LMFDB - Embedded newform 468.2.k.a.445.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [468,2,Mod(61,468)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("468.61"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(468, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 4, 4])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 468 = 2^{2} \cdot 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	468.k (of order $3$, degree $2$, 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:	$3.73699881460$
Analytic rank:	$0$
Dimension:	$28$
Relative dimension:	$14$ over $\Q(\zeta_{3})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			445.5
Character	$\chi$	$=$	468.445
Dual form			468.2.k.a.61.5

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-0.824641 - 1.52314i) q^{3} +(-0.913993 + 1.58308i) q^{5} +3.54787 q^{7} +(-1.63994 + 2.51209i) q^{9} +(-0.394633 + 0.683524i) q^{11} +(3.60317 + 0.131005i) q^{13} +(3.16498 + 0.0866688i) q^{15} +(-1.48507 + 2.57222i) q^{17} +(1.51934 - 2.63157i) q^{19} +(-2.92572 - 5.40392i) q^{21} +5.81581 q^{23} +(0.829233 + 1.43627i) q^{25} +(5.17864 + 0.426283i) q^{27} +(4.09470 - 7.09223i) q^{29} +(-0.129332 + 0.224010i) q^{31} +(1.36654 + 0.0374208i) q^{33} +(-3.24273 + 5.61657i) q^{35} +(-1.90793 - 3.30463i) q^{37} +(-2.77178 - 5.59618i) q^{39} +0.0912193 q^{41} +8.61643 q^{43} +(-2.47796 - 4.89219i) q^{45} +(1.16369 + 2.01557i) q^{47} +5.58740 q^{49} +(5.14252 + 0.140821i) q^{51} -6.58958 q^{53} +(-0.721383 - 1.24947i) q^{55} +(-5.26117 - 0.144070i) q^{57} +(0.605518 + 1.04879i) q^{59} -3.76705 q^{61} +(-5.81828 + 8.91259i) q^{63} +(-3.50066 + 5.58438i) q^{65} -8.47022 q^{67} +(-4.79595 - 8.85831i) q^{69} +(-6.51214 + 11.2794i) q^{71} -7.67537 q^{73} +(1.50383 - 2.44745i) q^{75} +(-1.40011 + 2.42506i) q^{77} +(5.76726 + 9.98919i) q^{79} +(-3.62122 - 8.23934i) q^{81} +(7.82491 + 13.5531i) q^{83} +(-2.71469 - 4.70199i) q^{85} +(-14.1791 - 0.388277i) q^{87} +(-2.17897 - 3.77409i) q^{89} +(12.7836 + 0.464789i) q^{91} +(0.447853 + 0.0122639i) q^{93} +(2.77733 + 4.81047i) q^{95} +0.113077 q^{97} +(-1.06990 - 2.11229i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q - 4 q^{7} - 2 q^{9} - 4 q^{11} + q^{13} - 4 q^{15} - 8 q^{17} - q^{19} + 14 q^{21} + 8 q^{23} - 14 q^{25} - 13 q^{29} + 2 q^{31} - 25 q^{33} + 3 q^{35} - q^{37} - 3 q^{39} - 8 q^{41} - 4 q^{43}+ \cdots - 29 q^{99}+O(q^{100}) $ 28 * q - 4 * q^7 - 2 * q^9 - 4 * q^11 + q^13 - 4 * q^15 - 8 * q^17 - q^19 + 14 * q^21 + 8 * q^23 - 14 * q^25 - 13 * q^29 + 2 * q^31 - 25 * q^33 + 3 * q^35 - q^37 - 3 * q^39 - 8 * q^41 - 4 * q^43 - 38 * q^45 + 11 * q^47 + 24 * q^49 + 5 * q^51 + 52 * q^53 - q^57 - 8 * q^59 + 14 * q^61 - 21 * q^63 + 38 * q^65 + 14 * q^67 - 21 * q^69 - 12 * q^71 + 14 * q^73 - 14 * q^75 - 28 * q^77 + 5 * q^79 + 10 * q^81 + 9 * q^83 + 18 * q^85 + 18 * q^87 - 11 * q^89 - q^91 - 9 * q^93 + 28 * q^95 + 50 * q^97 - 29 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/468\mathbb{Z}\right)^\times$.

$n$	$145$	$209$	$235$
$\chi(n)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{1}{3}\right)$	$1$

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 $

$2$		0			0
$2$		0			0
$3$	−0.824641	−	1.52314i	−0.476107	−	0.879388i
$3$	−0.824641	−	1.52314i	−0.476107	−	0.879388i
$4$		0			0
$5$	−0.913993	+	1.58308i	−0.408750	+	0.707976i	−0.994750	−	0.102335i	$-0.967369\pi$
$5$	−0.913993	+	1.58308i	−0.408750	+	0.707976i	0.586000	+	0.810311i	$0.300702\pi$
$6$		0			0
$7$	3.54787			1.34097			0.670485	−	0.741923i	$-0.266086\pi$
$7$	3.54787			1.34097			0.670485	+	0.741923i	$0.266086\pi$
$8$		0			0
$9$	−1.63994	+	2.51209i	−0.546645	+	0.837364i
$10$		0			0
$11$	−0.394633	+	0.683524i	−0.118986	+	0.206090i	−0.919366	−	0.393403i	$-0.871298\pi$
$11$	−0.394633	+	0.683524i	−0.118986	+	0.206090i	0.800380	+	0.599493i	$0.204631\pi$
$12$		0			0
$13$	3.60317	+	0.131005i	0.999340	+	0.0363342i
$13$	3.60317	+	0.131005i	0.999340	+	0.0363342i
$14$		0			0
$15$	3.16498	+	0.0866688i	0.817194	+	0.0223778i
$16$		0			0
$17$	−1.48507	+	2.57222i	−0.360183	+	0.623855i	−0.987991	−	0.154513i	$-0.950619\pi$
$17$	−1.48507	+	2.57222i	−0.360183	+	0.623855i	0.627808	+	0.778369i	$0.283952\pi$
$18$		0			0
$19$	1.51934	−	2.63157i	0.348560	−	0.603724i	−0.637434	−	0.770505i	$-0.720004\pi$
$19$	1.51934	−	2.63157i	0.348560	−	0.603724i	0.985994	+	0.166781i	$0.0533374\pi$
$20$		0			0
$21$	−2.92572	−	5.40392i	−0.638444	−	1.17923i
$22$		0			0
$23$	5.81581			1.21268			0.606340	−	0.795206i	$-0.292637\pi$
$23$	5.81581			1.21268			0.606340	+	0.795206i	$0.292637\pi$
$24$		0			0
$25$	0.829233	+	1.43627i	0.165847	+	0.287255i
$26$		0			0
$27$	5.17864	+	0.426283i	0.996629	+	0.0820383i
$28$		0			0
$29$	4.09470	−	7.09223i	0.760367	−	1.31699i	−0.182295	−	0.983244i	$-0.558352\pi$
$29$	4.09470	−	7.09223i	0.760367	−	1.31699i	0.942662	−	0.333750i	$-0.108314\pi$
$30$		0			0
$31$	−0.129332	+	0.224010i	−0.0232288	+	0.0402334i	−0.877406	−	0.479748i	$-0.840728\pi$
$31$	−0.129332	+	0.224010i	−0.0232288	+	0.0402334i	0.854177	+	0.519982i	$0.174061\pi$
$32$		0			0
$33$	1.36654	+	0.0374208i	0.237883	+	0.00651412i
$34$		0			0
$35$	−3.24273	+	5.61657i	−0.548122	+	0.949374i
$36$		0			0
$37$	−1.90793	−	3.30463i	−0.313661	−	0.543277i	0.665491	−	0.746406i	$-0.268222\pi$
$37$	−1.90793	−	3.30463i	−0.313661	−	0.543277i	−0.979152	+	0.203129i	$0.934889\pi$
$38$		0			0
$39$	−2.77178	−	5.59618i	−0.443840	−	0.896106i
$40$		0			0
$41$	0.0912193			0.0142461			0.00712303	−	0.999975i	$-0.497733\pi$
$41$	0.0912193			0.0142461			0.00712303	+	0.999975i	$0.497733\pi$
$42$		0			0
$43$	8.61643			1.31399			0.656997	−	0.753893i	$-0.271826\pi$
$43$	8.61643			1.31399			0.656997	+	0.753893i	$0.271826\pi$
$44$		0			0
$45$	−2.47796	−	4.89219i	−0.369393	−	0.729284i
$46$		0			0
$47$	1.16369	+	2.01557i	0.169741	+	0.294001i	0.938329	−	0.345744i	$-0.112373\pi$
$47$	1.16369	+	2.01557i	0.169741	+	0.294001i	−0.768587	+	0.639745i	$0.779040\pi$
$48$		0			0
$49$	5.58740			0.798200
$50$		0			0
$51$	5.14252	+	0.140821i	0.720096	+	0.0197189i
$52$		0			0
$53$	−6.58958			−0.905149			−0.452574	−	0.891727i	$-0.649494\pi$
$53$	−6.58958			−0.905149			−0.452574	+	0.891727i	$0.649494\pi$
$54$		0			0
$55$	−0.721383	−	1.24947i	−0.0972713	−	0.168479i
$56$		0			0
$57$	−5.26117	−	0.144070i	−0.696859	−	0.0190826i
$58$		0			0
$59$	0.605518	+	1.04879i	0.0788318	+	0.136541i	0.902746	−	0.430174i	$-0.141548\pi$
$59$	0.605518	+	1.04879i	0.0788318	+	0.136541i	−0.823914	+	0.566714i	$0.808214\pi$
$60$		0			0
$61$	−3.76705			−0.482321			−0.241161	−	0.970485i	$-0.577528\pi$
$61$	−3.76705			−0.482321			−0.241161	+	0.970485i	$0.577528\pi$
$62$		0			0
$63$	−5.81828	+	8.91259i	−0.733035	+	1.12288i
$64$		0			0
$65$	−3.50066	+	5.58438i	−0.434204	+	0.692657i
$66$		0			0
$67$	−8.47022			−1.03480			−0.517401	−	0.855743i	$-0.673100\pi$
$67$	−8.47022			−1.03480			−0.517401	+	0.855743i	$0.673100\pi$
$68$		0			0
$69$	−4.79595	−	8.85831i	−0.577365	−	1.06642i
$70$		0			0
$71$	−6.51214	+	11.2794i	−0.772849	+	1.33861i	0.163146	+	0.986602i	$0.447836\pi$
$71$	−6.51214	+	11.2794i	−0.772849	+	1.33861i	−0.935995	+	0.352012i	$0.885498\pi$
$72$		0			0
$73$	−7.67537			−0.898334			−0.449167	−	0.893448i	$-0.648279\pi$
$73$	−7.67537			−0.898334			−0.449167	+	0.893448i	$0.648279\pi$
$74$		0			0
$75$	1.50383	−	2.44745i	0.173648	−	0.282607i
$76$		0			0
$77$	−1.40011	+	2.42506i	−0.159557	+	0.276361i
$78$		0			0
$79$	5.76726	+	9.98919i	0.648868	+	1.12387i	0.983394	+	0.181485i	$0.0580905\pi$
$79$	5.76726	+	9.98919i	0.648868	+	1.12387i	−0.334526	+	0.942387i	$0.608576\pi$
$80$		0			0
$81$	−3.62122	−	8.23934i	−0.402358	−	0.915482i
$82$		0			0
$83$	7.82491	+	13.5531i	0.858896	+	1.48765i	0.872983	+	0.487750i	$0.162182\pi$
$83$	7.82491	+	13.5531i	0.858896	+	1.48765i	−0.0140878	+	0.999901i	$0.504484\pi$
$84$		0			0
$85$	−2.71469	−	4.70199i	−0.294450	−	0.510002i
$86$		0			0
$87$	−14.1791	−	0.388277i	−1.52016	−	0.0416277i
$88$		0			0
$89$	−2.17897	−	3.77409i	−0.230970	−	0.400053i	0.727124	−	0.686507i	$-0.240857\pi$
$89$	−2.17897	−	3.77409i	−0.230970	−	0.400053i	−0.958094	+	0.286454i	$0.907523\pi$
$90$		0			0
$91$	12.7836	+	0.464789i	1.34008	+	0.0487231i
$92$		0			0
$93$	0.447853	+	0.0122639i	0.0464401	+	0.00127170i
$94$		0			0
$95$	2.77733	+	4.81047i	0.284948	+	0.493544i
$96$		0			0
$97$	0.113077			0.0114813			0.00574063	−	0.999984i	$-0.498173\pi$
$97$	0.113077			0.0114813			0.00574063	+	0.999984i	$0.498173\pi$
$98$		0			0
$99$	−1.06990	−	2.11229i	−0.107529	−	0.212293i
$100$		0			0
$101$	−4.07159	+	7.05221i	−0.405139	+	0.701721i	−0.994338	−	0.106267i	$-0.966110\pi$
$101$	−4.07159	+	7.05221i	−0.405139	+	0.701721i	0.589199	+	0.807988i	$0.299443\pi$
$102$		0			0
$103$	−1.51452	+	2.62322i	−0.149230	+	0.258474i	−0.930943	−	0.365164i	$-0.881013\pi$
$103$	−1.51452	+	2.62322i	−0.149230	+	0.258474i	0.781713	+	0.623638i	$0.214346\pi$
$104$		0			0
$105$	11.2289	+	0.307490i	1.09583	+	0.0300079i
$106$		0			0
$107$	−9.94326	−	17.2222i	−0.961252	−	1.66494i	−0.719365	−	0.694632i	$-0.755567\pi$
$107$	−9.94326	−	17.2222i	−0.961252	−	1.66494i	−0.241886	−	0.970305i	$-0.577766\pi$
$108$		0			0
$109$	18.7499			1.79591			0.897957	−	0.440082i	$-0.145051\pi$
$109$	18.7499			1.79591			0.897957	+	0.440082i	$0.145051\pi$
$110$		0			0
$111$	−3.46007	+	5.63118i	−0.328415	+	0.534488i
$112$		0			0
$113$	1.25463	+	2.17308i	0.118025	+	0.204426i	0.918985	−	0.394292i	$-0.129010\pi$
$113$	1.25463	+	2.17308i	0.118025	+	0.204426i	−0.800960	+	0.598718i	$0.795677\pi$
$114$		0			0
$115$	−5.31561	+	9.20690i	−0.495683	+	0.858548i
$116$		0			0
$117$	−6.23806	+	8.83666i	−0.576709	+	0.816950i
$118$		0			0
$119$	−5.26885	+	9.12591i	−0.482995	+	0.836571i
$120$		0			0
$121$	5.18853	+	8.98680i	0.471685	+	0.816982i
$122$		0			0
$123$	−0.0752231	−	0.138940i	−0.00678264	−	0.0125278i
$124$		0			0
$125$	−12.1716			−1.08866
$126$		0			0
$127$	−10.9444	−	18.9563i	−0.971160	−	1.68210i	−0.692065	−	0.721836i	$-0.743299\pi$
$127$	−10.9444	−	18.9563i	−0.971160	−	1.68210i	−0.279096	−	0.960263i	$-0.590035\pi$
$128$		0			0
$129$	−7.10546	−	13.1241i	−0.625601	−	1.15551i
$130$		0			0
$131$	8.14623	−	14.1097i	0.711740	−	1.23277i	−0.252464	−	0.967606i	$-0.581241\pi$
$131$	8.14623	−	14.1097i	0.711740	−	1.23277i	0.964204	−	0.265163i	$-0.0854258\pi$
$132$		0			0
$133$	5.39042	−	9.33647i	0.467408	−	0.809575i
$134$		0			0
$135$	−5.40808	+	7.80859i	−0.465453	+	0.672056i
$136$		0			0
$137$	−15.1730			−1.29631			−0.648157	−	0.761507i	$-0.724460\pi$
$137$	−15.1730			−1.29631			−0.648157	+	0.761507i	$0.724460\pi$
$138$		0			0
$139$	−9.03063	−	15.6415i	−0.765968	−	1.32670i	−0.939733	−	0.341908i	$-0.888927\pi$
$139$	−9.03063	−	15.6415i	−0.765968	−	1.32670i	0.173765	−	0.984787i	$-0.444407\pi$
$140$		0			0
$141$	2.11038	−	3.43459i	0.177726	−	0.289244i
$142$		0			0
$143$	−1.51147	+	2.41115i	−0.126396	+	0.201631i
$144$		0			0
$145$	7.48506	+	12.9645i	0.621600	+	1.07664i
$146$		0			0
$147$	−4.60760	−	8.51041i	−0.380028	−	0.701927i
$148$		0			0
$149$	−5.00434	−	8.66778i	−0.409972	−	0.710092i	0.584914	−	0.811095i	$-0.301128\pi$
$149$	−5.00434	−	8.66778i	−0.409972	−	0.710092i	−0.994886	+	0.101003i	$0.967795\pi$
$150$		0			0
$151$	−9.52036	−	16.4898i	−0.774756	−	1.34192i	−0.934931	−	0.354829i	$-0.884539\pi$
$151$	−9.52036	−	16.4898i	−0.774756	−	1.34192i	0.160175	−	0.987089i	$-0.448794\pi$
$152$		0			0
$153$	−4.02624	−	7.94892i	−0.325502	−	0.642632i
$154$		0			0
$155$	−0.236418	−	0.409488i	−0.0189895	−	0.0328908i
$156$		0			0
$157$	−5.01398	+	8.68447i	−0.400159	+	0.693096i	−0.993745	−	0.111675i	$-0.964379\pi$
$157$	−5.01398	+	8.68447i	−0.400159	+	0.693096i	0.593585	+	0.804771i	$0.297712\pi$
$158$		0			0
$159$	5.43404	+	10.0369i	0.430947	+	0.795977i
$160$		0			0
$161$	20.6337			1.62617
$162$		0			0
$163$	−2.74172	+	4.74879i	−0.214748	+	0.371954i	−0.953195	−	0.302358i	$-0.902226\pi$
$163$	−2.74172	+	4.74879i	−0.214748	+	0.371954i	0.738447	+	0.674312i	$0.235560\pi$
$164$		0			0
$165$	−1.30824	+	2.12914i	−0.101847	+	0.165753i
$166$		0			0
$167$	15.1142			1.16957			0.584784	−	0.811189i	$-0.301179\pi$
$167$	15.1142			1.16957			0.584784	+	0.811189i	$0.301179\pi$
$168$		0			0
$169$	12.9657	+	0.944067i	0.997360	+	0.0726205i
$170$		0			0
$171$	4.11913	+	8.13232i	0.314998	+	0.621894i
$172$		0			0
$173$	−6.88965			−0.523811			−0.261905	−	0.965094i	$-0.584351\pi$
$173$	−6.88965			−0.523811			−0.261905	+	0.965094i	$0.584351\pi$
$174$		0			0
$175$	2.94201	+	5.09572i	0.222395	+	0.385200i
$176$		0			0
$177$	1.09812	−	1.78717i	0.0825398	−	0.134332i
$178$		0			0
$179$	−3.90439	−	6.76259i	−0.291827	−	0.505460i	0.682414	−	0.730966i	$-0.260930\pi$
$179$	−3.90439	−	6.76259i	−0.291827	−	0.505460i	−0.974242	+	0.225505i	$0.927597\pi$
$180$		0			0
$181$	−9.35311			−0.695211			−0.347606	−	0.937641i	$-0.613005\pi$
$181$	−9.35311			−0.695211			−0.347606	+	0.937641i	$0.613005\pi$
$182$		0			0
$183$	3.10646	+	5.73776i	0.229636	+	0.424147i
$184$		0			0
$185$	6.97533			0.512836
$186$		0			0
$187$	−1.17212	−	2.03017i	−0.0857137	−	0.148460i
$188$		0			0
$189$	18.3731	+	1.51240i	1.33645	+	0.110011i
$190$		0			0
$191$	19.3660			1.40127			0.700636	−	0.713519i	$-0.252900\pi$
$191$	19.3660			1.40127			0.700636	+	0.713519i	$0.252900\pi$
$192$		0			0
$193$	5.57023			0.400954			0.200477	−	0.979698i	$-0.435751\pi$
$193$	5.57023			0.400954			0.200477	+	0.979698i	$0.435751\pi$
$194$		0			0
$195$	11.3926	+	0.726911i	0.815841	+	0.0520551i
$196$		0			0
$197$	−11.3330	−	19.6294i	−0.807445	−	1.39854i	−0.914628	−	0.404297i	$-0.867516\pi$
$197$	−11.3330	−	19.6294i	−0.807445	−	1.39854i	0.107182	−	0.994239i	$-0.465817\pi$
$198$		0			0
$199$	−5.10112	+	8.83540i	−0.361609	+	0.626325i	−0.988226	−	0.153002i	$-0.951106\pi$
$199$	−5.10112	+	8.83540i	−0.361609	+	0.626325i	0.626617	+	0.779327i	$0.284439\pi$
$200$		0			0
$201$	6.98489	+	12.9014i	0.492676	+	0.909992i
$202$		0			0
$203$	14.5275	−	25.1623i	1.01963	−	1.76605i
$204$		0			0
$205$	−0.0833738	+	0.144408i	−0.00582308	+	0.0100859i
$206$		0			0
$207$	−9.53755	+	14.6099i	−0.662906	+	1.01545i
$208$		0			0
$209$	1.19916	+	2.07701i	0.0829477	+	0.143670i
$210$		0			0
$211$	4.13229			0.284478			0.142239	−	0.989832i	$-0.454570\pi$
$211$	4.13229			0.284478			0.142239	+	0.989832i	$0.454570\pi$
$212$		0			0
$213$	22.5503	+	0.617510i	1.54512	+	0.0423111i
$214$		0			0
$215$	−7.87536	+	13.6405i	−0.537095	+	0.930276i
$216$		0			0
$217$	−0.458855	+	0.794760i	−0.0311491	+	0.0539518i
$218$		0			0
$219$	6.32942	+	11.6907i	0.427703	+	0.789984i
$220$		0			0
$221$	−5.68795	+	9.07360i	−0.382613	+	0.610357i
$222$		0			0
$223$	−0.997068	+	1.72697i	−0.0667686	+	0.115647i	−0.897477	−	0.441061i	$-0.854602\pi$
$223$	−0.997068	+	1.72697i	−0.0667686	+	0.115647i	0.830709	+	0.556708i	$0.187936\pi$
$224$		0			0
$225$	−4.96794	−	0.272285i	−0.331196	−	0.0181524i
$226$		0			0
$227$	18.3994			1.22121			0.610607	−	0.791934i	$-0.290926\pi$
$227$	18.3994			1.22121			0.610607	+	0.791934i	$0.290926\pi$
$228$		0			0
$229$	−10.8242	+	18.7481i	−0.715284	+	1.23891i	0.247565	+	0.968871i	$0.420370\pi$
$229$	−10.8242	+	18.7481i	−0.715284	+	1.23891i	−0.962850	+	0.270038i	$0.912964\pi$
$230$		0			0
$231$	4.84829	+	0.132764i	0.318994	+	0.00873524i
$232$		0			0
$233$	10.5670			0.692265			0.346133	−	0.938186i	$-0.387495\pi$
$233$	10.5670			0.692265			0.346133	+	0.938186i	$0.387495\pi$
$234$		0			0
$235$	−4.25441			−0.277527
$236$		0			0
$237$	10.4591	−	17.0219i	0.679389	−	1.10569i
$238$		0			0
$239$	−1.75441	+	3.03873i	−0.113484	+	0.196559i	−0.917173	−	0.398490i	$-0.869534\pi$
$239$	−1.75441	+	3.03873i	−0.113484	+	0.196559i	0.803689	+	0.595050i	$0.202868\pi$
$240$		0			0
$241$	−26.1264			−1.68295			−0.841475	−	0.540296i	$-0.818312\pi$
$241$	−26.1264			−1.68295			−0.841475	+	0.540296i	$0.818312\pi$
$242$		0			0
$243$	−9.56349	+	12.3101i	−0.613498	+	0.789696i
$244$		0			0
$245$	−5.10684	+	8.84531i	−0.326264	+	0.565106i
$246$		0			0
$247$	5.81918	−	9.28295i	0.370266	−	0.590660i
$248$		0			0
$249$	14.1906	−	23.0949i	0.899296	−	1.46358i
$250$		0			0
$251$	8.00649	−	13.8677i	0.505365	−	0.875319i	−0.494615	−	0.869112i	$-0.664691\pi$
$251$	8.00649	−	13.8677i	0.505365	−	0.875319i	0.999981	−	0.00620652i	$-0.00197561\pi$
$252$		0			0
$253$	−2.29511	+	3.97524i	−0.144292	+	0.249921i
$254$		0			0
$255$	−4.92316	+	8.01232i	−0.308300	+	0.501751i
$256$		0			0
$257$	−4.38690			−0.273647			−0.136824	−	0.990595i	$-0.543689\pi$
$257$	−4.38690			−0.273647			−0.136824	+	0.990595i	$0.543689\pi$
$258$		0			0
$259$	−6.76908	−	11.7244i	−0.420610	−	0.728518i
$260$		0			0
$261$	11.1013	+	21.9171i	0.687153	+	1.35663i
$262$		0			0
$263$	11.3028	−	19.5770i	0.696958	−	1.20717i	−0.272558	−	0.962139i	$-0.587870\pi$
$263$	11.3028	−	19.5770i	0.696958	−	1.20717i	0.969516	−	0.245027i	$-0.0787970\pi$
$264$		0			0
$265$	6.02283	−	10.4319i	0.369980	−	0.640824i
$266$		0			0
$267$	−3.95161	+	6.43115i	−0.241835	+	0.393580i
$268$		0			0
$269$	−7.33389	+	12.7027i	−0.447155	+	0.774496i	−0.998200	−	0.0599804i	$-0.980896\pi$
$269$	−7.33389	+	12.7027i	−0.447155	+	0.774496i	0.551044	+	0.834476i	$0.314229\pi$
$270$		0			0
$271$	2.35377	+	4.07684i	0.142981	+	0.247651i	0.928618	−	0.371037i	$-0.120998\pi$
$271$	2.35377	+	4.07684i	0.142981	+	0.247651i	−0.785637	+	0.618688i	$0.787664\pi$
$272$		0			0
$273$	−9.83393	−	19.8545i	−0.595176	−	1.20165i
$274$		0			0
$275$	−1.30897			−0.0789339
$276$		0			0
$277$	−15.8732			−0.953727			−0.476863	−	0.878977i	$-0.658226\pi$
$277$	−15.8732			−0.953727			−0.476863	+	0.878977i	$0.658226\pi$
$278$		0			0
$279$	−0.350638	−	0.692257i	−0.0209921	−	0.0414444i
$280$		0			0
$281$	4.96870	+	8.60604i	0.296408	+	0.513393i	0.975311	−	0.220834i	$-0.0708780\pi$
$281$	4.96870	+	8.60604i	0.296408	+	0.513393i	−0.678904	+	0.734227i	$0.737545\pi$
$282$		0			0
$283$	−6.51285			−0.387149			−0.193574	−	0.981086i	$-0.562008\pi$
$283$	−6.51285			−0.387149			−0.193574	+	0.981086i	$0.562008\pi$
$284$		0			0
$285$	5.03675	−	8.19718i	0.298351	−	0.485559i
$286$		0			0
$287$	0.323634			0.0191035
$288$		0			0
$289$	4.08912	+	7.08256i	0.240536	+	0.416621i
$290$		0			0
$291$	−0.0932482	−	0.172233i	−0.00546630	−	0.0100965i
$292$		0			0
$293$	−15.7842	−	27.3390i	−0.922120	−	1.59716i	−0.796128	−	0.605128i	$-0.793122\pi$
$293$	−15.7842	−	27.3390i	−0.922120	−	1.59716i	−0.125993	−	0.992031i	$-0.540212\pi$
$294$		0			0
$295$	−2.21376			−0.128890
$296$		0			0
$297$	−2.33503	+	3.37150i	−0.135492	+	0.195634i
$298$		0			0
$299$	20.9553	+	0.761900i	1.21188	+	0.0440618i
$300$		0			0
$301$	30.5700			1.76203
$302$		0			0
$303$	14.0991	+	0.386086i	0.809974	+	0.0221801i
$304$		0			0
$305$	3.44306	−	5.96355i	0.197149	−	0.341472i
$306$		0			0
$307$	−9.62508			−0.549332			−0.274666	−	0.961540i	$-0.588567\pi$
$307$	−9.62508			−0.549332			−0.274666	+	0.961540i	$0.588567\pi$
$308$		0			0
$309$	5.24448	+	0.143613i	0.298348	+	0.00816987i
$310$		0			0
$311$	3.09612	−	5.36263i	0.175565	−	0.304087i	−0.764792	−	0.644277i	$-0.777158\pi$
$311$	3.09612	−	5.36263i	0.175565	−	0.304087i	0.940357	+	0.340190i	$0.110492\pi$
$312$		0			0
$313$	−8.25806	−	14.3034i	−0.466773	−	0.808475i	0.532506	−	0.846426i	$-0.321250\pi$
$313$	−8.25806	−	14.3034i	−0.466773	−	0.808475i	−0.999280	+	0.0379511i	$0.987917\pi$
$314$		0			0
$315$	−8.79149	−	17.3569i	−0.495344	−	0.977948i
$316$		0			0
$317$	11.2779	+	19.5340i	0.633432	+	1.09714i	0.986845	+	0.161669i	$0.0516876\pi$
$317$	11.2779	+	19.5340i	0.633432	+	1.09714i	−0.353413	+	0.935467i	$0.614979\pi$
$318$		0			0
$319$	3.23181	+	5.59765i	0.180946	+	0.313408i
$320$		0			0
$321$	−18.0323	+	29.3472i	−1.00647	+	1.63800i
$322$		0			0
$323$	4.51265	+	7.81615i	0.251091	+	0.434902i
$324$		0			0
$325$	2.79971	+	5.28377i	0.155300	+	0.293091i
$326$		0			0
$327$	−15.4619	−	28.5588i	−0.855047	−	1.57931i
$328$		0			0
$329$	4.12862	+	7.15098i	0.227618	+	0.394246i
$330$		0			0
$331$	13.9793			0.768374			0.384187	−	0.923255i	$-0.374482\pi$
$331$	13.9793			0.768374			0.384187	+	0.923255i	$0.374482\pi$
$332$		0			0
$333$	11.4304	+	0.626483i	0.626382	+	0.0343311i
$334$		0			0
$335$	7.74172	−	13.4091i	0.422976	−	0.732615i
$336$		0			0
$337$	0.207169	−	0.358827i	0.0112852	−	0.0195466i	−0.860328	−	0.509741i	$-0.829741\pi$
$337$	0.207169	−	0.358827i	0.0112852	−	0.0195466i	0.871613	+	0.490195i	$0.163074\pi$
$338$		0			0
$339$	2.27529	−	3.70299i	0.123577	−	0.201119i
$340$		0			0
$341$	−0.102078	−	0.176804i	−0.00552781	−	0.00957445i
$342$		0			0
$343$	−5.01173			−0.270608
$344$		0			0
$345$	18.4069	+	0.504049i	0.990995	+	0.0271371i
$346$		0			0
$347$	−5.06201	−	8.76765i	−0.271743	−	0.470672i	0.697565	−	0.716521i	$-0.254267\pi$
$347$	−5.06201	−	8.76765i	−0.271743	−	0.470672i	−0.969308	+	0.245849i	$0.920933\pi$
$348$		0			0
$349$	−7.41055	+	12.8355i	−0.396678	+	0.687066i	−0.993314	−	0.115446i	$-0.963170\pi$
$349$	−7.41055	+	12.8355i	−0.396678	+	0.687066i	0.596636	+	0.802512i	$0.296504\pi$
$350$		0			0
$351$	18.6037	+	2.21440i	0.992990	+	0.118196i
$352$		0			0
$353$	−15.0414	+	26.0525i	−0.800573	+	1.38663i	0.118666	+	0.992934i	$0.462138\pi$
$353$	−15.0414	+	26.0525i	−0.800573	+	1.38663i	−0.919239	+	0.393699i	$0.871195\pi$
$354$		0			0
$355$	−11.9041	−	20.6185i	−0.631804	−	1.09432i
$356$		0			0
$357$	18.2450	+	0.499615i	0.965627	+	0.0264424i
$358$		0			0
$359$	3.20722			0.169271			0.0846353	−	0.996412i	$-0.473027\pi$
$359$	3.20722			0.169271			0.0846353	+	0.996412i	$0.473027\pi$
$360$		0			0
$361$	4.88323	+	8.45800i	0.257012	+	0.445158i
$362$		0			0
$363$	9.40951	−	15.3138i	0.493871	−	0.803764i
$364$		0			0
$365$	7.01523	−	12.1507i	0.367194	−	0.635999i
$366$		0			0
$367$	1.08009	−	1.87076i	0.0563800	−	0.0976530i	−0.836458	−	0.548031i	$-0.815377\pi$
$367$	1.08009	−	1.87076i	0.0563800	−	0.0976530i	0.892838	+	0.450378i	$0.148711\pi$
$368$		0			0
$369$	−0.149594	+	0.229151i	−0.00778754	+	0.0119291i
$370$		0			0
$371$	−23.3790			−1.21378
$372$		0			0
$373$	−6.98479	−	12.0980i	−0.361659	−	0.626411i	0.626575	−	0.779361i	$-0.284456\pi$
$373$	−6.98479	−	12.0980i	−0.361659	−	0.626411i	−0.988234	+	0.152950i	$0.951123\pi$
$374$		0			0
$375$	10.0372	+	18.5391i	0.518318	+	0.957354i
$376$		0			0
$377$	15.6830	−	25.0181i	0.807717	−	1.28850i
$378$		0			0
$379$	−3.47398	−	6.01711i	−0.178446	−	0.309078i	0.762902	−	0.646514i	$-0.223774\pi$
$379$	−3.47398	−	6.01711i	−0.178446	−	0.309078i	−0.941349	+	0.337436i	$0.890440\pi$
$380$		0			0
$381$	−19.8479	+	32.3021i	−1.01684	+	1.65488i
$382$		0			0
$383$	−12.5333	−	21.7082i	−0.640420	−	1.10924i	−0.985339	−	0.170607i	$-0.945427\pi$
$383$	−12.5333	−	21.7082i	−0.640420	−	1.10924i	0.344919	−	0.938632i	$-0.387906\pi$
$384$		0			0
$385$	−2.55938	−	4.43297i	−0.130438	−	0.225925i
$386$		0			0
$387$	−14.1304	+	21.6453i	−0.718288	+	1.10029i
$388$		0			0
$389$	−14.9765	−	25.9401i	−0.759339	−	1.31521i	−0.943188	−	0.332259i	$-0.892189\pi$
$389$	−14.9765	−	25.9401i	−0.759339	−	1.31521i	0.183850	−	0.982954i	$-0.441144\pi$
$390$		0			0
$391$	−8.63690	+	14.9596i	−0.436787	+	0.756537i
$392$		0			0
$393$	−28.2088	−	0.772461i	−1.42295	−	0.0389655i
$394$		0			0
$395$	−21.0850			−1.06090
$396$		0			0
$397$	−14.4399	+	25.0106i	−0.724718	+	1.25525i	0.234372	+	0.972147i	$0.424696\pi$
$397$	−14.4399	+	25.0106i	−0.724718	+	1.25525i	−0.959090	+	0.283101i	$0.908637\pi$
$398$		0			0
$399$	−18.6660	−	0.511143i	−0.934466	−	0.0255891i
$400$		0			0
$401$	9.64590			0.481693			0.240847	−	0.970563i	$-0.422575\pi$
$401$	9.64590			0.481693			0.240847	+	0.970563i	$0.422575\pi$
$402$		0			0
$403$	−0.495353	+	0.790204i	−0.0246753	+	0.0393629i
$404$		0			0
$405$	16.3533	+	1.79800i	0.812604	+	0.0893435i
$406$		0			0
$407$	3.01172			0.149286
$408$		0			0
$409$	−13.2875	−	23.0146i	−0.657024	−	1.13800i	−0.981382	−	0.192065i	$-0.938482\pi$
$409$	−13.2875	−	23.0146i	−0.657024	−	1.13800i	0.324358	−	0.945934i	$-0.394852\pi$
$410$		0			0
$411$	12.5122	+	23.1106i	0.617184	+	1.13996i
$412$		0			0
$413$	2.14830	+	3.72097i	0.105711	+	0.183097i
$414$		0			0
$415$	−28.6077			−1.40429
$416$		0			0
$417$	−16.3772	+	26.6536i	−0.801997	+	1.30523i
$418$		0			0
$419$	−30.4742			−1.48876			−0.744380	−	0.667756i	$-0.767255\pi$
$419$	−30.4742			−1.48876			−0.744380	+	0.667756i	$0.767255\pi$
$420$		0			0
$421$	15.2202	+	26.3622i	0.741788	+	1.28481i	0.951681	+	0.307090i	$0.0993552\pi$
$421$	15.2202	+	26.3622i	0.741788	+	1.28481i	−0.209893	+	0.977724i	$0.567311\pi$
$422$		0			0
$423$	−6.97167	−	0.382107i	−0.338974	−	0.0185787i
$424$		0			0
$425$	−4.92589			−0.238941
$426$		0			0
$427$	−13.3650			−0.646778
$428$		0			0
$429$	4.91896	+	0.313857i	0.237490	+	0.0151531i
$430$		0			0
$431$	0.842335	+	1.45897i	0.0405738	+	0.0702759i	0.885599	−	0.464450i	$-0.153748\pi$
$431$	0.842335	+	1.45897i	0.0405738	+	0.0702759i	−0.845025	+	0.534726i	$0.820415\pi$
$432$		0			0
$433$	10.5289	−	18.2367i	0.505989	−	0.876398i	−0.493987	−	0.869469i	$-0.664461\pi$
$433$	10.5289	−	18.2367i	0.505989	−	0.876398i	0.999976	−	0.00692888i	$-0.00220555\pi$
$434$		0			0
$435$	13.5743	−	22.0919i	0.650839	−	1.05922i
$436$		0			0
$437$	8.83618	−	15.3047i	0.422692	−	0.732123i
$438$		0			0
$439$	−2.94942	+	5.10855i	−0.140768	+	0.243818i	−0.927786	−	0.373112i	$-0.878291\pi$
$439$	−2.94942	+	5.10855i	−0.140768	+	0.243818i	0.787018	+	0.616930i	$0.211624\pi$
$440$		0			0
$441$	−9.16297	+	14.0361i	−0.436332	+	0.668384i
$442$		0			0
$443$	−11.8651	−	20.5510i	−0.563728	−	0.976406i	−0.997167	−	0.0752229i	$-0.976033\pi$
$443$	−11.8651	−	20.5510i	−0.563728	−	0.976406i	0.433438	−	0.901183i	$-0.357300\pi$
$444$		0			0
$445$	7.96626			0.377637
$446$		0			0
$447$	−9.07549	+	14.7701i	−0.429256	+	0.698604i
$448$		0			0
$449$	12.3374	−	21.3689i	0.582236	−	1.00846i	−0.412977	−	0.910741i	$-0.635511\pi$
$449$	12.3374	−	21.3689i	0.582236	−	1.00846i	0.995214	−	0.0977216i	$-0.0311555\pi$
$450$		0			0
$451$	−0.0359981	+	0.0623506i	−0.00169509	+	0.00293597i
$452$		0			0
$453$	−17.2654	+	28.0990i	−0.811199	+	1.32021i
$454$		0			0
$455$	−12.4199	+	19.8127i	−0.582254	+	0.928832i
$456$		0			0
$457$	−7.24537	+	12.5493i	−0.338924	+	0.587033i	−0.984231	−	0.176890i	$-0.943396\pi$
$457$	−7.24537	+	12.5493i	−0.338924	+	0.587033i	0.645307	+	0.763924i	$0.276730\pi$
$458$		0			0
$459$	−8.78715	+	12.6875i	−0.410149	+	0.592204i
$460$		0			0
$461$	27.9740			1.30288			0.651439	−	0.758701i	$-0.274166\pi$
$461$	27.9740			1.30288			0.651439	+	0.758701i	$0.274166\pi$
$462$		0			0
$463$	−12.9541	+	22.4371i	−0.602027	+	1.04274i	0.390487	+	0.920608i	$0.372307\pi$
$463$	−12.9541	+	22.4371i	−0.602027	+	1.04274i	−0.992514	+	0.122132i	$0.961027\pi$
$464$		0			0
$465$	−0.428749	+	0.697778i	−0.0198828	+	0.0323587i
$466$		0			0
$467$	8.61901			0.398840			0.199420	−	0.979914i	$-0.436094\pi$
$467$	8.61901			0.398840			0.199420	+	0.979914i	$0.436094\pi$
$468$		0			0
$469$	−30.0513			−1.38764
$470$		0			0
$471$	17.3624	+	0.475448i	0.800019	+	0.0219075i
$472$		0			0
$473$	−3.40033	+	5.88954i	−0.156347	+	0.270801i
$474$		0			0
$475$	5.03954			0.231230
$476$		0			0
$477$	10.8065	−	16.5536i	0.494795	−	0.757940i
$478$		0			0
$479$	−9.15223	+	15.8521i	−0.418176	+	0.724302i	−0.995756	−	0.0920318i	$-0.970664\pi$
$479$	−9.15223	+	15.8521i	−0.418176	+	0.724302i	0.577580	+	0.816334i	$0.303997\pi$
$480$		0			0
$481$	−6.44166	−	12.1571i	−0.293715	−	0.554315i
$482$		0			0
$483$	−17.0154	−	31.4282i	−0.774229	−	1.43003i
$484$		0			0
$485$	−0.103352	+	0.179011i	−0.00469297	+	0.00812846i
$486$		0			0
$487$	16.8413	−	29.1699i	0.763150	−	1.32182i	−0.178069	−	0.984018i	$-0.556985\pi$
$487$	16.8413	−	29.1699i	0.763150	−	1.32182i	0.941219	−	0.337797i	$-0.109682\pi$
$488$		0			0
$489$	9.49403	+	0.259982i	0.429335	+	0.0117568i
$490$		0			0
$491$	30.8537			1.39241			0.696204	−	0.717844i	$-0.254871\pi$
$491$	30.8537			1.39241			0.696204	+	0.717844i	$0.254871\pi$
$492$		0			0
$493$	12.1619	+	21.0650i	0.547743	+	0.948718i
$494$		0			0
$495$	4.32181	+	0.236872i	0.194251	+	0.0106466i
$496$		0			0
$497$	−23.1042	+	40.0177i	−1.03637	+	1.79504i
$498$		0			0
$499$	−6.47626	+	11.2172i	−0.289917	+	0.502151i	−0.973790	−	0.227450i	$-0.926961\pi$
$499$	−6.47626	+	11.2172i	−0.289917	+	0.502151i	0.683873	+	0.729601i	$0.260294\pi$
$500$		0			0
$501$	−12.4637	−	23.0210i	−0.556839	−	1.02850i
$502$		0			0
$503$	−12.8940	+	22.3331i	−0.574916	+	0.995784i	0.421135	+	0.906998i	$0.361632\pi$
$503$	−12.8940	+	22.3331i	−0.574916	+	0.995784i	−0.996051	+	0.0887855i	$0.971701\pi$
$504$		0			0
$505$	−7.44282	−	12.8913i	−0.331201	−	0.573657i
$506$		0			0
$507$	−9.25407	−	20.5271i	−0.410988	−	0.911641i
$508$		0			0
$509$	−20.0195			−0.887350			−0.443675	−	0.896188i	$-0.646326\pi$
$509$	−20.0195			−0.887350			−0.443675	+	0.896188i	$0.646326\pi$
$510$		0			0
$511$	−27.2312			−1.20464
$512$		0			0
$513$	8.98989	−	12.9803i	0.396913	−	0.573093i
$514$		0			0
$515$	−2.76852	−	4.79522i	−0.121996	−	0.211302i
$516$		0			0
$517$	−1.83692			−0.0807876
$518$		0			0
$519$	5.68149	+	10.4939i	0.249390	+	0.460633i
$520$		0			0
$521$	24.1247			1.05692			0.528461	−	0.848958i	$-0.322769\pi$
$521$	24.1247			1.05692			0.528461	+	0.848958i	$0.322769\pi$
$522$		0			0
$523$	0.701127	+	1.21439i	0.0306581	+	0.0531015i	0.880947	−	0.473214i	$-0.156906\pi$
$523$	0.701127	+	1.21439i	0.0306581	+	0.0531015i	−0.850289	+	0.526316i	$0.823573\pi$
$524$		0			0
$525$	5.33541	−	8.68325i	0.232856	−	0.378968i
$526$		0			0
$527$	−0.384136	−	0.665343i	−0.0167332	−	0.0289828i
$528$		0			0
$529$	10.8236			0.470593
$530$		0			0
$531$	−3.62767	−	0.198827i	−0.157427	−	0.00862835i
$532$		0			0
$533$	0.328679	+	0.0119502i	0.0142367	+	0.000517620i
$534$		0			0
$535$	36.3523			1.57165
$536$		0			0
$537$	−7.08069	+	11.5237i	−0.305554	+	0.497282i
$538$		0			0
$539$	−2.20497	+	3.81912i	−0.0949748	+	0.164501i
$540$		0			0
$541$	16.6942			0.717741			0.358871	−	0.933387i	$-0.383162\pi$
$541$	16.6942			0.717741			0.358871	+	0.933387i	$0.383162\pi$
$542$		0			0
$543$	7.71296	+	14.2461i	0.330995	+	0.611360i
$544$		0			0
$545$	−17.1373	+	29.6826i	−0.734080	+	1.27146i
$546$		0			0
$547$	−9.02605	−	15.6336i	−0.385926	−	0.668443i	0.605971	−	0.795487i	$-0.292785\pi$
$547$	−9.02605	−	15.6336i	−0.385926	−	0.668443i	−0.991897	+	0.127043i	$0.959451\pi$
$548$		0			0
$549$	6.17772	−	9.46318i	0.263659	−	0.403879i
$550$		0			0
$551$	−12.4425	−	21.5510i	−0.530067	−	0.918103i
$552$		0			0
$553$	20.4615	+	35.4404i	0.870112	+	1.50708i
$554$		0			0
$555$	−5.75214	−	10.6244i	−0.244165	−	0.450982i
$556$		0			0
$557$	−9.46050	−	16.3861i	−0.400854	−	0.694300i	0.592975	−	0.805221i	$-0.297953\pi$
$557$	−9.46050	−	16.3861i	−0.400854	−	0.694300i	−0.993829	+	0.110921i	$0.964620\pi$
$558$		0			0
$559$	31.0465	+	1.12880i	1.31313	+	0.0477430i
$560$		0			0
$561$	−2.12566	+	3.45946i	−0.0897454	+	0.146059i
$562$		0			0
$563$	6.84531	+	11.8564i	0.288496	+	0.499689i	0.973451	−	0.228896i	$-0.0735116\pi$
$563$	6.84531	+	11.8564i	0.288496	+	0.499689i	−0.684955	+	0.728585i	$0.740178\pi$
$564$		0			0
$565$	−4.58688			−0.192972
$566$		0			0
$567$	−12.8476	−	29.2321i	−0.539550	−	1.22763i
$568$		0			0
$569$	8.12083	−	14.0657i	0.340443	−	0.589665i	−0.644072	−	0.764965i	$-0.722756\pi$
$569$	8.12083	−	14.0657i	0.340443	−	0.589665i	0.984515	+	0.175300i	$0.0560896\pi$
$570$		0			0
$571$	18.7269	−	32.4359i	0.783696	−	1.35740i	−0.146079	−	0.989273i	$-0.546665\pi$
$571$	18.7269	−	32.4359i	0.783696	−	1.35740i	0.929775	−	0.368129i	$-0.120001\pi$
$572$		0			0
$573$	−15.9700	−	29.4971i	−0.667155	−	1.23226i
$574$		0			0
$575$	4.82266	+	8.35309i	0.201119	+	0.348348i
$576$		0			0
$577$	24.9058			1.03684			0.518421	−	0.855126i	$-0.326520\pi$
$577$	24.9058			1.03684			0.518421	+	0.855126i	$0.326520\pi$
$578$		0			0
$579$	−4.59344	−	8.48427i	−0.190897	−	0.352594i
$580$		0			0
$581$	27.7618	+	48.0848i	1.15175	+	1.99489i
$582$		0			0
$583$	2.60047	−	4.50414i	0.107700	−	0.186542i
$584$		0			0
$585$	−8.28762	−	17.9520i	−0.342651	−	0.742225i
$586$		0			0
$587$	−13.9379	+	24.1412i	−0.575279	+	0.996413i	0.420732	+	0.907185i	$0.361773\pi$
$587$	−13.9379	+	24.1412i	−0.575279	+	0.996413i	−0.996011	+	0.0892281i	$0.971560\pi$
$588$		0			0
$589$	0.392999	+	0.680694i	0.0161932	+	0.0280475i
$590$		0			0
$591$	−20.5527	+	33.4490i	−0.845426	+	1.37591i
$592$		0			0
$593$	−39.1962			−1.60959			−0.804797	−	0.593550i	$-0.797726\pi$
$593$	−39.1962			−1.60959			−0.804797	+	0.593550i	$0.797726\pi$
$594$		0			0
$595$	−9.63138	−	16.6820i	−0.394848	−	0.683897i
$596$		0			0
$597$	17.6642	+	0.483710i	0.722947	+	0.0197970i
$598$		0			0
$599$	−5.21215	+	9.02771i	−0.212963	+	0.368862i	−0.952640	−	0.304099i	$-0.901645\pi$
$599$	−5.21215	+	9.02771i	−0.212963	+	0.368862i	0.739678	+	0.672961i	$0.234978\pi$
$600$		0			0
$601$	14.0755	−	24.3795i	0.574152	−	0.994460i	−0.421981	−	0.906605i	$-0.638665\pi$
$601$	14.0755	−	24.3795i	0.574152	−	0.994460i	0.996133	−	0.0878557i	$-0.0280014\pi$
$602$		0			0
$603$	13.8906	−	21.2780i	0.565670	−	0.866507i
$604$		0			0
$605$	−18.9691			−0.771205
$606$		0			0
$607$	−10.7141	−	18.5573i	−0.434871	−	0.753219i	0.562414	−	0.826856i	$-0.309873\pi$
$607$	−10.7141	−	18.5573i	−0.434871	−	0.753219i	−0.997285	+	0.0736371i	$0.976539\pi$
$608$		0			0
$609$	−50.3058	−	1.37756i	−2.03849	−	0.0558215i
$610$		0			0
$611$	3.92892	+	7.41488i	0.158947	+	0.299974i
$612$		0			0
$613$	17.1769	+	29.7512i	0.693766	+	1.20164i	0.970595	+	0.240719i	$0.0773833\pi$
$613$	17.1769	+	29.7512i	0.693766	+	1.20164i	−0.276828	+	0.960919i	$0.589283\pi$
$614$		0			0
$615$	0.288707	+	0.00790587i	0.0116418	+	0.000318795i
$616$		0			0
$617$	6.18881	+	10.7193i	0.249152	+	0.431544i	0.963291	−	0.268460i	$-0.0865148\pi$
$617$	6.18881	+	10.7193i	0.249152	+	0.431544i	−0.714139	+	0.700004i	$0.753181\pi$
$618$		0			0
$619$	−7.10653	−	12.3089i	−0.285635	−	0.494735i	0.687128	−	0.726537i	$-0.258871\pi$
$619$	−7.10653	−	12.3089i	−0.285635	−	0.494735i	−0.972763	+	0.231802i	$0.925538\pi$
$620$		0			0
$621$	30.1180	+	2.47918i	1.20859	+	0.0994861i
$622$		0			0
$623$	−7.73071	−	13.3900i	−0.309724	−	0.536458i
$624$		0			0
$625$	6.97858	−	12.0873i	0.279143	−	0.483490i
$626$		0			0
$627$	2.17470	−	3.53928i	0.0868493	−	0.141345i
$628$		0			0
$629$	11.3336			0.451902
$630$		0			0
$631$	−19.7301	+	34.1735i	−0.785441	+	1.36042i	0.143294	+	0.989680i	$0.454231\pi$
$631$	−19.7301	+	34.1735i	−0.785441	+	1.36042i	−0.928735	+	0.370744i	$0.879103\pi$
$632$		0			0
$633$	−3.40765	−	6.29407i	−0.135442	−	0.250167i
$634$		0			0
$635$	40.0125			1.58785
$636$		0			0
$637$	20.1323	+	0.731977i	0.797673	+	0.0290020i
$638$		0			0
$639$	−17.6553	−	34.8565i	−0.698433	−	1.37890i
$640$		0			0
$641$	38.5836			1.52396			0.761980	−	0.647600i	$-0.224227\pi$
$641$	38.5836			1.52396			0.761980	+	0.647600i	$0.224227\pi$
$642$		0			0
$643$	16.7399	+	28.9944i	0.660158	+	1.14343i	0.980574	+	0.196150i	$0.0628438\pi$
$643$	16.7399	+	28.9944i	0.660158	+	1.14343i	−0.320416	+	0.947277i	$0.603823\pi$
$644$		0			0
$645$	27.2708	+	0.746776i	1.07379	+	0.0294043i
$646$		0			0
$647$	−9.34123	−	16.1795i	−0.367242	−	0.636081i	0.621891	−	0.783103i	$-0.286364\pi$
$647$	−9.34123	−	16.1795i	−0.367242	−	0.636081i	−0.989133	+	0.147022i	$0.953031\pi$
$648$		0			0
$649$	−0.955829			−0.0375196
$650$		0			0
$651$	1.58892	+	0.0435106i	0.0622748	+	0.00170531i
$652$		0			0
$653$	21.8099			0.853486			0.426743	−	0.904373i	$-0.359661\pi$
$653$	21.8099			0.853486			0.426743	+	0.904373i	$0.359661\pi$
$654$		0			0
$655$	14.8912	+	25.7923i	0.581848	+	1.00779i
$656$		0			0
$657$	12.5871	−	19.2812i	0.491070	−	0.752233i
$658$		0			0
$659$	42.8462			1.66905			0.834526	−	0.550969i	$-0.185742\pi$
$659$	42.8462			1.66905			0.834526	+	0.550969i	$0.185742\pi$
$660$		0			0
$661$	−24.4668			−0.951650			−0.475825	−	0.879540i	$-0.657850\pi$
$661$	−24.4668			−0.951650			−0.475825	+	0.879540i	$0.657850\pi$
$662$		0			0
$663$	18.5109	+	1.18110i	0.718904	+	0.0458700i
$664$		0			0
$665$	9.85361	+	17.0669i	0.382106	+	0.661828i
$666$		0			0
$667$	23.8140	−	41.2471i	0.922082	−	1.59709i
$668$		0			0
$669$	3.45265	+	0.0945463i	0.133487	+	0.00365537i
$670$		0			0
$671$	1.48660	−	2.57487i	0.0573896	−	0.0994017i
$672$		0			0
$673$	4.26726	−	7.39111i	0.164491	−	0.284907i	−0.771983	−	0.635643i	$-0.780735\pi$
$673$	4.26726	−	7.39111i	0.164491	−	0.284907i	0.936474	+	0.350736i	$0.114069\pi$
$674$		0			0
$675$	3.68204	+	7.79143i	0.141722	+	0.299892i
$676$		0			0
$677$	−21.1789	−	36.6829i	−0.813970	−	1.40984i	−0.910065	−	0.414465i	$-0.863969\pi$
$677$	−21.1789	−	36.6829i	−0.813970	−	1.40984i	0.0960958	−	0.995372i	$-0.469364\pi$
$678$		0			0
$679$	0.401184			0.0153960
$680$		0			0
$681$	−15.1729	−	28.0250i	−0.581428	−	1.07392i
$682$		0			0
$683$	−16.5016	+	28.5816i	−0.631415	+	1.09364i	0.355847	+	0.934544i	$0.384192\pi$
$683$	−16.5016	+	28.5816i	−0.631415	+	1.09364i	−0.987263	+	0.159099i	$0.949141\pi$
$684$		0			0
$685$	13.8680	−	24.0201i	0.529869	−	0.917759i
$686$		0			0
$687$	37.4821	+	1.02640i	1.43003	+	0.0391596i
$688$		0			0
$689$	−23.7434	−	0.863268i	−0.904551	−	0.0328879i
$690$		0			0
$691$	5.50867	−	9.54129i	0.209560	−	0.362968i	−0.742016	−	0.670382i	$-0.766130\pi$
$691$	5.50867	−	9.54129i	0.209560	−	0.362968i	0.951576	+	0.307414i	$0.0994637\pi$
$692$		0			0
$693$	−3.79588	−	7.49413i	−0.144194	−	0.284679i
$694$		0			0
$695$	33.0157			1.25236
$696$		0			0
$697$	−0.135467	+	0.234636i	−0.00513119	+	0.00888748i
$698$		0			0
$699$	−8.71396	−	16.0950i	−0.329592	−	0.608770i
$700$		0			0
$701$	−3.41066			−0.128819			−0.0644094	−	0.997924i	$-0.520516\pi$
$701$	−3.41066			−0.128819			−0.0644094	+	0.997924i	$0.520516\pi$
$702$		0			0
$703$	−11.5951			−0.437319
$704$		0			0
$705$	3.50836	+	6.48009i	0.132133	+	0.244054i
$706$		0			0
$707$	−14.4455	+	25.0203i	−0.543279	+	0.940986i
$708$		0			0
$709$	28.7937			1.08137			0.540684	−	0.841226i	$-0.318165\pi$
$709$	28.7937			1.08137			0.540684	+	0.841226i	$0.318165\pi$
$710$		0			0
$711$	−34.5517	−	1.89373i	−1.29579	−	0.0710203i
$712$		0			0
$713$	−0.752172	+	1.30280i	−0.0281691	+	0.0487903i
$714$		0			0
$715$	−2.43558	−	4.59657i	−0.0910855	−	0.171902i
$716$		0			0
$717$	6.07519	+	0.166361i	0.226882	+	0.00621287i
$718$		0			0
$719$	−6.18739	+	10.7169i	−0.230751	+	0.399672i	−0.958029	−	0.286670i	$-0.907452\pi$
$719$	−6.18739	+	10.7169i	−0.230751	+	0.399672i	0.727278	+	0.686343i	$0.240785\pi$
$720$		0			0
$721$	−5.37332	+	9.30686i	−0.200113	+	0.346606i
$722$		0			0
$723$	21.5449	+	39.7943i	0.801264	+	1.47997i
$724$		0			0
$725$	13.5818			0.504417
$726$		0			0
$727$	0.0603226	+	0.104482i	0.00223724	+	0.00387502i	0.867142	−	0.498061i	$-0.165955\pi$
$727$	0.0603226	+	0.104482i	0.00223724	+	0.00387502i	−0.864905	+	0.501936i	$0.832621\pi$
$728$		0			0
$729$	26.6366	+	4.41513i	0.986539	+	0.163523i
$730$		0			0
$731$	−12.7960	+	22.1634i	−0.473278	+	0.819742i
$732$		0			0
$733$	−6.33410	+	10.9710i	−0.233955	+	0.405223i	−0.958969	−	0.283512i	$-0.908500\pi$
$733$	−6.33410	+	10.9710i	−0.233955	+	0.405223i	0.725013	+	0.688735i	$0.241834\pi$
$734$		0			0
$735$	17.6840	+	0.484253i	0.652284	+	0.0178619i
$736$		0			0
$737$	3.34263	−	5.78960i	0.123127	−	0.213263i
$738$		0			0
$739$	−1.93983	−	3.35988i	−0.0713578	−	0.123595i	0.828139	−	0.560523i	$-0.189400\pi$
$739$	−1.93983	−	3.35988i	−0.0713578	−	0.123595i	−0.899497	+	0.436928i	$0.856067\pi$
$740$		0			0
$741$	−18.9380	−	1.20835i	−0.695705	−	0.0443898i
$742$		0			0
$743$	−44.0182			−1.61487			−0.807435	−	0.589956i	$-0.799145\pi$
$743$	−44.0182			−1.61487			−0.807435	+	0.589956i	$0.799145\pi$
$744$		0			0
$745$	18.2957			0.670304
$746$		0			0
$747$	−46.8791	−	2.56937i	−1.71522	−	0.0940084i
$748$		0			0
$749$	−35.2774	−	61.1023i	−1.28901	−	2.23263i
$750$		0			0
$751$	−29.5451			−1.07812			−0.539058	−	0.842269i	$-0.681219\pi$
$751$	−29.5451			−1.07812			−0.539058	+	0.842269i	$0.681219\pi$
$752$		0			0
$753$	−27.7249	−	0.759211i	−1.01035	−	0.0276672i
$754$		0			0
$755$	34.8062			1.26673
$756$		0			0
$757$	−3.72486	−	6.45165i	−0.135382	−	0.234489i	0.790361	−	0.612641i	$-0.209893\pi$
$757$	−3.72486	−	6.45165i	−0.135382	−	0.234489i	−0.925743	+	0.378152i	$0.876560\pi$
$758$		0			0
$759$	7.94751	+	0.217632i	0.288476	+	0.00789955i
$760$		0			0
$761$	4.73158	+	8.19534i	0.171520	+	0.297081i	0.938951	−	0.344050i	$-0.111799\pi$
$761$	4.73158	+	8.19534i	0.171520	+	0.297081i	−0.767432	+	0.641131i	$0.778466\pi$
$762$		0			0
$763$	66.5223			2.40827
$764$		0			0
$765$	16.2637	+	0.891392i	0.588017	+	0.0322283i
$766$		0			0
$767$	2.04439	+	3.85829i	0.0738186	+	0.139315i
$768$		0			0
$769$	32.9002			1.18641			0.593205	−	0.805051i	$-0.297862\pi$
$769$	32.9002			1.18641			0.593205	+	0.805051i	$0.297862\pi$
$770$		0			0
$771$	3.61762	+	6.68188i	0.130285	+	0.240642i
$772$		0			0
$773$	−25.4419	+	44.0667i	−0.915082	+	1.58497i	−0.108302	+	0.994118i	$0.534541\pi$
$773$	−25.4419	+	44.0667i	−0.915082	+	1.58497i	−0.806781	+	0.590851i	$0.798792\pi$
$774$		0			0
$775$	−0.428987			−0.0154097
$776$		0			0
$777$	−12.2759	+	19.9787i	−0.440395	+	0.716732i
$778$		0			0
$779$	0.138593	−	0.240050i	0.00496561	−	0.00860068i
$780$		0			0
$781$	−5.13981	−	8.90241i	−0.183917	−	0.318553i
$782$		0			0
$783$	24.2283	−	34.9826i	0.865848	−	1.25018i
$784$		0			0
$785$	−9.16549	−	15.8751i	−0.327130	−	0.566606i
$786$		0			0
$787$	10.6565	+	18.4575i	0.379862	+	0.657940i	0.991042	−	0.133552i	$-0.0426382\pi$
$787$	10.6565	+	18.4575i	0.379862	+	0.657940i	−0.611180	+	0.791492i	$0.709305\pi$
$788$		0			0
$789$	−39.1392	−	1.07178i	−1.39339	−	0.0381563i
$790$		0			0
$791$	4.45126	+	7.70980i	0.158268	+	0.274129i
$792$		0			0
$793$	−13.5733	−	0.493502i	−0.482003	−	0.0175248i
$794$		0			0
$795$	−20.8559	−	0.571111i	−0.739682	−	0.0202552i
$796$		0			0
$797$	17.2569	+	29.8898i	0.611270	+	1.05875i	0.991027	+	0.133664i	$0.0426743\pi$
$797$	17.2569	+	29.8898i	0.611270	+	1.05875i	−0.379757	+	0.925086i	$0.623992\pi$
$798$		0			0
$799$	−6.91265			−0.244552
$800$		0			0
$801$	13.0542	+	0.715483i	0.461249	+	0.0252803i
$802$		0			0
$803$	3.02895	−	5.24630i	0.106889	−	0.185138i
$804$		0			0
$805$	−18.8591	+	32.6649i	−0.664696	+	1.15129i
$806$		0			0
$807$	25.3958	+	0.695431i	0.893975	+	0.0244803i
$808$		0			0
$809$	17.9703	+	31.1254i	0.631801	+	1.09431i	0.987183	+	0.159591i	$0.0510175\pi$
$809$	17.9703	+	31.1254i	0.631801	+	1.09431i	−0.355382	+	0.934721i	$0.615649\pi$
$810$		0			0
$811$	25.8673			0.908323			0.454161	−	0.890919i	$-0.349939\pi$
$811$	25.8673			0.908323			0.454161	+	0.890919i	$0.349939\pi$
$812$		0			0
$813$	4.26861	−	6.94706i	0.149707	−	0.243644i
$814$		0			0
$815$	−5.01182	−	8.68073i	−0.175556	−	0.304073i
$816$		0			0
$817$	13.0913	−	22.6747i	0.458006	−	0.793289i
$818$		0			0
$819$	−22.1319	+	31.3513i	−0.773349	+	1.09550i
$820$		0			0
$821$	6.61949	−	11.4653i	0.231022	−	0.400142i	−0.727087	−	0.686545i	$-0.759126\pi$
$821$	6.61949	−	11.4653i	0.231022	−	0.400142i	0.958109	+	0.286404i	$0.0924598\pi$
$822$		0			0
$823$	8.00285	+	13.8614i	0.278962	+	0.483176i	0.971127	−	0.238563i	$-0.0766763\pi$
$823$	8.00285	+	13.8614i	0.278962	+	0.483176i	−0.692165	+	0.721739i	$0.743343\pi$
$824$		0			0
$825$	1.07943	+	1.99375i	0.0375809	+	0.0694135i
$826$		0			0
$827$	32.7803			1.13988			0.569942	−	0.821685i	$-0.306966\pi$
$827$	32.7803			1.13988			0.569942	+	0.821685i	$0.306966\pi$
$828$		0			0
$829$	−6.77951	−	11.7424i	−0.235462	−	0.407832i	0.723945	−	0.689858i	$-0.242327\pi$
$829$	−6.77951	−	11.7424i	−0.235462	−	0.407832i	−0.959407	+	0.282026i	$0.908994\pi$
$830$		0			0
$831$	13.0897	+	24.1771i	0.454075	+	0.838695i
$832$		0			0
$833$	−8.29769	+	14.3720i	−0.287498	+	0.497961i
$834$		0			0
$835$	−13.8142	+	23.9269i	−0.478061	+	0.828026i
$836$		0			0
$837$	−0.765257	+	1.10494i	−0.0264512	+	0.0381922i
$838$		0			0
$839$	22.1091			0.763290			0.381645	−	0.924309i	$-0.375358\pi$
$839$	22.1091			0.763290			0.381645	+	0.924309i	$0.375358\pi$
$840$		0			0
$841$	−19.0332	−	32.9664i	−0.656316	−	1.13677i
$842$		0			0
$843$	9.01084	−	14.6649i	0.310350	−	0.505087i
$844$		0			0
$845$	−13.3451	+	19.6629i	−0.459084	+	0.676423i
$846$		0			0
$847$	18.4082	+	31.8840i	0.632515	+	1.09555i
$848$		0			0
$849$	5.37076	+	9.92001i	0.184324	+	0.340454i
$850$		0			0
$851$	−11.0961	−	19.2191i	−0.380371	−	0.658821i
$852$		0			0
$853$	25.0344	+	43.3608i	0.857160	+	1.48464i	0.874627	+	0.484797i	$0.161107\pi$
$853$	25.0344	+	43.3608i	0.857160	+	1.48464i	−0.0174667	+	0.999847i	$0.505560\pi$
$854$		0			0
$855$	−16.6390	−	0.911958i	−0.569042	−	0.0311883i
$856$		0			0
$857$	−20.7269	−	35.9000i	−0.708018	−	1.22632i	−0.965591	−	0.260064i	$-0.916256\pi$
$857$	−20.7269	−	35.9000i	−0.708018	−	1.22632i	0.257574	−	0.966259i	$-0.417077\pi$
$858$		0			0
$859$	−12.4494	+	21.5630i	−0.424769	+	0.735721i	−0.996399	−	0.0847904i	$-0.972978\pi$
$859$	−12.4494	+	21.5630i	−0.424769	+	0.735721i	0.571630	+	0.820511i	$0.306311\pi$
$860$		0			0
$861$	−0.266882	−	0.492942i	−0.00909532	−	0.0167994i
$862$		0			0
$863$	−15.8767			−0.540448			−0.270224	−	0.962797i	$-0.587098\pi$
$863$	−15.8767			−0.540448			−0.270224	+	0.962797i	$0.587098\pi$
$864$		0			0
$865$	6.29710	−	10.9069i	0.214108	−	0.370845i
$866$		0			0
$867$	7.41570	−	12.0689i	0.251850	−	0.409881i
$868$		0			0
$869$	−9.10380			−0.308825
$870$		0			0
$871$	−30.5197	−	1.10964i	−1.03412	−	0.0375988i
$872$		0			0
$873$	−0.185439	+	0.284061i	−0.00627618	+	0.00961400i
$874$		0			0
$875$	−43.1832			−1.45986
$876$		0			0
$877$	5.00766	+	8.67352i	0.169097	+	0.292884i	0.938103	−	0.346358i	$-0.112582\pi$
$877$	5.00766	+	8.67352i	0.169097	+	0.292884i	−0.769006	+	0.639242i	$0.779248\pi$
$878$		0			0
$879$	−28.6249	+	46.5864i	−0.965495	+	1.57132i
$880$		0			0
$881$	11.0336	+	19.1108i	0.371732	+	0.643859i	0.989832	−	0.142240i	$-0.0454306\pi$
$881$	11.0336	+	19.1108i	0.371732	+	0.643859i	−0.618100	+	0.786100i	$0.712097\pi$
$882$		0			0
$883$	46.2340			1.55590			0.777950	−	0.628326i	$-0.216260\pi$
$883$	46.2340			1.55590			0.777950	+	0.628326i	$0.216260\pi$
$884$		0			0
$885$	1.82556	+	3.37187i	0.0613654	+	0.113344i
$886$		0			0
$887$	−1.92642			−0.0646828			−0.0323414	−	0.999477i	$-0.510296\pi$
$887$	−1.92642			−0.0646828			−0.0323414	+	0.999477i	$0.510296\pi$
$888$		0			0
$889$	−38.8294	−	67.2545i	−1.30230	−	2.25564i
$890$		0			0
$891$	7.06084	+	0.776320i	0.236547	+	0.0260077i
$892$		0			0
$893$	7.07214			0.236660
$894$		0			0
$895$	14.2743			0.477138
$896$		0			0
$897$	−16.1202	−	32.5463i	−0.538236	−	1.08669i
$898$		0			0
$899$	1.05915	+	1.83451i	0.0353248	+	0.0611843i
$900$		0			0
$901$	9.78601	−	16.9499i	0.326019	−	0.564682i
$902$		0			0
$903$	−25.2093	−	46.5625i	−0.838912	−	1.54950i
$904$		0			0
$905$	8.54868	−	14.8067i	0.284168	−	0.492193i
$906$		0			0
$907$	−1.54206	+	2.67092i	−0.0512032	+	0.0886865i	−0.890491	−	0.455001i	$-0.849639\pi$
$907$	−1.54206	+	2.67092i	−0.0512032	+	0.0886865i	0.839288	+	0.543687i	$0.182972\pi$
$908$		0			0
$909$	−11.0386	−	21.7934i	−0.366129	−	0.722841i
$910$		0			0
$911$	−5.56268	−	9.63484i	−0.184300	−	0.319216i	0.759041	−	0.651043i	$-0.225668\pi$
$911$	−5.56268	−	9.63484i	−0.184300	−	0.319216i	−0.943340	+	0.331827i	$0.892335\pi$
$912$		0			0
$913$	−12.3519			−0.408787
$914$		0			0
$915$	−11.9226	−	0.326486i	−0.394150	−	0.0107933i
$916$		0			0
$917$	28.9018	−	50.0594i	0.954422	−	1.65311i
$918$		0			0
$919$	28.4673	−	49.3069i	0.939051	−	1.62648i	0.171804	−	0.985131i	$-0.445040\pi$
$919$	28.4673	−	49.3069i	0.939051	−	1.62648i	0.767247	−	0.641352i	$-0.221626\pi$
$920$		0			0
$921$	7.93723	+	14.6604i	0.261541	+	0.483076i
$922$		0			0
$923$	−24.9420	+	39.7883i	−0.820976	+	1.30965i
$924$		0			0
$925$	3.16423	−	5.48061i	0.104039	−	0.180201i
$926$		0			0
$927$	−4.10607	−	8.10653i	−0.134861	−	0.266253i
$928$		0			0
$929$	−2.34805			−0.0770370			−0.0385185	−	0.999258i	$-0.512264\pi$
$929$	−2.34805			−0.0770370			−0.0385185	+	0.999258i	$0.512264\pi$
$930$		0			0
$931$	8.48914	−	14.7036i	0.278220	−	0.481892i
$932$		0			0
$933$	−10.7212	−	0.293587i	−0.350998	−	0.00961162i
$934$		0			0
$935$	4.28523			0.140142
$936$		0			0
$937$	1.23201			0.0402482			0.0201241	−	0.999797i	$-0.493594\pi$
$937$	1.23201			0.0402482			0.0201241	+	0.999797i	$0.493594\pi$
$938$		0			0
$939$	−14.9762	+	24.3734i	−0.488729	+	0.795395i
$940$		0			0
$941$	−10.2312	+	17.7210i	−0.333529	+	0.577689i	−0.983201	−	0.182525i	$-0.941573\pi$
$941$	−10.2312	+	17.7210i	−0.333529	+	0.577689i	0.649672	+	0.760214i	$0.274906\pi$
$942$		0			0
$943$	0.530514			0.0172759
$944$		0			0
$945$	−19.1872	+	27.7039i	−0.624159	+	0.901207i
$946$		0			0
$947$	18.0932	−	31.3384i	0.587951	−	1.01836i	−0.406549	−	0.913629i	$-0.633268\pi$
$947$	18.0932	−	31.3384i	0.587951	−	1.01836i	0.994500	−	0.104733i	$-0.0333987\pi$
$948$		0			0
$949$	−27.6557	−	1.00551i	−0.897741	−	0.0326403i
$950$		0			0
$951$	20.4528	−	33.2864i	0.663227	−	1.07939i
$952$		0			0
$953$	10.3996	−	18.0126i	0.336875	−	0.583484i	−0.646969	−	0.762517i	$-0.723964\pi$
$953$	10.3996	−	18.0126i	0.336875	−	0.583484i	0.983843	+	0.179033i	$0.0572968\pi$
$954$		0			0
$955$	−17.7004	+	30.6579i	−0.572770	+	0.992067i
$956$		0			0
$957$	5.86095	−	9.53856i	0.189458	−	0.308338i
$958$		0			0
$959$	−53.8317			−1.73832
$960$		0			0
$961$	15.4665	+	26.7888i	0.498921	+	0.864156i
$962$		0			0
$963$	59.5702	+	3.26495i	1.91962	+	0.105212i
$964$		0			0
$965$	−5.09116	+	8.81814i	−0.163890	+	0.283866i
$966$		0			0
$967$	6.25537	−	10.8346i	0.201159	−	0.348418i	−0.747743	−	0.663988i	$-0.768862\pi$
$967$	6.25537	−	10.8346i	0.201159	−	0.348418i	0.948902	+	0.315570i	$0.102196\pi$
$968$		0			0
$969$	8.18380	−	13.3189i	0.262901	−	0.427866i
$970$		0			0
$971$	−2.25114	+	3.89909i	−0.0722425	+	0.125128i	−0.899884	−	0.436130i	$-0.856349\pi$
$971$	−2.25114	+	3.89909i	−0.0722425	+	0.125128i	0.827641	+	0.561257i	$0.189682\pi$
$972$		0			0
$973$	−32.0395	−	55.4941i	−1.02714	−	1.77906i
$974$		0			0
$975$	5.73919	−	8.62158i	0.183801	−	0.276111i
$976$		0			0
$977$	−47.5050			−1.51982			−0.759910	−	0.650029i	$-0.774757\pi$
$977$	−47.5050			−1.51982			−0.759910	+	0.650029i	$0.774757\pi$
$978$		0			0
$979$	3.43957			0.109929
$980$		0			0
$981$	−30.7486	+	47.1015i	−0.981728	+	1.50384i
$982$		0			0
$983$	6.94982	+	12.0374i	0.221665	+	0.383935i	0.955314	−	0.295594i	$-0.0955176\pi$
$983$	6.94982	+	12.0374i	0.221665	+	0.383935i	−0.733649	+	0.679529i	$0.762184\pi$
$984$		0			0
$985$	41.4333			1.32017
$986$		0			0
$987$	7.48734	−	12.1855i	0.238325	−	0.387868i
$988$		0			0
$989$	50.1115			1.59345
$990$		0			0
$991$	9.92979	+	17.1989i	0.315430	+	0.546341i	0.979529	−	0.201304i	$-0.0645178\pi$
$991$	9.92979	+	17.1989i	0.315430	+	0.546341i	−0.664099	+	0.747645i	$0.731185\pi$
$992$		0			0
$993$	−11.5279	−	21.2926i	−0.365828	−	0.675699i
$994$		0			0
$995$	−9.32478	−	16.1510i	−0.295615	−	0.512021i
$996$		0			0
$997$	−52.8484			−1.67372			−0.836862	−	0.547414i	$-0.815612\pi$
$997$	−52.8484			−1.67372			−0.836862	+	0.547414i	$0.815612\pi$
$998$		0			0
$999$	−8.47175	−	17.9268i	−0.268034	−	0.567178i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	468.2.k.a.445.5	yes	28
3.2	odd	2		1404.2.k.a.1225.11		28
9.2	odd	6		1404.2.j.a.289.11		28
9.7	even	3		468.2.j.a.133.6	✓	28
13.9	even	3		468.2.j.a.373.6	yes	28
39.35	odd	6		1404.2.j.a.685.11		28
117.61	even	3	inner	468.2.k.a.61.5	yes	28
117.74	odd	6		1404.2.k.a.1153.11		28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
468.2.j.a.133.6	✓	28	9.7	even	3
468.2.j.a.373.6	yes	28	13.9	even	3
468.2.k.a.61.5	yes	28	117.61	even	3	inner
468.2.k.a.445.5	yes	28	1.1	even	1	trivial
1404.2.j.a.289.11		28	9.2	odd	6
1404.2.j.a.685.11		28	39.35	odd	6
1404.2.k.a.1153.11		28	117.74	odd	6
1404.2.k.a.1225.11		28	3.2	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:	\( N \)	\(=\)	\( 468 = 2^{2} \cdot 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	468.k (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.824641 - 1.52314i) q^{3} +(-0.913993 + 1.58308i) q^{5} +3.54787 q^{7} +(-1.63994 + 2.51209i) q^{9} +(-0.394633 + 0.683524i) q^{11} +(3.60317 + 0.131005i) q^{13} +(3.16498 + 0.0866688i) q^{15} +(-1.48507 + 2.57222i) q^{17} +(1.51934 - 2.63157i) q^{19} +(-2.92572 - 5.40392i) q^{21} +5.81581 q^{23} +(0.829233 + 1.43627i) q^{25} +(5.17864 + 0.426283i) q^{27} +(4.09470 - 7.09223i) q^{29} +(-0.129332 + 0.224010i) q^{31} +(1.36654 + 0.0374208i) q^{33} +(-3.24273 + 5.61657i) q^{35} +(-1.90793 - 3.30463i) q^{37} +(-2.77178 - 5.59618i) q^{39} +0.0912193 q^{41} +8.61643 q^{43} +(-2.47796 - 4.89219i) q^{45} +(1.16369 + 2.01557i) q^{47} +5.58740 q^{49} +(5.14252 + 0.140821i) q^{51} -6.58958 q^{53} +(-0.721383 - 1.24947i) q^{55} +(-5.26117 - 0.144070i) q^{57} +(0.605518 + 1.04879i) q^{59} -3.76705 q^{61} +(-5.81828 + 8.91259i) q^{63} +(-3.50066 + 5.58438i) q^{65} -8.47022 q^{67} +(-4.79595 - 8.85831i) q^{69} +(-6.51214 + 11.2794i) q^{71} -7.67537 q^{73} +(1.50383 - 2.44745i) q^{75} +(-1.40011 + 2.42506i) q^{77} +(5.76726 + 9.98919i) q^{79} +(-3.62122 - 8.23934i) q^{81} +(7.82491 + 13.5531i) q^{83} +(-2.71469 - 4.70199i) q^{85} +(-14.1791 - 0.388277i) q^{87} +(-2.17897 - 3.77409i) q^{89} +(12.7836 + 0.464789i) q^{91} +(0.447853 + 0.0122639i) q^{93} +(2.77733 + 4.81047i) q^{95} +0.113077 q^{97} +(-1.06990 - 2.11229i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - 4 q^{7} - 2 q^{9} - 4 q^{11} + q^{13} - 4 q^{15} - 8 q^{17} - q^{19} + 14 q^{21} + 8 q^{23} - 14 q^{25} - 13 q^{29} + 2 q^{31} - 25 q^{33} + 3 q^{35} - q^{37} - 3 q^{39} - 8 q^{41} - 4 q^{43}+ \cdots - 29 q^{99}+O(q^{100}) \) 28 * q - 4 * q^7 - 2 * q^9 - 4 * q^11 + q^13 - 4 * q^15 - 8 * q^17 - q^19 + 14 * q^21 + 8 * q^23 - 14 * q^25 - 13 * q^29 + 2 * q^31 - 25 * q^33 + 3 * q^35 - q^37 - 3 * q^39 - 8 * q^41 - 4 * q^43 - 38 * q^45 + 11 * q^47 + 24 * q^49 + 5 * q^51 + 52 * q^53 - q^57 - 8 * q^59 + 14 * q^61 - 21 * q^63 + 38 * q^65 + 14 * q^67 - 21 * q^69 - 12 * q^71 + 14 * q^73 - 14 * q^75 - 28 * q^77 + 5 * q^79 + 10 * q^81 + 9 * q^83 + 18 * q^85 + 18 * q^87 - 11 * q^89 - q^91 - 9 * q^93 + 28 * q^95 + 50 * q^97 - 29 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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