LMFDB - Embedded newform 690.2.d.b.139.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [690,2,Mod(139,690)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(690, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("690.139");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$5.50967773947$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ 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			139.1
Root			$-0.707107 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	690.139
Dual form			690.2.d.b.139.3

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +(-0.707107 + 2.12132i) q^{5} +1.00000 q^{6} +2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +(-0.707107 + 2.12132i) q^{5} +1.00000 q^{6} +2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +(2.12132 + 0.707107i) q^{10} -4.24264 q^{11} -1.00000i q^{12} -0.828427i q^{13} +2.00000 q^{14} +(-2.12132 - 0.707107i) q^{15} +1.00000 q^{16} -6.82843i q^{17} +1.00000i q^{18} -6.24264 q^{19} +(0.707107 - 2.12132i) q^{20} -2.00000 q^{21} +4.24264i q^{22} -1.00000i q^{23} -1.00000 q^{24} +(-4.00000 - 3.00000i) q^{25} -0.828427 q^{26} -1.00000i q^{27} -2.00000i q^{28} -3.65685 q^{29} +(-0.707107 + 2.12132i) q^{30} +6.00000 q^{31} -1.00000i q^{32} -4.24264i q^{33} -6.82843 q^{34} +(-4.24264 - 1.41421i) q^{35} +1.00000 q^{36} -0.585786i q^{37} +6.24264i q^{38} +0.828427 q^{39} +(-2.12132 - 0.707107i) q^{40} -6.82843 q^{41} +2.00000i q^{42} +10.2426i q^{43} +4.24264 q^{44} +(0.707107 - 2.12132i) q^{45} -1.00000 q^{46} +0.828427i q^{47} +1.00000i q^{48} +3.00000 q^{49} +(-3.00000 + 4.00000i) q^{50} +6.82843 q^{51} +0.828427i q^{52} +10.5858i q^{53} -1.00000 q^{54} +(3.00000 - 9.00000i) q^{55} -2.00000 q^{56} -6.24264i q^{57} +3.65685i q^{58} -8.48528 q^{59} +(2.12132 + 0.707107i) q^{60} -0.585786 q^{61} -6.00000i q^{62} -2.00000i q^{63} -1.00000 q^{64} +(1.75736 + 0.585786i) q^{65} -4.24264 q^{66} +3.41421i q^{67} +6.82843i q^{68} +1.00000 q^{69} +(-1.41421 + 4.24264i) q^{70} -5.65685 q^{71} -1.00000i q^{72} +7.65685i q^{73} -0.585786 q^{74} +(3.00000 - 4.00000i) q^{75} +6.24264 q^{76} -8.48528i q^{77} -0.828427i q^{78} +3.65685 q^{79} +(-0.707107 + 2.12132i) q^{80} +1.00000 q^{81} +6.82843i q^{82} +1.41421i q^{83} +2.00000 q^{84} +(14.4853 + 4.82843i) q^{85} +10.2426 q^{86} -3.65685i q^{87} -4.24264i q^{88} -9.17157 q^{89} +(-2.12132 - 0.707107i) q^{90} +1.65685 q^{91} +1.00000i q^{92} +6.00000i q^{93} +0.828427 q^{94} +(4.41421 - 13.2426i) q^{95} +1.00000 q^{96} +6.00000i q^{97} -3.00000i q^{98} +4.24264 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4} + 4 q^{6} - 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^4 + 4 * q^6 - 4 * q^9	$ 4 q - 4 q^{4} + 4 q^{6} - 4 q^{9} + 8 q^{14} + 4 q^{16} - 8 q^{19} - 8 q^{21} - 4 q^{24} - 16 q^{25} + 8 q^{26} + 8 q^{29} + 24 q^{31} - 16 q^{34} + 4 q^{36} - 8 q^{39} - 16 q^{41} - 4 q^{46} + 12 q^{49} - 12 q^{50} + 16 q^{51} - 4 q^{54} + 12 q^{55} - 8 q^{56} - 8 q^{61} - 4 q^{64} + 24 q^{65} + 4 q^{69} - 8 q^{74} + 12 q^{75} + 8 q^{76} - 8 q^{79} + 4 q^{81} + 8 q^{84} + 24 q^{85} + 24 q^{86} - 48 q^{89} - 16 q^{91} - 8 q^{94} + 12 q^{95} + 4 q^{96}+O(q^{100}) $ 4 * q - 4 * q^4 + 4 * q^6 - 4 * q^9 + 8 * q^14 + 4 * q^16 - 8 * q^19 - 8 * q^21 - 4 * q^24 - 16 * q^25 + 8 * q^26 + 8 * q^29 + 24 * q^31 - 16 * q^34 + 4 * q^36 - 8 * q^39 - 16 * q^41 - 4 * q^46 + 12 * q^49 - 12 * q^50 + 16 * q^51 - 4 * q^54 + 12 * q^55 - 8 * q^56 - 8 * q^61 - 4 * q^64 + 24 * q^65 + 4 * q^69 - 8 * q^74 + 12 * q^75 + 8 * q^76 - 8 * q^79 + 4 * q^81 + 8 * q^84 + 24 * q^85 + 24 * q^86 - 48 * q^89 - 16 * q^91 - 8 * q^94 + 12 * q^95 + 4 * q^96

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	$0.123376\pi$
$7$			2.00000i			0.755929i	−0.925820	+	0.377964i	$0.876624\pi$
$8$			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	$-0.720901\pi$
$11$	−4.24264			−1.27920			−0.639602	+	0.768706i	$0.720901\pi$
$12$		−	1.00000i		−	0.288675i
$13$		−	0.828427i		−	0.229764i	−0.993379	−	0.114882i	$-0.963351\pi$
$13$		−	0.828427i		−	0.229764i	0.993379	−	0.114882i	$-0.0366490\pi$
$14$	2.00000			0.534522
$15$	−2.12132	−	0.707107i	−0.547723	−	0.182574i
$16$	1.00000			0.250000
$17$		−	6.82843i		−	1.65614i	−0.560627	−	0.828068i	$-0.689440\pi$
$17$		−	6.82843i		−	1.65614i	0.560627	−	0.828068i	$-0.310560\pi$
$18$			1.00000i			0.235702i
$19$	−6.24264			−1.43216			−0.716080	−	0.698018i	$-0.754065\pi$
$19$	−6.24264			−1.43216			−0.716080	+	0.698018i	$0.754065\pi$
$20$	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
$26$	−0.828427			−0.162468
$27$		−	1.00000i		−	0.192450i
$28$		−	2.00000i		−	0.377964i
$29$	−3.65685			−0.679061			−0.339530	−	0.940595i	$-0.610268\pi$
$29$	−3.65685			−0.679061			−0.339530	+	0.940595i	$0.610268\pi$
$30$	−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
$34$	−6.82843			−1.17107
$35$	−4.24264	−	1.41421i	−0.717137	−	0.239046i
$36$	1.00000			0.166667
$37$		−	0.585786i		−	0.0963027i	−0.998840	−	0.0481513i	$-0.984667\pi$
$37$		−	0.585786i		−	0.0963027i	0.998840	−	0.0481513i	$-0.0153330\pi$
$38$			6.24264i			1.01269i
$39$	0.828427			0.132655
$40$	−2.12132	−	0.707107i	−0.335410	−	0.111803i
$41$	−6.82843			−1.06642			−0.533211	−	0.845983i	$-0.679015\pi$
$41$	−6.82843			−1.06642			−0.533211	+	0.845983i	$0.679015\pi$
$42$			2.00000i			0.308607i
$43$			10.2426i			1.56199i	0.624538	+	0.780994i	$0.285287\pi$
$43$			10.2426i			1.56199i	−0.624538	+	0.780994i	$0.714713\pi$
$44$	4.24264			0.639602
$45$	0.707107	−	2.12132i	0.105409	−	0.316228i
$46$	−1.00000			−0.147442
$47$			0.828427i			0.120839i	0.998173	+	0.0604193i	$0.0192438\pi$
$47$			0.828427i			0.120839i	−0.998173	+	0.0604193i	$0.980756\pi$
$48$			1.00000i			0.144338i
$49$	3.00000			0.428571
$50$	−3.00000	+	4.00000i	−0.424264	+	0.565685i
$51$	6.82843			0.956171
$52$			0.828427i			0.114882i
$53$			10.5858i			1.45407i	0.686601	+	0.727035i	$0.259102\pi$
$53$			10.5858i			1.45407i	−0.686601	+	0.727035i	$0.740898\pi$
$54$	−1.00000			−0.136083
$55$	3.00000	−	9.00000i	0.404520	−	1.21356i
$56$	−2.00000			−0.267261
$57$		−	6.24264i		−	0.826858i
$58$			3.65685i			0.480168i
$59$	−8.48528			−1.10469			−0.552345	−	0.833616i	$-0.686267\pi$
$59$	−8.48528			−1.10469			−0.552345	+	0.833616i	$0.686267\pi$
$60$	2.12132	+	0.707107i	0.273861	+	0.0912871i
$61$	−0.585786			−0.0750023			−0.0375011	−	0.999297i	$-0.511940\pi$
$61$	−0.585786			−0.0750023			−0.0375011	+	0.999297i	$0.511940\pi$
$62$		−	6.00000i		−	0.762001i
$63$		−	2.00000i		−	0.251976i
$64$	−1.00000			−0.125000
$65$	1.75736	+	0.585786i	0.217974	+	0.0726579i
$66$	−4.24264			−0.522233
$67$			3.41421i			0.417113i	0.978010	+	0.208556i	$0.0668764\pi$
$67$			3.41421i			0.417113i	−0.978010	+	0.208556i	$0.933124\pi$
$68$			6.82843i			0.828068i
$69$	1.00000			0.120386
$70$	−1.41421	+	4.24264i	−0.169031	+	0.507093i
$71$	−5.65685			−0.671345			−0.335673	−	0.941979i	$-0.608964\pi$
$71$	−5.65685			−0.671345			−0.335673	+	0.941979i	$0.608964\pi$
$72$		−	1.00000i		−	0.117851i
$73$			7.65685i			0.896167i	0.893992	+	0.448084i	$0.147893\pi$
$73$			7.65685i			0.896167i	−0.893992	+	0.448084i	$0.852107\pi$
$74$	−0.585786			−0.0680963
$75$	3.00000	−	4.00000i	0.346410	−	0.461880i
$76$	6.24264			0.716080
$77$		−	8.48528i		−	0.966988i
$78$		−	0.828427i		−	0.0938009i
$79$	3.65685			0.411428			0.205714	−	0.978612i	$-0.434048\pi$
$79$	3.65685			0.411428			0.205714	+	0.978612i	$0.434048\pi$
$80$	−0.707107	+	2.12132i	−0.0790569	+	0.237171i
$81$	1.00000			0.111111
$82$			6.82843i			0.754074i
$83$			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
$85$	14.4853	+	4.82843i	1.57115	+	0.523716i
$86$	10.2426			1.10449
$87$		−	3.65685i		−	0.392056i
$88$		−	4.24264i		−	0.452267i
$89$	−9.17157			−0.972185			−0.486092	−	0.873907i	$-0.661578\pi$
$89$	−9.17157			−0.972185			−0.486092	+	0.873907i	$0.661578\pi$
$90$	−2.12132	−	0.707107i	−0.223607	−	0.0745356i
$91$	1.65685			0.173686
$92$			1.00000i			0.104257i
$93$			6.00000i			0.622171i
$94$	0.828427			0.0854457
$95$	4.41421	−	13.2426i	0.452889	−	1.35867i
$96$	1.00000			0.102062
$97$			6.00000i			0.609208i	0.952479	+	0.304604i	$0.0985241\pi$
$97$			6.00000i			0.609208i	−0.952479	+	0.304604i	$0.901476\pi$
$98$		−	3.00000i		−	0.303046i
$99$	4.24264			0.426401
$100$	4.00000	+	3.00000i	0.400000	+	0.300000i
$101$	13.3137			1.32476			0.662382	−	0.749166i	$-0.269546\pi$
$101$	13.3137			1.32476			0.662382	+	0.749166i	$0.269546\pi$
$102$		−	6.82843i		−	0.676115i
$103$			16.1421i			1.59053i	0.606261	+	0.795266i	$0.292669\pi$
$103$			16.1421i			1.59053i	−0.606261	+	0.795266i	$0.707331\pi$
$104$	0.828427			0.0812340
$105$	1.41421	−	4.24264i	0.138013	−	0.414039i
$106$	10.5858			1.02818
$107$		−	0.928932i		−	0.0898033i	−0.998991	−	0.0449016i	$-0.985703\pi$
$107$		−	0.928932i		−	0.0898033i	0.998991	−	0.0449016i	$-0.0142974\pi$
$108$			1.00000i			0.0962250i
$109$	−12.5858			−1.20550			−0.602750	−	0.797930i	$-0.705928\pi$
$109$	−12.5858			−1.20550			−0.602750	+	0.797930i	$0.705928\pi$
$110$	−9.00000	−	3.00000i	−0.858116	−	0.286039i
$111$	0.585786			0.0556004
$112$			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	$-0.294229\pi$
$114$	−6.24264			−0.584677
$115$	2.12132	+	0.707107i	0.197814	+	0.0659380i
$116$	3.65685			0.339530
$117$			0.828427i			0.0765881i
$118$			8.48528i			0.781133i
$119$	13.6569			1.25192
$120$	0.707107	−	2.12132i	0.0645497	−	0.193649i
$121$	7.00000			0.636364
$122$			0.585786i			0.0530346i
$123$		−	6.82843i		−	0.615699i
$124$	−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	$0.0282826\pi$
$127$			2.00000i			0.177471i	−0.996055	+	0.0887357i	$0.971717\pi$
$128$			1.00000i			0.0883883i
$129$	−10.2426			−0.901814
$130$	0.585786	−	1.75736i	0.0513769	−	0.154131i
$131$	−21.6569			−1.89217			−0.946084	−	0.323921i	$-0.894999\pi$
$131$	−21.6569			−1.89217			−0.946084	+	0.323921i	$0.894999\pi$
$132$			4.24264i			0.369274i
$133$		−	12.4853i		−	1.08261i
$134$	3.41421			0.294943
$135$	2.12132	+	0.707107i	0.182574	+	0.0608581i
$136$	6.82843			0.585533
$137$			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	$-0.244274\pi$
$139$	16.9706			1.43942			0.719712	+	0.694273i	$0.244274\pi$
$140$	4.24264	+	1.41421i	0.358569	+	0.119523i
$141$	−0.828427			−0.0697661
$142$			5.65685i			0.474713i
$143$			3.51472i			0.293916i
$144$	−1.00000			−0.0833333
$145$	2.58579	−	7.75736i	0.214738	−	0.644214i
$146$	7.65685			0.633686
$147$			3.00000i			0.247436i
$148$			0.585786i			0.0481513i
$149$	−12.7279			−1.04271			−0.521356	−	0.853339i	$-0.674574\pi$
$149$	−12.7279			−1.04271			−0.521356	+	0.853339i	$0.674574\pi$
$150$	−4.00000	−	3.00000i	−0.326599	−	0.244949i
$151$	−4.34315			−0.353440			−0.176720	−	0.984261i	$-0.556549\pi$
$151$	−4.34315			−0.353440			−0.176720	+	0.984261i	$0.556549\pi$
$152$		−	6.24264i		−	0.506345i
$153$			6.82843i			0.552046i
$154$	−8.48528			−0.683763
$155$	−4.24264	+	12.7279i	−0.340777	+	1.02233i
$156$	−0.828427			−0.0663273
$157$		−	21.0711i		−	1.68165i	−0.541304	−	0.840827i	$-0.682069\pi$
$157$		−	21.0711i		−	1.68165i	0.541304	−	0.840827i	$-0.317931\pi$
$158$		−	3.65685i		−	0.290924i
$159$	−10.5858			−0.839507
$160$	2.12132	+	0.707107i	0.167705	+	0.0559017i
$161$	2.00000			0.157622
$162$		−	1.00000i		−	0.0785674i
$163$			6.82843i			0.534844i	0.963580	+	0.267422i	$0.0861717\pi$
$163$			6.82843i			0.534844i	−0.963580	+	0.267422i	$0.913828\pi$
$164$	6.82843			0.533211
$165$	9.00000	+	3.00000i	0.700649	+	0.233550i
$166$	1.41421			0.109764
$167$			3.17157i			0.245424i	0.992442	+	0.122712i	$0.0391591\pi$
$167$			3.17157i			0.245424i	−0.992442	+	0.122712i	$0.960841\pi$
$168$		−	2.00000i		−	0.154303i
$169$	12.3137			0.947208
$170$	4.82843	−	14.4853i	0.370323	−	1.11097i
$171$	6.24264			0.477387
$172$		−	10.2426i		−	0.780994i
$173$			8.82843i			0.671213i	0.942002	+	0.335606i	$0.108941\pi$
$173$			8.82843i			0.671213i	−0.942002	+	0.335606i	$0.891059\pi$
$174$	−3.65685			−0.277225
$175$	6.00000	−	8.00000i	0.453557	−	0.604743i
$176$	−4.24264			−0.319801
$177$		−	8.48528i		−	0.637793i
$178$			9.17157i			0.687438i
$179$	21.6569			1.61871			0.809355	−	0.587320i	$-0.199817\pi$
$179$	21.6569			1.61871			0.809355	+	0.587320i	$0.199817\pi$
$180$	−0.707107	+	2.12132i	−0.0527046	+	0.158114i
$181$	−20.3848			−1.51519			−0.757594	−	0.652726i	$-0.773625\pi$
$181$	−20.3848			−1.51519			−0.757594	+	0.652726i	$0.773625\pi$
$182$		−	1.65685i		−	0.122814i
$183$		−	0.585786i		−	0.0433026i
$184$	1.00000			0.0737210
$185$	1.24264	+	0.414214i	0.0913608	+	0.0304536i
$186$	6.00000			0.439941
$187$			28.9706i			2.11854i
$188$		−	0.828427i		−	0.0604193i
$189$	2.00000			0.145479
$190$	−13.2426	−	4.41421i	−0.960722	−	0.320241i
$191$	−10.8284			−0.783517			−0.391759	−	0.920068i	$-0.628133\pi$
$191$	−10.8284			−0.783517			−0.391759	+	0.920068i	$0.628133\pi$
$192$		−	1.00000i		−	0.0721688i
$193$		−	20.9706i		−	1.50949i	−0.656016	−	0.754747i	$-0.727760\pi$
$193$		−	20.9706i		−	1.50949i	0.656016	−	0.754747i	$-0.272240\pi$
$194$	6.00000			0.430775
$195$	−0.585786	+	1.75736i	−0.0419490	+	0.125847i
$196$	−3.00000			−0.214286
$197$		−	20.6274i		−	1.46964i	−0.678261	−	0.734821i	$-0.737266\pi$
$197$		−	20.6274i		−	1.46964i	0.678261	−	0.734821i	$-0.262734\pi$
$198$		−	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
$201$	−3.41421			−0.240820
$202$		−	13.3137i		−	0.936749i
$203$		−	7.31371i		−	0.513322i
$204$	−6.82843			−0.478086
$205$	4.82843	−	14.4853i	0.337232	−	1.01170i
$206$	16.1421			1.12468
$207$			1.00000i			0.0695048i
$208$		−	0.828427i		−	0.0574411i
$209$	26.4853			1.83203
$210$	−4.24264	−	1.41421i	−0.292770	−	0.0975900i
$211$	−3.51472			−0.241963			−0.120982	−	0.992655i	$-0.538604\pi$
$211$	−3.51472			−0.241963			−0.120982	+	0.992655i	$0.538604\pi$
$212$		−	10.5858i		−	0.727035i
$213$		−	5.65685i		−	0.387601i
$214$	−0.928932			−0.0635005
$215$	−21.7279	−	7.24264i	−1.48183	−	0.493944i
$216$	1.00000			0.0680414
$217$			12.0000i			0.814613i
$218$			12.5858i			0.852417i
$219$	−7.65685			−0.517402
$220$	−3.00000	+	9.00000i	−0.202260	+	0.606780i
$221$	−5.65685			−0.380521
$222$		−	0.585786i		−	0.0393154i
$223$			16.0000i			1.07144i	0.844396	+	0.535720i	$0.179960\pi$
$223$			16.0000i			1.07144i	−0.844396	+	0.535720i	$0.820040\pi$
$224$	2.00000			0.133631
$225$	4.00000	+	3.00000i	0.266667	+	0.200000i
$226$	−16.9706			−1.12887
$227$			22.3848i			1.48573i	0.669441	+	0.742865i	$0.266534\pi$
$227$			22.3848i			1.48573i	−0.669441	+	0.742865i	$0.733466\pi$
$228$			6.24264i			0.413429i
$229$	−10.2426			−0.676853			−0.338426	−	0.940993i	$-0.609895\pi$
$229$	−10.2426			−0.676853			−0.338426	+	0.940993i	$0.609895\pi$
$230$	0.707107	−	2.12132i	0.0466252	−	0.139876i
$231$	8.48528			0.558291
$232$		−	3.65685i		−	0.240084i
$233$			2.14214i			0.140336i	0.997535	+	0.0701680i	$0.0223535\pi$
$233$			2.14214i			0.140336i	−0.997535	+	0.0701680i	$0.977646\pi$
$234$	0.828427			0.0541560
$235$	−1.75736	−	0.585786i	−0.114637	−	0.0382125i
$236$	8.48528			0.552345
$237$			3.65685i			0.237538i
$238$		−	13.6569i		−	0.885242i
$239$	16.8284			1.08854			0.544270	−	0.838910i	$-0.316807\pi$
$239$	16.8284			1.08854			0.544270	+	0.838910i	$0.316807\pi$
$240$	−2.12132	−	0.707107i	−0.136931	−	0.0456435i
$241$	6.48528			0.417754			0.208877	−	0.977942i	$-0.433019\pi$
$241$	6.48528			0.417754			0.208877	+	0.977942i	$0.433019\pi$
$242$		−	7.00000i		−	0.449977i
$243$			1.00000i			0.0641500i
$244$	0.585786			0.0375011
$245$	−2.12132	+	6.36396i	−0.135526	+	0.406579i
$246$	−6.82843			−0.435365
$247$			5.17157i			0.329059i
$248$			6.00000i			0.381000i
$249$	−1.41421			−0.0896221
$250$	−6.36396	−	9.19239i	−0.402492	−	0.581378i
$251$	0.727922			0.0459460			0.0229730	−	0.999736i	$-0.492687\pi$
$251$	0.727922			0.0459460			0.0229730	+	0.999736i	$0.492687\pi$
$252$			2.00000i			0.125988i
$253$			4.24264i			0.266733i
$254$	2.00000			0.125491
$255$	−4.82843	+	14.4853i	−0.302368	+	0.907104i
$256$	1.00000			0.0625000
$257$		−	18.0000i		−	1.12281i	−0.827541	−	0.561405i	$-0.810261\pi$
$257$		−	18.0000i		−	1.12281i	0.827541	−	0.561405i	$-0.189739\pi$
$258$			10.2426i			0.637679i
$259$	1.17157			0.0727980
$260$	−1.75736	−	0.585786i	−0.108987	−	0.0363289i
$261$	3.65685			0.226354
$262$			21.6569i			1.33796i
$263$		−	6.34315i		−	0.391135i	−0.980690	−	0.195568i	$-0.937345\pi$
$263$		−	6.34315i		−	0.391135i	0.980690	−	0.195568i	$-0.0626549\pi$
$264$	4.24264			0.261116
$265$	−22.4558	−	7.48528i	−1.37945	−	0.459817i
$266$	−12.4853			−0.765522
$267$		−	9.17157i		−	0.561291i
$268$		−	3.41421i		−	0.208556i
$269$	19.1716			1.16891			0.584456	−	0.811426i	$-0.301308\pi$
$269$	19.1716			1.16891			0.584456	+	0.811426i	$0.301308\pi$
$270$	0.707107	−	2.12132i	0.0430331	−	0.129099i
$271$	9.65685			0.586612			0.293306	−	0.956019i	$-0.405245\pi$
$271$	9.65685			0.586612			0.293306	+	0.956019i	$0.405245\pi$
$272$		−	6.82843i		−	0.414034i
$273$			1.65685i			0.100277i
$274$	8.48528			0.512615
$275$	16.9706	+	12.7279i	1.02336	+	0.767523i
$276$	−1.00000			−0.0601929
$277$		−	13.5147i		−	0.812021i	−0.913868	−	0.406010i	$-0.866920\pi$
$277$		−	13.5147i		−	0.812021i	0.913868	−	0.406010i	$-0.133080\pi$
$278$		−	16.9706i		−	1.01783i
$279$	−6.00000			−0.359211
$280$	1.41421	−	4.24264i	0.0845154	−	0.253546i
$281$	−1.85786			−0.110831			−0.0554154	−	0.998463i	$-0.517648\pi$
$281$	−1.85786			−0.110831			−0.0554154	+	0.998463i	$0.517648\pi$
$282$			0.828427i			0.0493321i
$283$			15.2132i			0.904331i	0.891934	+	0.452166i	$0.149348\pi$
$283$			15.2132i			0.904331i	−0.891934	+	0.452166i	$0.850652\pi$
$284$	5.65685			0.335673
$285$	13.2426	+	4.41421i	0.784426	+	0.261475i
$286$	3.51472			0.207830
$287$		−	13.6569i		−	0.806139i
$288$			1.00000i			0.0589256i
$289$	−29.6274			−1.74279
$290$	−7.75736	−	2.58579i	−0.455528	−	0.151843i
$291$	−6.00000			−0.351726
$292$		−	7.65685i		−	0.448084i
$293$			20.0416i			1.17084i	0.810729	+	0.585422i	$0.199071\pi$
$293$			20.0416i			1.17084i	−0.810729	+	0.585422i	$0.800929\pi$
$294$	3.00000			0.174964
$295$	6.00000	−	18.0000i	0.349334	−	1.04800i
$296$	0.585786			0.0340481
$297$			4.24264i			0.246183i
$298$			12.7279i			0.737309i
$299$	−0.828427			−0.0479092
$300$	−3.00000	+	4.00000i	−0.173205	+	0.230940i
$301$	−20.4853			−1.18075
$302$			4.34315i			0.249920i
$303$			13.3137i			0.764853i
$304$	−6.24264			−0.358040
$305$	0.414214	−	1.24264i	0.0237178	−	0.0711534i
$306$	6.82843			0.390355
$307$		−	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
$309$	−16.1421			−0.918294
$310$	12.7279	+	4.24264i	0.722897	+	0.240966i
$311$	−21.7990			−1.23611			−0.618054	−	0.786136i	$-0.712079\pi$
$311$	−21.7990			−1.23611			−0.618054	+	0.786136i	$0.712079\pi$
$312$			0.828427i			0.0469005i
$313$			18.0000i			1.01742i	0.860938	+	0.508710i	$0.169877\pi$
$313$			18.0000i			1.01742i	−0.860938	+	0.508710i	$0.830123\pi$
$314$	−21.0711			−1.18911
$315$	4.24264	+	1.41421i	0.239046	+	0.0796819i
$316$	−3.65685			−0.205714
$317$			14.4853i			0.813574i	0.913523	+	0.406787i	$0.133351\pi$
$317$			14.4853i			0.813574i	−0.913523	+	0.406787i	$0.866649\pi$
$318$			10.5858i			0.593621i
$319$	15.5147			0.868657
$320$	0.707107	−	2.12132i	0.0395285	−	0.118585i
$321$	0.928932			0.0518479
$322$		−	2.00000i		−	0.111456i
$323$			42.6274i			2.37185i
$324$	−1.00000			−0.0555556
$325$	−2.48528	+	3.31371i	−0.137859	+	0.183811i
$326$	6.82843			0.378192
$327$		−	12.5858i		−	0.695996i
$328$		−	6.82843i		−	0.377037i
$329$	−1.65685			−0.0913453
$330$	3.00000	−	9.00000i	0.165145	−	0.495434i
$331$	−20.4853			−1.12597			−0.562986	−	0.826466i	$-0.690348\pi$
$331$	−20.4853			−1.12597			−0.562986	+	0.826466i	$0.690348\pi$
$332$		−	1.41421i		−	0.0776151i
$333$			0.585786i			0.0321009i
$334$	3.17157			0.173541
$335$	−7.24264	−	2.41421i	−0.395708	−	0.131903i
$336$	−2.00000			−0.109109
$337$			26.4853i			1.44275i	0.692547	+	0.721373i	$0.256488\pi$
$337$			26.4853i			1.44275i	−0.692547	+	0.721373i	$0.743512\pi$
$338$		−	12.3137i		−	0.669777i
$339$	16.9706			0.921714
$340$	−14.4853	−	4.82843i	−0.785575	−	0.261858i
$341$	−25.4558			−1.37851
$342$		−	6.24264i		−	0.337563i
$343$			20.0000i			1.07990i
$344$	−10.2426			−0.552246
$345$	−0.707107	+	2.12132i	−0.0380693	+	0.114208i
$346$	8.82843			0.474619
$347$			31.7990i			1.70706i	0.521044	+	0.853530i	$0.325543\pi$
$347$			31.7990i			1.70706i	−0.521044	+	0.853530i	$0.674457\pi$
$348$			3.65685i			0.196028i
$349$	−2.68629			−0.143794			−0.0718969	−	0.997412i	$-0.522905\pi$
$349$	−2.68629			−0.143794			−0.0718969	+	0.997412i	$0.522905\pi$
$350$	−8.00000	−	6.00000i	−0.427618	−	0.320713i
$351$	−0.828427			−0.0442182
$352$			4.24264i			0.226134i
$353$			2.14214i			0.114014i	0.998374	+	0.0570072i	$0.0181558\pi$
$353$			2.14214i			0.114014i	−0.998374	+	0.0570072i	$0.981844\pi$
$354$	−8.48528			−0.450988
$355$	4.00000	−	12.0000i	0.212298	−	0.636894i
$356$	9.17157			0.486092
$357$			13.6569i			0.722797i
$358$		−	21.6569i		−	1.14460i
$359$	−19.7990			−1.04495			−0.522475	−	0.852654i	$-0.674991\pi$
$359$	−19.7990			−1.04495			−0.522475	+	0.852654i	$0.674991\pi$
$360$	2.12132	+	0.707107i	0.111803	+	0.0372678i
$361$	19.9706			1.05108
$362$			20.3848i			1.07140i
$363$			7.00000i			0.367405i
$364$	−1.65685			−0.0868428
$365$	−16.2426	−	5.41421i	−0.850179	−	0.283393i
$366$	−0.585786			−0.0306195
$367$		−	1.02944i		−	0.0537362i	−0.999639	−	0.0268681i	$-0.991447\pi$
$367$		−	1.02944i		−	0.0537362i	0.999639	−	0.0268681i	$-0.00855341\pi$
$368$		−	1.00000i		−	0.0521286i
$369$	6.82843			0.355474
$370$	0.414214	−	1.24264i	0.0215339	−	0.0646018i
$371$	−21.1716			−1.09917
$372$		−	6.00000i		−	0.311086i
$373$		−	23.2132i		−	1.20193i	−0.799274	−	0.600967i	$-0.794782\pi$
$373$		−	23.2132i		−	1.20193i	0.799274	−	0.600967i	$-0.205218\pi$
$374$	28.9706			1.49803
$375$	6.36396	+	9.19239i	0.328634	+	0.474693i
$376$	−0.828427			−0.0427229
$377$			3.02944i			0.156024i
$378$		−	2.00000i		−	0.102869i
$379$	−23.2132			−1.19238			−0.596191	−	0.802843i	$-0.703320\pi$
$379$	−23.2132			−1.19238			−0.596191	+	0.802843i	$0.703320\pi$
$380$	−4.41421	+	13.2426i	−0.226444	+	0.679333i
$381$	−2.00000			−0.102463
$382$			10.8284i			0.554031i
$383$			26.6274i			1.36060i	0.732935	+	0.680299i	$0.238150\pi$
$383$			26.6274i			1.36060i	−0.732935	+	0.680299i	$0.761850\pi$
$384$	−1.00000			−0.0510310
$385$	18.0000	+	6.00000i	0.917365	+	0.305788i
$386$	−20.9706			−1.06737
$387$		−	10.2426i		−	0.520663i
$388$		−	6.00000i		−	0.304604i
$389$	30.3848			1.54057			0.770285	−	0.637700i	$-0.220114\pi$
$389$	30.3848			1.54057			0.770285	+	0.637700i	$0.220114\pi$
$390$	1.75736	+	0.585786i	0.0889873	+	0.0296624i
$391$	−6.82843			−0.345328
$392$			3.00000i			0.151523i
$393$		−	21.6569i		−	1.09244i
$394$	−20.6274			−1.03919
$395$	−2.58579	+	7.75736i	−0.130105	+	0.390315i
$396$	−4.24264			−0.213201
$397$			19.6569i			0.986549i	0.869874	+	0.493275i	$0.164200\pi$
$397$			19.6569i			0.986549i	−0.869874	+	0.493275i	$0.835800\pi$
$398$		−	14.0000i		−	0.701757i
$399$	12.4853			0.625046
$400$	−4.00000	−	3.00000i	−0.200000	−	0.150000i
$401$	4.97056			0.248218			0.124109	−	0.992269i	$-0.460393\pi$
$401$	4.97056			0.248218			0.124109	+	0.992269i	$0.460393\pi$
$402$			3.41421i			0.170285i
$403$		−	4.97056i		−	0.247601i
$404$	−13.3137			−0.662382
$405$	−0.707107	+	2.12132i	−0.0351364	+	0.105409i
$406$	−7.31371			−0.362973
$407$			2.48528i			0.123191i
$408$			6.82843i			0.338058i
$409$	−18.0000			−0.890043			−0.445021	−	0.895520i	$-0.646804\pi$
$409$	−18.0000			−0.890043			−0.445021	+	0.895520i	$0.646804\pi$
$410$	−14.4853	−	4.82843i	−0.715377	−	0.238459i
$411$	−8.48528			−0.418548
$412$		−	16.1421i		−	0.795266i
$413$		−	16.9706i		−	0.835067i
$414$	1.00000			0.0491473
$415$	−3.00000	−	1.00000i	−0.147264	−	0.0490881i
$416$	−0.828427			−0.0406170
$417$			16.9706i			0.831052i
$418$		−	26.4853i		−	1.29544i
$419$	39.0711			1.90875			0.954373	−	0.298616i	$-0.0965250\pi$
$419$	39.0711			1.90875			0.954373	+	0.298616i	$0.0965250\pi$
$420$	−1.41421	+	4.24264i	−0.0690066	+	0.207020i
$421$	1.75736			0.0856485			0.0428242	−	0.999083i	$-0.486364\pi$
$421$	1.75736			0.0856485			0.0428242	+	0.999083i	$0.486364\pi$
$422$			3.51472i			0.171094i
$423$		−	0.828427i		−	0.0402795i
$424$	−10.5858			−0.514091
$425$	−20.4853	+	27.3137i	−0.993682	+	1.32491i
$426$	−5.65685			−0.274075
$427$		−	1.17157i		−	0.0566964i
$428$			0.928932i			0.0449016i
$429$	−3.51472			−0.169692
$430$	−7.24264	+	21.7279i	−0.349271	+	1.04781i
$431$	−8.48528			−0.408722			−0.204361	−	0.978896i	$-0.565512\pi$
$431$	−8.48528			−0.408722			−0.204361	+	0.978896i	$0.565512\pi$
$432$		−	1.00000i		−	0.0481125i
$433$		−	6.97056i		−	0.334984i	−0.985873	−	0.167492i	$-0.946433\pi$
$433$		−	6.97056i		−	0.334984i	0.985873	−	0.167492i	$-0.0535668\pi$
$434$	12.0000			0.576018
$435$	7.75736	+	2.58579i	0.371937	+	0.123979i
$436$	12.5858			0.602750
$437$			6.24264i			0.298626i
$438$			7.65685i			0.365859i
$439$		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
$443$		−	0.686292i		−	0.0326067i	−0.999867	−	0.0163033i	$-0.994810\pi$
$443$		−	0.686292i		−	0.0326067i	0.999867	−	0.0163033i	$-0.00518975\pi$
$444$	−0.585786			−0.0278002
$445$	6.48528	−	19.4558i	0.307432	−	0.922295i
$446$	16.0000			0.757622
$447$		−	12.7279i		−	0.602010i
$448$		−	2.00000i		−	0.0944911i
$449$	17.1716			0.810377			0.405188	−	0.914233i	$-0.367206\pi$
$449$	17.1716			0.810377			0.405188	+	0.914233i	$0.367206\pi$
$450$	3.00000	−	4.00000i	0.141421	−	0.188562i
$451$	28.9706			1.36417
$452$			16.9706i			0.798228i
$453$		−	4.34315i		−	0.204059i
$454$	22.3848			1.05057
$455$	−1.17157	+	3.51472i	−0.0549242	+	0.164773i
$456$	6.24264			0.292338
$457$		−	34.9706i		−	1.63585i	−0.575322	−	0.817927i	$-0.695123\pi$
$457$		−	34.9706i		−	1.63585i	0.575322	−	0.817927i	$-0.304877\pi$
$458$			10.2426i			0.478607i
$459$	−6.82843			−0.318724
$460$	−2.12132	−	0.707107i	−0.0989071	−	0.0329690i
$461$	−7.17157			−0.334013			−0.167007	−	0.985956i	$-0.553410\pi$
$461$	−7.17157			−0.334013			−0.167007	+	0.985956i	$0.553410\pi$
$462$		−	8.48528i		−	0.394771i
$463$		−	30.9706i		−	1.43932i	−0.694325	−	0.719662i	$-0.744297\pi$
$463$		−	30.9706i		−	1.43932i	0.694325	−	0.719662i	$-0.255703\pi$
$464$	−3.65685			−0.169765
$465$	−12.7279	−	4.24264i	−0.590243	−	0.196748i
$466$	2.14214			0.0992325
$467$			31.3553i			1.45095i	0.688247	+	0.725476i	$0.258380\pi$
$467$			31.3553i			1.45095i	−0.688247	+	0.725476i	$0.741620\pi$
$468$		−	0.828427i		−	0.0382941i
$469$	−6.82843			−0.315307
$470$	−0.585786	+	1.75736i	−0.0270203	+	0.0810609i
$471$	21.0711			0.970904
$472$		−	8.48528i		−	0.390567i
$473$		−	43.4558i		−	1.99810i
$474$	3.65685			0.167965
$475$	24.9706	+	18.7279i	1.14573	+	0.859296i
$476$	−13.6569			−0.625961
$477$		−	10.5858i		−	0.484690i
$478$		−	16.8284i		−	0.769714i
$479$	−27.3137			−1.24800			−0.623998	−	0.781426i	$-0.714493\pi$
$479$	−27.3137			−1.24800			−0.623998	+	0.781426i	$0.714493\pi$
$480$	−0.707107	+	2.12132i	−0.0322749	+	0.0968246i
$481$	−0.485281			−0.0221269
$482$		−	6.48528i		−	0.295396i
$483$			2.00000i			0.0910032i
$484$	−7.00000			−0.318182
$485$	−12.7279	−	4.24264i	−0.577945	−	0.192648i
$486$	1.00000			0.0453609
$487$		−	39.9411i		−	1.80991i	−0.425512	−	0.904953i	$-0.639906\pi$
$487$		−	39.9411i		−	1.80991i	0.425512	−	0.904953i	$-0.360094\pi$
$488$		−	0.585786i		−	0.0265173i
$489$	−6.82843			−0.308792
$490$	6.36396	+	2.12132i	0.287494	+	0.0958315i
$491$	−18.6274			−0.840644			−0.420322	−	0.907375i	$-0.638083\pi$
$491$	−18.6274			−0.840644			−0.420322	+	0.907375i	$0.638083\pi$
$492$			6.82843i			0.307849i
$493$			24.9706i			1.12462i
$494$	5.17157			0.232680
$495$	−3.00000	+	9.00000i	−0.134840	+	0.404520i
$496$	6.00000			0.269408
$497$		−	11.3137i		−	0.507489i
$498$			1.41421i			0.0633724i
$499$	37.4558			1.67675			0.838377	−	0.545091i	$-0.183505\pi$
$499$	37.4558			1.67675			0.838377	+	0.545091i	$0.183505\pi$
$500$	−9.19239	+	6.36396i	−0.411096	+	0.284605i
$501$	−3.17157			−0.141695
$502$		−	0.727922i		−	0.0324888i
$503$		−	6.14214i		−	0.273864i	−0.990580	−	0.136932i	$-0.956276\pi$
$503$		−	6.14214i		−	0.273864i	0.990580	−	0.136932i	$-0.0437242\pi$
$504$	2.00000			0.0890871
$505$	−9.41421	+	28.2426i	−0.418927	+	1.25678i
$506$	4.24264			0.188608
$507$			12.3137i			0.546871i
$508$		−	2.00000i		−	0.0887357i
$509$	10.9706			0.486262			0.243131	−	0.969994i	$-0.421826\pi$
$509$	10.9706			0.486262			0.243131	+	0.969994i	$0.421826\pi$
$510$	14.4853	+	4.82843i	0.641419	+	0.213806i
$511$	−15.3137			−0.677439
$512$		−	1.00000i		−	0.0441942i
$513$			6.24264i			0.275619i
$514$	−18.0000			−0.793946
$515$	−34.2426	−	11.4142i	−1.50891	−	0.502970i
$516$	10.2426			0.450907
$517$		−	3.51472i		−	0.154577i
$518$		−	1.17157i		−	0.0514760i
$519$	−8.82843			−0.387525
$520$	−0.585786	+	1.75736i	−0.0256884	+	0.0770653i
$521$	40.2843			1.76489			0.882443	−	0.470419i	$-0.155897\pi$
$521$	40.2843			1.76489			0.882443	+	0.470419i	$0.155897\pi$
$522$		−	3.65685i		−	0.160056i
$523$		−	17.0711i		−	0.746466i	−0.927738	−	0.373233i	$-0.878249\pi$
$523$		−	17.0711i		−	0.746466i	0.927738	−	0.373233i	$-0.121751\pi$
$524$	21.6569			0.946084
$525$	8.00000	+	6.00000i	0.349149	+	0.261861i
$526$	−6.34315			−0.276574
$527$		−	40.9706i		−	1.78471i
$528$		−	4.24264i		−	0.184637i
$529$	−1.00000			−0.0434783
$530$	−7.48528	+	22.4558i	−0.325140	+	0.975420i
$531$	8.48528			0.368230
$532$			12.4853i			0.541306i
$533$			5.65685i			0.245026i
$534$	−9.17157			−0.396893
$535$	1.97056	+	0.656854i	0.0851949	+	0.0283983i
$536$	−3.41421			−0.147472
$537$			21.6569i			0.934562i
$538$		−	19.1716i		−	0.826545i
$539$	−12.7279			−0.548230
$540$	−2.12132	−	0.707107i	−0.0912871	−	0.0304290i
$541$	−31.9411			−1.37326			−0.686628	−	0.727009i	$-0.740910\pi$
$541$	−31.9411			−1.37326			−0.686628	+	0.727009i	$0.740910\pi$
$542$		−	9.65685i		−	0.414797i
$543$		−	20.3848i		−	0.874794i
$544$	−6.82843			−0.292766
$545$	8.89949	−	26.6985i	0.381212	−	1.14364i
$546$	1.65685			0.0709068
$547$		−	12.4853i		−	0.533832i	−0.963720	−	0.266916i	$-0.913995\pi$
$547$		−	12.4853i		−	0.533832i	0.963720	−	0.266916i	$-0.0860046\pi$
$548$		−	8.48528i		−	0.362473i
$549$	0.585786			0.0250008
$550$	12.7279	−	16.9706i	0.542720	−	0.723627i
$551$	22.8284			0.972524
$552$			1.00000i			0.0425628i
$553$			7.31371i			0.311011i
$554$	−13.5147			−0.574185
$555$	−0.414214	+	1.24264i	−0.0175824	+	0.0527472i
$556$	−16.9706			−0.719712
$557$		−	38.8701i		−	1.64698i	−0.567333	−	0.823489i	$-0.692025\pi$
$557$		−	38.8701i		−	1.64698i	0.567333	−	0.823489i	$-0.307975\pi$
$558$			6.00000i			0.254000i
$559$	8.48528			0.358889
$560$	−4.24264	−	1.41421i	−0.179284	−	0.0597614i
$561$	−28.9706			−1.22314
$562$			1.85786i			0.0783693i
$563$		−	2.10051i		−	0.0885257i	−0.999020	−	0.0442629i	$-0.985906\pi$
$563$		−	2.10051i		−	0.0885257i	0.999020	−	0.0442629i	$-0.0140939\pi$
$564$	0.828427			0.0348831
$565$	36.0000	+	12.0000i	1.51453	+	0.504844i
$566$	15.2132			0.639459
$567$			2.00000i			0.0839921i
$568$		−	5.65685i		−	0.237356i
$569$	−14.3431			−0.601296			−0.300648	−	0.953735i	$-0.597203\pi$
$569$	−14.3431			−0.601296			−0.300648	+	0.953735i	$0.597203\pi$
$570$	4.41421	−	13.2426i	0.184891	−	0.554673i
$571$	−17.2721			−0.722814			−0.361407	−	0.932408i	$-0.617703\pi$
$571$	−17.2721			−0.722814			−0.361407	+	0.932408i	$0.617703\pi$
$572$		−	3.51472i		−	0.146958i
$573$		−	10.8284i		−	0.452364i
$574$	−13.6569			−0.570026
$575$	−3.00000	+	4.00000i	−0.125109	+	0.166812i
$576$	1.00000			0.0416667
$577$		−	24.6274i		−	1.02525i	−0.858612	−	0.512626i	$-0.828673\pi$
$577$		−	24.6274i		−	1.02525i	0.858612	−	0.512626i	$-0.171327\pi$
$578$			29.6274i			1.23234i
$579$	20.9706			0.871507
$580$	−2.58579	+	7.75736i	−0.107369	+	0.322107i
$581$	−2.82843			−0.117343
$582$			6.00000i			0.248708i
$583$		−	44.9117i		−	1.86005i
$584$	−7.65685			−0.316843
$585$	−1.75736	−	0.585786i	−0.0726579	−	0.0242193i
$586$	20.0416			0.827912
$587$		−	21.1716i		−	0.873844i	−0.899499	−	0.436922i	$-0.856069\pi$
$587$		−	21.1716i		−	0.873844i	0.899499	−	0.436922i	$-0.143931\pi$
$588$		−	3.00000i		−	0.123718i
$589$	−37.4558			−1.54334
$590$	−18.0000	−	6.00000i	−0.741048	−	0.247016i
$591$	20.6274			0.848499
$592$		−	0.585786i		−	0.0240757i
$593$		−	23.6569i		−	0.971471i	−0.874106	−	0.485735i	$-0.838552\pi$
$593$		−	23.6569i		−	0.971471i	0.874106	−	0.485735i	$-0.161448\pi$
$594$	4.24264			0.174078
$595$	−9.65685	+	28.9706i	−0.395892	+	1.18768i
$596$	12.7279			0.521356
$597$			14.0000i			0.572982i
$598$			0.828427i			0.0338769i
$599$	11.8579			0.484499			0.242250	−	0.970214i	$-0.422115\pi$
$599$	11.8579			0.484499			0.242250	+	0.970214i	$0.422115\pi$
$600$	4.00000	+	3.00000i	0.163299	+	0.122474i
$601$	12.9706			0.529080			0.264540	−	0.964375i	$-0.414780\pi$
$601$	12.9706			0.529080			0.264540	+	0.964375i	$0.414780\pi$
$602$			20.4853i			0.834918i
$603$		−	3.41421i		−	0.139038i
$604$	4.34315			0.176720
$605$	−4.94975	+	14.8492i	−0.201236	+	0.603708i
$606$	13.3137			0.540832
$607$		−	16.9706i		−	0.688814i	−0.938820	−	0.344407i	$-0.888080\pi$
$607$		−	16.9706i		−	0.688814i	0.938820	−	0.344407i	$-0.111920\pi$
$608$			6.24264i			0.253173i
$609$	7.31371			0.296366
$610$	−1.24264	−	0.414214i	−0.0503131	−	0.0167710i
$611$	0.686292			0.0277644
$612$		−	6.82843i		−	0.276023i
$613$		−	5.75736i		−	0.232538i	−0.993218	−	0.116269i	$-0.962907\pi$
$613$		−	5.75736i		−	0.232538i	0.993218	−	0.116269i	$-0.0370934\pi$
$614$	−25.4558			−1.02731
$615$	14.4853	+	4.82843i	0.584103	+	0.194701i
$616$	8.48528			0.341882
$617$			8.68629i			0.349697i	0.984595	+	0.174848i	$0.0559436\pi$
$617$			8.68629i			0.349697i	−0.984595	+	0.174848i	$0.944056\pi$
$618$			16.1421i			0.649332i
$619$	−23.2132			−0.933017			−0.466509	−	0.884517i	$-0.654488\pi$
$619$	−23.2132			−0.933017			−0.466509	+	0.884517i	$0.654488\pi$
$620$	4.24264	−	12.7279i	0.170389	−	0.511166i
$621$	−1.00000			−0.0401286
$622$			21.7990i			0.874060i
$623$		−	18.3431i		−	0.734903i
$624$	0.828427			0.0331636
$625$	7.00000	+	24.0000i	0.280000	+	0.960000i
$626$	18.0000			0.719425
$627$			26.4853i			1.05772i
$628$			21.0711i			0.840827i
$629$	−4.00000			−0.159490
$630$	1.41421	−	4.24264i	0.0563436	−	0.169031i
$631$	40.8284			1.62535			0.812677	−	0.582714i	$-0.198009\pi$
$631$	40.8284			1.62535			0.812677	+	0.582714i	$0.198009\pi$
$632$			3.65685i			0.145462i
$633$		−	3.51472i		−	0.139698i
$634$	14.4853			0.575284
$635$	−4.24264	−	1.41421i	−0.168364	−	0.0561214i
$636$	10.5858			0.419754
$637$		−	2.48528i		−	0.0984704i
$638$		−	15.5147i		−	0.614234i
$639$	5.65685			0.223782
$640$	−2.12132	−	0.707107i	−0.0838525	−	0.0279508i
$641$	48.2843			1.90711			0.953557	−	0.301213i	$-0.0973914\pi$
$641$	48.2843			1.90711			0.953557	+	0.301213i	$0.0973914\pi$
$642$		−	0.928932i		−	0.0366620i
$643$			4.38478i			0.172919i	0.996255	+	0.0864593i	$0.0275553\pi$
$643$			4.38478i			0.172919i	−0.996255	+	0.0864593i	$0.972445\pi$
$644$	−2.00000			−0.0788110
$645$	7.24264	−	21.7279i	0.285179	−	0.855536i
$646$	42.6274			1.67715
$647$			18.3431i			0.721143i	0.932731	+	0.360572i	$0.117418\pi$
$647$			18.3431i			0.721143i	−0.932731	+	0.360572i	$0.882582\pi$
$648$			1.00000i			0.0392837i
$649$	36.0000			1.41312
$650$	3.31371	+	2.48528i	0.129974	+	0.0974808i
$651$	−12.0000			−0.470317
$652$		−	6.82843i		−	0.267422i
$653$		−	7.45584i		−	0.291770i	−0.989302	−	0.145885i	$-0.953397\pi$
$653$		−	7.45584i		−	0.291770i	0.989302	−	0.145885i	$-0.0466029\pi$
$654$	−12.5858			−0.492143
$655$	15.3137	−	45.9411i	0.598356	−	1.79507i
$656$	−6.82843			−0.266605
$657$		−	7.65685i		−	0.298722i
$658$			1.65685i			0.0645909i
$659$	45.8995			1.78799			0.893995	−	0.448076i	$-0.147891\pi$
$659$	45.8995			1.78799			0.893995	+	0.448076i	$0.147891\pi$
$660$	−9.00000	−	3.00000i	−0.350325	−	0.116775i
$661$	34.7279			1.35076			0.675380	−	0.737470i	$-0.263980\pi$
$661$	34.7279			1.35076			0.675380	+	0.737470i	$0.263980\pi$
$662$			20.4853i			0.796183i
$663$		−	5.65685i		−	0.219694i
$664$	−1.41421			−0.0548821
$665$	26.4853	+	8.82843i	1.02706	+	0.342352i
$666$	0.585786			0.0226988
$667$			3.65685i			0.141594i
$668$		−	3.17157i		−	0.122712i
$669$	−16.0000			−0.618596
$670$	−2.41421	+	7.24264i	−0.0932692	+	0.279808i
$671$	2.48528			0.0959432
$672$			2.00000i			0.0771517i
$673$			27.9411i			1.07705i	0.842609	+	0.538526i	$0.181018\pi$
$673$			27.9411i			1.07705i	−0.842609	+	0.538526i	$0.818982\pi$
$674$	26.4853			1.02017
$675$	−3.00000	+	4.00000i	−0.115470	+	0.153960i
$676$	−12.3137			−0.473604
$677$			40.5269i			1.55758i	0.627287	+	0.778788i	$0.284165\pi$
$677$			40.5269i			1.55758i	−0.627287	+	0.778788i	$0.715835\pi$
$678$		−	16.9706i		−	0.651751i
$679$	−12.0000			−0.460518
$680$	−4.82843	+	14.4853i	−0.185162	+	0.555485i
$681$	−22.3848			−0.857786
$682$			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	$-0.0519042\pi$
$684$	−6.24264			−0.238693
$685$	−18.0000	−	6.00000i	−0.687745	−	0.229248i
$686$	20.0000			0.763604
$687$		−	10.2426i		−	0.390781i
$688$			10.2426i			0.390497i
$689$	8.76955			0.334093
$690$	2.12132	+	0.707107i	0.0807573	+	0.0269191i
$691$	−13.1716			−0.501070			−0.250535	−	0.968108i	$-0.580607\pi$
$691$	−13.1716			−0.501070			−0.250535	+	0.968108i	$0.580607\pi$
$692$		−	8.82843i		−	0.335606i
$693$			8.48528i			0.322329i
$694$	31.7990			1.20707
$695$	−12.0000	+	36.0000i	−0.455186	+	1.36556i
$696$	3.65685			0.138613
$697$			46.6274i			1.76614i
$698$			2.68629i			0.101678i
$699$	−2.14214			−0.0810230
$700$	−6.00000	+	8.00000i	−0.226779	+	0.302372i
$701$	−38.1838			−1.44218			−0.721090	−	0.692841i	$-0.756359\pi$
$701$	−38.1838			−1.44218			−0.721090	+	0.692841i	$0.756359\pi$
$702$			0.828427i			0.0312670i
$703$			3.65685i			0.137921i
$704$	4.24264			0.159901
$705$	0.585786	−	1.75736i	0.0220620	−	0.0661860i
$706$	2.14214			0.0806203
$707$			26.6274i			1.00143i
$708$			8.48528i			0.318896i
$709$	−32.5858			−1.22378			−0.611892	−	0.790941i	$-0.709591\pi$
$709$	−32.5858			−1.22378			−0.611892	+	0.790941i	$0.709591\pi$
$710$	−12.0000	−	4.00000i	−0.450352	−	0.150117i
$711$	−3.65685			−0.137143
$712$		−	9.17157i		−	0.343719i
$713$		−	6.00000i		−	0.224702i
$714$	13.6569			0.511095
$715$	−7.45584	−	2.48528i	−0.278833	−	0.0929443i
$716$	−21.6569			−0.809355
$717$			16.8284i			0.628469i
$718$			19.7990i			0.738892i
$719$	0.142136			0.00530076			0.00265038	−	0.999996i	$-0.499156\pi$
$719$	0.142136			0.00530076			0.00265038	+	0.999996i	$0.499156\pi$
$720$	0.707107	−	2.12132i	0.0263523	−	0.0790569i
$721$	−32.2843			−1.20233
$722$		−	19.9706i		−	0.743227i
$723$			6.48528i			0.241190i
$724$	20.3848			0.757594
$725$	14.6274	+	10.9706i	0.543249	+	0.407436i
$726$	7.00000			0.259794
$727$			20.3431i			0.754486i	0.926114	+	0.377243i	$0.123128\pi$
$727$			20.3431i			0.754486i	−0.926114	+	0.377243i	$0.876872\pi$
$728$			1.65685i			0.0614071i
$729$	−1.00000			−0.0370370
$730$	−5.41421	+	16.2426i	−0.200389	+	0.601167i
$731$	69.9411			2.58687
$732$			0.585786i			0.0216513i
$733$			11.4142i			0.421594i	0.977530	+	0.210797i	$0.0676058\pi$
$733$			11.4142i			0.421594i	−0.977530	+	0.210797i	$0.932394\pi$
$734$	−1.02944			−0.0379972
$735$	−6.36396	−	2.12132i	−0.234738	−	0.0782461i
$736$	−1.00000			−0.0368605
$737$		−	14.4853i		−	0.533572i
$738$		−	6.82843i		−	0.251358i
$739$	2.62742			0.0966511			0.0483255	−	0.998832i	$-0.484612\pi$
$739$	2.62742			0.0966511			0.0483255	+	0.998832i	$0.484612\pi$
$740$	−1.24264	−	0.414214i	−0.0456804	−	0.0152268i
$741$	−5.17157			−0.189982
$742$			21.1716i			0.777233i
$743$			13.6569i			0.501021i	0.968114	+	0.250511i	$0.0805985\pi$
$743$			13.6569i			0.501021i	−0.968114	+	0.250511i	$0.919401\pi$
$744$	−6.00000			−0.219971
$745$	9.00000	−	27.0000i	0.329734	−	0.989203i
$746$	−23.2132			−0.849896
$747$		−	1.41421i		−	0.0517434i
$748$		−	28.9706i		−	1.05927i
$749$	1.85786			0.0678849
$750$	9.19239	−	6.36396i	0.335659	−	0.232379i
$751$	−43.1716			−1.57535			−0.787677	−	0.616089i	$-0.788716\pi$
$751$	−43.1716			−1.57535			−0.787677	+	0.616089i	$0.788716\pi$
$752$			0.828427i			0.0302096i
$753$			0.727922i			0.0265270i
$754$	3.02944			0.110326
$755$	3.07107	−	9.21320i	0.111768	−	0.335303i
$756$	−2.00000			−0.0727393
$757$		−	33.7574i		−	1.22693i	−0.789721	−	0.613466i	$-0.789775\pi$
$757$		−	33.7574i		−	1.22693i	0.789721	−	0.613466i	$-0.210225\pi$
$758$			23.2132i			0.843142i
$759$	−4.24264			−0.153998
$760$	13.2426	+	4.41421i	0.480361	+	0.160120i
$761$	−35.6569			−1.29256			−0.646280	−	0.763100i	$-0.723676\pi$
$761$	−35.6569			−1.29256			−0.646280	+	0.763100i	$0.723676\pi$
$762$			2.00000i			0.0724524i
$763$		−	25.1716i		−	0.911272i
$764$	10.8284			0.391759
$765$	−14.4853	−	4.82843i	−0.523716	−	0.174572i
$766$	26.6274			0.962088
$767$			7.02944i			0.253818i
$768$			1.00000i			0.0360844i
$769$	−6.20101			−0.223614			−0.111807	−	0.993730i	$-0.535664\pi$
$769$	−6.20101			−0.223614			−0.111807	+	0.993730i	$0.535664\pi$
$770$	6.00000	−	18.0000i	0.216225	−	0.648675i
$771$	18.0000			0.648254
$772$			20.9706i			0.754747i
$773$		−	31.0711i		−	1.11755i	−0.829320	−	0.558774i	$-0.811272\pi$
$773$		−	31.0711i		−	1.11755i	0.829320	−	0.558774i	$-0.188728\pi$
$774$	−10.2426			−0.368164
$775$	−24.0000	−	18.0000i	−0.862105	−	0.646579i
$776$	−6.00000			−0.215387
$777$			1.17157i			0.0420299i
$778$		−	30.3848i		−	1.08935i
$779$	42.6274			1.52729
$780$	0.585786	−	1.75736i	0.0209745	−	0.0629236i
$781$	24.0000			0.858788
$782$			6.82843i			0.244184i
$783$			3.65685i			0.130685i
$784$	3.00000			0.107143
$785$	44.6985	+	14.8995i	1.59536	+	0.531786i
$786$	−21.6569			−0.772474
$787$			22.7279i			0.810163i	0.914281	+	0.405081i	$0.132757\pi$
$787$			22.7279i			0.810163i	−0.914281	+	0.405081i	$0.867243\pi$
$788$			20.6274i			0.734821i
$789$	6.34315			0.225822
$790$	7.75736	+	2.58579i	0.275994	+	0.0919982i
$791$	33.9411			1.20681
$792$			4.24264i			0.150756i
$793$			0.485281i			0.0172328i
$794$	19.6569			0.697596
$795$	7.48528	−	22.4558i	0.265476	−	0.796427i
$796$	−14.0000			−0.496217
$797$			18.5858i			0.658342i	0.944270	+	0.329171i	$0.106769\pi$
$797$			18.5858i			0.658342i	−0.944270	+	0.329171i	$0.893231\pi$
$798$		−	12.4853i		−	0.441974i
$799$	5.65685			0.200125
$800$	−3.00000	+	4.00000i	−0.106066	+	0.141421i
$801$	9.17157			0.324062
$802$		−	4.97056i		−	0.175517i
$803$		−	32.4853i		−	1.14638i
$804$	3.41421			0.120410
$805$	−1.41421	+	4.24264i	−0.0498445	+	0.149533i
$806$	−4.97056			−0.175081
$807$			19.1716i			0.674871i
$808$			13.3137i			0.468375i
$809$	9.31371			0.327453			0.163726	−	0.986506i	$-0.447649\pi$
$809$	9.31371			0.327453			0.163726	+	0.986506i	$0.447649\pi$
$810$	2.12132	+	0.707107i	0.0745356	+	0.0248452i
$811$	1.17157			0.0411395			0.0205697	−	0.999788i	$-0.493452\pi$
$811$	1.17157			0.0411395			0.0205697	+	0.999788i	$0.493452\pi$
$812$			7.31371i			0.256661i
$813$			9.65685i			0.338681i
$814$	2.48528			0.0871091
$815$	−14.4853	−	4.82843i	−0.507397	−	0.169132i
$816$	6.82843			0.239043
$817$		−	63.9411i		−	2.23702i
$818$			18.0000i			0.629355i
$819$	−1.65685			−0.0578952
$820$	−4.82843	+	14.4853i	−0.168616	+	0.505848i
$821$	−29.3137			−1.02306			−0.511528	−	0.859267i	$-0.670920\pi$
$821$	−29.3137			−1.02306			−0.511528	+	0.859267i	$0.670920\pi$
$822$			8.48528i			0.295958i
$823$		−	24.2843i		−	0.846496i	−0.906014	−	0.423248i	$-0.860890\pi$
$823$		−	24.2843i		−	0.846496i	0.906014	−	0.423248i	$-0.139110\pi$
$824$	−16.1421			−0.562338
$825$	−12.7279	+	16.9706i	−0.443129	+	0.590839i
$826$	−16.9706			−0.590481
$827$		−	37.8995i		−	1.31789i	−0.752189	−	0.658947i	$-0.771002\pi$
$827$		−	37.8995i		−	1.31789i	0.752189	−	0.658947i	$-0.228998\pi$
$828$		−	1.00000i		−	0.0347524i
$829$	6.28427			0.218262			0.109131	−	0.994027i	$-0.465193\pi$
$829$	6.28427			0.218262			0.109131	+	0.994027i	$0.465193\pi$
$830$	−1.00000	+	3.00000i	−0.0347105	+	0.104132i
$831$	13.5147			0.468820
$832$			0.828427i			0.0287205i
$833$		−	20.4853i		−	0.709773i
$834$	16.9706			0.587643
$835$	−6.72792	−	2.24264i	−0.232829	−	0.0776098i
$836$	−26.4853			−0.916013
$837$		−	6.00000i		−	0.207390i
$838$		−	39.0711i		−	1.34969i
$839$	−33.9411			−1.17178			−0.585889	−	0.810391i	$-0.699255\pi$
$839$	−33.9411			−1.17178			−0.585889	+	0.810391i	$0.699255\pi$
$840$	4.24264	+	1.41421i	0.146385	+	0.0487950i
$841$	−15.6274			−0.538876
$842$		−	1.75736i		−	0.0605626i
$843$		−	1.85786i		−	0.0639882i
$844$	3.51472			0.120982
$845$	−8.70711	+	26.1213i	−0.299534	+	0.898601i
$846$	−0.828427			−0.0284819
$847$			14.0000i			0.481046i
$848$			10.5858i			0.363517i
$849$	−15.2132			−0.522116
$850$	27.3137	+	20.4853i	0.936852	+	0.702639i
$851$	−0.585786			−0.0200805
$852$			5.65685i			0.193801i
$853$		−	21.3137i		−	0.729767i	−0.931053	−	0.364884i	$-0.881109\pi$
$853$		−	21.3137i		−	0.729767i	0.931053	−	0.364884i	$-0.118891\pi$
$854$	−1.17157			−0.0400904
$855$	−4.41421	+	13.2426i	−0.150963	+	0.452889i
$856$	0.928932			0.0317502
$857$			50.8284i			1.73627i	0.496332	+	0.868133i	$0.334680\pi$
$857$			50.8284i			1.73627i	−0.496332	+	0.868133i	$0.665320\pi$
$858$			3.51472i			0.119991i
$859$	−24.2843			−0.828569			−0.414284	−	0.910148i	$-0.635968\pi$
$859$	−24.2843			−0.828569			−0.414284	+	0.910148i	$0.635968\pi$
$860$	21.7279	+	7.24264i	0.740916	+	0.246972i
$861$	13.6569			0.465424
$862$			8.48528i			0.289010i
$863$		−	22.6274i		−	0.770246i	−0.922865	−	0.385123i	$-0.874159\pi$
$863$		−	22.6274i		−	0.770246i	0.922865	−	0.385123i	$-0.125841\pi$
$864$	−1.00000			−0.0340207
$865$	−18.7279	−	6.24264i	−0.636768	−	0.212256i
$866$	−6.97056			−0.236869
$867$		−	29.6274i		−	1.00620i
$868$		−	12.0000i		−	0.407307i
$869$	−15.5147			−0.526301
$870$	2.58579	−	7.75736i	0.0876664	−	0.262999i
$871$	2.82843			0.0958376
$872$		−	12.5858i		−	0.426209i
$873$		−	6.00000i		−	0.203069i
$874$	6.24264			0.211160
$875$	12.7279	+	18.3848i	0.430282	+	0.621519i
$876$	7.65685			0.258701
$877$			16.8284i			0.568256i	0.958786	+	0.284128i	$0.0917040\pi$
$877$			16.8284i			0.568256i	−0.958786	+	0.284128i	$0.908296\pi$
$878$		0			0
$879$	−20.0416			−0.675987
$880$	3.00000	−	9.00000i	0.101130	−	0.303390i
$881$	26.1421			0.880751			0.440375	−	0.897814i	$-0.354845\pi$
$881$	26.1421			0.880751			0.440375	+	0.897814i	$0.354845\pi$
$882$			3.00000i			0.101015i
$883$			18.8284i			0.633627i	0.948488	+	0.316814i	$0.102613\pi$
$883$			18.8284i			0.633627i	−0.948488	+	0.316814i	$0.897387\pi$
$884$	5.65685			0.190261
$885$	18.0000	+	6.00000i	0.605063	+	0.201688i
$886$	−0.686292			−0.0230564
$887$		−	36.4264i		−	1.22308i	−0.791214	−	0.611540i	$-0.790551\pi$
$887$		−	36.4264i		−	1.22308i	0.791214	−	0.611540i	$-0.209449\pi$
$888$			0.585786i			0.0196577i
$889$	−4.00000			−0.134156
$890$	−19.4558	−	6.48528i	−0.652161	−	0.217387i
$891$	−4.24264			−0.142134
$892$		−	16.0000i		−	0.535720i
$893$		−	5.17157i		−	0.173060i
$894$	−12.7279			−0.425685
$895$	−15.3137	+	45.9411i	−0.511881	+	1.53564i
$896$	−2.00000			−0.0668153
$897$		−	0.828427i		−	0.0276604i
$898$		−	17.1716i		−	0.573023i
$899$	−21.9411			−0.731778
$900$	−4.00000	−	3.00000i	−0.133333	−	0.100000i
$901$	72.2843			2.40814
$902$		−	28.9706i		−	0.964614i
$903$		−	20.4853i		−	0.681707i
$904$	16.9706			0.564433
$905$	14.4142	−	43.2426i	0.479145	−	1.43743i
$906$	−4.34315			−0.144291
$907$			15.2132i			0.505146i	0.967578	+	0.252573i	$0.0812768\pi$
$907$			15.2132i			0.505146i	−0.967578	+	0.252573i	$0.918723\pi$
$908$		−	22.3848i		−	0.742865i
$909$	−13.3137			−0.441588
$910$	3.51472	+	1.17157i	0.116512	+	0.0388373i
$911$	−39.7990			−1.31860			−0.659300	−	0.751880i	$-0.729147\pi$
$911$	−39.7990			−1.31860			−0.659300	+	0.751880i	$0.729147\pi$
$912$		−	6.24264i		−	0.206714i
$913$		−	6.00000i		−	0.198571i
$914$	−34.9706			−1.15672
$915$	1.24264	+	0.414214i	0.0410804	+	0.0136935i
$916$	10.2426			0.338426
$917$		−	43.3137i		−	1.43034i
$918$			6.82843i			0.225372i
$919$	−38.9706			−1.28552			−0.642760	−	0.766068i	$-0.722211\pi$
$919$	−38.9706			−1.28552			−0.642760	+	0.766068i	$0.722211\pi$
$920$	−0.707107	+	2.12132i	−0.0233126	+	0.0699379i
$921$	25.4558			0.838799
$922$			7.17157i			0.236183i
$923$			4.68629i			0.154251i
$924$	−8.48528			−0.279145
$925$	−1.75736	+	2.34315i	−0.0577816	+	0.0770422i
$926$	−30.9706			−1.01776
$927$		−	16.1421i		−	0.530177i
$928$			3.65685i			0.120042i
$929$	−13.8579			−0.454662			−0.227331	−	0.973818i	$-0.573000\pi$
$929$	−13.8579			−0.454662			−0.227331	+	0.973818i	$0.573000\pi$
$930$	−4.24264	+	12.7279i	−0.139122	+	0.417365i
$931$	−18.7279			−0.613783
$932$		−	2.14214i		−	0.0701680i
$933$		−	21.7990i		−	0.713667i
$934$	31.3553			1.02598
$935$	−61.4558	−	20.4853i	−2.00982	−	0.669940i
$936$	−0.828427			−0.0270780
$937$			40.4264i			1.32067i	0.750970	+	0.660337i	$0.229586\pi$
$937$			40.4264i			1.32067i	−0.750970	+	0.660337i	$0.770414\pi$
$938$			6.82843i			0.222956i
$939$	−18.0000			−0.587408
$940$	1.75736	+	0.585786i	0.0573187	+	0.0191062i
$941$	−0.443651			−0.0144626			−0.00723130	−	0.999974i	$-0.502302\pi$
$941$	−0.443651			−0.0144626			−0.00723130	+	0.999974i	$0.502302\pi$
$942$		−	21.0711i		−	0.686532i
$943$			6.82843i			0.222364i
$944$	−8.48528			−0.276172
$945$	−1.41421	+	4.24264i	−0.0460044	+	0.138013i
$946$	−43.4558			−1.41287
$947$			49.4558i			1.60710i	0.595238	+	0.803549i	$0.297058\pi$
$947$			49.4558i			1.60710i	−0.595238	+	0.803549i	$0.702942\pi$
$948$		−	3.65685i		−	0.118769i
$949$	6.34315			0.205907
$950$	18.7279	−	24.9706i	0.607614	−	0.810152i
$951$	−14.4853			−0.469717
$952$			13.6569i			0.442621i
$953$		−	56.0833i		−	1.81672i	−0.418194	−	0.908358i	$-0.637337\pi$
$953$		−	56.0833i		−	1.81672i	0.418194	−	0.908358i	$-0.362663\pi$
$954$	−10.5858			−0.342727
$955$	7.65685	−	22.9706i	0.247770	−	0.743310i
$956$	−16.8284			−0.544270
$957$			15.5147i			0.501520i
$958$			27.3137i			0.882466i
$959$	−16.9706			−0.548008
$960$	2.12132	+	0.707107i	0.0684653	+	0.0228218i
$961$	5.00000			0.161290
$962$			0.485281i			0.0156461i
$963$			0.928932i			0.0299344i
$964$	−6.48528			−0.208877
$965$	44.4853	+	14.8284i	1.43203	+	0.477344i
$966$	2.00000			0.0643489
$967$			28.9706i			0.931630i	0.884882	+	0.465815i	$0.154239\pi$
$967$			28.9706i			0.931630i	−0.884882	+	0.465815i	$0.845761\pi$
$968$			7.00000i			0.224989i
$969$	−42.6274			−1.36939
$970$	−4.24264	+	12.7279i	−0.136223	+	0.408669i
$971$	−54.3848			−1.74529			−0.872645	−	0.488355i	$-0.837597\pi$
$971$	−54.3848			−1.74529			−0.872645	+	0.488355i	$0.837597\pi$
$972$		−	1.00000i		−	0.0320750i
$973$			33.9411i			1.08810i
$974$	−39.9411			−1.27980
$975$	−3.31371	−	2.48528i	−0.106124	−	0.0795927i
$976$	−0.585786			−0.0187506
$977$			19.7990i			0.633426i	0.948521	+	0.316713i	$0.102579\pi$
$977$			19.7990i			0.633426i	−0.948521	+	0.316713i	$0.897421\pi$
$978$			6.82843i			0.218349i
$979$	38.9117			1.24362
$980$	2.12132	−	6.36396i	0.0677631	−	0.203289i
$981$	12.5858			0.401833
$982$			18.6274i			0.594425i
$983$		−	3.51472i		−	0.112102i	−0.998428	−	0.0560511i	$-0.982149\pi$
$983$		−	3.51472i		−	0.112102i	0.998428	−	0.0560511i	$-0.0178510\pi$
$984$	6.82843			0.217682
$985$	43.7574	+	14.5858i	1.39423	+	0.464742i
$986$	24.9706			0.795225
$987$		−	1.65685i		−	0.0527383i
$988$		−	5.17157i		−	0.164530i
$989$	10.2426			0.325697
$990$	9.00000	+	3.00000i	0.286039	+	0.0953463i
$991$	−15.9411			−0.506387			−0.253193	−	0.967416i	$-0.581481\pi$
$991$	−15.9411			−0.506387			−0.253193	+	0.967416i	$0.581481\pi$
$992$		−	6.00000i		−	0.190500i
$993$		−	20.4853i		−	0.650081i
$994$	−11.3137			−0.358849
$995$	−9.89949	+	29.6985i	−0.313835	+	0.941505i
$996$	1.41421			0.0448111
$997$			13.0294i			0.412646i	0.978484	+	0.206323i	$0.0661498\pi$
$997$			13.0294i			0.412646i	−0.978484	+	0.206323i	$0.933850\pi$
$998$		−	37.4558i		−	1.18564i
$999$	−0.585786			−0.0185335

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	690.2.d.b.139.1	✓	4
3.2	odd	2		2070.2.d.a.829.4		4
5.2	odd	4		3450.2.a.bk.1.1		2
5.3	odd	4		3450.2.a.bg.1.1		2
5.4	even	2	inner	690.2.d.b.139.3	yes	4
15.14	odd	2		2070.2.d.a.829.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
690.2.d.b.139.1	✓	4	1.1	even	1	trivial
690.2.d.b.139.3	yes	4	5.4	even	2	inner
2070.2.d.a.829.2		4	15.14	odd	2
2070.2.d.a.829.4		4	3.2	odd	2
3450.2.a.bg.1.1		2	5.3	odd	4
3450.2.a.bk.1.1		2	5.2	odd	4

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)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +(-0.707107 + 2.12132i) q^{5} +1.00000 q^{6} +2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +(-0.707107 + 2.12132i) q^{5} +1.00000 q^{6} +2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +(2.12132 + 0.707107i) q^{10} -4.24264 q^{11} -1.00000i q^{12} -0.828427i q^{13} +2.00000 q^{14} +(-2.12132 - 0.707107i) q^{15} +1.00000 q^{16} -6.82843i q^{17} +1.00000i q^{18} -6.24264 q^{19} +(0.707107 - 2.12132i) q^{20} -2.00000 q^{21} +4.24264i q^{22} -1.00000i q^{23} -1.00000 q^{24} +(-4.00000 - 3.00000i) q^{25} -0.828427 q^{26} -1.00000i q^{27} -2.00000i q^{28} -3.65685 q^{29} +(-0.707107 + 2.12132i) q^{30} +6.00000 q^{31} -1.00000i q^{32} -4.24264i q^{33} -6.82843 q^{34} +(-4.24264 - 1.41421i) q^{35} +1.00000 q^{36} -0.585786i q^{37} +6.24264i q^{38} +0.828427 q^{39} +(-2.12132 - 0.707107i) q^{40} -6.82843 q^{41} +2.00000i q^{42} +10.2426i q^{43} +4.24264 q^{44} +(0.707107 - 2.12132i) q^{45} -1.00000 q^{46} +0.828427i q^{47} +1.00000i q^{48} +3.00000 q^{49} +(-3.00000 + 4.00000i) q^{50} +6.82843 q^{51} +0.828427i q^{52} +10.5858i q^{53} -1.00000 q^{54} +(3.00000 - 9.00000i) q^{55} -2.00000 q^{56} -6.24264i q^{57} +3.65685i q^{58} -8.48528 q^{59} +(2.12132 + 0.707107i) q^{60} -0.585786 q^{61} -6.00000i q^{62} -2.00000i q^{63} -1.00000 q^{64} +(1.75736 + 0.585786i) q^{65} -4.24264 q^{66} +3.41421i q^{67} +6.82843i q^{68} +1.00000 q^{69} +(-1.41421 + 4.24264i) q^{70} -5.65685 q^{71} -1.00000i q^{72} +7.65685i q^{73} -0.585786 q^{74} +(3.00000 - 4.00000i) q^{75} +6.24264 q^{76} -8.48528i q^{77} -0.828427i q^{78} +3.65685 q^{79} +(-0.707107 + 2.12132i) q^{80} +1.00000 q^{81} +6.82843i q^{82} +1.41421i q^{83} +2.00000 q^{84} +(14.4853 + 4.82843i) q^{85} +10.2426 q^{86} -3.65685i q^{87} -4.24264i q^{88} -9.17157 q^{89} +(-2.12132 - 0.707107i) q^{90} +1.65685 q^{91} +1.00000i q^{92} +6.00000i q^{93} +0.828427 q^{94} +(4.41421 - 13.2426i) q^{95} +1.00000 q^{96} +6.00000i q^{97} -3.00000i q^{98} +4.24264 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4} + 4 q^{6} - 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^4 + 4 * q^6 - 4 * q^9	\( 4 q - 4 q^{4} + 4 q^{6} - 4 q^{9} + 8 q^{14} + 4 q^{16} - 8 q^{19} - 8 q^{21} - 4 q^{24} - 16 q^{25} + 8 q^{26} + 8 q^{29} + 24 q^{31} - 16 q^{34} + 4 q^{36} - 8 q^{39} - 16 q^{41} - 4 q^{46} + 12 q^{49} - 12 q^{50} + 16 q^{51} - 4 q^{54} + 12 q^{55} - 8 q^{56} - 8 q^{61} - 4 q^{64} + 24 q^{65} + 4 q^{69} - 8 q^{74} + 12 q^{75} + 8 q^{76} - 8 q^{79} + 4 q^{81} + 8 q^{84} + 24 q^{85} + 24 q^{86} - 48 q^{89} - 16 q^{91} - 8 q^{94} + 12 q^{95} + 4 q^{96}+O(q^{100}) \) 4 * q - 4 * q^4 + 4 * q^6 - 4 * q^9 + 8 * q^14 + 4 * q^16 - 8 * q^19 - 8 * q^21 - 4 * q^24 - 16 * q^25 + 8 * q^26 + 8 * q^29 + 24 * q^31 - 16 * q^34 + 4 * q^36 - 8 * q^39 - 16 * q^41 - 4 * q^46 + 12 * q^49 - 12 * q^50 + 16 * q^51 - 4 * q^54 + 12 * q^55 - 8 * q^56 - 8 * q^61 - 4 * q^64 + 24 * q^65 + 4 * q^69 - 8 * q^74 + 12 * q^75 + 8 * q^76 - 8 * q^79 + 4 * q^81 + 8 * q^84 + 24 * q^85 + 24 * q^86 - 48 * q^89 - 16 * q^91 - 8 * q^94 + 12 * q^95 + 4 * q^96

Introduction

L-functions

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