LMFDB - Embedded newform 69.3.b.a.47.8

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			47.8
Root			$0.634638i$ of defining polynomial
Character	$\chi$	$=$	69.47
Dual form			69.3.b.a.47.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.634638i q^{2} +(1.99717 + 2.23859i) q^{3} +3.59723 q^{4} -4.18173i q^{5} +(-1.42070 + 1.26748i) q^{6} -4.15888 q^{7} +4.82149i q^{8} +(-1.02261 + 8.94172i) q^{9} +O(q^{10})$	\(q+0.634638i q^{2} +(1.99717 + 2.23859i) q^{3} +3.59723 q^{4} -4.18173i q^{5} +(-1.42070 + 1.26748i) q^{6} -4.15888 q^{7} +4.82149i q^{8} +(-1.02261 + 8.94172i) q^{9} +2.65389 q^{10} -2.77055i q^{11} +(7.18430 + 8.05275i) q^{12} -4.48197 q^{13} -2.63938i q^{14} +(9.36120 - 8.35164i) q^{15} +11.3290 q^{16} -14.7798i q^{17} +(-5.67475 - 0.648985i) q^{18} -32.7597 q^{19} -15.0427i q^{20} +(-8.30600 - 9.31004i) q^{21} +1.75830 q^{22} +4.79583i q^{23} +(-10.7934 + 9.62935i) q^{24} +7.51312 q^{25} -2.84443i q^{26} +(-22.0592 + 15.5689i) q^{27} -14.9605 q^{28} -52.7534i q^{29} +(5.30027 + 5.94097i) q^{30} +17.7279 q^{31} +26.4758i q^{32} +(6.20213 - 5.53326i) q^{33} +9.37985 q^{34} +17.3913i q^{35} +(-3.67856 + 32.1654i) q^{36} +16.8735 q^{37} -20.7906i q^{38} +(-8.95127 - 10.0333i) q^{39} +20.1622 q^{40} +5.53854i q^{41} +(5.90851 - 5.27130i) q^{42} +9.31848 q^{43} -9.96631i q^{44} +(37.3919 + 4.27627i) q^{45} -3.04362 q^{46} +56.2845i q^{47} +(22.6260 + 25.3611i) q^{48} -31.7037 q^{49} +4.76811i q^{50} +(33.0861 - 29.5179i) q^{51} -16.1227 q^{52} +103.655i q^{53} +(-9.88064 - 13.9996i) q^{54} -11.5857 q^{55} -20.0520i q^{56} +(-65.4268 - 73.3357i) q^{57} +33.4793 q^{58} +53.2531i q^{59} +(33.6744 - 30.0428i) q^{60} +41.6427 q^{61} +11.2508i q^{62} +(4.25290 - 37.1875i) q^{63} +28.5136 q^{64} +18.7424i q^{65} +(3.51162 + 3.93611i) q^{66} -33.4560 q^{67} -53.1666i q^{68} +(-10.7359 + 9.57810i) q^{69} -11.0372 q^{70} -51.7423i q^{71} +(-43.1124 - 4.93049i) q^{72} -34.2559 q^{73} +10.7086i q^{74} +(15.0050 + 16.8188i) q^{75} -117.844 q^{76} +11.5224i q^{77} +(6.36752 - 5.68082i) q^{78} +100.858 q^{79} -47.3750i q^{80} +(-78.9086 - 18.2877i) q^{81} -3.51497 q^{82} +57.1240i q^{83} +(-29.8786 - 33.4904i) q^{84} -61.8054 q^{85} +5.91386i q^{86} +(118.093 - 105.358i) q^{87} +13.3582 q^{88} -92.6743i q^{89} +(-2.71388 + 23.7303i) q^{90} +18.6400 q^{91} +17.2517i q^{92} +(35.4056 + 39.6855i) q^{93} -35.7203 q^{94} +136.992i q^{95} +(-59.2686 + 52.8768i) q^{96} -148.605 q^{97} -20.1204i q^{98} +(24.7735 + 2.83318i) 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 $

$2$			0.634638i			0.317319i	0.987333	+	0.158659i	$0.0507172\pi$
$2$			0.634638i			0.317319i	−0.987333	+	0.158659i	$0.949283\pi$
$3$	1.99717	+	2.23859i	0.665724	+	0.746198i
$3$	1.99717	+	2.23859i	0.665724	+	0.746198i
$4$	3.59723			0.899309
$5$		−	4.18173i		−	0.836346i	−0.908367	−	0.418173i	$-0.862671\pi$
$5$		−	4.18173i		−	0.836346i	0.908367	−	0.418173i	$-0.137329\pi$
$6$	−1.42070	+	1.26748i	−0.236783	+	0.211247i
$7$	−4.15888			−0.594126			−0.297063	−	0.954858i	$-0.596007\pi$
$7$	−4.15888			−0.594126			−0.297063	+	0.954858i	$0.596007\pi$
$8$			4.82149i			0.602687i
$9$	−1.02261	+	8.94172i	−0.113623	+	0.993524i
$10$	2.65389			0.265389
$11$		−	2.77055i		−	0.251868i	−0.992039	−	0.125934i	$-0.959807\pi$
$11$		−	2.77055i		−	0.251868i	0.992039	−	0.125934i	$-0.0401928\pi$
$12$	7.18430	+	8.05275i	0.598691	+	0.671062i
$13$	−4.48197			−0.344767			−0.172384	−	0.985030i	$-0.555147\pi$
$13$	−4.48197			−0.344767			−0.172384	+	0.985030i	$0.555147\pi$
$14$		−	2.63938i		−	0.188527i
$15$	9.36120	−	8.35164i	0.624080	−	0.556776i
$16$	11.3290			0.708065
$17$		−	14.7798i		−	0.869403i	−0.900575	−	0.434701i	$-0.856854\pi$
$17$		−	14.7798i		−	0.869403i	0.900575	−	0.434701i	$-0.143146\pi$
$18$	−5.67475	−	0.648985i	−0.315264	−	0.0360547i
$19$	−32.7597			−1.72420			−0.862098	−	0.506741i	$-0.830850\pi$
$19$	−32.7597			−1.72420			−0.862098	+	0.506741i	$0.830850\pi$
$20$		−	15.0427i		−	0.752134i
$21$	−8.30600	−	9.31004i	−0.395524	−	0.443335i
$22$	1.75830			0.0799225
$23$			4.79583i			0.208514i
$23$			4.79583i			0.208514i
$24$	−10.7934	+	9.62935i	−0.449724	+	0.401223i
$25$	7.51312			0.300525
$26$		−	2.84443i		−	0.109401i
$27$	−22.0592	+	15.5689i	−0.817007	+	0.576628i
$28$	−14.9605			−0.534302
$29$		−	52.7534i		−	1.81908i	−0.415613	−	0.909542i	$-0.636433\pi$
$29$		−	52.7534i		−	1.81908i	0.415613	−	0.909542i	$-0.363567\pi$
$30$	5.30027	+	5.94097i	0.176676	+	0.198032i
$31$	17.7279			0.571866			0.285933	−	0.958250i	$-0.407696\pi$
$31$	17.7279			0.571866			0.285933	+	0.958250i	$0.407696\pi$
$32$			26.4758i			0.827369i
$33$	6.20213	−	5.53326i	0.187943	−	0.167675i
$34$	9.37985			0.275878
$35$			17.3913i			0.496895i
$36$	−3.67856	+	32.1654i	−0.102182	+	0.893485i
$37$	16.8735			0.456040			0.228020	−	0.973656i	$-0.426775\pi$
$37$	16.8735			0.456040			0.228020	+	0.973656i	$0.426775\pi$
$38$		−	20.7906i		−	0.547120i
$39$	−8.95127	−	10.0333i	−0.229520	−	0.257264i
$40$	20.1622			0.504055
$41$			5.53854i			0.135086i	0.997716	+	0.0675432i	$0.0215160\pi$
$41$			5.53854i			0.135086i	−0.997716	+	0.0675432i	$0.978484\pi$
$42$	5.90851	−	5.27130i	0.140679	−	0.125507i
$43$	9.31848			0.216709			0.108354	−	0.994112i	$-0.465442\pi$
$43$	9.31848			0.216709			0.108354	+	0.994112i	$0.465442\pi$
$44$		−	9.96631i		−	0.226507i
$45$	37.3919	+	4.27627i	0.830930	+	0.0950282i
$46$	−3.04362			−0.0661656
$47$			56.2845i			1.19754i	0.800920	+	0.598771i	$0.204344\pi$
$47$			56.2845i			1.19754i	−0.800920	+	0.598771i	$0.795656\pi$
$48$	22.6260	+	25.3611i	0.471376	+	0.528357i
$49$	−31.7037			−0.647015
$50$			4.76811i			0.0953622i
$51$	33.0861	−	29.5179i	0.648747	−	0.578782i
$52$	−16.1227			−0.310052
$53$			103.655i			1.95576i	0.209170	+	0.977879i	$0.432924\pi$
$53$			103.655i			1.95576i	−0.209170	+	0.977879i	$0.567076\pi$
$54$	−9.88064	−	13.9996i	−0.182975	−	0.259252i
$55$	−11.5857			−0.210649
$56$		−	20.0520i		−	0.358072i
$57$	−65.4268	−	73.3357i	−1.14784	−	1.28659i
$58$	33.4793			0.577230
$59$			53.2531i			0.902594i	0.892374	+	0.451297i	$0.149039\pi$
$59$			53.2531i			0.902594i	−0.892374	+	0.451297i	$0.850961\pi$
$60$	33.6744	−	30.0428i	0.561241	−	0.500713i
$61$	41.6427			0.682667			0.341334	−	0.939942i	$-0.389121\pi$
$61$	41.6427			0.682667			0.341334	+	0.939942i	$0.389121\pi$
$62$			11.2508i			0.181464i
$63$	4.25290	−	37.1875i	0.0675063	−	0.590278i
$64$	28.5136			0.445525
$65$			18.7424i			0.288345i
$66$	3.51162	+	3.93611i	0.0532063	+	0.0596380i
$67$	−33.4560			−0.499343			−0.249672	−	0.968331i	$-0.580323\pi$
$67$	−33.4560			−0.499343			−0.249672	+	0.968331i	$0.580323\pi$
$68$		−	53.1666i		−	0.781861i
$69$	−10.7359	+	9.57810i	−0.155593	+	0.138813i
$70$	−11.0372			−0.157674
$71$		−	51.7423i		−	0.728764i	−0.931250	−	0.364382i	$-0.881280\pi$
$71$		−	51.7423i		−	0.728764i	0.931250	−	0.364382i	$-0.118720\pi$
$72$	−43.1124	−	4.93049i	−0.598784	−	0.0684791i
$73$	−34.2559			−0.469259			−0.234630	−	0.972085i	$-0.575388\pi$
$73$	−34.2559			−0.469259			−0.234630	+	0.972085i	$0.575388\pi$
$74$			10.7086i			0.144710i
$75$	15.0050	+	16.8188i	0.200067	+	0.224251i
$76$	−117.844			−1.55058
$77$			11.5224i			0.149641i
$78$	6.36752	−	5.68082i	0.0816349	−	0.0728310i
$79$	100.858			1.27668			0.638339	−	0.769756i	$-0.279622\pi$
$79$	100.858			1.27668			0.638339	+	0.769756i	$0.279622\pi$
$80$		−	47.3750i		−	0.592187i
$81$	−78.9086	−	18.2877i	−0.974180	−	0.225774i
$82$	−3.51497			−0.0428655
$83$			57.1240i			0.688241i	0.938926	+	0.344120i	$0.111823\pi$
$83$			57.1240i			0.688241i	−0.938926	+	0.344120i	$0.888177\pi$
$84$	−29.8786	−	33.4904i	−0.355698	−	0.398695i
$85$	−61.8054			−0.727122
$86$			5.91386i			0.0687659i
$87$	118.093	−	105.358i	1.35740	−	1.21101i
$88$	13.3582			0.151798
$89$		−	92.6743i		−	1.04128i	−0.853775	−	0.520642i	$-0.825693\pi$
$89$		−	92.6743i		−	1.04128i	0.853775	−	0.520642i	$-0.174307\pi$
$90$	−2.71388	+	23.7303i	−0.0301542	+	0.263670i
$91$	18.6400			0.204835
$92$			17.2517i			0.187519i
$93$	35.4056	+	39.6855i	0.380705	+	0.426726i
$94$	−35.7203			−0.380003
$95$			136.992i			1.44203i
$96$	−59.2686	+	52.8768i	−0.617381	+	0.550799i
$97$	−148.605			−1.53201			−0.766003	−	0.642837i	$-0.777757\pi$
$97$	−148.605			−1.53201			−0.766003	+	0.642837i	$0.777757\pi$
$98$		−	20.1204i		−	0.205310i
$99$	24.7735	+	2.83318i	0.250237	+	0.0286180i
$100$	27.0264			0.270264
$101$		−	145.086i		−	1.43650i	−0.695787	−	0.718248i	$-0.744944\pi$
$101$		−	145.086i		−	1.43650i	0.695787	−	0.718248i	$-0.255056\pi$
$102$	18.7332	+	20.9977i	0.183659	+	0.205860i
$103$	152.787			1.48337			0.741683	−	0.670751i	$-0.234028\pi$
$103$	152.787			1.48337			0.741683	+	0.670751i	$0.234028\pi$
$104$		−	21.6098i		−	0.207787i
$105$	−38.9321	+	34.7335i	−0.370782	+	0.330795i
$106$	−65.7835			−0.620599
$107$			35.7030i			0.333673i	0.985985	+	0.166837i	$0.0533552\pi$
$107$			35.7030i			0.333673i	−0.985985	+	0.166837i	$0.946645\pi$
$108$	−79.3521	+	56.0051i	−0.734742	+	0.518566i
$109$	58.3684			0.535490			0.267745	−	0.963490i	$-0.413722\pi$
$109$	58.3684			0.535490			0.267745	+	0.963490i	$0.413722\pi$
$110$		−	7.35272i		−	0.0668429i
$111$	33.6993	+	37.7729i	0.303597	+	0.340296i
$112$	−47.1161			−0.420679
$113$			174.062i			1.54037i	0.637822	+	0.770184i	$0.279836\pi$
$113$			174.062i			1.54037i	−0.637822	+	0.770184i	$0.720164\pi$
$114$	46.5416	−	41.5223i	0.408260	−	0.364231i
$115$	20.0549			0.174390
$116$		−	189.766i		−	1.63592i
$117$	4.58329	−	40.0765i	0.0391735	−	0.342534i
$118$	−33.7964			−0.286410
$119$			61.4676i			0.516534i
$120$	40.2674	+	45.1350i	0.335561	+	0.376125i
$121$	113.324			0.936563
$122$			26.4280i			0.216623i
$123$	−12.3985	+	11.0614i	−0.100801	+	0.0899302i
$124$	63.7713			0.514284
$125$		−	135.961i		−	1.08769i
$126$	23.6006	+	2.69905i	0.187306	+	0.0214210i
$127$	155.777			1.22659			0.613294	−	0.789855i	$-0.289844\pi$
$127$	155.777			1.22659			0.613294	+	0.789855i	$0.289844\pi$
$128$			123.999i			0.968743i
$129$	18.6106	+	20.8603i	0.144268	+	0.161708i
$130$	−11.8946			−0.0914972
$131$			103.776i			0.792185i	0.918211	+	0.396093i	$0.129634\pi$
$131$			103.776i			0.792185i	−0.918211	+	0.396093i	$0.870366\pi$
$132$	22.3105	−	19.9044i	0.169019	−	0.150791i
$133$	136.244			1.02439
$134$		−	21.2325i		−	0.158451i
$135$	65.1051	+	92.2456i	0.482260	+	0.683301i
$136$	71.2609			0.523977
$137$		−	205.587i		−	1.50063i	−0.661079	−	0.750316i	$-0.729901\pi$
$137$		−	205.587i		−	1.50063i	0.661079	−	0.750316i	$-0.270099\pi$
$138$	−6.07863	−	6.81342i	−0.0440480	−	0.0493726i
$139$	28.4952			0.205002			0.102501	−	0.994733i	$-0.467316\pi$
$139$	28.4952			0.205002			0.102501	+	0.994733i	$0.467316\pi$
$140$			62.5607i			0.446862i
$141$	−125.998	+	112.410i	−0.893604	+	0.797233i
$142$	32.8376			0.231251
$143$			12.4175i			0.0868358i
$144$	−11.5851	+	101.301i	−0.0804524	+	0.703479i
$145$	−220.601			−1.52138
$146$		−	21.7401i		−	0.148905i
$147$	−63.3178	−	70.9718i	−0.430733	−	0.482801i
$148$	60.6979			0.410121
$149$			71.9395i			0.482815i	0.970424	+	0.241408i	$0.0776091\pi$
$149$			71.9395i			0.482815i	−0.970424	+	0.241408i	$0.922391\pi$
$150$	−10.6739	+	9.52274i	−0.0711591	+	0.0634849i
$151$	−283.013			−1.87426			−0.937131	−	0.348979i	$-0.886529\pi$
$151$	−283.013			−1.87426			−0.937131	+	0.348979i	$0.886529\pi$
$152$		−	157.951i		−	1.03915i
$153$	132.157	+	15.1140i	0.863772	+	0.0987841i
$154$	−7.31254			−0.0474840
$155$		−	74.1331i		−	0.478278i
$156$	−32.1998	−	36.0922i	−0.206409	−	0.231360i
$157$	−193.650			−1.23344			−0.616719	−	0.787183i	$-0.711538\pi$
$157$	−193.650			−1.23344			−0.616719	+	0.787183i	$0.711538\pi$
$158$			64.0080i			0.405114i
$159$	−232.042	+	207.017i	−1.45938	+	1.30200i
$160$	110.715			0.691967
$161$		−	19.9453i		−	0.123884i
$162$	11.6061	−	50.0784i	0.0716425	−	0.309126i
$163$	19.9356			0.122304			0.0611520	−	0.998128i	$-0.480523\pi$
$163$	19.9356			0.122304			0.0611520	+	0.998128i	$0.480523\pi$
$164$			19.9234i			0.121484i
$165$	−23.1386	−	25.9357i	−0.140234	−	0.157186i
$166$	−36.2531			−0.218392
$167$		−	31.7834i		−	0.190320i	−0.995462	−	0.0951598i	$-0.969664\pi$
$167$		−	31.7834i		−	0.190320i	0.995462	−	0.0951598i	$-0.0303362\pi$
$168$	44.8883	−	40.0473i	0.267192	−	0.238377i
$169$	−148.912			−0.881136
$170$		−	39.2240i		−	0.230730i
$171$	33.5003	−	292.928i	0.195908	−	1.71303i
$172$	33.5208			0.194888
$173$		−	236.777i		−	1.36865i	−0.729175	−	0.684327i	$-0.760096\pi$
$173$		−	236.777i		−	1.36865i	0.729175	−	0.684327i	$-0.239904\pi$
$174$	66.8640	+	74.9466i	0.384276	+	0.430728i
$175$	−31.2462			−0.178549
$176$		−	31.3876i		−	0.178339i
$177$	−119.212	+	106.356i	−0.673514	+	0.600879i
$178$	58.8146			0.330419
$179$		−	306.185i		−	1.71053i	−0.518190	−	0.855265i	$-0.673394\pi$
$179$		−	306.185i		−	1.71053i	0.518190	−	0.855265i	$-0.326606\pi$
$180$	134.507	+	15.3827i	0.747263	+	0.0854597i
$181$	−44.9088			−0.248115			−0.124057	−	0.992275i	$-0.539591\pi$
$181$	−44.9088			−0.248115			−0.124057	+	0.992275i	$0.539591\pi$
$182$			11.8296i			0.0649980i
$183$	83.1677	+	93.2211i	0.454468	+	0.509405i
$184$	−23.1231			−0.125669
$185$		−	70.5604i		−	0.381408i
$186$	−25.1859	+	22.4697i	−0.135408	+	0.120805i
$187$	−40.9483			−0.218975
$188$			202.469i			1.07696i
$189$	91.7415	−	64.7494i	0.485405	−	0.342589i
$190$	−86.9406			−0.457582
$191$			59.7882i			0.313027i	0.987676	+	0.156514i	$0.0500255\pi$
$191$			59.7882i			0.313027i	−0.987676	+	0.156514i	$0.949974\pi$
$192$	56.9465	+	63.8304i	0.296597	+	0.332450i
$193$	297.648			1.54222			0.771109	−	0.636703i	$-0.219702\pi$
$193$	297.648			1.54222			0.771109	+	0.636703i	$0.219702\pi$
$194$		−	94.3101i		−	0.486135i
$195$	−41.9566	+	37.4318i	−0.215162	+	0.191958i
$196$	−114.046			−0.581866
$197$		−	81.0833i		−	0.411590i	−0.978595	−	0.205795i	$-0.934022\pi$
$197$		−	81.0833i		−	0.411590i	0.978595	−	0.205795i	$-0.0659781\pi$
$198$	−1.79804	+	15.7222i	−0.00908103	+	0.0794049i
$199$	79.3026			0.398505			0.199253	−	0.979948i	$-0.436149\pi$
$199$	79.3026			0.398505			0.199253	+	0.979948i	$0.436149\pi$
$200$			36.2245i			0.181122i
$201$	−66.8174	−	74.8944i	−0.332425	−	0.372609i
$202$	92.0772			0.455828
$203$			219.395i			1.08076i
$204$	119.018	−	106.183i	0.583423	−	0.520504i
$205$	23.1607			0.112979
$206$			96.9642i			0.470700i
$207$	−42.8830	−	4.90425i	−0.207164	−	0.0236920i
$208$	−50.7764			−0.244117
$209$			90.7624i			0.434270i
$210$	−22.0432	−	24.7078i	−0.104967	−	0.117656i
$211$	−262.124			−1.24229			−0.621146	−	0.783695i	$-0.713333\pi$
$211$	−262.124			−1.24229			−0.621146	+	0.783695i	$0.713333\pi$
$212$			372.872i			1.75883i
$213$	115.830	−	103.338i	0.543802	−	0.485156i
$214$	−22.6585			−0.105881
$215$		−	38.9674i		−	0.181244i
$216$	−75.0656	−	106.358i	−0.347526	−	0.492399i
$217$	−73.7280			−0.339760
$218$			37.0428i			0.169921i
$219$	−68.4149	−	76.6851i	−0.312397	−	0.350160i
$220$	−41.6764			−0.189438
$221$			66.2428i			0.299741i
$222$	−23.9721	+	21.3868i	−0.107983	+	0.0963371i
$223$	273.083			1.22459			0.612294	−	0.790630i	$-0.290247\pi$
$223$	273.083			1.22459			0.612294	+	0.790630i	$0.290247\pi$
$224$		−	110.110i		−	0.491561i
$225$	−7.68297	+	67.1802i	−0.0341465	+	0.298579i
$226$	−110.466			−0.488788
$227$		−	99.6622i		−	0.439041i	−0.975608	−	0.219520i	$-0.929551\pi$
$227$		−	99.6622i		−	0.439041i	0.975608	−	0.219520i	$-0.0704492\pi$
$228$	−235.356	−	263.806i	−1.03226	−	1.15704i
$229$	−37.4849			−0.163690			−0.0818448	−	0.996645i	$-0.526081\pi$
$229$	−37.4849			−0.163690			−0.0818448	+	0.996645i	$0.526081\pi$
$230$			12.7276i			0.0553373i
$231$	−25.7939	+	23.0122i	−0.111662	+	0.0996198i
$232$	254.350			1.09634
$233$			226.832i			0.973528i	0.873533	+	0.486764i	$0.161823\pi$
$233$			226.832i			0.973528i	−0.873533	+	0.486764i	$0.838177\pi$
$234$	25.4341	+	2.90873i	0.108693	+	0.0124305i
$235$	235.367			1.00156
$236$			191.564i			0.811711i
$237$	201.430	+	225.779i	0.849915	+	0.952654i
$238$	−39.0097			−0.163906
$239$			201.816i			0.844420i	0.906498	+	0.422210i	$0.138746\pi$
$239$			201.816i			0.844420i	−0.906498	+	0.422210i	$0.861254\pi$
$240$	106.053	−	94.6160i	0.441889	−	0.394233i
$241$	32.0434			0.132960			0.0664800	−	0.997788i	$-0.478823\pi$
$241$	32.0434			0.132960			0.0664800	+	0.997788i	$0.478823\pi$
$242$			71.9198i			0.297189i
$243$	−116.655	−	213.168i	−0.480062	−	0.877234i
$244$	149.799			0.613929
$245$			132.576i			0.541128i
$246$	−7.01999	−	7.86858i	−0.0285366	−	0.0319861i
$247$	146.828			0.594446
$248$			85.4747i			0.344656i
$249$	−127.877	+	114.086i	−0.513564	+	0.458178i
$250$	86.2861			0.345144
$251$		−	25.8671i		−	0.103056i	−0.998672	−	0.0515281i	$-0.983591\pi$
$251$		−	25.8671i		−	0.103056i	0.998672	−	0.0515281i	$-0.0164092\pi$
$252$	15.2987	−	133.772i	0.0607090	−	0.530842i
$253$	13.2871			0.0525181
$254$			98.8617i			0.389219i
$255$	−123.436	−	138.357i	−0.484062	−	0.542577i
$256$	35.3599			0.138124
$257$			185.474i			0.721689i	0.932626	+	0.360845i	$0.117512\pi$
$257$			185.474i			0.721689i	−0.932626	+	0.360845i	$0.882488\pi$
$258$	−13.2387	+	11.8110i	−0.0513129	+	0.0457791i
$259$	−70.1748			−0.270945
$260$			67.4208i			0.259311i
$261$	471.706	+	53.9460i	1.80730	+	0.206690i
$262$	−65.8603			−0.251375
$263$		−	11.6167i		−	0.0441698i	−0.999756	−	0.0220849i	$-0.992970\pi$
$263$		−	11.6167i		−	0.0441698i	0.999756	−	0.0220849i	$-0.00703041\pi$
$264$	26.6786	+	29.9035i	0.101055	+	0.113271i
$265$	433.458			1.63569
$266$			86.4655i			0.325058i
$267$	207.460	−	185.087i	0.777004	−	0.693208i
$268$	−120.349			−0.449064
$269$		−	263.847i		−	0.980842i	−0.871486	−	0.490421i	$-0.836843\pi$
$269$		−	263.847i		−	0.980842i	0.871486	−	0.490421i	$-0.163157\pi$
$270$	−58.5426	+	41.3182i	−0.216824	+	0.153030i
$271$	−379.244			−1.39943			−0.699713	−	0.714424i	$-0.746689\pi$
$271$	−379.244			−1.39943			−0.699713	+	0.714424i	$0.746689\pi$
$272$		−	167.441i		−	0.615593i
$273$	37.2273	+	41.7274i	0.136364	+	0.152847i
$274$	130.473			0.476179
$275$		−	20.8155i		−	0.0756926i
$276$	−38.6196	+	34.4547i	−0.139926	+	0.124836i
$277$	−201.193			−0.726329			−0.363165	−	0.931725i	$-0.618304\pi$
$277$	−201.193			−0.726329			−0.363165	+	0.931725i	$0.618304\pi$
$278$			18.0841i			0.0650509i
$279$	−18.1286	+	158.517i	−0.0649772	+	0.568163i
$280$	−83.8521			−0.299472
$281$		−	439.366i		−	1.56358i	−0.623543	−	0.781789i	$-0.714307\pi$
$281$		−	439.366i		−	1.56358i	0.623543	−	0.781789i	$-0.285693\pi$
$282$	−71.3396	−	79.9632i	−0.252977	−	0.283558i
$283$	−361.821			−1.27852			−0.639260	−	0.768991i	$-0.720759\pi$
$283$	−361.821			−1.27852			−0.639260	+	0.768991i	$0.720759\pi$
$284$		−	186.129i		−	0.655384i
$285$	−306.670	+	273.597i	−1.07604	+	0.959991i
$286$	−7.88063			−0.0275546
$287$		−	23.0341i		−	0.0802582i
$288$	−236.739	−	27.0743i	−0.822011	−	0.0940081i
$289$	70.5562			0.244139
$290$		−	140.002i		−	0.482764i
$291$	−296.789	−	332.665i	−1.01989	−	1.14318i
$292$	−123.227			−0.422009
$293$		−	121.925i		−	0.416127i	−0.978115	−	0.208063i	$-0.933284\pi$
$293$		−	121.925i		−	0.416127i	0.978115	−	0.208063i	$-0.0667160\pi$
$294$	45.0414	−	40.1839i	0.153202	−	0.136680i
$295$	222.690			0.754881
$296$			81.3555i			0.274850i
$297$	43.1345	+	61.1161i	0.145234	+	0.205778i
$298$	−45.6555			−0.153206
$299$		−	21.4948i		−	0.0718889i
$300$	53.9765	+	60.5013i	0.179922	+	0.201671i
$301$	−38.7545			−0.128752
$302$		−	179.611i		−	0.594739i
$303$	324.789	−	289.762i	1.07191	−	0.956310i
$304$	−371.136			−1.22084
$305$		−	174.139i		−	0.570946i
$306$	−9.59190	+	83.8720i	−0.0313461	+	0.274091i
$307$	−400.358			−1.30410			−0.652049	−	0.758176i	$-0.726091\pi$
$307$	−400.358			−1.30410			−0.652049	+	0.758176i	$0.726091\pi$
$308$			41.4487i			0.134574i
$309$	305.141	+	342.027i	0.987512	+	1.10688i
$310$	47.0477			0.151767
$311$			226.177i			0.727256i	0.931544	+	0.363628i	$0.118462\pi$
$311$			226.177i			0.727256i	−0.931544	+	0.363628i	$0.881538\pi$
$312$	48.3756	−	43.1585i	0.155050	−	0.138328i
$313$	−174.364			−0.557073			−0.278536	−	0.960426i	$-0.589849\pi$
$313$	−174.364			−0.557073			−0.278536	+	0.960426i	$0.589849\pi$
$314$		−	122.897i		−	0.391393i
$315$	−155.508	−	17.7845i	−0.493677	−	0.0564587i
$316$	362.808			1.14813
$317$			36.6281i			0.115546i	0.998330	+	0.0577730i	$0.0184000\pi$
$317$			36.6281i			0.115546i	−0.998330	+	0.0577730i	$0.981600\pi$
$318$	−131.381	−	147.263i	−0.413148	−	0.463090i
$319$	−146.156			−0.458169
$320$		−	119.236i		−	0.372613i
$321$	−79.9246	+	71.3051i	−0.248986	+	0.222134i
$322$	12.6580			0.0393107
$323$			484.184i			1.49902i
$324$	−283.853	−	65.7852i	−0.876088	−	0.203041i
$325$	−33.6736			−0.103611
$326$			12.6519i			0.0388094i
$327$	116.572	+	130.663i	0.356488	+	0.399581i
$328$	−26.7040			−0.0814147
$329$		−	234.081i		−	0.711491i
$330$	16.4598	−	14.6846i	0.0498780	−	0.0444989i
$331$	389.635			1.17715			0.588573	−	0.808444i	$-0.299690\pi$
$331$	389.635			1.17715			0.588573	+	0.808444i	$0.299690\pi$
$332$			205.488i			0.618941i
$333$	−17.2550	+	150.878i	−0.0518167	+	0.453087i
$334$	20.1709			0.0603920
$335$			139.904i			0.417624i
$336$	−94.0990	−	105.474i	−0.280056	−	0.313910i
$337$	289.665			0.859540			0.429770	−	0.902938i	$-0.358595\pi$
$337$	289.665			0.859540			0.429770	+	0.902938i	$0.358595\pi$
$338$		−	94.5052i		−	0.279601i
$339$	−389.653	+	347.631i	−1.14942	+	1.02546i
$340$	−222.328			−0.653907
$341$		−	49.1159i		−	0.144035i
$342$	185.903	+	21.2606i	0.543577	+	0.0621654i
$343$	335.637			0.978534
$344$			44.9290i			0.130608i
$345$	40.0531	+	44.8947i	0.116096	+	0.130130i
$346$	150.268			0.434300
$347$		−	304.066i		−	0.876272i	−0.898909	−	0.438136i	$-0.855639\pi$
$347$		−	304.066i		−	0.876272i	0.898909	−	0.438136i	$-0.144361\pi$
$348$	424.810	−	378.996i	1.22072	−	1.08907i
$349$	−77.8624			−0.223102			−0.111551	−	0.993759i	$-0.535582\pi$
$349$	−77.8624			−0.223102			−0.111551	+	0.993759i	$0.535582\pi$
$350$		−	19.8300i		−	0.0566571i
$351$	98.8687	−	69.7796i	0.281677	−	0.198802i
$352$	73.3525			0.208388
$353$		−	113.970i		−	0.322860i	−0.986884	−	0.161430i	$-0.948389\pi$
$353$		−	113.970i		−	0.322860i	0.986884	−	0.161430i	$-0.0516106\pi$
$354$	−67.4973	−	75.6565i	−0.190670	−	0.213719i
$355$	−216.372			−0.609499
$356$		−	333.371i		−	0.936436i
$357$	−137.601	+	122.761i	−0.385437	+	0.343869i
$358$	194.317			0.542784
$359$			385.118i			1.07275i	0.843979	+	0.536377i	$0.180207\pi$
$359$			385.118i			1.07275i	−0.843979	+	0.536377i	$0.819793\pi$
$360$	−20.6180	+	180.285i	−0.0572722	+	0.500791i
$361$	712.200			1.97285
$362$		−	28.5008i		−	0.0787315i
$363$	226.328	+	253.687i	0.623492	+	0.698861i
$364$	67.0524			0.184210
$365$			143.249i			0.392463i
$366$	−59.1617	+	52.7814i	−0.161644	+	0.144211i
$367$	−521.461			−1.42088			−0.710438	−	0.703760i	$-0.751503\pi$
$367$	−521.461			−1.42088			−0.710438	+	0.703760i	$0.751503\pi$
$368$			54.3321i			0.147642i
$369$	−49.5240	−	5.66375i	−0.134211	−	0.0153489i
$370$	44.7803			0.121028
$371$		−	431.090i		−	1.16197i
$372$	127.362	+	142.758i	0.342371	+	0.383758i
$373$	−2.13177			−0.00571520			−0.00285760	−	0.999996i	$-0.500910\pi$
$373$	−2.13177			−0.00571520			−0.00285760	+	0.999996i	$0.500910\pi$
$374$		−	25.9873i		−	0.0694848i
$375$	304.362	−	271.538i	0.811631	−	0.724101i
$376$	−271.375			−0.721743
$377$			236.439i			0.627160i
$378$	41.0924	+	58.2227i	0.108710	+	0.154028i
$379$	−259.362			−0.684333			−0.342167	−	0.939639i	$-0.611161\pi$
$379$	−259.362			−0.684333			−0.342167	+	0.939639i	$0.611161\pi$
$380$			492.794i			1.29683i
$381$	311.113	+	348.721i	0.816569	+	0.915277i
$382$	−37.9439			−0.0993295
$383$		−	467.278i		−	1.22005i	−0.792384	−	0.610023i	$-0.791160\pi$
$383$		−	467.278i		−	1.22005i	0.792384	−	0.610023i	$-0.208840\pi$
$384$	−277.584	+	247.647i	−0.722874	+	0.644915i
$385$	48.1835			0.125152
$386$			188.899i			0.489375i
$387$	−9.52914	+	83.3232i	−0.0246231	+	0.215306i
$388$	−534.566			−1.37775
$389$		−	77.1304i		−	0.198279i	−0.995074	−	0.0991394i	$-0.968391\pi$
$389$		−	77.1304i		−	0.198279i	0.995074	−	0.0991394i	$-0.0316090\pi$
$390$	−23.7556	−	26.6273i	−0.0609119	−	0.0682751i
$391$	70.8816			0.181283
$392$		−	152.859i		−	0.389947i
$393$	−232.313	+	207.259i	−0.591127	+	0.527377i
$394$	51.4586			0.130605
$395$		−	421.759i		−	1.06774i
$396$	89.1159	+	10.1916i	0.225040	+	0.0257364i
$397$	156.469			0.394129			0.197065	−	0.980391i	$-0.436859\pi$
$397$	156.469			0.394129			0.197065	+	0.980391i	$0.436859\pi$
$398$			50.3284i			0.126453i
$399$	272.102	+	304.994i	0.681961	+	0.764397i
$400$	85.1164			0.212791
$401$			740.860i			1.84753i	0.382959	+	0.923765i	$0.374905\pi$
$401$			740.860i			1.84753i	−0.382959	+	0.923765i	$0.625095\pi$
$402$	47.5308	−	42.4049i	0.118236	−	0.105485i
$403$	−79.4558			−0.197161
$404$		−	521.909i		−	1.29185i
$405$	−76.4743	+	329.974i	−0.188826	+	0.814752i
$406$	−139.236			−0.342947
$407$		−	46.7488i		−	0.114862i
$408$	142.320	+	159.524i	0.348824	+	0.390991i
$409$	177.379			0.433689			0.216845	−	0.976206i	$-0.430424\pi$
$409$	177.379			0.433689			0.216845	+	0.976206i	$0.430424\pi$
$410$			14.6986i			0.0358504i
$411$	460.225	−	410.592i	1.11977	−	0.999007i
$412$	549.610			1.33400
$413$		−	221.473i		−	0.536254i
$414$	3.11242	−	27.2152i	0.00751793	−	0.0657371i
$415$	238.877			0.575608
$416$		−	118.664i		−	0.285250i
$417$	56.9098	+	63.7892i	0.136474	+	0.152972i
$418$	−57.6013			−0.137802
$419$		−	254.725i		−	0.607936i	−0.952682	−	0.303968i	$-0.901688\pi$
$419$		−	254.725i		−	0.607936i	0.952682	−	0.303968i	$-0.0983117\pi$
$420$	−140.048	+	124.944i	−0.333447	+	0.297487i
$421$	385.663			0.916064			0.458032	−	0.888936i	$-0.348555\pi$
$421$	385.663			0.916064			0.458032	+	0.888936i	$0.348555\pi$
$422$		−	166.354i		−	0.394203i
$423$	−503.280	−	57.5569i	−1.18979	−	0.136068i
$424$	−499.773			−1.17871
$425$		−	111.043i		−	0.261277i
$426$	65.5823	+	73.5101i	0.153949	+	0.172559i
$427$	−173.187			−0.405590
$428$			128.432i			0.300075i
$429$	−27.7978	+	24.7999i	−0.0647967	+	0.0578087i
$430$	24.7302			0.0575121
$431$			690.613i			1.60235i	0.598429	+	0.801176i	$0.295792\pi$
$431$			690.613i			1.60235i	−0.598429	+	0.801176i	$0.704208\pi$
$432$	−249.909	+	176.381i	−0.578494	+	0.408290i
$433$	380.430			0.878590			0.439295	−	0.898343i	$-0.355228\pi$
$433$	380.430			0.878590			0.439295	+	0.898343i	$0.355228\pi$
$434$		−	46.7906i		−	0.107812i
$435$	−440.577	−	493.835i	−1.01282	−	1.13525i
$436$	209.965			0.481571
$437$		−	157.110i		−	0.359520i
$438$	48.6673	−	43.4187i	0.111112	−	0.0991295i
$439$	319.346			0.727440			0.363720	−	0.931508i	$-0.381506\pi$
$439$	319.346			0.727440			0.363720	+	0.931508i	$0.381506\pi$
$440$		−	55.8603i		−	0.126955i
$441$	32.4204	−	283.486i	0.0735157	−	0.642825i
$442$	−42.0402			−0.0951136
$443$		−	136.251i		−	0.307564i	−0.988105	−	0.153782i	$-0.950855\pi$
$443$		−	136.251i		−	0.307564i	0.988105	−	0.153782i	$-0.0491453\pi$
$444$	121.224	+	135.878i	0.273028	+	0.306032i
$445$	−387.539			−0.870875
$446$			173.309i			0.388585i
$447$	−161.043	+	143.676i	−0.360276	+	0.321422i
$448$	−118.585			−0.264698
$449$		−	347.198i		−	0.773269i	−0.922233	−	0.386635i	$-0.873637\pi$
$449$		−	347.198i		−	0.773269i	0.922233	−	0.386635i	$-0.126363\pi$
$450$	−42.6351	−	4.87590i	−0.0947446	−	0.0108353i
$451$	15.3448			0.0340239
$452$			626.140i			1.38527i
$453$	−565.227	−	633.552i	−1.24774	−	1.39857i
$454$	63.2494			0.139316
$455$		−	77.9474i		−	0.171313i
$456$	353.588	−	315.455i	0.775412	−	0.691787i
$457$	−737.440			−1.61366			−0.806828	−	0.590787i	$-0.798817\pi$
$457$	−737.440			−1.61366			−0.806828	+	0.590787i	$0.798817\pi$
$458$		−	23.7894i		−	0.0519418i
$459$	230.107	+	326.031i	0.501322	+	0.710308i
$460$	72.1421			0.156831
$461$		−	175.172i		−	0.379983i	−0.981786	−	0.189992i	$-0.939154\pi$
$461$		−	175.172i		−	0.379983i	0.981786	−	0.189992i	$-0.0608461\pi$
$462$	−14.6044	−	16.3698i	−0.0316112	−	0.0354325i
$463$	−368.135			−0.795108			−0.397554	−	0.917579i	$-0.630141\pi$
$463$	−368.135			−0.795108			−0.397554	+	0.917579i	$0.630141\pi$
$464$		−	597.645i		−	1.28803i
$465$	165.954	−	148.057i	0.356890	−	0.318401i
$466$	−143.956			−0.308919
$467$		−	381.480i		−	0.816874i	−0.912787	−	0.408437i	$-0.866074\pi$
$467$		−	381.480i		−	0.816874i	0.912787	−	0.408437i	$-0.133926\pi$
$468$	16.4872	−	144.165i	0.0352290	−	0.308044i
$469$	139.140			0.296673
$470$			149.373i			0.317814i
$471$	−386.752	−	433.503i	−0.821129	−	0.920389i
$472$	−256.759			−0.543982
$473$		−	25.8173i		−	0.0545820i
$474$	−143.288	+	127.835i	−0.302295	+	0.269694i
$475$	−246.128			−0.518164
$476$			221.113i			0.464524i
$477$	−926.855	−	105.999i	−1.94309	−	0.222219i
$478$	−128.080			−0.267951
$479$		−	65.5723i		−	0.136894i	−0.997655	−	0.0684471i	$-0.978196\pi$
$479$		−	65.5723i		−	0.136894i	0.997655	−	0.0684471i	$-0.0218044\pi$
$480$	221.116	+	247.845i	0.460659	+	0.516345i
$481$	−75.6265			−0.157228
$482$			20.3359i			0.0421907i
$483$	44.6494	−	39.8342i	0.0924418	−	0.0824724i
$484$	407.653			0.842259
$485$			621.425i			1.28129i
$486$	135.284	−	74.0338i	0.278363	−	0.152333i
$487$	615.070			1.26298			0.631489	−	0.775385i	$-0.282444\pi$
$487$	615.070			1.26298			0.631489	+	0.775385i	$0.282444\pi$
$488$			200.780i			0.411435i
$489$	39.8147	+	44.6276i	0.0814208	+	0.0912630i
$490$	−84.1381			−0.171710
$491$			812.285i			1.65435i	0.561946	+	0.827174i	$0.310053\pi$
$491$			812.285i			1.65435i	−0.561946	+	0.827174i	$0.689947\pi$
$492$	−44.6005	+	39.7905i	−0.0906513	+	0.0808750i
$493$	−779.687			−1.58152
$494$			93.1827i			0.188629i
$495$	11.8476	−	103.596i	0.0239346	−	0.209285i
$496$	200.840			0.404918
$497$			215.190i			0.432978i
$498$	−72.4036	−	81.1559i	−0.145389	−	0.162964i
$499$	570.990			1.14427			0.572135	−	0.820160i	$-0.306115\pi$
$499$	570.990			1.14427			0.572135	+	0.820160i	$0.306115\pi$
$500$		−	489.084i		−	0.978168i
$501$	71.1500	−	63.4768i	0.142016	−	0.126700i
$502$	16.4163			0.0327017
$503$			306.074i			0.608497i	0.952593	+	0.304248i	$0.0984053\pi$
$503$			306.074i			0.608497i	−0.952593	+	0.304248i	$0.901595\pi$
$504$	179.299	+	20.5053i	0.355753	+	0.0406852i
$505$	−606.711			−1.20141
$506$			8.43249i			0.0166650i
$507$	−297.403	−	333.353i	−0.586593	−	0.657502i
$508$	560.365			1.10308
$509$		−	293.269i		−	0.576167i	−0.957605	−	0.288083i	$-0.906982\pi$
$509$		−	293.269i		−	0.576167i	0.957605	−	0.288083i	$-0.0930180\pi$
$510$	87.8067	−	78.3371i	0.172170	−	0.153602i
$511$	142.466			0.278799
$512$			518.437i			1.01257i
$513$	722.653	−	510.034i	1.40868	−	0.994219i
$514$	−117.709			−0.229006
$515$		−	638.913i		−	1.24061i
$516$	66.9468	+	75.0394i	0.129742	+	0.145425i
$517$	155.939			0.301623
$518$		−	44.5356i		−	0.0859761i
$519$	530.048	−	472.885i	1.02129	−	0.911146i
$520$	−90.3664			−0.173781
$521$			542.212i			1.04071i	0.853949	+	0.520357i	$0.174201\pi$
$521$			542.212i			1.04071i	−0.853949	+	0.520357i	$0.825799\pi$
$522$	−34.2362	+	299.363i	−0.0655865	+	0.573491i
$523$	−757.659			−1.44868			−0.724340	−	0.689443i	$-0.757855\pi$
$523$	−757.659			−1.44868			−0.724340	+	0.689443i	$0.757855\pi$
$524$			373.308i			0.712419i
$525$	−62.4040	−	69.9475i	−0.118865	−	0.133233i
$526$	7.37237			0.0140159
$527$		−	262.015i		−	0.497182i
$528$	70.2642	−	62.6865i	0.133076	−	0.118724i
$529$	−23.0000			−0.0434783
$530$			275.089i			0.519036i
$531$	−476.174	−	54.4569i	−0.896749	−	0.102555i
$532$	490.101			0.921242
$533$		−	24.8236i		−	0.0465733i
$534$	117.463	+	131.662i	0.219968	+	0.246558i
$535$	149.301			0.279066
$536$		−	161.308i		−	0.300948i
$537$	685.424	−	611.504i	1.27639	−	1.13874i
$538$	167.447			0.311240
$539$			87.8367i			0.162962i
$540$	234.198	+	331.829i	0.433701	+	0.614498i
$541$	508.006			0.939013			0.469507	−	0.882929i	$-0.344432\pi$
$541$	508.006			0.939013			0.469507	+	0.882929i	$0.344432\pi$
$542$		−	240.683i		−	0.444064i
$543$	−89.6905	−	100.533i	−0.165176	−	0.185143i
$544$	391.308			0.719317
$545$		−	244.081i		−	0.447855i
$546$	−26.4818	+	23.6258i	−0.0485014	+	0.0432707i
$547$	−409.299			−0.748262			−0.374131	−	0.927376i	$-0.622059\pi$
$547$	−409.299			−0.748262			−0.374131	+	0.927376i	$0.622059\pi$
$548$		−	739.543i		−	1.34953i
$549$	−42.5841	+	372.357i	−0.0775667	+	0.678246i
$550$	13.2103			0.0240187
$551$			1728.19i			3.13646i
$552$	−46.1808	−	51.7632i	−0.0836608	−	0.0937739i
$553$	−419.454			−0.758507
$554$		−	127.685i		−	0.230478i
$555$	157.956	−	140.921i	0.284606	−	0.253912i
$556$	102.504			0.184360
$557$			135.826i			0.243854i	0.992539	+	0.121927i	$0.0389073\pi$
$557$			135.826i			0.243854i	−0.992539	+	0.121927i	$0.961093\pi$
$558$	−100.601	−	11.5051i	−0.180289	−	0.0206185i
$559$	−41.7652			−0.0747141
$560$			197.027i			0.351834i
$561$	−81.7808	−	91.6666i	−0.145777	−	0.163399i
$562$	278.838			0.496153
$563$			815.521i			1.44853i	0.689523	+	0.724264i	$0.257820\pi$
$563$			815.521i			1.44853i	−0.689523	+	0.724264i	$0.742180\pi$
$564$	−453.245	+	404.365i	−0.803626	+	0.716959i
$565$	727.879			1.28828
$566$		−	229.625i		−	0.405699i
$567$	328.171	+	76.0564i	0.578785	+	0.134138i
$568$	249.475			0.439216
$569$			757.701i			1.33164i	0.746114	+	0.665818i	$0.231917\pi$
$569$			757.701i			1.33164i	−0.746114	+	0.665818i	$0.768083\pi$
$570$	−173.635	−	194.625i	−0.304623	−	0.341447i
$571$	−545.946			−0.956122			−0.478061	−	0.878327i	$-0.658660\pi$
$571$	−545.946			−0.956122			−0.478061	+	0.878327i	$0.658660\pi$
$572$			44.6687i			0.0780922i
$573$	−133.842	+	119.407i	−0.233580	+	0.208390i
$574$	14.6183			0.0254675
$575$			36.0317i			0.0626637i
$576$	−29.1582	+	254.960i	−0.0506219	+	0.442640i
$577$	153.900			0.266724			0.133362	−	0.991067i	$-0.457423\pi$
$577$	153.900			0.266724			0.133362	+	0.991067i	$0.457423\pi$
$578$			44.7776i			0.0774699i
$579$	594.455	+	666.314i	1.02669	+	1.15080i
$580$	−793.552			−1.36819
$581$		−	237.572i		−	0.408902i
$582$	211.122	−	188.354i	0.362753	−	0.323632i
$583$	287.182			0.492593
$584$		−	165.165i		−	0.282816i
$585$	−167.589	−	19.1661i	−0.286477	−	0.0327626i
$586$	77.3783			0.132045
$587$		−	267.020i		−	0.454890i	−0.973791	−	0.227445i	$-0.926963\pi$
$587$		−	267.020i		−	0.454890i	0.973791	−	0.227445i	$-0.0730372\pi$
$588$	−227.769	−	255.302i	−0.387362	−	0.434187i
$589$	−580.760			−0.986010
$590$			141.328i			0.239538i
$591$	181.513	−	161.937i	0.307128	−	0.274006i
$592$	191.160			0.322906
$593$			696.331i			1.17425i	0.809496	+	0.587125i	$0.199740\pi$
$593$			696.331i			1.17425i	−0.809496	+	0.587125i	$0.800260\pi$
$594$	−38.7866	+	27.3748i	−0.0652973	+	0.0460855i
$595$	257.041			0.432002
$596$			258.783i			0.434200i
$597$	158.381	+	177.526i	0.265295	+	0.297364i
$598$	13.6414			0.0228117
$599$		−	856.660i		−	1.43015i	−0.699047	−	0.715075i	$-0.746392\pi$
$599$		−	856.660i		−	1.43015i	0.699047	−	0.715075i	$-0.253608\pi$
$600$	−81.0918	+	72.3465i	−0.135153	+	0.120577i
$601$	781.504			1.30034			0.650170	−	0.759789i	$-0.274698\pi$
$601$	781.504			1.30034			0.650170	+	0.759789i	$0.274698\pi$
$602$		−	24.5950i		−	0.0408556i
$603$	34.2123	−	299.154i	0.0567369	−	0.496110i
$604$	−1018.07			−1.68554
$605$		−	473.891i		−	0.783291i
$606$	183.894	+	206.123i	0.303455	+	0.340138i
$607$	28.9236			0.0476501			0.0238250	−	0.999716i	$-0.492416\pi$
$607$	28.9236			0.0476501			0.0238250	+	0.999716i	$0.492416\pi$
$608$		−	867.340i		−	1.42655i
$609$	−491.137	+	438.170i	−0.806464	+	0.719491i
$610$	110.515			0.181172
$611$		−	252.266i		−	0.412873i
$612$	475.400	+	54.3685i	0.776798	+	0.0888374i
$613$	−92.5854			−0.151037			−0.0755183	−	0.997144i	$-0.524061\pi$
$613$	−92.5854			−0.151037			−0.0755183	+	0.997144i	$0.524061\pi$
$614$		−	254.083i		−	0.413815i
$615$	46.2559	+	51.8474i	0.0752128	+	0.0843047i
$616$	−55.5551			−0.0901868
$617$			75.8191i			0.122883i	0.998111	+	0.0614417i	$0.0195698\pi$
$617$			75.8191i			0.122883i	−0.998111	+	0.0614417i	$0.980430\pi$
$618$	−217.064	+	193.654i	−0.351236	+	0.313356i
$619$	819.311			1.32360			0.661802	−	0.749678i	$-0.269792\pi$
$619$	819.311			1.32360			0.661802	+	0.749678i	$0.269792\pi$
$620$		−	266.674i		−	0.430120i
$621$	−74.6660	−	105.792i	−0.120235	−	0.170358i
$622$	−143.540			−0.230772
$623$			385.421i			0.618654i
$624$	−101.409	−	113.668i	−0.162515	−	0.182160i
$625$	−380.725			−0.609160
$626$		−	110.658i		−	0.176770i
$627$	−203.180	+	181.268i	−0.324051	+	0.289104i
$628$	−696.604			−1.10924
$629$		−	249.388i		−	0.396483i
$630$	11.2867	−	98.6914i	0.0179154	−	0.156653i
$631$	−634.954			−1.00627			−0.503133	−	0.864209i	$-0.667820\pi$
$631$	−634.954			−1.00627			−0.503133	+	0.864209i	$0.667820\pi$
$632$			486.284i			0.769436i
$633$	−523.506	−	586.789i	−0.827024	−	0.926996i
$634$	−23.2456			−0.0366649
$635$		−	651.416i		−	1.02585i
$636$	−834.709	+	744.690i	−1.31244	+	1.17090i
$637$	142.095			0.223069
$638$		−	92.7561i		−	0.145386i
$639$	462.665	+	52.9120i	0.724045	+	0.0828044i
$640$	518.531			0.810204
$641$		−	549.228i		−	0.856830i	−0.903582	−	0.428415i	$-0.859072\pi$
$641$		−	549.228i		−	0.856830i	0.903582	−	0.428415i	$-0.140928\pi$
$642$	−45.2529	−	50.7232i	−0.0704874	−	0.0790081i
$643$	−399.323			−0.621031			−0.310515	−	0.950568i	$-0.600502\pi$
$643$	−399.323			−0.621031			−0.310515	+	0.950568i	$0.600502\pi$
$644$		−	71.7479i		−	0.111410i
$645$	87.2322	−	77.8246i	0.135244	−	0.120658i
$646$	−307.281			−0.475668
$647$			366.980i			0.567202i	0.958942	+	0.283601i	$0.0915291\pi$
$647$			366.980i			0.567202i	−0.958942	+	0.283601i	$0.908471\pi$
$648$	88.1741	−	380.457i	0.136071	−	0.587125i
$649$	147.540			0.227335
$650$		−	21.3705i		−	0.0328777i
$651$	−147.248	−	165.047i	−0.226187	−	0.253529i
$652$	71.7129			0.109989
$653$			212.732i			0.325777i	0.986645	+	0.162888i	$0.0520810\pi$
$653$			212.732i			0.325777i	−0.986645	+	0.162888i	$0.947919\pi$
$654$	−82.9238	+	73.9808i	−0.126795	+	0.113121i
$655$	433.964			0.662541
$656$			62.7463i			0.0956499i
$657$	35.0303	−	306.307i	0.0533186	−	0.466220i
$658$	148.556			0.225770
$659$		−	270.157i		−	0.409950i	−0.978767	−	0.204975i	$-0.934289\pi$
$659$		−	270.157i		−	0.409950i	0.978767	−	0.204975i	$-0.0657113\pi$
$660$	−83.2350	−	93.2966i	−0.126114	−	0.141359i
$661$	411.615			0.622716			0.311358	−	0.950293i	$-0.399216\pi$
$661$	411.615			0.622716			0.311358	+	0.950293i	$0.399216\pi$
$662$			247.277i			0.373531i
$663$	−148.291	+	132.298i	−0.223666	+	0.199545i
$664$	−275.423			−0.414794
$665$		−	569.735i		−	0.856744i
$666$	−95.7529	−	10.9506i	−0.143773	−	0.0164424i
$667$	252.996			0.379305
$668$		−	114.332i		−	0.171156i
$669$	545.394	+	611.322i	0.815237	+	0.913785i
$670$	−88.7884			−0.132520
$671$		−	115.373i		−	0.171942i
$672$	246.491	−	219.908i	0.366802	−	0.327244i
$673$	276.577			0.410962			0.205481	−	0.978661i	$-0.434124\pi$
$673$	276.577			0.410962			0.205481	+	0.978661i	$0.434124\pi$
$674$			183.832i			0.272749i
$675$	−165.733	+	116.971i	−0.245531	+	0.173291i
$676$	−535.671			−0.792413
$677$		−	103.160i		−	0.152378i	−0.997093	−	0.0761890i	$-0.975725\pi$
$677$		−	103.160i		−	0.152378i	0.997093	−	0.0761890i	$-0.0242752\pi$
$678$	−220.620	−	247.289i	−0.325398	−	0.364733i
$679$	618.029			0.910204
$680$		−	297.994i		−	0.438227i
$681$	223.103	−	199.043i	0.327611	−	0.292280i
$682$	31.1708			0.0457050
$683$			23.5837i			0.0345295i	0.999851	+	0.0172648i	$0.00549582\pi$
$683$			23.5837i			0.0345295i	−0.999851	+	0.0172648i	$0.994504\pi$
$684$	120.509	−	1053.73i	0.176182	−	1.54054i
$685$	−859.708			−1.25505
$686$			213.008i			0.310507i
$687$	−74.8638	−	83.9135i	−0.108972	−	0.122145i
$688$	105.569			0.153444
$689$		−	464.580i		−	0.674281i
$690$	−28.4919	+	25.4192i	−0.0412926	+	0.0368394i
$691$	−36.8586			−0.0533410			−0.0266705	−	0.999644i	$-0.508490\pi$
$691$	−36.8586			−0.0533410			−0.0266705	+	0.999644i	$0.508490\pi$
$692$		−	851.743i		−	1.23084i
$693$	−103.030	−	11.7829i	−0.148672	−	0.0170027i
$694$	192.972			0.278058
$695$		−	119.159i		−	0.171452i
$696$	507.981	+	569.387i	0.729858	+	0.818085i
$697$	81.8587			0.117444
$698$		−	49.4145i		−	0.0707943i
$699$	−507.785	+	453.023i	−0.726445	+	0.648101i
$700$	−112.400			−0.160571
$701$			278.701i			0.397577i	0.980042	+	0.198788i	$0.0637006\pi$
$701$			278.701i			0.397577i	−0.980042	+	0.198788i	$0.936299\pi$
$702$	44.2848	+	62.7458i	0.0630837	+	0.0893815i
$703$	−552.771			−0.786303
$704$		−	78.9983i		−	0.112213i
$705$	470.068	+	526.891i	0.666763	+	0.747362i
$706$	72.3294			0.102450
$707$			603.396i			0.853459i
$708$	−428.834	+	382.586i	−0.605697	+	0.540375i
$709$	−751.227			−1.05956			−0.529779	−	0.848135i	$-0.677725\pi$
$709$	−751.227			−1.05956			−0.529779	+	0.848135i	$0.677725\pi$
$710$		−	137.318i		−	0.193406i
$711$	−103.138	+	901.839i	−0.145060	+	1.26841i
$712$	446.829			0.627568
$713$			85.0198i			0.119242i
$714$	−77.9090	−	87.3268i	−0.109116	−	0.122306i
$715$	51.9267			0.0726248
$716$		−	1101.42i		−	1.53829i
$717$	−451.785	+	403.062i	−0.630105	+	0.562151i
$718$	−244.411			−0.340405
$719$			1028.57i			1.43056i	0.698839	+	0.715279i	$0.253701\pi$
$719$			1028.57i			1.43056i	−0.698839	+	0.715279i	$0.746299\pi$
$720$	423.614	+	48.4460i	0.588352	+	0.0672861i
$721$	−635.421			−0.881306
$722$			451.989i			0.626024i
$723$	63.9961	+	71.7321i	0.0885147	+	0.0992145i
$724$	−161.547			−0.223132
$725$		−	396.343i		−	0.546679i
$726$	−160.999	+	143.636i	−0.221762	+	0.197846i
$727$	276.064			0.379731			0.189865	−	0.981810i	$-0.439195\pi$
$727$	276.064			0.379731			0.189865	+	0.981810i	$0.439195\pi$
$728$			89.8725i			0.123451i
$729$	244.216	−	686.877i	0.335001	−	0.942218i
$730$	−90.9113			−0.124536
$731$		−	137.726i		−	0.188407i
$732$	299.174	+	335.338i	0.408707	+	0.458112i
$733$	422.867			0.576899			0.288450	−	0.957495i	$-0.406860\pi$
$733$	422.867			0.576899			0.288450	+	0.957495i	$0.406860\pi$
$734$		−	330.939i		−	0.450871i
$735$	−296.785	+	264.778i	−0.403789	+	0.360242i
$736$	−126.974			−0.172518
$737$			92.6915i			0.125769i
$738$	3.59443	−	31.4298i	0.00487050	−	0.0425879i
$739$	1.71966			0.00232701			0.00116351	−	0.999999i	$-0.499630\pi$
$739$	1.71966			0.00232701			0.00116351	+	0.999999i	$0.499630\pi$
$740$		−	253.822i		−	0.343003i
$741$	293.241	+	328.689i	0.395737	+	0.443574i
$742$	273.586			0.368714
$743$		−	823.464i		−	1.10830i	−0.832418	−	0.554148i	$-0.813044\pi$
$743$		−	823.464i		−	1.10830i	0.832418	−	0.554148i	$-0.186956\pi$
$744$	−191.343	+	170.708i	−0.257182	+	0.229446i
$745$	300.832			0.403801
$746$		−	1.35290i		−	0.00181354i
$747$	−510.786	−	58.4154i	−0.683784	−	0.0782000i
$748$	−147.301			−0.196926
$749$		−	148.485i		−	0.198244i
$750$	172.328	+	193.160i	0.229771	+	0.257546i
$751$	−60.3517			−0.0803618			−0.0401809	−	0.999192i	$-0.512793\pi$
$751$	−60.3517			−0.0803618			−0.0401809	+	0.999192i	$0.512793\pi$
$752$			637.649i			0.847938i
$753$	57.9060	−	51.6611i	0.0769003	−	0.0686070i
$754$	−150.053			−0.199010
$755$			1183.49i			1.56753i
$756$	330.016	−	232.919i	0.436529	−	0.308093i
$757$	671.970			0.887675			0.443837	−	0.896107i	$-0.353617\pi$
$757$	671.970			0.887675			0.443837	+	0.896107i	$0.353617\pi$
$758$		−	164.601i		−	0.217152i
$759$	26.5366	+	29.7444i	0.0349626	+	0.0391889i
$760$	−660.508			−0.869089
$761$			374.608i			0.492257i	0.969237	+	0.246129i	$0.0791586\pi$
$761$			374.608i			0.492257i	−0.969237	+	0.246129i	$0.920841\pi$
$762$	−221.311	+	197.444i	−0.290435	+	0.259113i
$763$	−242.747			−0.318148
$764$			215.072i			0.281508i
$765$	63.2026	−	552.646i	0.0826177	−	0.722413i
$766$	296.552			0.387144
$767$		−	238.679i		−	0.311185i
$768$	70.6197	+	79.1564i	0.0919527	+	0.103068i
$769$	1346.66			1.75119			0.875595	−	0.483046i	$-0.160470\pi$
$769$	1346.66			1.75119			0.875595	+	0.483046i	$0.160470\pi$
$770$			30.5791i			0.0397131i
$771$	−415.201	+	370.424i	−0.538523	+	0.480446i
$772$	1070.71			1.38693
$773$			192.648i			0.249221i	0.992206	+	0.124611i	$0.0397681\pi$
$773$			192.648i			0.249221i	−0.992206	+	0.124611i	$0.960232\pi$
$774$	−52.8801	−	6.04756i	−0.0683205	−	0.00781338i
$775$	133.191			0.171860
$776$		−	716.496i		−	0.923320i
$777$	−140.151	−	157.093i	−0.180375	−	0.202179i
$778$	48.9499			0.0629176
$779$		−	181.441i		−	0.232915i
$780$	−150.928	+	134.651i	−0.193497	+	0.172629i
$781$	−143.354			−0.183552
$782$			44.9842i			0.0575245i
$783$	821.315	+	1163.70i	1.04893	+	1.48620i
$784$	−359.173			−0.458128
$785$			809.791i			1.03158i
$786$	−131.534	−	147.435i	−0.167347	−	0.187576i
$787$	314.601			0.399747			0.199874	−	0.979822i	$-0.435947\pi$
$787$	314.601			0.399747			0.199874	+	0.979822i	$0.435947\pi$
$788$		−	291.676i		−	0.370147i
$789$	26.0050	−	23.2005i	0.0329594	−	0.0294049i
$790$	267.664			0.338816
$791$		−	723.901i		−	0.915172i
$792$	−13.6602	+	119.445i	−0.0172477	+	0.150814i
$793$	−186.641			−0.235361
$794$			99.3013i			0.125065i
$795$	865.691	+	970.337i	1.08892	+	1.22055i
$796$	285.270			0.358379
$797$		−	1586.72i		−	1.99087i	−0.0954643	−	0.995433i	$-0.530434\pi$
$797$		−	1586.72i		−	1.99087i	0.0954643	−	0.995433i	$-0.469566\pi$
$798$	−193.561	+	172.686i	−0.242558	+	0.216399i
$799$	831.876			1.04115
$800$			198.916i			0.248645i
$801$	828.667	+	94.7694i	1.03454	+	0.118314i
$802$	−470.178			−0.586257
$803$			94.9076i			0.118191i
$804$	−240.358	−	269.413i	−0.298953	−	0.335091i
$805$	−83.4058			−0.103610
$806$		−	50.4256i		−	0.0625628i
$807$	590.645	−	526.947i	0.731902	−	0.652970i
$808$	699.532			0.865757
$809$			361.771i			0.447183i	0.974683	+	0.223592i	$0.0717782\pi$
$809$			361.771i			0.447183i	−0.974683	+	0.223592i	$0.928222\pi$
$810$	−209.414	−	48.5335i	−0.258536	−	0.0599179i
$811$	535.262			0.660003			0.330001	−	0.943980i	$-0.392951\pi$
$811$	535.262			0.660003			0.330001	+	0.943980i	$0.392951\pi$
$812$			789.216i			0.971940i
$813$	−757.416	−	848.974i	−0.931631	−	1.04425i
$814$	29.6686			0.0364479
$815$		−	83.3652i		−	0.102289i
$816$	374.833	−	334.409i	0.459355	−	0.409815i
$817$	−305.271			−0.373649
$818$			112.571i			0.137618i
$819$	−19.0614	+	166.673i	−0.0232740	+	0.203508i
$820$	83.3144			0.101603
$821$		−	217.894i		−	0.265400i	−0.991156	−	0.132700i	$-0.957635\pi$
$821$		−	217.894i		−	0.265400i	0.991156	−	0.132700i	$-0.0423647\pi$
$822$	260.577	+	292.076i	0.317004	+	0.355324i
$823$	1218.53			1.48060			0.740300	−	0.672276i	$-0.234683\pi$
$823$	1218.53			1.48060			0.740300	+	0.672276i	$0.234683\pi$
$824$			736.660i			0.894005i
$825$	46.5974	−	41.5721i	0.0564816	−	0.0503904i
$826$	140.555			0.170164
$827$			653.177i			0.789815i	0.918721	+	0.394907i	$0.129223\pi$
$827$			653.177i			0.789815i	−0.918721	+	0.394907i	$0.870777\pi$
$828$	−154.260	−	17.6417i	−0.186304	−	0.0213064i
$829$	48.1583			0.0580921			0.0290460	−	0.999578i	$-0.490753\pi$
$829$	48.1583			0.0580921			0.0290460	+	0.999578i	$0.490753\pi$
$830$			151.601i			0.182651i
$831$	−401.817	−	450.390i	−0.483535	−	0.541985i
$832$	−127.797			−0.153602
$833$			468.576i			0.562516i
$834$	−40.4831	+	36.1171i	−0.0485408	+	0.0433059i
$835$	−132.909			−0.159173
$836$			326.494i			0.390543i
$837$	−391.062	+	276.004i	−0.467219	+	0.329754i
$838$	161.658			0.192910
$839$		−	1248.69i		−	1.48831i	−0.668007	−	0.744155i	$-0.732852\pi$
$839$		−	1248.69i		−	1.48831i	0.668007	−	0.744155i	$-0.267148\pi$
$840$	−167.467	−	187.711i	−0.199366	−	0.223465i
$841$	−1941.92			−2.30906
$842$			244.756i			0.290684i
$843$	983.561	−	877.489i	1.16674	−	1.04091i
$844$	−942.921			−1.11720
$845$			622.710i			0.736935i
$846$	36.5278	−	319.401i	0.0431771	−	0.377542i
$847$	−471.301			−0.556436
$848$			1174.31i			1.38480i
$849$	−722.619	−	809.971i	−0.851141	−	0.954029i
$850$	70.4719			0.0829082
$851$			80.9224i			0.0950910i
$852$	416.667	−	371.732i	0.489046	−	0.436305i
$853$	−1330.39			−1.55966			−0.779830	−	0.625991i	$-0.784695\pi$
$853$	−1330.39			−1.55966			−0.779830	+	0.625991i	$0.784695\pi$
$854$		−	109.911i		−	0.128701i
$855$	−1224.95	−	140.089i	−1.43269	−	0.163847i
$856$	−172.142			−0.201100
$857$		−	882.500i		−	1.02976i	−0.857264	−	0.514878i	$-0.827837\pi$
$857$		−	882.500i		−	1.02976i	0.857264	−	0.514878i	$-0.172163\pi$
$858$	−15.7390	−	17.6415i	−0.0183438	−	0.0205612i
$859$	−245.092			−0.285323			−0.142661	−	0.989772i	$-0.545566\pi$
$859$	−245.092			−0.285323			−0.142661	+	0.989772i	$0.545566\pi$
$860$		−	140.175i		−	0.162994i
$861$	51.5640	−	46.0031i	0.0598885	−	0.0534298i
$862$	−438.289			−0.508456
$863$			125.564i			0.145497i	0.997350	+	0.0727484i	$0.0231770\pi$
$863$			125.564i			0.145497i	−0.997350	+	0.0727484i	$0.976823\pi$
$864$	−412.200	−	584.035i	−0.477084	−	0.675966i
$865$	−990.138			−1.14467
$866$			241.435i			0.278793i
$867$	140.913	+	157.947i	0.162529	+	0.182176i
$868$	−265.217			−0.305550
$869$		−	279.431i		−	0.321554i
$870$	313.407	−	279.607i	0.360237	−	0.321388i
$871$	149.949			0.172157
$872$			281.423i			0.322733i
$873$	151.964	−	1328.78i	0.174071	−	1.52209i
$874$	99.7081			0.114082
$875$			565.446i			0.646224i
$876$	−246.105	−	275.854i	−0.280941	−	0.314902i
$877$	−1360.77			−1.55162			−0.775808	−	0.630969i	$-0.782657\pi$
$877$	−1360.77			−1.55162			−0.775808	+	0.630969i	$0.782657\pi$
$878$			202.669i			0.230831i
$879$	272.941	−	243.506i	0.310513	−	0.277026i
$880$	−131.255			−0.149153
$881$			1391.11i			1.57901i	0.613743	+	0.789506i	$0.289663\pi$
$881$			1391.11i			1.57901i	−0.613743	+	0.789506i	$0.710337\pi$
$882$	179.911	+	20.5752i	0.203980	+	0.0233279i
$883$	402.653			0.456006			0.228003	−	0.973660i	$-0.426780\pi$
$883$	402.653			0.456006			0.228003	+	0.973660i	$0.426780\pi$
$884$			238.291i			0.269560i
$885$	444.750	+	498.513i	0.502543	+	0.563291i
$886$	86.4700			0.0975959
$887$			1268.58i			1.43019i	0.699029	+	0.715094i	$0.253616\pi$
$887$			1268.58i			1.43019i	−0.699029	+	0.715094i	$0.746384\pi$
$888$	−182.122	+	162.481i	−0.205092	+	0.182974i
$889$	−647.856			−0.728747
$890$		−	245.947i		−	0.276345i
$891$	−50.6670	+	218.620i	−0.0568653	+	0.245365i
$892$	982.343			1.10128
$893$		−	1843.87i		−	2.06480i
$894$	−91.1820	−	102.204i	−0.101993	−	0.114322i
$895$	−1280.38			−1.43060
$896$		−	515.697i		−	0.575555i
$897$	48.1181	−	42.9288i	0.0536434	−	0.0478582i
$898$	220.345			0.245373
$899$		−	935.205i		−	1.04027i
$900$	−27.6374	+	241.663i	−0.0307083	+	0.268514i
$901$	1532.01			1.70034
$902$			9.73839i			0.0107964i
$903$	−77.3993	−	86.7555i	−0.0857135	−	0.0960747i
$904$	−839.237			−0.928359
$905$			187.796i			0.207510i
$906$	402.076	−	358.714i	0.443793	−	0.395932i
$907$	254.061			0.280111			0.140056	−	0.990144i	$-0.455272\pi$
$907$	254.061			0.280111			0.140056	+	0.990144i	$0.455272\pi$
$908$		−	358.508i		−	0.394833i
$909$	1297.32	+	148.366i	1.42719	+	0.163219i
$910$	49.4684			0.0543609
$911$		−	905.997i		−	0.994509i	−0.867605	−	0.497254i	$-0.834342\pi$
$911$		−	905.997i		−	0.994509i	0.867605	−	0.497254i	$-0.165658\pi$
$912$	−741.223	−	830.823i	−0.812744	−	0.910990i
$913$	158.265			0.173346
$914$		−	468.008i		−	0.512043i
$915$	389.826	−	347.785i	0.426039	−	0.380093i
$916$	−134.842			−0.147207
$917$		−	431.593i		−	0.470657i
$918$	−206.912	+	146.034i	−0.225394	+	0.159079i
$919$	−163.081			−0.177455			−0.0887273	−	0.996056i	$-0.528280\pi$
$919$	−163.081			−0.177455			−0.0887273	+	0.996056i	$0.528280\pi$
$920$			96.6945i			0.105103i
$921$	−799.585	−	896.240i	−0.868170	−	0.973116i
$922$	111.171			0.120576
$923$			231.907i			0.251254i
$924$	−92.7868	+	82.7802i	−0.100419	+	0.0895889i
$925$	126.773			0.137051
$926$		−	233.632i		−	0.252303i
$927$	−156.241	+	1366.18i	−0.168544	+	1.47376i
$928$	1396.69			1.50505
$929$			542.652i			0.584125i	0.956399	+	0.292063i	$0.0943416\pi$
$929$			542.652i			0.584125i	−0.956399	+	0.292063i	$0.905658\pi$
$930$	93.9624	+	105.321i	0.101035	+	0.113248i
$931$	1038.61			1.11558
$932$			815.968i			0.875502i
$933$	−506.318	+	451.714i	−0.542677	+	0.484152i
$934$	242.102			0.259210
$935$			171.235i			0.183139i
$936$	193.229	+	22.0983i	0.206441	+	0.0236093i
$937$	−730.620			−0.779743			−0.389872	−	0.920869i	$-0.627481\pi$
$937$	−730.620			−0.779743			−0.389872	+	0.920869i	$0.627481\pi$
$938$			88.3032i			0.0941399i
$939$	−348.234	−	390.330i	−0.370857	−	0.415687i
$940$	846.669			0.900712
$941$			1175.37i			1.24906i	0.781000	+	0.624531i	$0.214710\pi$
$941$			1175.37i			1.24906i	−0.781000	+	0.624531i	$0.785290\pi$
$942$	275.118	−	245.447i	0.292057	−	0.260560i
$943$	−26.5619			−0.0281674
$944$			603.306i			0.639095i
$945$	−270.764	−	383.638i	−0.286523	−	0.405967i
$946$	16.3846			0.0173199
$947$		−	312.749i		−	0.330253i	−0.986272	−	0.165126i	$-0.947197\pi$
$947$		−	312.749i		−	0.330253i	0.986272	−	0.165126i	$-0.0528032\pi$
$948$	724.590	+	812.180i	0.764336	+	0.856730i
$949$	153.534			0.161785
$950$		−	156.202i		−	0.164423i
$951$	−81.9954	+	73.1526i	−0.0862202	+	0.0769217i
$952$	−296.366			−0.311308
$953$			667.615i			0.700540i	0.936649	+	0.350270i	$0.113910\pi$
$953$			667.615i			0.700540i	−0.936649	+	0.350270i	$0.886090\pi$
$954$	67.2707	−	588.218i	0.0705143	−	0.616580i
$955$	250.018			0.261799
$956$			725.981i			0.759395i
$957$	−291.898	−	327.184i	−0.305014	−	0.341885i
$958$	41.6147			0.0434391
$959$			855.010i			0.891564i
$960$	266.921	−	238.135i	0.278043	−	0.248057i
$961$	−646.723			−0.672969
$962$		−	47.9955i		−	0.0498913i
$963$	−319.246	−	36.5102i	−0.331512	−	0.0379129i
$964$	115.268			0.119572
$965$		−	1244.69i		−	1.28983i
$966$	25.2803	+	28.3362i	0.0261701	+	0.0293335i
$967$	478.742			0.495079			0.247540	−	0.968878i	$-0.420378\pi$
$967$	478.742			0.495079			0.247540	+	0.968878i	$0.420378\pi$
$968$			546.391i			0.564454i
$969$	−1083.89	+	966.998i	−1.11857	+	0.997934i
$970$	−394.380			−0.406577
$971$		−	1246.25i		−	1.28347i	−0.766926	−	0.641736i	$-0.778215\pi$
$971$		−	1246.25i		−	1.28347i	0.766926	−	0.641736i	$-0.221785\pi$
$972$	−419.636	−	766.815i	−0.431724	−	0.788904i
$973$	−118.508			−0.121797
$974$			390.347i			0.400767i
$975$	−67.2519	−	75.3815i	−0.0689764	−	0.0773143i
$976$	471.772			0.483373
$977$		−	411.925i		−	0.421622i	−0.977527	−	0.210811i	$-0.932389\pi$
$977$		−	411.925i		−	0.421622i	0.977527	−	0.210811i	$-0.0676105\pi$
$978$	−28.3224	+	25.2680i	−0.0289595	+	0.0258363i
$979$	−256.759			−0.262266
$980$			476.909i			0.486641i
$981$	−59.6879	+	521.914i	−0.0608440	+	0.532022i
$982$	−515.507			−0.524956
$983$		−	184.275i		−	0.187462i	−0.995598	−	0.0937311i	$-0.970121\pi$
$983$		−	184.275i		−	0.187462i	0.995598	−	0.0937311i	$-0.0298794\pi$
$984$	−53.3325	−	59.7795i	−0.0541997	−	0.0607515i
$985$	−339.069			−0.344232
$986$		−	494.819i		−	0.501845i
$987$	524.011	−	467.499i	0.530913	−	0.473657i
$988$	528.175			0.534590
$989$			44.6899i			0.0451869i
$990$	65.7459	+	7.51894i	0.0664100	+	0.00759489i
$991$	−170.407			−0.171954			−0.0859772	−	0.996297i	$-0.527401\pi$
$991$	−170.407			−0.171954			−0.0859772	+	0.996297i	$0.527401\pi$
$992$			469.359i			0.473145i
$993$	778.169	+	872.236i	0.783655	+	0.878384i
$994$	−136.568			−0.137392
$995$		−	331.622i		−	0.333288i
$996$	−460.005	+	410.396i	−0.461853	+	0.412044i
$997$	191.040			0.191615			0.0958076	−	0.995400i	$-0.469457\pi$
$997$	191.040			0.191615			0.0958076	+	0.995400i	$0.469457\pi$
$998$			362.372i			0.363098i
$999$	−372.216	+	262.703i	−0.372588	+	0.262965i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.3.b.a.47.8	yes	14
3.2	odd	2	inner	69.3.b.a.47.7	✓	14
4.3	odd	2		1104.3.g.b.737.3		14
12.11	even	2		1104.3.g.b.737.4		14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.3.b.a.47.7	✓	14	3.2	odd	2	inner
69.3.b.a.47.8	yes	14	1.1	even	1	trivial
1104.3.g.b.737.3		14	4.3	odd	2
1104.3.g.b.737.4		14	12.11	even	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)

\(f(q)\)	\(=\)	\(q+0.634638i q^{2} +(1.99717 + 2.23859i) q^{3} +3.59723 q^{4} -4.18173i q^{5} +(-1.42070 + 1.26748i) q^{6} -4.15888 q^{7} +4.82149i q^{8} +(-1.02261 + 8.94172i) q^{9} +O(q^{10})\)	\(q+0.634638i q^{2} +(1.99717 + 2.23859i) q^{3} +3.59723 q^{4} -4.18173i q^{5} +(-1.42070 + 1.26748i) q^{6} -4.15888 q^{7} +4.82149i q^{8} +(-1.02261 + 8.94172i) q^{9} +2.65389 q^{10} -2.77055i q^{11} +(7.18430 + 8.05275i) q^{12} -4.48197 q^{13} -2.63938i q^{14} +(9.36120 - 8.35164i) q^{15} +11.3290 q^{16} -14.7798i q^{17} +(-5.67475 - 0.648985i) q^{18} -32.7597 q^{19} -15.0427i q^{20} +(-8.30600 - 9.31004i) q^{21} +1.75830 q^{22} +4.79583i q^{23} +(-10.7934 + 9.62935i) q^{24} +7.51312 q^{25} -2.84443i q^{26} +(-22.0592 + 15.5689i) q^{27} -14.9605 q^{28} -52.7534i q^{29} +(5.30027 + 5.94097i) q^{30} +17.7279 q^{31} +26.4758i q^{32} +(6.20213 - 5.53326i) q^{33} +9.37985 q^{34} +17.3913i q^{35} +(-3.67856 + 32.1654i) q^{36} +16.8735 q^{37} -20.7906i q^{38} +(-8.95127 - 10.0333i) q^{39} +20.1622 q^{40} +5.53854i q^{41} +(5.90851 - 5.27130i) q^{42} +9.31848 q^{43} -9.96631i q^{44} +(37.3919 + 4.27627i) q^{45} -3.04362 q^{46} +56.2845i q^{47} +(22.6260 + 25.3611i) q^{48} -31.7037 q^{49} +4.76811i q^{50} +(33.0861 - 29.5179i) q^{51} -16.1227 q^{52} +103.655i q^{53} +(-9.88064 - 13.9996i) q^{54} -11.5857 q^{55} -20.0520i q^{56} +(-65.4268 - 73.3357i) q^{57} +33.4793 q^{58} +53.2531i q^{59} +(33.6744 - 30.0428i) q^{60} +41.6427 q^{61} +11.2508i q^{62} +(4.25290 - 37.1875i) q^{63} +28.5136 q^{64} +18.7424i q^{65} +(3.51162 + 3.93611i) q^{66} -33.4560 q^{67} -53.1666i q^{68} +(-10.7359 + 9.57810i) q^{69} -11.0372 q^{70} -51.7423i q^{71} +(-43.1124 - 4.93049i) q^{72} -34.2559 q^{73} +10.7086i q^{74} +(15.0050 + 16.8188i) q^{75} -117.844 q^{76} +11.5224i q^{77} +(6.36752 - 5.68082i) q^{78} +100.858 q^{79} -47.3750i q^{80} +(-78.9086 - 18.2877i) q^{81} -3.51497 q^{82} +57.1240i q^{83} +(-29.8786 - 33.4904i) q^{84} -61.8054 q^{85} +5.91386i q^{86} +(118.093 - 105.358i) q^{87} +13.3582 q^{88} -92.6743i q^{89} +(-2.71388 + 23.7303i) q^{90} +18.6400 q^{91} +17.2517i q^{92} +(35.4056 + 39.6855i) q^{93} -35.7203 q^{94} +136.992i q^{95} +(-59.2686 + 52.8768i) q^{96} -148.605 q^{97} -20.1204i q^{98} +(24.7735 + 2.83318i) 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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