LMFDB - Embedded newform 69.3.b.a.47.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [69,3,Mod(47,69)]

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, 0]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("69.47");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 69 = 3 \cdot 23 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	69.b (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:	$1.88011382409$
Analytic rank:	$0$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} + 40x^{12} + 598x^{10} + 4207x^{8} + 14465x^{6} + 23786x^{4} + 17144x^{2} + 3887 $ x^14 + 40x^12 + 598x^10 + 4207x^8 + 14465x^6 + 23786x^4 + 17144x^2 + 3887
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{8}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			47.6
Root			$-1.13976i$ of defining polynomial
Character	$\chi$	$=$	69.47
Dual form			69.3.b.a.47.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.13976i q^{2} +(-2.53605 - 1.60264i) q^{3} +2.70094 q^{4} -4.29888i q^{5} +(-1.82663 + 2.89049i) q^{6} -9.25811 q^{7} -7.63748i q^{8} +(3.86309 + 8.12875i) q^{9} +O(q^{10})$	\(q-1.13976i q^{2} +(-2.53605 - 1.60264i) q^{3} +2.70094 q^{4} -4.29888i q^{5} +(-1.82663 + 2.89049i) q^{6} -9.25811 q^{7} -7.63748i q^{8} +(3.86309 + 8.12875i) q^{9} -4.89970 q^{10} -15.8448i q^{11} +(-6.84972 - 4.32864i) q^{12} +10.0233 q^{13} +10.5520i q^{14} +(-6.88956 + 10.9022i) q^{15} +2.09884 q^{16} +27.1126i q^{17} +(9.26485 - 4.40300i) q^{18} +27.5173 q^{19} -11.6110i q^{20} +(23.4790 + 14.8374i) q^{21} -18.0594 q^{22} +4.79583i q^{23} +(-12.2401 + 19.3690i) q^{24} +6.51963 q^{25} -11.4242i q^{26} +(3.23049 - 26.8060i) q^{27} -25.0056 q^{28} -19.8347i q^{29} +(12.4259 + 7.85247i) q^{30} +12.5473 q^{31} -32.9421i q^{32} +(-25.3936 + 40.1833i) q^{33} +30.9020 q^{34} +39.7995i q^{35} +(10.4340 + 21.9553i) q^{36} -20.8158 q^{37} -31.3632i q^{38} +(-25.4195 - 16.0637i) q^{39} -32.8326 q^{40} +46.8742i q^{41} +(16.9111 - 26.7605i) q^{42} -49.0913 q^{43} -42.7959i q^{44} +(34.9445 - 16.6069i) q^{45} +5.46611 q^{46} +7.46736i q^{47} +(-5.32276 - 3.36368i) q^{48} +36.7126 q^{49} -7.43084i q^{50} +(43.4518 - 68.7590i) q^{51} +27.0723 q^{52} -25.1269i q^{53} +(-30.5525 - 3.68199i) q^{54} -68.1150 q^{55} +70.7086i q^{56} +(-69.7853 - 44.1004i) q^{57} -22.6069 q^{58} +29.4515i q^{59} +(-18.6083 + 29.4461i) q^{60} +16.7591 q^{61} -14.3009i q^{62} +(-35.7649 - 75.2568i) q^{63} -29.1509 q^{64} -43.0889i q^{65} +(45.7994 + 28.9427i) q^{66} +38.4334 q^{67} +73.2296i q^{68} +(7.68599 - 12.1625i) q^{69} +45.3620 q^{70} +14.6898i q^{71} +(62.0832 - 29.5043i) q^{72} +139.301 q^{73} +23.7251i q^{74} +(-16.5341 - 10.4486i) q^{75} +74.3226 q^{76} +146.693i q^{77} +(-18.3088 + 28.9722i) q^{78} -103.267 q^{79} -9.02265i q^{80} +(-51.1531 + 62.8041i) q^{81} +53.4255 q^{82} +5.81691i q^{83} +(63.4154 + 40.0750i) q^{84} +116.554 q^{85} +55.9524i q^{86} +(-31.7880 + 50.3019i) q^{87} -121.015 q^{88} -48.4794i q^{89} +(-18.9280 - 39.8285i) q^{90} -92.7966 q^{91} +12.9533i q^{92} +(-31.8205 - 20.1088i) q^{93} +8.51102 q^{94} -118.294i q^{95} +(-52.7944 + 83.5428i) q^{96} -66.6277 q^{97} -41.8436i q^{98} +(128.799 - 61.2099i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 4 q^{3} - 24 q^{4} + 11 q^{6} - 4 q^{7} - 8 q^{9}+O(q^{10}) $ 14 * q - 4 * q^3 - 24 * q^4 + 11 * q^6 - 4 * q^7 - 8 * q^9	$ 14 q - 4 q^{3} - 24 q^{4} + 11 q^{6} - 4 q^{7} - 8 q^{9} - 8 q^{10} + 19 q^{12} - 14 q^{15} + 72 q^{16} - 31 q^{18} + 8 q^{19} - 2 q^{21} - 84 q^{22} - 44 q^{24} + 38 q^{25} + 62 q^{27} + 76 q^{28} + 62 q^{30} - 144 q^{31} + 90 q^{33} - 68 q^{34} + 3 q^{36} + 48 q^{37} - 78 q^{39} + 120 q^{40} - 76 q^{42} - 48 q^{43} - 18 q^{45} - 317 q^{48} - 30 q^{49} + 18 q^{51} - 6 q^{52} + 312 q^{54} + 232 q^{55} + 76 q^{57} + 66 q^{58} - 36 q^{60} - 140 q^{61} - 206 q^{63} - 346 q^{64} + 398 q^{66} + 204 q^{67} + 80 q^{70} + 384 q^{72} - 224 q^{73} - 80 q^{75} + 100 q^{76} - 341 q^{78} - 344 q^{79} - 232 q^{81} - 62 q^{82} - 330 q^{84} + 480 q^{85} + 86 q^{87} + 436 q^{88} - 514 q^{90} - 172 q^{91} + 62 q^{93} + 514 q^{94} + 609 q^{96} - 24 q^{97} + 234 q^{99}+O(q^{100}) $ 14 * q - 4 * q^3 - 24 * q^4 + 11 * q^6 - 4 * q^7 - 8 * q^9 - 8 * q^10 + 19 * q^12 - 14 * q^15 + 72 * q^16 - 31 * q^18 + 8 * q^19 - 2 * q^21 - 84 * q^22 - 44 * q^24 + 38 * q^25 + 62 * q^27 + 76 * q^28 + 62 * q^30 - 144 * q^31 + 90 * q^33 - 68 * q^34 + 3 * q^36 + 48 * q^37 - 78 * q^39 + 120 * q^40 - 76 * q^42 - 48 * q^43 - 18 * q^45 - 317 * q^48 - 30 * q^49 + 18 * q^51 - 6 * q^52 + 312 * q^54 + 232 * q^55 + 76 * q^57 + 66 * q^58 - 36 * q^60 - 140 * q^61 - 206 * q^63 - 346 * q^64 + 398 * q^66 + 204 * q^67 + 80 * q^70 + 384 * q^72 - 224 * q^73 - 80 * q^75 + 100 * q^76 - 341 * q^78 - 344 * q^79 - 232 * q^81 - 62 * q^82 - 330 * q^84 + 480 * q^85 + 86 * q^87 + 436 * q^88 - 514 * q^90 - 172 * q^91 + 62 * q^93 + 514 * q^94 + 609 * q^96 - 24 * q^97 + 234 * q^99

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 $

$2$		−	1.13976i		−	0.569882i	−0.958545	−	0.284941i	$-0.908026\pi$
$2$		−	1.13976i		−	0.569882i	0.958545	−	0.284941i	$-0.0919739\pi$
$3$	−2.53605	−	1.60264i	−0.845350	−	0.534214i
$3$	−2.53605	−	1.60264i	−0.845350	−	0.534214i
$4$	2.70094			0.675235
$5$		−	4.29888i		−	0.859776i	−0.902882	−	0.429888i	$-0.858553\pi$
$5$		−	4.29888i		−	0.859776i	0.902882	−	0.429888i	$-0.141447\pi$
$6$	−1.82663	+	2.89049i	−0.304438	+	0.481749i
$7$	−9.25811			−1.32259			−0.661293	−	0.750127i	$-0.729992\pi$
$7$	−9.25811			−1.32259			−0.661293	+	0.750127i	$0.729992\pi$
$8$		−	7.63748i		−	0.954686i
$9$	3.86309	+	8.12875i	0.429232	+	0.903194i
$10$	−4.89970			−0.489970
$11$		−	15.8448i		−	1.44044i	−0.693746	−	0.720220i	$-0.744041\pi$
$11$		−	15.8448i		−	1.44044i	0.693746	−	0.720220i	$-0.255959\pi$
$12$	−6.84972	−	4.32864i	−0.570810	−	0.360720i
$13$	10.0233			0.771021			0.385511	−	0.922703i	$-0.374025\pi$
$13$	10.0233			0.771021			0.385511	+	0.922703i	$0.374025\pi$
$14$			10.5520i			0.753718i
$15$	−6.88956	+	10.9022i	−0.459304	+	0.726811i
$16$	2.09884			0.131177
$17$			27.1126i			1.59486i	0.603411	+	0.797431i	$0.293808\pi$
$17$			27.1126i			1.59486i	−0.603411	+	0.797431i	$0.706192\pi$
$18$	9.26485	−	4.40300i	0.514714	−	0.244611i
$19$	27.5173			1.44828			0.724140	−	0.689653i	$-0.242237\pi$
$19$	27.5173			1.44828			0.724140	+	0.689653i	$0.242237\pi$
$20$		−	11.6110i		−	0.580551i
$21$	23.4790	+	14.8374i	1.11805	+	0.706544i
$22$	−18.0594			−0.820880
$23$			4.79583i			0.208514i
$23$			4.79583i			0.208514i
$24$	−12.2401	+	19.3690i	−0.510006	+	0.807043i
$25$	6.51963			0.260785
$26$		−	11.4242i		−	0.439391i
$27$	3.23049	−	26.8060i	0.119648	−	0.992816i
$28$	−25.0056			−0.893057
$29$		−	19.8347i		−	0.683957i	−0.939708	−	0.341978i	$-0.888903\pi$
$29$		−	19.8347i		−	0.683957i	0.939708	−	0.341978i	$-0.111097\pi$
$30$	12.4259	+	7.85247i	0.414196	+	0.261749i
$31$	12.5473			0.404750			0.202375	−	0.979308i	$-0.435134\pi$
$31$	12.5473			0.404750			0.202375	+	0.979308i	$0.435134\pi$
$32$		−	32.9421i		−	1.02944i
$33$	−25.3936	+	40.1833i	−0.769502	+	1.21767i
$34$	30.9020			0.908882
$35$			39.7995i			1.13713i
$36$	10.4340	+	21.9553i	0.289832	+	0.609868i
$37$	−20.8158			−0.562590			−0.281295	−	0.959621i	$-0.590764\pi$
$37$	−20.8158			−0.562590			−0.281295	+	0.959621i	$0.590764\pi$
$38$		−	31.3632i		−	0.825348i
$39$	−25.4195	−	16.0637i	−0.651783	−	0.411890i
$40$	−32.8326			−0.820816
$41$			46.8742i			1.14327i	0.820507	+	0.571637i	$0.193691\pi$
$41$			46.8742i			1.14327i	−0.820507	+	0.571637i	$0.806309\pi$
$42$	16.9111	−	26.7605i	0.402646	−	0.637155i
$43$	−49.0913			−1.14166			−0.570829	−	0.821069i	$-0.693378\pi$
$43$	−49.0913			−1.14166			−0.570829	+	0.821069i	$0.693378\pi$
$44$		−	42.7959i		−	0.972635i
$45$	34.9445	−	16.6069i	0.776545	−	0.369043i
$46$	5.46611			0.118829
$47$			7.46736i			0.158880i	0.996840	+	0.0794400i	$0.0253132\pi$
$47$			7.46736i			0.158880i	−0.996840	+	0.0794400i	$0.974687\pi$
$48$	−5.32276	−	3.36368i	−0.110891	−	0.0700767i
$49$	36.7126			0.749236
$50$		−	7.43084i		−	0.148617i
$51$	43.4518	−	68.7590i	0.851996	−	1.34822i
$52$	27.0723			0.520621
$53$		−	25.1269i		−	0.474092i	−0.971498	−	0.237046i	$-0.923821\pi$
$53$		−	25.1269i		−	0.474092i	0.971498	−	0.237046i	$-0.0761791\pi$
$54$	−30.5525	−	3.68199i	−0.565788	−	0.0681851i
$55$	−68.1150			−1.23846
$56$			70.7086i			1.26265i
$57$	−69.7853	−	44.1004i	−1.22430	−	0.773691i
$58$	−22.6069			−0.389774
$59$			29.4515i			0.499177i	0.968352	+	0.249589i	$0.0802954\pi$
$59$			29.4515i			0.499177i	−0.968352	+	0.249589i	$0.919705\pi$
$60$	−18.6083	+	29.4461i	−0.310138	+	0.490768i
$61$	16.7591			0.274740			0.137370	−	0.990520i	$-0.456135\pi$
$61$	16.7591			0.274740			0.137370	+	0.990520i	$0.456135\pi$
$62$		−	14.3009i		−	0.230660i
$63$	−35.7649	−	75.2568i	−0.567696	−	1.19455i
$64$	−29.1509			−0.455482
$65$		−	43.0889i		−	0.662906i
$66$	45.7994	+	28.9427i	0.693930	+	0.438525i
$67$	38.4334			0.573632			0.286816	−	0.957986i	$-0.407403\pi$
$67$	38.4334			0.573632			0.286816	+	0.957986i	$0.407403\pi$
$68$			73.2296i			1.07691i
$69$	7.68599	−	12.1625i	0.111391	−	0.176268i
$70$	45.3620			0.648028
$71$			14.6898i			0.206899i	0.994635	+	0.103449i	$0.0329880\pi$
$71$			14.6898i			0.206899i	−0.994635	+	0.103449i	$0.967012\pi$
$72$	62.0832	−	29.5043i	0.862267	−	0.409781i
$73$	139.301			1.90823			0.954115	−	0.299442i	$-0.0968003\pi$
$73$	139.301			1.90823			0.954115	+	0.299442i	$0.0968003\pi$
$74$			23.7251i			0.320609i
$75$	−16.5341	−	10.4486i	−0.220455	−	0.139315i
$76$	74.3226			0.977929
$77$			146.693i			1.90511i
$78$	−18.3088	+	28.9722i	−0.234729	+	0.371439i
$79$	−103.267			−1.30718			−0.653588	−	0.756851i	$-0.726737\pi$
$79$	−103.267			−1.30718			−0.653588	+	0.756851i	$0.726737\pi$
$80$		−	9.02265i		−	0.112783i
$81$	−51.1531	+	62.8041i	−0.631520	+	0.775359i
$82$	53.4255			0.651530
$83$			5.81691i			0.0700832i	0.999386	+	0.0350416i	$0.0111564\pi$
$83$			5.81691i			0.0700832i	−0.999386	+	0.0350416i	$0.988844\pi$
$84$	63.4154	+	40.0750i	0.754945	+	0.477083i
$85$	116.554			1.37122
$86$			55.9524i			0.650610i
$87$	−31.7880	+	50.3019i	−0.365379	+	0.578183i
$88$	−121.015			−1.37517
$89$		−	48.4794i		−	0.544712i	−0.962197	−	0.272356i	$-0.912197\pi$
$89$		−	48.4794i		−	0.544712i	0.962197	−	0.272356i	$-0.0878028\pi$
$90$	−18.9280	−	39.8285i	−0.210311	−	0.442539i
$91$	−92.7966			−1.01974
$92$			12.9533i			0.140796i
$93$	−31.8205	−	20.1088i	−0.342156	−	0.216223i
$94$	8.51102			0.0905428
$95$		−	118.294i		−	1.24520i
$96$	−52.7944	+	83.5428i	−0.549941	+	0.870238i
$97$	−66.6277			−0.686884			−0.343442	−	0.939174i	$-0.611593\pi$
$97$	−66.6277			−0.686884			−0.343442	+	0.939174i	$0.611593\pi$
$98$		−	41.8436i		−	0.426976i
$99$	128.799	−	61.2099i	1.30100	−	0.618282i
$100$	17.6091			0.176091
$101$			97.7943i			0.968260i	0.874996	+	0.484130i	$0.160864\pi$
$101$			97.7943i			0.968260i	−0.874996	+	0.484130i	$0.839136\pi$
$102$	−78.3689	−	49.5248i	−0.768323	−	0.485537i
$103$	−6.33568			−0.0615115			−0.0307557	−	0.999527i	$-0.509791\pi$
$103$	−6.33568			−0.0615115			−0.0307557	+	0.999527i	$0.509791\pi$
$104$		−	76.5526i		−	0.736083i
$105$	63.7843	−	100.933i	0.607469	−	0.961271i
$106$	−28.6387			−0.270176
$107$			102.090i			0.954116i	0.878872	+	0.477058i	$0.158297\pi$
$107$			102.090i			0.954116i	−0.878872	+	0.477058i	$0.841703\pi$
$108$	8.72536	−	72.4015i	0.0807904	−	0.670384i
$109$	−11.3038			−0.103704			−0.0518521	−	0.998655i	$-0.516512\pi$
$109$	−11.3038			−0.103704			−0.0518521	+	0.998655i	$0.516512\pi$
$110$			77.6350i			0.705773i
$111$	52.7899	+	33.3603i	0.475585	+	0.300543i
$112$	−19.4313			−0.173493
$113$		−	65.2458i		−	0.577397i	−0.957420	−	0.288698i	$-0.906778\pi$
$113$		−	65.2458i		−	0.577397i	0.957420	−	0.288698i	$-0.0932225\pi$
$114$	−50.2640	+	79.5387i	−0.440912	+	0.697707i
$115$	20.6167			0.179276
$116$		−	53.5725i		−	0.461832i
$117$	38.7208	+	81.4767i	0.330947	+	0.696382i
$118$	33.5677			0.284472
$119$		−	251.012i		−	2.10934i
$120$	83.2651	+	52.6189i	0.693876	+	0.438491i
$121$	−130.059			−1.07487
$122$		−	19.1014i		−	0.156569i
$123$	75.1225	−	118.875i	0.610752	−	0.966465i
$124$	33.8894			0.273302
$125$		−	135.499i		−	1.08399i
$126$	−85.7750	+	40.7635i	−0.680754	+	0.323520i
$127$	115.289			0.907789			0.453894	−	0.891056i	$-0.350034\pi$
$127$	115.289			0.907789			0.453894	+	0.891056i	$0.350034\pi$
$128$		−	98.5434i		−	0.769870i
$129$	124.498	+	78.6757i	0.965100	+	0.609889i
$130$	−49.1111			−0.377778
$131$			48.6952i			0.371719i	0.982576	+	0.185860i	$0.0595070\pi$
$131$			48.6952i			0.371719i	−0.982576	+	0.185860i	$0.940493\pi$
$132$	−68.5865	+	108.533i	−0.519595	+	0.822217i
$133$	−254.758			−1.91548
$134$		−	43.8049i		−	0.326903i
$135$	−115.236	−	13.8875i	−0.853600	−	0.102870i
$136$	207.072			1.52259
$137$			119.583i			0.872868i	0.899736	+	0.436434i	$0.143759\pi$
$137$			119.583i			0.872868i	−0.899736	+	0.436434i	$0.856241\pi$
$138$	−13.8623	−	8.76021i	−0.100452	−	0.0634798i
$139$	−131.728			−0.947686			−0.473843	−	0.880609i	$-0.657134\pi$
$139$	−131.728			−0.947686			−0.473843	+	0.880609i	$0.657134\pi$
$140$			107.496i			0.767829i
$141$	11.9675	−	18.9376i	0.0848759	−	0.134309i
$142$	16.7429			0.117908
$143$		−	158.817i		−	1.11061i
$144$	8.10799	+	17.0609i	0.0563055	+	0.118479i
$145$	−85.2672			−0.588050
$146$		−	158.770i		−	1.08746i
$147$	−93.1048	−	58.8370i	−0.633366	−	0.400252i
$148$	−56.2223			−0.379880
$149$			229.136i			1.53783i	0.639354	+	0.768913i	$0.279202\pi$
$149$			229.136i			1.53783i	−0.639354	+	0.768913i	$0.720798\pi$
$150$	−11.9090	+	18.8450i	−0.0793931	+	0.125633i
$151$	268.458			1.77787			0.888934	−	0.458034i	$-0.151446\pi$
$151$	268.458			1.77787			0.888934	+	0.458034i	$0.151446\pi$
$152$		−	210.163i		−	1.38265i
$153$	−220.392	+	104.738i	−1.44047	+	0.684565i
$154$	167.195			1.08568
$155$		−	53.9392i		−	0.347995i
$156$	−68.6566	−	43.3871i	−0.440106	−	0.278123i
$157$	145.452			0.926444			0.463222	−	0.886242i	$-0.346693\pi$
$157$	145.452			0.926444			0.463222	+	0.886242i	$0.346693\pi$
$158$			117.700i			0.744935i
$159$	−40.2693	+	63.7229i	−0.253266	+	0.400773i
$160$	−141.614			−0.885089
$161$		−	44.4003i		−	0.275778i
$162$	71.5818	+	58.3025i	0.441863	+	0.359892i
$163$	−21.5343			−0.132112			−0.0660562	−	0.997816i	$-0.521042\pi$
$163$	−21.5343			−0.132112			−0.0660562	+	0.997816i	$0.521042\pi$
$164$			126.604i			0.771978i
$165$	172.743	+	109.164i	1.04693	+	0.661599i
$166$	6.62990			0.0399391
$167$			308.454i			1.84703i	0.383565	+	0.923514i	$0.374696\pi$
$167$			308.454i			1.84703i	−0.383565	+	0.923514i	$0.625304\pi$
$168$	113.321	−	179.321i	0.674527	−	1.06738i
$169$	−68.5339			−0.405526
$170$		−	132.844i		−	0.781435i
$171$	106.302	+	223.681i	0.621648	+	1.30808i
$172$	−132.593			−0.770887
$173$		−	69.2587i		−	0.400339i	−0.979761	−	0.200170i	$-0.935851\pi$
$173$		−	69.2587i		−	0.400339i	0.979761	−	0.200170i	$-0.0641493\pi$
$174$	57.3322	+	36.2308i	0.329496	+	0.208223i
$175$	−60.3595			−0.344911
$176$		−	33.2557i		−	0.188953i
$177$	47.2001	−	74.6903i	0.266667	−	0.421979i
$178$	−55.2550			−0.310421
$179$			108.705i			0.607289i	0.952785	+	0.303645i	$0.0982035\pi$
$179$			108.705i			0.607289i	−0.952785	+	0.303645i	$0.901796\pi$
$180$	94.3830	−	44.8544i	0.524350	−	0.249191i
$181$	−9.02162			−0.0498432			−0.0249216	−	0.999689i	$-0.507934\pi$
$181$	−9.02162			−0.0498432			−0.0249216	+	0.999689i	$0.507934\pi$
$182$			105.766i			0.581133i
$183$	−42.5020	−	26.8589i	−0.232251	−	0.146770i
$184$	36.6281			0.199066
$185$			89.4847i			0.483701i
$186$	−22.9192	+	36.2678i	−0.123222	+	0.194988i
$187$	429.595			2.29730
$188$			20.1689i			0.107281i
$189$	−29.9082	+	248.173i	−0.158245	+	1.31309i
$190$	−134.827			−0.709614
$191$		−	139.748i		−	0.731666i	−0.930681	−	0.365833i	$-0.880784\pi$
$191$		−	139.748i		−	0.731666i	0.930681	−	0.365833i	$-0.119216\pi$
$192$	73.9280	+	46.7183i	0.385042	+	0.243325i
$193$	259.578			1.34496			0.672482	−	0.740113i	$-0.265228\pi$
$193$	259.578			1.34496			0.672482	+	0.740113i	$0.265228\pi$
$194$			75.9398i			0.391442i
$195$	−69.0560	+	109.275i	−0.354133	+	0.560387i
$196$	99.1584			0.505910
$197$		−	339.721i		−	1.72447i	−0.506506	−	0.862236i	$-0.669063\pi$
$197$		−	339.721i		−	1.72447i	0.506506	−	0.862236i	$-0.330937\pi$
$198$	−69.7648	−	146.800i	−0.352348	−	0.741414i
$199$	−181.151			−0.910307			−0.455153	−	0.890413i	$-0.650416\pi$
$199$	−181.151			−0.910307			−0.455153	+	0.890413i	$0.650416\pi$
$200$		−	49.7936i		−	0.248968i
$201$	−97.4689	−	61.5949i	−0.484920	−	0.306442i
$202$	111.462			0.551794
$203$			183.632i			0.904592i
$204$	117.361	−	185.714i	0.575298	−	0.910362i
$205$	201.507			0.982959
$206$			7.22117i			0.0350542i
$207$	−38.9841	+	18.5267i	−0.188329	+	0.0895010i
$208$	21.0372			0.101141
$209$		−	436.007i		−	2.08616i
$210$	−115.040	−	72.6990i	−0.547811	−	0.346186i
$211$	−144.244			−0.683621			−0.341810	−	0.939769i	$-0.611040\pi$
$211$	−144.244			−0.683621			−0.341810	+	0.939769i	$0.611040\pi$
$212$		−	67.8661i		−	0.320123i
$213$	23.5425	−	37.2541i	0.110528	−	0.174902i
$214$	116.359			0.543733
$215$			211.037i			0.981570i
$216$	−204.731	−	24.6728i	−0.947827	−	0.114226i
$217$	−116.164			−0.535318
$218$			12.8836i			0.0590992i
$219$	−353.273	−	223.249i	−1.61312	−	1.01940i
$220$	−183.975			−0.836248
$221$			271.758i			1.22967i
$222$	38.0228	−	60.1680i	0.171274	−	0.271027i
$223$	−222.377			−0.997207			−0.498603	−	0.866830i	$-0.666154\pi$
$223$	−222.377			−0.997207			−0.498603	+	0.866830i	$0.666154\pi$
$224$			304.982i			1.36153i
$225$	25.1859	+	52.9965i	0.111937	+	0.235540i
$226$	−74.3648			−0.329048
$227$			297.201i			1.30925i	0.755952	+	0.654627i	$0.227174\pi$
$227$			297.201i			1.30925i	−0.755952	+	0.654627i	$0.772826\pi$
$228$	−188.486	−	119.112i	−0.826692	−	0.522423i
$229$	−103.434			−0.451675			−0.225837	−	0.974165i	$-0.572512\pi$
$229$	−103.434			−0.451675			−0.225837	+	0.974165i	$0.572512\pi$
$230$		−	23.4982i		−	0.102166i
$231$	235.096	−	372.021i	1.01773	−	1.61048i
$232$	−151.488			−0.652964
$233$		−	90.3203i		−	0.387641i	−0.981037	−	0.193820i	$-0.937912\pi$
$233$		−	90.3203i		−	0.387641i	0.981037	−	0.193820i	$-0.0620879\pi$
$234$	92.8642	−	44.1325i	0.396855	−	0.188601i
$235$	32.1013			0.136601
$236$			79.5466i			0.337062i
$237$	261.890	+	165.500i	1.10502	+	0.698311i
$238$	−286.094			−1.20208
$239$		−	69.5986i		−	0.291207i	−0.989343	−	0.145604i	$-0.953488\pi$
$239$		−	69.5986i		−	0.291207i	0.989343	−	0.145604i	$-0.0465125\pi$
$240$	−14.4601	+	22.8819i	−0.0602503	+	0.0953412i
$241$	82.9954			0.344379			0.172190	−	0.985064i	$-0.444916\pi$
$241$	82.9954			0.344379			0.172190	+	0.985064i	$0.444916\pi$
$242$			148.236i			0.612546i
$243$	230.379	−	77.2942i	0.948063	−	0.318083i
$244$	45.2654			0.185514
$245$		−	157.823i		−	0.644175i
$246$	−135.490	−	85.6218i	−0.550771	−	0.348056i
$247$	275.814			1.11665
$248$		−	95.8295i		−	0.386409i
$249$	9.32242	−	14.7520i	0.0374394	−	0.0592448i
$250$	−154.437			−0.617748
$251$		−	84.7557i		−	0.337672i	−0.985644	−	0.168836i	$-0.945999\pi$
$251$		−	84.7557i		−	0.337672i	0.985644	−	0.168836i	$-0.0540008\pi$
$252$	−96.5988	−	203.264i	−0.383328	−	0.806604i
$253$	75.9891			0.300352
$254$		−	131.402i		−	0.517332i
$255$	−295.587	−	186.794i	−1.15916	−	0.732526i
$256$	−228.920			−0.894217
$257$		−	26.1604i		−	0.101791i	−0.998704	−	0.0508957i	$-0.983792\pi$
$257$		−	26.1604i		−	0.101791i	0.998704	−	0.0508957i	$-0.0162076\pi$
$258$	89.6716	−	141.898i	0.347564	−	0.549993i
$259$	192.715			0.744074
$260$		−	116.380i		−	0.447617i
$261$	161.232	−	76.6233i	0.617746	−	0.293576i
$262$	55.5010			0.211836
$263$			14.1868i			0.0539423i	0.999636	+	0.0269711i	$0.00858622\pi$
$263$			14.1868i			0.0539423i	−0.999636	+	0.0269711i	$0.991414\pi$
$264$	306.899	+	193.943i	1.16250	+	0.734633i
$265$	−108.017			−0.407613
$266$			290.364i			1.09159i
$267$	−77.6950	+	122.946i	−0.290992	+	0.460472i
$268$	103.806			0.387337
$269$			112.702i			0.418965i	0.977812	+	0.209483i	$0.0671780\pi$
$269$			112.702i			0.418965i	−0.977812	+	0.209483i	$0.932822\pi$
$270$	−15.8284	+	131.342i	−0.0586239	+	0.486451i
$271$	−283.857			−1.04744			−0.523721	−	0.851890i	$-0.675456\pi$
$271$	−283.857			−1.04744			−0.523721	+	0.851890i	$0.675456\pi$
$272$			56.9050i			0.209210i
$273$	235.337	+	148.720i	0.862039	+	0.544760i
$274$	136.296			0.497431
$275$		−	103.302i		−	0.375645i
$276$	20.7594	−	32.8501i	0.0752153	−	0.119022i
$277$	91.7694			0.331298			0.165649	−	0.986185i	$-0.447028\pi$
$277$	91.7694			0.331298			0.165649	+	0.986185i	$0.447028\pi$
$278$			150.139i			0.540069i
$279$	48.4712	+	101.994i	0.173732	+	0.365568i
$280$	303.968			1.08560
$281$			407.341i			1.44961i	0.688953	+	0.724806i	$0.258071\pi$
$281$			407.341i			1.44961i	−0.688953	+	0.724806i	$0.741929\pi$
$282$	−21.5844	−	13.6401i	−0.0765403	−	0.0483692i
$283$	−56.0662			−0.198114			−0.0990568	−	0.995082i	$-0.531583\pi$
$283$	−56.0662			−0.198114			−0.0990568	+	0.995082i	$0.531583\pi$
$284$			39.6763i			0.139705i
$285$	−189.582	+	299.998i	−0.665201	+	1.05263i
$286$	−181.014			−0.632916
$287$		−	433.966i		−	1.51208i
$288$	267.778	−	127.258i	0.929785	−	0.441869i
$289$	−446.095			−1.54358
$290$			97.1844i			0.335119i
$291$	168.971	+	106.780i	0.580657	+	0.366943i
$292$	376.243			1.28850
$293$		−	348.736i		−	1.19022i	−0.803643	−	0.595112i	$-0.797108\pi$
$293$		−	348.736i		−	1.19022i	0.803643	−	0.595112i	$-0.202892\pi$
$294$	−67.0603	+	106.117i	−0.228096	+	0.360944i
$295$	126.608			0.429181
$296$			158.980i			0.537096i
$297$	−424.737	−	51.1866i	−1.43009	−	0.172345i
$298$	261.161			0.876379
$299$			48.0700i			0.160769i
$300$	−44.6576	−	28.2211i	−0.148859	−	0.0940704i
$301$	454.492			1.50994
$302$		−	305.979i		−	1.01317i
$303$	156.729	−	248.011i	0.517258	−	0.818518i
$304$	57.7544			0.189982
$305$		−	72.0455i		−	0.236215i
$306$	119.377	+	251.195i	0.390121	+	0.820897i
$307$	−403.287			−1.31364			−0.656819	−	0.754049i	$-0.728098\pi$
$307$	−403.287			−1.31364			−0.656819	+	0.754049i	$0.728098\pi$
$308$			396.209i			1.28639i
$309$	16.0676	+	10.1538i	0.0519987	+	0.0328603i
$310$	−61.4779			−0.198316
$311$			319.082i			1.02599i	0.858392	+	0.512994i	$0.171464\pi$
$311$			319.082i			1.02599i	−0.858392	+	0.512994i	$0.828536\pi$
$312$	−122.686	+	194.141i	−0.393226	+	0.622247i
$313$	−265.431			−0.848024			−0.424012	−	0.905657i	$-0.639379\pi$
$313$	−265.431			−0.848024			−0.424012	+	0.905657i	$0.639379\pi$
$314$		−	165.780i		−	0.527963i
$315$	−323.520	+	153.749i	−1.02705	+	0.488092i
$316$	−278.918			−0.882651
$317$		−	559.221i		−	1.76411i	−0.471151	−	0.882053i	$-0.656161\pi$
$317$		−	559.221i		−	1.76411i	0.471151	−	0.882053i	$-0.343839\pi$
$318$	72.6290	+	45.8975i	0.228393	+	0.144332i
$319$	−314.278			−0.985198
$320$			125.316i			0.391613i
$321$	163.614	−	258.906i	0.509701	−	0.806561i
$322$	−50.6058			−0.157161
$323$			746.067i			2.30981i
$324$	−138.162	+	169.630i	−0.426425	+	0.523550i
$325$	65.3481			0.201071
$326$			24.5440i			0.0752884i
$327$	28.6669	+	18.1159i	0.0876664	+	0.0554002i
$328$	358.001			1.09147
$329$		−	69.1336i		−	0.210133i
$330$	124.421	−	196.886i	0.377033	−	0.596625i
$331$	534.218			1.61395			0.806976	−	0.590584i	$-0.201102\pi$
$331$	534.218			1.61395			0.806976	+	0.590584i	$0.201102\pi$
$332$			15.7111i			0.0473227i
$333$	−80.4133	−	169.207i	−0.241481	−	0.508128i
$334$	351.564			1.05259
$335$		−	165.220i		−	0.493195i
$336$	49.2786	+	31.1413i	0.146663	+	0.0926826i
$337$	102.202			0.303270			0.151635	−	0.988437i	$-0.451546\pi$
$337$	102.202			0.303270			0.151635	+	0.988437i	$0.451546\pi$
$338$			78.1124i			0.231102i
$339$	−104.566	+	165.467i	−0.308453	+	0.488102i
$340$	314.805			0.925898
$341$		−	198.809i		−	0.583018i
$342$	254.944	−	121.159i	0.745450	−	0.354266i
$343$	113.759			0.331657
$344$			374.934i			1.08992i
$345$	−52.2850	−	33.0412i	−0.151551	−	0.0957715i
$346$	−78.9385			−0.228146
$347$			347.278i			1.00080i	0.865794	+	0.500401i	$0.166814\pi$
$347$			347.278i			1.00080i	−0.865794	+	0.500401i	$0.833186\pi$
$348$	−85.8574	+	135.862i	−0.246717	+	0.390409i
$349$	−606.338			−1.73736			−0.868680	−	0.495374i	$-0.835031\pi$
$349$	−606.338			−1.73736			−0.868680	+	0.495374i	$0.835031\pi$
$350$			68.7955i			0.196559i
$351$	32.3801	−	268.684i	0.0922510	−	0.765483i
$352$	−521.962			−1.48285
$353$		−	435.366i		−	1.23333i	−0.787225	−	0.616666i	$-0.788483\pi$
$353$		−	435.366i		−	1.23333i	0.787225	−	0.616666i	$-0.211517\pi$
$354$	−85.1293	−	53.7969i	−0.240478	−	0.151969i
$355$	63.1498			0.177887
$356$		−	130.940i		−	0.367809i
$357$	−402.282	+	636.578i	−1.12684	+	1.78313i
$358$	123.898			0.346083
$359$		−	539.984i		−	1.50413i	−0.659086	−	0.752067i	$-0.729057\pi$
$359$		−	539.984i		−	1.50413i	0.659086	−	0.752067i	$-0.270943\pi$
$360$	−126.835	−	266.888i	−0.352320	−	0.741356i
$361$	396.203			1.09751
$362$			10.2825i			0.0284047i
$363$	329.835	+	208.437i	0.908637	+	0.574208i
$364$	−250.638			−0.688566
$365$		−	598.837i		−	1.64065i
$366$	−30.6127	+	48.4422i	−0.0836414	+	0.132356i
$367$	−167.972			−0.457689			−0.228845	−	0.973463i	$-0.573495\pi$
$367$	−167.972			−0.457689			−0.228845	+	0.973463i	$0.573495\pi$
$368$			10.0657i			0.0273524i
$369$	−381.029	+	181.079i	−1.03260	+	0.490729i
$370$	101.991			0.275652
$371$			232.627i			0.627027i
$372$	−85.9452	−	54.3125i	−0.231035	−	0.146001i
$373$	−351.079			−0.941231			−0.470616	−	0.882338i	$-0.655968\pi$
$373$	−351.079			−0.941231			−0.470616	+	0.882338i	$0.655968\pi$
$374$		−	489.637i		−	1.30919i
$375$	−217.156	+	343.632i	−0.579084	+	0.916353i
$376$	57.0318			0.151680
$377$		−	198.809i		−	0.527345i
$378$	282.859	+	34.0883i	0.748303	+	0.0901807i
$379$	504.592			1.33138			0.665688	−	0.746230i	$-0.268138\pi$
$379$	504.592			1.33138			0.665688	+	0.746230i	$0.268138\pi$
$380$		−	319.504i		−	0.840800i
$381$	−292.379	−	184.767i	−0.767399	−	0.484953i
$382$	−159.280			−0.416963
$383$		−	211.367i		−	0.551871i	−0.961176	−	0.275935i	$-0.911012\pi$
$383$		−	211.367i		−	0.551871i	0.961176	−	0.275935i	$-0.0889876\pi$
$384$	−157.930	+	249.911i	−0.411275	+	0.650809i
$385$	630.616			1.63796
$386$		−	295.858i		−	0.766470i
$387$	−189.644	−	399.051i	−0.490036	−	1.03114i
$388$	−179.957			−0.463808
$389$		−	504.503i		−	1.29692i	−0.761248	−	0.648461i	$-0.775413\pi$
$389$		−	504.503i		−	1.29692i	0.761248	−	0.648461i	$-0.224587\pi$
$390$	124.548	+	78.7075i	0.319354	+	0.201814i
$391$	−130.028			−0.332552
$392$		−	280.392i		−	0.715285i
$393$	78.0410	−	123.494i	0.198578	−	0.314233i
$394$	−387.202			−0.982745
$395$			443.932i			1.12388i
$396$	347.877	−	165.324i	0.878479	−	0.417486i
$397$	−559.014			−1.40810			−0.704048	−	0.710153i	$-0.748626\pi$
$397$	−559.014			−1.40810			−0.704048	+	0.710153i	$0.748626\pi$
$398$			206.469i			0.518767i
$399$	646.079	+	408.286i	1.61925	+	1.02327i
$400$	13.6837			0.0342091
$401$			77.0726i			0.192201i	0.995372	+	0.0961005i	$0.0306370\pi$
$401$			77.0726i			0.192201i	−0.995372	+	0.0961005i	$0.969363\pi$
$402$	−70.2036	+	111.091i	−0.174636	+	0.276347i
$403$	125.765			0.312071
$404$			264.136i			0.653803i
$405$	269.987	+	219.901i	0.666635	+	0.542966i
$406$	209.297			0.515510
$407$			329.823i			0.810376i
$408$	−525.146	−	331.863i	−1.28712	−	0.813389i
$409$	111.980			0.273789			0.136895	−	0.990586i	$-0.456288\pi$
$409$	111.980			0.273789			0.136895	+	0.990586i	$0.456288\pi$
$410$		−	229.670i		−	0.560170i
$411$	191.648	−	303.268i	0.466298	−	0.737878i
$412$	−17.1123			−0.0415347
$413$		−	272.665i		−	0.660205i
$414$	21.1161	+	44.4327i	0.0510050	+	0.107325i
$415$	25.0062			0.0602559
$416$		−	330.188i		−	0.793721i
$417$	334.070	+	211.113i	0.801126	+	0.506267i
$418$	−496.945			−1.18886
$419$		−	626.311i		−	1.49478i	−0.664388	−	0.747388i	$-0.731308\pi$
$419$		−	626.311i		−	1.49478i	0.664388	−	0.747388i	$-0.268692\pi$
$420$	172.278	−	272.615i	0.410185	−	0.649084i
$421$	−766.796			−1.82137			−0.910685	−	0.413102i	$-0.864445\pi$
$421$	−766.796			−1.82137			−0.910685	+	0.413102i	$0.864445\pi$
$422$			164.404i			0.389583i
$423$	−60.7003	+	28.8471i	−0.143500	+	0.0681963i
$424$	−191.906			−0.452608
$425$			176.764i			0.415916i
$426$	−42.4608	−	26.8329i	−0.0996733	−	0.0629880i
$427$	−155.158			−0.363367
$428$			275.740i			0.644252i
$429$	−254.527	+	402.768i	−0.593303	+	0.938853i
$430$	240.533			0.559378
$431$			96.8801i			0.224780i	0.993664	+	0.112390i	$0.0358506\pi$
$431$			96.8801i			0.224780i	−0.993664	+	0.112390i	$0.964149\pi$
$432$	6.78028	−	56.2615i	0.0156951	−	0.130235i
$433$	235.348			0.543528			0.271764	−	0.962364i	$-0.412393\pi$
$433$	235.348			0.543528			0.271764	+	0.962364i	$0.412393\pi$
$434$			132.399i			0.305068i
$435$	216.242	+	136.653i	0.497107	+	0.314144i
$436$	−30.5308			−0.0700248
$437$			131.968i			0.301987i
$438$	−254.451	+	402.648i	−0.580938	+	0.919288i
$439$	552.254			1.25798			0.628991	−	0.777412i	$-0.283468\pi$
$439$	552.254			1.25798			0.628991	+	0.777412i	$0.283468\pi$
$440$			520.227i			1.18234i
$441$	141.824	+	298.427i	0.321596	+	0.676706i
$442$	309.739			0.700767
$443$			568.873i			1.28414i	0.766647	+	0.642069i	$0.221924\pi$
$443$			568.873i			1.28414i	−0.766647	+	0.642069i	$0.778076\pi$
$444$	142.582	+	90.1041i	0.321132	+	0.202937i
$445$	−208.407			−0.468330
$446$			253.457i			0.568290i
$447$	367.223	−	581.100i	0.821527	−	1.30000i
$448$	269.882			0.602415
$449$			83.1369i			0.185160i	0.995705	+	0.0925801i	$0.0295114\pi$
$449$			83.1369i			0.185160i	−0.995705	+	0.0925801i	$0.970489\pi$
$450$	60.4034	−	28.7060i	0.134230	−	0.0637910i
$451$	742.714			1.64682
$452$		−	176.225i		−	0.389879i
$453$	−680.823	−	430.242i	−1.50292	−	0.949762i
$454$	338.738			0.746120
$455$			398.921i			0.876750i
$456$	−336.816	+	532.984i	−0.738631	+	1.16882i
$457$	222.465			0.486795			0.243398	−	0.969927i	$-0.421738\pi$
$457$	222.465			0.486795			0.243398	+	0.969927i	$0.421738\pi$
$458$			117.890i			0.257401i
$459$	726.783	+	87.5871i	1.58340	+	0.190822i
$460$	55.6845			0.121053
$461$		−	352.980i		−	0.765683i	−0.923814	−	0.382841i	$-0.874946\pi$
$461$		−	352.980i		−	0.765683i	0.923814	−	0.382841i	$-0.125054\pi$
$462$	−424.016	−	267.954i	−0.917783	−	0.579987i
$463$	−707.963			−1.52908			−0.764539	−	0.644577i	$-0.777033\pi$
$463$	−707.963			−1.52908			−0.764539	+	0.644577i	$0.777033\pi$
$464$		−	41.6299i		−	0.0897197i
$465$	−86.4451	+	136.792i	−0.185903	+	0.294177i
$466$	−102.944			−0.220909
$467$			751.106i			1.60836i	0.594383	+	0.804182i	$0.297396\pi$
$467$			751.106i			1.60836i	−0.594383	+	0.804182i	$0.702604\pi$
$468$	104.583	+	220.064i	0.223467	+	0.470222i
$469$	−355.820			−0.758679
$470$		−	36.5879i		−	0.0778465i
$471$	−368.873	−	233.107i	−0.783169	−	0.494919i
$472$	224.935			0.476557
$473$			777.843i			1.64449i
$474$	188.630	−	298.492i	0.397955	−	0.629731i
$475$	179.403			0.377690
$476$		−	677.968i		−	1.42430i
$477$	204.250	−	97.0672i	0.428197	−	0.203495i
$478$	−79.3259			−0.165954
$479$			165.358i			0.345214i	0.984991	+	0.172607i	$0.0552191\pi$
$479$			165.358i			0.345214i	−0.984991	+	0.172607i	$0.944781\pi$
$480$	359.140	+	226.957i	0.748209	+	0.472826i
$481$	−208.643			−0.433769
$482$		−	94.5951i		−	0.196255i
$483$	−71.1578	+	112.601i	−0.147325	+	0.233129i
$484$	−351.281			−0.725787
$485$			286.425i			0.590566i
$486$	−88.0971	−	262.578i	−0.181270	−	0.540284i
$487$	188.659			0.387391			0.193695	−	0.981062i	$-0.437953\pi$
$487$	188.659			0.387391			0.193695	+	0.981062i	$0.437953\pi$
$488$		−	127.998i		−	0.262290i
$489$	54.6121	+	34.5118i	0.111681	+	0.0705762i
$490$	−179.881			−0.367103
$491$			227.992i			0.464343i	0.972675	+	0.232171i	$0.0745830\pi$
$491$			227.992i			0.464343i	−0.972675	+	0.232171i	$0.925417\pi$
$492$	202.901	−	321.075i	0.412401	−	0.652591i
$493$	537.772			1.09082
$494$		−	314.362i		−	0.636361i
$495$	−263.134	−	553.690i	−0.531584	−	1.11857i
$496$	26.3347			0.0530941
$497$		−	136.000i		−	0.273642i
$498$	−16.8137	−	10.6253i	−0.0337625	−	0.0213360i
$499$	−394.469			−0.790519			−0.395260	−	0.918569i	$-0.629345\pi$
$499$	−394.469			−0.790519			−0.395260	+	0.918569i	$0.629345\pi$
$500$		−	365.975i		−	0.731950i
$501$	494.340	−	782.253i	0.986707	−	1.56138i
$502$	−96.6014			−0.192433
$503$			375.735i			0.746988i	0.927633	+	0.373494i	$0.121840\pi$
$503$			375.735i			0.746988i	−0.927633	+	0.373494i	$0.878160\pi$
$504$	−574.773	+	273.154i	−1.14042	+	0.541971i
$505$	420.406			0.832487
$506$		−	86.6096i		−	0.171165i
$507$	173.805	+	109.835i	0.342811	+	0.216637i
$508$	311.389			0.612971
$509$		−	178.295i		−	0.350284i	−0.984543	−	0.175142i	$-0.943961\pi$
$509$		−	178.295i		−	0.350284i	0.984543	−	0.175142i	$-0.0560385\pi$
$510$	−212.901	+	336.899i	−0.417453	+	0.660586i
$511$	−1289.66			−2.52380
$512$		−	133.260i		−	0.260273i
$513$	88.8944	−	737.630i	0.173283	−	1.43788i
$514$	−29.8166			−0.0580090
$515$			27.2363i			0.0528861i
$516$	336.261	+	212.498i	0.651669	+	0.411818i
$517$	118.319			0.228857
$518$		−	219.650i		−	0.424034i
$519$	−110.997	+	175.643i	−0.213867	+	0.338427i
$520$	−329.091			−0.632866
$521$			726.525i			1.39448i	0.716837	+	0.697241i	$0.245589\pi$
$521$			726.525i			1.39448i	−0.716837	+	0.697241i	$0.754411\pi$
$522$	−87.3324	−	183.766i	−0.167304	−	0.352042i
$523$	1011.96			1.93490			0.967452	−	0.253053i	$-0.0814347\pi$
$523$	1011.96			1.93490			0.967452	+	0.253053i	$0.0814347\pi$
$524$			131.523i			0.250998i
$525$	153.075	+	96.7345i	0.291571	+	0.184256i
$526$	16.1696			0.0307407
$527$			340.189i			0.645521i
$528$	−53.2970	+	84.3382i	−0.100941	+	0.159731i
$529$	−23.0000			−0.0434783
$530$			123.114i			0.232291i
$531$	−239.403	+	113.773i	−0.450854	+	0.214263i
$532$	−688.087			−1.29340
$533$			469.833i			0.881488i
$534$	140.129	+	88.5539i	0.262414	+	0.165831i
$535$	438.874			0.820326
$536$		−	293.534i		−	0.547639i
$537$	174.215	−	275.681i	0.324422	−	0.513372i
$538$	128.453			0.238761
$539$		−	581.704i		−	1.07923i
$540$	−311.245	−	37.5093i	−0.576380	−	0.0694616i
$541$	124.826			0.230732			0.115366	−	0.993323i	$-0.463196\pi$
$541$	124.826			0.230732			0.115366	+	0.993323i	$0.463196\pi$
$542$			323.529i			0.596917i
$543$	22.8793	+	14.4584i	0.0421349	+	0.0266269i
$544$	893.148			1.64182
$545$			48.5935i			0.0891624i
$546$	169.505	−	268.228i	0.310449	−	0.491260i
$547$	−228.544			−0.417813			−0.208907	−	0.977936i	$-0.566990\pi$
$547$	−228.544			−0.417813			−0.208907	+	0.977936i	$0.566990\pi$
$548$			322.986i			0.589391i
$549$	64.7420	+	136.231i	0.117927	+	0.248143i
$550$	−117.740			−0.214073
$551$		−	545.799i		−	0.990561i
$552$	−92.8906	−	58.7017i	−0.168280	−	0.106344i
$553$	956.056			1.72885
$554$		−	104.595i		−	0.188800i
$555$	143.412	−	226.938i	0.258400	−	0.408897i
$556$	−355.790			−0.639911
$557$			197.789i			0.355097i	0.984112	+	0.177548i	$0.0568166\pi$
$557$			197.789i			0.355097i	−0.984112	+	0.177548i	$0.943183\pi$
$558$	116.249	−	55.2456i	0.208331	−	0.0990065i
$559$	−492.056			−0.880243
$560$			83.5327i			0.149165i
$561$	−1089.47	−	688.487i	−1.94202	−	1.22725i
$562$	464.272			0.826107
$563$		−	157.256i		−	0.279318i	−0.990200	−	0.139659i	$-0.955399\pi$
$563$		−	157.256i		−	0.279318i	0.990200	−	0.139659i	$-0.0446007\pi$
$564$	32.3235	−	51.1493i	0.0573111	−	0.0906902i
$565$	−280.484			−0.496432
$566$			63.9021i			0.112901i
$567$	473.581	−	581.447i	0.835240	−	1.02548i
$568$	112.193			0.197523
$569$		−	97.4459i		−	0.171258i	−0.996327	−	0.0856291i	$-0.972710\pi$
$569$		−	97.4459i		−	0.171258i	0.996327	−	0.0856291i	$-0.0272900\pi$
$570$	341.927	+	216.079i	0.599872	+	0.379086i
$571$	−761.024			−1.33279			−0.666396	−	0.745598i	$-0.732164\pi$
$571$	−761.024			−1.33279			−0.666396	+	0.745598i	$0.732164\pi$
$572$		−	428.956i		−	0.749923i
$573$	−223.966	+	354.408i	−0.390866	+	0.618513i
$574$	−494.619			−0.861705
$575$			31.2671i			0.0543775i
$576$	−112.612	−	236.960i	−0.195507	−	0.411389i
$577$	−735.666			−1.27498			−0.637492	−	0.770457i	$-0.720028\pi$
$577$	−735.666			−1.27498			−0.637492	+	0.770457i	$0.720028\pi$
$578$			508.443i			0.879659i
$579$	−658.303	−	416.010i	−1.13696	−	0.718498i
$580$	−230.302			−0.397072
$581$		−	53.8536i		−	0.0926912i
$582$	121.704	−	192.587i	0.209114	−	0.330906i
$583$	−398.131			−0.682900
$584$		−	1063.91i		−	1.82176i
$585$	350.259	−	166.456i	0.598733	−	0.284540i
$586$	−397.476			−0.678287
$587$		−	490.856i		−	0.836212i	−0.908398	−	0.418106i	$-0.862694\pi$
$587$		−	490.856i		−	0.836212i	0.908398	−	0.418106i	$-0.137306\pi$
$588$	−251.471	−	158.915i	−0.427671	−	0.270264i
$589$	345.267			0.586192
$590$		−	144.303i		−	0.244582i
$591$	−544.451	+	861.549i	−0.921237	+	1.45778i
$592$	−43.6890			−0.0737990
$593$			1145.16i			1.93113i	0.260165	+	0.965564i	$0.416223\pi$
$593$			1145.16i			1.93113i	−0.260165	+	0.965564i	$0.583777\pi$
$594$	−58.3406	+	484.100i	−0.0982165	+	0.814983i
$595$	−1079.07			−1.81356
$596$			618.883i			1.03839i
$597$	459.408	+	290.320i	0.769527	+	0.486298i
$598$	54.7884			0.0916193
$599$		−	1076.26i		−	1.79675i	−0.439225	−	0.898377i	$-0.644747\pi$
$599$		−	1076.26i		−	1.79675i	0.439225	−	0.898377i	$-0.355253\pi$
$600$	−79.8012	+	126.279i	−0.133002	+	0.210465i
$601$	418.907			0.697016			0.348508	−	0.937306i	$-0.386688\pi$
$601$	418.907			0.697016			0.348508	+	0.937306i	$0.386688\pi$
$602$		−	518.014i		−	0.860488i
$603$	148.471	+	312.415i	0.246221	+	0.518102i
$604$	725.090			1.20048
$605$			559.107i			0.924143i
$606$	−282.674	−	178.634i	−0.466458	−	0.294776i
$607$	1048.47			1.72729			0.863646	−	0.504099i	$-0.168175\pi$
$607$	1048.47			1.72729			0.863646	+	0.504099i	$0.168175\pi$
$608$		−	906.479i		−	1.49092i
$609$	294.296	−	465.700i	0.483245	−	0.764697i
$610$	−82.1148			−0.134614
$611$			74.8474i			0.122500i
$612$	−595.265	+	282.892i	−0.972656	+	0.462242i
$613$	−498.397			−0.813046			−0.406523	−	0.913640i	$-0.633259\pi$
$613$	−498.397			−0.813046			−0.406523	+	0.913640i	$0.633259\pi$
$614$			459.651i			0.748618i
$615$	−511.030	−	322.943i	−0.830944	−	0.525110i
$616$	1120.37			1.81878
$617$			585.285i			0.948598i	0.880364	+	0.474299i	$0.157298\pi$
$617$			585.285i			0.948598i	−0.880364	+	0.474299i	$0.842702\pi$
$618$	11.5729	−	18.3133i	0.0187265	−	0.0296331i
$619$	365.160			0.589920			0.294960	−	0.955510i	$-0.404694\pi$
$619$	365.160			0.589920			0.294960	+	0.955510i	$0.404694\pi$
$620$		−	145.686i		−	0.234978i
$621$	128.557	+	15.4929i	0.207017	+	0.0249483i
$622$	363.678			0.584691
$623$			448.827i			0.720429i
$624$	−53.3515	−	33.7151i	−0.0854991	−	0.0540307i
$625$	−419.504			−0.671206
$626$			302.529i			0.483273i
$627$	−698.763	+	1105.74i	−1.11445	+	1.76353i
$628$	392.856			0.625567
$629$		−	564.372i		−	0.897252i
$630$	175.237	+	368.736i	0.278154	+	0.585296i
$631$	7.95591			0.0126084			0.00630421	−	0.999980i	$-0.497993\pi$
$631$	7.95591			0.0126084			0.00630421	+	0.999980i	$0.497993\pi$
$632$			788.699i			1.24794i
$633$	365.810	+	231.171i	0.577899	+	0.365200i
$634$	−637.380			−1.00533
$635$		−	495.614i		−	0.780495i
$636$	−108.765	+	172.112i	−0.171014	+	0.270616i
$637$	367.980			0.577677
$638$			358.203i			0.561446i
$639$	−119.410	+	56.7480i	−0.186870	+	0.0888076i
$640$	−423.626			−0.661916
$641$		−	1221.48i		−	1.90558i	−0.303627	−	0.952791i	$-0.598198\pi$
$641$		−	1221.48i		−	1.90558i	0.303627	−	0.952791i	$-0.401802\pi$
$642$	−295.092	−	186.481i	−0.459644	−	0.290469i
$643$	791.810			1.23143			0.615715	−	0.787969i	$-0.288867\pi$
$643$	791.810			1.23143			0.615715	+	0.787969i	$0.288867\pi$
$644$		−	119.923i		−	0.186215i
$645$	338.217	−	535.201i	0.524368	−	0.829770i
$646$	850.340			1.31632
$647$		−	149.186i		−	0.230582i	−0.993332	−	0.115291i	$-0.963220\pi$
$647$		−	149.186i		−	0.230582i	0.993332	−	0.115291i	$-0.0367800\pi$
$648$	479.665	+	390.681i	0.740224	+	0.602903i
$649$	466.653			0.719034
$650$		−	74.4814i		−	0.114587i
$651$	294.597	+	186.169i	0.452530	+	0.285974i
$652$	−58.1629			−0.0892069
$653$		−	129.263i		−	0.197952i	−0.995090	−	0.0989762i	$-0.968443\pi$
$653$		−	129.263i		−	0.197952i	0.995090	−	0.0989762i	$-0.0315568\pi$
$654$	20.6478	−	32.6735i	0.0315716	−	0.0499594i
$655$	209.335			0.319595
$656$			98.3813i			0.149972i
$657$	538.131	+	1132.34i	0.819073	+	1.72350i
$658$	−78.7960			−0.119751
$659$			518.274i			0.786456i	0.919441	+	0.393228i	$0.128642\pi$
$659$			518.274i			0.786456i	−0.919441	+	0.393228i	$0.871358\pi$
$660$	466.569	+	294.845i	0.706922	+	0.446735i
$661$	−97.6247			−0.147692			−0.0738462	−	0.997270i	$-0.523527\pi$
$661$	−97.6247			−0.147692			−0.0738462	+	0.997270i	$0.523527\pi$
$662$		−	608.882i		−	0.919762i
$663$	435.530	−	689.190i	0.656908	−	1.03950i
$664$	44.4266			0.0669075
$665$			1095.18i			1.64688i
$666$	−192.855	+	91.6521i	−0.289573	+	0.137616i
$667$	95.1241			0.142615
$668$			833.115i			1.24718i
$669$	563.959	+	356.391i	0.842988	+	0.532721i
$670$	−188.312			−0.281063
$671$		−	265.546i		−	0.395746i
$672$	488.776	−	773.448i	0.727345	−	1.15096i
$673$	−864.488			−1.28453			−0.642264	−	0.766483i	$-0.722005\pi$
$673$	−864.488			−1.28453			−0.642264	+	0.766483i	$0.722005\pi$
$674$		−	116.486i		−	0.172828i
$675$	21.0616	−	174.766i	0.0312024	−	0.258912i
$676$	−185.106			−0.273825
$677$			419.228i			0.619244i	0.950860	+	0.309622i	$0.100203\pi$
$677$			419.228i			0.619244i	−0.950860	+	0.309622i	$0.899797\pi$
$678$	188.593	+	119.180i	0.278160	+	0.175782i
$679$	616.847			0.908463
$680$		−	890.179i		−	1.30909i
$681$	476.306	−	753.716i	0.699421	−	1.10678i
$682$	−226.595			−0.332251
$683$			47.3262i			0.0692916i	0.999400	+	0.0346458i	$0.0110303\pi$
$683$			47.3262i			0.0692916i	−0.999400	+	0.0346458i	$0.988970\pi$
$684$	287.115	+	604.150i	0.419758	+	0.883260i
$685$	514.073			0.750471
$686$		−	129.658i		−	0.189005i
$687$	262.313	+	165.767i	0.381823	+	0.241291i
$688$	−103.035			−0.149760
$689$		−	251.853i		−	0.365535i
$690$	−37.6591	+	59.5925i	−0.0545784	+	0.0863659i
$691$	−798.224			−1.15517			−0.577586	−	0.816330i	$-0.696005\pi$
$691$	−798.224			−1.15517			−0.577586	+	0.816330i	$0.696005\pi$
$692$		−	187.064i		−	0.270323i
$693$	−1192.43	+	566.688i	−1.72068	+	0.817732i
$694$	395.815			0.570338
$695$			566.284i			0.814798i
$696$	384.180	+	242.780i	0.551983	+	0.348822i
$697$	−1270.88			−1.82336
$698$			691.082i			0.990089i
$699$	−144.751	+	229.057i	−0.207083	+	0.327692i
$700$	−163.027			−0.232896
$701$			560.432i			0.799475i	0.916630	+	0.399738i	$0.130899\pi$
$701$			560.432i			0.799475i	−0.916630	+	0.399738i	$0.869101\pi$
$702$	−306.237	−	36.9057i	−0.436234	−	0.0525722i
$703$	−572.795			−0.814787
$704$			461.890i			0.656094i
$705$	−81.4104	−	51.4468i	−0.115476	−	0.0729742i
$706$	−496.214			−0.702853
$707$		−	905.390i		−	1.28061i
$708$	127.485	−	201.734i	0.180063	−	0.284935i
$709$	620.664			0.875408			0.437704	−	0.899119i	$-0.355792\pi$
$709$	620.664			0.875408			0.437704	+	0.899119i	$0.355792\pi$
$710$		−	71.9758i		−	0.101374i
$711$	−398.929	−	839.431i	−0.561081	−	1.18063i
$712$	−370.260			−0.520029
$713$			60.1746i			0.0843963i
$714$	725.548	+	458.506i	1.01617	+	0.642165i
$715$	−682.736			−0.954875
$716$			293.605i			0.410063i
$717$	−111.542	+	176.505i	−0.155567	+	0.246172i
$718$	−615.454			−0.857178
$719$		−	484.256i		−	0.673513i	−0.941592	−	0.336756i	$-0.890670\pi$
$719$		−	484.256i		−	0.673513i	0.941592	−	0.336756i	$-0.109330\pi$
$720$	73.3429	−	34.8553i	0.101865	−	0.0484101i
$721$	58.6564			0.0813542
$722$		−	451.577i		−	0.625453i
$723$	−210.480	−	133.012i	−0.291121	−	0.183972i
$724$	−24.3668			−0.0336559
$725$		−	129.315i		−	0.178366i
$726$	237.569	−	375.934i	0.327230	−	0.517815i
$727$	−464.377			−0.638757			−0.319379	−	0.947627i	$-0.603474\pi$
$727$	−464.377			−0.638757			−0.319379	+	0.947627i	$0.603474\pi$
$728$			708.733i			0.973534i
$729$	−708.128	−	173.193i	−0.971369	−	0.237577i
$730$	−682.532			−0.934976
$731$		−	1330.99i		−	1.82079i
$732$	−114.795	−	72.5442i	−0.156824	−	0.0991041i
$733$	571.068			0.779083			0.389541	−	0.921009i	$-0.372634\pi$
$733$	571.068			0.779083			0.389541	+	0.921009i	$0.372634\pi$
$734$			191.448i			0.260829i
$735$	−252.933	+	400.246i	−0.344127	+	0.544553i
$736$	157.985			0.214653
$737$		−	608.970i		−	0.826283i
$738$	206.387	+	434.282i	0.279657	+	0.588458i
$739$	242.062			0.327554			0.163777	−	0.986497i	$-0.447632\pi$
$739$	242.062			0.327554			0.163777	+	0.986497i	$0.447632\pi$
$740$			241.693i			0.326612i
$741$	−699.477	−	442.030i	−0.943964	−	0.596532i
$742$	265.140			0.357331
$743$		−	973.403i		−	1.31010i	−0.755586	−	0.655049i	$-0.772648\pi$
$743$		−	973.403i		−	1.31010i	0.755586	−	0.655049i	$-0.227352\pi$
$744$	−153.580	+	243.028i	−0.206425	+	0.326651i
$745$	985.028			1.32219
$746$			400.147i			0.536390i
$747$	−47.2842	+	22.4712i	−0.0632988	+	0.0300820i
$748$	1160.31			1.55122
$749$		−	945.164i		−	1.26190i
$750$	391.659	+	247.507i	0.522213	+	0.330009i
$751$	−349.911			−0.465926			−0.232963	−	0.972486i	$-0.574842\pi$
$751$	−349.911			−0.465926			−0.232963	+	0.972486i	$0.574842\pi$
$752$			15.6728i			0.0208415i
$753$	−135.833	+	214.945i	−0.180389	+	0.285451i
$754$	−226.595			−0.300524
$755$		−	1154.07i		−	1.52857i
$756$	−80.7803	+	670.301i	−0.106852	+	0.886642i
$757$	612.339			0.808902			0.404451	−	0.914560i	$-0.367463\pi$
$757$	612.339			0.808902			0.404451	+	0.914560i	$0.367463\pi$
$758$		−	575.115i		−	0.758727i
$759$	−192.712	−	121.783i	−0.253903	−	0.160452i
$760$	−903.466			−1.18877
$761$		−	771.430i		−	1.01371i	−0.862032	−	0.506853i	$-0.830809\pi$
$761$		−	771.430i		−	1.01371i	0.862032	−	0.506853i	$-0.169191\pi$
$762$	−210.591	+	333.243i	−0.276366	+	0.437326i
$763$	104.651			0.137158
$764$		−	377.451i		−	0.494046i
$765$	450.258	+	947.438i	0.588573	+	1.23848i
$766$	−240.908			−0.314501
$767$			295.200i			0.384876i
$768$	580.551	+	366.876i	0.755926	+	0.477703i
$769$	99.4612			0.129338			0.0646692	−	0.997907i	$-0.479401\pi$
$769$	99.4612			0.129338			0.0646692	+	0.997907i	$0.479401\pi$
$770$		−	718.753i		−	0.933446i
$771$	−41.9257	+	66.3440i	−0.0543783	+	0.0860493i
$772$	701.105			0.908167
$773$			579.305i			0.749425i	0.927141	+	0.374712i	$0.122259\pi$
$773$			579.305i			0.749425i	−0.927141	+	0.374712i	$0.877741\pi$
$774$	−454.823	+	216.149i	−0.587627	+	0.279262i
$775$	81.8036			0.105553
$776$			508.868i			0.655758i
$777$	−488.735	−	308.853i	−0.629002	−	0.397494i
$778$	−575.013			−0.739092
$779$			1289.85i			1.65578i
$780$	−186.516	+	295.147i	−0.239123	+	0.378393i
$781$	232.758			0.298025
$782$			148.201i			0.189515i
$783$	−531.691	−	64.0760i	−0.679044	−	0.0818339i
$784$	77.0537			0.0982828
$785$		−	625.279i		−	0.796534i
$786$	−140.753	−	88.9482i	−0.179076	−	0.113166i
$787$	581.141			0.738425			0.369213	−	0.929345i	$-0.379627\pi$
$787$	581.141			0.738425			0.369213	+	0.929345i	$0.379627\pi$
$788$		−	917.566i		−	1.16442i
$789$	22.7364	−	35.9785i	0.0288167	−	0.0456001i
$790$	505.977			0.640477
$791$			604.053i			0.763657i
$792$	−467.490	−	983.698i	−0.590265	−	1.24204i
$793$	167.981			0.211830
$794$			637.143i			0.802448i
$795$	273.937	+	173.113i	0.344575	+	0.217752i
$796$	−489.278			−0.614671
$797$			101.951i			0.127918i	0.997953	+	0.0639592i	$0.0203728\pi$
$797$			101.951i			0.127918i	−0.997953	+	0.0639592i	$0.979627\pi$
$798$	465.349	−	736.377i	0.583144	−	0.922779i
$799$	−202.460			−0.253392
$800$		−	214.771i		−	0.268463i
$801$	394.077	−	187.280i	0.491981	−	0.233808i
$802$	87.8445			0.109532
$803$		−	2207.20i		−	2.74869i
$804$	−263.258	−	166.364i	−0.327435	−	0.206921i
$805$	−190.872			−0.237108
$806$		−	143.342i		−	0.177844i
$807$	180.620	−	285.817i	0.223817	−	0.354172i
$808$	746.902			0.924384
$809$			1321.86i			1.63394i	0.576680	+	0.816970i	$0.304348\pi$
$809$			1321.86i			1.63394i	−0.576680	+	0.816970i	$0.695652\pi$
$810$	250.635	−	307.722i	0.309426	−	0.379903i
$811$	−1112.26			−1.37146			−0.685732	−	0.727854i	$-0.740518\pi$
$811$	−1112.26			−1.37146			−0.685732	+	0.727854i	$0.740518\pi$
$812$			495.980i			0.610812i
$813$	719.874	+	454.920i	0.885454	+	0.559557i
$814$	375.920			0.461819
$815$			92.5735i			0.113587i
$816$	91.1983	−	144.314i	0.111763	−	0.176855i
$817$	−1350.86			−1.65344
$818$		−	127.630i		−	0.156027i
$819$	−358.481	−	754.320i	−0.437706	−	0.921026i
$820$	544.257			0.663728
$821$		−	399.025i		−	0.486023i	−0.970023	−	0.243012i	$-0.921865\pi$
$821$		−	399.025i		−	0.486023i	0.970023	−	0.243012i	$-0.0781354\pi$
$822$	−345.654	−	218.434i	−0.420503	−	0.265735i
$823$	−826.500			−1.00425			−0.502126	−	0.864794i	$-0.667449\pi$
$823$	−826.500			−1.00425			−0.502126	+	0.864794i	$0.667449\pi$
$824$			48.3887i			0.0587241i
$825$	−165.557	+	261.980i	−0.200675	+	0.317552i
$826$	−310.773			−0.376239
$827$			1151.23i			1.39205i	0.718016	+	0.696027i	$0.245051\pi$
$827$			1151.23i			1.39205i	−0.718016	+	0.696027i	$0.754949\pi$
$828$	−105.294	+	50.0395i	−0.127166	+	0.0604342i
$829$	125.694			0.151622			0.0758108	−	0.997122i	$-0.475845\pi$
$829$	125.694			0.151622			0.0758108	+	0.997122i	$0.475845\pi$
$830$		−	28.5011i		−	0.0343387i
$831$	−232.732	−	147.073i	−0.280062	−	0.176984i
$832$	−292.187			−0.351186
$833$			995.374i			1.19493i
$834$	240.619	−	380.760i	0.288512	−	0.456547i
$835$	1326.00			1.58803
$836$		−	1177.63i		−	1.40865i
$837$	40.5338	−	336.343i	0.0484275	−	0.401843i
$838$	−713.847			−0.851846
$839$			542.277i			0.646337i	0.946341	+	0.323169i	$0.104748\pi$
$839$			542.277i			0.646337i	−0.946341	+	0.323169i	$0.895252\pi$
$840$	−770.878	−	487.151i	−0.917711	−	0.579942i
$841$	447.583			0.532203
$842$			873.966i			1.03796i
$843$	652.821	−	1033.04i	0.774402	−	1.22543i
$844$	−389.594			−0.461605
$845$			294.619i			0.348661i
$846$	32.8788	+	69.1840i	0.0388638	+	0.0817777i
$847$	1204.10			1.42160
$848$		−	52.7372i		−	0.0621901i
$849$	142.186	+	89.8539i	0.167475	+	0.105835i
$850$	201.470			0.237023
$851$		−	99.8292i		−	0.117308i
$852$	63.5869	−	100.621i	0.0746325	−	0.118100i
$853$	−401.694			−0.470920			−0.235460	−	0.971884i	$-0.575660\pi$
$853$	−401.694			−0.470920			−0.235460	+	0.971884i	$0.575660\pi$
$854$			176.843i			0.207076i
$855$	961.579	−	456.978i	1.12465	−	0.534478i
$856$	779.714			0.910880
$857$		−	1094.12i		−	1.27668i	−0.769753	−	0.638342i	$-0.779620\pi$
$857$		−	1094.12i		−	1.27668i	0.769753	−	0.638342i	$-0.220380\pi$
$858$	459.060	+	290.100i	0.535035	+	0.338112i
$859$	337.853			0.393309			0.196655	−	0.980473i	$-0.436992\pi$
$859$	337.853			0.393309			0.196655	+	0.980473i	$0.436992\pi$
$860$			570.000i			0.662790i
$861$	−695.492	+	1100.56i	−0.807772	+	1.27823i
$862$	110.420			0.128098
$863$			39.7209i			0.0460265i	0.999735	+	0.0230133i	$0.00732600\pi$
$863$			39.7209i			0.0460265i	−0.999735	+	0.0230133i	$0.992674\pi$
$864$	−883.048	−	106.419i	−1.02205	−	0.123170i
$865$	−297.735			−0.344202
$866$		−	268.241i		−	0.309747i
$867$	1131.32	+	714.930i	1.30487	+	0.824602i
$868$	−313.752			−0.361465
$869$			1636.25i			1.88291i
$870$	155.752	−	246.464i	0.179025	−	0.283292i
$871$	385.228			0.442283
$872$			86.3323i			0.0990050i
$873$	−257.389	−	541.600i	−0.294832	−	0.620390i
$874$	150.413			0.172097
$875$			1254.47i			1.43367i
$876$	−954.170	−	602.982i	−1.08924	−	0.688336i
$877$	1045.03			1.19159			0.595796	−	0.803136i	$-0.296837\pi$
$877$	1045.03			1.19159			0.595796	+	0.803136i	$0.296837\pi$
$878$		−	629.439i		−	0.716901i
$879$	−558.898	+	884.411i	−0.635834	+	1.00616i
$880$	−142.962			−0.162457
$881$			1383.04i			1.56986i	0.619587	+	0.784928i	$0.287300\pi$
$881$			1383.04i			1.56986i	−0.619587	+	0.784928i	$0.712700\pi$
$882$	340.136	−	161.645i	0.385642	−	0.183272i
$883$	−813.052			−0.920784			−0.460392	−	0.887716i	$-0.652291\pi$
$883$	−813.052			−0.920784			−0.460392	+	0.887716i	$0.652291\pi$
$884$			734.001i			0.830318i
$885$	−321.085	−	202.908i	−0.362808	−	0.229274i
$886$	648.380			0.731806
$887$			908.330i			1.02405i	0.858971	+	0.512024i	$0.171104\pi$
$887$			908.330i			1.02405i	−0.858971	+	0.512024i	$0.828896\pi$
$888$	254.789	−	403.182i	0.286924	−	0.454034i
$889$	−1067.36			−1.20063
$890$			237.535i			0.266893i
$891$	995.121	+	810.513i	1.11686	+	0.909666i
$892$	−600.627			−0.673349
$893$			205.482i			0.230103i
$894$	−662.317	−	418.547i	−0.740846	−	0.468173i
$895$	467.309			0.522133
$896$			912.325i			1.01822i
$897$	77.0389	−	121.908i	0.0858850	−	0.135906i
$898$	94.7564			0.105519
$899$		−	248.872i		−	0.276832i
$900$	68.0256	+	143.140i	0.0755840	+	0.159045i
$901$	681.255			0.756110
$902$		−	846.518i		−	0.938490i
$903$	−1152.61	−	728.388i	−1.27643	−	0.806631i
$904$	−498.314			−0.551232
$905$			38.7828i			0.0428540i
$906$	−490.374	+	775.977i	−0.541252	+	0.856487i
$907$	430.841			0.475018			0.237509	−	0.971385i	$-0.423669\pi$
$907$	430.841			0.475018			0.237509	+	0.971385i	$0.423669\pi$
$908$			802.721i			0.884054i
$909$	−794.945	+	377.788i	−0.874527	+	0.415608i
$910$	454.676			0.499644
$911$			137.295i			0.150708i	0.997157	+	0.0753542i	$0.0240088\pi$
$911$			137.295i			0.150708i	−0.997157	+	0.0753542i	$0.975991\pi$
$912$	−146.468	−	92.5595i	−0.160601	−	0.101491i
$913$	92.1680			0.100951
$914$		−	253.558i		−	0.277416i
$915$	−115.463	+	182.711i	−0.126189	+	0.199684i
$916$	−279.368			−0.304987
$917$		−	450.826i		−	0.491631i
$918$	99.8286	−	828.360i	0.108746	−	0.902353i
$919$	−663.064			−0.721506			−0.360753	−	0.932661i	$-0.617480\pi$
$919$	−663.064			−0.721506			−0.360753	+	0.932661i	$0.617480\pi$
$920$		−	157.460i		−	0.171152i
$921$	1022.75	+	646.323i	1.11048	+	0.701763i
$922$	−402.313			−0.436348
$923$			147.240i			0.159523i
$924$	634.981	−	1004.81i	0.687209	−	1.08745i
$925$	−135.712			−0.146715
$926$			806.910i			0.871394i
$927$	−24.4753	−	51.5012i	−0.0264027	−	0.0555568i
$928$	−653.399			−0.704093
$929$			907.464i			0.976818i	0.872615	+	0.488409i	$0.162423\pi$
$929$			907.464i			0.976818i	−0.872615	+	0.488409i	$0.837577\pi$
$930$	155.911	+	98.5270i	0.167646	+	0.105943i
$931$	1010.23			1.08510
$932$		−	243.950i		−	0.261749i
$933$	511.374	−	809.208i	0.548096	−	0.867318i
$934$	856.083			0.916577
$935$		−	1846.78i		−	1.97516i
$936$	622.277	−	295.729i	0.664826	−	0.315950i
$937$	−224.080			−0.239146			−0.119573	−	0.992825i	$-0.538153\pi$
$937$	−224.080			−0.239146			−0.119573	+	0.992825i	$0.538153\pi$
$938$			405.551i			0.432357i
$939$	673.147	+	425.391i	0.716877	+	0.453026i
$940$	86.7036			0.0922379
$941$		−	234.587i		−	0.249296i	−0.992201	−	0.124648i	$-0.960220\pi$
$941$		−	234.587i		−	0.249296i	0.992201	−	0.124648i	$-0.0397801\pi$
$942$	−265.687	+	420.427i	−0.282045	+	0.446314i
$943$	−224.801			−0.238389
$944$			61.8138i			0.0654807i
$945$	1066.87	+	128.572i	1.12896	+	0.136055i
$946$	886.557			0.937164
$947$		−	229.675i		−	0.242529i	−0.992620	−	0.121265i	$-0.961305\pi$
$947$		−	229.675i		−	0.242529i	0.992620	−	0.121265i	$-0.0386950\pi$
$948$	707.349	+	447.005i	0.746148	+	0.471524i
$949$	1396.25			1.47129
$950$		−	204.477i		−	0.215239i
$951$	−896.231	+	1418.21i	−0.942409	+	1.49129i
$952$	−1917.10			−2.01376
$953$			816.441i			0.856706i	0.903611	+	0.428353i	$0.140906\pi$
$953$			816.441i			0.856706i	−0.903611	+	0.428353i	$0.859094\pi$
$954$	−110.634	−	232.797i	−0.115968	−	0.244021i
$955$	−600.760			−0.629069
$956$		−	187.982i		−	0.196633i
$957$	797.025	+	503.675i	0.832837	+	0.526306i
$958$	188.468			0.196731
$959$		−	1107.11i		−	1.15444i
$960$	200.837	−	317.808i	0.209205	−	0.331049i
$961$	−803.566			−0.836177
$962$			237.803i			0.247197i
$963$	−829.867	+	394.384i	−0.861752	+	0.409537i
$964$	224.166			0.232537
$965$		−	1115.89i		−	1.15637i
$966$	128.339	+	81.1030i	0.132856	+	0.0839575i
$967$	−576.100			−0.595760			−0.297880	−	0.954603i	$-0.596279\pi$
$967$	−576.100			−0.595760			−0.297880	+	0.954603i	$0.596279\pi$
$968$			993.321i			1.02616i
$969$	1195.68	−	1892.06i	1.23393	−	1.95259i
$970$	326.456			0.336553
$971$			422.300i			0.434912i	0.976070	+	0.217456i	$0.0697759\pi$
$971$			422.300i			0.434912i	−0.976070	+	0.217456i	$0.930224\pi$
$972$	622.241	−	208.767i	0.640165	−	0.214781i
$973$	1219.56			1.25340
$974$		−	215.027i		−	0.220767i
$975$	−165.726	−	104.730i	−0.169975	−	0.107415i
$976$	35.1747			0.0360396
$977$		−	1019.17i		−	1.04317i	−0.853200	−	0.521583i	$-0.825342\pi$
$977$		−	1019.17i		−	1.04317i	0.853200	−	0.521583i	$-0.174658\pi$
$978$	39.3353	−	62.2448i	0.0402201	−	0.0636450i
$979$	−768.147			−0.784624
$980$		−	426.270i		−	0.434969i
$981$	−43.6674	−	91.8855i	−0.0445132	−	0.0936651i
$982$	259.857			0.264620
$983$			340.527i			0.346416i	0.984885	+	0.173208i	$0.0554133\pi$
$983$			340.527i			0.346416i	−0.984885	+	0.173208i	$0.944587\pi$
$984$	−907.908	−	573.747i	−0.922670	−	0.583076i
$985$	−1460.42			−1.48266
$986$		−	612.933i		−	0.621636i
$987$	−110.796	+	175.326i	−0.112256	+	0.177635i
$988$	744.956			0.754004
$989$		−	235.433i		−	0.238052i
$990$	−631.075	+	299.911i	−0.637450	+	0.302940i
$991$	724.950			0.731534			0.365767	−	0.930706i	$-0.380807\pi$
$991$	724.950			0.731534			0.365767	+	0.930706i	$0.380807\pi$
$992$		−	413.333i		−	0.416667i
$993$	−1354.80	−	856.160i	−1.36435	−	0.862195i
$994$	−155.008			−0.155943
$995$			778.746i			0.782660i
$996$	25.1793	−	39.8442i	0.0252804	−	0.0400042i
$997$	−977.931			−0.980874			−0.490437	−	0.871477i	$-0.663163\pi$
$997$	−977.931			−0.980874			−0.490437	+	0.871477i	$0.663163\pi$
$998$			449.601i			0.450502i
$999$	−67.2453	+	557.990i	−0.0673126	+	0.558548i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.3.b.a.47.6	✓	14
3.2	odd	2	inner	69.3.b.a.47.9	yes	14
4.3	odd	2		1104.3.g.b.737.14		14
12.11	even	2		1104.3.g.b.737.13		14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.3.b.a.47.6	✓	14	1.1	even	1	trivial
69.3.b.a.47.9	yes	14	3.2	odd	2	inner
1104.3.g.b.737.13		14	12.11	even	2
1104.3.g.b.737.14		14	4.3	odd	2

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:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	69.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.13976i q^{2} +(-2.53605 - 1.60264i) q^{3} +2.70094 q^{4} -4.29888i q^{5} +(-1.82663 + 2.89049i) q^{6} -9.25811 q^{7} -7.63748i q^{8} +(3.86309 + 8.12875i) q^{9} +O(q^{10})\)	\(q-1.13976i q^{2} +(-2.53605 - 1.60264i) q^{3} +2.70094 q^{4} -4.29888i q^{5} +(-1.82663 + 2.89049i) q^{6} -9.25811 q^{7} -7.63748i q^{8} +(3.86309 + 8.12875i) q^{9} -4.89970 q^{10} -15.8448i q^{11} +(-6.84972 - 4.32864i) q^{12} +10.0233 q^{13} +10.5520i q^{14} +(-6.88956 + 10.9022i) q^{15} +2.09884 q^{16} +27.1126i q^{17} +(9.26485 - 4.40300i) q^{18} +27.5173 q^{19} -11.6110i q^{20} +(23.4790 + 14.8374i) q^{21} -18.0594 q^{22} +4.79583i q^{23} +(-12.2401 + 19.3690i) q^{24} +6.51963 q^{25} -11.4242i q^{26} +(3.23049 - 26.8060i) q^{27} -25.0056 q^{28} -19.8347i q^{29} +(12.4259 + 7.85247i) q^{30} +12.5473 q^{31} -32.9421i q^{32} +(-25.3936 + 40.1833i) q^{33} +30.9020 q^{34} +39.7995i q^{35} +(10.4340 + 21.9553i) q^{36} -20.8158 q^{37} -31.3632i q^{38} +(-25.4195 - 16.0637i) q^{39} -32.8326 q^{40} +46.8742i q^{41} +(16.9111 - 26.7605i) q^{42} -49.0913 q^{43} -42.7959i q^{44} +(34.9445 - 16.6069i) q^{45} +5.46611 q^{46} +7.46736i q^{47} +(-5.32276 - 3.36368i) q^{48} +36.7126 q^{49} -7.43084i q^{50} +(43.4518 - 68.7590i) q^{51} +27.0723 q^{52} -25.1269i q^{53} +(-30.5525 - 3.68199i) q^{54} -68.1150 q^{55} +70.7086i q^{56} +(-69.7853 - 44.1004i) q^{57} -22.6069 q^{58} +29.4515i q^{59} +(-18.6083 + 29.4461i) q^{60} +16.7591 q^{61} -14.3009i q^{62} +(-35.7649 - 75.2568i) q^{63} -29.1509 q^{64} -43.0889i q^{65} +(45.7994 + 28.9427i) q^{66} +38.4334 q^{67} +73.2296i q^{68} +(7.68599 - 12.1625i) q^{69} +45.3620 q^{70} +14.6898i q^{71} +(62.0832 - 29.5043i) q^{72} +139.301 q^{73} +23.7251i q^{74} +(-16.5341 - 10.4486i) q^{75} +74.3226 q^{76} +146.693i q^{77} +(-18.3088 + 28.9722i) q^{78} -103.267 q^{79} -9.02265i q^{80} +(-51.1531 + 62.8041i) q^{81} +53.4255 q^{82} +5.81691i q^{83} +(63.4154 + 40.0750i) q^{84} +116.554 q^{85} +55.9524i q^{86} +(-31.7880 + 50.3019i) q^{87} -121.015 q^{88} -48.4794i q^{89} +(-18.9280 - 39.8285i) q^{90} -92.7966 q^{91} +12.9533i q^{92} +(-31.8205 - 20.1088i) q^{93} +8.51102 q^{94} -118.294i q^{95} +(-52.7944 + 83.5428i) q^{96} -66.6277 q^{97} -41.8436i q^{98} +(128.799 - 61.2099i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 4 q^{3} - 24 q^{4} + 11 q^{6} - 4 q^{7} - 8 q^{9}+O(q^{10}) \) 14 * q - 4 * q^3 - 24 * q^4 + 11 * q^6 - 4 * q^7 - 8 * q^9	\( 14 q - 4 q^{3} - 24 q^{4} + 11 q^{6} - 4 q^{7} - 8 q^{9} - 8 q^{10} + 19 q^{12} - 14 q^{15} + 72 q^{16} - 31 q^{18} + 8 q^{19} - 2 q^{21} - 84 q^{22} - 44 q^{24} + 38 q^{25} + 62 q^{27} + 76 q^{28} + 62 q^{30} - 144 q^{31} + 90 q^{33} - 68 q^{34} + 3 q^{36} + 48 q^{37} - 78 q^{39} + 120 q^{40} - 76 q^{42} - 48 q^{43} - 18 q^{45} - 317 q^{48} - 30 q^{49} + 18 q^{51} - 6 q^{52} + 312 q^{54} + 232 q^{55} + 76 q^{57} + 66 q^{58} - 36 q^{60} - 140 q^{61} - 206 q^{63} - 346 q^{64} + 398 q^{66} + 204 q^{67} + 80 q^{70} + 384 q^{72} - 224 q^{73} - 80 q^{75} + 100 q^{76} - 341 q^{78} - 344 q^{79} - 232 q^{81} - 62 q^{82} - 330 q^{84} + 480 q^{85} + 86 q^{87} + 436 q^{88} - 514 q^{90} - 172 q^{91} + 62 q^{93} + 514 q^{94} + 609 q^{96} - 24 q^{97} + 234 q^{99}+O(q^{100}) \) 14 * q - 4 * q^3 - 24 * q^4 + 11 * q^6 - 4 * q^7 - 8 * q^9 - 8 * q^10 + 19 * q^12 - 14 * q^15 + 72 * q^16 - 31 * q^18 + 8 * q^19 - 2 * q^21 - 84 * q^22 - 44 * q^24 + 38 * q^25 + 62 * q^27 + 76 * q^28 + 62 * q^30 - 144 * q^31 + 90 * q^33 - 68 * q^34 + 3 * q^36 + 48 * q^37 - 78 * q^39 + 120 * q^40 - 76 * q^42 - 48 * q^43 - 18 * q^45 - 317 * q^48 - 30 * q^49 + 18 * q^51 - 6 * q^52 + 312 * q^54 + 232 * q^55 + 76 * q^57 + 66 * q^58 - 36 * q^60 - 140 * q^61 - 206 * q^63 - 346 * q^64 + 398 * q^66 + 204 * q^67 + 80 * q^70 + 384 * q^72 - 224 * q^73 - 80 * q^75 + 100 * q^76 - 341 * q^78 - 344 * q^79 - 232 * q^81 - 62 * q^82 - 330 * q^84 + 480 * q^85 + 86 * q^87 + 436 * q^88 - 514 * q^90 - 172 * q^91 + 62 * q^93 + 514 * q^94 + 609 * q^96 - 24 * q^97 + 234 * q^99

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