LMFDB - Embedded newform 42.4.d.a.41.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [42,4,Mod(41,42)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("42.41"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(42, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 42 = 2 \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	42.d (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

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

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

Self dual:	no
Analytic conductor:	$2.47808022024$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.116876510171136.13
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 15x^{6} + 455x^{4} + 5097x^{2} + 21904 $ x^8 + 15x^6 + 455x^4 + 5097*x^2 + 21904
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{5}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			41.2
Root			$-3.04373 - 3.17617i$ of defining polynomial
Character	$\chi$	$=$	42.41
Dual form			42.4.d.a.41.6

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.00000i q^{2} +(-0.985634 - 5.10182i) q^{3} -4.00000 q^{4} -14.1462 q^{5} +(-10.2036 + 1.97127i) q^{6} +(18.0570 - 4.11618i) q^{7} +8.00000i q^{8} +(-25.0570 + 10.0570i) q^{9} +28.2923i q^{10} -50.1141i q^{11} +(3.94254 + 20.4073i) q^{12} -50.6709i q^{13} +(-8.23236 - 36.1141i) q^{14} +(13.9430 + 72.1711i) q^{15} +16.0000 q^{16} +114.211 q^{17} +(20.1141 + 50.1141i) q^{18} +51.7127i q^{19} +56.5847 q^{20} +(-38.7976 - 88.0667i) q^{21} -100.228 q^{22} +13.7718i q^{23} +(40.8145 - 7.88507i) q^{24} +75.1141 q^{25} -101.342 q^{26} +(76.0063 + 117.924i) q^{27} +(-72.2282 + 16.4647i) q^{28} -128.114i q^{29} +(144.342 - 27.8859i) q^{30} +107.490i q^{31} -32.0000i q^{32} +(-255.673 + 49.3942i) q^{33} -228.422i q^{34} +(-255.438 + 58.2282i) q^{35} +(100.228 - 40.2282i) q^{36} -74.0000 q^{37} +103.425 q^{38} +(-258.513 + 49.9430i) q^{39} -113.169i q^{40} +10.3161 q^{41} +(-176.133 + 77.5953i) q^{42} +68.9128 q^{43} +200.456i q^{44} +(354.461 - 142.269i) q^{45} +27.5436 q^{46} -218.679 q^{47} +(-15.7701 - 81.6291i) q^{48} +(309.114 - 148.652i) q^{49} -150.228i q^{50} +(-112.570 - 582.685i) q^{51} +202.683i q^{52} +385.027i q^{53} +(235.848 - 152.013i) q^{54} +708.923i q^{55} +(32.9295 + 144.456i) q^{56} +(263.829 - 50.9698i) q^{57} -256.228 q^{58} +757.510 q^{59} +(-55.7718 - 288.685i) q^{60} -300.665i q^{61} +214.981 q^{62} +(-411.060 + 284.740i) q^{63} -64.0000 q^{64} +716.799i q^{65} +(98.7884 + 511.346i) q^{66} +132.228 q^{67} -456.845 q^{68} +(70.2612 - 13.5740i) q^{69} +(116.456 + 510.876i) q^{70} +147.483i q^{71} +(-80.4564 - 200.456i) q^{72} -970.274i q^{73} +148.000i q^{74} +(-74.0350 - 383.218i) q^{75} -206.851i q^{76} +(-206.279 - 904.913i) q^{77} +(99.8859 + 517.027i) q^{78} +669.027 q^{79} -226.339 q^{80} +(526.711 - 504.000i) q^{81} -20.6322i q^{82} -612.350 q^{83} +(155.191 + 352.267i) q^{84} -1615.65 q^{85} -137.826i q^{86} +(-653.615 + 126.274i) q^{87} +400.913 q^{88} +502.644 q^{89} +(-284.537 - 708.923i) q^{90} +(-208.570 - 914.966i) q^{91} -55.0872i q^{92} +(548.396 - 105.946i) q^{93} +437.357i q^{94} -731.537i q^{95} +(-163.258 + 31.5403i) q^{96} +1124.03i q^{97} +(-297.304 - 618.228i) q^{98} +(504.000 + 1255.71i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 32 q^{4} + 4 q^{7} - 60 q^{9} + 252 q^{15} + 128 q^{16} - 120 q^{18} - 168 q^{21} - 240 q^{22} + 320 q^{25} - 16 q^{28} + 312 q^{30} + 240 q^{36} - 592 q^{37} - 804 q^{39} - 216 q^{42} - 1696 q^{43}+ \cdots + 4032 q^{99}+O(q^{100}) $ 8 * q - 32 * q^4 + 4 * q^7 - 60 * q^9 + 252 * q^15 + 128 * q^16 - 120 * q^18 - 168 * q^21 - 240 * q^22 + 320 * q^25 - 16 * q^28 + 312 * q^30 + 240 * q^36 - 592 * q^37 - 804 * q^39 - 216 * q^42 - 1696 * q^43 + 1344 * q^46 + 2192 * q^49 + 504 * q^51 + 2532 * q^57 - 1488 * q^58 - 1008 * q^60 - 2496 * q^63 - 512 * q^64 + 496 * q^67 - 192 * q^70 + 480 * q^72 + 1080 * q^78 + 2824 * q^79 + 672 * q^84 - 3936 * q^85 + 960 * q^88 - 264 * q^91 - 1512 * q^93 + 4032 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/42\mathbb{Z}\right)^\times$.

$n$	$29$	$31$
$\chi(n)$	$-1$	$-1$

Coefficient data

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

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

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

$2$		−	2.00000i		−	0.707107i
$2$		−	2.00000i		−	0.707107i
$3$	−0.985634	−	5.10182i	−0.189685	−	0.981845i
$3$	−0.985634	−	5.10182i	−0.189685	−	0.981845i
$4$	−4.00000			−0.500000
$5$	−14.1462			−1.26527			−0.632636	−	0.774449i	$-0.718027\pi$
$5$	−14.1462			−1.26527			−0.632636	+	0.774449i	$0.718027\pi$
$6$	−10.2036	+	1.97127i	−0.694269	+	0.134128i
$7$	18.0570	−	4.11618i	0.974989	−	0.222253i
$7$	18.0570	−	4.11618i	0.974989	−	0.222253i
$8$			8.00000i			0.353553i
$9$	−25.0570	+	10.0570i	−0.928039	+	0.372483i
$10$			28.2923i			0.894682i
$11$		−	50.1141i		−	1.37363i	−0.726831	−	0.686817i	$-0.759007\pi$
$11$		−	50.1141i		−	1.37363i	0.726831	−	0.686817i	$-0.240993\pi$
$12$	3.94254	+	20.4073i	0.0948427	+	0.490922i
$13$		−	50.6709i		−	1.08104i	−0.841330	−	0.540522i	$-0.818227\pi$
$13$		−	50.6709i		−	1.08104i	0.841330	−	0.540522i	$-0.181773\pi$
$14$	−8.23236	−	36.1141i	−0.157157	−	0.689421i
$15$	13.9430	+	72.1711i	0.240004	+	1.24230i
$16$	16.0000			0.250000
$17$	114.211			1.62943			0.814714	−	0.579862i	$-0.196894\pi$
$17$	114.211			1.62943			0.814714	+	0.579862i	$0.196894\pi$
$18$	20.1141	+	50.1141i	0.263385	+	0.656223i
$19$			51.7127i			0.624406i	0.950015	+	0.312203i	$0.101067\pi$
$19$			51.7127i			0.624406i	−0.950015	+	0.312203i	$0.898933\pi$
$20$	56.5847			0.632636
$21$	−38.7976	−	88.0667i	−0.403159	−	0.915130i
$22$	−100.228			−0.971306
$23$			13.7718i			0.124853i	0.998050	+	0.0624265i	$0.0198839\pi$
$23$			13.7718i			0.124853i	−0.998050	+	0.0624265i	$0.980116\pi$
$24$	40.8145	−	7.88507i	0.347135	−	0.0670639i
$25$	75.1141			0.600913
$26$	−101.342			−0.764413
$27$	76.0063	+	117.924i	0.541756	+	0.840536i
$28$	−72.2282	+	16.4647i	−0.487495	+	0.111126i
$29$		−	128.114i		−	0.820351i	−0.912007	−	0.410176i	$-0.865467\pi$
$29$		−	128.114i		−	0.820351i	0.912007	−	0.410176i	$-0.134533\pi$
$30$	144.342	−	27.8859i	0.878439	−	0.169708i
$31$			107.490i			0.622769i	0.950284	+	0.311385i	$0.100793\pi$
$31$			107.490i			0.622769i	−0.950284	+	0.311385i	$0.899207\pi$
$32$		−	32.0000i		−	0.176777i
$33$	−255.673	+	49.3942i	−1.34870	+	0.260558i
$34$		−	228.422i		−	1.15218i
$35$	−255.438	+	58.2282i	−1.23363	+	0.281210i
$36$	100.228	−	40.2282i	0.464019	−	0.186242i
$37$	−74.0000			−0.328798			−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−74.0000			−0.328798			−0.164399	+	0.986394i	$0.552568\pi$
$38$	103.425			0.441522
$39$	−258.513	+	49.9430i	−1.06142	+	0.205058i
$40$		−	113.169i		−	0.447341i
$41$	10.3161			0.0392952			0.0196476	−	0.999807i	$-0.493746\pi$
$41$	10.3161			0.0392952			0.0196476	+	0.999807i	$0.493746\pi$
$42$	−176.133	+	77.5953i	−0.647095	+	0.285077i
$43$	68.9128			0.244398			0.122199	−	0.992506i	$-0.461005\pi$
$43$	68.9128			0.244398			0.122199	+	0.992506i	$0.461005\pi$
$44$			200.456i			0.686817i
$45$	354.461	−	142.269i	1.17422	−	0.471293i
$46$	27.5436			0.0882844
$47$	−218.679			−0.678671			−0.339336	−	0.940665i	$-0.610202\pi$
$47$	−218.679			−0.678671			−0.339336	+	0.940665i	$0.610202\pi$
$48$	−15.7701	−	81.6291i	−0.0474214	−	0.245461i
$49$	309.114	−	148.652i	0.901207	−	0.433388i
$50$		−	150.228i		−	0.424910i
$51$	−112.570	−	582.685i	−0.309079	−	1.59985i
$52$			202.683i			0.540522i
$53$			385.027i			0.997878i	0.866637	+	0.498939i	$0.166277\pi$
$53$			385.027i			0.997878i	−0.866637	+	0.498939i	$0.833723\pi$
$54$	235.848	−	152.013i	0.594348	−	0.383080i
$55$			708.923i			1.73802i
$56$	32.9295	+	144.456i	0.0785783	+	0.344711i
$57$	263.829	−	50.9698i	0.613070	−	0.118441i
$58$	−256.228			−0.580076
$59$	757.510			1.67151			0.835757	−	0.549099i	$-0.185029\pi$
$59$	757.510			1.67151			0.835757	+	0.549099i	$0.185029\pi$
$60$	−55.7718	−	288.685i	−0.120002	−	0.621150i
$61$		−	300.665i		−	0.631085i	−0.948911	−	0.315542i	$-0.897814\pi$
$61$		−	300.665i		−	0.631085i	0.948911	−	0.315542i	$-0.102186\pi$
$62$	214.981			0.440364
$63$	−411.060	+	284.740i	−0.822042	+	0.569427i
$64$	−64.0000			−0.125000
$65$			716.799i			1.36781i
$66$	98.7884	+	511.346i	0.184243	+	0.953671i
$67$	132.228			0.241108			0.120554	−	0.992707i	$-0.461533\pi$
$67$	132.228			0.241108			0.120554	+	0.992707i	$0.461533\pi$
$68$	−456.845			−0.814714
$69$	70.2612	−	13.5740i	0.122586	−	0.0236828i
$70$	116.456	+	510.876i	0.198846	+	0.872305i
$71$			147.483i			0.246522i	0.992374	+	0.123261i	$0.0393352\pi$
$71$			147.483i			0.246522i	−0.992374	+	0.123261i	$0.960665\pi$
$72$	−80.4564	−	200.456i	−0.131693	−	0.328111i
$73$		−	970.274i		−	1.55564i	−0.628484	−	0.777822i	$-0.716324\pi$
$73$		−	970.274i		−	1.55564i	0.628484	−	0.777822i	$-0.283676\pi$
$74$			148.000i			0.232495i
$75$	−74.0350	−	383.218i	−0.113984	−	0.590003i
$76$		−	206.851i		−	0.312203i
$77$	−206.279	−	904.913i	−0.305294	−	1.33928i
$78$	99.8859	+	517.027i	0.144998	+	0.750535i
$79$	669.027			0.952803			0.476401	−	0.879228i	$-0.341941\pi$
$79$	669.027			0.952803			0.476401	+	0.879228i	$0.341941\pi$
$80$	−226.339			−0.316318
$81$	526.711	−	504.000i	0.722512	−	0.691358i
$82$		−	20.6322i		−	0.0277859i
$83$	−612.350			−0.809809			−0.404905	−	0.914359i	$-0.632695\pi$
$83$	−612.350			−0.809809			−0.404905	+	0.914359i	$0.632695\pi$
$84$	155.191	+	352.267i	0.201580	+	0.457565i
$85$	−1615.65			−2.06167
$86$		−	137.826i		−	0.172815i
$87$	−653.615	+	126.274i	−0.805458	+	0.155609i
$88$	400.913			0.485653
$89$	502.644			0.598653			0.299327	−	0.954151i	$-0.403238\pi$
$89$	502.644			0.598653			0.299327	+	0.954151i	$0.403238\pi$
$90$	−284.537	−	708.923i	−0.333254	−	0.830300i
$91$	−208.570	−	914.966i	−0.240265	−	1.05401i
$92$		−	55.0872i		−	0.0624265i
$93$	548.396	−	105.946i	0.611463	−	0.118130i
$94$			437.357i			0.479893i
$95$		−	731.537i		−	0.790043i
$96$	−163.258	+	31.5403i	−0.173567	+	0.0335320i
$97$			1124.03i			1.17658i	0.808650	+	0.588290i	$0.200199\pi$
$97$			1124.03i			1.17658i	−0.808650	+	0.588290i	$0.799801\pi$
$98$	−297.304	−	618.228i	−0.306452	−	0.637250i
$99$	504.000	+	1255.71i	0.511656	+	1.27479i
$100$	−300.456			−0.300456
$101$	−1233.48			−1.21520			−0.607601	−	0.794243i	$-0.707868\pi$
$101$	−1233.48			−1.21520			−0.607601	+	0.794243i	$0.707868\pi$
$102$	−1165.37	+	225.141i	−1.13126	+	0.218552i
$103$			843.091i			0.806527i	0.915084	+	0.403263i	$0.132124\pi$
$103$			843.091i			0.806527i	−0.915084	+	0.403263i	$0.867876\pi$
$104$	405.367			0.382207
$105$	548.838	+	1245.81i	0.510106	+	1.15789i
$106$	770.054			0.705606
$107$			1030.57i			0.931112i	0.885018	+	0.465556i	$0.154146\pi$
$107$			1030.57i			0.931112i	−0.885018	+	0.465556i	$0.845854\pi$
$108$	−304.025	−	471.696i	−0.270878	−	0.420268i
$109$	−419.141			−0.368316			−0.184158	−	0.982897i	$-0.558956\pi$
$109$	−419.141			−0.368316			−0.184158	+	0.982897i	$0.558956\pi$
$110$	1417.85			1.22897
$111$	72.9369	+	377.534i	0.0623682	+	0.322829i
$112$	288.913	−	65.8589i	0.243747	−	0.0555632i
$113$		−	2177.13i		−	1.81246i	−0.422789	−	0.906228i	$-0.638949\pi$
$113$		−	2177.13i		−	1.81246i	0.422789	−	0.906228i	$-0.361051\pi$
$114$	−101.940	−	527.658i	−0.0837502	−	0.433506i
$115$		−	194.818i		−	0.157973i
$116$			512.456i			0.410176i
$117$	509.599	+	1269.66i	0.402671	+	1.00325i
$118$		−	1515.02i		−	1.18194i
$119$	2062.32	−	470.114i	1.58868	−	0.362145i
$120$	−577.369	+	111.544i	−0.439220	+	0.0848541i
$121$	−1180.42			−0.886869
$122$	−601.330			−0.446244
$123$	−10.1679	−	52.6308i	−0.00745373	−	0.0385818i
$124$		−	429.961i		−	0.311385i
$125$	705.694			0.504954
$126$	569.480	+	822.120i	0.402645	+	0.581272i
$127$	1508.56			1.05404			0.527021	−	0.849852i	$-0.323309\pi$
$127$	1508.56			1.05404			0.527021	+	0.849852i	$0.323309\pi$
$128$			128.000i			0.0883883i
$129$	−67.9228	−	351.580i	−0.0463587	−	0.239961i
$130$	1433.60			0.967191
$131$	741.250			0.494376			0.247188	−	0.968967i	$-0.420493\pi$
$131$	741.250			0.494376			0.247188	+	0.968967i	$0.420493\pi$
$132$	1022.69	−	197.577i	0.674348	−	0.130279i
$133$	212.859	+	933.779i	0.138776	+	0.608789i
$134$		−	264.456i		−	0.170489i
$135$	−1075.20	−	1668.17i	−0.685469	−	1.06351i
$136$			913.690i			0.576090i
$137$			2036.34i			1.26990i	0.772554	+	0.634949i	$0.218979\pi$
$137$			2036.34i			1.26990i	−0.772554	+	0.634949i	$0.781021\pi$
$138$	−27.1479	−	140.522i	−0.0167463	−	0.0866816i
$139$		−	1740.45i		−	1.06204i	−0.847361	−	0.531018i	$-0.821810\pi$
$139$		−	1740.45i		−	1.06204i	0.847361	−	0.531018i	$-0.178190\pi$
$140$	1021.75	−	232.913i	0.616813	−	0.140605i
$141$	215.537	+	1115.66i	0.128734	+	0.666350i
$142$	294.967			0.174317
$143$	−2539.33			−1.48496
$144$	−400.913	+	160.913i	−0.232010	+	0.0931208i
$145$			1812.32i			1.03797i
$146$	−1940.55			−1.10001
$147$	−1063.07	−	1430.53i	−0.596466	−	0.802638i
$148$	296.000			0.164399
$149$			402.866i			0.221504i	0.993848	+	0.110752i	$0.0353259\pi$
$149$			402.866i			0.221504i	−0.993848	+	0.110752i	$0.964674\pi$
$150$	−766.437	+	148.070i	−0.417195	+	0.0805991i
$151$	−2313.37			−1.24675			−0.623375	−	0.781923i	$-0.714239\pi$
$151$	−2313.37			−1.24675			−0.623375	+	0.781923i	$0.714239\pi$
$152$	−413.702			−0.220761
$153$	−2861.80	+	1148.63i	−1.51217	+	0.606935i
$154$	−1809.83	+	412.557i	−0.947012	+	0.215876i
$155$		−	1520.58i		−	0.787972i
$156$	1034.05	−	199.772i	0.530709	−	0.102529i
$157$		−	598.644i		−	0.304312i	−0.988356	−	0.152156i	$-0.951378\pi$
$157$		−	598.644i		−	0.304312i	0.988356	−	0.152156i	$-0.0486216\pi$
$158$		−	1338.05i		−	0.673733i
$159$	1964.34	−	379.496i	0.979761	−	0.189283i
$160$			452.677i			0.223671i
$161$	56.6872	+	248.678i	0.0277489	+	0.121730i
$162$	−1008.00	−	1053.42i	−0.488864	−	0.510893i
$163$	2423.42			1.16452			0.582261	−	0.813002i	$-0.302168\pi$
$163$	2423.42			1.16452			0.582261	+	0.813002i	$0.302168\pi$
$164$	−41.2644			−0.0196476
$165$	3616.79	−	698.738i	1.70647	−	0.329677i
$166$			1224.70i			0.572622i
$167$	−893.527			−0.414031			−0.207016	−	0.978338i	$-0.566375\pi$
$167$	−893.527			−0.414031			−0.207016	+	0.978338i	$0.566375\pi$
$168$	704.534	−	310.381i	0.323547	−	0.142538i
$169$	−370.537			−0.168656
$170$			3231.30i			1.45782i
$171$	−520.078	−	1295.77i	−0.232581	−	0.579473i
$172$	−275.651			−0.122199
$173$	−952.430			−0.418566			−0.209283	−	0.977855i	$-0.567113\pi$
$173$	−952.430			−0.418566			−0.209283	+	0.977855i	$0.567113\pi$
$174$	252.547	+	1307.23i	0.110032	+	0.569545i
$175$	1356.34	−	309.183i	0.585883	−	0.133555i
$176$		−	801.826i		−	0.343408i
$177$	−746.628	−	3864.67i	−0.317062	−	1.64117i
$178$		−	1005.29i		−	0.423312i
$179$			1289.54i			0.538461i	0.963076	+	0.269231i	$0.0867694\pi$
$179$			1289.54i			0.538461i	−0.963076	+	0.269231i	$0.913231\pi$
$180$	−1417.85	+	569.075i	−0.587111	+	0.235646i
$181$		−	1792.03i		−	0.735914i	−0.929843	−	0.367957i	$-0.880057\pi$
$181$		−	1792.03i		−	0.735914i	0.929843	−	0.367957i	$-0.119943\pi$
$182$	−1829.93	+	417.141i	−0.745295	+	0.169893i
$183$	−1533.94	+	296.346i	−0.619628	+	0.119708i
$184$	−110.174			−0.0441422
$185$	1046.82			0.416019
$186$	−211.892	−	1096.79i	−0.0835307	−	0.432369i
$187$		−	5723.59i		−	2.23824i
$188$	874.714			0.339336
$189$	1857.85	+	1816.50i	0.715018	+	0.699106i
$190$	−1463.07			−0.558645
$191$			3967.59i			1.50306i	0.659698	+	0.751531i	$0.270684\pi$
$191$			3967.59i			1.50306i	−0.659698	+	0.751531i	$0.729316\pi$
$192$	63.0806	+	326.516i	0.0237107	+	0.122731i
$193$	−2808.97			−1.04764			−0.523819	−	0.851829i	$-0.675493\pi$
$193$	−2808.97			−1.04764			−0.523819	+	0.851829i	$0.675493\pi$
$194$	2248.07			0.831968
$195$	3656.97	−	706.501i	1.34298	−	0.259454i
$196$	−1236.46	+	594.609i	−0.450604	+	0.216694i
$197$			3986.84i			1.44188i	0.692997	+	0.720940i	$0.256290\pi$
$197$			3986.84i			1.44188i	−0.692997	+	0.720940i	$0.743710\pi$
$198$	2511.42	−	1008.00i	0.901409	−	0.361795i
$199$			4407.40i			1.57001i	0.619489	+	0.785005i	$0.287340\pi$
$199$			4407.40i			1.57001i	−0.619489	+	0.785005i	$0.712660\pi$
$200$			600.913i			0.212455i
$201$	−130.329	−	674.604i	−0.0457347	−	0.236731i
$202$			2466.95i			0.859277i
$203$	−527.341	−	2313.36i	−0.182325	−	0.799834i
$204$	450.282	+	2330.74i	0.154539	+	0.799923i
$205$	−145.933			−0.0497191
$206$	1686.18			0.570300
$207$	−138.504	−	345.081i	−0.0465057	−	0.115868i
$208$		−	810.734i		−	0.270261i
$209$	2591.54			0.857705
$210$	2491.61	−	1097.68i	0.818751	−	0.360699i
$211$	3543.88			1.15626			0.578130	−	0.815945i	$-0.303783\pi$
$211$	3543.88			1.15626			0.578130	+	0.815945i	$0.303783\pi$
$212$		−	1540.11i		−	0.498939i
$213$	752.433	−	145.365i	0.242046	−	0.0467616i
$214$	2061.14			0.658396
$215$	−974.852			−0.309230
$216$	−943.391	+	608.050i	−0.297174	+	0.191540i
$217$	442.450	+	1940.96i	0.138412	+	0.607193i
$218$			838.282i			0.260439i
$219$	−4950.16	+	956.336i	−1.52740	+	0.295083i
$220$		−	2835.69i		−	0.869010i
$221$		−	5787.18i		−	1.76148i
$222$	755.069	−	145.874i	0.228274	−	0.0441010i
$223$			4919.74i			1.47736i	0.674059	+	0.738678i	$0.264549\pi$
$223$			4919.74i			1.47736i	−0.674059	+	0.738678i	$0.735451\pi$
$224$	−131.718	−	577.826i	−0.0392891	−	0.172355i
$225$	−1882.14	+	755.426i	−0.557670	+	0.223830i
$226$	−4354.27			−1.28160
$227$	4091.04			1.19618			0.598088	−	0.801430i	$-0.295927\pi$
$227$	4091.04			1.19618			0.598088	+	0.801430i	$0.295927\pi$
$228$	−1055.32	+	203.879i	−0.306535	+	0.0592204i
$229$			3252.08i			0.938443i	0.883081	+	0.469221i	$0.155465\pi$
$229$			3252.08i			0.938443i	−0.883081	+	0.469221i	$0.844535\pi$
$230$	−389.636			−0.111704
$231$	−4413.38	+	1944.31i	−1.25705	+	0.553793i
$232$	1024.91			0.290038
$233$		−	1063.09i		−	0.298906i	−0.988769	−	0.149453i	$-0.952249\pi$
$233$		−	1063.09i		−	0.298906i	0.988769	−	0.149453i	$-0.0477513\pi$
$234$	2539.33	−	1019.20i	0.709405	−	0.284731i
$235$	3093.46			0.858704
$236$	−3030.04			−0.835757
$237$	−659.416	−	3413.25i	−0.180733	−	0.935504i
$238$	−940.228	−	4124.64i	−0.256075	−	1.12336i
$239$		−	2013.87i		−	0.545047i	−0.962149	−	0.272523i	$-0.912142\pi$
$239$		−	2013.87i		−	0.545047i	0.962149	−	0.272523i	$-0.0878582\pi$
$240$	223.087	+	1154.74i	0.0600009	+	0.310575i
$241$		−	4055.69i		−	1.08403i	−0.840370	−	0.542013i	$-0.817662\pi$
$241$		−	4055.69i		−	1.08403i	0.840370	−	0.542013i	$-0.182338\pi$
$242$			2360.85i			0.627111i
$243$	−3090.46	−	2190.43i	−0.815856	−	0.578255i
$244$			1202.66i			0.315542i
$245$	−4372.78	+	2102.86i	−1.14027	+	0.548354i
$246$	−105.262	+	20.3358i	−0.0272814	+	0.00527058i
$247$	2620.33			0.675010
$248$	−859.923			−0.220182
$249$	603.553	+	3124.10i	0.153609	+	0.795107i
$250$		−	1411.39i		−	0.357056i
$251$	−2285.30			−0.574688			−0.287344	−	0.957828i	$-0.592772\pi$
$251$	−2285.30			−0.574688			−0.287344	+	0.957828i	$0.592772\pi$
$252$	1644.24	−	1138.96i	0.411021	−	0.284713i
$253$	690.161			0.171502
$254$		−	3017.13i		−	0.745321i
$255$	1592.44	+	8242.75i	0.391069	+	2.02424i
$256$	256.000			0.0625000
$257$	−2692.04			−0.653405			−0.326702	−	0.945127i	$-0.605937\pi$
$257$	−2692.04			−0.653405			−0.326702	+	0.945127i	$0.605937\pi$
$258$	−703.161	+	135.846i	−0.169678	+	0.0327805i
$259$	−1336.22	+	304.597i	−0.320574	+	0.0730763i
$260$		−	2867.19i		−	0.683907i
$261$	1288.45	+	3210.16i	0.305567	+	0.761318i
$262$		−	1482.50i		−	0.349577i
$263$			818.114i			0.191814i	0.995390	+	0.0959070i	$0.0305751\pi$
$263$			818.114i			0.191814i	−0.995390	+	0.0959070i	$0.969425\pi$
$264$	−395.153	−	2045.38i	−0.0921213	−	0.476836i
$265$		−	5446.66i		−	1.26259i
$266$	1867.56	−	425.718i	0.430479	−	0.0981295i
$267$	−495.423	−	2564.40i	−0.113556	−	0.587785i
$268$	−528.913			−0.120554
$269$	2394.81			0.542804			0.271402	−	0.962466i	$-0.412513\pi$
$269$	2394.81			0.542804			0.271402	+	0.962466i	$0.412513\pi$
$270$	−3336.34	+	2150.40i	−0.752012	+	0.484700i
$271$		−	229.464i		−	0.0514353i	−0.999669	−	0.0257176i	$-0.991813\pi$
$271$		−	229.464i		−	0.0514353i	0.999669	−	0.0257176i	$-0.00818708\pi$
$272$	1827.38			0.407357
$273$	−4462.42	+	1965.91i	−0.989296	+	0.435833i
$274$	4072.67			0.897953
$275$		−	3764.28i		−	0.825434i
$276$	−281.045	+	54.2958i	−0.0612931	+	0.0118414i
$277$	−5151.92			−1.11750			−0.558752	−	0.829334i	$-0.688720\pi$
$277$	−5151.92			−1.11750			−0.558752	+	0.829334i	$0.688720\pi$
$278$	−3480.90			−0.750972
$279$	−1081.04	−	2693.39i	−0.231971	−	0.577954i
$280$	−465.826	−	2043.50i	−0.0994229	−	0.436153i
$281$		−	5618.50i		−	1.19278i	−0.802694	−	0.596390i	$-0.796601\pi$
$281$		−	5618.50i		−	1.19278i	0.802694	−	0.596390i	$-0.203399\pi$
$282$	2231.32	−	431.074i	0.471180	−	0.0910287i
$283$			7727.21i			1.62309i	0.584288	+	0.811546i	$0.301374\pi$
$283$			7727.21i			1.62309i	−0.584288	+	0.811546i	$0.698626\pi$
$284$		−	589.933i		−	0.123261i
$285$	−3732.17	+	721.028i	−0.775700	+	0.149860i
$286$			5078.65i			1.05002i
$287$	186.278	−	42.4629i	0.0383124	−	0.00873347i
$288$	321.826	+	801.826i	0.0658464	+	0.164056i
$289$	8131.20			1.65504
$290$	3624.65			0.733954
$291$	5734.61	−	1107.89i	1.15522	−	0.223180i
$292$			3881.10i			0.777822i
$293$	4694.21			0.935968			0.467984	−	0.883737i	$-0.344980\pi$
$293$	4694.21			0.935968			0.467984	+	0.883737i	$0.344980\pi$
$294$	−2861.05	+	2126.14i	−0.567551	+	0.421765i
$295$	−10715.9			−2.11492
$296$		−	592.000i		−	0.116248i
$297$	5909.65	−	3808.99i	1.15459	−	0.744175i
$298$	805.731			0.156627
$299$	697.829			0.134972
$300$	296.140	+	1532.87i	0.0569922	+	0.295002i
$301$	1244.36	−	283.658i	0.238285	−	0.0543181i
$302$			4626.74i			0.881586i
$303$	1215.76	+	6292.96i	0.230506	+	1.19314i
$304$			827.404i			0.156102i
$305$			4253.26i			0.798494i
$306$	2297.26	+	5723.59i	0.429168	+	1.06927i
$307$		−	3854.92i		−	0.716651i	−0.933597	−	0.358325i	$-0.883348\pi$
$307$		−	3854.92i		−	0.716651i	0.933597	−	0.358325i	$-0.116652\pi$
$308$	825.115	+	3619.65i	0.152647	+	0.669639i
$309$	4301.30	−	830.980i	0.791884	−	0.152986i
$310$	−3041.15			−0.557180
$311$	3554.09			0.648020			0.324010	−	0.946054i	$-0.394969\pi$
$311$	3554.09			0.648020			0.324010	+	0.946054i	$0.394969\pi$
$312$	−399.544	−	2068.11i	−0.0724990	−	0.375268i
$313$		−	156.167i		−	0.0282015i	−0.999901	−	0.0141007i	$-0.995511\pi$
$313$		−	156.167i		−	0.0282015i	0.999901	−	0.0141007i	$-0.00448855\pi$
$314$	−1197.29			−0.215181
$315$	5814.92	−	4027.98i	1.04011	−	0.720479i
$316$	−2676.11			−0.476401
$317$		−	2199.44i		−	0.389694i	−0.980834	−	0.194847i	$-0.937579\pi$
$317$		−	2199.44i		−	0.389694i	0.980834	−	0.194847i	$-0.0624210\pi$
$318$	−758.991	−	3928.67i	−0.133843	−	0.692796i
$319$	−6420.32			−1.12686
$320$	905.355			0.158159
$321$	5257.78	−	1015.77i	0.914208	−	0.176618i
$322$	497.356	−	113.374i	0.0860763	−	0.0196215i
$323$			5906.17i			1.01743i
$324$	−2106.85	+	2016.00i	−0.361256	+	0.345679i
$325$		−	3806.10i		−	0.649613i
$326$		−	4846.85i		−	0.823442i
$327$	413.120	+	2138.38i	0.0698641	+	0.361629i
$328$			82.5287i			0.0138929i
$329$	−3948.69	+	900.121i	−0.661697	+	0.150837i
$330$	−1397.48	−	7233.58i	−0.233117	−	1.20665i
$331$	−3595.96			−0.597136			−0.298568	−	0.954388i	$-0.596509\pi$
$331$	−3595.96			−0.597136			−0.298568	+	0.954388i	$0.596509\pi$
$332$	2449.40			0.404905
$333$	1854.22	−	744.222i	0.305137	−	0.122472i
$334$			1787.05i			0.292764i
$335$	−1870.52			−0.305067
$336$	−620.762	−	1409.07i	−0.100790	−	0.228782i
$337$	8094.56			1.30842			0.654212	−	0.756312i	$-0.273001\pi$
$337$	8094.56			1.30842			0.654212	+	0.756312i	$0.273001\pi$
$338$			741.074i			0.119258i
$339$	−11107.3	+	2145.86i	−1.77955	+	0.343797i
$340$	6462.60			1.03084
$341$	5386.78			0.855457
$342$	−2591.54	+	1040.16i	−0.409749	+	0.164459i
$343$	4969.81	−	3956.59i	0.782345	−	0.622845i
$344$			551.302i			0.0864077i
$345$	−993.927	+	192.020i	−0.155105	+	0.0299652i
$346$			1904.86i			0.295971i
$347$			4849.95i			0.750315i	0.926961	+	0.375157i	$0.122411\pi$
$347$			4849.95i			0.750315i	−0.926961	+	0.375157i	$0.877589\pi$
$348$	2614.46	−	505.095i	0.402729	−	0.0778044i
$349$		−	7102.53i		−	1.08937i	−0.838641	−	0.544685i	$-0.816649\pi$
$349$		−	7102.53i		−	1.08937i	0.838641	−	0.544685i	$-0.183351\pi$
$350$	−618.367	−	2712.68i	−0.0944374	−	0.414282i
$351$	5975.31	−	3851.31i	0.908656	−	0.585662i
$352$	−1603.65			−0.242826
$353$	−6791.96			−1.02408			−0.512039	−	0.858962i	$-0.671110\pi$
$353$	−6791.96			−1.02408			−0.512039	+	0.858962i	$0.671110\pi$
$354$	−7729.35	+	1493.26i	−1.16048	+	0.224197i
$355$		−	2086.32i		−	0.311917i
$356$	−2010.58			−0.299327
$357$	−4431.13	−	10058.2i	−0.656919	−	1.49114i
$358$	2579.07			0.380749
$359$		−	12593.8i		−	1.85146i	−0.378188	−	0.925729i	$-0.623453\pi$
$359$		−	12593.8i		−	1.85146i	0.378188	−	0.925729i	$-0.376547\pi$
$360$	1138.15	+	2835.69i	0.166627	+	0.415150i
$361$	4184.79			0.610117
$362$	−3584.06			−0.520370
$363$	1163.47	+	6022.30i	0.168226	+	0.870768i
$364$	834.282	+	3659.87i	0.120133	+	0.527003i
$365$			13725.7i			1.96831i
$366$	592.691	+	3067.87i	0.0846461	+	0.438143i
$367$			5967.85i			0.848826i	0.905469	+	0.424413i	$0.139520\pi$
$367$			5967.85i			0.848826i	−0.905469	+	0.424413i	$0.860480\pi$
$368$			220.349i			0.0312132i
$369$	−258.491	+	103.749i	−0.0364675	+	0.0146368i
$370$		−	2093.63i		−	0.294170i
$371$	1584.84	+	6952.45i	0.221781	+	0.972920i
$372$	−2193.58	+	423.785i	−0.305731	+	0.0590651i
$373$	−13294.3			−1.84545			−0.922726	−	0.385455i	$-0.874044\pi$
$373$	−13294.3			−1.84545			−0.922726	+	0.385455i	$0.874044\pi$
$374$	−11447.2			−1.58267
$375$	−695.557	−	3600.32i	−0.0957824	−	0.495786i
$376$		−	1749.43i		−	0.239946i
$377$	−6491.65			−0.886836
$378$	3633.00	−	3715.69i	0.494343	−	0.505594i
$379$	−12488.3			−1.69256			−0.846282	−	0.532735i	$-0.821164\pi$
$379$	−12488.3			−1.69256			−0.846282	+	0.532735i	$0.821164\pi$
$380$			2926.15i			0.395022i
$381$	−1486.89	−	7696.42i	−0.199937	−	1.03491i
$382$	7935.18			1.06283
$383$	−770.555			−0.102803			−0.0514014	−	0.998678i	$-0.516369\pi$
$383$	−770.555			−0.102803			−0.0514014	+	0.998678i	$0.516369\pi$
$384$	653.032	−	126.161i	0.0867836	−	0.0167660i
$385$	2918.05	+	12801.0i	0.386280	+	1.69455i
$386$			5617.95i			0.740793i
$387$	−1726.75	+	693.059i	−0.226811	+	0.0910341i
$388$		−	4496.13i		−	0.588290i
$389$		−	7412.44i		−	0.966132i	−0.875584	−	0.483066i	$-0.839523\pi$
$389$		−	7412.44i		−	0.966132i	0.875584	−	0.483066i	$-0.160477\pi$
$390$	−1413.00	−	7313.95i	−0.183462	−	0.949631i
$391$			1572.89i			0.203439i
$392$	1189.22	+	2472.91i	0.153226	+	0.318625i
$393$	−730.602	−	3781.72i	−0.0937760	−	0.485401i
$394$	7973.68			1.01956
$395$	−9464.17			−1.20555
$396$	−2016.00	−	5022.85i	−0.255828	−	0.637393i
$397$			5000.42i			0.632151i	0.948734	+	0.316075i	$0.102365\pi$
$397$			5000.42i			0.632151i	−0.948734	+	0.316075i	$0.897635\pi$
$398$	8814.79			1.11016
$399$	4554.17	−	2006.33i	0.571413	−	0.251735i
$400$	1201.83			0.150228
$401$			7575.97i			0.943456i	0.881744	+	0.471728i	$0.156370\pi$
$401$			7575.97i			0.943456i	−0.881744	+	0.471728i	$0.843630\pi$
$402$	−1349.21	+	260.657i	−0.167394	+	0.0323393i
$403$	5446.63			0.673241
$404$	4933.90			0.607601
$405$	−7450.95	+	7129.67i	−0.914175	+	0.874756i
$406$	−4626.73	+	1054.68i	−0.565568	+	0.128924i
$407$			3708.44i			0.451648i
$408$	4661.48	−	900.564i	0.565631	−	0.109276i
$409$		−	6119.22i		−	0.739794i	−0.929073	−	0.369897i	$-0.879393\pi$
$409$		−	6119.22i		−	0.739794i	0.929073	−	0.369897i	$-0.120607\pi$
$410$			291.866i			0.0351567i
$411$	10389.0	−	2007.08i	1.24684	−	0.240881i
$412$		−	3372.36i		−	0.403263i
$413$	13678.4	−	3118.05i	1.62971	−	0.371499i
$414$	−690.161	+	277.007i	−0.0819314	+	0.0328845i
$415$	8662.41			1.02463
$416$	−1621.47			−0.191103
$417$	−8879.45	+	1715.45i	−1.04275	+	0.201453i
$418$		−	5183.07i		−	0.606489i
$419$	505.902			0.0589855			0.0294927	−	0.999565i	$-0.490611\pi$
$419$	505.902			0.0589855			0.0294927	+	0.999565i	$0.490611\pi$
$420$	−2195.35	−	4983.23i	−0.255053	−	0.578944i
$421$	−684.295			−0.0792173			−0.0396087	−	0.999215i	$-0.512611\pi$
$421$	−684.295			−0.0792173			−0.0396087	+	0.999215i	$0.512611\pi$
$422$		−	7087.76i		−	0.817599i
$423$	5479.44	−	2199.26i	0.629833	−	0.252794i
$424$	−3080.22			−0.352803
$425$	8578.87			0.979145
$426$	−290.729	−	1504.87i	−0.0330654	−	0.171152i
$427$	−1237.59	−	5429.12i	−0.140260	−	0.615301i
$428$		−	4122.28i		−	0.465556i
$429$	2502.85	+	12955.2i	0.281675	+	1.45800i
$430$			1949.70i			0.218658i
$431$		−	7452.00i		−	0.832831i	−0.909174	−	0.416416i	$-0.863286\pi$
$431$		−	7452.00i		−	0.832831i	0.909174	−	0.416416i	$-0.136714\pi$
$432$	1216.10	+	1886.78i	0.135439	+	0.210134i
$433$			16210.5i			1.79914i	0.436774	+	0.899571i	$0.356121\pi$
$433$			16210.5i			1.79914i	−0.436774	+	0.899571i	$0.643879\pi$
$434$	3881.92	−	884.900i	0.429350	−	0.0978722i
$435$	9246.14	−	1786.29i	1.01912	−	0.196887i
$436$	1676.56			0.184158
$437$	−712.177			−0.0779590
$438$	1912.67	+	9900.32i	0.208655	+	1.08004i
$439$		−	10076.8i		−	1.09554i	−0.836630	−	0.547769i	$-0.815477\pi$
$439$		−	10076.8i		−	1.09554i	0.836630	−	0.547769i	$-0.184523\pi$
$440$	−5671.38			−0.614483
$441$	−6250.48	+	6833.56i	−0.674925	+	0.737886i
$442$	−11574.4			−1.24556
$443$		−	7656.46i		−	0.821150i	−0.911827	−	0.410575i	$-0.865328\pi$
$443$		−	7656.46i		−	0.821150i	0.911827	−	0.410575i	$-0.134672\pi$
$444$	−291.748	−	1510.14i	−0.0311841	−	0.161414i
$445$	−7110.48			−0.757459
$446$	9839.48			1.04465
$447$	2055.35	−	397.078i	0.217482	−	0.0420160i
$448$	−1155.65	+	263.436i	−0.121874	+	0.0277816i
$449$		−	7945.09i		−	0.835082i	−0.908658	−	0.417541i	$-0.862892\pi$
$449$		−	7945.09i		−	0.835082i	0.908658	−	0.417541i	$-0.137108\pi$
$450$	1510.85	+	3764.28i	0.158272	+	0.394333i
$451$		−	516.982i		−	0.0539772i
$452$			8708.54i			0.906228i
$453$	2280.14	+	11802.4i	0.236490	+	1.22412i
$454$		−	8182.09i		−	0.845825i
$455$	2950.47	+	12943.3i	0.304001	+	1.33360i
$456$	407.759	+	2110.63i	0.0418751	+	0.216753i
$457$	3699.78			0.378705			0.189353	−	0.981909i	$-0.439361\pi$
$457$	3699.78			0.378705			0.189353	+	0.981909i	$0.439361\pi$
$458$	6504.16			0.663579
$459$	8680.77	+	13468.2i	0.882753	+	1.36959i
$460$			779.273i			0.0789865i
$461$	−6111.58			−0.617451			−0.308725	−	0.951151i	$-0.599902\pi$
$461$	−6111.58			−0.617451			−0.308725	+	0.951151i	$0.599902\pi$
$462$	3888.62	+	8826.77i	0.391591	+	0.888871i
$463$	−1113.82			−0.111800			−0.0559002	−	0.998436i	$-0.517803\pi$
$463$	−1113.82			−0.111800			−0.0559002	+	0.998436i	$0.517803\pi$
$464$		−	2049.83i		−	0.205088i
$465$	−7757.70	+	1498.73i	−0.773666	+	0.149467i
$466$	−2126.17			−0.211359
$467$	4761.73			0.471834			0.235917	−	0.971773i	$-0.424191\pi$
$467$	4761.73			0.471834			0.235917	+	0.971773i	$0.424191\pi$
$468$	−2038.40	−	5078.65i	−0.201335	−	0.501625i
$469$	2387.65	−	544.275i	0.235078	−	0.0535870i
$470$		−	6186.93i		−	0.607195i
$471$	−3054.17	+	590.044i	−0.298787	+	0.0577236i
$472$			6060.08i			0.590970i
$473$		−	3453.50i		−	0.335713i
$474$	−6826.50	+	1318.83i	−0.661502	+	0.127797i
$475$			3884.36i			0.375214i
$476$	−8249.27	+	1880.46i	−0.794338	+	0.181073i
$477$	−3872.23	−	9647.64i	−0.371693	−	0.926069i
$478$	−4027.73			−0.385406
$479$	−12183.2			−1.16214			−0.581068	−	0.813855i	$-0.697365\pi$
$479$	−12183.2			−1.16214			−0.581068	+	0.813855i	$0.697365\pi$
$480$	2309.48	−	446.174i	0.219610	−	0.0424270i
$481$			3749.64i			0.355445i
$482$	−8111.39			−0.766522
$483$	1212.84	−	534.313i	0.114257	−	0.0503356i
$484$	4721.69			0.443435
$485$		−	15900.8i		−	1.48869i
$486$	−4380.85	+	6180.92i	−0.408888	+	0.576898i
$487$	−3934.82			−0.366127			−0.183063	−	0.983101i	$-0.558601\pi$
$487$	−3934.82			−0.366127			−0.183063	+	0.983101i	$0.558601\pi$
$488$	2405.32			0.223122
$489$	−2388.61	−	12363.9i	−0.220893	−	1.14338i
$490$	4205.72	+	8745.56i	0.387745	+	0.806294i
$491$			751.711i			0.0690922i	0.999403	+	0.0345461i	$0.0109986\pi$
$491$			751.711i			0.0690922i	−0.999403	+	0.0345461i	$0.989001\pi$
$492$	40.6716	+	210.523i	0.00372686	+	0.0192909i
$493$		−	14632.1i		−	1.33670i
$494$		−	5240.66i		−	0.477304i
$495$	−7129.67	−	17763.5i	−0.647383	−	1.61295i
$496$			1719.85i			0.155692i
$497$	607.068	+	2663.11i	0.0547902	+	0.240356i
$498$	6248.20	−	1207.11i	0.562226	−	0.108618i
$499$	−9040.78			−0.811064			−0.405532	−	0.914081i	$-0.632914\pi$
$499$	−9040.78			−0.811064			−0.405532	+	0.914081i	$0.632914\pi$
$500$	−2822.78			−0.252477
$501$	880.691	+	4558.61i	0.0785357	+	0.406514i
$502$			4570.59i			0.406366i
$503$	11545.3			1.02342			0.511710	−	0.859158i	$-0.329012\pi$
$503$	11545.3			1.02342			0.511710	+	0.859158i	$0.329012\pi$
$504$	−2277.92	−	3288.48i	−0.201323	−	0.290636i
$505$	17448.9			1.53756
$506$		−	1380.32i		−	0.121270i
$507$	365.214	+	1890.41i	0.0319916	+	0.165594i
$508$	−6034.26			−0.527021
$509$	−18624.0			−1.62180			−0.810900	−	0.585184i	$-0.801022\pi$
$509$	−18624.0			−1.62180			−0.810900	+	0.585184i	$0.801022\pi$
$510$	16485.5	−	3184.88i	1.43135	−	0.276527i
$511$	−3993.83	−	17520.3i	−0.345746	−	1.51674i
$512$		−	512.000i		−	0.0441942i
$513$	−6098.17	+	3930.49i	−0.524836	+	0.338276i
$514$			5384.08i			0.462027i
$515$		−	11926.5i		−	1.02048i
$516$	271.691	+	1406.32i	0.0231793	+	0.119980i
$517$			10958.9i			0.932246i
$518$	609.195	+	2672.44i	0.0516728	+	0.226680i
$519$	938.748	+	4859.12i	0.0793959	+	0.410967i
$520$	−5734.39			−0.483595
$521$	10031.3			0.843532			0.421766	−	0.906705i	$-0.361410\pi$
$521$	10031.3			0.843532			0.421766	+	0.906705i	$0.361410\pi$
$522$	6420.32	−	2576.90i	0.538333	−	0.216069i
$523$			4074.36i			0.340649i	0.985388	+	0.170324i	$0.0544815\pi$
$523$			4074.36i			0.340649i	−0.985388	+	0.170324i	$0.945518\pi$
$524$	−2965.00			−0.247188
$525$	−2914.25	−	6615.05i	−0.242263	−	0.549913i
$526$	1636.23			0.135633
$527$			12276.6i			1.01476i
$528$	−4090.77	+	790.307i	−0.337174	+	0.0651396i
$529$	11977.3			0.984412
$530$	−10893.3			−0.892783
$531$	−18981.0	+	7618.31i	−1.55123	+	0.622611i
$532$	−851.436	−	3735.12i	−0.0693880	−	0.304395i
$533$		−	522.725i		−	0.0424798i
$534$	−5128.79	+	990.846i	−0.415626	+	0.0802961i
$535$		−	14578.6i		−	1.17811i
$536$			1057.83i			0.0852446i
$537$	6578.98	−	1271.01i	0.528685	−	0.102138i
$538$		−	4789.62i		−	0.383820i
$539$	−7449.57	−	15491.0i	−0.595317	−	1.23793i
$540$	4300.79	+	6672.68i	0.342734	+	0.531753i
$541$	8599.11			0.683373			0.341687	−	0.939814i	$-0.389002\pi$
$541$	8599.11			0.683373			0.341687	+	0.939814i	$0.389002\pi$
$542$	−458.929			−0.0363702
$543$	−9142.60	+	1766.29i	−0.722554	+	0.139592i
$544$		−	3654.76i		−	0.288045i
$545$	5929.24			0.466020
$546$	3931.82	+	8924.83i	0.308180	+	0.699538i
$547$	742.054			0.0580035			0.0290018	−	0.999579i	$-0.490767\pi$
$547$	742.054			0.0580035			0.0290018	+	0.999579i	$0.490767\pi$
$548$		−	8145.34i		−	0.634949i
$549$	3023.80	+	7533.77i	0.235069	+	0.585671i
$550$	−7528.55			−0.583670
$551$	6625.13			0.512232
$552$	108.592	+	562.090i	0.00837313	+	0.0433408i
$553$	12080.7	−	2753.84i	0.928972	−	0.211763i
$554$			10303.8i			0.790195i
$555$	−1031.78	−	5340.67i	−0.0789127	−	0.408466i
$556$			6961.79i			0.531018i
$557$			25095.2i			1.90901i	0.298202	+	0.954503i	$0.403613\pi$
$557$			25095.2i			1.90901i	−0.298202	+	0.954503i	$0.596387\pi$
$558$	−5386.78	+	2162.07i	−0.408675	+	0.164028i
$559$		−	3491.87i		−	0.264205i
$560$	−4087.01	+	931.651i	−0.308407	+	0.0703026i
$561$	−29200.7	+	5641.37i	−2.19760	+	0.424561i
$562$	−11237.0			−0.843423
$563$	18466.6			1.38237			0.691186	−	0.722677i	$-0.257088\pi$
$563$	18466.6			1.38237			0.691186	+	0.722677i	$0.257088\pi$
$564$	−862.148	−	4462.63i	−0.0643670	−	0.333175i
$565$			30798.1i			2.29325i
$566$	15454.4			1.14770
$567$	7436.30	−	11268.8i	0.550785	−	0.834647i
$568$	−1179.87			−0.0871586
$569$		−	281.578i		−	0.0207458i	−0.999946	−	0.0103729i	$-0.996698\pi$
$569$		−	281.578i		−	0.0207458i	0.999946	−	0.0103729i	$-0.00330186\pi$
$570$	1442.06	+	7464.34i	0.105967	+	0.548503i
$571$	7479.13			0.548147			0.274073	−	0.961709i	$-0.411629\pi$
$571$	7479.13			0.548147			0.274073	+	0.961709i	$0.411629\pi$
$572$	10157.3			0.742479
$573$	20241.9	−	3910.59i	1.47577	−	0.285109i
$574$	−84.9258	−	372.556i	−0.00617550	−	0.0270909i
$575$			1034.46i			0.0750258i
$576$	1603.65	−	643.651i	0.116005	−	0.0465604i
$577$			6635.32i			0.478738i	0.970929	+	0.239369i	$0.0769406\pi$
$577$			6635.32i			0.478738i	−0.970929	+	0.239369i	$0.923059\pi$
$578$		−	16262.4i		−	1.17029i
$579$	2768.62	+	14330.9i	0.198722	+	1.02862i
$580$		−	7249.29i		−	0.518984i
$581$	−11057.2	+	2520.54i	−0.789555	+	0.179982i
$582$	−2215.77	−	11469.2i	−0.157812	−	0.816864i
$583$	19295.3			1.37072
$584$	7762.20			0.550003
$585$	−7208.88	−	17960.9i	−0.509488	−	1.26938i
$586$		−	9388.42i		−	0.661829i
$587$	2998.97			0.210870			0.105435	−	0.994426i	$-0.466376\pi$
$587$	2998.97			0.210870			0.105435	+	0.994426i	$0.466376\pi$
$588$	4252.28	+	5722.11i	0.298233	+	0.401319i
$589$	−5558.62			−0.388861
$590$			21431.7i			1.49547i
$591$	20340.1	−	3929.57i	1.41570	−	0.273504i
$592$	−1184.00			−0.0821995
$593$	−21253.7			−1.47182			−0.735908	−	0.677082i	$-0.763244\pi$
$593$	−21253.7			−1.47182			−0.735908	+	0.677082i	$0.763244\pi$
$594$	−7617.98	−	11819.3i	−0.526211	−	0.816417i
$595$	−29173.9	+	6650.31i	−2.01011	+	0.458212i
$596$		−	1611.46i		−	0.110752i
$597$	22485.7	−	4344.08i	1.54151	−	0.297808i
$598$		−	1395.66i		−	0.0954393i
$599$			6521.42i			0.444838i	0.974951	+	0.222419i	$0.0713953\pi$
$599$			6521.42i			0.444838i	−0.974951	+	0.222419i	$0.928605\pi$
$600$	3065.75	−	592.280i	0.208598	−	0.0402996i
$601$		−	1701.95i		−	0.115514i	−0.998331	−	0.0577569i	$-0.981605\pi$
$601$		−	1701.95i		−	0.115514i	0.998331	−	0.0577569i	$-0.0183948\pi$
$602$	−567.315	−	2488.72i	−0.0384087	−	0.168493i
$603$	−3313.25	+	1329.83i	−0.223758	+	0.0898088i
$604$	9253.48			0.623375
$605$	16698.5			1.12213
$606$	12585.9	−	2431.51i	0.843677	−	0.162992i
$607$		−	8394.45i		−	0.561318i	−0.959808	−	0.280659i	$-0.909447\pi$
$607$		−	8394.45i		−	0.561318i	0.959808	−	0.280659i	$-0.0905530\pi$
$608$	1654.81			0.110380
$609$	−11282.6	+	4970.53i	−0.750728	+	0.330732i
$610$	8506.51			0.564621
$611$			11080.6i			0.733673i
$612$	11447.2	−	4594.51i	0.756087	−	0.303468i
$613$	2299.24			0.151493			0.0757465	−	0.997127i	$-0.475866\pi$
$613$	2299.24			0.151493			0.0757465	+	0.997127i	$0.475866\pi$
$614$	−7709.84			−0.506749
$615$	143.837	+	744.524i	0.00943099	+	0.0488164i
$616$	7239.30	−	1650.23i	0.473506	−	0.107938i
$617$			15066.4i			0.983066i	0.870859	+	0.491533i	$0.163563\pi$
$617$			15066.4i			0.983066i	−0.870859	+	0.491533i	$0.836437\pi$
$618$	−1661.96	−	8602.59i	−0.108178	−	0.559947i
$619$			25684.8i			1.66779i	0.551925	+	0.833894i	$0.313894\pi$
$619$			25684.8i			1.66779i	−0.551925	+	0.833894i	$0.686106\pi$
$620$			6082.31i			0.393986i
$621$	−1624.02	+	1046.74i	−0.104943	+	0.0676399i
$622$		−	7108.18i		−	0.458219i
$623$	9076.26	−	2068.97i	0.583680	−	0.133052i
$624$	−4136.22	+	799.087i	−0.265354	+	0.0512646i
$625$	−19372.1			−1.23982
$626$	−312.333			−0.0199415
$627$	−2554.31	−	13221.5i	−0.162694	−	0.842133i
$628$			2394.58i			0.152156i
$629$	−8451.63			−0.535753
$630$	−8055.96	−	11629.8i	−0.509456	−	0.735467i
$631$	−2.28850			−0.000144380			−7.21901e−5	−	1.00000i	$-0.500023\pi$
$631$	−2.28850			−0.000144380			−7.21901e−5		1.00000i	$0.500023\pi$
$632$			5352.22i			0.336867i
$633$	−3492.97	−	18080.2i	−0.219326	−	1.13527i
$634$	−4398.89			−0.275555
$635$	−21340.4			−1.33365
$636$	−7857.35	+	1517.98i	−0.489881	+	0.0946414i
$637$	−7532.34	−	15663.1i	−0.468512	−	0.974245i
$638$			12840.6i			0.796812i
$639$	−1483.25	−	3695.50i	−0.0918253	−	0.228782i
$640$		−	1810.71i		−	0.111835i
$641$		−	6916.37i		−	0.426178i	−0.977033	−	0.213089i	$-0.931648\pi$
$641$		−	6916.37i		−	0.426178i	0.977033	−	0.213089i	$-0.0683524\pi$
$642$	−2031.53	−	10515.6i	−0.124888	−	0.646443i
$643$			17869.3i			1.09595i	0.836494	+	0.547976i	$0.184601\pi$
$643$			17869.3i			1.09595i	−0.836494	+	0.547976i	$0.815399\pi$
$644$	−226.749	−	994.712i	−0.0138745	−	0.0608652i
$645$	960.848	+	4973.52i	0.0586563	+	0.303615i
$646$	11812.3			0.719428
$647$	−13311.2			−0.808838			−0.404419	−	0.914574i	$-0.632526\pi$
$647$	−13311.2			−0.808838			−0.404419	+	0.914574i	$0.632526\pi$
$648$	4032.00	+	4213.69i	0.244432	+	0.255447i
$649$		−	37961.9i		−	2.29605i
$650$	−7612.19			−0.459346
$651$	9466.32	−	4170.37i	0.569915	−	0.251075i
$652$	−9693.69			−0.582261
$653$			1074.66i			0.0644025i	0.999481	+	0.0322012i	$0.0102517\pi$
$653$			1074.66i			0.0644025i	−0.999481	+	0.0322012i	$0.989748\pi$
$654$	4276.76	−	826.240i	0.255710	−	0.0494014i
$655$	−10485.8			−0.625521
$656$	165.057			0.00982380
$657$	9758.10	+	24312.2i	0.579452	+	1.44370i
$658$	1800.24	+	7897.38i	0.106658	+	0.467890i
$659$		−	13000.7i		−	0.768493i	−0.923231	−	0.384246i	$-0.874461\pi$
$659$		−	13000.7i		−	0.768493i	0.923231	−	0.384246i	$-0.125539\pi$
$660$	−14467.2	+	2794.95i	−0.853233	+	0.164839i
$661$			5496.48i			0.323432i	0.986837	+	0.161716i	$0.0517028\pi$
$661$			5496.48i			0.323432i	−0.986837	+	0.161716i	$0.948297\pi$
$662$			7191.92i			0.422239i
$663$	−29525.1	+	5704.05i	−1.72950	+	0.334128i
$664$		−	4898.80i		−	0.286311i
$665$	−3011.14	−	13209.4i	−0.175589	−	0.770284i
$666$	−1488.44	−	3708.44i	−0.0866006	−	0.215765i
$667$	1764.36			0.102423
$668$	3574.11			0.207016
$669$	25099.6	−	4849.07i	1.45053	−	0.280233i
$670$			3741.04i			0.215715i
$671$	−15067.5			−0.866880
$672$	−2818.13	+	1241.52i	−0.161774	+	0.0712691i
$673$	58.8868			0.00337283			0.00168642	−	0.999999i	$-0.499463\pi$
$673$	58.8868			0.00337283			0.00168642	+	0.999999i	$0.499463\pi$
$674$		−	16189.1i		−	0.925195i
$675$	5709.15	+	8857.75i	0.325548	+	0.505089i
$676$	1482.15			0.0843280
$677$	9214.00			0.523076			0.261538	−	0.965193i	$-0.415770\pi$
$677$	9214.00			0.523076			0.261538	+	0.965193i	$0.415770\pi$
$678$	4291.72	+	22214.7i	0.243101	+	1.25833i
$679$	4626.73	+	20296.7i	0.261498	+	1.14715i
$680$		−	12925.2i		−	0.728911i
$681$	−4032.27	−	20871.8i	−0.226897	−	1.17446i
$682$		−	10773.6i		−	0.604899i
$683$			23014.1i			1.28933i	0.764466	+	0.644664i	$0.223003\pi$
$683$			23014.1i			1.28933i	−0.764466	+	0.644664i	$0.776997\pi$
$684$	2080.31	+	5183.07i	0.116290	+	0.289737i
$685$		−	28806.4i		−	1.60677i
$686$	−7913.18	−	9939.62i	−0.440418	−	0.553202i
$687$	16591.5	−	3205.36i	0.921405	−	0.178009i
$688$	1102.60			0.0610994
$689$	19509.6			1.07875
$690$	384.039	+	1987.85i	0.0211886	+	0.109676i
$691$			11815.5i			0.650483i	0.945631	+	0.325241i	$0.105446\pi$
$691$			11815.5i			0.650483i	−0.945631	+	0.325241i	$0.894554\pi$
$692$	3809.72			0.209283
$693$	14269.5	+	20599.9i	0.782183	+	1.12918i
$694$	9699.91			0.530553
$695$			24620.7i			1.34376i
$696$	−1010.19	−	5228.92i	−0.0550160	−	0.284772i
$697$	1178.21			0.0640287
$698$	−14205.1			−0.770301
$699$	−5423.68	+	1047.82i	−0.293480	+	0.0566982i
$700$	−5425.36	+	1236.73i	−0.292942	+	0.0667773i
$701$		−	27635.2i		−	1.48897i	−0.667639	−	0.744485i	$-0.732695\pi$
$701$		−	27635.2i		−	1.48897i	0.667639	−	0.744485i	$-0.267305\pi$
$702$	−7702.61	−	11950.6i	−0.414126	−	0.642517i
$703$		−	3826.74i		−	0.205303i
$704$			3207.30i			0.171704i
$705$	−3049.02	−	15782.3i	−0.162884	−	0.843114i
$706$			13583.9i			0.724133i
$707$	−22272.9	+	5077.21i	−1.18481	+	0.270082i
$708$	2986.51	+	15458.7i	0.158531	+	0.820584i
$709$	−31927.4			−1.69119			−0.845597	−	0.533821i	$-0.820756\pi$
$709$	−31927.4			−1.69119			−0.845597	+	0.533821i	$0.820756\pi$
$710$	−4172.65			−0.220559
$711$	−16763.8	+	6728.44i	−0.884238	+	0.354903i
$712$			4021.15i			0.211656i
$713$	−1480.34			−0.0777546
$714$	−20116.4	+	8862.25i	−1.05439	+	0.464512i
$715$	35921.7			1.87888
$716$		−	5158.15i		−	0.269231i
$717$	−10274.4	+	1984.94i	−0.535151	+	0.103387i
$718$	−25187.5			−1.30918
$719$	37176.2			1.92828			0.964142	−	0.265385i	$-0.0854992\pi$
$719$	37176.2			1.92828			0.964142	+	0.265385i	$0.0854992\pi$
$720$	5671.38	−	2276.30i	0.293555	−	0.117823i
$721$	3470.32	+	15223.7i	0.179253	+	0.786355i
$722$		−	8369.59i		−	0.431418i
$723$	−20691.4	+	3997.43i	−1.06435	+	0.205624i
$724$			7168.12i			0.367957i
$725$		−	9623.18i		−	0.492960i
$726$	12044.6	−	2326.93i	0.615726	−	0.118954i
$727$		−	16425.6i		−	0.837951i	−0.907997	−	0.418976i	$-0.862389\pi$
$727$		−	16425.6i		−	0.837951i	0.907997	−	0.418976i	$-0.137611\pi$
$728$	7319.73	−	1668.56i	0.372647	−	0.0849466i
$729$	−8129.08	+	17925.9i	−0.413000	+	0.910731i
$730$	27451.3			1.39181
$731$	7870.61			0.398229
$732$	6135.75	−	1185.38i	0.309814	−	0.0598538i
$733$		−	23066.1i		−	1.16230i	−0.813796	−	0.581150i	$-0.802603\pi$
$733$		−	23066.1i		−	1.16230i	0.813796	−	0.581150i	$-0.197397\pi$
$734$	11935.7			0.600211
$735$	15038.4	+	20236.5i	0.754692	+	1.01556i
$736$	440.698			0.0220711
$737$		−	6626.50i		−	0.331194i
$738$	207.499	+	516.982i	0.0103498	+	0.0257864i
$739$	−6577.18			−0.327396			−0.163698	−	0.986511i	$-0.552342\pi$
$739$	−6577.18			−0.327396			−0.163698	+	0.986511i	$0.552342\pi$
$740$	−4187.27			−0.208009
$741$	−2582.69	−	13368.4i	−0.128040	−	0.662755i
$742$	13904.9	−	3169.68i	0.687958	−	0.156823i
$743$		−	25611.8i		−	1.26461i	−0.774720	−	0.632305i	$-0.782109\pi$
$743$		−	25611.8i		−	1.26461i	0.774720	−	0.632305i	$-0.217891\pi$
$744$	847.570	+	4387.17i	0.0417653	+	0.216185i
$745$		−	5699.00i		−	0.280262i
$746$			26588.6i			1.30493i
$747$	15343.7	−	6158.44i	0.751534	−	0.301640i
$748$			22894.4i			1.11912i
$749$	4242.02	+	18609.1i	0.206942	+	0.907824i
$750$	−7200.65	+	1391.11i	−0.350574	+	0.0677284i
$751$	3648.38			0.177272			0.0886360	−	0.996064i	$-0.471749\pi$
$751$	3648.38			0.177272			0.0886360	+	0.996064i	$0.471749\pi$
$752$	−3498.86			−0.169668
$753$	2252.47	+	11659.2i	0.109010	+	0.564254i
$754$			12983.3i			0.627088i
$755$	32725.3			1.57748
$756$	−7431.38	−	7266.01i	−0.357509	−	0.349553i
$757$	−3215.18			−0.154370			−0.0771848	−	0.997017i	$-0.524593\pi$
$757$	−3215.18			−0.154370			−0.0771848	+	0.997017i	$0.524593\pi$
$758$			24976.6i			1.19682i
$759$	−680.247	−	3521.08i	−0.0325315	−	0.168389i
$760$	5852.30			0.279323
$761$	36437.5			1.73569			0.867843	−	0.496839i	$-0.165506\pi$
$761$	36437.5			1.73569			0.867843	+	0.496839i	$0.165506\pi$
$762$	−15392.8	+	2973.78i	−0.731789	+	0.141376i
$763$	−7568.45	+	1725.26i	−0.359104	+	0.0818593i
$764$		−	15870.4i		−	0.751531i
$765$	40483.5	−	16248.7i	1.91331	−	0.767938i
$766$			1541.11i			0.0726926i
$767$		−	38383.7i		−	1.80698i
$768$	−252.322	−	1306.06i	−0.0118553	−	0.0613653i
$769$			5867.99i			0.275169i	0.990490	+	0.137585i	$0.0439339\pi$
$769$			5867.99i			0.275169i	−0.990490	+	0.137585i	$0.956066\pi$
$770$	25602.1	−	5836.11i	1.19823	−	0.273141i
$771$	2653.37	+	13734.3i	0.123941	+	0.641542i
$772$	11235.9			0.523819
$773$	−23839.4			−1.10924			−0.554621	−	0.832103i	$-0.687137\pi$
$773$	−23839.4			−1.10924			−0.554621	+	0.832103i	$0.687137\pi$
$774$	1386.12	+	3453.50i	0.0643708	+	0.160379i
$775$			8074.04i			0.374230i
$776$	−8992.27			−0.415984
$777$	2871.03	+	6516.94i	0.132558	+	0.300893i
$778$	−14824.9			−0.683159
$779$			533.473i			0.0245362i
$780$	−14627.9	+	2826.01i	−0.671491	+	0.129727i
$781$	7390.99			0.338631
$782$	3145.79			0.143853
$783$	15107.7	−	9737.48i	0.689535	−	0.444431i
$784$	4945.83	−	2378.44i	0.225302	−	0.108347i
$785$			8468.52i			0.385037i
$786$	−7563.44	+	1461.20i	−0.343230	+	0.0663097i
$787$		−	12631.0i		−	0.572106i	−0.958214	−	0.286053i	$-0.907657\pi$
$787$		−	12631.0i		−	0.572106i	0.958214	−	0.286053i	$-0.0923434\pi$
$788$		−	15947.4i		−	0.720940i
$789$	4173.87	−	806.361i	0.188332	−	0.0363843i
$790$			18928.3i			0.852456i
$791$	−8961.48	−	39312.6i	−0.402824	−	1.76713i
$792$	−10045.7	+	4032.00i	−0.450705	+	0.180898i
$793$	−15234.9			−0.682231
$794$	10000.8			0.446998
$795$	−27787.8	+	5368.41i	−1.23966	+	0.239494i
$796$		−	17629.6i		−	0.785005i
$797$	−8778.22			−0.390139			−0.195069	−	0.980789i	$-0.562493\pi$
$797$	−8778.22			−0.390139			−0.195069	+	0.980789i	$0.562493\pi$
$798$	−4012.67	−	9108.34i	−0.178004	−	0.404050i
$799$	−24975.5			−1.10585
$800$		−	2403.65i		−	0.106227i
$801$	−12594.8	+	5055.11i	−0.555573	+	0.222988i
$802$	15151.9			0.667124
$803$	−48624.4			−2.13689
$804$	521.315	+	2698.42i	0.0228673	+	0.118365i
$805$	−801.907	−	3517.84i	−0.0351100	−	0.154022i
$806$		−	10893.3i		−	0.476053i
$807$	−2360.41	−	12217.9i	−0.102962	−	0.532949i
$808$		−	9867.80i		−	0.429639i
$809$			21329.9i			0.926969i	0.886105	+	0.463484i	$0.153401\pi$
$809$			21329.9i			0.926969i	−0.886105	+	0.463484i	$0.846599\pi$
$810$	14259.3	+	14901.9i	0.618546	+	0.646419i
$811$			6763.10i			0.292829i	0.989223	+	0.146415i	$0.0467734\pi$
$811$			6763.10i			0.292829i	−0.989223	+	0.146415i	$0.953227\pi$
$812$	2109.36	+	9253.45i	0.0911627	+	0.399917i
$813$	−1170.68	+	226.168i	−0.0505015	+	0.00975652i
$814$	7416.89			0.319363
$815$	−34282.2			−1.47344
$816$	−1801.13	−	9322.95i	−0.0772697	−	0.399962i
$817$			3563.67i			0.152603i
$818$	−12238.4			−0.523113
$819$	14428.0	+	20828.8i	0.615575	+	0.888664i
$820$	583.733			0.0248596
$821$			36836.2i			1.56589i	0.622093	+	0.782944i	$0.286283\pi$
$821$			36836.2i			1.56589i	−0.622093	+	0.782944i	$0.713717\pi$
$822$	−4014.16	−	20778.0i	−0.170329	−	0.881650i
$823$	31892.9			1.35081			0.675404	−	0.737448i	$-0.263969\pi$
$823$	31892.9			1.35081			0.675404	+	0.737448i	$0.263969\pi$
$824$	−6744.73			−0.285150
$825$	−19204.6	+	3710.20i	−0.810448	+	0.156573i
$826$	−6236.09	−	27356.8i	−0.262689	−	1.15238i
$827$			26589.3i			1.11802i	0.829162	+	0.559009i	$0.188818\pi$
$827$			26589.3i			1.11802i	−0.829162	+	0.559009i	$0.811182\pi$
$828$	554.015	+	1380.32i	0.0232528	+	0.0579342i
$829$			21815.0i			0.913953i	0.889479	+	0.456976i	$0.151068\pi$
$829$			21815.0i			0.913953i	−0.889479	+	0.456976i	$0.848932\pi$
$830$		−	17324.8i		−	0.724522i
$831$	5077.91	+	26284.1i	0.211974	+	1.09722i
$832$			3242.94i			0.135130i
$833$	35304.3	−	16977.7i	1.46845	−	0.706175i
$834$	3430.89	+	17758.9i	0.142449	+	0.737338i
$835$	12640.0			0.523862
$836$	−10366.1			−0.428853
$837$	−12675.7	+	8169.95i	−0.523460	+	0.337389i
$838$		−	1011.80i		−	0.0417090i
$839$	−6259.85			−0.257585			−0.128793	−	0.991672i	$-0.541110\pi$
$839$	−6259.85			−0.257585			−0.128793	+	0.991672i	$0.541110\pi$
$840$	−9966.45	+	4390.70i	−0.409375	+	0.180350i
$841$	7975.78			0.327024
$842$			1368.59i			0.0560151i
$843$	−28664.5	+	5537.78i	−1.17113	+	0.226253i
$844$	−14175.5			−0.578130
$845$	5241.68			0.213396
$846$	−4398.52	−	10958.9i	−0.178752	−	0.445359i
$847$	−21315.0	+	4858.84i	−0.864688	+	0.197109i
$848$			6160.43i			0.249469i
$849$	39422.8	−	7616.21i	1.59362	−	0.307877i
$850$		−	17157.7i		−	0.692360i
$851$		−	1019.11i		−	0.0410514i
$852$	−3009.73	+	581.458i	−0.121023	+	0.0233808i
$853$		−	29076.5i		−	1.16713i	−0.812066	−	0.583565i	$-0.801657\pi$
$853$		−	29076.5i		−	1.16713i	0.812066	−	0.583565i	$-0.198343\pi$
$854$	−10858.2	+	2475.18i	−0.435083	+	0.0991791i
$855$	7357.11	+	18330.2i	0.294278	+	0.733191i
$856$	−8244.56			−0.329198
$857$	−27423.2			−1.09307			−0.546534	−	0.837437i	$-0.684053\pi$
$857$	−27423.2			−1.09307			−0.546534	+	0.837437i	$0.684053\pi$
$858$	25910.3	−	5005.69i	1.03096	−	0.199174i
$859$			18019.1i			0.715719i	0.933775	+	0.357860i	$0.116493\pi$
$859$			18019.1i			0.715719i	−0.933775	+	0.357860i	$0.883507\pi$
$860$	3899.41			0.154615
$861$	−400.240	−	908.504i	−0.0158422	−	0.0359602i
$862$	−14904.0			−0.588901
$863$		−	10846.6i		−	0.427836i	−0.976852	−	0.213918i	$-0.931377\pi$
$863$		−	10846.6i		−	0.427836i	0.976852	−	0.213918i	$-0.0686226\pi$
$864$	3773.56	−	2432.20i	0.148587	−	0.0957699i
$865$	13473.2			0.529600
$866$	32421.1			1.27219
$867$	−8014.39	−	41483.9i	−0.313937	−	1.62499i
$868$	−1769.80	−	7763.84i	−0.0692061	−	0.303597i
$869$		−	33527.7i		−	1.30880i
$870$	−3572.58	−	18492.3i	−0.139220	−	0.720629i
$871$		−	6700.12i		−	0.260648i
$872$		−	3353.13i		−	0.130219i
$873$	−11304.5	−	28165.0i	−0.438257	−	1.09191i
$874$			1424.35i			0.0551253i
$875$	12742.8	−	2904.77i	0.492324	−	0.112227i
$876$	19800.6	−	3825.34i	0.763701	−	0.147542i
$877$	−45098.4			−1.73645			−0.868224	−	0.496173i	$-0.834738\pi$
$877$	−45098.4			−1.73645			−0.868224	+	0.496173i	$0.834738\pi$
$878$	−20153.7			−0.774663
$879$	−4626.77	−	23949.0i	−0.177539	−	0.918976i
$880$			11342.8i			0.434505i
$881$	−40251.5			−1.53928			−0.769640	−	0.638478i	$-0.779564\pi$
$881$	−40251.5			−1.53928			−0.769640	+	0.638478i	$0.779564\pi$
$882$	13667.1	+	12501.0i	0.521764	+	0.477244i
$883$	−30997.8			−1.18138			−0.590690	−	0.806899i	$-0.701144\pi$
$883$	−30997.8			−1.18138			−0.590690	+	0.806899i	$0.701144\pi$
$884$			23148.7i			0.880742i
$885$	10561.9	+	54670.3i	0.401170	+	2.07652i
$886$	−15312.9			−0.580641
$887$	48086.8			1.82029			0.910146	−	0.414289i	$-0.135970\pi$
$887$	48086.8			1.82029			0.910146	+	0.414289i	$0.135970\pi$
$888$	−3020.27	+	583.496i	−0.114137	+	0.0220505i
$889$	27240.2	−	6209.52i	1.02768	−	0.234264i
$890$			14221.0i			0.535604i
$891$	−25257.5	−	26395.7i	−0.949673	−	0.992467i
$892$		−	19679.0i		−	0.738678i
$893$		−	11308.5i		−	0.423766i
$894$	−794.156	−	4110.69i	−0.0297098	−	0.153783i
$895$		−	18242.0i		−	0.681300i
$896$	526.871	+	2311.30i	0.0196446	+	0.0861777i
$897$	−687.804	−	3560.20i	−0.0256021	−	0.132521i
$898$	−15890.2			−0.590492
$899$	13771.0			0.510889
$900$	7528.55	−	3021.70i	0.278835	−	0.111915i
$901$			43974.4i			1.62597i
$902$	−1033.96			−0.0381676
$903$	−2673.65	−	6068.92i	−0.0985312	−	0.223656i
$904$	17417.1			0.640800
$905$			25350.3i			0.931131i
$906$	23604.8	−	4560.27i	0.865580	−	0.167224i
$907$	44084.4			1.61389			0.806944	−	0.590628i	$-0.201120\pi$
$907$	44084.4			1.61389			0.806944	+	0.590628i	$0.201120\pi$
$908$	−16364.2			−0.598088
$909$	30907.2	−	12405.1i	1.12775	−	0.452642i
$910$	25886.5	−	5900.95i	0.943000	−	0.214961i
$911$			24699.6i			0.898282i	0.893461	+	0.449141i	$0.148270\pi$
$911$			24699.6i			0.898282i	−0.893461	+	0.449141i	$0.851730\pi$
$912$	4221.26	−	815.518i	0.153267	−	0.0296102i
$913$			30687.4i			1.11238i
$914$		−	7399.56i		−	0.267785i
$915$	21699.3	−	4192.15i	0.783997	−	0.151463i
$916$		−	13008.3i		−	0.469221i
$917$	13384.8	−	3051.12i	0.482012	−	0.109877i
$918$	26936.5	−	17361.5i	0.968448	−	0.624201i
$919$	12673.4			0.454906			0.227453	−	0.973789i	$-0.426960\pi$
$919$	12673.4			0.454906			0.227453	+	0.973789i	$0.426960\pi$
$920$	1558.55			0.0558519
$921$	−19667.1	+	3799.54i	−0.703640	+	0.135938i
$922$			12223.2i			0.436604i
$923$	7473.11			0.266501
$924$	17653.5	−	7777.24i	0.628527	−	0.276896i
$925$	−5558.44			−0.197579
$926$			2227.64i			0.0790548i
$927$	−8479.01	−	21125.4i	−0.300418	−	0.748488i
$928$	−4099.65			−0.145019
$929$	9761.49			0.344741			0.172370	−	0.985032i	$-0.444857\pi$
$929$	9761.49			0.344741			0.172370	+	0.985032i	$0.444857\pi$
$930$	2997.47	+	15515.4i	0.105689	+	0.547065i
$931$	7687.21	+	15985.1i	0.270610	+	0.562719i
$932$			4252.35i			0.149453i
$933$	−3503.04	−	18132.3i	−0.122920	−	0.636255i
$934$		−	9523.45i		−	0.333637i
$935$			80966.9i			2.83198i
$936$	−10157.3	+	4076.80i	−0.354703	+	0.142366i
$937$			42949.5i			1.49744i	0.662888	+	0.748718i	$0.269330\pi$
$937$			42949.5i			1.49744i	−0.662888	+	0.748718i	$0.730670\pi$
$938$	−1088.55	−	4775.30i	−0.0378917	−	0.166225i
$939$	−796.734	+	153.923i	−0.0276895	+	0.00534941i
$940$	−12373.9			−0.429352
$941$	−24907.1			−0.862856			−0.431428	−	0.902147i	$-0.641990\pi$
$941$	−24907.1			−0.862856			−0.431428	+	0.902147i	$0.641990\pi$
$942$	1180.09	+	6108.34i	0.0408167	+	0.211274i
$943$			142.071i			0.00490612i
$944$	12120.2			0.417879
$945$	−26281.4	−	25696.5i	−0.904692	−	0.884559i
$946$	−6907.01			−0.237385
$947$			16041.9i			0.550467i	0.961377	+	0.275234i	$0.0887552\pi$
$947$			16041.9i			0.550467i	−0.961377	+	0.275234i	$0.911245\pi$
$948$	2637.66	+	13653.0i	0.0903664	+	0.467752i
$949$	−49164.6			−1.68172
$950$	7768.71			0.265316
$951$	−11221.2	+	2167.85i	−0.382619	+	0.0739193i
$952$	3760.91	+	16498.5i	0.128038	+	0.561682i
$953$			20275.9i			0.689191i	0.938751	+	0.344596i	$0.111984\pi$
$953$			20275.9i			0.689191i	−0.938751	+	0.344596i	$0.888016\pi$
$954$	−19295.3	+	7744.47i	−0.654830	+	0.262826i
$955$		−	56126.2i		−	1.90178i
$956$			8055.47i			0.272523i
$957$	6328.09	+	32755.3i	0.213749	+	1.10640i
$958$			24366.3i			0.821754i
$959$	8381.93	+	36770.2i	0.282238	+	1.23814i
$960$	−892.349	−	4618.95i	−0.0300005	−	0.155288i
$961$	18236.8			0.612159
$962$	7499.29			0.251338
$963$	−10364.5	−	25823.1i	−0.346824	−	0.864108i
$964$			16222.8i			0.542013i
$965$	39736.2			1.32555
$966$	−1068.63	−	2425.67i	−0.0355927	−	0.0807917i
$967$	25645.7			0.852855			0.426428	−	0.904522i	$-0.359772\pi$
$967$	25645.7			0.852855			0.426428	+	0.904522i	$0.359772\pi$
$968$		−	9443.38i		−	0.313556i
$969$	30132.2	−	5821.33i	0.998954	−	0.192991i
$970$	−31801.5			−1.05267
$971$	−5194.16			−0.171667			−0.0858334	−	0.996310i	$-0.527355\pi$
$971$	−5194.16			−0.171667			−0.0858334	+	0.996310i	$0.527355\pi$
$972$	12361.8	+	8761.70i	0.407928	+	0.289127i
$973$	−7164.00	−	31427.4i	−0.236040	−	1.03547i
$974$			7869.64i			0.258891i
$975$	−19418.0	+	3751.42i	−0.637819	+	0.123222i
$976$		−	4810.64i		−	0.157771i
$977$		−	51245.0i		−	1.67807i	−0.544078	−	0.839035i	$-0.683120\pi$
$977$		−	51245.0i		−	1.67807i	0.544078	−	0.839035i	$-0.316880\pi$
$978$	−24727.7	+	4777.22i	−0.808492	+	0.156195i
$979$		−	25189.5i		−	0.822330i
$980$	17491.1	−	8411.44i	0.570136	−	0.274177i
$981$	10502.4	−	4215.32i	0.341811	−	0.137192i
$982$	1503.42			0.0488556
$983$	9485.53			0.307774			0.153887	−	0.988088i	$-0.450821\pi$
$983$	9485.53			0.307774			0.153887	+	0.988088i	$0.450821\pi$
$984$	421.046	−	81.3432i	0.0136407	−	0.00263529i
$985$		−	56398.5i		−	1.82437i
$986$	−29264.1			−0.945193
$987$	8484.21	+	19258.3i	0.273612	+	0.621072i
$988$	−10481.3			−0.337505
$989$			949.053i			0.0305138i
$990$	−35527.0	+	14259.3i	−1.14053	+	0.457769i
$991$	29495.6			0.945467			0.472734	−	0.881205i	$-0.343267\pi$
$991$	29495.6			0.945467			0.472734	+	0.881205i	$0.343267\pi$
$992$	3439.69			0.110091
$993$	3544.30	+	18345.9i	0.113268	+	0.586295i
$994$	5326.23	−	1214.14i	0.169957	−	0.0387425i
$995$		−	62347.8i		−	1.98649i
$996$	−2414.21	−	12496.4i	−0.0768045	−	0.397554i
$997$			55510.2i			1.76331i	0.471890	+	0.881657i	$0.343572\pi$
$997$			55510.2i			1.76331i	−0.471890	+	0.881657i	$0.656428\pi$
$998$			18081.6i			0.573509i
$999$	−5624.47	−	8726.37i	−0.178128	−	0.276366i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	42.4.d.a.41.2	✓	8
3.2	odd	2	inner	42.4.d.a.41.7	yes	8
4.3	odd	2		336.4.k.c.209.6		8
7.2	even	3		294.4.f.b.227.4		16
7.3	odd	6		294.4.f.b.215.7		16
7.4	even	3		294.4.f.b.215.6		16
7.5	odd	6		294.4.f.b.227.1		16
7.6	odd	2	inner	42.4.d.a.41.3	yes	8
12.11	even	2		336.4.k.c.209.4		8
21.2	odd	6		294.4.f.b.227.7		16
21.5	even	6		294.4.f.b.227.6		16
21.11	odd	6		294.4.f.b.215.1		16
21.17	even	6		294.4.f.b.215.4		16
21.20	even	2	inner	42.4.d.a.41.6	yes	8
28.27	even	2		336.4.k.c.209.3		8
84.83	odd	2		336.4.k.c.209.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
42.4.d.a.41.2	✓	8	1.1	even	1	trivial
42.4.d.a.41.3	yes	8	7.6	odd	2	inner
42.4.d.a.41.6	yes	8	21.20	even	2	inner
42.4.d.a.41.7	yes	8	3.2	odd	2	inner
294.4.f.b.215.1		16	21.11	odd	6
294.4.f.b.215.4		16	21.17	even	6
294.4.f.b.215.6		16	7.4	even	3
294.4.f.b.215.7		16	7.3	odd	6
294.4.f.b.227.1		16	7.5	odd	6
294.4.f.b.227.4		16	7.2	even	3
294.4.f.b.227.6		16	21.5	even	6
294.4.f.b.227.7		16	21.2	odd	6
336.4.k.c.209.3		8	28.27	even	2
336.4.k.c.209.4		8	12.11	even	2
336.4.k.c.209.5		8	84.83	odd	2
336.4.k.c.209.6		8	4.3	odd	2

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:	\( N \)	\(=\)	\( 42 = 2 \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	42.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.00000i q^{2} +(-0.985634 - 5.10182i) q^{3} -4.00000 q^{4} -14.1462 q^{5} +(-10.2036 + 1.97127i) q^{6} +(18.0570 - 4.11618i) q^{7} +8.00000i q^{8} +(-25.0570 + 10.0570i) q^{9} +28.2923i q^{10} -50.1141i q^{11} +(3.94254 + 20.4073i) q^{12} -50.6709i q^{13} +(-8.23236 - 36.1141i) q^{14} +(13.9430 + 72.1711i) q^{15} +16.0000 q^{16} +114.211 q^{17} +(20.1141 + 50.1141i) q^{18} +51.7127i q^{19} +56.5847 q^{20} +(-38.7976 - 88.0667i) q^{21} -100.228 q^{22} +13.7718i q^{23} +(40.8145 - 7.88507i) q^{24} +75.1141 q^{25} -101.342 q^{26} +(76.0063 + 117.924i) q^{27} +(-72.2282 + 16.4647i) q^{28} -128.114i q^{29} +(144.342 - 27.8859i) q^{30} +107.490i q^{31} -32.0000i q^{32} +(-255.673 + 49.3942i) q^{33} -228.422i q^{34} +(-255.438 + 58.2282i) q^{35} +(100.228 - 40.2282i) q^{36} -74.0000 q^{37} +103.425 q^{38} +(-258.513 + 49.9430i) q^{39} -113.169i q^{40} +10.3161 q^{41} +(-176.133 + 77.5953i) q^{42} +68.9128 q^{43} +200.456i q^{44} +(354.461 - 142.269i) q^{45} +27.5436 q^{46} -218.679 q^{47} +(-15.7701 - 81.6291i) q^{48} +(309.114 - 148.652i) q^{49} -150.228i q^{50} +(-112.570 - 582.685i) q^{51} +202.683i q^{52} +385.027i q^{53} +(235.848 - 152.013i) q^{54} +708.923i q^{55} +(32.9295 + 144.456i) q^{56} +(263.829 - 50.9698i) q^{57} -256.228 q^{58} +757.510 q^{59} +(-55.7718 - 288.685i) q^{60} -300.665i q^{61} +214.981 q^{62} +(-411.060 + 284.740i) q^{63} -64.0000 q^{64} +716.799i q^{65} +(98.7884 + 511.346i) q^{66} +132.228 q^{67} -456.845 q^{68} +(70.2612 - 13.5740i) q^{69} +(116.456 + 510.876i) q^{70} +147.483i q^{71} +(-80.4564 - 200.456i) q^{72} -970.274i q^{73} +148.000i q^{74} +(-74.0350 - 383.218i) q^{75} -206.851i q^{76} +(-206.279 - 904.913i) q^{77} +(99.8859 + 517.027i) q^{78} +669.027 q^{79} -226.339 q^{80} +(526.711 - 504.000i) q^{81} -20.6322i q^{82} -612.350 q^{83} +(155.191 + 352.267i) q^{84} -1615.65 q^{85} -137.826i q^{86} +(-653.615 + 126.274i) q^{87} +400.913 q^{88} +502.644 q^{89} +(-284.537 - 708.923i) q^{90} +(-208.570 - 914.966i) q^{91} -55.0872i q^{92} +(548.396 - 105.946i) q^{93} +437.357i q^{94} -731.537i q^{95} +(-163.258 + 31.5403i) q^{96} +1124.03i q^{97} +(-297.304 - 618.228i) q^{98} +(504.000 + 1255.71i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 32 q^{4} + 4 q^{7} - 60 q^{9} + 252 q^{15} + 128 q^{16} - 120 q^{18} - 168 q^{21} - 240 q^{22} + 320 q^{25} - 16 q^{28} + 312 q^{30} + 240 q^{36} - 592 q^{37} - 804 q^{39} - 216 q^{42} - 1696 q^{43}+ \cdots + 4032 q^{99}+O(q^{100}) \) 8 * q - 32 * q^4 + 4 * q^7 - 60 * q^9 + 252 * q^15 + 128 * q^16 - 120 * q^18 - 168 * q^21 - 240 * q^22 + 320 * q^25 - 16 * q^28 + 312 * q^30 + 240 * q^36 - 592 * q^37 - 804 * q^39 - 216 * q^42 - 1696 * q^43 + 1344 * q^46 + 2192 * q^49 + 504 * q^51 + 2532 * q^57 - 1488 * q^58 - 1008 * q^60 - 2496 * q^63 - 512 * q^64 + 496 * q^67 - 192 * q^70 + 480 * q^72 + 1080 * q^78 + 2824 * q^79 + 672 * q^84 - 3936 * q^85 + 960 * q^88 - 264 * q^91 - 1512 * q^93 + 4032 * q^99

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