LMFDB - Embedded newform 1050.2.o.c.499.2

Newspace parameters

Level:	$ N $	$=$	$ 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1050.o (of order $6$, degree $2$, not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$8.38429221223$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{12})$
Defining polynomial:	$x^{4} - x^{2} + 1$
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 210)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			499.2
Root			$-0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	1050.499
Dual form			1050.2.o.c.949.2

$q$-expansion

$f(q)$	$=$	$q+(0.866025 + 0.500000i) q^{2} +(-0.866025 + 0.500000i) q^{3} +(0.500000 + 0.866025i) q^{4} -1.00000 q^{6} +(1.73205 - 2.00000i) q^{7} +1.00000i q^{8} +(0.500000 - 0.866025i) q^{9} +O(q^{10})$	\(q+(0.866025 + 0.500000i) q^{2} +(-0.866025 + 0.500000i) q^{3} +(0.500000 + 0.866025i) q^{4} -1.00000 q^{6} +(1.73205 - 2.00000i) q^{7} +1.00000i q^{8} +(0.500000 - 0.866025i) q^{9} +(-1.50000 - 2.59808i) q^{11} +(-0.866025 - 0.500000i) q^{12} -5.00000i q^{13} +(2.50000 - 0.866025i) q^{14} +(-0.500000 + 0.866025i) q^{16} +(0.866025 - 0.500000i) q^{18} +(2.50000 - 4.33013i) q^{19} +(-0.500000 + 2.59808i) q^{21} -3.00000i q^{22} +(-7.79423 - 4.50000i) q^{23} +(-0.500000 - 0.866025i) q^{24} +(2.50000 - 4.33013i) q^{26} +1.00000i q^{27} +(2.59808 + 0.500000i) q^{28} +(5.00000 + 8.66025i) q^{31} +(-0.866025 + 0.500000i) q^{32} +(2.59808 + 1.50000i) q^{33} +1.00000 q^{36} +(0.866025 + 0.500000i) q^{37} +(4.33013 - 2.50000i) q^{38} +(2.50000 + 4.33013i) q^{39} +9.00000 q^{41} +(-1.73205 + 2.00000i) q^{42} -8.00000i q^{43} +(1.50000 - 2.59808i) q^{44} +(-4.50000 - 7.79423i) q^{46} +(-2.59808 - 1.50000i) q^{47} -1.00000i q^{48} +(-1.00000 - 6.92820i) q^{49} +(4.33013 - 2.50000i) q^{52} +(2.59808 - 1.50000i) q^{53} +(-0.500000 + 0.866025i) q^{54} +(2.00000 + 1.73205i) q^{56} +5.00000i q^{57} +(6.00000 + 10.3923i) q^{59} +(-4.00000 + 6.92820i) q^{61} +10.0000i q^{62} +(-0.866025 - 2.50000i) q^{63} -1.00000 q^{64} +(1.50000 + 2.59808i) q^{66} +(6.92820 - 4.00000i) q^{67} +9.00000 q^{69} -6.00000 q^{71} +(0.866025 + 0.500000i) q^{72} +(-1.73205 + 1.00000i) q^{73} +(0.500000 + 0.866025i) q^{74} +5.00000 q^{76} +(-7.79423 - 1.50000i) q^{77} +5.00000i q^{78} +(4.00000 - 6.92820i) q^{79} +(-0.500000 - 0.866025i) q^{81} +(7.79423 + 4.50000i) q^{82} +(-2.50000 + 0.866025i) q^{84} +(4.00000 - 6.92820i) q^{86} +(2.59808 - 1.50000i) q^{88} +(3.00000 - 5.19615i) q^{89} +(-10.0000 - 8.66025i) q^{91} -9.00000i q^{92} +(-8.66025 - 5.00000i) q^{93} +(-1.50000 - 2.59808i) q^{94} +(0.500000 - 0.866025i) q^{96} +8.00000i q^{97} +(2.59808 - 6.50000i) q^{98} -3.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4q + 2q^{4} - 4q^{6} + 2q^{9} + O(q^{10}) $	$ 4q + 2q^{4} - 4q^{6} + 2q^{9} - 6q^{11} + 10q^{14} - 2q^{16} + 10q^{19} - 2q^{21} - 2q^{24} + 10q^{26} + 20q^{31} + 4q^{36} + 10q^{39} + 36q^{41} + 6q^{44} - 18q^{46} - 4q^{49} - 2q^{54} + 8q^{56} + 24q^{59} - 16q^{61} - 4q^{64} + 6q^{66} + 36q^{69} - 24q^{71} + 2q^{74} + 20q^{76} + 16q^{79} - 2q^{81} - 10q^{84} + 16q^{86} + 12q^{89} - 40q^{91} - 6q^{94} + 2q^{96} - 12q^{99} + O(q^{100}) $

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1050\mathbb{Z}\right)^\times$.

$n$	$127$	$451$	$701$
$\chi(n)$	$-1$	$e\left(\frac{1}{3}\right)$	$1$

Coefficient data

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

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

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

$2$	0.866025	+	0.500000i	0.612372	+	0.353553i
$2$	0.866025	+	0.500000i	0.612372	+	0.353553i
$3$	−0.866025	+	0.500000i	−0.500000	+	0.288675i
$3$	−0.866025	+	0.500000i	−0.500000	+	0.288675i
$4$	0.500000	+	0.866025i	0.250000	+	0.433013i
$5$		0			0
$5$		0			0
$6$	−1.00000			−0.408248
$7$	1.73205	−	2.00000i	0.654654	−	0.755929i
$7$	1.73205	−	2.00000i	0.654654	−	0.755929i
$8$			1.00000i			0.353553i
$9$	0.500000	−	0.866025i	0.166667	−	0.288675i
$10$		0			0
$11$	−1.50000	−	2.59808i	−0.452267	−	0.783349i	0.546259	−	0.837616i	$-0.316051\pi$
$11$	−1.50000	−	2.59808i	−0.452267	−	0.783349i	−0.998526	+	0.0542666i	$0.982718\pi$
$12$	−0.866025	−	0.500000i	−0.250000	−	0.144338i
$13$		−	5.00000i		−	1.38675i	−0.720577	−	0.693375i	$-0.756123\pi$
$13$		−	5.00000i		−	1.38675i	0.720577	−	0.693375i	$-0.243877\pi$
$14$	2.50000	−	0.866025i	0.668153	−	0.231455i
$15$		0			0
$16$	−0.500000	+	0.866025i	−0.125000	+	0.216506i
$17$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$17$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$18$	0.866025	−	0.500000i	0.204124	−	0.117851i
$19$	2.50000	−	4.33013i	0.573539	−	0.993399i	−0.422659	−	0.906289i	$-0.638903\pi$
$19$	2.50000	−	4.33013i	0.573539	−	0.993399i	0.996199	−	0.0871106i	$-0.0277634\pi$
$20$		0			0
$21$	−0.500000	+	2.59808i	−0.109109	+	0.566947i
$22$		−	3.00000i		−	0.639602i
$23$	−7.79423	−	4.50000i	−1.62521	−	0.938315i	−0.985496	−	0.169701i	$-0.945720\pi$
$23$	−7.79423	−	4.50000i	−1.62521	−	0.938315i	−0.639713	−	0.768613i	$-0.720947\pi$
$24$	−0.500000	−	0.866025i	−0.102062	−	0.176777i
$25$		0			0
$26$	2.50000	−	4.33013i	0.490290	−	0.849208i
$27$			1.00000i			0.192450i
$28$	2.59808	+	0.500000i	0.490990	+	0.0944911i
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$		0			0
$31$	5.00000	+	8.66025i	0.898027	+	1.55543i	0.830014	+	0.557743i	$0.188333\pi$
$31$	5.00000	+	8.66025i	0.898027	+	1.55543i	0.0680129	+	0.997684i	$0.478334\pi$
$32$	−0.866025	+	0.500000i	−0.153093	+	0.0883883i
$33$	2.59808	+	1.50000i	0.452267	+	0.261116i
$34$		0			0
$35$		0			0
$36$	1.00000			0.166667
$37$	0.866025	+	0.500000i	0.142374	+	0.0821995i	0.569495	−	0.821995i	$-0.307139\pi$
$37$	0.866025	+	0.500000i	0.142374	+	0.0821995i	−0.427121	+	0.904194i	$0.640472\pi$
$38$	4.33013	−	2.50000i	0.702439	−	0.405554i
$39$	2.50000	+	4.33013i	0.400320	+	0.693375i
$40$		0			0
$41$	9.00000			1.40556			0.702782	−	0.711405i	$-0.251941\pi$
$41$	9.00000			1.40556			0.702782	+	0.711405i	$0.251941\pi$
$42$	−1.73205	+	2.00000i	−0.267261	+	0.308607i
$43$		−	8.00000i		−	1.21999i	−0.792406	−	0.609994i	$-0.791172\pi$
$43$		−	8.00000i		−	1.21999i	0.792406	−	0.609994i	$-0.208828\pi$
$44$	1.50000	−	2.59808i	0.226134	−	0.391675i
$45$		0			0
$46$	−4.50000	−	7.79423i	−0.663489	−	1.14920i
$47$	−2.59808	−	1.50000i	−0.378968	−	0.218797i	0.298401	−	0.954441i	$-0.403547\pi$
$47$	−2.59808	−	1.50000i	−0.378968	−	0.218797i	−0.677369	+	0.735643i	$0.736880\pi$
$48$		−	1.00000i		−	0.144338i
$49$	−1.00000	−	6.92820i	−0.142857	−	0.989743i
$50$		0			0
$51$		0			0
$52$	4.33013	−	2.50000i	0.600481	−	0.346688i
$53$	2.59808	−	1.50000i	0.356873	−	0.206041i	−0.310835	−	0.950464i	$-0.600609\pi$
$53$	2.59808	−	1.50000i	0.356873	−	0.206041i	0.667708	+	0.744423i	$0.267275\pi$
$54$	−0.500000	+	0.866025i	−0.0680414	+	0.117851i
$55$		0			0
$56$	2.00000	+	1.73205i	0.267261	+	0.231455i
$57$			5.00000i			0.662266i
$58$		0			0
$59$	6.00000	+	10.3923i	0.781133	+	1.35296i	0.931282	+	0.364299i	$0.118692\pi$
$59$	6.00000	+	10.3923i	0.781133	+	1.35296i	−0.150148	+	0.988663i	$0.547975\pi$
$60$		0			0
$61$	−4.00000	+	6.92820i	−0.512148	+	0.887066i	0.487753	+	0.872982i	$0.337817\pi$
$61$	−4.00000	+	6.92820i	−0.512148	+	0.887066i	−0.999901	+	0.0140840i	$0.995517\pi$
$62$			10.0000i			1.27000i
$63$	−0.866025	−	2.50000i	−0.109109	−	0.314970i
$64$	−1.00000			−0.125000
$65$		0			0
$66$	1.50000	+	2.59808i	0.184637	+	0.319801i
$67$	6.92820	−	4.00000i	0.846415	−	0.488678i	−0.0130248	−	0.999915i	$-0.504146\pi$
$67$	6.92820	−	4.00000i	0.846415	−	0.488678i	0.859440	+	0.511237i	$0.170813\pi$
$68$		0			0
$69$	9.00000			1.08347
$70$		0			0
$71$	−6.00000			−0.712069			−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000			−0.712069			−0.356034	+	0.934473i	$0.615871\pi$
$72$	0.866025	+	0.500000i	0.102062	+	0.0589256i
$73$	−1.73205	+	1.00000i	−0.202721	+	0.117041i	−0.597924	−	0.801553i	$-0.704008\pi$
$73$	−1.73205	+	1.00000i	−0.202721	+	0.117041i	0.395203	+	0.918594i	$0.370674\pi$
$74$	0.500000	+	0.866025i	0.0581238	+	0.100673i
$75$		0			0
$76$	5.00000			0.573539
$77$	−7.79423	−	1.50000i	−0.888235	−	0.170941i
$78$			5.00000i			0.566139i
$79$	4.00000	−	6.92820i	0.450035	−	0.779484i	−0.548352	−	0.836247i	$-0.684745\pi$
$79$	4.00000	−	6.92820i	0.450035	−	0.779484i	0.998388	+	0.0567635i	$0.0180781\pi$
$80$		0			0
$81$	−0.500000	−	0.866025i	−0.0555556	−	0.0962250i
$82$	7.79423	+	4.50000i	0.860729	+	0.496942i
$83$		0			0		1.00000			$0$
$83$		0			0		−1.00000			$\pi$
$84$	−2.50000	+	0.866025i	−0.272772	+	0.0944911i
$85$		0			0
$86$	4.00000	−	6.92820i	0.431331	−	0.747087i
$87$		0			0
$88$	2.59808	−	1.50000i	0.276956	−	0.159901i
$89$	3.00000	−	5.19615i	0.317999	−	0.550791i	−0.662071	−	0.749441i	$-0.730322\pi$
$89$	3.00000	−	5.19615i	0.317999	−	0.550791i	0.980071	+	0.198650i	$0.0636557\pi$
$90$		0			0
$91$	−10.0000	−	8.66025i	−1.04828	−	0.907841i
$92$		−	9.00000i		−	0.938315i
$93$	−8.66025	−	5.00000i	−0.898027	−	0.518476i
$94$	−1.50000	−	2.59808i	−0.154713	−	0.267971i
$95$		0			0
$96$	0.500000	−	0.866025i	0.0510310	−	0.0883883i
$97$			8.00000i			0.812277i	0.913812	+	0.406138i	$0.133125\pi$
$97$			8.00000i			0.812277i	−0.913812	+	0.406138i	$0.866875\pi$
$98$	2.59808	−	6.50000i	0.262445	−	0.656599i
$99$	−3.00000			−0.301511
$100$		0			0
$101$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$101$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$102$		0			0
$103$	6.92820	+	4.00000i	0.682656	+	0.394132i	0.800855	−	0.598858i	$-0.204379\pi$
$103$	6.92820	+	4.00000i	0.682656	+	0.394132i	−0.118199	+	0.992990i	$0.537712\pi$
$104$	5.00000			0.490290
$105$		0			0
$106$	3.00000			0.291386
$107$	−5.19615	−	3.00000i	−0.502331	−	0.290021i	0.227345	−	0.973814i	$-0.426996\pi$
$107$	−5.19615	−	3.00000i	−0.502331	−	0.290021i	−0.729676	+	0.683793i	$0.760329\pi$
$108$	−0.866025	+	0.500000i	−0.0833333	+	0.0481125i
$109$	7.00000	+	12.1244i	0.670478	+	1.16130i	0.977769	+	0.209687i	$0.0672444\pi$
$109$	7.00000	+	12.1244i	0.670478	+	1.16130i	−0.307290	+	0.951616i	$0.599422\pi$
$110$		0			0
$111$	−1.00000			−0.0949158
$112$	0.866025	+	2.50000i	0.0818317	+	0.236228i
$113$			18.0000i			1.69330i	0.532152	+	0.846649i	$0.321383\pi$
$113$			18.0000i			1.69330i	−0.532152	+	0.846649i	$0.678617\pi$
$114$	−2.50000	+	4.33013i	−0.234146	+	0.405554i
$115$		0			0
$116$		0			0
$117$	−4.33013	−	2.50000i	−0.400320	−	0.231125i
$118$			12.0000i			1.10469i
$119$		0			0
$120$		0			0
$121$	1.00000	−	1.73205i	0.0909091	−	0.157459i
$122$	−6.92820	+	4.00000i	−0.627250	+	0.362143i
$123$	−7.79423	+	4.50000i	−0.702782	+	0.405751i
$124$	−5.00000	+	8.66025i	−0.449013	+	0.777714i
$125$		0			0
$126$	0.500000	−	2.59808i	0.0445435	−	0.231455i
$127$		−	13.0000i		−	1.15356i	−0.816898	−	0.576782i	$-0.804308\pi$
$127$		−	13.0000i		−	1.15356i	0.816898	−	0.576782i	$-0.195692\pi$
$128$	−0.866025	−	0.500000i	−0.0765466	−	0.0441942i
$129$	4.00000	+	6.92820i	0.352180	+	0.609994i
$130$		0			0
$131$	4.50000	−	7.79423i	0.393167	−	0.680985i	−0.599699	−	0.800226i	$-0.704713\pi$
$131$	4.50000	−	7.79423i	0.393167	−	0.680985i	0.992865	+	0.119241i	$0.0380462\pi$
$132$			3.00000i			0.261116i
$133$	−4.33013	−	12.5000i	−0.375470	−	1.08389i
$134$	8.00000			0.691095
$135$		0			0
$136$		0			0
$137$	−15.5885	+	9.00000i	−1.33181	+	0.768922i	−0.985577	−	0.169226i	$-0.945873\pi$
$137$	−15.5885	+	9.00000i	−1.33181	+	0.768922i	−0.346235	+	0.938148i	$0.612540\pi$
$138$	7.79423	+	4.50000i	0.663489	+	0.383065i
$139$	−20.0000			−1.69638			−0.848189	−	0.529694i	$-0.822307\pi$
$139$	−20.0000			−1.69638			−0.848189	+	0.529694i	$0.822307\pi$
$140$		0			0
$141$	3.00000			0.252646
$142$	−5.19615	−	3.00000i	−0.436051	−	0.251754i
$143$	−12.9904	+	7.50000i	−1.08631	+	0.627182i
$144$	0.500000	+	0.866025i	0.0416667	+	0.0721688i
$145$		0			0
$146$	−2.00000			−0.165521
$147$	4.33013	+	5.50000i	0.357143	+	0.453632i
$148$			1.00000i			0.0821995i
$149$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$149$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$150$		0			0
$151$	5.00000	+	8.66025i	0.406894	+	0.704761i	0.994540	−	0.104357i	$-0.0332784\pi$
$151$	5.00000	+	8.66025i	0.406894	+	0.704761i	−0.587646	+	0.809118i	$0.699945\pi$
$152$	4.33013	+	2.50000i	0.351220	+	0.202777i
$153$		0			0
$154$	−6.00000	−	5.19615i	−0.483494	−	0.418718i
$155$		0			0
$156$	−2.50000	+	4.33013i	−0.200160	+	0.346688i
$157$	4.33013	−	2.50000i	0.345582	−	0.199522i	−0.317156	−	0.948373i	$-0.602728\pi$
$157$	4.33013	−	2.50000i	0.345582	−	0.199522i	0.662738	+	0.748852i	$0.269394\pi$
$158$	6.92820	−	4.00000i	0.551178	−	0.318223i
$159$	−1.50000	+	2.59808i	−0.118958	+	0.206041i
$160$		0			0
$161$	−22.5000	+	7.79423i	−1.77325	+	0.614271i
$162$		−	1.00000i		−	0.0785674i
$163$	−13.8564	−	8.00000i	−1.08532	−	0.626608i	−0.152992	−	0.988227i	$-0.548891\pi$
$163$	−13.8564	−	8.00000i	−1.08532	−	0.626608i	−0.932326	+	0.361619i	$0.882224\pi$
$164$	4.50000	+	7.79423i	0.351391	+	0.608627i
$165$		0			0
$166$		0			0
$167$		−	3.00000i		−	0.232147i	−0.993241	−	0.116073i	$-0.962969\pi$
$167$		−	3.00000i		−	0.232147i	0.993241	−	0.116073i	$-0.0370308\pi$
$168$	−2.59808	−	0.500000i	−0.200446	−	0.0385758i
$169$	−12.0000			−0.923077
$170$		0			0
$171$	−2.50000	−	4.33013i	−0.191180	−	0.331133i
$172$	6.92820	−	4.00000i	0.528271	−	0.304997i
$173$	7.79423	+	4.50000i	0.592584	+	0.342129i	0.766119	−	0.642699i	$-0.222185\pi$
$173$	7.79423	+	4.50000i	0.592584	+	0.342129i	−0.173534	+	0.984828i	$0.555519\pi$
$174$		0			0
$175$		0			0
$176$	3.00000			0.226134
$177$	−10.3923	−	6.00000i	−0.781133	−	0.450988i
$178$	5.19615	−	3.00000i	0.389468	−	0.224860i
$179$	−7.50000	−	12.9904i	−0.560576	−	0.970947i	−0.997446	−	0.0714220i	$-0.977246\pi$
$179$	−7.50000	−	12.9904i	−0.560576	−	0.970947i	0.436870	−	0.899525i	$-0.356087\pi$
$180$		0			0
$181$	8.00000			0.594635			0.297318	−	0.954779i	$-0.403908\pi$
$181$	8.00000			0.594635			0.297318	+	0.954779i	$0.403908\pi$
$182$	−4.33013	−	12.5000i	−0.320970	−	0.926562i
$183$		−	8.00000i		−	0.591377i
$184$	4.50000	−	7.79423i	0.331744	−	0.574598i
$185$		0			0
$186$	−5.00000	−	8.66025i	−0.366618	−	0.635001i
$187$		0			0
$188$		−	3.00000i		−	0.218797i
$189$	2.00000	+	1.73205i	0.145479	+	0.125988i
$190$		0			0
$191$	−9.00000	+	15.5885i	−0.651217	+	1.12794i	0.331611	+	0.943416i	$0.392408\pi$
$191$	−9.00000	+	15.5885i	−0.651217	+	1.12794i	−0.982828	+	0.184525i	$0.940925\pi$
$192$	0.866025	−	0.500000i	0.0625000	−	0.0360844i
$193$	8.66025	−	5.00000i	0.623379	−	0.359908i	−0.154805	−	0.987945i	$-0.549475\pi$
$193$	8.66025	−	5.00000i	0.623379	−	0.359908i	0.778183	+	0.628037i	$0.216141\pi$
$194$	−4.00000	+	6.92820i	−0.287183	+	0.497416i
$195$		0			0
$196$	5.50000	−	4.33013i	0.392857	−	0.309295i
$197$			15.0000i			1.06871i	0.845262	+	0.534353i	$0.179445\pi$
$197$			15.0000i			1.06871i	−0.845262	+	0.534353i	$0.820555\pi$
$198$	−2.59808	−	1.50000i	−0.184637	−	0.106600i
$199$	−8.00000	−	13.8564i	−0.567105	−	0.982255i	−0.996850	−	0.0793045i	$-0.974730\pi$
$199$	−8.00000	−	13.8564i	−0.567105	−	0.982255i	0.429745	−	0.902950i	$-0.358603\pi$
$200$		0			0
$201$	−4.00000	+	6.92820i	−0.282138	+	0.488678i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$	4.00000	+	6.92820i	0.278693	+	0.482711i
$207$	−7.79423	+	4.50000i	−0.541736	+	0.312772i
$208$	4.33013	+	2.50000i	0.300240	+	0.173344i
$209$	−15.0000			−1.03757
$210$		0			0
$211$	5.00000			0.344214			0.172107	−	0.985078i	$-0.444942\pi$
$211$	5.00000			0.344214			0.172107	+	0.985078i	$0.444942\pi$
$212$	2.59808	+	1.50000i	0.178437	+	0.103020i
$213$	5.19615	−	3.00000i	0.356034	−	0.205557i
$214$	−3.00000	−	5.19615i	−0.205076	−	0.355202i
$215$		0			0
$216$	−1.00000			−0.0680414
$217$	25.9808	+	5.00000i	1.76369	+	0.339422i
$218$			14.0000i			0.948200i
$219$	1.00000	−	1.73205i	0.0675737	−	0.117041i
$220$		0			0
$221$		0			0
$222$	−0.866025	−	0.500000i	−0.0581238	−	0.0335578i
$223$			28.0000i			1.87502i	0.347960	+	0.937509i	$0.386874\pi$
$223$			28.0000i			1.87502i	−0.347960	+	0.937509i	$0.613126\pi$
$224$	−0.500000	+	2.59808i	−0.0334077	+	0.173591i
$225$		0			0
$226$	−9.00000	+	15.5885i	−0.598671	+	1.03693i
$227$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$227$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$228$	−4.33013	+	2.50000i	−0.286770	+	0.165567i
$229$	7.00000	−	12.1244i	0.462573	−	0.801200i	−0.536515	−	0.843891i	$-0.680260\pi$
$229$	7.00000	−	12.1244i	0.462573	−	0.801200i	0.999088	+	0.0426906i	$0.0135930\pi$
$230$		0			0
$231$	7.50000	−	2.59808i	0.493464	−	0.170941i
$232$		0			0
$233$	−5.19615	−	3.00000i	−0.340411	−	0.196537i	0.320043	−	0.947403i	$-0.396303\pi$
$233$	−5.19615	−	3.00000i	−0.340411	−	0.196537i	−0.660454	+	0.750867i	$0.729636\pi$
$234$	−2.50000	−	4.33013i	−0.163430	−	0.283069i
$235$		0			0
$236$	−6.00000	+	10.3923i	−0.390567	+	0.676481i
$237$			8.00000i			0.519656i
$238$		0			0
$239$	−30.0000			−1.94054			−0.970269	−	0.242028i	$-0.922188\pi$
$239$	−30.0000			−1.94054			−0.970269	+	0.242028i	$0.922188\pi$
$240$		0			0
$241$	0.500000	+	0.866025i	0.0322078	+	0.0557856i	0.881680	−	0.471848i	$-0.156413\pi$
$241$	0.500000	+	0.866025i	0.0322078	+	0.0557856i	−0.849472	+	0.527633i	$0.823079\pi$
$242$	1.73205	−	1.00000i	0.111340	−	0.0642824i
$243$	0.866025	+	0.500000i	0.0555556	+	0.0320750i
$244$	−8.00000			−0.512148
$245$		0			0
$246$	−9.00000			−0.573819
$247$	−21.6506	−	12.5000i	−1.37760	−	0.795356i
$248$	−8.66025	+	5.00000i	−0.549927	+	0.317500i
$249$		0			0
$250$		0			0
$251$	9.00000			0.568075			0.284037	−	0.958813i	$-0.408326\pi$
$251$	9.00000			0.568075			0.284037	+	0.958813i	$0.408326\pi$
$252$	1.73205	−	2.00000i	0.109109	−	0.125988i
$253$			27.0000i			1.69748i
$254$	6.50000	−	11.2583i	0.407846	−	0.706410i
$255$		0			0
$256$	−0.500000	−	0.866025i	−0.0312500	−	0.0541266i
$257$	10.3923	+	6.00000i	0.648254	+	0.374270i	0.787787	−	0.615948i	$-0.211227\pi$
$257$	10.3923	+	6.00000i	0.648254	+	0.374270i	−0.139533	+	0.990217i	$0.544560\pi$
$258$			8.00000i			0.498058i
$259$	2.50000	−	0.866025i	0.155342	−	0.0538122i
$260$		0			0
$261$		0			0
$262$	7.79423	−	4.50000i	0.481529	−	0.278011i
$263$	20.7846	−	12.0000i	1.28163	−	0.739952i	0.304487	−	0.952517i	$-0.401515\pi$
$263$	20.7846	−	12.0000i	1.28163	−	0.739952i	0.977147	+	0.212565i	$0.0681817\pi$
$264$	−1.50000	+	2.59808i	−0.0923186	+	0.159901i
$265$		0			0
$266$	2.50000	−	12.9904i	0.153285	−	0.796491i
$267$			6.00000i			0.367194i
$268$	6.92820	+	4.00000i	0.423207	+	0.244339i
$269$	6.00000	+	10.3923i	0.365826	+	0.633630i	0.988908	−	0.148527i	$-0.0474530\pi$
$269$	6.00000	+	10.3923i	0.365826	+	0.633630i	−0.623082	+	0.782157i	$0.714120\pi$
$270$		0			0
$271$	8.00000	−	13.8564i	0.485965	−	0.841717i	−0.513905	−	0.857847i	$-0.671801\pi$
$271$	8.00000	−	13.8564i	0.485965	−	0.841717i	0.999870	+	0.0161307i	$0.00513477\pi$
$272$		0			0
$273$	12.9904	+	2.50000i	0.786214	+	0.151307i
$274$	−18.0000			−1.08742
$275$		0			0
$276$	4.50000	+	7.79423i	0.270868	+	0.469157i
$277$	22.5167	−	13.0000i	1.35290	−	0.781094i	0.364241	−	0.931305i	$-0.381328\pi$
$277$	22.5167	−	13.0000i	1.35290	−	0.781094i	0.988654	+	0.150210i	$0.0479951\pi$
$278$	−17.3205	−	10.0000i	−1.03882	−	0.599760i
$279$	10.0000			0.598684
$280$		0			0
$281$	−21.0000			−1.25275			−0.626377	−	0.779520i	$-0.715463\pi$
$281$	−21.0000			−1.25275			−0.626377	+	0.779520i	$0.715463\pi$
$282$	2.59808	+	1.50000i	0.154713	+	0.0893237i
$283$	19.0526	−	11.0000i	1.13256	−	0.653882i	0.187980	−	0.982173i	$-0.439806\pi$
$283$	19.0526	−	11.0000i	1.13256	−	0.653882i	0.944577	+	0.328291i	$0.106473\pi$
$284$	−3.00000	−	5.19615i	−0.178017	−	0.308335i
$285$		0			0
$286$	−15.0000			−0.886969
$287$	15.5885	−	18.0000i	0.920158	−	1.06251i
$288$			1.00000i			0.0589256i
$289$	−8.50000	+	14.7224i	−0.500000	+	0.866025i
$290$		0			0
$291$	−4.00000	−	6.92820i	−0.234484	−	0.406138i
$292$	−1.73205	−	1.00000i	−0.101361	−	0.0585206i
$293$			21.0000i			1.22683i	0.789760	+	0.613417i	$0.210205\pi$
$293$			21.0000i			1.22683i	−0.789760	+	0.613417i	$0.789795\pi$
$294$	1.00000	+	6.92820i	0.0583212	+	0.404061i
$295$		0			0
$296$	−0.500000	+	0.866025i	−0.0290619	+	0.0503367i
$297$	2.59808	−	1.50000i	0.150756	−	0.0870388i
$298$		0			0
$299$	−22.5000	+	38.9711i	−1.30121	+	2.25376i
$300$		0			0
$301$	−16.0000	−	13.8564i	−0.922225	−	0.798670i
$302$			10.0000i			0.575435i
$303$		0			0
$304$	2.50000	+	4.33013i	0.143385	+	0.248350i
$305$		0			0
$306$		0			0
$307$		−	28.0000i		−	1.59804i	−0.601302	−	0.799022i	$-0.705351\pi$
$307$		−	28.0000i		−	1.59804i	0.601302	−	0.799022i	$-0.294649\pi$
$308$	−2.59808	−	7.50000i	−0.148039	−	0.427352i
$309$	−8.00000			−0.455104
$310$		0			0
$311$	6.00000	+	10.3923i	0.340229	+	0.589294i	0.984475	−	0.175525i	$-0.0561621\pi$
$311$	6.00000	+	10.3923i	0.340229	+	0.589294i	−0.644246	+	0.764818i	$0.722829\pi$
$312$	−4.33013	+	2.50000i	−0.245145	+	0.141535i
$313$	−3.46410	−	2.00000i	−0.195803	−	0.113047i	0.398894	−	0.916997i	$-0.369394\pi$
$313$	−3.46410	−	2.00000i	−0.195803	−	0.113047i	−0.594696	+	0.803951i	$0.702728\pi$
$314$	5.00000			0.282166
$315$		0			0
$316$	8.00000			0.450035
$317$	15.5885	+	9.00000i	0.875535	+	0.505490i	0.869184	−	0.494489i	$-0.164645\pi$
$317$	15.5885	+	9.00000i	0.875535	+	0.505490i	0.00635137	+	0.999980i	$0.497978\pi$
$318$	−2.59808	+	1.50000i	−0.145693	+	0.0841158i
$319$		0			0
$320$		0			0
$321$	6.00000			0.334887
$322$	−23.3827	−	4.50000i	−1.30307	−	0.250775i
$323$		0			0
$324$	0.500000	−	0.866025i	0.0277778	−	0.0481125i
$325$		0			0
$326$	−8.00000	−	13.8564i	−0.443079	−	0.767435i
$327$	−12.1244	−	7.00000i	−0.670478	−	0.387101i
$328$			9.00000i			0.496942i
$329$	−7.50000	+	2.59808i	−0.413488	+	0.143237i
$330$		0			0
$331$	−5.50000	+	9.52628i	−0.302307	+	0.523612i	−0.976658	−	0.214799i	$-0.931090\pi$
$331$	−5.50000	+	9.52628i	−0.302307	+	0.523612i	0.674351	+	0.738411i	$0.264424\pi$
$332$		0			0
$333$	0.866025	−	0.500000i	0.0474579	−	0.0273998i
$334$	1.50000	−	2.59808i	0.0820763	−	0.142160i
$335$		0			0
$336$	−2.00000	−	1.73205i	−0.109109	−	0.0944911i
$337$			20.0000i			1.08947i	0.838608	+	0.544735i	$0.183370\pi$
$337$			20.0000i			1.08947i	−0.838608	+	0.544735i	$0.816630\pi$
$338$	−10.3923	−	6.00000i	−0.565267	−	0.326357i
$339$	−9.00000	−	15.5885i	−0.488813	−	0.846649i
$340$		0			0
$341$	15.0000	−	25.9808i	0.812296	−	1.40694i
$342$		−	5.00000i		−	0.270369i
$343$	−15.5885	−	10.0000i	−0.841698	−	0.539949i
$344$	8.00000			0.431331
$345$		0			0
$346$	4.50000	+	7.79423i	0.241921	+	0.419020i
$347$	25.9808	−	15.0000i	1.39472	−	0.805242i	0.400887	−	0.916127i	$-0.368702\pi$
$347$	25.9808	−	15.0000i	1.39472	−	0.805242i	0.993833	+	0.110885i	$0.0353686\pi$
$348$		0			0
$349$	28.0000			1.49881			0.749403	−	0.662114i	$-0.230341\pi$
$349$	28.0000			1.49881			0.749403	+	0.662114i	$0.230341\pi$
$350$		0			0
$351$	5.00000			0.266880
$352$	2.59808	+	1.50000i	0.138478	+	0.0799503i
$353$	−20.7846	+	12.0000i	−1.10625	+	0.638696i	−0.937856	−	0.347024i	$-0.887192\pi$
$353$	−20.7846	+	12.0000i	−1.10625	+	0.638696i	−0.168397	+	0.985719i	$0.553859\pi$
$354$	−6.00000	−	10.3923i	−0.318896	−	0.552345i
$355$		0			0
$356$	6.00000			0.317999
$357$		0			0
$358$		−	15.0000i		−	0.792775i
$359$	6.00000	−	10.3923i	0.316668	−	0.548485i	−0.663123	−	0.748511i	$-0.730769\pi$
$359$	6.00000	−	10.3923i	0.316668	−	0.548485i	0.979791	+	0.200026i	$0.0641026\pi$
$360$		0			0
$361$	−3.00000	−	5.19615i	−0.157895	−	0.273482i
$362$	6.92820	+	4.00000i	0.364138	+	0.210235i
$363$			2.00000i			0.104973i
$364$	2.50000	−	12.9904i	0.131036	−	0.680881i
$365$		0			0
$366$	4.00000	−	6.92820i	0.209083	−	0.362143i
$367$	−16.4545	+	9.50000i	−0.858917	+	0.495896i	−0.863649	−	0.504093i	$-0.831827\pi$
$367$	−16.4545	+	9.50000i	−0.858917	+	0.495896i	0.00473247	+	0.999989i	$0.498494\pi$
$368$	7.79423	−	4.50000i	0.406302	−	0.234579i
$369$	4.50000	−	7.79423i	0.234261	−	0.405751i
$370$		0			0
$371$	1.50000	−	7.79423i	0.0778761	−	0.404656i
$372$		−	10.0000i		−	0.518476i
$373$	−8.66025	−	5.00000i	−0.448411	−	0.258890i	0.258748	−	0.965945i	$-0.416690\pi$
$373$	−8.66025	−	5.00000i	−0.448411	−	0.258890i	−0.707159	+	0.707055i	$0.750023\pi$
$374$		0			0
$375$		0			0
$376$	1.50000	−	2.59808i	0.0773566	−	0.133986i
$377$		0			0
$378$	0.866025	+	2.50000i	0.0445435	+	0.128586i
$379$	19.0000			0.975964			0.487982	−	0.872854i	$-0.337733\pi$
$379$	19.0000			0.975964			0.487982	+	0.872854i	$0.337733\pi$
$380$		0			0
$381$	6.50000	+	11.2583i	0.333005	+	0.576782i
$382$	−15.5885	+	9.00000i	−0.797575	+	0.460480i
$383$	23.3827	+	13.5000i	1.19480	+	0.689818i	0.959391	−	0.282079i	$-0.0910240\pi$
$383$	23.3827	+	13.5000i	1.19480	+	0.689818i	0.235408	+	0.971897i	$0.424357\pi$
$384$	1.00000			0.0510310
$385$		0			0
$386$	10.0000			0.508987
$387$	−6.92820	−	4.00000i	−0.352180	−	0.203331i
$388$	−6.92820	+	4.00000i	−0.351726	+	0.203069i
$389$	−3.00000	−	5.19615i	−0.152106	−	0.263455i	0.779895	−	0.625910i	$-0.215272\pi$
$389$	−3.00000	−	5.19615i	−0.152106	−	0.263455i	−0.932002	+	0.362454i	$0.881939\pi$
$390$		0			0
$391$		0			0
$392$	6.92820	−	1.00000i	0.349927	−	0.0505076i
$393$			9.00000i			0.453990i
$394$	−7.50000	+	12.9904i	−0.377845	+	0.654446i
$395$		0			0
$396$	−1.50000	−	2.59808i	−0.0753778	−	0.130558i
$397$	−12.1244	−	7.00000i	−0.608504	−	0.351320i	0.163876	−	0.986481i	$-0.447600\pi$
$397$	−12.1244	−	7.00000i	−0.608504	−	0.351320i	−0.772380	+	0.635161i	$0.780934\pi$
$398$		−	16.0000i		−	0.802008i
$399$	10.0000	+	8.66025i	0.500626	+	0.433555i
$400$		0			0
$401$	−7.50000	+	12.9904i	−0.374532	+	0.648709i	−0.990257	−	0.139253i	$-0.955530\pi$
$401$	−7.50000	+	12.9904i	−0.374532	+	0.648709i	0.615725	+	0.787961i	$0.288863\pi$
$402$	−6.92820	+	4.00000i	−0.345547	+	0.199502i
$403$	43.3013	−	25.0000i	2.15699	−	1.24534i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	3.00000i		−	0.148704i
$408$		0			0
$409$	1.00000	+	1.73205i	0.0494468	+	0.0856444i	0.889689	−	0.456566i	$-0.150921\pi$
$409$	1.00000	+	1.73205i	0.0494468	+	0.0856444i	−0.840243	+	0.542211i	$0.817588\pi$
$410$		0			0
$411$	9.00000	−	15.5885i	0.443937	−	0.768922i
$412$			8.00000i			0.394132i
$413$	31.1769	+	6.00000i	1.53412	+	0.295241i
$414$	−9.00000			−0.442326
$415$		0			0
$416$	2.50000	+	4.33013i	0.122573	+	0.212302i
$417$	17.3205	−	10.0000i	0.848189	−	0.489702i
$418$	−12.9904	−	7.50000i	−0.635380	−	0.366837i
$419$	9.00000			0.439679			0.219839	−	0.975536i	$-0.429447\pi$
$419$	9.00000			0.439679			0.219839	+	0.975536i	$0.429447\pi$
$420$		0			0
$421$	14.0000			0.682318			0.341159	−	0.940006i	$-0.389181\pi$
$421$	14.0000			0.682318			0.341159	+	0.940006i	$0.389181\pi$
$422$	4.33013	+	2.50000i	0.210787	+	0.121698i
$423$	−2.59808	+	1.50000i	−0.126323	+	0.0729325i
$424$	1.50000	+	2.59808i	0.0728464	+	0.126174i
$425$		0			0
$426$	6.00000			0.290701
$427$	6.92820	+	20.0000i	0.335279	+	0.967868i
$428$		−	6.00000i		−	0.290021i
$429$	7.50000	−	12.9904i	0.362103	−	0.627182i
$430$		0			0
$431$	12.0000	+	20.7846i	0.578020	+	1.00116i	0.995706	+	0.0925683i	$0.0295076\pi$
$431$	12.0000	+	20.7846i	0.578020	+	1.00116i	−0.417687	+	0.908591i	$0.637159\pi$
$432$	−0.866025	−	0.500000i	−0.0416667	−	0.0240563i
$433$			16.0000i			0.768911i	0.923144	+	0.384455i	$0.125611\pi$
$433$			16.0000i			0.768911i	−0.923144	+	0.384455i	$0.874389\pi$
$434$	20.0000	+	17.3205i	0.960031	+	0.831411i
$435$		0			0
$436$	−7.00000	+	12.1244i	−0.335239	+	0.580651i
$437$	−38.9711	+	22.5000i	−1.86424	+	1.07632i
$438$	1.73205	−	1.00000i	0.0827606	−	0.0477818i
$439$	4.00000	−	6.92820i	0.190910	−	0.330665i	−0.754642	−	0.656136i	$-0.772190\pi$
$439$	4.00000	−	6.92820i	0.190910	−	0.330665i	0.945552	+	0.325471i	$0.105523\pi$
$440$		0			0
$441$	−6.50000	−	2.59808i	−0.309524	−	0.123718i
$442$		0			0
$443$	−20.7846	−	12.0000i	−0.987507	−	0.570137i	−0.0829786	−	0.996551i	$-0.526443\pi$
$443$	−20.7846	−	12.0000i	−0.987507	−	0.570137i	−0.904528	+	0.426414i	$0.859777\pi$
$444$	−0.500000	−	0.866025i	−0.0237289	−	0.0410997i
$445$		0			0
$446$	−14.0000	+	24.2487i	−0.662919	+	1.14821i
$447$		0			0
$448$	−1.73205	+	2.00000i	−0.0818317	+	0.0944911i
$449$	33.0000			1.55737			0.778683	−	0.627417i	$-0.215888\pi$
$449$	33.0000			1.55737			0.778683	+	0.627417i	$0.215888\pi$
$450$		0			0
$451$	−13.5000	−	23.3827i	−0.635690	−	1.10105i
$452$	−15.5885	+	9.00000i	−0.733219	+	0.423324i
$453$	−8.66025	−	5.00000i	−0.406894	−	0.234920i
$454$		0			0
$455$		0			0
$456$	−5.00000			−0.234146
$457$	8.66025	+	5.00000i	0.405110	+	0.233890i	0.688686	−	0.725059i	$-0.258188\pi$
$457$	8.66025	+	5.00000i	0.405110	+	0.233890i	−0.283577	+	0.958950i	$0.591521\pi$
$458$	12.1244	−	7.00000i	0.566534	−	0.327089i
$459$		0			0
$460$		0			0
$461$	12.0000			0.558896			0.279448	−	0.960161i	$-0.409849\pi$
$461$	12.0000			0.558896			0.279448	+	0.960161i	$0.409849\pi$
$462$	7.79423	+	1.50000i	0.362620	+	0.0697863i
$463$			1.00000i			0.0464739i	0.999730	+	0.0232370i	$0.00739722\pi$
$463$			1.00000i			0.0464739i	−0.999730	+	0.0232370i	$0.992603\pi$
$464$		0			0
$465$		0			0
$466$	−3.00000	−	5.19615i	−0.138972	−	0.240707i
$467$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$467$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$468$		−	5.00000i		−	0.231125i
$469$	4.00000	−	20.7846i	0.184703	−	0.959744i
$470$		0			0
$471$	−2.50000	+	4.33013i	−0.115194	+	0.199522i
$472$	−10.3923	+	6.00000i	−0.478345	+	0.276172i
$473$	−20.7846	+	12.0000i	−0.955677	+	0.551761i
$474$	−4.00000	+	6.92820i	−0.183726	+	0.318223i
$475$		0			0
$476$		0			0
$477$		−	3.00000i		−	0.137361i
$478$	−25.9808	−	15.0000i	−1.18833	−	0.686084i
$479$	−9.00000	−	15.5885i	−0.411220	−	0.712255i	0.583803	−	0.811895i	$-0.301564\pi$
$479$	−9.00000	−	15.5885i	−0.411220	−	0.712255i	−0.995023	+	0.0996406i	$0.968231\pi$
$480$		0			0
$481$	2.50000	−	4.33013i	0.113990	−	0.197437i
$482$			1.00000i			0.0455488i
$483$	15.5885	−	18.0000i	0.709299	−	0.819028i
$484$	2.00000			0.0909091
$485$		0			0
$486$	0.500000	+	0.866025i	0.0226805	+	0.0392837i
$487$	−34.6410	+	20.0000i	−1.56973	+	0.906287i	−0.573535	+	0.819181i	$0.694428\pi$
$487$	−34.6410	+	20.0000i	−1.56973	+	0.906287i	−0.996199	+	0.0871056i	$0.972238\pi$
$488$	−6.92820	−	4.00000i	−0.313625	−	0.181071i
$489$	16.0000			0.723545
$490$		0			0
$491$	12.0000			0.541552			0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000			0.541552			0.270776	+	0.962642i	$0.412720\pi$
$492$	−7.79423	−	4.50000i	−0.351391	−	0.202876i
$493$		0			0
$494$	−12.5000	−	21.6506i	−0.562402	−	0.974108i
$495$		0			0
$496$	−10.0000			−0.449013
$497$	−10.3923	+	12.0000i	−0.466159	+	0.538274i
$498$		0			0
$499$	−8.00000	+	13.8564i	−0.358129	+	0.620298i	−0.987648	−	0.156687i	$-0.949919\pi$
$499$	−8.00000	+	13.8564i	−0.358129	+	0.620298i	0.629519	+	0.776985i	$0.283252\pi$
$500$		0			0
$501$	1.50000	+	2.59808i	0.0670151	+	0.116073i
$502$	7.79423	+	4.50000i	0.347873	+	0.200845i
$503$			12.0000i			0.535054i	0.963550	+	0.267527i	$0.0862064\pi$
$503$			12.0000i			0.535054i	−0.963550	+	0.267527i	$0.913794\pi$
$504$	2.50000	−	0.866025i	0.111359	−	0.0385758i
$505$		0			0
$506$	−13.5000	+	23.3827i	−0.600148	+	1.03949i
$507$	10.3923	−	6.00000i	0.461538	−	0.266469i
$508$	11.2583	−	6.50000i	0.499508	−	0.288391i
$509$	3.00000	−	5.19615i	0.132973	−	0.230315i	−0.791849	−	0.610718i	$-0.790881\pi$
$509$	3.00000	−	5.19615i	0.132973	−	0.230315i	0.924821	+	0.380402i	$0.124214\pi$
$510$		0			0
$511$	−1.00000	+	5.19615i	−0.0442374	+	0.229864i
$512$		−	1.00000i		−	0.0441942i
$513$	4.33013	+	2.50000i	0.191180	+	0.110378i
$514$	6.00000	+	10.3923i	0.264649	+	0.458385i
$515$		0			0
$516$	−4.00000	+	6.92820i	−0.176090	+	0.304997i
$517$			9.00000i			0.395820i
$518$	2.59808	+	0.500000i	0.114153	+	0.0219687i
$519$	−9.00000			−0.395056
$520$		0			0
$521$	−13.5000	−	23.3827i	−0.591446	−	1.02441i	−0.994038	−	0.109035i	$-0.965224\pi$
$521$	−13.5000	−	23.3827i	−0.591446	−	1.02441i	0.402592	−	0.915379i	$-0.368109\pi$
$522$		0			0
$523$	−19.0526	−	11.0000i	−0.833110	−	0.480996i	0.0218062	−	0.999762i	$-0.493058\pi$
$523$	−19.0526	−	11.0000i	−0.833110	−	0.480996i	−0.854916	+	0.518766i	$0.826392\pi$
$524$	9.00000			0.393167
$525$		0			0
$526$	24.0000			1.04645
$527$		0			0
$528$	−2.59808	+	1.50000i	−0.113067	+	0.0652791i
$529$	29.0000	+	50.2295i	1.26087	+	2.18389i
$530$		0			0
$531$	12.0000			0.520756
$532$	8.66025	−	10.0000i	0.375470	−	0.433555i
$533$		−	45.0000i		−	1.94917i
$534$	−3.00000	+	5.19615i	−0.129823	+	0.224860i
$535$		0			0
$536$	4.00000	+	6.92820i	0.172774	+	0.299253i
$537$	12.9904	+	7.50000i	0.560576	+	0.323649i
$538$			12.0000i			0.517357i
$539$	−16.5000	+	12.9904i	−0.710705	+	0.559535i
$540$		0			0
$541$	5.00000	−	8.66025i	0.214967	−	0.372333i	−0.738296	−	0.674477i	$-0.764369\pi$
$541$	5.00000	−	8.66025i	0.214967	−	0.372333i	0.953262	+	0.302144i	$0.0977023\pi$
$542$	13.8564	−	8.00000i	0.595184	−	0.343629i
$543$	−6.92820	+	4.00000i	−0.297318	+	0.171656i
$544$		0			0
$545$		0			0
$546$	10.0000	+	8.66025i	0.427960	+	0.370625i
$547$			8.00000i			0.342055i	0.985266	+	0.171028i	$0.0547087\pi$
$547$			8.00000i			0.342055i	−0.985266	+	0.171028i	$0.945291\pi$
$548$	−15.5885	−	9.00000i	−0.665906	−	0.384461i
$549$	4.00000	+	6.92820i	0.170716	+	0.295689i
$550$		0			0
$551$		0			0
$552$			9.00000i			0.383065i
$553$	−6.92820	−	20.0000i	−0.294617	−	0.850487i
$554$	26.0000			1.10463
$555$		0			0
$556$	−10.0000	−	17.3205i	−0.424094	−	0.734553i
$557$	33.7750	−	19.5000i	1.43109	−	0.826242i	0.433888	−	0.900967i	$-0.357141\pi$
$557$	33.7750	−	19.5000i	1.43109	−	0.826242i	0.997204	+	0.0747252i	$0.0238080\pi$
$558$	8.66025	+	5.00000i	0.366618	+	0.211667i
$559$	−40.0000			−1.69182
$560$		0			0
$561$		0			0
$562$	−18.1865	−	10.5000i	−0.767153	−	0.442916i
$563$	5.19615	−	3.00000i	0.218992	−	0.126435i	−0.386492	−	0.922293i	$-0.626313\pi$
$563$	5.19615	−	3.00000i	0.218992	−	0.126435i	0.605483	+	0.795858i	$0.292980\pi$
$564$	1.50000	+	2.59808i	0.0631614	+	0.109399i
$565$		0			0
$566$	22.0000			0.924729
$567$	−2.59808	−	0.500000i	−0.109109	−	0.0209980i
$568$		−	6.00000i		−	0.251754i
$569$	13.5000	−	23.3827i	0.565949	−	0.980253i	−0.431011	−	0.902347i	$-0.641843\pi$
$569$	13.5000	−	23.3827i	0.565949	−	0.980253i	0.996961	−	0.0779066i	$-0.0248236\pi$
$570$		0			0
$571$	8.00000	+	13.8564i	0.334790	+	0.579873i	0.983444	−	0.181210i	$-0.0580014\pi$
$571$	8.00000	+	13.8564i	0.334790	+	0.579873i	−0.648655	+	0.761083i	$0.724668\pi$
$572$	−12.9904	−	7.50000i	−0.543155	−	0.313591i
$573$		−	18.0000i		−	0.751961i
$574$	22.5000	−	7.79423i	0.939132	−	0.325325i
$575$		0			0
$576$	−0.500000	+	0.866025i	−0.0208333	+	0.0360844i
$577$	−19.0526	+	11.0000i	−0.793168	+	0.457936i	−0.841077	−	0.540916i	$-0.818078\pi$
$577$	−19.0526	+	11.0000i	−0.793168	+	0.457936i	0.0479084	+	0.998852i	$0.484744\pi$
$578$	−14.7224	+	8.50000i	−0.612372	+	0.353553i
$579$	−5.00000	+	8.66025i	−0.207793	+	0.359908i
$580$		0			0
$581$		0			0
$582$		−	8.00000i		−	0.331611i
$583$	−7.79423	−	4.50000i	−0.322804	−	0.186371i
$584$	−1.00000	−	1.73205i	−0.0413803	−	0.0716728i
$585$		0			0
$586$	−10.5000	+	18.1865i	−0.433751	+	0.751279i
$587$		−	42.0000i		−	1.73353i	−0.498721	−	0.866763i	$-0.666197\pi$
$587$		−	42.0000i		−	1.73353i	0.498721	−	0.866763i	$-0.333803\pi$
$588$	−2.59808	+	6.50000i	−0.107143	+	0.268055i
$589$	50.0000			2.06021
$590$		0			0
$591$	−7.50000	−	12.9904i	−0.308509	−	0.534353i
$592$	−0.866025	+	0.500000i	−0.0355934	+	0.0205499i
$593$	20.7846	+	12.0000i	0.853522	+	0.492781i	0.861838	−	0.507184i	$-0.169314\pi$
$593$	20.7846	+	12.0000i	0.853522	+	0.492781i	−0.00831589	+	0.999965i	$0.502647\pi$
$594$	3.00000			0.123091
$595$		0			0
$596$		0			0
$597$	13.8564	+	8.00000i	0.567105	+	0.327418i
$598$	−38.9711	+	22.5000i	−1.59365	+	0.920093i
$599$	9.00000	+	15.5885i	0.367730	+	0.636927i	0.989210	−	0.146503i	$-0.0468017\pi$
$599$	9.00000	+	15.5885i	0.367730	+	0.636927i	−0.621480	+	0.783430i	$0.713468\pi$
$600$		0			0
$601$	38.0000			1.55005			0.775026	−	0.631929i	$-0.217737\pi$
$601$	38.0000			1.55005			0.775026	+	0.631929i	$0.217737\pi$
$602$	−6.92820	−	20.0000i	−0.282372	−	0.815139i
$603$		−	8.00000i		−	0.325785i
$604$	−5.00000	+	8.66025i	−0.203447	+	0.352381i
$605$		0			0
$606$		0			0
$607$	21.6506	+	12.5000i	0.878772	+	0.507359i	0.870253	−	0.492604i	$-0.163955\pi$
$607$	21.6506	+	12.5000i	0.878772	+	0.507359i	0.00851879	+	0.999964i	$0.497288\pi$
$608$			5.00000i			0.202777i
$609$		0			0
$610$		0			0
$611$	−7.50000	+	12.9904i	−0.303418	+	0.525535i
$612$		0			0
$613$	−9.52628	+	5.50000i	−0.384763	+	0.222143i	−0.679888	−	0.733316i	$-0.737972\pi$
$613$	−9.52628	+	5.50000i	−0.384763	+	0.222143i	0.295126	+	0.955458i	$0.404638\pi$
$614$	14.0000	−	24.2487i	0.564994	−	0.978598i
$615$		0			0
$616$	1.50000	−	7.79423i	0.0604367	−	0.314038i
$617$			12.0000i			0.483102i	0.970388	+	0.241551i	$0.0776561\pi$
$617$			12.0000i			0.483102i	−0.970388	+	0.241551i	$0.922344\pi$
$618$	−6.92820	−	4.00000i	−0.278693	−	0.160904i
$619$	14.5000	+	25.1147i	0.582804	+	1.00945i	0.995145	+	0.0984169i	$0.0313779\pi$
$619$	14.5000	+	25.1147i	0.582804	+	1.00945i	−0.412341	+	0.911030i	$0.635289\pi$
$620$		0			0
$621$	4.50000	−	7.79423i	0.180579	−	0.312772i
$622$			12.0000i			0.481156i
$623$	−5.19615	−	15.0000i	−0.208179	−	0.600962i
$624$	−5.00000			−0.200160
$625$		0			0
$626$	−2.00000	−	3.46410i	−0.0799361	−	0.138453i
$627$	12.9904	−	7.50000i	0.518786	−	0.299521i
$628$	4.33013	+	2.50000i	0.172791	+	0.0997609i
$629$		0			0
$630$		0			0
$631$	−34.0000			−1.35352			−0.676759	−	0.736204i	$-0.736616\pi$
$631$	−34.0000			−1.35352			−0.676759	+	0.736204i	$0.736616\pi$
$632$	6.92820	+	4.00000i	0.275589	+	0.159111i
$633$	−4.33013	+	2.50000i	−0.172107	+	0.0993661i
$634$	9.00000	+	15.5885i	0.357436	+	0.619097i
$635$		0			0
$636$	−3.00000			−0.118958
$637$	−34.6410	+	5.00000i	−1.37253	+	0.198107i
$638$		0			0
$639$	−3.00000	+	5.19615i	−0.118678	+	0.205557i
$640$		0			0
$641$	−16.5000	−	28.5788i	−0.651711	−	1.12880i	−0.982708	−	0.185164i	$-0.940718\pi$
$641$	−16.5000	−	28.5788i	−0.651711	−	1.12880i	0.330997	−	0.943632i	$-0.392615\pi$
$642$	5.19615	+	3.00000i	0.205076	+	0.118401i
$643$			34.0000i			1.34083i	0.741987	+	0.670415i	$0.233884\pi$
$643$			34.0000i			1.34083i	−0.741987	+	0.670415i	$0.766116\pi$
$644$	−18.0000	−	15.5885i	−0.709299	−	0.614271i
$645$		0			0
$646$		0			0
$647$	−7.79423	+	4.50000i	−0.306423	+	0.176913i	−0.645325	−	0.763908i	$-0.723278\pi$
$647$	−7.79423	+	4.50000i	−0.306423	+	0.176913i	0.338902	+	0.940822i	$0.389945\pi$
$648$	0.866025	−	0.500000i	0.0340207	−	0.0196419i
$649$	18.0000	−	31.1769i	0.706562	−	1.22380i
$650$		0			0
$651$	−25.0000	+	8.66025i	−0.979827	+	0.339422i
$652$		−	16.0000i		−	0.626608i
$653$	33.7750	+	19.5000i	1.32172	+	0.763094i	0.984003	−	0.178154i	$-0.0570127\pi$
$653$	33.7750	+	19.5000i	1.32172	+	0.763094i	0.337715	+	0.941248i	$0.390346\pi$
$654$	−7.00000	−	12.1244i	−0.273722	−	0.474100i
$655$		0			0
$656$	−4.50000	+	7.79423i	−0.175695	+	0.304314i
$657$			2.00000i			0.0780274i
$658$	−7.79423	−	1.50000i	−0.303851	−	0.0584761i
$659$	−12.0000			−0.467454			−0.233727	−	0.972302i	$-0.575092\pi$
$659$	−12.0000			−0.467454			−0.233727	+	0.972302i	$0.575092\pi$
$660$		0			0
$661$	20.0000	+	34.6410i	0.777910	+	1.34738i	0.933144	+	0.359502i	$0.117053\pi$
$661$	20.0000	+	34.6410i	0.777910	+	1.34738i	−0.155235	+	0.987878i	$0.549613\pi$
$662$	−9.52628	+	5.50000i	−0.370249	+	0.213764i
$663$		0			0
$664$		0			0
$665$		0			0
$666$	1.00000			0.0387492
$667$		0			0
$668$	2.59808	−	1.50000i	0.100523	−	0.0580367i
$669$	−14.0000	−	24.2487i	−0.541271	−	0.937509i
$670$		0			0
$671$	24.0000			0.926510
$672$	−0.866025	−	2.50000i	−0.0334077	−	0.0964396i
$673$		−	20.0000i		−	0.770943i	−0.922720	−	0.385472i	$-0.874039\pi$
$673$		−	20.0000i		−	0.770943i	0.922720	−	0.385472i	$-0.125961\pi$
$674$	−10.0000	+	17.3205i	−0.385186	+	0.667161i
$675$		0			0
$676$	−6.00000	−	10.3923i	−0.230769	−	0.399704i
$677$	−2.59808	−	1.50000i	−0.0998522	−	0.0576497i	0.449242	−	0.893410i	$-0.351694\pi$
$677$	−2.59808	−	1.50000i	−0.0998522	−	0.0576497i	−0.549095	+	0.835760i	$0.685027\pi$
$678$		−	18.0000i		−	0.691286i
$679$	16.0000	+	13.8564i	0.614024	+	0.531760i
$680$		0			0
$681$		0			0
$682$	25.9808	−	15.0000i	0.994855	−	0.574380i
$683$	10.3923	−	6.00000i	0.397650	−	0.229584i	−0.287819	−	0.957685i	$-0.592930\pi$
$683$	10.3923	−	6.00000i	0.397650	−	0.229584i	0.685470	+	0.728101i	$0.259597\pi$
$684$	2.50000	−	4.33013i	0.0955899	−	0.165567i
$685$		0			0
$686$	−8.50000	−	16.4545i	−0.324532	−	0.628235i
$687$			14.0000i			0.534133i
$688$	6.92820	+	4.00000i	0.264135	+	0.152499i
$689$	−7.50000	−	12.9904i	−0.285727	−	0.494894i
$690$		0			0
$691$	14.0000	−	24.2487i	0.532585	−	0.922464i	−0.466691	−	0.884420i	$-0.654554\pi$
$691$	14.0000	−	24.2487i	0.532585	−	0.922464i	0.999276	−	0.0380440i	$-0.0121127\pi$
$692$			9.00000i			0.342129i
$693$	−5.19615	+	6.00000i	−0.197386	+	0.227921i
$694$	30.0000			1.13878
$695$		0			0
$696$		0			0
$697$		0			0
$698$	24.2487	+	14.0000i	0.917827	+	0.529908i
$699$	6.00000			0.226941
$700$		0			0
$701$	−42.0000			−1.58632			−0.793159	−	0.609015i	$-0.791565\pi$
$701$	−42.0000			−1.58632			−0.793159	+	0.609015i	$0.791565\pi$
$702$	4.33013	+	2.50000i	0.163430	+	0.0943564i
$703$	4.33013	−	2.50000i	0.163314	−	0.0942893i
$704$	1.50000	+	2.59808i	0.0565334	+	0.0979187i
$705$		0			0
$706$	−24.0000			−0.903252
$707$		0			0
$708$		−	12.0000i		−	0.450988i
$709$	−14.0000	+	24.2487i	−0.525781	+	0.910679i	0.473768	+	0.880650i	$0.342894\pi$
$709$	−14.0000	+	24.2487i	−0.525781	+	0.910679i	−0.999549	+	0.0300298i	$0.990440\pi$
$710$		0			0
$711$	−4.00000	−	6.92820i	−0.150012	−	0.259828i
$712$	5.19615	+	3.00000i	0.194734	+	0.112430i
$713$		−	90.0000i		−	3.37053i
$714$		0			0
$715$		0			0
$716$	7.50000	−	12.9904i	0.280288	−	0.485473i
$717$	25.9808	−	15.0000i	0.970269	−	0.560185i
$718$	10.3923	−	6.00000i	0.387837	−	0.223918i
$719$	9.00000	−	15.5885i	0.335643	−	0.581351i	−0.647965	−	0.761670i	$-0.724380\pi$
$719$	9.00000	−	15.5885i	0.335643	−	0.581351i	0.983608	+	0.180319i	$0.0577130\pi$
$720$		0			0
$721$	20.0000	−	6.92820i	0.744839	−	0.258020i
$722$		−	6.00000i		−	0.223297i
$723$	−0.866025	−	0.500000i	−0.0322078	−	0.0185952i
$724$	4.00000	+	6.92820i	0.148659	+	0.257485i
$725$		0			0
$726$	−1.00000	+	1.73205i	−0.0371135	+	0.0642824i
$727$			23.0000i			0.853023i	0.904482	+	0.426511i	$0.140258\pi$
$727$			23.0000i			0.853023i	−0.904482	+	0.426511i	$0.859742\pi$
$728$	8.66025	−	10.0000i	0.320970	−	0.370625i
$729$	−1.00000			−0.0370370
$730$		0			0
$731$		0			0
$732$	6.92820	−	4.00000i	0.256074	−	0.147844i
$733$	−0.866025	−	0.500000i	−0.0319874	−	0.0184679i	0.483921	−	0.875112i	$-0.339212\pi$
$733$	−0.866025	−	0.500000i	−0.0319874	−	0.0184679i	−0.515908	+	0.856644i	$0.672546\pi$
$734$	−19.0000			−0.701303
$735$		0			0
$736$	9.00000			0.331744
$737$	−20.7846	−	12.0000i	−0.765611	−	0.442026i
$738$	7.79423	−	4.50000i	0.286910	−	0.165647i
$739$	2.50000	+	4.33013i	0.0919640	+	0.159286i	0.908337	−	0.418238i	$-0.137352\pi$
$739$	2.50000	+	4.33013i	0.0919640	+	0.159286i	−0.816373	+	0.577524i	$0.804019\pi$
$740$		0			0
$741$	25.0000			0.918398
$742$	5.19615	−	6.00000i	0.190757	−	0.220267i
$743$			39.0000i			1.43077i	0.698730	+	0.715386i	$0.253749\pi$
$743$			39.0000i			1.43077i	−0.698730	+	0.715386i	$0.746251\pi$
$744$	5.00000	−	8.66025i	0.183309	−	0.317500i
$745$		0			0
$746$	−5.00000	−	8.66025i	−0.183063	−	0.317074i
$747$		0			0
$748$		0			0
$749$	−15.0000	+	5.19615i	−0.548088	+	0.189863i
$750$		0			0
$751$	−1.00000	+	1.73205i	−0.0364905	+	0.0632034i	−0.883694	−	0.468065i	$-0.844951\pi$
$751$	−1.00000	+	1.73205i	−0.0364905	+	0.0632034i	0.847203	+	0.531269i	$0.178285\pi$
$752$	2.59808	−	1.50000i	0.0947421	−	0.0546994i
$753$	−7.79423	+	4.50000i	−0.284037	+	0.163989i
$754$		0			0
$755$		0			0
$756$	−0.500000	+	2.59808i	−0.0181848	+	0.0944911i
$757$			2.00000i			0.0726912i	0.999339	+	0.0363456i	$0.0115717\pi$
$757$			2.00000i			0.0726912i	−0.999339	+	0.0363456i	$0.988428\pi$
$758$	16.4545	+	9.50000i	0.597654	+	0.345056i
$759$	−13.5000	−	23.3827i	−0.490019	−	0.848738i
$760$		0			0
$761$	16.5000	−	28.5788i	0.598125	−	1.03598i	−0.394973	−	0.918693i	$-0.629246\pi$
$761$	16.5000	−	28.5788i	0.598125	−	1.03598i	0.993098	−	0.117289i	$-0.0374205\pi$
$762$			13.0000i			0.470940i
$763$	36.3731	+	7.00000i	1.31679	+	0.253417i
$764$	−18.0000			−0.651217
$765$		0			0
$766$	13.5000	+	23.3827i	0.487775	+	0.844851i
$767$	51.9615	−	30.0000i	1.87622	−	1.08324i
$768$	0.866025	+	0.500000i	0.0312500	+	0.0180422i
$769$	−35.0000			−1.26213			−0.631066	−	0.775729i	$-0.717382\pi$
$769$	−35.0000			−1.26213			−0.631066	+	0.775729i	$0.717382\pi$
$770$		0			0
$771$	−12.0000			−0.432169
$772$	8.66025	+	5.00000i	0.311689	+	0.179954i
$773$	−28.5788	+	16.5000i	−1.02791	+	0.593464i	−0.916385	−	0.400298i	$-0.868907\pi$
$773$	−28.5788	+	16.5000i	−1.02791	+	0.593464i	−0.111524	+	0.993762i	$0.535573\pi$
$774$	−4.00000	−	6.92820i	−0.143777	−	0.249029i
$775$		0			0
$776$	−8.00000			−0.287183
$777$	−1.73205	+	2.00000i	−0.0621370	+	0.0717496i
$778$		−	6.00000i		−	0.215110i
$779$	22.5000	−	38.9711i	0.806146	−	1.39629i
$780$		0			0
$781$	9.00000	+	15.5885i	0.322045	+	0.557799i
$782$		0			0
$783$		0			0
$784$	6.50000	+	2.59808i	0.232143	+	0.0927884i
$785$		0			0
$786$	−4.50000	+	7.79423i	−0.160510	+	0.278011i
$787$	12.1244	−	7.00000i	0.432187	−	0.249523i	−0.268091	−	0.963394i	$-0.586393\pi$
$787$	12.1244	−	7.00000i	0.432187	−	0.249523i	0.700278	+	0.713871i	$0.253059\pi$
$788$

Overview	Random
Universe	Knowledge

Degree 1	Degree 2
Degree 3	Degree 4
$\zeta$ zeros

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:	\( N \)	\(=\)	\( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1050.o (of order \(6\), degree \(2\), not minimal)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Properties

Related objects

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