LMFDB - Embedded newform 69.3.b.a.47.14

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			47.14
Root			$3.86310i$ of defining polynomial
Character	$\chi$	$=$	69.47
Dual form			69.3.b.a.47.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.86310i q^{2} +(-2.26727 - 1.96456i) q^{3} -10.9236 q^{4} +1.45241i q^{5} +(7.58928 - 8.75871i) q^{6} -6.75781 q^{7} -26.7464i q^{8} +(1.28104 + 8.90836i) q^{9} +O(q^{10})$	\(q+3.86310i q^{2} +(-2.26727 - 1.96456i) q^{3} -10.9236 q^{4} +1.45241i q^{5} +(7.58928 - 8.75871i) q^{6} -6.75781 q^{7} -26.7464i q^{8} +(1.28104 + 8.90836i) q^{9} -5.61080 q^{10} +15.0621i q^{11} +(24.7667 + 21.4599i) q^{12} -2.84541 q^{13} -26.1061i q^{14} +(2.85333 - 3.29300i) q^{15} +59.6300 q^{16} +7.16409i q^{17} +(-34.4139 + 4.94881i) q^{18} -21.2472 q^{19} -15.8655i q^{20} +(15.3218 + 13.2761i) q^{21} -58.1863 q^{22} +4.79583i q^{23} +(-52.5449 + 60.6415i) q^{24} +22.8905 q^{25} -10.9921i q^{26} +(14.5965 - 22.7144i) q^{27} +73.8194 q^{28} +17.3958i q^{29} +(12.7212 + 11.0227i) q^{30} -39.8708 q^{31} +123.371i q^{32} +(29.5902 - 34.1498i) q^{33} -27.6756 q^{34} -9.81509i q^{35} +(-13.9936 - 97.3111i) q^{36} +52.7869 q^{37} -82.0803i q^{38} +(6.45133 + 5.58997i) q^{39} +38.8467 q^{40} -0.0221360i q^{41} +(-51.2869 + 59.1897i) q^{42} +43.5885 q^{43} -164.531i q^{44} +(-12.9386 + 1.86060i) q^{45} -18.5268 q^{46} -5.52999i q^{47} +(-135.197 - 117.146i) q^{48} -3.33198 q^{49} +88.4284i q^{50} +(14.0742 - 16.2429i) q^{51} +31.0821 q^{52} +33.6486i q^{53} +(87.7479 + 56.3878i) q^{54} -21.8762 q^{55} +180.747i q^{56} +(48.1733 + 41.7414i) q^{57} -67.2018 q^{58} +15.3715i q^{59} +(-31.1686 + 35.9713i) q^{60} -72.5473 q^{61} -154.025i q^{62} +(-8.65705 - 60.2010i) q^{63} -238.075 q^{64} -4.13270i q^{65} +(131.924 + 114.310i) q^{66} -27.8923 q^{67} -78.2574i q^{68} +(9.42168 - 10.8735i) q^{69} +37.9167 q^{70} -8.13234i q^{71} +(238.267 - 34.2634i) q^{72} -48.5809 q^{73} +203.921i q^{74} +(-51.8990 - 44.9697i) q^{75} +232.096 q^{76} -101.787i q^{77} +(-21.5946 + 24.9221i) q^{78} -80.0237 q^{79} +86.6071i q^{80} +(-77.7179 + 22.8240i) q^{81} +0.0855135 q^{82} -23.8409i q^{83} +(-167.369 - 145.022i) q^{84} -10.4052 q^{85} +168.387i q^{86} +(34.1750 - 39.4410i) q^{87} +402.856 q^{88} +99.7735i q^{89} +(-7.18768 - 49.9830i) q^{90} +19.2288 q^{91} -52.3876i q^{92} +(90.3978 + 78.3283i) q^{93} +21.3629 q^{94} -30.8596i q^{95} +(242.369 - 279.716i) q^{96} +51.7420 q^{97} -12.8718i q^{98} +(-134.178 + 19.2952i) 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 $

$2$			3.86310i			1.93155i	0.259378	+	0.965776i	$0.416482\pi$
$2$			3.86310i			1.93155i	−0.259378	+	0.965776i	$0.583518\pi$
$3$	−2.26727	−	1.96456i	−0.755757	−	0.654852i
$3$	−2.26727	−	1.96456i	−0.755757	−	0.654852i
$4$	−10.9236			−2.73089
$5$			1.45241i			0.290481i	0.989396	+	0.145241i	$0.0463957\pi$
$5$			1.45241i			0.290481i	−0.989396	+	0.145241i	$0.953604\pi$
$6$	7.58928	−	8.75871i	1.26488	−	1.45978i
$7$	−6.75781			−0.965402			−0.482701	−	0.875785i	$-0.660344\pi$
$7$	−6.75781			−0.965402			−0.482701	+	0.875785i	$0.660344\pi$
$8$		−	26.7464i		−	3.34331i
$9$	1.28104	+	8.90836i	0.142338	+	0.989818i
$10$	−5.61080			−0.561080
$11$			15.0621i			1.36928i	0.728883	+	0.684639i	$0.240040\pi$
$11$			15.0621i			1.36928i	−0.728883	+	0.684639i	$0.759960\pi$
$12$	24.7667	+	21.4599i	2.06389	+	1.78833i
$13$	−2.84541			−0.218878			−0.109439	−	0.993994i	$-0.534905\pi$
$13$	−2.84541			−0.218878			−0.109439	+	0.993994i	$0.534905\pi$
$14$		−	26.1061i		−	1.86472i
$15$	2.85333	−	3.29300i	0.190222	−	0.219533i
$16$	59.6300			3.72688
$17$			7.16409i			0.421417i	0.977549	+	0.210708i	$0.0675771\pi$
$17$			7.16409i			0.421417i	−0.977549	+	0.210708i	$0.932423\pi$
$18$	−34.4139	+	4.94881i	−1.91188	+	0.274934i
$19$	−21.2472			−1.11828			−0.559138	−	0.829075i	$-0.688868\pi$
$19$	−21.2472			−1.11828			−0.559138	+	0.829075i	$0.688868\pi$
$20$		−	15.8655i		−	0.793273i
$21$	15.3218	+	13.2761i	0.729609	+	0.632195i
$22$	−58.1863			−2.64483
$23$			4.79583i			0.208514i
$23$			4.79583i			0.208514i
$24$	−52.5449	+	60.6415i	−2.18937	+	2.52673i
$25$	22.8905			0.915621
$26$		−	10.9921i		−	0.422774i
$27$	14.5965	−	22.7144i	0.540611	−	0.841273i
$28$	73.8194			2.63641
$29$			17.3958i			0.599855i	0.953962	+	0.299928i	$0.0969625\pi$
$29$			17.3958i			0.599855i	−0.953962	+	0.299928i	$0.903038\pi$
$30$	12.7212	+	11.0227i	0.424040	+	0.367424i
$31$	−39.8708			−1.28615			−0.643077	−	0.765802i	$-0.722342\pi$
$31$	−39.8708			−1.28615			−0.643077	+	0.765802i	$0.722342\pi$
$32$			123.371i			3.85535i
$33$	29.5902	−	34.1498i	0.896674	−	1.03484i
$34$	−27.6756			−0.813988
$35$		−	9.81509i		−	0.280431i
$36$	−13.9936	−	97.3111i	−0.388710	−	2.70309i
$37$	52.7869			1.42667			0.713336	−	0.700822i	$-0.247183\pi$
$37$	52.7869			1.42667			0.713336	+	0.700822i	$0.247183\pi$
$38$		−	82.0803i		−	2.16001i
$39$	6.45133	+	5.58997i	0.165419	+	0.143333i
$40$	38.8467			0.971168
$41$		−	0.0221360i		−	0.000539902i	−1.00000		0.000269951i	$-0.999914\pi$
$41$		−	0.0221360i		−	0.000539902i	1.00000		0.000269951i	$-8.59280e-5\pi$
$42$	−51.2869	+	59.1897i	−1.22112	+	1.40928i
$43$	43.5885			1.01369			0.506843	−	0.862038i	$-0.330812\pi$
$43$	43.5885			1.01369			0.506843	+	0.862038i	$0.330812\pi$
$44$		−	164.531i		−	3.73935i
$45$	−12.9386	+	1.86060i	−0.287524	+	0.0413466i
$46$	−18.5268			−0.402756
$47$		−	5.52999i		−	0.117659i	−0.998268	−	0.0588297i	$-0.981263\pi$
$47$		−	5.52999i		−	0.117659i	0.998268	−	0.0588297i	$-0.0187369\pi$
$48$	−135.197	−	117.146i	−2.81661	−	2.44055i
$49$	−3.33198			−0.0679996
$50$			88.4284i			1.76857i
$51$	14.0742	−	16.2429i	0.275966	−	0.318489i
$52$	31.0821			0.597732
$53$			33.6486i			0.634880i	0.948278	+	0.317440i	$0.102823\pi$
$53$			33.6486i			0.634880i	−0.948278	+	0.317440i	$0.897177\pi$
$54$	87.7479	+	56.3878i	1.62496	+	1.04422i
$55$	−21.8762			−0.397750
$56$			180.747i			3.22763i
$57$	48.1733	+	41.7414i	0.845145	+	0.732305i
$58$	−67.2018			−1.15865
$59$			15.3715i			0.260535i	0.991479	+	0.130267i	$0.0415835\pi$
$59$			15.3715i			0.260535i	−0.991479	+	0.130267i	$0.958416\pi$
$60$	−31.1686	+	35.9713i	−0.519476	+	0.599522i
$61$	−72.5473			−1.18930			−0.594650	−	0.803985i	$-0.702709\pi$
$61$	−72.5473			−1.18930			−0.594650	+	0.803985i	$0.702709\pi$
$62$		−	154.025i		−	2.48427i
$63$	−8.65705	−	60.2010i	−0.137414	−	0.955572i
$64$	−238.075			−3.71993
$65$		−	4.13270i		−	0.0635800i
$66$	131.924	+	114.310i	1.99885	+	1.73197i
$67$	−27.8923			−0.416303			−0.208152	−	0.978097i	$-0.566745\pi$
$67$	−27.8923			−0.416303			−0.208152	+	0.978097i	$0.566745\pi$
$68$		−	78.2574i		−	1.15084i
$69$	9.42168	−	10.8735i	0.136546	−	0.157586i
$70$	37.9167			0.541667
$71$		−	8.13234i		−	0.114540i	−0.998359	−	0.0572700i	$-0.981760\pi$
$71$		−	8.13234i		−	0.114540i	0.998359	−	0.0572700i	$-0.0182396\pi$
$72$	238.267	−	34.2634i	3.30926	−	0.475880i
$73$	−48.5809			−0.665492			−0.332746	−	0.943017i	$-0.607975\pi$
$73$	−48.5809			−0.665492			−0.332746	+	0.943017i	$0.607975\pi$
$74$			203.921i			2.75569i
$75$	−51.8990	−	44.9697i	−0.691987	−	0.599596i
$76$	232.096			3.05389
$77$		−	101.787i		−	1.32190i
$78$	−21.5946	+	24.9221i	−0.276854	+	0.319515i
$79$	−80.0237			−1.01296			−0.506479	−	0.862252i	$-0.669053\pi$
$79$	−80.0237			−1.01296			−0.506479	+	0.862252i	$0.669053\pi$
$80$			86.6071i			1.08259i
$81$	−77.7179	+	22.8240i	−0.959480	+	0.281778i
$82$	0.0855135			0.00104285
$83$		−	23.8409i		−	0.287239i	−0.989633	−	0.143620i	$-0.954126\pi$
$83$		−	23.8409i		−	0.287239i	0.989633	−	0.143620i	$-0.0458742\pi$
$84$	−167.369	−	145.022i	−1.99248	−	1.72646i
$85$	−10.4052			−0.122414
$86$			168.387i			1.95799i
$87$	34.1750	−	39.4410i	0.392816	−	0.453345i
$88$	402.856			4.57791
$89$			99.7735i			1.12105i	0.828137	+	0.560525i	$0.189401\pi$
$89$			99.7735i			1.12105i	−0.828137	+	0.560525i	$0.810599\pi$
$90$	−7.18768	−	49.9830i	−0.0798631	−	0.555367i
$91$	19.2288			0.211305
$92$		−	52.3876i		−	0.569430i
$93$	90.3978	+	78.3283i	0.972020	+	0.842240i
$94$	21.3629			0.227265
$95$		−	30.8596i		−	0.324838i
$96$	242.369	−	279.716i	2.52468	−	2.91371i
$97$	51.7420			0.533423			0.266712	−	0.963776i	$-0.414063\pi$
$97$	51.7420			0.533423			0.266712	+	0.963776i	$0.414063\pi$
$98$		−	12.8718i		−	0.131345i
$99$	−134.178	+	19.2952i	−1.35534	+	0.194901i
$100$	−250.046			−2.50046
$101$		−	38.4216i		−	0.380412i	−0.981744	−	0.190206i	$-0.939084\pi$
$101$		−	38.4216i		−	0.380412i	0.981744	−	0.190206i	$-0.0609156\pi$
$102$	62.7481	+	54.3703i	0.615178	+	0.533042i
$103$	44.7485			0.434451			0.217226	−	0.976121i	$-0.430299\pi$
$103$	44.7485			0.434451			0.217226	+	0.976121i	$0.430299\pi$
$104$			76.1047i			0.731776i
$105$	−19.2823	+	22.2535i	−0.183641	+	0.211938i
$106$	−129.988			−1.22630
$107$			50.6200i			0.473084i	0.971621	+	0.236542i	$0.0760141\pi$
$107$			50.6200i			0.473084i	−0.971621	+	0.236542i	$0.923986\pi$
$108$	−159.446	+	248.122i	−1.47635	+	2.29742i
$109$	121.109			1.11109			0.555544	−	0.831487i	$-0.312510\pi$
$109$	121.109			1.11109			0.555544	+	0.831487i	$0.312510\pi$
$110$		−	84.5101i		−	0.768274i
$111$	−119.682	−	103.703i	−1.07822	−	0.934259i
$112$	−402.968			−3.59793
$113$			207.417i			1.83555i	0.397105	+	0.917773i	$0.370015\pi$
$113$			207.417i			1.83555i	−0.397105	+	0.917773i	$0.629985\pi$
$114$	−161.251	+	186.098i	−1.41448	+	1.63244i
$115$	−6.96550			−0.0605696
$116$		−	190.024i		−	1.63814i
$117$	−3.64510	−	25.3480i	−0.0311547	−	0.216649i
$118$	−59.3818			−0.503236
$119$		−	48.4135i		−	0.406837i
$120$	−88.0761	−	76.3165i	−0.733967	−	0.635971i
$121$	−105.865			−0.874921
$122$		−	280.258i		−	2.29719i
$123$	−0.0434873	+	0.0501883i	−0.000353556	+	0.000408035i
$124$	435.531			3.51235
$125$			69.5565i			0.556452i
$126$	232.563	−	33.4431i	1.84574	−	0.265421i
$127$	−8.08377			−0.0636517			−0.0318259	−	0.999493i	$-0.510132\pi$
$127$	−8.08377			−0.0636517			−0.0318259	+	0.999493i	$0.510132\pi$
$128$		−	426.225i		−	3.32988i
$129$	−98.8270	−	85.6321i	−0.766101	−	0.663814i
$130$	15.9650			0.122808
$131$			165.371i			1.26237i	0.775631	+	0.631187i	$0.217432\pi$
$131$			165.371i			1.26237i	−0.775631	+	0.631187i	$0.782568\pi$
$132$	−323.231	+	373.037i	−2.44872	+	2.82604i
$133$	143.585			1.07959
$134$		−	107.751i		−	0.804111i
$135$	32.9905	+	21.2001i	0.244374	+	0.157037i
$136$	191.614			1.40893
$137$		−	200.517i		−	1.46363i	−0.681503	−	0.731815i	$-0.738674\pi$
$137$		−	200.517i		−	1.46363i	0.681503	−	0.731815i	$-0.261326\pi$
$138$	42.0053	+	36.3969i	0.304386	+	0.263746i
$139$	165.698			1.19207			0.596036	−	0.802958i	$-0.296741\pi$
$139$	165.698			1.19207			0.596036	+	0.802958i	$0.296741\pi$
$140$			107.216i			0.765827i
$141$	−10.8640	+	12.5380i	−0.0770495	+	0.0889220i
$142$	31.4161			0.221240
$143$		−	42.8578i		−	0.299705i
$144$	76.3887	+	531.206i	0.530477	+	3.68893i
$145$	−25.2658			−0.174247
$146$		−	187.673i		−	1.28543i
$147$	7.55451	+	6.54586i	0.0513912	+	0.0445297i
$148$	−576.621			−3.89609
$149$		−	33.9574i		−	0.227902i	−0.993486	−	0.113951i	$-0.963649\pi$
$149$		−	33.9574i		−	0.227902i	0.993486	−	0.113951i	$-0.0363507\pi$
$150$	173.723	−	200.491i	1.15815	−	1.33661i
$151$	82.4382			0.545948			0.272974	−	0.962021i	$-0.411993\pi$
$151$	82.4382			0.545948			0.272974	+	0.962021i	$0.411993\pi$
$152$			568.288i			3.73874i
$153$	−63.8203	+	9.17751i	−0.417126	+	0.0599837i
$154$	393.212			2.55332
$155$		−	57.9086i		−	0.373604i
$156$	−70.4715	−	61.0624i	−0.451740	−	0.391426i
$157$	79.3839			0.505630			0.252815	−	0.967515i	$-0.418644\pi$
$157$	79.3839			0.505630			0.252815	+	0.967515i	$0.418644\pi$
$158$		−	309.140i		−	1.95658i
$159$	66.1046	−	76.2906i	0.415752	−	0.479815i
$160$	−179.185			−1.11991
$161$		−	32.4093i		−	0.201300i
$162$	−88.1715	−	300.232i	−0.544269	−	1.85328i
$163$	141.690			0.869262			0.434631	−	0.900609i	$-0.356879\pi$
$163$	141.690			0.869262			0.434631	+	0.900609i	$0.356879\pi$
$164$			0.241804i			0.00147441i
$165$	49.5994	+	42.9771i	0.300602	+	0.260467i
$166$	92.0998			0.554818
$167$		−	283.496i		−	1.69758i	−0.528731	−	0.848789i	$-0.677332\pi$
$167$		−	283.496i		−	1.69758i	0.528731	−	0.848789i	$-0.322668\pi$
$168$	355.088	−	409.804i	2.11362	−	2.43931i
$169$	−160.904			−0.952092
$170$		−	40.1962i		−	0.236448i
$171$	−27.2186	−	189.278i	−0.159173	−	1.10689i
$172$	−476.142			−2.76827
$173$			107.869i			0.623519i	0.950161	+	0.311760i	$0.100918\pi$
$173$			107.869i			0.623519i	−0.950161	+	0.311760i	$0.899082\pi$
$174$	152.365	+	132.022i	0.875659	+	0.758745i
$175$	−154.690			−0.883942
$176$			898.151i			5.10313i
$177$	30.1982	−	34.8515i	0.170612	−	0.196901i
$178$	−385.435			−2.16537
$179$			165.365i			0.923827i	0.886925	+	0.461913i	$0.152837\pi$
$179$			165.365i			0.923827i	−0.886925	+	0.461913i	$0.847163\pi$
$180$	141.335	−	20.3244i	0.785196	−	0.112913i
$181$	337.645			1.86544			0.932721	−	0.360598i	$-0.117427\pi$
$181$	337.645			1.86544			0.932721	+	0.360598i	$0.117427\pi$
$182$			74.2827i			0.408147i
$183$	164.484	+	142.523i	0.898822	+	0.778815i
$184$	128.271			0.697127
$185$			76.6680i			0.414422i
$186$	−302.590	+	349.216i	−1.62683	+	1.87751i
$187$	−107.906			−0.577037
$188$			60.4072i			0.321315i
$189$	−98.6404	+	153.499i	−0.521907	+	0.812166i
$190$	119.214			0.627442
$191$			233.659i			1.22335i	0.791110	+	0.611674i	$0.209503\pi$
$191$			233.659i			1.22335i	−0.791110	+	0.611674i	$0.790497\pi$
$192$	539.781	+	467.712i	2.81136	+	2.43600i
$193$	−165.108			−0.855481			−0.427740	−	0.903902i	$-0.640690\pi$
$193$	−165.108			−0.855481			−0.427740	+	0.903902i	$0.640690\pi$
$194$			199.885i			1.03033i
$195$	−8.11892	+	9.36995i	−0.0416355	+	0.0480510i
$196$	36.3971			0.185700
$197$			58.2646i			0.295759i	0.989005	+	0.147880i	$0.0472448\pi$
$197$			58.2646i			0.295759i	−0.989005	+	0.147880i	$0.952755\pi$
$198$	−74.5392	−	518.344i	−0.376460	−	2.61790i
$199$	95.0636			0.477707			0.238853	−	0.971056i	$-0.423228\pi$
$199$	95.0636			0.477707			0.238853	+	0.971056i	$0.423228\pi$
$200$		−	612.240i		−	3.06120i
$201$	63.2394	+	54.7960i	0.314624	+	0.272617i
$202$	148.427			0.734786
$203$		−	117.558i		−	0.579101i
$204$	−153.741	+	177.431i	−0.753632	+	0.869758i
$205$	0.0321504			0.000156831
$206$			172.868i			0.839165i
$207$	−42.7230	+	6.14367i	−0.206391	+	0.0296796i
$208$	−169.672			−0.815731
$209$		−	320.027i		−	1.53123i
$210$	−85.9675	−	74.4895i	−0.409369	−	0.354712i
$211$	25.3868			0.120317			0.0601583	−	0.998189i	$-0.480839\pi$
$211$	25.3868			0.120317			0.0601583	+	0.998189i	$0.480839\pi$
$212$		−	367.563i		−	1.73379i
$213$	−15.9764	+	18.4382i	−0.0750067	+	0.0865645i
$214$	−195.550			−0.913787
$215$			63.3083i			0.294457i
$216$	−607.528	−	390.404i	−2.81263	−	1.80743i
$217$	269.439			1.24165
$218$			467.855i			2.14612i
$219$	110.146	+	95.4399i	0.502950	+	0.435799i
$220$	238.966			1.08621
$221$		−	20.3848i		−	0.0922389i
$222$	400.614	−	462.345i	1.80457	−	2.08263i
$223$	−391.673			−1.75638			−0.878190	−	0.478312i	$-0.841249\pi$
$223$	−391.673			−1.75638			−0.878190	+	0.478312i	$0.841249\pi$
$224$		−	833.719i		−	3.72196i
$225$	29.3238	+	203.917i	0.130328	+	0.906298i
$226$	−801.272			−3.54545
$227$			418.524i			1.84372i	0.387525	+	0.921859i	$0.373330\pi$
$227$			418.524i			1.84372i	−0.387525	+	0.921859i	$0.626670\pi$
$228$	−526.224	−	455.965i	−2.30800	−	1.99984i
$229$	−17.3507			−0.0757671			−0.0378836	−	0.999282i	$-0.512062\pi$
$229$	−17.3507			−0.0757671			−0.0378836	+	0.999282i	$0.512062\pi$
$230$		−	26.9084i		−	0.116993i
$231$	−199.965	+	230.778i	−0.865650	+	0.999038i
$232$	465.276			2.00550
$233$		−	305.778i		−	1.31235i	−0.754607	−	0.656177i	$-0.772173\pi$
$233$		−	305.778i		−	1.31235i	0.754607	−	0.656177i	$-0.227827\pi$
$234$	97.9219	−	14.0814i	0.418470	−	0.0601769i
$235$	8.03180			0.0341779
$236$		−	167.912i		−	0.711492i
$237$	181.436	+	157.211i	0.765551	+	0.663338i
$238$	187.027			0.785826
$239$		−	441.422i		−	1.84695i	−0.383656	−	0.923476i	$-0.625335\pi$
$239$		−	441.422i		−	1.84695i	0.383656	−	0.923476i	$-0.374665\pi$
$240$	170.144	−	196.362i	0.708935	−	0.818174i
$241$	92.1992			0.382569			0.191285	−	0.981535i	$-0.438735\pi$
$241$	92.1992			0.382569			0.191285	+	0.981535i	$0.438735\pi$
$242$		−	408.969i		−	1.68996i
$243$	221.047	+	100.933i	0.909657	+	0.415361i
$244$	792.475			3.24785
$245$		−	4.83939i		−	0.0197526i
$246$	−0.193882	−	0.167996i	−0.000788140	−	0.000682911i
$247$	60.4572			0.244766
$248$			1066.40i			4.30000i
$249$	−46.8367	+	54.0538i	−0.188099	+	0.217083i
$250$	−268.704			−1.07482
$251$			303.130i			1.20769i	0.797101	+	0.603846i	$0.206366\pi$
$251$			303.130i			1.20769i	−0.797101	+	0.603846i	$0.793634\pi$
$252$	94.5659	+	657.610i	0.375262	+	2.60956i
$253$	−72.2351			−0.285514
$254$		−	31.2284i		−	0.122947i
$255$	23.5913	+	20.4415i	0.0925151	+	0.0801629i
$256$	694.249			2.71191
$257$		−	38.5727i		−	0.150088i	−0.997180	−	0.0750442i	$-0.976090\pi$
$257$		−	38.5727i		−	0.150088i	0.997180	−	0.0750442i	$-0.0239098\pi$
$258$	330.805	−	381.779i	1.28219	−	1.47976i
$259$	−356.724			−1.37731
$260$			45.1438i			0.173630i
$261$	−154.968	+	22.2848i	−0.593747	+	0.0853823i
$262$	−638.845			−2.43834
$263$		−	190.216i		−	0.723256i	−0.932322	−	0.361628i	$-0.882221\pi$
$263$		−	190.216i		−	0.723256i	0.932322	−	0.361628i	$-0.117779\pi$
$264$	−913.385	−	791.434i	−3.45979	−	2.99786i
$265$	−48.8715			−0.184421
$266$			554.683i			2.08527i
$267$	196.011	−	226.214i	0.734122	−	0.847242i
$268$	304.683			1.13688
$269$		−	134.557i		−	0.500213i	−0.968218	−	0.250107i	$-0.919534\pi$
$269$		−	134.557i		−	0.500213i	0.968218	−	0.250107i	$-0.0804656\pi$
$270$	−81.8980	+	127.446i	−0.303326	+	0.472021i
$271$	−417.174			−1.53939			−0.769694	−	0.638413i	$-0.779591\pi$
$271$	−417.174			−1.53939			−0.769694	+	0.638413i	$0.779591\pi$
$272$			427.195i			1.57057i
$273$	−43.5969	−	37.7760i	−0.159695	−	0.138374i
$274$	774.619			2.82708
$275$			344.778i			1.25374i
$276$	−102.918	+	118.777i	−0.372892	+	0.430351i
$277$	−244.964			−0.884346			−0.442173	−	0.896930i	$-0.645792\pi$
$277$	−244.964			−0.884346			−0.442173	+	0.896930i	$0.645792\pi$
$278$			640.109i			2.30255i
$279$	−51.0762	−	355.183i	−0.183069	−	1.27306i
$280$	−262.519			−0.937567
$281$			213.400i			0.759432i	0.925103	+	0.379716i	$0.123978\pi$
$281$			213.400i			0.759432i	−0.925103	+	0.379716i	$0.876022\pi$
$282$	−48.4356	−	41.9687i	−0.171757	−	0.148825i
$283$	274.964			0.971604			0.485802	−	0.874069i	$-0.338528\pi$
$283$	274.964			0.971604			0.485802	+	0.874069i	$0.338528\pi$
$284$			88.8342i			0.312796i
$285$	−60.6255	+	69.9672i	−0.212721	+	0.245499i
$286$	165.564			0.578895
$287$			0.149591i			0.000521222i
$288$	−1099.03	+	158.044i	−3.81609	+	0.548763i
$289$	237.676			0.822408
$290$		−	97.6043i		−	0.336567i
$291$	−117.313	−	101.650i	−0.403138	−	0.349313i
$292$	530.677			1.81739
$293$		−	289.283i		−	0.987314i	−0.869657	−	0.493657i	$-0.835660\pi$
$293$		−	289.283i		−	0.987314i	0.869657	−	0.493657i	$-0.164340\pi$
$294$	−25.2873	+	29.1838i	−0.0860113	+	0.0992647i
$295$	−22.3257			−0.0756804
$296$		−	1411.86i		−	4.76980i
$297$	342.125	+	219.853i	1.15194	+	0.740247i
$298$	131.181			0.440205
$299$		−	13.6461i		−	0.0456392i
$300$	566.922	+	491.229i	1.88974	+	1.63743i
$301$	−294.563			−0.978615
$302$			318.467i			1.05453i
$303$	−75.4814	+	87.1123i	−0.249114	+	0.287499i
$304$	−1266.97			−4.16767
$305$		−	105.368i		−	0.345469i
$306$	−35.4537	−	246.544i	−0.115862	−	0.805700i
$307$	347.529			1.13202			0.566008	−	0.824400i	$-0.308487\pi$
$307$	347.529			1.13202			0.566008	+	0.824400i	$0.308487\pi$
$308$			1111.87i			3.60997i
$309$	−101.457	−	87.9109i	−0.328340	−	0.284501i
$310$	223.707			0.721635
$311$			203.389i			0.653983i	0.945027	+	0.326991i	$0.106035\pi$
$311$			203.389i			0.653983i	−0.945027	+	0.326991i	$0.893965\pi$
$312$	149.512	−	172.550i	0.479205	−	0.553045i
$313$	−558.607			−1.78469			−0.892344	−	0.451357i	$-0.850940\pi$
$313$	−558.607			−1.78469			−0.892344	+	0.451357i	$0.850940\pi$
$314$			306.668i			0.976650i
$315$	87.4364	−	12.5736i	0.277576	−	0.0399161i
$316$	874.144			2.76628
$317$		−	316.666i		−	0.998946i	−0.866330	−	0.499473i	$-0.833527\pi$
$317$		−	316.666i		−	0.998946i	0.866330	−	0.499473i	$-0.166473\pi$
$318$	294.719	+	255.369i	0.926788	+	0.803047i
$319$	−262.016			−0.821368
$320$		−	345.782i		−	1.08057i
$321$	99.4458	−	114.769i	0.309800	−	0.357537i
$322$	125.201			0.388822
$323$		−	152.217i		−	0.471260i
$324$	848.956	−	249.320i	2.62023	−	0.769505i
$325$	−65.1330			−0.200409
$326$			547.362i			1.67902i
$327$	−274.586	−	237.924i	−0.839712	−	0.727597i
$328$	−0.592058			−0.00180506
$329$			37.3707i			0.113589i
$330$	−166.025	+	191.607i	−0.503106	+	0.580629i
$331$	−21.3456			−0.0644882			−0.0322441	−	0.999480i	$-0.510265\pi$
$331$	−21.3456			−0.0644882			−0.0322441	+	0.999480i	$0.510265\pi$
$332$			260.427i			0.784420i
$333$	67.6223	+	470.245i	0.203070	+	1.41215i
$334$	1095.17			3.27896
$335$		−	40.5110i		−	0.120928i
$336$	913.639	+	791.654i	2.71916	+	2.35611i
$337$	−232.492			−0.689888			−0.344944	−	0.938623i	$-0.612102\pi$
$337$	−232.492			−0.689888			−0.344944	+	0.938623i	$0.612102\pi$
$338$		−	621.587i		−	1.83902i
$339$	407.482	−	470.270i	1.20201	−	1.38723i
$340$	113.662			0.334299
$341$		−	600.535i		−	1.76110i
$342$	731.201	−	105.148i	2.13801	−	0.307452i
$343$	353.650			1.03105
$344$		−	1165.84i		−	3.38906i
$345$	15.7927	+	13.6841i	0.0457759	+	0.0396641i
$346$	−416.708			−1.20436
$347$			438.068i			1.26244i	0.775603	+	0.631222i	$0.217446\pi$
$347$			438.068i			1.26244i	−0.775603	+	0.631222i	$0.782554\pi$
$348$	−373.313	+	430.836i	−1.07274	+	1.23804i
$349$	−35.1699			−0.100773			−0.0503866	−	0.998730i	$-0.516045\pi$
$349$	−35.1699			−0.100773			−0.0503866	+	0.998730i	$0.516045\pi$
$350$		−	597.583i		−	1.70738i
$351$	−41.5331	+	64.6318i	−0.118328	+	0.184136i
$352$	−1858.22			−5.27904
$353$		−	673.678i		−	1.90844i	−0.299111	−	0.954218i	$-0.596690\pi$
$353$		−	673.678i		−	1.90844i	0.299111	−	0.954218i	$-0.403310\pi$
$354$	134.635	+	116.659i	0.380324	+	0.329545i
$355$	11.8115			0.0332717
$356$		−	1089.88i		−	3.06147i
$357$	−95.1111	+	109.767i	−0.266418	+	0.307470i
$358$	−638.822			−1.78442
$359$		−	42.9978i		−	0.119771i	−0.998205	−	0.0598855i	$-0.980926\pi$
$359$		−	42.9978i		−	0.119771i	0.998205	−	0.0598855i	$-0.0190736\pi$
$360$	49.7644	+	346.061i	0.138234	+	0.961280i
$361$	90.4449			0.250540
$362$			1304.36i			3.60320i
$363$	240.026	+	207.979i	0.661228	+	0.572944i
$364$	−210.047			−0.577052
$365$		−	70.5592i		−	0.193313i
$366$	−550.581	+	635.420i	−1.50432	+	1.73612i
$367$	667.975			1.82009			0.910047	−	0.414504i	$-0.136045\pi$
$367$	667.975			1.82009			0.910047	+	0.414504i	$0.136045\pi$
$368$			285.976i			0.777107i
$369$	0.197195	−	0.0283571i	0.000534404	−	7.68486e-5i
$370$	−296.177			−0.800477
$371$		−	227.391i		−	0.612914i
$372$	−987.467	−	855.624i	−2.65448	−	2.30007i
$373$	−326.813			−0.876175			−0.438088	−	0.898932i	$-0.644344\pi$
$373$	−326.813			−0.876175			−0.438088	+	0.898932i	$0.644344\pi$
$374$		−	416.851i		−	1.11458i
$375$	136.648	−	157.704i	0.364394	−	0.420543i
$376$	−147.908			−0.393371
$377$		−	49.4982i		−	0.131295i
$378$	−592.984	−	381.058i	−1.56874	−	1.00809i
$379$	435.899			1.15013			0.575065	−	0.818108i	$-0.304977\pi$
$379$	435.899			1.15013			0.575065	+	0.818108i	$0.304977\pi$
$380$			337.097i			0.887098i
$381$	18.3281	+	15.8810i	0.0481052	+	0.0416824i
$382$	−902.650			−2.36296
$383$			53.1944i			0.138889i	0.997586	+	0.0694443i	$0.0221226\pi$
$383$			53.1944i			0.138889i	−0.997586	+	0.0694443i	$0.977877\pi$
$384$	−837.342	+	966.368i	−2.18058	+	2.51658i
$385$	147.835			0.383988
$386$		−	637.829i		−	1.65241i
$387$	55.8388	+	388.302i	0.144286	+	1.00337i
$388$	−565.208			−1.45672
$389$		−	167.333i		−	0.430161i	−0.976596	−	0.215081i	$-0.930999\pi$
$389$		−	167.333i		−	0.430161i	0.976596	−	0.215081i	$-0.0690014\pi$
$390$	−36.1971	−	31.3642i	−0.0928131	−	0.0804211i
$391$	−34.3578			−0.0878715
$392$			89.1186i			0.227343i
$393$	324.880	−	374.941i	0.826668	−	0.954048i
$394$	−225.082			−0.571274
$395$		−	116.227i		−	0.294246i
$396$	1465.70	−	210.772i	3.70127	−	0.532252i
$397$	−89.4522			−0.225320			−0.112660	−	0.993634i	$-0.535937\pi$
$397$	−89.4522			−0.225320			−0.112660	+	0.993634i	$0.535937\pi$
$398$			367.241i			0.922715i
$399$	−325.546	−	282.080i	−0.815904	−	0.706968i
$400$	1364.96			3.41240
$401$			329.380i			0.821397i	0.911771	+	0.410699i	$0.134715\pi$
$401$			329.380i			0.821397i	−0.911771	+	0.410699i	$0.865285\pi$
$402$	−211.683	+	244.300i	−0.526573	+	0.607713i
$403$	113.449			0.281511
$404$			419.701i			1.03886i
$405$	−33.1498	−	112.878i	−0.0818512	−	0.278711i
$406$	454.137			1.11856
$407$			795.079i			1.95351i
$408$	−434.441	−	376.436i	−1.06481	−	0.922637i
$409$	−262.356			−0.641457			−0.320729	−	0.947171i	$-0.603928\pi$
$409$	−262.356			−0.641457			−0.320729	+	0.947171i	$0.603928\pi$
$410$			0.124200i			0.000302928i
$411$	−393.927	+	454.627i	−0.958461	+	1.10615i
$412$	−488.813			−1.18644
$413$		−	103.878i		−	0.251521i
$414$	−23.7336	−	165.043i	−0.0573276	−	0.398656i
$415$	34.6267			0.0834377
$416$		−	351.042i		−	0.843851i
$417$	−375.683	−	325.523i	−0.900917	−	0.780631i
$418$	1236.30			2.95765
$419$			490.346i			1.17028i	0.810934	+	0.585138i	$0.198960\pi$
$419$			490.346i			1.17028i	−0.810934	+	0.585138i	$0.801040\pi$
$420$	210.631	−	243.087i	0.501503	−	0.578780i
$421$	−27.1659			−0.0645271			−0.0322636	−	0.999479i	$-0.510272\pi$
$421$	−27.1659			−0.0645271			−0.0322636	+	0.999479i	$0.510272\pi$
$422$			98.0719i			0.232398i
$423$	49.2632	−	7.08416i	0.116461	−	0.0167474i
$424$	899.982			2.12260
$425$			163.990i			0.385858i
$426$	−71.2288	−	61.7186i	−0.167204	−	0.144879i
$427$	490.261			1.14815
$428$		−	552.951i		−	1.29194i
$429$	−84.1965	+	97.1702i	−0.196262	+	0.226504i
$430$	−244.566			−0.568759
$431$			583.847i			1.35463i	0.735691	+	0.677317i	$0.236857\pi$
$431$			583.847i			1.35463i	−0.735691	+	0.677317i	$0.763143\pi$
$432$	870.389	−	1354.46i	2.01479	−	3.13532i
$433$	157.250			0.363165			0.181582	−	0.983376i	$-0.441878\pi$
$433$	157.250			0.363165			0.181582	+	0.983376i	$0.441878\pi$
$434$			1040.87i			2.39832i
$435$	57.2844	+	49.6360i	0.131688	+	0.114106i
$436$	−1322.94			−3.03426
$437$		−	101.898i		−	0.233177i
$438$	−368.694	+	425.506i	−0.841767	+	0.971474i
$439$	−561.980			−1.28014			−0.640069	−	0.768318i	$-0.721094\pi$
$439$	−561.980			−1.28014			−0.640069	+	0.768318i	$0.721094\pi$
$440$			585.111i			1.32980i
$441$	−4.26841	−	29.6825i	−0.00967894	−	0.0673072i
$442$	78.7486			0.178164
$443$			203.785i			0.460011i	0.973189	+	0.230006i	$0.0738745\pi$
$443$			203.785i			0.460011i	−0.973189	+	0.230006i	$0.926126\pi$
$444$	1307.36	+	1132.80i	2.94450	+	2.55136i
$445$	−144.912			−0.325644
$446$		−	1513.07i		−	3.39254i
$447$	−66.7113	+	76.9908i	−0.149242	+	0.172239i
$448$	1608.87			3.59122
$449$			355.653i			0.792100i	0.918229	+	0.396050i	$0.129619\pi$
$449$			355.653i			0.792100i	−0.918229	+	0.396050i	$0.870381\pi$
$450$	−787.752	+	113.281i	−1.75056	+	0.251735i
$451$	0.333413			0.000739275
$452$		−	2265.73i		−	5.01268i
$453$	−186.910	−	161.954i	−0.412604	−	0.357515i
$454$	−1616.80			−3.56124
$455$			27.9280i			0.0613802i
$456$	1116.43	−	1288.46i	2.44832	−	2.82558i
$457$	615.491			1.34681			0.673404	−	0.739275i	$-0.264832\pi$
$457$	615.491			1.34681			0.673404	+	0.739275i	$0.264832\pi$
$458$		−	67.0274i		−	0.146348i
$459$	162.728	+	104.571i	0.354526	+	0.227823i
$460$	76.0881			0.165409
$461$		−	760.615i		−	1.64993i	−0.565187	−	0.824963i	$-0.691196\pi$
$461$		−	760.615i		−	1.64993i	0.565187	−	0.824963i	$-0.308804\pi$
$462$	−891.518	−	772.486i	−1.92969	−	1.67205i
$463$	298.061			0.643761			0.321881	−	0.946780i	$-0.395685\pi$
$463$	298.061			0.643761			0.321881	+	0.946780i	$0.395685\pi$
$464$			1037.31i			2.23559i
$465$	−113.765	+	131.294i	−0.244655	+	0.282354i
$466$	1181.25			2.53488
$467$		−	359.852i		−	0.770562i	−0.922799	−	0.385281i	$-0.874105\pi$
$467$		−	359.852i		−	0.770562i	0.922799	−	0.385281i	$-0.125895\pi$
$468$	39.8175	+	276.890i	0.0850801	+	0.591646i
$469$	188.491			0.401900
$470$			31.0277i			0.0660163i
$471$	−179.985	−	155.954i	−0.382134	−	0.331113i
$472$	411.134			0.871047
$473$			656.533i			1.38802i
$474$	−607.322	+	700.904i	−1.28127	+	1.47870i
$475$	−486.360			−1.02392
$476$			528.849i			1.11103i
$477$	−299.754	+	43.1054i	−0.628416	+	0.0903677i
$478$	1705.26			3.56748
$479$		−	754.906i		−	1.57600i	−0.615673	−	0.788002i	$-0.711116\pi$
$479$		−	754.906i		−	1.57600i	0.615673	−	0.788002i	$-0.288884\pi$
$480$	406.261	+	352.019i	0.846378	+	0.733373i
$481$	−150.201			−0.312267
$482$			356.175i			0.738952i
$483$	−63.6699	+	73.4808i	−0.131822	+	0.152134i
$484$	1156.43			2.38931
$485$			75.1505i			0.154950i
$486$	−389.914	+	853.926i	−0.802292	+	1.75705i
$487$	253.731			0.521009			0.260505	−	0.965473i	$-0.416111\pi$
$487$	253.731			0.521009			0.260505	+	0.965473i	$0.416111\pi$
$488$			1940.38i			3.97619i
$489$	−321.249	−	278.357i	−0.656951	−	0.569238i
$490$	18.6951			0.0381532
$491$			729.103i			1.48494i	0.669882	+	0.742468i	$0.266345\pi$
$491$			729.103i			1.48494i	−0.669882	+	0.742468i	$0.733655\pi$
$492$	0.475037	−	0.548235i	0.000965522	−	0.00111430i
$493$	−124.625			−0.252789
$494$			233.552i			0.472778i
$495$	−28.0244	−	194.881i	−0.0566150	−	0.393700i
$496$	−2377.49			−4.79333
$497$			54.9568i			0.110577i
$498$	−208.815	−	180.935i	−0.419308	−	0.363323i
$499$	11.0699			0.0221841			0.0110921	−	0.999938i	$-0.496469\pi$
$499$	11.0699			0.0221841			0.0110921	+	0.999938i	$0.496469\pi$
$500$		−	759.805i		−	1.51961i
$501$	−556.943	+	642.762i	−1.11166	+	1.28296i
$502$	−1171.02			−2.33272
$503$			180.558i			0.358961i	0.983762	+	0.179481i	$0.0574417\pi$
$503$			180.558i			0.358961i	−0.983762	+	0.179481i	$0.942558\pi$
$504$	−1610.16	+	231.545i	−3.19477	+	0.459416i
$505$	55.8038			0.110503
$506$		−	279.052i		−	0.551485i
$507$	364.812	+	316.104i	0.719551	+	0.623479i
$508$	88.3035			0.173826
$509$			46.5796i			0.0915119i	0.998953	+	0.0457560i	$0.0145697\pi$
$509$			46.5796i			0.0915119i	−0.998953	+	0.0457560i	$0.985430\pi$
$510$	−78.9677	+	91.1358i	−0.154839	+	0.178698i
$511$	328.301			0.642467
$512$			977.058i			1.90832i
$513$	−310.135	+	482.617i	−0.604552	+	0.940775i
$514$	149.010			0.289903
$515$			64.9930i			0.126200i
$516$	1079.54	+	935.407i	2.09214	+	1.81280i
$517$	83.2930			0.161108
$518$		−	1378.06i		−	2.66035i
$519$	211.914	−	244.568i	0.408313	−	0.471229i
$520$	−110.535			−0.212567
$521$			714.631i			1.37165i	0.727765	+	0.685827i	$0.240559\pi$
$521$			714.631i			1.37165i	−0.727765	+	0.685827i	$0.759441\pi$
$522$	−86.0884	−	598.658i	−0.164920	−	1.14685i
$523$	98.5007			0.188338			0.0941689	−	0.995556i	$-0.469981\pi$
$523$	98.5007			0.188338			0.0941689	+	0.995556i	$0.469981\pi$
$524$		−	1806.44i		−	3.44740i
$525$	350.724	+	303.897i	0.668045	+	0.578851i
$526$	734.825			1.39701
$527$		−	285.638i		−	0.542007i
$528$	1764.47	−	2036.35i	3.34179	−	3.85673i
$529$	−23.0000			−0.0434783
$530$		−	188.796i		−	0.356218i
$531$	−136.935	+	19.6916i	−0.257882	+	0.0370840i
$532$	−1568.46			−2.94823
$533$			0.0629860i			0.000118173i
$534$	873.886	+	757.209i	1.63649	+	1.41799i
$535$	−73.5209			−0.137422
$536$			746.020i			1.39183i
$537$	324.869	−	374.927i	0.604970	−	0.698189i
$538$	519.809			0.966187
$539$		−	50.1865i		−	0.0931103i
$540$	−360.374	−	231.580i	−0.667359	−	0.428852i
$541$	−590.463			−1.09143			−0.545714	−	0.837971i	$-0.683742\pi$
$541$	−590.463			−1.09143			−0.545714	+	0.837971i	$0.683742\pi$
$542$		−	1611.59i		−	2.97341i
$543$	−765.533	−	663.323i	−1.40982	−	1.22159i
$544$	−883.841			−1.62471
$545$			175.899i			0.322750i
$546$	145.933	−	168.419i	0.267276	−	0.308460i
$547$	95.8571			0.175241			0.0876207	−	0.996154i	$-0.472074\pi$
$547$	95.8571			0.175241			0.0876207	+	0.996154i	$0.472074\pi$
$548$			2190.36i			3.99701i
$549$	−92.9362	−	646.277i	−0.169283	−	1.17719i
$550$	−1331.91			−2.42166
$551$		−	369.613i		−	0.670803i
$552$	−290.826	−	251.996i	−0.526859	−	0.456515i
$553$	540.785			0.977912
$554$		−	946.320i		−	1.70816i
$555$	150.619	−	173.827i	0.271385	−	0.313202i
$556$	−1810.01			−3.25542
$557$			510.048i			0.915706i	0.889028	+	0.457853i	$0.151381\pi$
$557$			510.048i			0.915706i	−0.889028	+	0.457853i	$0.848619\pi$
$558$	1372.11	−	197.313i	2.45898	−	0.353607i
$559$	−124.027			−0.221874
$560$		−	585.274i		−	1.04513i
$561$	244.652	+	211.987i	0.436100	+	0.377873i
$562$	−824.388			−1.46688
$563$		−	371.349i		−	0.659589i	−0.944053	−	0.329795i	$-0.893021\pi$
$563$		−	371.349i		−	0.659589i	0.944053	−	0.329795i	$-0.106979\pi$
$564$	118.673	−	136.960i	0.210414	−	0.242836i
$565$	−301.253			−0.533192
$566$			1062.21i			1.87670i
$567$	525.203	−	154.240i	0.926283	−	0.272029i
$568$	−217.511			−0.382942
$569$			178.514i			0.313733i	0.987620	+	0.156867i	$0.0501393\pi$
$569$			178.514i			0.313733i	−0.987620	+	0.156867i	$0.949861\pi$
$570$	−270.290	−	234.202i	−0.474194	−	0.410881i
$571$	502.763			0.880496			0.440248	−	0.897876i	$-0.354891\pi$
$571$	502.763			0.880496			0.440248	+	0.897876i	$0.354891\pi$
$572$			468.160i			0.818461i
$573$	459.037	−	529.769i	0.801111	−	0.924554i
$574$	−0.577884			−0.00100677
$575$			109.779i			0.190920i
$576$	−304.985	−	2120.86i	−0.529488	−	3.68205i
$577$	431.919			0.748560			0.374280	−	0.927316i	$-0.377890\pi$
$577$	431.919			0.748560			0.374280	+	0.927316i	$0.377890\pi$
$578$			918.166i			1.58852i
$579$	374.344	+	324.363i	0.646536	+	0.560213i
$580$	275.992			0.475849
$581$			161.112i			0.277301i
$582$	392.685	−	453.193i	0.674716	−	0.778683i
$583$	−506.818			−0.869327
$584$			1299.37i			2.22494i
$585$	36.8156	−	5.29417i	0.0629326	−	0.00904986i
$586$	1117.53			1.90705
$587$			284.521i			0.484703i	0.970188	+	0.242352i	$0.0779187\pi$
$587$			284.521i			0.484703i	−0.970188	+	0.242352i	$0.922081\pi$
$588$	−82.5221	−	71.5041i	−0.140344	−	0.121606i
$589$	847.143			1.43827
$590$		−	86.2466i		−	0.146181i
$591$	114.464	−	132.102i	0.193678	−	0.223522i
$592$	3147.68			5.31703
$593$		−	331.992i		−	0.559852i	−0.960022	−	0.279926i	$-0.909690\pi$
$593$		−	331.992i		−	0.559852i	0.960022	−	0.279926i	$-0.0903100\pi$
$594$	−849.316	+	1321.66i	−1.42982	+	2.22502i
$595$	70.3162			0.118178
$596$			370.936i			0.622376i
$597$	−215.535	−	186.758i	−0.361030	−	0.312827i
$598$	52.7164			0.0881545
$599$			179.307i			0.299344i	0.988736	+	0.149672i	$0.0478217\pi$
$599$			179.307i			0.299344i	−0.988736	+	0.149672i	$0.952178\pi$
$600$	−1202.78	+	1388.11i	−2.00463	+	2.31352i
$601$	994.416			1.65460			0.827301	−	0.561758i	$-0.189875\pi$
$601$	994.416			1.65460			0.827301	+	0.561758i	$0.189875\pi$
$602$		−	1137.93i		−	1.89024i
$603$	−35.7313	−	248.475i	−0.0592558	−	0.412064i
$604$	−900.519			−1.49093
$605$		−	153.760i		−	0.254148i
$606$	−336.524	−	291.592i	−0.555320	−	0.481176i
$607$	−361.463			−0.595491			−0.297746	−	0.954645i	$-0.596235\pi$
$607$	−361.463			−0.595491			−0.297746	+	0.954645i	$0.596235\pi$
$608$		−	2621.30i		−	4.31134i
$609$	−230.948	+	266.535i	−0.379225	+	0.437660i
$610$	407.048			0.667292
$611$			15.7351i			0.0257531i
$612$	697.145	−	100.251i	1.13913	−	0.163809i
$613$	−332.337			−0.542148			−0.271074	−	0.962559i	$-0.587379\pi$
$613$	−332.337			−0.542148			−0.271074	+	0.962559i	$0.587379\pi$
$614$			1342.54i			2.18655i
$615$	−0.0728938	−	0.0631613i	−0.000118526	−	0.000102701i
$616$	−2722.43			−4.41953
$617$		−	670.074i		−	1.08602i	−0.839727	−	0.543009i	$-0.817285\pi$
$617$		−	670.074i		−	1.08602i	0.839727	−	0.543009i	$-0.182715\pi$
$618$	339.609	−	391.939i	0.549529	−	0.634205i
$619$	354.125			0.572093			0.286046	−	0.958216i	$-0.407659\pi$
$619$	354.125			0.572093			0.286046	+	0.958216i	$0.407659\pi$
$620$			632.568i			1.02027i
$621$	108.934	+	70.0023i	0.175417	+	0.112725i
$622$	−785.711			−1.26320
$623$		−	674.250i		−	1.08226i
$624$	384.693	+	333.330i	0.616495	+	0.534183i
$625$	471.238			0.753982
$626$		−	2157.96i		−	3.44722i
$627$	−628.711	+	725.588i	−1.00273	+	1.15724i
$628$	−867.155			−1.38082
$629$			378.170i			0.601224i
$630$	48.5730	+	337.776i	0.0771000	+	0.536152i
$631$	314.364			0.498200			0.249100	−	0.968478i	$-0.419865\pi$
$631$	314.364			0.498200			0.249100	+	0.968478i	$0.419865\pi$
$632$			2140.35i			3.38663i
$633$	−57.5588	−	49.8738i	−0.0909302	−	0.0787896i
$634$	1223.31			1.92952
$635$		−	11.7409i		−	0.0184896i
$636$	−722.098	+	833.366i	−1.13537	+	1.31032i
$637$	9.48086			0.0148836
$638$		−	1012.20i		−	1.58651i
$639$	72.4458	−	10.4179i	0.113374	−	0.0163034i
$640$	619.052			0.967269
$641$			293.398i			0.457719i	0.973459	+	0.228860i	$0.0734997\pi$
$641$			293.398i			0.457719i	−0.973459	+	0.228860i	$0.926500\pi$
$642$	443.366	+	384.169i	0.690601	+	0.598395i
$643$	−1124.09			−1.74820			−0.874099	−	0.485747i	$-0.838548\pi$
$643$	−1124.09			−1.74820			−0.874099	+	0.485747i	$0.838548\pi$
$644$			354.025i			0.549729i
$645$	124.373	−	143.537i	0.192826	−	0.222538i
$646$	588.030			0.910263
$647$			125.359i			0.193755i	0.995296	+	0.0968773i	$0.0308854\pi$
$647$			125.359i			0.193755i	−0.995296	+	0.0968773i	$0.969115\pi$
$648$	610.461	+	2078.68i	0.942070	+	3.20783i
$649$	−231.527			−0.356744
$650$		−	251.615i		−	0.387101i
$651$	−610.892	−	529.328i	−0.938390	−	0.813100i
$652$	−1547.76			−2.37386
$653$			364.122i			0.557614i	0.960347	+	0.278807i	$0.0899390\pi$
$653$			364.122i			0.557614i	−0.960347	+	0.278807i	$0.910061\pi$
$654$	919.126	−	1060.75i	1.40539	−	1.62195i
$655$	−240.186			−0.366696
$656$		−	1.31997i		−	0.00201215i
$657$	−62.2343	−	432.776i	−0.0947249	−	0.658716i
$658$	−144.367			−0.219402
$659$		−	1195.30i		−	1.81380i	−0.421343	−	0.906901i	$-0.638441\pi$
$659$		−	1195.30i		−	1.81380i	0.421343	−	0.906901i	$-0.361559\pi$
$660$	−541.802	−	469.463i	−0.820912	−	0.711307i
$661$	445.126			0.673413			0.336707	−	0.941610i	$-0.390687\pi$
$661$	445.126			0.673413			0.336707	+	0.941610i	$0.390687\pi$
$662$		−	82.4603i		−	0.124562i
$663$	−40.0471	+	46.2179i	−0.0604028	+	0.0697102i
$664$	−637.659			−0.960329
$665$			208.544i			0.313599i
$666$	−1816.60	+	261.232i	−2.72763	+	0.392240i
$667$	−83.4273			−0.125078
$668$			3096.78i			4.63590i
$669$	888.029	+	769.463i	1.32740	+	1.15017i
$670$	156.498			0.233579
$671$		−	1092.71i		−	1.62848i
$672$	−1637.89	+	1890.27i	−2.43733	+	2.81290i
$673$	−765.350			−1.13722			−0.568611	−	0.822607i	$-0.692519\pi$
$673$	−765.350			−1.13722			−0.568611	+	0.822607i	$0.692519\pi$
$674$		−	898.142i		−	1.33255i
$675$	334.121	−	519.943i	0.494995	−	0.770287i
$676$	1757.64			2.60006
$677$			137.097i			0.202507i	0.994861	+	0.101254i	$0.0322854\pi$
$677$			137.097i			0.202507i	−0.994861	+	0.101254i	$0.967715\pi$
$678$	1816.70	+	1574.14i	2.67950	+	2.32175i
$679$	−349.663			−0.514968
$680$			278.301i			0.409267i
$681$	822.214	−	948.908i	1.20736	−	1.39340i
$682$	2319.93			3.40166
$683$			413.952i			0.606080i	0.952978	+	0.303040i	$0.0980015\pi$
$683$			413.952i			0.606080i	−0.952978	+	0.303040i	$0.901998\pi$
$684$	297.325	+	2067.59i	0.434685	+	3.02279i
$685$	291.233			0.425157
$686$			1366.19i			1.99152i
$687$	39.3387	+	34.0864i	0.0572615	+	0.0496162i
$688$	2599.18			3.77788
$689$		−	95.7443i		−	0.138961i
$690$	−52.8631	+	61.0088i	−0.0766132	+	0.0884185i
$691$	−951.419			−1.37687			−0.688436	−	0.725297i	$-0.741703\pi$
$691$	−951.419			−1.37687			−0.688436	+	0.725297i	$0.741703\pi$
$692$		−	1178.31i		−	1.70276i
$693$	906.751	−	130.393i	1.30844	−	0.188157i
$694$	−1692.30			−2.43847
$695$			240.661i			0.346275i
$696$	−1054.91	−	914.060i	−1.51567	−	1.31330i
$697$	0.158584			0.000227524
$698$		−	135.865i		−	0.194649i
$699$	−600.718	+	693.283i	−0.859397	+	0.991821i
$700$	1689.76			2.41395
$701$		−	957.275i		−	1.36558i	−0.730612	−	0.682792i	$-0.760765\pi$
$701$		−	957.275i		−	1.36558i	0.730612	−	0.682792i	$-0.239235\pi$
$702$	−249.679	−	160.447i	−0.355668	−	0.228556i
$703$	−1121.58			−1.59541
$704$		−	3585.90i		−	5.09361i
$705$	−18.2103	−	15.7789i	−0.0258302	−	0.0223814i
$706$	2602.49			3.68624
$707$			259.646i			0.367251i
$708$	−329.872	+	380.702i	−0.465922	+	0.537715i
$709$	−195.946			−0.276369			−0.138184	−	0.990407i	$-0.544127\pi$
$709$	−195.946			−0.276369			−0.138184	+	0.990407i	$0.544127\pi$
$710$			45.6289i			0.0642661i
$711$	−102.514	−	712.880i	−0.144183	−	1.00264i
$712$	2668.59			3.74801
$713$		−	191.213i		−	0.268182i
$714$	−424.040	−	367.424i	−0.593894	−	0.514599i
$715$	62.2469			0.0870587
$716$		−	1806.38i		−	2.52287i
$717$	−867.197	+	1000.82i	−1.20948	+	1.39585i
$718$	166.105			0.231344
$719$			1073.79i			1.49345i	0.665135	+	0.746723i	$0.268374\pi$
$719$			1073.79i			1.49345i	−0.665135	+	0.746723i	$0.731626\pi$
$720$	−771.527	+	110.947i	−1.07157	+	0.154094i
$721$	−302.402			−0.419420
$722$			349.398i			0.483931i
$723$	−209.041	−	181.130i	−0.289129	−	0.250526i
$724$	−3688.29			−5.09432
$725$			398.199i			0.549240i
$726$	−803.443	+	927.244i	−1.10667	+	1.27720i
$727$	1314.49			1.80810			0.904051	−	0.427426i	$-0.140579\pi$
$727$	1314.49			1.80810			0.904051	+	0.427426i	$0.140579\pi$
$728$		−	514.301i		−	0.706458i
$729$	−302.885	−	663.100i	−0.415480	−	0.909603i
$730$	272.578			0.373394
$731$			312.272i			0.427185i
$732$	−1796.76	−	1556.86i	−2.45458	−	2.12686i
$733$	−338.570			−0.461896			−0.230948	−	0.972966i	$-0.574183\pi$
$733$	−338.570			−0.461896			−0.230948	+	0.972966i	$0.574183\pi$
$734$			2580.46i			3.51561i
$735$	−9.50725	+	10.9722i	−0.0129350	+	0.0149282i
$736$	−591.667			−0.803896
$737$		−	420.115i		−	0.570034i
$738$	0.109547	+	0.761785i	0.000148437	+	0.00103223i
$739$	−474.230			−0.641719			−0.320859	−	0.947127i	$-0.603972\pi$
$739$	−474.230			−0.641719			−0.320859	+	0.947127i	$0.603972\pi$
$740$		−	837.488i		−	1.13174i
$741$	−137.073	−	118.771i	−0.184984	−	0.160285i
$742$	878.436			1.18388
$743$			230.056i			0.309631i	0.987943	+	0.154816i	$0.0494783\pi$
$743$			230.056i			0.309631i	−0.987943	+	0.154816i	$0.950522\pi$
$744$	2095.00	−	2417.82i	2.81587	−	3.24976i
$745$	49.3200			0.0662014
$746$		−	1262.51i		−	1.69238i
$747$	212.383	−	30.5412i	0.284315	−	0.0408852i
$748$	1178.72			1.57582
$749$		−	342.081i		−	0.456716i
$750$	609.225	+	527.884i	0.812300	+	0.703845i
$751$	−821.578			−1.09398			−0.546989	−	0.837140i	$-0.684226\pi$
$751$	−821.578			−1.09398			−0.546989	+	0.837140i	$0.684226\pi$
$752$		−	329.754i		−	0.438502i
$753$	595.517	−	687.279i	0.790859	−	0.912721i
$754$	191.217			0.253603
$755$			119.734i			0.158588i
$756$	1077.50	−	1676.76i	1.42527	−	2.21794i
$757$	−990.077			−1.30790			−0.653948	−	0.756540i	$-0.726888\pi$
$757$	−990.077			−1.30790			−0.653948	+	0.756540i	$0.726888\pi$
$758$			1683.92i			2.22153i
$759$	163.777	+	141.910i	0.215779	+	0.186969i
$760$	−825.385			−1.08603
$761$			1173.07i			1.54149i	0.637144	+	0.770745i	$0.280116\pi$
$761$			1173.07i			1.54149i	−0.637144	+	0.770745i	$0.719884\pi$
$762$	−61.3500	+	70.8033i	−0.0805118	+	0.0929177i
$763$	−818.428			−1.07265
$764$		−	2552.39i		−	3.34083i
$765$	−13.3295	−	92.6930i	−0.0174242	−	0.121167i
$766$	−205.495			−0.268271
$767$		−	43.7384i		−	0.0570253i
$768$	−1574.05	−	1363.89i	−2.04955	−	1.77590i
$769$	−904.938			−1.17677			−0.588386	−	0.808580i	$-0.700236\pi$
$769$	−904.938			−1.17677			−0.588386	+	0.808580i	$0.700236\pi$
$770$			571.104i			0.741693i
$771$	−75.7782	+	87.4548i	−0.0982857	+	0.113430i
$772$	1803.57			2.33623
$773$			967.847i			1.25207i	0.779796	+	0.626033i	$0.215323\pi$
$773$			967.847i			1.25207i	−0.779796	+	0.626033i	$0.784677\pi$
$774$	−1500.05	+	215.711i	−1.93805	+	0.278696i
$775$	−912.662			−1.17763
$776$		−	1383.92i		−	1.78340i
$777$	808.790	+	700.804i	1.04091	+	0.901935i
$778$	646.423			0.830879
$779$			0.470328i			0.000603759i
$780$	88.6875	−	102.353i	0.113702	−	0.131222i
$781$	122.490			0.156837
$782$		−	132.728i		−	0.169728i
$783$	395.134	+	253.918i	0.504642	+	0.324288i
$784$	−198.686			−0.253426
$785$			115.298i			0.146876i
$786$	1448.44	+	1255.05i	1.84279	+	1.59675i
$787$	−26.1652			−0.0332468			−0.0166234	−	0.999862i	$-0.505292\pi$
$787$	−26.1652			−0.0332468			−0.0166234	+	0.999862i	$0.505292\pi$
$788$		−	636.457i		−	0.807686i
$789$	−373.691	+	431.272i	−0.473626	+	0.546606i
$790$	448.997			0.568350
$791$		−	1401.68i		−	1.77204i
$792$	516.077	+	3588.79i	0.651612	+	4.53130i
$793$	206.427			0.260311
$794$		−	345.563i		−	0.435218i
$795$	110.805	+	96.0108i	0.139377	+	0.120768i
$796$	−1038.43			−1.30457
$797$			1344.44i			1.68687i	0.537231	+	0.843435i	$0.319470\pi$
$797$			1344.44i			1.68687i	−0.537231	+	0.843435i	$0.680530\pi$
$798$	1089.71	−	1257.62i	1.36555	−	1.57596i
$799$	39.6173			0.0495837
$800$			2824.03i			3.53004i
$801$	−888.818	+	127.814i	−1.10964	+	0.159568i
$802$	−1272.43			−1.58657
$803$		−	731.728i		−	0.911243i
$804$	−690.800	−	598.567i	−0.859204	−	0.744487i
$805$	47.0715			0.0584740
$806$			438.265i			0.543752i
$807$	−264.345	+	305.078i	−0.327565	+	0.378040i
$808$	−1027.64			−1.27183
$809$			1114.63i			1.37779i	0.724860	+	0.688896i	$0.241904\pi$
$809$			1114.63i			1.37779i	−0.724860	+	0.688896i	$0.758096\pi$
$810$	436.059	−	128.061i	0.538345	−	0.158100i
$811$	1177.17			1.45151			0.725754	−	0.687955i	$-0.241491\pi$
$811$	1177.17			1.45151			0.725754	+	0.687955i	$0.241491\pi$
$812$			1284.15i			1.58146i
$813$	945.848	+	819.562i	1.16340	+	1.00807i
$814$	−3071.47			−3.77331
$815$			205.791i			0.252505i
$816$	839.247	−	968.566i	1.02849	−	1.18697i
$817$	−926.135			−1.13358
$818$		−	1013.51i		−	1.23901i
$819$	24.6329	+	171.297i	0.0300768	+	0.209154i
$820$	−0.351197			−0.000428289
$821$		−	548.254i		−	0.667788i	−0.942611	−	0.333894i	$-0.891637\pi$
$821$		−	548.254i		−	0.667788i	0.942611	−	0.333894i	$-0.108363\pi$
$822$	−1756.27	−	1521.78i	−2.13658	−	1.85132i
$823$	−1407.69			−1.71043			−0.855217	−	0.518270i	$-0.826576\pi$
$823$	−1407.69			−1.71043			−0.855217	+	0.518270i	$0.826576\pi$
$824$		−	1196.86i		−	1.45250i
$825$	677.336	−	781.706i	0.821013	−	0.947522i
$826$	401.291			0.485825
$827$		−	927.502i		−	1.12153i	−0.827976	−	0.560763i	$-0.810508\pi$
$827$		−	927.502i		−	1.12153i	0.827976	−	0.560763i	$-0.189492\pi$
$828$	466.688	−	67.1108i	0.563632	−	0.0810517i
$829$	1472.18			1.77585			0.887923	−	0.459992i	$-0.152148\pi$
$829$	1472.18			1.77585			0.887923	+	0.459992i	$0.152148\pi$
$830$			133.766i			0.161164i
$831$	555.399	+	481.245i	0.668351	+	0.579115i
$832$	677.423			0.814210
$833$		−	23.8706i		−	0.0286562i
$834$	1257.53	−	1451.30i	1.50783	−	1.74017i
$835$	411.751			0.493115
$836$			3495.84i			4.18162i
$837$	−581.973	+	905.639i	−0.695309	+	1.08201i
$838$	−1894.26			−2.26045
$839$		−	437.243i		−	0.521148i	−0.965454	−	0.260574i	$-0.916088\pi$
$839$		−	437.243i		−	0.521148i	0.965454	−	0.260574i	$-0.0839118\pi$
$840$	595.202	+	515.733i	0.708573	+	0.613968i
$841$	538.386			0.640174
$842$		−	104.945i		−	0.124637i
$843$	419.237	−	483.837i	0.497315	−	0.573946i
$844$	−277.315			−0.328572
$845$		−	233.698i		−	0.276565i
$846$	27.3669	+	190.309i	0.0323485	+	0.224951i
$847$	715.419			0.844650
$848$			2006.47i			2.36612i
$849$	−623.418	−	540.182i	−0.734297	−	0.636257i
$850$	−633.509			−0.745304
$851$			253.157i			0.297482i
$852$	174.520	−	201.411i	0.204835	−	0.236398i
$853$	354.923			0.416088			0.208044	−	0.978119i	$-0.433290\pi$
$853$	354.923			0.416088			0.208044	+	0.978119i	$0.433290\pi$
$854$			1893.93i			2.21771i
$855$	274.909	−	39.5325i	0.321531	−	0.0462369i
$856$	1353.91			1.58167
$857$			861.558i			1.00532i	0.864485	+	0.502659i	$0.167645\pi$
$857$			861.558i			1.00532i	−0.864485	+	0.502659i	$0.832355\pi$
$858$	−375.379	−	325.260i	−0.437504	−	0.379091i
$859$	887.181			1.03281			0.516403	−	0.856346i	$-0.327271\pi$
$859$	887.181			1.03281			0.516403	+	0.856346i	$0.327271\pi$
$860$		−	691.552i		−	0.804130i
$861$	0.293879	−	0.339163i	0.000341323	−	0.000393917i
$862$	−2255.46			−2.61655
$863$		−	1495.61i		−	1.73304i	−0.499147	−	0.866518i	$-0.666353\pi$
$863$		−	1495.61i		−	1.73304i	0.499147	−	0.866518i	$-0.333647\pi$
$864$	2802.30	+	1800.79i	3.24340	+	2.08424i
$865$	−156.669			−0.181121
$866$			607.474i			0.701471i
$867$	−538.876	−	466.927i	−0.621541	−	0.538555i
$868$	−2943.24			−3.39082
$869$		−	1205.32i		−	1.38702i
$870$	−191.749	+	221.295i	−0.220401	+	0.254363i
$871$	79.3652			0.0911196
$872$		−	3239.22i		−	3.71470i
$873$	66.2838	+	460.937i	0.0759265	+	0.527992i
$874$	393.643			0.450393
$875$		−	470.050i		−	0.537200i
$876$	−1203.19	−	1042.54i	−1.37350	−	1.19012i
$877$	661.875			0.754703			0.377352	−	0.926070i	$-0.376835\pi$
$877$	661.875			0.754703			0.377352	+	0.926070i	$0.376835\pi$
$878$		−	2170.99i		−	2.47265i
$879$	−568.313	+	655.883i	−0.646545	+	0.746170i
$880$	−1304.48			−1.48236
$881$			290.110i			0.329297i	0.986352	+	0.164648i	$0.0526489\pi$
$881$			290.110i			0.329297i	−0.986352	+	0.164648i	$0.947351\pi$
$882$	114.667	−	16.4893i	0.130007	−	0.0186954i
$883$	971.834			1.10061			0.550303	−	0.834965i	$-0.314512\pi$
$883$	971.834			1.10061			0.550303	+	0.834965i	$0.314512\pi$
$884$			222.675i			0.251894i
$885$	50.6185	+	43.8601i	0.0571961	+	0.0495595i
$886$	−787.242			−0.888535
$887$		−	455.491i		−	0.513518i	−0.966475	−	0.256759i	$-0.917345\pi$
$887$		−	455.491i		−	0.513518i	0.966475	−	0.256759i	$-0.0826547\pi$
$888$	−2773.68	+	3201.07i	−3.12351	+	3.60481i
$889$	54.6286			0.0614495
$890$		−	559.809i		−	0.628999i
$891$	−343.776	−	1170.59i	−0.385832	−	1.31379i
$892$	4278.46			4.79648
$893$			117.497i			0.131576i
$894$	−297.423	−	257.713i	−0.332688	−	0.288269i
$895$	−240.177			−0.268354
$896$			2880.35i			3.21467i
$897$	−26.8086	+	30.9395i	−0.0298869	+	0.0344922i
$898$	−1373.92			−1.52998
$899$		−	693.584i		−	0.771506i
$900$	−320.320	−	2227.50i	−0.355911	−	2.47500i
$901$	−241.062			−0.267549
$902$			1.28801i			0.00142795i
$903$	667.854	+	578.685i	0.739595	+	0.640848i
$904$	5547.66			6.13679
$905$			490.398i			0.541876i
$906$	625.646	−	722.052i	0.690559	−	0.796966i
$907$	937.867			1.03403			0.517016	−	0.855976i	$-0.327043\pi$
$907$	937.867			1.03403			0.517016	+	0.855976i	$0.327043\pi$
$908$		−	4571.78i		−	5.03499i
$909$	342.274	−	49.2198i	0.376539	−	0.0541472i
$910$	−107.889			−0.118559
$911$			1199.27i			1.31643i	0.752831	+	0.658214i	$0.228688\pi$
$911$			1199.27i			1.31643i	−0.752831	+	0.658214i	$0.771312\pi$
$912$	2872.57	+	2489.04i	3.14975	+	2.72921i
$913$	359.093			0.393311
$914$			2377.71i			2.60143i
$915$	−207.002	+	238.898i	−0.226231	+	0.261091i
$916$	189.531			0.206912
$917$		−	1117.55i		−	1.21870i
$918$	−403.967	+	628.634i	−0.440051	+	0.684786i
$919$	535.279			0.582458			0.291229	−	0.956653i	$-0.405936\pi$
$919$	535.279			0.582458			0.291229	+	0.956653i	$0.405936\pi$
$920$			186.302i			0.202503i
$921$	−787.942	−	682.740i	−0.855529	−	0.741302i
$922$	2938.34			3.18692
$923$			23.1399i			0.0250703i
$924$	2184.33	−	2520.92i	2.36400	−	2.72826i
$925$	1208.32			1.30629
$926$			1151.44i			1.24346i
$927$	57.3248	+	398.636i	0.0618390	+	0.430028i
$928$	−2146.14			−2.31265
$929$		−	1272.51i		−	1.36977i	−0.728653	−	0.684883i	$-0.759853\pi$
$929$		−	1272.51i		−	1.36977i	0.728653	−	0.684883i	$-0.240147\pi$
$930$	−507.204	−	439.484i	−0.545381	−	0.472564i
$931$	70.7954			0.0760423
$932$			3340.19i			3.58389i
$933$	399.568	−	461.137i	0.428262	−	0.494252i
$934$	1390.15			1.48838
$935$		−	156.723i		−	0.167618i
$936$	−677.968	+	97.4935i	−0.724325	+	0.104160i
$937$	−807.050			−0.861313			−0.430656	−	0.902516i	$-0.641718\pi$
$937$	−807.050			−0.861313			−0.430656	+	0.902516i	$0.641718\pi$
$938$			728.160i			0.776290i
$939$	1266.51	+	1097.41i	1.34879	+	1.16871i
$940$	−87.7359			−0.0933361
$941$		−	61.4182i		−	0.0652691i	−0.999467	−	0.0326345i	$-0.989610\pi$
$941$		−	61.4182i		−	0.0652691i	0.999467	−	0.0326345i	$-0.0103897\pi$
$942$	602.467	−	695.300i	0.639561	−	0.738111i
$943$	0.106160			0.000112577
$944$			916.605i			0.970980i
$945$	−222.944	−	143.266i	−0.235919	−	0.151604i
$946$	−2536.25			−2.68103
$947$		−	730.787i		−	0.771687i	−0.922564	−	0.385843i	$-0.873911\pi$
$947$		−	730.787i		−	0.771687i	0.922564	−	0.385843i	$-0.126089\pi$
$948$	−1981.92	−	1717.30i	−2.09064	−	1.81150i
$949$	138.233			0.145662
$950$		−	1878.86i		−	1.97775i
$951$	−622.108	+	717.968i	−0.654161	+	0.754961i
$952$	−1294.89			−1.36018
$953$			1431.14i			1.50172i	0.660459	+	0.750862i	$0.270362\pi$
$953$			1431.14i			1.50172i	−0.660459	+	0.750862i	$0.729638\pi$
$954$	−166.521	−	1157.98i	−0.174550	−	1.21382i
$955$	−339.369			−0.355360
$956$			4821.90i			5.04383i
$957$	594.062	+	514.746i	0.620755	+	0.537874i
$958$	2916.28			3.04413
$959$			1355.06i			1.41299i
$960$	−679.308	+	783.982i	−0.707613	+	0.816648i
$961$	628.677			0.654191
$962$		−	580.240i		−	0.603160i
$963$	−450.941	+	64.8465i	−0.468267	+	0.0673380i
$964$	−1007.14			−1.04475
$965$		−	239.804i		−	0.248501i
$966$	−283.864	−	245.963i	−0.293855	−	0.254621i
$967$	681.976			0.705249			0.352625	−	0.935765i	$-0.385289\pi$
$967$	681.976			0.705249			0.352625	+	0.935765i	$0.385289\pi$
$968$			2831.52i			2.92513i
$969$	−299.039	+	345.117i	−0.308606	+	0.356158i
$970$	−290.314			−0.299293
$971$			487.120i			0.501668i	0.968030	+	0.250834i	$0.0807049\pi$
$971$			487.120i			0.501668i	−0.968030	+	0.250834i	$0.919295\pi$
$972$	−2414.62	−	1102.55i	−2.48417	−	1.13431i
$973$	−1119.76			−1.15083
$974$			980.191i			1.00636i
$975$	147.674	+	127.957i	0.151461	+	0.131238i
$976$	−4325.99			−4.43237
$977$		−	695.928i		−	0.712311i	−0.934427	−	0.356155i	$-0.884087\pi$
$977$		−	695.928i		−	0.712311i	0.934427	−	0.356155i	$-0.115913\pi$
$978$	1075.32	−	1241.02i	1.09951	−	1.26894i
$979$	−1502.79			−1.53503
$980$			52.8634i			0.0539422i
$981$	155.145	+	1078.88i	0.158150	+	1.09977i
$982$	−2816.60			−2.86823
$983$			1645.25i			1.67371i	0.547427	+	0.836854i	$0.315608\pi$
$983$			1645.25i			1.67371i	−0.547427	+	0.836854i	$0.684392\pi$
$984$	1.34236	+	1.16313i	0.00136418	+	0.00118204i
$985$	−84.6239			−0.0859126
$986$		−	481.439i		−	0.488275i
$987$	73.4167	−	84.7294i	0.0743837	−	0.0858454i
$988$	−660.408			−0.668429
$989$			209.043i			0.211368i
$990$	752.847	−	108.261i	0.760451	−	0.109355i
$991$	321.346			0.324265			0.162132	−	0.986769i	$-0.448163\pi$
$991$	321.346			0.324265			0.162132	+	0.986769i	$0.448163\pi$
$992$		−	4918.90i		−	4.95857i
$993$	48.3963	+	41.9346i	0.0487375	+	0.0422302i
$994$	−212.304			−0.213585
$995$			138.071i			0.138765i
$996$	511.624	−	590.460i	0.513679	−	0.592831i
$997$	787.495			0.789865			0.394932	−	0.918710i	$-0.370768\pi$
$997$	787.495			0.789865			0.394932	+	0.918710i	$0.370768\pi$
$998$			42.7641i			0.0428498i
$999$	770.504	−	1199.02i	0.771275	−	1.20022i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.3.b.a.47.14	yes	14
3.2	odd	2	inner	69.3.b.a.47.1	✓	14
4.3	odd	2		1104.3.g.b.737.12		14
12.11	even	2		1104.3.g.b.737.11		14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.3.b.a.47.1	✓	14	3.2	odd	2	inner
69.3.b.a.47.14	yes	14	1.1	even	1	trivial
1104.3.g.b.737.11		14	12.11	even	2
1104.3.g.b.737.12		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)

\(f(q)\)	\(=\)	\(q+3.86310i q^{2} +(-2.26727 - 1.96456i) q^{3} -10.9236 q^{4} +1.45241i q^{5} +(7.58928 - 8.75871i) q^{6} -6.75781 q^{7} -26.7464i q^{8} +(1.28104 + 8.90836i) q^{9} +O(q^{10})\)	\(q+3.86310i q^{2} +(-2.26727 - 1.96456i) q^{3} -10.9236 q^{4} +1.45241i q^{5} +(7.58928 - 8.75871i) q^{6} -6.75781 q^{7} -26.7464i q^{8} +(1.28104 + 8.90836i) q^{9} -5.61080 q^{10} +15.0621i q^{11} +(24.7667 + 21.4599i) q^{12} -2.84541 q^{13} -26.1061i q^{14} +(2.85333 - 3.29300i) q^{15} +59.6300 q^{16} +7.16409i q^{17} +(-34.4139 + 4.94881i) q^{18} -21.2472 q^{19} -15.8655i q^{20} +(15.3218 + 13.2761i) q^{21} -58.1863 q^{22} +4.79583i q^{23} +(-52.5449 + 60.6415i) q^{24} +22.8905 q^{25} -10.9921i q^{26} +(14.5965 - 22.7144i) q^{27} +73.8194 q^{28} +17.3958i q^{29} +(12.7212 + 11.0227i) q^{30} -39.8708 q^{31} +123.371i q^{32} +(29.5902 - 34.1498i) q^{33} -27.6756 q^{34} -9.81509i q^{35} +(-13.9936 - 97.3111i) q^{36} +52.7869 q^{37} -82.0803i q^{38} +(6.45133 + 5.58997i) q^{39} +38.8467 q^{40} -0.0221360i q^{41} +(-51.2869 + 59.1897i) q^{42} +43.5885 q^{43} -164.531i q^{44} +(-12.9386 + 1.86060i) q^{45} -18.5268 q^{46} -5.52999i q^{47} +(-135.197 - 117.146i) q^{48} -3.33198 q^{49} +88.4284i q^{50} +(14.0742 - 16.2429i) q^{51} +31.0821 q^{52} +33.6486i q^{53} +(87.7479 + 56.3878i) q^{54} -21.8762 q^{55} +180.747i q^{56} +(48.1733 + 41.7414i) q^{57} -67.2018 q^{58} +15.3715i q^{59} +(-31.1686 + 35.9713i) q^{60} -72.5473 q^{61} -154.025i q^{62} +(-8.65705 - 60.2010i) q^{63} -238.075 q^{64} -4.13270i q^{65} +(131.924 + 114.310i) q^{66} -27.8923 q^{67} -78.2574i q^{68} +(9.42168 - 10.8735i) q^{69} +37.9167 q^{70} -8.13234i q^{71} +(238.267 - 34.2634i) q^{72} -48.5809 q^{73} +203.921i q^{74} +(-51.8990 - 44.9697i) q^{75} +232.096 q^{76} -101.787i q^{77} +(-21.5946 + 24.9221i) q^{78} -80.0237 q^{79} +86.6071i q^{80} +(-77.7179 + 22.8240i) q^{81} +0.0855135 q^{82} -23.8409i q^{83} +(-167.369 - 145.022i) q^{84} -10.4052 q^{85} +168.387i q^{86} +(34.1750 - 39.4410i) q^{87} +402.856 q^{88} +99.7735i q^{89} +(-7.18768 - 49.9830i) q^{90} +19.2288 q^{91} -52.3876i q^{92} +(90.3978 + 78.3283i) q^{93} +21.3629 q^{94} -30.8596i q^{95} +(242.369 - 279.716i) q^{96} +51.7420 q^{97} -12.8718i q^{98} +(-134.178 + 19.2952i) 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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