LMFDB - Embedded newform 624.4.q.b.529.1

Newspace parameters

Level:	$ N $	$=$	$ 624 = 2^{4} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	624.q (of order $3$, degree $2$, not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$36.8171918436$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 39)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			529.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	624.529
Dual form			624.4.q.b.289.1

$q$-expansion

$f(q)$	$=$	$q+(1.50000 - 2.59808i) q^{3} -9.00000 q^{5} +(1.00000 + 1.73205i) q^{7} +(-4.50000 - 7.79423i) q^{9} +O(q^{10})$	\(q+(1.50000 - 2.59808i) q^{3} -9.00000 q^{5} +(1.00000 + 1.73205i) q^{7} +(-4.50000 - 7.79423i) q^{9} +(15.0000 - 25.9808i) q^{11} +(32.5000 + 33.7750i) q^{13} +(-13.5000 + 23.3827i) q^{15} +(55.5000 + 96.1288i) q^{17} +(-23.0000 - 39.8372i) q^{19} +6.00000 q^{21} +(-3.00000 + 5.19615i) q^{23} -44.0000 q^{25} -27.0000 q^{27} +(52.5000 - 90.9327i) q^{29} +100.000 q^{31} +(-45.0000 - 77.9423i) q^{33} +(-9.00000 - 15.5885i) q^{35} +(-8.50000 + 14.7224i) q^{37} +(136.500 - 33.7750i) q^{39} +(115.500 - 200.052i) q^{41} +(-257.000 - 445.137i) q^{43} +(40.5000 + 70.1481i) q^{45} +162.000 q^{47} +(169.500 - 293.583i) q^{49} +333.000 q^{51} +639.000 q^{53} +(-135.000 + 233.827i) q^{55} -138.000 q^{57} +(300.000 + 519.615i) q^{59} +(-116.500 - 201.784i) q^{61} +(9.00000 - 15.5885i) q^{63} +(-292.500 - 303.975i) q^{65} +(463.000 - 801.940i) q^{67} +(9.00000 + 15.5885i) q^{69} +(-465.000 - 805.404i) q^{71} -253.000 q^{73} +(-66.0000 + 114.315i) q^{75} +60.0000 q^{77} +1324.00 q^{79} +(-40.5000 + 70.1481i) q^{81} -810.000 q^{83} +(-499.500 - 865.159i) q^{85} +(-157.500 - 272.798i) q^{87} +(-249.000 + 431.281i) q^{89} +(-26.0000 + 90.0666i) q^{91} +(150.000 - 259.808i) q^{93} +(207.000 + 358.535i) q^{95} +(-679.000 - 1176.06i) q^{97} -270.000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 3 q^{3} - 18 q^{5} + 2 q^{7} - 9 q^{9}+O(q^{10}) $ 2 * q + 3 * q^3 - 18 * q^5 + 2 * q^7 - 9 * q^9	$ 2 q + 3 q^{3} - 18 q^{5} + 2 q^{7} - 9 q^{9} + 30 q^{11} + 65 q^{13} - 27 q^{15} + 111 q^{17} - 46 q^{19} + 12 q^{21} - 6 q^{23} - 88 q^{25} - 54 q^{27} + 105 q^{29} + 200 q^{31} - 90 q^{33} - 18 q^{35} - 17 q^{37} + 273 q^{39} + 231 q^{41} - 514 q^{43} + 81 q^{45} + 324 q^{47} + 339 q^{49} + 666 q^{51} + 1278 q^{53} - 270 q^{55} - 276 q^{57} + 600 q^{59} - 233 q^{61} + 18 q^{63} - 585 q^{65} + 926 q^{67} + 18 q^{69} - 930 q^{71} - 506 q^{73} - 132 q^{75} + 120 q^{77} + 2648 q^{79} - 81 q^{81} - 1620 q^{83} - 999 q^{85} - 315 q^{87} - 498 q^{89} - 52 q^{91} + 300 q^{93} + 414 q^{95} - 1358 q^{97} - 540 q^{99}+O(q^{100}) $ 2 * q + 3 * q^3 - 18 * q^5 + 2 * q^7 - 9 * q^9 + 30 * q^11 + 65 * q^13 - 27 * q^15 + 111 * q^17 - 46 * q^19 + 12 * q^21 - 6 * q^23 - 88 * q^25 - 54 * q^27 + 105 * q^29 + 200 * q^31 - 90 * q^33 - 18 * q^35 - 17 * q^37 + 273 * q^39 + 231 * q^41 - 514 * q^43 + 81 * q^45 + 324 * q^47 + 339 * q^49 + 666 * q^51 + 1278 * q^53 - 270 * q^55 - 276 * q^57 + 600 * q^59 - 233 * q^61 + 18 * q^63 - 585 * q^65 + 926 * q^67 + 18 * q^69 - 930 * q^71 - 506 * q^73 - 132 * q^75 + 120 * q^77 + 2648 * q^79 - 81 * q^81 - 1620 * q^83 - 999 * q^85 - 315 * q^87 - 498 * q^89 - 52 * q^91 + 300 * q^93 + 414 * q^95 - 1358 * q^97 - 540 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/624\mathbb{Z}\right)^\times$.

$n$	$79$	$145$	$209$	$469$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$	$1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	1.50000	−	2.59808i	0.288675	−	0.500000i
$3$	1.50000	−	2.59808i	0.288675	−	0.500000i
$4$		0			0
$5$	−9.00000			−0.804984			−0.402492	−	0.915423i	$-0.631856\pi$
$5$	−9.00000			−0.804984			−0.402492	+	0.915423i	$0.631856\pi$
$6$		0			0
$7$	1.00000	+	1.73205i	0.0539949	+	0.0935220i	0.891760	−	0.452510i	$-0.149471\pi$
$7$	1.00000	+	1.73205i	0.0539949	+	0.0935220i	−0.837765	+	0.546032i	$0.816138\pi$
$8$		0			0
$9$	−4.50000	−	7.79423i	−0.166667	−	0.288675i
$10$		0			0
$11$	15.0000	−	25.9808i	0.411152	−	0.712136i	−0.583864	−	0.811851i	$-0.698460\pi$
$11$	15.0000	−	25.9808i	0.411152	−	0.712136i	0.995016	+	0.0997155i	$0.0317933\pi$
$12$		0			0
$13$	32.5000	+	33.7750i	0.693375	+	0.720577i
$13$	32.5000	+	33.7750i	0.693375	+	0.720577i
$14$		0			0
$15$	−13.5000	+	23.3827i	−0.232379	+	0.402492i
$16$		0			0
$17$	55.5000	+	96.1288i	0.791807	+	1.37145i	0.924847	+	0.380340i	$0.124193\pi$
$17$	55.5000	+	96.1288i	0.791807	+	1.37145i	−0.133039	+	0.991111i	$0.542474\pi$
$18$		0			0
$19$	−23.0000	−	39.8372i	−0.277714	−	0.481014i	0.693102	−	0.720839i	$-0.256243\pi$
$19$	−23.0000	−	39.8372i	−0.277714	−	0.481014i	−0.970816	+	0.239825i	$0.922910\pi$
$20$		0			0
$21$	6.00000			0.0623480
$22$		0			0
$23$	−3.00000	+	5.19615i	−0.0271975	+	0.0471075i	−0.879304	−	0.476261i	$-0.841992\pi$
$23$	−3.00000	+	5.19615i	−0.0271975	+	0.0471075i	0.852106	+	0.523369i	$0.175325\pi$
$24$		0			0
$25$	−44.0000			−0.352000
$26$		0			0
$27$	−27.0000			−0.192450
$28$		0			0
$29$	52.5000	−	90.9327i	0.336173	−	0.582268i	−0.647537	−	0.762034i	$-0.724201\pi$
$29$	52.5000	−	90.9327i	0.336173	−	0.582268i	0.983709	+	0.179766i	$0.0575341\pi$
$30$		0			0
$31$	100.000			0.579372			0.289686	−	0.957122i	$-0.406449\pi$
$31$	100.000			0.579372			0.289686	+	0.957122i	$0.406449\pi$
$32$		0			0
$33$	−45.0000	−	77.9423i	−0.237379	−	0.411152i
$34$		0			0
$35$	−9.00000	−	15.5885i	−0.0434651	−	0.0752837i
$36$		0			0
$37$	−8.50000	+	14.7224i	−0.0377673	+	0.0654149i	−0.884291	−	0.466936i	$-0.845358\pi$
$37$	−8.50000	+	14.7224i	−0.0377673	+	0.0654149i	0.846524	+	0.532351i	$0.178691\pi$
$38$		0			0
$39$	136.500	−	33.7750i	0.560449	−	0.138675i
$40$		0			0
$41$	115.500	−	200.052i	0.439953	−	0.762021i	−0.557732	−	0.830021i	$-0.688328\pi$
$41$	115.500	−	200.052i	0.439953	−	0.762021i	0.997685	+	0.0680000i	$0.0216618\pi$
$42$		0			0
$43$	−257.000	−	445.137i	−0.911445	−	1.57867i	−0.812024	−	0.583623i	$-0.801634\pi$
$43$	−257.000	−	445.137i	−0.911445	−	1.57867i	−0.0994205	−	0.995046i	$-0.531699\pi$
$44$		0			0
$45$	40.5000	+	70.1481i	0.134164	+	0.232379i
$46$		0			0
$47$	162.000			0.502769			0.251384	−	0.967887i	$-0.419114\pi$
$47$	162.000			0.502769			0.251384	+	0.967887i	$0.419114\pi$
$48$		0			0
$49$	169.500	−	293.583i	0.494169	−	0.855926i
$50$		0			0
$51$	333.000			0.914301
$52$		0			0
$53$	639.000			1.65610			0.828051	−	0.560653i	$-0.189450\pi$
$53$	639.000			1.65610			0.828051	+	0.560653i	$0.189450\pi$
$54$		0			0
$55$	−135.000	+	233.827i	−0.330971	+	0.573258i
$56$		0			0
$57$	−138.000			−0.320676
$58$		0			0
$59$	300.000	+	519.615i	0.661978	+	1.14658i	0.980095	+	0.198527i	$0.0636159\pi$
$59$	300.000	+	519.615i	0.661978	+	1.14658i	−0.318118	+	0.948051i	$0.603051\pi$
$60$		0			0
$61$	−116.500	−	201.784i	−0.244529	−	0.423537i	0.717470	−	0.696590i	$-0.245300\pi$
$61$	−116.500	−	201.784i	−0.244529	−	0.423537i	−0.961999	+	0.273052i	$0.911967\pi$
$62$		0			0
$63$	9.00000	−	15.5885i	0.0179983	−	0.0311740i
$64$		0			0
$65$	−292.500	−	303.975i	−0.558156	−	0.580053i
$66$		0			0
$67$	463.000	−	801.940i	0.844246	−	1.46228i	−0.0420292	−	0.999116i	$-0.513382\pi$
$67$	463.000	−	801.940i	0.844246	−	1.46228i	0.886275	−	0.463160i	$-0.153284\pi$
$68$		0			0
$69$	9.00000	+	15.5885i	0.0157025	+	0.0271975i
$70$		0			0
$71$	−465.000	−	805.404i	−0.777258	−	1.34625i	−0.933516	−	0.358535i	$-0.883276\pi$
$71$	−465.000	−	805.404i	−0.777258	−	1.34625i	0.156258	−	0.987716i	$-0.450057\pi$
$72$		0			0
$73$	−253.000			−0.405636			−0.202818	−	0.979216i	$-0.565010\pi$
$73$	−253.000			−0.405636			−0.202818	+	0.979216i	$0.565010\pi$
$74$		0			0
$75$	−66.0000	+	114.315i	−0.101614	+	0.176000i
$76$		0			0
$77$	60.0000			0.0888004
$78$		0			0
$79$	1324.00			1.88559			0.942795	−	0.333373i	$-0.108187\pi$
$79$	1324.00			1.88559			0.942795	+	0.333373i	$0.108187\pi$
$80$		0			0
$81$	−40.5000	+	70.1481i	−0.0555556	+	0.0962250i
$82$		0			0
$83$	−810.000			−1.07119			−0.535597	−	0.844474i	$-0.679913\pi$
$83$	−810.000			−1.07119			−0.535597	+	0.844474i	$0.679913\pi$
$84$		0			0
$85$	−499.500	−	865.159i	−0.637393	−	1.10400i
$86$		0			0
$87$	−157.500	−	272.798i	−0.194089	−	0.336173i
$88$		0			0
$89$	−249.000	+	431.281i	−0.296561	+	0.513659i	−0.975347	−	0.220677i	$-0.929173\pi$
$89$	−249.000	+	431.281i	−0.296561	+	0.513659i	0.678786	+	0.734336i	$0.262507\pi$
$90$		0			0
$91$	−26.0000	+	90.0666i	−0.0299510	+	0.103753i
$92$		0			0
$93$	150.000	−	259.808i	0.167250	−	0.289686i
$94$		0			0
$95$	207.000	+	358.535i	0.223555	+	0.387209i
$96$		0			0
$97$	−679.000	−	1176.06i	−0.710742	−	1.23104i	−0.964579	−	0.263795i	$-0.915026\pi$
$97$	−679.000	−	1176.06i	−0.710742	−	1.23104i	0.253837	−	0.967247i	$-0.418307\pi$
$98$		0			0
$99$	−270.000			−0.274101
$100$		0			0
$101$	178.500	−	309.171i	0.175856	−	0.304591i	−0.764601	−	0.644503i	$-0.777064\pi$
$101$	178.500	−	309.171i	0.175856	−	0.304591i	0.940457	+	0.339913i	$0.110397\pi$
$102$		0			0
$103$	−1118.00			−1.06951			−0.534756	−	0.845006i	$-0.679597\pi$
$103$	−1118.00			−1.06951			−0.534756	+	0.845006i	$0.679597\pi$
$104$		0			0
$105$	−54.0000			−0.0501891
$106$		0			0
$107$	357.000	−	618.342i	0.322547	−	0.558667i	−0.658466	−	0.752610i	$-0.728794\pi$
$107$	357.000	−	618.342i	0.322547	−	0.558667i	0.981013	+	0.193943i	$0.0621277\pi$
$108$		0			0
$109$	2006.00			1.76275			0.881376	−	0.472416i	$-0.156618\pi$
$109$	2006.00			1.76275			0.881376	+	0.472416i	$0.156618\pi$
$110$		0			0
$111$	25.5000	+	44.1673i	0.0218050	+	0.0377673i
$112$		0			0
$113$	559.500	+	969.082i	0.465782	+	0.806758i	0.999236	−	0.0390710i	$-0.0124399\pi$
$113$	559.500	+	969.082i	0.465782	+	0.806758i	−0.533455	+	0.845829i	$0.679107\pi$
$114$		0			0
$115$	27.0000	−	46.7654i	0.0218936	−	0.0379208i
$116$		0			0
$117$	117.000	−	405.300i	0.0924500	−	0.320256i
$118$		0			0
$119$	−111.000	+	192.258i	−0.0855072	+	0.148103i
$120$		0			0
$121$	215.500	+	373.257i	0.161908	+	0.280433i
$122$		0			0
$123$	−346.500	−	600.156i	−0.254007	−	0.439953i
$124$		0			0
$125$	1521.00			1.08834
$126$		0			0
$127$	−302.000	+	523.079i	−0.211009	+	0.365479i	−0.952031	−	0.306003i	$-0.901008\pi$
$127$	−302.000	+	523.079i	−0.211009	+	0.365479i	0.741021	+	0.671481i	$0.234342\pi$
$128$		0			0
$129$	−1542.00			−1.05245
$130$		0			0
$131$	1584.00			1.05645			0.528224	−	0.849105i	$-0.322858\pi$
$131$	1584.00			1.05645			0.528224	+	0.849105i	$0.322858\pi$
$132$		0			0
$133$	46.0000	−	79.6743i	0.0299903	−	0.0519447i
$134$		0			0
$135$	243.000			0.154919
$136$		0			0
$137$	−358.500	−	620.940i	−0.223567	−	0.387230i	0.732321	−	0.680959i	$-0.238437\pi$
$137$	−358.500	−	620.940i	−0.223567	−	0.387230i	−0.955889	+	0.293729i	$0.905104\pi$
$138$		0			0
$139$	−410.000	−	710.141i	−0.250185	−	0.433334i	0.713391	−	0.700766i	$-0.247158\pi$
$139$	−410.000	−	710.141i	−0.250185	−	0.433334i	−0.963577	+	0.267432i	$0.913825\pi$
$140$		0			0
$141$	243.000	−	420.888i	0.145137	−	0.251384i
$142$		0			0
$143$	1365.00	−	337.750i	0.798231	−	0.197511i
$144$		0			0
$145$	−472.500	+	818.394i	−0.270614	+	0.468717i
$146$		0			0
$147$	−508.500	−	880.748i	−0.285309	−	0.494169i
$148$		0			0
$149$	874.500	+	1514.68i	0.480818	+	0.832801i	0.999758	−	0.0220100i	$-0.00700656\pi$
$149$	874.500	+	1514.68i	0.480818	+	0.832801i	−0.518940	+	0.854811i	$0.673673\pi$
$150$		0			0
$151$	370.000			0.199405			0.0997026	−	0.995017i	$-0.468211\pi$
$151$	370.000			0.199405			0.0997026	+	0.995017i	$0.468211\pi$
$152$		0			0
$153$	499.500	−	865.159i	0.263936	−	0.457150i
$154$		0			0
$155$	−900.000			−0.466385
$156$		0			0
$157$	−2611.00			−1.32726			−0.663632	−	0.748059i	$-0.730986\pi$
$157$	−2611.00			−1.32726			−0.663632	+	0.748059i	$0.730986\pi$
$158$		0			0
$159$	958.500	−	1660.17i	0.478075	−	0.828051i
$160$		0			0
$161$	−12.0000			−0.00587411
$162$		0			0
$163$	−818.000	−	1416.82i	−0.393072	−	0.680820i	0.599781	−	0.800164i	$-0.295254\pi$
$163$	−818.000	−	1416.82i	−0.393072	−	0.680820i	−0.992853	+	0.119344i	$0.961921\pi$
$164$		0			0
$165$	405.000	+	701.481i	0.191086	+	0.330971i
$166$		0			0
$167$	132.000	−	228.631i	0.0611645	−	0.105940i	−0.833822	−	0.552034i	$-0.813852\pi$
$167$	132.000	−	228.631i	0.0611645	−	0.105940i	0.894986	+	0.446094i	$0.147185\pi$
$168$		0			0
$169$	−84.5000	+	2195.37i	−0.0384615	+	0.999260i
$170$		0			0
$171$	−207.000	+	358.535i	−0.0925713	+	0.160338i
$172$		0			0
$173$	−705.000	−	1221.10i	−0.309827	−	0.536637i	0.668497	−	0.743715i	$-0.266938\pi$
$173$	−705.000	−	1221.10i	−0.309827	−	0.536637i	−0.978324	+	0.207078i	$0.933605\pi$
$174$		0			0
$175$	−44.0000	−	76.2102i	−0.0190062	−	0.0329197i
$176$		0			0
$177$	1800.00			0.764386
$178$		0			0
$179$	−237.000	+	410.496i	−0.0989621	+	0.171407i	−0.911255	−	0.411842i	$-0.864886\pi$
$179$	−237.000	+	410.496i	−0.0989621	+	0.171407i	0.812293	+	0.583249i	$0.198219\pi$
$180$		0			0
$181$	2249.00			0.923574			0.461787	−	0.886991i	$-0.347208\pi$
$181$	2249.00			0.923574			0.461787	+	0.886991i	$0.347208\pi$
$182$		0			0
$183$	−699.000			−0.282358
$184$		0			0
$185$	76.5000	−	132.502i	0.0304021	−	0.0526580i
$186$		0			0
$187$	3330.00			1.30221
$188$		0			0
$189$	−27.0000	−	46.7654i	−0.0103913	−	0.0179983i
$190$		0			0
$191$	1722.00	+	2982.59i	0.652354	+	1.12991i	0.982550	+	0.185997i	$0.0595516\pi$
$191$	1722.00	+	2982.59i	0.652354	+	1.12991i	−0.330197	+	0.943912i	$0.607115\pi$
$192$		0			0
$193$	2136.50	−	3700.53i	0.796832	−	1.38015i	−0.124837	−	0.992177i	$-0.539841\pi$
$193$	2136.50	−	3700.53i	0.796832	−	1.38015i	0.921669	−	0.387977i	$-0.126826\pi$
$194$		0			0
$195$	−1228.50	+	303.975i	−0.451152	+	0.111631i
$196$		0			0
$197$	993.000	−	1719.93i	0.359129	−	0.622029i	−0.628687	−	0.777658i	$-0.716407\pi$
$197$	993.000	−	1719.93i	0.359129	−	0.622029i	0.987815	+	0.155630i	$0.0497406\pi$
$198$		0			0
$199$	−1193.00	−	2066.34i	−0.424973	−	0.736074i	0.571445	−	0.820640i	$-0.306383\pi$
$199$	−1193.00	−	2066.34i	−0.424973	−	0.736074i	−0.996418	+	0.0845661i	$0.973050\pi$
$200$		0			0
$201$	−1389.00	−	2405.82i	−0.487425	−	0.844246i
$202$		0			0
$203$	210.000			0.0726065
$204$		0			0
$205$	−1039.50	+	1800.47i	−0.354155	+	0.613415i
$206$		0			0
$207$	54.0000			0.0181317
$208$		0			0
$209$	−1380.00			−0.456730
$210$		0			0
$211$	−800.000	+	1385.64i	−0.261016	+	0.452092i	−0.966512	−	0.256621i	$-0.917391\pi$
$211$	−800.000	+	1385.64i	−0.261016	+	0.452092i	0.705497	+	0.708713i	$0.250724\pi$
$212$		0			0
$213$	−2790.00			−0.897501
$214$		0			0
$215$	2313.00	+	4006.23i	0.733699	+	1.27080i
$216$		0			0
$217$	100.000	+	173.205i	0.0312831	+	0.0541840i
$218$		0			0
$219$	−379.500	+	657.313i	−0.117097	+	0.202818i
$220$		0			0
$221$	−1443.00	+	4998.70i	−0.439216	+	1.52149i
$222$		0			0
$223$	−1916.00	+	3318.61i	−0.575358	+	0.996549i	0.420645	+	0.907226i	$0.361804\pi$
$223$	−1916.00	+	3318.61i	−0.575358	+	0.996549i	−0.996003	+	0.0893239i	$0.971529\pi$
$224$		0			0
$225$	198.000	+	342.946i	0.0586667	+	0.101614i
$226$		0			0
$227$	−699.000	−	1210.70i	−0.204380	−	0.353997i	0.745555	−	0.666444i	$-0.232184\pi$
$227$	−699.000	−	1210.70i	−0.204380	−	0.353997i	−0.949935	+	0.312448i	$0.898851\pi$
$228$		0			0
$229$	4466.00			1.28874			0.644370	−	0.764714i	$-0.277120\pi$
$229$	4466.00			1.28874			0.644370	+	0.764714i	$0.277120\pi$
$230$		0			0
$231$	90.0000	−	155.885i	0.0256345	−	0.0444002i
$232$		0			0
$233$	−1638.00			−0.460553			−0.230277	−	0.973125i	$-0.573963\pi$
$233$	−1638.00			−0.460553			−0.230277	+	0.973125i	$0.573963\pi$
$234$		0			0
$235$	−1458.00			−0.404721
$236$		0			0
$237$	1986.00	−	3439.85i	0.544323	−	0.942795i
$238$		0			0
$239$	594.000			0.160764			0.0803821	−	0.996764i	$-0.474386\pi$
$239$	594.000			0.160764			0.0803821	+	0.996764i	$0.474386\pi$
$240$		0			0
$241$	−1151.50	−	1994.46i	−0.307779	−	0.533088i	0.670098	−	0.742273i	$-0.266252\pi$
$241$	−1151.50	−	1994.46i	−0.307779	−	0.533088i	−0.977876	+	0.209185i	$0.932919\pi$
$242$		0			0
$243$	121.500	+	210.444i	0.0320750	+	0.0555556i
$244$		0			0
$245$	−1525.50	+	2642.24i	−0.397798	+	0.689007i
$246$		0			0
$247$	598.000	−	2071.53i	0.154048	−	0.533638i
$248$		0			0
$249$	−1215.00	+	2104.44i	−0.309227	+	0.535597i
$250$		0			0
$251$	3162.00	+	5476.74i	0.795154	+	1.37725i	0.922741	+	0.385420i	$0.125943\pi$
$251$	3162.00	+	5476.74i	0.795154	+	1.37725i	−0.127587	+	0.991827i	$0.540723\pi$
$252$		0			0
$253$	90.0000	+	155.885i	0.0223646	+	0.0387367i
$254$		0			0
$255$	−2997.00			−0.735998
$256$		0			0
$257$	−3916.50	+	6783.58i	−0.950601	+	1.64649i	−0.206474	+	0.978452i	$0.566199\pi$
$257$	−3916.50	+	6783.58i	−0.950601	+	1.64649i	−0.744127	+	0.668038i	$0.767134\pi$
$258$		0			0
$259$	−34.0000			−0.00815698
$260$		0			0
$261$	−945.000			−0.224115
$262$		0			0
$263$	−1515.00	+	2624.06i	−0.355205	+	0.615233i	−0.987153	−	0.159778i	$-0.948922\pi$
$263$	−1515.00	+	2624.06i	−0.355205	+	0.615233i	0.631948	+	0.775011i	$0.282256\pi$
$264$		0			0
$265$	−5751.00			−1.33314
$266$		0			0
$267$	747.000	+	1293.84i	0.171220	+	0.296561i
$268$		0			0
$269$	267.000	+	462.458i	0.0605178	+	0.104820i	0.894697	−	0.446674i	$-0.147391\pi$
$269$	267.000	+	462.458i	0.0605178	+	0.104820i	−0.834179	+	0.551493i	$0.814058\pi$
$270$		0			0
$271$	−1844.00	+	3193.90i	−0.413340	+	0.715925i	−0.995253	−	0.0973259i	$-0.968971\pi$
$271$	−1844.00	+	3193.90i	−0.413340	+	0.715925i	0.581913	+	0.813251i	$0.302304\pi$
$272$		0			0
$273$	195.000	+	202.650i	0.0432305	+	0.0449265i
$274$		0			0
$275$	−660.000	+	1143.15i	−0.144725	+	0.250672i
$276$		0			0
$277$	−932.500	−	1615.14i	−0.202269	−	0.350340i	0.746990	−	0.664835i	$-0.231498\pi$
$277$	−932.500	−	1615.14i	−0.202269	−	0.350340i	−0.949259	+	0.314495i	$0.898165\pi$
$278$		0			0
$279$	−450.000	−	779.423i	−0.0965620	−	0.167250i
$280$		0			0
$281$	2997.00			0.636249			0.318125	−	0.948049i	$-0.396947\pi$
$281$	2997.00			0.636249			0.318125	+	0.948049i	$0.396947\pi$
$282$		0			0
$283$	−2057.00	+	3562.83i	−0.432071	+	0.748368i	−0.997051	−	0.0767359i	$-0.975550\pi$
$283$	−2057.00	+	3562.83i	−0.432071	+	0.748368i	0.564981	+	0.825104i	$0.308884\pi$
$284$		0			0
$285$	1242.00			0.258139
$286$		0			0
$287$	462.000			0.0950209
$288$		0			0
$289$	−3704.00	+	6415.52i	−0.753918	+	1.30582i
$290$		0			0
$291$	−4074.00			−0.820695
$292$		0			0
$293$	2332.50	+	4040.01i	0.465072	+	0.805528i	0.999205	−	0.0398722i	$-0.0126951\pi$
$293$	2332.50	+	4040.01i	0.465072	+	0.805528i	−0.534133	+	0.845401i	$0.679362\pi$
$294$		0			0
$295$	−2700.00	−	4676.54i	−0.532882	−	0.922978i
$296$		0			0
$297$	−405.000	+	701.481i	−0.0791262	+	0.137051i
$298$		0			0
$299$	−273.000	+	67.5500i	−0.0528027	+	0.0130653i
$300$		0			0
$301$	514.000	−	890.274i	0.0984268	−	0.170480i
$302$		0			0
$303$	−535.500	−	927.513i	−0.101530	−	0.175856i
$304$		0			0
$305$	1048.50	+	1816.06i	0.196842	+	0.340941i
$306$		0			0
$307$	−1502.00			−0.279230			−0.139615	−	0.990206i	$-0.544587\pi$
$307$	−1502.00			−0.279230			−0.139615	+	0.990206i	$0.544587\pi$
$308$		0			0
$309$	−1677.00	+	2904.65i	−0.308742	+	0.534756i
$310$		0			0
$311$	−2106.00			−0.383988			−0.191994	−	0.981396i	$-0.561495\pi$
$311$	−2106.00			−0.383988			−0.191994	+	0.981396i	$0.561495\pi$
$312$		0			0
$313$	−3898.00			−0.703923			−0.351962	−	0.936014i	$-0.614485\pi$
$313$	−3898.00			−0.703923			−0.351962	+	0.936014i	$0.614485\pi$
$314$		0			0
$315$	−81.0000	+	140.296i	−0.0144884	+	0.0250946i
$316$		0			0
$317$	9351.00			1.65680			0.828398	−	0.560140i	$-0.189253\pi$
$317$	9351.00			1.65680			0.828398	+	0.560140i	$0.189253\pi$
$318$		0			0
$319$	−1575.00	−	2727.98i	−0.276436	−	0.478801i
$320$		0			0
$321$	−1071.00	−	1855.03i	−0.186222	−	0.322547i
$322$		0			0
$323$	2553.00	−	4421.93i	0.439792	−	0.761742i
$324$		0			0
$325$	−1430.00	−	1486.10i	−0.244068	−	0.253643i
$326$		0			0
$327$	3009.00	−	5211.74i	0.508863	−	0.881376i
$328$		0			0
$329$	162.000	+	280.592i	0.0271470	+	0.0470199i
$330$		0			0
$331$	−4586.00	−	7943.19i	−0.761539	−	1.31902i	−0.942057	−	0.335452i	$-0.891111\pi$
$331$	−4586.00	−	7943.19i	−0.761539	−	1.31902i	0.180518	−	0.983572i	$-0.442222\pi$
$332$		0			0
$333$	153.000			0.0251782
$334$		0			0
$335$	−4167.00	+	7217.46i	−0.679605	+	1.17711i
$336$		0			0
$337$	−11089.0			−1.79245			−0.896226	−	0.443598i	$-0.853702\pi$
$337$	−11089.0			−1.79245			−0.896226	+	0.443598i	$0.853702\pi$
$338$		0			0
$339$	3357.00			0.537838
$340$		0			0
$341$	1500.00	−	2598.08i	0.238210	−	0.412592i
$342$		0			0
$343$	1364.00			0.214720
$344$		0			0
$345$	−81.0000	−	140.296i	−0.0126403	−	0.0218936i
$346$		0			0
$347$	4881.00	+	8454.14i	0.755118	+	1.30790i	0.945316	+	0.326156i	$0.105754\pi$
$347$	4881.00	+	8454.14i	0.755118	+	1.30790i	−0.190198	+	0.981746i	$0.560913\pi$
$348$		0			0
$349$	4145.00	−	7179.35i	0.635750	−	1.10115i	−0.350606	−	0.936523i	$-0.614024\pi$
$349$	4145.00	−	7179.35i	0.635750	−	1.10115i	0.986356	−	0.164628i	$-0.0526424\pi$
$350$		0			0
$351$	−877.500	−	911.925i	−0.133440	−	0.138675i
$352$		0			0
$353$	−6202.50	+	10743.0i	−0.935200	+	1.61981i	−0.160924	+	0.986967i	$0.551448\pi$
$353$	−6202.50	+	10743.0i	−0.935200	+	1.61981i	−0.774276	+	0.632848i	$0.781886\pi$
$354$		0			0
$355$	4185.00	+	7248.63i	0.625681	+	1.08371i
$356$		0			0
$357$	333.000	+	576.773i	0.0493676	+	0.0855072i
$358$		0			0
$359$	1098.00			0.161421			0.0807106	−	0.996738i	$-0.474281\pi$
$359$	1098.00			0.161421			0.0807106	+	0.996738i	$0.474281\pi$
$360$		0			0
$361$	2371.50	−	4107.56i	0.345750	−	0.598857i
$362$		0			0
$363$	1293.00			0.186956
$364$		0			0
$365$	2277.00			0.326530
$366$		0			0
$367$	−2867.00	+	4965.79i	−0.407783	+	0.706300i	−0.994641	−	0.103390i	$-0.967031\pi$
$367$	−2867.00	+	4965.79i	−0.407783	+	0.706300i	0.586858	+	0.809690i	$0.300365\pi$
$368$		0			0
$369$	−2079.00			−0.293302
$370$		0			0
$371$	639.000	+	1106.78i	0.0894211	+	0.154882i
$372$		0			0
$373$	4485.50	+	7769.11i	0.622655	+	1.07847i	0.988989	+	0.147987i	$0.0472794\pi$
$373$	4485.50	+	7769.11i	0.622655	+	1.07847i	−0.366334	+	0.930483i	$0.619387\pi$
$374$		0			0
$375$	2281.50	−	3951.67i	0.314176	−	0.544170i
$376$		0			0
$377$	4777.50	−	1182.12i	0.652663	−	0.161492i
$378$		0			0
$379$	3622.00	−	6273.49i	0.490896	−	0.850257i	−0.509049	−	0.860738i	$-0.670003\pi$
$379$	3622.00	−	6273.49i	0.490896	−	0.850257i	0.999945	+	0.0104805i	$0.00333611\pi$
$380$		0			0
$381$	906.000	+	1569.24i	0.121826	+	0.211009i
$382$		0			0
$383$	−3156.00	−	5466.35i	−0.421055	−	0.729289i	0.574988	−	0.818162i	$-0.305007\pi$
$383$	−3156.00	−	5466.35i	−0.421055	−	0.729289i	−0.996043	+	0.0888732i	$0.971673\pi$
$384$		0			0
$385$	−540.000			−0.0714830
$386$		0			0
$387$	−2313.00	+	4006.23i	−0.303815	+	0.526223i
$388$		0			0
$389$	3627.00			0.472741			0.236370	−	0.971663i	$-0.424042\pi$
$389$	3627.00			0.472741			0.236370	+	0.971663i	$0.424042\pi$
$390$		0			0
$391$	−666.000			−0.0861408
$392$		0			0
$393$	2376.00	−	4115.35i	0.304970	−	0.528224i
$394$		0			0
$395$	−11916.0			−1.51787
$396$		0			0
$397$	1949.00	+	3375.77i	0.246392	+	0.426763i	0.962522	−	0.271204i	$-0.0874217\pi$
$397$	1949.00	+	3375.77i	0.246392	+	0.426763i	−0.716130	+	0.697967i	$0.754088\pi$
$398$		0			0
$399$	−138.000	−	239.023i	−0.0173149	−	0.0299903i
$400$		0			0
$401$	2851.50	−	4938.94i	0.355105	−	0.615060i	−0.632031	−	0.774943i	$-0.717778\pi$
$401$	2851.50	−	4938.94i	0.355105	−	0.615060i	0.987136	+	0.159883i	$0.0511118\pi$
$402$		0			0
$403$	3250.00	+	3377.50i	0.401722	+	0.417482i
$404$		0			0
$405$	364.500	−	631.333i	0.0447214	−	0.0774597i
$406$		0			0
$407$	255.000	+	441.673i	0.0310562	+	0.0537909i
$408$		0			0
$409$	−3155.50	−	5465.49i	−0.381490	−	0.660760i	0.609785	−	0.792567i	$-0.291256\pi$
$409$	−3155.50	−	5465.49i	−0.381490	−	0.660760i	−0.991275	+	0.131806i	$0.957922\pi$
$410$		0			0
$411$	−2151.00			−0.258153
$412$		0			0
$413$	−600.000	+	1039.23i	−0.0714869	+	0.123819i
$414$		0			0
$415$	7290.00			0.862294
$416$		0			0
$417$	−2460.00			−0.288889
$418$		0			0
$419$	−1164.00	+	2016.11i	−0.135716	+	0.235067i	−0.925871	−	0.377840i	$-0.876667\pi$
$419$	−1164.00	+	2016.11i	−0.135716	+	0.235067i	0.790155	+	0.612908i	$0.210000\pi$
$420$		0			0
$421$	2045.00			0.236739			0.118370	−	0.992970i	$-0.462233\pi$
$421$	2045.00			0.236739			0.118370	+	0.992970i	$0.462233\pi$
$422$		0			0
$423$	−729.000	−	1262.67i	−0.0837948	−	0.145137i
$424$		0			0
$425$	−2442.00	−	4229.67i	−0.278716	−	0.482751i
$426$		0			0
$427$	233.000	−	403.568i	0.0264067	−	0.0457377i
$428$		0			0
$429$	1170.00	−	4053.00i	0.131674	−	0.456132i
$430$		0			0
$431$	2517.00	−	4359.57i	0.281298	−	0.487223i	−0.690406	−	0.723422i	$-0.742568\pi$
$431$	2517.00	−	4359.57i	0.281298	−	0.487223i	0.971705	+	0.236199i	$0.0759016\pi$
$432$		0			0
$433$	−2141.50	−	3709.19i	−0.237676	−	0.411668i	0.722371	−	0.691506i	$-0.243052\pi$
$433$	−2141.50	−	3709.19i	−0.237676	−	0.411668i	−0.960047	+	0.279838i	$0.909719\pi$
$434$		0			0
$435$	1417.50	+	2455.18i	0.156239	+	0.270614i
$436$		0			0
$437$	276.000			0.0302125
$438$		0			0
$439$	−653.000	+	1131.03i	−0.0709931	+	0.122964i	−0.899337	−	0.437257i	$-0.855950\pi$
$439$	−653.000	+	1131.03i	−0.0709931	+	0.122964i	0.828344	+	0.560220i	$0.189284\pi$
$440$		0			0
$441$	−3051.00			−0.329446
$442$		0			0
$443$	5796.00			0.621617			0.310808	−	0.950473i	$-0.399400\pi$
$443$	5796.00			0.621617			0.310808	+	0.950473i	$0.399400\pi$
$444$		0			0
$445$	2241.00	−	3881.53i	0.238727	−	0.413488i
$446$		0			0
$447$	5247.00			0.555200
$448$		0			0
$449$	−1353.00	−	2343.46i	−0.142209	−	0.246314i	0.786119	−	0.618075i	$-0.212087\pi$
$449$	−1353.00	−	2343.46i	−0.142209	−	0.246314i	−0.928328	+	0.371761i	$0.878754\pi$
$450$		0			0
$451$	−3465.00	−	6001.56i	−0.361775	−	0.626612i
$452$		0			0
$453$	555.000	−	961.288i	0.0575633	−	0.0997026i
$454$		0			0
$455$	234.000	−	810.600i	0.0241101	−	0.0835198i
$456$		0			0
$457$	414.500	−	717.935i	0.0424278	−	0.0734871i	−0.844032	−	0.536293i	$-0.819824\pi$
$457$	414.500	−	717.935i	0.0424278	−	0.0734871i	0.886459	+	0.462806i	$0.153157\pi$
$458$		0			0
$459$	−1498.50	−	2595.48i	−0.152383	−	0.263936i
$460$		0			0
$461$	2746.50	+	4757.08i	0.277478	+	0.480606i	0.970757	−	0.240063i	$-0.0771682\pi$
$461$	2746.50	+	4757.08i	0.277478	+	0.480606i	−0.693279	+	0.720669i	$0.743835\pi$
$462$		0			0
$463$	15346.0			1.54037			0.770183	−	0.637823i	$-0.220165\pi$
$463$	15346.0			1.54037			0.770183	+	0.637823i	$0.220165\pi$
$464$		0			0
$465$	−1350.00	+	2338.27i	−0.134634	+	0.233193i
$466$		0			0
$467$	9594.00			0.950658			0.475329	−	0.879808i	$-0.342329\pi$
$467$	9594.00			0.950658			0.475329	+	0.879808i	$0.342329\pi$
$468$		0			0
$469$	1852.00			0.182340
$470$		0			0
$471$	−3916.50	+	6783.58i	−0.383148	+	0.663632i
$472$		0			0
$473$	−15420.0			−1.49897
$474$		0			0
$475$	1012.00	+	1752.84i	0.0977553	+	0.169317i
$476$		0			0
$477$	−2875.50	−	4980.51i	−0.276017	−	0.478075i
$478$		0			0
$479$	−6420.00	+	11119.8i	−0.612395	+	1.06070i	0.378440	+	0.925626i	$0.376461\pi$
$479$	−6420.00	+	11119.8i	−0.612395	+	1.06070i	−0.990836	+	0.135074i	$0.956873\pi$
$480$		0			0
$481$	−773.500	+	191.392i	−0.0733234	+	0.0181428i
$482$		0			0
$483$	−18.0000	+	31.1769i	−0.00169571	+	0.00293706i
$484$		0			0
$485$	6111.00	+	10584.6i	0.572137	+	0.990970i
$486$		0			0
$487$	−7043.00	−	12198.8i	−0.655336	−	1.13508i	−0.981809	−	0.189869i	$-0.939194\pi$
$487$	−7043.00	−	12198.8i	−0.655336	−	1.13508i	0.326473	−	0.945207i	$-0.394140\pi$
$488$		0			0
$489$	−4908.00			−0.453880
$490$		0			0
$491$	5847.00	−	10127.3i	0.537416	−	0.930832i	−0.461626	−	0.887075i	$-0.652734\pi$
$491$	5847.00	−	10127.3i	0.537416	−	0.930832i	0.999042	−	0.0437577i	$-0.0139329\pi$
$492$		0			0
$493$	11655.0			1.06474
$494$		0			0
$495$	2430.00			0.220647
$496$		0			0
$497$	930.000	−	1610.81i	0.0839360	−	0.145381i
$498$		0			0
$499$	3688.00			0.330857			0.165428	−	0.986222i	$-0.447099\pi$
$499$	3688.00			0.330857			0.165428	+	0.986222i	$0.447099\pi$
$500$		0			0
$501$	−396.000	−	685.892i	−0.0353133	−	0.0611645i
$502$		0			0
$503$	−2373.00	−	4110.16i	−0.210352	−	0.364340i	0.741473	−	0.670983i	$-0.234128\pi$
$503$	−2373.00	−	4110.16i	−0.210352	−	0.364340i	−0.951825	+	0.306643i	$0.900794\pi$
$504$		0			0
$505$	−1606.50	+	2782.54i	−0.141561	+	0.245191i
$506$		0			0
$507$	5577.00	+	3512.60i	0.488527	+	0.307692i
$508$		0			0
$509$	7252.50	−	12561.7i	0.631555	−	1.09389i	−0.355679	−	0.934608i	$-0.615750\pi$
$509$	7252.50	−	12561.7i	0.631555	−	1.09389i	0.987234	−	0.159277i	$-0.0509163\pi$
$510$		0			0
$511$	−253.000	−	438.209i	−0.0219023	−	0.0379358i
$512$		0			0
$513$	621.000	+	1075.60i	0.0534460	+	0.0925713i
$514$		0			0
$515$	10062.0			0.860941
$516$		0			0
$517$	2430.00	−	4208.88i	0.206714	−	0.358040i
$518$		0			0
$519$	−4230.00			−0.357758
$520$		0			0
$521$	5085.00			0.427597			0.213798	−	0.976878i	$-0.431416\pi$
$521$	5085.00			0.427597			0.213798	+	0.976878i	$0.431416\pi$
$522$		0			0
$523$	−5441.00	+	9424.09i	−0.454911	+	0.787929i	−0.998683	−	0.0513043i	$-0.983662\pi$
$523$	−5441.00	+	9424.09i	−0.454911	+	0.787929i	0.543772	+	0.839233i	$0.316996\pi$
$524$		0			0
$525$	−264.000			−0.0219465
$526$		0			0
$527$	5550.00	+	9612.88i	0.458751	+	0.794580i
$528$		0			0
$529$	6065.50	+	10505.8i	0.498521	+	0.863463i
$530$		0			0
$531$	2700.00	−	4676.54i	0.220659	−	0.382193i
$532$		0			0
$533$	10510.5	−	2600.67i	0.854147	−	0.211347i
$534$		0			0
$535$	−3213.00	+	5565.08i	−0.259645	+	0.449718i
$536$		0			0
$537$	711.000	+	1231.49i	0.0571358	+	0.0989621i
$538$		0			0
$539$	−5085.00	−	8807.48i	−0.406357	−	0.703831i
$540$		0			0
$541$	−4699.00			−0.373430			−0.186715	−	0.982414i	$-0.559784\pi$
$541$	−4699.00			−0.373430			−0.186715	+	0.982414i	$0.559784\pi$
$542$		0			0
$543$	3373.50	−	5843.07i	0.266613	−	0.461787i
$544$		0			0
$545$	−18054.0			−1.41899
$546$		0			0
$547$	−8270.00			−0.646434			−0.323217	−	0.946325i	$-0.604764\pi$
$547$	−8270.00			−0.646434			−0.323217	+	0.946325i	$0.604764\pi$
$548$		0			0
$549$	−1048.50	+	1816.06i	−0.0815098	+	0.141179i
$550$		0			0
$551$	−4830.00			−0.373439
$552$		0			0
$553$	1324.00	+	2293.24i	0.101812	+	0.176344i
$554$		0			0
$555$	−229.500	−	397.506i	−0.0175527	−	0.0304021i
$556$		0			0
$557$	11392.5	−	19732.4i	0.866635	−	1.50106i	0.00122056	−	0.999999i	$-0.499611\pi$
$557$	11392.5	−	19732.4i	0.866635	−	1.50106i	0.865414	−	0.501057i	$-0.167055\pi$
$558$		0			0
$559$	6682.00	−	23147.1i	0.505579	−	1.75138i
$560$		0			0
$561$	4995.00	−	8651.59i	0.375916	−	0.651106i
$562$		0			0
$563$	−5964.00	−	10330.0i	−0.446452	−	0.773278i	0.551700	−	0.834043i	$-0.313979\pi$
$563$	−5964.00	−	10330.0i	−0.446452	−	0.773278i	−0.998152	+	0.0607647i	$0.980646\pi$
$564$		0			0
$565$	−5035.50	−	8721.74i	−0.374947	−	0.649427i
$566$		0			0
$567$	−162.000			−0.0119989
$568$		0			0
$569$	3981.00	−	6895.29i	0.293308	−	0.508024i	−0.681282	−	0.732021i	$-0.738577\pi$
$569$	3981.00	−	6895.29i	0.293308	−	0.508024i	0.974590	+	0.223997i	$0.0719106\pi$
$570$		0			0
$571$	−20618.0			−1.51110			−0.755549	−	0.655093i	$-0.772630\pi$
$571$	−20618.0			−1.51110			−0.755549	+	0.655093i	$0.772630\pi$
$572$		0			0
$573$	10332.0			0.753273
$574$		0			0
$575$	132.000	−	228.631i	0.00957353	−	0.0165818i
$576$		0			0
$577$	−3493.00			−0.252020			−0.126010	−	0.992029i	$-0.540217\pi$
$577$	−3493.00			−0.252020			−0.126010	+	0.992029i	$0.540217\pi$
$578$		0			0
$579$	−6409.50	−	11101.6i	−0.460051	−	0.796832i
$580$		0			0
$581$	−810.000	−	1402.96i	−0.0578390	−	0.100180i
$582$		0			0
$583$	9585.00	−	16601.7i	0.680909	−	1.17937i
$584$		0			0
$585$	−1053.00	+	3647.70i	−0.0744208	+	0.257801i
$586$		0			0
$587$	5208.00	−	9020.52i	0.366196	−	0.634270i	−0.622771	−	0.782404i	$-0.713993\pi$
$587$	5208.00	−	9020.52i	0.366196	−	0.634270i	0.988967	+	0.148134i	$0.0473266\pi$
$588$		0			0
$589$	−2300.00	−	3983.72i	−0.160900	−	0.278686i
$590$		0			0
$591$	−2979.00	−	5159.78i	−0.207343	−	0.359129i
$592$		0			0
$593$	2061.00			0.142724			0.0713618	−	0.997450i	$-0.477266\pi$
$593$	2061.00			0.142724			0.0713618	+	0.997450i	$0.477266\pi$
$594$		0			0
$595$	999.000	−	1730.32i	0.0688319	−	0.119220i
$596$		0			0
$597$	−7158.00			−0.490716
$598$		0			0
$599$	−12456.0			−0.849647			−0.424823	−	0.905276i	$-0.639664\pi$
$599$	−12456.0			−0.849647			−0.424823	+	0.905276i	$0.639664\pi$
$600$		0			0
$601$	390.500	−	676.366i	0.0265039	−	0.0459061i	−0.852469	−	0.522777i	$-0.824896\pi$
$601$	390.500	−	676.366i	0.0265039	−	0.0459061i	0.878973	+	0.476871i	$0.158229\pi$
$602$		0			0
$603$	−8334.00			−0.562830
$604$		0			0
$605$	−1939.50	−	3359.31i	−0.130334	−	0.225745i
$606$		0			0
$607$	9652.00	+	16717.8i	0.645408	+	1.11788i	0.984207	+	0.177021i	$0.0566459\pi$
$607$	9652.00	+	16717.8i	0.645408	+	1.11788i	−0.338799	+	0.940859i	$0.610021\pi$
$608$		0			0
$609$	315.000	−	545.596i	0.0209597	−	0.0363032i
$610$		0			0
$611$	5265.00	+	5471.55i	0.348607	+	0.362283i
$612$		0			0
$613$	−6020.50	+	10427.8i	−0.396681	+	0.687072i	−0.993314	−	0.115442i	$-0.963172\pi$
$613$	−6020.50	+	10427.8i	−0.396681	+	0.687072i	0.596633	+	0.802514i	$0.296505\pi$
$614$		0			0
$615$	3118.50	+	5401.40i	0.204472	+	0.354155i
$616$		0			0
$617$	−4858.50	−	8415.17i	−0.317011	−	0.549079i	0.662852	−	0.748751i	$-0.269346\pi$
$617$	−4858.50	−	8415.17i	−0.317011	−	0.549079i	−0.979863	+	0.199671i	$0.936013\pi$
$618$		0			0
$619$	21040.0			1.36619			0.683093	−	0.730332i	$-0.260634\pi$
$619$	21040.0			1.36619			0.683093	+	0.730332i	$0.260634\pi$
$620$		0			0
$621$	81.0000	−	140.296i	0.00523417	−	0.00906584i
$622$		0			0
$623$	−996.000			−0.0640512
$624$		0			0
$625$	−8189.00			−0.524096
$626$		0			0
$627$	−2070.00	+	3585.35i	−0.131847	+	0.228365i
$628$		0			0
$629$	−1887.00			−0.119618
$630$		0			0
$631$	−2534.00	−	4389.02i	−0.159868	−	0.276900i	0.774953	−	0.632019i	$-0.217774\pi$
$631$	−2534.00	−	4389.02i	−0.159868	−	0.276900i	−0.934821	+	0.355119i	$0.884440\pi$
$632$		0			0
$633$	2400.00	+	4156.92i	0.150697	+	0.261016i
$634$		0			0
$635$	2718.00	−	4707.71i	0.169859	−	0.294205i
$636$		0			0
$637$	15424.5	−	3816.57i	0.959405	−	0.237391i
$638$		0			0
$639$	−4185.00	+	7248.63i	−0.259086	+	0.448750i
$640$		0			0
$641$	−5092.50	−	8820.47i	−0.313794	−	0.543506i	0.665387	−	0.746499i	$-0.268267\pi$
$641$	−5092.50	−	8820.47i	−0.313794	−	0.543506i	−0.979180	+	0.202992i	$0.934933\pi$
$642$		0			0
$643$	12964.0	+	22454.3i	0.795101	+	1.37716i	0.922775	+	0.385340i	$0.125916\pi$
$643$	12964.0	+	22454.3i	0.795101	+	1.37716i	−0.127673	+	0.991816i	$0.540751\pi$
$644$		0			0
$645$	13878.0			0.847203
$646$		0			0
$647$	11580.0	−	20057.1i	0.703643	−	1.21874i	−0.263537	−	0.964649i	$-0.584889\pi$
$647$	11580.0	−	20057.1i	0.703643	−	1.21874i	0.967179	−	0.254095i	$-0.0817777\pi$
$648$		0			0
$649$	18000.0			1.08869
$650$		0			0
$651$	600.000			0.0361227
$652$		0			0
$653$	−8313.00	+	14398.5i	−0.498182	+	0.862876i	−0.999998	−	0.00209801i	$-0.999332\pi$
$653$	−8313.00	+	14398.5i	−0.498182	+	0.862876i	0.501816	+	0.864974i	$0.332666\pi$
$654$		0			0
$655$	−14256.0			−0.850424
$656$		0			0
$657$	1138.50	+	1971.94i	0.0676060	+	0.117097i
$658$		0			0
$659$	−7404.00	−	12824.1i	−0.437661	−	0.758052i	0.559847	−	0.828596i	$-0.310860\pi$
$659$	−7404.00	−	12824.1i	−0.437661	−	0.758052i	−0.997509	+	0.0705440i	$0.977526\pi$
$660$		0			0
$661$	−2426.50	+	4202.82i	−0.142784	+	0.247308i	−0.928544	−	0.371223i	$-0.878939\pi$
$661$	−2426.50	+	4202.82i	−0.142784	+	0.247308i	0.785760	+	0.618531i	$0.212272\pi$
$662$		0			0
$663$	10822.5	+	11247.1i	0.633953	+	0.658824i
$664$		0			0
$665$	−414.000	+	717.069i	−0.0241417	+	0.0418147i
$666$		0			0
$667$	315.000	+	545.596i	0.0182861	+	0.0316725i
$668$		0			0
$669$	5748.00	+	9955.83i	0.332183	+	0.575358i
$670$		0			0
$671$	−6990.00			−0.402155
$672$		0			0
$673$	8082.50	−	13999.3i	0.462938	−	0.801833i	−0.536168	−	0.844112i	$-0.680128\pi$
$673$	8082.50	−	13999.3i	0.462938	−	0.801833i	0.999106	+	0.0422789i	$0.0134618\pi$
$674$		0			0
$675$	1188.00			0.0677424
$676$		0			0
$677$	−25686.0			−1.45819			−0.729094	−	0.684414i	$-0.760058\pi$
$677$	−25686.0			−1.45819			−0.729094	+	0.684414i	$0.760058\pi$
$678$		0			0
$679$	1358.00	−	2352.12i	0.0767530	−	0.132940i
$680$		0			0
$681$	−4194.00			−0.235998
$682$		0			0
$683$	−9528.00	−	16503.0i	−0.533790	−	0.924552i	−0.999221	−	0.0394675i	$-0.987434\pi$
$683$	−9528.00	−	16503.0i	−0.533790	−	0.924552i	0.465431	−	0.885084i	$-0.345900\pi$
$684$		0			0
$685$	3226.50	+	5588.46i	0.179968	+	0.311714i
$686$		0			0
$687$	6699.00	−	11603.0i	0.372027	−	0.644370i
$688$		0			0
$689$	20767.5	+	21582.2i	1.14830	+	1.19335i
$690$		0			0
$691$	−8195.00	+	14194.2i	−0.451161	+	0.781434i	−0.998458	−	0.0555040i	$-0.982323\pi$
$691$	−8195.00	+	14194.2i	−0.451161	+	0.781434i	0.547297	+	0.836938i	$0.315657\pi$
$692$		0			0
$693$	−270.000	−	467.654i	−0.0148001	−	0.0256345i
$694$		0			0
$695$	3690.00	+	6391.27i	0.201395	+	0.348827i
$696$		0			0
$697$	25641.0			1.39343
$698$		0			0
$699$	−2457.00	+	4255.65i	−0.132950	+	0.230277i
$700$		0			0
$701$	−27846.0			−1.50033			−0.750163	−	0.661253i	$-0.770025\pi$
$701$	−27846.0			−1.50033			−0.750163	+	0.661253i	$0.770025\pi$
$702$		0			0
$703$	782.000			0.0419540
$704$		0			0
$705$	−2187.00	+	3788.00i	−0.116833	+	0.202360i
$706$		0			0
$707$	714.000			0.0379812
$708$		0			0
$709$	6141.50	+	10637.4i	0.325316	+	0.563463i	0.981576	−	0.191071i	$-0.0611961\pi$
$709$	6141.50	+	10637.4i	0.325316	+	0.563463i	−0.656260	+	0.754534i	$0.727863\pi$
$710$		0			0
$711$	−5958.00	−	10319.6i	−0.314265	−	0.544323i
$712$		0			0
$713$	−300.000	+	519.615i	−0.0157575	+	0.0272928i
$714$		0			0
$715$	−12285.0	+	3039.75i	−0.642564	+	0.158993i
$716$		0			0
$717$	891.000	−	1543.26i	0.0464087	−	0.0803821i
$718$		0			0
$719$	12756.0	+	22094.0i	0.661639	+	1.14599i	0.980185	+	0.198085i	$0.0634722\pi$
$719$	12756.0	+	22094.0i	0.661639	+	1.14599i	−0.318546	+	0.947908i	$0.603194\pi$
$720$		0			0
$721$	−1118.00	−	1936.43i	−0.0577483	−	0.100023i
$722$		0			0
$723$	−6909.00			−0.355392
$724$		0			0
$725$	−2310.00	+	4001.04i	−0.118333	+	0.204958i
$726$		0			0
$727$	−6110.00			−0.311702			−0.155851	−	0.987781i	$-0.549812\pi$
$727$	−6110.00			−0.311702			−0.155851	+	0.987781i	$0.549812\pi$
$728$		0			0
$729$	729.000			0.0370370
$730$		0			0
$731$	28527.0	−	49410.2i	1.44338	−	2.50000i
$732$		0			0
$733$	−27127.0			−1.36693			−0.683464	−	0.729984i	$-0.739527\pi$
$733$	−27127.0			−1.36693			−0.683464	+	0.729984i	$0.739527\pi$
$734$		0			0
$735$	4576.50	+	7926.73i	0.229669	+	0.397798i
$736$		0			0
$737$	−13890.0	−	24058.2i	−0.694226	−	1.20244i
$738$		0			0
$739$	−440.000	+	762.102i	−0.0219021	+	0.0379356i	−0.876769	−	0.480912i	$-0.840306\pi$
$739$	−440.000	+	762.102i	−0.0219021	+	0.0379356i	0.854867	+	0.518848i	$0.173639\pi$
$740$		0			0
$741$	−4485.00	−	4660.95i	−0.222349	−	0.231072i
$742$		0			0
$743$	−10938.0	+	18945.2i	−0.540076	+	0.935439i	0.458823	+	0.888528i	$0.348271\pi$
$743$	−10938.0	+	18945.2i	−0.540076	+	0.935439i	−0.998899	+	0.0469111i	$0.985062\pi$
$744$		0			0
$745$	−7870.50	−	13632.1i	−0.387051	−	0.670392i
$746$		0			0
$747$	3645.00	+	6313.33i	0.178532	+	0.309227i
$748$		0			0
$749$	1428.00			0.0696635
$750$		0			0
$751$	5899.00	−	10217.4i	0.286628	−	0.496454i	−0.686375	−	0.727248i	$-0.740799\pi$
$751$	5899.00	−	10217.4i	0.286628	−	0.496454i	0.973003	+	0.230794i	$0.0741323\pi$
$752$		0			0
$753$	18972.0			0.918165
$754$		0			0
$755$	−3330.00			−0.160518
$756$		0			0
$757$	4037.00	−	6992.29i	0.193827	−	0.335719i	−0.752688	−	0.658377i	$-0.771243\pi$
$757$	4037.00	−	6992.29i	0.193827	−	0.335719i	0.946515	+	0.322658i	$0.104577\pi$
$758$		0			0
$759$	540.000			0.0258245
$760$		0			0
$761$	−9777.00	−	16934.3i	−0.465724	−	0.806658i	0.533510	−	0.845794i	$-0.320873\pi$
$761$	−9777.00	−	16934.3i	−0.465724	−	0.806658i	−0.999234	+	0.0391362i	$0.987539\pi$
$762$		0			0
$763$	2006.00	+	3474.49i	0.0951797	+	0.164856i
$764$		0			0
$765$	−4495.50	+	7786.43i	−0.212464	+	0.367999i
$766$		0			0
$767$	−7800.00	+	27020.0i	−0.367199	+	1.27201i
$768$		0			0
$769$	−7015.00	+	12150.3i	−0.328956	+	0.569769i	−0.982305	−	0.187288i	$-0.940030\pi$
$769$	−7015.00	+	12150.3i	−0.328956	+	0.569769i	0.653349	+	0.757057i	$0.273364\pi$
$770$		0			0
$771$	11749.5	+	20350.7i	0.548830	+	0.950601i
$772$		0			0
$773$	−18021.0	−	31213.3i	−0.838513	−	1.45235i	−0.891138	−	0.453732i	$-0.850092\pi$
$773$	−18021.0	−	31213.3i	−0.838513	−	1.45235i	0.0526253	−	0.998614i	$-0.483241\pi$
$774$		0			0
$775$	−4400.00			−0.203939
$776$		0			0
$777$	−51.0000	+	88.3346i	−0.00235472	+	0.00407849i
$778$		0			0
$779$	−10626.0			−0.488724
$780$		0			0
$781$	−27900.0			−1.27828
$782$		0			0
$783$	−1417.50	+	2455.18i	−0.0646964	+	0.112058i
$784$		0			0
$785$	23499.0			1.06843
$786$		0			0
$787$	14314.0	+	24792.6i	0.648334	+	1.12295i	0.983521	+	0.180796i	$0.0578674\pi$
$787$	14314.0	+	24792.6i	0.648334	+	1.12295i	−0.335186	+	0.942152i	$0.608799\pi$
$788$		0			0
$789$	4545.00	+	7872.17i	0.205078	+	0.355205i
$790$		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:	\( N \)	\(=\)	\( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	624.q (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.50000 - 2.59808i) q^{3} -9.00000 q^{5} +(1.00000 + 1.73205i) q^{7} +(-4.50000 - 7.79423i) q^{9} +O(q^{10})\)	\(q+(1.50000 - 2.59808i) q^{3} -9.00000 q^{5} +(1.00000 + 1.73205i) q^{7} +(-4.50000 - 7.79423i) q^{9} +(15.0000 - 25.9808i) q^{11} +(32.5000 + 33.7750i) q^{13} +(-13.5000 + 23.3827i) q^{15} +(55.5000 + 96.1288i) q^{17} +(-23.0000 - 39.8372i) q^{19} +6.00000 q^{21} +(-3.00000 + 5.19615i) q^{23} -44.0000 q^{25} -27.0000 q^{27} +(52.5000 - 90.9327i) q^{29} +100.000 q^{31} +(-45.0000 - 77.9423i) q^{33} +(-9.00000 - 15.5885i) q^{35} +(-8.50000 + 14.7224i) q^{37} +(136.500 - 33.7750i) q^{39} +(115.500 - 200.052i) q^{41} +(-257.000 - 445.137i) q^{43} +(40.5000 + 70.1481i) q^{45} +162.000 q^{47} +(169.500 - 293.583i) q^{49} +333.000 q^{51} +639.000 q^{53} +(-135.000 + 233.827i) q^{55} -138.000 q^{57} +(300.000 + 519.615i) q^{59} +(-116.500 - 201.784i) q^{61} +(9.00000 - 15.5885i) q^{63} +(-292.500 - 303.975i) q^{65} +(463.000 - 801.940i) q^{67} +(9.00000 + 15.5885i) q^{69} +(-465.000 - 805.404i) q^{71} -253.000 q^{73} +(-66.0000 + 114.315i) q^{75} +60.0000 q^{77} +1324.00 q^{79} +(-40.5000 + 70.1481i) q^{81} -810.000 q^{83} +(-499.500 - 865.159i) q^{85} +(-157.500 - 272.798i) q^{87} +(-249.000 + 431.281i) q^{89} +(-26.0000 + 90.0666i) q^{91} +(150.000 - 259.808i) q^{93} +(207.000 + 358.535i) q^{95} +(-679.000 - 1176.06i) q^{97} -270.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 3 q^{3} - 18 q^{5} + 2 q^{7} - 9 q^{9}+O(q^{10}) \) 2 * q + 3 * q^3 - 18 * q^5 + 2 * q^7 - 9 * q^9	\( 2 q + 3 q^{3} - 18 q^{5} + 2 q^{7} - 9 q^{9} + 30 q^{11} + 65 q^{13} - 27 q^{15} + 111 q^{17} - 46 q^{19} + 12 q^{21} - 6 q^{23} - 88 q^{25} - 54 q^{27} + 105 q^{29} + 200 q^{31} - 90 q^{33} - 18 q^{35} - 17 q^{37} + 273 q^{39} + 231 q^{41} - 514 q^{43} + 81 q^{45} + 324 q^{47} + 339 q^{49} + 666 q^{51} + 1278 q^{53} - 270 q^{55} - 276 q^{57} + 600 q^{59} - 233 q^{61} + 18 q^{63} - 585 q^{65} + 926 q^{67} + 18 q^{69} - 930 q^{71} - 506 q^{73} - 132 q^{75} + 120 q^{77} + 2648 q^{79} - 81 q^{81} - 1620 q^{83} - 999 q^{85} - 315 q^{87} - 498 q^{89} - 52 q^{91} + 300 q^{93} + 414 q^{95} - 1358 q^{97} - 540 q^{99}+O(q^{100}) \) 2 * q + 3 * q^3 - 18 * q^5 + 2 * q^7 - 9 * q^9 + 30 * q^11 + 65 * q^13 - 27 * q^15 + 111 * q^17 - 46 * q^19 + 12 * q^21 - 6 * q^23 - 88 * q^25 - 54 * q^27 + 105 * q^29 + 200 * q^31 - 90 * q^33 - 18 * q^35 - 17 * q^37 + 273 * q^39 + 231 * q^41 - 514 * q^43 + 81 * q^45 + 324 * q^47 + 339 * q^49 + 666 * q^51 + 1278 * q^53 - 270 * q^55 - 276 * q^57 + 600 * q^59 - 233 * q^61 + 18 * q^63 - 585 * q^65 + 926 * q^67 + 18 * q^69 - 930 * q^71 - 506 * q^73 - 132 * q^75 + 120 * q^77 + 2648 * q^79 - 81 * q^81 - 1620 * q^83 - 999 * q^85 - 315 * q^87 - 498 * q^89 - 52 * q^91 + 300 * q^93 + 414 * q^95 - 1358 * q^97 - 540 * q^99

Introduction

L-functions

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