LMFDB - Embedded newform 432.2.be.b.47.1

Newspace parameters

Level:	$ N $	$=$	$ 432 = 2^{4} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	432.be (of order $18$, degree $6$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$3.44953736732$
Analytic rank:	$0$
Dimension:	$36$
Relative dimension:	$6$ over $\Q(\zeta_{18})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			47.1
Character	$\chi$	$=$	432.47
Dual form			432.2.be.b.239.1

$q$-expansion

$f(q)$	$=$	$q+(-1.65284 - 0.517813i) q^{3} +(-2.52917 + 0.445961i) q^{5} +(-1.40789 + 1.67786i) q^{7} +(2.46374 + 1.71172i) q^{9} +O(q^{10})$	\(q+(-1.65284 - 0.517813i) q^{3} +(-2.52917 + 0.445961i) q^{5} +(-1.40789 + 1.67786i) q^{7} +(2.46374 + 1.71172i) q^{9} +(0.751810 - 4.26373i) q^{11} +(5.33154 + 1.94052i) q^{13} +(4.41123 + 0.572537i) q^{15} +(5.53709 - 3.19684i) q^{17} +(-2.85493 - 1.64830i) q^{19} +(3.19584 - 2.04421i) q^{21} +(3.31503 - 2.78164i) q^{23} +(1.49936 - 0.545724i) q^{25} +(-3.18581 - 4.10495i) q^{27} +(0.131941 + 0.362506i) q^{29} +(4.37197 + 5.21032i) q^{31} +(-3.45043 + 6.65795i) q^{33} +(2.81254 - 4.87147i) q^{35} +(-3.81950 - 6.61557i) q^{37} +(-7.80734 - 5.96811i) q^{39} +(-0.138464 + 0.380428i) q^{41} +(9.35445 + 1.64944i) q^{43} +(-6.99458 - 3.23050i) q^{45} +(6.74409 + 5.65897i) q^{47} +(0.382480 + 2.16915i) q^{49} +(-10.8073 + 2.41668i) q^{51} -7.00741i q^{53} +11.1190i q^{55} +(3.86523 + 4.20269i) q^{57} +(0.296830 + 1.68341i) q^{59} +(8.66165 + 7.26798i) q^{61} +(-6.34071 + 1.72389i) q^{63} +(-14.3498 - 2.53025i) q^{65} +(-0.683922 + 1.87906i) q^{67} +(-6.91957 + 2.88103i) q^{69} +(-1.85419 - 3.21154i) q^{71} +(5.37828 - 9.31546i) q^{73} +(-2.76079 + 0.125602i) q^{75} +(6.09548 + 7.26431i) q^{77} +(-2.34025 - 6.42979i) q^{79} +(3.14002 + 8.43447i) q^{81} +(1.31616 - 0.479045i) q^{83} +(-12.5786 + 10.5547i) q^{85} +(-0.0303672 - 0.667485i) q^{87} +(-6.85289 - 3.95652i) q^{89} +(-10.7622 + 6.21354i) q^{91} +(-4.52819 - 10.8757i) q^{93} +(7.95569 + 2.89564i) q^{95} +(2.09088 - 11.8580i) q^{97} +(9.15058 - 9.21782i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 36 q + 3 q^{5} + 6 q^{9}+O(q^{10}) $ 36 * q + 3 * q^5 + 6 * q^9	$ 36 q + 3 q^{5} + 6 q^{9} - 18 q^{11} + 9 q^{15} + 18 q^{21} - 9 q^{25} + 30 q^{29} + 27 q^{31} + 27 q^{33} + 27 q^{35} - 45 q^{39} + 18 q^{41} + 27 q^{45} + 45 q^{47} - 63 q^{51} - 9 q^{57} + 54 q^{59} - 63 q^{63} - 57 q^{65} - 63 q^{69} + 36 q^{71} + 9 q^{73} - 45 q^{75} - 81 q^{77} - 54 q^{81} - 27 q^{83} - 36 q^{85} + 45 q^{87} - 63 q^{89} + 27 q^{91} - 63 q^{93} - 72 q^{95} + 99 q^{99}+O(q^{100}) $ 36 * q + 3 * q^5 + 6 * q^9 - 18 * q^11 + 9 * q^15 + 18 * q^21 - 9 * q^25 + 30 * q^29 + 27 * q^31 + 27 * q^33 + 27 * q^35 - 45 * q^39 + 18 * q^41 + 27 * q^45 + 45 * q^47 - 63 * q^51 - 9 * q^57 + 54 * q^59 - 63 * q^63 - 57 * q^65 - 63 * q^69 + 36 * q^71 + 9 * q^73 - 45 * q^75 - 81 * q^77 - 54 * q^81 - 27 * q^83 - 36 * q^85 + 45 * q^87 - 63 * q^89 + 27 * q^91 - 63 * q^93 - 72 * q^95 + 99 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/432\mathbb{Z}\right)^\times$.

$n$	$271$	$325$	$353$
$\chi(n)$	$-1$	$1$	$e\left(\frac{7}{18}\right)$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	−1.65284	−	0.517813i	−0.954266	−	0.298960i
$3$	−1.65284	−	0.517813i	−0.954266	−	0.298960i
$4$		0			0
$5$	−2.52917	+	0.445961i	−1.13108	+	0.199440i	−0.707700	−	0.706513i	$-0.750267\pi$
$5$	−2.52917	+	0.445961i	−1.13108	+	0.199440i	−0.423379	+	0.905952i	$0.639156\pi$
$6$		0			0
$7$	−1.40789	+	1.67786i	−0.532134	+	0.634172i	−0.963405	−	0.268050i	$-0.913621\pi$
$7$	−1.40789	+	1.67786i	−0.532134	+	0.634172i	0.431271	+	0.902222i	$0.358065\pi$
$8$		0			0
$9$	2.46374	+	1.71172i	0.821246	+	0.570574i
$10$		0			0
$11$	0.751810	−	4.26373i	0.226679	−	1.28556i	−0.632769	−	0.774340i	$-0.718082\pi$
$11$	0.751810	−	4.26373i	0.226679	−	1.28556i	0.859449	−	0.511222i	$-0.170807\pi$
$12$		0			0
$13$	5.33154	+	1.94052i	1.47870	+	0.538204i	0.950449	−	0.310882i	$-0.100624\pi$
$13$	5.33154	+	1.94052i	1.47870	+	0.538204i	0.528255	+	0.849086i	$0.322846\pi$
$14$		0			0
$15$	4.41123	+	0.572537i	1.13898	+	0.147828i
$16$		0			0
$17$	5.53709	−	3.19684i	1.34294	−	0.775347i	0.355703	−	0.934599i	$-0.384241\pi$
$17$	5.53709	−	3.19684i	1.34294	−	0.775347i	0.987238	+	0.159252i	$0.0509081\pi$
$18$		0			0
$19$	−2.85493	−	1.64830i	−0.654967	−	0.378145i	0.135390	−	0.990792i	$-0.456771\pi$
$19$	−2.85493	−	1.64830i	−0.654967	−	0.378145i	−0.790357	+	0.612647i	$0.790105\pi$
$20$		0			0
$21$	3.19584	−	2.04421i	0.697389	−	0.446082i
$22$		0			0
$23$	3.31503	−	2.78164i	0.691232	−	0.580012i	−0.228033	−	0.973654i	$-0.573229\pi$
$23$	3.31503	−	2.78164i	0.691232	−	0.580012i	0.919264	+	0.393641i	$0.128785\pi$
$24$		0			0
$25$	1.49936	−	0.545724i	0.299873	−	0.109145i
$26$		0			0
$27$	−3.18581	−	4.10495i	−0.613109	−	0.789998i
$28$		0			0
$29$	0.131941	+	0.362506i	0.0245009	+	0.0673157i	0.951340	−	0.308143i	$-0.0997075\pi$
$29$	0.131941	+	0.362506i	0.0245009	+	0.0673157i	−0.926839	+	0.375459i	$0.877485\pi$
$30$		0			0
$31$	4.37197	+	5.21032i	0.785230	+	0.935800i	0.999157	−	0.0410491i	$-0.0130700\pi$
$31$	4.37197	+	5.21032i	0.785230	+	0.935800i	−0.213927	+	0.976850i	$0.568626\pi$
$32$		0			0
$33$	−3.45043	+	6.65795i	−0.600644	+	1.15900i
$34$		0			0
$35$	2.81254	−	4.87147i	0.475407	−	0.823428i
$36$		0			0
$37$	−3.81950	−	6.61557i	−0.627922	−	1.08759i	−0.987968	−	0.154658i	$-0.950572\pi$
$37$	−3.81950	−	6.61557i	−0.627922	−	1.08759i	0.360046	−	0.932934i	$-0.382761\pi$
$38$		0			0
$39$	−7.80734	−	5.96811i	−1.25018	−	0.955662i
$40$		0			0
$41$	−0.138464	+	0.380428i	−0.0216245	+	0.0594128i	−0.950035	−	0.312143i	$-0.898953\pi$
$41$	−0.138464	+	0.380428i	−0.0216245	+	0.0594128i	0.928411	+	0.371556i	$0.121175\pi$
$42$		0			0
$43$	9.35445	+	1.64944i	1.42654	+	0.251538i	0.833003	−	0.553269i	$-0.186620\pi$
$43$	9.35445	+	1.64944i	1.42654	+	0.251538i	0.593538	+	0.804806i	$0.297731\pi$
$44$		0			0
$45$	−6.99458	−	3.23050i	−1.04269	−	0.481575i
$46$		0			0
$47$	6.74409	+	5.65897i	0.983727	+	0.825445i	0.984648	−	0.174554i	$-0.0558485\pi$
$47$	6.74409	+	5.65897i	0.983727	+	0.825445i	−0.000920318		1.00000i	$0.500293\pi$
$48$		0			0
$49$	0.382480	+	2.16915i	0.0546400	+	0.309879i
$50$		0			0
$51$	−10.8073	+	2.41668i	−1.51332	+	0.338402i
$52$		0			0
$53$		−	7.00741i		−	0.962542i	−0.876572	−	0.481271i	$-0.840175\pi$
$53$		−	7.00741i		−	0.962542i	0.876572	−	0.481271i	$-0.159825\pi$
$54$		0			0
$55$			11.1190i			1.49928i
$56$		0			0
$57$	3.86523	+	4.20269i	0.511962	+	0.556660i
$58$		0			0
$59$	0.296830	+	1.68341i	0.0386440	+	0.219161i	0.998014	−	0.0629893i	$-0.0200634\pi$
$59$	0.296830	+	1.68341i	0.0386440	+	0.219161i	−0.959370	+	0.282150i	$0.908952\pi$
$60$		0			0
$61$	8.66165	+	7.26798i	1.10901	+	0.930570i	0.997998	−	0.0632489i	$-0.0201462\pi$
$61$	8.66165	+	7.26798i	1.10901	+	0.930570i	0.111012	+	0.993819i	$0.464591\pi$
$62$		0			0
$63$	−6.34071	+	1.72389i	−0.798855	+	0.217190i
$64$		0			0
$65$	−14.3498	−	2.53025i	−1.77987	−	0.313839i
$66$		0			0
$67$	−0.683922	+	1.87906i	−0.0835543	+	0.229564i	−0.974433	−	0.224677i	$-0.927867\pi$
$67$	−0.683922	+	1.87906i	−0.0835543	+	0.229564i	0.890879	+	0.454241i	$0.150089\pi$
$68$		0			0
$69$	−6.91957	+	2.88103i	−0.833019	+	0.346835i
$70$		0			0
$71$	−1.85419	−	3.21154i	−0.220051	−	0.381140i	0.734772	−	0.678314i	$-0.237289\pi$
$71$	−1.85419	−	3.21154i	−0.220051	−	0.381140i	−0.954823	+	0.297174i	$0.903956\pi$
$72$		0			0
$73$	5.37828	−	9.31546i	0.629481	−	1.09029i	−0.358176	−	0.933654i	$-0.616601\pi$
$73$	5.37828	−	9.31546i	0.629481	−	1.09029i	0.987656	−	0.156638i	$-0.0500656\pi$
$74$		0			0
$75$	−2.76079	+	0.125602i	−0.318788	+	0.0145033i
$76$		0			0
$77$	6.09548	+	7.26431i	0.694644	+	0.827845i
$78$		0			0
$79$	−2.34025	−	6.42979i	−0.263299	−	0.723408i	−0.998940	−	0.0460362i	$-0.985341\pi$
$79$	−2.34025	−	6.42979i	−0.263299	−	0.723408i	0.735641	−	0.677372i	$-0.236881\pi$
$80$		0			0
$81$	3.14002	+	8.43447i	0.348891	+	0.937163i
$82$		0			0
$83$	1.31616	−	0.479045i	0.144468	−	0.0525820i	−0.268774	−	0.963203i	$-0.586619\pi$
$83$	1.31616	−	0.479045i	0.144468	−	0.0525820i	0.413242	+	0.910621i	$0.364396\pi$
$84$		0			0
$85$	−12.5786	+	10.5547i	−1.36434	+	1.14482i
$86$		0			0
$87$	−0.0303672	−	0.667485i	−0.00325571	−	0.0715619i
$88$		0			0
$89$	−6.85289	−	3.95652i	−0.726404	−	0.419390i	0.0907009	−	0.995878i	$-0.471089\pi$
$89$	−6.85289	−	3.95652i	−0.726404	−	0.419390i	−0.817105	+	0.576488i	$0.804423\pi$
$90$		0			0
$91$	−10.7622	+	6.21354i	−1.12818	+	0.651356i
$92$		0			0
$93$	−4.52819	−	10.8757i	−0.469551	−	1.12775i
$94$		0			0
$95$	7.95569	+	2.89564i	0.816237	+	0.297086i
$96$		0			0
$97$	2.09088	−	11.8580i	0.212297	−	1.20399i	−0.673239	−	0.739424i	$-0.735098\pi$
$97$	2.09088	−	11.8580i	0.212297	−	1.20399i	0.885536	−	0.464570i	$-0.153791\pi$
$98$		0			0
$99$	9.15058	−	9.21782i	0.919668	−	0.926426i
$100$		0			0
$101$	−1.06468	+	1.26884i	−0.105940	+	0.126254i	−0.816409	−	0.577474i	$-0.804038\pi$
$101$	−1.06468	+	1.26884i	−0.105940	+	0.126254i	0.710469	+	0.703728i	$0.248483\pi$
$102$		0			0
$103$	2.83638	−	0.500131i	0.279477	−	0.0492793i	−0.0321527	−	0.999483i	$-0.510236\pi$
$103$	2.83638	−	0.500131i	0.279477	−	0.0492793i	0.311630	+	0.950204i	$0.399125\pi$
$104$		0			0
$105$	−7.17118	+	6.59537i	−0.699836	+	0.643642i
$106$		0			0
$107$	−12.6955			−1.22732			−0.613659	−	0.789571i	$-0.710303\pi$
$107$	−12.6955			−1.22732			−0.613659	+	0.789571i	$0.710303\pi$
$108$		0			0
$109$	−6.54537			−0.626933			−0.313466	−	0.949599i	$-0.601490\pi$
$109$	−6.54537			−0.626933			−0.313466	+	0.949599i	$0.601490\pi$
$110$		0			0
$111$	2.88738	+	12.9122i	0.274058	+	1.22558i
$112$		0			0
$113$	−10.2726	+	1.81134i	−0.966367	+	0.170397i	−0.634494	−	0.772928i	$-0.718791\pi$
$113$	−10.2726	+	1.81134i	−0.966367	+	0.170397i	−0.331873	+	0.943324i	$0.607680\pi$
$114$		0			0
$115$	−7.14378	+	8.51362i	−0.666160	+	0.793899i
$116$		0			0
$117$	9.81389	+	13.9071i	0.907295	+	1.28571i
$118$		0			0
$119$	−2.43177	+	13.7913i	−0.222920	+	1.26424i
$120$		0			0
$121$	−7.27754	−	2.64881i	−0.661595	−	0.240801i
$122$		0			0
$123$	0.425850	−	0.557086i	0.0383976	−	0.0502308i
$124$		0			0
$125$	7.57180	−	4.37158i	0.677243	−	0.391006i
$126$		0			0
$127$	12.0771	+	6.97273i	1.07167	+	0.618730i	0.928638	−	0.370988i	$-0.120981\pi$
$127$	12.0771	+	6.97273i	1.07167	+	0.618730i	0.143034	+	0.989718i	$0.454314\pi$
$128$		0			0
$129$	−14.6073	−	7.57012i	−1.28610	−	0.666512i
$130$		0			0
$131$	12.8891	−	10.8153i	1.12613	−	0.944933i	0.127230	−	0.991873i	$-0.459392\pi$
$131$	12.8891	−	10.8153i	1.12613	−	0.944933i	0.998898	+	0.0469403i	$0.0149471\pi$
$132$		0			0
$133$	6.78506	−	2.46956i	0.588339	−	0.214138i
$134$		0			0
$135$	9.88810	+	8.96138i	0.851032	+	0.771273i
$136$		0			0
$137$	−7.20373	−	19.7921i	−0.615456	−	1.69095i	−0.717841	−	0.696207i	$-0.754869\pi$
$137$	−7.20373	−	19.7921i	−0.615456	−	1.69095i	0.102384	−	0.994745i	$-0.467353\pi$
$138$		0			0
$139$	5.34649	+	6.37169i	0.453483	+	0.540440i	0.943544	−	0.331248i	$-0.107470\pi$
$139$	5.34649	+	6.37169i	0.453483	+	0.540440i	−0.490061	+	0.871688i	$0.663025\pi$
$140$		0			0
$141$	−8.21660	−	12.8455i	−0.691962	−	1.08179i
$142$		0			0
$143$	12.2822	−	21.2733i	1.02709	−	1.77897i
$144$		0			0
$145$	−0.495366	−	0.858000i	−0.0411379	−	0.0712530i
$146$		0			0
$147$	0.491038	−	3.78331i	0.0405002	−	0.312042i
$148$		0			0
$149$	−2.34190	+	6.43432i	−0.191856	+	0.527120i	−0.997903	−	0.0647320i	$-0.979381\pi$
$149$	−2.34190	+	6.43432i	−0.191856	+	0.527120i	0.806047	+	0.591852i	$0.201603\pi$
$150$		0			0
$151$	−9.29194	−	1.63842i	−0.756167	−	0.133333i	−0.217742	−	0.976006i	$-0.569869\pi$
$151$	−9.29194	−	1.63842i	−0.756167	−	0.133333i	−0.538425	+	0.842674i	$0.680980\pi$
$152$		0			0
$153$	19.1140	+	1.60177i	1.54528	+	0.129496i
$154$		0			0
$155$	−13.3811	−	11.2281i	−1.07479	−	0.901859i
$156$		0			0
$157$	1.84720	+	10.4760i	0.147422	+	0.836074i	0.965390	+	0.260810i	$0.0839897\pi$
$157$	1.84720	+	10.4760i	0.147422	+	0.836074i	−0.817968	+	0.575264i	$0.804899\pi$
$158$		0			0
$159$	−3.62853	+	11.5821i	−0.287761	+	0.918521i
$160$		0			0
$161$			9.47842i			0.747004i
$162$		0			0
$163$		−	13.7337i		−	1.07570i	−0.843039	−	0.537852i	$-0.819236\pi$
$163$		−	13.7337i		−	1.07570i	0.843039	−	0.537852i	$-0.180764\pi$
$164$		0			0
$165$	5.75755	−	18.3779i	0.448225	−	1.43071i
$166$		0			0
$167$	−0.294930	−	1.67263i	−0.0228223	−	0.129432i	0.971268	−	0.237989i	$-0.0764881\pi$
$167$	−0.294930	−	1.67263i	−0.0228223	−	0.129432i	−0.994090	+	0.108557i	$0.965377\pi$
$168$		0			0
$169$	14.7011	+	12.3357i	1.13086	+	0.948901i
$170$		0			0
$171$	−4.21239	−	8.94782i	−0.322129	−	0.684257i
$172$		0			0
$173$	0.171533	+	0.0302458i	0.0130414	+	0.00229955i	0.180165	−	0.983636i	$-0.442337\pi$
$173$	0.171533	+	0.0302458i	0.0130414	+	0.00229955i	−0.167124	+	0.985936i	$0.553448\pi$
$174$		0			0
$175$	−1.19530	+	3.28405i	−0.0903558	+	0.248251i
$176$		0			0
$177$	0.381079	−	2.93610i	0.0286436	−	0.220691i
$178$		0			0
$179$	11.3351	+	19.6329i	0.847223	+	1.46743i	0.883677	+	0.468097i	$0.155060\pi$
$179$	11.3351	+	19.6329i	0.847223	+	1.46743i	−0.0364543	+	0.999335i	$0.511606\pi$
$180$		0			0
$181$	−5.78896	+	10.0268i	−0.430290	+	0.745284i	−0.996898	−	0.0787036i	$-0.974922\pi$
$181$	−5.78896	+	10.0268i	−0.430290	+	0.745284i	0.566608	+	0.823987i	$0.308255\pi$
$182$		0			0
$183$	−10.5528	−	16.4979i	−0.780088	−	1.21956i
$184$		0			0
$185$	12.6105	+	15.0286i	0.927139	+	1.10492i
$186$		0			0
$187$	−9.46761	−	26.0121i	−0.692340	−	1.90219i
$188$		0			0
$189$	11.3728	+	0.433992i	0.827251	+	0.0315682i
$190$		0			0
$191$	−18.0496	+	6.56951i	−1.30602	+	0.475353i	−0.898953	−	0.438044i	$-0.855671\pi$
$191$	−18.0496	+	6.56951i	−1.30602	+	0.475353i	−0.407069	+	0.913397i	$0.633449\pi$
$192$		0			0
$193$	6.28804	−	5.27629i	0.452623	−	0.379796i	−0.387785	−	0.921750i	$-0.626760\pi$
$193$	6.28804	−	5.27629i	0.452623	−	0.379796i	0.840408	+	0.541954i	$0.182315\pi$
$194$		0			0
$195$	22.4076	+	11.6126i	1.60465	+	0.831596i
$196$		0			0
$197$	13.9850	+	8.07425i	0.996391	+	0.575266i	0.907178	−	0.420746i	$-0.138232\pi$
$197$	13.9850	+	8.07425i	0.996391	+	0.575266i	0.0892122	+	0.996013i	$0.471565\pi$
$198$		0			0
$199$	1.42031	−	0.820018i	0.100683	−	0.0581295i	−0.448813	−	0.893626i	$-0.648153\pi$
$199$	1.42031	−	0.820018i	0.100683	−	0.0581295i	0.549496	+	0.835496i	$0.314820\pi$
$200$		0			0
$201$	2.10341	−	2.75163i	0.148363	−	0.194085i
$202$		0			0
$203$	−0.793995	−	0.288991i	−0.0557275	−	0.0202832i
$204$		0			0
$205$	0.180544	−	1.02392i	0.0126098	−	0.0715135i
$206$		0			0
$207$	12.9288	−	1.17883i	0.898611	−	0.0819343i
$208$		0			0
$209$	−9.17426	+	10.9335i	−0.634597	+	0.756283i
$210$		0			0
$211$	14.5381	−	2.56346i	1.00084	−	0.176476i	0.350864	−	0.936427i	$-0.385888\pi$
$211$	14.5381	−	2.56346i	1.00084	−	0.176476i	0.649981	+	0.759951i	$0.274777\pi$
$212$		0			0
$213$	1.40169	+	6.26828i	0.0960420	+	0.429495i
$214$		0			0
$215$	−24.3946			−1.66370
$216$		0			0
$217$	−14.8975			−1.01131
$218$		0			0
$219$	−13.7131	+	12.6120i	−0.926645	+	0.852239i
$220$		0			0
$221$	35.7248	−	6.29924i	2.40311	−	0.423733i
$222$		0			0
$223$	13.6087	−	16.2182i	0.911308	−	1.08605i	−0.0846660	−	0.996409i	$-0.526982\pi$
$223$	13.6087	−	16.2182i	0.911308	−	1.08605i	0.995974	−	0.0896449i	$-0.0285732\pi$
$224$		0			0
$225$	4.62817	+	1.22197i	0.308545	+	0.0814648i
$226$		0			0
$227$	−3.65627	+	20.7357i	−0.242675	+	1.37628i	0.583155	+	0.812361i	$0.301818\pi$
$227$	−3.65627	+	20.7357i	−0.242675	+	1.37628i	−0.825830	+	0.563919i	$0.809293\pi$
$228$		0			0
$229$	−21.1473	−	7.69700i	−1.39746	−	0.508632i	−0.470033	−	0.882649i	$-0.655758\pi$
$229$	−21.1473	−	7.69700i	−1.39746	−	0.508632i	−0.927422	+	0.374017i	$0.877980\pi$
$230$		0			0
$231$	−6.31328	−	15.1630i	−0.415383	−	0.997655i
$232$		0			0
$233$	22.3267	−	12.8903i	1.46267	−	0.844472i	0.463534	−	0.886079i	$-0.346581\pi$
$233$	22.3267	−	12.8903i	1.46267	−	0.844472i	0.999134	+	0.0416070i	$0.0132478\pi$
$234$		0			0
$235$	−19.5806	−	11.3049i	−1.27730	−	0.737450i
$236$		0			0
$237$	0.538625	+	11.8392i	0.0349875	+	0.769039i
$238$		0			0
$239$	−8.25623	+	6.92780i	−0.534051	+	0.448122i	−0.869498	−	0.493937i	$-0.835557\pi$
$239$	−8.25623	+	6.92780i	−0.534051	+	0.448122i	0.335446	+	0.942059i	$0.391113\pi$
$240$		0			0
$241$	23.7530	−	8.64537i	1.53006	−	0.556897i	0.566426	−	0.824112i	$-0.308326\pi$
$241$	23.7530	−	8.64537i	1.53006	−	0.556897i	0.963637	+	0.267215i	$0.0861034\pi$
$242$		0			0
$243$	−0.822462	−	15.5667i	−0.0527609	−	0.998607i
$244$		0			0
$245$	−1.93471	−	5.31558i	−0.123604	−	0.339600i
$246$		0			0
$247$	−12.0226	−	14.3280i	−0.764982	−	0.911671i
$248$		0			0
$249$	−2.42346	+	0.110255i	−0.153581	+	0.00698715i
$250$		0			0
$251$	−7.50990	+	13.0075i	−0.474021	+	0.821028i	−0.999558	−	0.0297426i	$-0.990531\pi$
$251$	−7.50990	+	13.0075i	−0.474021	+	0.821028i	0.525537	+	0.850771i	$0.323865\pi$
$252$		0			0
$253$	−9.36788	−	16.2257i	−0.588954	−	1.02010i
$254$		0			0
$255$	26.2557	−	10.9318i	1.64420	−	0.684577i
$256$		0			0
$257$	−6.91255	+	18.9921i	−0.431193	+	1.18469i	0.513888	+	0.857857i	$0.328205\pi$
$257$	−6.91255	+	18.9921i	−0.431193	+	1.18469i	−0.945081	+	0.326836i	$0.894018\pi$
$258$		0			0
$259$	16.4775	+	2.90542i	1.02386	+	0.180534i
$260$		0			0
$261$	−0.295440	+	1.11897i	−0.0182873	+	0.0692624i
$262$		0			0
$263$	3.94316	+	3.30870i	0.243146	+	0.204023i	0.756214	−	0.654324i	$-0.227047\pi$
$263$	3.94316	+	3.30870i	0.243146	+	0.204023i	−0.513068	+	0.858348i	$0.671491\pi$
$264$		0			0
$265$	3.12503	+	17.7229i	0.191969	+	1.08871i
$266$		0			0
$267$	9.27797	+	10.0880i	0.567802	+	0.617375i
$268$		0			0
$269$			5.82144i			0.354939i	0.984126	+	0.177470i	$0.0567912\pi$
$269$			5.82144i			0.354939i	−0.984126	+	0.177470i	$0.943209\pi$
$270$		0			0
$271$		−	4.39965i		−	0.267260i	−0.991031	−	0.133630i	$-0.957337\pi$
$271$		−	4.39965i		−	0.267260i	0.991031	−	0.133630i	$-0.0426634\pi$
$272$		0			0
$273$	21.0056	−	4.69718i	1.27131	−	0.284286i
$274$		0			0
$275$	−1.19958	−	6.80316i	−0.0723374	−	0.410246i
$276$		0			0
$277$	−7.99331	−	6.70718i	−0.480271	−	0.402995i	0.370253	−	0.928931i	$-0.379271\pi$
$277$	−7.99331	−	6.70718i	−0.480271	−	0.402995i	−0.850525	+	0.525935i	$0.823715\pi$
$278$		0			0
$279$	1.85279	+	20.3205i	0.110924	+	1.21655i
$280$		0			0
$281$	−4.88693	−	0.861697i	−0.291530	−	0.0514045i	0.0259707	−	0.999663i	$-0.491732\pi$
$281$	−4.88693	−	0.861697i	−0.291530	−	0.0514045i	−0.317500	+	0.948258i	$0.602843\pi$
$282$		0			0
$283$	−0.925706	+	2.54336i	−0.0550275	+	0.151187i	−0.964161	−	0.265318i	$-0.914523\pi$
$283$	−0.925706	+	2.54336i	−0.0550275	+	0.151187i	0.909133	+	0.416505i	$0.136745\pi$
$284$		0			0
$285$	−11.6501	−	8.90558i	−0.690090	−	0.527521i
$286$		0			0
$287$	−0.443362	−	0.767926i	−0.0261709	−	0.0453292i
$288$		0			0
$289$	11.9396	−	20.6799i	0.702327	−	1.21647i
$290$		0			0
$291$	−9.59610	+	18.5166i	−0.562533	+	1.08546i
$292$		0			0
$293$	1.38940	+	1.65583i	0.0811699	+	0.0967345i	0.805101	−	0.593138i	$-0.202111\pi$
$293$	1.38940	+	1.65583i	0.0811699	+	0.0967345i	−0.723931	+	0.689872i	$0.757667\pi$
$294$		0			0
$295$	−1.50147	−	4.12525i	−0.0874189	−	0.240182i
$296$		0			0
$297$	−19.8975	+	10.4973i	−1.15457	+	0.609113i
$298$		0			0
$299$	23.0721	−	8.39754i	1.33429	−	0.485642i
$300$		0			0
$301$	−15.9376	+	13.3732i	−0.918629	+	0.770821i
$302$		0			0
$303$	2.41677	−	1.54588i	0.138840	−	0.0888083i
$304$		0			0
$305$	−25.1480	−	14.5192i	−1.43997	−	0.831368i
$306$		0			0
$307$	2.33463	−	1.34790i	0.133244	−	0.0769287i	−0.431896	−	0.901923i	$-0.642155\pi$
$307$	2.33463	−	1.34790i	0.133244	−	0.0769287i	0.565140	+	0.824995i	$0.308822\pi$
$308$		0			0
$309$	−4.94705	−	0.642082i	−0.281428	−	0.0365267i
$310$		0			0
$311$	−0.211163	−	0.0768571i	−0.0119740	−	0.00435817i	0.336026	−	0.941853i	$-0.390917\pi$
$311$	−0.211163	−	0.0768571i	−0.0119740	−	0.00435817i	−0.348000	+	0.937494i	$0.613139\pi$
$312$		0			0
$313$	−1.54162	+	8.74295i	−0.0871374	+	0.494181i	0.909737	+	0.415184i	$0.136283\pi$
$313$	−1.54162	+	8.74295i	−0.0871374	+	0.494181i	−0.996875	+	0.0789969i	$0.974828\pi$
$314$		0			0
$315$	15.2680	−	7.18773i	0.860252	−	0.404983i
$316$		0			0
$317$	2.12018	−	2.52673i	0.119081	−	0.141915i	−0.703210	−	0.710982i	$-0.748251\pi$
$317$	2.12018	−	2.52673i	0.119081	−	0.141915i	0.822292	+	0.569066i	$0.192695\pi$
$318$		0			0
$319$	1.64482	−	0.290027i	0.0920924	−	0.0162384i
$320$		0			0
$321$	20.9836	+	6.57389i	1.17119	+	0.366919i
$322$		0			0
$323$	−21.0774			−1.17278
$324$		0			0
$325$	9.05291			0.502165
$326$		0			0
$327$	10.8184	+	3.38928i	0.598261	+	0.187428i
$328$		0			0
$329$	−18.9899	+	3.34844i	−1.04695	+	0.184605i
$330$		0			0
$331$	14.8545	−	17.7029i	0.816478	−	0.973041i	−0.183472	−	0.983025i	$-0.558734\pi$
$331$	14.8545	−	17.7029i	0.816478	−	0.973041i	0.999950	+	0.00998434i	$0.00317817\pi$
$332$		0			0
$333$	1.91376	−	22.8370i	0.104873	−	1.25146i
$334$		0			0
$335$	0.891768	−	5.05746i	0.0487225	−	0.276319i
$336$		0			0
$337$	9.67555	+	3.52161i	0.527061	+	0.191834i	0.591826	−	0.806066i	$-0.298407\pi$
$337$	9.67555	+	3.52161i	0.527061	+	0.191834i	−0.0647647	+	0.997901i	$0.520630\pi$
$338$		0			0
$339$	17.9169	+	2.32545i	0.973112	+	0.126301i
$340$		0			0
$341$	25.5023	−	14.7237i	1.38103	−	0.797335i
$342$		0			0
$343$	−17.4560	−	10.0782i	−0.942534	−	0.544172i
$344$		0			0
$345$	16.2160	−	10.3725i	0.873038	−	0.558436i
$346$		0			0
$347$	−27.5760	+	23.1390i	−1.48036	+	1.24217i	−0.574560	+	0.818462i	$0.694827\pi$
$347$	−27.5760	+	23.1390i	−1.48036	+	1.24217i	−0.905799	+	0.423707i	$0.860729\pi$
$348$		0			0
$349$	−21.1590	+	7.70125i	−1.13262	+	0.412238i	−0.839241	−	0.543759i	$-0.817000\pi$
$349$	−21.1590	+	7.70125i	−1.13262	+	0.412238i	−0.293374	+	0.955998i	$0.594778\pi$
$350$		0			0
$351$	−9.01951	−	28.0679i	−0.481426	−	1.49815i
$352$		0			0
$353$	9.62764	+	26.4517i	0.512428	+	1.40788i	0.878700	+	0.477375i	$0.158412\pi$
$353$	9.62764	+	26.4517i	0.512428	+	1.40788i	−0.366272	+	0.930508i	$0.619366\pi$
$354$		0			0
$355$	6.12178	+	7.29565i	0.324910	+	0.387213i
$356$		0			0
$357$	11.1606	−	21.5355i	0.590683	−	1.13978i
$358$		0			0
$359$	13.4479	−	23.2924i	0.709753	−	1.22933i	−0.255196	−	0.966889i	$-0.582140\pi$
$359$	13.4479	−	23.2924i	0.709753	−	1.22933i	0.964949	−	0.262439i	$-0.0845268\pi$
$360$		0			0
$361$	−4.06623	−	7.04293i	−0.214012	−	0.370680i
$362$		0			0
$363$	10.6570	+	8.14645i	0.559347	+	0.427578i
$364$		0			0
$365$	−9.44827	+	25.9589i	−0.494545	+	1.35875i
$366$		0			0
$367$	−27.9737	−	4.93252i	−1.46022	−	0.257475i	−0.613574	−	0.789638i	$-0.710269\pi$
$367$	−27.9737	−	4.93252i	−1.46022	−	0.257475i	−0.846642	+	0.532162i	$0.821380\pi$
$368$		0			0
$369$	−0.992327	+	0.700262i	−0.0516585	+	0.0364542i
$370$		0			0
$371$	11.7575	+	9.86569i	0.610418	+	0.512201i
$372$		0			0
$373$	2.05586	+	11.6594i	0.106448	+	0.603699i	0.990632	+	0.136559i	$0.0436045\pi$
$373$	2.05586	+	11.6594i	0.106448	+	0.603699i	−0.884183	+	0.467140i	$0.845284\pi$
$374$		0			0
$375$	−14.7786	+	3.30473i	−0.763165	+	0.170656i
$376$		0			0
$377$			2.18875i			0.112726i
$378$		0			0
$379$			21.1690i			1.08738i	0.839286	+	0.543690i	$0.182973\pi$
$379$			21.1690i			1.08738i	−0.839286	+	0.543690i	$0.817027\pi$
$380$		0			0
$381$	−16.3509	−	17.7785i	−0.837684	−	0.910819i
$382$		0			0
$383$	3.40920	+	19.3346i	0.174202	+	0.987950i	0.939061	+	0.343751i	$0.111698\pi$
$383$	3.40920	+	19.3346i	0.174202	+	0.987950i	−0.764859	+	0.644198i	$0.777191\pi$
$384$		0			0
$385$	−18.6561	−	15.6543i	−0.950804	−	0.797819i
$386$		0			0
$387$	20.2235	+	20.0760i	1.02802	+	1.02052i
$388$		0			0
$389$	−8.24025	−	1.45298i	−0.417798	−	0.0736690i	−0.0392022	−	0.999231i	$-0.512482\pi$
$389$	−8.24025	−	1.45298i	−0.417798	−	0.0736690i	−0.378595	+	0.925562i	$0.623593\pi$
$390$		0			0
$391$	9.46316	−	25.9998i	0.478572	−	1.31487i
$392$		0			0
$393$	−26.9039	+	11.2017i	−1.35712	+	0.565051i
$394$		0			0
$395$	8.78634	+	15.2184i	0.442089	+	0.765720i
$396$		0			0
$397$	−13.4307	+	23.2627i	−0.674068	+	1.16752i	0.302672	+	0.953095i	$0.402121\pi$
$397$	−13.4307	+	23.2627i	−0.674068	+	1.16752i	−0.976740	+	0.214426i	$0.931212\pi$
$398$		0			0
$399$	−12.4934	+	0.568386i	−0.625451	+	0.0284549i
$400$		0			0
$401$	−12.5634	−	14.9724i	−0.627385	−	0.747688i	0.354937	−	0.934890i	$-0.384502\pi$
$401$	−12.5634	−	14.9724i	−0.627385	−	0.747688i	−0.982321	+	0.187203i	$0.940058\pi$
$402$		0			0
$403$	13.1986	+	36.2629i	0.657471	+	1.80639i
$404$		0			0
$405$	−11.7031	−	19.9319i	−0.581531	−	0.990424i
$406$		0			0
$407$	−31.0785	+	11.3117i	−1.54051	+	0.560698i
$408$		0			0
$409$	−2.96469	+	2.48767i	−0.146594	+	0.123007i	−0.713136	−	0.701026i	$-0.752726\pi$
$409$	−2.96469	+	2.48767i	−0.146594	+	0.123007i	0.566541	+	0.824033i	$0.308281\pi$
$410$		0			0
$411$	1.65799	+	36.4433i	0.0817825	+	1.79761i
$412$		0			0
$413$	−3.24243	−	1.87202i	−0.159550	−	0.0921161i
$414$		0			0
$415$	−3.11517	+	1.79854i	−0.152918	+	0.0882871i
$416$		0			0
$417$	−5.53752	−	13.2999i	−0.271174	−	0.651297i
$418$		0			0
$419$	−5.55790	−	2.02291i	−0.271521	−	0.0988256i	0.202671	−	0.979247i	$-0.435038\pi$
$419$	−5.55790	−	2.02291i	−0.271521	−	0.0988256i	−0.474192	+	0.880421i	$0.657260\pi$
$420$		0			0
$421$	−2.43514	+	13.8103i	−0.118681	+	0.673075i	0.866180	+	0.499732i	$0.166568\pi$
$421$	−2.43514	+	13.8103i	−0.118681	+	0.673075i	−0.984862	+	0.173343i	$0.944543\pi$
$422$		0			0
$423$	6.92911	+	25.4862i	0.336905	+	1.23918i
$424$		0			0
$425$	6.55752	−	7.81495i	0.318086	−	0.379081i
$426$		0			0
$427$	−24.3894	+	4.30050i	−1.18028	+	0.208116i
$428$		0			0
$429$	−31.3160	+	28.8015i	−1.51195	+	1.39055i
$430$		0			0
$431$	−34.8272			−1.67757			−0.838784	−	0.544465i	$-0.816733\pi$
$431$	−34.8272			−1.67757			−0.838784	+	0.544465i	$0.816733\pi$
$432$		0			0
$433$	−13.7584			−0.661187			−0.330593	−	0.943773i	$-0.607249\pi$
$433$	−13.7584			−0.661187			−0.330593	+	0.943773i	$0.607249\pi$
$434$		0			0
$435$	0.374476	+	1.67464i	0.0179548	+	0.0802929i
$436$		0			0
$437$	−14.0492	+	2.47725i	−0.672062	+	0.118503i
$438$		0			0
$439$	−14.8296	+	17.6732i	−0.707779	+	0.843498i	−0.993383	−	0.114850i	$-0.963361\pi$
$439$	−14.8296	+	17.6732i	−0.707779	+	0.843498i	0.285604	+	0.958348i	$0.407806\pi$
$440$		0			0
$441$	−2.77065	+	5.99892i	−0.131936	+	0.285663i
$442$		0			0
$443$	4.37334	−	24.8025i	0.207784	−	1.17840i	−0.685215	−	0.728341i	$-0.740292\pi$
$443$	4.37334	−	24.8025i	0.207784	−	1.17840i	0.892999	−	0.450059i	$-0.148597\pi$
$444$		0			0
$445$	19.0966	+	6.95058i	0.905264	+	0.329489i
$446$		0			0
$447$	7.20256	−	9.42221i	0.340669	−	0.445655i
$448$		0			0
$449$	−11.0355	+	6.37137i	−0.520799	+	0.300683i	−0.737261	−	0.675608i	$-0.763881\pi$
$449$	−11.0355	+	6.37137i	−0.520799	+	0.300683i	0.216463	+	0.976291i	$0.430548\pi$
$450$		0			0
$451$	1.51794	+	0.876384i	0.0714771	+	0.0412673i
$452$		0			0
$453$	14.5097	+	7.51953i	0.681723	+	0.353298i
$454$		0			0
$455$	24.4484	−	20.5146i	1.14616	−	0.961741i
$456$		0			0
$457$	−17.7529	+	6.46151i	−0.830444	+	0.302257i	−0.722041	−	0.691850i	$-0.756796\pi$
$457$	−17.7529	+	6.46151i	−0.830444	+	0.302257i	−0.108403	+	0.994107i	$0.534574\pi$
$458$		0			0
$459$	−30.7630	−	12.5450i	−1.43589	−	0.585549i
$460$		0			0
$461$	2.34645	+	6.44682i	0.109285	+	0.300258i	0.982265	−	0.187498i	$-0.0600378\pi$
$461$	2.34645	+	6.44682i	0.109285	+	0.300258i	−0.872980	+	0.487756i	$0.837816\pi$
$462$		0			0
$463$	−4.22001	−	5.02921i	−0.196120	−	0.233727i	0.659018	−	0.752127i	$-0.270972\pi$
$463$	−4.22001	−	5.02921i	−0.196120	−	0.233727i	−0.855138	+	0.518400i	$0.826528\pi$
$464$		0			0
$465$	16.3027	+	25.4870i	0.756019	+	1.18193i
$466$		0			0
$467$	19.4184	−	33.6337i	0.898578	−	1.55638i	0.0692651	−	0.997598i	$-0.477935\pi$
$467$	19.4184	−	33.6337i	0.898578	−	1.55638i	0.829313	−	0.558784i	$-0.188732\pi$
$468$		0			0
$469$	−2.18991	−	3.79304i	−0.101121	−	0.175146i
$470$		0			0
$471$	2.37148	−	18.2716i	0.109272	−	0.841911i
$472$		0			0
$473$	14.0655	−	38.6448i	0.646735	−	1.77689i
$474$		0			0
$475$	−5.18010	−	0.913391i	−0.237679	−	0.0419093i
$476$		0			0
$477$	11.9947	−	17.2644i	0.549201	−	0.790484i
$478$		0			0
$479$	−1.85078	−	1.55299i	−0.0845645	−	0.0709580i	0.599525	−	0.800356i	$-0.295356\pi$
$479$	−1.85078	−	1.55299i	−0.0845645	−	0.0709580i	−0.684090	+	0.729398i	$0.739800\pi$
$480$		0			0
$481$	−7.52616	−	42.6830i	−0.343164	−	1.94618i
$482$		0			0
$483$	4.90805	−	15.6663i	0.223324	−	0.712840i
$484$		0			0
$485$			30.9233i			1.40415i
$486$		0			0
$487$			1.69185i			0.0766648i	0.999265	+	0.0383324i	$0.0122046\pi$
$487$			1.69185i			0.0766648i	−0.999265	+	0.0383324i	$0.987795\pi$
$488$		0			0
$489$	−7.11147	+	22.6995i	−0.321592	+	1.02651i
$490$		0			0
$491$	−1.07391	−	6.09047i	−0.0484650	−	0.274859i	0.950939	−	0.309378i	$-0.100121\pi$
$491$	−1.07391	−	6.09047i	−0.0484650	−	0.274859i	−0.999404	+	0.0345196i	$0.989010\pi$
$492$		0			0
$493$	1.88945	+	1.58543i	0.0850964	+	0.0714043i
$494$		0			0
$495$	−19.0326	+	27.3943i	−0.855451	+	1.23128i
$496$		0			0
$497$	7.99902	+	1.41044i	0.358805	+	0.0632670i
$498$		0			0
$499$	−7.06711	+	19.4167i	−0.316367	+	0.869213i	0.674967	+	0.737848i	$0.264158\pi$
$499$	−7.06711	+	19.4167i	−0.316367	+	0.869213i	−0.991334	+	0.131364i	$0.958064\pi$
$500$		0			0
$501$	−0.378639	+	2.91730i	−0.0169163	+	0.130335i
$502$		0			0
$503$	15.4292	+	26.7241i	0.687954	+	1.19157i	0.972499	+	0.232908i	$0.0748241\pi$
$503$	15.4292	+	26.7241i	0.687954	+	1.19157i	−0.284545	+	0.958663i	$0.591843\pi$
$504$		0			0
$505$	2.12691	−	3.68392i	0.0946463	−	0.163932i
$506$		0			0
$507$	−17.9110	−	28.0014i	−0.795455	−	1.24358i
$508$		0			0
$509$	−25.1823	−	30.0112i	−1.11619	−	1.33022i	−0.938162	−	0.346195i	$-0.887473\pi$
$509$	−25.1823	−	30.0112i	−1.11619	−	1.33022i	−0.178026	−	0.984026i	$-0.556971\pi$
$510$		0			0
$511$	8.05801	+	22.1392i	0.356465	+	0.979380i
$512$		0			0
$513$	2.32909	+	16.9705i	0.102832	+	0.749267i
$514$		0			0
$515$	−6.95066	+	2.52983i	−0.306283	+	0.111478i
$516$		0			0
$517$	29.1986	−	24.5005i	1.28415	−	1.07753i
$518$		0			0
$519$	−0.267854	−	0.138813i	−0.0117575	−	0.00609322i
$520$		0			0
$521$	−0.354649	−	0.204757i	−0.0155375	−	0.00897056i	0.492211	−	0.870476i	$-0.336189\pi$
$521$	−0.354649	−	0.204757i	−0.0155375	−	0.00897056i	−0.507749	+	0.861505i	$0.669522\pi$
$522$		0			0
$523$	11.5381	−	6.66155i	0.504528	−	0.291289i	−0.226054	−	0.974115i	$-0.572582\pi$
$523$	11.5381	−	6.66155i	0.504528	−	0.291289i	0.730581	+	0.682826i	$0.239249\pi$
$524$		0			0
$525$	3.67615	−	4.80905i	0.160440	−	0.209884i
$526$		0			0
$527$	40.8646	+	14.8735i	1.78009	+	0.647899i
$528$		0			0
$529$	−0.742008	+	4.20814i	−0.0322612	+	0.182962i
$530$		0			0
$531$	−2.15021	+	4.65557i	−0.0933113	+	0.202035i
$532$		0			0
$533$	−1.47646	+	1.75957i	−0.0639525	+	0.0762156i
$534$		0			0
$535$	32.1090	−	5.66169i	1.38820	−	0.244776i
$536$		0			0
$537$	−8.56884	−	38.3194i	−0.369773	−	1.65361i
$538$		0			0
$539$	9.53622			0.410754
$540$		0			0
$541$	27.7454			1.19287			0.596434	−	0.802662i	$-0.296584\pi$
$541$	27.7454			1.19287			0.596434	+	0.802662i	$0.296584\pi$
$542$		0			0
$543$	14.7602	−	13.5750i	0.633420	−	0.582559i
$544$		0			0
$545$	16.5544	−	2.91898i	0.709111	−	0.125035i
$546$		0			0
$547$	22.7506	−	27.1131i	0.972744	−	1.15927i	−0.0144745	−	0.999895i	$-0.504608\pi$
$547$	22.7506	−	27.1131i	0.972744	−	1.15927i	0.987218	−	0.159375i	$-0.0509480\pi$
$548$		0			0
$549$	8.89927	+	32.7327i	0.379812	+	1.39700i
$550$		0			0
$551$	0.220834	−	1.25241i	0.00940783	−	0.0533545i
$552$		0			0
$553$	14.0831	+	5.12584i	0.598876	+	0.217973i
$554$		0			0
$555$	−13.0610	−	31.3696i	−0.554410	−	1.33157i
$556$		0			0
$557$	4.22184	−	2.43748i	0.178885	−	0.103279i	−0.407884	−	0.913034i	$-0.633733\pi$
$557$	4.22184	−	2.43748i	0.178885	−	0.103279i	0.586769	+	0.809755i	$0.300400\pi$
$558$		0			0
$559$	46.6729	+	26.9466i	1.97405	+	1.13972i
$560$		0			0
$561$	2.17904	+	47.8961i	0.0919990	+	2.02218i
$562$		0			0
$563$	−17.8937	+	15.0146i	−0.754131	+	0.632791i	−0.936592	−	0.350422i	$-0.886038\pi$
$563$	−17.8937	+	15.0146i	−0.754131	+	0.632791i	0.182461	+	0.983213i	$0.441594\pi$
$564$		0			0
$565$	25.1734	−	9.16238i	1.05905	−	0.385464i
$566$		0			0
$567$	−18.5727	−	6.60631i	−0.779980	−	0.277439i
$568$		0			0
$569$	−1.89142	−	5.19662i	−0.0792923	−	0.217854i	0.893712	−	0.448641i	$-0.148092\pi$
$569$	−1.89142	−	5.19662i	−0.0792923	−	0.217854i	−0.973004	+	0.230788i	$0.925870\pi$
$570$		0			0
$571$	5.47700	+	6.52724i	0.229205	+	0.273156i	0.868374	−	0.495911i	$-0.165166\pi$
$571$	5.47700	+	6.52724i	0.229205	+	0.273156i	−0.639168	+	0.769067i	$0.720721\pi$
$572$		0			0
$573$	33.2348	−	1.51202i	1.38840	−	0.0631655i
$574$		0			0
$575$	3.45243	−	5.97978i	0.143976	−	0.249374i
$576$		0			0
$577$	11.9313	+	20.6656i	0.496706	+	0.860319i	0.999993	−	0.00379980i	$-0.00120952\pi$
$577$	11.9313	+	20.6656i	0.496706	+	0.860319i	−0.503287	+	0.864119i	$0.667876\pi$
$578$		0			0
$579$	−13.1252	+	5.46482i	−0.545466	+	0.227110i
$580$		0			0
$581$	−1.04925	+	2.88279i	−0.0435302	+	0.119598i
$582$		0			0
$583$	−29.8777	−	5.26824i	−1.23741	−	0.218188i
$584$		0			0
$585$	−31.0230	−	30.7967i	−1.28264	−	1.27329i
$586$		0			0
$587$	15.5765	+	13.0702i	0.642910	+	0.539466i	0.904910	−	0.425602i	$-0.139938\pi$
$587$	15.5765	+	13.0702i	0.642910	+	0.539466i	−0.262000	+	0.965068i	$0.584382\pi$
$588$		0			0
$589$	−3.89355	−	22.0814i	−0.160431	−	0.909849i
$590$		0			0
$591$	−18.9340	−	20.5870i	−0.778840	−	0.846838i
$592$		0			0
$593$		−	4.06230i		−	0.166819i	−0.996515	−	0.0834094i	$-0.973419\pi$
$593$		−	4.06230i		−	0.166819i	0.996515	−	0.0834094i	$-0.0265809\pi$
$594$		0			0
$595$		−	35.9650i		−	1.47442i
$596$		0			0
$597$	−2.77216	+	0.619899i	−0.113457	+	0.0253708i
$598$		0			0
$599$	−4.78180	−	27.1190i	−0.195379	−	1.10805i	−0.911878	−	0.410462i	$-0.865367\pi$
$599$	−4.78180	−	27.1190i	−0.195379	−	1.10805i	0.716499	−	0.697589i	$-0.245744\pi$
$600$		0			0
$601$	5.76201	+	4.83490i	0.235037	+	0.197220i	0.752697	−	0.658367i	$-0.228752\pi$
$601$	5.76201	+	4.83490i	0.235037	+	0.197220i	−0.517660	+	0.855586i	$0.673197\pi$
$602$		0			0
$603$	−4.90143	+	3.45883i	−0.199602	+	0.140854i
$604$		0			0
$605$	19.5874	+	3.45379i	0.796341	+	0.140416i
$606$		0			0
$607$	12.5106	−	34.3727i	0.507791	−	1.39514i	−0.375719	−	0.926733i	$-0.622604\pi$
$607$	12.5106	−	34.3727i	0.507791	−	1.39514i	0.883511	−	0.468411i	$-0.155173\pi$
$608$		0			0
$609$	1.16270	+	0.888795i	0.0471150	+	0.0360158i
$610$		0			0
$611$	24.9751	+	43.2581i	1.01038	+	1.75003i
$612$		0			0
$613$	13.0121	−	22.5376i	0.525552	−	0.910283i	−0.474005	−	0.880522i	$-0.657192\pi$
$613$	13.0121	−	22.5376i	0.525552	−	0.910283i	0.999557	−	0.0297610i	$-0.00947463\pi$
$614$		0			0
$615$	−0.828608	+	1.59888i	−0.0334127	+	0.0644730i
$616$		0			0
$617$	9.69439	+	11.5533i	0.390281	+	0.465119i	0.925031	−	0.379891i	$-0.124039\pi$
$617$	9.69439	+	11.5533i	0.390281	+	0.465119i	−0.534750	+	0.845010i	$0.679594\pi$
$618$		0			0
$619$	5.42776	+	14.9126i	0.218160	+	0.599390i	0.999701	−	0.0244646i	$-0.00778810\pi$
$619$	5.42776	+	14.9126i	0.218160	+	0.599390i	−0.781541	+	0.623854i	$0.785566\pi$
$620$		0			0
$621$	−21.9795	−	4.74627i	−0.882009	−	0.190461i
$622$		0			0
$623$	16.2866	−	5.92785i	0.652510	−	0.237494i
$624$		0			0
$625$	−23.3123	+	19.5613i	−0.932492	+	0.782454i
$626$		0			0
$627$	20.8250	−	13.3207i	0.831672	−	0.531976i
$628$		0			0
$629$	−42.2978	−	24.4207i	−1.68652	−	0.973715i
$630$		0			0
$631$	−6.79006	+	3.92024i	−0.270308	+	0.156062i	−0.629028	−	0.777383i	$-0.716547\pi$
$631$	−6.79006	+	3.92024i	−0.270308	+	0.156062i	0.358720	+	0.933445i	$0.383213\pi$
$632$		0			0
$633$	−25.3565	−	3.29104i	−1.00783	−	0.130807i
$634$		0			0
$635$	−33.6547	−	12.2493i	−1.33555	−	0.486099i
$636$		0			0
$637$	−2.17008	+	12.3071i	−0.0859817	+	0.487626i
$638$		0			0
$639$	0.929038	−	11.0863i	0.0367522	−	0.438565i
$640$		0			0
$641$	2.91672	−	3.47601i	0.115204	−	0.137294i	−0.705361	−	0.708849i	$-0.749215\pi$
$641$	2.91672	−	3.47601i	0.115204	−	0.137294i	0.820564	+	0.571554i	$0.193659\pi$
$642$		0			0
$643$	−34.2656	+	6.04196i	−1.35131	+	0.238272i	−0.801987	−	0.597342i	$-0.796224\pi$
$643$	−34.2656	+	6.04196i	−1.35131	+	0.238272i	−0.549318	+	0.835613i	$0.685113\pi$
$644$		0			0
$645$	40.3203	+	12.6318i	1.58761	+	0.497378i
$646$		0			0
$647$	−14.0701			−0.553153			−0.276576	−	0.960992i	$-0.589200\pi$
$647$	−14.0701			−0.553153			−0.276576	+	0.960992i	$0.589200\pi$
$648$		0			0
$649$	7.40076			0.290505
$650$		0			0
$651$	24.6231	+	7.71411i	0.965055	+	0.302340i
$652$		0			0
$653$	20.1254	−	3.54865i	0.787567	−	0.138869i	0.234622	−	0.972087i	$-0.424615\pi$
$653$	20.1254	−	3.54865i	0.787567	−	0.138869i	0.552945	+	0.833217i	$0.313504\pi$
$654$		0			0
$655$	−27.7756	+	33.1017i	−1.08528	+	1.29339i
$656$		0			0
$657$	29.1962	−	13.7447i	1.13905	−	0.536233i
$658$		0			0
$659$	6.56089	−	37.2087i	0.255576	−	1.44944i	−0.539014	−	0.842297i	$-0.681203\pi$
$659$	6.56089	−	37.2087i	0.255576	−	1.44944i	0.794590	−	0.607147i	$-0.207686\pi$
$660$		0			0
$661$	−7.02492	−	2.55686i	−0.273238	−	0.0994505i	0.201767	−	0.979434i	$-0.435332\pi$
$661$	−7.02492	−	2.55686i	−0.273238	−	0.0994505i	−0.475005	+	0.879983i	$0.657554\pi$
$662$		0			0
$663$	−62.3090	−	8.08714i	−2.41988	−	0.314078i
$664$		0			0
$665$	−16.0592	+	9.27181i	−0.622751	+	0.359545i
$666$		0			0
$667$	1.44575	+	0.834705i	0.0559797	+	0.0323199i
$668$		0			0
$669$	−30.8910	+	19.7593i	−1.19432	+	0.763940i
$670$		0			0
$671$	37.5006	−	31.4668i	1.44770	−	1.21476i
$672$		0			0
$673$	17.8914	−	6.51195i	0.689664	−	0.251017i	0.0266725	−	0.999644i	$-0.491509\pi$
$673$	17.8914	−	6.51195i	0.689664	−	0.251017i	0.662991	+	0.748627i	$0.269287\pi$
$674$		0			0
$675$	−7.01685	−	4.41625i	−0.270079	−	0.169981i
$676$		0			0
$677$	4.20531	+	11.5540i	0.161623	+	0.444056i	0.993897	−	0.110308i	$-0.0351837\pi$
$677$	4.20531	+	11.5540i	0.161623	+	0.444056i	−0.832274	+	0.554364i	$0.812961\pi$
$678$		0			0
$679$	16.9523	+	20.2030i	0.650570	+	0.775319i
$680$		0			0
$681$	16.7805	−	32.3795i	0.643028	−	1.24079i
$682$		0			0
$683$	14.1366	−	24.4853i	0.540921	−	0.936903i	−0.457931	−	0.888988i	$-0.651409\pi$
$683$	14.1366	−	24.4853i	0.540921	−	0.936903i	0.998851	−	0.0479145i	$-0.0152575\pi$
$684$		0			0
$685$	27.0460	+	46.8450i	1.03337	+	1.78986i
$686$		0			0
$687$	30.9675	+	23.6723i	1.18148	+	0.903153i
$688$		0			0
$689$	13.5980	−	37.3603i	0.518044	−	1.42331i
$690$		0			0
$691$	9.74960	+	1.71912i	0.370892	+	0.0653983i	0.355988	−	0.934491i	$-0.384144\pi$
$691$	9.74960	+	1.71912i	0.370892	+	0.0653983i	0.0149045	+	0.999889i	$0.495256\pi$
$692$		0			0
$693$	2.58320	+	28.3311i	0.0981275	+	1.07621i
$694$		0			0
$695$	−16.3637	−	13.7308i	−0.620711	−	0.520838i
$696$		0			0
$697$	0.449477	+	2.54911i	0.0170252	+	0.0965545i
$698$		0			0
$699$	−43.5771	+	9.74453i	−1.64824	+	0.368572i
$700$		0			0
$701$			9.95360i			0.375942i	0.982175	+	0.187971i	$0.0601911\pi$
$701$			9.95360i			0.375942i	−0.982175	+	0.187971i	$0.939809\pi$
$702$		0			0
$703$			25.1827i			0.949783i
$704$		0			0
$705$	26.5098	+	28.8243i	0.998417	+	1.08558i
$706$		0			0
$707$	−0.629978	−	3.57278i	−0.0236927	−	0.134368i
$708$		0			0
$709$	−17.3138	−	14.5280i	−0.650232	−	0.545610i	0.256909	−	0.966436i	$-0.417296\pi$
$709$	−17.3138	−	14.5280i	−0.650232	−	0.545610i	−0.907141	+	0.420826i	$0.861740\pi$
$710$		0			0
$711$	5.24024	−	19.8472i	0.196524	−	0.744328i
$712$		0			0
$713$	28.9865	+	5.11109i	1.08555	+	0.191412i
$714$		0			0
$715$	−21.5766	+	59.2813i	−0.806920	+	2.21699i
$716$		0			0
$717$	17.2335	−	7.17534i	0.643597	−	0.267968i
$718$		0			0
$719$	−16.3460	−	28.3121i	−0.609602	−	1.05586i	−0.991306	−	0.131577i	$-0.957996\pi$
$719$	−16.3460	−	28.3121i	−0.609602	−	1.05586i	0.381704	−	0.924285i	$-0.375337\pi$
$720$		0			0
$721$	−3.15417	+	5.46319i	−0.117468	+	0.203460i
$722$		0			0
$723$	−43.7365	+	1.98979i	−1.62658	+	0.0740011i
$724$		0			0
$725$	0.395657	+	0.471525i	0.0146943	+	0.0175120i
$726$		0			0
$727$	3.76774	+	10.3518i	0.139738	+	0.383926i	0.989745	−	0.142844i	$-0.0456248\pi$
$727$	3.76774	+	10.3518i	0.139738	+	0.383926i	−0.850007	+	0.526771i	$0.823403\pi$
$728$		0			0
$729$	−6.70127	+	26.1552i	−0.248195	+	0.968710i
$730$		0			0
$731$	57.0694	−	20.7716i	2.11079	−	0.768264i
$732$		0			0
$733$	24.1879	−	20.2961i	0.893402	−	0.749653i	−0.0754875	−	0.997147i	$-0.524051\pi$
$733$	24.1879	−	20.2961i	0.893402	−	0.749653i	0.968890	+	0.247493i	$0.0796068\pi$
$734$		0			0
$735$	0.445288	+	9.78761i	0.0164247	+	0.361022i
$736$		0			0
$737$	7.49762	+	4.32875i	0.276178	+	0.159452i
$738$		0			0
$739$	−6.54458	+	3.77851i	−0.240746	+	0.138995i	−0.615520	−	0.788122i	$-0.711054\pi$
$739$	−6.54458	+	3.77851i	−0.240746	+	0.138995i	0.374773	+	0.927116i	$0.377720\pi$
$740$		0			0
$741$	12.4522	+	29.9074i	0.457444	+	1.09867i
$742$		0			0
$743$	−27.9075	−	10.1575i	−1.02383	−	0.372643i	−0.225101	−	0.974336i	$-0.572271\pi$
$743$	−27.9075	−	10.1575i	−1.02383	−	0.372643i	−0.798728	+	0.601692i	$0.794493\pi$
$744$		0			0
$745$	3.05361	−	17.3179i	0.111876	−	0.634479i
$746$		0			0
$747$	4.06268	+	1.07267i	0.148646	+	0.0392468i
$748$		0			0
$749$	17.8739	−	21.3013i	0.653098	−	0.778331i
$750$		0			0
$751$	−35.8098	+	6.31424i	−1.30672	+	0.230410i	−0.783288	−	0.621659i	$-0.786459\pi$
$751$	−35.8098	+	6.31424i	−1.30672	+	0.230410i	−0.523430	+	0.852068i	$0.675348\pi$
$752$		0			0
$753$	19.1481	−	17.6106i	0.697796	−	0.641766i
$754$		0			0
$755$	24.2316			0.881877
$756$		0			0
$757$	20.1447			0.732170			0.366085	−	0.930581i	$-0.380698\pi$
$757$	20.1447			0.732170			0.366085	+	0.930581i	$0.380698\pi$
$758$		0			0
$759$	7.08173	+	31.6692i	0.257050	+	1.14952i
$760$		0			0
$761$	−17.1911	+	3.03125i	−0.623175	+	0.109883i	−0.476315	−	0.879275i	$-0.658028\pi$
$761$	−17.1911	+	3.03125i	−0.623175	+	0.109883i	−0.146860	+	0.989157i	$0.546917\pi$
$762$		0			0
$763$	9.21519	−	10.9822i	0.333612	−	0.397584i
$764$		0			0
$765$	−49.0570	+	4.47296i	−1.77366	+	0.161720i
$766$		0			0
$767$	−1.68413	+	9.55117i	−0.0608104	+	0.344873i
$768$		0			0
$769$	11.4962	+	4.18429i	0.414565	+	0.150889i	0.540878	−	0.841101i	$-0.318092\pi$
$769$	11.4962	+	4.18429i	0.414565	+	0.150889i	−0.126313	+	0.991990i	$0.540314\pi$
$770$		0			0
$771$	21.2597	−	27.8114i	0.765648	−	1.00160i
$772$		0			0
$773$	−28.3344	+	16.3589i	−1.01912	+	0.588389i	−0.913849	−	0.406055i	$-0.866904\pi$
$773$	−28.3344	+	16.3589i	−1.01912	+	0.588389i	−0.105270	+	0.994444i	$0.533571\pi$
$774$		0			0
$775$	9.39857	+	5.42627i	0.337607	+	0.194917i
$776$		0			0
$777$	−25.7301	−	13.3344i	−0.923062	−	0.478370i
$778$		0			0
$779$	1.02236	−	0.857866i	0.0366300	−	0.0307362i
$780$		0			0
$781$	−15.0871	+	5.49127i	−0.539860	+	0.196493i
$782$		0			0
$783$	1.06773	−	1.69649i	0.0381576	−	0.0606275i
$784$		0			0
$785$	−9.34376	−	25.6718i	−0.333493	−	0.916265i
$786$		0			0
$787$	−12.3621	−	14.7326i	−0.440662	−	0.525161i	0.499305	−	0.866427i	$-0.333589\pi$
$787$	−12.3621	−	14.7326i	−0.440662	−	0.525161i	−0.939967	+	0.341266i	$0.889144\pi$
$788$		0			0
$789$	−4.80411	−	7.51057i	−0.171031	−	0.267383i
$790$		0			0
$791$	11.4236	−	19.7862i	0.406176	−	0.703517i
$792$		0			0
$793$	32.0762	+	55.5577i	1.13906	+	1.97291i
$794$		0			0
$795$	4.01200

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 432 = 2^{4} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	432.be (of order \(18\), degree \(6\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.65284 - 0.517813i) q^{3} +(-2.52917 + 0.445961i) q^{5} +(-1.40789 + 1.67786i) q^{7} +(2.46374 + 1.71172i) q^{9} +O(q^{10})\)	\(q+(-1.65284 - 0.517813i) q^{3} +(-2.52917 + 0.445961i) q^{5} +(-1.40789 + 1.67786i) q^{7} +(2.46374 + 1.71172i) q^{9} +(0.751810 - 4.26373i) q^{11} +(5.33154 + 1.94052i) q^{13} +(4.41123 + 0.572537i) q^{15} +(5.53709 - 3.19684i) q^{17} +(-2.85493 - 1.64830i) q^{19} +(3.19584 - 2.04421i) q^{21} +(3.31503 - 2.78164i) q^{23} +(1.49936 - 0.545724i) q^{25} +(-3.18581 - 4.10495i) q^{27} +(0.131941 + 0.362506i) q^{29} +(4.37197 + 5.21032i) q^{31} +(-3.45043 + 6.65795i) q^{33} +(2.81254 - 4.87147i) q^{35} +(-3.81950 - 6.61557i) q^{37} +(-7.80734 - 5.96811i) q^{39} +(-0.138464 + 0.380428i) q^{41} +(9.35445 + 1.64944i) q^{43} +(-6.99458 - 3.23050i) q^{45} +(6.74409 + 5.65897i) q^{47} +(0.382480 + 2.16915i) q^{49} +(-10.8073 + 2.41668i) q^{51} -7.00741i q^{53} +11.1190i q^{55} +(3.86523 + 4.20269i) q^{57} +(0.296830 + 1.68341i) q^{59} +(8.66165 + 7.26798i) q^{61} +(-6.34071 + 1.72389i) q^{63} +(-14.3498 - 2.53025i) q^{65} +(-0.683922 + 1.87906i) q^{67} +(-6.91957 + 2.88103i) q^{69} +(-1.85419 - 3.21154i) q^{71} +(5.37828 - 9.31546i) q^{73} +(-2.76079 + 0.125602i) q^{75} +(6.09548 + 7.26431i) q^{77} +(-2.34025 - 6.42979i) q^{79} +(3.14002 + 8.43447i) q^{81} +(1.31616 - 0.479045i) q^{83} +(-12.5786 + 10.5547i) q^{85} +(-0.0303672 - 0.667485i) q^{87} +(-6.85289 - 3.95652i) q^{89} +(-10.7622 + 6.21354i) q^{91} +(-4.52819 - 10.8757i) q^{93} +(7.95569 + 2.89564i) q^{95} +(2.09088 - 11.8580i) q^{97} +(9.15058 - 9.21782i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q + 3 q^{5} + 6 q^{9}+O(q^{10}) \) 36 * q + 3 * q^5 + 6 * q^9	\( 36 q + 3 q^{5} + 6 q^{9} - 18 q^{11} + 9 q^{15} + 18 q^{21} - 9 q^{25} + 30 q^{29} + 27 q^{31} + 27 q^{33} + 27 q^{35} - 45 q^{39} + 18 q^{41} + 27 q^{45} + 45 q^{47} - 63 q^{51} - 9 q^{57} + 54 q^{59} - 63 q^{63} - 57 q^{65} - 63 q^{69} + 36 q^{71} + 9 q^{73} - 45 q^{75} - 81 q^{77} - 54 q^{81} - 27 q^{83} - 36 q^{85} + 45 q^{87} - 63 q^{89} + 27 q^{91} - 63 q^{93} - 72 q^{95} + 99 q^{99}+O(q^{100}) \) 36 * q + 3 * q^5 + 6 * q^9 - 18 * q^11 + 9 * q^15 + 18 * q^21 - 9 * q^25 + 30 * q^29 + 27 * q^31 + 27 * q^33 + 27 * q^35 - 45 * q^39 + 18 * q^41 + 27 * q^45 + 45 * q^47 - 63 * q^51 - 9 * q^57 + 54 * q^59 - 63 * q^63 - 57 * q^65 - 63 * q^69 + 36 * q^71 + 9 * q^73 - 45 * q^75 - 81 * q^77 - 54 * q^81 - 27 * q^83 - 36 * q^85 + 45 * q^87 - 63 * q^89 + 27 * q^91 - 63 * q^93 - 72 * q^95 + 99 * q^99

Introduction

L-functions

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