LMFDB - Embedded newform 168.4.q.f.121.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [168,4,Mod(25,168)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(168, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 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("168.25");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 168 = 2^{3} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	168.q (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:	$9.91232088096$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 173x^{6} + 9457x^{4} + 168048x^{2} + 746496 $ x^8 + 173x^6 + 9457x^4 + 168048*x^2 + 746496
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6}\cdot 7 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			121.3
Root			$-2.57353i$ of defining polynomial
Character	$\chi$	$=$	168.121
Dual form			168.4.q.f.25.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.50000 - 2.59808i) q^{3} +(-0.0642956 + 0.111363i) q^{5} +(-0.866259 + 18.5000i) q^{7} +(-4.50000 + 7.79423i) q^{9} +O(q^{10})$	\(q+(-1.50000 - 2.59808i) q^{3} +(-0.0642956 + 0.111363i) q^{5} +(-0.866259 + 18.5000i) q^{7} +(-4.50000 + 7.79423i) q^{9} +(-27.0033 - 46.7711i) q^{11} -50.2350 q^{13} +0.385773 q^{15} +(65.7436 + 113.871i) q^{17} +(-45.7547 + 79.2495i) q^{19} +(49.3638 - 25.4994i) q^{21} +(-89.7436 + 155.441i) q^{23} +(62.4917 + 108.239i) q^{25} +27.0000 q^{27} -69.8961 q^{29} +(-163.423 - 283.058i) q^{31} +(-81.0099 + 140.313i) q^{33} +(-2.00452 - 1.28594i) q^{35} +(-150.849 + 261.278i) q^{37} +(75.3525 + 130.514i) q^{39} +296.048 q^{41} -144.302 q^{43} +(-0.578660 - 1.00227i) q^{45} +(180.043 - 311.843i) q^{47} +(-341.499 - 32.0516i) q^{49} +(197.231 - 341.614i) q^{51} +(-0.917567 - 1.58927i) q^{53} +6.94477 q^{55} +274.528 q^{57} +(26.6193 + 46.1060i) q^{59} +(54.0605 - 93.6356i) q^{61} +(-140.295 - 90.0018i) q^{63} +(3.22989 - 5.59433i) q^{65} +(-421.004 - 729.199i) q^{67} +538.462 q^{69} -241.111 q^{71} +(103.492 + 179.253i) q^{73} +(187.475 - 324.717i) q^{75} +(888.656 - 459.045i) q^{77} +(-279.981 + 484.942i) q^{79} +(-40.5000 - 70.1481i) q^{81} +986.652 q^{83} -16.9081 q^{85} +(104.844 + 181.595i) q^{87} +(221.683 - 383.966i) q^{89} +(43.5165 - 929.348i) q^{91} +(-490.270 + 849.173i) q^{93} +(-5.88365 - 10.1908i) q^{95} -740.815 q^{97} +486.059 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 12 q^{3} - 4 q^{5} + 18 q^{7} - 36 q^{9}+O(q^{10}) $ 8 * q - 12 * q^3 - 4 * q^5 + 18 * q^7 - 36 * q^9	$ 8 q - 12 q^{3} - 4 q^{5} + 18 q^{7} - 36 q^{9} - 14 q^{11} + 44 q^{13} + 24 q^{15} - 96 q^{17} + 26 q^{19} - 36 q^{21} - 96 q^{23} - 110 q^{25} + 216 q^{27} - 152 q^{29} - 238 q^{31} - 42 q^{33} + 152 q^{35} - 562 q^{37} - 66 q^{39} + 856 q^{41} - 516 q^{43} - 36 q^{45} + 80 q^{47} + 156 q^{49} - 288 q^{51} + 2952 q^{55} - 156 q^{57} - 262 q^{59} + 276 q^{61} - 54 q^{63} - 2196 q^{65} - 150 q^{67} + 576 q^{69} - 1696 q^{71} + 218 q^{73} - 330 q^{75} - 764 q^{77} - 1762 q^{79} - 324 q^{81} + 6900 q^{83} + 2904 q^{85} + 228 q^{87} + 344 q^{89} - 2806 q^{91} - 714 q^{93} - 2004 q^{95} - 1244 q^{97} + 252 q^{99}+O(q^{100}) $ 8 * q - 12 * q^3 - 4 * q^5 + 18 * q^7 - 36 * q^9 - 14 * q^11 + 44 * q^13 + 24 * q^15 - 96 * q^17 + 26 * q^19 - 36 * q^21 - 96 * q^23 - 110 * q^25 + 216 * q^27 - 152 * q^29 - 238 * q^31 - 42 * q^33 + 152 * q^35 - 562 * q^37 - 66 * q^39 + 856 * q^41 - 516 * q^43 - 36 * q^45 + 80 * q^47 + 156 * q^49 - 288 * q^51 + 2952 * q^55 - 156 * q^57 - 262 * q^59 + 276 * q^61 - 54 * q^63 - 2196 * q^65 - 150 * q^67 + 576 * q^69 - 1696 * q^71 + 218 * q^73 - 330 * q^75 - 764 * q^77 - 1762 * q^79 - 324 * q^81 + 6900 * q^83 + 2904 * q^85 + 228 * q^87 + 344 * q^89 - 2806 * q^91 - 714 * q^93 - 2004 * q^95 - 1244 * q^97 + 252 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/168\mathbb{Z}\right)^\times$.

$n$	$73$	$85$	$113$	$127$
$\chi(n)$	$e\left(\frac{1}{3}\right)$	$1$	$1$	$1$

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 $

$2$		0			0
$2$		0			0
$3$	−1.50000	−	2.59808i	−0.288675	−	0.500000i
$3$	−1.50000	−	2.59808i	−0.288675	−	0.500000i
$4$		0			0
$5$	−0.0642956	+	0.111363i	−0.00575077	+	0.00996063i	−0.868886	−	0.495011i	$-0.835164\pi$
$5$	−0.0642956	+	0.111363i	−0.00575077	+	0.00996063i	0.863136	+	0.504972i	$0.168497\pi$
$6$		0			0
$7$	−0.866259	+	18.5000i	−0.0467736	+	0.998906i
$7$	−0.866259	+	18.5000i	−0.0467736	+	0.998906i
$8$		0			0
$9$	−4.50000	+	7.79423i	−0.166667	+	0.288675i
$10$		0			0
$11$	−27.0033	−	46.7711i	−0.740163	−	1.28200i	−0.952421	−	0.304787i	$-0.901415\pi$
$11$	−27.0033	−	46.7711i	−0.740163	−	1.28200i	0.212257	−	0.977214i	$-0.431919\pi$
$12$		0			0
$13$	−50.2350			−1.07175			−0.535873	−	0.844299i	$-0.680017\pi$
$13$	−50.2350			−1.07175			−0.535873	+	0.844299i	$0.680017\pi$
$14$		0			0
$15$	0.385773			0.00664042
$16$		0			0
$17$	65.7436	+	113.871i	0.937951	+	1.62458i	0.769285	+	0.638906i	$0.220613\pi$
$17$	65.7436	+	113.871i	0.937951	+	1.62458i	0.168666	+	0.985673i	$0.446054\pi$
$18$		0			0
$19$	−45.7547	+	79.2495i	−0.552466	+	0.956899i	0.445630	+	0.895217i	$0.352980\pi$
$19$	−45.7547	+	79.2495i	−0.552466	+	0.956899i	−0.998096	+	0.0616814i	$0.980354\pi$
$20$		0			0
$21$	49.3638	−	25.4994i	0.512955	−	0.264972i
$22$		0			0
$23$	−89.7436	+	155.441i	−0.813602	+	1.40920i	0.0967260	+	0.995311i	$0.469163\pi$
$23$	−89.7436	+	155.441i	−0.813602	+	1.40920i	−0.910328	+	0.413888i	$0.864170\pi$
$24$		0			0
$25$	62.4917	+	108.239i	0.499934	+	0.865911i
$26$		0			0
$27$	27.0000			0.192450
$28$		0			0
$29$	−69.8961			−0.447565			−0.223782	−	0.974639i	$-0.571841\pi$
$29$	−69.8961			−0.447565			−0.223782	+	0.974639i	$0.571841\pi$
$30$		0			0
$31$	−163.423	−	283.058i	−0.946829	−	1.63996i	−0.752046	−	0.659110i	$-0.770933\pi$
$31$	−163.423	−	283.058i	−0.946829	−	1.63996i	−0.194783	−	0.980846i	$-0.562400\pi$
$32$		0			0
$33$	−81.0099	+	140.313i	−0.427334	+	0.740163i
$34$		0			0
$35$	−2.00452	−	1.28594i	−0.00968074	−	0.00621037i
$36$		0			0
$37$	−150.849	+	261.278i	−0.670254	+	1.16091i	0.307578	+	0.951523i	$0.400481\pi$
$37$	−150.849	+	261.278i	−0.670254	+	1.16091i	−0.977832	+	0.209391i	$0.932852\pi$
$38$		0			0
$39$	75.3525	+	130.514i	0.309386	+	0.535873i
$40$		0			0
$41$	296.048			1.12768			0.563840	−	0.825884i	$-0.309324\pi$
$41$	296.048			1.12768			0.563840	+	0.825884i	$0.309324\pi$
$42$		0			0
$43$	−144.302			−0.511764			−0.255882	−	0.966708i	$-0.582366\pi$
$43$	−144.302			−0.511764			−0.255882	+	0.966708i	$0.582366\pi$
$44$		0			0
$45$	−0.578660	−	1.00227i	−0.00191692	−	0.00332021i
$46$		0			0
$47$	180.043	−	311.843i	0.558764	−	0.967808i	−0.438836	−	0.898567i	$-0.644609\pi$
$47$	180.043	−	311.843i	0.558764	−	0.967808i	0.997600	−	0.0692409i	$-0.0220577\pi$
$48$		0			0
$49$	−341.499	−	32.0516i	−0.995624	−	0.0934448i
$50$		0			0
$51$	197.231	−	341.614i	0.541526	−	0.937951i
$52$		0			0
$53$	−0.917567	−	1.58927i	−0.00237807	−	0.00411893i	0.864834	−	0.502058i	$-0.167424\pi$
$53$	−0.917567	−	1.58927i	−0.00237807	−	0.00411893i	−0.867212	+	0.497939i	$0.834090\pi$
$54$		0			0
$55$	6.94477			0.0170260
$56$		0			0
$57$	274.528			0.637932
$58$		0			0
$59$	26.6193	+	46.1060i	0.0587379	+	0.101737i	0.893899	−	0.448268i	$-0.147959\pi$
$59$	26.6193	+	46.1060i	0.0587379	+	0.101737i	−0.835161	+	0.550005i	$0.814626\pi$
$60$		0			0
$61$	54.0605	−	93.6356i	0.113471	−	0.196538i	−0.803696	−	0.595040i	$-0.797136\pi$
$61$	54.0605	−	93.6356i	0.113471	−	0.196538i	0.917168	+	0.398502i	$0.130470\pi$
$62$		0			0
$63$	−140.295	−	90.0018i	−0.280564	−	0.179987i
$64$		0			0
$65$	3.22989	−	5.59433i	0.00616336	−	0.0106753i
$66$		0			0
$67$	−421.004	−	729.199i	−0.767668	−	1.32964i	−0.938824	−	0.344396i	$-0.888084\pi$
$67$	−421.004	−	729.199i	−0.767668	−	1.32964i	0.171156	−	0.985244i	$-0.445250\pi$
$68$		0			0
$69$	538.462			0.939466
$70$		0			0
$71$	−241.111			−0.403023			−0.201511	−	0.979486i	$-0.564585\pi$
$71$	−241.111			−0.403023			−0.201511	+	0.979486i	$0.564585\pi$
$72$		0			0
$73$	103.492	+	179.253i	0.165929	+	0.287397i	0.936985	−	0.349370i	$-0.113605\pi$
$73$	103.492	+	179.253i	0.165929	+	0.287397i	−0.771056	+	0.636767i	$0.780271\pi$
$74$		0			0
$75$	187.475	−	324.717i	0.288637	−	0.499934i
$76$		0			0
$77$	888.656	−	459.045i	1.31522	−	0.679390i
$78$		0			0
$79$	−279.981	+	484.942i	−0.398738	+	0.690635i	−0.993571	−	0.113215i	$-0.963885\pi$
$79$	−279.981	+	484.942i	−0.398738	+	0.690635i	0.594832	+	0.803850i	$0.297219\pi$
$80$		0			0
$81$	−40.5000	−	70.1481i	−0.0555556	−	0.0962250i
$82$		0			0
$83$	986.652			1.30481			0.652404	−	0.757871i	$-0.273760\pi$
$83$	986.652			1.30481			0.652404	+	0.757871i	$0.273760\pi$
$84$		0			0
$85$	−16.9081			−0.0215758
$86$		0			0
$87$	104.844	+	181.595i	0.129201	+	0.223782i
$88$		0			0
$89$	221.683	−	383.966i	0.264026	−	0.457307i	−0.703282	−	0.710911i	$-0.748283\pi$
$89$	221.683	−	383.966i	0.264026	−	0.457307i	0.967308	+	0.253604i	$0.0816161\pi$
$90$		0			0
$91$	43.5165	−	929.348i	0.0501294	−	1.07057i
$92$		0			0
$93$	−490.270	+	849.173i	−0.546652	+	0.946829i
$94$		0			0
$95$	−5.88365	−	10.1908i	−0.00635421	−	0.0110058i
$96$		0			0
$97$	−740.815			−0.775447			−0.387723	−	0.921776i	$-0.626738\pi$
$97$	−740.815			−0.775447			−0.387723	+	0.921776i	$0.626738\pi$
$98$		0			0
$99$	486.059			0.493442
$100$		0			0
$101$	−371.888	−	644.130i	−0.366379	−	0.634587i	0.622617	−	0.782526i	$-0.286069\pi$
$101$	−371.888	−	644.130i	−0.366379	−	0.634587i	−0.988996	+	0.147939i	$0.952736\pi$
$102$		0			0
$103$	−52.3253	+	90.6300i	−0.0500559	+	0.0866994i	−0.889968	−	0.456024i	$-0.849273\pi$
$103$	−52.3253	+	90.6300i	−0.0500559	+	0.0866994i	0.839912	+	0.542723i	$0.182607\pi$
$104$		0			0
$105$	−0.334180	+	7.13680i	−0.000310596	+	0.00663315i
$106$		0			0
$107$	−256.850	+	444.878i	−0.232062	+	0.401943i	−0.958415	−	0.285379i	$-0.907881\pi$
$107$	−256.850	+	444.878i	−0.232062	+	0.401943i	0.726353	+	0.687322i	$0.241214\pi$
$108$		0			0
$109$	−487.519	−	844.408i	−0.428402	−	0.742015i	0.568329	−	0.822801i	$-0.307590\pi$
$109$	−487.519	−	844.408i	−0.428402	−	0.742015i	−0.996731	+	0.0807868i	$0.974257\pi$
$110$		0			0
$111$	905.093			0.773942
$112$		0			0
$113$	1926.07			1.60345			0.801723	−	0.597696i	$-0.203917\pi$
$113$	1926.07			1.60345			0.801723	+	0.597696i	$0.203917\pi$
$114$		0			0
$115$	−11.5402	−	19.9883i	−0.00935767	−	0.0162080i
$116$		0			0
$117$	226.058	−	391.543i	0.178624	−	0.309386i
$118$		0			0
$119$	−2163.57	+	1117.61i	−1.66667	+	0.860937i
$120$		0			0
$121$	−792.855	+	1373.27i	−0.595684	+	1.03175i
$122$		0			0
$123$	−444.071	−	769.154i	−0.325533	−	0.563840i
$124$		0			0
$125$	−32.1457			−0.0230016
$126$		0			0
$127$	−1125.95			−0.786709			−0.393355	−	0.919387i	$-0.628686\pi$
$127$	−1125.95			−0.786709			−0.393355	+	0.919387i	$0.628686\pi$
$128$		0			0
$129$	216.453	+	374.908i	0.147734	+	0.255882i
$130$		0			0
$131$	−746.621	+	1293.19i	−0.497959	+	0.862490i	−0.999997	−	0.00235541i	$-0.999250\pi$
$131$	−746.621	+	1293.19i	−0.497959	+	0.862490i	0.502038	+	0.864845i	$0.332584\pi$
$132$		0			0
$133$	−1426.48	−	915.112i	−0.930010	−	0.596619i
$134$		0			0
$135$	−1.73598	+	3.00681i	−0.00110674	+	0.00191692i
$136$		0			0
$137$	−730.386	−	1265.07i	−0.455483	−	0.788919i	0.543233	−	0.839582i	$-0.317200\pi$
$137$	−730.386	−	1265.07i	−0.455483	−	0.788919i	−0.998716	+	0.0506627i	$0.983867\pi$
$138$		0			0
$139$	2225.85			1.35823			0.679116	−	0.734031i	$-0.262363\pi$
$139$	2225.85			1.35823			0.679116	+	0.734031i	$0.262363\pi$
$140$		0			0
$141$	−1080.26			−0.645206
$142$		0			0
$143$	1356.51	+	2349.55i	0.793267	+	1.37398i
$144$		0			0
$145$	4.49401	−	7.78385i	0.00257384	−	0.00445803i
$146$		0			0
$147$	428.976	+	935.318i	0.240690	+	0.524787i
$148$		0			0
$149$	−197.340	+	341.804i	−0.108502	+	0.187930i	−0.915163	−	0.403083i	$-0.867939\pi$
$149$	−197.340	+	341.804i	−0.108502	+	0.187930i	0.806662	+	0.591013i	$0.201272\pi$
$150$		0			0
$151$	1562.10	+	2705.64i	0.841867	+	1.45816i	0.888314	+	0.459236i	$0.151877\pi$
$151$	1562.10	+	2705.64i	0.841867	+	1.45816i	−0.0464473	+	0.998921i	$0.514790\pi$
$152$		0			0
$153$	−1183.39			−0.625301
$154$		0			0
$155$	42.0296			0.0217800
$156$		0			0
$157$	1800.35	+	3118.30i	0.915183	+	1.58514i	0.806633	+	0.591053i	$0.201288\pi$
$157$	1800.35	+	3118.30i	0.915183	+	1.58514i	0.108551	+	0.994091i	$0.465379\pi$
$158$		0			0
$159$	−2.75270	+	4.76782i	−0.00137298	+	0.00237807i
$160$		0			0
$161$	−2797.91	−	1794.91i	−1.36960	−	0.878624i
$162$		0			0
$163$	−987.012	+	1709.55i	−0.474287	+	0.821489i	−0.999567	−	0.0294409i	$-0.990627\pi$
$163$	−987.012	+	1709.55i	−0.474287	+	0.821489i	0.525280	+	0.850930i	$0.323961\pi$
$164$		0			0
$165$	−10.4172	−	18.0430i	−0.00491499	−	0.00851302i
$166$		0			0
$167$	1067.09			0.494453			0.247227	−	0.968958i	$-0.420481\pi$
$167$	1067.09			0.494453			0.247227	+	0.968958i	$0.420481\pi$
$168$		0			0
$169$	326.559			0.148638
$170$		0			0
$171$	−411.792	−	713.245i	−0.184155	−	0.318966i
$172$		0			0
$173$	137.327	−	237.858i	0.0603515	−	0.104532i	−0.834271	−	0.551355i	$-0.814111\pi$
$173$	137.327	−	237.858i	0.0603515	−	0.104532i	0.894623	+	0.446823i	$0.147445\pi$
$174$		0			0
$175$	−2056.55	+	1062.33i	−0.888347	+	0.458885i
$176$		0			0
$177$	79.8578	−	138.318i	0.0339123	−	0.0587379i
$178$		0			0
$179$	−277.166	−	480.066i	−0.115734	−	0.200457i	0.802339	−	0.596869i	$-0.203589\pi$
$179$	−277.166	−	480.066i	−0.115734	−	0.200457i	−0.918073	+	0.396412i	$0.870255\pi$
$180$		0			0
$181$	−685.436			−0.281481			−0.140741	−	0.990047i	$-0.544948\pi$
$181$	−685.436			−0.281481			−0.140741	+	0.990047i	$0.544948\pi$
$182$		0			0
$183$	−324.363			−0.131025
$184$		0			0
$185$	−19.3978	−	33.5980i	−0.00770895	−	0.0133523i
$186$		0			0
$187$	3550.59	−	6149.80i	1.38847	−	2.40491i
$188$		0			0
$189$	−23.3890	+	499.500i	−0.00900158	+	0.192239i
$190$		0			0
$191$	−2449.44	+	4242.55i	−0.927932	+	1.60722i	−0.141155	+	0.989988i	$0.545082\pi$
$191$	−2449.44	+	4242.55i	−0.927932	+	1.60722i	−0.786777	+	0.617237i	$0.788252\pi$
$192$		0			0
$193$	−1570.13	−	2719.55i	−0.585598	−	1.01429i	−0.994801	−	0.101842i	$-0.967526\pi$
$193$	−1570.13	−	2719.55i	−0.585598	−	1.01429i	0.409202	−	0.912444i	$-0.365807\pi$
$194$		0			0
$195$	−19.3793			−0.00711684
$196$		0			0
$197$	−227.412			−0.0822460			−0.0411230	−	0.999154i	$-0.513094\pi$
$197$	−227.412			−0.0822460			−0.0411230	+	0.999154i	$0.513094\pi$
$198$		0			0
$199$	607.250	+	1051.79i	0.216316	+	0.374670i	0.953679	−	0.300827i	$-0.0972626\pi$
$199$	607.250	+	1051.79i	0.216316	+	0.374670i	−0.737363	+	0.675497i	$0.763929\pi$
$200$		0			0
$201$	−1263.01	+	2187.60i	−0.443213	+	0.767668i
$202$		0			0
$203$	60.5481	−	1293.08i	0.0209342	−	0.447075i
$204$		0			0
$205$	−19.0345	+	32.9688i	−0.00648502	+	0.0112324i
$206$		0			0
$207$	−807.693	−	1398.96i	−0.271201	−	0.469733i
$208$		0			0
$209$	4942.11			1.63566
$210$		0			0
$211$	5116.07			1.66922			0.834608	−	0.550844i	$-0.185694\pi$
$211$	5116.07			1.66922			0.834608	+	0.550844i	$0.185694\pi$
$212$		0			0
$213$	361.666	+	626.425i	0.116343	+	0.201511i
$214$		0			0
$215$	9.27799	−	16.0699i	0.00294304	−	0.00509749i
$216$		0			0
$217$	5378.13	−	2778.13i	1.68245	−	0.869087i
$218$		0			0
$219$	310.475	−	537.759i	0.0957989	−	0.165929i
$220$		0			0
$221$	−3302.63	−	5720.33i	−1.00524	−	1.74114i
$222$		0			0
$223$	119.384			0.0358499			0.0179250	−	0.999839i	$-0.494294\pi$
$223$	119.384			0.0358499			0.0179250	+	0.999839i	$0.494294\pi$
$224$		0			0
$225$	−1124.85			−0.333289
$226$		0			0
$227$	−996.000	−	1725.12i	−0.291220	−	0.504407i	0.682879	−	0.730532i	$-0.260728\pi$
$227$	−996.000	−	1725.12i	−0.291220	−	0.504407i	−0.974098	+	0.226124i	$0.927394\pi$
$228$		0			0
$229$	−351.738	+	609.228i	−0.101500	+	0.175803i	−0.912303	−	0.409516i	$-0.865698\pi$
$229$	−351.738	+	609.228i	−0.101500	+	0.175803i	0.810803	+	0.585319i	$0.199031\pi$
$230$		0			0
$231$	−2525.62	−	1620.23i	−0.719365	−	0.461486i
$232$		0			0
$233$	−480.892	+	832.929i	−0.135211	+	0.234193i	−0.925678	−	0.378312i	$-0.876505\pi$
$233$	−480.892	+	832.929i	−0.135211	+	0.234193i	0.790467	+	0.612505i	$0.209838\pi$
$234$		0			0
$235$	23.1519	+	40.1003i	0.00642665	+	0.0111313i
$236$		0			0
$237$	1679.89			0.460423
$238$		0			0
$239$	−4464.71			−1.20836			−0.604179	−	0.796848i	$-0.706499\pi$
$239$	−4464.71			−1.20836			−0.604179	+	0.796848i	$0.706499\pi$
$240$		0			0
$241$	217.656	+	376.991i	0.0581761	+	0.100764i	0.893647	−	0.448771i	$-0.148138\pi$
$241$	217.656	+	376.991i	0.0581761	+	0.100764i	−0.835471	+	0.549535i	$0.814805\pi$
$242$		0			0
$243$	−121.500	+	210.444i	−0.0320750	+	0.0555556i
$244$		0			0
$245$	25.5262	−	35.9697i	0.00665638	−	0.00937966i
$246$		0			0
$247$	2298.49	−	3981.10i	0.592103	−	1.02555i
$248$		0			0
$249$	−1479.98	−	2563.40i	−0.376666	−	0.652404i
$250$		0			0
$251$	863.003			0.217021			0.108510	−	0.994095i	$-0.465392\pi$
$251$	863.003			0.217021			0.108510	+	0.994095i	$0.465392\pi$
$252$		0			0
$253$	9693.49			2.40879
$254$		0			0
$255$	25.3621	+	43.9285i	0.00622839	+	0.0107879i
$256$		0			0
$257$	268.346	−	464.790i	0.0651322	−	0.112812i	−0.831620	−	0.555344i	$-0.812586\pi$
$257$	268.346	−	464.790i	0.0651322	−	0.112812i	0.896753	+	0.442532i	$0.145920\pi$
$258$		0			0
$259$	−4702.96	−	3017.03i	−1.12829	−	0.723820i
$260$		0			0
$261$	314.533	−	544.786i	0.0745942	−	0.129201i
$262$		0			0
$263$	−185.954	−	322.082i	−0.0435986	−	0.0755149i	0.843403	−	0.537282i	$-0.180549\pi$
$263$	−185.954	−	322.082i	−0.0435986	−	0.0755149i	−0.887001	+	0.461767i	$0.847216\pi$
$264$		0			0
$265$	0.235982			5.47029e−5
$266$		0			0
$267$	−1330.10			−0.304871
$268$		0			0
$269$	2252.80	+	3901.97i	0.510616	+	0.884414i	0.999924	+	0.0123024i	$0.00391609\pi$
$269$	2252.80	+	3901.97i	0.510616	+	0.884414i	−0.489308	+	0.872111i	$0.662751\pi$
$270$		0			0
$271$	198.058	−	343.046i	0.0443954	−	0.0768951i	−0.842974	−	0.537955i	$-0.819197\pi$
$271$	198.058	−	343.046i	0.0443954	−	0.0768951i	0.887369	+	0.461059i	$0.152531\pi$
$272$		0			0
$273$	−2479.79	+	1280.96i	−0.549757	+	0.283983i
$274$		0			0
$275$	3374.96	−	5845.61i	0.740066	−	1.28183i
$276$		0			0
$277$	3254.37	+	5636.73i	0.705907	+	1.22267i	0.966363	+	0.257181i	$0.0827935\pi$
$277$	3254.37	+	5636.73i	0.705907	+	1.22267i	−0.260457	+	0.965486i	$0.583873\pi$
$278$		0			0
$279$	2941.62			0.631220
$280$		0			0
$281$	2785.80			0.591413			0.295707	−	0.955279i	$-0.404445\pi$
$281$	2785.80			0.591413			0.295707	+	0.955279i	$0.404445\pi$
$282$		0			0
$283$	−0.305923	−	0.529874i	−6.42588e−5	−	0.000111299i	0.865993	−	0.500056i	$-0.166687\pi$
$283$	−0.305923	−	0.529874i	−6.42588e−5	−	0.000111299i	−0.866058	+	0.499944i	$0.833354\pi$
$284$		0			0
$285$	−17.6509	+	30.5723i	−0.00366860	+	0.00635421i
$286$		0			0
$287$	−256.454	+	5476.88i	−0.0527456	+	1.12645i
$288$		0			0
$289$	−6187.95	+	10717.8i	−1.25950	+	2.18153i
$290$		0			0
$291$	1111.22	+	1924.69i	0.223852	+	0.387723i
$292$		0			0
$293$	−4145.98			−0.826657			−0.413329	−	0.910582i	$-0.635634\pi$
$293$	−4145.98			−0.826657			−0.413329	+	0.910582i	$0.635634\pi$
$294$		0			0
$295$	−6.84601			−0.00135115
$296$		0			0
$297$	−729.089	−	1262.82i	−0.142445	−	0.246721i
$298$		0			0
$299$	4508.27	−	7808.56i	0.871974	−	1.51030i
$300$		0			0
$301$	125.003	−	2669.59i	0.0239371	−	0.511204i
$302$		0			0
$303$	−1115.67	+	1932.39i	−0.211529	+	0.366379i
$304$		0			0
$305$	6.95171	+	12.0407i	0.00130509	+	0.00226049i
$306$		0			0
$307$	1960.53			0.364473			0.182236	−	0.983255i	$-0.441666\pi$
$307$	1960.53			0.364473			0.182236	+	0.983255i	$0.441666\pi$
$308$		0			0
$309$	313.952			0.0577996
$310$		0			0
$311$	−2603.78	−	4509.89i	−0.474749	−	0.822290i	0.524833	−	0.851206i	$-0.324128\pi$
$311$	−2603.78	−	4509.89i	−0.474749	−	0.822290i	−0.999582	+	0.0289155i	$0.990795\pi$
$312$		0			0
$313$	1995.86	−	3456.92i	0.360423	−	0.624271i	−0.627607	−	0.778530i	$-0.715966\pi$
$313$	1995.86	−	3456.92i	0.360423	−	0.624271i	0.988030	+	0.154259i	$0.0492990\pi$
$314$		0			0
$315$	19.0432	−	9.83698i	0.00340624	−	0.00175953i
$316$		0			0
$317$	−902.519	+	1563.21i	−0.159907	+	0.276967i	−0.934835	−	0.355083i	$-0.884453\pi$
$317$	−902.519	+	1563.21i	−0.159907	+	0.276967i	0.774928	+	0.632050i	$0.217786\pi$
$318$		0			0
$319$	1887.43	+	3269.12i	0.331271	+	0.573779i
$320$		0			0
$321$	1541.10			0.267962
$322$		0			0
$323$	−12032.3			−2.07274
$324$		0			0
$325$	−3139.27	−	5437.38i	−0.535802	−	0.928036i
$326$		0			0
$327$	−1462.56	+	2533.22i	−0.247338	+	0.428402i
$328$		0			0
$329$	5613.13	+	3600.92i	0.940614	+	0.603421i
$330$		0			0
$331$	−3453.09	+	5980.93i	−0.573411	+	0.993177i	0.422801	+	0.906222i	$0.361047\pi$
$331$	−3453.09	+	5980.93i	−0.573411	+	0.993177i	−0.996212	+	0.0869547i	$0.972286\pi$
$332$		0			0
$333$	−1357.64	−	2351.50i	−0.223418	−	0.386971i
$334$		0			0
$335$	108.275			0.0176587
$336$		0			0
$337$	−6081.36			−0.983006			−0.491503	−	0.870876i	$-0.663552\pi$
$337$	−6081.36			−0.983006			−0.491503	+	0.870876i	$0.663552\pi$
$338$		0			0
$339$	−2889.10	−	5004.07i	−0.462875	−	0.801723i
$340$		0			0
$341$	−8825.94	+	15287.0i	−1.40162	+	2.42767i
$342$		0			0
$343$	888.780	−	6289.97i	0.139911	−	0.990164i
$344$		0			0
$345$	−34.6207	+	59.9648i	−0.00540265	+	0.00935767i
$346$		0			0
$347$	3893.51	+	6743.75i	0.602347	+	1.04330i	0.992465	+	0.122531i	$0.0391011\pi$
$347$	3893.51	+	6743.75i	0.602347	+	1.04330i	−0.390117	+	0.920765i	$0.627566\pi$
$348$		0			0
$349$	−1928.25			−0.295751			−0.147875	−	0.989006i	$-0.547243\pi$
$349$	−1928.25			−0.295751			−0.147875	+	0.989006i	$0.547243\pi$
$350$		0			0
$351$	−1356.35			−0.206258
$352$		0			0
$353$	1702.14	+	2948.18i	0.256645	+	0.444521i	0.965341	−	0.260992i	$-0.0840497\pi$
$353$	1702.14	+	2948.18i	0.256645	+	0.444521i	−0.708696	+	0.705514i	$0.750716\pi$
$354$		0			0
$355$	15.5024	−	26.8509i	0.00231769	−	0.00401436i
$356$		0			0
$357$	6149.00	+	3944.69i	0.911595	+	0.584805i
$358$		0			0
$359$	−2208.53	+	3825.28i	−0.324684	+	0.562369i	−0.981448	−	0.191727i	$-0.938591\pi$
$359$	−2208.53	+	3825.28i	−0.324684	+	0.562369i	0.656764	+	0.754096i	$0.271925\pi$
$360$		0			0
$361$	−757.484	−	1312.00i	−0.110436	−	0.191282i
$362$		0			0
$363$	4757.13			0.687837
$364$		0			0
$365$	−26.6162			−0.00381687
$366$		0			0
$367$	2538.33	+	4396.51i	0.361034	+	0.625329i	0.988131	−	0.153612i	$-0.0490905\pi$
$367$	2538.33	+	4396.51i	0.361034	+	0.625329i	−0.627097	+	0.778941i	$0.715757\pi$
$368$		0			0
$369$	−1332.21	+	2307.46i	−0.187947	+	0.325533i
$370$		0			0
$371$	30.1964	−	15.5983i	0.00422566	−	0.00218281i
$372$		0			0
$373$	3438.32	−	5955.35i	0.477291	−	0.826692i	−0.522370	−	0.852719i	$-0.674952\pi$
$373$	3438.32	−	5955.35i	0.477291	−	0.826692i	0.999661	+	0.0260264i	$0.00828540\pi$
$374$		0			0
$375$	48.2185	+	83.5169i	0.00663998	+	0.0115008i
$376$		0			0
$377$	3511.23			0.479676
$378$		0			0
$379$	−9285.61			−1.25850			−0.629248	−	0.777205i	$-0.716637\pi$
$379$	−9285.61			−1.25850			−0.629248	+	0.777205i	$0.716637\pi$
$380$		0			0
$381$	1688.93	+	2925.31i	0.227103	+	0.393355i
$382$		0			0
$383$	3840.83	−	6652.51i	0.512420	−	0.887538i	−0.487476	−	0.873136i	$-0.662082\pi$
$383$	3840.83	−	6652.51i	0.512420	−	0.887538i	0.999896	−	0.0144017i	$-0.00458437\pi$
$384$		0			0
$385$	−6.01597	+	128.478i	−0.000796369	+	0.0170074i
$386$		0			0
$387$	649.360	−	1124.72i	0.0852941	−	0.147734i
$388$		0			0
$389$	−4313.86	−	7471.83i	−0.562266	−	0.973873i	−0.997298	−	0.0734585i	$-0.976596\pi$
$389$	−4313.86	−	7471.83i	−0.562266	−	0.973873i	0.435032	−	0.900415i	$-0.356737\pi$
$390$		0			0
$391$	−23600.3			−3.05247
$392$		0			0
$393$	4479.73			0.574993
$394$		0			0
$395$	−36.0031	−	62.3592i	−0.00458611	−	0.00794337i
$396$		0			0
$397$	−2867.81	+	4967.19i	−0.362547	+	0.627949i	−0.988379	−	0.152008i	$-0.951426\pi$
$397$	−2867.81	+	4967.19i	−0.362547	+	0.627949i	0.625832	+	0.779957i	$0.284759\pi$
$398$		0			0
$399$	−237.812	+	5078.77i	−0.0298384	+	0.637234i
$400$		0			0
$401$	3608.89	−	6250.79i	0.449425	−	0.778427i	−0.548923	−	0.835873i	$-0.684962\pi$
$401$	3608.89	−	6250.79i	0.449425	−	0.778427i	0.998349	+	0.0574453i	$0.0182955\pi$
$402$		0			0
$403$	8209.58	+	14219.4i	1.01476	+	1.75762i
$404$		0			0
$405$	10.4159			0.00127795
$406$		0			0
$407$	16293.7			1.98439
$408$		0			0
$409$	5401.46	+	9355.60i	0.653019	+	1.13106i	0.982386	+	0.186861i	$0.0598313\pi$
$409$	5401.46	+	9355.60i	0.653019	+	1.13106i	−0.329367	+	0.944202i	$0.606835\pi$
$410$		0			0
$411$	−2191.16	+	3795.20i	−0.262973	+	0.455483i
$412$		0			0
$413$	−876.019	+	452.517i	−0.104373	+	0.0539150i
$414$		0			0
$415$	−63.4373	+	109.877i	−0.00750365	+	0.0129967i
$416$		0			0
$417$	−3338.78	−	5782.93i	−0.392088	−	0.679116i
$418$		0			0
$419$	13257.1			1.54571			0.772856	−	0.634582i	$-0.218828\pi$
$419$	13257.1			1.54571			0.772856	+	0.634582i	$0.218828\pi$
$420$		0			0
$421$	−6252.11			−0.723774			−0.361887	−	0.932222i	$-0.617867\pi$
$421$	−6252.11			−0.723774			−0.361887	+	0.932222i	$0.617867\pi$
$422$		0			0
$423$	1620.38	+	2806.59i	0.186255	+	0.322603i
$424$		0			0
$425$	−8216.87	+	14232.0i	−0.937827	+	1.62436i
$426$		0			0
$427$	1685.43	+	1081.23i	0.191015	+	0.122540i
$428$		0			0
$429$	4069.53	−	7048.64i	0.457993	−	0.793267i
$430$		0			0
$431$	−2474.54	−	4286.03i	−0.276553	−	0.479004i	0.693973	−	0.720001i	$-0.255859\pi$
$431$	−2474.54	−	4286.03i	−0.276553	−	0.479004i	−0.970526	+	0.240997i	$0.922526\pi$
$432$		0			0
$433$	−16602.8			−1.84267			−0.921337	−	0.388764i	$-0.872902\pi$
$433$	−16602.8			−1.84267			−0.921337	+	0.388764i	$0.872902\pi$
$434$		0			0
$435$	−26.9641			−0.00297202
$436$		0			0
$437$	−8212.38	−	14224.3i	−0.898974	−	1.55707i
$438$		0			0
$439$	2354.19	−	4077.57i	0.255944	−	0.443307i	−0.709208	−	0.705000i	$-0.750947\pi$
$439$	2354.19	−	4077.57i	0.255944	−	0.443307i	0.965151	+	0.261692i	$0.0842805\pi$
$440$		0			0
$441$	1786.56	−	2517.49i	0.192913	−	0.271838i
$442$		0			0
$443$	850.218	−	1472.62i	0.0911852	−	0.157937i	−0.816825	−	0.576886i	$-0.804268\pi$
$443$	850.218	−	1472.62i	0.0911852	−	0.157937i	0.908010	+	0.418948i	$0.137601\pi$
$444$		0			0
$445$	28.5065	+	49.3746i	0.00303671	+	0.00525973i
$446$		0			0
$447$	1184.04			0.125287
$448$		0			0
$449$	10050.2			1.05635			0.528173	−	0.849137i	$-0.322877\pi$
$449$	10050.2			1.05635			0.528173	+	0.849137i	$0.322877\pi$
$450$		0			0
$451$	−7994.26	−	13846.5i	−0.834667	−	1.44569i
$452$		0			0
$453$	4686.30	−	8116.91i	0.486052	−	0.841867i
$454$		0			0
$455$	100.697	+	64.5991i	0.0103753	+	0.00665594i
$456$		0			0
$457$	5495.95	−	9519.27i	0.562560	−	0.974382i	−0.434712	−	0.900569i	$-0.643150\pi$
$457$	5495.95	−	9519.27i	0.562560	−	0.974382i	0.997272	−	0.0738128i	$-0.0235167\pi$
$458$		0			0
$459$	1775.08	+	3074.52i	0.180509	+	0.312650i
$460$		0			0
$461$	−548.440			−0.0554086			−0.0277043	−	0.999616i	$-0.508820\pi$
$461$	−548.440			−0.0554086			−0.0277043	+	0.999616i	$0.508820\pi$
$462$		0			0
$463$	4028.04			0.404317			0.202159	−	0.979353i	$-0.435204\pi$
$463$	4028.04			0.404317			0.202159	+	0.979353i	$0.435204\pi$
$464$		0			0
$465$	−63.0444	−	109.196i	−0.00628734	−	0.0108900i
$466$		0			0
$467$	−1035.35	+	1793.27i	−0.102591	+	0.177693i	−0.912751	−	0.408515i	$-0.866047\pi$
$467$	−1035.35	+	1793.27i	−0.102591	+	0.177693i	0.810160	+	0.586208i	$0.199380\pi$
$468$		0			0
$469$	13854.9	−	7156.88i	1.36409	−	0.704636i
$470$		0			0
$471$	5401.06	−	9354.91i	0.528381	−	0.915183i
$472$		0			0
$473$	3896.63	+	6749.16i	0.378789	+	0.656082i
$474$		0			0
$475$	−11437.2			−1.10479
$476$		0			0
$477$	16.5162			0.00158538
$478$		0			0
$479$	89.3926	+	154.832i	0.00852704	+	0.0147693i	0.870257	−	0.492597i	$-0.163952\pi$
$479$	89.3926	+	154.832i	0.00852704	+	0.0147693i	−0.861730	+	0.507366i	$0.830619\pi$
$480$		0			0
$481$	7577.89	−	13125.3i	0.718341	−	1.24420i
$482$		0			0
$483$	−466.447	+	9961.54i	−0.0439422	+	0.938438i
$484$		0			0
$485$	47.6311	−	82.4995i	0.00445942	−	0.00772393i
$486$		0			0
$487$	8298.48	+	14373.4i	0.772156	+	1.33741i	0.936379	+	0.350991i	$0.114155\pi$
$487$	8298.48	+	14373.4i	0.772156	+	1.33741i	−0.164223	+	0.986423i	$0.552512\pi$
$488$		0			0
$489$	5922.07			0.547659
$490$		0			0
$491$	−4547.22			−0.417949			−0.208975	−	0.977921i	$-0.567013\pi$
$491$	−4547.22			−0.417949			−0.208975	+	0.977921i	$0.567013\pi$
$492$		0			0
$493$	−4595.22	−	7959.16i	−0.419794	−	0.727105i
$494$		0			0
$495$	−31.2515	+	54.1291i	−0.00283767	+	0.00491499i
$496$		0			0
$497$	208.864	−	4460.55i	0.0188508	−	0.402581i
$498$		0			0
$499$	−512.984	+	888.515i	−0.0460207	+	0.0797102i	−0.888118	−	0.459615i	$-0.847987\pi$
$499$	−512.984	+	888.515i	−0.0460207	+	0.0797102i	0.842097	+	0.539325i	$0.181321\pi$
$500$		0			0
$501$	−1600.63	−	2772.37i	−0.142736	−	0.247227i
$502$		0			0
$503$	6745.26			0.597925			0.298962	−	0.954265i	$-0.403359\pi$
$503$	6745.26			0.597925			0.298962	+	0.954265i	$0.403359\pi$
$504$		0			0
$505$	95.6431			0.00842785
$506$		0			0
$507$	−489.838	−	848.424i	−0.0429082	−	0.0743192i
$508$		0			0
$509$	3000.51	−	5197.03i	0.261287	−	0.452563i	−0.705297	−	0.708912i	$-0.749186\pi$
$509$	3000.51	−	5197.03i	0.261287	−	0.452563i	0.966584	+	0.256349i	$0.0825197\pi$
$510$		0			0
$511$	−3405.83	+	1759.32i	−0.294843	+	0.152304i
$512$		0			0
$513$	−1235.38	+	2139.74i	−0.106322	+	0.184155i
$514$		0			0
$515$	−6.72856	−	11.6542i	−0.000575720	−	0.000997177i
$516$		0			0
$517$	−19447.0			−1.65431
$518$		0			0
$519$	−823.964			−0.0696879
$520$		0			0
$521$	−2347.61	−	4066.17i	−0.197410	−	0.341924i	0.750278	−	0.661122i	$-0.229920\pi$
$521$	−2347.61	−	4066.17i	−0.197410	−	0.341924i	−0.947688	+	0.319199i	$0.896586\pi$
$522$		0			0
$523$	−385.884	+	668.370i	−0.0322629	+	0.0558810i	−0.881706	−	0.471799i	$-0.843605\pi$
$523$	−385.884	+	668.370i	−0.0322629	+	0.0558810i	0.849443	+	0.527680i	$0.176938\pi$
$524$		0			0
$525$	5844.85	+	3749.58i	0.485886	+	0.311705i
$526$		0			0
$527$	21488.1	−	37218.5i	1.77616	−	3.07640i
$528$		0			0
$529$	−10024.3	−	17362.7i	−0.823895	−	1.42703i
$530$		0			0
$531$	−479.147			−0.0391586
$532$		0			0
$533$	−14872.0			−1.20859
$534$		0			0
$535$	−33.0286	−	57.2073i	−0.00266907	−	0.00462297i
$536$		0			0
$537$	−831.498	+	1440.20i	−0.0668190	+	0.115734i
$538$		0			0
$539$	7722.52	+	16837.8i	0.617129	+	1.34556i
$540$		0			0
$541$	−1692.83	+	2932.07i	−0.134529	+	0.233012i	−0.925418	−	0.378949i	$-0.876286\pi$
$541$	−1692.83	+	2932.07i	−0.134529	+	0.233012i	0.790888	+	0.611961i	$0.209619\pi$
$542$		0			0
$543$	1028.15	+	1780.82i	0.0812566	+	0.140741i
$544$		0			0
$545$	125.381			0.00985457
$546$		0			0
$547$	4988.75			0.389952			0.194976	−	0.980808i	$-0.437537\pi$
$547$	4988.75			0.389952			0.194976	+	0.980808i	$0.437537\pi$
$548$		0			0
$549$	486.545	+	842.720i	0.0378237	+	0.0655126i
$550$		0			0
$551$	3198.08	−	5539.23i	0.247264	−	0.428274i
$552$		0			0
$553$	−8728.88	−	5599.73i	−0.671229	−	0.430606i
$554$		0			0
$555$	−58.1934	+	100.794i	−0.00445076	+	0.00770895i
$556$		0			0
$557$	10103.6	+	17500.0i	0.768591	+	1.33124i	0.938327	+	0.345749i	$0.112375\pi$
$557$	10103.6	+	17500.0i	0.768591	+	1.33124i	−0.169736	+	0.985490i	$0.554291\pi$
$558$		0			0
$559$	7249.02			0.548481
$560$		0			0
$561$	−21303.5			−1.60327
$562$		0			0
$563$	5845.20	+	10124.2i	0.437559	+	0.757875i	0.997501	−	0.0706574i	$-0.0225097\pi$
$563$	5845.20	+	10124.2i	0.437559	+	0.757875i	−0.559941	+	0.828532i	$0.689176\pi$
$564$		0			0
$565$	−123.838	+	214.493i	−0.00922104	+	0.0159713i
$566$		0			0
$567$	1332.82	−	688.483i	0.0987183	−	0.0509940i
$568$		0			0
$569$	8964.45	−	15526.9i	0.660473	−	1.14397i	−0.320018	−	0.947411i	$-0.603689\pi$
$569$	8964.45	−	15526.9i	0.660473	−	1.14397i	0.980491	−	0.196562i	$-0.0629777\pi$
$570$		0			0
$571$	7836.84	+	13573.8i	0.574364	+	0.994827i	0.996110	+	0.0881136i	$0.0280838\pi$
$571$	7836.84	+	13573.8i	0.574364	+	0.994827i	−0.421747	+	0.906714i	$0.638583\pi$
$572$		0			0
$573$	14696.6			1.07148
$574$		0			0
$575$	−22432.9			−1.62699
$576$		0			0
$577$	6826.50	+	11823.8i	0.492532	+	0.853090i	0.999963	−	0.00860205i	$-0.00273815\pi$
$577$	6826.50	+	11823.8i	0.492532	+	0.853090i	−0.507431	+	0.861692i	$0.669405\pi$
$578$		0			0
$579$	−4710.39	+	8158.64i	−0.338095	+	0.585598i
$580$		0			0
$581$	−854.696	+	18253.0i	−0.0610306	+	1.30338i
$582$		0			0
$583$	−49.5547	+	85.8312i	−0.00352032	+	0.00609737i
$584$		0			0
$585$	29.0690	+	50.3490i	0.00205445	+	0.00355842i
$586$		0			0
$587$	−18021.5			−1.26717			−0.633583	−	0.773675i	$-0.718417\pi$
$587$	−18021.5			−1.26717			−0.633583	+	0.773675i	$0.718417\pi$
$588$		0			0
$589$	29909.6			2.09236
$590$		0			0
$591$	341.118	+	590.834i	0.0237424	+	0.0411230i
$592$		0			0
$593$	−10568.7	+	18305.5i	−0.731877	+	1.26765i	0.224204	+	0.974542i	$0.428022\pi$
$593$	−10568.7	+	18305.5i	−0.731877	+	1.26765i	−0.956080	+	0.293105i	$0.905311\pi$
$594$		0			0
$595$	14.6468	−	312.800i	0.00100918	−	0.0215522i
$596$		0			0
$597$	1821.75	−	3155.36i	0.124890	−	0.216316i
$598$		0			0
$599$	−4684.80	−	8114.30i	−0.319559	−	0.553492i	0.660837	−	0.750529i	$-0.270201\pi$
$599$	−4684.80	−	8114.30i	−0.319559	−	0.553492i	−0.980396	+	0.197037i	$0.936868\pi$
$600$		0			0
$601$	−22750.0			−1.54408			−0.772040	−	0.635573i	$-0.780764\pi$
$601$	−22750.0			−1.54408			−0.772040	+	0.635573i	$0.780764\pi$
$602$		0			0
$603$	7578.06			0.511779
$604$		0			0
$605$	−101.954	−	176.590i	−0.00685128	−	0.0118668i
$606$		0			0
$607$	2986.82	−	5173.33i	0.199722	−	0.345929i	−0.748716	−	0.662891i	$-0.769329\pi$
$607$	2986.82	−	5173.33i	0.199722	−	0.345929i	0.948438	+	0.316962i	$0.102663\pi$
$608$		0			0
$609$	−3450.34	+	1782.31i	−0.229581	+	0.118592i
$610$		0			0
$611$	−9044.45	+	15665.4i	−0.598853	+	1.03724i
$612$		0			0
$613$	11673.7	+	20219.4i	0.769162	+	1.33223i	0.938018	+	0.346586i	$0.112659\pi$
$613$	11673.7	+	20219.4i	0.769162	+	1.33223i	−0.168856	+	0.985641i	$0.554007\pi$
$614$		0			0
$615$	114.207			0.00748826
$616$		0			0
$617$	28199.1			1.83996			0.919979	−	0.391968i	$-0.128206\pi$
$617$	28199.1			1.83996			0.919979	+	0.391968i	$0.128206\pi$
$618$		0			0
$619$	1487.78	+	2576.91i	0.0966057	+	0.167326i	0.910278	−	0.413998i	$-0.135868\pi$
$619$	1487.78	+	2576.91i	0.0966057	+	0.167326i	−0.813672	+	0.581324i	$0.802535\pi$
$620$		0			0
$621$	−2423.08	+	4196.89i	−0.156578	+	0.271201i
$622$		0			0
$623$	6911.33	+	4433.75i	0.444457	+	0.285127i
$624$		0			0
$625$	−7809.40	+	13526.3i	−0.499802	+	0.865682i
$626$		0			0
$627$	−7413.16	−	12840.0i	−0.472174	−	0.817830i
$628$		0			0
$629$	−39669.4			−2.51466
$630$		0			0
$631$	−2631.33			−0.166009			−0.0830044	−	0.996549i	$-0.526452\pi$
$631$	−2631.33			−0.166009			−0.0830044	+	0.996549i	$0.526452\pi$
$632$		0			0
$633$	−7674.10	−	13291.9i	−0.481861	−	0.834608i
$634$		0			0
$635$	72.3937	−	125.390i	0.00452418	−	0.00783611i
$636$		0			0
$637$	17155.2	+	1610.11i	1.06706	+	0.100149i
$638$		0			0
$639$	1085.00	−	1879.27i	0.0671704	−	0.116343i
$640$		0			0
$641$	−7247.12	−	12552.4i	−0.446559	−	0.773462i	0.551601	−	0.834108i	$-0.314017\pi$
$641$	−7247.12	−	12552.4i	−0.446559	−	0.773462i	−0.998159	+	0.0606460i	$0.980684\pi$
$642$		0			0
$643$	−15176.0			−0.930767			−0.465383	−	0.885109i	$-0.654084\pi$
$643$	−15176.0			−0.930767			−0.465383	+	0.885109i	$0.654084\pi$
$644$		0			0
$645$	−55.6679			−0.00339833
$646$		0			0
$647$	3784.59	+	6555.10i	0.229965	+	0.398312i	0.957798	−	0.287443i	$-0.0928053\pi$
$647$	3784.59	+	6555.10i	0.229965	+	0.398312i	−0.727832	+	0.685755i	$0.759472\pi$
$648$		0			0
$649$	1437.62	−	2490.02i	0.0869513	−	0.150604i
$650$		0			0
$651$	−15285.0	−	9805.60i	−0.920224	−	0.590341i
$652$		0			0
$653$	6966.75	−	12066.8i	0.417504	−	0.723138i	−0.578184	−	0.815906i	$-0.696238\pi$
$653$	6966.75	−	12066.8i	0.417504	−	0.723138i	0.995688	+	0.0927688i	$0.0295717\pi$
$654$		0			0
$655$	−96.0089	−	166.292i	−0.00572729	−	0.00991996i
$656$		0			0
$657$	−1862.85			−0.110619
$658$		0			0
$659$	−17015.5			−1.00581			−0.502904	−	0.864342i	$-0.667735\pi$
$659$	−17015.5			−1.00581			−0.502904	+	0.864342i	$0.667735\pi$
$660$		0			0
$661$	8267.70	+	14320.1i	0.486500	+	0.842642i	0.999880	−	0.0155194i	$-0.00494019\pi$
$661$	8267.70	+	14320.1i	0.486500	+	0.842642i	−0.513380	+	0.858161i	$0.671607\pi$
$662$		0			0
$663$	−9907.90	+	17161.0i	−0.580378	+	1.00524i
$664$		0			0
$665$	193.626	−	100.020i	0.0112910	−	0.00583247i
$666$		0			0
$667$	6272.73	−	10864.7i	0.364140	−	0.630708i
$668$		0			0
$669$	−179.076	−	310.168i	−0.0103490	−	0.0179250i
$670$		0			0
$671$	−5839.25			−0.335949
$672$		0			0
$673$	4571.05			0.261814			0.130907	−	0.991395i	$-0.458211\pi$
$673$	4571.05			0.261814			0.130907	+	0.991395i	$0.458211\pi$
$674$		0			0
$675$	1687.28	+	2922.45i	0.0962123	+	0.166645i
$676$		0			0
$677$	−13142.3	+	22763.1i	−0.746083	+	1.29225i	0.203604	+	0.979053i	$0.434735\pi$
$677$	−13142.3	+	22763.1i	−0.746083	+	1.29225i	−0.949687	+	0.313201i	$0.898599\pi$
$678$		0			0
$679$	641.737	−	13705.1i	0.0362704	−	0.774598i
$680$		0			0
$681$	−2988.00	+	5175.37i	−0.168136	+	0.291220i
$682$		0			0
$683$	−3398.61	−	5886.57i	−0.190402	−	0.329785i	0.754982	−	0.655746i	$-0.227646\pi$
$683$	−3398.61	−	5886.57i	−0.190402	−	0.329785i	−0.945383	+	0.325961i	$0.894312\pi$
$684$		0			0
$685$	187.842			0.0104775
$686$		0			0
$687$	2110.43			0.117202
$688$		0			0
$689$	46.0940	+	79.8372i	0.00254868	+	0.00441445i
$690$		0			0
$691$	−13879.3	+	24039.7i	−0.764103	+	1.32346i	0.176617	+	0.984280i	$0.443485\pi$
$691$	−13879.3	+	24039.7i	−0.764103	+	1.32346i	−0.940720	+	0.339185i	$0.889849\pi$
$692$		0			0
$693$	−421.053	+	8992.09i	−0.0230801	+	0.492902i
$694$		0			0
$695$	−143.112	+	247.878i	−0.00781088	+	0.0135288i
$696$		0			0
$697$	19463.2	+	33711.3i	1.05771	+	1.83200i
$698$		0			0
$699$	2885.35			0.156129
$700$		0			0
$701$	−21638.3			−1.16586			−0.582929	−	0.812523i	$-0.698093\pi$
$701$	−21638.3			−1.16586			−0.582929	+	0.812523i	$0.698093\pi$
$702$		0			0
$703$	−13804.1	−	23909.4i	−0.740584	−	1.28273i
$704$		0			0
$705$	69.4557	−	120.301i	0.00371043	−	0.00642665i
$706$		0			0
$707$	12238.5	−	6321.95i	0.651029	−	0.336296i
$708$		0			0
$709$	8144.36	−	14106.4i	0.431408	−	0.747220i	−0.565587	−	0.824688i	$-0.691350\pi$
$709$	8144.36	−	14106.4i	0.431408	−	0.747220i	0.996995	+	0.0774687i	$0.0246838\pi$
$710$		0			0
$711$	−2519.83	−	4364.47i	−0.132913	−	0.230212i
$712$		0			0
$713$	58664.8			3.08137
$714$		0			0
$715$	−348.871			−0.0182476
$716$		0			0
$717$	6697.06	+	11599.6i	0.348823	+	0.604179i
$718$		0			0
$719$	1695.87	−	2937.33i	0.0879628	−	0.152356i	−0.818687	−	0.574240i	$-0.805298\pi$
$719$	1695.87	−	2937.33i	0.0879628	−	0.152356i	0.906650	+	0.421884i	$0.138631\pi$
$720$		0			0
$721$	−1631.33	−	1046.53i	−0.0842632	−	0.0540564i
$722$		0			0
$723$	652.967	−	1130.97i	0.0335880	−	0.0581761i
$724$		0			0
$725$	−4367.93	−	7565.48i	−0.223753	−	0.387551i
$726$		0			0
$727$	23158.7			1.18144			0.590722	−	0.806875i	$-0.298843\pi$
$727$	23158.7			1.18144			0.590722	+	0.806875i	$0.298843\pi$
$728$		0			0
$729$	729.000			0.0370370
$730$		0			0
$731$	−9486.94	−	16431.9i	−0.480010	−	0.831402i
$732$		0			0
$733$	13814.6	−	23927.6i	0.696116	−	1.20571i	−0.273686	−	0.961819i	$-0.588243\pi$
$733$	13814.6	−	23927.6i	0.696116	−	1.20571i	0.969803	−	0.243890i	$-0.0784236\pi$
$734$		0			0
$735$	−131.741	−	12.3646i	−0.00661136	−	0.000620512i
$736$		0			0
$737$	−22737.0	+	39381.6i	−1.13640	+	1.96830i
$738$		0			0
$739$	4730.82	+	8194.03i	0.235489	+	0.407879i	0.959415	−	0.281999i	$-0.0909976\pi$
$739$	4730.82	+	8194.03i	0.235489	+	0.407879i	−0.723926	+	0.689878i	$0.757664\pi$
$740$		0			0
$741$	−13790.9			−0.683701
$742$		0			0
$743$	−14316.9			−0.706913			−0.353457	−	0.935451i	$-0.614994\pi$
$743$	−14316.9			−0.706913			−0.353457	+	0.935451i	$0.614994\pi$
$744$		0			0
$745$	−25.3762	−	43.9529i	−0.00124794	−	0.00216149i
$746$		0			0
$747$	−4439.93	+	7690.19i	−0.217468	+	0.376666i
$748$		0			0
$749$	−8007.73	−	5137.10i	−0.390649	−	0.250608i
$750$		0			0
$751$	3743.03	−	6483.13i	0.181871	−	0.315010i	−0.760647	−	0.649166i	$-0.775118\pi$
$751$	3743.03	−	6483.13i	0.181871	−	0.315010i	0.942518	+	0.334156i	$0.108451\pi$
$752$		0			0
$753$	−1294.50	−	2242.15i	−0.0626486	−	0.108510i
$754$		0			0
$755$	−401.744			−0.0193655
$756$		0			0
$757$	−17416.8			−0.836227			−0.418114	−	0.908395i	$-0.637309\pi$
$757$	−17416.8			−0.836227			−0.418114	+	0.908395i	$0.637309\pi$
$758$		0			0
$759$	−14540.2	−	25184.4i	−0.695359	−	1.20440i
$760$		0			0
$761$	16615.5	−	28778.9i	0.791474	−	1.37087i	−0.133580	−	0.991038i	$-0.542647\pi$
$761$	16615.5	−	28778.9i	0.791474	−	1.37087i	0.925054	−	0.379835i	$-0.124019\pi$
$762$		0			0
$763$	16043.9	−	8287.62i	0.761240	−	0.393227i
$764$		0			0
$765$	76.0864	−	131.786i	0.00359596	−	0.00622839i
$766$		0			0
$767$	−1337.22	−	2316.13i	−0.0629521	−	0.109036i
$768$		0			0
$769$	13714.2			0.643103			0.321552	−	0.946892i	$-0.395796\pi$
$769$	13714.2			0.643103			0.321552	+	0.946892i	$0.395796\pi$
$770$		0			0
$771$	−1610.08			−0.0752082
$772$		0			0
$773$	−7604.65	−	13171.6i	−0.353842	−	0.612873i	0.633077	−	0.774089i	$-0.281792\pi$
$773$	−7604.65	−	13171.6i	−0.353842	−	0.612873i	−0.986919	+	0.161216i	$0.948458\pi$
$774$		0			0
$775$	20425.2	−	35377.5i	0.946704	−	1.63974i
$776$		0			0
$777$	−784.045	+	16744.2i	−0.0362001	+	0.773095i
$778$		0			0
$779$	−13545.6	+	23461.6i	−0.623004	+	1.07907i
$780$		0			0
$781$	6510.79	+	11277.0i	0.298303	+	0.516675i
$782$		0			0
$783$	−1887.20			−0.0861339
$784$		0			0
$785$	−463.019			−0.0210520
$786$		0			0
$787$	6610.74	+	11450.1i	0.299425	+	0.518619i	0.976004	−	0.217750i	$-0.0698719\pi$
$787$	6610.74	+	11450.1i	0.299425	+	0.518619i	−0.676580	+	0.736369i	$0.736539\pi$
$788$		0			0
$789$	−557.862	+	966.246i	−0.0251716	+	0.0435986i
$790$		0			0
$791$	−1668.47	+	35632.3i	−0.0749989	+	1.60169i
$792$		0			0
$793$	−2715.73	+	4703.79i	−0.121612	+	0.210639i
$794$		0			0
$795$	−0.353973	−	0.613099i	−1.57914e−5	−	2.73514e-5i
$796$		0			0
$797$	13373.9			0.594387			0.297194	−	0.954817i	$-0.403949\pi$
$797$	13373.9			0.594387			0.297194	+	0.954817i	$0.403949\pi$
$798$		0			0
$799$	47346.6			2.09637
$800$		0			0
$801$	1995.15	+	3455.69i	0.0880088	+	0.152436i
$802$		0			0
$803$	5589.23	−	9680.84i	0.245629	−	0.425441i
$804$		0			0
$805$	379.780	−	196.179i	0.0166279	−	0.00858933i
$806$		0			0
$807$	6758.41	−	11705.9i	0.294805	−	0.510616i
$808$		0			0
$809$	5614.64	+	9724.84i	0.244005	+	0.422630i	0.961851	−	0.273572i	$-0.0882052\pi$
$809$	5614.64	+	9724.84i	0.244005	+	0.422630i	−0.717846	+	0.696202i	$0.754872\pi$
$810$		0			0
$811$	−4301.42			−0.186243			−0.0931215	−	0.995655i	$-0.529685\pi$
$811$	−4301.42			−0.186243			−0.0931215	+	0.995655i	$0.529685\pi$
$812$		0			0
$813$	−1188.35			−0.0512634
$814$		0			0
$815$	−126.921	−	219.834i	−0.00545503	−	0.00944838i
$816$		0			0
$817$	6602.50	−	11435.9i	0.282732	−	0.489707i
$818$		0			0
$819$	7047.72	+	4521.24i	0.300693	+	0.192900i
$820$		0			0
$821$	20455.3	−	35429.6i	0.869543	−	1.50609i	0.00707888	−	0.999975i	$-0.497747\pi$
$821$	20455.3	−	35429.6i	0.869543	−	1.50609i	0.862464	−	0.506118i	$-0.168920\pi$
$822$		0			0
$823$	−14021.6	−	24286.1i	−0.593879	−	1.02863i	−0.993704	−	0.112038i	$-0.964262\pi$
$823$	−14021.6	−	24286.1i	−0.593879	−	1.02863i	0.399825	−	0.916592i	$-0.369071\pi$
$824$		0			0
$825$	−20249.8			−0.854554
$826$		0			0
$827$	17598.9			0.739993			0.369996	−	0.929033i	$-0.379359\pi$
$827$	17598.9			0.739993			0.369996	+	0.929033i	$0.379359\pi$
$828$		0			0
$829$	7282.76	+	12614.1i	0.305116	+	0.528476i	0.977287	−	0.211920i	$-0.0679716\pi$
$829$	7282.76	+	12614.1i	0.305116	+	0.528476i	−0.672172	+	0.740395i	$0.734638\pi$
$830$		0			0
$831$	9763.11	−	16910.2i	0.407555	−	0.705907i
$832$		0			0
$833$	−18801.6	−	40994.1i	−0.782039	−	1.70512i
$834$		0			0
$835$	−68.6090	+	118.834i	−0.00284349	+	0.00492506i
$836$		0			0
$837$	−4412.43	−	7642.56i	−0.182217	−	0.315610i
$838$		0			0
$839$	711.056			0.0292591			0.0146295	−	0.999893i	$-0.495343\pi$
$839$	711.056			0.0292591			0.0146295	+	0.999893i	$0.495343\pi$
$840$		0			0
$841$	−19503.5			−0.799686
$842$		0			0
$843$	−4178.70	−	7237.73i	−0.170726	−	0.295707i
$844$		0			0
$845$	−20.9963	+	36.3666i	−0.000854785	+	0.00148053i
$846$		0			0
$847$	−24718.6	−	15857.4i	−1.00276	−	0.643291i
$848$		0			0
$849$	−0.917769	+	1.58962i	−3.70998e−5	+	6.42588e-5i
$850$		0			0
$851$	−27075.4	−	46896.0i	−1.09064	−	1.88904i
$852$		0			0
$853$	−7459.48			−0.299423			−0.149712	−	0.988730i	$-0.547834\pi$
$853$	−7459.48			−0.299423			−0.149712	+	0.988730i	$0.547834\pi$
$854$		0			0
$855$	105.906			0.00423614
$856$		0			0
$857$	−1780.46	−	3083.85i	−0.0709678	−	0.122920i	0.828358	−	0.560199i	$-0.189275\pi$
$857$	−1780.46	−	3083.85i	−0.0709678	−	0.122920i	−0.899326	+	0.437279i	$0.855942\pi$
$858$		0			0
$859$	−6759.57	+	11707.9i	−0.268491	+	0.465039i	−0.968472	−	0.249121i	$-0.919858\pi$
$859$	−6759.57	+	11707.9i	−0.268491	+	0.465039i	0.699982	+	0.714161i	$0.253191\pi$
$860$		0			0
$861$	14614.0	−	7549.03i	0.578449	−	0.298804i
$862$		0			0
$863$	−10198.0	+	17663.4i	−0.402251	+	0.696719i	−0.993997	−	0.109405i	$-0.965105\pi$
$863$	−10198.0	+	17663.4i	−0.402251	+	0.696719i	0.591746	+	0.806124i	$0.298439\pi$
$864$		0			0
$865$	17.6591	+	30.5864i	0.000694135	+	0.00120228i
$866$		0			0
$867$	37127.7			1.45435
$868$		0			0
$869$	30241.6			1.18053
$870$		0			0
$871$	21149.1	+	36631.4i	0.822745	+	1.42504i
$872$		0			0
$873$	3333.67	−	5774.08i	0.129241	−	0.223852i
$874$		0			0
$875$	27.8465	−	594.694i	0.00107587	−	0.0229764i
$876$		0			0
$877$	−3598.63	+	6233.01i	−0.138560	+	0.239993i	−0.926952	−	0.375181i	$-0.877581\pi$
$877$	−3598.63	+	6233.01i	−0.138560	+	0.239993i	0.788392	+	0.615174i	$0.210914\pi$
$878$		0			0
$879$	6218.96	+	10771.6i	0.238635	+	0.413329i
$880$		0			0
$881$	11588.3			0.443156			0.221578	−	0.975143i	$-0.428879\pi$
$881$	11588.3			0.443156			0.221578	+	0.975143i	$0.428879\pi$
$882$		0			0
$883$	21959.0			0.836898			0.418449	−	0.908240i	$-0.362574\pi$
$883$	21959.0			0.836898			0.418449	+	0.908240i	$0.362574\pi$
$884$		0			0
$885$	10.2690	+	17.7864i	0.000390044	+	0.000675576i
$886$		0			0
$887$	−6259.45	+	10841.7i	−0.236947	+	0.410404i	−0.959837	−	0.280559i	$-0.909480\pi$
$887$	−6259.45	+	10841.7i	−0.236947	+	0.410404i	0.722890	+	0.690963i	$0.242813\pi$
$888$		0			0
$889$	975.366	−	20830.1i	0.0367972	−	0.785848i
$890$		0			0
$891$	−2187.27	+	3788.46i	−0.0822404	+	0.142445i
$892$		0			0
$893$	16475.6	+	28536.6i	0.617396	+	1.06936i
$894$		0			0
$895$	71.2822			0.00266224
$896$		0			0
$897$	−27049.6			−1.00687
$898$		0			0
$899$	11422.7	+	19784.6i	0.423768	+	0.733987i
$900$		0			0
$901$	120.648	−	208.969i	0.00446102	−	0.00772672i
$902$		0			0
$903$	−7123.30	+	3679.61i	−0.262512	+	0.135603i
$904$		0			0
$905$	44.0705	−	76.3324i	0.00161873	−	0.00280373i
$906$		0			0
$907$	−14135.0	−	24482.6i	−0.517470	−	0.896285i	−0.999794	−	0.0202920i	$-0.993540\pi$
$907$	−14135.0	−	24482.6i	−0.517470	−	0.896285i	0.482324	−	0.875993i	$-0.339793\pi$
$908$		0			0
$909$	6693.99			0.244253
$910$		0			0
$911$	−7908.58			−0.287621			−0.143811	−	0.989605i	$-0.545936\pi$
$911$	−7908.58			−0.287621			−0.143811	+	0.989605i	$0.545936\pi$
$912$		0			0
$913$	−26642.8	−	46146.8i	−0.965771	−	1.67277i
$914$		0			0
$915$	20.8551	−	36.1221i	0.000753496	−	0.00130509i
$916$		0			0
$917$	−23277.2	−	14932.7i	−0.838255	−	0.537756i
$918$		0			0
$919$	−15725.3	+	27237.0i	−0.564451	+	0.977658i	0.432650	+	0.901562i	$0.357579\pi$
$919$	−15725.3	+	27237.0i	−0.564451	+	0.977658i	−0.997101	+	0.0760955i	$0.975755\pi$
$920$		0			0
$921$	−2940.79	−	5093.60i	−0.105214	−	0.182236i
$922$		0			0
$923$	12112.2			0.431938
$924$		0			0
$925$	−37707.2			−1.34033
$926$		0			0
$927$	−470.927	−	815.670i	−0.0166853	−	0.0288998i
$928$		0			0
$929$	1734.59	−	3004.40i	0.0612596	−	0.106105i	−0.833769	−	0.552113i	$-0.813822\pi$
$929$	1734.59	−	3004.40i	0.0612596	−	0.106105i	0.895029	+	0.446009i	$0.147155\pi$
$930$		0			0
$931$	18165.3	−	25597.1i	0.639465	−	0.901087i
$932$		0			0
$933$	−7811.35	+	13529.7i	−0.274097	+	0.474749i
$934$		0			0
$935$	456.574	+	790.810i	0.0159696	+	0.0276601i
$936$		0			0
$937$	50935.7			1.77588			0.887939	−	0.459962i	$-0.152137\pi$
$937$	50935.7			1.77588			0.887939	+	0.459962i	$0.152137\pi$
$938$		0			0
$939$	−11975.1			−0.416181
$940$		0			0
$941$	−23947.5	−	41478.2i	−0.829612	−	1.43693i	−0.898343	−	0.439295i	$-0.855228\pi$
$941$	−23947.5	−	41478.2i	−0.829612	−	1.43693i	0.0687311	−	0.997635i	$-0.478105\pi$
$942$		0			0
$943$	−26568.4	+	46017.8i	−0.917482	+	1.58913i
$944$		0			0
$945$	−54.1221	−	34.7203i	−0.00186306	−	0.00119519i
$946$		0			0
$947$	−5269.44	+	9126.94i	−0.180817	+	0.313184i	−0.942159	−	0.335166i	$-0.891208\pi$
$947$	−5269.44	+	9126.94i	−0.180817	+	0.313184i	0.761342	+	0.648351i	$0.224541\pi$
$948$		0			0
$949$	−5198.91	−	9004.78i	−0.177833	−	0.308016i
$950$		0			0
$951$	5415.11			0.184645
$952$		0			0
$953$	31123.5			1.05791			0.528955	−	0.848650i	$-0.322584\pi$
$953$	31123.5			1.05791			0.528955	+	0.848650i	$0.322584\pi$
$954$		0			0
$955$	−314.976	−	545.554i	−0.0106726	−	0.0184856i
$956$		0			0
$957$	5662.28	−	9807.35i	0.191260	−	0.331271i
$958$		0			0
$959$	24036.4	−	12416.3i	0.809360	−	0.418084i
$960$		0			0
$961$	−38518.9	+	66716.7i	−1.29297	+	2.23949i
$962$		0			0
$963$	−2311.65	−	4003.90i	−0.0773540	−	0.133981i
$964$		0			0
$965$	403.810			0.0134706
$966$		0			0
$967$	−1626.14			−0.0540776			−0.0270388	−	0.999634i	$-0.508608\pi$
$967$	−1626.14			−0.0540776			−0.0270388	+	0.999634i	$0.508608\pi$
$968$		0			0
$969$	18048.5	+	31260.9i	0.598349	+	1.03637i
$970$		0			0
$971$	−17750.0	+	30744.0i	−0.586638	+	1.01609i	0.408031	+	0.912968i	$0.366216\pi$
$971$	−17750.0	+	30744.0i	−0.586638	+	1.01609i	−0.994669	+	0.103119i	$0.967118\pi$
$972$		0			0
$973$	−1928.16	+	41178.2i	−0.0635294	+	1.35675i
$974$		0			0
$975$	−9417.82	+	16312.1i	−0.309345	+	0.535802i
$976$		0			0
$977$	−13285.1	−	23010.5i	−0.435034	−	0.753501i	0.562264	−	0.826958i	$-0.309930\pi$
$977$	−13285.1	−	23010.5i	−0.435034	−	0.753501i	−0.997298	+	0.0734563i	$0.976597\pi$
$978$		0			0
$979$	−23944.7			−0.781691
$980$		0			0
$981$	8775.34			0.285602
$982$		0			0
$983$	26925.8	+	46636.9i	0.873652	+	1.51321i	0.858192	+	0.513329i	$0.171588\pi$
$983$	26925.8	+	46636.9i	0.873652	+	1.51321i	0.0154597	+	0.999880i	$0.495079\pi$
$984$		0			0
$985$	14.6216	−	25.3254i	0.000472978	−	0.000819221i
$986$		0			0
$987$	935.782	−	19984.7i	0.0301786	−	0.644499i
$988$		0			0
$989$	12950.2	−	22430.4i	0.416372	−	0.721178i
$990$		0			0
$991$	−1414.87	−	2450.63i	−0.0453531	−	0.0785539i	0.842458	−	0.538762i	$-0.181108\pi$
$991$	−1414.87	−	2450.63i	−0.0453531	−	0.0785539i	−0.887811	+	0.460209i	$0.847775\pi$
$992$		0			0
$993$	20718.6			0.662118
$994$		0			0
$995$	−156.174			−0.00497593
$996$		0			0
$997$	−6452.14	−	11175.4i	−0.204956	−	0.354994i	0.745163	−	0.666883i	$-0.232372\pi$
$997$	−6452.14	−	11175.4i	−0.204956	−	0.354994i	−0.950119	+	0.311888i	$0.899038\pi$
$998$		0			0
$999$	−4072.92	+	7054.50i	−0.128990	+	0.223418i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	168.4.q.f.121.3	yes	8
3.2	odd	2		504.4.s.j.289.2		8
4.3	odd	2		336.4.q.m.289.3		8
7.2	even	3		1176.4.a.bd.1.2		4
7.4	even	3	inner	168.4.q.f.25.3	✓	8
7.5	odd	6		1176.4.a.ba.1.3		4
21.11	odd	6		504.4.s.j.361.2		8
28.11	odd	6		336.4.q.m.193.3		8
28.19	even	6		2352.4.a.cp.1.3		4
28.23	odd	6		2352.4.a.cm.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.4.q.f.25.3	✓	8	7.4	even	3	inner
168.4.q.f.121.3	yes	8	1.1	even	1	trivial
336.4.q.m.193.3		8	28.11	odd	6
336.4.q.m.289.3		8	4.3	odd	2
504.4.s.j.289.2		8	3.2	odd	2
504.4.s.j.361.2		8	21.11	odd	6
1176.4.a.ba.1.3		4	7.5	odd	6
1176.4.a.bd.1.2		4	7.2	even	3
2352.4.a.cm.1.2		4	28.23	odd	6
2352.4.a.cp.1.3		4	28.19	even	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:	\( N \)	\(=\)	\( 168 = 2^{3} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	168.q (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.50000 - 2.59808i) q^{3} +(-0.0642956 + 0.111363i) q^{5} +(-0.866259 + 18.5000i) q^{7} +(-4.50000 + 7.79423i) q^{9} +O(q^{10})\)	\(q+(-1.50000 - 2.59808i) q^{3} +(-0.0642956 + 0.111363i) q^{5} +(-0.866259 + 18.5000i) q^{7} +(-4.50000 + 7.79423i) q^{9} +(-27.0033 - 46.7711i) q^{11} -50.2350 q^{13} +0.385773 q^{15} +(65.7436 + 113.871i) q^{17} +(-45.7547 + 79.2495i) q^{19} +(49.3638 - 25.4994i) q^{21} +(-89.7436 + 155.441i) q^{23} +(62.4917 + 108.239i) q^{25} +27.0000 q^{27} -69.8961 q^{29} +(-163.423 - 283.058i) q^{31} +(-81.0099 + 140.313i) q^{33} +(-2.00452 - 1.28594i) q^{35} +(-150.849 + 261.278i) q^{37} +(75.3525 + 130.514i) q^{39} +296.048 q^{41} -144.302 q^{43} +(-0.578660 - 1.00227i) q^{45} +(180.043 - 311.843i) q^{47} +(-341.499 - 32.0516i) q^{49} +(197.231 - 341.614i) q^{51} +(-0.917567 - 1.58927i) q^{53} +6.94477 q^{55} +274.528 q^{57} +(26.6193 + 46.1060i) q^{59} +(54.0605 - 93.6356i) q^{61} +(-140.295 - 90.0018i) q^{63} +(3.22989 - 5.59433i) q^{65} +(-421.004 - 729.199i) q^{67} +538.462 q^{69} -241.111 q^{71} +(103.492 + 179.253i) q^{73} +(187.475 - 324.717i) q^{75} +(888.656 - 459.045i) q^{77} +(-279.981 + 484.942i) q^{79} +(-40.5000 - 70.1481i) q^{81} +986.652 q^{83} -16.9081 q^{85} +(104.844 + 181.595i) q^{87} +(221.683 - 383.966i) q^{89} +(43.5165 - 929.348i) q^{91} +(-490.270 + 849.173i) q^{93} +(-5.88365 - 10.1908i) q^{95} -740.815 q^{97} +486.059 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 12 q^{3} - 4 q^{5} + 18 q^{7} - 36 q^{9}+O(q^{10}) \) 8 * q - 12 * q^3 - 4 * q^5 + 18 * q^7 - 36 * q^9	\( 8 q - 12 q^{3} - 4 q^{5} + 18 q^{7} - 36 q^{9} - 14 q^{11} + 44 q^{13} + 24 q^{15} - 96 q^{17} + 26 q^{19} - 36 q^{21} - 96 q^{23} - 110 q^{25} + 216 q^{27} - 152 q^{29} - 238 q^{31} - 42 q^{33} + 152 q^{35} - 562 q^{37} - 66 q^{39} + 856 q^{41} - 516 q^{43} - 36 q^{45} + 80 q^{47} + 156 q^{49} - 288 q^{51} + 2952 q^{55} - 156 q^{57} - 262 q^{59} + 276 q^{61} - 54 q^{63} - 2196 q^{65} - 150 q^{67} + 576 q^{69} - 1696 q^{71} + 218 q^{73} - 330 q^{75} - 764 q^{77} - 1762 q^{79} - 324 q^{81} + 6900 q^{83} + 2904 q^{85} + 228 q^{87} + 344 q^{89} - 2806 q^{91} - 714 q^{93} - 2004 q^{95} - 1244 q^{97} + 252 q^{99}+O(q^{100}) \) 8 * q - 12 * q^3 - 4 * q^5 + 18 * q^7 - 36 * q^9 - 14 * q^11 + 44 * q^13 + 24 * q^15 - 96 * q^17 + 26 * q^19 - 36 * q^21 - 96 * q^23 - 110 * q^25 + 216 * q^27 - 152 * q^29 - 238 * q^31 - 42 * q^33 + 152 * q^35 - 562 * q^37 - 66 * q^39 + 856 * q^41 - 516 * q^43 - 36 * q^45 + 80 * q^47 + 156 * q^49 - 288 * q^51 + 2952 * q^55 - 156 * q^57 - 262 * q^59 + 276 * q^61 - 54 * q^63 - 2196 * q^65 - 150 * q^67 + 576 * q^69 - 1696 * q^71 + 218 * q^73 - 330 * q^75 - 764 * q^77 - 1762 * q^79 - 324 * q^81 + 6900 * q^83 + 2904 * q^85 + 228 * q^87 + 344 * q^89 - 2806 * q^91 - 714 * q^93 - 2004 * q^95 - 1244 * q^97 + 252 * q^99

Introduction

L-functions

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