LMFDB - Embedded newform 114.2.b.d.113.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [114,2,Mod(113,114)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("114.113"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(114, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 114 = 2 \cdot 3 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	114.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [2,2,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	no
Analytic conductor:	$0.910294583043$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-2}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 2 $ x^2 + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			113.2
Root			$1.41421i$ of defining polynomial
Character	$\chi$	$=$	114.113
Dual form			114.2.b.d.113.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000 q^{2} +(1.00000 + 1.41421i) q^{3} +1.00000 q^{4} -1.41421i q^{5} +(1.00000 + 1.41421i) q^{6} -4.00000 q^{7} +1.00000 q^{8} +(-1.00000 + 2.82843i) q^{9} -1.41421i q^{10} -5.65685i q^{11} +(1.00000 + 1.41421i) q^{12} +4.24264i q^{13} -4.00000 q^{14} +(2.00000 - 1.41421i) q^{15} +1.00000 q^{16} +2.82843i q^{17} +(-1.00000 + 2.82843i) q^{18} +(1.00000 - 4.24264i) q^{19} -1.41421i q^{20} +(-4.00000 - 5.65685i) q^{21} -5.65685i q^{22} -1.41421i q^{23} +(1.00000 + 1.41421i) q^{24} +3.00000 q^{25} +4.24264i q^{26} +(-5.00000 + 1.41421i) q^{27} -4.00000 q^{28} -6.00000 q^{29} +(2.00000 - 1.41421i) q^{30} +4.24264i q^{31} +1.00000 q^{32} +(8.00000 - 5.65685i) q^{33} +2.82843i q^{34} +5.65685i q^{35} +(-1.00000 + 2.82843i) q^{36} +4.24264i q^{37} +(1.00000 - 4.24264i) q^{38} +(-6.00000 + 4.24264i) q^{39} -1.41421i q^{40} +(-4.00000 - 5.65685i) q^{42} +2.00000 q^{43} -5.65685i q^{44} +(4.00000 + 1.41421i) q^{45} -1.41421i q^{46} -1.41421i q^{47} +(1.00000 + 1.41421i) q^{48} +9.00000 q^{49} +3.00000 q^{50} +(-4.00000 + 2.82843i) q^{51} +4.24264i q^{52} +6.00000 q^{53} +(-5.00000 + 1.41421i) q^{54} -8.00000 q^{55} -4.00000 q^{56} +(7.00000 - 2.82843i) q^{57} -6.00000 q^{58} +12.0000 q^{59} +(2.00000 - 1.41421i) q^{60} +2.00000 q^{61} +4.24264i q^{62} +(4.00000 - 11.3137i) q^{63} +1.00000 q^{64} +6.00000 q^{65} +(8.00000 - 5.65685i) q^{66} +2.82843i q^{68} +(2.00000 - 1.41421i) q^{69} +5.65685i q^{70} -12.0000 q^{71} +(-1.00000 + 2.82843i) q^{72} -4.00000 q^{73} +4.24264i q^{74} +(3.00000 + 4.24264i) q^{75} +(1.00000 - 4.24264i) q^{76} +22.6274i q^{77} +(-6.00000 + 4.24264i) q^{78} +4.24264i q^{79} -1.41421i q^{80} +(-7.00000 - 5.65685i) q^{81} +2.82843i q^{83} +(-4.00000 - 5.65685i) q^{84} +4.00000 q^{85} +2.00000 q^{86} +(-6.00000 - 8.48528i) q^{87} -5.65685i q^{88} +6.00000 q^{89} +(4.00000 + 1.41421i) q^{90} -16.9706i q^{91} -1.41421i q^{92} +(-6.00000 + 4.24264i) q^{93} -1.41421i q^{94} +(-6.00000 - 1.41421i) q^{95} +(1.00000 + 1.41421i) q^{96} -8.48528i q^{97} +9.00000 q^{98} +(16.0000 + 5.65685i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{6} - 8 q^{7} + 2 q^{8} - 2 q^{9} + 2 q^{12} - 8 q^{14} + 4 q^{15} + 2 q^{16} - 2 q^{18} + 2 q^{19} - 8 q^{21} + 2 q^{24} + 6 q^{25} - 10 q^{27} - 8 q^{28} - 12 q^{29}+ \cdots + 32 q^{99}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + 2 * q^6 - 8 * q^7 + 2 * q^8 - 2 * q^9 + 2 * q^12 - 8 * q^14 + 4 * q^15 + 2 * q^16 - 2 * q^18 + 2 * q^19 - 8 * q^21 + 2 * q^24 + 6 * q^25 - 10 * q^27 - 8 * q^28 - 12 * q^29 + 4 * q^30 + 2 * q^32 + 16 * q^33 - 2 * q^36 + 2 * q^38 - 12 * q^39 - 8 * q^42 + 4 * q^43 + 8 * q^45 + 2 * q^48 + 18 * q^49 + 6 * q^50 - 8 * q^51 + 12 * q^53 - 10 * q^54 - 16 * q^55 - 8 * q^56 + 14 * q^57 - 12 * q^58 + 24 * q^59 + 4 * q^60 + 4 * q^61 + 8 * q^63 + 2 * q^64 + 12 * q^65 + 16 * q^66 + 4 * q^69 - 24 * q^71 - 2 * q^72 - 8 * q^73 + 6 * q^75 + 2 * q^76 - 12 * q^78 - 14 * q^81 - 8 * q^84 + 8 * q^85 + 4 * q^86 - 12 * q^87 + 12 * q^89 + 8 * q^90 - 12 * q^93 - 12 * q^95 + 2 * q^96 + 18 * q^98 + 32 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/114\mathbb{Z}\right)^\times$.

$n$	$77$	$97$
$\chi(n)$	$-1$	$-1$

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 $

$2$	1.00000			0.707107
$2$	1.00000			0.707107
$3$	1.00000	+	1.41421i	0.577350	+	0.816497i
$3$	1.00000	+	1.41421i	0.577350	+	0.816497i
$4$	1.00000			0.500000
$5$		−	1.41421i		−	0.632456i	−0.948683	−	0.316228i	$-0.897584\pi$
$5$		−	1.41421i		−	0.632456i	0.948683	−	0.316228i	$-0.102416\pi$
$6$	1.00000	+	1.41421i	0.408248	+	0.577350i
$7$	−4.00000			−1.51186			−0.755929	−	0.654654i	$-0.772814\pi$
$7$	−4.00000			−1.51186			−0.755929	+	0.654654i	$0.772814\pi$
$8$	1.00000			0.353553
$9$	−1.00000	+	2.82843i	−0.333333	+	0.942809i
$10$		−	1.41421i		−	0.447214i
$11$		−	5.65685i		−	1.70561i	−0.522233	−	0.852803i	$-0.674901\pi$
$11$		−	5.65685i		−	1.70561i	0.522233	−	0.852803i	$-0.325099\pi$
$12$	1.00000	+	1.41421i	0.288675	+	0.408248i
$13$			4.24264i			1.17670i	0.808608	+	0.588348i	$0.200222\pi$
$13$			4.24264i			1.17670i	−0.808608	+	0.588348i	$0.799778\pi$
$14$	−4.00000			−1.06904
$15$	2.00000	−	1.41421i	0.516398	−	0.365148i
$16$	1.00000			0.250000
$17$			2.82843i			0.685994i	0.939336	+	0.342997i	$0.111442\pi$
$17$			2.82843i			0.685994i	−0.939336	+	0.342997i	$0.888558\pi$
$18$	−1.00000	+	2.82843i	−0.235702	+	0.666667i
$19$	1.00000	−	4.24264i	0.229416	−	0.973329i
$19$	1.00000	−	4.24264i	0.229416	−	0.973329i
$20$		−	1.41421i		−	0.316228i
$21$	−4.00000	−	5.65685i	−0.872872	−	1.23443i
$22$		−	5.65685i		−	1.20605i
$23$		−	1.41421i		−	0.294884i	−0.989071	−	0.147442i	$-0.952896\pi$
$23$		−	1.41421i		−	0.294884i	0.989071	−	0.147442i	$-0.0471040\pi$
$24$	1.00000	+	1.41421i	0.204124	+	0.288675i
$25$	3.00000			0.600000
$26$			4.24264i			0.832050i
$27$	−5.00000	+	1.41421i	−0.962250	+	0.272166i
$28$	−4.00000			−0.755929
$29$	−6.00000			−1.11417			−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000			−1.11417			−0.557086	+	0.830455i	$0.688081\pi$
$30$	2.00000	−	1.41421i	0.365148	−	0.258199i
$31$			4.24264i			0.762001i	0.924575	+	0.381000i	$0.124420\pi$
$31$			4.24264i			0.762001i	−0.924575	+	0.381000i	$0.875580\pi$
$32$	1.00000			0.176777
$33$	8.00000	−	5.65685i	1.39262	−	0.984732i
$34$			2.82843i			0.485071i
$35$			5.65685i			0.956183i
$36$	−1.00000	+	2.82843i	−0.166667	+	0.471405i
$37$			4.24264i			0.697486i	0.937218	+	0.348743i	$0.113391\pi$
$37$			4.24264i			0.697486i	−0.937218	+	0.348743i	$0.886609\pi$
$38$	1.00000	−	4.24264i	0.162221	−	0.688247i
$39$	−6.00000	+	4.24264i	−0.960769	+	0.679366i
$40$		−	1.41421i		−	0.223607i
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$	−4.00000	−	5.65685i	−0.617213	−	0.872872i
$43$	2.00000			0.304997			0.152499	−	0.988304i	$-0.451268\pi$
$43$	2.00000			0.304997			0.152499	+	0.988304i	$0.451268\pi$
$44$		−	5.65685i		−	0.852803i
$45$	4.00000	+	1.41421i	0.596285	+	0.210819i
$46$		−	1.41421i		−	0.208514i
$47$		−	1.41421i		−	0.206284i	−0.994667	−	0.103142i	$-0.967110\pi$
$47$		−	1.41421i		−	0.206284i	0.994667	−	0.103142i	$-0.0328896\pi$
$48$	1.00000	+	1.41421i	0.144338	+	0.204124i
$49$	9.00000			1.28571
$50$	3.00000			0.424264
$51$	−4.00000	+	2.82843i	−0.560112	+	0.396059i
$52$			4.24264i			0.588348i
$53$	6.00000			0.824163			0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000			0.824163			0.412082	+	0.911147i	$0.364802\pi$
$54$	−5.00000	+	1.41421i	−0.680414	+	0.192450i
$55$	−8.00000			−1.07872
$56$	−4.00000			−0.534522
$57$	7.00000	−	2.82843i	0.927173	−	0.374634i
$58$	−6.00000			−0.787839
$59$	12.0000			1.56227			0.781133	−	0.624364i	$-0.214642\pi$
$59$	12.0000			1.56227			0.781133	+	0.624364i	$0.214642\pi$
$60$	2.00000	−	1.41421i	0.258199	−	0.182574i
$61$	2.00000			0.256074			0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000			0.256074			0.128037	+	0.991769i	$0.459132\pi$
$62$			4.24264i			0.538816i
$63$	4.00000	−	11.3137i	0.503953	−	1.42539i
$64$	1.00000			0.125000
$65$	6.00000			0.744208
$66$	8.00000	−	5.65685i	0.984732	−	0.696311i
$67$		0			0		1.00000			$0$
$67$		0			0		−1.00000			$\pi$
$68$			2.82843i			0.342997i
$69$	2.00000	−	1.41421i	0.240772	−	0.170251i
$70$			5.65685i			0.676123i
$71$	−12.0000			−1.42414			−0.712069	−	0.702109i	$-0.752242\pi$
$71$	−12.0000			−1.42414			−0.712069	+	0.702109i	$0.752242\pi$
$72$	−1.00000	+	2.82843i	−0.117851	+	0.333333i
$73$	−4.00000			−0.468165			−0.234082	−	0.972217i	$-0.575209\pi$
$73$	−4.00000			−0.468165			−0.234082	+	0.972217i	$0.575209\pi$
$74$			4.24264i			0.493197i
$75$	3.00000	+	4.24264i	0.346410	+	0.489898i
$76$	1.00000	−	4.24264i	0.114708	−	0.486664i
$77$			22.6274i			2.57863i
$78$	−6.00000	+	4.24264i	−0.679366	+	0.480384i
$79$			4.24264i			0.477334i	0.971101	+	0.238667i	$0.0767105\pi$
$79$			4.24264i			0.477334i	−0.971101	+	0.238667i	$0.923290\pi$
$80$		−	1.41421i		−	0.158114i
$81$	−7.00000	−	5.65685i	−0.777778	−	0.628539i
$82$		0			0
$83$			2.82843i			0.310460i	0.987878	+	0.155230i	$0.0496119\pi$
$83$			2.82843i			0.310460i	−0.987878	+	0.155230i	$0.950388\pi$
$84$	−4.00000	−	5.65685i	−0.436436	−	0.617213i
$85$	4.00000			0.433861
$86$	2.00000			0.215666
$87$	−6.00000	−	8.48528i	−0.643268	−	0.909718i
$88$		−	5.65685i		−	0.603023i
$89$	6.00000			0.635999			0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000			0.635999			0.317999	+	0.948091i	$0.396989\pi$
$90$	4.00000	+	1.41421i	0.421637	+	0.149071i
$91$		−	16.9706i		−	1.77900i
$92$		−	1.41421i		−	0.147442i
$93$	−6.00000	+	4.24264i	−0.622171	+	0.439941i
$94$		−	1.41421i		−	0.145865i
$95$	−6.00000	−	1.41421i	−0.615587	−	0.145095i
$96$	1.00000	+	1.41421i	0.102062	+	0.144338i
$97$		−	8.48528i		−	0.861550i	−0.902459	−	0.430775i	$-0.858240\pi$
$97$		−	8.48528i		−	0.861550i	0.902459	−	0.430775i	$-0.141760\pi$
$98$	9.00000			0.909137
$99$	16.0000	+	5.65685i	1.60806	+	0.568535i
$100$	3.00000			0.300000
$101$		−	9.89949i		−	0.985037i	−0.870302	−	0.492518i	$-0.836076\pi$
$101$		−	9.89949i		−	0.985037i	0.870302	−	0.492518i	$-0.163924\pi$
$102$	−4.00000	+	2.82843i	−0.396059	+	0.280056i
$103$			12.7279i			1.25412i	0.778971	+	0.627060i	$0.215742\pi$
$103$			12.7279i			1.25412i	−0.778971	+	0.627060i	$0.784258\pi$
$104$			4.24264i			0.416025i
$105$	−8.00000	+	5.65685i	−0.780720	+	0.552052i
$106$	6.00000			0.582772
$107$	−12.0000			−1.16008			−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000			−1.16008			−0.580042	+	0.814587i	$0.696964\pi$
$108$	−5.00000	+	1.41421i	−0.481125	+	0.136083i
$109$		−	4.24264i		−	0.406371i	−0.979140	−	0.203186i	$-0.934871\pi$
$109$		−	4.24264i		−	0.406371i	0.979140	−	0.203186i	$-0.0651295\pi$
$110$	−8.00000			−0.762770
$111$	−6.00000	+	4.24264i	−0.569495	+	0.402694i
$112$	−4.00000			−0.377964
$113$	−12.0000			−1.12887			−0.564433	−	0.825479i	$-0.690905\pi$
$113$	−12.0000			−1.12887			−0.564433	+	0.825479i	$0.690905\pi$
$114$	7.00000	−	2.82843i	0.655610	−	0.264906i
$115$	−2.00000			−0.186501
$116$	−6.00000			−0.557086
$117$	−12.0000	−	4.24264i	−1.10940	−	0.392232i
$118$	12.0000			1.10469
$119$		−	11.3137i		−	1.03713i
$120$	2.00000	−	1.41421i	0.182574	−	0.129099i
$121$	−21.0000			−1.90909
$122$	2.00000			0.181071
$123$		0			0
$124$			4.24264i			0.381000i
$125$		−	11.3137i		−	1.01193i
$126$	4.00000	−	11.3137i	0.356348	−	1.00791i
$127$		−	21.2132i		−	1.88237i	−0.337895	−	0.941184i	$-0.609715\pi$
$127$		−	21.2132i		−	1.88237i	0.337895	−	0.941184i	$-0.390285\pi$
$128$	1.00000			0.0883883
$129$	2.00000	+	2.82843i	0.176090	+	0.249029i
$130$	6.00000			0.526235
$131$			19.7990i			1.72985i	0.501905	+	0.864923i	$0.332633\pi$
$131$			19.7990i			1.72985i	−0.501905	+	0.864923i	$0.667367\pi$
$132$	8.00000	−	5.65685i	0.696311	−	0.492366i
$133$	−4.00000	+	16.9706i	−0.346844	+	1.47153i
$134$		0			0
$135$	2.00000	+	7.07107i	0.172133	+	0.608581i
$136$			2.82843i			0.242536i
$137$		−	14.1421i		−	1.20824i	−0.796892	−	0.604122i	$-0.793524\pi$
$137$		−	14.1421i		−	1.20824i	0.796892	−	0.604122i	$-0.206476\pi$
$138$	2.00000	−	1.41421i	0.170251	−	0.120386i
$139$	−4.00000			−0.339276			−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000			−0.339276			−0.169638	+	0.985506i	$0.554260\pi$
$140$			5.65685i			0.478091i
$141$	2.00000	−	1.41421i	0.168430	−	0.119098i
$142$	−12.0000			−1.00702
$143$	24.0000			2.00698
$144$	−1.00000	+	2.82843i	−0.0833333	+	0.235702i
$145$			8.48528i			0.704664i
$146$	−4.00000			−0.331042
$147$	9.00000	+	12.7279i	0.742307	+	1.04978i
$148$			4.24264i			0.348743i
$149$		−	9.89949i		−	0.810998i	−0.914095	−	0.405499i	$-0.867098\pi$
$149$		−	9.89949i		−	0.810998i	0.914095	−	0.405499i	$-0.132902\pi$
$150$	3.00000	+	4.24264i	0.244949	+	0.346410i
$151$		−	12.7279i		−	1.03578i	−0.855446	−	0.517892i	$-0.826717\pi$
$151$		−	12.7279i		−	1.03578i	0.855446	−	0.517892i	$-0.173283\pi$
$152$	1.00000	−	4.24264i	0.0811107	−	0.344124i
$153$	−8.00000	−	2.82843i	−0.646762	−	0.228665i
$154$			22.6274i			1.82337i
$155$	6.00000			0.481932
$156$	−6.00000	+	4.24264i	−0.480384	+	0.339683i
$157$	14.0000			1.11732			0.558661	−	0.829396i	$-0.311315\pi$
$157$	14.0000			1.11732			0.558661	+	0.829396i	$0.311315\pi$
$158$			4.24264i			0.337526i
$159$	6.00000	+	8.48528i	0.475831	+	0.672927i
$160$		−	1.41421i		−	0.111803i
$161$			5.65685i			0.445823i
$162$	−7.00000	−	5.65685i	−0.549972	−	0.444444i
$163$	20.0000			1.56652			0.783260	−	0.621694i	$-0.213555\pi$
$163$	20.0000			1.56652			0.783260	+	0.621694i	$0.213555\pi$
$164$		0			0
$165$	−8.00000	−	11.3137i	−0.622799	−	0.880771i
$166$			2.82843i			0.219529i
$167$		0			0			−	1.00000i	$-0.5\pi$
$167$		0			0				1.00000i	$0.5\pi$
$168$	−4.00000	−	5.65685i	−0.308607	−	0.436436i
$169$	−5.00000			−0.384615
$170$	4.00000			0.306786
$171$	11.0000	+	7.07107i	0.841191	+	0.540738i
$172$	2.00000			0.152499
$173$	6.00000			0.456172			0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000			0.456172			0.228086	+	0.973641i	$0.426753\pi$
$174$	−6.00000	−	8.48528i	−0.454859	−	0.643268i
$175$	−12.0000			−0.907115
$176$		−	5.65685i		−	0.426401i
$177$	12.0000	+	16.9706i	0.901975	+	1.27559i
$178$	6.00000			0.449719
$179$	−18.0000			−1.34538			−0.672692	−	0.739923i	$-0.734862\pi$
$179$	−18.0000			−1.34538			−0.672692	+	0.739923i	$0.734862\pi$
$180$	4.00000	+	1.41421i	0.298142	+	0.105409i
$181$			12.7279i			0.946059i	0.881047	+	0.473029i	$0.156840\pi$
$181$			12.7279i			0.946059i	−0.881047	+	0.473029i	$0.843160\pi$
$182$		−	16.9706i		−	1.25794i
$183$	2.00000	+	2.82843i	0.147844	+	0.209083i
$184$		−	1.41421i		−	0.104257i
$185$	6.00000			0.441129
$186$	−6.00000	+	4.24264i	−0.439941	+	0.311086i
$187$	16.0000			1.17004
$188$		−	1.41421i		−	0.103142i
$189$	20.0000	−	5.65685i	1.45479	−	0.411476i
$190$	−6.00000	−	1.41421i	−0.435286	−	0.102598i
$191$		−	18.3848i		−	1.33028i	−0.746721	−	0.665138i	$-0.768373\pi$
$191$		−	18.3848i		−	1.33028i	0.746721	−	0.665138i	$-0.231627\pi$
$192$	1.00000	+	1.41421i	0.0721688	+	0.102062i
$193$			8.48528i			0.610784i	0.952227	+	0.305392i	$0.0987875\pi$
$193$			8.48528i			0.610784i	−0.952227	+	0.305392i	$0.901213\pi$
$194$		−	8.48528i		−	0.609208i
$195$	6.00000	+	8.48528i	0.429669	+	0.607644i
$196$	9.00000			0.642857
$197$			7.07107i			0.503793i	0.967754	+	0.251896i	$0.0810542\pi$
$197$			7.07107i			0.503793i	−0.967754	+	0.251896i	$0.918946\pi$
$198$	16.0000	+	5.65685i	1.13707	+	0.402015i
$199$	−4.00000			−0.283552			−0.141776	−	0.989899i	$-0.545281\pi$
$199$	−4.00000			−0.283552			−0.141776	+	0.989899i	$0.545281\pi$
$200$	3.00000			0.212132
$201$		0			0
$202$		−	9.89949i		−	0.696526i
$203$	24.0000			1.68447
$204$	−4.00000	+	2.82843i	−0.280056	+	0.198030i
$205$		0			0
$206$			12.7279i			0.886796i
$207$	4.00000	+	1.41421i	0.278019	+	0.0982946i
$208$			4.24264i			0.294174i
$209$	−24.0000	−	5.65685i	−1.66011	−	0.391293i
$210$	−8.00000	+	5.65685i	−0.552052	+	0.390360i
$211$		−	8.48528i		−	0.584151i	−0.956395	−	0.292075i	$-0.905654\pi$
$211$		−	8.48528i		−	0.584151i	0.956395	−	0.292075i	$-0.0943458\pi$
$212$	6.00000			0.412082
$213$	−12.0000	−	16.9706i	−0.822226	−	1.16280i
$214$	−12.0000			−0.820303
$215$		−	2.82843i		−	0.192897i
$216$	−5.00000	+	1.41421i	−0.340207	+	0.0962250i
$217$		−	16.9706i		−	1.15204i
$218$		−	4.24264i		−	0.287348i
$219$	−4.00000	−	5.65685i	−0.270295	−	0.382255i
$220$	−8.00000			−0.539360
$221$	−12.0000			−0.807207
$222$	−6.00000	+	4.24264i	−0.402694	+	0.284747i
$223$		−	4.24264i		−	0.284108i	−0.989859	−	0.142054i	$-0.954629\pi$
$223$		−	4.24264i		−	0.284108i	0.989859	−	0.142054i	$-0.0453707\pi$
$224$	−4.00000			−0.267261
$225$	−3.00000	+	8.48528i	−0.200000	+	0.565685i
$226$	−12.0000			−0.798228
$227$	−18.0000			−1.19470			−0.597351	−	0.801980i	$-0.703780\pi$
$227$	−18.0000			−1.19470			−0.597351	+	0.801980i	$0.703780\pi$
$228$	7.00000	−	2.82843i	0.463586	−	0.187317i
$229$	2.00000			0.132164			0.0660819	−	0.997814i	$-0.478950\pi$
$229$	2.00000			0.132164			0.0660819	+	0.997814i	$0.478950\pi$
$230$	−2.00000			−0.131876
$231$	−32.0000	+	22.6274i	−2.10545	+	1.48877i
$232$	−6.00000			−0.393919
$233$			11.3137i			0.741186i	0.928795	+	0.370593i	$0.120845\pi$
$233$			11.3137i			0.741186i	−0.928795	+	0.370593i	$0.879155\pi$
$234$	−12.0000	−	4.24264i	−0.784465	−	0.277350i
$235$	−2.00000			−0.130466
$236$	12.0000			0.781133
$237$	−6.00000	+	4.24264i	−0.389742	+	0.275589i
$238$		−	11.3137i		−	0.733359i
$239$			15.5563i			1.00626i	0.864212	+	0.503128i	$0.167818\pi$
$239$			15.5563i			1.00626i	−0.864212	+	0.503128i	$0.832182\pi$
$240$	2.00000	−	1.41421i	0.129099	−	0.0912871i
$241$			8.48528i			0.546585i	0.961931	+	0.273293i	$0.0881127\pi$
$241$			8.48528i			0.546585i	−0.961931	+	0.273293i	$0.911887\pi$
$242$	−21.0000			−1.34993
$243$	1.00000	−	15.5563i	0.0641500	−	0.997940i
$244$	2.00000			0.128037
$245$		−	12.7279i		−	0.813157i
$246$		0			0
$247$	18.0000	+	4.24264i	1.14531	+	0.269953i
$248$			4.24264i			0.269408i
$249$	−4.00000	+	2.82843i	−0.253490	+	0.179244i
$250$		−	11.3137i		−	0.715542i
$251$			19.7990i			1.24970i	0.780744	+	0.624851i	$0.214840\pi$
$251$			19.7990i			1.24970i	−0.780744	+	0.624851i	$0.785160\pi$
$252$	4.00000	−	11.3137i	0.251976	−	0.712697i
$253$	−8.00000			−0.502956
$254$		−	21.2132i		−	1.33103i
$255$	4.00000	+	5.65685i	0.250490	+	0.354246i
$256$	1.00000			0.0625000
$257$	6.00000			0.374270			0.187135	−	0.982334i	$-0.440080\pi$
$257$	6.00000			0.374270			0.187135	+	0.982334i	$0.440080\pi$
$258$	2.00000	+	2.82843i	0.124515	+	0.176090i
$259$		−	16.9706i		−	1.05450i
$260$	6.00000			0.372104
$261$	6.00000	−	16.9706i	0.371391	−	1.05045i
$262$			19.7990i			1.22319i
$263$			15.5563i			0.959246i	0.877475	+	0.479623i	$0.159226\pi$
$263$			15.5563i			0.959246i	−0.877475	+	0.479623i	$0.840774\pi$
$264$	8.00000	−	5.65685i	0.492366	−	0.348155i
$265$		−	8.48528i		−	0.521247i
$266$	−4.00000	+	16.9706i	−0.245256	+	1.04053i
$267$	6.00000	+	8.48528i	0.367194	+	0.519291i
$268$		0			0
$269$	18.0000			1.09748			0.548740	−	0.835993i	$-0.315108\pi$
$269$	18.0000			1.09748			0.548740	+	0.835993i	$0.315108\pi$
$270$	2.00000	+	7.07107i	0.121716	+	0.430331i
$271$	−16.0000			−0.971931			−0.485965	−	0.873978i	$-0.661532\pi$
$271$	−16.0000			−0.971931			−0.485965	+	0.873978i	$0.661532\pi$
$272$			2.82843i			0.171499i
$273$	24.0000	−	16.9706i	1.45255	−	1.02711i
$274$		−	14.1421i		−	0.854358i
$275$		−	16.9706i		−	1.02336i
$276$	2.00000	−	1.41421i	0.120386	−	0.0851257i
$277$	−22.0000			−1.32185			−0.660926	−	0.750451i	$-0.729836\pi$
$277$	−22.0000			−1.32185			−0.660926	+	0.750451i	$0.729836\pi$
$278$	−4.00000			−0.239904
$279$	−12.0000	−	4.24264i	−0.718421	−	0.254000i
$280$			5.65685i			0.338062i
$281$	−12.0000			−0.715860			−0.357930	−	0.933748i	$-0.616517\pi$
$281$	−12.0000			−0.715860			−0.357930	+	0.933748i	$0.616517\pi$
$282$	2.00000	−	1.41421i	0.119098	−	0.0842152i
$283$	20.0000			1.18888			0.594438	−	0.804141i	$-0.297374\pi$
$283$	20.0000			1.18888			0.594438	+	0.804141i	$0.297374\pi$
$284$	−12.0000			−0.712069
$285$	−4.00000	−	9.89949i	−0.236940	−	0.586395i
$286$	24.0000			1.41915
$287$		0			0
$288$	−1.00000	+	2.82843i	−0.0589256	+	0.166667i
$289$	9.00000			0.529412
$290$			8.48528i			0.498273i
$291$	12.0000	−	8.48528i	0.703452	−	0.497416i
$292$	−4.00000			−0.234082
$293$	−6.00000			−0.350524			−0.175262	−	0.984522i	$-0.556077\pi$
$293$	−6.00000			−0.350524			−0.175262	+	0.984522i	$0.556077\pi$
$294$	9.00000	+	12.7279i	0.524891	+	0.742307i
$295$		−	16.9706i		−	0.988064i
$296$			4.24264i			0.246598i
$297$	8.00000	+	28.2843i	0.464207	+	1.64122i
$298$		−	9.89949i		−	0.573462i
$299$	6.00000			0.346989
$300$	3.00000	+	4.24264i	0.173205	+	0.244949i
$301$	−8.00000			−0.461112
$302$		−	12.7279i		−	0.732410i
$303$	14.0000	−	9.89949i	0.804279	−	0.568711i
$304$	1.00000	−	4.24264i	0.0573539	−	0.243332i
$305$		−	2.82843i		−	0.161955i
$306$	−8.00000	−	2.82843i	−0.457330	−	0.161690i
$307$			16.9706i			0.968561i	0.874913	+	0.484281i	$0.160919\pi$
$307$			16.9706i			0.968561i	−0.874913	+	0.484281i	$0.839081\pi$
$308$			22.6274i			1.28932i
$309$	−18.0000	+	12.7279i	−1.02398	+	0.724066i
$310$	6.00000			0.340777
$311$		−	1.41421i		−	0.0801927i	−0.999196	−	0.0400963i	$-0.987234\pi$
$311$		−	1.41421i		−	0.0801927i	0.999196	−	0.0400963i	$-0.0127665\pi$
$312$	−6.00000	+	4.24264i	−0.339683	+	0.240192i
$313$	8.00000			0.452187			0.226093	−	0.974106i	$-0.427405\pi$
$313$	8.00000			0.452187			0.226093	+	0.974106i	$0.427405\pi$
$314$	14.0000			0.790066
$315$	−16.0000	−	5.65685i	−0.901498	−	0.318728i
$316$			4.24264i			0.238667i
$317$	18.0000			1.01098			0.505490	−	0.862832i	$-0.331312\pi$
$317$	18.0000			1.01098			0.505490	+	0.862832i	$0.331312\pi$
$318$	6.00000	+	8.48528i	0.336463	+	0.475831i
$319$			33.9411i			1.90034i
$320$		−	1.41421i		−	0.0790569i
$321$	−12.0000	−	16.9706i	−0.669775	−	0.947204i
$322$			5.65685i			0.315244i
$323$	12.0000	+	2.82843i	0.667698	+	0.157378i
$324$	−7.00000	−	5.65685i	−0.388889	−	0.314270i
$325$			12.7279i			0.706018i
$326$	20.0000			1.10770
$327$	6.00000	−	4.24264i	0.331801	−	0.234619i
$328$		0			0
$329$			5.65685i			0.311872i
$330$	−8.00000	−	11.3137i	−0.440386	−	0.622799i
$331$			16.9706i			0.932786i	0.884577	+	0.466393i	$0.154447\pi$
$331$			16.9706i			0.932786i	−0.884577	+	0.466393i	$0.845553\pi$
$332$			2.82843i			0.155230i
$333$	−12.0000	−	4.24264i	−0.657596	−	0.232495i
$334$		0			0
$335$		0			0
$336$	−4.00000	−	5.65685i	−0.218218	−	0.308607i
$337$		−	16.9706i		−	0.924445i	−0.886764	−	0.462223i	$-0.847052\pi$
$337$		−	16.9706i		−	0.924445i	0.886764	−	0.462223i	$-0.152948\pi$
$338$	−5.00000			−0.271964
$339$	−12.0000	−	16.9706i	−0.651751	−	0.921714i
$340$	4.00000			0.216930
$341$	24.0000			1.29967
$342$	11.0000	+	7.07107i	0.594812	+	0.382360i
$343$	−8.00000			−0.431959
$344$	2.00000			0.107833
$345$	−2.00000	−	2.82843i	−0.107676	−	0.152277i
$346$	6.00000			0.322562
$347$			11.3137i			0.607352i	0.952775	+	0.303676i	$0.0982140\pi$
$347$			11.3137i			0.607352i	−0.952775	+	0.303676i	$0.901786\pi$
$348$	−6.00000	−	8.48528i	−0.321634	−	0.454859i
$349$	2.00000			0.107058			0.0535288	−	0.998566i	$-0.482953\pi$
$349$	2.00000			0.107058			0.0535288	+	0.998566i	$0.482953\pi$
$350$	−12.0000			−0.641427
$351$	−6.00000	−	21.2132i	−0.320256	−	1.13228i
$352$		−	5.65685i		−	0.301511i
$353$			28.2843i			1.50542i	0.658352	+	0.752710i	$0.271254\pi$
$353$			28.2843i			1.50542i	−0.658352	+	0.752710i	$0.728746\pi$
$354$	12.0000	+	16.9706i	0.637793	+	0.901975i
$355$			16.9706i			0.900704i
$356$	6.00000			0.317999
$357$	16.0000	−	11.3137i	0.846810	−	0.598785i
$358$	−18.0000			−0.951330
$359$		−	18.3848i		−	0.970311i	−0.874428	−	0.485156i	$-0.838763\pi$
$359$		−	18.3848i		−	0.970311i	0.874428	−	0.485156i	$-0.161237\pi$
$360$	4.00000	+	1.41421i	0.210819	+	0.0745356i
$361$	−17.0000	−	8.48528i	−0.894737	−	0.446594i
$362$			12.7279i			0.668965i
$363$	−21.0000	−	29.6985i	−1.10221	−	1.55877i
$364$		−	16.9706i		−	0.889499i
$365$			5.65685i			0.296093i
$366$	2.00000	+	2.82843i	0.104542	+	0.147844i
$367$	−28.0000			−1.46159			−0.730794	−	0.682598i	$-0.760850\pi$
$367$	−28.0000			−1.46159			−0.730794	+	0.682598i	$0.760850\pi$
$368$		−	1.41421i		−	0.0737210i
$369$		0			0
$370$	6.00000			0.311925
$371$	−24.0000			−1.24602
$372$	−6.00000	+	4.24264i	−0.311086	+	0.219971i
$373$			29.6985i			1.53773i	0.639412	+	0.768865i	$0.279178\pi$
$373$			29.6985i			1.53773i	−0.639412	+	0.768865i	$0.720822\pi$
$374$	16.0000			0.827340
$375$	16.0000	−	11.3137i	0.826236	−	0.584237i
$376$		−	1.41421i		−	0.0729325i
$377$		−	25.4558i		−	1.31104i
$378$	20.0000	−	5.65685i	1.02869	−	0.290957i
$379$			33.9411i			1.74344i	0.490006	+	0.871719i	$0.336995\pi$
$379$			33.9411i			1.74344i	−0.490006	+	0.871719i	$0.663005\pi$
$380$	−6.00000	−	1.41421i	−0.307794	−	0.0725476i
$381$	30.0000	−	21.2132i	1.53695	−	1.08679i
$382$		−	18.3848i		−	0.940647i
$383$	−24.0000			−1.22634			−0.613171	−	0.789950i	$-0.710106\pi$
$383$	−24.0000			−1.22634			−0.613171	+	0.789950i	$0.710106\pi$
$384$	1.00000	+	1.41421i	0.0510310	+	0.0721688i
$385$	32.0000			1.63087
$386$			8.48528i			0.431889i
$387$	−2.00000	+	5.65685i	−0.101666	+	0.287554i
$388$		−	8.48528i		−	0.430775i
$389$			15.5563i			0.788738i	0.918952	+	0.394369i	$0.129037\pi$
$389$			15.5563i			0.788738i	−0.918952	+	0.394369i	$0.870963\pi$
$390$	6.00000	+	8.48528i	0.303822	+	0.429669i
$391$	4.00000			0.202289
$392$	9.00000			0.454569
$393$	−28.0000	+	19.7990i	−1.41241	+	0.998727i
$394$			7.07107i			0.356235i
$395$	6.00000			0.301893
$396$	16.0000	+	5.65685i	0.804030	+	0.284268i
$397$	38.0000			1.90717			0.953583	−	0.301131i	$-0.0973643\pi$
$397$	38.0000			1.90717			0.953583	+	0.301131i	$0.0973643\pi$
$398$	−4.00000			−0.200502
$399$	−28.0000	+	11.3137i	−1.40175	+	0.566394i
$400$	3.00000			0.150000
$401$	24.0000			1.19850			0.599251	−	0.800561i	$-0.295465\pi$
$401$	24.0000			1.19850			0.599251	+	0.800561i	$0.295465\pi$
$402$		0			0
$403$	−18.0000			−0.896644
$404$		−	9.89949i		−	0.492518i
$405$	−8.00000	+	9.89949i	−0.397523	+	0.491910i
$406$	24.0000			1.19110
$407$	24.0000			1.18964
$408$	−4.00000	+	2.82843i	−0.198030	+	0.140028i
$409$		−	8.48528i		−	0.419570i	−0.977748	−	0.209785i	$-0.932724\pi$
$409$		−	8.48528i		−	0.419570i	0.977748	−	0.209785i	$-0.0672764\pi$
$410$		0			0
$411$	20.0000	−	14.1421i	0.986527	−	0.697580i
$412$			12.7279i			0.627060i
$413$	−48.0000			−2.36193
$414$	4.00000	+	1.41421i	0.196589	+	0.0695048i
$415$	4.00000			0.196352
$416$			4.24264i			0.208013i
$417$	−4.00000	−	5.65685i	−0.195881	−	0.277017i
$418$	−24.0000	−	5.65685i	−1.17388	−	0.276686i
$419$		−	39.5980i		−	1.93449i	−0.253849	−	0.967244i	$-0.581697\pi$
$419$		−	39.5980i		−	1.93449i	0.253849	−	0.967244i	$-0.418303\pi$
$420$	−8.00000	+	5.65685i	−0.390360	+	0.276026i
$421$		−	12.7279i		−	0.620321i	−0.950684	−	0.310160i	$-0.899617\pi$
$421$		−	12.7279i		−	0.620321i	0.950684	−	0.310160i	$-0.100383\pi$
$422$		−	8.48528i		−	0.413057i
$423$	4.00000	+	1.41421i	0.194487	+	0.0687614i
$424$	6.00000			0.291386
$425$			8.48528i			0.411597i
$426$	−12.0000	−	16.9706i	−0.581402	−	0.822226i
$427$	−8.00000			−0.387147
$428$	−12.0000			−0.580042
$429$	24.0000	+	33.9411i	1.15873	+	1.63869i
$430$		−	2.82843i		−	0.136399i
$431$	12.0000			0.578020			0.289010	−	0.957326i	$-0.406674\pi$
$431$	12.0000			0.578020			0.289010	+	0.957326i	$0.406674\pi$
$432$	−5.00000	+	1.41421i	−0.240563	+	0.0680414i
$433$		−	25.4558i		−	1.22333i	−0.791117	−	0.611665i	$-0.790500\pi$
$433$		−	25.4558i		−	1.22333i	0.791117	−	0.611665i	$-0.209500\pi$
$434$		−	16.9706i		−	0.814613i
$435$	−12.0000	+	8.48528i	−0.575356	+	0.406838i
$436$		−	4.24264i		−	0.203186i
$437$	−6.00000	−	1.41421i	−0.287019	−	0.0676510i
$438$	−4.00000	−	5.65685i	−0.191127	−	0.270295i
$439$		−	12.7279i		−	0.607471i	−0.952756	−	0.303735i	$-0.901766\pi$
$439$		−	12.7279i		−	0.607471i	0.952756	−	0.303735i	$-0.0982338\pi$
$440$	−8.00000			−0.381385
$441$	−9.00000	+	25.4558i	−0.428571	+	1.21218i
$442$	−12.0000			−0.570782
$443$			28.2843i			1.34383i	0.740630	+	0.671913i	$0.234527\pi$
$443$			28.2843i			1.34383i	−0.740630	+	0.671913i	$0.765473\pi$
$444$	−6.00000	+	4.24264i	−0.284747	+	0.201347i
$445$		−	8.48528i		−	0.402241i
$446$		−	4.24264i		−	0.200895i
$447$	14.0000	−	9.89949i	0.662177	−	0.468230i
$448$	−4.00000			−0.188982
$449$	−30.0000			−1.41579			−0.707894	−	0.706319i	$-0.750354\pi$
$449$	−30.0000			−1.41579			−0.707894	+	0.706319i	$0.750354\pi$
$450$	−3.00000	+	8.48528i	−0.141421	+	0.400000i
$451$		0			0
$452$	−12.0000			−0.564433
$453$	18.0000	−	12.7279i	0.845714	−	0.598010i
$454$	−18.0000			−0.844782
$455$	−24.0000			−1.12514
$456$	7.00000	−	2.82843i	0.327805	−	0.132453i
$457$	−10.0000			−0.467780			−0.233890	−	0.972263i	$-0.575146\pi$
$457$	−10.0000			−0.467780			−0.233890	+	0.972263i	$0.575146\pi$
$458$	2.00000			0.0934539
$459$	−4.00000	−	14.1421i	−0.186704	−	0.660098i
$460$	−2.00000			−0.0932505
$461$		−	1.41421i		−	0.0658665i	−0.999458	−	0.0329332i	$-0.989515\pi$
$461$		−	1.41421i		−	0.0658665i	0.999458	−	0.0329332i	$-0.0104849\pi$
$462$	−32.0000	+	22.6274i	−1.48877	+	1.05272i
$463$	20.0000			0.929479			0.464739	−	0.885448i	$-0.346148\pi$
$463$	20.0000			0.929479			0.464739	+	0.885448i	$0.346148\pi$
$464$	−6.00000			−0.278543
$465$	6.00000	+	8.48528i	0.278243	+	0.393496i
$466$			11.3137i			0.524097i
$467$			2.82843i			0.130884i	0.997856	+	0.0654420i	$0.0208457\pi$
$467$			2.82843i			0.130884i	−0.997856	+	0.0654420i	$0.979154\pi$
$468$	−12.0000	−	4.24264i	−0.554700	−	0.196116i
$469$		0			0
$470$	−2.00000			−0.0922531
$471$	14.0000	+	19.7990i	0.645086	+	0.912289i
$472$	12.0000			0.552345
$473$		−	11.3137i		−	0.520205i
$474$	−6.00000	+	4.24264i	−0.275589	+	0.194871i
$475$	3.00000	−	12.7279i	0.137649	−	0.583997i
$476$		−	11.3137i		−	0.518563i
$477$	−6.00000	+	16.9706i	−0.274721	+	0.777029i
$478$			15.5563i			0.711531i
$479$		−	9.89949i		−	0.452319i	−0.974090	−	0.226160i	$-0.927383\pi$
$479$		−	9.89949i		−	0.452319i	0.974090	−	0.226160i	$-0.0726171\pi$
$480$	2.00000	−	1.41421i	0.0912871	−	0.0645497i
$481$	−18.0000			−0.820729
$482$			8.48528i			0.386494i
$483$	−8.00000	+	5.65685i	−0.364013	+	0.257396i
$484$	−21.0000			−0.954545
$485$	−12.0000			−0.544892
$486$	1.00000	−	15.5563i	0.0453609	−	0.705650i
$487$		−	12.7279i		−	0.576757i	−0.957516	−	0.288379i	$-0.906884\pi$
$487$		−	12.7279i		−	0.576757i	0.957516	−	0.288379i	$-0.0931162\pi$
$488$	2.00000			0.0905357
$489$	20.0000	+	28.2843i	0.904431	+	1.27906i
$490$		−	12.7279i		−	0.574989i
$491$		−	14.1421i		−	0.638226i	−0.947717	−	0.319113i	$-0.896615\pi$
$491$		−	14.1421i		−	0.638226i	0.947717	−	0.319113i	$-0.103385\pi$
$492$		0			0
$493$		−	16.9706i		−	0.764316i
$494$	18.0000	+	4.24264i	0.809858	+	0.190885i
$495$	8.00000	−	22.6274i	0.359573	−	1.01703i
$496$			4.24264i			0.190500i
$497$	48.0000			2.15309
$498$	−4.00000	+	2.82843i	−0.179244	+	0.126745i
$499$	−10.0000			−0.447661			−0.223831	−	0.974628i	$-0.571856\pi$
$499$	−10.0000			−0.447661			−0.223831	+	0.974628i	$0.571856\pi$
$500$		−	11.3137i		−	0.505964i
$501$		0			0
$502$			19.7990i			0.883672i
$503$			7.07107i			0.315283i	0.987496	+	0.157642i	$0.0503891\pi$
$503$			7.07107i			0.315283i	−0.987496	+	0.157642i	$0.949611\pi$
$504$	4.00000	−	11.3137i	0.178174	−	0.503953i
$505$	−14.0000			−0.622992
$506$	−8.00000			−0.355643
$507$	−5.00000	−	7.07107i	−0.222058	−	0.314037i
$508$		−	21.2132i		−	0.941184i
$509$	6.00000			0.265945			0.132973	−	0.991120i	$-0.457548\pi$
$509$	6.00000			0.265945			0.132973	+	0.991120i	$0.457548\pi$
$510$	4.00000	+	5.65685i	0.177123	+	0.250490i
$511$	16.0000			0.707798
$512$	1.00000			0.0441942
$513$	1.00000	+	22.6274i	0.0441511	+	0.999025i
$514$	6.00000			0.264649
$515$	18.0000			0.793175
$516$	2.00000	+	2.82843i	0.0880451	+	0.124515i
$517$	−8.00000			−0.351840
$518$		−	16.9706i		−	0.745644i
$519$	6.00000	+	8.48528i	0.263371	+	0.372463i
$520$	6.00000			0.263117
$521$	6.00000			0.262865			0.131432	−	0.991325i	$-0.458042\pi$
$521$	6.00000			0.262865			0.131432	+	0.991325i	$0.458042\pi$
$522$	6.00000	−	16.9706i	0.262613	−	0.742781i
$523$			16.9706i			0.742071i	0.928619	+	0.371035i	$0.120997\pi$
$523$			16.9706i			0.742071i	−0.928619	+	0.371035i	$0.879003\pi$
$524$			19.7990i			0.864923i
$525$	−12.0000	−	16.9706i	−0.523723	−	0.740656i
$526$			15.5563i			0.678289i
$527$	−12.0000			−0.522728
$528$	8.00000	−	5.65685i	0.348155	−	0.246183i
$529$	21.0000			0.913043
$530$		−	8.48528i		−	0.368577i
$531$	−12.0000	+	33.9411i	−0.520756	+	1.47292i
$532$	−4.00000	+	16.9706i	−0.173422	+	0.735767i
$533$		0			0
$534$	6.00000	+	8.48528i	0.259645	+	0.367194i
$535$			16.9706i			0.733701i
$536$		0			0
$537$	−18.0000	−	25.4558i	−0.776757	−	1.09850i
$538$	18.0000			0.776035
$539$		−	50.9117i		−	2.19292i
$540$	2.00000	+	7.07107i	0.0860663	+	0.304290i
$541$	38.0000			1.63375			0.816874	−	0.576816i	$-0.195705\pi$
$541$	38.0000			1.63375			0.816874	+	0.576816i	$0.195705\pi$
$542$	−16.0000			−0.687259
$543$	−18.0000	+	12.7279i	−0.772454	+	0.546207i
$544$			2.82843i			0.121268i
$545$	−6.00000			−0.257012
$546$	24.0000	−	16.9706i	1.02711	−	0.726273i
$547$		−	16.9706i		−	0.725609i	−0.931865	−	0.362804i	$-0.881819\pi$
$547$		−	16.9706i		−	0.725609i	0.931865	−	0.362804i	$-0.118181\pi$
$548$		−	14.1421i		−	0.604122i
$549$	−2.00000	+	5.65685i	−0.0853579	+	0.241429i
$550$		−	16.9706i		−	0.723627i
$551$	−6.00000	+	25.4558i	−0.255609	+	1.08446i
$552$	2.00000	−	1.41421i	0.0851257	−	0.0601929i
$553$		−	16.9706i		−	0.721662i
$554$	−22.0000			−0.934690
$555$	6.00000	+	8.48528i	0.254686	+	0.360180i
$556$	−4.00000			−0.169638
$557$			24.0416i			1.01868i	0.860566	+	0.509338i	$0.170110\pi$
$557$			24.0416i			1.01868i	−0.860566	+	0.509338i	$0.829890\pi$
$558$	−12.0000	−	4.24264i	−0.508001	−	0.179605i
$559$			8.48528i			0.358889i
$560$			5.65685i			0.239046i
$561$	16.0000	+	22.6274i	0.675521	+	0.955330i
$562$	−12.0000			−0.506189
$563$	36.0000			1.51722			0.758610	−	0.651546i	$-0.225879\pi$
$563$	36.0000			1.51722			0.758610	+	0.651546i	$0.225879\pi$
$564$	2.00000	−	1.41421i	0.0842152	−	0.0595491i
$565$			16.9706i			0.713957i
$566$	20.0000			0.840663
$567$	28.0000	+	22.6274i	1.17589	+	0.950262i
$568$	−12.0000			−0.503509
$569$	−6.00000			−0.251533			−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000			−0.251533			−0.125767	+	0.992060i	$0.540139\pi$
$570$	−4.00000	−	9.89949i	−0.167542	−	0.414644i
$571$	14.0000			0.585882			0.292941	−	0.956131i	$-0.405366\pi$
$571$	14.0000			0.585882			0.292941	+	0.956131i	$0.405366\pi$
$572$	24.0000			1.00349
$573$	26.0000	−	18.3848i	1.08617	−	0.768035i
$574$		0			0
$575$		−	4.24264i		−	0.176930i
$576$	−1.00000	+	2.82843i	−0.0416667	+	0.117851i
$577$	−10.0000			−0.416305			−0.208153	−	0.978096i	$-0.566745\pi$
$577$	−10.0000			−0.416305			−0.208153	+	0.978096i	$0.566745\pi$
$578$	9.00000			0.374351
$579$	−12.0000	+	8.48528i	−0.498703	+	0.352636i
$580$			8.48528i			0.352332i
$581$		−	11.3137i		−	0.469372i
$582$	12.0000	−	8.48528i	0.497416	−	0.351726i
$583$		−	33.9411i		−	1.40570i
$584$	−4.00000			−0.165521
$585$	−6.00000	+	16.9706i	−0.248069	+	0.701646i
$586$	−6.00000			−0.247858
$587$		−	22.6274i		−	0.933933i	−0.884275	−	0.466967i	$-0.845347\pi$
$587$		−	22.6274i		−	0.933933i	0.884275	−	0.466967i	$-0.154653\pi$
$588$	9.00000	+	12.7279i	0.371154	+	0.524891i
$589$	18.0000	+	4.24264i	0.741677	+	0.174815i
$590$		−	16.9706i		−	0.698667i
$591$	−10.0000	+	7.07107i	−0.411345	+	0.290865i
$592$			4.24264i			0.174371i
$593$		−	5.65685i		−	0.232299i	−0.993232	−	0.116150i	$-0.962945\pi$
$593$		−	5.65685i		−	0.232299i	0.993232	−	0.116150i	$-0.0370552\pi$
$594$	8.00000	+	28.2843i	0.328244	+	1.16052i
$595$	−16.0000			−0.655936
$596$		−	9.89949i		−	0.405499i
$597$	−4.00000	−	5.65685i	−0.163709	−	0.231520i
$598$	6.00000			0.245358
$599$	12.0000			0.490307			0.245153	−	0.969484i	$-0.421162\pi$
$599$	12.0000			0.490307			0.245153	+	0.969484i	$0.421162\pi$
$600$	3.00000	+	4.24264i	0.122474	+	0.173205i
$601$			16.9706i			0.692244i	0.938190	+	0.346122i	$0.112502\pi$
$601$			16.9706i			0.692244i	−0.938190	+	0.346122i	$0.887498\pi$
$602$	−8.00000			−0.326056
$603$		0			0
$604$		−	12.7279i		−	0.517892i
$605$			29.6985i			1.20742i
$606$	14.0000	−	9.89949i	0.568711	−	0.402139i
$607$			38.1838i			1.54983i	0.632065	+	0.774916i	$0.282208\pi$
$607$			38.1838i			1.54983i	−0.632065	+	0.774916i	$0.717792\pi$
$608$	1.00000	−	4.24264i	0.0405554	−	0.172062i
$609$	24.0000	+	33.9411i	0.972529	+	1.37536i
$610$		−	2.82843i		−	0.114520i
$611$	6.00000			0.242734
$612$	−8.00000	−	2.82843i	−0.323381	−	0.114332i
$613$	−22.0000			−0.888572			−0.444286	−	0.895885i	$-0.646543\pi$
$613$	−22.0000			−0.888572			−0.444286	+	0.895885i	$0.646543\pi$
$614$			16.9706i			0.684876i
$615$		0			0
$616$			22.6274i			0.911685i
$617$		−	22.6274i		−	0.910946i	−0.890250	−	0.455473i	$-0.849470\pi$
$617$		−	22.6274i		−	0.910946i	0.890250	−	0.455473i	$-0.150530\pi$
$618$	−18.0000	+	12.7279i	−0.724066	+	0.511992i
$619$	−46.0000			−1.84890			−0.924448	−	0.381308i	$-0.875474\pi$
$619$	−46.0000			−1.84890			−0.924448	+	0.381308i	$0.875474\pi$
$620$	6.00000			0.240966
$621$	2.00000	+	7.07107i	0.0802572	+	0.283752i
$622$		−	1.41421i		−	0.0567048i
$623$	−24.0000			−0.961540
$624$	−6.00000	+	4.24264i	−0.240192	+	0.169842i
$625$	−1.00000			−0.0400000
$626$	8.00000			0.319744
$627$	−16.0000	−	39.5980i	−0.638978	−	1.58139i
$628$	14.0000			0.558661
$629$	−12.0000			−0.478471
$630$	−16.0000	−	5.65685i	−0.637455	−	0.225374i
$631$	−4.00000			−0.159237			−0.0796187	−	0.996825i	$-0.525370\pi$
$631$	−4.00000			−0.159237			−0.0796187	+	0.996825i	$0.525370\pi$
$632$			4.24264i			0.168763i
$633$	12.0000	−	8.48528i	0.476957	−	0.337260i
$634$	18.0000			0.714871
$635$	−30.0000			−1.19051
$636$	6.00000	+	8.48528i	0.237915	+	0.336463i
$637$			38.1838i			1.51290i
$638$			33.9411i			1.34374i
$639$	12.0000	−	33.9411i	0.474713	−	1.34269i
$640$		−	1.41421i		−	0.0559017i
$641$	30.0000			1.18493			0.592464	−	0.805597i	$-0.298155\pi$
$641$	30.0000			1.18493			0.592464	+	0.805597i	$0.298155\pi$
$642$	−12.0000	−	16.9706i	−0.473602	−	0.669775i
$643$	−28.0000			−1.10421			−0.552106	−	0.833774i	$-0.686176\pi$
$643$	−28.0000			−1.10421			−0.552106	+	0.833774i	$0.686176\pi$
$644$			5.65685i			0.222911i
$645$	4.00000	−	2.82843i	0.157500	−	0.111369i
$646$	12.0000	+	2.82843i	0.472134	+	0.111283i
$647$			24.0416i			0.945174i	0.881284	+	0.472587i	$0.156680\pi$
$647$			24.0416i			0.945174i	−0.881284	+	0.472587i	$0.843320\pi$
$648$	−7.00000	−	5.65685i	−0.274986	−	0.222222i
$649$		−	67.8823i		−	2.66461i
$650$			12.7279i			0.499230i
$651$	24.0000	−	16.9706i	0.940634	−	0.665129i
$652$	20.0000			0.783260
$653$		−	35.3553i		−	1.38356i	−0.722108	−	0.691781i	$-0.756827\pi$
$653$		−	35.3553i		−	1.38356i	0.722108	−	0.691781i	$-0.243173\pi$
$654$	6.00000	−	4.24264i	0.234619	−	0.165900i
$655$	28.0000			1.09405
$656$		0			0
$657$	4.00000	−	11.3137i	0.156055	−	0.441390i
$658$			5.65685i			0.220527i
$659$	−18.0000			−0.701180			−0.350590	−	0.936529i	$-0.614019\pi$
$659$	−18.0000			−0.701180			−0.350590	+	0.936529i	$0.614019\pi$
$660$	−8.00000	−	11.3137i	−0.311400	−	0.440386i
$661$		−	29.6985i		−	1.15514i	−0.816342	−	0.577569i	$-0.804002\pi$
$661$		−	29.6985i		−	1.15514i	0.816342	−	0.577569i	$-0.195998\pi$
$662$			16.9706i			0.659580i
$663$	−12.0000	−	16.9706i	−0.466041	−	0.659082i
$664$			2.82843i			0.109764i
$665$	24.0000	+	5.65685i	0.930680	+	0.219363i
$666$	−12.0000	−	4.24264i	−0.464991	−	0.164399i
$667$			8.48528i			0.328551i
$668$		0			0
$669$	6.00000	−	4.24264i	0.231973	−	0.164030i
$670$		0			0
$671$		−	11.3137i		−	0.436761i
$672$	−4.00000	−	5.65685i	−0.154303	−	0.218218i
$673$		−	8.48528i		−	0.327084i	−0.986536	−	0.163542i	$-0.947708\pi$
$673$		−	8.48528i		−	0.327084i	0.986536	−	0.163542i	$-0.0522919\pi$
$674$		−	16.9706i		−	0.653682i
$675$	−15.0000	+	4.24264i	−0.577350	+	0.163299i
$676$	−5.00000			−0.192308
$677$	−30.0000			−1.15299			−0.576497	−	0.817099i	$-0.695581\pi$
$677$	−30.0000			−1.15299			−0.576497	+	0.817099i	$0.695581\pi$
$678$	−12.0000	−	16.9706i	−0.460857	−	0.651751i
$679$			33.9411i			1.30254i
$680$	4.00000			0.153393
$681$	−18.0000	−	25.4558i	−0.689761	−	0.975470i
$682$	24.0000			0.919007
$683$	−18.0000			−0.688751			−0.344375	−	0.938832i	$-0.611909\pi$
$683$	−18.0000			−0.688751			−0.344375	+	0.938832i	$0.611909\pi$
$684$	11.0000	+	7.07107i	0.420596	+	0.270369i
$685$	−20.0000			−0.764161
$686$	−8.00000			−0.305441
$687$	2.00000	+	2.82843i	0.0763048	+	0.107911i
$688$	2.00000			0.0762493
$689$			25.4558i			0.969790i
$690$	−2.00000	−	2.82843i	−0.0761387	−	0.107676i
$691$	−10.0000			−0.380418			−0.190209	−	0.981744i	$-0.560917\pi$
$691$	−10.0000			−0.380418			−0.190209	+	0.981744i	$0.560917\pi$
$692$	6.00000			0.228086
$693$	−64.0000	−	22.6274i	−2.43116	−	0.859544i
$694$			11.3137i			0.429463i
$695$			5.65685i			0.214577i
$696$	−6.00000	−	8.48528i	−0.227429	−	0.321634i
$697$		0			0
$698$	2.00000			0.0757011
$699$	−16.0000	+	11.3137i	−0.605176	+	0.427924i
$700$	−12.0000			−0.453557
$701$			7.07107i			0.267071i	0.991044	+	0.133535i	$0.0426329\pi$
$701$			7.07107i			0.267071i	−0.991044	+	0.133535i	$0.957367\pi$
$702$	−6.00000	−	21.2132i	−0.226455	−	0.800641i
$703$	18.0000	+	4.24264i	0.678883	+	0.160014i
$704$		−	5.65685i		−	0.213201i
$705$	−2.00000	−	2.82843i	−0.0753244	−	0.106525i
$706$			28.2843i			1.06449i
$707$			39.5980i			1.48924i
$708$	12.0000	+	16.9706i	0.450988	+	0.637793i
$709$	26.0000			0.976450			0.488225	−	0.872718i	$-0.337644\pi$
$709$	26.0000			0.976450			0.488225	+	0.872718i	$0.337644\pi$
$710$			16.9706i			0.636894i
$711$	−12.0000	−	4.24264i	−0.450035	−	0.159111i
$712$	6.00000			0.224860
$713$	6.00000			0.224702
$714$	16.0000	−	11.3137i	0.598785	−	0.423405i
$715$		−	33.9411i		−	1.26933i
$716$	−18.0000			−0.672692
$717$	−22.0000	+	15.5563i	−0.821605	+	0.580963i
$718$		−	18.3848i		−	0.686114i
$719$			7.07107i			0.263706i	0.991269	+	0.131853i	$0.0420927\pi$
$719$			7.07107i			0.263706i	−0.991269	+	0.131853i	$0.957907\pi$
$720$	4.00000	+	1.41421i	0.149071	+	0.0527046i
$721$		−	50.9117i		−	1.89605i
$722$	−17.0000	−	8.48528i	−0.632674	−	0.315789i
$723$	−12.0000	+	8.48528i	−0.446285	+	0.315571i
$724$			12.7279i			0.473029i
$725$	−18.0000			−0.668503
$726$	−21.0000	−	29.6985i	−0.779383	−	1.10221i
$727$	8.00000			0.296704			0.148352	−	0.988935i	$-0.452603\pi$
$727$	8.00000			0.296704			0.148352	+	0.988935i	$0.452603\pi$
$728$		−	16.9706i		−	0.628971i
$729$	23.0000	−	14.1421i	0.851852	−	0.523783i
$730$			5.65685i			0.209370i
$731$			5.65685i			0.209226i
$732$	2.00000	+	2.82843i	0.0739221	+	0.104542i
$733$	−22.0000			−0.812589			−0.406294	−	0.913742i	$-0.633179\pi$
$733$	−22.0000			−0.812589			−0.406294	+	0.913742i	$0.633179\pi$
$734$	−28.0000			−1.03350
$735$	18.0000	−	12.7279i	0.663940	−	0.469476i
$736$		−	1.41421i		−	0.0521286i
$737$		0			0
$738$		0			0
$739$	20.0000			0.735712			0.367856	−	0.929883i	$-0.380092\pi$
$739$	20.0000			0.735712			0.367856	+	0.929883i	$0.380092\pi$
$740$	6.00000			0.220564
$741$	12.0000	+	29.6985i	0.440831	+	1.09100i
$742$	−24.0000			−0.881068
$743$		0			0			−	1.00000i	$-0.5\pi$
$743$		0			0				1.00000i	$0.5\pi$
$744$	−6.00000	+	4.24264i	−0.219971	+	0.155543i
$745$	−14.0000			−0.512920
$746$			29.6985i			1.08734i
$747$	−8.00000	−	2.82843i	−0.292705	−	0.103487i
$748$	16.0000			0.585018
$749$	48.0000			1.75388
$750$	16.0000	−	11.3137i	0.584237	−	0.413118i
$751$		−	38.1838i		−	1.39335i	−0.717389	−	0.696673i	$-0.754663\pi$
$751$		−	38.1838i		−	1.39335i	0.717389	−	0.696673i	$-0.245337\pi$
$752$		−	1.41421i		−	0.0515711i
$753$	−28.0000	+	19.7990i	−1.02038	+	0.721515i
$754$		−	25.4558i		−	0.927047i
$755$	−18.0000			−0.655087
$756$	20.0000	−	5.65685i	0.727393	−	0.205738i
$757$	−34.0000			−1.23575			−0.617876	−	0.786276i	$-0.712006\pi$
$757$	−34.0000			−1.23575			−0.617876	+	0.786276i	$0.712006\pi$
$758$			33.9411i			1.23280i
$759$	−8.00000	−	11.3137i	−0.290382	−	0.410662i
$760$	−6.00000	−	1.41421i	−0.217643	−	0.0512989i
$761$		−	14.1421i		−	0.512652i	−0.966590	−	0.256326i	$-0.917488\pi$
$761$		−	14.1421i		−	0.512652i	0.966590	−	0.256326i	$-0.0825121\pi$
$762$	30.0000	−	21.2132i	1.08679	−	0.768473i
$763$			16.9706i			0.614376i
$764$		−	18.3848i		−	0.665138i
$765$	−4.00000	+	11.3137i	−0.144620	+	0.409048i
$766$	−24.0000			−0.867155
$767$			50.9117i			1.83831i
$768$	1.00000	+	1.41421i	0.0360844	+	0.0510310i
$769$	−16.0000			−0.576975			−0.288487	−	0.957484i	$-0.593152\pi$
$769$	−16.0000			−0.576975			−0.288487	+	0.957484i	$0.593152\pi$
$770$	32.0000			1.15320
$771$	6.00000	+	8.48528i	0.216085	+	0.305590i
$772$			8.48528i			0.305392i
$773$	−18.0000			−0.647415			−0.323708	−	0.946157i	$-0.604929\pi$
$773$	−18.0000			−0.647415			−0.323708	+	0.946157i	$0.604929\pi$
$774$	−2.00000	+	5.65685i	−0.0718885	+	0.203331i
$775$			12.7279i			0.457200i
$776$		−	8.48528i		−	0.304604i
$777$	24.0000	−	16.9706i	0.860995	−	0.608816i
$778$			15.5563i			0.557722i
$779$		0			0
$780$	6.00000	+	8.48528i	0.214834	+	0.303822i
$781$			67.8823i			2.42902i
$782$	4.00000			0.143040
$783$	30.0000	−	8.48528i	1.07211	−	0.303239i
$784$	9.00000			0.321429
$785$		−	19.7990i		−	0.706656i
$786$	−28.0000	+	19.7990i	−0.998727	+	0.706207i
$787$		−	8.48528i		−	0.302468i	−0.988498	−	0.151234i	$-0.951675\pi$
$787$		−	8.48528i		−	0.302468i	0.988498	−	0.151234i	$-0.0483246\pi$
$788$			7.07107i			0.251896i
$789$	−22.0000	+	15.5563i	−0.783221	+	0.553821i
$790$	6.00000			0.213470
$791$	48.0000			1.70668
$792$	16.0000	+	5.65685i	0.568535	+	0.201008i
$793$			8.48528i			0.301321i
$794$	38.0000			1.34857
$795$	12.0000	−	8.48528i	0.425596	−	0.300942i
$796$	−4.00000			−0.141776
$797$	−6.00000			−0.212531			−0.106265	−	0.994338i	$-0.533889\pi$
$797$	−6.00000			−0.212531			−0.106265	+	0.994338i	$0.533889\pi$
$798$	−28.0000	+	11.3137i	−0.991189	+	0.400501i
$799$	4.00000			0.141510
$800$	3.00000			0.106066
$801$	−6.00000	+	16.9706i	−0.212000	+	0.599625i
$802$	24.0000			0.847469
$803$			22.6274i			0.798504i
$804$		0			0
$805$	8.00000			0.281963
$806$	−18.0000			−0.634023
$807$	18.0000	+	25.4558i	0.633630	+	0.896088i
$808$		−	9.89949i		−	0.348263i
$809$		−	31.1127i		−	1.09386i	−0.837177	−	0.546932i	$-0.815796\pi$
$809$		−	31.1127i		−	1.09386i	0.837177	−	0.546932i	$-0.184204\pi$
$810$	−8.00000	+	9.89949i	−0.281091	+	0.347833i
$811$			33.9411i			1.19183i	0.803046	+	0.595917i	$0.203211\pi$
$811$			33.9411i			1.19183i	−0.803046	+	0.595917i	$0.796789\pi$
$812$	24.0000			0.842235
$813$	−16.0000	−	22.6274i	−0.561144	−	0.793578i
$814$	24.0000			0.841200
$815$		−	28.2843i		−	0.990755i
$816$	−4.00000	+	2.82843i	−0.140028	+	0.0990148i
$817$	2.00000	−	8.48528i	0.0699711	−	0.296862i
$818$		−	8.48528i		−	0.296681i
$819$	48.0000	+	16.9706i	1.67726	+	0.592999i
$820$		0			0
$821$			7.07107i			0.246782i	0.992358	+	0.123391i	$0.0393769\pi$
$821$			7.07107i			0.246782i	−0.992358	+	0.123391i	$0.960623\pi$
$822$	20.0000	−	14.1421i	0.697580	−	0.493264i
$823$	32.0000			1.11545			0.557725	−	0.830026i	$-0.311674\pi$
$823$	32.0000			1.11545			0.557725	+	0.830026i	$0.311674\pi$
$824$			12.7279i			0.443398i
$825$	24.0000	−	16.9706i	0.835573	−	0.590839i
$826$	−48.0000			−1.67013
$827$	42.0000			1.46048			0.730242	−	0.683189i	$-0.239408\pi$
$827$	42.0000			1.46048			0.730242	+	0.683189i	$0.239408\pi$
$828$	4.00000	+	1.41421i	0.139010	+	0.0491473i
$829$		−	46.6690i		−	1.62088i	−0.585820	−	0.810442i	$-0.699227\pi$
$829$		−	46.6690i		−	1.62088i	0.585820	−	0.810442i	$-0.300773\pi$
$830$	4.00000			0.138842
$831$	−22.0000	−	31.1127i	−0.763172	−	1.07929i
$832$			4.24264i			0.147087i
$833$			25.4558i			0.881993i
$834$	−4.00000	−	5.65685i	−0.138509	−	0.195881i
$835$		0			0
$836$	−24.0000	−	5.65685i	−0.830057	−	0.195646i
$837$	−6.00000	−	21.2132i	−0.207390	−	0.733236i
$838$		−	39.5980i		−	1.36789i
$839$	12.0000			0.414286			0.207143	−	0.978311i	$-0.433583\pi$
$839$	12.0000			0.414286			0.207143	+	0.978311i	$0.433583\pi$
$840$	−8.00000	+	5.65685i	−0.276026	+	0.195180i
$841$	7.00000			0.241379
$842$		−	12.7279i		−	0.438633i
$843$	−12.0000	−	16.9706i	−0.413302	−	0.584497i
$844$		−	8.48528i		−	0.292075i
$845$			7.07107i			0.243252i
$846$	4.00000	+	1.41421i	0.137523	+	0.0486217i
$847$	84.0000			2.88627
$848$	6.00000			0.206041
$849$	20.0000	+	28.2843i	0.686398	+	0.970714i
$850$			8.48528i			0.291043i
$851$	6.00000			0.205677
$852$	−12.0000	−	16.9706i	−0.411113	−	0.581402i
$853$	−46.0000			−1.57501			−0.787505	−	0.616308i	$-0.788628\pi$
$853$	−46.0000			−1.57501			−0.787505	+	0.616308i	$0.788628\pi$
$854$	−8.00000			−0.273754
$855$	10.0000	−	15.5563i	0.341993	−	0.532016i
$856$	−12.0000			−0.410152
$857$	−54.0000			−1.84460			−0.922302	−	0.386469i	$-0.873695\pi$
$857$	−54.0000			−1.84460			−0.922302	+	0.386469i	$0.873695\pi$
$858$	24.0000	+	33.9411i	0.819346	+	1.15873i
$859$	−10.0000			−0.341196			−0.170598	−	0.985341i	$-0.554570\pi$
$859$	−10.0000			−0.341196			−0.170598	+	0.985341i	$0.554570\pi$
$860$		−	2.82843i		−	0.0964486i
$861$		0			0
$862$	12.0000			0.408722
$863$	−36.0000			−1.22545			−0.612727	−	0.790295i	$-0.709928\pi$
$863$	−36.0000			−1.22545			−0.612727	+	0.790295i	$0.709928\pi$
$864$	−5.00000	+	1.41421i	−0.170103	+	0.0481125i
$865$		−	8.48528i		−	0.288508i
$866$		−	25.4558i		−	0.865025i
$867$	9.00000	+	12.7279i	0.305656	+	0.432263i
$868$		−	16.9706i		−	0.576018i
$869$	24.0000			0.814144
$870$	−12.0000	+	8.48528i	−0.406838	+	0.287678i
$871$		0			0
$872$		−	4.24264i		−	0.143674i
$873$	24.0000	+	8.48528i	0.812277	+	0.287183i
$874$	−6.00000	−	1.41421i	−0.202953	−	0.0478365i
$875$			45.2548i			1.52989i
$876$	−4.00000	−	5.65685i	−0.135147	−	0.191127i
$877$		−	38.1838i		−	1.28937i	−0.764447	−	0.644687i	$-0.776988\pi$
$877$		−	38.1838i		−	1.28937i	0.764447	−	0.644687i	$-0.223012\pi$
$878$		−	12.7279i		−	0.429547i
$879$	−6.00000	−	8.48528i	−0.202375	−	0.286201i
$880$	−8.00000			−0.269680
$881$			28.2843i			0.952921i	0.879196	+	0.476461i	$0.158081\pi$
$881$			28.2843i			0.952921i	−0.879196	+	0.476461i	$0.841919\pi$
$882$	−9.00000	+	25.4558i	−0.303046	+	0.857143i
$883$	26.0000			0.874970			0.437485	−	0.899226i	$-0.355869\pi$
$883$	26.0000			0.874970			0.437485	+	0.899226i	$0.355869\pi$
$884$	−12.0000			−0.403604
$885$	24.0000	−	16.9706i	0.806751	−	0.570459i
$886$			28.2843i			0.950229i
$887$	24.0000			0.805841			0.402921	−	0.915235i	$-0.367995\pi$
$887$	24.0000			0.805841			0.402921	+	0.915235i	$0.367995\pi$
$888$	−6.00000	+	4.24264i	−0.201347	+	0.142374i
$889$			84.8528i			2.84587i
$890$		−	8.48528i		−	0.284427i
$891$	−32.0000	+	39.5980i	−1.07204	+	1.32658i
$892$		−	4.24264i		−	0.142054i
$893$	−6.00000	−	1.41421i	−0.200782	−	0.0473249i
$894$	14.0000	−	9.89949i	0.468230	−	0.331089i
$895$			25.4558i			0.850895i
$896$	−4.00000			−0.133631
$897$	6.00000	+	8.48528i	0.200334	+	0.283315i
$898$	−30.0000			−1.00111
$899$		−	25.4558i		−	0.849000i
$900$	−3.00000	+	8.48528i	−0.100000	+	0.282843i
$901$			16.9706i			0.565371i
$902$		0			0
$903$	−8.00000	−	11.3137i	−0.266223	−	0.376497i
$904$	−12.0000			−0.399114
$905$	18.0000			0.598340
$906$	18.0000	−	12.7279i	0.598010	−	0.422857i
$907$		0			0		1.00000			$0$
$907$		0			0		−1.00000			$\pi$
$908$	−18.0000			−0.597351
$909$	28.0000	+	9.89949i	0.928701	+	0.328346i
$910$	−24.0000			−0.795592
$911$	60.0000			1.98789			0.993944	−	0.109885i	$-0.0350482\pi$
$911$	60.0000			1.98789			0.993944	+	0.109885i	$0.0350482\pi$
$912$	7.00000	−	2.82843i	0.231793	−	0.0936586i
$913$	16.0000			0.529523
$914$	−10.0000			−0.330771
$915$	4.00000	−	2.82843i	0.132236	−	0.0935049i
$916$	2.00000			0.0660819
$917$		−	79.1960i		−	2.61528i
$918$	−4.00000	−	14.1421i	−0.132020	−	0.466760i
$919$	−16.0000			−0.527791			−0.263896	−	0.964551i	$-0.585007\pi$
$919$	−16.0000			−0.527791			−0.263896	+	0.964551i	$0.585007\pi$
$920$	−2.00000			−0.0659380
$921$	−24.0000	+	16.9706i	−0.790827	+	0.559199i
$922$		−	1.41421i		−	0.0465746i
$923$		−	50.9117i		−	1.67578i
$924$	−32.0000	+	22.6274i	−1.05272	+	0.744387i
$925$			12.7279i			0.418491i
$926$	20.0000			0.657241
$927$	−36.0000	−	12.7279i	−1.18240	−	0.418040i
$928$	−6.00000			−0.196960
$929$			53.7401i			1.76316i	0.472038	+	0.881578i	$0.343518\pi$
$929$			53.7401i			1.76316i	−0.472038	+	0.881578i	$0.656482\pi$
$930$	6.00000	+	8.48528i	0.196748	+	0.278243i
$931$	9.00000	−	38.1838i	0.294963	−	1.25142i
$932$			11.3137i			0.370593i
$933$	2.00000	−	1.41421i	0.0654771	−	0.0462993i
$934$			2.82843i			0.0925490i
$935$		−	22.6274i		−	0.739996i
$936$	−12.0000	−	4.24264i	−0.392232	−	0.138675i
$937$	−52.0000			−1.69877			−0.849383	−	0.527777i	$-0.823026\pi$
$937$	−52.0000			−1.69877			−0.849383	+	0.527777i	$0.823026\pi$
$938$		0			0
$939$	8.00000	+	11.3137i	0.261070	+	0.369209i
$940$	−2.00000			−0.0652328
$941$	−6.00000			−0.195594			−0.0977972	−	0.995206i	$-0.531180\pi$
$941$	−6.00000			−0.195594			−0.0977972	+	0.995206i	$0.531180\pi$
$942$	14.0000	+	19.7990i	0.456145	+	0.645086i
$943$		0			0
$944$	12.0000			0.390567
$945$	−8.00000	−	28.2843i	−0.260240	−	0.920087i
$946$		−	11.3137i		−	0.367840i
$947$		−	48.0833i		−	1.56250i	−0.624221	−	0.781248i	$-0.714583\pi$
$947$		−	48.0833i		−	1.56250i	0.624221	−	0.781248i	$-0.285417\pi$
$948$	−6.00000	+	4.24264i	−0.194871	+	0.137795i
$949$		−	16.9706i		−	0.550888i
$950$	3.00000	−	12.7279i	0.0973329	−	0.412948i
$951$	18.0000	+	25.4558i	0.583690	+	0.825462i
$952$		−	11.3137i		−	0.366679i
$953$	−24.0000			−0.777436			−0.388718	−	0.921357i	$-0.627082\pi$
$953$	−24.0000			−0.777436			−0.388718	+	0.921357i	$0.627082\pi$
$954$	−6.00000	+	16.9706i	−0.194257	+	0.549442i
$955$	−26.0000			−0.841340
$956$			15.5563i			0.503128i
$957$	−48.0000	+	33.9411i	−1.55162	+	1.09716i
$958$		−	9.89949i		−	0.319838i
$959$			56.5685i			1.82669i
$960$	2.00000	−	1.41421i	0.0645497	−	0.0456435i
$961$	13.0000			0.419355
$962$	−18.0000			−0.580343
$963$	12.0000	−	33.9411i	0.386695	−	1.09374i
$964$			8.48528i			0.273293i
$965$	12.0000			0.386294
$966$	−8.00000	+	5.65685i	−0.257396	+	0.182006i
$967$	−4.00000			−0.128631			−0.0643157	−	0.997930i	$-0.520486\pi$
$967$	−4.00000			−0.128631			−0.0643157	+	0.997930i	$0.520486\pi$
$968$	−21.0000			−0.674966
$969$	8.00000	+	19.7990i	0.256997	+	0.636035i
$970$	−12.0000			−0.385297
$971$	−42.0000			−1.34784			−0.673922	−	0.738802i	$-0.735392\pi$
$971$	−42.0000			−1.34784			−0.673922	+	0.738802i	$0.735392\pi$
$972$	1.00000	−	15.5563i	0.0320750	−	0.498970i
$973$	16.0000			0.512936
$974$		−	12.7279i		−	0.407829i
$975$	−18.0000	+	12.7279i	−0.576461	+	0.407620i
$976$	2.00000			0.0640184
$977$		0			0			−	1.00000i	$-0.5\pi$
$977$		0			0				1.00000i	$0.5\pi$
$978$	20.0000	+	28.2843i	0.639529	+	0.904431i
$979$		−	33.9411i		−	1.08476i
$980$		−	12.7279i		−	0.406579i
$981$	12.0000	+	4.24264i	0.383131	+	0.135457i
$982$		−	14.1421i		−	0.451294i
$983$		0			0			−	1.00000i	$-0.5\pi$
$983$		0			0				1.00000i	$0.5\pi$
$984$		0			0
$985$	10.0000			0.318626
$986$		−	16.9706i		−	0.540453i
$987$	−8.00000	+	5.65685i	−0.254643	+	0.180060i
$988$	18.0000	+	4.24264i	0.572656	+	0.134976i
$989$		−	2.82843i		−	0.0899388i
$990$	8.00000	−	22.6274i	0.254257	−	0.719147i
$991$			29.6985i			0.943403i	0.881758	+	0.471702i	$0.156360\pi$
$991$			29.6985i			0.943403i	−0.881758	+	0.471702i	$0.843640\pi$
$992$			4.24264i			0.134704i
$993$	−24.0000	+	16.9706i	−0.761617	+	0.538545i
$994$	48.0000			1.52247
$995$			5.65685i			0.179334i
$996$	−4.00000	+	2.82843i	−0.126745	+	0.0896221i
$997$	62.0000			1.96356			0.981780	−	0.190022i	$-0.0608559\pi$
$997$	62.0000			1.96356			0.981780	+	0.190022i	$0.0608559\pi$
$998$	−10.0000			−0.316544
$999$	−6.00000	−	21.2132i	−0.189832	−	0.671156i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	114.2.b.d.113.2	yes	2
3.2	odd	2		114.2.b.a.113.2	yes	2
4.3	odd	2		912.2.f.b.113.1		2
12.11	even	2		912.2.f.d.113.1		2
19.18	odd	2		114.2.b.a.113.1	✓	2
57.56	even	2	inner	114.2.b.d.113.1	yes	2
76.75	even	2		912.2.f.d.113.2		2
228.227	odd	2		912.2.f.b.113.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
114.2.b.a.113.1	✓	2	19.18	odd	2
114.2.b.a.113.2	yes	2	3.2	odd	2
114.2.b.d.113.1	yes	2	57.56	even	2	inner
114.2.b.d.113.2	yes	2	1.1	even	1	trivial
912.2.f.b.113.1		2	4.3	odd	2
912.2.f.b.113.2		2	228.227	odd	2
912.2.f.d.113.1		2	12.11	even	2
912.2.f.d.113.2		2	76.75	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 114 = 2 \cdot 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	114.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +(1.00000 + 1.41421i) q^{3} +1.00000 q^{4} -1.41421i q^{5} +(1.00000 + 1.41421i) q^{6} -4.00000 q^{7} +1.00000 q^{8} +(-1.00000 + 2.82843i) q^{9} -1.41421i q^{10} -5.65685i q^{11} +(1.00000 + 1.41421i) q^{12} +4.24264i q^{13} -4.00000 q^{14} +(2.00000 - 1.41421i) q^{15} +1.00000 q^{16} +2.82843i q^{17} +(-1.00000 + 2.82843i) q^{18} +(1.00000 - 4.24264i) q^{19} -1.41421i q^{20} +(-4.00000 - 5.65685i) q^{21} -5.65685i q^{22} -1.41421i q^{23} +(1.00000 + 1.41421i) q^{24} +3.00000 q^{25} +4.24264i q^{26} +(-5.00000 + 1.41421i) q^{27} -4.00000 q^{28} -6.00000 q^{29} +(2.00000 - 1.41421i) q^{30} +4.24264i q^{31} +1.00000 q^{32} +(8.00000 - 5.65685i) q^{33} +2.82843i q^{34} +5.65685i q^{35} +(-1.00000 + 2.82843i) q^{36} +4.24264i q^{37} +(1.00000 - 4.24264i) q^{38} +(-6.00000 + 4.24264i) q^{39} -1.41421i q^{40} +(-4.00000 - 5.65685i) q^{42} +2.00000 q^{43} -5.65685i q^{44} +(4.00000 + 1.41421i) q^{45} -1.41421i q^{46} -1.41421i q^{47} +(1.00000 + 1.41421i) q^{48} +9.00000 q^{49} +3.00000 q^{50} +(-4.00000 + 2.82843i) q^{51} +4.24264i q^{52} +6.00000 q^{53} +(-5.00000 + 1.41421i) q^{54} -8.00000 q^{55} -4.00000 q^{56} +(7.00000 - 2.82843i) q^{57} -6.00000 q^{58} +12.0000 q^{59} +(2.00000 - 1.41421i) q^{60} +2.00000 q^{61} +4.24264i q^{62} +(4.00000 - 11.3137i) q^{63} +1.00000 q^{64} +6.00000 q^{65} +(8.00000 - 5.65685i) q^{66} +2.82843i q^{68} +(2.00000 - 1.41421i) q^{69} +5.65685i q^{70} -12.0000 q^{71} +(-1.00000 + 2.82843i) q^{72} -4.00000 q^{73} +4.24264i q^{74} +(3.00000 + 4.24264i) q^{75} +(1.00000 - 4.24264i) q^{76} +22.6274i q^{77} +(-6.00000 + 4.24264i) q^{78} +4.24264i q^{79} -1.41421i q^{80} +(-7.00000 - 5.65685i) q^{81} +2.82843i q^{83} +(-4.00000 - 5.65685i) q^{84} +4.00000 q^{85} +2.00000 q^{86} +(-6.00000 - 8.48528i) q^{87} -5.65685i q^{88} +6.00000 q^{89} +(4.00000 + 1.41421i) q^{90} -16.9706i q^{91} -1.41421i q^{92} +(-6.00000 + 4.24264i) q^{93} -1.41421i q^{94} +(-6.00000 - 1.41421i) q^{95} +(1.00000 + 1.41421i) q^{96} -8.48528i q^{97} +9.00000 q^{98} +(16.0000 + 5.65685i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{6} - 8 q^{7} + 2 q^{8} - 2 q^{9} + 2 q^{12} - 8 q^{14} + 4 q^{15} + 2 q^{16} - 2 q^{18} + 2 q^{19} - 8 q^{21} + 2 q^{24} + 6 q^{25} - 10 q^{27} - 8 q^{28} - 12 q^{29}+ \cdots + 32 q^{99}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + 2 * q^6 - 8 * q^7 + 2 * q^8 - 2 * q^9 + 2 * q^12 - 8 * q^14 + 4 * q^15 + 2 * q^16 - 2 * q^18 + 2 * q^19 - 8 * q^21 + 2 * q^24 + 6 * q^25 - 10 * q^27 - 8 * q^28 - 12 * q^29 + 4 * q^30 + 2 * q^32 + 16 * q^33 - 2 * q^36 + 2 * q^38 - 12 * q^39 - 8 * q^42 + 4 * q^43 + 8 * q^45 + 2 * q^48 + 18 * q^49 + 6 * q^50 - 8 * q^51 + 12 * q^53 - 10 * q^54 - 16 * q^55 - 8 * q^56 + 14 * q^57 - 12 * q^58 + 24 * q^59 + 4 * q^60 + 4 * q^61 + 8 * q^63 + 2 * q^64 + 12 * q^65 + 16 * q^66 + 4 * q^69 - 24 * q^71 - 2 * q^72 - 8 * q^73 + 6 * q^75 + 2 * q^76 - 12 * q^78 - 14 * q^81 - 8 * q^84 + 8 * q^85 + 4 * q^86 - 12 * q^87 + 12 * q^89 + 8 * q^90 - 12 * q^93 - 12 * q^95 + 2 * q^96 + 18 * q^98 + 32 * q^99

Introduction

L-functions

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