LMFDB - Embedded newform 378.3.q.a.71.9

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [378,3,Mod(71,378)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(378, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([5, 0])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("378.71"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 378 = 2 \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	378.q (of order $6$, degree $2$, not minimal)

Newform invariants

comment:select newform

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

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

Self dual:	no
Analytic conductor:	$10.2997539928$
Analytic rank:	$0$
Dimension:	$24$
Relative dimension:	$12$ over $\Q(\zeta_{6})$
Twist minimal:	no (minimal twist has level 126)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			71.9
Character	$\chi$	$=$	378.71
Dual form			378.3.q.a.197.9

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(1.22474 - 0.707107i) q^{2} +(1.00000 - 1.73205i) q^{4} +(-2.76866 - 1.59849i) q^{5} +(-1.32288 - 2.29129i) q^{7} -2.82843i q^{8} -4.52120 q^{10} +(16.3294 - 9.42781i) q^{11} +(-5.18593 + 8.98229i) q^{13} +(-3.24037 - 1.87083i) q^{14} +(-2.00000 - 3.46410i) q^{16} -24.4251i q^{17} -26.5935 q^{19} +(-5.53732 + 3.19697i) q^{20} +(13.3329 - 23.0933i) q^{22} +(-35.3991 - 20.4377i) q^{23} +(-7.38968 - 12.7993i) q^{25} +14.6680i q^{26} -5.29150 q^{28} +(18.8212 - 10.8664i) q^{29} +(1.37716 - 2.38532i) q^{31} +(-4.89898 - 2.82843i) q^{32} +(-17.2712 - 29.9145i) q^{34} +8.45840i q^{35} +45.6422 q^{37} +(-32.5702 + 18.8044i) q^{38} +(-4.52120 + 7.83095i) q^{40} +(12.8400 + 7.41318i) q^{41} +(14.4554 + 25.0374i) q^{43} -37.7112i q^{44} -57.8065 q^{46} +(25.7813 - 14.8849i) q^{47} +(-3.50000 + 6.06218i) q^{49} +(-18.1009 - 10.4506i) q^{50} +(10.3719 + 17.9646i) q^{52} +18.1401i q^{53} -60.2809 q^{55} +(-6.48074 + 3.74166i) q^{56} +(15.3674 - 26.6172i) q^{58} +(46.4112 + 26.7955i) q^{59} +(13.9337 + 24.1339i) q^{61} -3.89520i q^{62} -8.00000 q^{64} +(28.7161 - 16.5793i) q^{65} +(46.2776 - 80.1552i) q^{67} +(-42.3056 - 24.4251i) q^{68} +(5.98099 + 10.3594i) q^{70} +81.9005i q^{71} -81.2899 q^{73} +(55.9000 - 32.2739i) q^{74} +(-26.5935 + 46.0612i) q^{76} +(-43.2036 - 24.9436i) q^{77} +(42.3065 + 73.2770i) q^{79} +12.7879i q^{80} +20.9677 q^{82} +(99.9040 - 57.6796i) q^{83} +(-39.0432 + 67.6249i) q^{85} +(35.4083 + 20.4430i) q^{86} +(-26.6659 - 46.1866i) q^{88} +13.7346i q^{89} +27.4413 q^{91} +(-70.7982 + 40.8754i) q^{92} +(21.0504 - 36.4603i) q^{94} +(73.6282 + 42.5093i) q^{95} +(55.7929 + 96.6361i) q^{97} +9.89949i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 24 q^{4} - 36 q^{5} - 48 q^{16} + 24 q^{19} - 72 q^{20} + 24 q^{22} + 72 q^{23} + 72 q^{25} + 108 q^{29} - 60 q^{31} - 48 q^{34} - 168 q^{37} - 144 q^{38} - 108 q^{41} + 60 q^{43} + 324 q^{47} - 84 q^{49}+ \cdots + 48 q^{97}+O(q^{100}) $ 24 * q + 24 * q^4 - 36 * q^5 - 48 * q^16 + 24 * q^19 - 72 * q^20 + 24 * q^22 + 72 * q^23 + 72 * q^25 + 108 * q^29 - 60 * q^31 - 48 * q^34 - 168 * q^37 - 144 * q^38 - 108 * q^41 + 60 * q^43 + 324 * q^47 - 84 * q^49 - 144 * q^50 + 264 * q^55 + 432 * q^59 - 192 * q^64 + 180 * q^65 + 72 * q^67 - 72 * q^68 + 24 * q^73 - 288 * q^74 + 24 * q^76 + 12 * q^79 - 288 * q^82 - 756 * q^83 - 156 * q^85 - 360 * q^86 - 48 * q^88 - 168 * q^91 + 144 * q^92 + 936 * q^95 + 48 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/378\mathbb{Z}\right)^\times$.

$n$	$29$	$325$
$\chi(n)$	$e\left(\frac{5}{6}\right)$	$1$

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 $

$2$	1.22474	−	0.707107i	0.612372	−	0.353553i
$2$	1.22474	−	0.707107i	0.612372	−	0.353553i
$3$		0			0
$3$		0			0
$4$	1.00000	−	1.73205i	0.250000	−	0.433013i
$5$	−2.76866	−	1.59849i	−0.553732	−	0.319697i	0.196894	−	0.980425i	$-0.436915\pi$
$5$	−2.76866	−	1.59849i	−0.553732	−	0.319697i	−0.750626	+	0.660728i	$0.770248\pi$
$6$		0			0
$7$	−1.32288	−	2.29129i	−0.188982	−	0.327327i
$7$	−1.32288	−	2.29129i	−0.188982	−	0.327327i
$8$		−	2.82843i		−	0.353553i
$9$		0			0
$10$	−4.52120			−0.452120
$11$	16.3294	−	9.42781i	1.48449	−	0.857073i	0.484650	−	0.874708i	$-0.338947\pi$
$11$	16.3294	−	9.42781i	1.48449	−	0.857073i	0.999844	+	0.0176350i	$0.00561367\pi$
$12$		0			0
$13$	−5.18593	+	8.98229i	−0.398917	+	0.690945i	−0.993593	−	0.113021i	$-0.963947\pi$
$13$	−5.18593	+	8.98229i	−0.398917	+	0.690945i	0.594675	+	0.803966i	$0.297281\pi$
$14$	−3.24037	−	1.87083i	−0.231455	−	0.133631i
$15$		0			0
$16$	−2.00000	−	3.46410i	−0.125000	−	0.216506i
$17$		−	24.4251i		−	1.43677i	−0.695645	−	0.718386i	$-0.744881\pi$
$17$		−	24.4251i		−	1.43677i	0.695645	−	0.718386i	$-0.255119\pi$
$18$		0			0
$19$	−26.5935			−1.39966			−0.699828	−	0.714312i	$-0.746740\pi$
$19$	−26.5935			−1.39966			−0.699828	+	0.714312i	$0.746740\pi$
$20$	−5.53732	+	3.19697i	−0.276866	+	0.159849i
$21$		0			0
$22$	13.3329	−	23.0933i	0.606042	−	1.04970i
$23$	−35.3991	−	20.4377i	−1.53909	−	0.888595i	−0.998892	−	0.0470582i	$-0.985015\pi$
$23$	−35.3991	−	20.4377i	−1.53909	−	0.888595i	−0.540200	−	0.841537i	$-0.681651\pi$
$24$		0			0
$25$	−7.38968	−	12.7993i	−0.295587	−	0.511972i
$26$			14.6680i			0.564154i
$27$		0			0
$28$	−5.29150			−0.188982
$29$	18.8212	−	10.8664i	0.649006	−	0.374704i	−0.139069	−	0.990283i	$-0.544411\pi$
$29$	18.8212	−	10.8664i	0.649006	−	0.374704i	0.788075	+	0.615579i	$0.211078\pi$
$30$		0			0
$31$	1.37716	−	2.38532i	0.0444246	−	0.0769457i	−0.842958	−	0.537979i	$-0.819188\pi$
$31$	1.37716	−	2.38532i	0.0444246	−	0.0769457i	0.887383	+	0.461034i	$0.152521\pi$
$32$	−4.89898	−	2.82843i	−0.153093	−	0.0883883i
$33$		0			0
$34$	−17.2712	−	29.9145i	−0.507976	−	0.879840i
$35$			8.45840i			0.241668i
$36$		0			0
$37$	45.6422			1.23357			0.616786	−	0.787131i	$-0.288434\pi$
$37$	45.6422			1.23357			0.616786	+	0.787131i	$0.288434\pi$
$38$	−32.5702	+	18.8044i	−0.857111	+	0.494853i
$39$		0			0
$40$	−4.52120	+	7.83095i	−0.113030	+	0.195774i
$41$	12.8400	+	7.41318i	0.313171	+	0.180809i	0.648345	−	0.761347i	$-0.275462\pi$
$41$	12.8400	+	7.41318i	0.313171	+	0.180809i	−0.335174	+	0.942156i	$0.608795\pi$
$42$		0			0
$43$	14.4554	+	25.0374i	0.336171	+	0.582266i	0.983709	−	0.179767i	$-0.0575345\pi$
$43$	14.4554	+	25.0374i	0.336171	+	0.582266i	−0.647538	+	0.762033i	$0.724201\pi$
$44$		−	37.7112i		−	0.857073i
$45$		0			0
$46$	−57.8065			−1.25666
$47$	25.7813	−	14.8849i	0.548539	−	0.316699i	−0.199993	−	0.979797i	$-0.564092\pi$
$47$	25.7813	−	14.8849i	0.548539	−	0.316699i	0.748533	+	0.663098i	$0.230759\pi$
$48$		0			0
$49$	−3.50000	+	6.06218i	−0.0714286	+	0.123718i
$50$	−18.1009	−	10.4506i	−0.362019	−	0.209012i
$51$		0			0
$52$	10.3719	+	17.9646i	0.199459	+	0.345473i
$53$			18.1401i			0.342266i	0.985248	+	0.171133i	$0.0547428\pi$
$53$			18.1401i			0.342266i	−0.985248	+	0.171133i	$0.945257\pi$
$54$		0			0
$55$	−60.2809			−1.09602
$56$	−6.48074	+	3.74166i	−0.115728	+	0.0668153i
$57$		0			0
$58$	15.3674	−	26.6172i	0.264956	−	0.458917i
$59$	46.4112	+	26.7955i	0.786631	+	0.454162i	0.838775	−	0.544478i	$-0.183272\pi$
$59$	46.4112	+	26.7955i	0.786631	+	0.454162i	−0.0521440	+	0.998640i	$0.516605\pi$
$60$		0			0
$61$	13.9337	+	24.1339i	0.228422	+	0.395638i	0.957341	−	0.288962i	$-0.0933102\pi$
$61$	13.9337	+	24.1339i	0.228422	+	0.395638i	−0.728919	+	0.684600i	$0.759977\pi$
$62$		−	3.89520i		−	0.0628259i
$63$		0			0
$64$	−8.00000			−0.125000
$65$	28.7161	−	16.5793i	0.441787	−	0.255066i
$66$		0			0
$67$	46.2776	−	80.1552i	0.690711	−	1.19635i	−0.280895	−	0.959739i	$-0.590631\pi$
$67$	46.2776	−	80.1552i	0.690711	−	1.19635i	0.971605	−	0.236607i	$-0.0760355\pi$
$68$	−42.3056	−	24.4251i	−0.622141	−	0.359193i
$69$		0			0
$70$	5.98099	+	10.3594i	0.0854427	+	0.147991i
$71$			81.9005i			1.15353i	0.816911	+	0.576764i	$0.195685\pi$
$71$			81.9005i			1.15353i	−0.816911	+	0.576764i	$0.804315\pi$
$72$		0			0
$73$	−81.2899			−1.11356			−0.556780	−	0.830660i	$-0.687964\pi$
$73$	−81.2899			−1.11356			−0.556780	+	0.830660i	$0.687964\pi$
$74$	55.9000	−	32.2739i	0.755406	−	0.436134i
$75$		0			0
$76$	−26.5935	+	46.0612i	−0.349914	+	0.606069i
$77$	−43.2036	−	24.9436i	−0.561086	−	0.323943i
$78$		0			0
$79$	42.3065	+	73.2770i	0.535525	+	0.927557i	0.999138	+	0.0415187i	$0.0132196\pi$
$79$	42.3065	+	73.2770i	0.535525	+	0.927557i	−0.463613	+	0.886038i	$0.653447\pi$
$80$			12.7879i			0.159849i
$81$		0			0
$82$	20.9677			0.255703
$83$	99.9040	−	57.6796i	1.20366	−	0.694935i	0.242295	−	0.970203i	$-0.422100\pi$
$83$	99.9040	−	57.6796i	1.20366	−	0.694935i	0.961368	+	0.275268i	$0.0887665\pi$
$84$		0			0
$85$	−39.0432	+	67.6249i	−0.459332	+	0.795587i
$86$	35.4083	+	20.4430i	0.411724	+	0.237709i
$87$		0			0
$88$	−26.6659	−	46.1866i	−0.303021	−	0.524848i
$89$			13.7346i			0.154321i	0.997019	+	0.0771606i	$0.0245854\pi$
$89$			13.7346i			0.154321i	−0.997019	+	0.0771606i	$0.975415\pi$
$90$		0			0
$91$	27.4413			0.301553
$92$	−70.7982	+	40.8754i	−0.769546	+	0.444298i
$93$		0			0
$94$	21.0504	−	36.4603i	0.223940	−	0.387876i
$95$	73.6282	+	42.5093i	0.775034	+	0.447466i
$96$		0			0
$97$	55.7929	+	96.6361i	0.575184	+	0.996248i	0.996022	+	0.0891122i	$0.0284030\pi$
$97$	55.7929	+	96.6361i	0.575184	+	0.996248i	−0.420837	+	0.907136i	$0.638264\pi$
$98$			9.89949i			0.101015i
$99$		0			0
$100$	−29.5587			−0.295587
$101$	−125.514	+	72.4658i	−1.24272	+	0.717483i	−0.969647	−	0.244510i	$-0.921373\pi$
$101$	−125.514	+	72.4658i	−1.24272	+	0.717483i	−0.273071	+	0.961994i	$0.588039\pi$
$102$		0			0
$103$	22.4051	−	38.8068i	0.217525	−	0.376765i	−0.736525	−	0.676410i	$-0.763535\pi$
$103$	22.4051	−	38.8068i	0.217525	−	0.376765i	0.954051	+	0.299645i	$0.0968681\pi$
$104$	25.4057	+	14.6680i	0.244286	+	0.141039i
$105$		0			0
$106$	12.8270	+	22.2170i	0.121009	+	0.209594i
$107$		−	100.220i		−	0.936634i	−0.883560	−	0.468317i	$-0.844861\pi$
$107$		−	100.220i		−	0.936634i	0.883560	−	0.468317i	$-0.155139\pi$
$108$		0			0
$109$	−60.7449			−0.557292			−0.278646	−	0.960394i	$-0.589886\pi$
$109$	−60.7449			−0.557292			−0.278646	+	0.960394i	$0.589886\pi$
$110$	−73.8287	+	42.6250i	−0.671170	+	0.387500i
$111$		0			0
$112$	−5.29150	+	9.16515i	−0.0472456	+	0.0818317i
$113$	−12.6128	−	7.28199i	−0.111617	−	0.0644424i	0.443152	−	0.896446i	$-0.353860\pi$
$113$	−12.6128	−	7.28199i	−0.111617	−	0.0644424i	−0.554769	+	0.832004i	$0.687194\pi$
$114$		0			0
$115$	65.3387	+	113.170i	0.568163	+	0.984087i
$116$		−	43.4656i		−	0.374704i
$117$		0			0
$118$	75.7892			0.642282
$119$	−55.9650	+	32.3114i	−0.470294	+	0.271524i
$120$		0			0
$121$	117.267	−	203.112i	0.969149	−	1.67862i
$122$	34.1305	+	19.7053i	0.279758	+	0.161519i
$123$		0			0
$124$	−2.75433	−	4.77063i	−0.0222123	−	0.0384728i
$125$			127.174i			1.01739i
$126$		0			0
$127$	221.852			1.74687			0.873433	−	0.486944i	$-0.161888\pi$
$127$	221.852			1.74687			0.873433	+	0.486944i	$0.161888\pi$
$128$	−9.79796	+	5.65685i	−0.0765466	+	0.0441942i
$129$		0			0
$130$	23.4466	−	40.6107i	0.180359	−	0.312390i
$131$	−2.93262	−	1.69315i	−0.0223864	−	0.0129248i	0.488765	−	0.872415i	$-0.337448\pi$
$131$	−2.93262	−	1.69315i	−0.0223864	−	0.0129248i	−0.511151	+	0.859491i	$0.670781\pi$
$132$		0			0
$133$	35.1798	+	60.9333i	0.264510	+	0.458145i
$134$		−	130.893i		−	0.976812i
$135$		0			0
$136$	−69.0847			−0.507976
$137$	18.6956	−	10.7939i	0.136464	−	0.0787877i	−0.430214	−	0.902727i	$-0.641562\pi$
$137$	18.6956	−	10.7939i	0.136464	−	0.0787877i	0.566678	+	0.823939i	$0.308228\pi$
$138$		0			0
$139$	30.1614	−	52.2410i	0.216988	−	0.375835i	−0.736897	−	0.676005i	$-0.763710\pi$
$139$	30.1614	−	52.2410i	0.216988	−	0.375835i	0.953886	+	0.300170i	$0.0970433\pi$
$140$	14.6504	+	8.45840i	0.104646	+	0.0604171i
$141$		0			0
$142$	57.9124	+	100.307i	0.407834	+	0.706389i
$143$			195.568i			1.36761i
$144$		0			0
$145$	−69.4792			−0.479167
$146$	−99.5594	+	57.4807i	−0.681914	+	0.393703i
$147$		0			0
$148$	45.6422	−	79.0546i	0.308393	−	0.534153i
$149$	−13.4563	−	7.76899i	−0.0903107	−	0.0521409i	0.454165	−	0.890918i	$-0.349938\pi$
$149$	−13.4563	−	7.76899i	−0.0903107	−	0.0521409i	−0.544475	+	0.838777i	$0.683271\pi$
$150$		0			0
$151$	−13.4117	−	23.2297i	−0.0888192	−	0.153839i	0.818193	−	0.574944i	$-0.194976\pi$
$151$	−13.4117	−	23.2297i	−0.0888192	−	0.153839i	−0.907012	+	0.421104i	$0.861643\pi$
$152$			75.2177i			0.494853i
$153$		0			0
$154$	−70.5512			−0.458125
$155$	−7.62579	+	4.40275i	−0.0491987	+	0.0284049i
$156$		0			0
$157$	−53.0669	+	91.9146i	−0.338006	+	0.585444i	−0.984058	−	0.177850i	$-0.943086\pi$
$157$	−53.0669	+	91.9146i	−0.338006	+	0.585444i	0.646052	+	0.763294i	$0.276419\pi$
$158$	103.629	+	59.8304i	0.655882	+	0.378673i
$159$		0			0
$160$	9.04241	+	15.6619i	0.0565150	+	0.0978869i
$161$			108.146i			0.671715i
$162$		0			0
$163$	11.8953			0.0729774			0.0364887	−	0.999334i	$-0.488383\pi$
$163$	11.8953			0.0729774			0.0364887	+	0.999334i	$0.488383\pi$
$164$	25.6800	−	14.8264i	0.156586	−	0.0904047i
$165$		0			0
$166$	81.5713	−	141.286i	0.491393	−	0.851118i
$167$	−120.719	−	69.6971i	−0.722868	−	0.417348i	0.0929395	−	0.995672i	$-0.470374\pi$
$167$	−120.719	−	69.6971i	−0.722868	−	0.417348i	−0.815807	+	0.578324i	$0.803707\pi$
$168$		0			0
$169$	30.7123	+	53.1953i	0.181730	+	0.314765i
$170$			110.431i			0.649594i
$171$		0			0
$172$	57.8215			0.336171
$173$	198.663	−	114.698i	1.14834	−	0.662994i	0.199858	−	0.979825i	$-0.435952\pi$
$173$	198.663	−	114.698i	1.14834	−	0.662994i	0.948482	+	0.316830i	$0.102619\pi$
$174$		0			0
$175$	−19.5513	+	33.8638i	−0.111721	+	0.193507i
$176$	−65.3178	−	37.7112i	−0.371124	−	0.214268i
$177$		0			0
$178$	9.71182	+	16.8214i	0.0545608	+	0.0945021i
$179$		−	202.541i		−	1.13152i	−0.824571	−	0.565758i	$-0.808584\pi$
$179$		−	202.541i		−	1.13152i	0.824571	−	0.565758i	$-0.191416\pi$
$180$		0			0
$181$	4.20404			0.0232267			0.0116134	−	0.999933i	$-0.496303\pi$
$181$	4.20404			0.0232267			0.0116134	+	0.999933i	$0.496303\pi$
$182$	33.6086	−	19.4040i	0.184663	−	0.106615i
$183$		0			0
$184$	−57.8065	+	100.124i	−0.314166	+	0.544151i
$185$	−126.368	−	72.9584i	−0.683069	−	0.394370i
$186$		0			0
$187$	−230.275	−	398.849i	−1.23142	−	2.13288i
$188$		−	59.5395i		−	0.316699i
$189$		0			0
$190$	120.234			0.632813
$191$	235.481	−	135.955i	1.23288	−	0.711806i	0.265255	−	0.964178i	$-0.414544\pi$
$191$	235.481	−	135.955i	1.23288	−	0.711806i	0.967630	+	0.252372i	$0.0812106\pi$
$192$		0			0
$193$	72.2766	−	125.187i	0.374490	−	0.648636i	−0.615760	−	0.787934i	$-0.711151\pi$
$193$	72.2766	−	125.187i	0.374490	−	0.648636i	0.990251	+	0.139297i	$0.0444843\pi$
$194$	136.664	+	78.9030i	0.704454	+	0.406717i
$195$		0			0
$196$	7.00000	+	12.1244i	0.0357143	+	0.0618590i
$197$		−	53.0627i		−	0.269354i	−0.990890	−	0.134677i	$-0.957000\pi$
$197$		−	53.0627i		−	0.269354i	0.990890	−	0.134677i	$-0.0429996\pi$
$198$		0			0
$199$	−114.808			−0.576923			−0.288461	−	0.957491i	$-0.593144\pi$
$199$	−114.808			−0.576923			−0.288461	+	0.957491i	$0.593144\pi$
$200$	−36.2019	+	20.9012i	−0.181009	+	0.104506i
$201$		0			0
$202$	−102.482	+	177.504i	−0.507337	+	0.878734i
$203$	−49.7962	−	28.7498i	−0.245301	−	0.141625i
$204$		0			0
$205$	−23.6997	−	41.0492i	−0.115609	−	0.200240i
$206$		−	63.3713i		−	0.307628i
$207$		0			0
$208$	41.4874			0.199459
$209$	−434.256	+	250.718i	−2.07778	+	1.19961i
$210$		0			0
$211$	−180.480	+	312.600i	−0.855353	+	1.48152i	0.0209634	+	0.999780i	$0.493327\pi$
$211$	−180.480	+	312.600i	−0.855353	+	1.48152i	−0.876317	+	0.481735i	$0.840007\pi$
$212$	31.4196	+	18.1401i	0.148206	+	0.0855665i
$213$		0			0
$214$	−70.8661	−	122.744i	−0.331150	−	0.573569i
$215$		−	92.4269i		−	0.429892i
$216$		0			0
$217$	−7.28726			−0.0335818
$218$	−74.3970	+	42.9531i	−0.341271	+	0.197033i
$219$		0			0
$220$	−60.2809	+	104.410i	−0.274004	+	0.474589i
$221$	219.393	+	126.667i	0.992731	+	0.573153i
$222$		0			0
$223$	103.429	+	179.144i	0.463805	+	0.803334i	0.999147	−	0.0413015i	$-0.0131504\pi$
$223$	103.429	+	179.144i	0.463805	+	0.803334i	−0.535342	+	0.844636i	$0.679817\pi$
$224$			14.9666i			0.0668153i
$225$		0			0
$226$	−20.5966			−0.0911353
$227$	−114.004	+	65.8205i	−0.502222	+	0.289958i	−0.729631	−	0.683841i	$-0.760308\pi$
$227$	−114.004	+	65.8205i	−0.502222	+	0.289958i	0.227409	+	0.973799i	$0.426975\pi$
$228$		0			0
$229$	−118.264	+	204.839i	−0.516436	+	0.894494i	0.483381	+	0.875410i	$0.339408\pi$
$229$	−118.264	+	204.839i	−0.516436	+	0.894494i	−0.999818	+	0.0190842i	$0.993925\pi$
$230$	160.047	+	92.4029i	0.695855	+	0.401752i
$231$		0			0
$232$	−30.7349	−	53.2343i	−0.132478	−	0.229458i
$233$			155.574i			0.667699i	0.942626	+	0.333849i	$0.108348\pi$
$233$			155.574i			0.667699i	−0.942626	+	0.333849i	$0.891652\pi$
$234$		0			0
$235$	−95.1730			−0.404992
$236$	92.8225	−	53.5911i	0.393316	−	0.227081i
$237$		0			0
$238$	−45.6952	+	79.1465i	−0.191997	+	0.332548i
$239$	51.6119	+	29.7981i	0.215949	+	0.124678i	0.604073	−	0.796929i	$-0.293543\pi$
$239$	51.6119	+	29.7981i	0.215949	+	0.124678i	−0.388124	+	0.921607i	$0.626877\pi$
$240$		0			0
$241$	59.7453	+	103.482i	0.247906	+	0.429385i	0.962945	−	0.269699i	$-0.0869243\pi$
$241$	59.7453	+	103.482i	0.247906	+	0.429385i	−0.715039	+	0.699085i	$0.753591\pi$
$242$		−	331.681i		−	1.37058i
$243$		0			0
$244$	55.7349			0.228422
$245$	19.3806	−	11.1894i	0.0791046	−	0.0456710i
$246$		0			0
$247$	137.912	−	238.870i	0.558347	−	0.967085i
$248$	−6.74669	−	3.89520i	−0.0272044	−	0.0157065i
$249$		0			0
$250$	89.9253	+	155.755i	0.359701	+	0.623021i
$251$		−	483.177i		−	1.92501i	−0.271267	−	0.962504i	$-0.587443\pi$
$251$		−	483.177i		−	1.92501i	0.271267	−	0.962504i	$-0.412557\pi$
$252$		0			0
$253$	−770.730			−3.04636
$254$	271.712	−	156.873i	1.06973	−	0.617611i
$255$		0			0
$256$	−8.00000	+	13.8564i	−0.0312500	+	0.0541266i
$257$	−136.833	−	79.0003i	−0.532422	−	0.307394i	0.209580	−	0.977792i	$-0.432790\pi$
$257$	−136.833	−	79.0003i	−0.532422	−	0.307394i	−0.742002	+	0.670397i	$0.766124\pi$
$258$		0			0
$259$	−60.3789	−	104.579i	−0.233123	−	0.403781i
$260$		−	66.3171i		−	0.255066i
$261$		0			0
$262$	−4.78895			−0.0182784
$263$	244.980	−	141.439i	0.931483	−	0.537792i	0.0442024	−	0.999023i	$-0.485925\pi$
$263$	244.980	−	141.439i	0.931483	−	0.537792i	0.887280	+	0.461231i	$0.152592\pi$
$264$		0			0
$265$	28.9967	−	50.2238i	0.109422	−	0.189524i
$266$	86.1726	+	49.7518i	0.323957	+	0.187037i
$267$		0			0
$268$	−92.5552	−	160.310i	−0.345355	−	0.598173i
$269$			125.436i			0.466303i	0.972440	+	0.233152i	$0.0749039\pi$
$269$			125.436i			0.466303i	−0.972440	+	0.233152i	$0.925096\pi$
$270$		0			0
$271$	418.065			1.54268			0.771338	−	0.636426i	$-0.219588\pi$
$271$	418.065			1.54268			0.771338	+	0.636426i	$0.219588\pi$
$272$	−84.6111	+	48.8502i	−0.311070	+	0.179597i
$273$		0			0
$274$	15.2649	−	26.4396i	0.0557113	−	0.0964948i
$275$	−241.339	−	139.337i	−0.877595	−	0.506680i
$276$		0			0
$277$	−228.356	−	395.524i	−0.824389	−	1.42788i	−0.902386	−	0.430929i	$-0.858186\pi$
$277$	−228.356	−	395.524i	−0.824389	−	1.42788i	0.0779971	−	0.996954i	$-0.475148\pi$
$278$		−	85.3093i		−	0.306868i
$279$		0			0
$280$	23.9240			0.0854427
$281$	−221.123	+	127.665i	−0.786913	+	0.454324i	−0.838875	−	0.544325i	$-0.816786\pi$
$281$	−221.123	+	127.665i	−0.786913	+	0.454324i	0.0519617	+	0.998649i	$0.483453\pi$
$282$		0			0
$283$	−4.11688	+	7.13065i	−0.0145473	+	0.0251966i	−0.873207	−	0.487349i	$-0.837964\pi$
$283$	−4.11688	+	7.13065i	−0.0145473	+	0.0251966i	0.858660	+	0.512545i	$0.171297\pi$
$284$	141.856	+	81.9005i	0.499492	+	0.288382i
$285$		0			0
$286$	138.287	+	239.520i	0.483522	+	0.837484i
$287$		−	39.2269i		−	0.136679i
$288$		0			0
$289$	−307.587			−1.06431
$290$	−85.0944	+	49.1292i	−0.293429	+	0.169411i
$291$		0			0
$292$	−81.2899	+	140.798i	−0.278390	+	0.482186i
$293$	−92.6650	−	53.5002i	−0.316263	−	0.182594i	0.333463	−	0.942763i	$-0.391783\pi$
$293$	−92.6650	−	53.5002i	−0.316263	−	0.182594i	−0.649726	+	0.760169i	$0.725116\pi$
$294$		0			0
$295$	−85.6646	−	148.375i	−0.290389	−	0.502968i
$296$		−	129.096i		−	0.436134i
$297$		0			0
$298$	−21.9740			−0.0737384
$299$	367.154	−	211.977i	1.22794	−	0.708952i
$300$		0			0
$301$	38.2453	−	66.2428i	0.127061	−	0.220076i
$302$	−32.8518	−	18.9670i	−0.108781	−	0.0628046i
$303$		0			0
$304$	53.1869	+	92.1224i	0.174957	+	0.303034i
$305$		−	89.0915i		−	0.292103i
$306$		0			0
$307$	540.799			1.76156			0.880780	−	0.473526i	$-0.157019\pi$
$307$	540.799			1.76156			0.880780	+	0.473526i	$0.157019\pi$
$308$	−86.4073	+	49.8873i	−0.280543	+	0.161972i
$309$		0			0
$310$	−6.22643	+	10.7845i	−0.0200853	+	0.0347887i
$311$	−226.920	−	131.012i	−0.729646	−	0.421261i	0.0886465	−	0.996063i	$-0.471746\pi$
$311$	−226.920	−	131.012i	−0.729646	−	0.421261i	−0.818293	+	0.574802i	$0.805079\pi$
$312$		0			0
$313$	−302.440	−	523.842i	−0.966262	−	1.67362i	−0.706185	−	0.708027i	$-0.749585\pi$
$313$	−302.440	−	523.842i	−0.966262	−	1.67362i	−0.260077	−	0.965588i	$-0.583748\pi$
$314$			150.096i			0.478013i
$315$		0			0
$316$	169.226			0.535525
$317$	−75.7622	+	43.7413i	−0.238997	+	0.137985i	−0.614716	−	0.788749i	$-0.710729\pi$
$317$	−75.7622	+	43.7413i	−0.238997	+	0.137985i	0.375718	+	0.926734i	$0.377396\pi$
$318$		0			0
$319$	204.893	−	354.885i	0.642297	−	1.11249i
$320$	22.1493	+	12.7879i	0.0692165	+	0.0399622i
$321$		0			0
$322$	76.4708	+	132.451i	0.237487	+	0.411340i
$323$			649.549i			2.01099i
$324$		0			0
$325$	153.289			0.471660
$326$	14.5687	−	8.41126i	0.0446894	−	0.0258014i
$327$		0			0
$328$	20.9677	−	36.3170i	0.0639258	−	0.110723i
$329$	−68.2110	−	39.3816i	−0.207328	−	0.119701i
$330$		0			0
$331$	−68.1379	−	118.018i	−0.205855	−	0.356551i	0.744550	−	0.667567i	$-0.232664\pi$
$331$	−68.1379	−	118.018i	−0.205855	−	0.356551i	−0.950405	+	0.311016i	$0.899331\pi$
$332$		−	230.718i		−	0.694935i
$333$		0			0
$334$	−197.133			−0.590219
$335$	−256.254	+	147.948i	−0.764937	+	0.441637i
$336$		0			0
$337$	−121.241	+	209.996i	−0.359767	+	0.623135i	−0.987922	−	0.154954i	$-0.950477\pi$
$337$	−121.241	+	209.996i	−0.359767	+	0.623135i	0.628155	+	0.778088i	$0.283810\pi$
$338$	75.2296	+	43.4338i	0.222573	+	0.128502i
$339$		0			0
$340$	78.0865	+	135.250i	0.229666	+	0.397793i
$341$		−	51.9345i		−	0.152301i
$342$		0			0
$343$	18.5203			0.0539949
$344$	70.8166	−	40.8860i	0.205862	−	0.118855i
$345$		0			0
$346$	162.208	−	280.952i	0.468808	−	0.811999i
$347$	−558.304	−	322.337i	−1.60894	−	0.928924i	−0.989607	−	0.143800i	$-0.954068\pi$
$347$	−558.304	−	322.337i	−1.60894	−	0.928924i	−0.619338	−	0.785125i	$-0.712599\pi$
$348$		0			0
$349$	−302.915	−	524.665i	−0.867952	−	1.50334i	−0.864086	−	0.503345i	$-0.832103\pi$
$349$	−302.915	−	524.665i	−0.867952	−	1.50334i	−0.00386633	−	0.999993i	$-0.501231\pi$
$350$			55.2993i			0.157998i
$351$		0			0
$352$	−106.663			−0.303021
$353$	−141.198	+	81.5210i	−0.399996	+	0.230938i	−0.686482	−	0.727147i	$-0.740846\pi$
$353$	−141.198	+	81.5210i	−0.399996	+	0.230938i	0.286487	+	0.958084i	$0.407513\pi$
$354$		0			0
$355$	130.917	−	226.755i	0.368780	−	0.638745i
$356$	23.7890	+	13.7346i	0.0668231	+	0.0385803i
$357$		0			0
$358$	−143.218	−	248.062i	−0.400051	−	0.692909i
$359$			227.325i			0.633217i	0.948556	+	0.316608i	$0.102544\pi$
$359$			227.325i			0.633217i	−0.948556	+	0.316608i	$0.897456\pi$
$360$		0			0
$361$	346.212			0.959036
$362$	5.14888	−	2.97270i	0.0142234	−	0.00821189i
$363$		0			0
$364$	27.4413	−	47.5298i	0.0753883	−	0.130576i
$365$	225.064	+	129.941i	0.616614	+	0.356002i
$366$		0			0
$367$	337.753	+	585.006i	0.920308	+	1.59402i	0.798938	+	0.601413i	$0.205395\pi$
$367$	337.753	+	585.006i	0.920308	+	1.59402i	0.121370	+	0.992607i	$0.461271\pi$
$368$			163.501i			0.444298i
$369$		0			0
$370$	−206.358			−0.557723
$371$	41.5642	−	23.9971i	0.112033	−	0.0646822i
$372$		0			0
$373$	−222.392	+	385.194i	−0.596225	+	1.03269i	0.397148	+	0.917755i	$0.370000\pi$
$373$	−222.392	+	385.194i	−0.596225	+	1.03269i	−0.993373	+	0.114937i	$0.963333\pi$
$374$	−564.057	−	325.658i	−1.50817	−	0.870745i
$375$		0			0
$376$	−42.1007	−	72.9206i	−0.111970	−	0.193938i
$377$			225.410i			0.597903i
$378$		0			0
$379$	586.241			1.54681			0.773405	−	0.633912i	$-0.218552\pi$
$379$	586.241			1.54681			0.773405	+	0.633912i	$0.218552\pi$
$380$	147.256	−	85.0186i	0.387517	−	0.223733i
$381$		0			0
$382$	192.269	−	333.020i	0.503323	−	0.871781i
$383$	130.125	+	75.1280i	0.339753	+	0.196157i	0.660163	−	0.751122i	$-0.270487\pi$
$383$	130.125	+	75.1280i	0.339753	+	0.196157i	−0.320410	+	0.947279i	$0.603821\pi$
$384$		0			0
$385$	79.7441	+	138.121i	0.207128	+	0.358755i
$386$		−	204.429i		−	0.529609i
$387$		0			0
$388$	223.171			0.575184
$389$	467.627	−	269.985i	1.20213	−	0.694048i	0.241099	−	0.970500i	$-0.422492\pi$
$389$	467.627	−	269.985i	1.20213	−	0.694048i	0.961028	+	0.276452i	$0.0891587\pi$
$390$		0			0
$391$	−499.193	+	864.628i	−1.27671	+	2.21132i
$392$	17.1464	+	9.89949i	0.0437409	+	0.0252538i
$393$		0			0
$394$	−37.5210	−	64.9882i	−0.0952309	−	0.164945i
$395$		−	270.505i		−	0.684824i
$396$		0			0
$397$	491.223			1.23734			0.618669	−	0.785652i	$-0.287672\pi$
$397$	491.223			1.23734			0.618669	+	0.785652i	$0.287672\pi$
$398$	−140.610	+	81.1813i	−0.353292	+	0.203973i
$399$		0			0
$400$	−29.5587	+	51.1972i	−0.0738968	+	0.127993i
$401$	187.085	+	108.014i	0.466546	+	0.269361i	0.714793	−	0.699336i	$-0.246521\pi$
$401$	187.085	+	108.014i	0.466546	+	0.269361i	−0.248247	+	0.968697i	$0.579854\pi$
$402$		0			0
$403$	14.2837	+	24.7401i	0.0354435	+	0.0613899i
$404$			289.863i			0.717483i
$405$		0			0
$406$	−81.3168			−0.200288
$407$	745.311	−	430.306i	1.83123	−	1.05726i
$408$		0			0
$409$	−52.8761	+	91.5842i	−0.129282	+	0.223922i	−0.923398	−	0.383843i	$-0.874600\pi$
$409$	−52.8761	+	91.5842i	−0.129282	+	0.223922i	0.794117	+	0.607765i	$0.207934\pi$
$410$	−58.0523	−	33.5165i	−0.141591	−	0.0817476i
$411$		0			0
$412$	−44.8103	−	77.6136i	−0.108763	−	0.188383i
$413$		−	141.789i		−	0.343314i
$414$		0			0
$415$	−368.800			−0.888675
$416$	50.8115	−	29.3360i	0.122143	−	0.0705193i
$417$		0			0
$418$	−354.569	+	614.131i	−0.848251	+	1.46921i
$419$	−11.5619	−	6.67526i	−0.0275940	−	0.0159314i	0.486140	−	0.873881i	$-0.338405\pi$
$419$	−11.5619	−	6.67526i	−0.0275940	−	0.0159314i	−0.513734	+	0.857950i	$0.671738\pi$
$420$		0			0
$421$	237.620	+	411.570i	0.564419	+	0.977602i	0.997104	+	0.0760565i	$0.0242329\pi$
$421$	237.620	+	411.570i	0.564419	+	0.977602i	−0.432685	+	0.901545i	$0.642434\pi$
$422$			510.473i			1.20965i
$423$		0			0
$424$	51.3080			0.121009
$425$	−312.625	+	180.494i	−0.735587	+	0.424692i
$426$		0			0
$427$	36.8652	−	63.8523i	0.0863353	−	0.149537i
$428$	−173.586	−	100.220i	−0.405574	−	0.234159i
$429$		0			0
$430$	−65.3557	−	113.199i	−0.151990	−	0.263254i
$431$			235.192i			0.545690i	0.962058	+	0.272845i	$0.0879646\pi$
$431$			235.192i			0.545690i	−0.962058	+	0.272845i	$0.912035\pi$
$432$		0			0
$433$	−21.0400			−0.0485913			−0.0242957	−	0.999705i	$-0.507734\pi$
$433$	−21.0400			−0.0485913			−0.0242957	+	0.999705i	$0.507734\pi$
$434$	−8.92504	+	5.15287i	−0.0205646	+	0.0118730i
$435$		0			0
$436$	−60.7449	+	105.213i	−0.139323	+	0.241315i
$437$	941.385	+	543.509i	2.15420	+	1.24373i
$438$		0			0
$439$	124.576	+	215.771i	0.283772	+	0.491507i	0.972311	−	0.233693i	$-0.0750809\pi$
$439$	124.576	+	215.771i	0.283772	+	0.491507i	−0.688539	+	0.725199i	$0.741748\pi$
$440$			170.500i			0.387500i
$441$		0			0
$442$	358.268			0.810561
$443$	−282.967	+	163.371i	−0.638751	+	0.368783i	−0.784133	−	0.620592i	$-0.786892\pi$
$443$	−282.967	+	163.371i	−0.638751	+	0.368783i	0.145382	+	0.989376i	$0.453559\pi$
$444$		0			0
$445$	21.9546	−	38.0264i	0.0493361	−	0.0854526i
$446$	253.347	+	146.270i	0.568043	+	0.327960i
$447$		0			0
$448$	10.5830	+	18.3303i	0.0236228	+	0.0409159i
$449$			510.632i			1.13727i	0.822592	+	0.568633i	$0.192527\pi$
$449$			510.632i			1.13727i	−0.822592	+	0.568633i	$0.807473\pi$
$450$		0			0
$451$	279.560			0.619868
$452$	−25.2255	+	14.5640i	−0.0558087	+	0.0322212i
$453$		0			0
$454$	−93.0842	+	161.227i	−0.205031	+	0.355125i
$455$	−75.9757	−	43.8646i	−0.166980	−	0.0964057i
$456$		0			0
$457$	90.5487	+	156.835i	0.198137	+	0.343184i	0.947924	−	0.318495i	$-0.103178\pi$
$457$	90.5487	+	156.835i	0.198137	+	0.343184i	−0.749787	+	0.661679i	$0.769844\pi$
$458$			334.501i			0.730351i
$459$		0			0
$460$	261.355			0.568163
$461$	−256.511	+	148.097i	−0.556423	+	0.321251i	−0.751709	−	0.659495i	$-0.770770\pi$
$461$	−256.511	+	148.097i	−0.556423	+	0.321251i	0.195286	+	0.980746i	$0.437437\pi$
$462$		0			0
$463$	−268.155	+	464.458i	−0.579169	+	1.00315i	0.416406	+	0.909179i	$0.363289\pi$
$463$	−268.155	+	464.458i	−0.579169	+	1.00315i	−0.995575	+	0.0939712i	$0.970044\pi$
$464$	−75.2847	−	43.4656i	−0.162252	−	0.0936760i
$465$		0			0
$466$	110.007	+	190.538i	0.236067	+	0.408880i
$467$		−	87.9169i		−	0.188259i	−0.995560	−	0.0941294i	$-0.969993\pi$
$467$		−	87.9169i		−	0.188259i	0.995560	−	0.0941294i	$-0.0300068\pi$
$468$		0			0
$469$	−244.878			−0.522128
$470$	−116.563	+	67.2975i	−0.248006	+	0.143186i
$471$		0			0
$472$	75.7892	−	131.271i	0.160570	−	0.278116i
$473$	472.096	+	272.565i	0.998089	+	0.576247i
$474$		0			0
$475$	196.517	+	340.378i	0.413720	+	0.716585i
$476$			129.246i			0.271524i
$477$		0			0
$478$	84.2818			0.176322
$479$	257.970	−	148.939i	0.538559	−	0.310937i	−0.205936	−	0.978565i	$-0.566024\pi$
$479$	257.970	−	148.939i	0.538559	−	0.310937i	0.744495	+	0.667628i	$0.232691\pi$
$480$		0			0
$481$	−236.697	+	409.971i	−0.492094	+	0.852331i
$482$	146.345	+	84.4926i	0.303621	+	0.175296i
$483$		0			0
$484$	−234.534	−	406.225i	−0.484575	−	0.839308i
$485$		−	356.737i		−	0.735539i
$486$		0			0
$487$	−441.814			−0.907215			−0.453608	−	0.891202i	$-0.649863\pi$
$487$	−441.814			−0.907215			−0.453608	+	0.891202i	$0.649863\pi$
$488$	68.2610	−	39.4105i	0.139879	−	0.0807593i
$489$		0			0
$490$	15.8242	−	27.4083i	0.0322943	−	0.0559354i
$491$	−288.654	−	166.655i	−0.587891	−	0.339419i	0.176372	−	0.984324i	$-0.443564\pi$
$491$	−288.654	−	166.655i	−0.587891	−	0.339419i	−0.764263	+	0.644905i	$0.776897\pi$
$492$		0			0
$493$	−265.413	−	459.710i	−0.538364	−	0.932474i
$494$		−	390.073i		−	0.789622i
$495$		0			0
$496$	−11.0173			−0.0222123
$497$	187.658	−	108.344i	0.377581	−	0.217996i
$498$		0			0
$499$	422.243	−	731.347i	0.846179	−	1.46563i	−0.0384141	−	0.999262i	$-0.512231\pi$
$499$	422.243	−	731.347i	0.846179	−	1.46563i	0.884593	−	0.466363i	$-0.154436\pi$
$500$	220.271	+	127.174i	0.440542	+	0.254347i
$501$		0			0
$502$	−341.658	−	591.769i	−0.680593	−	1.17882i
$503$			60.3243i			0.119929i	0.998201	+	0.0599645i	$0.0190987\pi$
$503$			60.3243i			0.119929i	−0.998201	+	0.0599645i	$0.980901\pi$
$504$		0			0
$505$	463.343			0.917510
$506$	−943.948	+	544.989i	−1.86551	+	1.07705i
$507$		0			0
$508$	221.852	−	384.259i	0.436717	−	0.756415i
$509$	−339.888	−	196.234i	−0.667756	−	0.385529i	0.127470	−	0.991842i	$-0.459314\pi$
$509$	−339.888	−	196.234i	−0.667756	−	0.385529i	−0.795226	+	0.606313i	$0.792648\pi$
$510$		0			0
$511$	107.536	+	186.259i	0.210443	+	0.364498i
$512$			22.6274i			0.0441942i
$513$		0			0
$514$	−223.447			−0.434721
$515$	−124.064	+	71.6286i	−0.240902	+	0.139085i
$516$		0			0
$517$	280.663	−	486.123i	0.542869	−	0.940276i
$518$	−147.898	−	85.3887i	−0.285517	−	0.164843i
$519$		0			0
$520$	−46.8932	−	81.2215i	−0.0901793	−	0.156195i
$521$		−	45.5654i		−	0.0874577i	−0.999043	−	0.0437288i	$-0.986076\pi$
$521$		−	45.5654i		−	0.0874577i	0.999043	−	0.0437288i	$-0.0139238\pi$
$522$		0			0
$523$	445.063			0.850980			0.425490	−	0.904963i	$-0.360102\pi$
$523$	445.063			0.850980			0.425490	+	0.904963i	$0.360102\pi$
$524$	−5.86524	+	3.38630i	−0.0111932	+	0.00646240i
$525$		0			0
$526$	200.025	−	346.454i	0.380276	−	0.658658i
$527$	−58.2616	−	33.6374i	−0.110553	−	0.0638280i
$528$		0			0
$529$	570.898	+	988.825i	1.07920	+	1.86923i
$530$		−	82.0151i		−	0.154745i
$531$		0			0
$532$	140.719			0.264510
$533$	−133.175	+	76.8884i	−0.249859	+	0.144256i
$534$		0			0
$535$	−160.200	+	277.475i	−0.299439	+	0.518644i
$536$	−226.713	−	130.893i	−0.422972	−	0.244203i
$537$		0			0
$538$	88.6964	+	153.627i	0.164863	+	0.285551i
$539$			131.989i			0.244878i
$540$		0			0
$541$	302.399			0.558964			0.279482	−	0.960151i	$-0.409837\pi$
$541$	302.399			0.558964			0.279482	+	0.960151i	$0.409837\pi$
$542$	512.023	−	295.617i	0.944692	−	0.545418i
$543$		0			0
$544$	−69.0847	+	119.658i	−0.126994	+	0.219960i
$545$	168.182	+	97.0999i	0.308591	+	0.178165i
$546$		0			0
$547$	−169.176	−	293.021i	−0.309279	−	0.535688i	0.668926	−	0.743329i	$-0.266754\pi$
$547$	−169.176	−	293.021i	−0.309279	−	0.535688i	−0.978205	+	0.207642i	$0.933421\pi$
$548$		−	43.1756i		−	0.0787877i
$549$		0			0
$550$	−394.104			−0.716554
$551$	−500.520	+	288.975i	−0.908385	+	0.524456i
$552$		0			0
$553$	111.932	−	193.873i	0.202409	−	0.350583i
$554$	−559.355	−	322.944i	−1.00967	−	0.582931i
$555$		0			0
$556$	−60.3228	−	104.482i	−0.108494	−	0.187917i
$557$			250.404i			0.449559i	0.974410	+	0.224780i	$0.0721662\pi$
$557$			250.404i			0.449559i	−0.974410	+	0.224780i	$0.927834\pi$
$558$		0			0
$559$	−299.858			−0.536418
$560$	29.3007	−	16.9168i	0.0523228	−	0.0302086i
$561$		0			0
$562$	−180.546	+	312.714i	−0.321256	+	0.556431i
$563$	721.716	+	416.683i	1.28191	+	0.740112i	0.977197	−	0.212332i	$-0.0681060\pi$
$563$	721.716	+	416.683i	1.28191	+	0.740112i	0.304713	+	0.952444i	$0.401439\pi$
$564$		0			0
$565$	23.2803	+	40.3227i	0.0412041	+	0.0713676i
$566$			11.6443i			0.0205730i
$567$		0			0
$568$	231.650			0.407834
$569$	205.714	−	118.769i	0.361536	−	0.208733i	−0.308218	−	0.951316i	$-0.599733\pi$
$569$	205.714	−	118.769i	0.361536	−	0.208733i	0.669754	+	0.742583i	$0.266399\pi$
$570$		0			0
$571$	−396.747	+	687.186i	−0.694829	+	1.20348i	0.275410	+	0.961327i	$0.411186\pi$
$571$	−396.747	+	687.186i	−0.694829	+	1.20348i	−0.970238	+	0.242152i	$0.922147\pi$
$572$	338.733	+	195.568i	0.592191	+	0.341901i
$573$		0			0
$574$	−27.7376	−	48.0429i	−0.0483233	−	0.0836985i
$575$			604.112i			1.05063i
$576$		0			0
$577$	−164.298			−0.284745			−0.142372	−	0.989813i	$-0.545473\pi$
$577$	−164.298			−0.284745			−0.142372	+	0.989813i	$0.545473\pi$
$578$	−376.715	+	217.497i	−0.651757	+	0.376292i
$579$		0			0
$580$	−69.4792	+	120.342i	−0.119792	+	0.207485i
$581$	−264.321	−	152.606i	−0.454942	−	0.262661i
$582$		0			0
$583$	171.021	+	296.218i	0.293347	+	0.508092i
$584$			229.923i			0.393703i
$585$		0			0
$586$	−151.321			−0.258227
$587$	796.533	−	459.879i	1.35696	−	0.783439i	0.367744	−	0.929927i	$-0.380130\pi$
$587$	796.533	−	459.879i	1.35696	−	0.783439i	0.989212	+	0.146488i	$0.0467969\pi$
$588$		0			0
$589$	−36.6235	+	63.4338i	−0.0621792	+	0.107697i
$590$	−209.835	−	121.148i	−0.355652	−	0.205336i
$591$		0			0
$592$	−91.2844	−	158.109i	−0.154197	−	0.267076i
$593$		−	1156.90i		−	1.95092i	−0.220178	−	0.975460i	$-0.570664\pi$
$593$		−	1156.90i		−	1.95092i	0.220178	−	0.975460i	$-0.429336\pi$
$594$		0			0
$595$	206.597			0.347222
$596$	−26.9126	+	15.5380i	−0.0451553	+	0.0260704i
$597$		0			0
$598$	299.780	−	519.235i	0.501305	−	0.868285i
$599$	784.703	+	453.049i	1.31002	+	0.756342i	0.982100	−	0.188363i	$-0.0603181\pi$
$599$	784.703	+	453.049i	1.31002	+	0.756342i	0.327923	+	0.944705i	$0.393651\pi$
$600$		0			0
$601$	−529.218	−	916.632i	−0.880562	−	1.52518i	−0.850718	−	0.525623i	$-0.823832\pi$
$601$	−529.218	−	916.632i	−0.880562	−	1.52518i	−0.0298440	−	0.999555i	$-0.509501\pi$
$602$		−	108.174i		−	0.179691i
$603$		0			0
$604$	−53.6468			−0.0888192
$605$	−649.345	+	374.900i	−1.07330	+	0.619669i
$606$		0			0
$607$	461.576	−	799.472i	0.760421	−	1.31709i	−0.182213	−	0.983259i	$-0.558326\pi$
$607$	461.576	−	799.472i	0.760421	−	1.31709i	0.942634	−	0.333829i	$-0.108341\pi$
$608$	130.281	+	75.2177i	0.214278	+	0.123713i
$609$		0			0
$610$	−62.9972	−	109.114i	−0.103274	−	0.178876i
$611$			308.767i			0.505347i
$612$		0			0
$613$	−613.947			−1.00154			−0.500772	−	0.865579i	$-0.666951\pi$
$613$	−613.947			−1.00154			−0.500772	+	0.865579i	$0.666951\pi$
$614$	662.341	−	382.403i	1.07873	−	0.622805i
$615$		0			0
$616$	−70.5512	+	122.198i	−0.114531	+	0.198374i
$617$	−460.026	−	265.596i	−0.745585	−	0.430464i	0.0785114	−	0.996913i	$-0.474983\pi$
$617$	−460.026	−	265.596i	−0.745585	−	0.430464i	−0.824096	+	0.566450i	$0.808317\pi$
$618$		0			0
$619$	−210.891	−	365.273i	−0.340695	−	0.590102i	0.643867	−	0.765138i	$-0.277329\pi$
$619$	−210.891	−	365.273i	−0.340695	−	0.590102i	−0.984562	+	0.175036i	$0.943996\pi$
$620$			17.6110i			0.0284049i
$621$		0			0
$622$	−370.559			−0.595754
$623$	31.4699	−	18.1692i	0.0505135	−	0.0291640i
$624$		0			0
$625$	18.5432	−	32.1177i	0.0296691	−	0.0513883i
$626$	−740.824	−	427.715i	−1.18342	−	0.683250i
$627$		0			0
$628$	106.134	+	183.829i	0.169003	+	0.292722i
$629$		−	1114.82i		−	1.77236i
$630$		0			0
$631$	−552.102			−0.874964			−0.437482	−	0.899227i	$-0.644130\pi$
$631$	−552.102			−0.874964			−0.437482	+	0.899227i	$0.644130\pi$
$632$	207.259	−	119.661i	0.327941	−	0.189337i
$633$		0			0
$634$	−61.8595	+	107.144i	−0.0975703	+	0.168997i
$635$	−614.233	−	354.628i	−0.967296	−	0.558469i
$636$		0			0
$637$	−36.3015	−	62.8760i	−0.0569882	−	0.0987064i
$638$		−	579.524i		−	0.908346i
$639$		0			0
$640$	36.1696			0.0565150
$641$	−391.148	+	225.829i	−0.610215	+	0.352308i	−0.773050	−	0.634345i	$-0.781270\pi$
$641$	−391.148	+	225.829i	−0.610215	+	0.352308i	0.162834	+	0.986653i	$0.447936\pi$
$642$		0			0
$643$	409.507	−	709.287i	0.636869	−	1.10309i	−0.349247	−	0.937031i	$-0.613563\pi$
$643$	409.507	−	709.287i	0.636869	−	1.10309i	0.986116	−	0.166059i	$-0.0531041\pi$
$644$	187.314	+	108.146i	0.290861	+	0.167929i
$645$		0			0
$646$	459.300	+	795.531i	0.710991	+	1.23147i
$647$		−	908.400i		−	1.40402i	−0.712168	−	0.702009i	$-0.752286\pi$
$647$		−	908.400i		−	1.40402i	0.712168	−	0.702009i	$-0.247714\pi$
$648$		0			0
$649$	1010.49			1.55700
$650$	187.740	−	108.392i	0.288831	−	0.166757i
$651$		0			0
$652$	11.8953	−	20.6033i	0.0182444	−	0.0316002i
$653$	199.877	+	115.399i	0.306090	+	0.176721i	0.645176	−	0.764034i	$-0.276784\pi$
$653$	199.877	+	115.399i	0.306090	+	0.176721i	−0.339085	+	0.940756i	$0.610118\pi$
$654$		0			0
$655$	5.41295	+	9.37551i	0.00826405	+	0.0143137i
$656$		−	59.3055i		−	0.0904047i
$657$		0			0
$658$	−111.388			−0.169283
$659$	39.7641	−	22.9578i	0.0603400	−	0.0348373i	−0.469526	−	0.882918i	$-0.655575\pi$
$659$	39.7641	−	22.9578i	0.0603400	−	0.0348373i	0.529867	+	0.848081i	$0.322242\pi$
$660$		0			0
$661$	−169.751	+	294.018i	−0.256810	+	0.444808i	−0.965386	−	0.260827i	$-0.916005\pi$
$661$	−169.751	+	294.018i	−0.256810	+	0.444808i	0.708576	+	0.705635i	$0.249338\pi$
$662$	−166.903	−	96.3615i	−0.252119	−	0.145561i
$663$		0			0
$664$	−163.143	−	282.571i	−0.245697	−	0.425559i
$665$		−	224.938i		−	0.338253i
$666$		0			0
$667$	−888.337			−1.33184
$668$	−241.438	+	139.394i	−0.361434	+	0.208674i
$669$		0			0
$670$	−209.230	+	362.398i	−0.312284	+	0.540892i
$671$	455.060	+	262.729i	0.678181	+	0.391548i
$672$		0			0
$673$	263.886	+	457.063i	0.392104	+	0.679143i	0.992727	−	0.120389i	$-0.0384142\pi$
$673$	263.886	+	457.063i	0.392104	+	0.679143i	−0.600623	+	0.799532i	$0.705081\pi$
$674$			342.923i			0.508787i
$675$		0			0
$676$	122.849			0.181730
$677$	−162.175	+	93.6317i	−0.239549	+	0.138304i	−0.614970	−	0.788551i	$-0.710832\pi$
$677$	−162.175	+	93.6317i	−0.239549	+	0.138304i	0.375420	+	0.926855i	$0.377498\pi$
$678$		0			0
$679$	147.614	−	255.675i	0.217399	−	0.376546i
$680$	191.272	+	110.431i	0.281282	+	0.162398i
$681$		0			0
$682$	−36.7232	−	63.6065i	−0.0538464	−	0.0932647i
$683$		−	575.317i		−	0.842338i	−0.906982	−	0.421169i	$-0.861620\pi$
$683$		−	575.317i		−	0.842338i	0.906982	−	0.421169i	$-0.138380\pi$
$684$		0			0
$685$	−69.0157			−0.100753
$686$	22.6826	−	13.0958i	0.0330650	−	0.0190901i
$687$		0			0
$688$	57.8215	−	100.150i	0.0840428	−	0.145566i
$689$	−162.940	−	94.0733i	−0.236487	−	0.136536i
$690$		0			0
$691$	−269.115	−	466.122i	−0.389458	−	0.674561i	0.602919	−	0.797803i	$-0.294004\pi$
$691$	−269.115	−	466.122i	−0.389458	−	0.674561i	−0.992377	+	0.123242i	$0.960671\pi$
$692$		−	458.792i		−	0.662994i
$693$		0			0
$694$	−911.706			−1.31370
$695$	−167.013	+	96.4251i	−0.240307	+	0.138741i
$696$		0			0
$697$	181.068	−	313.619i	0.259782	−	0.449955i
$698$	−741.988	−	428.387i	−1.06302	−	0.613735i
$699$		0			0
$700$	39.1025	+	67.7276i	0.0558607	+	0.0967536i
$701$		−	356.636i		−	0.508753i	−0.967105	−	0.254376i	$-0.918130\pi$
$701$		−	356.636i		−	0.508753i	0.967105	−	0.254376i	$-0.0818702\pi$
$702$		0			0
$703$	−1213.78			−1.72658
$704$	−130.636	+	75.4224i	−0.185562	+	0.107134i
$705$		0			0
$706$	−115.288	+	199.685i	−0.163298	+	0.282840i
$707$	332.080	+	191.727i	0.469703	+	0.271183i
$708$		0			0
$709$	200.807	+	347.809i	0.283226	+	0.490562i	0.972177	−	0.234245i	$-0.0752619\pi$
$709$	200.807	+	347.809i	0.283226	+	0.490562i	−0.688951	+	0.724808i	$0.741929\pi$
$710$		−	370.289i		−	0.521533i
$711$		0			0
$712$	38.8473			0.0545608
$713$	−97.5007	+	56.2920i	−0.136747	+	0.0789510i
$714$		0			0
$715$	312.612	−	541.460i	0.437220	−	0.757287i
$716$	−350.812	−	202.541i	−0.489961	−	0.282879i
$717$		0			0
$718$	160.743	+	278.415i	0.223876	+	0.387764i
$719$			916.413i			1.27457i	0.770630	+	0.637283i	$0.219942\pi$
$719$			916.413i			1.27457i	−0.770630	+	0.637283i	$0.780058\pi$
$720$		0			0
$721$	−118.557			−0.164434
$722$	424.021	−	244.809i	0.587287	−	0.339070i
$723$		0			0
$724$	4.20404	−	7.28161i	0.00580668	−	0.0100575i
$725$	−278.165	−	160.599i	−0.383676	−	0.221515i
$726$		0			0
$727$	−354.343	−	613.739i	−0.487404	−	0.844208i	0.512491	−	0.858692i	$-0.328723\pi$
$727$	−354.343	−	613.739i	−0.487404	−	0.844208i	−0.999895	+	0.0144843i	$0.995389\pi$
$728$		−	77.6158i		−	0.106615i
$729$		0			0
$730$	367.528			0.503463
$731$	611.542	−	353.074i	0.836583	−	0.483002i
$732$		0			0
$733$	−640.861	+	1110.00i	−0.874298	+	1.51433i	−0.0167898	+	0.999859i	$0.505345\pi$
$733$	−640.861	+	1110.00i	−0.874298	+	1.51433i	−0.857508	+	0.514470i	$0.827989\pi$
$734$	827.323	+	477.655i	1.12714	+	0.650756i
$735$		0			0
$736$	115.613	+	200.248i	0.157083	+	0.272076i
$737$		−	1745.19i		−	2.36796i
$738$		0			0
$739$	−1077.96			−1.45867			−0.729336	−	0.684155i	$-0.760171\pi$
$739$	−1077.96			−1.45867			−0.729336	+	0.684155i	$0.760171\pi$
$740$	−252.735	+	145.917i	−0.341534	+	0.197185i
$741$		0			0
$742$	33.9370	−	58.7807i	0.0457372	−	0.0792192i
$743$	229.237	+	132.350i	0.308529	+	0.178130i	0.646268	−	0.763110i	$-0.276329\pi$
$743$	229.237	+	132.350i	0.308529	+	0.178130i	−0.337739	+	0.941240i	$0.609662\pi$
$744$		0			0
$745$	24.8373	+	43.0194i	0.0333386	+	0.0577442i
$746$			629.019i			0.843189i
$747$		0			0
$748$	−921.101			−1.23142
$749$	−229.633	+	132.578i	−0.306585	+	0.177007i
$750$		0			0
$751$	18.2733	−	31.6504i	0.0243320	−	0.0421443i	−0.853603	−	0.520924i	$-0.825587\pi$
$751$	18.2733	−	31.6504i	0.0243320	−	0.0421443i	0.877935	+	0.478780i	$0.158921\pi$
$752$	−103.125	−	59.5395i	−0.137135	−	0.0791748i
$753$		0			0
$754$	159.389	+	276.069i	0.211391	+	0.366140i
$755$			85.7536i			0.113581i
$756$		0			0
$757$	299.469			0.395600			0.197800	−	0.980242i	$-0.436620\pi$
$757$	299.469			0.395600			0.197800	+	0.980242i	$0.436620\pi$
$758$	717.996	−	414.535i	0.947224	−	0.546880i
$759$		0			0
$760$	120.234	−	208.252i	0.158203	−	0.274016i
$761$	437.315	+	252.484i	0.574658	+	0.331779i	0.759008	−	0.651082i	$-0.225685\pi$
$761$	437.315	+	252.484i	0.574658	+	0.331779i	−0.184350	+	0.982861i	$0.559018\pi$
$762$		0			0
$763$	80.3579	+	139.184i	0.105318	+	0.182417i
$764$		−	543.820i		−	0.711806i
$765$		0			0
$766$	212.494			0.277407
$767$	−481.371	+	277.919i	−0.627602	+	0.362346i
$768$		0			0
$769$	376.324	−	651.812i	0.489368	−	0.847610i	−0.510558	−	0.859844i	$-0.670561\pi$
$769$	376.324	−	651.812i	0.489368	−	0.847610i	0.999925	+	0.0122340i	$0.00389431\pi$
$770$	195.332	+	112.775i	0.253678	+	0.146461i
$771$		0			0
$772$	−144.553	−	250.374i	−0.187245	−	0.324318i
$773$			4.76817i			0.00616840i	0.999995	+	0.00308420i	$0.000981732\pi$
$773$			4.76817i			0.00616840i	−0.999995	+	0.00308420i	$0.999018\pi$
$774$		0			0
$775$	−40.7072			−0.0525254
$776$	273.328	−	157.806i	0.352227	−	0.203358i
$777$		0			0
$778$	381.816	−	661.325i	0.490766	−	0.850032i
$779$	−341.460	−	197.142i	−0.438332	−	0.253071i
$780$		0			0
$781$	772.142	+	1337.39i	0.988658	+	1.71241i
$782$			1411.93i			1.80554i
$783$		0			0
$784$	28.0000			0.0357143
$785$	293.849	−	169.654i	0.374329	−	0.216119i
$786$		0			0
$787$	424.871	−	735.899i	0.539862	−	0.935068i	−0.459049	−	0.888411i	$-0.651810\pi$
$787$	424.871	−	735.899i	0.539862	−	0.935068i	0.998911	−	0.0466572i	$-0.0148569\pi$
$788$	−91.9072	−	53.0627i	−0.116634	−	0.0673384i
$789$		0			0
$790$	−191.276	−	331.300i	−0.242122	−	0.419367i
$791$			38.5327i			0.0487139i
$792$		0			0
$793$	−289.037			−0.364485
$794$	601.623	−	347.347i	0.757711	−	0.437465i
$795$		0			0
$796$	−114.808	+	198.853i	−0.144231	+	0.249815i
$797$	166.295	+	96.0104i	0.208651	+	0.120465i	0.600684	−	0.799486i	$-0.294895\pi$
$797$	166.295	+	96.0104i	0.208651	+	0.120465i	−0.392033	+	0.919951i	$0.628228\pi$
$798$		0			0
$799$	−363.565	−	629.712i	−0.455025	−	0.788126i
$800$			83.6047i			0.104506i
$801$		0			0
$802$	305.508			0.380933
$803$	−1327.42	+	766.386i	−1.65307	+	0.954403i
$804$		0			0
$805$	172.870	−	299.420i	0.214745	−	0.371950i
$806$	34.9878	+	20.2002i	0.0434092	+	0.0250623i
$807$		0			0
$808$	204.964	+	355.009i	0.253669	+	0.439367i
$809$			1015.55i			1.25532i	0.778488	+	0.627660i	$0.215987\pi$
$809$			1015.55i			1.25532i	−0.778488	+	0.627660i	$0.784013\pi$
$810$		0			0
$811$	481.576			0.593805			0.296902	−	0.954908i	$-0.404046\pi$
$811$	481.576			0.593805			0.296902	+	0.954908i	$0.404046\pi$
$812$	−99.5923	+	57.4996i	−0.122651	+	0.0708124i
$813$		0			0
$814$	608.544	−	1054.03i	0.747597	−	1.29488i
$815$	−32.9341	−	19.0145i	−0.0404099	−	0.0233307i
$816$		0			0
$817$	−384.418	−	665.832i	−0.470524	−	0.814972i
$818$			149.556i			0.182832i
$819$		0			0
$820$	−94.7990			−0.115609
$821$	−415.980	+	240.166i	−0.506675	+	0.292529i	−0.731466	−	0.681878i	$-0.761164\pi$
$821$	−415.980	+	240.166i	−0.506675	+	0.292529i	0.224791	+	0.974407i	$0.427830\pi$
$822$		0			0
$823$	−294.402	+	509.919i	−0.357718	+	0.619586i	−0.987579	−	0.157122i	$-0.949779\pi$
$823$	−294.402	+	509.919i	−0.357718	+	0.619586i	0.629861	+	0.776708i	$0.283112\pi$
$824$	−109.762	−	63.3713i	−0.133207	−	0.0769069i
$825$		0			0
$826$	−100.260	−	173.655i	−0.121380	−	0.210236i
$827$		−	280.409i		−	0.339068i	−0.985524	−	0.169534i	$-0.945774\pi$
$827$		−	280.409i		−	0.339068i	0.985524	−	0.169534i	$-0.0542262\pi$
$828$		0			0
$829$	190.758			0.230106			0.115053	−	0.993359i	$-0.463296\pi$
$829$	190.758			0.230106			0.115053	+	0.993359i	$0.463296\pi$
$830$	−451.686	+	260.781i	−0.544200	+	0.314194i
$831$		0			0
$832$	41.4874	−	71.8583i	0.0498647	−	0.0863681i
$833$	148.069	+	85.4879i	0.177754	+	0.102627i
$834$		0			0
$835$	222.820	+	385.935i	0.266850	+	0.462198i
$836$			1002.87i			1.19961i
$837$		0			0
$838$	−18.8805			−0.0225304
$839$	929.306	−	536.535i	1.10763	−	0.639493i	0.169419	−	0.985544i	$-0.445811\pi$
$839$	929.306	−	536.535i	1.10763	−	0.639493i	0.938216	+	0.346051i	$0.112478\pi$
$840$		0			0
$841$	−184.342	+	319.290i	−0.219194	+	0.379655i
$842$	582.048	+	336.046i	0.691269	+	0.399104i
$843$		0			0
$844$	360.959	+	625.200i	0.427677	+	0.740758i
$845$		−	196.373i		−	0.232394i
$846$		0			0
$847$	−620.519			−0.732608
$848$	62.8392	−	36.2802i	0.0741028	−	0.0427833i
$849$		0			0
$850$	−255.257	+	442.118i	−0.300302	+	0.520139i
$851$	−1615.69	−	932.821i	−1.89858	−	1.09615i
$852$		0			0
$853$	504.339	+	873.540i	0.591253	+	1.02408i	0.994064	+	0.108797i	$0.0346998\pi$
$853$	504.339	+	873.540i	0.591253	+	1.02408i	−0.402811	+	0.915283i	$0.631967\pi$
$854$		−	104.270i		−	0.122097i
$855$		0			0
$856$	−283.465			−0.331150
$857$	261.331	−	150.880i	0.304937	−	0.176055i	−0.339722	−	0.940526i	$-0.610333\pi$
$857$	261.331	−	150.880i	0.304937	−	0.176055i	0.644659	+	0.764471i	$0.277000\pi$
$858$		0			0
$859$	−544.251	+	942.670i	−0.633587	+	1.09740i	0.353226	+	0.935538i	$0.385085\pi$
$859$	−544.251	+	942.670i	−0.633587	+	1.09740i	−0.986813	+	0.161866i	$0.948249\pi$
$860$	−160.088	−	92.4269i	−0.186149	−	0.107473i
$861$		0			0
$862$	166.306	+	288.051i	0.192930	+	0.334165i
$863$			1120.50i			1.29838i	0.760627	+	0.649189i	$0.224892\pi$
$863$			1120.50i			1.29838i	−0.760627	+	0.649189i	$0.775108\pi$
$864$		0			0
$865$	−733.373			−0.847830
$866$	−25.7687	+	14.8776i	−0.0297560	+	0.0171796i
$867$		0			0
$868$	−7.28726	+	12.6219i	−0.00839546	+	0.0145414i
$869$	1381.68	+	797.715i	1.58997	+	0.917969i
$870$		0			0
$871$	479.985	+	831.358i	0.551073	+	0.954486i
$872$			171.812i			0.197033i
$873$		0			0
$874$	1537.27			1.75890
$875$	291.391	−	168.235i	0.333019	−	0.192268i
$876$		0			0
$877$	−22.9814	+	39.8049i	−0.0262046	+	0.0453876i	−0.878830	−	0.477134i	$-0.841675\pi$
$877$	−22.9814	+	39.8049i	−0.0262046	+	0.0453876i	0.852626	+	0.522522i	$0.175009\pi$
$878$	305.147	+	176.177i	0.347548	+	0.200657i
$879$		0			0
$880$	120.562	+	208.819i	0.137002	+	0.237294i
$881$			881.775i			1.00088i	0.865771	+	0.500440i	$0.166828\pi$
$881$			881.775i			1.00088i	−0.865771	+	0.500440i	$0.833172\pi$
$882$		0			0
$883$	216.488			0.245174			0.122587	−	0.992458i	$-0.460881\pi$
$883$	216.488			0.245174			0.122587	+	0.992458i	$0.460881\pi$
$884$	438.787	−	253.334i	0.496365	−	0.286577i
$885$		0			0
$886$	−231.041	+	400.175i	−0.260769	+	0.451665i
$887$	−123.029	−	71.0306i	−0.138702	−	0.0800796i	0.429043	−	0.903284i	$-0.358851\pi$
$887$	−123.029	−	71.0306i	−0.138702	−	0.0800796i	−0.567745	+	0.823204i	$0.692184\pi$
$888$		0			0
$889$	−293.483	−	508.327i	−0.330127	−	0.571796i
$890$		−	62.0969i		−	0.0697718i
$891$		0			0
$892$	413.714			0.463805
$893$	−685.615	+	395.840i	−0.767766	+	0.443270i
$894$		0			0
$895$	−323.760	+	560.768i	−0.361743	+	0.626557i
$896$	25.9230	+	14.9666i	0.0289319	+	0.0167038i
$897$		0			0
$898$	361.071	+	625.394i	0.402084	+	0.696430i
$899$		−	59.8593i		−	0.0665843i
$900$		0			0
$901$	443.074			0.491758
$902$	342.390	−	197.679i	0.379590	−	0.219156i
$903$		0			0
$904$	−20.5966	+	35.6743i	−0.0227838	+	0.0394627i
$905$	−11.6396	−	6.72010i	−0.0128614	−	0.00742552i
$906$		0			0
$907$	225.530	+	390.629i	0.248655	+	0.430683i	0.963153	−	0.268955i	$-0.0866782\pi$
$907$	225.530	+	390.629i	0.248655	+	0.430683i	−0.714498	+	0.699637i	$0.753345\pi$
$908$			263.282i			0.289958i
$909$		0			0
$910$	−124.068			−0.136338
$911$	−856.971	+	494.772i	−0.940692	+	0.543109i	−0.890177	−	0.455614i	$-0.849420\pi$
$911$	−856.971	+	494.772i	−0.940692	+	0.543109i	−0.0505152	+	0.998723i	$0.516086\pi$
$912$		0			0
$913$	1087.58	−	1883.75i	1.19122	−	2.06325i
$914$	221.798	+	128.055i	0.242667	+	0.140104i
$915$		0			0
$916$	236.528	+	409.678i	0.258218	+	0.447247i
$917$			8.95930i			0.00977023i
$918$		0			0
$919$	942.500			1.02557			0.512786	−	0.858517i	$-0.328614\pi$
$919$	942.500			1.02557			0.512786	+	0.858517i	$0.328614\pi$
$920$	320.093	−	184.806i	0.347927	−	0.200876i
$921$		0			0
$922$	−209.440	+	362.761i	−0.227159	+	0.393451i
$923$	−735.654	−	424.730i	−0.797025	−	0.460162i
$924$		0			0
$925$	−337.281	−	584.188i	−0.364628	−	0.631555i
$926$			758.457i			0.819068i
$927$		0			0
$928$	−122.939			−0.132478
$929$	545.923	−	315.189i	0.587646	−	0.339278i	−0.176520	−	0.984297i	$-0.556484\pi$
$929$	545.923	−	315.189i	0.587646	−	0.339278i	0.764166	+	0.645019i	$0.223151\pi$
$930$		0			0
$931$	93.0771	−	161.214i	0.0999754	−	0.173162i
$932$	269.462	+	155.574i	0.289122	+	0.166925i
$933$		0			0
$934$	−62.1666	−	107.676i	−0.0665596	−	0.115285i
$935$			1472.37i			1.57473i
$936$		0			0
$937$	236.083			0.251956			0.125978	−	0.992033i	$-0.459793\pi$
$937$	236.083			0.251956			0.125978	+	0.992033i	$0.459793\pi$
$938$	−299.913	+	173.155i	−0.319737	+	0.184600i
$939$		0			0
$940$	−95.1730	+	164.844i	−0.101248	+	0.175366i
$941$	−721.947	−	416.816i	−0.767213	−	0.442950i	0.0646667	−	0.997907i	$-0.479402\pi$
$941$	−721.947	−	416.816i	−0.767213	−	0.442950i	−0.831879	+	0.554956i	$0.812735\pi$
$942$		0			0
$943$	−303.017	−	524.840i	−0.321333	−	0.556564i
$944$		−	214.364i		−	0.227081i
$945$		0			0
$946$	770.930			0.814936
$947$	−680.541	+	392.911i	−0.718628	+	0.414900i	−0.814248	−	0.580518i	$-0.802850\pi$
$947$	−680.541	+	392.911i	−0.718628	+	0.414900i	0.0956192	+	0.995418i	$0.469517\pi$
$948$		0			0
$949$	421.564	−	730.170i	0.444219	−	0.769409i
$950$	481.367	+	277.917i	0.506702	+	0.292544i
$951$		0			0
$952$	91.3904	+	158.293i	0.0959984	+	0.166274i
$953$		−	1096.70i		−	1.15079i	−0.817875	−	0.575396i	$-0.804848\pi$
$953$		−	1096.70i		−	1.15079i	0.817875	−	0.575396i	$-0.195152\pi$
$954$		0			0
$955$	−869.289			−0.910250
$956$	103.224	−	59.5962i	0.107975	−	0.0623392i
$957$		0			0
$958$	210.631	−	364.824i	0.219866	−	0.380818i
$959$	−49.4639	−	28.5580i	−0.0515786	−	0.0297789i
$960$		0			0
$961$	476.707	+	825.680i	0.496053	+	0.859189i
$962$			669.480i			0.695925i
$963$		0			0
$964$	238.981			0.247906
$965$	−400.219	+	231.066i	−0.414735	+	0.239447i
$966$		0			0
$967$	−647.975	+	1122.33i	−0.670088	+	1.16063i	0.307791	+	0.951454i	$0.400410\pi$
$967$	−647.975	+	1122.33i	−0.670088	+	1.16063i	−0.977879	+	0.209172i	$0.932923\pi$
$968$	−574.489	−	331.681i	−0.593480	−	0.342646i
$969$		0			0
$970$	−252.251	−	436.911i	−0.260052	−	0.450424i
$971$			830.704i			0.855514i	0.903894	+	0.427757i	$0.140696\pi$
$971$			830.704i			0.855514i	−0.903894	+	0.427757i	$0.859304\pi$
$972$		0			0
$973$	−159.599			−0.164028
$974$	−541.109	+	312.410i	−0.555554	+	0.320749i
$975$		0			0
$976$	55.7349	−	96.5357i	0.0571054	−	0.0989095i
$977$	444.638	+	256.712i	0.455105	+	0.262755i	0.709984	−	0.704218i	$-0.248702\pi$
$977$	444.638	+	256.712i	0.455105	+	0.262755i	−0.254879	+	0.966973i	$0.582036\pi$
$978$		0			0
$979$	129.487	+	224.278i	0.132265	+	0.229089i
$980$		−	44.7576i		−	0.0456710i
$981$		0			0
$982$	−471.371			−0.480011
$983$	−765.485	+	441.953i	−0.778723	+	0.449596i	−0.835978	−	0.548764i	$-0.815099\pi$
$983$	−765.485	+	441.953i	−0.778723	+	0.449596i	0.0572543	+	0.998360i	$0.481765\pi$
$984$		0			0
$985$	−84.8199	+	146.912i	−0.0861116	+	0.149150i
$986$	−650.128	−	375.351i	−0.659359	−	0.380681i
$987$		0			0
$988$	−275.823	−	477.740i	−0.279173	−	0.483543i
$989$		−	1181.74i		−	1.19488i
$990$		0			0
$991$	1014.86			1.02407			0.512036	−	0.858964i	$-0.328891\pi$
$991$	1014.86			1.02407			0.512036	+	0.858964i	$0.328891\pi$
$992$	−13.4934	+	7.79041i	−0.0136022	+	0.00785324i
$993$		0			0
$994$	153.222	−	265.388i	0.154147	−	0.266990i
$995$	317.863	+	183.519i	0.319461	+	0.184441i
$996$		0			0
$997$	−617.454	−	1069.46i	−0.619312	−	1.07268i	−0.989611	−	0.143768i	$-0.954078\pi$
$997$	−617.454	−	1069.46i	−0.619312	−	1.07268i	0.370299	−	0.928913i	$-0.379255\pi$
$998$		−	1194.28i		−	1.19668i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	378.3.q.a.71.9		24
3.2	odd	2		126.3.q.a.113.1	yes	24
9.2	odd	6	inner	378.3.q.a.197.9		24
9.4	even	3		1134.3.b.c.323.20		24
9.5	odd	6		1134.3.b.c.323.5		24
9.7	even	3		126.3.q.a.29.1	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
126.3.q.a.29.1	✓	24	9.7	even	3
126.3.q.a.113.1	yes	24	3.2	odd	2
378.3.q.a.71.9		24	1.1	even	1	trivial
378.3.q.a.197.9		24	9.2	odd	6	inner
1134.3.b.c.323.5		24	9.5	odd	6
1134.3.b.c.323.20		24	9.4	even	3

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:	\( N \)	\(=\)	\( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	378.q (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.22474 - 0.707107i) q^{2} +(1.00000 - 1.73205i) q^{4} +(-2.76866 - 1.59849i) q^{5} +(-1.32288 - 2.29129i) q^{7} -2.82843i q^{8} -4.52120 q^{10} +(16.3294 - 9.42781i) q^{11} +(-5.18593 + 8.98229i) q^{13} +(-3.24037 - 1.87083i) q^{14} +(-2.00000 - 3.46410i) q^{16} -24.4251i q^{17} -26.5935 q^{19} +(-5.53732 + 3.19697i) q^{20} +(13.3329 - 23.0933i) q^{22} +(-35.3991 - 20.4377i) q^{23} +(-7.38968 - 12.7993i) q^{25} +14.6680i q^{26} -5.29150 q^{28} +(18.8212 - 10.8664i) q^{29} +(1.37716 - 2.38532i) q^{31} +(-4.89898 - 2.82843i) q^{32} +(-17.2712 - 29.9145i) q^{34} +8.45840i q^{35} +45.6422 q^{37} +(-32.5702 + 18.8044i) q^{38} +(-4.52120 + 7.83095i) q^{40} +(12.8400 + 7.41318i) q^{41} +(14.4554 + 25.0374i) q^{43} -37.7112i q^{44} -57.8065 q^{46} +(25.7813 - 14.8849i) q^{47} +(-3.50000 + 6.06218i) q^{49} +(-18.1009 - 10.4506i) q^{50} +(10.3719 + 17.9646i) q^{52} +18.1401i q^{53} -60.2809 q^{55} +(-6.48074 + 3.74166i) q^{56} +(15.3674 - 26.6172i) q^{58} +(46.4112 + 26.7955i) q^{59} +(13.9337 + 24.1339i) q^{61} -3.89520i q^{62} -8.00000 q^{64} +(28.7161 - 16.5793i) q^{65} +(46.2776 - 80.1552i) q^{67} +(-42.3056 - 24.4251i) q^{68} +(5.98099 + 10.3594i) q^{70} +81.9005i q^{71} -81.2899 q^{73} +(55.9000 - 32.2739i) q^{74} +(-26.5935 + 46.0612i) q^{76} +(-43.2036 - 24.9436i) q^{77} +(42.3065 + 73.2770i) q^{79} +12.7879i q^{80} +20.9677 q^{82} +(99.9040 - 57.6796i) q^{83} +(-39.0432 + 67.6249i) q^{85} +(35.4083 + 20.4430i) q^{86} +(-26.6659 - 46.1866i) q^{88} +13.7346i q^{89} +27.4413 q^{91} +(-70.7982 + 40.8754i) q^{92} +(21.0504 - 36.4603i) q^{94} +(73.6282 + 42.5093i) q^{95} +(55.7929 + 96.6361i) q^{97} +9.89949i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 24 q^{4} - 36 q^{5} - 48 q^{16} + 24 q^{19} - 72 q^{20} + 24 q^{22} + 72 q^{23} + 72 q^{25} + 108 q^{29} - 60 q^{31} - 48 q^{34} - 168 q^{37} - 144 q^{38} - 108 q^{41} + 60 q^{43} + 324 q^{47} - 84 q^{49}+ \cdots + 48 q^{97}+O(q^{100}) \) 24 * q + 24 * q^4 - 36 * q^5 - 48 * q^16 + 24 * q^19 - 72 * q^20 + 24 * q^22 + 72 * q^23 + 72 * q^25 + 108 * q^29 - 60 * q^31 - 48 * q^34 - 168 * q^37 - 144 * q^38 - 108 * q^41 + 60 * q^43 + 324 * q^47 - 84 * q^49 - 144 * q^50 + 264 * q^55 + 432 * q^59 - 192 * q^64 + 180 * q^65 + 72 * q^67 - 72 * q^68 + 24 * q^73 - 288 * q^74 + 24 * q^76 + 12 * q^79 - 288 * q^82 - 756 * q^83 - 156 * q^85 - 360 * q^86 - 48 * q^88 - 168 * q^91 + 144 * q^92 + 936 * q^95 + 48 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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