LMFDB - Embedded newform 177.5.b.a.119.54

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [177,5,Mod(119,177)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(177, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("177.119");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 177 = 3 \cdot 59 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	177.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$18.2964834658$
Analytic rank:	$0$
Dimension:	$78$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			119.54
Character	$\chi$	$=$	177.119
Dual form			177.5.b.a.119.25

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.45868i q^{2} +(-8.90633 + 1.29509i) q^{3} +4.03751 q^{4} +17.0185i q^{5} +(-4.47930 - 30.8042i) q^{6} +51.0370 q^{7} +69.3034i q^{8} +(77.6455 - 23.0690i) q^{9} +O(q^{10})$	\(q+3.45868i q^{2} +(-8.90633 + 1.29509i) q^{3} +4.03751 q^{4} +17.0185i q^{5} +(-4.47930 - 30.8042i) q^{6} +51.0370 q^{7} +69.3034i q^{8} +(77.6455 - 23.0690i) q^{9} -58.8618 q^{10} -165.031i q^{11} +(-35.9594 + 5.22894i) q^{12} +256.549 q^{13} +176.521i q^{14} +(-22.0405 - 151.573i) q^{15} -175.098 q^{16} -282.155i q^{17} +(79.7883 + 268.551i) q^{18} +436.069 q^{19} +68.7126i q^{20} +(-454.552 + 66.0974i) q^{21} +570.790 q^{22} +949.775i q^{23} +(-89.7540 - 617.239i) q^{24} +335.369 q^{25} +887.323i q^{26} +(-661.660 + 306.018i) q^{27} +206.063 q^{28} -21.4628i q^{29} +(524.242 - 76.2312i) q^{30} -839.439 q^{31} +503.245i q^{32} +(213.730 + 1469.82i) q^{33} +975.885 q^{34} +868.576i q^{35} +(313.495 - 93.1413i) q^{36} -1677.08 q^{37} +1508.22i q^{38} +(-2284.91 + 332.254i) q^{39} -1179.44 q^{40} -427.147i q^{41} +(-228.610 - 1572.15i) q^{42} +2506.01 q^{43} -666.316i q^{44} +(392.600 + 1321.41i) q^{45} -3284.97 q^{46} +734.130i q^{47} +(1559.48 - 226.768i) q^{48} +203.776 q^{49} +1159.94i q^{50} +(365.416 + 2512.97i) q^{51} +1035.82 q^{52} +777.792i q^{53} +(-1058.42 - 2288.47i) q^{54} +2808.59 q^{55} +3537.04i q^{56} +(-3883.77 + 564.747i) q^{57} +74.2329 q^{58} +453.188i q^{59} +(-88.9889 - 611.977i) q^{60} +1577.76 q^{61} -2903.35i q^{62} +(3962.79 - 1177.37i) q^{63} -4542.14 q^{64} +4366.10i q^{65} +(-5083.65 + 739.224i) q^{66} -797.230 q^{67} -1139.21i q^{68} +(-1230.04 - 8459.01i) q^{69} -3004.13 q^{70} -4907.36i q^{71} +(1598.76 + 5381.10i) q^{72} -5266.90 q^{73} -5800.50i q^{74} +(-2986.91 + 434.333i) q^{75} +1760.63 q^{76} -8422.69i q^{77} +(-1149.16 - 7902.79i) q^{78} +8295.79 q^{79} -2979.92i q^{80} +(5496.64 - 3582.40i) q^{81} +1477.37 q^{82} +13452.6i q^{83} +(-1835.26 + 266.869i) q^{84} +4801.87 q^{85} +8667.51i q^{86} +(27.7962 + 191.154i) q^{87} +11437.2 q^{88} +3659.18i q^{89} +(-4570.35 + 1357.88i) q^{90} +13093.5 q^{91} +3834.73i q^{92} +(7476.33 - 1087.15i) q^{93} -2539.12 q^{94} +7421.25i q^{95} +(-651.747 - 4482.07i) q^{96} +5030.26 q^{97} +704.796i q^{98} +(-3807.10 - 12813.9i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 78 q - 612 q^{4} + 64 q^{6} + 76 q^{7} - 100 q^{9}+O(q^{10}) $ 78 * q - 612 * q^4 + 64 * q^6 + 76 * q^7 - 100 * q^9	$ 78 q - 612 q^{4} + 64 q^{6} + 76 q^{7} - 100 q^{9} - 156 q^{10} - 4 q^{13} + 13 q^{15} + 4948 q^{16} + 22 q^{18} + 812 q^{19} - 173 q^{21} - 1644 q^{22} - 678 q^{24} - 8238 q^{25} + 777 q^{27} - 3764 q^{28} + 1374 q^{30} + 4664 q^{31} - 1042 q^{33} + 3244 q^{34} + 3648 q^{36} - 3960 q^{37} - 7078 q^{39} - 1576 q^{40} + 4934 q^{42} - 1492 q^{43} - 2063 q^{45} - 2036 q^{46} - 2620 q^{48} + 24274 q^{49} + 7300 q^{51} + 8408 q^{52} - 14766 q^{54} + 9780 q^{55} + 6939 q^{57} - 3856 q^{58} + 4712 q^{60} - 212 q^{61} - 7438 q^{63} - 45760 q^{64} + 3048 q^{66} - 12972 q^{67} + 21672 q^{69} + 5828 q^{70} - 866 q^{72} - 5240 q^{73} - 20922 q^{75} + 12368 q^{76} - 16508 q^{78} - 14976 q^{79} + 25524 q^{81} - 14484 q^{82} + 9540 q^{84} + 11572 q^{85} + 5695 q^{87} + 62160 q^{88} - 31672 q^{90} + 8284 q^{91} - 9590 q^{93} - 10992 q^{94} + 34102 q^{96} - 55000 q^{97} - 14254 q^{99}+O(q^{100}) $ 78 * q - 612 * q^4 + 64 * q^6 + 76 * q^7 - 100 * q^9 - 156 * q^10 - 4 * q^13 + 13 * q^15 + 4948 * q^16 + 22 * q^18 + 812 * q^19 - 173 * q^21 - 1644 * q^22 - 678 * q^24 - 8238 * q^25 + 777 * q^27 - 3764 * q^28 + 1374 * q^30 + 4664 * q^31 - 1042 * q^33 + 3244 * q^34 + 3648 * q^36 - 3960 * q^37 - 7078 * q^39 - 1576 * q^40 + 4934 * q^42 - 1492 * q^43 - 2063 * q^45 - 2036 * q^46 - 2620 * q^48 + 24274 * q^49 + 7300 * q^51 + 8408 * q^52 - 14766 * q^54 + 9780 * q^55 + 6939 * q^57 - 3856 * q^58 + 4712 * q^60 - 212 * q^61 - 7438 * q^63 - 45760 * q^64 + 3048 * q^66 - 12972 * q^67 + 21672 * q^69 + 5828 * q^70 - 866 * q^72 - 5240 * q^73 - 20922 * q^75 + 12368 * q^76 - 16508 * q^78 - 14976 * q^79 + 25524 * q^81 - 14484 * q^82 + 9540 * q^84 + 11572 * q^85 + 5695 * q^87 + 62160 * q^88 - 31672 * q^90 + 8284 * q^91 - 9590 * q^93 - 10992 * q^94 + 34102 * q^96 - 55000 * q^97 - 14254 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/177\mathbb{Z}\right)^\times$.

$n$	$61$	$119$
$\chi(n)$	$1$	$-1$

Coefficient data

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

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

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

$2$			3.45868i			0.864671i	0.901713	+	0.432335i	$0.142310\pi$
$2$			3.45868i			0.864671i	−0.901713	+	0.432335i	$0.857690\pi$
$3$	−8.90633	+	1.29509i	−0.989592	+	0.143899i
$3$	−8.90633	+	1.29509i	−0.989592	+	0.143899i
$4$	4.03751			0.252345
$5$			17.0185i			0.680742i	0.940291	+	0.340371i	$0.110553\pi$
$5$			17.0185i			0.680742i	−0.940291	+	0.340371i	$0.889447\pi$
$6$	−4.47930	−	30.8042i	−0.124425	−	0.855672i
$7$	51.0370			1.04157			0.520786	−	0.853687i	$-0.325639\pi$
$7$	51.0370			1.04157			0.520786	+	0.853687i	$0.325639\pi$
$8$			69.3034i			1.08287i
$9$	77.6455	−	23.0690i	0.958586	−	0.284802i
$10$	−58.8618			−0.588618
$11$		−	165.031i		−	1.36389i	−0.731402	−	0.681947i	$-0.761134\pi$
$11$		−	165.031i		−	1.36389i	0.731402	−	0.681947i	$-0.238866\pi$
$12$	−35.9594	+	5.22894i	−0.249718	+	0.0363121i
$13$	256.549			1.51804			0.759022	−	0.651065i	$-0.225678\pi$
$13$	256.549			1.51804			0.759022	+	0.651065i	$0.225678\pi$
$14$			176.521i			0.900616i
$15$	−22.0405	−	151.573i	−0.0979579	−	0.673657i
$16$	−175.098			−0.683978
$17$		−	282.155i		−	0.976315i	−0.872756	−	0.488157i	$-0.837669\pi$
$17$		−	282.155i		−	0.976315i	0.872756	−	0.488157i	$-0.162331\pi$
$18$	79.7883	+	268.551i	0.246260	+	0.828861i
$19$	436.069			1.20795			0.603973	−	0.797005i	$-0.293583\pi$
$19$	436.069			1.20795			0.603973	+	0.797005i	$0.293583\pi$
$20$			68.7126i			0.171782i
$21$	−454.552	+	66.0974i	−1.03073	+	0.149881i
$22$	570.790			1.17932
$23$			949.775i			1.79542i	0.440592	+	0.897708i	$0.354769\pi$
$23$			949.775i			1.79542i	−0.440592	+	0.897708i	$0.645231\pi$
$24$	−89.7540	−	617.239i	−0.155823	−	1.07160i
$25$	335.369			0.536591
$26$			887.323i			1.31261i
$27$	−661.660	+	306.018i	−0.907627	+	0.419777i
$28$	206.063			0.262835
$29$		−	21.4628i		−	0.0255205i	−0.999919	−	0.0127603i	$-0.995938\pi$
$29$		−	21.4628i		−	0.0255205i	0.999919	−	0.0127603i	$-0.00406183\pi$
$30$	524.242	−	76.2312i	0.582491	−	0.0847013i
$31$	−839.439			−0.873506			−0.436753	−	0.899581i	$-0.643872\pi$
$31$	−839.439			−0.873506			−0.436753	+	0.899581i	$0.643872\pi$
$32$			503.245i			0.491450i
$33$	213.730	+	1469.82i	0.196263	+	1.34970i
$34$	975.885			0.844191
$35$			868.576i			0.709041i
$36$	313.495	−	93.1413i	0.241894	−	0.0718683i
$37$	−1677.08			−1.22504			−0.612521	−	0.790454i	$-0.709845\pi$
$37$	−1677.08			−1.22504			−0.612521	+	0.790454i	$0.709845\pi$
$38$			1508.22i			1.04448i
$39$	−2284.91	+	332.254i	−1.50224	+	0.218445i
$40$	−1179.44			−0.737152
$41$		−	427.147i		−	0.254103i	−0.991896	−	0.127052i	$-0.959449\pi$
$41$		−	427.147i		−	0.254103i	0.991896	−	0.127052i	$-0.0405514\pi$
$42$	−228.610	−	1572.15i	−0.129598	−	0.891243i
$43$	2506.01			1.35533			0.677667	−	0.735369i	$-0.262991\pi$
$43$	2506.01			1.35533			0.677667	+	0.735369i	$0.262991\pi$
$44$		−	666.316i		−	0.344171i
$45$	392.600	+	1321.41i	0.193877	+	0.652550i
$46$	−3284.97			−1.55244
$47$			734.130i			0.332336i	0.986097	+	0.166168i	$0.0531394\pi$
$47$			734.130i			0.332336i	−0.986097	+	0.166168i	$0.946861\pi$
$48$	1559.48	−	226.768i	0.676859	−	0.0984235i
$49$	203.776			0.0848712
$50$			1159.94i			0.463974i
$51$	365.416	+	2512.97i	0.140490	+	0.966154i
$52$	1035.82			0.383070
$53$			777.792i			0.276893i	0.990370	+	0.138446i	$0.0442109\pi$
$53$			777.792i			0.276893i	−0.990370	+	0.138446i	$0.955789\pi$
$54$	−1058.42	−	2288.47i	−0.362969	−	0.784799i
$55$	2808.59			0.928460
$56$			3537.04i			1.12788i
$57$	−3883.77	+	564.747i	−1.19537	+	0.173822i
$58$	74.2329			0.0220668
$59$			453.188i			0.130189i
$59$			453.188i			0.130189i
$60$	−88.9889	−	611.977i	−0.0247191	−	0.169994i
$61$	1577.76			0.424014			0.212007	−	0.977268i	$-0.432000\pi$
$61$	1577.76			0.424014			0.212007	+	0.977268i	$0.432000\pi$
$62$		−	2903.35i		−	0.755295i
$63$	3962.79	−	1177.37i	0.998436	−	0.296642i
$64$	−4542.14			−1.10892
$65$			4366.10i			1.03340i
$66$	−5083.65	+	739.224i	−1.16705	+	0.169702i
$67$	−797.230			−0.177596			−0.0887982	−	0.996050i	$-0.528303\pi$
$67$	−797.230			−0.177596			−0.0887982	+	0.996050i	$0.528303\pi$
$68$		−	1139.21i		−	0.246368i
$69$	−1230.04	−	8459.01i	−0.258358	−	1.77673i
$70$	−3004.13			−0.613087
$71$		−	4907.36i		−	0.973489i	−0.873545	−	0.486744i	$-0.838184\pi$
$71$		−	4907.36i		−	0.973489i	0.873545	−	0.486744i	$-0.161816\pi$
$72$	1598.76	+	5381.10i	0.308402	+	1.03802i
$73$	−5266.90			−0.988347			−0.494174	−	0.869363i	$-0.664529\pi$
$73$	−5266.90			−0.988347			−0.494174	+	0.869363i	$0.664529\pi$
$74$		−	5800.50i		−	1.05926i
$75$	−2986.91	+	434.333i	−0.531006	+	0.0772147i
$76$	1760.63			0.304819
$77$		−	8422.69i		−	1.42059i
$78$	−1149.16	−	7902.79i	−0.188883	−	1.29895i
$79$	8295.79			1.32924			0.664620	−	0.747181i	$-0.268593\pi$
$79$	8295.79			1.32924			0.664620	+	0.747181i	$0.268593\pi$
$80$		−	2979.92i		−	0.465612i
$81$	5496.64	−	3582.40i	0.837775	−	0.546015i
$82$	1477.37			0.219715
$83$			13452.6i			1.95277i	0.216035	+	0.976386i	$0.430688\pi$
$83$			13452.6i			1.95277i	−0.216035	+	0.976386i	$0.569312\pi$
$84$	−1835.26	+	266.869i	−0.260100	+	0.0378216i
$85$	4801.87			0.664618
$86$			8667.51i			1.17192i
$87$	27.7962	+	191.154i	0.00367237	+	0.0252549i
$88$	11437.2			1.47691
$89$			3659.18i			0.461959i	0.972959	+	0.230979i	$0.0741930\pi$
$89$			3659.18i			0.461959i	−0.972959	+	0.230979i	$0.925807\pi$
$90$	−4570.35	+	1357.88i	−0.564241	+	0.167640i
$91$	13093.5			1.58115
$92$			3834.73i			0.453063i
$93$	7476.33	−	1087.15i	0.864415	−	0.125696i
$94$	−2539.12			−0.287361
$95$			7421.25i			0.822299i
$96$	−651.747	−	4482.07i	−0.0707191	−	0.486336i
$97$	5030.26			0.534622			0.267311	−	0.963610i	$-0.413865\pi$
$97$	5030.26			0.534622			0.267311	+	0.963610i	$0.413865\pi$
$98$			704.796i			0.0733857i
$99$	−3807.10	−	12813.9i	−0.388440	−	1.30741i
$100$	1354.06			0.135406
$101$		−	10986.7i		−	1.07702i	−0.842618	−	0.538512i	$-0.818987\pi$
$101$		−	10986.7i		−	1.07702i	0.842618	−	0.538512i	$-0.181013\pi$
$102$	−8691.55	+	1263.86i	−0.835405	+	0.121478i
$103$	9533.97			0.898668			0.449334	−	0.893364i	$-0.351661\pi$
$103$	9533.97			0.898668			0.449334	+	0.893364i	$0.351661\pi$
$104$			17779.7i			1.64384i
$105$	−1124.88	−	7735.82i	−0.102030	−	0.701662i
$106$	−2690.14			−0.239421
$107$		−	12106.3i		−	1.05741i	−0.848805	−	0.528706i	$-0.822677\pi$
$107$		−	12106.3i		−	1.05741i	0.848805	−	0.528706i	$-0.177323\pi$
$108$	−2671.46	+	1235.55i	−0.229035	+	0.105929i
$109$	−12832.9			−1.08012			−0.540061	−	0.841626i	$-0.681599\pi$
$109$	−12832.9			−1.08012			−0.540061	+	0.841626i	$0.681599\pi$
$110$			9714.02i			0.802812i
$111$	14936.7	−	2171.97i	1.21229	−	0.176282i
$112$	−8936.49			−0.712412
$113$		−	6092.41i		−	0.477125i	−0.971127	−	0.238562i	$-0.923324\pi$
$113$		−	6092.41i		−	0.477125i	0.971127	−	0.238562i	$-0.0766762\pi$
$114$	−1953.28	−	13432.7i	−0.150299	−	1.03361i
$115$	−16163.8			−1.22221
$116$		−	86.6562i		−	0.00643997i
$117$	19919.9	−	5918.33i	1.45518	−	0.432342i
$118$	−1567.43			−0.112571
$119$		−	14400.3i		−	1.01690i
$120$	10504.5	−	1527.48i	0.729480	−	0.106075i
$121$	−12594.3			−0.860206
$122$			5456.96i			0.366633i
$123$	553.193	+	3804.31i	0.0365651	+	0.251458i
$124$	−3389.25			−0.220425
$125$			16344.1i			1.04602i
$126$	4072.15	+	13706.0i	0.256497	+	0.863318i
$127$	−13837.4			−0.857918			−0.428959	−	0.903324i	$-0.641119\pi$
$127$	−13837.4			−0.857918			−0.428959	+	0.903324i	$0.641119\pi$
$128$		−	7657.89i		−	0.467400i
$129$	−22319.4	+	3245.51i	−1.34123	+	0.195031i
$130$	−15100.9			−0.893547
$131$			1750.33i			0.101995i	0.998699	+	0.0509973i	$0.0162400\pi$
$131$			1750.33i			0.101995i	−0.998699	+	0.0509973i	$0.983760\pi$
$132$	862.938	+	5934.43i	0.0495258	+	0.340589i
$133$	22255.6			1.25816
$134$		−	2757.37i		−	0.153562i
$135$	−5207.98	−	11260.5i	−0.285760	−	0.617860i
$136$	19554.3			1.05722
$137$		−	33347.5i		−	1.77674i	−0.459133	−	0.888368i	$-0.651840\pi$
$137$		−	33347.5i		−	1.77674i	0.459133	−	0.888368i	$-0.348160\pi$
$138$	29257.0	−	4254.33i	1.53629	−	0.223395i
$139$	−23791.4			−1.23138			−0.615688	−	0.787990i	$-0.711122\pi$
$139$	−23791.4			−1.23138			−0.615688	+	0.787990i	$0.711122\pi$
$140$			3506.89i			0.178923i
$141$	−950.763	−	6538.40i	−0.0478227	−	0.328877i
$142$	16973.0			0.841747
$143$		−	42338.6i		−	2.07045i
$144$	−13595.6	+	4039.34i	−0.655651	+	0.194798i
$145$	365.265			0.0173729
$146$		−	18216.5i		−	0.854595i
$147$	−1814.90	+	263.908i	−0.0839879	+	0.0122129i
$148$	−6771.25			−0.309133
$149$			16410.4i			0.739176i	0.929196	+	0.369588i	$0.120501\pi$
$149$			16410.4i			0.739176i	−0.929196	+	0.369588i	$0.879499\pi$
$150$	−1502.22	−	10330.8i	−0.0667653	−	0.459145i
$151$	−33809.2			−1.48279			−0.741397	−	0.671066i	$-0.765837\pi$
$151$	−33809.2			−1.48279			−0.741397	+	0.671066i	$0.765837\pi$
$152$			30221.0i			1.30804i
$153$	−6509.03	−	21908.1i	−0.278057	−	0.935882i
$154$	29131.4			1.22834
$155$		−	14286.0i		−	0.594632i
$156$	−9225.37	+	1341.48i	−0.379083	+	0.0551233i
$157$	35896.8			1.45632			0.728160	−	0.685407i	$-0.240376\pi$
$157$	35896.8			1.45632			0.728160	+	0.685407i	$0.240376\pi$
$158$			28692.5i			1.14936i
$159$	−1007.31	−	6927.27i	−0.0398445	−	0.274011i
$160$	−8564.50			−0.334551
$161$			48473.7i			1.87005i
$162$	12390.4	+	19011.1i	0.472123	+	0.724400i
$163$	15259.5			0.574335			0.287168	−	0.957880i	$-0.407286\pi$
$163$	15259.5			0.574335			0.287168	+	0.957880i	$0.407286\pi$
$164$		−	1724.61i		−	0.0641215i
$165$	−25014.2	+	3637.37i	−0.918797	+	0.133604i
$166$	−46528.4			−1.68850
$167$			27414.2i			0.982976i	0.870884	+	0.491488i	$0.163547\pi$
$167$			27414.2i			0.982976i	−0.870884	+	0.491488i	$0.836453\pi$
$168$	−4580.78	−	31502.0i	−0.162301	−	1.11614i
$169$	37256.6			1.30446
$170$			16608.1i			0.574676i
$171$	33858.8	−	10059.7i	1.15792	−	0.344026i
$172$	10118.1			0.342011
$173$			3533.86i			0.118075i	0.998256	+	0.0590374i	$0.0188031\pi$
$173$			3533.86i			0.118075i	−0.998256	+	0.0590374i	$0.981197\pi$
$174$	−661.142	+	96.1381i	−0.0218372	+	0.00317539i
$175$	17116.2			0.558897
$176$			28896.7i			0.932873i
$177$	−586.918	−	4036.24i	−0.0187340	−	0.128834i
$178$	−12655.9			−0.399442
$179$			20091.4i			0.627053i	0.949579	+	0.313526i	$0.101510\pi$
$179$			20091.4i			0.627053i	−0.949579	+	0.313526i	$0.898490\pi$
$180$	1585.13	+	5335.23i	0.0489238	+	0.164667i
$181$	−47818.4			−1.45961			−0.729806	−	0.683654i	$-0.760390\pi$
$181$	−47818.4			−1.45961			−0.729806	+	0.683654i	$0.760390\pi$
$182$			45286.3i			1.36717i
$183$	−14052.0	+	2043.34i	−0.419601	+	0.0610151i
$184$	−65822.6			−1.94419
$185$		−	28541.5i		−	0.833938i
$186$	3760.10	+	25858.2i	0.108686	+	0.747434i
$187$	−46564.4			−1.33159
$188$			2964.06i			0.0838632i
$189$	−33769.2	+	15618.2i	−0.945359	+	0.437228i
$190$	−25667.8			−0.711018
$191$			61518.6i			1.68632i	0.537664	+	0.843159i	$0.319307\pi$
$191$			61518.6i			1.68632i	−0.537664	+	0.843159i	$0.680693\pi$
$192$	40453.8	−	5882.47i	1.09738	−	0.159572i
$193$	−5049.33			−0.135556			−0.0677781	−	0.997700i	$-0.521591\pi$
$193$	−5049.33			−0.135556			−0.0677781	+	0.997700i	$0.521591\pi$
$194$			17398.1i			0.462272i
$195$	−5654.48	−	38885.9i	−0.148704	−	1.02264i
$196$	822.748			0.0214168
$197$		−	7707.45i		−	0.198600i	−0.995058	−	0.0992998i	$-0.968340\pi$
$197$		−	7707.45i		−	0.198600i	0.995058	−	0.0992998i	$-0.0316603\pi$
$198$	44319.3	−	13167.5i	1.13048	−	0.335873i
$199$	47756.6			1.20594			0.602972	−	0.797762i	$-0.293983\pi$
$199$	47756.6			1.20594			0.602972	+	0.797762i	$0.293983\pi$
$200$			23242.2i			0.581055i
$201$	7100.40	−	1032.48i	0.175748	−	0.0255559i
$202$	37999.6			0.931271
$203$		−	1095.39i		−	0.0265814i
$204$	1475.37	+	10146.1i	0.0354520	+	0.243804i
$205$	7269.42			0.172979
$206$			32975.0i			0.777052i
$207$	21910.3	+	73745.7i	0.511338	+	1.72106i
$208$	−44921.3			−1.03831
$209$		−	71964.9i		−	1.64751i
$210$	26755.8	−	3890.61i	0.606706	−	0.0882225i
$211$	−38950.6			−0.874882			−0.437441	−	0.899247i	$-0.644115\pi$
$211$	−38950.6			−0.874882			−0.437441	+	0.899247i	$0.644115\pi$
$212$			3140.35i			0.0698724i
$213$	6355.46	+	43706.5i	0.140084	+	0.963357i
$214$	41871.9			0.914313
$215$			42648.7i			0.922633i
$216$	−21208.1	−	45855.3i	−0.454563	−	0.982838i
$217$	−42842.5			−0.909819
$218$		−	44385.0i		−	0.933949i
$219$	46908.8	−	6821.10i	0.978061	−	0.142222i
$220$	11339.7			0.234292
$221$		−	72386.7i		−	1.48209i
$222$	7512.16	+	51661.2i	0.152426	+	1.04823i
$223$	8965.05			0.180278			0.0901391	−	0.995929i	$-0.471269\pi$
$223$	8965.05			0.180278			0.0901391	+	0.995929i	$0.471269\pi$
$224$			25684.1i			0.511881i
$225$	26039.9	−	7736.62i	0.514368	−	0.152822i
$226$	21071.7			0.412556
$227$			12349.3i			0.239657i	0.992795	+	0.119828i	$0.0382345\pi$
$227$			12349.3i			0.239657i	−0.992795	+	0.119828i	$0.961766\pi$
$228$	−15680.8	+	2280.18i	−0.301646	+	0.0438630i
$229$	75002.0			1.43022			0.715109	−	0.699013i	$-0.246377\pi$
$229$	75002.0			1.43022			0.715109	+	0.699013i	$0.246377\pi$
$230$		−	55905.4i		−	1.05681i
$231$	10908.1	+	75015.3i	0.204421	+	1.40581i
$232$	1487.44			0.0276353
$233$		−	73116.3i		−	1.34680i	−0.739279	−	0.673399i	$-0.764834\pi$
$233$		−	73116.3i		−	1.34680i	0.739279	−	0.673399i	$-0.235166\pi$
$234$	20469.6	+	68896.6i	0.373834	+	1.25825i
$235$	−12493.8			−0.226235
$236$			1829.75i			0.0328525i
$237$	−73885.0	+	10743.8i	−1.31541	+	0.191276i
$238$	49806.2			0.879285
$239$			13917.8i			0.243655i	0.992551	+	0.121828i	$0.0388755\pi$
$239$			13917.8i			0.243655i	−0.992551	+	0.121828i	$0.961125\pi$
$240$	3859.26	+	26540.1i	0.0670010	+	0.460766i
$241$	19942.9			0.343364			0.171682	−	0.985152i	$-0.445080\pi$
$241$	19942.9			0.343364			0.171682	+	0.985152i	$0.445080\pi$
$242$		−	43559.6i		−	0.743795i
$243$	−44315.4	+	39024.7i	−0.750485	+	0.660887i
$244$	6370.22			0.106998
$245$			3467.97i			0.0577754i
$246$	−13157.9	+	1913.32i	−0.217429	+	0.0316168i
$247$	111873.			1.83371
$248$		−	58176.0i		−	0.945890i
$249$	−17422.4	−	119814.i	−0.281001	−	1.93245i
$250$	−56529.0			−0.904464
$251$			115941.i			1.84031i	0.391553	+	0.920156i	$0.371938\pi$
$251$			115941.i			1.84031i	−0.391553	+	0.920156i	$0.628062\pi$
$252$	15999.8	−	4753.65i	0.251950	−	0.0748560i
$253$	156742.			2.44876
$254$		−	47859.0i		−	0.741816i
$255$	−42767.0	+	6218.84i	−0.657701	+	0.0956378i
$256$	−46188.0			−0.704773
$257$		−	17361.0i		−	0.262850i	−0.991326	−	0.131425i	$-0.958045\pi$
$257$		−	17361.0i		−	0.262850i	0.991326	−	0.131425i	$-0.0419553\pi$
$258$	−11225.2	−	77195.7i	−0.168638	−	1.15972i
$259$	−85593.3			−1.27597
$260$			17628.2i			0.260772i
$261$	−495.124	−	1666.49i	−0.00726830	−	0.0244636i
$262$	−6053.84			−0.0881918
$263$		−	28844.9i		−	0.417021i	−0.978020	−	0.208510i	$-0.933139\pi$
$263$		−	28844.9i		−	0.417021i	0.978020	−	0.208510i	$-0.0668615\pi$
$264$	−101864.	+	14812.2i	−1.46154	+	0.212526i
$265$	−13236.9			−0.188493
$266$			76975.2i			1.08790i
$267$	−4738.96	−	32589.8i	−0.0664753	−	0.457151i
$268$	−3218.83			−0.0448155
$269$		−	143251.i		−	1.97967i	−0.142217	−	0.989836i	$-0.545423\pi$
$269$		−	143251.i		−	1.97967i	0.142217	−	0.989836i	$-0.454577\pi$
$270$	38946.5	−	18012.7i	0.534245	−	0.247088i
$271$	−20873.9			−0.284227			−0.142113	−	0.989850i	$-0.545390\pi$
$271$	−20873.9			−0.284227			−0.142113	+	0.989850i	$0.545390\pi$
$272$			49404.9i			0.667778i
$273$	−116615.	+	16957.3i	−1.56470	+	0.227526i
$274$	115339.			1.53629
$275$		−	55346.3i		−	0.731852i
$276$	−4966.31	−	34153.4i	−0.0651953	−	0.448348i
$277$	−103634.			−1.35064			−0.675322	−	0.737523i	$-0.735996\pi$
$277$	−103634.			−1.35064			−0.675322	+	0.737523i	$0.735996\pi$
$278$		−	82287.0i		−	1.06473i
$279$	−65178.7	+	19365.0i	−0.837331	+	0.248776i
$280$	−60195.2			−0.767797
$281$		−	121070.i		−	1.53329i	−0.642071	−	0.766645i	$-0.721924\pi$
$281$		−	121070.i		−	1.53329i	0.642071	−	0.766645i	$-0.278076\pi$
$282$	22614.3	−	3288.39i	0.284370	−	0.0413509i
$283$	50369.4			0.628917			0.314459	−	0.949271i	$-0.398177\pi$
$283$	50369.4			0.628917			0.314459	+	0.949271i	$0.398177\pi$
$284$		−	19813.5i		−	0.245655i
$285$	−9611.18	−	66096.1i	−0.118328	−	0.813741i
$286$	146436.			1.79026
$287$		−	21800.3i		−	0.264666i
$288$	11609.4	+	39074.7i	0.139966	+	0.471098i
$289$	3909.54			0.0468090
$290$			1263.34i			0.0150218i
$291$	−44801.2	+	6514.63i	−0.529058	+	0.0769314i
$292$	−21265.2			−0.249404
$293$			5356.78i			0.0623977i	0.999513	+	0.0311988i	$0.00993251\pi$
$293$			5356.78i			0.0623977i	−0.999513	+	0.0311988i	$0.990067\pi$
$294$	−912.773	−	6277.15i	−0.0105601	−	0.0726219i
$295$	−7712.59			−0.0886250
$296$		−	116228.i		−	1.32656i
$297$	50502.5	+	109195.i	0.572532	+	1.23791i
$298$	−56758.5			−0.639144
$299$			243664.i			2.72552i
$300$	−12059.7	+	1753.62i	−0.133997	+	0.0194847i
$301$	127899.			1.41168
$302$		−	116935.i		−	1.28213i
$303$	14228.8	+	97851.4i	0.154982	+	1.06581i
$304$	−76354.8			−0.826208
$305$			26851.1i			0.288644i
$306$	75773.0	−	22512.7i	0.809230	−	0.240427i
$307$	−127052.			−1.34805			−0.674024	−	0.738709i	$-0.735436\pi$
$307$	−127052.			−1.34805			−0.674024	+	0.738709i	$0.735436\pi$
$308$		−	34006.8i		−	0.358479i
$309$	−84912.7	+	12347.3i	−0.889315	+	0.129317i
$310$	49410.9			0.514161
$311$		−	127672.i		−	1.32000i	−0.751266	−	0.659999i	$-0.770557\pi$
$311$		−	127672.i		−	1.32000i	0.751266	−	0.659999i	$-0.229443\pi$
$312$	−23026.3	−	158352.i	−0.236546	−	1.62673i
$313$	−95812.3			−0.977986			−0.488993	−	0.872288i	$-0.662636\pi$
$313$	−95812.3			−0.977986			−0.488993	+	0.872288i	$0.662636\pi$
$314$			124156.i			1.25924i
$315$	20037.2	+	67441.0i	0.201937	+	0.679677i
$316$	33494.4			0.335427
$317$		−	56340.0i		−	0.560658i	−0.959904	−	0.280329i	$-0.909556\pi$
$317$		−	56340.0i		−	0.560658i	0.959904	−	0.280329i	$-0.0904436\pi$
$318$	23959.2	−	3483.96i	0.236929	−	0.0344524i
$319$	−3542.02			−0.0348073
$320$		−	77300.6i		−	0.754888i
$321$	15678.7	+	107823.i	0.152160	+	1.04641i
$322$	−167655.			−1.61698
$323$		−	123039.i		−	1.17934i
$324$	22192.8	−	14464.0i	0.211408	−	0.137784i
$325$	86038.7			0.814568
$326$			52777.8i			0.496611i
$327$	114294.	−	16619.8i	1.06888	−	0.155428i
$328$	29602.8			0.275159
$329$			37467.8i			0.346151i
$330$	−12580.5	−	86516.3i	−0.115524	−	0.794456i
$331$	−64386.7			−0.587679			−0.293839	−	0.955855i	$-0.594933\pi$
$331$	−64386.7			−0.587679			−0.293839	+	0.955855i	$0.594933\pi$
$332$			54315.2i			0.492771i
$333$	−130218.	+	38688.6i	−1.17431	+	0.348895i
$334$	−94817.1			−0.849950
$335$		−	13567.7i		−	0.120897i
$336$	79591.3	−	11573.5i	0.704997	−	0.102515i
$337$	208234.			1.83355			0.916774	−	0.399406i	$-0.130784\pi$
$337$	208234.			1.83355			0.916774	+	0.399406i	$0.130784\pi$
$338$			128859.i			1.12793i
$339$	7890.21	+	54261.0i	0.0686577	+	0.472159i
$340$	19387.6			0.167713
$341$			138534.i			1.19137i
$342$	34793.2	+	117107.i	0.297469	+	1.00122i
$343$	−112140.			−0.953172
$344$			173675.i			1.46765i
$345$	143960.	−	20933.5i	1.20949	−	0.175875i
$346$	−12222.5			−0.102096
$347$		−	207826.i		−	1.72600i	−0.505202	−	0.863001i	$-0.668582\pi$
$347$		−	207826.i		−	1.72600i	0.505202	−	0.863001i	$-0.331418\pi$
$348$	112.227	+	771.789i	0.000926703	+	0.00637294i
$349$	44805.3			0.367857			0.183928	−	0.982940i	$-0.441119\pi$
$349$	44805.3			0.367857			0.183928	+	0.982940i	$0.441119\pi$
$350$			59199.6i			0.483262i
$351$	−169748.	+	78508.7i	−1.37782	+	0.637240i
$352$	83051.1			0.670286
$353$		−	169439.i		−	1.35977i	−0.733320	−	0.679884i	$-0.762030\pi$
$353$		−	169439.i		−	1.35977i	0.733320	−	0.679884i	$-0.237970\pi$
$354$	13960.1	−	2029.96i	0.111399	−	0.0161988i
$355$	83516.1			0.662694
$356$			14774.0i			0.116573i
$357$	18649.7	+	128254.i	0.146331	+	1.00632i
$358$	−69489.8			−0.542194
$359$		−	72104.4i		−	0.559465i	−0.960078	−	0.279733i	$-0.909754\pi$
$359$		−	72104.4i		−	0.559465i	0.960078	−	0.279733i	$-0.0902458\pi$
$360$	−91578.4	+	27208.5i	−0.706624	+	0.209942i
$361$	59834.8			0.459134
$362$		−	165389.i		−	1.26208i
$363$	112169.	−	16310.7i	0.851253	−	0.123783i
$364$	52865.2			0.398995
$365$		−	89635.0i		−	0.672809i
$366$	−7067.25	−	48601.5i	−0.0527580	−	0.362817i
$367$	111311.			0.826429			0.413215	−	0.910634i	$-0.364406\pi$
$367$	111311.			0.826429			0.413215	+	0.910634i	$0.364406\pi$
$368$		−	166304.i		−	1.22802i
$369$	−9853.85	−	33166.1i	−0.0723691	−	0.243580i
$370$	98716.1			0.721082
$371$			39696.2i			0.288404i
$372$	30185.8	−	4389.38i	0.218131	−	0.0317188i
$373$	−177766.			−1.27771			−0.638853	−	0.769329i	$-0.720591\pi$
$373$	−177766.			−1.27771			−0.638853	+	0.769329i	$0.720591\pi$
$374$		−	161051.i		−	1.15139i
$375$	−21167.0	−	145566.i	−0.150521	−	1.03513i
$376$	−50877.7			−0.359875
$377$		−	5506.26i		−	0.0387413i
$378$	−54018.5	−	116797.i	−0.378058	−	0.817424i
$379$	60539.5			0.421464			0.210732	−	0.977544i	$-0.432415\pi$
$379$	60539.5			0.421464			0.210732	+	0.977544i	$0.432415\pi$
$380$			29963.4i			0.207503i
$381$	123240.	−	17920.6i	0.848989	−	0.123453i
$382$	−212773.			−1.45811
$383$		−	24697.0i		−	0.168363i	−0.996450	−	0.0841816i	$-0.973172\pi$
$383$		−	24697.0i		−	0.168363i	0.996450	−	0.0841816i	$-0.0268276\pi$
$384$	9917.64	+	68203.7i	0.0672583	+	0.462536i
$385$	143342.			0.967057
$386$		−	17464.0i		−	0.117211i
$387$	194581.	−	57811.2i	1.29921	−	0.386002i
$388$	20309.7			0.134909
$389$			219906.i			1.45324i	0.687038	+	0.726621i	$0.258911\pi$
$389$			219906.i			1.45324i	−0.687038	+	0.726621i	$0.741089\pi$
$390$	134494.	−	19557.1i	0.884247	−	0.128580i
$391$	267984.			1.75289
$392$			14122.4i			0.0919041i
$393$	−2266.83	−	15589.0i	−0.0146769	−	0.100933i
$394$	26657.6			0.171723
$395$			141182.i			0.904869i
$396$	−15371.2	−	51736.4i	−0.0980207	−	0.329918i
$397$	−8019.77			−0.0508840			−0.0254420	−	0.999676i	$-0.508099\pi$
$397$	−8019.77			−0.0508840			−0.0254420	+	0.999676i	$0.508099\pi$
$398$			165175.i			1.04274i
$399$	−198216.	+	28823.0i	−1.24507	+	0.181048i
$400$	−58722.5			−0.367016
$401$		−	190817.i		−	1.18667i	−0.804957	−	0.593334i	$-0.797811\pi$
$401$		−	190817.i		−	1.18667i	0.804957	−	0.593334i	$-0.202189\pi$
$402$	3571.03	+	24558.0i	0.0220974	+	0.151964i
$403$	−215358.			−1.32602
$404$		−	44359.0i		−	0.271781i
$405$	60967.3	+	93544.9i	0.371695	+	0.570309i
$406$	3788.62			0.0229842
$407$			276771.i			1.67083i
$408$	−174157.	+	25324.6i	−1.04621	+	0.152132i
$409$	−245378.			−1.46686			−0.733430	−	0.679765i	$-0.762082\pi$
$409$	−245378.			−1.46686			−0.733430	+	0.679765i	$0.762082\pi$
$410$			25142.6i			0.149570i
$411$	43188.0	+	297004.i	0.255670	+	1.75824i
$412$	38493.5			0.226774
$413$			23129.3i			0.135601i
$414$	−255063.	+	75780.9i	−1.48815	+	0.442139i
$415$	−228944.			−1.32933
$416$			129107.i			0.746043i
$417$	211894.	−	30812.0i	1.21856	−	0.177193i
$418$	248904.			1.42455
$419$			210228.i			1.19746i	0.800950	+	0.598731i	$0.204328\pi$
$419$			210228.i			1.19746i	−0.800950	+	0.598731i	$0.795672\pi$
$420$	−4541.73	−	31233.5i	−0.0257468	−	0.177061i
$421$	−92966.7			−0.524521			−0.262261	−	0.964997i	$-0.584468\pi$
$421$	−92966.7			−0.524521			−0.262261	+	0.964997i	$0.584468\pi$
$422$		−	134718.i		−	0.756484i
$423$	16935.6	+	57001.9i	0.0946499	+	0.318573i
$424$	−53903.6			−0.299838
$425$		−	94626.1i		−	0.523881i
$426$	−151167.	+	21981.5i	−0.832986	+	0.121126i
$427$	80524.0			0.441641
$428$		−	48879.4i		−	0.266832i
$429$	54832.3	+	377082.i	0.297935	+	2.04890i
$430$	−147508.			−0.797774
$431$		−	332694.i		−	1.79098i	−0.445084	−	0.895489i	$-0.646826\pi$
$431$		−	332694.i		−	1.79098i	0.445084	−	0.895489i	$-0.353174\pi$
$432$	115856.	−	53583.2i	0.620797	−	0.287118i
$433$	−84824.8			−0.452425			−0.226213	−	0.974078i	$-0.572634\pi$
$433$	−84824.8			−0.452425			−0.226213	+	0.974078i	$0.572634\pi$
$434$		−	148179.i		−	0.786694i
$435$	−3253.17	+	473.050i	−0.0171921	+	0.00249994i
$436$	−51813.1			−0.272563
$437$			414167.i			2.16876i
$438$	23592.0	+	162243.i	0.122975	+	0.845700i
$439$	187294.			0.971842			0.485921	−	0.874003i	$-0.338484\pi$
$439$	187294.			0.971842			0.485921	+	0.874003i	$0.338484\pi$
$440$			194645.i			1.00540i
$441$	15822.3	−	4700.90i	0.0813564	−	0.0241715i
$442$	250363.			1.28152
$443$			71225.5i			0.362934i	0.983397	+	0.181467i	$0.0580846\pi$
$443$			71225.5i			0.362934i	−0.983397	+	0.181467i	$0.941915\pi$
$444$	60307.0	−	8769.37i	0.305916	−	0.0444839i
$445$	−62273.9			−0.314475
$446$			31007.3i			0.155881i
$447$	−21253.0	−	146157.i	−0.106366	−	0.731483i
$448$	−231817.			−1.15502
$449$		−	242237.i		−	1.20157i	−0.799412	−	0.600783i	$-0.794856\pi$
$449$		−	242237.i		−	1.20157i	0.799412	−	0.600783i	$-0.205144\pi$
$450$	26758.5	+	90063.7i	0.132141	+	0.444759i
$451$	−70492.6			−0.346570
$452$		−	24598.2i		−	0.120400i
$453$	301116.	−	43785.9i	1.46736	−	0.213372i
$454$	−42712.2			−0.207224
$455$			222833.i			1.07636i
$456$	−39138.9	−	269159.i	−0.188226	−	1.29443i
$457$	348206.			1.66726			0.833630	−	0.552324i	$-0.186259\pi$
$457$	348206.			1.66726			0.833630	+	0.552324i	$0.186259\pi$
$458$			259408.i			1.23667i
$459$	86344.4	+	186691.i	0.409835	+	0.886130i
$460$	−65261.5			−0.308419
$461$		−	31096.2i		−	0.146321i	−0.997320	−	0.0731604i	$-0.976692\pi$
$461$		−	31096.2i		−	0.146321i	0.997320	−	0.0731604i	$-0.0233085\pi$
$462$	−259454.	+	37727.8i	−1.21556	+	0.176757i
$463$	837.205			0.00390544			0.00195272	−	0.999998i	$-0.499378\pi$
$463$	837.205			0.00390544			0.00195272	+	0.999998i	$0.499378\pi$
$464$			3758.09i			0.0174555i
$465$	18501.7	+	127236.i	0.0855668	+	0.588444i
$466$	252886.			1.16454
$467$			118252.i			0.542218i	0.962549	+	0.271109i	$0.0873903\pi$
$467$			118252.i			0.542218i	−0.962549	+	0.271109i	$0.912610\pi$
$468$	80426.9	−	23895.3i	0.367206	−	0.109099i
$469$	−40688.2			−0.184979
$470$		−	43212.2i		−	0.195619i
$471$	−319709.	+	46489.6i	−1.44116	+	0.209563i
$472$	−31407.4			−0.140977
$473$		−	413570.i		−	1.84853i
$474$	−37159.3	−	255545.i	−0.165391	−	1.13739i
$475$	146244.			0.648172
$476$		−	58141.6i		−	0.256610i
$477$	17942.9	+	60392.0i	0.0788597	+	0.265426i
$478$	−48137.3			−0.210681
$479$			72016.9i			0.313880i	0.987608	+	0.156940i	$0.0501629\pi$
$479$			72016.9i			0.313880i	−0.987608	+	0.156940i	$0.949837\pi$
$480$	76278.3	−	11091.8i	0.331069	−	0.0481414i
$481$	−430255.			−1.85967
$482$			68976.2i			0.296896i
$483$	−62777.7	−	431722.i	−0.269098	−	1.85059i
$484$	−50849.6			−0.217068
$485$			85607.7i			0.363940i
$486$	−134974.	−	153273.i	−0.571450	−	0.648923i
$487$	−79780.9			−0.336388			−0.168194	−	0.985754i	$-0.553794\pi$
$487$	−79780.9			−0.336388			−0.168194	+	0.985754i	$0.553794\pi$
$488$			109344.i			0.459151i
$489$	−135906.	+	19762.4i	−0.568358	+	0.0826461i
$490$	−11994.6			−0.0499567
$491$			229258.i			0.950960i	0.879727	+	0.475480i	$0.157726\pi$
$491$			229258.i			0.950960i	−0.879727	+	0.475480i	$0.842274\pi$
$492$	2233.53	+	15360.0i	0.00922701	+	0.0634542i
$493$	−6055.82			−0.0249161
$494$			386934.i			1.58556i
$495$	218074.	−	64791.3i	0.890009	−	0.264427i
$496$	146984.			0.597459
$497$		−	250457.i		−	1.01396i
$498$	414398.	−	60258.4i	1.67093	−	0.242974i
$499$	−362360.			−1.45526			−0.727628	−	0.685972i	$-0.759377\pi$
$499$	−362360.			−1.45526			−0.727628	+	0.685972i	$0.759377\pi$
$500$			65989.5i			0.263958i
$501$	−35503.8	−	244160.i	−0.141449	−	0.972746i
$502$	−401005.			−1.59126
$503$		−	176237.i		−	0.696565i	−0.937390	−	0.348283i	$-0.886765\pi$
$503$		−	176237.i		−	0.696565i	0.937390	−	0.348283i	$-0.113235\pi$
$504$	81595.8	+	274635.i	0.321223	+	1.08117i
$505$	186978.			0.733175
$506$			542122.i			2.11737i
$507$	−331819.	+	48250.6i	−1.29088	+	0.187710i
$508$	−55868.5			−0.216491
$509$			217838.i			0.840812i	0.907336	+	0.420406i	$0.138112\pi$
$509$			217838.i			0.840812i	−0.907336	+	0.420406i	$0.861888\pi$
$510$	−21509.0	−	147918.i	−0.0826952	−	0.568695i
$511$	−268807.			−1.02943
$512$		−	282276.i		−	1.07680i
$513$	−288529.	+	133445.i	−1.09636	+	0.507068i
$514$	60046.2			0.227279
$515$			162254.i			0.611761i
$516$	−90114.9	+	13103.8i	−0.338452	+	0.0492150i
$517$	121154.			0.453271
$518$		−	296040.i		−	1.10329i
$519$	−4576.67	−	31473.8i	−0.0169908	−	0.116846i
$520$	−302585.			−1.11903
$521$			314301.i			1.15790i	0.815363	+	0.578950i	$0.196537\pi$
$521$			314301.i			1.15790i	−0.815363	+	0.578950i	$0.803463\pi$
$522$	5763.85	−	1712.48i	0.0211530	−	0.00628468i
$523$	−102505.			−0.374750			−0.187375	−	0.982288i	$-0.559998\pi$
$523$	−102505.			−0.374750			−0.187375	+	0.982288i	$0.559998\pi$
$524$			7066.98i			0.0257378i
$525$	−152443.	+	22167.0i	−0.553081	+	0.0804246i
$526$	99765.4			0.360586
$527$			236852.i			0.852817i
$528$	−37423.7	−	257363.i	−0.134239	−	0.923164i
$529$	−622231.			−2.22352
$530$		−	45782.2i		−	0.162984i
$531$	10454.6	+	35188.0i	0.0370781	+	0.124797i
$532$	89857.4			0.317491
$533$		−	109584.i		−	0.385740i
$534$	112718.	−	16390.6i	0.395285	−	0.0574793i
$535$	206032.			0.719824
$536$		−	55250.8i		−	0.192313i
$537$	−26020.1	−	178941.i	−0.0902321	−	0.620527i
$538$	495460.			1.71176
$539$		−	33629.4i		−	0.115755i
$540$	−21027.3	−	45464.4i	−0.0721100	−	0.155914i
$541$	−236784.			−0.809017			−0.404508	−	0.914534i	$-0.632557\pi$
$541$	−236784.			−0.809017			−0.404508	+	0.914534i	$0.632557\pi$
$542$		−	72196.2i		−	0.245763i
$543$	425886.	−	61929.0i	1.44442	−	0.210036i
$544$	141993.			0.479810
$545$		−	218398.i		−	0.735284i
$546$	−58649.8	−	403335.i	−0.196735	−	1.35295i
$547$	−254727.			−0.851336			−0.425668	−	0.904879i	$-0.639961\pi$
$547$	−254727.			−0.851336			−0.425668	+	0.904879i	$0.639961\pi$
$548$		−	134641.i		−	0.448350i
$549$	122506.	−	36397.2i	0.406454	−	0.120760i
$550$	191425.			0.632811
$551$		−	9359.23i		−	0.0308274i
$552$	586238.	−	85246.1i	1.92396	−	0.279767i
$553$	423392.			1.38450
$554$		−	358436.i		−	1.16786i
$555$	36963.8	+	254200.i	0.120003	+	0.825259i
$556$	−96058.2			−0.310731
$557$			119486.i			0.385128i	0.981284	+	0.192564i	$0.0616803\pi$
$557$			119486.i			0.385128i	−0.981284	+	0.192564i	$0.938320\pi$
$558$	−66977.4	−	225432.i	−0.215110	−	0.724016i
$559$	642916.			2.05746
$560$		−	152086.i		−	0.484968i
$561$	414718.	−	60305.0i	1.31773	−	0.191614i
$562$	418743.			1.32579
$563$		−	470604.i		−	1.48470i	−0.670013	−	0.742350i	$-0.733711\pi$
$563$		−	470604.i		−	1.48470i	0.670013	−	0.742350i	$-0.266289\pi$
$564$	−3838.72	−	26398.9i	−0.0120678	−	0.0829903i
$565$	103684.			0.324799
$566$			174212.i			0.543806i
$567$	280532.	−	182835.i	0.872603	−	0.568714i
$568$	340096.			1.05416
$569$		−	360992.i		−	1.11500i	−0.830178	−	0.557498i	$-0.811761\pi$
$569$		−	360992.i		−	1.11500i	0.830178	−	0.557498i	$-0.188239\pi$
$570$	228606.	−	33242.0i	0.703618	−	0.102315i
$571$	−103378.			−0.317069			−0.158535	−	0.987353i	$-0.550677\pi$
$571$	−103378.			−0.317069			−0.158535	+	0.987353i	$0.550677\pi$
$572$		−	170943.i		−	0.522467i
$573$	−79672.0	−	547905.i	−0.242659	−	1.66877i
$574$	75400.4			0.228849
$575$			318525.i			0.963403i
$576$	−352676.	+	104782.i	−1.06300	+	0.315823i
$577$	87029.9			0.261407			0.130703	−	0.991422i	$-0.458276\pi$
$577$	87029.9			0.261407			0.130703	+	0.991422i	$0.458276\pi$
$578$			13521.8i			0.0404744i
$579$	44971.0	−	6539.33i	0.134145	−	0.0195064i
$580$	1474.76			0.00438395
$581$			686583.i			2.03395i
$582$	−22532.0	−	154953.i	−0.0665203	−	0.457461i
$583$	128360.			0.377652
$584$		−	365014.i		−	1.07025i
$585$	100721.	+	339008.i	0.294313	+	0.990599i
$586$	−18527.4			−0.0539535
$587$		−	38519.5i		−	0.111790i	−0.998437	−	0.0558952i	$-0.982199\pi$
$587$		−	38519.5i		−	0.111790i	0.998437	−	0.0558952i	$-0.0178013\pi$
$588$	−7327.67	+	1065.53i	−0.0211939	+	0.00308185i
$589$	−366053.			−1.05515
$590$		−	26675.4i		−	0.0766315i
$591$	9981.83	+	68645.1i	0.0285782	+	0.196533i
$592$	293654.			0.837902
$593$			273778.i			0.778555i	0.921120	+	0.389278i	$0.127275\pi$
$593$			273778.i			0.778555i	−0.921120	+	0.389278i	$0.872725\pi$
$594$	−377669.	+	174672.i	−1.07038	+	0.495051i
$595$	245073.			0.692248
$596$			66257.4i			0.186527i
$597$	−425336.	+	61849.0i	−1.19339	+	0.173534i
$598$	−842757.			−2.35668
$599$		−	394240.i		−	1.09877i	−0.835569	−	0.549385i	$-0.814862\pi$
$599$		−	394240.i		−	1.09877i	0.835569	−	0.549385i	$-0.185138\pi$
$600$	−30100.7	−	207003.i	−0.0836131	−	0.575008i
$601$	−356771.			−0.987735			−0.493868	−	0.869537i	$-0.664417\pi$
$601$	−356771.			−0.987735			−0.493868	+	0.869537i	$0.664417\pi$
$602$			442364.i			1.22064i
$603$	−61901.3	+	18391.3i	−0.170241	+	0.0505798i
$604$	−136505.			−0.374175
$605$		−	214336.i		−	0.585578i
$606$	−338437.	+	49212.8i	−0.921578	+	0.134009i
$607$	323835.			0.878915			0.439458	−	0.898263i	$-0.355171\pi$
$607$	323835.			0.878915			0.439458	+	0.898263i	$0.355171\pi$
$608$			219449.i			0.593646i
$609$	1418.63	+	9755.95i	0.00382504	+	0.0263048i
$610$	−92869.6			−0.249582
$611$			188341.i			0.504500i
$612$	−26280.3	−	88454.1i	−0.0701661	−	0.236165i
$613$	−192418.			−0.512064			−0.256032	−	0.966668i	$-0.582415\pi$
$613$	−192418.			−0.512064			−0.256032	+	0.966668i	$0.582415\pi$
$614$		−	439433.i		−	1.16562i
$615$	−64743.9	+	9414.55i	−0.171178	+	0.0248914i
$616$	583721.			1.53831
$617$		−	509857.i		−	1.33930i	−0.742676	−	0.669650i	$-0.766444\pi$
$617$		−	509857.i		−	1.33930i	0.742676	−	0.669650i	$-0.233556\pi$
$618$	−42705.5	−	293686.i	−0.111817	−	0.768965i
$619$	−40634.3			−0.106050			−0.0530250	−	0.998593i	$-0.516886\pi$
$619$	−40634.3			−0.106050			−0.0530250	+	0.998593i	$0.516886\pi$
$620$		−	57680.1i		−	0.150052i
$621$	−290648.	−	628428.i	−0.753675	−	1.62957i
$622$	441575.			1.14136
$623$			186753.i			0.481163i
$624$	400084.	−	58177.1i	1.02750	−	0.149411i
$625$	−68546.9			−0.175480
$626$		−	331384.i		−	0.845636i
$627$	93200.9	+	640943.i	0.237075	+	1.63036i
$628$	144934.			0.367495
$629$			473198.i			1.19603i
$630$	−233257.	+	69302.1i	−0.587697	+	0.174609i
$631$	163133.			0.409716			0.204858	−	0.978792i	$-0.434327\pi$
$631$	163133.			0.409716			0.204858	+	0.978792i	$0.434327\pi$
$632$			574926.i			1.43939i
$633$	346907.	−	50444.5i	0.865776	−	0.125894i
$634$	194862.			0.484785
$635$		−	235492.i		−	0.584020i
$636$	−4067.03	−	27969.0i	−0.0100546	−	0.0691452i
$637$	52278.6			0.128838
$638$		−	12250.7i		−	0.0300968i
$639$	−113208.	−	381034.i	−0.277252	−	0.933173i
$640$	130326.			0.318179
$641$			201073.i			0.489370i	0.969603	+	0.244685i	$0.0786846\pi$
$641$			201073.i			0.489370i	−0.969603	+	0.244685i	$0.921315\pi$
$642$	−372925.	+	54227.8i	−0.904797	+	0.131568i
$643$	−288080.			−0.696773			−0.348386	−	0.937351i	$-0.613270\pi$
$643$	−288080.			−0.696773			−0.348386	+	0.937351i	$0.613270\pi$
$644$			195713.i			0.471898i
$645$	−55233.9	−	379844.i	−0.132766	−	0.913031i
$646$	425553.			1.01974
$647$		−	334926.i		−	0.800092i	−0.916495	−	0.400046i	$-0.868994\pi$
$647$		−	334926.i		−	0.800092i	0.916495	−	0.400046i	$-0.131006\pi$
$648$	248273.	+	380936.i	0.591261	+	0.907198i
$649$	74790.1			0.177564
$650$			297581.i			0.704333i
$651$	381569.	−	55484.8i	0.900350	−	0.130922i
$652$	61610.5			0.144930
$653$		−	522679.i		−	1.22577i	−0.790172	−	0.612885i	$-0.790009\pi$
$653$		−	522679.i		−	1.22577i	0.790172	−	0.612885i	$-0.209991\pi$
$654$	57482.5	+	395308.i	0.134394	+	0.924229i
$655$	−29788.1			−0.0694320
$656$			74792.7i			0.173801i
$657$	−408951.	+	121502.i	−0.947416	+	0.281483i
$658$	−129589.			−0.299307
$659$		−	510220.i		−	1.17486i	−0.809275	−	0.587430i	$-0.800140\pi$
$659$		−	510220.i		−	1.17486i	0.809275	−	0.587430i	$-0.199860\pi$
$660$	−100995.	+	14685.9i	−0.231853	+	0.0337143i
$661$	−150703.			−0.344921			−0.172460	−	0.985016i	$-0.555172\pi$
$661$	−150703.			−0.344921			−0.172460	+	0.985016i	$0.555172\pi$
$662$		−	222693.i		−	0.508148i
$663$	93747.2	+	644700.i	0.213271	+	1.46666i
$664$	−932314.			−2.11459
$665$			378758.i			0.856484i
$666$	−133812.	−	450383.i	−0.301679	−	1.01539i
$667$	20384.8			0.0458199
$668$			110685.i			0.248049i
$669$	−79845.7	+	11610.5i	−0.178402	+	0.0259418i
$670$	46926.4			0.104536
$671$		−	260379.i		−	0.578310i
$672$	−33263.2	−	228751.i	−0.0736590	−	0.506553i
$673$	−291819.			−0.644293			−0.322147	−	0.946690i	$-0.604404\pi$
$673$	−291819.			−0.644293			−0.322147	+	0.946690i	$0.604404\pi$
$674$			720216.i			1.58542i
$675$	−221900.	+	102629.i	−0.487024	+	0.225249i
$676$	150424.			0.329173
$677$			566524.i			1.23607i	0.786153	+	0.618033i	$0.212070\pi$
$677$			566524.i			1.23607i	−0.786153	+	0.618033i	$0.787930\pi$
$678$	−187672.	+	27289.7i	−0.408262	+	0.0593663i
$679$	256729.			0.556847
$680$			332786.i			0.719693i
$681$	−15993.4	−	109987.i	−0.0344863	−	0.237163i
$682$	−479144.			−1.03014
$683$			816248.i			1.74977i	0.484332	+	0.874884i	$0.339063\pi$
$683$			816248.i			1.74977i	−0.484332	+	0.874884i	$0.660937\pi$
$684$	136705.	−	40616.0i	0.292195	−	0.0868130i
$685$	567527.			1.20950
$686$		−	387856.i		−	0.824180i
$687$	−667993.	+	97134.3i	−1.41533	+	0.205807i
$688$	−438799.			−0.927018
$689$			199542.i			0.420335i
$690$	72402.4	+	497912.i	0.152074	+	1.04581i
$691$	−489750.			−1.02570			−0.512848	−	0.858479i	$-0.671410\pi$
$691$	−489750.			−1.02570			−0.512848	+	0.858479i	$0.671410\pi$
$692$			14268.0i			0.0297956i
$693$	−194303.	−	653984.i	−0.404588	−	1.36176i
$694$	718805.			1.49242
$695$		−	404895.i		−	0.838249i
$696$	−13247.7	+	1926.37i	−0.0273477	+	0.00397668i
$697$	−120522.			−0.248085
$698$			154967.i			0.318075i
$699$	94692.1	+	651198.i	0.193802	+	1.33278i
$700$	69107.0			0.141035
$701$		−	389452.i		−	0.792534i	−0.918135	−	0.396267i	$-0.870306\pi$
$701$		−	389452.i		−	0.792534i	0.918135	−	0.396267i	$-0.129694\pi$
$702$	−271537.	−	587106.i	−0.551003	−	1.19136i
$703$	−731323.			−1.47979
$704$			749594.i			1.51245i
$705$	111274.	−	16180.6i	0.223880	−	0.0325549i
$706$	586037.			1.17575
$707$		−	560729.i		−	1.12180i
$708$	−2369.69	−	16296.4i	−0.00472743	−	0.0325106i
$709$	225101.			0.447800			0.223900	−	0.974612i	$-0.428121\pi$
$709$	225101.			0.447800			0.223900	+	0.974612i	$0.428121\pi$
$710$			288856.i			0.573012i
$711$	644131.	−	191375.i	1.27419	−	0.378571i
$712$	−253593.			−0.500240
$713$		−	797278.i		−	1.56831i
$714$	−443591.	+	64503.5i	−0.870134	+	0.126528i
$715$	720542.			1.40944
$716$			81119.3i			0.158233i
$717$	−18024.8	−	123957.i	−0.0350617	−	0.241119i
$718$	249386.			0.483753
$719$			848932.i			1.64216i	0.570814	+	0.821079i	$0.306628\pi$
$719$			848932.i			1.64216i	−0.570814	+	0.821079i	$0.693372\pi$
$720$	−68743.6	−	231377.i	−0.132607	−	0.446329i
$721$	486585.			0.936027
$722$			206949.i			0.396999i
$723$	−177618.	+	25827.8i	−0.339790	+	0.0494096i
$724$	−193067.			−0.368325
$725$		−	7197.94i		−	0.0136941i
$726$	56413.5	+	387956.i	0.107031	+	0.736054i
$727$	−93551.0			−0.177003			−0.0885013	−	0.996076i	$-0.528208\pi$
$727$	−93551.0			−0.177003			−0.0885013	+	0.996076i	$0.528208\pi$
$728$			907425.i			1.71217i
$729$	344147.	−	404959.i	0.647574	−	0.762003i
$730$	310019.			0.581758
$731$		−	707084.i		−	1.32323i
$732$	−56735.3	+	8250.00i	−0.105884	+	0.0153968i
$733$	820608.			1.52731			0.763656	−	0.645624i	$-0.223402\pi$
$733$	820608.			1.52731			0.763656	+	0.645624i	$0.223402\pi$
$734$			384989.i			0.714589i
$735$	−4491.33	−	30886.9i	−0.00831381	−	0.0571741i
$736$	−477970.			−0.882358
$737$			131568.i			0.242223i
$738$	114711.	−	34081.3i	0.210616	−	0.0625754i
$739$	−745272.			−1.36466			−0.682332	−	0.731043i	$-0.739034\pi$
$739$	−745272.			−1.36466			−0.682332	+	0.731043i	$0.739034\pi$
$740$		−	115237.i		−	0.210440i
$741$	−996379.	+	144886.i	−1.81463	+	0.263869i
$742$	−137296.			−0.249374
$743$			121713.i			0.220475i	0.993905	+	0.110238i	$0.0351612\pi$
$743$			121713.i			0.220475i	−0.993905	+	0.110238i	$0.964839\pi$
$744$	75343.1	+	518135.i	0.136112	+	0.936045i
$745$	−279282.			−0.503188
$746$		−	614836.i		−	1.10479i
$747$	310339.	+	1.04454e6i	0.556154	+	1.87190i
$748$	−188004.			−0.336020
$749$		−	617870.i		−	1.10137i
$750$	503466.	−	73210.1i	0.895051	−	0.130151i
$751$	542939.			0.962655			0.481328	−	0.876541i	$-0.340155\pi$
$751$	542939.			0.962655			0.481328	+	0.876541i	$0.340155\pi$
$752$		−	128545.i		−	0.227310i
$753$	−150154.	−	1.03261e6i	−0.264818	−	1.82116i
$754$	19044.4			0.0334984
$755$		−	575383.i		−	1.00940i
$756$	−136343.	+	63058.8i	−0.238556	+	0.110332i
$757$	210021.			0.366498			0.183249	−	0.983067i	$-0.441339\pi$
$757$	210021.			0.366498			0.183249	+	0.983067i	$0.441339\pi$
$758$			209387.i			0.364427i
$759$	−1.39600e6	+	202995.i	−2.42327	+	0.352373i
$760$	−514318.			−0.890440
$761$		−	841089.i		−	1.45235i	−0.687508	−	0.726177i	$-0.741295\pi$
$761$		−	841089.i		−	1.45235i	0.687508	−	0.726177i	$-0.258705\pi$
$762$	61981.7	+	426248.i	0.106746	+	0.734096i
$763$	−654954.			−1.12502
$764$			248382.i			0.425533i
$765$	372843.	−	110774.i	0.637094	−	0.189285i
$766$	85419.2			0.145579
$767$			116265.i			0.197632i
$768$	411366.	−	59817.5i	0.697438	−	0.101416i
$769$	264110.			0.446614			0.223307	−	0.974748i	$-0.428315\pi$
$769$	264110.			0.446614			0.223307	+	0.974748i	$0.428315\pi$
$770$			495775.i			0.836186i
$771$	22484.0	+	154623.i	0.0378238	+	0.260115i
$772$	−20386.8			−0.0342069
$773$		−	219678.i		−	0.367643i	−0.982960	−	0.183822i	$-0.941153\pi$
$773$		−	219678.i		−	0.367643i	0.982960	−	0.183822i	$-0.0588469\pi$
$774$	199950.	+	672993.i	0.333765	+	1.12338i
$775$	−281522.			−0.468715
$776$			348614.i			0.578924i
$777$	762323.	−	110851.i	1.26269	−	0.183610i
$778$	−760586.			−1.25658
$779$		−	186265.i		−	0.306943i
$780$	−22830.1	−	157002.i	−0.0375247	−	0.258058i
$781$	−809867.			−1.32773
$782$			926871.i			1.51567i
$783$	6567.98	+	14201.0i	0.0107129	+	0.0231631i
$784$	−35680.8			−0.0580500
$785$			610912.i			0.991378i
$786$	53917.5	−	7840.26i	0.0872739	−	0.0126907i
$787$	20632.5			0.0333121			0.0166560	−	0.999861i	$-0.494698\pi$
$787$	20632.5			0.0333121			0.0166560	+	0.999861i	$0.494698\pi$
$788$		−	31119.0i		−	0.0501156i
$789$	37356.7	+	256902.i	0.0600087	+	0.412680i
$790$	−488305.			−0.782414
$791$		−	310938.i		−	0.496960i
$792$	888049.	−	263845.i	1.41575	−	0.420628i
$793$	404773.			0.643672
$794$		−	27737.8i		−	0.0439979i
$795$	117892.	−	17142.9i	0.186531	−	0.0271238i
$796$	192818.			0.304314
$797$		−	282817.i		−	0.445234i	−0.974906	−	0.222617i	$-0.928540\pi$
$797$		−	282817.i		−	0.445234i	0.974906	−	0.222617i	$-0.0714599\pi$
$798$	−99689.7	−	685566.i	−0.156547	−	1.07657i
$799$	207138.			0.324464
$800$			168773.i			0.263708i
$801$	84413.5	+	284119.i	0.131567	+	0.442828i
$802$	659977.			1.02608
$803$			869203.i			1.34800i
$804$	28667.9	−	4168.67i	0.0443491	−	0.00644889i
$805$	−824951.			−1.27302
$806$		−	744854.i		−	1.14657i
$807$	185523.	+	1.27584e6i	0.284872	+	1.95907i
$808$	761417.			1.16627
$809$			599598.i			0.916143i	0.888915	+	0.458071i	$0.151460\pi$
$809$			599598.i			0.916143i	−0.888915	+	0.458071i	$0.848540\pi$
$810$	−323542.	+	210867.i	−0.493129	+	0.321394i
$811$	208205.			0.316555			0.158277	−	0.987395i	$-0.449406\pi$
$811$	208205.			0.316555			0.158277	+	0.987395i	$0.449406\pi$
$812$		−	4422.67i		−	0.00670769i
$813$	185910.	−	27033.6i	0.281269	−	0.0408999i
$814$	−957263.			−1.44472
$815$			259695.i			0.390974i
$816$	−63983.7	−	440016.i	−0.0960923	−	0.660828i
$817$	1.09279e6			1.63717
$818$		−	848684.i		−	1.26835i
$819$	1.01665e6	−	302054.i	1.51567	−	0.450315i
$820$	29350.4			0.0436502
$821$		−	387333.i		−	0.574644i	−0.957834	−	0.287322i	$-0.907235\pi$
$821$		−	387333.i		−	0.574644i	0.957834	−	0.287322i	$-0.0927649\pi$
$822$	−1.02724e6	+	149374.i	−1.52030	+	0.221070i
$823$	596678.			0.880928			0.440464	−	0.897770i	$-0.354814\pi$
$823$	596678.			0.880928			0.440464	+	0.897770i	$0.354814\pi$
$824$			660736.i			0.973137i
$825$	71678.4	+	492933.i	0.105313	+	0.724236i
$826$	−79997.0			−0.117250
$827$		−	546491.i		−	0.799046i	−0.916723	−	0.399523i	$-0.869176\pi$
$827$		−	546491.i		−	0.799046i	0.916723	−	0.399523i	$-0.130824\pi$
$828$	88463.3	+	297749.i	0.129033	+	0.434300i
$829$	−255163.			−0.371286			−0.185643	−	0.982617i	$-0.559437\pi$
$829$	−255163.			−0.371286			−0.185643	+	0.982617i	$0.559437\pi$
$830$		−	791846.i		−	1.14944i
$831$	922995.	−	134215.i	1.33659	−	0.194356i
$832$	−1.16528e6			−1.68339
$833$		−	57496.4i		−	0.0828611i
$834$	106569.	+	732875.i	0.153214	+	1.05365i
$835$	−466550.			−0.669153
$836$		−	290559.i		−	0.415740i
$837$	555424.	−	256883.i	0.792818	−	0.366678i
$838$	−727111.			−1.03541
$839$		−	135472.i		−	0.192454i	−0.995359	−	0.0962270i	$-0.969323\pi$
$839$		−	135472.i		−	0.192454i	0.995359	−	0.0962270i	$-0.0306775\pi$
$840$	536119.	−	77958.2i	0.759806	−	0.110485i
$841$	706820.			0.999349
$842$		−	321542.i		−	0.453538i
$843$	156797.	+	1.07829e6i	0.220639	+	1.51733i
$844$	−157264.			−0.220772
$845$			634053.i			0.887998i
$846$	−197151.	+	58574.9i	−0.275460	+	0.0818410i
$847$	−642774.			−0.895966
$848$		−	136190.i		−	0.189388i
$849$	−448606.	+	65232.8i	−0.622372	+	0.0905004i
$850$	327282.			0.452985
$851$		−	1.59285e6i		−	2.19946i
$852$	25660.3	+	176466.i	0.0353494	+	0.243098i
$853$	1.36924e6			1.88183			0.940916	−	0.338640i	$-0.109967\pi$
$853$	1.36924e6			1.88183			0.940916	+	0.338640i	$0.109967\pi$
$854$			278507.i			0.381874i
$855$	171201.	+	576227.i	0.234193	+	0.788245i
$856$	839008.			1.14503
$857$			1.17065e6i			1.59392i	0.604032	+	0.796960i	$0.293560\pi$
$857$			1.17065e6i			1.59392i	−0.604032	+	0.796960i	$0.706440\pi$
$858$	−1.30421e6	+	189647.i	−1.77163	+	0.257616i
$859$	1.26206e6			1.71039			0.855195	−	0.518307i	$-0.173438\pi$
$859$	1.26206e6			1.71039			0.855195	+	0.518307i	$0.173438\pi$
$860$			172195.i			0.232822i
$861$	28233.3	+	194161.i	0.0380852	+	0.261912i
$862$	1.15068e6			1.54861
$863$			133363.i			0.179067i	0.995984	+	0.0895334i	$0.0285376\pi$
$863$			133363.i			0.179067i	−0.995984	+	0.0895334i	$0.971462\pi$
$864$	−154002.	−	332977.i	−0.206300	−	0.446054i
$865$	−60141.2			−0.0803785
$866$		−	293382.i		−	0.391199i
$867$	−34819.6	+	5063.20i	−0.0463219	+	0.00673576i
$868$	−172977.			−0.229588
$869$		−	1.36906e6i		−	1.81294i
$870$	−1636.13	−	11251.7i	−0.00216162	−	0.0148655i
$871$	−204529.			−0.269599
$872$		−	889365.i		−	1.16963i
$873$	390577.	−	116043.i	0.512481	−	0.152262i
$874$	−1.43247e6			−1.87527
$875$			834153.i			1.08951i
$876$	189395.	−	27540.3i	0.246808	−	0.0358889i
$877$	−554061.			−0.720374			−0.360187	−	0.932880i	$-0.617287\pi$
$877$	−554061.			−0.720374			−0.360187	+	0.932880i	$0.617287\pi$
$878$			647792.i			0.840323i
$879$	−6937.50	−	47709.3i	−0.00897895	−	0.0617483i
$880$	−491779.			−0.635045
$881$		−	228827.i		−	0.294819i	−0.989076	−	0.147410i	$-0.952906\pi$
$881$		−	228827.i		−	0.294819i	0.989076	−	0.147410i	$-0.0470935\pi$
$882$	16258.9	+	54724.2i	0.0209004	+	0.0703465i
$883$	36783.2			0.0471767			0.0235884	−	0.999722i	$-0.492491\pi$
$883$	36783.2			0.0471767			0.0235884	+	0.999722i	$0.492491\pi$
$884$		−	292262.i		−	0.373997i
$885$	68690.9	−	9988.49i	0.0877027	−	0.0127530i
$886$	−246346.			−0.313819
$887$			1.38652e6i			1.76229i	0.472845	+	0.881146i	$0.343227\pi$
$887$			1.38652e6i			1.76229i	−0.472845	+	0.881146i	$0.656773\pi$
$888$	150525.	+	1.03516e6i	0.190890	+	1.31275i
$889$	−706217.			−0.893583
$890$		−	215386.i		−	0.271917i
$891$	−591208.	−	907118.i	−0.744706	−	1.14264i
$892$	36196.5			0.0454922
$893$			320131.i			0.401444i
$894$	505510.	−	73507.3i	0.632492	−	0.0919720i
$895$	−341926.			−0.426861
$896$		−	390836.i		−	0.486831i
$897$	−315567.	−	2.17015e6i	−0.392199	−	2.69715i
$898$	837821.			1.03896
$899$			18016.7i			0.0222923i
$900$	105136.	−	31236.7i	0.129798	−	0.0385639i
$901$	219458.			0.270335
$902$		−	243812.i		−	0.299669i
$903$	−1.13911e6	+	165641.i	−1.39699	+	0.203139i
$904$	422225.			0.516662
$905$		−	813799.i		−	0.993619i
$906$	151442.	+	1.04146e6i	0.184497	+	1.26879i
$907$	−669563.			−0.813910			−0.406955	−	0.913448i	$-0.633409\pi$
$907$	−669563.			−0.813910			−0.406955	+	0.913448i	$0.633409\pi$
$908$			49860.4i			0.0604761i
$909$	−253452.	−	853069.i	−0.306739	−	1.03242i
$910$	−770707.			−0.930693
$911$			1.01613e6i			1.22437i	0.790713	+	0.612187i	$0.209710\pi$
$911$			1.01613e6i			1.22437i	−0.790713	+	0.612187i	$0.790290\pi$
$912$	680041.	−	98886.3i	0.817609	−	0.118890i
$913$	2.22010e6			2.66337
$914$			1.20433e6i			1.44163i
$915$	−34774.6	−	239145.i	−0.0415355	−	0.285640i
$916$	302822.			0.360908
$917$			89331.6i			0.106235i
$918$	−645704.	+	298638.i	−0.766211	+	0.354372i
$919$	−1.51307e6			−1.79155			−0.895773	−	0.444511i	$-0.853377\pi$
$919$	−1.51307e6			−1.79155			−0.895773	+	0.444511i	$0.853377\pi$
$920$		−	1.12021e6i		−	1.32349i
$921$	1.13157e6	−	164544.i	1.33402	−	0.193983i
$922$	107552.			0.126519
$923$		−	1.25898e6i		−	1.47780i
$924$	44041.8	+	302875.i	0.0515847	+	0.354748i
$925$	−562442.			−0.657346
$926$			2895.63i			0.00337692i
$927$	740270.	−	219939.i	0.861451	−	0.255943i
$928$	10801.0			0.0125421
$929$		−	1.28065e6i		−	1.48389i	−0.670464	−	0.741943i	$-0.733905\pi$
$929$		−	1.28065e6i		−	1.48389i	0.670464	−	0.741943i	$-0.266095\pi$
$930$	−440070.	+	63991.5i	−0.508810	+	0.0739871i
$931$	88860.2			0.102520
$932$		−	295208.i		−	0.339857i
$933$	165346.	+	1.13709e6i	0.189946	+	1.30626i
$934$	−408995.			−0.468840
$935$		−	792458.i		−	0.906469i
$936$	410161.	+	1.38052e6i	0.468168	+	1.57576i
$937$	−442375.			−0.503862			−0.251931	−	0.967745i	$-0.581066\pi$
$937$	−442375.			−0.503862			−0.251931	+	0.967745i	$0.581066\pi$
$938$		−	140728.i		−	0.159946i
$939$	853336.	−	124085.i	0.967807	−	0.140731i
$940$	−50444.0			−0.0570892
$941$			230903.i			0.260765i	0.991464	+	0.130383i	$0.0416206\pi$
$941$			230903.i			0.260765i	−0.991464	+	0.130383i	$0.958379\pi$
$942$	−160793.	−	1.10577e6i	−0.181203	−	1.24613i
$943$	405694.			0.456221
$944$		−	79352.4i		−	0.0890463i
$945$	−265800.	−	574702.i	−0.297640	−	0.643545i
$946$	1.43041e6			1.59837
$947$			641893.i			0.715752i	0.933769	+	0.357876i	$0.116499\pi$
$947$			641893.i			0.715752i	−0.933769	+	0.357876i	$0.883501\pi$
$948$	−298312.	+	43378.2i	−0.331936	+	0.0482675i
$949$	−1.35122e6			−1.50035
$950$			505811.i			0.560456i
$951$	72965.3	+	501783.i	0.0806780	+	0.554823i
$952$	997993.			1.10117
$953$		−	990558.i		−	1.09067i	−0.838218	−	0.545336i	$-0.816402\pi$
$953$		−	990558.i		−	1.09067i	0.838218	−	0.545336i	$-0.183598\pi$
$954$	−208877.	+	62058.7i	−0.229506	+	0.0681877i
$955$	−1.04696e6			−1.14795
$956$			56193.4i			0.0614851i
$957$	31546.4	−	4587.23i	0.0344450	−	0.00500872i
$958$	−249084.			−0.271403
$959$		−	1.70196e6i		−	1.85060i
$960$	100111.	+	688465.i	0.108627	+	0.747032i
$961$	−218862.			−0.236987
$962$		−	1.48811e6i		−	1.60800i
$963$	−279280.	−	940000.i	−0.301153	−	1.01362i
$964$	80519.7			0.0866460
$965$		−	85932.3i		−	0.0922788i
$966$	1.49319e6	−	217128.i	1.60015	−	0.232681i
$967$	−779696.			−0.833820			−0.416910	−	0.908948i	$-0.636887\pi$
$967$	−779696.			−0.833820			−0.416910	+	0.908948i	$0.636887\pi$
$968$		−	872826.i		−	0.931488i
$969$	159346.	+	1.09583e6i	0.169705	+	1.16706i
$970$	−296090.			−0.314688
$971$		−	1.31172e6i		−	1.39124i	−0.718410	−	0.695620i	$-0.755130\pi$
$971$		−	1.31172e6i		−	1.39124i	0.718410	−	0.695620i	$-0.244870\pi$
$972$	−178924.	+	157563.i	−0.189381	+	0.166771i
$973$	−1.21424e6			−1.28257
$974$		−	275937.i		−	0.290865i
$975$	−766289.	+	111428.i	−0.806090	+	0.117215i
$976$	−276263.			−0.290016
$977$		−	125227.i		−	0.131193i	−0.997846	−	0.0655963i	$-0.979105\pi$
$977$		−	125227.i		−	0.131193i	0.997846	−	0.0655963i	$-0.0208949\pi$
$978$	−68352.0	−	470057.i	−0.0714617	−	0.491443i
$979$	603878.			0.630063
$980$			14002.0i			0.0145793i
$981$	−996419.	+	296042.i	−1.03539	+	0.307621i
$982$	−792932.			−0.822267
$983$			1.09083e6i			1.12888i	0.825473	+	0.564442i	$0.190908\pi$
$983$			1.09083e6i			1.12888i	−0.825473	+	0.564442i	$0.809092\pi$
$984$	−263652.	+	38338.2i	−0.272296	+	0.0395951i
$985$	131170.			0.135195
$986$		−	20945.2i		−	0.0215442i
$987$	−48524.1	−	333700.i	−0.0498108	−	0.342549i
$988$	451689.			0.462728
$989$			2.38015e6i			2.43339i
$990$	224093.	+	754250.i	0.228643	+	0.769564i
$991$	1.36511e6			1.39002			0.695010	−	0.719000i	$-0.255400\pi$
$991$	1.36511e6			1.39002			0.695010	+	0.719000i	$0.255400\pi$
$992$		−	422444.i		−	0.429285i
$993$	573449.	−	83386.4i	0.581562	−	0.0845662i
$994$	866250.			0.876740
$995$			812748.i			0.820937i
$996$	−70343.0	−	483750.i	−0.0709092	−	0.487643i
$997$	884601.			0.889933			0.444966	−	0.895547i	$-0.353216\pi$
$997$	884601.			0.889933			0.444966	+	0.895547i	$0.353216\pi$
$998$		−	1.25329e6i		−	1.25832i
$999$	1.10966e6	−	513217.i	1.11188	−	0.514245i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.5.b.a.119.54	yes	78
3.2	odd	2	inner	177.5.b.a.119.25	✓	78

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.5.b.a.119.25	✓	78	3.2	odd	2	inner
177.5.b.a.119.54	yes	78	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:	\( N \)	\(=\)	\( 177 = 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	177.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+3.45868i q^{2} +(-8.90633 + 1.29509i) q^{3} +4.03751 q^{4} +17.0185i q^{5} +(-4.47930 - 30.8042i) q^{6} +51.0370 q^{7} +69.3034i q^{8} +(77.6455 - 23.0690i) q^{9} +O(q^{10})\)	\(q+3.45868i q^{2} +(-8.90633 + 1.29509i) q^{3} +4.03751 q^{4} +17.0185i q^{5} +(-4.47930 - 30.8042i) q^{6} +51.0370 q^{7} +69.3034i q^{8} +(77.6455 - 23.0690i) q^{9} -58.8618 q^{10} -165.031i q^{11} +(-35.9594 + 5.22894i) q^{12} +256.549 q^{13} +176.521i q^{14} +(-22.0405 - 151.573i) q^{15} -175.098 q^{16} -282.155i q^{17} +(79.7883 + 268.551i) q^{18} +436.069 q^{19} +68.7126i q^{20} +(-454.552 + 66.0974i) q^{21} +570.790 q^{22} +949.775i q^{23} +(-89.7540 - 617.239i) q^{24} +335.369 q^{25} +887.323i q^{26} +(-661.660 + 306.018i) q^{27} +206.063 q^{28} -21.4628i q^{29} +(524.242 - 76.2312i) q^{30} -839.439 q^{31} +503.245i q^{32} +(213.730 + 1469.82i) q^{33} +975.885 q^{34} +868.576i q^{35} +(313.495 - 93.1413i) q^{36} -1677.08 q^{37} +1508.22i q^{38} +(-2284.91 + 332.254i) q^{39} -1179.44 q^{40} -427.147i q^{41} +(-228.610 - 1572.15i) q^{42} +2506.01 q^{43} -666.316i q^{44} +(392.600 + 1321.41i) q^{45} -3284.97 q^{46} +734.130i q^{47} +(1559.48 - 226.768i) q^{48} +203.776 q^{49} +1159.94i q^{50} +(365.416 + 2512.97i) q^{51} +1035.82 q^{52} +777.792i q^{53} +(-1058.42 - 2288.47i) q^{54} +2808.59 q^{55} +3537.04i q^{56} +(-3883.77 + 564.747i) q^{57} +74.2329 q^{58} +453.188i q^{59} +(-88.9889 - 611.977i) q^{60} +1577.76 q^{61} -2903.35i q^{62} +(3962.79 - 1177.37i) q^{63} -4542.14 q^{64} +4366.10i q^{65} +(-5083.65 + 739.224i) q^{66} -797.230 q^{67} -1139.21i q^{68} +(-1230.04 - 8459.01i) q^{69} -3004.13 q^{70} -4907.36i q^{71} +(1598.76 + 5381.10i) q^{72} -5266.90 q^{73} -5800.50i q^{74} +(-2986.91 + 434.333i) q^{75} +1760.63 q^{76} -8422.69i q^{77} +(-1149.16 - 7902.79i) q^{78} +8295.79 q^{79} -2979.92i q^{80} +(5496.64 - 3582.40i) q^{81} +1477.37 q^{82} +13452.6i q^{83} +(-1835.26 + 266.869i) q^{84} +4801.87 q^{85} +8667.51i q^{86} +(27.7962 + 191.154i) q^{87} +11437.2 q^{88} +3659.18i q^{89} +(-4570.35 + 1357.88i) q^{90} +13093.5 q^{91} +3834.73i q^{92} +(7476.33 - 1087.15i) q^{93} -2539.12 q^{94} +7421.25i q^{95} +(-651.747 - 4482.07i) q^{96} +5030.26 q^{97} +704.796i q^{98} +(-3807.10 - 12813.9i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 78 q - 612 q^{4} + 64 q^{6} + 76 q^{7} - 100 q^{9}+O(q^{10}) \) 78 * q - 612 * q^4 + 64 * q^6 + 76 * q^7 - 100 * q^9	\( 78 q - 612 q^{4} + 64 q^{6} + 76 q^{7} - 100 q^{9} - 156 q^{10} - 4 q^{13} + 13 q^{15} + 4948 q^{16} + 22 q^{18} + 812 q^{19} - 173 q^{21} - 1644 q^{22} - 678 q^{24} - 8238 q^{25} + 777 q^{27} - 3764 q^{28} + 1374 q^{30} + 4664 q^{31} - 1042 q^{33} + 3244 q^{34} + 3648 q^{36} - 3960 q^{37} - 7078 q^{39} - 1576 q^{40} + 4934 q^{42} - 1492 q^{43} - 2063 q^{45} - 2036 q^{46} - 2620 q^{48} + 24274 q^{49} + 7300 q^{51} + 8408 q^{52} - 14766 q^{54} + 9780 q^{55} + 6939 q^{57} - 3856 q^{58} + 4712 q^{60} - 212 q^{61} - 7438 q^{63} - 45760 q^{64} + 3048 q^{66} - 12972 q^{67} + 21672 q^{69} + 5828 q^{70} - 866 q^{72} - 5240 q^{73} - 20922 q^{75} + 12368 q^{76} - 16508 q^{78} - 14976 q^{79} + 25524 q^{81} - 14484 q^{82} + 9540 q^{84} + 11572 q^{85} + 5695 q^{87} + 62160 q^{88} - 31672 q^{90} + 8284 q^{91} - 9590 q^{93} - 10992 q^{94} + 34102 q^{96} - 55000 q^{97} - 14254 q^{99}+O(q^{100}) \) 78 * q - 612 * q^4 + 64 * q^6 + 76 * q^7 - 100 * q^9 - 156 * q^10 - 4 * q^13 + 13 * q^15 + 4948 * q^16 + 22 * q^18 + 812 * q^19 - 173 * q^21 - 1644 * q^22 - 678 * q^24 - 8238 * q^25 + 777 * q^27 - 3764 * q^28 + 1374 * q^30 + 4664 * q^31 - 1042 * q^33 + 3244 * q^34 + 3648 * q^36 - 3960 * q^37 - 7078 * q^39 - 1576 * q^40 + 4934 * q^42 - 1492 * q^43 - 2063 * q^45 - 2036 * q^46 - 2620 * q^48 + 24274 * q^49 + 7300 * q^51 + 8408 * q^52 - 14766 * q^54 + 9780 * q^55 + 6939 * q^57 - 3856 * q^58 + 4712 * q^60 - 212 * q^61 - 7438 * q^63 - 45760 * q^64 + 3048 * q^66 - 12972 * q^67 + 21672 * q^69 + 5828 * q^70 - 866 * q^72 - 5240 * q^73 - 20922 * q^75 + 12368 * q^76 - 16508 * q^78 - 14976 * q^79 + 25524 * q^81 - 14484 * q^82 + 9540 * q^84 + 11572 * q^85 + 5695 * q^87 + 62160 * q^88 - 31672 * q^90 + 8284 * q^91 - 9590 * q^93 - 10992 * q^94 + 34102 * q^96 - 55000 * q^97 - 14254 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more