LMFDB - Embedded newform 69.6.c.b.68.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [69,6,Mod(68,69)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(69, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1]))

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("69.68");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 69 = 3 \cdot 23 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	69.c (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$11.0664835671$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			68.5
Character	$\chi$	$=$	69.68
Dual form			69.6.c.b.68.27

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-8.75944i q^{2} +(10.6432 + 11.3896i) q^{3} -44.7279 q^{4} -43.1688 q^{5} +(99.7663 - 93.2285i) q^{6} -151.444i q^{7} +111.489i q^{8} +(-16.4449 + 242.443i) q^{9} +O(q^{10})$	\(q-8.75944i q^{2} +(10.6432 + 11.3896i) q^{3} -44.7279 q^{4} -43.1688 q^{5} +(99.7663 - 93.2285i) q^{6} -151.444i q^{7} +111.489i q^{8} +(-16.4449 + 242.443i) q^{9} +378.135i q^{10} -610.914 q^{11} +(-476.047 - 509.431i) q^{12} +132.232 q^{13} -1326.57 q^{14} +(-459.454 - 491.674i) q^{15} -454.710 q^{16} -1720.33 q^{17} +(2123.67 + 144.048i) q^{18} +554.256i q^{19} +1930.85 q^{20} +(1724.89 - 1611.85i) q^{21} +5351.26i q^{22} +(2536.22 - 62.8693i) q^{23} +(-1269.81 + 1186.60i) q^{24} -1261.46 q^{25} -1158.28i q^{26} +(-2936.35 + 2393.07i) q^{27} +6773.79i q^{28} -7005.84i q^{29} +(-4306.79 + 4024.56i) q^{30} +4819.56 q^{31} +7550.66i q^{32} +(-6502.07 - 6958.05i) q^{33} +15069.2i q^{34} +6537.67i q^{35} +(735.543 - 10844.0i) q^{36} -1324.61i q^{37} +4854.97 q^{38} +(1407.37 + 1506.07i) q^{39} -4812.84i q^{40} +231.107i q^{41} +(-14118.9 - 15109.1i) q^{42} -18413.9i q^{43} +27324.9 q^{44} +(709.905 - 10466.0i) q^{45} +(-550.700 - 22215.8i) q^{46} -11782.0i q^{47} +(-4839.57 - 5178.96i) q^{48} -6128.42 q^{49} +11049.7i q^{50} +(-18309.8 - 19593.9i) q^{51} -5914.46 q^{52} +1891.55 q^{53} +(20961.9 + 25720.8i) q^{54} +26372.4 q^{55} +16884.4 q^{56} +(-6312.74 + 5899.05i) q^{57} -61367.2 q^{58} +7712.06i q^{59} +(20550.4 + 21991.5i) q^{60} -1208.84i q^{61} -42216.7i q^{62} +(36716.6 + 2490.48i) q^{63} +51588.8 q^{64} -5708.30 q^{65} +(-60948.6 + 56954.5i) q^{66} +41089.5i q^{67} +76946.8 q^{68} +(27709.5 + 28217.3i) q^{69} +57266.4 q^{70} -8193.96i q^{71} +(-27029.7 - 1833.42i) q^{72} -78243.7 q^{73} -11602.9 q^{74} +(-13425.9 - 14367.5i) q^{75} -24790.7i q^{76} +92519.5i q^{77} +(13192.3 - 12327.8i) q^{78} -77477.3i q^{79} +19629.3 q^{80} +(-58508.1 - 7973.88i) q^{81} +2024.37 q^{82} +17093.4 q^{83} +(-77150.5 + 72094.7i) q^{84} +74264.7 q^{85} -161296. q^{86} +(79793.5 - 74564.5i) q^{87} -68110.1i q^{88} +135956. q^{89} +(-91676.0 - 6218.37i) q^{90} -20025.8i q^{91} +(-113439. + 2812.01i) q^{92} +(51295.5 + 54892.7i) q^{93} -103204. q^{94} -23926.5i q^{95} +(-85998.8 + 80363.1i) q^{96} +75181.6i q^{97} +53681.6i q^{98} +(10046.4 - 148112. i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q - 408 q^{4} - 528 q^{6} - 444 q^{9}+O(q^{10}) $ 32 * q - 408 * q^4 - 528 * q^6 - 444 * q^9	$ 32 q - 408 q^{4} - 528 q^{6} - 444 q^{9} - 2484 q^{12} + 520 q^{13} + 4936 q^{16} + 7188 q^{18} + 18660 q^{24} + 36032 q^{25} - 22032 q^{27} + 6544 q^{31} - 33912 q^{36} - 63912 q^{39} + 54328 q^{46} + 88284 q^{48} - 207664 q^{49} + 46296 q^{52} - 38628 q^{54} - 139296 q^{55} - 184144 q^{58} + 486584 q^{64} - 113580 q^{69} + 37176 q^{70} - 15504 q^{72} - 93896 q^{73} + 249840 q^{75} + 368028 q^{78} - 339372 q^{81} - 23512 q^{82} + 259584 q^{85} + 509928 q^{87} + 82740 q^{93} - 562000 q^{94} + 1404 q^{96}+O(q^{100}) $ 32 * q - 408 * q^4 - 528 * q^6 - 444 * q^9 - 2484 * q^12 + 520 * q^13 + 4936 * q^16 + 7188 * q^18 + 18660 * q^24 + 36032 * q^25 - 22032 * q^27 + 6544 * q^31 - 33912 * q^36 - 63912 * q^39 + 54328 * q^46 + 88284 * q^48 - 207664 * q^49 + 46296 * q^52 - 38628 * q^54 - 139296 * q^55 - 184144 * q^58 + 486584 * q^64 - 113580 * q^69 + 37176 * q^70 - 15504 * q^72 - 93896 * q^73 + 249840 * q^75 + 368028 * q^78 - 339372 * q^81 - 23512 * q^82 + 259584 * q^85 + 509928 * q^87 + 82740 * q^93 - 562000 * q^94 + 1404 * q^96

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/69\mathbb{Z}\right)^\times$.

$n$	$28$	$47$
$\chi(n)$	$-1$	$-1$

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 $

$2$		−	8.75944i		−	1.54847i	−0.632901	−	0.774233i	$-0.718136\pi$
$2$		−	8.75944i		−	1.54847i	0.632901	−	0.774233i	$-0.281864\pi$
$3$	10.6432	+	11.3896i	0.682761	+	0.730642i
$3$	10.6432	+	11.3896i	0.682761	+	0.730642i
$4$	−44.7279			−1.39775
$5$	−43.1688			−0.772227			−0.386113	−	0.922451i	$-0.626183\pi$
$5$	−43.1688			−0.772227			−0.386113	+	0.922451i	$0.626183\pi$
$6$	99.7663	−	93.2285i	1.13137	−	1.05723i
$7$		−	151.444i		−	1.16818i	−0.811690	−	0.584088i	$-0.801452\pi$
$7$		−	151.444i		−	1.16818i	0.811690	−	0.584088i	$-0.198548\pi$
$8$			111.489i			0.615895i
$9$	−16.4449	+	242.443i	−0.0676743	+	0.997707i
$10$			378.135i			1.19577i
$11$	−610.914			−1.52229			−0.761146	−	0.648580i	$-0.775363\pi$
$11$	−610.914			−1.52229			−0.761146	+	0.648580i	$0.775363\pi$
$12$	−476.047	−	509.431i	−0.954326	−	1.02125i
$13$	132.232			0.217009			0.108505	−	0.994096i	$-0.465394\pi$
$13$	132.232			0.217009			0.108505	+	0.994096i	$0.465394\pi$
$14$	−1326.57			−1.80888
$15$	−459.454	−	491.674i	−0.527246	−	0.564221i
$16$	−454.710			−0.444053
$17$	−1720.33			−1.44374			−0.721872	−	0.692026i	$-0.756718\pi$
$17$	−1720.33			−1.44374			−0.721872	+	0.692026i	$0.756718\pi$
$18$	2123.67	+	144.048i	1.54492	+	0.104791i
$19$			554.256i			0.352230i	0.984370	+	0.176115i	$0.0563530\pi$
$19$			554.256i			0.352230i	−0.984370	+	0.176115i	$0.943647\pi$
$20$	1930.85			1.07938
$21$	1724.89	−	1611.85i	0.853518	−	0.797585i
$22$			5351.26i			2.35722i
$23$	2536.22	−	62.8693i	0.999693	−	0.0247810i
$23$	2536.22	−	62.8693i	0.999693	−	0.0247810i
$24$	−1269.81	+	1186.60i	−0.449999	+	0.420509i
$25$	−1261.46			−0.403666
$26$		−	1158.28i		−	0.336031i
$27$	−2936.35	+	2393.07i	−0.775172	+	0.631750i
$28$			6773.79i			1.63281i
$29$		−	7005.84i		−	1.54691i	−0.633851	−	0.773455i	$-0.718527\pi$
$29$		−	7005.84i		−	1.54691i	0.633851	−	0.773455i	$-0.281473\pi$
$30$	−4306.79	+	4024.56i	−0.873677	+	0.816423i
$31$	4819.56			0.900748			0.450374	−	0.892840i	$-0.351291\pi$
$31$	4819.56			0.900748			0.450374	+	0.892840i	$0.351291\pi$
$32$			7550.66i			1.30350i
$33$	−6502.07	−	6958.05i	−1.03936	−	1.11225i
$34$			15069.2i			2.23559i
$35$			6537.67i			0.902097i
$36$	735.543	−	10844.0i	0.0945915	−	1.39454i
$37$		−	1324.61i		−	0.159069i	−0.996832	−	0.0795343i	$-0.974657\pi$
$37$		−	1324.61i		−	0.159069i	0.996832	−	0.0795343i	$-0.0253433\pi$
$38$	4854.97			0.545416
$39$	1407.37	+	1506.07i	0.148166	+	0.158556i
$40$		−	4812.84i		−	0.475611i
$41$			231.107i			0.0214710i	0.999942	+	0.0107355i	$0.00341729\pi$
$41$			231.107i			0.0214710i	−0.999942	+	0.0107355i	$0.996583\pi$
$42$	−14118.9	−	15109.1i	−1.23503	−	1.32164i
$43$		−	18413.9i		−	1.51871i	−0.650675	−	0.759357i	$-0.725514\pi$
$43$		−	18413.9i		−	1.51871i	0.650675	−	0.759357i	$-0.274486\pi$
$44$	27324.9			2.12778
$45$	709.905	−	10466.0i	0.0522599	−	0.770456i
$46$	−550.700	−	22215.8i	−0.0383726	−	1.54799i
$47$		−	11782.0i		−	0.777990i	−0.921240	−	0.388995i	$-0.872822\pi$
$47$		−	11782.0i		−	0.777990i	0.921240	−	0.388995i	$-0.127178\pi$
$48$	−4839.57	−	5178.96i	−0.303182	−	0.324444i
$49$	−6128.42			−0.364635
$50$			11049.7i			0.625063i
$51$	−18309.8	−	19593.9i	−0.985733	−	1.05486i
$52$	−5914.46			−0.303324
$53$	1891.55			0.0924973			0.0462487	−	0.998930i	$-0.485273\pi$
$53$	1891.55			0.0924973			0.0462487	+	0.998930i	$0.485273\pi$
$54$	20961.9	+	25720.8i	0.978244	+	1.20033i
$55$	26372.4			1.17556
$56$	16884.4			0.719474
$57$	−6312.74	+	5899.05i	−0.257354	+	0.240489i
$58$	−61367.2			−2.39534
$59$			7712.06i			0.288430i	0.989546	+	0.144215i	$0.0460657\pi$
$59$			7712.06i			0.288430i	−0.989546	+	0.144215i	$0.953934\pi$
$60$	20550.4	+	21991.5i	0.736956	+	0.788637i
$61$		−	1208.84i		−	0.0415953i	−0.999784	−	0.0207977i	$-0.993379\pi$
$61$		−	1208.84i		−	0.0415953i	0.999784	−	0.0207977i	$-0.00662058\pi$
$62$		−	42216.7i		−	1.39478i
$63$	36716.6	+	2490.48i	1.16550	+	0.0790555i
$64$	51588.8			1.57437
$65$	−5708.30			−0.167580
$66$	−60948.6	+	56954.5i	−1.72228	+	1.60942i
$67$			41089.5i			1.11826i	0.829080	+	0.559131i	$0.188865\pi$
$67$			41089.5i			1.11826i	−0.829080	+	0.559131i	$0.811135\pi$
$68$	76946.8			2.01799
$69$	27709.5	+	28217.3i	0.700658	+	0.713498i
$70$	57266.4			1.39687
$71$		−	8193.96i		−	0.192907i	−0.995337	−	0.0964535i	$-0.969250\pi$
$71$		−	8193.96i		−	0.192907i	0.995337	−	0.0964535i	$-0.0307499\pi$
$72$	−27029.7	−	1833.42i	−0.614483	−	0.0416803i
$73$	−78243.7			−1.71847			−0.859235	−	0.511581i	$-0.829060\pi$
$73$	−78243.7			−1.71847			−0.859235	+	0.511581i	$0.829060\pi$
$74$	−11602.9			−0.246312
$75$	−13425.9	−	14367.5i	−0.275607	−	0.294935i
$76$		−	24790.7i		−	0.492328i
$77$			92519.5i			1.77831i
$78$	13192.3	−	12327.8i	0.245519	−	0.229429i
$79$		−	77477.3i		−	1.39671i	−0.715751	−	0.698356i	$-0.753915\pi$
$79$		−	77477.3i		−	1.39671i	0.715751	−	0.698356i	$-0.246085\pi$
$80$	19629.3			0.342910
$81$	−58508.1	−	7973.88i	−0.990840	−	0.135038i
$82$	2024.37			0.0332472
$83$	17093.4			0.272354			0.136177	−	0.990685i	$-0.456518\pi$
$83$	17093.4			0.272354			0.136177	+	0.990685i	$0.456518\pi$
$84$	−77150.5	+	72094.7i	−1.19300	+	1.11482i
$85$	74264.7			1.11490
$86$	−161296.			−2.35168
$87$	79793.5	−	74564.5i	1.13024	−	1.05617i
$88$		−	68110.1i		−	0.937573i
$89$	135956.			1.81938			0.909691	−	0.415286i	$-0.136319\pi$
$89$	135956.			1.81938			0.909691	+	0.415286i	$0.136319\pi$
$90$	−91676.0	−	6218.37i	−1.19303	−	0.0809227i
$91$		−	20025.8i		−	0.253505i
$92$	−113439.	+	2812.01i	−1.39732	+	0.0346376i
$93$	51295.5	+	54892.7i	0.614996	+	0.658124i
$94$	−103204.			−1.20469
$95$		−	23926.5i		−	0.272001i
$96$	−85998.8	+	80363.1i	−0.952388	+	0.889976i
$97$			75181.6i			0.811302i	0.914028	+	0.405651i	$0.132955\pi$
$97$			75181.6i			0.811302i	−0.914028	+	0.405651i	$0.867045\pi$
$98$			53681.6i			0.564625i
$99$	10046.4	−	148112.i	0.103020	−	1.51880i
$100$	56422.2			0.564222
$101$		−	182646.i		−	1.78159i	−0.454410	−	0.890793i	$-0.650150\pi$
$101$		−	182646.i		−	1.78159i	0.454410	−	0.890793i	$-0.349850\pi$
$102$	−171631.	+	160384.i	−1.63341	+	1.52637i
$103$			60433.1i			0.561283i	0.959813	+	0.280642i	$0.0905472\pi$
$103$			60433.1i			0.561283i	−0.959813	+	0.280642i	$0.909453\pi$
$104$			14742.4i			0.133655i
$105$	−74461.3	+	69581.7i	−0.659109	+	0.615917i
$106$		−	16569.0i		−	0.143229i
$107$	−19007.4			−0.160495			−0.0802477	−	0.996775i	$-0.525571\pi$
$107$	−19007.4			−0.160495			−0.0802477	+	0.996775i	$0.525571\pi$
$108$	131337.	−	107037.i	1.08349	−	0.883026i
$109$			95078.3i			0.766505i	0.923644	+	0.383253i	$0.125196\pi$
$109$			95078.3i			0.766505i	−0.923644	+	0.383253i	$0.874804\pi$
$110$		−	231008.i		−	1.82031i
$111$	15086.8	−	14098.1i	0.116222	−	0.108606i
$112$			68863.3i			0.518732i
$113$	68259.2			0.502881			0.251440	−	0.967873i	$-0.419096\pi$
$113$	68259.2			0.502881			0.251440	+	0.967873i	$0.419096\pi$
$114$	51672.4	+	55296.1i	0.372389	+	0.398503i
$115$	−109485.	+	2713.99i	−0.771989	+	0.0191366i
$116$			313356.i			2.16219i
$117$	−2174.54	+	32058.7i	−0.0146860	+	0.216512i
$118$	67553.3			0.446624
$119$			260535.i			1.68655i
$120$	54816.2	−	51224.0i	0.347501	−	0.324729i
$121$	212165.			1.31738
$122$	−10588.8			−0.0644089
$123$	−2632.21	+	2459.71i	−0.0156876	+	0.0146596i
$124$	−215569.			−1.25902
$125$	189358.			1.08395
$126$	21815.2	−	321617.i	0.122415	−	1.80473i
$127$	−162337.			−0.893116			−0.446558	−	0.894755i	$-0.647350\pi$
$127$	−162337.			−0.893116			−0.446558	+	0.894755i	$0.647350\pi$
$128$		−	210268.i		−	1.13435i
$129$	209727.	−	195983.i	1.10964	−	1.03692i
$130$			50001.5i			0.259492i
$131$		−	120535.i		−	0.613669i	−0.951763	−	0.306835i	$-0.900730\pi$
$131$		−	120535.i		−	0.613669i	0.951763	−	0.306835i	$-0.0992699\pi$
$132$	290824.	+	311219.i	1.45276	+	1.55464i
$133$	83939.0			0.411466
$134$	359921.			1.73159
$135$	126759.	−	103306.i	0.598608	−	0.487854i
$136$		−	191798.i		−	0.889195i
$137$	−324305.			−1.47623			−0.738113	−	0.674677i	$-0.764283\pi$
$137$	−324305.			−1.47623			−0.738113	+	0.674677i	$0.764283\pi$
$138$	247168.	−	242720.i	1.10483	−	1.08494i
$139$	168785.			0.740964			0.370482	−	0.928840i	$-0.379193\pi$
$139$	168785.			0.740964			0.370482	+	0.928840i	$0.379193\pi$
$140$		−	292416.i		−	1.26090i
$141$	134192.	−	125398.i	0.568432	−	0.531182i
$142$	−71774.5			−0.298710
$143$	−80782.4			−0.330352
$144$	7477.65	−	110241.i	0.0300510	−	0.443035i
$145$			302433.i			1.19456i
$146$			685371.i			2.66099i
$147$	−65226.0	−	69800.1i	−0.248959	−	0.266417i
$148$			59247.1i			0.222338i
$149$	−346489.			−1.27857			−0.639283	−	0.768971i	$-0.720769\pi$
$149$	−346489.			−1.27857			−0.639283	+	0.768971i	$0.720769\pi$
$150$	−125851.	+	117604.i	−0.456697	+	0.426769i
$151$	−484291.			−1.72848			−0.864239	−	0.503082i	$-0.832199\pi$
$151$	−484291.			−1.72848			−0.864239	+	0.503082i	$0.832199\pi$
$152$	−61793.4			−0.216937
$153$	28290.6	−	417083.i	0.0977044	−	1.44043i
$154$	810419.			2.75365
$155$	−208055.			−0.695581
$156$	−62948.7	−	67363.1i	−0.207098	−	0.221621i
$157$			129167.i			0.418218i	0.977892	+	0.209109i	$0.0670564\pi$
$157$			129167.i			0.418218i	−0.977892	+	0.209109i	$0.932944\pi$
$158$	−678658.			−2.16276
$159$	20132.2	+	21544.0i	0.0631536	+	0.0675824i
$160$		−	325953.i		−	1.00659i
$161$	−9521.21	−	384096.i	−0.0289486	−	1.16782i
$162$	−69846.7	+	512499.i	−0.209102	+	1.53428i
$163$	−54105.7			−0.159505			−0.0797525	−	0.996815i	$-0.525413\pi$
$163$	−54105.7			−0.159505			−0.0797525	+	0.996815i	$0.525413\pi$
$164$		−	10336.9i		−	0.0300110i
$165$	280687.	+	300370.i	0.802623	+	0.858909i
$166$		−	149729.i		−	0.421730i
$167$		−	46358.5i		−	0.128629i	−0.997930	−	0.0643144i	$-0.979514\pi$
$167$		−	46358.5i		−	0.128629i	0.997930	−	0.0643144i	$-0.0204861\pi$
$168$	179704.	+	192306.i	0.491229	+	0.525678i
$169$	−353808.			−0.952907
$170$		−	650517.i		−	1.72638i
$171$	−134375.	−	9114.66i	−0.351422	−	0.0238369i
$172$			823617.i			2.12277i
$173$			371854.i			0.944620i	0.881433	+	0.472310i	$0.156580\pi$
$173$			371854.i			0.944620i	−0.881433	+	0.472310i	$0.843420\pi$
$174$	−653143.	−	698947.i	−1.63544	−	1.75013i
$175$			191041.i			0.471553i
$176$	277789.			0.675979
$177$	−87837.0	+	82080.9i	−0.210739	+	0.196929i
$178$		−	1.19090e6i		−	2.81725i
$179$			320039.i			0.746570i	0.927717	+	0.373285i	$0.121769\pi$
$179$			320039.i			0.746570i	−0.927717	+	0.373285i	$0.878231\pi$
$180$	−31752.5	+	468120.i	−0.0730461	+	1.07690i
$181$		−	512556.i		−	1.16291i	−0.813580	−	0.581453i	$-0.802484\pi$
$181$		−	512556.i		−	1.16291i	0.813580	−	0.581453i	$-0.197516\pi$
$182$	−175415.			−0.392544
$183$	13768.2	−	12865.9i	0.0303913	−	0.0283997i
$184$	7009.23	+	282760.i	0.0152625	+	0.615706i
$185$			57181.9i			0.122837i
$186$	480830.	−	449320.i	1.01908	−	0.952299i
$187$	1.05098e6			2.19780
$188$			526983.i			1.08743i
$189$	362417.	+	444694.i	0.737995	+	0.905537i
$190$	−209583.			−0.421185
$191$	−269982.			−0.535489			−0.267745	−	0.963490i	$-0.586278\pi$
$191$	−269982.			−0.535489			−0.267745	+	0.963490i	$0.586278\pi$
$192$	549070.	+	587575.i	1.07492	+	1.15030i
$193$	−624538.			−1.20688			−0.603442	−	0.797407i	$-0.706205\pi$
$193$	−624538.			−1.20688			−0.603442	+	0.797407i	$0.706205\pi$
$194$	658549.			1.25627
$195$	−60754.5	−	65015.1i	−0.114417	−	0.122441i
$196$	274111.			0.509667
$197$		−	733978.i		−	1.34747i	−0.738975	−	0.673733i	$-0.764690\pi$
$197$		−	733978.i		−	1.34747i	0.738975	−	0.673733i	$-0.235310\pi$
$198$	−1.29738e6	−	88000.8i	−2.35181	−	0.159523i
$199$			896250.i			1.60434i	0.597095	+	0.802170i	$0.296321\pi$
$199$			896250.i			1.60434i	−0.597095	+	0.802170i	$0.703679\pi$
$200$		−	140638.i		−	0.248616i
$201$	−467991.	+	437323.i	−0.817048	+	0.763505i
$202$	−1.59988e6			−2.75872
$203$	−1.06099e6			−1.80706
$204$	818960.	+	876392.i	1.37780	+	1.47443i
$205$		−	9976.60i		−	0.0165805i
$206$	529360.			0.869127
$207$	−26465.5	+	615921.i	−0.0429293	+	0.999078i
$208$	−60127.3			−0.0963636
$209$		−	338602.i		−	0.536197i
$210$	609497.	+	652240.i	0.953726	+	1.02061i
$211$	1.08224e6			1.67347			0.836737	−	0.547605i	$-0.184460\pi$
$211$	1.08224e6			1.67347			0.836737	+	0.547605i	$0.184460\pi$
$212$	−84605.2			−0.129288
$213$	93325.7	−	87209.9i	0.140946	−	0.131709i
$214$			166494.i			0.248522i
$215$			794908.i			1.17279i
$216$	−266801.	−	327370.i	−0.389092	−	0.477425i
$217$		−	729896.i		−	1.05223i
$218$	832833.			1.18691
$219$	−832763.	−	891162.i	−1.17330	−	1.25559i
$220$	−1.17958e6			−1.64313
$221$	−227483.			−0.313306
$222$	−123492.	−	132152.i	−0.168173	−	0.179966i
$223$	472939.			0.636859			0.318430	−	0.947947i	$-0.396845\pi$
$223$	472939.			0.636859			0.318430	+	0.947947i	$0.396845\pi$
$224$	1.14350e6			1.52271
$225$	20744.5	−	305831.i	0.0273178	−	0.402741i
$226$		−	597913.i		−	0.778694i
$227$	−410057.			−0.528177			−0.264089	−	0.964498i	$-0.585071\pi$
$227$	−410057.			−0.528177			−0.264089	+	0.964498i	$0.585071\pi$
$228$	282355.	−	263852.i	0.359715	−	0.336142i
$229$			395268.i			0.498085i	0.968493	+	0.249042i	$0.0801158\pi$
$229$			395268.i			0.498085i	−0.968493	+	0.249042i	$0.919884\pi$
$230$	23773.1	+	959031.i	0.0296323	+	1.19540i
$231$	−1.05376e6	+	984703.i	−1.29930	+	1.21416i
$232$	781073.			0.952734
$233$			1.00983e6i			1.21859i	0.792944	+	0.609294i	$0.208547\pi$
$233$			1.00983e6i			1.21859i	−0.792944	+	0.609294i	$0.791453\pi$
$234$	280817.	+	19047.7i	0.335261	+	0.0227407i
$235$			508614.i			0.600785i
$236$		−	344944.i		−	0.403151i
$237$	882434.	−	824606.i	1.02050	−	0.953621i
$238$	2.28214e6			2.61156
$239$		−	1.07223e6i		−	1.21421i	−0.794623	−	0.607103i	$-0.792332\pi$
$239$		−	1.07223e6i		−	1.21421i	0.794623	−	0.607103i	$-0.207668\pi$
$240$	208918.	+	223569.i	0.234125	+	0.250544i
$241$		−	1.41714e6i		−	1.57170i	−0.618415	−	0.785851i	$-0.712225\pi$
$241$		−	1.41714e6i		−	1.57170i	0.618415	−	0.785851i	$-0.287775\pi$
$242$		−	1.85844e6i		−	2.03991i
$243$	−531894.	−	751250.i	−0.577843	−	0.816148i
$244$			54068.8i			0.0581397i
$245$	264556.			0.281581
$246$	21545.7	+	23056.7i	0.0226999	+	0.0242918i
$247$			73290.4i			0.0764372i
$248$			537328.i			0.554766i
$249$	181928.	+	194687.i	0.185952	+	0.198993i
$250$		−	1.65867e6i		−	1.67846i
$251$	−387993.			−0.388722			−0.194361	−	0.980930i	$-0.562263\pi$
$251$	−387993.			−0.388722			−0.194361	+	0.980930i	$0.562263\pi$
$252$	−1.64226e6	−	111394.i	−1.62907	−	0.110499i
$253$	−1.54941e6	+	38407.7i	−1.52183	+	0.0377240i
$254$			1.42198e6i			1.38296i
$255$	790414.	+	845843.i	0.761209	+	0.814591i
$256$	−190992.			−0.182144
$257$		−	972339.i		−	0.918301i	−0.888359	−	0.459150i	$-0.848154\pi$
$257$		−	972339.i		−	0.918301i	0.888359	−	0.459150i	$-0.151846\pi$
$258$	−1.71670e6	−	1.83709e6i	−1.60563	−	1.71823i
$259$	−200605.			−0.185820
$260$	255320.			0.234235
$261$	1.69852e6	+	115210.i	1.54336	+	0.104686i
$262$	−1.05582e6			−0.950246
$263$	−1.47839e6			−1.31795			−0.658977	−	0.752163i	$-0.729011\pi$
$263$	−1.47839e6			−1.31795			−0.658977	+	0.752163i	$0.729011\pi$
$264$	775746.	−	724909.i	0.685030	−	0.640138i
$265$	−81656.1			−0.0714289
$266$		−	735259.i		−	0.637142i
$267$	1.44701e6	+	1.54848e6i	1.24220	+	1.32932i
$268$		−	1.83784e6i		−	1.56304i
$269$			1.39260e6i			1.17340i	0.809805	+	0.586699i	$0.199573\pi$
$269$			1.39260e6i			1.17340i	−0.809805	+	0.586699i	$0.800427\pi$
$270$	−904901.	−	1.11033e6i	−0.755426	−	0.926925i
$271$	−1.26005e6			−1.04223			−0.521115	−	0.853486i	$-0.674484\pi$
$271$	−1.26005e6			−1.04223			−0.521115	+	0.853486i	$0.674484\pi$
$272$	782253.			0.641099
$273$	228085.	−	213139.i	0.185221	−	0.173083i
$274$			2.84073e6i			2.28588i
$275$	770641.			0.614498
$276$	−1.23939e6	−	1.26210e6i	−0.979341	−	0.997288i
$277$	1.23028e6			0.963397			0.481699	−	0.876337i	$-0.340020\pi$
$277$	1.23028e6			0.963397			0.481699	+	0.876337i	$0.340020\pi$
$278$		−	1.47846e6i		−	1.14736i
$279$	−79257.0	+	1.16847e6i	−0.0609575	+	0.898683i
$280$	−728878.			−0.555597
$281$	2.37975e6			1.79790			0.898951	−	0.438050i	$-0.144331\pi$
$281$	2.37975e6			1.79790			0.898951	+	0.438050i	$0.144331\pi$
$282$	−1.09842e6	−	1.17545e6i	−0.822516	−	0.880198i
$283$			1.01299e6i			0.751863i	0.926647	+	0.375932i	$0.122677\pi$
$283$			1.01299e6i			0.751863i	−0.926647	+	0.375932i	$0.877323\pi$
$284$			366498.i			0.269635i
$285$	272513.	−	254655.i	0.198735	−	0.185712i
$286$			707609.i			0.511538i
$287$	34999.8			0.0250819
$288$	−1.83060e6	−	124169.i	−1.30051	−	0.0882132i
$289$	1.53969e6			1.08440
$290$	2.64915e6			1.84974
$291$	−856287.	+	800173.i	−0.592771	+	0.553925i
$292$	3.49967e6			2.40198
$293$	1.28525e6			0.874618			0.437309	−	0.899311i	$-0.355932\pi$
$293$	1.28525e6			0.874618			0.437309	+	0.899311i	$0.355932\pi$
$294$	−611410.	+	571343.i	−0.412538	+	0.385504i
$295$		−	332920.i		−	0.222733i
$296$	147680.			0.0979696
$297$	1.79386e6	−	1.46196e6i	1.18004	−	0.961709i
$298$			3.03505e6i			1.97982i
$299$	335369.	−	8313.34i	0.216943	−	0.00537771i
$300$	600513.	+	642625.i	0.385229	+	0.412244i
$301$	−2.78869e6			−1.77412
$302$			4.24212e6i			2.67649i
$303$	2.08026e6	−	1.94394e6i	1.30170	−	1.21640i
$304$		−	252026.i		−	0.156409i
$305$			52184.2i			0.0321210i
$306$	−3.65341e6	−	247810.i	−2.23046	−	0.151292i
$307$	440102.			0.266507			0.133253	−	0.991082i	$-0.457458\pi$
$307$	440102.			0.266507			0.133253	+	0.991082i	$0.457458\pi$
$308$		−	4.13820e6i		−	2.48562i
$309$	−688308.	+	643201.i	−0.410097	+	0.383222i
$310$			1.82244e6i			1.07708i
$311$		−	459889.i		−	0.269620i	−0.990871	−	0.134810i	$-0.956958\pi$
$311$		−	459889.i		−	0.269620i	0.990871	−	0.134810i	$-0.0430424\pi$
$312$	−167910.	+	156906.i	−0.0976539	+	0.0912544i
$313$		−	2.79451e6i		−	1.61230i	−0.591712	−	0.806149i	$-0.701548\pi$
$313$		−	2.79451e6i		−	1.61230i	0.591712	−	0.806149i	$-0.298452\pi$
$314$	1.13143e6			0.647597
$315$	−1.58501e6	−	107511.i	−0.900028	−	0.0610488i
$316$			3.46539e6i			1.95225i
$317$			2.26703e6i			1.26709i	0.773704	+	0.633547i	$0.218402\pi$
$317$			2.26703e6i			1.26709i	−0.773704	+	0.633547i	$0.781598\pi$
$318$	188713.	−	176347.i	0.104649	−	0.0977911i
$319$			4.27996e6i			2.35485i
$320$	−2.22703e6			−1.21577
$321$	−202299.	−	216486.i	−0.109580	−	0.117265i
$322$	−3.36447e6	+	83400.5i	−1.80832	+	0.0448259i
$323$		−	953505.i		−	0.508530i
$324$	2.61694e6	+	356655.i	1.38494	+	0.188749i
$325$	−166805.			−0.0875993
$326$			473936.i			0.246988i
$327$	−1.08290e6	+	1.01194e6i	−0.560041	+	0.523340i
$328$	−25765.8			−0.0132239
$329$	−1.78432e6			−0.908830
$330$	2.63108e6	−	2.45866e6i	1.32999	−	1.24283i
$331$	85093.5			0.0426900			0.0213450	−	0.999772i	$-0.493205\pi$
$331$	85093.5			0.0426900			0.0213450	+	0.999772i	$0.493205\pi$
$332$	−764551.			−0.380681
$333$	321143.	+	21783.1i	0.158704	+	0.0107649i
$334$	−406075.			−0.199177
$335$		−	1.77378e6i		−	0.863551i
$336$	−784324.	+	732926.i	−0.379007	+	0.354170i
$337$		−	2.25704e6i		−	1.08259i	−0.840832	−	0.541297i	$-0.817934\pi$
$337$		−	2.25704e6i		−	1.08259i	0.840832	−	0.541297i	$-0.182066\pi$
$338$			3.09916e6i			1.47554i
$339$	726496.	+	777443.i	0.343347	+	0.367426i
$340$	−3.32170e6			−1.55834
$341$	−2.94434e6			−1.37120
$342$	−79839.3	+	1.17705e6i	−0.0369106	+	0.544165i
$343$		−	1.61721e6i		−	0.742218i
$344$	2.05295e6			0.935368
$345$	−1.19618e6	−	1.21811e6i	−0.541066	−	0.550982i
$346$	3.25723e6			1.46271
$347$		−	895437.i		−	0.399219i	−0.979876	−	0.199610i	$-0.936033\pi$
$347$		−	895437.i		−	0.399219i	0.979876	−	0.199610i	$-0.0639674\pi$
$348$	−3.56899e6	+	3.33511e6i	−1.57978	+	1.47626i
$349$	1.69659e6			0.745614			0.372807	−	0.927909i	$-0.378395\pi$
$349$	1.69659e6			0.745614			0.372807	+	0.927909i	$0.378395\pi$
$350$	1.67341e6			0.730183
$351$	−388279.	+	316440.i	−0.168220	+	0.137096i
$352$		−	4.61280e6i		−	1.98430i
$353$		−	603075.i		−	0.257593i	−0.991671	−	0.128797i	$-0.958889\pi$
$353$		−	603075.i		−	0.257593i	0.991671	−	0.128797i	$-0.0411114\pi$
$354$	718983.	+	769404.i	0.304937	+	0.326322i
$355$			353723.i			0.148968i
$356$	−6.08103e6			−2.54303
$357$	−2.96738e6	+	2.77292e6i	−1.23226	+	1.15151i
$358$	2.80337e6			1.15604
$359$	−3.74407e6			−1.53323			−0.766617	−	0.642105i	$-0.778061\pi$
$359$	−3.74407e6			−1.53323			−0.766617	+	0.642105i	$0.778061\pi$
$360$	1.16684e6	+	79146.5i	0.474520	+	0.0321866i
$361$	2.16890e6			0.875934
$362$	−4.48970e6			−1.80072
$363$	2.25811e6	+	2.41647e6i	0.899453	+	0.962529i
$364$			895712.i			0.354336i
$365$	3.37768e6			1.32705
$366$	−112698.	−	120602.i	−0.0439759	−	0.0470598i
$367$		−	4.16068e6i		−	1.61250i	−0.591577	−	0.806249i	$-0.701494\pi$
$367$		−	4.16068e6i		−	1.61250i	0.591577	−	0.806249i	$-0.298506\pi$
$368$	−1.15324e6	+	28587.3i	−0.443917	+	0.0110041i
$369$	−56030.2	−	3800.52i	−0.0214218	−	0.00145304i
$370$	500882.			0.190209
$371$		−	286465.i		−	0.108053i
$372$	−2.29434e6	−	2.45523e6i	−0.859607	−	0.919890i
$373$		−	3.99193e6i		−	1.48563i	−0.669496	−	0.742816i	$-0.733490\pi$
$373$		−	3.99193e6i		−	1.48563i	0.669496	−	0.742816i	$-0.266510\pi$
$374$		−	9.20596e6i		−	3.40322i
$375$	2.01537e6	+	2.15671e6i	0.740078	+	0.791978i
$376$	1.31356e6			0.479161
$377$		−	926396.i		−	0.335694i
$378$	3.89527e6	−	3.17457e6i	1.40219	−	1.14276i
$379$		−	98283.1i		−	0.0351464i	−0.999846	−	0.0175732i	$-0.994406\pi$
$379$		−	98283.1i		−	0.0351464i	0.999846	−	0.0175732i	$-0.00559401\pi$
$380$			1.07018e6i			0.380189i
$381$	−1.72778e6	−	1.84895e6i	−0.609785	−	0.652547i
$382$			2.36489e6i			0.829186i
$383$	−2.81372e6			−0.980130			−0.490065	−	0.871686i	$-0.663027\pi$
$383$	−2.81372e6			−0.980130			−0.490065	+	0.871686i	$0.663027\pi$
$384$	2.39487e6	−	2.23793e6i	0.828807	−	0.774494i
$385$		−	3.99395e6i		−	1.37326i
$386$			5.47061e6i			1.86882i
$387$	4.46433e6	+	302815.i	1.51523	+	0.102778i
$388$		−	3.36271e6i		−	1.13399i
$389$	2.68573e6			0.899889			0.449944	−	0.893057i	$-0.351444\pi$
$389$	2.68573e6			0.899889			0.449944	+	0.893057i	$0.351444\pi$
$390$	−569496.	+	532176.i	−0.189596	+	0.177171i
$391$	−4.36314e6	+	108156.i	−1.44330	+	0.0357775i
$392$		−	683251.i		−	0.224577i
$393$	1.37284e6	−	1.28288e6i	0.448372	−	0.418990i
$394$	−6.42924e6			−2.08650
$395$			3.34460e6i			1.07858i
$396$	−449354.	+	6.62472e6i	−0.143996	+	2.12290i
$397$	3.49044e6			1.11149			0.555743	−	0.831354i	$-0.312434\pi$
$397$	3.49044e6			1.11149			0.555743	+	0.831354i	$0.312434\pi$
$398$	7.85065e6			2.48427
$399$	893379.	+	956029.i	0.280933	+	0.300635i
$400$	573597.			0.179249
$401$	1.45743e6			0.452613			0.226307	−	0.974056i	$-0.427335\pi$
$401$	1.45743e6			0.452613			0.226307	+	0.974056i	$0.427335\pi$
$402$	3.83071e6	+	4.09934e6i	1.18226	+	1.26517i
$403$	637300.			0.195471
$404$			8.16936e6i			2.49020i
$405$	2.52572e6	+	344223.i	0.765153	+	0.104280i
$406$			9.29372e6i			2.79817i
$407$			809224.i			0.242149i
$408$	2.18450e6	−	2.04134e6i	0.649683	−	0.607108i
$409$	−397052.			−0.117365			−0.0586825	−	0.998277i	$-0.518690\pi$
$409$	−397052.			−0.117365			−0.0586825	+	0.998277i	$0.518690\pi$
$410$	−87389.4			−0.0256743
$411$	−3.45164e6	−	3.69370e6i	−1.00791	−	1.07859i
$412$		−	2.70304e6i		−	0.784531i
$413$	1.16795e6			0.336937
$414$	5.39513e6	+	231823.i	1.54704	+	0.0664746i
$415$	−737901.			−0.210319
$416$			998439.i			0.282871i
$417$	1.79641e6	+	1.92239e6i	0.505901	+	0.541379i
$418$	−2.96597e6			−0.830283
$419$	−4.90111e6			−1.36383			−0.681913	−	0.731433i	$-0.738852\pi$
$419$	−4.90111e6			−1.36383			−0.681913	+	0.731433i	$0.738852\pi$
$420$	3.33049e6	−	3.11224e6i	0.921267	−	0.860895i
$421$			4.82268e6i			1.32612i	0.748566	+	0.663061i	$0.230743\pi$
$421$			4.82268e6i			1.32612i	−0.748566	+	0.663061i	$0.769257\pi$
$422$		−	9.47985e6i		−	2.59132i
$423$	2.85646e6	+	193753.i	0.776207	+	0.0526500i
$424$			210887.i			0.0569687i
$425$	2.17013e6			0.582791
$426$	−763910.	−	817481.i	−0.203947	−	0.218250i
$427$	−183072.			−0.0485906
$428$	850159.			0.224332
$429$	−859783.	−	920077.i	−0.225551	−	0.241369i
$430$	6.96295e6			1.81603
$431$	3.02169e6			0.783533			0.391767	−	0.920065i	$-0.371864\pi$
$431$	3.02169e6			0.783533			0.391767	+	0.920065i	$0.371864\pi$
$432$	1.33519e6	−	1.08815e6i	0.344217	−	0.280531i
$433$		−	3.30312e6i		−	0.846650i	−0.905978	−	0.423325i	$-0.860863\pi$
$433$		−	3.30312e6i		−	0.846650i	0.905978	−	0.423325i	$-0.139137\pi$
$434$	−6.39348e6			−1.62934
$435$	−3.44459e6	+	3.21886e6i	−0.872799	+	0.815602i
$436$		−	4.25265e6i		−	1.07138i
$437$	34845.7	+	1.40571e6i	0.00872862	+	0.352122i
$438$	−7.80609e6	+	7.29454e6i	−1.94423	+	1.81682i
$439$	−3.90756e6			−0.967708			−0.483854	−	0.875149i	$-0.660763\pi$
$439$	−3.90756e6			−0.967708			−0.483854	+	0.875149i	$0.660763\pi$
$440$			2.94023e6i			0.724019i
$441$	100781.	−	1.48579e6i	0.0246764	−	0.363799i
$442$			1.99263e6i			0.485143i
$443$		−	2.12284e6i		−	0.513934i	−0.966420	−	0.256967i	$-0.917277\pi$
$443$		−	2.12284e6i		−	0.513934i	0.966420	−	0.256967i	$-0.0827231\pi$
$444$	−674799.	+	630578.i	−0.162449	+	0.151803i
$445$	−5.86906e6			−1.40497
$446$		−	4.14269e6i		−	0.986154i
$447$	−3.68775e6	−	3.94636e6i	−0.872956	−	0.934174i
$448$		−	7.81284e6i		−	1.83914i
$449$		−	551494.i		−	0.129100i	−0.997914	−	0.0645499i	$-0.979439\pi$
$449$		−	551494.i		−	0.129100i	0.997914	−	0.0645499i	$-0.0205612\pi$
$450$	−2.67891e6	−	181710.i	−0.623630	−	0.0423007i
$451$		−	141186.i		−	0.0326852i
$452$	−3.05309e6			−0.702899
$453$	−5.15440e6	−	5.51586e6i	−1.18014	−	1.26290i
$454$			3.59187e6i			0.817864i
$455$			864490.i			0.195763i
$456$	−657679.	−	703800.i	−0.148116	−	0.158503i
$457$			146258.i			0.0327590i	0.999866	+	0.0163795i	$0.00521398\pi$
$457$			146258.i			0.0327590i	−0.999866	+	0.0163795i	$0.994786\pi$
$458$	3.46233e6			0.771267
$459$	5.05150e6	−	4.11687e6i	1.11915	−	0.912086i
$460$	4.89704e6	−	121391.i	1.07904	−	0.0267480i
$461$		−	5.82484e6i		−	1.27653i	−0.769816	−	0.638266i	$-0.779652\pi$
$461$		−	5.82484e6i		−	1.27653i	0.769816	−	0.638266i	$-0.220348\pi$
$462$	8.62545e6	+	9.23033e6i	1.88008	+	2.01193i
$463$	4.42318e6			0.958920			0.479460	−	0.877564i	$-0.340833\pi$
$463$	4.42318e6			0.958920			0.479460	+	0.877564i	$0.340833\pi$
$464$			3.18563e6i			0.686910i
$465$	−2.21436e6	−	2.36965e6i	−0.474916	−	0.508221i
$466$	8.84552e6			1.88694
$467$	−1.31195e6			−0.278371			−0.139185	−	0.990266i	$-0.544448\pi$
$467$	−1.31195e6			−0.278371			−0.139185	+	0.990266i	$0.544448\pi$
$468$	97262.4	−	1.43392e6i	0.0205272	−	0.302628i
$469$	6.22277e6			1.30633
$470$	4.45518e6			0.930295
$471$	−1.47116e6	+	1.37475e6i	−0.305568	+	0.285543i
$472$	−859809.			−0.177643
$473$			1.12493e7i			2.31193i
$474$	−7.22309e6	−	7.72963e6i	−1.47665	−	1.58020i
$475$		−	699169.i		−	0.142183i
$476$		−	1.16532e7i		−	2.35736i
$477$	−31106.3	+	458594.i	−0.00625969	+	0.0922853i
$478$	−9.39212e6			−1.88016
$479$	−491382.			−0.0978545			−0.0489273	−	0.998802i	$-0.515580\pi$
$479$	−491382.			−0.0978545			−0.0489273	+	0.998802i	$0.515580\pi$
$480$	3.71246e6	−	3.46918e6i	0.735460	−	0.687264i
$481$		−	175156.i		−	0.0345194i
$482$	−1.24134e7			−2.43373
$483$	4.27335e6	−	4.19645e6i	0.833491	−	0.818491i
$484$	−9.48967e6			−1.84136
$485$		−	3.24550e6i		−	0.626509i
$486$	−6.58054e6	+	4.65910e6i	−1.26378	+	0.894770i
$487$	−2.13183e6			−0.407315			−0.203657	−	0.979042i	$-0.565283\pi$
$487$	−2.13183e6			−0.407315			−0.203657	+	0.979042i	$0.565283\pi$
$488$	134772.			0.0256184
$489$	−575858.	−	616241.i	−0.108904	−	0.116541i
$490$		−	2.31737e6i		−	0.436018i
$491$			4.98864e6i			0.933854i	0.884296	+	0.466927i	$0.154639\pi$
$491$			4.98864e6i			0.933854i	−0.884296	+	0.466927i	$0.845361\pi$
$492$	117733.	−	110018.i	0.0219273	−	0.0204904i
$493$			1.20524e7i			2.23334i
$494$	641983.			0.118360
$495$	−433690.	+	6.39380e6i	−0.0795549	+	1.17286i
$496$	−2.19150e6			−0.399980
$497$	−1.24093e6			−0.225349
$498$	1.70535e6	−	1.59359e6i	0.308134	−	0.287941i
$499$	3.54759e6			0.637797			0.318898	−	0.947789i	$-0.396687\pi$
$499$	3.54759e6			0.637797			0.318898	+	0.947789i	$0.396687\pi$
$500$	−8.46958e6			−1.51508
$501$	528004.	−	493403.i	0.0939816	−	0.0878228i
$502$			3.39860e6i			0.601923i
$503$	−442972.			−0.0780651			−0.0390325	−	0.999238i	$-0.512428\pi$
$503$	−442972.			−0.0780651			−0.0390325	+	0.999238i	$0.512428\pi$
$504$	−277661.	+	4.09350e6i	−0.0486899	+	0.717825i
$505$			7.88460e6i			1.37579i
$506$	336430.	+	1.35720e7i	0.0584143	+	2.35649i
$507$	−3.76564e6	−	4.02972e6i	−0.650608	−	0.696233i
$508$	7.26098e6			1.24835
$509$		−	5.88309e6i		−	1.00649i	−0.864143	−	0.503247i	$-0.832139\pi$
$509$		−	5.88309e6i		−	1.00649i	0.864143	−	0.503247i	$-0.167861\pi$
$510$	7.40912e6	−	6.92358e6i	1.26137	−	1.17871i
$511$			1.18496e7i			2.00748i
$512$		−	5.05561e6i		−	0.852311i
$513$	−1.32637e6	−	1.62749e6i	−0.222521	−	0.273039i
$514$	−8.51715e6			−1.42196
$515$		−	2.60882e6i		−	0.433438i
$516$	−9.38064e6	+	8.76591e6i	−1.55099	+	1.44935i
$517$			7.19778e6i			1.18433i
$518$			1.75719e6i			0.287736i
$519$	−4.23526e6	+	3.95771e6i	−0.690179	+	0.644950i
$520$		−	636412.i		−	0.103212i
$521$	8.70774e6			1.40544			0.702718	−	0.711468i	$-0.251969\pi$
$521$	8.70774e6			1.40544			0.702718	+	0.711468i	$0.251969\pi$
$522$	1.00918e6	−	1.48780e7i	0.162103	−	2.38984i
$523$			4.34219e6i			0.694153i	0.937837	+	0.347076i	$0.112826\pi$
$523$			4.34219e6i			0.694153i	−0.937837	+	0.347076i	$0.887174\pi$
$524$			5.39127e6i			0.857754i
$525$	−2.17587e6	+	2.03328e6i	−0.344536	+	0.321958i
$526$			1.29499e7i			2.04081i
$527$	−8.29125e6			−1.30045
$528$	2.95656e6	+	3.16390e6i	0.461532	+	0.493898i
$529$	6.42844e6	−	318900.i	0.998772	−	0.0495468i
$530$			715262.i			0.110605i
$531$	−1.86973e6	−	126824.i	−0.287769	−	0.0195193i
$532$	−3.75441e6			−0.575125
$533$			30559.7i			0.00465941i
$534$	1.35638e7	−	1.26750e7i	2.05840	−	1.92351i
$535$	820525.			0.123939
$536$	−4.58102e6			−0.688732
$537$	−3.64511e6	+	3.40624e6i	−0.545475	+	0.509729i
$538$	1.21984e7			1.81697
$539$	3.74394e6			0.555081
$540$	−5.66964e6	+	4.62065e6i	−0.836702	+	0.681896i
$541$	9.64282e6			1.41648			0.708241	−	0.705971i	$-0.249489\pi$
$541$	9.64282e6			1.41648			0.708241	+	0.705971i	$0.249489\pi$
$542$			1.10373e7i			1.61386i
$543$	5.83779e6	−	5.45523e6i	0.849668	−	0.793987i
$544$		−	1.29896e7i		−	1.88192i
$545$		−	4.10441e6i		−	0.591916i
$546$	−1.86698e6	−	1.99790e6i	−0.268014	−	0.286809i
$547$	−4.65385e6			−0.665034			−0.332517	−	0.943097i	$-0.607898\pi$
$547$	−4.65385e6			−0.665034			−0.332517	+	0.943097i	$0.607898\pi$
$548$	1.45055e7			2.06339
$549$	293075.	+	19879.2i	0.0415000	+	0.00281493i
$550$		−	6.75039e6i		−	0.951529i
$551$	3.88302e6			0.544868
$552$	−3.14592e6	+	3.08930e6i	−0.439440	+	0.431532i
$553$	−1.17335e7			−1.63161
$554$		−	1.07766e7i		−	1.49179i
$555$	−651278.	+	608598.i	−0.0897499	+	0.0838684i
$556$	−7.54940e6			−1.03568
$557$	1.19450e7			1.63136			0.815680	−	0.578504i	$-0.196363\pi$
$557$	1.19450e7			1.63136			0.815680	+	0.578504i	$0.196363\pi$
$558$	1.02351e7	+	694247.i	1.39158	+	0.0943906i
$559$		−	2.43491e6i		−	0.329575i
$560$		−	2.97275e6i		−	0.400579i
$561$	1.11857e7	+	1.19702e7i	1.50057	+	1.60581i
$562$		−	2.08453e7i		−	2.78399i
$563$	5.15603e6			0.685558			0.342779	−	0.939416i	$-0.388632\pi$
$563$	5.15603e6			0.685558			0.342779	+	0.939416i	$0.388632\pi$
$564$	−6.00212e6	+	5.60879e6i	−0.794523	+	0.742457i
$565$	−2.94667e6			−0.388338
$566$	8.87322e6			1.16423
$567$	−1.20760e6	+	8.86073e6i	−0.157749	+	1.15748i
$568$	913536.			0.118810
$569$	−1.87013e6			−0.242154			−0.121077	−	0.992643i	$-0.538635\pi$
$569$	−1.87013e6			−0.242154			−0.121077	+	0.992643i	$0.538635\pi$
$570$	−2.23063e6	−	2.38706e6i	−0.287569	−	0.307735i
$571$			7.63524e6i			0.980015i	0.871718	+	0.490007i	$0.163006\pi$
$571$			7.63524e6i			0.980015i	−0.871718	+	0.490007i	$0.836994\pi$
$572$	3.61322e6			0.461748
$573$	−2.87347e6	−	3.07497e6i	−0.365611	−	0.391251i
$574$		−	306579.i		−	0.0388385i
$575$	−3.19933e6	+	79306.9i	−0.403542	+	0.0100033i
$576$	−848371.	+	1.25073e7i	−0.106544	+	1.57076i
$577$	−3.94420e6			−0.493196			−0.246598	−	0.969118i	$-0.579313\pi$
$577$	−3.94420e6			−0.493196			−0.246598	+	0.969118i	$0.579313\pi$
$578$		−	1.34868e7i		−	1.67915i
$579$	−6.64708e6	−	7.11322e6i	−0.824014	−	0.881800i
$580$		−	1.35272e7i		−	1.66970i
$581$		−	2.58870e6i		−	0.318157i
$582$	7.00907e6	+	7.50060e6i	0.857735	+	0.917885i
$583$	−1.15558e6			−0.140808
$584$		−	8.72330e6i		−	1.05840i
$585$	93872.1	−	1.38394e6i	0.0113409	−	0.167196i
$586$		−	1.12581e7i		−	1.35432i
$587$			1.46795e7i			1.75839i	0.476464	+	0.879194i	$0.341918\pi$
$587$			1.46795e7i			1.75839i	−0.476464	+	0.879194i	$0.658082\pi$
$588$	2.91742e6	+	3.12201e6i	0.347981	+	0.372384i
$589$			2.67127e6i			0.317270i
$590$	−2.91619e6			−0.344895
$591$	8.35970e6	−	7.81187e6i	0.984514	−	0.919997i
$592$			602315.i			0.0706349i
$593$			9.13831e6i			1.06716i	0.845750	+	0.533580i	$0.179154\pi$
$593$			9.13831e6i			1.06716i	−0.845750	+	0.533580i	$0.820846\pi$
$594$	−1.28059e7	−	1.57132e7i	−1.48917	−	1.82725i
$595$		−	1.12470e7i		−	1.30240i
$596$	1.54977e7			1.78711
$597$	−1.02079e7	+	9.53896e6i	−1.17220	+	1.09538i
$598$	−72820.2	−	2.93765e6i	−0.00832720	−	0.335928i
$599$		−	7.63288e6i		−	0.869203i	−0.900623	−	0.434602i	$-0.856889\pi$
$599$		−	7.63288e6i		−	0.869203i	0.900623	−	0.434602i	$-0.143111\pi$
$600$	1.60181e6	−	1.49684e6i	0.181649	−	0.169745i
$601$	−7.47363e6			−0.844006			−0.422003	−	0.906594i	$-0.638673\pi$
$601$	−7.47363e6			−0.844006			−0.422003	+	0.906594i	$0.638673\pi$
$602$			2.44274e7i			2.74717i
$603$	−9.96185e6	−	675710.i	−1.11570	−	0.0756776i
$604$	2.16613e7			2.41597
$605$	−9.15889e6			−1.01731
$606$	−1.70278e7	−	1.82219e7i	−1.88355	−	2.01564i
$607$	7.40886e6			0.816168			0.408084	−	0.912944i	$-0.366197\pi$
$607$	7.40886e6			0.816168			0.408084	+	0.912944i	$0.366197\pi$
$608$	−4.18499e6			−0.459130
$609$	−1.12924e7	−	1.20843e7i	−1.23379	−	1.32032i
$610$	457104.			0.0497383
$611$		−	1.55796e6i		−	0.168831i
$612$	−1.26538e6	+	1.86552e7i	−0.136566	+	2.01336i
$613$		−	1.25160e7i		−	1.34528i	−0.739969	−	0.672641i	$-0.765160\pi$
$613$		−	1.25160e7i		−	1.34528i	0.739969	−	0.672641i	$-0.234840\pi$
$614$		−	3.85505e6i		−	0.412676i
$615$	113629.	−	106183.i	0.0121144	−	0.0113205i
$616$	−1.03149e7			−1.09525
$617$	−4.97604e6			−0.526225			−0.263112	−	0.964765i	$-0.584749\pi$
$617$	−4.97604e6			−0.526225			−0.263112	+	0.964765i	$0.584749\pi$
$618$	5.63409e6	+	6.02919e6i	0.593406	+	0.635021i
$619$			5.54618e6i			0.581792i	0.956755	+	0.290896i	$0.0939534\pi$
$619$			5.54618e6i			0.581792i	−0.956755	+	0.290896i	$0.906047\pi$
$620$	9.30583e6			0.972246
$621$	−7.29676e6	+	6.25394e6i	−0.759279	+	0.650766i
$622$	−4.02838e6			−0.417498
$623$		−	2.05898e7i		−	2.12536i
$624$	−639946.	−	684824.i	−0.0657933	−	0.0704073i
$625$	−4.23230e6			−0.433388
$626$	−2.44784e7			−2.49659
$627$	3.85654e6	−	3.60381e6i	0.391768	−	0.366095i
$628$		−	5.77737e6i		−	0.584563i
$629$			2.27878e6i			0.229654i
$630$	−941737.	+	1.38838e7i	−0.0945319	+	1.39366i
$631$			3.82572e6i			0.382507i	0.981541	+	0.191254i	$0.0612553\pi$
$631$			3.82572e6i			0.382507i	−0.981541	+	0.191254i	$0.938745\pi$
$632$	8.63786e6			0.860228
$633$	1.15185e7	+	1.23263e7i	1.14258	+	1.22271i
$634$	1.98579e7			1.96205
$635$	7.00788e6			0.689688
$636$	−900469.	−	963617.i	−0.0882726	−	0.0944630i
$637$	−810374.			−0.0791292
$638$	3.74901e7			3.64640
$639$	1.98657e6	+	134748.i	0.192465	+	0.0130548i
$640$			9.07703e6i			0.875979i
$641$	4.33390e6			0.416614			0.208307	−	0.978063i	$-0.433205\pi$
$641$	4.33390e6			0.416614			0.208307	+	0.978063i	$0.433205\pi$
$642$	−1.89630e6	+	1.77203e6i	−0.181580	+	0.169681i
$643$		−	1.11605e7i		−	1.06453i	−0.846578	−	0.532265i	$-0.821341\pi$
$643$		−	1.11605e7i		−	1.06453i	0.846578	−	0.532265i	$-0.178659\pi$
$644$	425863.	+	1.71798e7i	0.0404628	+	1.63231i
$645$	−9.05366e6	+	8.46036e6i	−0.856890	+	0.800736i
$646$	−8.35217e6			−0.787441
$647$		−	3.89565e6i		−	0.365863i	−0.983126	−	0.182932i	$-0.941441\pi$
$647$		−	3.89565e6i		−	0.365863i	0.983126	−	0.182932i	$-0.0585587\pi$
$648$	888999.	−	6.52301e6i	0.0831695	−	0.610254i
$649$		−	4.71140e6i		−	0.439075i
$650$			1.46112e6i			0.135644i
$651$	8.31320e6	−	7.76842e6i	0.768804	−	0.718423i
$652$	2.42003e6			0.222947
$653$			1.74450e6i			0.160099i	0.996791	+	0.0800495i	$0.0255078\pi$
$653$			1.74450e6i			0.160099i	−0.996791	+	0.0800495i	$0.974492\pi$
$654$	8.86400e6	+	9.48561e6i	0.810374	+	0.867204i
$655$			5.20334e6i			0.473892i
$656$		−	105087.i		−	0.00953428i
$657$	1.28671e6	−	1.89696e7i	0.116296	−	1.71453i
$658$			1.56296e7i			1.40729i
$659$	−732273.			−0.0656839			−0.0328420	−	0.999461i	$-0.510456\pi$
$659$	−732273.			−0.0656839			−0.0328420	+	0.999461i	$0.510456\pi$
$660$	−1.25545e7	−	1.34349e7i	−1.12186	−	1.20054i
$661$		−	6.40243e6i		−	0.569956i	−0.958534	−	0.284978i	$-0.908014\pi$
$661$		−	6.40243e6i		−	0.569956i	0.958534	−	0.284978i	$-0.0919863\pi$
$662$		−	745372.i		−	0.0661041i
$663$	−2.42115e6	−	2.59094e6i	−0.213913	−	0.228914i
$664$			1.90572e6i			0.167741i
$665$	−3.62354e6			−0.317745
$666$	190808.	−	2.81303e6i	0.0166690	−	0.245748i
$667$	−440452.	−	1.77683e7i	−0.0383340	−	1.54643i
$668$			2.07352e6i			0.179790i
$669$	5.03359e6	+	5.38658e6i	0.434823	+	0.465316i
$670$	−1.55373e7			−1.33718
$671$			738497.i			0.0633203i
$672$	1.21705e7	+	1.30240e7i	1.03965	+	1.11256i
$673$	−1.55698e7			−1.32509			−0.662546	−	0.749021i	$-0.730524\pi$
$673$	−1.55698e7			−1.32509			−0.662546	+	0.749021i	$0.730524\pi$
$674$	−1.97704e7			−1.67636
$675$	3.70407e6	−	3.01875e6i	0.312911	−	0.255016i
$676$	1.58251e7			1.33192
$677$	445662.			0.0373710			0.0186855	−	0.999825i	$-0.494052\pi$
$677$	445662.			0.0373710			0.0186855	+	0.999825i	$0.494052\pi$
$678$	6.80997e6	−	6.36370e6i	0.568946	−	0.531662i
$679$	1.13858e7			0.947743
$680$			8.27969e6i			0.686660i
$681$	−4.36432e6	−	4.67038e6i	−0.360619	−	0.385908i
$682$			2.57907e7i			2.12326i
$683$			5.85494e6i			0.480254i	0.970741	+	0.240127i	$0.0771891\pi$
$683$			5.85494e6i			0.480254i	−0.970741	+	0.240127i	$0.922811\pi$
$684$	6.01032e6	+	407679.i	0.491199	+	0.0333179i
$685$	1.39999e7			1.13998
$686$	−1.41659e7			−1.14930
$687$	−4.50194e6	+	4.20692e6i	−0.363921	+	0.340073i
$688$			8.37301e6i			0.674389i
$689$	250124.			0.0200728
$690$	−1.06699e7	+	1.04779e7i	−0.853177	+	0.837823i
$691$	801751.			0.0638770			0.0319385	−	0.999490i	$-0.489832\pi$
$691$	801751.			0.0638770			0.0319385	+	0.999490i	$0.489832\pi$
$692$		−	1.66322e7i		−	1.32034i
$693$	−2.24307e7	−	1.52147e6i	−1.77423	−	0.120346i
$694$	−7.84353e6			−0.618177
$695$	−7.28625e6			−0.572192
$696$	8.31311e6	+	8.89609e6i	0.650490	+	0.696107i
$697$		−	397581.i		−	0.0309987i
$698$		−	1.48612e7i		−	1.15456i
$699$	−1.15015e7	+	1.07478e7i	−0.890351	+	0.832004i
$700$		−	8.54483e6i		−	0.659111i
$701$	2.12413e7			1.63263			0.816313	−	0.577610i	$-0.196015\pi$
$701$	2.12413e7			1.63263			0.816313	+	0.577610i	$0.196015\pi$
$702$	2.77184e6	+	3.40111e6i	0.212288	+	0.260482i
$703$	734174.			0.0560287
$704$	−3.15163e7			−2.39665
$705$	−5.79290e6	+	5.41328e6i	−0.438958	+	0.410193i
$706$	−5.28260e6			−0.398874
$707$	−2.76607e7			−2.08121
$708$	3.92876e6	−	3.67130e6i	0.294559	−	0.275256i
$709$			3.90485e6i			0.291735i	0.989304	+	0.145868i	$0.0465974\pi$
$709$			3.90485e6i			0.291735i	−0.989304	+	0.145868i	$0.953403\pi$
$710$	3.09842e6			0.230672
$711$	1.87838e7	+	1.27410e6i	1.39351	+	0.0945215i
$712$			1.51576e7i			1.12055i
$713$	1.22234e7	−	303002.i	0.900471	−	0.0223214i
$714$	2.42893e7	+	2.59926e7i	1.78307	+	1.90811i
$715$	3.48728e6			0.255106
$716$		−	1.43147e7i		−	1.04352i
$717$	1.22122e7	−	1.14119e7i	0.887149	−	0.829013i
$718$			3.27960e7i			2.37416i
$719$			1.90699e7i			1.37571i	0.725848	+	0.687855i	$0.241448\pi$
$719$			1.90699e7i			1.37571i	−0.725848	+	0.687855i	$0.758552\pi$
$720$	−322801.	+	4.75898e6i	−0.0232062	+	0.342123i
$721$	9.15226e6			0.655677
$722$		−	1.89984e7i		−	1.35635i
$723$	1.61406e7	−	1.50829e7i	1.14835	−	1.07310i
$724$			2.29255e7i			1.62545i
$725$			8.83756e6i			0.624435i
$726$	2.11669e7	−	1.97798e7i	1.49044	−	1.39277i
$727$			1.09662e7i			0.769522i	0.923016	+	0.384761i	$0.125716\pi$
$727$			1.09662e7i			0.769522i	−0.923016	+	0.384761i	$0.874284\pi$
$728$	2.23266e6			0.156133
$729$	2.89537e6	−	1.40538e7i	0.201783	−	0.979430i
$730$		−	2.95866e7i		−	2.05489i
$731$			3.16781e7i			2.19263i
$732$	−615821.	+	575465.i	−0.0424793	+	0.0396955i
$733$			945012.i			0.0649647i	0.999472	+	0.0324824i	$0.0103413\pi$
$733$			945012.i			0.0649647i	−0.999472	+	0.0324824i	$0.989659\pi$
$734$	−3.64452e7			−2.49690
$735$	2.81573e6	+	3.01319e6i	0.192252	+	0.205735i
$736$	474705.	+	1.91501e7i	0.0323020	+	1.30310i
$737$		−	2.51021e7i		−	1.70232i
$738$	−33290.4	+	490793.i	−0.00224998	+	0.0331709i
$739$	−2.68786e7			−1.81049			−0.905243	−	0.424894i	$-0.860311\pi$
$739$	−2.68786e7			−1.81049			−0.905243	+	0.424894i	$0.860311\pi$
$740$		−	2.55762e6i		−	0.171695i
$741$	−834746.	+	780044.i	−0.0558482	+	0.0521883i
$742$	−2.50928e6			−0.167317
$743$	1.24569e7			0.827825			0.413912	−	0.910317i	$-0.364162\pi$
$743$	1.24569e7			0.827825			0.413912	+	0.910317i	$0.364162\pi$
$744$	−6.11993e6	+	5.71888e6i	−0.405335	+	0.378773i
$745$	1.49575e7			0.987343
$746$	−3.49671e7			−2.30045
$747$	−281098.	+	4.14417e6i	−0.0184313	+	0.271729i
$748$	−4.70079e7			−3.07197
$749$			2.87856e6i			0.187487i
$750$	1.88916e7	−	1.76536e7i	1.22635	−	1.14599i
$751$		−	5.77627e6i		−	0.373721i	−0.982386	−	0.186861i	$-0.940169\pi$
$751$		−	5.77627e6i		−	0.373721i	0.982386	−	0.186861i	$-0.0598313\pi$
$752$			5.35739e6i			0.345469i
$753$	−4.12948e6	−	4.41907e6i	−0.265404	−	0.284017i
$754$	−8.11471e6			−0.519810
$755$	2.09062e7			1.33478
$756$	−1.62101e7	−	1.98902e7i	−1.03153	−	1.26571i
$757$		−	8.56985e6i		−	0.543542i	−0.962362	−	0.271771i	$-0.912391\pi$
$757$		−	8.56985e6i		−	0.543542i	0.962362	−	0.271771i	$-0.0876094\pi$
$758$	−860906.			−0.0544230
$759$	−1.69281e7	−	1.72383e7i	−1.06661	−	1.08615i
$760$	2.66755e6			0.167524
$761$		−	1.26939e7i		−	0.794570i	−0.917695	−	0.397285i	$-0.869952\pi$
$761$		−	1.26939e7i		−	0.794570i	0.917695	−	0.397285i	$-0.130048\pi$
$762$	−1.61957e7	+	1.51344e7i	−1.01045	+	0.944230i
$763$	1.43991e7			0.895413
$764$	1.20757e7			0.748477
$765$	−1.22127e6	+	1.80049e7i	−0.0754500	+	1.11234i
$766$			2.46466e7i			1.51770i
$767$			1.01978e6i			0.0625919i
$768$	−2.03276e6	−	2.17531e6i	−0.124361	−	0.133082i
$769$		−	216230.i		−	0.0131856i	−0.999978	−	0.00659281i	$-0.997901\pi$
$769$		−	216230.i		−	0.0131856i	0.999978	−	0.00659281i	$-0.00209857\pi$
$770$	−3.49848e7			−2.12644
$771$	1.10745e7	−	1.03488e7i	0.670949	−	0.626980i
$772$	2.79343e7			1.68692
$773$	−2.26652e7			−1.36430			−0.682150	−	0.731212i	$-0.738955\pi$
$773$	−2.26652e7			−1.36430			−0.682150	+	0.731212i	$0.738955\pi$
$774$	2.65249e6	−	3.91051e7i	0.159148	−	2.34628i
$775$	−6.07966e6			−0.363601
$776$	−8.38192e6			−0.499677
$777$	−2.13508e6	−	2.28481e6i	−0.126871	−	0.135768i
$778$		−	2.35255e7i		−	1.39345i
$779$	−128092.			−0.00756274
$780$	2.71742e6	+	2.90798e6i	0.159926	+	0.171142i
$781$			5.00580e6i			0.293661i
$782$	947388.	+	3.82187e7i	0.0554002	+	2.23490i
$783$	1.67654e7	+	2.05716e7i	0.977261	+	1.19912i
$784$	2.78666e6			0.161917
$785$		−	5.57599e6i		−	0.322959i
$786$	−1.12373e7	−	1.20253e7i	−0.648791	−	0.694289i
$787$		−	3.17556e7i		−	1.82761i	−0.406154	−	0.913805i	$-0.633130\pi$
$787$		−	3.17556e7i		−	1.82761i	0.406154	−	0.913805i	$-0.366870\pi$
$788$			3.28293e7i			1.88341i
$789$	−1.57348e7	−	1.68383e7i	−0.899848	−	0.962953i
$790$	2.92968e7			1.67014
$791$		−	1.03375e7i		−	0.587453i
$792$	1.65128e7	+	1.12006e6i	0.935423	+	0.0634496i
$793$		−	159847.i		−	0.00902657i
$794$		−	3.05743e7i		−	1.72110i
$795$	−869082.	−	930028.i	−0.0487689	−	0.0521889i
$796$		−	4.00874e7i		−	2.24246i
$797$	1.43657e7			0.801091			0.400546	−	0.916277i	$-0.368821\pi$
$797$	1.43657e7			0.801091			0.400546	+	0.916277i	$0.368821\pi$
$798$	8.37428e6	−	7.82550e6i	0.465522	−	0.435016i
$799$			2.02690e7i			1.12322i
$800$		−	9.52482e6i		−	0.526177i
$801$	−2.23578e6	+	3.29616e7i	−0.123125	+	1.81521i
$802$		−	1.27663e7i		−	0.700856i
$803$	4.78001e7			2.61602
$804$	2.09323e7	−	1.95605e7i	1.14203	−	1.06719i
$805$	411019.	+	1.65809e7i	0.0223549	+	0.901820i
$806$		−	5.58240e6i		−	0.302680i
$807$	−1.58611e7	+	1.48217e7i	−0.857333	+	0.801150i
$808$	2.03630e7			1.09727
$809$		−	1.33708e7i		−	0.718266i	−0.933286	−	0.359133i	$-0.883072\pi$
$809$		−	1.33708e7i		−	0.718266i	0.933286	−	0.359133i	$-0.116928\pi$
$810$	3.01520e6	−	2.21239e7i	0.161474	−	1.18481i
$811$	−2.90401e7			−1.55041			−0.775203	−	0.631712i	$-0.782352\pi$
$811$	−2.90401e7			−1.55041			−0.775203	+	0.631712i	$0.782352\pi$
$812$	4.74560e7			2.52581
$813$	−1.34109e7	−	1.43514e7i	−0.711595	−	0.761497i
$814$	7.08835e6			0.374960
$815$	2.33568e6			0.123174
$816$	8.32567e6	+	8.90953e6i	0.437718	+	0.468414i
$817$	1.02060e7			0.534936
$818$			3.47795e6i			0.181736i
$819$	4.85512e6	+	329322.i	0.252924	+	0.0171558i
$820$			446232.i			0.0231753i
$821$			6.75457e6i			0.349736i	0.984592	+	0.174868i	$0.0559498\pi$
$821$			6.75457e6i			0.349736i	−0.984592	+	0.174868i	$0.944050\pi$
$822$	−3.23548e7	+	3.02345e7i	−1.67016	+	1.56071i
$823$	8.26650e6			0.425424			0.212712	−	0.977115i	$-0.431770\pi$
$823$	8.26650e6			0.425424			0.212712	+	0.977115i	$0.431770\pi$
$824$	−6.73762e6			−0.345692
$825$	8.20208e6	+	8.77727e6i	0.419555	+	0.448978i
$826$		−	1.02306e7i		−	0.521735i
$827$	−2.07606e7			−1.05554			−0.527771	−	0.849387i	$-0.676972\pi$
$827$	−2.07606e7			−1.05554			−0.527771	+	0.849387i	$0.676972\pi$
$828$	1.18374e6	−	2.75488e7i	0.0600043	−	1.39646i
$829$	−1.74380e7			−0.881274			−0.440637	−	0.897685i	$-0.645247\pi$
$829$	−1.74380e7			−0.881274			−0.440637	+	0.897685i	$0.645247\pi$
$830$			6.46360e6i			0.325671i
$831$	1.30941e7	+	1.40124e7i	0.657770	+	0.703898i
$832$	6.82169e6			0.341652
$833$	1.05429e7			0.526440
$834$	1.68391e7	−	1.57356e7i	0.838307	−	0.783371i
$835$			2.00124e6i			0.0993307i
$836$			1.51450e7i			0.749467i
$837$	−1.41519e7	+	1.15335e7i	−0.698234	+	0.569048i
$838$			4.29310e7i			2.11184i
$839$	−1.80301e7			−0.884288			−0.442144	−	0.896944i	$-0.645782\pi$
$839$	−1.80301e7			−0.884288			−0.442144	+	0.896944i	$0.645782\pi$
$840$	−7.75759e6	−	8.30161e6i	−0.379340	−	0.405942i
$841$	−2.85706e7			−1.39293
$842$	4.22440e7			2.05345
$843$	2.53282e7	+	2.71044e7i	1.22754	+	1.31362i
$844$	−4.84064e7			−2.33909
$845$	1.52734e7			0.735860
$846$	1.69717e6	−	2.50210e7i	0.0815267	−	1.20193i
$847$		−	3.21312e7i		−	1.53893i
$848$	−860109.			−0.0410737
$849$	−1.15375e7	+	1.07814e7i	−0.549343	+	0.513343i
$850$		−	1.90091e7i		−	0.902431i
$851$	−83277.5	−	3.35950e6i	−0.00394188	−	0.159020i
$852$	−4.17426e6	+	3.90071e6i	−0.197006	+	0.184096i
$853$	2.85948e7			1.34560			0.672799	−	0.739826i	$-0.265092\pi$
$853$	2.85948e7			1.34560			0.672799	+	0.739826i	$0.265092\pi$
$854$			1.60361e6i			0.0752409i
$855$	5.80082e6	+	393469.i	0.271378	+	0.0184075i
$856$		−	2.11911e6i		−	0.0988484i
$857$		−	5.96691e6i		−	0.277522i	−0.990326	−	0.138761i	$-0.955688\pi$
$857$		−	5.96691e6i		−	0.277522i	0.990326	−	0.138761i	$-0.0443120\pi$
$858$	−8.05936e6	+	7.53122e6i	−0.373751	+	0.349258i
$859$	3.92119e7			1.81315			0.906577	−	0.422041i	$-0.138686\pi$
$859$	3.92119e7			1.81315			0.906577	+	0.422041i	$0.138686\pi$
$860$		−	3.55545e7i		−	1.63926i
$861$	372510.	+	398633.i	0.0171250	+	0.0183259i
$862$		−	2.64684e7i		−	1.21327i
$863$		−	1.60530e6i		−	0.0733716i	−0.999327	−	0.0366858i	$-0.988320\pi$
$863$		−	1.60530e6i		−	0.0733716i	0.999327	−	0.0366858i	$-0.0116801\pi$
$864$	−1.80692e7	−	2.21713e7i	−0.823484	−	1.01043i
$865$		−	1.60525e7i		−	0.729461i
$866$	−2.89335e7			−1.31101
$867$	1.63872e7	+	1.75364e7i	0.740385	+	0.792306i
$868$			3.26467e7i			1.47075i
$869$			4.73320e7i			2.12620i
$870$	2.81954e7	+	3.01727e7i	1.26293	+	1.35150i
$871$			5.43334e6i			0.242673i
$872$	−1.06002e7			−0.472087
$873$	−1.82273e7	−	1.23635e6i	−0.809442	−	0.0549043i
$874$	1.23133e7	−	305229.i	0.545248	−	0.0135160i
$875$		−	2.86772e7i		−	1.26624i
$876$	3.72477e7	+	3.98598e7i	1.63998	+	1.75499i
$877$	1.49689e7			0.657192			0.328596	−	0.944471i	$-0.393425\pi$
$877$	1.49689e7			0.657192			0.328596	+	0.944471i	$0.393425\pi$
$878$			3.42280e7i			1.49846i
$879$	1.36792e7	+	1.46384e7i	0.597155	+	0.639032i
$880$	−1.19918e7			−0.522009
$881$	1.06909e6			0.0464062			0.0232031	−	0.999731i	$-0.492614\pi$
$881$	1.06909e6			0.0464062			0.0232031	+	0.999731i	$0.492614\pi$
$882$	−1.30147e7	−	882786.i	−0.563330	−	0.0382106i
$883$	303833.			0.0131139			0.00655696	−	0.999979i	$-0.497913\pi$
$883$	303833.			0.0131139			0.00655696	+	0.999979i	$0.497913\pi$
$884$	1.01748e7			0.437922
$885$	3.79182e6	−	3.54333e6i	0.162738	−	0.152074i
$886$	−1.85949e7			−0.795808
$887$		−	4.42729e7i		−	1.88942i	−0.327906	−	0.944710i	$-0.606343\pi$
$887$		−	4.42729e7i		−	1.88942i	0.327906	−	0.944710i	$-0.393657\pi$
$888$	1.57178e6	+	1.68201e6i	0.0668899	+	0.0715807i
$889$			2.45850e7i			1.04332i
$890$			5.14097e7i			2.17556i
$891$	3.57434e7	+	4.87135e6i	1.50835	+	0.205568i
$892$	−2.11536e7			−0.890167
$893$	6.53024e6			0.274031
$894$	−3.45679e7	+	3.23026e7i	−1.44654	+	1.35174i
$895$		−	1.38157e7i		−	0.576521i
$896$	−3.18440e7			−1.32513
$897$	3.66408e6	+	3.73123e6i	0.152049	+	0.154836i
$898$	−4.83078e6			−0.199906
$899$		−	3.37650e7i		−	1.39338i
$900$	−927856.	+	1.36792e7i	−0.0381834	+	0.562929i
$901$	−3.25410e6			−0.133542
$902$	−1.23671e6			−0.0506119
$903$	−2.96806e7	−	3.17620e7i	−1.21130	−	1.29625i
$904$			7.61015e6i			0.309722i
$905$			2.21264e7i			0.898027i
$906$	−4.83159e7	+	4.51497e7i	−1.95555	+	1.82740i
$907$		−	2.24802e7i		−	0.907365i	−0.891163	−	0.453682i	$-0.850110\pi$
$907$		−	2.24802e7i		−	0.907365i	0.891163	−	0.453682i	$-0.149890\pi$
$908$	1.83410e7			0.738257
$909$	4.42812e7	+	3.00359e6i	1.77750	+	0.120568i
$910$	7.57245e6			0.303133
$911$	1.79023e7			0.714681			0.357341	−	0.933974i	$-0.383684\pi$
$911$	1.79023e7			0.714681			0.357341	+	0.933974i	$0.383684\pi$
$912$	2.87047e6	−	2.68236e6i	0.114279	−	0.106790i
$913$	−1.04426e7			−0.414602
$914$	1.28114e6			0.0507261
$915$	−594356.	+	555406.i	−0.0234689	+	0.0219310i
$916$		−	1.76795e7i		−	0.696196i
$917$	−1.82543e7			−0.716874
$918$	−3.60615e7	−	4.42483e7i	−1.41233	−	1.73297i
$919$		−	1.24506e7i		−	0.486297i	−0.969989	−	0.243149i	$-0.921820\pi$
$919$		−	1.24506e7i		−	0.486297i	0.969989	−	0.243149i	$-0.0781803\pi$
$920$	−302580.	−	1.22064e7i	−0.0117861	−	0.475465i
$921$	4.68410e6	+	5.01258e6i	0.181960	+	0.194721i
$922$	−5.10223e7			−1.97667
$923$		−	1.08350e6i		−	0.0418626i
$924$	4.71323e7	−	4.40437e7i	1.81610	−	1.69708i
$925$			1.67094e6i			0.0642106i
$926$		−	3.87446e7i		−	1.48485i
$927$	−1.46516e7	−	993814.i	−0.559996	−	0.0379844i
$928$	5.28986e7			2.01639
$929$			2.16576e7i			0.823325i	0.911336	+	0.411662i	$0.135052\pi$
$929$			2.16576e7i			0.823325i	−0.911336	+	0.411662i	$0.864948\pi$
$930$	−2.07568e7	+	1.93966e7i	−0.786962	+	0.735391i
$931$		−	3.39671e6i		−	0.128435i
$932$		−	4.51674e7i		−	1.70328i
$933$	5.23795e6	−	4.89469e6i	0.196996	−	0.184086i
$934$			1.14919e7i			0.431048i
$935$	−4.53693e7			−1.69720
$936$	−3.57419e6	−	242437.i	−0.133349	−	0.00904501i
$937$		−	3.14740e7i		−	1.17112i	−0.810628	−	0.585562i	$-0.800874\pi$
$937$		−	3.14740e7i		−	1.17112i	0.810628	−	0.585562i	$-0.199126\pi$
$938$		−	5.45080e7i		−	2.02280i
$939$	3.18283e7	−	2.97425e7i	1.17801	−	1.10081i
$940$		−	2.27492e7i		−	0.839744i
$941$	−3.65269e7			−1.34474			−0.672370	−	0.740215i	$-0.734724\pi$
$941$	−3.65269e7			−1.34474			−0.672370	+	0.740215i	$0.734724\pi$
$942$	1.20421e7	+	1.28865e7i	0.442154	+	0.473161i
$943$	14529.5	+	586137.i	0.000532074	+	0.0214644i
$944$		−	3.50675e6i		−	0.128078i
$945$	−1.56451e7	−	1.91969e7i	−0.569900	−	0.699280i
$946$	9.85379e7			3.57994
$947$			4.50703e7i			1.63311i	0.577266	+	0.816556i	$0.304119\pi$
$947$			4.50703e7i			1.63311i	−0.577266	+	0.816556i	$0.695881\pi$
$948$	−3.94694e7	+	3.68829e7i	−1.42639	+	1.33292i
$949$	−1.03463e7			−0.372924
$950$	−6.12433e6			−0.220166
$951$	−2.58205e7	+	2.41284e7i	−0.925791	+	0.865122i
$952$	−2.90468e7			−1.03874
$953$	−2.82396e7			−1.00723			−0.503613	−	0.863930i	$-0.667996\pi$
$953$	−2.82396e7			−1.00723			−0.503613	+	0.863930i	$0.667996\pi$
$954$	4.01703e6	+	272474.i	0.142901	+	0.00969292i
$955$	1.16548e7			0.413519
$956$			4.79585e7i			1.69715i
$957$	−4.87469e7	+	4.55525e7i	−1.72055	+	1.60780i
$958$			4.30424e6i			0.151524i
$959$			4.91142e7i			1.72449i
$960$	−2.37027e7	−	2.53649e7i	−0.830079	−	0.888290i
$961$	−5.40099e6			−0.188654
$962$	−1.53427e6			−0.0534521
$963$	312574.	−	4.60820e6i	0.0108614	−	0.160128i
$964$			6.33857e7i			2.19684i
$965$	2.69606e7			0.931988
$966$	−3.67586e7	−	3.74322e7i	−1.26741	−	1.29063i
$967$	−2.39847e7			−0.824838			−0.412419	−	0.910994i	$-0.635316\pi$
$967$	−2.39847e7			−0.824838			−0.412419	+	0.910994i	$0.635316\pi$
$968$			2.36540e7i			0.811365i
$969$	1.08600e7	−	1.01483e7i	0.371553	−	0.347204i
$970$	−2.84288e7			−0.970128
$971$	−1.37433e7			−0.467780			−0.233890	−	0.972263i	$-0.575146\pi$
$971$	−1.37433e7			−0.467780			−0.233890	+	0.972263i	$0.575146\pi$
$972$	2.37905e7	+	3.36018e7i	0.807677	+	1.14077i
$973$		−	2.55616e7i		−	0.865576i
$974$			1.86737e7i			0.630713i
$975$	−1.77534e6	−	1.89984e6i	−0.0598094	−	0.0640037i
$976$			549672.i			0.0184705i
$977$	−1.51346e7			−0.507266			−0.253633	−	0.967301i	$-0.581626\pi$
$977$	−1.51346e7			−0.507266			−0.253633	+	0.967301i	$0.581626\pi$
$978$	−5.39793e6	+	5.04419e6i	−0.180460	+	0.168634i
$979$	−8.30575e7			−2.76963
$980$	−1.18330e7			−0.393578
$981$	−2.30511e7	−	1.56355e6i	−0.764748	−	0.0518727i
$982$	4.36978e7			1.44604
$983$	−3.85715e7			−1.27316			−0.636580	−	0.771210i	$-0.719652\pi$
$983$	−3.85715e7			−1.27316			−0.636580	+	0.771210i	$0.719652\pi$
$984$	−274231.	−	293462.i	−0.00902877	−	0.00966194i
$985$			3.16849e7i			1.04055i
$986$	1.05572e8			3.45825
$987$	−1.89908e7	−	2.03226e7i	−0.620514	−	0.664029i
$988$		−	3.27812e6i		−	0.106840i
$989$	−1.15767e6	−	4.67017e7i	−0.0376353	−	1.51825i
$990$	5.60062e7	+	3.79889e6i	1.81613	+	0.123188i
$991$	1.69044e7			0.546784			0.273392	−	0.961903i	$-0.411854\pi$
$991$	1.69044e7			0.546784			0.273392	+	0.961903i	$0.411854\pi$
$992$			3.63908e7i			1.17412i
$993$	905667.	+	969179.i	0.0291471	+	0.0311911i
$994$			1.08699e7i			0.348946i
$995$		−	3.86900e7i		−	1.23891i
$996$	−8.13726e6	−	8.70791e6i	−0.259914	−	0.278141i
$997$	−1.14588e7			−0.365090			−0.182545	−	0.983197i	$-0.558434\pi$
$997$	−1.14588e7			−0.365090			−0.182545	+	0.983197i	$0.558434\pi$
$998$		−	3.10749e7i		−	0.987606i
$999$	3.16989e6	+	3.88952e6i	0.100492	+	0.123306i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.6.c.b.68.5	✓	32
3.2	odd	2	inner	69.6.c.b.68.28	yes	32
23.22	odd	2	inner	69.6.c.b.68.6	yes	32
69.68	even	2	inner	69.6.c.b.68.27	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.6.c.b.68.5	✓	32	1.1	even	1	trivial
69.6.c.b.68.6	yes	32	23.22	odd	2	inner
69.6.c.b.68.27	yes	32	69.68	even	2	inner
69.6.c.b.68.28	yes	32	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 69 = 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	69.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-8.75944i q^{2} +(10.6432 + 11.3896i) q^{3} -44.7279 q^{4} -43.1688 q^{5} +(99.7663 - 93.2285i) q^{6} -151.444i q^{7} +111.489i q^{8} +(-16.4449 + 242.443i) q^{9} +O(q^{10})\)	\(q-8.75944i q^{2} +(10.6432 + 11.3896i) q^{3} -44.7279 q^{4} -43.1688 q^{5} +(99.7663 - 93.2285i) q^{6} -151.444i q^{7} +111.489i q^{8} +(-16.4449 + 242.443i) q^{9} +378.135i q^{10} -610.914 q^{11} +(-476.047 - 509.431i) q^{12} +132.232 q^{13} -1326.57 q^{14} +(-459.454 - 491.674i) q^{15} -454.710 q^{16} -1720.33 q^{17} +(2123.67 + 144.048i) q^{18} +554.256i q^{19} +1930.85 q^{20} +(1724.89 - 1611.85i) q^{21} +5351.26i q^{22} +(2536.22 - 62.8693i) q^{23} +(-1269.81 + 1186.60i) q^{24} -1261.46 q^{25} -1158.28i q^{26} +(-2936.35 + 2393.07i) q^{27} +6773.79i q^{28} -7005.84i q^{29} +(-4306.79 + 4024.56i) q^{30} +4819.56 q^{31} +7550.66i q^{32} +(-6502.07 - 6958.05i) q^{33} +15069.2i q^{34} +6537.67i q^{35} +(735.543 - 10844.0i) q^{36} -1324.61i q^{37} +4854.97 q^{38} +(1407.37 + 1506.07i) q^{39} -4812.84i q^{40} +231.107i q^{41} +(-14118.9 - 15109.1i) q^{42} -18413.9i q^{43} +27324.9 q^{44} +(709.905 - 10466.0i) q^{45} +(-550.700 - 22215.8i) q^{46} -11782.0i q^{47} +(-4839.57 - 5178.96i) q^{48} -6128.42 q^{49} +11049.7i q^{50} +(-18309.8 - 19593.9i) q^{51} -5914.46 q^{52} +1891.55 q^{53} +(20961.9 + 25720.8i) q^{54} +26372.4 q^{55} +16884.4 q^{56} +(-6312.74 + 5899.05i) q^{57} -61367.2 q^{58} +7712.06i q^{59} +(20550.4 + 21991.5i) q^{60} -1208.84i q^{61} -42216.7i q^{62} +(36716.6 + 2490.48i) q^{63} +51588.8 q^{64} -5708.30 q^{65} +(-60948.6 + 56954.5i) q^{66} +41089.5i q^{67} +76946.8 q^{68} +(27709.5 + 28217.3i) q^{69} +57266.4 q^{70} -8193.96i q^{71} +(-27029.7 - 1833.42i) q^{72} -78243.7 q^{73} -11602.9 q^{74} +(-13425.9 - 14367.5i) q^{75} -24790.7i q^{76} +92519.5i q^{77} +(13192.3 - 12327.8i) q^{78} -77477.3i q^{79} +19629.3 q^{80} +(-58508.1 - 7973.88i) q^{81} +2024.37 q^{82} +17093.4 q^{83} +(-77150.5 + 72094.7i) q^{84} +74264.7 q^{85} -161296. q^{86} +(79793.5 - 74564.5i) q^{87} -68110.1i q^{88} +135956. q^{89} +(-91676.0 - 6218.37i) q^{90} -20025.8i q^{91} +(-113439. + 2812.01i) q^{92} +(51295.5 + 54892.7i) q^{93} -103204. q^{94} -23926.5i q^{95} +(-85998.8 + 80363.1i) q^{96} +75181.6i q^{97} +53681.6i q^{98} +(10046.4 - 148112. i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 408 q^{4} - 528 q^{6} - 444 q^{9}+O(q^{10}) \) 32 * q - 408 * q^4 - 528 * q^6 - 444 * q^9	\( 32 q - 408 q^{4} - 528 q^{6} - 444 q^{9} - 2484 q^{12} + 520 q^{13} + 4936 q^{16} + 7188 q^{18} + 18660 q^{24} + 36032 q^{25} - 22032 q^{27} + 6544 q^{31} - 33912 q^{36} - 63912 q^{39} + 54328 q^{46} + 88284 q^{48} - 207664 q^{49} + 46296 q^{52} - 38628 q^{54} - 139296 q^{55} - 184144 q^{58} + 486584 q^{64} - 113580 q^{69} + 37176 q^{70} - 15504 q^{72} - 93896 q^{73} + 249840 q^{75} + 368028 q^{78} - 339372 q^{81} - 23512 q^{82} + 259584 q^{85} + 509928 q^{87} + 82740 q^{93} - 562000 q^{94} + 1404 q^{96}+O(q^{100}) \) 32 * q - 408 * q^4 - 528 * q^6 - 444 * q^9 - 2484 * q^12 + 520 * q^13 + 4936 * q^16 + 7188 * q^18 + 18660 * q^24 + 36032 * q^25 - 22032 * q^27 + 6544 * q^31 - 33912 * q^36 - 63912 * q^39 + 54328 * q^46 + 88284 * q^48 - 207664 * q^49 + 46296 * q^52 - 38628 * q^54 - 139296 * q^55 - 184144 * q^58 + 486584 * q^64 - 113580 * q^69 + 37176 * q^70 - 15504 * q^72 - 93896 * q^73 + 249840 * q^75 + 368028 * q^78 - 339372 * q^81 - 23512 * q^82 + 259584 * q^85 + 509928 * q^87 + 82740 * q^93 - 562000 * q^94 + 1404 * q^96

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