LMFDB - Embedded newform 690.2.d.b.139.4

Newspace parameters

Level:	$ N $	$=$	$ 690 = 2 \cdot 3 \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	690.d (of order $2$, degree $1$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$5.50967773947$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
Defining polynomial:	$x^{4} + 1$
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			139.4
Root			$0.707107 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	690.139
Dual form			690.2.d.b.139.2

$q$-expansion

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +(0.707107 + 2.12132i) q^{5} +1.00000 q^{6} -2.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +(0.707107 + 2.12132i) q^{5} +1.00000 q^{6} -2.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +(-2.12132 + 0.707107i) q^{10} +4.24264 q^{11} +1.00000i q^{12} -4.82843i q^{13} +2.00000 q^{14} +(2.12132 - 0.707107i) q^{15} +1.00000 q^{16} +1.17157i q^{17} -1.00000i q^{18} +2.24264 q^{19} +(-0.707107 - 2.12132i) q^{20} -2.00000 q^{21} +4.24264i q^{22} +1.00000i q^{23} -1.00000 q^{24} +(-4.00000 + 3.00000i) q^{25} +4.82843 q^{26} +1.00000i q^{27} +2.00000i q^{28} +7.65685 q^{29} +(0.707107 + 2.12132i) q^{30} +6.00000 q^{31} +1.00000i q^{32} -4.24264i q^{33} -1.17157 q^{34} +(4.24264 - 1.41421i) q^{35} +1.00000 q^{36} +3.41421i q^{37} +2.24264i q^{38} -4.82843 q^{39} +(2.12132 - 0.707107i) q^{40} -1.17157 q^{41} -2.00000i q^{42} -1.75736i q^{43} -4.24264 q^{44} +(-0.707107 - 2.12132i) q^{45} -1.00000 q^{46} +4.82843i q^{47} -1.00000i q^{48} +3.00000 q^{49} +(-3.00000 - 4.00000i) q^{50} +1.17157 q^{51} +4.82843i q^{52} -13.4142i q^{53} -1.00000 q^{54} +(3.00000 + 9.00000i) q^{55} -2.00000 q^{56} -2.24264i q^{57} +7.65685i q^{58} +8.48528 q^{59} +(-2.12132 + 0.707107i) q^{60} -3.41421 q^{61} +6.00000i q^{62} +2.00000i q^{63} -1.00000 q^{64} +(10.2426 - 3.41421i) q^{65} +4.24264 q^{66} -0.585786i q^{67} -1.17157i q^{68} +1.00000 q^{69} +(1.41421 + 4.24264i) q^{70} +5.65685 q^{71} +1.00000i q^{72} +3.65685i q^{73} -3.41421 q^{74} +(3.00000 + 4.00000i) q^{75} -2.24264 q^{76} -8.48528i q^{77} -4.82843i q^{78} -7.65685 q^{79} +(0.707107 + 2.12132i) q^{80} +1.00000 q^{81} -1.17157i q^{82} +1.41421i q^{83} +2.00000 q^{84} +(-2.48528 + 0.828427i) q^{85} +1.75736 q^{86} -7.65685i q^{87} -4.24264i q^{88} -14.8284 q^{89} +(2.12132 - 0.707107i) q^{90} -9.65685 q^{91} -1.00000i q^{92} -6.00000i q^{93} -4.82843 q^{94} +(1.58579 + 4.75736i) q^{95} +1.00000 q^{96} -6.00000i q^{97} +3.00000i q^{98} -4.24264 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4q - 4q^{4} + 4q^{6} - 4q^{9} + O(q^{10}) $	$ 4q - 4q^{4} + 4q^{6} - 4q^{9} + 8q^{14} + 4q^{16} - 8q^{19} - 8q^{21} - 4q^{24} - 16q^{25} + 8q^{26} + 8q^{29} + 24q^{31} - 16q^{34} + 4q^{36} - 8q^{39} - 16q^{41} - 4q^{46} + 12q^{49} - 12q^{50} + 16q^{51} - 4q^{54} + 12q^{55} - 8q^{56} - 8q^{61} - 4q^{64} + 24q^{65} + 4q^{69} - 8q^{74} + 12q^{75} + 8q^{76} - 8q^{79} + 4q^{81} + 8q^{84} + 24q^{85} + 24q^{86} - 48q^{89} - 16q^{91} - 8q^{94} + 12q^{95} + 4q^{96} + O(q^{100}) $

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/690\mathbb{Z}\right)^\times$.

$n$	$277$	$461$	$511$
$\chi(n)$	$-1$	$1$	$1$

Coefficient data

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

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

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

$2$			1.00000i			0.707107i
$2$			1.00000i			0.707107i
$3$		−	1.00000i		−	0.577350i
$3$		−	1.00000i		−	0.577350i
$4$	−1.00000			−0.500000
$5$	0.707107	+	2.12132i	0.316228	+	0.948683i
$5$	0.707107	+	2.12132i	0.316228	+	0.948683i
$6$	1.00000			0.408248
$7$		−	2.00000i		−	0.755929i	−0.925820	−	0.377964i	$-0.876624\pi$
$7$		−	2.00000i		−	0.755929i	0.925820	−	0.377964i	$-0.123376\pi$
$8$		−	1.00000i		−	0.353553i
$9$	−1.00000			−0.333333
$10$	−2.12132	+	0.707107i	−0.670820	+	0.223607i
$11$	4.24264			1.27920			0.639602	−	0.768706i	$-0.279099\pi$
$11$	4.24264			1.27920			0.639602	+	0.768706i	$0.279099\pi$
$12$			1.00000i			0.288675i
$13$		−	4.82843i		−	1.33916i	−0.742738	−	0.669582i	$-0.766473\pi$
$13$		−	4.82843i		−	1.33916i	0.742738	−	0.669582i	$-0.233527\pi$
$14$	2.00000			0.534522
$15$	2.12132	−	0.707107i	0.547723	−	0.182574i
$16$	1.00000			0.250000
$17$			1.17157i			0.284148i	0.989856	+	0.142074i	$0.0453771\pi$
$17$			1.17157i			0.284148i	−0.989856	+	0.142074i	$0.954623\pi$
$18$		−	1.00000i		−	0.235702i
$19$	2.24264			0.514497			0.257249	−	0.966345i	$-0.417184\pi$
$19$	2.24264			0.514497			0.257249	+	0.966345i	$0.417184\pi$
$20$	−0.707107	−	2.12132i	−0.158114	−	0.474342i
$21$	−2.00000			−0.436436
$22$			4.24264i			0.904534i
$23$			1.00000i			0.208514i
$23$			1.00000i			0.208514i
$24$	−1.00000			−0.204124
$25$	−4.00000	+	3.00000i	−0.800000	+	0.600000i
$26$	4.82843			0.946932
$27$			1.00000i			0.192450i
$28$			2.00000i			0.377964i
$29$	7.65685			1.42184			0.710921	−	0.703272i	$-0.248278\pi$
$29$	7.65685			1.42184			0.710921	+	0.703272i	$0.248278\pi$
$30$	0.707107	+	2.12132i	0.129099	+	0.387298i
$31$	6.00000			1.07763			0.538816	−	0.842424i	$-0.318872\pi$
$31$	6.00000			1.07763			0.538816	+	0.842424i	$0.318872\pi$
$32$			1.00000i			0.176777i
$33$		−	4.24264i		−	0.738549i
$34$	−1.17157			−0.200923
$35$	4.24264	−	1.41421i	0.717137	−	0.239046i
$36$	1.00000			0.166667
$37$			3.41421i			0.561293i	0.959811	+	0.280647i	$0.0905489\pi$
$37$			3.41421i			0.561293i	−0.959811	+	0.280647i	$0.909451\pi$
$38$			2.24264i			0.363804i
$39$	−4.82843			−0.773167
$40$	2.12132	−	0.707107i	0.335410	−	0.111803i
$41$	−1.17157			−0.182969			−0.0914845	−	0.995807i	$-0.529161\pi$
$41$	−1.17157			−0.182969			−0.0914845	+	0.995807i	$0.529161\pi$
$42$		−	2.00000i		−	0.308607i
$43$		−	1.75736i		−	0.267995i	−0.990982	−	0.133997i	$-0.957219\pi$
$43$		−	1.75736i		−	0.267995i	0.990982	−	0.133997i	$-0.0427814\pi$
$44$	−4.24264			−0.639602
$45$	−0.707107	−	2.12132i	−0.105409	−	0.316228i
$46$	−1.00000			−0.147442
$47$			4.82843i			0.704298i	0.935944	+	0.352149i	$0.114549\pi$
$47$			4.82843i			0.704298i	−0.935944	+	0.352149i	$0.885451\pi$
$48$		−	1.00000i		−	0.144338i
$49$	3.00000			0.428571
$50$	−3.00000	−	4.00000i	−0.424264	−	0.565685i
$51$	1.17157			0.164053
$52$			4.82843i			0.669582i
$53$		−	13.4142i		−	1.84258i	−0.388872	−	0.921292i	$-0.627135\pi$
$53$		−	13.4142i		−	1.84258i	0.388872	−	0.921292i	$-0.372865\pi$
$54$	−1.00000			−0.136083
$55$	3.00000	+	9.00000i	0.404520	+	1.21356i
$56$	−2.00000			−0.267261
$57$		−	2.24264i		−	0.297045i
$58$			7.65685i			1.00539i
$59$	8.48528			1.10469			0.552345	−	0.833616i	$-0.313733\pi$
$59$	8.48528			1.10469			0.552345	+	0.833616i	$0.313733\pi$
$60$	−2.12132	+	0.707107i	−0.273861	+	0.0912871i
$61$	−3.41421			−0.437145			−0.218573	−	0.975821i	$-0.570140\pi$
$61$	−3.41421			−0.437145			−0.218573	+	0.975821i	$0.570140\pi$
$62$			6.00000i			0.762001i
$63$			2.00000i			0.251976i
$64$	−1.00000			−0.125000
$65$	10.2426	−	3.41421i	1.27044	−	0.423481i
$66$	4.24264			0.522233
$67$		−	0.585786i		−	0.0715652i	−0.999360	−	0.0357826i	$-0.988608\pi$
$67$		−	0.585786i		−	0.0715652i	0.999360	−	0.0357826i	$-0.0113924\pi$
$68$		−	1.17157i		−	0.142074i
$69$	1.00000			0.120386
$70$	1.41421	+	4.24264i	0.169031	+	0.507093i
$71$	5.65685			0.671345			0.335673	−	0.941979i	$-0.391036\pi$
$71$	5.65685			0.671345			0.335673	+	0.941979i	$0.391036\pi$
$72$			1.00000i			0.117851i
$73$			3.65685i			0.428002i	0.976833	+	0.214001i	$0.0686496\pi$
$73$			3.65685i			0.428002i	−0.976833	+	0.214001i	$0.931350\pi$
$74$	−3.41421			−0.396894
$75$	3.00000	+	4.00000i	0.346410	+	0.461880i
$76$	−2.24264			−0.257249
$77$		−	8.48528i		−	0.966988i
$78$		−	4.82843i		−	0.546712i
$79$	−7.65685			−0.861463			−0.430732	−	0.902480i	$-0.641744\pi$
$79$	−7.65685			−0.861463			−0.430732	+	0.902480i	$0.641744\pi$
$80$	0.707107	+	2.12132i	0.0790569	+	0.237171i
$81$	1.00000			0.111111
$82$		−	1.17157i		−	0.129379i
$83$			1.41421i			0.155230i	0.996983	+	0.0776151i	$0.0247305\pi$
$83$			1.41421i			0.155230i	−0.996983	+	0.0776151i	$0.975269\pi$
$84$	2.00000			0.218218
$85$	−2.48528	+	0.828427i	−0.269567	+	0.0898555i
$86$	1.75736			0.189501
$87$		−	7.65685i		−	0.820901i
$88$		−	4.24264i		−	0.452267i
$89$	−14.8284			−1.57181			−0.785905	−	0.618347i	$-0.787803\pi$
$89$	−14.8284			−1.57181			−0.785905	+	0.618347i	$0.787803\pi$
$90$	2.12132	−	0.707107i	0.223607	−	0.0745356i
$91$	−9.65685			−1.01231
$92$		−	1.00000i		−	0.104257i
$93$		−	6.00000i		−	0.622171i
$94$	−4.82843			−0.498014
$95$	1.58579	+	4.75736i	0.162698	+	0.488095i
$96$	1.00000			0.102062
$97$		−	6.00000i		−	0.609208i	−0.952479	−	0.304604i	$-0.901476\pi$
$97$		−	6.00000i		−	0.609208i	0.952479	−	0.304604i	$-0.0985241\pi$
$98$			3.00000i			0.303046i
$99$	−4.24264			−0.426401
$100$	4.00000	−	3.00000i	0.400000	−	0.300000i
$101$	−9.31371			−0.926749			−0.463374	−	0.886163i	$-0.653361\pi$
$101$	−9.31371			−0.926749			−0.463374	+	0.886163i	$0.653361\pi$
$102$			1.17157i			0.116003i
$103$			12.1421i			1.19640i	0.801347	+	0.598200i	$0.204117\pi$
$103$			12.1421i			1.19640i	−0.801347	+	0.598200i	$0.795883\pi$
$104$	−4.82843			−0.473466
$105$	−1.41421	−	4.24264i	−0.138013	−	0.414039i
$106$	13.4142			1.30290
$107$			15.0711i			1.45698i	0.685059	+	0.728488i	$0.259776\pi$
$107$			15.0711i			1.45698i	−0.685059	+	0.728488i	$0.740224\pi$
$108$		−	1.00000i		−	0.0962250i
$109$	−15.4142			−1.47641			−0.738207	−	0.674574i	$-0.764327\pi$
$109$	−15.4142			−1.47641			−0.738207	+	0.674574i	$0.764327\pi$
$110$	−9.00000	+	3.00000i	−0.858116	+	0.286039i
$111$	3.41421			0.324063
$112$		−	2.00000i		−	0.188982i
$113$		−	16.9706i		−	1.59646i	−0.602355	−	0.798228i	$-0.705771\pi$
$113$		−	16.9706i		−	1.59646i	0.602355	−	0.798228i	$-0.294229\pi$
$114$	2.24264			0.210043
$115$	−2.12132	+	0.707107i	−0.197814	+	0.0659380i
$116$	−7.65685			−0.710921
$117$			4.82843i			0.446388i
$118$			8.48528i			0.781133i
$119$	2.34315			0.214796
$120$	−0.707107	−	2.12132i	−0.0645497	−	0.193649i
$121$	7.00000			0.636364
$122$		−	3.41421i		−	0.309108i
$123$			1.17157i			0.105637i
$124$	−6.00000			−0.538816
$125$	−9.19239	−	6.36396i	−0.822192	−	0.569210i
$126$	−2.00000			−0.178174
$127$		−	2.00000i		−	0.177471i	−0.996055	−	0.0887357i	$-0.971717\pi$
$127$		−	2.00000i		−	0.177471i	0.996055	−	0.0887357i	$-0.0282826\pi$
$128$		−	1.00000i		−	0.0883883i
$129$	−1.75736			−0.154727
$130$	3.41421	+	10.2426i	0.299446	+	0.898339i
$131$	−10.3431			−0.903685			−0.451842	−	0.892098i	$-0.649233\pi$
$131$	−10.3431			−0.903685			−0.451842	+	0.892098i	$0.649233\pi$
$132$			4.24264i			0.369274i
$133$		−	4.48528i		−	0.388923i
$134$	0.585786			0.0506042
$135$	−2.12132	+	0.707107i	−0.182574	+	0.0608581i
$136$	1.17157			0.100462
$137$			8.48528i			0.724947i	0.931994	+	0.362473i	$0.118068\pi$
$137$			8.48528i			0.724947i	−0.931994	+	0.362473i	$0.881932\pi$
$138$			1.00000i			0.0851257i
$139$	−16.9706			−1.43942			−0.719712	−	0.694273i	$-0.755726\pi$
$139$	−16.9706			−1.43942			−0.719712	+	0.694273i	$0.755726\pi$
$140$	−4.24264	+	1.41421i	−0.358569	+	0.119523i
$141$	4.82843			0.406627
$142$			5.65685i			0.474713i
$143$		−	20.4853i		−	1.71307i
$144$	−1.00000			−0.0833333
$145$	5.41421	+	16.2426i	0.449626	+	1.34888i
$146$	−3.65685			−0.302643
$147$		−	3.00000i		−	0.247436i
$148$		−	3.41421i		−	0.280647i
$149$	12.7279			1.04271			0.521356	−	0.853339i	$-0.325426\pi$
$149$	12.7279			1.04271			0.521356	+	0.853339i	$0.325426\pi$
$150$	−4.00000	+	3.00000i	−0.326599	+	0.244949i
$151$	−15.6569			−1.27414			−0.637068	−	0.770807i	$-0.719853\pi$
$151$	−15.6569			−1.27414			−0.637068	+	0.770807i	$0.719853\pi$
$152$		−	2.24264i		−	0.181902i
$153$		−	1.17157i		−	0.0947161i
$154$	8.48528			0.683763
$155$	4.24264	+	12.7279i	0.340777	+	1.02233i
$156$	4.82843			0.386584
$157$			6.92893i			0.552989i	0.961016	+	0.276494i	$0.0891728\pi$
$157$			6.92893i			0.552989i	−0.961016	+	0.276494i	$0.910827\pi$
$158$		−	7.65685i		−	0.609147i
$159$	−13.4142			−1.06382
$160$	−2.12132	+	0.707107i	−0.167705	+	0.0559017i
$161$	2.00000			0.157622
$162$			1.00000i			0.0785674i
$163$		−	1.17157i		−	0.0917647i	−0.998947	−	0.0458823i	$-0.985390\pi$
$163$		−	1.17157i		−	0.0917647i	0.998947	−	0.0458823i	$-0.0146099\pi$
$164$	1.17157			0.0914845
$165$	9.00000	−	3.00000i	0.700649	−	0.233550i
$166$	−1.41421			−0.109764
$167$		−	8.82843i		−	0.683164i	−0.939852	−	0.341582i	$-0.889037\pi$
$167$		−	8.82843i		−	0.683164i	0.939852	−	0.341582i	$-0.110963\pi$
$168$			2.00000i			0.154303i
$169$	−10.3137			−0.793362
$170$	−0.828427	−	2.48528i	−0.0635375	−	0.190612i
$171$	−2.24264			−0.171499
$172$			1.75736i			0.133997i
$173$		−	3.17157i		−	0.241130i	−0.992705	−	0.120565i	$-0.961529\pi$
$173$		−	3.17157i		−	0.241130i	0.992705	−	0.120565i	$-0.0384707\pi$
$174$	7.65685			0.580465
$175$	6.00000	+	8.00000i	0.453557	+	0.604743i
$176$	4.24264			0.319801
$177$		−	8.48528i		−	0.637793i
$178$		−	14.8284i		−	1.11144i
$179$	10.3431			0.773083			0.386542	−	0.922272i	$-0.373670\pi$
$179$	10.3431			0.773083			0.386542	+	0.922272i	$0.373670\pi$
$180$	0.707107	+	2.12132i	0.0527046	+	0.158114i
$181$	16.3848			1.21787			0.608935	−	0.793220i	$-0.291597\pi$
$181$	16.3848			1.21787			0.608935	+	0.793220i	$0.291597\pi$
$182$		−	9.65685i		−	0.715814i
$183$			3.41421i			0.252386i
$184$	1.00000			0.0737210
$185$	−7.24264	+	2.41421i	−0.532490	+	0.177497i
$186$	6.00000			0.439941
$187$			4.97056i			0.363484i
$188$		−	4.82843i		−	0.352149i
$189$	2.00000			0.145479
$190$	−4.75736	+	1.58579i	−0.345135	+	0.115045i
$191$	−5.17157			−0.374202			−0.187101	−	0.982341i	$-0.559909\pi$
$191$	−5.17157			−0.374202			−0.187101	+	0.982341i	$0.559909\pi$
$192$			1.00000i			0.0721688i
$193$		−	12.9706i		−	0.933642i	−0.884352	−	0.466821i	$-0.845399\pi$
$193$		−	12.9706i		−	0.933642i	0.884352	−	0.466821i	$-0.154601\pi$
$194$	6.00000			0.430775
$195$	−3.41421	−	10.2426i	−0.244497	−	0.733491i
$196$	−3.00000			−0.214286
$197$		−	24.6274i		−	1.75463i	−0.479914	−	0.877315i	$-0.659332\pi$
$197$		−	24.6274i		−	1.75463i	0.479914	−	0.877315i	$-0.340668\pi$
$198$		−	4.24264i		−	0.301511i
$199$	14.0000			0.992434			0.496217	−	0.868199i	$-0.334722\pi$
$199$	14.0000			0.992434			0.496217	+	0.868199i	$0.334722\pi$
$200$	3.00000	+	4.00000i	0.212132	+	0.282843i
$201$	−0.585786			−0.0413182
$202$		−	9.31371i		−	0.655310i
$203$		−	15.3137i		−	1.07481i
$204$	−1.17157			−0.0820265
$205$	−0.828427	−	2.48528i	−0.0578599	−	0.173580i
$206$	−12.1421			−0.845983
$207$		−	1.00000i		−	0.0695048i
$208$		−	4.82843i		−	0.334791i
$209$	9.51472			0.658147
$210$	4.24264	−	1.41421i	0.292770	−	0.0975900i
$211$	−20.4853			−1.41026			−0.705132	−	0.709076i	$-0.749112\pi$
$211$	−20.4853			−1.41026			−0.705132	+	0.709076i	$0.749112\pi$
$212$			13.4142i			0.921292i
$213$		−	5.65685i		−	0.387601i
$214$	−15.0711			−1.03024
$215$	3.72792	−	1.24264i	0.254242	−	0.0847474i
$216$	1.00000			0.0680414
$217$		−	12.0000i		−	0.814613i
$218$		−	15.4142i		−	1.04398i
$219$	3.65685			0.247107
$220$	−3.00000	−	9.00000i	−0.202260	−	0.606780i
$221$	5.65685			0.380521
$222$			3.41421i			0.229147i
$223$		−	16.0000i		−	1.07144i	−0.844396	−	0.535720i	$-0.820040\pi$
$223$		−	16.0000i		−	1.07144i	0.844396	−	0.535720i	$-0.179960\pi$
$224$	2.00000			0.133631
$225$	4.00000	−	3.00000i	0.266667	−	0.200000i
$226$	16.9706			1.12887
$227$			14.3848i			0.954751i	0.878699	+	0.477376i	$0.158412\pi$
$227$			14.3848i			0.954751i	−0.878699	+	0.477376i	$0.841588\pi$
$228$			2.24264i			0.148523i
$229$	−1.75736			−0.116130			−0.0580648	−	0.998313i	$-0.518493\pi$
$229$	−1.75736			−0.116130			−0.0580648	+	0.998313i	$0.518493\pi$
$230$	−0.707107	−	2.12132i	−0.0466252	−	0.139876i
$231$	−8.48528			−0.558291
$232$		−	7.65685i		−	0.502697i
$233$			26.1421i			1.71263i	0.516455	+	0.856314i	$0.327251\pi$
$233$			26.1421i			1.71263i	−0.516455	+	0.856314i	$0.672749\pi$
$234$	−4.82843			−0.315644
$235$	−10.2426	+	3.41421i	−0.668156	+	0.222719i
$236$	−8.48528			−0.552345
$237$			7.65685i			0.497366i
$238$			2.34315i			0.151884i
$239$	11.1716			0.722629			0.361314	−	0.932444i	$-0.382328\pi$
$239$	11.1716			0.722629			0.361314	+	0.932444i	$0.382328\pi$
$240$	2.12132	−	0.707107i	0.136931	−	0.0456435i
$241$	−10.4853			−0.675416			−0.337708	−	0.941251i	$-0.609652\pi$
$241$	−10.4853			−0.675416			−0.337708	+	0.941251i	$0.609652\pi$
$242$			7.00000i			0.449977i
$243$		−	1.00000i		−	0.0641500i
$244$	3.41421			0.218573
$245$	2.12132	+	6.36396i	0.135526	+	0.406579i
$246$	−1.17157			−0.0746968
$247$		−	10.8284i		−	0.688996i
$248$		−	6.00000i		−	0.381000i
$249$	1.41421			0.0896221
$250$	6.36396	−	9.19239i	0.402492	−	0.581378i
$251$	−24.7279			−1.56081			−0.780406	−	0.625273i	$-0.784988\pi$
$251$	−24.7279			−1.56081			−0.780406	+	0.625273i	$0.784988\pi$
$252$		−	2.00000i		−	0.125988i
$253$			4.24264i			0.266733i
$254$	2.00000			0.125491
$255$	0.828427	+	2.48528i	0.0518781	+	0.155634i
$256$	1.00000			0.0625000
$257$			18.0000i			1.12281i	0.827541	+	0.561405i	$0.189739\pi$
$257$			18.0000i			1.12281i	−0.827541	+	0.561405i	$0.810261\pi$
$258$		−	1.75736i		−	0.109408i
$259$	6.82843			0.424298
$260$	−10.2426	+	3.41421i	−0.635222	+	0.211741i
$261$	−7.65685			−0.473947
$262$		−	10.3431i		−	0.639002i
$263$			17.6569i			1.08877i	0.838836	+	0.544384i	$0.183237\pi$
$263$			17.6569i			1.08877i	−0.838836	+	0.544384i	$0.816763\pi$
$264$	−4.24264			−0.261116
$265$	28.4558	−	9.48528i	1.74803	−	0.582676i
$266$	4.48528			0.275010
$267$			14.8284i			0.907485i
$268$			0.585786i			0.0357826i
$269$	24.8284			1.51382			0.756908	−	0.653521i	$-0.226709\pi$
$269$	24.8284			1.51382			0.756908	+	0.653521i	$0.226709\pi$
$270$	−0.707107	−	2.12132i	−0.0430331	−	0.129099i
$271$	−1.65685			−0.100647			−0.0503234	−	0.998733i	$-0.516025\pi$
$271$	−1.65685			−0.100647			−0.0503234	+	0.998733i	$0.516025\pi$
$272$			1.17157i			0.0710370i
$273$			9.65685i			0.584459i
$274$	−8.48528			−0.512615
$275$	−16.9706	+	12.7279i	−1.02336	+	0.767523i
$276$	−1.00000			−0.0601929
$277$			30.4853i			1.83168i	0.401540	+	0.915842i	$0.368475\pi$
$277$			30.4853i			1.83168i	−0.401540	+	0.915842i	$0.631525\pi$
$278$		−	16.9706i		−	1.01783i
$279$	−6.00000			−0.359211
$280$	−1.41421	−	4.24264i	−0.0845154	−	0.253546i
$281$	−30.1421			−1.79813			−0.899065	−	0.437816i	$-0.855752\pi$
$281$	−30.1421			−1.79813			−0.899065	+	0.437816i	$0.855752\pi$
$282$			4.82843i			0.287529i
$283$			27.2132i			1.61766i	0.588045	+	0.808829i	$0.299898\pi$
$283$			27.2132i			1.61766i	−0.588045	+	0.808829i	$0.700102\pi$
$284$	−5.65685			−0.335673
$285$	4.75736	−	1.58579i	0.281802	−	0.0939339i
$286$	20.4853			1.21132
$287$			2.34315i			0.138312i
$288$		−	1.00000i		−	0.0589256i
$289$	15.6274			0.919260
$290$	−16.2426	+	5.41421i	−0.953801	+	0.317934i
$291$	−6.00000			−0.351726
$292$		−	3.65685i		−	0.214001i
$293$			28.0416i			1.63821i	0.573644	+	0.819105i	$0.305529\pi$
$293$			28.0416i			1.63821i	−0.573644	+	0.819105i	$0.694471\pi$
$294$	3.00000			0.174964
$295$	6.00000	+	18.0000i	0.349334	+	1.04800i
$296$	3.41421			0.198447
$297$			4.24264i			0.246183i
$298$			12.7279i			0.737309i
$299$	4.82843			0.279235
$300$	−3.00000	−	4.00000i	−0.173205	−	0.230940i
$301$	−3.51472			−0.202585
$302$		−	15.6569i		−	0.900951i
$303$			9.31371i			0.535059i
$304$	2.24264			0.128624
$305$	−2.41421	−	7.24264i	−0.138237	−	0.414712i
$306$	1.17157			0.0669744
$307$		−	25.4558i		−	1.45284i	−0.687250	−	0.726421i	$-0.741182\pi$
$307$		−	25.4558i		−	1.45284i	0.687250	−	0.726421i	$-0.258818\pi$
$308$			8.48528i			0.483494i
$309$	12.1421			0.690742
$310$	−12.7279	+	4.24264i	−0.722897	+	0.240966i
$311$	17.7990			1.00929			0.504644	−	0.863328i	$-0.331624\pi$
$311$	17.7990			1.00929			0.504644	+	0.863328i	$0.331624\pi$
$312$			4.82843i			0.273356i
$313$		−	18.0000i		−	1.01742i	−0.860938	−	0.508710i	$-0.830123\pi$
$313$		−	18.0000i		−	1.01742i	0.860938	−	0.508710i	$-0.169877\pi$
$314$	−6.92893			−0.391022
$315$	−4.24264	+	1.41421i	−0.239046	+	0.0796819i
$316$	7.65685			0.430732
$317$			2.48528i			0.139587i	0.997561	+	0.0697937i	$0.0222341\pi$
$317$			2.48528i			0.139587i	−0.997561	+	0.0697937i	$0.977766\pi$
$318$		−	13.4142i		−	0.752232i
$319$	32.4853			1.81883
$320$	−0.707107	−	2.12132i	−0.0395285	−	0.118585i
$321$	15.0711			0.841185
$322$			2.00000i			0.111456i
$323$			2.62742i			0.146193i
$324$	−1.00000			−0.0555556
$325$	14.4853	+	19.3137i	0.803499	+	1.07133i
$326$	1.17157			0.0648874
$327$			15.4142i			0.852408i
$328$			1.17157i			0.0646893i
$329$	9.65685			0.532400
$330$	3.00000	+	9.00000i	0.165145	+	0.495434i
$331$	−3.51472			−0.193186			−0.0965932	−	0.995324i	$-0.530795\pi$
$331$	−3.51472			−0.193186			−0.0965932	+	0.995324i	$0.530795\pi$
$332$		−	1.41421i		−	0.0776151i
$333$		−	3.41421i		−	0.187098i
$334$	8.82843			0.483070
$335$	1.24264	−	0.414214i	0.0678927	−	0.0226309i
$336$	−2.00000			−0.109109
$337$		−	9.51472i		−	0.518300i	−0.965837	−	0.259150i	$-0.916558\pi$
$337$		−	9.51472i		−	0.518300i	0.965837	−	0.259150i	$-0.0834424\pi$
$338$		−	10.3137i		−	0.560992i
$339$	−16.9706			−0.921714
$340$	2.48528	−	0.828427i	0.134783	−	0.0449278i
$341$	25.4558			1.37851
$342$		−	2.24264i		−	0.121268i
$343$		−	20.0000i		−	1.07990i
$344$	−1.75736			−0.0947505
$345$	0.707107	+	2.12132i	0.0380693	+	0.114208i
$346$	3.17157			0.170505
$347$			7.79899i			0.418672i	0.977844	+	0.209336i	$0.0671302\pi$
$347$			7.79899i			0.418672i	−0.977844	+	0.209336i	$0.932870\pi$
$348$			7.65685i			0.410450i
$349$	−25.3137			−1.35501			−0.677506	−	0.735517i	$-0.736939\pi$
$349$	−25.3137			−1.35501			−0.677506	+	0.735517i	$0.736939\pi$
$350$	−8.00000	+	6.00000i	−0.427618	+	0.320713i
$351$	4.82843			0.257722
$352$			4.24264i			0.226134i
$353$			26.1421i			1.39141i	0.718330	+	0.695703i	$0.244907\pi$
$353$			26.1421i			1.39141i	−0.718330	+	0.695703i	$0.755093\pi$
$354$	8.48528			0.450988
$355$	4.00000	+	12.0000i	0.212298	+	0.636894i
$356$	14.8284			0.785905
$357$		−	2.34315i		−	0.124012i
$358$			10.3431i			0.546652i
$359$	19.7990			1.04495			0.522475	−	0.852654i	$-0.325009\pi$
$359$	19.7990			1.04495			0.522475	+	0.852654i	$0.325009\pi$
$360$	−2.12132	+	0.707107i	−0.111803	+	0.0372678i
$361$	−13.9706			−0.735293
$362$			16.3848i			0.861165i
$363$		−	7.00000i		−	0.367405i
$364$	9.65685			0.506157
$365$	−7.75736	+	2.58579i	−0.406039	+	0.135346i
$366$	−3.41421			−0.178464
$367$			34.9706i			1.82545i	0.408576	+	0.912724i	$0.366025\pi$
$367$			34.9706i			1.82545i	−0.408576	+	0.912724i	$0.633975\pi$
$368$			1.00000i			0.0521286i
$369$	1.17157			0.0609896
$370$	−2.41421	−	7.24264i	−0.125509	−	0.376527i
$371$	−26.8284			−1.39286
$372$			6.00000i			0.311086i
$373$		−	19.2132i		−	0.994822i	−0.867515	−	0.497411i	$-0.834284\pi$
$373$		−	19.2132i		−	0.994822i	0.867515	−	0.497411i	$-0.165716\pi$
$374$	−4.97056			−0.257022
$375$	−6.36396	+	9.19239i	−0.328634	+	0.474693i
$376$	4.82843			0.249007
$377$		−	36.9706i		−	1.90408i
$378$			2.00000i			0.102869i
$379$	19.2132			0.986916			0.493458	−	0.869770i	$-0.335733\pi$
$379$	19.2132			0.986916			0.493458	+	0.869770i	$0.335733\pi$
$380$	−1.58579	−	4.75736i	−0.0813491	−	0.244047i
$381$	−2.00000			−0.102463
$382$		−	5.17157i		−	0.264601i
$383$			18.6274i			0.951817i	0.879495	+	0.475908i	$0.157881\pi$
$383$			18.6274i			0.951817i	−0.879495	+	0.475908i	$0.842119\pi$
$384$	−1.00000			−0.0510310
$385$	18.0000	−	6.00000i	0.917365	−	0.305788i
$386$	12.9706			0.660184
$387$			1.75736i			0.0893316i
$388$			6.00000i			0.304604i
$389$	−6.38478			−0.323721			−0.161861	−	0.986814i	$-0.551749\pi$
$389$	−6.38478			−0.323721			−0.161861	+	0.986814i	$0.551749\pi$
$390$	10.2426	−	3.41421i	0.518656	−	0.172885i
$391$	−1.17157			−0.0592490
$392$		−	3.00000i		−	0.151523i
$393$			10.3431i			0.521743i
$394$	24.6274			1.24071
$395$	−5.41421	−	16.2426i	−0.272419	−	0.817256i
$396$	4.24264			0.213201
$397$		−	8.34315i		−	0.418730i	−0.977838	−	0.209365i	$-0.932860\pi$
$397$		−	8.34315i		−	0.418730i	0.977838	−	0.209365i	$-0.0671398\pi$
$398$			14.0000i			0.701757i
$399$	−4.48528			−0.224545
$400$	−4.00000	+	3.00000i	−0.200000	+	0.150000i
$401$	−28.9706			−1.44672			−0.723360	−	0.690471i	$-0.757403\pi$
$401$	−28.9706			−1.44672			−0.723360	+	0.690471i	$0.757403\pi$
$402$		−	0.585786i		−	0.0292164i
$403$		−	28.9706i		−	1.44313i
$404$	9.31371			0.463374
$405$	0.707107	+	2.12132i	0.0351364	+	0.105409i
$406$	15.3137			0.760007
$407$			14.4853i			0.718009i
$408$		−	1.17157i		−	0.0580015i
$409$	−18.0000			−0.890043			−0.445021	−	0.895520i	$-0.646804\pi$
$409$	−18.0000			−0.890043			−0.445021	+	0.895520i	$0.646804\pi$
$410$	2.48528	−	0.828427i	0.122739	−	0.0409131i
$411$	8.48528			0.418548
$412$		−	12.1421i		−	0.598200i
$413$		−	16.9706i		−	0.835067i
$414$	1.00000			0.0491473
$415$	−3.00000	+	1.00000i	−0.147264	+	0.0490881i
$416$	4.82843			0.236733
$417$			16.9706i			0.831052i
$418$			9.51472i			0.465380i
$419$	24.9289			1.21786			0.608929	−	0.793225i	$-0.291599\pi$
$419$	24.9289			1.21786			0.608929	+	0.793225i	$0.291599\pi$
$420$	1.41421	+	4.24264i	0.0690066	+	0.207020i
$421$	10.2426			0.499196			0.249598	−	0.968350i	$-0.419702\pi$
$421$	10.2426			0.499196			0.249598	+	0.968350i	$0.419702\pi$
$422$		−	20.4853i		−	0.997208i
$423$		−	4.82843i		−	0.234766i
$424$	−13.4142			−0.651452
$425$	−3.51472	−	4.68629i	−0.170489	−	0.227319i
$426$	5.65685			0.274075
$427$			6.82843i			0.330451i
$428$		−	15.0711i		−	0.728488i
$429$	−20.4853			−0.989039
$430$	1.24264	+	3.72792i	0.0599255	+	0.179776i
$431$	8.48528			0.408722			0.204361	−	0.978896i	$-0.434488\pi$
$431$	8.48528			0.408722			0.204361	+	0.978896i	$0.434488\pi$
$432$			1.00000i			0.0481125i
$433$		−	26.9706i		−	1.29612i	−0.761588	−	0.648061i	$-0.775580\pi$
$433$		−	26.9706i		−	1.29612i	0.761588	−	0.648061i	$-0.224420\pi$
$434$	12.0000			0.576018
$435$	16.2426	−	5.41421i	0.778775	−	0.259592i
$436$	15.4142			0.738207
$437$			2.24264i			0.107280i
$438$			3.65685i			0.174731i
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$	9.00000	−	3.00000i	0.429058	−	0.143019i
$441$	−3.00000			−0.142857
$442$			5.65685i			0.269069i
$443$			23.3137i			1.10767i	0.832627	+	0.553834i	$0.186836\pi$
$443$			23.3137i			1.10767i	−0.832627	+	0.553834i	$0.813164\pi$
$444$	−3.41421			−0.162031
$445$	−10.4853	−	31.4558i	−0.497050	−	1.49115i
$446$	16.0000			0.757622
$447$		−	12.7279i		−	0.602010i
$448$			2.00000i			0.0944911i
$449$	22.8284			1.07734			0.538670	−	0.842517i	$-0.318927\pi$
$449$	22.8284			1.07734			0.538670	+	0.842517i	$0.318927\pi$
$450$	3.00000	+	4.00000i	0.141421	+	0.188562i
$451$	−4.97056			−0.234055
$452$			16.9706i			0.798228i
$453$			15.6569i			0.735623i
$454$	−14.3848			−0.675111
$455$	−6.82843	−	20.4853i	−0.320122	−	0.960365i
$456$	−2.24264			−0.105021
$457$			1.02944i			0.0481550i	0.999710	+	0.0240775i	$0.00766485\pi$
$457$			1.02944i			0.0481550i	−0.999710	+	0.0240775i	$0.992335\pi$
$458$		−	1.75736i		−	0.0821160i
$459$	−1.17157			−0.0546843
$460$	2.12132	−	0.707107i	0.0989071	−	0.0329690i
$461$	−12.8284			−0.597479			−0.298740	−	0.954335i	$-0.596566\pi$
$461$	−12.8284			−0.597479			−0.298740	+	0.954335i	$0.596566\pi$
$462$		−	8.48528i		−	0.394771i
$463$		−	2.97056i		−	0.138054i	−0.997615	−	0.0690269i	$-0.978011\pi$
$463$		−	2.97056i		−	0.138054i	0.997615	−	0.0690269i	$-0.0219894\pi$
$464$	7.65685			0.355461
$465$	12.7279	−	4.24264i	0.590243	−	0.196748i
$466$	−26.1421			−1.21101
$467$			39.3553i			1.82115i	0.413346	+	0.910574i	$0.364360\pi$
$467$			39.3553i			1.82115i	−0.413346	+	0.910574i	$0.635640\pi$
$468$		−	4.82843i		−	0.223194i
$469$	−1.17157			−0.0540982
$470$	−3.41421	−	10.2426i	−0.157486	−	0.472458i
$471$	6.92893			0.319268
$472$		−	8.48528i		−	0.390567i
$473$		−	7.45584i		−	0.342820i
$474$	−7.65685			−0.351691
$475$	−8.97056	+	6.72792i	−0.411598	+	0.308698i
$476$	−2.34315			−0.107398
$477$			13.4142i			0.614195i
$478$			11.1716i			0.510976i
$479$	−4.68629			−0.214122			−0.107061	−	0.994252i	$-0.534144\pi$
$479$	−4.68629			−0.214122			−0.107061	+	0.994252i	$0.534144\pi$
$480$	0.707107	+	2.12132i	0.0322749	+	0.0968246i
$481$	16.4853			0.751664
$482$		−	10.4853i		−	0.477591i
$483$		−	2.00000i		−	0.0910032i
$484$	−7.00000			−0.318182
$485$	12.7279	−	4.24264i	0.577945	−	0.192648i
$486$	1.00000			0.0453609
$487$		−	27.9411i		−	1.26613i	−0.774097	−	0.633067i	$-0.781796\pi$
$487$		−	27.9411i		−	1.26613i	0.774097	−	0.633067i	$-0.218204\pi$
$488$			3.41421i			0.154554i
$489$	−1.17157			−0.0529804
$490$	−6.36396	+	2.12132i	−0.287494	+	0.0958315i
$491$	26.6274			1.20168			0.600839	−	0.799370i	$-0.294833\pi$
$491$	26.6274			1.20168			0.600839	+	0.799370i	$0.294833\pi$
$492$		−	1.17157i		−	0.0528186i
$493$			8.97056i			0.404014i
$494$	10.8284			0.487194
$495$	−3.00000	−	9.00000i	−0.134840	−	0.404520i
$496$	6.00000			0.269408
$497$		−	11.3137i		−	0.507489i
$498$			1.41421i			0.0633724i
$499$	−13.4558			−0.602366			−0.301183	−	0.953566i	$-0.597382\pi$
$499$	−13.4558			−0.602366			−0.301183	+	0.953566i	$0.597382\pi$
$500$	9.19239	+	6.36396i	0.411096	+	0.284605i
$501$	−8.82843			−0.394425
$502$		−	24.7279i		−	1.10366i
$503$		−	22.1421i		−	0.987269i	−0.869669	−	0.493635i	$-0.835668\pi$
$503$		−	22.1421i		−	0.987269i	0.869669	−	0.493635i	$-0.164332\pi$
$504$	2.00000			0.0890871
$505$	−6.58579	−	19.7574i	−0.293064	−	0.879191i
$506$	−4.24264			−0.188608
$507$			10.3137i			0.458048i
$508$			2.00000i			0.0887357i
$509$	−22.9706			−1.01815			−0.509076	−	0.860721i	$-0.670013\pi$
$509$	−22.9706			−1.01815			−0.509076	+	0.860721i	$0.670013\pi$
$510$	−2.48528	+	0.828427i	−0.110050	+	0.0366834i
$511$	7.31371			0.323539
$512$			1.00000i			0.0441942i
$513$			2.24264i			0.0990150i
$514$	−18.0000			−0.793946
$515$	−25.7574	+	8.58579i	−1.13500	+	0.378335i
$516$	1.75736			0.0773634
$517$			20.4853i			0.900942i
$518$			6.82843i			0.300024i
$519$	−3.17157			−0.139217
$520$	−3.41421	−	10.2426i	−0.149723	−	0.449170i
$521$	−16.2843			−0.713427			−0.356713	−	0.934214i	$-0.616103\pi$
$521$	−16.2843			−0.713427			−0.356713	+	0.934214i	$0.616103\pi$
$522$		−	7.65685i		−	0.335131i
$523$			2.92893i			0.128073i	0.997948	+	0.0640366i	$0.0203974\pi$
$523$			2.92893i			0.128073i	−0.997948	+	0.0640366i	$0.979603\pi$
$524$	10.3431			0.451842
$525$	8.00000	−	6.00000i	0.349149	−	0.261861i
$526$	−17.6569			−0.769875
$527$			7.02944i			0.306207i
$528$		−	4.24264i		−	0.184637i
$529$	−1.00000			−0.0434783
$530$	9.48528	+	28.4558i	0.412014	+	1.23604i
$531$	−8.48528			−0.368230
$532$			4.48528i			0.194462i
$533$			5.65685i			0.245026i
$534$	−14.8284			−0.641689
$535$	−31.9706	+	10.6569i	−1.38221	+	0.460736i
$536$	−0.585786			−0.0253021
$537$		−	10.3431i		−	0.446340i
$538$			24.8284i			1.07043i
$539$	12.7279			0.548230
$540$	2.12132	−	0.707107i	0.0912871	−	0.0304290i
$541$	35.9411			1.54523			0.772615	−	0.634875i	$-0.218948\pi$
$541$	35.9411			1.54523			0.772615	+	0.634875i	$0.218948\pi$
$542$		−	1.65685i		−	0.0711680i
$543$		−	16.3848i		−	0.703138i
$544$	−1.17157			−0.0502308
$545$	−10.8995	−	32.6985i	−0.466883	−	1.40065i
$546$	−9.65685			−0.413275
$547$		−	4.48528i		−	0.191777i	−0.995392	−	0.0958884i	$-0.969431\pi$
$547$		−	4.48528i		−	0.191777i	0.995392	−	0.0958884i	$-0.0305692\pi$
$548$		−	8.48528i		−	0.362473i
$549$	3.41421			0.145715
$550$	−12.7279	−	16.9706i	−0.542720	−	0.723627i
$551$	17.1716			0.731534
$552$		−	1.00000i		−	0.0425628i
$553$			15.3137i			0.651205i
$554$	−30.4853			−1.29520
$555$	2.41421	+	7.24264i	0.102478	+	0.307433i
$556$	16.9706			0.719712
$557$		−	14.8701i		−	0.630065i	−0.949081	−	0.315032i	$-0.897985\pi$
$557$		−	14.8701i		−	0.630065i	0.949081	−	0.315032i	$-0.102015\pi$
$558$		−	6.00000i		−	0.254000i
$559$	−8.48528			−0.358889
$560$	4.24264	−	1.41421i	0.179284	−	0.0597614i
$561$	4.97056			0.209857
$562$		−	30.1421i		−	1.27147i
$563$			21.8995i			0.922954i	0.887152	+	0.461477i	$0.152680\pi$
$563$			21.8995i			0.922954i	−0.887152	+	0.461477i	$0.847320\pi$
$564$	−4.82843			−0.203313
$565$	36.0000	−	12.0000i	1.51453	−	0.504844i
$566$	−27.2132			−1.14386
$567$		−	2.00000i		−	0.0839921i
$568$		−	5.65685i		−	0.237356i
$569$	−25.6569			−1.07559			−0.537796	−	0.843075i	$-0.680743\pi$
$569$	−25.6569			−1.07559			−0.537796	+	0.843075i	$0.680743\pi$
$570$	1.58579	+	4.75736i	0.0664213	+	0.199264i
$571$	−42.7279			−1.78811			−0.894054	−	0.447959i	$-0.852151\pi$
$571$	−42.7279			−1.78811			−0.894054	+	0.447959i	$0.852151\pi$
$572$			20.4853i			0.856533i
$573$			5.17157i			0.216046i
$574$	−2.34315			−0.0978010
$575$	−3.00000	−	4.00000i	−0.125109	−	0.166812i
$576$	1.00000			0.0416667
$577$		−	20.6274i		−	0.858731i	−0.903131	−	0.429365i	$-0.858737\pi$
$577$		−	20.6274i		−	0.858731i	0.903131	−	0.429365i	$-0.141263\pi$
$578$			15.6274i			0.650015i
$579$	−12.9706			−0.539038
$580$	−5.41421	−	16.2426i	−0.224813	−	0.674439i
$581$	2.82843			0.117343
$582$		−	6.00000i		−	0.248708i
$583$		−	56.9117i		−	2.35704i
$584$	3.65685			0.151322
$585$	−10.2426	+	3.41421i	−0.423481	+	0.141160i
$586$	−28.0416			−1.15839
$587$			26.8284i			1.10733i	0.832740	+	0.553664i	$0.186771\pi$
$587$			26.8284i			1.10733i	−0.832740	+	0.553664i	$0.813229\pi$
$588$			3.00000i			0.123718i
$589$	13.4558			0.554438
$590$	−18.0000	+	6.00000i	−0.741048	+	0.247016i
$591$	−24.6274			−1.01304
$592$			3.41421i			0.140323i
$593$			12.3431i			0.506872i	0.967352	+	0.253436i	$0.0815608\pi$
$593$			12.3431i			0.506872i	−0.967352	+	0.253436i	$0.918439\pi$
$594$	−4.24264			−0.174078
$595$	1.65685	+	4.97056i	0.0679244	+	0.203773i
$596$	−12.7279			−0.521356
$597$		−	14.0000i		−	0.572982i
$598$			4.82843i			0.197449i
$599$	40.1421			1.64016			0.820082	−	0.572247i	$-0.193928\pi$
$599$	40.1421			1.64016			0.820082	+	0.572247i	$0.193928\pi$
$600$	4.00000	−	3.00000i	0.163299	−	0.122474i
$601$	−20.9706			−0.855407			−0.427704	−	0.903919i	$-0.640677\pi$
$601$	−20.9706			−0.855407			−0.427704	+	0.903919i	$0.640677\pi$
$602$		−	3.51472i		−	0.143249i
$603$			0.585786i			0.0238551i
$604$	15.6569			0.637068
$605$	4.94975	+	14.8492i	0.201236	+	0.603708i
$606$	−9.31371			−0.378344
$607$		−	16.9706i		−	0.688814i	−0.938820	−	0.344407i	$-0.888080\pi$
$607$		−	16.9706i		−	0.688814i	0.938820	−	0.344407i	$-0.111920\pi$
$608$			2.24264i			0.0909511i
$609$	−15.3137			−0.620543
$610$	7.24264	−	2.41421i	0.293246	−	0.0977486i
$611$	23.3137			0.943172
$612$			1.17157i			0.0473580i
$613$			14.2426i			0.575255i	0.957742	+	0.287627i	$0.0928665\pi$
$613$			14.2426i			0.575255i	−0.957742	+	0.287627i	$0.907134\pi$
$614$	25.4558			1.02731
$615$	−2.48528	+	0.828427i	−0.100216	+	0.0334054i
$616$	−8.48528			−0.341882
$617$		−	31.3137i		−	1.26064i	−0.776334	−	0.630321i	$-0.782923\pi$
$617$		−	31.3137i		−	1.26064i	0.776334	−	0.630321i	$-0.217077\pi$
$618$			12.1421i			0.488428i
$619$	19.2132			0.772244			0.386122	−	0.922448i	$-0.373814\pi$
$619$	19.2132			0.772244			0.386122	+	0.922448i	$0.373814\pi$
$620$	−4.24264	−	12.7279i	−0.170389	−	0.511166i
$621$	−1.00000			−0.0401286
$622$			17.7990i			0.713674i
$623$			29.6569i			1.18818i
$624$	−4.82843			−0.193292
$625$	7.00000	−	24.0000i	0.280000	−	0.960000i
$626$	18.0000			0.719425
$627$		−	9.51472i		−	0.379981i
$628$		−	6.92893i		−	0.276494i
$629$	−4.00000			−0.159490
$630$	−1.41421	−	4.24264i	−0.0563436	−	0.169031i
$631$	35.1716			1.40016			0.700079	−	0.714065i	$-0.253148\pi$
$631$	35.1716			1.40016			0.700079	+	0.714065i	$0.253148\pi$
$632$			7.65685i			0.304573i
$633$			20.4853i			0.814217i
$634$	−2.48528			−0.0987031
$635$	4.24264	−	1.41421i	0.168364	−	0.0561214i
$636$	13.4142			0.531908
$637$		−	14.4853i		−	0.573928i
$638$			32.4853i			1.28610i
$639$	−5.65685			−0.223782
$640$	2.12132	−	0.707107i	0.0838525	−	0.0279508i
$641$	−8.28427			−0.327209			−0.163605	−	0.986526i	$-0.552312\pi$
$641$	−8.28427			−0.327209			−0.163605	+	0.986526i	$0.552312\pi$
$642$			15.0711i			0.594808i
$643$			32.3848i			1.27713i	0.769568	+	0.638565i	$0.220472\pi$
$643$			32.3848i			1.27713i	−0.769568	+	0.638565i	$0.779528\pi$
$644$	−2.00000			−0.0788110
$645$	−1.24264	−	3.72792i	−0.0489289	−	0.146787i
$646$	−2.62742			−0.103374
$647$		−	29.6569i		−	1.16593i	−0.812497	−	0.582966i	$-0.801892\pi$
$647$		−	29.6569i		−	1.16593i	0.812497	−	0.582966i	$-0.198108\pi$
$648$		−	1.00000i		−	0.0392837i
$649$	36.0000			1.41312
$650$	−19.3137	+	14.4853i	−0.757546	+	0.568159i
$651$	−12.0000			−0.470317
$652$			1.17157i			0.0458823i
$653$		−	43.4558i		−	1.70056i	−0.526332	−	0.850279i	$-0.676433\pi$
$653$		−	43.4558i		−	1.70056i	0.526332	−	0.850279i	$-0.323567\pi$
$654$	−15.4142			−0.602743
$655$	−7.31371	−	21.9411i	−0.285770	−	0.857311i
$656$	−1.17157			−0.0457422
$657$		−	3.65685i		−	0.142667i
$658$			9.65685i			0.376463i
$659$	26.1005			1.01673			0.508366	−	0.861141i	$-0.330250\pi$
$659$	26.1005			1.01673			0.508366	+	0.861141i	$0.330250\pi$
$660$	−9.00000	+	3.00000i	−0.350325	+	0.116775i
$661$	9.27208			0.360642			0.180321	−	0.983608i	$-0.442286\pi$
$661$	9.27208			0.360642			0.180321	+	0.983608i	$0.442286\pi$
$662$		−	3.51472i		−	0.136603i
$663$		−	5.65685i		−	0.219694i
$664$	1.41421			0.0548821
$665$	9.51472	−	3.17157i	0.368965	−	0.122988i
$666$	3.41421			0.132298
$667$			7.65685i			0.296475i
$668$			8.82843i			0.341582i
$669$	−16.0000			−0.618596
$670$	0.414214	+	1.24264i	0.0160025	+	0.0480074i
$671$	−14.4853			−0.559198
$672$		−	2.00000i		−	0.0771517i
$673$			39.9411i			1.53962i	0.638275	+	0.769809i	$0.279648\pi$
$673$			39.9411i			1.53962i	−0.638275	+	0.769809i	$0.720352\pi$
$674$	9.51472			0.366493
$675$	−3.00000	−	4.00000i	−0.115470	−	0.153960i
$676$	10.3137			0.396681
$677$			24.5269i			0.942646i	0.881961	+	0.471323i	$0.156223\pi$
$677$			24.5269i			0.942646i	−0.881961	+	0.471323i	$0.843777\pi$
$678$		−	16.9706i		−	0.651751i
$679$	−12.0000			−0.460518
$680$	0.828427	+	2.48528i	0.0317687	+	0.0953062i
$681$	14.3848			0.551226
$682$			25.4558i			0.974755i
$683$		−	8.48528i		−	0.324680i	−0.986735	−	0.162340i	$-0.948096\pi$
$683$		−	8.48528i		−	0.324680i	0.986735	−	0.162340i	$-0.0519042\pi$
$684$	2.24264			0.0857495
$685$	−18.0000	+	6.00000i	−0.687745	+	0.229248i
$686$	20.0000			0.763604
$687$			1.75736i			0.0670474i
$688$		−	1.75736i		−	0.0669987i
$689$	−64.7696			−2.46752
$690$	−2.12132	+	0.707107i	−0.0807573	+	0.0269191i
$691$	−18.8284			−0.716267			−0.358134	−	0.933670i	$-0.616587\pi$
$691$	−18.8284			−0.716267			−0.358134	+	0.933670i	$0.616587\pi$
$692$			3.17157i			0.120565i
$693$			8.48528i			0.322329i
$694$	−7.79899			−0.296046
$695$	−12.0000	−	36.0000i	−0.455186	−	1.36556i
$696$	−7.65685			−0.290232
$697$		−	1.37258i		−	0.0519903i
$698$		−	25.3137i		−	0.958138i
$699$	26.1421			0.988786
$700$	−6.00000	−	8.00000i	−0.226779	−	0.302372i
$701$	38.1838			1.44218			0.721090	−	0.692841i	$-0.243641\pi$
$701$	38.1838			1.44218			0.721090	+	0.692841i	$0.243641\pi$
$702$			4.82843i			0.182237i
$703$			7.65685i			0.288784i
$704$	−4.24264			−0.159901
$705$	3.41421	+	10.2426i	0.128587	+	0.385760i
$706$	−26.1421			−0.983872
$707$			18.6274i			0.700556i
$708$			8.48528i			0.318896i
$709$	−35.4142			−1.33001			−0.665004	−	0.746839i	$-0.731570\pi$
$709$	−35.4142			−1.33001			−0.665004	+	0.746839i	$0.731570\pi$
$710$	−12.0000	+	4.00000i	−0.450352	+	0.150117i
$711$	7.65685			0.287154
$712$			14.8284i			0.555719i
$713$			6.00000i			0.224702i
$714$	2.34315			0.0876900
$715$	43.4558	−	14.4853i	1.62516	−	0.541719i
$716$	−10.3431			−0.386542
$717$		−	11.1716i		−	0.417210i
$718$			19.7990i			0.738892i
$719$	−28.1421			−1.04952			−0.524762	−	0.851249i	$-0.675846\pi$
$719$	−28.1421			−1.04952			−0.524762	+	0.851249i	$0.675846\pi$
$720$	−0.707107	−	2.12132i	−0.0263523	−	0.0790569i
$721$	24.2843			0.904394
$722$		−	13.9706i		−	0.519931i
$723$			10.4853i			0.389952i
$724$	−16.3848			−0.608935
$725$	−30.6274	+	22.9706i	−1.13747	+	0.853105i
$726$	7.00000			0.259794
$727$		−	31.6569i		−	1.17409i	−0.809555	−	0.587044i	$-0.800292\pi$
$727$		−	31.6569i		−	1.17409i	0.809555	−	0.587044i	$-0.199708\pi$
$728$			9.65685i			0.357907i
$729$	−1.00000			−0.0370370
$730$	−2.58579	−	7.75736i	−0.0957042	−	0.287113i
$731$	2.05887			0.0761502
$732$		−	3.41421i		−	0.126193i
$733$		−	8.58579i		−	0.317123i	−0.987349	−	0.158562i	$-0.949314\pi$
$733$		−	8.58579i		−	0.317123i	0.987349	−	0.158562i	$-0.0506857\pi$
$734$	−34.9706			−1.29079
$735$	6.36396	−	2.12132i	0.234738	−	0.0782461i
$736$	−1.00000			−0.0368605
$737$		−	2.48528i		−	0.0915465i
$738$			1.17157i			0.0431262i
$739$	−42.6274			−1.56807			−0.784037	−	0.620714i	$-0.786843\pi$
$739$	−42.6274			−1.56807			−0.784037	+	0.620714i	$0.786843\pi$
$740$	7.24264	−	2.41421i	0.266245	−	0.0887483i
$741$	−10.8284			−0.397792
$742$		−	26.8284i		−	0.984903i
$743$		−	2.34315i		−	0.0859617i	−0.999076	−	0.0429808i	$-0.986315\pi$
$743$		−	2.34315i		−	0.0859617i	0.999076	−	0.0429808i	$-0.0136854\pi$
$744$	−6.00000			−0.219971
$745$	9.00000	+	27.0000i	0.329734	+	0.989203i
$746$	19.2132			0.703445
$747$		−	1.41421i		−	0.0517434i
$748$		−	4.97056i		−	0.181742i
$749$	30.1421			1.10137
$750$	−9.19239	−	6.36396i	−0.335659	−	0.232379i
$751$	−48.8284			−1.78177			−0.890887	−	0.454224i	$-0.849916\pi$
$751$	−48.8284			−1.78177			−0.890887	+	0.454224i	$0.849916\pi$
$752$			4.82843i			0.176075i
$753$			24.7279i			0.901136i
$754$	36.9706			1.34639
$755$	−11.0711	−	33.2132i	−0.402917	−	1.20875i
$756$	−2.00000			−0.0727393
$757$			42.2426i			1.53533i	0.640848	+	0.767667i	$0.278583\pi$
$757$			42.2426i			1.53533i	−0.640848	+	0.767667i	$0.721417\pi$
$758$			19.2132i			0.697855i
$759$	4.24264			0.153998
$760$	4.75736	−	1.58579i	0.172568	−	0.0575225i
$761$	−24.3431			−0.882438			−0.441219	−	0.897399i	$-0.645454\pi$
$761$	−24.3431			−0.882438			−0.441219	+	0.897399i	$0.645454\pi$
$762$		−	2.00000i		−	0.0724524i
$763$			30.8284i			1.11606i
$764$	5.17157			0.187101
$765$	2.48528	−	0.828427i	0.0898555	−	0.0299518i
$766$	−18.6274			−0.673036
$767$		−	40.9706i		−	1.47936i
$768$		−	1.00000i		−	0.0360844i
$769$	−45.7990			−1.65155			−0.825777	−	0.563997i	$-0.809263\pi$
$769$	−45.7990			−1.65155			−0.825777	+	0.563997i	$0.809263\pi$
$770$	6.00000	+	18.0000i	0.216225	+	0.648675i
$771$	18.0000			0.648254
$772$			12.9706i			0.466821i
$773$			16.9289i			0.608891i	0.952530	+	0.304446i	$0.0984712\pi$
$773$			16.9289i			0.608891i	−0.952530	+	0.304446i	$0.901529\pi$
$774$	−1.75736			−0.0631670
$775$	−24.0000	+	18.0000i	−0.862105	+	0.646579i
$776$	−6.00000			−0.215387
$777$		−	6.82843i		−	0.244968i
$778$		−	6.38478i		−	0.228905i
$779$	−2.62742			−0.0941370
$780$	3.41421	+	10.2426i	0.122248	+	0.366745i
$781$	24.0000			0.858788
$782$		−	1.17157i		−	0.0418954i
$783$			7.65685i			0.273634i
$784$	3.00000			0.107143
$785$	−14.6985	+	4.89949i	−0.524611	+	0.174870i
$786$	−10.3431			−0.368928
$787$			2.72792i			0.0972399i	0.998817	+	0.0486200i	$0.0154823\pi$
$787$			2.72792i			0.0972399i	−0.998817	+	0.0486200i	$0.984518\pi$
$788$			24.6274i			0.877315i

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:	\( N \)	\(=\)	\( 690 = 2 \cdot 3 \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	690.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +(0.707107 + 2.12132i) q^{5} +1.00000 q^{6} -2.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +(0.707107 + 2.12132i) q^{5} +1.00000 q^{6} -2.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +(-2.12132 + 0.707107i) q^{10} +4.24264 q^{11} +1.00000i q^{12} -4.82843i q^{13} +2.00000 q^{14} +(2.12132 - 0.707107i) q^{15} +1.00000 q^{16} +1.17157i q^{17} -1.00000i q^{18} +2.24264 q^{19} +(-0.707107 - 2.12132i) q^{20} -2.00000 q^{21} +4.24264i q^{22} +1.00000i q^{23} -1.00000 q^{24} +(-4.00000 + 3.00000i) q^{25} +4.82843 q^{26} +1.00000i q^{27} +2.00000i q^{28} +7.65685 q^{29} +(0.707107 + 2.12132i) q^{30} +6.00000 q^{31} +1.00000i q^{32} -4.24264i q^{33} -1.17157 q^{34} +(4.24264 - 1.41421i) q^{35} +1.00000 q^{36} +3.41421i q^{37} +2.24264i q^{38} -4.82843 q^{39} +(2.12132 - 0.707107i) q^{40} -1.17157 q^{41} -2.00000i q^{42} -1.75736i q^{43} -4.24264 q^{44} +(-0.707107 - 2.12132i) q^{45} -1.00000 q^{46} +4.82843i q^{47} -1.00000i q^{48} +3.00000 q^{49} +(-3.00000 - 4.00000i) q^{50} +1.17157 q^{51} +4.82843i q^{52} -13.4142i q^{53} -1.00000 q^{54} +(3.00000 + 9.00000i) q^{55} -2.00000 q^{56} -2.24264i q^{57} +7.65685i q^{58} +8.48528 q^{59} +(-2.12132 + 0.707107i) q^{60} -3.41421 q^{61} +6.00000i q^{62} +2.00000i q^{63} -1.00000 q^{64} +(10.2426 - 3.41421i) q^{65} +4.24264 q^{66} -0.585786i q^{67} -1.17157i q^{68} +1.00000 q^{69} +(1.41421 + 4.24264i) q^{70} +5.65685 q^{71} +1.00000i q^{72} +3.65685i q^{73} -3.41421 q^{74} +(3.00000 + 4.00000i) q^{75} -2.24264 q^{76} -8.48528i q^{77} -4.82843i q^{78} -7.65685 q^{79} +(0.707107 + 2.12132i) q^{80} +1.00000 q^{81} -1.17157i q^{82} +1.41421i q^{83} +2.00000 q^{84} +(-2.48528 + 0.828427i) q^{85} +1.75736 q^{86} -7.65685i q^{87} -4.24264i q^{88} -14.8284 q^{89} +(2.12132 - 0.707107i) q^{90} -9.65685 q^{91} -1.00000i q^{92} -6.00000i q^{93} -4.82843 q^{94} +(1.58579 + 4.75736i) q^{95} +1.00000 q^{96} -6.00000i q^{97} +3.00000i q^{98} -4.24264 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4q - 4q^{4} + 4q^{6} - 4q^{9} + O(q^{10}) \)	\( 4q - 4q^{4} + 4q^{6} - 4q^{9} + 8q^{14} + 4q^{16} - 8q^{19} - 8q^{21} - 4q^{24} - 16q^{25} + 8q^{26} + 8q^{29} + 24q^{31} - 16q^{34} + 4q^{36} - 8q^{39} - 16q^{41} - 4q^{46} + 12q^{49} - 12q^{50} + 16q^{51} - 4q^{54} + 12q^{55} - 8q^{56} - 8q^{61} - 4q^{64} + 24q^{65} + 4q^{69} - 8q^{74} + 12q^{75} + 8q^{76} - 8q^{79} + 4q^{81} + 8q^{84} + 24q^{85} + 24q^{86} - 48q^{89} - 16q^{91} - 8q^{94} + 12q^{95} + 4q^{96} + O(q^{100}) \)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Properties

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