LMFDB - Embedded newform 1734.2.f.e.1483.2

Newspace parameters

Level:	$ N $	$=$	$ 1734 = 2 \cdot 3 \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1734.f (of order $4$, degree $2$, not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$13.8460597105$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(\zeta_{8})$
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 102)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			1483.2
Root			$0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1734.1483
Dual form			1734.2.f.e.829.2

$q$-expansion

$f(q)$	$=$	$q+1.00000i q^{2} +(0.707107 - 0.707107i) q^{3} -1.00000 q^{4} +(1.41421 - 1.41421i) q^{5} +(0.707107 + 0.707107i) q^{6} -1.00000i q^{8} -1.00000i q^{9} +O(q^{10})$	\(q+1.00000i q^{2} +(0.707107 - 0.707107i) q^{3} -1.00000 q^{4} +(1.41421 - 1.41421i) q^{5} +(0.707107 + 0.707107i) q^{6} -1.00000i q^{8} -1.00000i q^{9} +(1.41421 + 1.41421i) q^{10} +(-2.82843 - 2.82843i) q^{11} +(-0.707107 + 0.707107i) q^{12} +2.00000 q^{13} -2.00000i q^{15} +1.00000 q^{16} +1.00000 q^{18} +4.00000i q^{19} +(-1.41421 + 1.41421i) q^{20} +(2.82843 - 2.82843i) q^{22} +(-0.707107 - 0.707107i) q^{24} +1.00000i q^{25} +2.00000i q^{26} +(-0.707107 - 0.707107i) q^{27} +(7.07107 - 7.07107i) q^{29} +2.00000 q^{30} +(5.65685 - 5.65685i) q^{31} +1.00000i q^{32} -4.00000 q^{33} +1.00000i q^{36} +(-1.41421 + 1.41421i) q^{37} -4.00000 q^{38} +(1.41421 - 1.41421i) q^{39} +(-1.41421 - 1.41421i) q^{40} +(-7.07107 - 7.07107i) q^{41} -12.0000i q^{43} +(2.82843 + 2.82843i) q^{44} +(-1.41421 - 1.41421i) q^{45} +(0.707107 - 0.707107i) q^{48} -7.00000i q^{49} -1.00000 q^{50} -2.00000 q^{52} +6.00000i q^{53} +(0.707107 - 0.707107i) q^{54} -8.00000 q^{55} +(2.82843 + 2.82843i) q^{57} +(7.07107 + 7.07107i) q^{58} -12.0000i q^{59} +2.00000i q^{60} +(7.07107 + 7.07107i) q^{61} +(5.65685 + 5.65685i) q^{62} -1.00000 q^{64} +(2.82843 - 2.82843i) q^{65} -4.00000i q^{66} -12.0000 q^{67} -1.00000 q^{72} +(-7.07107 + 7.07107i) q^{73} +(-1.41421 - 1.41421i) q^{74} +(0.707107 + 0.707107i) q^{75} -4.00000i q^{76} +(1.41421 + 1.41421i) q^{78} +(-5.65685 - 5.65685i) q^{79} +(1.41421 - 1.41421i) q^{80} -1.00000 q^{81} +(7.07107 - 7.07107i) q^{82} +4.00000i q^{83} +12.0000 q^{86} -10.0000i q^{87} +(-2.82843 + 2.82843i) q^{88} +6.00000 q^{89} +(1.41421 - 1.41421i) q^{90} -8.00000i q^{93} +(5.65685 + 5.65685i) q^{95} +(0.707107 + 0.707107i) q^{96} +(9.89949 - 9.89949i) q^{97} +7.00000 q^{98} +(-2.82843 + 2.82843i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4}+O(q^{10}) $ 4 * q - 4 * q^4	$ 4 q - 4 q^{4} + 8 q^{13} + 4 q^{16} + 4 q^{18} + 8 q^{30} - 16 q^{33} - 16 q^{38} - 4 q^{50} - 8 q^{52} - 32 q^{55} - 4 q^{64} - 48 q^{67} - 4 q^{72} - 4 q^{81} + 48 q^{86} + 24 q^{89} + 28 q^{98}+O(q^{100}) $ 4 * q - 4 * q^4 + 8 * q^13 + 4 * q^16 + 4 * q^18 + 8 * q^30 - 16 * q^33 - 16 * q^38 - 4 * q^50 - 8 * q^52 - 32 * q^55 - 4 * q^64 - 48 * q^67 - 4 * q^72 - 4 * q^81 + 48 * q^86 + 24 * q^89 + 28 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1734\mathbb{Z}\right)^\times$.

$n$	$1157$	$1159$
$\chi(n)$	$1$	$e\left(\frac{3}{4}\right)$

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 $

$2$			1.00000i			0.707107i
$2$			1.00000i			0.707107i
$3$	0.707107	−	0.707107i	0.408248	−	0.408248i
$3$	0.707107	−	0.707107i	0.408248	−	0.408248i
$4$	−1.00000			−0.500000
$5$	1.41421	−	1.41421i	0.632456	−	0.632456i	−0.316228	−	0.948683i	$-0.602416\pi$
$5$	1.41421	−	1.41421i	0.632456	−	0.632456i	0.948683	+	0.316228i	$0.102416\pi$
$6$	0.707107	+	0.707107i	0.288675	+	0.288675i
$7$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$7$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$8$		−	1.00000i		−	0.353553i
$9$		−	1.00000i		−	0.333333i
$10$	1.41421	+	1.41421i	0.447214	+	0.447214i
$11$	−2.82843	−	2.82843i	−0.852803	−	0.852803i	0.137675	−	0.990478i	$-0.456037\pi$
$11$	−2.82843	−	2.82843i	−0.852803	−	0.852803i	−0.990478	+	0.137675i	$0.956037\pi$
$12$	−0.707107	+	0.707107i	−0.204124	+	0.204124i
$13$	2.00000			0.554700			0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000			0.554700			0.277350	+	0.960769i	$0.410544\pi$
$14$		0			0
$15$		−	2.00000i		−	0.516398i
$16$	1.00000			0.250000
$17$		0			0
$17$		0			0
$18$	1.00000			0.235702
$19$			4.00000i			0.917663i	0.888523	+	0.458831i	$0.151732\pi$
$19$			4.00000i			0.917663i	−0.888523	+	0.458831i	$0.848268\pi$
$20$	−1.41421	+	1.41421i	−0.316228	+	0.316228i
$21$		0			0
$22$	2.82843	−	2.82843i	0.603023	−	0.603023i
$23$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$23$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$24$	−0.707107	−	0.707107i	−0.144338	−	0.144338i
$25$			1.00000i			0.200000i
$26$			2.00000i			0.392232i
$27$	−0.707107	−	0.707107i	−0.136083	−	0.136083i
$28$		0			0
$29$	7.07107	−	7.07107i	1.31306	−	1.31306i	0.393919	−	0.919145i	$-0.371119\pi$
$29$	7.07107	−	7.07107i	1.31306	−	1.31306i	0.919145	−	0.393919i	$-0.128881\pi$
$30$	2.00000			0.365148
$31$	5.65685	−	5.65685i	1.01600	−	1.01600i	0.0161311	−	0.999870i	$-0.494865\pi$
$31$	5.65685	−	5.65685i	1.01600	−	1.01600i	0.999870	−	0.0161311i	$-0.00513492\pi$
$32$			1.00000i			0.176777i
$33$	−4.00000			−0.696311
$34$		0			0
$35$		0			0
$36$			1.00000i			0.166667i
$37$	−1.41421	+	1.41421i	−0.232495	+	0.232495i	−0.813733	−	0.581238i	$-0.802568\pi$
$37$	−1.41421	+	1.41421i	−0.232495	+	0.232495i	0.581238	+	0.813733i	$0.302568\pi$
$38$	−4.00000			−0.648886
$39$	1.41421	−	1.41421i	0.226455	−	0.226455i
$40$	−1.41421	−	1.41421i	−0.223607	−	0.223607i
$41$	−7.07107	−	7.07107i	−1.10432	−	1.10432i	−0.993884	−	0.110432i	$-0.964777\pi$
$41$	−7.07107	−	7.07107i	−1.10432	−	1.10432i	−0.110432	−	0.993884i	$-0.535223\pi$
$42$		0			0
$43$		−	12.0000i		−	1.82998i	−0.403473	−	0.914991i	$-0.632197\pi$
$43$		−	12.0000i		−	1.82998i	0.403473	−	0.914991i	$-0.367803\pi$
$44$	2.82843	+	2.82843i	0.426401	+	0.426401i
$45$	−1.41421	−	1.41421i	−0.210819	−	0.210819i
$46$		0			0
$47$		0			0			−	1.00000i	$-0.5\pi$
$47$		0			0				1.00000i	$0.5\pi$
$48$	0.707107	−	0.707107i	0.102062	−	0.102062i
$49$		−	7.00000i		−	1.00000i
$50$	−1.00000			−0.141421
$51$		0			0
$52$	−2.00000			−0.277350
$53$			6.00000i			0.824163i	0.911147	+	0.412082i	$0.135198\pi$
$53$			6.00000i			0.824163i	−0.911147	+	0.412082i	$0.864802\pi$
$54$	0.707107	−	0.707107i	0.0962250	−	0.0962250i
$55$	−8.00000			−1.07872
$56$		0			0
$57$	2.82843	+	2.82843i	0.374634	+	0.374634i
$58$	7.07107	+	7.07107i	0.928477	+	0.928477i
$59$		−	12.0000i		−	1.56227i	−0.624364	−	0.781133i	$-0.714642\pi$
$59$		−	12.0000i		−	1.56227i	0.624364	−	0.781133i	$-0.285358\pi$
$60$			2.00000i			0.258199i
$61$	7.07107	+	7.07107i	0.905357	+	0.905357i	0.995893	−	0.0905357i	$-0.0288579\pi$
$61$	7.07107	+	7.07107i	0.905357	+	0.905357i	−0.0905357	+	0.995893i	$0.528858\pi$
$62$	5.65685	+	5.65685i	0.718421	+	0.718421i
$63$		0			0
$64$	−1.00000			−0.125000
$65$	2.82843	−	2.82843i	0.350823	−	0.350823i
$66$		−	4.00000i		−	0.492366i
$67$	−12.0000			−1.46603			−0.733017	−	0.680211i	$-0.761888\pi$
$67$	−12.0000			−1.46603			−0.733017	+	0.680211i	$0.761888\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$71$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$72$	−1.00000			−0.117851
$73$	−7.07107	+	7.07107i	−0.827606	+	0.827606i	−0.987185	−	0.159579i	$-0.948986\pi$
$73$	−7.07107	+	7.07107i	−0.827606	+	0.827606i	0.159579	+	0.987185i	$0.448986\pi$
$74$	−1.41421	−	1.41421i	−0.164399	−	0.164399i
$75$	0.707107	+	0.707107i	0.0816497	+	0.0816497i
$76$		−	4.00000i		−	0.458831i
$77$		0			0
$78$	1.41421	+	1.41421i	0.160128	+	0.160128i
$79$	−5.65685	−	5.65685i	−0.636446	−	0.636446i	0.313231	−	0.949677i	$-0.398589\pi$
$79$	−5.65685	−	5.65685i	−0.636446	−	0.636446i	−0.949677	+	0.313231i	$0.898589\pi$
$80$	1.41421	−	1.41421i	0.158114	−	0.158114i
$81$	−1.00000			−0.111111
$82$	7.07107	−	7.07107i	0.780869	−	0.780869i
$83$			4.00000i			0.439057i	0.975606	+	0.219529i	$0.0704519\pi$
$83$			4.00000i			0.439057i	−0.975606	+	0.219529i	$0.929548\pi$
$84$		0			0
$85$		0			0
$86$	12.0000			1.29399
$87$		−	10.0000i		−	1.07211i
$88$	−2.82843	+	2.82843i	−0.301511	+	0.301511i
$89$	6.00000			0.635999			0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000			0.635999			0.317999	+	0.948091i	$0.396989\pi$
$90$	1.41421	−	1.41421i	0.149071	−	0.149071i
$91$		0			0
$92$		0			0
$93$		−	8.00000i		−	0.829561i
$94$		0			0
$95$	5.65685	+	5.65685i	0.580381	+	0.580381i
$96$	0.707107	+	0.707107i	0.0721688	+	0.0721688i
$97$	9.89949	−	9.89949i	1.00514	−	1.00514i	0.00515471	−	0.999987i	$-0.498359\pi$
$97$	9.89949	−	9.89949i	1.00514	−	1.00514i	0.999987	−	0.00515471i	$-0.00164080\pi$
$98$	7.00000			0.707107
$99$	−2.82843	+	2.82843i	−0.284268	+	0.284268i
$100$		−	1.00000i		−	0.100000i
$101$	−10.0000			−0.995037			−0.497519	−	0.867453i	$-0.665755\pi$
$101$	−10.0000			−0.995037			−0.497519	+	0.867453i	$0.665755\pi$
$102$		0			0
$103$	−8.00000			−0.788263			−0.394132	−	0.919054i	$-0.628955\pi$
$103$	−8.00000			−0.788263			−0.394132	+	0.919054i	$0.628955\pi$
$104$		−	2.00000i		−	0.196116i
$105$		0			0
$106$	−6.00000			−0.582772
$107$	2.82843	−	2.82843i	0.273434	−	0.273434i	−0.557047	−	0.830481i	$-0.688066\pi$
$107$	2.82843	−	2.82843i	0.273434	−	0.273434i	0.830481	+	0.557047i	$0.188066\pi$
$108$	0.707107	+	0.707107i	0.0680414	+	0.0680414i
$109$	7.07107	+	7.07107i	0.677285	+	0.677285i	0.959385	−	0.282100i	$-0.0910309\pi$
$109$	7.07107	+	7.07107i	0.677285	+	0.677285i	−0.282100	+	0.959385i	$0.591031\pi$
$110$		−	8.00000i		−	0.762770i
$111$			2.00000i			0.189832i
$112$		0			0
$113$	1.41421	+	1.41421i	0.133038	+	0.133038i	0.770490	−	0.637452i	$-0.220012\pi$
$113$	1.41421	+	1.41421i	0.133038	+	0.133038i	−0.637452	+	0.770490i	$0.720012\pi$
$114$	−2.82843	+	2.82843i	−0.264906	+	0.264906i
$115$		0			0
$116$	−7.07107	+	7.07107i	−0.656532	+	0.656532i
$117$		−	2.00000i		−	0.184900i
$118$	12.0000			1.10469
$119$		0			0
$120$	−2.00000			−0.182574
$121$			5.00000i			0.454545i
$122$	−7.07107	+	7.07107i	−0.640184	+	0.640184i
$123$	−10.0000			−0.901670
$124$	−5.65685	+	5.65685i	−0.508001	+	0.508001i
$125$	8.48528	+	8.48528i	0.758947	+	0.758947i
$126$		0			0
$127$		0			0		1.00000			$0$
$127$		0			0		−1.00000			$\pi$
$128$		−	1.00000i		−	0.0883883i
$129$	−8.48528	−	8.48528i	−0.747087	−	0.747087i
$130$	2.82843	+	2.82843i	0.248069	+	0.248069i
$131$	8.48528	−	8.48528i	0.741362	−	0.741362i	−0.231478	−	0.972840i	$-0.574356\pi$
$131$	8.48528	−	8.48528i	0.741362	−	0.741362i	0.972840	+	0.231478i	$0.0743560\pi$
$132$	4.00000			0.348155
$133$		0			0
$134$		−	12.0000i		−	1.03664i
$135$	−2.00000			−0.172133
$136$		0			0
$137$	10.0000			0.854358			0.427179	−	0.904167i	$-0.359507\pi$
$137$	10.0000			0.854358			0.427179	+	0.904167i	$0.359507\pi$
$138$		0			0
$139$	−2.82843	+	2.82843i	−0.239904	+	0.239904i	−0.816810	−	0.576906i	$-0.804260\pi$
$139$	−2.82843	+	2.82843i	−0.239904	+	0.239904i	0.576906	+	0.816810i	$0.304260\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−5.65685	−	5.65685i	−0.473050	−	0.473050i
$144$		−	1.00000i		−	0.0833333i
$145$		−	20.0000i		−	1.66091i
$146$	−7.07107	−	7.07107i	−0.585206	−	0.585206i
$147$	−4.94975	−	4.94975i	−0.408248	−	0.408248i
$148$	1.41421	−	1.41421i	0.116248	−	0.116248i
$149$	10.0000			0.819232			0.409616	−	0.912258i	$-0.365663\pi$
$149$	10.0000			0.819232			0.409616	+	0.912258i	$0.365663\pi$
$150$	−0.707107	+	0.707107i	−0.0577350	+	0.0577350i
$151$			24.0000i			1.95309i	0.215308	+	0.976546i	$0.430924\pi$
$151$			24.0000i			1.95309i	−0.215308	+	0.976546i	$0.569076\pi$
$152$	4.00000			0.324443
$153$		0			0
$154$		0			0
$155$		−	16.0000i		−	1.28515i
$156$	−1.41421	+	1.41421i	−0.113228	+	0.113228i
$157$	2.00000			0.159617			0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000			0.159617			0.0798087	+	0.996810i	$0.474569\pi$
$158$	5.65685	−	5.65685i	0.450035	−	0.450035i
$159$	4.24264	+	4.24264i	0.336463	+	0.336463i
$160$	1.41421	+	1.41421i	0.111803	+	0.111803i
$161$		0			0
$162$		−	1.00000i		−	0.0785674i
$163$	−2.82843	−	2.82843i	−0.221540	−	0.221540i	0.587607	−	0.809146i	$-0.300070\pi$
$163$	−2.82843	−	2.82843i	−0.221540	−	0.221540i	−0.809146	+	0.587607i	$0.800070\pi$
$164$	7.07107	+	7.07107i	0.552158	+	0.552158i
$165$	−5.65685	+	5.65685i	−0.440386	+	0.440386i
$166$	−4.00000			−0.310460
$167$	11.3137	−	11.3137i	0.875481	−	0.875481i	−0.117582	−	0.993063i	$-0.537514\pi$
$167$	11.3137	−	11.3137i	0.875481	−	0.875481i	0.993063	+	0.117582i	$0.0375143\pi$
$168$		0			0
$169$	−9.00000			−0.692308
$170$		0			0
$171$	4.00000			0.305888
$172$			12.0000i			0.914991i
$173$	4.24264	−	4.24264i	0.322562	−	0.322562i	−0.527187	−	0.849749i	$-0.676753\pi$
$173$	4.24264	−	4.24264i	0.322562	−	0.322562i	0.849749	+	0.527187i	$0.176753\pi$
$174$	10.0000			0.758098
$175$		0			0
$176$	−2.82843	−	2.82843i	−0.213201	−	0.213201i
$177$	−8.48528	−	8.48528i	−0.637793	−	0.637793i
$178$			6.00000i			0.449719i
$179$			12.0000i			0.896922i	0.893802	+	0.448461i	$0.148028\pi$
$179$			12.0000i			0.896922i	−0.893802	+	0.448461i	$0.851972\pi$
$180$	1.41421	+	1.41421i	0.105409	+	0.105409i
$181$	9.89949	+	9.89949i	0.735824	+	0.735824i	0.971767	−	0.235943i	$-0.0758179\pi$
$181$	9.89949	+	9.89949i	0.735824	+	0.735824i	−0.235943	+	0.971767i	$0.575818\pi$
$182$		0			0
$183$	10.0000			0.739221
$184$		0			0
$185$			4.00000i			0.294086i
$186$	8.00000			0.586588
$187$		0			0
$188$		0			0
$189$		0			0
$190$	−5.65685	+	5.65685i	−0.410391	+	0.410391i
$191$	16.0000			1.15772			0.578860	−	0.815427i	$-0.303498\pi$
$191$	16.0000			1.15772			0.578860	+	0.815427i	$0.303498\pi$
$192$	−0.707107	+	0.707107i	−0.0510310	+	0.0510310i
$193$	12.7279	+	12.7279i	0.916176	+	0.916176i	0.996749	−	0.0805728i	$-0.0256750\pi$
$193$	12.7279	+	12.7279i	0.916176	+	0.916176i	−0.0805728	+	0.996749i	$0.525675\pi$
$194$	9.89949	+	9.89949i	0.710742	+	0.710742i
$195$		−	4.00000i		−	0.286446i
$196$			7.00000i			0.500000i
$197$	−9.89949	−	9.89949i	−0.705310	−	0.705310i	0.260235	−	0.965545i	$-0.416200\pi$
$197$	−9.89949	−	9.89949i	−0.705310	−	0.705310i	−0.965545	+	0.260235i	$0.916200\pi$
$198$	−2.82843	−	2.82843i	−0.201008	−	0.201008i
$199$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$199$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$200$	1.00000			0.0707107
$201$	−8.48528	+	8.48528i	−0.598506	+	0.598506i
$202$		−	10.0000i		−	0.703598i
$203$		0			0
$204$		0			0
$205$	−20.0000			−1.39686
$206$		−	8.00000i		−	0.557386i
$207$		0			0
$208$	2.00000			0.138675
$209$	11.3137	−	11.3137i	0.782586	−	0.782586i
$210$		0			0
$211$	19.7990	+	19.7990i	1.36302	+	1.36302i	0.870043	+	0.492975i	$0.164091\pi$
$211$	19.7990	+	19.7990i	1.36302	+	1.36302i	0.492975	+	0.870043i	$0.335909\pi$
$212$		−	6.00000i		−	0.412082i
$213$		0			0
$214$	2.82843	+	2.82843i	0.193347	+	0.193347i
$215$	−16.9706	−	16.9706i	−1.15738	−	1.15738i
$216$	−0.707107	+	0.707107i	−0.0481125	+	0.0481125i
$217$		0			0
$218$	−7.07107	+	7.07107i	−0.478913	+	0.478913i
$219$			10.0000i			0.675737i
$220$	8.00000			0.539360
$221$		0			0
$222$	−2.00000			−0.134231
$223$			16.0000i			1.07144i	0.844396	+	0.535720i	$0.179960\pi$
$223$			16.0000i			1.07144i	−0.844396	+	0.535720i	$0.820040\pi$
$224$		0			0
$225$	1.00000			0.0666667
$226$	−1.41421	+	1.41421i	−0.0940721	+	0.0940721i
$227$	2.82843	+	2.82843i	0.187729	+	0.187729i	0.794714	−	0.606984i	$-0.207621\pi$
$227$	2.82843	+	2.82843i	0.187729	+	0.187729i	−0.606984	+	0.794714i	$0.707621\pi$
$228$	−2.82843	−	2.82843i	−0.187317	−	0.187317i
$229$			26.0000i			1.71813i	0.511868	+	0.859064i	$0.328954\pi$
$229$			26.0000i			1.71813i	−0.511868	+	0.859064i	$0.671046\pi$
$230$		0			0
$231$		0			0
$232$	−7.07107	−	7.07107i	−0.464238	−	0.464238i
$233$	−18.3848	+	18.3848i	−1.20443	+	1.20443i	−0.231621	+	0.972806i	$0.574403\pi$
$233$	−18.3848	+	18.3848i	−1.20443	+	1.20443i	−0.972806	+	0.231621i	$0.925597\pi$
$234$	2.00000			0.130744
$235$		0			0
$236$			12.0000i			0.781133i
$237$	−8.00000			−0.519656
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		−	2.00000i		−	0.129099i
$241$	1.41421	−	1.41421i	0.0910975	−	0.0910975i	−0.660089	−	0.751187i	$-0.729482\pi$
$241$	1.41421	−	1.41421i	0.0910975	−	0.0910975i	0.751187	+	0.660089i	$0.229482\pi$
$242$	−5.00000			−0.321412
$243$	−0.707107	+	0.707107i	−0.0453609	+	0.0453609i
$244$	−7.07107	−	7.07107i	−0.452679	−	0.452679i
$245$	−9.89949	−	9.89949i	−0.632456	−	0.632456i
$246$		−	10.0000i		−	0.637577i
$247$			8.00000i			0.509028i
$248$	−5.65685	−	5.65685i	−0.359211	−	0.359211i
$249$	2.82843	+	2.82843i	0.179244	+	0.179244i
$250$	−8.48528	+	8.48528i	−0.536656	+	0.536656i
$251$	−28.0000			−1.76734			−0.883672	−	0.468106i	$-0.844936\pi$
$251$	−28.0000			−1.76734			−0.883672	+	0.468106i	$0.844936\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	1.00000			0.0625000
$257$			2.00000i			0.124757i	0.998053	+	0.0623783i	$0.0198685\pi$
$257$			2.00000i			0.124757i	−0.998053	+	0.0623783i	$0.980131\pi$
$258$	8.48528	−	8.48528i	0.528271	−	0.528271i
$259$		0			0
$260$	−2.82843	+	2.82843i	−0.175412	+	0.175412i
$261$	−7.07107	−	7.07107i	−0.437688	−	0.437688i
$262$	8.48528	+	8.48528i	0.524222	+	0.524222i
$263$			8.00000i			0.493301i	0.969104	+	0.246651i	$0.0793300\pi$
$263$			8.00000i			0.493301i	−0.969104	+	0.246651i	$0.920670\pi$
$264$			4.00000i			0.246183i
$265$	8.48528	+	8.48528i	0.521247	+	0.521247i
$266$		0			0
$267$	4.24264	−	4.24264i	0.259645	−	0.259645i
$268$	12.0000			0.733017
$269$	4.24264	−	4.24264i	0.258678	−	0.258678i	−0.565838	−	0.824516i	$-0.691447\pi$
$269$	4.24264	−	4.24264i	0.258678	−	0.258678i	0.824516	+	0.565838i	$0.191447\pi$
$270$		−	2.00000i		−	0.121716i
$271$	16.0000			0.971931			0.485965	−	0.873978i	$-0.338468\pi$
$271$	16.0000			0.971931			0.485965	+	0.873978i	$0.338468\pi$
$272$		0			0
$273$		0			0
$274$			10.0000i			0.604122i
$275$	2.82843	−	2.82843i	0.170561	−	0.170561i
$276$		0			0
$277$	−21.2132	+	21.2132i	−1.27458	+	1.27458i	−0.330919	+	0.943659i	$0.607359\pi$
$277$	−21.2132	+	21.2132i	−1.27458	+	1.27458i	−0.943659	+	0.330919i	$0.892641\pi$
$278$	−2.82843	−	2.82843i	−0.169638	−	0.169638i
$279$	−5.65685	−	5.65685i	−0.338667	−	0.338667i
$280$		0			0
$281$			6.00000i			0.357930i	0.983855	+	0.178965i	$0.0572749\pi$
$281$			6.00000i			0.357930i	−0.983855	+	0.178965i	$0.942725\pi$
$282$		0			0
$283$	8.48528	+	8.48528i	0.504398	+	0.504398i	0.912801	−	0.408404i	$-0.133914\pi$
$283$	8.48528	+	8.48528i	0.504398	+	0.504398i	−0.408404	+	0.912801i	$0.633914\pi$
$284$		0			0
$285$	8.00000			0.473879
$286$	5.65685	−	5.65685i	0.334497	−	0.334497i
$287$		0			0
$288$	1.00000			0.0589256
$289$		0			0
$290$	20.0000			1.17444
$291$		−	14.0000i		−	0.820695i
$292$	7.07107	−	7.07107i	0.413803	−	0.413803i
$293$	26.0000			1.51894			0.759468	−	0.650545i	$-0.225459\pi$
$293$	26.0000			1.51894			0.759468	+	0.650545i	$0.225459\pi$
$294$	4.94975	−	4.94975i	0.288675	−	0.288675i
$295$	−16.9706	−	16.9706i	−0.988064	−	0.988064i
$296$	1.41421	+	1.41421i	0.0821995	+	0.0821995i
$297$			4.00000i			0.232104i
$298$			10.0000i			0.579284i
$299$		0			0
$300$	−0.707107	−	0.707107i	−0.0408248	−	0.0408248i
$301$		0			0
$302$	−24.0000			−1.38104
$303$	−7.07107	+	7.07107i	−0.406222	+	0.406222i
$304$			4.00000i			0.229416i
$305$	20.0000			1.14520
$306$		0			0
$307$	−12.0000			−0.684876			−0.342438	−	0.939540i	$-0.611253\pi$
$307$	−12.0000			−0.684876			−0.342438	+	0.939540i	$0.611253\pi$
$308$		0			0
$309$	−5.65685	+	5.65685i	−0.321807	+	0.321807i
$310$	16.0000			0.908739
$311$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$311$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$312$	−1.41421	−	1.41421i	−0.0800641	−	0.0800641i
$313$	−7.07107	−	7.07107i	−0.399680	−	0.399680i	0.478440	−	0.878120i	$-0.341202\pi$
$313$	−7.07107	−	7.07107i	−0.399680	−	0.399680i	−0.878120	+	0.478440i	$0.841202\pi$
$314$			2.00000i			0.112867i
$315$		0			0
$316$	5.65685	+	5.65685i	0.318223	+	0.318223i
$317$	4.24264	+	4.24264i	0.238290	+	0.238290i	0.816142	−	0.577851i	$-0.196109\pi$
$317$	4.24264	+	4.24264i	0.238290	+	0.238290i	−0.577851	+	0.816142i	$0.696109\pi$
$318$	−4.24264	+	4.24264i	−0.237915	+	0.237915i
$319$	−40.0000			−2.23957
$320$	−1.41421	+	1.41421i	−0.0790569	+	0.0790569i
$321$		−	4.00000i		−	0.223258i
$322$		0			0
$323$		0			0
$324$	1.00000			0.0555556
$325$			2.00000i			0.110940i
$326$	2.82843	−	2.82843i	0.156652	−	0.156652i
$327$	10.0000			0.553001
$328$	−7.07107	+	7.07107i	−0.390434	+	0.390434i
$329$		0			0
$330$	−5.65685	−	5.65685i	−0.311400	−	0.311400i
$331$			20.0000i			1.09930i	0.835395	+	0.549650i	$0.185239\pi$
$331$			20.0000i			1.09930i	−0.835395	+	0.549650i	$0.814761\pi$
$332$		−	4.00000i		−	0.219529i
$333$	1.41421	+	1.41421i	0.0774984	+	0.0774984i
$334$	11.3137	+	11.3137i	0.619059	+	0.619059i
$335$	−16.9706	+	16.9706i	−0.927201	+	0.927201i
$336$		0			0
$337$	−9.89949	+	9.89949i	−0.539260	+	0.539260i	−0.923312	−	0.384052i	$-0.874528\pi$
$337$	−9.89949	+	9.89949i	−0.539260	+	0.539260i	0.384052	+	0.923312i	$0.374528\pi$
$338$		−	9.00000i		−	0.489535i
$339$	2.00000			0.108625
$340$		0			0
$341$	−32.0000			−1.73290
$342$			4.00000i			0.216295i
$343$		0			0
$344$	−12.0000			−0.646997
$345$		0			0
$346$	4.24264	+	4.24264i	0.228086	+	0.228086i
$347$	−19.7990	−	19.7990i	−1.06287	−	1.06287i	−0.997887	−	0.0649788i	$-0.979302\pi$
$347$	−19.7990	−	19.7990i	−1.06287	−	1.06287i	−0.0649788	−	0.997887i	$-0.520698\pi$
$348$			10.0000i			0.536056i
$349$		−	14.0000i		−	0.749403i	−0.927146	−	0.374701i	$-0.877745\pi$
$349$		−	14.0000i		−	0.749403i	0.927146	−	0.374701i	$-0.122255\pi$
$350$		0			0
$351$	−1.41421	−	1.41421i	−0.0754851	−	0.0754851i
$352$	2.82843	−	2.82843i	0.150756	−	0.150756i
$353$	30.0000			1.59674			0.798369	−	0.602168i	$-0.205696\pi$
$353$	30.0000			1.59674			0.798369	+	0.602168i	$0.205696\pi$
$354$	8.48528	−	8.48528i	0.450988	−	0.450988i
$355$		0			0
$356$	−6.00000			−0.317999
$357$		0			0
$358$	−12.0000			−0.634220
$359$		−	24.0000i		−	1.26667i	−0.773877	−	0.633336i	$-0.781685\pi$
$359$		−	24.0000i		−	1.26667i	0.773877	−	0.633336i	$-0.218315\pi$
$360$	−1.41421	+	1.41421i	−0.0745356	+	0.0745356i
$361$	3.00000			0.157895
$362$	−9.89949	+	9.89949i	−0.520306	+	0.520306i
$363$	3.53553	+	3.53553i	0.185567	+	0.185567i
$364$		0			0
$365$			20.0000i			1.04685i
$366$			10.0000i			0.522708i
$367$	−16.9706	−	16.9706i	−0.885856	−	0.885856i	0.108266	−	0.994122i	$-0.465470\pi$
$367$	−16.9706	−	16.9706i	−0.885856	−	0.885856i	−0.994122	+	0.108266i	$0.965470\pi$
$368$		0			0
$369$	−7.07107	+	7.07107i	−0.368105	+	0.368105i
$370$	−4.00000			−0.207950
$371$		0			0
$372$			8.00000i			0.414781i
$373$	6.00000			0.310668			0.155334	−	0.987862i	$-0.450355\pi$
$373$	6.00000			0.310668			0.155334	+	0.987862i	$0.450355\pi$
$374$		0			0
$375$	12.0000			0.619677
$376$		0			0
$377$	14.1421	−	14.1421i	0.728357	−	0.728357i
$378$		0			0
$379$	2.82843	−	2.82843i	0.145287	−	0.145287i	−0.630722	−	0.776009i	$-0.717241\pi$
$379$	2.82843	−	2.82843i	0.145287	−	0.145287i	0.776009	+	0.630722i	$0.217241\pi$
$380$	−5.65685	−	5.65685i	−0.290191	−	0.290191i
$381$		0			0
$382$			16.0000i			0.818631i
$383$			16.0000i			0.817562i	0.912633	+	0.408781i	$0.134046\pi$
$383$			16.0000i			0.817562i	−0.912633	+	0.408781i	$0.865954\pi$
$384$	−0.707107	−	0.707107i	−0.0360844	−	0.0360844i
$385$		0			0
$386$	−12.7279	+	12.7279i	−0.647834	+	0.647834i
$387$	−12.0000			−0.609994
$388$	−9.89949	+	9.89949i	−0.502571	+	0.502571i
$389$		−	26.0000i		−	1.31825i	−0.752032	−	0.659126i	$-0.770926\pi$
$389$		−	26.0000i		−	1.31825i	0.752032	−	0.659126i	$-0.229074\pi$
$390$	4.00000			0.202548
$391$		0			0
$392$	−7.00000			−0.353553
$393$		−	12.0000i		−	0.605320i
$394$	9.89949	−	9.89949i	0.498729	−	0.498729i
$395$	−16.0000			−0.805047
$396$	2.82843	−	2.82843i	0.142134	−	0.142134i
$397$	−18.3848	−	18.3848i	−0.922705	−	0.922705i	0.0745145	−	0.997220i	$-0.476259\pi$
$397$	−18.3848	−	18.3848i	−0.922705	−	0.922705i	−0.997220	+	0.0745145i	$0.976259\pi$
$398$		0			0
$399$		0			0
$400$			1.00000i			0.0500000i
$401$	9.89949	+	9.89949i	0.494357	+	0.494357i	0.909676	−	0.415319i	$-0.136330\pi$
$401$	9.89949	+	9.89949i	0.494357	+	0.494357i	−0.415319	+	0.909676i	$0.636330\pi$
$402$	−8.48528	−	8.48528i	−0.423207	−	0.423207i
$403$	11.3137	−	11.3137i	0.563576	−	0.563576i
$404$	10.0000			0.497519
$405$	−1.41421	+	1.41421i	−0.0702728	+	0.0702728i
$406$		0			0
$407$	8.00000			0.396545
$408$		0			0
$409$	26.0000			1.28562			0.642809	−	0.766027i	$-0.277769\pi$
$409$	26.0000			1.28562			0.642809	+	0.766027i	$0.277769\pi$
$410$		−	20.0000i		−	0.987730i
$411$	7.07107	−	7.07107i	0.348790	−	0.348790i
$412$	8.00000			0.394132
$413$		0			0
$414$		0			0
$415$	5.65685	+	5.65685i	0.277684	+	0.277684i
$416$			2.00000i			0.0980581i
$417$			4.00000i			0.195881i
$418$	11.3137	+	11.3137i	0.553372	+	0.553372i
$419$	2.82843	+	2.82843i	0.138178	+	0.138178i	0.772812	−	0.634635i	$-0.218849\pi$
$419$	2.82843	+	2.82843i	0.138178	+	0.138178i	−0.634635	+	0.772812i	$0.718849\pi$
$420$		0			0
$421$	−22.0000			−1.07221			−0.536107	−	0.844150i	$-0.680106\pi$
$421$	−22.0000			−1.07221			−0.536107	+	0.844150i	$0.680106\pi$
$422$	−19.7990	+	19.7990i	−0.963800	+	0.963800i
$423$		0			0
$424$	6.00000			0.291386
$425$		0			0
$426$		0			0
$427$		0			0
$428$	−2.82843	+	2.82843i	−0.136717	+	0.136717i
$429$	−8.00000			−0.386244
$430$	16.9706	−	16.9706i	0.818393	−	0.818393i
$431$	−5.65685	−	5.65685i	−0.272481	−	0.272481i	0.557617	−	0.830098i	$-0.311716\pi$
$431$	−5.65685	−	5.65685i	−0.272481	−	0.272481i	−0.830098	+	0.557617i	$0.811716\pi$
$432$	−0.707107	−	0.707107i	−0.0340207	−	0.0340207i
$433$			14.0000i			0.672797i	0.941720	+	0.336399i	$0.109209\pi$
$433$			14.0000i			0.672797i	−0.941720	+	0.336399i	$0.890791\pi$
$434$		0			0
$435$	−14.1421	−	14.1421i	−0.678064	−	0.678064i
$436$	−7.07107	−	7.07107i	−0.338643	−	0.338643i
$437$		0			0
$438$	−10.0000			−0.477818
$439$	11.3137	−	11.3137i	0.539974	−	0.539974i	−0.383547	−	0.923521i	$-0.625298\pi$
$439$	11.3137	−	11.3137i	0.539974	−	0.539974i	0.923521	+	0.383547i	$0.125298\pi$
$440$			8.00000i			0.381385i
$441$	−7.00000			−0.333333
$442$		0			0
$443$	−4.00000			−0.190046			−0.0950229	−	0.995475i	$-0.530292\pi$
$443$	−4.00000			−0.190046			−0.0950229	+	0.995475i	$0.530292\pi$
$444$		−	2.00000i		−	0.0949158i
$445$	8.48528	−	8.48528i	0.402241	−	0.402241i
$446$	−16.0000			−0.757622
$447$	7.07107	−	7.07107i	0.334450	−	0.334450i
$448$		0			0
$449$	−1.41421	−	1.41421i	−0.0667409	−	0.0667409i	0.672948	−	0.739689i	$-0.265028\pi$
$449$	−1.41421	−	1.41421i	−0.0667409	−	0.0667409i	−0.739689	+	0.672948i	$0.765028\pi$
$450$			1.00000i			0.0471405i
$451$			40.0000i			1.88353i
$452$	−1.41421	−	1.41421i	−0.0665190	−	0.0665190i
$453$	16.9706	+	16.9706i	0.797347	+	0.797347i
$454$	−2.82843	+	2.82843i	−0.132745	+	0.132745i
$455$		0			0
$456$	2.82843	−	2.82843i	0.132453	−	0.132453i
$457$			10.0000i			0.467780i	0.972263	+	0.233890i	$0.0751456\pi$
$457$			10.0000i			0.467780i	−0.972263	+	0.233890i	$0.924854\pi$
$458$	−26.0000			−1.21490
$459$		0			0
$460$		0			0
$461$			30.0000i			1.39724i	0.715493	+	0.698620i	$0.246202\pi$
$461$			30.0000i			1.39724i	−0.715493	+	0.698620i	$0.753798\pi$
$462$		0			0
$463$	−32.0000			−1.48717			−0.743583	−	0.668644i	$-0.766875\pi$
$463$	−32.0000			−1.48717			−0.743583	+	0.668644i	$0.766875\pi$
$464$	7.07107	−	7.07107i	0.328266	−	0.328266i
$465$	−11.3137	−	11.3137i	−0.524661	−	0.524661i
$466$	−18.3848	−	18.3848i	−0.851658	−	0.851658i
$467$			12.0000i			0.555294i	0.960683	+	0.277647i	$0.0895545\pi$
$467$			12.0000i			0.555294i	−0.960683	+	0.277647i	$0.910445\pi$
$468$			2.00000i			0.0924500i
$469$		0			0
$470$		0			0
$471$	1.41421	−	1.41421i	0.0651635	−	0.0651635i
$472$	−12.0000			−0.552345
$473$	−33.9411	+	33.9411i	−1.56061	+	1.56061i
$474$		−	8.00000i		−	0.367452i
$475$	−4.00000			−0.183533
$476$		0			0
$477$	6.00000			0.274721
$478$		0			0
$479$	16.9706	−	16.9706i	0.775405	−	0.775405i	−0.203641	−	0.979046i	$-0.565277\pi$
$479$	16.9706	−	16.9706i	0.775405	−	0.775405i	0.979046	+	0.203641i	$0.0652775\pi$
$480$	2.00000			0.0912871
$481$	−2.82843	+	2.82843i	−0.128965	+	0.128965i
$482$	1.41421	+	1.41421i	0.0644157	+	0.0644157i
$483$		0			0
$484$		−	5.00000i		−	0.227273i
$485$		−	28.0000i		−	1.27141i
$486$	−0.707107	−	0.707107i	−0.0320750	−	0.0320750i
$487$	−11.3137	−	11.3137i	−0.512673	−	0.512673i	0.402671	−	0.915345i	$-0.368082\pi$
$487$	−11.3137	−	11.3137i	−0.512673	−	0.512673i	−0.915345	+	0.402671i	$0.868082\pi$
$488$	7.07107	−	7.07107i	0.320092	−	0.320092i
$489$	−4.00000			−0.180886
$490$	9.89949	−	9.89949i	0.447214	−	0.447214i
$491$			12.0000i			0.541552i	0.962642	+	0.270776i	$0.0872803\pi$
$491$			12.0000i			0.541552i	−0.962642	+	0.270776i	$0.912720\pi$
$492$	10.0000			0.450835
$493$		0			0
$494$	−8.00000			−0.359937
$495$			8.00000i			0.359573i
$496$	5.65685	−	5.65685i	0.254000	−	0.254000i
$497$		0			0
$498$	−2.82843	+	2.82843i	−0.126745	+	0.126745i
$499$	2.82843	+	2.82843i	0.126618	+	0.126618i	0.767576	−	0.640958i	$-0.221463\pi$
$499$	2.82843	+	2.82843i	0.126618	+	0.126618i	−0.640958	+	0.767576i	$0.721463\pi$
$500$	−8.48528	−	8.48528i	−0.379473	−	0.379473i
$501$		−	16.0000i		−	0.714827i
$502$		−	28.0000i		−	1.24970i
$503$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$503$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$504$		0			0
$505$	−14.1421	+	14.1421i	−0.629317	+	0.629317i
$506$		0			0
$507$	−6.36396	+	6.36396i	−0.282633	+	0.282633i
$508$		0			0
$509$	30.0000			1.32973			0.664863	−	0.746965i	$-0.268490\pi$
$509$	30.0000			1.32973			0.664863	+	0.746965i	$0.268490\pi$
$510$		0			0
$511$		0			0
$512$			1.00000i			0.0441942i
$513$	2.82843	−	2.82843i	0.124878	−	0.124878i
$514$	−2.00000			−0.0882162
$515$	−11.3137	+	11.3137i	−0.498542	+	0.498542i
$516$	8.48528	+	8.48528i	0.373544	+	0.373544i
$517$		0			0
$518$		0			0
$519$		−	6.00000i		−	0.263371i
$520$	−2.82843	−	2.82843i	−0.124035	−	0.124035i
$521$	−15.5563	−	15.5563i	−0.681536	−	0.681536i	0.278810	−	0.960346i	$-0.410060\pi$
$521$	−15.5563	−	15.5563i	−0.681536	−	0.681536i	−0.960346	+	0.278810i	$0.910060\pi$
$522$	7.07107	−	7.07107i	0.309492	−	0.309492i
$523$	4.00000			0.174908			0.0874539	−	0.996169i	$-0.472127\pi$
$523$	4.00000			0.174908			0.0874539	+	0.996169i	$0.472127\pi$
$524$	−8.48528	+	8.48528i	−0.370681	+	0.370681i
$525$		0			0
$526$	−8.00000			−0.348817
$527$		0			0
$528$	−4.00000			−0.174078
$529$		−	23.0000i		−	1.00000i
$530$	−8.48528	+	8.48528i	−0.368577	+	0.368577i
$531$	−12.0000			−0.520756
$532$		0			0
$533$	−14.1421	−	14.1421i	−0.612564	−	0.612564i
$534$	4.24264	+	4.24264i	0.183597	+	0.183597i
$535$		−	8.00000i		−	0.345870i
$536$			12.0000i			0.518321i
$537$	8.48528	+	8.48528i	0.366167	+	0.366167i
$538$	4.24264	+	4.24264i	0.182913	+	0.182913i
$539$	−19.7990	+	19.7990i	−0.852803	+	0.852803i
$540$	2.00000			0.0860663
$541$	−7.07107	+	7.07107i	−0.304009	+	0.304009i	−0.842580	−	0.538571i	$-0.818964\pi$
$541$	−7.07107	+	7.07107i	−0.304009	+	0.304009i	0.538571	+	0.842580i	$0.318964\pi$
$542$			16.0000i			0.687259i
$543$	14.0000			0.600798
$544$		0			0
$545$	20.0000			0.856706
$546$		0			0
$547$	−8.48528	+	8.48528i	−0.362804	+	0.362804i	−0.864844	−	0.502040i	$-0.832583\pi$
$547$	−8.48528	+	8.48528i	−0.362804	+	0.362804i	0.502040	+	0.864844i	$0.332583\pi$
$548$	−10.0000			−0.427179
$549$	7.07107	−	7.07107i	0.301786	−	0.301786i
$550$	2.82843	+	2.82843i	0.120605	+	0.120605i
$551$	28.2843	+	28.2843i	1.20495	+	1.20495i
$552$		0			0
$553$		0			0
$554$	−21.2132	−	21.2132i	−0.901263	−	0.901263i
$555$	2.82843	+	2.82843i	0.120060	+	0.120060i
$556$	2.82843	−	2.82843i	0.119952	−	0.119952i
$557$	34.0000			1.44063			0.720313	−	0.693649i	$-0.243998\pi$
$557$	34.0000			1.44063			0.720313	+	0.693649i	$0.243998\pi$
$558$	5.65685	−	5.65685i	0.239474	−	0.239474i
$559$		−	24.0000i		−	1.01509i
$560$		0			0
$561$		0			0
$562$	−6.00000			−0.253095
$563$			36.0000i			1.51722i	0.651546	+	0.758610i	$0.274121\pi$
$563$			36.0000i			1.51722i	−0.651546	+	0.758610i	$0.725879\pi$
$564$		0			0
$565$	4.00000			0.168281
$566$	−8.48528	+	8.48528i	−0.356663	+	0.356663i
$567$		0			0
$568$		0			0
$569$			6.00000i			0.251533i	0.992060	+	0.125767i	$0.0401390\pi$
$569$			6.00000i			0.251533i	−0.992060	+	0.125767i	$0.959861\pi$
$570$			8.00000i			0.335083i
$571$	−19.7990	−	19.7990i	−0.828562	−	0.828562i	0.158756	−	0.987318i	$-0.449252\pi$
$571$	−19.7990	−	19.7990i	−0.828562	−	0.828562i	−0.987318	+	0.158756i	$0.949252\pi$
$572$	5.65685	+	5.65685i	0.236525	+	0.236525i
$573$	11.3137	−	11.3137i	0.472637	−	0.472637i
$574$		0			0
$575$		0			0
$576$			1.00000i			0.0416667i
$577$	2.00000			0.0832611			0.0416305	−	0.999133i	$-0.486745\pi$
$577$	2.00000			0.0832611			0.0416305	+	0.999133i	$0.486745\pi$
$578$		0			0
$579$	18.0000			0.748054
$580$			20.0000i			0.830455i
$581$		0			0
$582$	14.0000			0.580319
$583$	16.9706	−	16.9706i	0.702849	−	0.702849i
$584$	7.07107	+	7.07107i	0.292603	+	0.292603i
$585$	−2.82843	−	2.82843i	−0.116941	−	0.116941i
$586$			26.0000i			1.07405i
$587$			20.0000i			0.825488i	0.910847	+	0.412744i	$0.135430\pi$
$587$			20.0000i			0.825488i	−0.910847	+	0.412744i	$0.864570\pi$
$588$	4.94975	+	4.94975i	0.204124	+	0.204124i
$589$	22.6274	+	22.6274i	0.932346	+	0.932346i
$590$	16.9706	−	16.9706i	0.698667	−	0.698667i
$591$	−14.0000			−0.575883
$592$	−1.41421	+	1.41421i	−0.0581238	+	0.0581238i
$593$		−	14.0000i		−	0.574911i	−0.957794	−	0.287456i	$-0.907191\pi$
$593$		−	14.0000i		−	0.574911i	0.957794	−	0.287456i	$-0.0928094\pi$
$594$	−4.00000			−0.164122
$595$		0			0
$596$	−10.0000			−0.409616
$597$		0			0
$598$		0			0
$599$	8.00000			0.326871			0.163436	−	0.986554i	$-0.447742\pi$
$599$	8.00000			0.326871			0.163436	+	0.986554i	$0.447742\pi$
$600$	0.707107	−	0.707107i	0.0288675	−	0.0288675i
$601$	−4.24264	−	4.24264i	−0.173061	−	0.173061i	0.615262	−	0.788323i	$-0.289050\pi$
$601$	−4.24264	−	4.24264i	−0.173061	−	0.173061i	−0.788323	+	0.615262i	$0.789050\pi$
$602$		0			0
$603$			12.0000i			0.488678i
$604$		−	24.0000i		−	0.976546i
$605$	7.07107	+	7.07107i	0.287480	+	0.287480i
$606$	−7.07107	−	7.07107i	−0.287242	−	0.287242i
$607$	16.9706	−	16.9706i	0.688814	−	0.688814i	−0.273156	−	0.961970i	$-0.588067\pi$
$607$	16.9706	−	16.9706i	0.688814	−	0.688814i	0.961970	+	0.273156i	$0.0880675\pi$
$608$	−4.00000			−0.162221
$609$		0			0
$610$			20.0000i			0.809776i
$611$		0			0
$612$		0			0
$613$	38.0000			1.53481			0.767403	−	0.641165i	$-0.221549\pi$
$613$	38.0000			1.53481			0.767403	+	0.641165i	$0.221549\pi$
$614$		−	12.0000i		−	0.484281i
$615$	−14.1421	+	14.1421i	−0.570266	+	0.570266i
$616$		0			0
$617$	4.24264	−	4.24264i	0.170802	−	0.170802i	−0.616530	−	0.787332i	$-0.711462\pi$
$617$	4.24264	−	4.24264i	0.170802	−	0.170802i	0.787332	+	0.616530i	$0.211462\pi$
$618$	−5.65685	−	5.65685i	−0.227552	−	0.227552i
$619$	14.1421	+	14.1421i	0.568420	+	0.568420i	0.931686	−	0.363265i	$-0.118338\pi$
$619$	14.1421	+	14.1421i	0.568420	+	0.568420i	−0.363265	+	0.931686i	$0.618338\pi$
$620$			16.0000i			0.642575i
$621$		0			0
$622$		0			0
$623$		0			0
$624$	1.41421	−	1.41421i	0.0566139	−	0.0566139i
$625$	19.0000			0.760000
$626$	7.07107	−	7.07107i	0.282617	−	0.282617i
$627$		−	16.0000i		−	0.638978i
$628$	−2.00000			−0.0798087
$629$		0			0
$630$		0			0
$631$		−	24.0000i		−	0.955425i	−0.878516	−	0.477712i	$-0.841466\pi$
$631$		−	24.0000i		−	0.955425i	0.878516	−	0.477712i	$-0.158534\pi$
$632$	−5.65685	+	5.65685i	−0.225018	+	0.225018i
$633$	28.0000			1.11290
$634$	−4.24264	+	4.24264i	−0.168497	+	0.168497i
$635$		0			0
$636$	−4.24264	−	4.24264i	−0.168232	−	0.168232i
$637$		−	14.0000i		−	0.554700i
$638$		−	40.0000i		−	1.58362i
$639$		0			0
$640$	−1.41421	−	1.41421i	−0.0559017	−	0.0559017i
$641$	−1.41421	+	1.41421i	−0.0558581	+	0.0558581i	−0.734484	−	0.678626i	$-0.762576\pi$
$641$	−1.41421	+	1.41421i	−0.0558581	+	0.0558581i	0.678626	+	0.734484i	$0.262576\pi$
$642$	4.00000			0.157867
$643$	−19.7990	+	19.7990i	−0.780796	+	0.780796i	−0.979965	−	0.199169i	$-0.936176\pi$
$643$	−19.7990	+	19.7990i	−0.780796	+	0.780796i	0.199169	+	0.979965i	$0.436176\pi$
$644$		0			0
$645$	−24.0000			−0.944999
$646$		0			0
$647$	−40.0000			−1.57256			−0.786281	−	0.617869i	$-0.787996\pi$
$647$	−40.0000			−1.57256			−0.786281	+	0.617869i	$0.787996\pi$
$648$			1.00000i			0.0392837i
$649$	−33.9411	+	33.9411i	−1.33231	+	1.33231i
$650$	−2.00000			−0.0784465
$651$		0			0
$652$	2.82843	+	2.82843i	0.110770	+	0.110770i
$653$	−4.24264	−	4.24264i	−0.166027	−	0.166027i	0.619203	−	0.785231i	$-0.287456\pi$
$653$	−4.24264	−	4.24264i	−0.166027	−	0.166027i	−0.785231	+	0.619203i	$0.787456\pi$
$654$			10.0000i			0.391031i
$655$		−	24.0000i		−	0.937758i
$656$	−7.07107	−	7.07107i	−0.276079	−	0.276079i
$657$	7.07107	+	7.07107i	0.275869	+	0.275869i
$658$		0			0
$659$	−36.0000			−1.40236			−0.701180	−	0.712984i	$-0.747343\pi$
$659$	−36.0000			−1.40236			−0.701180	+	0.712984i	$0.747343\pi$
$660$	5.65685	−	5.65685i	0.220193	−	0.220193i
$661$			6.00000i			0.233373i	0.993169	+	0.116686i	$0.0372273\pi$
$661$			6.00000i			0.233373i	−0.993169	+	0.116686i	$0.962773\pi$
$662$	−20.0000			−0.777322
$663$		0			0
$664$	4.00000			0.155230
$665$		0			0
$666$	−1.41421	+	1.41421i	−0.0547997	+	0.0547997i
$667$		0			0
$668$	−11.3137	+	11.3137i	−0.437741	+	0.437741i
$669$	11.3137	+	11.3137i	0.437413	+	0.437413i
$670$	−16.9706	−	16.9706i	−0.655630	−	0.655630i
$671$		−	40.0000i		−	1.54418i
$672$		0			0
$673$	32.5269	+	32.5269i	1.25382	+	1.25382i	0.953994	+	0.299827i	$0.0969288\pi$
$673$	32.5269	+	32.5269i	1.25382	+	1.25382i	0.299827	+	0.953994i	$0.403071\pi$
$674$	−9.89949	−	9.89949i	−0.381314	−	0.381314i
$675$	0.707107	−	0.707107i	0.0272166	−	0.0272166i
$676$	9.00000			0.346154
$677$	32.5269	−	32.5269i	1.25011	−	1.25011i	0.294441	−	0.955670i	$-0.404866\pi$
$677$	32.5269	−	32.5269i	1.25011	−	1.25011i	0.955670	−	0.294441i	$-0.0951335\pi$
$678$			2.00000i			0.0768095i
$679$		0			0
$680$		0			0
$681$	4.00000			0.153280
$682$		−	32.0000i		−	1.22534i
$683$	−14.1421	+	14.1421i	−0.541134	+	0.541134i	−0.923861	−	0.382727i	$-0.874985\pi$
$683$	−14.1421	+	14.1421i	−0.541134	+	0.541134i	0.382727	+	0.923861i	$0.374985\pi$
$684$	−4.00000			−0.152944
$685$	14.1421	−	14.1421i	0.540343	−	0.540343i
$686$		0			0
$687$	18.3848	+	18.3848i	0.701423	+	0.701423i
$688$		−	12.0000i		−	0.457496i
$689$			12.0000i			0.457164i
$690$		0			0
$691$	−19.7990	−	19.7990i	−0.753189	−	0.753189i	0.221884	−	0.975073i	$-0.428779\pi$
$691$	−19.7990	−	19.7990i	−0.753189	−	0.753189i	−0.975073	+	0.221884i	$0.928779\pi$
$692$	−4.24264	+	4.24264i	−0.161281	+	0.161281i
$693$		0			0
$694$	19.7990	−	19.7990i	0.751559	−	0.751559i
$695$			8.00000i			0.303457i
$696$	−10.0000			−0.379049
$697$		0			0
$698$	14.0000			0.529908
$699$			26.0000i			0.983410i
$700$		0			0
$701$	−46.0000			−1.73740			−0.868698	−	0.495342i	$-0.835043\pi$
$701$	−46.0000			−1.73740			−0.868698	+	0.495342i	$0.835043\pi$
$702$	1.41421	−	1.41421i	0.0533761	−	0.0533761i
$703$	−5.65685	−	5.65685i	−0.213352	−	0.213352i
$704$	2.82843	+	2.82843i	0.106600	+	0.106600i
$705$		0			0
$706$			30.0000i			1.12906i
$707$		0			0
$708$	8.48528	+	8.48528i	0.318896	+	0.318896i
$709$	−32.5269	+	32.5269i	−1.22157	+	1.22157i	−0.254501	+	0.967072i	$0.581911\pi$
$709$	−32.5269	+	32.5269i	−1.22157	+	1.22157i	−0.967072	+	0.254501i	$0.918089\pi$
$710$		0			0
$711$	−5.65685	+	5.65685i	−0.212149	+	0.212149i
$712$		−	6.00000i		−	0.224860i
$713$		0			0
$714$		0			0
$715$	−16.0000			−0.598366
$716$		−	12.0000i		−	0.448461i
$717$		0			0
$718$	24.0000			0.895672
$719$	−28.2843	+	28.2843i	−1.05483	+	1.05483i	−0.0564181	+	0.998407i	$0.517968\pi$
$719$	−28.2843	+	28.2843i	−1.05483	+	1.05483i	−0.998407	+	0.0564181i	$0.982032\pi$
$720$	−1.41421	−	1.41421i	−0.0527046	−	0.0527046i
$721$		0			0
$722$			3.00000i			0.111648i
$723$		−	2.00000i		−	0.0743808i
$724$	−9.89949	−	9.89949i	−0.367912	−	0.367912i
$725$	7.07107	+	7.07107i	0.262613	+	0.262613i
$726$	−3.53553	+	3.53553i	−0.131216	+	0.131216i
$727$	8.00000			0.296704			0.148352	−	0.988935i	$-0.452603\pi$
$727$	8.00000			0.296704			0.148352	+	0.988935i	$0.452603\pi$
$728$		0			0
$729$			1.00000i			0.0370370i
$730$	−20.0000			−0.740233
$731$		0			0
$732$	−10.0000			−0.369611
$733$			46.0000i			1.69905i	0.527549	+	0.849524i	$0.323111\pi$
$733$			46.0000i			1.69905i	−0.527549	+	0.849524i	$0.676889\pi$
$734$	16.9706	−	16.9706i	0.626395	−	0.626395i
$735$	−14.0000			−0.516398
$736$		0			0
$737$	33.9411	+	33.9411i	1.25024	+	1.25024i
$738$	−7.07107	−	7.07107i	−0.260290	−	0.260290i
$739$		−	52.0000i		−	1.91285i	−0.291977	−	0.956425i	$-0.594313\pi$
$739$		−	52.0000i		−	1.91285i	0.291977	−	0.956425i	$-0.405687\pi$
$740$		−	4.00000i		−	0.147043i
$741$	5.65685	+	5.65685i	0.207810	+	0.207810i
$742$		0			0
$743$	11.3137	−	11.3137i	0.415060	−	0.415060i	−0.468437	−	0.883497i	$-0.655183\pi$
$743$	11.3137	−	11.3137i	0.415060	−	0.415060i	0.883497	+	0.468437i	$0.155183\pi$
$744$	−8.00000			−0.293294
$745$	14.1421	−	14.1421i	0.518128	−	0.518128i
$746$			6.00000i			0.219676i
$747$	4.00000			0.146352
$748$		0			0
$749$		0			0
$750$			12.0000i			0.438178i
$751$	5.65685	−	5.65685i	0.206422	−	0.206422i	−0.596323	−	0.802745i	$-0.703372\pi$
$751$	5.65685	−	5.65685i	0.206422	−	0.206422i	0.802745	+	0.596323i	$0.203372\pi$
$752$		0			0
$753$	−19.7990	+	19.7990i	−0.721515	+	0.721515i
$754$	14.1421	+	14.1421i	0.515026	+	0.515026i
$755$	33.9411	+	33.9411i	1.23524	+	1.23524i
$756$		0			0
$757$			10.0000i			0.363456i	0.983349	+	0.181728i	$0.0581691\pi$
$757$			10.0000i			0.363456i	−0.983349	+	0.181728i	$0.941831\pi$
$758$	2.82843	+	2.82843i	0.102733	+	0.102733i
$759$		0			0
$760$	5.65685	−	5.65685i	0.205196	−	0.205196i
$761$	6.00000			0.217500			0.108750	−	0.994069i	$-0.465315\pi$
$761$	6.00000			0.217500			0.108750	+	0.994069i	$0.465315\pi$
$762$		0			0
$763$		0			0
$764$	−16.0000			−0.578860
$765$		0			0
$766$	−16.0000			−0.578103
$767$		−	24.0000i		−	0.866590i
$768$	0.707107	−	0.707107i	0.0255155	−	0.0255155i
$769$	30.0000			1.08183			0.540914	−	0.841078i	$-0.318079\pi$
$769$	30.0000			1.08183			0.540914	+	0.841078i	$0.318079\pi$
$770$		0			0
$771$	1.41421	+	1.41421i	0.0509317	+	0.0509317i
$772$	−12.7279	−	12.7279i	−0.458088	−	0.458088i
$773$		−	38.0000i		−	1.36677i	−0.730061	−	0.683383i	$-0.760508\pi$
$773$		−	38.0000i		−	1.36677i	0.730061	−	0.683383i	$-0.239492\pi$
$774$		−	12.0000i		−	0.431331i
$775$	5.65685	+	5.65685i	0.203200	+	0.203200i
$776$	−9.89949	−	9.89949i	−0.355371	−	0.355371i
$777$		0			0
$778$	26.0000			0.932145
$779$	28.2843	−	28.2843i	1.01339	−	1.01339i
$780$			4.00000i			0.143223i
$781$		0			0
$782$		0			0
$783$	−10.0000			−0.357371
$784$		−	7.00000i		−	0.250000i
$785$	2.82843	−	2.82843i	0.100951	−	0.100951i
$786$	12.0000			0.428026
$787$	−2.82843	+	2.82843i	−0.100823	+	0.100823i	−0.755719	−	0.654896i	$-0.772712\pi$
$787$	−2.82843	+	2.82843i	−0.100823	+	0.100823i	0.654896	+	0.755719i	$0.272712\pi$
$788$	9.89949	+	9.89949i	0.352655	+	0.352655i
$789$	5.65685	+	5.65685i	0.201389

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:	\( N \)	\(=\)	\( 1734 = 2 \cdot 3 \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1734.f (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} +(0.707107 - 0.707107i) q^{3} -1.00000 q^{4} +(1.41421 - 1.41421i) q^{5} +(0.707107 + 0.707107i) q^{6} -1.00000i q^{8} -1.00000i q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} +(0.707107 - 0.707107i) q^{3} -1.00000 q^{4} +(1.41421 - 1.41421i) q^{5} +(0.707107 + 0.707107i) q^{6} -1.00000i q^{8} -1.00000i q^{9} +(1.41421 + 1.41421i) q^{10} +(-2.82843 - 2.82843i) q^{11} +(-0.707107 + 0.707107i) q^{12} +2.00000 q^{13} -2.00000i q^{15} +1.00000 q^{16} +1.00000 q^{18} +4.00000i q^{19} +(-1.41421 + 1.41421i) q^{20} +(2.82843 - 2.82843i) q^{22} +(-0.707107 - 0.707107i) q^{24} +1.00000i q^{25} +2.00000i q^{26} +(-0.707107 - 0.707107i) q^{27} +(7.07107 - 7.07107i) q^{29} +2.00000 q^{30} +(5.65685 - 5.65685i) q^{31} +1.00000i q^{32} -4.00000 q^{33} +1.00000i q^{36} +(-1.41421 + 1.41421i) q^{37} -4.00000 q^{38} +(1.41421 - 1.41421i) q^{39} +(-1.41421 - 1.41421i) q^{40} +(-7.07107 - 7.07107i) q^{41} -12.0000i q^{43} +(2.82843 + 2.82843i) q^{44} +(-1.41421 - 1.41421i) q^{45} +(0.707107 - 0.707107i) q^{48} -7.00000i q^{49} -1.00000 q^{50} -2.00000 q^{52} +6.00000i q^{53} +(0.707107 - 0.707107i) q^{54} -8.00000 q^{55} +(2.82843 + 2.82843i) q^{57} +(7.07107 + 7.07107i) q^{58} -12.0000i q^{59} +2.00000i q^{60} +(7.07107 + 7.07107i) q^{61} +(5.65685 + 5.65685i) q^{62} -1.00000 q^{64} +(2.82843 - 2.82843i) q^{65} -4.00000i q^{66} -12.0000 q^{67} -1.00000 q^{72} +(-7.07107 + 7.07107i) q^{73} +(-1.41421 - 1.41421i) q^{74} +(0.707107 + 0.707107i) q^{75} -4.00000i q^{76} +(1.41421 + 1.41421i) q^{78} +(-5.65685 - 5.65685i) q^{79} +(1.41421 - 1.41421i) q^{80} -1.00000 q^{81} +(7.07107 - 7.07107i) q^{82} +4.00000i q^{83} +12.0000 q^{86} -10.0000i q^{87} +(-2.82843 + 2.82843i) q^{88} +6.00000 q^{89} +(1.41421 - 1.41421i) q^{90} -8.00000i q^{93} +(5.65685 + 5.65685i) q^{95} +(0.707107 + 0.707107i) q^{96} +(9.89949 - 9.89949i) q^{97} +7.00000 q^{98} +(-2.82843 + 2.82843i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4}+O(q^{10}) \) 4 * q - 4 * q^4	\( 4 q - 4 q^{4} + 8 q^{13} + 4 q^{16} + 4 q^{18} + 8 q^{30} - 16 q^{33} - 16 q^{38} - 4 q^{50} - 8 q^{52} - 32 q^{55} - 4 q^{64} - 48 q^{67} - 4 q^{72} - 4 q^{81} + 48 q^{86} + 24 q^{89} + 28 q^{98}+O(q^{100}) \) 4 * q - 4 * q^4 + 8 * q^13 + 4 * q^16 + 4 * q^18 + 8 * q^30 - 16 * q^33 - 16 * q^38 - 4 * q^50 - 8 * q^52 - 32 * q^55 - 4 * q^64 - 48 * q^67 - 4 * q^72 - 4 * q^81 + 48 * q^86 + 24 * q^89 + 28 * q^98

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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