LMFDB - Embedded newform 2142.2.p.g.1135.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2142,2,Mod(1135,2142)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2142, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2142.1135"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 2142 = 2 \cdot 3^{2} \cdot 7 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2142.p (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [12,0,0,-12,4,0,0,0,0,4,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$17.1039561130$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 38x^{10} + 509x^{8} + 2748x^{6} + 4804x^{4} + 2496x^{2} + 256 $ x^12 + 38x^10 + 509x^8 + 2748x^6 + 4804x^4 + 2496*x^2 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 714)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			1135.5
Root			$3.26061i$ of defining polynomial
Character	$\chi$	$=$	2142.1135
Dual form			2142.2.p.g.1891.5

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000i q^{2} -1.00000 q^{4} +(2.30560 + 2.30560i) q^{5} +(-0.707107 + 0.707107i) q^{7} +1.00000i q^{8} +(2.30560 - 2.30560i) q^{10} +(-1.30560 + 1.30560i) q^{11} +5.71315 q^{13} +(0.707107 + 0.707107i) q^{14} +1.00000 q^{16} +(-1.64722 + 3.77977i) q^{17} +5.00953i q^{19} +(-2.30560 - 2.30560i) q^{20} +(1.30560 + 1.30560i) q^{22} +(-4.58060 + 4.58060i) q^{23} +5.63155i q^{25} -5.71315i q^{26} +(0.707107 - 0.707107i) q^{28} +(-6.80737 - 6.80737i) q^{29} +(-0.764801 - 0.764801i) q^{31} -1.00000i q^{32} +(3.77977 + 1.64722i) q^{34} -3.26061 q^{35} +(-1.17305 - 1.17305i) q^{37} +5.00953 q^{38} +(-2.30560 + 2.30560i) q^{40} +(-8.19179 + 8.19179i) q^{41} -6.49371i q^{43} +(1.30560 - 1.30560i) q^{44} +(4.58060 + 4.58060i) q^{46} +4.87629 q^{47} -1.00000i q^{49} +5.63155 q^{50} -5.71315 q^{52} -12.9777i q^{53} -6.02036 q^{55} +(-0.707107 - 0.707107i) q^{56} +(-6.80737 + 6.80737i) q^{58} +10.9980i q^{59} +(5.77747 - 5.77747i) q^{61} +(-0.764801 + 0.764801i) q^{62} -1.00000 q^{64} +(13.1722 + 13.1722i) q^{65} -10.1865 q^{67} +(1.64722 - 3.77977i) q^{68} +3.26061i q^{70} +(-1.96941 - 1.96941i) q^{71} +(6.98042 + 6.98042i) q^{73} +(-1.17305 + 1.17305i) q^{74} -5.00953i q^{76} -1.84639i q^{77} +(-5.84293 + 5.84293i) q^{79} +(2.30560 + 2.30560i) q^{80} +(8.19179 + 8.19179i) q^{82} +13.5467i q^{83} +(-12.5125 + 4.91679i) q^{85} -6.49371 q^{86} +(-1.30560 - 1.30560i) q^{88} +17.6188 q^{89} +(-4.03980 + 4.03980i) q^{91} +(4.58060 - 4.58060i) q^{92} -4.87629i q^{94} +(-11.5500 + 11.5500i) q^{95} +(3.89882 + 3.89882i) q^{97} -1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{4} + 4 q^{5} + 4 q^{10} + 8 q^{11} + 12 q^{13} + 12 q^{16} + 12 q^{17} - 4 q^{20} - 8 q^{22} - 8 q^{23} - 12 q^{29} + 12 q^{31} + 4 q^{34} + 4 q^{35} + 24 q^{38} - 4 q^{40} - 4 q^{41} - 8 q^{44}+ \cdots - 12 q^{98}+O(q^{100}) $ 12 * q - 12 * q^4 + 4 * q^5 + 4 * q^10 + 8 * q^11 + 12 * q^13 + 12 * q^16 + 12 * q^17 - 4 * q^20 - 8 * q^22 - 8 * q^23 - 12 * q^29 + 12 * q^31 + 4 * q^34 + 4 * q^35 + 24 * q^38 - 4 * q^40 - 4 * q^41 - 8 * q^44 + 8 * q^46 - 8 * q^47 + 16 * q^50 - 12 * q^52 - 68 * q^55 - 12 * q^58 + 40 * q^61 + 12 * q^62 - 12 * q^64 - 20 * q^65 - 12 * q^67 - 12 * q^68 - 24 * q^71 - 8 * q^79 + 4 * q^80 + 4 * q^82 - 12 * q^85 - 20 * q^86 + 8 * q^88 + 44 * q^89 - 4 * q^91 + 8 * q^92 - 92 * q^95 - 32 * q^97 - 12 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2142\mathbb{Z}\right)^\times$.

$n$	$1261$	$1667$	$1837$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$1$	$1$

Coefficient data

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

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

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

$2$		−	1.00000i		−	0.707107i
$2$		−	1.00000i		−	0.707107i
$3$		0			0
$3$		0			0
$4$	−1.00000			−0.500000
$5$	2.30560	+	2.30560i	1.03109	+	1.03109i	0.999501	+	0.0315935i	$0.0100582\pi$
$5$	2.30560	+	2.30560i	1.03109	+	1.03109i	0.0315935	+	0.999501i	$0.489942\pi$
$6$		0			0
$7$	−0.707107	+	0.707107i	−0.267261	+	0.267261i
$7$	−0.707107	+	0.707107i	−0.267261	+	0.267261i
$8$			1.00000i			0.353553i
$9$		0			0
$10$	2.30560	−	2.30560i	0.729094	−	0.729094i
$11$	−1.30560	+	1.30560i	−0.393652	+	0.393652i	−0.875987	−	0.482335i	$-0.839789\pi$
$11$	−1.30560	+	1.30560i	−0.393652	+	0.393652i	0.482335	+	0.875987i	$0.339789\pi$
$12$		0			0
$13$	5.71315			1.58454			0.792271	−	0.610170i	$-0.208899\pi$
$13$	5.71315			1.58454			0.792271	+	0.610170i	$0.208899\pi$
$14$	0.707107	+	0.707107i	0.188982	+	0.188982i
$15$		0			0
$16$	1.00000			0.250000
$17$	−1.64722	+	3.77977i	−0.399510	+	0.916729i
$17$	−1.64722	+	3.77977i	−0.399510	+	0.916729i
$18$		0			0
$19$			5.00953i			1.14927i	0.818411	+	0.574633i	$0.194855\pi$
$19$			5.00953i			1.14927i	−0.818411	+	0.574633i	$0.805145\pi$
$20$	−2.30560	−	2.30560i	−0.515547	−	0.515547i
$21$		0			0
$22$	1.30560	+	1.30560i	0.278354	+	0.278354i
$23$	−4.58060	+	4.58060i	−0.955121	+	0.955121i	−0.999035	−	0.0439142i	$-0.986017\pi$
$23$	−4.58060	+	4.58060i	−0.955121	+	0.955121i	0.0439142	+	0.999035i	$0.486017\pi$
$24$		0			0
$25$			5.63155i			1.12631i
$26$		−	5.71315i		−	1.12044i
$27$		0			0
$28$	0.707107	−	0.707107i	0.133631	−	0.133631i
$29$	−6.80737	−	6.80737i	−1.26410	−	1.26410i	−0.949089	−	0.315007i	$-0.897993\pi$
$29$	−6.80737	−	6.80737i	−1.26410	−	1.26410i	−0.315007	−	0.949089i	$-0.602007\pi$
$30$		0			0
$31$	−0.764801	−	0.764801i	−0.137362	−	0.137362i	0.635082	−	0.772445i	$-0.280966\pi$
$31$	−0.764801	−	0.764801i	−0.137362	−	0.137362i	−0.772445	+	0.635082i	$0.780966\pi$
$32$		−	1.00000i		−	0.176777i
$33$		0			0
$34$	3.77977	+	1.64722i	0.648225	+	0.282496i
$35$	−3.26061			−0.551143
$36$		0			0
$37$	−1.17305	−	1.17305i	−0.192848	−	0.192848i	0.604077	−	0.796926i	$-0.293542\pi$
$37$	−1.17305	−	1.17305i	−0.192848	−	0.192848i	−0.796926	+	0.604077i	$0.793542\pi$
$38$	5.00953			0.812653
$39$		0			0
$40$	−2.30560	+	2.30560i	−0.364547	+	0.364547i
$41$	−8.19179	+	8.19179i	−1.27934	+	1.27934i	−0.338308	+	0.941036i	$0.609854\pi$
$41$	−8.19179	+	8.19179i	−1.27934	+	1.27934i	−0.941036	+	0.338308i	$0.890146\pi$
$42$		0			0
$43$		−	6.49371i		−	0.990282i	−0.868813	−	0.495141i	$-0.835116\pi$
$43$		−	6.49371i		−	0.990282i	0.868813	−	0.495141i	$-0.164884\pi$
$44$	1.30560	−	1.30560i	0.196826	−	0.196826i
$45$		0			0
$46$	4.58060	+	4.58060i	0.675373	+	0.675373i
$47$	4.87629			0.711279			0.355640	−	0.934623i	$-0.384263\pi$
$47$	4.87629			0.711279			0.355640	+	0.934623i	$0.384263\pi$
$48$		0			0
$49$		−	1.00000i		−	0.142857i
$50$	5.63155			0.796422
$51$		0			0
$52$	−5.71315			−0.792271
$53$		−	12.9777i		−	1.78262i	−0.453396	−	0.891309i	$-0.649788\pi$
$53$		−	12.9777i		−	1.78262i	0.453396	−	0.891309i	$-0.350212\pi$
$54$		0			0
$55$	−6.02036			−0.811785
$56$	−0.707107	−	0.707107i	−0.0944911	−	0.0944911i
$57$		0			0
$58$	−6.80737	+	6.80737i	−0.893851	+	0.893851i
$59$			10.9980i			1.43182i	0.698193	+	0.715910i	$0.253988\pi$
$59$			10.9980i			1.43182i	−0.698193	+	0.715910i	$0.746012\pi$
$60$		0			0
$61$	5.77747	−	5.77747i	0.739730	−	0.739730i	−0.232796	−	0.972526i	$-0.574787\pi$
$61$	5.77747	−	5.77747i	0.739730	−	0.739730i	0.972526	+	0.232796i	$0.0747874\pi$
$62$	−0.764801	+	0.764801i	−0.0971298	+	0.0971298i
$63$		0			0
$64$	−1.00000			−0.125000
$65$	13.1722	+	13.1722i	1.63381	+	1.63381i
$66$		0			0
$67$	−10.1865			−1.24448			−0.622240	−	0.782827i	$-0.713777\pi$
$67$	−10.1865			−1.24448			−0.622240	+	0.782827i	$0.713777\pi$
$68$	1.64722	−	3.77977i	0.199755	−	0.458364i
$69$		0			0
$70$			3.26061i			0.389717i
$71$	−1.96941	−	1.96941i	−0.233726	−	0.233726i	0.580520	−	0.814246i	$-0.302849\pi$
$71$	−1.96941	−	1.96941i	−0.233726	−	0.233726i	−0.814246	+	0.580520i	$0.802849\pi$
$72$		0			0
$73$	6.98042	+	6.98042i	0.816996	+	0.816996i	0.985672	−	0.168676i	$-0.0539490\pi$
$73$	6.98042	+	6.98042i	0.816996	+	0.816996i	−0.168676	+	0.985672i	$0.553949\pi$
$74$	−1.17305	+	1.17305i	−0.136364	+	0.136364i
$75$		0			0
$76$		−	5.00953i		−	0.574633i
$77$		−	1.84639i		−	0.210416i
$78$		0			0
$79$	−5.84293	+	5.84293i	−0.657381	+	0.657381i	−0.954760	−	0.297379i	$-0.903888\pi$
$79$	−5.84293	+	5.84293i	−0.657381	+	0.657381i	0.297379	+	0.954760i	$0.403888\pi$
$80$	2.30560	+	2.30560i	0.257774	+	0.257774i
$81$		0			0
$82$	8.19179	+	8.19179i	0.904632	+	0.904632i
$83$			13.5467i			1.48695i	0.668766	+	0.743473i	$0.266823\pi$
$83$			13.5467i			1.48695i	−0.668766	+	0.743473i	$0.733177\pi$
$84$		0			0
$85$	−12.5125	+	4.91679i	−1.35717	+	0.533301i
$86$	−6.49371			−0.700235
$87$		0			0
$88$	−1.30560	−	1.30560i	−0.139177	−	0.139177i
$89$	17.6188			1.86759			0.933794	−	0.357812i	$-0.116477\pi$
$89$	17.6188			1.86759			0.933794	+	0.357812i	$0.116477\pi$
$90$		0			0
$91$	−4.03980	+	4.03980i	−0.423487	+	0.423487i
$92$	4.58060	−	4.58060i	0.477561	−	0.477561i
$93$		0			0
$94$		−	4.87629i		−	0.502950i
$95$	−11.5500	+	11.5500i	−1.18500	+	1.18500i
$96$		0			0
$97$	3.89882	+	3.89882i	0.395866	+	0.395866i	0.876772	−	0.480906i	$-0.159692\pi$
$97$	3.89882	+	3.89882i	0.395866	+	0.395866i	−0.480906	+	0.876772i	$0.659692\pi$
$98$	−1.00000			−0.101015
$99$		0			0
$100$		−	5.63155i		−	0.563155i
$101$	−13.6813			−1.36134			−0.680669	−	0.732591i	$-0.738311\pi$
$101$	−13.6813			−1.36134			−0.680669	+	0.732591i	$0.738311\pi$
$102$		0			0
$103$	0.545951			0.0537941			0.0268971	−	0.999638i	$-0.491437\pi$
$103$	0.545951			0.0537941			0.0268971	+	0.999638i	$0.491437\pi$
$104$			5.71315i			0.560220i
$105$		0			0
$106$	−12.9777			−1.26050
$107$	−2.70541	−	2.70541i	−0.261542	−	0.261542i	0.564138	−	0.825680i	$-0.309209\pi$
$107$	−2.70541	−	2.70541i	−0.261542	−	0.261542i	−0.825680	+	0.564138i	$0.809209\pi$
$108$		0			0
$109$	2.89425	−	2.89425i	0.277219	−	0.277219i	−0.554779	−	0.831998i	$-0.687197\pi$
$109$	2.89425	−	2.89425i	0.277219	−	0.277219i	0.831998	+	0.554779i	$0.187197\pi$
$110$			6.02036i			0.574019i
$111$		0			0
$112$	−0.707107	+	0.707107i	−0.0668153	+	0.0668153i
$113$	−2.18078	+	2.18078i	−0.205151	+	0.205151i	−0.802203	−	0.597052i	$-0.796339\pi$
$113$	−2.18078	+	2.18078i	−0.205151	+	0.205151i	0.597052	+	0.802203i	$0.296339\pi$
$114$		0			0
$115$	−21.1220			−1.96964
$116$	6.80737	+	6.80737i	0.632048	+	0.632048i
$117$		0			0
$118$	10.9980			1.01245
$119$	−1.50794	−	3.83746i	−0.138232	−	0.351780i
$120$		0			0
$121$			7.59083i			0.690076i
$122$	−5.77747	−	5.77747i	−0.523068	−	0.523068i
$123$		0			0
$124$	0.764801	+	0.764801i	0.0686812	+	0.0686812i
$125$	−1.45611	+	1.45611i	−0.130238	+	0.130238i
$126$		0			0
$127$			3.77239i			0.334746i	0.985894	+	0.167373i	$0.0535284\pi$
$127$			3.77239i			0.334746i	−0.985894	+	0.167373i	$0.946472\pi$
$128$			1.00000i			0.0883883i
$129$		0			0
$130$	13.1722	−	13.1722i	1.15528	−	1.15528i
$131$	7.76100	+	7.76100i	0.678082	+	0.678082i	0.959566	−	0.281484i	$-0.0908266\pi$
$131$	7.76100	+	7.76100i	0.678082	+	0.678082i	−0.281484	+	0.959566i	$0.590827\pi$
$132$		0			0
$133$	−3.54227	−	3.54227i	−0.307154	−	0.307154i
$134$			10.1865i			0.879980i
$135$		0			0
$136$	−3.77977	−	1.64722i	−0.324113	−	0.141248i
$137$	8.64712			0.738774			0.369387	−	0.929276i	$-0.379568\pi$
$137$	8.64712			0.738774			0.369387	+	0.929276i	$0.379568\pi$
$138$		0			0
$139$	2.08193	+	2.08193i	0.176587	+	0.176587i	0.789866	−	0.613279i	$-0.210150\pi$
$139$	2.08193	+	2.08193i	0.176587	+	0.176587i	−0.613279	+	0.789866i	$0.710150\pi$
$140$	3.26061			0.275572
$141$		0			0
$142$	−1.96941	+	1.96941i	−0.165269	+	0.165269i
$143$	−7.45906	+	7.45906i	−0.623758	+	0.623758i
$144$		0			0
$145$		−	31.3901i		−	2.60680i
$146$	6.98042	−	6.98042i	0.577703	−	0.577703i
$147$		0			0
$148$	1.17305	+	1.17305i	0.0964242	+	0.0964242i
$149$	6.48200			0.531026			0.265513	−	0.964107i	$-0.414459\pi$
$149$	6.48200			0.531026			0.265513	+	0.964107i	$0.414459\pi$
$150$		0			0
$151$			13.0424i			1.06138i	0.847567	+	0.530689i	$0.178067\pi$
$151$			13.0424i			1.06138i	−0.847567	+	0.530689i	$0.821933\pi$
$152$	−5.00953			−0.406327
$153$		0			0
$154$	−1.84639			−0.148787
$155$		−	3.52665i		−	0.283267i
$156$		0			0
$157$	16.4243			1.31080			0.655401	−	0.755281i	$-0.272500\pi$
$157$	16.4243			1.31080			0.655401	+	0.755281i	$0.272500\pi$
$158$	5.84293	+	5.84293i	0.464839	+	0.464839i
$159$		0			0
$160$	2.30560	−	2.30560i	0.182273	−	0.182273i
$161$		−	6.47795i		−	0.510534i
$162$		0			0
$163$	4.40677	−	4.40677i	0.345165	−	0.345165i	−0.513140	−	0.858305i	$-0.671518\pi$
$163$	4.40677	−	4.40677i	0.345165	−	0.345165i	0.858305	+	0.513140i	$0.171518\pi$
$164$	8.19179	−	8.19179i	0.639672	−	0.639672i
$165$		0			0
$166$	13.5467			1.05143
$167$	−11.3979	−	11.3979i	−0.881994	−	0.881994i	0.111743	−	0.993737i	$-0.464357\pi$
$167$	−11.3979	−	11.3979i	−0.881994	−	0.881994i	−0.993737	+	0.111743i	$0.964357\pi$
$168$		0			0
$169$	19.6400			1.51077
$170$	4.91679	+	12.5125i	0.377101	+	0.959662i
$171$		0			0
$172$			6.49371i			0.495141i
$173$	10.4886	+	10.4886i	0.797436	+	0.797436i	0.982691	−	0.185254i	$-0.0593109\pi$
$173$	10.4886	+	10.4886i	0.797436	+	0.797436i	−0.185254	+	0.982691i	$0.559311\pi$
$174$		0			0
$175$	−3.98211	−	3.98211i	−0.301019	−	0.301019i
$176$	−1.30560	+	1.30560i	−0.0984131	+	0.0984131i
$177$		0			0
$178$		−	17.6188i		−	1.32058i
$179$			5.48452i			0.409932i	0.978769	+	0.204966i	$0.0657085\pi$
$179$			5.48452i			0.409932i	−0.978769	+	0.204966i	$0.934292\pi$
$180$		0			0
$181$	−5.75711	+	5.75711i	−0.427923	+	0.427923i	−0.887920	−	0.459997i	$-0.847850\pi$
$181$	−5.75711	+	5.75711i	−0.427923	+	0.427923i	0.459997	+	0.887920i	$0.347850\pi$
$182$	4.03980	+	4.03980i	0.299450	+	0.299450i
$183$		0			0
$184$	−4.58060	−	4.58060i	−0.337686	−	0.337686i
$185$		−	5.40917i		−	0.397690i
$186$		0			0
$187$	−2.78424	−	7.08546i	−0.203604	−	0.518140i
$188$	−4.87629			−0.355640
$189$		0			0
$190$	11.5500	+	11.5500i	0.837922	+	0.837922i
$191$	24.2852			1.75721			0.878607	−	0.477546i	$-0.158474\pi$
$191$	24.2852			1.75721			0.878607	+	0.477546i	$0.158474\pi$
$192$		0			0
$193$	11.3040	−	11.3040i	0.813678	−	0.813678i	−0.171505	−	0.985183i	$-0.554863\pi$
$193$	11.3040	−	11.3040i	0.813678	−	0.813678i	0.985183	+	0.171505i	$0.0548630\pi$
$194$	3.89882	−	3.89882i	0.279919	−	0.279919i
$195$		0			0
$196$			1.00000i			0.0714286i
$197$	−1.29953	+	1.29953i	−0.0925875	+	0.0925875i	−0.751883	−	0.659296i	$-0.770854\pi$
$197$	−1.29953	+	1.29953i	−0.0925875	+	0.0925875i	0.659296	+	0.751883i	$0.270854\pi$
$198$		0			0
$199$	−6.09112	−	6.09112i	−0.431788	−	0.431788i	0.457448	−	0.889236i	$-0.348764\pi$
$199$	−6.09112	−	6.09112i	−0.431788	−	0.431788i	−0.889236	+	0.457448i	$0.848764\pi$
$200$	−5.63155			−0.398211
$201$		0			0
$202$			13.6813i			0.962611i
$203$	9.62707			0.675688
$204$		0			0
$205$	−37.7739			−2.63825
$206$		−	0.545951i		−	0.0380382i
$207$		0			0
$208$	5.71315			0.396135
$209$	−6.54043	−	6.54043i	−0.452411	−	0.452411i
$210$		0			0
$211$	−11.4696	+	11.4696i	−0.789597	+	0.789597i	−0.981428	−	0.191831i	$-0.938557\pi$
$211$	−11.4696	+	11.4696i	−0.789597	+	0.789597i	0.191831	+	0.981428i	$0.438557\pi$
$212$			12.9777i			0.891309i
$213$		0			0
$214$	−2.70541	+	2.70541i	−0.184938	+	0.184938i
$215$	14.9719	−	14.9719i	1.02107	−	1.02107i
$216$		0			0
$217$	1.08159			0.0734232
$218$	−2.89425	−	2.89425i	−0.196023	−	0.196023i
$219$		0			0
$220$	6.02036			0.405893
$221$	−9.41083	+	21.5944i	−0.633041	+	1.45259i
$222$		0			0
$223$			14.2184i			0.952133i	0.879409	+	0.476066i	$0.157938\pi$
$223$			14.2184i			0.952133i	−0.879409	+	0.476066i	$0.842062\pi$
$224$	0.707107	+	0.707107i	0.0472456	+	0.0472456i
$225$		0			0
$226$	2.18078	+	2.18078i	0.145064	+	0.145064i
$227$	12.4143	−	12.4143i	0.823967	−	0.823967i	−0.162707	−	0.986674i	$-0.552023\pi$
$227$	12.4143	−	12.4143i	0.823967	−	0.823967i	0.986674	+	0.162707i	$0.0520225\pi$
$228$		0			0
$229$			7.48179i			0.494411i	0.968963	+	0.247205i	$0.0795122\pi$
$229$			7.48179i			0.494411i	−0.968963	+	0.247205i	$0.920488\pi$
$230$			21.1220i			1.39275i
$231$		0			0
$232$	6.80737	−	6.80737i	0.446925	−	0.446925i
$233$	−12.7751	−	12.7751i	−0.836925	−	0.836925i	0.151528	−	0.988453i	$-0.451581\pi$
$233$	−12.7751	−	12.7751i	−0.836925	−	0.836925i	−0.988453	+	0.151528i	$0.951581\pi$
$234$		0			0
$235$	11.2427	+	11.2427i	0.733396	+	0.733396i
$236$		−	10.9980i		−	0.715910i
$237$		0			0
$238$	−3.83746	+	1.50794i	−0.248746	+	0.0977451i
$239$	24.8996			1.61062			0.805311	−	0.592853i	$-0.201999\pi$
$239$	24.8996			1.61062			0.805311	+	0.592853i	$0.201999\pi$
$240$		0			0
$241$	−7.66021	−	7.66021i	−0.493437	−	0.493437i	0.415950	−	0.909387i	$-0.363449\pi$
$241$	−7.66021	−	7.66021i	−0.493437	−	0.493437i	−0.909387	+	0.415950i	$0.863449\pi$
$242$	7.59083			0.487957
$243$		0			0
$244$	−5.77747	+	5.77747i	−0.369865	+	0.369865i
$245$	2.30560	−	2.30560i	0.147299	−	0.147299i
$246$		0			0
$247$			28.6202i			1.82106i
$248$	0.764801	−	0.764801i	0.0485649	−	0.0485649i
$249$		0			0
$250$	1.45611	+	1.45611i	0.0920924	+	0.0920924i
$251$	−8.75005			−0.552298			−0.276149	−	0.961115i	$-0.589058\pi$
$251$	−8.75005			−0.552298			−0.276149	+	0.961115i	$0.589058\pi$
$252$		0			0
$253$		−	11.9608i		−	0.751971i
$254$	3.77239			0.236701
$255$		0			0
$256$	1.00000			0.0625000
$257$			22.9937i			1.43431i	0.696916	+	0.717153i	$0.254555\pi$
$257$			22.9937i			1.43431i	−0.696916	+	0.717153i	$0.745445\pi$
$258$		0			0
$259$	1.65894			0.103082
$260$	−13.1722	−	13.1722i	−0.816906	−	0.816906i
$261$		0			0
$262$	7.76100	−	7.76100i	0.479476	−	0.479476i
$263$		−	7.87376i		−	0.485517i	−0.970087	−	0.242759i	$-0.921948\pi$
$263$		−	7.87376i		−	0.485517i	0.970087	−	0.242759i	$-0.0780523\pi$
$264$		0			0
$265$	29.9212	−	29.9212i	1.83805	−	1.83805i
$266$	−3.54227	+	3.54227i	−0.217191	+	0.217191i
$267$		0			0
$268$	10.1865			0.622240
$269$	9.05635	+	9.05635i	0.552175	+	0.552175i	0.927068	−	0.374893i	$-0.122320\pi$
$269$	9.05635	+	9.05635i	0.552175	+	0.552175i	−0.374893	+	0.927068i	$0.622320\pi$
$270$		0			0
$271$	25.4839			1.54804			0.774018	−	0.633163i	$-0.218244\pi$
$271$	25.4839			1.54804			0.774018	+	0.633163i	$0.218244\pi$
$272$	−1.64722	+	3.77977i	−0.0998776	+	0.229182i
$273$		0			0
$274$		−	8.64712i		−	0.522392i
$275$	−7.35254	−	7.35254i	−0.443375	−	0.443375i
$276$		0			0
$277$	−12.2501	−	12.2501i	−0.736036	−	0.736036i	0.235772	−	0.971808i	$-0.424238\pi$
$277$	−12.2501	−	12.2501i	−0.736036	−	0.736036i	−0.971808	+	0.235772i	$0.924238\pi$
$278$	2.08193	−	2.08193i	0.124866	−	0.124866i
$279$		0			0
$280$		−	3.26061i		−	0.194858i
$281$		−	15.2279i		−	0.908421i	−0.890894	−	0.454210i	$-0.849921\pi$
$281$		−	15.2279i		−	0.908421i	0.890894	−	0.454210i	$-0.150079\pi$
$282$		0			0
$283$	8.18153	−	8.18153i	0.486341	−	0.486341i	−0.420808	−	0.907150i	$-0.638253\pi$
$283$	8.18153	−	8.18153i	0.486341	−	0.486341i	0.907150	+	0.420808i	$0.138253\pi$
$284$	1.96941	+	1.96941i	0.116863	+	0.116863i
$285$		0			0
$286$	7.45906	+	7.45906i	0.441064	+	0.441064i
$287$		−	11.5849i		−	0.683838i
$288$		0			0
$289$	−11.5733	−	12.4522i	−0.680783	−	0.732485i
$290$	−31.3901			−1.84329
$291$		0			0
$292$	−6.98042	−	6.98042i	−0.408498	−	0.408498i
$293$	11.4033			0.666187			0.333093	−	0.942894i	$-0.391908\pi$
$293$	11.4033			0.666187			0.333093	+	0.942894i	$0.391908\pi$
$294$		0			0
$295$	−25.3570	+	25.3570i	−1.47634	+	1.47634i
$296$	1.17305	−	1.17305i	0.0681822	−	0.0681822i
$297$		0			0
$298$		−	6.48200i		−	0.375492i
$299$	−26.1696	+	26.1696i	−1.51343	+	1.51343i
$300$		0			0
$301$	4.59175	+	4.59175i	0.264664	+	0.264664i
$302$	13.0424			0.750507
$303$		0			0
$304$			5.00953i			0.287316i
$305$	26.6410			1.52546
$306$		0			0
$307$	12.3314			0.703787			0.351894	−	0.936040i	$-0.385538\pi$
$307$	12.3314			0.703787			0.351894	+	0.936040i	$0.385538\pi$
$308$			1.84639i			0.105208i
$309$		0			0
$310$	−3.52665			−0.200300
$311$	−23.3587	−	23.3587i	−1.32455	−	1.32455i	−0.910048	−	0.414503i	$-0.863955\pi$
$311$	−23.3587	−	23.3587i	−1.32455	−	1.32455i	−0.414503	−	0.910048i	$-0.636045\pi$
$312$		0			0
$313$	−10.0953	+	10.0953i	−0.570621	+	0.570621i	−0.932302	−	0.361681i	$-0.882203\pi$
$313$	−10.0953	+	10.0953i	−0.570621	+	0.570621i	0.361681	+	0.932302i	$0.382203\pi$
$314$		−	16.4243i		−	0.926877i
$315$		0			0
$316$	5.84293	−	5.84293i	0.328691	−	0.328691i
$317$	−6.57821	+	6.57821i	−0.369469	+	0.369469i	−0.867283	−	0.497815i	$-0.834136\pi$
$317$	−6.57821	+	6.57821i	−0.369469	+	0.369469i	0.497815	+	0.867283i	$0.334136\pi$
$318$		0			0
$319$	17.7754			0.995229
$320$	−2.30560	−	2.30560i	−0.128887	−	0.128887i
$321$		0			0
$322$	−6.47795			−0.361002
$323$	−18.9349	−	8.25182i	−1.05356	−	0.459144i
$324$		0			0
$325$			32.1739i			1.78469i
$326$	−4.40677	−	4.40677i	−0.244069	−	0.244069i
$327$		0			0
$328$	−8.19179	−	8.19179i	−0.452316	−	0.452316i
$329$	−3.44805	+	3.44805i	−0.190097	+	0.190097i
$330$		0			0
$331$			7.63186i			0.419485i	0.977757	+	0.209742i	$0.0672625\pi$
$331$			7.63186i			0.419485i	−0.977757	+	0.209742i	$0.932737\pi$
$332$		−	13.5467i		−	0.743473i
$333$		0			0
$334$	−11.3979	+	11.3979i	−0.623664	+	0.623664i
$335$	−23.4860	−	23.4860i	−1.28318	−	1.28318i
$336$		0			0
$337$	2.31022	+	2.31022i	0.125845	+	0.125845i	0.767224	−	0.641379i	$-0.221637\pi$
$337$	2.31022	+	2.31022i	0.125845	+	0.125845i	−0.641379	+	0.767224i	$0.721637\pi$
$338$		−	19.6400i		−	1.06828i
$339$		0			0
$340$	12.5125	−	4.91679i	0.678583	−	0.266650i
$341$	1.99704			0.108146
$342$		0			0
$343$	0.707107	+	0.707107i	0.0381802	+	0.0381802i
$344$	6.49371			0.350118
$345$		0			0
$346$	10.4886	−	10.4886i	0.563873	−	0.563873i
$347$	5.30560	−	5.30560i	0.284819	−	0.284819i	−0.550208	−	0.835028i	$-0.685452\pi$
$347$	5.30560	−	5.30560i	0.284819	−	0.284819i	0.835028	+	0.550208i	$0.185452\pi$
$348$		0			0
$349$		−	21.0965i		−	1.12927i	−0.825340	−	0.564635i	$-0.809017\pi$
$349$		−	21.0965i		−	1.12927i	0.825340	−	0.564635i	$-0.190983\pi$
$350$	−3.98211	+	3.98211i	−0.212853	+	0.212853i
$351$		0			0
$352$	1.30560	+	1.30560i	0.0695885	+	0.0695885i
$353$	1.42983			0.0761022			0.0380511	−	0.999276i	$-0.487885\pi$
$353$	1.42983			0.0761022			0.0380511	+	0.999276i	$0.487885\pi$
$354$		0			0
$355$		−	9.08131i		−	0.481986i
$356$	−17.6188			−0.933794
$357$		0			0
$358$	5.48452			0.289866
$359$		−	1.53180i		−	0.0808452i	−0.999183	−	0.0404226i	$-0.987130\pi$
$359$		−	1.53180i		−	0.0808452i	0.999183	−	0.0404226i	$-0.0128704\pi$
$360$		0			0
$361$	−6.09541			−0.320811
$362$	5.75711	+	5.75711i	0.302587	+	0.302587i
$363$		0			0
$364$	4.03980	−	4.03980i	0.211743	−	0.211743i
$365$			32.1881i			1.68480i
$366$		0			0
$367$	−16.7992	+	16.7992i	−0.876914	+	0.876914i	−0.993214	−	0.116300i	$-0.962896\pi$
$367$	−16.7992	+	16.7992i	−0.876914	+	0.876914i	0.116300	+	0.993214i	$0.462896\pi$
$368$	−4.58060	+	4.58060i	−0.238780	+	0.238780i
$369$		0			0
$370$	−5.40917			−0.281209
$371$	9.17659	+	9.17659i	0.476425	+	0.476425i
$372$		0			0
$373$	28.2473			1.46259			0.731295	−	0.682061i	$-0.238916\pi$
$373$	28.2473			1.46259			0.731295	+	0.682061i	$0.238916\pi$
$374$	−7.08546	+	2.78424i	−0.366381	+	0.143970i
$375$		0			0
$376$			4.87629i			0.251475i
$377$	−38.8915	−	38.8915i	−2.00301	−	2.00301i
$378$		0			0
$379$	−11.1432	−	11.1432i	−0.572387	−	0.572387i	0.360408	−	0.932795i	$-0.382638\pi$
$379$	−11.1432	−	11.1432i	−0.572387	−	0.572387i	−0.932795	+	0.360408i	$0.882638\pi$
$380$	11.5500	−	11.5500i	0.592501	−	0.592501i
$381$		0			0
$382$		−	24.2852i		−	1.24254i
$383$		−	32.2696i		−	1.64890i	−0.565935	−	0.824450i	$-0.691485\pi$
$383$		−	32.2696i		−	1.64890i	0.565935	−	0.824450i	$-0.308515\pi$
$384$		0			0
$385$	4.25704	−	4.25704i	0.216959	−	0.216959i
$386$	−11.3040	−	11.3040i	−0.575357	−	0.575357i
$387$		0			0
$388$	−3.89882	−	3.89882i	−0.197933	−	0.197933i
$389$		−	35.3387i		−	1.79174i	−0.444314	−	0.895871i	$-0.646552\pi$
$389$		−	35.3387i		−	1.79174i	0.444314	−	0.895871i	$-0.353448\pi$
$390$		0			0
$391$	−9.76834	−	24.8589i	−0.494006	−	1.25717i
$392$	1.00000			0.0505076
$393$		0			0
$394$	1.29953	+	1.29953i	0.0654692	+	0.0654692i
$395$	−26.9429			−1.35564
$396$		0			0
$397$	8.58575	−	8.58575i	0.430907	−	0.430907i	−0.458030	−	0.888937i	$-0.651445\pi$
$397$	8.58575	−	8.58575i	0.430907	−	0.430907i	0.888937	+	0.458030i	$0.151445\pi$
$398$	−6.09112	+	6.09112i	−0.305320	+	0.305320i
$399$		0			0
$400$			5.63155i			0.281578i
$401$	−0.915055	+	0.915055i	−0.0456956	+	0.0456956i	−0.729585	−	0.683890i	$-0.760287\pi$
$401$	−0.915055	+	0.915055i	−0.0456956	+	0.0456956i	0.683890	+	0.729585i	$0.260287\pi$
$402$		0			0
$403$	−4.36942	−	4.36942i	−0.217656	−	0.217656i
$404$	13.6813			0.680669
$405$		0			0
$406$		−	9.62707i		−	0.477783i
$407$	3.06306			0.151830
$408$		0			0
$409$	−6.73985			−0.333264			−0.166632	−	0.986019i	$-0.553289\pi$
$409$	−6.73985			−0.333264			−0.166632	+	0.986019i	$0.553289\pi$
$410$			37.7739i			1.86552i
$411$		0			0
$412$	−0.545951			−0.0268971
$413$	−7.77677	−	7.77677i	−0.382670	−	0.382670i
$414$		0			0
$415$	−31.2333	+	31.2333i	−1.53318	+	1.53318i
$416$		−	5.71315i		−	0.280110i
$417$		0			0
$418$	−6.54043	+	6.54043i	−0.319903	+	0.319903i
$419$	15.2800	−	15.2800i	0.746478	−	0.746478i	−0.227338	−	0.973816i	$-0.573002\pi$
$419$	15.2800	−	15.2800i	0.746478	−	0.746478i	0.973816	+	0.227338i	$0.0730023\pi$
$420$		0			0
$421$	−23.1022			−1.12593			−0.562966	−	0.826480i	$-0.690340\pi$
$421$	−23.1022			−1.12593			−0.562966	+	0.826480i	$0.690340\pi$
$422$	11.4696	+	11.4696i	0.558329	+	0.558329i
$423$		0			0
$424$	12.9777			0.630251
$425$	−21.2860	−	9.27643i	−1.03252	−	0.449973i
$426$		0			0
$427$			8.17058i			0.395402i
$428$	2.70541	+	2.70541i	0.130771	+	0.130771i
$429$		0			0
$430$	−14.9719	−	14.9719i	−0.722009	−	0.722009i
$431$	8.55830	−	8.55830i	0.412239	−	0.412239i	−0.470279	−	0.882518i	$-0.655847\pi$
$431$	8.55830	−	8.55830i	0.412239	−	0.412239i	0.882518	+	0.470279i	$0.155847\pi$
$432$		0			0
$433$			9.88742i			0.475159i	0.971368	+	0.237579i	$0.0763540\pi$
$433$			9.88742i			0.475159i	−0.971368	+	0.237579i	$0.923646\pi$
$434$		−	1.08159i		−	0.0519181i
$435$		0			0
$436$	−2.89425	+	2.89425i	−0.138609	+	0.138609i
$437$	−22.9467	−	22.9467i	−1.09769	−	1.09769i
$438$		0			0
$439$	4.48946	+	4.48946i	0.214270	+	0.214270i	0.806079	−	0.591808i	$-0.201586\pi$
$439$	4.48946	+	4.48946i	0.214270	+	0.214270i	−0.591808	+	0.806079i	$0.701586\pi$
$440$		−	6.02036i		−	0.287009i
$441$		0			0
$442$	21.5944	+	9.41083i	1.02714	+	0.447627i
$443$	6.90922			0.328267			0.164134	−	0.986438i	$-0.447517\pi$
$443$	6.90922			0.328267			0.164134	+	0.986438i	$0.447517\pi$
$444$		0			0
$445$	40.6218	+	40.6218i	1.92566	+	1.92566i
$446$	14.2184			0.673260
$447$		0			0
$448$	0.707107	−	0.707107i	0.0334077	−	0.0334077i
$449$	22.2387	−	22.2387i	1.04951	−	1.04951i	0.0508021	−	0.998709i	$-0.483822\pi$
$449$	22.2387	−	22.2387i	1.04951	−	1.04951i	0.998709	−	0.0508021i	$-0.0161778\pi$
$450$		0			0
$451$		−	21.3904i		−	1.00723i
$452$	2.18078	−	2.18078i	0.102575	−	0.102575i
$453$		0			0
$454$	−12.4143	−	12.4143i	−0.582633	−	0.582633i
$455$	−18.6283			−0.873309
$456$		0			0
$457$			39.8882i			1.86589i	0.360018	+	0.932945i	$0.382771\pi$
$457$			39.8882i			1.86589i	−0.360018	+	0.932945i	$0.617229\pi$
$458$	7.48179			0.349601
$459$		0			0
$460$	21.1220			0.984820
$461$			6.78516i			0.316016i	0.987438	+	0.158008i	$0.0505073\pi$
$461$			6.78516i			0.316016i	−0.987438	+	0.158008i	$0.949493\pi$
$462$		0			0
$463$	−27.7289			−1.28867			−0.644335	−	0.764743i	$-0.722866\pi$
$463$	−27.7289			−1.28867			−0.644335	+	0.764743i	$0.722866\pi$
$464$	−6.80737	−	6.80737i	−0.316024	−	0.316024i
$465$		0			0
$466$	−12.7751	+	12.7751i	−0.591796	+	0.591796i
$467$			2.92533i			0.135368i	0.997707	+	0.0676841i	$0.0215610\pi$
$467$			2.92533i			0.135368i	−0.997707	+	0.0676841i	$0.978439\pi$
$468$		0			0
$469$	7.20294	−	7.20294i	0.332601	−	0.332601i
$470$	11.2427	−	11.2427i	0.518589	−	0.518589i
$471$		0			0
$472$	−10.9980			−0.506225
$473$	8.47817	+	8.47817i	0.389827	+	0.389827i
$474$		0			0
$475$	−28.2114			−1.29443
$476$	1.50794	+	3.83746i	0.0691162	+	0.175890i
$477$		0			0
$478$		−	24.8996i		−	1.13888i
$479$	16.9751	+	16.9751i	0.775613	+	0.775613i	0.979082	−	0.203468i	$-0.0652214\pi$
$479$	16.9751	+	16.9751i	0.775613	+	0.775613i	−0.203468	+	0.979082i	$0.565221\pi$
$480$		0			0
$481$	−6.70181	−	6.70181i	−0.305576	−	0.305576i
$482$	−7.66021	+	7.66021i	−0.348913	+	0.348913i
$483$		0			0
$484$		−	7.59083i		−	0.345038i
$485$			17.9782i			0.816350i
$486$		0			0
$487$	7.39788	−	7.39788i	0.335230	−	0.335230i	−0.519339	−	0.854569i	$-0.673822\pi$
$487$	7.39788	−	7.39788i	0.335230	−	0.335230i	0.854569	+	0.519339i	$0.173822\pi$
$488$	5.77747	+	5.77747i	0.261534	+	0.261534i
$489$		0			0
$490$	−2.30560	−	2.30560i	−0.104156	−	0.104156i
$491$			17.3434i			0.782699i	0.920242	+	0.391349i	$0.127992\pi$
$491$			17.3434i			0.782699i	−0.920242	+	0.391349i	$0.872008\pi$
$492$		0			0
$493$	36.9435	−	14.5170i	1.66385	−	0.653814i
$494$	28.6202			1.28768
$495$		0			0
$496$	−0.764801	−	0.764801i	−0.0343406	−	0.0343406i
$497$	2.78516			0.124932
$498$		0			0
$499$	6.44426	−	6.44426i	0.288485	−	0.288485i	−0.547996	−	0.836481i	$-0.684609\pi$
$499$	6.44426	−	6.44426i	0.288485	−	0.288485i	0.836481	+	0.547996i	$0.184609\pi$
$500$	1.45611	−	1.45611i	0.0651191	−	0.0651191i
$501$		0			0
$502$			8.75005i			0.390534i
$503$	−25.7142	+	25.7142i	−1.14654	+	1.14654i	−0.159309	+	0.987229i	$0.550927\pi$
$503$	−25.7142	+	25.7142i	−1.14654	+	1.14654i	−0.987229	+	0.159309i	$0.949073\pi$
$504$		0			0
$505$	−31.5435	−	31.5435i	−1.40367	−	1.40367i
$506$	−11.9608			−0.531724
$507$		0			0
$508$		−	3.77239i		−	0.167373i
$509$	12.1989			0.540709			0.270354	−	0.962761i	$-0.412859\pi$
$509$	12.1989			0.540709			0.270354	+	0.962761i	$0.412859\pi$
$510$		0			0
$511$	−9.87180			−0.436703
$512$		−	1.00000i		−	0.0441942i
$513$		0			0
$514$	22.9937			1.01421
$515$	1.25874	+	1.25874i	0.0554668	+	0.0554668i
$516$		0			0
$517$	−6.36646	+	6.36646i	−0.279997	+	0.279997i
$518$		−	1.65894i		−	0.0728899i
$519$		0			0
$520$	−13.1722	+	13.1722i	−0.577640	+	0.577640i
$521$	−1.56576	+	1.56576i	−0.0685972	+	0.0685972i	−0.740573	−	0.671976i	$-0.765446\pi$
$521$	−1.56576	+	1.56576i	−0.0685972	+	0.0685972i	0.671976	+	0.740573i	$0.265446\pi$
$522$		0			0
$523$	−30.1951			−1.32034			−0.660169	−	0.751117i	$-0.729515\pi$
$523$	−30.1951			−1.32034			−0.660169	+	0.751117i	$0.729515\pi$
$524$	−7.76100	−	7.76100i	−0.339041	−	0.339041i
$525$		0			0
$526$	−7.87376			−0.343312
$527$	4.15057	−	1.63097i	0.180802	−	0.0710463i
$528$		0			0
$529$		−	18.9638i		−	0.824513i
$530$	−29.9212	−	29.9212i	−1.29970	−	1.29970i
$531$		0			0
$532$	3.54227	+	3.54227i	0.153577	+	0.153577i
$533$	−46.8009	+	46.8009i	−2.02717	+	2.02717i
$534$		0			0
$535$		−	12.4752i		−	0.539349i
$536$		−	10.1865i		−	0.439990i
$537$		0			0
$538$	9.05635	−	9.05635i	0.390447	−	0.390447i
$539$	1.30560	+	1.30560i	0.0562360	+	0.0562360i
$540$		0			0
$541$	−1.33483	−	1.33483i	−0.0573889	−	0.0573889i	0.677830	−	0.735219i	$-0.262921\pi$
$541$	−1.33483	−	1.33483i	−0.0573889	−	0.0573889i	−0.735219	+	0.677830i	$0.762921\pi$
$542$		−	25.4839i		−	1.09463i
$543$		0			0
$544$	3.77977	+	1.64722i	0.162056	+	0.0706241i
$545$	13.3460			0.571678
$546$		0			0
$547$	17.4935	+	17.4935i	0.747969	+	0.747969i	0.974097	−	0.226129i	$-0.0726070\pi$
$547$	17.4935	+	17.4935i	0.747969	+	0.747969i	−0.226129	+	0.974097i	$0.572607\pi$
$548$	−8.64712			−0.369387
$549$		0			0
$550$	−7.35254	+	7.35254i	−0.313513	+	0.313513i
$551$	34.1017	−	34.1017i	1.45278	−	1.45278i
$552$		0			0
$553$		−	8.26315i		−	0.351385i
$554$	−12.2501	+	12.2501i	−0.520456	+	0.520456i
$555$		0			0
$556$	−2.08193	−	2.08193i	−0.0882934	−	0.0882934i
$557$	4.50376			0.190831			0.0954153	−	0.995438i	$-0.469582\pi$
$557$	4.50376			0.190831			0.0954153	+	0.995438i	$0.469582\pi$
$558$		0			0
$559$		−	37.0995i		−	1.56914i
$560$	−3.26061			−0.137786
$561$		0			0
$562$	−15.2279			−0.642351
$563$		−	3.34929i		−	0.141156i	−0.997506	−	0.0705779i	$-0.977516\pi$
$563$		−	3.34929i		−	0.141156i	0.997506	−	0.0705779i	$-0.0224844\pi$
$564$		0			0
$565$	−10.0560			−0.423060
$566$	−8.18153	−	8.18153i	−0.343895	−	0.343895i
$567$		0			0
$568$	1.96941	−	1.96941i	0.0826344	−	0.0826344i
$569$		−	12.5973i		−	0.528106i	−0.964508	−	0.264053i	$-0.914941\pi$
$569$		−	12.5973i		−	0.528106i	0.964508	−	0.264053i	$-0.0850593\pi$
$570$		0			0
$571$	−11.6715	+	11.6715i	−0.488439	+	0.488439i	−0.907813	−	0.419375i	$-0.862249\pi$
$571$	−11.6715	+	11.6715i	−0.488439	+	0.488439i	0.419375	+	0.907813i	$0.362249\pi$
$572$	7.45906	−	7.45906i	0.311879	−	0.311879i
$573$		0			0
$574$	−11.5849			−0.483546
$575$	−25.7959	−	25.7959i	−1.07576	−	1.07576i
$576$		0			0
$577$	−12.5655			−0.523107			−0.261554	−	0.965189i	$-0.584235\pi$
$577$	−12.5655			−0.523107			−0.261554	+	0.965189i	$0.584235\pi$
$578$	−12.4522	+	11.5733i	−0.517945	+	0.481386i
$579$		0			0
$580$			31.3901i			1.30340i
$581$	−9.57898	−	9.57898i	−0.397403	−	0.397403i
$582$		0			0
$583$	16.9436	+	16.9436i	0.701732	+	0.701732i
$584$	−6.98042	+	6.98042i	−0.288852	+	0.288852i
$585$		0			0
$586$		−	11.4033i		−	0.471065i
$587$			29.3030i			1.20946i	0.796429	+	0.604732i	$0.206720\pi$
$587$			29.3030i			1.20946i	−0.796429	+	0.604732i	$0.793280\pi$
$588$		0			0
$589$	3.83129	−	3.83129i	0.157866	−	0.157866i
$590$	25.3570	+	25.3570i	1.04393	+	1.04393i
$591$		0			0
$592$	−1.17305	−	1.17305i	−0.0482121	−	0.0482121i
$593$			39.8483i			1.63637i	0.574953	+	0.818187i	$0.305020\pi$
$593$			39.8483i			1.63637i	−0.574953	+	0.818187i	$0.694980\pi$
$594$		0			0
$595$	5.37095	−	12.3243i	0.220187	−	0.505249i
$596$	−6.48200			−0.265513
$597$		0			0
$598$	26.1696	+	26.1696i	1.07016	+	1.07016i
$599$	14.8083			0.605049			0.302525	−	0.953142i	$-0.402171\pi$
$599$	14.8083			0.605049			0.302525	+	0.953142i	$0.402171\pi$
$600$		0			0
$601$	2.68294	−	2.68294i	0.109439	−	0.109439i	−0.650267	−	0.759706i	$-0.725343\pi$
$601$	2.68294	−	2.68294i	0.109439	−	0.109439i	0.759706	+	0.650267i	$0.225343\pi$
$602$	4.59175	−	4.59175i	0.187146	−	0.187146i
$603$		0			0
$604$		−	13.0424i		−	0.530689i
$605$	−17.5014	+	17.5014i	−0.711533	+	0.711533i
$606$		0			0
$607$	−12.8746	−	12.8746i	−0.522566	−	0.522566i	0.395780	−	0.918346i	$-0.370474\pi$
$607$	−12.8746	−	12.8746i	−0.522566	−	0.522566i	−0.918346	+	0.395780i	$0.870474\pi$
$608$	5.00953			0.203163
$609$		0			0
$610$		−	26.6410i		−	1.07866i
$611$	27.8589			1.12705
$612$		0			0
$613$	28.5428			1.15283			0.576416	−	0.817156i	$-0.304451\pi$
$613$	28.5428			1.15283			0.576416	+	0.817156i	$0.304451\pi$
$614$		−	12.3314i		−	0.497653i
$615$		0			0
$616$	1.84639			0.0743933
$617$	14.8324	+	14.8324i	0.597131	+	0.597131i	0.939548	−	0.342417i	$-0.111246\pi$
$617$	14.8324	+	14.8324i	0.597131	+	0.597131i	−0.342417	+	0.939548i	$0.611246\pi$
$618$		0			0
$619$	9.95097	−	9.95097i	0.399963	−	0.399963i	−0.478257	−	0.878220i	$-0.658731\pi$
$619$	9.95097	−	9.95097i	0.399963	−	0.399963i	0.878220	+	0.478257i	$0.158731\pi$
$620$			3.52665i			0.141633i
$621$		0			0
$622$	−23.3587	+	23.3587i	−0.936599	+	0.936599i
$623$	−12.4584	+	12.4584i	−0.499134	+	0.499134i
$624$		0			0
$625$	21.4434			0.857735
$626$	10.0953	+	10.0953i	0.403490	+	0.403490i
$627$		0			0
$628$	−16.4243			−0.655401
$629$	6.36614	−	2.50159i	0.253835	−	0.0997447i
$630$		0			0
$631$		−	3.61242i		−	0.143808i	−0.997412	−	0.0719041i	$-0.977092\pi$
$631$		−	3.61242i		−	0.143808i	0.997412	−	0.0719041i	$-0.0229075\pi$
$632$	−5.84293	−	5.84293i	−0.232419	−	0.232419i
$633$		0			0
$634$	6.57821	+	6.57821i	0.261254	+	0.261254i
$635$	−8.69762	+	8.69762i	−0.345154	+	0.345154i
$636$		0			0
$637$		−	5.71315i		−	0.226363i
$638$		−	17.7754i		−	0.703733i
$639$		0			0
$640$	−2.30560	+	2.30560i	−0.0911367	+	0.0911367i
$641$	18.7362	+	18.7362i	0.740036	+	0.740036i	0.972585	−	0.232549i	$-0.0747064\pi$
$641$	18.7362	+	18.7362i	0.740036	+	0.740036i	−0.232549	+	0.972585i	$0.574706\pi$
$642$		0			0
$643$	−19.8197	−	19.8197i	−0.781614	−	0.781614i	0.198489	−	0.980103i	$-0.436397\pi$
$643$	−19.8197	−	19.8197i	−0.781614	−	0.781614i	−0.980103	+	0.198489i	$0.936397\pi$
$644$			6.47795i			0.255267i
$645$		0			0
$646$	−8.25182	+	18.9349i	−0.324663	+	0.744983i
$647$	37.8110			1.48650			0.743252	−	0.669012i	$-0.233282\pi$
$647$	37.8110			1.48650			0.743252	+	0.669012i	$0.233282\pi$
$648$		0			0
$649$	−14.3590	−	14.3590i	−0.563639	−	0.563639i
$650$	32.1739			1.26196
$651$		0			0
$652$	−4.40677	+	4.40677i	−0.172583	+	0.172583i
$653$	0.561081	−	0.561081i	0.0219568	−	0.0219568i	−0.696043	−	0.718000i	$-0.745058\pi$
$653$	0.561081	−	0.561081i	0.0219568	−	0.0219568i	0.718000	+	0.696043i	$0.245058\pi$
$654$		0			0
$655$			35.7875i			1.39833i
$656$	−8.19179	+	8.19179i	−0.319836	+	0.319836i
$657$		0			0
$658$	3.44805	+	3.44805i	0.134419	+	0.134419i
$659$	−22.7898			−0.887766			−0.443883	−	0.896085i	$-0.646399\pi$
$659$	−22.7898			−0.887766			−0.443883	+	0.896085i	$0.646399\pi$
$660$		0			0
$661$			7.14793i			0.278022i	0.990291	+	0.139011i	$0.0443924\pi$
$661$			7.14793i			0.278022i	−0.990291	+	0.139011i	$0.955608\pi$
$662$	7.63186			0.296621
$663$		0			0
$664$	−13.5467			−0.525715
$665$		−	16.3341i		−	0.633410i
$666$		0			0
$667$	62.3636			2.41473
$668$	11.3979	+	11.3979i	0.440997	+	0.440997i
$669$		0			0
$670$	−23.4860	+	23.4860i	−0.907342	+	0.907342i
$671$			15.0861i			0.582393i
$672$		0			0
$673$	12.8708	−	12.8708i	0.496133	−	0.496133i	−0.414099	−	0.910232i	$-0.635903\pi$
$673$	12.8708	−	12.8708i	0.496133	−	0.496133i	0.910232	+	0.414099i	$0.135903\pi$
$674$	2.31022	−	2.31022i	0.0889862	−	0.0889862i
$675$		0			0
$676$	−19.6400			−0.755386
$677$	−4.79047	−	4.79047i	−0.184113	−	0.184113i	0.609033	−	0.793145i	$-0.291558\pi$
$677$	−4.79047	−	4.79047i	−0.184113	−	0.184113i	−0.793145	+	0.609033i	$0.791558\pi$
$678$		0			0
$679$	−5.51377			−0.211599
$680$	−4.91679	−	12.5125i	−0.188550	−	0.479831i
$681$		0			0
$682$		−	1.99704i		−	0.0764707i
$683$	2.41071	+	2.41071i	0.0922431	+	0.0922431i	0.751723	−	0.659479i	$-0.229223\pi$
$683$	2.41071	+	2.41071i	0.0922431	+	0.0922431i	−0.659479	+	0.751723i	$0.729223\pi$
$684$		0			0
$685$	19.9368	+	19.9368i	0.761745	+	0.761745i
$686$	0.707107	−	0.707107i	0.0269975	−	0.0269975i
$687$		0			0
$688$		−	6.49371i		−	0.247571i
$689$		−	74.1432i		−	2.82463i
$690$		0			0
$691$	2.25589	−	2.25589i	0.0858183	−	0.0858183i	−0.662895	−	0.748713i	$-0.730672\pi$
$691$	2.25589	−	2.25589i	0.0858183	−	0.0858183i	0.748713	+	0.662895i	$0.230672\pi$
$692$	−10.4886	−	10.4886i	−0.398718	−	0.398718i
$693$		0			0
$694$	−5.30560	−	5.30560i	−0.201398	−	0.201398i
$695$			9.60017i			0.364155i
$696$		0			0
$697$	−17.4694	−	44.4568i	−0.661700	−	1.68392i
$698$	−21.0965			−0.798515
$699$		0			0
$700$	3.98211	+	3.98211i	0.150510	+	0.150510i
$701$	−19.8302			−0.748978			−0.374489	−	0.927231i	$-0.622182\pi$
$701$	−19.8302			−0.748978			−0.374489	+	0.927231i	$0.622182\pi$
$702$		0			0
$703$	5.87644	−	5.87644i	0.221634	−	0.221634i
$704$	1.30560	−	1.30560i	0.0492065	−	0.0492065i
$705$		0			0
$706$		−	1.42983i		−	0.0538124i
$707$	9.67412	−	9.67412i	0.363833	−	0.363833i
$708$		0			0
$709$	−15.3033	−	15.3033i	−0.574727	−	0.574727i	0.358718	−	0.933446i	$-0.383214\pi$
$709$	−15.3033	−	15.3033i	−0.574727	−	0.574727i	−0.933446	+	0.358718i	$0.883214\pi$
$710$	−9.08131			−0.340816
$711$		0			0
$712$			17.6188i			0.660292i
$713$	7.00649			0.262395
$714$		0			0
$715$	−34.3952			−1.28631
$716$		−	5.48452i		−	0.204966i
$717$		0			0
$718$	−1.53180			−0.0571662
$719$	12.6098	+	12.6098i	0.470266	+	0.470266i	0.902001	−	0.431735i	$-0.142098\pi$
$719$	12.6098	+	12.6098i	0.470266	+	0.470266i	−0.431735	+	0.902001i	$0.642098\pi$
$720$		0			0
$721$	−0.386046	+	0.386046i	−0.0143771	+	0.0143771i
$722$			6.09541i			0.226848i
$723$		0			0
$724$	5.75711	−	5.75711i	0.213961	−	0.213961i
$725$	38.3360	−	38.3360i	1.42376	−	1.42376i
$726$		0			0
$727$	−32.4589			−1.20383			−0.601916	−	0.798559i	$-0.705596\pi$
$727$	−32.4589			−1.20383			−0.601916	+	0.798559i	$0.705596\pi$
$728$	−4.03980	−	4.03980i	−0.149725	−	0.149725i
$729$		0			0
$730$	32.1881			1.19133
$731$	24.5447	+	10.6966i	0.907820	+	0.395628i
$732$		0			0
$733$			50.1810i			1.85348i	0.375708	+	0.926738i	$0.377400\pi$
$733$			50.1810i			1.85348i	−0.375708	+	0.926738i	$0.622600\pi$
$734$	16.7992	+	16.7992i	0.620072	+	0.620072i
$735$		0			0
$736$	4.58060	+	4.58060i	0.168843	+	0.168843i
$737$	13.2995	−	13.2995i	0.489892	−	0.489892i
$738$		0			0
$739$		−	20.8199i		−	0.765872i	−0.923775	−	0.382936i	$-0.874913\pi$
$739$		−	20.8199i		−	0.765872i	0.923775	−	0.382936i	$-0.125087\pi$
$740$			5.40917i			0.198845i
$741$		0			0
$742$	9.17659	−	9.17659i	0.336883	−	0.336883i
$743$	−16.4365	−	16.4365i	−0.602996	−	0.602996i	0.338110	−	0.941106i	$-0.390212\pi$
$743$	−16.4365	−	16.4365i	−0.602996	−	0.602996i	−0.941106	+	0.338110i	$0.890212\pi$
$744$		0			0
$745$	14.9449	+	14.9449i	0.547538	+	0.547538i
$746$		−	28.2473i		−	1.03421i
$747$		0			0
$748$	2.78424	+	7.08546i	0.101802	+	0.259070i
$749$	3.82603			0.139800
$750$		0			0
$751$	15.5986	+	15.5986i	0.569202	+	0.569202i	0.931905	−	0.362703i	$-0.118146\pi$
$751$	15.5986	+	15.5986i	0.569202	+	0.569202i	−0.362703	+	0.931905i	$0.618146\pi$
$752$	4.87629			0.177820
$753$		0			0
$754$	−38.8915	+	38.8915i	−1.41634	+	1.41634i
$755$	−30.0706	+	30.0706i	−1.09438	+	1.09438i
$756$		0			0
$757$		−	14.6134i		−	0.531133i	−0.964093	−	0.265566i	$-0.914441\pi$
$757$		−	14.6134i		−	0.531133i	0.964093	−	0.265566i	$-0.0855589\pi$
$758$	−11.1432	+	11.1432i	−0.404739	+	0.404739i
$759$		0			0
$760$	−11.5500	−	11.5500i	−0.418961	−	0.418961i
$761$	22.4035			0.812125			0.406063	−	0.913845i	$-0.366901\pi$
$761$	22.4035			0.812125			0.406063	+	0.913845i	$0.366901\pi$
$762$		0			0
$763$			4.09309i			0.148180i
$764$	−24.2852			−0.878607
$765$		0			0
$766$	−32.2696			−1.16595
$767$			62.8333i			2.26878i
$768$		0			0
$769$	4.26500			0.153800			0.0769000	−	0.997039i	$-0.475498\pi$
$769$	4.26500			0.153800			0.0769000	+	0.997039i	$0.475498\pi$
$770$	−4.25704	−	4.25704i	−0.153413	−	0.153413i
$771$		0			0
$772$	−11.3040	+	11.3040i	−0.406839	+	0.406839i
$773$			29.5875i			1.06419i	0.846685	+	0.532094i	$0.178595\pi$
$773$			29.5875i			1.06419i	−0.846685	+	0.532094i	$0.821405\pi$
$774$		0			0
$775$	4.30702	−	4.30702i	0.154713	−	0.154713i
$776$	−3.89882	+	3.89882i	−0.139960	+	0.139960i
$777$		0			0
$778$	−35.3387			−1.26695
$779$	−41.0371	−	41.0371i	−1.47031	−	1.47031i
$780$		0			0
$781$	5.14250			0.184013
$782$	−24.8589	+	9.76834i	−0.888952	+	0.349315i
$783$		0			0
$784$		−	1.00000i		−	0.0357143i
$785$	37.8678	+	37.8678i	1.35156	+	1.35156i
$786$		0			0
$787$	22.9911	+	22.9911i	0.819544	+	0.819544i	0.986042	−	0.166497i	$-0.0532458\pi$
$787$	22.9911	+	22.9911i	0.819544	+	0.819544i	−0.166497	+	0.986042i	$0.553246\pi$
$788$	1.29953	−	1.29953i	0.0462937	−	0.0462937i
$789$		0			0
$790$			26.9429i			0.958585i
$791$		−	3.08409i		−	0.109658i
$792$		0			0
$793$	33.0075	−	33.0075i	1.17213	−	1.17213i
$794$	−8.58575	−	8.58575i	−0.304697	−	0.304697i
$795$		0			0
$796$	6.09112	+	6.09112i	0.215894	+	0.215894i
$797$		−	49.3210i		−	1.74704i	−0.486788	−	0.873520i	$-0.661832\pi$
$797$		−	49.3210i		−	1.74704i	0.486788	−	0.873520i	$-0.338168\pi$
$798$		0			0
$799$	−8.03233	+	18.4312i	−0.284163	+	0.652050i
$800$	5.63155			0.199105
$801$		0			0
$802$	0.915055	+	0.915055i	0.0323117	+	0.0323117i
$803$	−18.2272			−0.643225
$804$		0			0
$805$	14.9355	−	14.9355i	0.526408	−	0.526408i
$806$	−4.36942	+	4.36942i	−0.153906	+	0.153906i
$807$		0			0
$808$		−	13.6813i		−	0.481305i
$809$	−4.65408	+	4.65408i	−0.163629	+	0.163629i	−0.784172	−	0.620543i	$-0.786912\pi$
$809$	−4.65408	+	4.65408i	−0.163629	+	0.163629i	0.620543	+	0.784172i	$0.286912\pi$
$810$		0			0
$811$	−15.8229	−	15.8229i	−0.555616	−	0.555616i	0.372440	−	0.928056i	$-0.378521\pi$
$811$	−15.8229	−	15.8229i	−0.555616	−	0.555616i	−0.928056	+	0.372440i	$0.878521\pi$
$812$	−9.62707			−0.337844
$813$		0			0
$814$		−	3.06306i		−	0.107360i
$815$	20.3205			0.711795
$816$		0			0
$817$	32.5305			1.13810
$818$			6.73985i			0.235653i
$819$		0			0
$820$	37.7739			1.31912
$821$	1.50011	+	1.50011i	0.0523541	+	0.0523541i	0.732799	−	0.680445i	$-0.238213\pi$
$821$	1.50011	+	1.50011i	0.0523541	+	0.0523541i	−0.680445	+	0.732799i	$0.738213\pi$
$822$		0			0
$823$	14.8960	−	14.8960i	0.519243	−	0.519243i	−0.398099	−	0.917342i	$-0.630330\pi$
$823$	14.8960	−	14.8960i	0.519243	−	0.519243i	0.917342	+	0.398099i	$0.130330\pi$
$824$			0.545951i			0.0190191i
$825$		0			0
$826$	−7.77677	+	7.77677i	−0.270588	+	0.270588i
$827$	19.1898	−	19.1898i	0.667294	−	0.667294i	−0.289795	−	0.957089i	$-0.593587\pi$
$827$	19.1898	−	19.1898i	0.667294	−	0.667294i	0.957089	+	0.289795i	$0.0935871\pi$
$828$		0			0
$829$	−22.5520			−0.783265			−0.391632	−	0.920122i	$-0.628089\pi$
$829$	−22.5520			−0.783265			−0.391632	+	0.920122i	$0.628089\pi$
$830$	31.2333	+	31.2333i	1.08412	+	1.08412i
$831$		0			0
$832$	−5.71315			−0.198068
$833$	3.77977	+	1.64722i	0.130961	+	0.0570729i
$834$		0			0
$835$		−	52.5578i		−	1.81884i
$836$	6.54043	+	6.54043i	0.226205	+	0.226205i
$837$		0			0
$838$	−15.2800	−	15.2800i	−0.527839	−	0.527839i
$839$	12.7959	−	12.7959i	0.441762	−	0.441762i	−0.450842	−	0.892604i	$-0.648876\pi$
$839$	12.7959	−	12.7959i	0.441762	−	0.441762i	0.892604	+	0.450842i	$0.148876\pi$
$840$		0			0
$841$			63.6805i			2.19588i
$842$			23.1022i			0.796155i
$843$		0			0
$844$	11.4696	−	11.4696i	0.394798	−	0.394798i
$845$	45.2820	+	45.2820i	1.55775	+	1.55775i
$846$		0			0
$847$	−5.36753	−	5.36753i	−0.184431	−	0.184431i
$848$		−	12.9777i		−	0.445655i
$849$		0			0
$850$	−9.27643	+	21.2860i	−0.318179	+	0.730103i
$851$	10.7466			0.368387
$852$		0			0
$853$	−7.76900	−	7.76900i	−0.266005	−	0.266005i	0.561483	−	0.827488i	$-0.310231\pi$
$853$	−7.76900	−	7.76900i	−0.266005	−	0.266005i	−0.827488	+	0.561483i	$0.810231\pi$
$854$	8.17058			0.279592
$855$		0			0
$856$	2.70541	−	2.70541i	0.0924691	−	0.0924691i
$857$	14.3093	−	14.3093i	0.488795	−	0.488795i	−0.419131	−	0.907926i	$-0.637665\pi$
$857$	14.3093	−	14.3093i	0.488795	−	0.488795i	0.907926	+	0.419131i	$0.137665\pi$
$858$		0			0
$859$		−	57.9569i		−	1.97746i	−0.149696	−	0.988732i	$-0.547829\pi$
$859$		−	57.9569i		−	1.97746i	0.149696	−	0.988732i	$-0.452171\pi$
$860$	−14.9719	+	14.9719i	−0.510537	+	0.510537i
$861$		0			0
$862$	−8.55830	−	8.55830i	−0.291497	−	0.291497i
$863$	−14.7330			−0.501518			−0.250759	−	0.968050i	$-0.580680\pi$
$863$	−14.7330			−0.501518			−0.250759	+	0.968050i	$0.580680\pi$
$864$		0			0
$865$			48.3651i			1.64446i
$866$	9.88742			0.335988
$867$		0			0
$868$	−1.08159			−0.0367116
$869$		−	15.2570i		−	0.517559i
$870$		0			0
$871$	−58.1970			−1.97193
$872$	2.89425	+	2.89425i	0.0980117	+	0.0980117i
$873$		0			0
$874$	−22.9467	+	22.9467i	−0.776182	+	0.776182i
$875$		−	2.05925i		−	0.0696153i
$876$		0			0
$877$	20.2500	−	20.2500i	0.683794	−	0.683794i	−0.277059	−	0.960853i	$-0.589360\pi$
$877$	20.2500	−	20.2500i	0.683794	−	0.683794i	0.960853	+	0.277059i	$0.0893597\pi$
$878$	4.48946	−	4.48946i	0.151512	−	0.151512i
$879$		0			0
$880$	−6.02036			−0.202946
$881$	−18.4800	−	18.4800i	−0.622607	−	0.622607i	0.323590	−	0.946197i	$-0.395110\pi$
$881$	−18.4800	−	18.4800i	−0.622607	−	0.622607i	−0.946197	+	0.323590i	$0.895110\pi$
$882$		0			0
$883$	31.5159			1.06060			0.530298	−	0.847811i	$-0.322080\pi$
$883$	31.5159			1.06060			0.530298	+	0.847811i	$0.322080\pi$
$884$	9.41083	−	21.5944i	0.316520	−	0.726297i
$885$		0			0
$886$		−	6.90922i		−	0.232120i
$887$	−13.1259	−	13.1259i	−0.440723	−	0.440723i	0.451532	−	0.892255i	$-0.350878\pi$
$887$	−13.1259	−	13.1259i	−0.440723	−	0.440723i	−0.892255	+	0.451532i	$0.850878\pi$
$888$		0			0
$889$	−2.66748	−	2.66748i	−0.0894646	−	0.0894646i
$890$	40.6218	−	40.6218i	1.36165	−	1.36165i
$891$		0			0
$892$		−	14.2184i		−	0.476066i
$893$			24.4279i			0.817449i
$894$		0			0
$895$	−12.6451	+	12.6451i	−0.422679	+	0.422679i
$896$	−0.707107	−	0.707107i	−0.0236228	−	0.0236228i
$897$		0			0
$898$	−22.2387	−	22.2387i	−0.742116	−	0.742116i
$899$			10.4126i			0.347278i
$900$		0			0
$901$	49.0525	+	21.3771i	1.63418	+	0.712174i
$902$	−21.3904			−0.712221
$903$		0			0
$904$	−2.18078	−	2.18078i	−0.0725318	−	0.0725318i
$905$	−26.5472			−0.882458
$906$		0			0
$907$	27.2074	−	27.2074i	0.903408	−	0.903408i	−0.0923216	−	0.995729i	$-0.529429\pi$
$907$	27.2074	−	27.2074i	0.903408	−	0.903408i	0.995729	+	0.0923216i	$0.0294288\pi$
$908$	−12.4143	+	12.4143i	−0.411984	+	0.411984i
$909$		0			0
$910$			18.6283i			0.617523i
$911$	−1.27720	+	1.27720i	−0.0423156	+	0.0423156i	−0.727948	−	0.685632i	$-0.759526\pi$
$911$	−1.27720	+	1.27720i	−0.0423156	+	0.0423156i	0.685632	+	0.727948i	$0.259526\pi$
$912$		0			0
$913$	−17.6866	−	17.6866i	−0.585340	−	0.585340i
$914$	39.8882			1.31938
$915$		0			0
$916$		−	7.48179i		−	0.247205i
$917$	−10.9757			−0.362450
$918$		0			0
$919$	9.80140			0.323318			0.161659	−	0.986847i	$-0.448315\pi$
$919$	9.80140			0.323318			0.161659	+	0.986847i	$0.448315\pi$
$920$		−	21.1220i		−	0.696373i
$921$		0			0
$922$	6.78516			0.223457
$923$	−11.2515	−	11.2515i	−0.370348	−	0.370348i
$924$		0			0
$925$	6.60610	−	6.60610i	0.217207	−	0.217207i
$926$			27.7289i			0.911227i
$927$		0			0
$928$	−6.80737	+	6.80737i	−0.223463	+	0.223463i
$929$	−11.0678	+	11.0678i	−0.363124	+	0.363124i	−0.864962	−	0.501838i	$-0.832658\pi$
$929$	−11.0678	+	11.0678i	−0.363124	+	0.363124i	0.501838	+	0.864962i	$0.332658\pi$
$930$		0			0
$931$	5.00953			0.164181
$932$	12.7751	+	12.7751i	0.418463	+	0.418463i
$933$		0			0
$934$	2.92533			0.0957198
$935$	9.91688	−	22.7556i	0.324317	−	0.744187i
$936$		0			0
$937$			48.1706i			1.57366i	0.617167	+	0.786832i	$0.288280\pi$
$937$			48.1706i			1.57366i	−0.617167	+	0.786832i	$0.711720\pi$
$938$	−7.20294	−	7.20294i	−0.235184	−	0.235184i
$939$		0			0
$940$	−11.2427	−	11.2427i	−0.366698	−	0.366698i
$941$	−11.7802	+	11.7802i	−0.384024	+	0.384024i	−0.872549	−	0.488526i	$-0.837535\pi$
$941$	−11.7802	+	11.7802i	−0.384024	+	0.384024i	0.488526	+	0.872549i	$0.337535\pi$
$942$		0			0
$943$		−	75.0467i		−	2.44386i
$944$			10.9980i			0.357955i
$945$		0			0
$946$	8.47817	−	8.47817i	0.275649	−	0.275649i
$947$	−34.6747	−	34.6747i	−1.12678	−	1.12678i	−0.990698	−	0.136079i	$-0.956550\pi$
$947$	−34.6747	−	34.6747i	−1.12678	−	1.12678i	−0.136079	−	0.990698i	$-0.543450\pi$
$948$		0			0
$949$	39.8801	+	39.8801i	1.29456	+	1.29456i
$950$			28.2114i			0.915300i
$951$		0			0
$952$	3.83746	−	1.50794i	0.124373	−	0.0488725i
$953$	−4.64325			−0.150409			−0.0752047	−	0.997168i	$-0.523961\pi$
$953$	−4.64325			−0.150409			−0.0752047	+	0.997168i	$0.523961\pi$
$954$		0			0
$955$	55.9918	+	55.9918i	1.81185	+	1.81185i
$956$	−24.8996			−0.805311
$957$		0			0
$958$	16.9751	−	16.9751i	0.548441	−	0.548441i
$959$	−6.11444	+	6.11444i	−0.197446	+	0.197446i
$960$		0			0
$961$		−	29.8302i		−	0.962263i
$962$	−6.70181	+	6.70181i	−0.216075	+	0.216075i
$963$		0			0
$964$	7.66021	+	7.66021i	0.246719	+	0.246719i
$965$	52.1248			1.67796
$966$		0			0
$967$		−	6.82813i		−	0.219578i	−0.993955	−	0.109789i	$-0.964983\pi$
$967$		−	6.82813i		−	0.219578i	0.993955	−	0.109789i	$-0.0350175\pi$
$968$	−7.59083			−0.243979
$969$		0			0
$970$	17.9782			0.577246
$971$		−	31.0144i		−	0.995301i	−0.867378	−	0.497650i	$-0.834196\pi$
$971$		−	31.0144i		−	0.995301i	0.867378	−	0.497650i	$-0.165804\pi$
$972$		0			0
$973$	−2.94429			−0.0943896
$974$	−7.39788	−	7.39788i	−0.237043	−	0.237043i
$975$		0			0
$976$	5.77747	−	5.77747i	0.184932	−	0.184932i
$977$			17.3520i			0.555138i	0.960706	+	0.277569i	$0.0895287\pi$
$977$			17.3520i			0.555138i	−0.960706	+	0.277569i	$0.910471\pi$
$978$		0			0
$979$	−23.0030	+	23.0030i	−0.735180	+	0.735180i
$980$	−2.30560	+	2.30560i	−0.0736496	+	0.0736496i
$981$		0			0
$982$	17.3434			0.553452
$983$	−37.3167	−	37.3167i	−1.19022	−	1.19022i	−0.977006	−	0.213210i	$-0.931608\pi$
$983$	−37.3167	−	37.3167i	−1.19022	−	1.19022i	−0.213210	−	0.977006i	$-0.568392\pi$
$984$		0			0
$985$	−5.99237			−0.190933
$986$	−14.5170	−	36.9435i	−0.462316	−	1.17652i
$987$		0			0
$988$		−	28.6202i		−	0.910529i
$989$	29.7451	+	29.7451i	0.945839	+	0.945839i
$990$		0			0
$991$	3.71089	+	3.71089i	0.117880	+	0.117880i	0.763586	−	0.645706i	$-0.223437\pi$
$991$	3.71089	+	3.71089i	0.117880	+	0.117880i	−0.645706	+	0.763586i	$0.723437\pi$
$992$	−0.764801	+	0.764801i	−0.0242825	+	0.0242825i
$993$		0			0
$994$		−	2.78516i		−	0.0883399i
$995$		−	28.0874i		−	0.890429i
$996$		0			0
$997$	5.23392	−	5.23392i	0.165760	−	0.165760i	−0.619353	−	0.785113i	$-0.712605\pi$
$997$	5.23392	−	5.23392i	0.165760	−	0.165760i	0.785113	+	0.619353i	$0.212605\pi$
$998$	−6.44426	−	6.44426i	−0.203989	−	0.203989i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2142.2.p.g.1135.5		12
3.2	odd	2		714.2.m.d.421.4	✓	12
17.4	even	4	inner	2142.2.p.g.1891.5		12
51.38	odd	4		714.2.m.d.463.4	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
714.2.m.d.421.4	✓	12	3.2	odd	2
714.2.m.d.463.4	yes	12	51.38	odd	4
2142.2.p.g.1135.5		12	1.1	even	1	trivial
2142.2.p.g.1891.5		12	17.4	even	4	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2142 = 2 \cdot 3^{2} \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2142.p (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{4} +(2.30560 + 2.30560i) q^{5} +(-0.707107 + 0.707107i) q^{7} +1.00000i q^{8} +(2.30560 - 2.30560i) q^{10} +(-1.30560 + 1.30560i) q^{11} +5.71315 q^{13} +(0.707107 + 0.707107i) q^{14} +1.00000 q^{16} +(-1.64722 + 3.77977i) q^{17} +5.00953i q^{19} +(-2.30560 - 2.30560i) q^{20} +(1.30560 + 1.30560i) q^{22} +(-4.58060 + 4.58060i) q^{23} +5.63155i q^{25} -5.71315i q^{26} +(0.707107 - 0.707107i) q^{28} +(-6.80737 - 6.80737i) q^{29} +(-0.764801 - 0.764801i) q^{31} -1.00000i q^{32} +(3.77977 + 1.64722i) q^{34} -3.26061 q^{35} +(-1.17305 - 1.17305i) q^{37} +5.00953 q^{38} +(-2.30560 + 2.30560i) q^{40} +(-8.19179 + 8.19179i) q^{41} -6.49371i q^{43} +(1.30560 - 1.30560i) q^{44} +(4.58060 + 4.58060i) q^{46} +4.87629 q^{47} -1.00000i q^{49} +5.63155 q^{50} -5.71315 q^{52} -12.9777i q^{53} -6.02036 q^{55} +(-0.707107 - 0.707107i) q^{56} +(-6.80737 + 6.80737i) q^{58} +10.9980i q^{59} +(5.77747 - 5.77747i) q^{61} +(-0.764801 + 0.764801i) q^{62} -1.00000 q^{64} +(13.1722 + 13.1722i) q^{65} -10.1865 q^{67} +(1.64722 - 3.77977i) q^{68} +3.26061i q^{70} +(-1.96941 - 1.96941i) q^{71} +(6.98042 + 6.98042i) q^{73} +(-1.17305 + 1.17305i) q^{74} -5.00953i q^{76} -1.84639i q^{77} +(-5.84293 + 5.84293i) q^{79} +(2.30560 + 2.30560i) q^{80} +(8.19179 + 8.19179i) q^{82} +13.5467i q^{83} +(-12.5125 + 4.91679i) q^{85} -6.49371 q^{86} +(-1.30560 - 1.30560i) q^{88} +17.6188 q^{89} +(-4.03980 + 4.03980i) q^{91} +(4.58060 - 4.58060i) q^{92} -4.87629i q^{94} +(-11.5500 + 11.5500i) q^{95} +(3.89882 + 3.89882i) q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{4} + 4 q^{5} + 4 q^{10} + 8 q^{11} + 12 q^{13} + 12 q^{16} + 12 q^{17} - 4 q^{20} - 8 q^{22} - 8 q^{23} - 12 q^{29} + 12 q^{31} + 4 q^{34} + 4 q^{35} + 24 q^{38} - 4 q^{40} - 4 q^{41} - 8 q^{44}+ \cdots - 12 q^{98}+O(q^{100}) \) 12 * q - 12 * q^4 + 4 * q^5 + 4 * q^10 + 8 * q^11 + 12 * q^13 + 12 * q^16 + 12 * q^17 - 4 * q^20 - 8 * q^22 - 8 * q^23 - 12 * q^29 + 12 * q^31 + 4 * q^34 + 4 * q^35 + 24 * q^38 - 4 * q^40 - 4 * q^41 - 8 * q^44 + 8 * q^46 - 8 * q^47 + 16 * q^50 - 12 * q^52 - 68 * q^55 - 12 * q^58 + 40 * q^61 + 12 * q^62 - 12 * q^64 - 20 * q^65 - 12 * q^67 - 12 * q^68 - 24 * q^71 - 8 * q^79 + 4 * q^80 + 4 * q^82 - 12 * q^85 - 20 * q^86 + 8 * q^88 + 44 * q^89 - 4 * q^91 + 8 * q^92 - 92 * q^95 - 32 * q^97 - 12 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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