LMFDB - Embedded newform 378.3.i.a.359.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [378,3,Mod(179,378)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(378, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([5, 4])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("378.179"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 378 = 2 \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	378.i (of order $6$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)

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

Self dual:	no
Analytic conductor:	$10.2997539928$
Analytic rank:	$0$
Dimension:	$32$
Relative dimension:	$16$ over $\Q(\zeta_{6})$
Twist minimal:	no (minimal twist has level 126)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			359.2
Character	$\chi$	$=$	378.359
Dual form			378.3.i.a.179.7

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-1.22474 + 0.707107i) q^{2} +(1.00000 - 1.73205i) q^{4} -8.31909i q^{5} +(0.934868 - 6.93729i) q^{7} +2.82843i q^{8} +(5.88249 + 10.1888i) q^{10} -10.5446i q^{11} +(1.24066 + 2.14889i) q^{13} +(3.76043 + 9.15746i) q^{14} +(-2.00000 - 3.46410i) q^{16} +(-27.5318 + 15.8955i) q^{17} +(6.85989 - 11.8817i) q^{19} +(-14.4091 - 8.31909i) q^{20} +(7.45618 + 12.9145i) q^{22} +32.1357i q^{23} -44.2073 q^{25} +(-3.03899 - 1.75456i) q^{26} +(-11.0809 - 8.55653i) q^{28} +(-2.05673 - 1.18745i) q^{29} +(-10.3843 + 17.9861i) q^{31} +(4.89898 + 2.82843i) q^{32} +(22.4796 - 38.9358i) q^{34} +(-57.7120 - 7.77726i) q^{35} +(5.23692 - 9.07061i) q^{37} +19.4027i q^{38} +23.5300 q^{40} +(43.1999 - 24.9415i) q^{41} +(16.3930 - 28.3936i) q^{43} +(-18.2638 - 10.5446i) q^{44} +(-22.7234 - 39.3581i) q^{46} +(-36.0009 + 20.7851i) q^{47} +(-47.2520 - 12.9709i) q^{49} +(54.1427 - 31.2593i) q^{50} +4.96266 q^{52} +(67.8875 - 39.1948i) q^{53} -87.7218 q^{55} +(19.6216 + 2.64421i) q^{56} +3.35862 q^{58} +(-62.1285 - 35.8699i) q^{59} +(-33.0727 - 57.2835i) q^{61} -29.3711i q^{62} -8.00000 q^{64} +(17.8768 - 10.3212i) q^{65} +(12.4724 - 21.6028i) q^{67} +63.5820i q^{68} +(76.1818 - 31.2834i) q^{70} +23.3875i q^{71} +(19.5871 + 33.9258i) q^{73} +14.8122i q^{74} +(-13.7198 - 23.7633i) q^{76} +(-73.1512 - 9.85784i) q^{77} +(5.03687 + 8.72412i) q^{79} +(-28.8182 + 16.6382i) q^{80} +(-35.2726 + 61.0939i) q^{82} +(-95.0843 - 54.8969i) q^{83} +(132.236 + 229.040i) q^{85} +46.3665i q^{86} +29.8247 q^{88} +(36.4975 + 21.0718i) q^{89} +(16.0674 - 6.59792i) q^{91} +(55.6607 + 32.1357i) q^{92} +(29.3946 - 50.9129i) q^{94} +(-98.8447 - 57.0680i) q^{95} +(-37.7643 + 65.4097i) q^{97} +(67.0435 - 17.5262i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q + 32 q^{4} + 2 q^{7} + 10 q^{13} - 36 q^{14} - 64 q^{16} - 54 q^{17} + 28 q^{19} - 160 q^{25} - 72 q^{26} - 4 q^{28} - 36 q^{29} - 8 q^{31} - 90 q^{35} + 22 q^{37} - 72 q^{41} + 16 q^{43} + 72 q^{44}+ \cdots + 288 q^{98}+O(q^{100}) $ 32 * q + 32 * q^4 + 2 * q^7 + 10 * q^13 - 36 * q^14 - 64 * q^16 - 54 * q^17 + 28 * q^19 - 160 * q^25 - 72 * q^26 - 4 * q^28 - 36 * q^29 - 8 * q^31 - 90 * q^35 + 22 * q^37 - 72 * q^41 + 16 * q^43 + 72 * q^44 - 12 * q^46 + 108 * q^47 + 74 * q^49 + 288 * q^50 + 40 * q^52 - 72 * q^53 - 24 * q^55 + 48 * q^58 + 90 * q^59 - 62 * q^61 - 256 * q^64 - 378 * q^65 + 70 * q^67 - 108 * q^70 + 196 * q^73 - 56 * q^76 - 630 * q^77 - 38 * q^79 + 60 * q^85 - 486 * q^89 - 122 * q^91 - 252 * q^92 + 168 * q^94 + 72 * q^95 - 38 * q^97 + 288 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/378\mathbb{Z}\right)^\times$.

$n$	$29$	$325$
$\chi(n)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{1}{3}\right)$

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 $

$2$	−1.22474	+	0.707107i	−0.612372	+	0.353553i
$2$	−1.22474	+	0.707107i	−0.612372	+	0.353553i
$3$		0			0
$3$		0			0
$4$	1.00000	−	1.73205i	0.250000	−	0.433013i
$5$		−	8.31909i		−	1.66382i	−0.554911	−	0.831909i	$-0.687248\pi$
$5$		−	8.31909i		−	1.66382i	0.554911	−	0.831909i	$-0.312752\pi$
$6$		0			0
$7$	0.934868	−	6.93729i	0.133553	−	0.991042i
$7$	0.934868	−	6.93729i	0.133553	−	0.991042i
$8$			2.82843i			0.353553i
$9$		0			0
$10$	5.88249	+	10.1888i	0.588249	+	1.01888i
$11$		−	10.5446i		−	0.958603i	−0.877650	−	0.479301i	$-0.840890\pi$
$11$		−	10.5446i		−	0.958603i	0.877650	−	0.479301i	$-0.159110\pi$
$12$		0			0
$13$	1.24066	+	2.14889i	0.0954357	+	0.165300i	0.909790	−	0.415068i	$-0.136242\pi$
$13$	1.24066	+	2.14889i	0.0954357	+	0.165300i	−0.814355	+	0.580368i	$0.802909\pi$
$14$	3.76043	+	9.15746i	0.268602	+	0.654105i
$15$		0			0
$16$	−2.00000	−	3.46410i	−0.125000	−	0.216506i
$17$	−27.5318	+	15.8955i	−1.61952	+	0.935029i	−0.632473	+	0.774582i	$0.717960\pi$
$17$	−27.5318	+	15.8955i	−1.61952	+	0.935029i	−0.987045	+	0.160446i	$0.948707\pi$
$18$		0			0
$19$	6.85989	−	11.8817i	0.361047	−	0.625351i	−0.627087	−	0.778949i	$-0.715753\pi$
$19$	6.85989	−	11.8817i	0.361047	−	0.625351i	0.988133	+	0.153598i	$0.0490862\pi$
$20$	−14.4091	−	8.31909i	−0.720455	−	0.415955i
$21$		0			0
$22$	7.45618	+	12.9145i	0.338917	+	0.587022i
$23$			32.1357i			1.39721i	0.715509	+	0.698603i	$0.246195\pi$
$23$			32.1357i			1.39721i	−0.715509	+	0.698603i	$0.753805\pi$
$24$		0			0
$25$	−44.2073			−1.76829
$26$	−3.03899	−	1.75456i	−0.116884	−	0.0674832i
$27$		0			0
$28$	−11.0809	−	8.55653i	−0.395746	−	0.305590i
$29$	−2.05673	−	1.18745i	−0.0709216	−	0.0409466i	0.464120	−	0.885772i	$-0.346371\pi$
$29$	−2.05673	−	1.18745i	−0.0709216	−	0.0409466i	−0.535041	+	0.844826i	$0.679704\pi$
$30$		0			0
$31$	−10.3843	+	17.9861i	−0.334976	+	0.580196i	−0.983480	−	0.181015i	$-0.942062\pi$
$31$	−10.3843	+	17.9861i	−0.334976	+	0.580196i	0.648504	+	0.761211i	$0.275395\pi$
$32$	4.89898	+	2.82843i	0.153093	+	0.0883883i
$33$		0			0
$34$	22.4796	−	38.9358i	0.661165	−	1.14517i
$35$	−57.7120	−	7.77726i	−1.64891	−	0.222207i
$36$		0			0
$37$	5.23692	−	9.07061i	0.141538	−	0.245152i	−0.786538	−	0.617542i	$-0.788128\pi$
$37$	5.23692	−	9.07061i	0.141538	−	0.245152i	0.928076	+	0.372391i	$0.121462\pi$
$38$			19.4027i			0.510597i
$39$		0			0
$40$	23.5300			0.588249
$41$	43.1999	−	24.9415i	1.05366	−	0.608329i	0.129986	−	0.991516i	$-0.458507\pi$
$41$	43.1999	−	24.9415i	1.05366	−	0.608329i	0.923671	+	0.383187i	$0.125173\pi$
$42$		0			0
$43$	16.3930	−	28.3936i	0.381234	−	0.660316i	−0.610005	−	0.792397i	$-0.708833\pi$
$43$	16.3930	−	28.3936i	0.381234	−	0.660316i	0.991239	+	0.132081i	$0.0421660\pi$
$44$	−18.2638	−	10.5446i	−0.415087	−	0.239651i
$45$		0			0
$46$	−22.7234	−	39.3581i	−0.493987	−	0.855610i
$47$	−36.0009	+	20.7851i	−0.765976	+	0.442236i	−0.831437	−	0.555619i	$-0.812481\pi$
$47$	−36.0009	+	20.7851i	−0.765976	+	0.442236i	0.0654613	+	0.997855i	$0.479148\pi$
$48$		0			0
$49$	−47.2520	−	12.9709i	−0.964327	−	0.264712i
$50$	54.1427	−	31.2593i	1.08285	−	0.625186i
$51$		0			0
$52$	4.96266			0.0954357
$53$	67.8875	−	39.1948i	1.28090	−	0.739525i	0.303883	−	0.952709i	$-0.401717\pi$
$53$	67.8875	−	39.1948i	1.28090	−	0.739525i	0.977012	+	0.213184i	$0.0683834\pi$
$54$		0			0
$55$	−87.7218			−1.59494
$56$	19.6216	+	2.64421i	0.350386	+	0.0472180i
$57$		0			0
$58$	3.35862			0.0579072
$59$	−62.1285	−	35.8699i	−1.05303	−	0.607965i	−0.129531	−	0.991575i	$-0.541347\pi$
$59$	−62.1285	−	35.8699i	−1.05303	−	0.607965i	−0.923495	+	0.383610i	$0.874681\pi$
$60$		0			0
$61$	−33.0727	−	57.2835i	−0.542175	−	0.939074i	−0.998779	−	0.0494044i	$-0.984268\pi$
$61$	−33.0727	−	57.2835i	−0.542175	−	0.939074i	0.456604	−	0.889670i	$-0.349066\pi$
$62$		−	29.3711i		−	0.473728i
$63$		0			0
$64$	−8.00000			−0.125000
$65$	17.8768	−	10.3212i	0.275028	−	0.158788i
$66$		0			0
$67$	12.4724	−	21.6028i	0.186155	−	0.322431i	−0.757810	−	0.652475i	$-0.773731\pi$
$67$	12.4724	−	21.6028i	0.186155	−	0.322431i	0.943965	+	0.330045i	$0.107064\pi$
$68$			63.5820i			0.935029i
$69$		0			0
$70$	76.1818	−	31.2834i	1.08831	−	0.446905i
$71$			23.3875i			0.329402i	0.986344	+	0.164701i	$0.0526659\pi$
$71$			23.3875i			0.329402i	−0.986344	+	0.164701i	$0.947334\pi$
$72$		0			0
$73$	19.5871	+	33.9258i	0.268316	+	0.464737i	0.968427	−	0.249297i	$-0.0801996\pi$
$73$	19.5871	+	33.9258i	0.268316	+	0.464737i	−0.700111	+	0.714034i	$0.746866\pi$
$74$			14.8122i			0.200166i
$75$		0			0
$76$	−13.7198	−	23.7633i	−0.180523	−	0.312676i
$77$	−73.1512	−	9.85784i	−0.950015	−	0.128024i
$78$		0			0
$79$	5.03687	+	8.72412i	0.0637579	+	0.110432i	0.896142	−	0.443767i	$-0.146358\pi$
$79$	5.03687	+	8.72412i	0.0637579	+	0.110432i	−0.832384	+	0.554199i	$0.813025\pi$
$80$	−28.8182	+	16.6382i	−0.360227	+	0.207977i
$81$		0			0
$82$	−35.2726	+	61.0939i	−0.430154	+	0.745048i
$83$	−95.0843	−	54.8969i	−1.14559	−	0.661409i	−0.197784	−	0.980246i	$-0.563375\pi$
$83$	−95.0843	−	54.8969i	−1.14559	−	0.661409i	−0.947810	+	0.318837i	$0.896708\pi$
$84$		0			0
$85$	132.236	+	229.040i	1.55572	+	2.69458i
$86$			46.3665i			0.539146i
$87$		0			0
$88$	29.8247			0.338917
$89$	36.4975	+	21.0718i	0.410084	+	0.236762i	0.690826	−	0.723021i	$-0.257247\pi$
$89$	36.4975	+	21.0718i	0.410084	+	0.236762i	−0.280742	+	0.959783i	$0.590580\pi$
$90$		0			0
$91$	16.0674	−	6.59792i	0.176564	−	0.0725046i
$92$	55.6607	+	32.1357i	0.605008	+	0.349302i
$93$		0			0
$94$	29.3946	−	50.9129i	0.312708	−	0.541627i
$95$	−98.8447	−	57.0680i	−1.04047	−	0.600716i
$96$		0			0
$97$	−37.7643	+	65.4097i	−0.389322	+	0.674326i	−0.992359	−	0.123388i	$-0.960624\pi$
$97$	−37.7643	+	65.4097i	−0.389322	+	0.674326i	0.603036	+	0.797714i	$0.293957\pi$
$98$	67.0435	−	17.5262i	0.684117	−	0.178839i
$99$		0			0
$100$	−44.2073	+	76.5693i	−0.442073	+	0.765693i
$101$		−	131.004i		−	1.29707i	−0.761186	−	0.648534i	$-0.775382\pi$
$101$		−	131.004i		−	1.29707i	0.761186	−	0.648534i	$-0.224618\pi$
$102$		0			0
$103$	−33.2224			−0.322548			−0.161274	−	0.986910i	$-0.551560\pi$
$103$	−33.2224			−0.322548			−0.161274	+	0.986910i	$0.551560\pi$
$104$	−6.07799	+	3.50913i	−0.0584422	+	0.0337416i
$105$		0			0
$106$	−55.4299	+	96.0074i	−0.522923	+	0.905730i
$107$	46.8532	+	27.0507i	0.437881	+	0.252810i	0.702698	−	0.711488i	$-0.251978\pi$
$107$	46.8532	+	27.0507i	0.437881	+	0.252810i	−0.264818	+	0.964299i	$0.585312\pi$
$108$		0			0
$109$	−14.9320	−	25.8630i	−0.136991	−	0.237275i	0.789365	−	0.613924i	$-0.210410\pi$
$109$	−14.9320	−	25.8630i	−0.136991	−	0.237275i	−0.926356	+	0.376648i	$0.877076\pi$
$110$	107.437	−	62.0287i	0.976698	−	0.563897i
$111$		0			0
$112$	−25.9012	+	10.6361i	−0.231261	+	0.0949652i
$113$	160.151	−	92.4632i	1.41726	−	0.818258i	0.421207	−	0.906964i	$-0.361607\pi$
$113$	160.151	−	92.4632i	1.41726	−	0.818258i	0.996058	+	0.0887062i	$0.0282732\pi$
$114$		0			0
$115$	267.340			2.32470
$116$	−4.11345	+	2.37490i	−0.0354608	+	0.0204733i
$117$		0			0
$118$	101.455			0.859792
$119$	84.5331	+	205.856i	0.710362	+	1.72988i
$120$		0			0
$121$	9.81073			0.0810804
$122$	81.0112	+	46.7718i	0.664026	+	0.383376i
$123$		0			0
$124$	20.7685	+	35.9721i	0.167488	+	0.290098i
$125$			159.788i			1.27830i
$126$		0			0
$127$	−111.511			−0.878041			−0.439020	−	0.898477i	$-0.644674\pi$
$127$	−111.511			−0.878041			−0.439020	+	0.898477i	$0.644674\pi$
$128$	9.79796	−	5.65685i	0.0765466	−	0.0441942i
$129$		0			0
$130$	−14.5964	+	25.2817i	−0.112280	+	0.194474i
$131$			56.5812i			0.431918i	0.976402	+	0.215959i	$0.0692877\pi$
$131$			56.5812i			0.431918i	−0.976402	+	0.215959i	$0.930712\pi$
$132$		0			0
$133$	−76.0135	−	58.6968i	−0.571530	−	0.441329i
$134$			35.2773i			0.263263i
$135$		0			0
$136$	−44.9592	−	77.8717i	−0.330583	−	0.572586i
$137$		−	66.0338i		−	0.481998i	−0.970525	−	0.240999i	$-0.922525\pi$
$137$		−	66.0338i		−	0.481998i	0.970525	−	0.240999i	$-0.0774751\pi$
$138$		0			0
$139$	45.2312	+	78.3427i	0.325404	+	0.563617i	0.981594	−	0.190979i	$-0.0611663\pi$
$139$	45.2312	+	78.3427i	0.325404	+	0.563617i	−0.656190	+	0.754596i	$0.727833\pi$
$140$	−71.1826	+	92.1828i	−0.508447	+	0.658449i
$141$		0			0
$142$	−16.5375	−	28.6437i	−0.116461	−	0.201717i
$143$	22.6593	−	13.0823i	0.158457	−	0.0914850i
$144$		0			0
$145$	−9.87852	+	17.1101i	−0.0681277	+	0.118001i
$146$	−47.9783	−	27.7003i	−0.328618	−	0.189728i
$147$		0			0
$148$	−10.4738	−	18.1412i	−0.0707692	−	0.122576i
$149$			98.5488i			0.661401i	0.943736	+	0.330701i	$0.107285\pi$
$149$			98.5488i			0.661401i	−0.943736	+	0.330701i	$0.892715\pi$
$150$		0			0
$151$	−39.5943			−0.262214			−0.131107	−	0.991368i	$-0.541853\pi$
$151$	−39.5943			−0.262214			−0.131107	+	0.991368i	$0.541853\pi$
$152$	33.6064	+	19.4027i	0.221095	+	0.127649i
$153$		0			0
$154$	96.5621	−	39.6524i	0.627027	−	0.257483i
$155$	149.628	+	86.3876i	0.965341	+	0.557340i
$156$		0			0
$157$	124.779	−	216.124i	0.794772	−	1.37658i	−0.128213	−	0.991747i	$-0.540924\pi$
$157$	124.779	−	216.124i	0.794772	−	1.37658i	0.922984	−	0.384838i	$-0.125743\pi$
$158$	−12.3378	−	7.12321i	−0.0780872	−	0.0450836i
$159$		0			0
$160$	23.5300	−	40.7551i	0.147062	−	0.254719i
$161$	222.935	+	30.0427i	1.38469	+	0.186600i
$162$		0			0
$163$	−121.137	+	209.816i	−0.743175	+	1.28722i	0.207868	+	0.978157i	$0.433348\pi$
$163$	−121.137	+	209.816i	−0.743175	+	1.28722i	−0.951043	+	0.309060i	$0.899986\pi$
$164$		−	99.7660i		−	0.608329i
$165$		0			0
$166$	155.272			0.935373
$167$	84.4167	−	48.7380i	0.505489	−	0.291844i	−0.225488	−	0.974246i	$-0.572398\pi$
$167$	84.4167	−	48.7380i	0.505489	−	0.291844i	0.730977	+	0.682402i	$0.239064\pi$
$168$		0			0
$169$	81.4215	−	141.026i	0.481784	−	0.834474i
$170$	−323.911	−	187.010i	−1.90536	−	1.10006i
$171$		0			0
$172$	−32.7861	−	56.7872i	−0.190617	−	0.330158i
$173$	46.4195	−	26.8003i	0.268321	−	0.154915i	−0.359803	−	0.933028i	$-0.617156\pi$
$173$	46.4195	−	26.8003i	0.268321	−	0.154915i	0.628124	+	0.778113i	$0.283823\pi$
$174$		0			0
$175$	−41.3280	+	306.679i	−0.236160	+	1.75245i
$176$	−36.5277	+	21.0893i	−0.207544	+	0.119825i
$177$		0			0
$178$	−59.6001			−0.334832
$179$	122.940	−	70.9794i	0.686815	−	0.396533i	−0.115603	−	0.993296i	$-0.536880\pi$
$179$	122.940	−	70.9794i	0.686815	−	0.396533i	0.802418	+	0.596763i	$0.203547\pi$
$180$		0			0
$181$	−207.905			−1.14864			−0.574322	−	0.818629i	$-0.694734\pi$
$181$	−207.905			−1.14864			−0.574322	+	0.818629i	$0.694734\pi$
$182$	−15.0130	+	19.4421i	−0.0824889	+	0.106825i
$183$		0			0
$184$	−90.8936			−0.493987
$185$	−75.4593	−	43.5664i	−0.407888	−	0.235494i
$186$		0			0
$187$	167.612	+	290.313i	0.896321	+	1.55247i
$188$			83.1404i			0.442236i
$189$		0			0
$190$	161.413			0.849541
$191$	−266.638	+	153.943i	−1.39601	+	0.805987i	−0.993972	−	0.109637i	$-0.965031\pi$
$191$	−266.638	+	153.943i	−1.39601	+	0.805987i	−0.402038	+	0.915623i	$0.631698\pi$
$192$		0			0
$193$	82.5670	−	143.010i	0.427808	−	0.740986i	−0.568870	−	0.822428i	$-0.692619\pi$
$193$	82.5670	−	143.010i	0.427808	−	0.740986i	0.996678	+	0.0814418i	$0.0259525\pi$
$194$		−	106.814i		−	0.550585i
$195$		0			0
$196$	−69.7183	+	68.8720i	−0.355706	+	0.351388i
$197$			86.2870i			0.438005i	0.975724	+	0.219003i	$0.0702803\pi$
$197$			86.2870i			0.438005i	−0.975724	+	0.219003i	$0.929720\pi$
$198$		0			0
$199$	−164.720	−	285.304i	−0.827739	−	1.43369i	−0.899808	−	0.436287i	$-0.856293\pi$
$199$	−164.720	−	285.304i	−0.827739	−	1.43369i	0.0720681	−	0.997400i	$-0.477040\pi$
$200$		−	125.037i		−	0.625186i
$201$		0			0
$202$	92.6337	+	160.446i	0.458583	+	0.794289i
$203$	−10.1605	+	13.1580i	−0.0500515	+	0.0648177i
$204$		0			0
$205$	−207.491	−	359.384i	−1.01215	−	1.75309i
$206$	40.6890	−	23.4918i	0.197519	−	0.114038i
$207$		0			0
$208$	4.96266	−	8.59557i	0.0238589	−	0.0413249i
$209$	−125.288	−	72.3350i	−0.599463	−	0.346100i
$210$		0			0
$211$	−40.7068	−	70.5063i	−0.192923	−	0.334153i	0.753294	−	0.657683i	$-0.228464\pi$
$211$	−40.7068	−	70.5063i	−0.192923	−	0.334153i	−0.946218	+	0.323530i	$0.895130\pi$
$212$		−	156.779i		−	0.739525i
$213$		0			0
$214$	−76.5110			−0.357528
$215$	−236.209	−	136.375i	−1.09865	−	0.634304i
$216$		0			0
$217$	115.067	+	88.8533i	0.530261	+	0.409462i
$218$	36.5758	+	21.1171i	0.167779	+	0.0968673i
$219$		0			0
$220$	−87.7218	+	151.939i	−0.398735	+	0.690630i
$221$	−68.3154	−	39.4419i	−0.309120	−	0.178470i
$222$		0			0
$223$	93.2204	−	161.462i	0.418029	−	0.724047i	−0.577712	−	0.816240i	$-0.696054\pi$
$223$	93.2204	−	161.462i	0.418029	−	0.724047i	0.995741	+	0.0921934i	$0.0293878\pi$
$224$	24.2015	−	31.3414i	0.108043	−	0.139917i
$225$		0			0
$226$	−130.763	+	226.488i	−0.578596	+	1.00216i
$227$		−	40.8357i		−	0.179893i	−0.995947	−	0.0899465i	$-0.971330\pi$
$227$		−	40.8357i		−	0.179893i	0.995947	−	0.0899465i	$-0.0286696\pi$
$228$		0			0
$229$	−71.2022			−0.310926			−0.155463	−	0.987842i	$-0.549687\pi$
$229$	−71.2022			−0.310926			−0.155463	+	0.987842i	$0.549687\pi$
$230$	−327.424	+	189.038i	−1.42358	+	0.821905i
$231$		0			0
$232$	3.35862	−	5.81730i	0.0144768	−	0.0250746i
$233$	−6.66414	−	3.84755i	−0.0286015	−	0.0165131i	0.485631	−	0.874164i	$-0.338590\pi$
$233$	−6.66414	−	3.84755i	−0.0286015	−	0.0165131i	−0.514233	+	0.857651i	$0.671923\pi$
$234$		0			0
$235$	172.913	+	299.495i	0.735801	+	1.27445i
$236$	−124.257	+	71.7399i	−0.526513	+	0.303982i
$237$		0			0
$238$	−249.094	−	192.348i	−1.04661	−	0.808183i
$239$	204.678	−	118.171i	0.856392	−	0.494438i	−0.00641057	−	0.999979i	$-0.502041\pi$
$239$	204.678	−	118.171i	0.856392	−	0.494438i	0.862802	+	0.505541i	$0.168707\pi$
$240$		0			0
$241$	126.585			0.525250			0.262625	−	0.964898i	$-0.415412\pi$
$241$	126.585			0.525250			0.262625	+	0.964898i	$0.415412\pi$
$242$	−12.0156	+	6.93724i	−0.0496514	+	0.0286663i
$243$		0			0
$244$	−132.291			−0.542175
$245$	−107.906	+	393.094i	−0.440433	+	1.60447i
$246$		0			0
$247$	34.0433			0.137827
$248$	−50.8723	−	29.3711i	−0.205130	−	0.118432i
$249$		0			0
$250$	−112.987	−	195.699i	−0.451948	−	0.782796i
$251$		−	318.476i		−	1.26883i	−0.772993	−	0.634414i	$-0.781241\pi$
$251$		−	318.476i		−	1.26883i	0.772993	−	0.634414i	$-0.218759\pi$
$252$		0			0
$253$	338.860			1.33937
$254$	136.573	−	78.8503i	0.537688	−	0.310434i
$255$		0			0
$256$	−8.00000	+	13.8564i	−0.0312500	+	0.0541266i
$257$		−	486.137i		−	1.89158i	−0.324773	−	0.945792i	$-0.605288\pi$
$257$		−	486.137i		−	1.89158i	0.324773	−	0.945792i	$-0.394712\pi$
$258$		0			0
$259$	−58.0297	−	44.8099i	−0.224053	−	0.173011i
$260$		−	41.2848i		−	0.158788i
$261$		0			0
$262$	−40.0090	−	69.2976i	−0.152706	−	0.264495i
$263$			348.891i			1.32658i	0.748361	+	0.663291i	$0.230841\pi$
$263$			348.891i			1.32658i	−0.748361	+	0.663291i	$0.769159\pi$
$264$		0			0
$265$	−326.066	−	564.762i	−1.23044	−	2.13118i
$266$	134.602	+	18.1390i	0.506023	+	0.0681915i
$267$		0			0
$268$	−24.9448	−	43.2057i	−0.0930777	−	0.161215i
$269$	335.922	−	193.945i	1.24878	−	0.720985i	0.277916	−	0.960606i	$-0.410356\pi$
$269$	335.922	−	193.945i	1.24878	−	0.720985i	0.970867	+	0.239621i	$0.0770231\pi$
$270$		0			0
$271$	118.336	−	204.964i	0.436664	−	0.756324i	−0.560766	−	0.827974i	$-0.689493\pi$
$271$	118.336	−	204.964i	0.436664	−	0.756324i	0.997430	+	0.0716504i	$0.0228266\pi$
$272$	110.127	+	63.5820i	0.404879	+	0.233757i
$273$		0			0
$274$	46.6929	+	80.8745i	0.170412	+	0.295162i
$275$			466.150i			1.69509i
$276$		0			0
$277$	−203.511			−0.734697			−0.367348	−	0.930083i	$-0.619734\pi$
$277$	−203.511			−0.734697			−0.367348	+	0.930083i	$0.619734\pi$
$278$	−110.793	−	63.9665i	−0.398537	−	0.230095i
$279$		0			0
$280$	21.9974	−	163.234i	0.0785621	−	0.582979i
$281$	−45.9455	−	26.5266i	−0.163507	−	0.0944008i	0.416014	−	0.909358i	$-0.363427\pi$
$281$	−45.9455	−	26.5266i	−0.163507	−	0.0944008i	−0.579521	+	0.814958i	$0.696760\pi$
$282$		0			0
$283$	23.9055	−	41.4056i	0.0844718	−	0.146309i	−0.820694	−	0.571368i	$-0.806413\pi$
$283$	23.9055	−	41.4056i	0.0844718	−	0.146309i	0.905166	+	0.425058i	$0.139746\pi$
$284$	40.5084	+	23.3875i	0.142635	+	0.0823504i
$285$		0			0
$286$	−18.5012	+	32.0451i	−0.0646896	+	0.112046i
$287$	−132.640	−	323.008i	−0.462161	−	1.12546i
$288$		0			0
$289$	360.833	−	624.982i	1.24856	−	2.16257i
$290$		−	27.9407i		−	0.0963471i
$291$		0			0
$292$	78.3482			0.268316
$293$	10.7293	−	6.19459i	0.0366189	−	0.0211419i	−0.481579	−	0.876403i	$-0.659936\pi$
$293$	10.7293	−	6.19459i	0.0366189	−	0.0211419i	0.518198	+	0.855261i	$0.326603\pi$
$294$		0			0
$295$	−298.405	+	516.853i	−1.01154	+	1.75204i
$296$	25.6556	+	14.8122i	0.0866742	+	0.0500414i
$297$		0			0
$298$	−69.6845	−	120.697i	−0.233841	−	0.405024i
$299$	−69.0563	+	39.8697i	−0.230957	+	0.133343i
$300$		0			0
$301$	−181.649	−	140.268i	−0.603486	−	0.466005i
$302$	48.4930	−	27.9974i	0.160573	−	0.0927067i
$303$		0			0
$304$	−54.8791			−0.180523
$305$	−476.547	+	275.135i	−1.56245	+	0.902081i
$306$		0			0
$307$	420.839			1.37081			0.685406	−	0.728161i	$-0.259625\pi$
$307$	420.839			1.37081			0.685406	+	0.728161i	$0.259625\pi$
$308$	−90.2255	+	116.844i	−0.292940	+	0.379363i
$309$		0			0
$310$	−244.341			−0.788197
$311$	−402.800	−	232.557i	−1.29518	−	0.747771i	−0.315610	−	0.948889i	$-0.602209\pi$
$311$	−402.800	−	232.557i	−1.29518	−	0.747771i	−0.979567	+	0.201118i	$0.935542\pi$
$312$		0			0
$313$	182.485	+	316.073i	0.583019	+	1.00982i	0.995119	+	0.0986800i	$0.0314620\pi$
$313$	182.485	+	316.073i	0.583019	+	1.00982i	−0.412100	+	0.911139i	$0.635205\pi$
$314$			352.929i			1.12398i
$315$		0			0
$316$	20.1475			0.0637579
$317$	20.6537	−	11.9244i	0.0651536	−	0.0376164i	−0.467069	−	0.884221i	$-0.654690\pi$
$317$	20.6537	−	11.9244i	0.0651536	−	0.0376164i	0.532223	+	0.846604i	$0.321357\pi$
$318$		0			0
$319$	−12.5212	+	21.6874i	−0.0392515	+	0.0679856i
$320$			66.5528i			0.207977i
$321$		0			0
$322$	−294.282	+	120.844i	−0.913919	+	0.375293i
$323$			436.165i			1.35036i
$324$		0			0
$325$	−54.8465	−	94.9969i	−0.168758	−	0.292298i
$326$		−	342.629i		−	1.05101i
$327$		0			0
$328$	70.5452	+	122.188i	0.215077	+	0.372524i
$329$	110.536	+	269.180i	0.335977	+	0.818176i
$330$		0			0
$331$	−113.068	−	195.839i	−0.341594	−	0.591658i	0.643135	−	0.765753i	$-0.277633\pi$
$331$	−113.068	−	195.839i	−0.341594	−	0.591658i	−0.984729	+	0.174095i	$0.944300\pi$
$332$	−190.169	+	109.794i	−0.572797	+	0.330704i
$333$		0			0
$334$	−68.9259	+	119.383i	−0.206365	+	0.357435i
$335$	−179.716	−	103.759i	−0.536466	−	0.309729i
$336$		0			0
$337$	151.164	+	261.824i	0.448558	+	0.776926i	0.998292	−	0.0584139i	$-0.0186043\pi$
$337$	151.164	+	261.824i	0.448558	+	0.776926i	−0.549734	+	0.835340i	$0.685271\pi$
$338$			230.295i			0.681346i
$339$		0			0
$340$	528.944			1.55572
$341$	189.656	+	109.498i	0.556177	+	0.321109i
$342$		0			0
$343$	−134.157	+	315.675i	−0.391129	+	0.920336i
$344$	80.3092	+	46.3665i	0.233457	+	0.134786i
$345$		0			0
$346$	−37.9014	+	65.6471i	−0.109542	+	0.189732i
$347$	152.785	+	88.2104i	0.440302	+	0.254209i	0.703726	−	0.710472i	$-0.251518\pi$
$347$	152.785	+	88.2104i	0.440302	+	0.254209i	−0.263423	+	0.964680i	$0.584852\pi$
$348$		0			0
$349$	−87.3168	+	151.237i	−0.250191	+	0.433344i	−0.963578	−	0.267426i	$-0.913827\pi$
$349$	−87.3168	+	151.237i	−0.250191	+	0.433344i	0.713387	+	0.700770i	$0.247160\pi$
$350$	−166.239	−	404.827i	−0.474968	−	1.15665i
$351$		0			0
$352$	29.8247	−	51.6579i	0.0847293	−	0.146756i
$353$			142.647i			0.404099i	0.979375	+	0.202050i	$0.0647602\pi$
$353$			142.647i			0.404099i	−0.979375	+	0.202050i	$0.935240\pi$
$354$		0			0
$355$	194.563			0.548065
$356$	72.9950	−	42.1437i	0.205042	−	0.118381i
$357$		0			0
$358$	−100.380	+	173.863i	−0.280391	+	0.485652i
$359$	529.406	+	305.653i	1.47467	+	0.851401i	0.999593	−	0.0285432i	$-0.00908683\pi$
$359$	529.406	+	305.653i	1.47467	+	0.851401i	0.475077	+	0.879944i	$0.342420\pi$
$360$		0			0
$361$	86.3839	+	149.621i	0.239291	+	0.414464i
$362$	254.630	−	147.011i	0.703398	−	0.406107i
$363$		0			0
$364$	4.63943	−	34.4274i	0.0127457	−	0.0945808i
$365$	282.232	−	162.947i	0.773238	−	0.446429i
$366$		0			0
$367$	168.853			0.460090			0.230045	−	0.973180i	$-0.426113\pi$
$367$	168.853			0.460090			0.230045	+	0.973180i	$0.426113\pi$
$368$	111.321	−	64.2715i	0.302504	−	0.174651i
$369$		0			0
$370$	123.224			0.333039
$371$	−208.440	−	507.597i	−0.561834	−	1.36819i
$372$		0			0
$373$	325.465			0.872559			0.436280	−	0.899811i	$-0.356296\pi$
$373$	325.465			0.872559			0.436280	+	0.899811i	$0.356296\pi$
$374$	−410.564	−	237.039i	−1.09777	−	0.633795i
$375$		0			0
$376$	−58.7892	−	101.826i	−0.156354	−	0.270813i
$377$		−	5.89291i		−	0.0156311i
$378$		0			0
$379$	7.72342			0.0203784			0.0101892	−	0.999948i	$-0.496757\pi$
$379$	7.72342			0.0203784			0.0101892	+	0.999948i	$0.496757\pi$
$380$	−197.689	+	114.136i	−0.520235	+	0.300358i
$381$		0			0
$382$	217.709	−	377.083i	0.569919	−	0.987128i
$383$			647.886i			1.69161i	0.533494	+	0.845804i	$0.320879\pi$
$383$			647.886i			1.69161i	−0.533494	+	0.845804i	$0.679121\pi$
$384$		0			0
$385$	−82.0083	+	608.552i	−0.213009	+	1.58065i
$386$			233.535i			0.605012i
$387$		0			0
$388$	75.5286	+	130.819i	0.194661	+	0.337163i
$389$		−	149.368i		−	0.383979i	−0.981397	−	0.191990i	$-0.938506\pi$
$389$		−	149.368i		−	0.383979i	0.981397	−	0.191990i	$-0.0614940\pi$
$390$		0			0
$391$	−510.813	−	884.755i	−1.30643	−	2.26280i
$392$	36.6873	−	133.649i	0.0935900	−	0.340941i
$393$		0			0
$394$	−61.0141	−	105.680i	−0.154858	−	0.268222i
$395$	72.5768	−	41.9022i	0.183739	−	0.106082i
$396$		0			0
$397$	47.6251	−	82.4890i	0.119962	−	0.207781i	−0.799790	−	0.600280i	$-0.795056\pi$
$397$	47.6251	−	82.4890i	0.119962	−	0.207781i	0.919753	+	0.392499i	$0.128389\pi$
$398$	403.480	+	232.949i	1.01377	+	0.585300i
$399$		0			0
$400$	88.4147	+	153.139i	0.221037	+	0.382847i
$401$		−	364.458i		−	0.908874i	−0.890779	−	0.454437i	$-0.849840\pi$
$401$		−	364.458i		−	0.908874i	0.890779	−	0.454437i	$-0.150160\pi$
$402$		0			0
$403$	−51.5335			−0.127875
$404$	−226.905	−	131.004i	−0.561647	−	0.324267i
$405$		0			0
$406$	3.13987	−	23.2997i	0.00773366	−	0.0573885i
$407$	−95.6463	−	55.2214i	−0.235003	−	0.135679i
$408$		0			0
$409$	−390.666	+	676.654i	−0.955174	+	1.65441i	−0.221206	+	0.975227i	$0.570999\pi$
$409$	−390.666	+	676.654i	−0.955174	+	1.65441i	−0.733969	+	0.679183i	$0.762334\pi$
$410$	508.246	+	293.436i	1.23962	+	0.715698i
$411$		0			0
$412$	−33.2224	+	57.5429i	−0.0806369	+	0.139667i
$413$	−306.922	+	397.470i	−0.743153	+	0.962398i
$414$		0			0
$415$	−456.693	+	791.015i	−1.10046	+	1.90606i
$416$			14.0365i			0.0337416i
$417$		0			0
$418$	204.594			0.489460
$419$	401.472	−	231.790i	0.958167	−	0.553198i	0.0625584	−	0.998041i	$-0.480074\pi$
$419$	401.472	−	231.790i	0.958167	−	0.553198i	0.895608	+	0.444843i	$0.146741\pi$
$420$		0			0
$421$	299.652	−	519.012i	0.711762	−	1.23281i	−0.252433	−	0.967614i	$-0.581231\pi$
$421$	299.652	−	519.012i	0.711762	−	1.23281i	0.964195	−	0.265194i	$-0.0854359\pi$
$422$	99.7110	+	57.5682i	0.236282	+	0.136417i
$423$		0			0
$424$	110.860	+	192.015i	0.261462	+	0.452865i
$425$	1217.11	−	702.697i	2.86378	−	1.65341i
$426$		0			0
$427$	−428.311	+	175.882i	−1.00307	+	0.411902i
$428$	93.7064	−	54.1014i	0.218940	−	0.126405i
$429$		0			0
$430$	385.728			0.897041
$431$	−261.400	+	150.919i	−0.606496	+	0.350161i	−0.771593	−	0.636117i	$-0.780540\pi$
$431$	−261.400	+	150.919i	−0.606496	+	0.350161i	0.165097	+	0.986277i	$0.447206\pi$
$432$		0			0
$433$	582.634			1.34557			0.672787	−	0.739836i	$-0.265097\pi$
$433$	582.634			1.34557			0.672787	+	0.739836i	$0.265097\pi$
$434$	−203.756	−	27.4581i	−0.469484	−	0.0632676i
$435$		0			0
$436$	−59.7281			−0.136991
$437$	381.826	+	220.447i	0.873744	+	0.504456i
$438$		0			0
$439$	−28.1868	−	48.8210i	−0.0642069	−	0.111210i	0.832135	−	0.554573i	$-0.187118\pi$
$439$	−28.1868	−	48.8210i	−0.0642069	−	0.111210i	−0.896342	+	0.443363i	$0.853785\pi$
$440$		−	248.115i		−	0.563897i
$441$		0			0
$442$	111.559			0.252395
$443$	165.291	−	95.4305i	0.373116	−	0.215419i	−0.301703	−	0.953402i	$-0.597555\pi$
$443$	165.291	−	95.4305i	0.373116	−	0.215419i	0.674819	+	0.737983i	$0.264222\pi$
$444$		0			0
$445$	175.299	−	303.626i	0.393929	−	0.682306i
$446$			263.667i			0.591182i
$447$		0			0
$448$	−7.47894	+	55.4983i	−0.0166941	+	0.123880i
$449$		−	4.42392i		−	0.00985283i	−0.999988	−	0.00492642i	$-0.998432\pi$
$449$		−	4.42392i		−	0.00985283i	0.999988	−	0.00492642i	$-0.00156813\pi$
$450$		0			0
$451$	−262.999	−	455.527i	−0.583146	−	1.01004i
$452$		−	369.853i		−	0.818258i
$453$		0			0
$454$	28.8752	+	50.0133i	0.0636018	+	0.110162i
$455$	−54.8887	−	133.666i	−0.120635	−	0.293771i
$456$		0			0
$457$	337.437	+	584.459i	0.738375	+	1.27890i	0.953227	+	0.302257i	$0.0977400\pi$
$457$	337.437	+	584.459i	0.738375	+	1.27890i	−0.214851	+	0.976647i	$0.568927\pi$
$458$	87.2045	−	50.3475i	0.190403	−	0.109929i
$459$		0			0
$460$	267.340	−	463.047i	0.581174	−	1.00662i
$461$	−365.804	−	211.197i	−0.793501	−	0.458128i	0.0476924	−	0.998862i	$-0.484813\pi$
$461$	−365.804	−	211.197i	−0.793501	−	0.458128i	−0.841194	+	0.540734i	$0.818147\pi$
$462$		0			0
$463$	40.8889	+	70.8216i	0.0883129	+	0.152962i	0.906798	−	0.421565i	$-0.138519\pi$
$463$	40.8889	+	70.8216i	0.0883129	+	0.152962i	−0.818485	+	0.574528i	$0.805186\pi$
$464$			9.49961i			0.0204733i
$465$		0			0
$466$	10.8825			0.0233530
$467$	−484.152	−	279.525i	−1.03673	−	0.598555i	−0.117823	−	0.993035i	$-0.537591\pi$
$467$	−484.152	−	279.525i	−1.03673	−	0.598555i	−0.918905	+	0.394480i	$0.870925\pi$
$468$		0			0
$469$	−138.205	−	106.721i	−0.294681	−	0.227549i
$470$	−423.549	−	244.536i	−0.901169	−	0.520290i
$471$		0			0
$472$	101.455	−	175.726i	0.214948	−	0.372301i
$473$	−299.400	−	172.859i	−0.632981	−	0.365452i
$474$		0			0
$475$	−303.257	+	525.257i	−0.638436	+	1.10580i
$476$	441.087	+	59.4407i	0.926653	+	0.124876i
$477$		0			0
$478$	−167.119	+	289.458i	−0.349620	+	0.605560i
$479$		−	80.4381i		−	0.167929i	−0.996469	−	0.0839646i	$-0.973242\pi$
$479$		−	80.4381i		−	0.167929i	0.996469	−	0.0839646i	$-0.0267583\pi$
$480$		0			0
$481$	25.9890			0.0540313
$482$	−155.035	+	89.5092i	−0.321648	+	0.185704i
$483$		0			0
$484$	9.81073	−	16.9927i	0.0202701	−	0.0351089i
$485$	544.149	+	314.165i	1.12196	+	0.647762i
$486$		0			0
$487$	201.883	+	349.672i	0.414544	+	0.718011i	0.995380	−	0.0960090i	$-0.0306078\pi$
$487$	201.883	+	349.672i	0.414544	+	0.718011i	−0.580836	+	0.814020i	$0.697274\pi$
$488$	162.022	−	93.5436i	0.332013	−	0.191688i
$489$		0			0
$490$	−145.802	−	557.741i	−0.297555	−	1.13825i
$491$	−708.736	+	409.189i	−1.44345	+	0.833378i	−0.998078	−	0.0619657i	$-0.980263\pi$
$491$	−708.736	+	409.189i	−1.44345	+	0.833378i	−0.445375	+	0.895344i	$0.646930\pi$
$492$		0			0
$493$	75.5005			0.153145
$494$	−41.6943	+	24.0722i	−0.0844014	+	0.0487292i
$495$		0			0
$496$	83.0741			0.167488
$497$	162.246	+	21.8642i	0.326451	+	0.0439924i
$498$		0			0
$499$	375.029			0.751561			0.375780	−	0.926709i	$-0.377375\pi$
$499$	375.029			0.751561			0.375780	+	0.926709i	$0.377375\pi$
$500$	276.760	+	159.788i	0.553520	+	0.319575i
$501$		0			0
$502$	225.196	+	390.052i	0.448599	+	0.776995i
$503$		−	434.043i		−	0.862909i	−0.902135	−	0.431454i	$-0.858001\pi$
$503$		−	434.043i		−	0.862909i	0.902135	−	0.431454i	$-0.141999\pi$
$504$		0			0
$505$	−1089.83			−2.15809
$506$	−415.016	+	239.610i	−0.820191	+	0.473537i
$507$		0			0
$508$	−111.511	+	193.143i	−0.219510	+	0.380203i
$509$		−	180.371i		−	0.354364i	−0.984178	−	0.177182i	$-0.943302\pi$
$509$		−	180.371i		−	0.354364i	0.984178	−	0.177182i	$-0.0566980\pi$
$510$		0			0
$511$	253.664	−	104.165i	0.496408	−	0.203845i
$512$		−	22.6274i		−	0.0441942i
$513$		0			0
$514$	343.751	+	595.394i	0.668776	+	1.15835i
$515$			276.380i			0.536661i
$516$		0			0
$517$	219.171	+	379.616i	0.423929	+	0.734267i
$518$	102.757	+	13.8475i	0.198372	+	0.0267326i
$519$		0			0
$520$	29.1928	+	50.5634i	0.0561399	+	0.0972372i
$521$	296.089	−	170.947i	0.568310	−	0.328114i	−0.188164	−	0.982138i	$-0.560254\pi$
$521$	296.089	−	170.947i	0.568310	−	0.328114i	0.756474	+	0.654024i	$0.226920\pi$
$522$		0			0
$523$	−189.342	+	327.951i	−0.362031	+	0.627057i	−0.988295	−	0.152555i	$-0.951250\pi$
$523$	−189.342	+	327.951i	−0.362031	+	0.627057i	0.626264	+	0.779611i	$0.284583\pi$
$524$	98.0016	+	56.5812i	0.187026	+	0.107979i
$525$		0			0
$526$	−246.703	−	427.303i	−0.469018	−	0.812363i
$527$		−	660.252i		−	1.25285i
$528$		0			0
$529$	−503.706			−0.952185
$530$	798.694	+	461.126i	1.50697	+	0.870050i
$531$		0			0
$532$	−177.679	+	72.9625i	−0.333984	+	0.137147i
$533$	107.193	+	61.8880i	0.201113	+	0.116113i
$534$		0			0
$535$	225.037	−	389.776i	0.420631	−	0.728554i
$536$	61.1021	+	35.2773i	0.113996	+	0.0658159i
$537$		0			0
$538$	−274.279	+	475.066i	−0.509813	+	0.883022i
$539$	−136.773	+	498.255i	−0.253754	+	0.924407i
$540$		0			0
$541$	143.032	−	247.739i	0.264385	−	0.457928i	−0.703017	−	0.711173i	$-0.748164\pi$
$541$	143.032	−	247.739i	0.264385	−	0.457928i	0.967402	+	0.253244i	$0.0814977\pi$
$542$			334.704i			0.617536i
$543$		0			0
$544$	−179.837			−0.330583
$545$	−215.157	+	124.221i	−0.394783	+	0.227928i
$546$		0			0
$547$	−318.027	+	550.839i	−0.581402	+	1.00702i	0.413911	+	0.910317i	$0.364162\pi$
$547$	−318.027	+	550.839i	−0.581402	+	1.00702i	−0.995313	+	0.0967009i	$0.969171\pi$
$548$	−114.374	−	66.0338i	−0.208711	−	0.120500i
$549$		0			0
$550$	−329.618	−	570.915i	−0.599305	−	1.03803i
$551$	−28.2178	+	16.2916i	−0.0512120	+	0.0295673i
$552$		0			0
$553$	65.2306	−	26.7864i	0.117958	−	0.0484383i
$554$	249.249	−	143.904i	0.449908	−	0.259754i
$555$		0			0
$556$	180.925			0.325404
$557$	2.15468	−	1.24401i	0.00386837	−	0.00223341i	−0.498065	−	0.867140i	$-0.665956\pi$
$557$	2.15468	−	1.24401i	0.00386837	−	0.00223341i	0.501933	+	0.864907i	$0.332622\pi$
$558$		0			0
$559$	81.3531			0.145533
$560$	88.4828	+	215.475i	0.158005	+	0.384776i
$561$		0			0
$562$	75.0286			0.133503
$563$	−299.745	−	173.058i	−0.532406	−	0.307385i	0.209590	−	0.977789i	$-0.432787\pi$
$563$	−299.745	−	173.058i	−0.532406	−	0.307385i	−0.741996	+	0.670405i	$0.766121\pi$
$564$		0			0
$565$	−769.210	−	1332.31i	−1.36143	−	2.35807i
$566$			67.6150i			0.119461i
$567$		0			0
$568$	−66.1499			−0.116461
$569$	−836.465	+	482.933i	−1.47006	+	0.848740i	−0.999436	−	0.0335928i	$-0.989305\pi$
$569$	−836.465	+	482.933i	−1.47006	+	0.848740i	−0.470626	+	0.882333i	$0.655972\pi$
$570$		0			0
$571$	−82.5549	+	142.989i	−0.144579	+	0.250419i	−0.929216	−	0.369537i	$-0.879516\pi$
$571$	−82.5549	+	142.989i	−0.144579	+	0.250419i	0.784637	+	0.619956i	$0.212850\pi$
$572$		−	52.3294i		−	0.0914850i
$573$		0			0
$574$	390.851	+	301.811i	0.680925	+	0.525803i
$575$		−	1420.64i		−	2.47067i
$576$		0			0
$577$	−449.998	−	779.419i	−0.779892	−	1.35081i	−0.932004	−	0.362449i	$-0.881941\pi$
$577$	−449.998	−	779.419i	−0.779892	−	1.35081i	0.152112	−	0.988363i	$-0.451393\pi$
$578$			1020.59i			1.76573i
$579$		0			0
$580$	19.7570	+	34.2202i	0.0340639	+	0.0590003i
$581$	−469.727	+	608.306i	−0.808481	+	1.04700i
$582$		0			0
$583$	−413.295	−	715.848i	−0.708911	−	1.22787i
$584$	−95.9566	+	55.4006i	−0.164309	+	0.0948640i
$585$		0			0
$586$	−8.76047	+	15.1736i	−0.0149496	+	0.0258935i
$587$	829.734	+	479.047i	1.41352	+	0.816094i	0.995718	−	0.0924455i	$-0.0294684\pi$
$587$	829.734	+	479.047i	1.41352	+	0.816094i	0.417799	+	0.908540i	$0.362802\pi$
$588$		0			0
$589$	142.470	+	246.765i	0.241884	+	0.418955i
$590$		−	844.018i		−	1.43054i
$591$		0			0
$592$	−41.8954			−0.0707692
$593$	−362.413	−	209.239i	−0.611151	−	0.352848i	0.162265	−	0.986747i	$-0.448120\pi$
$593$	−362.413	−	209.239i	−0.611151	−	0.352848i	−0.773416	+	0.633899i	$0.781453\pi$
$594$		0			0
$595$	1712.54	−	703.239i	2.87822	−	1.18191i
$596$	170.691	+	98.5488i	0.286395	+	0.165350i
$597$		0			0
$598$	56.3842	−	97.6603i	0.0942880	−	0.163312i
$599$	−10.9571	−	6.32609i	−0.0182923	−	0.0105611i	0.490826	−	0.871258i	$-0.336695\pi$
$599$	−10.9571	−	6.32609i	−0.0182923	−	0.0105611i	−0.509118	+	0.860697i	$0.670028\pi$
$600$		0			0
$601$	77.9185	−	134.959i	0.129648	−	0.224557i	−0.793892	−	0.608059i	$-0.791949\pi$
$601$	77.9185	−	134.959i	0.129648	−	0.224557i	0.923540	+	0.383502i	$0.125282\pi$
$602$	321.658	+	43.3466i	0.534316	+	0.0720043i
$603$		0			0
$604$	−39.5943	+	68.5794i	−0.0655535	+	0.113542i
$605$		−	81.6164i		−	0.134903i
$606$		0			0
$607$	−431.015			−0.710075			−0.355037	−	0.934852i	$-0.615532\pi$
$607$	−431.015			−0.710075			−0.355037	+	0.934852i	$0.615532\pi$
$608$	67.2129	−	38.8054i	0.110547	−	0.0638246i
$609$		0			0
$610$	389.099	−	673.939i	0.637867	−	1.10482i
$611$	−89.3300	−	51.5747i	−0.146203	−	0.0844103i
$612$		0			0
$613$	438.082	+	758.781i	0.714653	+	1.23781i	0.963093	+	0.269168i	$0.0867486\pi$
$613$	438.082	+	758.781i	0.714653	+	1.23781i	−0.248441	+	0.968647i	$0.579918\pi$
$614$	−515.421	+	297.578i	−0.839447	+	0.484655i
$615$		0			0
$616$	27.8822	−	206.903i	0.0452633	−	0.335881i
$617$	311.826	−	180.033i	0.505390	−	0.291787i	−0.225546	−	0.974232i	$-0.572417\pi$
$617$	311.826	−	180.033i	0.505390	−	0.291787i	0.730937	+	0.682445i	$0.239083\pi$
$618$		0			0
$619$	56.5368			0.0913357			0.0456679	−	0.998957i	$-0.485458\pi$
$619$	56.5368			0.0913357			0.0456679	+	0.998957i	$0.485458\pi$
$620$	299.256	−	172.775i	0.482670	−	0.278670i
$621$		0			0
$622$	657.770			1.05751
$623$	180.302	−	233.494i	0.289409	−	0.374790i
$624$		0			0
$625$	224.105			0.358568
$626$	−446.995	−	258.073i	−0.714050	−	0.412257i
$627$		0			0
$628$	−249.558	−	432.248i	−0.397386	−	0.688292i
$629$			332.974i			0.529370i
$630$		0			0
$631$	−632.353			−1.00214			−0.501072	−	0.865406i	$-0.667061\pi$
$631$	−632.353			−1.00214			−0.501072	+	0.865406i	$0.667061\pi$
$632$	−24.6755	+	14.2464i	−0.0390436	+	0.0225418i
$633$		0			0
$634$	−16.8637	+	29.2087i	−0.0265988	+	0.0460705i
$635$			927.672i			1.46090i
$636$		0			0
$637$	−30.7508	−	117.632i	−0.0482745	−	0.184666i
$638$		−	35.4154i		−	0.0555100i
$639$		0			0
$640$	−47.0599	−	81.5101i	−0.0735311	−	0.127360i
$641$		−	516.035i		−	0.805047i	−0.915410	−	0.402523i	$-0.868133\pi$
$641$		−	516.035i		−	0.805047i	0.915410	−	0.402523i	$-0.131867\pi$
$642$		0			0
$643$	273.812	+	474.257i	0.425836	+	0.737569i	0.996498	−	0.0836157i	$-0.0266468\pi$
$643$	273.812	+	474.257i	0.425836	+	0.737569i	−0.570662	+	0.821185i	$0.693313\pi$
$644$	274.970	−	356.092i	0.426973	−	0.552938i
$645$		0			0
$646$	−308.415	−	534.191i	−0.477423	−	0.826921i
$647$	186.510	−	107.681i	0.288268	−	0.166432i	−0.348892	−	0.937163i	$-0.613442\pi$
$647$	186.510	−	107.681i	0.288268	−	0.166432i	0.637161	+	0.770731i	$0.280109\pi$
$648$		0			0
$649$	−378.235	+	655.123i	−0.582797	+	1.00943i
$650$	134.346	+	77.5646i	0.206686	+	0.119330i
$651$		0			0
$652$	242.275	+	419.633i	0.371587	+	0.643608i
$653$		−	321.278i		−	0.492004i	−0.969269	−	0.246002i	$-0.920883\pi$
$653$		−	321.278i		−	0.492004i	0.969269	−	0.246002i	$-0.0791169\pi$
$654$		0			0
$655$	470.705			0.718633
$656$	−172.800	−	99.7660i	−0.263414	−	0.152082i
$657$		0			0
$658$	−325.718	−	251.516i	−0.495012	−	0.382243i
$659$	346.818	+	200.235i	0.526279	+	0.303847i	0.739500	−	0.673157i	$-0.235062\pi$
$659$	346.818	+	200.235i	0.526279	+	0.303847i	−0.213221	+	0.977004i	$0.568395\pi$
$660$		0			0
$661$	−287.726	+	498.356i	−0.435289	+	0.753942i	−0.997319	−	0.0731743i	$-0.976687\pi$
$661$	−287.726	+	498.356i	−0.435289	+	0.753942i	0.562030	+	0.827117i	$0.310020\pi$
$662$	276.958	+	159.902i	0.418365	+	0.241543i
$663$		0			0
$664$	155.272	−	268.939i	0.233843	−	0.405029i
$665$	−488.304	+	632.364i	−0.734292	+	0.950923i
$666$		0			0
$667$	38.1596	−	66.0944i	0.0572108	−	0.0990921i
$668$		−	194.952i		−	0.291844i
$669$		0			0
$670$	293.475			0.438023
$671$	−604.034	+	348.739i	−0.900199	+	0.519730i
$672$		0			0
$673$	473.931	−	820.873i	0.704207	−	1.21972i	−0.262770	−	0.964859i	$-0.584636\pi$
$673$	473.931	−	820.873i	0.704207	−	1.21972i	0.966977	−	0.254864i	$-0.0820307\pi$
$674$	−370.275	−	213.778i	−0.549369	−	0.317179i
$675$		0			0
$676$	−162.843	−	282.052i	−0.240892	−	0.417237i
$677$	−1131.25	+	653.129i	−1.67098	+	0.964741i	−0.703889	+	0.710310i	$0.748555\pi$
$677$	−1131.25	+	653.129i	−1.67098	+	0.964741i	−0.967091	+	0.254430i	$0.918112\pi$
$678$		0			0
$679$	418.461	+	323.131i	0.616290	+	0.475893i
$680$	−647.822	+	374.020i	−0.952679	+	0.550030i
$681$		0			0
$682$	−309.708			−0.454117
$683$	−617.670	+	356.612i	−0.904349	+	0.522126i	−0.878609	−	0.477543i	$-0.841528\pi$
$683$	−617.670	+	356.612i	−0.904349	+	0.522126i	−0.0257402	+	0.999669i	$0.508194\pi$
$684$		0			0
$685$	−549.341			−0.801958
$686$	−58.9075	−	481.485i	−0.0858709	−	0.701873i
$687$		0			0
$688$	−131.144			−0.190617
$689$	168.451	+	97.2553i	0.244486	+	0.141154i
$690$		0			0
$691$	−182.396	−	315.919i	−0.263959	−	0.457191i	0.703331	−	0.710862i	$-0.251695\pi$
$691$	−182.396	−	315.919i	−0.263959	−	0.457191i	−0.967290	+	0.253672i	$0.918362\pi$
$692$		−	107.201i		−	0.154915i
$693$		0			0
$694$	−249.497			−0.359505
$695$	651.740	−	376.282i	0.937756	−	0.541414i
$696$		0			0
$697$	−792.915	+	1373.37i	−1.13761	+	1.97040i
$698$		−	246.969i		−	0.353824i
$699$		0			0
$700$	489.856	+	378.261i	0.699794	+	0.540373i
$701$		−	944.475i		−	1.34732i	−0.739039	−	0.673662i	$-0.764720\pi$
$701$		−	944.475i		−	1.34732i	0.739039	−	0.673662i	$-0.235280\pi$
$702$		0			0
$703$	−71.8494	−	124.447i	−0.102204	−	0.177022i
$704$			84.3571i			0.119825i
$705$		0			0
$706$	−100.867	−	174.706i	−0.142871	−	0.247459i
$707$	−908.812	−	122.471i	−1.28545	−	0.173227i
$708$		0			0
$709$	515.772	+	893.343i	0.727464	+	1.26000i	0.957952	+	0.286929i	$0.0926345\pi$
$709$	515.772	+	893.343i	0.727464	+	1.26000i	−0.230488	+	0.973075i	$0.574032\pi$
$710$	−238.290	+	137.577i	−0.335620	+	0.193770i
$711$		0			0
$712$	−59.6001	+	103.230i	−0.0837081	+	0.144987i
$713$	−577.996	−	333.706i	−0.810653	−	0.468031i
$714$		0			0
$715$	−108.833	−	188.505i	−0.152214	−	0.263643i
$716$		−	283.918i		−	0.396533i
$717$		0			0
$718$	−864.517			−1.20406
$719$	−27.9770	−	16.1525i	−0.0389109	−	0.0224652i	0.480418	−	0.877039i	$-0.340485\pi$
$719$	−27.9770	−	16.1525i	−0.0389109	−	0.0224652i	−0.519329	+	0.854574i	$0.673818\pi$
$720$		0			0
$721$	−31.0586	+	230.474i	−0.0430771	+	0.319658i
$722$	−211.597	−	122.165i	−0.293070	−	0.169204i
$723$		0			0
$724$	−207.905	+	360.102i	−0.287161	+	0.497378i
$725$	90.9224	+	52.4940i	0.125410	+	0.0724056i
$726$		0			0
$727$	−102.737	+	177.945i	−0.141316	+	0.244767i	−0.927992	−	0.372599i	$-0.878467\pi$
$727$	−102.737	+	177.945i	−0.141316	+	0.244767i	0.786676	+	0.617366i	$0.211800\pi$
$728$	18.6617	+	45.4454i	0.0256342	+	0.0624249i
$729$		0			0
$730$	−230.441	+	399.136i	−0.315673	+	0.546762i
$731$			1042.30i			1.42586i
$732$		0			0
$733$	−587.743			−0.801832			−0.400916	−	0.916115i	$-0.631308\pi$
$733$	−587.743			−0.801832			−0.400916	+	0.916115i	$0.631308\pi$
$734$	−206.802	+	119.397i	−0.281746	+	0.162666i
$735$		0			0
$736$	−90.8936	+	157.432i	−0.123497	+	0.213903i
$737$	−227.794	−	131.517i	−0.309083	−	0.178449i
$738$		0			0
$739$	−395.335	−	684.741i	−0.534960	−	0.926578i	−0.999165	−	0.0408504i	$-0.986993\pi$
$739$	−395.335	−	684.741i	−0.534960	−	0.926578i	0.464205	−	0.885728i	$-0.346340\pi$
$740$	−150.919	+	87.1329i	−0.203944	+	0.117747i
$741$		0			0
$742$	614.211	+	474.287i	0.827778	+	0.639201i
$743$	15.5592	−	8.98312i	0.0209411	−	0.0120903i	−0.489493	−	0.872007i	$-0.662818\pi$
$743$	15.5592	−	8.98312i	0.0209411	−	0.0120903i	0.510434	+	0.859917i	$0.329485\pi$
$744$		0			0
$745$	819.836			1.10045
$746$	−398.611	+	230.138i	−0.534331	+	0.308496i
$747$		0			0
$748$	670.448			0.896321
$749$	231.460	−	299.746i	0.309026	−	0.400194i
$750$		0			0
$751$	794.238			1.05757			0.528787	−	0.848755i	$-0.322647\pi$
$751$	794.238			1.05757			0.528787	+	0.848755i	$0.322647\pi$
$752$	144.003	+	83.1404i	0.191494	+	0.110559i
$753$		0			0
$754$	4.16692	+	7.21731i	0.00552642	+	0.00957204i
$755$			329.389i			0.436277i
$756$		0			0
$757$	99.2649			0.131129			0.0655647	−	0.997848i	$-0.479115\pi$
$757$	99.2649			0.131129			0.0655647	+	0.997848i	$0.479115\pi$
$758$	−9.45922	+	5.46129i	−0.0124792	+	0.00720486i
$759$		0			0
$760$	161.413	−	279.575i	0.212385	−	0.367862i
$761$			735.539i			0.966543i	0.875471	+	0.483272i	$0.160552\pi$
$761$			735.539i			0.966543i	−0.875471	+	0.483272i	$0.839448\pi$
$762$		0			0
$763$	−193.379	+	79.4093i	−0.253445	+	0.104075i
$764$			615.774i			0.805987i
$765$		0			0
$766$	−458.124	−	793.495i	−0.598074	−	1.03589i
$767$		−	178.010i		−	0.232086i
$768$		0			0
$769$	−239.582	−	414.969i	−0.311550	−	0.539621i	0.667148	−	0.744925i	$-0.267515\pi$
$769$	−239.582	−	414.969i	−0.311550	−	0.539621i	−0.978698	+	0.205304i	$0.934182\pi$
$770$	−329.872	−	803.309i	−0.428405	−	1.04326i
$771$		0			0
$772$	−165.134	−	286.021i	−0.213904	−	0.370493i
$773$	−100.386	+	57.9577i	−0.129865	+	0.0749776i	−0.563525	−	0.826099i	$-0.690555\pi$
$773$	−100.386	+	57.9577i	−0.129865	+	0.0749776i	0.433660	+	0.901077i	$0.357222\pi$
$774$		0			0
$775$	459.060	−	795.116i	0.592336	−	1.02596i
$776$	−185.006	−	106.814i	−0.238410	−	0.137646i
$777$		0			0
$778$	105.619	+	182.938i	0.135757	+	0.235138i
$779$		−	684.383i		−	0.878541i
$780$		0			0
$781$	246.613			0.315765
$782$	1251.23	+	722.399i	1.60004	+	0.923784i
$783$		0			0
$784$	49.5716	+	189.628i	0.0632290	+	0.241872i
$785$	−1797.95	−	1038.05i	−2.29039	−	1.32236i
$786$		0			0
$787$	−488.719	+	846.487i	−0.620990	+	1.07559i	0.368312	+	0.929702i	$0.379936\pi$
$787$	−488.719	+	846.487i	−0.620990	+	1.07559i	−0.989302	+	0.145884i	$0.953397\pi$
$788$	149.453	+	86.2870i	0.189662	+	0.109501i
$789$		0			0
$790$	−59.2587	+	102.639i	−0.0750110	+	0.129923i
$791$	−491.724	−	1197.45i	−0.621649	−	1.51385i
$792$		0			0
$793$	82.0642	−	142.139i	0.103486	−	0.179242i
$794$			134.704i			0.169652i
$795$		0			0
$796$	−658.881			−0.827739
$797$	982.477	−	567.234i	1.23272	−	0.711711i	0.265124	−	0.964214i	$-0.414587\pi$
$797$	982.477	−	567.234i	1.23272	−	0.711711i	0.967596	+	0.252503i	$0.0812539\pi$
$798$		0			0
$799$	660.779	−	1144.50i	0.827008	−	1.43242i
$800$	−216.571	−	125.037i	−0.270714	−	0.156297i
$801$		0			0
$802$	257.711	+	446.369i	0.321335	+	0.556569i
$803$	357.735	−	206.538i	0.445498	−	0.257208i
$804$		0			0
$805$	249.928	−	1854.62i	0.310469	−	2.30387i
$806$	63.1154	−	36.4397i	0.0783070	−	0.0452106i
$807$		0			0
$808$	370.535			0.458583
$809$	751.325	−	433.778i	0.928708	−	0.536190i	0.0423053	−	0.999105i	$-0.486530\pi$
$809$	751.325	−	433.778i	0.928708	−	0.536190i	0.886403	+	0.462915i	$0.153196\pi$
$810$		0			0
$811$	−329.692			−0.406525			−0.203263	−	0.979124i	$-0.565155\pi$
$811$	−329.692			−0.406525			−0.203263	+	0.979124i	$0.565155\pi$
$812$	12.6299	+	30.7564i	0.0155540	+	0.0378774i
$813$		0			0
$814$	156.190			0.191879
$815$	1745.48	+	1007.75i	2.14170	+	1.23651i
$816$		0			0
$817$	−224.909	−	389.553i	−0.275286	−	0.476810i
$818$		−	1104.97i		−	1.35082i
$819$		0			0
$820$	−829.963			−1.01215
$821$	890.753	−	514.277i	1.08496	−	0.626403i	0.152731	−	0.988268i	$-0.451193\pi$
$821$	890.753	−	514.277i	1.08496	−	0.626403i	0.932230	+	0.361865i	$0.117860\pi$
$822$		0			0
$823$	624.999	−	1082.53i	0.759415	−	1.31535i	−0.183734	−	0.982976i	$-0.558819\pi$
$823$	624.999	−	1082.53i	0.759415	−	1.31535i	0.943149	−	0.332369i	$-0.107848\pi$
$824$		−	93.9672i		−	0.114038i
$825$		0			0
$826$	94.8475	−	703.826i	0.114827	−	0.852090i
$827$			1582.29i			1.91329i	0.291248	+	0.956647i	$0.405929\pi$
$827$			1582.29i			1.91329i	−0.291248	+	0.956647i	$0.594071\pi$
$828$		0			0
$829$	−275.656	−	477.450i	−0.332516	−	0.575935i	0.650488	−	0.759516i	$-0.274564\pi$
$829$	−275.656	−	477.450i	−0.332516	−	0.575935i	−0.983004	+	0.183581i	$0.941231\pi$
$830$		−	1291.72i		−	1.55629i
$831$		0			0
$832$	−9.92531	−	17.1911i	−0.0119295	−	0.0206624i
$833$	1507.11	−	393.982i	1.80926	−	0.472968i
$834$		0			0
$835$	−405.456	−	702.270i	−0.485576	−	0.841042i
$836$	−250.576	+	144.670i	−0.299732	+	0.173050i
$837$		0			0
$838$	−327.800	+	567.767i	−0.391170	+	0.677526i
$839$	1220.53	+	704.672i	1.45474	+	0.839895i	0.998745	−	0.0500860i	$-0.0159495\pi$
$839$	1220.53	+	704.672i	1.45474	+	0.839895i	0.455997	+	0.889981i	$0.349283\pi$
$840$		0			0
$841$	−417.680	−	723.443i	−0.496647	−	0.860217i
$842$			847.543i			1.00658i
$843$		0			0
$844$	−162.827			−0.192923
$845$	−1173.21	−	677.353i	−1.38841	−	0.801601i
$846$		0			0
$847$	9.17174	−	68.0599i	0.0108285	−	0.0803541i
$848$	−271.550	−	156.779i	−0.320224	−	0.184881i
$849$		0			0
$850$	−993.764	+	1721.25i	−1.16913	+	2.02500i
$851$	291.491	+	168.292i	0.342527	+	0.197758i
$852$		0			0
$853$	−305.025	+	528.318i	−0.357590	+	0.619365i	−0.987558	−	0.157257i	$-0.949735\pi$
$853$	−305.025	+	528.318i	−0.357590	+	0.619365i	0.629967	+	0.776622i	$0.283068\pi$
$854$	400.204	−	518.273i	0.468624	−	0.606877i
$855$		0			0
$856$	−76.5110	+	132.521i	−0.0893820	+	0.154814i
$857$			1083.39i			1.26417i	0.774901	+	0.632083i	$0.217800\pi$
$857$			1083.39i			1.26417i	−0.774901	+	0.632083i	$0.782200\pi$
$858$		0			0
$859$	1396.05			1.62520			0.812601	−	0.582821i	$-0.198051\pi$
$859$	1396.05			1.62520			0.812601	+	0.582821i	$0.198051\pi$
$860$	−472.418	+	272.751i	−0.549323	+	0.317152i
$861$		0			0
$862$	213.432	−	369.675i	0.247601	−	0.428857i
$863$	33.5922	+	19.3944i	0.0389249	+	0.0224733i	0.519336	−	0.854570i	$-0.326179\pi$
$863$	33.5922	+	19.3944i	0.0389249	+	0.0224733i	−0.480411	+	0.877043i	$0.659513\pi$
$864$		0			0
$865$	−222.954	−	386.168i	−0.257751	−	0.446438i
$866$	−713.578	+	411.984i	−0.823993	+	0.475733i
$867$		0			0
$868$	268.965	−	110.448i	0.309868	−	0.127244i
$869$	91.9926	−	53.1120i	0.105860	−	0.0611185i
$870$		0			0
$871$	61.8963			0.0710635
$872$	73.1517	−	42.2341i	0.0838895	−	0.0484336i
$873$		0			0
$874$	−623.520			−0.713409
$875$	1108.49	+	149.380i	1.26685	+	0.170720i
$876$		0			0
$877$	1250.57			1.42596			0.712982	−	0.701182i	$-0.247344\pi$
$877$	1250.57			1.42596			0.712982	+	0.701182i	$0.247344\pi$
$878$	69.0434	+	39.8622i	0.0786371	+	0.0454011i
$879$		0			0
$880$	175.444	+	303.877i	0.199368	+	0.345315i
$881$			844.579i			0.958659i	0.877635	+	0.479329i	$0.159120\pi$
$881$			844.579i			0.958659i	−0.877635	+	0.479329i	$0.840880\pi$
$882$		0			0
$883$	−378.496			−0.428648			−0.214324	−	0.976763i	$-0.568755\pi$
$883$	−378.496			−0.428648			−0.214324	+	0.976763i	$0.568755\pi$
$884$	−136.631	+	78.8839i	−0.154560	+	0.0892352i
$885$		0			0
$886$	−134.959	+	233.756i	−0.152324	+	0.263833i
$887$		−	1158.64i		−	1.30624i	−0.757253	−	0.653121i	$-0.773459\pi$
$887$		−	1158.64i		−	1.30624i	0.757253	−	0.653121i	$-0.226541\pi$
$888$		0			0
$889$	−104.248	+	773.586i	−0.117265	+	0.870175i
$890$			495.819i			0.557100i
$891$		0			0
$892$	−186.441	−	322.925i	−0.209014	−	0.362023i
$893$			570.334i			0.638672i
$894$		0			0
$895$	−590.484	−	1022.75i	−0.659759	−	1.14274i
$896$	−30.0835	−	73.2597i	−0.0335753	−	0.0817631i
$897$		0			0
$898$	3.12818	+	5.41817i	0.00348350	+	0.00603360i
$899$	42.7152	−	24.6616i	0.0475141	−	0.0274323i
$900$		0			0
$901$	−1246.04	+	2158.21i	−1.38296	+	2.39535i
$902$	644.213	+	371.937i	0.714205	+	0.412347i
$903$		0			0
$904$	261.525	+	452.975i	0.289298	+	0.501079i
$905$			1729.58i			1.91114i
$906$		0			0
$907$	−281.301			−0.310144			−0.155072	−	0.987903i	$-0.549561\pi$
$907$	−281.301			−0.310144			−0.155072	+	0.987903i	$0.549561\pi$
$908$	−70.7295	−	40.8357i	−0.0778960	−	0.0449733i
$909$		0			0
$910$	161.741	+	124.894i	0.177737	+	0.137247i
$911$	−176.629	−	101.977i	−0.193885	−	0.111939i	0.399915	−	0.916552i	$-0.369040\pi$
$911$	−176.629	−	101.977i	−0.193885	−	0.111939i	−0.593800	+	0.804613i	$0.702373\pi$
$912$		0			0
$913$	−578.868	+	1002.63i	−0.634029	+	1.09817i
$914$	−826.550	−	477.209i	−0.904321	−	0.522110i
$915$		0			0
$916$	−71.2022	+	123.326i	−0.0777316	+	0.134635i
$917$	392.521	+	52.8960i	0.428049	+	0.0576838i
$918$		0			0
$919$	−499.667	+	865.449i	−0.543708	+	0.941730i	0.454979	+	0.890502i	$0.349647\pi$
$919$	−499.667	+	865.449i	−0.543708	+	0.941730i	−0.998687	+	0.0512275i	$0.983687\pi$
$920$			756.152i			0.821905i
$921$		0			0
$922$	597.356			0.647891
$923$	−50.2573	+	29.0161i	−0.0544499	+	0.0314367i
$924$		0			0
$925$	−231.510	+	400.988i	−0.250281	+	0.433500i
$926$	−100.157	−	57.8256i	−0.108161	−	0.0624466i
$927$		0			0
$928$	−6.71724	−	11.6346i	−0.00723840	−	0.0125373i
$929$	−52.9181	+	30.5523i	−0.0569625	+	0.0328873i	−0.528211	−	0.849113i	$-0.677137\pi$
$929$	−52.9181	+	30.5523i	−0.0569625	+	0.0328873i	0.471248	+	0.882001i	$0.343804\pi$
$930$		0			0
$931$	−478.260	+	472.454i	−0.513705	+	0.507470i
$932$	−13.3283	+	7.69509i	−0.0143007	+	0.00825653i
$933$		0			0
$934$	790.616			0.846484
$935$	2415.14	−	1394.38i	2.58304	−	1.49132i
$936$		0			0
$937$	−233.228			−0.248909			−0.124455	−	0.992225i	$-0.539718\pi$
$937$	−233.228			−0.248909			−0.124455	+	0.992225i	$0.539718\pi$
$938$	244.729	+	32.9796i	0.260905	+	0.0351595i
$939$		0			0
$940$	691.653			0.735801
$941$	−712.127	−	411.147i	−0.756777	−	0.436925i	0.0713605	−	0.997451i	$-0.477266\pi$
$941$	−712.127	−	411.147i	−0.756777	−	0.436925i	−0.828137	+	0.560525i	$0.810599\pi$
$942$		0			0
$943$	801.513	+	1388.26i	0.849961	+	1.47218i
$944$			286.959i			0.303982i
$945$		0			0
$946$	488.918			0.516827
$947$	1356.61	−	783.240i	1.43254	−	0.827075i	0.435222	−	0.900323i	$-0.356670\pi$
$947$	1356.61	−	783.240i	1.43254	−	0.827075i	0.997314	+	0.0732485i	$0.0233366\pi$
$948$		0			0
$949$	−48.6019	+	84.1810i	−0.0512138	+	0.0887049i
$950$		−	857.741i		−	0.902885i
$951$		0			0
$952$	−582.250	+	239.096i	−0.611607	+	0.251151i
$953$			852.327i			0.894362i	0.894443	+	0.447181i	$0.147572\pi$
$953$			852.327i			0.894362i	−0.894443	+	0.447181i	$0.852428\pi$
$954$		0			0
$955$	1280.67	+	2218.19i	1.34102	+	2.32271i
$956$		−	472.683i		−	0.494438i
$957$		0			0
$958$	56.8783	+	98.5161i	0.0593719	+	0.102835i
$959$	−458.096	−	61.7329i	−0.477680	−	0.0643721i
$960$		0			0
$961$	264.834	+	458.706i	0.275582	+	0.477322i
$962$	−31.8299	+	18.3770i	−0.0330873	+	0.0191029i
$963$		0			0
$964$	126.585	−	219.252i	0.131312	−	0.227440i
$965$	−1189.72	−	686.883i	−1.23287	−	0.711796i
$966$		0			0
$967$	−553.241	−	958.241i	−0.572121	−	0.990942i	−0.996348	−	0.0853865i	$-0.972787\pi$
$967$	−553.241	−	958.241i	−0.572121	−	0.990942i	0.424227	−	0.905556i	$-0.360546\pi$
$968$			27.7489i			0.0286663i
$969$		0			0
$970$	−888.592			−0.916074
$971$	479.817	+	277.023i	0.494148	+	0.285296i	0.726293	−	0.687385i	$-0.241241\pi$
$971$	479.817	+	277.023i	0.494148	+	0.285296i	−0.232146	+	0.972681i	$0.574575\pi$
$972$		0			0
$973$	585.771	−	240.542i	0.602026	−	0.247217i
$974$	−494.510	−	285.506i	−0.507711	−	0.293127i
$975$		0			0
$976$	−132.291	+	229.134i	−0.135544	+	0.234769i
$977$	−150.343	−	86.8005i	−0.153882	−	0.0888439i	0.421082	−	0.907023i	$-0.361651\pi$
$977$	−150.343	−	86.8005i	−0.153882	−	0.0888439i	−0.574964	+	0.818179i	$0.694984\pi$
$978$		0			0
$979$	222.195	−	384.853i	0.226961	−	0.393108i
$980$	572.953	+	579.993i	0.584646	+	0.591830i
$981$		0			0
$982$	578.680	−	1002.30i	0.589287	−	1.02068i
$983$		−	1373.08i		−	1.39683i	−0.715693	−	0.698415i	$-0.753889\pi$
$983$		−	1373.08i		−	1.39683i	0.715693	−	0.698415i	$-0.246111\pi$
$984$		0			0
$985$	717.830			0.728761
$986$	−92.4688	+	53.3869i	−0.0937818	+	0.0541449i
$987$		0			0
$988$	34.0433	−	58.9647i	0.0344567	−	0.0596808i
$989$	912.449	+	526.803i	0.922597	+	0.532662i
$990$		0			0
$991$	−523.608	−	906.916i	−0.528364	−	0.915153i	−0.999453	−	0.0330672i	$-0.989472\pi$
$991$	−523.608	−	906.916i	−0.528364	−	0.915153i	0.471090	−	0.882085i	$-0.343861\pi$
$992$	−101.745	+	58.7423i	−0.102565	+	0.0592160i
$993$		0			0
$994$	−214.170	+	87.9472i	−0.215463	+	0.0884780i
$995$	−2373.47	+	1370.32i	−2.38540	+	1.37721i
$996$		0			0
$997$	189.359			0.189929			0.0949643	−	0.995481i	$-0.469726\pi$
$997$	189.359			0.189929			0.0949643	+	0.995481i	$0.469726\pi$
$998$	−459.315	+	265.185i	−0.460235	+	0.265717i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	378.3.i.a.359.2		32
3.2	odd	2		126.3.i.a.65.13	✓	32
7.4	even	3		378.3.r.a.305.7		32
9.4	even	3		126.3.r.a.23.7	yes	32
9.5	odd	6		378.3.r.a.233.15		32
21.11	odd	6		126.3.r.a.11.15	yes	32
63.4	even	3		126.3.i.a.95.13	yes	32
63.32	odd	6	inner	378.3.i.a.179.7		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
126.3.i.a.65.13	✓	32	3.2	odd	2
126.3.i.a.95.13	yes	32	63.4	even	3
126.3.r.a.11.15	yes	32	21.11	odd	6
126.3.r.a.23.7	yes	32	9.4	even	3
378.3.i.a.179.7		32	63.32	odd	6	inner
378.3.i.a.359.2		32	1.1	even	1	trivial
378.3.r.a.233.15		32	9.5	odd	6
378.3.r.a.305.7		32	7.4	even	3

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:	\( N \)	\(=\)	\( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	378.i (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.22474 + 0.707107i) q^{2} +(1.00000 - 1.73205i) q^{4} -8.31909i q^{5} +(0.934868 - 6.93729i) q^{7} +2.82843i q^{8} +(5.88249 + 10.1888i) q^{10} -10.5446i q^{11} +(1.24066 + 2.14889i) q^{13} +(3.76043 + 9.15746i) q^{14} +(-2.00000 - 3.46410i) q^{16} +(-27.5318 + 15.8955i) q^{17} +(6.85989 - 11.8817i) q^{19} +(-14.4091 - 8.31909i) q^{20} +(7.45618 + 12.9145i) q^{22} +32.1357i q^{23} -44.2073 q^{25} +(-3.03899 - 1.75456i) q^{26} +(-11.0809 - 8.55653i) q^{28} +(-2.05673 - 1.18745i) q^{29} +(-10.3843 + 17.9861i) q^{31} +(4.89898 + 2.82843i) q^{32} +(22.4796 - 38.9358i) q^{34} +(-57.7120 - 7.77726i) q^{35} +(5.23692 - 9.07061i) q^{37} +19.4027i q^{38} +23.5300 q^{40} +(43.1999 - 24.9415i) q^{41} +(16.3930 - 28.3936i) q^{43} +(-18.2638 - 10.5446i) q^{44} +(-22.7234 - 39.3581i) q^{46} +(-36.0009 + 20.7851i) q^{47} +(-47.2520 - 12.9709i) q^{49} +(54.1427 - 31.2593i) q^{50} +4.96266 q^{52} +(67.8875 - 39.1948i) q^{53} -87.7218 q^{55} +(19.6216 + 2.64421i) q^{56} +3.35862 q^{58} +(-62.1285 - 35.8699i) q^{59} +(-33.0727 - 57.2835i) q^{61} -29.3711i q^{62} -8.00000 q^{64} +(17.8768 - 10.3212i) q^{65} +(12.4724 - 21.6028i) q^{67} +63.5820i q^{68} +(76.1818 - 31.2834i) q^{70} +23.3875i q^{71} +(19.5871 + 33.9258i) q^{73} +14.8122i q^{74} +(-13.7198 - 23.7633i) q^{76} +(-73.1512 - 9.85784i) q^{77} +(5.03687 + 8.72412i) q^{79} +(-28.8182 + 16.6382i) q^{80} +(-35.2726 + 61.0939i) q^{82} +(-95.0843 - 54.8969i) q^{83} +(132.236 + 229.040i) q^{85} +46.3665i q^{86} +29.8247 q^{88} +(36.4975 + 21.0718i) q^{89} +(16.0674 - 6.59792i) q^{91} +(55.6607 + 32.1357i) q^{92} +(29.3946 - 50.9129i) q^{94} +(-98.8447 - 57.0680i) q^{95} +(-37.7643 + 65.4097i) q^{97} +(67.0435 - 17.5262i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 32 q^{4} + 2 q^{7} + 10 q^{13} - 36 q^{14} - 64 q^{16} - 54 q^{17} + 28 q^{19} - 160 q^{25} - 72 q^{26} - 4 q^{28} - 36 q^{29} - 8 q^{31} - 90 q^{35} + 22 q^{37} - 72 q^{41} + 16 q^{43} + 72 q^{44}+ \cdots + 288 q^{98}+O(q^{100}) \) 32 * q + 32 * q^4 + 2 * q^7 + 10 * q^13 - 36 * q^14 - 64 * q^16 - 54 * q^17 + 28 * q^19 - 160 * q^25 - 72 * q^26 - 4 * q^28 - 36 * q^29 - 8 * q^31 - 90 * q^35 + 22 * q^37 - 72 * q^41 + 16 * q^43 + 72 * q^44 - 12 * q^46 + 108 * q^47 + 74 * q^49 + 288 * q^50 + 40 * q^52 - 72 * q^53 - 24 * q^55 + 48 * q^58 + 90 * q^59 - 62 * q^61 - 256 * q^64 - 378 * q^65 + 70 * q^67 - 108 * q^70 + 196 * q^73 - 56 * q^76 - 630 * q^77 - 38 * q^79 + 60 * q^85 - 486 * q^89 - 122 * q^91 - 252 * q^92 + 168 * q^94 + 72 * q^95 - 38 * q^97 + 288 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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