LMFDB - Embedded newform 342.4.g.h.163.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [342,4,Mod(163,342)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(342, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 4]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("342.163");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 342 = 2 \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	342.g (of order $3$, degree $2$, 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:	$20.1786532220$
Analytic rank:	$0$
Dimension:	$6$
Relative dimension:	$3$ over $\Q(\zeta_{3})$
Coefficient field:	6.0.627014547.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 26x^{4} + 169x^{2} + 147 $ x^6 + 26x^4 + 169x^2 + 147
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{3} $
Twist minimal:	no (minimal twist has level 114)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			163.3
Root			$-2.99107i$ of defining polynomial
Character	$\chi$	$=$	342.163
Dual form			342.4.g.h.235.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.00000 - 1.73205i) q^{2} +(-2.00000 - 3.46410i) q^{4} +(7.12716 - 12.3446i) q^{5} +13.2543 q^{7} -8.00000 q^{8} +O(q^{10})$	\(q+(1.00000 - 1.73205i) q^{2} +(-2.00000 - 3.46410i) q^{4} +(7.12716 - 12.3446i) q^{5} +13.2543 q^{7} -8.00000 q^{8} +(-14.2543 - 24.6892i) q^{10} +65.9138 q^{11} +(34.2629 + 59.3451i) q^{13} +(13.2543 - 22.9571i) q^{14} +(-8.00000 + 13.8564i) q^{16} +(49.8470 - 86.3375i) q^{17} +(-80.3556 + 20.0493i) q^{19} -57.0172 q^{20} +(65.9138 - 114.166i) q^{22} +(1.76943 + 3.06475i) q^{23} +(-39.0927 - 67.7105i) q^{25} +137.052 q^{26} +(-26.5086 - 45.9143i) q^{28} +(-40.9504 - 70.9282i) q^{29} -247.190 q^{31} +(16.0000 + 27.7128i) q^{32} +(-99.6940 - 172.675i) q^{34} +(94.4655 - 163.619i) q^{35} +421.065 q^{37} +(-45.6293 + 159.229i) q^{38} +(-57.0172 + 98.7568i) q^{40} +(-172.629 + 299.003i) q^{41} +(183.084 - 317.111i) q^{43} +(-131.828 - 228.332i) q^{44} +7.07773 q^{46} +(-45.4699 - 78.7562i) q^{47} -167.323 q^{49} -156.371 q^{50} +(137.052 - 237.381i) q^{52} +(-344.246 - 596.251i) q^{53} +(469.778 - 813.680i) q^{55} -106.034 q^{56} -163.802 q^{58} +(91.1444 - 157.867i) q^{59} +(0.258685 + 0.448055i) q^{61} +(-247.190 + 428.145i) q^{62} +64.0000 q^{64} +976.789 q^{65} +(79.9310 + 138.445i) q^{67} -398.776 q^{68} +(-188.931 - 327.238i) q^{70} +(-395.942 + 685.792i) q^{71} +(-161.065 + 278.972i) q^{73} +(421.065 - 729.306i) q^{74} +(230.164 + 238.262i) q^{76} +873.642 q^{77} +(-159.185 + 275.717i) q^{79} +(114.034 + 197.514i) q^{80} +(345.259 + 598.005i) q^{82} -684.160 q^{83} +(-710.535 - 1230.68i) q^{85} +(-366.168 - 634.222i) q^{86} -527.311 q^{88} +(360.203 + 623.890i) q^{89} +(454.132 + 786.579i) q^{91} +(7.07773 - 12.2590i) q^{92} -181.880 q^{94} +(-325.207 + 1134.85i) q^{95} +(-207.696 + 359.740i) q^{97} +(-167.323 + 289.812i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{2} - 12 q^{4} + 10 q^{5} + 14 q^{7} - 48 q^{8}+O(q^{10}) $ 6 * q + 6 * q^2 - 12 * q^4 + 10 * q^5 + 14 * q^7 - 48 * q^8	$ 6 q + 6 q^{2} - 12 q^{4} + 10 q^{5} + 14 q^{7} - 48 q^{8} - 20 q^{10} + 88 q^{11} + 9 q^{13} + 14 q^{14} - 48 q^{16} - 84 q^{17} + 32 q^{19} - 80 q^{20} + 88 q^{22} - 2 q^{23} + 83 q^{25} + 36 q^{26} - 28 q^{28} + 92 q^{29} - 218 q^{31} + 96 q^{32} + 168 q^{34} + 282 q^{35} + 490 q^{37} + 74 q^{38} - 80 q^{40} - 688 q^{41} + 103 q^{43} - 176 q^{44} - 8 q^{46} + 322 q^{47} - 1508 q^{49} + 332 q^{50} + 36 q^{52} - 1322 q^{53} + 248 q^{55} - 112 q^{56} + 368 q^{58} + 252 q^{59} + 435 q^{61} - 218 q^{62} + 384 q^{64} + 3164 q^{65} + 719 q^{67} + 672 q^{68} - 564 q^{70} - 62 q^{71} + 581 q^{73} + 490 q^{74} + 20 q^{76} + 408 q^{77} + 489 q^{79} + 160 q^{80} + 1376 q^{82} - 4992 q^{83} - 1632 q^{85} - 206 q^{86} - 704 q^{88} + 1584 q^{89} + 1573 q^{91} - 8 q^{92} + 1288 q^{94} - 2362 q^{95} - 974 q^{97} - 1508 q^{98}+O(q^{100}) $ 6 * q + 6 * q^2 - 12 * q^4 + 10 * q^5 + 14 * q^7 - 48 * q^8 - 20 * q^10 + 88 * q^11 + 9 * q^13 + 14 * q^14 - 48 * q^16 - 84 * q^17 + 32 * q^19 - 80 * q^20 + 88 * q^22 - 2 * q^23 + 83 * q^25 + 36 * q^26 - 28 * q^28 + 92 * q^29 - 218 * q^31 + 96 * q^32 + 168 * q^34 + 282 * q^35 + 490 * q^37 + 74 * q^38 - 80 * q^40 - 688 * q^41 + 103 * q^43 - 176 * q^44 - 8 * q^46 + 322 * q^47 - 1508 * q^49 + 332 * q^50 + 36 * q^52 - 1322 * q^53 + 248 * q^55 - 112 * q^56 + 368 * q^58 + 252 * q^59 + 435 * q^61 - 218 * q^62 + 384 * q^64 + 3164 * q^65 + 719 * q^67 + 672 * q^68 - 564 * q^70 - 62 * q^71 + 581 * q^73 + 490 * q^74 + 20 * q^76 + 408 * q^77 + 489 * q^79 + 160 * q^80 + 1376 * q^82 - 4992 * q^83 - 1632 * q^85 - 206 * q^86 - 704 * q^88 + 1584 * q^89 + 1573 * q^91 - 8 * q^92 + 1288 * q^94 - 2362 * q^95 - 974 * q^97 - 1508 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/342\mathbb{Z}\right)^\times$.

$n$	$191$	$325$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$

Coefficient data

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

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

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

$2$	1.00000	−	1.73205i	0.353553	−	0.612372i
$2$	1.00000	−	1.73205i	0.353553	−	0.612372i
$3$		0			0
$3$		0			0
$4$	−2.00000	−	3.46410i	−0.250000	−	0.433013i
$5$	7.12716	−	12.3446i	0.637472	−	1.10413i	−0.348513	−	0.937304i	$-0.613313\pi$
$5$	7.12716	−	12.3446i	0.637472	−	1.10413i	0.985986	−	0.166830i	$-0.0533532\pi$
$6$		0			0
$7$	13.2543			0.715666			0.357833	−	0.933786i	$-0.383516\pi$
$7$	13.2543			0.715666			0.357833	+	0.933786i	$0.383516\pi$
$8$	−8.00000			−0.353553
$9$		0			0
$10$	−14.2543	−	24.6892i	−0.450761	−	0.780741i
$11$	65.9138			1.80671			0.903353	−	0.428898i	$-0.141098\pi$
$11$	65.9138			1.80671			0.903353	+	0.428898i	$0.141098\pi$
$12$		0			0
$13$	34.2629	+	59.3451i	0.730987	+	1.26611i	0.956462	+	0.291857i	$0.0942732\pi$
$13$	34.2629	+	59.3451i	0.730987	+	1.26611i	−0.225475	+	0.974249i	$0.572393\pi$
$14$	13.2543	−	22.9571i	0.253026	−	0.438254i
$15$		0			0
$16$	−8.00000	+	13.8564i	−0.125000	+	0.216506i
$17$	49.8470	−	86.3375i	0.711157	−	1.23176i	−0.253266	−	0.967397i	$-0.581505\pi$
$17$	49.8470	−	86.3375i	0.711157	−	1.23176i	0.964423	−	0.264364i	$-0.0851620\pi$
$18$		0			0
$19$	−80.3556	+	20.0493i	−0.970255	+	0.242085i
$19$	−80.3556	+	20.0493i	−0.970255	+	0.242085i
$20$	−57.0172			−0.637472
$21$		0			0
$22$	65.9138	−	114.166i	0.638767	−	1.10638i
$23$	1.76943	+	3.06475i	0.0160414	+	0.0277845i	0.873935	−	0.486043i	$-0.161560\pi$
$23$	1.76943	+	3.06475i	0.0160414	+	0.0277845i	−0.857893	+	0.513828i	$0.828227\pi$
$24$		0			0
$25$	−39.0927	−	67.7105i	−0.312742	−	0.541684i
$26$	137.052			1.03377
$27$		0			0
$28$	−26.5086	−	45.9143i	−0.178916	−	0.309892i
$29$	−40.9504	−	70.9282i	−0.262217	−	0.454174i	0.704614	−	0.709591i	$-0.251120\pi$
$29$	−40.9504	−	70.9282i	−0.262217	−	0.454174i	−0.966831	+	0.255418i	$0.917787\pi$
$30$		0			0
$31$	−247.190			−1.43215			−0.716074	−	0.698025i	$-0.754063\pi$
$31$	−247.190			−1.43215			−0.716074	+	0.698025i	$0.754063\pi$
$32$	16.0000	+	27.7128i	0.0883883	+	0.153093i
$33$		0			0
$34$	−99.6940	−	172.675i	−0.502864	−	0.870986i
$35$	94.4655	−	163.619i	0.456217	−	0.790191i
$36$		0			0
$37$	421.065			1.87088			0.935441	−	0.353483i	$-0.115003\pi$
$37$	421.065			1.87088			0.935441	+	0.353483i	$0.115003\pi$
$38$	−45.6293	+	159.229i	−0.194791	+	0.679747i
$39$		0			0
$40$	−57.0172	+	98.7568i	−0.225380	+	0.390370i
$41$	−172.629	+	299.003i	−0.657565	+	1.13894i	0.323679	+	0.946167i	$0.395080\pi$
$41$	−172.629	+	299.003i	−0.657565	+	1.13894i	−0.981244	+	0.192769i	$0.938253\pi$
$42$		0			0
$43$	183.084	−	317.111i	0.649304	−	1.12463i	−0.333986	−	0.942578i	$-0.608394\pi$
$43$	183.084	−	317.111i	0.649304	−	1.12463i	0.983289	−	0.182049i	$-0.0582730\pi$
$44$	−131.828	−	228.332i	−0.451677	−	0.782327i
$45$		0			0
$46$	7.07773			0.0226860
$47$	−45.4699	−	78.7562i	−0.141116	−	0.244421i	0.786801	−	0.617207i	$-0.211736\pi$
$47$	−45.4699	−	78.7562i	−0.141116	−	0.244421i	−0.927917	+	0.372786i	$0.878403\pi$
$48$		0			0
$49$	−167.323			−0.487823
$50$	−156.371			−0.442283
$51$		0			0
$52$	137.052	−	237.381i	0.365493	−	0.633053i
$53$	−344.246	−	596.251i	−0.892185	−	1.54531i	−0.837251	−	0.546819i	$-0.815839\pi$
$53$	−344.246	−	596.251i	−0.892185	−	1.54531i	−0.0549339	−	0.998490i	$-0.517495\pi$
$54$		0			0
$55$	469.778	−	813.680i	1.15172	−	1.99485i
$56$	−106.034			−0.253026
$57$		0			0
$58$	−163.802			−0.370831
$59$	91.1444	−	157.867i	0.201118	−	0.348347i	−0.747771	−	0.663957i	$-0.768876\pi$
$59$	91.1444	−	157.867i	0.201118	−	0.348347i	0.948889	+	0.315610i	$0.102209\pi$
$60$		0			0
$61$	0.258685	+	0.448055i	0.000542971	+	0.000940453i	0.866297	−	0.499530i	$-0.166494\pi$
$61$	0.258685	+	0.448055i	0.000542971	+	0.000940453i	−0.865754	+	0.500470i	$0.833161\pi$
$62$	−247.190	+	428.145i	−0.506341	+	0.877008i
$63$		0			0
$64$	64.0000			0.125000
$65$	976.789			1.86393
$66$		0			0
$67$	79.9310	+	138.445i	0.145748	+	0.252443i	0.929652	−	0.368439i	$-0.120108\pi$
$67$	79.9310	+	138.445i	0.145748	+	0.252443i	−0.783904	+	0.620883i	$0.786774\pi$
$68$	−398.776			−0.711157
$69$		0			0
$70$	−188.931	−	327.238i	−0.322594	−	0.558749i
$71$	−395.942	+	685.792i	−0.661826	+	1.14632i	0.318309	+	0.947987i	$0.396885\pi$
$71$	−395.942	+	685.792i	−0.661826	+	1.14632i	−0.980135	+	0.198330i	$0.936448\pi$
$72$		0			0
$73$	−161.065	+	278.972i	−0.258235	+	0.447277i	−0.965769	−	0.259403i	$-0.916474\pi$
$73$	−161.065	+	278.972i	−0.258235	+	0.447277i	0.707534	+	0.706679i	$0.249808\pi$
$74$	421.065	−	729.306i	0.661457	−	1.14568i
$75$		0			0
$76$	230.164	+	238.262i	0.347390	+	0.359611i
$77$	873.642			1.29300
$78$		0			0
$79$	−159.185	+	275.717i	−0.226706	+	0.392666i	−0.956830	−	0.290649i	$-0.906129\pi$
$79$	−159.185	+	275.717i	−0.226706	+	0.392666i	0.730124	+	0.683315i	$0.239462\pi$
$80$	114.034	+	197.514i	0.159368	+	0.276034i
$81$		0			0
$82$	345.259	+	598.005i	0.464969	+	0.805349i
$83$	−684.160			−0.904774			−0.452387	−	0.891822i	$-0.649427\pi$
$83$	−684.160			−0.904774			−0.452387	+	0.891822i	$0.649427\pi$
$84$		0			0
$85$	−710.535	−	1230.68i	−0.906686	−	1.57043i
$86$	−366.168	−	634.222i	−0.459127	−	0.795232i
$87$		0			0
$88$	−527.311			−0.638767
$89$	360.203	+	623.890i	0.429005	+	0.743058i	0.996785	−	0.0801222i	$-0.0255310\pi$
$89$	360.203	+	623.890i	0.429005	+	0.743058i	−0.567780	+	0.823180i	$0.692198\pi$
$90$		0			0
$91$	454.132	+	786.579i	0.523142	+	0.906109i
$92$	7.07773	−	12.2590i	0.00802070	−	0.0138923i
$93$		0			0
$94$	−181.880			−0.199569
$95$	−325.207	+	1134.85i	−0.351216	+	1.22561i
$96$		0			0
$97$	−207.696	+	359.740i	−0.217406	+	0.376558i	−0.954014	−	0.299762i	$-0.903093\pi$
$97$	−207.696	+	359.740i	−0.217406	+	0.376558i	0.736608	+	0.676319i	$0.236426\pi$
$98$	−167.323	+	289.812i	−0.172471	+	0.298729i
$99$		0			0
$100$	−156.371	+	270.842i	−0.156371	+	0.270842i
$101$	168.670	+	292.146i	0.166172	+	0.287818i	0.937071	−	0.349140i	$-0.113526\pi$
$101$	168.670	+	292.146i	0.166172	+	0.287818i	−0.770899	+	0.636957i	$0.780193\pi$
$102$		0			0
$103$	−1467.05			−1.40342			−0.701711	−	0.712462i	$-0.747580\pi$
$103$	−1467.05			−1.40342			−0.701711	+	0.712462i	$0.747580\pi$
$104$	−274.103	−	474.761i	−0.258443	−	0.447636i
$105$		0			0
$106$	−1376.98			−1.26174
$107$	−382.014			−0.345146			−0.172573	−	0.984997i	$-0.555208\pi$
$107$	−382.014			−0.345146			−0.172573	+	0.984997i	$0.555208\pi$
$108$		0			0
$109$	66.4287	−	115.058i	0.0583736	−	0.101106i	−0.835362	−	0.549700i	$-0.814742\pi$
$109$	66.4287	−	115.058i	0.0583736	−	0.101106i	0.893735	+	0.448594i	$0.148075\pi$
$110$	−939.556	−	1627.36i	−0.814393	−	1.41057i
$111$		0			0
$112$	−106.034	+	183.657i	−0.0894582	+	0.154946i
$113$	1137.94			0.947331			0.473665	−	0.880705i	$-0.342931\pi$
$113$	1137.94			0.947331			0.473665	+	0.880705i	$0.342931\pi$
$114$		0			0
$115$	50.4441			0.0409038
$116$	−163.802	+	283.713i	−0.131109	+	0.227087i
$117$		0			0
$118$	−182.289	−	315.733i	−0.142212	−	0.246319i
$119$	660.688	−	1144.34i	0.508951	−	0.881529i
$120$		0			0
$121$	3013.63			2.26419
$122$	1.03474			0.000767877
$123$		0			0
$124$	494.379	+	856.290i	0.358037	+	0.620138i
$125$	667.310			0.477488
$126$		0			0
$127$	754.987	+	1307.68i	0.527514	+	0.913681i	0.999486	+	0.0320676i	$0.0102092\pi$
$127$	754.987	+	1307.68i	0.527514	+	0.913681i	−0.471971	+	0.881614i	$0.656457\pi$
$128$	64.0000	−	110.851i	0.0441942	−	0.0765466i
$129$		0			0
$130$	976.789	−	1691.85i	0.659001	−	1.14142i
$131$	−185.472	+	321.247i	−0.123701	+	0.214256i	−0.921224	−	0.389032i	$-0.872810\pi$
$131$	−185.472	+	321.247i	−0.123701	+	0.214256i	0.797524	+	0.603288i	$0.206143\pi$
$132$		0			0
$133$	−1065.06	+	265.739i	−0.694378	+	0.173252i
$134$	319.724			0.206119
$135$		0			0
$136$	−398.776	+	690.700i	−0.251432	+	0.435493i
$137$	−845.187	−	1463.91i	−0.527075	−	0.912920i	−0.999502	−	0.0315505i	$-0.989955\pi$
$137$	−845.187	−	1463.91i	−0.527075	−	0.912920i	0.472428	−	0.881370i	$-0.343378\pi$
$138$		0			0
$139$	1355.76	+	2348.25i	0.827296	+	1.43292i	0.900152	+	0.435576i	$0.143455\pi$
$139$	1355.76	+	2348.25i	0.827296	+	1.43292i	−0.0728556	+	0.997343i	$0.523211\pi$
$140$	−755.724			−0.456217
$141$		0			0
$142$	791.884	+	1371.58i	0.467982	+	0.810568i
$143$	2258.40	+	3911.67i	1.32068	+	2.28748i
$144$		0			0
$145$	−1167.44			−0.668625
$146$	322.129	+	557.944i	0.182600	+	0.316273i
$147$		0			0
$148$	−842.130	−	1458.61i	−0.467720	−	0.810116i
$149$	1299.45	−	2250.71i	0.714464	−	1.23749i	−0.248702	−	0.968580i	$-0.580004\pi$
$149$	1299.45	−	2250.71i	0.714464	−	1.23749i	0.963166	−	0.268908i	$-0.0866628\pi$
$150$		0			0
$151$	594.863			0.320591			0.160296	−	0.987069i	$-0.448755\pi$
$151$	594.863			0.320591			0.160296	+	0.987069i	$0.448755\pi$
$152$	642.845	−	160.394i	0.343037	−	0.0855900i
$153$		0			0
$154$	873.642	−	1513.19i	0.457144	−	0.791796i
$155$	−1761.76	+	3051.46i	−0.912954	+	1.58128i
$156$		0			0
$157$	29.1274	−	50.4502i	0.0148065	−	0.0256456i	−0.858527	−	0.512768i	$-0.828620\pi$
$157$	29.1274	−	50.4502i	0.0148065	−	0.0256456i	0.873334	+	0.487122i	$0.161953\pi$
$158$	318.371	+	551.434i	0.160305	+	0.277657i
$159$		0			0
$160$	456.138			0.225380
$161$	23.4526	+	40.6211i	0.0114803	+	0.0198844i
$162$		0			0
$163$	1152.53			0.553825			0.276912	−	0.960895i	$-0.410689\pi$
$163$	1152.53			0.553825			0.276912	+	0.960895i	$0.410689\pi$
$164$	1381.03			0.657565
$165$		0			0
$166$	−684.160	+	1185.00i	−0.319886	+	0.554059i
$167$	629.554	+	1090.42i	0.291715	+	0.505265i	0.974215	−	0.225620i	$-0.0724408\pi$
$167$	629.554	+	1090.42i	0.291715	+	0.505265i	−0.682500	+	0.730885i	$0.739107\pi$
$168$		0			0
$169$	−1249.40	+	2164.02i	−0.568683	+	0.984988i
$170$	−2842.14			−1.28225
$171$		0			0
$172$	−1464.67			−0.649304
$173$	−637.050	+	1103.40i	−0.279965	+	0.484914i	−0.971376	−	0.237548i	$-0.923656\pi$
$173$	−637.050	+	1103.40i	−0.279965	+	0.484914i	0.691411	+	0.722462i	$0.256990\pi$
$174$		0			0
$175$	−518.147	−	897.456i	−0.223818	−	0.387665i
$176$	−527.311	+	913.329i	−0.225838	+	0.391163i
$177$		0			0
$178$	1440.81			0.606704
$179$	−2034.50			−0.849528			−0.424764	−	0.905304i	$-0.639643\pi$
$179$	−2034.50			−0.849528			−0.424764	+	0.905304i	$0.639643\pi$
$180$		0			0
$181$	−516.657	−	894.876i	−0.212170	−	0.367490i	0.740223	−	0.672361i	$-0.234720\pi$
$181$	−516.657	−	894.876i	−0.212170	−	0.367490i	−0.952393	+	0.304872i	$0.901386\pi$
$182$	1816.53			0.739835
$183$		0			0
$184$	−14.1555	−	24.5180i	−0.00567149	−	0.00982331i
$185$	3000.99	−	5197.87i	1.19264	−	2.06570i
$186$		0			0
$187$	3285.61	−	5690.84i	1.28485	−	2.22543i
$188$	−181.880	+	315.025i	−0.0705582	+	0.122210i
$189$		0			0
$190$	1640.41	+	1698.13i	0.626359	+	0.648395i
$191$	1135.46			0.430154			0.215077	−	0.976597i	$-0.431000\pi$
$191$	1135.46			0.430154			0.215077	+	0.976597i	$0.431000\pi$
$192$		0			0
$193$	−545.536	+	944.897i	−0.203464	+	0.352410i	−0.949642	−	0.313336i	$-0.898553\pi$
$193$	−545.536	+	944.897i	−0.203464	+	0.352410i	0.746178	+	0.665746i	$0.231887\pi$
$194$	415.392	+	719.480i	0.153729	+	0.266266i
$195$		0			0
$196$	334.646	+	579.625i	0.121956	+	0.211233i
$197$	−4138.08			−1.49658			−0.748289	−	0.663373i	$-0.769124\pi$
$197$	−4138.08			−1.49658			−0.748289	+	0.663373i	$0.769124\pi$
$198$		0			0
$199$	1367.12	+	2367.91i	0.486997	+	0.843503i	0.999888	−	0.0149506i	$-0.00475910\pi$
$199$	1367.12	+	2367.91i	0.486997	+	0.843503i	−0.512892	+	0.858453i	$0.671426\pi$
$200$	312.742	+	541.684i	0.110571	+	0.191514i
$201$		0			0
$202$	674.682			0.235002
$203$	−542.770	−	940.104i	−0.187660	−	0.325036i
$204$		0			0
$205$	2460.71	+	4262.08i	0.838359	+	1.45208i
$206$	−1467.05	+	2541.00i	−0.496185	+	0.859417i
$207$		0			0
$208$	−1096.41			−0.365493
$209$	−5296.55	+	1321.52i	−1.75297	+	0.437377i
$210$		0			0
$211$	−1529.00	+	2648.31i	−0.498867	+	0.864063i	−0.999999	−	0.00130803i	$-0.999584\pi$
$211$	−1529.00	+	2648.31i	−0.498867	+	0.864063i	0.501132	+	0.865371i	$0.332917\pi$
$212$	−1376.98	+	2385.00i	−0.446092	+	0.772655i
$213$		0			0
$214$	−382.014	+	661.667i	−0.122028	+	0.211358i
$215$	−2609.74	−	4520.20i	−0.827826	−	1.43384i
$216$		0			0
$217$	−3276.33			−1.02494
$218$	−132.857	−	230.116i	−0.0412764	−	0.0714927i
$219$		0			0
$220$	−3758.23			−1.15172
$221$	6831.62			2.07939
$222$		0			0
$223$	−809.133	+	1401.46i	−0.242976	+	0.420846i	−0.961561	−	0.274593i	$-0.911457\pi$
$223$	−809.133	+	1401.46i	−0.242976	+	0.420846i	0.718585	+	0.695439i	$0.244790\pi$
$224$	212.069	+	367.314i	0.0632565	+	0.109563i
$225$		0			0
$226$	1137.94	−	1970.97i	0.334932	−	0.580119i
$227$	6065.43			1.77347			0.886733	−	0.462282i	$-0.152969\pi$
$227$	6065.43			1.77347			0.886733	+	0.462282i	$0.152969\pi$
$228$		0			0
$229$	−1916.36			−0.552999			−0.276500	−	0.961014i	$-0.589174\pi$
$229$	−1916.36			−0.552999			−0.276500	+	0.961014i	$0.589174\pi$
$230$	50.4441	−	87.3717i	0.0144617	−	0.0250483i
$231$		0			0
$232$	327.603	+	567.426i	0.0927078	+	0.160575i
$233$	−2004.27	+	3471.50i	−0.563537	+	0.976074i	0.433647	+	0.901083i	$0.357226\pi$
$233$	−2004.27	+	3471.50i	−0.563537	+	0.976074i	−0.997184	+	0.0749917i	$0.976107\pi$
$234$		0			0
$235$	−1296.29			−0.359831
$236$	−729.155			−0.201118
$237$		0			0
$238$	−1321.38	−	2288.69i	−0.359883	−	0.623335i
$239$	−530.292			−0.143522			−0.0717610	−	0.997422i	$-0.522862\pi$
$239$	−530.292			−0.143522			−0.0717610	+	0.997422i	$0.522862\pi$
$240$		0			0
$241$	−635.613	−	1100.91i	−0.169890	−	0.294258i	0.768491	−	0.639860i	$-0.221008\pi$
$241$	−635.613	−	1100.91i	−0.169890	−	0.294258i	−0.938381	+	0.345603i	$0.887674\pi$
$242$	3013.63	−	5219.77i	0.800511	−	1.38653i
$243$		0			0
$244$	1.03474	−	1.79222i	0.000271485	−	0.000470226i
$245$	−1192.54	+	2065.54i	−0.310973	+	0.538622i
$246$		0			0
$247$	−3943.05	−	4081.77i	−1.01575	−	1.05148i
$248$	1977.52			0.506341
$249$		0			0
$250$	667.310	−	1155.81i	0.168818	−	0.292401i
$251$	275.994	+	478.036i	0.0694048	+	0.120213i	0.898639	−	0.438688i	$-0.144557\pi$
$251$	275.994	+	478.036i	0.0694048	+	0.120213i	−0.829235	+	0.558901i	$0.811223\pi$
$252$		0			0
$253$	116.630	+	202.009i	0.0289821	+	0.0501984i
$254$	3019.95			0.746018
$255$		0			0
$256$	−128.000	−	221.703i	−0.0312500	−	0.0541266i
$257$	−2617.92	−	4534.37i	−0.635414	−	1.10057i	−0.986427	−	0.164198i	$-0.947496\pi$
$257$	−2617.92	−	4534.37i	−0.635414	−	1.10057i	0.351014	−	0.936370i	$-0.385837\pi$
$258$		0			0
$259$	5580.92			1.33893
$260$	−1953.58	−	3383.70i	−0.465984	−	0.807107i
$261$		0			0
$262$	370.944	+	642.494i	0.0874695	+	0.151502i
$263$	−907.129	+	1571.19i	−0.212684	+	0.368380i	−0.952554	−	0.304371i	$-0.901554\pi$
$263$	−907.129	+	1571.19i	−0.212684	+	0.368380i	0.739869	+	0.672750i	$0.234887\pi$
$264$		0			0
$265$	−9813.97			−2.27497
$266$	−604.785	+	2110.47i	−0.139405	+	0.486472i
$267$		0			0
$268$	319.724	−	553.779i	0.0728741	−	0.126222i
$269$	506.699	−	877.629i	0.114848	−	0.198922i	−0.802871	−	0.596153i	$-0.796695\pi$
$269$	506.699	−	877.629i	0.114848	−	0.198922i	0.917719	+	0.397231i	$0.130029\pi$
$270$		0			0
$271$	959.258	−	1661.48i	0.215021	−	0.372428i	−0.738258	−	0.674519i	$-0.764351\pi$
$271$	959.258	−	1661.48i	0.215021	−	0.372428i	0.953279	+	0.302091i	$0.0976845\pi$
$272$	797.552	+	1381.40i	0.177789	+	0.307940i
$273$		0			0
$274$	−3380.75			−0.745396
$275$	−2576.75	−	4463.06i	−0.565032	−	0.978664i
$276$		0			0
$277$	6283.45			1.36295			0.681473	−	0.731844i	$-0.261340\pi$
$277$	6283.45			1.36295			0.681473	+	0.731844i	$0.261340\pi$
$278$	5423.04			1.16997
$279$		0			0
$280$	−755.724	+	1308.95i	−0.161297	+	0.279375i
$281$	−4028.68	−	6977.88i	−0.855270	−	1.48137i	−0.876394	−	0.481594i	$-0.840058\pi$
$281$	−4028.68	−	6977.88i	−0.855270	−	1.48137i	0.0211241	−	0.999777i	$-0.493276\pi$
$282$		0			0
$283$	96.7083	−	167.504i	0.0203135	−	0.0351840i	−0.855690	−	0.517489i	$-0.826867\pi$
$283$	96.7083	−	167.504i	0.0203135	−	0.0351840i	0.876003	+	0.482305i	$0.160200\pi$
$284$	3167.54			0.661826
$285$		0			0
$286$	9033.61			1.86772
$287$	−2288.08	+	3963.07i	−0.470597	+	0.815097i
$288$		0			0
$289$	−2512.95	−	4352.55i	−0.511489	−	0.885926i
$290$	−1167.44	+	2022.07i	−0.236395	+	0.409447i
$291$		0			0
$292$	1288.52			0.258235
$293$	4735.47			0.944194			0.472097	−	0.881547i	$-0.343497\pi$
$293$	4735.47			0.944194			0.472097	+	0.881547i	$0.343497\pi$
$294$		0			0
$295$	−1299.20	−	2250.28i	−0.256415	−	0.444124i
$296$	−3368.52			−0.661457
$297$		0			0
$298$	−2598.90	−	4501.43i	−0.505202	−	0.875036i
$299$	−121.252	+	210.014i	−0.0234521	+	0.0406202i
$300$		0			0
$301$	2426.65	−	4203.09i	0.464684	−	0.804857i
$302$	594.863	−	1030.33i	0.113346	−	0.196321i
$303$		0			0
$304$	365.034	−	1273.83i	0.0688689	−	0.240327i
$305$	7.37475			0.00138451
$306$		0			0
$307$	14.3561	−	24.8655i	0.00266888	−	0.00462263i	−0.864688	−	0.502310i	$-0.832484\pi$
$307$	14.3561	−	24.8655i	0.00266888	−	0.00462263i	0.867357	+	0.497687i	$0.165817\pi$
$308$	−1747.28	−	3026.39i	−0.323249	−	0.559884i
$309$		0			0
$310$	3523.52	+	6102.91i	0.645556	+	1.11814i
$311$	1165.69			0.212540			0.106270	−	0.994337i	$-0.466109\pi$
$311$	1165.69			0.212540			0.106270	+	0.994337i	$0.466109\pi$
$312$		0			0
$313$	−1256.38	−	2176.11i	−0.226884	−	0.392974i	0.729999	−	0.683448i	$-0.239520\pi$
$313$	−1256.38	−	2176.11i	−0.226884	−	0.392974i	−0.956883	+	0.290474i	$0.906187\pi$
$314$	−58.2549	−	100.900i	−0.0104698	−	0.0181342i
$315$		0			0
$316$	1273.48			0.226706
$317$	−863.783	−	1496.12i	−0.153044	−	0.265080i	0.779301	−	0.626649i	$-0.215574\pi$
$317$	−863.783	−	1496.12i	−0.153044	−	0.265080i	−0.932345	+	0.361570i	$0.882241\pi$
$318$		0			0
$319$	−2699.20	−	4675.15i	−0.473750	−	0.820558i
$320$	456.138	−	790.054i	0.0796840	−	0.138017i
$321$		0			0
$322$	93.8104			0.0162356
$323$	−2274.48	+	7937.10i	−0.391813	+	1.36728i
$324$		0			0
$325$	2678.86	−	4639.92i	0.457220	−	0.791928i
$326$	1152.53	−	1996.25i	0.195807	−	0.339147i
$327$		0			0
$328$	1381.03	−	2392.02i	0.232484	−	0.402675i
$329$	−602.673	−	1043.86i	−0.100992	−	0.174924i
$330$		0			0
$331$	−6816.13			−1.13187			−0.565934	−	0.824450i	$-0.691484\pi$
$331$	−6816.13			−1.13187			−0.565934	+	0.824450i	$0.691484\pi$
$332$	1368.32	+	2370.00i	0.226194	+	0.391779i
$333$		0			0
$334$	2518.22			0.412547
$335$	2278.72			0.371642
$336$		0			0
$337$	5204.19	−	9013.93i	0.841218	−	1.45703i	−0.0476473	−	0.998864i	$-0.515172\pi$
$337$	5204.19	−	9013.93i	0.841218	−	1.45703i	0.888865	−	0.458168i	$-0.151494\pi$
$338$	2498.79	+	4328.04i	0.402120	+	0.696492i
$339$		0			0
$340$	−2842.14	+	4922.73i	−0.453343	+	0.785213i
$341$	−16293.2			−2.58747
$342$		0			0
$343$	−6763.98			−1.06478
$344$	−1464.67	+	2536.89i	−0.229564	+	0.397616i
$345$		0			0
$346$	1274.10	+	2206.80i	0.197965	+	0.342886i
$347$	−2792.02	+	4835.92i	−0.431940	+	0.748143i	−0.997040	−	0.0768797i	$-0.975504\pi$
$347$	−2792.02	+	4835.92i	−0.431940	+	0.748143i	0.565100	+	0.825022i	$0.308838\pi$
$348$		0			0
$349$	5505.82			0.844470			0.422235	−	0.906486i	$-0.361246\pi$
$349$	5505.82			0.844470			0.422235	+	0.906486i	$0.361246\pi$
$350$	−2072.59			−0.316527
$351$		0			0
$352$	1054.62	+	1826.66i	0.159692	+	0.276594i
$353$	5782.46			0.871868			0.435934	−	0.899979i	$-0.356418\pi$
$353$	5782.46			0.871868			0.435934	+	0.899979i	$0.356418\pi$
$354$		0			0
$355$	5643.88	+	9775.49i	0.843792	+	1.46149i
$356$	1440.81	−	2495.56i	0.214502	−	0.371529i
$357$		0			0
$358$	−2034.50	+	3523.85i	−0.300353	+	0.520227i
$359$	−951.820	+	1648.60i	−0.139931	+	0.242367i	−0.927470	−	0.373897i	$-0.878021\pi$
$359$	−951.820	+	1648.60i	−0.139931	+	0.242367i	0.787539	+	0.616264i	$0.211355\pi$
$360$		0			0
$361$	6055.05	−	3222.14i	0.882790	−	0.469769i
$362$	−2066.63			−0.300054
$363$		0			0
$364$	1816.53	−	3146.32i	0.261571	−	0.453054i
$365$	2295.87	+	3976.55i	0.329236	+	0.570253i
$366$		0			0
$367$	6374.02	+	11040.1i	0.906597	+	1.57027i	0.818759	+	0.574137i	$0.194662\pi$
$367$	6374.02	+	11040.1i	0.906597	+	1.57027i	0.0878374	+	0.996135i	$0.472004\pi$
$368$	−56.6218			−0.00802070
$369$		0			0
$370$	−6001.99	−	10395.7i	−0.843320	−	1.46067i
$371$	−4562.74	−	7902.90i	−0.638506	−	1.10592i
$372$		0			0
$373$	−736.660			−0.102260			−0.0511298	−	0.998692i	$-0.516282\pi$
$373$	−736.660			−0.102260			−0.0511298	+	0.998692i	$0.516282\pi$
$374$	−6571.21	−	11381.7i	−0.908528	−	1.57362i
$375$		0			0
$376$	363.760	+	630.050i	0.0498922	+	0.0864158i
$377$	2806.16	−	4860.42i	0.383355	−	0.663990i
$378$		0			0
$379$	1543.57			0.209202			0.104601	−	0.994514i	$-0.466643\pi$
$379$	1543.57			0.209202			0.104601	+	0.994514i	$0.466643\pi$
$380$	4581.66	−	1143.15i	0.618511	−	0.154323i
$381$		0			0
$382$	1135.46	−	1966.68i	0.152082	−	0.263414i
$383$	3282.18	−	5684.90i	0.437889	−	0.758446i	−0.559637	−	0.828738i	$-0.689060\pi$
$383$	3282.18	−	5684.90i	0.437889	−	0.758446i	0.997527	+	0.0702912i	$0.0223929\pi$
$384$		0			0
$385$	6226.59	−	10784.8i	0.824250	−	1.42764i
$386$	1091.07	+	1889.79i	0.143871	+	0.249192i
$387$		0			0
$388$	1661.57			0.217406
$389$	−1749.13	−	3029.59i	−0.227981	−	0.394875i	0.729229	−	0.684270i	$-0.239879\pi$
$389$	−1749.13	−	3029.59i	−0.227981	−	0.394875i	−0.957210	+	0.289395i	$0.906546\pi$
$390$		0			0
$391$	352.803			0.0456318
$392$	1338.59			0.172471
$393$		0			0
$394$	−4138.08	+	7167.36i	−0.529120	+	0.916463i
$395$	2269.08	+	3930.16i	0.289037	+	0.500627i
$396$		0			0
$397$	−4305.30	+	7457.00i	−0.544274	+	0.942710i	0.454378	+	0.890809i	$0.349861\pi$
$397$	−4305.30	+	7457.00i	−0.544274	+	0.942710i	−0.998652	+	0.0519013i	$0.983472\pi$
$398$	5468.47			0.688717
$399$		0			0
$400$	1250.97			0.156371
$401$	882.872	−	1529.18i	0.109946	−	0.190433i	−0.805802	−	0.592185i	$-0.798265\pi$
$401$	882.872	−	1529.18i	0.109946	−	0.190433i	0.915748	+	0.401752i	$0.131599\pi$
$402$		0			0
$403$	−8469.44	−	14669.5i	−1.04688	−	1.81325i
$404$	674.682	−	1168.58i	0.0830858	−	0.143909i
$405$		0			0
$406$	−2171.08			−0.265391
$407$	27754.0			3.38013
$408$		0			0
$409$	−6697.86	−	11601.0i	−0.809751	−	1.40253i	−0.913037	−	0.407877i	$-0.866269\pi$
$409$	−6697.86	−	11601.0i	−0.809751	−	1.40253i	0.103286	−	0.994652i	$-0.467064\pi$
$410$	9842.85			1.18562
$411$		0			0
$412$	2934.10	+	5082.00i	0.350856	+	0.607700i
$413$	1208.06	−	2092.41i	0.143934	−	0.249300i
$414$		0			0
$415$	−4876.11	+	8445.67i	−0.576769	+	0.998992i
$416$	−1096.41	+	1899.04i	−0.129221	+	0.223818i
$417$		0			0
$418$	−3007.60	+	10495.4i	−0.351930	+	1.22810i
$419$	−9117.71			−1.06308			−0.531539	−	0.847034i	$-0.678386\pi$
$419$	−9117.71			−1.06308			−0.531539	+	0.847034i	$0.678386\pi$
$420$		0			0
$421$	−4102.23	+	7105.27i	−0.474894	+	0.822540i	−0.999587	−	0.0287514i	$-0.990847\pi$
$421$	−4102.23	+	7105.27i	−0.474894	+	0.822540i	0.524693	+	0.851292i	$0.324180\pi$
$422$	3058.01	+	5296.62i	0.352752	+	0.610985i
$423$		0			0
$424$	2753.97	+	4770.01i	0.315435	+	0.546349i
$425$	−7794.61			−0.889634
$426$		0			0
$427$	3.42869	+	5.93867i	0.000388585	+	0.000673050i
$428$	764.027	+	1323.33i	0.0862865	+	0.149453i
$429$		0			0
$430$	−10439.0			−1.17072
$431$	−635.672	−	1101.02i	−0.0710423	−	0.123049i	0.828316	−	0.560261i	$-0.189299\pi$
$431$	−635.672	−	1101.02i	−0.0710423	−	0.123049i	−0.899358	+	0.437212i	$0.855966\pi$
$432$		0			0
$433$	1983.79	+	3436.03i	0.220173	+	0.381351i	0.954860	−	0.297055i	$-0.0960044\pi$
$433$	1983.79	+	3436.03i	0.220173	+	0.381351i	−0.734687	+	0.678406i	$0.762671\pi$
$434$	−3276.33	+	5674.77i	−0.362370	+	0.627644i
$435$		0			0
$436$	−531.430			−0.0583736
$437$	−203.630	−	210.794i	−0.0222905	−	0.0230747i
$438$		0			0
$439$	−104.519	+	181.032i	−0.0113632	+	0.0196816i	−0.871651	−	0.490127i	$-0.836950\pi$
$439$	−104.519	+	181.032i	−0.0113632	+	0.0196816i	0.860288	+	0.509808i	$0.170284\pi$
$440$	−3758.23	+	6509.44i	−0.407196	+	0.705285i
$441$		0			0
$442$	6831.62	−	11832.7i	0.735174	−	1.27336i
$443$	2867.02	+	4965.83i	0.307486	+	0.532581i	0.977812	−	0.209486i	$-0.0671789\pi$
$443$	2867.02	+	4965.83i	0.307486	+	0.532581i	−0.670326	+	0.742067i	$0.733846\pi$
$444$		0			0
$445$	10268.9			1.09391
$446$	1618.27	+	2802.92i	0.171810	+	0.297583i
$447$		0			0
$448$	848.276			0.0894582
$449$	−9741.13			−1.02386			−0.511929	−	0.859028i	$-0.671069\pi$
$449$	−9741.13			−1.02386			−0.511929	+	0.859028i	$0.671069\pi$
$450$		0			0
$451$	−11378.7	+	19708.4i	−1.18803	+	2.05772i
$452$	−2275.88	−	3941.94i	−0.236833	−	0.410206i
$453$		0			0
$454$	6065.43	−	10505.6i	0.627015	−	1.08602i
$455$	12946.7			1.33395
$456$		0			0
$457$	−17233.8			−1.76403			−0.882016	−	0.471220i	$-0.843814\pi$
$457$	−17233.8			−1.76403			−0.882016	+	0.471220i	$0.843814\pi$
$458$	−1916.36	+	3319.24i	−0.195515	+	0.338641i
$459$		0			0
$460$	−100.888	−	174.743i	−0.0102259	−	0.0177118i
$461$	832.617	−	1442.14i	0.0841190	−	0.145698i	−0.820896	−	0.571077i	$-0.806526\pi$
$461$	832.617	−	1442.14i	0.0841190	−	0.145698i	0.905015	+	0.425379i	$0.139859\pi$
$462$		0			0
$463$	−10694.0			−1.07342			−0.536710	−	0.843767i	$-0.680333\pi$
$463$	−10694.0			−1.07342			−0.536710	+	0.843767i	$0.680333\pi$
$464$	1310.41			0.131109
$465$		0			0
$466$	4008.54	+	6942.99i	0.398481	+	0.690189i
$467$	−13212.7			−1.30923			−0.654614	−	0.755963i	$-0.727169\pi$
$467$	−13212.7			−1.30923			−0.654614	+	0.755963i	$0.727169\pi$
$468$		0			0
$469$	1059.43	+	1834.99i	0.104307	+	0.180665i
$470$	−1296.29	+	2245.23i	−0.127220	+	0.220351i
$471$		0			0
$472$	−729.155	+	1262.93i	−0.0711061	+	0.123159i
$473$	12067.8	−	20902.0i	1.17310	−	2.03187i
$474$		0			0
$475$	4498.86	+	4657.14i	0.434573	+	0.449862i
$476$	−5285.50			−0.508951
$477$		0			0
$478$	−530.292	+	918.493i	−0.0507427	+	0.0878889i
$479$	2704.77	+	4684.79i	0.258004	+	0.446876i	0.965707	−	0.259634i	$-0.0836019\pi$
$479$	2704.77	+	4684.79i	0.258004	+	0.446876i	−0.707703	+	0.706510i	$0.750269\pi$
$480$		0			0
$481$	14426.9	+	24988.1i	1.36759	+	2.36874i
$482$	−2542.45			−0.240260
$483$		0			0
$484$	−6027.27	−	10439.5i	−0.566047	−	0.980422i
$485$	2960.56	+	5127.85i	0.277180	+	0.480090i
$486$		0			0
$487$	14297.0			1.33031			0.665154	−	0.746706i	$-0.268366\pi$
$487$	14297.0			1.33031			0.665154	+	0.746706i	$0.268366\pi$
$488$	−2.06948	−	3.58444i	−0.000191969	−	0.000332500i
$489$		0			0
$490$	2385.08	+	4131.08i	0.219891	+	0.380863i
$491$	1401.21	−	2426.97i	0.128790	−	0.223070i	−0.794418	−	0.607371i	$-0.792224\pi$
$491$	1401.21	−	2426.97i	0.128790	−	0.223070i	0.923208	+	0.384301i	$0.125557\pi$
$492$		0			0
$493$	−8165.02			−0.745911
$494$	−11012.9	+	2747.79i	−1.00302	+	0.250261i
$495$		0			0
$496$	1977.52	−	3425.16i	0.179018	−	0.310069i
$497$	−5247.94	+	9089.70i	−0.473646	+	0.820380i
$498$		0			0
$499$	−7089.13	+	12278.7i	−0.635978	+	1.10155i	0.350329	+	0.936627i	$0.386070\pi$
$499$	−7089.13	+	12278.7i	−0.635978	+	1.10155i	−0.986307	+	0.164919i	$0.947264\pi$
$500$	−1334.62	−	2311.63i	−0.119372	−	0.206758i
$501$		0			0
$502$	1103.98			0.0981533
$503$	−2768.93	−	4795.92i	−0.245448	−	0.425128i	0.716810	−	0.697269i	$-0.245602\pi$
$503$	−2768.93	−	4795.92i	−0.245448	−	0.425128i	−0.962257	+	0.272141i	$0.912268\pi$
$504$		0			0
$505$	4808.56			0.423719
$506$	466.520			0.0409869
$507$		0			0
$508$	3019.95	−	5230.71i	0.263757	−	0.456841i
$509$	4557.55	+	7893.90i	0.396876	+	0.687409i	0.993339	−	0.115232i	$-0.0367611\pi$
$509$	4557.55	+	7893.90i	0.396876	+	0.687409i	−0.596463	+	0.802641i	$0.703428\pi$
$510$		0			0
$511$	−2134.80	+	3697.58i	−0.184810	+	0.320101i
$512$	−512.000			−0.0441942
$513$		0			0
$514$	−10471.7			−0.898610
$515$	−10455.9	+	18110.1i	−0.894643	+	1.54957i
$516$		0			0
$517$	−2997.10	−	5191.13i	−0.254956	−	0.441597i
$518$	5580.92	−	9666.44i	0.473382	−	0.819921i
$519$		0			0
$520$	−7814.31			−0.659001
$521$	2705.46			0.227502			0.113751	−	0.993509i	$-0.463713\pi$
$521$	2705.46			0.227502			0.113751	+	0.993509i	$0.463713\pi$
$522$		0			0
$523$	11564.2	+	20029.7i	0.966856	+	1.67464i	0.704544	+	0.709660i	$0.251151\pi$
$523$	11564.2	+	20029.7i	0.966856	+	1.67464i	0.262311	+	0.964983i	$0.415515\pi$
$524$	1483.78			0.123701
$525$		0			0
$526$	1814.26	+	3142.39i	0.150390	+	0.260484i
$527$	−12321.7	+	21341.7i	−1.01848	+	1.76406i
$528$		0			0
$529$	6077.24	−	10526.1i	0.499485	−	0.865134i
$530$	−9813.97	+	16998.3i	−0.804324	+	1.39313i
$531$		0			0
$532$	3050.66	+	3157.99i	0.248615	+	0.257362i
$533$	−23659.1			−1.92269
$534$		0			0
$535$	−2722.67	+	4715.80i	−0.220021	+	0.381088i
$536$	−639.448	−	1107.56i	−0.0515298	−	0.0892522i
$537$		0			0
$538$	−1013.40	−	1755.26i	−0.0812095	−	0.140659i
$539$	−11028.9			−0.881353
$540$		0			0
$541$	4820.58	+	8349.50i	0.383093	+	0.663536i	0.991503	−	0.130087i	$-0.0415257\pi$
$541$	4820.58	+	8349.50i	0.383093	+	0.663536i	−0.608410	+	0.793623i	$0.708192\pi$
$542$	−1918.52	−	3322.97i	−0.152043	−	0.263346i
$543$		0			0
$544$	3190.21			0.251432
$545$	−946.896	−	1640.07i	−0.0744231	−	0.128905i
$546$		0			0
$547$	8688.50	+	15048.9i	0.679147	+	1.17632i	0.975238	+	0.221158i	$0.0709835\pi$
$547$	8688.50	+	15048.9i	0.679147	+	1.17632i	−0.296091	+	0.955160i	$0.595683\pi$
$548$	−3380.75	+	5855.63i	−0.263537	+	0.456460i
$549$		0			0
$550$	−10307.0			−0.799076
$551$	4712.66	+	4878.45i	0.364366	+	0.377185i
$552$		0			0
$553$	−2109.89	+	3654.44i	−0.162246	+	0.281018i
$554$	6283.45	−	10883.3i	0.481874	−	0.834630i
$555$		0			0
$556$	5423.04	−	9392.99i	0.413648	−	0.716459i
$557$	9372.14	+	16233.0i	0.712945	+	1.23486i	0.963747	+	0.266819i	$0.0859725\pi$
$557$	9372.14	+	16233.0i	0.712945	+	1.23486i	−0.250802	+	0.968038i	$0.580694\pi$
$558$		0			0
$559$	25092.0			1.89853
$560$	1511.45	+	2617.91i	0.114054	+	0.197548i
$561$		0			0
$562$	−16114.7			−1.20953
$563$	7618.70			0.570320			0.285160	−	0.958480i	$-0.407953\pi$
$563$	7618.70			0.570320			0.285160	+	0.958480i	$0.407953\pi$
$564$		0			0
$565$	8110.28	−	14047.4i	0.603897	−	1.04598i
$566$	−193.417	−	335.007i	−0.0143638	−	0.0248788i
$567$		0			0
$568$	3167.54	−	5486.33i	0.233991	−	0.405284i
$569$	13081.6			0.963816			0.481908	−	0.876222i	$-0.339944\pi$
$569$	13081.6			0.963816			0.481908	+	0.876222i	$0.339944\pi$
$570$		0			0
$571$	8643.49			0.633483			0.316741	−	0.948512i	$-0.397411\pi$
$571$	8643.49			0.633483			0.316741	+	0.948512i	$0.397411\pi$
$572$	9033.61	−	15646.7i	0.660339	−	1.14374i
$573$		0			0
$574$	4576.16	+	7926.15i	0.332762	+	0.576361i
$575$	138.344	−	239.618i	0.0100336	−	0.0173787i
$576$		0			0
$577$	9841.26			0.710047			0.355024	−	0.934857i	$-0.384473\pi$
$577$	9841.26			0.710047			0.355024	+	0.934857i	$0.384473\pi$
$578$	−10051.8			−0.723355
$579$		0			0
$580$	2334.88	+	4044.13i	0.167156	+	0.289523i
$581$	−9068.06			−0.647516
$582$		0			0
$583$	−22690.6	−	39301.2i	−1.61192	−	2.79192i
$584$	1288.52	−	2231.78i	0.0913000	−	0.158136i
$585$		0			0
$586$	4735.47	−	8202.07i	0.333823	−	0.578198i
$587$	9949.24	−	17232.6i	0.699573	−	1.21170i	−0.269042	−	0.963128i	$-0.586707\pi$
$587$	9949.24	−	17232.6i	0.699573	−	1.21170i	0.968615	−	0.248567i	$-0.0799596\pi$
$588$		0			0
$589$	19863.1	−	4955.97i	1.38955	−	0.346702i
$590$	−5196.80			−0.362625
$591$		0			0
$592$	−3368.52	+	5834.44i	−0.233860	+	0.405058i
$593$	11097.5	+	19221.4i	0.768497	+	1.33108i	0.938378	+	0.345611i	$0.112328\pi$
$593$	11097.5	+	19221.4i	0.768497	+	1.33108i	−0.169881	+	0.985465i	$0.554338\pi$
$594$		0			0
$595$	−9417.65	−	16311.8i	−0.648884	−	1.12390i
$596$	−10395.6			−0.714464
$597$		0			0
$598$	242.504	+	420.029i	0.0165831	+	0.0287228i
$599$	9848.11	+	17057.4i	0.671758	+	1.16352i	0.977405	+	0.211374i	$0.0677938\pi$
$599$	9848.11	+	17057.4i	0.671758	+	1.16352i	−0.305647	+	0.952145i	$0.598873\pi$
$600$		0			0
$601$	−14626.5			−0.992724			−0.496362	−	0.868116i	$-0.665331\pi$
$601$	−14626.5			−0.992724			−0.496362	+	0.868116i	$0.665331\pi$
$602$	−4853.31	−	8406.17i	−0.328581	−	0.569120i
$603$		0			0
$604$	−1189.73	−	2060.67i	−0.0801478	−	0.138820i
$605$	21478.6	−	37202.1i	1.44336	−	2.49997i
$606$		0			0
$607$	−16312.0			−1.09075			−0.545375	−	0.838192i	$-0.683613\pi$
$607$	−16312.0			−1.09075			−0.545375	+	0.838192i	$0.683613\pi$
$608$	−1841.31	−	1906.09i	−0.122821	−	0.127142i
$609$		0			0
$610$	7.37475	−	12.7734i	0.000489500	−	0.000847839i
$611$	3115.87	−	5396.84i	0.206308	−	0.357337i
$612$		0			0
$613$	−8874.36	+	15370.8i	−0.584718	+	1.01276i	0.410192	+	0.911999i	$0.365462\pi$
$613$	−8874.36	+	15370.8i	−0.584718	+	1.01276i	−0.994911	+	0.100762i	$0.967872\pi$
$614$	−28.7122	−	49.7309i	−0.00188718	−	0.00326869i
$615$		0			0
$616$	−6989.14			−0.457144
$617$	1864.90	+	3230.10i	0.121682	+	0.210760i	0.920431	−	0.390905i	$-0.127838\pi$
$617$	1864.90	+	3230.10i	0.121682	+	0.210760i	−0.798749	+	0.601665i	$0.794504\pi$
$618$		0			0
$619$	−1003.67			−0.0651714			−0.0325857	−	0.999469i	$-0.510374\pi$
$619$	−1003.67			−0.0651714			−0.0325857	+	0.999469i	$0.510374\pi$
$620$	14094.1			0.912954
$621$		0			0
$622$	1165.69	−	2019.03i	0.0751444	−	0.130154i
$623$	4774.24	+	8269.23i	0.307024	+	0.531781i
$624$		0			0
$625$	9642.61	−	16701.5i	0.617127	−	1.06890i
$626$	−5025.51			−0.320862
$627$		0			0
$628$	−233.019			−0.0148065
$629$	20988.8	−	36353.7i	1.33049	−	2.30448i
$630$		0			0
$631$	−8423.38	−	14589.7i	−0.531426	−	0.920456i	−0.999327	−	0.0366756i	$-0.988323\pi$
$631$	−8423.38	−	14589.7i	−0.531426	−	0.920456i	0.467902	−	0.883781i	$-0.345010\pi$
$632$	1273.48	−	2205.74i	0.0801526	−	0.138828i
$633$		0			0
$634$	−3455.13			−0.216437
$635$	21523.7			1.34510
$636$		0			0
$637$	−5732.98	−	9929.82i	−0.356592	−	0.617636i
$638$	−10796.8			−0.669983
$639$		0			0
$640$	−912.276	−	1580.11i	−0.0563451	−	0.0975926i
$641$	−80.9977	+	140.292i	−0.00499098	+	0.00864463i	−0.868510	−	0.495671i	$-0.834922\pi$
$641$	−80.9977	+	140.292i	−0.00499098	+	0.00864463i	0.863519	+	0.504316i	$0.168255\pi$
$642$		0			0
$643$	−3145.45	+	5448.09i	−0.192915	+	0.334139i	−0.946215	−	0.323538i	$-0.895128\pi$
$643$	−3145.45	+	5448.09i	−0.192915	+	0.334139i	0.753300	+	0.657677i	$0.228461\pi$
$644$	93.8104	−	162.484i	0.00574014	−	0.00994221i
$645$		0			0
$646$	11473.0	+	11876.6i	0.698759	+	0.723343i
$647$	−28249.8			−1.71656			−0.858280	−	0.513181i	$-0.828467\pi$
$647$	−28249.8			−1.71656			−0.858280	+	0.513181i	$0.828467\pi$
$648$		0			0
$649$	6007.68	−	10405.6i	0.363362	−	0.629361i
$650$	−5357.72	−	9279.85i	−0.323303	−	0.559978i
$651$		0			0
$652$	−2305.07	−	3992.50i	−0.138456	−	0.239813i
$653$	−11423.5			−0.684587			−0.342293	−	0.939593i	$-0.611204\pi$
$653$	−11423.5			−0.684587			−0.342293	+	0.939593i	$0.611204\pi$
$654$		0			0
$655$	2643.78	+	4579.16i	0.157711	+	0.273164i
$656$	−2762.07	−	4784.04i	−0.164391	−	0.284734i
$657$		0			0
$658$	−2410.69			−0.142824
$659$	100.069	+	173.324i	0.00591522	+	0.0102455i	0.868968	−	0.494869i	$-0.164784\pi$
$659$	100.069	+	173.324i	0.00591522	+	0.0102455i	−0.863053	+	0.505114i	$0.831450\pi$
$660$		0			0
$661$	1303.50	+	2257.73i	0.0767026	+	0.132853i	0.901825	−	0.432101i	$-0.142227\pi$
$661$	1303.50	+	2257.73i	0.0767026	+	0.132853i	−0.825123	+	0.564953i	$0.808894\pi$
$662$	−6816.13	+	11805.9i	−0.400176	+	0.693125i
$663$		0			0
$664$	5473.28			0.319886
$665$	−4310.39	+	15041.7i	−0.251353	+	0.877130i
$666$		0			0
$667$	144.918	−	251.005i	0.00841266	−	0.0145712i
$668$	2518.22	−	4361.68i	0.145857	−	0.252632i
$669$		0			0
$670$	2278.72	−	3946.87i	0.131395	−	0.227583i
$671$	17.0509	+	29.5331i	0.000980989	+	0.00169912i
$672$		0			0
$673$	14561.2			0.834015			0.417007	−	0.908903i	$-0.363079\pi$
$673$	14561.2			0.834015			0.417007	+	0.908903i	$0.363079\pi$
$674$	−10408.4	−	18027.9i	−0.594831	−	1.03028i
$675$		0			0
$676$	9995.18			0.568683
$677$	−12684.6			−0.720100			−0.360050	−	0.932933i	$-0.617240\pi$
$677$	−12684.6			−0.720100			−0.360050	+	0.932933i	$0.617240\pi$
$678$		0			0
$679$	−2752.87	+	4768.11i	−0.155590	+	0.269489i
$680$	5684.28	+	9845.46i	0.320562	+	0.555230i
$681$		0			0
$682$	−16293.2	+	28220.7i	−0.914809	+	1.58450i
$683$	−12054.5			−0.675332			−0.337666	−	0.941266i	$-0.609637\pi$
$683$	−12054.5			−0.675332			−0.337666	+	0.941266i	$0.609637\pi$
$684$		0			0
$685$	−24095.1			−1.34398
$686$	−6763.98	+	11715.6i	−0.376458	+	0.652044i
$687$		0			0
$688$	2929.35	+	5073.78i	0.162326	+	0.281157i
$689$	23589.7	−	40858.6i	1.30435	−	2.25920i
$690$		0			0
$691$	3338.42			0.183791			0.0918953	−	0.995769i	$-0.470707\pi$
$691$	3338.42			0.183791			0.0918953	+	0.995769i	$0.470707\pi$
$692$	5096.40			0.279965
$693$		0			0
$694$	5584.03	+	9671.83i	0.305428	+	0.529017i
$695$	38650.9			2.10951
$696$		0			0
$697$	17210.1	+	29808.8i	0.935264	+	1.61993i
$698$	5505.82	−	9536.37i	0.298565	−	0.517130i
$699$		0			0
$700$	−2072.59	+	3589.83i	−0.111909	+	0.193832i
$701$	−11891.4	+	20596.5i	−0.640702	+	1.10973i	0.344575	+	0.938759i	$0.388023\pi$
$701$	−11891.4	+	20596.5i	−0.640702	+	1.10973i	−0.985276	+	0.170969i	$0.945310\pi$
$702$		0			0
$703$	−33834.9	+	8442.04i	−1.81523	+	0.452913i
$704$	4218.49			0.225838
$705$		0			0
$706$	5782.46	−	10015.5i	0.308252	−	0.533908i
$707$	2235.61	+	3872.19i	0.118923	+	0.205981i
$708$		0			0
$709$	−16092.6	−	27873.1i	−0.852424	−	1.47644i	−0.879015	−	0.476795i	$-0.841798\pi$
$709$	−16092.6	−	27873.1i	−0.852424	−	1.47644i	0.0265907	−	0.999646i	$-0.491535\pi$
$710$	22575.5			1.19330
$711$		0			0
$712$	−2881.62	−	4991.12i	−0.151676	−	0.262711i
$713$	−437.385	−	757.573i	−0.0229736	−	0.0397915i
$714$		0			0
$715$	64383.9			3.36758
$716$	4068.99	+	7047.70i	0.212382	+	0.367856i
$717$		0			0
$718$	1903.64	+	3297.20i	0.0989459	+	0.171379i
$719$	−19159.7	+	33185.5i	−0.993791	+	1.72130i	−0.400536	+	0.916281i	$0.631176\pi$
$719$	−19159.7	+	33185.5i	−0.993791	+	1.72130i	−0.593255	+	0.805015i	$0.702157\pi$
$720$		0			0
$721$	−19444.7			−1.00438
$722$	474.139	−	13709.8i	0.0244399	−	0.706684i
$723$		0			0
$724$	−2066.63	+	3579.50i	−0.106085	+	0.183745i
$725$	−3201.72	+	5545.55i	−0.164012	+	0.284078i
$726$		0			0
$727$	18271.6	−	31647.3i	0.932125	−	1.61449i	0.152442	−	0.988312i	$-0.451286\pi$
$727$	18271.6	−	31647.3i	0.932125	−	1.61449i	0.779683	−	0.626175i	$-0.215380\pi$
$728$	−3633.05	−	6292.63i	−0.184959	−	0.320358i
$729$		0			0
$730$	9183.46			0.465610
$731$	−18252.4	−	31614.1i	−0.923514	−	1.59957i
$732$		0			0
$733$	−15689.3			−0.790584			−0.395292	−	0.918555i	$-0.629357\pi$
$733$	−15689.3			−0.790584			−0.395292	+	0.918555i	$0.629357\pi$
$734$	25496.1			1.28212
$735$		0			0
$736$	−56.6218	+	98.0719i	−0.00283574	+	0.00491165i
$737$	5268.56	+	9125.42i	0.263324	+	0.456091i
$738$		0			0
$739$	18222.8	−	31562.9i	0.907088	−	1.57112i	0.0889991	−	0.996032i	$-0.471633\pi$
$739$	18222.8	−	31562.9i	0.907088	−	1.57112i	0.818089	−	0.575091i	$-0.195033\pi$
$740$	−24008.0			−1.19264
$741$		0			0
$742$	−18251.0			−0.902984
$743$	839.144	−	1453.44i	0.0414337	−	0.0717652i	−0.844565	−	0.535453i	$-0.820141\pi$
$743$	839.144	−	1453.44i	0.0414337	−	0.0717652i	0.885999	+	0.463688i	$0.153474\pi$
$744$		0			0
$745$	−18522.8	−	32082.4i	−0.910902	−	1.57773i
$746$	−736.660	+	1275.93i	−0.0361542	+	0.0626209i
$747$		0			0
$748$	−26284.9			−1.28485
$749$	−5063.33			−0.247009
$750$		0			0
$751$	−4320.84	−	7483.91i	−0.209946	−	0.363637i	0.741751	−	0.670675i	$-0.233996\pi$
$751$	−4320.84	−	7483.91i	−0.209946	−	0.363637i	−0.951697	+	0.307038i	$0.900662\pi$
$752$	1455.04			0.0705582
$753$		0			0
$754$	−5612.33	−	9720.83i	−0.271073	−	0.469512i
$755$	4239.68	−	7343.34i	0.204368	−	0.353976i
$756$		0			0
$757$	−1793.40	+	3106.25i	−0.0861058	+	0.149140i	−0.905862	−	0.423573i	$-0.860776\pi$
$757$	−1793.40	+	3106.25i	−0.0861058	+	0.149140i	0.819756	+	0.572713i	$0.194109\pi$
$758$	1543.57	−	2673.54i	0.0739642	−	0.128110i
$759$		0			0
$760$	2601.66	−	9078.82i	0.124174	−	0.433320i
$761$	11615.4			0.553297			0.276649	−	0.960971i	$-0.410776\pi$
$761$	11615.4			0.553297			0.276649	+	0.960971i	$0.410776\pi$
$762$		0			0
$763$	880.467	−	1525.01i	0.0417760	−	0.0723581i
$764$	−2270.93	−	3933.37i	−0.107538	−	0.186262i
$765$		0			0
$766$	−6564.36	−	11369.8i	−0.309634	−	0.536303i
$767$	12491.5			0.588060
$768$		0			0
$769$	−3010.91	−	5215.06i	−0.141192	−	0.244551i	0.786754	−	0.617267i	$-0.211760\pi$
$769$	−3010.91	−	5215.06i	−0.141192	−	0.244551i	−0.927946	+	0.372716i	$0.878427\pi$
$770$	−12453.2	−	21569.5i	−0.582833	−	1.00950i
$771$		0			0
$772$	4364.29			0.203464
$773$	−754.493	−	1306.82i	−0.0351064	−	0.0608061i	0.847938	−	0.530095i	$-0.177844\pi$
$773$	−754.493	−	1306.82i	−0.0351064	−	0.0608061i	−0.883045	+	0.469289i	$0.844510\pi$
$774$		0			0
$775$	9663.31	+	16737.3i	0.447892	+	0.775772i
$776$	1661.57	−	2877.92i	0.0768645	−	0.133133i
$777$		0			0
$778$	−6996.54			−0.322414
$779$	7876.95	−	27487.6i	0.362286	−	1.26425i
$780$		0			0
$781$	−26098.1	+	45203.2i	−1.19573	+	2.07106i
$782$	352.803	−	611.074i	0.0161333	−	0.0279437i
$783$		0			0
$784$	1338.59	−	2318.50i	0.0609779	−	0.105617i
$785$	−415.192	−	719.133i	−0.0188775	−	0.0326968i
$786$		0			0
$787$	3255.84			0.147469			0.0737346	−	0.997278i	$-0.476508\pi$
$787$	3255.84			0.147469			0.0737346	+	0.997278i	$0.476508\pi$
$788$	8276.16	+	14334.7i	0.374144	+	0.648037i
$789$		0			0
$790$	9076.31			0.408760
$791$	15082.6			0.677972
$792$		0			0
$793$	−17.7266	+	30.7034i	−0.000793809	+	0.00137492i
$794$	8610.60	+	14914.0i	0.384860	+	0.666597i
$795$		0			0
$796$	5468.47	−	9471.66i	0.243498	−	0.421751i
$797$	−31732.8			−1.41033			−0.705164	−	0.709044i	$-0.749127\pi$
$797$	−31732.8			−1.41033			−0.705164	+	0.709044i	$0.749127\pi$
$798$		0			0
$799$	−9066.16			−0.401424
$800$	1250.97	−	2166.74i	0.0552854	−	0.0957572i
$801$		0			0
$802$	−1765.74	−	3058.36i	−0.0777439	−	0.134656i
$803$	−10616.4	+	18388.1i	−0.466556	+	0.808098i
$804$		0			0
$805$	668.601			0.0292734
$806$	−33877.8			−1.48051
$807$		0			0
$808$	−1349.36	−	2337.17i	−0.0587506	−	0.101759i
$809$	−3853.96			−0.167488			−0.0837441	−	0.996487i	$-0.526688\pi$
$809$	−3853.96			−0.167488			−0.0837441	+	0.996487i	$0.526688\pi$
$810$		0			0
$811$	−15935.9	−	27601.8i	−0.689995	−	1.19511i	−0.971839	−	0.235646i	$-0.924279\pi$
$811$	−15935.9	−	27601.8i	−0.689995	−	1.19511i	0.281844	−	0.959460i	$-0.409054\pi$
$812$	−2171.08	+	3760.42i	−0.0938299	+	0.162518i
$813$		0			0
$814$	27754.0	−	48071.3i	1.19506	−	2.06990i
$815$	8214.29	−	14227.6i	0.353048	−	0.611497i
$816$		0			0
$817$	−8354.00	+	29152.4i	−0.357735	+	1.24836i
$818$	−26791.4			−1.14516
$819$		0			0
$820$	9842.85	−	17048.3i	0.419179	−	0.726040i
$821$	−19585.0	−	33922.2i	−0.832547	−	1.44201i	−0.896012	−	0.444030i	$-0.853548\pi$
$821$	−19585.0	−	33922.2i	−0.832547	−	1.44201i	0.0634643	−	0.997984i	$-0.479785\pi$
$822$		0			0
$823$	9419.41	+	16314.9i	0.398955	+	0.691010i	0.993597	−	0.112980i	$-0.0360396\pi$
$823$	9419.41	+	16314.9i	0.398955	+	0.691010i	−0.594642	+	0.803990i	$0.702706\pi$
$824$	11736.4			0.496185
$825$		0			0
$826$	−2416.11	−	4184.83i	−0.101776	−	0.176282i
$827$	443.653	+	768.430i	0.0186546	+	0.0323107i	0.875202	−	0.483758i	$-0.160728\pi$
$827$	443.653	+	768.430i	0.0186546	+	0.0323107i	−0.856547	+	0.516068i	$0.827395\pi$
$828$		0			0
$829$	32877.0			1.37740			0.688700	−	0.725046i	$-0.258182\pi$
$829$	32877.0			1.37740			0.688700	+	0.725046i	$0.258182\pi$
$830$	9752.22	+	16891.3i	0.407837	+	0.706394i
$831$		0			0
$832$	2192.83	+	3798.09i	0.0913733	+	0.158263i
$833$	−8340.56	+	14446.3i	−0.346919	+	0.600881i
$834$		0			0
$835$	17947.7			0.743840
$836$	15171.0	+	15704.7i	0.627631	+	0.649712i
$837$		0			0
$838$	−9117.71	+	15792.3i	−0.375855	+	0.650999i
$839$	−20756.4	+	35951.1i	−0.854099	+	1.47934i	0.0233792	+	0.999727i	$0.492557\pi$
$839$	−20756.4	+	35951.1i	−0.854099	+	1.47934i	−0.877478	+	0.479616i	$0.840776\pi$
$840$		0			0
$841$	8840.63	−	15312.4i	0.362484	−	0.627841i
$842$	8204.45	+	14210.5i	0.335801	+	0.581624i
$843$		0			0
$844$	12232.0			0.498867
$845$	17809.3	+	30846.6i	0.725040	+	1.25581i
$846$		0			0
$847$	39943.6			1.62040
$848$	11015.9			0.446092
$849$		0			0
$850$	−7794.61	+	13500.7i	−0.314533	+	0.544787i
$851$	745.045	+	1290.46i	0.0300115	+	0.0519815i
$852$		0			0
$853$	−743.876	+	1288.43i	−0.0298591	+	0.0517175i	−0.880569	−	0.473918i	$-0.842839\pi$
$853$	−743.876	+	1288.43i	−0.0298591	+	0.0517175i	0.850710	+	0.525636i	$0.176173\pi$
$854$	13.7148			0.000549543
$855$		0			0
$856$	3056.11			0.122028
$857$	10138.0	−	17559.6i	0.404094	−	0.699911i	−0.590122	−	0.807314i	$-0.700920\pi$
$857$	10138.0	−	17559.6i	0.404094	−	0.699911i	0.994216	+	0.107403i	$0.0342536\pi$
$858$		0			0
$859$	−3718.43	−	6440.51i	−0.147696	−	0.255818i	0.782679	−	0.622425i	$-0.213853\pi$
$859$	−3718.43	−	6440.51i	−0.147696	−	0.255818i	−0.930376	+	0.366608i	$0.880519\pi$
$860$	−10439.0	+	18080.8i	−0.413913	+	0.716919i
$861$		0			0
$862$	−2542.69			−0.100469
$863$	−35570.4			−1.40305			−0.701524	−	0.712646i	$-0.747497\pi$
$863$	−35570.4			−1.40305			−0.701524	+	0.712646i	$0.747497\pi$
$864$		0			0
$865$	9080.70	+	15728.2i	0.356940	+	0.618238i
$866$	7935.18			0.311372
$867$		0			0
$868$	6552.66	+	11349.5i	0.256235	+	0.443811i
$869$	−10492.5	+	18173.6i	−0.409591	+	0.709432i
$870$		0			0
$871$	−5477.34	+	9487.04i	−0.213080	+	0.369065i
$872$	−531.430	+	920.464i	−0.0206382	+	0.0357464i
$873$		0			0
$874$	−568.735	+	141.903i	−0.0220112	+	0.00549193i
$875$	8844.74			0.341722
$876$		0			0
$877$	−1934.21	+	3350.15i	−0.0744740	+	0.128993i	−0.900857	−	0.434115i	$-0.857061\pi$
$877$	−1934.21	+	3350.15i	−0.0744740	+	0.128993i	0.826383	+	0.563108i	$0.190394\pi$
$878$	209.038	+	362.065i	0.00803496	+	0.0139170i
$879$		0			0
$880$	7516.45	+	13018.9i	0.287931	+	0.498712i
$881$	20877.4			0.798384			0.399192	−	0.916867i	$-0.369291\pi$
$881$	20877.4			0.798384			0.399192	+	0.916867i	$0.369291\pi$
$882$		0			0
$883$	12103.4	+	20963.7i	0.461283	+	0.798965i	0.999025	−	0.0441440i	$-0.0140561\pi$
$883$	12103.4	+	20963.7i	0.461283	+	0.798965i	−0.537742	+	0.843109i	$0.680723\pi$
$884$	−13663.2	−	23665.4i	−0.519847	−	0.900401i
$885$		0			0
$886$	11468.1			0.434851
$887$	−13558.8	−	23484.5i	−0.513256	−	0.888986i	−0.999882	−	0.0153754i	$-0.995106\pi$
$887$	−13558.8	−	23484.5i	−0.513256	−	0.888986i	0.486625	−	0.873611i	$-0.338228\pi$
$888$		0			0
$889$	10006.8	+	17332.4i	0.377524	+	0.653890i
$890$	10268.9	−	17786.2i	0.386757	−	0.669883i
$891$		0			0
$892$	6473.07			0.242976
$893$	5232.77	+	5416.87i	0.196090	+	0.202988i
$894$		0			0
$895$	−14500.2	+	25115.0i	−0.541550	+	0.937993i
$896$	848.276	−	1469.26i	0.0316282	−	0.0547817i
$897$		0			0
$898$	−9741.13	+	16872.1i	−0.361989	+	0.626983i
$899$	10122.5	+	17532.7i	0.375534	+	0.650444i
$900$		0			0
$901$	−68638.5			−2.53793
$902$	22757.3	+	39416.8i	0.840062	+	1.45503i
$903$		0			0
$904$	−9103.52			−0.334932
$905$	−14729.2			−0.541010
$906$		0			0
$907$	17797.4	−	30826.0i	0.651548	−	1.12851i	−0.331199	−	0.943561i	$-0.607453\pi$
$907$	17797.4	−	30826.0i	0.651548	−	1.12851i	0.982747	−	0.184953i	$-0.0592134\pi$
$908$	−12130.9	−	21011.3i	−0.443367	−	0.767933i
$909$		0			0
$910$	12946.7	−	22424.3i	0.471624	−	0.816877i
$911$	−27909.5			−1.01502			−0.507510	−	0.861646i	$-0.669434\pi$
$911$	−27909.5			−1.01502			−0.507510	+	0.861646i	$0.669434\pi$
$912$		0			0
$913$	−45095.6			−1.63466
$914$	−17233.8	+	29849.8i	−0.623679	+	1.08024i
$915$		0			0
$916$	3832.73	+	6638.47i	0.138250	+	0.239456i
$917$	−2458.30	+	4257.91i	−0.0885282	+	0.153335i
$918$		0			0
$919$	−48960.0			−1.75739			−0.878696	−	0.477383i	$-0.841586\pi$
$919$	−48960.0			−1.75739			−0.878696	+	0.477383i	$0.841586\pi$
$920$	−403.552			−0.0144617
$921$		0			0
$922$	−1665.23	−	2884.27i	−0.0594811	−	0.103024i
$923$	−54264.5			−1.93515
$924$		0			0
$925$	−16460.6	−	28510.5i	−0.585102	−	1.01343i
$926$	−10694.0	+	18522.6i	−0.379511	+	0.657333i
$927$		0			0
$928$	1310.41	−	2269.70i	0.0463539	−	0.0802873i
$929$	−5985.87	+	10367.8i	−0.211399	+	0.366155i	−0.952153	−	0.305623i	$-0.901135\pi$
$929$	−5985.87	+	10367.8i	−0.211399	+	0.366155i	0.740753	+	0.671777i	$0.234469\pi$
$930$		0			0
$931$	13445.4	−	3354.71i	0.473313	−	0.118095i
$932$	16034.2			0.563537
$933$		0			0
$934$	−13212.7	+	22885.0i	−0.462882	+	0.801736i
$935$	−46834.1	−	81119.0i	−1.63812	−	2.83730i
$936$		0			0
$937$	25217.1	+	43677.3i	0.879197	+	1.52281i	0.852224	+	0.523177i	$0.175253\pi$
$937$	25217.1	+	43677.3i	0.879197	+	1.52281i	0.0269728	+	0.999636i	$0.491413\pi$
$938$	4237.72			0.147512
$939$		0			0
$940$	2592.57	+	4490.46i	0.0899578	+	0.155811i
$941$	6297.13	+	10906.9i	0.218151	+	0.377849i	0.954243	−	0.299033i	$-0.0966640\pi$
$941$	6297.13	+	10906.9i	0.218151	+	0.377849i	−0.736091	+	0.676882i	$0.763331\pi$
$942$		0			0
$943$	−1221.82			−0.0421930
$944$	1458.31	+	2525.87i	0.0502796	+	0.0870869i
$945$		0			0
$946$	−24135.5	−	41804.0i	−0.829508	−	1.43675i
$947$	13816.3	−	23930.5i	0.474096	−	0.821158i	−0.525464	−	0.850816i	$-0.676108\pi$
$947$	13816.3	−	23930.5i	0.474096	−	0.821158i	0.999560	+	0.0296574i	$0.00944162\pi$
$948$		0			0
$949$	−22074.2			−0.755067
$950$	12565.3	−	3135.12i	0.429128	−	0.107070i
$951$		0			0
$952$	−5285.50	+	9154.76i	−0.179941	+	0.311667i
$953$	16010.9	−	27731.6i	0.544221	−	0.942619i	−0.454434	−	0.890780i	$-0.650158\pi$
$953$	16010.9	−	27731.6i	0.544221	−	0.942619i	0.998655	−	0.0518388i	$-0.0165082\pi$
$954$		0			0
$955$	8092.64	−	14016.9i	0.274211	−	0.474947i
$956$	1060.58	+	1836.99i	0.0358805	+	0.0621468i
$957$		0			0
$958$	10819.1			0.364873
$959$	−11202.4	−	19403.1i	−0.377209	−	0.653345i
$960$		0			0
$961$	31311.7			1.05105
$962$	57707.7			1.93406
$963$		0			0
$964$	−2542.45	+	4403.65i	−0.0849448	+	0.147129i
$965$	7776.24	+	13468.9i	0.259405	+	0.449303i
$966$		0			0
$967$	6438.21	−	11151.3i	0.214104	−	0.370840i	−0.738891	−	0.673825i	$-0.764650\pi$
$967$	6438.21	−	11151.3i	0.214104	−	0.370840i	0.952995	+	0.302985i	$0.0979834\pi$
$968$	−24109.1			−0.800511
$969$		0			0
$970$	11842.3			0.391992
$971$	1207.26	−	2091.03i	0.0398998	−	0.0691085i	−0.845386	−	0.534156i	$-0.820629\pi$
$971$	1207.26	−	2091.03i	0.0398998	−	0.0691085i	0.885286	+	0.465048i	$0.153963\pi$
$972$		0			0
$973$	17969.7	+	31124.4i	0.592067	+	1.02549i
$974$	14297.0	−	24763.2i	0.470335	−	0.814644i
$975$		0			0
$976$	−8.27792			−0.000271485
$977$	14131.4			0.462746			0.231373	−	0.972865i	$-0.425678\pi$
$977$	14131.4			0.462746			0.231373	+	0.972865i	$0.425678\pi$
$978$		0			0
$979$	23742.3	+	41122.9i	0.775085	+	1.34249i
$980$	9540.31			0.310973
$981$		0			0
$982$	−2802.42	−	4853.94i	−0.0910681	−	0.157735i
$983$	−10134.1	+	17552.7i	−0.328817	+	0.569528i	−0.982277	−	0.187433i	$-0.939983\pi$
$983$	−10134.1	+	17552.7i	−0.328817	+	0.569528i	0.653460	+	0.756961i	$0.273317\pi$
$984$		0			0
$985$	−29492.7	+	51082.9i	−0.954027	+	1.65242i
$986$	−8165.02	+	14142.2i	−0.263719	+	0.456775i
$987$		0			0
$988$	−6253.57	+	21822.7i	−0.201369	+	0.702703i
$989$	1295.82			0.0416629
$990$		0			0
$991$	−7775.08	+	13466.8i	−0.249226	+	0.431673i	−0.963311	−	0.268386i	$-0.913510\pi$
$991$	−7775.08	+	13466.8i	−0.249226	+	0.431673i	0.714085	+	0.700059i	$0.246843\pi$
$992$	−3955.03	−	6850.32i	−0.126585	−	0.219252i
$993$		0			0
$994$	10495.9	+	18179.4i	0.334919	+	0.580096i
$995$	38974.6			1.24179
$996$		0			0
$997$	−6357.89	−	11012.2i	−0.201962	−	0.349809i	0.747198	−	0.664601i	$-0.231399\pi$
$997$	−6357.89	−	11012.2i	−0.201962	−	0.349809i	−0.949161	+	0.314792i	$0.898065\pi$
$998$	14178.3	+	24557.5i	0.449704	+	0.778910i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	342.4.g.h.163.3		6
3.2	odd	2		114.4.e.d.49.1	yes	6
19.7	even	3	inner	342.4.g.h.235.3		6
57.8	even	6		2166.4.a.t.1.3		3
57.11	odd	6		2166.4.a.u.1.3		3
57.26	odd	6		114.4.e.d.7.1	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
114.4.e.d.7.1	✓	6	57.26	odd	6
114.4.e.d.49.1	yes	6	3.2	odd	2
342.4.g.h.163.3		6	1.1	even	1	trivial
342.4.g.h.235.3		6	19.7	even	3	inner
2166.4.a.t.1.3		3	57.8	even	6
2166.4.a.u.1.3		3	57.11	odd	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 342 = 2 \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	342.g (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.00000 - 1.73205i) q^{2} +(-2.00000 - 3.46410i) q^{4} +(7.12716 - 12.3446i) q^{5} +13.2543 q^{7} -8.00000 q^{8} +O(q^{10})\)	\(q+(1.00000 - 1.73205i) q^{2} +(-2.00000 - 3.46410i) q^{4} +(7.12716 - 12.3446i) q^{5} +13.2543 q^{7} -8.00000 q^{8} +(-14.2543 - 24.6892i) q^{10} +65.9138 q^{11} +(34.2629 + 59.3451i) q^{13} +(13.2543 - 22.9571i) q^{14} +(-8.00000 + 13.8564i) q^{16} +(49.8470 - 86.3375i) q^{17} +(-80.3556 + 20.0493i) q^{19} -57.0172 q^{20} +(65.9138 - 114.166i) q^{22} +(1.76943 + 3.06475i) q^{23} +(-39.0927 - 67.7105i) q^{25} +137.052 q^{26} +(-26.5086 - 45.9143i) q^{28} +(-40.9504 - 70.9282i) q^{29} -247.190 q^{31} +(16.0000 + 27.7128i) q^{32} +(-99.6940 - 172.675i) q^{34} +(94.4655 - 163.619i) q^{35} +421.065 q^{37} +(-45.6293 + 159.229i) q^{38} +(-57.0172 + 98.7568i) q^{40} +(-172.629 + 299.003i) q^{41} +(183.084 - 317.111i) q^{43} +(-131.828 - 228.332i) q^{44} +7.07773 q^{46} +(-45.4699 - 78.7562i) q^{47} -167.323 q^{49} -156.371 q^{50} +(137.052 - 237.381i) q^{52} +(-344.246 - 596.251i) q^{53} +(469.778 - 813.680i) q^{55} -106.034 q^{56} -163.802 q^{58} +(91.1444 - 157.867i) q^{59} +(0.258685 + 0.448055i) q^{61} +(-247.190 + 428.145i) q^{62} +64.0000 q^{64} +976.789 q^{65} +(79.9310 + 138.445i) q^{67} -398.776 q^{68} +(-188.931 - 327.238i) q^{70} +(-395.942 + 685.792i) q^{71} +(-161.065 + 278.972i) q^{73} +(421.065 - 729.306i) q^{74} +(230.164 + 238.262i) q^{76} +873.642 q^{77} +(-159.185 + 275.717i) q^{79} +(114.034 + 197.514i) q^{80} +(345.259 + 598.005i) q^{82} -684.160 q^{83} +(-710.535 - 1230.68i) q^{85} +(-366.168 - 634.222i) q^{86} -527.311 q^{88} +(360.203 + 623.890i) q^{89} +(454.132 + 786.579i) q^{91} +(7.07773 - 12.2590i) q^{92} -181.880 q^{94} +(-325.207 + 1134.85i) q^{95} +(-207.696 + 359.740i) q^{97} +(-167.323 + 289.812i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{2} - 12 q^{4} + 10 q^{5} + 14 q^{7} - 48 q^{8}+O(q^{10}) \) 6 * q + 6 * q^2 - 12 * q^4 + 10 * q^5 + 14 * q^7 - 48 * q^8	\( 6 q + 6 q^{2} - 12 q^{4} + 10 q^{5} + 14 q^{7} - 48 q^{8} - 20 q^{10} + 88 q^{11} + 9 q^{13} + 14 q^{14} - 48 q^{16} - 84 q^{17} + 32 q^{19} - 80 q^{20} + 88 q^{22} - 2 q^{23} + 83 q^{25} + 36 q^{26} - 28 q^{28} + 92 q^{29} - 218 q^{31} + 96 q^{32} + 168 q^{34} + 282 q^{35} + 490 q^{37} + 74 q^{38} - 80 q^{40} - 688 q^{41} + 103 q^{43} - 176 q^{44} - 8 q^{46} + 322 q^{47} - 1508 q^{49} + 332 q^{50} + 36 q^{52} - 1322 q^{53} + 248 q^{55} - 112 q^{56} + 368 q^{58} + 252 q^{59} + 435 q^{61} - 218 q^{62} + 384 q^{64} + 3164 q^{65} + 719 q^{67} + 672 q^{68} - 564 q^{70} - 62 q^{71} + 581 q^{73} + 490 q^{74} + 20 q^{76} + 408 q^{77} + 489 q^{79} + 160 q^{80} + 1376 q^{82} - 4992 q^{83} - 1632 q^{85} - 206 q^{86} - 704 q^{88} + 1584 q^{89} + 1573 q^{91} - 8 q^{92} + 1288 q^{94} - 2362 q^{95} - 974 q^{97} - 1508 q^{98}+O(q^{100}) \) 6 * q + 6 * q^2 - 12 * q^4 + 10 * q^5 + 14 * q^7 - 48 * q^8 - 20 * q^10 + 88 * q^11 + 9 * q^13 + 14 * q^14 - 48 * q^16 - 84 * q^17 + 32 * q^19 - 80 * q^20 + 88 * q^22 - 2 * q^23 + 83 * q^25 + 36 * q^26 - 28 * q^28 + 92 * q^29 - 218 * q^31 + 96 * q^32 + 168 * q^34 + 282 * q^35 + 490 * q^37 + 74 * q^38 - 80 * q^40 - 688 * q^41 + 103 * q^43 - 176 * q^44 - 8 * q^46 + 322 * q^47 - 1508 * q^49 + 332 * q^50 + 36 * q^52 - 1322 * q^53 + 248 * q^55 - 112 * q^56 + 368 * q^58 + 252 * q^59 + 435 * q^61 - 218 * q^62 + 384 * q^64 + 3164 * q^65 + 719 * q^67 + 672 * q^68 - 564 * q^70 - 62 * q^71 + 581 * q^73 + 490 * q^74 + 20 * q^76 + 408 * q^77 + 489 * q^79 + 160 * q^80 + 1376 * q^82 - 4992 * q^83 - 1632 * q^85 - 206 * q^86 - 704 * q^88 + 1584 * q^89 + 1573 * q^91 - 8 * q^92 + 1288 * q^94 - 2362 * q^95 - 974 * q^97 - 1508 * q^98

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