LMFDB - Embedded newform 87.5.d.c.86.8

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [87,5,Mod(86,87)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(87, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 5, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("87.86"); S:= CuspForms(chi, 5); N := Newforms(S);

Level:	$ N $	$=$	$ 87 = 3 \cdot 29 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	87.d (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [32,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	no
Analytic conductor:	$8.99318678829$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			86.8
Character	$\chi$	$=$	87.86
Dual form			87.5.d.c.86.7

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-4.29093 q^{2} +(-7.78044 + 4.52380i) q^{3} +2.41206 q^{4} -28.6540i q^{5} +(33.3853 - 19.4113i) q^{6} +23.9440 q^{7} +58.3049 q^{8} +(40.0705 - 70.3943i) q^{9} +122.952i q^{10} -59.0487 q^{11} +(-18.7669 + 10.9117i) q^{12} -28.6931 q^{13} -102.742 q^{14} +(129.625 + 222.941i) q^{15} -288.775 q^{16} -333.946 q^{17} +(-171.940 + 302.057i) q^{18} -184.247i q^{19} -69.1152i q^{20} +(-186.295 + 108.318i) q^{21} +253.374 q^{22} +632.903i q^{23} +(-453.638 + 263.759i) q^{24} -196.051 q^{25} +123.120 q^{26} +(6.68323 + 728.969i) q^{27} +57.7544 q^{28} +(86.2163 + 836.569i) q^{29} +(-556.211 - 956.623i) q^{30} +461.077i q^{31} +306.235 q^{32} +(459.425 - 267.124i) q^{33} +1432.94 q^{34} -686.091i q^{35} +(96.6526 - 169.795i) q^{36} -351.914i q^{37} +790.590i q^{38} +(223.245 - 129.802i) q^{39} -1670.67i q^{40} -2020.92 q^{41} +(799.378 - 464.784i) q^{42} -26.2743i q^{43} -142.429 q^{44} +(-2017.08 - 1148.18i) q^{45} -2715.74i q^{46} +2574.36 q^{47} +(2246.80 - 1306.36i) q^{48} -1827.68 q^{49} +841.243 q^{50} +(2598.25 - 1510.71i) q^{51} -69.2095 q^{52} +4053.17i q^{53} +(-28.6773 - 3127.95i) q^{54} +1691.98i q^{55} +1396.05 q^{56} +(833.496 + 1433.52i) q^{57} +(-369.948 - 3589.66i) q^{58} +986.035i q^{59} +(312.663 + 537.747i) q^{60} +1531.87i q^{61} -1978.45i q^{62} +(959.448 - 1685.52i) q^{63} +3306.37 q^{64} +822.172i q^{65} +(-1971.36 + 1146.21i) q^{66} -5183.12 q^{67} -805.499 q^{68} +(-2863.12 - 4924.26i) q^{69} +2943.97i q^{70} +5463.33i q^{71} +(2336.31 - 4104.33i) q^{72} -2759.31i q^{73} +1510.04i q^{74} +(1525.37 - 886.897i) q^{75} -444.415i q^{76} -1413.86 q^{77} +(-957.928 + 556.970i) q^{78} +11511.4i q^{79} +8274.56i q^{80} +(-3349.71 - 5641.47i) q^{81} +8671.61 q^{82} -3854.87i q^{83} +(-449.355 + 261.269i) q^{84} +9568.90i q^{85} +112.741i q^{86} +(-4455.27 - 6118.85i) q^{87} -3442.82 q^{88} -12181.9 q^{89} +(8655.13 + 4926.76i) q^{90} -687.028 q^{91} +1526.60i q^{92} +(-2085.82 - 3587.38i) q^{93} -11046.4 q^{94} -5279.41 q^{95} +(-2382.64 + 1385.34i) q^{96} +15846.2i q^{97} +7842.46 q^{98} +(-2366.11 + 4156.69i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q + 188 q^{4} - 36 q^{6} - 84 q^{7} - 452 q^{9} - 224 q^{13} - 52 q^{16} - 4216 q^{22} - 832 q^{24} - 7684 q^{25} - 396 q^{28} + 3384 q^{30} - 3308 q^{33} + 9124 q^{34} - 3680 q^{36} + 19764 q^{42} + 44 q^{45}+ \cdots + 13884 q^{96}+O(q^{100}) $ 32 * q + 188 * q^4 - 36 * q^6 - 84 * q^7 - 452 * q^9 - 224 * q^13 - 52 * q^16 - 4216 * q^22 - 832 * q^24 - 7684 * q^25 - 396 * q^28 + 3384 * q^30 - 3308 * q^33 + 9124 * q^34 - 3680 * q^36 + 19764 * q^42 + 44 * q^45 + 4796 * q^49 - 7052 * q^51 + 34048 * q^52 + 3008 * q^54 + 7976 * q^57 + 5936 * q^58 - 22848 * q^63 - 14668 * q^64 - 1872 * q^67 + 15192 * q^78 - 41088 * q^81 - 27028 * q^82 - 35572 * q^87 - 60032 * q^88 - 34272 * q^91 + 38000 * q^93 + 59636 * q^94 + 13884 * q^96

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/87\mathbb{Z}\right)^\times$.

$n$	$31$	$59$
$\chi(n)$	$-1$	$-1$

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 $

$2$	−4.29093			−1.07273			−0.536366	−	0.843986i	$-0.680203\pi$
$2$	−4.29093			−1.07273			−0.536366	+	0.843986i	$0.680203\pi$
$3$	−7.78044	+	4.52380i	−0.864493	+	0.502644i
$3$	−7.78044	+	4.52380i	−0.864493	+	0.502644i
$4$	2.41206			0.150754
$5$		−	28.6540i		−	1.14616i	−0.819499	−	0.573080i	$-0.805748\pi$
$5$		−	28.6540i		−	1.14616i	0.819499	−	0.573080i	$-0.194252\pi$
$6$	33.3853	−	19.4113i	0.927370	−	0.539202i
$7$	23.9440			0.488653			0.244327	−	0.969693i	$-0.421433\pi$
$7$	23.9440			0.488653			0.244327	+	0.969693i	$0.421433\pi$
$8$	58.3049			0.911013
$9$	40.0705	−	70.3943i	0.494698	−	0.869065i
$10$			122.952i			1.22952i
$11$	−59.0487			−0.488005			−0.244003	−	0.969775i	$-0.578461\pi$
$11$	−59.0487			−0.488005			−0.244003	+	0.969775i	$0.578461\pi$
$12$	−18.7669	+	10.9117i	−0.130326	+	0.0757756i
$13$	−28.6931			−0.169782			−0.0848908	−	0.996390i	$-0.527054\pi$
$13$	−28.6931			−0.169782			−0.0848908	+	0.996390i	$0.527054\pi$
$14$	−102.742			−0.524194
$15$	129.625	+	222.941i	0.576111	+	0.990848i
$16$	−288.775			−1.12803
$17$	−333.946			−1.15552			−0.577762	−	0.816205i	$-0.696074\pi$
$17$	−333.946			−1.15552			−0.577762	+	0.816205i	$0.696074\pi$
$18$	−171.940	+	302.057i	−0.530678	+	0.932274i
$19$		−	184.247i		−	0.510379i	−0.966891	−	0.255190i	$-0.917862\pi$
$19$		−	184.247i		−	0.510379i	0.966891	−	0.255190i	$-0.0821379\pi$
$20$		−	69.1152i		−	0.172788i
$21$	−186.295	+	108.318i	−0.422437	+	0.245619i
$22$	253.374			0.523499
$23$			632.903i			1.19641i	0.801342	+	0.598207i	$0.204120\pi$
$23$			632.903i			1.19641i	−0.801342	+	0.598207i	$0.795880\pi$
$24$	−453.638	+	263.759i	−0.787565	+	0.457916i
$25$	−196.051			−0.313682
$26$	123.120			0.182130
$27$	6.68323	+	728.969i	0.00916767	+	0.999958i
$28$	57.7544			0.0736664
$29$	86.2163	+	836.569i	0.102516	+	0.994731i
$29$	86.2163	+	836.569i	0.102516	+	0.994731i
$30$	−556.211	−	956.623i	−0.618012	−	1.06291i
$31$			461.077i			0.479789i	0.970799	+	0.239894i	$0.0771128\pi$
$31$			461.077i			0.479789i	−0.970799	+	0.239894i	$0.922887\pi$
$32$	306.235			0.299057
$33$	459.425	−	267.124i	0.421877	−	0.245293i
$34$	1432.94			1.23957
$35$		−	686.091i		−	0.560075i
$36$	96.6526	−	169.795i	0.0745776	−	0.131015i
$37$		−	351.914i		−	0.257059i	−0.991706	−	0.128529i	$-0.958974\pi$
$37$		−	351.914i		−	0.257059i	0.991706	−	0.128529i	$-0.0410257\pi$
$38$			790.590i			0.547500i
$39$	223.245	−	129.802i	0.146775	−	0.0853398i
$40$		−	1670.67i		−	1.04417i
$41$	−2020.92			−1.20221			−0.601106	−	0.799169i	$-0.705273\pi$
$41$	−2020.92			−1.20221			−0.601106	+	0.799169i	$0.705273\pi$
$42$	799.378	−	464.784i	0.453162	−	0.263483i
$43$		−	26.2743i		−	0.0142100i	−0.999975	−	0.00710501i	$-0.997738\pi$
$43$		−	26.2743i		−	0.0142100i	0.999975	−	0.00710501i	$-0.00226161\pi$
$44$	−142.429			−0.0735687
$45$	−2017.08	−	1148.18i	−0.996088	−	0.567003i
$46$		−	2715.74i		−	1.28343i
$47$	2574.36			1.16540			0.582699	−	0.812688i	$-0.301997\pi$
$47$	2574.36			1.16540			0.582699	+	0.812688i	$0.301997\pi$
$48$	2246.80	−	1306.36i	0.975172	−	0.566996i
$49$	−1827.68			−0.761218
$50$	841.243			0.336497
$51$	2598.25	−	1510.71i	0.998943	−	0.580817i
$52$	−69.2095			−0.0255952
$53$			4053.17i			1.44292i	0.692454	+	0.721462i	$0.256529\pi$
$53$			4053.17i			1.44292i	−0.692454	+	0.721462i	$0.743471\pi$
$54$	−28.6773	−	3127.95i	−0.00983445	−	1.07269i
$55$			1691.98i			0.559332i
$56$	1396.05			0.445170
$57$	833.496	+	1433.52i	0.256539	+	0.441219i
$58$	−369.948	−	3589.66i	−0.109973	−	1.06708i
$59$			986.035i			0.283262i	0.989920	+	0.141631i	$0.0452347\pi$
$59$			986.035i			0.283262i	−0.989920	+	0.141631i	$0.954765\pi$
$60$	312.663	+	537.747i	0.0868509	+	0.149374i
$61$			1531.87i			0.411683i	0.978585	+	0.205841i	$0.0659931\pi$
$61$			1531.87i			0.411683i	−0.978585	+	0.205841i	$0.934007\pi$
$62$		−	1978.45i		−	0.514685i
$63$	959.448	−	1685.52i	0.241736	−	0.424671i
$64$	3306.37			0.807219
$65$			822.172i			0.194597i
$66$	−1971.36	+	1146.21i	−0.452561	+	0.263134i
$67$	−5183.12			−1.15463			−0.577313	−	0.816523i	$-0.695899\pi$
$67$	−5183.12			−1.15463			−0.577313	+	0.816523i	$0.695899\pi$
$68$	−805.499			−0.174200
$69$	−2863.12	−	4924.26i	−0.601370	−	1.03429i
$70$			2943.97i			0.600810i
$71$			5463.33i			1.08378i	0.840450	+	0.541889i	$0.182291\pi$
$71$			5463.33i			1.08378i	−0.840450	+	0.541889i	$0.817709\pi$
$72$	2336.31	−	4104.33i	0.450676	−	0.791730i
$73$		−	2759.31i		−	0.517792i	−0.965905	−	0.258896i	$-0.916641\pi$
$73$		−	2759.31i		−	0.517792i	0.965905	−	0.258896i	$-0.0833586\pi$
$74$			1510.04i			0.275755i
$75$	1525.37	−	886.897i	0.271176	−	0.157671i
$76$		−	444.415i		−	0.0769417i
$77$	−1413.86			−0.238465
$78$	−957.928	+	556.970i	−0.157450	+	0.0915467i
$79$			11511.4i			1.84448i	0.386612	+	0.922242i	$0.373645\pi$
$79$			11511.4i			1.84448i	−0.386612	+	0.922242i	$0.626355\pi$
$80$			8274.56i			1.29290i
$81$	−3349.71	−	5641.47i	−0.510548	−	0.859849i
$82$	8671.61			1.28965
$83$		−	3854.87i		−	0.559569i	−0.960063	−	0.279784i	$-0.909737\pi$
$83$		−	3854.87i		−	0.559569i	0.960063	−	0.279784i	$-0.0902630\pi$
$84$	−449.355	+	261.269i	−0.0636841	+	0.0370280i
$85$			9568.90i			1.32441i
$86$			112.741i			0.0152435i
$87$	−4455.27	−	6118.85i	−0.588621	−	0.808409i
$88$	−3442.82			−0.444580
$89$	−12181.9			−1.53792			−0.768961	−	0.639295i	$-0.779226\pi$
$89$	−12181.9			−1.53792			−0.768961	+	0.639295i	$0.779226\pi$
$90$	8655.13	+	4926.76i	1.06853	+	0.608242i
$91$	−687.028			−0.0829643
$92$			1526.60i			0.180364i
$93$	−2085.82	−	3587.38i	−0.241163	−	0.414774i
$94$	−11046.4			−1.25016
$95$	−5279.41			−0.584976
$96$	−2382.64	+	1385.34i	−0.258533	+	0.150319i
$97$			15846.2i			1.68415i	0.539358	+	0.842076i	$0.318667\pi$
$97$			15846.2i			1.68415i	−0.539358	+	0.842076i	$0.681333\pi$
$98$	7842.46			0.816583
$99$	−2366.11	+	4156.69i	−0.241415	+	0.424109i
$100$	−472.888			−0.0472888
$101$	1249.36			0.122474			0.0612370	−	0.998123i	$-0.480495\pi$
$101$	1249.36			0.122474			0.0612370	+	0.998123i	$0.480495\pi$
$102$	−11148.9	+	6482.33i	−1.07160	+	0.623061i
$103$	13962.3			1.31608			0.658039	−	0.752984i	$-0.271386\pi$
$103$	13962.3			1.31608			0.658039	+	0.752984i	$0.271386\pi$
$104$	−1672.95			−0.154673
$105$	3103.74	+	5338.09i	0.281518	+	0.484181i
$106$		−	17391.9i		−	1.54787i
$107$		−	15955.1i		−	1.39358i	−0.717275	−	0.696790i	$-0.754611\pi$
$107$		−	15955.1i		−	1.39358i	0.717275	−	0.696790i	$-0.245389\pi$
$108$	16.1204	+	1758.32i	0.00138206	+	0.150748i
$109$	544.299			0.0458126			0.0229063	−	0.999738i	$-0.492708\pi$
$109$	544.299			0.0458126			0.0229063	+	0.999738i	$0.492708\pi$
$110$		−	7260.16i		−	0.600014i
$111$	1591.99	+	2738.04i	0.129209	+	0.222226i
$112$	−6914.43			−0.551214
$113$	14094.8			1.10383			0.551914	−	0.833901i	$-0.313898\pi$
$113$	14094.8			1.10383			0.551914	+	0.833901i	$0.313898\pi$
$114$	−3576.47	−	6151.14i	−0.275198	−	0.473310i
$115$	18135.2			1.37128
$116$	207.959	+	2017.86i	0.0154548	+	0.149960i
$117$	−1149.75	+	2019.83i	−0.0839906	+	0.147551i
$118$		−	4231.01i		−	0.303864i
$119$	−7996.01			−0.564650
$120$	7557.76	+	12998.5i	0.524844	+	0.902676i
$121$	−11154.3			−0.761851
$122$		−	6573.15i		−	0.441625i
$123$	15723.6	−	9142.22i	1.03930	−	0.604285i
$124$			1112.15i			0.0723300i
$125$		−	12291.1i		−	0.786630i
$126$	−4116.92	+	7232.45i	−0.259317	+	0.455559i
$127$		−	8801.98i		−	0.545724i	−0.962053	−	0.272862i	$-0.912030\pi$
$127$		−	8801.98i		−	0.545724i	0.962053	−	0.272862i	$-0.0879702\pi$
$128$	−19087.1			−1.16499
$129$	118.860	+	204.426i	0.00714258	+	0.0122845i
$130$		−	3527.88i		−	0.208750i
$131$	10443.3			0.608549			0.304275	−	0.952584i	$-0.401586\pi$
$131$	10443.3			0.608549			0.304275	+	0.952584i	$0.401586\pi$
$132$	1108.16	−	644.320i	0.0635997	−	0.0369789i
$133$		−	4411.61i		−	0.249398i
$134$	22240.4			1.23860
$135$	20887.9	−	191.501i	1.14611	−	0.0105076i
$136$	−19470.7			−1.05270
$137$	−3122.40			−0.166359			−0.0831797	−	0.996535i	$-0.526508\pi$
$137$	−3122.40			−0.166359			−0.0831797	+	0.996535i	$0.526508\pi$
$138$	12285.5	+	21129.7i	0.645109	+	1.10952i
$139$	−9302.10			−0.481450			−0.240725	−	0.970593i	$-0.577385\pi$
$139$	−9302.10			−0.481450			−0.240725	+	0.970593i	$0.577385\pi$
$140$		−	1654.90i		−	0.0844334i
$141$	−20029.7	+	11645.9i	−1.00748	+	0.585780i
$142$		−	23442.7i		−	1.16260i
$143$	1694.29			0.0828544
$144$	−11571.4	+	20328.1i	−0.558032	+	0.980329i
$145$	23971.0	−	2470.44i	1.14012	−	0.117500i
$146$			11840.0i			0.555452i
$147$	14220.2	−	8268.08i	0.658068	−	0.382622i
$148$		−	848.838i		−	0.0387526i
$149$			38141.2i			1.71799i	0.511980	+	0.858997i	$0.328912\pi$
$149$			38141.2i			1.71799i	−0.511980	+	0.858997i	$0.671088\pi$
$150$	−6545.24	+	3805.61i	−0.290900	+	0.169138i
$151$	−15274.1			−0.669889			−0.334945	−	0.942238i	$-0.608718\pi$
$151$	−15274.1			−0.669889			−0.334945	+	0.942238i	$0.608718\pi$
$152$		−	10742.5i		−	0.464962i
$153$	−13381.4	+	23507.9i	−0.571635	+	1.00423i
$154$	6066.78			0.255809
$155$	13211.7			0.549915
$156$	538.481	−	313.090i	0.0221269	−	0.0128653i
$157$		−	22784.1i		−	0.924340i	−0.886791	−	0.462170i	$-0.847071\pi$
$157$		−	22784.1i		−	0.924340i	0.886791	−	0.462170i	$-0.152929\pi$
$158$		−	49394.7i		−	1.97864i
$159$	−18335.7	−	31535.5i	−0.725277	−	1.24740i
$160$		−	8774.85i		−	0.342768i
$161$			15154.2i			0.584631i
$162$	14373.4	+	24207.1i	0.547682	+	0.922388i
$163$		−	41262.0i		−	1.55301i	−0.630110	−	0.776506i	$-0.716990\pi$
$163$		−	41262.0i		−	1.55301i	0.630110	−	0.776506i	$-0.283010\pi$
$164$	−4874.58			−0.181238
$165$	−7654.18	−	13164.4i	−0.281145	−	0.483539i
$166$			16541.0i			0.600267i
$167$		−	30.0039i		−	0.00107583i	−1.00000		0.000537916i	$-0.999829\pi$
$167$		−	30.0039i		−	0.00107583i	1.00000		0.000537916i	$-0.000171224\pi$
$168$	−10861.9	+	6315.46i	−0.384846	+	0.223762i
$169$	−27737.7			−0.971174
$170$		−	41059.4i		−	1.42074i
$171$	−12969.9	−	7382.87i	−0.443553	−	0.252483i
$172$		−	63.3753i		−	0.00214221i
$173$			3628.84i			0.121248i	0.998161	+	0.0606241i	$0.0193091\pi$
$173$			3628.84i			0.121248i	−0.998161	+	0.0606241i	$0.980691\pi$
$174$	19117.2	+	26255.5i	0.631432	+	0.867207i
$175$	−4694.26			−0.153282
$176$	17051.8			0.550483
$177$	−4460.62	−	7671.79i	−0.142380	−	0.244878i
$178$	52271.6			1.64978
$179$			10638.6i			0.332032i	0.986123	+	0.166016i	$0.0530904\pi$
$179$			10638.6i			0.332032i	−0.986123	+	0.166016i	$0.946910\pi$
$180$	−4865.32	−	2769.48i	−0.150164	−	0.0854778i
$181$	40651.0			1.24084			0.620418	−	0.784271i	$-0.286963\pi$
$181$	40651.0			1.24084			0.620418	+	0.784271i	$0.286963\pi$
$182$	2947.99			0.0889985
$183$	−6929.88	−	11918.6i	−0.206930	−	0.355897i
$184$			36901.3i			1.08995i
$185$	−10083.7			−0.294631
$186$	8950.10	+	15393.2i	0.258703	+	0.444942i
$187$	19719.1			0.563902
$188$	6209.53			0.175688
$189$	160.023	+	17454.4i	0.00447981	+	0.488633i
$190$	22653.6			0.627523
$191$	3937.08			0.107921			0.0539606	−	0.998543i	$-0.482815\pi$
$191$	3937.08			0.107921			0.0539606	+	0.998543i	$0.482815\pi$
$192$	−25725.0	+	14957.3i	−0.697835	+	0.405744i
$193$			51041.1i			1.37027i	0.728418	+	0.685133i	$0.240256\pi$
$193$			51041.1i			1.37027i	−0.728418	+	0.685133i	$0.759744\pi$
$194$		−	67994.9i		−	1.80664i
$195$	−3719.34	−	6396.86i	−0.0978130	−	0.168228i
$196$	−4408.49			−0.114757
$197$		−	56908.7i		−	1.46638i	−0.680024	−	0.733190i	$-0.738031\pi$
$197$		−	56908.7i		−	1.46638i	0.680024	−	0.733190i	$-0.261969\pi$
$198$	10152.8	−	17836.0i	0.258974	−	0.454955i
$199$	−54939.8			−1.38733			−0.693666	−	0.720297i	$-0.744006\pi$
$199$	−54939.8			−1.38733			−0.693666	+	0.720297i	$0.744006\pi$
$200$	−11430.8			−0.285769
$201$	40326.9	−	23447.4i	0.998166	−	0.580366i
$202$	−5360.91			−0.131382
$203$	2064.36	+	20030.8i	0.0500950	+	0.486079i
$204$	6267.14	−	3643.92i	0.150594	−	0.0875605i
$205$			57907.4i			1.37793i
$206$	−59911.1			−1.41180
$207$	44552.7	+	25360.7i	1.03976	+	0.591863i
$208$	8285.85			0.191518
$209$			10879.5i			0.249068i
$210$	−13317.9	−	22905.4i	−0.301994	−	0.519396i
$211$		−	9604.93i		−	0.215739i	−0.994165	−	0.107870i	$-0.965597\pi$
$211$		−	9604.93i		−	0.215739i	0.994165	−	0.107870i	$-0.0344029\pi$
$212$			9776.50i			0.217526i
$213$	−24715.0	−	42507.1i	−0.544755	−	0.936919i
$214$			68462.2i			1.49494i
$215$	−752.864			−0.0162869
$216$	389.665	+	42502.5i	0.00835187	+	0.910975i
$217$			11040.0i			0.234450i
$218$	−2335.55			−0.0491446
$219$	12482.6	+	21468.7i	0.260265	+	0.447627i
$220$			4081.16i			0.0843215i
$221$	9581.96			0.196187
$222$	−6831.10	−	11748.7i	−0.138607	−	0.238389i
$223$	−83733.8			−1.68380			−0.841901	−	0.539632i	$-0.818563\pi$
$223$	−83733.8			−1.68380			−0.841901	+	0.539632i	$0.818563\pi$
$224$	7332.49			0.146135
$225$	−7855.88	+	13800.9i	−0.155178	+	0.272610i
$226$	−60479.6			−1.18411
$227$			74816.2i			1.45193i	0.687734	+	0.725963i	$0.258605\pi$
$227$			74816.2i			1.45193i	−0.687734	+	0.725963i	$0.741395\pi$
$228$	2010.44	+	3457.74i	0.0386743	+	0.0665156i
$229$			11165.8i			0.212921i	0.994317	+	0.106460i	$0.0339517\pi$
$229$			11165.8i			0.212921i	−0.994317	+	0.106460i	$0.966048\pi$
$230$	−77816.8			−1.47102
$231$	11000.5	−	6396.02i	0.206152	−	0.119863i
$232$	5026.83	+	48776.0i	0.0933939	+	0.906214i
$233$			59855.5i			1.10253i	0.834329	+	0.551267i	$0.185856\pi$
$233$			59855.5i			1.10253i	−0.834329	+	0.551267i	$0.814144\pi$
$234$	4933.48	−	8666.94i	0.0900994	−	0.158283i
$235$		−	73765.8i		−	1.33573i
$236$			2378.38i			0.0427029i
$237$	−52075.4	−	89564.0i	−0.927120	−	1.59455i
$238$	34310.3			0.605718
$239$			48135.2i			0.842688i	0.906901	+	0.421344i	$0.138441\pi$
$239$			48135.2i			0.842688i	−0.906901	+	0.421344i	$0.861559\pi$
$240$	−37432.4	−	64379.7i	−0.649868	−	1.11770i
$241$	−52874.3			−0.910354			−0.455177	−	0.890401i	$-0.650424\pi$
$241$	−52874.3			−0.910354			−0.455177	+	0.890401i	$0.650424\pi$
$242$	47862.1			0.817262
$243$	51583.1	+	28739.7i	0.873564	+	0.486710i
$244$			3694.97i			0.0620628i
$245$			52370.5i			0.872478i
$246$	−67469.0	+	39228.6i	−1.11489	+	0.648236i
$247$			5286.61i			0.0866530i
$248$			26883.0i			0.437094i
$249$	17438.6	+	29992.6i	0.281264	+	0.483744i
$250$			52740.2i			0.843843i
$251$	−11928.2			−0.189333			−0.0946665	−	0.995509i	$-0.530178\pi$
$251$	−11928.2			−0.189333			−0.0946665	+	0.995509i	$0.530178\pi$
$252$	2314.25	−	4065.58i	0.0364426	−	0.0640209i
$253$		−	37372.1i		−	0.583856i
$254$			37768.7i			0.585415i
$255$	−43287.8	−	74450.2i	−0.665709	−	1.14495i
$256$	28999.7			0.442500
$257$		−	110094.i		−	1.66686i	−0.552625	−	0.833430i	$-0.686374\pi$
$257$		−	110094.i		−	1.66686i	0.552625	−	0.833430i	$-0.313626\pi$
$258$	−510.018	−	877.176i	−0.00766207	−	0.0131779i
$259$		−	8426.22i		−	0.125613i
$260$			1983.13i			0.0293362i
$261$	62344.4	+	27452.6i	0.915201	+	0.402998i
$262$	−44811.5			−0.652810
$263$	98146.3			1.41893			0.709467	−	0.704738i	$-0.248936\pi$
$263$	98146.3			1.41893			0.709467	+	0.704738i	$0.248936\pi$
$264$	26786.7	−	15574.6i	0.384336	−	0.223465i
$265$	116140.			1.65382
$266$			18929.9i			0.267538i
$267$	94780.4	−	55108.4i	1.32952	−	0.773028i
$268$	−12502.0			−0.174064
$269$	−32067.8			−0.443163			−0.221582	−	0.975142i	$-0.571122\pi$
$269$	−32067.8			−0.443163			−0.221582	+	0.975142i	$0.571122\pi$
$270$	−89628.4	+	821.718i	−1.22947	+	0.0112719i
$271$		−	130168.i		−	1.77241i	−0.463289	−	0.886207i	$-0.653331\pi$
$271$		−	130168.i		−	1.77241i	0.463289	−	0.886207i	$-0.346669\pi$
$272$	96435.3			1.30346
$273$	5345.38	−	3107.97i	0.0717221	−	0.0417015i
$274$	13398.0			0.178459
$275$	11576.6			0.153079
$276$	−6906.03	−	11877.6i	−0.0906589	−	0.155923i
$277$	96915.3			1.26309			0.631543	−	0.775341i	$-0.282422\pi$
$277$	96915.3			1.26309			0.631543	+	0.775341i	$0.282422\pi$
$278$	39914.6			0.516467
$279$	32457.2	+	18475.6i	0.416968	+	0.237350i
$280$		−	40002.5i		−	0.510235i
$281$		−	10583.1i		−	0.134030i	−0.997752	−	0.0670149i	$-0.978653\pi$
$281$		−	10583.1i		−	0.134030i	0.997752	−	0.0670149i	$-0.0213475\pi$
$282$	85945.9	−	49971.7i	1.08075	−	0.628385i
$283$	64072.7			0.800018			0.400009	−	0.916511i	$-0.369007\pi$
$283$	64072.7			0.800018			0.400009	+	0.916511i	$0.369007\pi$
$284$			13177.9i			0.163384i
$285$	41076.1	−	23883.0i	0.505708	−	0.294035i
$286$	−7270.07			−0.0888805
$287$	−48388.9			−0.587465
$288$	12271.0	−	21557.2i	0.147943	−	0.259900i
$289$	27999.2			0.335235
$290$	−102858.	+	10600.5i	−1.22304	+	0.126046i
$291$	−71685.0	−	123290.i	−0.846530	−	1.45594i
$292$		−	6655.63i		−	0.0780591i
$293$	−146649.			−1.70822			−0.854109	−	0.520094i	$-0.825897\pi$
$293$	−146649.			−1.70822			−0.854109	+	0.520094i	$0.825897\pi$
$294$	−61017.8	+	35477.7i	−0.705931	+	0.410451i
$295$	28253.8			0.324664
$296$		−	20518.3i		−	0.234184i
$297$	−394.636	−	43044.7i	−0.00447387	−	0.487985i
$298$		−	163661.i		−	1.84295i
$299$		−	18159.9i		−	0.203129i
$300$	3679.28	−	2139.25i	0.0408809	−	0.0237695i
$301$		−	629.112i		−	0.00694377i
$302$	65540.2			0.718611
$303$	−9720.56	+	5651.84i	−0.105878	+	0.0615609i
$304$			53205.9i			0.575722i
$305$	43894.2			0.471854
$306$	57418.6	−	100871.i	0.613211	−	1.07726i
$307$		−	27609.9i		−	0.292946i	−0.989215	−	0.146473i	$-0.953208\pi$
$307$		−	27609.9i		−	0.292946i	0.989215	−	0.146473i	$-0.0467922\pi$
$308$	−3410.32			−0.0359496
$309$	−108633.	+	63162.5i	−1.13774	+	0.661519i
$310$	−56690.5			−0.589911
$311$	129451.			1.33840			0.669200	−	0.743082i	$-0.266637\pi$
$311$	129451.			1.33840			0.669200	+	0.743082i	$0.266637\pi$
$312$	13016.3	−	7568.07i	0.133714	−	0.0777457i
$313$	−43255.8			−0.441525			−0.220763	−	0.975328i	$-0.570855\pi$
$313$	−43255.8			−0.441525			−0.220763	+	0.975328i	$0.570855\pi$
$314$			97764.8i			0.991569i
$315$	−48296.9	−	27492.0i	−0.486741	−	0.277068i
$316$			27766.3i			0.278063i
$317$	−66527.1			−0.662034			−0.331017	−	0.943625i	$-0.607392\pi$
$317$	−66527.1			−0.662034			−0.331017	+	0.943625i	$0.607392\pi$
$318$	78677.3	+	135316.i	0.778028	+	1.33812i
$319$	−5090.96	−	49398.3i	−0.0500286	−	0.485434i
$320$		−	94740.7i		−	0.925202i
$321$	72177.6	+	124138.i	0.700475	+	1.20474i
$322$		−	65025.7i		−	0.627153i
$323$			61528.6i			0.589755i
$324$	−8079.70	−	13607.6i	−0.0769672	−	0.129626i
$325$	5625.32			0.0532575
$326$			177052.i			1.66597i
$327$	−4234.89	+	2462.30i	−0.0396047	+	0.0230274i
$328$	−117829.			−1.09523
$329$	61640.6			0.569475
$330$	32843.5	+	56487.3i	0.301593	+	0.518708i
$331$		−	77219.6i		−	0.704809i	−0.935848	−	0.352404i	$-0.885364\pi$
$331$		−	77219.6i		−	0.704809i	0.935848	−	0.352404i	$-0.114636\pi$
$332$		−	9298.18i		−	0.0843572i
$333$	−24772.7	−	14101.4i	−0.223401	−	0.127166i
$334$			128.745i			0.00115408i
$335$			148517.i			1.32339i
$336$	53797.3	−	31279.5i	0.476521	−	0.277065i
$337$			138891.i			1.22296i	0.791259	+	0.611481i	$0.209426\pi$
$337$			138891.i			1.22296i	−0.791259	+	0.611481i	$0.790574\pi$
$338$	119020.			1.04181
$339$	−109664.	+	63761.9i	−0.954251	+	0.554832i
$340$			23080.8i			0.199661i
$341$		−	27226.0i		−	0.234140i
$342$	55653.0	+	31679.4i	0.475813	+	0.270847i
$343$	−101252.			−0.860625
$344$		−	1531.92i		−	0.0129455i
$345$	−141100.	+	82039.9i	−1.18546	+	0.689266i
$346$		−	15571.1i		−	0.130067i
$347$		−	128832.i		−	1.06995i	−0.844867	−	0.534977i	$-0.820320\pi$
$347$		−	128832.i		−	1.06995i	0.844867	−	0.534977i	$-0.179680\pi$
$348$	−10746.4	−	14759.0i	−0.0887369	−	0.121871i
$349$	−210819.			−1.73085			−0.865425	−	0.501038i	$-0.832952\pi$
$349$	−210819.			−1.73085			−0.865425	+	0.501038i	$0.832952\pi$
$350$	20142.7			0.164430
$351$	−191.763	−	20916.4i	−0.00155650	−	0.169775i
$352$	−18082.7			−0.145942
$353$			63648.9i			0.510789i	0.966837	+	0.255395i	$0.0822053\pi$
$353$			63648.9i			0.510789i	−0.966837	+	0.255395i	$0.917795\pi$
$354$	19140.2	+	32919.1i	0.152736	+	0.262689i
$355$	156546.			1.24218
$356$	−29383.5			−0.231848
$357$	62212.5	−	36172.3i	0.488136	−	0.283818i
$358$		−	45649.7i		−	0.356182i
$359$	−159532.			−1.23783			−0.618913	−	0.785460i	$-0.712426\pi$
$359$	−159532.			−1.23783			−0.618913	+	0.785460i	$0.712426\pi$
$360$	−117605.	−	66944.5i	−0.907449	−	0.516547i
$361$	96374.1			0.739513
$362$	−174431.			−1.33108
$363$	86785.0	−	50459.6i	0.658615	−	0.382940i
$364$	−1657.15			−0.0125072
$365$	−79065.3			−0.593472
$366$	29735.6	+	51142.0i	0.221980	+	0.381782i
$367$			197925.i			1.46950i	0.678339	+	0.734749i	$0.262700\pi$
$367$			197925.i			1.46950i	−0.678339	+	0.734749i	$0.737300\pi$
$368$		−	182766.i		−	1.34959i
$369$	−80979.2	+	142261.i	−0.594731	+	1.04480i
$370$	43268.6			0.316060
$371$			97049.2i			0.705089i
$372$	−5031.13	−	8652.99i	−0.0363563	−	0.0625288i
$373$	−110399.			−0.793498			−0.396749	−	0.917927i	$-0.629862\pi$
$373$	−110399.			−0.793498			−0.396749	+	0.917927i	$0.629862\pi$
$374$	−84613.2			−0.604916
$375$	55602.4	+	95630.1i	0.395395	+	0.680036i
$376$	150098.			1.06169
$377$	−2473.81	−	24003.8i	−0.0174054	−	0.168887i
$378$	−686.649	−	74895.8i	−0.00480564	−	0.524172i
$379$			140164.i			0.975792i	0.872902	+	0.487896i	$0.162236\pi$
$379$			140164.i			0.975792i	−0.872902	+	0.487896i	$0.837764\pi$
$380$	−12734.3			−0.0881874
$381$	39818.4	+	68483.3i	0.274305	+	0.471775i
$382$	−16893.7			−0.115771
$383$			163063.i			1.11163i	0.831307	+	0.555813i	$0.187593\pi$
$383$			163063.i			1.11163i	−0.831307	+	0.555813i	$0.812407\pi$
$384$	148506.	−	86346.4i	1.00712	−	0.585574i
$385$			40512.8i			0.273319i
$386$		−	219014.i		−	1.46993i
$387$	−1849.56	−	1052.83i	−0.0123494	−	0.00702966i
$388$			38222.0i			0.253893i
$389$	−190261.			−1.25733			−0.628666	−	0.777675i	$-0.716399\pi$
$389$	−190261.			−1.25733			−0.628666	+	0.777675i	$0.716399\pi$
$390$	15959.4	+	27448.5i	0.104927	+	0.180463i
$391$		−	211356.i		−	1.38248i
$392$	−106563.			−0.693480
$393$	−81253.6	+	47243.4i	−0.526087	+	0.305884i
$394$			244191.i			1.57303i
$395$	329848.			2.11407
$396$	−5707.20	+	10026.2i	−0.0363943	+	0.0639360i
$397$	101185.			0.642001			0.321000	−	0.947079i	$-0.395981\pi$
$397$	101185.			0.642001			0.321000	+	0.947079i	$0.395981\pi$
$398$	235743.			1.48824
$399$	19957.2	+	34324.3i	0.125359	+	0.215603i
$400$	56614.8			0.353842
$401$		−	266487.i		−	1.65725i	−0.559806	−	0.828624i	$-0.689124\pi$
$401$		−	266487.i		−	1.65725i	0.559806	−	0.828624i	$-0.310876\pi$
$402$	−173040.	+	100611.i	−1.07076	+	0.622577i
$403$		−	13229.7i		−	0.0814594i
$404$	3013.53			0.0184634
$405$	−161651.	+	95982.5i	−0.985524	+	0.585170i
$406$	−8858.04	−	85950.8i	−0.0537385	−	0.521432i
$407$			20780.0i			0.125446i
$408$	151491.	−	88081.5i	0.910050	−	0.529132i
$409$			190411.i			1.13827i	0.822243	+	0.569136i	$0.192722\pi$
$409$			190411.i			1.13827i	−0.822243	+	0.569136i	$0.807278\pi$
$410$		−	248476.i		−	1.47815i
$411$	24293.6	−	14125.1i	0.143817	−	0.0836195i
$412$	33677.9			0.198404
$413$			23609.6i			0.138417i
$414$	−191173.	−	108821.i	−1.11539	−	0.634910i
$415$	−110457.			−0.641355
$416$	−8786.82			−0.0507744
$417$	72374.4	−	42080.8i	0.416210	−	0.241998i
$418$		−	46683.3i		−	0.267183i
$419$		−	109918.i		−	0.626096i	−0.949737	−	0.313048i	$-0.898650\pi$
$419$		−	109918.i		−	0.626096i	0.949737	−	0.313048i	$-0.101350\pi$
$420$	7486.41	+	12875.8i	0.0424400	+	0.0729921i
$421$		−	15136.1i		−	0.0853984i	−0.999088	−	0.0426992i	$-0.986404\pi$
$421$		−	15136.1i		−	0.0853984i	0.999088	−	0.0426992i	$-0.0135957\pi$
$422$			41214.1i			0.231431i
$423$	103156.	−	181220.i	0.576520	−	1.01281i
$424$			236320.i			1.31452i
$425$	65470.7			0.362467
$426$	106050.	+	182395.i	0.584376	+	1.00506i
$427$			36679.1i			0.201170i
$428$		−	38484.7i		−	0.210088i
$429$	−13182.3	+	7664.62i	−0.0716271	+	0.0416463i
$430$	3230.49			0.0174715
$431$		−	59722.0i		−	0.321499i	−0.986995	−	0.160750i	$-0.948609\pi$
$431$		−	59722.0i		−	0.321499i	0.986995	−	0.160750i	$-0.0513911\pi$
$432$	−1929.95	−	210508.i	−0.0103414	−	1.12798i
$433$		−	177385.i		−	0.946111i	−0.881033	−	0.473056i	$-0.843151\pi$
$433$		−	177385.i		−	0.946111i	0.881033	−	0.473056i	$-0.156849\pi$
$434$		−	47372.0i		−	0.251502i
$435$	−175330.	+	127661.i	−0.926566	+	0.674653i
$436$	1312.88			0.00690642
$437$	116610.			0.610625
$438$	−53561.8	−	92120.4i	−0.279194	−	0.480184i
$439$	149110.			0.773711			0.386856	−	0.922140i	$-0.373561\pi$
$439$	149110.			0.773711			0.386856	+	0.922140i	$0.373561\pi$
$440$			98650.7i			0.509559i
$441$	−73236.3	+	128659.i	−0.376573	+	0.661548i
$442$	−41115.5			−0.210456
$443$	−170322.			−0.867886			−0.433943	−	0.900940i	$-0.642878\pi$
$443$	−170322.			−0.867886			−0.433943	+	0.900940i	$0.642878\pi$
$444$	3839.97	+	6604.33i	0.0194788	+	0.0335014i
$445$			349060.i			1.76270i
$446$	359296.			1.80627
$447$	−172543.	−	296755.i	−0.863540	−	1.48520i
$448$	79167.7			0.394450
$449$	−267267.			−1.32572			−0.662862	−	0.748742i	$-0.730658\pi$
$449$	−267267.			−1.32572			−0.662862	+	0.748742i	$0.730658\pi$
$450$	33709.0	−	59218.7i	0.166464	−	0.292438i
$451$	119333.			0.586686
$452$	33997.5			0.166406
$453$	118840.	−	69097.1i	0.579115	−	0.336716i
$454$		−	321031.i		−	1.55753i
$455$			19686.1i			0.0950904i
$456$	48596.9	+	83581.3i	0.233711	+	0.401957i
$457$	−212540.			−1.01767			−0.508837	−	0.860863i	$-0.669924\pi$
$457$	−212540.			−1.01767			−0.508837	+	0.860863i	$0.669924\pi$
$458$		−	47911.5i		−	0.228407i
$459$	−2231.84	−	243437.i	−0.0105935	−	1.15548i
$460$	43743.2			0.206726
$461$	90782.6			0.427170			0.213585	−	0.976924i	$-0.431486\pi$
$461$	90782.6			0.427170			0.213585	+	0.976924i	$0.431486\pi$
$462$	−47202.2	+	27444.9i	−0.221146	+	0.128581i
$463$	−196899.			−0.918504			−0.459252	−	0.888306i	$-0.651882\pi$
$463$	−196899.			−0.918504			−0.459252	+	0.888306i	$0.651882\pi$
$464$	−24897.1	−	241580.i	−0.115641	−	1.12208i
$465$	−102793.	+	59767.1i	−0.475398	+	0.276411i
$466$		−	256836.i		−	1.18272i
$467$	−185037.			−0.848447			−0.424223	−	0.905558i	$-0.639453\pi$
$467$	−185037.			−0.848447			−0.424223	+	0.905558i	$0.639453\pi$
$468$	−2773.26	+	4871.96i	−0.0126619	+	0.0222439i
$469$	−124105.			−0.564212
$470$			316524.i			1.43288i
$471$	103071.	+	177270.i	0.464614	+	0.799086i
$472$			57490.6i			0.258056i
$473$			1551.46i			0.00693456i
$474$	223452.	+	384313.i	0.994551	+	1.71052i
$475$			36121.9i			0.160097i
$476$	−19286.9			−0.0851232
$477$	285320.	+	162413.i	1.25399	+	0.713811i
$478$		−	206544.i		−	0.903978i
$479$	−180523.			−0.786794			−0.393397	−	0.919369i	$-0.628700\pi$
$479$	−180523.			−0.786794			−0.393397	+	0.919369i	$0.628700\pi$
$480$	39695.6	+	68272.2i	0.172290	+	0.296320i
$481$			10097.5i			0.0436439i
$482$	226880.			0.976566
$483$	−68554.7	−	117907.i	−0.293861	−	0.505410i
$484$	−26904.8			−0.114852
$485$	454057.			1.93031
$486$	−221339.	−	123320.i	−0.937100	−	0.522109i
$487$	14658.4			0.0618058			0.0309029	−	0.999522i	$-0.490162\pi$
$487$	14658.4			0.0618058			0.0309029	+	0.999522i	$0.490162\pi$
$488$			89315.6i			0.375049i
$489$	186661.	+	321036.i	0.780613	+	1.34257i
$490$		−	224718.i		−	0.935935i
$491$	−187436.			−0.777480			−0.388740	−	0.921348i	$-0.627090\pi$
$491$	−187436.			−0.777480			−0.388740	+	0.921348i	$0.627090\pi$
$492$	37926.4	−	22051.6i	0.156679	−	0.0910983i
$493$	−28791.6	−	279369.i	−0.118460	−	1.14944i
$494$		−	22684.5i		−	0.0929555i
$495$	119106.	+	67798.5i	0.486096	+	0.276700i
$496$		−	133148.i		−	0.541215i
$497$			130814.i			0.529592i
$498$	−74828.0	−	128696.i	−0.301721	−	0.518927i
$499$	202059.			0.811478			0.405739	−	0.913989i	$-0.367014\pi$
$499$	202059.			0.811478			0.405739	+	0.913989i	$0.367014\pi$
$500$		−	29646.9i		−	0.118587i
$501$	135.732	+	233.444i	0.000540761	+	0.000930051i
$502$	51182.9			0.203103
$503$	140653.			0.555920			0.277960	−	0.960593i	$-0.410342\pi$
$503$	140653.			0.555920			0.277960	+	0.960593i	$0.410342\pi$
$504$	55940.5	−	98274.1i	0.220224	−	0.386881i
$505$		−	35799.1i		−	0.140375i
$506$			160361.i			0.626321i
$507$	215812.	−	125480.i	0.839574	−	0.488155i
$508$		−	21230.9i		−	0.0822700i
$509$		−	144771.i		−	0.558785i	−0.960177	−	0.279393i	$-0.909867\pi$
$509$		−	144771.i		−	0.558785i	0.960177	−	0.279393i	$-0.0901331\pi$
$510$	185745.	+	319461.i	0.714128	+	1.22822i
$511$		−	66069.0i		−	0.253020i
$512$	180959.			0.690303
$513$	134310.	−	1231.36i	0.510358	−	0.00467899i
$514$			472407.i			1.78809i
$515$		−	400075.i		−	1.50844i
$516$	286.697	+	493.088i	0.00107677	+	0.00185193i
$517$	−152013.			−0.568721
$518$			36156.3i			0.134749i
$519$	−16416.1	−	28234.0i	−0.0609447	−	0.104818i
$520$			47936.6i			0.177280i
$521$		−	162019.i		−	0.596885i	−0.954428	−	0.298442i	$-0.903533\pi$
$521$		−	162019.i		−	0.596885i	0.954428	−	0.298442i	$-0.0964671\pi$
$522$	−267515.	−	117797.i	−0.981765	−	0.432309i
$523$	422145.			1.54333			0.771665	−	0.636029i	$-0.219424\pi$
$523$	422145.			1.54333			0.771665	+	0.636029i	$0.219424\pi$
$524$	25189.9			0.0917411
$525$	36523.4	−	21235.9i	0.132511	−	0.0770462i
$526$	−421139.			−1.52214
$527$		−	153975.i		−	0.554407i
$528$	−132670.	+	77138.8i	−0.475889	+	0.276697i
$529$	−120725.			−0.431405
$530$	−498347.			−1.77411
$531$	69411.2	+	39510.9i	0.246173	+	0.140129i
$532$		−	10641.1i		−	0.0375978i
$533$	57986.4			0.204114
$534$	−406696.	+	236466.i	−1.42622	+	0.829252i
$535$	−457177.			−1.59727
$536$	−302201.			−1.05188
$537$	−48127.1	−	82773.3i	−0.166894	−	0.287040i
$538$	137600.			0.475396
$539$	107922.			0.371479
$540$	50382.9	−	461.913i	0.172781	−	0.00158406i
$541$			46793.7i			0.159879i	0.996800	+	0.0799397i	$0.0254728\pi$
$541$			46793.7i			0.159879i	−0.996800	+	0.0799397i	$0.974527\pi$
$542$			558541.i			1.90133i
$543$	−316283.	+	183897.i	−1.07269	+	0.623699i
$544$	−102266.			−0.345568
$545$		−	15596.3i		−	0.0525085i
$546$	−22936.6	+	13336.1i	−0.0769386	+	0.0447346i
$547$	392202.			1.31080			0.655398	−	0.755284i	$-0.272501\pi$
$547$	392202.			1.31080			0.655398	+	0.755284i	$0.272501\pi$
$548$	−7531.42			−0.0250793
$549$	107835.	+	61382.9i	0.357779	+	0.203658i
$550$	−49674.3			−0.164212
$551$	154135.	−	15885.1i	0.507690	−	0.0523223i
$552$	−166934.	−	287108.i	−0.547856	−	0.942254i
$553$			275630.i			0.901313i
$554$	−415857.			−1.35495
$555$	78455.9	−	45616.8i	0.254706	−	0.148094i
$556$	−22437.2			−0.0725805
$557$		−	319490.i		−	1.02979i	−0.857254	−	0.514893i	$-0.827832\pi$
$557$		−	319490.i		−	1.02979i	0.857254	−	0.514893i	$-0.172168\pi$
$558$	−139271.	−	79277.5i	−0.447295	−	0.254613i
$559$			753.891i			0.00241260i
$560$			198126.i			0.631779i
$561$	−153423.	+	89205.1i	−0.487489	+	0.283442i
$562$			45411.4i			0.143778i
$563$	191137.			0.603014			0.301507	−	0.953464i	$-0.402510\pi$
$563$	191137.			0.603014			0.301507	+	0.953464i	$0.402510\pi$
$564$	−48312.9	+	28090.6i	−0.151881	+	0.0883087i
$565$		−	403871.i		−	1.26516i
$566$	−274931.			−0.858205
$567$	−80205.4	−	135079.i	−0.249481	−	0.420168i
$568$			318539.i			0.987337i
$569$	429862.			1.32771			0.663857	−	0.747860i	$-0.268918\pi$
$569$	429862.			1.32771			0.663857	+	0.747860i	$0.268918\pi$
$570$	−176255.	+	102480.i	−0.542489	+	0.315421i
$571$	225577.			0.691866			0.345933	−	0.938259i	$-0.387562\pi$
$571$	225577.			0.691866			0.345933	+	0.938259i	$0.387562\pi$
$572$	4086.73			0.0124906
$573$	−30632.2	+	17810.5i	−0.0932972	+	0.0542460i
$574$	207633.			0.630192
$575$		−	124082.i		−	0.375294i
$576$	132488.	−	232749.i	0.399329	−	0.701526i
$577$			13815.4i			0.0414967i	0.999785	+	0.0207483i	$0.00660487\pi$
$577$			13815.4i			0.0414967i	−0.999785	+	0.0207483i	$0.993395\pi$
$578$	−120142.			−0.359617
$579$	−230899.	−	397122.i	−0.688757	−	1.18459i
$580$	57819.7	−	5958.86i	0.171878	−	0.0177136i
$581$		−	92301.0i		−	0.273435i
$582$	307595.	+	529030.i	0.908099	+	1.56183i
$583$		−	239334.i		−	0.704155i
$584$		−	160881.i		−	0.471715i
$585$	57876.2	+	32944.8i	0.169117	+	0.0962666i
$586$	629259.			1.83246
$587$			632828.i			1.83658i	0.395910	+	0.918289i	$0.370429\pi$
$587$			632828.i			1.83658i	−0.395910	+	0.918289i	$0.629571\pi$
$588$	34300.0	−	19943.1i	0.0992063	−	0.0576817i
$589$	84952.0			0.244874
$590$	−121235.			−0.348277
$591$	257444.	+	442775.i	0.737067	+	1.26768i
$592$			101624.i			0.289969i
$593$		−	370627.i		−	1.05397i	−0.849875	−	0.526984i	$-0.823323\pi$
$593$		−	370627.i		−	1.05397i	0.849875	−	0.526984i	$-0.176677\pi$
$594$	1693.35	+	184702.i	0.00479927	+	0.523477i
$595$			229118.i			0.647179i
$596$			91998.9i			0.258994i
$597$	427455.	−	248536.i	1.19934	−	0.697335i
$598$			77923.0i			0.217903i
$599$	399602.			1.11371			0.556857	−	0.830608i	$-0.312007\pi$
$599$	399602.			1.11371			0.556857	+	0.830608i	$0.312007\pi$
$600$	88936.3	−	51710.4i	0.247045	−	0.143640i
$601$		−	24873.1i		−	0.0688621i	−0.999407	−	0.0344311i	$-0.989038\pi$
$601$		−	24873.1i		−	0.0688621i	0.999407	−	0.0344311i	$-0.0109619\pi$
$602$			2699.48i			0.00744880i
$603$	−207690.	+	364862.i	−0.571191	+	1.00345i
$604$	−36842.2			−0.100988
$605$			319614.i			0.873203i
$606$	41710.2	−	24251.7i	0.113579	−	0.0660383i
$607$		−	219241.i		−	0.595037i	−0.954716	−	0.297518i	$-0.903841\pi$
$607$		−	219241.i		−	0.595037i	0.954716	−	0.297518i	$-0.0961590\pi$
$608$		−	56422.8i		−	0.152633i
$609$	−106677.	−	146510.i	−0.287631	−	0.395032i
$610$	−188347.			−0.506173
$611$	−73866.5			−0.197863
$612$	−32276.8	+	56702.5i	−0.0861762	+	0.151391i
$613$	−224.775			−0.000598174			−0.000299087	−	1.00000i	$-0.500095\pi$
$613$	−224.775			−0.000598174			−0.000299087		1.00000i	$0.500095\pi$
$614$			118472.i			0.314253i
$615$	−261961.	−	450545.i	−0.692607	−	1.19121i
$616$	−82435.0			−0.217245
$617$	−610605.			−1.60395			−0.801973	−	0.597360i	$-0.796216\pi$
$617$	−610605.			−1.60395			−0.801973	+	0.597360i	$0.796216\pi$
$618$	466135.	−	271026.i	1.22049	−	0.709633i
$619$			157766.i			0.411748i	0.978579	+	0.205874i	$0.0660037\pi$
$619$			157766.i			0.411748i	−0.978579	+	0.205874i	$0.933996\pi$
$620$	31867.4			0.0829018
$621$	−461367.	+	4229.84i	−1.19636	+	0.0109683i
$622$	−555467.			−1.43574
$623$	−291683.			−0.751511
$624$	−64467.5	+	37483.5i	−0.165566	+	0.0962656i
$625$	−474721.			−1.21529
$626$	185607.			0.473638
$627$	−49216.8	−	84647.6i	−0.125193	−	0.215318i
$628$		−	54956.6i		−	0.139348i
$629$			117520.i			0.297038i
$630$	207239.	+	117966.i	0.522143	+	0.297219i
$631$	113226.			0.284373			0.142186	−	0.989840i	$-0.454587\pi$
$631$	113226.			0.284373			0.142186	+	0.989840i	$0.454587\pi$
$632$			671172.i			1.68035i
$633$	43450.8	+	74730.6i	0.108440	+	0.186505i
$634$	285463.			0.710185
$635$	−252212.			−0.625487
$636$	−44226.9	−	76065.5i	−0.109338	−	0.188050i
$637$	52441.9			0.129241
$638$	21844.9	+	211964.i	0.0536673	+	0.520741i
$639$	384587.	+	218918.i	0.941874	+	0.536143i
$640$			546923.i			1.33526i
$641$	−734203.			−1.78690			−0.893450	−	0.449162i	$-0.851723\pi$
$641$	−734203.			−1.78690			−0.893450	+	0.449162i	$0.851723\pi$
$642$	−309709.	−	532666.i	−0.751422	−	1.29236i
$643$	455201.			1.10099			0.550493	−	0.834840i	$-0.314440\pi$
$643$	455201.			1.10099			0.550493	+	0.834840i	$0.314440\pi$
$644$			36552.9i			0.0881354i
$645$	5857.61	−	3405.80i	0.0140800	−	0.00818654i
$646$		−	264015.i		−	0.632649i
$647$			796438.i			1.90258i	0.308293	+	0.951291i	$0.400242\pi$
$647$			796438.i			1.90258i	−0.308293	+	0.951291i	$0.599758\pi$
$648$	−195304.	−	328925.i	−0.465117	−	0.783334i
$649$		−	58224.1i		−	0.138233i
$650$	−24137.9			−0.0571310
$651$	−49942.9	−	85896.3i	−0.117845	−	0.202681i
$652$		−	99526.5i		−	0.234123i
$653$	256299.			0.601064			0.300532	−	0.953772i	$-0.402836\pi$
$653$	256299.			0.601064			0.300532	+	0.953772i	$0.402836\pi$
$654$	18171.6	−	10565.5i	0.0424852	−	0.0247022i
$655$		−	299243.i		−	0.697494i
$656$	583591.			1.35613
$657$	−194240.	−	110567.i	−0.449995	−	0.256150i
$658$	−264495.			−0.610894
$659$	286452.			0.659601			0.329801	−	0.944051i	$-0.393018\pi$
$659$	286452.			0.659601			0.329801	+	0.944051i	$0.393018\pi$
$660$	−18462.3	−	31753.2i	−0.0423837	−	0.0728954i
$661$	131899.			0.301884			0.150942	−	0.988543i	$-0.451769\pi$
$661$	131899.			0.301884			0.150942	+	0.988543i	$0.451769\pi$
$662$			331344.i			0.756071i
$663$	−74551.8	+	43346.8i	−0.169602	+	0.0986121i
$664$		−	224758.i		−	0.509775i
$665$	−126410.			−0.285850
$666$	106298.	+	60507.9i	0.239649	+	0.136416i
$667$	−529467.	+	54566.6i	−1.19011	+	0.122652i
$668$		−	72.3713i		−	0.000162186i
$669$	651486.	−	378795.i	1.45564	−	0.846353i
$670$		−	637276.i		−	1.41964i
$671$		−	90455.0i		−	0.200903i
$672$	−57050.0	+	33170.7i	−0.126333	+	0.0734541i
$673$	−713133.			−1.57449			−0.787246	−	0.616639i	$-0.788494\pi$
$673$	−713133.			−1.57449			−0.787246	+	0.616639i	$0.788494\pi$
$674$		−	595970.i		−	1.31191i
$675$	−1310.26	−	142916.i	−0.00287574	−	0.313669i
$676$	−66905.1			−0.146408
$677$	825670.			1.80148			0.900739	−	0.434361i	$-0.143026\pi$
$677$	825670.			1.80148			0.900739	+	0.434361i	$0.143026\pi$
$678$	470558.	−	273598.i	1.02366	−	0.595186i
$679$			379421.i			0.822967i
$680$			557913.i			1.20656i
$681$	−338454.	−	582103.i	−0.729802	−	1.25518i
$682$			116825.i			0.251169i
$683$		−	264646.i		−	0.567315i	−0.958926	−	0.283657i	$-0.908452\pi$
$683$		−	264646.i		−	0.567315i	0.958926	−	0.283657i	$-0.0915478\pi$
$684$	−31284.3	−	17807.9i	−0.0668673	−	0.0380629i
$685$			89469.2i			0.190674i
$686$	434463.			0.923220
$687$	−50511.7	−	86874.6i	−0.107023	−	0.184068i
$688$			7587.36i			0.0160293i
$689$		−	116298.i		−	0.244982i
$690$	605449.	−	352027.i	1.27168	−	0.739398i
$691$	−715631.			−1.49876			−0.749382	−	0.662138i	$-0.769649\pi$
$691$	−715631.			−1.49876			−0.749382	+	0.662138i	$0.769649\pi$
$692$			8752.98i			0.0182786i
$693$	−56654.1	+	99527.7i	−0.117968	+	0.207242i
$694$			552809.i			1.14777i
$695$			266542.i			0.551819i
$696$	−259764.	−	356759.i	−0.536241	−	0.736472i
$697$	674878.			1.38918
$698$	904610.			1.85674
$699$	−270774.	−	465702.i	−0.554183	−	0.953134i
$700$	−11322.8			−0.0231078
$701$		−	30161.6i		−	0.0613787i	−0.999529	−	0.0306894i	$-0.990230\pi$
$701$		−	30161.6i		−	0.0613787i	0.999529	−	0.0306894i	$-0.00977027\pi$
$702$	822.840	+	89750.7i	0.00166971	+	0.182123i
$703$	−64839.0			−0.131198
$704$	−195237.			−0.393927
$705$	333702.	+	573931.i	0.671398	+	1.15473i
$706$		−	273113.i		−	0.547940i
$707$	29914.6			0.0598473
$708$	−10759.3	−	18504.8i	−0.0214643	−	0.0369163i
$709$	415024.			0.825621			0.412811	−	0.910817i	$-0.364547\pi$
$709$	415024.			0.825621			0.412811	+	0.910817i	$0.364547\pi$
$710$	−671728.			−1.33253
$711$	810339.	+	461269.i	1.60298	+	0.912462i
$712$	−710263.			−1.40107
$713$	−291817.			−0.574026
$714$	−266949.	+	155213.i	−0.523640	+	0.304461i
$715$		−	48548.1i		−	0.0949644i
$716$			25661.1i			0.0500551i
$717$	−217754.	−	374513.i	−0.423572	−	0.728498i
$718$	684541.			1.32785
$719$		−	505661.i		−	0.978141i	−0.872244	−	0.489070i	$-0.837336\pi$
$719$		−	505661.i		−	0.978141i	0.872244	−	0.489070i	$-0.162664\pi$
$720$	582481.	+	331566.i	1.12361	+	0.639594i
$721$	334313.			0.643106
$722$	−413534.			−0.793299
$723$	411385.	−	239193.i	0.786995	−	0.457584i
$724$	98052.8			0.187061
$725$	−16902.8	−	164011.i	−0.0321576	−	0.312030i
$726$	−372388.	+	216518.i	−0.706517	+	0.410792i
$727$			697013.i			1.31878i	0.751802	+	0.659389i	$0.229185\pi$
$727$			697013.i			1.31878i	−0.751802	+	0.659389i	$0.770815\pi$
$728$	−40057.1			−0.0755816
$729$	−531352.	+	9743.74i	−0.999832	+	0.0183346i
$730$	339263.			0.636636
$731$			8774.21i			0.0164200i
$732$	−16715.3	−	28748.5i	−0.0311955	−	0.0536529i
$733$		−	880230.i		−	1.63828i	−0.573593	−	0.819141i	$-0.694451\pi$
$733$		−	880230.i		−	1.63828i	0.573593	−	0.819141i	$-0.305549\pi$
$734$		−	849283.i		−	1.57638i
$735$	−236913.	−	407465.i	−0.438546	−	0.754251i
$736$			193817.i			0.357796i
$737$	306056.			0.563464
$738$	347476.	−	610432.i	0.637987	−	1.12079i
$739$			876649.i			1.60523i	0.596499	+	0.802614i	$0.296558\pi$
$739$			876649.i			1.60523i	−0.596499	+	0.802614i	$0.703442\pi$
$740$	−24322.6			−0.0444167
$741$	−23915.6	−	41132.2i	−0.0435556	−	0.0749110i
$742$		−	416431.i		−	0.756372i
$743$	−6796.65			−0.0123117			−0.00615584	−	0.999981i	$-0.501959\pi$
$743$	−6796.65			−0.0123117			−0.00615584	+	0.999981i	$0.501959\pi$
$744$	−121613.	−	209162.i	−0.219703	−	0.377865i
$745$	1.09290e6			1.96910
$746$	473712.			0.851211
$747$	−271361.	−	154467.i	−0.486302	−	0.276817i
$748$	47563.7			0.0850104
$749$		−	382029.i		−	0.680977i
$750$	−238586.	−	410342.i	−0.424153	−	0.729497i
$751$		−	187486.i		−	0.332421i	−0.986090	−	0.166211i	$-0.946847\pi$
$751$		−	187486.i		−	0.332421i	0.986090	−	0.166211i	$-0.0531531\pi$
$752$	−743412.			−1.31460
$753$	92806.4	−	53960.6i	0.163677	−	0.0951671i
$754$	10615.0	+	102998.i	0.0186713	+	0.181171i
$755$			437665.i			0.767800i
$756$	385.986	+	42101.2i	0.000675349	+	0.0736633i
$757$		−	71328.3i		−	0.124472i	−0.998061	−	0.0622358i	$-0.980177\pi$
$757$		−	71328.3i		−	0.124472i	0.998061	−	0.0622358i	$-0.0198231\pi$
$758$		−	601433.i		−	1.04676i
$759$	169064.	+	290771.i	0.293472	+	0.504740i
$760$	−307815.			−0.532921
$761$		−	803433.i		−	1.38733i	−0.720297	−	0.693666i	$-0.755995\pi$
$761$		−	803433.i		−	1.38733i	0.720297	−	0.693666i	$-0.244005\pi$
$762$	−170858.	−	293857.i	−0.294256	−	0.506088i
$763$	13032.7			0.0223864
$764$	9496.47			0.0162696
$765$	673596.	+	383431.i	1.15100	+	0.655185i
$766$		−	699693.i		−	1.19248i
$767$		−	28292.4i		−	0.0480927i
$768$	−225630.	+	131189.i	−0.382538	+	0.222420i
$769$			108583.i			0.183616i	0.995777	+	0.0918078i	$0.0292645\pi$
$769$			108583.i			0.183616i	−0.995777	+	0.0918078i	$0.970735\pi$
$770$		−	173837.i		−	0.293199i
$771$	498045.	+	856583.i	0.837838	+	1.44099i
$772$			123114.i			0.206573i
$773$	161657.			0.270542			0.135271	−	0.990809i	$-0.456809\pi$
$773$	161657.			0.270542			0.135271	+	0.990809i	$0.456809\pi$
$774$	7936.33	+	4517.60i	0.0132476	+	0.00754094i
$775$		−	90394.9i		−	0.150501i
$776$			923910.i			1.53429i
$777$	38118.5	+	65559.7i	0.0631385	+	0.108591i
$778$	816395.			1.34878
$779$			372348.i			0.613584i
$780$	−8971.28	−	15429.6i	−0.0147457	−	0.0253610i
$781$		−	322602.i		−	0.528890i
$782$			906911.i			1.48303i
$783$	−609257.	+	68440.1i	−0.993750	+	0.111632i
$784$	527790.			0.858675
$785$	−652855.			−1.05944
$786$	348653.	−	202718.i	0.564350	−	0.328131i
$787$	142643.			0.230304			0.115152	−	0.993348i	$-0.463264\pi$
$787$	142643.			0.230304			0.115152	+	0.993348i	$0.463264\pi$
$788$		−	137267.i		−	0.221062i
$789$	−763621.	+	443994.i	−1.22666	+	0.713219i
$790$	−1.41536e6			−2.26784
$791$	337485.			0.539389
$792$	−137956.	+	242355.i	−0.219932	+	0.386369i
$793$		−	43954.1i		−	0.0698962i
$794$	−434178.			−0.688695
$795$	−903617.	+	525392.i	−1.42972	+	0.831284i
$796$	−132518.			−0.209146
$797$	1.11465e6			1.75477			0.877387	−	0.479784i	$-0.159285\pi$
$797$	1.11465e6			1.75477			0.877387	+	0.479784i	$0.159285\pi$
$798$	−85635.0	−	147283.i	−0.134476	−	0.231285i
$799$	−859699.			−1.34664
$800$	−60037.8			−0.0938090
$801$	−488134.	+	857535.i	−0.760807	+	1.33655i
$802$			1.14348e6i			1.77778i
$803$			162934.i			0.252685i
$804$	97271.0	−	56556.5i	0.150477	−	0.0874924i
$805$	434229.			0.670081
$806$			56767.8i			0.0873841i
$807$	249501.	−	145068.i	0.383112	−	0.222754i
$808$	72843.7			0.111576
$809$	−126139.			−0.192731			−0.0963657	−	0.995346i	$-0.530722\pi$
$809$	−126139.			−0.192731			−0.0963657	+	0.995346i	$0.530722\pi$
$810$	693631.	−	411854.i	1.05720	−	0.627731i
$811$	−705180.			−1.07216			−0.536078	−	0.844168i	$-0.680095\pi$
$811$	−705180.			−1.07216			−0.536078	+	0.844168i	$0.680095\pi$
$812$	4979.38	+	48315.6i	0.00755201	+	0.0732782i
$813$	588853.	+	1.01276e6i	0.890894	+	1.53224i
$814$		−	89165.6i		−	0.134570i
$815$	−1.18232e6			−1.78000
$816$	−750309.	+	436254.i	−1.12683	+	0.655178i
$817$	−4840.96			−0.00725249
$818$		−	817042.i		−	1.22106i
$819$	−27529.5	+	48362.8i	−0.0410423	+	0.0721014i
$820$			139676.i			0.207728i
$821$			25388.7i			0.0376664i	0.999823	+	0.0188332i	$0.00599516\pi$
$821$			25388.7i			0.0376664i	−0.999823	+	0.0188332i	$0.994005\pi$
$822$	−104242.	+	60609.8i	−0.154277	+	0.0897014i
$823$			1.33438e6i			1.97006i	0.172370	+	0.985032i	$0.444857\pi$
$823$			1.33438e6i			1.97006i	−0.172370	+	0.985032i	$0.555143\pi$
$824$	814069.			1.19897
$825$	−90070.9	+	52370.1i	−0.132336	+	0.0769441i
$826$		−	101307.i		−	0.148484i
$827$	−647656.			−0.946964			−0.473482	−	0.880803i	$-0.657003\pi$
$827$	−647656.			−0.946964			−0.473482	+	0.880803i	$0.657003\pi$
$828$	107464.	+	61171.7i	0.156748	+	0.0892256i
$829$			598022.i			0.870178i	0.900387	+	0.435089i	$0.143283\pi$
$829$			598022.i			0.870178i	−0.900387	+	0.435089i	$0.856717\pi$
$830$	473965.			0.688002
$831$	−754044.	+	438425.i	−1.09193	+	0.634883i
$832$	−94869.9			−0.137051
$833$	610349.			0.879606
$834$	−310553.	+	180566.i	−0.446482	+	0.259599i
$835$	−859.732			−0.00123308
$836$			26242.1i			0.0375479i
$837$	−336111.	+	3081.49i	−0.479769	+	0.00439855i
$838$			471650.i			0.671633i
$839$	−1.01661e6			−1.44421			−0.722106	−	0.691783i	$-0.756826\pi$
$839$	−1.01661e6			−1.44421			−0.722106	+	0.691783i	$0.756826\pi$
$840$	180963.	+	311237.i	0.256467	+	0.441095i
$841$	−692414.	+	144252.i	−0.978981	+	0.203953i
$842$			64947.9i			0.0916096i
$843$	47875.9	+	82341.4i	0.0673693	+	0.115868i
$844$		−	23167.7i		−	0.0325236i
$845$			794796.i			1.11312i
$846$	−442635.	+	777604.i	−0.618451	+	1.08647i
$847$	−267078.			−0.372281
$848$		−	1.17045e6i		−	1.62766i
$849$	−498514.	+	289852.i	−0.691611	+	0.402125i
$850$	−280930.			−0.388830
$851$	222727.			0.307549
$852$	−59614.1	−	102530.i	−0.0821239	−	0.141244i
$853$		−	1.14072e6i		−	1.56777i	−0.620907	−	0.783885i	$-0.713235\pi$
$853$		−	1.14072e6i		−	1.56777i	0.620907	−	0.783885i	$-0.286765\pi$
$854$		−	157388.i		−	0.215802i
$855$	−211549.	+	371640.i	−0.289386	+	0.508382i
$856$		−	930260.i		−	1.26957i
$857$		−	983521.i		−	1.33913i	−0.742755	−	0.669564i	$-0.766481\pi$
$857$		−	983521.i		−	1.33913i	0.742755	−	0.669564i	$-0.233519\pi$
$858$	56564.4	−	32888.3i	0.0768366	−	0.0446753i
$859$		−	476850.i		−	0.646242i	−0.946358	−	0.323121i	$-0.895268\pi$
$859$		−	476850.i		−	0.646242i	0.946358	−	0.323121i	$-0.104732\pi$
$860$	−1815.95			−0.00245532
$861$	376487.	−	218901.i	0.507859	−	0.295286i
$862$			256263.i			0.344882i
$863$			804061.i			1.07961i	0.841790	+	0.539806i	$0.181502\pi$
$863$			804061.i			1.07961i	−0.841790	+	0.539806i	$0.818498\pi$
$864$	2046.64	+	223236.i	0.00274166	+	0.299045i
$865$	103981.			0.138970
$866$			761148.i			1.01492i
$867$	−217846.	+	126663.i	−0.289808	+	0.168504i
$868$			26629.2i			0.0353443i
$869$		−	679735.i		−	0.900119i
$870$	752326.	−	547785.i	0.993957	−	0.723722i
$871$	148720.			0.196034
$872$	31735.3			0.0417359
$873$	1.11548e6	+	634965.i	1.46364	+	0.833146i
$874$	−500367.			−0.655037
$875$		−	294298.i		−	0.384389i
$876$	30108.7	+	51783.7i	0.0392359	+	0.0674816i
$877$	9236.02			0.0120084			0.00600421	−	0.999982i	$-0.498089\pi$
$877$	9236.02			0.0120084			0.00600421	+	0.999982i	$0.498089\pi$
$878$	−639822.			−0.829985
$879$	1.14099e6	−	663409.i	1.47674	−	0.858626i
$880$		−	488601.i		−	0.630942i
$881$	137677.			0.177382			0.0886911	−	0.996059i	$-0.471732\pi$
$881$	137677.			0.177382			0.0886911	+	0.996059i	$0.471732\pi$
$882$	314252.	−	552065.i	0.403962	−	0.709664i
$883$	−92524.8			−0.118669			−0.0593344	−	0.998238i	$-0.518898\pi$
$883$	−92524.8			−0.118669			−0.0593344	+	0.998238i	$0.518898\pi$
$884$	23112.3			0.0295759
$885$	−219827.	+	127815.i	−0.280670	+	0.163190i
$886$	730838.			0.931009
$887$	−430783.			−0.547534			−0.273767	−	0.961796i	$-0.588270\pi$
$887$	−430783.			−0.547534			−0.273767	+	0.961796i	$0.588270\pi$
$888$	92820.5	+	159641.i	0.117711	+	0.202451i
$889$		−	210755.i		−	0.266670i
$890$		−	1.49779e6i		−	1.89091i
$891$	197796.	+	333121.i	0.249150	+	0.419611i
$892$	−201971.			−0.253840
$893$		−	474319.i		−	0.594795i
$894$	740370.	+	1.27336e6i	0.926347	+	1.59322i
$895$	304840.			0.380562
$896$	−457023.			−0.569274
$897$	82151.9	+	141292.i	0.102102	+	0.175604i
$898$	1.14682e6			1.42215
$899$	−385723.	+	39752.4i	−0.477261	+	0.0491863i
$900$	−18948.9	+	33288.6i	−0.0233937	+	0.0410971i
$901$		−	1.35354e6i		−	1.66733i
$902$	−512047.			−0.629357
$903$	2845.98	+	4894.77i	0.00349024	+	0.00600284i
$904$	821793.			1.00560
$905$		−	1.16481e6i		−	1.42220i
$906$	−509932.	+	296491.i	−0.621235	+	0.361206i
$907$		−	168211.i		−	0.204475i	−0.994760	−	0.102238i	$-0.967400\pi$
$907$		−	168211.i		−	0.204475i	0.994760	−	0.102238i	$-0.0326002\pi$
$908$			180461.i			0.218883i
$909$	50062.4	−	87947.7i	0.0605876	−	0.106438i
$910$		−	84471.6i		−	0.102006i
$911$	205670.			0.247819			0.123909	−	0.992294i	$-0.460457\pi$
$911$	205670.			0.247819			0.123909	+	0.992294i	$0.460457\pi$
$912$	−240693.	−	413965.i	−0.289383	−	0.497708i
$913$			227625.i			0.273073i
$914$	911994.			1.09169
$915$	−341517.	+	198569.i	−0.407915	+	0.237175i
$916$			26932.5i			0.0320986i
$917$	250055.			0.297369
$918$	9576.67	+	1.04457e6i	0.0113639	+	1.23952i
$919$	−980095.			−1.16048			−0.580239	−	0.814446i	$-0.697041\pi$
$919$	−980095.			−1.16048			−0.580239	+	0.814446i	$0.697041\pi$
$920$	1.05737e6			1.24926
$921$	124902.	+	214817.i	0.147248	+	0.253250i
$922$	−389541.			−0.458239
$923$		−	156760.i		−	0.184006i
$924$	26533.8	−	15427.6i	0.0310782	−	0.0180699i
$925$			68993.2i			0.0806349i
$926$	844878.			0.985308
$927$	559476.	−	982865.i	0.651061	−	1.14376i
$928$	26402.4	+	256186.i	0.0306583	+	0.297482i
$929$			373925.i			0.433265i	0.976253	+	0.216633i	$0.0695073\pi$
$929$			373925.i			0.433265i	−0.976253	+	0.216633i	$0.930493\pi$
$930$	441077.	−	256456.i	0.509974	−	0.296515i
$931$			336745.i			0.388510i
$932$			144375.i			0.166211i
$933$	−1.00719e6	+	585612.i	−1.15704	+	0.672739i
$934$	793980.			0.910156
$935$		−	565031.i		−	0.646322i
$936$	−67035.8	+	117766.i	−0.0765166	+	0.134421i
$937$	1.17888e6			1.34273			0.671365	−	0.741126i	$-0.265708\pi$
$937$	1.17888e6			1.34273			0.671365	+	0.741126i	$0.265708\pi$
$938$	532524.			0.605248
$939$	336549.	−	195680.i	0.381696	−	0.221930i
$940$		−	177928.i		−	0.201367i
$941$		−	683086.i		−	0.771430i	−0.922618	−	0.385715i	$-0.873955\pi$
$941$		−	683086.i		−	0.771430i	0.922618	−	0.385715i	$-0.126045\pi$
$942$	−442268.	−	760653.i	−0.498407	−	0.857205i
$943$		−	1.27904e6i		−	1.43834i
$944$		−	284742.i		−	0.319527i
$945$	500140.	−	4585.31i	0.560051	−	0.00513458i
$946$		−	6657.22i		−	0.00743893i
$947$	104514.			0.116540			0.0582700	−	0.998301i	$-0.481442\pi$
$947$	104514.			0.116540			0.0582700	+	0.998301i	$0.481442\pi$
$948$	−125609.	−	216034.i	−0.139767	−	0.240384i
$949$			79173.2i			0.0879115i
$950$		−	154996.i		−	0.171741i
$951$	517610.	−	300955.i	0.572324	−	0.332768i
$952$	−466206.			−0.514404
$953$		−	213724.i		−	0.235325i	−0.993054	−	0.117663i	$-0.962460\pi$
$953$		−	213724.i		−	0.235325i	0.993054	−	0.117663i	$-0.0375401\pi$
$954$	−1.22429e6	−	696901.i	−1.34520	−	0.765728i
$955$		−	112813.i		−	0.123695i
$956$			116105.i			0.127038i
$957$	263078.	+	361310.i	0.287250	+	0.394508i
$958$	774610.			0.844019
$959$	−74762.7			−0.0812920
$960$	428588.	+	737124.i	0.465047	+	0.799831i
$961$	710929.			0.769803
$962$		−	43327.6i		−	0.0468182i
$963$	−1.12315e6	−	639329.i	−1.21111	−	0.689401i
$964$	−127536.			−0.137239
$965$	1.46253e6			1.57054
$966$	294163.	+	505929.i	0.315235	+	0.542169i
$967$			1.36465e6i			1.45938i	0.683777	+	0.729691i	$0.260336\pi$
$967$			1.36465e6i			1.45938i	−0.683777	+	0.729691i	$0.739664\pi$
$968$	−650347.			−0.694056
$969$	−278343.	−	478719.i	−0.296437	−	0.509840i
$970$	−1.94832e6			−2.07070
$971$	−1.18833e6			−1.26037			−0.630187	−	0.776443i	$-0.717022\pi$
$971$	−1.18833e6			−1.26037			−0.630187	+	0.776443i	$0.717022\pi$
$972$	124422.	+	69322.0i	0.131693	+	0.0733734i
$973$	−222729.			−0.235262
$974$	−62898.2			−0.0663011
$975$	−43767.5	+	25447.8i	−0.0460408	+	0.0267696i
$976$		−	442366.i		−	0.464389i
$977$			1.70862e6i			1.79001i	0.446054	+	0.895006i	$0.352829\pi$
$977$			1.70862e6i			1.79001i	−0.446054	+	0.895006i	$0.647171\pi$
$978$	−800948.	−	1.37754e6i	−0.837388	−	1.44022i
$979$	719324.			0.750515
$980$			126321.i			0.131529i
$981$	21810.3	−	38315.5i	0.0226634	−	0.0398141i
$982$	804273.			0.834027
$983$	1.09990e6			1.13828			0.569139	−	0.822242i	$-0.307277\pi$
$983$	1.09990e6			1.13828			0.569139	+	0.822242i	$0.307277\pi$
$984$	916764.	−	533036.i	0.946820	−	0.550512i
$985$	−1.63066e6			−1.68071
$986$	123543.	+	1.19875e6i	0.127076	+	1.23304i
$987$	−479591.	+	278850.i	−0.492308	+	0.286243i
$988$			12751.6i			0.0130633i
$989$	16629.1			0.0170010
$990$	−511074.	−	290918.i	−0.521451	−	0.296825i
$991$	575548.			0.586050			0.293025	−	0.956105i	$-0.405338\pi$
$991$	575548.			0.586050			0.293025	+	0.956105i	$0.405338\pi$
$992$			141198.i			0.143484i
$993$	349326.	+	600802.i	0.354268	+	0.609303i
$994$		−	561313.i		−	0.568110i
$995$			1.57424e6i			1.59010i
$996$	42063.1	+	72344.0i	0.0424016	+	0.0729262i
$997$		−	1.13198e6i		−	1.13881i	−0.822058	−	0.569403i	$-0.807174\pi$
$997$		−	1.13198e6i		−	1.13881i	0.822058	−	0.569403i	$-0.192826\pi$
$998$	−867019.			−0.870498
$999$	256534.	−	2351.92i	0.257048	−	0.00235663i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	87.5.d.c.86.8	yes	32
3.2	odd	2	inner	87.5.d.c.86.26	yes	32
29.28	even	2	inner	87.5.d.c.86.25	yes	32
87.86	odd	2	inner	87.5.d.c.86.7	✓	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
87.5.d.c.86.7	✓	32	87.86	odd	2	inner
87.5.d.c.86.8	yes	32	1.1	even	1	trivial
87.5.d.c.86.25	yes	32	29.28	even	2	inner
87.5.d.c.86.26	yes	32	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 87 = 3 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	87.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-4.29093 q^{2} +(-7.78044 + 4.52380i) q^{3} +2.41206 q^{4} -28.6540i q^{5} +(33.3853 - 19.4113i) q^{6} +23.9440 q^{7} +58.3049 q^{8} +(40.0705 - 70.3943i) q^{9} +122.952i q^{10} -59.0487 q^{11} +(-18.7669 + 10.9117i) q^{12} -28.6931 q^{13} -102.742 q^{14} +(129.625 + 222.941i) q^{15} -288.775 q^{16} -333.946 q^{17} +(-171.940 + 302.057i) q^{18} -184.247i q^{19} -69.1152i q^{20} +(-186.295 + 108.318i) q^{21} +253.374 q^{22} +632.903i q^{23} +(-453.638 + 263.759i) q^{24} -196.051 q^{25} +123.120 q^{26} +(6.68323 + 728.969i) q^{27} +57.7544 q^{28} +(86.2163 + 836.569i) q^{29} +(-556.211 - 956.623i) q^{30} +461.077i q^{31} +306.235 q^{32} +(459.425 - 267.124i) q^{33} +1432.94 q^{34} -686.091i q^{35} +(96.6526 - 169.795i) q^{36} -351.914i q^{37} +790.590i q^{38} +(223.245 - 129.802i) q^{39} -1670.67i q^{40} -2020.92 q^{41} +(799.378 - 464.784i) q^{42} -26.2743i q^{43} -142.429 q^{44} +(-2017.08 - 1148.18i) q^{45} -2715.74i q^{46} +2574.36 q^{47} +(2246.80 - 1306.36i) q^{48} -1827.68 q^{49} +841.243 q^{50} +(2598.25 - 1510.71i) q^{51} -69.2095 q^{52} +4053.17i q^{53} +(-28.6773 - 3127.95i) q^{54} +1691.98i q^{55} +1396.05 q^{56} +(833.496 + 1433.52i) q^{57} +(-369.948 - 3589.66i) q^{58} +986.035i q^{59} +(312.663 + 537.747i) q^{60} +1531.87i q^{61} -1978.45i q^{62} +(959.448 - 1685.52i) q^{63} +3306.37 q^{64} +822.172i q^{65} +(-1971.36 + 1146.21i) q^{66} -5183.12 q^{67} -805.499 q^{68} +(-2863.12 - 4924.26i) q^{69} +2943.97i q^{70} +5463.33i q^{71} +(2336.31 - 4104.33i) q^{72} -2759.31i q^{73} +1510.04i q^{74} +(1525.37 - 886.897i) q^{75} -444.415i q^{76} -1413.86 q^{77} +(-957.928 + 556.970i) q^{78} +11511.4i q^{79} +8274.56i q^{80} +(-3349.71 - 5641.47i) q^{81} +8671.61 q^{82} -3854.87i q^{83} +(-449.355 + 261.269i) q^{84} +9568.90i q^{85} +112.741i q^{86} +(-4455.27 - 6118.85i) q^{87} -3442.82 q^{88} -12181.9 q^{89} +(8655.13 + 4926.76i) q^{90} -687.028 q^{91} +1526.60i q^{92} +(-2085.82 - 3587.38i) q^{93} -11046.4 q^{94} -5279.41 q^{95} +(-2382.64 + 1385.34i) q^{96} +15846.2i q^{97} +7842.46 q^{98} +(-2366.11 + 4156.69i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 188 q^{4} - 36 q^{6} - 84 q^{7} - 452 q^{9} - 224 q^{13} - 52 q^{16} - 4216 q^{22} - 832 q^{24} - 7684 q^{25} - 396 q^{28} + 3384 q^{30} - 3308 q^{33} + 9124 q^{34} - 3680 q^{36} + 19764 q^{42} + 44 q^{45}+ \cdots + 13884 q^{96}+O(q^{100}) \) 32 * q + 188 * q^4 - 36 * q^6 - 84 * q^7 - 452 * q^9 - 224 * q^13 - 52 * q^16 - 4216 * q^22 - 832 * q^24 - 7684 * q^25 - 396 * q^28 + 3384 * q^30 - 3308 * q^33 + 9124 * q^34 - 3680 * q^36 + 19764 * q^42 + 44 * q^45 + 4796 * q^49 - 7052 * q^51 + 34048 * q^52 + 3008 * q^54 + 7976 * q^57 + 5936 * q^58 - 22848 * q^63 - 14668 * q^64 - 1872 * q^67 + 15192 * q^78 - 41088 * q^81 - 27028 * q^82 - 35572 * q^87 - 60032 * q^88 - 34272 * q^91 + 38000 * q^93 + 59636 * q^94 + 13884 * q^96

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more