LMFDB - Embedded newform 63.4.p.a.17.1

Newspace parameters

Level:	$ N $	$=$	$ 63 = 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	63.p (of order $6$, degree $2$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$3.71712033036$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
Defining polynomial:	$x^{16} - 48 x^{14} + 1647 x^{12} - 27620 x^{10} + 336765 x^{8} - 1200006 x^{6} + 3242464 x^{4} - 1762200 x^{2} + 810000$
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{8} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			17.1
Root			$-4.21355 + 2.43270i$ of defining polynomial
Character	$\chi$	$=$	63.17
Dual form			63.4.p.a.26.1

$q$-expansion

$f(q)$	$=$	$q+(-4.21355 + 2.43270i) q^{2} +(7.83601 - 13.5724i) q^{4} +(6.38217 + 11.0542i) q^{5} +(2.53897 + 18.3454i) q^{7} +37.3274i q^{8} +O(q^{10})$	\(q+(-4.21355 + 2.43270i) q^{2} +(7.83601 - 13.5724i) q^{4} +(6.38217 + 11.0542i) q^{5} +(2.53897 + 18.3454i) q^{7} +37.3274i q^{8} +(-53.7832 - 31.0517i) q^{10} +(-46.8633 - 27.0565i) q^{11} +8.85528i q^{13} +(-55.3268 - 71.1228i) q^{14} +(-28.1181 - 48.7020i) q^{16} +(-34.4587 + 59.6841i) q^{17} +(-141.898 + 81.9246i) q^{19} +200.043 q^{20} +263.281 q^{22} +(81.3807 - 46.9852i) q^{23} +(-18.9642 + 32.8469i) q^{25} +(-21.5422 - 37.3122i) q^{26} +(268.886 + 109.295i) q^{28} +119.620i q^{29} +(85.6311 + 49.4391i) q^{31} +(-21.6577 - 12.5041i) q^{32} -335.310i q^{34} +(-186.590 + 145.150i) q^{35} +(-47.0949 - 81.5708i) q^{37} +(398.595 - 690.387i) q^{38} +(-412.626 + 238.230i) q^{40} +259.347 q^{41} +5.01418 q^{43} +(-734.443 + 424.031i) q^{44} +(-228.601 + 395.949i) q^{46} +(28.6747 + 49.6660i) q^{47} +(-330.107 + 93.1568i) q^{49} -184.536i q^{50} +(120.187 + 69.3901i) q^{52} +(407.058 + 235.015i) q^{53} -690.718i q^{55} +(-684.786 + 94.7731i) q^{56} +(-290.999 - 504.025i) q^{58} +(-112.979 + 195.685i) q^{59} +(370.650 - 213.995i) q^{61} -481.082 q^{62} +571.564 q^{64} +(-97.8884 + 56.5159i) q^{65} +(-81.9267 + 141.901i) q^{67} +(540.037 + 935.372i) q^{68} +(433.103 - 1065.51i) q^{70} +79.8529i q^{71} +(666.447 + 384.774i) q^{73} +(396.874 + 229.135i) q^{74} +2567.85i q^{76} +(377.379 - 928.422i) q^{77} +(-267.408 - 463.165i) q^{79} +(358.909 - 621.648i) q^{80} +(-1092.77 + 630.912i) q^{82} +438.520 q^{83} -879.684 q^{85} +(-21.1275 + 12.1980i) q^{86} +(1009.95 - 1749.29i) q^{88} +(12.8242 + 22.2121i) q^{89} +(-162.454 + 22.4833i) q^{91} -1472.71i q^{92} +(-241.644 - 139.513i) q^{94} +(-1811.23 - 1045.71i) q^{95} +1381.00i q^{97} +(1164.30 - 1195.57i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16q + 32q^{4} + 56q^{7} + O(q^{10}) $	$ 16q + 32q^{4} + 56q^{7} - 72q^{10} - 188q^{16} - 612q^{19} + 528q^{22} - 20q^{25} + 1036q^{28} + 1128q^{31} - 1196q^{37} - 3204q^{40} + 328q^{43} - 1392q^{46} + 784q^{49} + 4452q^{52} - 3372q^{58} - 1632q^{61} + 5432q^{64} + 308q^{67} + 3612q^{70} + 4068q^{73} - 2176q^{79} - 10188q^{82} - 4608q^{85} + 708q^{88} + 924q^{91} - 2916q^{94} + O(q^{100}) $

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/63\mathbb{Z}\right)^\times$.

$n$	$10$	$29$
$\chi(n)$	$e\left(\frac{1}{6}\right)$	$-1$

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 $

$2$	−4.21355	+	2.43270i	−1.48972	+	0.860088i	−0.999931	−	0.0117558i	$-0.996258\pi$
$2$	−4.21355	+	2.43270i	−1.48972	+	0.860088i	−0.489785	+	0.871843i	$0.662925\pi$
$3$		0			0
$3$		0			0
$4$	7.83601	−	13.5724i	0.979502	−	1.69655i
$5$	6.38217	+	11.0542i	0.570839	+	0.988721i	0.996480	+	0.0838295i	$0.0267151\pi$
$5$	6.38217	+	11.0542i	0.570839	+	0.988721i	−0.425642	+	0.904892i	$0.639952\pi$
$6$		0			0
$7$	2.53897	+	18.3454i	0.137091	+	0.990558i
$7$	2.53897	+	18.3454i	0.137091	+	0.990558i
$8$			37.3274i			1.64965i
$9$		0			0
$10$	−53.7832	−	31.0517i	−1.70077	−	0.981942i
$11$	−46.8633	−	27.0565i	−1.28453	−	0.741623i	−0.306856	−	0.951756i	$-0.599277\pi$
$11$	−46.8633	−	27.0565i	−1.28453	−	0.741623i	−0.977673	+	0.210133i	$0.932610\pi$
$12$		0			0
$13$			8.85528i			0.188924i	0.995528	+	0.0944620i	$0.0301131\pi$
$13$			8.85528i			0.188924i	−0.995528	+	0.0944620i	$0.969887\pi$
$14$	−55.3268	−	71.1228i	−1.05619	−	1.35774i
$15$		0			0
$16$	−28.1181	−	48.7020i	−0.439345	−	0.760968i
$17$	−34.4587	+	59.6841i	−0.491615	+	0.851502i	−0.999953	−	0.00965543i	$-0.996927\pi$
$17$	−34.4587	+	59.6841i	−0.491615	+	0.851502i	0.508339	+	0.861157i	$0.330260\pi$
$18$		0			0
$19$	−141.898	+	81.9246i	−1.71334	+	0.989199i	−0.783383	+	0.621540i	$0.786508\pi$
$19$	−141.898	+	81.9246i	−1.71334	+	0.989199i	−0.929960	+	0.367660i	$0.880159\pi$
$20$	200.043			2.23655
$21$		0			0
$22$	263.281			2.55144
$23$	81.3807	−	46.9852i	0.737785	−	0.425960i	−0.0834783	−	0.996510i	$-0.526603\pi$
$23$	81.3807	−	46.9852i	0.737785	−	0.425960i	0.821263	+	0.570549i	$0.193270\pi$
$24$		0			0
$25$	−18.9642	+	32.8469i	−0.151713	+	0.262775i
$26$	−21.5422	−	37.3122i	−0.162491	−	0.281443i
$27$		0			0
$28$	268.886	+	109.295i	1.81481	+	0.737672i
$29$			119.620i			0.765961i	0.923756	+	0.382981i	$0.125102\pi$
$29$			119.620i			0.765961i	−0.923756	+	0.382981i	$0.874898\pi$
$30$		0			0
$31$	85.6311	+	49.4391i	0.496123	+	0.286437i	0.727111	−	0.686520i	$-0.240863\pi$
$31$	85.6311	+	49.4391i	0.496123	+	0.286437i	−0.230988	+	0.972957i	$0.574196\pi$
$32$	−21.6577	−	12.5041i	−0.119643	−	0.0690760i
$33$		0			0
$34$		−	335.310i		−	1.69133i
$35$	−186.590	+	145.150i	−0.901129	+	0.700994i
$36$		0			0
$37$	−47.0949	−	81.5708i	−0.209253	−	0.362437i	0.742227	−	0.670149i	$-0.233770\pi$
$37$	−47.0949	−	81.5708i	−0.209253	−	0.362437i	−0.951479	+	0.307712i	$0.900437\pi$
$38$	398.595	−	690.387i	1.70160	−	2.94725i
$39$		0			0
$40$	−412.626	+	238.230i	−1.63105	+	0.941686i
$41$	259.347			0.987883			0.493941	−	0.869495i	$-0.335556\pi$
$41$	259.347			0.987883			0.493941	+	0.869495i	$0.335556\pi$
$42$		0			0
$43$	5.01418			0.0177827			0.00889133	−	0.999960i	$-0.497170\pi$
$43$	5.01418			0.0177827			0.00889133	+	0.999960i	$0.497170\pi$
$44$	−734.443	+	424.031i	−2.51640	+	1.45284i
$45$		0			0
$46$	−228.601	+	395.949i	−0.732727	+	1.26912i
$47$	28.6747	+	49.6660i	0.0889921	+	0.154139i	0.907085	−	0.420947i	$-0.138302\pi$
$47$	28.6747	+	49.6660i	0.0889921	+	0.154139i	−0.818093	+	0.575086i	$0.804969\pi$
$48$		0			0
$49$	−330.107	+	93.1568i	−0.962412	+	0.271594i
$50$		−	184.536i		−	0.521947i
$51$		0			0
$52$	120.187	+	69.3901i	0.320518	+	0.185051i
$53$	407.058	+	235.015i	1.05497	+	0.609090i	0.924038	−	0.382301i	$-0.124868\pi$
$53$	407.058	+	235.015i	1.05497	+	0.609090i	0.130937	+	0.991391i	$0.458202\pi$
$54$		0			0
$55$		−	690.718i		−	1.69339i
$56$	−684.786	+	94.7731i	−1.63408	+	0.226153i
$57$		0			0
$58$	−290.999	−	504.025i	−0.658794	−	1.14106i
$59$	−112.979	+	195.685i	−0.249299	+	0.431798i	−0.963331	−	0.268314i	$-0.913533\pi$
$59$	−112.979	+	195.685i	−0.249299	+	0.431798i	0.714033	+	0.700112i	$0.246867\pi$
$60$		0			0
$61$	370.650	−	213.995i	0.777982	−	0.449168i	−0.0577325	−	0.998332i	$-0.518387\pi$
$61$	370.650	−	213.995i	0.777982	−	0.449168i	0.835715	+	0.549164i	$0.185054\pi$
$62$	−481.082			−0.985442
$63$		0			0
$64$	571.564			1.11634
$65$	−97.8884	+	56.5159i	−0.186793	+	0.107845i
$66$		0			0
$67$	−81.9267	+	141.901i	−0.149387	+	0.258746i	−0.931001	−	0.365016i	$-0.881063\pi$
$67$	−81.9267	+	141.901i	−0.149387	+	0.258746i	0.781614	+	0.623762i	$0.214397\pi$
$68$	540.037	+	935.372i	0.963075	+	1.66809i
$69$		0			0
$70$	433.103	−	1065.51i	0.739510	−	1.81933i
$71$			79.8529i			0.133476i	0.997771	+	0.0667380i	$0.0212592\pi$
$71$			79.8529i			0.133476i	−0.997771	+	0.0667380i	$0.978741\pi$
$72$		0			0
$73$	666.447	+	384.774i	1.06852	+	0.616909i	0.927776	−	0.373138i	$-0.121718\pi$
$73$	666.447	+	384.774i	1.06852	+	0.616909i	0.140741	+	0.990046i	$0.455051\pi$
$74$	396.874	+	229.135i	0.623454	+	0.359952i
$75$		0			0
$76$			2567.85i			3.87569i
$77$	377.379	−	928.422i	0.558523	−	1.37407i
$78$		0			0
$79$	−267.408	−	463.165i	−0.380833	−	0.659622i	0.610349	−	0.792133i	$-0.291029\pi$
$79$	−267.408	−	463.165i	−0.380833	−	0.659622i	−0.991182	+	0.132511i	$0.957696\pi$
$80$	358.909	−	621.648i	0.501590	−	0.868780i
$81$		0			0
$82$	−1092.77	+	630.912i	−1.47166	+	0.849666i
$83$	438.520			0.579926			0.289963	−	0.957038i	$-0.406357\pi$
$83$	438.520			0.579926			0.289963	+	0.957038i	$0.406357\pi$
$84$		0			0
$85$	−879.684			−1.12253
$86$	−21.1275	+	12.1980i	−0.0264911	+	0.0152947i
$87$		0			0
$88$	1009.95	−	1749.29i	1.22342	−	2.11903i
$89$	12.8242	+	22.2121i	0.0152737	+	0.0264548i	0.873561	−	0.486714i	$-0.161805\pi$
$89$	12.8242	+	22.2121i	0.0152737	+	0.0264548i	−0.858288	+	0.513169i	$0.828471\pi$
$90$		0			0
$91$	−162.454	+	22.4833i	−0.187140	+	0.0258999i
$92$		−	1472.71i		−	1.66892i
$93$		0			0
$94$	−241.644	−	139.513i	−0.265146	−	0.153082i
$95$	−1811.23	−	1045.71i	−1.95608	−	1.12935i
$96$		0			0
$97$			1381.00i			1.44555i	0.691081	+	0.722777i	$0.257135\pi$
$97$			1381.00i			1.44555i	−0.691081	+	0.722777i	$0.742865\pi$
$98$	1164.30	−	1195.57i	1.20013	−	1.23236i
$99$		0			0
$100$	297.207	+	514.777i	0.297207	+	0.514777i
$101$	−356.808	+	618.009i	−0.351522	+	0.608854i	−0.986516	−	0.163663i	$-0.947669\pi$
$101$	−356.808	+	618.009i	−0.351522	+	0.608854i	0.634994	+	0.772517i	$0.281002\pi$
$102$		0			0
$103$	1552.42	−	896.288i	1.48509	−	0.857416i	0.485232	−	0.874385i	$-0.338735\pi$
$103$	1552.42	−	896.288i	1.48509	−	0.857416i	0.999856	+	0.0169695i	$0.00540182\pi$
$104$	−330.544			−0.311659
$105$		0			0
$106$	−2286.88			−2.09548
$107$	−19.5366	+	11.2794i	−0.0176511	+	0.0101909i	−0.508800	−	0.860885i	$-0.669911\pi$
$107$	−19.5366	+	11.2794i	−0.0176511	+	0.0101909i	0.491148	+	0.871076i	$0.336577\pi$
$108$		0			0
$109$	−476.210	+	824.820i	−0.418465	+	0.724802i	−0.995785	−	0.0917154i	$-0.970765\pi$
$109$	−476.210	+	824.820i	−0.418465	+	0.724802i	0.577320	+	0.816518i	$0.304098\pi$
$110$	1680.31	+	2910.37i	1.45646	+	2.52267i
$111$		0			0
$112$	822.066	−	639.490i	0.693553	−	0.539519i
$113$		−	120.145i		−	0.100020i	−0.998749	−	0.0500102i	$-0.984075\pi$
$113$		−	120.145i		−	0.100020i	0.998749	−	0.0500102i	$-0.0159254\pi$
$114$		0			0
$115$	1038.77	+	599.735i	0.842312	+	0.486309i
$116$	1623.53	+	937.344i	1.29949	+	0.750260i
$117$		0			0
$118$		−	1099.37i		−	0.857674i
$119$	−1182.42	−	480.622i	−0.910859	−	0.370240i
$120$		0			0
$121$	798.612	+	1383.24i	0.600009	+	1.03925i
$122$	−1041.17	+	1803.36i	−0.772648	+	1.33827i
$123$		0			0
$124$	1342.01	−	774.812i	0.971906	−	0.561130i
$125$	1111.41			0.795262
$126$		0			0
$127$	884.302			0.617867			0.308934	−	0.951084i	$-0.400028\pi$
$127$	884.302			0.617867			0.308934	+	0.951084i	$0.400028\pi$
$128$	−2235.05	+	1290.41i	−1.54338	+	0.891071i
$129$		0			0
$130$	274.972	−	476.265i	0.185512	−	0.321317i
$131$	−803.439	−	1391.60i	−0.535853	−	0.928125i	−0.999122	−	0.0419070i	$-0.986657\pi$
$131$	−803.439	−	1391.60i	−0.535853	−	0.928125i	0.463268	−	0.886218i	$-0.346677\pi$
$132$		0			0
$133$	−1863.21	−	2395.16i	−1.21474	−	1.56156i
$134$		−	797.210i		−	0.513944i
$135$		0			0
$136$	−2227.85	−	1286.25i	−1.40468	−	0.810994i
$137$	−615.297	−	355.242i	−0.383711	−	0.221535i	0.295721	−	0.955274i	$-0.404440\pi$
$137$	−615.297	−	355.242i	−0.383711	−	0.221535i	−0.679431	+	0.733739i	$0.737773\pi$
$138$		0			0
$139$		−	1531.91i		−	0.934782i	−0.884051	−	0.467391i	$-0.845194\pi$
$139$		−	1531.91i		−	0.934782i	0.884051	−	0.467391i	$-0.154806\pi$
$140$	507.903	+	3669.87i	0.306612	+	2.21543i
$141$		0			0
$142$	−194.258	−	336.464i	−0.114801	−	0.198841i
$143$	239.593	−	414.987i	0.140110	−	0.242678i
$144$		0			0
$145$	−1322.31	+	763.435i	−0.757322	+	0.437240i
$146$	−3744.15			−2.12238
$147$		0			0
$148$	−1476.15			−0.819854
$149$	2079.39	−	1200.54i	1.14329	−	0.660079i	0.196046	−	0.980595i	$-0.437190\pi$
$149$	2079.39	−	1200.54i	1.14329	−	0.660079i	0.947243	+	0.320516i	$0.103856\pi$
$150$		0			0
$151$	−1233.99	+	2137.33i	−0.665035	+	1.15188i	0.314240	+	0.949343i	$0.398250\pi$
$151$	−1233.99	+	2137.33i	−0.665035	+	1.15188i	−0.979276	+	0.202532i	$0.935083\pi$
$152$	−3058.03	−	5296.67i	−1.63184	−	2.82642i
$153$		0			0
$154$	668.463	+	4830.00i	0.349781	+	2.52735i
$155$			1262.12i			0.654036i
$156$		0			0
$157$	2109.74	+	1218.06i	1.07246	+	0.619184i	0.928852	−	0.370451i	$-0.120797\pi$
$157$	2109.74	+	1218.06i	1.07246	+	0.619184i	0.143606	+	0.989635i	$0.454130\pi$
$158$	2253.48	+	1301.05i	1.13466	+	0.655099i
$159$		0			0
$160$		−	319.213i		−	0.157725i
$161$	1068.59	+	1373.67i	0.523083	+	0.672424i
$162$		0			0
$163$	−1638.50	−	2837.97i	−0.787347	−	1.36372i	−0.927587	−	0.373607i	$-0.878121\pi$
$163$	−1638.50	−	2837.97i	−0.787347	−	1.36372i	0.140240	−	0.990118i	$-0.455213\pi$
$164$	2032.25	−	3519.95i	0.967633	−	1.67599i
$165$		0			0
$166$	−1847.73	+	1066.79i	−0.863924	+	0.498787i
$167$	−365.585			−0.169400			−0.0847000	−	0.996406i	$-0.526993\pi$
$167$	−365.585			−0.169400			−0.0847000	+	0.996406i	$0.526993\pi$
$168$		0			0
$169$	2118.58			0.964308
$170$	3706.59	−	2140.00i	1.67225	−	0.965475i
$171$		0			0
$172$	39.2912	−	68.0543i	0.0174182	−	0.0301691i
$173$	1046.47	+	1812.53i	0.459892	+	0.796557i	0.998955	−	0.0457089i	$-0.0145547\pi$
$173$	1046.47	+	1812.53i	0.459892	+	0.796557i	−0.539062	+	0.842266i	$0.681221\pi$
$174$		0			0
$175$	−650.739	−	264.508i	−0.281093	−	0.114257i
$176$			3043.11i			1.30331i
$177$		0			0
$178$	−108.071	−	62.3946i	−0.0455070	−	0.0262735i
$179$	−1524.01	−	879.890i	−0.636370	−	0.367408i	0.146845	−	0.989160i	$-0.453088\pi$
$179$	−1524.01	−	879.890i	−0.636370	−	0.367408i	−0.783215	+	0.621751i	$0.786422\pi$
$180$		0			0
$181$		−	3197.54i		−	1.31310i	−0.754282	−	0.656551i	$-0.772015\pi$
$181$		−	3197.54i		−	1.31310i	0.754282	−	0.656551i	$-0.227985\pi$
$182$	629.812	−	489.934i	0.256510	−	0.199540i
$183$		0			0
$184$	1753.84	+	3037.73i	0.702687	+	1.21709i
$185$	601.135	−	1041.20i	0.238899	−	0.413786i
$186$		0			0
$187$	3229.69	−	1864.66i	1.26299	−	0.729186i
$188$	898.780			0.348672
$189$		0			0
$190$	10175.6			3.88535
$191$	−475.772	+	274.687i	−0.180239	+	0.104061i	−0.587405	−	0.809293i	$-0.699850\pi$
$191$	−475.772	+	274.687i	−0.180239	+	0.104061i	0.407166	+	0.913354i	$0.366517\pi$
$192$		0			0
$193$	352.238	−	610.094i	0.131371	−	0.227542i	−0.792834	−	0.609437i	$-0.791395\pi$
$193$	352.238	−	610.094i	0.131371	−	0.227542i	0.924205	+	0.381896i	$0.124729\pi$
$194$	−3359.54	−	5818.90i	−1.24330	−	2.15347i
$195$		0			0
$196$	−1322.37	+	5210.32i	−0.481912	+	1.89880i
$197$			5317.81i			1.92324i	0.274384	+	0.961620i	$0.411526\pi$
$197$			5317.81i			1.92324i	−0.274384	+	0.961620i	$0.588474\pi$
$198$		0			0
$199$	−2155.80	−	1244.65i	−0.767942	−	0.443371i	0.0641982	−	0.997937i	$-0.479551\pi$
$199$	−2155.80	−	1244.65i	−0.767942	−	0.443371i	−0.832140	+	0.554566i	$0.812884\pi$
$200$	−1226.09	−	707.883i	−0.433488	−	0.250274i
$201$		0			0
$202$		−	3472.02i		−	1.20936i
$203$	−2194.48	+	303.711i	−0.758729	+	0.105007i
$204$		0			0
$205$	1655.20	+	2866.88i	0.563922	+	0.976741i
$206$	−4360.79	+	7553.11i	−1.47491	+	2.55461i
$207$		0			0
$208$	431.269	−	248.993i	0.143765	−	0.0830029i
$209$	8866.38			2.93445
$210$		0			0
$211$	−3454.31			−1.12704			−0.563519	−	0.826103i	$-0.690553\pi$
$211$	−3454.31			−1.12704			−0.563519	+	0.826103i	$0.690553\pi$
$212$	6379.42	−	3683.16i	2.06670	−	1.19321i
$213$		0			0
$214$	54.8789	−	95.0530i	0.0175301	−	0.0303630i
$215$	32.0013	+	55.4279i	0.0101510	+	0.0175821i
$216$		0			0
$217$	−689.566	+	1696.46i	−0.215718	+	0.530706i
$218$		−	4633.90i		−	1.43967i
$219$		0			0
$220$	−9374.68	−	5412.47i	−2.87291	−	1.65868i
$221$	−528.520	−	305.141i	−0.160869	−	0.0928778i
$222$		0			0
$223$			3896.38i			1.17005i	0.811016	+	0.585024i	$0.198915\pi$
$223$			3896.38i			1.17005i	−0.811016	+	0.585024i	$0.801085\pi$
$224$	174.404	−	429.067i	0.0520218	−	0.127983i
$225$		0			0
$226$	292.277	+	506.238i	0.0860263	+	0.149002i
$227$	−302.747	+	524.374i	−0.0885201	+	0.153321i	−0.906886	−	0.421377i	$-0.861547\pi$
$227$	−302.747	+	524.374i	−0.0885201	+	0.153321i	0.818366	+	0.574698i	$0.194880\pi$
$228$		0			0
$229$	−1912.98	+	1104.46i	−0.552023	+	0.318711i	−0.749938	−	0.661509i	$-0.769917\pi$
$229$	−1912.98	+	1104.46i	−0.552023	+	0.318711i	0.197914	+	0.980219i	$0.436583\pi$
$230$	−5835.89			−1.67307
$231$		0			0
$232$	−4465.10			−1.26357
$233$	−2065.77	+	1192.67i	−0.580829	+	0.335342i	−0.761463	−	0.648209i	$-0.775518\pi$
$233$	−2065.77	+	1192.67i	−0.580829	+	0.335342i	0.180634	+	0.983550i	$0.442185\pi$
$234$		0			0
$235$	−366.013	+	633.953i	−0.101600	+	0.175977i
$236$	1770.61	+	3066.79i	0.488377	+	0.845893i
$237$		0			0
$238$	6151.39	−	851.341i	1.67536	−	0.231866i
$239$			3017.95i			0.816798i	0.912803	+	0.408399i	$0.133913\pi$
$239$			3017.95i			0.816798i	−0.912803	+	0.408399i	$0.866087\pi$
$240$		0			0
$241$	−2178.48	−	1257.75i	−0.582275	−	0.336176i	0.179762	−	0.983710i	$-0.442467\pi$
$241$	−2178.48	−	1257.75i	−0.582275	−	0.336176i	−0.762037	+	0.647534i	$0.775800\pi$
$242$	−6729.99	−	3885.56i	−1.78769	−	1.03212i
$243$		0			0
$244$		−	6707.47i		−	1.75984i
$245$	−3136.58	−	3054.54i	−0.817913	−	0.796521i
$246$		0			0
$247$	−725.465	−	1256.54i	−0.186883	−	0.323692i
$248$	−1845.44	+	3196.39i	−0.472521	+	0.818431i
$249$		0			0
$250$	−4682.99	+	2703.73i	−1.18471	+	0.683995i
$251$	−1306.11			−0.328451			−0.164226	−	0.986423i	$-0.552512\pi$
$251$	−1306.11			−0.328451			−0.164226	+	0.986423i	$0.552512\pi$
$252$		0			0
$253$	−5085.03			−1.26361
$254$	−3726.05	+	2151.24i	−0.920447	+	0.531420i
$255$		0			0
$256$	3992.09	−	6914.50i	0.974630	−	1.68811i
$257$	−3735.91	−	6470.79i	−0.906770	−	1.57057i	−0.818524	−	0.574473i	$-0.805207\pi$
$257$	−3735.91	−	6470.79i	−0.906770	−	1.57057i	−0.0882460	−	0.996099i	$-0.528126\pi$
$258$		0			0
$259$	1376.88	−	1071.08i	0.330328	−	0.256964i
$260$			1771.44i			0.422538i
$261$		0			0
$262$	6770.66	+	3909.04i	1.59654	+	0.921762i
$263$	1330.77	+	768.318i	0.312010	+	0.180139i	0.647825	−	0.761789i	$-0.275679\pi$
$263$	1330.77	+	768.318i	0.312010	+	0.180139i	−0.335816	+	0.941928i	$0.609012\pi$
$264$		0			0
$265$			5999.62i			1.39077i
$266$	13677.4	+	5559.51i	3.15270	+	1.28149i
$267$		0			0
$268$	1283.96	+	2223.88i	0.292650	+	0.506884i
$269$	−1958.09	+	3391.52i	−0.443818	+	0.768715i	−0.997969	−	0.0637010i	$-0.979710\pi$
$269$	−1958.09	+	3391.52i	−0.443818	+	0.768715i	0.554151	+	0.832416i	$0.313043\pi$
$270$		0			0
$271$	3117.42	−	1799.84i	0.698780	−	0.403441i	−0.108113	−	0.994139i	$-0.534481\pi$
$271$	3117.42	−	1799.84i	0.698780	−	0.403441i	0.806893	+	0.590698i	$0.201147\pi$
$272$	3875.65			0.863955
$273$		0			0
$274$	3456.78			0.762159
$275$	1777.45	−	1026.21i	0.389760	−	0.225028i
$276$		0			0
$277$	−142.040	+	246.021i	−0.0308100	+	0.0533645i	−0.881019	−	0.473080i	$-0.843142\pi$
$277$	−142.040	+	246.021i	−0.0308100	+	0.0533645i	0.850209	+	0.526445i	$0.176475\pi$
$278$	3726.67	+	6454.77i	0.803995	+	1.39256i
$279$		0			0
$280$	−5418.07	−	6964.93i	−1.15640	−	1.48655i
$281$		−	321.256i		−	0.0682011i	−0.999418	−	0.0341006i	$-0.989143\pi$
$281$		−	321.256i		−	0.0682011i	0.999418	−	0.0341006i	$-0.0108566\pi$
$282$		0			0
$283$	5891.40	+	3401.40i	1.23748	+	0.714460i	0.968579	−	0.248707i	$-0.0800057\pi$
$283$	5891.40	+	3401.40i	1.23748	+	0.714460i	0.268903	+	0.963167i	$0.413339\pi$
$284$	1083.79	+	625.728i	0.226448	+	0.130740i
$285$		0			0
$286$			2331.43i			0.482029i
$287$	658.474	+	4757.82i	0.135430	+	0.978556i
$288$		0			0
$289$	81.7017	+	141.511i	0.0166297	+	0.0288035i
$290$	3714.41	−	6433.55i	0.752130	−	1.30273i
$291$		0			0
$292$	10444.6	−	6030.18i	2.09323	−	1.20853i
$293$	−3180.05			−0.634063			−0.317031	−	0.948415i	$-0.602686\pi$
$293$	−3180.05			−0.634063			−0.317031	+	0.948415i	$0.602686\pi$
$294$		0			0
$295$	−2884.20			−0.569237
$296$	3044.83	−	1757.93i	0.597895	−	0.345195i
$297$		0			0
$298$	−5841.07	+	10117.0i	−1.13545	+	1.96666i
$299$	416.067	+	720.649i	0.0804741	+	0.139385i
$300$		0			0
$301$	12.7308	+	91.9871i	0.00243785	+	0.0176148i
$302$		−	12007.6i		−	2.28795i
$303$		0			0
$304$	7979.78	+	4607.13i	1.50550	+	0.869200i
$305$	4731.11	+	2731.51i	0.888204	+	0.512805i
$306$		0			0
$307$			2976.39i			0.553328i	0.960967	+	0.276664i	$0.0892289\pi$
$307$			2976.39i			0.553328i	−0.960967	+	0.276664i	$0.910771\pi$
$308$	−9643.74	−	12397.0i	−1.78410	−	2.29346i
$309$		0			0
$310$	−3070.34	−	5317.99i	−0.562528	−	0.974328i
$311$	2340.89	−	4054.55i	0.426817	−	0.739268i	−0.569772	−	0.821803i	$-0.692968\pi$
$311$	2340.89	−	4054.55i	0.426817	−	0.739268i	0.996588	+	0.0825352i	$0.0263017\pi$
$312$		0			0
$313$	−850.477	+	491.023i	−0.153584	+	0.0886718i	−0.574823	−	0.818278i	$-0.694929\pi$
$313$	−850.477	+	491.023i	−0.153584	+	0.0886718i	0.421238	+	0.906950i	$0.361596\pi$
$314$	−11852.7			−2.13021
$315$		0			0
$316$	−8381.66			−1.49211
$317$	−4269.80	+	2465.17i	−0.756516	+	0.436775i	−0.828044	−	0.560664i	$-0.810546\pi$
$317$	−4269.80	+	2465.17i	−0.756516	+	0.436775i	0.0715272	+	0.997439i	$0.477213\pi$
$318$		0			0
$319$	3236.50	−	5605.79i	0.568055	−	0.983899i
$320$	3647.82	+	6318.21i	0.637248	+	1.10375i
$321$		0			0
$322$	−7844.26	−	3188.48i	−1.35759	−	0.551823i
$323$		−	11292.0i		−	1.94522i
$324$		0			0
$325$	−290.868	−	167.933i	−0.0496445	−	0.0286623i
$326$	13807.8	+	7971.96i	2.34585	+	1.35437i
$327$		0			0
$328$			9680.75i			1.62966i
$329$	−838.338	+	652.148i	−0.140483	+	0.109283i
$330$		0			0
$331$	2017.25	+	3493.99i	0.334980	+	0.580202i	0.983481	−	0.181012i	$-0.0579372\pi$
$331$	2017.25	+	3493.99i	0.334980	+	0.580202i	−0.648501	+	0.761214i	$0.724604\pi$
$332$	3436.25	−	5951.76i	0.568038	−	0.983871i
$333$		0			0
$334$	1540.41	−	889.356i	0.252358	−	0.145699i
$335$	−2091.48			−0.341104
$336$		0			0
$337$	2771.62			0.448011			0.224006	−	0.974588i	$-0.428087\pi$
$337$	2771.62			0.448011			0.224006	+	0.974588i	$0.428087\pi$
$338$	−8926.76	+	5153.87i	−1.43654	+	0.829389i
$339$		0			0
$340$	−6893.21	+	11939.4i	−1.09952	+	1.90443i
$341$	−2675.30	−	4633.76i	−0.424856	−	0.735872i
$342$		0			0
$343$	−2547.13	−	5819.43i	−0.400968	−	0.916092i
$344$			187.166i			0.0293352i
$345$		0			0
$346$	−8818.68	−	5091.47i	−1.37022	−	0.791095i
$347$	3743.15	+	2161.11i	0.579085	+	0.334335i	0.760770	−	0.649022i	$-0.224822\pi$
$347$	3743.15	+	2161.11i	0.579085	+	0.334335i	−0.181685	+	0.983357i	$0.558155\pi$
$348$		0			0
$349$		−	1331.65i		−	0.204245i	−0.994772	−	0.102122i	$-0.967437\pi$
$349$		−	1331.65i		−	0.204245i	0.994772	−	0.102122i	$-0.0325634\pi$
$350$	3385.39	−	468.531i	0.517019	−	0.0715545i
$351$		0			0
$352$	676.635	+	1171.97i	0.102457	+	0.177460i
$353$	5674.26	−	9828.10i	0.855553	−	1.48186i	−0.0205782	−	0.999788i	$-0.506551\pi$
$353$	5674.26	−	9828.10i	0.855553	−	1.48186i	0.876131	−	0.482073i	$-0.160116\pi$
$354$		0			0
$355$	−882.713	+	509.634i	−0.131970	+	0.0761932i
$356$	401.962			0.0598425
$357$		0			0
$358$	8562.02			1.26401
$359$	7247.49	−	4184.34i	1.06548	−	0.615156i	0.138538	−	0.990357i	$-0.455760\pi$
$359$	7247.49	−	4184.34i	1.06548	−	0.615156i	0.926943	+	0.375201i	$0.122426\pi$
$360$		0			0
$361$	9993.77	−	17309.7i	1.45703	−	2.52365i
$362$	7778.64	+	13473.0i	1.12938	+	1.95615i
$363$		0			0
$364$	−967.837	+	2381.06i	−0.139364	+	0.342861i
$365$			9822.76i			1.40862i
$366$		0			0
$367$	2351.31	+	1357.53i	0.334434	+	0.193086i	0.657808	−	0.753186i	$-0.271484\pi$
$367$	2351.31	+	1357.53i	0.334434	+	0.193086i	−0.323374	+	0.946271i	$0.604817\pi$
$368$	−4576.54	−	2642.27i	−0.648285	−	0.374287i
$369$		0			0
$370$			5849.52i			0.821897i
$371$	−3277.93	+	8064.33i	−0.458711	+	1.12851i
$372$		0			0
$373$	−3048.56	−	5280.25i	−0.423186	−	0.732979i	0.573063	−	0.819511i	$-0.305755\pi$
$373$	−3048.56	−	5280.25i	−0.423186	−	0.732979i	−0.996249	+	0.0865320i	$0.972422\pi$
$374$	−9072.32	+	15713.7i	−1.25433	+	2.17256i
$375$		0			0
$376$	−1853.90	+	1070.35i	−0.254276	+	0.146806i
$377$	−1059.27			−0.144708
$378$		0			0
$379$	−9922.24			−1.34478			−0.672389	−	0.740198i	$-0.734732\pi$
$379$	−9922.24			−1.34478			−0.672389	+	0.740198i	$0.734732\pi$
$380$	−28385.6	+	16388.4i	−3.83198	+	2.21239i
$381$		0			0
$382$	1336.46	−	2314.82i	0.179003	−	0.310043i
$383$	609.532	+	1055.74i	0.0813202	+	0.140851i	0.903817	−	0.427919i	$-0.140753\pi$
$383$	609.532	+	1055.74i	0.0813202	+	0.140851i	−0.822497	+	0.568769i	$0.807420\pi$
$384$		0			0
$385$	12671.5	−	1753.71i	1.67740	−	0.232149i
$386$			3427.55i			0.451963i
$387$		0			0
$388$	18743.4	+	10821.5i	2.45245	+	1.41592i
$389$	−11374.6	−	6567.15i	−1.48256	−	0.855958i	−0.482759	−	0.875753i	$-0.660365\pi$
$389$	−11374.6	−	6567.15i	−1.48256	−	0.855958i	−0.999804	+	0.0197949i	$0.993699\pi$
$390$		0			0
$391$			6476.19i			0.837634i
$392$	−3477.30	−	12322.0i	−0.448036	−	1.58765i
$393$		0			0
$394$	−12936.6	−	22406.9i	−1.65416	−	2.86508i
$395$	3413.29	−	5911.99i	0.434788	−	0.753075i
$396$		0			0
$397$	2697.68	−	1557.51i	0.341040	−	0.196900i	−0.319692	−	0.947522i	$-0.603579\pi$
$397$	2697.68	−	1557.51i	0.341040	−	0.196900i	0.660732	+	0.750622i	$0.270246\pi$
$398$	12111.4			1.52535
$399$		0			0
$400$	2132.94			0.266618
$401$	9724.47	−	5614.42i	1.21101	−	0.699179i	0.248034	−	0.968751i	$-0.420216\pi$
$401$	9724.47	−	5614.42i	1.21101	−	0.699179i	0.962980	+	0.269572i	$0.0868822\pi$
$402$		0			0
$403$	−437.797	+	758.287i	−0.0541147	+	0.0937295i
$404$	5591.90	+	9685.46i	0.688633	+	1.19275i
$405$		0			0
$406$	8507.70	−	6618.20i	1.03998	−	0.809004i
$407$			5096.90i			0.620747i
$408$		0			0
$409$	−10739.4	−	6200.41i	−1.29836	−	0.749610i	−0.318242	−	0.948010i	$-0.603092\pi$
$409$	−10739.4	−	6200.41i	−1.29836	−	0.749610i	−0.980121	+	0.198399i	$0.936426\pi$
$410$	−13948.5	−	8053.18i	−1.68017	−	0.970044i
$411$		0			0
$412$		−	28093.3i		−	3.35936i
$413$	−3876.78	−	1575.81i	−0.461898	−	0.187749i
$414$		0			0
$415$	2798.71	+	4847.51i	0.331044	+	0.573385i
$416$	110.727	−	191.785i	0.0130501	−	0.0226035i
$417$		0			0
$418$	−37359.0	+	21569.2i	−4.37150	+	2.52389i
$419$	8260.19			0.963095			0.481547	−	0.876420i	$-0.340075\pi$
$419$	8260.19			0.963095			0.481547	+	0.876420i	$0.340075\pi$
$420$		0			0
$421$	5571.81			0.645020			0.322510	−	0.946566i	$-0.395473\pi$
$421$	5571.81			0.645020			0.322510	+	0.946566i	$0.395473\pi$
$422$	14554.9	−	8403.30i	1.67896	−	0.969351i
$423$		0			0
$424$	−8772.49	+	15194.4i	−1.00479	+	1.74034i
$425$	−1306.96	−	2263.72i	−0.149169	−	0.258368i
$426$		0			0
$427$	4866.89	+	6256.40i	0.551582	+	0.709060i
$428$			353.543i			0.0399279i
$429$		0			0
$430$	−269.678	−	155.699i	−0.0302443	−	0.0174616i
$431$	6094.88	+	3518.88i	0.681160	+	0.393268i	0.800292	−	0.599611i	$-0.204678\pi$
$431$	6094.88	+	3518.88i	0.681160	+	0.393268i	−0.119132	+	0.992878i	$0.538011\pi$
$432$		0			0
$433$			9212.26i			1.02243i	0.859452	+	0.511216i	$0.170805\pi$
$433$			9212.26i			1.02243i	−0.859452	+	0.511216i	$0.829195\pi$
$434$	−1221.45	−	8825.63i	−0.135096	−	0.976138i
$435$		0			0
$436$	7463.18	+	12926.6i	0.819774	+	1.41989i
$437$	−7698.48	+	13334.2i	−0.842719	+	1.45963i
$438$		0			0
$439$	−7345.30	+	4240.81i	−0.798570	+	0.461054i	−0.842971	−	0.537959i	$-0.819195\pi$
$439$	−7345.30	+	4240.81i	−0.798570	+	0.461054i	0.0444012	+	0.999014i	$0.485862\pi$
$440$	25782.7			2.79350
$441$		0			0
$442$	2969.26			0.319532
$443$	−11305.9	+	6527.49i	−1.21255	+	0.700068i	−0.963315	−	0.268374i	$-0.913514\pi$
$443$	−11305.9	+	6527.49i	−1.21255	+	0.700068i	−0.249239	+	0.968442i	$0.580180\pi$
$444$		0			0
$445$	−163.692	+	283.523i	−0.0174376	+	0.0302029i
$446$	−9478.70	−	16417.6i	−1.00634	−	1.74304i
$447$		0			0
$448$	1451.18	+	10485.6i	0.153040	+	1.10580i
$449$		−	14114.0i		−	1.48348i	−0.670689	−	0.741738i	$-0.734002\pi$
$449$		−	14114.0i		−	1.48348i	0.670689	−	0.741738i	$-0.265998\pi$
$450$		0			0
$451$	−12153.9	−	7017.03i	−1.26896	−	0.732637i
$452$	−1630.66	−	941.459i	−0.169689	−	0.0979702i
$453$		0			0
$454$		−	2945.97i		−	0.304540i
$455$	−1285.34	−	1652.31i	−0.132435	−	0.170245i
$456$		0			0
$457$	4486.87	+	7771.49i	0.459271	+	0.795481i	0.998923	−	0.0464073i	$-0.0147772\pi$
$457$	4486.87	+	7771.49i	0.459271	+	0.795481i	−0.539651	+	0.841889i	$0.681444\pi$
$458$	5373.63	−	9307.40i	0.548238	−	0.949577i
$459$		0			0
$460$	16279.7	−	9399.06i	1.65009	−	0.952682i
$461$	−955.010			−0.0964842			−0.0482421	−	0.998836i	$-0.515362\pi$
$461$	−955.010			−0.0964842			−0.0482421	+	0.998836i	$0.515362\pi$
$462$		0			0
$463$	12004.5			1.20496			0.602479	−	0.798135i	$-0.294180\pi$
$463$	12004.5			1.20496			0.602479	+	0.798135i	$0.294180\pi$
$464$	5825.73	−	3363.49i	0.582872	−	0.336521i
$465$		0			0
$466$	5802.82	−	10050.8i	0.576846	−	0.999127i
$467$	−2532.46	−	4386.34i	−0.250938	−	0.434638i	0.712846	−	0.701320i	$-0.247406\pi$
$467$	−2532.46	−	4386.34i	−0.250938	−	0.434638i	−0.963784	+	0.266683i	$0.914072\pi$
$468$		0			0
$469$	−2811.24	−	1142.69i	−0.276783	−	0.112505i
$470$		−	3561.59i		−	0.349540i
$471$		0			0
$472$	−7304.43	−	4217.21i	−0.712317	−	0.411256i
$473$	−234.981	−	135.666i	−0.0228423	−	0.0131880i
$474$		0			0
$475$		−	6214.52i		−	0.600299i
$476$	−15788.6	+	12282.1i	−1.52032	+	1.18266i
$477$		0			0
$478$	−7341.75	−	12716.3i	−0.702518	−	1.21680i
$479$	7606.85	−	13175.5i	0.725607	−	1.25679i	−0.233116	−	0.972449i	$-0.574892\pi$
$479$	7606.85	−	13175.5i	0.725607	−	1.25679i	0.958724	−	0.284340i	$-0.0917744\pi$
$480$		0			0
$481$	722.332	−	417.038i	0.0684730	−	0.0395329i
$482$	12238.8			1.15656
$483$		0			0
$484$	25031.7			2.35084
$485$	−15265.9	+	8813.75i	−1.42925	+	0.825179i
$486$		0			0
$487$	−7905.92	+	13693.5i	−0.735629	+	1.27415i	0.218817	+	0.975766i	$0.429780\pi$
$487$	−7905.92	+	13693.5i	−0.735629	+	1.27415i	−0.954447	+	0.298382i	$0.903553\pi$
$488$	7987.88	+	13835.4i	0.740972	+	1.28340i
$489$		0			0
$490$	20646.9	+	5240.14i	1.90354	+	0.483113i
$491$			18064.2i			1.66034i	0.557512	+	0.830169i	$0.311756\pi$
$491$			18064.2i			1.66034i	−0.557512	+	0.830169i	$0.688244\pi$
$492$		0			0
$493$	−7139.42	−	4121.94i	−0.652217	−	0.376558i
$494$	6113.57	+	3529.67i	0.556806	+	0.321472i
$495$		0			0
$496$		−	5560.54i		−	0.503378i
$497$	−1464.93	+	202.744i	−0.132216	+	0.0182984i
$498$		0			0
$499$	5262.33	+	9114.62i	0.472092	+	0.817688i	0.999490	−	0.0319305i	$-0.0101655\pi$
$499$	5262.33	+	9114.62i	0.472092	+	0.817688i	−0.527398	+	0.849619i	$0.676832\pi$
$500$	8709.04	−	15084.5i	0.778960	−	1.34920i
$501$		0			0
$502$	5503.38	−	3177.38i	0.489299	−	0.282497i
$503$	−7790.82			−0.690607			−0.345304	−	0.938491i	$-0.612224\pi$
$503$	−7790.82			−0.690607			−0.345304	+	0.938491i	$0.612224\pi$
$504$		0			0
$505$	−9108.83			−0.802649
$506$	21426.0	−	12370.3i	1.88242	−	1.08681i
$507$		0			0
$508$	6929.41	−	12002.1i	0.605202	−	1.04824i
$509$	5098.24	+	8830.41i	0.443960	+	0.768961i	0.997979	−	0.0635430i	$-0.0202400\pi$
$509$	5098.24	+	8830.41i	0.443960	+	0.768961i	−0.554019	+	0.832504i	$0.686907\pi$
$510$		0			0
$511$	−5366.74	+	13203.2i	−0.464600	+	1.14300i
$512$			18199.6i			1.57093i
$513$		0			0
$514$	31482.9	+	18176.7i	2.70166	+	1.55980i
$515$	19815.6	+	11440.5i	1.69549	+	0.978892i
$516$		0			0
$517$		−	3103.35i		−	0.263994i
$518$	−3195.93	+	7862.57i	−0.271083	+	0.666914i
$519$		0			0
$520$	−2109.59	−	3653.92i	−0.177907	−	0.308144i
$521$	963.789	−	1669.33i	0.0810449	−	0.140374i	−0.822654	−	0.568542i	$-0.807508\pi$
$521$	963.789	−	1669.33i	0.0810449	−	0.140374i	0.903699	+	0.428168i	$0.140841\pi$
$522$		0			0
$523$	−6716.70	+	3877.89i	−0.561569	+	0.324222i	−0.753775	−	0.657133i	$-0.771769\pi$
$523$	−6716.70	+	3877.89i	−0.561569	+	0.324222i	0.192206	+	0.981355i	$0.438436\pi$
$524$	−25183.0			−2.09948
$525$		0			0
$526$	−7476.33			−0.619741
$527$	−5901.47	+	3407.21i	−0.487803	+	0.281633i
$528$		0			0
$529$	−1668.28	+	2889.55i	−0.137115	+	0.237491i
$530$	−14595.2	−	25279.7i	−1.19618	−	2.07185i
$531$		0			0
$532$	−47108.2	+	6519.68i	−3.83910	+	0.531324i
$533$			2296.59i			0.186635i
$534$		0			0
$535$	−249.371	−	143.975i	−0.0201519	−	0.0116347i
$536$	−5296.80	−	3058.11i	−0.426841	−	0.246437i
$537$		0			0
$538$		−	19053.8i		−	1.52689i
$539$	17990.4	+	4565.93i	1.43767	+	0.364876i
$540$		0			0
$541$	−8380.42	−	14515.3i	−0.665993	−	1.15353i	−0.979015	−	0.203788i	$-0.934675\pi$
$541$	−8380.42	−	14515.3i	−0.665993	−	1.15353i	0.313022	−	0.949746i	$-0.398659\pi$
$542$	−8756.93	+	15167.4i	−0.693989	+	1.20202i
$543$		0			0
$544$	1492.59	−	861.748i	0.117637	−	0.0679176i
$545$	−12157.0			−0.955503
$546$		0			0
$547$	5869.79			0.458819			0.229410	−	0.973330i	$-0.426320\pi$
$547$	5869.79			0.458819			0.229410	+	0.973330i	$0.426320\pi$
$548$	−9642.95	+	5567.36i	−0.751690	+	0.433989i
$549$		0			0
$550$	−4992.91	+	8647.97i	−0.387088	+	0.670456i
$551$	−9799.82	−	16973.8i	−0.757688	−	1.31235i
$552$		0			0
$553$	7818.00	−	6081.67i	0.601185	−	0.467666i
$554$		−	1382.16i		−	0.105997i
$555$		0			0
$556$	−20791.6	−	12004.1i	−1.58590	−	0.915621i
$557$	−18756.5	−	10829.1i	−1.42682	−	0.823774i	−0.429951	−	0.902852i	$-0.641469\pi$
$557$	−18756.5	−	10829.1i	−1.42682	−	0.823774i	−0.996868	+	0.0790779i	$0.974802\pi$
$558$		0			0
$559$			44.4019i			0.00335957i
$560$	12315.6	+	5005.98i	0.929341	+	0.377752i
$561$		0			0
$562$	781.517	+	1353.63i	0.0586589	+	0.101600i
$563$	4795.36	−	8305.80i	0.358970	−	0.621755i	−0.628819	−	0.777552i	$-0.716461\pi$
$563$	4795.36	−	8305.80i	0.358970	−	0.621755i	0.987789	+	0.155797i	$0.0497946\pi$
$564$		0			0
$565$	1328.11	−	766.787i	0.0988923	−	0.0570955i
$566$	−33098.3			−2.45799
$567$		0			0
$568$	−2980.70			−0.220189
$569$	14405.5	−	8317.01i	1.06135	−	0.612772i	0.135546	−	0.990771i	$-0.456721\pi$
$569$	14405.5	−	8317.01i	1.06135	−	0.612772i	0.925806	+	0.377999i	$0.123388\pi$
$570$		0			0
$571$	−3165.51	+	5482.83i	−0.232001	+	0.401838i	−0.958397	−	0.285439i	$-0.907861\pi$
$571$	−3165.51	+	5482.83i	−0.232001	+	0.401838i	0.726396	+	0.687277i	$0.241194\pi$
$572$	−3754.91	−	6503.69i	−0.274477	−	0.475408i
$573$		0			0
$574$	−14348.8	−	18445.5i	−1.04340	−	1.34129i
$575$			3564.14i			0.258495i
$576$		0			0
$577$	8431.94	+	4868.18i	0.608364	+	0.351239i	0.772325	−	0.635228i	$-0.219094\pi$
$577$	8431.94	+	4868.18i	0.608364	+	0.351239i	−0.163961	+	0.986467i	$0.552427\pi$
$578$	−688.509	−	397.511i	−0.0495470	−	0.0286060i
$579$		0			0
$580$			23929.1i			1.71311i
$581$	1113.39	+	8044.83i	0.0795028	+	0.574450i
$582$		0			0
$583$	−12717.4	−	22027.1i	−0.903430	−	1.56479i
$584$	−14362.6	+	24876.8i	−1.01769	+	1.76268i
$585$		0			0
$586$	13399.3	−	7736.09i	0.944573	−	0.545350i
$587$	10940.3			0.769255			0.384628	−	0.923072i	$-0.374330\pi$
$587$	10940.3			0.769255			0.384628	+	0.923072i	$0.374330\pi$
$588$		0			0
$589$	−16201.1			−1.13337
$590$	12152.7	−	7016.39i	0.848001	−	0.489594i
$591$		0			0
$592$	−2648.44	+	4587.23i	−0.183869	+	0.318470i
$593$	12430.4	+	21530.0i	0.860799	+	1.49095i	0.871159	+	0.491000i	$0.163369\pi$
$593$	12430.4	+	21530.0i	0.860799	+	1.49095i	−0.0103608	+	0.999946i	$0.503298\pi$
$594$		0			0
$595$	−2233.49	−	16138.2i	−0.153889	−	1.11193i
$596$		−	37629.6i		−	2.58619i
$597$		0			0
$598$	−3506.24	−	2024.33i	−0.239767	−	0.138430i
$599$	20207.8	+	11667.0i	1.37841	+	0.795825i	0.991968	−	0.126488i	$-0.0403706\pi$
$599$	20207.8	+	11667.0i	1.37841	+	0.795825i	0.386442	+	0.922314i	$0.373704\pi$
$600$		0			0
$601$			13012.4i			0.883175i	0.897218	+	0.441587i	$0.145584\pi$
$601$			13012.4i			0.883175i	−0.897218	+	0.441587i	$0.854416\pi$
$602$	−277.419	−	356.622i	−0.0187820	−	0.0241442i
$603$		0			0
$604$	19339.1	+	33496.2i	1.30281	+	2.25653i
$605$	−10193.8	+	17656.1i	−0.685017	+	1.18648i
$606$		0			0
$607$	7355.69	−	4246.81i	0.491859	−	0.283975i	−0.233486	−	0.972360i	$-0.575013\pi$
$607$	7355.69	−	4246.81i	0.491859	−	0.283975i	0.725345	+	0.688385i	$0.241680\pi$
$608$	4097.57			0.273320
$609$		0			0
$610$	−26579.7			−1.76423
$611$	−439.806	+	253.922i	−0.0291205	+	0.0168127i
$612$		0			0
$613$	4569.79	−	7915.11i	0.301097	−	0.521514i	−0.675288	−	0.737554i	$-0.735981\pi$
$613$	4569.79	−	7915.11i	0.301097	−	0.521514i	0.976385	+	0.216040i	$0.0693140\pi$
$614$	−7240.66	−	12541.2i	−0.475911	−	0.824302i
$615$		0			0
$616$	34655.6	+	14086.6i	2.26674	+	0.921370i
$617$		−	7360.91i		−	0.480290i	−0.970737	−	0.240145i	$-0.922805\pi$
$617$		−	7360.91i		−	0.480290i	0.970737	−	0.240145i	$-0.0771950\pi$
$618$		0			0
$619$	19878.5	+	11476.9i	1.29077	+	0.745225i	0.978790	−	0.204864i	$-0.0656753\pi$
$619$	19878.5	+	11476.9i	1.29077	+	0.745225i	0.311978	+	0.950089i	$0.399009\pi$
$620$	17129.9	+	9889.96i	1.10960	+	0.640629i
$621$		0			0
$622$			22778.7i			1.46840i
$623$	−374.930	+	291.660i	−0.0241112	+	0.0187562i
$624$		0			0
$625$	9463.74	+	16391.7i	0.605679	+	1.04907i
$626$	2389.02	−	4137.91i	0.152531	−	0.264192i
$627$		0			0
$628$	33064.0	−	19089.5i	2.10095	−	1.21298i
$629$	6491.31			0.411487
$630$		0			0
$631$	21126.0			1.33282			0.666412	−	0.745584i	$-0.267829\pi$
$631$	21126.0			1.33282			0.666412	+	0.745584i	$0.267829\pi$
$632$	17288.7	−	9981.66i	1.08815	−	0.628242i
$633$		0			0
$634$	11994.0	−	20774.2i	0.751330	−	1.30134i
$635$	5643.77	+	9775.29i	0.352703	+	0.610899i
$636$		0			0
$637$	−824.929	−	2923.19i	−0.0513106	−	0.181823i
$638$			31493.7i			1.95431i
$639$		0			0
$640$	−28529.0	−	16471.2i	−1.76204	−	1.01732i
$641$	−15447.5	−	8918.61i	−0.951855	−	0.549554i	−0.0581985	−	0.998305i	$-0.518536\pi$
$641$	−15447.5	−	8918.61i	−0.951855	−	0.549554i	−0.893657	+	0.448751i	$0.851869\pi$
$642$		0			0
$643$		−	25449.0i		−	1.56082i	−0.625266	−	0.780412i	$-0.715009\pi$
$643$		−	25449.0i		−	1.56082i	0.625266	−	0.780412i	$-0.284991\pi$
$644$	27017.4	−	3739.16i	1.65316	−	0.228794i
$645$		0			0
$646$	27470.1	+	47579.6i	1.67306	+	2.89782i
$647$	−13149.2	+	22775.1i	−0.798993	+	1.38390i	0.121280	+	0.992618i	$0.461300\pi$
$647$	−13149.2	+	22775.1i	−0.798993	+	1.38390i	−0.920273	+	0.391277i	$0.872033\pi$
$648$		0			0
$649$	10589.1	−	6113.64i	0.640462	−	0.369771i
$650$	1634.12			0.0986083
$651$		0			0
$652$	−51357.4			−3.08483
$653$	8203.79	−	4736.46i	0.491637	−	0.283847i	−0.233616	−	0.972329i	$-0.575056\pi$
$653$	8203.79	−	4736.46i	0.491637	−	0.283847i	0.725253	+	0.688482i	$0.241723\pi$
$654$		0			0
$655$	10255.4	−	17762.8i	0.611771	−	1.05962i
$656$	−7292.34	−	12630.7i	−0.434022	−	0.751747i
$657$		0			0
$658$	1945.90	−	4787.28i	0.115287	−	0.283629i
$659$		−	22384.5i		−	1.32318i	−0.749865	−	0.661591i	$-0.769882\pi$
$659$		−	22384.5i		−	1.32318i	0.749865	−	0.661591i	$-0.230118\pi$
$660$		0			0
$661$	−19857.3	−	11464.6i	−1.16847	−	0.674618i	−0.215153	−	0.976580i	$-0.569025\pi$
$661$	−19857.3	−	11464.6i	−1.16847	−	0.674618i	−0.953320	+	0.301962i	$0.902359\pi$
$662$	−16999.6	−	9814.73i	−0.998049	−	0.576224i
$663$		0			0
$664$			16368.8i			0.956677i
$665$	14585.4	−	35882.7i	0.850521	−	2.09244i
$666$		0			0
$667$	5620.37	+	9734.77i	0.326269	+	0.565115i
$668$	−2864.73	+	4961.85i	−0.165928	+	0.287395i
$669$		0			0
$670$	8812.56	−	5087.93i	0.508147	−	0.293379i
$671$	−23159.9			−1.33245
$672$		0			0
$673$	−4873.86			−0.279158			−0.139579	−	0.990211i	$-0.544575\pi$
$673$	−4873.86			−0.279158			−0.139579	+	0.990211i	$0.544575\pi$
$674$	−11678.4	+	6742.51i	−0.667409	+	0.385329i
$675$		0			0
$676$	16601.3	−	28754.2i	0.944541	−	1.63599i
$677$	8123.71	+	14070.7i	0.461181	+	0.798789i	0.999020	−	0.0442583i	$-0.0140925\pi$
$677$	8123.71	+	14070.7i	0.461181	+	0.798789i	−0.537839	+	0.843048i	$0.680759\pi$
$678$		0			0
$679$	−25334.9	+	3506.30i	−1.43191	+	0.198173i
$680$		−	32836.3i		−	1.85179i
$681$		0			0
$682$	22545.1	+	13016.4i	1.26583	+	0.730827i
$683$	−18786.2	−	10846.2i	−1.05247	−	0.607641i	−0.129127	−	0.991628i	$-0.541217\pi$
$683$	−18786.2	−	10846.2i	−1.05247	−	0.607641i	−0.923339	+	0.383987i	$0.874551\pi$
$684$		0			0
$685$		−	9068.85i		−	0.505844i
$686$	24889.4	+	18324.1i	1.38525	+	1.01985i
$687$		0			0
$688$	−140.989	−	244.200i	−0.00781273	−	0.0135320i
$689$	−2081.12	+	3604.61i	−0.115072	+	0.199310i
$690$		0			0
$691$	−19499.9	+	11258.3i	−1.07353	+	0.619803i	−0.929144	−	0.369718i	$-0.879454\pi$
$691$	−19499.9	+	11258.3i	−1.07353	+	0.619803i	−0.144387	+	0.989521i	$0.546121\pi$
$692$	32800.5			1.80186
$693$		0			0
$694$	−21029.2			−1.15023
$695$	16934.1	−	9776.90i	0.924239	−	0.533610i
$696$		0			0
$697$	−8936.75	+	15478.9i	−0.485658	+	0.841184i
$698$	3239.49	+	5610.97i	0.175669	+	0.304267i
$699$		0			0
$700$	−8689.20	+	6759.38i	−0.469173	+	0.364972i
$701$		−	33929.0i		−	1.82807i	−0.405631	−	0.914037i	$-0.632948\pi$
$701$		−	33929.0i		−	1.82807i	0.405631	−	0.914037i	$-0.367052\pi$
$702$		0			0
$703$	13365.3	+	7716.46i	0.717044	+	0.413986i
$704$	−26785.4	−	15464.5i	−1.43397	−	0.827901i
$705$		0			0
$706$			55214.9i			2.94340i
$707$	−12243.6	−	4976.68i	−0.651296	−	0.264734i
$708$		0			0
$709$	−9593.62	−	16616.6i	−0.508175	−	0.880185i	−0.999955	−	0.00946553i	$-0.996987\pi$
$709$	−9593.62	−	16616.6i	−0.508175	−	0.880185i	0.491780	−	0.870719i	$-0.336346\pi$
$710$	2479.57	−	4294.74i	0.131066	−	0.227012i
$711$		0			0
$712$	−829.121	+	478.693i	−0.0436413	+	0.0251963i
$713$	9291.63			0.488043
$714$		0			0
$715$	6116.49			0.319922
$716$	−23884.4	+	13789.7i	−1.24665	+	0.719754i
$717$		0			0
$718$	−20358.4	+	35261.8i	−1.05818	+	1.83281i
$719$	6883.43	+	11922.4i	0.357036	+	0.618404i	0.987464	−	0.157844i	$-0.0504542\pi$
$719$	6883.43	+	11922.4i	0.357036	+	0.618404i	−0.630429	+	0.776247i	$0.717121\pi$
$720$		0			0
$721$	20384.3	+	26204.0i	1.05291	+	1.35352i
$722$			97247.2i			5.01269i
$723$		0			0
$724$	−43398.2	−	25056.0i	−2.22774	−	1.28618i
$725$	−3929.15	−	2268.49i	−0.201276	−	0.116207i
$726$		0			0
$727$		−	12226.1i		−	0.623717i	−0.950129	−	0.311858i	$-0.899049\pi$
$727$		−	12226.1i		−	0.623717i	0.950129	−	0.311858i	$-0.100951\pi$
$728$	−839.242	−	6063.97i	−0.0427258	−	0.308717i
$729$		0			0
$730$	−23895.8	−	41388.7i	−1.21154	−	2.09845i
$731$	−172.782	+	299.267i	−0.00874222	+	0.0151420i
$732$		0			0
$733$	2256.76	−	1302.94i	0.113718	−	0.0656553i	−0.442062	−	0.896984i	$-0.645753\pi$
$733$	2256.76	−	1302.94i	0.113718	−	0.0656553i	0.555780	+	0.831329i	$0.312420\pi$
$734$	−13209.8			−0.664282
$735$		0			0
$736$	−2350.03			−0.117695
$737$	7678.71	−	4433.30i	0.383784	−	0.221578i
$738$		0			0
$739$	−16871.7	+	29222.7i	−0.839833	+	1.45463i	0.0502016	+	0.998739i	$0.484014\pi$
$739$	−16871.7	+	29222.7i	−0.839833	+	1.45463i	−0.890034	+	0.455894i	$0.849320\pi$
$740$	−9421.01	−	16317.7i	−0.468004	−	0.810607i
$741$		0			0
$742$	−5806.31	−	41953.7i	−0.287273	−	2.07570i
$743$		−	14586.9i		−	0.720244i	−0.932905	−	0.360122i	$-0.882735\pi$
$743$		−	14586.9i		−	0.720244i	0.932905	−	0.360122i	$-0.117265\pi$
$744$		0			0
$745$	26542.0	+	15324.0i	1.30527	+	0.753597i
$746$	25690.5	+	14832.4i	1.26085	+	0.727953i
$747$		0			0
$748$		−	58446.1i		−	2.85695i
$749$	−256.528	−	329.768i	−0.0125145	−	0.0160874i
$750$		0			0
$751$	3757.62	+	6508.39i	0.182580	+	0.316238i	0.942758	−	0.333477i	$-0.108222\pi$
$751$	3757.62	+	6508.39i	0.182580	+	0.316238i	−0.760178	+	0.649714i	$0.774889\pi$
$752$	1612.55	−	2793.02i	0.0781965	−	0.135440i
$753$		0			0
$754$	4463.28	−	2576.88i	0.215574	−	0.124462i
$755$	−31502.0			−1.51851
$756$		0			0
$757$	23917.4			1.14834			0.574169	−	0.818737i	$-0.305325\pi$
$757$	23917.4			1.14834			0.574169	+	0.818737i	$0.305325\pi$
$758$	41807.9	−	24137.8i	2.00334	−	1.15663i
$759$		0			0
$760$	39033.7	−	67608.4i	1.86303	−	3.22686i
$761$	−6099.27	−	10564.2i	−0.290537	−	0.503224i	0.683400	−	0.730044i	$-0.260500\pi$
$761$	−6099.27	−	10564.2i	−0.290537	−	0.503224i	−0.973937	+	0.226820i	$0.927167\pi$
$762$		0			0
$763$	−16340.7	−	6642.07i	−0.775327	−	0.315150i
$764$			8609.80i			0.407712i
$765$		0			0
$766$	−5136.59	−	2965.61i	−0.242288	−	0.139885i
$767$	−1732.85	−	1000.46i	−0.0815769	−	0.0470985i
$768$		0			0
$769$		−	2013.08i		−	0.0943999i	−0.998885	−	0.0471999i	$-0.984970\pi$
$769$		−	2013.08i		−	0.0943999i	0.998885	−	0.0471999i	$-0.0150298\pi$
$770$	−49125.7	+	38215.2i	−2.29918	+	1.78855i
$771$		0			0
$772$	−5520.28	−	9561.41i	−0.257357	−	0.445755i
$773$	−15139.1	+	26221.7i	−0.704418	+	1.22009i	0.262483	+	0.964937i	$0.415459\pi$
$773$	−15139.1	+	26221.7i	−0.704418	+	1.22009i	−0.966901	+	0.255152i	$0.917875\pi$
$774$		0			0
$775$	−3247.85	+	1875.14i	−0.150537	+	0.0869125i
$776$	−51549.0			−2.38467
$777$		0			0
$778$	63903.5			2.94480
$779$	−36800.7	+	21246.9i	−1.69258	+	0.977213i
$780$		0			0
$781$	2160.54	−	3742.17i	0.0989888	−	0.171454i
$782$	−15754.6	−	27287.8i	−0.720439	−	1.24784i
$783$		0			0
$784$	13818.9	+	13457.5i	0.629506	+	0.613041i
$785$			31095.5i			1.41382i
$786$		0			0

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 63 = 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	63.p (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-4.21355 + 2.43270i) q^{2} +(7.83601 - 13.5724i) q^{4} +(6.38217 + 11.0542i) q^{5} +(2.53897 + 18.3454i) q^{7} +37.3274i q^{8} +O(q^{10})\)	\(q+(-4.21355 + 2.43270i) q^{2} +(7.83601 - 13.5724i) q^{4} +(6.38217 + 11.0542i) q^{5} +(2.53897 + 18.3454i) q^{7} +37.3274i q^{8} +(-53.7832 - 31.0517i) q^{10} +(-46.8633 - 27.0565i) q^{11} +8.85528i q^{13} +(-55.3268 - 71.1228i) q^{14} +(-28.1181 - 48.7020i) q^{16} +(-34.4587 + 59.6841i) q^{17} +(-141.898 + 81.9246i) q^{19} +200.043 q^{20} +263.281 q^{22} +(81.3807 - 46.9852i) q^{23} +(-18.9642 + 32.8469i) q^{25} +(-21.5422 - 37.3122i) q^{26} +(268.886 + 109.295i) q^{28} +119.620i q^{29} +(85.6311 + 49.4391i) q^{31} +(-21.6577 - 12.5041i) q^{32} -335.310i q^{34} +(-186.590 + 145.150i) q^{35} +(-47.0949 - 81.5708i) q^{37} +(398.595 - 690.387i) q^{38} +(-412.626 + 238.230i) q^{40} +259.347 q^{41} +5.01418 q^{43} +(-734.443 + 424.031i) q^{44} +(-228.601 + 395.949i) q^{46} +(28.6747 + 49.6660i) q^{47} +(-330.107 + 93.1568i) q^{49} -184.536i q^{50} +(120.187 + 69.3901i) q^{52} +(407.058 + 235.015i) q^{53} -690.718i q^{55} +(-684.786 + 94.7731i) q^{56} +(-290.999 - 504.025i) q^{58} +(-112.979 + 195.685i) q^{59} +(370.650 - 213.995i) q^{61} -481.082 q^{62} +571.564 q^{64} +(-97.8884 + 56.5159i) q^{65} +(-81.9267 + 141.901i) q^{67} +(540.037 + 935.372i) q^{68} +(433.103 - 1065.51i) q^{70} +79.8529i q^{71} +(666.447 + 384.774i) q^{73} +(396.874 + 229.135i) q^{74} +2567.85i q^{76} +(377.379 - 928.422i) q^{77} +(-267.408 - 463.165i) q^{79} +(358.909 - 621.648i) q^{80} +(-1092.77 + 630.912i) q^{82} +438.520 q^{83} -879.684 q^{85} +(-21.1275 + 12.1980i) q^{86} +(1009.95 - 1749.29i) q^{88} +(12.8242 + 22.2121i) q^{89} +(-162.454 + 22.4833i) q^{91} -1472.71i q^{92} +(-241.644 - 139.513i) q^{94} +(-1811.23 - 1045.71i) q^{95} +1381.00i q^{97} +(1164.30 - 1195.57i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16q + 32q^{4} + 56q^{7} + O(q^{10}) \)	\( 16q + 32q^{4} + 56q^{7} - 72q^{10} - 188q^{16} - 612q^{19} + 528q^{22} - 20q^{25} + 1036q^{28} + 1128q^{31} - 1196q^{37} - 3204q^{40} + 328q^{43} - 1392q^{46} + 784q^{49} + 4452q^{52} - 3372q^{58} - 1632q^{61} + 5432q^{64} + 308q^{67} + 3612q^{70} + 4068q^{73} - 2176q^{79} - 10188q^{82} - 4608q^{85} + 708q^{88} + 924q^{91} - 2916q^{94} + O(q^{100}) \)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Properties

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