LMFDB - Embedded newform 684.2.r.b.559.10

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [684,2,Mod(487,684)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(684, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 0, 5])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("684.487"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 684 = 2^{2} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	684.r (of order $6$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [20,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

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

Self dual:	no
Analytic conductor:	$5.46176749826$
Analytic rank:	$0$
Dimension:	$20$
Relative dimension:	$10$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 2 x^{19} + 3 x^{18} - 2 x^{17} + x^{16} + 3 x^{14} - 12 x^{13} + 28 x^{12} - 24 x^{11} + \cdots + 1024 $ x^20 - 2x^19 + 3x^18 - 2x^17 + x^16 + 3x^14 - 12x^13 + 28x^12 - 24x^11 + 40x^10 - 48x^9 + 112x^8 - 96x^7 + 48x^6 + 64x^4 - 256x^3 + 768x^2 - 1024x + 1024
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 228)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			559.10
Root			$-0.874835 - 1.11115i$ of defining polynomial
Character	$\chi$	$=$	684.559
Dual form			684.2.r.b.487.10

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(1.39971 - 0.202053i) q^{2} +(1.91835 - 0.565628i) q^{4} +(0.490176 - 0.849009i) q^{5} +4.28530i q^{7} +(2.57084 - 1.17932i) q^{8} +(0.514557 - 1.28740i) q^{10} -3.77820i q^{11} +(4.37962 - 2.52858i) q^{13} +(0.865857 + 5.99816i) q^{14} +(3.36013 - 2.17015i) q^{16} +(-0.354457 + 0.613937i) q^{17} +(-2.31504 + 3.69332i) q^{19} +(0.460105 - 1.90595i) q^{20} +(-0.763397 - 5.28837i) q^{22} +(-4.89773 + 2.82770i) q^{23} +(2.01946 + 3.49780i) q^{25} +(5.61928 - 4.42418i) q^{26} +(2.42389 + 8.22071i) q^{28} +(3.94046 - 2.27503i) q^{29} +3.03527 q^{31} +(4.26471 - 3.71649i) q^{32} +(-0.372087 + 0.930950i) q^{34} +(3.63826 + 2.10055i) q^{35} -5.22254i q^{37} +(-2.49412 + 5.63732i) q^{38} +(0.258908 - 2.76074i) q^{40} +(-1.10560 - 0.638319i) q^{41} +(-4.92588 - 2.84396i) q^{43} +(-2.13706 - 7.24792i) q^{44} +(-6.28403 + 4.94755i) q^{46} +(-0.130104 + 0.0751157i) q^{47} -11.3638 q^{49} +(3.53338 + 4.48785i) q^{50} +(6.97141 - 7.32794i) q^{52} +(-9.82290 + 5.67125i) q^{53} +(-3.20773 - 1.85198i) q^{55} +(5.05375 + 11.0168i) q^{56} +(5.05581 - 3.98055i) q^{58} +(3.36409 - 5.82677i) q^{59} +(-6.30501 - 10.9206i) q^{61} +(4.24849 - 0.613285i) q^{62} +(5.21840 - 6.06368i) q^{64} -4.95779i q^{65} +(-5.08009 - 8.79898i) q^{67} +(-0.332712 + 1.37824i) q^{68} +(5.51692 + 2.20503i) q^{70} +(-1.60313 + 2.77670i) q^{71} +(-7.73383 + 13.3954i) q^{73} +(-1.05523 - 7.31002i) q^{74} +(-2.35200 + 8.39453i) q^{76} +16.1908 q^{77} +(-3.16218 + 5.47706i) q^{79} +(-0.195421 - 3.91653i) q^{80} +(-1.67649 - 0.670069i) q^{82} +15.8553i q^{83} +(0.347492 + 0.601874i) q^{85} +(-7.46942 - 2.98542i) q^{86} +(-4.45572 - 9.71315i) q^{88} +(-3.25153 + 1.87727i) q^{89} +(10.8357 + 18.7680i) q^{91} +(-7.79612 + 8.19482i) q^{92} +(-0.166930 + 0.131428i) q^{94} +(2.00089 + 3.77586i) q^{95} +(1.29749 + 0.749106i) q^{97} +(-15.9060 + 2.29609i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - q^{2} + q^{4} - 4 q^{8} + 6 q^{10} + 6 q^{13} - 9 q^{14} - 11 q^{16} + 12 q^{19} + 14 q^{20} + 8 q^{22} - 10 q^{25} + 7 q^{28} + 12 q^{31} + 29 q^{32} - 6 q^{34} - 25 q^{38} - 46 q^{40} - 12 q^{41}+ \cdots - 12 q^{97}+O(q^{100}) $ 20 * q - q^2 + q^4 - 4 * q^8 + 6 * q^10 + 6 * q^13 - 9 * q^14 - 11 * q^16 + 12 * q^19 + 14 * q^20 + 8 * q^22 - 10 * q^25 + 7 * q^28 + 12 * q^31 + 29 * q^32 - 6 * q^34 - 25 * q^38 - 46 * q^40 - 12 * q^41 - 18 * q^43 + 12 * q^44 - 32 * q^46 + 36 * q^47 - 24 * q^49 + 34 * q^50 + 3 * q^52 + 12 * q^53 - 38 * q^56 + 16 * q^58 + 10 * q^61 + 55 * q^62 - 14 * q^64 + 14 * q^67 - 40 * q^68 + 26 * q^70 - 8 * q^71 - 6 * q^73 - 45 * q^74 - 69 * q^76 + 32 * q^77 - 6 * q^79 - 22 * q^80 + 24 * q^82 + 8 * q^85 - 7 * q^86 + 44 * q^88 + 24 * q^89 + 26 * q^91 + 18 * q^92 - 52 * q^94 - 4 * q^95 - 12 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/684\mathbb{Z}\right)^\times$.

$n$	$325$	$343$	$533$
$\chi(n)$	$e\left(\frac{1}{6}\right)$	$-1$	$1$

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 $

$2$	1.39971	−	0.202053i	0.989741	−	0.142873i
$2$	1.39971	−	0.202053i	0.989741	−	0.142873i
$3$		0			0
$3$		0			0
$4$	1.91835	−	0.565628i	0.959175	−	0.282814i
$5$	0.490176	−	0.849009i	0.219213	−	0.379689i	−0.735354	−	0.677683i	$-0.762984\pi$
$5$	0.490176	−	0.849009i	0.219213	−	0.379689i	0.954568	+	0.297994i	$0.0963176\pi$
$6$		0			0
$7$			4.28530i			1.61969i	0.586642	+	0.809846i	$0.300449\pi$
$7$			4.28530i			1.61969i	−0.586642	+	0.809846i	$0.699551\pi$
$8$	2.57084	−	1.17932i	0.908928	−	0.416953i
$9$		0			0
$10$	0.514557	−	1.28740i	0.162717	−	0.407113i
$11$		−	3.77820i		−	1.13917i	−0.821932	−	0.569586i	$-0.807104\pi$
$11$		−	3.77820i		−	1.13917i	0.821932	−	0.569586i	$-0.192896\pi$
$12$		0			0
$13$	4.37962	−	2.52858i	1.21469	−	0.701301i	0.250912	−	0.968010i	$-0.419269\pi$
$13$	4.37962	−	2.52858i	1.21469	−	0.701301i	0.963777	+	0.266709i	$0.0859362\pi$
$14$	0.865857	+	5.99816i	0.231410	+	1.60308i
$15$		0			0
$16$	3.36013	−	2.17015i	0.840032	−	0.542537i
$17$	−0.354457	+	0.613937i	−0.0859684	+	0.148902i	−0.905803	−	0.423698i	$-0.860732\pi$
$17$	−0.354457	+	0.613937i	−0.0859684	+	0.148902i	0.819835	+	0.572600i	$0.194065\pi$
$18$		0			0
$19$	−2.31504	+	3.69332i	−0.531106	+	0.847306i
$19$	−2.31504	+	3.69332i	−0.531106	+	0.847306i
$20$	0.460105	−	1.90595i	0.102883	−	0.426184i
$21$		0			0
$22$	−0.763397	−	5.28837i	−0.162757	−	1.12748i
$23$	−4.89773	+	2.82770i	−1.02125	+	0.589617i	−0.914465	−	0.404666i	$-0.867388\pi$
$23$	−4.89773	+	2.82770i	−1.02125	+	0.589617i	−0.106782	+	0.994282i	$0.534055\pi$
$24$		0			0
$25$	2.01946	+	3.49780i	0.403891	+	0.699560i
$26$	5.61928	−	4.42418i	1.10203	−	0.867653i
$27$		0			0
$28$	2.42389	+	8.22071i	0.458072	+	1.55357i
$29$	3.94046	−	2.27503i	0.731726	−	0.422462i	−0.0873273	−	0.996180i	$-0.527833\pi$
$29$	3.94046	−	2.27503i	0.731726	−	0.422462i	0.819053	+	0.573717i	$0.194499\pi$
$30$		0			0
$31$	3.03527			0.545151			0.272575	−	0.962134i	$-0.412125\pi$
$31$	3.03527			0.545151			0.272575	+	0.962134i	$0.412125\pi$
$32$	4.26471	−	3.71649i	0.753901	−	0.656988i
$33$		0			0
$34$	−0.372087	+	0.930950i	−0.0638124	+	0.159657i
$35$	3.63826	+	2.10055i	0.614979	+	0.355058i
$36$		0			0
$37$		−	5.22254i		−	0.858581i	−0.903166	−	0.429290i	$-0.858764\pi$
$37$		−	5.22254i		−	0.858581i	0.903166	−	0.429290i	$-0.141236\pi$
$38$	−2.49412	+	5.63732i	−0.404600	+	0.914494i
$39$		0			0
$40$	0.258908	−	2.76074i	0.0409369	−	0.436511i
$41$	−1.10560	−	0.638319i	−0.172666	−	0.0996887i	0.411176	−	0.911556i	$-0.365118\pi$
$41$	−1.10560	−	0.638319i	−0.172666	−	0.0996887i	−0.583842	+	0.811867i	$0.698451\pi$
$42$		0			0
$43$	−4.92588	−	2.84396i	−0.751190	−	0.433700i	0.0749335	−	0.997189i	$-0.476126\pi$
$43$	−4.92588	−	2.84396i	−0.751190	−	0.433700i	−0.826124	+	0.563489i	$0.809459\pi$
$44$	−2.13706	−	7.24792i	−0.322174	−	1.09266i
$45$		0			0
$46$	−6.28403	+	4.94755i	−0.926529	+	0.729476i
$47$	−0.130104	+	0.0751157i	−0.0189776	+	0.0109568i	−0.509459	−	0.860495i	$-0.670154\pi$
$47$	−0.130104	+	0.0751157i	−0.0189776	+	0.0109568i	0.490481	+	0.871452i	$0.336821\pi$
$48$		0			0
$49$	−11.3638			−1.62340
$50$	3.53338	+	4.48785i	0.499696	+	0.634678i
$51$		0			0
$52$	6.97141	−	7.32794i	0.966761	−	1.01620i
$53$	−9.82290	+	5.67125i	−1.34928	+	0.779007i	−0.988147	−	0.153508i	$-0.950943\pi$
$53$	−9.82290	+	5.67125i	−1.34928	+	0.779007i	−0.361132	+	0.932515i	$0.617610\pi$
$54$		0			0
$55$	−3.20773	−	1.85198i	−0.432530	−	0.249722i
$56$	5.05375	+	11.0168i	0.675335	+	1.47218i
$57$		0			0
$58$	5.05581	−	3.98055i	0.663861	−	0.522672i
$59$	3.36409	−	5.82677i	0.437967	−	0.758581i	−0.559566	−	0.828786i	$-0.689032\pi$
$59$	3.36409	−	5.82677i	0.437967	−	0.758581i	0.997533	+	0.0702051i	$0.0223654\pi$
$60$		0			0
$61$	−6.30501	−	10.9206i	−0.807274	−	1.39824i	−0.914745	−	0.404031i	$-0.867609\pi$
$61$	−6.30501	−	10.9206i	−0.807274	−	1.39824i	0.107472	−	0.994208i	$-0.465724\pi$
$62$	4.24849	−	0.613285i	0.539558	−	0.0778873i
$63$		0			0
$64$	5.21840	−	6.06368i	0.652301	−	0.757960i
$65$		−	4.95779i		−	0.614938i
$66$		0			0
$67$	−5.08009	−	8.79898i	−0.620632	−	1.07497i	−0.989368	−	0.145432i	$-0.953543\pi$
$67$	−5.08009	−	8.79898i	−0.620632	−	1.07497i	0.368736	−	0.929534i	$-0.379791\pi$
$68$	−0.332712	+	1.37824i	−0.0403472	+	0.167136i
$69$		0			0
$70$	5.51692	+	2.20503i	0.659398	+	0.263552i
$71$	−1.60313	+	2.77670i	−0.190256	+	0.329534i	−0.945335	−	0.326100i	$-0.894265\pi$
$71$	−1.60313	+	2.77670i	−0.190256	+	0.329534i	0.755079	+	0.655634i	$0.227599\pi$
$72$		0			0
$73$	−7.73383	+	13.3954i	−0.905177	+	1.56781i	−0.0844965	+	0.996424i	$0.526928\pi$
$73$	−7.73383	+	13.3954i	−0.905177	+	1.56781i	−0.820680	+	0.571388i	$0.806405\pi$
$74$	−1.05523	−	7.31002i	−0.122668	−	0.849773i
$75$		0			0
$76$	−2.35200	+	8.39453i	−0.269793	+	0.962918i
$77$	16.1908			1.84511
$78$		0			0
$79$	−3.16218	+	5.47706i	−0.355773	+	0.616217i	−0.987250	−	0.159178i	$-0.949116\pi$
$79$	−3.16218	+	5.47706i	−0.355773	+	0.616217i	0.631477	+	0.775395i	$0.282449\pi$
$80$	−0.195421	−	3.91653i	−0.0218487	−	0.437882i
$81$		0			0
$82$	−1.67649	−	0.670069i	−0.185137	−	0.0739967i
$83$			15.8553i			1.74035i	0.492747	+	0.870173i	$0.335993\pi$
$83$			15.8553i			1.74035i	−0.492747	+	0.870173i	$0.664007\pi$
$84$		0			0
$85$	0.347492	+	0.601874i	0.0376908	+	0.0652824i
$86$	−7.46942	−	2.98542i	−0.805448	−	0.321926i
$87$		0			0
$88$	−4.45572	−	9.71315i	−0.474981	−	1.03543i
$89$	−3.25153	+	1.87727i	−0.344662	+	0.198991i	−0.662332	−	0.749211i	$-0.730433\pi$
$89$	−3.25153	+	1.87727i	−0.344662	+	0.198991i	0.317670	+	0.948201i	$0.397100\pi$
$90$		0			0
$91$	10.8357	+	18.7680i	1.13589	+	1.96742i
$92$	−7.79612	+	8.19482i	−0.812802	+	0.854369i
$93$		0			0
$94$	−0.166930	+	0.131428i	−0.0172175	+	0.0135557i
$95$	2.00089	+	3.77586i	0.205287	+	0.387395i
$96$		0			0
$97$	1.29749	+	0.749106i	0.131740	+	0.0760601i	0.564422	−	0.825487i	$-0.309099\pi$
$97$	1.29749	+	0.749106i	0.131740	+	0.0760601i	−0.432682	+	0.901547i	$0.642433\pi$
$98$	−15.9060	+	2.29609i	−1.60675	+	0.231940i
$99$		0			0
$100$	5.85248	+	5.56774i	0.585248	+	0.556774i
$101$	−6.30734	−	10.9246i	−0.627604	−	1.08704i	−0.988031	−	0.154254i	$-0.950702\pi$
$101$	−6.30734	−	10.9246i	−0.627604	−	1.08704i	0.360427	−	0.932787i	$-0.382631\pi$
$102$		0			0
$103$	−0.862870			−0.0850211			−0.0425105	−	0.999096i	$-0.513536\pi$
$103$	−0.862870			−0.0850211			−0.0425105	+	0.999096i	$0.513536\pi$
$104$	8.27730	−	11.6655i	0.811656	−	1.14390i
$105$		0			0
$106$	−12.6033	+	9.92283i	−1.22414	+	0.963790i
$107$	−12.8583			−1.24306			−0.621528	−	0.783392i	$-0.713488\pi$
$107$	−12.8583			−1.24306			−0.621528	+	0.783392i	$0.713488\pi$
$108$		0			0
$109$	5.94933	+	3.43485i	0.569843	+	0.328999i	0.757086	−	0.653315i	$-0.226622\pi$
$109$	5.94933	+	3.43485i	0.569843	+	0.328999i	−0.187244	+	0.982313i	$0.559955\pi$
$110$	−4.86408	−	1.94410i	−0.463772	−	0.185363i
$111$		0			0
$112$	9.29973	+	14.3992i	0.878742	+	1.36059i
$113$			14.6160i			1.37496i	0.726205	+	0.687478i	$0.241282\pi$
$113$			14.6160i			1.37496i	−0.726205	+	0.687478i	$0.758718\pi$
$114$		0			0
$115$			5.54429i			0.517007i
$116$	6.27237	−	6.59314i	0.582375	−	0.612158i
$117$		0			0
$118$	3.53142	−	8.83548i	0.325093	−	0.813372i
$119$	−2.63091	−	1.51895i	−0.241175	−	0.139242i
$120$		0			0
$121$	−3.27483			−0.297712
$122$	−11.0317	−	14.0117i	−0.998762	−	1.26856i
$123$		0			0
$124$	5.82271	−	1.71684i	0.522895	−	0.154176i
$125$	8.86131			0.792580
$126$		0			0
$127$	−4.03813	−	6.99425i	−0.358326	−	0.620639i	0.629355	−	0.777118i	$-0.283319\pi$
$127$	−4.03813	−	6.99425i	−0.358326	−	0.620639i	−0.987681	+	0.156479i	$0.949986\pi$
$128$	6.07905	−	9.54176i	0.537317	−	0.843381i
$129$		0			0
$130$	−1.00173	−	6.93944i	−0.0878580	−	0.608630i
$131$	5.04606	+	2.91335i	0.440877	+	0.254540i	0.703969	−	0.710230i	$-0.251409\pi$
$131$	5.04606	+	2.91335i	0.440877	+	0.254540i	−0.263093	+	0.964771i	$0.584743\pi$
$132$		0			0
$133$	−15.8270	−	9.92063i	−1.37237	−	0.860227i
$134$	−8.88849	−	11.2895i	−0.767848	−	0.975267i
$135$		0			0
$136$	−0.187222	+	1.99635i	−0.0160541	+	0.171186i
$137$	−5.39030	−	9.33627i	−0.460524	−	0.797651i	0.538463	−	0.842649i	$-0.319005\pi$
$137$	−5.39030	−	9.33627i	−0.460524	−	0.797651i	−0.998987	+	0.0449981i	$0.985672\pi$
$138$		0			0
$139$	9.31646	−	5.37886i	0.790212	−	0.456229i	−0.0498253	−	0.998758i	$-0.515866\pi$
$139$	9.31646	−	5.37886i	0.790212	−	0.456229i	0.840037	+	0.542529i	$0.182533\pi$
$140$	8.16759	+	1.97169i	0.690287	+	0.166638i
$141$		0			0
$142$	−1.68287	+	4.21048i	−0.141223	+	0.353336i
$143$	−9.55348	−	16.5471i	−0.798902	−	1.38374i
$144$		0			0
$145$		−	4.46066i		−	0.370437i
$146$	−8.11851	+	20.3122i	−0.671893	+	1.68105i
$147$		0			0
$148$	−2.95402	−	10.0187i	−0.242819	−	0.823529i
$149$	−10.6555	+	18.4559i	−0.872934	+	1.51197i	−0.0139857	+	0.999902i	$0.504452\pi$
$149$	−10.6555	+	18.4559i	−0.872934	+	1.51197i	−0.858948	+	0.512063i	$0.828881\pi$
$150$		0			0
$151$	−8.79259			−0.715531			−0.357765	−	0.933812i	$-0.616461\pi$
$151$	−8.79259			−0.715531			−0.357765	+	0.933812i	$0.616461\pi$
$152$	−1.59597	+	12.2251i	−0.129450	+	0.991586i
$153$		0			0
$154$	22.6623	−	3.27139i	1.82618	−	0.263616i
$155$	1.48782	−	2.57697i	0.119504	−	0.206988i
$156$		0			0
$157$	0.270402	−	0.468350i	0.0215805	−	0.0373784i	−0.855033	−	0.518573i	$-0.826464\pi$
$157$	0.270402	−	0.468350i	0.0215805	−	0.0373784i	0.876614	+	0.481194i	$0.159797\pi$
$158$	−3.31947	+	8.30519i	−0.264083	+	0.660726i
$159$		0			0
$160$	−1.06488	−	5.44251i	−0.0841860	−	0.430268i
$161$	−12.1176	−	20.9882i	−0.954998	−	1.65410i
$162$		0			0
$163$		−	9.53202i		−	0.746605i	−0.927710	−	0.373303i	$-0.878225\pi$
$163$		−	9.53202i		−	0.746605i	0.927710	−	0.373303i	$-0.121775\pi$
$164$	−2.48198	−	0.599159i	−0.193810	−	0.0467865i
$165$		0			0
$166$	3.20361	+	22.1927i	0.248648	+	1.72249i
$167$	5.42333	+	9.39349i	0.419670	+	0.726890i	0.995906	−	0.0903933i	$-0.0288124\pi$
$167$	5.42333	+	9.39349i	0.419670	+	0.726890i	−0.576236	+	0.817283i	$0.695479\pi$
$168$		0			0
$169$	6.28741	−	10.8901i	0.483647	−	0.837701i
$170$	0.607997	+	0.772235i	0.0466312	+	0.0592277i
$171$		0			0
$172$	−11.0582	−	2.66949i	−0.843179	−	0.203547i
$173$	−2.60358	−	1.50318i	−0.197946	−	0.114284i	0.397751	−	0.917493i	$-0.369791\pi$
$173$	−2.60358	−	1.50318i	−0.197946	−	0.114284i	−0.595697	+	0.803209i	$0.703124\pi$
$174$		0			0
$175$	−14.9891	+	8.65398i	−1.13307	+	0.654179i
$176$	−8.19926	−	12.6953i	−0.618042	−	0.956941i
$177$		0			0
$178$	−4.17188	+	3.28461i	−0.312696	+	0.246192i
$179$	6.11802			0.457282			0.228641	−	0.973511i	$-0.426572\pi$
$179$	6.11802			0.457282			0.228641	+	0.973511i	$0.426572\pi$
$180$		0			0
$181$	2.86502	−	1.65412i	0.212956	−	0.122950i	−0.389729	−	0.920930i	$-0.627431\pi$
$181$	2.86502	−	1.65412i	0.212956	−	0.122950i	0.602684	+	0.797980i	$0.294098\pi$
$182$	18.9589	+	24.0803i	1.40533	+	1.78495i
$183$		0			0
$184$	−9.25649	+	13.0456i	−0.682397	+	0.961731i
$185$	−4.43399	−	2.55997i	−0.325993	−	0.188212i
$186$		0			0
$187$	2.31958	+	1.33921i	0.169624	+	0.0979327i
$188$	−0.207098	+	0.217689i	−0.0151042	+	0.0158766i
$189$		0			0
$190$	3.56358	+	4.88081i	0.258529	+	0.354091i
$191$		−	8.20235i		−	0.593501i	−0.954955	−	0.296751i	$-0.904097\pi$
$191$		−	8.20235i		−	0.593501i	0.954955	−	0.296751i	$-0.0959030\pi$
$192$		0			0
$193$	15.4678	+	8.93034i	1.11340	+	0.642820i	0.939707	−	0.341981i	$-0.111098\pi$
$193$	15.4678	+	8.93034i	1.11340	+	0.642820i	0.173690	+	0.984800i	$0.444431\pi$
$194$	1.96746	+	0.786366i	0.141255	+	0.0564578i
$195$		0			0
$196$	−21.7998	+	6.42770i	−1.55713	+	0.459121i
$197$	−11.5559			−0.823323			−0.411661	−	0.911337i	$-0.635051\pi$
$197$	−11.5559			−0.823323			−0.411661	+	0.911337i	$0.635051\pi$
$198$		0			0
$199$	19.5628	−	11.2946i	1.38677	−	0.800653i	0.393821	−	0.919187i	$-0.371153\pi$
$199$	19.5628	−	11.2946i	1.38677	−	0.800653i	0.992950	+	0.118534i	$0.0378196\pi$
$200$	9.31672	+	6.61069i	0.658791	+	0.467446i
$201$		0			0
$202$	−11.0358	−	14.0169i	−0.776474	−	0.986222i
$203$	9.74919	+	16.8861i	0.684259	+	1.18517i
$204$		0			0
$205$	−1.08388	+	0.625777i	−0.0757013	+	0.0437062i
$206$	−1.20776	+	0.174345i	−0.0841488	+	0.0121472i
$207$		0			0
$208$	9.22872	−	18.0008i	0.639897	−	1.24813i
$209$	13.9541	+	8.74668i	0.965227	+	0.605020i
$210$		0			0
$211$	0.533846	−	0.924649i	0.0367515	−	0.0636554i	−0.847065	−	0.531490i	$-0.821632\pi$
$211$	0.533846	−	0.924649i	0.0367515	−	0.0636554i	0.883816	+	0.467835i	$0.154966\pi$
$212$	−15.6359	+	16.4356i	−1.07388	+	1.12880i
$213$		0			0
$214$	−17.9978	+	2.59805i	−1.23030	+	0.177599i
$215$	−4.82910	+	2.78808i	−0.329342	+	0.190146i
$216$		0			0
$217$			13.0071i			0.882977i
$218$	9.02133	+	3.60570i	0.611002	+	0.244209i
$219$		0			0
$220$	−7.20109	−	1.73837i	−0.485497	−	0.117201i
$221$			3.58508i			0.241159i
$222$		0			0
$223$	7.85315	−	13.6020i	0.525886	−	0.910861i	−0.473660	−	0.880708i	$-0.657067\pi$
$223$	7.85315	−	13.6020i	0.525886	−	0.910861i	0.999545	−	0.0301526i	$-0.00959933\pi$
$224$	15.9263	+	18.2756i	1.06412	+	1.22109i
$225$		0			0
$226$	2.95320	+	20.4581i	0.196444	+	1.36085i
$227$	6.85339			0.454875			0.227438	−	0.973793i	$-0.426965\pi$
$227$	6.85339			0.454875			0.227438	+	0.973793i	$0.426965\pi$
$228$		0			0
$229$	3.91451			0.258678			0.129339	−	0.991600i	$-0.458714\pi$
$229$	3.91451			0.258678			0.129339	+	0.991600i	$0.458714\pi$
$230$	1.12024	+	7.76037i	0.0738663	+	0.511704i
$231$		0			0
$232$	7.44730	−	10.4958i	0.488939	−	0.689083i
$233$	7.38806	−	12.7965i	0.484008	−	0.838326i	−0.515823	−	0.856695i	$-0.672514\pi$
$233$	7.38806	−	12.7965i	0.484008	−	0.838326i	0.999831	+	0.0183686i	$0.00584723\pi$
$234$		0			0
$235$			0.147280i			0.00960746i
$236$	3.15771	−	13.0806i	0.205549	−	0.851475i
$237$		0			0
$238$	−3.98940	−	1.59451i	−0.258594	−	0.103357i
$239$		−	16.8531i		−	1.09014i	−0.838391	−	0.545069i	$-0.816503\pi$
$239$		−	16.8531i		−	1.09014i	0.838391	−	0.545069i	$-0.183497\pi$
$240$		0			0
$241$	10.7076	−	6.18203i	0.689737	−	0.398220i	−0.113776	−	0.993506i	$-0.536295\pi$
$241$	10.7076	−	6.18203i	0.689737	−	0.398220i	0.803514	+	0.595287i	$0.202961\pi$
$242$	−4.58380	+	0.661689i	−0.294658	+	0.0425350i
$243$		0			0
$244$	−18.2722	−	17.3832i	−1.16976	−	1.11285i
$245$	−5.57027	+	9.64799i	−0.355872	+	0.616388i
$246$		0			0
$247$	−0.800140	+	22.0291i	−0.0509117	+	1.40168i
$248$	7.80319	−	3.57956i	0.495503	−	0.227302i
$249$		0			0
$250$	12.4032	−	1.79045i	0.784449	−	0.113238i
$251$	20.6917	−	11.9464i	1.30605	−	0.754047i	0.324614	−	0.945847i	$-0.394766\pi$
$251$	20.6917	−	11.9464i	1.30605	−	0.754047i	0.981434	+	0.191800i	$0.0614324\pi$
$252$		0			0
$253$	10.6836	+	18.5046i	0.671675	+	1.16338i
$254$	−7.06540	−	8.97397i	−0.443323	−	0.563077i
$255$		0			0
$256$	6.58093	−	14.5839i	0.411308	−	0.911496i
$257$	2.89737	−	1.67280i	0.180733	−	0.104346i	−0.406904	−	0.913471i	$-0.633392\pi$
$257$	2.89737	−	1.67280i	0.180733	−	0.104346i	0.587637	+	0.809125i	$0.300058\pi$
$258$		0			0
$259$	22.3802			1.39064
$260$	−2.80427	−	9.51077i	−0.173913	−	0.589833i
$261$		0			0
$262$	7.65165	+	3.05825i	0.472721	+	0.188940i
$263$	−22.7935	−	13.1598i	−1.40551	−	0.811470i	−0.410556	−	0.911835i	$-0.634666\pi$
$263$	−22.7935	−	13.1598i	−1.40551	−	0.811470i	−0.994951	+	0.100365i	$0.967999\pi$
$264$		0			0
$265$			11.1196i			0.683075i
$266$	−24.1576	−	10.6881i	−1.48120	−	0.655327i
$267$		0			0
$268$	−14.7223	−	14.0061i	−0.899310	−	0.855557i
$269$	14.8066	+	8.54860i	0.902775	+	0.521217i	0.878100	−	0.478478i	$-0.158811\pi$
$269$	14.8066	+	8.54860i	0.902775	+	0.521217i	0.0246756	+	0.999696i	$0.492145\pi$
$270$		0			0
$271$	−4.23109	−	2.44282i	−0.257020	−	0.148391i	0.365954	−	0.930633i	$-0.380743\pi$
$271$	−4.23109	−	2.44282i	−0.257020	−	0.148391i	−0.622975	+	0.782242i	$0.714076\pi$
$272$	0.141313	+	2.83213i	0.00856834	+	0.171723i
$273$		0			0
$274$	−9.43124	−	11.9789i	−0.569762	−	0.723672i
$275$	13.2154	−	7.62992i	0.796919	−	0.460101i
$276$		0			0
$277$	−1.47561			−0.0886607			−0.0443303	−	0.999017i	$-0.514115\pi$
$277$	−1.47561			−0.0886607			−0.0443303	+	0.999017i	$0.514115\pi$
$278$	11.9535	−	9.41123i	0.716922	−	0.564448i
$279$		0			0
$280$	11.8306	+	1.10950i	0.707014	+	0.0663052i
$281$	22.8573	−	13.1967i	1.36355	−	0.787248i	0.373458	−	0.927647i	$-0.378172\pi$
$281$	22.8573	−	13.1967i	1.36355	−	0.787248i	0.990095	+	0.140400i	$0.0448387\pi$
$282$		0			0
$283$	2.66671	+	1.53963i	0.158520	+	0.0915214i	0.577161	−	0.816630i	$-0.304160\pi$
$283$	2.66671	+	1.53963i	0.158520	+	0.0915214i	−0.418642	+	0.908151i	$0.637494\pi$
$284$	−1.50478	+	6.23346i	−0.0892923	+	0.369888i
$285$		0			0
$286$	−16.7155	−	21.2308i	−0.988405	−	1.25540i
$287$	2.73539	−	4.73783i	0.161465	−	0.279665i
$288$		0			0
$289$	8.24872	+	14.2872i	0.485219	+	0.840424i
$290$	−0.901288	−	6.24360i	−0.0529254	−	0.366637i
$291$		0			0
$292$	−7.25938	+	30.0715i	−0.424823	+	1.75980i
$293$			22.5211i			1.31570i	0.753151	+	0.657848i	$0.228533\pi$
$293$			22.5211i			1.31570i	−0.753151	+	0.657848i	$0.771467\pi$
$294$		0			0
$295$	−3.29799	−	5.71228i	−0.192016	−	0.332582i
$296$	−6.15906	−	13.4263i	−0.357988	−	0.780388i
$297$		0			0
$298$	−11.1855	+	27.9858i	−0.647959	+	1.62117i
$299$	−14.3001	+	24.7686i	−0.826998	+	1.43240i
$300$		0			0
$301$	12.1872	−	21.1089i	0.702460	−	1.21670i
$302$	−12.3070	+	1.77657i	−0.708190	+	0.102230i
$303$		0			0
$304$	0.236227	+	17.4340i	0.0135485	+	0.999908i
$305$	−12.3622			−0.707860
$306$		0			0
$307$	−0.623630	+	1.08016i	−0.0355924	+	0.0616479i	−0.883273	−	0.468859i	$-0.844665\pi$
$307$	−0.623630	+	1.08016i	−0.0355924	+	0.0616479i	0.847680	+	0.530507i	$0.177998\pi$
$308$	31.0595	−	9.15795i	1.76978	−	0.521823i
$309$		0			0
$310$	1.56182	−	3.90762i	0.0887054	−	0.221938i
$311$		−	7.29013i		−	0.413385i	−0.978406	−	0.206693i	$-0.933730\pi$
$311$		−	7.29013i		−	0.413385i	0.978406	−	0.206693i	$-0.0662700\pi$
$312$		0			0
$313$	−6.28690	−	10.8892i	−0.355357	−	0.615496i	0.631822	−	0.775113i	$-0.282307\pi$
$313$	−6.28690	−	10.8892i	−0.355357	−	0.615496i	−0.987179	+	0.159617i	$0.948974\pi$
$314$	0.283852	−	0.710188i	0.0160187	−	0.0400782i
$315$		0			0
$316$	−2.96819	+	12.2955i	−0.166974	+	0.691678i
$317$	−21.7262	+	12.5436i	−1.22027	+	0.704521i	−0.964975	−	0.262343i	$-0.915505\pi$
$317$	−21.7262	+	12.5436i	−1.22027	+	0.704521i	−0.255292	+	0.966864i	$0.582172\pi$
$318$		0			0
$319$	−8.59552	−	14.8879i	−0.481257	−	0.833561i
$320$	−2.59019	−	7.40275i	−0.144796	−	0.413826i
$321$		0			0
$322$	−21.2017	−	26.9290i	−1.18153	−	1.50069i
$323$	−1.44689	−	2.73041i	−0.0805069	−	0.151924i
$324$		0			0
$325$	17.6889	+	10.2127i	0.981204	+	0.566499i
$326$	−1.92597	−	13.3420i	−0.106670	−	0.738946i
$327$		0			0
$328$	−3.59510	−	0.337156i	−0.198506	−	0.0186163i
$329$	−0.321894	−	0.557536i	−0.0177466	−	0.0307380i
$330$		0			0
$331$	−21.8003			−1.19825			−0.599127	−	0.800654i	$-0.704485\pi$
$331$	−21.8003			−1.19825			−0.599127	+	0.800654i	$0.704485\pi$
$332$	8.96821	+	30.4160i	0.492194	+	1.66930i
$333$		0			0
$334$	9.48905	+	12.0523i	0.519218	+	0.659473i
$335$	−9.96055			−0.544203
$336$		0			0
$337$	16.2854	+	9.40240i	0.887124	+	0.512182i	0.873001	−	0.487719i	$-0.162171\pi$
$337$	16.2854	+	9.40240i	0.887124	+	0.512182i	0.0141236	+	0.999900i	$0.495504\pi$
$338$	6.60014	−	16.5133i	0.359000	−	0.898207i
$339$		0			0
$340$	1.00705	+	0.958053i	0.0546149	+	0.0519577i
$341$		−	11.4679i		−	0.621021i
$342$		0			0
$343$		−	18.7003i		−	1.00972i
$344$	−16.0176	−	1.50216i	−0.863610	−	0.0809911i
$345$		0			0
$346$	−3.94796	−	1.57794i	−0.212244	−	0.0848308i
$347$	27.9162	+	16.1175i	1.49862	+	0.865230i	0.999999	−	0.00158848i	$-0.000505629\pi$
$347$	27.9162	+	16.1175i	1.49862	+	0.865230i	0.498624	+	0.866819i	$0.333839\pi$
$348$		0			0
$349$	−12.8141			−0.685923			−0.342961	−	0.939350i	$-0.611430\pi$
$349$	−12.8141			−0.685923			−0.342961	+	0.939350i	$0.611430\pi$
$350$	−19.2318	+	15.1416i	−1.02798	+	0.809353i
$351$		0			0
$352$	−14.0417	−	16.1129i	−0.748423	−	0.858822i
$353$	21.8474			1.16282			0.581410	−	0.813611i	$-0.302501\pi$
$353$	21.8474			1.16282			0.581410	+	0.813611i	$0.302501\pi$
$354$		0			0
$355$	1.57163	+	2.72214i	0.0834135	+	0.144476i
$356$	−5.17574	+	5.44043i	−0.274314	+	0.288342i
$357$		0			0
$358$	8.56342	−	1.23616i	0.452591	−	0.0653332i
$359$	−5.88277	−	3.39642i	−0.310481	−	0.179256i	0.336661	−	0.941626i	$-0.390702\pi$
$359$	−5.88277	−	3.39642i	−0.310481	−	0.179256i	−0.647142	+	0.762370i	$0.724036\pi$
$360$		0			0
$361$	−8.28122	−	17.1003i	−0.435854	−	0.900017i
$362$	3.67597	−	2.89417i	0.193205	−	0.152114i
$363$		0			0
$364$	31.4024	+	29.8746i	1.64593	+	1.56586i
$365$	7.58187	+	13.1322i	0.396853	+	0.687370i
$366$		0			0
$367$	14.8232	−	8.55817i	0.773764	−	0.446733i	−0.0604514	−	0.998171i	$-0.519254\pi$
$367$	14.8232	−	8.55817i	0.773764	−	0.446733i	0.834216	+	0.551438i	$0.185921\pi$
$368$	−10.3205	+	20.1302i	−0.537991	+	1.04936i
$369$		0			0
$370$	−6.72353	−	2.68730i	−0.349539	−	0.139706i
$371$	−24.3030	−	42.0941i	−1.26175	−	2.18542i
$372$		0			0
$373$			19.8642i			1.02853i	0.857631	+	0.514265i	$0.171935\pi$
$373$			19.8642i			1.02853i	−0.857631	+	0.514265i	$0.828065\pi$
$374$	3.51732	+	1.40582i	0.181876	+	0.0726933i
$375$		0			0
$376$	−0.245891	+	0.346545i	−0.0126809	+	0.0178717i
$377$	11.5052	−	19.9275i	0.592546	−	1.02632i
$378$		0			0
$379$	−10.3716			−0.532754			−0.266377	−	0.963869i	$-0.585827\pi$
$379$	−10.3716			−0.532754			−0.266377	+	0.963869i	$0.585827\pi$
$380$	5.97414	+	6.11166i	0.306467	+	0.313522i
$381$		0			0
$382$	−1.65731	−	11.4809i	−0.0847952	−	0.587413i
$383$	12.1885	−	21.1112i	0.622805	−	1.07873i	−0.366156	−	0.930554i	$-0.619326\pi$
$383$	12.1885	−	21.1112i	0.622805	−	1.07873i	0.988961	−	0.148176i	$-0.0473403\pi$
$384$		0			0
$385$	7.93632	−	13.7461i	0.404472	−	0.700566i
$386$	23.4548	+	9.37453i	1.19382	+	0.477151i
$387$		0			0
$388$	2.91275	+	0.703150i	0.147873	+	0.0356970i
$389$	2.87971	+	4.98780i	0.146007	+	0.252892i	0.929748	−	0.368196i	$-0.120024\pi$
$389$	2.87971	+	4.98780i	0.146007	+	0.252892i	−0.783741	+	0.621088i	$0.786691\pi$
$390$		0			0
$391$		−	4.00919i		−	0.202754i
$392$	−29.2145	+	13.4016i	−1.47556	+	0.676882i
$393$		0			0
$394$	−16.1748	+	2.33490i	−0.814876	+	0.117630i
$395$	3.10005	+	5.36944i	0.155980	+	0.270166i
$396$		0			0
$397$	−14.0559	+	24.3455i	−0.705444	+	1.22187i	0.261087	+	0.965315i	$0.415919\pi$
$397$	−14.0559	+	24.3455i	−0.705444	+	1.22187i	−0.966531	+	0.256550i	$0.917414\pi$
$398$	25.1001	−	19.7618i	1.25815	−	0.990571i
$399$		0			0
$400$	14.3764	+	7.37054i	0.718818	+	0.368527i
$401$	10.8958	+	6.29068i	0.544109	+	0.314142i	0.746743	−	0.665113i	$-0.231617\pi$
$401$	10.8958	+	6.29068i	0.544109	+	0.314142i	−0.202633	+	0.979255i	$0.564950\pi$
$402$		0			0
$403$	13.2934	−	7.67492i	0.662189	−	0.382315i
$404$	−18.2790	−	17.3897i	−0.909412	−	0.865168i
$405$		0			0
$406$	17.0579	+	21.6657i	0.846568	+	1.07525i
$407$	−19.7318			−0.978071
$408$		0			0
$409$	−12.6235	+	7.28819i	−0.624193	+	0.360378i	−0.778500	−	0.627645i	$-0.784019\pi$
$409$	−12.6235	+	7.28819i	−0.624193	+	0.360378i	0.154307	+	0.988023i	$0.450686\pi$
$410$	−1.39067	+	1.09490i	−0.0686802	+	0.0540734i
$411$		0			0
$412$	−1.65529	+	0.488064i	−0.0815501	+	0.0240452i
$413$	24.9695	+	14.4161i	1.22867	+	0.709372i
$414$		0			0
$415$	13.4613	+	7.77189i	0.660789	+	0.381507i
$416$	9.28038	−	27.0605i	0.455008	−	1.32675i
$417$		0			0
$418$	21.2989	+	9.42330i	1.04177	+	0.460909i
$419$			8.26594i			0.403817i	0.979404	+	0.201909i	$0.0647144\pi$
$419$			8.26594i			0.403817i	−0.979404	+	0.201909i	$0.935286\pi$
$420$		0			0
$421$	−32.0531	−	18.5059i	−1.56217	−	0.901921i	−0.997037	−	0.0769222i	$-0.975491\pi$
$421$	−32.0531	−	18.5059i	−1.56217	−	0.901921i	−0.565135	−	0.824998i	$-0.691176\pi$
$422$	0.560399	−	1.40210i	0.0272798	−	0.0682532i
$423$		0			0
$424$	−18.5648	+	26.1642i	−0.901589	+	1.27065i
$425$	−2.86324			−0.138887
$426$		0			0
$427$	46.7981	−	27.0189i	2.26472	−	1.30753i
$428$	−24.6667	+	7.27300i	−1.19231	+	0.351554i
$429$		0			0
$430$	−6.19598	+	4.87822i	−0.298796	+	0.235249i
$431$	19.1283	+	33.1311i	0.921376	+	1.59587i	0.797289	+	0.603598i	$0.206267\pi$
$431$	19.1283	+	33.1311i	0.921376	+	1.59587i	0.124087	+	0.992271i	$0.460400\pi$
$432$		0			0
$433$	−15.3878	+	8.88418i	−0.739493	+	0.426946i	−0.821885	−	0.569654i	$-0.807077\pi$
$433$	−15.3878	+	8.88418i	−0.739493	+	0.426946i	0.0823921	+	0.996600i	$0.473744\pi$
$434$	2.62811	+	18.2060i	0.126153	+	0.873918i
$435$		0			0
$436$	13.3557	+	3.22413i	0.639624	+	0.154408i
$437$	0.894795	−	24.6351i	0.0428039	−	1.17846i
$438$		0			0
$439$	1.17417	−	2.03372i	0.0560401	−	0.0970643i	−0.836644	−	0.547746i	$-0.815486\pi$
$439$	1.17417	−	2.03372i	0.0560401	−	0.0970643i	0.892685	+	0.450682i	$0.148819\pi$
$440$	−10.4306	−	0.978206i	−0.497261	−	0.0466341i
$441$		0			0
$442$	0.724376	+	5.01806i	0.0344551	+	0.238685i
$443$	−1.59306	+	0.919751i	−0.0756884	+	0.0436987i	−0.537367	−	0.843349i	$-0.680581\pi$
$443$	−1.59306	+	0.919751i	−0.0756884	+	0.0436987i	0.461678	+	0.887047i	$0.347247\pi$
$444$		0			0
$445$			3.68078i			0.174486i
$446$	8.24376	−	20.6256i	0.390353	−	0.976651i
$447$		0			0
$448$	25.9847	+	22.3624i	1.22766	+	1.05653i
$449$		−	1.49190i		−	0.0704073i	−0.999380	−	0.0352037i	$-0.988792\pi$
$449$		−	1.49190i		−	0.0704073i	0.999380	−	0.0352037i	$-0.0112080\pi$
$450$		0			0
$451$	−2.41170	+	4.17719i	−0.113562	+	0.196696i
$452$	8.26722	+	28.0386i	0.388857	+	1.31882i
$453$		0			0
$454$	9.59273	−	1.38475i	0.450209	−	0.0649894i
$455$	21.2456			0.996011
$456$		0			0
$457$	16.8982			0.790465			0.395233	−	0.918581i	$-0.370664\pi$
$457$	16.8982			0.790465			0.395233	+	0.918581i	$0.370664\pi$
$458$	5.47916	−	0.790937i	0.256024	−	0.0369581i
$459$		0			0
$460$	3.13601	+	10.6359i	0.146217	+	0.495901i
$461$	10.7302	−	18.5853i	0.499757	−	0.865604i	−0.500243	−	0.865885i	$-0.666756\pi$
$461$	10.7302	−	18.5853i	0.499757	−	0.865604i	1.00000		0.000280614i	$8.93223e-5\pi$
$462$		0			0
$463$			6.33305i			0.294322i	0.989113	+	0.147161i	$0.0470135\pi$
$463$			6.33305i			0.294322i	−0.989113	+	0.147161i	$0.952987\pi$
$464$	8.30333	−	16.1958i	0.385472	−	0.751870i
$465$		0			0
$466$	7.75554	−	19.4041i	0.359268	−	0.898878i
$467$			11.4220i			0.528548i	0.964448	+	0.264274i	$0.0851323\pi$
$467$			11.4220i			0.528548i	−0.964448	+	0.264274i	$0.914868\pi$
$468$		0			0
$469$	37.7063	−	21.7697i	1.74111	−	1.00523i
$470$	0.0297582	+	0.206148i	0.00137265	+	0.00950890i
$471$		0			0
$472$	1.77689	−	18.9470i	0.0817879	−	0.872107i
$473$	−10.7451	+	18.6110i	−0.494059	+	0.855735i
$474$		0			0
$475$	−17.5936	−	0.639034i	−0.807250	−	0.0293209i
$476$	−5.90616	−	1.42577i	−0.270708	−	0.0653500i
$477$		0			0
$478$	−3.40522	−	23.5894i	−0.155751	−	1.07896i
$479$	−28.1985	+	16.2804i	−1.28842	+	0.743870i	−0.978372	−	0.206851i	$-0.933678\pi$
$479$	−28.1985	+	16.2804i	−1.28842	+	0.743870i	−0.310048	+	0.950721i	$0.600345\pi$
$480$		0			0
$481$	−13.2056	−	22.8728i	−0.602124	−	1.04291i
$482$	13.7384	−	10.8165i	0.625766	−	0.492679i
$483$		0			0
$484$	−6.28227	+	1.85234i	−0.285558	+	0.0841972i
$485$	1.27200	−	0.734387i	0.0577583	−	0.0333468i
$486$		0			0
$487$	−12.5326			−0.567906			−0.283953	−	0.958838i	$-0.591646\pi$
$487$	−12.5326			−0.567906			−0.283953	+	0.958838i	$0.591646\pi$
$488$	−29.0880	−	20.6394i	−1.31675	−	0.934303i
$489$		0			0
$490$	−5.84733	+	14.6298i	−0.264156	+	0.660908i
$491$	−14.4609	−	8.34899i	−0.652610	−	0.376785i	0.136845	−	0.990592i	$-0.456304\pi$
$491$	−14.4609	−	8.34899i	−0.652610	−	0.376785i	−0.789456	+	0.613808i	$0.789637\pi$
$492$		0			0
$493$			3.22560i			0.145274i
$494$	3.33108	+	30.9959i	0.149872	+	1.39457i
$495$		0			0
$496$	10.1989	−	6.58698i	0.457944	−	0.295764i
$497$	−11.8990	−	6.86990i	−0.533743	−	0.308157i
$498$		0			0
$499$	7.54758	+	4.35759i	0.337876	+	0.195073i	0.659332	−	0.751852i	$-0.270839\pi$
$499$	7.54758	+	4.35759i	0.337876	+	0.195073i	−0.321456	+	0.946924i	$0.604172\pi$
$500$	16.9991	−	5.01221i	0.760222	−	0.224153i
$501$		0			0
$502$	26.5485	−	20.9022i	1.18492	−	0.932910i
$503$	16.3287	−	9.42738i	0.728061	−	0.420346i	−0.0896513	−	0.995973i	$-0.528575\pi$
$503$	16.3287	−	9.42738i	0.728061	−	0.420346i	0.817712	+	0.575627i	$0.195242\pi$
$504$		0			0
$505$	−12.3668			−0.550316
$506$	18.6929	+	23.7423i	0.830999	+	1.05548i
$507$		0			0
$508$	−11.7027	−	11.1333i	−0.519223	−	0.493962i
$509$	34.8313	−	20.1099i	1.54387	−	0.891354i	0.545281	−	0.838254i	$-0.316423\pi$
$509$	34.8313	−	20.1099i	1.54387	−	0.891354i	0.998589	−	0.0531001i	$-0.0169102\pi$
$510$		0			0
$511$	−57.4033	−	33.1418i	−2.53937	−	1.46611i
$512$	6.26464	−	21.7429i	0.276861	−	0.960910i
$513$		0			0
$514$	3.71747	−	2.92685i	0.163971	−	0.129098i
$515$	−0.422958	+	0.732585i	−0.0186378	+	0.0322815i
$516$		0			0
$517$	0.283803	+	0.491560i	0.0124816	+	0.0216188i
$518$	31.3257	−	4.52198i	1.37637	−	0.198684i
$519$		0			0
$520$	−5.84683	−	12.7457i	−0.256400	−	0.558935i
$521$			30.5495i			1.33840i	0.743082	+	0.669200i	$0.233363\pi$
$521$			30.5495i			1.33840i	−0.743082	+	0.669200i	$0.766637\pi$
$522$		0			0
$523$	1.09831	+	1.90233i	0.0480257	+	0.0831829i	0.889039	−	0.457832i	$-0.151374\pi$
$523$	1.09831	+	1.90233i	0.0480257	+	0.0831829i	−0.841013	+	0.541015i	$0.818040\pi$
$524$	11.3280	+	2.73462i	0.494865	+	0.119462i
$525$		0			0
$526$	−34.5632	−	13.8144i	−1.50703	−	0.602336i
$527$	−1.07587	+	1.86347i	−0.0468657	+	0.0811738i
$528$		0			0
$529$	4.49181	−	7.78005i	0.195296	−	0.338263i
$530$	2.24676	+	15.5642i	0.0975928	+	0.676067i
$531$		0			0
$532$	−35.9731	−	10.0790i	−1.55963	−	0.436981i
$533$	−6.45615			−0.279647
$534$		0			0
$535$	−6.30281	+	10.9168i	−0.272494	+	0.471974i
$536$	−23.4369	−	16.6297i	−1.01232	−	0.718293i
$537$		0			0
$538$	22.4522	+	8.97381i	0.967981	+	0.386888i
$539$			42.9348i			1.84933i
$540$		0			0
$541$	−1.16936	−	2.02540i	−0.0502748	−	0.0870786i	0.839793	−	0.542907i	$-0.182676\pi$
$541$	−1.16936	−	2.02540i	−0.0502748	−	0.0870786i	−0.890068	+	0.455828i	$0.849343\pi$
$542$	−6.41585	−	2.56432i	−0.275585	−	0.110147i
$543$		0			0
$544$	0.770036	+	3.93559i	0.0330150	+	0.168737i
$545$	5.83244	−	3.36736i	0.249834	−	0.144242i
$546$		0			0
$547$	12.9835	+	22.4880i	0.555133	+	0.961519i	0.997893	+	0.0648785i	$0.0206660\pi$
$547$	12.9835	+	22.4880i	0.555133	+	0.961519i	−0.442760	+	0.896640i	$0.646001\pi$
$548$	−15.6213	−	14.8613i	−0.667310	−	0.634844i
$549$		0			0
$550$	16.9560	−	13.3498i	0.723007	−	0.569239i
$551$	−0.719907	+	19.8202i	−0.0306691	+	0.844368i
$552$		0			0
$553$	−23.4709	−	13.5509i	−0.998082	−	0.576243i
$554$	−2.06542	+	0.298151i	−0.0877511	+	0.0126672i
$555$		0			0
$556$	14.8298	−	15.5882i	0.628923	−	0.661086i
$557$	−14.0655	−	24.3621i	−0.595974	−	1.03226i	−0.993409	−	0.114626i	$-0.963433\pi$
$557$	−14.0655	−	24.3621i	−0.595974	−	1.03226i	0.397435	−	0.917630i	$-0.369900\pi$
$558$		0			0
$559$	−28.7647			−1.21662
$560$	16.7835	−	0.837436i	0.709234	−	0.0353881i
$561$		0			0
$562$	29.3271	−	23.0898i	1.23709	−	0.973986i
$563$	39.3721			1.65934			0.829668	−	0.558257i	$-0.188530\pi$
$563$	39.3721			1.65934			0.829668	+	0.558257i	$0.188530\pi$
$564$		0			0
$565$	12.4091	+	7.16440i	0.522055	+	0.301409i
$566$	4.04370	+	1.61621i	0.169969	+	0.0679343i
$567$		0			0
$568$	−0.846762	+	9.02905i	−0.0355294	+	0.378851i
$569$			6.02447i			0.252559i	0.991995	+	0.126279i	$0.0403036\pi$
$569$			6.02447i			0.252559i	−0.991995	+	0.126279i	$0.959696\pi$
$570$		0			0
$571$			33.1860i			1.38879i	0.719594	+	0.694395i	$0.244328\pi$
$571$			33.1860i			1.38879i	−0.719594	+	0.694395i	$0.755672\pi$
$572$	−27.6864	−	26.3394i	−1.15763	−	1.10131i
$573$		0			0
$574$	2.87145	−	7.18426i	0.119852	−	0.299865i
$575$	−19.7815	−	11.4208i	−0.824945	−	0.476282i
$576$		0			0
$577$	34.9768			1.45610			0.728051	−	0.685523i	$-0.240426\pi$
$577$	34.9768			1.45610			0.728051	+	0.685523i	$0.240426\pi$
$578$	14.4325	+	18.3312i	0.600315	+	0.762477i
$579$		0			0
$580$	−2.52307	−	8.55710i	−0.104765	−	0.355314i
$581$	−67.9448			−2.81882
$582$		0			0
$583$	21.4272	+	37.1129i	0.887422	+	1.53706i
$584$	−4.08496	+	43.5580i	−0.169037	+	1.80244i
$585$		0			0
$586$	4.55045	+	31.5229i	0.187977	+	1.30220i
$587$	28.6111	+	16.5186i	1.18091	+	0.681796i	0.956224	−	0.292636i	$-0.0945325\pi$
$587$	28.6111	+	16.5186i	1.18091	+	0.681796i	0.224682	+	0.974432i	$0.427866\pi$
$588$		0			0
$589$	−7.02676	+	11.2102i	−0.289533	+	0.461910i
$590$	−5.77040	−	7.32915i	−0.237563	−	0.301736i
$591$		0			0
$592$	−11.3337	−	17.5484i	−0.465812	−	0.721236i
$593$	12.7775	+	22.1313i	0.524709	+	0.908822i	0.999586	+	0.0287702i	$0.00915911\pi$
$593$	12.7775	+	22.1313i	0.524709	+	0.908822i	−0.474877	+	0.880052i	$0.657508\pi$
$594$		0			0
$595$	−2.57921	+	1.48911i	−0.105737	+	0.0610475i
$596$	−10.0018	+	41.4319i	−0.409690	+	1.69712i
$597$		0			0
$598$	−15.0114	+	37.5581i	−0.613863	+	1.53586i
$599$	−2.25614	−	3.90775i	−0.0921835	−	0.159667i	0.816246	−	0.577704i	$-0.196051\pi$
$599$	−2.25614	−	3.90775i	−0.0921835	−	0.159667i	−0.908430	+	0.418038i	$0.862718\pi$
$600$		0			0
$601$			40.6603i			1.65857i	0.558826	+	0.829285i	$0.311252\pi$
$601$			40.6603i			1.65857i	−0.558826	+	0.829285i	$0.688748\pi$
$602$	12.7934	−	32.0087i	0.521421	−	1.30458i
$603$		0			0
$604$	−16.8673	+	4.97334i	−0.686319	+	0.202362i
$605$	−1.60524	+	2.78036i	−0.0652624	+	0.113038i
$606$		0			0
$607$	10.0709			0.408767			0.204384	−	0.978891i	$-0.434481\pi$
$607$	10.0709			0.408767			0.204384	+	0.978891i	$0.434481\pi$
$608$	3.85323	+	24.3547i	0.156269	+	0.987714i
$609$		0			0
$610$	−17.3035	+	2.49783i	−0.700598	+	0.101134i
$611$	−0.379872	+	0.657957i	−0.0153680	+	0.0266181i
$612$		0			0
$613$	21.4108	−	37.0847i	0.864776	−	1.49784i	−0.00249333	−	0.999997i	$-0.500794\pi$
$613$	21.4108	−	37.0847i	0.864776	−	1.49784i	0.867269	−	0.497839i	$-0.165873\pi$
$614$	−0.654649	+	1.63791i	−0.0264195	+	0.0661007i
$615$		0			0
$616$	41.6238	−	19.0941i	1.67707	−	0.769323i
$617$	−3.97251	−	6.88059i	−0.159927	−	0.277002i	0.774915	−	0.632065i	$-0.217793\pi$
$617$	−3.97251	−	6.88059i	−0.159927	−	0.277002i	−0.934842	+	0.355063i	$0.884459\pi$
$618$		0			0
$619$			33.5206i			1.34731i	0.739047	+	0.673653i	$0.235276\pi$
$619$			33.5206i			1.34731i	−0.739047	+	0.673653i	$0.764724\pi$
$620$	1.39654	−	5.78509i	0.0560865	−	0.232335i
$621$		0			0
$622$	−1.47299	−	10.2040i	−0.0590615	−	0.409144i
$623$	−8.04469	−	13.9338i	−0.322304	−	0.558246i
$624$		0			0
$625$	−5.75368	+	9.96566i	−0.230147	+	0.398626i
$626$	−11.0000	−	13.9714i	−0.439649	−	0.558411i
$627$		0			0
$628$	0.253814	−	1.05141i	0.0101283	−	0.0419557i
$629$	3.20631	+	1.85117i	0.127844	+	0.0738108i
$630$		0			0
$631$	17.9583	−	10.3682i	0.714909	−	0.412753i	−0.0979670	−	0.995190i	$-0.531234\pi$
$631$	17.9583	−	10.3682i	0.714909	−	0.412753i	0.812876	+	0.582437i	$0.197901\pi$
$632$	−1.67024	+	17.8098i	−0.0664387	+	0.708438i
$633$		0			0
$634$	−27.8758	+	21.9473i	−1.10709	+	0.871637i
$635$	−7.91758			−0.314200
$636$		0			0
$637$	−49.7693	+	28.7343i	−1.97193	+	1.13849i
$638$	−15.0393	−	19.1019i	−0.595413	−	0.756251i
$639$		0			0
$640$	−5.12124	−	9.83831i	−0.202435	−	0.388893i
$641$	−41.5249	−	23.9744i	−1.64013	−	0.946931i	−0.980785	−	0.195094i	$-0.937499\pi$
$641$	−41.5249	−	23.9744i	−1.64013	−	0.946931i	−0.659349	−	0.751837i	$-0.729168\pi$
$642$		0			0
$643$	−20.5401	−	11.8588i	−0.810023	−	0.467667i	0.0369410	−	0.999317i	$-0.488239\pi$
$643$	−20.5401	−	11.8588i	−0.810023	−	0.467667i	−0.846964	+	0.531651i	$0.821572\pi$
$644$	−35.1173	−	33.4087i	−1.38381	−	1.31649i
$645$		0			0
$646$	−2.57690	−	3.52942i	−0.101387	−	0.138863i
$647$		−	31.0790i		−	1.22184i	−0.791692	−	0.610921i	$-0.790799\pi$
$647$		−	31.0790i		−	1.22184i	0.791692	−	0.610921i	$-0.209201\pi$
$648$		0			0
$649$	−22.0147	−	12.7102i	−0.864154	−	0.498920i
$650$	26.8228	+	10.7207i	1.05208	+	0.420499i
$651$		0			0
$652$	−5.39158	−	18.2857i	−0.211151	−	0.716125i
$653$	−32.4012			−1.26796			−0.633979	−	0.773350i	$-0.718579\pi$
$653$	−32.4012			−1.26796			−0.633979	+	0.773350i	$0.718579\pi$
$654$		0			0
$655$	4.94692	−	2.85610i	0.193292	−	0.111597i
$656$	−5.10021	+	0.254481i	−0.199130	+	0.00993583i
$657$		0			0
$658$	−0.563208	−	0.715347i	−0.0219561	−	0.0278871i
$659$	−12.0633	−	20.8942i	−0.469918	−	0.813923i	0.529490	−	0.848316i	$-0.322383\pi$
$659$	−12.0633	−	20.8942i	−0.469918	−	0.813923i	−0.999408	+	0.0343936i	$0.989050\pi$
$660$		0			0
$661$	35.4425	−	20.4627i	1.37855	−	0.795907i	0.386567	−	0.922261i	$-0.373661\pi$
$661$	35.4425	−	20.4627i	1.37855	−	0.795907i	0.991985	+	0.126354i	$0.0403275\pi$
$662$	−30.5140	+	4.40481i	−1.18596	+	0.171198i
$663$		0			0
$664$	18.6985	+	40.7614i	0.725642	+	1.58185i
$665$	−16.1807	+	8.57442i	−0.627461	+	0.332502i
$666$		0			0
$667$	−12.8662	+	22.2849i	−0.498182	+	0.862876i
$668$	15.7171	+	14.9524i	0.608112	+	0.578526i
$669$		0			0
$670$	−13.9418	+	2.01256i	−0.538620	+	0.0777519i
$671$	−41.2602	+	23.8216i	−1.59283	+	0.919623i
$672$		0			0
$673$		−	21.5578i		−	0.830993i	−0.909595	−	0.415496i	$-0.863608\pi$
$673$		−	21.5578i		−	0.830993i	0.909595	−	0.415496i	$-0.136392\pi$
$674$	24.6946	+	9.87008i	0.951200	+	0.380181i
$675$		0			0
$676$	5.90169	−	24.4474i	0.226988	−	0.940283i
$677$		−	19.5655i		−	0.751961i	−0.926627	−	0.375981i	$-0.877306\pi$
$677$		−	19.5655i		−	0.751961i	0.926627	−	0.375981i	$-0.122694\pi$
$678$		0			0
$679$	−3.21014	+	5.56013i	−0.123194	+	0.213378i
$680$	1.60315	+	1.13752i	0.0614779	+	0.0436217i
$681$		0			0
$682$	−2.31712	−	16.0517i	−0.0887270	−	0.614650i
$683$	−33.7851			−1.29275			−0.646375	−	0.763020i	$-0.723716\pi$
$683$	−33.7851			−1.29275			−0.646375	+	0.763020i	$0.723716\pi$
$684$		0			0
$685$	−10.5688			−0.403812
$686$	−3.77844	−	26.1749i	−0.144262	−	0.999362i
$687$		0			0
$688$	−22.7234	+	1.13381i	−0.866322	+	0.0432263i
$689$	−28.6804	+	49.6759i	−1.09264	+	1.89250i
$690$		0			0
$691$		−	6.96281i		−	0.264878i	−0.991191	−	0.132439i	$-0.957719\pi$
$691$		−	6.96281i		−	0.264878i	0.991191	−	0.132439i	$-0.0422808\pi$
$692$	−5.84481	−	1.41096i	−0.222186	−	0.0536366i
$693$		0			0
$694$	42.3311	+	16.9191i	1.60687	+	0.642241i
$695$		−	10.5463i		−	0.400046i
$696$		0			0
$697$	0.783775	−	0.452513i	0.0296876	−	0.0171401i
$698$	−17.9359	+	2.58912i	−0.678886	+	0.0979997i
$699$		0			0
$700$	−23.8594	+	25.0796i	−0.901802	+	0.947921i
$701$	1.46064	−	2.52991i	0.0551677	−	0.0955532i	−0.837123	−	0.547015i	$-0.815764\pi$
$701$	1.46064	−	2.52991i	0.0551677	−	0.0955532i	0.892290	+	0.451462i	$0.149097\pi$
$702$		0			0
$703$	19.2885	+	12.0904i	0.727481	+	0.455997i
$704$	−22.9098	−	19.7162i	−0.863447	−	0.743082i
$705$		0			0
$706$	30.5799	−	4.41433i	1.15089	−	0.166135i
$707$	46.8154	−	27.0289i	1.76067	−	1.01652i
$708$		0			0
$709$	−2.62608	−	4.54851i	−0.0986247	−	0.170823i	0.812491	−	0.582974i	$-0.198111\pi$
$709$	−2.62608	−	4.54851i	−0.0986247	−	0.170823i	−0.911116	+	0.412151i	$0.864778\pi$
$710$	2.74984	+	3.49265i	0.103200	+	0.131077i
$711$		0			0
$712$	−6.14526	+	8.66077i	−0.230303	+	0.324576i
$713$	−14.8659	+	8.58285i	−0.556733	+	0.321430i
$714$		0			0
$715$	−18.7315			−0.700520
$716$	11.7365	−	3.46053i	0.438614	−	0.129326i
$717$		0			0
$718$	−8.92040	−	3.56536i	−0.332906	−	0.133058i
$719$	−25.8616	−	14.9312i	−0.964473	−	0.556839i	−0.0669265	−	0.997758i	$-0.521319\pi$
$719$	−25.8616	−	14.9312i	−0.964473	−	0.556839i	−0.897547	+	0.440919i	$0.854653\pi$
$720$		0			0
$721$		−	3.69766i		−	0.137708i
$722$	−15.0464	−	22.2622i	−0.559971	−	0.828513i
$723$		0			0
$724$	4.56050	−	4.79372i	0.169490	−	0.178157i
$725$	15.9152	+	9.18864i	0.591075	+	0.341257i
$726$		0			0
$727$	6.22247	+	3.59255i	0.230779	+	0.133240i	0.610931	−	0.791684i	$-0.290795\pi$
$727$	6.22247	+	3.59255i	0.230779	+	0.133240i	−0.380152	+	0.924924i	$0.624128\pi$
$728$	49.9904	+	35.4707i	1.85277	+	1.31463i
$729$		0			0
$730$	13.2658	+	16.8493i	0.490989	+	0.623619i
$731$	3.49202	−	2.01612i	0.129157	−	0.0745689i
$732$		0			0
$733$	−26.6597			−0.984698			−0.492349	−	0.870398i	$-0.663862\pi$
$733$	−26.6597			−0.984698			−0.492349	+	0.870398i	$0.663862\pi$
$734$	19.0189	−	14.9740i	0.702000	−	0.552700i
$735$		0			0
$736$	−10.3782	+	30.2617i	−0.382547	+	1.11546i
$737$	−33.2443	+	19.1936i	−1.22457	+	0.707006i
$738$		0			0
$739$	12.1059	+	6.98933i	0.445322	+	0.257107i	0.705852	−	0.708359i	$-0.250564\pi$
$739$	12.1059	+	6.98933i	0.445322	+	0.257107i	−0.260531	+	0.965466i	$0.583897\pi$
$740$	−9.95393	−	2.40292i	−0.365914	−	0.0883330i
$741$		0			0
$742$	−42.5223	−	54.0088i	−1.56104	−	1.98273i
$743$	0.420962	−	0.729128i	0.0154436	−	0.0267491i	−0.858200	−	0.513315i	$-0.828417\pi$
$743$	0.420962	−	0.729128i	0.0154436	−	0.0267491i	0.873644	+	0.486566i	$0.161751\pi$
$744$		0			0
$745$	10.4461	+	18.0933i	0.382717	+	0.662886i
$746$	4.01362	+	27.8040i	0.146949	+	1.01798i
$747$		0			0
$748$	5.20726	+	1.25705i	0.190396	+	0.0459624i
$749$		−	55.1016i		−	2.01337i
$750$		0			0
$751$	0.619203	+	1.07249i	0.0225950	+	0.0391358i	0.877102	−	0.480304i	$-0.159474\pi$
$751$	0.619203	+	1.07249i	0.0225950	+	0.0391358i	−0.854507	+	0.519440i	$0.826140\pi$
$752$	−0.274155	+	0.534744i	−0.00999740	+	0.0195001i
$753$		0			0
$754$	12.0774	−	30.2173i	0.439834	−	1.10045i
$755$	−4.30991	+	7.46499i	−0.156854	+	0.271679i
$756$		0			0
$757$	−26.6473	+	46.1545i	−0.968512	+	1.67751i	−0.268645	+	0.963239i	$0.586576\pi$
$757$	−26.6473	+	46.1545i	−0.968512	+	1.67751i	−0.699867	+	0.714273i	$0.746758\pi$
$758$	−14.5172	+	2.09561i	−0.527289	+	0.0761161i
$759$		0			0
$760$	9.59691	+	7.34744i	0.348117	+	0.266520i
$761$	−4.46182			−0.161741			−0.0808704	−	0.996725i	$-0.525770\pi$
$761$	−4.46182			−0.161741			−0.0808704	+	0.996725i	$0.525770\pi$
$762$		0			0
$763$	−14.7194	+	25.4947i	−0.532877	+	0.922970i
$764$	−4.63948	−	15.7350i	−0.167851	−	0.569272i
$765$		0			0
$766$	12.7948	−	32.0121i	0.462294	−	1.15665i
$767$		−	34.0254i		−	1.22859i
$768$		0			0
$769$	−18.3414	−	31.7682i	−0.661407	−	1.14559i	−0.980246	−	0.197782i	$-0.936626\pi$
$769$	−18.3414	−	31.7682i	−0.661407	−	1.14559i	0.318839	−	0.947809i	$-0.396707\pi$
$770$	8.33107	−	20.8440i	0.300231	−	0.751167i
$771$		0			0
$772$	34.7239	+	8.38248i	1.24974	+	0.301692i
$773$	41.4667	−	23.9408i	1.49145	−	0.861091i	0.491501	−	0.870877i	$-0.336448\pi$
$773$	41.4667	−	23.9408i	1.49145	−	0.861091i	0.999952	+	0.00978619i	$0.00311509\pi$
$774$		0			0
$775$	6.12960	+	10.6168i	0.220182	+	0.381366i
$776$	4.21907	+	0.395673i	0.151456	+	0.0142038i
$777$		0			0
$778$	5.03854	+	6.39960i	0.180640	+	0.229437i
$779$	4.91702	−	2.60561i	0.176171	−	0.0933555i
$780$		0			0
$781$	10.4909	+	6.05695i	0.375396	+	0.216735i
$782$	−0.810068	−	5.61169i	−0.0289680	−	0.200674i
$783$		0			0
$784$	−38.1839	+	24.6611i	−1.36371	+	0.880755i
$785$	−0.265089	−	0.459148i	−0.00946144	−	0.0163877i
$786$		0			0
$787$	−44.3265			−1.58007			−0.790035	−	0.613062i	$-0.789938\pi$
$787$	−44.3265			−1.58007			−0.790035	+	0.613062i	$0.789938\pi$
$788$	−22.1682	+	6.53634i	−0.789710	+	0.232847i
$789$		0			0
$790$	5.42407	+	6.88926i	0.192980	+	0.245109i
$791$	−62.6339			−2.22701
$792$		0			0
$793$	−55.2271	−	31.8854i	−1.96117	−	1.13228i
$794$	−14.7550	+	36.9165i	−0.523636	+	1.31012i
$795$		0			0
$796$	31.1398	−	32.7323i	1.10372	−	1.16016i
$797$		−	14.2943i		−	0.506329i	−0.967423	−	0.253165i	$-0.918529\pi$
$797$		−	14.2943i		−	0.506329i	0.967423	−	0.253165i	$-0.0814714\pi$
$798$		0			0
$799$		−	0.106501i		−	0.00376774i
$800$	21.6119	+	7.41180i	0.764096	+	0.262047i
$801$		0			0
$802$	16.5219	+	6.60358i	0.583410	+	0.233181i
$803$	50.6105	+	29.2200i	1.78601	+	1.03115i
$804$		0			0
$805$	−23.7590			−0.837393
$806$	17.0560	−	13.4286i	0.600773	−	0.473002i
$807$		0			0
$808$	−29.0988	−	20.6471i	−1.02369	−	0.726361i
$809$	−15.7026			−0.552074			−0.276037	−	0.961147i	$-0.589021\pi$
$809$	−15.7026			−0.552074			−0.276037	+	0.961147i	$0.589021\pi$
$810$		0			0
$811$	−2.70921	−	4.69248i	−0.0951331	−	0.164775i	0.814531	−	0.580120i	$-0.196994\pi$
$811$	−2.70921	−	4.69248i	−0.0951331	−	0.164775i	−0.909664	+	0.415345i	$0.863661\pi$
$812$	28.2536	+	26.8790i	0.991507	+	0.943268i
$813$		0			0
$814$	−27.6188	+	3.98687i	−0.968037	+	0.139740i
$815$	−8.09277	−	4.67236i	−0.283477	−	0.163666i
$816$		0			0
$817$	21.9073	−	11.6090i	0.766438	−	0.406147i
$818$	−16.1966	+	12.7519i	−0.566301	+	0.445861i
$819$		0			0
$820$	−1.72530	+	1.81353i	−0.0602500	+	0.0633312i
$821$	−8.93735	−	15.4799i	−0.311916	−	0.540254i	0.666861	−	0.745182i	$-0.267637\pi$
$821$	−8.93735	−	15.4799i	−0.311916	−	0.540254i	−0.978777	+	0.204928i	$0.934304\pi$
$822$		0			0
$823$	−25.1381	+	14.5135i	−0.876260	+	0.505909i	−0.869423	−	0.494068i	$-0.835509\pi$
$823$	−25.1381	+	14.5135i	−0.876260	+	0.505909i	−0.00683644	+	0.999977i	$0.502176\pi$
$824$	−2.21830	+	1.01760i	−0.0772780	+	0.0354498i
$825$		0			0
$826$	37.8627	+	15.1332i	1.31741	+	0.526551i
$827$	13.8666	+	24.0177i	0.482190	+	0.835178i	0.999791	−	0.0204443i	$-0.00650809\pi$
$827$	13.8666	+	24.0177i	0.482190	+	0.835178i	−0.517601	+	0.855622i	$0.673175\pi$
$828$		0			0
$829$		−	21.3004i		−	0.739795i	−0.929073	−	0.369897i	$-0.879393\pi$
$829$		−	21.3004i		−	0.739795i	0.929073	−	0.369897i	$-0.120607\pi$
$830$	20.4122	+	8.15846i	0.708517	+	0.283184i
$831$		0			0
$832$	7.52216	−	39.7518i	0.260784	−	1.37815i
$833$	4.02798	−	6.97667i	0.139561	−	0.241727i
$834$		0			0
$835$	10.6335			0.367989
$836$	31.7162	+	8.88634i	1.09693	+	0.307340i
$837$		0			0
$838$	1.67015	+	11.5699i	0.0576945	+	0.399675i
$839$	−16.9556	+	29.3680i	−0.585372	+	1.01389i	0.409457	+	0.912330i	$0.365718\pi$
$839$	−16.9556	+	29.3680i	−0.585372	+	1.01389i	−0.994829	+	0.101565i	$0.967615\pi$
$840$		0			0
$841$	−4.14849	+	7.18540i	−0.143051	+	0.247772i
$842$	−48.6040	−	19.4263i	−1.67501	−	0.669476i
$843$		0			0
$844$	0.501096	−	2.07576i	0.0172484	−	0.0714505i
$845$	−6.16387	−	10.6761i	−0.212044	−	0.367270i
$846$		0			0
$847$		−	14.0336i		−	0.482202i
$848$	−20.6988	+	40.3733i	−0.710798	+	1.38642i
$849$		0			0
$850$	−4.00769	+	0.578525i	−0.137463	+	0.0198432i
$851$	14.7678	+	25.5786i	0.506234	+	0.876823i
$852$		0			0
$853$	−0.360150	+	0.623798i	−0.0123313	+	0.0213584i	−0.872125	−	0.489283i	$-0.837259\pi$
$853$	−0.360150	+	0.623798i	−0.0123313	+	0.0213584i	0.859794	+	0.510641i	$0.170592\pi$
$854$	60.0442	−	47.2741i	2.05467	−	1.61769i
$855$		0			0
$856$	−33.0565	+	15.1640i	−1.12985	+	0.518296i
$857$	22.8485	+	13.1916i	0.780489	+	0.450616i	0.836604	−	0.547809i	$-0.184538\pi$
$857$	22.8485	+	13.1916i	0.780489	+	0.450616i	−0.0561143	+	0.998424i	$0.517871\pi$
$858$		0			0
$859$	18.1975	−	10.5063i	0.620890	−	0.358471i	−0.156326	−	0.987706i	$-0.549965\pi$
$859$	18.1975	−	10.5063i	0.620890	−	0.358471i	0.777215	+	0.629235i	$0.216632\pi$
$860$	−7.68688	+	8.07999i	−0.262120	+	0.275525i
$861$		0			0
$862$	33.4682	+	42.5089i	1.13993	+	1.44786i
$863$	33.4307			1.13799			0.568997	−	0.822340i	$-0.307332\pi$
$863$	33.4307			1.13799			0.568997	+	0.822340i	$0.307332\pi$
$864$		0			0
$865$	−2.55242	+	1.47364i	−0.0867850	+	0.0501053i
$866$	−19.7434	+	15.5444i	−0.670907	+	0.528220i
$867$		0			0
$868$	7.35716	+	24.9521i	0.249718	+	0.846929i
$869$	20.6934	+	11.9474i	0.701977	+	0.405287i
$870$		0			0
$871$	−44.4978	−	25.6908i	−1.50775	−	0.870500i
$872$	19.3455	+	1.81426i	0.655123	+	0.0614387i
$873$		0			0
$874$	−3.72514	−	34.6627i	−0.126005	−	1.17248i
$875$			37.9734i			1.28374i
$876$		0			0
$877$	2.95806	+	1.70784i	0.0998866	+	0.0576696i	0.549111	−	0.835749i	$-0.314966\pi$
$877$	2.95806	+	1.70784i	0.0998866	+	0.0576696i	−0.449225	+	0.893419i	$0.648300\pi$
$878$	1.23257	−	3.08386i	0.0415973	−	0.104075i
$879$		0			0
$880$	−14.7975	+	0.738339i	−0.498823	+	0.0248894i
$881$	28.9200			0.974340			0.487170	−	0.873307i	$-0.338029\pi$
$881$	28.9200			0.974340			0.487170	+	0.873307i	$0.338029\pi$
$882$		0			0
$883$	30.2498	−	17.4647i	1.01799	−	0.587734i	0.104466	−	0.994529i	$-0.466687\pi$
$883$	30.2498	−	17.4647i	1.01799	−	0.587734i	0.913520	+	0.406794i	$0.133353\pi$
$884$	2.02783	+	6.87744i	0.0682032	+	0.231313i
$885$		0			0
$886$	−2.04397	+	1.60926i	−0.0686685	+	0.0540642i
$887$	21.8812	+	37.8993i	0.734698	+	1.27253i	0.954856	+	0.297070i	$0.0960094\pi$
$887$	21.8812	+	37.8993i	0.734698	+	1.27253i	−0.220158	+	0.975464i	$0.570657\pi$
$888$		0			0
$889$	29.9725	−	17.3046i	1.00524	−	0.580378i
$890$	0.743711	+	5.15200i	0.0249293	+	0.172696i
$891$		0			0
$892$	7.37137	−	30.5354i	0.246812	−	1.02240i
$893$	0.0237695	−	0.654412i	0.000795417	−	0.0218991i
$894$		0			0
$895$	2.99891	−	5.19426i	0.100242	−	0.173625i
$896$	40.8893	+	26.0505i	1.36602	+	0.870288i
$897$		0			0
$898$	−0.301443	−	2.08823i	−0.0100593	−	0.0696850i
$899$	11.9604	−	6.90533i	0.398901	−	0.230306i
$900$		0			0
$901$		−	8.04086i		−	0.267880i
$902$	−2.53166	+	6.33412i	−0.0842949	+	0.210903i
$903$		0			0
$904$	17.2369	+	37.5753i	0.573292	+	1.24974i
$905$		−	3.24324i		−	0.107809i
$906$		0			0
$907$	−9.97196	+	17.2719i	−0.331113	+	0.573505i	−0.982730	−	0.185043i	$-0.940758\pi$
$907$	−9.97196	+	17.2719i	−0.331113	+	0.573505i	0.651617	+	0.758548i	$0.274091\pi$
$908$	13.1472	−	3.87647i	0.436305	−	0.128645i
$909$		0			0
$910$	29.7376	−	4.29274i	0.985793	−	0.142303i
$911$	−1.26063			−0.0417665			−0.0208833	−	0.999782i	$-0.506648\pi$
$911$	−1.26063			−0.0417665			−0.0208833	+	0.999782i	$0.506648\pi$
$912$		0			0
$913$	59.9046			1.98255
$914$	23.6525	−	3.41433i	0.782356	−	0.112936i
$915$		0			0
$916$	7.50940	−	2.21416i	0.248117	−	0.0731578i
$917$	−12.4846	+	21.6239i	−0.412277	+	0.714084i
$918$		0			0
$919$		−	37.8973i		−	1.25012i	−0.780578	−	0.625058i	$-0.785075\pi$
$919$		−	37.8973i		−	1.25012i	0.780578	−	0.625058i	$-0.214925\pi$
$920$	6.53849	+	14.2535i	0.215568	+	0.469923i
$921$		0			0
$922$	11.2640	−	28.1820i	0.370959	−	0.928126i
$923$			16.2145i			0.533708i
$924$		0			0
$925$	18.2674	−	10.5467i	0.600629	−	0.346773i
$926$	1.27961	+	8.86440i	0.0420506	+	0.291302i
$927$		0			0
$928$	8.34981	−	24.3470i	0.274096	−	0.799230i
$929$	−11.1030	+	19.2310i	−0.364278	+	0.630949i	−0.988660	−	0.150171i	$-0.952018\pi$
$929$	−11.1030	+	19.2310i	−0.364278	+	0.630949i	0.624382	+	0.781119i	$0.285351\pi$
$930$		0			0
$931$	26.3076	−	41.9702i	0.862198	−	1.37552i
$932$	6.93482	−	28.7271i	0.227158	−	0.940986i
$933$		0			0
$934$	2.30785	+	15.9875i	0.0755152	+	0.523126i
$935$	2.27400	−	1.31290i	0.0743679	−	0.0429363i
$936$		0			0
$937$	−19.6950	−	34.1127i	−0.643406	−	1.11441i	−0.984667	−	0.174444i	$-0.944187\pi$
$937$	−19.6950	−	34.1127i	−0.643406	−	1.11441i	0.341261	−	0.939969i	$-0.389146\pi$
$938$	48.3790	−	38.0899i	1.57963	−	1.24368i
$939$		0			0
$940$	0.0833055	+	0.282534i	0.00271713	+	0.00921523i
$941$	0.888923	−	0.513220i	0.0289781	−	0.0167305i	−0.485441	−	0.874269i	$-0.661341\pi$
$941$	0.888923	−	0.513220i	0.0289781	−	0.0167305i	0.514419	+	0.857539i	$0.328008\pi$
$942$		0			0
$943$	7.21991			0.235112
$944$	−1.34118	−	26.8793i	−0.0436515	−	0.874846i
$945$		0			0
$946$	−11.2795	+	28.2210i	−0.366729	+	0.917543i
$947$	−30.9889	−	17.8915i	−1.00701	−	0.581395i	−0.0966919	−	0.995314i	$-0.530826\pi$
$947$	−30.9889	−	17.8915i	−1.00701	−	0.581395i	−0.910313	+	0.413920i	$0.864159\pi$
$948$		0			0
$949$			78.2224i			2.53921i
$950$	−24.7550	+	2.66038i	−0.803157	+	0.0863140i
$951$		0			0
$952$	−8.55496	−	0.802302i	−0.277268	−	0.0260027i
$953$	48.8654	+	28.2125i	1.58291	+	0.913891i	0.994433	+	0.105374i	$0.0336041\pi$
$953$	48.8654	+	28.2125i	1.58291	+	0.913891i	0.588473	+	0.808517i	$0.299729\pi$
$954$		0			0
$955$	−6.96387	−	4.02060i	−0.225346	−	0.130103i
$956$	−9.53261	−	32.3302i	−0.308307	−	1.04563i
$957$		0			0
$958$	−36.1800	+	28.4853i	−1.16892	+	0.920319i
$959$	40.0087	−	23.0990i	1.29195	−	0.745907i
$960$		0			0
$961$	−21.7871			−0.702810
$962$	−23.1055	−	29.3469i	−0.744950	−	0.946183i
$963$		0			0
$964$	17.0442	−	17.9158i	0.548956	−	0.577030i
$965$	15.1639	−	8.75487i	0.488143	−	0.281829i
$966$		0			0
$967$	−2.89715	−	1.67267i	−0.0931661	−	0.0537895i	0.452693	−	0.891666i	$-0.350463\pi$
$967$	−2.89715	−	1.67267i	−0.0931661	−	0.0537895i	−0.545859	+	0.837877i	$0.683797\pi$
$968$	−8.41906	+	3.86208i	−0.270599	+	0.124132i
$969$		0			0
$970$	1.63203	−	1.28494i	0.0524014	−	0.0412568i
$971$	−7.62452	+	13.2061i	−0.244683	+	0.423803i	−0.962042	−	0.272900i	$-0.912017\pi$
$971$	−7.62452	+	13.2061i	−0.244683	+	0.423803i	0.717360	+	0.696703i	$0.245350\pi$
$972$		0			0
$973$	23.0500	+	39.9238i	0.738950	+	1.27990i
$974$	−17.5419	+	2.53225i	−0.562080	+	0.0811384i
$975$		0			0
$976$	−44.8849	−	23.0118i	−1.43673	−	0.736590i
$977$			9.19538i			0.294186i	0.989123	+	0.147093i	$0.0469917\pi$
$977$			9.19538i			0.294186i	−0.989123	+	0.147093i	$0.953008\pi$
$978$		0			0
$979$	7.09273	+	12.2850i	0.226684	+	0.392629i
$980$	−5.22855	+	21.6589i	−0.167020	+	0.691869i
$981$		0			0
$982$	−21.9279	−	8.76427i	−0.699748	−	0.279679i
$983$	−12.6703	+	21.9456i	−0.404119	+	0.699955i	−0.994219	−	0.107375i	$-0.965755\pi$
$983$	−12.6703	+	21.9456i	−0.404119	+	0.699955i	0.590099	+	0.807331i	$0.299089\pi$
$984$		0			0
$985$	−5.66442	+	9.81106i	−0.180483	+	0.312606i
$986$	0.651740	+	4.51488i	0.0207556	+	0.143783i
$987$		0			0
$988$	10.9253	+	42.7121i	0.347581	+	1.35885i
$989$	32.1675			1.02287
$990$		0			0
$991$	−13.6238	+	23.5971i	−0.432775	+	0.749587i	−0.997111	−	0.0759574i	$-0.975799\pi$
$991$	−13.6238	+	23.5971i	−0.432775	+	0.749587i	0.564337	+	0.825545i	$0.309132\pi$
$992$	12.9445	−	11.2806i	0.410990	−	0.358158i
$993$		0			0
$994$	−18.0432	−	7.21160i	−0.572295	−	0.228738i
$995$		−	22.1454i		−	0.702055i
$996$		0			0
$997$	4.14522	+	7.17973i	0.131280	+	0.227384i	0.924170	−	0.381981i	$-0.124758\pi$
$997$	4.14522	+	7.17973i	0.131280	+	0.227384i	−0.792890	+	0.609365i	$0.791425\pi$
$998$	11.4448	+	4.57434i	0.362280	+	0.144798i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	684.2.r.b.559.10		20
3.2	odd	2		228.2.k.b.103.1	yes	20
4.3	odd	2		684.2.r.c.559.7		20
12.11	even	2		228.2.k.a.103.4	yes	20
19.12	odd	6		684.2.r.c.487.7		20
57.50	even	6		228.2.k.a.31.4	✓	20
76.31	even	6	inner	684.2.r.b.487.10		20
228.107	odd	6		228.2.k.b.31.1	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
228.2.k.a.31.4	✓	20	57.50	even	6
228.2.k.a.103.4	yes	20	12.11	even	2
228.2.k.b.31.1	yes	20	228.107	odd	6
228.2.k.b.103.1	yes	20	3.2	odd	2
684.2.r.b.487.10		20	76.31	even	6	inner
684.2.r.b.559.10		20	1.1	even	1	trivial
684.2.r.c.487.7		20	19.12	odd	6
684.2.r.c.559.7		20	4.3	odd	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:	\( N \)	\(=\)	\( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	684.r (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.39971 - 0.202053i) q^{2} +(1.91835 - 0.565628i) q^{4} +(0.490176 - 0.849009i) q^{5} +4.28530i q^{7} +(2.57084 - 1.17932i) q^{8} +(0.514557 - 1.28740i) q^{10} -3.77820i q^{11} +(4.37962 - 2.52858i) q^{13} +(0.865857 + 5.99816i) q^{14} +(3.36013 - 2.17015i) q^{16} +(-0.354457 + 0.613937i) q^{17} +(-2.31504 + 3.69332i) q^{19} +(0.460105 - 1.90595i) q^{20} +(-0.763397 - 5.28837i) q^{22} +(-4.89773 + 2.82770i) q^{23} +(2.01946 + 3.49780i) q^{25} +(5.61928 - 4.42418i) q^{26} +(2.42389 + 8.22071i) q^{28} +(3.94046 - 2.27503i) q^{29} +3.03527 q^{31} +(4.26471 - 3.71649i) q^{32} +(-0.372087 + 0.930950i) q^{34} +(3.63826 + 2.10055i) q^{35} -5.22254i q^{37} +(-2.49412 + 5.63732i) q^{38} +(0.258908 - 2.76074i) q^{40} +(-1.10560 - 0.638319i) q^{41} +(-4.92588 - 2.84396i) q^{43} +(-2.13706 - 7.24792i) q^{44} +(-6.28403 + 4.94755i) q^{46} +(-0.130104 + 0.0751157i) q^{47} -11.3638 q^{49} +(3.53338 + 4.48785i) q^{50} +(6.97141 - 7.32794i) q^{52} +(-9.82290 + 5.67125i) q^{53} +(-3.20773 - 1.85198i) q^{55} +(5.05375 + 11.0168i) q^{56} +(5.05581 - 3.98055i) q^{58} +(3.36409 - 5.82677i) q^{59} +(-6.30501 - 10.9206i) q^{61} +(4.24849 - 0.613285i) q^{62} +(5.21840 - 6.06368i) q^{64} -4.95779i q^{65} +(-5.08009 - 8.79898i) q^{67} +(-0.332712 + 1.37824i) q^{68} +(5.51692 + 2.20503i) q^{70} +(-1.60313 + 2.77670i) q^{71} +(-7.73383 + 13.3954i) q^{73} +(-1.05523 - 7.31002i) q^{74} +(-2.35200 + 8.39453i) q^{76} +16.1908 q^{77} +(-3.16218 + 5.47706i) q^{79} +(-0.195421 - 3.91653i) q^{80} +(-1.67649 - 0.670069i) q^{82} +15.8553i q^{83} +(0.347492 + 0.601874i) q^{85} +(-7.46942 - 2.98542i) q^{86} +(-4.45572 - 9.71315i) q^{88} +(-3.25153 + 1.87727i) q^{89} +(10.8357 + 18.7680i) q^{91} +(-7.79612 + 8.19482i) q^{92} +(-0.166930 + 0.131428i) q^{94} +(2.00089 + 3.77586i) q^{95} +(1.29749 + 0.749106i) q^{97} +(-15.9060 + 2.29609i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - q^{2} + q^{4} - 4 q^{8} + 6 q^{10} + 6 q^{13} - 9 q^{14} - 11 q^{16} + 12 q^{19} + 14 q^{20} + 8 q^{22} - 10 q^{25} + 7 q^{28} + 12 q^{31} + 29 q^{32} - 6 q^{34} - 25 q^{38} - 46 q^{40} - 12 q^{41}+ \cdots - 12 q^{97}+O(q^{100}) \) 20 * q - q^2 + q^4 - 4 * q^8 + 6 * q^10 + 6 * q^13 - 9 * q^14 - 11 * q^16 + 12 * q^19 + 14 * q^20 + 8 * q^22 - 10 * q^25 + 7 * q^28 + 12 * q^31 + 29 * q^32 - 6 * q^34 - 25 * q^38 - 46 * q^40 - 12 * q^41 - 18 * q^43 + 12 * q^44 - 32 * q^46 + 36 * q^47 - 24 * q^49 + 34 * q^50 + 3 * q^52 + 12 * q^53 - 38 * q^56 + 16 * q^58 + 10 * q^61 + 55 * q^62 - 14 * q^64 + 14 * q^67 - 40 * q^68 + 26 * q^70 - 8 * q^71 - 6 * q^73 - 45 * q^74 - 69 * q^76 + 32 * q^77 - 6 * q^79 - 22 * q^80 + 24 * q^82 + 8 * q^85 - 7 * q^86 + 44 * q^88 + 24 * q^89 + 26 * q^91 + 18 * q^92 - 52 * q^94 - 4 * q^95 - 12 * q^97

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