LMFDB - Embedded newform 420.2.l.a.239.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [420,2,Mod(239,420)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(420, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 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("420.239");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 420 = 2^{2} \cdot 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	420.l (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:	$3.35371688489$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			239.3
Root			$-0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	420.239
Dual form			420.2.l.a.239.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421i q^{2} +(-1.00000 - 1.41421i) q^{3} -2.00000 q^{4} +(-2.12132 - 0.707107i) q^{5} +(2.00000 - 1.41421i) q^{6} +1.00000 q^{7} -2.82843i q^{8} +(-1.00000 + 2.82843i) q^{9} +O(q^{10})$	\(q+1.41421i q^{2} +(-1.00000 - 1.41421i) q^{3} -2.00000 q^{4} +(-2.12132 - 0.707107i) q^{5} +(2.00000 - 1.41421i) q^{6} +1.00000 q^{7} -2.82843i q^{8} +(-1.00000 + 2.82843i) q^{9} +(1.00000 - 3.00000i) q^{10} +4.24264 q^{11} +(2.00000 + 2.82843i) q^{12} +6.00000i q^{13} +1.41421i q^{14} +(1.12132 + 3.70711i) q^{15} +4.00000 q^{16} -4.24264 q^{17} +(-4.00000 - 1.41421i) q^{18} +6.00000i q^{19} +(4.24264 + 1.41421i) q^{20} +(-1.00000 - 1.41421i) q^{21} +6.00000i q^{22} -1.41421i q^{23} +(-4.00000 + 2.82843i) q^{24} +(4.00000 + 3.00000i) q^{25} -8.48528 q^{26} +(5.00000 - 1.41421i) q^{27} -2.00000 q^{28} +2.82843i q^{29} +(-5.24264 + 1.58579i) q^{30} +5.65685i q^{32} +(-4.24264 - 6.00000i) q^{33} -6.00000i q^{34} +(-2.12132 - 0.707107i) q^{35} +(2.00000 - 5.65685i) q^{36} +6.00000i q^{37} -8.48528 q^{38} +(8.48528 - 6.00000i) q^{39} +(-2.00000 + 6.00000i) q^{40} -1.41421i q^{41} +(2.00000 - 1.41421i) q^{42} +8.00000 q^{43} -8.48528 q^{44} +(4.12132 - 5.29289i) q^{45} +2.00000 q^{46} +2.82843i q^{47} +(-4.00000 - 5.65685i) q^{48} +1.00000 q^{49} +(-4.24264 + 5.65685i) q^{50} +(4.24264 + 6.00000i) q^{51} -12.0000i q^{52} +8.48528 q^{53} +(2.00000 + 7.07107i) q^{54} +(-9.00000 - 3.00000i) q^{55} -2.82843i q^{56} +(8.48528 - 6.00000i) q^{57} -4.00000 q^{58} +(-2.24264 - 7.41421i) q^{60} -10.0000 q^{61} +(-1.00000 + 2.82843i) q^{63} -8.00000 q^{64} +(4.24264 - 12.7279i) q^{65} +(8.48528 - 6.00000i) q^{66} -4.00000 q^{67} +8.48528 q^{68} +(-2.00000 + 1.41421i) q^{69} +(1.00000 - 3.00000i) q^{70} -12.7279 q^{71} +(8.00000 + 2.82843i) q^{72} -6.00000i q^{73} -8.48528 q^{74} +(0.242641 - 8.65685i) q^{75} -12.0000i q^{76} +4.24264 q^{77} +(8.48528 + 12.0000i) q^{78} +(-8.48528 - 2.82843i) q^{80} +(-7.00000 - 5.65685i) q^{81} +2.00000 q^{82} +2.82843i q^{83} +(2.00000 + 2.82843i) q^{84} +(9.00000 + 3.00000i) q^{85} +11.3137i q^{86} +(4.00000 - 2.82843i) q^{87} -12.0000i q^{88} +7.07107i q^{89} +(7.48528 + 5.82843i) q^{90} +6.00000i q^{91} +2.82843i q^{92} -4.00000 q^{94} +(4.24264 - 12.7279i) q^{95} +(8.00000 - 5.65685i) q^{96} -6.00000i q^{97} +1.41421i q^{98} +(-4.24264 + 12.0000i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} - 8 q^{4} + 8 q^{6} + 4 q^{7} - 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^3 - 8 * q^4 + 8 * q^6 + 4 * q^7 - 4 * q^9	$ 4 q - 4 q^{3} - 8 q^{4} + 8 q^{6} + 4 q^{7} - 4 q^{9} + 4 q^{10} + 8 q^{12} - 4 q^{15} + 16 q^{16} - 16 q^{18} - 4 q^{21} - 16 q^{24} + 16 q^{25} + 20 q^{27} - 8 q^{28} - 4 q^{30} + 8 q^{36} - 8 q^{40} + 8 q^{42} + 32 q^{43} + 8 q^{45} + 8 q^{46} - 16 q^{48} + 4 q^{49} + 8 q^{54} - 36 q^{55} - 16 q^{58} + 8 q^{60} - 40 q^{61} - 4 q^{63} - 32 q^{64} - 16 q^{67} - 8 q^{69} + 4 q^{70} + 32 q^{72} - 16 q^{75} - 28 q^{81} + 8 q^{82} + 8 q^{84} + 36 q^{85} + 16 q^{87} - 4 q^{90} - 16 q^{94} + 32 q^{96}+O(q^{100}) $ 4 * q - 4 * q^3 - 8 * q^4 + 8 * q^6 + 4 * q^7 - 4 * q^9 + 4 * q^10 + 8 * q^12 - 4 * q^15 + 16 * q^16 - 16 * q^18 - 4 * q^21 - 16 * q^24 + 16 * q^25 + 20 * q^27 - 8 * q^28 - 4 * q^30 + 8 * q^36 - 8 * q^40 + 8 * q^42 + 32 * q^43 + 8 * q^45 + 8 * q^46 - 16 * q^48 + 4 * q^49 + 8 * q^54 - 36 * q^55 - 16 * q^58 + 8 * q^60 - 40 * q^61 - 4 * q^63 - 32 * q^64 - 16 * q^67 - 8 * q^69 + 4 * q^70 + 32 * q^72 - 16 * q^75 - 28 * q^81 + 8 * q^82 + 8 * q^84 + 36 * q^85 + 16 * q^87 - 4 * q^90 - 16 * q^94 + 32 * q^96

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/420\mathbb{Z}\right)^\times$.

$n$	$211$	$241$	$281$	$337$
$\chi(n)$	$-1$	$1$	$-1$	$-1$

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 $

$2$			1.41421i			1.00000i
$2$			1.41421i			1.00000i
$3$	−1.00000	−	1.41421i	−0.577350	−	0.816497i
$3$	−1.00000	−	1.41421i	−0.577350	−	0.816497i
$4$	−2.00000			−1.00000
$5$	−2.12132	−	0.707107i	−0.948683	−	0.316228i
$5$	−2.12132	−	0.707107i	−0.948683	−	0.316228i
$6$	2.00000	−	1.41421i	0.816497	−	0.577350i
$7$	1.00000			0.377964
$7$	1.00000			0.377964
$8$		−	2.82843i		−	1.00000i
$9$	−1.00000	+	2.82843i	−0.333333	+	0.942809i
$10$	1.00000	−	3.00000i	0.316228	−	0.948683i
$11$	4.24264			1.27920			0.639602	−	0.768706i	$-0.279099\pi$
$11$	4.24264			1.27920			0.639602	+	0.768706i	$0.279099\pi$
$12$	2.00000	+	2.82843i	0.577350	+	0.816497i
$13$			6.00000i			1.66410i	0.554700	+	0.832050i	$0.312833\pi$
$13$			6.00000i			1.66410i	−0.554700	+	0.832050i	$0.687167\pi$
$14$			1.41421i			0.377964i
$15$	1.12132	+	3.70711i	0.289524	+	0.957171i
$16$	4.00000			1.00000
$17$	−4.24264			−1.02899			−0.514496	−	0.857493i	$-0.672021\pi$
$17$	−4.24264			−1.02899			−0.514496	+	0.857493i	$0.672021\pi$
$18$	−4.00000	−	1.41421i	−0.942809	−	0.333333i
$19$			6.00000i			1.37649i	0.725476	+	0.688247i	$0.241620\pi$
$19$			6.00000i			1.37649i	−0.725476	+	0.688247i	$0.758380\pi$
$20$	4.24264	+	1.41421i	0.948683	+	0.316228i
$21$	−1.00000	−	1.41421i	−0.218218	−	0.308607i
$22$			6.00000i			1.27920i
$23$		−	1.41421i		−	0.294884i	−0.989071	−	0.147442i	$-0.952896\pi$
$23$		−	1.41421i		−	0.294884i	0.989071	−	0.147442i	$-0.0471040\pi$
$24$	−4.00000	+	2.82843i	−0.816497	+	0.577350i
$25$	4.00000	+	3.00000i	0.800000	+	0.600000i
$26$	−8.48528			−1.66410
$27$	5.00000	−	1.41421i	0.962250	−	0.272166i
$28$	−2.00000			−0.377964
$29$			2.82843i			0.525226i	0.964901	+	0.262613i	$0.0845842\pi$
$29$			2.82843i			0.525226i	−0.964901	+	0.262613i	$0.915416\pi$
$30$	−5.24264	+	1.58579i	−0.957171	+	0.289524i
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$			5.65685i			1.00000i
$33$	−4.24264	−	6.00000i	−0.738549	−	1.04447i
$34$		−	6.00000i		−	1.02899i
$35$	−2.12132	−	0.707107i	−0.358569	−	0.119523i
$36$	2.00000	−	5.65685i	0.333333	−	0.942809i
$37$			6.00000i			0.986394i	0.869918	+	0.493197i	$0.164172\pi$
$37$			6.00000i			0.986394i	−0.869918	+	0.493197i	$0.835828\pi$
$38$	−8.48528			−1.37649
$39$	8.48528	−	6.00000i	1.35873	−	0.960769i
$40$	−2.00000	+	6.00000i	−0.316228	+	0.948683i
$41$		−	1.41421i		−	0.220863i	−0.993884	−	0.110432i	$-0.964777\pi$
$41$		−	1.41421i		−	0.220863i	0.993884	−	0.110432i	$-0.0352233\pi$
$42$	2.00000	−	1.41421i	0.308607	−	0.218218i
$43$	8.00000			1.21999			0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000			1.21999			0.609994	+	0.792406i	$0.291172\pi$
$44$	−8.48528			−1.27920
$45$	4.12132	−	5.29289i	0.614370	−	0.789018i
$46$	2.00000			0.294884
$47$			2.82843i			0.412568i	0.978492	+	0.206284i	$0.0661372\pi$
$47$			2.82843i			0.412568i	−0.978492	+	0.206284i	$0.933863\pi$
$48$	−4.00000	−	5.65685i	−0.577350	−	0.816497i
$49$	1.00000			0.142857
$50$	−4.24264	+	5.65685i	−0.600000	+	0.800000i
$51$	4.24264	+	6.00000i	0.594089	+	0.840168i
$52$		−	12.0000i		−	1.66410i
$53$	8.48528			1.16554			0.582772	−	0.812636i	$-0.301968\pi$
$53$	8.48528			1.16554			0.582772	+	0.812636i	$0.301968\pi$
$54$	2.00000	+	7.07107i	0.272166	+	0.962250i
$55$	−9.00000	−	3.00000i	−1.21356	−	0.404520i
$56$		−	2.82843i		−	0.377964i
$57$	8.48528	−	6.00000i	1.12390	−	0.794719i
$58$	−4.00000			−0.525226
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$	−2.24264	−	7.41421i	−0.289524	−	0.957171i
$61$	−10.0000			−1.28037			−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000			−1.28037			−0.640184	+	0.768221i	$0.721142\pi$
$62$		0			0
$63$	−1.00000	+	2.82843i	−0.125988	+	0.356348i
$64$	−8.00000			−1.00000
$65$	4.24264	−	12.7279i	0.526235	−	1.57870i
$66$	8.48528	−	6.00000i	1.04447	−	0.738549i
$67$	−4.00000			−0.488678			−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000			−0.488678			−0.244339	+	0.969690i	$0.578571\pi$
$68$	8.48528			1.02899
$69$	−2.00000	+	1.41421i	−0.240772	+	0.170251i
$70$	1.00000	−	3.00000i	0.119523	−	0.358569i
$71$	−12.7279			−1.51053			−0.755263	−	0.655422i	$-0.772491\pi$
$71$	−12.7279			−1.51053			−0.755263	+	0.655422i	$0.772491\pi$
$72$	8.00000	+	2.82843i	0.942809	+	0.333333i
$73$		−	6.00000i		−	0.702247i	−0.936329	−	0.351123i	$-0.885800\pi$
$73$		−	6.00000i		−	0.702247i	0.936329	−	0.351123i	$-0.114200\pi$
$74$	−8.48528			−0.986394
$75$	0.242641	−	8.65685i	0.0280177	−	0.999607i
$76$		−	12.0000i		−	1.37649i
$77$	4.24264			0.483494
$78$	8.48528	+	12.0000i	0.960769	+	1.35873i
$79$		0			0		1.00000			$0$
$79$		0			0		−1.00000			$\pi$
$80$	−8.48528	−	2.82843i	−0.948683	−	0.316228i
$81$	−7.00000	−	5.65685i	−0.777778	−	0.628539i
$82$	2.00000			0.220863
$83$			2.82843i			0.310460i	0.987878	+	0.155230i	$0.0496119\pi$
$83$			2.82843i			0.310460i	−0.987878	+	0.155230i	$0.950388\pi$
$84$	2.00000	+	2.82843i	0.218218	+	0.308607i
$85$	9.00000	+	3.00000i	0.976187	+	0.325396i
$86$			11.3137i			1.21999i
$87$	4.00000	−	2.82843i	0.428845	−	0.303239i
$88$		−	12.0000i		−	1.27920i
$89$			7.07107i			0.749532i	0.927119	+	0.374766i	$0.122277\pi$
$89$			7.07107i			0.749532i	−0.927119	+	0.374766i	$0.877723\pi$
$90$	7.48528	+	5.82843i	0.789018	+	0.614370i
$91$			6.00000i			0.628971i
$92$			2.82843i			0.294884i
$93$		0			0
$94$	−4.00000			−0.412568
$95$	4.24264	−	12.7279i	0.435286	−	1.30586i
$96$	8.00000	−	5.65685i	0.816497	−	0.577350i
$97$		−	6.00000i		−	0.609208i	−0.952479	−	0.304604i	$-0.901476\pi$
$97$		−	6.00000i		−	0.609208i	0.952479	−	0.304604i	$-0.0985241\pi$
$98$			1.41421i			0.142857i
$99$	−4.24264	+	12.0000i	−0.426401	+	1.20605i
$100$	−8.00000	−	6.00000i	−0.800000	−	0.600000i
$101$			15.5563i			1.54791i	0.633238	+	0.773957i	$0.281726\pi$
$101$			15.5563i			1.54791i	−0.633238	+	0.773957i	$0.718274\pi$
$102$	−8.48528	+	6.00000i	−0.840168	+	0.594089i
$103$	14.0000			1.37946			0.689730	−	0.724066i	$-0.257729\pi$
$103$	14.0000			1.37946			0.689730	+	0.724066i	$0.257729\pi$
$104$	16.9706			1.66410
$105$	1.12132	+	3.70711i	0.109430	+	0.361777i
$106$			12.0000i			1.16554i
$107$			7.07107i			0.683586i	0.939775	+	0.341793i	$0.111034\pi$
$107$			7.07107i			0.683586i	−0.939775	+	0.341793i	$0.888966\pi$
$108$	−10.0000	+	2.82843i	−0.962250	+	0.272166i
$109$	2.00000			0.191565			0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000			0.191565			0.0957826	+	0.995402i	$0.469465\pi$
$110$	4.24264	−	12.7279i	0.404520	−	1.21356i
$111$	8.48528	−	6.00000i	0.805387	−	0.569495i
$112$	4.00000			0.377964
$113$	−8.48528			−0.798228			−0.399114	−	0.916901i	$-0.630682\pi$
$113$	−8.48528			−0.798228			−0.399114	+	0.916901i	$0.630682\pi$
$114$	8.48528	+	12.0000i	0.794719	+	1.12390i
$115$	−1.00000	+	3.00000i	−0.0932505	+	0.279751i
$116$		−	5.65685i		−	0.525226i
$117$	−16.9706	−	6.00000i	−1.56893	−	0.554700i
$118$		0			0
$119$	−4.24264			−0.388922
$120$	10.4853	−	3.17157i	0.957171	−	0.289524i
$121$	7.00000			0.636364
$122$		−	14.1421i		−	1.28037i
$123$	−2.00000	+	1.41421i	−0.180334	+	0.127515i
$124$		0			0
$125$	−6.36396	−	9.19239i	−0.569210	−	0.822192i
$126$	−4.00000	−	1.41421i	−0.356348	−	0.125988i
$127$	−16.0000			−1.41977			−0.709885	−	0.704317i	$-0.751253\pi$
$127$	−16.0000			−1.41977			−0.709885	+	0.704317i	$0.751253\pi$
$128$		−	11.3137i		−	1.00000i
$129$	−8.00000	−	11.3137i	−0.704361	−	0.996116i
$130$	18.0000	+	6.00000i	1.57870	+	0.526235i
$131$	8.48528			0.741362			0.370681	−	0.928760i	$-0.379124\pi$
$131$	8.48528			0.741362			0.370681	+	0.928760i	$0.379124\pi$
$132$	8.48528	+	12.0000i	0.738549	+	1.04447i
$133$			6.00000i			0.520266i
$134$		−	5.65685i		−	0.488678i
$135$	−11.6066	−	0.535534i	−0.998937	−	0.0460914i
$136$			12.0000i			1.02899i
$137$	16.9706			1.44989			0.724947	−	0.688805i	$-0.241864\pi$
$137$	16.9706			1.44989			0.724947	+	0.688805i	$0.241864\pi$
$138$	−2.00000	−	2.82843i	−0.170251	−	0.240772i
$139$		−	12.0000i		−	1.01783i	−0.860818	−	0.508913i	$-0.830047\pi$
$139$		−	12.0000i		−	1.01783i	0.860818	−	0.508913i	$-0.169953\pi$
$140$	4.24264	+	1.41421i	0.358569	+	0.119523i
$141$	4.00000	−	2.82843i	0.336861	−	0.238197i
$142$		−	18.0000i		−	1.51053i
$143$			25.4558i			2.12872i
$144$	−4.00000	+	11.3137i	−0.333333	+	0.942809i
$145$	2.00000	−	6.00000i	0.166091	−	0.498273i
$146$	8.48528			0.702247
$147$	−1.00000	−	1.41421i	−0.0824786	−	0.116642i
$148$		−	12.0000i		−	0.986394i
$149$			11.3137i			0.926855i	0.886135	+	0.463428i	$0.153381\pi$
$149$			11.3137i			0.926855i	−0.886135	+	0.463428i	$0.846619\pi$
$150$	12.2426	+	0.343146i	0.999607	+	0.0280177i
$151$		−	12.0000i		−	0.976546i	−0.872691	−	0.488273i	$-0.837627\pi$
$151$		−	12.0000i		−	0.976546i	0.872691	−	0.488273i	$-0.162373\pi$
$152$	16.9706			1.37649
$153$	4.24264	−	12.0000i	0.342997	−	0.970143i
$154$			6.00000i			0.483494i
$155$		0			0
$156$	−16.9706	+	12.0000i	−1.35873	+	0.960769i
$157$			18.0000i			1.43656i	0.695756	+	0.718278i	$0.255069\pi$
$157$			18.0000i			1.43656i	−0.695756	+	0.718278i	$0.744931\pi$
$158$		0			0
$159$	−8.48528	−	12.0000i	−0.672927	−	0.951662i
$160$	4.00000	−	12.0000i	0.316228	−	0.948683i
$161$		−	1.41421i		−	0.111456i
$162$	8.00000	−	9.89949i	0.628539	−	0.777778i
$163$	20.0000			1.56652			0.783260	−	0.621694i	$-0.213555\pi$
$163$	20.0000			1.56652			0.783260	+	0.621694i	$0.213555\pi$
$164$			2.82843i			0.220863i
$165$	4.75736	+	15.7279i	0.370360	+	1.22442i
$166$	−4.00000			−0.310460
$167$		−	14.1421i		−	1.09435i	−0.837018	−	0.547176i	$-0.815703\pi$
$167$		−	14.1421i		−	1.09435i	0.837018	−	0.547176i	$-0.184297\pi$
$168$	−4.00000	+	2.82843i	−0.308607	+	0.218218i
$169$	−23.0000			−1.76923
$170$	−4.24264	+	12.7279i	−0.325396	+	0.976187i
$171$	−16.9706	−	6.00000i	−1.29777	−	0.458831i
$172$	−16.0000			−1.21999
$173$	−12.7279			−0.967686			−0.483843	−	0.875155i	$-0.660759\pi$
$173$	−12.7279			−0.967686			−0.483843	+	0.875155i	$0.660759\pi$
$174$	4.00000	+	5.65685i	0.303239	+	0.428845i
$175$	4.00000	+	3.00000i	0.302372	+	0.226779i
$176$	16.9706			1.27920
$177$		0			0
$178$	−10.0000			−0.749532
$179$	−4.24264			−0.317110			−0.158555	−	0.987350i	$-0.550683\pi$
$179$	−4.24264			−0.317110			−0.158555	+	0.987350i	$0.550683\pi$
$180$	−8.24264	+	10.5858i	−0.614370	+	0.789018i
$181$	2.00000			0.148659			0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000			0.148659			0.0743294	+	0.997234i	$0.476318\pi$
$182$	−8.48528			−0.628971
$183$	10.0000	+	14.1421i	0.739221	+	1.04542i
$184$	−4.00000			−0.294884
$185$	4.24264	−	12.7279i	0.311925	−	0.935775i
$186$		0			0
$187$	−18.0000			−1.31629
$188$		−	5.65685i		−	0.412568i
$189$	5.00000	−	1.41421i	0.363696	−	0.102869i
$190$	18.0000	+	6.00000i	1.30586	+	0.435286i
$191$	−12.7279			−0.920960			−0.460480	−	0.887670i	$-0.652323\pi$
$191$	−12.7279			−0.920960			−0.460480	+	0.887670i	$0.652323\pi$
$192$	8.00000	+	11.3137i	0.577350	+	0.816497i
$193$			18.0000i			1.29567i	0.761781	+	0.647834i	$0.224325\pi$
$193$			18.0000i			1.29567i	−0.761781	+	0.647834i	$0.775675\pi$
$194$	8.48528			0.609208
$195$	−22.2426	+	6.72792i	−1.59283	+	0.481797i
$196$	−2.00000			−0.142857
$197$	−25.4558			−1.81365			−0.906827	−	0.421503i	$-0.861503\pi$
$197$	−25.4558			−1.81365			−0.906827	+	0.421503i	$0.861503\pi$
$198$	−16.9706	−	6.00000i	−1.20605	−	0.426401i
$199$		−	6.00000i		−	0.425329i	−0.977125	−	0.212664i	$-0.931786\pi$
$199$		−	6.00000i		−	0.425329i	0.977125	−	0.212664i	$-0.0682141\pi$
$200$	8.48528	−	11.3137i	0.600000	−	0.800000i
$201$	4.00000	+	5.65685i	0.282138	+	0.399004i
$202$	−22.0000			−1.54791
$203$			2.82843i			0.198517i
$204$	−8.48528	−	12.0000i	−0.594089	−	0.840168i
$205$	−1.00000	+	3.00000i	−0.0698430	+	0.209529i
$206$			19.7990i			1.37946i
$207$	4.00000	+	1.41421i	0.278019	+	0.0982946i
$208$			24.0000i			1.66410i
$209$			25.4558i			1.76082i
$210$	−5.24264	+	1.58579i	−0.361777	+	0.109430i
$211$		−	12.0000i		−	0.826114i	−0.910705	−	0.413057i	$-0.864461\pi$
$211$		−	12.0000i		−	0.826114i	0.910705	−	0.413057i	$-0.135539\pi$
$212$	−16.9706			−1.16554
$213$	12.7279	+	18.0000i	0.872103	+	1.23334i
$214$	−10.0000			−0.683586
$215$	−16.9706	−	5.65685i	−1.15738	−	0.385794i
$216$	−4.00000	−	14.1421i	−0.272166	−	0.962250i
$217$		0			0
$218$			2.82843i			0.191565i
$219$	−8.48528	+	6.00000i	−0.573382	+	0.405442i
$220$	18.0000	+	6.00000i	1.21356	+	0.404520i
$221$		−	25.4558i		−	1.71235i
$222$	8.48528	+	12.0000i	0.569495	+	0.805387i
$223$	8.00000			0.535720			0.267860	−	0.963458i	$-0.413684\pi$
$223$	8.00000			0.535720			0.267860	+	0.963458i	$0.413684\pi$
$224$			5.65685i			0.377964i
$225$	−12.4853	+	8.31371i	−0.832352	+	0.554247i
$226$		−	12.0000i		−	0.798228i
$227$		−	22.6274i		−	1.50183i	−0.660396	−	0.750917i	$-0.729612\pi$
$227$		−	22.6274i		−	1.50183i	0.660396	−	0.750917i	$-0.270388\pi$
$228$	−16.9706	+	12.0000i	−1.12390	+	0.794719i
$229$	14.0000			0.925146			0.462573	−	0.886581i	$-0.346926\pi$
$229$	14.0000			0.925146			0.462573	+	0.886581i	$0.346926\pi$
$230$	−4.24264	−	1.41421i	−0.279751	−	0.0932505i
$231$	−4.24264	−	6.00000i	−0.279145	−	0.394771i
$232$	8.00000			0.525226
$233$	16.9706			1.11178			0.555889	−	0.831256i	$-0.312378\pi$
$233$	16.9706			1.11178			0.555889	+	0.831256i	$0.312378\pi$
$234$	8.48528	−	24.0000i	0.554700	−	1.56893i
$235$	2.00000	−	6.00000i	0.130466	−	0.391397i
$236$		0			0
$237$		0			0
$238$		−	6.00000i		−	0.388922i
$239$	4.24264			0.274434			0.137217	−	0.990541i	$-0.456184\pi$
$239$	4.24264			0.274434			0.137217	+	0.990541i	$0.456184\pi$
$240$	4.48528	+	14.8284i	0.289524	+	0.957171i
$241$	26.0000			1.67481			0.837404	−	0.546585i	$-0.184072\pi$
$241$	26.0000			1.67481			0.837404	+	0.546585i	$0.184072\pi$
$242$			9.89949i			0.636364i
$243$	−1.00000	+	15.5563i	−0.0641500	+	0.997940i
$244$	20.0000			1.28037
$245$	−2.12132	−	0.707107i	−0.135526	−	0.0451754i
$246$	−2.00000	−	2.82843i	−0.127515	−	0.180334i
$247$	−36.0000			−2.29063
$248$		0			0
$249$	4.00000	−	2.82843i	0.253490	−	0.179244i
$250$	13.0000	−	9.00000i	0.822192	−	0.569210i
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$	2.00000	−	5.65685i	0.125988	−	0.356348i
$253$		−	6.00000i		−	0.377217i
$254$		−	22.6274i		−	1.41977i
$255$	−4.75736	−	15.7279i	−0.297917	−	0.984921i
$256$	16.0000			1.00000
$257$	−4.24264			−0.264649			−0.132324	−	0.991206i	$-0.542244\pi$
$257$	−4.24264			−0.264649			−0.132324	+	0.991206i	$0.542244\pi$
$258$	16.0000	−	11.3137i	0.996116	−	0.704361i
$259$			6.00000i			0.372822i
$260$	−8.48528	+	25.4558i	−0.526235	+	1.57870i
$261$	−8.00000	−	2.82843i	−0.495188	−	0.175075i
$262$			12.0000i			0.741362i
$263$			15.5563i			0.959246i	0.877475	+	0.479623i	$0.159226\pi$
$263$			15.5563i			0.959246i	−0.877475	+	0.479623i	$0.840774\pi$
$264$	−16.9706	+	12.0000i	−1.04447	+	0.738549i
$265$	−18.0000	−	6.00000i	−1.10573	−	0.368577i
$266$	−8.48528			−0.520266
$267$	10.0000	−	7.07107i	0.611990	−	0.432742i
$268$	8.00000			0.488678
$269$		−	18.3848i		−	1.12094i	−0.828175	−	0.560470i	$-0.810621\pi$
$269$		−	18.3848i		−	1.12094i	0.828175	−	0.560470i	$-0.189379\pi$
$270$	0.757359	−	16.4142i	0.0460914	−	0.998937i
$271$			24.0000i			1.45790i	0.684569	+	0.728948i	$0.259990\pi$
$271$			24.0000i			1.45790i	−0.684569	+	0.728948i	$0.740010\pi$
$272$	−16.9706			−1.02899
$273$	8.48528	−	6.00000i	0.513553	−	0.363137i
$274$			24.0000i			1.44989i
$275$	16.9706	+	12.7279i	1.02336	+	0.767523i
$276$	4.00000	−	2.82843i	0.240772	−	0.170251i
$277$			6.00000i			0.360505i	0.983620	+	0.180253i	$0.0576915\pi$
$277$			6.00000i			0.360505i	−0.983620	+	0.180253i	$0.942309\pi$
$278$	16.9706			1.01783
$279$		0			0
$280$	−2.00000	+	6.00000i	−0.119523	+	0.358569i
$281$			2.82843i			0.168730i	0.996435	+	0.0843649i	$0.0268861\pi$
$281$			2.82843i			0.168730i	−0.996435	+	0.0843649i	$0.973114\pi$
$282$	4.00000	+	5.65685i	0.238197	+	0.336861i
$283$	−4.00000			−0.237775			−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000			−0.237775			−0.118888	+	0.992908i	$0.537933\pi$
$284$	25.4558			1.51053
$285$	−22.2426	+	6.72792i	−1.31754	+	0.398528i
$286$	−36.0000			−2.12872
$287$		−	1.41421i		−	0.0834784i
$288$	−16.0000	−	5.65685i	−0.942809	−	0.333333i
$289$	1.00000			0.0588235
$290$	8.48528	+	2.82843i	0.498273	+	0.166091i
$291$	−8.48528	+	6.00000i	−0.497416	+	0.351726i
$292$			12.0000i			0.702247i
$293$	−21.2132			−1.23929			−0.619644	−	0.784883i	$-0.712723\pi$
$293$	−21.2132			−1.23929			−0.619644	+	0.784883i	$0.712723\pi$
$294$	2.00000	−	1.41421i	0.116642	−	0.0824786i
$295$		0			0
$296$	16.9706			0.986394
$297$	21.2132	−	6.00000i	1.23091	−	0.348155i
$298$	−16.0000			−0.926855
$299$	8.48528			0.490716
$300$	−0.485281	+	17.3137i	−0.0280177	+	0.999607i
$301$	8.00000			0.461112
$302$	16.9706			0.976546
$303$	22.0000	−	15.5563i	1.26387	−	0.893689i
$304$			24.0000i			1.37649i
$305$	21.2132	+	7.07107i	1.21466	+	0.404888i
$306$	16.9706	+	6.00000i	0.970143	+	0.342997i
$307$	20.0000			1.14146			0.570730	−	0.821138i	$-0.306660\pi$
$307$	20.0000			1.14146			0.570730	+	0.821138i	$0.306660\pi$
$308$	−8.48528			−0.483494
$309$	−14.0000	−	19.7990i	−0.796432	−	1.12633i
$310$		0			0
$311$	25.4558			1.44347			0.721734	−	0.692170i	$-0.243345\pi$
$311$	25.4558			1.44347			0.721734	+	0.692170i	$0.243345\pi$
$312$	−16.9706	−	24.0000i	−0.960769	−	1.35873i
$313$		−	18.0000i		−	1.01742i	−0.860938	−	0.508710i	$-0.830123\pi$
$313$		−	18.0000i		−	1.01742i	0.860938	−	0.508710i	$-0.169877\pi$
$314$	−25.4558			−1.43656
$315$	4.12132	−	5.29289i	0.232210	−	0.298221i
$316$		0			0
$317$	25.4558			1.42974			0.714871	−	0.699256i	$-0.246485\pi$
$317$	25.4558			1.42974			0.714871	+	0.699256i	$0.246485\pi$
$318$	16.9706	−	12.0000i	0.951662	−	0.672927i
$319$			12.0000i			0.671871i
$320$	16.9706	+	5.65685i	0.948683	+	0.316228i
$321$	10.0000	−	7.07107i	0.558146	−	0.394669i
$322$	2.00000			0.111456
$323$		−	25.4558i		−	1.41640i
$324$	14.0000	+	11.3137i	0.777778	+	0.628539i
$325$	−18.0000	+	24.0000i	−0.998460	+	1.33128i
$326$			28.2843i			1.56652i
$327$	−2.00000	−	2.82843i	−0.110600	−	0.156412i
$328$	−4.00000			−0.220863
$329$			2.82843i			0.155936i
$330$	−22.2426	+	6.72792i	−1.22442	+	0.370360i
$331$		0			0		1.00000			$0$
$331$		0			0		−1.00000			$\pi$
$332$		−	5.65685i		−	0.310460i
$333$	−16.9706	−	6.00000i	−0.929981	−	0.328798i
$334$	20.0000			1.09435
$335$	8.48528	+	2.82843i	0.463600	+	0.154533i
$336$	−4.00000	−	5.65685i	−0.218218	−	0.308607i
$337$		−	12.0000i		−	0.653682i	−0.945079	−	0.326841i	$-0.894016\pi$
$337$		−	12.0000i		−	0.653682i	0.945079	−	0.326841i	$-0.105984\pi$
$338$		−	32.5269i		−	1.76923i
$339$	8.48528	+	12.0000i	0.460857	+	0.651751i
$340$	−18.0000	−	6.00000i	−0.976187	−	0.325396i
$341$		0			0
$342$	8.48528	−	24.0000i	0.458831	−	1.29777i
$343$	1.00000			0.0539949
$344$		−	22.6274i		−	1.21999i
$345$	5.24264	−	1.58579i	0.282254	−	0.0853759i
$346$		−	18.0000i		−	0.967686i
$347$		−	1.41421i		−	0.0759190i	−0.999279	−	0.0379595i	$-0.987914\pi$
$347$		−	1.41421i		−	0.0759190i	0.999279	−	0.0379595i	$-0.0120858\pi$
$348$	−8.00000	+	5.65685i	−0.428845	+	0.303239i
$349$	26.0000			1.39175			0.695874	−	0.718164i	$-0.255017\pi$
$349$	26.0000			1.39175			0.695874	+	0.718164i	$0.255017\pi$
$350$	−4.24264	+	5.65685i	−0.226779	+	0.302372i
$351$	8.48528	+	30.0000i	0.452911	+	1.60128i
$352$			24.0000i			1.27920i
$353$	4.24264			0.225813			0.112906	−	0.993606i	$-0.463984\pi$
$353$	4.24264			0.225813			0.112906	+	0.993606i	$0.463984\pi$
$354$		0			0
$355$	27.0000	+	9.00000i	1.43301	+	0.477670i
$356$		−	14.1421i		−	0.749532i
$357$	4.24264	+	6.00000i	0.224544	+	0.317554i
$358$		−	6.00000i		−	0.317110i
$359$	4.24264			0.223918			0.111959	−	0.993713i	$-0.464287\pi$
$359$	4.24264			0.223918			0.111959	+	0.993713i	$0.464287\pi$
$360$	−14.9706	−	11.6569i	−0.789018	−	0.614370i
$361$	−17.0000			−0.894737
$362$			2.82843i			0.148659i
$363$	−7.00000	−	9.89949i	−0.367405	−	0.519589i
$364$		−	12.0000i		−	0.628971i
$365$	−4.24264	+	12.7279i	−0.222070	+	0.666210i
$366$	−20.0000	+	14.1421i	−1.04542	+	0.739221i
$367$	8.00000			0.417597			0.208798	−	0.977959i	$-0.433045\pi$
$367$	8.00000			0.417597			0.208798	+	0.977959i	$0.433045\pi$
$368$		−	5.65685i		−	0.294884i
$369$	4.00000	+	1.41421i	0.208232	+	0.0736210i
$370$	18.0000	+	6.00000i	0.935775	+	0.311925i
$371$	8.48528			0.440534
$372$		0			0
$373$			12.0000i			0.621336i	0.950518	+	0.310668i	$0.100553\pi$
$373$			12.0000i			0.621336i	−0.950518	+	0.310668i	$0.899447\pi$
$374$		−	25.4558i		−	1.31629i
$375$	−6.63604	+	18.1924i	−0.342684	+	0.939451i
$376$	8.00000			0.412568
$377$	−16.9706			−0.874028
$378$	2.00000	+	7.07107i	0.102869	+	0.363696i
$379$		−	36.0000i		−	1.84920i	−0.380945	−	0.924598i	$-0.624401\pi$
$379$		−	36.0000i		−	1.84920i	0.380945	−	0.924598i	$-0.375599\pi$
$380$	−8.48528	+	25.4558i	−0.435286	+	1.30586i
$381$	16.0000	+	22.6274i	0.819705	+	1.15924i
$382$		−	18.0000i		−	0.920960i
$383$			11.3137i			0.578103i	0.957313	+	0.289052i	$0.0933400\pi$
$383$			11.3137i			0.578103i	−0.957313	+	0.289052i	$0.906660\pi$
$384$	−16.0000	+	11.3137i	−0.816497	+	0.577350i
$385$	−9.00000	−	3.00000i	−0.458682	−	0.152894i
$386$	−25.4558			−1.29567
$387$	−8.00000	+	22.6274i	−0.406663	+	1.15022i
$388$			12.0000i			0.609208i
$389$			2.82843i			0.143407i	0.997426	+	0.0717035i	$0.0228435\pi$
$389$			2.82843i			0.143407i	−0.997426	+	0.0717035i	$0.977156\pi$
$390$	−9.51472	−	31.4558i	−0.481797	−	1.59283i
$391$			6.00000i			0.303433i
$392$		−	2.82843i		−	0.142857i
$393$	−8.48528	−	12.0000i	−0.428026	−	0.605320i
$394$		−	36.0000i		−	1.81365i
$395$		0			0
$396$	8.48528	−	24.0000i	0.426401	−	1.20605i
$397$		−	30.0000i		−	1.50566i	−0.658217	−	0.752828i	$-0.728689\pi$
$397$		−	30.0000i		−	1.50566i	0.658217	−	0.752828i	$-0.271311\pi$
$398$	8.48528			0.425329
$399$	8.48528	−	6.00000i	0.424795	−	0.300376i
$400$	16.0000	+	12.0000i	0.800000	+	0.600000i
$401$		−	14.1421i		−	0.706225i	−0.935581	−	0.353112i	$-0.885123\pi$
$401$		−	14.1421i		−	0.706225i	0.935581	−	0.353112i	$-0.114877\pi$
$402$	−8.00000	+	5.65685i	−0.399004	+	0.282138i
$403$		0			0
$404$		−	31.1127i		−	1.54791i
$405$	10.8492	+	16.9497i	0.539103	+	0.842240i
$406$	−4.00000			−0.198517
$407$			25.4558i			1.26180i
$408$	16.9706	−	12.0000i	0.840168	−	0.594089i
$409$	−22.0000			−1.08783			−0.543915	−	0.839140i	$-0.683059\pi$
$409$	−22.0000			−1.08783			−0.543915	+	0.839140i	$0.683059\pi$
$410$	−4.24264	−	1.41421i	−0.209529	−	0.0698430i
$411$	−16.9706	−	24.0000i	−0.837096	−	1.18383i
$412$	−28.0000			−1.37946
$413$		0			0
$414$	−2.00000	+	5.65685i	−0.0982946	+	0.278019i
$415$	2.00000	−	6.00000i	0.0981761	−	0.294528i
$416$	−33.9411			−1.66410
$417$	−16.9706	+	12.0000i	−0.831052	+	0.587643i
$418$	−36.0000			−1.76082
$419$	16.9706			0.829066			0.414533	−	0.910034i	$-0.363945\pi$
$419$	16.9706			0.829066			0.414533	+	0.910034i	$0.363945\pi$
$420$	−2.24264	−	7.41421i	−0.109430	−	0.361777i
$421$	8.00000			0.389896			0.194948	−	0.980814i	$-0.437546\pi$
$421$	8.00000			0.389896			0.194948	+	0.980814i	$0.437546\pi$
$422$	16.9706			0.826114
$423$	−8.00000	−	2.82843i	−0.388973	−	0.137523i
$424$		−	24.0000i		−	1.16554i
$425$	−16.9706	−	12.7279i	−0.823193	−	0.617395i
$426$	−25.4558	+	18.0000i	−1.23334	+	0.872103i
$427$	−10.0000			−0.483934
$428$		−	14.1421i		−	0.683586i
$429$	36.0000	−	25.4558i	1.73810	−	1.22902i
$430$	8.00000	−	24.0000i	0.385794	−	1.15738i
$431$	−4.24264			−0.204361			−0.102180	−	0.994766i	$-0.532582\pi$
$431$	−4.24264			−0.204361			−0.102180	+	0.994766i	$0.532582\pi$
$432$	20.0000	−	5.65685i	0.962250	−	0.272166i
$433$		−	18.0000i		−	0.865025i	−0.901628	−	0.432512i	$-0.857627\pi$
$433$		−	18.0000i		−	0.865025i	0.901628	−	0.432512i	$-0.142373\pi$
$434$		0			0
$435$	−10.4853	+	3.17157i	−0.502731	+	0.152065i
$436$	−4.00000			−0.191565
$437$	8.48528			0.405906
$438$	−8.48528	−	12.0000i	−0.405442	−	0.573382i
$439$			18.0000i			0.859093i	0.903045	+	0.429547i	$0.141327\pi$
$439$			18.0000i			0.859093i	−0.903045	+	0.429547i	$0.858673\pi$
$440$	−8.48528	+	25.4558i	−0.404520	+	1.21356i
$441$	−1.00000	+	2.82843i	−0.0476190	+	0.134687i
$442$	36.0000			1.71235
$443$		−	35.3553i		−	1.67978i	−0.542754	−	0.839891i	$-0.682619\pi$
$443$		−	35.3553i		−	1.67978i	0.542754	−	0.839891i	$-0.317381\pi$
$444$	−16.9706	+	12.0000i	−0.805387	+	0.569495i
$445$	5.00000	−	15.0000i	0.237023	−	0.711068i
$446$			11.3137i			0.535720i
$447$	16.0000	−	11.3137i	0.756774	−	0.535120i
$448$	−8.00000			−0.377964
$449$		−	5.65685i		−	0.266963i	−0.991051	−	0.133482i	$-0.957384\pi$
$449$		−	5.65685i		−	0.266963i	0.991051	−	0.133482i	$-0.0426157\pi$
$450$	−11.7574	−	17.6569i	−0.554247	−	0.832352i
$451$		−	6.00000i		−	0.282529i
$452$	16.9706			0.798228
$453$	−16.9706	+	12.0000i	−0.797347	+	0.563809i
$454$	32.0000			1.50183
$455$	4.24264	−	12.7279i	0.198898	−	0.596694i
$456$	−16.9706	−	24.0000i	−0.794719	−	1.12390i
$457$		0			0		1.00000			$0$
$457$		0			0		−1.00000			$\pi$
$458$			19.7990i			0.925146i
$459$	−21.2132	+	6.00000i	−0.990148	+	0.280056i
$460$	2.00000	−	6.00000i	0.0932505	−	0.279751i
$461$			15.5563i			0.724531i	0.932075	+	0.362266i	$0.117997\pi$
$461$			15.5563i			0.724531i	−0.932075	+	0.362266i	$0.882003\pi$
$462$	8.48528	−	6.00000i	0.394771	−	0.279145i
$463$	−4.00000			−0.185896			−0.0929479	−	0.995671i	$-0.529629\pi$
$463$	−4.00000			−0.185896			−0.0929479	+	0.995671i	$0.529629\pi$
$464$			11.3137i			0.525226i
$465$		0			0
$466$			24.0000i			1.11178i
$467$		−	5.65685i		−	0.261768i	−0.991398	−	0.130884i	$-0.958218\pi$
$467$		−	5.65685i		−	0.261768i	0.991398	−	0.130884i	$-0.0417815\pi$
$468$	33.9411	+	12.0000i	1.56893	+	0.554700i
$469$	−4.00000			−0.184703
$470$	8.48528	+	2.82843i	0.391397	+	0.130466i
$471$	25.4558	−	18.0000i	1.17294	−	0.829396i
$472$		0			0
$473$	33.9411			1.56061
$474$		0			0
$475$	−18.0000	+	24.0000i	−0.825897	+	1.10120i
$476$	8.48528			0.388922
$477$	−8.48528	+	24.0000i	−0.388514	+	1.09888i
$478$			6.00000i			0.274434i
$479$	−8.48528			−0.387702			−0.193851	−	0.981031i	$-0.562098\pi$
$479$	−8.48528			−0.387702			−0.193851	+	0.981031i	$0.562098\pi$
$480$	−20.9706	+	6.34315i	−0.957171	+	0.289524i
$481$	−36.0000			−1.64146
$482$			36.7696i			1.67481i
$483$	−2.00000	+	1.41421i	−0.0910032	+	0.0643489i
$484$	−14.0000			−0.636364
$485$	−4.24264	+	12.7279i	−0.192648	+	0.577945i
$486$	−22.0000	−	1.41421i	−0.997940	−	0.0641500i
$487$	20.0000			0.906287			0.453143	−	0.891438i	$-0.350303\pi$
$487$	20.0000			0.906287			0.453143	+	0.891438i	$0.350303\pi$
$488$			28.2843i			1.28037i
$489$	−20.0000	−	28.2843i	−0.904431	−	1.27906i
$490$	1.00000	−	3.00000i	0.0451754	−	0.135526i
$491$	29.6985			1.34027			0.670137	−	0.742237i	$-0.266235\pi$
$491$	29.6985			1.34027			0.670137	+	0.742237i	$0.266235\pi$
$492$	4.00000	−	2.82843i	0.180334	−	0.127515i
$493$		−	12.0000i		−	0.540453i
$494$		−	50.9117i		−	2.29063i
$495$	17.4853	−	22.4558i	0.785905	−	1.00932i
$496$		0			0
$497$	−12.7279			−0.570925
$498$	4.00000	+	5.65685i	0.179244	+	0.253490i
$499$			12.0000i			0.537194i	0.963253	+	0.268597i	$0.0865599\pi$
$499$			12.0000i			0.537194i	−0.963253	+	0.268597i	$0.913440\pi$
$500$	12.7279	+	18.3848i	0.569210	+	0.822192i
$501$	−20.0000	+	14.1421i	−0.893534	+	0.631824i
$502$		0			0
$503$		−	5.65685i		−	0.252227i	−0.992016	−	0.126113i	$-0.959750\pi$
$503$		−	5.65685i		−	0.252227i	0.992016	−	0.126113i	$-0.0402503\pi$
$504$	8.00000	+	2.82843i	0.356348	+	0.125988i
$505$	11.0000	−	33.0000i	0.489494	−	1.46848i
$506$	8.48528			0.377217
$507$	23.0000	+	32.5269i	1.02147	+	1.44457i
$508$	32.0000			1.41977
$509$			24.0416i			1.06563i	0.846233	+	0.532813i	$0.178865\pi$
$509$			24.0416i			1.06563i	−0.846233	+	0.532813i	$0.821135\pi$
$510$	22.2426	−	6.72792i	0.984921	−	0.297917i
$511$		−	6.00000i		−	0.265424i
$512$			22.6274i			1.00000i
$513$	8.48528	+	30.0000i	0.374634	+	1.32453i
$514$		−	6.00000i		−	0.264649i
$515$	−29.6985	−	9.89949i	−1.30867	−	0.436224i
$516$	16.0000	+	22.6274i	0.704361	+	0.996116i
$517$			12.0000i			0.527759i
$518$	−8.48528			−0.372822
$519$	12.7279	+	18.0000i	0.558694	+	0.790112i
$520$	−36.0000	−	12.0000i	−1.57870	−	0.526235i
$521$			7.07107i			0.309789i	0.987931	+	0.154895i	$0.0495038\pi$
$521$			7.07107i			0.309789i	−0.987931	+	0.154895i	$0.950496\pi$
$522$	4.00000	−	11.3137i	0.175075	−	0.495188i
$523$	−34.0000			−1.48672			−0.743358	−	0.668894i	$-0.766768\pi$
$523$	−34.0000			−1.48672			−0.743358	+	0.668894i	$0.766768\pi$
$524$	−16.9706			−0.741362
$525$	0.242641	−	8.65685i	0.0105897	−	0.377816i
$526$	−22.0000			−0.959246
$527$		0			0
$528$	−16.9706	−	24.0000i	−0.738549	−	1.04447i
$529$	21.0000			0.913043
$530$	8.48528	−	25.4558i	0.368577	−	1.10573i
$531$		0			0
$532$		−	12.0000i		−	0.520266i
$533$	8.48528			0.367538
$534$	10.0000	+	14.1421i	0.432742	+	0.611990i
$535$	5.00000	−	15.0000i	0.216169	−	0.648507i
$536$			11.3137i			0.488678i
$537$	4.24264	+	6.00000i	0.183083	+	0.258919i
$538$	26.0000			1.12094
$539$	4.24264			0.182743
$540$	23.2132	+	1.07107i	0.998937	+	0.0460914i
$541$	−16.0000			−0.687894			−0.343947	−	0.938989i	$-0.611764\pi$
$541$	−16.0000			−0.687894			−0.343947	+	0.938989i	$0.611764\pi$
$542$	−33.9411			−1.45790
$543$	−2.00000	−	2.82843i	−0.0858282	−	0.121379i
$544$		−	24.0000i		−	1.02899i
$545$	−4.24264	−	1.41421i	−0.181735	−	0.0605783i
$546$	8.48528	+	12.0000i	0.363137	+	0.513553i
$547$	−28.0000			−1.19719			−0.598597	−	0.801050i	$-0.704275\pi$
$547$	−28.0000			−1.19719			−0.598597	+	0.801050i	$0.704275\pi$
$548$	−33.9411			−1.44989
$549$	10.0000	−	28.2843i	0.426790	−	1.20714i
$550$	−18.0000	+	24.0000i	−0.767523	+	1.02336i
$551$	−16.9706			−0.722970
$552$	4.00000	+	5.65685i	0.170251	+	0.240772i
$553$		0			0
$554$	−8.48528			−0.360505
$555$	−22.2426	+	6.72792i	−0.944148	+	0.285584i
$556$			24.0000i			1.01783i
$557$	8.48528			0.359533			0.179766	−	0.983709i	$-0.442466\pi$
$557$	8.48528			0.359533			0.179766	+	0.983709i	$0.442466\pi$
$558$		0			0
$559$			48.0000i			2.03018i
$560$	−8.48528	−	2.82843i	−0.358569	−	0.119523i
$561$	18.0000	+	25.4558i	0.759961	+	1.07475i
$562$	−4.00000			−0.168730
$563$			11.3137i			0.476816i	0.971165	+	0.238408i	$0.0766255\pi$
$563$			11.3137i			0.476816i	−0.971165	+	0.238408i	$0.923374\pi$
$564$	−8.00000	+	5.65685i	−0.336861	+	0.238197i
$565$	18.0000	+	6.00000i	0.757266	+	0.252422i
$566$		−	5.65685i		−	0.237775i
$567$	−7.00000	−	5.65685i	−0.293972	−	0.237566i
$568$			36.0000i			1.51053i
$569$			2.82843i			0.118574i	0.998241	+	0.0592869i	$0.0188827\pi$
$569$			2.82843i			0.118574i	−0.998241	+	0.0592869i	$0.981117\pi$
$570$	−9.51472	−	31.4558i	−0.398528	−	1.31754i
$571$			36.0000i			1.50655i	0.657704	+	0.753277i	$0.271528\pi$
$571$			36.0000i			1.50655i	−0.657704	+	0.753277i	$0.728472\pi$
$572$		−	50.9117i		−	2.12872i
$573$	12.7279	+	18.0000i	0.531717	+	0.751961i
$574$	2.00000			0.0834784
$575$	4.24264	−	5.65685i	0.176930	−	0.235907i
$576$	8.00000	−	22.6274i	0.333333	−	0.942809i
$577$		−	42.0000i		−	1.74848i	−0.485491	−	0.874241i	$-0.661359\pi$
$577$		−	42.0000i		−	1.74848i	0.485491	−	0.874241i	$-0.338641\pi$
$578$			1.41421i			0.0588235i
$579$	25.4558	−	18.0000i	1.05791	−	0.748054i
$580$	−4.00000	+	12.0000i	−0.166091	+	0.498273i
$581$			2.82843i			0.117343i
$582$	−8.48528	−	12.0000i	−0.351726	−	0.497416i
$583$	36.0000			1.49097
$584$	−16.9706			−0.702247
$585$	31.7574	+	24.7279i	1.31301	+	1.02237i
$586$		−	30.0000i		−	1.23929i
$587$			28.2843i			1.16742i	0.811963	+	0.583708i	$0.198399\pi$
$587$			28.2843i			1.16742i	−0.811963	+	0.583708i	$0.801601\pi$
$588$	2.00000	+	2.82843i	0.0824786	+	0.116642i
$589$		0			0
$590$		0			0
$591$	25.4558	+	36.0000i	1.04711	+	1.48084i
$592$			24.0000i			0.986394i
$593$	4.24264			0.174224			0.0871122	−	0.996199i	$-0.472236\pi$
$593$	4.24264			0.174224			0.0871122	+	0.996199i	$0.472236\pi$
$594$	8.48528	+	30.0000i	0.348155	+	1.23091i
$595$	9.00000	+	3.00000i	0.368964	+	0.122988i
$596$		−	22.6274i		−	0.926855i
$597$	−8.48528	+	6.00000i	−0.347279	+	0.245564i
$598$			12.0000i			0.490716i
$599$	−12.7279			−0.520049			−0.260024	−	0.965602i	$-0.583731\pi$
$599$	−12.7279			−0.520049			−0.260024	+	0.965602i	$0.583731\pi$
$600$	−24.4853	−	0.686292i	−0.999607	−	0.0280177i
$601$	−10.0000			−0.407909			−0.203954	−	0.978980i	$-0.565379\pi$
$601$	−10.0000			−0.407909			−0.203954	+	0.978980i	$0.565379\pi$
$602$			11.3137i			0.461112i
$603$	4.00000	−	11.3137i	0.162893	−	0.460730i
$604$			24.0000i			0.976546i
$605$	−14.8492	−	4.94975i	−0.603708	−	0.201236i
$606$	22.0000	+	31.1127i	0.893689	+	1.26387i
$607$	32.0000			1.29884			0.649420	−	0.760430i	$-0.275012\pi$
$607$	32.0000			1.29884			0.649420	+	0.760430i	$0.275012\pi$
$608$	−33.9411			−1.37649
$609$	4.00000	−	2.82843i	0.162088	−	0.114614i
$610$	−10.0000	+	30.0000i	−0.404888	+	1.21466i
$611$	−16.9706			−0.686555
$612$	−8.48528	+	24.0000i	−0.342997	+	0.970143i
$613$		−	24.0000i		−	0.969351i	−0.874694	−	0.484675i	$-0.838938\pi$
$613$		−	24.0000i		−	0.969351i	0.874694	−	0.484675i	$-0.161062\pi$
$614$			28.2843i			1.14146i
$615$	5.24264	−	1.58579i	0.211404	−	0.0639451i
$616$		−	12.0000i		−	0.483494i
$617$	16.9706			0.683209			0.341605	−	0.939844i	$-0.389030\pi$
$617$	16.9706			0.683209			0.341605	+	0.939844i	$0.389030\pi$
$618$	28.0000	−	19.7990i	1.12633	−	0.796432i
$619$		−	12.0000i		−	0.482321i	−0.970485	−	0.241160i	$-0.922472\pi$
$619$		−	12.0000i		−	0.482321i	0.970485	−	0.241160i	$-0.0775280\pi$
$620$		0			0
$621$	−2.00000	−	7.07107i	−0.0802572	−	0.283752i
$622$			36.0000i			1.44347i
$623$			7.07107i			0.283296i
$624$	33.9411	−	24.0000i	1.35873	−	0.960769i
$625$	7.00000	+	24.0000i	0.280000	+	0.960000i
$626$	25.4558			1.01742
$627$	36.0000	−	25.4558i	1.43770	−	1.01661i
$628$		−	36.0000i		−	1.43656i
$629$		−	25.4558i		−	1.01499i
$630$	7.48528	+	5.82843i	0.298221	+	0.232210i
$631$		−	36.0000i		−	1.43314i	−0.697517	−	0.716569i	$-0.745712\pi$
$631$		−	36.0000i		−	1.43314i	0.697517	−	0.716569i	$-0.254288\pi$
$632$		0			0
$633$	−16.9706	+	12.0000i	−0.674519	+	0.476957i
$634$			36.0000i			1.42974i
$635$	33.9411	+	11.3137i	1.34691	+	0.448971i
$636$	16.9706	+	24.0000i	0.672927	+	0.951662i
$637$			6.00000i			0.237729i
$638$	−16.9706			−0.671871
$639$	12.7279	−	36.0000i	0.503509	−	1.42414i
$640$	−8.00000	+	24.0000i	−0.316228	+	0.948683i
$641$		−	48.0833i		−	1.89917i	−0.313503	−	0.949587i	$-0.601502\pi$
$641$		−	48.0833i		−	1.89917i	0.313503	−	0.949587i	$-0.398498\pi$
$642$	10.0000	+	14.1421i	0.394669	+	0.558146i
$643$	−4.00000			−0.157745			−0.0788723	−	0.996885i	$-0.525132\pi$
$643$	−4.00000			−0.157745			−0.0788723	+	0.996885i	$0.525132\pi$
$644$			2.82843i			0.111456i
$645$	8.97056	+	29.6569i	0.353216	+	1.16774i
$646$	36.0000			1.41640
$647$		−	31.1127i		−	1.22317i	−0.791180	−	0.611583i	$-0.790533\pi$
$647$		−	31.1127i		−	1.22317i	0.791180	−	0.611583i	$-0.209467\pi$
$648$	−16.0000	+	19.7990i	−0.628539	+	0.777778i
$649$		0			0
$650$	−33.9411	−	25.4558i	−1.33128	−	0.998460i
$651$		0			0
$652$	−40.0000			−1.56652
$653$	−16.9706			−0.664109			−0.332055	−	0.943260i	$-0.607742\pi$
$653$	−16.9706			−0.664109			−0.332055	+	0.943260i	$0.607742\pi$
$654$	4.00000	−	2.82843i	0.156412	−	0.110600i
$655$	−18.0000	−	6.00000i	−0.703318	−	0.234439i
$656$		−	5.65685i		−	0.220863i
$657$	16.9706	+	6.00000i	0.662085	+	0.234082i
$658$	−4.00000			−0.155936
$659$	−4.24264			−0.165270			−0.0826349	−	0.996580i	$-0.526334\pi$
$659$	−4.24264			−0.165270			−0.0826349	+	0.996580i	$0.526334\pi$
$660$	−9.51472	−	31.4558i	−0.370360	−	1.22442i
$661$	14.0000			0.544537			0.272268	−	0.962221i	$-0.412226\pi$
$661$	14.0000			0.544537			0.272268	+	0.962221i	$0.412226\pi$
$662$		0			0
$663$	−36.0000	+	25.4558i	−1.39812	+	0.988623i
$664$	8.00000			0.310460
$665$	4.24264	−	12.7279i	0.164523	−	0.493568i
$666$	8.48528	−	24.0000i	0.328798	−	0.929981i
$667$	4.00000			0.154881
$668$			28.2843i			1.09435i
$669$	−8.00000	−	11.3137i	−0.309298	−	0.437413i
$670$	−4.00000	+	12.0000i	−0.154533	+	0.463600i
$671$	−42.4264			−1.63785
$672$	8.00000	−	5.65685i	0.308607	−	0.218218i
$673$			30.0000i			1.15642i	0.815890	+	0.578208i	$0.196248\pi$
$673$			30.0000i			1.15642i	−0.815890	+	0.578208i	$0.803752\pi$
$674$	16.9706			0.653682
$675$	24.2426	+	9.34315i	0.933100	+	0.359618i
$676$	46.0000			1.76923
$677$	−12.7279			−0.489174			−0.244587	−	0.969627i	$-0.578652\pi$
$677$	−12.7279			−0.489174			−0.244587	+	0.969627i	$0.578652\pi$
$678$	−16.9706	+	12.0000i	−0.651751	+	0.460857i
$679$		−	6.00000i		−	0.230259i
$680$	8.48528	−	25.4558i	0.325396	−	0.976187i
$681$	−32.0000	+	22.6274i	−1.22624	+	0.867085i
$682$		0			0
$683$			32.5269i			1.24461i	0.782776	+	0.622304i	$0.213803\pi$
$683$			32.5269i			1.24461i	−0.782776	+	0.622304i	$0.786197\pi$
$684$	33.9411	+	12.0000i	1.29777	+	0.458831i
$685$	−36.0000	−	12.0000i	−1.37549	−	0.458496i
$686$			1.41421i			0.0539949i
$687$	−14.0000	−	19.7990i	−0.534133	−	0.755379i
$688$	32.0000			1.21999
$689$			50.9117i			1.93958i
$690$	2.24264	+	7.41421i	0.0853759	+	0.282254i
$691$		−	36.0000i		−	1.36950i	−0.728776	−	0.684752i	$-0.759910\pi$
$691$		−	36.0000i		−	1.36950i	0.728776	−	0.684752i	$-0.240090\pi$
$692$	25.4558			0.967686
$693$	−4.24264	+	12.0000i	−0.161165	+	0.455842i
$694$	2.00000			0.0759190
$695$	−8.48528	+	25.4558i	−0.321865	+	0.965595i
$696$	−8.00000	−	11.3137i	−0.303239	−	0.428845i
$697$			6.00000i			0.227266i
$698$			36.7696i			1.39175i
$699$	−16.9706	−	24.0000i	−0.641886	−	0.907763i
$700$	−8.00000	−	6.00000i	−0.302372	−	0.226779i
$701$			19.7990i			0.747798i	0.927470	+	0.373899i	$0.121979\pi$
$701$			19.7990i			0.747798i	−0.927470	+	0.373899i	$0.878021\pi$
$702$	−42.4264	+	12.0000i	−1.60128	+	0.452911i
$703$	−36.0000			−1.35777
$704$	−33.9411			−1.27920
$705$	−10.4853	+	3.17157i	−0.394899	+	0.119448i
$706$			6.00000i			0.225813i
$707$			15.5563i			0.585057i
$708$		0			0
$709$	−10.0000			−0.375558			−0.187779	−	0.982211i	$-0.560129\pi$
$709$	−10.0000			−0.375558			−0.187779	+	0.982211i	$0.560129\pi$
$710$	−12.7279	+	38.1838i	−0.477670	+	1.43301i
$711$		0			0
$712$	20.0000			0.749532
$713$		0			0
$714$	−8.48528	+	6.00000i	−0.317554	+	0.224544i
$715$	18.0000	−	54.0000i	0.673162	−	2.01949i
$716$	8.48528			0.317110
$717$	−4.24264	−	6.00000i	−0.158444	−	0.224074i
$718$			6.00000i			0.223918i
$719$	−33.9411			−1.26579			−0.632895	−	0.774237i	$-0.718134\pi$
$719$	−33.9411			−1.26579			−0.632895	+	0.774237i	$0.718134\pi$
$720$	16.4853	−	21.1716i	0.614370	−	0.789018i
$721$	14.0000			0.521387
$722$		−	24.0416i		−	0.894737i
$723$	−26.0000	−	36.7696i	−0.966950	−	1.36747i
$724$	−4.00000			−0.148659
$725$	−8.48528	+	11.3137i	−0.315135	+	0.420181i
$726$	14.0000	−	9.89949i	0.519589	−	0.367405i
$727$	26.0000			0.964287			0.482143	−	0.876092i	$-0.339858\pi$
$727$	26.0000			0.964287			0.482143	+	0.876092i	$0.339858\pi$
$728$	16.9706			0.628971
$729$	23.0000	−	14.1421i	0.851852	−	0.523783i
$730$	−18.0000	−	6.00000i	−0.666210	−	0.222070i
$731$	−33.9411			−1.25536
$732$	−20.0000	−	28.2843i	−0.739221	−	1.04542i
$733$		−	6.00000i		−	0.221615i	−0.993842	−	0.110808i	$-0.964656\pi$
$733$		−	6.00000i		−	0.221615i	0.993842	−	0.110808i	$-0.0353437\pi$
$734$			11.3137i			0.417597i
$735$	1.12132	+	3.70711i	0.0413605	+	0.136739i
$736$	8.00000			0.294884
$737$	−16.9706			−0.625119
$738$	−2.00000	+	5.65685i	−0.0736210	+	0.208232i
$739$		−	24.0000i		−	0.882854i	−0.897297	−	0.441427i	$-0.854472\pi$
$739$		−	24.0000i		−	0.882854i	0.897297	−	0.441427i	$-0.145528\pi$
$740$	−8.48528	+	25.4558i	−0.311925	+	0.935775i
$741$	36.0000	+	50.9117i	1.32249	+	1.87029i
$742$			12.0000i			0.440534i
$743$		−	1.41421i		−	0.0518825i	−0.999663	−	0.0259412i	$-0.991742\pi$
$743$		−	1.41421i		−	0.0518825i	0.999663	−	0.0259412i	$-0.00825828\pi$
$744$		0			0
$745$	8.00000	−	24.0000i	0.293097	−	0.879292i
$746$	−16.9706			−0.621336
$747$	−8.00000	−	2.82843i	−0.292705	−	0.103487i
$748$	36.0000			1.31629
$749$			7.07107i			0.258371i
$750$	−25.7279	−	9.38478i	−0.939451	−	0.342684i
$751$			12.0000i			0.437886i	0.975738	+	0.218943i	$0.0702609\pi$
$751$			12.0000i			0.437886i	−0.975738	+	0.218943i	$0.929739\pi$
$752$			11.3137i			0.412568i
$753$		0			0
$754$		−	24.0000i		−	0.874028i
$755$	−8.48528	+	25.4558i	−0.308811	+	0.926433i
$756$	−10.0000	+	2.82843i	−0.363696	+	0.102869i
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$	50.9117			1.84920
$759$	−8.48528	+	6.00000i	−0.307996	+	0.217786i
$760$	−36.0000	−	12.0000i	−1.30586	−	0.435286i
$761$		−	52.3259i		−	1.89681i	−0.317058	−	0.948406i	$-0.602695\pi$
$761$		−	52.3259i		−	1.89681i	0.317058	−	0.948406i	$-0.397305\pi$
$762$	−32.0000	+	22.6274i	−1.15924	+	0.819705i
$763$	2.00000			0.0724049
$764$	25.4558			0.920960
$765$	−17.4853	+	22.4558i	−0.632182	+	0.811893i
$766$	−16.0000			−0.578103
$767$		0			0
$768$	−16.0000	−	22.6274i	−0.577350	−	0.816497i
$769$	14.0000			0.504853			0.252426	−	0.967616i	$-0.418771\pi$
$769$	14.0000			0.504853			0.252426	+	0.967616i	$0.418771\pi$
$770$	4.24264	−	12.7279i	0.152894	−	0.458682i
$771$	4.24264	+	6.00000i	0.152795	+	0.216085i
$772$		−	36.0000i		−	1.29567i
$773$	−29.6985			−1.06818			−0.534090	−	0.845428i	$-0.679346\pi$
$773$	−29.6985			−1.06818			−0.534090	+	0.845428i	$0.679346\pi$
$774$	−32.0000	−	11.3137i	−1.15022	−	0.406663i
$775$		0			0
$776$	−16.9706			−0.609208
$777$	8.48528	−	6.00000i	0.304408	−	0.215249i
$778$	−4.00000			−0.143407
$779$	8.48528			0.304017
$780$	44.4853	−	13.4558i	1.59283	−	0.481797i
$781$	−54.0000			−1.93227
$782$	−8.48528			−0.303433
$783$	4.00000	+	14.1421i	0.142948	+	0.505399i
$784$	4.00000			0.142857
$785$	12.7279	−	38.1838i	0.454279	−	1.36284i
$786$	16.9706	−	12.0000i	0.605320	−	0.428026i
$787$	−22.0000			−0.784215			−0.392108	−	0.919919i	$-0.628254\pi$
$787$	−22.0000			−0.784215			−0.392108	+	0.919919i	$0.628254\pi$
$788$	50.9117			1.81365
$789$	22.0000	−	15.5563i	0.783221	−	0.553821i
$790$		0			0
$791$	−8.48528			−0.301702
$792$	33.9411	+	12.0000i	1.20605	+	0.426401i
$793$		−	60.0000i		−	2.13066i
$794$	42.4264			1.50566
$795$	9.51472	+	31.4558i	0.337452	+	1.11562i
$796$			12.0000i			0.425329i
$797$	4.24264			0.150282			0.0751410	−	0.997173i	$-0.476059\pi$
$797$	4.24264			0.150282			0.0751410	+	0.997173i	$0.476059\pi$
$798$	8.48528	+	12.0000i	0.300376	+	0.424795i
$799$		−	12.0000i		−	0.424529i
$800$	−16.9706	+	22.6274i	−0.600000	+	0.800000i
$801$	−20.0000	−	7.07107i	−0.706665	−	0.249844i
$802$	20.0000			0.706225
$803$		−	25.4558i		−	0.898317i
$804$	−8.00000	−	11.3137i	−0.282138	−	0.399004i
$805$	−1.00000	+	3.00000i	−0.0352454	+	0.105736i
$806$		0			0
$807$	−26.0000	+	18.3848i	−0.915243	+	0.647175i
$808$	44.0000			1.54791
$809$			45.2548i			1.59108i	0.605904	+	0.795538i	$0.292811\pi$
$809$			45.2548i			1.59108i	−0.605904	+	0.795538i	$0.707189\pi$
$810$	−23.9706	+	15.3431i	−0.842240	+	0.539103i
$811$			36.0000i			1.26413i	0.774915	+	0.632065i	$0.217793\pi$
$811$			36.0000i			1.26413i	−0.774915	+	0.632065i	$0.782207\pi$
$812$		−	5.65685i		−	0.198517i
$813$	33.9411	−	24.0000i	1.19037	−	0.841717i
$814$	−36.0000			−1.26180
$815$	−42.4264	−	14.1421i	−1.48613	−	0.495377i
$816$	16.9706	+	24.0000i	0.594089	+	0.840168i
$817$			48.0000i			1.67931i
$818$		−	31.1127i		−	1.08783i
$819$	−16.9706	−	6.00000i	−0.592999	−	0.209657i
$820$	2.00000	−	6.00000i	0.0698430	−	0.209529i
$821$		−	22.6274i		−	0.789702i	−0.918745	−	0.394851i	$-0.870796\pi$
$821$		−	22.6274i		−	0.789702i	0.918745	−	0.394851i	$-0.129204\pi$
$822$	33.9411	−	24.0000i	1.18383	−	0.837096i
$823$	−40.0000			−1.39431			−0.697156	−	0.716919i	$-0.745552\pi$
$823$	−40.0000			−1.39431			−0.697156	+	0.716919i	$0.745552\pi$
$824$		−	39.5980i		−	1.37946i
$825$	1.02944	−	36.7279i	0.0358404	−	1.27870i
$826$		0			0
$827$			32.5269i			1.13107i	0.824724	+	0.565536i	$0.191331\pi$
$827$			32.5269i			1.13107i	−0.824724	+	0.565536i	$0.808669\pi$
$828$	−8.00000	−	2.82843i	−0.278019	−	0.0982946i
$829$	2.00000			0.0694629			0.0347314	−	0.999397i	$-0.488942\pi$
$829$	2.00000			0.0694629			0.0347314	+	0.999397i	$0.488942\pi$
$830$	8.48528	+	2.82843i	0.294528	+	0.0981761i
$831$	8.48528	−	6.00000i	0.294351	−	0.208138i
$832$		−	48.0000i		−	1.66410i
$833$	−4.24264			−0.146999
$834$	−16.9706	−	24.0000i	−0.587643	−	0.831052i
$835$	−10.0000	+	30.0000i	−0.346064	+	1.03819i
$836$		−	50.9117i		−	1.76082i
$837$		0			0
$838$			24.0000i			0.829066i
$839$	−42.4264			−1.46472			−0.732361	−	0.680916i	$-0.761582\pi$
$839$	−42.4264			−1.46472			−0.732361	+	0.680916i	$0.761582\pi$
$840$	10.4853	−	3.17157i	0.361777	−	0.109430i
$841$	21.0000			0.724138
$842$			11.3137i			0.389896i
$843$	4.00000	−	2.82843i	0.137767	−	0.0974162i
$844$			24.0000i			0.826114i
$845$	48.7904	+	16.2635i	1.67844	+	0.559480i
$846$	4.00000	−	11.3137i	0.137523	−	0.388973i
$847$	7.00000			0.240523
$848$	33.9411			1.16554
$849$	4.00000	+	5.65685i	0.137280	+	0.194143i
$850$	18.0000	−	24.0000i	0.617395	−	0.823193i
$851$	8.48528			0.290872
$852$	−25.4558	−	36.0000i	−0.872103	−	1.23334i
$853$			30.0000i			1.02718i	0.858036	+	0.513590i	$0.171685\pi$
$853$			30.0000i			1.02718i	−0.858036	+	0.513590i	$0.828315\pi$
$854$		−	14.1421i		−	0.483934i
$855$	31.7574	+	24.7279i	1.08608	+	0.845677i
$856$	20.0000			0.683586
$857$	−21.2132			−0.724629			−0.362315	−	0.932056i	$-0.618013\pi$
$857$	−21.2132			−0.724629			−0.362315	+	0.932056i	$0.618013\pi$
$858$	36.0000	+	50.9117i	1.22902	+	1.73810i
$859$		−	6.00000i		−	0.204717i	−0.994748	−	0.102359i	$-0.967361\pi$
$859$		−	6.00000i		−	0.204717i	0.994748	−	0.102359i	$-0.0326389\pi$
$860$	33.9411	+	11.3137i	1.15738	+	0.385794i
$861$	−2.00000	+	1.41421i	−0.0681598	+	0.0481963i
$862$		−	6.00000i		−	0.204361i
$863$		−	18.3848i		−	0.625825i	−0.949782	−	0.312913i	$-0.898695\pi$
$863$		−	18.3848i		−	0.625825i	0.949782	−	0.312913i	$-0.101305\pi$
$864$	8.00000	+	28.2843i	0.272166	+	0.962250i
$865$	27.0000	+	9.00000i	0.918028	+	0.306009i
$866$	25.4558			0.865025
$867$	−1.00000	−	1.41421i	−0.0339618	−	0.0480292i
$868$		0			0
$869$		0			0
$870$	−4.48528	−	14.8284i	−0.152065	−	0.502731i
$871$		−	24.0000i		−	0.813209i
$872$		−	5.65685i		−	0.191565i
$873$	16.9706	+	6.00000i	0.574367	+	0.203069i
$874$			12.0000i			0.405906i
$875$	−6.36396	−	9.19239i	−0.215141	−	0.310759i
$876$	16.9706	−	12.0000i	0.573382	−	0.405442i
$877$		−	24.0000i		−	0.810422i	−0.914223	−	0.405211i	$-0.867198\pi$
$877$		−	24.0000i		−	0.810422i	0.914223	−	0.405211i	$-0.132802\pi$
$878$	−25.4558			−0.859093
$879$	21.2132	+	30.0000i	0.715504	+	1.01187i
$880$	−36.0000	−	12.0000i	−1.21356	−	0.404520i
$881$			7.07107i			0.238230i	0.992880	+	0.119115i	$0.0380058\pi$
$881$			7.07107i			0.238230i	−0.992880	+	0.119115i	$0.961994\pi$
$882$	−4.00000	−	1.41421i	−0.134687	−	0.0476190i
$883$	−16.0000			−0.538443			−0.269221	−	0.963078i	$-0.586766\pi$
$883$	−16.0000			−0.538443			−0.269221	+	0.963078i	$0.586766\pi$
$884$			50.9117i			1.71235i
$885$		0			0
$886$	50.0000			1.67978
$887$		−	14.1421i		−	0.474846i	−0.971406	−	0.237423i	$-0.923697\pi$
$887$		−	14.1421i		−	0.474846i	0.971406	−	0.237423i	$-0.0763028\pi$
$888$	−16.9706	−	24.0000i	−0.569495	−	0.805387i
$889$	−16.0000			−0.536623
$890$	21.2132	+	7.07107i	0.711068	+	0.237023i
$891$	−29.6985	−	24.0000i	−0.994937	−	0.804030i
$892$	−16.0000			−0.535720
$893$	−16.9706			−0.567898
$894$	16.0000	+	22.6274i	0.535120	+	0.756774i
$895$	9.00000	+	3.00000i	0.300837	+	0.100279i
$896$		−	11.3137i		−	0.377964i
$897$	−8.48528	−	12.0000i	−0.283315	−	0.400668i
$898$	8.00000			0.266963
$899$		0			0
$900$	24.9706	−	16.6274i	0.832352	−	0.554247i
$901$	−36.0000			−1.19933
$902$	8.48528			0.282529
$903$	−8.00000	−	11.3137i	−0.266223	−	0.376497i
$904$			24.0000i			0.798228i
$905$	−4.24264	−	1.41421i	−0.141030	−	0.0470100i
$906$	−16.9706	−	24.0000i	−0.563809	−	0.797347i
$907$	8.00000			0.265636			0.132818	−	0.991140i	$-0.457597\pi$
$907$	8.00000			0.265636			0.132818	+	0.991140i	$0.457597\pi$
$908$			45.2548i			1.50183i
$909$	−44.0000	−	15.5563i	−1.45939	−	0.515972i
$910$	18.0000	+	6.00000i	0.596694	+	0.198898i
$911$	−4.24264			−0.140565			−0.0702825	−	0.997527i	$-0.522390\pi$
$911$	−4.24264			−0.140565			−0.0702825	+	0.997527i	$0.522390\pi$
$912$	33.9411	−	24.0000i	1.12390	−	0.794719i
$913$			12.0000i			0.397142i
$914$		0			0
$915$	−11.2132	−	37.0711i	−0.370697	−	1.22553i
$916$	−28.0000			−0.925146
$917$	8.48528			0.280209
$918$	−8.48528	−	30.0000i	−0.280056	−	0.990148i
$919$		−	12.0000i		−	0.395843i	−0.980218	−	0.197922i	$-0.936581\pi$
$919$		−	12.0000i		−	0.395843i	0.980218	−	0.197922i	$-0.0634192\pi$
$920$	8.48528	+	2.82843i	0.279751	+	0.0932505i
$921$	−20.0000	−	28.2843i	−0.659022	−	0.931998i
$922$	−22.0000			−0.724531
$923$		−	76.3675i		−	2.51367i
$924$	8.48528	+	12.0000i	0.279145	+	0.394771i
$925$	−18.0000	+	24.0000i	−0.591836	+	0.789115i
$926$		−	5.65685i		−	0.185896i
$927$	−14.0000	+	39.5980i	−0.459820	+	1.30057i
$928$	−16.0000			−0.525226
$929$			41.0122i			1.34557i	0.739840	+	0.672783i	$0.234901\pi$
$929$			41.0122i			1.34557i	−0.739840	+	0.672783i	$0.765099\pi$
$930$		0			0
$931$			6.00000i			0.196642i
$932$	−33.9411			−1.11178
$933$	−25.4558	−	36.0000i	−0.833387	−	1.17859i
$934$	8.00000			0.261768
$935$	38.1838	+	12.7279i	1.24874	+	0.416248i
$936$	−16.9706	+	48.0000i	−0.554700	+	1.56893i
$937$		−	18.0000i		−	0.588034i	−0.955800	−	0.294017i	$-0.905008\pi$
$937$		−	18.0000i		−	0.588034i	0.955800	−	0.294017i	$-0.0949923\pi$
$938$		−	5.65685i		−	0.184703i
$939$	−25.4558	+	18.0000i	−0.830720	+	0.587408i
$940$	−4.00000	+	12.0000i	−0.130466	+	0.391397i
$941$		−	26.8701i		−	0.875939i	−0.898990	−	0.437969i	$-0.855698\pi$
$941$		−	26.8701i		−	0.875939i	0.898990	−	0.437969i	$-0.144302\pi$
$942$	25.4558	+	36.0000i	0.829396	+	1.17294i
$943$	−2.00000			−0.0651290
$944$		0			0
$945$	−11.6066	−	0.535534i	−0.377563	−	0.0174209i
$946$			48.0000i			1.56061i
$947$		−	9.89949i		−	0.321690i	−0.986980	−	0.160845i	$-0.948578\pi$
$947$		−	9.89949i		−	0.321690i	0.986980	−	0.160845i	$-0.0514220\pi$
$948$		0			0
$949$	36.0000			1.16861
$950$	−33.9411	−	25.4558i	−1.10120	−	0.825897i
$951$	−25.4558	−	36.0000i	−0.825462	−	1.16738i
$952$			12.0000i			0.388922i
$953$	−25.4558			−0.824596			−0.412298	−	0.911049i	$-0.635274\pi$
$953$	−25.4558			−0.824596			−0.412298	+	0.911049i	$0.635274\pi$
$954$	−33.9411	−	12.0000i	−1.09888	−	0.388514i
$955$	27.0000	+	9.00000i	0.873699	+	0.291233i
$956$	−8.48528			−0.274434
$957$	16.9706	−	12.0000i	0.548580	−	0.387905i
$958$		−	12.0000i		−	0.387702i
$959$	16.9706			0.548008
$960$	−8.97056	−	29.6569i	−0.289524	−	0.957171i
$961$	31.0000			1.00000
$962$		−	50.9117i		−	1.64146i
$963$	−20.0000	−	7.07107i	−0.644491	−	0.227862i
$964$	−52.0000			−1.67481
$965$	12.7279	−	38.1838i	0.409726	−	1.22918i
$966$	−2.00000	−	2.82843i	−0.0643489	−	0.0910032i
$967$	−4.00000			−0.128631			−0.0643157	−	0.997930i	$-0.520486\pi$
$967$	−4.00000			−0.128631			−0.0643157	+	0.997930i	$0.520486\pi$
$968$		−	19.7990i		−	0.636364i
$969$	−36.0000	+	25.4558i	−1.15649	+	0.817760i
$970$	−18.0000	−	6.00000i	−0.577945	−	0.192648i
$971$	42.4264			1.36153			0.680764	−	0.732503i	$-0.261648\pi$
$971$	42.4264			1.36153			0.680764	+	0.732503i	$0.261648\pi$
$972$	2.00000	−	31.1127i	0.0641500	−	0.997940i
$973$		−	12.0000i		−	0.384702i
$974$			28.2843i			0.906287i
$975$	51.9411	+	1.45584i	1.66345	+	0.0466243i
$976$	−40.0000			−1.28037
$977$		0			0			−	1.00000i	$-0.5\pi$
$977$		0			0				1.00000i	$0.5\pi$
$978$	40.0000	−	28.2843i	1.27906	−	0.904431i
$979$			30.0000i			0.958804i
$980$	4.24264	+	1.41421i	0.135526	+	0.0451754i
$981$	−2.00000	+	5.65685i	−0.0638551	+	0.180609i
$982$			42.0000i			1.34027i
$983$			28.2843i			0.902128i	0.892492	+	0.451064i	$0.148955\pi$
$983$			28.2843i			0.902128i	−0.892492	+	0.451064i	$0.851045\pi$
$984$	4.00000	+	5.65685i	0.127515	+	0.180334i
$985$	54.0000	+	18.0000i	1.72058	+	0.573528i
$986$	16.9706			0.540453
$987$	4.00000	−	2.82843i	0.127321	−	0.0900298i
$988$	72.0000			2.29063
$989$		−	11.3137i		−	0.359755i
$990$	31.7574	+	24.7279i	1.00932	+	0.785905i
$991$		−	60.0000i		−	1.90596i	−0.303029	−	0.952981i	$-0.597998\pi$
$991$		−	60.0000i		−	1.90596i	0.303029	−	0.952981i	$-0.402002\pi$
$992$		0			0
$993$		0			0
$994$		−	18.0000i		−	0.570925i
$995$	−4.24264	+	12.7279i	−0.134501	+	0.403502i
$996$	−8.00000	+	5.65685i	−0.253490	+	0.179244i
$997$			42.0000i			1.33015i	0.746775	+	0.665077i	$0.231601\pi$
$997$			42.0000i			1.33015i	−0.746775	+	0.665077i	$0.768399\pi$
$998$	−16.9706			−0.537194
$999$	8.48528	+	30.0000i	0.268462	+	0.949158i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	420.2.l.a.239.3	yes	4
3.2	odd	2	inner	420.2.l.a.239.2	yes	4
4.3	odd	2		420.2.l.b.239.1	yes	4
5.4	even	2		420.2.l.b.239.2	yes	4
12.11	even	2		420.2.l.b.239.4	yes	4
15.14	odd	2		420.2.l.b.239.3	yes	4
20.19	odd	2	inner	420.2.l.a.239.4	yes	4
60.59	even	2	inner	420.2.l.a.239.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
420.2.l.a.239.1	✓	4	60.59	even	2	inner
420.2.l.a.239.2	yes	4	3.2	odd	2	inner
420.2.l.a.239.3	yes	4	1.1	even	1	trivial
420.2.l.a.239.4	yes	4	20.19	odd	2	inner
420.2.l.b.239.1	yes	4	4.3	odd	2
420.2.l.b.239.2	yes	4	5.4	even	2
420.2.l.b.239.3	yes	4	15.14	odd	2
420.2.l.b.239.4	yes	4	12.11	even	2

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:	\( N \)	\(=\)	\( 420 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	420.l (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.41421i q^{2} +(-1.00000 - 1.41421i) q^{3} -2.00000 q^{4} +(-2.12132 - 0.707107i) q^{5} +(2.00000 - 1.41421i) q^{6} +1.00000 q^{7} -2.82843i q^{8} +(-1.00000 + 2.82843i) q^{9} +O(q^{10})\)	\(q+1.41421i q^{2} +(-1.00000 - 1.41421i) q^{3} -2.00000 q^{4} +(-2.12132 - 0.707107i) q^{5} +(2.00000 - 1.41421i) q^{6} +1.00000 q^{7} -2.82843i q^{8} +(-1.00000 + 2.82843i) q^{9} +(1.00000 - 3.00000i) q^{10} +4.24264 q^{11} +(2.00000 + 2.82843i) q^{12} +6.00000i q^{13} +1.41421i q^{14} +(1.12132 + 3.70711i) q^{15} +4.00000 q^{16} -4.24264 q^{17} +(-4.00000 - 1.41421i) q^{18} +6.00000i q^{19} +(4.24264 + 1.41421i) q^{20} +(-1.00000 - 1.41421i) q^{21} +6.00000i q^{22} -1.41421i q^{23} +(-4.00000 + 2.82843i) q^{24} +(4.00000 + 3.00000i) q^{25} -8.48528 q^{26} +(5.00000 - 1.41421i) q^{27} -2.00000 q^{28} +2.82843i q^{29} +(-5.24264 + 1.58579i) q^{30} +5.65685i q^{32} +(-4.24264 - 6.00000i) q^{33} -6.00000i q^{34} +(-2.12132 - 0.707107i) q^{35} +(2.00000 - 5.65685i) q^{36} +6.00000i q^{37} -8.48528 q^{38} +(8.48528 - 6.00000i) q^{39} +(-2.00000 + 6.00000i) q^{40} -1.41421i q^{41} +(2.00000 - 1.41421i) q^{42} +8.00000 q^{43} -8.48528 q^{44} +(4.12132 - 5.29289i) q^{45} +2.00000 q^{46} +2.82843i q^{47} +(-4.00000 - 5.65685i) q^{48} +1.00000 q^{49} +(-4.24264 + 5.65685i) q^{50} +(4.24264 + 6.00000i) q^{51} -12.0000i q^{52} +8.48528 q^{53} +(2.00000 + 7.07107i) q^{54} +(-9.00000 - 3.00000i) q^{55} -2.82843i q^{56} +(8.48528 - 6.00000i) q^{57} -4.00000 q^{58} +(-2.24264 - 7.41421i) q^{60} -10.0000 q^{61} +(-1.00000 + 2.82843i) q^{63} -8.00000 q^{64} +(4.24264 - 12.7279i) q^{65} +(8.48528 - 6.00000i) q^{66} -4.00000 q^{67} +8.48528 q^{68} +(-2.00000 + 1.41421i) q^{69} +(1.00000 - 3.00000i) q^{70} -12.7279 q^{71} +(8.00000 + 2.82843i) q^{72} -6.00000i q^{73} -8.48528 q^{74} +(0.242641 - 8.65685i) q^{75} -12.0000i q^{76} +4.24264 q^{77} +(8.48528 + 12.0000i) q^{78} +(-8.48528 - 2.82843i) q^{80} +(-7.00000 - 5.65685i) q^{81} +2.00000 q^{82} +2.82843i q^{83} +(2.00000 + 2.82843i) q^{84} +(9.00000 + 3.00000i) q^{85} +11.3137i q^{86} +(4.00000 - 2.82843i) q^{87} -12.0000i q^{88} +7.07107i q^{89} +(7.48528 + 5.82843i) q^{90} +6.00000i q^{91} +2.82843i q^{92} -4.00000 q^{94} +(4.24264 - 12.7279i) q^{95} +(8.00000 - 5.65685i) q^{96} -6.00000i q^{97} +1.41421i q^{98} +(-4.24264 + 12.0000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} - 8 q^{4} + 8 q^{6} + 4 q^{7} - 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^3 - 8 * q^4 + 8 * q^6 + 4 * q^7 - 4 * q^9	\( 4 q - 4 q^{3} - 8 q^{4} + 8 q^{6} + 4 q^{7} - 4 q^{9} + 4 q^{10} + 8 q^{12} - 4 q^{15} + 16 q^{16} - 16 q^{18} - 4 q^{21} - 16 q^{24} + 16 q^{25} + 20 q^{27} - 8 q^{28} - 4 q^{30} + 8 q^{36} - 8 q^{40} + 8 q^{42} + 32 q^{43} + 8 q^{45} + 8 q^{46} - 16 q^{48} + 4 q^{49} + 8 q^{54} - 36 q^{55} - 16 q^{58} + 8 q^{60} - 40 q^{61} - 4 q^{63} - 32 q^{64} - 16 q^{67} - 8 q^{69} + 4 q^{70} + 32 q^{72} - 16 q^{75} - 28 q^{81} + 8 q^{82} + 8 q^{84} + 36 q^{85} + 16 q^{87} - 4 q^{90} - 16 q^{94} + 32 q^{96}+O(q^{100}) \) 4 * q - 4 * q^3 - 8 * q^4 + 8 * q^6 + 4 * q^7 - 4 * q^9 + 4 * q^10 + 8 * q^12 - 4 * q^15 + 16 * q^16 - 16 * q^18 - 4 * q^21 - 16 * q^24 + 16 * q^25 + 20 * q^27 - 8 * q^28 - 4 * q^30 + 8 * q^36 - 8 * q^40 + 8 * q^42 + 32 * q^43 + 8 * q^45 + 8 * q^46 - 16 * q^48 + 4 * q^49 + 8 * q^54 - 36 * q^55 - 16 * q^58 + 8 * q^60 - 40 * q^61 - 4 * q^63 - 32 * q^64 - 16 * q^67 - 8 * q^69 + 4 * q^70 + 32 * q^72 - 16 * q^75 - 28 * q^81 + 8 * q^82 + 8 * q^84 + 36 * q^85 + 16 * q^87 - 4 * q^90 - 16 * q^94 + 32 * q^96

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more