.17119e8	+	5.49266e8i	−0.0167952	+	0.0290901i	0.857506	+	0.514475i	$0.172013\pi$
$864$		0			0
$865$	1.69200e8	−	1.41976e8i	0.00888882	−	0.00745860i
$866$		0			0
$867$	7.36912e7	+	1.27637e8i	0.00384015	+	0.00665134i
$868$		0			0
$869$	3.17924e8	−	1.80304e9i	0.0164344	−	0.0932042i
$870$		0			0
$871$	−4.15143e10	+	1.51100e10i	−2.12880	+	0.774819i
$872$		0			0
$873$	−1.90336e9			−0.0968214
$874$		0			0
$875$	−146899.			−7.41294e−6
$876$		0			0
$877$	3.55898e10	−	1.29536e10i	1.78167	−	0.648474i	0.781985	−	0.623297i	$-0.214207\pi$
$877$	3.55898e10	−	1.29536e10i	1.78167	−	0.648474i	0.999683	−	0.0251767i	$-0.00801483\pi$
$878$		0			0
$879$	3.62079e9	−	2.05345e10i	0.179822	−	1.01982i
$880$		0			0
$881$	1.86647e9	+	3.23282e9i	0.0919613	+	0.159282i	0.908336	−	0.418240i	$-0.137353\pi$
$881$	1.86647e9	+	3.23282e9i	0.0919613	+	0.159282i	−0.816375	+	0.577522i	$0.804020\pi$
$882$		0			0
$883$	3.61130e9	−	3.03024e9i	0.176523	−	0.148121i	−0.550245	−	0.835003i	$-0.685466\pi$
$883$	3.61130e9	−	3.03024e9i	0.176523	−	0.148121i	0.726768	+	0.686883i	$0.241021\pi$
$884$		0			0
$885$	1.40009e8	−	2.42503e8i	0.00678977	−	0.0117602i
$886$		0			0
$887$	3.46870e10	+	1.26250e10i	1.66891	+	0.607435i	0.991726	−	0.128374i	$-0.0409757\pi$
$887$	3.46870e10	+	1.26250e10i	1.66891	+	0.607435i	0.677189	+	0.735809i	$0.263198\pi$
$888$		0			0
$889$	684266.	+	3.88066e6i	3.26639e−5	+	0.000185246i
$890$		0			0
$891$	2.34334e10	+	1.96629e10i	1.10985	+	0.931272i
$892$		0			0
$893$	2.53413e10	−	2.87650e10i	1.19083	−	1.35171i
$894$		0			0
$895$	−4.29445e8	−	3.60347e8i	−0.0200229	−	0.0168012i
$896$		0			0
$897$	−5.77108e9	−	3.27294e10i	−0.266983	−	1.51414i
$898$		0			0
$899$	−2.79810e8	−	1.01842e8i	−0.0128441	−	0.00467487i
$900$		0			0
$901$	1.45579e10	−	2.52150e10i	0.663074	−	1.14848i
$902$		0			0
$903$	4.81219e6	−	4.03791e6i	0.000217488	−	0.000182494i
$904$		0			0
$905$	−2.05196e8	−	3.55410e8i	−0.00920237	−	0.0159390i
$906$		0			0
$907$	6.38183e9	−	3.61931e10i	0.284001	−	1.61065i	−0.424832	−	0.905272i	$-0.639667\pi$
$907$	6.38183e9	−	3.61931e10i	0.284001	−	1.61065i	0.708833	−	0.705376i	$-0.249222\pi$
$908$		0			0
$909$	1.39800e9	−	5.08829e8i	0.0617352	−	0.0224698i
$910$		0			0
$911$	−1.48764e9			−0.0651904			−0.0325952	−	0.999469i	$-0.510377\pi$
$911$	−1.48764e9			−0.0651904			−0.0325952	+	0.999469i	$0.510377\pi$
$912$		0			0
$913$	2.82599e10			1.22892
$914$		0			0
$915$	−3.19232e8	+	1.16191e8i	−0.0137763	+	0.00501416i
$916$		0			0
$917$	−1.46887e6	+	8.33038e6i	−6.29058e−5	+	0.000356757i
$918$		0			0
$919$	−7.87644e9	−	1.36424e10i	−0.334754	−	0.579810i	0.648684	−	0.761058i	$-0.275320\pi$
$919$	−7.87644e9	−	1.36424e10i	−0.334754	−	0.579810i	−0.983437	+	0.181248i	$0.941986\pi$
$920$		0			0
$921$	−2.24916e10	+	1.88727e10i	−0.948664	+	0.796024i
$922$		0			0
$923$	−1.32291e10	+	2.29135e10i	−0.553765	+	0.959149i
$924$		0			0
$925$	2.49228e10	+	9.07114e9i	1.03538	+	0.376848i
$926$		0			0
$927$	−2.28101e8	−	1.29362e9i	−0.00940475	−	0.0533370i
$928$		0			0
$929$	2.60660e10	+	2.18719e10i	1.06664	+	0.895019i	0.994744	−	0.102393i	$-0.0326498\pi$
$929$	2.60660e10	+	2.18719e10i	1.06664	+	0.895019i	0.0718985	+	0.997412i	$0.477094\pi$
$930$		0			0
$931$	8.97573e9	−	2.29277e10i	0.364540	−	0.931187i
$932$		0			0
$933$	−2.77260e10	−	2.32649e10i	−1.11764	−	0.937810i
$934$		0			0
$935$	−1.04831e8	−	5.94525e8i	−0.00419419	−	0.0237865i
$936$		0			0
$937$	3.63210e9	+	1.32198e9i	0.144234	+	0.0524970i	0.413129	−	0.910673i	$-0.364436\pi$
$937$	3.63210e9	+	1.32198e9i	0.144234	+	0.0524970i	−0.268894	+	0.963170i	$0.586658\pi$
$938$		0			0
$939$	−8.51432e9	+	1.47472e10i	−0.335599	+	0.581274i
$940$		0			0
$941$	−1.54911e10	+	1.29986e10i	−0.606064	+	0.508548i	−0.893388	−	0.449285i	$-0.851679\pi$
$941$	−1.54911e10	+	1.29986e10i	−0.606064	+	0.508548i	0.287324	+	0.957833i	$0.407234\pi$
$942$		0			0
$943$	2.81813e9	+	4.88115e9i	0.109439	+	0.189553i
$944$		0			0
$945$	17195.6	−	97520.8i	6.62834e−7	−	3.75912e-6i
$946$		0			0
$947$	−1.11794e9	+	4.06896e8i	−0.0427753	+	0.0155689i	−0.363319	−	0.931665i	$-0.618357\pi$
$947$	−1.11794e9	+	4.06896e8i	−0.0427753	+	0.0155689i	0.320544	+	0.947234i	$0.396134\pi$
$948$		0			0
$949$	−5.29059e10			−2.00943
$950$		0			0
$951$	2.38270e10			0.898335
$952$		0			0
$953$	−4.00944e10	+	1.45932e10i	−1.50058	+	0.546166i	−0.956210	−	0.292681i	$-0.905453\pi$
$953$	−4.00944e10	+	1.45932e10i	−1.50058	+	0.546166i	−0.544368	+	0.838847i	$0.683230\pi$
$954$		0			0
$955$	−3.38824e7	+	1.92157e8i	−0.00125882	+	0.00713910i
$956$		0			0
$957$	−2.67072e8	−	4.62583e8i	−0.00985002	−	0.0170607i
$958$		0			0
$959$	2.18659e6	−	1.83477e6i	8.00575e−5	−	6.71762e-5i
$960$		0			0
$961$	−1.24170e9	+	2.15068e9i	−0.0451319	+	0.0781708i
$962$		0			0
$963$	−1.37978e9	−	5.02200e8i	−0.0497873	−	0.0181211i
$964$		0			0
$965$	−5.68688e7	−	3.22519e8i	−0.00203718	−	0.0115534i
$966$		0			0
$967$	2.09200e10	+	1.75540e10i	0.743993	+	0.624284i	0.933907	−	0.357516i	$-0.116376\pi$
$967$	2.09200e10	+	1.75540e10i	0.743993	+	0.624284i	−0.189914	+	0.981801i	$0.560821\pi$
$968$		0			0
$969$	−2.69574e10	−	5.42544e9i	−0.951799	−	0.191559i
$970$		0			0
$971$	−3.61638e9	−	3.03450e9i	−0.126767	−	0.106370i	0.577200	−	0.816603i	$-0.304145\pi$
$971$	−3.61638e9	−	3.03450e9i	−0.126767	−	0.106370i	−0.703967	+	0.710233i	$0.748590\pi$
$972$		0			0
$973$	1.11809e6	+	6.34099e6i	3.89118e−5	+	0.000220680i
$974$		0			0
$975$	3.57611e10	+	1.30160e10i	1.23565	+	0.449739i
$976$		0			0
$977$	6.71579e9	−	1.16321e10i	0.230391	−	0.399050i	−0.727532	−	0.686074i	$-0.759333\pi$
$977$	6.71579e9	−	1.16321e10i	0.230391	−	0.399050i	0.957923	+	0.287024i	$0.0926660\pi$
$978$		0			0
$979$	−2.17998e10	+	1.82922e10i	−0.742527	+	0.623054i
$980$		0			0
$981$	3.20141e8	+	5.54500e8i	0.0108268	+	0.0187525i
$982$		0			0
$983$	5.88540e9	−	3.33778e10i	0.197624	−	1.12078i	−0.711009	−	0.703183i	$-0.751762\pi$
$983$	5.88540e9	−	3.33778e10i	0.197624	−	1.12078i	0.908633	−	0.417596i	$-0.137127\pi$
$984$		0			0
$985$	6.71675e7	−	2.44470e7i	0.00223940	−	0.000815076i
$986$		0			0
$987$	1.26182e7			0.000417721
$988$		0			0
$989$	−4.35415e10			−1.43125
$990$		0			0
$991$	−1.44999e10	+	5.27754e9i	−0.473269	+	0.172256i	−0.567633	−	0.823282i	$-0.692141\pi$
$991$	−1.44999e10	+	5.27754e9i	−0.473269	+	0.172256i	0.0943636	+	0.995538i	$0.469918\pi$
$992$		0			0
$993$	−5.81211e9	+	3.29621e10i	−0.188370	+	1.06830i
$994$		0			0
$995$	2.64693e8	+	4.58462e8i	0.00851847	+	0.0147544i
$996$		0			0
$997$	2.27293e10	−	1.90722e10i	0.726362	−	0.609490i	−0.202775	−	0.979225i	$-0.564996\pi$
$997$	2.27293e10	−	1.90722e10i	0.726362	−	0.609490i	0.929137	+	0.369735i	$0.120552\pi$
$998$		0			0
$999$	−1.78809e10	+	3.09706e10i	−0.567427	+	0.982813i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	76.8.i.a.73.10	yes	72
19.6	even	9	inner	76.8.i.a.25.10	✓	72

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.8.i.a.25.10	✓	72	19.6	even	9	inner
76.8.i.a.73.10	yes	72	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 76 = 2^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	76.i (of order \(9\), degree \(6\), minimal)

\(f(q)\)	\(=\)	\(q+(42.4972 - 15.4677i) q^{3} +(-0.750345 + 4.25542i) q^{5} +(0.108800 + 0.188448i) q^{7} +(-108.581 + 91.1103i) q^{9} +O(q^{10})\)	\(q+(42.4972 - 15.4677i) q^{3} +(-0.750345 + 4.25542i) q^{5} +(0.108800 + 0.188448i) q^{7} +(-108.581 + 91.1103i) q^{9} +(3434.86 - 5949.36i) q^{11} +(10124.0 + 3684.82i) q^{13} +(33.9340 + 192.449i) q^{15} +(-15579.1 - 13072.4i) q^{17} +(29559.9 - 4481.44i) q^{19} +(7.53855 + 6.32560i) q^{21} +(-11844.6 - 67173.8i) q^{23} +(73395.9 + 26713.9i) q^{25} +(-52658.1 + 91206.6i) q^{27} +(1317.04 - 1105.13i) q^{29} +(-86596.8 - 149990. i) q^{31} +(53949.0 - 305960. i) q^{33} +(-0.883561 + 0.321590i) q^{35} +339566. q^{37} +487235. q^{39} +(-77647.7 + 28261.5i) q^{41} +(110847. - 628646. i) q^{43} +(-306.239 - 530.421i) q^{45} +(982238. - 824195. i) q^{47} +(411771. - 713209. i) q^{49} +(-864269. - 314568. i) q^{51} +(248605. + 1.40991e6i) q^{53} +(22739.7 + 19080.8i) q^{55} +(1.18689e6 - 647672. i) q^{57} +(-1.09768e6 - 921063. i) q^{59} +(301874. + 1.71201e6i) q^{61} +(-28.9831 - 10.5490i) q^{63} +(-23276.9 + 40316.8i) q^{65} +(-3.14124e6 + 2.63582e6i) q^{67} +(-1.54238e6 - 2.67149e6i) q^{69} +(-426449. + 2.41851e6i) q^{71} +(-4.61451e6 + 1.67954e6i) q^{73} +3.53232e6 q^{75} +1494.86 q^{77} +(250438. - 91151.9i) q^{79} +(-773236. + 4.38524e6i) q^{81} +(2.05684e6 + 3.56255e6i) q^{83} +(67318.4 - 56486.8i) q^{85} +(38876.7 - 67336.4i) q^{87} +(-3.89264e6 - 1.41680e6i) q^{89} +(407.094 + 2308.74i) q^{91} +(-6.00012e6 - 5.03470e6i) q^{93} +(-3109.75 + 129152. i) q^{95} +(1.02867e7 + 8.63156e6i) q^{97} +(169087. + 958938. i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q - 39 q^{3} + 591 q^{7} - 1677 q^{9}+O(q^{10}) \) 72 * q - 39 * q^3 + 591 * q^7 - 1677 * q^9	\( 72 q - 39 q^{3} + 591 q^{7} - 1677 q^{9} - 5793 q^{11} + 22083 q^{13} - 70707 q^{15} + 29409 q^{17} + 12756 q^{19} - 40419 q^{21} - 116796 q^{23} + 26934 q^{25} + 439347 q^{27} - 148005 q^{29} + 141366 q^{31} + 1206654 q^{33} - 531651 q^{35} - 1508004 q^{37} - 2140644 q^{39} + 494157 q^{41} + 923511 q^{43} + 1790028 q^{45} - 662955 q^{47} - 4191369 q^{49} + 3753363 q^{51} - 246513 q^{53} + 2650185 q^{55} + 5831286 q^{57} - 2244432 q^{59} - 10302306 q^{61} - 12102117 q^{63} + 3985635 q^{65} + 11688288 q^{67} + 12632535 q^{69} + 10541736 q^{71} + 560730 q^{73} - 17718750 q^{75} - 22266192 q^{77} + 9424767 q^{79} + 16585437 q^{81} + 2359839 q^{83} - 20859762 q^{85} + 2348574 q^{87} - 4397130 q^{89} - 12653112 q^{91} + 52259766 q^{93} + 46248669 q^{95} - 34112766 q^{97} - 93757926 q^{99}+O(q^{100}) \) 72 * q - 39 * q^3 + 591 * q^7 - 1677 * q^9 - 5793 * q^11 + 22083 * q^13 - 70707 * q^15 + 29409 * q^17 + 12756 * q^19 - 40419 * q^21 - 116796 * q^23 + 26934 * q^25 + 439347 * q^27 - 148005 * q^29 + 141366 * q^31 + 1206654 * q^33 - 531651 * q^35 - 1508004 * q^37 - 2140644 * q^39 + 494157 * q^41 + 923511 * q^43 + 1790028 * q^45 - 662955 * q^47 - 4191369 * q^49 + 3753363 * q^51 - 246513 * q^53 + 2650185 * q^55 + 5831286 * q^57 - 2244432 * q^59 - 10302306 * q^61 - 12102117 * q^63 + 3985635 * q^65 + 11688288 * q^67 + 12632535 * q^69 + 10541736 * q^71 + 560730 * q^73 - 17718750 * q^75 - 22266192 * q^77 + 9424767 * q^79 + 16585437 * q^81 + 2359839 * q^83 - 20859762 * q^85 + 2348574 * q^87 - 4397130 * q^89 - 12653112 * q^91 + 52259766 * q^93 + 46248669 * q^95 - 34112766 * q^97 - 93757926 * q^99

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