LMFDB - Embedded newform 810.4.e.bf.541.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [810,4,Mod(271,810)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("810.271"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(810, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([2, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 810 = 2 \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	810.e (of order $3$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,4,0,-8,10,0,26,-32,0,40,-36] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$47.7915471046$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\zeta_{12})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			541.1
Root			$-0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	810.541
Dual form			810.4.e.bf.271.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(1.00000 - 1.73205i) q^{2} +(-2.00000 - 3.46410i) q^{4} +(2.50000 + 4.33013i) q^{5} +(5.63397 - 9.75833i) q^{7} -8.00000 q^{8} +10.0000 q^{10} +(0.526279 - 0.911543i) q^{11} +(6.73205 + 11.6603i) q^{13} +(-11.2679 - 19.5167i) q^{14} +(-8.00000 + 13.8564i) q^{16} +136.799 q^{17} -46.7128 q^{19} +(10.0000 - 17.3205i) q^{20} +(-1.05256 - 1.82309i) q^{22} +(-10.9474 - 18.9615i) q^{23} +(-12.5000 + 21.6506i) q^{25} +26.9282 q^{26} -45.0718 q^{28} +(-54.1532 + 93.7961i) q^{29} +(28.5526 + 49.4545i) q^{31} +(16.0000 + 27.7128i) q^{32} +(136.799 - 236.942i) q^{34} +56.3397 q^{35} +223.181 q^{37} +(-46.7128 + 80.9090i) q^{38} +(-20.0000 - 34.6410i) q^{40} +(-34.4212 - 59.6192i) q^{41} +(177.380 - 307.231i) q^{43} -4.21024 q^{44} -43.7898 q^{46} +(241.322 - 417.983i) q^{47} +(108.017 + 187.090i) q^{49} +(25.0000 + 43.3013i) q^{50} +(26.9282 - 46.6410i) q^{52} -110.445 q^{53} +5.26279 q^{55} +(-45.0718 + 78.0666i) q^{56} +(108.306 + 187.592i) q^{58} +(-328.903 - 569.677i) q^{59} +(19.2539 - 33.3487i) q^{61} +114.210 q^{62} +64.0000 q^{64} +(-33.6603 + 58.3013i) q^{65} +(339.004 + 587.172i) q^{67} +(-273.597 - 473.885i) q^{68} +(56.3397 - 97.5833i) q^{70} +572.065 q^{71} +107.458 q^{73} +(223.181 - 386.560i) q^{74} +(93.4256 + 161.818i) q^{76} +(-5.93009 - 10.2712i) q^{77} +(51.8653 - 89.8334i) q^{79} -80.0000 q^{80} -137.685 q^{82} +(472.804 - 818.921i) q^{83} +(341.997 + 592.356i) q^{85} +(-354.760 - 614.463i) q^{86} +(-4.21024 + 7.29234i) q^{88} +577.235 q^{89} +151.713 q^{91} +(-43.7898 + 75.8461i) q^{92} +(-482.645 - 835.965i) q^{94} +(-116.782 - 202.272i) q^{95} +(593.643 - 1028.22i) q^{97} +432.067 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{2} - 8 q^{4} + 10 q^{5} + 26 q^{7} - 32 q^{8} + 40 q^{10} - 36 q^{11} + 20 q^{13} - 52 q^{14} - 32 q^{16} + 180 q^{17} - 76 q^{19} + 40 q^{20} + 72 q^{22} - 120 q^{23} - 50 q^{25} + 80 q^{26}+ \cdots + 1368 q^{98}+O(q^{100}) $ 4 * q + 4 * q^2 - 8 * q^4 + 10 * q^5 + 26 * q^7 - 32 * q^8 + 40 * q^10 - 36 * q^11 + 20 * q^13 - 52 * q^14 - 32 * q^16 + 180 * q^17 - 76 * q^19 + 40 * q^20 + 72 * q^22 - 120 * q^23 - 50 * q^25 + 80 * q^26 - 208 * q^28 - 324 * q^29 + 38 * q^31 + 64 * q^32 + 180 * q^34 + 260 * q^35 - 292 * q^37 - 76 * q^38 - 80 * q^40 - 252 * q^41 + 422 * q^43 + 288 * q^44 - 480 * q^46 + 366 * q^47 + 342 * q^49 + 100 * q^50 + 80 * q^52 - 324 * q^53 - 360 * q^55 - 208 * q^56 + 648 * q^58 - 72 * q^59 + 368 * q^61 + 152 * q^62 + 256 * q^64 - 100 * q^65 + 296 * q^67 - 360 * q^68 + 260 * q^70 + 480 * q^71 - 1420 * q^73 - 292 * q^74 + 152 * q^76 + 534 * q^77 - 520 * q^79 - 320 * q^80 - 1008 * q^82 - 66 * q^83 + 450 * q^85 - 844 * q^86 + 288 * q^88 + 2496 * q^89 + 496 * q^91 - 480 * q^92 - 732 * q^94 - 190 * q^95 + 1706 * q^97 + 1368 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/810\mathbb{Z}\right)^\times$.

$n$	$487$	$731$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$

Coefficient data

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

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

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

$2$	1.00000	−	1.73205i	0.353553	−	0.612372i
$2$	1.00000	−	1.73205i	0.353553	−	0.612372i
$3$		0			0
$3$		0			0
$4$	−2.00000	−	3.46410i	−0.250000	−	0.433013i
$5$	2.50000	+	4.33013i	0.223607	+	0.387298i
$5$	2.50000	+	4.33013i	0.223607	+	0.387298i
$6$		0			0
$7$	5.63397	−	9.75833i	0.304206	−	0.526900i	−0.672878	−	0.739753i	$-0.734942\pi$
$7$	5.63397	−	9.75833i	0.304206	−	0.526900i	0.977084	+	0.212853i	$0.0682755\pi$
$8$	−8.00000			−0.353553
$9$		0			0
$10$	10.0000			0.316228
$11$	0.526279	−	0.911543i	0.0144254	−	0.0249855i	−0.858723	−	0.512441i	$-0.828741\pi$
$11$	0.526279	−	0.911543i	0.0144254	−	0.0249855i	0.873148	+	0.487455i	$0.162075\pi$
$12$		0			0
$13$	6.73205	+	11.6603i	0.143626	+	0.248767i	0.928859	−	0.370432i	$-0.120791\pi$
$13$	6.73205	+	11.6603i	0.143626	+	0.248767i	−0.785234	+	0.619200i	$0.787457\pi$
$14$	−11.2679	−	19.5167i	−0.215106	−	0.372575i
$15$		0			0
$16$	−8.00000	+	13.8564i	−0.125000	+	0.216506i
$17$	136.799			1.95168			0.975840	−	0.218487i	$-0.0701121\pi$
$17$	136.799			1.95168			0.975840	+	0.218487i	$0.0701121\pi$
$18$		0			0
$19$	−46.7128			−0.564034			−0.282017	−	0.959409i	$-0.591004\pi$
$19$	−46.7128			−0.564034			−0.282017	+	0.959409i	$0.591004\pi$
$20$	10.0000	−	17.3205i	0.111803	−	0.193649i
$21$		0			0
$22$	−1.05256	−	1.82309i	−0.0102003	−	0.0176674i
$23$	−10.9474	−	18.9615i	−0.0992478	−	0.171902i	0.812126	−	0.583482i	$-0.198310\pi$
$23$	−10.9474	−	18.9615i	−0.0992478	−	0.171902i	−0.911374	+	0.411580i	$0.864977\pi$
$24$		0			0
$25$	−12.5000	+	21.6506i	−0.100000	+	0.173205i
$26$	26.9282			0.203118
$27$		0			0
$28$	−45.0718			−0.304206
$29$	−54.1532	+	93.7961i	−0.346759	+	0.600603i	−0.985672	−	0.168675i	$-0.946051\pi$
$29$	−54.1532	+	93.7961i	−0.346759	+	0.600603i	0.638913	+	0.769279i	$0.279384\pi$
$30$		0			0
$31$	28.5526	+	49.4545i	0.165426	+	0.286525i	0.936806	−	0.349848i	$-0.113767\pi$
$31$	28.5526	+	49.4545i	0.165426	+	0.286525i	−0.771381	+	0.636374i	$0.780434\pi$
$32$	16.0000	+	27.7128i	0.0883883	+	0.153093i
$33$		0			0
$34$	136.799	−	236.942i	0.690023	−	1.19515i
$35$	56.3397			0.272090
$36$		0			0
$37$	223.181			0.991640			0.495820	−	0.868425i	$-0.334868\pi$
$37$	223.181			0.991640			0.495820	+	0.868425i	$0.334868\pi$
$38$	−46.7128	+	80.9090i	−0.199416	+	0.345399i
$39$		0			0
$40$	−20.0000	−	34.6410i	−0.0790569	−	0.136931i
$41$	−34.4212	−	59.6192i	−0.131114	−	0.227096i	0.792992	−	0.609232i	$-0.208522\pi$
$41$	−34.4212	−	59.6192i	−0.131114	−	0.227096i	−0.924106	+	0.382135i	$0.875189\pi$
$42$		0			0
$43$	177.380	−	307.231i	0.629075	−	1.08959i	−0.358663	−	0.933467i	$-0.616767\pi$
$43$	177.380	−	307.231i	0.629075	−	1.08959i	0.987738	−	0.156122i	$-0.0498994\pi$
$44$	−4.21024			−0.0144254
$45$		0			0
$46$	−43.7898			−0.140358
$47$	241.322	−	417.983i	0.748947	−	1.29721i	−0.199381	−	0.979922i	$-0.563893\pi$
$47$	241.322	−	417.983i	0.748947	−	1.29721i	0.948328	−	0.317292i	$-0.102773\pi$
$48$		0			0
$49$	108.017	+	187.090i	0.314917	+	0.545453i
$50$	25.0000	+	43.3013i	0.0707107	+	0.122474i
$51$		0			0
$52$	26.9282	−	46.6410i	0.0718129	−	0.124384i
$53$	−110.445			−0.286241			−0.143120	−	0.989705i	$-0.545714\pi$
$53$	−110.445			−0.286241			−0.143120	+	0.989705i	$0.545714\pi$
$54$		0			0
$55$	5.26279			0.0129025
$56$	−45.0718	+	78.0666i	−0.107553	+	0.186287i
$57$		0			0
$58$	108.306	+	187.592i	0.245195	+	0.424691i
$59$	−328.903	−	569.677i	−0.725755	−	1.25704i	−0.958663	−	0.284545i	$-0.908157\pi$
$59$	−328.903	−	569.677i	−0.725755	−	1.25704i	0.232908	−	0.972499i	$-0.425176\pi$
$60$		0			0
$61$	19.2539	−	33.3487i	0.0404132	−	0.0699977i	−0.845111	−	0.534590i	$-0.820466\pi$
$61$	19.2539	−	33.3487i	0.0404132	−	0.0699977i	0.885525	+	0.464593i	$0.153799\pi$
$62$	114.210			0.233947
$63$		0			0
$64$	64.0000			0.125000
$65$	−33.6603	+	58.3013i	−0.0642314	+	0.111252i
$66$		0			0
$67$	339.004	+	587.172i	0.618148	+	1.07066i	0.989823	+	0.142301i	$0.0454500\pi$
$67$	339.004	+	587.172i	0.618148	+	1.07066i	−0.371676	+	0.928363i	$0.621217\pi$
$68$	−273.597	−	473.885i	−0.487920	−	0.845102i
$69$		0			0
$70$	56.3397	−	97.5833i	0.0961984	−	0.166621i
$71$	572.065			0.956221			0.478110	−	0.878300i	$-0.341322\pi$
$71$	572.065			0.956221			0.478110	+	0.878300i	$0.341322\pi$
$72$		0			0
$73$	107.458			0.172287			0.0861435	−	0.996283i	$-0.472546\pi$
$73$	107.458			0.172287			0.0861435	+	0.996283i	$0.472546\pi$
$74$	223.181	−	386.560i	0.350598	−	0.607253i
$75$		0			0
$76$	93.4256	+	161.818i	0.141009	+	0.244234i
$77$	−5.93009	−	10.2712i	−0.00877658	−	0.0152015i
$78$		0			0
$79$	51.8653	−	89.8334i	0.0738646	−	0.127937i	−0.826727	−	0.562603i	$-0.809800\pi$
$79$	51.8653	−	89.8334i	0.0738646	−	0.127937i	0.900592	+	0.434665i	$0.143133\pi$
$80$	−80.0000			−0.111803
$81$		0			0
$82$	−137.685			−0.185423
$83$	472.804	−	818.921i	0.625265	−	1.08299i	−0.363224	−	0.931702i	$-0.618324\pi$
$83$	472.804	−	818.921i	0.625265	−	1.08299i	0.988489	−	0.151290i	$-0.0483426\pi$
$84$		0			0
$85$	341.997	+	592.356i	0.436409	+	0.755882i
$86$	−354.760	−	614.463i	−0.444823	−	0.770456i
$87$		0			0
$88$	−4.21024	+	7.29234i	−0.00510014	+	0.00883371i
$89$	577.235			0.687492			0.343746	−	0.939063i	$-0.388304\pi$
$89$	577.235			0.687492			0.343746	+	0.939063i	$0.388304\pi$
$90$		0			0
$91$	151.713			0.174767
$92$	−43.7898	+	75.8461i	−0.0496239	+	0.0859511i
$93$		0			0
$94$	−482.645	−	835.965i	−0.529585	−	0.917268i
$95$	−116.782	−	202.272i	−0.126122	−	0.218450i
$96$		0			0
$97$	593.643	−	1028.22i	0.621395	−	1.07629i	−0.367831	−	0.929893i	$-0.619900\pi$
$97$	593.643	−	1028.22i	0.621395	−	1.07629i	0.989226	−	0.146395i	$-0.0467670\pi$
$98$	432.067			0.445360
$99$		0			0
$100$	100.000			0.100000
$101$	644.418	−	1116.17i	0.634872	−	1.09963i	−0.351671	−	0.936124i	$-0.614386\pi$
$101$	644.418	−	1116.17i	0.634872	−	1.09963i	0.986542	−	0.163506i	$-0.0522803\pi$
$102$		0			0
$103$	−109.904	−	190.359i	−0.105137	−	0.182103i	0.808657	−	0.588280i	$-0.200195\pi$
$103$	−109.904	−	190.359i	−0.105137	−	0.182103i	−0.913794	+	0.406177i	$0.866862\pi$
$104$	−53.8564	−	93.2820i	−0.0507794	−	0.0879525i
$105$		0			0
$106$	−110.445	+	191.296i	−0.101201	+	0.175286i
$107$	−917.782			−0.829209			−0.414604	−	0.910002i	$-0.636080\pi$
$107$	−917.782			−0.829209			−0.414604	+	0.910002i	$0.636080\pi$
$108$		0			0
$109$	470.882			0.413783			0.206891	−	0.978364i	$-0.433665\pi$
$109$	470.882			0.413783			0.206891	+	0.978364i	$0.433665\pi$
$110$	5.26279	−	9.11543i	0.00456171	−	0.00790111i
$111$		0			0
$112$	90.1436	+	156.133i	0.0760515	+	0.131725i
$113$	−243.934	−	422.506i	−0.203074	−	0.351735i	0.746443	−	0.665449i	$-0.231760\pi$
$113$	−243.934	−	422.506i	−0.203074	−	0.351735i	−0.949517	+	0.313714i	$0.898427\pi$
$114$		0			0
$115$	54.7372	−	94.8076i	0.0443850	−	0.0768770i
$116$	433.226			0.346759
$117$		0			0
$118$	−1315.61			−1.02637
$119$	770.720	−	1334.93i	0.593713	−	1.02834i
$120$		0			0
$121$	664.946	+	1151.72i	0.499584	+	0.865305i
$122$	−38.5077	−	66.6973i	−0.0285764	−	0.0494959i
$123$		0			0
$124$	114.210	−	197.818i	0.0827128	−	0.143263i
$125$	−125.000			−0.0894427
$126$		0			0
$127$	−1158.20			−0.809239			−0.404619	−	0.914485i	$-0.632596\pi$
$127$	−1158.20			−0.809239			−0.404619	+	0.914485i	$0.632596\pi$
$128$	64.0000	−	110.851i	0.0441942	−	0.0765466i
$129$		0			0
$130$	67.3205	+	116.603i	0.0454185	+	0.0786671i
$131$	−543.480	−	941.335i	−0.362474	−	0.627823i	0.625894	−	0.779908i	$-0.284734\pi$
$131$	−543.480	−	941.335i	−0.362474	−	0.627823i	−0.988367	+	0.152086i	$0.951401\pi$
$132$		0			0
$133$	−263.179	+	455.839i	−0.171583	+	0.297190i
$134$	1356.02			0.874193
$135$		0			0
$136$	−1094.39			−0.690023
$137$	708.042	−	1226.37i	0.441548	−	0.764784i	−0.556256	−	0.831011i	$-0.687763\pi$
$137$	708.042	−	1226.37i	0.441548	−	0.764784i	0.997805	+	0.0662266i	$0.0210960\pi$
$138$		0			0
$139$	265.792	+	460.366i	0.162189	+	0.280919i	0.935653	−	0.352920i	$-0.114811\pi$
$139$	265.792	+	460.366i	0.162189	+	0.280919i	−0.773465	+	0.633839i	$0.781478\pi$
$140$	−112.679	−	195.167i	−0.0680225	−	0.117818i
$141$		0			0
$142$	572.065	−	990.846i	0.338075	−	0.585563i
$143$	14.1718			0.00828743
$144$		0			0
$145$	−541.532			−0.310150
$146$	107.458	−	186.122i	0.0609127	−	0.105504i
$147$		0			0
$148$	−446.361	−	773.121i	−0.247910	−	0.429393i
$149$	−865.892	−	1499.77i	−0.476085	−	0.824603i	0.523540	−	0.852001i	$-0.324611\pi$
$149$	−865.892	−	1499.77i	−0.476085	−	0.824603i	−0.999625	+	0.0273981i	$0.991278\pi$
$150$		0			0
$151$	−277.592	+	480.804i	−0.149603	+	0.259121i	−0.931081	−	0.364813i	$-0.881133\pi$
$151$	−277.592	+	480.804i	−0.149603	+	0.259121i	0.781477	+	0.623933i	$0.214466\pi$
$152$	373.703			0.199416
$153$		0			0
$154$	−23.7204			−0.0124120
$155$	−142.763	+	247.272i	−0.0739805	+	0.128138i
$156$		0			0
$157$	743.649	+	1288.04i	0.378023	+	0.654756i	0.990775	−	0.135520i	$-0.0432705\pi$
$157$	743.649	+	1288.04i	0.378023	+	0.654756i	−0.612751	+	0.790276i	$0.709937\pi$
$158$	−103.731	−	179.667i	−0.0522302	−	0.0904653i
$159$		0			0
$160$	−80.0000	+	138.564i	−0.0395285	+	0.0684653i
$161$	−246.710			−0.120767
$162$		0			0
$163$	−1501.78			−0.721646			−0.360823	−	0.932634i	$-0.617504\pi$
$163$	−1501.78			−0.721646			−0.360823	+	0.932634i	$0.617504\pi$
$164$	−137.685	+	238.477i	−0.0655571	+	0.113548i
$165$		0			0
$166$	−945.609	−	1637.84i	−0.442129	−	0.765790i
$167$	1704.08	+	2951.56i	0.789615	+	1.36765i	0.926203	+	0.377026i	$0.123053\pi$
$167$	1704.08	+	2951.56i	0.789615	+	1.36765i	−0.136587	+	0.990628i	$0.543613\pi$
$168$		0			0
$169$	1007.86	−	1745.66i	0.458743	−	0.794567i
$170$	1367.99			0.617175
$171$		0			0
$172$	−1419.04			−0.629075
$173$	1212.52	−	2100.15i	0.532870	−	0.922958i	−0.466393	−	0.884578i	$-0.654447\pi$
$173$	1212.52	−	2100.15i	0.532870	−	0.922958i	0.999263	−	0.0383804i	$-0.0122199\pi$
$174$		0			0
$175$	140.849	+	243.958i	0.0608412	+	0.105380i
$176$	8.42047	+	14.5847i	0.00360635	+	0.00624637i
$177$		0			0
$178$	577.235	−	999.800i	0.243065	−	0.421001i
$179$	−4124.59			−1.72227			−0.861134	−	0.508378i	$-0.830245\pi$
$179$	−4124.59			−1.72227			−0.861134	+	0.508378i	$0.830245\pi$
$180$		0			0
$181$	−4053.64			−1.66467			−0.832334	−	0.554274i	$-0.812996\pi$
$181$	−4053.64			−1.66467			−0.832334	+	0.554274i	$0.812996\pi$
$182$	151.713	−	262.774i	0.0617896	−	0.107023i
$183$		0			0
$184$	87.5795	+	151.692i	0.0350894	+	0.0607766i
$185$	557.952	+	966.401i	0.221737	+	0.384061i
$186$		0			0
$187$	71.9943	−	124.698i	0.0281537	−	0.0487637i
$188$	−1930.58			−0.748947
$189$		0			0
$190$	−467.128			−0.178363
$191$	739.828	−	1281.42i	0.280272	−	0.485446i	−0.691179	−	0.722683i	$-0.742909\pi$
$191$	739.828	−	1281.42i	0.280272	−	0.485446i	0.971452	+	0.237237i	$0.0762418\pi$
$192$		0			0
$193$	1183.84	+	2050.48i	0.441528	+	0.764749i	0.997803	−	0.0662490i	$-0.0211032\pi$
$193$	1183.84	+	2050.48i	0.441528	+	0.764749i	−0.556275	+	0.830998i	$0.687770\pi$
$194$	−1187.29	−	2056.44i	−0.439393	−	0.761050i
$195$		0			0
$196$	432.067	−	748.361i	0.157459	−	0.272726i
$197$	3603.59			1.30327			0.651637	−	0.758531i	$-0.274082\pi$
$197$	3603.59			1.30327			0.651637	+	0.758531i	$0.274082\pi$
$198$		0			0
$199$	1842.45			0.656321			0.328160	−	0.944622i	$-0.393571\pi$
$199$	1842.45			0.656321			0.328160	+	0.944622i	$0.393571\pi$
$200$	100.000	−	173.205i	0.0353553	−	0.0612372i
$201$		0			0
$202$	−1288.84	−	2232.33i	−0.448922	−	0.777556i
$203$	610.196	+	1056.89i	0.210972	+	0.365414i
$204$		0			0
$205$	172.106	−	298.096i	0.0586361	−	0.101561i
$206$	−439.615			−0.148687
$207$		0			0
$208$	−215.426			−0.0718129
$209$	−24.5840	+	42.5807i	−0.00813641	+	0.0140927i
$210$		0			0
$211$	1604.90	+	2779.77i	0.523630	+	0.906954i	0.999622	+	0.0275040i	$0.00875590\pi$
$211$	1604.90	+	2779.77i	0.523630	+	0.906954i	−0.475992	+	0.879450i	$0.657911\pi$
$212$	220.890	+	382.592i	0.0715602	+	0.123946i
$213$		0			0
$214$	−917.782	+	1589.65i	−0.293170	+	0.507785i
$215$	1773.80			0.562662
$216$		0			0
$217$	643.458			0.201294
$218$	470.882	−	815.591i	0.146294	−	0.253389i
$219$		0			0
$220$	−10.5256	−	18.2309i	−0.00322561	−	0.00558693i
$221$	920.936	+	1595.11i	0.280312	+	0.485514i
$222$		0			0
$223$	−3189.34	+	5524.10i	−0.957732	+	1.65884i	−0.229743	+	0.973251i	$0.573789\pi$
$223$	−3189.34	+	5524.10i	−0.957732	+	1.65884i	−0.727989	+	0.685589i	$0.759545\pi$
$224$	360.574			0.107553
$225$		0			0
$226$	−975.735			−0.287190
$227$	535.429	−	927.390i	0.156554	−	0.271159i	−0.777070	−	0.629414i	$-0.783295\pi$
$227$	535.429	−	927.390i	0.156554	−	0.271159i	0.933624	+	0.358255i	$0.116628\pi$
$228$		0			0
$229$	1278.95	+	2215.20i	0.369062	+	0.639235i	0.989419	−	0.145085i	$-0.0463456\pi$
$229$	1278.95	+	2215.20i	0.369062	+	0.639235i	−0.620357	+	0.784320i	$0.713012\pi$
$230$	−109.474	−	189.615i	−0.0313849	−	0.0543603i
$231$		0			0
$232$	433.226	−	750.369i	0.122598	−	0.212345i
$233$	−2532.68			−0.712109			−0.356055	−	0.934465i	$-0.615878\pi$
$233$	−2532.68			−0.712109			−0.356055	+	0.934465i	$0.615878\pi$
$234$		0			0
$235$	2413.22			0.669878
$236$	−1315.61	+	2278.71i	−0.362877	+	0.628522i
$237$		0			0
$238$	−1541.44	−	2669.85i	−0.419818	−	0.727147i
$239$	−576.245	−	998.085i	−0.155959	−	0.270129i	0.777449	−	0.628946i	$-0.216513\pi$
$239$	−576.245	−	998.085i	−0.155959	−	0.270129i	−0.933408	+	0.358817i	$0.883180\pi$
$240$		0			0
$241$	−3472.89	+	6015.21i	−0.928250	+	1.60778i	−0.142000	+	0.989867i	$0.545353\pi$
$241$	−3472.89	+	6015.21i	−0.928250	+	1.60778i	−0.786249	+	0.617909i	$0.787980\pi$
$242$	2659.78			0.706518
$243$		0			0
$244$	−154.031			−0.0404132
$245$	−540.083	+	935.452i	−0.140835	+	0.243934i
$246$		0			0
$247$	−314.473	−	544.683i	−0.0810099	−	0.140313i
$248$	−228.420	−	395.636i	−0.0584868	−	0.101302i
$249$		0			0
$250$	−125.000	+	216.506i	−0.0316228	+	0.0547723i
$251$	−3096.59			−0.778704			−0.389352	−	0.921089i	$-0.627301\pi$
$251$	−3096.59			−0.778704			−0.389352	+	0.921089i	$0.627301\pi$
$252$		0			0
$253$	−23.0457			−0.00572675
$254$	−1158.20	+	2006.05i	−0.286109	+	0.495555i
$255$		0			0
$256$	−128.000	−	221.703i	−0.0312500	−	0.0541266i
$257$	1075.59	+	1862.98i	0.261064	+	0.452176i	0.966525	−	0.256573i	$-0.0825932\pi$
$257$	1075.59	+	1862.98i	0.261064	+	0.452176i	−0.705461	+	0.708749i	$0.749260\pi$
$258$		0			0
$259$	1257.39	−	2177.87i	0.301663	−	0.522495i
$260$	269.282			0.0642314
$261$		0			0
$262$	−2173.92			−0.512615
$263$	1029.92	−	1783.87i	0.241473	−	0.418243i	−0.719661	−	0.694326i	$-0.755703\pi$
$263$	1029.92	−	1783.87i	0.241473	−	0.418243i	0.961134	+	0.276082i	$0.0890361\pi$
$264$		0			0
$265$	−276.112	−	478.240i	−0.0640054	−	0.110861i
$266$	526.358	+	911.678i	0.121327	+	0.210145i
$267$		0			0
$268$	1356.02	−	2348.69i	0.309074	−	0.535332i
$269$	−2163.57			−0.490390			−0.245195	−	0.969474i	$-0.578852\pi$
$269$	−2163.57			−0.490390			−0.245195	+	0.969474i	$0.578852\pi$
$270$		0			0
$271$	6995.32			1.56803			0.784014	−	0.620743i	$-0.213169\pi$
$271$	6995.32			1.56803			0.784014	+	0.620743i	$0.213169\pi$
$272$	−1094.39	+	1895.54i	−0.243960	+	0.422551i
$273$		0			0
$274$	−1416.08	−	2452.73i	−0.312222	−	0.540784i
$275$	13.1570	+	22.7886i	0.00288508	+	0.00499710i
$276$		0			0
$277$	919.839	−	1593.21i	0.199523	−	0.345583i	−0.748851	−	0.662738i	$-0.769394\pi$
$277$	919.839	−	1593.21i	0.199523	−	0.345583i	0.948374	+	0.317155i	$0.102728\pi$
$278$	1063.17			0.229369
$279$		0			0
$280$	−450.718			−0.0961984
$281$	−64.9832	+	112.554i	−0.0137956	+	0.0238947i	−0.872841	−	0.488005i	$-0.837725\pi$
$281$	−64.9832	+	112.554i	−0.0137956	+	0.0238947i	0.859045	+	0.511900i	$0.171058\pi$
$282$		0			0
$283$	−2193.86	−	3799.88i	−0.460818	−	0.798161i	0.538184	−	0.842828i	$-0.319111\pi$
$283$	−2193.86	−	3799.88i	−0.460818	−	0.798161i	−0.999002	+	0.0446668i	$0.985777\pi$
$284$	−1144.13	−	1981.69i	−0.239055	−	0.414056i
$285$		0			0
$286$	14.1718	−	24.5462i	0.00293005	−	0.00507499i
$287$	−775.712			−0.159543
$288$		0			0
$289$	13800.9			2.80905
$290$	−541.532	+	937.961i	−0.109655	+	0.189928i
$291$		0			0
$292$	−214.915	−	372.244i	−0.0430718	−	0.0746025i
$293$	2390.20	+	4139.95i	0.476577	+	0.825456i	0.999640	−	0.0268385i	$-0.00854399\pi$
$293$	2390.20	+	4139.95i	0.476577	+	0.825456i	−0.523063	+	0.852294i	$0.675211\pi$
$294$		0			0
$295$	1644.52	−	2848.38i	0.324567	−	0.562167i
$296$	−1785.45			−0.350598
$297$		0			0
$298$	−3463.57			−0.673286
$299$	147.397	−	255.300i	0.0285091	−	0.0493792i
$300$		0			0
$301$	−1998.71	−	3461.87i	−0.382737	−	0.662919i
$302$	555.184	+	961.607i	0.105786	+	0.183226i
$303$		0			0
$304$	373.703	−	647.272i	0.0705043	−	0.122117i
$305$	192.539			0.0361467
$306$		0			0
$307$	−6471.01			−1.20300			−0.601499	−	0.798874i	$-0.705429\pi$
$307$	−6471.01			−1.20300			−0.601499	+	0.798874i	$0.705429\pi$
$308$	−23.7204	+	41.0849i	−0.00438829	+	0.00760074i
$309$		0			0
$310$	285.526	+	494.545i	0.0523121	+	0.0906073i
$311$	−975.580	−	1689.75i	−0.177878	−	0.308094i	0.763276	−	0.646073i	$-0.223590\pi$
$311$	−975.580	−	1689.75i	−0.177878	−	0.308094i	−0.941154	+	0.337979i	$0.890257\pi$
$312$		0			0
$313$	−2673.20	+	4630.11i	−0.482741	+	0.836132i	−0.999804	−	0.0198153i	$-0.993692\pi$
$313$	−2673.20	+	4630.11i	−0.482741	+	0.836132i	0.517062	+	0.855948i	$0.327026\pi$
$314$	2974.60			0.534606
$315$		0			0
$316$	−414.923			−0.0738646
$317$	−4873.07	+	8440.40i	−0.863403	+	1.49546i	0.00522201	+	0.999986i	$0.498338\pi$
$317$	−4873.07	+	8440.40i	−0.863403	+	1.49546i	−0.868625	+	0.495471i	$0.834996\pi$
$318$		0			0
$319$	56.9994	+	98.7259i	0.0100043	+	0.0173279i
$320$	160.000	+	277.128i	0.0279508	+	0.0484123i
$321$		0			0
$322$	−246.710	+	427.315i	−0.0426976	+	0.0739545i
$323$	−6390.25			−1.10081
$324$		0			0
$325$	−336.603			−0.0574503
$326$	−1501.78	+	2601.16i	−0.255140	+	0.441916i
$327$		0			0
$328$	275.369	+	476.954i	0.0463559	+	0.0802907i
$329$	−2719.21	−	4709.81i	−0.455668	−	0.789240i
$330$		0			0
$331$	2704.31	−	4684.00i	0.449070	−	0.777812i	−0.549256	−	0.835654i	$-0.685089\pi$
$331$	2704.31	−	4684.00i	0.449070	−	0.777812i	0.998326	+	0.0578423i	$0.0184221\pi$
$332$	−3782.43			−0.625265
$333$		0			0
$334$	6816.32			1.11668
$335$	−1695.02	+	2935.86i	−0.276444	+	0.478815i
$336$		0			0
$337$	3937.34	+	6819.67i	0.636440	+	1.10235i	0.986208	+	0.165511i	$0.0529272\pi$
$337$	3937.34	+	6819.67i	0.636440	+	1.10235i	−0.349768	+	0.936836i	$0.613739\pi$
$338$	−2015.72	−	3491.33i	−0.324380	−	0.561843i
$339$		0			0
$340$	1367.99	−	2369.42i	0.218204	−	0.377941i
$341$	60.1065			0.00954531
$342$		0			0
$343$	6299.16			0.991611
$344$	−1419.04	+	2457.85i	−0.222412	+	0.385228i
$345$		0			0
$346$	−2425.05	−	4200.31i	−0.376796	−	0.652630i
$347$	4255.25	+	7370.32i	0.658311	+	1.14023i	0.981053	+	0.193741i	$0.0620623\pi$
$347$	4255.25	+	7370.32i	0.658311	+	1.14023i	−0.322741	+	0.946487i	$0.604604\pi$
$348$		0			0
$349$	−4136.43	+	7164.51i	−0.634436	+	1.09887i	0.352199	+	0.935925i	$0.385434\pi$
$349$	−4136.43	+	7164.51i	−0.634436	+	1.09887i	−0.986634	+	0.162950i	$0.947899\pi$
$350$	563.397			0.0860425
$351$		0			0
$352$	33.6819			0.00510014
$353$	−2717.27	+	4706.46i	−0.409705	+	0.709630i	−0.994857	−	0.101294i	$-0.967702\pi$
$353$	−2717.27	+	4706.46i	−0.409705	+	0.709630i	0.585151	+	0.810924i	$0.301035\pi$
$354$		0			0
$355$	1430.16	+	2477.12i	0.213817	+	0.370343i
$356$	−1154.47	−	1999.60i	−0.171873	−	0.297693i
$357$		0			0
$358$	−4124.59	+	7143.99i	−0.608914	+	1.05467i
$359$	8737.11			1.28448			0.642238	−	0.766505i	$-0.278006\pi$
$359$	8737.11			1.28448			0.642238	+	0.766505i	$0.278006\pi$
$360$		0			0
$361$	−4676.91			−0.681865
$362$	−4053.64	+	7021.12i	−0.588549	+	1.01940i
$363$		0			0
$364$	−303.426	−	525.549i	−0.0436918	−	0.0756765i
$365$	268.644	+	465.305i	0.0385246	+	0.0667265i
$366$		0			0
$367$	1145.64	−	1984.31i	0.162948	−	0.282235i	−0.772976	−	0.634435i	$-0.781233\pi$
$367$	1145.64	−	1984.31i	0.162948	−	0.282235i	0.935925	+	0.352200i	$0.114566\pi$
$368$	350.318			0.0496239
$369$		0			0
$370$	2231.81			0.313584
$371$	−622.244	+	1077.76i	−0.0870762	+	0.150820i
$372$		0			0
$373$	−3174.83	−	5498.96i	−0.440714	−	0.763339i	0.557029	−	0.830493i	$-0.311941\pi$
$373$	−3174.83	−	5498.96i	−0.440714	−	0.763339i	−0.997743	+	0.0671545i	$0.978608\pi$
$374$	−143.989	−	249.396i	−0.0199077	−	0.0344811i
$375$		0			0
$376$	−1930.58	+	3343.86i	−0.264793	+	0.458634i
$377$	−1458.25			−0.199214
$378$		0			0
$379$	13681.6			1.85429			0.927146	−	0.374701i	$-0.122255\pi$
$379$	13681.6			1.85429			0.927146	+	0.374701i	$0.122255\pi$
$380$	−467.128	+	809.090i	−0.0630610	+	0.109225i
$381$		0			0
$382$	−1479.66	−	2562.84i	−0.198183	−	0.343262i
$383$	3737.27	+	6473.15i	0.498605	+	0.863610i	0.999999	−	0.00160974i	$-0.000512398\pi$
$383$	3737.27	+	6473.15i	0.498605	+	0.863610i	−0.501393	+	0.865219i	$0.667179\pi$
$384$		0			0
$385$	29.6505	−	51.3561i	0.00392500	−	0.00679831i
$386$	4735.38			0.624415
$387$		0			0
$388$	−4749.14			−0.621395
$389$	−6550.19	+	11345.3i	−0.853748	+	1.47873i	0.0240537	+	0.999711i	$0.492343\pi$
$389$	−6550.19	+	11345.3i	−0.853748	+	1.47873i	−0.877802	+	0.479024i	$0.840991\pi$
$390$		0			0
$391$	−1497.60	−	2593.91i	−0.193700	−	0.335498i
$392$	−864.133	−	1496.72i	−0.111340	−	0.192847i
$393$		0			0
$394$	3603.59	−	6241.60i	0.460777	−	0.798089i
$395$	518.653			0.0660665
$396$		0			0
$397$	5955.26			0.752862			0.376431	−	0.926445i	$-0.377151\pi$
$397$	5955.26			0.752862			0.376431	+	0.926445i	$0.377151\pi$
$398$	1842.45	−	3191.22i	0.232044	−	0.401913i
$399$		0			0
$400$	−200.000	−	346.410i	−0.0250000	−	0.0433013i
$401$	5880.23	+	10184.9i	0.732281	+	1.26835i	0.955906	+	0.293673i	$0.0948775\pi$
$401$	5880.23	+	10184.9i	0.732281	+	1.26835i	−0.223625	+	0.974675i	$0.571789\pi$
$402$		0			0
$403$	−384.435	+	665.860i	−0.0475187	+	0.0823049i
$404$	−5155.35			−0.634872
$405$		0			0
$406$	2440.78			0.298360
$407$	117.455	−	203.439i	0.0143048	−	0.0247766i
$408$		0			0
$409$	876.309	+	1517.81i	0.105943	+	0.183499i	0.914123	−	0.405437i	$-0.132881\pi$
$409$	876.309	+	1517.81i	0.105943	+	0.183499i	−0.808180	+	0.588935i	$0.799547\pi$
$410$	−344.212	−	596.192i	−0.0414620	−	0.0718142i
$411$		0			0
$412$	−439.615	+	761.436i	−0.0525687	+	0.0910516i
$413$	−7412.13			−0.883116
$414$		0			0
$415$	4728.04			0.559254
$416$	−215.426	+	373.128i	−0.0253897	+	0.0439762i
$417$		0			0
$418$	49.1680	+	85.1615i	0.00575331	+	0.00996503i
$419$	−2535.45	−	4391.52i	−0.295620	−	0.512029i	0.679509	−	0.733667i	$-0.262193\pi$
$419$	−2535.45	−	4391.52i	−0.295620	−	0.512029i	−0.975129	+	0.221638i	$0.928860\pi$
$420$		0			0
$421$	208.370	−	360.907i	0.0241219	−	0.0417804i	−0.853712	−	0.520745i	$-0.825654\pi$
$421$	208.370	−	360.907i	0.0241219	−	0.0417804i	0.877834	+	0.478964i	$0.158988\pi$
$422$	6419.60			0.740525
$423$		0			0
$424$	883.559			0.101201
$425$	−1709.98	+	2961.78i	−0.195168	+	0.338041i
$426$		0			0
$427$	−216.952	−	375.771i	−0.0245879	−	0.0425875i
$428$	1835.56	+	3179.29i	0.207302	+	0.359058i
$429$		0			0
$430$	1773.80	−	3072.31i	0.198931	−	0.344558i
$431$	−8870.50			−0.991361			−0.495681	−	0.868505i	$-0.665081\pi$
$431$	−8870.50			−0.991361			−0.495681	+	0.868505i	$0.665081\pi$
$432$		0			0
$433$	−12172.1			−1.35094			−0.675468	−	0.737389i	$-0.736058\pi$
$433$	−12172.1			−1.35094			−0.675468	+	0.737389i	$0.736058\pi$
$434$	643.458	−	1114.50i	0.0711681	−	0.123267i
$435$		0			0
$436$	−941.764	−	1631.18i	−0.103446	−	0.179173i
$437$	511.386	+	885.746i	0.0559792	+	0.0969588i
$438$		0			0
$439$	2794.96	−	4841.01i	0.303864	−	0.526307i	−0.673144	−	0.739511i	$-0.735057\pi$
$439$	2794.96	−	4841.01i	0.303864	−	0.526307i	0.977008	+	0.213204i	$0.0683899\pi$
$440$	−42.1024			−0.00456171
$441$		0			0
$442$	3683.74			0.396420
$443$	−1743.46	+	3019.76i	−0.186985	+	0.323867i	−0.944244	−	0.329248i	$-0.893205\pi$
$443$	−1743.46	+	3019.76i	−0.186985	+	0.323867i	0.757259	+	0.653115i	$0.226538\pi$
$444$		0			0
$445$	1443.09	+	2499.50i	0.153728	+	0.266264i
$446$	6378.69	+	11048.2i	0.677219	+	1.17298i
$447$		0			0
$448$	360.574	−	624.533i	0.0380258	−	0.0658625i
$449$	6661.15			0.700131			0.350066	−	0.936725i	$-0.386159\pi$
$449$	6661.15			0.700131			0.350066	+	0.936725i	$0.386159\pi$
$450$		0			0
$451$	−72.4606			−0.00756549
$452$	−975.735	+	1690.02i	−0.101537	+	0.175867i
$453$		0			0
$454$	−1070.86	−	1854.78i	−0.110700	−	0.191738i
$455$	379.282	+	656.936i	0.0390792	+	0.0676871i
$456$		0			0
$457$	−5821.95	+	10083.9i	−0.595928	+	1.03218i	0.397487	+	0.917608i	$0.369882\pi$
$457$	−5821.95	+	10083.9i	−0.595928	+	1.03218i	−0.993415	+	0.114570i	$0.963451\pi$
$458$	5115.79			0.521933
$459$		0			0
$460$	−437.898			−0.0443850
$461$	7751.45	−	13425.9i	0.783126	−	1.35641i	−0.146987	−	0.989138i	$-0.546957\pi$
$461$	7751.45	−	13425.9i	0.783126	−	1.35641i	0.930112	−	0.367275i	$-0.119709\pi$
$462$		0			0
$463$	−3260.36	−	5647.10i	−0.327261	−	0.566832i	0.654707	−	0.755883i	$-0.272792\pi$
$463$	−3260.36	−	5647.10i	−0.327261	−	0.566832i	−0.981967	+	0.189051i	$0.939459\pi$
$464$	−866.451	−	1500.74i	−0.0866896	−	0.150151i
$465$		0			0
$466$	−2532.68	+	4386.73i	−0.251769	+	0.436076i
$467$	11488.0			1.13833			0.569166	−	0.822222i	$-0.307266\pi$
$467$	11488.0			1.13833			0.569166	+	0.822222i	$0.307266\pi$
$468$		0			0
$469$	7639.75			0.752177
$470$	2413.22	−	4179.83i	0.236838	−	0.410215i
$471$		0			0
$472$	2631.22	+	4557.42i	0.256593	+	0.444432i
$473$	−186.703	−	323.379i	−0.0181493	−	0.0314355i
$474$		0			0
$475$	583.910	−	1011.36i	0.0564034	−	0.0976936i
$476$	−6165.76			−0.593713
$477$		0			0
$478$	−2304.98			−0.220559
$479$	−8819.69	+	15276.2i	−0.841299	+	1.45717i	0.0474988	+	0.998871i	$0.484875\pi$
$479$	−8819.69	+	15276.2i	−0.841299	+	1.45717i	−0.888797	+	0.458300i	$0.848458\pi$
$480$		0			0
$481$	1502.46	+	2602.34i	0.142425	+	0.246687i
$482$	6945.77	+	12030.4i	0.656372	+	1.13687i
$483$		0			0
$484$	2659.78	−	4606.88i	0.249792	−	0.432652i
$485$	5936.43			0.555792
$486$		0			0
$487$	−1182.88			−0.110065			−0.0550323	−	0.998485i	$-0.517526\pi$
$487$	−1182.88			−0.110065			−0.0550323	+	0.998485i	$0.517526\pi$
$488$	−154.031	+	266.789i	−0.0142882	+	0.0247479i
$489$		0			0
$490$	1080.17	+	1870.90i	0.0995856	+	0.172487i
$491$	−248.604	−	430.594i	−0.0228499	−	0.0395773i	0.854374	−	0.519658i	$-0.173941\pi$
$491$	−248.604	−	430.594i	−0.0228499	−	0.0395773i	−0.877224	+	0.480081i	$0.840607\pi$
$492$		0			0
$493$	−7408.09	+	12831.2i	−0.676762	+	1.17219i
$494$	−1257.89			−0.114565
$495$		0			0
$496$	−913.682			−0.0827128
$497$	3223.00	−	5582.40i	0.290888	−	0.503833i
$498$		0			0
$499$	2638.02	+	4569.18i	0.236661	+	0.409909i	0.959754	−	0.280842i	$-0.0906136\pi$
$499$	2638.02	+	4569.18i	0.236661	+	0.409909i	−0.723093	+	0.690751i	$0.757280\pi$
$500$	250.000	+	433.013i	0.0223607	+	0.0387298i
$501$		0			0
$502$	−3096.59	+	5363.45i	−0.275314	+	0.476857i
$503$	−19553.3			−1.73327			−0.866637	−	0.498940i	$-0.833723\pi$
$503$	−19553.3			−1.73327			−0.866637	+	0.498940i	$0.833723\pi$
$504$		0			0
$505$	6444.18			0.567846
$506$	−23.0457	+	39.9162i	−0.00202471	+	0.00350690i
$507$		0			0
$508$	2316.39	+	4012.11i	0.202310	+	0.350411i
$509$	−10182.1	−	17635.9i	−0.886667	−	1.53575i	−0.843792	−	0.536671i	$-0.819682\pi$
$509$	−10182.1	−	17635.9i	−0.886667	−	1.53575i	−0.0428750	−	0.999080i	$-0.513652\pi$
$510$		0			0
$511$	605.413	−	1048.61i	0.0524108	−	0.0907781i
$512$	−512.000			−0.0441942
$513$		0			0
$514$	4302.36			0.369200
$515$	549.519	−	951.795i	0.0470188	−	0.0814390i
$516$		0			0
$517$	−254.006	−	439.951i	−0.0216077	−	0.0374256i
$518$	−2514.79	−	4355.74i	−0.213308	−	0.369460i
$519$		0			0
$520$	269.282	−	466.410i	0.0227092	−	0.0393335i
$521$	−2074.16			−0.174416			−0.0872079	−	0.996190i	$-0.527794\pi$
$521$	−2074.16			−0.174416			−0.0872079	+	0.996190i	$0.527794\pi$
$522$		0			0
$523$	−20489.2			−1.71306			−0.856529	−	0.516099i	$-0.827383\pi$
$523$	−20489.2			−1.71306			−0.856529	+	0.516099i	$0.827383\pi$
$524$	−2173.92	+	3765.34i	−0.181237	+	0.313911i
$525$		0			0
$526$	−2059.83	−	3567.74i	−0.170747	−	0.295743i
$527$	3905.95	+	6765.31i	0.322858	+	0.559206i
$528$		0			0
$529$	5843.81	−	10121.8i	0.480300	−	0.831904i
$530$	−1104.45			−0.0905173
$531$		0			0
$532$	2105.43			0.171583
$533$	463.450	−	802.719i	0.0376628	−	0.0652338i
$534$		0			0
$535$	−2294.46	−	3974.11i	−0.185417	−	0.321151i
$536$	−2712.03	−	4697.37i	−0.218548	−	0.378537i
$537$		0			0
$538$	−2163.57	+	3747.41i	−0.173379	+	0.300302i
$539$	227.388			0.0181712
$540$		0			0
$541$	−16846.1			−1.33876			−0.669381	−	0.742919i	$-0.733441\pi$
$541$	−16846.1			−1.33876			−0.669381	+	0.742919i	$0.733441\pi$
$542$	6995.32	−	12116.3i	0.554382	−	0.960217i
$543$		0			0
$544$	2188.78	+	3791.08i	0.172506	+	0.298789i
$545$	1177.20	+	2038.98i	0.0925246	+	0.160257i
$546$		0			0
$547$	−103.176	+	178.706i	−0.00806489	+	0.0139688i	−0.870030	−	0.492999i	$-0.835900\pi$
$547$	−103.176	+	178.706i	−0.00806489	+	0.0139688i	0.861965	+	0.506968i	$0.169234\pi$
$548$	−5664.34			−0.441548
$549$		0			0
$550$	52.6279			0.00408011
$551$	2529.65	−	4381.48i	0.195584	−	0.338761i
$552$		0			0
$553$	−584.416	−	1012.24i	−0.0449401	−	0.0778386i
$554$	−1839.68	−	3186.42i	−0.141084	−	0.244364i
$555$		0			0
$556$	1063.17	−	1841.46i	0.0810943	−	0.140459i
$557$	−2773.08			−0.210950			−0.105475	−	0.994422i	$-0.533636\pi$
$557$	−2773.08			−0.210950			−0.105475	+	0.994422i	$0.533636\pi$
$558$		0			0
$559$	4776.53			0.361405
$560$	−450.718	+	780.666i	−0.0340113	+	0.0589092i
$561$		0			0
$562$	129.966	+	225.108i	0.00975498	+	0.0168961i
$563$	9206.82	+	15946.7i	0.689203	+	1.19373i	0.972096	+	0.234583i	$0.0753725\pi$
$563$	9206.82	+	15946.7i	0.689203	+	1.19373i	−0.282893	+	0.959151i	$0.591294\pi$
$564$		0			0
$565$	1219.67	−	2112.53i	0.0908175	−	0.157300i
$566$	−8775.45			−0.651696
$567$		0			0
$568$	−4576.52			−0.338075
$569$	2411.52	−	4176.88i	0.177674	−	0.307740i	−0.763410	−	0.645915i	$-0.776476\pi$
$569$	2411.52	−	4176.88i	0.177674	−	0.307740i	0.941083	+	0.338175i	$0.109810\pi$
$570$		0			0
$571$	−11204.7	−	19407.1i	−0.821195	−	1.42235i	−0.904793	−	0.425853i	$-0.859974\pi$
$571$	−11204.7	−	19407.1i	−0.821195	−	1.42235i	0.0835972	−	0.996500i	$-0.473359\pi$
$572$	−28.3435	−	49.0924i	−0.00207186	−	0.00358856i
$573$		0			0
$574$	−775.712	+	1343.57i	−0.0564069	+	0.0976997i
$575$	547.372			0.0396991
$576$		0			0
$577$	−18690.4			−1.34851			−0.674254	−	0.738499i	$-0.735535\pi$
$577$	−18690.4			−1.34851			−0.674254	+	0.738499i	$0.735535\pi$
$578$	13800.9	−	23903.8i	0.993151	−	1.72019i
$579$		0			0
$580$	1083.06	+	1875.92i	0.0775376	+	0.134299i
$581$	−5327.54	−	9227.56i	−0.380419	−	0.658905i
$582$		0			0
$583$	−58.1249	+	100.675i	−0.00412914	+	0.00715187i
$584$	−859.661			−0.0609127
$585$		0			0
$586$	9560.81			0.673982
$587$	3842.27	−	6655.01i	0.270166	−	0.467941i	−0.698738	−	0.715378i	$-0.746255\pi$
$587$	3842.27	−	6655.01i	0.270166	−	0.467941i	0.968904	+	0.247436i	$0.0795881\pi$
$588$		0			0
$589$	−1333.77	−	2310.16i	−0.0933057	−	0.161610i
$590$	−3289.03	−	5696.77i	−0.229504	−	0.397512i
$591$		0			0
$592$	−1785.45	+	3092.48i	−0.123955	+	0.214696i
$593$	−1520.59			−0.105301			−0.0526504	−	0.998613i	$-0.516767\pi$
$593$	−1520.59			−0.105301			−0.0526504	+	0.998613i	$0.516767\pi$
$594$		0			0
$595$	7707.20			0.531033
$596$	−3463.57	+	5999.08i	−0.238042	+	0.412302i
$597$		0			0
$598$	−294.795	−	510.600i	−0.0201590	−	0.0349164i
$599$	−2831.30	−	4903.96i	−0.193128	−	0.334508i	0.753157	−	0.657841i	$-0.228530\pi$
$599$	−2831.30	−	4903.96i	−0.193128	−	0.334508i	−0.946285	+	0.323333i	$0.895197\pi$
$600$		0			0
$601$	−1156.24	+	2002.67i	−0.0784761	+	0.135925i	−0.902593	−	0.430496i	$-0.858339\pi$
$601$	−1156.24	+	2002.67i	−0.0784761	+	0.135925i	0.824117	+	0.566420i	$0.191672\pi$
$602$	−7994.84			−0.541271
$603$		0			0
$604$	2220.74			0.149603
$605$	−3324.73	+	5758.60i	−0.223421	+	0.386976i
$606$		0			0
$607$	2977.30	+	5156.84i	0.199086	+	0.344826i	0.948232	−	0.317578i	$-0.102869\pi$
$607$	2977.30	+	5156.84i	0.199086	+	0.344826i	−0.749147	+	0.662404i	$0.769536\pi$
$608$	−747.405	−	1294.54i	−0.0498541	−	0.0863498i
$609$		0			0
$610$	192.539	−	333.487i	0.0127798	−	0.0221352i
$611$	6498.38			0.430272
$612$		0			0
$613$	−19170.4			−1.26310			−0.631552	−	0.775333i	$-0.717582\pi$
$613$	−19170.4			−1.26310			−0.631552	+	0.775333i	$0.717582\pi$
$614$	−6471.01	+	11208.1i	−0.425324	+	0.736682i
$615$		0			0
$616$	47.4407	+	82.1697i	0.00310299	+	0.00537453i
$617$	−14072.3	−	24373.9i	−0.918197	−	1.59036i	−0.802152	−	0.597120i	$-0.796312\pi$
$617$	−14072.3	−	24373.9i	−0.918197	−	1.59036i	−0.116045	−	0.993244i	$-0.537022\pi$
$618$		0			0
$619$	10541.7	−	18258.8i	0.684505	−	1.18560i	−0.289087	−	0.957303i	$-0.593352\pi$
$619$	10541.7	−	18258.8i	0.684505	−	1.18560i	0.973592	−	0.228294i	$-0.0733149\pi$
$620$	1142.10			0.0739805
$621$		0			0
$622$	−3902.32			−0.251557
$623$	3252.13	−	5632.85i	0.209139	−	0.362240i
$624$		0			0
$625$	−312.500	−	541.266i	−0.0200000	−	0.0346410i
$626$	5346.39	+	9260.23i	0.341350	+	0.591235i
$627$		0			0
$628$	2974.60	−	5152.15i	0.189012	−	0.327378i
$629$	30530.8			1.93536
$630$		0			0
$631$	−26107.2			−1.64709			−0.823544	−	0.567252i	$-0.808007\pi$
$631$	−26107.2			−1.64709			−0.823544	+	0.567252i	$0.808007\pi$
$632$	−414.923	+	718.667i	−0.0261151	+	0.0452327i
$633$		0			0
$634$	9746.13	+	16880.8i	0.610518	+	1.05745i
$635$	−2895.49	−	5015.14i	−0.180951	−	0.313417i
$636$		0			0
$637$	−1454.35	+	2519.00i	−0.0904605	+	0.156682i
$638$	227.998			0.0141481
$639$		0			0
$640$	640.000			0.0395285
$641$	−882.303	+	1528.19i	−0.0543664	+	0.0941654i	−0.891928	−	0.452178i	$-0.850647\pi$
$641$	−882.303	+	1528.19i	−0.0543664	+	0.0941654i	0.837561	+	0.546343i	$0.183981\pi$
$642$		0			0
$643$	−9141.93	−	15834.3i	−0.560688	−	0.971140i	−0.997437	−	0.0715565i	$-0.977203\pi$
$643$	−9141.93	−	15834.3i	−0.560688	−	0.971140i	0.436749	−	0.899584i	$-0.356130\pi$
$644$	493.421	+	854.630i	0.0301918	+	0.0522937i
$645$		0			0
$646$	−6390.25	+	11068.2i	−0.389197	+	0.674109i
$647$	−14271.6			−0.867192			−0.433596	−	0.901107i	$-0.642756\pi$
$647$	−14271.6			−0.867192			−0.433596	+	0.901107i	$0.642756\pi$
$648$		0			0
$649$	−692.380			−0.0418772
$650$	−336.603	+	583.013i	−0.0203118	+	0.0351810i
$651$		0			0
$652$	3003.56	+	5202.31i	0.180412	+	0.312482i
$653$	−3409.38	−	5905.22i	−0.204317	−	0.353888i	0.745598	−	0.666396i	$-0.232164\pi$
$653$	−3409.38	−	5905.22i	−0.204317	−	0.353888i	−0.949915	+	0.312508i	$0.898831\pi$
$654$		0			0
$655$	2717.40	−	4706.67i	0.162103	−	0.280771i
$656$	1101.48			0.0655571
$657$		0			0
$658$	−10876.8			−0.644412
$659$	14840.4	−	25704.3i	0.877236	−	1.51942i	0.0228754	−	0.999738i	$-0.492718\pi$
$659$	14840.4	−	25704.3i	0.877236	−	1.51942i	0.854361	−	0.519680i	$-0.173949\pi$
$660$		0			0
$661$	13752.9	+	23820.7i	0.809267	+	1.40169i	0.913372	+	0.407125i	$0.133469\pi$
$661$	13752.9	+	23820.7i	0.809267	+	1.40169i	−0.104106	+	0.994566i	$0.533198\pi$
$662$	−5408.61	−	9367.99i	−0.317540	−	0.549996i
$663$		0			0
$664$	−3782.43	+	6551.37i	−0.221065	+	0.382895i
$665$	−2631.79			−0.153468
$666$		0			0
$667$	2371.36			0.137660
$668$	6816.32	−	11806.2i	0.394808	−	0.683827i
$669$		0			0
$670$	3390.04	+	5871.72i	0.195475	+	0.338573i
$671$	−20.2658	−	35.1014i	−0.00116595	−	0.00201949i
$672$		0			0
$673$	−15610.2	+	27037.6i	−0.894099	+	1.54863i	−0.0591836	+	0.998247i	$0.518850\pi$
$673$	−15610.2	+	27037.6i	−0.894099	+	1.54863i	−0.834916	+	0.550378i	$0.814484\pi$
$674$	15749.3			0.900063
$675$		0			0
$676$	−8062.87			−0.458743
$677$	4343.28	−	7522.79i	0.246567	−	0.427067i	−0.716004	−	0.698096i	$-0.754031\pi$
$677$	4343.28	−	7522.79i	0.246567	−	0.427067i	0.962571	+	0.271029i	$0.0873641\pi$
$678$		0			0
$679$	−6689.14	−	11585.9i	−0.378064	−	0.654826i
$680$	−2735.97	−	4738.85i	−0.154294	−	0.267245i
$681$		0			0
$682$	60.1065	−	104.108i	0.00337478	−	0.00584528i
$683$	11939.8			0.668905			0.334452	−	0.942413i	$-0.391449\pi$
$683$	11939.8			0.668905			0.334452	+	0.942413i	$0.391449\pi$
$684$		0			0
$685$	7080.42			0.394933
$686$	6299.16	−	10910.5i	0.350587	−	0.607235i
$687$		0			0
$688$	2838.08	+	4915.70i	0.157269	+	0.272397i
$689$	−743.520	−	1287.82i	−0.0411116	−	0.0712073i
$690$		0			0
$691$	4890.07	−	8469.85i	0.269214	−	0.466293i	−0.699445	−	0.714687i	$-0.746569\pi$
$691$	4890.07	−	8469.85i	0.269214	−	0.466293i	0.968659	+	0.248394i	$0.0799027\pi$
$692$	−9700.19			−0.532870
$693$		0			0
$694$	17021.0			0.930993
$695$	−1328.96	+	2301.83i	−0.0725329	+	0.125631i
$696$		0			0
$697$	−4708.77	−	8155.83i	−0.255893	−	0.443220i
$698$	8272.86	+	14329.0i	0.448614	+	0.777022i
$699$		0			0
$700$	563.397	−	975.833i	0.0304206	−	0.0526900i
$701$	11963.4			0.644581			0.322290	−	0.946641i	$-0.395547\pi$
$701$	11963.4			0.644581			0.322290	+	0.946641i	$0.395547\pi$
$702$		0			0
$703$	−10425.4			−0.559319
$704$	33.6819	−	58.3387i	0.00180317	−	0.00312319i
$705$		0			0
$706$	5434.55	+	9412.91i	0.289705	+	0.501784i
$707$	−7261.27	−	12576.9i	−0.386264	−	0.669028i
$708$		0			0
$709$	6451.96	−	11175.1i	0.341761	−	0.591947i	−0.642999	−	0.765867i	$-0.722310\pi$
$709$	6451.96	−	11175.1i	0.341761	−	0.591947i	0.984760	+	0.173920i	$0.0556434\pi$
$710$	5720.65			0.302383
$711$		0			0
$712$	−4617.88			−0.243065
$713$	625.155	−	1082.80i	0.0328362	−	0.0568740i
$714$		0			0
$715$	35.4294	+	61.3655i	0.00185313	+	0.00320971i
$716$	8249.17	+	14288.0i	0.430567	+	0.745764i
$717$		0			0
$718$	8737.11	−	15133.1i	0.454131	−	0.786578i
$719$	19351.9			1.00376			0.501881	−	0.864936i	$-0.332641\pi$
$719$	19351.9			1.00376			0.501881	+	0.864936i	$0.332641\pi$
$720$		0			0
$721$	−2476.78			−0.127934
$722$	−4676.91	+	8100.65i	−0.241076	+	0.417555i
$723$		0			0
$724$	8107.29	+	14042.2i	0.416167	+	0.720822i
$725$	−1353.83	−	2344.90i	−0.0693517	−	0.120121i
$726$		0			0
$727$	16894.4	−	29261.9i	0.861868	−	1.49280i	−0.00825650	−	0.999966i	$-0.502628\pi$
$727$	16894.4	−	29261.9i	0.861868	−	1.49280i	0.870124	−	0.492833i	$-0.164039\pi$
$728$	−1213.70			−0.0617896
$729$		0			0
$730$	1074.58			0.0544820
$731$	24265.4	−	42028.8i	1.22775	−	2.12653i
$732$		0			0
$733$	−10498.6	−	18184.0i	−0.529022	−	0.916292i	−0.999427	−	0.0338421i	$-0.989226\pi$
$733$	−10498.6	−	18184.0i	−0.529022	−	0.916292i	0.470405	−	0.882450i	$-0.344108\pi$
$734$	−2291.28	−	3968.62i	−0.115222	−	0.199570i
$735$		0			0
$736$	350.318	−	606.769i	0.0175447	−	0.0303883i
$737$	713.643			0.0356681
$738$		0			0
$739$	27556.6			1.37170			0.685849	−	0.727744i	$-0.259431\pi$
$739$	27556.6			1.37170			0.685849	+	0.727744i	$0.259431\pi$
$740$	2231.81	−	3865.60i	0.110869	−	0.192030i
$741$		0			0
$742$	1244.49	+	2155.51i	0.0615722	+	0.106646i
$743$	5936.90	+	10283.0i	0.293141	+	0.507735i	0.974551	−	0.224167i	$-0.0719662\pi$
$743$	5936.90	+	10283.0i	0.293141	+	0.507735i	−0.681410	+	0.731902i	$0.738633\pi$
$744$		0			0
$745$	4329.46	−	7498.85i	0.212912	−	0.368774i
$746$	−12699.3			−0.623263
$747$		0			0
$748$	−575.955			−0.0281537
$749$	−5170.76	+	8956.02i	−0.252250	+	0.436910i
$750$		0			0
$751$	661.879	+	1146.41i	0.0321602	+	0.0557031i	0.881658	−	0.471890i	$-0.156428\pi$
$751$	661.879	+	1146.41i	0.0321602	+	0.0557031i	−0.849497	+	0.527593i	$0.823095\pi$
$752$	3861.16	+	6687.72i	0.187237	+	0.324303i
$753$		0			0
$754$	−1458.25	+	2525.76i	−0.0704327	+	0.121993i
$755$	−2775.92			−0.133809
$756$		0			0
$757$	−29436.0			−1.41330			−0.706650	−	0.707563i	$-0.749795\pi$
$757$	−29436.0			−1.41330			−0.706650	+	0.707563i	$0.749795\pi$
$758$	13681.6	−	23697.2i	0.655591	−	1.13552i
$759$		0			0
$760$	934.256	+	1618.18i	0.0445908	+	0.0772336i
$761$	−14505.6	−	25124.5i	−0.690970	−	1.19680i	−0.971520	−	0.236956i	$-0.923850\pi$
$761$	−14505.6	−	25124.5i	−0.690970	−	1.19680i	0.280550	−	0.959839i	$-0.409483\pi$
$762$		0			0
$763$	2652.94	−	4595.02i	0.125875	−	0.218022i
$764$	−5918.62			−0.280272
$765$		0			0
$766$	14949.1			0.705134
$767$	4428.39	−	7670.19i	0.208474	−	0.361088i
$768$		0			0
$769$	15169.5	+	26274.4i	0.711348	+	1.23209i	0.964351	+	0.264625i	$0.0852482\pi$
$769$	15169.5	+	26274.4i	0.711348	+	1.23209i	−0.253004	+	0.967465i	$0.581418\pi$
$770$	−59.3009	−	102.712i	−0.00277540	−	0.00480713i
$771$		0			0
$772$	4735.38	−	8201.91i	0.220764	−	0.382375i
$773$	16100.2			0.749139			0.374569	−	0.927199i	$-0.377791\pi$
$773$	16100.2			0.749139			0.374569	+	0.927199i	$0.377791\pi$
$774$		0			0
$775$	−1427.63			−0.0661702
$776$	−4749.14	+	8225.76i	−0.219696	+	0.380525i
$777$		0			0
$778$	13100.4	+	22690.5i	0.603691	+	1.04562i
$779$	1607.91	+	2784.98i	0.0739529	+	0.128090i
$780$		0			0
$781$	301.066	−	521.462i	0.0137938	−	0.0238916i
$782$	−5990.38			−0.273933
$783$		0			0
$784$	−3456.53			−0.157459
$785$	−3718.25	+	6440.19i	−0.169057	+	0.292816i
$786$		0			0
$787$	−9295.05	−	16099.5i	−0.421007	−	0.729206i	0.575031	−	0.818132i	$-0.304990\pi$
$787$	−9295.05	−	16099.5i	−0.421007	−	0.729206i	−0.996038	+	0.0889256i	$0.971657\pi$
$788$	−7207.18	−	12483.2i	−0.325819	−	0.564334i
$789$		0			0
$790$	518.653	−	898.334i	0.0233580	−	0.0404573i
$791$	−5497.27			−0.247105
$792$		0			0
$793$	518.472			0.0232175
$794$	5955.26	−	10314.8i	0.266177	−	0.461032i
$795$		0			0
$796$	−3684.90	−	6382.44i	−0.164080	−	0.284195i
$797$	−11051.8	−	19142.3i	−0.491186	−	0.850759i	0.508762	−	0.860907i	$-0.330103\pi$
$797$	−11051.8	−	19142.3i	−0.491186	−	0.850759i	−0.999949	+	0.0101477i	$0.996770\pi$
$798$		0			0
$799$	33012.6	−	57179.5i	1.46170	−	2.53175i
$800$	−800.000			−0.0353553
$801$		0			0
$802$	23520.9			1.03560
$803$	56.5527	−	97.9522i	0.00248531	−	0.00430468i
$804$		0			0
$805$	−616.776	−	1068.29i	−0.0270043	−	0.0467729i
$806$	768.869	+	1331.72i	0.0336008	+	0.0581983i
$807$		0			0
$808$	−5155.35	+	8929.32i	−0.224461	+	0.388778i
$809$	12985.4			0.564329			0.282165	−	0.959366i	$-0.408948\pi$
$809$	12985.4			0.564329			0.282165	+	0.959366i	$0.408948\pi$
$810$		0			0
$811$	−41541.9			−1.79868			−0.899342	−	0.437247i	$-0.855954\pi$
$811$	−41541.9			−1.79868			−0.899342	+	0.437247i	$0.855954\pi$
$812$	2440.78	−	4227.56i	0.105486	−	0.182707i
$813$		0			0
$814$	−234.911	−	406.877i	−0.0101150	−	0.0175197i
$815$	−3754.44	−	6502.89i	−0.161365	−	0.279492i
$816$		0			0
$817$	−8285.92	+	14351.6i	−0.354820	+	0.614566i
$818$	3505.24			0.149826
$819$		0			0
$820$	−1376.85			−0.0586361
$821$	−17699.9	+	30657.1i	−0.752413	+	1.30322i	0.194238	+	0.980954i	$0.437777\pi$
$821$	−17699.9	+	30657.1i	−0.752413	+	1.30322i	−0.946650	+	0.322262i	$0.895557\pi$
$822$		0			0
$823$	12200.0	+	21131.1i	0.516727	+	0.894998i	0.999811	+	0.0194239i	$0.00618322\pi$
$823$	12200.0	+	21131.1i	0.516727	+	0.894998i	−0.483084	+	0.875574i	$0.660483\pi$
$824$	879.230	+	1522.87i	0.0371717	+	0.0643832i
$825$		0			0
$826$	−7412.13	+	12838.2i	−0.312229	+	0.540796i
$827$	20292.5			0.853252			0.426626	−	0.904428i	$-0.359702\pi$
$827$	20292.5			0.853252			0.426626	+	0.904428i	$0.359702\pi$
$828$		0			0
$829$	26651.2			1.11657			0.558283	−	0.829651i	$-0.311460\pi$
$829$	26651.2			1.11657			0.558283	+	0.829651i	$0.311460\pi$
$830$	4728.04	−	8189.21i	0.197726	−	0.342472i
$831$		0			0
$832$	430.851	+	746.256i	0.0179532	+	0.0310959i
$833$	14776.5	+	25593.7i	0.614618	+	1.06455i
$834$		0			0
$835$	−8520.41	+	14757.8i	−0.353127	+	0.611633i
$836$	196.672			0.00813641
$837$		0			0
$838$	−10141.8			−0.418070
$839$	−21885.1	+	37906.2i	−0.900547	+	1.55979i	−0.0737604	+	0.997276i	$0.523500\pi$
$839$	−21885.1	+	37906.2i	−0.900547	+	1.55979i	−0.826786	+	0.562516i	$0.809833\pi$
$840$		0			0
$841$	6329.36	+	10962.8i	0.259517	+	0.449497i
$842$	−416.740	−	721.814i	−0.0170568	−	0.0295432i
$843$		0			0
$844$	6419.60	−	11119.1i	0.261815	−	0.453477i
$845$	10078.6			0.410312
$846$		0			0
$847$	14985.2			0.607906
$848$	883.559	−	1530.37i	0.0357801	−	0.0619730i
$849$		0			0
$850$	3419.97	+	5923.56i	0.138005	+	0.239031i
$851$	−2443.26	−	4231.85i	−0.0984181	−	0.170465i
$852$		0			0
$853$	−1742.56	+	3018.20i	−0.0699461	+	0.121150i	−0.898877	−	0.438200i	$-0.855616\pi$
$853$	−1742.56	+	3018.20i	−0.0699461	+	0.121150i	0.828931	+	0.559350i	$0.188949\pi$
$854$	−867.806			−0.0347725
$855$		0			0
$856$	7342.26			0.293170
$857$	5054.46	−	8754.58i	0.201467	−	0.348951i	−0.747535	−	0.664223i	$-0.768763\pi$
$857$	5054.46	−	8754.58i	0.201467	−	0.348951i	0.949001	+	0.315272i	$0.102096\pi$
$858$		0			0
$859$	20554.9	+	35602.1i	0.816441	+	1.41412i	0.908288	+	0.418344i	$0.137389\pi$
$859$	20554.9	+	35602.1i	0.816441	+	1.41412i	−0.0918474	+	0.995773i	$0.529277\pi$
$860$	−3547.60	−	6144.63i	−0.140665	−	0.243640i
$861$		0			0
$862$	−8870.50	+	15364.1i	−0.350499	+	0.607082i
$863$	−31278.0			−1.23374			−0.616869	−	0.787066i	$-0.711599\pi$
$863$	−31278.0			−1.23374			−0.616869	+	0.787066i	$0.711599\pi$
$864$		0			0
$865$	12125.2			0.476613
$866$	−12172.1	+	21082.8i	−0.477628	+	0.827276i
$867$		0			0
$868$	−1286.92	−	2229.00i	−0.0503234	−	0.0871628i
$869$	−54.5913	−	94.5549i	−0.00213105	−	0.00369109i
$870$		0			0
$871$	−4564.38	+	7905.74i	−0.177564	+	0.307550i
$872$	−3767.05			−0.146294
$873$		0			0
$874$	2045.54			0.0791665
$875$	−704.247	+	1219.79i	−0.0272090	+	0.0471274i
$876$		0			0
$877$	−3505.23	−	6071.23i	−0.134964	−	0.233764i	0.790620	−	0.612307i	$-0.209758\pi$
$877$	−3505.23	−	6071.23i	−0.134964	−	0.233764i	−0.925584	+	0.378543i	$0.876425\pi$
$878$	−5589.92	−	9682.03i	−0.214864	−	0.372156i
$879$		0			0
$880$	−42.1024	+	72.9234i	−0.00161281	+	0.00279346i
$881$	40740.1			1.55797			0.778983	−	0.627045i	$-0.215736\pi$
$881$	40740.1			1.55797			0.778983	+	0.627045i	$0.215736\pi$
$882$		0			0
$883$	−19342.0			−0.737158			−0.368579	−	0.929596i	$-0.620156\pi$
$883$	−19342.0			−0.737158			−0.368579	+	0.929596i	$0.620156\pi$
$884$	3683.74	−	6380.43i	0.140156	−	0.242757i
$885$		0			0
$886$	3486.92	+	6039.52i	0.132218	+	0.229009i
$887$	−7262.88	−	12579.7i	−0.274931	−	0.476194i	0.695187	−	0.718829i	$-0.255322\pi$
$887$	−7262.88	−	12579.7i	−0.274931	−	0.476194i	−0.970118	+	0.242635i	$0.921988\pi$
$888$		0			0
$889$	−6525.25	+	11302.1i	−0.246175	+	0.426388i
$890$	5772.35			0.217404
$891$		0			0
$892$	25514.7			0.957732
$893$	−11272.8	+	19525.1i	−0.422432	+	0.731673i
$894$		0			0
$895$	−10311.5	−	17860.0i	−0.385111	−	0.667032i
$896$	−721.149	−	1249.07i	−0.0268883	−	0.0465718i
$897$		0			0
$898$	6661.15	−	11537.4i	0.247534	−	0.428741i
$899$	−6184.85			−0.229451
$900$		0			0
$901$	−15108.7			−0.558651
$902$	−72.4606	+	125.505i	−0.00267480	+	0.00463290i
$903$		0			0
$904$	1951.47	+	3380.05i	0.0717975	+	0.124357i
$905$	−10134.1	−	17552.8i	−0.372231	−	0.644723i
$906$		0			0
$907$	11827.6	−	20486.1i	0.432999	−	0.749977i	−0.564131	−	0.825686i	$-0.690789\pi$
$907$	11827.6	−	20486.1i	0.432999	−	0.749977i	0.997130	+	0.0757086i	$0.0241219\pi$
$908$	−4283.43			−0.156554
$909$		0			0
$910$	1517.13			0.0552663
$911$	25239.3	−	43715.8i	0.917910	−	1.58987i	0.115327	−	0.993328i	$-0.463209\pi$
$911$	25239.3	−	43715.8i	0.917910	−	1.58987i	0.802584	−	0.596540i	$-0.203458\pi$
$912$		0			0
$913$	−497.654	−	861.963i	−0.0180394	−	0.0312451i
$914$	11643.9	+	20167.8i	0.421385	+	0.729860i
$915$		0			0
$916$	5115.79	−	8860.81i	0.184531	−	0.319617i
$917$	−12247.8			−0.441067
$918$		0			0
$919$	−14457.9			−0.518958			−0.259479	−	0.965749i	$-0.583551\pi$
$919$	−14457.9			−0.518958			−0.259479	+	0.965749i	$0.583551\pi$
$920$	−437.898	+	758.461i	−0.0156925	+	0.0271801i
$921$		0			0
$922$	−15502.9	−	26851.8i	−0.553753	−	0.959129i
$923$	3851.17	+	6670.43i	0.137338	+	0.237876i
$924$		0			0
$925$	−2789.76	+	4832.00i	−0.0991640	+	0.171757i
$926$	−13041.4			−0.462816
$927$		0			0
$928$	−3465.81			−0.122598
$929$	−13994.7	+	24239.5i	−0.494242	+	0.856052i	−0.999978	−	0.00663591i	$-0.997888\pi$
$929$	−13994.7	+	24239.5i	−0.494242	+	0.856052i	0.505736	+	0.862688i	$0.331221\pi$
$930$		0			0
$931$	−5045.76	−	8739.52i	−0.177624	−	0.307654i
$932$	5065.36	+	8773.47i	0.178027	+	0.308352i
$933$		0			0
$934$	11488.0	−	19897.8i	0.402461	−	0.697083i
$935$	719.943			0.0251815
$936$		0			0
$937$	33622.8			1.17226			0.586130	−	0.810217i	$-0.300651\pi$
$937$	33622.8			1.17226			0.586130	+	0.810217i	$0.300651\pi$
$938$	7639.75	−	13232.4i	0.265935	−	0.460613i
$939$		0			0
$940$	−4826.45	−	8359.65i	−0.167470	−	0.290066i
$941$	−11910.6	−	20629.8i	−0.412621	−	0.714680i	0.582555	−	0.812791i	$-0.302053\pi$
$941$	−11910.6	−	20629.8i	−0.412621	−	0.714680i	−0.995175	+	0.0981115i	$0.968720\pi$
$942$		0			0
$943$	−753.647	+	1305.36i	−0.0260256	+	0.0450776i
$944$	10524.9			0.362877
$945$		0			0
$946$	−746.812			−0.0256670
$947$	11035.5	−	19114.0i	0.378675	−	0.655884i	−0.612195	−	0.790707i	$-0.709713\pi$
$947$	11035.5	−	19114.0i	0.378675	−	0.655884i	0.990870	+	0.134823i	$0.0430465\pi$
$948$		0			0
$949$	723.410	+	1252.98i	0.0247449	+	0.0428594i
$950$	−1167.82	−	2022.72i	−0.0398833	−	0.0690798i
$951$		0			0
$952$	−6165.76	+	10679.4i	−0.209909	+	0.363573i
$953$	−7748.53			−0.263378			−0.131689	−	0.991291i	$-0.542040\pi$
$953$	−7748.53			−0.263378			−0.131689	+	0.991291i	$0.542040\pi$
$954$		0			0
$955$	7398.28			0.250683
$956$	−2304.98	+	3992.34i	−0.0779794	+	0.135064i
$957$		0			0
$958$	17639.4	+	30552.3i	0.594888	+	1.03038i
$959$	−7978.18	−	13818.6i	−0.268643	−	0.465304i
$960$		0			0
$961$	13265.0	−	22975.7i	0.445269	−	0.771228i
$962$	6009.85			0.201419
$963$		0			0
$964$	27783.1			0.928250
$965$	−5919.22	+	10252.4i	−0.197457	+	0.342006i
$966$		0			0
$967$	−20402.4	−	35337.9i	−0.678486	−	1.17517i	−0.975437	−	0.220279i	$-0.929303\pi$
$967$	−20402.4	−	35337.9i	−0.678486	−	1.17517i	0.296951	−	0.954893i	$-0.404030\pi$
$968$	−5319.57	−	9213.76i	−0.176630	−	0.305931i
$969$		0			0
$970$	5936.43	−	10282.2i	0.196502	−	0.340352i
$971$	−28616.3			−0.945768			−0.472884	−	0.881125i	$-0.656787\pi$
$971$	−28616.3			−0.945768			−0.472884	+	0.881125i	$0.656787\pi$
$972$		0			0
$973$	5989.87			0.197355
$974$	−1182.88	+	2048.81i	−0.0389137	+	0.0674005i
$975$		0			0
$976$	308.062	+	533.579i	0.0101033	+	0.0174994i
$977$	−6507.19	−	11270.8i	−0.213085	−	0.369073i	0.739594	−	0.673053i	$-0.235018\pi$
$977$	−6507.19	−	11270.8i	−0.213085	−	0.369073i	−0.952678	+	0.303980i	$0.901684\pi$
$978$		0			0
$979$	303.787	−	526.174i	0.00991733	−	0.0171773i
$980$	4320.67			0.140835
$981$		0			0
$982$	−994.414			−0.0323147
$983$	−16704.7	+	28933.4i	−0.542012	+	0.938792i	0.456777	+	0.889581i	$0.349004\pi$
$983$	−16704.7	+	28933.4i	−0.542012	+	0.938792i	−0.998788	+	0.0492105i	$0.984329\pi$
$984$		0			0
$985$	9008.97	+	15604.0i	0.291421	+	0.504756i
$986$	14816.2	+	25662.4i	0.478543	+	0.828860i
$987$		0			0
$988$	−1257.89	+	2178.73i	−0.0405049	+	0.0701566i
$989$	−7767.43			−0.249737
$990$		0			0
$991$	51837.9			1.66164			0.830820	−	0.556541i	$-0.187872\pi$
$991$	51837.9			1.66164			0.830820	+	0.556541i	$0.187872\pi$
$992$	−913.682	+	1582.54i	−0.0292434	+	0.0506510i
$993$		0			0
$994$	−6446.00	−	11164.8i	−0.205689	−	0.356264i
$995$	4606.13	+	7978.05i	0.146758	+	0.254192i
$996$		0			0
$997$	−13694.7	+	23719.9i	−0.435020	+	0.753476i	−0.997297	−	0.0734724i	$-0.976592\pi$
$997$	−13694.7	+	23719.9i	−0.435020	+	0.753476i	0.562278	+	0.826949i	$0.309925\pi$
$998$	10552.1			0.334689
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	810.4.e.bf.541.1		4
3.2	odd	2		810.4.e.bb.541.1		4
9.2	odd	6		810.4.a.m.1.2	yes	2
9.4	even	3	inner	810.4.e.bf.271.1		4
9.5	odd	6		810.4.e.bb.271.1		4
9.7	even	3		810.4.a.g.1.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
810.4.a.g.1.2	✓	2	9.7	even	3
810.4.a.m.1.2	yes	2	9.2	odd	6
810.4.e.bb.271.1		4	9.5	odd	6
810.4.e.bb.541.1		4	3.2	odd	2
810.4.e.bf.271.1		4	9.4	even	3	inner
810.4.e.bf.541.1		4	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 810 = 2 \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	810.e (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.00000 - 1.73205i) q^{2} +(-2.00000 - 3.46410i) q^{4} +(2.50000 + 4.33013i) q^{5} +(5.63397 - 9.75833i) q^{7} -8.00000 q^{8} +10.0000 q^{10} +(0.526279 - 0.911543i) q^{11} +(6.73205 + 11.6603i) q^{13} +(-11.2679 - 19.5167i) q^{14} +(-8.00000 + 13.8564i) q^{16} +136.799 q^{17} -46.7128 q^{19} +(10.0000 - 17.3205i) q^{20} +(-1.05256 - 1.82309i) q^{22} +(-10.9474 - 18.9615i) q^{23} +(-12.5000 + 21.6506i) q^{25} +26.9282 q^{26} -45.0718 q^{28} +(-54.1532 + 93.7961i) q^{29} +(28.5526 + 49.4545i) q^{31} +(16.0000 + 27.7128i) q^{32} +(136.799 - 236.942i) q^{34} +56.3397 q^{35} +223.181 q^{37} +(-46.7128 + 80.9090i) q^{38} +(-20.0000 - 34.6410i) q^{40} +(-34.4212 - 59.6192i) q^{41} +(177.380 - 307.231i) q^{43} -4.21024 q^{44} -43.7898 q^{46} +(241.322 - 417.983i) q^{47} +(108.017 + 187.090i) q^{49} +(25.0000 + 43.3013i) q^{50} +(26.9282 - 46.6410i) q^{52} -110.445 q^{53} +5.26279 q^{55} +(-45.0718 + 78.0666i) q^{56} +(108.306 + 187.592i) q^{58} +(-328.903 - 569.677i) q^{59} +(19.2539 - 33.3487i) q^{61} +114.210 q^{62} +64.0000 q^{64} +(-33.6603 + 58.3013i) q^{65} +(339.004 + 587.172i) q^{67} +(-273.597 - 473.885i) q^{68} +(56.3397 - 97.5833i) q^{70} +572.065 q^{71} +107.458 q^{73} +(223.181 - 386.560i) q^{74} +(93.4256 + 161.818i) q^{76} +(-5.93009 - 10.2712i) q^{77} +(51.8653 - 89.8334i) q^{79} -80.0000 q^{80} -137.685 q^{82} +(472.804 - 818.921i) q^{83} +(341.997 + 592.356i) q^{85} +(-354.760 - 614.463i) q^{86} +(-4.21024 + 7.29234i) q^{88} +577.235 q^{89} +151.713 q^{91} +(-43.7898 + 75.8461i) q^{92} +(-482.645 - 835.965i) q^{94} +(-116.782 - 202.272i) q^{95} +(593.643 - 1028.22i) q^{97} +432.067 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{2} - 8 q^{4} + 10 q^{5} + 26 q^{7} - 32 q^{8} + 40 q^{10} - 36 q^{11} + 20 q^{13} - 52 q^{14} - 32 q^{16} + 180 q^{17} - 76 q^{19} + 40 q^{20} + 72 q^{22} - 120 q^{23} - 50 q^{25} + 80 q^{26}+ \cdots + 1368 q^{98}+O(q^{100}) \) 4 * q + 4 * q^2 - 8 * q^4 + 10 * q^5 + 26 * q^7 - 32 * q^8 + 40 * q^10 - 36 * q^11 + 20 * q^13 - 52 * q^14 - 32 * q^16 + 180 * q^17 - 76 * q^19 + 40 * q^20 + 72 * q^22 - 120 * q^23 - 50 * q^25 + 80 * q^26 - 208 * q^28 - 324 * q^29 + 38 * q^31 + 64 * q^32 + 180 * q^34 + 260 * q^35 - 292 * q^37 - 76 * q^38 - 80 * q^40 - 252 * q^41 + 422 * q^43 + 288 * q^44 - 480 * q^46 + 366 * q^47 + 342 * q^49 + 100 * q^50 + 80 * q^52 - 324 * q^53 - 360 * q^55 - 208 * q^56 + 648 * q^58 - 72 * q^59 + 368 * q^61 + 152 * q^62 + 256 * q^64 - 100 * q^65 + 296 * q^67 - 360 * q^68 + 260 * q^70 + 480 * q^71 - 1420 * q^73 - 292 * q^74 + 152 * q^76 + 534 * q^77 - 520 * q^79 - 320 * q^80 - 1008 * q^82 - 66 * q^83 + 450 * q^85 - 844 * q^86 + 288 * q^88 + 2496 * q^89 + 496 * q^91 - 480 * q^92 - 732 * q^94 - 190 * q^95 + 1706 * q^97 + 1368 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more