LMFDB - Embedded newform 324.3.o.a.65.1

Newspace parameters

Level:	$ N $	$=$	$ 324 = 2^{2} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	324.o (of order $54$, degree $18$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$8.82836056527$
Analytic rank:	$0$
Dimension:	$324$
Relative dimension:	$18$ over $\Q(\zeta_{54})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{54}]$

Embedding invariants

Embedding label			65.1
Character	$\chi$	$=$	324.65
Dual form			324.3.o.a.5.1

$q$-expansion

$f(q)$	$=$	$q+(-2.85677 - 0.915910i) q^{3} +(-4.55987 - 3.39470i) q^{5} +(9.74231 + 6.40762i) q^{7} +(7.32222 + 5.23308i) q^{9} +O(q^{10})$	\(q+(-2.85677 - 0.915910i) q^{3} +(-4.55987 - 3.39470i) q^{5} +(9.74231 + 6.40762i) q^{7} +(7.32222 + 5.23308i) q^{9} +(0.713033 + 6.10039i) q^{11} +(4.08156 - 13.6334i) q^{13} +(9.91723 + 13.8743i) q^{15} +(-21.2831 + 3.75278i) q^{17} +(6.10801 - 34.6402i) q^{19} +(-21.9627 - 27.2281i) q^{21} +(-11.5087 - 17.4982i) q^{23} +(2.09835 + 7.00898i) q^{25} +(-16.1248 - 21.6562i) q^{27} +(4.94737 + 4.66760i) q^{29} +(-2.02555 - 34.7774i) q^{31} +(3.55044 - 18.0805i) q^{33} +(-22.6717 - 62.2901i) q^{35} +(14.3291 + 5.21535i) q^{37} +(-24.1470 + 35.2090i) q^{39} +(-0.792837 - 3.34524i) q^{41} +(-5.34616 + 12.3938i) q^{43} +(-15.6236 - 48.7188i) q^{45} +(29.6405 + 1.72636i) q^{47} +(34.4471 + 79.8574i) q^{49} +(64.2379 + 8.77257i) q^{51} +(58.2609 - 33.6369i) q^{53} +(17.4576 - 30.2375i) q^{55} +(-49.1765 + 93.3646i) q^{57} +(5.87325 - 50.2489i) q^{59} +(-19.7711 + 9.92944i) q^{61} +(37.8037 + 97.9003i) q^{63} +(-64.8925 + 48.3107i) q^{65} +(-72.8296 - 77.1949i) q^{67} +(16.8510 + 60.5292i) q^{69} +(-63.7955 - 76.0285i) q^{71} +(-40.8467 - 34.2745i) q^{73} +(0.425101 - 21.9449i) q^{75} +(-32.1424 + 64.0007i) q^{77} +(70.7096 + 16.7585i) q^{79} +(26.2297 + 76.6355i) q^{81} +(20.3016 - 85.6592i) q^{83} +(109.787 + 55.1373i) q^{85} +(-9.85837 - 17.8656i) q^{87} +(-17.1945 + 20.4917i) q^{89} +(127.121 - 106.667i) q^{91} +(-26.0664 + 101.206i) q^{93} +(-145.445 + 137.220i) q^{95} +(25.5532 + 34.3239i) q^{97} +(-26.7029 + 48.3997i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 324 q+O(q^{10}) $ 324 * q	$ 324 q - 135 q^{21} - 81 q^{23} + 27 q^{27} + 81 q^{29} + 189 q^{33} + 243 q^{35} + 216 q^{41} + 432 q^{45} + 324 q^{47} + 126 q^{51} - 216 q^{57} - 378 q^{59} - 540 q^{63} - 108 q^{65} - 351 q^{67} + 504 q^{69} + 648 q^{71} + 450 q^{75} + 432 q^{77} - 54 q^{79} - 72 q^{81} - 216 q^{83} + 270 q^{85} - 1008 q^{87} - 648 q^{89} - 684 q^{93} - 432 q^{95} + 459 q^{97} - 252 q^{99}+O(q^{100}) $ 324 * q - 135 * q^21 - 81 * q^23 + 27 * q^27 + 81 * q^29 + 189 * q^33 + 243 * q^35 + 216 * q^41 + 432 * q^45 + 324 * q^47 + 126 * q^51 - 216 * q^57 - 378 * q^59 - 540 * q^63 - 108 * q^65 - 351 * q^67 + 504 * q^69 + 648 * q^71 + 450 * q^75 + 432 * q^77 - 54 * q^79 - 72 * q^81 - 216 * q^83 + 270 * q^85 - 1008 * q^87 - 648 * q^89 - 684 * q^93 - 432 * q^95 + 459 * q^97 - 252 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/324\mathbb{Z}\right)^\times$.

$n$	$163$	$245$
$\chi(n)$	$1$	$e\left(\frac{31}{54}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	−2.85677	−	0.915910i	−0.952255	−	0.305303i
$3$	−2.85677	−	0.915910i	−0.952255	−	0.305303i
$4$		0			0
$5$	−4.55987	−	3.39470i	−0.911973	−	0.678939i	0.0352962	−	0.999377i	$-0.488763\pi$
$5$	−4.55987	−	3.39470i	−0.911973	−	0.678939i	−0.947270	+	0.320438i	$0.896170\pi$
$6$		0			0
$7$	9.74231	+	6.40762i	1.39176	+	0.915374i	0.999989	−	0.00468942i	$-0.00149269\pi$
$7$	9.74231	+	6.40762i	1.39176	+	0.915374i	0.391770	+	0.920063i	$0.371863\pi$
$8$		0			0
$9$	7.32222	+	5.23308i	0.813580	+	0.581454i
$10$		0			0
$11$	0.713033	+	6.10039i	0.0648212	+	0.554581i	0.986157	+	0.165813i	$0.0530250\pi$
$11$	0.713033	+	6.10039i	0.0648212	+	0.554581i	−0.921336	+	0.388767i	$0.872901\pi$
$12$		0			0
$13$	4.08156	−	13.6334i	0.313966	−	1.04872i	−0.644759	−	0.764386i	$-0.723042\pi$
$13$	4.08156	−	13.6334i	0.313966	−	1.04872i	0.958725	−	0.284334i	$-0.0917725\pi$
$14$		0			0
$15$	9.91723	+	13.8743i	0.661149	+	0.924952i
$16$		0			0
$17$	−21.2831	+	3.75278i	−1.25194	+	0.220752i	−0.760028	−	0.649891i	$-0.774815\pi$
$17$	−21.2831	+	3.75278i	−1.25194	+	0.220752i	−0.491917	+	0.870642i	$0.663704\pi$
$18$		0			0
$19$	6.10801	−	34.6402i	0.321474	−	1.82317i	−0.211901	−	0.977291i	$-0.567965\pi$
$19$	6.10801	−	34.6402i	0.321474	−	1.82317i	0.533375	−	0.845879i	$-0.320924\pi$
$20$		0			0
$21$	−21.9627	−	27.2281i	−1.04584	−	1.29658i
$22$		0			0
$23$	−11.5087	−	17.4982i	−0.500380	−	0.760791i	0.493678	−	0.869645i	$-0.335652\pi$
$23$	−11.5087	−	17.4982i	−0.500380	−	0.760791i	−0.994058	+	0.108854i	$0.965282\pi$
$24$		0			0
$25$	2.09835	+	7.00898i	0.0839340	+	0.280359i
$26$		0			0
$27$	−16.1248	−	21.6562i	−0.597216	−	0.802081i
$28$		0			0
$29$	4.94737	+	4.66760i	0.170599	+	0.160952i	0.766806	−	0.641879i	$-0.221845\pi$
$29$	4.94737	+	4.66760i	0.170599	+	0.160952i	−0.596207	+	0.802831i	$0.703326\pi$
$30$		0			0
$31$	−2.02555	−	34.7774i	−0.0653404	−	1.12185i	−0.858492	−	0.512827i	$-0.828598\pi$
$31$	−2.02555	−	34.7774i	−0.0653404	−	1.12185i	0.793152	−	0.609024i	$-0.208439\pi$
$32$		0			0
$33$	3.55044	−	18.0805i	0.107589	−	0.547893i
$34$		0			0
$35$	−22.6717	−	62.2901i	−0.647764	−	1.77972i
$36$		0			0
$37$	14.3291	+	5.21535i	0.387272	+	0.140955i	0.528316	−	0.849048i	$-0.322824\pi$
$37$	14.3291	+	5.21535i	0.387272	+	0.140955i	−0.141044	+	0.990003i	$0.545046\pi$
$38$		0			0
$39$	−24.1470	+	35.2090i	−0.619154	+	0.902794i
$40$		0			0
$41$	−0.792837	−	3.34524i	−0.0193375	−	0.0815913i	0.962521	−	0.271209i	$-0.0874234\pi$
$41$	−0.792837	−	3.34524i	−0.0193375	−	0.0815913i	−0.981858	+	0.189617i	$0.939275\pi$
$42$		0			0
$43$	−5.34616	+	12.3938i	−0.124329	+	0.288228i	−0.969001	−	0.247057i	$-0.920536\pi$
$43$	−5.34616	+	12.3938i	−0.124329	+	0.288228i	0.844672	+	0.535285i	$0.179796\pi$
$44$		0			0
$45$	−15.6236	−	48.7188i	−0.347191	−	1.08264i
$46$		0			0
$47$	29.6405	+	1.72636i	0.630648	+	0.0367311i	0.370492	−	0.928836i	$-0.379189\pi$
$47$	29.6405	+	1.72636i	0.630648	+	0.0367311i	0.260156	+	0.965567i	$0.416226\pi$
$48$		0			0
$49$	34.4471	+	79.8574i	0.703002	+	1.62974i
$50$		0			0
$51$	64.2379	+	8.77257i	1.25957	+	0.172011i
$52$		0			0
$53$	58.2609	−	33.6369i	1.09926	−	0.634659i	0.163235	−	0.986587i	$-0.447807\pi$
$53$	58.2609	−	33.6369i	1.09926	−	0.634659i	0.936027	+	0.351928i	$0.114474\pi$
$54$		0			0
$55$	17.4576	−	30.2375i	0.317411	−	0.549773i
$56$		0			0
$57$	−49.1765	+	93.3646i	−0.862746	+	1.63798i
$58$		0			0
$59$	5.87325	−	50.2489i	0.0995467	−	0.851676i	−0.847547	−	0.530720i	$-0.821922\pi$
$59$	5.87325	−	50.2489i	0.0995467	−	0.851676i	0.947094	−	0.320956i	$-0.104004\pi$
$60$		0			0
$61$	−19.7711	+	9.92944i	−0.324117	+	0.162778i	−0.603416	−	0.797426i	$-0.706194\pi$
$61$	−19.7711	+	9.92944i	−0.324117	+	0.162778i	0.279299	+	0.960204i	$0.409898\pi$
$62$		0			0
$63$	37.8037	+	97.9003i	0.600059	+	1.55397i
$64$		0			0
$65$	−64.8925	+	48.3107i	−0.998346	+	0.743241i
$66$		0			0
$67$	−72.8296	−	77.1949i	−1.08701	−	1.15216i	−0.987580	−	0.157118i	$-0.949780\pi$
$67$	−72.8296	−	77.1949i	−1.08701	−	1.15216i	−0.0994298	−	0.995045i	$-0.531702\pi$
$68$		0			0
$69$	16.8510	+	60.5292i	0.244217	+	0.877235i
$70$		0			0
$71$	−63.7955	−	76.0285i	−0.898528	−	1.07082i	−0.997131	−	0.0756971i	$-0.975882\pi$
$71$	−63.7955	−	76.0285i	−0.898528	−	1.07082i	0.0986029	−	0.995127i	$-0.468563\pi$
$72$		0			0
$73$	−40.8467	−	34.2745i	−0.559544	−	0.469514i	0.318613	−	0.947885i	$-0.396783\pi$
$73$	−40.8467	−	34.2745i	−0.559544	−	0.469514i	−0.878158	+	0.478371i	$0.841227\pi$
$74$		0			0
$75$	0.425101	−	21.9449i	0.00566802	−	0.292599i
$76$		0			0
$77$	−32.1424	+	64.0007i	−0.417433	+	0.831178i
$78$		0			0
$79$	70.7096	+	16.7585i	0.895059	+	0.212133i	0.652306	−	0.757955i	$-0.273802\pi$
$79$	70.7096	+	16.7585i	0.895059	+	0.212133i	0.242752	+	0.970088i	$0.421950\pi$
$80$		0			0
$81$	26.2297	+	76.6355i	0.323823	+	0.946118i
$82$		0			0
$83$	20.3016	−	85.6592i	0.244598	−	1.03204i	−0.702650	−	0.711536i	$-0.748000\pi$
$83$	20.3016	−	85.6592i	0.244598	−	1.03204i	0.947248	−	0.320503i	$-0.103852\pi$
$84$		0			0
$85$	109.787	+	55.1373i	1.29162	+	0.648674i
$86$		0			0
$87$	−9.85837	−	17.8656i	−0.113315	−	0.205352i
$88$		0			0
$89$	−17.1945	+	20.4917i	−0.193197	+	0.230243i	−0.853943	−	0.520366i	$-0.825796\pi$
$89$	−17.1945	+	20.4917i	−0.193197	+	0.230243i	0.660746	+	0.750609i	$0.270240\pi$
$90$		0			0
$91$	127.121	−	106.667i	1.39694	−	1.17217i
$92$		0			0
$93$	−26.0664	+	101.206i	−0.280284	+	1.08824i
$94$		0			0
$95$	−145.445	+	137.220i	−1.53100	+	1.44442i
$96$		0			0
$97$	25.5532	+	34.3239i	0.263435	+	0.353855i	0.914085	−	0.405523i	$-0.132911\pi$
$97$	25.5532	+	34.3239i	0.263435	+	0.353855i	−0.650650	+	0.759378i	$0.725503\pi$
$98$		0			0
$99$	−26.7029	+	48.3997i	−0.269726	+	0.488886i
$100$		0			0
$101$	−62.6480	−	124.742i	−0.620277	−	1.23507i	−0.956070	−	0.293138i	$-0.905301\pi$
$101$	−62.6480	−	124.742i	−0.620277	−	1.23507i	0.335793	−	0.941936i	$-0.390996\pi$
$102$		0			0
$103$	65.2925	+	7.63160i	0.633908	+	0.0740932i	0.426977	−	0.904262i	$-0.359578\pi$
$103$	65.2925	+	7.63160i	0.633908	+	0.0740932i	0.206931	+	0.978356i	$0.433653\pi$
$104$		0			0
$105$	7.71569	+	198.713i	0.0734828	+	1.89251i
$106$		0			0
$107$	181.287	+	104.666i	1.69427	+	0.978189i	0.950994	+	0.309209i	$0.100064\pi$
$107$	181.287	+	104.666i	1.69427	+	0.978189i	0.743280	+	0.668981i	$0.233269\pi$
$108$		0			0
$109$	20.8349	+	36.0872i	0.191146	+	0.331075i	0.945630	−	0.325243i	$-0.105446\pi$
$109$	20.8349	+	36.0872i	0.191146	+	0.331075i	−0.754484	+	0.656318i	$0.772113\pi$
$110$		0			0
$111$	−36.1580	−	28.0232i	−0.325747	−	0.252461i
$112$		0			0
$113$	−122.722	+	52.9372i	−1.08604	+	0.468471i	−0.862413	−	0.506205i	$-0.831048\pi$
$113$	−122.722	+	52.9372i	−1.08604	+	0.468471i	−0.223624	+	0.974676i	$0.571789\pi$
$114$		0			0
$115$	−6.92269	+	118.858i	−0.0601973	+	1.03355i
$116$		0			0
$117$	101.231	−	78.4673i	0.865219	−	0.670661i
$118$		0			0
$119$	−231.393	−	99.8130i	−1.94447	−	0.838765i
$120$		0			0
$121$	81.0321	−	19.2050i	0.669687	−	0.158719i
$122$		0			0
$123$	−0.798993	+	10.2827i	−0.00649588	+	0.0835995i
$124$		0			0
$125$	−34.3823	+	94.4646i	−0.275058	+	0.755717i
$126$		0			0
$127$	−94.9637	+	34.5640i	−0.747746	+	0.272157i	−0.687657	−	0.726036i	$-0.741361\pi$
$127$	−94.9637	+	34.5640i	−0.747746	+	0.272157i	−0.0600888	+	0.998193i	$0.519138\pi$
$128$		0			0
$129$	26.6243	−	30.5095i	0.206390	−	0.236508i
$130$		0			0
$131$	252.920	−	14.7309i	1.93068	−	0.112450i	0.950523	−	0.310654i	$-0.100548\pi$
$131$	252.920	−	14.7309i	1.93068	−	0.112450i	0.980161	+	0.198204i	$0.0635110\pi$
$132$		0			0
$133$	281.467	−	298.338i	2.11630	−	2.24314i
$134$		0			0
$135$	0.0108963	+	153.488i	8.07134e−5	+	1.13695i
$136$		0			0
$137$	10.5100	−	3.14647i	0.0767150	−	0.0229670i	−0.248216	−	0.968705i	$-0.579844\pi$
$137$	10.5100	−	3.14647i	0.0767150	−	0.0229670i	0.324931	+	0.945738i	$0.394659\pi$
$138$		0			0
$139$	−147.347	+	96.9117i	−1.06005	+	0.697206i	−0.954803	−	0.297240i	$-0.903934\pi$
$139$	−147.347	+	96.9117i	−1.06005	+	0.697206i	−0.105248	+	0.994446i	$0.533564\pi$
$140$		0			0
$141$	−83.0946	−	32.0798i	−0.589324	−	0.227516i
$142$		0			0
$143$	86.0791	+	15.1781i	0.601952	+	0.106140i
$144$		0			0
$145$	−6.71426	−	38.0785i	−0.0463052	−	0.262610i
$146$		0			0
$147$	−25.2651	−	259.684i	−0.171872	−	1.76656i
$148$		0			0
$149$	−145.353	−	43.5159i	−0.975525	−	0.292053i	−0.240918	−	0.970546i	$-0.577448\pi$
$149$	−145.353	−	43.5159i	−0.975525	−	0.292053i	−0.734607	+	0.678493i	$0.762634\pi$
$150$		0			0
$151$	242.653	−	28.3621i	1.60697	−	0.187828i	0.735239	−	0.677808i	$-0.237070\pi$
$151$	242.653	−	28.3621i	1.60697	−	0.187828i	0.871734	+	0.489980i	$0.162996\pi$
$152$		0			0
$153$	−175.478	−	83.8973i	−1.14691	−	0.548349i
$154$		0			0
$155$	−108.822	+	165.456i	−0.702080	+	1.06746i
$156$		0			0
$157$	−169.034	+	227.051i	−1.07665	+	1.44619i	−0.191139	+	0.981563i	$0.561218\pi$
$157$	−169.034	+	227.051i	−1.07665	+	1.44619i	−0.885508	+	0.464625i	$0.846189\pi$
$158$		0			0
$159$	−197.246	+	42.7311i	−1.24054	+	0.268749i
$160$		0			0
$161$		−	244.216i		−	1.51687i
$162$		0			0
$163$	−240.338			−1.47447			−0.737234	−	0.675637i	$-0.763869\pi$
$163$	−240.338			−1.47447			−0.737234	+	0.675637i	$0.763869\pi$
$164$		0			0
$165$	−77.5672	+	70.3918i	−0.470104	+	0.426617i
$166$		0			0
$167$	217.785	+	162.135i	1.30410	+	0.970868i	0.999833	+	0.0182817i	$0.00581957\pi$
$167$	217.785	+	162.135i	1.30410	+	0.970868i	0.304269	+	0.952586i	$0.401588\pi$
$168$		0			0
$169$	−28.0120	−	18.4238i	−0.165751	−	0.109016i
$170$		0			0
$171$	225.999	−	221.680i	1.32163	−	1.29637i
$172$		0			0
$173$	−32.6034	−	278.939i	−0.188459	−	1.61237i	−0.674394	−	0.738372i	$-0.735595\pi$
$173$	−32.6034	−	278.939i	−0.188459	−	1.61237i	0.485935	−	0.873995i	$-0.338479\pi$
$174$		0			0
$175$	−24.4681	+	81.7290i	−0.139818	+	0.467023i
$176$		0			0
$177$	−62.8020	+	138.170i	−0.354813	+	0.780621i
$178$		0			0
$179$	123.722	−	21.8155i	0.691183	−	0.121874i	0.182987	−	0.983115i	$-0.441423\pi$
$179$	123.722	−	21.8155i	0.691183	−	0.121874i	0.508197	+	0.861241i	$0.330312\pi$
$180$		0			0
$181$	−17.3397	+	98.3386i	−0.0957997	+	0.543307i	0.898700	+	0.438565i	$0.144513\pi$
$181$	−17.3397	+	98.3386i	−0.0957997	+	0.543307i	−0.994499	+	0.104743i	$0.966598\pi$
$182$		0			0
$183$	65.5760	−	10.2575i	0.358339	−	0.0560518i
$184$		0			0
$185$	−47.6341	−	72.4241i	−0.257481	−	0.391481i
$186$		0			0
$187$	−38.0689	−	127.159i	−0.203577	−	0.679995i
$188$		0			0
$189$	−18.3284	−	314.303i	−0.0969759	−	1.66298i
$190$		0			0
$191$	−2.52109	−	2.37852i	−0.0131994	−	0.0124530i	0.679614	−	0.733570i	$-0.262147\pi$
$191$	−2.52109	−	2.37852i	−0.0131994	−	0.0124530i	−0.692813	+	0.721117i	$0.743629\pi$
$192$		0			0
$193$	9.65273	+	165.731i	0.0500142	+	0.858710i	0.926269	+	0.376862i	$0.122997\pi$
$193$	9.65273	+	165.731i	0.0500142	+	0.858710i	−0.876255	+	0.481848i	$0.839966\pi$
$194$		0			0
$195$	229.631	−	78.5765i	1.17759	−	0.402957i
$196$		0			0
$197$	35.0480	+	96.2936i	0.177909	+	0.488800i	0.996308	−	0.0858498i	$-0.0273605\pi$
$197$	35.0480	+	96.2936i	0.177909	+	0.488800i	−0.818399	+	0.574650i	$0.805138\pi$
$198$		0			0
$199$	30.5394	+	11.1154i	0.153464	+	0.0558564i	0.417610	−	0.908626i	$-0.362868\pi$
$199$	30.5394	+	11.1154i	0.153464	+	0.0558564i	−0.264146	+	0.964483i	$0.585090\pi$
$200$		0			0
$201$	137.354	+	287.233i	0.683351	+	1.42902i
$202$		0			0
$203$	18.2906	+	77.1741i	0.0901014	+	0.380168i
$204$		0			0
$205$	−7.74085	+	17.9453i	−0.0377602	+	0.0875381i
$206$		0			0
$207$	7.29998	−	188.352i	0.0352656	−	0.909912i
$208$		0			0
$209$	215.674	+	12.5616i	1.03193	+	0.0601033i
$210$		0			0
$211$	−77.0783	−	178.688i	−0.365300	−	0.846861i	−0.997406	−	0.0719788i	$-0.977069\pi$
$211$	−77.0783	−	178.688i	−0.365300	−	0.846861i	0.632106	−	0.774882i	$-0.282191\pi$
$212$		0			0
$213$	112.613	+	275.627i	0.528702	+	1.29402i
$214$		0			0
$215$	66.4509	−	38.3654i	0.309074	−	0.178444i
$216$		0			0
$217$	203.107	−	351.791i	0.935975	−	1.62116i
$218$		0			0
$219$	85.2972	+	135.326i	0.389485	+	0.617928i
$220$		0			0
$221$	−35.7051	+	305.477i	−0.161562	+	1.38225i
$222$		0			0
$223$	−112.471	+	56.4849i	−0.504353	+	0.253295i	−0.682735	−	0.730666i	$-0.739210\pi$
$223$	−112.471	+	56.4849i	−0.504353	+	0.253295i	0.178383	+	0.983961i	$0.442914\pi$
$224$		0			0
$225$	−21.3140	+	62.3021i	−0.0947288	+	0.276898i
$226$		0			0
$227$	−354.671	+	264.043i	−1.56243	+	1.16319i	−0.645758	+	0.763542i	$0.723458\pi$
$227$	−354.671	+	264.043i	−1.56243	+	1.16319i	−0.916671	+	0.399643i	$0.869134\pi$
$228$		0			0
$229$	−69.5855	−	73.7563i	−0.303867	−	0.322080i	0.557389	−	0.830251i	$-0.311803\pi$
$229$	−69.5855	−	73.7563i	−0.303867	−	0.322080i	−0.861256	+	0.508171i	$0.830322\pi$
$230$		0			0
$231$	150.442	−	153.396i	0.651265	−	0.664050i
$232$		0			0
$233$	−63.5879	−	75.7811i	−0.272909	−	0.325241i	0.612130	−	0.790757i	$-0.290313\pi$
$233$	−63.5879	−	75.7811i	−0.272909	−	0.325241i	−0.885039	+	0.465517i	$0.845869\pi$
$234$		0			0
$235$	−129.296	−	108.492i	−0.550196	−	0.461669i
$236$		0			0
$237$	−186.652	−	112.639i	−0.787559	−	0.475269i
$238$		0			0
$239$	6.13576	−	12.2173i	0.0256726	−	0.0511184i	−0.880431	−	0.474175i	$-0.842746\pi$
$239$	6.13576	−	12.2173i	0.0256726	−	0.0511184i	0.906103	+	0.423056i	$0.139043\pi$
$240$		0			0
$241$	96.4899	+	22.8685i	0.400373	+	0.0948902i	0.425869	−	0.904785i	$-0.359968\pi$
$241$	96.4899	+	22.8685i	0.400373	+	0.0948902i	−0.0254961	+	0.999675i	$0.508117\pi$
$242$		0			0
$243$	−4.74084	−	242.954i	−0.0195096	−	0.999810i
$244$		0			0
$245$	114.017	−	481.077i	0.465377	−	1.96358i
$246$		0			0
$247$	−447.333	−	224.659i	−1.81106	−	0.909550i
$248$		0			0
$249$	−136.453	+	226.114i	−0.548004	+	0.908087i
$250$		0			0
$251$	157.220	−	187.368i	0.626375	−	0.746485i	−0.355777	−	0.934571i	$-0.615784\pi$
$251$	157.220	−	187.368i	0.626375	−	0.746485i	0.982153	+	0.188086i	$0.0602283\pi$
$252$		0			0
$253$	98.5397	−	82.6846i	0.389485	−	0.326817i
$254$		0			0
$255$	−263.136	−	258.070i	−1.03191	−	1.01204i
$256$		0			0
$257$	256.249	−	241.758i	0.997076	−	0.940693i	−0.00116387	−	0.999999i	$-0.500370\pi$
$257$	256.249	−	241.758i	0.997076	−	0.940693i	0.998240	+	0.0593067i	$0.0188890\pi$
$258$		0			0
$259$	106.180	+	142.625i	0.409962	+	0.550674i
$260$		0			0
$261$	11.7998	+	60.0672i	0.0452098	+	0.230142i
$262$		0			0
$263$	−196.319	−	390.904i	−0.746461	−	1.48633i	−0.869504	−	0.493925i	$-0.835562\pi$
$263$	−196.319	−	390.904i	−0.746461	−	1.48633i	0.123043	−	0.992401i	$-0.460735\pi$
$264$		0			0
$265$	−379.849	−	44.3980i	−1.43339	−	0.167540i
$266$		0			0
$267$	67.8893	−	42.7912i	0.254267	−	0.160267i
$268$		0			0
$269$	166.520	+	96.1402i	0.619033	+	0.357399i	0.776492	−	0.630127i	$-0.216997\pi$
$269$	166.520	+	96.1402i	0.619033	+	0.357399i	−0.157460	+	0.987525i	$0.550330\pi$
$270$		0			0
$271$	−60.1759	−	104.228i	−0.222051	−	0.384604i	0.733379	−	0.679820i	$-0.237942\pi$
$271$	−60.1759	−	104.228i	−0.222051	−	0.384604i	−0.955431	+	0.295215i	$0.904609\pi$
$272$		0			0
$273$	−460.853	+	188.292i	−1.68811	+	0.689714i
$274$		0			0
$275$	−41.2613	+	17.7984i	−0.150041	+	0.0647214i
$276$		0			0
$277$	−13.4243	+	230.487i	−0.0484633	+	0.832082i	0.883253	+	0.468896i	$0.155348\pi$
$277$	−13.4243	+	230.487i	−0.0484633	+	0.832082i	−0.931717	+	0.363186i	$0.881689\pi$
$278$		0			0
$279$	167.161	−	265.247i	0.599145	−	0.950707i
$280$		0			0
$281$	245.678	+	105.975i	0.874300	+	0.377136i	0.785462	−	0.618910i	$-0.212425\pi$
$281$	245.678	+	105.975i	0.874300	+	0.377136i	0.0888383	+	0.996046i	$0.471685\pi$
$282$		0			0
$283$	−15.5619	+	3.68823i	−0.0549889	+	0.0130326i	−0.258018	−	0.966140i	$-0.583069\pi$
$283$	−15.5619	+	3.68823i	−0.0549889	+	0.0130326i	0.203029	+	0.979173i	$0.434921\pi$
$284$		0			0
$285$	541.183	−	258.791i	1.89889	−	0.908039i
$286$		0			0
$287$	13.7110	−	37.6706i	0.0477734	−	0.131256i
$288$		0			0
$289$	167.314	−	60.8974i	0.578942	−	0.210717i
$290$		0			0
$291$	−41.5619	−	121.460i	−0.142824	−	0.417388i
$292$		0			0
$293$	−509.342	+	29.6658i	−1.73837	+	0.101248i	−0.897786	−	0.440433i	$-0.854825\pi$
$293$	−509.342	+	29.6658i	−1.73837	+	0.101248i	−0.840584	+	0.541681i	$0.817788\pi$
$294$		0			0
$295$	−197.361	+	209.190i	−0.669020	+	0.709120i
$296$		0			0
$297$	120.614	−	113.809i	0.406106	−	0.383196i
$298$		0			0
$299$	−285.533	+	85.4829i	−0.954959	+	0.285896i
$300$		0			0
$301$	−131.499	+	86.4880i	−0.436872	+	0.287335i
$302$		0			0
$303$	64.7177	+	413.740i	0.213590	+	1.36548i
$304$		0			0
$305$	123.861	+	21.8401i	0.406102	+	0.0716068i
$306$		0			0
$307$	−11.0876	−	62.8808i	−0.0361159	−	0.204823i	0.961410	−	0.275118i	$-0.0887171\pi$
$307$	−11.0876	−	62.8808i	−0.0361159	−	0.204823i	−0.997526	+	0.0702950i	$0.977606\pi$
$308$		0			0
$309$	−179.535	−	81.6038i	−0.581021	−	0.264090i
$310$		0			0
$311$	540.937	+	161.946i	1.73935	+	0.520726i	0.989953	−	0.141399i	$-0.0451600\pi$
$311$	540.937	+	161.946i	1.73935	+	0.520726i	0.749393	+	0.662125i	$0.230345\pi$
$312$		0			0
$313$	−127.811	+	14.9390i	−0.408343	+	0.0477285i	−0.317785	−	0.948163i	$-0.602939\pi$
$313$	−127.811	+	14.9390i	−0.408343	+	0.0477285i	−0.0905580	+	0.995891i	$0.528865\pi$
$314$		0			0
$315$	159.962	−	574.744i	0.507815	−	1.82458i
$316$		0			0
$317$	−170.224	+	258.812i	−0.536983	+	0.816443i	−0.997460	−	0.0712288i	$-0.977308\pi$
$317$	−170.224	+	258.812i	−0.536983	+	0.816443i	0.460477	+	0.887672i	$0.347678\pi$
$318$		0			0
$319$	−24.9466	+	33.5090i	−0.0782024	+	0.105044i
$320$		0			0
$321$	−422.030	−	465.050i	−1.31474	−	1.44875i
$322$		0			0
$323$			760.172i			2.35347i
$324$		0			0
$325$	104.120			0.320371
$326$		0			0
$327$	−26.4679	−	122.175i	−0.0809416	−	0.373625i
$328$		0			0
$329$	277.705	+	206.743i	0.844087	+	0.628399i
$330$		0			0
$331$	208.163	+	136.911i	0.628891	+	0.413628i	0.823551	−	0.567243i	$-0.191990\pi$
$331$	208.163	+	136.911i	0.628891	+	0.413628i	−0.194660	+	0.980871i	$0.562360\pi$
$332$		0			0
$333$	77.6281	+	113.173i	0.233117	+	0.339859i
$334$		0			0
$335$	70.0403	+	599.233i	0.209075	+	1.78876i
$336$		0			0
$337$	−22.4371	+	74.9452i	−0.0665790	+	0.222389i	−0.984804	−	0.173670i	$-0.944437\pi$
$337$	−22.4371	+	74.9452i	−0.0665790	+	0.222389i	0.918225	+	0.396059i	$0.129623\pi$
$338$		0			0
$339$	399.074	−	38.8266i	1.17721	−	0.114533i
$340$		0			0
$341$	210.711	−	37.1541i	0.617922	−	0.108956i
$342$		0			0
$343$	−76.8837	+	436.029i	−0.224151	+	1.27122i
$344$		0			0
$345$	128.640	−	333.209i	0.372869	−	0.965823i
$346$		0			0
$347$	85.9382	+	130.663i	0.247660	+	0.376549i	0.937828	−	0.347100i	$-0.112834\pi$
$347$	85.9382	+	130.663i	0.247660	+	0.376549i	−0.690168	+	0.723649i	$0.742463\pi$
$348$		0			0
$349$	142.642	+	476.457i	0.408716	+	1.36521i	0.876516	+	0.481372i	$0.159861\pi$
$349$	142.642	+	476.457i	0.408716	+	1.36521i	−0.467800	+	0.883834i	$0.654953\pi$
$350$		0			0
$351$	−361.061	+	131.444i	−1.02866	+	0.374486i
$352$		0			0
$353$	83.6354	+	78.9059i	0.236927	+	0.223529i	0.795543	−	0.605897i	$-0.207186\pi$
$353$	83.6354	+	78.9059i	0.236927	+	0.223529i	−0.558615	+	0.829427i	$0.688667\pi$
$354$		0			0
$355$	32.8053	+	563.246i	0.0924094	+	1.58661i
$356$		0			0
$357$	569.614	+	497.077i	1.59556	+	1.39237i
$358$		0			0
$359$	−40.0128	−	109.934i	−0.111456	−	0.306224i	0.871407	−	0.490561i	$-0.163208\pi$
$359$	−40.0128	−	109.934i	−0.111456	−	0.306224i	−0.982863	+	0.184338i	$0.940986\pi$
$360$		0			0
$361$	−823.409	−	299.696i	−2.28091	−	0.830184i
$362$		0			0
$363$	−249.080	−	19.3541i	−0.686170	−	0.0533170i
$364$		0			0
$365$	69.9043	+	294.949i	0.191519	+	0.808080i
$366$		0			0
$367$	−84.6431	+	196.225i	−0.230635	+	0.534672i	−0.993755	−	0.111585i	$-0.964407\pi$
$367$	−84.6431	+	196.225i	−0.230635	+	0.534672i	0.763120	+	0.646257i	$0.223667\pi$
$368$		0			0
$369$	11.7006	−	28.6436i	0.0317090	−	0.0776249i
$370$		0			0
$371$	783.128	+	45.6120i	2.11086	+	0.122943i
$372$		0			0
$373$	88.1077	+	204.257i	0.236214	+	0.547605i	0.994550	−	0.104258i	$-0.0332469\pi$
$373$	88.1077	+	204.257i	0.236214	+	0.547605i	−0.758337	+	0.651863i	$0.773988\pi$
$374$		0			0
$375$	184.743	−	238.372i	0.492649	−	0.635659i
$376$		0			0
$377$	83.8281	−	48.3982i	0.222356	−	0.128377i
$378$		0			0
$379$	289.707	−	501.788i	0.764399	−	1.32398i	−0.176164	−	0.984361i	$-0.556369\pi$
$379$	289.707	−	501.788i	0.764399	−	1.32398i	0.940564	−	0.339618i	$-0.110298\pi$
$380$		0			0
$381$	302.946	−	11.7629i	0.795135	−	0.0308737i
$382$		0			0
$383$	−73.9675	+	632.833i	−0.193127	+	1.65230i	0.454794	+	0.890597i	$0.349713\pi$
$383$	−73.9675	+	632.833i	−0.193127	+	1.65230i	−0.647920	+	0.761708i	$0.724361\pi$
$384$		0			0
$385$	363.828	−	182.721i	0.945008	−	0.474601i
$386$		0			0
$387$	−104.003	+	62.7731i	−0.268743	+	0.162204i
$388$		0			0
$389$	−78.9710	+	58.7917i	−0.203010	+	0.151136i	−0.693863	−	0.720107i	$-0.744093\pi$
$389$	−78.9710	+	58.7917i	−0.203010	+	0.151136i	0.490853	+	0.871243i	$0.336685\pi$
$390$		0			0
$391$	310.608	+	329.225i	0.794394	+	0.842008i
$392$		0			0
$393$	−736.024	−	189.569i	−1.87283	−	0.482364i
$394$		0			0
$395$	−265.537	−	316.454i	−0.672244	−	0.801150i
$396$		0			0
$397$	−349.614	−	293.361i	−0.880640	−	0.738945i	0.0856707	−	0.996324i	$-0.472697\pi$
$397$	−349.614	−	293.361i	−0.880640	−	0.738945i	−0.966311	+	0.257379i	$0.917141\pi$
$398$		0			0
$399$	−1077.34	+	594.483i	−2.70009	+	1.48993i
$400$		0			0
$401$	69.8376	−	139.058i	0.174159	−	0.346778i	−0.789571	−	0.613659i	$-0.789697\pi$
$401$	69.8376	−	139.058i	0.174159	−	0.346778i	0.963729	+	0.266881i	$0.0859931\pi$
$402$		0			0
$403$	−482.400	−	114.331i	−1.19702	−	0.283700i
$404$		0			0
$405$	140.550	−	438.490i	0.347038	−	1.08269i
$406$		0			0
$407$	−21.5986	+	91.1315i	−0.0530677	+	0.223910i
$408$		0			0
$409$	415.908	+	208.877i	1.01689	+	0.510701i	0.877521	−	0.479539i	$-0.159196\pi$
$409$	415.908	+	208.877i	1.01689	+	0.510701i	0.139370	+	0.990240i	$0.455492\pi$
$410$		0			0
$411$	−32.9064	−	0.637439i	−0.0800642	−	0.00155095i
$412$		0			0
$413$	379.195	−	451.907i	0.918147	−	1.09421i
$414$		0			0
$415$	−383.359	+	321.677i	−0.923758	+	0.775125i
$416$		0			0
$417$	509.698	−	141.897i	1.22230	−	0.340281i
$418$		0			0
$419$	188.971	−	178.285i	0.451005	−	0.425501i	−0.426927	−	0.904286i	$-0.640404\pi$
$419$	188.971	−	178.285i	0.451005	−	0.425501i	0.877932	+	0.478785i	$0.158922\pi$
$420$		0			0
$421$	102.618	+	137.840i	0.243748	+	0.327410i	0.907114	−	0.420885i	$-0.138280\pi$
$421$	102.618	+	137.840i	0.243748	+	0.327410i	−0.663366	+	0.748295i	$0.730873\pi$
$422$		0			0
$423$	208.000	+	167.752i	0.491725	+	0.396576i
$424$		0			0
$425$	−70.9624	−	141.298i	−0.166970	−	0.332465i
$426$		0			0
$427$	−256.241	−	29.9502i	−0.600095	−	0.0701411i
$428$		0			0
$429$	−232.006	−	122.201i	−0.540807	−	0.284851i
$430$		0			0
$431$	−85.6611	−	49.4564i	−0.198750	−	0.114748i	0.397323	−	0.917679i	$-0.369939\pi$
$431$	−85.6611	−	49.4564i	−0.198750	−	0.114748i	−0.596072	+	0.802931i	$0.703273\pi$
$432$		0			0
$433$	366.778	+	635.278i	0.847062	+	1.46715i	0.883819	+	0.467830i	$0.154964\pi$
$433$	366.778	+	635.278i	0.847062	+	1.46715i	−0.0367569	+	0.999324i	$0.511703\pi$
$434$		0			0
$435$	−15.6954	+	114.931i	−0.0360814	+	0.264209i
$436$		0			0
$437$	−676.437	+	291.786i	−1.54791	+	0.667703i
$438$		0			0
$439$	17.3442	−	297.789i	0.0395085	−	0.678335i	−0.919173	−	0.393854i	$-0.871142\pi$
$439$	17.3442	−	297.789i	0.0395085	−	0.678335i	0.958682	−	0.284481i	$-0.0918214\pi$
$440$		0			0
$441$	−165.671	+	764.998i	−0.375671	+	1.73469i
$442$		0			0
$443$	246.735	+	106.431i	0.556964	+	0.240251i	0.655906	−	0.754842i	$-0.272287\pi$
$443$	246.735	+	106.431i	0.556964	+	0.240251i	−0.0989420	+	0.995093i	$0.531546\pi$
$444$		0			0
$445$	147.968	−	35.0690i	0.332512	−	0.0788068i
$446$		0			0
$447$	375.383	+	257.445i	0.839784	+	0.575940i
$448$		0			0
$449$	116.174	−	319.185i	0.258739	−	0.710881i	−0.740506	−	0.672049i	$-0.765414\pi$
$449$	116.174	−	319.185i	0.258739	−	0.710881i	0.999246	−	0.0388313i	$-0.0123635\pi$
$450$		0			0
$451$	19.8420	−	7.22189i	0.0439955	−	0.0160131i
$452$		0			0
$453$	−719.179	−	141.225i	−1.58759	−	0.311754i
$454$		0			0
$455$	−941.759	+	54.8512i	−2.06980	+	0.120552i
$456$		0			0
$457$	230.720	−	244.549i	0.504858	−	0.535118i	−0.423947	−	0.905687i	$-0.639356\pi$
$457$	230.720	−	244.549i	0.504858	−	0.535118i	0.928805	+	0.370569i	$0.120837\pi$
$458$		0			0
$459$	424.456	+	400.397i	0.924741	+	0.872324i
$460$		0			0
$461$	−547.795	+	163.999i	−1.18827	+	0.355746i	−0.819120	−	0.573622i	$-0.805538\pi$
$461$	−547.795	+	163.999i	−1.18827	+	0.355746i	−0.369154	+	0.929368i	$0.620353\pi$
$462$		0			0
$463$	−468.978	+	308.451i	−1.01291	+	0.666202i	−0.943495	−	0.331386i	$-0.892484\pi$
$463$	−468.978	+	308.451i	−1.01291	+	0.666202i	−0.0694153	+	0.997588i	$0.522113\pi$
$464$		0			0
$465$	462.423	−	372.998i	0.994458	−	0.802147i
$466$		0			0
$467$	858.425	+	151.363i	1.83817	+	0.324119i	0.981458	−	0.191678i	$-0.0613930\pi$
$467$	858.425	+	151.363i	1.83817	+	0.324119i	0.856711	+	0.515797i	$0.172504\pi$
$468$		0			0
$469$	−214.893	−	1218.72i	−0.458195	−	2.59855i
$470$		0			0
$471$	690.848	−	493.813i	1.46677	−	1.04844i
$472$		0			0
$473$	−79.4189	−	23.7765i	−0.167905	−	0.0502674i
$474$		0			0
$475$	255.609	−	29.8764i	0.538125	−	0.0628978i
$476$		0			0
$477$	602.624	+	58.5871i	1.26336	+	0.122824i
$478$		0			0
$479$	22.1952	−	33.7461i	0.0463365	−	0.0704512i	−0.811577	−	0.584246i	$-0.801390\pi$
$479$	22.1952	−	33.7461i	0.0463365	−	0.0704512i	0.857913	+	0.513795i	$0.171761\pi$
$480$		0			0
$481$	129.588	−	174.066i	0.269413	−	0.361884i
$482$		0			0
$483$	−223.680	+	697.669i	−0.463106	+	1.44445i
$484$		0			0
$485$		−	243.258i		−	0.501562i
$486$		0			0
$487$	−31.4352			−0.0645487			−0.0322743	−	0.999479i	$-0.510275\pi$
$487$	−31.4352			−0.0645487			−0.0322743	+	0.999479i	$0.510275\pi$
$488$		0			0
$489$	686.590	+	220.128i	1.40407	+	0.450160i
$490$		0			0
$491$	−331.401	−	246.719i	−0.674951	−	0.502482i	0.204259	−	0.978917i	$-0.434522\pi$
$491$	−331.401	−	246.719i	−0.674951	−	0.502482i	−0.879210	+	0.476434i	$0.841929\pi$
$492$		0			0
$493$	−122.812	−	80.7745i	−0.249111	−	0.163843i
$494$		0			0
$495$	286.064	−	130.048i	0.577907	−	0.262724i
$496$		0			0
$497$	−134.354	−	1149.47i	−0.270330	−	2.31282i
$498$		0			0
$499$	241.974	−	808.249i	0.484917	−	1.61974i	−0.269261	−	0.963067i	$-0.586779\pi$
$499$	241.974	−	808.249i	0.484917	−	1.61974i	0.754178	−	0.656670i	$-0.228035\pi$
$500$		0			0
$501$	−473.660	−	662.653i	−0.945428	−	1.32266i
$502$		0			0
$503$	−253.898	+	44.7690i	−0.504767	+	0.0890040i	−0.420232	−	0.907416i	$-0.638051\pi$
$503$	−253.898	+	44.7690i	−0.504767	+	0.0890040i	−0.0845346	+	0.996421i	$0.526940\pi$
$504$		0			0
$505$	−137.796	+	781.480i	−0.272863	+	1.54748i
$506$		0			0
$507$	63.1491	+	78.2889i	0.124555	+	0.154416i
$508$		0			0
$509$	−12.7762	−	19.4252i	−0.0251005	−	0.0381634i	0.822720	−	0.568447i	$-0.192456\pi$
$509$	−12.7762	−	19.4252i	−0.0251005	−	0.0381634i	−0.847820	+	0.530284i	$0.822085\pi$
$510$		0			0
$511$	−178.324	−	595.643i	−0.348970	−	1.16564i
$512$		0			0
$513$	−848.666	+	426.291i	−1.65432	+	0.830977i
$514$		0			0
$515$	−271.818	−	256.447i	−0.527802	−	0.497956i
$516$		0			0
$517$	10.6032	+	182.049i	0.0205090	+	0.352126i
$518$		0			0
$519$	−162.343	+	826.726i	−0.312800	+	1.59292i
$520$		0			0
$521$	−62.8514	−	172.683i	−0.120636	−	0.331445i	0.864646	−	0.502382i	$-0.167543\pi$
$521$	−62.8514	−	172.683i	−0.120636	−	0.331445i	−0.985282	+	0.170937i	$0.945321\pi$
$522$		0			0
$523$	−671.395	−	244.368i	−1.28374	−	0.467242i	−0.392071	−	0.919935i	$-0.628241\pi$
$523$	−671.395	−	244.368i	−1.28374	−	0.467242i	−0.891667	+	0.452693i	$0.850463\pi$
$524$		0			0
$525$	144.756	−	211.070i	0.275726	−	0.402038i
$526$		0			0
$527$	173.622	+	732.568i	0.329453	+	1.39007i
$528$		0			0
$529$	35.7907	−	82.9721i	0.0676573	−	0.156847i
$530$		0			0
$531$	305.962	−	337.198i	0.576199	−	0.635025i
$532$		0			0
$533$	−48.8429	−	2.84478i	−0.0916378	−	0.00533729i
$534$		0			0
$535$	−471.336	−	1092.68i	−0.881002	−	2.04239i
$536$		0			0
$537$	−373.425	−	50.9964i	−0.695392	−	0.0949653i
$538$		0			0
$539$	−462.599	+	267.082i	−0.858255	+	0.495514i
$540$		0			0
$541$	225.360	−	390.335i	0.416561	−	0.721506i	−0.579030	−	0.815307i	$-0.696568\pi$
$541$	225.360	−	390.335i	0.416561	−	0.721506i	0.995591	+	0.0938010i	$0.0299017\pi$
$542$		0			0
$543$	139.605	−	265.049i	0.257099	−	0.488119i
$544$		0			0
$545$	27.5004	−	235.281i	0.0504594	−	0.431708i
$546$		0			0
$547$	−733.120	+	368.187i	−1.34026	+	0.673102i	−0.967036	−	0.254640i	$-0.918043\pi$
$547$	−733.120	+	368.187i	−1.34026	+	0.673102i	−0.373221	+	0.927743i	$0.621747\pi$
$548$		0			0
$549$	−196.730	−	30.7585i	−0.358343	−	0.0560264i
$550$		0			0
$551$	191.905	−	142.868i	0.348286	−	0.259289i
$552$		0			0
$553$	581.493	+	616.347i	1.05152	+	1.11455i
$554$		0			0
$555$	69.7454	+	250.527i	0.125667	+	0.451400i
$556$		0			0
$557$	431.034	+	513.686i	0.773848	+	0.922237i	0.998638	−	0.0521707i	$-0.0166140\pi$
$557$	431.034	+	513.686i	0.773848	+	0.922237i	−0.224790	+	0.974407i	$0.572170\pi$
$558$		0			0
$559$	147.148	+	123.472i	0.263235	+	0.220880i
$560$		0			0
$561$	−7.71233	+	398.131i	−0.0137475	+	0.709682i
$562$		0			0
$563$	322.900	−	642.947i	0.573535	−	1.14200i	−0.400646	−	0.916233i	$-0.631214\pi$
$563$	322.900	−	642.947i	0.573535	−	1.14200i	0.974181	−	0.225769i	$-0.0724896\pi$
$564$		0			0
$565$	739.303	+	175.218i	1.30850	+	0.310120i
$566$		0			0
$567$	−235.513	+	914.677i	−0.415367	+	1.61319i
$568$		0			0
$569$	180.519	−	761.670i	0.317257	−	1.33861i	−0.547661	−	0.836700i	$-0.684482\pi$
$569$	180.519	−	761.670i	0.317257	−	1.33861i	0.864918	−	0.501912i	$-0.167370\pi$
$570$		0			0
$571$	946.211	+	475.205i	1.65711	+	0.832233i	0.996686	+	0.0813421i	$0.0259206\pi$
$571$	946.211	+	475.205i	1.65711	+	0.832233i	0.660426	+	0.750891i	$0.270376\pi$
$572$		0			0
$573$	5.02364	+	9.10398i	0.00876726	+	0.0158883i
$574$		0			0
$575$	98.4950	−	117.382i	0.171296	−	0.204142i
$576$		0			0
$577$	636.586	−	534.159i	1.10327	−	0.925752i	0.105628	−	0.994406i	$-0.466315\pi$
$577$	636.586	−	534.159i	1.10327	−	0.925752i	0.997641	+	0.0686535i	$0.0218703\pi$
$578$		0			0
$579$	124.219	−	482.296i	0.214541	−	0.832981i
$580$		0			0
$581$	746.656	−	704.433i	1.28512	−	1.21245i
$582$		0			0
$583$	246.740	+	331.430i	0.423225	+	0.568490i
$584$		0			0
$585$	−727.970	+	14.1534i	−1.24439	+	0.0241939i
$586$		0			0
$587$	−54.2594	−	108.039i	−0.0924350	−	0.184053i	0.842664	−	0.538440i	$-0.180986\pi$
$587$	−54.2594	−	108.039i	−0.0924350	−	0.184053i	−0.935099	+	0.354386i	$0.884690\pi$
$588$		0			0
$589$	−1217.07	−	142.255i	−2.06633	−	0.241519i
$590$		0			0
$591$	−11.9276	−	307.189i	−0.0201821	−	0.519778i
$592$		0			0
$593$	313.520	+	181.011i	0.528702	+	0.305246i	0.740488	−	0.672070i	$-0.234595\pi$
$593$	313.520	+	181.011i	0.528702	+	0.305246i	−0.211786	+	0.977316i	$0.567928\pi$
$594$		0			0
$595$	716.284	+	1240.64i	1.20384	+	2.08511i
$596$		0			0
$597$	−77.0631	−	59.7255i	−0.129084	−	0.100043i
$598$		0			0
$599$	−356.686	+	153.859i	−0.595469	+	0.256860i	−0.672414	−	0.740175i	$-0.734743\pi$
$599$	−356.686	+	153.859i	−0.595469	+	0.256860i	0.0769452	+	0.997035i	$0.475483\pi$
$600$		0			0
$601$	15.2482	−	261.801i	0.0253714	−	0.435610i	−0.961573	−	0.274549i	$-0.911471\pi$
$601$	15.2482	−	261.801i	0.0253714	−	0.435610i	0.986945	−	0.161061i	$-0.0514915\pi$
$602$		0			0
$603$	−129.307	−	946.361i	−0.214440	−	1.56942i
$604$		0			0
$605$	−434.690	−	187.507i	−0.718497	−	0.309929i
$606$		0			0
$607$	−83.7301	+	19.8444i	−0.137941	+	0.0326926i	−0.299006	−	0.954251i	$-0.596655\pi$
$607$	−83.7301	+	19.8444i	−0.137941	+	0.0326926i	0.161065	+	0.986944i	$0.448507\pi$
$608$		0			0
$609$	18.4326	−	237.221i	0.0302670	−	0.389525i
$610$		0			0
$611$	144.515	−	397.053i	0.236523	−	0.649841i
$612$		0			0
$613$	12.8212	−	4.66653i	0.0209155	−	0.00761261i	−0.331541	−	0.943441i	$-0.607569\pi$
$613$	12.8212	−	4.66653i	0.0209155	−	0.00761261i	0.352457	+	0.935828i	$0.385346\pi$
$614$		0			0
$615$	38.5501	−	44.1756i	0.0626831	−	0.0718302i
$616$		0			0
$617$	−313.342	+	18.2501i	−0.507847	+	0.0295787i	−0.310156	−	0.950686i	$-0.600381\pi$
$617$	−313.342	+	18.2501i	−0.507847	+	0.0295787i	−0.197691	+	0.980264i	$0.563344\pi$
$618$		0			0
$619$	−102.967	+	109.139i	−0.166344	+	0.176314i	−0.805184	−	0.593026i	$-0.797933\pi$
$619$	−102.967	+	109.139i	−0.166344	+	0.176314i	0.638840	+	0.769340i	$0.279415\pi$
$620$		0			0
$621$	−193.368	+	531.390i	−0.311381	+	0.855701i
$622$		0			0
$623$	−298.817	+	89.4600i	−0.479643	+	0.143596i
$624$		0			0
$625$	630.275	−	414.538i	1.00844	−	0.663261i
$626$		0			0
$627$	−604.625	−	233.424i	−0.964315	−	0.372287i
$628$		0			0
$629$	−324.538	−	57.2248i	−0.515959	−	0.0909775i
$630$		0			0
$631$	147.025	+	833.822i	0.233004	+	1.32143i	0.846777	+	0.531948i	$0.178540\pi$
$631$	147.025	+	833.822i	0.233004	+	1.32143i	−0.613773	+	0.789482i	$0.710349\pi$
$632$		0			0
$633$	56.5328	+	581.065i	0.0893093	+	0.917955i
$634$		0			0
$635$	550.356	+	164.766i	0.866702	+	0.259474i
$636$		0			0
$637$	1229.32	−	143.687i	1.92986	−	0.225569i
$638$		0			0
$639$	−69.2610	−	890.544i	−0.108390	−	1.39365i
$640$		0			0
$641$	−433.403	+	658.957i	−0.676136	+	1.02801i	0.320595	+	0.947216i	$0.396117\pi$
$641$	−433.403	+	658.957i	−0.676136	+	1.02801i	−0.996730	+	0.0807981i	$0.974253\pi$
$642$		0			0
$643$	435.168	−	584.531i	0.676777	−	0.909069i	−0.322520	−	0.946563i	$-0.604530\pi$
$643$	435.168	−	584.531i	0.676777	−	0.909069i	0.999297	+	0.0374936i	$0.0119374\pi$
$644$		0			0
$645$	−224.974	+	48.7380i	−0.348797	+	0.0755628i
$646$		0			0
$647$			540.276i			0.835048i	0.908666	+	0.417524i	$0.137102\pi$
$647$			540.276i			0.835048i	−0.908666	+	0.417524i	$0.862898\pi$
$648$		0			0
$649$	310.726			0.478776
$650$		0			0
$651$	−902.437	+	818.957i	−1.38623	+	1.25800i
$652$		0			0
$653$	217.491	+	161.916i	0.333065	+	0.247958i	0.750679	−	0.660667i	$-0.229726\pi$
$653$	217.491	+	161.916i	0.333065	+	0.247958i	−0.417614	+	0.908624i	$0.637134\pi$
$654$		0			0
$655$	−1203.29	−	791.414i	−1.83708	−	1.20827i
$656$		0			0
$657$	−119.727	−	464.720i	−0.182234	−	0.707336i
$658$		0			0
$659$	96.9441	+	829.409i	0.147108	+	1.25859i	0.842001	+	0.539475i	$0.181377\pi$
$659$	96.9441	+	829.409i	0.147108	+	1.25859i	−0.694894	+	0.719113i	$0.744549\pi$
$660$		0			0
$661$	138.086	−	461.240i	0.208905	−	0.697792i	−0.787744	−	0.616002i	$-0.788751\pi$
$661$	138.086	−	461.240i	0.208905	−	0.697792i	0.996650	−	0.0817896i	$-0.0260636\pi$
$662$		0			0
$663$	381.791	−	839.973i	0.575853	−	1.26693i
$664$		0			0
$665$	−2296.22	+	404.886i	−3.45296	+	0.608851i
$666$		0			0
$667$	24.7366	−	140.288i	0.0370863	−	0.210327i
$668$		0			0
$669$	373.037	−	58.3510i	0.557604	−	0.0872213i
$670$		0			0
$671$	−74.6709	−	113.532i	−0.111283	−	0.169198i
$672$		0			0
$673$	−186.905	−	624.307i	−0.277720	−	0.927648i	−0.977079	−	0.212878i	$-0.931716\pi$
$673$	−186.905	−	624.307i	−0.277720	−	0.927648i	0.699359	−	0.714770i	$-0.253469\pi$
$674$		0			0
$675$	117.952	−	158.461i	0.174744	−	0.234757i
$676$		0			0
$677$	−745.201	−	703.061i	−1.10074	−	1.03849i	−0.999046	−	0.0436787i	$-0.986092\pi$
$677$	−745.201	−	703.061i	−1.10074	−	1.03849i	−0.101694	−	0.994816i	$-0.532426\pi$
$678$		0			0
$679$	29.0127	+	498.129i	0.0427286	+	0.733622i
$680$		0			0
$681$	1255.05	−	429.462i	1.84296	−	0.630634i
$682$		0			0
$683$	−226.518	−	622.353i	−0.331652	−	0.911205i	−0.987683	−	0.156471i	$-0.949988\pi$
$683$	−226.518	−	622.353i	−0.331652	−	0.911205i	0.656031	−	0.754734i	$-0.272234\pi$
$684$		0			0
$685$	−58.6053	−	21.3306i	−0.0855552	−	0.0311396i
$686$		0			0
$687$	131.235	+	274.439i	0.191027	+	0.399474i
$688$		0			0
$689$	−220.789	−	931.583i	−0.320449	−	1.35208i
$690$		0			0
$691$	207.987	−	482.167i	0.300994	−	0.697782i	−0.698853	−	0.715265i	$-0.746306\pi$
$691$	207.987	−	482.167i	0.300994	−	0.697782i	0.999847	+	0.0174831i	$0.00556532\pi$
$692$		0			0
$693$	−570.275	+	300.424i	−0.822907	+	0.433512i
$694$		0			0
$695$	1000.87	+	58.2940i	1.44010	+	0.0838762i
$696$		0			0
$697$	29.4280	+	68.2217i	0.0422209	+	0.0978790i
$698$		0			0
$699$	112.247	+	274.730i	0.160582	+	0.393032i
$700$		0			0
$701$	−646.931	+	373.506i	−0.922868	+	0.532818i	−0.884549	−	0.466447i	$-0.845534\pi$
$701$	−646.931	+	373.506i	−0.922868	+	0.532818i	−0.0383193	+	0.999266i	$0.512200\pi$
$702$		0			0
$703$	268.183	−	464.506i	0.381484	−	0.660749i
$704$		0			0
$705$	269.999	+	428.361i	0.382978	+	0.607604i
$706$		0			0
$707$	188.966	−	1616.70i	0.267278	−	2.28671i
$708$		0			0
$709$	−224.708	+	112.853i	−0.316936	+	0.159171i	−0.600154	−	0.799884i	$-0.704894\pi$
$709$	−224.708	+	112.853i	−0.316936	+	0.159171i	0.283218	+	0.959055i	$0.408598\pi$
$710$		0			0
$711$	430.053	+	492.739i	0.604856	+	0.693022i
$712$		0			0
$713$	−585.230	+	435.687i	−0.820799	+	0.611062i
$714$		0			0
$715$	−340.984	−	361.422i	−0.476901	−	0.505486i
$716$		0			0
$717$	−28.7184	+	29.2822i	−0.0400535	+	0.0408398i
$718$		0			0
$719$	76.7779	+	91.5004i	0.106784	+	0.127261i	0.816789	−	0.576936i	$-0.195752\pi$
$719$	76.7779	+	91.5004i	0.106784	+	0.127261i	−0.710005	+	0.704197i	$0.751307\pi$
$720$		0			0
$721$	587.199	+	492.719i	0.814423	+	0.683382i
$722$		0			0
$723$	−254.704	−	153.706i	−0.352287	−	0.212595i
$724$		0			0
$725$	−22.3338	+	44.4703i	−0.0308052	+	0.0613383i
$726$		0			0
$727$	−3.58303	−	0.849193i	−0.00492851	−	0.00116808i	0.228151	−	0.973626i	$-0.426732\pi$
$727$	−3.58303	−	0.849193i	−0.00492851	−	0.00116808i	−0.233079	+	0.972458i	$0.574880\pi$
$728$		0			0
$729$	−208.980	+	698.404i	−0.286667	+	0.958030i
$730$		0			0
$731$	67.2715	−	283.841i	0.0920266	−	0.388291i
$732$		0			0
$733$	−325.858	−	163.652i	−0.444554	−	0.223263i	0.212417	−	0.977179i	$-0.431866\pi$
$733$	−325.858	−	163.652i	−0.444554	−	0.223263i	−0.656971	+	0.753916i	$0.728163\pi$
$734$		0			0
$735$	−766.344	+	1269.89i	−1.04264	+	1.72775i
$736$		0			0
$737$	418.989	−	499.332i	0.568506	−	0.677519i
$738$		0			0
$739$	410.682	−	344.603i	0.555726	−	0.466310i	−0.321148	−	0.947029i	$-0.604069\pi$
$739$	410.682	−	344.603i	0.555726	−	0.466310i	0.876875	+	0.480719i	$0.159624\pi$
$740$		0			0
$741$	1072.16	+	1051.51i	1.44691	+	1.41905i
$742$		0			0
$743$	530.708	−	500.697i	0.714277	−	0.673886i	−0.240688	−	0.970603i	$-0.577373\pi$
$743$	530.708	−	500.697i	0.714277	−	0.673886i	0.954966	+	0.296716i	$0.0958916\pi$
$744$		0			0
$745$	515.068	+	691.857i	0.691367	+	0.928666i
$746$		0			0
$747$	596.914	−	520.975i	0.799082	−	0.697423i
$748$		0			0
$749$	1095.50	+	2181.31i	1.46261	+	2.91230i
$750$		0			0
$751$	572.563	+	66.9230i	0.762400	+	0.0891118i	0.488402	−	0.872619i	$-0.337580\pi$
$751$	572.563	+	66.9230i	0.762400	+	0.0891118i	0.273998	+	0.961730i	$0.411654\pi$
$752$		0			0
$753$	−620.753	+	391.266i	−0.824373	+	0.519610i
$754$		0			0
$755$	−1202.75	−	694.405i	−1.59304	−	0.919742i
$756$		0			0
$757$	76.0153	+	131.662i	0.100416	+	0.173926i	0.911856	−	0.410510i	$-0.134649\pi$
$757$	76.0153	+	131.662i	0.100416	+	0.173926i	−0.811440	+	0.584436i	$0.801316\pi$
$758$		0			0
$759$	−357.236	+	145.957i	−0.470667	+	0.192302i
$760$		0			0
$761$	−18.5423	+	7.99837i	−0.0243657	+	0.0105103i	−0.408228	−	0.912880i	$-0.633853\pi$
$761$	−18.5423	+	7.99837i	−0.0243657	+	0.0105103i	0.383862	+	0.923390i	$0.374594\pi$
$762$		0			0
$763$	−28.2524	+	485.075i	−0.0370280	+	0.635746i
$764$		0			0
$765$	515.349	+	978.254i	0.673659	+	1.27876i
$766$		0			0
$767$	−661.089	−	285.166i	−0.861916	−	0.371794i
$768$		0			0
$769$	−30.1241	+	7.13955i	−0.0391731	+	0.00928420i	−0.250156	−	0.968206i	$-0.580482\pi$
$769$	−30.1241	+	7.13955i	−0.0391731	+	0.00928420i	0.210983	+	0.977490i	$0.432334\pi$
$770$		0			0
$771$	−953.471	+	455.945i	−1.23667	+	0.591369i
$772$		0			0
$773$	−400.057	+	1099.15i	−0.517538	+	1.42192i	0.355687	+	0.934605i	$0.384247\pi$
$773$	−400.057	+	1099.15i	−0.517538	+	1.42192i	−0.873225	+	0.487318i	$0.837975\pi$
$774$		0			0
$775$	239.504	−	87.1721i	0.309037	−	0.112480i
$776$		0			0
$777$	−172.700	−	504.697i	−0.222266	−	0.649545i
$778$		0			0
$779$	−120.723	+	7.03130i	−0.154971	+	0.00902605i
$780$		0			0
$781$	418.315	−	443.388i	0.535615	−	0.567719i
$782$		0			0
$783$	21.3070	−	182.405i	0.0272120	−	0.232957i
$784$		0			0
$785$	1541.54	−	461.507i	1.96375	−	0.587907i
$786$		0			0
$787$	928.845	−	610.911i	1.18024	−	0.776253i	0.201122	−	0.979566i	$-0.435541\pi$
$787$	928.845	−	610.911i	1.18024	−	0.776253i	0.979114	+	0.203313i	$0.0651710\pi$
$788$		0			0
$789$	202.805	+	1296.53i	0.257041	+	1.64326i
$790$		0			0
$791$	−1534.80	−	270.626i	−1.94033	−	0.342132i
$792$		0			0
$793$	54.6745	+	310.075i	0.0689465	+	0.391015i
$794$		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:	\( N \)	\(=\)	\( 324 = 2^{2} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	324.o (of order \(54\), degree \(18\), minimal)

\(f(q)\)	\(=\)	\(q+(-2.85677 - 0.915910i) q^{3} +(-4.55987 - 3.39470i) q^{5} +(9.74231 + 6.40762i) q^{7} +(7.32222 + 5.23308i) q^{9} +O(q^{10})\)	\(q+(-2.85677 - 0.915910i) q^{3} +(-4.55987 - 3.39470i) q^{5} +(9.74231 + 6.40762i) q^{7} +(7.32222 + 5.23308i) q^{9} +(0.713033 + 6.10039i) q^{11} +(4.08156 - 13.6334i) q^{13} +(9.91723 + 13.8743i) q^{15} +(-21.2831 + 3.75278i) q^{17} +(6.10801 - 34.6402i) q^{19} +(-21.9627 - 27.2281i) q^{21} +(-11.5087 - 17.4982i) q^{23} +(2.09835 + 7.00898i) q^{25} +(-16.1248 - 21.6562i) q^{27} +(4.94737 + 4.66760i) q^{29} +(-2.02555 - 34.7774i) q^{31} +(3.55044 - 18.0805i) q^{33} +(-22.6717 - 62.2901i) q^{35} +(14.3291 + 5.21535i) q^{37} +(-24.1470 + 35.2090i) q^{39} +(-0.792837 - 3.34524i) q^{41} +(-5.34616 + 12.3938i) q^{43} +(-15.6236 - 48.7188i) q^{45} +(29.6405 + 1.72636i) q^{47} +(34.4471 + 79.8574i) q^{49} +(64.2379 + 8.77257i) q^{51} +(58.2609 - 33.6369i) q^{53} +(17.4576 - 30.2375i) q^{55} +(-49.1765 + 93.3646i) q^{57} +(5.87325 - 50.2489i) q^{59} +(-19.7711 + 9.92944i) q^{61} +(37.8037 + 97.9003i) q^{63} +(-64.8925 + 48.3107i) q^{65} +(-72.8296 - 77.1949i) q^{67} +(16.8510 + 60.5292i) q^{69} +(-63.7955 - 76.0285i) q^{71} +(-40.8467 - 34.2745i) q^{73} +(0.425101 - 21.9449i) q^{75} +(-32.1424 + 64.0007i) q^{77} +(70.7096 + 16.7585i) q^{79} +(26.2297 + 76.6355i) q^{81} +(20.3016 - 85.6592i) q^{83} +(109.787 + 55.1373i) q^{85} +(-9.85837 - 17.8656i) q^{87} +(-17.1945 + 20.4917i) q^{89} +(127.121 - 106.667i) q^{91} +(-26.0664 + 101.206i) q^{93} +(-145.445 + 137.220i) q^{95} +(25.5532 + 34.3239i) q^{97} +(-26.7029 + 48.3997i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 324 q+O(q^{10}) \) 324 * q	\( 324 q - 135 q^{21} - 81 q^{23} + 27 q^{27} + 81 q^{29} + 189 q^{33} + 243 q^{35} + 216 q^{41} + 432 q^{45} + 324 q^{47} + 126 q^{51} - 216 q^{57} - 378 q^{59} - 540 q^{63} - 108 q^{65} - 351 q^{67} + 504 q^{69} + 648 q^{71} + 450 q^{75} + 432 q^{77} - 54 q^{79} - 72 q^{81} - 216 q^{83} + 270 q^{85} - 1008 q^{87} - 648 q^{89} - 684 q^{93} - 432 q^{95} + 459 q^{97} - 252 q^{99}+O(q^{100}) \) 324 * q - 135 * q^21 - 81 * q^23 + 27 * q^27 + 81 * q^29 + 189 * q^33 + 243 * q^35 + 216 * q^41 + 432 * q^45 + 324 * q^47 + 126 * q^51 - 216 * q^57 - 378 * q^59 - 540 * q^63 - 108 * q^65 - 351 * q^67 + 504 * q^69 + 648 * q^71 + 450 * q^75 + 432 * q^77 - 54 * q^79 - 72 * q^81 - 216 * q^83 + 270 * q^85 - 1008 * q^87 - 648 * q^89 - 684 * q^93 - 432 * q^95 + 459 * q^97 - 252 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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