LMFDB - Embedded newform 432.2.be.b.191.5

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			191.5
Character	$\chi$	$=$	432.191
Dual form			432.2.be.b.95.5

$q$-expansion

$f(q)$	$=$	$q+(1.63553 - 0.570126i) q^{3} +(2.34697 + 2.79701i) q^{5} +(0.550750 + 1.51317i) q^{7} +(2.34991 - 1.86492i) q^{9} +O(q^{10})$	\(q+(1.63553 - 0.570126i) q^{3} +(2.34697 + 2.79701i) q^{5} +(0.550750 + 1.51317i) q^{7} +(2.34991 - 1.86492i) q^{9} +(-2.21396 - 1.85773i) q^{11} +(0.121792 + 0.690719i) q^{13} +(5.43318 + 3.23652i) q^{15} +(-5.56131 + 3.21082i) q^{17} +(-1.62624 - 0.938910i) q^{19} +(1.76347 + 2.16084i) q^{21} +(-2.18387 - 0.794864i) q^{23} +(-1.44675 + 8.20494i) q^{25} +(2.78011 - 4.38987i) q^{27} +(5.54988 + 0.978594i) q^{29} +(3.63921 - 9.99865i) q^{31} +(-4.68013 - 1.77614i) q^{33} +(-2.93977 + 5.09183i) q^{35} +(-4.10464 - 7.10944i) q^{37} +(0.592992 + 1.06025i) q^{39} +(6.18442 - 1.09048i) q^{41} +(5.65989 - 6.74520i) q^{43} +(10.7314 + 2.19583i) q^{45} +(-1.67459 + 0.609501i) q^{47} +(3.37594 - 2.83275i) q^{49} +(-7.26511 + 8.42204i) q^{51} +0.849889i q^{53} -10.5525i q^{55} +(-3.19506 - 0.608453i) q^{57} +(-4.54459 + 3.81337i) q^{59} +(-9.95979 + 3.62507i) q^{61} +(4.11616 + 2.52872i) q^{63} +(-1.64610 + 1.96175i) q^{65} +(-14.5520 + 2.56590i) q^{67} +(-4.02496 - 0.0549417i) q^{69} +(5.80783 + 10.0595i) q^{71} +(3.12044 - 5.40475i) q^{73} +(2.31165 + 14.2443i) q^{75} +(1.59173 - 4.37325i) q^{77} +(4.47831 + 0.789647i) q^{79} +(2.04418 - 8.76478i) q^{81} +(-1.28300 + 7.27624i) q^{83} +(-22.0329 - 8.01933i) q^{85} +(9.63492 - 1.56361i) q^{87} +(-6.02277 - 3.47725i) q^{89} +(-0.978101 + 0.564707i) q^{91} +(0.251545 - 18.4279i) q^{93} +(-1.19060 - 6.75220i) q^{95} +(1.35281 + 1.13514i) q^{97} +(-8.66711 - 0.236660i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/432\mathbb{Z}\right)^\times$.

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

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
$3$	1.63553	−	0.570126i	0.944273	−	0.329162i
$3$	1.63553	−	0.570126i	0.944273	−	0.329162i
$4$		0			0
$5$	2.34697	+	2.79701i	1.04960	+	1.25086i	0.967134	+	0.254266i	$0.0818339\pi$
$5$	2.34697	+	2.79701i	1.04960	+	1.25086i	0.0824620	+	0.996594i	$0.473722\pi$
$6$		0			0
$7$	0.550750	+	1.51317i	0.208164	+	0.571926i	0.999206	−	0.0398355i	$-0.0126834\pi$
$7$	0.550750	+	1.51317i	0.208164	+	0.571926i	−0.791042	+	0.611762i	$0.790461\pi$
$8$		0			0
$9$	2.34991	−	1.86492i	0.783304	−	0.621639i
$10$		0			0
$11$	−2.21396	−	1.85773i	−0.667533	−	0.560127i	0.244801	−	0.969573i	$-0.421277\pi$
$11$	−2.21396	−	1.85773i	−0.667533	−	0.560127i	−0.912334	+	0.409447i	$0.865722\pi$
$12$		0			0
$13$	0.121792	+	0.690719i	0.0337791	+	0.191571i	0.997028	−	0.0770394i	$-0.0245467\pi$
$13$	0.121792	+	0.690719i	0.0337791	+	0.191571i	−0.963249	+	0.268610i	$0.913436\pi$
$14$		0			0
$15$	5.43318	+	3.23652i	1.40284	+	0.835666i
$16$		0			0
$17$	−5.56131	+	3.21082i	−1.34882	+	0.778739i	−0.988082	−	0.153930i	$-0.950807\pi$
$17$	−5.56131	+	3.21082i	−1.34882	+	0.778739i	−0.360734	+	0.932669i	$0.617474\pi$
$18$		0			0
$19$	−1.62624	−	0.938910i	−0.373085	−	0.215401i	0.301720	−	0.953396i	$-0.402439\pi$
$19$	−1.62624	−	0.938910i	−0.373085	−	0.215401i	−0.674805	+	0.737996i	$0.735772\pi$
$20$		0			0
$21$	1.76347	+	2.16084i	0.384820	+	0.471535i
$22$		0			0
$23$	−2.18387	−	0.794864i	−0.455369	−	0.165741i	0.104144	−	0.994562i	$-0.466790\pi$
$23$	−2.18387	−	0.794864i	−0.455369	−	0.165741i	−0.559513	+	0.828822i	$0.689012\pi$
$24$		0			0
$25$	−1.44675	+	8.20494i	−0.289351	+	1.64099i
$26$		0			0
$27$	2.78011	−	4.38987i	0.535033	−	0.844831i
$28$		0			0
$29$	5.54988	+	0.978594i	1.03059	+	0.181720i	0.663274	−	0.748377i	$-0.269167\pi$
$29$	5.54988	+	0.978594i	1.03059	+	0.181720i	0.367313	+	0.930097i	$0.380278\pi$
$30$		0			0
$31$	3.63921	−	9.99865i	0.653622	−	1.79581i	0.0497045	−	0.998764i	$-0.484172\pi$
$31$	3.63921	−	9.99865i	0.653622	−	1.79581i	0.603917	−	0.797047i	$-0.293606\pi$
$32$		0			0
$33$	−4.68013	−	1.77614i	−0.814706	−	0.309186i
$34$		0			0
$35$	−2.93977	+	5.09183i	−0.496911	+	0.860676i
$36$		0			0
$37$	−4.10464	−	7.10944i	−0.674798	−	1.16878i	−0.976528	−	0.215391i	$-0.930897\pi$
$37$	−4.10464	−	7.10944i	−0.674798	−	1.16878i	0.301730	−	0.953393i	$-0.402436\pi$
$38$		0			0
$39$	0.592992	+	1.06025i	0.0949547	+	0.169777i
$40$		0			0
$41$	6.18442	−	1.09048i	0.965843	−	0.170304i	0.331585	−	0.943425i	$-0.392417\pi$
$41$	6.18442	−	1.09048i	0.965843	−	0.170304i	0.634258	+	0.773121i	$0.281306\pi$
$42$		0			0
$43$	5.65989	−	6.74520i	0.863125	−	1.02863i	−0.136154	−	0.990688i	$-0.543474\pi$
$43$	5.65989	−	6.74520i	0.863125	−	1.02863i	0.999280	−	0.0379452i	$-0.0120812\pi$
$44$		0			0
$45$	10.7314	+	2.19583i	1.59974	+	0.327334i
$46$		0			0
$47$	−1.67459	+	0.609501i	−0.244264	+	0.0889049i	−0.461251	−	0.887270i	$-0.652599\pi$
$47$	−1.67459	+	0.609501i	−0.244264	+	0.0889049i	0.216987	+	0.976174i	$0.430377\pi$
$48$		0			0
$49$	3.37594	−	2.83275i	0.482277	−	0.404679i
$50$		0			0
$51$	−7.26511	+	8.42204i	−1.01732	+	1.17932i
$52$		0			0
$53$			0.849889i			0.116741i	0.998295	+	0.0583706i	$0.0185905\pi$
$53$			0.849889i			0.116741i	−0.998295	+	0.0583706i	$0.981409\pi$
$54$		0			0
$55$		−	10.5525i		−	1.42290i
$56$		0			0
$57$	−3.19506	−	0.608453i	−0.423196	−	0.0805916i
$58$		0			0
$59$	−4.54459	+	3.81337i	−0.591656	+	0.496458i	−0.888751	−	0.458390i	$-0.848426\pi$
$59$	−4.54459	+	3.81337i	−0.591656	+	0.496458i	0.297096	+	0.954848i	$0.403982\pi$
$60$		0			0
$61$	−9.95979	+	3.62507i	−1.27522	+	0.464142i	−0.888849	−	0.458201i	$-0.848494\pi$
$61$	−9.95979	+	3.62507i	−1.27522	+	0.464142i	−0.386372	+	0.922343i	$0.626272\pi$
$62$		0			0
$63$	4.11616	+	2.52872i	0.518587	+	0.318589i
$64$		0			0
$65$	−1.64610	+	1.96175i	−0.204174	+	0.243325i
$66$		0			0
$67$	−14.5520	+	2.56590i	−1.77781	+	0.313475i	−0.963649	−	0.267171i	$-0.913911\pi$
$67$	−14.5520	+	2.56590i	−1.77781	+	0.313475i	−0.814156	+	0.580646i	$0.802800\pi$
$68$		0			0
$69$	−4.02496	−	0.0549417i	−0.484548	−	0.00661420i
$70$		0			0
$71$	5.80783	+	10.0595i	0.689263	+	1.19384i	0.972077	+	0.234663i	$0.0753988\pi$
$71$	5.80783	+	10.0595i	0.689263	+	1.19384i	−0.282814	+	0.959175i	$0.591268\pi$
$72$		0			0
$73$	3.12044	−	5.40475i	0.365219	−	0.632579i	−0.623592	−	0.781750i	$-0.714327\pi$
$73$	3.12044	−	5.40475i	0.365219	−	0.632579i	0.988811	+	0.149171i	$0.0476606\pi$
$74$		0			0
$75$	2.31165	+	14.2443i	0.266926	+	1.64479i
$76$		0			0
$77$	1.59173	−	4.37325i	0.181395	−	0.498378i
$78$		0			0
$79$	4.47831	+	0.789647i	0.503849	+	0.0888422i	0.419795	−	0.907619i	$-0.362102\pi$
$79$	4.47831	+	0.789647i	0.503849	+	0.0888422i	0.0840543	+	0.996461i	$0.473213\pi$
$80$		0			0
$81$	2.04418	−	8.76478i	0.227131	−	0.973864i
$82$		0			0
$83$	−1.28300	+	7.27624i	−0.140827	+	0.798671i	0.829796	+	0.558067i	$0.188457\pi$
$83$	−1.28300	+	7.27624i	−0.140827	+	0.798671i	−0.970623	+	0.240604i	$0.922654\pi$
$84$		0			0
$85$	−22.0329	−	8.01933i	−2.38981	−	0.869818i
$86$		0			0
$87$	9.63492	−	1.56361i	1.03297	−	0.167637i
$88$		0			0
$89$	−6.02277	−	3.47725i	−0.638412	−	0.368588i	0.145590	−	0.989345i	$-0.453492\pi$
$89$	−6.02277	−	3.47725i	−0.638412	−	0.368588i	−0.784003	+	0.620757i	$0.786825\pi$
$90$		0			0
$91$	−0.978101	+	0.564707i	−0.102533	+	0.0591974i
$92$		0			0
$93$	0.251545	−	18.4279i	0.0260840	−	1.91088i
$94$		0			0
$95$	−1.19060	−	6.75220i	−0.122152	−	0.692761i
$96$		0			0
$97$	1.35281	+	1.13514i	0.137357	+	0.115256i	0.708878	−	0.705331i	$-0.249202\pi$
$97$	1.35281	+	1.13514i	0.137357	+	0.115256i	−0.571521	+	0.820588i	$0.693646\pi$
$98$		0			0
$99$	−8.66711	−	0.236660i	−0.871078	−	0.0237853i
$100$		0			0
$101$	3.88953	+	10.6864i	0.387023	+	1.06334i	0.968335	+	0.249656i	$0.0803175\pi$
$101$	3.88953	+	10.6864i	0.387023	+	1.06334i	−0.581312	+	0.813681i	$0.697460\pi$
$102$		0			0
$103$	10.6329	+	12.6718i	1.04769	+	1.24859i	0.967783	+	0.251784i	$0.0810174\pi$
$103$	10.6329	+	12.6718i	1.04769	+	1.24859i	0.0799054	+	0.996802i	$0.474538\pi$
$104$		0			0
$105$	−1.90509	+	10.0039i	−0.185918	+	0.976278i
$106$		0			0
$107$	−16.6875			−1.61324			−0.806619	−	0.591071i	$-0.798705\pi$
$107$	−16.6875			−1.61324			−0.806619	+	0.591071i	$0.798705\pi$
$108$		0			0
$109$	10.2128			0.978212			0.489106	−	0.872224i	$-0.337323\pi$
$109$	10.2128			0.978212			0.489106	+	0.872224i	$0.337323\pi$
$110$		0			0
$111$	−10.7665	−	9.28753i	−1.02191	−	0.881534i
$112$		0			0
$113$	−9.06215	−	10.7998i	−0.852495	−	1.01596i	−0.999639	−	0.0268492i	$-0.991453\pi$
$113$	−9.06215	−	10.7998i	−0.852495	−	1.01596i	0.147144	−	0.989115i	$-0.452992\pi$
$114$		0			0
$115$	−2.90224	−	7.97383i	−0.270635	−	0.743564i
$116$		0			0
$117$	1.57433	+	1.39600i	0.145547	+	0.129060i
$118$		0			0
$119$	−7.92143	−	6.64687i	−0.726156	−	0.609317i
$120$		0			0
$121$	−0.459688	−	2.60702i	−0.0417898	−	0.237002i
$122$		0			0
$123$	9.49308	−	5.30941i	0.855962	−	0.478733i
$124$		0			0
$125$	−10.5345	+	6.08208i	−0.942232	+	0.543998i
$126$		0			0
$127$	−6.11030	−	3.52779i	−0.542202	−	0.313040i	0.203769	−	0.979019i	$-0.434681\pi$
$127$	−6.11030	−	3.52779i	−0.542202	−	0.313040i	−0.745971	+	0.665979i	$0.768014\pi$
$128$		0			0
$129$	5.41131	−	14.2588i	0.476439	−	1.25542i
$130$		0			0
$131$	−13.6909	−	4.98310i	−1.19618	−	0.435375i	−0.334293	−	0.942469i	$-0.608497\pi$
$131$	−13.6909	−	4.98310i	−1.19618	−	0.435375i	−0.861891	+	0.507094i	$0.830720\pi$
$132$		0			0
$133$	0.525082	−	2.97789i	0.0455304	−	0.258216i
$134$		0			0
$135$	18.8033	−	2.52689i	1.61833	−	0.217480i
$136$		0			0
$137$	14.3046	+	2.52229i	1.22212	+	0.215493i	0.747240	−	0.664555i	$-0.231379\pi$
$137$	14.3046	+	2.52229i	1.22212	+	0.215493i	0.474884	+	0.880048i	$0.342490\pi$
$138$		0			0
$139$	−3.14344	+	8.63654i	−0.266624	+	0.732542i	0.732060	+	0.681240i	$0.238559\pi$
$139$	−3.14344	+	8.63654i	−0.266624	+	0.732542i	−0.998683	+	0.0513016i	$0.983663\pi$
$140$		0			0
$141$	−2.39135	+	1.95158i	−0.201388	+	0.164353i
$142$		0			0
$143$	1.01353	−	1.75548i	0.0847553	−	0.146801i
$144$		0			0
$145$	10.2883	+	17.8198i	0.854394	+	1.47985i
$146$		0			0
$147$	3.90642	−	6.55776i	0.322196	−	0.540875i
$148$		0			0
$149$	−1.24126	+	0.218868i	−0.101688	+	0.0179304i	−0.224261	−	0.974529i	$-0.571997\pi$
$149$	−1.24126	+	0.218868i	−0.101688	+	0.0179304i	0.122573	+	0.992460i	$0.460886\pi$
$150$		0			0
$151$	−7.74987	+	9.23593i	−0.630675	+	0.751609i	−0.982867	−	0.184319i	$-0.940992\pi$
$151$	−7.74987	+	9.23593i	−0.630675	+	0.751609i	0.352191	+	0.935928i	$0.385437\pi$
$152$		0			0
$153$	−7.08067	+	17.9165i	−0.572439	+	1.44847i
$154$		0			0
$155$	36.5074	−	13.2876i	2.93235	−	1.06729i
$156$		0			0
$157$	−2.80380	+	2.35267i	−0.223767	+	0.187763i	−0.747779	−	0.663948i	$-0.768879\pi$
$157$	−2.80380	+	2.35267i	−0.223767	+	0.187763i	0.524011	+	0.851711i	$0.324435\pi$
$158$		0			0
$159$	0.484544	+	1.39002i	0.0384268	+	0.110236i
$160$		0			0
$161$		−	3.74235i		−	0.294939i
$162$		0			0
$163$		−	2.44194i		−	0.191267i	−0.995417	−	0.0956336i	$-0.969512\pi$
$163$		−	2.44194i		−	0.191267i	0.995417	−	0.0956336i	$-0.0304877\pi$
$164$		0			0
$165$	−6.01625	−	17.2589i	−0.468364	−	1.34360i
$166$		0			0
$167$	−4.06934	+	3.41458i	−0.314895	+	0.264229i	−0.786512	−	0.617575i	$-0.788115\pi$
$167$	−4.06934	+	3.41458i	−0.314895	+	0.264229i	0.471617	+	0.881804i	$0.343671\pi$
$168$		0			0
$169$	11.7537	−	4.27801i	0.904134	−	0.329078i
$170$		0			0
$171$	−5.57251	+	0.826444i	−0.426141	+	0.0631998i
$172$		0			0
$173$	−16.8612	+	20.0945i	−1.28194	+	1.52775i	−0.583377	+	0.812202i	$0.698269\pi$
$173$	−16.8612	+	20.0945i	−1.28194	+	1.52775i	−0.698560	+	0.715551i	$0.746176\pi$
$174$		0			0
$175$	−13.2123	+	2.32969i	−0.998757	+	0.176108i
$176$		0			0
$177$	−5.25872	+	8.82786i	−0.395269	+	0.663543i
$178$		0			0
$179$	−5.42787	−	9.40134i	−0.405698	−	0.702689i	0.588705	−	0.808348i	$-0.299638\pi$
$179$	−5.42787	−	9.40134i	−0.405698	−	0.702689i	−0.994402	+	0.105659i	$0.966305\pi$
$180$		0			0
$181$	2.12621	−	3.68270i	0.158040	−	0.273733i	−0.776122	−	0.630583i	$-0.782816\pi$
$181$	2.12621	−	3.68270i	0.158040	−	0.273733i	0.934162	+	0.356850i	$0.116149\pi$
$182$		0			0
$183$	−14.2228	+	11.6072i	−1.05138	+	0.858032i
$184$		0			0
$185$	10.2517	−	28.1663i	0.753720	−	2.07083i
$186$		0			0
$187$	18.2773	+	3.22279i	1.33657	+	0.235674i
$188$		0			0
$189$	8.17379	+	1.78907i	0.594556	+	0.130136i
$190$		0			0
$191$	−1.25388	+	7.11111i	−0.0907276	+	0.514542i	0.905245	+	0.424889i	$0.139687\pi$
$191$	−1.25388	+	7.11111i	−0.0907276	+	0.514542i	−0.995973	+	0.0896529i	$0.971424\pi$
$192$		0			0
$193$	−15.5914	−	5.67481i	−1.12229	−	0.408482i	−0.286805	−	0.957989i	$-0.592593\pi$
$193$	−15.5914	−	5.67481i	−1.12229	−	0.408482i	−0.835489	+	0.549507i	$0.814815\pi$
$194$		0			0
$195$	−1.57381	+	4.14699i	−0.112703	+	0.296972i
$196$		0			0
$197$	4.35620	+	2.51505i	0.310367	+	0.179190i	0.647091	−	0.762413i	$-0.275986\pi$
$197$	4.35620	+	2.51505i	0.310367	+	0.179190i	−0.336724	+	0.941603i	$0.609319\pi$
$198$		0			0
$199$	12.8199	−	7.40159i	0.908781	−	0.524685i	0.0287420	−	0.999587i	$-0.490850\pi$
$199$	12.8199	−	7.40159i	0.908781	−	0.524685i	0.880039	+	0.474902i	$0.157517\pi$
$200$		0			0
$201$	−22.3373	+	12.4931i	−1.57555	+	0.881193i
$202$		0			0
$203$	1.57582	+	8.93690i	0.110601	+	0.627247i
$204$		0			0
$205$	17.5647	+	14.7385i	1.22677	+	1.02938i
$206$		0			0
$207$	−6.61426	+	2.20488i	−0.459723	+	0.153249i
$208$		0			0
$209$	1.85618	+	5.09982i	0.128395	+	0.352762i
$210$		0			0
$211$	7.82727	+	9.32818i	0.538851	+	0.642178i	0.964930	−	0.262508i	$-0.0845497\pi$
$211$	7.82727	+	9.32818i	0.538851	+	0.642178i	−0.426078	+	0.904686i	$0.640105\pi$
$212$		0			0
$213$	15.2340	+	13.1413i	1.04382	+	0.900430i
$214$		0			0
$215$	32.1500			2.19261
$216$		0			0
$217$	17.1340			1.16313
$218$		0			0
$219$	2.02217	−	10.6187i	0.136646	−	0.717544i
$220$		0			0
$221$	−2.89510	−	3.45025i	−0.194746	−	0.232089i
$222$		0			0
$223$	3.42628	+	9.41364i	0.229441	+	0.630384i	0.999975	−	0.00701461i	$-0.00223284\pi$
$223$	3.42628	+	9.41364i	0.229441	+	0.630384i	−0.770535	+	0.637398i	$0.780011\pi$
$224$		0			0
$225$	11.9018	+	21.9790i	0.793453	+	1.46526i
$226$		0			0
$227$	7.69615	+	6.45783i	0.510811	+	0.428622i	0.861415	−	0.507903i	$-0.169579\pi$
$227$	7.69615	+	6.45783i	0.510811	+	0.428622i	−0.350603	+	0.936524i	$0.614023\pi$
$228$		0			0
$229$	−2.02551	−	11.4872i	−0.133849	−	0.759096i	−0.975654	−	0.219315i	$-0.929618\pi$
$229$	−2.02551	−	11.4872i	−0.133849	−	0.759096i	0.841805	−	0.539782i	$-0.181493\pi$
$230$		0			0
$231$	0.110022	−	8.06006i	0.00723890	−	0.530313i
$232$		0			0
$233$	11.1464	−	6.43535i	0.730222	−	0.421594i	−0.0882815	−	0.996096i	$-0.528137\pi$
$233$	11.1464	−	6.43535i	0.730222	−	0.421594i	0.818503	+	0.574502i	$0.194804\pi$
$234$		0			0
$235$	−5.63499	−	3.25336i	−0.367586	−	0.212226i
$236$		0			0
$237$	7.77461	−	1.26171i	0.505015	−	0.0819569i
$238$		0			0
$239$	3.79461	+	1.38112i	0.245453	+	0.0893375i	0.461817	−	0.886975i	$-0.347198\pi$
$239$	3.79461	+	1.38112i	0.245453	+	0.0893375i	−0.216364	+	0.976313i	$0.569420\pi$
$240$		0			0
$241$	−3.73127	+	21.1611i	−0.240352	+	1.36311i	0.590691	+	0.806898i	$0.298855\pi$
$241$	−3.73127	+	21.1611i	−0.240352	+	1.36311i	−0.831043	+	0.556208i	$0.812256\pi$
$242$		0			0
$243$	−1.65372	−	15.5005i	−0.106086	−	0.994357i
$244$		0			0
$245$	15.8465	+	2.79416i	1.01239	+	0.178512i
$246$		0			0
$247$	0.450459	−	1.23763i	0.0286620	−	0.0787483i
$248$		0			0
$249$	2.04999	+	12.6320i	0.129913	+	0.800519i
$250$		0			0
$251$	−2.12275	+	3.67671i	−0.133987	+	0.232072i	−0.925210	−	0.379455i	$-0.876111\pi$
$251$	−2.12275	+	3.67671i	−0.133987	+	0.232072i	0.791223	+	0.611528i	$0.209445\pi$
$252$		0			0
$253$	3.35835	+	5.81684i	0.211138	+	0.365702i
$254$		0			0
$255$	−40.6075	−	0.554302i	−2.54294	−	0.0347117i
$256$		0			0
$257$	5.12260	−	0.903252i	0.319539	−	0.0563433i	−0.0115784	−	0.999933i	$-0.503686\pi$
$257$	5.12260	−	0.903252i	0.319539	−	0.0563433i	0.331117	+	0.943590i	$0.392574\pi$
$258$		0			0
$259$	8.49719	−	10.1266i	0.527990	−	0.629233i
$260$		0			0
$261$	14.8667	−	8.05045i	0.920228	−	0.498310i
$262$		0			0
$263$	22.9602	−	8.35683i	1.41579	−	0.515304i	0.482964	−	0.875640i	$-0.339560\pi$
$263$	22.9602	−	8.35683i	1.41579	−	0.515304i	0.932823	+	0.360336i	$0.117338\pi$
$264$		0			0
$265$	−2.37715	+	1.99466i	−0.146027	+	0.122531i
$266$		0			0
$267$	−11.8329	−	2.25340i	−0.724161	−	0.137906i
$268$		0			0
$269$			0.485873i			0.0296242i	0.999890	+	0.0148121i	$0.00471501\pi$
$269$			0.485873i			0.0296242i	−0.999890	+	0.0148121i	$0.995285\pi$
$270$		0			0
$271$			15.4958i			0.941306i	0.882318	+	0.470653i	$0.155982\pi$
$271$			15.4958i			0.941306i	−0.882318	+	0.470653i	$0.844018\pi$
$272$		0			0
$273$	−1.27776	+	1.48124i	−0.0773335	+	0.0896485i
$274$		0			0
$275$	18.4456	−	15.4777i	1.11231	−	0.933341i
$276$		0			0
$277$	−13.7884	+	5.01856i	−0.828463	+	0.301536i	−0.721228	−	0.692698i	$-0.756422\pi$
$277$	−13.7884	+	5.01856i	−0.828463	+	0.301536i	−0.107235	+	0.994234i	$0.534200\pi$
$278$		0			0
$279$	−10.0948	−	30.2828i	−0.604361	−	1.81298i
$280$		0			0
$281$	0.732164	−	0.872559i	0.0436772	−	0.0520525i	−0.743764	−	0.668443i	$-0.766961\pi$
$281$	0.732164	−	0.872559i	0.0436772	−	0.0520525i	0.787441	+	0.616390i	$0.211406\pi$
$282$		0			0
$283$	4.21444	−	0.743119i	0.250522	−	0.0441739i	−0.0469766	−	0.998896i	$-0.514959\pi$
$283$	4.21444	−	0.743119i	0.250522	−	0.0441739i	0.297499	+	0.954722i	$0.403848\pi$
$284$		0			0
$285$	−5.79686	−	10.3646i	−0.343376	−	0.613948i
$286$		0			0
$287$	5.05615	+	8.75752i	0.298455	+	0.516940i
$288$		0			0
$289$	12.1188	−	20.9903i	0.712869	−	1.23472i
$290$		0			0
$291$	2.85974	+	1.08529i	0.167641	+	0.0636207i
$292$		0			0
$293$	−2.29569	+	6.30736i	−0.134116	+	0.368480i	−0.988512	−	0.151142i	$-0.951705\pi$
$293$	−2.29569	+	6.30736i	−0.134116	+	0.368480i	0.854396	+	0.519622i	$0.173927\pi$
$294$		0			0
$295$	−21.3320	−	3.76141i	−1.24200	−	0.218998i
$296$		0			0
$297$	−14.3102	+	4.55428i	−0.830365	+	0.264266i
$298$		0			0
$299$	0.283049	−	1.60525i	0.0163691	−	0.0928340i
$300$		0			0
$301$	13.3238	+	4.84948i	0.767974	+	0.279520i
$302$		0			0
$303$	12.4540	+	15.2604i	0.715466	+	0.876687i
$304$		0			0
$305$	−33.5147	−	19.3497i	−1.91904	−	1.10796i
$306$		0			0
$307$	−9.47236	+	5.46887i	−0.540616	+	0.312125i	−0.745329	−	0.666697i	$-0.767707\pi$
$307$	−9.47236	+	5.46887i	−0.540616	+	0.312125i	0.204713	+	0.978822i	$0.434374\pi$
$308$		0			0
$309$	24.6149	+	14.6630i	1.40029	+	0.834147i
$310$		0			0
$311$	−1.99845	−	11.3338i	−0.113322	−	0.642678i	−0.987568	−	0.157195i	$-0.949755\pi$
$311$	−1.99845	−	11.3338i	−0.113322	−	0.642678i	0.874246	−	0.485483i	$-0.161356\pi$
$312$		0			0
$313$	−5.50981	−	4.62328i	−0.311433	−	0.261323i	0.473651	−	0.880713i	$-0.342936\pi$
$313$	−5.50981	−	4.62328i	−0.311433	−	0.261323i	−0.785084	+	0.619389i	$0.787380\pi$
$314$		0			0
$315$	2.58763	+	17.4478i	0.145797	+	0.983070i
$316$		0			0
$317$	−5.58824	−	15.3536i	−0.313867	−	0.862342i	−0.991867	−	0.127280i	$-0.959375\pi$
$317$	−5.58824	−	15.3536i	−0.313867	−	0.862342i	0.678000	−	0.735062i	$-0.262847\pi$
$318$		0			0
$319$	−10.4692	−	12.4767i	−0.586164	−	0.698564i
$320$		0			0
$321$	−27.2928	+	9.51396i	−1.52334	+	0.531018i
$322$		0			0
$323$	12.0587			0.670964
$324$		0			0
$325$	−5.84351			−0.324140
$326$		0			0
$327$	16.7034	−	5.82260i	0.923700	−	0.321991i
$328$		0			0
$329$	−1.84456	−	2.19826i	−0.101694	−	0.121194i
$330$		0			0
$331$	8.42268	+	23.1411i	0.462952	+	1.27195i	0.923255	+	0.384189i	$0.125519\pi$
$331$	8.42268	+	23.1411i	0.462952	+	1.27195i	−0.460302	+	0.887762i	$0.652259\pi$
$332$		0			0
$333$	−22.9040	−	9.05175i	−1.25513	−	0.496033i
$334$		0			0
$335$	−41.3299	−	34.6799i	−2.25809	−	1.89476i
$336$		0			0
$337$	0.871464	+	4.94232i	0.0474717	+	0.269225i	0.999300	−	0.0374058i	$-0.0119094\pi$
$337$	0.871464	+	4.94232i	0.0474717	+	0.269225i	−0.951829	+	0.306631i	$0.900798\pi$
$338$		0			0
$339$	−20.9787	−	12.4969i	−1.13941	−	0.678738i
$340$		0			0
$341$	−26.6319	+	15.3759i	−1.44220	+	0.832652i
$342$		0			0
$343$	15.9076	+	9.18425i	0.858929	+	0.495903i
$344$		0			0
$345$	−9.29278	−	11.3868i	−0.500307	−	0.613044i
$346$		0			0
$347$	14.1651	+	5.15569i	0.760424	+	0.276772i	0.692985	−	0.720952i	$-0.256295\pi$
$347$	14.1651	+	5.15569i	0.760424	+	0.276772i	0.0674388	+	0.997723i	$0.478517\pi$
$348$		0			0
$349$	−0.0306727	+	0.173953i	−0.00164187	+	0.00931151i	−0.985618	−	0.168990i	$-0.945949\pi$
$349$	−0.0306727	+	0.173953i	−0.00164187	+	0.00931151i	0.983976	+	0.178302i	$0.0570604\pi$
$350$		0			0
$351$	3.37076	+	1.38562i	0.179918	+	0.0739591i
$352$		0			0
$353$	4.03523	+	0.711520i	0.214774	+	0.0378704i	0.280000	−	0.960000i	$-0.409666\pi$
$353$	4.03523	+	0.711520i	0.214774	+	0.0378704i	−0.0652259	+	0.997871i	$0.520777\pi$
$354$		0			0
$355$	−14.5056	+	39.8538i	−0.769877	+	2.11522i
$356$		0			0
$357$	−16.7453	−	6.35493i	−0.886254	−	0.336339i
$358$		0			0
$359$	0.130804	−	0.226558i	0.00690355	−	0.0119573i	−0.862553	−	0.505967i	$-0.831136\pi$
$359$	0.130804	−	0.226558i	0.00690355	−	0.0119573i	0.869457	+	0.494009i	$0.164469\pi$
$360$		0			0
$361$	−7.73689	−	13.4007i	−0.407205	−	0.705300i
$362$		0			0
$363$	−2.23816	−	4.00178i	−0.117473	−	0.210039i
$364$		0			0
$365$	22.4407	−	3.95690i	1.17460	−	0.207114i
$366$		0			0
$367$	−9.25704	+	11.0321i	−0.483214	+	0.575872i	−0.951478	−	0.307717i	$-0.900435\pi$
$367$	−9.25704	+	11.0321i	−0.483214	+	0.575872i	0.468264	+	0.883588i	$0.344880\pi$
$368$		0			0
$369$	12.4992	−	14.0959i	0.650681	−	0.733806i
$370$		0			0
$371$	−1.28603	+	0.468077i	−0.0667674	+	0.0243013i
$372$		0			0
$373$	0.506826	−	0.425278i	0.0262425	−	0.0220200i	−0.629572	−	0.776942i	$-0.716770\pi$
$373$	0.506826	−	0.425278i	0.0262425	−	0.0220200i	0.655815	+	0.754922i	$0.272325\pi$
$374$		0			0
$375$	−13.7619	+	15.9534i	−0.710661	+	0.823830i
$376$		0			0
$377$			3.95259i			0.203569i
$378$		0			0
$379$		−	1.73288i		−	0.0890121i	−0.999009	−	0.0445061i	$-0.985829\pi$
$379$		−	1.73288i		−	0.0890121i	0.999009	−	0.0445061i	$-0.0141714\pi$
$380$		0			0
$381$	−12.0049	−	2.28615i	−0.615028	−	0.117123i
$382$		0			0
$383$	−17.1332	+	14.3765i	−0.875466	+	0.734603i	−0.965242	−	0.261359i	$-0.915829\pi$
$383$	−17.1332	+	14.3765i	−0.875466	+	0.734603i	0.0897757	+	0.995962i	$0.471385\pi$
$384$		0			0
$385$	15.9678	−	5.81179i	0.813792	−	0.296196i
$386$		0			0
$387$	0.721026	−	26.4058i	0.0366518	−	1.34228i
$388$		0			0
$389$	2.05959	−	2.45453i	0.104426	−	0.124450i	−0.711300	−	0.702888i	$-0.751893\pi$
$389$	2.05959	−	2.45453i	0.104426	−	0.124450i	0.815726	+	0.578439i	$0.196338\pi$
$390$		0			0
$391$	14.6974	−	2.59154i	0.743277	−	0.131060i
$392$		0			0
$393$	−25.2329	−	0.344435i	−1.27283	−	0.0173745i
$394$		0			0
$395$	8.30181	+	14.3791i	0.417709	+	0.723494i
$396$		0			0
$397$	6.32972	−	10.9634i	0.317680	−	0.550237i	−0.662324	−	0.749218i	$-0.730430\pi$
$397$	6.32972	−	10.9634i	0.317680	−	0.550237i	0.980003	+	0.198981i	$0.0637630\pi$
$398$		0			0
$399$	−0.838985	−	5.16979i	−0.0420018	−	0.258813i
$400$		0			0
$401$	−4.72777	+	12.9894i	−0.236094	+	0.648662i	0.763901	+	0.645334i	$0.223282\pi$
$401$	−4.72777	+	12.9894i	−0.236094	+	0.648662i	−0.999994	+	0.00332813i	$0.998941\pi$
$402$		0			0
$403$	7.34949	+	1.29591i	0.366104	+	0.0645540i
$404$		0			0
$405$	29.3128	−	14.8531i	1.45656	−	0.738055i
$406$		0			0
$407$	−4.11993	+	23.3653i	−0.204217	+	1.15817i
$408$		0			0
$409$	35.2902	+	12.8446i	1.74499	+	0.635124i	0.999507	−	0.0314018i	$-0.00999714\pi$
$409$	35.2902	+	12.8446i	1.74499	+	0.635124i	0.745482	+	0.666526i	$0.232219\pi$
$410$		0			0
$411$	24.8336	−	4.03015i	1.22495	−	0.198793i
$412$		0			0
$413$	−8.27323	−	4.77655i	−0.407099	−	0.235039i
$414$		0			0
$415$	−23.3629	+	13.4885i	−1.14684	+	0.662127i
$416$		0			0
$417$	−0.217277	+	15.9175i	−0.0106401	+	0.779482i
$418$		0			0
$419$	−6.05967	−	34.3661i	−0.296034	−	1.67889i	−0.662969	−	0.748647i	$-0.730704\pi$
$419$	−6.05967	−	34.3661i	−0.296034	−	1.67889i	0.366934	−	0.930247i	$-0.380407\pi$
$420$		0			0
$421$	−3.66182	−	3.07264i	−0.178466	−	0.149751i	0.549179	−	0.835705i	$-0.314941\pi$
$421$	−3.66182	−	3.07264i	−0.178466	−	0.149751i	−0.727645	+	0.685954i	$0.759385\pi$
$422$		0			0
$423$	−2.79847	+	4.55524i	−0.136066	+	0.221484i
$424$		0			0
$425$	−18.2988	−	50.2755i	−0.887621	−	2.43872i
$426$		0			0
$427$	−10.9707	−	13.0744i	−0.530910	−	0.632714i
$428$		0			0
$429$	0.656808	−	3.44897i	0.0317110	−	0.166518i
$430$		0			0
$431$	−27.7539			−1.33686			−0.668430	−	0.743775i	$-0.733033\pi$
$431$	−27.7539			−1.33686			−0.668430	+	0.743775i	$0.733033\pi$
$432$		0			0
$433$	2.25395			0.108318			0.0541589	−	0.998532i	$-0.482752\pi$
$433$	2.25395			0.108318			0.0541589	+	0.998532i	$0.482752\pi$
$434$		0			0
$435$	26.9863	+	23.2792i	1.29389	+	1.11615i
$436$		0			0
$437$	2.80519	+	3.34310i	0.134191	+	0.159922i
$438$		0			0
$439$	5.90435	+	16.2221i	0.281799	+	0.774237i	0.997148	+	0.0754693i	$0.0240455\pi$
$439$	5.90435	+	16.2221i	0.281799	+	0.774237i	−0.715349	+	0.698767i	$0.753732\pi$
$440$		0			0
$441$	2.65032	−	12.9526i	0.126206	−	0.616789i
$442$		0			0
$443$	−5.44673	−	4.57035i	−0.258782	−	0.217144i	0.504161	−	0.863610i	$-0.331802\pi$
$443$	−5.44673	−	4.57035i	−0.258782	−	0.217144i	−0.762943	+	0.646466i	$0.776246\pi$
$444$		0			0
$445$	−4.40936	−	25.0067i	−0.209024	−	1.18543i
$446$		0			0
$447$	−1.90534	+	1.06564i	−0.0901194	+	0.0504031i
$448$		0			0
$449$	15.7138	−	9.07236i	0.741579	−	0.428151i	−0.0810641	−	0.996709i	$-0.525832\pi$
$449$	15.7138	−	9.07236i	0.741579	−	0.428151i	0.822643	+	0.568558i	$0.192499\pi$
$450$		0			0
$451$	−15.7178	−	9.07470i	−0.740124	−	0.427311i
$452$		0			0
$453$	−7.40949	+	19.5240i	−0.348128	+	0.917319i
$454$		0			0
$455$	−3.87506	−	1.41041i	−0.181666	−	0.0661209i
$456$		0			0
$457$	−4.70335	+	26.6740i	−0.220014	+	1.24776i	0.651978	+	0.758238i	$0.273939\pi$
$457$	−4.70335	+	26.6740i	−0.220014	+	1.24776i	−0.871991	+	0.489521i	$0.837172\pi$
$458$		0			0
$459$	−1.36597	+	33.3399i	−0.0637581	+	1.55617i
$460$		0			0
$461$	12.6352	+	2.22793i	0.588482	+	0.103765i	0.459957	−	0.887941i	$-0.347865\pi$
$461$	12.6352	+	2.22793i	0.588482	+	0.103765i	0.128525	+	0.991706i	$0.458976\pi$
$462$		0			0
$463$	0.136338	−	0.374586i	0.00633617	−	0.0174085i	−0.936484	−	0.350710i	$-0.885940\pi$
$463$	0.136338	−	0.374586i	0.00633617	−	0.0174085i	0.942820	+	0.333301i	$0.108163\pi$
$464$		0			0
$465$	52.1334	−	42.5461i	2.41763	−	1.97303i
$466$		0			0
$467$	−4.12333	+	7.14182i	−0.190805	+	0.330484i	−0.945517	−	0.325572i	$-0.894443\pi$
$467$	−4.12333	+	7.14182i	−0.190805	+	0.330484i	0.754712	+	0.656056i	$0.227777\pi$
$468$		0			0
$469$	−11.8972	−	20.6065i	−0.549360	−	0.951519i
$470$		0			0
$471$	−3.24438	+	5.44637i	−0.149493	+	0.250956i
$472$		0			0
$473$	−25.0615	+	4.41902i	−1.15233	+	0.203187i
$474$		0			0
$475$	10.0565	−	11.9848i	0.461423	−	0.549902i
$476$		0			0
$477$	1.58497	+	1.99717i	0.0725709	+	0.0914439i
$478$		0			0
$479$	23.0690	−	8.39642i	1.05405	−	0.383642i	0.243859	−	0.969811i	$-0.421587\pi$
$479$	23.0690	−	8.39642i	1.05405	−	0.383642i	0.810189	+	0.586169i	$0.199364\pi$
$480$		0			0
$481$	4.41071	−	3.70102i	0.201111	−	0.168752i
$482$		0			0
$483$	−2.13361	−	6.12073i	−0.0970827	−	0.278503i
$484$		0			0
$485$			6.44797i			0.292787i
$486$		0			0
$487$			5.60176i			0.253840i	0.991913	+	0.126920i	$0.0405092\pi$
$487$			5.60176i			0.253840i	−0.991913	+	0.126920i	$0.959491\pi$
$488$		0			0
$489$	−1.39221	−	3.99386i	−0.0629580	−	0.180609i
$490$		0			0
$491$	27.3869	−	22.9803i	1.23595	−	1.03709i	0.238122	−	0.971235i	$-0.423468\pi$
$491$	27.3869	−	22.9803i	1.23595	−	1.03709i	0.997829	−	0.0658512i	$-0.0209763\pi$
$492$		0			0
$493$	−34.0067	+	12.3774i	−1.53158	+	0.557451i
$494$		0			0
$495$	−19.6795	−	24.7974i	−0.884528	−	1.11456i
$496$		0			0
$497$	−12.0231	+	14.3285i	−0.539308	+	0.642722i
$498$		0			0
$499$	40.9591	−	7.22219i	1.83358	−	0.323310i	0.853376	−	0.521297i	$-0.174551\pi$
$499$	40.9591	−	7.22219i	1.83358	−	0.323310i	0.980205	+	0.197987i	$0.0634404\pi$
$500$		0			0
$501$	−4.70879	+	7.90469i	−0.210373	+	0.353156i
$502$		0			0
$503$	−10.1863	−	17.6431i	−0.454183	−	0.786668i	0.544458	−	0.838788i	$-0.316735\pi$
$503$	−10.1863	−	17.6431i	−0.454183	−	0.786668i	−0.998641	+	0.0521202i	$0.983402\pi$
$504$		0			0
$505$	−20.7614	+	35.9597i	−0.923868	+	1.60019i
$506$		0			0
$507$	16.7846	−	13.6979i	0.745430	−	0.608347i
$508$		0			0
$509$	2.66835	−	7.33124i	0.118273	−	0.324952i	−0.866403	−	0.499345i	$-0.833574\pi$
$509$	2.66835	−	7.33124i	0.118273	−	0.324952i	0.984676	+	0.174393i	$0.0557963\pi$
$510$		0			0
$511$	9.89692	+	1.74509i	0.437814	+	0.0771984i
$512$		0			0
$513$	−8.64283	+	4.52871i	−0.381590	+	0.199947i
$514$		0			0
$515$	−10.4880	+	59.4805i	−0.462157	+	2.62102i
$516$		0			0
$517$	4.83976	+	1.76153i	0.212852	+	0.0774719i
$518$		0			0
$519$	−16.1207	+	42.4781i	−0.707620	+	1.86458i
$520$		0			0
$521$	10.5688	+	6.10193i	0.463030	+	0.267330i	0.713317	−	0.700841i	$-0.247192\pi$
$521$	10.5688	+	6.10193i	0.463030	+	0.267330i	−0.250288	+	0.968172i	$0.580525\pi$
$522$		0			0
$523$	−1.11506	+	0.643783i	−0.0487584	+	0.0281507i	−0.524181	−	0.851607i	$-0.675628\pi$
$523$	−1.11506	+	0.643783i	−0.0487584	+	0.0281507i	0.475423	+	0.879758i	$0.342295\pi$
$524$		0			0
$525$	−20.2809	+	11.3430i	−0.885131	+	0.495047i
$526$		0			0
$527$	11.8651	+	67.2905i	0.516853	+	2.93122i
$528$		0			0
$529$	−13.4815	−	11.3124i	−0.586154	−	0.491841i
$530$		0			0
$531$	−3.56779	+	17.4364i	−0.154829	+	0.756674i
$532$		0			0
$533$	1.50643	+	4.13888i	0.0652507	+	0.179275i
$534$		0			0
$535$	−39.1650	−	46.6750i	−1.69325	−	2.01794i
$536$		0			0
$537$	−14.2374	−	12.2816i	−0.614389	−	0.529990i
$538$		0			0
$539$	−12.7367			−0.548607
$540$		0			0
$541$	−33.3756			−1.43493			−0.717465	−	0.696594i	$-0.754698\pi$
$541$	−33.3756			−1.43493			−0.717465	+	0.696594i	$0.754698\pi$
$542$		0			0
$543$	1.37787	−	7.23537i	0.0591302	−	0.310500i
$544$		0			0
$545$	23.9692	+	28.5654i	1.02673	+	1.22361i
$546$		0			0
$547$	−15.6515	−	43.0020i	−0.669208	−	1.83863i	−0.529228	−	0.848480i	$-0.677518\pi$
$547$	−15.6515	−	43.0020i	−0.669208	−	1.83863i	−0.139980	−	0.990154i	$-0.544704\pi$
$548$		0			0
$549$	−16.6442	+	27.0928i	−0.710357	+	1.15629i
$550$		0			0
$551$	−8.10663	−	6.80227i	−0.345354	−	0.289786i
$552$		0			0
$553$	1.27156	+	7.21136i	0.0540721	+	0.306658i
$554$		0			0
$555$	0.708606	−	51.9116i	0.0300786	−	2.20353i
$556$		0			0
$557$	18.3056	−	10.5687i	0.775631	−	0.447811i	−0.0592484	−	0.998243i	$-0.518870\pi$
$557$	18.3056	−	10.5687i	0.775631	−	0.447811i	0.834880	+	0.550432i	$0.185537\pi$
$558$		0			0
$559$	5.34837	+	3.08788i	0.226212	+	0.130603i
$560$		0			0
$561$	31.7305	−	5.14942i	1.33966	−	0.217409i
$562$		0			0
$563$	−14.9675	−	5.44771i	−0.630803	−	0.229594i	0.00677752	−	0.999977i	$-0.497843\pi$
$563$	−14.9675	−	5.44771i	−0.630803	−	0.229594i	−0.637581	+	0.770383i	$0.720065\pi$
$564$		0			0
$565$	8.93869	−	50.6938i	0.376053	−	2.13270i
$566$		0			0
$567$	14.3885	−	1.73401i	0.604259	−	0.0728215i
$568$		0			0
$569$	5.38391	+	0.949329i	0.225705	+	0.0397979i	0.285357	−	0.958421i	$-0.407888\pi$
$569$	5.38391	+	0.949329i	0.225705	+	0.0397979i	−0.0596516	+	0.998219i	$0.518999\pi$
$570$		0			0
$571$	5.95709	−	16.3670i	0.249297	−	0.684937i	−0.750416	−	0.660966i	$-0.770147\pi$
$571$	5.95709	−	16.3670i	0.249297	−	0.684937i	0.999713	−	0.0239711i	$-0.00763096\pi$
$572$		0			0
$573$	2.00347	+	12.3453i	0.0836962	+	0.515732i
$574$		0			0
$575$	9.68134	−	16.7686i	0.403740	−	0.699298i
$576$		0			0
$577$	−6.89556	−	11.9435i	−0.287066	−	0.497213i	0.686042	−	0.727562i	$-0.259347\pi$
$577$	−6.89556	−	11.9435i	−0.287066	−	0.497213i	−0.973108	+	0.230349i	$0.926013\pi$
$578$		0			0
$579$	−28.7356	−	0.392247i	−1.19421	−	0.0163012i
$580$		0			0
$581$	−11.7168	+	2.06599i	−0.486096	+	0.0857118i
$582$		0			0
$583$	1.57886	−	1.88162i	0.0653899	−	0.0779286i
$584$		0			0
$585$	−0.209701	+	7.67979i	−0.00867006	+	0.317520i
$586$		0			0
$587$	21.4195	−	7.79604i	0.884075	−	0.321777i	0.140222	−	0.990120i	$-0.455218\pi$
$587$	21.4195	−	7.79604i	0.884075	−	0.321777i	0.743853	+	0.668343i	$0.232996\pi$
$588$		0			0
$589$	−15.3061	+	12.8433i	−0.630676	+	0.529200i
$590$		0			0
$591$	8.55859	+	1.62986i	0.352054	+	0.0670435i
$592$		0			0
$593$			34.0284i			1.39738i	0.715426	+	0.698689i	$0.246233\pi$
$593$			34.0284i			1.39738i	−0.715426	+	0.698689i	$0.753767\pi$
$594$		0			0
$595$		−	37.7563i		−	1.54786i
$596$		0			0
$597$	16.7475	−	19.4145i	0.685431	−	0.794582i
$598$		0			0
$599$	−25.6710	+	21.5405i	−1.04889	+	0.880122i	−0.992976	−	0.118313i	$-0.962251\pi$
$599$	−25.6710	+	21.5405i	−1.04889	+	0.880122i	−0.0559128	+	0.998436i	$0.517807\pi$
$600$		0			0
$601$	33.9607	−	12.3607i	1.38529	−	0.504203i	0.461510	−	0.887135i	$-0.347308\pi$
$601$	33.9607	−	12.3607i	1.38529	−	0.504203i	0.923777	+	0.382931i	$0.125085\pi$
$602$		0			0
$603$	−29.4107	+	33.1678i	−1.19769	+	1.35070i
$604$		0			0
$605$	6.21298	−	7.40434i	0.252594	−	0.301029i
$606$		0			0
$607$	20.9011	−	3.68542i	0.848348	−	0.149587i	0.267459	−	0.963569i	$-0.413816\pi$
$607$	20.9011	−	3.68542i	0.848348	−	0.149587i	0.580889	+	0.813983i	$0.302705\pi$
$608$		0			0
$609$	7.67245	+	13.7181i	0.310904	+	0.555887i
$610$		0			0
$611$	−0.624946	−	1.08244i	−0.0252826	−	0.0437908i
$612$		0			0
$613$	−17.4558	+	30.2343i	−0.705032	+	1.22115i	0.261648	+	0.965163i	$0.415734\pi$
$613$	−17.4558	+	30.2343i	−0.705032	+	1.22115i	−0.966680	+	0.255987i	$0.917599\pi$
$614$		0			0
$615$	37.1304	+	14.0912i	1.49724	+	0.568213i
$616$		0			0
$617$	−4.96269	+	13.6349i	−0.199790	+	0.548920i	−0.998613	−	0.0526468i	$-0.983234\pi$
$617$	−4.96269	+	13.6349i	−0.199790	+	0.548920i	0.798823	+	0.601566i	$0.205456\pi$
$618$		0			0
$619$	2.94818	+	0.519843i	0.118497	+	0.0208943i	0.232582	−	0.972577i	$-0.425283\pi$
$619$	2.94818	+	0.519843i	0.118497	+	0.0208943i	−0.114085	+	0.993471i	$0.536394\pi$
$620$		0			0
$621$	−9.56076	+	7.37710i	−0.383660	+	0.296033i
$622$		0			0
$623$	1.94464	−	11.0286i	0.0779103	−	0.441851i
$624$		0			0
$625$	−2.59037	−	0.942816i	−0.103615	−	0.0377126i
$626$		0			0
$627$	5.94338	+	7.28265i	0.237356	+	0.290841i
$628$		0			0
$629$	45.6543	+	26.3585i	1.82036	+	1.05098i
$630$		0			0
$631$	42.0709	−	24.2896i	1.67481	−	0.966955i	0.709933	−	0.704269i	$-0.248725\pi$
$631$	42.0709	−	24.2896i	1.67481	−	0.966955i	0.964882	−	0.262685i	$-0.0846080\pi$
$632$		0			0
$633$	18.1200	+	10.7940i	0.720204	+	0.429022i
$634$		0			0
$635$	−4.47345	−	25.3702i	−0.177523	−	1.00678i
$636$		0			0
$637$	2.36780	+	1.98682i	0.0938156	+	0.0787206i
$638$		0			0
$639$	32.4079	+	12.8077i	1.28204	+	0.506666i
$640$		0			0
$641$	5.70164	+	15.6651i	0.225201	+	0.618735i	0.999908	−	0.0135858i	$-0.00432462\pi$
$641$	5.70164	+	15.6651i	0.225201	+	0.618735i	−0.774707	+	0.632321i	$0.782102\pi$
$642$		0			0
$643$	24.0202	+	28.6262i	0.947265	+	1.12891i	0.991529	+	0.129887i	$0.0414613\pi$
$643$	24.0202	+	28.6262i	0.947265	+	1.12891i	−0.0442639	+	0.999020i	$0.514094\pi$
$644$		0			0
$645$	52.5822	−	18.3295i	2.07042	−	0.721725i
$646$		0			0
$647$	30.1624			1.18580			0.592902	−	0.805274i	$-0.297982\pi$
$647$	30.1624			1.18580			0.592902	+	0.805274i	$0.297982\pi$
$648$		0			0
$649$	17.1457			0.673029
$650$		0			0
$651$	28.0232	−	9.76854i	1.09831	−	0.382859i
$652$		0			0
$653$	−1.61055	−	1.91938i	−0.0630256	−	0.0751110i	0.733611	−	0.679570i	$-0.237834\pi$
$653$	−1.61055	−	1.91938i	−0.0630256	−	0.0751110i	−0.796636	+	0.604459i	$0.793389\pi$
$654$		0			0
$655$	−18.1945	−	49.9888i	−0.710916	−	1.95323i
$656$		0			0
$657$	−2.74666	−	18.5200i	−0.107157	−	0.722536i
$658$		0			0
$659$	−26.7567	−	22.4515i	−1.04229	−	0.874587i	−0.0500301	−	0.998748i	$-0.515932\pi$
$659$	−26.7567	−	22.4515i	−1.04229	−	0.874587i	−0.992262	+	0.124161i	$0.960376\pi$
$660$		0			0
$661$	5.01298	+	28.4300i	0.194982	+	1.10580i	0.912444	+	0.409202i	$0.134193\pi$
$661$	5.01298	+	28.4300i	0.194982	+	1.10580i	−0.717461	+	0.696598i	$0.754696\pi$
$662$		0			0
$663$	−6.70210	−	3.99241i	−0.260288	−	0.155052i
$664$		0			0
$665$	9.56154	−	5.52036i	0.370781	−	0.214070i
$666$		0			0
$667$	−11.3424	−	6.54853i	−0.439179	−	0.253560i
$668$		0			0
$669$	10.9707	+	13.4429i	0.424154	+	0.519731i
$670$		0			0
$671$	28.7849	+	10.4769i	1.11123	+	0.404455i
$672$		0			0
$673$	5.70280	−	32.3422i	0.219827	−	1.24670i	−0.652504	−	0.757785i	$-0.726282\pi$
$673$	5.70280	−	32.3422i	0.219827	−	1.24670i	0.872331	−	0.488915i	$-0.162607\pi$
$674$		0			0
$675$	31.9965	+	29.1617i	1.23155	+	1.12244i
$676$		0			0
$677$	−4.92975	−	0.869249i	−0.189466	−	0.0334079i	0.0781100	−	0.996945i	$-0.475111\pi$
$677$	−4.92975	−	0.869249i	−0.189466	−	0.0334079i	−0.267576	+	0.963537i	$0.586223\pi$
$678$		0			0
$679$	−0.972608	+	2.67222i	−0.0373253	+	0.102550i
$680$		0			0
$681$	16.2691	+	6.17420i	0.623431	+	0.236596i
$682$		0			0
$683$	−15.9017	+	27.5426i	−0.608463	+	1.05389i	0.383031	+	0.923735i	$0.374880\pi$
$683$	−15.9017	+	27.5426i	−0.608463	+	1.05389i	−0.991494	+	0.130153i	$0.958453\pi$
$684$		0			0
$685$	26.5176	+	45.9298i	1.01318	+	1.75489i
$686$		0			0
$687$	−9.86193	−	17.6329i	−0.376256	−	0.672736i
$688$		0			0
$689$	−0.587035	+	0.103510i	−0.0223642	+	0.00394342i
$690$		0			0
$691$	−17.8704	+	21.2971i	−0.679822	+	0.810180i	−0.990085	−	0.140471i	$-0.955138\pi$
$691$	−17.8704	+	21.2971i	−0.679822	+	0.810180i	0.310263	+	0.950651i	$0.399583\pi$
$692$		0			0
$693$	−4.41531	−	13.2452i	−0.167724	−	0.503143i
$694$		0			0
$695$	−31.5341	+	11.4775i	−1.19615	+	0.435365i
$696$		0			0
$697$	−30.8921	+	25.9216i	−1.17012	+	0.981849i
$698$		0			0
$699$	14.5612	−	16.8800i	0.550756	−	0.638461i
$700$		0			0
$701$		−	50.2148i		−	1.89659i	−0.317393	−	0.948294i	$-0.602807\pi$
$701$		−	50.2148i		−	1.89659i	0.317393	−	0.948294i	$-0.397193\pi$
$702$		0			0
$703$			15.4155i			0.581408i
$704$		0			0
$705$	−11.0710	−	2.10832i	−0.416959	−	0.0794038i
$706$		0			0
$707$	−14.0282	+	11.7711i	−0.527586	+	0.442697i
$708$		0			0
$709$	1.61463	−	0.587679i	0.0606388	−	0.0220707i	−0.311523	−	0.950239i	$-0.600839\pi$
$709$	1.61463	−	0.587679i	0.0606388	−	0.0220707i	0.372161	+	0.928168i	$0.378617\pi$
$710$		0			0
$711$	11.9963	−	6.49607i	0.449895	−	0.243622i
$712$		0			0
$713$	−15.8951	+	18.9431i	−0.595278	+	0.709425i
$714$		0			0
$715$	7.28880	−	1.28521i	0.272586	−	0.0480642i
$716$		0			0
$717$	6.99361	+	0.0954644i	0.261181	+	0.00356518i
$718$		0			0
$719$	−24.3767	−	42.2217i	−0.909099	−	1.57461i	−0.815319	−	0.579013i	$-0.803438\pi$
$719$	−24.3767	−	42.2217i	−0.909099	−	1.57461i	−0.0937803	−	0.995593i	$-0.529895\pi$
$720$		0			0
$721$	−13.3185	+	23.0684i	−0.496008	+	0.859112i
$722$		0			0
$723$	5.96189	+	36.7369i	0.221725	+	1.36626i
$724$		0			0
$725$	−16.0586	+	44.1207i	−0.596402	+	1.63860i
$726$		0			0
$727$	−39.8914	−	7.03393i	−1.47949	−	0.260874i	−0.625115	−	0.780532i	$-0.714948\pi$
$727$	−39.8914	−	7.03393i	−1.47949	−	0.260874i	−0.854374	+	0.519659i	$0.826059\pi$
$728$		0			0
$729$	−11.5419	−	24.4087i	−0.427479	−	0.904025i
$730$		0			0
$731$	−9.81877	+	55.6850i	−0.363160	+	2.05959i
$732$		0			0
$733$	16.7735	+	6.10505i	0.619543	+	0.225495i	0.632674	−	0.774418i	$-0.281957\pi$
$733$	16.7735	+	6.10505i	0.619543	+	0.225495i	−0.0131304	+	0.999914i	$0.504180\pi$
$734$		0			0
$735$	27.5104	−	4.46455i	1.01473	−	0.164677i
$736$		0			0
$737$	36.9842	+	21.3528i	1.36233	+	0.786541i
$738$		0			0
$739$	−1.22996	+	0.710118i	−0.0452448	+	0.0261221i	−0.522452	−	0.852669i	$-0.674983\pi$
$739$	−1.22996	+	0.710118i	−0.0452448	+	0.0261221i	0.477207	+	0.878791i	$0.341649\pi$
$740$		0			0
$741$	0.0311361	−	2.28099i	0.00114381	−	0.0837944i
$742$		0			0
$743$	1.09307	+	6.19911i	0.0401009	+	0.227423i	0.998271	−	0.0587751i	$-0.0187195\pi$
$743$	1.09307	+	6.19911i	0.0401009	+	0.227423i	−0.958170	+	0.286198i	$0.907608\pi$
$744$		0			0
$745$	−3.52538	−	2.95814i	−0.129160	−	0.108378i
$746$		0			0
$747$	10.5546	+	19.4912i	0.386174	+	0.713146i
$748$		0			0
$749$	−9.19063	−	25.2511i	−0.335818	−	0.922653i
$750$		0			0
$751$	−11.8427	−	14.1136i	−0.432146	−	0.515012i	0.505394	−	0.862889i	$-0.331347\pi$
$751$	−11.8427	−	14.1136i	−0.432146	−	0.515012i	−0.937540	+	0.347877i	$0.886903\pi$
$752$		0			0
$753$	−1.37563	+	7.22361i	−0.0501308	+	0.263243i
$754$		0			0
$755$	−44.0217			−1.60211
$756$		0			0
$757$	10.3039			0.374500			0.187250	−	0.982312i	$-0.440043\pi$
$757$	10.3039			0.374500			0.187250	+	0.982312i	$0.440043\pi$
$758$		0			0
$759$	8.80902	+	7.59893i	0.319747	+	0.275824i
$760$		0			0
$761$	−12.5156	−	14.9156i	−0.453692	−	0.540689i	0.489909	−	0.871773i	$-0.337030\pi$
$761$	−12.5156	−	14.9156i	−0.453692	−	0.540689i	−0.943601	+	0.331085i	$0.892585\pi$
$762$		0			0
$763$	5.62472	+	15.4538i	0.203629	+	0.559465i
$764$		0			0
$765$	−66.7308	+	22.2448i	−2.41266	+	0.804263i
$766$		0			0
$767$	−3.18746	−	2.67460i	−0.115093	−	0.0965741i
$768$		0			0
$769$	−4.85410	−	27.5290i	−0.175043	−	0.992721i	−0.938094	−	0.346382i	$-0.887410\pi$
$769$	−4.85410	−	27.5290i	−0.175043	−	0.992721i	0.763050	−	0.646339i	$-0.223701\pi$
$770$		0			0
$771$	7.86319	−	4.39782i	0.283186	−	0.158384i
$772$		0			0
$773$	15.1159	−	8.72717i	0.543681	−	0.313894i	−0.202888	−	0.979202i	$-0.565033\pi$
$773$	15.1159	−	8.72717i	0.543681	−	0.313894i	0.746569	+	0.665307i	$0.231700\pi$
$774$		0			0
$775$	76.7733	+	44.3251i	2.75778	+	1.59220i
$776$		0			0
$777$	8.12399	−	21.4067i	0.291446	−	0.767963i
$778$		0			0
$779$	−11.0812	−	4.03323i	−0.397025	−	0.144505i
$780$		0			0
$781$	5.82947	−	33.0606i	0.208595	−	1.18300i
$782$		0			0
$783$	19.7252	−	21.6427i	0.704921	−	0.773446i
$784$		0			0
$785$	−13.1609	−	2.32061i	−0.469731	−	0.0828263i
$786$		0			0
$787$	−0.407887	+	1.12066i	−0.0145396	+	0.0399472i	−0.946750	−	0.321969i	$-0.895655\pi$
$787$	−0.407887	+	1.12066i	−0.0145396	+	0.0399472i	0.932211	+	0.361916i	$0.117877\pi$
$788$		0			0
$789$	32.7876	−	26.7581i	1.16727	−	0.952612i
$790$		0			0
$791$	11.3511	−	19.6606i	0.403598	−	0.699052i
$792$		0			0
$793$	−3.71693	−	6.43791i	−0.131992	−	0.228617i
$794$		0			0

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)

\(f(q)\)	\(=\)	\(q+(1.63553 - 0.570126i) q^{3} +(2.34697 + 2.79701i) q^{5} +(0.550750 + 1.51317i) q^{7} +(2.34991 - 1.86492i) q^{9} +O(q^{10})\)	\(q+(1.63553 - 0.570126i) q^{3} +(2.34697 + 2.79701i) q^{5} +(0.550750 + 1.51317i) q^{7} +(2.34991 - 1.86492i) q^{9} +(-2.21396 - 1.85773i) q^{11} +(0.121792 + 0.690719i) q^{13} +(5.43318 + 3.23652i) q^{15} +(-5.56131 + 3.21082i) q^{17} +(-1.62624 - 0.938910i) q^{19} +(1.76347 + 2.16084i) q^{21} +(-2.18387 - 0.794864i) q^{23} +(-1.44675 + 8.20494i) q^{25} +(2.78011 - 4.38987i) q^{27} +(5.54988 + 0.978594i) q^{29} +(3.63921 - 9.99865i) q^{31} +(-4.68013 - 1.77614i) q^{33} +(-2.93977 + 5.09183i) q^{35} +(-4.10464 - 7.10944i) q^{37} +(0.592992 + 1.06025i) q^{39} +(6.18442 - 1.09048i) q^{41} +(5.65989 - 6.74520i) q^{43} +(10.7314 + 2.19583i) q^{45} +(-1.67459 + 0.609501i) q^{47} +(3.37594 - 2.83275i) q^{49} +(-7.26511 + 8.42204i) q^{51} +0.849889i q^{53} -10.5525i q^{55} +(-3.19506 - 0.608453i) q^{57} +(-4.54459 + 3.81337i) q^{59} +(-9.95979 + 3.62507i) q^{61} +(4.11616 + 2.52872i) q^{63} +(-1.64610 + 1.96175i) q^{65} +(-14.5520 + 2.56590i) q^{67} +(-4.02496 - 0.0549417i) q^{69} +(5.80783 + 10.0595i) q^{71} +(3.12044 - 5.40475i) q^{73} +(2.31165 + 14.2443i) q^{75} +(1.59173 - 4.37325i) q^{77} +(4.47831 + 0.789647i) q^{79} +(2.04418 - 8.76478i) q^{81} +(-1.28300 + 7.27624i) q^{83} +(-22.0329 - 8.01933i) q^{85} +(9.63492 - 1.56361i) q^{87} +(-6.02277 - 3.47725i) q^{89} +(-0.978101 + 0.564707i) q^{91} +(0.251545 - 18.4279i) q^{93} +(-1.19060 - 6.75220i) q^{95} +(1.35281 + 1.13514i) q^{97} +(-8.66711 - 0.236660i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q + 3 q^{5} + 6 q^{9}+O(q^{10}) \) 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

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