LMFDB - Embedded newform 405.3.i.d.296.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [405,3,Mod(26,405)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(405, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("405.26"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 405 = 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	405.i (of order $6$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,12,0,0,-16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$11.0354507066$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	8.0.3317760000.8
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 4x^{6} + 7x^{4} + 36x^{2} + 81 $ x^8 + 4x^6 + 7x^4 + 36*x^2 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 45)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			296.3
Root			$0.178197 + 1.72286i$ of defining polynomial
Character	$\chi$	$=$	405.296
Dual form			405.3.i.d.26.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(0.711747 - 0.410927i) q^{2} +(-1.66228 + 2.87915i) q^{4} +(-1.93649 - 1.11803i) q^{5} +(-0.418861 - 0.725489i) q^{7} +6.01972i q^{8} -1.83772 q^{10} +(-12.4462 + 7.18582i) q^{11} +(10.9057 - 18.8892i) q^{13} +(-0.596246 - 0.344243i) q^{14} +(-4.17544 - 7.23208i) q^{16} -23.5454i q^{17} -6.32456 q^{19} +(6.43797 - 3.71697i) q^{20} +(-5.90569 + 10.2290i) q^{22} +(-33.6644 - 19.4361i) q^{23} +(2.50000 + 4.33013i) q^{25} -17.9258i q^{26} +2.78505 q^{28} +(0.231001 - 0.133369i) q^{29} +(-15.1359 + 26.2162i) q^{31} +(-26.7966 - 15.4710i) q^{32} +(-9.67544 - 16.7584i) q^{34} +1.87320i q^{35} +9.53950 q^{37} +(-4.50148 + 2.59893i) q^{38} +(6.73025 - 11.6571i) q^{40} +(-16.7167 - 9.65138i) q^{41} +(-9.81139 - 16.9938i) q^{43} -47.7793i q^{44} -31.9473 q^{46} +(-19.1984 + 11.0842i) q^{47} +(24.1491 - 41.8275i) q^{49} +(3.55873 + 2.05464i) q^{50} +(36.2566 + 62.7982i) q^{52} -49.0012i q^{53} +32.1359 q^{55} +(4.36724 - 2.52143i) q^{56} +(0.109610 - 0.189849i) q^{58} +(-63.4235 - 36.6176i) q^{59} +(24.1623 + 41.8503i) q^{61} +24.8791i q^{62} +7.97367 q^{64} +(-42.2376 + 24.3859i) q^{65} +(-38.6491 + 66.9422i) q^{67} +(67.7907 + 39.1390i) q^{68} +(0.769751 + 1.33325i) q^{70} +104.044i q^{71} +47.6754 q^{73} +(6.78971 - 3.92004i) q^{74} +(10.5132 - 18.2093i) q^{76} +(10.4265 + 6.01972i) q^{77} +(-34.1096 - 59.0796i) q^{79} +18.6732i q^{80} -15.8641 q^{82} +(-24.4304 + 14.1049i) q^{83} +(-26.3246 + 45.5955i) q^{85} +(-13.9664 - 8.06353i) q^{86} +(-43.2566 - 74.9226i) q^{88} +53.7774i q^{89} -18.2719 q^{91} +(111.919 - 64.6165i) q^{92} +(-9.10961 + 15.7783i) q^{94} +(12.2474 + 7.07107i) q^{95} +(57.4605 + 99.5245i) q^{97} -39.6941i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 12 q^{4} - 16 q^{7} - 40 q^{10} + 24 q^{13} - 84 q^{16} + 16 q^{22} + 20 q^{25} - 256 q^{28} + 56 q^{31} - 128 q^{34} + 304 q^{37} - 60 q^{40} + 48 q^{43} + 48 q^{46} + 92 q^{49} + 328 q^{52} + 80 q^{55}+ \cdots + 232 q^{97}+O(q^{100}) $ 8 * q + 12 * q^4 - 16 * q^7 - 40 * q^10 + 24 * q^13 - 84 * q^16 + 16 * q^22 + 20 * q^25 - 256 * q^28 + 56 * q^31 - 128 * q^34 + 304 * q^37 - 60 * q^40 + 48 * q^43 + 48 * q^46 + 92 * q^49 + 328 * q^52 + 80 * q^55 - 328 * q^58 + 168 * q^61 - 88 * q^64 - 208 * q^67 + 120 * q^70 + 432 * q^73 + 160 * q^76 + 56 * q^79 - 304 * q^82 - 160 * q^85 - 384 * q^88 + 208 * q^91 + 256 * q^94 + 232 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/405\mathbb{Z}\right)^\times$.

$n$	$82$	$326$
$\chi(n)$	$1$	$e\left(\frac{5}{6}\right)$

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 $

$2$	0.711747	−	0.410927i	0.355873	−	0.205464i	−0.311396	−	0.950280i	$-0.600796\pi$
$2$	0.711747	−	0.410927i	0.355873	−	0.205464i	0.667269	+	0.744817i	$0.267463\pi$
$3$		0			0
$3$		0			0
$4$	−1.66228	+	2.87915i	−0.415569	+	0.719787i
$5$	−1.93649	−	1.11803i	−0.387298	−	0.223607i
$5$	−1.93649	−	1.11803i	−0.387298	−	0.223607i
$6$		0			0
$7$	−0.418861	−	0.725489i	−0.0598373	−	0.103641i	0.834555	−	0.550925i	$-0.185725\pi$
$7$	−0.418861	−	0.725489i	−0.0598373	−	0.103641i	−0.894392	+	0.447283i	$0.852392\pi$
$8$			6.01972i			0.752465i
$9$		0			0
$10$	−1.83772			−0.183772
$11$	−12.4462	+	7.18582i	−1.13147	+	0.653256i	−0.944305	−	0.329072i	$-0.893264\pi$
$11$	−12.4462	+	7.18582i	−1.13147	+	0.653256i	−0.187168	+	0.982328i	$0.559931\pi$
$12$		0			0
$13$	10.9057	−	18.8892i	0.838900	−	1.45302i	−0.0519160	−	0.998651i	$-0.516533\pi$
$13$	10.9057	−	18.8892i	0.838900	−	1.45302i	0.890816	−	0.454365i	$-0.150134\pi$
$14$	−0.596246	−	0.344243i	−0.0425890	−	0.0245888i
$15$		0			0
$16$	−4.17544	−	7.23208i	−0.260965	−	0.452005i
$17$		−	23.5454i		−	1.38502i	−0.721407	−	0.692512i	$-0.756504\pi$
$17$		−	23.5454i		−	1.38502i	0.721407	−	0.692512i	$-0.243496\pi$
$18$		0			0
$19$	−6.32456			−0.332871			−0.166436	−	0.986052i	$-0.553226\pi$
$19$	−6.32456			−0.332871			−0.166436	+	0.986052i	$0.553226\pi$
$20$	6.43797	−	3.71697i	0.321899	−	0.185848i
$21$		0			0
$22$	−5.90569	+	10.2290i	−0.268441	+	0.464953i
$23$	−33.6644	−	19.4361i	−1.46367	−	0.845049i	−0.464489	−	0.885579i	$-0.653762\pi$
$23$	−33.6644	−	19.4361i	−1.46367	−	0.845049i	−0.999178	+	0.0405297i	$0.987095\pi$
$24$		0			0
$25$	2.50000	+	4.33013i	0.100000	+	0.173205i
$26$		−	17.9258i		−	0.689453i
$27$		0			0
$28$	2.78505			0.0994662
$29$	0.231001	−	0.133369i	0.00796556	−	0.00459892i	−0.496012	−	0.868316i	$-0.665203\pi$
$29$	0.231001	−	0.133369i	0.00796556	−	0.00459892i	0.503977	+	0.863717i	$0.331869\pi$
$30$		0			0
$31$	−15.1359	+	26.2162i	−0.488256	+	0.845685i	−0.999909	−	0.0135078i	$-0.995700\pi$
$31$	−15.1359	+	26.2162i	−0.488256	+	0.845685i	0.511653	+	0.859192i	$0.329034\pi$
$32$	−26.7966	−	15.4710i	−0.837395	−	0.483470i
$33$		0			0
$34$	−9.67544	−	16.7584i	−0.284572	−	0.492893i
$35$			1.87320i			0.0535201i
$36$		0			0
$37$	9.53950			0.257824			0.128912	−	0.991656i	$-0.458851\pi$
$37$	9.53950			0.257824			0.128912	+	0.991656i	$0.458851\pi$
$38$	−4.50148	+	2.59893i	−0.118460	+	0.0683929i
$39$		0			0
$40$	6.73025	−	11.6571i	0.168256	−	0.291428i
$41$	−16.7167	−	9.65138i	−0.407724	−	0.235399i	0.282087	−	0.959389i	$-0.408973\pi$
$41$	−16.7167	−	9.65138i	−0.407724	−	0.235399i	−0.689811	+	0.723989i	$0.742307\pi$
$42$		0			0
$43$	−9.81139	−	16.9938i	−0.228172	−	0.395205i	0.729095	−	0.684413i	$-0.239941\pi$
$43$	−9.81139	−	16.9938i	−0.228172	−	0.395205i	−0.957266	+	0.289208i	$0.906608\pi$
$44$		−	47.7793i		−	1.08589i
$45$		0			0
$46$	−31.9473			−0.694507
$47$	−19.1984	+	11.0842i	−0.408477	+	0.235834i	−0.690135	−	0.723681i	$-0.742449\pi$
$47$	−19.1984	+	11.0842i	−0.408477	+	0.235834i	0.281658	+	0.959515i	$0.409116\pi$
$48$		0			0
$49$	24.1491	−	41.8275i	0.492839	−	0.853622i
$50$	3.55873	+	2.05464i	0.0711747	+	0.0410927i
$51$		0			0
$52$	36.2566	+	62.7982i	0.697242	+	1.20766i
$53$		−	49.0012i		−	0.924552i	−0.886736	−	0.462276i	$-0.847033\pi$
$53$		−	49.0012i		−	0.924552i	0.886736	−	0.462276i	$-0.152967\pi$
$54$		0			0
$55$	32.1359			0.584290
$56$	4.36724	−	2.52143i	0.0779864	−	0.0450255i
$57$		0			0
$58$	0.109610	−	0.189849i	0.00188982	−	0.00327326i
$59$	−63.4235	−	36.6176i	−1.07497	−	0.620637i	−0.145438	−	0.989367i	$-0.546459\pi$
$59$	−63.4235	−	36.6176i	−1.07497	−	0.620637i	−0.929536	+	0.368731i	$0.879792\pi$
$60$		0			0
$61$	24.1623	+	41.8503i	0.396103	+	0.686070i	0.993241	−	0.116068i	$-0.0370290\pi$
$61$	24.1623	+	41.8503i	0.396103	+	0.686070i	−0.597138	+	0.802138i	$0.703696\pi$
$62$			24.8791i			0.401276i
$63$		0			0
$64$	7.97367			0.124589
$65$	−42.2376	+	24.3859i	−0.649809	+	0.375167i
$66$		0			0
$67$	−38.6491	+	66.9422i	−0.576852	+	0.999138i	0.418985	+	0.907993i	$0.362386\pi$
$67$	−38.6491	+	66.9422i	−0.576852	+	0.999138i	−0.995838	+	0.0911446i	$0.970947\pi$
$68$	67.7907	+	39.1390i	0.996922	+	0.575573i
$69$		0			0
$70$	0.769751	+	1.33325i	0.0109964	+	0.0190464i
$71$			104.044i			1.46541i	0.680548	+	0.732703i	$0.261742\pi$
$71$			104.044i			1.46541i	−0.680548	+	0.732703i	$0.738258\pi$
$72$		0			0
$73$	47.6754			0.653088			0.326544	−	0.945182i	$-0.394116\pi$
$73$	47.6754			0.653088			0.326544	+	0.945182i	$0.394116\pi$
$74$	6.78971	−	3.92004i	0.0917528	−	0.0529735i
$75$		0			0
$76$	10.5132	−	18.2093i	0.138331	−	0.239597i
$77$	10.4265	+	6.01972i	0.135409	+	0.0781782i
$78$		0			0
$79$	−34.1096	−	59.0796i	−0.431767	−	0.747843i	0.565258	−	0.824914i	$-0.308776\pi$
$79$	−34.1096	−	59.0796i	−0.431767	−	0.747843i	−0.997026	+	0.0770712i	$0.975443\pi$
$80$			18.6732i			0.233414i
$81$		0			0
$82$	−15.8641			−0.193464
$83$	−24.4304	+	14.1049i	−0.294342	+	0.169938i	−0.639898	−	0.768460i	$-0.721024\pi$
$83$	−24.4304	+	14.1049i	−0.294342	+	0.169938i	0.345556	+	0.938398i	$0.387690\pi$
$84$		0			0
$85$	−26.3246	+	45.5955i	−0.309701	+	0.536417i
$86$	−13.9664	−	8.06353i	−0.162401	−	0.0937620i
$87$		0			0
$88$	−43.2566	−	74.9226i	−0.491552	−	0.851393i
$89$			53.7774i			0.604240i	0.953270	+	0.302120i	$0.0976943\pi$
$89$			53.7774i			0.604240i	−0.953270	+	0.302120i	$0.902306\pi$
$90$		0			0
$91$	−18.2719			−0.200790
$92$	111.919	−	64.6165i	1.21651	−	0.702353i
$93$		0			0
$94$	−9.10961	+	15.7783i	−0.0969107	+	0.167854i
$95$	12.2474	+	7.07107i	0.128921	+	0.0744323i
$96$		0			0
$97$	57.4605	+	99.5245i	0.592376	+	1.02603i	0.993911	+	0.110182i	$0.0351434\pi$
$97$	57.4605	+	99.5245i	0.592376	+	1.02603i	−0.401535	+	0.915844i	$0.631523\pi$
$98$		−	39.6941i		−	0.405042i
$99$		0			0
$100$	−16.6228			−0.166228
$101$	−15.1964	+	8.77366i	−0.150460	+	0.0868679i	−0.573340	−	0.819318i	$-0.694352\pi$
$101$	−15.1964	+	8.77366i	−0.150460	+	0.0868679i	0.422880	+	0.906186i	$0.361019\pi$
$102$		0			0
$103$	35.7698	−	61.9550i	0.347279	−	0.601505i	−0.638486	−	0.769633i	$-0.720439\pi$
$103$	35.7698	−	61.9550i	0.347279	−	0.601505i	0.985765	+	0.168128i	$0.0537723\pi$
$104$	113.708	+	65.6492i	1.09334	+	0.631242i
$105$		0			0
$106$	−20.1359	−	34.8765i	−0.189962	−	0.329023i
$107$		−	76.3675i		−	0.713715i	−0.934159	−	0.356858i	$-0.883848\pi$
$107$		−	76.3675i		−	0.713715i	0.934159	−	0.356858i	$-0.116152\pi$
$108$		0			0
$109$	−126.921			−1.16441			−0.582206	−	0.813041i	$-0.697810\pi$
$109$	−126.921			−1.16441			−0.582206	+	0.813041i	$0.697810\pi$
$110$	22.8727	−	13.2055i	0.207933	−	0.120050i
$111$		0			0
$112$	−3.49786	+	6.05848i	−0.0312309	+	0.0540935i
$113$	13.0424	+	7.53006i	0.115420	+	0.0666377i	0.556599	−	0.830782i	$-0.312106\pi$
$113$	13.0424	+	7.53006i	0.115420	+	0.0666377i	−0.441179	+	0.897419i	$0.645440\pi$
$114$		0			0
$115$	43.4605	+	75.2758i	0.377917	+	0.654572i
$116$			0.886783i			0.00764468i
$117$		0			0
$118$	−60.1886			−0.510073
$119$	−17.0819	+	9.86225i	−0.143546	+	0.0828761i
$120$		0			0
$121$	42.7719	−	74.0831i	0.353487	−	0.612257i
$122$	34.3948	+	19.8579i	0.281925	+	0.162769i
$123$		0			0
$124$	−50.3203	−	87.1573i	−0.405809	−	0.702881i
$125$		−	11.1803i		−	0.0894427i
$126$		0			0
$127$	158.031			1.24434			0.622168	−	0.782884i	$-0.286252\pi$
$127$	158.031			1.24434			0.622168	+	0.782884i	$0.286252\pi$
$128$	112.862	−	65.1608i	0.881733	−	0.509069i
$129$		0			0
$130$	−20.0416	+	34.7131i	−0.154166	+	0.267024i
$131$	182.922	+	105.610i	1.39635	+	0.806184i	0.994008	−	0.109306i	$-0.0348627\pi$
$131$	182.922	+	105.610i	1.39635	+	0.806184i	0.402343	+	0.915489i	$0.368196\pi$
$132$		0			0
$133$	2.64911	+	4.58839i	0.0199181	+	0.0344992i
$134$			63.5279i			0.474089i
$135$		0			0
$136$	141.737			1.04218
$137$	−59.9802	+	34.6296i	−0.437812	+	0.252771i	−0.702669	−	0.711517i	$-0.748009\pi$
$137$	−59.9802	+	34.6296i	−0.437812	+	0.252771i	0.264857	+	0.964288i	$0.414675\pi$
$138$		0			0
$139$	79.9210	−	138.427i	0.574971	−	0.995879i	−0.421074	−	0.907026i	$-0.638347\pi$
$139$	79.9210	−	138.427i	0.574971	−	0.995879i	0.996045	−	0.0888529i	$-0.0283201\pi$
$140$	−5.39323	−	3.11379i	−0.0385231	−	0.0222413i
$141$		0			0
$142$	42.7544	+	74.0529i	0.301088	+	0.521499i
$143$			313.465i			2.19206i
$144$		0			0
$145$	−0.596443			−0.00411340
$146$	33.9328	−	19.5911i	0.232417	−	0.134186i
$147$		0			0
$148$	−15.8573	+	27.4656i	−0.107144	+	0.185579i
$149$	8.50348	+	4.90948i	0.0570703	+	0.0329496i	0.528264	−	0.849080i	$-0.322843\pi$
$149$	8.50348	+	4.90948i	0.0570703	+	0.0329496i	−0.471193	+	0.882030i	$0.656177\pi$
$150$		0			0
$151$	−105.325	−	182.427i	−0.697514	−	1.20813i	−0.969326	−	0.245779i	$-0.920956\pi$
$151$	−105.325	−	182.427i	−0.697514	−	1.20813i	0.271812	−	0.962350i	$-0.412377\pi$
$152$		−	38.0720i		−	0.250474i
$153$		0			0
$154$	9.89466			0.0642511
$155$	58.6213	−	33.8450i	0.378202	−	0.218355i
$156$		0			0
$157$	−105.638	+	182.971i	−0.672854	+	1.16542i	0.304237	+	0.952596i	$0.401598\pi$
$157$	−105.638	+	182.971i	−0.672854	+	1.16542i	−0.977091	+	0.212821i	$0.931735\pi$
$158$	−48.5548	−	28.0331i	−0.307309	−	0.177425i
$159$		0			0
$160$	34.5943	+	59.9191i	0.216214	+	0.374494i
$161$			32.5642i			0.202262i
$162$		0			0
$163$	−222.763			−1.36664			−0.683322	−	0.730117i	$-0.739465\pi$
$163$	−222.763			−1.36664			−0.683322	+	0.730117i	$0.739465\pi$
$164$	55.5755	−	32.0865i	0.338875	−	0.195650i
$165$		0			0
$166$	−11.5922	+	20.0782i	−0.0698323	+	0.120953i
$167$	28.8944	+	16.6822i	0.173020	+	0.0998933i	0.584009	−	0.811747i	$-0.301483\pi$
$167$	28.8944	+	16.6822i	0.173020	+	0.0998933i	−0.410989	+	0.911640i	$0.634817\pi$
$168$		0			0
$169$	−153.368	−	265.642i	−0.907505	−	1.57184i
$170$			43.2699i			0.254529i
$171$		0			0
$172$	65.2370			0.379285
$173$	−25.8164	+	14.9051i	−0.149228	+	0.0861567i	−0.572755	−	0.819727i	$-0.694125\pi$
$173$	−25.8164	+	14.9051i	−0.149228	+	0.0861567i	0.423527	+	0.905884i	$0.360792\pi$
$174$		0			0
$175$	2.09431	−	3.62744i	0.0119675	−	0.0207283i
$176$	103.937	+	60.0079i	0.590550	+	0.340954i
$177$		0			0
$178$	22.0986	+	38.2759i	0.124149	+	0.215033i
$179$		−	111.841i		−	0.624808i	−0.949949	−	0.312404i	$-0.898866\pi$
$179$		−	111.841i		−	0.624808i	0.949949	−	0.312404i	$-0.101134\pi$
$180$		0			0
$181$	−49.0790			−0.271155			−0.135577	−	0.990767i	$-0.543289\pi$
$181$	−49.0790			−0.271155			−0.135577	+	0.990767i	$0.543289\pi$
$182$	−13.0050	+	7.50842i	−0.0714558	+	0.0412550i
$183$		0			0
$184$	117.000	−	202.650i	0.635870	−	1.10136i
$185$	−18.4732	−	10.6655i	−0.0998549	−	0.0576513i
$186$		0			0
$187$	169.193	+	293.051i	0.904775	+	1.56712i
$188$		−	73.7002i		−	0.392022i
$189$		0			0
$190$	11.6228			0.0611725
$191$	−241.575	+	139.474i	−1.26479	+	0.730229i	−0.973998	−	0.226557i	$-0.927253\pi$
$191$	−241.575	+	139.474i	−1.26479	+	0.730229i	−0.290795	+	0.956785i	$0.593920\pi$
$192$		0			0
$193$	−44.9473	+	77.8511i	−0.232888	+	0.403373i	−0.958657	−	0.284565i	$-0.908151\pi$
$193$	−44.9473	+	77.8511i	−0.232888	+	0.403373i	0.725769	+	0.687939i	$0.241484\pi$
$194$	81.7947	+	47.2242i	0.421622	+	0.243424i
$195$		0			0
$196$	80.2851	+	139.058i	0.409618	+	0.709479i
$197$			212.709i			1.07974i	0.841748	+	0.539870i	$0.181527\pi$
$197$			212.709i			1.07974i	−0.841748	+	0.539870i	$0.818473\pi$
$198$		0			0
$199$	96.4911			0.484880			0.242440	−	0.970166i	$-0.422052\pi$
$199$	96.4911			0.484880			0.242440	+	0.970166i	$0.422052\pi$
$200$	−26.0661	+	15.0493i	−0.130331	+	0.0752465i
$201$		0			0
$202$	−7.21067	+	12.4893i	−0.0356964	+	0.0618280i
$203$	−0.193515	−	0.111726i	−0.000953275	−	0.000550374i
$204$		0			0
$205$	21.5811	+	37.3796i	0.105274	+	0.182340i
$206$		−	58.7951i		−	0.285413i
$207$		0			0
$208$	−182.144			−0.875695
$209$	78.7167	−	45.4471i	0.376635	−	0.217450i
$210$		0			0
$211$	32.8947	−	56.9752i	0.155899	−	0.270025i	−0.777487	−	0.628899i	$-0.783506\pi$
$211$	32.8947	−	56.9752i	0.155899	−	0.270025i	0.933386	+	0.358874i	$0.116839\pi$
$212$	141.082	+	81.4537i	0.665481	+	0.384215i
$213$		0			0
$214$	−31.3815	−	54.3543i	−0.146643	−	0.253992i
$215$			43.8779i			0.204083i
$216$		0			0
$217$	25.3594			0.116864
$218$	−90.3356	+	52.1553i	−0.414384	+	0.239244i
$219$		0			0
$220$	−53.4189	+	92.5242i	−0.242813	+	0.420564i
$221$	−444.754	−	256.779i	−2.01246	−	1.16190i
$222$		0			0
$223$	−51.1512	−	88.5966i	−0.229378	−	0.397294i	0.728246	−	0.685316i	$-0.240336\pi$
$223$	−51.1512	−	88.5966i	−0.229378	−	0.397294i	−0.957624	+	0.288022i	$0.907002\pi$
$224$			25.9209i			0.115718i
$225$		0			0
$226$	12.3772			0.0547665
$227$	10.8510	−	6.26481i	0.0478016	−	0.0275983i	−0.475909	−	0.879495i	$-0.657881\pi$
$227$	10.8510	−	6.26481i	0.0478016	−	0.0275983i	0.523710	+	0.851896i	$0.324547\pi$
$228$		0			0
$229$	−11.6491	+	20.1769i	−0.0508695	+	0.0881085i	−0.890339	−	0.455298i	$-0.849533\pi$
$229$	−11.6491	+	20.1769i	−0.0508695	+	0.0881085i	0.839469	+	0.543407i	$0.182866\pi$
$230$	61.8657	+	35.7182i	0.268981	+	0.155297i
$231$		0			0
$232$	0.802841	+	1.39056i	0.00346052	+	0.00599380i
$233$		−	356.382i		−	1.52954i	−0.644306	−	0.764768i	$-0.722854\pi$
$233$		−	356.382i		−	1.52954i	0.644306	−	0.764768i	$-0.277146\pi$
$234$		0			0
$235$	49.5701			0.210937
$236$	210.855	−	121.737i	0.893453	−	0.515835i
$237$		0			0
$238$	−8.10534	+	14.0389i	−0.0340560	+	0.0589868i
$239$	−152.008	−	87.7618i	−0.636016	−	0.367204i	0.147062	−	0.989127i	$-0.453018\pi$
$239$	−152.008	−	87.7618i	−0.636016	−	0.367204i	−0.783078	+	0.621923i	$0.786352\pi$
$240$		0			0
$241$	−52.2192	−	90.4463i	−0.216677	−	0.375296i	0.737113	−	0.675770i	$-0.236189\pi$
$241$	−52.2192	−	90.4463i	−0.216677	−	0.375296i	−0.953790	+	0.300474i	$0.902855\pi$
$242$		−	70.3045i		−	0.290515i
$243$		0			0
$244$	−160.658			−0.658433
$245$	−93.5291	+	53.9991i	−0.381751	+	0.220404i
$246$		0			0
$247$	−68.9737	+	119.466i	−0.279246	+	0.483668i
$248$	−157.814	−	91.1141i	−0.636348	−	0.367396i
$249$		0			0
$250$	−4.59431	−	7.95757i	−0.0183772	−	0.0318303i
$251$		−	130.945i		−	0.521694i	−0.965380	−	0.260847i	$-0.915998\pi$
$251$		−	130.945i		−	0.521694i	0.965380	−	0.260847i	$-0.0840018\pi$
$252$		0			0
$253$	558.658			2.20813
$254$	112.478	−	64.9391i	0.442826	−	0.255666i
$255$		0			0
$256$	37.6053	−	65.1344i	0.146896	−	0.254431i
$257$	−368.616	−	212.821i	−1.43430	−	0.828095i	−0.436858	−	0.899530i	$-0.643909\pi$
$257$	−368.616	−	212.821i	−1.43430	−	0.828095i	−0.997445	+	0.0714350i	$0.977242\pi$
$258$		0			0
$259$	−3.99573	−	6.92080i	−0.0154275	−	0.0267212i
$260$		−	162.144i		−	0.623632i
$261$		0			0
$262$	173.592			0.662566
$263$	−64.5192	+	37.2502i	−0.245320	+	0.141636i	−0.617620	−	0.786477i	$-0.711903\pi$
$263$	−64.5192	+	37.2502i	−0.245320	+	0.141636i	0.372299	+	0.928113i	$0.378569\pi$
$264$		0			0
$265$	−54.7851	+	94.8905i	−0.206736	+	0.358077i
$266$	3.77099	+	2.17718i	0.0141767	+	0.00818490i
$267$		0			0
$268$	−128.491	−	222.553i	−0.479444	−	0.830422i
$269$		−	205.067i		−	0.762331i	−0.924507	−	0.381165i	$-0.875523\pi$
$269$		−	205.067i		−	0.762331i	0.924507	−	0.381165i	$-0.124477\pi$
$270$		0			0
$271$	−233.351			−0.861073			−0.430537	−	0.902573i	$-0.641676\pi$
$271$	−233.351			−0.861073			−0.430537	+	0.902573i	$0.641676\pi$
$272$	−170.282	+	98.3125i	−0.626038	+	0.361443i
$273$		0			0
$274$	−28.4605	+	49.2950i	−0.103870	+	0.179909i
$275$	−62.2310	−	35.9291i	−0.226294	−	0.130651i
$276$		0			0
$277$	211.857	+	366.948i	0.764828	+	1.32472i	0.940338	+	0.340243i	$0.110509\pi$
$277$	211.857	+	366.948i	0.764828	+	1.32472i	−0.175510	+	0.984478i	$0.556157\pi$
$278$		−	131.367i		−	0.472543i
$279$		0			0
$280$	−11.2762			−0.0402720
$281$	348.665	−	201.302i	1.24080	−	0.716377i	0.271544	−	0.962426i	$-0.412466\pi$
$281$	348.665	−	201.302i	1.24080	−	0.716377i	0.969257	+	0.246049i	$0.0791324\pi$
$282$		0			0
$283$	136.167	−	235.847i	0.481154	−	0.833383i	−0.518612	−	0.855010i	$-0.673551\pi$
$283$	136.167	−	235.847i	0.481154	−	0.833383i	0.999766	+	0.0216266i	$0.00688449\pi$
$284$	−299.558	−	172.950i	−1.05478	−	0.608978i
$285$		0			0
$286$	128.811	+	223.108i	0.450389	+	0.780097i
$287$			16.1704i			0.0563427i
$288$		0			0
$289$	−265.386			−0.918290
$290$	−0.424516	+	0.245094i	−0.00146385	+	0.000845153i
$291$		0			0
$292$	−79.2498	+	137.265i	−0.271404	+	0.470085i
$293$	383.812	+	221.594i	1.30994	+	0.756294i	0.982086	−	0.188434i	$-0.0603412\pi$
$293$	383.812	+	221.594i	1.30994	+	0.756294i	0.327854	+	0.944728i	$0.393675\pi$
$294$		0			0
$295$	81.8794	+	141.819i	0.277557	+	0.480743i
$296$			57.4251i			0.194004i
$297$		0			0
$298$	8.06976			0.0270797
$299$	−734.266	+	423.929i	−2.45574	+	1.41782i
$300$		0			0
$301$	−8.21922	+	14.2361i	−0.0273064	+	0.0472960i
$302$	−149.929	−	86.5615i	−0.496453	−	0.286627i
$303$		0			0
$304$	26.4078	+	45.7397i	0.0868679	+	0.150460i
$305$		−	108.057i		−	0.354285i
$306$		0			0
$307$	−390.824			−1.27304			−0.636522	−	0.771259i	$-0.719627\pi$
$307$	−390.824			−1.27304			−0.636522	+	0.771259i	$0.719627\pi$
$308$	−34.6633	+	20.0129i	−0.112543	+	0.0649769i
$309$		0			0
$310$	27.8157	−	48.1781i	0.0897279	−	0.155413i
$311$	2.57850	+	1.48870i	0.00829100	+	0.00478681i	0.504140	−	0.863622i	$-0.331810\pi$
$311$	2.57850	+	1.48870i	0.00829100	+	0.00478681i	−0.495849	+	0.868409i	$0.665143\pi$
$312$		0			0
$313$	−65.0527	−	112.675i	−0.207836	−	0.359983i	0.743197	−	0.669073i	$-0.233309\pi$
$313$	−65.0527	−	112.675i	−0.207836	−	0.359983i	−0.951033	+	0.309091i	$0.899975\pi$
$314$			173.638i			0.552988i
$315$		0			0
$316$	226.799			0.717717
$317$	114.036	−	65.8384i	0.359733	−	0.207692i	−0.309230	−	0.950987i	$-0.600071\pi$
$317$	114.036	−	65.8384i	0.359733	−	0.207692i	0.668964	+	0.743295i	$0.266738\pi$
$318$		0			0
$319$	−1.91672	+	3.31986i	−0.00600854	+	0.0104071i
$320$	−15.4409	−	8.91483i	−0.0482529	−	0.0278588i
$321$		0			0
$322$	13.3815	+	23.1774i	0.0415574	+	0.0719796i
$323$			148.914i			0.461035i
$324$		0			0
$325$	109.057			0.335560
$326$	−158.551	+	91.5394i	−0.486352	+	0.280796i
$327$		0			0
$328$	58.0986	−	100.630i	0.177130	−	0.306798i
$329$	16.0829	+	9.28549i	0.0488843	+	0.0282234i
$330$		0			0
$331$	−80.2413	−	138.982i	−0.242421	−	0.419885i	0.718983	−	0.695028i	$-0.244608\pi$
$331$	−80.2413	−	138.982i	−0.242421	−	0.419885i	−0.961403	+	0.275143i	$0.911275\pi$
$332$		−	93.7850i		−	0.282485i
$333$		0			0
$334$	27.4207			0.0820978
$335$	149.687	−	86.4220i	0.446828	−	0.257976i
$336$		0			0
$337$	64.0569	−	110.950i	0.190080	−	0.329228i	−0.755197	−	0.655498i	$-0.772459\pi$
$337$	64.0569	−	110.950i	0.190080	−	0.329228i	0.945277	+	0.326270i	$0.105792\pi$
$338$	−218.319	−	126.046i	−0.645914	−	0.372918i
$339$		0			0
$340$	−87.5174	−	151.585i	−0.257404	−	0.445837i
$341$		−	435.056i		−	1.27583i
$342$		0			0
$343$	−81.5089			−0.237635
$344$	102.298	−	59.0618i	0.297378	−	0.171691i
$345$		0			0
$346$	−12.2498	+	21.2173i	−0.0354041	+	0.0613217i
$347$	190.211	+	109.818i	0.548159	+	0.316480i	0.748379	−	0.663271i	$-0.230832\pi$
$347$	190.211	+	109.818i	0.548159	+	0.316480i	−0.200220	+	0.979751i	$0.564166\pi$
$348$		0			0
$349$	−201.732	−	349.411i	−0.578030	−	1.00118i	−0.995705	−	0.0925806i	$-0.970488\pi$
$349$	−201.732	−	349.411i	−0.578030	−	1.00118i	0.417675	−	0.908596i	$-0.362845\pi$
$350$		−	3.44243i		−	0.00983551i
$351$		0			0
$352$	444.688			1.26332
$353$	47.4748	−	27.4096i	0.134489	−	0.0776475i	−0.431246	−	0.902235i	$-0.641926\pi$
$353$	47.4748	−	27.4096i	0.134489	−	0.0776475i	0.565735	+	0.824587i	$0.308592\pi$
$354$		0			0
$355$	116.325	−	201.480i	0.327675	−	0.567549i
$356$	−154.833	−	89.3929i	−0.434924	−	0.251104i
$357$		0			0
$358$	−45.9584	−	79.6022i	−0.128375	−	0.222353i
$359$			480.460i			1.33833i	0.743114	+	0.669165i	$0.233348\pi$
$359$			480.460i			1.33833i	−0.743114	+	0.669165i	$0.766652\pi$
$360$		0			0
$361$	−321.000			−0.889197
$362$	−34.9318	+	20.1679i	−0.0964967	+	0.0557124i
$363$		0			0
$364$	30.3729	−	52.6075i	0.0834422	−	0.144526i
$365$	−92.3231	−	53.3028i	−0.252940	−	0.146035i
$366$		0			0
$367$	−261.182	−	452.380i	−0.711667	−	1.23264i	−0.964231	−	0.265063i	$-0.914607\pi$
$367$	−261.182	−	452.380i	−0.711667	−	1.23264i	0.252564	−	0.967580i	$-0.418726\pi$
$368$			324.618i			0.882114i
$369$		0			0
$370$	−17.5310			−0.0473810
$371$	−35.5499	+	20.5247i	−0.0958217	+	0.0553227i
$372$		0			0
$373$	116.642	−	202.030i	0.312714	−	0.541637i	−0.666235	−	0.745742i	$-0.732095\pi$
$373$	116.642	−	202.030i	0.312714	−	0.541637i	0.978949	+	0.204105i	$0.0654285\pi$
$374$	240.845	+	139.052i	0.643971	+	0.371797i
$375$		0			0
$376$	−66.7238	−	115.569i	−0.177457	−	0.307365i
$377$		−	5.81791i		−	0.0154321i
$378$		0			0
$379$	248.596			0.655927			0.327964	−	0.944690i	$-0.393638\pi$
$379$	248.596			0.655927			0.327964	+	0.944690i	$0.393638\pi$
$380$	−40.7173	+	23.5082i	−0.107151	+	0.0618636i
$381$		0			0
$382$	−114.627	+	198.540i	−0.300071	+	0.519738i
$383$	−405.552	−	234.145i	−1.05888	−	0.611346i	−0.133759	−	0.991014i	$-0.542705\pi$
$383$	−405.552	−	234.145i	−1.05888	−	0.611346i	−0.925123	+	0.379668i	$0.876038\pi$
$384$		0			0
$385$	−13.4605	−	23.3143i	−0.0349623	−	0.0605565i
$386$			73.8803i			0.191400i
$387$		0			0
$388$	−382.061			−0.984694
$389$	419.362	−	242.119i	1.07805	−	0.622414i	0.147682	−	0.989035i	$-0.452819\pi$
$389$	419.362	−	242.119i	1.07805	−	0.622414i	0.930370	+	0.366621i	$0.119486\pi$
$390$		0			0
$391$	−457.631	+	792.641i	−1.17041	+	2.02721i
$392$	251.790	+	145.371i	0.642321	+	0.370844i
$393$		0			0
$394$	87.4078	+	151.395i	0.221847	+	0.384251i
$395$			152.543i			0.386184i
$396$		0			0
$397$	−298.943			−0.753005			−0.376503	−	0.926416i	$-0.622873\pi$
$397$	−298.943			−0.753005			−0.376503	+	0.926416i	$0.622873\pi$
$398$	68.6772	−	39.6508i	0.172556	−	0.0996252i
$399$		0			0
$400$	20.8772	−	36.1604i	0.0521931	−	0.0904010i
$401$	404.875	+	233.754i	1.00966	+	0.582929i	0.911093	−	0.412201i	$-0.135240\pi$
$401$	404.875	+	233.754i	1.00966	+	0.582929i	0.0985694	+	0.995130i	$0.468573\pi$
$402$		0			0
$403$	330.136	+	571.812i	0.819196	+	1.41889i
$404$		−	58.3370i		−	0.144399i
$405$		0			0
$406$	−0.183645			−0.000452327
$407$	−118.731	+	68.5491i	−0.291721	+	0.168425i
$408$		0			0
$409$	92.0790	−	159.486i	0.225132	−	0.389940i	−0.731227	−	0.682134i	$-0.761052\pi$
$409$	92.0790	−	159.486i	0.225132	−	0.389940i	0.956359	+	0.292194i	$0.0943853\pi$
$410$	30.7206	+	17.7366i	0.0749283	+	0.0432599i
$411$		0			0
$412$	118.919	+	205.973i	0.288637	+	0.499934i
$413$			61.3507i			0.148549i
$414$		0			0
$415$	63.0790			0.151998
$416$	−584.472	+	337.445i	−1.40498	+	0.811166i
$417$		0			0
$418$	37.3509	−	64.6936i	0.0893562	−	0.154769i
$419$	372.253	+	214.920i	0.888431	+	0.512936i	0.873429	−	0.486951i	$-0.161891\pi$
$419$	372.253	+	214.920i	0.888431	+	0.512936i	0.0150022	+	0.999887i	$0.495224\pi$
$420$		0			0
$421$	152.517	+	264.168i	0.362274	+	0.627477i	0.988335	−	0.152297i	$-0.0486671\pi$
$421$	152.517	+	264.168i	0.362274	+	0.627477i	−0.626061	+	0.779774i	$0.715334\pi$
$422$		−	54.0692i		−	0.128126i
$423$		0			0
$424$	294.974			0.695693
$425$	101.955	−	58.8635i	0.239893	−	0.138502i
$426$		0			0
$427$	20.2413	−	35.0589i	0.0474035	−	0.0821052i
$428$	219.874	+	126.944i	0.513723	+	0.296598i
$429$		0			0
$430$	18.0306	+	31.2299i	0.0419316	+	0.0726277i
$431$		−	128.880i		−	0.299025i	−0.988760	−	0.149512i	$-0.952230\pi$
$431$		−	128.880i		−	0.299025i	0.988760	−	0.149512i	$-0.0477704\pi$
$432$		0			0
$433$	243.886			0.563247			0.281624	−	0.959525i	$-0.409127\pi$
$433$	243.886			0.563247			0.281624	+	0.959525i	$0.409127\pi$
$434$	18.0495	−	10.4209i	0.0415887	−	0.0240112i
$435$		0			0
$436$	210.978	−	365.425i	0.483894	−	0.838130i
$437$	212.912	+	122.925i	0.487213	+	0.281293i
$438$		0			0
$439$	129.807	+	224.833i	0.295688	+	0.512147i	0.975145	−	0.221569i	$-0.0711177\pi$
$439$	129.807	+	224.833i	0.295688	+	0.512147i	−0.679457	+	0.733716i	$0.737784\pi$
$440$			193.449i			0.439658i
$441$		0			0
$442$	−422.070			−0.954909
$443$	468.529	−	270.505i	1.05763	−	0.610622i	0.132852	−	0.991136i	$-0.457586\pi$
$443$	468.529	−	270.505i	1.05763	−	0.610622i	0.924775	+	0.380514i	$0.124253\pi$
$444$		0			0
$445$	60.1249	−	104.139i	0.135112	−	0.234021i
$446$	−72.8135	−	42.0389i	−0.163259	−	0.0942576i
$447$		0			0
$448$	−3.33986	−	5.78481i	−0.00745504	−	0.0129125i
$449$		−	791.947i		−	1.76380i	−0.471436	−	0.881900i	$-0.656264\pi$
$449$		−	791.947i		−	1.76380i	0.471436	−	0.881900i	$-0.343736\pi$
$450$		0			0
$451$	277.412			0.615104
$452$	−43.3603	+	25.0341i	−0.0959299	+	0.0553852i
$453$		0			0
$454$	5.14876	−	8.91792i	0.0113409	−	0.0196430i
$455$	35.3834	+	20.4286i	0.0777656	+	0.0448980i
$456$		0			0
$457$	−305.680	−	529.453i	−0.668883	−	1.15854i	−0.978217	−	0.207586i	$-0.933439\pi$
$457$	−305.680	−	529.453i	−0.668883	−	1.15854i	0.309333	−	0.950954i	$-0.399894\pi$
$458$			19.1477i			0.0418073i
$459$		0			0
$460$	−288.974			−0.628204
$461$	508.349	−	293.496i	1.10271	−	0.636650i	0.165779	−	0.986163i	$-0.446986\pi$
$461$	508.349	−	293.496i	1.10271	−	0.636650i	0.936931	+	0.349513i	$0.113653\pi$
$462$		0			0
$463$	−97.6424	+	169.122i	−0.210891	+	0.365273i	−0.951994	−	0.306118i	$-0.900970\pi$
$463$	−97.6424	+	169.122i	−0.210891	+	0.365273i	0.741103	+	0.671391i	$0.234303\pi$
$464$	−1.92907	−	1.11375i	−0.00415747	−	0.00240032i
$465$		0			0
$466$	−146.447	−	253.654i	−0.314264	−	0.544321i
$467$			753.763i			1.61405i	0.590515	+	0.807027i	$0.298925\pi$
$467$			753.763i			1.61405i	−0.590515	+	0.807027i	$0.701075\pi$
$468$		0			0
$469$	64.7544			0.138069
$470$	35.2814	−	20.3697i	0.0750667	−	0.0433398i
$471$		0			0
$472$	220.427	−	381.791i	0.467007	−	0.808880i
$473$	244.229	+	141.006i	0.516340	+	0.298109i
$474$		0			0
$475$	−15.8114	−	27.3861i	−0.0332871	−	0.0576550i
$476$		−	65.5752i		−	0.137763i
$477$		0			0
$478$	−144.255			−0.301788
$479$	532.474	−	307.424i	1.11164	−	0.641803i	0.172384	−	0.985030i	$-0.444853\pi$
$479$	532.474	−	307.424i	1.11164	−	0.641803i	0.939253	+	0.343226i	$0.111520\pi$
$480$		0			0
$481$	104.035	−	180.194i	0.216289	−	0.374623i
$482$	−74.3337	−	42.9166i	−0.154219	−	0.0890386i
$483$		0			0
$484$	142.198	+	246.293i	0.293796	+	0.508870i
$485$		−	256.971i		−	0.529837i
$486$		0			0
$487$	478.197			0.981924			0.490962	−	0.871181i	$-0.336645\pi$
$487$	478.197			0.981924			0.490962	+	0.871181i	$0.336645\pi$
$488$	−251.927	+	145.450i	−0.516244	+	0.298053i
$489$		0			0
$490$	−44.3794	+	76.8673i	−0.0905701	+	0.156872i
$491$	534.531	+	308.612i	1.08866	+	0.628537i	0.933219	−	0.359309i	$-0.116987\pi$
$491$	534.531	+	308.612i	1.08866	+	0.628537i	0.155439	+	0.987845i	$0.450321\pi$
$492$		0			0
$493$	−3.14022	−	5.43902i	−0.00636961	−	0.0110325i
$494$			113.373i			0.229499i
$495$		0			0
$496$	252.797			0.509672
$497$	75.4826	−	43.5799i	0.151877	−	0.0876860i
$498$		0			0
$499$	−391.548	+	678.181i	−0.784665	+	1.35908i	0.144533	+	0.989500i	$0.453832\pi$
$499$	−391.548	+	678.181i	−0.784665	+	1.35908i	−0.929199	+	0.369580i	$0.879501\pi$
$500$	32.1899	+	18.5848i	0.0643797	+	0.0371697i
$501$		0			0
$502$	−53.8089	−	93.1998i	−0.107189	−	0.185657i
$503$			369.395i			0.734383i	0.930145	+	0.367191i	$0.119681\pi$
$503$			369.395i			0.734383i	−0.930145	+	0.367191i	$0.880319\pi$
$504$		0			0
$505$	39.2370			0.0776970
$506$	397.623	−	229.568i	0.785816	−	0.453691i
$507$		0			0
$508$	−262.691	+	454.994i	−0.517108	+	0.895657i
$509$	−195.406	−	112.818i	−0.383901	−	0.221645i	0.295613	−	0.955308i	$-0.404476\pi$
$509$	−195.406	−	112.818i	−0.383901	−	0.221645i	−0.679514	+	0.733662i	$0.737809\pi$
$510$		0			0
$511$	−19.9694	−	34.5880i	−0.0390790	−	0.0676869i
$512$			459.474i			0.897410i
$513$		0			0
$514$	−349.815			−0.680574
$515$	−138.536	+	79.9836i	−0.269001	+	0.155308i
$516$		0			0
$517$	159.298	−	275.913i	0.308120	−	0.533680i
$518$	−5.68789	−	3.28391i	−0.0109805	−	0.00633959i
$519$		0			0
$520$	−146.796	−	254.258i	−0.282300	−	0.488958i
$521$			1007.73i			1.93422i	0.254367	+	0.967108i	$0.418133\pi$
$521$			1007.73i			1.93422i	−0.254367	+	0.967108i	$0.581867\pi$
$522$		0			0
$523$	935.517			1.78875			0.894376	−	0.447316i	$-0.147620\pi$
$523$	935.517			1.78875			0.894376	+	0.447316i	$0.147620\pi$
$524$	−608.134	+	351.106i	−1.16056	+	0.670050i
$525$		0			0
$526$	−30.6142	+	53.0254i	−0.0582020	+	0.100809i
$527$	617.271	+	356.382i	1.17129	+	0.676246i
$528$		0			0
$529$	491.026	+	850.482i	0.928215	+	1.60772i
$530$			90.0507i			0.169907i
$531$		0			0
$532$	−17.6142			−0.0331095
$533$	−364.614	+	210.510i	−0.684079	+	0.394953i
$534$		0			0
$535$	−85.3815	+	147.885i	−0.159592	+	0.276421i
$536$	−402.973	−	232.657i	−0.751816	−	0.434061i
$537$		0			0
$538$	−84.2676	−	145.956i	−0.156631	−	0.271293i
$539$			694.124i			1.28780i
$540$		0			0
$541$	−399.149			−0.737798			−0.368899	−	0.929469i	$-0.620265\pi$
$541$	−399.149			−0.737798			−0.368899	+	0.929469i	$0.620265\pi$
$542$	−166.087	+	95.8902i	−0.306433	+	0.176919i
$543$		0			0
$544$	−364.272	+	630.937i	−0.669617	+	1.15981i
$545$	245.781	+	141.902i	0.450975	+	0.260371i
$546$		0			0
$547$	−240.416	−	416.413i	−0.439518	−	0.761268i	0.558134	−	0.829751i	$-0.311517\pi$
$547$	−240.416	−	416.413i	−0.439518	−	0.761268i	−0.997652	+	0.0684831i	$0.978184\pi$
$548$		−	230.256i		−	0.420175i
$549$		0			0
$550$	−59.0569			−0.107376
$551$	−1.46098	+	0.843497i	−0.00265151	+	0.00153085i
$552$		0			0
$553$	−28.5744	+	49.4923i	−0.0516716	+	0.0894978i
$554$	301.578	+	174.116i	0.544364	+	0.314289i
$555$		0			0
$556$	265.702	+	460.209i	0.477881	+	0.827714i
$557$		−	751.542i		−	1.34927i	−0.738152	−	0.674634i	$-0.764301\pi$
$557$		−	751.542i		−	1.34927i	0.738152	−	0.674634i	$-0.235699\pi$
$558$		0			0
$559$	−428.000			−0.765653
$560$	13.5472	−	7.82146i	0.0241914	−	0.0139669i
$561$		0			0
$562$	165.441	−	286.552i	0.294379	−	0.509879i
$563$	580.948	+	335.410i	1.03188	+	0.595755i	0.917522	−	0.397685i	$-0.130186\pi$
$563$	580.948	+	335.410i	1.03188	+	0.595755i	0.114356	+	0.993440i	$0.463520\pi$
$564$		0			0
$565$	−16.8377	−	29.1638i	−0.0298013	−	0.0516173i
$566$		−	223.818i		−	0.395438i
$567$		0			0
$568$	−626.315			−1.10267
$569$	238.438	−	137.662i	0.419048	−	0.241937i	−0.275622	−	0.961266i	$-0.588884\pi$
$569$	238.438	−	137.662i	0.419048	−	0.241937i	0.694670	+	0.719329i	$0.255550\pi$
$570$		0			0
$571$	450.144	−	779.673i	0.788344	−	1.36545i	−0.138637	−	0.990343i	$-0.544272\pi$
$571$	450.144	−	779.673i	0.788344	−	1.36545i	0.926981	−	0.375109i	$-0.122395\pi$
$572$	−902.513	−	521.066i	−1.57782	−	0.910955i
$573$		0			0
$574$	6.64484	+	11.5092i	0.0115764	+	0.0200509i
$575$		−	194.361i		−	0.338020i
$576$		0			0
$577$	−596.236			−1.03334			−0.516669	−	0.856185i	$-0.672828\pi$
$577$	−596.236			−1.03334			−0.516669	+	0.856185i	$0.672828\pi$
$578$	−188.887	+	109.054i	−0.326795	+	0.188675i
$579$		0			0
$580$	0.991453	−	1.71725i	0.00170940	−	0.00296077i
$581$	20.4659	+	11.8160i	0.0352253	+	0.0203373i
$582$		0			0
$583$	352.114	+	609.879i	0.603969	+	1.04610i
$584$			286.993i			0.491426i
$585$		0			0
$586$	364.236			0.621564
$587$	430.788	−	248.715i	0.733880	−	0.423706i	−0.0859598	−	0.996299i	$-0.527396\pi$
$587$	430.788	−	248.715i	0.733880	−	0.423706i	0.819840	+	0.572593i	$0.194062\pi$
$588$		0			0
$589$	95.7281	−	165.806i	0.162527	−	0.281504i
$590$	116.555	+	67.2929i	0.197550	+	0.114056i
$591$		0			0
$592$	−39.8317	−	68.9905i	−0.0672832	−	0.116538i
$593$		−	898.856i		−	1.51578i	−0.652384	−	0.757889i	$-0.726231\pi$
$593$		−	898.856i		−	1.51578i	0.652384	−	0.757889i	$-0.273769\pi$
$594$		0			0
$595$	44.1053			0.0741266
$596$	−28.2703	+	16.3219i	−0.0474333	+	0.0273857i
$597$		0			0
$598$	−348.408	+	603.460i	−0.582622	+	1.00913i
$599$	−24.7424	−	14.2851i	−0.0413063	−	0.0238482i	0.479205	−	0.877703i	$-0.340925\pi$
$599$	−24.7424	−	14.2851i	−0.0413063	−	0.0238482i	−0.520511	+	0.853855i	$0.674258\pi$
$600$		0			0
$601$	10.1053	+	17.5030i	0.0168142	+	0.0291231i	0.874310	−	0.485368i	$-0.161314\pi$
$601$	10.1053	+	17.5030i	0.0168142	+	0.0291231i	−0.857496	+	0.514491i	$0.827981\pi$
$602$			13.5100i			0.0224419i
$603$		0			0
$604$	700.315			1.15946
$605$	−165.655	+	95.6408i	−0.273810	+	0.158084i
$606$		0			0
$607$	356.813	−	618.019i	0.587831	−	1.01815i	−0.406685	−	0.913568i	$-0.633315\pi$
$607$	356.813	−	618.019i	0.587831	−	1.01815i	0.994516	−	0.104584i	$-0.0333512\pi$
$608$	169.477	+	97.8475i	0.278745	+	0.160933i
$609$		0			0
$610$	−44.4036	−	76.9092i	−0.0727927	−	0.126081i
$611$			483.524i			0.791365i
$612$		0			0
$613$	−76.1530			−0.124230			−0.0621150	−	0.998069i	$-0.519785\pi$
$613$	−76.1530			−0.124230			−0.0621150	+	0.998069i	$0.519785\pi$
$614$	−278.168	+	160.600i	−0.453042	+	0.261564i
$615$		0			0
$616$	−36.2370	+	62.7643i	−0.0588263	+	0.101890i
$617$	−174.590	−	100.800i	−0.282966	−	0.163371i	0.351799	−	0.936075i	$-0.385570\pi$
$617$	−174.590	−	100.800i	−0.282966	−	0.163371i	−0.634766	+	0.772705i	$0.718903\pi$
$618$		0			0
$619$	102.132	+	176.897i	0.164995	+	0.285779i	0.936653	−	0.350258i	$-0.113906\pi$
$619$	102.132	+	176.897i	0.164995	+	0.285779i	−0.771659	+	0.636037i	$0.780573\pi$
$620$			225.039i			0.362966i
$621$		0			0
$622$	2.44699			0.00393406
$623$	39.0149	−	22.5252i	0.0626242	−	0.0361561i
$624$		0			0
$625$	−12.5000	+	21.6506i	−0.0200000	+	0.0346410i
$626$	−92.6021	−	53.4638i	−0.147927	−	0.0854055i
$627$		0			0
$628$	−351.200	−	608.296i	−0.559235	−	0.968624i
$629$		−	224.611i		−	0.357093i
$630$		0			0
$631$	−639.903			−1.01411			−0.507055	−	0.861914i	$-0.669266\pi$
$631$	−639.903			−1.01411			−0.507055	+	0.861914i	$0.669266\pi$
$632$	355.642	−	205.330i	0.562725	−	0.324890i
$633$		0			0
$634$	54.1096	−	93.7206i	0.0853464	−	0.147824i
$635$	−306.025	−	176.684i	−0.481929	−	0.278242i
$636$		0			0
$637$	−526.726	−	912.316i	−0.826885	−	1.43221i
$638$			3.15054i			0.00493815i
$639$		0			0
$640$	−291.408			−0.455325
$641$	−494.249	+	285.355i	−0.771059	+	0.445171i	−0.833252	−	0.552893i	$-0.813524\pi$
$641$	−494.249	+	285.355i	−0.771059	+	0.445171i	0.0621934	+	0.998064i	$0.480190\pi$
$642$		0			0
$643$	−226.846	+	392.909i	−0.352794	+	0.611056i	−0.986738	−	0.162322i	$-0.948102\pi$
$643$	−226.846	+	392.909i	−0.352794	+	0.611056i	0.633944	+	0.773379i	$0.281435\pi$
$644$	−93.7571	−	54.1307i	−0.145586	−	0.0840538i
$645$		0			0
$646$	61.1929	+	105.989i	0.0947258	+	0.164070i
$647$		−	983.536i		−	1.52015i	−0.649837	−	0.760074i	$-0.725163\pi$
$647$		−	983.536i		−	1.52015i	0.649837	−	0.760074i	$-0.274837\pi$
$648$		0			0
$649$	1052.51			1.62174
$650$	77.6209	−	44.8145i	0.119417	−	0.0689453i
$651$		0			0
$652$	370.294	−	641.368i	0.567936	−	0.983693i
$653$	−471.532	−	272.239i	−0.722101	−	0.416905i	0.0934244	−	0.995626i	$-0.470219\pi$
$653$	−471.532	−	272.239i	−0.722101	−	0.416905i	−0.815526	+	0.578721i	$0.803552\pi$
$654$		0			0
$655$	−236.151	−	409.026i	−0.360536	−	0.624467i
$656$			161.195i			0.245724i
$657$		0			0
$658$	15.2626			0.0231955
$659$	−93.4729	+	53.9666i	−0.141840	+	0.0818916i	−0.569241	−	0.822171i	$-0.692763\pi$
$659$	−93.4729	+	53.9666i	−0.141840	+	0.0818916i	0.427400	+	0.904062i	$0.359429\pi$
$660$		0			0
$661$	−172.864	+	299.409i	−0.261519	+	0.452964i	−0.966646	−	0.256117i	$-0.917557\pi$
$661$	−172.864	+	299.409i	−0.261519	+	0.452964i	0.705127	+	0.709081i	$0.250890\pi$
$662$	−114.223	−	65.9467i	−0.172542	−	0.0996173i
$663$		0			0
$664$	−84.9075	−	147.064i	−0.127873	−	0.221482i
$665$		−	11.8472i		−	0.0178153i
$666$		0			0
$667$	−10.3687			−0.0155452
$668$	−96.0610	+	55.4608i	−0.143804	+	0.0830252i
$669$		0			0
$670$	71.0263	−	123.021i	0.106009	−	0.183614i
$671$	−601.457	−	347.251i	−0.896359	−	0.517513i
$672$		0			0
$673$	602.390	+	1043.37i	0.895082	+	1.55033i	0.833703	+	0.552213i	$0.186216\pi$
$673$	602.390	+	1043.37i	0.895082	+	1.55033i	0.0613784	+	0.998115i	$0.480450\pi$
$674$		−	105.291i		−	0.156218i
$675$		0			0
$676$	1019.76			1.50853
$677$	497.617	−	287.299i	0.735032	−	0.424371i	−0.0852280	−	0.996361i	$-0.527162\pi$
$677$	497.617	−	287.299i	0.735032	−	0.424371i	0.820260	+	0.571990i	$0.193829\pi$
$678$		0			0
$679$	48.1359	−	83.3739i	0.0708924	−	0.122789i
$680$	−274.472	−	158.466i	−0.403635	−	0.233039i
$681$		0			0
$682$	−178.777	−	309.650i	−0.262136	−	0.454032i
$683$			334.387i			0.489585i	0.969575	+	0.244792i	$0.0787198\pi$
$683$			334.387i			0.489585i	−0.969575	+	0.244792i	$0.921280\pi$
$684$		0			0
$685$	154.868			0.226085
$686$	−58.0137	+	33.4942i	−0.0845681	+	0.0488254i
$687$		0			0
$688$	−81.9338	+	141.914i	−0.119090	+	0.206270i
$689$	−925.595	−	534.393i	−1.34339	−	0.775606i
$690$		0			0
$691$	−450.574	−	780.418i	−0.652061	−	1.12940i	−0.982622	−	0.185619i	$-0.940571\pi$
$691$	−450.574	−	780.418i	−0.652061	−	1.12940i	0.330561	−	0.943785i	$-0.392762\pi$
$692$		−	99.1057i		−	0.143216i
$693$		0			0
$694$	180.510			0.260100
$695$	−309.533	+	178.709i	−0.445371	+	0.257135i
$696$		0			0
$697$	−227.246	+	393.601i	−0.326034	+	0.564707i
$698$	−287.165	−	165.795i	−0.411411	−	0.237528i
$699$		0			0
$700$	6.96264	+	12.0596i	0.00994662	+	0.0172281i
$701$		−	887.934i		−	1.26667i	−0.773879	−	0.633334i	$-0.781686\pi$
$701$		−	887.934i		−	1.26667i	0.773879	−	0.633334i	$-0.218314\pi$
$702$		0			0
$703$	−60.3331			−0.0858223
$704$	−99.2418	+	57.2973i	−0.140968	+	0.0813882i
$705$		0			0
$706$	22.5267	−	39.0174i	0.0319075	−	0.0552654i
$707$	12.7304	+	7.34989i	0.0180062	+	0.0103959i
$708$		0			0
$709$	513.175	+	888.845i	0.723801	+	1.25366i	0.959466	+	0.281826i	$0.0909401\pi$
$709$	513.175	+	888.845i	0.723801	+	1.25366i	−0.235664	+	0.971835i	$0.575727\pi$
$710$		−	191.204i		−	0.269301i
$711$		0			0
$712$	−323.725			−0.454669
$713$	1019.08	−	588.368i	1.42929	−	0.825201i
$714$		0			0
$715$	350.465	−	607.023i	0.490161	−	0.848983i
$716$	322.006	+	185.910i	0.449729	+	0.259651i
$717$		0			0
$718$	197.434	+	341.966i	0.274978	+	0.476276i
$719$		−	740.080i		−	1.02932i	−0.857395	−	0.514659i	$-0.827919\pi$
$719$		−	740.080i		−	1.02932i	0.857395	−	0.514659i	$-0.172081\pi$
$720$		0			0
$721$	−59.9302			−0.0831210
$722$	−228.471	+	131.908i	−0.316441	+	0.182698i
$723$		0			0
$724$	81.5829	−	141.306i	0.112684	−	0.195174i
$725$	1.15501	+	0.666843i	0.00159311	+	0.000919784i
$726$		0			0
$727$	−531.875	−	921.235i	−0.731603	−	1.26717i	−0.956198	−	0.292721i	$-0.905439\pi$
$727$	−531.875	−	921.235i	−0.731603	−	1.26717i	0.224595	−	0.974452i	$-0.427894\pi$
$728$		−	109.992i		−	0.151087i
$729$		0			0
$730$	−87.6142			−0.120019
$731$	−400.126	+	231.013i	−0.547368	+	0.316023i
$732$		0			0
$733$	−67.3747	+	116.696i	−0.0919164	+	0.159204i	−0.908317	−	0.418281i	$-0.862633\pi$
$733$	−67.3747	+	116.696i	−0.0919164	+	0.159204i	0.816401	+	0.577485i	$0.195966\pi$
$734$	−371.791	−	214.653i	−0.506527	−	0.292443i
$735$		0			0
$736$	601.394	+	1041.65i	0.817112	+	1.41528i
$737$		−	1110.90i		−	1.50733i
$738$		0			0
$739$	−711.429			−0.962692			−0.481346	−	0.876531i	$-0.659852\pi$
$739$	−711.429			−0.962692			−0.481346	+	0.876531i	$0.659852\pi$
$740$	61.4151	−	35.4580i	0.0829933	−	0.0479162i
$741$		0			0
$742$	−16.8683	+	29.2168i	−0.0227336	+	0.0393757i
$743$	−403.854	−	233.165i	−0.543545	−	0.313816i	0.202970	−	0.979185i	$-0.434941\pi$
$743$	−403.854	−	233.165i	−0.543545	−	0.313816i	−0.746514	+	0.665369i	$0.768274\pi$
$744$		0			0
$745$	−10.9779	−	19.0143i	−0.0147355	−	0.0255226i
$746$		−	191.726i		−	0.257005i
$747$		0			0
$748$	−1124.98			−1.50399
$749$	−55.4038	+	31.9874i	−0.0739703	+	0.0427068i
$750$		0			0
$751$	−113.680	+	196.899i	−0.151371	+	0.262182i	−0.931732	−	0.363147i	$-0.881702\pi$
$751$	−113.680	+	196.899i	−0.151371	+	0.262182i	0.780361	+	0.625330i	$0.215036\pi$
$752$	160.324	+	92.5630i	0.213197	+	0.123089i
$753$		0			0
$754$	−2.39074	−	4.14088i	−0.00317074	−	0.00549188i
$755$			471.026i			0.623875i
$756$		0			0
$757$	−552.258			−0.729535			−0.364768	−	0.931099i	$-0.618852\pi$
$757$	−552.258			−0.729535			−0.364768	+	0.931099i	$0.618852\pi$
$758$	176.938	−	102.155i	0.233427	−	0.134769i
$759$		0			0
$760$	−42.5658	+	73.7262i	−0.0560077	+	0.0970081i
$761$	262.450	+	151.526i	0.344875	+	0.199114i	0.662426	−	0.749127i	$-0.269527\pi$
$761$	262.450	+	151.526i	0.344875	+	0.199114i	−0.317550	+	0.948241i	$0.602860\pi$
$762$		0			0
$763$	53.1623	+	92.0798i	0.0696753	+	0.120681i
$764$		−	927.376i		−	1.21384i
$765$		0			0
$766$	−384.867			−0.502437
$767$	−1383.35	+	798.680i	−1.80359	+	1.04130i
$768$		0			0
$769$	−369.842	+	640.585i	−0.480939	+	0.833011i	−0.999761	−	0.0218718i	$-0.993037\pi$
$769$	−369.842	+	640.585i	−0.480939	+	0.833011i	0.518822	+	0.854882i	$0.326371\pi$
$770$	−19.1609	−	11.0626i	−0.0248843	−	0.0143670i
$771$		0			0
$772$	−149.430	−	258.820i	−0.193562	−	0.335259i
$773$			603.479i			0.780697i	0.920667	+	0.390348i	$0.127645\pi$
$773$			603.479i			0.780697i	−0.920667	+	0.390348i	$0.872355\pi$
$774$		0			0
$775$	−151.359			−0.195302
$776$	−599.109	+	345.896i	−0.772048	+	0.445742i
$777$		0			0
$778$	198.986	−	344.655i	0.255767	−	0.443001i
$779$	105.726	+	61.0407i	0.135720	+	0.0783577i
$780$		0			0
$781$	−747.640	−	1294.95i	−0.957285	−	1.65807i
$782$			752.213i			0.961909i
$783$		0			0
$784$	−403.333			−0.514455
$785$	409.135	−	236.214i	0.521190	−	0.300909i
$786$		0			0
$787$	330.241	−	571.995i	0.419620	−	0.726804i	−0.576281	−	0.817252i	$-0.695497\pi$
$787$	330.241	−	571.995i	0.419620	−	0.726804i	0.995901	+	0.0904479i	$0.0288299\pi$
$788$	−612.420	−	353.581i	−0.777183	−	0.448707i
$789$		0			0
$790$	62.6840	+	108.572i	0.0793468	+	0.137433i
$791$		−	12.6162i		−	0.0159497i
$792$		0			0
$793$	1054.03			1.32916
$794$	−212.772	+	122.844i	−0.267975	+	0.154715i
$795$		0			0
$796$	−160.395	+	277.812i	−0.201501	+	0.349010i
$797$	−549.899	−	317.484i	−0.689961	−	0.398349i	0.113636	−	0.993522i	$-0.463750\pi$
$797$	−549.899	−	317.484i	−0.689961	−	0.398349i	−0.803598	+	0.595173i	$0.797083\pi$
$798$		0			0
$799$	260.982	+	452.034i	0.326636	+	0.565750i
$800$		−	154.710i		−	0.193388i
$801$		0			0
$802$	384.224			0.479083
$803$	−593.378	+	342.587i	−0.738951	+	0.426634i
$804$		0			0
$805$	36.4078	−	63.0602i	0.0452271	−	0.0783357i
$806$	469.946	+	271.324i	0.583060	+	0.336630i
$807$		0			0
$808$	−52.8150	−	91.4782i	−0.0653651	−	0.113216i
$809$			405.048i			0.500677i	0.968158	+	0.250339i	$0.0805419\pi$
$809$			405.048i			0.500677i	−0.968158	+	0.250339i	$0.919458\pi$
$810$		0			0
$811$	406.034			0.500659			0.250329	−	0.968161i	$-0.419461\pi$
$811$	406.034			0.500659			0.250329	+	0.968161i	$0.419461\pi$
$812$	0.643351	−	0.371439i	0.000792304	−	0.000457437i
$813$		0			0
$814$	−56.3374	+	97.5792i	−0.0692105	+	0.119876i
$815$	431.379	+	249.057i	0.529299	+	0.305591i
$816$		0			0
$817$	62.0527	+	107.478i	0.0759519	+	0.131552i
$818$		−	151.351i		−	0.185026i
$819$		0			0
$820$	−143.495			−0.174994
$821$	476.377	−	275.036i	0.580240	−	0.335002i	−0.180989	−	0.983485i	$-0.557930\pi$
$821$	476.377	−	275.036i	0.580240	−	0.335002i	0.761229	+	0.648483i	$0.224596\pi$
$822$		0			0
$823$	736.256	−	1275.23i	0.894600	−	1.54949i	0.0603014	−	0.998180i	$-0.480794\pi$
$823$	736.256	−	1275.23i	0.894600	−	1.54949i	0.834299	−	0.551313i	$-0.185873\pi$
$824$	372.952	+	215.324i	0.452611	+	0.261315i
$825$		0			0
$826$	25.2107	+	43.6662i	0.0305214	+	0.0528646i
$827$			1510.47i			1.82644i	0.407466	+	0.913220i	$0.366413\pi$
$827$			1510.47i			1.82644i	−0.407466	+	0.913220i	$0.633587\pi$
$828$		0			0
$829$	712.692			0.859701			0.429850	−	0.902900i	$-0.358566\pi$
$829$	712.692			0.859701			0.429850	+	0.902900i	$0.358566\pi$
$830$	44.8963	−	25.9209i	0.0540919	−	0.0312300i
$831$		0			0
$832$	86.9584	−	150.616i	0.104517	−	0.181029i
$833$	−984.845	−	568.600i	−1.18229	−	0.682594i
$834$		0			0
$835$	−37.3025	−	64.6098i	−0.0446736	−	0.0773770i
$836$			302.183i			0.361463i
$837$		0			0
$838$	353.266			0.421559
$839$	459.495	−	265.289i	0.547669	−	0.316197i	−0.200512	−	0.979691i	$-0.564261\pi$
$839$	459.495	−	265.289i	0.547669	−	0.316197i	0.748181	+	0.663494i	$0.230927\pi$
$840$		0			0
$841$	−420.464	+	728.266i	−0.499958	+	0.865952i
$842$	217.108	+	125.347i	0.257848	+	0.148868i
$843$		0			0
$844$	109.360	+	189.417i	0.129574	+	0.224428i
$845$			685.884i			0.811697i
$846$		0			0
$847$	−71.6619			−0.0846068
$848$	−354.381	+	204.602i	−0.417902	+	0.241276i
$849$		0			0
$850$	48.3772	−	83.7918i	0.0569144	−	0.0985786i
$851$	−321.141	−	185.411i	−0.377369	−	0.217874i
$852$		0			0
$853$	−107.616	−	186.396i	−0.126162	−	0.218519i	0.796025	−	0.605264i	$-0.206933\pi$
$853$	−107.616	−	186.396i	−0.126162	−	0.218519i	−0.922186	+	0.386746i	$0.873599\pi$
$854$		−	33.2708i		−	0.0389587i
$855$		0			0
$856$	459.711			0.537046
$857$	−4.30797	+	2.48721i	−0.00502680	+	0.00290222i	−0.502511	−	0.864571i	$-0.667590\pi$
$857$	−4.30797	+	2.48721i	−0.00502680	+	0.00290222i	0.497484	+	0.867473i	$0.334257\pi$
$858$		0			0
$859$	−826.140	+	1430.92i	−0.961746	+	1.66579i	−0.243633	+	0.969867i	$0.578339\pi$
$859$	−826.140	+	1430.92i	−0.961746	+	1.66579i	−0.718113	+	0.695926i	$0.754994\pi$
$860$	−126.331	−	72.9372i	−0.146896	−	0.0848107i
$861$		0			0
$862$	−52.9602	−	91.7297i	−0.0614387	−	0.106415i
$863$		−	379.077i		−	0.439255i	−0.975584	−	0.219627i	$-0.929516\pi$
$863$		−	379.077i		−	0.439255i	0.975584	−	0.219627i	$-0.0704841\pi$
$864$		0			0
$865$	66.6577			0.0770609
$866$	173.585	−	100.219i	0.200445	−	0.115727i
$867$		0			0
$868$	−42.1544	+	73.0136i	−0.0485650	+	0.0841171i
$869$	849.070	+	490.211i	0.977065	+	0.564109i
$870$		0			0
$871$	842.991	+	1460.10i	0.967842	+	1.67635i
$872$		−	764.029i		−	0.876180i
$873$		0			0
$874$	202.053			0.231182
$875$	−8.11121	+	4.68301i	−0.00926996	+	0.00535201i
$876$		0			0
$877$	−550.800	+	954.014i	−0.628051	+	1.08782i	0.359892	+	0.932994i	$0.382814\pi$
$877$	−550.800	+	954.014i	−0.628051	+	1.08782i	−0.987942	+	0.154821i	$0.950520\pi$
$878$	184.780	+	106.683i	0.210455	+	0.121506i
$879$		0			0
$880$	−134.182	−	232.410i	−0.152479	−	0.264102i
$881$		−	184.877i		−	0.209850i	−0.994480	−	0.104925i	$-0.966540\pi$
$881$		−	184.877i		−	0.209850i	0.994480	−	0.104925i	$-0.0334602\pi$
$882$		0			0
$883$	978.236			1.10786			0.553928	−	0.832565i	$-0.313128\pi$
$883$	978.236			1.10786			0.553928	+	0.832565i	$0.313128\pi$
$884$	1478.61	−	853.676i	1.67264	−	0.965696i
$885$		0			0
$886$	222.316	−	385.063i	0.250921	−	0.434608i
$887$	−578.450	−	333.968i	−0.652142	−	0.376514i	0.137134	−	0.990552i	$-0.456211\pi$
$887$	−578.450	−	333.968i	−0.652142	−	0.376514i	−0.789276	+	0.614038i	$0.789544\pi$
$888$		0			0
$889$	−66.1929	−	114.649i	−0.0744577	−	0.128964i
$890$		−	98.8279i		−	0.111043i
$891$		0			0
$892$	340.110			0.381290
$893$	121.421	−	70.1027i	0.135970	−	0.0785025i
$894$		0			0
$895$	−125.042	+	216.578i	−0.139711	+	0.241987i
$896$	−94.5468	−	54.5866i	−0.105521	−	0.0609226i
$897$		0			0
$898$	−325.432	−	563.665i	−0.362397	−	0.627690i
$899$			8.07464i			0.00898180i
$900$		0			0
$901$	−1153.75			−1.28053
$902$	197.447	−	113.996i	0.218899	−	0.126382i
$903$		0			0
$904$	−45.3288	+	78.5118i	−0.0501425	+	0.0868494i
$905$	95.0411	+	54.8720i	0.105018	+	0.0606320i
$906$		0			0
$907$	416.311	+	721.072i	0.458998	+	0.795008i	0.998908	−	0.0467149i	$-0.0148752\pi$
$907$	416.311	+	721.072i	0.458998	+	0.795008i	−0.539910	+	0.841722i	$0.681542\pi$
$908$			41.6554i			0.0458760i
$909$		0			0
$910$	33.5787			0.0368996
$911$	−489.532	+	282.631i	−0.537357	+	0.310243i	−0.744007	−	0.668172i	$-0.767077\pi$
$911$	−489.532	+	282.631i	−0.537357	+	0.310243i	0.206650	+	0.978415i	$0.433744\pi$
$912$		0			0
$913$	202.710	−	351.105i	0.222027	−	0.384561i
$914$	−435.133	−	251.224i	−0.476076	−	0.274862i
$915$		0			0
$916$	−38.7281	−	67.0791i	−0.0422796	−	0.0732304i
$917$		−	176.944i		−	0.192959i
$918$		0			0
$919$	−1185.75			−1.29026			−0.645128	−	0.764075i	$-0.723196\pi$
$919$	−1185.75			−1.29026			−0.645128	+	0.764075i	$0.723196\pi$
$920$	−453.139	+	261.620i	−0.492542	+	0.284370i
$921$		0			0
$922$	241.211	−	417.789i	0.261617	−	0.453134i
$923$	1965.31	+	1134.67i	2.12926	+	1.22933i
$924$		0			0
$925$	23.8488	+	41.3073i	0.0257824	+	0.0446565i
$926$			160.496i			0.173321i
$927$		0			0
$928$	−8.25341			−0.00889376
$929$	−816.657	+	471.497i	−0.879071	+	0.507532i	−0.870352	−	0.492430i	$-0.836109\pi$
$929$	−816.657	+	471.497i	−0.879071	+	0.507532i	−0.00871924	+	0.999962i	$0.502775\pi$
$930$		0			0
$931$	−152.732	+	264.540i	−0.164052	+	0.284146i
$932$	1026.08	+	592.406i	1.10094	+	0.635628i
$933$		0			0
$934$	309.742	+	536.488i	0.331629	+	0.574399i
$935$		−	756.654i		−	0.809255i
$936$		0			0
$937$	−531.281			−0.567002			−0.283501	−	0.958972i	$-0.591496\pi$
$937$	−531.281			−0.567002			−0.283501	+	0.958972i	$0.591496\pi$
$938$	46.0888	−	26.6094i	0.0491351	−	0.0283682i
$939$		0			0
$940$	−82.3993	+	142.720i	−0.0876588	+	0.151830i
$941$	−515.848	−	297.825i	−0.548191	−	0.316498i	0.200201	−	0.979755i	$-0.435840\pi$
$941$	−515.848	−	297.825i	−0.548191	−	0.316498i	−0.748392	+	0.663257i	$0.769174\pi$
$942$		0			0
$943$	375.171	+	649.815i	0.397848	+	0.689093i
$944$			611.578i			0.647859i
$945$		0			0
$946$	231.772			0.245002
$947$	677.445	−	391.123i	0.715359	−	0.413013i	−0.0976830	−	0.995218i	$-0.531143\pi$
$947$	677.445	−	391.123i	0.715359	−	0.413013i	0.813042	+	0.582205i	$0.197810\pi$
$948$		0			0
$949$	519.934	−	900.552i	0.547875	−	0.948948i
$950$	−22.5074	−	12.9947i	−0.0236920	−	0.0136786i
$951$		0			0
$952$	−59.3680	−	102.828i	−0.0623613	−	0.108013i
$953$		−	426.708i		−	0.447752i	−0.974618	−	0.223876i	$-0.928129\pi$
$953$		−	426.708i		−	0.447752i	0.974618	−	0.223876i	$-0.0718711\pi$
$954$		0			0
$955$	623.745			0.653136
$956$	505.358	−	291.769i	0.528618	−	0.305198i
$957$		0			0
$958$	252.658	−	437.616i	0.263735	−	0.456802i
$959$	50.2468	+	29.0100i	0.0523950	+	0.0302503i
$960$		0			0
$961$	22.3064	+	38.6359i	0.0232117	+	0.0402038i
$962$		−	171.003i		−	0.177758i
$963$		0			0
$964$	347.211			0.360178
$965$	174.080	−	100.505i	0.180394	−	0.104151i
$966$		0			0
$967$	605.454	−	1048.68i	0.626116	−	1.08446i	−0.362208	−	0.932097i	$-0.617977\pi$
$967$	605.454	−	1048.68i	0.626116	−	1.08446i	0.988324	−	0.152367i	$-0.0486895\pi$
$968$	445.959	+	257.475i	0.460702	+	0.265986i
$969$		0			0
$970$	−105.596	−	182.898i	−0.108862	−	0.188555i
$971$			1193.69i			1.22934i	0.788784	+	0.614670i	$0.210711\pi$
$971$			1193.69i			1.22934i	−0.788784	+	0.614670i	$0.789289\pi$
$972$		0			0
$973$	−133.903			−0.137619
$974$	340.355	−	196.504i	0.349441	−	0.201750i
$975$		0			0
$976$	201.777	−	349.487i	0.206738	−	0.358081i
$977$	787.629	+	454.738i	0.806171	+	0.465443i	0.845624	−	0.533779i	$-0.179228\pi$
$977$	787.629	+	454.738i	0.806171	+	0.465443i	−0.0394537	+	0.999221i	$0.512562\pi$
$978$		0			0
$979$	−386.434	−	669.324i	−0.394723	−	0.683681i
$980$		−	359.046i		−	0.366373i
$981$		0			0
$982$	507.268			0.516566
$983$	−1351.76	+	780.441i	−1.37514	+	0.793938i	−0.991570	−	0.129574i	$-0.958639\pi$
$983$	−1351.76	+	780.441i	−1.37514	+	0.793938i	−0.383570	+	0.923512i	$0.625306\pi$
$984$		0			0
$985$	237.816	−	411.909i	0.241437	−	0.418182i
$986$	−4.47008	−	2.58080i	−0.00453355	−	0.00261745i
$987$		0			0
$988$	−229.307	−	397.171i	−0.232092	−	0.401995i
$989$			762.782i			0.771265i
$990$		0			0
$991$	−1692.63			−1.70800			−0.854001	−	0.520271i	$-0.825831\pi$
$991$	−1692.63			−1.70800			−0.854001	+	0.520271i	$0.825831\pi$
$992$	811.185	−	468.338i	0.817726	−	0.472115i
$993$		0			0
$994$	35.8164	−	62.0357i	0.0360326	−	0.0624102i
$995$	−186.854	−	107.880i	−0.187793	−	0.108422i
$996$		0			0
$997$	−8.28719	−	14.3538i	−0.00831213	−	0.0143970i	0.861839	−	0.507181i	$-0.169313\pi$
$997$	−8.28719	−	14.3538i	−0.00831213	−	0.0143970i	−0.870152	+	0.492784i	$0.835979\pi$
$998$			643.591i			0.644881i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	405.3.i.d.296.3		8
3.2	odd	2	inner	405.3.i.d.296.2		8
9.2	odd	6	inner	405.3.i.d.26.3		8
9.4	even	3		45.3.c.a.26.3	yes	4
9.5	odd	6		45.3.c.a.26.2	✓	4
9.7	even	3	inner	405.3.i.d.26.2		8
36.23	even	6		720.3.l.a.161.2		4
36.31	odd	6		720.3.l.a.161.4		4
45.4	even	6		225.3.c.c.26.2		4
45.13	odd	12		225.3.d.b.224.5		8
45.14	odd	6		225.3.c.c.26.3		4
45.22	odd	12		225.3.d.b.224.4		8
45.23	even	12		225.3.d.b.224.3		8
45.32	even	12		225.3.d.b.224.6		8
72.5	odd	6		2880.3.l.g.1601.3		4
72.13	even	6		2880.3.l.g.1601.1		4
72.59	even	6		2880.3.l.c.1601.4		4
72.67	odd	6		2880.3.l.c.1601.2		4
180.23	odd	12		3600.3.c.i.449.5		8
180.59	even	6		3600.3.l.v.1601.1		4
180.67	even	12		3600.3.c.i.449.4		8
180.103	even	12		3600.3.c.i.449.6		8
180.139	odd	6		3600.3.l.v.1601.2		4
180.167	odd	12		3600.3.c.i.449.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.3.c.a.26.2	✓	4	9.5	odd	6
45.3.c.a.26.3	yes	4	9.4	even	3
225.3.c.c.26.2		4	45.4	even	6
225.3.c.c.26.3		4	45.14	odd	6
225.3.d.b.224.3		8	45.23	even	12
225.3.d.b.224.4		8	45.22	odd	12
225.3.d.b.224.5		8	45.13	odd	12
225.3.d.b.224.6		8	45.32	even	12
405.3.i.d.26.2		8	9.7	even	3	inner
405.3.i.d.26.3		8	9.2	odd	6	inner
405.3.i.d.296.2		8	3.2	odd	2	inner
405.3.i.d.296.3		8	1.1	even	1	trivial
720.3.l.a.161.2		4	36.23	even	6
720.3.l.a.161.4		4	36.31	odd	6
2880.3.l.c.1601.2		4	72.67	odd	6
2880.3.l.c.1601.4		4	72.59	even	6
2880.3.l.g.1601.1		4	72.13	even	6
2880.3.l.g.1601.3		4	72.5	odd	6
3600.3.c.i.449.3		8	180.167	odd	12
3600.3.c.i.449.4		8	180.67	even	12
3600.3.c.i.449.5		8	180.23	odd	12
3600.3.c.i.449.6		8	180.103	even	12
3600.3.l.v.1601.1		4	180.59	even	6
3600.3.l.v.1601.2		4	180.139	odd	6

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:	\( N \)	\(=\)	\( 405 = 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	405.i (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.711747 - 0.410927i) q^{2} +(-1.66228 + 2.87915i) q^{4} +(-1.93649 - 1.11803i) q^{5} +(-0.418861 - 0.725489i) q^{7} +6.01972i q^{8} -1.83772 q^{10} +(-12.4462 + 7.18582i) q^{11} +(10.9057 - 18.8892i) q^{13} +(-0.596246 - 0.344243i) q^{14} +(-4.17544 - 7.23208i) q^{16} -23.5454i q^{17} -6.32456 q^{19} +(6.43797 - 3.71697i) q^{20} +(-5.90569 + 10.2290i) q^{22} +(-33.6644 - 19.4361i) q^{23} +(2.50000 + 4.33013i) q^{25} -17.9258i q^{26} +2.78505 q^{28} +(0.231001 - 0.133369i) q^{29} +(-15.1359 + 26.2162i) q^{31} +(-26.7966 - 15.4710i) q^{32} +(-9.67544 - 16.7584i) q^{34} +1.87320i q^{35} +9.53950 q^{37} +(-4.50148 + 2.59893i) q^{38} +(6.73025 - 11.6571i) q^{40} +(-16.7167 - 9.65138i) q^{41} +(-9.81139 - 16.9938i) q^{43} -47.7793i q^{44} -31.9473 q^{46} +(-19.1984 + 11.0842i) q^{47} +(24.1491 - 41.8275i) q^{49} +(3.55873 + 2.05464i) q^{50} +(36.2566 + 62.7982i) q^{52} -49.0012i q^{53} +32.1359 q^{55} +(4.36724 - 2.52143i) q^{56} +(0.109610 - 0.189849i) q^{58} +(-63.4235 - 36.6176i) q^{59} +(24.1623 + 41.8503i) q^{61} +24.8791i q^{62} +7.97367 q^{64} +(-42.2376 + 24.3859i) q^{65} +(-38.6491 + 66.9422i) q^{67} +(67.7907 + 39.1390i) q^{68} +(0.769751 + 1.33325i) q^{70} +104.044i q^{71} +47.6754 q^{73} +(6.78971 - 3.92004i) q^{74} +(10.5132 - 18.2093i) q^{76} +(10.4265 + 6.01972i) q^{77} +(-34.1096 - 59.0796i) q^{79} +18.6732i q^{80} -15.8641 q^{82} +(-24.4304 + 14.1049i) q^{83} +(-26.3246 + 45.5955i) q^{85} +(-13.9664 - 8.06353i) q^{86} +(-43.2566 - 74.9226i) q^{88} +53.7774i q^{89} -18.2719 q^{91} +(111.919 - 64.6165i) q^{92} +(-9.10961 + 15.7783i) q^{94} +(12.2474 + 7.07107i) q^{95} +(57.4605 + 99.5245i) q^{97} -39.6941i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 12 q^{4} - 16 q^{7} - 40 q^{10} + 24 q^{13} - 84 q^{16} + 16 q^{22} + 20 q^{25} - 256 q^{28} + 56 q^{31} - 128 q^{34} + 304 q^{37} - 60 q^{40} + 48 q^{43} + 48 q^{46} + 92 q^{49} + 328 q^{52} + 80 q^{55}+ \cdots + 232 q^{97}+O(q^{100}) \) 8 * q + 12 * q^4 - 16 * q^7 - 40 * q^10 + 24 * q^13 - 84 * q^16 + 16 * q^22 + 20 * q^25 - 256 * q^28 + 56 * q^31 - 128 * q^34 + 304 * q^37 - 60 * q^40 + 48 * q^43 + 48 * q^46 + 92 * q^49 + 328 * q^52 + 80 * q^55 - 328 * q^58 + 168 * q^61 - 88 * q^64 - 208 * q^67 + 120 * q^70 + 432 * q^73 + 160 * q^76 + 56 * q^79 - 304 * q^82 - 160 * q^85 - 384 * q^88 + 208 * q^91 + 256 * q^94 + 232 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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