LMFDB - Embedded newform 432.3.bc.a.113.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [432,3,Mod(65,432)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(432, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([0, 0, 13]))

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

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

chi := DirichletCharacter("432.65");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 432 = 2^{4} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	432.bc (of order $18$, degree $6$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$11.7711474204$
Analytic rank:	$0$
Dimension:	$30$
Relative dimension:	$5$ over $\Q(\zeta_{18})$
Twist minimal:	no (minimal twist has level 27)
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			113.2
Character	$\chi$	$=$	432.113
Dual form			432.3.bc.a.65.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.428198 + 2.96928i) q^{3} +(2.62195 + 7.20376i) q^{5} +(-0.231638 - 1.31369i) q^{7} +(-8.63329 + 2.54288i) q^{9} +O(q^{10})$	\(q+(0.428198 + 2.96928i) q^{3} +(2.62195 + 7.20376i) q^{5} +(-0.231638 - 1.31369i) q^{7} +(-8.63329 + 2.54288i) q^{9} +(-0.367624 + 1.01004i) q^{11} +(16.0318 + 13.4523i) q^{13} +(-20.2673 + 10.8700i) q^{15} +(-12.1867 - 7.03599i) q^{17} +(9.79833 + 16.9712i) q^{19} +(3.80152 - 1.25032i) q^{21} +(1.76488 + 0.311197i) q^{23} +(-25.8684 + 21.7062i) q^{25} +(-11.2473 - 24.5458i) q^{27} +(-25.8129 - 30.7626i) q^{29} +(2.95677 - 16.7687i) q^{31} +(-3.15651 - 0.659083i) q^{33} +(8.85613 - 5.11309i) q^{35} +(1.80012 - 3.11791i) q^{37} +(-33.0788 + 53.3631i) q^{39} +(-3.09265 + 3.68568i) q^{41} +(-16.1462 - 5.87675i) q^{43} +(-40.9544 - 55.5248i) q^{45} +(-45.1184 + 7.95560i) q^{47} +(44.3728 - 16.1504i) q^{49} +(15.6735 - 39.1985i) q^{51} +51.2852i q^{53} -8.23996 q^{55} +(-46.1967 + 36.3611i) q^{57} +(32.0502 + 88.0571i) q^{59} +(-3.88855 - 22.0530i) q^{61} +(5.34035 + 10.7524i) q^{63} +(-54.8722 + 150.760i) q^{65} +(14.9624 + 12.5549i) q^{67} +(-0.168312 + 5.37369i) q^{69} +(-74.9736 - 43.2860i) q^{71} +(18.0755 + 31.3076i) q^{73} +(-75.5285 - 67.5161i) q^{75} +(1.41203 + 0.248979i) q^{77} +(22.3922 - 18.7892i) q^{79} +(68.0675 - 43.9069i) q^{81} +(76.1905 + 90.8003i) q^{83} +(18.7326 - 106.238i) q^{85} +(80.2898 - 89.8182i) q^{87} +(104.884 - 60.5550i) q^{89} +(13.9585 - 24.1768i) q^{91} +(51.0571 + 1.59918i) q^{93} +(-96.5657 + 115.083i) q^{95} +(9.05396 + 3.29537i) q^{97} +(0.605396 - 9.65478i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 30 q + 6 q^{3} - 15 q^{5} + 6 q^{7}+O(q^{10}) $ 30 * q + 6 * q^3 - 15 * q^5 + 6 * q^7	$ 30 q + 6 q^{3} - 15 q^{5} + 6 q^{7} + 6 q^{11} - 6 q^{13} + 9 q^{15} - 9 q^{17} + 3 q^{19} + 132 q^{21} - 120 q^{23} - 15 q^{25} + 90 q^{27} - 168 q^{29} - 39 q^{31} - 207 q^{33} + 252 q^{35} - 3 q^{37} - 15 q^{39} + 228 q^{41} + 96 q^{43} + 477 q^{45} - 399 q^{47} - 78 q^{49} - 36 q^{51} + 12 q^{55} - 192 q^{57} + 474 q^{59} + 138 q^{61} + 585 q^{63} - 411 q^{65} - 354 q^{67} + 99 q^{69} - 315 q^{71} - 66 q^{73} - 255 q^{75} + 201 q^{77} - 30 q^{79} + 36 q^{81} + 33 q^{83} - 261 q^{85} + 279 q^{87} + 72 q^{89} - 96 q^{91} + 591 q^{93} - 681 q^{95} - 582 q^{97} - 513 q^{99}+O(q^{100}) $ 30 * q + 6 * q^3 - 15 * q^5 + 6 * q^7 + 6 * q^11 - 6 * q^13 + 9 * q^15 - 9 * q^17 + 3 * q^19 + 132 * q^21 - 120 * q^23 - 15 * q^25 + 90 * q^27 - 168 * q^29 - 39 * q^31 - 207 * q^33 + 252 * q^35 - 3 * q^37 - 15 * q^39 + 228 * q^41 + 96 * q^43 + 477 * q^45 - 399 * q^47 - 78 * q^49 - 36 * q^51 + 12 * q^55 - 192 * q^57 + 474 * q^59 + 138 * q^61 + 585 * q^63 - 411 * q^65 - 354 * q^67 + 99 * q^69 - 315 * q^71 - 66 * q^73 - 255 * q^75 + 201 * q^77 - 30 * q^79 + 36 * q^81 + 33 * q^83 - 261 * q^85 + 279 * q^87 + 72 * q^89 - 96 * q^91 + 591 * q^93 - 681 * q^95 - 582 * q^97 - 513 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/432\mathbb{Z}\right)^\times$.

$n$	$271$	$325$	$353$
$\chi(n)$	$1$	$1$	$e\left(\frac{5}{18}\right)$

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 $

$2$		0			0
$2$		0			0
$3$	0.428198	+	2.96928i	0.142733	+	0.989761i
$3$	0.428198	+	2.96928i	0.142733	+	0.989761i
$4$		0			0
$5$	2.62195	+	7.20376i	0.524391	+	1.44075i	0.865586	+	0.500760i	$0.166946\pi$
$5$	2.62195	+	7.20376i	0.524391	+	1.44075i	−0.341196	+	0.939992i	$0.610832\pi$
$6$		0			0
$7$	−0.231638	−	1.31369i	−0.0330912	−	0.187669i	0.963782	−	0.266693i	$-0.0859309\pi$
$7$	−0.231638	−	1.31369i	−0.0330912	−	0.187669i	−0.996873	+	0.0790234i	$0.974820\pi$
$8$		0			0
$9$	−8.63329	+	2.54288i	−0.959255	+	0.282542i
$10$		0			0
$11$	−0.367624	+	1.01004i	−0.0334203	+	0.0918217i	−0.955281	−	0.295701i	$-0.904447\pi$
$11$	−0.367624	+	1.01004i	−0.0334203	+	0.0918217i	0.921860	+	0.387523i	$0.126669\pi$
$12$		0			0
$13$	16.0318	+	13.4523i	1.23321	+	1.03479i	0.998024	+	0.0628396i	$0.0200156\pi$
$13$	16.0318	+	13.4523i	1.23321	+	1.03479i	0.235190	+	0.971949i	$0.424429\pi$
$14$		0			0
$15$	−20.2673	+	10.8700i	−1.35115	+	0.724664i
$16$		0			0
$17$	−12.1867	−	7.03599i	−0.716864	−	0.413881i	0.0967335	−	0.995310i	$-0.469161\pi$
$17$	−12.1867	−	7.03599i	−0.716864	−	0.413881i	−0.813597	+	0.581429i	$0.802494\pi$
$18$		0			0
$19$	9.79833	+	16.9712i	0.515702	+	0.893221i	0.999834	+	0.0182265i	$0.00580199\pi$
$19$	9.79833	+	16.9712i	0.515702	+	0.893221i	−0.484132	+	0.874995i	$0.660865\pi$
$20$		0			0
$21$	3.80152	−	1.25032i	0.181025	−	0.0595389i
$22$		0			0
$23$	1.76488	+	0.311197i	0.0767341	+	0.0135303i	0.211883	−	0.977295i	$-0.432040\pi$
$23$	1.76488	+	0.311197i	0.0767341	+	0.0135303i	−0.135149	+	0.990825i	$0.543151\pi$
$24$		0			0
$25$	−25.8684	+	21.7062i	−1.03474	+	0.868246i
$26$		0			0
$27$	−11.2473	−	24.5458i	−0.416567	−	0.909105i
$28$		0			0
$29$	−25.8129	−	30.7626i	−0.890099	−	1.06078i	−0.997780	−	0.0665930i	$-0.978787\pi$
$29$	−25.8129	−	30.7626i	−0.890099	−	1.06078i	0.107681	−	0.994185i	$-0.465657\pi$
$30$		0			0
$31$	2.95677	−	16.7687i	0.0953798	−	0.540926i	−0.899250	−	0.437434i	$-0.855887\pi$
$31$	2.95677	−	16.7687i	0.0953798	−	0.540926i	0.994630	−	0.103492i	$-0.0330016\pi$
$32$		0			0
$33$	−3.15651	−	0.659083i	−0.0956517	−	0.0199722i
$34$		0			0
$35$	8.85613	−	5.11309i	0.253032	−	0.146088i
$36$		0			0
$37$	1.80012	−	3.11791i	0.0486520	−	0.0842678i	−0.840674	−	0.541542i	$-0.817841\pi$
$37$	1.80012	−	3.11791i	0.0486520	−	0.0842678i	0.889326	+	0.457274i	$0.151174\pi$
$38$		0			0
$39$	−33.0788	+	53.3631i	−0.848174	+	1.36829i
$40$		0			0
$41$	−3.09265	+	3.68568i	−0.0754305	+	0.0898946i	−0.802440	−	0.596733i	$-0.796465\pi$
$41$	−3.09265	+	3.68568i	−0.0754305	+	0.0898946i	0.727009	+	0.686627i	$0.240910\pi$
$42$		0			0
$43$	−16.1462	−	5.87675i	−0.375494	−	0.136669i	0.147377	−	0.989080i	$-0.452917\pi$
$43$	−16.1462	−	5.87675i	−0.375494	−	0.136669i	−0.522871	+	0.852412i	$0.675139\pi$
$44$		0			0
$45$	−40.9544	−	55.5248i	−0.910098	−	1.23389i
$46$		0			0
$47$	−45.1184	+	7.95560i	−0.959966	+	0.169268i	−0.631610	−	0.775286i	$-0.717606\pi$
$47$	−45.1184	+	7.95560i	−0.959966	+	0.169268i	−0.328356	+	0.944554i	$0.606495\pi$
$48$		0			0
$49$	44.3728	−	16.1504i	0.905568	−	0.329600i
$50$		0			0
$51$	15.6735	−	39.1985i	0.307324	−	0.768598i
$52$		0			0
$53$			51.2852i			0.967645i	0.875166	+	0.483822i	$0.160752\pi$
$53$			51.2852i			0.967645i	−0.875166	+	0.483822i	$0.839248\pi$
$54$		0			0
$55$	−8.23996			−0.149818
$56$		0			0
$57$	−46.1967	+	36.3611i	−0.810468	+	0.637913i
$58$		0			0
$59$	32.0502	+	88.0571i	0.543223	+	1.49249i	0.842697	+	0.538389i	$0.180967\pi$
$59$	32.0502	+	88.0571i	0.543223	+	1.49249i	−0.299473	+	0.954105i	$0.596811\pi$
$60$		0			0
$61$	−3.88855	−	22.0530i	−0.0637467	−	0.361525i	−0.999949	−	0.0100647i	$-0.996796\pi$
$61$	−3.88855	−	22.0530i	−0.0637467	−	0.361525i	0.936203	−	0.351461i	$-0.114315\pi$
$62$		0			0
$63$	5.34035	+	10.7524i	0.0847675	+	0.170673i
$64$		0			0
$65$	−54.8722	+	150.760i	−0.844188	+	2.31939i
$66$		0			0
$67$	14.9624	+	12.5549i	0.223319	+	0.187387i	0.747582	−	0.664169i	$-0.231215\pi$
$67$	14.9624	+	12.5549i	0.223319	+	0.187387i	−0.524263	+	0.851556i	$0.675659\pi$
$68$		0			0
$69$	−0.168312	+	5.37369i	−0.00243930	+	0.0778796i
$70$		0			0
$71$	−74.9736	−	43.2860i	−1.05597	−	0.609663i	−0.131653	−	0.991296i	$-0.542028\pi$
$71$	−74.9736	−	43.2860i	−1.05597	−	0.609663i	−0.924314	+	0.381633i	$0.875362\pi$
$72$		0			0
$73$	18.0755	+	31.3076i	0.247609	+	0.428871i	0.962862	−	0.269994i	$-0.0870218\pi$
$73$	18.0755	+	31.3076i	0.247609	+	0.428871i	−0.715253	+	0.698866i	$0.753689\pi$
$74$		0			0
$75$	−75.5285	−	67.5161i	−1.00705	−	0.900214i
$76$		0			0
$77$	1.41203	+	0.248979i	0.0183380	+	0.00323349i
$78$		0			0
$79$	22.3922	−	18.7892i	0.283445	−	0.237839i	−0.489969	−	0.871740i	$-0.662992\pi$
$79$	22.3922	−	18.7892i	0.283445	−	0.237839i	0.773414	+	0.633901i	$0.218547\pi$
$80$		0			0
$81$	68.0675	−	43.9069i	0.840340	−	0.542060i
$82$		0			0
$83$	76.1905	+	90.8003i	0.917957	+	1.09398i	0.995287	+	0.0969733i	$0.0309162\pi$
$83$	76.1905	+	90.8003i	0.917957	+	1.09398i	−0.0773297	+	0.997006i	$0.524639\pi$
$84$		0			0
$85$	18.7326	−	106.238i	0.220384	−	1.24986i
$86$		0			0
$87$	80.2898	−	89.8182i	0.922871	−	1.03239i
$88$		0			0
$89$	104.884	−	60.5550i	1.17848	−	0.680394i	0.222815	−	0.974861i	$-0.428476\pi$
$89$	104.884	−	60.5550i	1.17848	−	0.680394i	0.955662	+	0.294467i	$0.0951422\pi$
$90$		0			0
$91$	13.9585	−	24.1768i	0.153390	−	0.265679i
$92$		0			0
$93$	51.0571	+	1.59918i	0.549001	+	0.0171955i
$94$		0			0
$95$	−96.5657	+	115.083i	−1.01648	+	1.21139i
$96$		0			0
$97$	9.05396	+	3.29537i	0.0933398	+	0.0339729i	0.388268	−	0.921547i	$-0.373074\pi$
$97$	9.05396	+	3.29537i	0.0933398	+	0.0339729i	−0.294928	+	0.955519i	$0.595296\pi$
$98$		0			0
$99$	0.605396	−	9.65478i	0.00611511	−	0.0975230i
$100$		0			0
$101$	3.79771	−	0.669639i	0.0376011	−	0.00663009i	−0.154816	−	0.987943i	$-0.549478\pi$
$101$	3.79771	−	0.669639i	0.0376011	−	0.00663009i	0.192417	+	0.981313i	$0.438367\pi$
$102$		0			0
$103$	115.073	−	41.8832i	1.11722	−	0.406633i	0.283580	−	0.958948i	$-0.408478\pi$
$103$	115.073	−	41.8832i	1.11722	−	0.406633i	0.833635	+	0.552315i	$0.186256\pi$
$104$		0			0
$105$	18.9744	+	24.1070i	0.180709	+	0.229590i
$106$		0			0
$107$		−	80.0804i		−	0.748415i	−0.927345	−	0.374207i	$-0.877915\pi$
$107$		−	80.0804i		−	0.748415i	0.927345	−	0.374207i	$-0.122085\pi$
$108$		0			0
$109$	−96.8098			−0.888163			−0.444081	−	0.895986i	$-0.646470\pi$
$109$	−96.8098			−0.888163			−0.444081	+	0.895986i	$0.646470\pi$
$110$		0			0
$111$	10.0288	+	4.01000i	0.0903492	+	0.0361261i
$112$		0			0
$113$	22.1070	+	60.7386i	0.195637	+	0.537509i	0.998259	−	0.0589789i	$-0.0187845\pi$
$113$	22.1070	+	60.7386i	0.195637	+	0.537509i	−0.802622	+	0.596488i	$0.796562\pi$
$114$		0			0
$115$	2.38566	+	13.5297i	0.0207449	+	0.117650i
$116$		0			0
$117$	−172.615	−	75.3704i	−1.47534	−	0.644191i
$118$		0			0
$119$	−6.42017	+	17.6393i	−0.0539510	+	0.148229i
$120$		0			0
$121$	91.8063	+	77.0347i	0.758730	+	0.636650i
$122$		0			0
$123$	−12.2681	−	7.60476i	−0.0997406	−	0.0618273i
$124$		0			0
$125$	−58.2161	−	33.6111i	−0.465729	−	0.268889i
$126$		0			0
$127$	82.7295	+	143.292i	0.651414	+	1.12828i	0.982780	+	0.184779i	$0.0591570\pi$
$127$	82.7295	+	143.292i	0.651414	+	1.12828i	−0.331366	+	0.943502i	$0.607510\pi$
$128$		0			0
$129$	10.5360	−	50.4592i	0.0816741	−	0.391157i
$130$		0			0
$131$	−36.1521	−	6.37458i	−0.275970	−	0.0486610i	0.0339502	−	0.999424i	$-0.489191\pi$
$131$	−36.1521	−	6.37458i	−0.275970	−	0.0486610i	−0.309920	+	0.950763i	$0.600302\pi$
$132$		0			0
$133$	20.0252	−	16.8031i	0.150565	−	0.126339i
$134$		0			0
$135$	147.332	−	145.381i	1.09135	−	1.07690i
$136$		0			0
$137$	74.9644	+	89.3391i	0.547185	+	0.652110i	0.966783	−	0.255600i	$-0.0822731\pi$
$137$	74.9644	+	89.3391i	0.547185	+	0.652110i	−0.419597	+	0.907710i	$0.637829\pi$
$138$		0			0
$139$	17.1095	−	97.0328i	0.123090	−	0.698078i	−0.859334	−	0.511415i	$-0.829122\pi$
$139$	17.1095	−	97.0328i	0.123090	−	0.698078i	0.982424	−	0.186663i	$-0.0597673\pi$
$140$		0			0
$141$	−42.9420	−	130.563i	−0.304553	−	0.925978i
$142$		0			0
$143$	−19.4810	+	11.2473i	−0.136230	+	0.0786527i
$144$		0			0
$145$	153.926	−	266.608i	1.06156	−	1.83867i
$146$		0			0
$147$	66.9554	+	124.840i	0.455479	+	0.849251i
$148$		0			0
$149$	−89.3217	+	106.449i	−0.599474	+	0.714426i	−0.977397	−	0.211411i	$-0.932194\pi$
$149$	−89.3217	+	106.449i	−0.599474	+	0.714426i	0.377923	+	0.925837i	$0.376638\pi$
$150$		0			0
$151$	122.547	+	44.6035i	0.811571	+	0.295388i	0.714272	−	0.699868i	$-0.246758\pi$
$151$	122.547	+	44.6035i	0.811571	+	0.295388i	0.0972982	+	0.995255i	$0.468980\pi$
$152$		0			0
$153$	123.103	+	29.7544i	0.804594	+	0.194473i
$154$		0			0
$155$	128.550	−	22.6669i	0.829356	−	0.146238i
$156$		0			0
$157$	219.522	−	79.8994i	1.39823	−	0.508913i	0.470576	−	0.882359i	$-0.344046\pi$
$157$	219.522	−	79.8994i	1.39823	−	0.508913i	0.927652	+	0.373446i	$0.121824\pi$
$158$		0			0
$159$	−152.280	+	21.9602i	−0.957737	+	0.138114i
$160$		0			0
$161$		−	2.39059i		−	0.0148484i
$162$		0			0
$163$	−65.8701			−0.404111			−0.202056	−	0.979374i	$-0.564762\pi$
$163$	−65.8701			−0.404111			−0.202056	+	0.979374i	$0.564762\pi$
$164$		0			0
$165$	−3.52834	−	24.4668i	−0.0213839	−	0.148284i
$166$		0			0
$167$	−57.3115	−	157.462i	−0.343182	−	0.942886i	−0.984465	−	0.175581i	$-0.943820\pi$
$167$	−57.3115	−	157.462i	−0.343182	−	0.942886i	0.641283	−	0.767305i	$-0.278403\pi$
$168$		0			0
$169$	46.7081	+	264.895i	0.276379	+	1.56742i
$170$		0			0
$171$	−127.748	−	121.601i	−0.747062	−	0.711119i
$172$		0			0
$173$	90.8802	−	249.691i	0.525319	−	1.44330i	−0.339206	−	0.940712i	$-0.610158\pi$
$173$	90.8802	−	249.691i	0.525319	−	1.44330i	0.864525	−	0.502590i	$-0.167620\pi$
$174$		0			0
$175$	34.5072	+	28.9550i	0.197184	+	0.165457i
$176$		0			0
$177$	−247.743	+	132.872i	−1.39968	+	0.750689i
$178$		0			0
$179$	213.546	+	123.291i	1.19299	+	0.688776i	0.958985	−	0.283458i	$-0.0914818\pi$
$179$	213.546	+	123.291i	1.19299	+	0.688776i	0.234010	+	0.972234i	$0.424815\pi$
$180$		0			0
$181$	0.670719	+	1.16172i	0.00370563	+	0.00641834i	0.867872	−	0.496787i	$-0.165487\pi$
$181$	0.670719	+	1.16172i	0.00370563	+	0.00641834i	−0.864167	+	0.503206i	$0.832154\pi$
$182$		0			0
$183$	63.8167	−	20.9893i	0.348725	−	0.114695i
$184$		0			0
$185$	27.1805	+	4.79266i	0.146922	+	0.0259062i
$186$		0			0
$187$	11.5867	−	9.72242i	0.0619611	−	0.0519916i
$188$		0			0
$189$	−29.6402	+	20.4612i	−0.156827	+	0.108260i
$190$		0			0
$191$	−29.8515	−	35.5757i	−0.156291	−	0.186260i	0.682217	−	0.731150i	$-0.261016\pi$
$191$	−29.8515	−	35.5757i	−0.156291	−	0.186260i	−0.838508	+	0.544890i	$0.816572\pi$
$192$		0			0
$193$	−24.7371	+	140.291i	−0.128171	+	0.726896i	0.851202	+	0.524838i	$0.175874\pi$
$193$	−24.7371	+	140.291i	−0.128171	+	0.726896i	−0.979374	+	0.202058i	$0.935237\pi$
$194$		0			0
$195$	−471.146	−	98.3760i	−2.41613	−	0.504493i
$196$		0			0
$197$	−46.8442	+	27.0455i	−0.237788	+	0.137287i	−0.614160	−	0.789182i	$-0.710505\pi$
$197$	−46.8442	+	27.0455i	−0.237788	+	0.137287i	0.376372	+	0.926469i	$0.377172\pi$
$198$		0			0
$199$	164.437	−	284.813i	0.826317	−	1.43122i	−0.0745915	−	0.997214i	$-0.523765\pi$
$199$	164.437	−	284.813i	0.826317	−	1.43122i	0.900909	−	0.434009i	$-0.142901\pi$
$200$		0			0
$201$	−30.8723	+	49.8035i	−0.153593	+	0.247779i
$202$		0			0
$203$	−34.4331	+	41.0358i	−0.169621	+	0.202147i
$204$		0			0
$205$	−34.6595	−	12.6150i	−0.169071	−	0.0615368i
$206$		0			0
$207$	−16.0281	+	1.80124i	−0.0774304	+	0.00870164i
$208$		0			0
$209$	−20.7437	+	3.65767i	−0.0992520	+	0.0175008i
$210$		0			0
$211$	89.0532	−	32.4127i	0.422053	−	0.153615i	−0.122258	−	0.992498i	$-0.539013\pi$
$211$	89.0532	−	32.4127i	0.422053	−	0.153615i	0.544311	+	0.838884i	$0.316791\pi$
$212$		0			0
$213$	96.4250	−	241.153i	0.452700	−	1.13217i
$214$		0			0
$215$		−	131.722i		−	0.612662i
$216$		0			0
$217$	−22.7137			−0.104671
$218$		0			0
$219$	−85.2213	+	67.0770i	−0.389138	+	0.306288i
$220$		0			0
$221$	−100.724	−	276.738i	−0.455766	−	1.25221i
$222$		0			0
$223$	14.1117	+	80.0312i	0.0632810	+	0.358884i	0.999962	+	0.00870098i	$0.00276964\pi$
$223$	14.1117	+	80.0312i	0.0632810	+	0.358884i	−0.936681	+	0.350183i	$0.886119\pi$
$224$		0			0
$225$	168.133	−	253.176i	0.747259	−	1.12523i
$226$		0			0
$227$	−57.3885	+	157.674i	−0.252813	+	0.694597i	0.746752	+	0.665102i	$0.231612\pi$
$227$	−57.3885	+	157.674i	−0.252813	+	0.694597i	−0.999565	+	0.0294950i	$0.990610\pi$
$228$		0			0
$229$	56.2458	+	47.1958i	0.245615	+	0.206095i	0.757281	−	0.653089i	$-0.226527\pi$
$229$	56.2458	+	47.1958i	0.245615	+	0.206095i	−0.511666	+	0.859184i	$0.670972\pi$
$230$		0			0
$231$	−0.134661	+	4.29933i	−0.000582948	+	0.0186118i
$232$		0			0
$233$	−159.041	−	91.8223i	−0.682579	−	0.394087i	0.118247	−	0.992984i	$-0.462272\pi$
$233$	−159.041	−	91.8223i	−0.682579	−	0.394087i	−0.800826	+	0.598897i	$0.795606\pi$
$234$		0			0
$235$	−175.609	−	304.163i	−0.747271	−	1.29431i
$236$		0			0
$237$	65.3789	+	58.4431i	0.275860	+	0.246596i
$238$		0			0
$239$	8.71524	+	1.53673i	0.0364655	+	0.00642984i	0.191851	−	0.981424i	$-0.438551\pi$
$239$	8.71524	+	1.53673i	0.0364655	+	0.00642984i	−0.155386	+	0.987854i	$0.549662\pi$
$240$		0			0
$241$	−12.1911	+	10.2295i	−0.0505853	+	0.0424461i	−0.667730	−	0.744404i	$-0.732734\pi$
$241$	−12.1911	+	10.2295i	−0.0505853	+	0.0424461i	0.617145	+	0.786850i	$0.288289\pi$
$242$		0			0
$243$	159.518	+	183.311i	0.656454	+	0.754366i
$244$		0			0
$245$	232.687	+	277.306i	0.949743	+	1.13186i
$246$		0			0
$247$	−71.2164	+	403.888i	−0.288325	+	1.63517i
$248$		0			0
$249$	−236.987	+	265.112i	−0.951756	+	1.06471i
$250$		0			0
$251$	−228.852	+	132.128i	−0.911762	+	0.526406i	−0.880998	−	0.473121i	$-0.843128\pi$
$251$	−228.852	+	132.128i	−0.911762	+	0.526406i	−0.0307645	+	0.999527i	$0.509794\pi$
$252$		0			0
$253$	−0.963134	+	1.66820i	−0.00380685	+	0.00659366i
$254$		0			0
$255$	323.472	+	10.1316i	1.26852	+	0.0397317i
$256$		0			0
$257$	−219.915	+	262.084i	−0.855699	+	1.01978i	0.143846	+	0.989600i	$0.454053\pi$
$257$	−219.915	+	262.084i	−0.855699	+	1.01978i	−0.999544	+	0.0301818i	$0.990391\pi$
$258$		0			0
$259$	−4.51293	−	1.64257i	−0.0174244	−	0.00634198i
$260$		0			0
$261$	301.076	+	199.943i	1.15355	+	0.766066i
$262$		0			0
$263$	374.609	−	66.0536i	1.42437	−	0.251154i	0.592250	−	0.805754i	$-0.298240\pi$
$263$	374.609	−	66.0536i	1.42437	−	0.251154i	0.832117	+	0.554600i	$0.187129\pi$
$264$		0			0
$265$	−369.446	+	134.467i	−1.39414	+	0.507424i
$266$		0			0
$267$	224.716	+	285.502i	0.841634	+	1.06930i
$268$		0			0
$269$			88.9301i			0.330595i	0.986244	+	0.165297i	$0.0528584\pi$
$269$			88.9301i			0.330595i	−0.986244	+	0.165297i	$0.947142\pi$
$270$		0			0
$271$	−487.123			−1.79750			−0.898751	−	0.438460i	$-0.855524\pi$
$271$	−487.123			−1.79750			−0.898751	+	0.438460i	$0.855524\pi$
$272$		0			0
$273$	77.7647	+	31.0942i	0.284852	+	0.113898i
$274$		0			0
$275$	−12.4142	−	34.1078i	−0.0451426	−	0.124028i
$276$		0			0
$277$	−15.1883	−	86.1369i	−0.0548312	−	0.310963i	0.945041	−	0.326952i	$-0.106022\pi$
$277$	−15.1883	−	86.1369i	−0.0548312	−	0.310963i	−0.999872	+	0.0159886i	$0.994910\pi$
$278$		0			0
$279$	17.1141	+	152.288i	0.0613410	+	0.545835i
$280$		0			0
$281$	−55.1861	+	151.623i	−0.196392	+	0.539582i	−0.998326	−	0.0578296i	$-0.981582\pi$
$281$	−55.1861	+	151.623i	−0.196392	+	0.539582i	0.801935	+	0.597412i	$0.203804\pi$
$282$		0			0
$283$	105.120	+	88.2058i	0.371447	+	0.311681i	0.809334	−	0.587349i	$-0.199828\pi$
$283$	105.120	+	88.2058i	0.371447	+	0.311681i	−0.437886	+	0.899030i	$0.644273\pi$
$284$		0			0
$285$	−383.062	−	237.453i	−1.34408	−	0.833168i
$286$		0			0
$287$	5.55820	+	3.20903i	0.0193666	+	0.0111813i
$288$		0			0
$289$	−45.4898	−	78.7907i	−0.157404	−	0.272632i
$290$		0			0
$291$	−5.90801	+	28.2948i	−0.0203024	+	0.0972332i
$292$		0			0
$293$	−382.393	−	67.4262i	−1.30510	−	0.230124i	−0.522492	−	0.852644i	$-0.674998\pi$
$293$	−382.393	−	67.4262i	−1.30510	−	0.230124i	−0.782604	+	0.622520i	$0.786109\pi$
$294$		0			0
$295$	−550.308	+	461.763i	−1.86545	+	1.56530i
$296$		0			0
$297$	28.9270	−	2.33656i	0.0973973	−	0.00786721i
$298$		0			0
$299$	24.1079	+	28.7307i	0.0806285	+	0.0960893i
$300$		0			0
$301$	−3.98012	+	22.5724i	−0.0132230	+	0.0749913i
$302$		0			0
$303$	3.61452	+	10.9897i	0.0119291	+	0.0362698i
$304$		0			0
$305$	148.669	−	85.8342i	0.487440	−	0.281424i
$306$		0			0
$307$	207.323	−	359.093i	0.675318	−	1.16969i	−0.301057	−	0.953606i	$-0.597340\pi$
$307$	207.323	−	359.093i	0.675318	−	1.16969i	0.976376	−	0.216080i	$-0.0693271\pi$
$308$		0			0
$309$	173.637	+	323.751i	0.561933	+	1.04774i
$310$		0			0
$311$	−280.533	+	334.326i	−0.902036	+	1.07500i	0.0947982	+	0.995497i	$0.469779\pi$
$311$	−280.533	+	334.326i	−0.902036	+	1.07500i	−0.996834	+	0.0795082i	$0.974665\pi$
$312$		0			0
$313$	386.486	+	140.669i	1.23478	+	0.449423i	0.875232	−	0.483703i	$-0.160709\pi$
$313$	386.486	+	140.669i	1.23478	+	0.449423i	0.359548	+	0.933127i	$0.382931\pi$
$314$		0			0
$315$	−63.4556	+	66.6629i	−0.201446	+	0.211628i
$316$		0			0
$317$	402.773	−	71.0197i	1.27058	−	0.224037i	0.502602	−	0.864518i	$-0.332376\pi$
$317$	402.773	−	71.0197i	1.27058	−	0.224037i	0.767974	+	0.640481i	$0.221265\pi$
$318$		0			0
$319$	40.5608	−	14.7629i	0.127150	−	0.0462788i
$320$		0			0
$321$	237.781	−	34.2902i	0.740752	−	0.106823i
$322$		0			0
$323$		−	275.764i		−	0.853757i
$324$		0			0
$325$	−706.713			−2.17450
$326$		0			0
$327$	−41.4537	−	287.456i	−0.126770	−	0.879069i
$328$		0			0
$329$	20.9023	+	57.4286i	0.0635329	+	0.174555i
$330$		0			0
$331$	56.8942	+	322.663i	0.171886	+	0.974812i	0.941677	+	0.336517i	$0.109249\pi$
$331$	56.8942	+	322.663i	0.171886	+	0.974812i	−0.769792	+	0.638295i	$0.779640\pi$
$332$		0			0
$333$	−7.61253	+	31.4953i	−0.0228605	+	0.0945805i
$334$		0			0
$335$	−51.2120	+	140.704i	−0.152872	+	0.420011i
$336$		0			0
$337$	−430.977	−	361.633i	−1.27886	−	1.07309i	−0.993400	−	0.114705i	$-0.963408\pi$
$337$	−430.977	−	361.633i	−1.27886	−	1.07309i	−0.285464	−	0.958389i	$-0.592148\pi$
$338$		0			0
$339$	−170.884	+	91.6502i	−0.504082	+	0.270354i
$340$		0			0
$341$	15.8500	+	9.15103i	0.0464811	+	0.0268359i
$342$		0			0
$343$	−64.1768	−	111.157i	−0.187104	−	0.324074i
$344$		0			0
$345$	−39.1521	+	12.8771i	−0.113484	+	0.0373249i
$346$		0			0
$347$	−240.215	−	42.3563i	−0.692261	−	0.122064i	−0.183561	−	0.983008i	$-0.558763\pi$
$347$	−240.215	−	42.3563i	−0.692261	−	0.122064i	−0.508700	+	0.860944i	$0.669874\pi$
$348$		0			0
$349$	−282.886	+	237.369i	−0.810561	+	0.680141i	−0.950742	−	0.309985i	$-0.899676\pi$
$349$	−282.886	+	237.369i	−0.810561	+	0.680141i	0.140181	+	0.990126i	$0.455232\pi$
$350$		0			0
$351$	149.883	−	544.815i	0.427017	−	1.55218i
$352$		0			0
$353$	60.3702	+	71.9464i	0.171020	+	0.203814i	0.844746	−	0.535167i	$-0.179751\pi$
$353$	60.3702	+	71.9464i	0.171020	+	0.203814i	−0.673726	+	0.738982i	$0.735307\pi$
$354$		0			0
$355$	115.245	−	653.586i	0.324633	−	1.84109i
$356$		0			0
$357$	−55.1251	−	11.5102i	−0.154412	−	0.0322415i
$358$		0			0
$359$	−89.5202	+	51.6845i	−0.249360	+	0.143968i	−0.619471	−	0.785019i	$-0.712653\pi$
$359$	−89.5202	+	51.6845i	−0.249360	+	0.143968i	0.370111	+	0.928987i	$0.379320\pi$
$360$		0			0
$361$	−11.5145	+	19.9437i	−0.0318962	+	0.0552458i
$362$		0			0
$363$	−189.427	+	305.585i	−0.521836	+	0.841832i
$364$		0			0
$365$	−178.139	+	212.298i	−0.488053	+	0.581639i
$366$		0			0
$367$	−85.2083	−	31.0133i	−0.232175	−	0.0845048i	0.223313	−	0.974747i	$-0.428313\pi$
$367$	−85.2083	−	31.0133i	−0.232175	−	0.0845048i	−0.455488	+	0.890242i	$0.650535\pi$
$368$		0			0
$369$	17.3275	−	39.6838i	0.0469580	−	0.107544i
$370$		0			0
$371$	67.3726	−	11.8796i	0.181597	−	0.0320205i
$372$		0			0
$373$	406.838	−	148.077i	1.09072	−	0.396989i	0.266831	−	0.963743i	$-0.414023\pi$
$373$	406.838	−	148.077i	1.09072	−	0.396989i	0.823887	+	0.566754i	$0.191801\pi$
$374$		0			0
$375$	74.8729	−	187.252i	0.199661	−	0.499340i
$376$		0			0
$377$		−	840.420i		−	2.22923i
$378$		0			0
$379$	−333.700			−0.880476			−0.440238	−	0.897881i	$-0.645106\pi$
$379$	−333.700			−0.880476			−0.440238	+	0.897881i	$0.645106\pi$
$380$		0			0
$381$	−390.049	+	307.005i	−1.02375	+	0.805787i
$382$		0			0
$383$	−203.470	−	559.030i	−0.531254	−	1.45961i	−0.857579	−	0.514352i	$-0.828033\pi$
$383$	−203.470	−	559.030i	−0.531254	−	1.45961i	0.326325	−	0.945258i	$-0.394190\pi$
$384$		0			0
$385$	1.90869	+	10.8247i	0.00495764	+	0.0281162i
$386$		0			0
$387$	154.339	+	9.67773i	0.398809	+	0.0250071i
$388$		0			0
$389$	106.062	−	291.404i	0.272654	−	0.749111i	−0.725491	−	0.688232i	$-0.758387\pi$
$389$	106.062	−	291.404i	0.272654	−	0.749111i	0.998145	−	0.0608795i	$-0.0193905\pi$
$390$		0			0
$391$	−19.3185	−	16.2102i	−0.0494079	−	0.0414582i
$392$		0			0
$393$	3.44771	−	110.075i	0.00877281	−	0.280090i
$394$		0			0
$395$	194.064	+	112.043i	0.491302	+	0.283653i
$396$		0			0
$397$	−69.0323	−	119.567i	−0.173885	−	0.301177i	0.765890	−	0.642972i	$-0.222299\pi$
$397$	−69.0323	−	119.567i	−0.173885	−	0.301177i	−0.939775	+	0.341794i	$0.888965\pi$
$398$		0			0
$399$	58.4679	+	52.2653i	0.146536	+	0.130991i
$400$		0			0
$401$	266.978	+	47.0754i	0.665780	+	0.117395i	0.496317	−	0.868142i	$-0.334686\pi$
$401$	266.978	+	47.0754i	0.665780	+	0.117395i	0.169463	+	0.985537i	$0.445797\pi$
$402$		0			0
$403$	272.979	−	229.057i	0.677368	−	0.568379i
$404$		0			0
$405$	494.765	+	375.220i	1.22164	+	0.926469i
$406$		0			0
$407$	2.48744	+	2.96441i	0.00611164	+	0.00728357i
$408$		0			0
$409$	50.3565	−	285.586i	0.123121	−	0.698254i	−0.859285	−	0.511497i	$-0.829091\pi$
$409$	50.3565	−	285.586i	0.123121	−	0.698254i	0.982406	−	0.186757i	$-0.0597976\pi$
$410$		0			0
$411$	−233.174	+	260.845i	−0.567332	+	0.634660i
$412$		0			0
$413$	108.255	−	62.5013i	0.262120	−	0.151335i
$414$		0			0
$415$	−454.335	+	786.932i	−1.09478	+	1.89622i
$416$		0			0
$417$	295.444	+	9.25373i	0.708499	+	0.0221912i
$418$		0			0
$419$	215.521	−	256.848i	0.514371	−	0.613003i	−0.444870	−	0.895595i	$-0.646750\pi$
$419$	215.521	−	256.848i	0.514371	−	0.613003i	0.959240	+	0.282592i	$0.0911944\pi$
$420$		0			0
$421$	−566.754	−	206.282i	−1.34621	−	0.489980i	−0.434447	−	0.900697i	$-0.643056\pi$
$421$	−566.754	−	206.282i	−1.34621	−	0.489980i	−0.911763	+	0.410717i	$0.865278\pi$
$422$		0			0
$423$	369.290	−	183.414i	0.873027	−	0.433602i
$424$		0			0
$425$	467.974	−	82.5165i	1.10112	−	0.194156i
$426$		0			0
$427$	−28.0700	+	10.2167i	−0.0657378	+	0.0239266i
$428$		0			0
$429$	−41.7382	−	53.0284i	−0.0972919	−	0.123609i
$430$		0			0
$431$		−	773.546i		−	1.79477i	−0.441247	−	0.897385i	$-0.645464\pi$
$431$		−	773.546i		−	1.79477i	0.441247	−	0.897385i	$-0.354536\pi$
$432$		0			0
$433$	71.6603			0.165497			0.0827486	−	0.996570i	$-0.473630\pi$
$433$	71.6603			0.165497			0.0827486	+	0.996570i	$0.473630\pi$
$434$		0			0
$435$	857.545	+	342.889i	1.97137	+	0.788251i
$436$		0			0
$437$	12.0115	+	33.0014i	0.0274863	+	0.0755181i
$438$		0			0
$439$	118.933	+	674.503i	0.270918	+	1.53645i	0.751636	+	0.659579i	$0.229265\pi$
$439$	118.933	+	674.503i	0.270918	+	1.53645i	−0.480717	+	0.876876i	$0.659624\pi$
$440$		0			0
$441$	−342.015	+	252.266i	−0.775544	+	0.572031i
$442$		0			0
$443$	155.729	−	427.863i	0.351534	−	0.965831i	−0.630344	−	0.776316i	$-0.717086\pi$
$443$	155.729	−	427.863i	0.351534	−	0.965831i	0.981878	−	0.189515i	$-0.0606916\pi$
$444$		0			0
$445$	711.226	+	596.789i	1.59826	+	1.34110i
$446$		0			0
$447$	−354.326	−	219.640i	−0.792675	−	0.491365i
$448$		0			0
$449$	−456.689	−	263.670i	−1.01713	−	0.587238i	−0.103856	−	0.994592i	$-0.533118\pi$
$449$	−456.689	−	263.670i	−1.01713	−	0.587238i	−0.913270	+	0.407355i	$0.866451\pi$
$450$		0			0
$451$	−2.58574	−	4.47864i	−0.00573336	−	0.00993046i
$452$		0			0
$453$	−79.9661	+	382.976i	−0.176526	+	0.845423i
$454$		0			0
$455$	210.762	+	37.1631i	0.463214	+	0.0816770i
$456$		0			0
$457$	−112.672	+	94.5433i	−0.246548	+	0.206878i	−0.757684	−	0.652622i	$-0.773669\pi$
$457$	−112.672	+	94.5433i	−0.246548	+	0.206878i	0.511136	+	0.859500i	$0.329225\pi$
$458$		0			0
$459$	−35.6369	+	378.268i	−0.0776404	+	0.824114i
$460$		0			0
$461$	−230.109	−	274.233i	−0.499152	−	0.594866i	0.456368	−	0.889791i	$-0.349150\pi$
$461$	−230.109	−	274.233i	−0.499152	−	0.594866i	−0.955521	+	0.294924i	$0.904706\pi$
$462$		0			0
$463$	−71.1591	+	403.563i	−0.153691	+	0.871627i	0.806281	+	0.591533i	$0.201477\pi$
$463$	−71.1591	+	403.563i	−0.153691	+	0.871627i	−0.959972	+	0.280095i	$0.909634\pi$
$464$		0			0
$465$	122.349	+	371.996i	0.263117	+	0.799992i
$466$		0			0
$467$	778.946	−	449.725i	1.66798	−	0.963008i	0.699252	−	0.714876i	$-0.253517\pi$
$467$	778.946	−	449.725i	1.66798	−	0.963008i	0.968726	−	0.248132i	$-0.0798167\pi$
$468$		0			0
$469$	13.0274	−	22.5641i	0.0277769	−	0.0481110i
$470$		0			0
$471$	331.243	+	617.610i	0.703276	+	1.31127i
$472$		0			0
$473$	11.8715	−	14.1479i	0.0250983	−	0.0299110i
$474$		0			0
$475$	−621.847	−	226.334i	−1.30915	−	0.476492i
$476$		0			0
$477$	−130.412	−	442.760i	−0.273401	−	0.928218i
$478$		0			0
$479$	−822.620	+	145.050i	−1.71737	+	0.302819i	−0.943708	−	0.330780i	$-0.892688\pi$
$479$	−822.620	+	145.050i	−1.71737	+	0.302819i	−0.773662	+	0.633599i	$0.781577\pi$
$480$		0			0
$481$	70.8021	−	25.7699i	0.147198	−	0.0535756i
$482$		0			0
$483$	7.09834	−	1.02364i	0.0146963	−	0.00211935i
$484$		0			0
$485$			73.8629i			0.152295i
$486$		0			0
$487$	601.667			1.23546			0.617728	−	0.786392i	$-0.288053\pi$
$487$	601.667			1.23546			0.617728	+	0.786392i	$0.288053\pi$
$488$		0			0
$489$	−28.2054	−	195.587i	−0.0576798	−	0.399973i
$490$		0			0
$491$	146.639	+	402.888i	0.298655	+	0.820547i	0.994725	+	0.102574i	$0.0327077\pi$
$491$	146.639	+	402.888i	0.298655	+	0.820547i	−0.696071	+	0.717973i	$0.745070\pi$
$492$		0			0
$493$	98.1282	+	556.513i	0.199043	+	1.12883i
$494$		0			0
$495$	71.1380	−	20.9533i	0.143713	−	0.0423298i
$496$		0			0
$497$	−39.4975	+	108.519i	−0.0794719	+	0.218347i
$498$		0			0
$499$	435.651	+	365.554i	0.873048	+	0.732574i	0.964738	−	0.263214i	$-0.0847824\pi$
$499$	435.651	+	365.554i	0.873048	+	0.732574i	−0.0916899	+	0.995788i	$0.529227\pi$
$500$		0			0
$501$	443.009	−	237.599i	0.884249	−	0.474249i
$502$		0			0
$503$	−178.281	−	102.930i	−0.354435	−	0.204633i	0.312202	−	0.950016i	$-0.398933\pi$
$503$	−178.281	−	102.930i	−0.354435	−	0.204633i	−0.666637	+	0.745383i	$0.732267\pi$
$504$		0			0
$505$	14.7813	+	25.6020i	0.0292700	+	0.0506971i
$506$		0			0
$507$	−766.547	+	252.117i	−1.51193	+	0.497272i
$508$		0			0
$509$	663.268	+	116.952i	1.30308	+	0.229768i	0.781753	−	0.623589i	$-0.214326\pi$
$509$	663.268	+	116.952i	1.30308	+	0.229768i	0.521327	+	0.853357i	$0.325437\pi$
$510$		0			0
$511$	36.9414	−	30.9975i	0.0722924	−	0.0606605i
$512$		0			0
$513$	306.368	−	431.388i	0.597208	−	0.840913i
$514$		0			0
$515$	603.433	+	719.144i	1.17172	+	1.39640i
$516$		0			0
$517$	8.55115	−	48.4960i	0.0165399	−	0.0938027i
$518$		0			0
$519$	780.319	+	162.932i	1.50351	+	0.313934i
$520$		0			0
$521$	678.655	−	391.821i	1.30260	−	0.752056i	0.321751	−	0.946824i	$-0.395729\pi$
$521$	678.655	−	391.821i	1.30260	−	0.752056i	0.980849	+	0.194768i	$0.0623954\pi$
$522$		0			0
$523$	231.463	−	400.906i	0.442569	−	0.766551i	−0.555311	−	0.831643i	$-0.687401\pi$
$523$	231.463	−	400.906i	0.442569	−	0.766551i	0.997879	+	0.0650916i	$0.0207340\pi$
$524$		0			0
$525$	−71.1996	+	114.860i	−0.135618	+	0.218781i
$526$		0			0
$527$	−154.018	+	183.551i	−0.292254	+	0.348294i
$528$		0			0
$529$	−494.079	−	179.830i	−0.933988	−	0.339944i
$530$		0			0
$531$	−500.617	−	678.723i	−0.942782	−	1.27820i
$532$		0			0
$533$	−99.1614	+	17.4848i	−0.186044	+	0.0328046i
$534$		0			0
$535$	576.880	−	209.967i	1.07828	−	0.392462i
$536$		0			0
$537$	−274.646	+	686.872i	−0.511444	+	1.27909i
$538$		0			0
$539$			50.7555i			0.0941661i
$540$		0			0
$541$	595.277			1.10033			0.550164	−	0.835057i	$-0.314565\pi$
$541$	595.277			1.10033			0.550164	+	0.835057i	$0.314565\pi$
$542$		0			0
$543$	−3.16228	+	2.48900i	−0.00582371	+	0.00458380i
$544$		0			0
$545$	−253.831	−	697.394i	−0.465744	−	1.27962i
$546$		0			0
$547$	−175.350	−	994.459i	−0.320567	−	1.81802i	−0.539154	−	0.842207i	$-0.681256\pi$
$547$	−175.350	−	994.459i	−0.320567	−	1.81802i	0.218588	−	0.975817i	$-0.429855\pi$
$548$		0			0
$549$	89.6492	+	180.502i	0.163296	+	0.328784i
$550$		0			0
$551$	269.155	−	739.497i	0.488485	−	1.34210i
$552$		0			0
$553$	−29.8701	−	25.0639i	−0.0540146	−	0.0453236i
$554$		0			0
$555$	−2.59212	+	82.7588i	−0.00467049	+	0.149115i
$556$		0			0
$557$	53.4658	+	30.8685i	0.0959889	+	0.0554192i	0.547226	−	0.836985i	$-0.315684\pi$
$557$	53.4658	+	30.8685i	0.0959889	+	0.0554192i	−0.451237	+	0.892404i	$0.649017\pi$
$558$		0			0
$559$	−179.797	−	311.418i	−0.321641	−	0.557099i
$560$		0			0
$561$	33.8300	+	30.2412i	0.0603031	+	0.0539058i
$562$		0			0
$563$	955.335	+	168.451i	1.69686	+	0.299203i	0.936597	−	0.350408i	$-0.113957\pi$
$563$	955.335	+	168.451i	1.69686	+	0.299203i	0.760267	+	0.649611i	$0.225068\pi$
$564$		0			0
$565$	−379.582	+	318.507i	−0.671827	+	0.563730i
$566$		0			0
$567$	−73.4469	−	79.2488i	−0.129536	−	0.139769i
$568$		0			0
$569$	−61.7612	−	73.6041i	−0.108543	−	0.129357i	0.709037	−	0.705171i	$-0.249130\pi$
$569$	−61.7612	−	73.6041i	−0.108543	−	0.129357i	−0.817581	+	0.575814i	$0.804685\pi$
$570$		0			0
$571$	112.283	−	636.787i	0.196642	−	1.11521i	−0.713419	−	0.700738i	$-0.752854\pi$
$571$	112.283	−	636.787i	0.196642	−	1.11521i	0.910061	−	0.414475i	$-0.136035\pi$
$572$		0			0
$573$	92.8519	−	103.871i	0.162045	−	0.181276i
$574$		0			0
$575$	−52.4096	+	30.2587i	−0.0911471	+	0.0526238i
$576$		0			0
$577$	−126.992	+	219.957i	−0.220091	+	0.381208i	−0.954835	−	0.297136i	$-0.903969\pi$
$577$	−126.992	+	219.957i	−0.220091	+	0.381208i	0.734745	+	0.678344i	$0.237302\pi$
$578$		0			0
$579$	−427.156	−	13.3791i	−0.737748	−	0.0231073i
$580$		0			0
$581$	101.634	−	121.123i	0.174930	−	0.208474i
$582$		0			0
$583$	−51.8000	−	18.8537i	−0.0888507	−	0.0323390i
$584$		0			0
$585$	90.3626	−	1441.09i	0.154466	−	2.46340i
$586$		0			0
$587$	220.174	−	38.8225i	0.375083	−	0.0661372i	0.0170710	−	0.999854i	$-0.494566\pi$
$587$	220.174	−	38.8225i	0.375083	−	0.0661372i	0.358012	+	0.933717i	$0.383455\pi$
$588$		0			0
$589$	313.557	−	114.125i	0.532354	−	0.193761i
$590$		0			0
$591$	−100.364	−	127.513i	−0.169821	−	0.215758i
$592$		0			0
$593$			69.9992i			0.118042i	0.998257	+	0.0590212i	$0.0187980\pi$
$593$			69.9992i			0.118042i	−0.998257	+	0.0590212i	$0.981202\pi$
$594$		0			0
$595$	−143.903			−0.241853
$596$		0			0
$597$	916.103	+	366.304i	1.53451	+	0.613574i
$598$		0			0
$599$	−143.261	−	393.606i	−0.239167	−	0.657105i	−0.999967	−	0.00811910i	$-0.997416\pi$
$599$	−143.261	−	393.606i	−0.239167	−	0.657105i	0.760800	−	0.648986i	$-0.224807\pi$
$600$		0			0
$601$	−137.993	−	782.595i	−0.229605	−	1.30216i	−0.853683	−	0.520793i	$-0.825636\pi$
$601$	−137.993	−	782.595i	−0.229605	−	1.30216i	0.624078	−	0.781362i	$-0.285475\pi$
$602$		0			0
$603$	−161.100	−	70.3428i	−0.267165	−	0.116655i
$604$		0			0
$605$	−314.227	+	863.332i	−0.519384	+	1.42700i
$606$		0			0
$607$	−251.711	−	211.211i	−0.414681	−	0.347959i	0.411454	−	0.911430i	$-0.365021\pi$
$607$	−251.711	−	211.211i	−0.414681	−	0.347959i	−0.826135	+	0.563472i	$0.809465\pi$
$608$		0			0
$609$	−136.591	−	84.6703i	−0.224288	−	0.139032i
$610$		0			0
$611$	−830.349	−	479.402i	−1.35900	−	0.784619i
$612$		0			0
$613$	−367.664	−	636.812i	−0.599778	−	1.03885i	−0.992853	−	0.119340i	$-0.961922\pi$
$613$	−367.664	−	636.812i	−0.599778	−	1.03885i	0.393076	−	0.919506i	$-0.371411\pi$
$614$		0			0
$615$	22.6165	−	108.316i	0.0367748	−	0.176123i
$616$		0			0
$617$	−331.915	−	58.5257i	−0.537951	−	0.0948552i	−0.101929	−	0.994792i	$-0.532502\pi$
$617$	−331.915	−	58.5257i	−0.537951	−	0.0948552i	−0.436021	+	0.899936i	$0.643613\pi$
$618$		0			0
$619$	−900.806	+	755.866i	−1.45526	+	1.22111i	−0.526633	+	0.850093i	$0.676546\pi$
$619$	−900.806	+	755.866i	−1.45526	+	1.22111i	−0.928627	+	0.371015i	$0.879010\pi$
$620$		0			0
$621$	−12.2116	−	46.8207i	−0.0196644	−	0.0753956i
$622$		0			0
$623$	−103.846	−	123.758i	−0.166686	−	0.198649i
$624$		0			0
$625$	−57.1109	+	323.892i	−0.0913775	+	0.518227i
$626$		0			0
$627$	−19.7430	−	60.0276i	−0.0314881	−	0.0957378i
$628$		0			0
$629$	−43.8751	+	25.3313i	−0.0697537	+	0.0402723i
$630$		0			0
$631$	−107.380	+	185.988i	−0.170175	+	0.294751i	−0.938481	−	0.345332i	$-0.887766\pi$
$631$	−107.380	+	185.988i	−0.170175	+	0.294751i	0.768306	+	0.640082i	$0.221100\pi$
$632$		0			0
$633$	134.375	+	250.545i	0.212283	+	0.395806i
$634$		0			0
$635$	−815.326	+	971.668i	−1.28398	+	1.53019i
$636$		0			0
$637$	928.634	+	337.995i	1.45782	+	0.530605i
$638$		0			0
$639$	757.341	+	183.052i	1.18520	+	0.286466i
$640$		0			0
$641$	−979.635	+	172.736i	−1.52829	+	0.269479i	−0.873685	−	0.486491i	$-0.838276\pi$
$641$	−979.635	+	172.736i	−1.52829	+	0.269479i	−0.654606	+	0.755970i	$0.727165\pi$
$642$		0			0
$643$	−797.989	+	290.444i	−1.24104	+	0.451702i	−0.877365	−	0.479824i	$-0.840700\pi$
$643$	−797.989	+	290.444i	−1.24104	+	0.451702i	−0.363675	+	0.931526i	$0.618478\pi$
$644$		0			0
$645$	391.121	−	56.4032i	0.606389	−	0.0874468i
$646$		0			0
$647$			350.755i			0.542125i	0.962562	+	0.271063i	$0.0873751\pi$
$647$			350.755i			0.542125i	−0.962562	+	0.271063i	$0.912625\pi$
$648$		0			0
$649$	−100.723			−0.155198
$650$		0			0
$651$	−9.72596	−	67.4435i	−0.0149400	−	0.103600i
$652$		0			0
$653$	96.2871	+	264.547i	0.147453	+	0.405125i	0.991327	−	0.131417i	$-0.0419526\pi$
$653$	96.2871	+	264.547i	0.147453	+	0.405125i	−0.843874	+	0.536542i	$0.819730\pi$
$654$		0			0
$655$	−48.8681	−	277.145i	−0.0746078	−	0.423122i
$656$		0			0
$657$	−235.662	−	224.324i	−0.358694	−	0.341437i
$658$		0			0
$659$	−34.4733	+	94.7146i	−0.0523115	+	0.143725i	−0.963097	−	0.269156i	$-0.913255\pi$
$659$	−34.4733	+	94.7146i	−0.0523115	+	0.143725i	0.910785	+	0.412881i	$0.135477\pi$
$660$		0			0
$661$	−555.119	−	465.801i	−0.839818	−	0.704691i	0.117705	−	0.993049i	$-0.462446\pi$
$661$	−555.119	−	465.801i	−0.839818	−	0.704691i	−0.957523	+	0.288358i	$0.906891\pi$
$662$		0			0
$663$	778.583	−	417.578i	1.17433	−	0.629830i
$664$		0			0
$665$	173.551	+	100.199i	0.260978	+	0.150676i
$666$		0			0
$667$	−35.9835	−	62.3252i	−0.0539483	−	0.0934412i
$668$		0			0
$669$	−231.593	+	76.1707i	−0.346178	+	0.113858i
$670$		0			0
$671$	23.7039	+	4.17964i	0.0353263	+	0.00622898i
$672$		0			0
$673$	313.428	−	262.997i	0.465717	−	0.390783i	−0.379512	−	0.925187i	$-0.623908\pi$
$673$	313.428	−	262.997i	0.465717	−	0.390783i	0.845229	+	0.534404i	$0.179464\pi$
$674$		0			0
$675$	823.745	+	390.826i	1.22036	+	0.579001i
$676$		0			0
$677$	−217.703	−	259.448i	−0.321570	−	0.383232i	0.580907	−	0.813970i	$-0.302698\pi$
$677$	−217.703	−	259.448i	−0.321570	−	0.383232i	−0.902477	+	0.430738i	$0.858253\pi$
$678$		0			0
$679$	2.23184	−	12.6574i	0.00328695	−	0.0186412i
$680$		0			0
$681$	−492.751	−	102.887i	−0.723570	−	0.151083i
$682$		0			0
$683$	269.651	−	155.683i	0.394804	−	0.227940i	−0.289436	−	0.957197i	$-0.593468\pi$
$683$	269.651	−	155.683i	0.394804	−	0.227940i	0.684239	+	0.729257i	$0.260134\pi$
$684$		0			0
$685$	−447.024	+	774.268i	−0.652590	+	1.13032i
$686$		0			0
$687$	−116.053	+	187.219i	−0.168928	+	0.272517i
$688$		0			0
$689$	−689.901	+	822.192i	−1.00131	+	1.19331i
$690$		0			0
$691$	777.773	+	283.086i	1.12558	+	0.409676i	0.836684	−	0.547686i	$-0.184491\pi$
$691$	777.773	+	283.086i	1.12558	+	0.409676i	0.288891	+	0.957362i	$0.406713\pi$
$692$		0			0
$693$	−12.8236	+	1.44112i	−0.0185044	+	0.00207953i
$694$		0			0
$695$	743.861	−	131.163i	1.07030	−	0.188723i
$696$		0			0
$697$	63.6215	−	23.1563i	0.0912791	−	0.0332229i
$698$		0			0
$699$	204.545	−	511.556i	0.292626	−	0.731839i
$700$		0			0
$701$			215.718i			0.307729i	0.988092	+	0.153864i	$0.0491718\pi$
$701$			215.718i			0.307729i	−0.988092	+	0.153864i	$0.950828\pi$
$702$		0			0
$703$	70.5529			0.100360
$704$		0			0
$705$	827.951	−	651.674i	1.17440	−	0.924360i
$706$		0			0
$707$	−1.75939	−	4.83389i	−0.00248853	−	0.00683718i
$708$		0			0
$709$	179.173	+	1016.14i	0.252712	+	1.43320i	0.801878	+	0.597487i	$0.203834\pi$
$709$	179.173	+	1016.14i	0.252712	+	1.43320i	−0.549167	+	0.835713i	$0.685055\pi$
$710$		0			0
$711$	−145.539	+	219.154i	−0.204696	+	0.308233i
$712$		0			0
$713$	10.4367	−	28.6747i	0.0146378	−	0.0402169i
$714$		0			0
$715$	−132.101	−	110.846i	−0.184757	−	0.155030i
$716$		0			0
$717$	−0.831146	+	26.5361i	−0.00115920	+	0.0370098i
$718$		0			0
$719$	−539.953	−	311.742i	−0.750978	−	0.433578i	0.0750689	−	0.997178i	$-0.476082\pi$
$719$	−539.953	−	311.742i	−0.750978	−	0.433578i	−0.826047	+	0.563601i	$0.809416\pi$
$720$		0			0
$721$	−81.6768	−	141.468i	−0.113283	−	0.196211i
$722$		0			0
$723$	−35.5945	−	31.8184i	−0.0492317	−	0.0440089i
$724$		0			0
$725$	1335.47	+	235.480i	1.84203	+	0.324800i
$726$		0			0
$727$	575.569	−	482.959i	0.791704	−	0.664318i	−0.154463	−	0.987999i	$-0.549365\pi$
$727$	575.569	−	482.959i	0.791704	−	0.664318i	0.946167	+	0.323680i	$0.104920\pi$
$728$		0			0
$729$	−475.997	+	552.149i	−0.652945	+	0.757406i
$730$		0			0
$731$	155.420	+	185.223i	0.212614	+	0.253383i
$732$		0			0
$733$	130.291	−	738.919i	0.177751	−	1.00807i	−0.757170	−	0.653218i	$-0.773419\pi$
$733$	130.291	−	738.919i	0.177751	−	1.00807i	0.934921	−	0.354857i	$-0.115470\pi$
$734$		0			0
$735$	−723.763	+	809.655i	−0.984711	+	1.10157i
$736$		0			0
$737$	−18.1815	+	10.4971i	−0.0246696	+	0.0142430i
$738$		0			0
$739$	−676.597	+	1171.90i	−0.915557	+	1.58579i	−0.109474	+	0.993990i	$0.534917\pi$
$739$	−676.597	+	1171.90i	−0.915557	+	1.58579i	−0.806083	+	0.591802i	$0.798417\pi$
$740$		0			0
$741$	−1229.75	−	38.5176i	−1.65959	−	0.0519806i
$742$		0			0
$743$	−545.577	+	650.194i	−0.734290	+	0.875092i	−0.995935	−	0.0900717i	$-0.971290\pi$
$743$	−545.577	+	650.194i	−0.734290	+	0.875092i	0.261646	+	0.965164i	$0.415735\pi$
$744$		0			0
$745$	−1001.03	−	364.346i	−1.34367	−	0.489056i
$746$		0			0
$747$	−888.669	−	590.162i	−1.18965	−	0.790043i
$748$		0			0
$749$	−105.200	+	18.5497i	−0.140455	+	0.0247659i
$750$		0			0
$751$	757.303	−	275.636i	1.00839	−	0.367025i	0.215577	−	0.976487i	$-0.430837\pi$
$751$	757.303	−	275.636i	1.00839	−	0.367025i	0.792816	+	0.609462i	$0.208614\pi$
$752$		0			0
$753$	−490.319	−	622.951i	−0.651155	−	0.827292i
$754$		0			0
$755$			999.749i			1.32417i
$756$		0			0
$757$	−171.631			−0.226725			−0.113363	−	0.993554i	$-0.536162\pi$
$757$	−171.631			−0.226725			−0.113363	+	0.993554i	$0.536162\pi$
$758$		0			0
$759$	−5.36576	−	2.14550i	−0.00706951	−	0.00282674i
$760$		0			0
$761$	−462.136	−	1269.71i	−0.607275	−	1.66847i	−0.736154	−	0.676814i	$-0.763360\pi$
$761$	−462.136	−	1269.71i	−0.607275	−	1.66847i	0.128880	−	0.991660i	$-0.458862\pi$
$762$		0			0
$763$	22.4248	+	127.178i	0.0293904	+	0.166681i
$764$		0			0
$765$	108.426	+	964.818i	0.141734	+	1.26120i
$766$		0			0
$767$	−670.746	+	1842.86i	−0.874506	+	2.40268i
$768$		0			0
$769$	770.979	+	646.928i	1.00257	+	0.841259i	0.987339	−	0.158625i	$-0.0507060\pi$
$769$	770.979	+	646.928i	1.00257	+	0.841259i	0.0152349	+	0.999884i	$0.495150\pi$
$770$		0			0
$771$	−872.369	−	540.765i	−1.13148	−	0.701381i
$772$		0			0
$773$	24.6722	+	14.2445i	0.0319175	+	0.0184276i	0.515874	−	0.856665i	$-0.327467\pi$
$773$	24.6722	+	14.2445i	0.0319175	+	0.0184276i	−0.483956	+	0.875092i	$0.660801\pi$
$774$		0			0
$775$	287.497	+	497.960i	0.370964	+	0.642528i
$776$		0			0
$777$	2.94484	−	14.1035i	0.00379001	−	0.0181512i
$778$		0			0
$779$	−92.8532	−	16.3725i	−0.119195	−	0.0210174i
$780$		0			0
$781$	71.2827	−	59.8133i	0.0912710	−	0.0765855i
$782$		0			0
$783$	−464.768	+	979.594i	−0.593574	+	1.25108i
$784$		0			0
$785$	1151.15	+	1371.89i	1.46644	+	1.74763i
$786$		0			0
$787$	−34.7237	+	196.928i	−0.0441216	+	0.250226i	−0.998889	−	0.0471280i	$-0.984993\pi$
$787$	−34.7237	+	196.928i	−0.0441216	+	0.250226i	0.954767	+	0.297354i	$0.0961042\pi$
$788$		0			0
$789$	356.539	+	1084.04i	0.451887	+	1.37394i
$790$		0			0
$791$	74.6706	−	43.1111i	0.0944002	−	0.0545020i
$792$		0			0
$793$	234.323	−	405.859i	0.295489	−	0.511802i
$794$		0			0
$795$	−557.468	−	1039.41i	−0.701217	−	1.30744i
$796$		0			0
$797$	−675.871	+	805.471i	−0.848018	+	1.01063i	0.151735	+	0.988421i	$0.451514\pi$
$797$	−675.871	+	805.471i	−0.848018	+	1.01063i	−0.999753	+	0.0222076i	$0.992931\pi$
$798$		0			0
$799$	605.819	+	220.500i	0.758222	+	0.275970i
$800$		0			0
$801$	−751.513	+	789.498i	−0.938219	+	0.985640i
$802$		0			0
$803$	−38.2669	+	6.74748i	−0.0476549	+	0.00840284i
$804$		0			0
$805$	17.2212	−	6.26801i	0.0213928	−	0.00778635i
$806$		0			0
$807$	−264.059	+	38.0797i	−0.327210	+	0.0471867i
$808$		0			0
$809$			482.349i			0.596229i	0.954530	+	0.298114i	$0.0963577\pi$
$809$			482.349i			0.596229i	−0.954530	+	0.298114i	$0.903642\pi$
$810$		0			0
$811$	1202.48			1.48272			0.741358	−	0.671110i	$-0.234182\pi$
$811$	1202.48			1.48272			0.741358	+	0.671110i	$0.234182\pi$
$812$		0			0
$813$	−208.585	−	1446.41i	−0.256562	−	1.77910i
$814$		0			0
$815$	−172.708	−	474.512i	−0.211912	−	0.582224i
$816$		0			0
$817$	−58.4707	−	331.604i	−0.0715675	−	0.405880i
$818$		0			0
$819$	−59.0289	+	244.220i	−0.0720743	+	0.298193i
$820$		0			0
$821$	443.863	−	1219.50i	0.540637	−	1.48539i	−0.305380	−	0.952231i	$-0.598783\pi$
$821$	443.863	−	1219.50i	0.540637	−	1.48539i	0.846016	−	0.533157i	$-0.178994\pi$
$822$		0			0
$823$	−810.125	−	679.776i	−0.984356	−	0.825973i	0.000384609	−	1.00000i	$-0.499878\pi$
$823$	−810.125	−	679.776i	−0.984356	−	0.825973i	−0.984741	+	0.174027i	$0.944322\pi$
$824$		0			0
$825$	95.9599	−	51.4662i	0.116315	−	0.0623833i
$826$		0			0
$827$	767.759	+	443.266i	0.928367	+	0.535993i	0.886295	−	0.463122i	$-0.153271\pi$
$827$	767.759	+	443.266i	0.928367	+	0.535993i	0.0420722	+	0.999115i	$0.486604\pi$
$828$		0			0
$829$	−406.625	−	704.295i	−0.490500	−	0.849572i	0.509440	−	0.860506i	$-0.329853\pi$
$829$	−406.625	−	704.295i	−0.490500	−	0.849572i	−0.999940	+	0.0109347i	$0.996519\pi$
$830$		0			0
$831$	249.261	−	81.9819i	0.299953	−	0.0986545i
$832$		0			0
$833$	−654.391	−	115.387i	−0.785584	−	0.138520i
$834$		0			0
$835$	984.050	−	825.716i	1.17850	−	0.988881i
$836$		0			0
$837$	−444.858	+	116.026i	−0.531491	+	0.138621i
$838$		0			0
$839$	−960.723	−	1144.94i	−1.14508	−	1.36465i	−0.920756	−	0.390140i	$-0.872427\pi$
$839$	−960.723	−	1144.94i	−1.14508	−	1.36465i	−0.224325	−	0.974514i	$-0.572018\pi$
$840$		0			0
$841$	−133.994	+	759.917i	−0.159327	+	0.903588i
$842$		0			0
$843$	−473.841	−	98.9388i	−0.562089	−	0.117365i
$844$		0			0
$845$	−1785.77	+	1031.02i	−2.11334	+	1.22014i
$846$		0			0
$847$	79.9335	−	138.449i	0.0943725	−	0.163458i
$848$		0			0
$849$	−216.896	+	349.899i	−0.255472	+	0.412131i
$850$		0			0
$851$	4.14729	−	4.94255i	0.00487344	−	0.00580793i
$852$		0			0
$853$	786.440	+	286.241i	0.921969	+	0.335569i	0.759022	−	0.651065i	$-0.225678\pi$
$853$	786.440	+	286.241i	0.921969	+	0.335569i	0.162947	+	0.986635i	$0.447900\pi$
$854$		0			0
$855$	541.039	−	1239.10i	0.632794	−	1.44924i
$856$		0			0
$857$	781.650	−	137.826i	0.912077	−	0.160824i	0.302131	−	0.953266i	$-0.402302\pi$
$857$	781.650	−	137.826i	0.912077	−	0.160824i	0.609947	+	0.792443i	$0.291191\pi$
$858$		0			0
$859$	−925.160	+	336.731i	−1.07702	+	0.392003i	−0.818798	−	0.574082i	$-0.805359\pi$
$859$	−925.160	+	336.731i	−1.07702	+	0.392003i	−0.258222	+	0.966086i	$0.583137\pi$
$860$		0			0
$861$	−7.14851	+	17.8780i	−0.00830256	+	0.0207642i
$862$		0			0
$863$		−	217.563i		−	0.252101i	−0.992024	−	0.126050i	$-0.959770\pi$
$863$		−	217.563i		−	0.252101i	0.992024	−	0.126050i	$-0.0402301\pi$
$864$		0			0
$865$	2037.00			2.35491
$866$		0			0
$867$	214.473	−	168.810i	0.247374	−	0.194706i
$868$		0			0
$869$	10.7460	+	29.5243i	0.0123659	+	0.0339750i
$870$		0			0
$871$	70.9814	+	402.555i	0.0814941	+	0.462176i
$872$		0			0
$873$	−86.5452	−	5.42676i	−0.0991354	−	0.00621622i
$874$		0			0
$875$	−30.6694	+	84.2634i	−0.0350507	+	0.0963010i
$876$		0			0
$877$	−237.821	−	199.556i	−0.271176	−	0.227544i	0.497051	−	0.867721i	$-0.334416\pi$
$877$	−237.821	−	199.556i	−0.271176	−	0.227544i	−0.768227	+	0.640178i	$0.778861\pi$
$878$		0			0
$879$	36.4677	−	1164.31i	0.0414877	−	1.32458i
$880$		0			0
$881$	1349.01	+	778.850i	1.53122	+	0.884053i	0.999306	+	0.0372567i	$0.0118619\pi$
$881$	1349.01	+	778.850i	1.53122	+	0.884053i	0.531918	+	0.846796i	$0.321471\pi$
$882$		0			0
$883$	−412.908	−	715.177i	−0.467619	−	0.809940i	0.531696	−	0.846935i	$-0.321555\pi$
$883$	−412.908	−	715.177i	−0.467619	−	0.809940i	−0.999315	+	0.0369949i	$0.988221\pi$
$884$		0			0
$885$	−1606.75	−	1436.30i	−1.81553	−	1.62293i
$886$		0			0
$887$	−402.913	−	71.0445i	−0.454243	−	0.0800953i	−0.0581543	−	0.998308i	$-0.518522\pi$
$887$	−402.913	−	71.0445i	−0.454243	−	0.0800953i	−0.396089	+	0.918212i	$0.629633\pi$
$888$		0			0
$889$	169.077	−	141.872i	0.190188	−	0.159587i
$890$		0			0
$891$	19.3244	+	84.8920i	0.0216884	+	0.0952772i
$892$		0			0
$893$	−577.101	−	687.762i	−0.646250	−	0.770171i
$894$		0			0
$895$	−328.250	+	1861.60i	−0.366760	+	2.08000i
$896$		0			0
$897$	−74.9867	+	83.8857i	−0.0835972	+	0.0935181i
$898$		0			0
$899$	−592.171	+	341.890i	−0.658700	+	0.380301i
$900$		0			0
$901$	360.842	−	624.996i	0.400490	−	0.693669i
$902$		0			0
$903$	−68.7281	−	2.15266i	−0.0761108	−	0.00238390i
$904$		0			0
$905$	−6.61015	+	7.87768i	−0.00730404	+	0.00870461i
$906$		0			0
$907$	370.613	+	134.892i	0.408614	+	0.148723i	0.538145	−	0.842852i	$-0.319125\pi$
$907$	370.613	+	134.892i	0.408614	+	0.148723i	−0.129532	+	0.991575i	$0.541347\pi$
$908$		0			0
$909$	−31.0839	+	15.4383i	−0.0341958	+	0.0169839i
$910$		0			0
$911$	217.128	−	38.2855i	0.238340	−	0.0420258i	−0.0532022	−	0.998584i	$-0.516943\pi$
$911$	217.128	−	38.2855i	0.238340	−	0.0420258i	0.291542	+	0.956558i	$0.405832\pi$
$912$		0			0
$913$	−119.721	+	43.5749i	−0.131129	+	0.0477272i
$914$		0			0
$915$	318.526	+	404.687i	0.348116	+	0.442281i
$916$		0			0
$917$			48.9691i			0.0534014i
$918$		0			0
$919$	873.180			0.950142			0.475071	−	0.879948i	$-0.342422\pi$
$919$	873.180			0.950142			0.475071	+	0.879948i	$0.342422\pi$
$920$		0			0
$921$	1155.03	+	461.837i	1.25410	+	0.501452i
$922$		0			0
$923$	−619.665	−	1702.52i	−0.671360	−	1.84455i
$924$		0			0
$925$	21.1115	+	119.729i	0.0228232	+	0.129437i
$926$		0			0
$927$	−886.957	+	654.208i	−0.956803	+	0.705726i
$928$		0			0
$929$	13.2277	−	36.3427i	0.0142386	−	0.0391202i	−0.932370	−	0.361506i	$-0.882263\pi$
$929$	13.2277	−	36.3427i	0.0142386	−	0.0391202i	0.946608	+	0.322386i	$0.104485\pi$
$930$		0			0
$931$	708.871	+	594.813i	0.761408	+	0.638897i
$932$		0			0
$933$	−1112.83	−	689.825i	−1.19275	−	0.739362i
$934$		0			0
$935$	100.418	+	57.9763i	0.107399	+	0.0620067i
$936$		0			0
$937$	402.676	+	697.456i	0.429750	+	0.744350i	0.996851	−	0.0792991i	$-0.0252682\pi$
$937$	402.676	+	697.456i	0.429750	+	0.744350i	−0.567100	+	0.823649i	$0.691935\pi$
$938$		0			0
$939$	−252.195	+	1207.82i	−0.268578	+	1.28628i
$940$		0			0
$941$	1320.90	+	232.910i	1.40372	+	0.247513i	0.823670	−	0.567070i	$-0.191923\pi$
$941$	1320.90	+	232.910i	1.40372	+	0.247513i	0.580047	+	0.814583i	$0.303034\pi$
$942$		0			0
$943$	−6.60514	+	5.54237i	−0.00700439	+	0.00587738i
$944$		0			0
$945$	−225.113	−	159.873i	−0.238214	−	0.169178i
$946$		0			0
$947$	−379.824	−	452.656i	−0.401081	−	0.477990i	0.527268	−	0.849699i	$-0.323216\pi$
$947$	−379.824	−	452.656i	−0.401081	−	0.477990i	−0.928349	+	0.371709i	$0.878772\pi$
$948$		0			0
$949$	−131.376	+	745.072i	−0.138437	+	0.785113i
$950$		0			0
$951$	383.344	+	1165.54i	0.403096	+	1.22559i
$952$		0			0
$953$	−314.082	+	181.335i	−0.329572	+	0.190278i	−0.655651	−	0.755064i	$-0.727606\pi$
$953$	−314.082	+	181.335i	−0.329572	+	0.190278i	0.326079	+	0.945342i	$0.394272\pi$
$954$		0			0
$955$	178.009	−	308.321i	0.186397	−	0.322849i
$956$		0			0
$957$	61.2034	+	114.115i	0.0639534	+	0.119243i
$958$		0			0
$959$	99.9989	−	119.174i	0.104274	−	0.124269i
$960$		0			0
$961$	630.598	+	229.519i	0.656189	+	0.238833i
$962$		0			0
$963$	203.635	+	691.357i	0.211459	+	0.717920i
$964$		0			0
$965$	−1075.48	+	189.636i	−1.11449	+	0.196514i
$966$		0			0
$967$	27.9723	−	10.1811i	0.0289269	−	0.0105285i	−0.327516	−	0.944846i	$-0.606212\pi$
$967$	27.9723	−	10.1811i	0.0289269	−	0.0105285i	0.356443	+	0.934317i	$0.383989\pi$
$968$		0			0
$969$	818.820	−	118.081i	0.845016	−	0.121859i
$970$		0			0
$971$			1095.66i			1.12839i	0.825643	+	0.564193i	$0.190812\pi$
$971$			1095.66i			1.12839i	−0.825643	+	0.564193i	$0.809188\pi$
$972$		0			0
$973$	−131.434			−0.135081
$974$		0			0
$975$	−302.613	−	2098.43i	−0.310372	−	2.15224i
$976$		0			0
$977$	−212.328	−	583.365i	−0.217326	−	0.597099i	0.782342	−	0.622849i	$-0.214025\pi$
$977$	−212.328	−	583.365i	−0.217326	−	0.597099i	−0.999668	+	0.0257502i	$0.991803\pi$
$978$		0			0
$979$	22.6049	+	128.199i	0.0230898	+	0.130949i
$980$		0			0
$981$	835.787	−	246.176i	0.851975	−	0.250944i
$982$		0			0
$983$	207.764	−	570.827i	0.211357	−	0.580699i	−0.788032	−	0.615634i	$-0.788900\pi$
$983$	207.764	−	570.827i	0.211357	−	0.580699i	0.999390	+	0.0349345i	$0.0111223\pi$
$984$		0			0
$985$	−317.653	−	266.542i	−0.322490	−	0.270601i
$986$		0			0
$987$	−161.572	+	86.6557i	−0.163700	+	0.0877971i
$988$		0			0
$989$	−26.6674	−	15.3964i	−0.0269640	−	0.0155677i
$990$		0			0
$991$	−269.816	−	467.335i	−0.272266	−	0.471579i	0.697176	−	0.716900i	$-0.254440\pi$
$991$	−269.816	−	467.335i	−0.272266	−	0.471579i	−0.969442	+	0.245322i	$0.921106\pi$
$992$		0			0
$993$	−933.716	+	307.098i	−0.940298	+	0.309263i
$994$		0			0
$995$	2482.87	+	437.798i	2.49535	+	0.439998i
$996$		0			0
$997$	508.600	−	426.766i	0.510130	−	0.428050i	−0.351045	−	0.936359i	$-0.614174\pi$
$997$	508.600	−	426.766i	0.510130	−	0.428050i	0.861175	+	0.508309i	$0.169729\pi$
$998$		0			0
$999$	−96.7782	−	9.11755i	−0.0968751	−	0.00912667i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	432.3.bc.a.113.2		30
4.3	odd	2		27.3.f.a.5.3	✓	30
12.11	even	2		81.3.f.a.44.3		30
27.11	odd	18	inner	432.3.bc.a.65.2		30
36.7	odd	6		243.3.f.d.53.3		30
36.11	even	6		243.3.f.a.53.3		30
36.23	even	6		243.3.f.b.215.3		30
36.31	odd	6		243.3.f.c.215.3		30
108.7	odd	18		243.3.f.b.26.3		30
108.11	even	18		27.3.f.a.11.3	yes	30
108.23	even	18		729.3.b.a.728.18		30
108.31	odd	18		729.3.b.a.728.13		30
108.43	odd	18		81.3.f.a.35.3		30
108.47	even	18		243.3.f.c.26.3		30
108.79	odd	18		243.3.f.a.188.3		30
108.83	even	18		243.3.f.d.188.3		30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.3.f.a.5.3	✓	30	4.3	odd	2
27.3.f.a.11.3	yes	30	108.11	even	18
81.3.f.a.35.3		30	108.43	odd	18
81.3.f.a.44.3		30	12.11	even	2
243.3.f.a.53.3		30	36.11	even	6
243.3.f.a.188.3		30	108.79	odd	18
243.3.f.b.26.3		30	108.7	odd	18
243.3.f.b.215.3		30	36.23	even	6
243.3.f.c.26.3		30	108.47	even	18
243.3.f.c.215.3		30	36.31	odd	6
243.3.f.d.53.3		30	36.7	odd	6
243.3.f.d.188.3		30	108.83	even	18
432.3.bc.a.65.2		30	27.11	odd	18	inner
432.3.bc.a.113.2		30	1.1	even	1	trivial
729.3.b.a.728.13		30	108.31	odd	18
729.3.b.a.728.18		30	108.23	even	18

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 432 = 2^{4} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	432.bc (of order \(18\), degree \(6\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.428198 + 2.96928i) q^{3} +(2.62195 + 7.20376i) q^{5} +(-0.231638 - 1.31369i) q^{7} +(-8.63329 + 2.54288i) q^{9} +O(q^{10})\)	\(q+(0.428198 + 2.96928i) q^{3} +(2.62195 + 7.20376i) q^{5} +(-0.231638 - 1.31369i) q^{7} +(-8.63329 + 2.54288i) q^{9} +(-0.367624 + 1.01004i) q^{11} +(16.0318 + 13.4523i) q^{13} +(-20.2673 + 10.8700i) q^{15} +(-12.1867 - 7.03599i) q^{17} +(9.79833 + 16.9712i) q^{19} +(3.80152 - 1.25032i) q^{21} +(1.76488 + 0.311197i) q^{23} +(-25.8684 + 21.7062i) q^{25} +(-11.2473 - 24.5458i) q^{27} +(-25.8129 - 30.7626i) q^{29} +(2.95677 - 16.7687i) q^{31} +(-3.15651 - 0.659083i) q^{33} +(8.85613 - 5.11309i) q^{35} +(1.80012 - 3.11791i) q^{37} +(-33.0788 + 53.3631i) q^{39} +(-3.09265 + 3.68568i) q^{41} +(-16.1462 - 5.87675i) q^{43} +(-40.9544 - 55.5248i) q^{45} +(-45.1184 + 7.95560i) q^{47} +(44.3728 - 16.1504i) q^{49} +(15.6735 - 39.1985i) q^{51} +51.2852i q^{53} -8.23996 q^{55} +(-46.1967 + 36.3611i) q^{57} +(32.0502 + 88.0571i) q^{59} +(-3.88855 - 22.0530i) q^{61} +(5.34035 + 10.7524i) q^{63} +(-54.8722 + 150.760i) q^{65} +(14.9624 + 12.5549i) q^{67} +(-0.168312 + 5.37369i) q^{69} +(-74.9736 - 43.2860i) q^{71} +(18.0755 + 31.3076i) q^{73} +(-75.5285 - 67.5161i) q^{75} +(1.41203 + 0.248979i) q^{77} +(22.3922 - 18.7892i) q^{79} +(68.0675 - 43.9069i) q^{81} +(76.1905 + 90.8003i) q^{83} +(18.7326 - 106.238i) q^{85} +(80.2898 - 89.8182i) q^{87} +(104.884 - 60.5550i) q^{89} +(13.9585 - 24.1768i) q^{91} +(51.0571 + 1.59918i) q^{93} +(-96.5657 + 115.083i) q^{95} +(9.05396 + 3.29537i) q^{97} +(0.605396 - 9.65478i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q + 6 q^{3} - 15 q^{5} + 6 q^{7}+O(q^{10}) \) 30 * q + 6 * q^3 - 15 * q^5 + 6 * q^7	\( 30 q + 6 q^{3} - 15 q^{5} + 6 q^{7} + 6 q^{11} - 6 q^{13} + 9 q^{15} - 9 q^{17} + 3 q^{19} + 132 q^{21} - 120 q^{23} - 15 q^{25} + 90 q^{27} - 168 q^{29} - 39 q^{31} - 207 q^{33} + 252 q^{35} - 3 q^{37} - 15 q^{39} + 228 q^{41} + 96 q^{43} + 477 q^{45} - 399 q^{47} - 78 q^{49} - 36 q^{51} + 12 q^{55} - 192 q^{57} + 474 q^{59} + 138 q^{61} + 585 q^{63} - 411 q^{65} - 354 q^{67} + 99 q^{69} - 315 q^{71} - 66 q^{73} - 255 q^{75} + 201 q^{77} - 30 q^{79} + 36 q^{81} + 33 q^{83} - 261 q^{85} + 279 q^{87} + 72 q^{89} - 96 q^{91} + 591 q^{93} - 681 q^{95} - 582 q^{97} - 513 q^{99}+O(q^{100}) \) 30 * q + 6 * q^3 - 15 * q^5 + 6 * q^7 + 6 * q^11 - 6 * q^13 + 9 * q^15 - 9 * q^17 + 3 * q^19 + 132 * q^21 - 120 * q^23 - 15 * q^25 + 90 * q^27 - 168 * q^29 - 39 * q^31 - 207 * q^33 + 252 * q^35 - 3 * q^37 - 15 * q^39 + 228 * q^41 + 96 * q^43 + 477 * q^45 - 399 * q^47 - 78 * q^49 - 36 * q^51 + 12 * q^55 - 192 * q^57 + 474 * q^59 + 138 * q^61 + 585 * q^63 - 411 * q^65 - 354 * q^67 + 99 * q^69 - 315 * q^71 - 66 * q^73 - 255 * q^75 + 201 * q^77 - 30 * q^79 + 36 * q^81 + 33 * q^83 - 261 * q^85 + 279 * q^87 + 72 * q^89 - 96 * q^91 + 591 * q^93 - 681 * q^95 - 582 * q^97 - 513 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more