LMFDB - Embedded newform 690.3.g.a.461.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [690,3,Mod(461,690)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(690, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 0]))

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

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

chi := DirichletCharacter("690.461");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 690 = 2 \cdot 3 \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	690.g (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:	$18.8011382409$
Analytic rank:	$0$
Dimension:	$56$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			461.4
Character	$\chi$	$=$	690.461
Dual form			690.3.g.a.461.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421i q^{2} +(-2.87483 - 0.857518i) q^{3} -2.00000 q^{4} -2.23607i q^{5} +(1.21271 - 4.06563i) q^{6} +5.06417 q^{7} -2.82843i q^{8} +(7.52932 + 4.93044i) q^{9} +O(q^{10})$	\(q+1.41421i q^{2} +(-2.87483 - 0.857518i) q^{3} -2.00000 q^{4} -2.23607i q^{5} +(1.21271 - 4.06563i) q^{6} +5.06417 q^{7} -2.82843i q^{8} +(7.52932 + 4.93044i) q^{9} +3.16228 q^{10} -18.8188i q^{11} +(5.74967 + 1.71504i) q^{12} -3.24812 q^{13} +7.16182i q^{14} +(-1.91747 + 6.42832i) q^{15} +4.00000 q^{16} +19.0579i q^{17} +(-6.97270 + 10.6481i) q^{18} -8.49496 q^{19} +4.47214i q^{20} +(-14.5586 - 4.34262i) q^{21} +26.6138 q^{22} +4.79583i q^{23} +(-2.42543 + 8.13125i) q^{24} -5.00000 q^{25} -4.59354i q^{26} +(-17.4176 - 20.6307i) q^{27} -10.1283 q^{28} +18.1025i q^{29} +(-9.09102 - 2.71171i) q^{30} -7.67595 q^{31} +5.65685i q^{32} +(-16.1375 + 54.1009i) q^{33} -26.9519 q^{34} -11.3238i q^{35} +(-15.0586 - 9.86089i) q^{36} +5.74178 q^{37} -12.0137i q^{38} +(9.33781 + 2.78532i) q^{39} -6.32456 q^{40} -48.7570i q^{41} +(6.14139 - 20.5890i) q^{42} +9.50005 q^{43} +37.6376i q^{44} +(11.0248 - 16.8361i) q^{45} -6.78233 q^{46} -52.9480i q^{47} +(-11.4993 - 3.43007i) q^{48} -23.3542 q^{49} -7.07107i q^{50} +(16.3425 - 54.7882i) q^{51} +6.49624 q^{52} -89.4628i q^{53} +(29.1763 - 24.6322i) q^{54} -42.0801 q^{55} -14.3236i q^{56} +(24.4216 + 7.28458i) q^{57} -25.6009 q^{58} -58.9584i q^{59} +(3.83494 - 12.8566i) q^{60} -94.2459 q^{61} -10.8554i q^{62} +(38.1298 + 24.9686i) q^{63} -8.00000 q^{64} +7.26302i q^{65} +(-76.5102 - 22.8218i) q^{66} -4.76905 q^{67} -38.1157i q^{68} +(4.11251 - 13.7872i) q^{69} +16.0143 q^{70} +16.3306i q^{71} +(13.9454 - 21.2961i) q^{72} -91.7883 q^{73} +8.12011i q^{74} +(14.3742 + 4.28759i) q^{75} +16.9899 q^{76} -95.3015i q^{77} +(-3.93904 + 13.2057i) q^{78} -110.541 q^{79} -8.94427i q^{80} +(32.3815 + 74.2458i) q^{81} +68.9528 q^{82} -15.5961i q^{83} +(29.1173 + 8.68523i) q^{84} +42.6147 q^{85} +13.4351i q^{86} +(15.5233 - 52.0418i) q^{87} -53.2276 q^{88} -98.9857i q^{89} +(23.8098 + 15.5914i) q^{90} -16.4490 q^{91} -9.59166i q^{92} +(22.0671 + 6.58227i) q^{93} +74.8798 q^{94} +18.9953i q^{95} +(4.85086 - 16.2625i) q^{96} -41.8721 q^{97} -33.0278i q^{98} +(92.7850 - 141.693i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 56 q - 8 q^{3} - 112 q^{4} + 16 q^{6} - 16 q^{7}+O(q^{10}) $ 56 * q - 8 * q^3 - 112 * q^4 + 16 * q^6 - 16 * q^7	$ 56 q - 8 q^{3} - 112 q^{4} + 16 q^{6} - 16 q^{7} + 16 q^{12} + 80 q^{13} - 40 q^{15} + 224 q^{16} - 32 q^{18} - 64 q^{19} + 56 q^{21} - 96 q^{22} - 32 q^{24} - 280 q^{25} + 40 q^{27} + 32 q^{28} - 80 q^{31} + 32 q^{33} + 192 q^{34} + 240 q^{37} - 56 q^{39} - 144 q^{43} - 32 q^{48} + 72 q^{49} - 24 q^{51} - 160 q^{52} + 16 q^{54} - 16 q^{57} + 80 q^{60} + 112 q^{61} - 64 q^{63} - 448 q^{64} + 160 q^{66} + 832 q^{67} + 64 q^{72} - 608 q^{73} + 40 q^{75} + 128 q^{76} - 320 q^{78} + 48 q^{79} - 32 q^{81} - 448 q^{82} - 112 q^{84} + 240 q^{85} + 200 q^{87} + 192 q^{88} + 80 q^{91} - 232 q^{93} + 160 q^{94} + 64 q^{96} - 448 q^{97} + 464 q^{99}+O(q^{100}) $ 56 * q - 8 * q^3 - 112 * q^4 + 16 * q^6 - 16 * q^7 + 16 * q^12 + 80 * q^13 - 40 * q^15 + 224 * q^16 - 32 * q^18 - 64 * q^19 + 56 * q^21 - 96 * q^22 - 32 * q^24 - 280 * q^25 + 40 * q^27 + 32 * q^28 - 80 * q^31 + 32 * q^33 + 192 * q^34 + 240 * q^37 - 56 * q^39 - 144 * q^43 - 32 * q^48 + 72 * q^49 - 24 * q^51 - 160 * q^52 + 16 * q^54 - 16 * q^57 + 80 * q^60 + 112 * q^61 - 64 * q^63 - 448 * q^64 + 160 * q^66 + 832 * q^67 + 64 * q^72 - 608 * q^73 + 40 * q^75 + 128 * q^76 - 320 * q^78 + 48 * q^79 - 32 * q^81 - 448 * q^82 - 112 * q^84 + 240 * q^85 + 200 * q^87 + 192 * q^88 + 80 * q^91 - 232 * q^93 + 160 * q^94 + 64 * q^96 - 448 * q^97 + 464 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/690\mathbb{Z}\right)^\times$.

$n$	$277$	$461$	$511$
$\chi(n)$	$1$	$-1$	$1$

Coefficient data

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

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

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

$2$			1.41421i			0.707107i
$2$			1.41421i			0.707107i
$3$	−2.87483	−	0.857518i	−0.958278	−	0.285839i
$3$	−2.87483	−	0.857518i	−0.958278	−	0.285839i
$4$	−2.00000			−0.500000
$5$		−	2.23607i		−	0.447214i
$5$		−	2.23607i		−	0.447214i
$6$	1.21271	−	4.06563i	0.202119	−	0.677605i
$7$	5.06417			0.723453			0.361726	−	0.932284i	$-0.382187\pi$
$7$	5.06417			0.723453			0.361726	+	0.932284i	$0.382187\pi$
$8$		−	2.82843i		−	0.353553i
$9$	7.52932	+	4.93044i	0.836592	+	0.547827i
$10$	3.16228			0.316228
$11$		−	18.8188i		−	1.71080i	−0.517969	−	0.855400i	$-0.673312\pi$
$11$		−	18.8188i		−	1.71080i	0.517969	−	0.855400i	$-0.326688\pi$
$12$	5.74967	+	1.71504i	0.479139	+	0.142920i
$13$	−3.24812			−0.249856			−0.124928	−	0.992166i	$-0.539870\pi$
$13$	−3.24812			−0.249856			−0.124928	+	0.992166i	$0.539870\pi$
$14$			7.16182i			0.511558i
$15$	−1.91747	+	6.42832i	−0.127831	+	0.428555i
$16$	4.00000			0.250000
$17$			19.0579i			1.12105i	0.828137	+	0.560526i	$0.189401\pi$
$17$			19.0579i			1.12105i	−0.828137	+	0.560526i	$0.810599\pi$
$18$	−6.97270	+	10.6481i	−0.387372	+	0.591560i
$19$	−8.49496			−0.447103			−0.223552	−	0.974692i	$-0.571765\pi$
$19$	−8.49496			−0.447103			−0.223552	+	0.974692i	$0.571765\pi$
$20$			4.47214i			0.223607i
$21$	−14.5586	−	4.34262i	−0.693268	−	0.206791i
$22$	26.6138			1.20972
$23$			4.79583i			0.208514i
$23$			4.79583i			0.208514i
$24$	−2.42543	+	8.13125i	−0.101060	+	0.338802i
$25$	−5.00000			−0.200000
$26$		−	4.59354i		−	0.176675i
$27$	−17.4176	−	20.6307i	−0.645096	−	0.764101i
$28$	−10.1283			−0.361726
$29$			18.1025i			0.624226i	0.950045	+	0.312113i	$0.101037\pi$
$29$			18.1025i			0.624226i	−0.950045	+	0.312113i	$0.898963\pi$
$30$	−9.09102	−	2.71171i	−0.303034	−	0.0903904i
$31$	−7.67595			−0.247611			−0.123806	−	0.992306i	$-0.539510\pi$
$31$	−7.67595			−0.247611			−0.123806	+	0.992306i	$0.539510\pi$
$32$			5.65685i			0.176777i
$33$	−16.1375	+	54.1009i	−0.489014	+	1.63942i
$34$	−26.9519			−0.792703
$35$		−	11.3238i		−	0.323538i
$36$	−15.0586	−	9.86089i	−0.418296	−	0.273914i
$37$	5.74178			0.155183			0.0775917	−	0.996985i	$-0.475277\pi$
$37$	5.74178			0.155183			0.0775917	+	0.996985i	$0.475277\pi$
$38$		−	12.0137i		−	0.316150i
$39$	9.33781	+	2.78532i	0.239431	+	0.0714186i
$40$	−6.32456			−0.158114
$41$		−	48.7570i		−	1.18920i	−0.804023	−	0.594598i	$-0.797311\pi$
$41$		−	48.7570i		−	1.18920i	0.804023	−	0.594598i	$-0.202689\pi$
$42$	6.14139	−	20.5890i	0.146224	−	0.490215i
$43$	9.50005			0.220931			0.110466	−	0.993880i	$-0.464766\pi$
$43$	9.50005			0.220931			0.110466	+	0.993880i	$0.464766\pi$
$44$			37.6376i			0.855400i
$45$	11.0248	−	16.8361i	0.244996	−	0.374135i
$46$	−6.78233			−0.147442
$47$		−	52.9480i		−	1.12655i	−0.826268	−	0.563277i	$-0.809541\pi$
$47$		−	52.9480i		−	1.12655i	0.826268	−	0.563277i	$-0.190459\pi$
$48$	−11.4993	−	3.43007i	−0.239569	−	0.0714599i
$49$	−23.3542			−0.476616
$50$		−	7.07107i		−	0.141421i
$51$	16.3425	−	54.7882i	0.320441	−	1.07428i
$52$	6.49624			0.124928
$53$		−	89.4628i		−	1.68798i	−0.536361	−	0.843989i	$-0.680201\pi$
$53$		−	89.4628i		−	1.68798i	0.536361	−	0.843989i	$-0.319799\pi$
$54$	29.1763	−	24.6322i	0.540301	−	0.456152i
$55$	−42.0801			−0.765093
$56$		−	14.3236i		−	0.255779i
$57$	24.4216	+	7.28458i	0.428449	+	0.127800i
$58$	−25.6009			−0.441394
$59$		−	58.9584i		−	0.999295i	−0.866229	−	0.499648i	$-0.833463\pi$
$59$		−	58.9584i		−	0.999295i	0.866229	−	0.499648i	$-0.166537\pi$
$60$	3.83494	−	12.8566i	0.0639156	−	0.214277i
$61$	−94.2459			−1.54501			−0.772507	−	0.635006i	$-0.780998\pi$
$61$	−94.2459			−1.54501			−0.772507	+	0.635006i	$0.780998\pi$
$62$		−	10.8554i		−	0.175088i
$63$	38.1298	+	24.9686i	0.605234	+	0.396327i
$64$	−8.00000			−0.125000
$65$			7.26302i			0.111739i
$66$	−76.5102	−	22.8218i	−1.15925	−	0.345785i
$67$	−4.76905			−0.0711799			−0.0355900	−	0.999366i	$-0.511331\pi$
$67$	−4.76905			−0.0711799			−0.0355900	+	0.999366i	$0.511331\pi$
$68$		−	38.1157i		−	0.560526i
$69$	4.11251	−	13.7872i	0.0596016	−	0.199815i
$70$	16.0143			0.228776
$71$			16.3306i			0.230008i	0.993365	+	0.115004i	$0.0366881\pi$
$71$			16.3306i			0.230008i	−0.993365	+	0.115004i	$0.963312\pi$
$72$	13.9454	−	21.2961i	0.193686	−	0.295780i
$73$	−91.7883			−1.25737			−0.628687	−	0.777659i	$-0.716407\pi$
$73$	−91.7883			−1.25737			−0.628687	+	0.777659i	$0.716407\pi$
$74$			8.12011i			0.109731i
$75$	14.3742	+	4.28759i	0.191656	+	0.0571679i
$76$	16.9899			0.223552
$77$		−	95.3015i		−	1.23768i
$78$	−3.93904	+	13.2057i	−0.0505006	+	0.169303i
$79$	−110.541			−1.39926			−0.699628	−	0.714507i	$-0.746651\pi$
$79$	−110.541			−1.39926			−0.699628	+	0.714507i	$0.746651\pi$
$80$		−	8.94427i		−	0.111803i
$81$	32.3815	+	74.2458i	0.399771	+	0.916615i
$82$	68.9528			0.840888
$83$		−	15.5961i		−	0.187905i	−0.995577	−	0.0939526i	$-0.970050\pi$
$83$		−	15.5961i		−	0.187905i	0.995577	−	0.0939526i	$-0.0299502\pi$
$84$	29.1173	+	8.68523i	0.346634	+	0.103396i
$85$	42.6147			0.501349
$86$			13.4351i			0.156222i
$87$	15.5233	−	52.0418i	0.178428	−	0.598181i
$88$	−53.2276			−0.604859
$89$		−	98.9857i		−	1.11220i	−0.831116	−	0.556100i	$-0.812297\pi$
$89$		−	98.9857i		−	1.11220i	0.831116	−	0.556100i	$-0.187703\pi$
$90$	23.8098	+	15.5914i	0.264553	+	0.173238i
$91$	−16.4490			−0.180759
$92$		−	9.59166i		−	0.104257i
$93$	22.0671	+	6.58227i	0.237280	+	0.0707771i
$94$	74.8798			0.796593
$95$			18.9953i			0.199951i
$96$	4.85086	−	16.2625i	0.0505298	−	0.169401i
$97$	−41.8721			−0.431671			−0.215835	−	0.976430i	$-0.569247\pi$
$97$	−41.8721			−0.431671			−0.215835	+	0.976430i	$0.569247\pi$
$98$		−	33.0278i		−	0.337019i
$99$	92.7850	−	141.693i	0.937222	−	1.43124i
$100$	10.0000			0.100000
$101$		−	17.7056i		−	0.175303i	−0.996151	−	0.0876517i	$-0.972064\pi$
$101$		−	17.7056i		−	0.175303i	0.996151	−	0.0876517i	$-0.0279363\pi$
$102$	77.4822	+	23.1118i	0.759629	+	0.226586i
$103$	69.5747			0.675482			0.337741	−	0.941239i	$-0.390337\pi$
$103$	69.5747			0.675482			0.337741	+	0.941239i	$0.390337\pi$
$104$			9.18708i			0.0883373i
$105$	−9.71039	+	32.5541i	−0.0924799	+	0.310039i
$106$	126.520			1.19358
$107$		−	56.0057i		−	0.523418i	−0.965147	−	0.261709i	$-0.915714\pi$
$107$		−	56.0057i		−	0.523418i	0.965147	−	0.261709i	$-0.0842861\pi$
$108$	34.8352	+	41.2615i	0.322548	+	0.382051i
$109$	−23.3967			−0.214648			−0.107324	−	0.994224i	$-0.534228\pi$
$109$	−23.3967			−0.214648			−0.107324	+	0.994224i	$0.534228\pi$
$110$		−	59.5102i		−	0.541002i
$111$	−16.5067	−	4.92368i	−0.148709	−	0.0443575i
$112$	20.2567			0.180863
$113$			189.931i			1.68080i	0.541966	+	0.840401i	$0.317680\pi$
$113$			189.931i			1.68080i	−0.541966	+	0.840401i	$0.682320\pi$
$114$	−10.3020	+	34.5373i	−0.0903680	+	0.302959i
$115$	10.7238			0.0932505
$116$		−	36.2051i		−	0.312113i
$117$	−24.4562	−	16.0147i	−0.209027	−	0.136878i
$118$	83.3798			0.706609
$119$			96.5123i			0.811027i
$120$	18.1820	+	5.42342i	0.151517	+	0.0451952i
$121$	−233.147			−1.92683
$122$		−	133.284i		−	1.09249i
$123$	−41.8100	+	140.168i	−0.339919	+	1.13958i
$124$	15.3519			0.123806
$125$			11.1803i			0.0894427i
$126$	−35.3109	+	53.9236i	−0.280245	+	0.427965i
$127$	201.992			1.59049			0.795245	−	0.606288i	$-0.207342\pi$
$127$	201.992			1.59049			0.795245	+	0.606288i	$0.207342\pi$
$128$		−	11.3137i		−	0.0883883i
$129$	−27.3111	−	8.14647i	−0.211714	−	0.0631509i
$130$	−10.2715			−0.0790113
$131$			131.815i			1.00622i	0.864223	+	0.503110i	$0.167811\pi$
$131$			131.815i			1.00622i	−0.864223	+	0.503110i	$0.832189\pi$
$132$	32.2749	−	108.202i	0.244507	−	0.819710i
$133$	−43.0199			−0.323458
$134$		−	6.74446i		−	0.0503318i
$135$	−46.1317	+	38.9469i	−0.341716	+	0.288496i
$136$	53.9038			0.396351
$137$			69.1354i			0.504638i	0.967644	+	0.252319i	$0.0811932\pi$
$137$			69.1354i			0.504638i	−0.967644	+	0.252319i	$0.918807\pi$
$138$	19.4981	+	5.81597i	0.141290	+	0.0421447i
$139$	51.2040			0.368374			0.184187	−	0.982891i	$-0.441035\pi$
$139$	51.2040			0.368374			0.184187	+	0.982891i	$0.441035\pi$
$140$			22.6476i			0.161769i
$141$	−45.4039	+	152.217i	−0.322013	+	1.07955i
$142$	−23.0949			−0.162640
$143$			61.1257i			0.427453i
$144$	30.1173	+	19.7218i	0.209148	+	0.136957i
$145$	40.4785			0.279162
$146$		−	129.808i		−	0.889097i
$147$	67.1394	+	20.0267i	0.456731	+	0.136236i
$148$	−11.4836			−0.0775917
$149$		−	219.870i		−	1.47564i	−0.674998	−	0.737820i	$-0.735856\pi$
$149$		−	219.870i		−	1.47564i	0.674998	−	0.737820i	$-0.264144\pi$
$150$	−6.06357	+	20.3281i	−0.0404238	+	0.135521i
$151$	−214.804			−1.42255			−0.711273	−	0.702916i	$-0.751881\pi$
$151$	−214.804			−1.42255			−0.711273	+	0.702916i	$0.751881\pi$
$152$			24.0274i			0.158075i
$153$	−93.9638	+	143.493i	−0.614142	+	0.937862i
$154$	134.777			0.875174
$155$			17.1640i			0.110735i
$156$	−18.6756	−	5.57065i	−0.119715	−	0.0357093i
$157$	99.1915			0.631793			0.315896	−	0.948794i	$-0.397695\pi$
$157$	99.1915			0.631793			0.315896	+	0.948794i	$0.397695\pi$
$158$		−	156.329i		−	0.989423i
$159$	−76.7160	+	257.191i	−0.482491	+	1.61755i
$160$	12.6491			0.0790569
$161$			24.2869i			0.150850i
$162$	−104.999	+	45.7943i	−0.648145	+	0.282681i
$163$	−123.345			−0.756716			−0.378358	−	0.925659i	$-0.623511\pi$
$163$	−123.345			−0.756716			−0.378358	+	0.925659i	$0.623511\pi$
$164$			97.5140i			0.594598i
$165$	120.973	+	36.0845i	0.733171	+	0.218694i
$166$	22.0563			0.132869
$167$			11.3950i			0.0682337i	0.999418	+	0.0341169i	$0.0108618\pi$
$167$			11.3950i			0.0682337i	−0.999418	+	0.0341169i	$0.989138\pi$
$168$	−12.2828	+	41.1780i	−0.0731118	+	0.245107i
$169$	−158.450			−0.937572
$170$			60.2663i			0.354508i
$171$	−63.9613	−	41.8839i	−0.374043	−	0.244935i
$172$	−19.0001			−0.110466
$173$			59.2460i			0.342462i	0.985231	+	0.171231i	$0.0547745\pi$
$173$			59.2460i			0.342462i	−0.985231	+	0.171231i	$0.945225\pi$
$174$	73.5982	+	21.9532i	0.422978	+	0.126168i
$175$	−25.3208			−0.144691
$176$		−	75.2752i		−	0.427700i
$177$	−50.5579	+	169.496i	−0.285638	+	0.957602i
$178$	139.987			0.786444
$179$		−	137.958i		−	0.770712i	−0.922768	−	0.385356i	$-0.874079\pi$
$179$		−	137.958i		−	0.770712i	0.922768	−	0.385356i	$-0.125921\pi$
$180$	−22.0496	+	33.6722i	−0.122498	+	0.187068i
$181$	−272.565			−1.50588			−0.752941	−	0.658088i	$-0.771366\pi$
$181$	−272.565			−1.50588			−0.752941	+	0.658088i	$0.771366\pi$
$182$		−	23.2625i		−	0.127816i
$183$	270.941	+	80.8176i	1.48055	+	0.441626i
$184$	13.5647			0.0737210
$185$		−	12.8390i		−	0.0694001i
$186$	−9.30874	+	31.2076i	−0.0500470	+	0.167783i
$187$	358.646			1.91789
$188$			105.896i			0.563277i
$189$	−88.2057	−	104.478i	−0.466697	−	0.552791i
$190$	−26.8634			−0.141386
$191$		−	85.7231i		−	0.448812i	−0.974496	−	0.224406i	$-0.927956\pi$
$191$		−	85.7231i		−	0.448812i	0.974496	−	0.224406i	$-0.0720442\pi$
$192$	22.9987	+	6.86015i	0.119785	+	0.0357299i
$193$	197.316			1.02236			0.511182	−	0.859472i	$-0.329208\pi$
$193$	197.316			1.02236			0.511182	+	0.859472i	$0.329208\pi$
$194$		−	59.2160i		−	0.305237i
$195$	6.22817	−	20.8800i	0.0319394	−	0.107077i
$196$	46.7084			0.238308
$197$		−	196.588i		−	0.997910i	−0.866628	−	0.498955i	$-0.833717\pi$
$197$		−	196.588i		−	0.997910i	0.866628	−	0.498955i	$-0.166283\pi$
$198$	200.384	+	131.218i	1.01204	+	0.662716i
$199$	−13.6075			−0.0683794			−0.0341897	−	0.999415i	$-0.510885\pi$
$199$	−13.6075			−0.0683794			−0.0341897	+	0.999415i	$0.510885\pi$
$200$			14.1421i			0.0707107i
$201$	13.7102	+	4.08955i	0.0682101	+	0.0203460i
$202$	25.0396			0.123958
$203$			91.6743i			0.451598i
$204$	−32.6850	+	109.576i	−0.160220	+	0.537139i
$205$	−109.024			−0.531824
$206$			98.3935i			0.477638i
$207$	−23.6456	+	36.1094i	−0.114230	+	0.174441i
$208$	−12.9925			−0.0624639
$209$			159.865i			0.764904i
$210$	−46.0384	−	13.7326i	−0.219231	−	0.0653932i
$211$	318.759			1.51071			0.755353	−	0.655318i	$-0.227465\pi$
$211$	318.759			1.51071			0.755353	+	0.655318i	$0.227465\pi$
$212$			178.926i			0.843989i
$213$	14.0038	−	46.9477i	0.0657454	−	0.220412i
$214$	79.2041			0.370113
$215$		−	21.2428i		−	0.0988035i
$216$	−58.3525	+	49.2644i	−0.270151	+	0.228076i
$217$	−38.8723			−0.179135
$218$		−	33.0879i		−	0.151779i
$219$	263.876	+	78.7101i	1.20491	+	0.359407i
$220$	84.1602			0.382546
$221$		−	61.9023i		−	0.280101i
$222$	6.96314	−	23.3439i	0.0313655	−	0.105153i
$223$	−109.529			−0.491161			−0.245580	−	0.969376i	$-0.578979\pi$
$223$	−109.529			−0.491161			−0.245580	+	0.969376i	$0.578979\pi$
$224$			28.6473i			0.127890i
$225$	−37.6466	−	24.6522i	−0.167318	−	0.109565i
$226$	−268.602			−1.18851
$227$		−	123.236i		−	0.542891i	−0.962454	−	0.271446i	$-0.912498\pi$
$227$		−	123.236i		−	0.542891i	0.962454	−	0.271446i	$-0.0875018\pi$
$228$	−48.8432	−	14.5692i	−0.214224	−	0.0638999i
$229$	−408.810			−1.78520			−0.892598	−	0.450853i	$-0.851120\pi$
$229$	−408.810			−1.78520			−0.892598	+	0.450853i	$0.851120\pi$
$230$			15.1658i			0.0659380i
$231$	−81.7228	+	273.976i	−0.353778	+	1.18604i
$232$	51.2017			0.220697
$233$		−	310.433i		−	1.33233i	−0.745804	−	0.666165i	$-0.767935\pi$
$233$		−	310.433i		−	1.33233i	0.745804	−	0.666165i	$-0.232065\pi$
$234$	22.6482	−	34.5862i	0.0967871	−	0.147804i
$235$	−118.395			−0.503810
$236$			117.917i			0.499648i
$237$	317.787	+	94.7911i	1.34088	+	0.399963i
$238$	−136.489			−0.573483
$239$			306.119i			1.28083i	0.768028	+	0.640417i	$0.221238\pi$
$239$			306.119i			1.28083i	−0.768028	+	0.640417i	$0.778762\pi$
$240$	−7.66988	+	25.7133i	−0.0319578	+	0.107139i
$241$	−336.911			−1.39797			−0.698986	−	0.715136i	$-0.746365\pi$
$241$	−336.911			−1.39797			−0.698986	+	0.715136i	$0.746365\pi$
$242$		−	329.720i		−	1.36248i
$243$	−29.4241	−	241.212i	−0.121087	−	0.992642i
$244$	188.492			0.772507
$245$			52.2216i			0.213149i
$246$	−198.228	−	59.1283i	−0.805804	−	0.240359i
$247$	27.5927			0.111711
$248$			21.7109i			0.0875439i
$249$	−13.3740	+	44.8363i	−0.0537107	+	0.180065i
$250$	−15.8114			−0.0632456
$251$		−	384.416i		−	1.53154i	−0.643115	−	0.765770i	$-0.722358\pi$
$251$		−	384.416i		−	1.53154i	0.643115	−	0.765770i	$-0.277642\pi$
$252$	−76.2595	−	49.9372i	−0.302617	−	0.198163i
$253$	90.2518			0.356726
$254$			285.660i			1.12465i
$255$	−122.510	−	36.5429i	−0.480432	−	0.143305i
$256$	16.0000			0.0625000
$257$			291.537i			1.13439i	0.823585	+	0.567193i	$0.191971\pi$
$257$			291.537i			1.13439i	−0.823585	+	0.567193i	$0.808029\pi$
$258$	11.5208	−	38.6237i	0.0446544	−	0.149704i
$259$	29.0774			0.112268
$260$		−	14.5260i		−	0.0558694i
$261$	−89.2536	+	136.300i	−0.341968	+	0.522222i
$262$	−186.414			−0.711504
$263$		−	48.4412i		−	0.184187i	−0.995750	−	0.0920935i	$-0.970644\pi$
$263$		−	48.4412i		−	0.184187i	0.995750	−	0.0920935i	$-0.0293559\pi$
$264$	153.020	+	45.6436i	0.579623	+	0.172893i
$265$	−200.045			−0.754887
$266$		−	60.8393i		−	0.228719i
$267$	−84.8821	+	284.567i	−0.317910	+	1.06580i
$268$	9.53811			0.0355900
$269$		−	200.718i		−	0.746165i	−0.927798	−	0.373082i	$-0.878301\pi$
$269$		−	200.718i		−	0.746165i	0.927798	−	0.373082i	$-0.121699\pi$
$270$	−55.0793	−	65.2401i	−0.203997	−	0.241630i
$271$	394.542			1.45587			0.727936	−	0.685645i	$-0.240480\pi$
$271$	394.542			1.45587			0.727936	+	0.685645i	$0.240480\pi$
$272$			76.2315i			0.280263i
$273$	47.2882	+	14.1054i	0.173217	+	0.0516680i
$274$	−97.7722			−0.356833
$275$			94.0940i			0.342160i
$276$	−8.22503	+	27.5744i	−0.0298008	+	0.0999073i
$277$	167.087			0.603204			0.301602	−	0.953434i	$-0.402479\pi$
$277$	167.087			0.603204			0.301602	+	0.953434i	$0.402479\pi$
$278$			72.4133i			0.260480i
$279$	−57.7948	−	37.8459i	−0.207150	−	0.135648i
$280$	−32.0286			−0.114388
$281$			282.594i			1.00567i	0.864381	+	0.502837i	$0.167710\pi$
$281$			282.594i			1.00567i	−0.864381	+	0.502837i	$0.832290\pi$
$282$	−215.267	−	64.2108i	−0.763357	−	0.227698i
$283$	−116.160			−0.410461			−0.205230	−	0.978714i	$-0.565794\pi$
$283$	−116.160			−0.410461			−0.205230	+	0.978714i	$0.565794\pi$
$284$		−	32.6612i		−	0.115004i
$285$	16.2888	−	54.6083i	0.0571538	−	0.191608i
$286$	−86.4448			−0.302255
$287$		−	246.914i		−	0.860327i
$288$	−27.8908	+	42.5923i	−0.0968431	+	0.147890i
$289$	−74.2025			−0.256756
$290$			57.2453i			0.197398i
$291$	120.375	+	35.9061i	0.413660	+	0.123389i
$292$	183.577			0.628687
$293$			437.333i			1.49261i	0.665607	+	0.746303i	$0.268173\pi$
$293$			437.333i			1.49261i	−0.665607	+	0.746303i	$0.731827\pi$
$294$	−28.3220	+	94.9495i	−0.0963332	+	0.322957i
$295$	−131.835			−0.446898
$296$		−	16.2402i		−	0.0548656i
$297$	−388.246	+	327.778i	−1.30722	+	1.10363i
$298$	310.943			1.04343
$299$		−	15.5774i		−	0.0520985i
$300$	−28.7483	−	8.57518i	−0.0958278	−	0.0285839i
$301$	48.1099			0.159833
$302$		−	303.779i		−	1.00589i
$303$	−15.1829	+	50.9008i	−0.0501086	+	0.167989i
$304$	−33.9798			−0.111776
$305$			210.740i			0.690951i
$306$	−202.930	−	132.885i	−0.663169	−	0.434264i
$307$	72.6694			0.236708			0.118354	−	0.992971i	$-0.462238\pi$
$307$	72.6694			0.236708			0.118354	+	0.992971i	$0.462238\pi$
$308$			190.603i			0.618841i
$309$	−200.016	−	59.6616i	−0.647300	−	0.193080i
$310$	−24.2735			−0.0783016
$311$		−	260.179i		−	0.836589i	−0.908312	−	0.418294i	$-0.862628\pi$
$311$		−	260.179i		−	0.836589i	0.908312	−	0.418294i	$-0.137372\pi$
$312$	7.87809	−	26.4113i	0.0252503	−	0.0846516i
$313$	376.440			1.20268			0.601342	−	0.798992i	$-0.294633\pi$
$313$	376.440			1.20268			0.601342	+	0.798992i	$0.294633\pi$
$314$			140.278i			0.446745i
$315$	55.8315	−	85.2607i	0.177243	−	0.270669i
$316$	221.082			0.699628
$317$			312.850i			0.986907i	0.869772	+	0.493454i	$0.164266\pi$
$317$			312.850i			0.986907i	−0.869772	+	0.493454i	$0.835734\pi$
$318$	−363.722	−	108.493i	−1.14378	−	0.341172i
$319$	340.668			1.06792
$320$			17.8885i			0.0559017i
$321$	−48.0260	+	161.007i	−0.149614	+	0.501580i
$322$	−34.3469			−0.106667
$323$		−	161.896i		−	0.501225i
$324$	−64.7629	−	148.492i	−0.199886	−	0.458308i
$325$	16.2406			0.0499711
$326$		−	174.436i		−	0.535079i
$327$	67.2615	+	20.0631i	0.205693	+	0.0613550i
$328$	−137.906			−0.420444
$329$		−	268.138i		−	0.815008i
$330$	−51.0311	+	171.082i	−0.154640	+	0.518430i
$331$	104.128			0.314585			0.157292	−	0.987552i	$-0.449723\pi$
$331$	104.128			0.314585			0.157292	+	0.987552i	$0.449723\pi$
$332$			31.1923i			0.0939526i
$333$	43.2317	+	28.3095i	0.129825	+	0.0850136i
$334$	−16.1150			−0.0482485
$335$			10.6639i			0.0318326i
$336$	−58.2345	−	17.3705i	−0.173317	−	0.0516978i
$337$	114.015			0.338324			0.169162	−	0.985588i	$-0.445894\pi$
$337$	114.015			0.338324			0.169162	+	0.985588i	$0.445894\pi$
$338$		−	224.082i		−	0.662964i
$339$	162.869	−	546.018i	0.480439	−	1.61067i
$340$	−85.2294			−0.250675
$341$			144.452i			0.423614i
$342$	59.2328	−	90.4549i	0.173195	−	0.264488i
$343$	−366.414			−1.06826
$344$		−	26.8702i		−	0.0781111i
$345$	−30.8291	−	9.19586i	−0.0893598	−	0.0266547i
$346$	−83.7865			−0.242157
$347$			15.3420i			0.0442131i	0.999756	+	0.0221066i	$0.00703731\pi$
$347$			15.3420i			0.0442131i	−0.999756	+	0.0221066i	$0.992963\pi$
$348$	−31.0465	+	104.084i	−0.0892142	+	0.299091i
$349$	645.031			1.84823			0.924113	−	0.382119i	$-0.124806\pi$
$349$	645.031			1.84823			0.924113	+	0.382119i	$0.124806\pi$
$350$		−	35.8091i		−	0.102312i
$351$	56.5745	+	67.0111i	0.161181	+	0.190915i
$352$	106.455			0.302429
$353$			261.860i			0.741813i	0.928670	+	0.370907i	$0.120953\pi$
$353$			261.860i			0.741813i	−0.928670	+	0.370907i	$0.879047\pi$
$354$	−239.703	−	71.4997i	−0.677127	−	0.201977i
$355$	36.5163			0.102863
$356$			197.971i			0.556100i
$357$	82.7610	−	277.457i	0.231824	−	0.777189i
$358$	195.101			0.544976
$359$		−	500.283i		−	1.39354i	−0.717292	−	0.696772i	$-0.754619\pi$
$359$		−	500.283i		−	1.39354i	0.717292	−	0.696772i	$-0.245381\pi$
$360$	−47.6196	−	31.1829i	−0.132277	−	0.0866191i
$361$	−288.836			−0.800099
$362$		−	385.465i		−	1.06482i
$363$	670.258	+	199.928i	1.84644	+	0.550765i
$364$	32.8981			0.0903793
$365$			205.245i			0.562315i
$366$	−114.293	+	383.169i	−0.312277	+	1.04691i
$367$	664.129			1.80962			0.904809	−	0.425819i	$-0.140014\pi$
$367$	664.129			1.80962			0.904809	+	0.425819i	$0.140014\pi$
$368$			19.1833i			0.0521286i
$369$	240.394	−	367.107i	0.651474	−	0.994871i
$370$	18.1571			0.0490733
$371$		−	453.055i		−	1.22117i
$372$	−44.1342	−	13.1645i	−0.118640	−	0.0353886i
$373$	−120.804			−0.323871			−0.161936	−	0.986801i	$-0.551774\pi$
$373$	−120.804			−0.323871			−0.161936	+	0.986801i	$0.551774\pi$
$374$			507.202i			1.35616i
$375$	9.58735	−	32.1416i	0.0255663	−	0.0857109i
$376$	−149.760			−0.398297
$377$		−	58.7993i		−	0.155966i
$378$	147.754	−	124.742i	0.390882	−	0.330004i
$379$	346.819			0.915089			0.457545	−	0.889187i	$-0.348729\pi$
$379$	346.819			0.915089			0.457545	+	0.889187i	$0.348729\pi$
$380$		−	37.9906i		−	0.0999753i
$381$	−580.694	−	173.212i	−1.52413	−	0.454625i
$382$	121.231			0.317358
$383$		−	340.037i		−	0.887825i	−0.896070	−	0.443913i	$-0.853590\pi$
$383$		−	340.037i		−	0.887825i	0.896070	−	0.443913i	$-0.146410\pi$
$384$	−9.70171	+	32.5250i	−0.0252649	+	0.0847006i
$385$	−213.101			−0.553508
$386$			279.047i			0.722921i
$387$	71.5290	+	46.8395i	0.184829	+	0.121032i
$388$	83.7441			0.215835
$389$			182.872i			0.470108i	0.971982	+	0.235054i	$0.0755267\pi$
$389$			182.872i			0.470108i	−0.971982	+	0.235054i	$0.924473\pi$
$390$	29.5287	+	8.80797i	0.0757147	+	0.0225845i
$391$	−91.3983			−0.233755
$392$			66.0557i			0.168509i
$393$	113.034	−	378.945i	0.287617	−	0.964237i
$394$	278.018			0.705629
$395$			247.178i			0.625766i
$396$	−185.570	+	283.386i	−0.468611	+	0.715620i
$397$	148.814			0.374845			0.187423	−	0.982279i	$-0.439987\pi$
$397$	148.814			0.374845			0.187423	+	0.982279i	$0.439987\pi$
$398$		−	19.2439i		−	0.0483515i
$399$	123.675	+	36.8904i	0.309962	+	0.0924570i
$400$	−20.0000			−0.0500000
$401$			597.537i			1.49012i	0.666999	+	0.745058i	$0.267578\pi$
$401$			597.537i			1.49012i	−0.666999	+	0.745058i	$0.732422\pi$
$402$	−5.78350	+	19.3892i	−0.0143868	+	0.0482318i
$403$	24.9324			0.0618671
$404$			35.4113i			0.0876517i
$405$	166.019	−	72.4071i	0.409923	−	0.178783i
$406$	−129.647			−0.319328
$407$		−	108.053i		−	0.265488i
$408$	−154.964	−	46.2235i	−0.379815	−	0.113293i
$409$	−272.829			−0.667065			−0.333532	−	0.942739i	$-0.608241\pi$
$409$	−272.829			−0.667065			−0.333532	+	0.942739i	$0.608241\pi$
$410$		−	154.183i		−	0.376057i
$411$	59.2849	−	198.753i	0.144245	−	0.483583i
$412$	−139.149			−0.337741
$413$		−	298.575i		−	0.722943i
$414$	−51.0664	−	33.4399i	−0.123349	−	0.0807727i
$415$	−34.8740			−0.0840337
$416$		−	18.3742i		−	0.0441686i
$417$	−147.203	−	43.9083i	−0.353004	−	0.105296i
$418$	−226.083			−0.540869
$419$			262.152i			0.625661i	0.949809	+	0.312831i	$0.101277\pi$
$419$			262.152i			0.625661i	−0.949809	+	0.312831i	$0.898723\pi$
$420$	19.4208	−	65.1082i	0.0462399	−	0.155020i
$421$	630.483			1.49758			0.748792	−	0.662806i	$-0.230634\pi$
$421$	630.483			1.49758			0.748792	+	0.662806i	$0.230634\pi$
$422$			450.794i			1.06823i
$423$	261.057	−	398.663i	0.617156	−	0.942465i
$424$	−253.039			−0.596790
$425$		−	95.2894i		−	0.224210i
$426$	66.3940	+	19.8043i	0.155855	+	0.0464890i
$427$	−477.277			−1.11774
$428$			112.011i			0.261709i
$429$	52.4164	−	175.726i	0.122183	−	0.409618i
$430$	30.0418			0.0698646
$431$		−	69.6557i		−	0.161614i	−0.996730	−	0.0808071i	$-0.974250\pi$
$431$		−	69.6557i		−	0.161614i	0.996730	−	0.0808071i	$-0.0257498\pi$
$432$	−69.6704	−	82.5229i	−0.161274	−	0.191025i
$433$	−372.470			−0.860209			−0.430104	−	0.902779i	$-0.641523\pi$
$433$	−372.470			−0.860209			−0.430104	+	0.902779i	$0.641523\pi$
$434$		−	54.9738i		−	0.126668i
$435$	−116.369	−	34.7111i	−0.267515	−	0.0797956i
$436$	46.7934			0.107324
$437$		−	40.7404i		−	0.0932274i
$438$	−111.313	+	373.177i	−0.254139	+	0.852002i
$439$	337.965			0.769851			0.384926	−	0.922948i	$-0.374227\pi$
$439$	337.965			0.769851			0.384926	+	0.922948i	$0.374227\pi$
$440$			119.020i			0.270501i
$441$	−175.841	−	115.147i	−0.398733	−	0.261103i
$442$	87.5431			0.198061
$443$		−	715.113i		−	1.61425i	−0.590380	−	0.807126i	$-0.701022\pi$
$443$		−	715.113i		−	1.61425i	0.590380	−	0.807126i	$-0.298978\pi$
$444$	33.0133	+	9.84737i	0.0743543	+	0.0221788i
$445$	−221.339			−0.497391
$446$		−	154.897i		−	0.347303i
$447$	−188.543	+	632.090i	−0.421796	+	1.41407i
$448$	−40.5133			−0.0904316
$449$		−	684.244i		−	1.52393i	−0.647619	−	0.761964i	$-0.724235\pi$
$449$		−	684.244i		−	1.52393i	0.647619	−	0.761964i	$-0.275765\pi$
$450$	34.8635	−	53.2404i	0.0774744	−	0.118312i
$451$	−917.548			−2.03448
$452$		−	379.861i		−	0.840401i
$453$	617.527	+	184.199i	1.36319	+	0.406620i
$454$	174.283			0.383882
$455$			36.7812i			0.0808377i
$456$	20.6039	−	69.0747i	0.0451840	−	0.151480i
$457$	277.770			0.607813			0.303906	−	0.952702i	$-0.401709\pi$
$457$	277.770			0.607813			0.303906	+	0.952702i	$0.401709\pi$
$458$		−	578.145i		−	1.26232i
$459$	393.178	−	331.942i	0.856597	−	0.723186i
$460$	−21.4476			−0.0466252
$461$		−	51.6571i		−	0.112054i	−0.998429	−	0.0560272i	$-0.982157\pi$
$461$		−	51.6571i		−	0.112054i	0.998429	−	0.0560272i	$-0.0178433\pi$
$462$	−387.461	−	115.574i	−0.838659	−	0.250159i
$463$	274.835			0.593595			0.296798	−	0.954940i	$-0.404081\pi$
$463$	274.835			0.593595			0.296798	+	0.954940i	$0.404081\pi$
$464$			72.4102i			0.156056i
$465$	14.7184	−	49.3435i	0.0316525	−	0.106115i
$466$	439.018			0.942100
$467$			908.495i			1.94539i	0.232096	+	0.972693i	$0.425442\pi$
$467$			908.495i			1.94539i	−0.232096	+	0.972693i	$0.574558\pi$
$468$	48.9123	+	32.0294i	0.104514	+	0.0684388i
$469$	−24.1513			−0.0514953
$470$		−	167.436i		−	0.356247i
$471$	−285.159	−	85.0585i	−0.605433	−	0.180591i
$472$	−166.760			−0.353304
$473$		−	178.779i		−	0.377969i
$474$	−134.055	+	449.419i	−0.282816	+	0.948142i
$475$	42.4748			0.0894206
$476$		−	193.025i		−	0.405514i
$477$	441.091	−	673.595i	0.924720	−	1.41215i
$478$	−432.918			−0.905686
$479$		−	59.4517i		−	0.124116i	−0.998073	−	0.0620582i	$-0.980234\pi$
$479$		−	59.4517i		−	0.124116i	0.998073	−	0.0620582i	$-0.0197664\pi$
$480$	−36.3641	−	10.8468i	−0.0757585	−	0.0225976i
$481$	−18.6500			−0.0387734
$482$		−	476.464i		−	0.988515i
$483$	20.8265	−	69.8208i	0.0431190	−	0.144556i
$484$	466.294			0.963417
$485$			93.6288i			0.193049i
$486$	341.125	−	41.6120i	0.701904	−	0.0856213i
$487$	706.752			1.45124			0.725618	−	0.688097i	$-0.241554\pi$
$487$	706.752			1.45124			0.725618	+	0.688097i	$0.241554\pi$
$488$			266.568i			0.546245i
$489$	354.595	+	105.770i	0.725144	+	0.216299i
$490$	−73.8525			−0.150719
$491$			785.728i			1.60026i	0.599827	+	0.800130i	$0.295236\pi$
$491$			785.728i			1.60026i	−0.599827	+	0.800130i	$0.704764\pi$
$492$	83.6201	−	280.337i	0.169960	−	0.569790i
$493$	−344.996			−0.699789
$494$			39.0219i			0.0789917i
$495$	−316.835	−	207.474i	−0.640070	−	0.419139i
$496$	−30.7038			−0.0619029
$497$			82.7008i			0.166400i
$498$	−63.4080	−	18.9136i	−0.127325	−	0.0379792i
$499$	628.387			1.25929			0.629646	−	0.776882i	$-0.283200\pi$
$499$	628.387			1.25929			0.629646	+	0.776882i	$0.283200\pi$
$500$		−	22.3607i		−	0.0447214i
$501$	9.77145	−	32.7588i	0.0195039	−	0.0653868i
$502$	543.647			1.08296
$503$		−	436.393i		−	0.867581i	−0.901014	−	0.433791i	$-0.857176\pi$
$503$		−	436.393i		−	0.867581i	0.901014	−	0.433791i	$-0.142824\pi$
$504$	70.6219	−	107.847i	0.140123	−	0.213983i
$505$	−39.5910			−0.0783981
$506$			127.635i			0.252244i
$507$	455.516	+	135.874i	0.898454	+	0.267995i
$508$	−403.984			−0.795245
$509$		−	152.541i		−	0.299687i	−0.988710	−	0.149844i	$-0.952123\pi$
$509$		−	152.541i		−	0.299687i	0.988710	−	0.149844i	$-0.0478771\pi$
$510$	51.6794	−	173.255i	0.101332	−	0.339717i
$511$	−464.831			−0.909650
$512$			22.6274i			0.0441942i
$513$	147.962	+	175.257i	0.288425	+	0.341632i
$514$	−412.296			−0.802133
$515$		−	155.574i		−	0.302085i
$516$	54.6221	+	16.2929i	0.105857	+	0.0315755i
$517$	−996.417			−1.92731
$518$			41.1216i			0.0793853i
$519$	50.8045	−	170.322i	0.0978893	−	0.328174i
$520$	20.5429			0.0395056
$521$		−	686.243i		−	1.31717i	−0.752508	−	0.658583i	$-0.771156\pi$
$521$		−	686.243i		−	1.31717i	0.752508	−	0.658583i	$-0.228844\pi$
$522$	−192.757	−	126.224i	−0.369267	−	0.241808i
$523$	−759.604			−1.45240			−0.726199	−	0.687485i	$-0.758715\pi$
$523$	−759.604			−1.45240			−0.726199	+	0.687485i	$0.758715\pi$
$524$		−	263.629i		−	0.503110i
$525$	72.7932	+	21.7131i	0.138654	+	0.0413583i
$526$	68.5062			0.130240
$527$		−	146.287i		−	0.277585i
$528$	−64.5498	+	216.404i	−0.122253	+	0.409855i
$529$	−23.0000			−0.0434783
$530$		−	282.906i		−	0.533785i
$531$	290.691	−	443.917i	0.547441	−	0.836002i
$532$	86.0398			0.161729
$533$			158.369i			0.297127i
$534$	−402.439	−	120.041i	−0.753631	−	0.224797i
$535$	−125.233			−0.234080
$536$			13.4889i			0.0251659i
$537$	−118.301	+	396.605i	−0.220300	+	0.738556i
$538$	283.859			0.527618
$539$			439.498i			0.815395i
$540$	92.2634	−	77.8939i	0.170858	−	0.144248i
$541$	913.857			1.68920			0.844600	−	0.535399i	$-0.179839\pi$
$541$	913.857			1.68920			0.844600	+	0.535399i	$0.179839\pi$
$542$			557.966i			1.02946i
$543$	783.578	+	233.729i	1.44305	+	0.430441i
$544$	−107.808			−0.198176
$545$			52.3166i			0.0959937i
$546$	−19.9480	+	66.8757i	−0.0365348	+	0.122483i
$547$	343.498			0.627967			0.313983	−	0.949428i	$-0.398336\pi$
$547$	343.498			0.627967			0.313983	+	0.949428i	$0.398336\pi$
$548$		−	138.271i		−	0.252319i
$549$	−709.608	−	464.674i	−1.29255	−	0.846401i
$550$	−133.069			−0.241944
$551$		−	153.780i		−	0.279093i
$552$	−38.9961	−	11.6319i	−0.0706452	−	0.0210724i
$553$	−559.799			−1.01230
$554$			236.297i			0.426529i
$555$	−11.0097	+	36.9100i	−0.0198373	+	0.0665045i
$556$	−102.408			−0.184187
$557$			332.247i			0.596493i	0.954489	+	0.298247i	$0.0964017\pi$
$557$			332.247i			0.596493i	−0.954489	+	0.298247i	$0.903598\pi$
$558$	53.5221	−	81.7341i	0.0959178	−	0.146477i
$559$	−30.8573			−0.0552009
$560$		−	45.2953i		−	0.0808845i
$561$	−1031.05	−	307.546i	−1.83787	−	0.548210i
$562$	−399.649			−0.711119
$563$		−	741.449i		−	1.31696i	−0.752598	−	0.658481i	$-0.771199\pi$
$563$		−	741.449i		−	1.31696i	0.752598	−	0.658481i	$-0.228801\pi$
$564$	90.8078	−	304.433i	0.161007	−	0.539775i
$565$	424.698			0.751677
$566$		−	164.276i		−	0.290240i
$567$	163.985	+	375.993i	0.289215	+	0.663128i
$568$	46.1898			0.0813201
$569$			299.724i			0.526755i	0.964693	+	0.263378i	$0.0848366\pi$
$569$			299.724i			0.526755i	−0.964693	+	0.263378i	$0.915163\pi$
$570$	77.2278	+	23.0359i	0.135487	+	0.0404138i
$571$	−599.952			−1.05070			−0.525352	−	0.850885i	$-0.676066\pi$
$571$	−599.952			−1.05070			−0.525352	+	0.850885i	$0.676066\pi$
$572$		−	122.251i		−	0.213726i
$573$	−73.5092	+	246.440i	−0.128288	+	0.430087i
$574$	349.189			0.608343
$575$		−	23.9792i		−	0.0417029i
$576$	−60.2346	−	39.4435i	−0.104574	−	0.0684784i
$577$	−687.038			−1.19071			−0.595354	−	0.803464i	$-0.702988\pi$
$577$	−687.038			−1.19071			−0.595354	+	0.803464i	$0.702988\pi$
$578$		−	104.938i		−	0.181554i
$579$	−567.252	−	169.202i	−0.979709	−	0.292232i
$580$	−80.9570			−0.139581
$581$		−	78.9814i		−	0.135940i
$582$	−50.7788	+	170.236i	−0.0872489	+	0.292502i
$583$	−1683.58			−2.88779
$584$			259.616i			0.444549i
$585$	−35.8099	+	54.6856i	−0.0612135	+	0.0934797i
$586$	−618.483			−1.05543
$587$		−	124.548i		−	0.212177i	−0.994357	−	0.106089i	$-0.966167\pi$
$587$		−	124.548i		−	0.212177i	0.994357	−	0.106089i	$-0.0338328\pi$
$588$	−134.279	−	40.0533i	−0.228365	−	0.0681179i
$589$	65.2069			0.110708
$590$		−	186.443i		−	0.316005i
$591$	−168.578	+	565.159i	−0.285242	+	0.956275i
$592$	22.9671			0.0387958
$593$		−	289.161i		−	0.487625i	−0.969822	−	0.243812i	$-0.921602\pi$
$593$		−	289.161i		−	0.487625i	0.969822	−	0.243812i	$-0.0783981\pi$
$594$	−463.548	−	549.062i	−0.780385	−	0.924347i
$595$	215.808			0.362703
$596$			439.740i			0.737820i
$597$	39.1193	+	11.6687i	0.0655264	+	0.0195455i
$598$	22.0298			0.0368392
$599$		−	162.616i		−	0.271479i	−0.990745	−	0.135739i	$-0.956659\pi$
$599$		−	162.616i		−	0.271479i	0.990745	−	0.135739i	$-0.0433410\pi$
$600$	12.1271	−	40.6563i	0.0202119	−	0.0677605i
$601$	815.650			1.35715			0.678577	−	0.734529i	$-0.262597\pi$
$601$	815.650			1.35715			0.678577	+	0.734529i	$0.262597\pi$
$602$			68.0376i			0.113019i
$603$	−35.9078	−	23.5136i	−0.0595485	−	0.0389943i
$604$	429.609			0.711273
$605$			521.332i			0.861707i
$606$	−71.9845	−	21.4719i	−0.118786	−	0.0354322i
$607$	−771.203			−1.27052			−0.635258	−	0.772300i	$-0.719106\pi$
$607$	−771.203			−1.27052			−0.635258	+	0.772300i	$0.719106\pi$
$608$		−	48.0547i		−	0.0790374i
$609$	78.6124	−	263.548i	0.129084	−	0.432756i
$610$	−298.032			−0.488576
$611$			171.982i			0.281476i
$612$	187.928	−	286.986i	0.307071	−	0.468931i
$613$	149.236			0.243451			0.121726	−	0.992564i	$-0.461157\pi$
$613$	149.236			0.243451			0.121726	+	0.992564i	$0.461157\pi$
$614$			102.770i			0.167378i
$615$	313.426	+	93.4901i	0.509635	+	0.152016i
$616$	−269.553			−0.437587
$617$			91.1721i			0.147767i	0.997267	+	0.0738834i	$0.0235393\pi$
$617$			91.1721i			0.147767i	−0.997267	+	0.0738834i	$0.976461\pi$
$618$	84.3742	−	282.865i	0.136528	−	0.457710i
$619$	278.475			0.449879			0.224940	−	0.974373i	$-0.427782\pi$
$619$	278.475			0.449879			0.224940	+	0.974373i	$0.427782\pi$
$620$		−	34.3279i		−	0.0553676i
$621$	98.9415	−	83.5319i	0.159326	−	0.134512i
$622$	367.949			0.591558
$623$		−	501.280i		−	0.804623i
$624$	37.3512	+	11.1413i	0.0598577	+	0.0178546i
$625$	25.0000			0.0400000
$626$			532.366i			0.850425i
$627$	137.087	−	459.585i	0.218640	−	0.732990i
$628$	−198.383			−0.315896
$629$			109.426i			0.173968i
$630$	120.577	+	78.9576i	0.191392	+	0.125330i
$631$	−488.140			−0.773597			−0.386798	−	0.922164i	$-0.626419\pi$
$631$	−488.140			−0.773597			−0.386798	+	0.922164i	$0.626419\pi$
$632$			312.658i			0.494712i
$633$	−916.379	−	273.342i	−1.44768	−	0.431820i
$634$	−442.436			−0.697849
$635$		−	451.668i		−	0.711289i
$636$	153.432	−	514.381i	0.241245	−	0.808776i
$637$	75.8573			0.119085
$638$			481.777i			0.755137i
$639$	−80.5170	+	122.958i	−0.126005	+	0.192423i
$640$	−25.2982			−0.0395285
$641$		−	353.083i		−	0.550832i	−0.961325	−	0.275416i	$-0.911184\pi$
$641$		−	353.083i		−	0.550832i	0.961325	−	0.275416i	$-0.0888155\pi$
$642$	−227.698	−	67.9190i	−0.354671	−	0.105793i
$643$	675.054			1.04985			0.524926	−	0.851148i	$-0.324093\pi$
$643$	675.054			1.04985			0.524926	+	0.851148i	$0.324093\pi$
$644$		−	48.5738i		−	0.0754251i
$645$	−18.2161	+	61.0694i	−0.0282419	+	0.0946812i
$646$	228.955			0.354420
$647$			752.235i			1.16265i	0.813671	+	0.581325i	$0.197466\pi$
$647$			752.235i			1.16265i	−0.813671	+	0.581325i	$0.802534\pi$
$648$	209.999	−	91.5886i	0.324072	−	0.141340i
$649$	−1109.53			−1.70959
$650$			22.9677i			0.0353349i
$651$	111.751	+	33.3337i	0.171661	+	0.0512039i
$652$	246.689			0.378358
$653$		−	90.4569i		−	0.138525i	−0.997598	−	0.0692626i	$-0.977935\pi$
$653$		−	90.4569i		−	0.138525i	0.997598	−	0.0692626i	$-0.0220646\pi$
$654$	−28.3735	+	95.1222i	−0.0433845	+	0.145447i
$655$	294.747			0.449995
$656$		−	195.028i		−	0.297299i
$657$	−691.104	−	452.557i	−1.05191	−	0.688823i
$658$	379.204			0.576298
$659$			352.118i			0.534322i	0.963652	+	0.267161i	$0.0860855\pi$
$659$			352.118i			0.534322i	−0.963652	+	0.267161i	$0.913915\pi$
$660$	−241.946	−	72.1689i	−0.366586	−	0.109347i
$661$	699.250			1.05787			0.528933	−	0.848664i	$-0.322592\pi$
$661$	699.250			1.05787			0.528933	+	0.848664i	$0.322592\pi$
$662$			147.259i			0.222445i
$663$	−53.0824	+	177.959i	−0.0800639	+	0.268414i
$664$	−44.1125			−0.0664345
$665$			96.1954i			0.144655i
$666$	−40.0357	+	61.1389i	−0.0601137	+	0.0918002i
$667$	−86.8168			−0.130160
$668$		−	22.7901i		−	0.0341169i
$669$	314.877	+	93.9230i	0.470668	+	0.140393i
$670$	−15.0811			−0.0225091
$671$			1773.59i			2.64321i
$672$	24.5656	−	82.3561i	0.0365559	−	0.122554i
$673$	677.167			1.00619			0.503096	−	0.864231i	$-0.332194\pi$
$673$	677.167			1.00619			0.503096	+	0.864231i	$0.332194\pi$
$674$			161.242i			0.239231i
$675$	87.0880	+	103.154i	0.129019	+	0.152820i
$676$	316.899			0.468786
$677$		−	615.623i		−	0.909340i	−0.890660	−	0.454670i	$-0.849757\pi$
$677$		−	615.623i		−	0.909340i	0.890660	−	0.454670i	$-0.150243\pi$
$678$	772.187	+	230.331i	1.13892	+	0.339722i
$679$	−212.047			−0.312293
$680$		−	120.533i		−	0.177254i
$681$	−105.677	+	354.284i	−0.155180	+	0.520241i
$682$	−204.286			−0.299540
$683$			221.790i			0.324729i	0.986731	+	0.162364i	$0.0519120\pi$
$683$			221.790i			0.324729i	−0.986731	+	0.162364i	$0.948088\pi$
$684$	127.923	+	83.7678i	0.187021	+	0.122468i
$685$	154.591			0.225681
$686$		−	518.187i		−	0.755375i
$687$	1175.26	+	350.562i	1.71071	+	0.510280i
$688$	38.0002			0.0552329
$689$			290.586i			0.421751i
$690$	13.0049	−	43.5990i	0.0188477	−	0.0631869i
$691$	−61.9651			−0.0896745			−0.0448373	−	0.998994i	$-0.514277\pi$
$691$	−61.9651			−0.0896745			−0.0448373	+	0.998994i	$0.514277\pi$
$692$		−	118.492i		−	0.171231i
$693$	469.879	−	717.556i	0.678036	−	1.03543i
$694$	−21.6968			−0.0312634
$695$		−	114.496i		−	0.164742i
$696$	−147.196	−	43.9064i	−0.211489	−	0.0630839i
$697$	929.205			1.33315
$698$			912.211i			1.30689i
$699$	−266.202	+	892.443i	−0.380833	+	1.27674i
$700$	50.6417			0.0723453
$701$			420.912i			0.600445i	0.953869	+	0.300223i	$0.0970610\pi$
$701$			420.912i			0.600445i	−0.953869	+	0.300223i	$0.902939\pi$
$702$	−94.7681	+	80.0084i	−0.134997	+	0.113972i
$703$	−48.7762			−0.0693829
$704$			150.550i			0.213850i
$705$	340.367	+	101.526i	0.482790	+	0.144009i
$706$	−370.326			−0.524541
$707$		−	89.6644i		−	0.126824i
$708$	101.116	−	338.991i	0.142819	−	0.478801i
$709$	−560.407			−0.790418			−0.395209	−	0.918591i	$-0.629328\pi$
$709$	−560.407			−0.790418			−0.395209	+	0.918591i	$0.629328\pi$
$710$			51.6418i			0.0727350i
$711$	−832.301	−	545.017i	−1.17061	−	0.766550i
$712$	−279.974			−0.393222
$713$		−	36.8126i		−	0.0516306i
$714$	392.383	+	117.042i	0.549556	+	0.163924i
$715$	136.681			0.191163
$716$			275.915i			0.385356i
$717$	262.503	−	880.041i	0.366113	−	1.22739i
$718$	707.506			0.985385
$719$			60.7661i			0.0845148i	0.999107	+	0.0422574i	$0.0134550\pi$
$719$			60.7661i			0.0845148i	−0.999107	+	0.0422574i	$0.986545\pi$
$720$	44.0992	−	67.3443i	0.0612489	−	0.0935338i
$721$	352.338			0.488680
$722$		−	408.475i		−	0.565755i
$723$	968.563	+	288.907i	1.33964	+	0.399595i
$724$	545.130			0.752941
$725$		−	90.5127i		−	0.124845i
$726$	−282.741	+	947.889i	−0.389450	+	1.30563i
$727$	−305.359			−0.420026			−0.210013	−	0.977699i	$-0.567351\pi$
$727$	−305.359			−0.420026			−0.210013	+	0.977699i	$0.567351\pi$
$728$			46.5249i			0.0639078i
$729$	−122.254	+	718.676i	−0.167701	+	0.985838i
$730$	−290.260			−0.397616
$731$			181.051i			0.247675i
$732$	−541.882	−	161.635i	−0.740276	−	0.220813i
$733$	−820.155			−1.11890			−0.559451	−	0.828864i	$-0.688988\pi$
$733$	−820.155			−1.11890			−0.559451	+	0.828864i	$0.688988\pi$
$734$			939.221i			1.27959i
$735$	44.7810	−	150.128i	0.0609265	−	0.204256i
$736$	−27.1293			−0.0368605
$737$			89.7478i			0.121775i
$738$	519.168	+	339.968i	0.703480	+	0.460661i
$739$	−442.131			−0.598282			−0.299141	−	0.954209i	$-0.596700\pi$
$739$	−442.131			−0.598282			−0.299141	+	0.954209i	$0.596700\pi$
$740$			25.6780i			0.0347000i
$741$	−79.3243	−	23.6612i	−0.107050	−	0.0319315i
$742$	640.716			0.863499
$743$		−	604.855i		−	0.814072i	−0.913412	−	0.407036i	$-0.866562\pi$
$743$		−	604.855i		−	0.814072i	0.913412	−	0.407036i	$-0.133438\pi$
$744$	18.6175	−	62.4151i	0.0250235	−	0.0838913i
$745$	−491.645			−0.659926
$746$		−	170.843i		−	0.229012i
$747$	76.8958	−	117.428i	0.102940	−	0.157200i
$748$	−717.292			−0.958947
$749$		−	283.623i		−	0.378668i
$750$	45.4551	+	13.5586i	0.0606068	+	0.0180781i
$751$	1199.13			1.59671			0.798355	−	0.602187i	$-0.205704\pi$
$751$	1199.13			1.59671			0.798355	+	0.602187i	$0.205704\pi$
$752$		−	211.792i		−	0.281638i
$753$	−329.644	+	1105.13i	−0.437774	+	1.46764i
$754$	83.1547			0.110285
$755$			480.317i			0.636182i
$756$	176.411	+	208.955i	0.233348	+	0.276396i
$757$	561.572			0.741839			0.370919	−	0.928665i	$-0.379043\pi$
$757$	561.572			0.741839			0.370919	+	0.928665i	$0.379043\pi$
$758$			490.476i			0.647066i
$759$	−259.459	−	77.3925i	−0.341843	−	0.101966i
$760$	53.7268			0.0706932
$761$			1338.92i			1.75943i	0.475505	+	0.879713i	$0.342265\pi$
$761$			1338.92i			1.75943i	−0.475505	+	0.879713i	$0.657735\pi$
$762$	244.959	−	821.225i	0.321468	−	1.07772i
$763$	−118.485			−0.155288
$764$			171.446i			0.224406i
$765$	320.860	+	210.109i	0.419425	+	0.274653i
$766$	480.885			0.627787
$767$			191.504i			0.249679i
$768$	−45.9973	−	13.7203i	−0.0598923	−	0.0178650i
$769$	0.701536			0.000912271			0.000456135	−	1.00000i	$-0.499855\pi$
$769$	0.701536			0.000912271			0.000456135		1.00000i	$0.499855\pi$
$770$		−	301.370i		−	0.391389i
$771$	249.999	−	838.121i	0.324252	−	1.08706i
$772$	−394.633			−0.511182
$773$		−	835.888i		−	1.08136i	−0.841230	−	0.540678i	$-0.818168\pi$
$773$		−	835.888i		−	1.08136i	0.841230	−	0.540678i	$-0.181832\pi$
$774$	−66.2410	+	101.157i	−0.0855827	+	0.130694i
$775$	38.3798			0.0495223
$776$			118.432i			0.152619i
$777$	−83.5925	−	24.9344i	−0.107584	−	0.0320906i
$778$	−258.620			−0.332417
$779$			414.189i			0.531693i
$780$	−12.4563	+	41.7599i	−0.0159697	+	0.0535384i
$781$	307.322			0.393498
$782$		−	129.257i		−	0.165290i
$783$	373.469	−	315.303i	0.476972	−	0.402686i
$784$	−93.4168			−0.119154
$785$		−	221.799i		−	0.282546i
$786$	535.910	+	159.854i	0.681819	+	0.203376i
$787$	104.317			0.132550			0.0662749	−	0.997801i	$-0.478889\pi$
$787$	104.317			0.132550			0.0662749	+	0.997801i	$0.478889\pi$
$788$			393.177i			0.498955i
$789$	−41.5392	+	139.260i	−0.0526479	+	0.176502i
$790$	−349.562			−0.442484
$791$			961.840i			1.21598i
$792$	−400.768	−	262.436i	−0.506020	−	0.331358i
$793$	306.122			0.386030
$794$			210.454i			0.265056i
$795$	575.096	+	171.542i	0.723391	+	0.215776i
$796$	27.2150			0.0341897
$797$			295.295i			0.370508i	0.982691	+	0.185254i	$0.0593109\pi$
$797$			295.295i			0.370508i	−0.982691	+	0.185254i	$0.940689\pi$
$798$	−52.1708	+	174.903i	−0.0653770	+	0.219177i
$799$	1009.08			1.26292
$800$		−	28.2843i		−	0.0353553i
$801$	488.044	−	745.296i	0.609293	−	0.930457i
$802$	−845.045			−1.05367
$803$			1727.34i			2.15111i
$804$	−27.4205	−	8.17910i	−0.0341051	−	0.0101730i
$805$	54.3072			0.0674623
$806$			35.2598i			0.0437466i
$807$	−172.120	+	577.032i	−0.213283	+	0.715033i
$808$	−50.0791			−0.0619791
$809$		−	1487.81i		−	1.83908i	−0.393000	−	0.919538i	$-0.628563\pi$
$809$		−	1487.81i		−	1.83908i	0.393000	−	0.919538i	$-0.371437\pi$
$810$	102.399	+	234.786i	0.126419	+	0.289859i
$811$	733.626			0.904595			0.452297	−	0.891867i	$-0.350605\pi$
$811$	733.626			0.904595			0.452297	+	0.891867i	$0.350605\pi$
$812$		−	183.349i		−	0.225799i
$813$	−1134.24	−	338.327i	−1.39513	−	0.416146i
$814$	152.811			0.187728
$815$			275.807i			0.338414i
$816$	65.3699	−	219.153i	0.0801102	−	0.268570i
$817$	−80.7025			−0.0987791
$818$		−	385.839i		−	0.471686i
$819$	−123.850	−	81.1011i	−0.151221	−	0.0990245i
$820$	218.048			0.265912
$821$			1465.71i			1.78528i	0.450774	+	0.892638i	$0.351148\pi$
$821$			1465.71i			1.78528i	−0.450774	+	0.892638i	$0.648852\pi$
$822$	281.079	+	83.8415i	0.341945	+	0.101997i
$823$	−992.957			−1.20651			−0.603255	−	0.797549i	$-0.706130\pi$
$823$	−992.957			−1.20651			−0.603255	+	0.797549i	$0.706130\pi$
$824$		−	196.787i		−	0.238819i
$825$	80.6873	−	270.504i	0.0978028	−	0.327884i
$826$	422.249			0.511198
$827$			617.883i			0.747138i	0.927602	+	0.373569i	$0.121866\pi$
$827$			617.883i			0.747138i	−0.927602	+	0.373569i	$0.878134\pi$
$828$	47.2912	−	72.2187i	0.0571149	−	0.0872207i
$829$	−931.101			−1.12316			−0.561581	−	0.827422i	$-0.689807\pi$
$829$	−931.101			−1.12316			−0.561581	+	0.827422i	$0.689807\pi$
$830$		−	49.3193i		−	0.0594208i
$831$	−480.348	−	143.281i	−0.578036	−	0.172419i
$832$	25.9850			0.0312319
$833$		−	445.081i		−	0.534311i
$834$	62.0958	−	208.176i	0.0744554	−	0.249612i
$835$	25.4801			0.0305150
$836$		−	319.730i		−	0.382452i
$837$	133.697	+	158.361i	0.159733	+	0.189200i
$838$	−370.739			−0.442409
$839$			1177.29i			1.40320i	0.712570	+	0.701601i	$0.247531\pi$
$839$			1177.29i			1.40320i	−0.712570	+	0.701601i	$0.752469\pi$
$840$	92.0769	+	27.4651i	0.109615	+	0.0326966i
$841$	513.298			0.610342
$842$			891.637i			1.05895i
$843$	242.330	−	812.412i	0.287461	−	0.963715i
$844$	−637.518			−0.755353
$845$			354.304i			0.419295i
$846$	563.794	+	369.190i	0.666423	+	0.436395i
$847$	−1180.70			−1.39397
$848$		−	357.851i		−	0.421994i
$849$	333.942	+	99.6097i	0.393335	+	0.117326i
$850$	134.760			0.158541
$851$			27.5366i			0.0323580i
$852$	−28.0075	+	93.8953i	−0.0328727	+	0.110206i
$853$	−1192.04			−1.39747			−0.698733	−	0.715383i	$-0.746252\pi$
$853$	−1192.04			−1.39747			−0.698733	+	0.715383i	$0.746252\pi$
$854$		−	674.971i		−	0.790365i
$855$	−93.6553	+	143.022i	−0.109538	+	0.167277i
$856$	−158.408			−0.185056
$857$		−	776.967i		−	0.906613i	−0.891355	−	0.453306i	$-0.850244\pi$
$857$		−	776.967i		−	0.906613i	0.891355	−	0.453306i	$-0.149756\pi$
$858$	248.514	+	74.1280i	0.289644	+	0.0863963i
$859$	−1329.69			−1.54795			−0.773974	−	0.633217i	$-0.781734\pi$
$859$	−1329.69			−1.54795			−0.773974	+	0.633217i	$0.781734\pi$
$860$			42.4855i			0.0494018i
$861$	−211.733	+	709.836i	−0.245915	+	0.824432i
$862$	98.5081			0.114279
$863$			531.988i			0.616441i	0.951315	+	0.308220i	$0.0997334\pi$
$863$			531.988i			0.616441i	−0.951315	+	0.308220i	$0.900267\pi$
$864$	116.705	−	98.5288i	0.135075	−	0.114038i
$865$	132.478			0.153154
$866$		−	526.753i		−	0.608260i
$867$	213.320	+	63.6300i	0.246043	+	0.0733910i
$868$	77.7447			0.0895676
$869$			2080.25i			2.39385i
$870$	49.0889	−	164.571i	0.0564240	−	0.189162i
$871$	15.4905			0.0177847
$872$			66.1758i			0.0758897i
$873$	−315.268	−	206.448i	−0.361132	−	0.236481i
$874$	57.6156			0.0659218
$875$			56.6191i			0.0647076i
$876$	−527.752	−	157.420i	−0.602456	−	0.179704i
$877$	1640.32			1.87038			0.935191	−	0.354144i	$-0.115228\pi$
$877$	1640.32			1.87038			0.935191	+	0.354144i	$0.115228\pi$
$878$			477.954i			0.544367i
$879$	375.021	−	1257.26i	0.426645	−	1.43033i
$880$	−168.320			−0.191273
$881$		−	1020.38i		−	1.15820i	−0.815255	−	0.579102i	$-0.803403\pi$
$881$		−	1020.38i		−	1.15820i	0.815255	−	0.579102i	$-0.196597\pi$
$882$	162.842	−	248.677i	0.184628	−	0.281947i
$883$	−342.090			−0.387418			−0.193709	−	0.981059i	$-0.562052\pi$
$883$	−342.090			−0.387418			−0.193709	+	0.981059i	$0.562052\pi$
$884$			123.805i			0.140050i
$885$	379.004	+	113.051i	0.428253	+	0.127741i
$886$	1011.32			1.14145
$887$		−	32.6885i		−	0.0368529i	−0.999830	−	0.0184265i	$-0.994134\pi$
$887$		−	32.6885i		−	0.0368529i	0.999830	−	0.0184265i	$-0.00586565\pi$
$888$	−13.9263	+	46.6879i	−0.0156827	+	0.0525765i
$889$	1022.92			1.15064
$890$		−	313.020i		−	0.351708i
$891$	1397.22	−	609.380i	1.56814	−	0.683928i
$892$	219.058			0.245580
$893$			449.791i			0.503685i
$894$	−893.910	−	266.640i	−0.999900	−	0.298255i
$895$	−308.482			−0.344673
$896$		−	57.2945i		−	0.0639448i
$897$	−13.3579	+	44.7826i	−0.0148918	+	0.0499248i
$898$	967.667			1.07758
$899$		−	138.954i		−	0.154565i
$900$	75.2932	+	49.3044i	0.0836592	+	0.0547827i
$901$	1704.97			1.89231
$902$		−	1297.61i		−	1.43859i
$903$	−138.308	−	41.2551i	−0.153165	−	0.0456867i
$904$	537.205			0.594253
$905$			609.473i			0.673451i
$906$	−260.496	+	873.315i	−0.287524	+	0.963923i
$907$	1219.87			1.34495			0.672477	−	0.740118i	$-0.265230\pi$
$907$	1219.87			1.34495			0.672477	+	0.740118i	$0.265230\pi$
$908$			246.473i			0.271446i
$909$	87.2967	−	133.312i	0.0960359	−	0.146657i
$910$	−52.0164			−0.0571609
$911$		−	522.247i		−	0.573268i	−0.958040	−	0.286634i	$-0.907464\pi$
$911$		−	522.247i		−	0.573268i	0.958040	−	0.286634i	$-0.0925364\pi$
$912$	97.6863	+	29.1383i	0.107112	+	0.0319499i
$913$	−293.500			−0.321468
$914$			392.827i			0.429788i
$915$	180.714	−	605.843i	0.197501	−	0.662123i
$916$	817.620			0.892598
$917$			667.532i			0.727952i
$918$	469.437	+	556.037i	0.511370	+	0.605705i
$919$	310.847			0.338245			0.169123	−	0.985595i	$-0.445907\pi$
$919$	310.847			0.338245			0.169123	+	0.985595i	$0.445907\pi$
$920$		−	30.3315i		−	0.0329690i
$921$	−208.912	−	62.3154i	−0.226832	−	0.0676606i
$922$	73.0541			0.0792344
$923$		−	53.0437i		−	0.0574688i
$924$	163.446	−	547.952i	0.176889	−	0.593022i
$925$	−28.7089			−0.0310367
$926$			388.675i			0.419735i
$927$	523.850	+	343.034i	0.565103	+	0.370048i
$928$	−102.403			−0.110349
$929$			1114.04i			1.19918i	0.800308	+	0.599589i	$0.204669\pi$
$929$			1114.04i			1.19918i	−0.800308	+	0.599589i	$0.795331\pi$
$930$	69.7823	+	20.8150i	0.0750347	+	0.0223817i
$931$	198.393			0.213097
$932$			620.866i			0.666165i
$933$	−223.108	+	747.971i	−0.239130	+	0.801684i
$934$	−1284.81			−1.37560
$935$		−	801.957i		−	0.857708i
$936$	−45.2964	+	69.1725i	−0.0483936	+	0.0739022i
$937$	−1129.07			−1.20498			−0.602491	−	0.798125i	$-0.705825\pi$
$937$	−1129.07			−1.20498			−0.602491	+	0.798125i	$0.705825\pi$
$938$		−	34.1551i		−	0.0364127i
$939$	−1082.20	−	322.804i	−1.15250	−	0.343774i
$940$	236.791			0.251905
$941$		−	279.863i		−	0.297411i	−0.988882	−	0.148705i	$-0.952489\pi$
$941$		−	279.863i		−	0.297411i	0.988882	−	0.148705i	$-0.0475106\pi$
$942$	120.291	−	403.276i	0.127697	−	0.428106i
$943$	233.830			0.247964
$944$		−	235.834i		−	0.249824i
$945$	−233.619	+	197.234i	−0.247216	+	0.208713i
$946$	252.832			0.267265
$947$			873.948i			0.922860i	0.887177	+	0.461430i	$0.152663\pi$
$947$			873.948i			0.922860i	−0.887177	+	0.461430i	$0.847337\pi$
$948$	−635.575	−	189.582i	−0.670438	−	0.199981i
$949$	298.140			0.314162
$950$			60.0684i			0.0632299i
$951$	268.274	−	899.390i	0.282097	−	0.945731i
$952$	272.978			0.286742
$953$			1612.71i			1.69225i	0.532986	+	0.846124i	$0.321070\pi$
$953$			1612.71i			1.69225i	−0.532986	+	0.846124i	$0.678930\pi$
$954$	952.607	+	623.797i	0.998539	+	0.653876i
$955$	−191.683			−0.200715
$956$		−	612.238i		−	0.640417i
$957$	−979.364	−	292.129i	−1.02337	−	0.305255i
$958$	84.0774			0.0877635
$959$			350.113i			0.365082i
$960$	15.3398	−	51.4266i	0.0159789	−	0.0535693i
$961$	−902.080			−0.938689
$962$		−	26.3751i		−	0.0274169i
$963$	276.133	−	421.685i	0.286743	−	0.437887i
$964$	673.822			0.698986
$965$		−	441.213i		−	0.457215i
$966$	98.7415	+	29.4531i	0.102217	+	0.0304897i
$967$	434.311			0.449132			0.224566	−	0.974459i	$-0.427904\pi$
$967$	434.311			0.449132			0.224566	+	0.974459i	$0.427904\pi$
$968$			659.439i			0.681239i
$969$	−138.829	+	465.423i	−0.143270	+	0.480313i
$970$	−132.411			−0.136506
$971$		−	1408.35i		−	1.45041i	−0.688531	−	0.725207i	$-0.741744\pi$
$971$		−	1408.35i		−	1.45041i	0.688531	−	0.725207i	$-0.258256\pi$
$972$	58.8482	+	482.424i	0.0605434	+	0.496321i
$973$	259.306			0.266501
$974$			999.499i			1.02618i
$975$	−46.6890	−	13.9266i	−0.0478862	−	0.0142837i
$976$	−376.983			−0.386254
$977$		−	1211.82i		−	1.24035i	−0.784465	−	0.620174i	$-0.787062\pi$
$977$		−	1211.82i		−	1.24035i	0.784465	−	0.620174i	$-0.212938\pi$
$978$	−149.582	+	501.473i	−0.152947	+	0.512754i
$979$	−1862.79			−1.90275
$980$		−	104.443i		−	0.106575i
$981$	−176.161	−	115.356i	−0.179573	−	0.117590i
$982$	−1111.19			−1.13155
$983$		−	1274.30i		−	1.29633i	−0.761498	−	0.648167i	$-0.775536\pi$
$983$		−	1274.30i		−	1.29633i	0.761498	−	0.648167i	$-0.224464\pi$
$984$	396.456	+	118.257i	0.402902	+	0.120180i
$985$	−439.585			−0.446279
$986$		−	487.898i		−	0.494826i
$987$	−229.933	+	770.851i	−0.232961	+	0.781004i
$988$	−55.1853			−0.0558556
$989$			45.5606i			0.0460674i
$990$	293.412	−	448.072i	0.296376	−	0.452598i
$991$	934.842			0.943332			0.471666	−	0.881777i	$-0.343653\pi$
$991$	934.842			0.943332			0.471666	+	0.881777i	$0.343653\pi$
$992$		−	43.4218i		−	0.0437719i
$993$	−299.349	−	89.2913i	−0.301459	−	0.0899207i
$994$	−116.957			−0.117663
$995$			30.4273i			0.0305802i
$996$	26.7479	−	89.6725i	0.0268554	−	0.0900326i
$997$	−125.191			−0.125568			−0.0627840	−	0.998027i	$-0.519998\pi$
$997$	−125.191			−0.125568			−0.0627840	+	0.998027i	$0.519998\pi$
$998$			888.673i			0.890454i
$999$	−100.008	−	118.457i	−0.100108	−	0.118576i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	690.3.g.a.461.4	yes	56
3.2	odd	2	inner	690.3.g.a.461.3	✓	56

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
690.3.g.a.461.3	✓	56	3.2	odd	2	inner
690.3.g.a.461.4	yes	56	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 690 = 2 \cdot 3 \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	690.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.41421i q^{2} +(-2.87483 - 0.857518i) q^{3} -2.00000 q^{4} -2.23607i q^{5} +(1.21271 - 4.06563i) q^{6} +5.06417 q^{7} -2.82843i q^{8} +(7.52932 + 4.93044i) q^{9} +O(q^{10})\)	\(q+1.41421i q^{2} +(-2.87483 - 0.857518i) q^{3} -2.00000 q^{4} -2.23607i q^{5} +(1.21271 - 4.06563i) q^{6} +5.06417 q^{7} -2.82843i q^{8} +(7.52932 + 4.93044i) q^{9} +3.16228 q^{10} -18.8188i q^{11} +(5.74967 + 1.71504i) q^{12} -3.24812 q^{13} +7.16182i q^{14} +(-1.91747 + 6.42832i) q^{15} +4.00000 q^{16} +19.0579i q^{17} +(-6.97270 + 10.6481i) q^{18} -8.49496 q^{19} +4.47214i q^{20} +(-14.5586 - 4.34262i) q^{21} +26.6138 q^{22} +4.79583i q^{23} +(-2.42543 + 8.13125i) q^{24} -5.00000 q^{25} -4.59354i q^{26} +(-17.4176 - 20.6307i) q^{27} -10.1283 q^{28} +18.1025i q^{29} +(-9.09102 - 2.71171i) q^{30} -7.67595 q^{31} +5.65685i q^{32} +(-16.1375 + 54.1009i) q^{33} -26.9519 q^{34} -11.3238i q^{35} +(-15.0586 - 9.86089i) q^{36} +5.74178 q^{37} -12.0137i q^{38} +(9.33781 + 2.78532i) q^{39} -6.32456 q^{40} -48.7570i q^{41} +(6.14139 - 20.5890i) q^{42} +9.50005 q^{43} +37.6376i q^{44} +(11.0248 - 16.8361i) q^{45} -6.78233 q^{46} -52.9480i q^{47} +(-11.4993 - 3.43007i) q^{48} -23.3542 q^{49} -7.07107i q^{50} +(16.3425 - 54.7882i) q^{51} +6.49624 q^{52} -89.4628i q^{53} +(29.1763 - 24.6322i) q^{54} -42.0801 q^{55} -14.3236i q^{56} +(24.4216 + 7.28458i) q^{57} -25.6009 q^{58} -58.9584i q^{59} +(3.83494 - 12.8566i) q^{60} -94.2459 q^{61} -10.8554i q^{62} +(38.1298 + 24.9686i) q^{63} -8.00000 q^{64} +7.26302i q^{65} +(-76.5102 - 22.8218i) q^{66} -4.76905 q^{67} -38.1157i q^{68} +(4.11251 - 13.7872i) q^{69} +16.0143 q^{70} +16.3306i q^{71} +(13.9454 - 21.2961i) q^{72} -91.7883 q^{73} +8.12011i q^{74} +(14.3742 + 4.28759i) q^{75} +16.9899 q^{76} -95.3015i q^{77} +(-3.93904 + 13.2057i) q^{78} -110.541 q^{79} -8.94427i q^{80} +(32.3815 + 74.2458i) q^{81} +68.9528 q^{82} -15.5961i q^{83} +(29.1173 + 8.68523i) q^{84} +42.6147 q^{85} +13.4351i q^{86} +(15.5233 - 52.0418i) q^{87} -53.2276 q^{88} -98.9857i q^{89} +(23.8098 + 15.5914i) q^{90} -16.4490 q^{91} -9.59166i q^{92} +(22.0671 + 6.58227i) q^{93} +74.8798 q^{94} +18.9953i q^{95} +(4.85086 - 16.2625i) q^{96} -41.8721 q^{97} -33.0278i q^{98} +(92.7850 - 141.693i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q - 8 q^{3} - 112 q^{4} + 16 q^{6} - 16 q^{7}+O(q^{10}) \) 56 * q - 8 * q^3 - 112 * q^4 + 16 * q^6 - 16 * q^7	\( 56 q - 8 q^{3} - 112 q^{4} + 16 q^{6} - 16 q^{7} + 16 q^{12} + 80 q^{13} - 40 q^{15} + 224 q^{16} - 32 q^{18} - 64 q^{19} + 56 q^{21} - 96 q^{22} - 32 q^{24} - 280 q^{25} + 40 q^{27} + 32 q^{28} - 80 q^{31} + 32 q^{33} + 192 q^{34} + 240 q^{37} - 56 q^{39} - 144 q^{43} - 32 q^{48} + 72 q^{49} - 24 q^{51} - 160 q^{52} + 16 q^{54} - 16 q^{57} + 80 q^{60} + 112 q^{61} - 64 q^{63} - 448 q^{64} + 160 q^{66} + 832 q^{67} + 64 q^{72} - 608 q^{73} + 40 q^{75} + 128 q^{76} - 320 q^{78} + 48 q^{79} - 32 q^{81} - 448 q^{82} - 112 q^{84} + 240 q^{85} + 200 q^{87} + 192 q^{88} + 80 q^{91} - 232 q^{93} + 160 q^{94} + 64 q^{96} - 448 q^{97} + 464 q^{99}+O(q^{100}) \) 56 * q - 8 * q^3 - 112 * q^4 + 16 * q^6 - 16 * q^7 + 16 * q^12 + 80 * q^13 - 40 * q^15 + 224 * q^16 - 32 * q^18 - 64 * q^19 + 56 * q^21 - 96 * q^22 - 32 * q^24 - 280 * q^25 + 40 * q^27 + 32 * q^28 - 80 * q^31 + 32 * q^33 + 192 * q^34 + 240 * q^37 - 56 * q^39 - 144 * q^43 - 32 * q^48 + 72 * q^49 - 24 * q^51 - 160 * q^52 + 16 * q^54 - 16 * q^57 + 80 * q^60 + 112 * q^61 - 64 * q^63 - 448 * q^64 + 160 * q^66 + 832 * q^67 + 64 * q^72 - 608 * q^73 + 40 * q^75 + 128 * q^76 - 320 * q^78 + 48 * q^79 - 32 * q^81 - 448 * q^82 - 112 * q^84 + 240 * q^85 + 200 * q^87 + 192 * q^88 + 80 * q^91 - 232 * q^93 + 160 * q^94 + 64 * q^96 - 448 * q^97 + 464 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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