LMFDB - Embedded newform 87.7.b.a.59.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [87,7,Mod(59,87)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(87, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 7, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("87.59"); S:= CuspForms(chi, 7); N := Newforms(S);

Level:	$ N $	$=$	$ 87 = 3 \cdot 29 $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	87.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)

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

Self dual:	no
Analytic conductor:	$20.0147052749$
Analytic rank:	$0$
Dimension:	$56$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			59.6
Character	$\chi$	$=$	87.59
Dual form			87.7.b.a.59.51

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-13.5783i q^{2} +(0.291464 + 26.9984i) q^{3} -120.371 q^{4} +197.511i q^{5} +(366.594 - 3.95760i) q^{6} -18.6779 q^{7} +765.430i q^{8} +(-728.830 + 15.7381i) q^{9} +2681.87 q^{10} -1344.96i q^{11} +(-35.0839 - 3249.84i) q^{12} -175.989 q^{13} +253.614i q^{14} +(-5332.48 + 57.5672i) q^{15} +2689.50 q^{16} -5213.61i q^{17} +(213.698 + 9896.30i) q^{18} -3661.66 q^{19} -23774.6i q^{20} +(-5.44392 - 504.273i) q^{21} -18262.3 q^{22} -17120.6i q^{23} +(-20665.4 + 223.095i) q^{24} -23385.5 q^{25} +2389.63i q^{26} +(-637.333 - 19672.7i) q^{27} +2248.28 q^{28} -4528.92i q^{29} +(781.668 + 72406.2i) q^{30} +51010.2 q^{31} +12468.6i q^{32} +(36311.8 - 392.008i) q^{33} -70792.2 q^{34} -3689.08i q^{35} +(87730.3 - 1894.42i) q^{36} +27789.6 q^{37} +49719.2i q^{38} +(-51.2944 - 4751.42i) q^{39} -151181. q^{40} +4376.58i q^{41} +(-6847.19 + 73.9195i) q^{42} -127892. q^{43} +161895. i q^{44} +(-3108.45 - 143952. i) q^{45} -232470. q^{46} -71052.6i q^{47} +(783.892 + 72612.2i) q^{48} -117300. q^{49} +317536. i q^{50} +(140759. - 1519.58i) q^{51} +21184.0 q^{52} +156646. i q^{53} +(-267122. + 8653.92i) q^{54} +265644. q^{55} -14296.6i q^{56} +(-1067.24 - 98859.0i) q^{57} -61495.3 q^{58} -153093. i q^{59} +(641878. - 6929.45i) q^{60} -137572. q^{61} -692634. i q^{62} +(13613.0 - 293.955i) q^{63} +341430. q^{64} -34759.6i q^{65} +(-5322.81 - 493054. i) q^{66} -425359. q^{67} +627569. i q^{68} +(462230. - 4990.05i) q^{69} -50091.5 q^{70} -340435. i q^{71} +(-12046.4 - 557868. i) q^{72} +369916. q^{73} -377336. i q^{74} +(-6816.02 - 631371. i) q^{75} +440759. q^{76} +25121.0i q^{77} +(-64516.4 + 696.492i) q^{78} -725628. q^{79} +531204. i q^{80} +(530946. - 22940.9i) q^{81} +59426.8 q^{82} -394483. i q^{83} +(655.293 + 60700.0i) q^{84} +1.02974e6 q^{85} +1.73656e6i q^{86} +(122274. - 1320.02i) q^{87} +1.02947e6 q^{88} +536899. i q^{89} +(-1.95463e6 + 42207.6i) q^{90} +3287.09 q^{91} +2.06083e6i q^{92} +(14867.6 + 1.37719e6i) q^{93} -964777. q^{94} -723216. i q^{95} +(-336632. + 3634.14i) q^{96} -1.55559e6 q^{97} +1.59274e6i q^{98} +(21167.2 + 980248. i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 56 q + 52 q^{3} - 1924 q^{4} - 160 q^{6} + 160 q^{7} - 1060 q^{9} - 3588 q^{10} - 2166 q^{12} - 1400 q^{13} - 6240 q^{15} + 56588 q^{16} - 5978 q^{18} + 25000 q^{19} + 7520 q^{21} + 20970 q^{22} + 1238 q^{24}+ \cdots + 4793544 q^{99}+O(q^{100}) $ 56 * q + 52 * q^3 - 1924 * q^4 - 160 * q^6 + 160 * q^7 - 1060 * q^9 - 3588 * q^10 - 2166 * q^12 - 1400 * q^13 - 6240 * q^15 + 56588 * q^16 - 5978 * q^18 + 25000 * q^19 + 7520 * q^21 + 20970 * q^22 + 1238 * q^24 - 213304 * q^25 + 25492 * q^27 + 7870 * q^28 + 105092 * q^30 - 78656 * q^31 + 18952 * q^33 - 96642 * q^34 - 40224 * q^36 + 52120 * q^37 - 112096 * q^39 + 343440 * q^40 - 341758 * q^42 - 286760 * q^43 + 68312 * q^45 + 255516 * q^46 + 386086 * q^48 + 1051872 * q^49 - 256656 * q^51 + 58606 * q^52 + 450124 * q^54 - 409632 * q^55 - 1141736 * q^57 + 817806 * q^60 + 309976 * q^61 + 567632 * q^63 - 2844730 * q^64 + 958426 * q^66 + 748696 * q^67 - 1323704 * q^69 - 2123748 * q^70 - 1026242 * q^72 - 333728 * q^73 - 331316 * q^75 + 2846728 * q^76 + 292446 * q^78 - 661184 * q^79 + 1614852 * q^81 + 376908 * q^82 - 1670584 * q^84 + 3342000 * q^85 - 927168 * q^88 + 835882 * q^90 + 2248016 * q^91 + 3960776 * q^93 - 1368294 * q^94 - 5811456 * q^96 - 3057296 * q^97 + 4793544 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/87\mathbb{Z}\right)^\times$.

$n$	$31$	$59$
$\chi(n)$	$1$	$-1$

Coefficient data

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

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

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

$2$		−	13.5783i		−	1.69729i	−0.528961	−	0.848646i	$-0.677418\pi$
$2$		−	13.5783i		−	1.69729i	0.528961	−	0.848646i	$-0.322582\pi$
$3$	0.291464	+	26.9984i	0.0107950	+	0.999942i
$3$	0.291464	+	26.9984i	0.0107950	+	0.999942i
$4$	−120.371			−1.88080
$5$			197.511i			1.58009i	0.613052	+	0.790043i	$0.289942\pi$
$5$			197.511i			1.58009i	−0.613052	+	0.790043i	$0.710058\pi$
$6$	366.594	−	3.95760i	1.69719	−	0.0183222i
$7$	−18.6779			−0.0544544			−0.0272272	−	0.999629i	$-0.508668\pi$
$7$	−18.6779			−0.0544544			−0.0272272	+	0.999629i	$0.508668\pi$
$8$			765.430i			1.49498i
$9$	−728.830	+	15.7381i	−0.999767	+	0.0215887i
$10$	2681.87			2.68187
$11$		−	1344.96i		−	1.01049i	−0.862976	−	0.505245i	$-0.831402\pi$
$11$		−	1344.96i		−	1.01049i	0.862976	−	0.505245i	$-0.168598\pi$
$12$	−35.0839	−	3249.84i	−0.0203032	−	1.88069i
$13$	−175.989			−0.0801041			−0.0400520	−	0.999198i	$-0.512752\pi$
$13$	−175.989			−0.0801041			−0.0400520	+	0.999198i	$0.512752\pi$
$14$			253.614i			0.0924251i
$15$	−5332.48	+	57.5672i	−1.57999	+	0.0170570i
$16$	2689.50			0.656616
$17$		−	5213.61i		−	1.06119i	−0.847626	−	0.530593i	$-0.821969\pi$
$17$		−	5213.61i		−	1.06119i	0.847626	−	0.530593i	$-0.178031\pi$
$18$	213.698	+	9896.30i	0.0366423	+	1.69690i
$19$	−3661.66			−0.533847			−0.266924	−	0.963718i	$-0.586007\pi$
$19$	−3661.66			−0.533847			−0.266924	+	0.963718i	$0.586007\pi$
$20$		−	23774.6i		−	2.97183i
$21$	−5.44392	−	504.273i	−0.000587833	−	0.0544512i
$22$	−18262.3			−1.71510
$23$		−	17120.6i		−	1.40714i	−0.710628	−	0.703568i	$-0.751589\pi$
$23$		−	17120.6i		−	1.40714i	0.710628	−	0.703568i	$-0.248411\pi$
$24$	−20665.4	+	223.095i	−1.49489	+	0.0161382i
$25$	−23385.5			−1.49667
$26$			2389.63i			0.135960i
$27$	−637.333	−	19672.7i	−0.0323799	−	0.999476i
$28$	2248.28			0.102418
$29$		−	4528.92i		−	0.185695i
$29$		−	4528.92i		−	0.185695i
$30$	781.668	+	72406.2i	0.0289507	+	2.68171i
$31$	51010.2			1.71227			0.856134	−	0.516754i	$-0.172860\pi$
$31$	51010.2			1.71227			0.856134	+	0.516754i	$0.172860\pi$
$32$			12468.6i			0.380511i
$33$	36311.8	−	392.008i	1.01043	−	0.0109082i
$34$	−70792.2			−1.80114
$35$		−	3689.08i		−	0.0860426i
$36$	87730.3	−	1894.42i	1.88036	−	0.0406040i
$37$	27789.6			0.548626			0.274313	−	0.961640i	$-0.411550\pi$
$37$	27789.6			0.548626			0.274313	+	0.961640i	$0.411550\pi$
$38$			49719.2i			0.906095i
$39$	−51.2944	−	4751.42i	−0.000864721	−	0.0800994i
$40$	−151181.			−2.36220
$41$			4376.58i			0.0635015i	0.999496	+	0.0317507i	$0.0101083\pi$
$41$			4376.58i			0.0635015i	−0.999496	+	0.0317507i	$0.989892\pi$
$42$	−6847.19	+	73.9195i	−0.0924197	+	0.000997725i
$43$	−127892.			−1.60856			−0.804280	−	0.594250i	$-0.797449\pi$
$43$	−127892.			−1.60856			−0.804280	+	0.594250i	$0.797449\pi$
$44$			161895.i			1.90053i
$45$	−3108.45	−	143952.i	−0.0341119	−	1.57972i
$46$	−232470.			−2.38832
$47$		−	71052.6i		−	0.684363i	−0.939634	−	0.342182i	$-0.888834\pi$
$47$		−	71052.6i		−	0.684363i	0.939634	−	0.342182i	$-0.111166\pi$
$48$	783.892	+	72612.2i	0.00708814	+	0.656577i
$49$	−117300.			−0.997035
$50$			317536.i			2.54029i
$51$	140759.	−	1519.58i	1.06112	−	0.0114555i
$52$	21184.0			0.150660
$53$			156646.i			1.05219i	0.850427	+	0.526093i	$0.176344\pi$
$53$			156646.i			1.05219i	−0.850427	+	0.526093i	$0.823656\pi$
$54$	−267122.	+	8653.92i	−1.69640	+	0.0549581i
$55$	265644.			1.59666
$56$		−	14296.6i		−	0.0814082i
$57$	−1067.24	−	98859.0i	−0.00576286	−	0.533816i
$58$	−61495.3			−0.315179
$59$		−	153093.i		−	0.745419i	−0.927948	−	0.372709i	$-0.878429\pi$
$59$		−	153093.i		−	0.745419i	0.927948	−	0.372709i	$-0.121571\pi$
$60$	641878.	−	6929.45i	2.97166	−	0.0320808i
$61$	−137572.			−0.606096			−0.303048	−	0.952975i	$-0.598004\pi$
$61$	−137572.			−0.606096			−0.303048	+	0.952975i	$0.598004\pi$
$62$		−	692634.i		−	2.90622i
$63$	13613.0	−	293.955i	0.0544417	−	0.00117560i
$64$	341430.			1.30245
$65$		−	34759.6i		−	0.126571i
$66$	−5322.81	−	493054.i	−0.0185144	−	1.71500i
$67$	−425359.			−1.41427			−0.707133	−	0.707081i	$-0.750012\pi$
$67$	−425359.			−1.41427			−0.707133	+	0.707081i	$0.750012\pi$
$68$			627569.i			1.99588i
$69$	462230.	−	4990.05i	1.40705	−	0.0151900i
$70$	−50091.5			−0.146039
$71$		−	340435.i		−	0.951173i	−0.879669	−	0.475586i	$-0.842236\pi$
$71$		−	340435.i		−	0.951173i	0.879669	−	0.475586i	$-0.157764\pi$
$72$	−12046.4	−	557868.i	−0.0322746	−	1.49463i
$73$	369916.			0.950898			0.475449	−	0.879743i	$-0.342286\pi$
$73$	369916.			0.950898			0.475449	+	0.879743i	$0.342286\pi$
$74$		−	377336.i		−	0.931179i
$75$	−6816.02	−	631371.i	−0.0161565	−	1.49658i
$76$	440759.			1.00406
$77$			25121.0i			0.0550256i
$78$	−64516.4	+	696.492i	−0.135952	+	0.00146768i
$79$	−725628.			−1.47174			−0.735872	−	0.677120i	$-0.763228\pi$
$79$	−725628.			−1.47174			−0.735872	+	0.677120i	$0.763228\pi$
$80$			531204.i			1.03751i
$81$	530946.	−	22940.9i	0.999068	−	0.0431673i
$82$	59426.8			0.107781
$83$		−	394483.i		−	0.689912i	−0.938619	−	0.344956i	$-0.887894\pi$
$83$		−	394483.i		−	0.689912i	0.938619	−	0.344956i	$-0.112106\pi$
$84$	655.293	+	60700.0i	0.00110560	+	0.102412i
$85$	1.02974e6			1.67677
$86$			1.73656e6i			2.73020i
$87$	122274.	−	1320.02i	0.185685	−	0.00200457i
$88$	1.02947e6			1.51066
$89$			536899.i			0.761592i	0.924659	+	0.380796i	$0.124350\pi$
$89$			536899.i			0.761592i	−0.924659	+	0.380796i	$0.875650\pi$
$90$	−1.95463e6	+	42207.6i	−2.68124	+	0.0578979i
$91$	3287.09			0.00436202
$92$			2.06083e6i			2.64655i
$93$	14867.6	+	1.37719e6i	0.0184839	+	1.71217i
$94$	−964777.			−1.16156
$95$		−	723216.i		−	0.843524i
$96$	−336632.	+	3634.14i	−0.380489	+	0.00410760i
$97$	−1.55559e6			−1.70443			−0.852215	−	0.523193i	$-0.824741\pi$
$97$	−1.55559e6			−1.70443			−0.852215	+	0.523193i	$0.824741\pi$
$98$			1.59274e6i			1.69226i
$99$	21167.2	+	980248.i	0.0218151	+	1.01025i
$100$	2.81494e6			2.81494
$101$		−	2.01502e6i		−	1.95576i	−0.209169	−	0.977880i	$-0.567076\pi$
$101$		−	2.01502e6i		−	1.95576i	0.209169	−	0.977880i	$-0.432924\pi$
$102$	−20633.4	−	1.91128e6i	−0.0194433	−	1.80104i
$103$	314575.			0.287881			0.143940	−	0.989586i	$-0.454023\pi$
$103$	314575.			0.287881			0.143940	+	0.989586i	$0.454023\pi$
$104$		−	134707.i		−	0.119754i
$105$	99599.3	−	1075.23i	0.0860376	−	0.000928827i
$106$	2.12700e6			1.78587
$107$		−	36311.5i		−	0.0296410i	−0.999890	−	0.0148205i	$-0.995282\pi$
$107$		−	36311.5i		−	0.0296410i	0.999890	−	0.0148205i	$-0.00471768\pi$
$108$	76716.6	+	2.36803e6i	0.0609001	+	1.87982i
$109$	−927066.			−0.715865			−0.357932	−	0.933747i	$-0.616518\pi$
$109$	−927066.			−0.715865			−0.357932	+	0.933747i	$0.616518\pi$
$110$		−	3.60701e6i		−	2.71000i
$111$	8099.66	+	750275.i	0.00592240	+	0.548594i
$112$	−50234.1			−0.0357556
$113$			1.53691e6i			1.06516i	0.846381	+	0.532578i	$0.178777\pi$
$113$			1.53691e6i			1.06516i	−0.846381	+	0.532578i	$0.821223\pi$
$114$	−1.34234e6	+	14491.4i	−0.906042	+	0.00978126i
$115$	3.38151e6			2.22340
$116$			545153.i			0.349256i
$117$	128266.	−	2769.73i	0.0800854	−	0.00172934i
$118$	−2.07875e6			−1.26519
$119$			97379.1i			0.0577863i
$120$	−44063.7	−	4.08164e6i	−0.0254998	−	2.36206i
$121$	−37358.9			−0.0210881
$122$			1.86800e6i			1.02872i
$123$	−118161.	+	1275.62i	−0.0634978	+	0.000685496i
$124$	−6.14017e6			−3.22044
$125$		−	1.53277e6i		−	0.784781i
$126$	−3991.42	−	184842.i	−0.00199533	−	0.0924035i
$127$	−1.15266e6			−0.562718			−0.281359	−	0.959603i	$-0.590785\pi$
$127$	−1.15266e6			−0.562718			−0.281359	+	0.959603i	$0.590785\pi$
$128$		−	3.83807e6i		−	1.83013i
$129$	−37275.9	−	3.45288e6i	−0.0173643	−	1.60847i
$130$	−471978.			−0.214829
$131$			1.29218e6i			0.574790i	0.957812	+	0.287395i	$0.0927893\pi$
$131$			1.29218e6i			0.574790i	−0.957812	+	0.287395i	$0.907211\pi$
$132$	−4.37090e6	+	47186.5i	−1.90042	+	0.0205162i
$133$	68391.9			0.0290703
$134$			5.77567e6i			2.40042i
$135$	3.88556e6	−	125880.i	1.57926	−	0.0511629i
$136$	3.99065e6			1.58645
$137$			4.29547e6i			1.67051i	0.549863	+	0.835255i	$0.314680\pi$
$137$			4.29547e6i			1.67051i	−0.549863	+	0.835255i	$0.685320\pi$
$138$	−67756.6	−	6.27632e6i	−0.0257819	−	2.38818i
$139$	1.88254e6			0.700972			0.350486	−	0.936568i	$-0.386016\pi$
$139$	1.88254e6			0.700972			0.350486	+	0.936568i	$0.386016\pi$
$140$			444059.i			0.161829i
$141$	1.91831e6	−	20709.3i	0.684323	−	0.00738768i
$142$	−4.62254e6			−1.61442
$143$			236698.i			0.0809443i
$144$	−1.96019e6	+	42327.7i	−0.656463	+	0.0141755i
$145$	894511.			0.293414
$146$		−	5.02284e6i		−	1.61395i
$147$	−34188.8	−	3.16692e6i	−0.0107630	−	0.996977i
$148$	−3.34507e6			−1.03186
$149$		−	4.10368e6i		−	1.24055i	−0.784384	−	0.620276i	$-0.787021\pi$
$149$		−	4.10368e6i		−	1.24055i	0.784384	−	0.620276i	$-0.212979\pi$
$150$	−8.57297e6	+	92550.3i	−2.54014	+	0.0274223i
$151$	279509.			0.0811829			0.0405914	−	0.999176i	$-0.487076\pi$
$151$	279509.			0.0811829			0.0405914	+	0.999176i	$0.487076\pi$
$152$		−	2.80274e6i		−	0.798091i
$153$	82052.5	+	3.79984e6i	0.0229096	+	1.06094i
$154$	341101.			0.0933945
$155$			1.00751e7i			2.70553i
$156$	6174.37	+	571935.i	0.00162637	+	0.150651i
$157$	1.21865e6			0.314905			0.157453	−	0.987527i	$-0.449672\pi$
$157$	1.21865e6			0.314905			0.157453	+	0.987527i	$0.449672\pi$
$158$			9.85282e6i			2.49798i
$159$	−4.22920e6	+	45656.8i	−1.05212	+	0.0113583i
$160$	−2.46268e6			−0.601240
$161$			319777.i			0.0766248i
$162$	−311499.	−	7.20936e6i	−0.0732675	−	1.69571i
$163$	8.28605e6			1.91331			0.956654	−	0.291226i	$-0.0940632\pi$
$163$	8.28605e6			1.91331			0.956654	+	0.291226i	$0.0940632\pi$
$164$		−	526815.i		−	0.119434i
$165$	77425.7	+	7.17197e6i	0.0172359	+	1.59657i
$166$	−5.35642e6			−1.17098
$167$			4.82345e6i			1.03564i	0.855490	+	0.517819i	$0.173256\pi$
$167$			4.82345e6i			1.03564i	−0.855490	+	0.517819i	$0.826744\pi$
$168$	385985.	−	4166.94i	0.0814035	−	0.000878799i
$169$	−4.79584e6			−0.993583
$170$		−	1.39822e7i		−	2.84596i
$171$	2.66873e6	−	57627.7i	0.533723	−	0.0115250i
$172$	1.53945e7			3.02538
$173$		−	1.16122e6i		−	0.224272i	−0.993693	−	0.112136i	$-0.964231\pi$
$173$		−	1.16122e6i		−	0.224272i	0.993693	−	0.112136i	$-0.0357693\pi$
$174$	−17923.7	−	1.66028e6i	−0.00340235	−	0.315161i
$175$	436790.			0.0815003
$176$		−	3.61727e6i		−	0.663503i
$177$	4.13328e6	−	44621.2i	0.745375	−	0.00804677i
$178$	7.29019e6			1.29264
$179$		−	6.21686e6i		−	1.08396i	−0.840392	−	0.541979i	$-0.817675\pi$
$179$		−	6.21686e6i		−	1.08396i	0.840392	−	0.541979i	$-0.182325\pi$
$180$	374168.	+	1.73277e7i	0.0641578	+	2.97114i
$181$	816129.			0.137633			0.0688165	−	0.997629i	$-0.478078\pi$
$181$	816129.			0.137633			0.0688165	+	0.997629i	$0.478078\pi$
$182$		−	44633.3i		−	0.00740363i
$183$	−40097.3	−	3.71423e6i	−0.00654278	−	0.606060i
$184$	1.31046e7			2.10364
$185$			5.48874e6i			0.866876i
$186$	1.87000e7	−	201878.i	2.90605	−	0.0313725i
$187$	−7.01210e6			−1.07232
$188$			8.55270e6i			1.28715i
$189$	11904.0	+	367444.i	0.00176323	+	0.0544259i
$190$	−9.82008e6			−1.43171
$191$			1.66237e6i			0.238577i	0.992860	+	0.119288i	$0.0380613\pi$
$191$			1.66237e6i			0.238577i	−0.992860	+	0.119288i	$0.961939\pi$
$192$	99514.7	+	9.21809e6i	0.0140599	+	1.30238i
$193$	−8.31671e6			−1.15686			−0.578428	−	0.815733i	$-0.696334\pi$
$193$	−8.31671e6			−1.15686			−0.578428	+	0.815733i	$0.696334\pi$
$194$			2.11223e7i			2.89292i
$195$	938456.	−	10131.2i	0.126564	−	0.00136633i
$196$	1.41196e7			1.87523
$197$		−	6.39302e6i		−	0.836195i	−0.908402	−	0.418097i	$-0.862697\pi$
$197$		−	6.39302e6i		−	0.836195i	0.908402	−	0.418097i	$-0.137303\pi$
$198$	1.33101e7	−	287415.i	1.71470	−	0.0370266i
$199$	−1.51157e7			−1.91809			−0.959045	−	0.283254i	$-0.908586\pi$
$199$	−1.51157e7			−1.91809			−0.959045	+	0.283254i	$0.908586\pi$
$200$		−	1.78999e7i		−	2.23749i
$201$	−123977.	−	1.14840e7i	−0.0152669	−	1.41418i
$202$	−2.73606e7			−3.31950
$203$			84590.6i			0.0101119i
$204$	−1.69434e7	+	182914.i	−1.99577	+	0.0215455i
$205$	−864422.			−0.100338
$206$		−	4.27141e6i		−	0.488618i
$207$	269447.	+	1.24780e7i	0.0303782	+	1.40681i
$208$	−473321.			−0.0525976
$209$			4.92479e6i			0.539447i
$210$	−14599.9	−	1.35239e6i	−0.00157649	−	0.146031i
$211$	513037.			0.0546137			0.0273069	−	0.999627i	$-0.491307\pi$
$211$	513037.			0.0546137			0.0273069	+	0.999627i	$0.491307\pi$
$212$		−	1.88557e7i		−	1.97895i
$213$	9.19121e6	−	99224.6i	0.951117	−	0.0102679i
$214$	−493050.			−0.0503094
$215$		−	2.52600e7i		−	2.54166i
$216$	1.50581e7	−	487833.i	1.49420	−	0.0484072i
$217$	−952761.			−0.0932406
$218$			1.25880e7i			1.21503i
$219$	107817.	+	9.98714e6i	0.0102649	+	0.950843i
$220$	−3.19759e7			−3.00300
$221$			917536.i			0.0850054i
$222$	1.01875e7	−	109980.i	0.931125	−	0.0100520i
$223$	1.65192e7			1.48961			0.744807	−	0.667280i	$-0.232541\pi$
$223$	1.65192e7			1.48961			0.744807	+	0.667280i	$0.232541\pi$
$224$		−	232886.i		−	0.0207205i
$225$	1.70440e7	−	368044.i	1.49632	−	0.0323111i
$226$	2.08687e7			1.80788
$227$			4.13401e6i			0.353422i	0.984263	+	0.176711i	$0.0565458\pi$
$227$			4.13401e6i			0.353422i	−0.984263	+	0.176711i	$0.943454\pi$
$228$	128465.	+	1.18998e7i	0.0108388	+	1.00400i
$229$	6.02263e6			0.501510			0.250755	−	0.968051i	$-0.419321\pi$
$229$	6.02263e6			0.501510			0.250755	+	0.968051i	$0.419321\pi$
$230$		−	4.59153e7i		−	3.77375i
$231$	−678227.	+	7321.86i	−0.0550224	+	0.000593999i
$232$	3.46657e6			0.277611
$233$		−	2.20350e7i		−	1.74199i	−0.491292	−	0.870995i	$-0.663475\pi$
$233$		−	2.20350e7i		−	1.74199i	0.491292	−	0.870995i	$-0.336525\pi$
$234$	−37608.4	−	1.74164e6i	−0.00293520	−	0.135928i
$235$	1.40337e7			1.08135
$236$			1.84281e7i			1.40199i
$237$	−211494.	−	1.95908e7i	−0.0158874	−	1.47166i
$238$	1.32225e6			0.0980803
$239$		−	527177.i		−	0.0386156i	−0.999814	−	0.0193078i	$-0.993854\pi$
$239$		−	527177.i		−	0.0386156i	0.999814	−	0.0193078i	$-0.00614624\pi$
$240$	−1.43417e7	+	154827.i	−1.03745	+	0.0111999i
$241$	597854.			0.0427114			0.0213557	−	0.999772i	$-0.493202\pi$
$241$	597854.			0.0427114			0.0213557	+	0.999772i	$0.493202\pi$
$242$			507272.i			0.0357927i
$243$	774119.	+	1.43280e7i	0.0539497	+	0.998544i
$244$	1.65598e7			1.13995
$245$		−	2.31680e7i		−	1.57540i
$246$	17320.8	+	1.60443e6i	0.00116349	+	0.107774i
$247$	644410.			0.0427633
$248$			3.90447e7i			2.55981i
$249$	1.06504e7	−	114977.i	0.689872	−	0.00744757i
$250$	−2.08125e7			−1.33200
$251$			1.75598e7i			1.11045i	0.831700	+	0.555226i	$0.187368\pi$
$251$			1.75598e7i			1.11045i	−0.831700	+	0.555226i	$0.812632\pi$
$252$	−1.63861e6	+	35383.7i	−0.102394	+	0.00221107i
$253$	−2.30266e7			−1.42190
$254$			1.56512e7i			0.955097i
$255$	300133.	+	2.78015e7i	0.0181006	+	1.67667i
$256$	−3.02631e7			−1.80382
$257$		−	1.97693e7i		−	1.16464i	−0.812960	−	0.582319i	$-0.802145\pi$
$257$		−	1.97693e7i		−	1.16464i	0.812960	−	0.582319i	$-0.197855\pi$
$258$	−4.68844e7	+	506144.i	−2.73004	+	0.0294724i
$259$	−519050.			−0.0298751
$260$			4.18407e6i			0.238056i
$261$	71276.8	+	3.30082e6i	0.00400891	+	0.185652i
$262$	1.75457e7			0.975587
$263$		−	1.33040e7i		−	0.731330i	−0.930747	−	0.365665i	$-0.880842\pi$
$263$		−	1.33040e7i		−	0.731330i	0.930747	−	0.365665i	$-0.119158\pi$
$264$	300054.	+	2.77941e7i	0.0163075	+	1.51057i
$265$	−3.09393e7			−1.66254
$266$		−	928649.i		−	0.0493408i
$267$	−1.44954e7	+	156487.i	−0.761548	+	0.00822136i
$268$	5.12010e7			2.65995
$269$		−	7.67421e6i		−	0.394254i	−0.980378	−	0.197127i	$-0.936839\pi$
$269$		−	7.67421e6i		−	0.394254i	0.980378	−	0.197127i	$-0.0631612\pi$
$270$	−1.70924e6	−	5.27595e7i	−0.0868385	−	2.68046i
$271$	5.68712e6			0.285749			0.142874	−	0.989741i	$-0.454366\pi$
$271$	5.68712e6			0.285749			0.142874	+	0.989741i	$0.454366\pi$
$272$		−	1.40220e7i		−	0.696792i
$273$	958.069	+	88746.3i	4.70878e−5	+	0.00436177i
$274$	5.83253e7			2.83534
$275$			3.14525e7i			1.51237i
$276$	−5.56393e7	+	600659.i	−2.64639	+	0.0285694i
$277$	−2.31723e7			−1.09026			−0.545129	−	0.838352i	$-0.683519\pi$
$277$	−2.31723e7			−1.09026			−0.545129	+	0.838352i	$0.683519\pi$
$278$		−	2.55618e7i		−	1.18976i
$279$	−3.71778e7	+	802805.i	−1.71187	+	0.0369656i
$280$	2.82373e6			0.128632
$281$			3.94818e7i			1.77942i	0.456528	+	0.889709i	$0.349093\pi$
$281$			3.94818e7i			1.77942i	−0.456528	+	0.889709i	$0.650907\pi$
$282$	−281198.	−	2.60475e7i	−0.0125390	−	1.16150i
$283$	−3.37696e7			−1.48993			−0.744966	−	0.667102i	$-0.767534\pi$
$283$	−3.37696e7			−1.48993			−0.744966	+	0.667102i	$0.767534\pi$
$284$			4.09786e7i			1.78897i
$285$	1.95257e7	−	210792.i	0.843475	−	0.00910581i
$286$	3.21396e6			0.137386
$287$		−	81745.2i		−	0.00345793i
$288$	−196232.	−	9.08747e6i	−0.00821472	−	0.380422i
$289$	−3.04417e6			−0.126117
$290$		−	1.21460e7i		−	0.498010i
$291$	−453397.	−	4.19984e7i	−0.0183992	−	1.70433i
$292$	−4.45273e7			−1.78845
$293$		−	378760.i		−	0.0150578i	−0.999972	−	0.00752890i	$-0.997603\pi$
$293$		−	378760.i		−	0.0150578i	0.999972	−	0.00752890i	$-0.00239654\pi$
$294$	−4.30015e7	+	464227.i	−1.69216	+	0.0182679i
$295$	3.02376e7			1.17783
$296$			2.12710e7i			0.820185i
$297$	−2.64590e7	+	857188.i	−1.00996	+	0.0327195i
$298$	−5.57212e7			−2.10558
$299$			3.01304e6i			0.112717i
$300$	820454.	+	7.59990e7i	0.0303872	+	2.81478i
$301$	2.38875e6			0.0875932
$302$		−	3.79526e6i		−	0.137791i
$303$	5.44024e7	−	587306.i	1.95565	−	0.0211123i
$304$	−9.84802e6			−0.350532
$305$		−	2.71720e7i		−	0.957683i
$306$	5.15955e7	−	1.11414e6i	1.80072	−	0.0388843i
$307$	1.14595e7			0.396052			0.198026	−	0.980197i	$-0.436547\pi$
$307$	1.14595e7			0.396052			0.198026	+	0.980197i	$0.436547\pi$
$308$		−	3.02385e6i		−	0.103492i
$309$	91687.4	+	8.49304e6i	0.00310766	+	0.287864i
$310$	1.36803e8			4.59208
$311$			5.82988e7i			1.93811i	0.246843	+	0.969056i	$0.420607\pi$
$311$			5.82988e7i			1.93811i	−0.246843	+	0.969056i	$0.579393\pi$
$312$	3.63688e6	−	39262.2i	0.119747	−	0.00129274i
$313$	−3.90492e7			−1.27344			−0.636721	−	0.771094i	$-0.719710\pi$
$313$	−3.90492e7			−1.27344			−0.636721	+	0.771094i	$0.719710\pi$
$314$		−	1.65472e7i		−	0.534486i
$315$	58059.2	+	2.68871e6i	0.00185755	+	0.0860226i
$316$	8.73448e7			2.76806
$317$		−	3.36043e7i		−	1.05491i	−0.849582	−	0.527457i	$-0.823146\pi$
$317$		−	3.36043e7i		−	1.05491i	0.849582	−	0.527457i	$-0.176854\pi$
$318$	619943.	+	5.74256e7i	0.0192784	+	1.78576i
$319$	−6.09123e6			−0.187643
$320$			6.74362e7i			2.05799i
$321$	980353.	−	10583.5i	0.0296393	−	0.000319973i
$322$	4.34204e6			0.130055
$323$			1.90905e7i			0.566511i
$324$	−6.39106e7	+	2.76142e6i	−1.87905	+	0.0811891i
$325$	4.11558e6			0.119889
$326$		−	1.12511e8i		−	3.24744i
$327$	−270206.	−	2.50293e7i	−0.00772773	−	0.715823i
$328$	−3.34997e6			−0.0949334
$329$			1.32711e6i			0.0372666i
$330$	9.73835e7	−	1.05131e6i	2.70984	−	0.0292543i
$331$	−5.02864e6			−0.138665			−0.0693325	−	0.997594i	$-0.522087\pi$
$331$	−5.02864e6			−0.138665			−0.0693325	+	0.997594i	$0.522087\pi$
$332$			4.74844e7i			1.29759i
$333$	−2.02539e7	+	437356.i	−0.548498	+	0.0118441i
$334$	6.54944e7			1.75778
$335$		−	8.40129e7i		−	2.23466i
$336$	−14641.4	−	1.35624e6i	−0.000385981	−	0.0357535i
$337$	−4.87659e7			−1.27417			−0.637084	−	0.770794i	$-0.719860\pi$
$337$	−4.87659e7			−1.27417			−0.637084	+	0.770794i	$0.719860\pi$
$338$			6.51195e7i			1.68640i
$339$	−4.14941e7	+	447954.i	−1.06509	+	0.0114983i
$340$	−1.23952e8			−3.15367
$341$		−	6.86067e7i		−	1.73023i
$342$	−782488.	−	3.62369e7i	−0.0195614	−	0.905884i
$343$	4.38835e6			0.108747
$344$		−	9.78922e7i		−	2.40476i
$345$	985588.	+	9.12954e7i	0.0240015	+	2.22327i
$346$	−1.57674e7			−0.380655
$347$		−	8.13294e7i		−	1.94652i	−0.229702	−	0.973261i	$-0.573775\pi$
$347$		−	8.13294e7i		−	1.94652i	0.229702	−	0.973261i	$-0.426225\pi$
$348$	−1.47183e7	+	158892.i	−0.349236	+	0.00377021i
$349$	3.86015e7			0.908087			0.454044	−	0.890979i	$-0.349981\pi$
$349$	3.86015e7			0.908087			0.454044	+	0.890979i	$0.349981\pi$
$350$		−	5.93089e6i		−	0.138330i
$351$	112163.	+	3.46217e6i	0.00259376	+	0.0800621i
$352$	1.67697e7			0.384502
$353$			1.33384e7i			0.303235i	0.988439	+	0.151618i	$0.0484482\pi$
$353$			1.33384e7i			0.303235i	−0.988439	+	0.151618i	$0.951552\pi$
$354$	−605882.	−	5.61231e7i	−0.0136577	−	1.26512i
$355$	6.72396e7			1.50293
$356$		−	6.46272e7i		−	1.43240i
$357$	−2.62908e6	+	28382.5i	−0.0577829	+	0.000623801i
$358$	−8.44147e7			−1.83979
$359$			4.24232e7i			0.916896i	0.888721	+	0.458448i	$0.151594\pi$
$359$			4.24232e7i			0.916896i	−0.888721	+	0.458448i	$0.848406\pi$
$360$	1.10185e8	−	2.37930e6i	2.36164	−	0.0509967i
$361$	−3.36381e7			−0.715007
$362$		−	1.10817e7i		−	0.233604i
$363$	−10888.8	−	1.00863e6i	−0.000227646	−	0.0210869i
$364$	−395672.			−0.00820410
$365$			7.30623e7i			1.50250i
$366$	−5.04331e7	+	544455.i	−1.02866	+	0.0111050i
$367$	−8.14651e6			−0.164806			−0.0824031	−	0.996599i	$-0.526259\pi$
$367$	−8.14651e6			−0.164806			−0.0824031	+	0.996599i	$0.526259\pi$
$368$		−	4.60459e7i		−	0.923948i
$369$	−68879.3	−	3.18979e6i	−0.00137091	−	0.0634867i
$370$	7.45279e7			1.47134
$371$		−	2.92582e6i		−	0.0572962i
$372$	−1.78964e6	−	1.65775e8i	−0.0347645	−	3.22025i
$373$	−6.46998e7			−1.24674			−0.623371	−	0.781927i	$-0.714237\pi$
$373$	−6.46998e7			−1.24674			−0.623371	+	0.781927i	$0.714237\pi$
$374$			9.52127e7i			1.82004i
$375$	4.13825e7	−	446749.i	0.784735	−	0.00847168i
$376$	5.43858e7			1.02311
$377$			797039.i			0.0148750i
$378$	4.98927e6	−	161637.i	0.0923766	−	0.00299271i
$379$	2.93057e7			0.538313			0.269156	−	0.963097i	$-0.413255\pi$
$379$	2.93057e7			0.538313			0.269156	+	0.963097i	$0.413255\pi$
$380$			8.70545e7i			1.58650i
$381$	−335959.	−	3.11201e7i	−0.00607452	−	0.562685i
$382$	2.25723e7			0.404935
$383$			4.26896e7i			0.759846i	0.925018	+	0.379923i	$0.124049\pi$
$383$			4.26896e7i			0.759846i	−0.925018	+	0.379923i	$0.875951\pi$
$384$	1.03622e8	−	1.11866e6i	1.83003	−	0.0197562i
$385$	−4.96166e6			−0.0869451
$386$			1.12927e8i			1.96352i
$387$	9.32114e7	−	2.01278e6i	1.60819	−	0.0347267i
$388$	1.87248e8			3.20569
$389$			2.25880e7i			0.383733i	0.981421	+	0.191867i	$0.0614541\pi$
$389$			2.25880e7i			0.383733i	−0.981421	+	0.191867i	$0.938546\pi$
$390$	−137565.	−	1.27427e7i	−0.00231907	−	0.214816i
$391$	−8.92603e7			−1.49323
$392$		−	8.97850e7i		−	1.49055i
$393$	−3.48868e7	+	376624.i	−0.574757	+	0.00620484i
$394$	−8.68066e7			−1.41927
$395$		−	1.43319e8i		−	2.32548i
$396$	−2.54792e6	−	1.17994e8i	−0.0410299	−	1.90009i
$397$	5.94055e7			0.949412			0.474706	−	0.880144i	$-0.342554\pi$
$397$	5.94055e7			0.949412			0.474706	+	0.880144i	$0.342554\pi$
$398$			2.05246e8i			3.25556i
$399$	19933.8	+	1.84647e6i	0.000313813	+	0.0290686i
$400$	−6.28952e7			−0.982737
$401$			2.90133e7i			0.449950i	0.974365	+	0.224975i	$0.0722300\pi$
$401$			2.90133e7i			0.449950i	−0.974365	+	0.224975i	$0.927770\pi$
$402$	−1.55934e8	+	1.68340e6i	−2.40028	+	0.0259125i
$403$	−8.97722e6			−0.137160
$404$			2.42551e8i			3.67840i
$405$	4.53106e6	+	1.04867e8i	0.0682080	+	1.57861i
$406$	1.14860e6			0.0171629
$407$		−	3.73759e7i		−	0.554381i
$408$	1.16313e6	+	1.07741e8i	0.0171257	+	1.58636i
$409$	7.41317e6			0.108351			0.0541757	−	0.998531i	$-0.482747\pi$
$409$	7.41317e6			0.108351			0.0541757	+	0.998531i	$0.482747\pi$
$410$			1.17374e7i			0.170303i
$411$	−1.15971e8	+	1.25197e6i	−1.67041	+	0.0180331i
$412$	−3.78659e7			−0.541447
$413$			2.85946e6i			0.0405913i
$414$	1.69431e8	−	3.65864e6i	2.38777	−	0.0515607i
$415$	7.79145e7			1.09012
$416$		−	2.19433e6i		−	0.0304805i
$417$	548694.	+	5.08258e7i	0.00756697	+	0.700932i
$418$	6.68704e7			0.915599
$419$		−	3.91080e7i		−	0.531647i	−0.964022	−	0.265824i	$-0.914356\pi$
$419$		−	3.91080e7i		−	0.531647i	0.964022	−	0.265824i	$-0.0856439\pi$
$420$	−1.19889e7	+	129427.i	−0.161820	+	0.00174694i
$421$	−3.08155e7			−0.412974			−0.206487	−	0.978449i	$-0.566203\pi$
$421$	−3.08155e7			−0.412974			−0.206487	+	0.978449i	$0.566203\pi$
$422$		−	6.96620e6i		−	0.0926955i
$423$	1.11824e6	+	5.17853e7i	0.0147745	+	0.684204i
$424$	−1.19902e8			−1.57300
$425$			1.21923e8i			1.58825i
$426$	−1.34731e6	−	1.24801e8i	−0.0174276	−	1.61432i
$427$	2.56955e6			0.0330046
$428$			4.37086e6i			0.0557489i
$429$	−6.39047e6	+	68988.9i	−0.0809396	+	0.000873791i
$430$	−3.42989e8			−4.31395
$431$			8.56880e7i			1.07026i	0.844771	+	0.535129i	$0.179737\pi$
$431$			8.56880e7i			1.07026i	−0.844771	+	0.535129i	$0.820263\pi$
$432$	−1.71410e6	−	5.29096e7i	−0.0212611	−	0.656271i
$433$	−5.22248e7			−0.643299			−0.321649	−	0.946859i	$-0.604237\pi$
$433$	−5.22248e7			−0.643299			−0.321649	+	0.946859i	$0.604237\pi$
$434$			1.29369e7i			0.158257i
$435$	260718.	+	2.41504e7i	0.00316740	+	0.293397i
$436$	1.11592e8			1.34640
$437$			6.26899e7i			0.751196i
$438$	1.35609e8	−	1.46398e6i	1.61386	−	0.0174226i
$439$	1.30499e7			0.154246			0.0771230	−	0.997022i	$-0.475427\pi$
$439$	1.30499e7			0.154246			0.0771230	+	0.997022i	$0.475427\pi$
$440$			2.03332e8i			2.38697i
$441$	8.54919e7	−	1.84609e6i	0.996802	−	0.0215247i
$442$	1.24586e7			0.144279
$443$		−	1.01385e8i		−	1.16617i	−0.812410	−	0.583086i	$-0.801845\pi$
$443$		−	1.01385e8i		−	1.16617i	0.812410	−	0.583086i	$-0.198155\pi$
$444$	−974967.	−	9.03116e7i	−0.0111389	−	1.03180i
$445$	−1.06043e8			−1.20338
$446$		−	2.24303e8i		−	2.52831i
$447$	1.10793e8	−	1.19608e6i	1.24048	−	0.0133917i
$448$	−6.37719e6			−0.0709243
$449$			5.68747e7i			0.628319i	0.949370	+	0.314160i	$0.101723\pi$
$449$			5.68747e7i			0.628319i	−0.949370	+	0.314160i	$0.898277\pi$
$450$	−4.99742e6	−	2.31430e8i	−0.0548414	−	2.53969i
$451$	5.88634e6			0.0641675
$452$		−	1.85000e8i		−	2.00335i
$453$	81466.7	+	7.54629e6i	0.000876366	+	0.0811781i
$454$	5.61330e7			0.599861
$455$			649236.i			0.00689236i
$456$	7.56696e7	−	816898.i	0.798044	−	0.00861536i
$457$	1.08129e8			1.13290			0.566450	−	0.824096i	$-0.308316\pi$
$457$	1.08129e8			1.13290			0.566450	+	0.824096i	$0.308316\pi$
$458$		−	8.17773e7i		−	0.851209i
$459$	−1.02566e8	+	3.32280e6i	−1.06063	+	0.0343611i
$460$	−4.07037e8			−4.18177
$461$			4.16392e7i			0.425010i	0.977160	+	0.212505i	$0.0681622\pi$
$461$			4.16392e7i			0.425010i	−0.977160	+	0.212505i	$0.931838\pi$
$462$	99418.8	+	9.20920e6i	0.00100819	+	0.0933891i
$463$	1.69101e8			1.70374			0.851869	−	0.523756i	$-0.175470\pi$
$463$	1.69101e8			1.70374			0.851869	+	0.523756i	$0.175470\pi$
$464$		−	1.21805e7i		−	0.121930i
$465$	−2.72011e8	+	2.93652e6i	−2.70537	+	0.0292061i
$466$	−2.99199e8			−2.95667
$467$		−	7.43172e7i		−	0.729690i	−0.931068	−	0.364845i	$-0.881122\pi$
$467$		−	7.43172e7i		−	0.729690i	0.931068	−	0.364845i	$-0.118878\pi$
$468$	−1.54395e7	+	333397.i	−0.150625	+	0.00325255i
$469$	7.94479e6			0.0770130
$470$		−	1.90554e8i		−	1.83537i
$471$	355192.	+	3.29016e7i	0.00339939	+	0.314887i
$472$	1.17182e8			1.11439
$473$			1.72009e8i			1.62543i
$474$	−2.66011e8	+	2.87174e6i	−2.49784	+	0.0269656i
$475$	8.56295e7			0.798993
$476$		−	1.17217e7i		−	0.108685i
$477$	−2.46532e6	−	1.14169e8i	−0.0227153	−	1.05194i
$478$	−7.15818e6			−0.0655419
$479$			1.68099e8i			1.52953i	0.644309	+	0.764766i	$0.277145\pi$
$479$			1.68099e8i			1.52953i	−0.644309	+	0.764766i	$0.722855\pi$
$480$	−717782.	−	6.64884e7i	−0.00649036	−	0.601204i
$481$	−4.89065e6			−0.0439472
$482$		−	8.11787e6i		−	0.0724938i
$483$	−8.63347e6	+	93203.4i	−0.0766203	+	0.000827162i
$484$	4.49694e6			0.0396626
$485$		−	3.07245e8i		−	2.69314i
$486$	1.94551e8	−	1.05112e7i	1.69482	−	0.0915684i
$487$	1.03798e7			0.0898672			0.0449336	−	0.998990i	$-0.485692\pi$
$487$	1.03798e7			0.0898672			0.0449336	+	0.998990i	$0.485692\pi$
$488$		−	1.05302e8i		−	0.906101i
$489$	2.41509e6	+	2.23710e8i	0.0206541	+	1.91320i
$490$	−3.14583e8			−2.67391
$491$		−	1.70348e8i		−	1.43911i	−0.694436	−	0.719554i	$-0.744346\pi$
$491$		−	1.70348e8i		−	1.43911i	0.694436	−	0.719554i	$-0.255654\pi$
$492$	1.42232e7	−	153548.i	0.119427	−	0.00128928i
$493$	−2.36120e7			−0.197057
$494$		−	8.75002e6i		−	0.0725819i
$495$	−1.93609e8	+	4.18074e6i	−1.59629	+	0.0344697i
$496$	1.37192e8			1.12430
$497$			6.35860e6i			0.0517955i
$498$	−1.56120e6	−	1.44615e8i	−0.0126407	−	1.17091i
$499$	2.28803e8			1.84145			0.920724	−	0.390214i	$-0.127599\pi$
$499$	2.28803e8			1.84145			0.920724	+	0.390214i	$0.127599\pi$
$500$			1.84502e8i			1.47602i
$501$	−1.30225e8	+	1.40586e6i	−1.03558	+	0.0111797i
$502$	2.38434e8			1.88476
$503$		−	1.70112e8i		−	1.33669i	−0.743850	−	0.668347i	$-0.767002\pi$
$503$		−	1.70112e8i		−	1.33669i	0.743850	−	0.668347i	$-0.232998\pi$
$504$	225002.	+	1.04198e7i	0.00175750	+	0.0813893i
$505$	3.97988e8			3.09027
$506$			3.12663e8i			2.41337i
$507$	−1.39781e6	−	1.29480e8i	−0.0107257	−	0.993525i
$508$	1.38747e8			1.05836
$509$		−	1.57876e8i		−	1.19719i	−0.801052	−	0.598595i	$-0.795726\pi$
$509$		−	1.57876e8i		−	1.19719i	0.801052	−	0.598595i	$-0.204274\pi$
$510$	3.77498e8	−	4.07531e6i	2.84580	−	0.0307221i
$511$	−6.90923e6			−0.0517806
$512$			1.65286e8i			1.23148i
$513$	2.33369e6	+	7.20346e7i	0.0172859	+	0.533567i
$514$	−2.68434e8			−1.97673
$515$			6.21320e7i			0.454876i
$516$	4.48695e6	+	4.15628e8i	0.0326589	+	3.02521i
$517$	−9.55630e7			−0.691542
$518$			7.04783e6i			0.0507068i
$519$	3.13511e7	−	338453.i	0.224259	−	0.00242101i
$520$	2.66061e7			0.189222
$521$			1.23877e8i			0.875948i	0.898988	+	0.437974i	$0.144304\pi$
$521$			1.23877e8i			0.875948i	−0.898988	+	0.437974i	$0.855696\pi$
$522$	4.48196e7	−	967821.i	0.315106	−	0.00680430i
$523$	1.98518e8			1.38770			0.693848	−	0.720122i	$-0.255914\pi$
$523$	1.98518e8			1.38770			0.693848	+	0.720122i	$0.255914\pi$
$524$		−	1.55542e8i		−	1.08107i
$525$	127309.	+	1.17927e7i	0.000879792	+	0.0814955i
$526$	−1.80646e8			−1.24128
$527$		−	2.65947e8i		−	1.81704i
$528$	9.76606e7	−	1.05430e6i	0.663464	−	0.00716249i
$529$	−1.45080e8			−0.980033
$530$			4.20105e8i			2.82182i
$531$	2.40940e6	+	1.11579e8i	0.0160926	+	0.745245i
$532$	−8.23243e6			−0.0546755
$533$		−	770229.i		−	0.00508673i
$534$	2.12483e6	+	1.96824e8i	0.0139540	+	1.29257i
$535$	7.17191e6			0.0468353
$536$		−	3.25582e8i		−	2.11430i
$537$	1.67845e8	−	1.81199e6i	1.08389	−	0.0117013i
$538$	−1.04203e8			−0.669165
$539$			1.57764e8i			1.00749i
$540$	−4.67711e8	+	1.51523e7i	−2.97027	+	0.0962274i
$541$	−1.24422e8			−0.785789			−0.392894	−	0.919584i	$-0.628526\pi$
$541$	−1.24422e8			−0.785789			−0.392894	+	0.919584i	$0.628526\pi$
$542$		−	7.72216e7i		−	0.484999i
$543$	237872.	+	2.20342e7i	0.00148574	+	0.137625i
$544$	6.50063e7			0.403793
$545$		−	1.83105e8i		−	1.13113i
$546$	1.20503e6	−	13009.0i	0.00740319	−	7.99219e-5i
$547$	2.51668e8			1.53768			0.768841	−	0.639439i	$-0.220834\pi$
$547$	2.51668e8			1.53768			0.768841	+	0.639439i	$0.220834\pi$
$548$		−	5.17051e8i		−	3.14190i
$549$	1.00267e8	−	2.16513e6i	0.605954	−	0.0130848i
$550$	4.27073e8			2.56693
$551$			1.65834e7i			0.0991329i
$552$	3.81953e6	+	3.53805e8i	0.0227087	+	2.10352i
$553$	1.35532e7			0.0801430
$554$			3.14641e8i			1.85049i
$555$	−1.48187e8	+	1.59977e6i	−0.866826	+	0.00935790i
$556$	−2.26605e8			−1.31839
$557$		−	2.27023e8i		−	1.31372i	−0.754012	−	0.656861i	$-0.771884\pi$
$557$		−	2.27023e8i		−	1.31372i	0.754012	−	0.656861i	$-0.228116\pi$
$558$	1.09008e7	+	5.04812e8i	0.0627414	+	2.90554i
$559$	2.25075e7			0.128852
$560$		−	9.92176e6i		−	0.0564969i
$561$	−2.04378e6	−	1.89316e8i	−0.0115756	−	1.07226i
$562$	5.36097e8			3.02019
$563$		−	2.20513e8i		−	1.23569i	−0.786301	−	0.617844i	$-0.788006\pi$
$563$		−	2.20513e8i		−	1.23569i	0.786301	−	0.617844i	$-0.211994\pi$
$564$	−2.30910e8	+	2.49281e6i	−1.28708	+	0.0138948i
$565$	−3.03556e8			−1.68304
$566$			4.58535e8i			2.52885i
$567$	−9.91693e6	+	428486.i	−0.0544036	+	0.00235065i
$568$	2.60579e8			1.42198
$569$		−	8.15870e7i		−	0.442878i	−0.975174	−	0.221439i	$-0.928925\pi$
$569$		−	8.15870e7i		−	0.442878i	0.975174	−	0.221439i	$-0.0710754\pi$
$570$	−2.86220e6	−	2.65127e8i	−0.0154552	−	1.43162i
$571$	1.89117e8			1.01583			0.507915	−	0.861407i	$-0.330416\pi$
$571$	1.89117e8			1.01583			0.507915	+	0.861407i	$0.330416\pi$
$572$		−	2.84916e7i		−	0.152240i
$573$	−4.48814e7	+	484522.i	−0.238563	+	0.00257543i
$574$	−1.10996e6			−0.00586913
$575$			4.00374e8i			2.10602i
$576$	−2.48845e8	+	5.37348e6i	−1.30215	+	0.0281182i
$577$	1.65004e8			0.858949			0.429475	−	0.903079i	$-0.358699\pi$
$577$	1.65004e8			0.858949			0.429475	+	0.903079i	$0.358699\pi$
$578$			4.13347e7i			0.214058i
$579$	−2.42402e6	−	2.24538e8i	−0.0124882	−	1.15679i
$580$	−1.07673e8			−0.551855
$581$			7.36809e6i			0.0375687i
$582$	−5.70268e8	+	6.15638e6i	−2.89275	+	0.0312289i
$583$	2.10683e8			1.06322
$584$			2.83144e8i			1.42157i
$585$	547052.	+	2.53339e7i	0.00273251	+	0.126542i
$586$	−5.14293e6			−0.0255575
$587$			3.40867e8i			1.68528i	0.538481	+	0.842638i	$0.318998\pi$
$587$			3.40867e8i			1.68528i	−0.538481	+	0.842638i	$0.681002\pi$
$588$	4.11535e6	+	3.81206e8i	0.0202430	+	1.87512i
$589$	−1.86782e8			−0.914089
$590$		−	4.10576e8i		−	1.99911i
$591$	1.72602e8	−	1.86334e6i	0.836146	−	0.00902669i
$592$	7.47400e7			0.360237
$593$			8.15842e7i			0.391239i	0.980680	+	0.195619i	$0.0626717\pi$
$593$			8.15842e7i			0.391239i	−0.980680	+	0.195619i	$0.937328\pi$
$594$	1.16392e7	+	3.59269e8i	0.0555346	+	1.71420i
$595$	−1.92334e7			−0.0913073
$596$			4.93966e8i			2.33323i
$597$	−4.40568e6	−	4.08100e8i	−0.0207057	−	1.91798i
$598$	4.09120e7			0.191314
$599$		−	1.22349e8i		−	0.569272i	−0.958636	−	0.284636i	$-0.908127\pi$
$599$		−	1.22349e8i		−	0.569272i	0.958636	−	0.284636i	$-0.0918727\pi$
$600$	4.83270e8	−	5.21718e6i	2.23736	−	0.0241536i
$601$	−9.85220e7			−0.453847			−0.226924	−	0.973913i	$-0.572867\pi$
$601$	−9.85220e7			−0.453847			−0.226924	+	0.973913i	$0.572867\pi$
$602$		−	3.24352e7i		−	0.148671i
$603$	3.10014e8	−	6.69435e6i	1.41394	−	0.0305321i
$604$	−3.36448e7			−0.152689
$605$		−	7.37878e6i		−	0.0333210i
$606$	−7.97464e6	−	7.38694e8i	−0.0358338	−	3.31930i
$607$	8.21188e7			0.367178			0.183589	−	0.983003i	$-0.441228\pi$
$607$	8.21188e7			0.367178			0.183589	+	0.983003i	$0.441228\pi$
$608$		−	4.56557e7i		−	0.203135i
$609$	−2.28381e6	+	24655.1i	−0.0101113	+	0.000109158i
$610$	−3.68950e8			−1.62547
$611$			1.25045e7i			0.0548203i
$612$	−9.87677e6	−	4.57391e8i	−0.0430884	−	1.99542i
$613$	4.12186e8			1.78942			0.894709	−	0.446650i	$-0.147383\pi$
$613$	4.12186e8			1.78942			0.894709	+	0.446650i	$0.147383\pi$
$614$		−	1.55602e8i		−	0.672216i
$615$	−251948.	−	2.33380e7i	−0.00108314	−	0.100332i
$616$	−1.92284e7			−0.0822621
$617$			7.46215e7i			0.317694i	0.987303	+	0.158847i	$0.0507776\pi$
$617$			7.46215e7i			0.317694i	−0.987303	+	0.158847i	$0.949222\pi$
$618$	1.15321e8	−	1.24496e6i	0.488590	−	0.00527462i
$619$	2.37501e7			0.100137			0.0500684	−	0.998746i	$-0.484056\pi$
$619$	2.37501e7			0.100137			0.0500684	+	0.998746i	$0.484056\pi$
$620$		−	1.21275e9i		−	5.08857i
$621$	−3.36809e8	+	1.09115e7i	−1.40640	+	0.0455629i
$622$	7.91601e8			3.28954
$623$		−	1.00281e7i		−	0.0414720i
$624$	−137956.	−	1.27789e7i	−0.000567789	−	0.0525945i
$625$	−6.26586e7			−0.256650
$626$			5.30223e8i			2.16140i
$627$	−1.32961e8	+	1.43540e6i	−0.539415	+	0.00582331i
$628$	−1.46690e8			−0.592274
$629$		−	1.44884e8i		−	0.582195i
$630$	3.65082e7	−	788348.i	0.146005	−	0.00315280i
$631$	−2.27078e8			−0.903829			−0.451914	−	0.892061i	$-0.649259\pi$
$631$	−2.27078e8			−0.903829			−0.451914	+	0.892061i	$0.649259\pi$
$632$		−	5.55417e8i		−	2.20023i
$633$	149532.	+	1.38512e7i	0.000589553	+	0.0546105i
$634$	−4.56290e8			−1.79050
$635$		−	2.27663e8i		−	0.889142i
$636$	5.09075e8	−	5.49577e6i	1.97884	−	0.0213627i
$637$	2.06435e7			0.0798666
$638$			8.27087e7i			0.318485i
$639$	5.35781e6	+	2.48119e8i	0.0205345	+	0.950951i
$640$	7.58060e8			2.89177
$641$		−	4.24289e8i		−	1.61097i	−0.592616	−	0.805485i	$-0.701905\pi$
$641$		−	4.24289e8i		−	1.61097i	0.592616	−	0.805485i	$-0.298095\pi$
$642$	−143706.	−	1.33116e7i	−0.000543089	−	0.0503065i
$643$	−1.53087e8			−0.575846			−0.287923	−	0.957654i	$-0.592965\pi$
$643$	−1.53087e8			−0.575846			−0.287923	+	0.957654i	$0.592965\pi$
$644$		−	3.84920e7i		−	0.144116i
$645$	6.81980e8	−	7.36238e6i	2.54151	−	0.0274372i
$646$	2.59217e8			0.961536
$647$			3.83609e8i			1.41637i	0.706029	+	0.708183i	$0.250485\pi$
$647$			3.83609e8i			1.41637i	−0.706029	+	0.708183i	$0.749515\pi$
$648$	1.75596e7	+	4.06402e8i	0.0645342	+	1.49359i
$649$	−2.05905e8			−0.753237
$650$		−	5.58827e7i		−	0.203487i
$651$	−277696.	−	2.57231e7i	−0.00100653	−	0.0932351i
$652$	−9.97404e8			−3.59856
$653$		−	1.47876e8i		−	0.531078i	−0.964100	−	0.265539i	$-0.914450\pi$
$653$		−	1.47876e8i		−	0.531078i	0.964100	−	0.265539i	$-0.0855499\pi$
$654$	−3.39857e8	+	3.66895e6i	−1.21496	+	0.0131162i
$655$	−2.55219e8			−0.908218
$656$			1.17708e7i			0.0416961i
$657$	−2.69606e8	+	5.82178e6i	−0.950677	+	0.0205286i
$658$	1.80200e7			0.0632523
$659$		−	2.12554e8i		−	0.742699i	−0.928493	−	0.371350i	$-0.878895\pi$
$659$		−	2.12554e8i		−	0.742699i	0.928493	−	0.371350i	$-0.121105\pi$
$660$	−9.31984e6	−	8.63300e8i	−0.0324173	−	3.00283i
$661$	−4.46406e8			−1.54570			−0.772850	−	0.634589i	$-0.781170\pi$
$661$	−4.46406e8			−1.54570			−0.772850	+	0.634589i	$0.781170\pi$
$662$			6.82806e7i			0.235355i
$663$	−2.47720e7	+	267429.i	−0.0850004	+	0.000917630i
$664$	3.01949e8			1.03140
$665$			1.35081e7i			0.0459336i
$666$	5.93857e6	+	2.75014e8i	0.0201029	+	0.930962i
$667$	−7.75380e7			−0.261299
$668$		−	5.80605e8i		−	1.94783i
$669$	4.81475e6	+	4.45992e8i	0.0160803	+	1.48953i
$670$	−1.14076e9			−3.79287
$671$			1.85029e8i			0.612453i
$672$	6.28757e6	−	67878.0i	0.0207193	−	0.000223677i
$673$	−2.08664e8			−0.684544			−0.342272	−	0.939601i	$-0.611196\pi$
$673$	−2.08664e8			−0.684544			−0.342272	+	0.939601i	$0.611196\pi$
$674$			6.62160e8i			2.16264i
$675$	1.49043e7	+	4.60055e8i	0.0484619	+	1.49588i
$676$	5.77281e8			1.86873
$677$			6.00935e7i			0.193670i	0.995300	+	0.0968348i	$0.0308718\pi$
$677$			6.00935e7i			0.193670i	−0.995300	+	0.0968348i	$0.969128\pi$
$678$	6.08247e6	+	5.63422e8i	0.0195160	+	1.80778i
$679$	2.90550e7			0.0928137
$680$			7.88196e8i			2.50673i
$681$	−1.11612e8	+	1.20491e6i	−0.353401	+	0.00381518i
$682$	−9.31565e8			−2.93670
$683$		−	2.66812e7i		−	0.0837421i	−0.999123	−	0.0418710i	$-0.986668\pi$
$683$		−	2.66812e7i		−	0.0837421i	0.999123	−	0.0418710i	$-0.0133319\pi$
$684$	−3.21238e8	+	6.93672e6i	−1.00383	+	0.0216763i
$685$	−8.48401e8			−2.63955
$686$		−	5.95865e7i		−	0.184576i
$687$	1.75538e6	+	1.62602e8i	0.00541378	+	0.501481i
$688$	−3.43965e8			−1.05621
$689$		−	2.75680e7i		−	0.0842844i
$690$	1.23964e9	−	1.33826e7i	3.77353	−	0.0407375i
$691$	1.17465e8			0.356019			0.178009	−	0.984029i	$-0.443034\pi$
$691$	1.17465e8			0.356019			0.178009	+	0.984029i	$0.443034\pi$
$692$			1.39777e8i			0.421812i
$693$	−395358.	−	1.83089e7i	−0.00118793	−	0.0550128i
$694$	−1.10432e9			−3.30382
$695$			3.71823e8i			1.10760i
$696$	1.01038e6	+	9.35920e7i	0.00299680	+	0.277595i
$697$	2.28178e7			0.0673869
$698$		−	5.24144e8i		−	1.54129i
$699$	5.94911e8	−	6.42241e6i	1.74189	−	0.0188047i
$700$	−5.25771e7			−0.153286
$701$			3.35071e8i			0.972708i	0.873762	+	0.486354i	$0.161673\pi$
$701$			3.35071e8i			0.972708i	−0.873762	+	0.486354i	$0.838327\pi$
$702$	4.70105e7	−	1.52299e6i	0.135889	−	0.00440237i
$703$	−1.01756e8			−0.292883
$704$		−	4.59211e8i		−	1.31612i
$705$	4.09030e6	+	3.78887e8i	0.0116732	+	1.08129i
$706$	1.81113e8			0.514679
$707$			3.76363e7i			0.106500i
$708$	−4.97528e8	+	5.37111e6i	−1.40190	+	0.0151344i
$709$	3.57900e8			1.00421			0.502103	−	0.864808i	$-0.332560\pi$
$709$	3.57900e8			1.00421			0.502103	+	0.864808i	$0.332560\pi$
$710$		−	9.13002e8i		−	2.55092i
$711$	5.28859e8	−	1.14200e7i	1.47140	−	0.0317730i
$712$	−4.10958e8			−1.13856
$713$		−	8.73327e8i		−	2.40940i
$714$	385387.	+	3.56986e7i	0.00105877	+	0.0980745i
$715$	−4.67504e7			−0.127899
$716$			7.48332e8i			2.03871i
$717$	1.42329e7	−	153653.i	0.0386133	−	0.000416854i
$718$	5.76037e8			1.55624
$719$			5.38743e7i			0.144942i	0.997371	+	0.0724711i	$0.0230885\pi$
$719$			5.38743e7i			0.144942i	−0.997371	+	0.0724711i	$0.976911\pi$
$720$	−8.36017e6	−	3.87158e8i	−0.0223984	−	1.03727i
$721$	−5.87559e6			−0.0156764
$722$			4.56750e8i			1.21358i
$723$	174253.	+	1.61411e7i	0.000461068	+	0.0427090i
$724$	−9.82385e7			−0.258861
$725$			1.05911e8i			0.277925i
$726$	−1.36955e7	+	147851.i	−0.0357906	+	0.000386381i
$727$	6.38366e8			1.66137			0.830685	−	0.556743i	$-0.187949\pi$
$727$	6.38366e8			1.66137			0.830685	+	0.556743i	$0.187949\pi$
$728$			2.51604e6i			0.00652113i
$729$	−3.86608e8	+	2.50761e7i	−0.997903	+	0.0647258i
$730$	9.92065e8			2.55018
$731$			6.66778e8i			1.70698i
$732$	4.82657e6	+	4.47087e8i	0.0123057	+	1.13988i
$733$	1.79490e8			0.455751			0.227876	−	0.973690i	$-0.426822\pi$
$733$	1.79490e8			0.455751			0.227876	+	0.973690i	$0.426822\pi$
$734$			1.10616e8i			0.279724i
$735$	6.25500e8	−	6.75265e6i	1.57531	−	0.0170064i
$736$	2.13470e8			0.535431
$737$			5.72091e8i			1.42910i
$738$	−4.33120e7	+	935267.i	−0.107755	+	0.00232684i
$739$	−2.17172e7			−0.0538110			−0.0269055	−	0.999638i	$-0.508565\pi$
$739$	−2.17172e7			−0.0538110			−0.0269055	+	0.999638i	$0.508565\pi$
$740$		−	6.60687e8i		−	1.63042i
$741$	187822.	+	1.73981e7i	0.000461629	+	0.0427608i
$742$	−3.97278e7			−0.0972484
$743$			2.87628e8i			0.701238i	0.936518	+	0.350619i	$0.114029\pi$
$743$			2.87628e8i			0.701238i	−0.936518	+	0.350619i	$0.885971\pi$
$744$	−1.05415e9	+	1.13801e7i	−2.55966	+	0.0276330i
$745$	8.10521e8			1.96018
$746$			8.78516e8i			2.11608i
$747$	6.20842e6	+	2.87511e8i	0.0148943	+	0.689751i
$748$	8.44056e8			2.01682
$749$			678221.i			0.00161408i
$750$	−6.06610e6	−	5.61906e8i	−0.0143789	−	1.33192i
$751$	−2.76752e8			−0.653389			−0.326694	−	0.945130i	$-0.605935\pi$
$751$	−2.76752e8			−0.653389			−0.326694	+	0.945130i	$0.605935\pi$
$752$		−	1.91096e8i		−	0.449364i
$753$	−4.74088e8	+	5.11806e6i	−1.11039	+	0.0119873i
$754$	1.08225e7			0.0252472
$755$			5.52059e7i			0.128276i
$756$	−1.43290e6	−	4.42297e7i	−0.00331628	−	0.102364i
$757$	−2.47607e8			−0.570789			−0.285395	−	0.958410i	$-0.592125\pi$
$757$	−2.47607e8			−0.570789			−0.285395	+	0.958410i	$0.592125\pi$
$758$		−	3.97923e8i		−	0.913674i
$759$	−6.71142e6	−	6.21681e8i	−0.0153493	−	1.42181i
$760$	5.53571e8			1.26105
$761$		−	4.84437e8i		−	1.09922i	−0.835423	−	0.549608i	$-0.814777\pi$
$761$		−	4.84437e8i		−	1.09922i	0.835423	−	0.549608i	$-0.185223\pi$
$762$	−4.22559e8	+	4.56177e6i	−0.955041	+	0.0103102i
$763$	1.73156e7			0.0389820
$764$		−	2.00102e8i		−	0.448716i
$765$	−7.50508e8	+	1.62062e7i	−1.67637	+	0.0361991i
$766$	5.79654e8			1.28968
$767$			2.69427e7i			0.0597111i
$768$	−8.82060e6	−	8.17056e8i	−0.0194722	−	1.80372i
$769$	−2.09455e8			−0.460587			−0.230294	−	0.973121i	$-0.573969\pi$
$769$	−2.09455e8			−0.460587			−0.230294	+	0.973121i	$0.573969\pi$
$770$			6.73712e7i			0.147571i
$771$	5.33739e8	−	5.76203e6i	1.16457	−	0.0125722i
$772$	1.00109e9			2.17582
$773$		−	2.55411e8i		−	0.552970i	−0.961018	−	0.276485i	$-0.910830\pi$
$773$		−	2.55411e8i		−	0.552970i	0.961018	−	0.276485i	$-0.0891697\pi$
$774$	−2.73302e7	−	1.26566e9i	−0.0589413	−	2.72956i
$775$	−1.19290e9			−2.56270
$776$		−	1.19069e9i		−	2.54809i
$777$	−151284.	−	1.40135e7i	−0.000322501	−	0.0298734i
$778$	3.06708e8			0.651307
$779$		−	1.60256e7i		−	0.0339001i
$780$	−1.12963e8	+	1.21950e6i	−0.238042	+	0.00256980i
$781$	−4.57872e8			−0.961149
$782$			1.21201e9i			2.53446i
$783$	−8.90961e7	+	2.88643e6i	−0.185598	+	0.00601279i
$784$	−3.15478e8			−0.654669
$785$			2.40696e8i			0.497577i
$786$	5.11393e6	+	4.73705e8i	0.0105314	+	0.975530i
$787$	−2.95068e8			−0.605339			−0.302669	−	0.953096i	$-0.597878\pi$
$787$	−2.95068e8			−0.605339			−0.302669	+	0.953096i	$0.597878\pi$
$788$			7.69537e8i			1.57272i
$789$	3.59186e8	−	3.87762e6i	0.731288	−	0.00789468i
$790$	−1.94604e9			−3.94702
$791$		−	2.87062e7i		−	0.0580024i
$792$	−7.50311e8	+	1.62020e7i	−1.51031	+	0.0326131i
$793$	2.42112e7			0.0485507
$794$		−	8.06628e8i		−	1.61143i
$795$	−9.01770e6	−	8.35313e8i	−0.0179471	−	1.66245i
$796$	1.81950e9			3.60755
$797$			4.45213e8i			0.879414i	0.898141	+	0.439707i	$0.144918\pi$
$797$			4.45213e8i			0.879414i	−0.898141	+	0.439707i	$0.855082\pi$
$798$	2.50721e7	−	270668.i	0.0493380	−	0.000532633i
$799$	−3.70441e8			−0.726237
$800$		−	2.91583e8i		−	0.569499i
$801$	−8.44979e6	−	3.91308e8i	−0.0164418	−	0.761415i
$802$	3.93952e8			0.763696
$803$		−	4.97522e8i		−	0.960872i
$804$	1.49232e7	+	1.38235e9i	0.0287141	+	2.65980i
$805$	−6.31593e7			−0.121074
$806$			1.21896e8i			0.232800i
$807$	2.07191e8	−	2.23675e6i	0.394231	−	0.00425596i
$808$	1.54236e9			2.92382
$809$			4.19216e8i			0.791759i	0.918303	+	0.395879i	$0.129560\pi$
$809$			4.19216e8i			0.791759i	−0.918303	+	0.395879i	$0.870440\pi$
$810$	1.42393e9	−	6.15243e7i	2.67937	−	0.115769i
$811$	−2.21258e8			−0.414798			−0.207399	−	0.978256i	$-0.566500\pi$
$811$	−2.21258e8			−0.414798			−0.207399	+	0.978256i	$0.566500\pi$
$812$		−	1.01823e7i		−	0.0190185i
$813$	1.65759e6	+	1.53543e8i	0.00308465	+	0.285732i
$814$	−5.07502e8			−0.940947
$815$			1.63658e9i			3.02319i
$816$	3.78572e8	−	4.08691e6i	0.696751	−	0.00752184i
$817$	4.68296e8			0.858725
$818$		−	1.00659e8i		−	0.183904i
$819$	−2.39573e6	+	51732.7i	−0.00436100	+	9.41702e-5i
$820$	1.04052e8			0.188715
$821$			1.17998e8i			0.213228i	0.994300	+	0.106614i	$0.0340009\pi$
$821$			1.17998e8i			0.213228i	−0.994300	+	0.106614i	$0.965999\pi$
$822$	1.69997e7	+	1.57469e9i	0.0306074	+	2.83518i
$823$	−1.85580e8			−0.332913			−0.166457	−	0.986049i	$-0.553233\pi$
$823$	−1.85580e8			−0.332913			−0.166457	+	0.986049i	$0.553233\pi$
$824$			2.40785e8i			0.430376i
$825$	−8.49169e8	+	9.16728e6i	−1.51228	+	0.0163260i
$826$	3.88267e7			0.0688954
$827$		−	4.77130e8i		−	0.843569i	−0.906696	−	0.421784i	$-0.861404\pi$
$827$		−	4.77130e8i		−	0.843569i	0.906696	−	0.421784i	$-0.138596\pi$
$828$	−3.24337e7	−	1.50200e9i	−0.0571354	−	2.64593i
$829$	6.65299e8			1.16776			0.583880	−	0.811840i	$-0.301534\pi$
$829$	6.65299e8			1.16776			0.583880	+	0.811840i	$0.301534\pi$
$830$		−	1.05795e9i		−	1.85025i
$831$	−6.75388e6	−	6.25614e8i	−0.0117693	−	1.09019i
$832$	−6.00879e7			−0.104332
$833$			6.11557e8i			1.05804i
$834$	6.90129e8	−	7.45036e6i	1.18969	−	0.0128434i
$835$	−9.52682e8			−1.63640
$836$		−	5.92803e8i		−	1.01459i
$837$	−3.25105e7	−	1.00351e9i	−0.0554430	−	1.71137i
$838$	−5.31022e8			−0.902361
$839$		−	1.97867e8i		−	0.335033i	−0.985869	−	0.167517i	$-0.946425\pi$
$839$		−	1.97867e8i		−	0.335033i	0.985869	−	0.167517i	$-0.0535748\pi$
$840$	823015.	+	7.62362e7i	0.00138858	+	0.128624i
$841$	−2.05111e7			−0.0344828
$842$			4.18423e8i			0.700938i
$843$	−1.06595e9	+	1.15075e7i	−1.77931	+	0.0192087i
$844$	−6.17550e7			−0.102718
$845$		−	9.47229e8i		−	1.56995i
$846$	7.03159e8	−	1.51838e7i	1.16129	−	0.0250766i
$847$	697784.			0.00114834
$848$			4.21300e8i			0.690882i
$849$	−9.84262e6	−	9.11726e8i	−0.0160838	−	1.48985i
$850$	1.65551e9			2.69572
$851$		−	4.75775e8i		−	0.771992i
$852$	−1.10636e9	+	1.19438e7i	−1.78886	+	0.0193118i
$853$	1.68464e8			0.271432			0.135716	−	0.990748i	$-0.456667\pi$
$853$	1.68464e8			0.271432			0.135716	+	0.990748i	$0.456667\pi$
$854$		−	3.48903e7i		−	0.0560184i
$855$	1.13821e7	+	5.27102e8i	0.0182106	+	0.843327i
$856$	2.77939e7			0.0443127
$857$		−	4.17440e8i		−	0.663211i	−0.943418	−	0.331606i	$-0.892410\pi$
$857$		−	4.17440e8i		−	0.663211i	0.943418	−	0.331606i	$-0.107590\pi$
$858$	936755.	+	8.67720e7i	0.00148308	+	0.137378i
$859$	4.62926e8			0.730352			0.365176	−	0.930938i	$-0.381009\pi$
$859$	4.62926e8			0.730352			0.365176	+	0.930938i	$0.381009\pi$
$860$			3.04058e9i			4.78037i
$861$	2.20699e6	−	23825.8i	0.00345773	−	3.73283e-5i
$862$	1.16350e9			1.81654
$863$			3.65865e8i			0.569231i	0.958642	+	0.284616i	$0.0918660\pi$
$863$			3.65865e8i			0.569231i	−0.958642	+	0.284616i	$0.908134\pi$
$864$	2.45290e8	−	7.94663e6i	0.380311	−	0.0123209i
$865$	2.29353e8			0.354369
$866$			7.09126e8i			1.09187i
$867$	−887265.	−	8.21877e7i	−0.00136143	−	0.126110i
$868$	1.14685e8			0.175367
$869$			9.75941e8i			1.48718i
$870$	3.27922e8	−	3.54011e6i	0.497981	−	0.00537600i
$871$	7.48583e7			0.113288
$872$		−	7.09604e8i		−	1.07020i
$873$	1.13376e9	−	2.44820e7i	1.70403	−	0.0367964i
$874$	8.51225e8			1.27500
$875$			2.86289e7i			0.0427348i
$876$	−1.29781e7	−	1.20217e9i	−0.0193063	−	1.78835i
$877$	−6.15176e7			−0.0912012			−0.0456006	−	0.998960i	$-0.514520\pi$
$877$	−6.15176e7			−0.0912012			−0.0456006	+	0.998960i	$0.514520\pi$
$878$		−	1.77196e8i		−	0.261801i
$879$	1.02259e7	−	110395.i	0.0150569	−	0.000162548i
$880$	7.14449e8			1.04839
$881$		−	3.87838e7i		−	0.0567182i	−0.999598	−	0.0283591i	$-0.990972\pi$
$881$		−	3.87838e7i		−	0.0567182i	0.999598	−	0.0283591i	$-0.00902819\pi$
$882$	−2.50668e7	−	1.16084e9i	−0.0365336	−	1.69187i
$883$	−5.49322e8			−0.797893			−0.398947	−	0.916974i	$-0.630624\pi$
$883$	−5.49322e8			−0.797893			−0.398947	+	0.916974i	$0.630624\pi$
$884$		−	1.10445e8i		−	0.159878i
$885$	8.81316e6	+	8.16367e8i	0.0127146	+	1.17776i
$886$	−1.37664e9			−1.97934
$887$		−	4.14885e8i		−	0.594507i	−0.954799	−	0.297254i	$-0.903929\pi$
$887$		−	4.14885e8i		−	0.594507i	0.954799	−	0.297254i	$-0.0960707\pi$
$888$	−5.74282e8	+	6.19972e6i	−0.820137	+	0.00885387i
$889$	2.15293e7			0.0306425
$890$			1.43989e9i			2.04249i
$891$	−3.08546e7	−	7.14101e8i	−0.0436201	−	1.00955i
$892$	−1.98844e9			−2.80167
$893$			2.60170e8i			0.365345i
$894$	−1.62407e7	−	1.50438e9i	−0.0227297	−	2.10546i
$895$	1.22790e9			1.71274
$896$			7.16869e7i			0.0996589i
$897$	−8.13473e7	+	878192.i	−0.112711	+	0.00121678i
$898$	7.72265e8			1.06644
$899$		−	2.31021e8i		−	0.317960i
$900$	−2.05161e9	+	4.43019e7i	−2.81428	+	0.0607708i
$901$	8.16693e8			1.11657
$902$		−	7.99267e7i		−	0.108911i
$903$	696233.	+	6.44924e7i	0.000945565	+	0.0875881i
$904$	−1.17640e9			−1.59239
$905$			1.61194e8i			0.217472i
$906$	1.02466e8	−	1.10618e6i	0.137783	−	0.00148745i
$907$	−6.83297e8			−0.915772			−0.457886	−	0.889011i	$-0.651393\pi$
$907$	−6.83297e8			−0.915772			−0.457886	+	0.889011i	$0.651393\pi$
$908$		−	4.97616e8i		−	0.664717i
$909$	3.17127e7	+	1.46861e9i	0.0422222	+	1.95530i
$910$	8.81555e6			0.0116984
$911$		−	2.14298e7i		−	0.0283442i	−0.999900	−	0.0141721i	$-0.995489\pi$
$911$		−	2.14298e7i		−	0.0283442i	0.999900	−	0.0141721i	$-0.00451126\pi$
$912$	−2.87034e6	−	2.65881e8i	−0.00378398	−	0.350512i
$913$	−5.30564e8			−0.697148
$914$		−	1.46821e9i		−	1.92286i
$915$	7.33601e8	−	7.91965e6i	0.957627	−	0.0103382i
$916$	−7.24952e8			−0.943241
$917$		−	2.41352e7i		−	0.0312999i
$918$	4.51182e7	+	1.39267e9i	0.0583208	+	1.80020i
$919$	−3.16031e8			−0.407176			−0.203588	−	0.979057i	$-0.565260\pi$
$919$	−3.16031e8			−0.407176			−0.203588	+	0.979057i	$0.565260\pi$
$920$			2.58831e9i			3.32393i
$921$	3.34004e6	+	3.09390e8i	0.00427537	+	0.396029i
$922$	5.65391e8			0.721367
$923$			5.99127e7i			0.0761928i
$924$	8.16391e7	−	881343.i	0.103486	−	0.00111719i
$925$	−6.49872e8			−0.821112
$926$		−	2.29611e9i		−	2.89174i
$927$	−2.29272e8	+	4.95083e6i	−0.287814	+	0.00621497i
$928$	5.64692e7			0.0706591
$929$		−	1.05925e9i		−	1.32115i	−0.750762	−	0.660573i	$-0.770314\pi$
$929$		−	1.05925e9i		−	1.32115i	0.750762	−	0.660573i	$-0.229686\pi$
$930$	3.98730e7	+	3.69345e9i	0.0495713	+	4.59181i
$931$	4.29513e8			0.532264
$932$			2.65238e9i			3.27634i
$933$	−1.57398e9	+	1.69920e7i	−1.93800	+	0.0209218i
$934$	−1.00910e9			−1.23850
$935$		−	1.38497e9i		−	1.69435i
$936$	2.12004e6	+	9.81785e7i	0.00258533	+	0.119726i
$937$	−1.43586e9			−1.74539			−0.872697	−	0.488263i	$-0.837631\pi$
$937$	−1.43586e9			−1.74539			−0.872697	+	0.488263i	$0.837631\pi$
$938$		−	1.07877e8i		−	0.130714i
$939$	−1.13814e7	−	1.05427e9i	−0.0137468	−	1.27337i
$940$	−1.68925e9			−2.03381
$941$			4.06906e8i			0.488343i	0.969732	+	0.244172i	$0.0785160\pi$
$941$			4.06906e8i			0.488343i	−0.969732	+	0.244172i	$0.921484\pi$
$942$	4.46749e8	−	4.82292e6i	0.534455	−	0.00576976i
$943$	7.49299e7			0.0893552
$944$		−	4.11744e8i		−	0.489454i
$945$	−7.25740e7	+	2.35117e6i	−0.0859975	+	0.00278605i
$946$	2.33560e9			2.75883
$947$			3.29812e8i			0.388344i	0.980968	+	0.194172i	$0.0622020\pi$
$947$			3.29812e8i			0.388344i	−0.980968	+	0.194172i	$0.937798\pi$
$948$	2.54579e7	+	2.35817e9i	0.0298811	+	2.76790i
$949$	−6.51010e7			−0.0761709
$950$		−	1.16271e9i		−	1.35612i
$951$	9.07262e8	−	9.79443e6i	1.05485	−	0.0113877i
$952$	−7.45368e7			−0.0863893
$953$			2.13750e8i			0.246961i	0.992347	+	0.123480i	$0.0394056\pi$
$953$			2.13750e8i			0.246961i	−0.992347	+	0.123480i	$0.960594\pi$
$954$	−1.55022e9	+	3.34750e7i	−1.78545	+	0.0385545i
$955$	−3.28336e8			−0.376972
$956$			6.34570e7i			0.0726283i
$957$	−1.77537e6	−	1.64454e8i	−0.00202560	−	0.187632i
$958$	2.28250e9			2.59606
$959$		−	8.02302e7i		−	0.0909666i
$960$	−1.82067e9	+	1.96552e7i	−2.05787	+	0.0222159i
$961$	1.71454e9			1.93186
$962$			6.64069e7i			0.0745913i
$963$	571475.	+	2.64649e7i	0.000639910	+	0.0296341i
$964$	−7.19645e7			−0.0803318
$965$		−	1.64264e9i		−	1.82793i
$966$	1.26555e6	+	1.17228e8i	0.00140394	+	0.130047i
$967$	1.12700e9			1.24637			0.623184	−	0.782075i	$-0.285839\pi$
$967$	1.12700e9			1.24637			0.623184	+	0.782075i	$0.285839\pi$
$968$		−	2.85956e7i		−	0.0315263i
$969$	−5.15412e8	+	5.56418e6i	−0.566478	+	0.00611547i
$970$	−4.17188e9			−4.57105
$971$			1.15248e9i			1.25886i	0.777058	+	0.629428i	$0.216711\pi$
$971$			1.15248e9i			1.25886i	−0.777058	+	0.629428i	$0.783289\pi$
$972$	−9.31817e7	−	1.72468e9i	−0.101469	−	1.87806i
$973$	−3.51619e7			−0.0381710
$974$		−	1.40940e8i		−	0.152531i
$975$	1.19954e6	+	1.11114e8i	0.00129420	+	0.119882i
$976$	−3.70000e8			−0.397972
$977$		−	7.93051e8i		−	0.850388i	−0.905102	−	0.425194i	$-0.860206\pi$
$977$		−	7.93051e8i		−	0.850388i	0.905102	−	0.425194i	$-0.139794\pi$
$978$	3.03762e9	−	3.27929e7i	3.24726	−	0.0350560i
$979$	7.22108e8			0.769580
$980$			2.78877e9i			2.96302i
$981$	6.75673e8	−	1.45903e7i	0.715698	−	0.0154546i
$982$	−2.31305e9			−2.44259
$983$		−	1.36914e9i		−	1.44141i	−0.693244	−	0.720703i	$-0.743819\pi$
$983$		−	1.36914e9i		−	1.44141i	0.693244	−	0.720703i	$-0.256181\pi$
$984$	−976395.	−	9.04439e7i	−0.00102480	−	0.0949279i
$985$	1.26269e9			1.32126
$986$			3.20612e8i			0.334464i
$987$	−3.58299e7	+	386805.i	−0.0372644	+	0.000402291i
$988$	−7.75685e7			−0.0804294
$989$			2.18959e9i			2.26346i
$990$	5.67676e7	+	2.62890e9i	0.0585052	+	2.70937i
$991$	−4.51892e8			−0.464316			−0.232158	−	0.972678i	$-0.574579\pi$
$991$	−4.51892e8			−0.464316			−0.232158	+	0.972678i	$0.574579\pi$
$992$			6.36024e8i			0.651537i
$993$	−1.46567e6	−	1.35765e8i	−0.00149688	−	0.138657i
$994$	8.63392e7			0.0879122
$995$		−	2.98551e9i		−	3.03075i
$996$	−1.28200e9	+	1.38400e7i	−1.29751	+	0.0140074i
$997$	−8.81793e8			−0.889777			−0.444888	−	0.895586i	$-0.646757\pi$
$997$	−8.81793e8			−0.889777			−0.444888	+	0.895586i	$0.646757\pi$
$998$		−	3.10676e9i		−	3.12548i
$999$	−1.77112e7	−	5.46695e8i	−0.0177644	−	0.548339i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	87.7.b.a.59.6	✓	56
3.2	odd	2	inner	87.7.b.a.59.51	yes	56

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
87.7.b.a.59.6	✓	56	1.1	even	1	trivial
87.7.b.a.59.51	yes	56	3.2	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 87 = 3 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	87.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-13.5783i q^{2} +(0.291464 + 26.9984i) q^{3} -120.371 q^{4} +197.511i q^{5} +(366.594 - 3.95760i) q^{6} -18.6779 q^{7} +765.430i q^{8} +(-728.830 + 15.7381i) q^{9} +2681.87 q^{10} -1344.96i q^{11} +(-35.0839 - 3249.84i) q^{12} -175.989 q^{13} +253.614i q^{14} +(-5332.48 + 57.5672i) q^{15} +2689.50 q^{16} -5213.61i q^{17} +(213.698 + 9896.30i) q^{18} -3661.66 q^{19} -23774.6i q^{20} +(-5.44392 - 504.273i) q^{21} -18262.3 q^{22} -17120.6i q^{23} +(-20665.4 + 223.095i) q^{24} -23385.5 q^{25} +2389.63i q^{26} +(-637.333 - 19672.7i) q^{27} +2248.28 q^{28} -4528.92i q^{29} +(781.668 + 72406.2i) q^{30} +51010.2 q^{31} +12468.6i q^{32} +(36311.8 - 392.008i) q^{33} -70792.2 q^{34} -3689.08i q^{35} +(87730.3 - 1894.42i) q^{36} +27789.6 q^{37} +49719.2i q^{38} +(-51.2944 - 4751.42i) q^{39} -151181. q^{40} +4376.58i q^{41} +(-6847.19 + 73.9195i) q^{42} -127892. q^{43} +161895. i q^{44} +(-3108.45 - 143952. i) q^{45} -232470. q^{46} -71052.6i q^{47} +(783.892 + 72612.2i) q^{48} -117300. q^{49} +317536. i q^{50} +(140759. - 1519.58i) q^{51} +21184.0 q^{52} +156646. i q^{53} +(-267122. + 8653.92i) q^{54} +265644. q^{55} -14296.6i q^{56} +(-1067.24 - 98859.0i) q^{57} -61495.3 q^{58} -153093. i q^{59} +(641878. - 6929.45i) q^{60} -137572. q^{61} -692634. i q^{62} +(13613.0 - 293.955i) q^{63} +341430. q^{64} -34759.6i q^{65} +(-5322.81 - 493054. i) q^{66} -425359. q^{67} +627569. i q^{68} +(462230. - 4990.05i) q^{69} -50091.5 q^{70} -340435. i q^{71} +(-12046.4 - 557868. i) q^{72} +369916. q^{73} -377336. i q^{74} +(-6816.02 - 631371. i) q^{75} +440759. q^{76} +25121.0i q^{77} +(-64516.4 + 696.492i) q^{78} -725628. q^{79} +531204. i q^{80} +(530946. - 22940.9i) q^{81} +59426.8 q^{82} -394483. i q^{83} +(655.293 + 60700.0i) q^{84} +1.02974e6 q^{85} +1.73656e6i q^{86} +(122274. - 1320.02i) q^{87} +1.02947e6 q^{88} +536899. i q^{89} +(-1.95463e6 + 42207.6i) q^{90} +3287.09 q^{91} +2.06083e6i q^{92} +(14867.6 + 1.37719e6i) q^{93} -964777. q^{94} -723216. i q^{95} +(-336632. + 3634.14i) q^{96} -1.55559e6 q^{97} +1.59274e6i q^{98} +(21167.2 + 980248. i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q + 52 q^{3} - 1924 q^{4} - 160 q^{6} + 160 q^{7} - 1060 q^{9} - 3588 q^{10} - 2166 q^{12} - 1400 q^{13} - 6240 q^{15} + 56588 q^{16} - 5978 q^{18} + 25000 q^{19} + 7520 q^{21} + 20970 q^{22} + 1238 q^{24}+ \cdots + 4793544 q^{99}+O(q^{100}) \) 56 * q + 52 * q^3 - 1924 * q^4 - 160 * q^6 + 160 * q^7 - 1060 * q^9 - 3588 * q^10 - 2166 * q^12 - 1400 * q^13 - 6240 * q^15 + 56588 * q^16 - 5978 * q^18 + 25000 * q^19 + 7520 * q^21 + 20970 * q^22 + 1238 * q^24 - 213304 * q^25 + 25492 * q^27 + 7870 * q^28 + 105092 * q^30 - 78656 * q^31 + 18952 * q^33 - 96642 * q^34 - 40224 * q^36 + 52120 * q^37 - 112096 * q^39 + 343440 * q^40 - 341758 * q^42 - 286760 * q^43 + 68312 * q^45 + 255516 * q^46 + 386086 * q^48 + 1051872 * q^49 - 256656 * q^51 + 58606 * q^52 + 450124 * q^54 - 409632 * q^55 - 1141736 * q^57 + 817806 * q^60 + 309976 * q^61 + 567632 * q^63 - 2844730 * q^64 + 958426 * q^66 + 748696 * q^67 - 1323704 * q^69 - 2123748 * q^70 - 1026242 * q^72 - 333728 * q^73 - 331316 * q^75 + 2846728 * q^76 + 292446 * q^78 - 661184 * q^79 + 1614852 * q^81 + 376908 * q^82 - 1670584 * q^84 + 3342000 * q^85 - 927168 * q^88 + 835882 * q^90 + 2248016 * q^91 + 3960776 * q^93 - 1368294 * q^94 - 5811456 * q^96 - 3057296 * q^97 + 4793544 * q^99

Introduction

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