LMFDB - Embedded newform 177.5.c.a.58.1

Newspace parameters

Level:	$ N $	$=$	$ 177 = 3 \cdot 59 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	177.c (of order $2$, degree $1$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$18.2964834658$
Analytic rank:	$0$
Dimension:	$40$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			58.1
Character	$\chi$	$=$	177.58
Dual form			177.5.c.a.58.40

$q$-expansion

$f(q)$	$=$	$q-7.81880i q^{2} +5.19615 q^{3} -45.1337 q^{4} +21.3172 q^{5} -40.6277i q^{6} -30.7266 q^{7} +227.790i q^{8} +27.0000 q^{9} +O(q^{10})$	\(q-7.81880i q^{2} +5.19615 q^{3} -45.1337 q^{4} +21.3172 q^{5} -40.6277i q^{6} -30.7266 q^{7} +227.790i q^{8} +27.0000 q^{9} -166.675i q^{10} +207.480i q^{11} -234.521 q^{12} +206.020i q^{13} +240.245i q^{14} +110.768 q^{15} +1058.91 q^{16} -391.490 q^{17} -211.108i q^{18} -321.627 q^{19} -962.125 q^{20} -159.660 q^{21} +1622.24 q^{22} +287.967i q^{23} +1183.63i q^{24} -170.575 q^{25} +1610.83 q^{26} +140.296 q^{27} +1386.80 q^{28} +1020.53 q^{29} -866.070i q^{30} +560.747i q^{31} -4634.75i q^{32} +1078.10i q^{33} +3060.98i q^{34} -655.006 q^{35} -1218.61 q^{36} -1257.89i q^{37} +2514.74i q^{38} +1070.51i q^{39} +4855.86i q^{40} -1524.79 q^{41} +1248.35i q^{42} -2155.53i q^{43} -9364.31i q^{44} +575.565 q^{45} +2251.55 q^{46} -245.854i q^{47} +5502.25 q^{48} -1456.88 q^{49} +1333.69i q^{50} -2034.24 q^{51} -9298.43i q^{52} -1142.98 q^{53} -1096.95i q^{54} +4422.89i q^{55} -6999.22i q^{56} -1671.22 q^{57} -7979.34i q^{58} +(-3465.52 - 327.946i) q^{59} -4999.35 q^{60} -518.066i q^{61} +4384.37 q^{62} -829.618 q^{63} -19295.6 q^{64} +4391.77i q^{65} +8429.41 q^{66} +2495.71i q^{67} +17669.4 q^{68} +1496.32i q^{69} +5121.36i q^{70} +9311.79 q^{71} +6150.34i q^{72} +8005.70i q^{73} -9835.18 q^{74} -886.336 q^{75} +14516.2 q^{76} -6375.14i q^{77} +8370.11 q^{78} +5130.68 q^{79} +22573.0 q^{80} +729.000 q^{81} +11922.1i q^{82} +13428.1i q^{83} +7206.04 q^{84} -8345.49 q^{85} -16853.7 q^{86} +5302.84 q^{87} -47261.8 q^{88} -3141.70i q^{89} -4500.23i q^{90} -6330.29i q^{91} -12997.0i q^{92} +2913.73i q^{93} -1922.28 q^{94} -6856.20 q^{95} -24082.9i q^{96} +5322.39i q^{97} +11391.0i q^{98} +5601.95i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 40q - 320q^{4} + 80q^{7} + 1080q^{9} + O(q^{10}) $	$ 40q - 320q^{4} + 80q^{7} + 1080q^{9} + 360q^{12} + 144q^{15} + 3944q^{16} - 528q^{17} + 444q^{19} + 444q^{20} + 1304q^{22} + 4880q^{25} - 1452q^{26} - 1160q^{28} - 996q^{29} + 10320q^{35} - 8640q^{36} - 5196q^{41} - 10476q^{46} + 576q^{48} + 5104q^{49} + 936q^{51} - 2184q^{53} - 2520q^{57} - 11736q^{59} - 11448q^{60} + 15240q^{62} + 2160q^{63} - 81012q^{64} + 17352q^{66} + 29568q^{68} - 5964q^{71} + 14376q^{74} - 2736q^{75} + 3480q^{76} + 37692q^{78} + 19020q^{79} + 33096q^{80} + 29160q^{81} + 25128q^{84} + 20220q^{85} - 65880q^{86} + 1512q^{87} - 14932q^{88} - 17864q^{94} + 11004q^{95} + O(q^{100}) $

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/177\mathbb{Z}\right)^\times$.

$n$	$61$	$119$
$\chi(n)$	$-1$	$1$

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 $

$2$		−	7.81880i		−	1.95470i	−0.211629	−	0.977350i	$-0.567877\pi$
$2$		−	7.81880i		−	1.95470i	0.211629	−	0.977350i	$-0.432123\pi$
$3$	5.19615			0.577350
$3$	5.19615			0.577350
$4$	−45.1337			−2.82085
$5$	21.3172			0.852690			0.426345	−	0.904561i	$-0.359801\pi$
$5$	21.3172			0.852690			0.426345	+	0.904561i	$0.359801\pi$
$6$		−	40.6277i		−	1.12855i
$7$	−30.7266			−0.627073			−0.313537	−	0.949576i	$-0.601514\pi$
$7$	−30.7266			−0.627073			−0.313537	+	0.949576i	$0.601514\pi$
$8$			227.790i			3.55922i
$9$	27.0000			0.333333
$10$		−	166.675i		−	1.66675i
$11$			207.480i			1.71471i	0.514728	+	0.857353i	$0.327893\pi$
$11$			207.480i			1.71471i	−0.514728	+	0.857353i	$0.672107\pi$
$12$	−234.521			−1.62862
$13$			206.020i			1.21905i	0.792766	+	0.609526i	$0.208640\pi$
$13$			206.020i			1.21905i	−0.792766	+	0.609526i	$0.791360\pi$
$14$			240.245i			1.22574i
$15$	110.768			0.492301
$16$	1058.91			4.13636
$17$	−391.490			−1.35464			−0.677318	−	0.735690i	$-0.736858\pi$
$17$	−391.490			−1.35464			−0.677318	+	0.735690i	$0.736858\pi$
$18$		−	211.108i		−	0.651567i
$19$	−321.627			−0.890934			−0.445467	−	0.895298i	$-0.646962\pi$
$19$	−321.627			−0.890934			−0.445467	+	0.895298i	$0.646962\pi$
$20$	−962.125			−2.40531
$21$	−159.660			−0.362041
$22$	1622.24			3.35174
$23$			287.967i			0.544360i	0.962246	+	0.272180i	$0.0877447\pi$
$23$			287.967i			0.544360i	−0.962246	+	0.272180i	$0.912255\pi$
$24$			1183.63i			2.05492i
$25$	−170.575			−0.272921
$26$	1610.83			2.38288
$27$	140.296			0.192450
$28$	1386.80			1.76888
$29$	1020.53			1.21347			0.606737	−	0.794902i	$-0.292478\pi$
$29$	1020.53			1.21347			0.606737	+	0.794902i	$0.292478\pi$
$30$		−	866.070i		−	0.962300i
$31$			560.747i			0.583504i	0.956494	+	0.291752i	$0.0942381\pi$
$31$			560.747i			0.583504i	−0.956494	+	0.291752i	$0.905762\pi$
$32$		−	4634.75i		−	4.52612i
$33$			1078.10i			0.989987i
$34$			3060.98i			2.64791i
$35$	−655.006			−0.534699
$36$	−1218.61			−0.940284
$37$		−	1257.89i		−	0.918838i	−0.888220	−	0.459419i	$-0.848058\pi$
$37$		−	1257.89i		−	0.918838i	0.888220	−	0.459419i	$-0.151942\pi$
$38$			2514.74i			1.74151i
$39$			1070.51i			0.703820i
$40$			4855.86i			3.03491i
$41$	−1524.79			−0.907076			−0.453538	−	0.891237i	$-0.649838\pi$
$41$	−1524.79			−0.907076			−0.453538	+	0.891237i	$0.649838\pi$
$42$			1248.35i			0.707681i
$43$		−	2155.53i		−	1.16578i	−0.812550	−	0.582892i	$-0.801921\pi$
$43$		−	2155.53i		−	1.16578i	0.812550	−	0.582892i	$-0.198079\pi$
$44$		−	9364.31i		−	4.83694i
$45$	575.565			0.284230
$46$	2251.55			1.06406
$47$		−	245.854i		−	0.111296i	−0.998450	−	0.0556482i	$-0.982277\pi$
$47$		−	245.854i		−	0.111296i	0.998450	−	0.0556482i	$-0.0177225\pi$
$48$	5502.25			2.38813
$49$	−1456.88			−0.606779
$50$			1333.69i			0.533478i
$51$	−2034.24			−0.782100
$52$		−	9298.43i		−	3.43877i
$53$	−1142.98			−0.406899			−0.203449	−	0.979085i	$-0.565215\pi$
$53$	−1142.98			−0.406899			−0.203449	+	0.979085i	$0.565215\pi$
$54$		−	1096.95i		−	0.376182i
$55$			4422.89i			1.46211i
$56$		−	6999.22i		−	2.23189i
$57$	−1671.22			−0.514381
$58$		−	7979.34i		−	2.37198i
$59$	−3465.52	−	327.946i	−0.995552	−	0.0942101i
$59$	−3465.52	−	327.946i	−0.995552	−	0.0942101i
$60$	−4999.35			−1.38871
$61$		−	518.066i		−	0.139228i	−0.997574	−	0.0696138i	$-0.977823\pi$
$61$		−	518.066i		−	0.139228i	0.997574	−	0.0696138i	$-0.0221767\pi$
$62$	4384.37			1.14057
$63$	−829.618			−0.209024
$64$	−19295.6			−4.71085
$65$			4391.77i			1.03947i
$66$	8429.41			1.93513
$67$			2495.71i			0.555961i	0.960587	+	0.277980i	$0.0896650\pi$
$67$			2495.71i			0.555961i	−0.960587	+	0.277980i	$0.910335\pi$
$68$	17669.4			3.82123
$69$			1496.32i			0.314287i
$70$			5121.36i			1.04518i
$71$	9311.79			1.84721			0.923606	−	0.383344i	$-0.125228\pi$
$71$	9311.79			1.84721			0.923606	+	0.383344i	$0.125228\pi$
$72$			6150.34i			1.18641i
$73$			8005.70i			1.50229i	0.660138	+	0.751144i	$0.270498\pi$
$73$			8005.70i			1.50229i	−0.660138	+	0.751144i	$0.729502\pi$
$74$	−9835.18			−1.79605
$75$	−886.336			−0.157571
$76$	14516.2			2.51319
$77$		−	6375.14i		−	1.07525i
$78$	8370.11			1.37576
$79$	5130.68			0.822093			0.411047	−	0.911614i	$-0.365163\pi$
$79$	5130.68			0.822093			0.411047	+	0.911614i	$0.365163\pi$
$80$	22573.0			3.52703
$81$	729.000			0.111111
$82$			11922.1i			1.77306i
$83$			13428.1i			1.94921i	0.223939	+	0.974603i	$0.428108\pi$
$83$			13428.1i			1.94921i	−0.223939	+	0.974603i	$0.571892\pi$
$84$	7206.04			1.02126
$85$	−8345.49			−1.15508
$86$	−16853.7			−2.27876
$87$	5302.84			0.700600
$88$	−47261.8			−6.10302
$89$		−	3141.70i		−	0.396630i	−0.980138	−	0.198315i	$-0.936453\pi$
$89$		−	3141.70i		−	0.396630i	0.980138	−	0.198315i	$-0.0635468\pi$
$90$		−	4500.23i		−	0.555584i
$91$		−	6330.29i		−	0.764435i
$92$		−	12997.0i		−	1.53556i
$93$			2913.73i			0.336886i
$94$	−1922.28			−0.217551
$95$	−6856.20			−0.759690
$96$		−	24082.9i		−	2.61316i
$97$			5322.39i			0.565670i	0.959169	+	0.282835i	$0.0912749\pi$
$97$			5322.39i			0.565670i	−0.959169	+	0.282835i	$0.908725\pi$
$98$			11391.0i			1.18607i
$99$			5601.95i			0.571569i
$100$	7698.69			0.769869
$101$		−	6022.91i		−	0.590424i	−0.955432	−	0.295212i	$-0.904610\pi$
$101$		−	6022.91i		−	0.590424i	0.955432	−	0.295212i	$-0.0953902\pi$
$102$			15905.3i			1.52877i
$103$		−	3738.09i		−	0.352350i	−0.984359	−	0.176175i	$-0.943627\pi$
$103$		−	3738.09i		−	0.352350i	0.984359	−	0.176175i	$-0.0563725\pi$
$104$	−46929.3			−4.33888
$105$	−3403.51			−0.308708
$106$			8936.72i			0.795365i
$107$	10702.1			0.934764			0.467382	−	0.884056i	$-0.345197\pi$
$107$	10702.1			0.934764			0.467382	+	0.884056i	$0.345197\pi$
$108$	−6332.08			−0.542873
$109$			21298.0i			1.79261i	0.443442	+	0.896303i	$0.353757\pi$
$109$			21298.0i			1.79261i	−0.443442	+	0.896303i	$0.646243\pi$
$110$	34581.7			2.85799
$111$		−	6536.18i		−	0.530491i
$112$	−32536.6			−2.59380
$113$		−	16771.3i		−	1.31344i	−0.754135	−	0.656720i	$-0.771943\pi$
$113$		−	16771.3i		−	1.31344i	0.754135	−	0.656720i	$-0.228057\pi$
$114$			13067.0i			1.00546i
$115$			6138.65i			0.464170i
$116$	−46060.4			−3.42303
$117$			5562.54i			0.406351i
$118$	−2564.14	+	27096.2i	−0.184153	+	1.94601i
$119$	12029.2			0.849456
$120$			25231.8i			1.75221i
$121$	−28406.8			−1.94022
$122$	−4050.65			−0.272148
$123$	−7923.06			−0.523700
$124$		−	25308.6i		−	1.64598i
$125$	−16959.5			−1.08541
$126$			6486.62i			0.408580i
$127$	27235.6			1.68861			0.844305	−	0.535863i	$-0.180014\pi$
$127$	27235.6			1.68861			0.844305	+	0.535863i	$0.180014\pi$
$128$			76712.8i			4.68218i
$129$		−	11200.5i		−	0.673066i
$130$	34338.4			2.03186
$131$			1718.72i			0.100153i	0.998745	+	0.0500763i	$0.0159465\pi$
$131$			1718.72i			0.100153i	−0.998745	+	0.0500763i	$0.984054\pi$
$132$		−	48658.4i		−	2.79261i
$133$	9882.51			0.558681
$134$	19513.4			1.08674
$135$	2990.73			0.164100
$136$		−	89177.6i		−	4.82145i
$137$	8888.26			0.473560			0.236780	−	0.971563i	$-0.423908\pi$
$137$	8888.26			0.473560			0.236780	+	0.971563i	$0.423908\pi$
$138$	11699.4			0.614336
$139$	2771.10			0.143424			0.0717121	−	0.997425i	$-0.477154\pi$
$139$	2771.10			0.143424			0.0717121	+	0.997425i	$0.477154\pi$
$140$	29562.8			1.50831
$141$		−	1277.49i		−	0.0642570i
$142$		−	72807.1i		−	3.61074i
$143$	−42744.9			−2.09032
$144$	28590.5			1.37879
$145$	21754.9			1.03472
$146$	62594.9			2.93652
$147$	−7570.15			−0.350324
$148$			56773.1i			2.59191i
$149$			8123.08i			0.365888i	0.983123	+	0.182944i	$0.0585627\pi$
$149$			8123.08i			0.365888i	−0.983123	+	0.182944i	$0.941437\pi$
$150$			6930.08i			0.308004i
$151$		−	2438.11i		−	0.106930i	−0.998570	−	0.0534651i	$-0.982973\pi$
$151$		−	2438.11i		−	0.106930i	0.998570	−	0.0534651i	$-0.0170266\pi$
$152$		−	73263.5i		−	3.17103i
$153$	−10570.2			−0.451546
$154$	−49845.9			−2.10179
$155$			11953.6i			0.497547i
$156$		−	48316.1i		−	1.98537i
$157$			13811.6i			0.560332i	0.959952	+	0.280166i	$0.0903895\pi$
$157$			13811.6i			0.560332i	−0.959952	+	0.280166i	$0.909611\pi$
$158$		−	40115.8i		−	1.60695i
$159$	−5939.09			−0.234923
$160$		−	98800.0i		−	3.85938i
$161$		−	8848.23i		−	0.341354i
$162$		−	5699.91i		−	0.217189i
$163$	−14070.7			−0.529593			−0.264796	−	0.964304i	$-0.585305\pi$
$163$	−14070.7			−0.529593			−0.264796	+	0.964304i	$0.585305\pi$
$164$	68819.5			2.55873
$165$			22982.0i			0.844151i
$166$	104991.			3.81011
$167$	−37863.4			−1.35765			−0.678823	−	0.734302i	$-0.737510\pi$
$167$	−37863.4			−1.35765			−0.678823	+	0.734302i	$0.737510\pi$
$168$		−	36369.0i		−	1.28858i
$169$	−13883.2			−0.486089
$170$			65251.7i			2.25784i
$171$	−8683.93			−0.296978
$172$			97287.2i			3.28851i
$173$			13221.6i			0.441765i	0.975300	+	0.220883i	$0.0708938\pi$
$173$			13221.6i			0.441765i	−0.975300	+	0.220883i	$0.929106\pi$
$174$		−	41461.9i		−	1.36946i
$175$	5241.20			0.171141
$176$			219702.i			7.09264i
$177$	−18007.4	−	1704.05i	−0.574782	−	0.0543923i
$178$	−24564.3			−0.775292
$179$		−	398.257i		−	0.0124296i	−0.999981	−	0.00621480i	$-0.998022\pi$
$179$		−	398.257i		−	0.0124296i	0.999981	−	0.00621480i	$-0.00197825\pi$
$180$	−25977.4			−0.801771
$181$	46013.5			1.40452			0.702260	−	0.711920i	$-0.252174\pi$
$181$	46013.5			1.40452			0.702260	+	0.711920i	$0.252174\pi$
$182$	−49495.3			−1.49424
$183$		−	2691.95i		−	0.0803831i
$184$	−65596.0			−1.93750
$185$		−	26814.7i		−	0.783483i
$186$	22781.8			0.658511
$187$		−	81226.2i		−	2.32281i
$188$			11096.3i			0.313951i
$189$	−4310.82			−0.120680
$190$			53607.3i			1.48497i
$191$		−	60179.4i		−	1.64961i	−0.565418	−	0.824804i	$-0.691285\pi$
$191$		−	60179.4i		−	1.64961i	0.565418	−	0.824804i	$-0.308715\pi$
$192$	−100263.			−2.71981
$193$	−31955.1			−0.857879			−0.428939	−	0.903333i	$-0.641113\pi$
$193$	−31955.1			−0.857879			−0.428939	+	0.903333i	$0.641113\pi$
$194$	41614.7			1.10572
$195$			22820.3i			0.600140i
$196$	65754.2			1.71163
$197$	−34993.6			−0.901689			−0.450844	−	0.892603i	$-0.648877\pi$
$197$	−34993.6			−0.901689			−0.450844	+	0.892603i	$0.648877\pi$
$198$	43800.5			1.11725
$199$	3731.67			0.0942317			0.0471159	−	0.998889i	$-0.484997\pi$
$199$	3731.67			0.0942317			0.0471159	+	0.998889i	$0.484997\pi$
$200$		−	38855.4i		−	0.971385i
$201$			12968.1i			0.320984i
$202$	−47092.0			−1.15410
$203$	−31357.5			−0.760938
$204$	91812.8			2.20619
$205$	−32504.4			−0.773454
$206$	−29227.4			−0.688740
$207$			7775.10i			0.181453i
$208$			218156.i			5.04244i
$209$		−	66731.1i		−	1.52769i
$210$			26611.4i			0.603433i
$211$			13733.7i			0.308476i	0.988034	+	0.154238i	$0.0492923\pi$
$211$			13733.7i			0.308476i	−0.988034	+	0.154238i	$0.950708\pi$
$212$	51586.8			1.14780
$213$	48385.5			1.06649
$214$		−	83677.7i		−	1.82718i
$215$		−	45950.0i		−	0.994052i
$216$			31958.1i			0.684973i
$217$		−	17229.8i		−	0.365899i
$218$	166524.			3.50401
$219$			41598.8i			0.867347i
$220$		−	199621.i		−	4.12441i
$221$		−	80654.7i		−	1.65137i
$222$	−51105.1			−1.03695
$223$	−62492.9			−1.25667			−0.628335	−	0.777943i	$-0.716263\pi$
$223$	−62492.9			−1.25667			−0.628335	+	0.777943i	$0.716263\pi$
$224$			142410.i			2.83821i
$225$	−4605.54			−0.0909735
$226$	−131132.			−2.56738
$227$			15746.2i			0.305579i	0.988259	+	0.152790i	$0.0488257\pi$
$227$			15746.2i			0.305579i	−0.988259	+	0.152790i	$0.951174\pi$
$228$	75428.4			1.45099
$229$			50347.4i			0.960078i	0.877247	+	0.480039i	$0.159377\pi$
$229$			50347.4i			0.960078i	−0.877247	+	0.480039i	$0.840623\pi$
$230$	47996.9			0.907314
$231$		−	33126.2i		−	0.620794i
$232$			232467.i			4.31903i
$233$		−	54304.7i		−	1.00029i	−0.865942	−	0.500144i	$-0.833280\pi$
$233$		−	54304.7i		−	1.00029i	0.865942	−	0.500144i	$-0.166720\pi$
$234$	43492.4			0.794294
$235$		−	5240.92i		−	0.0949012i
$236$	156411.	+	14801.4i	2.80831	+	0.265753i
$237$	26659.8			0.474636
$238$		−	94053.6i		−	1.66043i
$239$	44247.5			0.774628			0.387314	−	0.921948i	$-0.373403\pi$
$239$	44247.5			0.774628			0.387314	+	0.921948i	$0.373403\pi$
$240$	117293.			2.03633
$241$	93615.1			1.61180			0.805902	−	0.592050i	$-0.201681\pi$
$241$	93615.1			1.61180			0.805902	+	0.592050i	$0.201681\pi$
$242$			222107.i			3.79255i
$243$	3788.00			0.0641500
$244$			23382.2i			0.392741i
$245$	−31056.6			−0.517394
$246$			61948.8i			1.02368i
$247$		−	66261.6i		−	1.08610i
$248$	−127733.			−2.07682
$249$			69774.4i			1.12537i
$250$			132603.i			2.12164i
$251$	−31025.2			−0.492455			−0.246228	−	0.969212i	$-0.579191\pi$
$251$	−31025.2			−0.492455			−0.246228	+	0.969212i	$0.579191\pi$
$252$	37443.7			0.589627
$253$	−59747.2			−0.933419
$254$		−	212950.i		−	3.30073i
$255$	−43364.4			−0.666888
$256$	291072.			4.44141
$257$	−51107.2			−0.773777			−0.386889	−	0.922126i	$-0.626450\pi$
$257$	−51107.2			−0.773777			−0.386889	+	0.922126i	$0.626450\pi$
$258$	−87574.4			−1.31564
$259$			38650.6i			0.576179i
$260$		−	198217.i		−	2.93220i
$261$	27554.4			0.404492
$262$	13438.3			0.195768
$263$	83386.8			1.20555			0.602776	−	0.797911i	$-0.294061\pi$
$263$	83386.8			1.20555			0.602776	+	0.797911i	$0.294061\pi$
$264$	−245580.			−3.52358
$265$	−24365.1			−0.346958
$266$		−	77269.3i		−	1.09205i
$267$		−	16324.8i		−	0.228994i
$268$		−	112640.i		−	1.56828i
$269$			55398.4i			0.765584i	0.923835	+	0.382792i	$0.125037\pi$
$269$			55398.4i			0.765584i	−0.923835	+	0.382792i	$0.874963\pi$
$270$		−	23383.9i		−	0.320767i
$271$	33799.6			0.460228			0.230114	−	0.973164i	$-0.426090\pi$
$271$	33799.6			0.460228			0.230114	+	0.973164i	$0.426090\pi$
$272$	−414552.			−5.60326
$273$		−	32893.1i		−	0.441347i
$274$		−	69495.5i		−	0.925669i
$275$		−	35390.9i		−	0.467979i
$276$		−	67534.3i		−	0.886557i
$277$	71567.4			0.932730			0.466365	−	0.884592i	$-0.345563\pi$
$277$	71567.4			0.932730			0.466365	+	0.884592i	$0.345563\pi$
$278$		−	21666.7i		−	0.280351i
$279$			15140.2i			0.194501i
$280$		−	149204.i		−	1.90311i
$281$	121564.			1.53955			0.769774	−	0.638316i	$-0.220369\pi$
$281$	121564.			1.53955			0.769774	+	0.638316i	$0.220369\pi$
$282$	−9988.46			−0.125603
$283$			142151.i			1.77491i	0.460890	+	0.887457i	$0.347530\pi$
$283$			142151.i			1.77491i	−0.460890	+	0.887457i	$0.652470\pi$
$284$	−420275.			−5.21071
$285$	−35625.9			−0.438607
$286$			334214.i			4.08594i
$287$	46851.7			0.568803
$288$		−	125138.i		−	1.50871i
$289$	69743.4			0.835041
$290$		−	170097.i		−	2.02256i
$291$			27655.9i			0.326590i
$292$		−	361326.i		−	4.23774i
$293$	−96135.1			−1.11982			−0.559908	−	0.828555i	$-0.689164\pi$
$293$	−96135.1			−1.11982			−0.559908	+	0.828555i	$0.689164\pi$
$294$			59189.5i			0.684779i
$295$	−73875.3	−	6990.89i	−0.848897	−	0.0803320i
$296$	286535.			3.27035
$297$			29108.6i			0.329996i
$298$	63512.8			0.715202
$299$	−59326.9			−0.663604
$300$	40003.6			0.444484
$301$			66232.2i			0.731032i
$302$	−19063.1			−0.209016
$303$		−	31296.0i		−	0.340881i
$304$	−340574.			−3.68522
$305$		−	11043.7i		−	0.118718i
$306$			82646.5i			0.882636i
$307$	35837.3			0.380240			0.190120	−	0.981761i	$-0.439112\pi$
$307$	35837.3			0.380240			0.190120	+	0.981761i	$0.439112\pi$
$308$			287733.i			3.03311i
$309$		−	19423.7i		−	0.203430i
$310$	93462.6			0.972556
$311$	−55265.8			−0.571394			−0.285697	−	0.958320i	$-0.592225\pi$
$311$	−55265.8			−0.571394			−0.285697	+	0.958320i	$0.592225\pi$
$312$	−243852.			−2.50505
$313$		−	96714.2i		−	0.987192i	−0.869691	−	0.493596i	$-0.835682\pi$
$313$		−	96714.2i		−	0.987192i	0.869691	−	0.493596i	$-0.164318\pi$
$314$	107990.			1.09528
$315$	−17685.2			−0.178233
$316$	−231566.			−2.31900
$317$	−132005.			−1.31363			−0.656813	−	0.754053i	$-0.728096\pi$
$317$	−132005.			−1.31363			−0.656813	+	0.754053i	$0.728096\pi$
$318$			46436.5i			0.459204i
$319$			211740.i			2.08075i
$320$	−411330.			−4.01689
$321$	55609.8			0.539686
$322$	−69182.6			−0.667245
$323$	125914.			1.20689
$324$	−32902.4			−0.313428
$325$		−	35141.9i		−	0.332705i
$326$			110016.i			1.03519i
$327$			110667.i			1.03496i
$328$		−	347333.i		−	3.22848i
$329$			7554.24i			0.0697909i
$330$	179692.			1.65006
$331$	53983.3			0.492724			0.246362	−	0.969178i	$-0.420765\pi$
$331$	53983.3			0.492724			0.246362	+	0.969178i	$0.420765\pi$
$332$		−	606058.i		−	5.49842i
$333$		−	33963.0i		−	0.306279i
$334$			296046.i			2.65379i
$335$			53201.6i			0.474062i
$336$	−169065.			−1.49753
$337$		−	87020.3i		−	0.766233i	−0.923700	−	0.383116i	$-0.874851\pi$
$337$		−	87020.3i		−	0.766233i	0.923700	−	0.383116i	$-0.125149\pi$
$338$			108550.i			0.950158i
$339$		−	87146.3i		−	0.758315i
$340$	376662.			3.25832
$341$	−116344.			−1.00054
$342$			67897.9i			0.580503i
$343$	118539.			1.00757
$344$	491010.			4.14928
$345$			31897.4i			0.267989i
$346$	103377.			0.863519
$347$		−	74094.0i		−	0.615353i	−0.951491	−	0.307677i	$-0.900449\pi$
$347$		−	74094.0i		−	0.615353i	0.951491	−	0.307677i	$-0.0995515\pi$
$348$	−239337.			−1.97629
$349$		−	27934.6i		−	0.229347i	−0.993403	−	0.114673i	$-0.963418\pi$
$349$		−	27934.6i		−	0.229347i	0.993403	−	0.114673i	$-0.0365821\pi$
$350$		−	40979.9i		−	0.334530i
$351$			28903.8i			0.234607i
$352$	961615.			7.76097
$353$		−	140969.i		−	1.13129i	−0.824648	−	0.565647i	$-0.808627\pi$
$353$		−	140969.i		−	1.13129i	0.824648	−	0.565647i	$-0.191373\pi$
$354$	−13323.7	+	140796.i	−0.106321	+	1.12353i
$355$	198502.			1.57510
$356$			141797.i			1.11883i
$357$	62505.3			0.490434
$358$	−3113.89			−0.0242961
$359$	−29736.8			−0.230731			−0.115365	−	0.993323i	$-0.536804\pi$
$359$	−29736.8			−0.230731			−0.115365	+	0.993323i	$0.536804\pi$
$360$			131108.i			1.01164i
$361$	−26877.0			−0.206237
$362$		−	359770.i		−	2.74542i
$363$	−147606.			−1.12019
$364$			285709.i			2.15636i
$365$			170659.i			1.28099i
$366$	−21047.8			−0.157125
$367$			52026.5i			0.386272i	0.981172	+	0.193136i	$0.0618658\pi$
$367$			52026.5i			0.386272i	−0.981172	+	0.193136i	$0.938134\pi$
$368$			304930.i			2.25167i
$369$	−41169.4			−0.302359
$370$	−209659.			−1.53148
$371$	35119.8			0.255155
$372$		−	131507.i		−	0.950306i
$373$	70424.5			0.506181			0.253091	−	0.967443i	$-0.418553\pi$
$373$	70424.5			0.506181			0.253091	+	0.967443i	$0.418553\pi$
$374$	−635091.			−4.54039
$375$	−88124.0			−0.626659
$376$	56003.0			0.396128
$377$			210250.i			1.47929i
$378$			33705.5i			0.235894i
$379$	−39937.6			−0.278038			−0.139019	−	0.990290i	$-0.544395\pi$
$379$	−39937.6			−0.278038			−0.139019	+	0.990290i	$0.544395\pi$
$380$	309445.			2.14297
$381$	141520.			0.974919
$382$	−470531.			−3.22449
$383$	155977.			1.06332			0.531659	−	0.846959i	$-0.321569\pi$
$383$	155977.			1.06332			0.531659	+	0.846959i	$0.321569\pi$
$384$			398612.i			2.70326i
$385$		−	135900.i		−	0.916852i
$386$			249851.i			1.67690i
$387$		−	58199.4i		−	0.388595i
$388$		−	240219.i		−	1.59567i
$389$	−184572.			−1.21974			−0.609870	−	0.792501i	$-0.708778\pi$
$389$	−184572.			−1.21974			−0.609870	+	0.792501i	$0.708778\pi$
$390$	178428.			1.17309
$391$		−	112736.i		−	0.737411i
$392$		−	331862.i		−	2.15966i
$393$			8930.73i			0.0578232i
$394$			273608.i			1.76253i
$395$	109372.			0.700990
$396$		−	252836.i		−	1.61231i
$397$			216054.i			1.37082i	0.728155	+	0.685412i	$0.240378\pi$
$397$			216054.i			1.37082i	−0.728155	+	0.685412i	$0.759622\pi$
$398$		−	29177.2i		−	0.184195i
$399$	51351.0			0.322555
$400$	−180624.			−1.12890
$401$			186079.i			1.15720i	0.815611	+	0.578601i	$0.196401\pi$
$401$			186079.i			1.15720i	−0.815611	+	0.578601i	$0.803599\pi$
$402$	101395.			0.627428
$403$	−115525.			−0.711321
$404$			271836.i			1.66550i
$405$	15540.3			0.0947433
$406$			245178.i			1.48741i
$407$	260986.			1.57554
$408$		−	463380.i		−	2.78367i
$409$			142032.i			0.849064i	0.905413	+	0.424532i	$0.139561\pi$
$409$			142032.i			0.849064i	−0.905413	+	0.424532i	$0.860439\pi$
$410$			254145.i			1.51187i
$411$	46184.7			0.273410
$412$			168713.i			0.993929i
$413$	106484.	+	10076.6i	0.624284	+	0.0590767i
$414$	60792.0			0.354687
$415$			286250.i			1.66207i
$416$	954850.			5.51758
$417$	14399.0			0.0828060
$418$	−521757.			−2.98618
$419$		−	305281.i		−	1.73889i	−0.494032	−	0.869444i	$-0.664477\pi$
$419$		−	305281.i		−	1.73889i	0.494032	−	0.869444i	$-0.335523\pi$
$420$	153613.			0.870821
$421$			40020.9i			0.225799i	0.993606	+	0.112900i	$0.0360139\pi$
$421$			40020.9i			0.225799i	−0.993606	+	0.112900i	$0.963986\pi$
$422$	107381.			0.602978
$423$		−	6638.05i		−	0.0370988i
$424$		−	260359.i		−	1.44824i
$425$	66778.6			0.369708
$426$		−	378317.i		−	2.08466i
$427$			15918.4i			0.0873059i
$428$	−483025.			−2.63683
$429$	−222109.			−1.20685
$430$	−359274.			−1.94307
$431$		−	199954.i		−	1.07641i	−0.842815	−	0.538203i	$-0.819103\pi$
$431$		−	199954.i		−	1.07641i	0.842815	−	0.538203i	$-0.180897\pi$
$432$	148561.			0.796043
$433$	−173422.			−0.924972			−0.462486	−	0.886627i	$-0.653042\pi$
$433$	−173422.			−0.924972			−0.462486	+	0.886627i	$0.653042\pi$
$434$	−134717.			−0.715224
$435$	113042.			0.597394
$436$		−	961254.i		−	5.05668i
$437$		−	92617.9i		−	0.484989i
$438$	325253.			1.69540
$439$	236050.			1.22483			0.612415	−	0.790536i	$-0.290198\pi$
$439$	236050.			1.22483			0.612415	+	0.790536i	$0.290198\pi$
$440$	−1.00749e6			−5.20398
$441$	−39335.7			−0.202260
$442$	−630623.			−3.22794
$443$			390579.i			1.99022i	0.0987629	+	0.995111i	$0.468511\pi$
$443$			390579.i			1.99022i	−0.0987629	+	0.995111i	$0.531489\pi$
$444$			295002.i			1.49644i
$445$		−	66972.4i		−	0.338202i
$446$			488620.i			2.45641i
$447$			42208.8i			0.211246i
$448$	592889.			2.95405
$449$	297658.			1.47647			0.738236	−	0.674543i	$-0.235659\pi$
$449$	297658.			1.47647			0.738236	+	0.674543i	$0.235659\pi$
$450$			36009.8i			0.177826i
$451$		−	316364.i		−	1.55537i
$452$			756951.i			3.70502i
$453$		−	12668.8i		−	0.0617361i
$454$	123116.			0.597316
$455$		−	134944.i		−	0.651826i
$456$		−	380688.i		−	1.83080i
$457$			138561.i			0.663449i	0.943376	+	0.331725i	$0.107630\pi$
$457$			138561.i			0.663449i	−0.943376	+	0.331725i	$0.892370\pi$
$458$	393657.			1.87666
$459$	−54924.5			−0.260700
$460$		−	277060.i		−	1.30936i
$461$	−249929.			−1.17602			−0.588010	−	0.808853i	$-0.700089\pi$
$461$	−249929.			−1.17602			−0.588010	+	0.808853i	$0.700089\pi$
$462$	−259007.			−1.21347
$463$		−	101021.i		−	0.471248i	−0.971844	−	0.235624i	$-0.924287\pi$
$463$		−	101021.i		−	0.471248i	0.971844	−	0.235624i	$-0.0757134\pi$
$464$	1.08065e6			5.01937
$465$			62112.6i			0.287259i
$466$	−424597.			−1.95526
$467$			290164.i			1.33048i	0.746628	+	0.665242i	$0.231672\pi$
$467$			290164.i			1.33048i	−0.746628	+	0.665242i	$0.768328\pi$
$468$		−	251058.i		−	1.14626i
$469$		−	76684.6i		−	0.348628i
$470$	−40977.7			−0.185503
$471$			71767.3i			0.323508i
$472$	74702.8	−	789411.i	0.335315	−	3.54339i
$473$	447229.			1.99898
$474$		−	208448.i		−	0.927771i
$475$	54861.7			0.243154
$476$	−542920.			−2.39619
$477$	−30860.4			−0.135633
$478$		−	345963.i		−	1.51417i
$479$	−90383.7			−0.393930			−0.196965	−	0.980411i	$-0.563109\pi$
$479$	−90383.7			−0.393930			−0.196965	+	0.980411i	$0.563109\pi$
$480$		−	513380.i		−	2.22821i
$481$	259150.			1.12011
$482$		−	731958.i		−	3.15059i
$483$		−	45976.8i		−	0.197081i
$484$	1.28210e6			5.47308
$485$			113459.i			0.482341i
$486$		−	29617.6i		−	0.125394i
$487$	151820.			0.640132			0.320066	−	0.947395i	$-0.396295\pi$
$487$	151820.			0.640132			0.320066	+	0.947395i	$0.396295\pi$
$488$	118010.			0.495542
$489$	−73113.7			−0.305760
$490$			242825.i			1.01135i
$491$	−103785.			−0.430497			−0.215249	−	0.976559i	$-0.569056\pi$
$491$	−103785.			−0.430497			−0.215249	+	0.976559i	$0.569056\pi$
$492$	357597.			1.47728
$493$	−399528.			−1.64382
$494$	−518086.			−2.12299
$495$			119418.i			0.487371i
$496$			593779.i			2.41358i
$497$	−286120.			−1.15834
$498$	545552.			2.19977
$499$	469746.			1.88652			0.943262	−	0.332049i	$-0.107740\pi$
$499$	469746.			1.88652			0.943262	+	0.332049i	$0.107740\pi$
$500$	765443.			3.06177
$501$	−196744.			−0.783837
$502$			242580.i			0.962602i
$503$			138093.i			0.545801i	0.962042	+	0.272901i	$0.0879831\pi$
$503$			138093.i			0.545801i	−0.962042	+	0.272901i	$0.912017\pi$
$504$		−	188979.i		−	0.743964i
$505$		−	128392.i		−	0.503448i
$506$			467151.i			1.82455i
$507$	−72139.2			−0.280644
$508$	−1.22924e6			−4.76332
$509$			123336.i			0.476051i	0.971259	+	0.238026i	$0.0765002\pi$
$509$			123336.i			0.476051i	−0.971259	+	0.238026i	$0.923500\pi$
$510$			339058.i			1.30357i
$511$		−	245988.i		−	0.942045i
$512$		−	1.04843e6i		−	3.99944i
$513$	−45123.0			−0.171460
$514$			399597.i			1.51250i
$515$		−	79685.7i		−	0.300446i
$516$			505519.i			1.89862i
$517$	51009.6			0.190841
$518$	302202.			1.12626
$519$			68701.4i			0.255053i
$520$	−1.00040e6			−3.69972
$521$	−68858.9			−0.253679			−0.126840	−	0.991923i	$-0.540483\pi$
$521$	−68858.9			−0.253679			−0.126840	+	0.991923i	$0.540483\pi$
$522$		−	215442.i		−	0.790660i
$523$	269285.			0.984482			0.492241	−	0.870459i	$-0.336178\pi$
$523$	269285.			0.984482			0.492241	+	0.870459i	$0.336178\pi$
$524$		−	77572.1i		−	0.282516i
$525$	27234.1			0.0988084
$526$		−	651985.i		−	2.35649i
$527$		−	219527.i		−	0.790435i
$528$			1.14160e6i			4.09494i
$529$	196916.			0.703672
$530$			190506.i			0.678199i
$531$	−93569.0	−	8854.53i	−0.331851	−	0.0314034i
$532$	−446034.			−1.57596
$533$		−	314138.i		−	1.10577i
$534$	−127640.			−0.447615
$535$	228139.			0.797063
$536$	−568498.			−1.97879
$537$		−	2069.40i		−	0.00717623i
$538$	433149.			1.49649
$539$		−	302272.i		−	1.04045i
$540$	−134982.			−0.462902
$541$			110046.i			0.375993i	0.982170	+	0.187997i	$0.0601994\pi$
$541$			110046.i			0.375993i	−0.982170	+	0.187997i	$0.939801\pi$
$542$		−	264272.i		−	0.899607i
$543$	239093.			0.810900
$544$			1.81446e6i			6.13125i
$545$			454014.i			1.52854i
$546$	−257185.			−0.862701
$547$	163036.			0.544889			0.272444	−	0.962172i	$-0.412168\pi$
$547$	163036.			0.544889			0.272444	+	0.962172i	$0.412168\pi$
$548$	−401159.			−1.33584
$549$		−	13987.8i		−	0.0464092i
$550$	−276714.			−0.914758
$551$	−328231.			−1.08113
$552$	−340847.			−1.11862
$553$	−157648.			−0.515513
$554$		−	559572.i		−	1.82321i
$555$		−	139333.i		−	0.452344i
$556$	−125070.			−0.404578
$557$	−447290.			−1.44171			−0.720857	−	0.693084i	$-0.756251\pi$
$557$	−447290.			−1.44171			−0.720857	+	0.693084i	$0.756251\pi$
$558$	118378.			0.380192
$559$	444083.			1.42115
$560$	−693591.			−2.21171
$561$		−	422064.i		−	1.34107i
$562$		−	950487.i		−	3.00936i
$563$			50595.3i			0.159622i	0.996810	+	0.0798111i	$0.0254317\pi$
$563$			50595.3i			0.159622i	−0.996810	+	0.0798111i	$0.974568\pi$
$564$			57657.9i			0.181259i
$565$		−	357518.i		−	1.11996i
$566$	1.11145e6			3.46943
$567$	−22399.7			−0.0696748
$568$			2.12114e6i			6.57464i
$569$			596739.i			1.84315i	0.388204	+	0.921573i	$0.373096\pi$
$569$			596739.i			1.84315i	−0.388204	+	0.921573i	$0.626904\pi$
$570$			278552.i			0.857346i
$571$		−	2297.80i		−	0.00704759i	−0.999994	−	0.00352379i	$-0.998878\pi$
$571$		−	2297.80i		−	0.00704759i	0.999994	−	0.00352379i	$-0.00112166\pi$
$572$	1.92923e6			5.89648
$573$		−	312701.i		−	0.952402i
$574$		−	366324.i		−	1.11184i
$575$		−	49120.0i		−	0.148567i
$576$	−520982.			−1.57028
$577$	−635182.			−1.90786			−0.953931	−	0.300027i	$-0.903004\pi$
$577$	−635182.			−1.90786			−0.953931	+	0.300027i	$0.903004\pi$
$578$		−	545310.i		−	1.63225i
$579$	−166044.			−0.495296
$580$	−981880.			−2.91879
$581$		−	412599.i		−	1.22229i
$582$	216236.			0.638385
$583$		−	237145.i		−	0.697712i
$584$	−1.82362e6			−5.34698
$585$			118578.i			0.346491i
$586$			751661.i			2.18891i
$587$		−	46453.2i		−	0.134815i	−0.997726	−	0.0674077i	$-0.978527\pi$
$587$		−	46453.2i		−	0.134815i	0.997726	−	0.0674077i	$-0.0214728\pi$
$588$	341669.			0.988213
$589$		−	180351.i		−	0.519863i
$590$	−54660.4	+	577616.i	−0.157025	+	1.65934i
$591$	−181832.			−0.520590
$592$		−	1.33199e6i		−	3.80064i
$593$	−292298.			−0.831221			−0.415611	−	0.909543i	$-0.636432\pi$
$593$	−292298.			−0.831221			−0.415611	+	0.909543i	$0.636432\pi$
$594$	227594.			0.645042
$595$	256428.			0.724323
$596$		−	366624.i		−	1.03212i
$597$	19390.3			0.0544047
$598$			463865.i			1.29715i
$599$	239592.			0.667758			0.333879	−	0.942616i	$-0.391642\pi$
$599$	239592.			0.667758			0.333879	+	0.942616i	$0.391642\pi$
$600$		−	201899.i		−	0.560829i
$601$		−	235688.i		−	0.652512i	−0.945281	−	0.326256i	$-0.894213\pi$
$601$		−	235688.i		−	0.652512i	0.945281	−	0.326256i	$-0.105787\pi$
$602$	517857.			1.42895
$603$			67384.1i			0.185320i
$604$			110041.i			0.301634i
$605$	−605554.			−1.65441
$606$	−244697.			−0.666321
$607$	423021.			1.14811			0.574056	−	0.818816i	$-0.305369\pi$
$607$	423021.			1.14811			0.574056	+	0.818816i	$0.305369\pi$
$608$			1.49066e6i			4.03247i
$609$	−162938.			−0.439328
$610$	−86348.7			−0.232058
$611$	50650.7			0.135676
$612$	477073.			1.27374
$613$			210522.i			0.560243i	0.959965	+	0.280122i	$0.0903748\pi$
$613$			210522.i			0.560243i	−0.959965	+	0.280122i	$0.909625\pi$
$614$		−	280205.i		−	0.743256i
$615$	−168898.			−0.446554
$616$	1.45219e6			3.82704
$617$	−342587.			−0.899914			−0.449957	−	0.893050i	$-0.648561\pi$
$617$	−342587.			−0.899914			−0.449957	+	0.893050i	$0.648561\pi$
$618$	−151870.			−0.397644
$619$	−202870.			−0.529463			−0.264732	−	0.964322i	$-0.585283\pi$
$619$	−202870.			−0.529463			−0.264732	+	0.964322i	$0.585283\pi$
$620$		−	539508.i		−	1.40351i
$621$			40400.6i			0.104762i
$622$			432112.i			1.11690i
$623$			96533.8i			0.248716i
$624$			1.13357e6i			2.91125i
$625$	−254919.			−0.652594
$626$	−756189.			−1.92966
$627$		−	346745.i		−	0.882013i
$628$		−	623369.i		−	1.58061i
$629$			492451.i			1.24469i
$630$			138277.i			0.348392i
$631$	−41966.6			−0.105401			−0.0527006	−	0.998610i	$-0.516783\pi$
$631$	−41966.6			−0.105401			−0.0527006	+	0.998610i	$0.516783\pi$
$632$			1.16872e6i			2.92601i
$633$			71362.2i			0.178099i
$634$			1.03212e6i			2.56775i
$635$	580587.			1.43986
$636$	268053.			0.662683
$637$		−	300146.i		−	0.739696i
$638$	1.65555e6			4.06725
$639$	251418.			0.615737
$640$			1.63531e6i			3.99245i
$641$	−336370.			−0.818656			−0.409328	−	0.912387i	$-0.634237\pi$
$641$	−336370.			−0.818656			−0.409328	+	0.912387i	$0.634237\pi$
$642$		−	434802.i		−	1.05492i
$643$	−387046.			−0.936140			−0.468070	−	0.883691i	$-0.655051\pi$
$643$	−387046.			−0.936140			−0.468070	+	0.883691i	$0.655051\pi$
$644$			399353.i			0.962909i
$645$		−	238763.i		−	0.573916i
$646$		−	984495.i		−	2.35911i
$647$	640773.			1.53072			0.765360	−	0.643602i	$-0.222561\pi$
$647$	640773.			1.53072			0.765360	+	0.643602i	$0.222561\pi$
$648$			166059.i			0.395469i
$649$	68042.0	−	719024.i	0.161543	−	1.70708i
$650$	−274768.			−0.650338
$651$		−	89528.9i		−	0.211252i
$652$	635064.			1.49390
$653$	59695.3			0.139995			0.0699977	−	0.997547i	$-0.477701\pi$
$653$	59695.3			0.139995			0.0699977	+	0.997547i	$0.477701\pi$
$654$	865287.			2.02304
$655$			36638.4i			0.0853991i
$656$	−1.61462e6			−3.75199
$657$			216154.i			0.500763i
$658$	59065.1			0.136420
$659$			277090.i			0.638043i	0.947747	+	0.319022i	$0.103354\pi$
$659$			277090.i			0.638043i	−0.947747	+	0.319022i	$0.896646\pi$
$660$		−	1.03726e6i		−	2.38123i
$661$	−170596.			−0.390450			−0.195225	−	0.980758i	$-0.562544\pi$
$661$	−170596.			−0.390450			−0.195225	+	0.980758i	$0.562544\pi$
$662$		−	422085.i		−	0.963128i
$663$		−	419094.i		−	0.953421i
$664$	−3.05879e6			−6.93766
$665$	210668.			0.476381
$666$	−265550.			−0.598684
$667$			293879.i			0.660568i
$668$	1.70891e6			3.82972
$669$	−324723.			−0.725539
$670$	415973.			0.926649
$671$	107488.			0.238734
$672$			739984.i			1.63864i
$673$		−	124494.i		−	0.274864i	−0.990511	−	0.137432i	$-0.956115\pi$
$673$		−	124494.i		−	0.274864i	0.990511	−	0.137432i	$-0.0438848\pi$
$674$	−680394.			−1.49775
$675$	−23931.1			−0.0525236
$676$	626599.			1.37119
$677$	470982.			1.02761			0.513804	−	0.857908i	$-0.328236\pi$
$677$	470982.			1.02761			0.513804	+	0.857908i	$0.328236\pi$
$678$	−681380.			−1.48228
$679$		−	163539.i		−	0.354716i
$680$		−	1.90102e6i		−	4.11120i
$681$			81819.6i			0.176426i
$682$			909667.i			1.95575i
$683$			543532.i			1.16516i	0.812775	+	0.582578i	$0.197956\pi$
$683$			543532.i			1.16516i	−0.812775	+	0.582578i	$0.802044\pi$
$684$	391938.			0.837731
$685$	189473.			0.403800
$686$		−	926836.i		−	1.96949i
$687$			261613.i			0.554301i
$688$		−	2.28251e6i		−	4.82210i
$689$		−	235476.i		−	0.496031i
$690$	249399.			0.523838
$691$		−	466845.i		−	0.977725i	−0.872361	−	0.488862i	$-0.837412\pi$
$691$		−	466845.i		−	0.977725i	0.872361	−	0.488862i	$-0.162588\pi$
$692$		−	596739.i		−	1.24615i
$693$		−	172129.i		−	0.358416i
$694$	−579327.			−1.20283
$695$	59072.1			0.122296
$696$			1.20794e6i			2.49359i
$697$	596942.			1.22876
$698$	−218415.			−0.448304
$699$		−	282175.i		−	0.577517i
$700$	−236554.			−0.482764
$701$		−	661106.i		−	1.34535i	−0.739939	−	0.672674i	$-0.765145\pi$
$701$		−	661106.i		−	1.34535i	0.739939	−	0.672674i	$-0.234855\pi$
$702$	225993.			0.458586
$703$			404571.i			0.818624i
$704$		−	4.00345e6i		−	8.07773i
$705$		−	27232.6i		−	0.0547912i
$706$	−1.10221e6			−2.21134
$707$			185064.i			0.370239i
$708$	812738.	+	76910.2i	1.62138	+	0.153433i
$709$	−64729.8			−0.128769			−0.0643845	−	0.997925i	$-0.520508\pi$
$709$	−64729.8			−0.128769			−0.0643845	+	0.997925i	$0.520508\pi$
$710$		−	1.55205e6i		−	3.07884i
$711$	138528.			0.274031
$712$	715649.			1.41169
$713$	−161476.			−0.317636
$714$		−	488717.i		−	0.958651i
$715$	−911203.			−1.78239
$716$			17974.8i			0.0350621i
$717$	229917.			0.447232
$718$			232506.i			0.451009i
$719$			648569.i			1.25458i	0.778786	+	0.627290i	$0.215836\pi$
$719$			648569.i			1.25458i	−0.778786	+	0.627290i	$0.784164\pi$
$720$	609471.			1.17568
$721$			114859.i			0.220950i
$722$			210146.i			0.403131i
$723$	486439.			0.930575
$724$	−2.07676e6			−3.96195
$725$	−174078.			−0.331182
$726$			1.15410e6i			2.18963i
$727$	201809.			0.381831			0.190916	−	0.981606i	$-0.438854\pi$
$727$	201809.			0.381831			0.190916	+	0.981606i	$0.438854\pi$
$728$	1.44198e6			2.72079
$729$	19683.0			0.0370370
$730$	1.33435e6			2.50394
$731$			843870.i			1.57921i
$732$			121497.i			0.226749i
$733$	−517718.			−0.963575			−0.481787	−	0.876288i	$-0.660012\pi$
$733$	−517718.			−0.963575			−0.481787	+	0.876288i	$0.660012\pi$
$734$	406785.			0.755045
$735$	−161375.			−0.298718
$736$	1.33465e6			2.46384
$737$	−517808.			−0.953310
$738$			321896.i			0.591020i
$739$		−	1.01019e6i		−	1.84976i	−0.380263	−	0.924879i	$-0.624166\pi$
$739$		−	1.01019e6i		−	1.84976i	0.380263	−	0.924879i	$-0.375834\pi$
$740$			1.21025e6i			2.21009i
$741$		−	344305.i		−	0.627057i
$742$		−	274595.i		−	0.498752i
$743$	−521556.			−0.944765			−0.472382	−	0.881394i	$-0.656606\pi$
$743$	−521556.			−0.944765			−0.472382	+	0.881394i	$0.656606\pi$
$744$	−663718.			−1.19905
$745$			173162.i			0.311989i
$746$		−	550635.i		−	0.989432i
$747$			362558.i			0.649735i
$748$			3.66603e6i			6.55229i
$749$	−328839.			−0.586165
$750$			689024.i			1.22493i
$751$			879845.i			1.56001i	0.625776	+	0.780003i	$0.284783\pi$
$751$			879845.i			1.56001i	−0.625776	+	0.780003i	$0.715217\pi$
$752$		−	260336.i		−	0.460362i
$753$	−161212.			−0.284319
$754$	1.64390e6			2.89157
$755$		−	51973.9i		−	0.0911782i
$756$	194563.			0.340421
$757$	−1.05871e6			−1.84750			−0.923748	−	0.383000i	$-0.874891\pi$
$757$	−1.05871e6			−1.84750			−0.923748	+	0.383000i	$0.874891\pi$
$758$			312264.i			0.543481i
$759$	−310456.			−0.538910
$760$		−	1.56178e6i		−	2.70391i
$761$	−319446.			−0.551604			−0.275802	−	0.961214i	$-0.588943\pi$
$761$	−319446.			−0.551604			−0.275802	+	0.961214i	$0.588943\pi$
$762$		−	1.10652e6i		−	1.90567i
$763$		−	654414.i		−	1.12410i
$764$			2.71612e6i			4.65330i
$765$	−225328.			−0.385028
$766$		−	1.21955e6i		−	2.07847i
$767$	67563.3	−	713966.i	0.114847	−	1.21363i
$768$	1.51245e6			2.56425
$769$			547254.i			0.925414i	0.886511	+	0.462707i	$0.153122\pi$
$769$			547254.i			0.925414i	−0.886511	+	0.462707i	$0.846878\pi$
$770$	−1.06258e6			−1.79217
$771$	−265561.			−0.446741
$772$	1.44225e6			2.41995
$773$		−	385413.i		−	0.645011i	−0.946568	−	0.322505i	$-0.895475\pi$
$773$		−	385413.i		−	0.645011i	0.946568	−	0.322505i	$-0.104525\pi$
$774$	−455050.			−0.759586
$775$		−	95649.6i		−	0.159250i
$776$	−1.21239e6			−2.01335
$777$			200835.i			0.332657i
$778$			1.44313e6i			2.38423i
$779$	490415.			0.808144
$780$		−	1.02996e6i		−	1.69291i
$781$			1.93201e6i			3.16743i
$782$	−881461.			−1.44142
$783$	143177.			0.233533
$784$	−1.54270e6			−2.50986
$785$			294426.i			0.477789i
$786$	69827.6			0.113027
$787$	1.03990e6			1.67896			0.839481	−	0.543389i	$-0.182859\pi$
$787$	1.03990e6			1.67896			0.839481	+	0.543389i	$0.182859\pi$
$788$	1.57939e6			2.54353
$789$	433290.			0.696025

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:	\( N \)	\(=\)	\( 177 = 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	177.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-7.81880i q^{2} +5.19615 q^{3} -45.1337 q^{4} +21.3172 q^{5} -40.6277i q^{6} -30.7266 q^{7} +227.790i q^{8} +27.0000 q^{9} +O(q^{10})\)	\(q-7.81880i q^{2} +5.19615 q^{3} -45.1337 q^{4} +21.3172 q^{5} -40.6277i q^{6} -30.7266 q^{7} +227.790i q^{8} +27.0000 q^{9} -166.675i q^{10} +207.480i q^{11} -234.521 q^{12} +206.020i q^{13} +240.245i q^{14} +110.768 q^{15} +1058.91 q^{16} -391.490 q^{17} -211.108i q^{18} -321.627 q^{19} -962.125 q^{20} -159.660 q^{21} +1622.24 q^{22} +287.967i q^{23} +1183.63i q^{24} -170.575 q^{25} +1610.83 q^{26} +140.296 q^{27} +1386.80 q^{28} +1020.53 q^{29} -866.070i q^{30} +560.747i q^{31} -4634.75i q^{32} +1078.10i q^{33} +3060.98i q^{34} -655.006 q^{35} -1218.61 q^{36} -1257.89i q^{37} +2514.74i q^{38} +1070.51i q^{39} +4855.86i q^{40} -1524.79 q^{41} +1248.35i q^{42} -2155.53i q^{43} -9364.31i q^{44} +575.565 q^{45} +2251.55 q^{46} -245.854i q^{47} +5502.25 q^{48} -1456.88 q^{49} +1333.69i q^{50} -2034.24 q^{51} -9298.43i q^{52} -1142.98 q^{53} -1096.95i q^{54} +4422.89i q^{55} -6999.22i q^{56} -1671.22 q^{57} -7979.34i q^{58} +(-3465.52 - 327.946i) q^{59} -4999.35 q^{60} -518.066i q^{61} +4384.37 q^{62} -829.618 q^{63} -19295.6 q^{64} +4391.77i q^{65} +8429.41 q^{66} +2495.71i q^{67} +17669.4 q^{68} +1496.32i q^{69} +5121.36i q^{70} +9311.79 q^{71} +6150.34i q^{72} +8005.70i q^{73} -9835.18 q^{74} -886.336 q^{75} +14516.2 q^{76} -6375.14i q^{77} +8370.11 q^{78} +5130.68 q^{79} +22573.0 q^{80} +729.000 q^{81} +11922.1i q^{82} +13428.1i q^{83} +7206.04 q^{84} -8345.49 q^{85} -16853.7 q^{86} +5302.84 q^{87} -47261.8 q^{88} -3141.70i q^{89} -4500.23i q^{90} -6330.29i q^{91} -12997.0i q^{92} +2913.73i q^{93} -1922.28 q^{94} -6856.20 q^{95} -24082.9i q^{96} +5322.39i q^{97} +11391.0i q^{98} +5601.95i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40q - 320q^{4} + 80q^{7} + 1080q^{9} + O(q^{10}) \)	\( 40q - 320q^{4} + 80q^{7} + 1080q^{9} + 360q^{12} + 144q^{15} + 3944q^{16} - 528q^{17} + 444q^{19} + 444q^{20} + 1304q^{22} + 4880q^{25} - 1452q^{26} - 1160q^{28} - 996q^{29} + 10320q^{35} - 8640q^{36} - 5196q^{41} - 10476q^{46} + 576q^{48} + 5104q^{49} + 936q^{51} - 2184q^{53} - 2520q^{57} - 11736q^{59} - 11448q^{60} + 15240q^{62} + 2160q^{63} - 81012q^{64} + 17352q^{66} + 29568q^{68} - 5964q^{71} + 14376q^{74} - 2736q^{75} + 3480q^{76} + 37692q^{78} + 19020q^{79} + 33096q^{80} + 29160q^{81} + 25128q^{84} + 20220q^{85} - 65880q^{86} + 1512q^{87} - 14932q^{88} - 17864q^{94} + 11004q^{95} + O(q^{100}) \)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Properties

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