LMFDB - Embedded newform 680.2.z.d.89.8

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [680,2,Mod(89,680)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(680, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 0, 2, 3])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("680.89"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 680 = 2^{3} \cdot 5 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	680.z (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [26,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	no
Analytic conductor:	$5.42982733745$
Analytic rank:	$0$
Dimension:	$26$
Relative dimension:	$13$ over $\Q(i)$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			89.8
Character	$\chi$	$=$	680.89
Dual form			680.2.z.d.489.8

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(0.513320 + 0.513320i) q^{3} +(-2.10308 + 0.759643i) q^{5} +(-0.542760 + 0.542760i) q^{7} -2.47300i q^{9} +(-2.81475 - 2.81475i) q^{11} -1.01011i q^{13} +(-1.46949 - 0.689613i) q^{15} +(-4.11972 + 0.167144i) q^{17} -2.31195i q^{19} -0.557219 q^{21} +(-4.31128 + 4.31128i) q^{23} +(3.84589 - 3.19518i) q^{25} +(2.80940 - 2.80940i) q^{27} +(2.92939 - 2.92939i) q^{29} +(5.35048 - 5.35048i) q^{31} -2.88974i q^{33} +(0.729164 - 1.55377i) q^{35} +(-7.83837 - 7.83837i) q^{37} +(0.518508 - 0.518508i) q^{39} +(-6.53610 - 6.53610i) q^{41} -2.05627 q^{43} +(1.87860 + 5.20093i) q^{45} +12.2560i q^{47} +6.41082i q^{49} +(-2.20053 - 2.02893i) q^{51} +6.90494 q^{53} +(8.05785 + 3.78144i) q^{55} +(1.18677 - 1.18677i) q^{57} -6.13169i q^{59} +(-10.2985 - 10.2985i) q^{61} +(1.34225 + 1.34225i) q^{63} +(0.767320 + 2.12433i) q^{65} +8.92915i q^{67} -4.42614 q^{69} +(-8.76366 + 8.76366i) q^{71} +(-6.18757 - 6.18757i) q^{73} +(3.61432 + 0.334022i) q^{75} +3.05547 q^{77} +(10.0433 + 10.0433i) q^{79} -4.53477 q^{81} +3.11595 q^{83} +(8.53712 - 3.48103i) q^{85} +3.00743 q^{87} -1.08068 q^{89} +(0.548245 + 0.548245i) q^{91} +5.49302 q^{93} +(1.75626 + 4.86221i) q^{95} +(-3.34215 - 3.34215i) q^{97} +(-6.96090 + 6.96090i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 26 q + 2 q^{3} - 2 q^{5} + 2 q^{7} + 2 q^{11} - 2 q^{15} + 2 q^{17} - 12 q^{21} - 2 q^{23} + 10 q^{25} + 20 q^{27} + 10 q^{29} - 10 q^{31} + 22 q^{35} - 6 q^{37} + 8 q^{39} - 10 q^{41} + 32 q^{43} + 28 q^{45}+ \cdots - 22 q^{99}+O(q^{100}) $ 26 * q + 2 * q^3 - 2 * q^5 + 2 * q^7 + 2 * q^11 - 2 * q^15 + 2 * q^17 - 12 * q^21 - 2 * q^23 + 10 * q^25 + 20 * q^27 + 10 * q^29 - 10 * q^31 + 22 * q^35 - 6 * q^37 + 8 * q^39 - 10 * q^41 + 32 * q^43 + 28 * q^45 - 6 * q^51 - 32 * q^53 - 26 * q^55 + 20 * q^57 + 6 * q^61 - 54 * q^63 + 24 * q^65 - 20 * q^69 - 30 * q^71 + 42 * q^73 - 78 * q^75 - 4 * q^77 + 14 * q^79 - 54 * q^81 - 16 * q^83 - 26 * q^85 + 8 * q^87 + 40 * q^89 + 24 * q^91 + 88 * q^93 + 44 * q^95 + 66 * q^97 - 22 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/680\mathbb{Z}\right)^\times$.

$n$	$137$	$241$	$341$	$511$
$\chi(n)$	$-1$	$e\left(\frac{3}{4}\right)$	$1$	$1$

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 $

$2$		0			0
$2$		0			0
$3$	0.513320	+	0.513320i	0.296365	+	0.296365i	0.839588	−	0.543223i	$-0.182796\pi$
$3$	0.513320	+	0.513320i	0.296365	+	0.296365i	−0.543223	+	0.839588i	$0.682796\pi$
$4$		0			0
$5$	−2.10308	+	0.759643i	−0.940526	+	0.339722i
$5$	−2.10308	+	0.759643i	−0.940526	+	0.339722i
$6$		0			0
$7$	−0.542760	+	0.542760i	−0.205144	+	0.205144i	−0.802200	−	0.597056i	$-0.796337\pi$
$7$	−0.542760	+	0.542760i	−0.205144	+	0.205144i	0.597056	+	0.802200i	$0.296337\pi$
$8$		0			0
$9$		−	2.47300i		−	0.824335i
$10$		0			0
$11$	−2.81475	−	2.81475i	−0.848680	−	0.848680i	0.141289	−	0.989968i	$-0.454875\pi$
$11$	−2.81475	−	2.81475i	−0.848680	−	0.848680i	−0.989968	+	0.141289i	$0.954875\pi$
$12$		0			0
$13$		−	1.01011i		−	0.280153i	−0.990141	−	0.140077i	$-0.955265\pi$
$13$		−	1.01011i		−	0.280153i	0.990141	−	0.140077i	$-0.0447349\pi$
$14$		0			0
$15$	−1.46949	−	0.689613i	−0.379421	−	0.178057i
$16$		0			0
$17$	−4.11972	+	0.167144i	−0.999178	+	0.0405383i
$17$	−4.11972	+	0.167144i	−0.999178	+	0.0405383i
$18$		0			0
$19$		−	2.31195i		−	0.530398i	−0.964194	−	0.265199i	$-0.914562\pi$
$19$		−	2.31195i		−	0.530398i	0.964194	−	0.265199i	$-0.0854376\pi$
$20$		0			0
$21$	−0.557219			−0.121595
$22$		0			0
$23$	−4.31128	+	4.31128i	−0.898965	+	0.898965i	−0.995345	−	0.0963799i	$-0.969274\pi$
$23$	−4.31128	+	4.31128i	−0.898965	+	0.898965i	0.0963799	+	0.995345i	$0.469274\pi$
$24$		0			0
$25$	3.84589	−	3.19518i	0.769177	−	0.639035i
$26$		0			0
$27$	2.80940	−	2.80940i	0.540670	−	0.540670i
$28$		0			0
$29$	2.92939	−	2.92939i	0.543973	−	0.543973i	−0.380718	−	0.924691i	$-0.624323\pi$
$29$	2.92939	−	2.92939i	0.543973	−	0.543973i	0.924691	+	0.380718i	$0.124323\pi$
$30$		0			0
$31$	5.35048	−	5.35048i	0.960975	−	0.960975i	−0.0382914	−	0.999267i	$-0.512192\pi$
$31$	5.35048	−	5.35048i	0.960975	−	0.960975i	0.999267	+	0.0382914i	$0.0121915\pi$
$32$		0			0
$33$		−	2.88974i		−	0.503039i
$34$		0			0
$35$	0.729164	−	1.55377i	0.123251	−	0.262635i
$36$		0			0
$37$	−7.83837	−	7.83837i	−1.28862	−	1.28862i	−0.935625	−	0.352995i	$-0.885163\pi$
$37$	−7.83837	−	7.83837i	−1.28862	−	1.28862i	−0.352995	−	0.935625i	$-0.614837\pi$
$38$		0			0
$39$	0.518508	−	0.518508i	0.0830277	−	0.0830277i
$40$		0			0
$41$	−6.53610	−	6.53610i	−1.02077	−	1.02077i	−0.999780	−	0.0209873i	$-0.993319\pi$
$41$	−6.53610	−	6.53610i	−1.02077	−	1.02077i	−0.0209873	−	0.999780i	$-0.506681\pi$
$42$		0			0
$43$	−2.05627			−0.313578			−0.156789	−	0.987632i	$-0.550114\pi$
$43$	−2.05627			−0.313578			−0.156789	+	0.987632i	$0.550114\pi$
$44$		0			0
$45$	1.87860	+	5.20093i	0.280045	+	0.775308i
$46$		0			0
$47$			12.2560i			1.78772i	0.448351	+	0.893858i	$0.352011\pi$
$47$			12.2560i			1.78772i	−0.448351	+	0.893858i	$0.647989\pi$
$48$		0			0
$49$			6.41082i			0.915832i
$50$		0			0
$51$	−2.20053	−	2.02893i	−0.308136	−	0.284108i
$52$		0			0
$53$	6.90494			0.948466			0.474233	−	0.880399i	$-0.342725\pi$
$53$	6.90494			0.948466			0.474233	+	0.880399i	$0.342725\pi$
$54$		0			0
$55$	8.05785	+	3.78144i	1.08652	+	0.509890i
$56$		0			0
$57$	1.18677	−	1.18677i	0.157192	−	0.157192i
$58$		0			0
$59$		−	6.13169i		−	0.798277i	−0.916891	−	0.399139i	$-0.869309\pi$
$59$		−	6.13169i		−	0.798277i	0.916891	−	0.399139i	$-0.130691\pi$
$60$		0			0
$61$	−10.2985	−	10.2985i	−1.31859	−	1.31859i	−0.914896	−	0.403690i	$-0.867727\pi$
$61$	−10.2985	−	10.2985i	−1.31859	−	1.31859i	−0.403690	−	0.914896i	$-0.632273\pi$
$62$		0			0
$63$	1.34225	+	1.34225i	0.169107	+	0.169107i
$64$		0			0
$65$	0.767320	+	2.12433i	0.0951743	+	0.263491i
$66$		0			0
$67$			8.92915i			1.09087i	0.838153	+	0.545435i	$0.183635\pi$
$67$			8.92915i			1.09087i	−0.838153	+	0.545435i	$0.816365\pi$
$68$		0			0
$69$	−4.42614			−0.532844
$70$		0			0
$71$	−8.76366	+	8.76366i	−1.04006	+	1.04006i	−0.0408921	+	0.999164i	$0.513020\pi$
$71$	−8.76366	+	8.76366i	−1.04006	+	1.04006i	−0.999164	+	0.0408921i	$0.986980\pi$
$72$		0			0
$73$	−6.18757	−	6.18757i	−0.724200	−	0.724200i	0.245258	−	0.969458i	$-0.421127\pi$
$73$	−6.18757	−	6.18757i	−0.724200	−	0.724200i	−0.969458	+	0.245258i	$0.921127\pi$
$74$		0			0
$75$	3.61432	+	0.334022i	0.417346	+	0.0385695i
$76$		0			0
$77$	3.05547			0.348203
$78$		0			0
$79$	10.0433	+	10.0433i	1.12996	+	1.12996i	0.990183	+	0.139776i	$0.0446384\pi$
$79$	10.0433	+	10.0433i	1.12996	+	1.12996i	0.139776	+	0.990183i	$0.455362\pi$
$80$		0			0
$81$	−4.53477			−0.503863
$82$		0			0
$83$	3.11595			0.342020			0.171010	−	0.985269i	$-0.445297\pi$
$83$	3.11595			0.342020			0.171010	+	0.985269i	$0.445297\pi$
$84$		0			0
$85$	8.53712	−	3.48103i	0.925981	−	0.377571i
$86$		0			0
$87$	3.00743			0.322430
$88$		0			0
$89$	−1.08068			−0.114551			−0.0572757	−	0.998358i	$-0.518241\pi$
$89$	−1.08068			−0.114551			−0.0572757	+	0.998358i	$0.518241\pi$
$90$		0			0
$91$	0.548245	+	0.548245i	0.0574717	+	0.0574717i
$92$		0			0
$93$	5.49302			0.569600
$94$		0			0
$95$	1.75626	+	4.86221i	0.180188	+	0.498853i
$96$		0			0
$97$	−3.34215	−	3.34215i	−0.339344	−	0.339344i	0.516777	−	0.856120i	$-0.327132\pi$
$97$	−3.34215	−	3.34215i	−0.339344	−	0.339344i	−0.856120	+	0.516777i	$0.827132\pi$
$98$		0			0
$99$	−6.96090	+	6.96090i	−0.699597	+	0.699597i
$100$		0			0
$101$	0.957755			0.0953001			0.0476501	−	0.998864i	$-0.484827\pi$
$101$	0.957755			0.0953001			0.0476501	+	0.998864i	$0.484827\pi$
$102$		0			0
$103$			5.64097i			0.555821i	0.960607	+	0.277911i	$0.0896419\pi$
$103$			5.64097i			0.555821i	−0.960607	+	0.277911i	$0.910358\pi$
$104$		0			0
$105$	1.17188	−	0.423287i	0.114363	−	0.0413086i
$106$		0			0
$107$	−9.94418	−	9.94418i	−0.961340	−	0.961340i	0.0379398	−	0.999280i	$-0.487920\pi$
$107$	−9.94418	−	9.94418i	−0.961340	−	0.961340i	−0.999280	+	0.0379398i	$0.987920\pi$
$108$		0			0
$109$	11.8500	+	11.8500i	1.13502	+	1.13502i	0.989330	+	0.145694i	$0.0465414\pi$
$109$	11.8500	+	11.8500i	1.13502	+	1.13502i	0.145694	+	0.989330i	$0.453459\pi$
$110$		0			0
$111$		−	8.04719i		−	0.763805i
$112$		0			0
$113$	−1.45022	+	1.45022i	−0.136426	+	0.136426i	−0.772022	−	0.635596i	$-0.780754\pi$
$113$	−1.45022	+	1.45022i	−0.136426	+	0.136426i	0.635596	+	0.772022i	$0.280754\pi$
$114$		0			0
$115$	5.79194	−	12.3420i	0.540101	−	1.15090i
$116$		0			0
$117$	−2.49800			−0.230940
$118$		0			0
$119$	2.14530	−	2.32674i	0.196659	−	0.213292i
$120$		0			0
$121$			4.84567i			0.440515i
$122$		0			0
$123$		−	6.71022i		−	0.605040i
$124$		0			0
$125$	−5.66101	+	9.64121i	−0.506336	+	0.862336i
$126$		0			0
$127$	17.3653			1.54092			0.770462	−	0.637486i	$-0.220025\pi$
$127$	17.3653			1.54092			0.770462	+	0.637486i	$0.220025\pi$
$128$		0			0
$129$	−1.05553	−	1.05553i	−0.0929338	−	0.0929338i
$130$		0			0
$131$	1.26700	−	1.26700i	0.110699	−	0.110699i	−0.649588	−	0.760287i	$-0.725059\pi$
$131$	1.26700	−	1.26700i	0.110699	−	0.110699i	0.760287	+	0.649588i	$0.225059\pi$
$132$		0			0
$133$	1.25483	+	1.25483i	0.108808	+	0.108808i
$134$		0			0
$135$	−3.77426	+	8.04254i	−0.324836	+	0.692192i
$136$		0			0
$137$		−	8.60549i		−	0.735216i	−0.929981	−	0.367608i	$-0.880177\pi$
$137$		−	8.60549i		−	0.735216i	0.929981	−	0.367608i	$-0.119823\pi$
$138$		0			0
$139$	−7.34182	+	7.34182i	−0.622725	+	0.622725i	−0.946227	−	0.323502i	$-0.895140\pi$
$139$	−7.34182	+	7.34182i	−0.622725	+	0.622725i	0.323502	+	0.946227i	$0.395140\pi$
$140$		0			0
$141$	−6.29123	+	6.29123i	−0.529817	+	0.529817i
$142$		0			0
$143$	−2.84320	+	2.84320i	−0.237760	+	0.237760i
$144$		0			0
$145$	−3.93545	+	8.38602i	−0.326821	+	0.696421i
$146$		0			0
$147$	−3.29080	+	3.29080i	−0.271421	+	0.271421i
$148$		0			0
$149$	−0.655665			−0.0537142			−0.0268571	−	0.999639i	$-0.508550\pi$
$149$	−0.655665			−0.0537142			−0.0268571	+	0.999639i	$0.508550\pi$
$150$		0			0
$151$		−	8.33507i		−	0.678298i	−0.940733	−	0.339149i	$-0.889861\pi$
$151$		−	8.33507i		−	0.678298i	0.940733	−	0.339149i	$-0.110139\pi$
$152$		0			0
$153$	0.413347	+	10.1881i	0.0334172	+	0.823657i
$154$		0			0
$155$	−7.18804	+	15.3169i	−0.577357	+	1.23029i
$156$		0			0
$157$		−	1.35959i		−	0.108507i	−0.998527	−	0.0542537i	$-0.982722\pi$
$157$		−	1.35959i		−	0.108507i	0.998527	−	0.0542537i	$-0.0172780\pi$
$158$		0			0
$159$	3.54444	+	3.54444i	0.281093	+	0.281093i
$160$		0			0
$161$		−	4.67998i		−	0.368834i
$162$		0			0
$163$	15.6484	−	15.6484i	1.22568	−	1.22568i	0.260098	−	0.965582i	$-0.416245\pi$
$163$	15.6484	−	15.6484i	1.22568	−	1.22568i	0.965582	−	0.260098i	$-0.0837549\pi$
$164$		0			0
$165$	2.19517	+	6.07735i	0.170894	+	0.473121i
$166$		0			0
$167$	1.87725	+	1.87725i	0.145266	+	0.145266i	0.776000	−	0.630733i	$-0.217246\pi$
$167$	1.87725	+	1.87725i	0.145266	+	0.145266i	−0.630733	+	0.776000i	$0.717246\pi$
$168$		0			0
$169$	11.9797			0.921514
$170$		0			0
$171$	−5.71746			−0.437225
$172$		0			0
$173$	−2.63953	−	2.63953i	−0.200680	−	0.200680i	0.599611	−	0.800291i	$-0.295322\pi$
$173$	−2.63953	−	2.63953i	−0.200680	−	0.200680i	−0.800291	+	0.599611i	$0.795322\pi$
$174$		0			0
$175$	−0.353179	+	3.82161i	−0.0266978	+	0.288886i
$176$		0			0
$177$	3.14752	−	3.14752i	0.236582	−	0.236582i
$178$		0			0
$179$			11.4469i			0.855581i	0.903878	+	0.427790i	$0.140708\pi$
$179$			11.4469i			0.855581i	−0.903878	+	0.427790i	$0.859292\pi$
$180$		0			0
$181$	1.63999	+	1.63999i	0.121899	+	0.121899i	0.765425	−	0.643525i	$-0.222529\pi$
$181$	1.63999	+	1.63999i	0.121899	+	0.121899i	−0.643525	+	0.765425i	$0.722529\pi$
$182$		0			0
$183$		−	10.5728i		−	0.781567i
$184$		0			0
$185$	22.4391	+	10.5304i	1.64975	+	0.774207i
$186$		0			0
$187$	12.0665	+	11.1255i	0.882386	+	0.813578i
$188$		0			0
$189$			3.04966i			0.221830i
$190$		0			0
$191$	−7.99129			−0.578230			−0.289115	−	0.957294i	$-0.593361\pi$
$191$	−7.99129			−0.578230			−0.289115	+	0.957294i	$0.593361\pi$
$192$		0			0
$193$	−8.98527	+	8.98527i	−0.646774	+	0.646774i	−0.952212	−	0.305438i	$-0.901197\pi$
$193$	−8.98527	+	8.98527i	−0.646774	+	0.646774i	0.305438	+	0.952212i	$0.401197\pi$
$194$		0			0
$195$	−0.696582	+	1.48434i	−0.0498833	+	0.106296i
$196$		0			0
$197$	5.50877	−	5.50877i	0.392484	−	0.392484i	−0.483088	−	0.875572i	$-0.660485\pi$
$197$	5.50877	−	5.50877i	0.392484	−	0.392484i	0.875572	+	0.483088i	$0.160485\pi$
$198$		0			0
$199$	5.30971	−	5.30971i	0.376395	−	0.376395i	−0.493405	−	0.869800i	$-0.664248\pi$
$199$	5.30971	−	5.30971i	0.376395	−	0.376395i	0.869800	+	0.493405i	$0.164248\pi$
$200$		0			0
$201$	−4.58351	+	4.58351i	−0.323296	+	0.323296i
$202$		0			0
$203$			3.17991i			0.223186i
$204$		0			0
$205$	18.7110	+	8.78084i	1.30684	+	0.613280i
$206$		0			0
$207$	10.6618	+	10.6618i	0.741048	+	0.741048i
$208$		0			0
$209$	−6.50757	+	6.50757i	−0.450138	+	0.450138i
$210$		0			0
$211$	−2.55631	−	2.55631i	−0.175984	−	0.175984i	0.613619	−	0.789603i	$-0.289713\pi$
$211$	−2.55631	−	2.55631i	−0.175984	−	0.175984i	−0.789603	+	0.613619i	$0.789713\pi$
$212$		0			0
$213$	−8.99713			−0.616473
$214$		0			0
$215$	4.32450	−	1.56203i	0.294929	−	0.106530i
$216$		0			0
$217$			5.80806i			0.394277i
$218$		0			0
$219$		−	6.35240i		−	0.429256i
$220$		0			0
$221$	0.168833	+	4.16135i	0.0113569	+	0.279923i
$222$		0			0
$223$	−13.7656			−0.921811			−0.460906	−	0.887449i	$-0.652475\pi$
$223$	−13.7656			−0.921811			−0.460906	+	0.887449i	$0.652475\pi$
$224$		0			0
$225$	−7.90169	−	9.51090i	−0.526779	−	0.634060i
$226$		0			0
$227$	0.652728	−	0.652728i	0.0433231	−	0.0433231i	−0.685113	−	0.728436i	$-0.740247\pi$
$227$	0.652728	−	0.652728i	0.0433231	−	0.0433231i	0.728436	+	0.685113i	$0.240247\pi$
$228$		0			0
$229$		−	6.92309i		−	0.457491i	−0.973486	−	0.228745i	$-0.926538\pi$
$229$		−	6.92309i		−	0.457491i	0.973486	−	0.228745i	$-0.0734623\pi$
$230$		0			0
$231$	1.56843	+	1.56843i	0.103195	+	0.103195i
$232$		0			0
$233$	18.6098	+	18.6098i	1.21917	+	1.21917i	0.967923	+	0.251248i	$0.0808409\pi$
$233$	18.6098	+	18.6098i	1.21917	+	1.21917i	0.251248	+	0.967923i	$0.419159\pi$
$234$		0			0
$235$	−9.31015	−	25.7753i	−0.607327	−	1.68139i
$236$		0			0
$237$			10.3109i			0.669762i
$238$		0			0
$239$	3.92551			0.253920			0.126960	−	0.991908i	$-0.459478\pi$
$239$	3.92551			0.253920			0.126960	+	0.991908i	$0.459478\pi$
$240$		0			0
$241$	9.26068	−	9.26068i	0.596533	−	0.596533i	−0.342855	−	0.939388i	$-0.611394\pi$
$241$	9.26068	−	9.26068i	0.596533	−	0.596533i	0.939388	+	0.342855i	$0.111394\pi$
$242$		0			0
$243$	−10.7560	−	10.7560i	−0.689998	−	0.689998i
$244$		0			0
$245$	−4.86993	−	13.4825i	−0.311129	−	0.861363i
$246$		0			0
$247$	−2.33531			−0.148593
$248$		0			0
$249$	1.59948	+	1.59948i	0.101363	+	0.101363i
$250$		0			0
$251$	−23.2963			−1.47045			−0.735225	−	0.677823i	$-0.762924\pi$
$251$	−23.2963			−1.47045			−0.735225	+	0.677823i	$0.762924\pi$
$252$		0			0
$253$	24.2704			1.52587
$254$		0			0
$255$	6.16916	+	2.59539i	0.386328	+	0.162530i
$256$		0			0
$257$	4.37385			0.272833			0.136416	−	0.990652i	$-0.456441\pi$
$257$	4.37385			0.272833			0.136416	+	0.990652i	$0.456441\pi$
$258$		0			0
$259$	8.50871			0.528705
$260$		0			0
$261$	−7.24439	−	7.24439i	−0.448416	−	0.448416i
$262$		0			0
$263$	−3.26609			−0.201396			−0.100698	−	0.994917i	$-0.532108\pi$
$263$	−3.26609			−0.201396			−0.100698	+	0.994917i	$0.532108\pi$
$264$		0			0
$265$	−14.5216	+	5.24529i	−0.892057	+	0.322215i
$266$		0			0
$267$	−0.554733	−	0.554733i	−0.0339491	−	0.0339491i
$268$		0			0
$269$	11.0085	−	11.0085i	0.671199	−	0.671199i	−0.286794	−	0.957992i	$-0.592589\pi$
$269$	11.0085	−	11.0085i	0.671199	−	0.671199i	0.957992	+	0.286794i	$0.0925895\pi$
$270$		0			0
$271$	−7.12361			−0.432728			−0.216364	−	0.976313i	$-0.569420\pi$
$271$	−7.12361			−0.432728			−0.216364	+	0.976313i	$0.569420\pi$
$272$		0			0
$273$			0.562850i			0.0340653i
$274$		0			0
$275$	−19.8189	−	1.83158i	−1.19512	−	0.110449i
$276$		0			0
$277$	−22.5702	−	22.5702i	−1.35611	−	1.35611i	−0.878658	−	0.477451i	$-0.841561\pi$
$277$	−22.5702	−	22.5702i	−1.35611	−	1.35611i	−0.477451	−	0.878658i	$-0.658439\pi$
$278$		0			0
$279$	−13.2318	−	13.2318i	−0.792165	−	0.792165i
$280$		0			0
$281$		−	14.5037i		−	0.865217i	−0.901582	−	0.432608i	$-0.857593\pi$
$281$		−	14.5037i		−	0.865217i	0.901582	−	0.432608i	$-0.142407\pi$
$282$		0			0
$283$	−2.49571	+	2.49571i	−0.148355	+	0.148355i	−0.777383	−	0.629028i	$-0.783453\pi$
$283$	−2.49571	+	2.49571i	−0.148355	+	0.148355i	0.629028	+	0.777383i	$0.283453\pi$
$284$		0			0
$285$	−1.59435	+	3.39739i	−0.0944412	+	0.201244i
$286$		0			0
$287$	7.09506			0.418808
$288$		0			0
$289$	16.9441	−	1.37717i	0.996713	−	0.0810100i
$290$		0			0
$291$		−	3.43118i		−	0.201140i
$292$		0			0
$293$		−	0.507918i		−	0.0296729i	−0.999890	−	0.0148364i	$-0.995277\pi$
$293$		−	0.507918i		−	0.0296729i	0.999890	−	0.0148364i	$-0.00472276\pi$
$294$		0			0
$295$	4.65789	+	12.8954i	0.271193	+	0.750800i
$296$		0			0
$297$	−15.8156			−0.917711
$298$		0			0
$299$	4.35485	+	4.35485i	0.251848	+	0.251848i
$300$		0			0
$301$	1.11606	−	1.11606i	0.0643287	−	0.0643287i
$302$		0			0
$303$	0.491635	+	0.491635i	0.0282437	+	0.0282437i
$304$		0			0
$305$	29.4817	+	13.8354i	1.68812	+	0.792211i
$306$		0			0
$307$		−	25.8970i		−	1.47802i	−0.673694	−	0.739010i	$-0.735293\pi$
$307$		−	25.8970i		−	1.47802i	0.673694	−	0.739010i	$-0.264707\pi$
$308$		0			0
$309$	−2.89562	+	2.89562i	−0.164726	+	0.164726i
$310$		0			0
$311$	1.97345	−	1.97345i	0.111904	−	0.111904i	−0.648938	−	0.760842i	$-0.724786\pi$
$311$	1.97345	−	1.97345i	0.111904	−	0.111904i	0.760842	+	0.648938i	$0.224786\pi$
$312$		0			0
$313$	−2.97609	+	2.97609i	−0.168219	+	0.168219i	−0.786196	−	0.617977i	$-0.787952\pi$
$313$	−2.97609	+	2.97609i	−0.168219	+	0.168219i	0.617977	+	0.786196i	$0.287952\pi$
$314$		0			0
$315$	−3.84248	−	1.80323i	−0.216499	−	0.101600i
$316$		0			0
$317$	−14.1844	+	14.1844i	−0.796674	+	0.796674i	−0.982570	−	0.185895i	$-0.940482\pi$
$317$	−14.1844	+	14.1844i	−0.796674	+	0.796674i	0.185895	+	0.982570i	$0.440482\pi$
$318$		0			0
$319$	−16.4910			−0.923319
$320$		0			0
$321$		−	10.2091i		−	0.569816i
$322$		0			0
$323$	0.386428	+	9.52458i	0.0215014	+	0.529962i
$324$		0			0
$325$	−3.22747	−	3.88475i	−0.179028	−	0.215487i
$326$		0			0
$327$			12.1657i			0.672764i
$328$		0			0
$329$	−6.65204	−	6.65204i	−0.366739	−	0.366739i
$330$		0			0
$331$		−	20.9239i		−	1.15008i	−0.818124	−	0.575042i	$-0.804986\pi$
$331$		−	20.9239i		−	1.15008i	0.818124	−	0.575042i	$-0.195014\pi$
$332$		0			0
$333$	−19.3843	+	19.3843i	−1.06225	+	1.06225i
$334$		0			0
$335$	−6.78297	−	18.7787i	−0.370593	−	1.02599i
$336$		0			0
$337$	−4.17667	−	4.17667i	−0.227518	−	0.227518i	0.584137	−	0.811655i	$-0.301433\pi$
$337$	−4.17667	−	4.17667i	−0.227518	−	0.227518i	−0.811655	+	0.584137i	$0.801433\pi$
$338$		0			0
$339$	−1.48886			−0.0808637
$340$		0			0
$341$	−30.1206			−1.63112
$342$		0			0
$343$	−7.27886	−	7.27886i	−0.393021	−	0.393021i
$344$		0			0
$345$	9.30852	−	3.36228i	0.501154	−	0.181019i
$346$		0			0
$347$	20.0134	−	20.0134i	1.07437	−	1.07437i	0.0773707	−	0.997002i	$-0.475347\pi$
$347$	20.0134	−	20.0134i	1.07437	−	1.07437i	0.997002	−	0.0773707i	$-0.0246525\pi$
$348$		0			0
$349$			2.29215i			0.122696i	0.998116	+	0.0613480i	$0.0195399\pi$
$349$			2.29215i			0.122696i	−0.998116	+	0.0613480i	$0.980460\pi$
$350$		0			0
$351$	−2.83780	−	2.83780i	−0.151470	−	0.151470i
$352$		0			0
$353$		−	31.7369i		−	1.68918i	−0.535412	−	0.844591i	$-0.679844\pi$
$353$		−	31.7369i		−	1.68918i	0.535412	−	0.844591i	$-0.320156\pi$
$354$		0			0
$355$	11.7734	−	25.0879i	0.624869	−	1.33153i
$356$		0			0
$357$	2.29558	−	0.0931357i	0.121495	−	0.00492927i
$358$		0			0
$359$		−	0.0923824i		−	0.00487576i	−0.999997	−	0.00243788i	$-0.999224\pi$
$359$		−	0.0923824i		−	0.00487576i	0.999997	−	0.00243788i	$-0.000776002\pi$
$360$		0			0
$361$	13.6549			0.718678
$362$		0			0
$363$	−2.48738	+	2.48738i	−0.130553	+	0.130553i
$364$		0			0
$365$	17.7133	+	8.31260i	0.927155	+	0.435102i
$366$		0			0
$367$	17.8025	−	17.8025i	0.929283	−	0.929283i	−0.0683761	−	0.997660i	$-0.521782\pi$
$367$	17.8025	−	17.8025i	0.929283	−	0.929283i	0.997660	+	0.0683761i	$0.0217818\pi$
$368$		0			0
$369$	−16.1638	+	16.1638i	−0.841454	+	0.841454i
$370$		0			0
$371$	−3.74772	+	3.74772i	−0.194572	+	0.194572i
$372$		0			0
$373$			6.20258i			0.321157i	0.987023	+	0.160579i	$0.0513361\pi$
$373$			6.20258i			0.321157i	−0.987023	+	0.160579i	$0.948664\pi$
$374$		0			0
$375$	−7.85494	+	2.04312i	−0.405627	+	0.105506i
$376$		0			0
$377$	−2.95899	−	2.95899i	−0.152396	−	0.152396i
$378$		0			0
$379$	12.7850	−	12.7850i	0.656723	−	0.656723i	−0.297880	−	0.954603i	$-0.596280\pi$
$379$	12.7850	−	12.7850i	0.656723	−	0.656723i	0.954603	+	0.297880i	$0.0962796\pi$
$380$		0			0
$381$	8.91398	+	8.91398i	0.456677	+	0.456677i
$382$		0			0
$383$	−36.0180			−1.84043			−0.920217	−	0.391410i	$-0.871988\pi$
$383$	−36.0180			−1.84043			−0.920217	+	0.391410i	$0.871988\pi$
$384$		0			0
$385$	−6.42590	+	2.32107i	−0.327494	+	0.118292i
$386$		0			0
$387$			5.08517i			0.258494i
$388$		0			0
$389$		−	21.6773i		−	1.09908i	−0.835466	−	0.549542i	$-0.814802\pi$
$389$		−	21.6773i		−	1.09908i	0.835466	−	0.549542i	$-0.185198\pi$
$390$		0			0
$391$	17.0407	−	18.4819i	0.861783	−	0.934668i
$392$		0			0
$393$	1.30076			0.0656145
$394$		0			0
$395$	−28.7512	−	13.4925i	−1.44663	−	0.678883i
$396$		0			0
$397$	−21.3015	+	21.3015i	−1.06909	+	1.06909i	−0.0716629	+	0.997429i	$0.522831\pi$
$397$	−21.3015	+	21.3015i	−1.06909	+	1.06909i	−0.997429	+	0.0716629i	$0.977169\pi$
$398$		0			0
$399$			1.28826i			0.0644938i
$400$		0			0
$401$	−0.317358	−	0.317358i	−0.0158481	−	0.0158481i	0.699138	−	0.714986i	$-0.253567\pi$
$401$	−0.317358	−	0.317358i	−0.0158481	−	0.0158481i	−0.714986	+	0.699138i	$0.753567\pi$
$402$		0			0
$403$	−5.40456	−	5.40456i	−0.269220	−	0.269220i
$404$		0			0
$405$	9.53698	−	3.44480i	0.473896	−	0.171174i
$406$		0			0
$407$			44.1262i			2.18725i
$408$		0			0
$409$	−13.8309			−0.683896			−0.341948	−	0.939719i	$-0.611087\pi$
$409$	−13.8309			−0.683896			−0.341948	+	0.939719i	$0.611087\pi$
$410$		0			0
$411$	4.41737	−	4.41737i	0.217893	−	0.217893i
$412$		0			0
$413$	3.32803	+	3.32803i	0.163762	+	0.163762i
$414$		0			0
$415$	−6.55310	+	2.36701i	−0.321679	+	0.116192i
$416$		0			0
$417$	−7.53741			−0.369109
$418$		0			0
$419$	−10.3186	−	10.3186i	−0.504098	−	0.504098i	0.408610	−	0.912709i	$-0.366013\pi$
$419$	−10.3186	−	10.3186i	−0.504098	−	0.504098i	−0.912709	+	0.408610i	$0.866013\pi$
$420$		0			0
$421$	15.5241			0.756601			0.378300	−	0.925683i	$-0.376509\pi$
$421$	15.5241			0.756601			0.378300	+	0.925683i	$0.376509\pi$
$422$		0			0
$423$	30.3090			1.47368
$424$		0			0
$425$	−15.3099	+	13.8060i	−0.742640	+	0.669691i
$426$		0			0
$427$	11.1792			0.541000
$428$		0			0
$429$	−2.91894			−0.140928
$430$		0			0
$431$	22.5522	+	22.5522i	1.08630	+	1.08630i	0.995906	+	0.0903937i	$0.0288125\pi$
$431$	22.5522	+	22.5522i	1.08630	+	1.08630i	0.0903937	+	0.995906i	$0.471187\pi$
$432$		0			0
$433$	23.9756			1.15219			0.576097	−	0.817382i	$-0.304575\pi$
$433$	23.9756			1.15219			0.576097	+	0.817382i	$0.304575\pi$
$434$		0			0
$435$	−6.32486	+	2.28457i	−0.303254	+	0.109537i
$436$		0			0
$437$	9.96747	+	9.96747i	0.476809	+	0.476809i
$438$		0			0
$439$	−17.8069	+	17.8069i	−0.849878	+	0.849878i	−0.990118	−	0.140240i	$-0.955213\pi$
$439$	−17.8069	+	17.8069i	−0.849878	+	0.849878i	0.140240	+	0.990118i	$0.455213\pi$
$440$		0			0
$441$	15.8540			0.754952
$442$		0			0
$443$		−	19.7784i		−	0.939700i	−0.882746	−	0.469850i	$-0.844308\pi$
$443$		−	19.7784i		−	0.939700i	0.882746	−	0.469850i	$-0.155692\pi$
$444$		0			0
$445$	2.27275	−	0.820928i	0.107739	−	0.0389157i
$446$		0			0
$447$	−0.336566	−	0.336566i	−0.0159190	−	0.0159190i
$448$		0			0
$449$	−0.857444	−	0.857444i	−0.0404653	−	0.0404653i	0.686585	−	0.727050i	$-0.259109\pi$
$449$	−0.857444	−	0.857444i	−0.0404653	−	0.0404653i	−0.727050	+	0.686585i	$0.759109\pi$
$450$		0			0
$451$			36.7950i			1.73261i
$452$		0			0
$453$	4.27856	−	4.27856i	0.201024	−	0.201024i
$454$		0			0
$455$	−1.56947	−	0.736533i	−0.0735780	−	0.0345292i
$456$		0			0
$457$	−7.32015			−0.342422			−0.171211	−	0.985234i	$-0.554768\pi$
$457$	−7.32015			−0.342422			−0.171211	+	0.985234i	$0.554768\pi$
$458$		0			0
$459$	−11.1044	+	12.0435i	−0.518308	+	0.562143i
$460$		0			0
$461$		−	19.1994i		−	0.894206i	−0.894482	−	0.447103i	$-0.852456\pi$
$461$		−	19.1994i		−	0.894206i	0.894482	−	0.447103i	$-0.147544\pi$
$462$		0			0
$463$		−	22.5776i		−	1.04927i	−0.851327	−	0.524635i	$-0.824202\pi$
$463$		−	22.5776i		−	1.04927i	0.851327	−	0.524635i	$-0.175798\pi$
$464$		0			0
$465$	−11.5523	+	4.17273i	−0.535723	+	0.193506i
$466$		0			0
$467$	−16.4171			−0.759691			−0.379845	−	0.925050i	$-0.624023\pi$
$467$	−16.4171			−0.759691			−0.379845	+	0.925050i	$0.624023\pi$
$468$		0			0
$469$	−4.84639	−	4.84639i	−0.223785	−	0.223785i
$470$		0			0
$471$	0.697907	−	0.697907i	0.0321579	−	0.0321579i
$472$		0			0
$473$	5.78790	+	5.78790i	0.266128	+	0.266128i
$474$		0			0
$475$	−7.38709	−	8.89150i	−0.338943	−	0.407970i
$476$		0			0
$477$		−	17.0759i		−	0.781854i
$478$		0			0
$479$	27.0509	−	27.0509i	1.23599	−	1.23599i	0.274357	−	0.961628i	$-0.411535\pi$
$479$	27.0509	−	27.0509i	1.23599	−	1.23599i	0.961628	−	0.274357i	$-0.0884651\pi$
$480$		0			0
$481$	−7.91759	+	7.91759i	−0.361011	+	0.361011i
$482$		0			0
$483$	2.40233	−	2.40233i	0.109310	−	0.109310i
$484$		0			0
$485$	9.56764	+	4.48997i	0.434444	+	0.203879i
$486$		0			0
$487$	−6.76648	+	6.76648i	−0.306619	+	0.306619i	−0.843596	−	0.536978i	$-0.819566\pi$
$487$	−6.76648	+	6.76648i	−0.306619	+	0.306619i	0.536978	+	0.843596i	$0.319566\pi$
$488$		0			0
$489$	16.0653			0.726499
$490$		0			0
$491$		−	1.50223i		−	0.0677946i	−0.999425	−	0.0338973i	$-0.989208\pi$
$491$		−	1.50223i		−	0.0677946i	0.999425	−	0.0338973i	$-0.0107919\pi$
$492$		0			0
$493$	−11.5786	+	12.5579i	−0.521475	+	0.565578i
$494$		0			0
$495$	9.35153	−	19.9271i	0.420320	−	0.895657i
$496$		0			0
$497$		−	9.51313i		−	0.426722i
$498$		0			0
$499$	16.9850	+	16.9850i	0.760353	+	0.760353i	0.976386	−	0.216033i	$-0.0693118\pi$
$499$	16.9850	+	16.9850i	0.760353	+	0.760353i	−0.216033	+	0.976386i	$0.569312\pi$
$500$		0			0
$501$			1.92726i			0.0861038i
$502$		0			0
$503$	−10.2153	+	10.2153i	−0.455476	+	0.455476i	−0.897167	−	0.441691i	$-0.854379\pi$
$503$	−10.2153	+	10.2153i	−0.455476	+	0.455476i	0.441691	+	0.897167i	$0.354379\pi$
$504$		0			0
$505$	−2.01423	+	0.727551i	−0.0896322	+	0.0323756i
$506$		0			0
$507$	6.14941	+	6.14941i	0.273105	+	0.273105i
$508$		0			0
$509$	−6.25617			−0.277300			−0.138650	−	0.990341i	$-0.544276\pi$
$509$	−6.25617			−0.277300			−0.138650	+	0.990341i	$0.544276\pi$
$510$		0			0
$511$	6.71673			0.297130
$512$		0			0
$513$	−6.49520	−	6.49520i	−0.286770	−	0.286770i
$514$		0			0
$515$	−4.28512	−	11.8634i	−0.188825	−	0.522764i
$516$		0			0
$517$	34.4975	−	34.4975i	1.51720	−	1.51720i
$518$		0			0
$519$		−	2.70985i		−	0.118949i
$520$		0			0
$521$	24.7142	+	24.7142i	1.08275	+	1.08275i	0.996252	+	0.0864967i	$0.0275672\pi$
$521$	24.7142	+	24.7142i	1.08275	+	1.08275i	0.0864967	+	0.996252i	$0.472433\pi$
$522$		0			0
$523$			39.5509i			1.72944i	0.502253	+	0.864721i	$0.332505\pi$
$523$			39.5509i			1.72944i	−0.502253	+	0.864721i	$0.667495\pi$
$524$		0			0
$525$	−2.14300	+	1.78041i	−0.0935282	+	0.0777036i
$526$		0			0
$527$	−21.1482	+	22.9368i	−0.921229	+	0.999142i
$528$		0			0
$529$		−	14.1743i		−	0.616275i
$530$		0			0
$531$	−15.1637			−0.658048
$532$		0			0
$533$	−6.60215	+	6.60215i	−0.285971	+	0.285971i
$534$		0			0
$535$	28.4674	+	13.3594i	1.23075	+	0.577576i
$536$		0			0
$537$	−5.87592	+	5.87592i	−0.253565	+	0.253565i
$538$		0			0
$539$	18.0449	−	18.0449i	0.777248	−	0.777248i
$540$		0			0
$541$	14.8850	−	14.8850i	0.639955	−	0.639955i	−0.310589	−	0.950544i	$-0.600526\pi$
$541$	14.8850	−	14.8850i	0.639955	−	0.639955i	0.950544	+	0.310589i	$0.100526\pi$
$542$		0			0
$543$			1.68368i			0.0722535i
$544$		0			0
$545$	−33.9232	−	15.9197i	−1.45311	−	0.681926i
$546$		0			0
$547$	−17.2857	−	17.2857i	−0.739084	−	0.739084i	0.233317	−	0.972401i	$-0.425042\pi$
$547$	−17.2857	−	17.2857i	−0.739084	−	0.739084i	−0.972401	+	0.233317i	$0.925042\pi$
$548$		0			0
$549$	−25.4682	+	25.4682i	−1.08696	+	1.08696i
$550$		0			0
$551$	−6.77260	−	6.77260i	−0.288522	−	0.288522i
$552$		0			0
$553$	−10.9022			−0.463609
$554$		0			0
$555$	6.11299	+	16.9239i	0.259482	+	0.718378i
$556$		0			0
$557$			4.34727i			0.184200i	0.995750	+	0.0920999i	$0.0293579\pi$
$557$			4.34727i			0.184200i	−0.995750	+	0.0920999i	$0.970642\pi$
$558$		0			0
$559$			2.07705i			0.0878500i
$560$		0			0
$561$	0.483002	+	11.9049i	0.0203924	+	0.502625i
$562$		0			0
$563$	33.9959			1.43276			0.716379	−	0.697712i	$-0.245798\pi$
$563$	33.9959			1.43276			0.716379	+	0.697712i	$0.245798\pi$
$564$		0			0
$565$	1.94828	−	4.15159i	0.0819649	−	0.174659i
$566$		0			0
$567$	2.46129	−	2.46129i	0.103364	−	0.103364i
$568$		0			0
$569$		−	16.8690i		−	0.707185i	−0.935400	−	0.353592i	$-0.884960\pi$
$569$		−	16.8690i		−	0.707185i	0.935400	−	0.353592i	$-0.115040\pi$
$570$		0			0
$571$	25.7752	+	25.7752i	1.07866	+	1.07866i	0.996630	+	0.0820281i	$0.0261397\pi$
$571$	25.7752	+	25.7752i	1.07866	+	1.07866i	0.0820281	+	0.996630i	$0.473860\pi$
$572$		0			0
$573$	−4.10209	−	4.10209i	−0.171367	−	0.171367i
$574$		0			0
$575$	−2.80539	+	30.3560i	−0.116993	+	1.26593i
$576$		0			0
$577$		−	16.4536i		−	0.684971i	−0.939523	−	0.342485i	$-0.888731\pi$
$577$		−	16.4536i		−	0.684971i	0.939523	−	0.342485i	$-0.111269\pi$
$578$		0			0
$579$	−9.22464			−0.383363
$580$		0			0
$581$	−1.69121	+	1.69121i	−0.0701634	+	0.0701634i
$582$		0			0
$583$	−19.4357	−	19.4357i	−0.804944	−	0.804944i
$584$		0			0
$585$	5.25349	−	1.89759i	0.217205	−	0.0784555i
$586$		0			0
$587$	0.0736793			0.00304107			0.00152054	−	0.999999i	$-0.499516\pi$
$587$	0.0736793			0.00304107			0.00152054	+	0.999999i	$0.499516\pi$
$588$		0			0
$589$	−12.3700	−	12.3700i	−0.509699	−	0.509699i
$590$		0			0
$591$	5.65553			0.232637
$592$		0			0
$593$	21.0562			0.864675			0.432337	−	0.901712i	$-0.357689\pi$
$593$	21.0562			0.864675			0.432337	+	0.901712i	$0.357689\pi$
$594$		0			0
$595$	−2.74424	+	6.52297i	−0.112503	+	0.267416i
$596$		0			0
$597$	5.45116			0.223101
$598$		0			0
$599$	24.0976			0.984601			0.492301	−	0.870425i	$-0.336156\pi$
$599$	24.0976			0.984601			0.492301	+	0.870425i	$0.336156\pi$
$600$		0			0
$601$	15.2285	+	15.2285i	0.621185	+	0.621185i	0.945834	−	0.324650i	$-0.105246\pi$
$601$	15.2285	+	15.2285i	0.621185	+	0.621185i	−0.324650	+	0.945834i	$0.605246\pi$
$602$		0			0
$603$	22.0818			0.899242
$604$		0			0
$605$	−3.68097	−	10.1908i	−0.149653	−	0.414316i
$606$		0			0
$607$	21.6109	+	21.6109i	0.877160	+	0.877160i	0.993240	−	0.116080i	$-0.0370330\pi$
$607$	21.6109	+	21.6109i	0.877160	+	0.877160i	−0.116080	+	0.993240i	$0.537033\pi$
$608$		0			0
$609$	−1.63231	+	1.63231i	−0.0661446	+	0.0661446i
$610$		0			0
$611$	12.3798			0.500834
$612$		0			0
$613$		−	29.0645i		−	1.17390i	−0.809622	−	0.586952i	$-0.800328\pi$
$613$		−	29.0645i		−	1.17390i	0.809622	−	0.586952i	$-0.199672\pi$
$614$		0			0
$615$	5.09737	+	14.1121i	0.205546	+	0.569056i
$616$		0			0
$617$	−15.7172	−	15.7172i	−0.632749	−	0.632749i	0.316008	−	0.948757i	$-0.397658\pi$
$617$	−15.7172	−	15.7172i	−0.632749	−	0.632749i	−0.948757	+	0.316008i	$0.897658\pi$
$618$		0			0
$619$	−19.0808	−	19.0808i	−0.766923	−	0.766923i	0.210641	−	0.977564i	$-0.432445\pi$
$619$	−19.0808	−	19.0808i	−0.766923	−	0.766923i	−0.977564	+	0.210641i	$0.932445\pi$
$620$		0			0
$621$			24.2243i			0.972086i
$622$		0			0
$623$	0.586548	−	0.586548i	0.0234995	−	0.0234995i
$624$		0			0
$625$	4.58168	−	24.5766i	0.183267	−	0.983063i
$626$		0			0
$627$	−6.68093			−0.266811
$628$		0			0
$629$	33.6020	+	30.9817i	1.33980	+	1.23532i
$630$		0			0
$631$			28.3829i			1.12991i	0.825123	+	0.564953i	$0.191106\pi$
$631$			28.3829i			1.12991i	−0.825123	+	0.564953i	$0.808894\pi$
$632$		0			0
$633$		−	2.62441i		−	0.104311i
$634$		0			0
$635$	−36.5207	+	13.1915i	−1.44928	+	0.523487i
$636$		0			0
$637$	6.47561			0.256573
$638$		0			0
$639$	21.6726	+	21.6726i	0.857354	+	0.857354i
$640$		0			0
$641$	−25.0393	+	25.0393i	−0.988994	+	0.988994i	−0.999940	−	0.0109456i	$-0.996516\pi$
$641$	−25.0393	+	25.0393i	−0.988994	+	0.988994i	0.0109456	+	0.999940i	$0.496516\pi$
$642$		0			0
$643$	26.0861	+	26.0861i	1.02873	+	1.02873i	0.999575	+	0.0291592i	$0.00928299\pi$
$643$	26.0861	+	26.0861i	1.02873	+	1.02873i	0.0291592	+	0.999575i	$0.490717\pi$
$644$		0			0
$645$	3.02168	+	1.41803i	0.118978	+	0.0558350i
$646$		0			0
$647$		−	14.1949i		−	0.558061i	−0.960282	−	0.279030i	$-0.909987\pi$
$647$		−	14.1949i		−	0.558061i	0.960282	−	0.279030i	$-0.0900130\pi$
$648$		0			0
$649$	−17.2592	+	17.2592i	−0.677482	+	0.677482i
$650$		0			0
$651$	−2.98139	+	2.98139i	−0.116850	+	0.116850i
$652$		0			0
$653$	0.793576	−	0.793576i	0.0310550	−	0.0310550i	−0.691409	−	0.722464i	$-0.743010\pi$
$653$	0.793576	−	0.793576i	0.0310550	−	0.0310550i	0.722464	+	0.691409i	$0.243010\pi$
$654$		0			0
$655$	−1.70214	+	3.62708i	−0.0665081	+	0.141722i
$656$		0			0
$657$	−15.3019	+	15.3019i	−0.596983	+	0.596983i
$658$		0			0
$659$	−19.8424			−0.772951			−0.386476	−	0.922300i	$-0.626308\pi$
$659$	−19.8424			−0.772951			−0.386476	+	0.922300i	$0.626308\pi$
$660$		0			0
$661$			27.7888i			1.08086i	0.841389	+	0.540430i	$0.181738\pi$
$661$			27.7888i			1.08086i	−0.841389	+	0.540430i	$0.818262\pi$
$662$		0			0
$663$	−2.04944	+	2.22277i	−0.0795936	+	0.0863252i
$664$		0			0
$665$	−3.59224	−	1.68579i	−0.139301	−	0.0653721i
$666$		0			0
$667$			25.2588i			0.978026i
$668$		0			0
$669$	−7.06615	−	7.06615i	−0.273193	−	0.273193i
$670$		0			0
$671$			57.9754i			2.23811i
$672$		0			0
$673$	30.7340	−	30.7340i	1.18471	−	1.18471i	0.206200	−	0.978510i	$-0.433890\pi$
$673$	30.7340	−	30.7340i	1.18471	−	1.18471i	0.978510	−	0.206200i	$-0.0661097\pi$
$674$		0			0
$675$	1.82810	−	19.7812i	0.0703637	−	0.761378i
$676$		0			0
$677$	−22.0723	−	22.0723i	−0.848308	−	0.848308i	0.141614	−	0.989922i	$-0.454771\pi$
$677$	−22.0723	−	22.0723i	−0.848308	−	0.848308i	−0.989922	+	0.141614i	$0.954771\pi$
$678$		0			0
$679$	3.62797			0.139229
$680$		0			0
$681$	0.670117			0.0256789
$682$		0			0
$683$	18.1924	+	18.1924i	0.696111	+	0.696111i	0.963569	−	0.267458i	$-0.0861838\pi$
$683$	18.1924	+	18.1924i	0.696111	+	0.696111i	−0.267458	+	0.963569i	$0.586184\pi$
$684$		0			0
$685$	6.53709	+	18.0980i	0.249770	+	0.691490i
$686$		0			0
$687$	3.55376	−	3.55376i	0.135584	−	0.135584i
$688$		0			0
$689$		−	6.97472i		−	0.265716i
$690$		0			0
$691$	−6.41681	−	6.41681i	−0.244107	−	0.244107i	0.574440	−	0.818547i	$-0.305220\pi$
$691$	−6.41681	−	6.41681i	−0.244107	−	0.244107i	−0.818547	+	0.574440i	$0.805220\pi$
$692$		0			0
$693$		−	7.55619i		−	0.287036i
$694$		0			0
$695$	9.86328	−	21.0176i	0.374135	−	0.797243i
$696$		0			0
$697$	28.0193	+	25.8344i	1.06131	+	0.978548i
$698$		0			0
$699$			19.1056i			0.722640i
$700$		0			0
$701$	−24.5605			−0.927639			−0.463819	−	0.885930i	$-0.653521\pi$
$701$	−24.5605			−0.927639			−0.463819	+	0.885930i	$0.653521\pi$
$702$		0			0
$703$	−18.1219	+	18.1219i	−0.683481	+	0.683481i
$704$		0			0
$705$	8.45187	−	18.0100i	0.318316	−	0.678297i
$706$		0			0
$707$	−0.519831	+	0.519831i	−0.0195502	+	0.0195502i
$708$		0			0
$709$	−23.2572	+	23.2572i	−0.873441	+	0.873441i	−0.992846	−	0.119404i	$-0.961902\pi$
$709$	−23.2572	+	23.2572i	−0.873441	+	0.873441i	0.119404	+	0.992846i	$0.461902\pi$
$710$		0			0
$711$	24.8371	−	24.8371i	0.931465	−	0.931465i
$712$		0			0
$713$			46.1349i			1.72777i
$714$		0			0
$715$	3.81966	−	8.13929i	0.142847	−	0.304392i
$716$		0			0
$717$	2.01505	+	2.01505i	0.0752532	+	0.0752532i
$718$		0			0
$719$	−27.1660	+	27.1660i	−1.01312	+	1.01312i	−0.0132092	+	0.999913i	$0.504205\pi$
$719$	−27.1660	+	27.1660i	−1.01312	+	1.01312i	−0.999913	+	0.0132092i	$0.995795\pi$
$720$		0			0
$721$	−3.06169	−	3.06169i	−0.114023	−	0.114023i
$722$		0			0
$723$	9.50739			0.353584
$724$		0			0
$725$	1.90618	−	20.6260i	0.0707937	−	0.766030i
$726$		0			0
$727$			22.5490i			0.836297i	0.908379	+	0.418148i	$0.137321\pi$
$727$			22.5490i			0.836297i	−0.908379	+	0.418148i	$0.862679\pi$
$728$		0			0
$729$			2.56177i			0.0948802i
$730$		0			0
$731$	8.47126	−	0.343693i	0.313321	−	0.0127119i
$732$		0			0
$733$	−2.15899			−0.0797440			−0.0398720	−	0.999205i	$-0.512695\pi$
$733$	−2.15899			−0.0797440			−0.0398720	+	0.999205i	$0.512695\pi$
$734$		0			0
$735$	4.42099	−	9.42066i	0.163071	−	0.347486i
$736$		0			0
$737$	25.1334	−	25.1334i	0.925799	−	0.925799i
$738$		0			0
$739$		−	26.5102i		−	0.975193i	−0.873069	−	0.487597i	$-0.837874\pi$
$739$		−	26.5102i		−	0.975193i	0.873069	−	0.487597i	$-0.162126\pi$
$740$		0			0
$741$	−1.19876	−	1.19876i	−0.0440377	−	0.0440377i
$742$		0			0
$743$	−27.2833	−	27.2833i	−1.00093	−	1.00093i	−1.00000		0.000926218i	$-0.999705\pi$
$743$	−27.2833	−	27.2833i	−1.00093	−	1.00093i	−0.000926218	−	1.00000i	$-0.500295\pi$
$744$		0			0
$745$	1.37892	−	0.498071i	0.0505196	−	0.0182479i
$746$		0			0
$747$		−	7.70577i		−	0.281939i
$748$		0			0
$749$	10.7946			0.394426
$750$		0			0
$751$	−14.6118	+	14.6118i	−0.533192	+	0.533192i	−0.921521	−	0.388329i	$-0.873052\pi$
$751$	−14.6118	+	14.6118i	−0.533192	+	0.533192i	0.388329	+	0.921521i	$0.373052\pi$
$752$		0			0
$753$	−11.9585	−	11.9585i	−0.435791	−	0.435791i
$754$		0			0
$755$	6.33167	+	17.5293i	0.230433	+	0.637957i
$756$		0			0
$757$	−31.1450			−1.13199			−0.565993	−	0.824410i	$-0.691507\pi$
$757$	−31.1450			−1.13199			−0.565993	+	0.824410i	$0.691507\pi$
$758$		0			0
$759$	12.4585	+	12.4585i	0.452214	+	0.452214i
$760$		0			0
$761$	34.9225			1.26594			0.632970	−	0.774177i	$-0.281836\pi$
$761$	34.9225			1.26594			0.632970	+	0.774177i	$0.281836\pi$
$762$		0			0
$763$	−12.8634			−0.465686
$764$		0			0
$765$	−8.60860	−	21.1123i	−0.311245	−	0.763318i
$766$		0			0
$767$	−6.19365			−0.223640
$768$		0			0
$769$	−12.1337			−0.437554			−0.218777	−	0.975775i	$-0.570207\pi$
$769$	−12.1337			−0.437554			−0.218777	+	0.975775i	$0.570207\pi$
$770$		0			0
$771$	2.24518	+	2.24518i	0.0808583	+	0.0808583i
$772$		0			0
$773$	−24.3094			−0.874350			−0.437175	−	0.899376i	$-0.644021\pi$
$773$	−24.3094			−0.874350			−0.437175	+	0.899376i	$0.644021\pi$
$774$		0			0
$775$	3.48161	−	37.6731i	0.125063	−	1.35326i
$776$		0			0
$777$	4.36769	+	4.36769i	0.156690	+	0.156690i
$778$		0			0
$779$	−15.1111	+	15.1111i	−0.541412	+	0.541412i
$780$		0			0
$781$	49.3351			1.76535
$782$		0			0
$783$		−	16.4597i		−	0.588220i
$784$		0			0
$785$	1.03281	+	2.85934i	0.0368624	+	0.102054i
$786$		0			0
$787$	−8.51407	−	8.51407i	−0.303494	−	0.303494i	0.538885	−	0.842379i	$-0.318846\pi$
$787$	−8.51407	−	8.51407i	−0.303494	−	0.303494i	−0.842379	+	0.538885i	$0.818846\pi$
$788$		0			0
$789$	−1.67655	−	1.67655i	−0.0596867	−	0.0596867i
$790$		0			0
$791$		−	1.57425i		−	0.0559738i
$792$		0			0
$793$	−10.4026	+	10.4026i	−0.369406	+	0.369406i
$794$		0			0
$795$	−10.1468	−	4.76174i	−0.359868	−	0.168881i
$796$		0			0
$797$	49.6699			1.75940			0.879699	−	0.475531i	$-0.157744\pi$
$797$	49.6699			1.75940			0.879699	+	0.475531i	$0.157744\pi$
$798$		0			0
$799$	−2.04851	−	50.4911i	−0.0724710	−	1.78625i
$800$		0			0
$801$			2.67252i			0.0944288i
$802$		0			0
$803$			34.8329i			1.22923i
$804$		0			0
$805$	3.55511	+	9.84238i	0.125301	+	0.346898i
$806$		0			0
$807$	11.3017			0.397840
$808$		0			0
$809$	−2.77212	−	2.77212i	−0.0974627	−	0.0974627i	0.656694	−	0.754157i	$-0.271954\pi$
$809$	−2.77212	−	2.77212i	−0.0974627	−	0.0974627i	−0.754157	+	0.656694i	$0.771954\pi$
$810$		0			0
$811$	16.0507	−	16.0507i	0.563616	−	0.563616i	−0.366717	−	0.930333i	$-0.619518\pi$
$811$	16.0507	−	16.0507i	0.563616	−	0.563616i	0.930333	+	0.366717i	$0.119518\pi$
$812$		0			0
$813$	−3.65669	−	3.65669i	−0.128246	−	0.128246i
$814$		0			0
$815$	−21.0227	+	44.7971i	−0.736393	+	1.56918i
$816$		0			0
$817$			4.75400i			0.166321i
$818$		0			0
$819$	1.35581	−	1.35581i	0.0473759	−	0.0473759i
$820$		0			0
$821$	0.287700	−	0.287700i	0.0100408	−	0.0100408i	−0.702069	−	0.712109i	$-0.747740\pi$
$821$	0.287700	−	0.287700i	0.0100408	−	0.0100408i	0.712109	+	0.702069i	$0.247740\pi$
$822$		0			0
$823$	−3.92073	+	3.92073i	−0.136668	+	0.136668i	−0.772131	−	0.635463i	$-0.780809\pi$
$823$	−3.92073	+	3.92073i	−0.136668	+	0.136668i	0.635463	+	0.772131i	$0.280809\pi$
$824$		0			0
$825$	−9.23323	−	11.1136i	−0.321460	−	0.386926i
$826$		0			0
$827$	−4.94680	+	4.94680i	−0.172017	+	0.172017i	−0.787865	−	0.615848i	$-0.788814\pi$
$827$	−4.94680	+	4.94680i	−0.172017	+	0.172017i	0.615848	+	0.787865i	$0.288814\pi$
$828$		0			0
$829$	25.8818			0.898912			0.449456	−	0.893302i	$-0.351618\pi$
$829$	25.8818			0.898912			0.449456	+	0.893302i	$0.351618\pi$
$830$		0			0
$831$		−	23.1714i		−	0.803808i
$832$		0			0
$833$	−1.07153	−	26.4108i	−0.0371263	−	0.915079i
$834$		0			0
$835$	−5.37405	−	2.52197i	−0.185977	−	0.0872764i
$836$		0			0
$837$		−	30.0633i		−	1.03914i
$838$		0			0
$839$	−2.22482	−	2.22482i	−0.0768093	−	0.0768093i	0.667658	−	0.744468i	$-0.267297\pi$
$839$	−2.22482	−	2.22482i	−0.0768093	−	0.0768093i	−0.744468	+	0.667658i	$0.767297\pi$
$840$		0			0
$841$			11.8374i			0.408186i
$842$		0			0
$843$	7.44503	−	7.44503i	0.256420	−	0.256420i
$844$		0			0
$845$	−25.1942	+	9.10028i	−0.866708	+	0.313059i
$846$		0			0
$847$	−2.63003	−	2.63003i	−0.0903690	−	0.0903690i
$848$		0			0
$849$	−2.56220			−0.0879345
$850$		0			0
$851$	67.5869			2.31685
$852$		0			0
$853$	−8.62929	−	8.62929i	−0.295461	−	0.295461i	0.543772	−	0.839233i	$-0.316996\pi$
$853$	−8.62929	−	8.62929i	−0.295461	−	0.295461i	−0.839233	+	0.543772i	$0.816996\pi$
$854$		0			0
$855$	12.0243	−	4.34323i	0.411222	−	0.148535i
$856$		0			0
$857$	2.93117	−	2.93117i	0.100127	−	0.100127i	−0.655269	−	0.755396i	$-0.727445\pi$
$857$	2.93117	−	2.93117i	0.100127	−	0.100127i	0.755396	+	0.655269i	$0.227445\pi$
$858$		0			0
$859$			23.4075i			0.798655i	0.916808	+	0.399327i	$0.130756\pi$
$859$			23.4075i			0.798655i	−0.916808	+	0.399327i	$0.869244\pi$
$860$		0			0
$861$	3.64204	+	3.64204i	0.124120	+	0.124120i
$862$		0			0
$863$		−	46.4610i		−	1.58155i	−0.612106	−	0.790776i	$-0.709677\pi$
$863$		−	46.4610i		−	1.58155i	0.612106	−	0.790776i	$-0.290323\pi$
$864$		0			0
$865$	7.55624	+	3.54604i	0.256920	+	0.120569i
$866$		0			0
$867$	9.40469	+	7.99083i	0.319400	+	0.271383i
$868$		0			0
$869$		−	56.5388i		−	1.91795i
$870$		0			0
$871$	9.01939			0.305610
$872$		0			0
$873$	−8.26515	+	8.26515i	−0.279733	+	0.279733i
$874$		0			0
$875$	−2.16029	−	8.30543i	−0.0730312	−	0.280775i
$876$		0			0
$877$	−21.0150	+	21.0150i	−0.709627	+	0.709627i	−0.966457	−	0.256829i	$-0.917322\pi$
$877$	−21.0150	+	21.0150i	−0.709627	+	0.709627i	0.256829	+	0.966457i	$0.417322\pi$
$878$		0			0
$879$	0.260724	−	0.260724i	0.00879401	−	0.00879401i
$880$		0			0
$881$	−6.42119	+	6.42119i	−0.216335	+	0.216335i	−0.806952	−	0.590617i	$-0.798885\pi$
$881$	−6.42119	+	6.42119i	−0.216335	+	0.216335i	0.590617	+	0.806952i	$0.298885\pi$
$882$		0			0
$883$			34.0674i			1.14646i	0.819394	+	0.573230i	$0.194310\pi$
$883$			34.0674i			1.14646i	−0.819394	+	0.573230i	$0.805690\pi$
$884$		0			0
$885$	−4.22849	+	9.01047i	−0.142139	+	0.302884i
$886$		0			0
$887$	−30.5748	−	30.5748i	−1.02660	−	1.02660i	−0.999636	−	0.0269645i	$-0.991416\pi$
$887$	−30.5748	−	30.5748i	−1.02660	−	1.02660i	−0.0269645	−	0.999636i	$-0.508584\pi$
$888$		0			0
$889$	−9.42521	+	9.42521i	−0.316111	+	0.316111i
$890$		0			0
$891$	12.7643	+	12.7643i	0.427619	+	0.427619i
$892$		0			0
$893$	28.3352			0.948200
$894$		0			0
$895$	−8.69554	−	24.0737i	−0.290660	−	0.804696i
$896$		0			0
$897$			4.47087i			0.149278i
$898$		0			0
$899$		−	31.3473i		−	1.04549i
$900$		0			0
$901$	−28.4464	+	1.15412i	−0.947687	+	0.0384492i
$902$		0			0
$903$	1.14579			0.0381296
$904$		0			0
$905$	−4.69483	−	2.20322i	−0.156061	−	0.0732375i
$906$		0			0
$907$	10.4775	−	10.4775i	0.347898	−	0.347898i	−0.511428	−	0.859326i	$-0.670883\pi$
$907$	10.4775	−	10.4775i	0.347898	−	0.347898i	0.859326	+	0.511428i	$0.170883\pi$
$908$		0			0
$909$		−	2.36853i		−	0.0785592i
$910$		0			0
$911$	−13.6008	−	13.6008i	−0.450614	−	0.450614i	0.444945	−	0.895558i	$-0.353223\pi$
$911$	−13.6008	−	13.6008i	−0.450614	−	0.450614i	−0.895558	+	0.444945i	$0.853223\pi$
$912$		0			0
$913$	−8.77064	−	8.77064i	−0.290266	−	0.290266i
$914$		0			0
$915$	8.03158	+	22.2355i	0.265516	+	0.735084i
$916$		0			0
$917$			1.37536i			0.0454183i
$918$		0			0
$919$	−29.2119			−0.963612			−0.481806	−	0.876278i	$-0.660019\pi$
$919$	−29.2119			−0.963612			−0.481806	+	0.876278i	$0.660019\pi$
$920$		0			0
$921$	13.2935	−	13.2935i	0.438034	−	0.438034i
$922$		0			0
$923$	8.85223	+	8.85223i	0.291375	+	0.291375i
$924$		0			0
$925$	−55.1905	−	5.10050i	−1.81465	−	0.167703i
$926$		0			0
$927$	13.9501			0.458183
$928$		0			0
$929$	−25.4859	−	25.4859i	−0.836165	−	0.836165i	0.152186	−	0.988352i	$-0.451369\pi$
$929$	−25.4859	−	25.4859i	−0.836165	−	0.836165i	−0.988352	+	0.152186i	$0.951369\pi$
$930$		0			0
$931$	14.8215			0.485755
$932$		0			0
$933$	2.02602			0.0663289
$934$		0			0
$935$	−33.8281	−	14.2317i	−1.10630	−	0.465425i
$936$		0			0
$937$	−40.5097			−1.32339			−0.661697	−	0.749771i	$-0.730164\pi$
$937$	−40.5097			−1.32339			−0.661697	+	0.749771i	$0.730164\pi$
$938$		0			0
$939$	−3.05538			−0.0997085
$940$		0			0
$941$	22.2677	+	22.2677i	0.725905	+	0.725905i	0.969801	−	0.243896i	$-0.0784256\pi$
$941$	22.2677	+	22.2677i	0.725905	+	0.725905i	−0.243896	+	0.969801i	$0.578426\pi$
$942$		0			0
$943$	56.3579			1.83527
$944$		0			0
$945$	−2.31665	−	6.41368i	−0.0753608	−	0.208637i
$946$		0			0
$947$	32.3931	+	32.3931i	1.05263	+	1.05263i	0.998536	+	0.0540971i	$0.0172280\pi$
$947$	32.3931	+	32.3931i	1.05263	+	1.05263i	0.0540971	+	0.998536i	$0.482772\pi$
$948$		0			0
$949$	−6.25010	+	6.25010i	−0.202887	+	0.202887i
$950$		0			0
$951$	−14.5623			−0.472214
$952$		0			0
$953$			57.2852i			1.85565i	0.373014	+	0.927826i	$0.378324\pi$
$953$			57.2852i			1.85565i	−0.373014	+	0.927826i	$0.621676\pi$
$954$		0			0
$955$	16.8063	−	6.07053i	0.543840	−	0.196438i
$956$		0			0
$957$	−8.46516	−	8.46516i	−0.273640	−	0.273640i
$958$		0			0
$959$	4.67071	+	4.67071i	0.150825	+	0.150825i
$960$		0			0
$961$		−	26.2553i		−	0.846947i
$962$		0			0
$963$	−24.5920	+	24.5920i	−0.792466	+	0.792466i
$964$		0			0
$965$	12.0711	−	25.7223i	0.388584	−	0.828031i
$966$		0			0
$967$	−3.21512			−0.103391			−0.0516956	−	0.998663i	$-0.516463\pi$
$967$	−3.21512			−0.103391			−0.0516956	+	0.998663i	$0.516463\pi$
$968$		0			0
$969$	−4.69080	+	5.08752i	−0.150690	+	0.163435i
$970$		0			0
$971$			5.32370i			0.170846i	0.996345	+	0.0854229i	$0.0272241\pi$
$971$			5.32370i			0.170846i	−0.996345	+	0.0854229i	$0.972776\pi$
$972$		0			0
$973$		−	7.96970i		−	0.255497i
$974$		0			0
$975$	0.337398	−	3.65085i	0.0108054	−	0.116921i
$976$		0			0
$977$	−13.4000			−0.428704			−0.214352	−	0.976756i	$-0.568764\pi$
$977$	−13.4000			−0.428704			−0.214352	+	0.976756i	$0.568764\pi$
$978$		0			0
$979$	3.04184	+	3.04184i	0.0972176	+	0.0972176i
$980$		0			0
$981$	29.3051	−	29.3051i	0.935640	−	0.935640i
$982$		0			0
$983$	−28.9134	−	28.9134i	−0.922196	−	0.922196i	0.0749887	−	0.997184i	$-0.476108\pi$
$983$	−28.9134	−	28.9134i	−0.922196	−	0.922196i	−0.997184	+	0.0749887i	$0.976108\pi$
$984$		0			0
$985$	−7.40069	+	15.7701i	−0.235806	+	0.502477i
$986$		0			0
$987$		−	6.82925i		−	0.217378i
$988$		0			0
$989$	8.86517	−	8.86517i	0.281896	−	0.281896i
$990$		0			0
$991$	0.269862	−	0.269862i	0.00857244	−	0.00857244i	−0.702808	−	0.711380i	$-0.748071\pi$
$991$	0.269862	−	0.269862i	0.00857244	−	0.00857244i	0.711380	+	0.702808i	$0.248071\pi$
$992$		0			0
$993$	10.7407	−	10.7407i	0.340845	−	0.340845i
$994$		0			0
$995$	−7.13325	+	15.2002i	−0.226139	+	0.481879i
$996$		0			0
$997$	−22.6148	+	22.6148i	−0.716219	+	0.716219i	−0.967829	−	0.251609i	$-0.919040\pi$
$997$	−22.6148	+	22.6148i	−0.716219	+	0.716219i	0.251609	+	0.967829i	$0.419040\pi$
$998$		0			0
$999$	−44.0423			−1.39344

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	680.2.z.d.89.8	yes	26
5.4	even	2		680.2.z.c.89.6	✓	26
17.13	even	4		680.2.z.c.489.6	yes	26
85.64	even	4	inner	680.2.z.d.489.8	yes	26

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
680.2.z.c.89.6	✓	26	5.4	even	2
680.2.z.c.489.6	yes	26	17.13	even	4
680.2.z.d.89.8	yes	26	1.1	even	1	trivial
680.2.z.d.489.8	yes	26	85.64	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:	\( N \)	\(=\)	\( 680 = 2^{3} \cdot 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	680.z (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(0.513320 + 0.513320i) q^{3} +(-2.10308 + 0.759643i) q^{5} +(-0.542760 + 0.542760i) q^{7} -2.47300i q^{9} +(-2.81475 - 2.81475i) q^{11} -1.01011i q^{13} +(-1.46949 - 0.689613i) q^{15} +(-4.11972 + 0.167144i) q^{17} -2.31195i q^{19} -0.557219 q^{21} +(-4.31128 + 4.31128i) q^{23} +(3.84589 - 3.19518i) q^{25} +(2.80940 - 2.80940i) q^{27} +(2.92939 - 2.92939i) q^{29} +(5.35048 - 5.35048i) q^{31} -2.88974i q^{33} +(0.729164 - 1.55377i) q^{35} +(-7.83837 - 7.83837i) q^{37} +(0.518508 - 0.518508i) q^{39} +(-6.53610 - 6.53610i) q^{41} -2.05627 q^{43} +(1.87860 + 5.20093i) q^{45} +12.2560i q^{47} +6.41082i q^{49} +(-2.20053 - 2.02893i) q^{51} +6.90494 q^{53} +(8.05785 + 3.78144i) q^{55} +(1.18677 - 1.18677i) q^{57} -6.13169i q^{59} +(-10.2985 - 10.2985i) q^{61} +(1.34225 + 1.34225i) q^{63} +(0.767320 + 2.12433i) q^{65} +8.92915i q^{67} -4.42614 q^{69} +(-8.76366 + 8.76366i) q^{71} +(-6.18757 - 6.18757i) q^{73} +(3.61432 + 0.334022i) q^{75} +3.05547 q^{77} +(10.0433 + 10.0433i) q^{79} -4.53477 q^{81} +3.11595 q^{83} +(8.53712 - 3.48103i) q^{85} +3.00743 q^{87} -1.08068 q^{89} +(0.548245 + 0.548245i) q^{91} +5.49302 q^{93} +(1.75626 + 4.86221i) q^{95} +(-3.34215 - 3.34215i) q^{97} +(-6.96090 + 6.96090i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 26 q + 2 q^{3} - 2 q^{5} + 2 q^{7} + 2 q^{11} - 2 q^{15} + 2 q^{17} - 12 q^{21} - 2 q^{23} + 10 q^{25} + 20 q^{27} + 10 q^{29} - 10 q^{31} + 22 q^{35} - 6 q^{37} + 8 q^{39} - 10 q^{41} + 32 q^{43} + 28 q^{45}+ \cdots - 22 q^{99}+O(q^{100}) \) 26 * q + 2 * q^3 - 2 * q^5 + 2 * q^7 + 2 * q^11 - 2 * q^15 + 2 * q^17 - 12 * q^21 - 2 * q^23 + 10 * q^25 + 20 * q^27 + 10 * q^29 - 10 * q^31 + 22 * q^35 - 6 * q^37 + 8 * q^39 - 10 * q^41 + 32 * q^43 + 28 * q^45 - 6 * q^51 - 32 * q^53 - 26 * q^55 + 20 * q^57 + 6 * q^61 - 54 * q^63 + 24 * q^65 - 20 * q^69 - 30 * q^71 + 42 * q^73 - 78 * q^75 - 4 * q^77 + 14 * q^79 - 54 * q^81 - 16 * q^83 - 26 * q^85 + 8 * q^87 + 40 * q^89 + 24 * q^91 + 88 * q^93 + 44 * q^95 + 66 * q^97 - 22 * q^99

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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