LMFDB - Embedded newform 690.3.g.a.461.19

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			461.19
Character	$\chi$	$=$	690.461
Dual form			690.3.g.a.461.20

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421i q^{2} +(-2.91669 - 0.702099i) q^{3} -2.00000 q^{4} -2.23607i q^{5} +(-0.992918 + 4.12482i) q^{6} -8.52828 q^{7} +2.82843i q^{8} +(8.01411 + 4.09561i) q^{9} +O(q^{10})$	\(q-1.41421i q^{2} +(-2.91669 - 0.702099i) q^{3} -2.00000 q^{4} -2.23607i q^{5} +(-0.992918 + 4.12482i) q^{6} -8.52828 q^{7} +2.82843i q^{8} +(8.01411 + 4.09561i) q^{9} -3.16228 q^{10} +11.2465i q^{11} +(5.83337 + 1.40420i) q^{12} +18.3730 q^{13} +12.0608i q^{14} +(-1.56994 + 6.52191i) q^{15} +4.00000 q^{16} -0.800569i q^{17} +(5.79206 - 11.3337i) q^{18} +13.2492 q^{19} +4.47214i q^{20} +(24.8743 + 5.98770i) q^{21} +15.9050 q^{22} -4.79583i q^{23} +(1.98584 - 8.24963i) q^{24} -5.00000 q^{25} -25.9833i q^{26} +(-20.4991 - 17.5723i) q^{27} +17.0566 q^{28} -10.9760i q^{29} +(9.22337 + 2.22023i) q^{30} -42.6501 q^{31} -5.65685i q^{32} +(7.89618 - 32.8026i) q^{33} -1.13218 q^{34} +19.0698i q^{35} +(-16.0282 - 8.19121i) q^{36} +42.8724 q^{37} -18.7371i q^{38} +(-53.5882 - 12.8996i) q^{39} +6.32456 q^{40} +59.2804i q^{41} +(8.46788 - 35.1776i) q^{42} +10.6963 q^{43} -22.4931i q^{44} +(9.15805 - 17.9201i) q^{45} -6.78233 q^{46} -65.8685i q^{47} +(-11.6667 - 2.80840i) q^{48} +23.7316 q^{49} +7.07107i q^{50} +(-0.562079 + 2.33501i) q^{51} -36.7459 q^{52} -68.3687i q^{53} +(-24.8510 + 28.9901i) q^{54} +25.1480 q^{55} -24.1216i q^{56} +(-38.6436 - 9.30223i) q^{57} -15.5224 q^{58} -54.6341i q^{59} +(3.13988 - 13.0438i) q^{60} +34.1922 q^{61} +60.3163i q^{62} +(-68.3466 - 34.9285i) q^{63} -8.00000 q^{64} -41.0832i q^{65} +(-46.3899 - 11.1669i) q^{66} +54.8459 q^{67} +1.60114i q^{68} +(-3.36715 + 13.9879i) q^{69} +26.9688 q^{70} +32.3338i q^{71} +(-11.5841 + 22.6673i) q^{72} -39.7792 q^{73} -60.6307i q^{74} +(14.5834 + 3.51050i) q^{75} -26.4983 q^{76} -95.9136i q^{77} +(-18.2429 + 75.7851i) q^{78} +14.3142 q^{79} -8.94427i q^{80} +(47.4520 + 65.6453i) q^{81} +83.8352 q^{82} -42.0770i q^{83} +(-49.7486 - 11.9754i) q^{84} -1.79013 q^{85} -15.1268i q^{86} +(-7.70625 + 32.0136i) q^{87} -31.8100 q^{88} -77.1399i q^{89} +(-25.3429 - 12.9514i) q^{90} -156.690 q^{91} +9.59166i q^{92} +(124.397 + 29.9446i) q^{93} -93.1521 q^{94} -29.6260i q^{95} +(-3.97167 + 16.4993i) q^{96} +139.910 q^{97} -33.5615i q^{98} +(-46.0614 + 90.1310i) 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
$3$	−2.91669	−	0.702099i	−0.972229	−	0.234033i
$3$	−2.91669	−	0.702099i	−0.972229	−	0.234033i
$4$	−2.00000			−0.500000
$5$		−	2.23607i		−	0.447214i
$5$		−	2.23607i		−	0.447214i
$6$	−0.992918	+	4.12482i	−0.165486	+	0.687469i
$7$	−8.52828			−1.21833			−0.609163	−	0.793045i	$-0.708494\pi$
$7$	−8.52828			−1.21833			−0.609163	+	0.793045i	$0.708494\pi$
$8$			2.82843i			0.353553i
$9$	8.01411	+	4.09561i	0.890457	+	0.455067i
$10$	−3.16228			−0.316228
$11$			11.2465i			1.02241i	0.859458	+	0.511206i	$0.170801\pi$
$11$			11.2465i			1.02241i	−0.859458	+	0.511206i	$0.829199\pi$
$12$	5.83337	+	1.40420i	0.486114	+	0.117017i
$13$	18.3730			1.41331			0.706653	−	0.707561i	$-0.250204\pi$
$13$	18.3730			1.41331			0.706653	+	0.707561i	$0.250204\pi$
$14$			12.0608i			0.861486i
$15$	−1.56994	+	6.52191i	−0.104663	+	0.434794i
$16$	4.00000			0.250000
$17$		−	0.800569i		−	0.0470923i	−0.999723	−	0.0235462i	$-0.992504\pi$
$17$		−	0.800569i		−	0.0470923i	0.999723	−	0.0235462i	$-0.00749567\pi$
$18$	5.79206	−	11.3337i	0.321781	−	0.629648i
$19$	13.2492			0.697324			0.348662	−	0.937248i	$-0.386636\pi$
$19$	13.2492			0.697324			0.348662	+	0.937248i	$0.386636\pi$
$20$			4.47214i			0.223607i
$21$	24.8743	+	5.98770i	1.18449	+	0.285129i
$22$	15.9050			0.722954
$23$		−	4.79583i		−	0.208514i
$23$		−	4.79583i		−	0.208514i
$24$	1.98584	−	8.24963i	0.0827432	−	0.343735i
$25$	−5.00000			−0.200000
$26$		−	25.9833i		−	0.999358i
$27$	−20.4991	−	17.5723i	−0.759227	−	0.650826i
$28$	17.0566			0.609163
$29$		−	10.9760i		−	0.378483i	−0.981931	−	0.189241i	$-0.939397\pi$
$29$		−	10.9760i		−	0.378483i	0.981931	−	0.189241i	$-0.0606029\pi$
$30$	9.22337	+	2.22023i	0.307446	+	0.0740078i
$31$	−42.6501			−1.37581			−0.687905	−	0.725801i	$-0.741469\pi$
$31$	−42.6501			−1.37581			−0.687905	+	0.725801i	$0.741469\pi$
$32$		−	5.65685i		−	0.176777i
$33$	7.89618	−	32.8026i	0.239278	−	0.994018i
$34$	−1.13218			−0.0332993
$35$			19.0698i			0.544852i
$36$	−16.0282	−	8.19121i	−0.445229	−	0.227534i
$37$	42.8724			1.15871			0.579357	−	0.815074i	$-0.303304\pi$
$37$	42.8724			1.15871			0.579357	+	0.815074i	$0.303304\pi$
$38$		−	18.7371i		−	0.493083i
$39$	−53.5882	−	12.8996i	−1.37406	−	0.330760i
$40$	6.32456			0.158114
$41$			59.2804i			1.44586i	0.690919	+	0.722932i	$0.257206\pi$
$41$			59.2804i			1.44586i	−0.690919	+	0.722932i	$0.742794\pi$
$42$	8.46788	−	35.1776i	0.201616	−	0.837562i
$43$	10.6963			0.248750			0.124375	−	0.992235i	$-0.460307\pi$
$43$	10.6963			0.248750			0.124375	+	0.992235i	$0.460307\pi$
$44$		−	22.4931i		−	0.511206i
$45$	9.15805	−	17.9201i	0.203512	−	0.398224i
$46$	−6.78233			−0.147442
$47$		−	65.8685i		−	1.40146i	−0.713428	−	0.700729i	$-0.752858\pi$
$47$		−	65.8685i		−	1.40146i	0.713428	−	0.700729i	$-0.247142\pi$
$48$	−11.6667	−	2.80840i	−0.243057	−	0.0585083i
$49$	23.7316			0.484317
$50$			7.07107i			0.141421i
$51$	−0.562079	+	2.33501i	−0.0110212	+	0.0457845i
$52$	−36.7459			−0.706653
$53$		−	68.3687i		−	1.28998i	−0.764193	−	0.644988i	$-0.776862\pi$
$53$		−	68.3687i		−	1.28998i	0.764193	−	0.644988i	$-0.223138\pi$
$54$	−24.8510	+	28.9901i	−0.460203	+	0.536855i
$55$	25.1480			0.457237
$56$		−	24.1216i		−	0.430743i
$57$	−38.6436	−	9.30223i	−0.677959	−	0.163197i
$58$	−15.5224			−0.267628
$59$		−	54.6341i		−	0.926002i	−0.886358	−	0.463001i	$-0.846773\pi$
$59$		−	54.6341i		−	0.926002i	0.886358	−	0.463001i	$-0.153227\pi$
$60$	3.13988	−	13.0438i	0.0523314	−	0.217397i
$61$	34.1922			0.560528			0.280264	−	0.959923i	$-0.409578\pi$
$61$	34.1922			0.560528			0.280264	+	0.959923i	$0.409578\pi$
$62$			60.3163i			0.972844i
$63$	−68.3466	−	34.9285i	−1.08487	−	0.554420i
$64$	−8.00000			−0.125000
$65$		−	41.0832i		−	0.632049i
$66$	−46.3899	−	11.1669i	−0.702877	−	0.169195i
$67$	54.8459			0.818595			0.409298	−	0.912401i	$-0.365774\pi$
$67$	54.8459			0.818595			0.409298	+	0.912401i	$0.365774\pi$
$68$			1.60114i			0.0235462i
$69$	−3.36715	+	13.9879i	−0.0487993	+	0.202724i
$70$	26.9688			0.385268
$71$			32.3338i			0.455405i	0.973731	+	0.227703i	$0.0731214\pi$
$71$			32.3338i			0.455405i	−0.973731	+	0.227703i	$0.926879\pi$
$72$	−11.5841	+	22.6673i	−0.160891	+	0.314824i
$73$	−39.7792			−0.544920			−0.272460	−	0.962167i	$-0.587837\pi$
$73$	−39.7792			−0.544920			−0.272460	+	0.962167i	$0.587837\pi$
$74$		−	60.6307i		−	0.819334i
$75$	14.5834	+	3.51050i	0.194446	+	0.0468066i
$76$	−26.4983			−0.348662
$77$		−	95.9136i		−	1.24563i
$78$	−18.2429	+	75.7851i	−0.233883	+	0.971604i
$79$	14.3142			0.181193			0.0905964	−	0.995888i	$-0.471123\pi$
$79$	14.3142			0.181193			0.0905964	+	0.995888i	$0.471123\pi$
$80$		−	8.94427i		−	0.111803i
$81$	47.4520	+	65.6453i	0.585827	+	0.810436i
$82$	83.8352			1.02238
$83$		−	42.0770i		−	0.506952i	−0.967342	−	0.253476i	$-0.918426\pi$
$83$		−	42.0770i		−	0.506952i	0.967342	−	0.253476i	$-0.0815738\pi$
$84$	−49.7486	−	11.9754i	−0.592246	−	0.142564i
$85$	−1.79013			−0.0210603
$86$		−	15.1268i		−	0.175893i
$87$	−7.70625	+	32.0136i	−0.0885775	+	0.367972i
$88$	−31.8100			−0.361477
$89$		−	77.1399i		−	0.866740i	−0.901216	−	0.433370i	$-0.857324\pi$
$89$		−	77.1399i		−	0.866740i	0.901216	−	0.433370i	$-0.142676\pi$
$90$	−25.3429	−	12.9514i	−0.281587	−	0.143905i
$91$	−156.690			−1.72187
$92$			9.59166i			0.104257i
$93$	124.397	+	29.9446i	1.33760	+	0.321985i
$94$	−93.1521			−0.990980
$95$		−	29.6260i		−	0.311853i
$96$	−3.97167	+	16.4993i	−0.0413716	+	0.171867i
$97$	139.910			1.44237			0.721187	−	0.692740i	$-0.243597\pi$
$97$	139.910			1.44237			0.721187	+	0.692740i	$0.243597\pi$
$98$		−	33.5615i		−	0.342464i
$99$	−46.0614	+	90.1310i	−0.465266	+	0.910414i
$100$	10.0000			0.100000
$101$		−	185.003i		−	1.83171i	−0.401504	−	0.915857i	$-0.631512\pi$
$101$		−	185.003i		−	1.83171i	0.401504	−	0.915857i	$-0.368488\pi$
$102$	3.30220	+	0.794900i	0.0323745	+	0.00779313i
$103$	−164.164			−1.59383			−0.796913	−	0.604093i	$-0.793535\pi$
$103$	−164.164			−1.59383			−0.796913	+	0.604093i	$0.793535\pi$
$104$			51.9666i			0.499679i
$105$	13.3889	−	55.6207i	0.127513	−	0.529721i
$106$	−96.6880			−0.912151
$107$		−	106.013i		−	0.990772i	−0.868673	−	0.495386i	$-0.835027\pi$
$107$		−	106.013i		−	0.990772i	0.868673	−	0.495386i	$-0.164973\pi$
$108$	40.9983	+	35.1446i	0.379614	+	0.325413i
$109$	58.8011			0.539459			0.269730	−	0.962936i	$-0.413066\pi$
$109$	58.8011			0.539459			0.269730	+	0.962936i	$0.413066\pi$
$110$		−	35.5647i		−	0.323315i
$111$	−125.045	−	30.1007i	−1.12653	−	0.271177i
$112$	−34.1131			−0.304581
$113$		−	15.8528i		−	0.140290i	−0.997537	−	0.0701450i	$-0.977654\pi$
$113$		−	15.8528i		−	0.140290i	0.997537	−	0.0701450i	$-0.0223462\pi$
$114$	−13.1553	+	54.6504i	−0.115398	+	0.479389i
$115$	−10.7238			−0.0932505
$116$			21.9520i			0.189241i
$117$	147.243	+	75.2484i	1.25849	+	0.643149i
$118$	−77.2643			−0.654782
$119$			6.82748i			0.0573738i
$120$	−18.4467	−	4.44047i	−0.153723	−	0.0370039i
$121$	−5.48447			−0.0453262
$122$		−	48.3551i		−	0.396353i
$123$	41.6207	−	172.902i	0.338380	−	1.40571i
$124$	85.3002			0.687905
$125$			11.1803i			0.0894427i
$126$	−49.3963	+	96.6567i	−0.392034	+	0.767117i
$127$	201.548			1.58699			0.793495	−	0.608577i	$-0.208259\pi$
$127$	201.548			1.58699			0.793495	+	0.608577i	$0.208259\pi$
$128$			11.3137i			0.0883883i
$129$	−31.1976	−	7.50983i	−0.241842	−	0.0582158i
$130$	−58.1004			−0.446926
$131$			137.815i			1.05202i	0.850477	+	0.526012i	$0.176313\pi$
$131$			137.815i			1.05202i	−0.850477	+	0.526012i	$0.823687\pi$
$132$	−15.7924	+	65.6052i	−0.119639	+	0.497009i
$133$	−112.993			−0.849568
$134$		−	77.5638i		−	0.578834i
$135$	−39.2929	+	45.8374i	−0.291058	+	0.339537i
$136$	2.26435			0.0166496
$137$		−	12.5145i		−	0.0913465i	−0.998956	−	0.0456733i	$-0.985457\pi$
$137$		−	12.5145i		−	0.0913465i	0.998956	−	0.0456733i	$-0.0145433\pi$
$138$	19.7819	+	4.76187i	0.143347	+	0.0345063i
$139$	271.132			1.95059			0.975296	−	0.220900i	$-0.0708996\pi$
$139$	271.132			1.95059			0.975296	+	0.220900i	$0.0708996\pi$
$140$		−	38.1396i		−	0.272426i
$141$	−46.2462	+	192.118i	−0.327987	+	1.36254i
$142$	45.7268			0.322020
$143$			206.632i			1.44498i
$144$	32.0565	+	16.3824i	0.222614	+	0.113767i
$145$	−24.5431			−0.169263
$146$			56.2562i			0.385317i
$147$	−69.2175	−	16.6619i	−0.470867	−	0.113346i
$148$	−85.7448			−0.579357
$149$			83.1004i			0.557721i	0.960332	+	0.278861i	$0.0899567\pi$
$149$			83.1004i			0.557721i	−0.960332	+	0.278861i	$0.910043\pi$
$150$	4.96459	−	20.6241i	0.0330973	−	0.137494i
$151$	170.027			1.12601			0.563003	−	0.826455i	$-0.309646\pi$
$151$	170.027			1.12601			0.563003	+	0.826455i	$0.309646\pi$
$152$			37.4743i			0.246541i
$153$	3.27882	−	6.41585i	0.0214302	−	0.0419337i
$154$	−135.642			−0.880794
$155$			95.3685i			0.615281i
$156$	107.176	+	25.7993i	0.687028	+	0.165380i
$157$	148.967			0.948833			0.474417	−	0.880300i	$-0.342659\pi$
$157$	148.967			0.948833			0.474417	+	0.880300i	$0.342659\pi$
$158$		−	20.2434i		−	0.128123i
$159$	−48.0016	+	199.410i	−0.301897	+	1.25415i
$160$	−12.6491			−0.0790569
$161$			40.9002i			0.254038i
$162$	92.8365	−	67.1073i	0.573065	−	0.414243i
$163$	−233.872			−1.43480			−0.717398	−	0.696664i	$-0.754667\pi$
$163$	−233.872			−1.43480			−0.717398	+	0.696664i	$0.754667\pi$
$164$		−	118.561i		−	0.722932i
$165$	−73.3488	−	17.6564i	−0.444538	−	0.107008i
$166$	−59.5059			−0.358469
$167$		−	140.840i		−	0.843356i	−0.906746	−	0.421678i	$-0.861441\pi$
$167$		−	140.840i		−	0.843356i	0.906746	−	0.421678i	$-0.138559\pi$
$168$	−16.9358	+	70.3552i	−0.100808	+	0.418781i
$169$	168.566			0.997432
$170$			2.53162i			0.0148919i
$171$	106.180	+	54.2633i	0.620937	+	0.317330i
$172$	−21.3925			−0.124375
$173$		−	112.577i		−	0.650735i	−0.945588	−	0.325368i	$-0.894512\pi$
$173$		−	112.577i		−	0.650735i	0.945588	−	0.325368i	$-0.105488\pi$
$174$	45.2740	+	10.8983i	0.260195	+	0.0626338i
$175$	42.6414			0.243665
$176$			44.9861i			0.255603i
$177$	−38.3586	+	159.350i	−0.216715	+	0.900285i
$178$	−109.092			−0.612878
$179$			265.456i			1.48300i	0.670954	+	0.741499i	$0.265885\pi$
$179$			265.456i			1.48300i	−0.670954	+	0.741499i	$0.734115\pi$
$180$	−18.3161	+	35.8402i	−0.101756	+	0.199112i
$181$	−181.424			−1.00234			−0.501172	−	0.865348i	$-0.667098\pi$
$181$	−181.424			−1.00234			−0.501172	+	0.865348i	$0.667098\pi$
$182$			221.593i			1.21754i
$183$	−99.7279	−	24.0063i	−0.544961	−	0.131182i
$184$	13.5647			0.0737210
$185$		−	95.8656i		−	0.518192i
$186$	42.3480	−	175.924i	0.227678	−	0.945827i
$187$	9.00363			0.0481477
$188$			131.737i			0.700729i
$189$	174.822	+	149.861i	0.924986	+	0.792918i
$190$	−41.8975			−0.220513
$191$		−	192.654i		−	1.00866i	−0.863511	−	0.504331i	$-0.831739\pi$
$191$		−	192.654i		−	1.00866i	0.863511	−	0.504331i	$-0.168261\pi$
$192$	23.3335	+	5.61679i	0.121529	+	0.0292541i
$193$	204.556			1.05988			0.529938	−	0.848036i	$-0.322215\pi$
$193$	204.556			1.05988			0.529938	+	0.848036i	$0.322215\pi$
$194$		−	197.863i		−	1.01991i
$195$	−28.8445	+	119.827i	−0.147920	+	0.614496i
$196$	−47.4631			−0.242159
$197$		−	41.7064i		−	0.211708i	−0.994382	−	0.105854i	$-0.966242\pi$
$197$		−	41.7064i		−	0.211708i	0.994382	−	0.105854i	$-0.0337576\pi$
$198$	127.464	+	65.1406i	0.643760	+	0.328993i
$199$	−7.69267			−0.0386566			−0.0193283	−	0.999813i	$-0.506153\pi$
$199$	−7.69267			−0.0386566			−0.0193283	+	0.999813i	$0.506153\pi$
$200$		−	14.1421i		−	0.0707107i
$201$	−159.968	−	38.5073i	−0.795862	−	0.191578i
$202$	−261.634			−1.29522
$203$			93.6065i			0.461116i
$204$	1.12416	−	4.67002i	0.00551058	−	0.0228922i
$205$	132.555			0.646610
$206$			232.163i			1.12701i
$207$	19.6418	−	38.4343i	0.0948881	−	0.185673i
$208$	73.4919			0.353326
$209$			149.007i			0.712953i
$210$	−78.6595	−	18.9348i	−0.374569	−	0.0901655i
$211$	348.331			1.65086			0.825429	−	0.564507i	$-0.190934\pi$
$211$	348.331			1.65086			0.825429	+	0.564507i	$0.190934\pi$
$212$			136.737i			0.644988i
$213$	22.7015	−	94.3074i	0.106580	−	0.442758i
$214$	−149.924			−0.700581
$215$		−	23.9176i		−	0.111244i
$216$	49.7020	−	57.9803i	0.230102	−	0.268427i
$217$	363.732			1.67618
$218$		−	83.1573i		−	0.381455i
$219$	116.023	+	27.9289i	0.529787	+	0.127529i
$220$	−50.2960			−0.228618
$221$		−	14.7088i		−	0.0665558i
$222$	−42.5688	+	176.841i	−0.191751	+	0.796580i
$223$	411.231			1.84409			0.922044	−	0.387086i	$-0.126518\pi$
$223$	411.231			1.84409			0.922044	+	0.387086i	$0.126518\pi$
$224$			48.2432i			0.215372i
$225$	−40.0706	−	20.4780i	−0.178091	−	0.0910135i
$226$	−22.4192			−0.0992000
$227$		−	97.7055i		−	0.430421i	−0.976568	−	0.215210i	$-0.930956\pi$
$227$		−	97.7055i		−	0.430421i	0.976568	−	0.215210i	$-0.0690437\pi$
$228$	77.2873	+	18.6045i	0.338979	+	0.0815985i
$229$	88.0463			0.384482			0.192241	−	0.981348i	$-0.438425\pi$
$229$	88.0463			0.384482			0.192241	+	0.981348i	$0.438425\pi$
$230$			15.1658i			0.0659380i
$231$	−67.3408	+	279.750i	−0.291519	+	1.21104i
$232$	31.0448			0.133814
$233$		−	113.860i		−	0.488671i	−0.969691	−	0.244336i	$-0.921430\pi$
$233$		−	113.860i		−	0.488671i	0.969691	−	0.244336i	$-0.0785699\pi$
$234$	106.417	−	208.233i	0.454775	−	0.889885i
$235$	−147.286			−0.626751
$236$			109.268i			0.463001i
$237$	−41.7501	−	10.0500i	−0.176161	−	0.0424051i
$238$	9.65551			0.0405694
$239$			88.4422i			0.370051i	0.982734	+	0.185026i	$0.0592368\pi$
$239$			88.4422i			0.370051i	−0.982734	+	0.185026i	$0.940763\pi$
$240$	−6.27977	+	26.0876i	−0.0261657	+	0.108698i
$241$	−297.884			−1.23603			−0.618017	−	0.786164i	$-0.712064\pi$
$241$	−297.884			−1.23603			−0.618017	+	0.786164i	$0.712064\pi$
$242$			7.75622i			0.0320505i
$243$	−92.3131	−	224.783i	−0.379889	−	0.925032i
$244$	−68.3844			−0.280264
$245$		−	53.0654i		−	0.216593i
$246$	−244.521	−	58.8606i	−0.993988	−	0.239271i
$247$	243.426			0.985532
$248$		−	120.633i		−	0.486422i
$249$	−29.5422	+	122.725i	−0.118643	+	0.492873i
$250$	15.8114			0.0632456
$251$			252.585i			1.00632i	0.864195	+	0.503158i	$0.167829\pi$
$251$			252.585i			1.00632i	−0.864195	+	0.503158i	$0.832171\pi$
$252$	136.693	+	69.8569i	0.542433	+	0.277210i
$253$	53.9365			0.213188
$254$		−	285.032i		−	1.12217i
$255$	5.22124	+	1.25685i	0.0204754	+	0.00492881i
$256$	16.0000			0.0625000
$257$			108.572i			0.422460i	0.977436	+	0.211230i	$0.0677469\pi$
$257$			108.572i			0.422460i	−0.977436	+	0.211230i	$0.932253\pi$
$258$	−10.6205	+	44.1201i	−0.0411648	+	0.171008i
$259$	−365.628			−1.41169
$260$			82.1664i			0.316025i
$261$	44.9534	−	87.9630i	0.172235	−	0.337023i
$262$	194.900			0.743893
$263$		−	411.174i		−	1.56340i	−0.623655	−	0.781700i	$-0.714353\pi$
$263$		−	411.174i		−	1.56340i	0.623655	−	0.781700i	$-0.285647\pi$
$264$	92.7798	+	22.3338i	0.351439	+	0.0845976i
$265$	−152.877			−0.576895
$266$			159.796i			0.600735i
$267$	−54.1598	+	224.993i	−0.202846	+	0.842670i
$268$	−109.692			−0.409298
$269$		−	270.452i		−	1.00540i	−0.864462	−	0.502698i	$-0.832341\pi$
$269$		−	270.452i		−	1.00540i	0.864462	−	0.502698i	$-0.167659\pi$
$270$	64.8239	+	55.5685i	0.240089	+	0.205809i
$271$	−138.939			−0.512689			−0.256345	−	0.966585i	$-0.582518\pi$
$271$	−138.939			−0.512689			−0.256345	+	0.966585i	$0.582518\pi$
$272$		−	3.20228i		−	0.0117731i
$273$	457.015	+	110.012i	1.67405	+	0.402974i
$274$	−17.6981			−0.0645917
$275$		−	56.2327i		−	0.204482i
$276$	6.73430	−	27.9759i	0.0243996	−	0.101362i
$277$	−64.2222			−0.231849			−0.115925	−	0.993258i	$-0.536983\pi$
$277$	−64.2222			−0.231849			−0.115925	+	0.993258i	$0.536983\pi$
$278$		−	383.439i		−	1.37928i
$279$	−341.803	−	174.678i	−1.22510	−	0.626086i
$280$	−53.9376			−0.192634
$281$		−	298.203i		−	1.06122i	−0.847616	−	0.530611i	$-0.821963\pi$
$281$		−	298.203i		−	1.06122i	0.847616	−	0.530611i	$-0.178037\pi$
$282$	271.695	+	65.4020i	0.963459	+	0.231922i
$283$	−118.877			−0.420061			−0.210031	−	0.977695i	$-0.567356\pi$
$283$	−118.877			−0.420061			−0.210031	+	0.977695i	$0.567356\pi$
$284$		−	64.6675i		−	0.227703i
$285$	−20.8004	+	86.4098i	−0.0729839	+	0.303192i
$286$	292.222			1.02176
$287$		−	505.560i		−	1.76153i
$288$	23.1682	−	45.3347i	0.0804453	−	0.157412i
$289$	288.359			0.997782
$290$			34.7092i			0.119687i
$291$	−408.074	−	98.2309i	−1.40232	−	0.337563i
$292$	79.5583			0.272460
$293$			338.920i			1.15672i	0.815781	+	0.578361i	$0.196308\pi$
$293$			338.920i			1.15672i	−0.815781	+	0.578361i	$0.803692\pi$
$294$	−23.5635	+	97.8883i	−0.0801479	+	0.332953i
$295$	−122.166			−0.414120
$296$			121.261i			0.409667i
$297$	197.627	−	230.544i	0.665412	−	0.776243i
$298$	117.522			0.394368
$299$		−	88.1137i		−	0.294695i
$300$	−29.1669	−	7.02099i	−0.0972229	−	0.0234033i
$301$	−91.2207			−0.303059
$302$		−	240.455i		−	0.796207i
$303$	−129.891	+	539.596i	−0.428682	+	1.78085i
$304$	52.9966			0.174331
$305$		−	76.4561i		−	0.250676i
$306$	−9.07338	−	4.63695i	−0.0296516	−	0.0151534i
$307$	194.037			0.632042			0.316021	−	0.948752i	$-0.397653\pi$
$307$	194.037			0.632042			0.316021	+	0.948752i	$0.397653\pi$
$308$			191.827i			0.622815i
$309$	478.815	+	115.260i	1.54956	+	0.373008i
$310$	134.871			0.435069
$311$			145.543i			0.467985i	0.972238	+	0.233992i	$0.0751791\pi$
$311$			145.543i			0.467985i	−0.972238	+	0.233992i	$0.924821\pi$
$312$	36.4857	−	151.570i	0.116941	−	0.485802i
$313$	−93.3372			−0.298202			−0.149101	−	0.988822i	$-0.547638\pi$
$313$	−93.3372			−0.298202			−0.149101	+	0.988822i	$0.547638\pi$
$314$		−	210.671i		−	0.670926i
$315$	−78.1024	+	152.828i	−0.247944	+	0.485167i
$316$	−28.6285			−0.0905964
$317$			282.185i			0.890174i	0.895487	+	0.445087i	$0.146827\pi$
$317$			282.185i			0.890174i	−0.895487	+	0.445087i	$0.853173\pi$
$318$	282.008	+	67.8845i	0.886819	+	0.213473i
$319$	123.442			0.386966
$320$			17.8885i			0.0559017i
$321$	−74.4313	+	309.205i	−0.231873	+	0.963257i
$322$	57.8416			0.179632
$323$		−	10.6069i		−	0.0328386i
$324$	−94.9041	−	131.291i	−0.292914	−	0.405218i
$325$	−91.8648			−0.282661
$326$			330.745i			1.01455i
$327$	−171.504	−	41.2842i	−0.524478	−	0.126251i
$328$	−167.670			−0.511190
$329$			561.745i			1.70743i
$330$	−24.9699	+	103.731i	−0.0756664	+	0.314336i
$331$	−384.366			−1.16123			−0.580613	−	0.814179i	$-0.697187\pi$
$331$	−384.366			−1.16123			−0.580613	+	0.814179i	$0.697187\pi$
$332$			84.1540i			0.253476i
$333$	343.584	+	175.588i	1.03178	+	0.527293i
$334$	−199.178			−0.596343
$335$		−	122.639i		−	0.366087i
$336$	99.4973	+	23.9508i	0.296123	+	0.0712821i
$337$	−267.563			−0.793954			−0.396977	−	0.917829i	$-0.629941\pi$
$337$	−267.563			−0.793954			−0.396977	+	0.917829i	$0.629941\pi$
$338$		−	238.388i		−	0.705291i
$339$	−11.1302	+	46.2376i	−0.0328325	+	0.136394i
$340$	3.58025			0.0105302
$341$		−	479.666i		−	1.40664i
$342$	76.7400	−	150.162i	0.224386	−	0.439069i
$343$	215.496			0.628269
$344$			30.2536i			0.0879465i
$345$	31.2780	+	7.52918i	0.0906608	+	0.0218237i
$346$	−159.208			−0.460139
$347$			518.074i			1.49301i	0.665380	+	0.746505i	$0.268270\pi$
$347$			518.074i			1.49301i	−0.665380	+	0.746505i	$0.731730\pi$
$348$	15.4125	−	64.0271i	0.0442888	−	0.183986i
$349$	−375.123			−1.07485			−0.537426	−	0.843311i	$-0.680603\pi$
$349$	−375.123			−1.07485			−0.537426	+	0.843311i	$0.680603\pi$
$350$		−	60.3040i		−	0.172297i
$351$	−376.630	−	322.855i	−1.07302	−	0.919816i
$352$	63.6200			0.180739
$353$			101.654i			0.287970i	0.989580	+	0.143985i	$0.0459918\pi$
$353$			101.654i			0.287970i	−0.989580	+	0.143985i	$0.954008\pi$
$354$	225.356	+	54.2472i	0.636598	+	0.153241i
$355$	72.3005			0.203663
$356$			154.280i			0.433370i
$357$	4.79357	−	19.9136i	0.0134274	−	0.0557804i
$358$	375.412			1.04864
$359$			147.915i			0.412019i	0.978550	+	0.206010i	$0.0660478\pi$
$359$			147.915i			0.412019i	−0.978550	+	0.206010i	$0.933952\pi$
$360$	50.6857	+	25.9029i	0.140794	+	0.0719525i
$361$	−185.460			−0.513739
$362$			256.573i			0.708764i
$363$	15.9965	+	3.85064i	0.0440674	+	0.0106078i
$364$	313.380			0.860933
$365$			88.9489i			0.243696i
$366$	−33.9501	+	141.037i	−0.0927597	+	0.385346i
$367$	−398.761			−1.08654			−0.543271	−	0.839557i	$-0.682814\pi$
$367$	−398.761			−1.08654			−0.543271	+	0.839557i	$0.682814\pi$
$368$		−	19.1833i		−	0.0521286i
$369$	−242.789	+	475.080i	−0.657966	+	1.28748i
$370$	−135.574			−0.366417
$371$			583.068i			1.57161i
$372$	−248.794	−	59.8892i	−0.668801	−	0.160992i
$373$	349.777			0.937739			0.468869	−	0.883268i	$-0.344662\pi$
$373$	349.777			0.937739			0.468869	+	0.883268i	$0.344662\pi$
$374$		−	12.7331i		−	0.0340456i
$375$	7.84971	−	32.6095i	0.0209326	−	0.0869588i
$376$	186.304			0.495490
$377$		−	201.662i		−	0.534912i
$378$	211.936	−	247.236i	0.560678	−	0.654064i
$379$	−568.739			−1.50063			−0.750315	−	0.661080i	$-0.770098\pi$
$379$	−568.739			−1.50063			−0.750315	+	0.661080i	$0.770098\pi$
$380$			59.2521i			0.155926i
$381$	−587.851	−	141.507i	−1.54292	−	0.371408i
$382$	−272.454			−0.713231
$383$		−	116.352i		−	0.303791i	−0.988397	−	0.151896i	$-0.951462\pi$
$383$		−	116.352i		−	0.303791i	0.988397	−	0.151896i	$-0.0485378\pi$
$384$	7.94335	−	32.9985i	0.0206858	−	0.0859337i
$385$	−214.469			−0.557063
$386$		−	289.286i		−	0.749445i
$387$	85.7210	+	43.8077i	0.221501	+	0.113198i
$388$	−279.821			−0.721187
$389$		−	718.880i		−	1.84802i	−0.382368	−	0.924010i	$-0.624891\pi$
$389$		−	718.880i		−	1.84802i	0.382368	−	0.924010i	$-0.375109\pi$
$390$	169.461	+	40.7923i	0.434515	+	0.104596i
$391$	−3.83939			−0.00981942
$392$			67.1230i			0.171232i
$393$	96.7599	−	401.963i	0.246208	−	1.02281i
$394$	−58.9818			−0.149700
$395$		−	32.0076i		−	0.0810319i
$396$	92.1227	−	180.262i	0.232633	−	0.455207i
$397$	−187.368			−0.471961			−0.235980	−	0.971758i	$-0.575830\pi$
$397$	−187.368			−0.471961			−0.235980	+	0.971758i	$0.575830\pi$
$398$			10.8791i			0.0273344i
$399$	329.564	+	79.3320i	0.825974	+	0.198827i
$400$	−20.0000			−0.0500000
$401$			95.8953i			0.239141i	0.992826	+	0.119570i	$0.0381517\pi$
$401$			95.8953i			0.239141i	−0.992826	+	0.119570i	$0.961848\pi$
$402$	−54.4575	+	226.229i	−0.135466	+	0.562759i
$403$	−783.609			−1.94444
$404$			370.006i			0.915857i
$405$	146.787	−	106.106i	0.362438	−	0.261990i
$406$	132.380			0.326058
$407$			482.166i			1.18468i
$408$	−6.60440	−	1.58980i	−0.0161873	−	0.00389657i
$409$	678.263			1.65835			0.829173	−	0.558993i	$-0.188812\pi$
$409$	678.263			1.65835			0.829173	+	0.558993i	$0.188812\pi$
$410$		−	187.461i		−	0.457222i
$411$	−8.78640	+	36.5008i	−0.0213781	+	0.0888097i
$412$	328.328			0.796913
$413$			465.935i			1.12817i
$414$	−54.3544	−	27.7777i	−0.131291	−	0.0670960i
$415$	−94.0870			−0.226716
$416$		−	103.933i		−	0.249839i
$417$	−790.808	−	190.362i	−1.89642	−	0.456503i
$418$	210.728			0.504134
$419$		−	149.069i		−	0.355774i	−0.984051	−	0.177887i	$-0.943074\pi$
$419$		−	149.069i		−	0.355774i	0.984051	−	0.177887i	$-0.0569262\pi$
$420$	−26.7778	+	111.241i	−0.0637567	+	0.264860i
$421$	−61.2688			−0.145532			−0.0727658	−	0.997349i	$-0.523183\pi$
$421$	−61.2688			−0.145532			−0.0727658	+	0.997349i	$0.523183\pi$
$422$		−	492.614i		−	1.16733i
$423$	269.771	−	527.877i	0.637757	−	1.24794i
$424$	193.376			0.456075
$425$			4.00285i			0.00941846i
$426$	−133.371	−	32.1048i	−0.313077	−	0.0753633i
$427$	−291.601			−0.682906
$428$			212.025i			0.495386i
$429$	145.076	−	602.681i	0.338173	−	1.40485i
$430$	−33.8245			−0.0786617
$431$		−	325.729i		−	0.755751i	−0.925856	−	0.377875i	$-0.876655\pi$
$431$		−	325.729i		−	0.755751i	0.925856	−	0.377875i	$-0.123345\pi$
$432$	−81.9965	−	70.2892i	−0.189807	−	0.162706i
$433$	406.666			0.939183			0.469592	−	0.882884i	$-0.344401\pi$
$433$	406.666			0.939183			0.469592	+	0.882884i	$0.344401\pi$
$434$		−	514.395i		−	1.18524i
$435$	71.5845	+	17.2317i	0.164562	+	0.0396131i
$436$	−117.602			−0.269730
$437$		−	63.5407i		−	0.145402i
$438$	39.4975	−	164.082i	0.0901769	−	0.374616i
$439$	583.212			1.32850			0.664250	−	0.747510i	$-0.268751\pi$
$439$	583.212			1.32850			0.664250	+	0.747510i	$0.268751\pi$
$440$			71.1293i			0.161658i
$441$	190.187	+	97.1951i	0.431264	+	0.220397i
$442$	−20.8014			−0.0470621
$443$			203.714i			0.459850i	0.973208	+	0.229925i	$0.0738482\pi$
$443$			203.714i			0.459850i	−0.973208	+	0.229925i	$0.926152\pi$
$444$	250.091	+	60.2013i	0.563267	+	0.135589i
$445$	−172.490			−0.387618
$446$		−	581.569i		−	1.30397i
$447$	58.3448	−	242.378i	0.130525	−	0.542232i
$448$	68.2262			0.152291
$449$		−	843.006i		−	1.87752i	−0.344574	−	0.938759i	$-0.611977\pi$
$449$		−	843.006i		−	1.87752i	0.344574	−	0.938759i	$-0.388023\pi$
$450$	−28.9603	+	56.6683i	−0.0643562	+	0.125930i
$451$	−666.699			−1.47827
$452$			31.7055i			0.0701450i
$453$	−495.915	−	119.376i	−1.09474	−	0.263523i
$454$	−138.176			−0.304353
$455$			350.369i			0.770042i
$456$	26.3107	−	109.301i	0.0576988	−	0.239695i
$457$	−209.735			−0.458938			−0.229469	−	0.973316i	$-0.573699\pi$
$457$	−209.735			−0.458938			−0.229469	+	0.973316i	$0.573699\pi$
$458$		−	124.516i		−	0.271870i
$459$	−14.0678	+	16.4110i	−0.0306489	+	0.0357538i
$460$	21.4476			0.0466252
$461$		−	131.916i		−	0.286153i	−0.989712	−	0.143076i	$-0.954301\pi$
$461$		−	131.916i		−	0.286153i	0.989712	−	0.143076i	$-0.0456995\pi$
$462$	395.626	+	95.2343i	0.856333	+	0.206135i
$463$	−483.695			−1.04470			−0.522349	−	0.852732i	$-0.674944\pi$
$463$	−483.695			−1.04470			−0.522349	+	0.852732i	$0.674944\pi$
$464$		−	43.9040i		−	0.0946207i
$465$	66.9581	−	278.160i	0.143996	−	0.598193i
$466$	−161.023			−0.345543
$467$		−	468.672i		−	1.00358i	−0.864989	−	0.501790i	$-0.832675\pi$
$467$		−	468.672i		−	1.00358i	0.864989	−	0.501790i	$-0.167325\pi$
$468$	−294.486	−	150.497i	−0.629244	−	0.321575i
$469$	−467.741			−0.997316
$470$			208.294i			0.443180i
$471$	−434.489	−	104.589i	−0.922483	−	0.222058i
$472$	154.529			0.327391
$473$			120.296i			0.254325i
$474$	−14.2129	+	59.0436i	−0.0299849	+	0.124565i
$475$	−66.2458			−0.139465
$476$		−	13.6550i		−	0.0286869i
$477$	280.011	−	547.915i	0.587026	−	1.14867i
$478$	125.076			0.261666
$479$		−	661.943i		−	1.38193i	−0.722890	−	0.690964i	$-0.757187\pi$
$479$		−	661.943i		−	1.38193i	0.722890	−	0.690964i	$-0.242813\pi$
$480$	36.8935	+	8.88093i	0.0768614	+	0.0185019i
$481$	787.693			1.63762
$482$			421.272i			0.874009i
$483$	28.7160	−	119.293i	0.0594534	−	0.246983i
$484$	10.9689			0.0226631
$485$		−	312.849i		−	0.645049i
$486$	−317.891	+	130.551i	−0.654096	+	0.268622i
$487$	−26.7852			−0.0550003			−0.0275002	−	0.999622i	$-0.508755\pi$
$487$	−26.7852			−0.0550003			−0.0275002	+	0.999622i	$0.508755\pi$
$488$			96.7102i			0.198177i
$489$	682.130	+	164.201i	1.39495	+	0.335790i
$490$	−75.0458			−0.153155
$491$			616.828i			1.25627i	0.778105	+	0.628135i	$0.216181\pi$
$491$			616.828i			1.25627i	−0.778105	+	0.628135i	$0.783819\pi$
$492$	−83.2415	+	345.805i	−0.169190	+	0.702855i
$493$	−8.78705			−0.0178236
$494$		−	344.257i		−	0.696876i
$495$	201.539	+	102.996i	0.407149	+	0.208073i
$496$	−170.600			−0.343952
$497$		−	275.751i		−	0.554832i
$498$	173.560	+	41.7790i	0.348514	+	0.0838936i
$499$	985.407			1.97476			0.987382	−	0.158359i	$-0.0506202\pi$
$499$	985.407			1.97476			0.987382	+	0.158359i	$0.0506202\pi$
$500$		−	22.3607i		−	0.0447214i
$501$	−98.8839	+	410.787i	−0.197373	+	0.819935i
$502$	357.209			0.711573
$503$			571.158i			1.13550i	0.823200	+	0.567752i	$0.192187\pi$
$503$			571.158i			1.13550i	−0.823200	+	0.567752i	$0.807813\pi$
$504$	98.7926	−	193.313i	0.196017	−	0.383558i
$505$	−413.680			−0.819168
$506$		−	76.2777i		−	0.150746i
$507$	−491.654	−	118.350i	−0.969732	−	0.233432i
$508$	−403.095			−0.793495
$509$			885.134i			1.73897i	0.493962	+	0.869484i	$0.335548\pi$
$509$			885.134i			1.73897i	−0.493962	+	0.869484i	$0.664452\pi$
$510$	1.77745	−	7.38395i	0.00348520	−	0.0144783i
$511$	339.248			0.663890
$512$		−	22.6274i		−	0.0441942i
$513$	−271.596	−	232.818i	−0.529427	−	0.453837i
$514$	153.544			0.298724
$515$			367.082i			0.712781i
$516$	62.3952	+	15.0197i	0.120921	+	0.0291079i
$517$	740.792			1.43287
$518$			517.076i			0.998216i
$519$	−79.0404	+	328.352i	−0.152294	+	0.632663i
$520$	116.201			0.223463
$521$		−	578.831i		−	1.11100i	−0.831516	−	0.555500i	$-0.812527\pi$
$521$		−	578.831i		−	1.11100i	0.831516	−	0.555500i	$-0.187473\pi$
$522$	−124.398	−	63.5737i	−0.238311	−	0.121789i
$523$	−918.390			−1.75600			−0.878002	−	0.478657i	$-0.841124\pi$
$523$	−918.390			−1.75600			−0.878002	+	0.478657i	$0.841124\pi$
$524$		−	275.630i		−	0.526012i
$525$	−124.372	−	29.9385i	−0.236898	−	0.0570257i
$526$	−581.488			−1.10549
$527$			34.1443i			0.0647900i
$528$	31.5847	−	131.210i	0.0598196	−	0.248505i
$529$	−23.0000			−0.0434783
$530$			216.201i			0.407926i
$531$	223.760	−	437.844i	0.421393	−	0.824565i
$532$	225.985			0.424784
$533$			1089.16i			2.04345i
$534$	318.188	+	76.5936i	0.595857	+	0.143434i
$535$	−237.051			−0.443087
$536$			155.128i			0.289417i
$537$	186.377	−	774.253i	0.347070	−	1.44181i
$538$	−382.476			−0.710922
$539$			266.898i			0.495172i
$540$	78.5857	−	91.6749i	0.145529	−	0.169768i
$541$	635.953			1.17551			0.587757	−	0.809037i	$-0.300011\pi$
$541$	635.953			1.17551			0.587757	+	0.809037i	$0.300011\pi$
$542$			196.489i			0.362526i
$543$	529.158	+	127.378i	0.974507	+	0.234582i
$544$	−4.52870			−0.00832482
$545$		−	131.483i		−	0.241254i
$546$	155.580	−	646.317i	0.284945	−	1.18373i
$547$	−147.805			−0.270209			−0.135105	−	0.990831i	$-0.543137\pi$
$547$	−147.805			−0.270209			−0.135105	+	0.990831i	$0.543137\pi$
$548$			25.0289i			0.0456733i
$549$	274.020	+	140.038i	0.499126	+	0.255078i
$550$	−79.5250			−0.144591
$551$		−	145.423i		−	0.263925i
$552$	−39.5639	−	9.52374i	−0.0716736	−	0.0172531i
$553$	−122.076			−0.220752
$554$			90.8239i			0.163942i
$555$	−67.3072	+	279.610i	−0.121274	+	0.503801i
$556$	−542.265			−0.975296
$557$		−	301.911i		−	0.542031i	−0.962575	−	0.271015i	$-0.912641\pi$
$557$		−	301.911i		−	0.542031i	0.962575	−	0.271015i	$-0.0873594\pi$
$558$	−247.032	+	483.382i	−0.442709	+	0.866276i
$559$	196.522			0.351560
$560$			76.2793i			0.136213i
$561$	−26.2608	−	6.32144i	−0.0468106	−	0.0112682i
$562$	−421.723			−0.750397
$563$		−	1059.79i		−	1.88240i	−0.337855	−	0.941198i	$-0.609701\pi$
$563$		−	1059.79i		−	1.88240i	0.337855	−	0.941198i	$-0.390299\pi$
$564$	92.4924	−	384.235i	0.163994	−	0.681268i
$565$	−35.4479			−0.0627396
$566$			168.118i			0.297028i
$567$	−404.684	−	559.841i	−0.713729	−	0.987375i
$568$	−91.4537			−0.161010
$569$			149.837i			0.263335i	0.991294	+	0.131667i	$0.0420330\pi$
$569$			149.837i			0.263335i	−0.991294	+	0.131667i	$0.957967\pi$
$570$	122.202	+	29.4162i	0.214389	+	0.0516074i
$571$	1037.30			1.81665			0.908323	−	0.418271i	$-0.137364\pi$
$571$	1037.30			1.81665			0.908323	+	0.418271i	$0.137364\pi$
$572$		−	413.264i		−	0.722490i
$573$	−135.262	+	561.912i	−0.236060	+	0.980649i
$574$	−714.970			−1.24559
$575$			23.9792i			0.0417029i
$576$	−64.1129	−	32.7648i	−0.111307	−	0.0568834i
$577$	580.551			1.00615			0.503077	−	0.864242i	$-0.332201\pi$
$577$	580.551			1.00615			0.503077	+	0.864242i	$0.332201\pi$
$578$		−	407.801i		−	0.705539i
$579$	−596.626	−	143.619i	−1.03044	−	0.248046i
$580$	49.0862			0.0846314
$581$			358.844i			0.617632i
$582$	−138.919	+	577.104i	−0.238693	+	0.991588i
$583$	768.911			1.31889
$584$		−	112.512i		−	0.192658i
$585$	168.261	−	329.245i	0.287625	−	0.562813i
$586$	479.305			0.817926
$587$		−	1007.65i		−	1.71662i	−0.513136	−	0.858308i	$-0.671516\pi$
$587$		−	1007.65i		−	1.71662i	0.513136	−	0.858308i	$-0.328484\pi$
$588$	138.435	+	33.3238i	0.235434	+	0.0566732i
$589$	−565.078			−0.959385
$590$			172.768i			0.292827i
$591$	−29.2820	+	121.644i	−0.0495466	+	0.205828i
$592$	171.490			0.289678
$593$			181.266i			0.305677i	0.988251	+	0.152838i	$0.0488414\pi$
$593$			181.266i			0.305677i	−0.988251	+	0.152838i	$0.951159\pi$
$594$	−326.039	−	279.487i	−0.548887	−	0.470517i
$595$	15.2667			0.0256583
$596$		−	166.201i		−	0.278861i
$597$	22.4371	+	5.40101i	0.0375831	+	0.00904693i
$598$	−124.612			−0.208381
$599$		−	75.2652i		−	0.125651i	−0.998025	−	0.0628257i	$-0.979989\pi$
$599$		−	75.2652i		−	0.125651i	0.998025	−	0.0628257i	$-0.0200112\pi$
$600$	−9.92918	+	41.2482i	−0.0165486	+	0.0687469i
$601$	−206.231			−0.343146			−0.171573	−	0.985171i	$-0.554885\pi$
$601$	−206.231			−0.343146			−0.171573	+	0.985171i	$0.554885\pi$
$602$			129.006i			0.214295i
$603$	439.541	+	224.627i	0.728924	+	0.372516i
$604$	−340.054			−0.563003
$605$			12.2637i			0.0202705i
$606$	763.104	+	183.693i	1.25925	+	0.303124i
$607$	649.464			1.06996			0.534978	−	0.844866i	$-0.320320\pi$
$607$	649.464			1.06996			0.534978	+	0.844866i	$0.320320\pi$
$608$		−	74.9486i		−	0.123271i
$609$	65.7210	−	273.021i	0.107916	−	0.448310i
$610$	−108.125			−0.177255
$611$		−	1210.20i		−	1.98069i
$612$	−6.55763	+	12.8317i	−0.0107151	+	0.0209668i
$613$	230.058			0.375299			0.187650	−	0.982236i	$-0.439913\pi$
$613$	230.058			0.375299			0.187650	+	0.982236i	$0.439913\pi$
$614$		−	274.410i		−	0.446921i
$615$	−386.622	−	93.0668i	−0.628653	−	0.151328i
$616$	271.285			0.440397
$617$			486.305i			0.788176i	0.919073	+	0.394088i	$0.128940\pi$
$617$			486.305i			0.788176i	−0.919073	+	0.394088i	$0.871060\pi$
$618$	163.002	−	677.147i	0.263757	−	1.09571i
$619$	−176.546			−0.285211			−0.142605	−	0.989780i	$-0.545548\pi$
$619$	−176.546			−0.285211			−0.142605	+	0.989780i	$0.545548\pi$
$620$		−	190.737i		−	0.307640i
$621$	−84.2738	+	98.3104i	−0.135707	+	0.158310i
$622$	205.829			0.330915
$623$			657.870i			1.05597i
$624$	−214.353	−	51.5986i	−0.343514	−	0.0826900i
$625$	25.0000			0.0400000
$626$			131.999i			0.210861i
$627$	104.618	−	434.607i	0.166855	−	0.693153i
$628$	−297.934			−0.474417
$629$		−	34.3223i		−	0.0545665i
$630$	216.131	+	110.454i	0.343065	+	0.175323i
$631$	−594.149			−0.941599			−0.470800	−	0.882240i	$-0.656035\pi$
$631$	−594.149			−0.941599			−0.470800	+	0.882240i	$0.656035\pi$
$632$			40.4868i			0.0640613i
$633$	−1015.97	−	244.563i	−1.60501	−	0.386355i
$634$	399.070			0.629448
$635$		−	450.674i		−	0.709724i
$636$	96.0033	−	398.820i	0.150949	−	0.627076i
$637$	436.019			0.684488
$638$		−	174.573i		−	0.273626i
$639$	−132.426	+	259.126i	−0.207240	+	0.405519i
$640$	25.2982			0.0395285
$641$			457.503i			0.713734i	0.934155	+	0.356867i	$0.116155\pi$
$641$			457.503i			0.713734i	−0.934155	+	0.356867i	$0.883845\pi$
$642$	437.282	+	105.262i	0.681125	+	0.163959i
$643$	−713.769			−1.11006			−0.555030	−	0.831830i	$-0.687293\pi$
$643$	−713.769			−1.11006			−0.555030	+	0.831830i	$0.687293\pi$
$644$		−	81.8004i		−	0.127019i
$645$	−16.7925	+	69.7600i	−0.0260349	+	0.108155i
$646$	−15.0004			−0.0232204
$647$		−	784.509i		−	1.21253i	−0.795261	−	0.606267i	$-0.792666\pi$
$647$		−	784.509i		−	1.21253i	0.795261	−	0.606267i	$-0.207334\pi$
$648$	−185.673	+	134.215i	−0.286532	+	0.207121i
$649$	614.444			0.946755
$650$			129.917i			0.199872i
$651$	−1060.89	−	255.376i	−1.62963	−	0.392282i
$652$	467.743			0.717398
$653$		−	400.746i		−	0.613700i	−0.951758	−	0.306850i	$-0.900725\pi$
$653$		−	400.746i		−	0.613700i	0.951758	−	0.306850i	$-0.0992750\pi$
$654$	−58.3847	+	242.544i	−0.0892732	+	0.370862i
$655$	308.164			0.470479
$656$			237.122i			0.361466i
$657$	−318.795	−	162.920i	−0.485228	−	0.247975i
$658$	794.427			1.20734
$659$			1057.28i			1.60437i	0.597073	+	0.802187i	$0.296330\pi$
$659$			1057.28i			1.60437i	−0.597073	+	0.802187i	$0.703670\pi$
$660$	146.698	+	35.3128i	0.222269	+	0.0535042i
$661$	−266.071			−0.402528			−0.201264	−	0.979537i	$-0.564505\pi$
$661$	−266.071			−0.402528			−0.201264	+	0.979537i	$0.564505\pi$
$662$			543.576i			0.821111i
$663$	−10.3271	+	42.9010i	−0.0155763	+	0.0647075i
$664$	119.012			0.179234
$665$			252.659i			0.379938i
$666$	248.320	−	485.901i	0.372852	−	0.729582i
$667$	−52.6391			−0.0789192
$668$			281.681i			0.421678i
$669$	−1199.43	−	288.725i	−1.79287	−	0.431577i
$670$	−173.438			−0.258863
$671$			384.544i			0.573091i
$672$	33.8715	−	140.710i	0.0504041	−	0.209390i
$673$	−10.7379			−0.0159552			−0.00797761	−	0.999968i	$-0.502539\pi$
$673$	−10.7379			−0.0159552			−0.00797761	+	0.999968i	$0.502539\pi$
$674$			378.391i			0.561410i
$675$	102.496	+	87.8615i	0.151845	+	0.130165i
$676$	−337.132			−0.498716
$677$			245.177i			0.362152i	0.983469	+	0.181076i	$0.0579581\pi$
$677$			245.177i			0.362152i	−0.983469	+	0.181076i	$0.942042\pi$
$678$	65.3898	+	15.7405i	0.0964451	+	0.0232161i
$679$	−1193.19			−1.75728
$680$		−	5.06324i		−	0.00744595i
$681$	−68.5990	+	284.976i	−0.100733	+	0.418467i
$682$	−678.349			−0.994647
$683$			965.369i			1.41342i	0.707501	+	0.706712i	$0.249822\pi$
$683$			965.369i			1.41342i	−0.707501	+	0.706712i	$0.750178\pi$
$684$	−212.361	−	108.527i	−0.310469	−	0.158665i
$685$	−27.9832			−0.0408514
$686$		−	304.758i		−	0.444253i
$687$	−256.803	−	61.8172i	−0.373804	−	0.0899814i
$688$	42.7850			0.0621875
$689$		−	1256.14i		−	1.82313i
$690$	10.6479	−	44.2337i	0.0154317	−	0.0641069i
$691$	−143.140			−0.207148			−0.103574	−	0.994622i	$-0.533028\pi$
$691$	−143.140			−0.207148			−0.103574	+	0.994622i	$0.533028\pi$
$692$			225.154i			0.325368i
$693$	392.824	−	768.662i	0.566846	−	1.10918i
$694$	732.668			1.05572
$695$		−	606.270i		−	0.872332i
$696$	−90.5480	−	21.7966i	−0.130098	−	0.0313169i
$697$	47.4581			0.0680891
$698$			530.504i			0.760034i
$699$	−79.9413	+	332.095i	−0.114365	+	0.475100i
$700$	−85.2828			−0.121833
$701$			880.642i			1.25627i	0.778106	+	0.628133i	$0.216181\pi$
$701$			880.642i			1.25627i	−0.778106	+	0.628133i	$0.783819\pi$
$702$	−456.586	+	532.635i	−0.650408	+	0.758739i
$703$	568.023			0.807999
$704$		−	89.9723i		−	0.127801i
$705$	429.588	+	103.410i	0.609345	+	0.146680i
$706$	143.760			0.203626
$707$			1577.76i			2.23163i
$708$	76.7171	−	318.701i	0.108357	−	0.450143i
$709$	384.075			0.541713			0.270857	−	0.962620i	$-0.412693\pi$
$709$	384.075			0.541713			0.270857	+	0.962620i	$0.412693\pi$
$710$		−	102.248i		−	0.144012i
$711$	114.716	+	58.6255i	0.161344	+	0.0824549i
$712$	218.184			0.306439
$713$			204.543i			0.286876i
$714$	−28.1621	−	6.77913i	−0.0394427	−	0.00949458i
$715$	462.044			0.646215
$716$		−	530.913i		−	0.741499i
$717$	62.0952	−	257.958i	0.0866042	−	0.359774i
$718$	209.183			0.291342
$719$			366.200i			0.509318i	0.967031	+	0.254659i	$0.0819632\pi$
$719$			366.200i			0.509318i	−0.967031	+	0.254659i	$0.918037\pi$
$720$	36.6322	−	71.6804i	0.0508781	−	0.0995561i
$721$	1400.04			1.94180
$722$			262.280i			0.363268i
$723$	868.835	+	209.144i	1.20171	+	0.289273i
$724$	362.848			0.501172
$725$			54.8800i			0.0756966i
$726$	5.44563	−	22.6224i	0.00750087	−	0.0311604i
$727$	46.5086			0.0639733			0.0319867	−	0.999488i	$-0.489817\pi$
$727$	46.5086			0.0639733			0.0319867	+	0.999488i	$0.489817\pi$
$728$		−	443.186i		−	0.608772i
$729$	111.429	+	720.434i	0.152851	+	0.988249i
$730$	125.793			0.172319
$731$		−	8.56309i		−	0.0117142i
$732$	199.456	+	48.0126i	0.272481	+	0.0655910i
$733$	−1189.43			−1.62269			−0.811344	−	0.584569i	$-0.801264\pi$
$733$	−1189.43			−1.62269			−0.811344	+	0.584569i	$0.801264\pi$
$734$			563.933i			0.768301i
$735$	−37.2572	+	154.775i	−0.0506900	+	0.210578i
$736$	−27.1293			−0.0368605
$737$			616.826i			0.836942i
$738$	671.865	+	343.356i	0.910386	+	0.465252i
$739$	91.0727			0.123238			0.0616189	−	0.998100i	$-0.480374\pi$
$739$	91.0727			0.123238			0.0616189	+	0.998100i	$0.480374\pi$
$740$			191.731i			0.259096i
$741$	−709.998	−	170.910i	−0.958163	−	0.230647i
$742$	824.582			1.11130
$743$			1075.51i			1.44753i	0.690048	+	0.723764i	$0.257589\pi$
$743$			1075.51i			1.44753i	−0.690048	+	0.723764i	$0.742411\pi$
$744$	−84.6961	+	351.848i	−0.113839	+	0.472913i
$745$	185.818			0.249420
$746$		−	494.659i		−	0.663081i
$747$	172.331	−	337.210i	0.230697	−	0.451419i
$748$	−18.0073			−0.0240739
$749$			904.105i			1.20708i
$750$	−46.1169	−	11.1012i	−0.0614891	−	0.0148016i
$751$	848.304			1.12957			0.564783	−	0.825240i	$-0.308960\pi$
$751$	848.304			1.12957			0.564783	+	0.825240i	$0.308960\pi$
$752$		−	263.474i		−	0.350364i
$753$	177.340	−	736.712i	0.235511	−	0.978369i
$754$	−285.193			−0.378240
$755$		−	380.192i		−	0.503566i
$756$	−349.645	−	299.723i	−0.462493	−	0.396459i
$757$	244.126			0.322491			0.161245	−	0.986914i	$-0.448449\pi$
$757$	244.126			0.322491			0.161245	+	0.986914i	$0.448449\pi$
$758$			804.318i			1.06111i
$759$	−157.316	−	37.8688i	−0.207267	−	0.0498930i
$760$	83.7951			0.110257
$761$		−	926.253i		−	1.21715i	−0.793495	−	0.608576i	$-0.791741\pi$
$761$		−	926.253i		−	1.21715i	0.793495	−	0.608576i	$-0.208259\pi$
$762$	−200.120	+	831.348i	−0.262625	+	1.09101i
$763$	−501.472			−0.657237
$764$			385.309i			0.504331i
$765$	−14.3463	−	7.33165i	−0.0187533	−	0.00958386i
$766$	−164.547			−0.214813
$767$		−	1003.79i		−	1.30872i
$768$	−46.6670	−	11.2336i	−0.0607643	−	0.0146271i
$769$	39.9471			0.0519468			0.0259734	−	0.999663i	$-0.491731\pi$
$769$	39.9471			0.0519468			0.0259734	+	0.999663i	$0.491731\pi$
$770$			303.305i			0.393903i
$771$	76.2284	−	316.671i	0.0988696	−	0.410728i
$772$	−409.112			−0.529938
$773$			697.995i			0.902969i	0.892279	+	0.451485i	$0.149105\pi$
$773$			697.995i			0.902969i	−0.892279	+	0.451485i	$0.850895\pi$
$774$	61.9534	−	121.228i	0.0800431	−	0.156625i
$775$	213.250			0.275162
$776$			395.726i			0.509956i
$777$	1066.42	+	256.707i	1.37249	+	0.330382i
$778$	−1016.65			−1.30675
$779$			785.416i			1.00824i
$780$	57.6890	−	239.654i	0.0739602	−	0.307248i
$781$	−363.643			−0.465612
$782$			5.42972i			0.00694338i
$783$	−192.874	+	224.999i	−0.246327	+	0.287355i
$784$	94.9262			0.121079
$785$		−	333.100i		−	0.424331i
$786$	−568.462	−	136.839i	−0.723234	−	0.174096i
$787$	831.189			1.05615			0.528075	−	0.849198i	$-0.322914\pi$
$787$	831.189			1.05615			0.528075	+	0.849198i	$0.322914\pi$
$788$			83.4128i			0.105854i
$789$	−288.685	+	1199.27i	−0.365887	+	1.51998i
$790$	−45.2656			−0.0572982
$791$			135.197i			0.170919i
$792$	−254.929	−	130.281i	−0.321880	−	0.164496i
$793$	628.212			0.792197
$794$			264.979i			0.333727i
$795$	445.895	+	107.335i	0.560874	+	0.135012i
$796$	15.3853			0.0193283
$797$			797.786i			1.00099i	0.865740	+	0.500493i	$0.166848\pi$
$797$			797.786i			1.00099i	−0.865740	+	0.500493i	$0.833152\pi$
$798$	112.192	−	466.074i	0.140592	−	0.584052i
$799$	−52.7323			−0.0659978
$800$			28.2843i			0.0353553i
$801$	315.934	−	618.208i	0.394425	−	0.771795i
$802$	135.617			0.169098
$803$		−	447.378i		−	0.557133i
$804$	319.936	+	77.0145i	0.397931	+	0.0957892i
$805$	91.4556			0.113609
$806$			1108.19i			1.37493i
$807$	−189.884	+	788.822i	−0.235296	+	0.977475i
$808$	523.268			0.647609
$809$			518.102i			0.640423i	0.947346	+	0.320211i	$0.103754\pi$
$809$			518.102i			0.640423i	−0.947346	+	0.320211i	$0.896246\pi$
$810$	−150.056	−	207.589i	−0.185255	−	0.256282i
$811$	465.165			0.573570			0.286785	−	0.957995i	$-0.407413\pi$
$811$	465.165			0.573570			0.286785	+	0.957995i	$0.407413\pi$
$812$		−	187.213i		−	0.230558i
$813$	405.241	+	97.5488i	0.498451	+	0.119986i
$814$	681.885			0.837697
$815$			522.953i			0.641660i
$816$	−2.24832	+	9.34004i	−0.00275529	+	0.0114461i
$817$	141.716			0.173460
$818$		−	959.209i		−	1.17263i
$819$	−1255.73	−	641.740i	−1.53325	−	0.783565i
$820$	−265.110			−0.323305
$821$			539.166i			0.656718i	0.944553	+	0.328359i	$0.106496\pi$
$821$			539.166i			0.656718i	−0.944553	+	0.328359i	$0.893504\pi$
$822$	51.6199	+	12.4258i	0.0627979	+	0.0151166i
$823$	253.501			0.308021			0.154010	−	0.988069i	$-0.450781\pi$
$823$	253.501			0.308021			0.154010	+	0.988069i	$0.450781\pi$
$824$		−	464.326i		−	0.563503i
$825$	−39.4809	+	164.013i	−0.0478556	+	0.198804i
$826$	658.931			0.797738
$827$		−	478.558i		−	0.578668i	−0.957228	−	0.289334i	$-0.906566\pi$
$827$		−	478.558i		−	0.578668i	0.957228	−	0.289334i	$-0.0934338\pi$
$828$	−39.2837	+	76.8687i	−0.0474440	+	0.0928366i
$829$	−671.600			−0.810133			−0.405067	−	0.914287i	$-0.632752\pi$
$829$	−671.600			−0.810133			−0.405067	+	0.914287i	$0.632752\pi$
$830$			133.059i			0.160312i
$831$	187.316	+	45.0904i	0.225410	+	0.0542604i
$832$	−146.984			−0.176663
$833$		−	18.9988i		−	0.0228076i
$834$	−269.212	+	1118.37i	−0.322797	+	1.34097i
$835$	−314.929			−0.377160
$836$		−	298.014i		−	0.356476i
$837$	874.290	+	749.460i	1.04455	+	0.895412i
$838$	−210.816			−0.251570
$839$		−	1490.36i		−	1.77636i	−0.459498	−	0.888179i	$-0.651970\pi$
$839$		−	1490.36i		−	1.77636i	0.459498	−	0.888179i	$-0.348030\pi$
$840$	157.319	+	37.8695i	0.187284	+	0.0450828i
$841$	720.527			0.856751
$842$			86.6471i			0.102906i
$843$	−209.368	+	869.765i	−0.248361	+	1.03175i
$844$	−696.662			−0.825429
$845$		−	376.925i		−	0.446065i
$846$	−746.531	−	381.514i	−0.882425	−	0.450962i
$847$	46.7731			0.0552221
$848$		−	273.475i		−	0.322494i
$849$	346.728	+	83.4637i	0.408396	+	0.0983082i
$850$	5.66088			0.00665986
$851$		−	205.609i		−	0.241608i
$852$	−45.4030	+	188.615i	−0.0532899	+	0.221379i
$853$	222.549			0.260901			0.130451	−	0.991455i	$-0.458358\pi$
$853$	222.549			0.260901			0.130451	+	0.991455i	$0.458358\pi$
$854$			412.386i			0.482887i
$855$	121.337	−	237.426i	0.141914	−	0.277692i
$856$	299.849			0.350291
$857$			1415.07i			1.65119i	0.564267	+	0.825593i	$0.309159\pi$
$857$			1415.07i			1.65119i	−0.564267	+	0.825593i	$0.690841\pi$
$858$	−852.320	−	205.169i	−0.993380	−	0.239125i
$859$	954.539			1.11122			0.555610	−	0.831443i	$-0.312485\pi$
$859$	954.539			1.11122			0.555610	+	0.831443i	$0.312485\pi$
$860$			47.8351i			0.0556222i
$861$	−354.953	+	1474.56i	−0.412257	+	1.71261i
$862$	−460.650			−0.534397
$863$			1414.59i			1.63915i	0.572969	+	0.819577i	$0.305792\pi$
$863$			1414.59i			1.63915i	−0.572969	+	0.819577i	$0.694208\pi$
$864$	−99.4039	+	115.961i	−0.115051	+	0.134214i
$865$	−251.730			−0.291018
$866$		−	575.113i		−	0.664103i
$867$	−841.053	−	202.457i	−0.970073	−	0.233514i
$868$	−727.464			−0.838092
$869$			160.985i			0.185254i
$870$	24.3693	−	101.236i	0.0280107	−	0.116363i
$871$	1007.68			1.15693
$872$			166.315i			0.190728i
$873$	1121.26	+	573.017i	1.28437	+	0.656377i
$874$	−89.8602			−0.102815
$875$		−	95.3491i		−	0.108970i
$876$	−232.047	−	55.8579i	−0.264894	−	0.0637647i
$877$	−1052.66			−1.20029			−0.600147	−	0.799889i	$-0.704891\pi$
$877$	−1052.66			−1.20029			−0.600147	+	0.799889i	$0.704891\pi$
$878$		−	824.786i		−	0.939392i
$879$	237.955	−	988.522i	0.270711	−	1.12460i
$880$	100.592			0.114309
$881$		−	279.941i		−	0.317754i	−0.987298	−	0.158877i	$-0.949213\pi$
$881$		−	279.941i		−	0.317754i	0.987298	−	0.158877i	$-0.0507873\pi$
$882$	137.455	−	268.966i	0.155844	−	0.304950i
$883$	545.266			0.617515			0.308758	−	0.951141i	$-0.400087\pi$
$883$	545.266			0.617515			0.308758	+	0.951141i	$0.400087\pi$
$884$			29.4177i			0.0332779i
$885$	356.319	+	85.7723i	0.402620	+	0.0969179i
$886$	288.095			0.325163
$887$		−	1247.29i		−	1.40619i	−0.711098	−	0.703093i	$-0.751802\pi$
$887$		−	1247.29i		−	1.40619i	0.711098	−	0.703093i	$-0.248198\pi$
$888$	85.1376	−	353.682i	0.0958756	−	0.398290i
$889$	−1718.86			−1.93347
$890$			243.938i			0.274087i
$891$	−738.282	+	533.671i	−0.828599	+	0.598957i
$892$	−822.463			−0.922044
$893$		−	872.702i		−	0.977270i
$894$	−342.774	−	82.5119i	−0.383416	−	0.0922952i
$895$	593.579			0.663216
$896$		−	96.4865i		−	0.107686i
$897$	−61.8645	+	257.000i	−0.0689683	+	0.286510i
$898$	−1192.19			−1.32761
$899$			468.128i			0.520720i
$900$	80.1411	+	40.9561i	0.0890457	+	0.0455067i
$901$	−54.7339			−0.0607479
$902$			942.855i			1.04529i
$903$	266.062	+	64.0460i	0.294642	+	0.0709258i
$904$	44.8384			0.0496000
$905$			405.677i			0.448262i
$906$	−168.823	+	701.330i	−0.186339	+	0.774095i
$907$	−1507.05			−1.66158			−0.830789	−	0.556588i	$-0.812110\pi$
$907$	−1507.05			−1.66158			−0.830789	+	0.556588i	$0.812110\pi$
$908$			195.411i			0.215210i
$909$	757.700	−	1482.64i	0.833554	−	1.63106i
$910$	495.497			0.544502
$911$			1565.44i			1.71838i	0.511656	+	0.859190i	$0.329032\pi$
$911$			1565.44i			1.71838i	−0.511656	+	0.859190i	$0.670968\pi$
$912$	−154.575	−	37.2089i	−0.169490	−	0.0407992i
$913$	473.220			0.518313
$914$			296.610i			0.324518i
$915$	−53.6798	+	222.998i	−0.0586664	+	0.243714i
$916$	−176.093			−0.192241
$917$		−	1175.33i		−	1.28171i
$918$	23.2086	+	19.8949i	0.0252817	+	0.0216720i
$919$	−250.103			−0.272147			−0.136073	−	0.990699i	$-0.543448\pi$
$919$	−250.103			−0.272147			−0.136073	+	0.990699i	$0.543448\pi$
$920$		−	30.3315i		−	0.0329690i
$921$	−565.945	−	136.233i	−0.614489	−	0.147919i
$922$	−186.558			−0.202341
$923$			594.067i			0.643626i
$924$	134.682	−	559.500i	0.145759	−	0.605519i
$925$	−214.362			−0.231743
$926$			684.049i			0.738713i
$927$	−1315.63	−	672.352i	−1.41923	−	0.725299i
$928$	−62.0897			−0.0669070
$929$			1690.98i			1.82021i	0.414375	+	0.910106i	$0.364000\pi$
$929$			1690.98i			1.82021i	−0.414375	+	0.910106i	$0.636000\pi$
$930$	−393.378	−	94.6931i	−0.422987	−	0.101821i
$931$	314.423			0.337726
$932$			227.721i			0.244336i
$933$	102.186	−	424.504i	0.109524	−	0.454988i
$934$	−662.802			−0.709638
$935$		−	20.1327i		−	0.0215323i
$936$	−212.835	+	416.466i	−0.227388	+	0.444943i
$937$	−554.337			−0.591608			−0.295804	−	0.955249i	$-0.595588\pi$
$937$	−554.337			−0.591608			−0.295804	+	0.955249i	$0.595588\pi$
$938$			661.486i			0.705209i
$939$	272.235	+	65.5320i	0.289921	+	0.0697891i
$940$	294.573			0.313375
$941$			1116.35i			1.18634i	0.805077	+	0.593171i	$0.202124\pi$
$941$			1116.35i			1.18634i	−0.805077	+	0.593171i	$0.797876\pi$
$942$	−147.912	+	614.461i	−0.157019	+	0.652294i
$943$	284.299			0.301484
$944$		−	218.536i		−	0.231500i
$945$	335.100	−	390.915i	0.354604	−	0.413666i
$946$	170.124			0.179835
$947$			347.859i			0.367328i	0.982989	+	0.183664i	$0.0587958\pi$
$947$			347.859i			0.367328i	−0.982989	+	0.183664i	$0.941204\pi$
$948$	83.5002	+	20.1000i	0.0880804	+	0.0212026i
$949$	−730.862			−0.770139
$950$			93.6857i			0.0986166i
$951$	198.122	−	823.046i	0.208330	−	0.865453i
$952$	−19.3110			−0.0202847
$953$			321.261i			0.337105i	0.985693	+	0.168552i	$0.0539092\pi$
$953$			321.261i			0.337105i	−0.985693	+	0.168552i	$0.946091\pi$
$954$	−774.868	−	395.996i	−0.812231	−	0.415090i
$955$	−430.788			−0.451087
$956$		−	176.884i		−	0.185026i
$957$	−360.042	−	86.6685i	−0.376219	−	0.0905627i
$958$	−936.129			−0.977170
$959$			106.727i			0.111290i
$960$	12.5595	−	52.1753i	0.0130828	−	0.0543492i
$961$	858.030			0.892851
$962$		−	1113.97i		−	1.15797i
$963$	434.186	−	849.597i	0.450868	−	0.882240i
$964$	595.769			0.618017
$965$		−	457.401i		−	0.473991i
$966$	−168.706	−	40.6105i	−0.174644	−	0.0420399i
$967$	−1469.81			−1.51997			−0.759985	−	0.649940i	$-0.774794\pi$
$967$	−1469.81			−1.51997			−0.759985	+	0.649940i	$0.774794\pi$
$968$		−	15.5124i		−	0.0160252i
$969$	−7.44708	+	30.9369i	−0.00768532	+	0.0319266i
$970$	−442.435			−0.456119
$971$		−	231.452i		−	0.238365i	−0.992872	−	0.119182i	$-0.961973\pi$
$971$		−	231.452i		−	0.238365i	0.992872	−	0.119182i	$-0.0380273\pi$
$972$	184.626	+	449.565i	0.189945	+	0.462516i
$973$	−2312.29			−2.37646
$974$			37.8799i			0.0388911i
$975$	267.941	+	64.4982i	0.274811	+	0.0661520i
$976$	136.769			0.140132
$977$			128.340i			0.131362i	0.997841	+	0.0656808i	$0.0209219\pi$
$977$			128.340i			0.131362i	−0.997841	+	0.0656808i	$0.979078\pi$
$978$	232.215	−	964.678i	0.237439	−	0.986378i
$979$	867.556			0.886165
$980$			106.131i			0.108297i
$981$	471.238	+	240.826i	0.480365	+	0.245490i
$982$	872.327			0.888316
$983$			547.128i			0.556590i	0.960496	+	0.278295i	$0.0897692\pi$
$983$			547.128i			0.556590i	−0.960496	+	0.278295i	$0.910231\pi$
$984$	489.042	+	117.721i	0.496994	+	0.119635i
$985$	−93.2584			−0.0946785
$986$			12.4268i			0.0126032i
$987$	394.401	−	1638.43i	0.399595	−	1.66001i
$988$	−486.853			−0.492766
$989$		−	51.2974i		−	0.0518680i
$990$	145.659	−	285.019i	0.147130	−	0.287898i
$991$	−792.171			−0.799365			−0.399683	−	0.916654i	$-0.630880\pi$
$991$	−792.171			−0.799365			−0.399683	+	0.916654i	$0.630880\pi$
$992$			241.265i			0.243211i
$993$	1121.08	+	269.863i	1.12898	+	0.271766i
$994$	−389.971			−0.392325
$995$			17.2013i			0.0172878i
$996$	59.0844	−	245.451i	0.0593217	−	0.246436i
$997$	−74.0339			−0.0742566			−0.0371283	−	0.999311i	$-0.511821\pi$
$997$	−74.0339			−0.0742566			−0.0371283	+	0.999311i	$0.511821\pi$
$998$		−	1393.58i		−	1.39637i
$999$	−878.847	−	753.366i	−0.879726	−	0.754121i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	690.3.g.a.461.19	✓	56
3.2	odd	2	inner	690.3.g.a.461.20	yes	56

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
690.3.g.a.461.19	✓	56	1.1	even	1	trivial
690.3.g.a.461.20	yes	56	3.2	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-1.41421i q^{2} +(-2.91669 - 0.702099i) q^{3} -2.00000 q^{4} -2.23607i q^{5} +(-0.992918 + 4.12482i) q^{6} -8.52828 q^{7} +2.82843i q^{8} +(8.01411 + 4.09561i) q^{9} +O(q^{10})\)	\(q-1.41421i q^{2} +(-2.91669 - 0.702099i) q^{3} -2.00000 q^{4} -2.23607i q^{5} +(-0.992918 + 4.12482i) q^{6} -8.52828 q^{7} +2.82843i q^{8} +(8.01411 + 4.09561i) q^{9} -3.16228 q^{10} +11.2465i q^{11} +(5.83337 + 1.40420i) q^{12} +18.3730 q^{13} +12.0608i q^{14} +(-1.56994 + 6.52191i) q^{15} +4.00000 q^{16} -0.800569i q^{17} +(5.79206 - 11.3337i) q^{18} +13.2492 q^{19} +4.47214i q^{20} +(24.8743 + 5.98770i) q^{21} +15.9050 q^{22} -4.79583i q^{23} +(1.98584 - 8.24963i) q^{24} -5.00000 q^{25} -25.9833i q^{26} +(-20.4991 - 17.5723i) q^{27} +17.0566 q^{28} -10.9760i q^{29} +(9.22337 + 2.22023i) q^{30} -42.6501 q^{31} -5.65685i q^{32} +(7.89618 - 32.8026i) q^{33} -1.13218 q^{34} +19.0698i q^{35} +(-16.0282 - 8.19121i) q^{36} +42.8724 q^{37} -18.7371i q^{38} +(-53.5882 - 12.8996i) q^{39} +6.32456 q^{40} +59.2804i q^{41} +(8.46788 - 35.1776i) q^{42} +10.6963 q^{43} -22.4931i q^{44} +(9.15805 - 17.9201i) q^{45} -6.78233 q^{46} -65.8685i q^{47} +(-11.6667 - 2.80840i) q^{48} +23.7316 q^{49} +7.07107i q^{50} +(-0.562079 + 2.33501i) q^{51} -36.7459 q^{52} -68.3687i q^{53} +(-24.8510 + 28.9901i) q^{54} +25.1480 q^{55} -24.1216i q^{56} +(-38.6436 - 9.30223i) q^{57} -15.5224 q^{58} -54.6341i q^{59} +(3.13988 - 13.0438i) q^{60} +34.1922 q^{61} +60.3163i q^{62} +(-68.3466 - 34.9285i) q^{63} -8.00000 q^{64} -41.0832i q^{65} +(-46.3899 - 11.1669i) q^{66} +54.8459 q^{67} +1.60114i q^{68} +(-3.36715 + 13.9879i) q^{69} +26.9688 q^{70} +32.3338i q^{71} +(-11.5841 + 22.6673i) q^{72} -39.7792 q^{73} -60.6307i q^{74} +(14.5834 + 3.51050i) q^{75} -26.4983 q^{76} -95.9136i q^{77} +(-18.2429 + 75.7851i) q^{78} +14.3142 q^{79} -8.94427i q^{80} +(47.4520 + 65.6453i) q^{81} +83.8352 q^{82} -42.0770i q^{83} +(-49.7486 - 11.9754i) q^{84} -1.79013 q^{85} -15.1268i q^{86} +(-7.70625 + 32.0136i) q^{87} -31.8100 q^{88} -77.1399i q^{89} +(-25.3429 - 12.9514i) q^{90} -156.690 q^{91} +9.59166i q^{92} +(124.397 + 29.9446i) q^{93} -93.1521 q^{94} -29.6260i q^{95} +(-3.97167 + 16.4993i) q^{96} +139.910 q^{97} -33.5615i q^{98} +(-46.0614 + 90.1310i) 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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