LMFDB - Embedded newform 441.4.e.k.361.1

Newspace parameters

Level:	$ N $	$=$	$ 441 = 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	441.e (of order $3$, degree $2$, not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$26.0198423125$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
Defining polynomial:	$x^{2} - x + 1$
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 7)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			361.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	441.361
Dual form			441.4.e.k.226.1

$q$-expansion

$f(q)$	$=$	$q+(1.00000 + 1.73205i) q^{2} +(2.00000 - 3.46410i) q^{4} +(-3.50000 - 6.06218i) q^{5} +24.0000 q^{8} +O(q^{10})$	\(q+(1.00000 + 1.73205i) q^{2} +(2.00000 - 3.46410i) q^{4} +(-3.50000 - 6.06218i) q^{5} +24.0000 q^{8} +(7.00000 - 12.1244i) q^{10} +(-2.50000 + 4.33013i) q^{11} +14.0000 q^{13} +(8.00000 + 13.8564i) q^{16} +(10.5000 - 18.1865i) q^{17} +(24.5000 + 42.4352i) q^{19} -28.0000 q^{20} -10.0000 q^{22} +(-79.5000 - 137.698i) q^{23} +(38.0000 - 65.8179i) q^{25} +(14.0000 + 24.2487i) q^{26} -58.0000 q^{29} +(73.5000 - 127.306i) q^{31} +(80.0000 - 138.564i) q^{32} +42.0000 q^{34} +(-109.500 - 189.660i) q^{37} +(-49.0000 + 84.8705i) q^{38} +(-84.0000 - 145.492i) q^{40} +350.000 q^{41} -124.000 q^{43} +(10.0000 + 17.3205i) q^{44} +(159.000 - 275.396i) q^{46} +(-262.500 - 454.663i) q^{47} +152.000 q^{50} +(28.0000 - 48.4974i) q^{52} +(151.500 - 262.406i) q^{53} +35.0000 q^{55} +(-58.0000 - 100.459i) q^{58} +(52.5000 - 90.9327i) q^{59} +(-206.500 - 357.668i) q^{61} +294.000 q^{62} +448.000 q^{64} +(-49.0000 - 84.8705i) q^{65} +(-207.500 + 359.401i) q^{67} +(-42.0000 - 72.7461i) q^{68} +432.000 q^{71} +(-556.500 + 963.886i) q^{73} +(219.000 - 379.319i) q^{74} +196.000 q^{76} +(51.5000 + 89.2006i) q^{79} +(56.0000 - 96.9948i) q^{80} +(350.000 + 606.218i) q^{82} +1092.00 q^{83} -147.000 q^{85} +(-124.000 - 214.774i) q^{86} +(-60.0000 + 103.923i) q^{88} +(164.500 + 284.922i) q^{89} -636.000 q^{92} +(525.000 - 909.327i) q^{94} +(171.500 - 297.047i) q^{95} +882.000 q^{97} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2q + 2q^{2} + 4q^{4} - 7q^{5} + 48q^{8} + O(q^{10}) $	$ 2q + 2q^{2} + 4q^{4} - 7q^{5} + 48q^{8} + 14q^{10} - 5q^{11} + 28q^{13} + 16q^{16} + 21q^{17} + 49q^{19} - 56q^{20} - 20q^{22} - 159q^{23} + 76q^{25} + 28q^{26} - 116q^{29} + 147q^{31} + 160q^{32} + 84q^{34} - 219q^{37} - 98q^{38} - 168q^{40} + 700q^{41} - 248q^{43} + 20q^{44} + 318q^{46} - 525q^{47} + 304q^{50} + 56q^{52} + 303q^{53} + 70q^{55} - 116q^{58} + 105q^{59} - 413q^{61} + 588q^{62} + 896q^{64} - 98q^{65} - 415q^{67} - 84q^{68} + 864q^{71} - 1113q^{73} + 438q^{74} + 392q^{76} + 103q^{79} + 112q^{80} + 700q^{82} + 2184q^{83} - 294q^{85} - 248q^{86} - 120q^{88} + 329q^{89} - 1272q^{92} + 1050q^{94} + 343q^{95} + 1764q^{97} + O(q^{100}) $

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/441\mathbb{Z}\right)^\times$.

$n$	$199$	$344$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$1$

Coefficient data

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

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

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

$2$	1.00000	+	1.73205i	0.353553	+	0.612372i	0.986869	−	0.161521i	$-0.0516399\pi$
$2$	1.00000	+	1.73205i	0.353553	+	0.612372i	−0.633316	+	0.773893i	$0.718307\pi$
$3$		0			0
$3$		0			0
$4$	2.00000	−	3.46410i	0.250000	−	0.433013i
$5$	−3.50000	−	6.06218i	−0.313050	−	0.542218i	0.665971	−	0.745977i	$-0.268017\pi$
$5$	−3.50000	−	6.06218i	−0.313050	−	0.542218i	−0.979021	+	0.203760i	$0.934684\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$	24.0000			1.06066
$9$		0			0
$10$	7.00000	−	12.1244i	0.221359	−	0.383406i
$11$	−2.50000	+	4.33013i	−0.0685253	+	0.118689i	−0.898252	−	0.439480i	$-0.855163\pi$
$11$	−2.50000	+	4.33013i	−0.0685253	+	0.118689i	0.829727	+	0.558169i	$0.188496\pi$
$12$		0			0
$13$	14.0000			0.298685			0.149342	−	0.988786i	$-0.452284\pi$
$13$	14.0000			0.298685			0.149342	+	0.988786i	$0.452284\pi$
$14$		0			0
$15$		0			0
$16$	8.00000	+	13.8564i	0.125000	+	0.216506i
$17$	10.5000	−	18.1865i	0.149801	−	0.259464i	−0.781353	−	0.624090i	$-0.785470\pi$
$17$	10.5000	−	18.1865i	0.149801	−	0.259464i	0.931154	+	0.364626i	$0.118803\pi$
$18$		0			0
$19$	24.5000	+	42.4352i	0.295826	+	0.512385i	0.975177	−	0.221429i	$-0.0710720\pi$
$19$	24.5000	+	42.4352i	0.295826	+	0.512385i	−0.679351	+	0.733813i	$0.737739\pi$
$20$	−28.0000			−0.313050
$21$		0			0
$22$	−10.0000			−0.0969094
$23$	−79.5000	−	137.698i	−0.720735	−	1.24835i	−0.960706	−	0.277569i	$-0.910471\pi$
$23$	−79.5000	−	137.698i	−0.720735	−	1.24835i	0.239971	−	0.970780i	$-0.422862\pi$
$24$		0			0
$25$	38.0000	−	65.8179i	0.304000	−	0.526543i
$26$	14.0000	+	24.2487i	0.105601	+	0.182906i
$27$		0			0
$28$		0			0
$29$	−58.0000			−0.371391			−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−58.0000			−0.371391			−0.185695	+	0.982607i	$0.559454\pi$
$30$		0			0
$31$	73.5000	−	127.306i	0.425838	−	0.737574i	−0.570660	−	0.821186i	$-0.693313\pi$
$31$	73.5000	−	127.306i	0.425838	−	0.737574i	0.996498	+	0.0836128i	$0.0266459\pi$
$32$	80.0000	−	138.564i	0.441942	−	0.765466i
$33$		0			0
$34$	42.0000			0.211851
$35$		0			0
$36$		0			0
$37$	−109.500	−	189.660i	−0.486532	−	0.842698i	0.513348	−	0.858181i	$-0.328405\pi$
$37$	−109.500	−	189.660i	−0.486532	−	0.842698i	−0.999880	+	0.0154821i	$0.995072\pi$
$38$	−49.0000	+	84.8705i	−0.209180	+	0.362311i
$39$		0			0
$40$	−84.0000	−	145.492i	−0.332039	−	0.575109i
$41$	350.000			1.33319			0.666595	−	0.745420i	$-0.267751\pi$
$41$	350.000			1.33319			0.666595	+	0.745420i	$0.267751\pi$
$42$		0			0
$43$	−124.000			−0.439763			−0.219882	−	0.975527i	$-0.570567\pi$
$43$	−124.000			−0.439763			−0.219882	+	0.975527i	$0.570567\pi$
$44$	10.0000	+	17.3205i	0.0342627	+	0.0593447i
$45$		0			0
$46$	159.000	−	275.396i	0.509636	−	0.882716i
$47$	−262.500	−	454.663i	−0.814671	−	1.41105i	−0.909564	−	0.415565i	$-0.863584\pi$
$47$	−262.500	−	454.663i	−0.814671	−	1.41105i	0.0948921	−	0.995488i	$-0.469749\pi$
$48$		0			0
$49$		0			0
$50$	152.000			0.429921
$51$		0			0
$52$	28.0000	−	48.4974i	0.0746712	−	0.129334i
$53$	151.500	−	262.406i	0.392644	−	0.680079i	−0.600153	−	0.799885i	$-0.704894\pi$
$53$	151.500	−	262.406i	0.392644	−	0.680079i	0.992797	+	0.119806i	$0.0382272\pi$
$54$		0			0
$55$	35.0000			0.0858073
$56$		0			0
$57$		0			0
$58$	−58.0000	−	100.459i	−0.131306	−	0.227429i
$59$	52.5000	−	90.9327i	0.115846	−	0.200651i	−0.802272	−	0.596959i	$-0.796375\pi$
$59$	52.5000	−	90.9327i	0.115846	−	0.200651i	0.918118	+	0.396308i	$0.129709\pi$
$60$		0			0
$61$	−206.500	−	357.668i	−0.433436	−	0.750734i	0.563730	−	0.825959i	$-0.309366\pi$
$61$	−206.500	−	357.668i	−0.433436	−	0.750734i	−0.997167	+	0.0752252i	$0.976032\pi$
$62$	294.000			0.602226
$63$		0			0
$64$	448.000			0.875000
$65$	−49.0000	−	84.8705i	−0.0935031	−	0.161952i
$66$		0			0
$67$	−207.500	+	359.401i	−0.378361	+	0.655340i	−0.990824	−	0.135159i	$-0.956845\pi$
$67$	−207.500	+	359.401i	−0.378361	+	0.655340i	0.612463	+	0.790499i	$0.290179\pi$
$68$	−42.0000	−	72.7461i	−0.0749007	−	0.129732i
$69$		0			0
$70$		0			0
$71$	432.000			0.722098			0.361049	−	0.932547i	$-0.382419\pi$
$71$	432.000			0.722098			0.361049	+	0.932547i	$0.382419\pi$
$72$		0			0
$73$	−556.500	+	963.886i	−0.892238	+	1.54540i	−0.0550526	+	0.998483i	$0.517533\pi$
$73$	−556.500	+	963.886i	−0.892238	+	1.54540i	−0.837186	+	0.546919i	$0.815801\pi$
$74$	219.000	−	379.319i	0.344030	−	0.595878i
$75$		0			0
$76$	196.000			0.295826
$77$		0			0
$78$		0			0
$79$	51.5000	+	89.2006i	0.0733443	+	0.127036i	0.900365	−	0.435135i	$-0.143299\pi$
$79$	51.5000	+	89.2006i	0.0733443	+	0.127036i	−0.827021	+	0.562171i	$0.809966\pi$
$80$	56.0000	−	96.9948i	0.0782624	−	0.135554i
$81$		0			0
$82$	350.000	+	606.218i	0.471354	+	0.816409i
$83$	1092.00			1.44413			0.722064	−	0.691827i	$-0.243194\pi$
$83$	1092.00			1.44413			0.722064	+	0.691827i	$0.243194\pi$
$84$		0			0
$85$	−147.000			−0.187581
$86$	−124.000	−	214.774i	−0.155480	−	0.269299i
$87$		0			0
$88$	−60.0000	+	103.923i	−0.0726821	+	0.125889i
$89$	164.500	+	284.922i	0.195921	+	0.339345i	0.947202	−	0.320637i	$-0.103897\pi$
$89$	164.500	+	284.922i	0.195921	+	0.339345i	−0.751281	+	0.659982i	$0.770564\pi$
$90$		0			0
$91$		0			0
$92$	−636.000			−0.720735
$93$		0			0
$94$	525.000	−	909.327i	0.576060	−	0.997765i
$95$	171.500	−	297.047i	0.185216	−	0.320804i
$96$		0			0
$97$	882.000			0.923232			0.461616	−	0.887080i	$-0.347270\pi$
$97$	882.000			0.923232			0.461616	+	0.887080i	$0.347270\pi$
$98$		0			0
$99$		0			0
$100$	−152.000	−	263.272i	−0.152000	−	0.263272i
$101$	−689.500	+	1194.25i	−0.679285	+	1.17656i	0.295911	+	0.955215i	$0.404377\pi$
$101$	−689.500	+	1194.25i	−0.679285	+	1.17656i	−0.975196	+	0.221341i	$0.928957\pi$
$102$		0			0
$103$	−339.500	−	588.031i	−0.324776	−	0.562529i	0.656691	−	0.754160i	$-0.271956\pi$
$103$	−339.500	−	588.031i	−0.324776	−	0.562529i	−0.981467	+	0.191631i	$0.938622\pi$
$104$	336.000			0.316803
$105$		0			0
$106$	606.000			0.555282
$107$	228.500	+	395.774i	0.206448	+	0.357578i	0.950593	−	0.310440i	$-0.100476\pi$
$107$	228.500	+	395.774i	0.206448	+	0.357578i	−0.744145	+	0.668018i	$0.767143\pi$
$108$		0			0
$109$	562.500	−	974.279i	0.494291	−	0.856137i	−0.505687	−	0.862717i	$-0.668761\pi$
$109$	562.500	−	974.279i	0.494291	−	0.856137i	0.999978	+	0.00657959i	$0.00209436\pi$
$110$	35.0000	+	60.6218i	0.0303374	+	0.0525460i
$111$		0			0
$112$		0			0
$113$	1538.00			1.28038			0.640190	−	0.768217i	$-0.278856\pi$
$113$	1538.00			1.28038			0.640190	+	0.768217i	$0.278856\pi$
$114$		0			0
$115$	−556.500	+	963.886i	−0.451251	+	0.781590i
$116$	−116.000	+	200.918i	−0.0928477	+	0.160817i
$117$		0			0
$118$	210.000			0.163831
$119$		0			0
$120$		0			0
$121$	653.000	+	1131.03i	0.490609	+	0.849759i
$122$	413.000	−	715.337i	0.306486	−	0.530849i
$123$		0			0
$124$	−294.000	−	509.223i	−0.212919	−	0.368787i
$125$	−1407.00			−1.00677
$126$		0			0
$127$	72.0000			0.0503068			0.0251534	−	0.999684i	$-0.491993\pi$
$127$	72.0000			0.0503068			0.0251534	+	0.999684i	$0.491993\pi$
$128$	−192.000	−	332.554i	−0.132583	−	0.229640i
$129$		0			0
$130$	98.0000	−	169.741i	0.0661167	−	0.114517i
$131$	−1074.50	−	1861.09i	−0.716637	−	1.24125i	−0.962325	−	0.271903i	$-0.912347\pi$
$131$	−1074.50	−	1861.09i	−0.716637	−	1.24125i	0.245687	−	0.969349i	$-0.420986\pi$
$132$		0			0
$133$		0			0
$134$	−830.000			−0.535083
$135$		0			0
$136$	252.000	−	436.477i	0.158888	−	0.275203i
$137$	−562.500	+	974.279i	−0.350786	+	0.607578i	−0.986387	−	0.164439i	$-0.947419\pi$
$137$	−562.500	+	974.279i	−0.350786	+	0.607578i	0.635602	+	0.772017i	$0.280752\pi$
$138$		0			0
$139$	−252.000			−0.153772			−0.0768862	−	0.997040i	$-0.524498\pi$
$139$	−252.000			−0.153772			−0.0768862	+	0.997040i	$0.524498\pi$
$140$		0			0
$141$		0			0
$142$	432.000	+	748.246i	0.255300	+	0.442193i
$143$	−35.0000	+	60.6218i	−0.0204675	+	0.0354507i
$144$		0			0
$145$	203.000	+	351.606i	0.116264	+	0.201375i
$146$	−2226.00			−1.26182
$147$		0			0
$148$	−876.000			−0.486532
$149$	−100.500	−	174.071i	−0.0552569	−	0.0957078i	0.837074	−	0.547090i	$-0.184264\pi$
$149$	−100.500	−	174.071i	−0.0552569	−	0.0957078i	−0.892331	+	0.451382i	$0.850931\pi$
$150$		0			0
$151$	−809.500	+	1402.10i	−0.436266	+	0.755635i	−0.997398	−	0.0720914i	$-0.977033\pi$
$151$	−809.500	+	1402.10i	−0.436266	+	0.755635i	0.561132	+	0.827726i	$0.310366\pi$
$152$	588.000	+	1018.45i	0.313770	+	0.543466i
$153$		0			0
$154$		0			0
$155$	−1029.00			−0.533234
$156$		0			0
$157$	339.500	−	588.031i	0.172580	−	0.298917i	−0.766741	−	0.641956i	$-0.778123\pi$
$157$	339.500	−	588.031i	0.172580	−	0.298917i	0.939321	+	0.343039i	$0.111456\pi$
$158$	−103.000	+	178.401i	−0.0518623	+	0.0898281i
$159$		0			0
$160$	−1120.00			−0.553399
$161$		0			0
$162$		0			0
$163$	233.500	+	404.434i	0.112203	+	0.194342i	0.916658	−	0.399672i	$-0.130876\pi$
$163$	233.500	+	404.434i	0.112203	+	0.194342i	−0.804455	+	0.594014i	$0.797543\pi$
$164$	700.000	−	1212.44i	0.333298	−	0.577288i
$165$		0			0
$166$	1092.00	+	1891.40i	0.510576	+	0.884344i
$167$	1204.00			0.557894			0.278947	−	0.960306i	$-0.410015\pi$
$167$	1204.00			0.557894			0.278947	+	0.960306i	$0.410015\pi$
$168$		0			0
$169$	−2001.00			−0.910787
$170$	−147.000	−	254.611i	−0.0663199	−	0.114869i
$171$		0			0
$172$	−248.000	+	429.549i	−0.109941	+	0.190423i
$173$	1410.50	+	2443.06i	0.619875	+	1.07365i	0.989508	+	0.144477i	$0.0461499\pi$
$173$	1410.50	+	2443.06i	0.619875	+	1.07365i	−0.369633	+	0.929178i	$0.620517\pi$
$174$		0			0
$175$		0			0
$176$	−80.0000			−0.0342627
$177$		0			0
$178$	−329.000	+	569.845i	−0.138537	+	0.239953i
$179$	−1626.50	+	2817.18i	−0.679164	+	1.17635i	0.296069	+	0.955166i	$0.404324\pi$
$179$	−1626.50	+	2817.18i	−0.679164	+	1.17635i	−0.975233	+	0.221180i	$0.929009\pi$
$180$		0			0
$181$	−1582.00			−0.649664			−0.324832	−	0.945772i	$-0.605308\pi$
$181$	−1582.00			−0.649664			−0.324832	+	0.945772i	$0.605308\pi$
$182$		0			0
$183$		0			0
$184$	−1908.00	−	3304.75i	−0.764454	−	1.32407i
$185$	−766.500	+	1327.62i	−0.304617	+	0.527613i
$186$		0			0
$187$	52.5000	+	90.9327i	0.0205304	+	0.0355597i
$188$	−2100.00			−0.814671
$189$		0			0
$190$	686.000			0.261935
$191$	1278.50	+	2214.43i	0.484340	+	0.838902i	0.999838	−	0.0179887i	$-0.00572630\pi$
$191$	1278.50	+	2214.43i	0.484340	+	0.838902i	−0.515498	+	0.856891i	$0.672393\pi$
$192$		0			0
$193$	198.500	−	343.812i	0.0740329	−	0.128229i	−0.826632	−	0.562742i	$-0.809746\pi$
$193$	198.500	−	343.812i	0.0740329	−	0.128229i	0.900665	+	0.434514i	$0.143080\pi$
$194$	882.000	+	1527.67i	0.326412	+	0.565362i
$195$		0			0
$196$		0			0
$197$	−2914.00			−1.05388			−0.526939	−	0.849903i	$-0.676660\pi$
$197$	−2914.00			−1.05388			−0.526939	+	0.849903i	$0.676660\pi$
$198$		0			0
$199$	1669.50	−	2891.66i	0.594712	−	1.03007i	−0.398875	−	0.917005i	$-0.630599\pi$
$199$	1669.50	−	2891.66i	0.594712	−	1.03007i	0.993587	−	0.113066i	$-0.0360673\pi$
$200$	912.000	−	1579.63i	0.322441	−	0.558484i
$201$		0			0
$202$	−2758.00			−0.960654
$203$		0			0
$204$		0			0
$205$	−1225.00	−	2121.76i	−0.417355	−	0.722880i
$206$	679.000	−	1176.06i	0.229651	−	0.397768i
$207$		0			0
$208$	112.000	+	193.990i	0.0373356	+	0.0646671i
$209$	−245.000			−0.0810861
$210$		0			0
$211$	1780.00			0.580759			0.290380	−	0.956911i	$-0.406218\pi$
$211$	1780.00			0.580759			0.290380	+	0.956911i	$0.406218\pi$
$212$	−606.000	−	1049.62i	−0.196322	−	0.340040i
$213$		0			0
$214$	−457.000	+	791.547i	−0.145981	+	0.252846i
$215$	434.000	+	751.710i	0.137668	+	0.238447i
$216$		0			0
$217$		0			0
$218$	2250.00			0.699033
$219$		0			0
$220$	70.0000	−	121.244i	0.0214518	−	0.0371556i
$221$	147.000	−	254.611i	0.0447434	−	0.0774978i
$222$		0			0
$223$	1400.00			0.420408			0.210204	−	0.977658i	$-0.432587\pi$
$223$	1400.00			0.420408			0.210204	+	0.977658i	$0.432587\pi$
$224$		0			0
$225$		0			0
$226$	1538.00	+	2663.89i	0.452682	+	0.784069i
$227$	1102.50	−	1909.59i	0.322359	−	0.558342i	−0.658615	−	0.752480i	$-0.728858\pi$
$227$	1102.50	−	1909.59i	0.322359	−	0.558342i	0.980974	+	0.194138i	$0.0621908\pi$
$228$		0			0
$229$	143.500	+	248.549i	0.0414094	+	0.0717231i	0.885987	−	0.463710i	$-0.153482\pi$
$229$	143.500	+	248.549i	0.0414094	+	0.0717231i	−0.844578	+	0.535433i	$0.820149\pi$
$230$	−2226.00			−0.638166
$231$		0			0
$232$	−1392.00			−0.393919
$233$	2293.50	+	3972.46i	0.644859	+	1.11693i	0.984334	+	0.176314i	$0.0564173\pi$
$233$	2293.50	+	3972.46i	0.644859	+	1.11693i	−0.339475	+	0.940615i	$0.610249\pi$
$234$		0			0
$235$	−1837.50	+	3182.64i	−0.510065	+	0.883459i
$236$	−210.000	−	363.731i	−0.0579230	−	0.100326i
$237$		0			0
$238$		0			0
$239$	−1668.00			−0.451439			−0.225720	−	0.974192i	$-0.572473\pi$
$239$	−1668.00			−0.451439			−0.225720	+	0.974192i	$0.572473\pi$
$240$		0			0
$241$	−1704.50	+	2952.28i	−0.455587	+	0.789100i	−0.998722	−	0.0505456i	$-0.983904\pi$
$241$	−1704.50	+	2952.28i	−0.455587	+	0.789100i	0.543135	+	0.839646i	$0.317237\pi$
$242$	−1306.00	+	2262.06i	−0.346913	+	0.600870i
$243$		0			0
$244$	−1652.00			−0.433436
$245$		0			0
$246$		0			0
$247$	343.000	+	594.093i	0.0883586	+	0.153042i
$248$	1764.00	−	3055.34i	0.451670	−	0.782315i
$249$		0			0
$250$	−1407.00	−	2437.00i	−0.355946	−	0.616517i
$251$	−4760.00			−1.19701			−0.598503	−	0.801121i	$-0.704238\pi$
$251$	−4760.00			−1.19701			−0.598503	+	0.801121i	$0.704238\pi$
$252$		0			0
$253$	795.000			0.197554
$254$	72.0000	+	124.708i	0.0177861	+	0.0308065i
$255$		0			0
$256$	2176.00	−	3768.94i	0.531250	−	0.920152i
$257$	402.500	+	697.150i	0.0976936	+	0.169210i	0.910730	−	0.413003i	$-0.135520\pi$
$257$	402.500	+	697.150i	0.0976936	+	0.169210i	−0.813036	+	0.582213i	$0.802187\pi$
$258$		0			0
$259$		0			0
$260$	−392.000			−0.0935031
$261$		0			0
$262$	2149.00	−	3722.18i	0.506739	−	0.877698i
$263$	−128.500	+	222.569i	−0.0301279	+	0.0521831i	−0.880696	−	0.473681i	$-0.842925\pi$
$263$	−128.500	+	222.569i	−0.0301279	+	0.0521831i	0.850568	+	0.525865i	$0.176258\pi$
$264$		0			0
$265$	−2121.00			−0.491668
$266$		0			0
$267$		0			0
$268$	830.000	+	1437.60i	0.189180	+	0.327670i
$269$	−1795.50	+	3109.90i	−0.406965	+	0.704884i	−0.994548	−	0.104280i	$-0.966746\pi$
$269$	−1795.50	+	3109.90i	−0.406965	+	0.704884i	0.587583	+	0.809164i	$0.300080\pi$
$270$		0			0
$271$	696.500	+	1206.37i	0.156123	+	0.270413i	0.933467	−	0.358662i	$-0.116767\pi$
$271$	696.500	+	1206.37i	0.156123	+	0.270413i	−0.777344	+	0.629075i	$0.783434\pi$
$272$	336.000			0.0749007
$273$		0			0
$274$	−2250.00			−0.496086
$275$	190.000	+	329.090i	0.0416634	+	0.0721631i
$276$		0			0
$277$	−207.500	+	359.401i	−0.0450089	+	0.0779577i	−0.887652	−	0.460514i	$-0.847665\pi$
$277$	−207.500	+	359.401i	−0.0450089	+	0.0779577i	0.842643	+	0.538472i	$0.180998\pi$
$278$	−252.000	−	436.477i	−0.0543667	−	0.0941660i
$279$		0			0
$280$		0			0
$281$	4954.00			1.05171			0.525856	−	0.850574i	$-0.323745\pi$
$281$	4954.00			1.05171			0.525856	+	0.850574i	$0.323745\pi$
$282$		0			0
$283$	−2138.50	+	3703.99i	−0.449190	+	0.778019i	−0.998333	−	0.0577087i	$-0.981621\pi$
$283$	−2138.50	+	3703.99i	−0.449190	+	0.778019i	0.549144	+	0.835728i	$0.314954\pi$
$284$	864.000	−	1496.49i	0.180525	−	0.312678i
$285$		0			0
$286$	−140.000			−0.0289454
$287$		0			0
$288$		0			0
$289$	2236.00	+	3872.87i	0.455119	+	0.788289i
$290$	−406.000	+	703.213i	−0.0822108	+	0.142393i
$291$		0			0
$292$	2226.00	+	3855.55i	0.446119	+	0.772701i
$293$	7742.00			1.54366			0.771830	−	0.635829i	$-0.219342\pi$
$293$	7742.00			1.54366			0.771830	+	0.635829i	$0.219342\pi$
$294$		0			0
$295$	−735.000			−0.145062
$296$	−2628.00	−	4551.83i	−0.516045	−	0.893817i
$297$		0			0
$298$	201.000	−	348.142i	0.0390725	−	0.0676756i
$299$	−1113.00	−	1927.77i	−0.215272	−	0.372863i
$300$		0			0
$301$		0			0
$302$	−3238.00			−0.616973
$303$		0			0
$304$	−392.000	+	678.964i	−0.0739564	+	0.128096i
$305$	−1445.50	+	2503.68i	−0.271374	+	0.470034i
$306$		0			0
$307$	7364.00			1.36901			0.684504	−	0.729009i	$-0.260019\pi$
$307$	7364.00			1.36901			0.684504	+	0.729009i	$0.260019\pi$
$308$		0			0
$309$		0			0
$310$	−1029.00	−	1782.28i	−0.188527	−	0.326538i
$311$	−4987.50	+	8638.60i	−0.909374	+	1.57508i	−0.0944372	+	0.995531i	$0.530105\pi$
$311$	−4987.50	+	8638.60i	−0.909374	+	1.57508i	−0.814936	+	0.579550i	$0.803228\pi$
$312$		0			0
$313$	−2376.50	−	4116.22i	−0.429162	−	0.743330i	0.567637	−	0.823279i	$-0.307858\pi$
$313$	−2376.50	−	4116.22i	−0.429162	−	0.743330i	−0.996799	+	0.0799485i	$0.974524\pi$
$314$	1358.00			0.244065
$315$		0			0
$316$	412.000			0.0733443
$317$	−1738.50	−	3011.17i	−0.308025	−	0.533515i	0.669905	−	0.742447i	$-0.266335\pi$
$317$	−1738.50	−	3011.17i	−0.308025	−	0.533515i	−0.977930	+	0.208932i	$0.933001\pi$
$318$		0			0
$319$	145.000	−	251.147i	0.0254497	−	0.0440801i
$320$	−1568.00	−	2715.86i	−0.273918	−	0.474440i
$321$		0			0
$322$		0			0
$323$	1029.00			0.177260
$324$		0			0
$325$	532.000	−	921.451i	0.0908002	−	0.157270i
$326$	−467.000	+	808.868i	−0.0793397	+	0.137420i
$327$		0			0
$328$	8400.00			1.41406
$329$		0			0
$330$		0			0
$331$	−1670.50	−	2893.39i	−0.277399	−	0.480469i	0.693339	−	0.720612i	$-0.256139\pi$
$331$	−1670.50	−	2893.39i	−0.277399	−	0.480469i	−0.970738	+	0.240143i	$0.922806\pi$
$332$	2184.00	−	3782.80i	0.361032	−	0.625325i
$333$		0			0
$334$	1204.00	+	2085.39i	0.197245	+	0.341639i
$335$	2905.00			0.473782
$336$		0			0
$337$	7366.00			1.19066			0.595329	−	0.803482i	$-0.297022\pi$
$337$	7366.00			1.19066			0.595329	+	0.803482i	$0.297022\pi$
$338$	−2001.00	−	3465.83i	−0.322012	−	0.557741i
$339$		0			0
$340$	−294.000	+	509.223i	−0.0468953	+	0.0812250i
$341$	367.500	+	636.529i	0.0583614	+	0.101085i
$342$		0			0
$343$		0			0
$344$	−2976.00			−0.466439
$345$		0			0
$346$	−2821.00	+	4886.12i	−0.438318	+	0.759188i
$347$	3707.50	−	6421.58i	0.573571	−	0.993454i	−0.422625	−	0.906305i	$-0.638891\pi$
$347$	3707.50	−	6421.58i	0.573571	−	0.993454i	0.996195	−	0.0871487i	$-0.0277755\pi$
$348$		0			0
$349$	3878.00			0.594798			0.297399	−	0.954753i	$-0.403881\pi$
$349$	3878.00			0.594798			0.297399	+	0.954753i	$0.403881\pi$
$350$		0			0
$351$		0			0
$352$	400.000	+	692.820i	0.0605684	+	0.104908i
$353$	−633.500	+	1097.25i	−0.0955179	+	0.165442i	−0.909825	−	0.414993i	$-0.863784\pi$
$353$	−633.500	+	1097.25i	−0.0955179	+	0.165442i	0.814307	+	0.580435i	$0.197117\pi$
$354$		0			0
$355$	−1512.00	−	2618.86i	−0.226052	−	0.391534i
$356$	1316.00			0.195921
$357$		0			0
$358$	−6506.00			−0.960483
$359$	2342.50	+	4057.33i	0.344380	+	0.596484i	0.985241	−	0.171173i	$-0.0547558\pi$
$359$	2342.50	+	4057.33i	0.344380	+	0.596484i	−0.640861	+	0.767657i	$0.721422\pi$
$360$		0			0
$361$	2229.00	−	3860.74i	0.324974	−	0.562872i
$362$	−1582.00	−	2740.10i	−0.229691	−	0.397836i
$363$		0			0
$364$		0			0
$365$	7791.00			1.11726
$366$		0			0
$367$	−2320.50	+	4019.22i	−0.330052	+	0.571667i	−0.982522	−	0.186148i	$-0.940400\pi$
$367$	−2320.50	+	4019.22i	−0.330052	+	0.571667i	0.652470	+	0.757815i	$0.273733\pi$
$368$	1272.00	−	2203.17i	0.180184	−	0.312087i
$369$		0			0
$370$	−3066.00			−0.430794
$371$		0			0
$372$		0			0
$373$	4398.50	+	7618.43i	0.610578	+	1.05755i	0.991143	+	0.132798i	$0.0423963\pi$
$373$	4398.50	+	7618.43i	0.610578	+	1.05755i	−0.380565	+	0.924754i	$0.624270\pi$
$374$	−105.000	+	181.865i	−0.0145172	+	0.0251445i
$375$		0			0
$376$	−6300.00	−	10911.9i	−0.864090	−	1.49665i
$377$	−812.000			−0.110929
$378$		0			0
$379$	13680.0			1.85407			0.927037	−	0.374969i	$-0.122347\pi$
$379$	13680.0			1.85407			0.927037	+	0.374969i	$0.122347\pi$
$380$	−686.000	−	1188.19i	−0.0926080	−	0.160402i
$381$		0			0
$382$	−2557.00	+	4428.85i	−0.342480	+	0.593193i
$383$	−4882.50	−	8456.74i	−0.651395	−	1.12825i	−0.982785	−	0.184755i	$-0.940851\pi$
$383$	−4882.50	−	8456.74i	−0.651395	−	1.12825i	0.331390	−	0.943494i	$-0.392482\pi$
$384$		0			0
$385$		0			0
$386$	794.000			0.104698
$387$		0			0
$388$	1764.00	−	3055.34i	0.230808	−	0.399771i
$389$	865.500	−	1499.09i	0.112809	−	0.195390i	−0.804093	−	0.594504i	$-0.797349\pi$
$389$	865.500	−	1499.09i	0.112809	−	0.195390i	0.916902	+	0.399113i	$0.130682\pi$
$390$		0			0
$391$	−3339.00			−0.431868
$392$		0			0
$393$		0			0
$394$	−2914.00	−	5047.20i	−0.372602	−	0.645366i
$395$	360.500	−	624.404i	0.0459208	−	0.0795372i
$396$		0			0
$397$	5491.50	+	9511.56i	0.694233	+	1.20245i	0.970439	+	0.241348i	$0.0775896\pi$
$397$	5491.50	+	9511.56i	0.694233	+	1.20245i	−0.276206	+	0.961099i	$0.589077\pi$
$398$	6678.00			0.841050
$399$		0			0
$400$	1216.00			0.152000
$401$	3301.50	+	5718.37i	0.411145	+	0.712124i	0.995015	−	0.0997232i	$-0.0317957\pi$
$401$	3301.50	+	5718.37i	0.411145	+	0.712124i	−0.583870	+	0.811847i	$0.698462\pi$
$402$		0			0
$403$	1029.00	−	1782.28i	0.127191	−	0.220302i
$404$	2758.00	+	4777.00i	0.339643	+	0.588278i
$405$		0			0
$406$		0			0
$407$	1095.00			0.133359
$408$		0			0
$409$	5477.50	−	9487.31i	0.662213	−	1.14699i	−0.317820	−	0.948151i	$-0.602951\pi$
$409$	5477.50	−	9487.31i	0.662213	−	1.14699i	0.980033	−	0.198835i	$-0.0637158\pi$
$410$	2450.00	−	4243.52i	0.295114	−	0.511153i
$411$		0			0
$412$	−2716.00			−0.324776
$413$		0			0
$414$		0			0
$415$	−3822.00	−	6619.90i	−0.452083	−	0.783031i
$416$	1120.00	−	1939.90i	0.132001	−	0.228633i
$417$		0			0
$418$	−245.000	−	424.352i	−0.0286683	−	0.0496549i
$419$	6636.00			0.773723			0.386861	−	0.922138i	$-0.373559\pi$
$419$	6636.00			0.773723			0.386861	+	0.922138i	$0.373559\pi$
$420$		0			0
$421$	−16630.0			−1.92517			−0.962585	−	0.270980i	$-0.912652\pi$
$421$	−16630.0			−1.92517			−0.962585	+	0.270980i	$0.912652\pi$
$422$	1780.00	+	3083.05i	0.205329	+	0.355641i
$423$		0			0
$424$	3636.00	−	6297.74i	0.416462	−	0.721333i
$425$	−798.000	−	1382.18i	−0.0910793	−	0.157754i
$426$		0			0
$427$		0			0
$428$	1828.00			0.206448
$429$		0			0
$430$	−868.000	+	1503.42i	−0.0973458	+	0.168608i
$431$	2461.50	−	4263.44i	0.275096	−	0.476480i	−0.695064	−	0.718948i	$-0.744624\pi$
$431$	2461.50	−	4263.44i	0.275096	−	0.476480i	0.970159	+	0.242468i	$0.0779571\pi$
$432$		0			0
$433$	−8974.00			−0.995988			−0.497994	−	0.867180i	$-0.665930\pi$
$433$	−8974.00			−0.995988			−0.497994	+	0.867180i	$0.665930\pi$
$434$		0			0
$435$		0			0
$436$	−2250.00	−	3897.11i	−0.247146	−	0.428069i
$437$	3895.50	−	6747.20i	0.426423	−	0.738587i
$438$		0			0
$439$	−2089.50	−	3619.12i	−0.227167	−	0.393465i	0.729800	−	0.683660i	$-0.239613\pi$
$439$	−2089.50	−	3619.12i	−0.227167	−	0.393465i	−0.956967	+	0.290195i	$0.906280\pi$
$440$	840.000			0.0910123
$441$		0			0
$442$	588.000			0.0632767
$443$	−6463.50	−	11195.1i	−0.693206	−	1.20067i	−0.970782	−	0.239964i	$-0.922864\pi$
$443$	−6463.50	−	11195.1i	−0.693206	−	1.20067i	0.277576	−	0.960704i	$-0.410469\pi$
$444$		0			0
$445$	1151.50	−	1994.46i	0.122666	−	0.212464i
$446$	1400.00	+	2424.87i	0.148637	+	0.257446i
$447$		0			0
$448$		0			0
$449$	2826.00			0.297032			0.148516	−	0.988910i	$-0.452550\pi$
$449$	2826.00			0.297032			0.148516	+	0.988910i	$0.452550\pi$
$450$		0			0
$451$	−875.000	+	1515.54i	−0.0913573	+	0.158235i
$452$	3076.00	−	5327.79i	0.320095	−	0.554421i
$453$		0			0
$454$	4410.00			0.455884
$455$		0			0
$456$		0			0
$457$	−4239.50	−	7343.03i	−0.433951	−	0.751625i	0.563259	−	0.826281i	$-0.309547\pi$
$457$	−4239.50	−	7343.03i	−0.433951	−	0.751625i	−0.997209	+	0.0746560i	$0.976214\pi$
$458$	−287.000	+	497.099i	−0.0292808	+	0.0507159i
$459$		0			0
$460$	2226.00	+	3855.55i	0.225626	+	0.390795i
$461$	9338.00			0.943414			0.471707	−	0.881755i	$-0.343638\pi$
$461$	9338.00			0.943414			0.471707	+	0.881755i	$0.343638\pi$
$462$		0			0
$463$	−4016.00			−0.403109			−0.201554	−	0.979477i	$-0.564599\pi$
$463$	−4016.00			−0.403109			−0.201554	+	0.979477i	$0.564599\pi$
$464$	−464.000	−	803.672i	−0.0464238	−	0.0804084i
$465$		0			0
$466$	−4587.00	+	7944.92i	−0.455984	+	0.789788i
$467$	2929.50	+	5074.04i	0.290281	+	0.502781i	0.973876	−	0.227080i	$-0.0729180\pi$
$467$	2929.50	+	5074.04i	0.290281	+	0.502781i	−0.683595	+	0.729861i	$0.739585\pi$
$468$		0			0
$469$		0			0
$470$	−7350.00			−0.721341
$471$		0			0
$472$	1260.00	−	2182.38i	0.122873	−	0.212823i
$473$	310.000	−	536.936i	0.0301349	−	0.0521952i
$474$		0			0
$475$	3724.00			0.359724
$476$		0			0
$477$		0			0
$478$	−1668.00	−	2889.06i	−0.159608	−	0.276449i
$479$	−3251.50	+	5631.76i	−0.310156	+	0.537206i	−0.978396	−	0.206740i	$-0.933715\pi$
$479$	−3251.50	+	5631.76i	−0.310156	+	0.537206i	0.668240	+	0.743946i	$0.267048\pi$
$480$		0			0
$481$	−1533.00	−	2655.23i	−0.145320	−	0.251701i
$482$	−6818.00			−0.644297
$483$		0			0
$484$	5224.00			0.490609
$485$	−3087.00	−	5346.84i	−0.289017	−	0.500593i
$486$		0			0
$487$	8024.50	−	13898.8i	0.746663	−	1.29326i	−0.202751	−	0.979230i	$-0.564988\pi$
$487$	8024.50	−	13898.8i	0.746663	−	1.29326i	0.949414	−	0.314028i	$-0.101678\pi$
$488$	−4956.00	−	8584.04i	−0.459729	−	0.796273i
$489$		0			0
$490$		0			0
$491$	−8864.00			−0.814718			−0.407359	−	0.913268i	$-0.633550\pi$
$491$	−8864.00			−0.814718			−0.407359	+	0.913268i	$0.633550\pi$
$492$		0			0
$493$	−609.000	+	1054.82i	−0.0556348	+	0.0963624i
$494$	−686.000	+	1188.19i	−0.0624789	+	0.108217i
$495$		0			0
$496$	2352.00			0.212919
$497$		0			0
$498$		0			0
$499$	5105.50	+	8842.99i	0.458023	+	0.793319i	0.998856	−	0.0478104i	$-0.0152243\pi$
$499$	5105.50	+	8842.99i	0.458023	+	0.793319i	−0.540833	+	0.841130i	$0.681891\pi$
$500$	−2814.00	+	4873.99i	−0.251692	+	0.435943i
$501$		0			0
$502$	−4760.00	−	8244.56i	−0.423206	−	0.733014i
$503$	−1680.00			−0.148921			−0.0744607	−	0.997224i	$-0.523724\pi$
$503$	−1680.00			−0.148921			−0.0744607	+	0.997224i	$0.523724\pi$
$504$		0			0
$505$	9653.00			0.850600
$506$	795.000	+	1376.98i	0.0698460	+	0.120977i
$507$		0			0
$508$	144.000	−	249.415i	0.0125767	−	0.0217835i
$509$	4728.50	+	8190.00i	0.411762	+	0.713193i	0.995083	−	0.0990489i	$-0.0315800\pi$
$509$	4728.50	+	8190.00i	0.411762	+	0.713193i	−0.583320	+	0.812242i	$0.698247\pi$
$510$		0			0
$511$		0			0
$512$	5632.00			0.486136
$513$		0			0
$514$	−805.000	+	1394.30i	−0.0690798	+	0.119650i
$515$	−2376.50	+	4116.22i	−0.203342	+	0.352199i
$516$		0			0
$517$	2625.00			0.223302
$518$		0			0
$519$		0			0
$520$	−1176.00	−	2036.89i	−0.0991750	−	0.171776i
$521$	9040.50	−	15658.6i	0.760214	−	1.31673i	−0.182526	−	0.983201i	$-0.558427\pi$
$521$	9040.50	−	15658.6i	0.760214	−	1.31673i	0.942740	−	0.333528i	$-0.108239\pi$
$522$		0			0
$523$	10188.5	+	17647.0i	0.851839	+	1.47543i	0.879546	+	0.475813i	$0.157846\pi$
$523$	10188.5	+	17647.0i	0.851839	+	1.47543i	−0.0277071	+	0.999616i	$0.508821\pi$
$524$	−8596.00			−0.716637
$525$		0			0
$526$	−514.000			−0.0426073
$527$	−1543.50	−	2673.42i	−0.127582	−	0.220979i
$528$		0			0
$529$	−6557.00	+	11357.1i	−0.538917	+	0.933431i
$530$	−2121.00	−	3673.68i	−0.173831	−	0.301084i
$531$		0			0
$532$		0			0
$533$	4900.00			0.398204
$534$		0			0
$535$	1599.50	−	2770.42i	0.129257	−	0.223879i
$536$	−4980.00	+	8625.61i	−0.401312	+	0.695093i
$537$		0			0
$538$	−7182.00			−0.575535
$539$		0			0
$540$		0			0
$541$	3096.50	+	5363.30i	0.246079	+	0.426222i	0.962435	−	0.271514i	$-0.0875243\pi$
$541$	3096.50	+	5363.30i	0.246079	+	0.426222i	−0.716355	+	0.697736i	$0.754191\pi$
$542$	−1393.00	+	2412.75i	−0.110396	+	0.191211i
$543$		0			0
$544$	−1680.00	−	2909.85i	−0.132407	−	0.229336i
$545$	−7875.00			−0.618950
$546$		0			0
$547$	−18464.0			−1.44326			−0.721630	−	0.692279i	$-0.756607\pi$
$547$	−18464.0			−1.44326			−0.721630	+	0.692279i	$0.756607\pi$
$548$	2250.00	+	3897.11i	0.175393	+	0.303789i
$549$		0			0
$550$	−380.000	+	658.179i	−0.0294605	+	0.0510270i
$551$	−1421.00	−	2461.24i	−0.109867	−	0.190295i
$552$		0			0
$553$		0			0
$554$	−830.000			−0.0636522
$555$		0			0
$556$	−504.000	+	872.954i	−0.0384431	+	0.0665854i
$557$	−4706.50	+	8151.90i	−0.358027	+	0.620120i	−0.987631	−	0.156796i	$-0.949884\pi$
$557$	−4706.50	+	8151.90i	−0.358027	+	0.620120i	0.629604	+	0.776916i	$0.283217\pi$
$558$		0			0
$559$	−1736.00			−0.131351
$560$		0			0
$561$		0			0
$562$	4954.00	+	8580.58i	0.371836	+	0.644039i
$563$	−1599.50	+	2770.42i	−0.119735	+	0.207387i	−0.919663	−	0.392709i	$-0.871538\pi$
$563$	−1599.50	+	2770.42i	−0.119735	+	0.207387i	0.799928	+	0.600097i	$0.204871\pi$
$564$		0			0
$565$	−5383.00	−	9323.63i	−0.400822	−	0.694244i
$566$	−8554.00			−0.635250
$567$		0			0
$568$	10368.0			0.765901
$569$	10791.5	+	18691.4i	0.795085	+	1.37713i	0.922785	+	0.385314i	$0.125907\pi$
$569$	10791.5	+	18691.4i	0.795085	+	1.37713i	−0.127701	+	0.991813i	$0.540760\pi$
$570$		0			0
$571$	−10133.5	+	17551.7i	−0.742686	+	1.28637i	0.208582	+	0.978005i	$0.433115\pi$
$571$	−10133.5	+	17551.7i	−0.742686	+	1.28637i	−0.951268	+	0.308365i	$0.900218\pi$
$572$	140.000	+	242.487i	0.0102337	+	0.0177253i
$573$		0			0
$574$		0			0
$575$	−12084.0			−0.876413
$576$		0			0
$577$	6975.50	−	12081.9i	0.503282	−	0.871710i	−0.496711	−	0.867916i	$-0.665459\pi$
$577$	6975.50	−	12081.9i	0.503282	−	0.871710i	0.999993	−	0.00379418i	$-0.00120773\pi$
$578$	−4472.00	+	7745.73i	−0.321818	+	0.557405i
$579$		0			0
$580$	1624.00			0.116264
$581$		0			0
$582$		0			0
$583$	757.500	+	1312.03i	0.0538121	+	0.0932053i
$584$	−13356.0	+	23133.3i	−0.946362	+	1.63915i
$585$		0			0
$586$	7742.00	+	13409.5i	0.545766	+	0.945295i
$587$	−20972.0			−1.47463			−0.737314	−	0.675550i	$-0.763906\pi$
$587$	−20972.0			−1.47463			−0.737314	+	0.675550i	$0.763906\pi$
$588$		0			0
$589$	7203.00			0.503895
$590$	−735.000	−	1273.06i	−0.0512872	−	0.0888321i
$591$		0			0
$592$	1752.00	−	3034.55i	0.121633	−	0.210675i
$593$	94.5000	+	163.679i	0.00654410	+	0.0113347i	0.869279	−	0.494322i	$-0.164584\pi$
$593$	94.5000	+	163.679i	0.00654410	+	0.0113347i	−0.862735	+	0.505657i	$0.831250\pi$
$594$		0			0
$595$		0			0
$596$	−804.000			−0.0552569
$597$		0			0
$598$	2226.00	−	3855.55i	0.152221	−	0.263654i
$599$	−5140.50	+	8903.61i	−0.350643	+	0.607331i	−0.986362	−	0.164589i	$-0.947370\pi$
$599$	−5140.50	+	8903.61i	−0.350643	+	0.607331i	0.635719	+	0.771920i	$0.280704\pi$
$600$		0			0
$601$	6090.00			0.413338			0.206669	−	0.978411i	$-0.433738\pi$
$601$	6090.00			0.413338			0.206669	+	0.978411i	$0.433738\pi$
$602$		0			0
$603$		0			0
$604$	3238.00	+	5608.38i	0.218133	+	0.377817i
$605$	4571.00	−	7917.20i	0.307170	−	0.532033i
$606$		0			0
$607$	2474.50	+	4285.96i	0.165464	+	0.286593i	0.936820	−	0.349812i	$-0.113754\pi$
$607$	2474.50	+	4285.96i	0.165464	+	0.286593i	−0.771356	+	0.636404i	$0.780421\pi$
$608$	7840.00			0.522951
$609$		0			0
$610$	−5782.00			−0.383781
$611$	−3675.00	−	6365.29i	−0.243330	−	0.421460i
$612$		0			0
$613$	7898.50	−	13680.6i	0.520420	−	0.901394i	−0.479298	−	0.877652i	$-0.659109\pi$
$613$	7898.50	−	13680.6i	0.520420	−	0.901394i	0.999718	−	0.0237416i	$-0.00755791\pi$
$614$	7364.00	+	12754.8i	0.484018	+	0.838343i
$615$		0			0
$616$		0			0
$617$	9378.00			0.611903			0.305951	−	0.952047i	$-0.401025\pi$
$617$	9378.00			0.611903			0.305951	+	0.952047i	$0.401025\pi$
$618$		0			0
$619$	−12176.5	+	21090.3i	−0.790654	+	1.36945i	0.134908	+	0.990858i	$0.456926\pi$
$619$	−12176.5	+	21090.3i	−0.790654	+	1.36945i	−0.925562	+	0.378595i	$0.876407\pi$
$620$	−2058.00	+	3564.56i	−0.133308	+	0.230897i
$621$		0			0
$622$	−19950.0			−1.28605
$623$		0			0
$624$		0			0
$625$	174.500	+	302.243i	0.0111680	+	0.0193435i
$626$	4753.00	−	8232.44i	0.303463	−	0.525614i
$627$		0			0
$628$	−1358.00	−	2352.12i	−0.0862900	−	0.149459i
$629$	−4599.00			−0.291533
$630$		0			0
$631$	−12640.0			−0.797449			−0.398725	−	0.917071i	$-0.630547\pi$
$631$	−12640.0			−0.797449			−0.398725	+	0.917071i	$0.630547\pi$
$632$	1236.00	+	2140.81i	0.0777934	+	0.134742i
$633$		0			0
$634$	3477.00	−	6022.34i	0.217806	−	0.377252i
$635$	−252.000	−	436.477i	−0.0157485	−	0.0272772i
$636$		0			0
$637$		0			0
$638$	580.000			0.0359913
$639$		0			0
$640$	−1344.00	+	2327.88i	−0.0830098	+	0.143777i
$641$	−520.500	+	901.532i	−0.0320726	+	0.0555513i	−0.881616	−	0.471967i	$-0.843544\pi$
$641$	−520.500	+	901.532i	−0.0320726	+	0.0555513i	0.849544	+	0.527518i	$0.176877\pi$
$642$		0			0
$643$	−9548.00			−0.585593			−0.292797	−	0.956175i	$-0.594586\pi$
$643$	−9548.00			−0.585593			−0.292797	+	0.956175i	$0.594586\pi$
$644$		0			0
$645$		0			0
$646$	1029.00	+	1782.28i	0.0626710	+	0.108549i
$647$	1620.50	−	2806.79i	0.0984674	−	0.170551i	−0.812583	−	0.582845i	$-0.801939\pi$
$647$	1620.50	−	2806.79i	0.0984674	−	0.170551i	0.911050	+	0.412295i	$0.135273\pi$
$648$		0			0
$649$	262.500	+	454.663i	0.0158768	+	0.0274994i
$650$	2128.00			0.128411
$651$		0			0
$652$	1868.00			0.112203
$653$	−4426.50	−	7666.92i	−0.265272	−	0.459464i	0.702363	−	0.711819i	$-0.252128\pi$
$653$	−4426.50	−	7666.92i	−0.265272	−	0.459464i	−0.967635	+	0.252355i	$0.918795\pi$
$654$		0			0
$655$	−7521.50	+	13027.6i	−0.448686	+	0.777147i
$656$	2800.00	+	4849.74i	0.166649	+	0.288644i
$657$		0			0
$658$		0			0
$659$	−7044.00			−0.416381			−0.208191	−	0.978088i	$-0.566757\pi$
$659$	−7044.00			−0.416381			−0.208191	+	0.978088i	$0.566757\pi$
$660$		0			0
$661$	−6044.50	+	10469.4i	−0.355679	+	0.616054i	−0.987234	−	0.159277i	$-0.949084\pi$
$661$	−6044.50	+	10469.4i	−0.355679	+	0.616054i	0.631555	+	0.775331i	$0.282417\pi$
$662$	3341.00	−	5786.78i	0.196151	−	0.339743i
$663$		0			0
$664$	26208.0			1.53173
$665$		0			0
$666$		0			0
$667$	4611.00	+	7986.49i	0.267674	+	0.463625i
$668$	2408.00	−	4170.78i	0.139474	−	0.241575i
$669$		0			0
$670$	2905.00	+	5031.61i	0.167507	+	0.290131i
$671$	2065.00			0.118805
$672$		0			0
$673$	982.000			0.0562456			0.0281228	−	0.999604i	$-0.491047\pi$
$673$	982.000			0.0562456			0.0281228	+	0.999604i	$0.491047\pi$
$674$	7366.00	+	12758.3i	0.420961	+	0.729126i
$675$		0			0
$676$	−4002.00	+	6931.67i	−0.227697	+	0.394383i
$677$	15256.5	+	26425.0i	0.866108	+	1.50014i	0.865943	+	0.500143i	$0.166719\pi$
$677$	15256.5	+	26425.0i	0.866108	+	1.50014i	0.000164659		1.00000i	$0.499948\pi$
$678$		0			0
$679$		0			0
$680$	−3528.00			−0.198960
$681$		0			0
$682$	−735.000	+	1273.06i	−0.0412677	+	0.0714778i
$683$	5737.50	−	9937.64i	0.321434	−	0.556740i	−0.659350	−	0.751836i	$-0.729169\pi$
$683$	5737.50	−	9937.64i	0.321434	−	0.556740i	0.980784	+	0.195096i	$0.0625019\pi$
$684$		0			0
$685$	7875.00			0.439253
$686$		0			0
$687$		0			0
$688$	−992.000	−	1718.19i	−0.0549704	−	0.0952116i
$689$	2121.00	−	3673.68i	0.117277	−	0.203129i
$690$		0			0
$691$	−14157.5	−	24521.5i	−0.779416	−	1.34999i	−0.932279	−	0.361741i	$-0.882182\pi$
$691$	−14157.5	−	24521.5i	−0.779416	−	1.34999i	0.152862	−	0.988248i	$-0.451151\pi$
$692$	11284.0			0.619875
$693$		0			0
$694$	14830.0			0.811151
$695$	882.000	+	1527.67i	0.0481384	+	0.0833781i
$696$		0			0
$697$	3675.00	−	6365.29i	0.199714	−	0.345915i
$698$	3878.00	+	6716.89i	0.210293	+	0.364238i
$699$		0			0
$700$		0			0
$701$	−10614.0			−0.571876			−0.285938	−	0.958248i	$-0.592305\pi$
$701$	−10614.0			−0.571876			−0.285938	+	0.958248i	$0.592305\pi$
$702$		0			0
$703$	5365.50	−	9293.32i	0.287857	−	0.498583i
$704$	−1120.00	+	1939.90i	−0.0599596	+	0.103853i
$705$		0			0
$706$	−2534.00			−0.135083
$707$		0			0
$708$		0			0
$709$	−5149.50	−	8919.20i	−0.272769	−	0.472451i	0.696801	−	0.717265i	$-0.254606\pi$
$709$	−5149.50	−	8919.20i	−0.272769	−	0.472451i	−0.969570	+	0.244814i	$0.921273\pi$
$710$	3024.00	−	5237.72i	0.159843	−	0.276857i
$711$		0			0
$712$	3948.00	+	6838.14i	0.207806	+	0.359930i
$713$	−23373.0			−1.22767
$714$		0			0
$715$	490.000			0.0256293
$716$	6506.00	+	11268.7i	0.339582	+	0.588173i
$717$		0			0
$718$	−4685.00	+	8114.66i	−0.243513	+	0.421778i
$719$	−16264.5	−	28170.9i	−0.843621	−	1.46119i	−0.886813	−	0.462128i	$-0.847086\pi$
$719$	−16264.5	−	28170.9i	−0.843621	−	1.46119i	0.0431924	−	0.999067i	$-0.486247\pi$
$720$		0			0
$721$		0			0
$722$	8916.00			0.459583
$723$		0			0
$724$	−3164.00	+	5480.21i	−0.162416	+	0.281313i
$725$	−2204.00	+	3817.44i	−0.112903	+	0.195553i
$726$		0			0
$727$	−29456.0			−1.50270			−0.751350	−	0.659904i	$-0.770597\pi$
$727$	−29456.0			−1.50270			−0.751350	+	0.659904i	$0.770597\pi$
$728$		0			0
$729$		0			0
$730$	7791.00	+	13494.4i	0.395011	+	0.684179i
$731$	−1302.00	+	2255.13i	−0.0658772	+	0.114103i
$732$		0			0
$733$	13933.5	+	24133.5i	0.702109	+	1.21609i	0.967725	+	0.252009i	$0.0810912\pi$
$733$	13933.5	+	24133.5i	0.702109	+	1.21609i	−0.265616	+	0.964079i	$0.585575\pi$
$734$	−9282.00			−0.466764
$735$		0			0
$736$	−25440.0			−1.27409
$737$	−1037.50	−	1797.00i	−0.0518546	−	0.0898147i
$738$		0			0
$739$	−9769.50	+	16921.3i	−0.486302	+	0.842299i	−0.999876	−	0.0157460i	$-0.994988\pi$
$739$	−9769.50	+	16921.3i	−0.486302	+	0.842299i	0.513574	+	0.858045i	$0.328321\pi$
$740$	3066.00	+	5310.47i	0.152309	+	0.263806i
$741$		0			0
$742$		0			0
$743$	−1248.00			−0.0616214			−0.0308107	−	0.999525i	$-0.509809\pi$
$743$	−1248.00			−0.0616214			−0.0308107	+	0.999525i	$0.509809\pi$
$744$		0			0
$745$	−703.500	+	1218.50i	−0.0345963	+	0.0599226i
$746$	−8797.00	+	15236.9i	−0.431744	+	0.747803i
$747$		0			0
$748$	420.000			0.0205304
$749$		0			0
$750$		0			0
$751$	−14046.5	−	24329.3i	−0.682509	−	1.18214i	−0.974213	−	0.225631i	$-0.927556\pi$
$751$	−14046.5	−	24329.3i	−0.682509	−	1.18214i	0.291704	−	0.956509i	$-0.405778\pi$
$752$	4200.00	−	7274.61i	0.203668	−	0.352763i
$753$		0			0
$754$	−812.000	−	1406.43i	−0.0392192	−	0.0679297i
$755$	11333.0			0.546292
$756$		0			0
$757$	35954.0			1.72625			0.863124	−	0.504991i	$-0.168504\pi$
$757$	35954.0			1.72625			0.863124	+	0.504991i	$0.168504\pi$
$758$	13680.0	+	23694.5i	0.655514	+	1.13538i
$759$		0			0
$760$	4116.00	−	7129.12i	0.196451	−	0.340264i
$761$	430.500	+	745.648i	0.0205067	+	0.0355187i	0.876097	−	0.482136i	$-0.160139\pi$
$761$	430.500	+	745.648i	0.0205067	+	0.0355187i	−0.855590	+	0.517654i	$0.826805\pi$
$762$		0			0
$763$		0			0
$764$	10228.0			0.484340
$765$		0			0
$766$	9765.00	−	16913.5i	0.460605	−	0.797792i
$767$	735.000	−	1273.06i	0.0346014	−	0.0599315i
$768$		0			0
$769$	−24710.0			−1.15873			−0.579366	−	0.815067i	$-0.696700\pi$
$769$	−24710.0			−1.15873			−0.579366	+	0.815067i	$0.696700\pi$
$770$		0			0
$771$		0			0
$772$	−794.000	−	1375.25i	−0.0370164	−	0.0641143i
$773$	−8249.50	+	14288.6i	−0.383847	+	0.664843i	−0.991609	−	0.129277i	$-0.958734\pi$
$773$	−8249.50	+	14288.6i	−0.383847	+	0.664843i	0.607761	+	0.794120i	$0.292068\pi$
$774$		0			0
$775$	−5586.00	−	9675.24i	−0.258910	−	0.448445i
$776$	21168.0			0.979236
$777$		0			0
$778$	3462.00			0.159536
$779$	8575.00	+	14852.3i	0.394392	+	0.683107i
$780$		0			0
$781$	−1080.00	+	1870.61i	−0.0494820	+	0.0857053i
$782$	−3339.00	−	5783.32i	−0.152688	−	0.264464i
$783$		0			0
$784$		0			0
$785$	−4753.00			−0.216104
$786$		0			0
$787$	8235.50	−	14264.3i	0.373016	−	0.646083i	−0.617012	−	0.786954i	$-0.711657\pi$
$787$	8235.50	−	14264.3i	0.373016	−	0.646083i	0.990028	+	0.140871i	$0.0449902\pi$
$788$	−5828.00	+	10094.4i	−0.263469	+	0.456342i
$789$		0			0
$790$	1442.00			0.0649418
$791$		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:	\( N \)	\(=\)	\( 441 = 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	441.e (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.00000 + 1.73205i) q^{2} +(2.00000 - 3.46410i) q^{4} +(-3.50000 - 6.06218i) q^{5} +24.0000 q^{8} +O(q^{10})\)	\(q+(1.00000 + 1.73205i) q^{2} +(2.00000 - 3.46410i) q^{4} +(-3.50000 - 6.06218i) q^{5} +24.0000 q^{8} +(7.00000 - 12.1244i) q^{10} +(-2.50000 + 4.33013i) q^{11} +14.0000 q^{13} +(8.00000 + 13.8564i) q^{16} +(10.5000 - 18.1865i) q^{17} +(24.5000 + 42.4352i) q^{19} -28.0000 q^{20} -10.0000 q^{22} +(-79.5000 - 137.698i) q^{23} +(38.0000 - 65.8179i) q^{25} +(14.0000 + 24.2487i) q^{26} -58.0000 q^{29} +(73.5000 - 127.306i) q^{31} +(80.0000 - 138.564i) q^{32} +42.0000 q^{34} +(-109.500 - 189.660i) q^{37} +(-49.0000 + 84.8705i) q^{38} +(-84.0000 - 145.492i) q^{40} +350.000 q^{41} -124.000 q^{43} +(10.0000 + 17.3205i) q^{44} +(159.000 - 275.396i) q^{46} +(-262.500 - 454.663i) q^{47} +152.000 q^{50} +(28.0000 - 48.4974i) q^{52} +(151.500 - 262.406i) q^{53} +35.0000 q^{55} +(-58.0000 - 100.459i) q^{58} +(52.5000 - 90.9327i) q^{59} +(-206.500 - 357.668i) q^{61} +294.000 q^{62} +448.000 q^{64} +(-49.0000 - 84.8705i) q^{65} +(-207.500 + 359.401i) q^{67} +(-42.0000 - 72.7461i) q^{68} +432.000 q^{71} +(-556.500 + 963.886i) q^{73} +(219.000 - 379.319i) q^{74} +196.000 q^{76} +(51.5000 + 89.2006i) q^{79} +(56.0000 - 96.9948i) q^{80} +(350.000 + 606.218i) q^{82} +1092.00 q^{83} -147.000 q^{85} +(-124.000 - 214.774i) q^{86} +(-60.0000 + 103.923i) q^{88} +(164.500 + 284.922i) q^{89} -636.000 q^{92} +(525.000 - 909.327i) q^{94} +(171.500 - 297.047i) q^{95} +882.000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2q + 2q^{2} + 4q^{4} - 7q^{5} + 48q^{8} + O(q^{10}) \)	\( 2q + 2q^{2} + 4q^{4} - 7q^{5} + 48q^{8} + 14q^{10} - 5q^{11} + 28q^{13} + 16q^{16} + 21q^{17} + 49q^{19} - 56q^{20} - 20q^{22} - 159q^{23} + 76q^{25} + 28q^{26} - 116q^{29} + 147q^{31} + 160q^{32} + 84q^{34} - 219q^{37} - 98q^{38} - 168q^{40} + 700q^{41} - 248q^{43} + 20q^{44} + 318q^{46} - 525q^{47} + 304q^{50} + 56q^{52} + 303q^{53} + 70q^{55} - 116q^{58} + 105q^{59} - 413q^{61} + 588q^{62} + 896q^{64} - 98q^{65} - 415q^{67} - 84q^{68} + 864q^{71} - 1113q^{73} + 438q^{74} + 392q^{76} + 103q^{79} + 112q^{80} + 700q^{82} + 2184q^{83} - 294q^{85} - 248q^{86} - 120q^{88} + 329q^{89} - 1272q^{92} + 1050q^{94} + 343q^{95} + 1764q^{97} + O(q^{100}) \)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Properties

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