LMFDB - Embedded newform 51.5.c.c.50.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [51,5,Mod(50,51)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(51, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1]))

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("51.50");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 51 = 3 \cdot 17 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	51.c (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$5.27186811728$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} + 200 x^{18} + 22051 x^{16} + 1226808 x^{14} + 5013252 x^{12} - 3569195664 x^{10} + \cdots + 12\!\cdots\!01 $ x^20 + 200x^18 + 22051x^16 + 1226808x^14 + 5013252x^12 - 3569195664x^10 + 32891946372x^8 + 52810061696568x^6 + 6227853708942531x^4 + 370604037770368200*x^2 + 12157665459056928801
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{10} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			50.10
Root			$-5.02953 + 7.46350i$ of defining polynomial
Character	$\chi$	$=$	51.50
Dual form			51.5.c.c.50.12

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.93178i q^{2} +(5.02953 + 7.46350i) q^{3} +12.2682 q^{4} -4.62649 q^{5} +(14.4178 - 9.71595i) q^{6} +58.0263i q^{7} -54.6080i q^{8} +(-30.4076 + 75.0758i) q^{9} +8.93735i q^{10} +163.508 q^{11} +(61.7034 + 91.5639i) q^{12} -81.9293 q^{13} +112.094 q^{14} +(-23.2691 - 34.5298i) q^{15} +90.8010 q^{16} +(44.7750 - 285.510i) q^{17} +(145.030 + 58.7408i) q^{18} +524.957 q^{19} -56.7588 q^{20} +(-433.079 + 291.845i) q^{21} -315.862i q^{22} -767.157 q^{23} +(407.567 - 274.653i) q^{24} -603.596 q^{25} +158.269i q^{26} +(-713.264 + 150.649i) q^{27} +711.880i q^{28} -462.049 q^{29} +(-66.7039 + 44.9507i) q^{30} -637.845i q^{31} -1049.14i q^{32} +(822.369 + 1220.34i) q^{33} +(-551.543 - 86.4954i) q^{34} -268.458i q^{35} +(-373.047 + 921.047i) q^{36} -1178.23i q^{37} -1014.10i q^{38} +(-412.066 - 611.479i) q^{39} +252.643i q^{40} -1299.93 q^{41} +(563.781 + 836.614i) q^{42} +1446.33 q^{43} +2005.95 q^{44} +(140.680 - 347.337i) q^{45} +1481.98i q^{46} -3326.42i q^{47} +(456.687 + 677.693i) q^{48} -966.054 q^{49} +1166.01i q^{50} +(2356.10 - 1101.81i) q^{51} -1005.13 q^{52} -257.237i q^{53} +(291.021 + 1377.87i) q^{54} -756.468 q^{55} +3168.70 q^{56} +(2640.29 + 3918.02i) q^{57} +892.577i q^{58} +4907.54i q^{59} +(-285.470 - 423.619i) q^{60} +3608.73i q^{61} -1232.18 q^{62} +(-4356.37 - 1764.44i) q^{63} -573.882 q^{64} +379.045 q^{65} +(2357.43 - 1588.64i) q^{66} +1636.77 q^{67} +(549.309 - 3502.71i) q^{68} +(-3858.44 - 5725.67i) q^{69} -518.602 q^{70} +1219.27 q^{71} +(4099.74 + 1660.50i) q^{72} -7484.07i q^{73} -2276.08 q^{74} +(-3035.80 - 4504.93i) q^{75} +6440.29 q^{76} +9487.77i q^{77} +(-1181.24 + 796.021i) q^{78} +2592.01i q^{79} -420.090 q^{80} +(-4711.75 - 4565.75i) q^{81} +2511.18i q^{82} +4314.37i q^{83} +(-5313.12 + 3580.42i) q^{84} +(-207.151 + 1320.91i) q^{85} -2793.99i q^{86} +(-2323.89 - 3448.50i) q^{87} -8928.85i q^{88} +1095.43i q^{89} +(-670.979 - 271.763i) q^{90} -4754.05i q^{91} -9411.65 q^{92} +(4760.55 - 3208.06i) q^{93} -6425.91 q^{94} -2428.71 q^{95} +(7830.22 - 5276.66i) q^{96} -4255.19i q^{97} +1866.20i q^{98} +(-4971.89 + 12275.5i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 168 q^{4} - 400 q^{9} - 308 q^{13} - 72 q^{15} + 600 q^{16} - 124 q^{18} - 548 q^{19} + 16 q^{21} + 3200 q^{25} - 2580 q^{30} + 4088 q^{33} - 2072 q^{34} + 4588 q^{36} - 464 q^{42} + 9028 q^{43}+ \cdots - 13736 q^{94}+O(q^{100}) $ 20 * q - 168 * q^4 - 400 * q^9 - 308 * q^13 - 72 * q^15 + 600 * q^16 - 124 * q^18 - 548 * q^19 + 16 * q^21 + 3200 * q^25 - 2580 * q^30 + 4088 * q^33 - 2072 * q^34 + 4588 * q^36 - 464 * q^42 + 9028 * q^43 - 5348 * q^49 - 8608 * q^51 - 24792 * q^52 + 7492 * q^55 + 7860 * q^60 - 23776 * q^64 - 14980 * q^66 + 11408 * q^67 + 14044 * q^69 - 12264 * q^70 - 26868 * q^72 + 68976 * q^76 - 8204 * q^81 + 53720 * q^84 + 7124 * q^85 + 45876 * q^87 - 68216 * q^93 - 13736 * q^94

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/51\mathbb{Z}\right)^\times$.

$n$	$35$	$37$
$\chi(n)$	$-1$	$-1$

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 $

$2$		−	1.93178i		−	0.482945i	−0.970408	−	0.241472i	$-0.922370\pi$
$2$		−	1.93178i		−	0.482945i	0.970408	−	0.241472i	$-0.0776304\pi$
$3$	5.02953	+	7.46350i	0.558837	+	0.829278i
$3$	5.02953	+	7.46350i	0.558837	+	0.829278i
$4$	12.2682			0.766764
$5$	−4.62649			−0.185059			−0.0925297	−	0.995710i	$-0.529495\pi$
$5$	−4.62649			−0.185059			−0.0925297	+	0.995710i	$0.529495\pi$
$6$	14.4178	−	9.71595i	0.400495	−	0.269887i
$7$			58.0263i			1.18421i	0.805861	+	0.592105i	$0.201703\pi$
$7$			58.0263i			1.18421i	−0.805861	+	0.592105i	$0.798297\pi$
$8$		−	54.6080i		−	0.853250i
$9$	−30.4076	+	75.0758i	−0.375403	+	0.926862i
$10$			8.93735i			0.0893735i
$11$	163.508			1.35131			0.675653	−	0.737220i	$-0.263862\pi$
$11$	163.508			1.35131			0.675653	+	0.737220i	$0.263862\pi$
$12$	61.7034	+	91.5639i	0.428496	+	0.635860i
$13$	−81.9293			−0.484789			−0.242394	−	0.970178i	$-0.577933\pi$
$13$	−81.9293			−0.484789			−0.242394	+	0.970178i	$0.577933\pi$
$14$	112.094			0.571909
$15$	−23.2691	−	34.5298i	−0.103418	−	0.153466i
$16$	90.8010			0.354691
$17$	44.7750	−	285.510i	0.154931	−	0.987925i
$17$	44.7750	−	285.510i	0.154931	−	0.987925i
$18$	145.030	+	58.7408i	0.447623	+	0.181299i
$19$	524.957			1.45417			0.727087	−	0.686545i	$-0.240873\pi$
$19$	524.957			1.45417			0.727087	+	0.686545i	$0.240873\pi$
$20$	−56.7588			−0.141897
$21$	−433.079	+	291.845i	−0.982039	+	0.661781i
$22$		−	315.862i		−	0.652606i
$23$	−767.157			−1.45020			−0.725101	−	0.688643i	$-0.758207\pi$
$23$	−767.157			−1.45020			−0.725101	+	0.688643i	$0.758207\pi$
$24$	407.567	−	274.653i	0.707581	−	0.476828i
$25$	−603.596			−0.965753
$26$			158.269i			0.234126i
$27$	−713.264	+	150.649i	−0.978415	+	0.206652i
$28$			711.880i			0.908010i
$29$	−462.049			−0.549405			−0.274702	−	0.961529i	$-0.588579\pi$
$29$	−462.049			−0.549405			−0.274702	+	0.961529i	$0.588579\pi$
$30$	−66.7039	+	44.9507i	−0.0741154	+	0.0499452i
$31$		−	637.845i		−	0.663730i	−0.943327	−	0.331865i	$-0.892322\pi$
$31$		−	637.845i		−	0.663730i	0.943327	−	0.331865i	$-0.107678\pi$
$32$		−	1049.14i		−	1.02455i
$33$	822.369	+	1220.34i	0.755160	+	1.12061i
$34$	−551.543	−	86.4954i	−0.477114	−	0.0748230i
$35$		−	268.458i		−	0.219149i
$36$	−373.047	+	921.047i	−0.287845	+	0.710684i
$37$		−	1178.23i		−	0.860651i	−0.902674	−	0.430325i	$-0.858399\pi$
$37$		−	1178.23i		−	0.860651i	0.902674	−	0.430325i	$-0.141601\pi$
$38$		−	1014.10i		−	0.702286i
$39$	−412.066	−	611.479i	−0.270918	−	0.402024i
$40$			252.643i			0.157902i
$41$	−1299.93			−0.773307			−0.386654	−	0.922225i	$-0.626369\pi$
$41$	−1299.93			−0.773307			−0.386654	+	0.922225i	$0.626369\pi$
$42$	563.781	+	836.614i	0.319604	+	0.474271i
$43$	1446.33			0.782224			0.391112	−	0.920343i	$-0.372091\pi$
$43$	1446.33			0.782224			0.391112	+	0.920343i	$0.372091\pi$
$44$	2005.95			1.03613
$45$	140.680	−	347.337i	0.0694718	−	0.171525i
$46$			1481.98i			0.700368i
$47$		−	3326.42i		−	1.50585i	−0.658108	−	0.752924i	$-0.728643\pi$
$47$		−	3326.42i		−	1.50585i	0.658108	−	0.752924i	$-0.271357\pi$
$48$	456.687	+	677.693i	0.198215	+	0.294138i
$49$	−966.054			−0.402355
$50$			1166.01i			0.466406i
$51$	2356.10	−	1101.81i	0.905845	−	0.423609i
$52$	−1005.13			−0.371718
$53$		−	257.237i		−	0.0915758i	−0.998951	−	0.0457879i	$-0.985420\pi$
$53$		−	257.237i		−	0.0915758i	0.998951	−	0.0457879i	$-0.0145798\pi$
$54$	291.021	+	1377.87i	0.0998014	+	0.472520i
$55$	−756.468			−0.250072
$56$	3168.70			1.01043
$57$	2640.29	+	3918.02i	0.812647	+	1.20591i
$58$			892.577i			0.265332i
$59$			4907.54i			1.40981i	0.709303	+	0.704904i	$0.249010\pi$
$59$			4907.54i			1.40981i	−0.709303	+	0.704904i	$0.750990\pi$
$60$	−285.470	−	423.619i	−0.0792972	−	0.117672i
$61$			3608.73i			0.969829i	0.874562	+	0.484915i	$0.161149\pi$
$61$			3608.73i			0.969829i	−0.874562	+	0.484915i	$0.838851\pi$
$62$	−1232.18			−0.320545
$63$	−4356.37	−	1764.44i	−1.09760	−	0.444556i
$64$	−573.882			−0.140108
$65$	379.045			0.0897147
$66$	2357.43	−	1588.64i	0.541192	−	0.364701i
$67$	1636.77			0.364618			0.182309	−	0.983241i	$-0.441643\pi$
$67$	1636.77			0.364618			0.182309	+	0.983241i	$0.441643\pi$
$68$	549.309	−	3502.71i	0.118795	−	0.757506i
$69$	−3858.44	−	5725.67i	−0.810426	−	1.20262i
$70$	−518.602			−0.105837
$71$	1219.27			0.241871			0.120935	−	0.992660i	$-0.461411\pi$
$71$	1219.27			0.241871			0.120935	+	0.992660i	$0.461411\pi$
$72$	4099.74	+	1660.50i	0.790845	+	0.320312i
$73$		−	7484.07i		−	1.40440i	−0.711977	−	0.702202i	$-0.752200\pi$
$73$		−	7484.07i		−	1.40440i	0.711977	−	0.702202i	$-0.247800\pi$
$74$	−2276.08			−0.415647
$75$	−3035.80	−	4504.93i	−0.539698	−	0.800877i
$76$	6440.29			1.11501
$77$			9487.77i			1.60023i
$78$	−1181.24	+	796.021i	−0.194156	+	0.130838i
$79$			2592.01i			0.415319i	0.978201	+	0.207660i	$0.0665847\pi$
$79$			2592.01i			0.415319i	−0.978201	+	0.207660i	$0.933415\pi$
$80$	−420.090			−0.0656390
$81$	−4711.75	−	4565.75i	−0.718146	−	0.695893i
$82$			2511.18i			0.373465i
$83$			4314.37i			0.626269i	0.949709	+	0.313135i	$0.101379\pi$
$83$			4314.37i			0.626269i	−0.949709	+	0.313135i	$0.898621\pi$
$84$	−5313.12	+	3580.42i	−0.752993	+	0.507430i
$85$	−207.151	+	1320.91i	−0.0286714	+	0.182825i
$86$		−	2793.99i		−	0.377771i
$87$	−2323.89	−	3448.50i	−0.307028	−	0.455609i
$88$		−	8928.85i		−	1.15300i
$89$			1095.43i			0.138294i	0.997606	+	0.0691471i	$0.0220278\pi$
$89$			1095.43i			0.138294i	−0.997606	+	0.0691471i	$0.977972\pi$
$90$	−670.979	−	271.763i	−0.0828369	−	0.0335510i
$91$		−	4754.05i		−	0.574092i
$92$	−9411.65			−1.11196
$93$	4760.55	−	3208.06i	0.550417	−	0.370917i
$94$	−6425.91			−0.727242
$95$	−2428.71			−0.269109
$96$	7830.22	−	5276.66i	0.849633	−	0.572554i
$97$		−	4255.19i		−	0.452246i	−0.974099	−	0.226123i	$-0.927395\pi$
$97$		−	4255.19i		−	0.452246i	0.974099	−	0.226123i	$-0.0726052\pi$
$98$			1866.20i			0.194315i
$99$	−4971.89	+	12275.5i	−0.507284	+	1.25247i
$100$	−7405.05			−0.740505
$101$			2593.06i			0.254196i	0.991890	+	0.127098i	$0.0405663\pi$
$101$			2593.06i			0.254196i	−0.991890	+	0.127098i	$0.959434\pi$
$102$	−2128.45	−	4551.47i	−0.204580	−	0.437473i
$103$	7610.50			0.717362			0.358681	−	0.933460i	$-0.383227\pi$
$103$	7610.50			0.717362			0.358681	+	0.933460i	$0.383227\pi$
$104$			4473.99i			0.413646i
$105$	2003.64	−	1350.22i	0.181736	−	0.122469i
$106$	−496.924			−0.0442261
$107$	−7007.07			−0.612025			−0.306012	−	0.952028i	$-0.598995\pi$
$107$	−7007.07			−0.612025			−0.306012	+	0.952028i	$0.598995\pi$
$108$	−8750.49	+	1848.20i	−0.750213	+	0.158453i
$109$			19775.5i			1.66446i	0.554427	+	0.832232i	$0.312937\pi$
$109$			19775.5i			1.66446i	−0.554427	+	0.832232i	$0.687063\pi$
$110$			1461.33i			0.120771i
$111$	8793.73	−	5925.95i	0.713719	−	0.480964i
$112$			5268.85i			0.420029i
$113$	22646.0			1.77352			0.886758	−	0.462234i	$-0.152952\pi$
$113$	22646.0			1.77352			0.886758	+	0.462234i	$0.152952\pi$
$114$	7568.75	−	5100.46i	0.582390	−	0.392464i
$115$	3549.24			0.268373
$116$	−5668.53			−0.421264
$117$	2491.27	−	6150.91i	0.181991	−	0.449332i
$118$	9480.29			0.680859
$119$	16567.1	+	2598.13i	1.16991	+	0.183471i
$120$	−1885.60	+	1270.68i	−0.130945	+	0.0882414i
$121$	12093.9			0.826028
$122$	6971.28			0.468374
$123$	−6538.04	−	9702.02i	−0.432153	−	0.641287i
$124$		−	7825.22i		−	0.508924i
$125$	5684.08			0.363781
$126$	−3408.51	+	8415.55i	−0.214696	+	0.530080i
$127$	−27486.1			−1.70414			−0.852071	−	0.523426i	$-0.824654\pi$
$127$	−27486.1			−1.70414			−0.852071	+	0.523426i	$0.824654\pi$
$128$		−	15677.6i		−	0.956882i
$129$	7274.37	+	10794.7i	0.437135	+	0.648681i
$130$		−	732.231i		−	0.0433273i
$131$	217.363			0.0126661			0.00633304	−	0.999980i	$-0.497984\pi$
$131$	217.363			0.0126661			0.00633304	+	0.999980i	$0.497984\pi$
$132$	10089.0	+	14971.4i	0.579029	+	0.859242i
$133$			30461.3i			1.72205i
$134$		−	3161.88i		−	0.176090i
$135$	3299.91	−	696.976i	0.181065	−	0.0382428i
$136$	−15591.2	−	2445.07i	−0.842947	−	0.132195i
$137$			20151.7i			1.07367i	0.843688	+	0.536834i	$0.180380\pi$
$137$			20151.7i			1.07367i	−0.843688	+	0.536834i	$0.819620\pi$
$138$	−11060.7	+	7453.65i	−0.580799	+	0.391391i
$139$		−	22261.3i		−	1.15218i	−0.817386	−	0.576090i	$-0.804578\pi$
$139$		−	22261.3i		−	1.15218i	0.817386	−	0.576090i	$-0.195422\pi$
$140$		−	3293.50i		−	0.168036i
$141$	24826.7	−	16730.3i	1.24877	−	0.841523i
$142$		−	2355.36i		−	0.116810i
$143$	−13396.1			−0.655098
$144$	−2761.04	+	6816.96i	−0.133152	+	0.328750i
$145$	2137.66			0.101672
$146$	−14457.6			−0.678250
$147$	−4858.80	−	7210.14i	−0.224851	−	0.333664i
$148$		−	14454.8i		−	0.659916i
$149$		−	2236.95i		−	0.100759i	−0.998730	−	0.0503796i	$-0.983957\pi$
$149$		−	2236.95i		−	0.100759i	0.998730	−	0.0503796i	$-0.0160431\pi$
$150$	−8702.54	+	5864.50i	−0.386780	+	0.260645i
$151$	−4703.53			−0.206286			−0.103143	−	0.994667i	$-0.532890\pi$
$151$	−4703.53			−0.206286			−0.103143	+	0.994667i	$0.532890\pi$
$152$		−	28666.9i		−	1.24077i
$153$	20073.4	+	12043.2i	0.857509	+	0.514469i
$154$	18328.3			0.772823
$155$			2950.98i			0.122830i
$156$	−5055.32	−	7501.76i	−0.207730	−	0.308258i
$157$	4228.69			0.171556			0.0857781	−	0.996314i	$-0.472662\pi$
$157$	4228.69			0.171556			0.0857781	+	0.996314i	$0.472662\pi$
$158$	5007.19			0.200576
$159$	1919.88	−	1293.78i	0.0759418	−	0.0511760i
$160$			4853.81i			0.189602i
$161$		−	44515.3i		−	1.71734i
$162$	−8820.03	+	9102.07i	−0.336078	+	0.346825i
$163$			24397.3i			0.918261i	0.888369	+	0.459131i	$0.151839\pi$
$163$			24397.3i			0.918261i	−0.888369	+	0.459131i	$0.848161\pi$
$164$	−15947.8			−0.592944
$165$	−3804.68	−	5645.89i	−0.139749	−	0.207379i
$166$	8334.41			0.302454
$167$	−54558.2			−1.95626			−0.978131	−	0.207988i	$-0.933309\pi$
$167$	−54558.2			−1.95626			−0.978131	+	0.207988i	$0.933309\pi$
$168$	15937.1	+	23649.6i	0.564664	+	0.837925i
$169$	−21848.6			−0.764980
$170$	2551.71	+	400.170i	0.0882944	+	0.0138467i
$171$	−15962.7	+	39411.6i	−0.545901	+	1.34782i
$172$	17743.9			0.599781
$173$	21855.7			0.730252			0.365126	−	0.930958i	$-0.381026\pi$
$173$	21855.7			0.730252			0.365126	+	0.930958i	$0.381026\pi$
$174$	−6661.75	+	4489.25i	−0.220034	+	0.148277i
$175$		−	35024.4i		−	1.14365i
$176$	14846.7			0.479297
$177$	−36627.4	+	24682.6i	−1.16912	+	0.787852i
$178$	2116.13			0.0667885
$179$		−	14369.1i		−	0.448459i	−0.974536	−	0.224230i	$-0.928013\pi$
$179$		−	14369.1i		−	0.448459i	0.974536	−	0.224230i	$-0.0719866\pi$
$180$	1725.90	−	4261.21i	0.0532685	−	0.131519i
$181$		−	59229.9i		−	1.80794i	−0.427597	−	0.903969i	$-0.640640\pi$
$181$		−	59229.9i		−	1.80794i	0.427597	−	0.903969i	$-0.359360\pi$
$182$	−9183.79			−0.277255
$183$	−26933.8	+	18150.2i	−0.804258	+	0.541976i
$184$			41892.9i			1.23738i
$185$			5451.07i			0.159272i
$186$	−6197.27	−	9196.34i	−0.179132	−	0.265821i
$187$	7321.07	−	46683.2i	0.209359	−	1.33499i
$188$		−	40809.2i		−	1.15463i
$189$	−8741.61	−	41388.1i	−0.244719	−	1.15865i
$190$			4691.73i			0.129965i
$191$			57570.2i			1.57809i	0.614337	+	0.789043i	$0.289423\pi$
$191$			57570.2i			1.57809i	−0.614337	+	0.789043i	$0.710577\pi$
$192$	−2886.36	−	4283.17i	−0.0782975	−	0.116188i
$193$			50923.2i			1.36710i	0.729902	+	0.683551i	$0.239566\pi$
$193$			50923.2i			1.36710i	−0.729902	+	0.683551i	$0.760434\pi$
$194$	−8220.08			−0.218410
$195$	1906.42	+	2829.00i	0.0501359	+	0.0743984i
$196$	−11851.8			−0.308511
$197$	−16795.4			−0.432770			−0.216385	−	0.976308i	$-0.569427\pi$
$197$	−16795.4			−0.432770			−0.216385	+	0.976308i	$0.569427\pi$
$198$	23713.6	+	9604.59i	0.604876	+	0.244990i
$199$			18089.3i			0.456789i	0.973569	+	0.228394i	$0.0733475\pi$
$199$			18089.3i			0.456789i	−0.973569	+	0.228394i	$0.926652\pi$
$200$			32961.1i			0.824029i
$201$	8232.19	+	12216.0i	0.203762	+	0.302370i
$202$	5009.21			0.122763
$203$		−	26811.0i		−	0.650611i
$204$	28905.2	−	13517.2i	0.694570	−	0.324808i
$205$	6014.11			0.143108
$206$		−	14701.8i		−	0.346447i
$207$	23327.4	−	57594.9i	0.544409	−	1.34414i
$208$	−7439.26			−0.171950
$209$	85834.7			1.96504
$210$	−2608.32	−	3870.58i	−0.0591457	−	0.0877683i
$211$			35276.8i			0.792364i	0.918172	+	0.396182i	$0.129665\pi$
$211$			35276.8i			0.792364i	−0.918172	+	0.396182i	$0.870335\pi$
$212$		−	3155.84i		−	0.0702171i
$213$	6132.36	+	9100.03i	0.135166	+	0.200578i
$214$			13536.1i			0.295574i
$215$	−6691.43			−0.144758
$216$	8226.64	+	38949.9i	0.176326	+	0.834832i
$217$	37011.8			0.785996
$218$	38201.9			0.803845
$219$	55857.4	−	37641.4i	1.16464	−	0.784833i
$220$	−9280.51			−0.191746
$221$	−3668.38	+	23391.7i	−0.0751086	+	0.478935i
$222$	−11447.6	−	16987.5i	−0.232279	−	0.344687i
$223$	−51868.5			−1.04302			−0.521511	−	0.853244i	$-0.674632\pi$
$223$	−51868.5			−1.04302			−0.521511	+	0.853244i	$0.674632\pi$
$224$	60877.5			1.21328
$225$	18353.9	−	45315.4i	0.362546	−	0.895120i
$226$		−	43747.1i		−	0.856511i
$227$	−84661.3			−1.64298			−0.821491	−	0.570221i	$-0.806858\pi$
$227$	−84661.3			−1.64298			−0.821491	+	0.570221i	$0.806858\pi$
$228$	32391.7	+	48067.1i	0.623108	+	0.924652i
$229$	70220.4			1.33904			0.669518	−	0.742796i	$-0.266501\pi$
$229$	70220.4			1.33904			0.669518	+	0.742796i	$0.266501\pi$
$230$		−	6856.35i		−	0.129610i
$231$	−70812.0	+	47719.0i	−1.32704	+	0.894268i
$232$			25231.6i			0.468779i
$233$	36195.4			0.666716			0.333358	−	0.942800i	$-0.391818\pi$
$233$	36195.4			0.666716			0.333358	+	0.942800i	$0.391818\pi$
$234$	−11882.2	−	4812.59i	−0.217003	−	0.0878916i
$235$			15389.6i			0.278671i
$236$			60206.8i			1.08099i
$237$	−19345.5	+	13036.6i	−0.344415	+	0.232096i
$238$	5019.01	−	32004.0i	0.0886062	−	0.565003i
$239$			33333.6i			0.583561i	0.956485	+	0.291780i	$0.0942477\pi$
$239$			33333.6i			0.583561i	−0.956485	+	0.291780i	$0.905752\pi$
$240$	−2112.85	−	3135.34i	−0.0366815	−	0.0544329i
$241$		−	3970.76i		−	0.0683660i	−0.999416	−	0.0341830i	$-0.989117\pi$
$241$		−	3970.76i		−	0.0683660i	0.999416	−	0.0341830i	$-0.0108829\pi$
$242$		−	23362.7i		−	0.398926i
$243$	10378.6	−	58129.8i	0.175762	−	0.984433i
$244$			44272.8i			0.743630i
$245$	4469.43			0.0744595
$246$	−18742.2	+	12630.1i	−0.309706	+	0.208706i
$247$	−43009.4			−0.704967
$248$	−34831.4			−0.566328
$249$	−32200.3	+	21699.3i	−0.519351	+	0.349982i
$250$		−	10980.4i		−	0.175686i
$251$		−	84826.3i		−	1.34643i	−0.739448	−	0.673214i	$-0.764913\pi$
$251$		−	84826.3i		−	1.34643i	0.739448	−	0.673214i	$-0.235087\pi$
$252$	−53445.0	−	21646.6i	−0.841600	−	0.340869i
$253$	−125436.			−1.95967
$254$			53097.1i			0.823007i
$255$	−10900.5	+	5097.49i	−0.167635	+	0.0783928i
$256$	−39467.7			−0.602229
$257$			65272.2i			0.988239i	0.869394	+	0.494119i	$0.164509\pi$
$257$			65272.2i			0.988239i	−0.869394	+	0.494119i	$0.835491\pi$
$258$	20853.0	−	14052.5i	0.313277	−	0.211112i
$259$	68368.4			1.01919
$260$	4650.20			0.0687900
$261$	14049.8	−	34688.7i	0.206248	−	0.509222i
$262$		−	419.897i		−	0.00611702i
$263$			50611.7i			0.731711i	0.930672	+	0.365855i	$0.119224\pi$
$263$			50611.7i			0.731711i	−0.930672	+	0.365855i	$0.880776\pi$
$264$	66640.4	−	44907.9i	0.956158	−	0.644340i
$265$			1190.10i			0.0169470i
$266$	58844.6			0.831655
$267$	−8175.73	+	5509.49i	−0.114684	+	0.0772839i
$268$	20080.3			0.279576
$269$	−74743.0			−1.03292			−0.516459	−	0.856312i	$-0.672750\pi$
$269$	−74743.0			−1.03292			−0.516459	+	0.856312i	$0.672750\pi$
$270$	−1346.40	−	6374.69i	−0.0184692	−	0.0874443i
$271$	41504.3			0.565138			0.282569	−	0.959247i	$-0.408813\pi$
$271$	41504.3			0.565138			0.282569	+	0.959247i	$0.408813\pi$
$272$	4065.61	−	25924.6i	0.0549526	−	0.350409i
$273$	35481.9	−	23910.7i	0.476081	−	0.320824i
$274$	38928.6			0.518523
$275$	−98692.7			−1.30503
$276$	−47336.2	−	70243.8i	−0.621406	−	0.922126i
$277$		−	23671.9i		−	0.308513i	−0.988031	−	0.154256i	$-0.950702\pi$
$277$		−	23671.9i		−	0.308513i	0.988031	−	0.154256i	$-0.0492982\pi$
$278$	−43003.9			−0.556440
$279$	47886.7	+	19395.3i	0.615186	+	0.249166i
$280$	−14659.9			−0.186989
$281$			84730.4i			1.07307i	0.843879	+	0.536533i	$0.180266\pi$
$281$			84730.4i			1.07307i	−0.843879	+	0.536533i	$0.819734\pi$
$282$	−32319.3	−	47959.7i	−0.406409	−	0.603085i
$283$		−	110801.i		−	1.38348i	−0.722149	−	0.691738i	$-0.756845\pi$
$283$		−	110801.i		−	1.38348i	0.722149	−	0.691738i	$-0.243155\pi$
$284$	14958.3			0.185458
$285$	−12215.3	−	18126.6i	−0.150388	−	0.223166i
$286$			25878.3i			0.316376i
$287$		−	75430.1i		−	0.915759i
$288$	78764.7	+	31901.7i	0.949613	+	0.384617i
$289$	−79511.4	−	25567.4i	−0.951993	−	0.306120i
$290$		−	4129.50i		−	0.0491022i
$291$	31758.6	−	21401.6i	0.375038	−	0.252732i
$292$		−	91816.3i		−	1.07685i
$293$			64767.6i			0.754436i	0.926125	+	0.377218i	$0.123119\pi$
$293$			64767.6i			0.754436i	−0.926125	+	0.377218i	$0.876881\pi$
$294$	−13928.4	+	9386.13i	−0.161141	+	0.108591i
$295$		−	22704.7i		−	0.260898i
$296$	−64340.8			−0.734350
$297$	−116624.	+	24632.3i	−1.32214	+	0.279250i
$298$	−4321.30			−0.0486611
$299$	62852.6			0.703041
$300$	−37243.9	−	55267.6i	−0.413821	−	0.614084i
$301$			83925.3i			0.926318i
$302$			9086.18i			0.0996248i
$303$	−19353.3	+	13041.9i	−0.210799	+	0.142054i
$304$	47666.6			0.515783
$305$		−	16695.8i		−	0.179476i
$306$	23264.8	−	38777.4i	0.248460	−	0.414130i
$307$	78685.6			0.834869			0.417435	−	0.908707i	$-0.362929\pi$
$307$	78685.6			0.834869			0.417435	+	0.908707i	$0.362929\pi$
$308$			116398.i			1.22700i
$309$	38277.2	+	56800.9i	0.400889	+	0.594893i
$310$	5700.64			0.0593199
$311$	97228.4			1.00525			0.502623	−	0.864506i	$-0.332368\pi$
$311$	97228.4			1.00525			0.502623	+	0.864506i	$0.332368\pi$
$312$	−33391.6	+	22502.1i	−0.343027	+	0.231161i
$313$			87641.4i			0.894583i	0.894388	+	0.447291i	$0.147611\pi$
$313$			87641.4i			0.894583i	−0.894388	+	0.447291i	$0.852389\pi$
$314$		−	8168.90i		−	0.0828522i
$315$	20154.7	+	8163.16i	0.203121	+	0.0822692i
$316$			31799.4i			0.318452i
$317$	−78037.3			−0.776576			−0.388288	−	0.921538i	$-0.626933\pi$
$317$	−78037.3			−0.776576			−0.388288	+	0.921538i	$0.626933\pi$
$318$	−2499.30	−	3708.79i	−0.0247152	−	0.0366757i
$319$	−75548.8			−0.742414
$320$	2655.06			0.0259283
$321$	−35242.3	−	52297.3i	−0.342022	−	0.507538i
$322$	−85993.7			−0.829383
$323$	23504.9	−	149881.i	0.225296	−	1.43662i
$324$	−57804.9	−	56013.7i	−0.550648	−	0.533586i
$325$	49452.1			0.468186
$326$	47130.2			0.443470
$327$	−147594.	+	99461.5i	−1.38030	+	0.930164i
$328$			70986.6i			0.659824i
$329$	193020.			1.78324
$330$	−10906.6	+	7349.80i	−0.100153	+	0.0674913i
$331$	156868.			1.43178			0.715891	−	0.698212i	$-0.246021\pi$
$331$	156868.			1.43178			0.715891	+	0.698212i	$0.246021\pi$
$332$			52929.7i			0.480201i
$333$	88456.7	+	35827.2i	0.797705	+	0.323091i
$334$			105394.i			0.944767i
$335$	−7572.50			−0.0674760
$336$	−39324.0	+	26499.8i	−0.348321	+	0.234728i
$337$		−	471.180i		−	0.00414884i	−0.999998	−	0.00207442i	$-0.999340\pi$
$337$		−	471.180i		−	0.00414884i	0.999998	−	0.00207442i	$-0.000660309\pi$
$338$			42206.7i			0.369443i
$339$	113899.	+	169019.i	0.991106	+	1.47074i
$340$	−2541.37	+	16205.2i	−0.0219842	+	0.140184i
$341$		−	104293.i		−	0.896903i
$342$	76134.5	+	30836.4i	0.650923	+	0.263640i
$343$			83264.6i			0.707738i
$344$		−	78981.3i		−	0.667432i
$345$	17851.0	+	26489.7i	0.149977	+	0.222556i
$346$		−	42220.4i		−	0.352671i
$347$	73350.4			0.609177			0.304589	−	0.952484i	$-0.401481\pi$
$347$	73350.4			0.609177			0.304589	+	0.952484i	$0.401481\pi$
$348$	−28510.0	−	42307.0i	−0.235418	−	0.349345i
$349$	−27633.5			−0.226874			−0.113437	−	0.993545i	$-0.536186\pi$
$349$	−27633.5			−0.226874			−0.113437	+	0.993545i	$0.536186\pi$
$350$	−67659.5			−0.552322
$351$	58437.2	−	12342.6i	0.474324	−	0.100182i
$352$		−	171542.i		−	1.38448i
$353$		−	147168.i		−	1.18104i	−0.807023	−	0.590521i	$-0.798922\pi$
$353$		−	147168.i		−	1.18104i	0.807023	−	0.590521i	$-0.201078\pi$
$354$	47681.4	+	70756.1i	0.380489	+	0.564621i
$355$	−5640.94			−0.0447605
$356$			13439.0i			0.106039i
$357$	63933.7	+	136716.i	0.501642	+	1.07271i
$358$	−27757.9			−0.216581
$359$		−	185845.i		−	1.44199i	−0.692941	−	0.720994i	$-0.743686\pi$
$359$		−	185845.i		−	1.44199i	0.692941	−	0.720994i	$-0.256314\pi$
$360$	−18967.4	−	7682.27i	−0.146353	−	0.0592768i
$361$	145259.			1.11462
$362$	−114419.			−0.873135
$363$	60826.6	+	90262.6i	0.461615	+	0.685007i
$364$		−	58323.8i		−	0.440193i
$365$			34625.0i			0.259898i
$366$	35062.3	+	52030.1i	0.261745	+	0.388412i
$367$		−	41786.8i		−	0.310246i	−0.987895	−	0.155123i	$-0.950423\pi$
$367$		−	41786.8i		−	0.310246i	0.987895	−	0.155123i	$-0.0495774\pi$
$368$	−69658.6			−0.514374
$369$	39527.8	−	97593.3i	0.290302	−	0.716749i
$370$	10530.3			0.0769194
$371$	14926.5			0.108445
$372$	58403.5	−	39357.2i	0.422040	−	0.284406i
$373$	−63558.2			−0.456829			−0.228415	−	0.973564i	$-0.573354\pi$
$373$	−63558.2			−0.456829			−0.228415	+	0.973564i	$0.573354\pi$
$374$	−90181.8	−	14142.7i	−0.644726	−	0.101109i
$375$	28588.3	+	42423.1i	0.203294	+	0.301675i
$376$	−181649.			−1.28486
$377$	37855.4			0.266345
$378$	−79952.7	+	16886.9i	−0.559564	+	0.118186i
$379$			47660.7i			0.331805i	0.986142	+	0.165902i	$0.0530536\pi$
$379$			47660.7i			0.331805i	−0.986142	+	0.165902i	$0.946946\pi$
$380$	−29795.9			−0.206343
$381$	−138242.	−	205143.i	−0.952338	−	1.41321i
$382$	111213.			0.762129
$383$		−	27055.7i		−	0.184442i	−0.995739	−	0.0922212i	$-0.970603\pi$
$383$		−	27055.7i		−	0.184442i	0.995739	−	0.0922212i	$-0.0293967\pi$
$384$	117009.	−	78850.8i	0.793521	−	0.534741i
$385$		−	43895.0i		−	0.296138i
$386$	98372.4			0.660235
$387$	−43979.5	+	108585.i	−0.293649	+	0.725013i
$388$		−	52203.6i		−	0.346766i
$389$		−	13739.0i		−	0.0907934i	−0.998969	−	0.0453967i	$-0.985545\pi$
$389$		−	13739.0i		−	0.0907934i	0.998969	−	0.0453967i	$-0.0144552\pi$
$390$	5465.00	−	3682.78i	0.0359303	−	0.0242129i
$391$	−34349.4	+	219031.i	−0.224681	+	1.43269i
$392$			52754.3i			0.343309i
$393$	1093.23	+	1622.29i	0.00707827	+	0.0105037i
$394$			32445.0i			0.209004i
$395$		−	11991.9i		−	0.0768588i
$396$	−60996.3	+	150599.i	−0.388967	+	0.960352i
$397$		−	196373.i		−	1.24595i	−0.782240	−	0.622977i	$-0.785923\pi$
$397$		−	196373.i		−	1.24595i	0.782240	−	0.622977i	$-0.214077\pi$
$398$	34944.5			0.220604
$399$	−227348.	+	153206.i	−1.42806	+	0.962345i
$400$	−54807.1			−0.342544
$401$	−139747.			−0.869065			−0.434533	−	0.900656i	$-0.643086\pi$
$401$	−139747.			−0.869065			−0.434533	+	0.900656i	$0.643086\pi$
$402$	23598.7	−	15902.8i	0.146028	−	0.0984059i
$403$			52258.1i			0.321769i
$404$			31812.2i			0.194909i
$405$	21798.9	+	21123.4i	0.132900	+	0.128781i
$406$	−51793.0			−0.314209
$407$		−	192650.i		−	1.16300i
$408$	−60167.4	−	128662.i	−0.361444	−	0.772912i
$409$	−203317.			−1.21542			−0.607710	−	0.794159i	$-0.707912\pi$
$409$	−203317.			−1.21542			−0.607710	+	0.794159i	$0.707912\pi$
$410$		−	11617.9i		−	0.0691132i
$411$	−150402.	+	101354.i	−0.890369	+	0.600005i
$412$	93367.3			0.550048
$413$	−284766.			−1.66951
$414$	−111261.	−	45063.4i	−0.649144	−	0.262920i
$415$		−	19960.4i		−	0.115897i
$416$			85954.9i			0.496688i
$417$	166147.	−	111964.i	0.955477	−	0.643881i
$418$		−	165814.i		−	0.949004i
$419$	234819.			1.33753			0.668767	−	0.743472i	$-0.266822\pi$
$419$	234819.			1.33753			0.668767	+	0.743472i	$0.266822\pi$
$420$	24581.0	−	16564.8i	0.139348	−	0.0939046i
$421$	193384.			1.09108			0.545540	−	0.838085i	$-0.316325\pi$
$421$	193384.			1.09108			0.545540	+	0.838085i	$0.316325\pi$
$422$	68147.1			0.382668
$423$	249733.	+	101148.i	1.39571	+	0.565299i
$424$	−14047.2			−0.0781371
$425$	−27026.0	+	172333.i	−0.149625	+	0.954092i
$426$	17579.2	−	11846.4i	0.0968682	−	0.0652779i
$427$	−209402.			−1.14848
$428$	−85964.3			−0.469279
$429$	−67376.1	−	99981.7i	−0.366093	−	0.543258i
$430$			12926.4i			0.0699101i
$431$	19605.1			0.105539			0.0527696	−	0.998607i	$-0.483195\pi$
$431$	19605.1			0.105539			0.0527696	+	0.998607i	$0.483195\pi$
$432$	−64765.1	+	13679.1i	−0.347035	+	0.0732976i
$433$	234164.			1.24895			0.624474	−	0.781046i	$-0.285314\pi$
$433$	234164.			1.24895			0.624474	+	0.781046i	$0.285314\pi$
$434$		−	71498.6i		−	0.379593i
$435$	10751.5	+	15954.5i	0.0568183	+	0.0843147i
$436$			242610.i			1.27625i
$437$	−402724.			−2.10885
$438$	−72714.9	−	107904.i	−0.379031	−	0.562458i
$439$		−	267368.i		−	1.38733i	−0.720296	−	0.693667i	$-0.755994\pi$
$439$		−	267368.i		−	1.38733i	0.720296	−	0.693667i	$-0.244006\pi$
$440$			41309.2i			0.213374i
$441$	29375.4	−	72527.3i	0.151045	−	0.372927i
$442$	45187.5	+	7086.50i	0.231299	+	0.0362733i
$443$			141083.i			0.718896i	0.933165	+	0.359448i	$0.117035\pi$
$443$			141083.i			0.718896i	−0.933165	+	0.359448i	$0.882965\pi$
$444$	107883.	−	72700.9i	0.547254	−	0.368786i
$445$		−	5067.98i		−	0.0255927i
$446$			100198.i			0.503722i
$447$	16695.5	−	11250.8i	0.0835573	−	0.0563079i
$448$		−	33300.3i		−	0.165917i
$449$	121188.			0.601129			0.300564	−	0.953762i	$-0.402825\pi$
$449$	121188.			0.601129			0.300564	+	0.953762i	$0.402825\pi$
$450$	−87539.4	−	35455.7i	−0.432294	−	0.175090i
$451$	−212549.			−1.04498
$452$	277827.			1.35987
$453$	−23656.5	−	35104.8i	−0.115280	−	0.171068i
$454$			163547.i			0.793470i
$455$			21994.6i			0.106241i
$456$	213955.	−	144181.i	1.02895	−	0.693391i
$457$	−172678.			−0.826808			−0.413404	−	0.910548i	$-0.635660\pi$
$457$	−172678.			−0.826808			−0.413404	+	0.910548i	$0.635660\pi$
$458$		−	135650.i		−	0.646680i
$459$	11075.5	+	210390.i	0.0525700	+	0.998617i
$460$	43542.9			0.205779
$461$			217522.i			1.02353i	0.859125	+	0.511766i	$0.171008\pi$
$461$			217522.i			1.02353i	−0.859125	+	0.511766i	$0.828992\pi$
$462$	92182.7	+	136793.i	0.431882	+	0.640885i
$463$	189076.			0.882010			0.441005	−	0.897505i	$-0.354622\pi$
$463$	189076.			0.882010			0.441005	+	0.897505i	$0.354622\pi$
$464$	−41954.5			−0.194869
$465$	−22024.6	+	14842.0i	−0.101860	+	0.0686417i
$466$		−	69921.5i		−	0.321987i
$467$		−	328940.i		−	1.50829i	−0.656711	−	0.754143i	$-0.728053\pi$
$467$		−	328940.i		−	1.50829i	0.656711	−	0.754143i	$-0.271947\pi$
$468$	30563.5	−	75460.7i	0.139544	−	0.344532i
$469$			94975.8i			0.431785i
$470$	29729.4			0.134583
$471$	21268.3	+	31560.8i	0.0958720	+	0.142268i
$472$	267991.			1.20292
$473$	236487.			1.05702
$474$	25183.8	+	37371.2i	0.112090	+	0.166334i
$475$	−316862.			−1.40437
$476$	203249.	+	31874.4i	0.897046	+	0.140679i
$477$	19312.2	+	7821.95i	0.0848782	+	0.0343778i
$478$	64393.1			0.281828
$479$	−190153.			−0.828766			−0.414383	−	0.910102i	$-0.636003\pi$
$479$	−190153.			−0.828766			−0.414383	+	0.910102i	$0.636003\pi$
$480$	−36226.4	+	24412.4i	−0.157233	+	0.105957i
$481$			96531.6i			0.417234i
$482$	−7670.64			−0.0330170
$483$	332240.	−	223891.i	1.42415	−	0.959715i
$484$	148370.			0.633369
$485$			19686.6i			0.0836924i
$486$	−112294.	−	20049.1i	−0.475427	−	0.0848833i
$487$			214235.i			0.903302i	0.892195	+	0.451651i	$0.149165\pi$
$487$			214235.i			0.903302i	−0.892195	+	0.451651i	$0.850835\pi$
$488$	197066.			0.827507
$489$	−182089.	+	122707.i	−0.761493	+	0.513158i
$490$		−	8633.96i		−	0.0359599i
$491$			187012.i			0.775723i	0.921718	+	0.387862i	$0.126786\pi$
$491$			187012.i			0.775723i	−0.921718	+	0.387862i	$0.873214\pi$
$492$	−80210.1	−	119027.i	−0.331359	−	0.491716i
$493$	−20688.2	+	131920.i	−0.0851196	+	0.542771i
$494$			83084.6i			0.340460i
$495$	23002.4	−	56792.4i	0.0938776	−	0.231782i
$496$		−	57916.9i		−	0.235419i
$497$			70749.8i			0.286426i
$498$	41918.2	+	62203.9i	0.169022	+	0.250818i
$499$			7806.07i			0.0313496i	0.999877	+	0.0156748i	$0.00498964\pi$
$499$			7806.07i			0.0313496i	−0.999877	+	0.0156748i	$0.995010\pi$
$500$	69733.6			0.278934
$501$	−274402.	−	407195.i	−1.09323	−	1.62228i
$502$	−163866.			−0.650251
$503$	403115.			1.59328			0.796641	−	0.604453i	$-0.206608\pi$
$503$	403115.			1.59328			0.796641	+	0.604453i	$0.206608\pi$
$504$	−96352.6	+	237893.i	−0.379317	+	0.936527i
$505$		−	11996.7i		−	0.0470414i
$506$			242315.i			0.946411i
$507$	−109888.	−	163067.i	−0.427499	−	0.634381i
$508$	−337206.			−1.30668
$509$			239308.i			0.923680i	0.886963	+	0.461840i	$0.152811\pi$
$509$			239308.i			0.923680i	−0.886963	+	0.461840i	$0.847189\pi$
$510$	9847.23	+	21057.3i	0.0378594	+	0.0809586i
$511$	434273.			1.66311
$512$		−	174598.i		−	0.666038i
$513$	−374433.	+	79084.3i	−1.42279	+	0.300508i
$514$	126091.			0.477265
$515$	−35209.9			−0.132755
$516$	89243.6	+	132432.i	0.335180	+	0.497385i
$517$		−	543896.i		−	2.03486i
$518$		−	132073.i		−	0.492214i
$519$	109924.	+	163120.i	0.408092	+	0.605581i
$520$		−	20698.9i		−	0.0765490i
$521$	−283337.			−1.04383			−0.521913	−	0.852999i	$-0.674781\pi$
$521$	−283337.			−1.04383			−0.521913	+	0.852999i	$0.674781\pi$
$522$	−67011.0	−	27141.1i	−0.245926	−	0.0996064i
$523$	67135.1			0.245441			0.122720	−	0.992441i	$-0.460838\pi$
$523$	67135.1			0.245441			0.122720	+	0.992441i	$0.460838\pi$
$524$	2666.65			0.00971190
$525$	261405.	−	176157.i	0.948407	−	0.639117i
$526$	97770.7			0.353376
$527$	−182111.	−	28559.5i	−0.655716	−	0.102832i
$528$	74671.9	+	110808.i	0.267849	+	0.397470i
$529$	308688.			1.10308
$530$	2299.01			0.00818445
$531$	−368438.	−	149227.i	−1.30670	−	0.529245i
$532$			373707.i			1.32041i
$533$	106502.			0.374891
$534$	10643.1	+	15793.7i	0.0373239	+	0.0553862i
$535$	32418.1			0.113261
$536$		−	89380.8i		−	0.311110i
$537$	107244.	−	72269.8i	0.371897	−	0.250616i
$538$			144387.i			0.498843i
$539$	−157958.			−0.543704
$540$	40484.0	−	8550.66i	0.138834	−	0.0293232i
$541$		−	537673.i		−	1.83706i	−0.395348	−	0.918531i	$-0.629376\pi$
$541$		−	537673.i		−	1.83706i	0.395348	−	0.918531i	$-0.370624\pi$
$542$		−	80177.2i		−	0.272931i
$543$	442062.	−	297899.i	1.49928	−	1.01034i
$544$	−299539.	−	46975.0i	−1.01218	−	0.158734i
$545$		−	91491.1i		−	0.308025i
$546$	−46190.1	−	68543.2i	−0.154940	−	0.229921i
$547$			581155.i			1.94230i	0.238460	+	0.971152i	$0.423358\pi$
$547$			581155.i			1.94230i	−0.238460	+	0.971152i	$0.576642\pi$
$548$			247225.i			0.823250i
$549$	−270929.	−	109733.i	−0.898898	−	0.364076i
$550$			190653.i			0.630257i
$551$	−242556.			−0.798930
$552$	−312667.	+	210702.i	−1.02614	+	0.691496i
$553$	−150405.			−0.491826
$554$	−45728.9			−0.148995
$555$	−40684.0	+	27416.3i	−0.132080	+	0.0890068i
$556$		−	273106.i		−	0.883450i
$557$			528535.i			1.70358i	0.523881	+	0.851791i	$0.324483\pi$
$557$			528535.i			1.70358i	−0.523881	+	0.851791i	$0.675517\pi$
$558$	37467.5	−	92506.6i	0.120333	−	0.297101i
$559$	−118497.			−0.379213
$560$		−	24376.2i		−	0.0777304i
$561$	385242.	−	180154.i	1.22407	−	0.572425i
$562$	163680.			0.518232
$563$			188127.i			0.593520i	0.954952	+	0.296760i	$0.0959061\pi$
$563$			188127.i			0.593520i	−0.954952	+	0.296760i	$0.904094\pi$
$564$	304580.	−	205251.i	0.957509	−	0.645250i
$565$	−104772.			−0.328206
$566$	−214044.			−0.668143
$567$	264934.	−	273406.i	0.824083	−	0.850436i
$568$		−	66581.9i		−	0.206376i
$569$			108905.i			0.336374i	0.985755	+	0.168187i	$0.0537913\pi$
$569$			108905.i			0.336374i	−0.985755	+	0.168187i	$0.946209\pi$
$570$	−35016.7	+	23597.2i	−0.107777	+	0.0726291i
$571$		−	311068.i		−	0.954076i	−0.878883	−	0.477038i	$-0.841710\pi$
$571$		−	311068.i		−	0.954076i	0.878883	−	0.477038i	$-0.158290\pi$
$572$	−164346.			−0.502305
$573$	−429675.	+	289551.i	−1.30867	+	0.881893i
$574$	−145714.			−0.442261
$575$	463052.			1.40054
$576$	17450.4	−	43084.7i	0.0525969	−	0.129861i
$577$	−279948.			−0.840863			−0.420432	−	0.907324i	$-0.638121\pi$
$577$	−279948.			−0.840863			−0.420432	+	0.907324i	$0.638121\pi$
$578$	−49390.7	+	153599.i	−0.147839	+	0.459760i
$579$	−380065.	+	256120.i	−1.13371	+	0.763988i
$580$	26225.3			0.0779588
$581$	−250347.			−0.741635
$582$	−41343.2	−	61350.6i	−0.122056	−	0.181123i
$583$		−	42060.2i		−	0.123747i
$584$	−408690.			−1.19831
$585$	−11525.8	+	28457.1i	−0.0336791	+	0.0831531i
$586$	125117.			0.364351
$587$		−	268348.i		−	0.778794i	−0.921070	−	0.389397i	$-0.872683\pi$
$587$		−	268348.i		−	0.778794i	0.921070	−	0.389397i	$-0.127317\pi$
$588$	−59608.8	−	88455.6i	−0.172407	−	0.255841i
$589$		−	334841.i		−	0.965180i
$590$	−43860.4			−0.125999
$591$	−84472.9	−	125352.i	−0.241848	−	0.358887i
$592$		−	106985.i		−	0.305266i
$593$		−	15008.5i		−	0.0426804i	−0.999772	−	0.0213402i	$-0.993207\pi$
$593$		−	15008.5i		−	0.0426804i	0.999772	−	0.0213402i	$-0.00679332\pi$
$594$	47584.2	+	225293.i	0.134862	+	0.638520i
$595$	−76647.5	−	12020.2i	−0.216503	−	0.0339530i
$596$		−	27443.4i		−	0.0772585i
$597$	−135009.	+	90980.6i	−0.378804	+	0.255270i
$598$		−	121417.i		−	0.339530i
$599$		−	518440.i		−	1.44492i	−0.691411	−	0.722462i	$-0.743010\pi$
$599$		−	518440.i		−	1.44492i	0.691411	−	0.722462i	$-0.256990\pi$
$600$	−246005.	+	165779.i	−0.683348	+	0.460498i
$601$			72059.1i			0.199499i	0.995013	+	0.0997493i	$0.0318041\pi$
$601$			72059.1i			0.199499i	−0.995013	+	0.0997493i	$0.968196\pi$
$602$	162125.			0.447360
$603$	−49770.3	+	122882.i	−0.136879	+	0.337951i
$604$	−57703.9			−0.158173
$605$	−55952.1			−0.152864
$606$	25194.0	+	37386.2i	0.0686044	+	0.101804i
$607$			461533.i			1.25264i	0.779567	+	0.626318i	$0.215439\pi$
$607$			461533.i			1.25264i	−0.779567	+	0.626318i	$0.784561\pi$
$608$		−	550751.i		−	1.48987i
$609$	200104.	−	134847.i	0.539537	−	0.363585i
$610$	−32252.5			−0.0866770
$611$			272531.i			0.730018i
$612$	246265.	+	147749.i	0.657507	+	0.394476i
$613$	−331132.			−0.881211			−0.440606	−	0.897701i	$-0.645236\pi$
$613$	−331132.			−0.881211			−0.440606	+	0.897701i	$0.645236\pi$
$614$		−	152003.i		−	0.403196i
$615$	30248.1	+	44886.3i	0.0799739	+	0.118676i
$616$	518108.			1.36540
$617$	648965.			1.70471			0.852355	−	0.522963i	$-0.175173\pi$
$617$	648965.			1.70471			0.852355	+	0.522963i	$0.175173\pi$
$618$	109727.	−	73943.2i	0.287300	−	0.193607i
$619$		−	336455.i		−	0.878104i	−0.898462	−	0.439052i	$-0.855314\pi$
$619$		−	336455.i		−	0.878104i	0.898462	−	0.439052i	$-0.144686\pi$
$620$			36203.3i			0.0941813i
$621$	547185.	−	115571.i	1.41890	−	0.299687i
$622$		−	187824.i		−	0.485478i
$623$	−63563.7			−0.163770
$624$	−37416.0	−	55522.9i	−0.0960922	−	0.142595i
$625$	350950.			0.898432
$626$	169304.			0.432034
$627$	431708.	+	640627.i	1.09813	+	1.62956i
$628$	51878.5			0.131543
$629$	−336397.	−	52755.3i	−0.850259	−	0.133341i
$630$	15769.4	−	38934.4i	0.0397315	−	0.0980963i
$631$	−206498.			−0.518630			−0.259315	−	0.965793i	$-0.583497\pi$
$631$	−206498.			−0.518630			−0.259315	+	0.965793i	$0.583497\pi$
$632$	141544.			0.354371
$633$	−263289.	+	177426.i	−0.657090	+	0.442802i
$634$			150751.i			0.375043i
$635$	127164.			0.315368
$636$	23553.6	−	15872.4i	0.0582294	−	0.0392399i
$637$	79148.1			0.195057
$638$			145944.i			0.358545i
$639$	−37075.1	+	91537.7i	−0.0907989	+	0.224181i
$640$			72532.0i			0.177080i
$641$	603583.			1.46900			0.734499	−	0.678610i	$-0.237417\pi$
$641$	603583.			1.46900			0.734499	+	0.678610i	$0.237417\pi$
$642$	−101027.	+	68080.3i	−0.245113	+	0.165178i
$643$			635208.i			1.53636i	0.640232	+	0.768181i	$0.278838\pi$
$643$			635208.i			1.53636i	−0.640232	+	0.768181i	$0.721162\pi$
$644$		−	546123.i		−	1.31680i
$645$	−33654.8	−	49941.5i	−0.0808960	−	0.120044i
$646$	−289537.	−	45406.4i	−0.693807	−	0.108806i
$647$		−	394267.i		−	0.941850i	−0.882173	−	0.470925i	$-0.843920\pi$
$647$		−	394267.i		−	0.941850i	0.882173	−	0.470925i	$-0.156080\pi$
$648$	−249327.	+	257299.i	−0.593770	+	0.612758i
$649$			802422.i			1.90508i
$650$		−	95530.7i		−	0.226108i
$651$	186152.	+	276237.i	0.439244	+	0.651809i
$652$			299311.i			0.704090i
$653$	328888.			0.771297			0.385649	−	0.922646i	$-0.373978\pi$
$653$	328888.			0.771297			0.385649	+	0.922646i	$0.373978\pi$
$654$	192138.	+	285120.i	0.449218	+	0.666610i
$655$	−1005.62			−0.00234398
$656$	−118035.			−0.274286
$657$	561873.	+	227573.i	1.30169	+	0.527217i
$658$		−	372872.i		−	0.861207i
$659$		−	542143.i		−	1.24837i	−0.781277	−	0.624185i	$-0.785431\pi$
$659$		−	542143.i		−	1.24837i	0.781277	−	0.624185i	$-0.214569\pi$
$660$	−46676.6	−	69265.1i	−0.107155	−	0.159011i
$661$	−425926.			−0.974835			−0.487418	−	0.873169i	$-0.662061\pi$
$661$	−425926.			−0.974835			−0.487418	+	0.873169i	$0.662061\pi$
$662$		−	303033.i		−	0.691472i
$663$	−193034.	+	90270.2i	−0.439143	+	0.205361i
$664$	235599.			0.534364
$665$		−	140929.i		−	0.318681i
$666$	69210.2	−	170879.i	0.156035	−	0.385247i
$667$	354464.			0.796747
$668$	−669333.			−1.49999
$669$	−260874.	−	387120.i	−0.582879	−	0.864955i
$670$			14628.4i			0.0325872i
$671$			590057.i			1.31054i
$672$	306185.	+	454359.i	0.678025	+	1.00614i
$673$			448839.i			0.990970i	0.868617	+	0.495485i	$0.165010\pi$
$673$			448839.i			0.990970i	−0.868617	+	0.495485i	$0.834990\pi$
$674$	−910.215			−0.00200366
$675$	430523.	−	90931.1i	0.944907	−	0.199574i
$676$	−268044.			−0.586559
$677$	−464891.			−1.01432			−0.507159	−	0.861853i	$-0.669304\pi$
$677$	−464891.			−1.01432			−0.507159	+	0.861853i	$0.669304\pi$
$678$	326507.	−	220028.i	0.710285	−	0.478650i
$679$	246913.			0.535555
$680$	72132.2	+	11312.1i	0.155995	+	0.0244639i
$681$	−425807.	−	631869.i	−0.918159	−	1.36249i
$682$	−201471.			−0.433155
$683$	35930.1			0.0770223			0.0385112	−	0.999258i	$-0.487738\pi$
$683$	35930.1			0.0770223			0.0385112	+	0.999258i	$0.487738\pi$
$684$	−195834.	+	483510.i	−0.418577	+	1.03346i
$685$		−	93231.4i		−	0.198692i
$686$	160849.			0.341798
$687$	353176.	+	524090.i	0.748302	+	1.11043i
$688$	131328.			0.277448
$689$			21075.2i			0.0443949i
$690$	51172.3	−	34484.2i	0.107482	−	0.0724306i
$691$		−	452399.i		−	0.947469i	−0.880668	−	0.473735i	$-0.842906\pi$
$691$		−	452399.i		−	0.947469i	0.880668	−	0.473735i	$-0.157094\pi$
$692$	268131.			0.559931
$693$	−712302.	−	288500.i	−1.48319	−	0.600731i
$694$		−	141697.i		−	0.294199i
$695$			102991.i			0.213222i
$696$	−188316.	+	126903.i	−0.388748	+	0.261971i
$697$	−58204.3	+	371144.i	−0.119809	+	0.763970i
$698$			53381.8i			0.109568i
$699$	182046.	+	270144.i	0.372586	+	0.552893i
$700$		−	429688.i		−	0.876914i
$701$			363829.i			0.740392i	0.928954	+	0.370196i	$0.120709\pi$
$701$			363829.i			0.740392i	−0.928954	+	0.370196i	$0.879291\pi$
$702$	−23843.1	−	112888.i	−0.0483826	−	0.229072i
$703$		−	618521.i		−	1.25154i
$704$	−93834.4			−0.189329
$705$	−114860.	+	77402.6i	−0.231096	+	0.155732i
$706$	−284297.			−0.570378
$707$	−150465.			−0.301022
$708$	−449353.	+	302812.i	−0.896441	+	0.604097i
$709$		−	169327.i		−	0.336847i	−0.985715	−	0.168424i	$-0.946132\pi$
$709$		−	169327.i		−	0.336847i	0.985715	−	0.168424i	$-0.0538677\pi$
$710$			10897.1i			0.0216168i
$711$	−194597.	−	78816.8i	−0.384944	−	0.155912i
$712$	59819.2			0.118000
$713$			489327.i			0.962543i
$714$	264105.	−	123506.i	0.518061	−	0.242265i
$715$	61976.8			0.121232
$716$		−	176283.i		−	0.343863i
$717$	−248785.	+	167652.i	−0.483934	+	0.326115i
$718$	−359011.			−0.696401
$719$	−72185.0			−0.139633			−0.0698167	−	0.997560i	$-0.522241\pi$
$719$	−72185.0			−0.139633			−0.0698167	+	0.997560i	$0.522241\pi$
$720$	12773.9	−	31538.6i	0.0246410	−	0.0608383i
$721$			441609.i			0.849508i
$722$		−	280608.i		−	0.538302i
$723$	29635.8	−	19971.1i	0.0566944	−	0.0382054i
$724$		−	726646.i		−	1.38626i
$725$	278891.			0.530589
$726$	174368.	−	117504.i	0.330821	−	0.222935i
$727$	659191.			1.24722			0.623609	−	0.781736i	$-0.285666\pi$
$727$	659191.			1.24722			0.623609	+	0.781736i	$0.285666\pi$
$728$	−259609.			−0.489844
$729$	486051.	−	214905.i	0.914590	−	0.404382i
$730$	66887.8			0.125517
$731$	64759.4	−	412943.i	0.121190	−	0.772779i
$732$	−330430.	+	222671.i	−0.616676	+	0.415568i
$733$	591647.			1.10117			0.550586	−	0.834779i	$-0.314404\pi$
$733$	591647.			1.10117			0.550586	+	0.834779i	$0.314404\pi$
$734$	−80722.8			−0.149832
$735$	22479.2	+	33357.6i	0.0416107	+	0.0617476i
$736$			804851.i			1.48580i
$737$	267625.			0.492711
$738$	−188529.	−	76358.9i	−0.346150	−	0.140200i
$739$	899639.			1.64733			0.823663	−	0.567080i	$-0.191927\pi$
$739$	899639.			1.64733			0.823663	+	0.567080i	$0.191927\pi$
$740$			66874.9i			0.122124i
$741$	−216317.	−	321000.i	−0.393962	−	0.584614i
$742$		−	28834.7i		−	0.0523730i
$743$	−475668.			−0.861641			−0.430821	−	0.902438i	$-0.641776\pi$
$743$	−475668.			−0.861641			−0.430821	+	0.902438i	$0.641776\pi$
$744$	−175186.	−	259964.i	−0.316485	−	0.469643i
$745$			10349.2i			0.0186464i
$746$			122780.i			0.220623i
$747$	−323905.	−	131190.i	−0.580465	−	0.235103i
$748$	89816.5	−	572721.i	0.160529	−	1.02362i
$749$		−	406595.i		−	0.724766i
$750$	81952.1	−	55226.2i	0.145693	−	0.0981800i
$751$		−	706267.i		−	1.25224i	−0.779725	−	0.626122i	$-0.784641\pi$
$751$		−	706267.i		−	1.25224i	0.779725	−	0.626122i	$-0.215359\pi$
$752$		−	302042.i		−	0.534111i
$753$	633101.	−	426637.i	1.11656	−	0.752434i
$754$		−	73128.2i		−	0.128630i
$755$	21760.8			0.0381752
$756$	−107244.	−	507759.i	−0.187642	−	0.888410i
$757$	−607747.			−1.06055			−0.530275	−	0.847826i	$-0.677911\pi$
$757$	−607747.			−1.06055			−0.530275	+	0.847826i	$0.677911\pi$
$758$	92070.1			0.160243
$759$	−630886.	−	936193.i	−1.09513	−	1.62511i
$760$			132627.i			0.229617i
$761$			634919.i			1.09635i	0.836364	+	0.548174i	$0.184677\pi$
$761$			634919.i			1.09635i	−0.836364	+	0.548174i	$0.815323\pi$
$762$	−396290.	+	267054.i	−0.682501	+	0.459927i
$763$	−1.14750e6			−1.97108
$764$			706284.i			1.21002i
$765$	−92869.4	−	55717.7i	−0.158690	−	0.0952073i
$766$	−52265.6			−0.0890755
$767$		−	402071.i		−	0.683459i
$768$	−198504.	−	294567.i	−0.336548	−	0.499415i
$769$	178949.			0.302605			0.151302	−	0.988488i	$-0.451653\pi$
$769$	178949.			0.302605			0.151302	+	0.988488i	$0.451653\pi$
$770$	−84795.5			−0.143018
$771$	−487159.	+	328289.i	−0.819524	+	0.552264i
$772$			624738.i			1.04825i
$773$			781464.i			1.30783i	0.756570	+	0.653913i	$0.226874\pi$
$773$			781464.i			1.30783i	−0.756570	+	0.653913i	$0.773126\pi$
$774$	209761.	+	84958.7i	0.350141	+	0.141816i
$775$			385000.i			0.640999i
$776$	−232367.			−0.385879
$777$	343861.	+	510268.i	0.569562	+	0.845193i
$778$	−26540.6			−0.0438482
$779$	−682407.			−1.12452
$780$	23388.4	+	34706.8i	0.0384424	+	0.0570460i
$781$	199361.			0.326842
$782$	423120.	+	66355.5i	0.691911	+	0.108508i
$783$	329563.	−	69607.3i	0.537545	−	0.113535i
$784$	−87718.7			−0.142712
$785$	−19564.0			−0.0317481
$786$	3133.90	−	2111.88i	0.00507271	−	0.00341842i
$787$		−	669529.i		−	1.08099i	−0.841349	−	0.540493i	$-0.818238\pi$
$787$		−	669529.i		−	1.08099i	0.841349	−	0.540493i	$-0.181762\pi$
$788$	−206050.			−0.331833
$789$	−377740.	+	254553.i	−0.606791	+	0.408907i
$790$	−23165.7			−0.0371186
$791$			1.31407e6i			2.10022i
$792$	670340.	+	271505.i	1.06867	+	0.432840i
$793$		−	295661.i		−	0.470162i
$794$	−379350.			−0.601727
$795$	−8882.32	+	5985.65i	−0.0140537	+	0.00947059i
$796$			221923.i			0.350249i
$797$		−	295124.i		−	0.464610i	−0.972643	−	0.232305i	$-0.925373\pi$
$797$		−	295124.i		−	0.464610i	0.972643	−	0.232305i	$-0.0746267\pi$
$798$	295961.	+	439186.i	0.464760	+	0.689673i
$799$	−949727.	−	148940.i	−1.48767	−	0.233302i
$800$			633254.i			0.989459i
$801$	−82240.2	−	33309.4i	−0.128180	−	0.0519160i
$802$			269960.i			0.419711i
$803$		−	1.22371e6i		−	1.89778i
$804$	100994.	+	149869.i	0.156237	+	0.231846i
$805$			205949.i			0.317811i
$806$	100951.			0.155397
$807$	−375923.	−	557845.i	−0.577233	−	0.856576i
$808$	141602.			0.216893
$809$	−473318.			−0.723196			−0.361598	−	0.932334i	$-0.617769\pi$
$809$	−473318.			−0.723196			−0.361598	+	0.932334i	$0.617769\pi$
$810$	40805.7	−	42110.6i	0.0621944	−	0.0641832i
$811$			88499.7i			0.134555i	0.997734	+	0.0672775i	$0.0214313\pi$
$811$			88499.7i			0.134555i	−0.997734	+	0.0672775i	$0.978569\pi$
$812$		−	328924.i		−	0.498865i
$813$	208747.	+	309767.i	0.315820	+	0.468657i
$814$	−372158.			−0.561666
$815$		−	112874.i		−	0.169933i
$816$	213937.	−	100045.i	0.321296	−	0.150250i
$817$	759262.			1.13749
$818$			392763.i			0.586981i
$819$	356914.	+	144559.i	0.532104	+	0.215516i
$820$	73782.4			0.109730
$821$	610628.			0.905921			0.452960	−	0.891531i	$-0.350368\pi$
$821$	610628.			0.905921			0.452960	+	0.891531i	$0.350368\pi$
$822$	195793.	+	290544.i	0.289770	+	0.429999i
$823$		−	9305.04i		−	0.0137378i	−0.999976	−	0.00686892i	$-0.997814\pi$
$823$		−	9305.04i		−	0.0137378i	0.999976	−	0.00686892i	$-0.00218646\pi$
$824$		−	415594.i		−	0.612089i
$825$	−496378.	−	736593.i	−0.729298	−	1.08223i
$826$			550106.i			0.806281i
$827$	402805.			0.588957			0.294478	−	0.955658i	$-0.404854\pi$
$827$	402805.			0.588957			0.294478	+	0.955658i	$0.404854\pi$
$828$	286186.	−	706587.i	0.417434	−	1.03064i
$829$	474627.			0.690627			0.345313	−	0.938487i	$-0.387773\pi$
$829$	474627.			0.690627			0.345313	+	0.938487i	$0.387773\pi$
$830$	−38559.0			−0.0559719
$831$	176675.	−	119059.i	0.255843	−	0.172408i
$832$	47017.8			0.0679228
$833$	−43255.0	+	275818.i	−0.0623371	+	0.397496i
$834$	−216289.	−	320959.i	−0.310959	−	0.461443i
$835$	252413.			0.362025
$836$	1.05304e6			1.50672
$837$	96090.7	+	454952.i	0.137161	+	0.649403i
$838$		−	453619.i		−	0.645956i
$839$	533860.			0.758408			0.379204	−	0.925313i	$-0.376198\pi$
$839$	533860.			0.758408			0.379204	+	0.925313i	$0.376198\pi$
$840$	−73732.7	−	109414.i	−0.104496	−	0.155066i
$841$	−493791.			−0.698155
$842$		−	373575.i		−	0.526932i
$843$	−632385.	+	426154.i	−0.889870	+	0.599669i
$844$			432784.i			0.607556i
$845$	101082.			0.141567
$846$	195396.	−	482430.i	0.273008	−	0.674052i
$847$			701763.i			0.978191i
$848$		−	23357.3i		−	0.0324812i
$849$	826965.	−	557278.i	1.14729	−	0.773137i
$850$	332909.	+	52208.2i	0.460774	+	0.0722605i
$851$			903888.i			1.24812i
$852$	75233.2	+	111641.i	0.103641	+	0.153796i
$853$		−	263569.i		−	0.362240i	−0.983461	−	0.181120i	$-0.942028\pi$
$853$		−	263569.i		−	0.362240i	0.983461	−	0.181120i	$-0.0579722\pi$
$854$			404518.i			0.554654i
$855$	73851.2	−	182337.i	0.101024	−	0.249427i
$856$			382642.i			0.522210i
$857$	444768.			0.605580			0.302790	−	0.953057i	$-0.402082\pi$
$857$	444768.			0.605580			0.302790	+	0.953057i	$0.402082\pi$
$858$	−193143.	+	130156.i	−0.262364	+	0.176803i
$859$	83933.7			0.113750			0.0568748	−	0.998381i	$-0.481886\pi$
$859$	83933.7			0.113750			0.0568748	+	0.998381i	$0.481886\pi$
$860$	−82092.0			−0.110995
$861$	562973.	−	379378.i	0.759418	−	0.511760i
$862$		−	37872.7i		−	0.0509696i
$863$			632283.i			0.848965i	0.905436	+	0.424482i	$0.139544\pi$
$863$			632283.i			0.848965i	−0.905436	+	0.424482i	$0.860456\pi$
$864$	158051.	+	748311.i	0.211724	+	1.00243i
$865$	−101115.			−0.135140
$866$		−	452353.i		−	0.603173i
$867$	−209083.	−	722025.i	−0.278150	−	0.960538i
$868$	454069.			0.602674
$869$			423814.i			0.561224i
$870$	30820.5	−	20769.4i	0.0407194	−	0.0274401i
$871$	−134099.			−0.176763
$872$	1.07990e6			1.42020
$873$	319462.	+	129390.i	0.419170	+	0.169774i
$874$			777975.i			1.01846i
$875$			329826.i			0.430793i
$876$	685271.	−	461793.i	0.893005	−	0.601782i
$877$			124932.i			0.162433i	0.996696	+	0.0812166i	$0.0258805\pi$
$877$			124932.i			0.162433i	−0.996696	+	0.0812166i	$0.974119\pi$
$878$	−516497.			−0.670006
$879$	−483393.	+	325751.i	−0.625637	+	0.421607i
$880$	−68688.0			−0.0886984
$881$	−95751.6			−0.123366			−0.0616828	−	0.998096i	$-0.519647\pi$
$881$	−95751.6			−0.123366			−0.0616828	+	0.998096i	$0.519647\pi$
$882$	−140107.	−	56746.8i	−0.180103	−	0.0729464i
$883$	−1.06600e6			−1.36722			−0.683608	−	0.729849i	$-0.739590\pi$
$883$	−1.06600e6			−1.36722			−0.683608	+	0.729849i	$0.739590\pi$
$884$	−45004.5	+	286974.i	−0.0575906	+	0.367230i
$885$	169456.	−	114194.i	0.216357	−	0.145800i
$886$	272540.			0.347187
$887$	376657.			0.478740			0.239370	−	0.970928i	$-0.423059\pi$
$887$	376657.			0.478740			0.239370	+	0.970928i	$0.423059\pi$
$888$	−323604.	−	480208.i	−0.410382	−	0.608980i
$889$		−	1.59492e6i		−	2.01806i
$890$	−9790.23			−0.0123598
$891$	−770410.	−	746537.i	−0.970435	−	0.940364i
$892$	−636334.			−0.799752
$893$		−	1.74623e6i		−	2.18977i
$894$	−21734.1	−	32252.0i	−0.0271936	−	0.0403536i
$895$			66478.4i			0.0829916i
$896$	909711.			1.13315
$897$	316119.	+	469100.i	0.392885	+	0.583016i
$898$		−	234109.i		−	0.290312i
$899$			294716.i			0.364656i
$900$	225170.	−	555940.i	0.277987	−	0.686346i
$901$	−73443.7	−	11517.8i	−0.0904701	−	0.0141879i
$902$			410598.i			0.504665i
$903$	−626376.	+	422105.i	−0.768174	+	0.517660i
$904$		−	1.23665e6i		−	1.51325i
$905$			274026.i			0.334576i
$906$	−67814.7	+	45699.2i	−0.0826166	+	0.0556740i
$907$		−	295686.i		−	0.359432i	−0.983719	−	0.179716i	$-0.942482\pi$
$907$		−	295686.i		−	0.359432i	0.983719	−	0.179716i	$-0.0575179\pi$
$908$	−1.03864e6			−1.25978
$909$	−194676.	−	78848.6i	−0.235605	−	0.0954259i
$910$	42488.6			0.0513086
$911$	−1.13846e6			−1.37177			−0.685883	−	0.727712i	$-0.740584\pi$
$911$	−1.13846e6			−1.37177			−0.685883	+	0.727712i	$0.740584\pi$
$912$	239741.	+	355760.i	0.288239	+	0.427728i
$913$			705434.i			0.846281i
$914$			333576.i			0.399303i
$915$	124609.	−	83971.8i	0.148835	−	0.100298i
$916$	861479.			1.02672
$917$			12612.8i			0.0149993i
$918$	406427.	−	21395.4i	0.482277	−	0.0253884i
$919$	−949089.			−1.12377			−0.561883	−	0.827217i	$-0.689923\pi$
$919$	−949089.			−1.12377			−0.561883	+	0.827217i	$0.689923\pi$
$920$		−	193817.i		−	0.228990i
$921$	395752.	+	587270.i	0.466556	+	0.692338i
$922$	420204.			0.494309
$923$	−99894.0			−0.117256
$924$	−868737.	+	585428.i	−1.01752	+	0.685693i
$925$			711175.i			0.831176i
$926$		−	365252.i		−	0.425962i
$927$	−231417.	+	571364.i	−0.269300	+	0.664896i
$928$			484752.i			0.562890i
$929$	503990.			0.583970			0.291985	−	0.956423i	$-0.405684\pi$
$929$	503990.			0.583970			0.291985	+	0.956423i	$0.405684\pi$
$930$	28671.6	+	42546.7i	0.0331501	+	0.0491927i
$931$	−507137.			−0.585094
$932$	444053.			0.511214
$933$	489013.	+	725664.i	0.561769	+	0.833628i
$934$	−635440.			−0.728419
$935$	−33870.8	+	215979.i	−0.0387438	+	0.247052i
$936$	−335889.	−	136043.i	−0.383392	−	0.155284i
$937$	−333432.			−0.379776			−0.189888	−	0.981806i	$-0.560812\pi$
$937$	−333432.			−0.379776			−0.189888	+	0.981806i	$0.560812\pi$
$938$	183472.			0.208528
$939$	−654111.	+	440795.i	−0.741858	+	0.499926i
$940$			188803.i			0.213675i
$941$	−1.38397e6			−1.56295			−0.781477	−	0.623934i	$-0.785533\pi$
$941$	−1.38397e6			−1.56295			−0.781477	+	0.623934i	$0.785533\pi$
$942$	60968.6	−	41085.7i	0.0687075	−	0.0463009i
$943$	997250.			1.12145
$944$			445610.i			0.500047i
$945$	40442.9	+	191481.i	0.0452876	+	0.214419i
$946$		−	456840.i		−	0.510484i
$947$	−1.38608e6			−1.54556			−0.772782	−	0.634672i	$-0.781135\pi$
$947$	−1.38608e6			−1.54556			−0.772782	+	0.634672i	$0.781135\pi$
$948$	−237334.	+	159936.i	−0.264085	+	0.177963i
$949$			613165.i			0.680839i
$950$			612107.i			0.678235i
$951$	−392491.	−	582432.i	−0.433979	−	0.643997i
$952$	141878.	−	904697.i	0.156546	−	0.998227i
$953$		−	547126.i		−	0.602423i	−0.953557	−	0.301211i	$-0.902609\pi$
$953$		−	547126.i		−	0.602423i	0.953557	−	0.301211i	$-0.0973910\pi$
$954$	15110.3	−	37307.0i	0.0166026	−	0.0409915i
$955$		−	266348.i		−	0.292040i
$956$			408944.i			0.447454i
$957$	−379975.	−	563858.i	−0.414888	−	0.615667i
$958$			367334.i			0.400249i
$959$	−1.16933e6			−1.27145
$960$	13353.7	+	19816.0i	0.0144897	+	0.0215018i
$961$	516675.			0.559462
$962$	186478.			0.201501
$963$	213068.	−	526061.i	0.229756	−	0.567262i
$964$		−	48714.2i		−	0.0524206i
$965$		−	235596.i		−	0.252995i
$966$	−432508.	−	641814.i	−0.463490	−	0.687788i
$967$	508950.			0.544279			0.272140	−	0.962258i	$-0.412269\pi$
$967$	508950.			0.544279			0.272140	+	0.962258i	$0.412269\pi$
$968$		−	660422.i		−	0.704808i
$969$	1.23685e6	−	578401.i	1.31726	−	0.616001i
$970$	38030.1			0.0404188
$971$		−	1.17767e6i		−	1.24907i	−0.780997	−	0.624534i	$-0.785289\pi$
$971$		−	1.17767e6i		−	1.24907i	0.780997	−	0.624534i	$-0.214711\pi$
$972$	127327.	−	713149.i	0.134768	−	0.754828i
$973$	1.29174e6			1.36442
$974$	413855.			0.436245
$975$	248721.	+	369086.i	0.261640	+	0.388256i
$976$			327677.i			0.343990i
$977$		−	1.47205e6i		−	1.54217i	−0.636731	−	0.771086i	$-0.719714\pi$
$977$		−	1.47205e6i		−	1.54217i	0.636731	−	0.771086i	$-0.280286\pi$
$978$	237043.	+	351756.i	0.247827	+	0.367759i
$979$			179111.i			0.186878i
$980$	54832.0			0.0570929
$981$	−1.48466e6	−	601326.i	−1.54273	−	0.624844i
$982$	361266.			0.374632
$983$	−80357.7			−0.0831611			−0.0415806	−	0.999135i	$-0.513239\pi$
$983$	−80357.7			−0.0831611			−0.0415806	+	0.999135i	$0.513239\pi$
$984$	−529808.	+	357029.i	−0.547178	+	0.368734i
$985$	77703.6			0.0800882
$986$	254840.	+	39965.1i	0.262128	+	0.0411081i
$987$	970799.	+	1.44060e6i	0.996541	+	1.47880i
$988$	−527649.			−0.540544
$989$	−1.10956e6			−1.13438
$990$	−109710.	−	44435.5i	−0.111938	−	0.0453377i
$991$		−	1.79395e6i		−	1.82669i	−0.407192	−	0.913343i	$-0.633492\pi$
$991$		−	1.79395e6i		−	1.82669i	0.407192	−	0.913343i	$-0.366508\pi$
$992$	−669185.			−0.680022
$993$	788970.	+	1.17078e6i	0.800133	+	1.18734i
$994$	136673.			0.138328
$995$		−	83689.8i		−	0.0845330i
$996$	−395040.	+	266211.i	−0.398220	+	0.268354i
$997$			562411.i			0.565800i	0.959149	+	0.282900i	$0.0912965\pi$
$997$			562411.i			0.565800i	−0.959149	+	0.282900i	$0.908703\pi$
$998$	15079.6			0.0151401
$999$	177499.	+	840390.i	0.177855	+	0.842073i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	51.5.c.c.50.10	yes	20
3.2	odd	2	inner	51.5.c.c.50.11	yes	20
17.16	even	2	inner	51.5.c.c.50.9	✓	20
51.50	odd	2	inner	51.5.c.c.50.12	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
51.5.c.c.50.9	✓	20	17.16	even	2	inner
51.5.c.c.50.10	yes	20	1.1	even	1	trivial
51.5.c.c.50.11	yes	20	3.2	odd	2	inner
51.5.c.c.50.12	yes	20	51.50	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 51 = 3 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	51.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.93178i q^{2} +(5.02953 + 7.46350i) q^{3} +12.2682 q^{4} -4.62649 q^{5} +(14.4178 - 9.71595i) q^{6} +58.0263i q^{7} -54.6080i q^{8} +(-30.4076 + 75.0758i) q^{9} +8.93735i q^{10} +163.508 q^{11} +(61.7034 + 91.5639i) q^{12} -81.9293 q^{13} +112.094 q^{14} +(-23.2691 - 34.5298i) q^{15} +90.8010 q^{16} +(44.7750 - 285.510i) q^{17} +(145.030 + 58.7408i) q^{18} +524.957 q^{19} -56.7588 q^{20} +(-433.079 + 291.845i) q^{21} -315.862i q^{22} -767.157 q^{23} +(407.567 - 274.653i) q^{24} -603.596 q^{25} +158.269i q^{26} +(-713.264 + 150.649i) q^{27} +711.880i q^{28} -462.049 q^{29} +(-66.7039 + 44.9507i) q^{30} -637.845i q^{31} -1049.14i q^{32} +(822.369 + 1220.34i) q^{33} +(-551.543 - 86.4954i) q^{34} -268.458i q^{35} +(-373.047 + 921.047i) q^{36} -1178.23i q^{37} -1014.10i q^{38} +(-412.066 - 611.479i) q^{39} +252.643i q^{40} -1299.93 q^{41} +(563.781 + 836.614i) q^{42} +1446.33 q^{43} +2005.95 q^{44} +(140.680 - 347.337i) q^{45} +1481.98i q^{46} -3326.42i q^{47} +(456.687 + 677.693i) q^{48} -966.054 q^{49} +1166.01i q^{50} +(2356.10 - 1101.81i) q^{51} -1005.13 q^{52} -257.237i q^{53} +(291.021 + 1377.87i) q^{54} -756.468 q^{55} +3168.70 q^{56} +(2640.29 + 3918.02i) q^{57} +892.577i q^{58} +4907.54i q^{59} +(-285.470 - 423.619i) q^{60} +3608.73i q^{61} -1232.18 q^{62} +(-4356.37 - 1764.44i) q^{63} -573.882 q^{64} +379.045 q^{65} +(2357.43 - 1588.64i) q^{66} +1636.77 q^{67} +(549.309 - 3502.71i) q^{68} +(-3858.44 - 5725.67i) q^{69} -518.602 q^{70} +1219.27 q^{71} +(4099.74 + 1660.50i) q^{72} -7484.07i q^{73} -2276.08 q^{74} +(-3035.80 - 4504.93i) q^{75} +6440.29 q^{76} +9487.77i q^{77} +(-1181.24 + 796.021i) q^{78} +2592.01i q^{79} -420.090 q^{80} +(-4711.75 - 4565.75i) q^{81} +2511.18i q^{82} +4314.37i q^{83} +(-5313.12 + 3580.42i) q^{84} +(-207.151 + 1320.91i) q^{85} -2793.99i q^{86} +(-2323.89 - 3448.50i) q^{87} -8928.85i q^{88} +1095.43i q^{89} +(-670.979 - 271.763i) q^{90} -4754.05i q^{91} -9411.65 q^{92} +(4760.55 - 3208.06i) q^{93} -6425.91 q^{94} -2428.71 q^{95} +(7830.22 - 5276.66i) q^{96} -4255.19i q^{97} +1866.20i q^{98} +(-4971.89 + 12275.5i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 168 q^{4} - 400 q^{9} - 308 q^{13} - 72 q^{15} + 600 q^{16} - 124 q^{18} - 548 q^{19} + 16 q^{21} + 3200 q^{25} - 2580 q^{30} + 4088 q^{33} - 2072 q^{34} + 4588 q^{36} - 464 q^{42} + 9028 q^{43}+ \cdots - 13736 q^{94}+O(q^{100}) \) 20 * q - 168 * q^4 - 400 * q^9 - 308 * q^13 - 72 * q^15 + 600 * q^16 - 124 * q^18 - 548 * q^19 + 16 * q^21 + 3200 * q^25 - 2580 * q^30 + 4088 * q^33 - 2072 * q^34 + 4588 * q^36 - 464 * q^42 + 9028 * q^43 - 5348 * q^49 - 8608 * q^51 - 24792 * q^52 + 7492 * q^55 + 7860 * q^60 - 23776 * q^64 - 14980 * q^66 + 11408 * q^67 + 14044 * q^69 - 12264 * q^70 - 26868 * q^72 + 68976 * q^76 - 8204 * q^81 + 53720 * q^84 + 7124 * q^85 + 45876 * q^87 - 68216 * q^93 - 13736 * q^94

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