LMFDB - Embedded newform 147.6.c.c.146.9

Newspace parameters

Level:	$ N $	$=$	$ 147 = 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	147.c (of order $2$, degree $1$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$23.5764215125$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
Defining polynomial:	$ x^{16} - 171 x^{14} + 21495 x^{12} - 1128902 x^{10} + 42970860 x^{8} - 655075344 x^{6} + 7244325760 x^{4} - 29387167488 x^{2} + 90230547456 $ x^16 - 171x^14 + 21495x^12 - 1128902x^10 + 42970860x^8 - 655075344x^6 + 7244325760x^4 - 29387167488*x^2 + 90230547456
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{12}\cdot 3^{8}\cdot 7^{4} $
Twist minimal:	no (minimal twist has level 21)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			146.9
Root			$1.84692 - 1.06632i$ of defining polynomial
Character	$\chi$	$=$	147.146
Dual form			147.6.c.c.146.7

$q$-expansion

$f(q)$	$=$	$q+2.13264i q^{2} +(-9.16236 - 12.6115i) q^{3} +27.4518 q^{4} -95.0525 q^{5} +(26.8959 - 19.5400i) q^{6} +126.790i q^{8} +(-75.1022 + 231.103i) q^{9} +O(q^{10})$	\(q+2.13264i q^{2} +(-9.16236 - 12.6115i) q^{3} +27.4518 q^{4} -95.0525 q^{5} +(26.8959 - 19.5400i) q^{6} +126.790i q^{8} +(-75.1022 + 231.103i) q^{9} -202.713i q^{10} +130.679i q^{11} +(-251.524 - 346.210i) q^{12} +14.6710i q^{13} +(870.905 + 1198.76i) q^{15} +608.062 q^{16} +640.492 q^{17} +(-492.860 - 160.166i) q^{18} +542.677i q^{19} -2609.37 q^{20} -278.692 q^{22} -4094.31i q^{23} +(1599.01 - 1161.69i) q^{24} +5909.98 q^{25} -31.2880 q^{26} +(3602.68 - 1170.29i) q^{27} -3594.70i q^{29} +(-2556.52 + 1857.33i) q^{30} -2900.94i q^{31} +5354.04i q^{32} +(1648.06 - 1197.33i) q^{33} +1365.94i q^{34} +(-2061.69 + 6344.20i) q^{36} -4342.05 q^{37} -1157.34 q^{38} +(185.024 - 134.421i) q^{39} -12051.7i q^{40} +10945.3 q^{41} +20541.6 q^{43} +3587.38i q^{44} +(7138.66 - 21966.9i) q^{45} +8731.70 q^{46} +8816.03 q^{47} +(-5571.28 - 7668.60i) q^{48} +12603.9i q^{50} +(-5868.42 - 8077.60i) q^{51} +402.745i q^{52} +6977.45i q^{53} +(2495.82 + 7683.23i) q^{54} -12421.4i q^{55} +(6844.00 - 4972.20i) q^{57} +7666.22 q^{58} +40020.1 q^{59} +(23908.0 + 32908.1i) q^{60} +33210.2i q^{61} +6186.66 q^{62} +8039.72 q^{64} -1394.51i q^{65} +(2553.47 + 3514.73i) q^{66} +6436.18 q^{67} +17582.7 q^{68} +(-51635.6 + 37513.6i) q^{69} +32493.9i q^{71} +(-29301.5 - 9522.18i) q^{72} -12737.8i q^{73} -9260.03i q^{74} +(-54149.3 - 74533.9i) q^{75} +14897.5i q^{76} +(286.672 + 394.590i) q^{78} +8597.89 q^{79} -57797.8 q^{80} +(-47768.3 - 34712.7i) q^{81} +23342.3i q^{82} +31669.6 q^{83} -60880.4 q^{85} +43807.8i q^{86} +(-45334.8 + 32936.0i) q^{87} -16568.7 q^{88} -31945.9 q^{89} +(46847.6 + 15224.2i) q^{90} -112396. i q^{92} +(-36585.3 + 26579.4i) q^{93} +18801.4i q^{94} -51582.8i q^{95} +(67522.8 - 49055.7i) q^{96} -143107. i q^{97} +(-30200.3 - 9814.28i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 172 q^{4} + 1212 q^{9}+O(q^{10}) $ 16 * q - 172 * q^4 + 1212 * q^9	$ 16 q - 172 q^{4} + 1212 q^{9} + 1188 q^{15} + 5716 q^{16} + 876 q^{18} - 21900 q^{22} + 13156 q^{25} - 900 q^{30} - 15132 q^{36} + 20932 q^{37} + 34836 q^{39} + 111052 q^{43} - 163392 q^{46} - 63192 q^{51} - 31368 q^{57} + 83412 q^{58} - 120132 q^{60} - 158884 q^{64} + 204404 q^{67} - 661728 q^{72} - 277512 q^{78} + 502616 q^{79} - 358524 q^{81} + 205152 q^{85} + 719028 q^{88} - 35352 q^{93} + 215472 q^{99}+O(q^{100}) $ 16 * q - 172 * q^4 + 1212 * q^9 + 1188 * q^15 + 5716 * q^16 + 876 * q^18 - 21900 * q^22 + 13156 * q^25 - 900 * q^30 - 15132 * q^36 + 20932 * q^37 + 34836 * q^39 + 111052 * q^43 - 163392 * q^46 - 63192 * q^51 - 31368 * q^57 + 83412 * q^58 - 120132 * q^60 - 158884 * q^64 + 204404 * q^67 - 661728 * q^72 - 277512 * q^78 + 502616 * q^79 - 358524 * q^81 + 205152 * q^85 + 719028 * q^88 - 35352 * q^93 + 215472 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/147\mathbb{Z}\right)^\times$.

$n$	$50$	$52$
$\chi(n)$	$-1$	$-1$

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 $

$2$			2.13264i			0.377001i	0.982073	+	0.188501i	$0.0603628\pi$
$2$			2.13264i			0.377001i	−0.982073	+	0.188501i	$0.939637\pi$
$3$	−9.16236	−	12.6115i	−0.587766	−	0.809031i
$3$	−9.16236	−	12.6115i	−0.587766	−	0.809031i
$4$	27.4518			0.857870
$5$	−95.0525			−1.70035			−0.850175	−	0.526500i	$-0.823504\pi$
$5$	−95.0525			−1.70035			−0.850175	+	0.526500i	$0.823504\pi$
$6$	26.8959	−	19.5400i	0.305006	−	0.221589i
$7$		0			0
$7$		0			0
$8$			126.790i			0.700420i
$9$	−75.1022	+	231.103i	−0.309063	+	0.951042i
$10$		−	202.713i		−	0.641035i
$11$			130.679i			0.325630i	0.986657	+	0.162815i	$0.0520573\pi$
$11$			130.679i			0.325630i	−0.986657	+	0.162815i	$0.947943\pi$
$12$	−251.524	−	346.210i	−0.504227	−	0.694043i
$13$			14.6710i			0.0240769i	0.999928	+	0.0120385i	$0.00383205\pi$
$13$			14.6710i			0.0240769i	−0.999928	+	0.0120385i	$0.996168\pi$
$14$		0			0
$15$	870.905	+	1198.76i	0.999408	+	1.37564i
$16$	608.062			0.593811
$17$	640.492			0.537516			0.268758	−	0.963208i	$-0.413387\pi$
$17$	640.492			0.537516			0.268758	+	0.963208i	$0.413387\pi$
$18$	−492.860	−	160.166i	−0.358544	−	0.116517i
$19$			542.677i			0.344872i	0.985021	+	0.172436i	$0.0551637\pi$
$19$			542.677i			0.344872i	−0.985021	+	0.172436i	$0.944836\pi$
$20$	−2609.37			−1.45868
$21$		0			0
$22$	−278.692			−0.122763
$23$		−	4094.31i		−	1.61384i	−0.590659	−	0.806922i	$-0.701132\pi$
$23$		−	4094.31i		−	1.61384i	0.590659	−	0.806922i	$-0.298868\pi$
$24$	1599.01	−	1161.69i	0.566661	−	0.411683i
$25$	5909.98			1.89119
$26$	−31.2880			−0.00907703
$27$	3602.68	−	1170.29i	0.951079	−	0.308948i
$28$		0			0
$29$		−	3594.70i		−	0.793721i	−0.917879	−	0.396861i	$-0.870100\pi$
$29$		−	3594.70i		−	0.793721i	0.917879	−	0.396861i	$-0.129900\pi$
$30$	−2556.52	+	1857.33i	−0.518617	+	0.376778i
$31$		−	2900.94i		−	0.542168i	−0.962556	−	0.271084i	$-0.912618\pi$
$31$		−	2900.94i		−	0.542168i	0.962556	−	0.271084i	$-0.0873821\pi$
$32$			5354.04i			0.924287i
$33$	1648.06	−	1197.33i	0.263445	−	0.191394i
$34$			1365.94i			0.202644i
$35$		0			0
$36$	−2061.69	+	6344.20i	−0.265136	+	0.815870i
$37$	−4342.05			−0.521423			−0.260712	−	0.965417i	$-0.583957\pi$
$37$	−4342.05			−0.521423			−0.260712	+	0.965417i	$0.583957\pi$
$38$	−1157.34			−0.130017
$39$	185.024	−	134.421i	0.0194790	−	0.0141516i
$40$		−	12051.7i		−	1.19096i
$41$	10945.3			1.01687			0.508436	−	0.861100i	$-0.330224\pi$
$41$	10945.3			1.01687			0.508436	+	0.861100i	$0.330224\pi$
$42$		0			0
$43$	20541.6			1.69419			0.847096	−	0.531440i	$-0.178349\pi$
$43$	20541.6			1.69419			0.847096	+	0.531440i	$0.178349\pi$
$44$			3587.38i			0.279348i
$45$	7138.66	−	21966.9i	0.525515	−	1.61710i
$46$	8731.70			0.608421
$47$	8816.03			0.582141			0.291071	−	0.956702i	$-0.405989\pi$
$47$	8816.03			0.582141			0.291071	+	0.956702i	$0.405989\pi$
$48$	−5571.28	−	7668.60i	−0.349022	−	0.480411i
$49$		0			0
$50$			12603.9i			0.712982i
$51$	−5868.42	−	8077.60i	−0.315934	−	0.434867i
$52$			402.745i			0.0206549i
$53$			6977.45i			0.341198i	0.985341	+	0.170599i	$0.0545703\pi$
$53$			6977.45i			0.341198i	−0.985341	+	0.170599i	$0.945430\pi$
$54$	2495.82	+	7683.23i	0.116474	+	0.358558i
$55$		−	12421.4i		−	0.553685i
$56$		0			0
$57$	6844.00	−	4972.20i	0.279012	−	0.202704i
$58$	7666.22			0.299234
$59$	40020.1			1.49675			0.748373	−	0.663278i	$-0.230835\pi$
$59$	40020.1			1.49675			0.748373	+	0.663278i	$0.230835\pi$
$60$	23908.0	+	32908.1i	0.857362	+	1.18012i
$61$			33210.2i			1.14274i	0.820693	+	0.571370i	$0.193588\pi$
$61$			33210.2i			1.14274i	−0.820693	+	0.571370i	$0.806412\pi$
$62$	6186.66			0.204398
$63$		0			0
$64$	8039.72			0.245353
$65$		−	1394.51i		−	0.0409392i
$66$	2553.47	+	3514.73i	0.0721558	+	0.0993190i
$67$	6436.18			0.175163			0.0875813	−	0.996157i	$-0.472086\pi$
$67$	6436.18			0.175163			0.0875813	+	0.996157i	$0.472086\pi$
$68$	17582.7			0.461119
$69$	−51635.6	+	37513.6i	−1.30565	+	0.948562i
$70$		0			0
$71$			32493.9i			0.764990i	0.923958	+	0.382495i	$0.124935\pi$
$71$			32493.9i			0.764990i	−0.923958	+	0.382495i	$0.875065\pi$
$72$	−29301.5	−	9522.18i	−0.666128	−	0.216474i
$73$		−	12737.8i		−	0.279761i	−0.990168	−	0.139880i	$-0.955328\pi$
$73$		−	12737.8i		−	0.279761i	0.990168	−	0.139880i	$-0.0446718\pi$
$74$		−	9260.03i		−	0.196577i
$75$	−54149.3	−	74533.9i	−1.11158	−	1.53003i
$76$			14897.5i			0.295855i
$77$		0			0
$78$	286.672	+	394.590i	0.00533517	+	0.00734360i
$79$	8597.89			0.154997			0.0774986	−	0.996992i	$-0.475307\pi$
$79$	8597.89			0.154997			0.0774986	+	0.996992i	$0.475307\pi$
$80$	−57797.8			−1.00969
$81$	−47768.3	−	34712.7i	−0.808960	−	0.587863i
$82$			23342.3i			0.383362i
$83$	31669.6			0.504601			0.252300	−	0.967649i	$-0.418813\pi$
$83$	31669.6			0.504601			0.252300	+	0.967649i	$0.418813\pi$
$84$		0			0
$85$	−60880.4			−0.913966
$86$			43807.8i			0.638713i
$87$	−45334.8	+	32936.0i	−0.642145	+	0.466522i
$88$	−16568.7			−0.228077
$89$	−31945.9			−0.427504			−0.213752	−	0.976888i	$-0.568568\pi$
$89$	−31945.9			−0.427504			−0.213752	+	0.976888i	$0.568568\pi$
$90$	46847.6	+	15224.2i	0.609651	+	0.198120i
$91$		0			0
$92$		−	112396.i		−	1.38447i
$93$	−36585.3	+	26579.4i	−0.438631	+	0.318668i
$94$			18801.4i			0.219468i
$95$		−	51582.8i		−	0.586403i
$96$	67522.8	−	49055.7i	0.747777	−	0.543264i
$97$		−	143107.i		−	1.54430i	−0.635441	−	0.772149i	$-0.719182\pi$
$97$		−	143107.i		−	1.54430i	0.635441	−	0.772149i	$-0.280818\pi$
$98$		0			0
$99$	−30200.3	−	9814.28i	−0.309687	−	0.100640i
$100$	162240.			1.62240
$101$	−53587.7			−0.522711			−0.261355	−	0.965243i	$-0.584169\pi$
$101$	−53587.7			−0.522711			−0.261355	+	0.965243i	$0.584169\pi$
$102$	17226.6	−	12515.2i	0.163946	−	0.119107i
$103$			153816.i			1.42860i	0.699842	+	0.714298i	$0.253254\pi$
$103$			153816.i			1.42860i	−0.699842	+	0.714298i	$0.746746\pi$
$104$	−1860.13			−0.0168639
$105$		0			0
$106$	−14880.4			−0.128632
$107$		−	153413.i		−	1.29540i	−0.761897	−	0.647698i	$-0.775732\pi$
$107$		−	153413.i		−	1.29540i	0.761897	−	0.647698i	$-0.224268\pi$
$108$	98900.2	−	32126.7i	0.815902	−	0.265038i
$109$	−127739.			−1.02981			−0.514907	−	0.857246i	$-0.672174\pi$
$109$	−127739.			−1.02981			−0.514907	+	0.857246i	$0.672174\pi$
$110$	26490.3			0.208740
$111$	39783.4	+	54759.9i	0.306475	+	0.421847i
$112$		0			0
$113$		−	90418.2i		−	0.666131i	−0.942904	−	0.333066i	$-0.891917\pi$
$113$		−	90418.2i		−	0.666131i	0.942904	−	0.333066i	$-0.108083\pi$
$114$	10603.9	+	14595.8i	0.0764196	+	0.105188i
$115$			389175.i			2.74410i
$116$		−	98681.2i		−	0.680910i
$117$	−3390.51	−	1101.82i	−0.0228981	−	0.00744128i
$118$			85348.6i			0.564276i
$119$		0			0
$120$	−151990.	+	110422.i	−0.963523	+	0.700005i
$121$	143974.			0.893965
$122$	−70825.5			−0.430814
$123$	−100284.	−	138037.i	−0.597683	−	0.822681i
$124$		−	79636.0i		−	0.465110i
$125$	−264719.			−1.51534
$126$		0			0
$127$	132321.			0.727982			0.363991	−	0.931402i	$-0.381414\pi$
$127$	132321.			0.727982			0.363991	+	0.931402i	$0.381414\pi$
$128$			188475.i			1.01679i
$129$	−188209.	−	259061.i	−0.995788	−	1.37065i
$130$	2974.00			0.0154341
$131$	−147778.			−0.752368			−0.376184	−	0.926545i	$-0.622764\pi$
$131$	−147778.			−0.752368			−0.376184	+	0.926545i	$0.622764\pi$
$132$	45242.4	−	32868.8i	0.226001	−	0.164191i
$133$		0			0
$134$			13726.1i			0.0660366i
$135$	−342444.	+	111239.i	−1.61717	+	0.525321i
$136$			81207.7i			0.376487i
$137$			226682.i			1.03185i	0.856634	+	0.515925i	$0.172552\pi$
$137$			226682.i			1.03185i	−0.856634	+	0.515925i	$0.827448\pi$
$138$	−80003.0	−	110120.i	−0.357609	−	0.492232i
$139$			230678.i			1.01267i	0.862335	+	0.506337i	$0.169001\pi$
$139$			230678.i			1.01267i	−0.862335	+	0.506337i	$0.830999\pi$
$140$		0			0
$141$	−80775.7	−	111184.i	−0.342163	−	0.470971i
$142$	−69297.8			−0.288402
$143$	−1917.19			−0.00784016
$144$	−45666.8	+	140525.i	−0.183525	+	0.564739i
$145$			341685.i			1.34960i
$146$	27165.1			0.105470
$147$		0			0
$148$	−119197.			−0.447313
$149$			9357.83i			0.0345310i	0.999851	+	0.0172655i	$0.00549605\pi$
$149$			9357.83i			0.0345310i	−0.999851	+	0.0172655i	$0.994504\pi$
$150$	158954.	−	115481.i	0.576825	−	0.419067i
$151$	138457.			0.494166			0.247083	−	0.968994i	$-0.420528\pi$
$151$	138457.			0.494166			0.247083	+	0.968994i	$0.420528\pi$
$152$	−68805.8			−0.241555
$153$	−48102.4	+	148020.i	−0.166126	+	0.511200i
$154$		0			0
$155$			275741.i			0.921876i
$156$	5079.24	−	3690.10i	0.0167104	−	0.0121402i
$157$		−	15464.9i		−	0.0500724i	−0.999687	−	0.0250362i	$-0.992030\pi$
$157$		−	15464.9i		−	0.0500724i	0.999687	−	0.0250362i	$-0.00797010\pi$
$158$			18336.2i			0.0584342i
$159$	87996.4	−	63929.9i	0.276040	−	0.200545i
$160$		−	508915.i		−	1.57161i
$161$		0			0
$162$	74029.8	−	101873.i	0.221625	−	0.304979i
$163$	289645.			0.853880			0.426940	−	0.904280i	$-0.359592\pi$
$163$	289645.			0.853880			0.426940	+	0.904280i	$0.359592\pi$
$164$	300467.			0.872344
$165$	−156653.	+	113809.i	−0.447948	+	0.325437i
$166$			67540.0i			0.190235i
$167$	347837.			0.965128			0.482564	−	0.875861i	$-0.339706\pi$
$167$	347837.			0.965128			0.482564	+	0.875861i	$0.339706\pi$
$168$		0			0
$169$	371078.			0.999420
$170$		−	129836.i		−	0.344567i
$171$	−125414.	−	40756.3i	−0.327987	−	0.106587i
$172$	563904.			1.45340
$173$	508138.			1.29082			0.645411	−	0.763836i	$-0.276686\pi$
$173$	508138.			1.29082			0.645411	+	0.763836i	$0.276686\pi$
$174$	−70240.7	−	96682.9i	−0.175880	−	0.242090i
$175$		0			0
$176$			79460.9i			0.193362i
$177$	−366679.	−	504716.i	−0.879736	−	1.21091i
$178$		−	68129.1i		−	0.161169i
$179$		−	400409.i		−	0.934053i	−0.884243	−	0.467026i	$-0.845325\pi$
$179$		−	400409.i		−	0.934053i	0.884243	−	0.467026i	$-0.154675\pi$
$180$	195969.	−	603032.i	0.450824	−	1.38727i
$181$		−	10694.7i		−	0.0242646i	−0.999926	−	0.0121323i	$-0.996138\pi$
$181$		−	10694.7i		−	0.0242646i	0.999926	−	0.0121323i	$-0.00386192\pi$
$182$		0			0
$183$	418832.	−	304284.i	0.924511	−	0.671663i
$184$	519116.			1.13037
$185$	412722.			0.886602
$186$	−56684.4	−	78023.3i	−0.120138	−	0.165364i
$187$			83698.9i			0.175031i
$188$	242016.			0.499402
$189$		0			0
$190$	110008.			0.221075
$191$			197252.i			0.391236i	0.980680	+	0.195618i	$0.0626712\pi$
$191$			197252.i			0.391236i	−0.980680	+	0.195618i	$0.937329\pi$
$192$	−73662.9	−	101393.i	−0.144210	−	0.198498i
$193$	−125511.			−0.242542			−0.121271	−	0.992619i	$-0.538697\pi$
$193$	−125511.			−0.242542			−0.121271	+	0.992619i	$0.538697\pi$
$194$	305196.			0.582203
$195$	−17587.0	+	12777.0i	−0.0331211	+	0.0240627i
$196$		0			0
$197$			631690.i			1.15968i	0.814730	+	0.579841i	$0.196885\pi$
$197$			631690.i			1.15968i	−0.814730	+	0.579841i	$0.803115\pi$
$198$	20930.4	−	64406.5i	0.0379414	−	0.116753i
$199$		−	622380.i		−	1.11410i	−0.830480	−	0.557049i	$-0.811934\pi$
$199$		−	622380.i		−	1.11410i	0.830480	−	0.557049i	$-0.188066\pi$
$200$			749323.i			1.32463i
$201$	−58970.6	−	81170.2i	−0.102955	−	0.141712i
$202$		−	114283.i		−	0.197063i
$203$		0			0
$204$	−161099.	−	221745.i	−0.271030	−	0.373060i
$205$	−1.04037e6			−1.72904
$206$	−328035.			−0.538583
$207$	946208.	+	307492.i	1.53483	+	0.498779i
$208$			8920.87i			0.0142971i
$209$	−70916.5			−0.112300
$210$		0			0
$211$	−271742.			−0.420195			−0.210098	−	0.977680i	$-0.567378\pi$
$211$	−271742.			−0.420195			−0.210098	+	0.977680i	$0.567378\pi$
$212$			191544.i			0.292704i
$213$	409798.	−	297721.i	0.618900	−	0.449635i
$214$	327175.			0.488366
$215$	−1.95253e6			−2.88072
$216$	148381.	+	456782.i	0.216394	+	0.666154i
$217$		0			0
$218$		−	272423.i		−	0.388241i
$219$	−160643.	+	116708.i	−0.226335	+	0.164434i
$220$		−	340989.i		−	0.474989i
$221$			9396.65i			0.0129417i
$222$	−116783.	+	84843.8i	−0.159037	+	0.115541i
$223$		−	256397.i		−	0.345264i	−0.984986	−	0.172632i	$-0.944773\pi$
$223$		−	256397.i		−	0.345264i	0.984986	−	0.172632i	$-0.0552271\pi$
$224$		0			0
$225$	−443852.	+	1.36581e6i	−0.584497	+	1.79860i
$226$	192830.			0.251133
$227$	−117895.			−0.151856			−0.0759278	−	0.997113i	$-0.524192\pi$
$227$	−117895.			−0.151856			−0.0759278	+	0.997113i	$0.524192\pi$
$228$	187880.	−	136496.i	0.239356	−	0.173893i
$229$		−	373781.i		−	0.471008i	−0.971873	−	0.235504i	$-0.924326\pi$
$229$		−	373781.i		−	0.471008i	0.971873	−	0.235504i	$-0.0756741\pi$
$230$	−829970.			−1.03453
$231$		0			0
$232$	455771.			0.555938
$233$		−	1.26480e6i		−	1.52627i	−0.646240	−	0.763134i	$-0.723660\pi$
$233$		−	1.26480e6i		−	1.52627i	0.646240	−	0.763134i	$-0.276340\pi$
$234$	2349.80	−	7230.74i	0.00280537	−	0.00863264i
$235$	−837986.			−0.989845
$236$	1.09863e6			1.28401
$237$	−78776.9	−	108433.i	−0.0911021	−	0.125398i
$238$		0			0
$239$		−	520817.i		−	0.589780i	−0.955531	−	0.294890i	$-0.904717\pi$
$239$		−	520817.i		−	0.589780i	0.955531	−	0.294890i	$-0.0952830\pi$
$240$	529564.	+	728920.i	0.593459	+	0.816868i
$241$			954289.i			1.05837i	0.848507	+	0.529185i	$0.177502\pi$
$241$			954289.i			1.05837i	−0.848507	+	0.529185i	$0.822498\pi$
$242$			307045.i			0.337026i
$243$	−110.689	+	920483.i	−0.000120251	+	1.00000i
$244$			911681.i			0.980321i
$245$		0			0
$246$	294383.	−	213871.i	0.310152	−	0.225327i
$247$	−7961.60			−0.00830344
$248$	367808.			0.379745
$249$	−290169.	−	399403.i	−0.296587	−	0.408238i
$250$		−	564551.i		−	0.571285i
$251$	−789792.			−0.791277			−0.395638	−	0.918406i	$-0.629477\pi$
$251$	−789792.			−0.791277			−0.395638	+	0.918406i	$0.629477\pi$
$252$		0			0
$253$	535040.			0.525515
$254$			282194.i			0.274450i
$255$	557808.	+	767796.i	0.537198	+	0.739427i
$256$	−144679.			−0.137977
$257$	1.10769e6			1.04613			0.523064	−	0.852293i	$-0.324789\pi$
$257$	1.10769e6			1.04613			0.523064	+	0.852293i	$0.324789\pi$
$258$	552485.	−	401383.i	0.516739	−	0.375414i
$259$		0			0
$260$		−	38281.9i		−	0.0351205i
$261$	830747.	+	269970.i	0.754862	+	0.245310i
$262$		−	315157.i		−	0.283644i
$263$		−	1.05399e6i		−	0.939608i	−0.882771	−	0.469804i	$-0.844325\pi$
$263$		−	1.05399e6i		−	0.939608i	0.882771	−	0.469804i	$-0.155675\pi$
$264$	151809.	+	208957.i	0.134056	+	0.184522i
$265$		−	663224.i		−	0.580157i
$266$		0			0
$267$	292700.	+	402887.i	0.251272	+	0.345864i
$268$	176685.			0.150267
$269$	−459896.			−0.387507			−0.193753	−	0.981050i	$-0.562066\pi$
$269$	−459896.			−0.387507			−0.193753	+	0.981050i	$0.562066\pi$
$270$	−237234.	−	730310.i	−0.198047	−	0.609675i
$271$			885691.i			0.732586i	0.930499	+	0.366293i	$0.119373\pi$
$271$			885691.i			0.732586i	−0.930499	+	0.366293i	$0.880627\pi$
$272$	389459.			0.319183
$273$		0			0
$274$	−483433.			−0.389009
$275$			772310.i			0.615828i
$276$	−1.41749e6	+	1.02982e6i	−1.12008	+	0.813743i
$277$	−770456.			−0.603321			−0.301660	−	0.953415i	$-0.597541\pi$
$277$	−770456.			−0.603321			−0.301660	+	0.953415i	$0.597541\pi$
$278$	−491955.			−0.381780
$279$	670415.	+	217867.i	0.515624	+	0.167564i
$280$		0			0
$281$		−	2.25084e6i		−	1.70051i	−0.526374	−	0.850253i	$-0.676449\pi$
$281$		−	2.25084e6i		−	1.70051i	0.526374	−	0.850253i	$-0.323551\pi$
$282$	237115.	−	172266.i	0.177557	−	0.128996i
$283$			5068.25i			0.00376177i	0.999998	+	0.00188088i	$0.000598704\pi$
$283$			5068.25i			0.00376177i	−0.999998	+	0.00188088i	$0.999401\pi$
$284$			892016.i			0.656262i
$285$	−650539.	+	472620.i	−0.474418	+	0.344667i
$286$		−	4088.68i		−	0.00295575i
$287$		0			0
$288$	−1.23734e6	−	402101.i	−0.879036	−	0.285663i
$289$	−1.00963e6			−0.711076
$290$	−728693.			−0.508803
$291$	−1.80480e6	+	1.31120e6i	−1.24939	+	0.907686i
$292$		−	349675.i		−	0.239998i
$293$	2.51767e6			1.71329			0.856643	−	0.515909i	$-0.172546\pi$
$293$	2.51767e6			1.71329			0.856643	+	0.515909i	$0.172546\pi$
$294$		0			0
$295$	−3.80401e6			−2.54499
$296$		−	550526.i		−	0.365215i
$297$	152933.	+	470795.i	0.100603	+	0.309699i
$298$	−19956.9			−0.0130182
$299$	60067.6			0.0388564
$300$	−1.48650e6	−	2.04609e6i	−0.953589	−	1.31257i
$301$		0			0
$302$			295280.i			0.186301i
$303$	490989.	+	675823.i	0.307231	+	0.422889i
$304$			329981.i			0.204788i
$305$		−	3.15671e6i		−	1.94306i
$306$	−315673.	−	102585.i	−0.192723	−	0.0626299i
$307$			2.31011e6i			1.39890i	0.714681	+	0.699451i	$0.246572\pi$
$307$			2.31011e6i			1.39890i	−0.714681	+	0.699451i	$0.753428\pi$
$308$		0			0
$309$	1.93986e6	−	1.40932e6i	1.15578	−	0.839680i
$310$	−588057.			−0.347549
$311$	−2.35286e6			−1.37941			−0.689707	−	0.724089i	$-0.742261\pi$
$311$	−2.35286e6			−1.37941			−0.689707	+	0.724089i	$0.742261\pi$
$312$	17043.2	+	23459.1i	0.00991205	+	0.0136435i
$313$		−	1.99117e6i		−	1.14881i	−0.818572	−	0.574404i	$-0.805234\pi$
$313$		−	1.99117e6i		−	1.14881i	0.818572	−	0.574404i	$-0.194766\pi$
$314$	32981.1			0.0188774
$315$		0			0
$316$	236028.			0.132967
$317$			2.09384e6i			1.17029i	0.810927	+	0.585147i	$0.198963\pi$
$317$			2.09384e6i			1.17029i	−0.810927	+	0.585147i	$0.801037\pi$
$318$	136340.	+	187665.i	0.0756056	+	0.104068i
$319$	469752.			0.258459
$320$	−764196.			−0.417186
$321$	−1.93478e6	+	1.40563e6i	−1.04802	+	0.761390i
$322$		0			0
$323$			347581.i			0.185374i
$324$	−1.31133e6	−	952928.i	−0.693983	−	0.504310i
$325$			86705.1i			0.0455341i
$326$			617709.i			0.321914i
$327$	1.17040e6	+	1.61099e6i	0.605289	+	0.833152i
$328$			1.38774e6i			0.712237i
$329$		0			0
$330$	−242714.	−	334084.i	−0.122690	−	0.168877i
$331$	275243.			0.138085			0.0690426	−	0.997614i	$-0.478006\pi$
$331$	275243.			0.138085			0.0690426	+	0.997614i	$0.478006\pi$
$332$	869390.			0.432882
$333$	326097.	−	1.00346e6i	0.161152	−	0.495895i
$334$			741812.i			0.363855i
$335$	−611775.			−0.297838
$336$		0			0
$337$	455651.			0.218553			0.109277	−	0.994011i	$-0.465147\pi$
$337$	455651.			0.218553			0.109277	+	0.994011i	$0.465147\pi$
$338$			791376.i			0.376783i
$339$	−1.14031e6	+	828445.i	−0.538921	+	0.391529i
$340$	−1.67128e6			−0.784064
$341$	379091.			0.176546
$342$	86918.5	−	267464.i	0.0401834	−	0.123652i
$343$		0			0
$344$			2.60446e6i			1.18665i
$345$	4.90809e6	−	3.56576e6i	2.22006	−	1.61289i
$346$			1.08368e6i			0.486642i
$347$			1.71199e6i			0.763271i	0.924313	+	0.381636i	$0.124639\pi$
$347$			1.71199e6i			0.763271i	−0.924313	+	0.381636i	$0.875361\pi$
$348$	−1.24452e6	+	904153.i	−0.550877	+	0.400215i
$349$		−	1.94736e6i		−	0.855822i	−0.903821	−	0.427911i	$-0.859250\pi$
$349$		−	1.94736e6i		−	0.855822i	0.903821	−	0.427911i	$-0.140750\pi$
$350$		0			0
$351$	17169.4	+	52854.9i	0.00743852	+	0.0228990i
$352$	−699661.			−0.300975
$353$	3.59350e6			1.53490			0.767452	−	0.641106i	$-0.221524\pi$
$353$	3.59350e6			1.53490			0.767452	+	0.641106i	$0.221524\pi$
$354$	1.07638e6	−	781995.i	0.456517	−	0.331662i
$355$		−	3.08862e6i		−	1.30075i
$356$	−876973.			−0.366742
$357$		0			0
$358$	853930.			0.352139
$359$		−	2.58351e6i		−	1.05797i	−0.848630	−	0.528987i	$-0.822572\pi$
$359$		−	2.58351e6i		−	1.05797i	0.848630	−	0.528987i	$-0.177428\pi$
$360$	2.78518e6	+	905107.i	1.13265	+	0.368081i
$361$	2.18160e6			0.881064
$362$	22808.0			0.00914779
$363$	−1.31914e6	−	1.81574e6i	−0.525442	−	0.723246i
$364$		0			0
$365$			1.21076e6i			0.475691i
$366$	648929.	+	893219.i	0.253218	+	0.348542i
$367$		−	3.76563e6i		−	1.45939i	−0.683772	−	0.729696i	$-0.739662\pi$
$367$		−	3.76563e6i		−	1.45939i	0.683772	−	0.729696i	$-0.260338\pi$
$368$		−	2.48960e6i		−	0.958317i
$369$	−822014.	+	2.52948e6i	−0.314277	+	0.967088i
$370$			880189.i			0.334250i
$371$		0			0
$372$	−1.00433e6	+	729654.i	−0.376288	+	0.273376i
$373$	−1.93722e6			−0.720952			−0.360476	−	0.932768i	$-0.617386\pi$
$373$	−1.93722e6			−0.720952			−0.360476	+	0.932768i	$0.617386\pi$
$374$	−178500.			−0.0659870
$375$	2.42545e6	+	3.33852e6i	0.890665	+	1.22596i
$376$			1.11778e6i			0.407743i
$377$	52737.8			0.0191104
$378$		0			0
$379$	2.04113e6			0.729917			0.364959	−	0.931024i	$-0.381083\pi$
$379$	2.04113e6			0.729917			0.364959	+	0.931024i	$0.381083\pi$
$380$		−	1.41604e6i		−	0.503057i
$381$	−1.21238e6	−	1.66878e6i	−0.427883	−	0.588960i
$382$	−420668.			−0.147496
$383$	−470701.			−0.163964			−0.0819819	−	0.996634i	$-0.526125\pi$
$383$	−470701.			−0.163964			−0.0819819	+	0.996634i	$0.526125\pi$
$384$	2.37696e6	−	1.72688e6i	0.822611	−	0.597632i
$385$		0			0
$386$		−	267670.i		−	0.0914388i
$387$	−1.54272e6	+	4.74722e6i	−0.523612	+	1.61125i
$388$		−	3.92855e6i		−	1.32481i
$389$			4.59134e6i			1.53839i	0.639017	+	0.769193i	$0.279341\pi$
$389$			4.59134e6i			1.53839i	−0.639017	+	0.769193i	$0.720659\pi$
$390$	−27248.8	−	37506.7i	−0.00907166	−	0.0124867i
$391$		−	2.62238e6i		−	0.867467i
$392$		0			0
$393$	1.35399e6	+	1.86371e6i	0.442216	+	0.608690i
$394$	−1.34717e6			−0.437202
$395$	−817250.			−0.263550
$396$	−829054.	−	269420.i	−0.265671	−	0.0863360i
$397$		−	4.55566e6i		−	1.45069i	−0.688384	−	0.725346i	$-0.741680\pi$
$397$		−	4.55566e6i		−	1.45069i	0.688384	−	0.725346i	$-0.258320\pi$
$398$	1.32731e6			0.420016
$399$		0			0
$400$	3.59363e6			1.12301
$401$			2.41750e6i			0.750766i	0.926870	+	0.375383i	$0.122489\pi$
$401$			2.41750e6i			0.750766i	−0.926870	+	0.375383i	$0.877511\pi$
$402$	173107.	−	125763.i	0.0534256	−	0.0388140i
$403$	42559.6			0.0130537
$404$	−1.47108e6			−0.448418
$405$	4.54050e6	+	3.29953e6i	1.37552	+	0.999573i
$406$		0			0
$407$		−	567414.i		−	0.169791i
$408$	1.02415e6	−	744054.i	0.304590	−	0.221286i
$409$			64985.1i			0.0192090i	0.999954	+	0.00960452i	$0.00305726\pi$
$409$			64985.1i			0.0192090i	−0.999954	+	0.00960452i	$0.996943\pi$
$410$		−	2.21875e6i		−	0.651851i
$411$	2.85882e6	−	2.07695e6i	0.834798	−	0.606486i
$412$			4.22254e6i			1.22555i
$413$		0			0
$414$	−655771.	+	2.01792e6i	−0.188040	+	0.578634i
$415$	−3.01028e6			−0.857998
$416$	−78549.1			−0.0222540
$417$	2.90921e6	−	2.11356e6i	0.819285	−	0.595216i
$418$		−	151239.i		−	0.0423374i
$419$	−4.52870e6			−1.26020			−0.630098	−	0.776515i	$-0.716985\pi$
$419$	−4.52870e6			−1.26020			−0.630098	+	0.776515i	$0.716985\pi$
$420$		0			0
$421$	−3.77935e6			−1.03923			−0.519615	−	0.854401i	$-0.673925\pi$
$421$	−3.77935e6			−1.03923			−0.519615	+	0.854401i	$0.673925\pi$
$422$		−	579529.i		−	0.158414i
$423$	−662104.	+	2.03741e6i	−0.179918	+	0.553641i
$424$	−884667.			−0.238982
$425$	3.78529e6			1.01655
$426$	634932.	+	873953.i	0.169513	+	0.233326i
$427$		0			0
$428$		−	4.21147e6i		−	1.11128i
$429$	17566.0	+	24178.7i	0.00460818	+	0.00634293i
$430$		−	4.16404e6i		−	1.08604i
$431$			6.61895e6i			1.71631i	0.513391	+	0.858155i	$0.328389\pi$
$431$			6.61895e6i			1.71631i	−0.513391	+	0.858155i	$0.671611\pi$
$432$	2.19065e6	−	711612.i	0.564761	−	0.183457i
$433$		−	5.25855e6i		−	1.34786i	−0.738794	−	0.673932i	$-0.764604\pi$
$433$		−	5.25855e6i		−	1.34786i	0.738794	−	0.673932i	$-0.235396\pi$
$434$		0			0
$435$	4.30918e6	−	3.13065e6i	1.09187	−	0.793251i
$436$	−3.50668e6			−0.883446
$437$	2.22189e6			0.556569
$438$	−248897.	−	342594.i	−0.0619917	−	0.0853286i
$439$			416451.i			0.103134i	0.998670	+	0.0515671i	$0.0164216\pi$
$439$			416451.i			0.103134i	−0.998670	+	0.0515671i	$0.983578\pi$
$440$	1.57490e6			0.387812
$441$		0			0
$442$	−20039.7			−0.00487905
$443$		−	7.00727e6i		−	1.69644i	−0.529641	−	0.848222i	$-0.677673\pi$
$443$		−	7.00727e6i		−	1.69644i	0.529641	−	0.848222i	$-0.322327\pi$
$444$	1.09213e6	+	1.50326e6i	0.262915	+	0.361890i
$445$	3.03653e6			0.726906
$446$	546803.			0.130165
$447$	118017.	−	85739.8i	0.0279367	−	0.0202961i
$448$		0			0
$449$			1.63032e6i			0.381643i	0.981625	+	0.190822i	$0.0611152\pi$
$449$			1.63032e6i			0.381643i	−0.981625	+	0.190822i	$0.938885\pi$
$450$	−2.91279e6	−	946579.i	−0.678076	−	0.220356i
$451$			1.43032e6i			0.331124i
$452$		−	2.48215e6i		−	0.571454i
$453$	−1.26860e6	−	1.74616e6i	−0.290454	−	0.399796i
$454$		−	251428.i		−	0.0572498i
$455$		0			0
$456$	630423.	+	867747.i	0.141978	+	0.195425i
$457$	8.38895e6			1.87896			0.939479	−	0.342606i	$-0.111310\pi$
$457$	8.38895e6			1.87896			0.939479	+	0.342606i	$0.111310\pi$
$458$	797141.			0.177571
$459$	2.30749e6	−	749565.i	0.511220	−	0.166065i
$460$			1.06836e7i			2.35408i
$461$	−186426.			−0.0408560			−0.0204280	−	0.999791i	$-0.506503\pi$
$461$	−186426.			−0.0408560			−0.0204280	+	0.999791i	$0.506503\pi$
$462$		0			0
$463$	−5.23908e6			−1.13580			−0.567901	−	0.823097i	$-0.692244\pi$
$463$	−5.23908e6			−1.13580			−0.567901	+	0.823097i	$0.692244\pi$
$464$		−	2.18580e6i		−	0.471320i
$465$	3.47752e6	−	2.52644e6i	0.745826	−	0.541847i
$466$	2.69736e6			0.575405
$467$	564223.			0.119718			0.0598589	−	0.998207i	$-0.480935\pi$
$467$	564223.			0.119718			0.0598589	+	0.998207i	$0.480935\pi$
$468$	−93075.7	−	30247.1i	−0.0196436	−	0.00638365i
$469$		0			0
$470$		−	1.78712e6i		−	0.373173i
$471$	−195036.	+	141695.i	−0.0405101	+	0.0294308i
$472$			5.07413e6i			1.04835i
$473$			2.68435e6i			0.551679i
$474$	231248.	−	168003.i	0.0472751	−	0.0343456i
$475$			3.20721e6i			0.652219i
$476$		0			0
$477$	−1.61251e6	−	524022.i	−0.324494	−	0.105452i
$478$	1.11072e6			0.222348
$479$	2.69727e6			0.537137			0.268568	−	0.963261i	$-0.413449\pi$
$479$	2.69727e6			0.537137			0.268568	+	0.963261i	$0.413449\pi$
$480$	−6.41821e6	+	4.66286e6i	−1.27148	+	0.923740i
$481$		−	63702.1i		−	0.0125543i
$482$	−2.03516e6			−0.399007
$483$		0			0
$484$	3.95235e6			0.766906
$485$			1.36027e7i			2.62585i
$486$	−1.96306e6	−	236.061i	−0.377001	−	4.53350e-5i
$487$	−2.84365e6			−0.543317			−0.271658	−	0.962394i	$-0.587572\pi$
$487$	−2.84365e6			−0.543317			−0.271658	+	0.962394i	$0.587572\pi$
$488$	−4.21071e6			−0.800397
$489$	−2.65383e6	−	3.65287e6i	−0.501881	−	0.690815i
$490$		0			0
$491$			7.38962e6i			1.38331i	0.722229	+	0.691654i	$0.243118\pi$
$491$			7.38962e6i			1.38331i	−0.722229	+	0.691654i	$0.756882\pi$
$492$	−2.75299e6	−	3.78936e6i	−0.512734	−	0.705754i
$493$		−	2.30238e6i		−	0.426638i
$494$		−	16979.3i		−	0.00313041i
$495$	2.87062e6	+	932872.i	0.526577	+	0.171123i
$496$		−	1.76395e6i		−	0.321945i
$497$		0			0
$498$	851784.	−	618826.i	0.153906	−	0.111814i
$499$	2.13675e6			0.384152			0.192076	−	0.981380i	$-0.438478\pi$
$499$	2.13675e6			0.384152			0.192076	+	0.981380i	$0.438478\pi$
$500$	−7.26702e6			−1.29996
$501$	−3.18701e6	−	4.38677e6i	−0.567269	−	0.780818i
$502$		−	1.68434e6i		−	0.298313i
$503$	2.60933e6			0.459842			0.229921	−	0.973209i	$-0.426153\pi$
$503$	2.60933e6			0.459842			0.229921	+	0.973209i	$0.426153\pi$
$504$		0			0
$505$	5.09364e6			0.888791
$506$			1.14105e6i			0.198120i
$507$	−3.39995e6	−	4.67986e6i	−0.587425	−	0.808562i
$508$	3.63246e6			0.624514
$509$	−5.78866e6			−0.990338			−0.495169	−	0.868797i	$-0.664894\pi$
$509$	−5.78866e6			−0.990338			−0.495169	+	0.868797i	$0.664894\pi$
$510$	−1.63743e6	+	1.18961e6i	−0.278765	+	0.202524i
$511$		0			0
$512$			5.72266e6i			0.964768i
$513$	635092.	+	1.95509e6i	0.106548	+	0.328000i
$514$			2.36231e6i			0.394392i
$515$		−	1.46206e7i		−	2.42911i
$516$	−5.16669e6	−	7.11170e6i	−0.854257	−	1.17584i
$517$			1.15207e6i			0.189563i
$518$		0			0
$519$	−4.65574e6	−	6.40840e6i	−0.758701	−	1.04431i
$520$	176810.			0.0286746
$521$	−306226.			−0.0494250			−0.0247125	−	0.999695i	$-0.507867\pi$
$521$	−306226.			−0.0494250			−0.0247125	+	0.999695i	$0.507867\pi$
$522$	−575750.	+	1.77169e6i	−0.0924821	+	0.284584i
$523$		−	7.18513e6i		−	1.14863i	−0.818634	−	0.574315i	$-0.805268\pi$
$523$		−	7.18513e6i		−	1.14863i	0.818634	−	0.574315i	$-0.194732\pi$
$524$	−4.05677e6			−0.645434
$525$		0			0
$526$	2.24778e6			0.354234
$527$		−	1.85803e6i		−	0.291424i
$528$	1.00213e6	−	728050.i	0.156436	−	0.113652i
$529$	−1.03270e7			−1.60449
$530$	1.41442e6			0.218720
$531$	−3.00560e6	+	9.24877e6i	−0.462589	+	1.42347i
$532$		0			0
$533$			160578.i			0.0244831i
$534$	−859214.	+	624224.i	−0.130391	+	0.0947299i
$535$			1.45823e7i			2.20263i
$536$			816040.i			0.122687i
$537$	−5.04978e6	+	3.66869e6i	−0.755678	+	0.549004i
$538$		−	980795.i		−	0.146091i
$539$		0			0
$540$	−9.40071e6	+	3.05373e6i	−1.38732	+	0.450657i
$541$	7.76483e6			1.14061			0.570307	−	0.821431i	$-0.306824\pi$
$541$	7.76483e6			1.14061			0.570307	+	0.821431i	$0.306824\pi$
$542$	−1.88886e6			−0.276186
$543$	−134877.	+	97988.9i	−0.0196308	+	0.0142619i
$544$			3.42922e6i			0.496819i
$545$	1.21420e7			1.75104
$546$		0			0
$547$	−7.87821e6			−1.12580			−0.562898	−	0.826527i	$-0.690313\pi$
$547$	−7.87821e6			−1.12580			−0.562898	+	0.826527i	$0.690313\pi$
$548$			6.22285e6i			0.885193i
$549$	−7.67498e6	−	2.49416e6i	−1.08679	−	0.353178i
$550$	−1.64706e6			−0.232168
$551$	1.95076e6			0.273732
$552$	−4.75633e6	−	6.54685e6i	−0.664391	−	0.914503i
$553$		0			0
$554$		−	1.64311e6i		−	0.227453i
$555$	−3.78151e6	−	5.20507e6i	−0.521114	−	0.717289i
$556$			6.33255e6i			0.868743i
$557$			4.59766e6i			0.627913i	0.949437	+	0.313956i	$0.101655\pi$
$557$			4.59766e6i			0.627913i	−0.949437	+	0.313956i	$0.898345\pi$
$558$	−464632.	+	1.42976e6i	−0.0631719	+	0.194391i
$559$			301365.i			0.0407909i
$560$		0			0
$561$	1.05557e6	−	766879.i	0.141606	−	0.102877i
$562$	4.80023e6			0.641094
$563$	2.72419e6			0.362215			0.181107	−	0.983463i	$-0.442032\pi$
$563$	2.72419e6			0.362215			0.181107	+	0.983463i	$0.442032\pi$
$564$	−2.21744e6	−	3.05220e6i	−0.293531	−	0.404031i
$565$			8.59448e6i			1.13266i
$566$	−10808.8			−0.00141819
$567$		0			0
$568$	−4.11988e6			−0.535814
$569$		−	8.67813e6i		−	1.12369i	−0.827243	−	0.561844i	$-0.810092\pi$
$569$		−	8.67813e6i		−	1.12369i	0.827243	−	0.561844i	$-0.189908\pi$
$570$	−1.00793e6	−	1.38737e6i	−0.129940	−	0.178856i
$571$	1.07384e7			1.37832			0.689161	−	0.724608i	$-0.257979\pi$
$571$	1.07384e7			1.37832			0.689161	+	0.724608i	$0.257979\pi$
$572$	−52630.3			−0.00672583
$573$	2.48765e6	−	1.80730e6i	0.316522	−	0.229955i
$574$		0			0
$575$		−	2.41973e7i		−	3.05209i
$576$	−603801.	+	1.85801e6i	−0.0758294	+	0.233341i
$577$			7.06870e6i			0.883894i	0.897041	+	0.441947i	$0.145712\pi$
$577$			7.06870e6i			0.883894i	−0.897041	+	0.441947i	$0.854288\pi$
$578$		−	2.15317e6i		−	0.268077i
$579$	1.14997e6	+	1.58288e6i	0.142558	+	0.196224i
$580$			9.37989e6i			1.15779i
$581$		0			0
$582$	−2.79631e6	−	3.84899e6i	−0.342199	−	0.471020i
$583$	−911805.			−0.111104
$584$	1.61502e6			0.195950
$585$	322276.	+	104731.i	0.0389349	+	0.0126528i
$586$			5.36929e6i			0.645911i
$587$	−3.96818e6			−0.475331			−0.237666	−	0.971347i	$-0.576382\pi$
$587$	−3.96818e6			−0.475331			−0.237666	+	0.971347i	$0.576382\pi$
$588$		0			0
$589$	1.57427e6			0.186978
$590$		−	8.11260e6i		−	0.959467i
$591$	7.96659e6	−	5.78777e6i	0.938218	−	0.681621i
$592$	−2.64023e6			−0.309626
$593$	1.67421e7			1.95512			0.977562	−	0.210649i	$-0.0675576\pi$
$593$	1.67421e7			1.95512			0.977562	+	0.210649i	$0.0675576\pi$
$594$	−1.00404e6	+	326151.i	−0.116757	+	0.0379274i
$595$		0			0
$596$			256889.i			0.0296231i
$597$	−7.84918e6	+	5.70247e6i	−0.901339	+	0.654828i
$598$			128103.i			0.0146489i
$599$		−	1.08814e7i		−	1.23913i	−0.784944	−	0.619567i	$-0.787308\pi$
$599$		−	1.08814e7i		−	1.23913i	0.784944	−	0.619567i	$-0.212692\pi$
$600$	9.45012e6	−	6.86557e6i	1.07167	−	0.778571i
$601$		−	1.72048e7i		−	1.94296i	−0.237128	−	0.971478i	$-0.576206\pi$
$601$		−	1.72048e7i		−	1.94296i	0.237128	−	0.971478i	$-0.423794\pi$
$602$		0			0
$603$	−483372.	+	1.48742e6i	−0.0541362	+	0.166587i
$604$	3.80090e6			0.423930
$605$	−1.36851e7			−1.52005
$606$	−1.44129e6	+	1.04711e6i	−0.159430	+	0.115827i
$607$			5.56535e6i			0.613085i	0.951857	+	0.306543i	$0.0991722\pi$
$607$			5.56535e6i			0.613085i	−0.951857	+	0.306543i	$0.900828\pi$
$608$	−2.90552e6			−0.318760
$609$		0			0
$610$	6.73214e6			0.732535
$611$			129340.i			0.0140162i
$612$	−1.32050e6	+	4.06341e6i	−0.142515	+	0.438543i
$613$	−1.18075e7			−1.26914			−0.634568	−	0.772867i	$-0.718822\pi$
$613$	−1.18075e7			−1.26914			−0.634568	+	0.772867i	$0.718822\pi$
$614$	−4.92664e6			−0.527388
$615$	9.53229e6	+	1.31207e7i	1.01627	+	1.39885i
$616$		0			0
$617$			5.91582e6i			0.625608i	0.949818	+	0.312804i	$0.101268\pi$
$617$			5.91582e6i			0.625608i	−0.949818	+	0.312804i	$0.898732\pi$
$618$	3.00558e6	+	4.13703e6i	0.316560	+	0.435730i
$619$			2.82876e6i			0.296736i	0.988932	+	0.148368i	$0.0474020\pi$
$619$			2.82876e6i			0.296736i	−0.988932	+	0.148368i	$0.952598\pi$
$620$			7.56960e6i			0.790849i
$621$	−4.79155e6	−	1.47505e7i	−0.498594	−	1.53489i
$622$		−	5.01780e6i		−	0.520041i
$623$		0			0
$624$	112506.	−	81736.2i	0.0115668	−	0.00840336i
$625$	6.69352e6			0.685416
$626$	4.24646e6			0.433103
$627$	649762.	+	894366.i	0.0660063	+	0.0908545i
$628$		−	424540.i		−	0.0429556i
$629$	−2.78105e6			−0.280273
$630$		0			0
$631$	−9.60301e6			−0.960139			−0.480069	−	0.877231i	$-0.659389\pi$
$631$	−9.60301e6			−0.960139			−0.480069	+	0.877231i	$0.659389\pi$
$632$			1.09012e6i			0.108563i
$633$	2.48980e6	+	3.42709e6i	0.246976	+	0.339951i
$634$	−4.46541e6			−0.441203
$635$	−1.25775e7			−1.23782
$636$	2.41566e6	−	1.75499e6i	0.236806	−	0.172041i
$637$		0			0
$638$			1.00181e6i			0.0974395i
$639$	−7.50943e6	−	2.44036e6i	−0.727537	−	0.236430i
$640$		−	1.79150e7i		−	1.72889i
$641$			9.00157e6i			0.865312i	0.901559	+	0.432656i	$0.142424\pi$
$641$			9.00157e6i			0.865312i	−0.901559	+	0.432656i	$0.857576\pi$
$642$	−2.99770e6	−	4.12618e6i	−0.287045	−	0.395104i
$643$			6.92082e6i			0.660131i	0.943958	+	0.330065i	$0.107071\pi$
$643$			6.92082e6i			0.660131i	−0.943958	+	0.330065i	$0.892929\pi$
$644$		0			0
$645$	1.78898e7	+	2.46244e7i	1.69319	+	2.33059i
$646$	−741265.			−0.0698863
$647$	−1.66850e6			−0.156699			−0.0783495	−	0.996926i	$-0.524965\pi$
$647$	−1.66850e6			−0.156699			−0.0783495	+	0.996926i	$0.524965\pi$
$648$	4.40121e6	−	6.05652e6i	0.411751	−	0.566612i
$649$			5.22979e6i			0.487385i
$650$	−184911.			−0.0171664
$651$		0			0
$652$	7.95128e6			0.732518
$653$			1.81810e7i			1.66853i	0.551364	+	0.834265i	$0.314108\pi$
$653$			1.81810e7i			1.66853i	−0.551364	+	0.834265i	$0.685892\pi$
$654$	−3.43567e6	+	2.49603e6i	−0.314099	+	0.228195i
$655$	1.40466e7			1.27929
$656$	6.65540e6			0.603830
$657$	2.94374e6	+	956635.i	0.266064	+	0.0864635i
$658$		0			0
$659$			5.58660e6i			0.501111i	0.968102	+	0.250556i	$0.0806133\pi$
$659$			5.58660e6i			0.501111i	−0.968102	+	0.250556i	$0.919387\pi$
$660$	−4.30040e6	+	3.12427e6i	−0.384281	+	0.279182i
$661$		−	1.30855e7i		−	1.16489i	−0.812870	−	0.582445i	$-0.802096\pi$
$661$		−	1.30855e7i		−	1.16489i	0.812870	−	0.582445i	$-0.197904\pi$
$662$			586996.i			0.0520583i
$663$	118506.	−	86095.5i	0.0104703	−	0.00760671i
$664$			4.01538e6i			0.353432i
$665$		0			0
$666$	2.14002e6	+	695449.i	0.186953	+	0.0607547i
$667$	−1.47178e7			−1.28094
$668$	9.54877e6			0.827954
$669$	−3.23356e6	+	2.34920e6i	−0.279329	+	0.202934i
$670$		−	1.30470e6i		−	0.112285i
$671$	−4.33988e6			−0.372110
$672$		0			0
$673$	9.79316e6			0.833461			0.416730	−	0.909030i	$-0.363176\pi$
$673$	9.79316e6			0.833461			0.416730	+	0.909030i	$0.363176\pi$
$674$			971740.i			0.0823949i
$675$	2.12918e7	−	6.91642e6i	1.79867	−	0.584281i
$676$	1.01868e7			0.857373
$677$	−8.00917e6			−0.671608			−0.335804	−	0.941932i	$-0.609008\pi$
$677$	−8.00917e6			−0.671608			−0.335804	+	0.941932i	$0.609008\pi$
$678$	−1.76678e6	−	2.43188e6i	−0.147607	−	0.203174i
$679$		0			0
$680$		−	7.71900e6i		−	0.640160i
$681$	1.08020e6	+	1.48684e6i	0.0892555	+	0.122856i
$682$			808466.i			0.0665581i
$683$			2.00990e7i			1.64863i	0.566133	+	0.824314i	$0.308439\pi$
$683$			2.00990e7i			1.64863i	−0.566133	+	0.824314i	$0.691561\pi$
$684$	−3.44285e6	−	1.11883e6i	−0.281370	−	0.0914377i
$685$		−	2.15467e7i		−	1.75451i
$686$		0			0
$687$	−4.71395e6	+	3.42471e6i	−0.381060	+	0.276842i
$688$	1.24906e7			1.00603
$689$	−102366.			−0.00821500
$690$	7.60449e6	+	1.04672e7i	0.608061	+	0.836967i
$691$			7.78435e6i			0.620193i	0.950705	+	0.310097i	$0.100361\pi$
$691$			7.78435e6i			0.620193i	−0.950705	+	0.310097i	$0.899639\pi$
$692$	1.39493e7			1.10736
$693$		0			0
$694$	−3.65107e6			−0.287754
$695$		−	2.19266e7i		−	1.72190i
$696$	−4.17594e6	−	5.74797e6i	−0.326761	−	0.449771i
$697$	7.01036e6			0.546586
$698$	4.15303e6			0.322646
$699$	−1.59510e7	+	1.15885e7i	−1.23480	+	0.897088i
$700$		0			0
$701$			1.12400e7i			0.863912i	0.901895	+	0.431956i	$0.142176\pi$
$701$			1.12400e7i			0.863912i	−0.901895	+	0.431956i	$0.857824\pi$
$702$	−112721.	+	36616.1i	−0.00863297	+	0.00280433i
$703$		−	2.35633e6i		−	0.179824i
$704$			1.05062e6i			0.0798942i
$705$	7.67793e6	+	1.05683e7i	0.581797	+	0.800815i
$706$			7.66366e6i			0.578661i
$707$		0			0
$708$	−1.00660e7	−	1.38554e7i	−0.754699	−	1.03881i
$709$	−1.15696e7			−0.864376			−0.432188	−	0.901783i	$-0.642258\pi$
$709$	−1.15696e7			−0.864376			−0.432188	+	0.901783i	$0.642258\pi$
$710$	6.58693e6			0.490385
$711$	−645721.	+	1.98700e6i	−0.0479039	+	0.147409i
$712$		−	4.05040e6i		−	0.299432i
$713$	−1.18773e7			−0.874974
$714$		0			0
$715$	182234.			0.0133310
$716$		−	1.09920e7i		−	0.801296i
$717$	−6.56830e6	+	4.77191e6i	−0.477150	+	0.346653i
$718$	5.50971e6			0.398857
$719$	461291.			0.0332777			0.0166388	−	0.999862i	$-0.494703\pi$
$719$	461291.			0.0332777			0.0166388	+	0.999862i	$0.494703\pi$
$720$	4.34075e6	−	1.33573e7i	0.312056	−	0.960254i
$721$		0			0
$722$			4.65257e6i			0.332162i
$723$	1.20351e7	−	8.74354e6i	0.856254	−	0.622073i
$724$		−	293590.i		−	0.0208159i
$725$		−	2.12446e7i		−	1.50108i
$726$	3.87231e6	−	2.81326e6i	0.272665	−	0.198093i
$727$			2.83046e6i			0.198619i	0.995057	+	0.0993096i	$0.0316634\pi$
$727$			2.83046e6i			0.198619i	−0.995057	+	0.0993096i	$0.968337\pi$
$728$		0			0
$729$	1.16097e7	−	8.43240e6i	0.809102	−	0.587669i
$730$	−2.58211e6			−0.179336
$731$	1.31567e7			0.910656
$732$	1.14977e7	−	8.35315e6i	0.793111	−	0.576199i
$733$		−	2.05309e7i		−	1.41140i	−0.708513	−	0.705698i	$-0.750634\pi$
$733$		−	2.05309e7i		−	1.41140i	0.708513	−	0.705698i	$-0.249366\pi$
$734$	8.03073e6			0.550193
$735$		0			0
$736$	2.19211e7			1.49165
$737$			841074.i			0.0570381i
$738$	−5.39448e6	−	1.75306e6i	−0.364594	−	0.118483i
$739$	−1.51479e7			−1.02033			−0.510167	−	0.860075i	$-0.670416\pi$
$739$	−1.51479e7			−1.02033			−0.510167	+	0.860075i	$0.670416\pi$
$740$	1.13300e7			0.760589
$741$	72947.1	+	100408.i	0.00488048	+	0.00671774i
$742$		0			0
$743$			3.14063e6i			0.208711i	0.994540	+	0.104355i	$0.0332779\pi$
$743$			3.14063e6i			0.208711i	−0.994540	+	0.104355i	$0.966722\pi$
$744$	−3.36999e6	−	4.63863e6i	−0.223201	−	0.307226i
$745$		−	889485.i		−	0.0587148i
$746$		−	4.13140e6i		−	0.271800i
$747$	−2.37846e6	+	7.31895e6i	−0.155953	+	0.479896i
$748$			2.29769e6i			0.150154i
$749$		0			0
$750$	−7.11986e6	+	5.17262e6i	−0.462188	+	0.335782i
$751$	−1.43376e6			−0.0927634			−0.0463817	−	0.998924i	$-0.514769\pi$
$751$	−1.43376e6			−0.0927634			−0.0463817	+	0.998924i	$0.514769\pi$
$752$	5.36069e6			0.345682
$753$	7.23636e6	+	9.96050e6i	0.465085	+	0.640168i
$754$			112471.i			0.00720463i
$755$	−1.31607e7			−0.840256
$756$		0			0
$757$	−6.81110e6			−0.431994			−0.215997	−	0.976394i	$-0.569300\pi$
$757$	−6.81110e6			−0.431994			−0.215997	+	0.976394i	$0.569300\pi$
$758$			4.35301e6i			0.275180i
$759$	−4.90223e6	−	6.74769e6i	−0.308880	−	0.425158i
$760$	6.54016e6			0.410728
$761$	110270.			0.00690235			0.00345117	−	0.999994i	$-0.498901\pi$
$761$	110270.			0.00690235			0.00345117	+	0.999994i	$0.498901\pi$
$762$	3.55890e6	−	2.58557e6i	0.222039	−	0.161313i
$763$		0			0
$764$			5.41493e6i			0.335629i
$765$	4.57225e6	−	1.40696e7i	0.282473	−	0.869220i
$766$		−	1.00384e6i		−	0.0618146i
$767$			587134.i			0.0360370i
$768$	1.32560e6	+	1.82463e6i	0.0810980	+	0.111628i
$769$			1.61688e7i			0.985968i	0.870038	+	0.492984i	$0.164094\pi$
$769$			1.61688e7i			0.985968i	−0.870038	+	0.492984i	$0.835906\pi$
$770$		0			0
$771$	−1.01491e7	−	1.39697e7i	−0.614879	−	0.846351i
$772$	−3.44550e6			−0.208070
$773$	−1.67611e7			−1.00892			−0.504458	−	0.863436i	$-0.668308\pi$
$773$	−1.67611e7			−1.00892			−0.504458	+	0.863436i	$0.668308\pi$
$774$	−1.01241e7	−	3.29007e6i	−0.607443	−	0.197402i
$775$		−	1.71445e7i		−	1.02534i
$776$	1.81445e7			1.08166
$777$		0			0
$778$	−9.79168e6			−0.579974
$779$			5.93974e6i			0.350690i
$780$

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:	\( N \)	\(=\)	\( 147 = 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	147.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.13264i q^{2} +(-9.16236 - 12.6115i) q^{3} +27.4518 q^{4} -95.0525 q^{5} +(26.8959 - 19.5400i) q^{6} +126.790i q^{8} +(-75.1022 + 231.103i) q^{9} +O(q^{10})\)	\(q+2.13264i q^{2} +(-9.16236 - 12.6115i) q^{3} +27.4518 q^{4} -95.0525 q^{5} +(26.8959 - 19.5400i) q^{6} +126.790i q^{8} +(-75.1022 + 231.103i) q^{9} -202.713i q^{10} +130.679i q^{11} +(-251.524 - 346.210i) q^{12} +14.6710i q^{13} +(870.905 + 1198.76i) q^{15} +608.062 q^{16} +640.492 q^{17} +(-492.860 - 160.166i) q^{18} +542.677i q^{19} -2609.37 q^{20} -278.692 q^{22} -4094.31i q^{23} +(1599.01 - 1161.69i) q^{24} +5909.98 q^{25} -31.2880 q^{26} +(3602.68 - 1170.29i) q^{27} -3594.70i q^{29} +(-2556.52 + 1857.33i) q^{30} -2900.94i q^{31} +5354.04i q^{32} +(1648.06 - 1197.33i) q^{33} +1365.94i q^{34} +(-2061.69 + 6344.20i) q^{36} -4342.05 q^{37} -1157.34 q^{38} +(185.024 - 134.421i) q^{39} -12051.7i q^{40} +10945.3 q^{41} +20541.6 q^{43} +3587.38i q^{44} +(7138.66 - 21966.9i) q^{45} +8731.70 q^{46} +8816.03 q^{47} +(-5571.28 - 7668.60i) q^{48} +12603.9i q^{50} +(-5868.42 - 8077.60i) q^{51} +402.745i q^{52} +6977.45i q^{53} +(2495.82 + 7683.23i) q^{54} -12421.4i q^{55} +(6844.00 - 4972.20i) q^{57} +7666.22 q^{58} +40020.1 q^{59} +(23908.0 + 32908.1i) q^{60} +33210.2i q^{61} +6186.66 q^{62} +8039.72 q^{64} -1394.51i q^{65} +(2553.47 + 3514.73i) q^{66} +6436.18 q^{67} +17582.7 q^{68} +(-51635.6 + 37513.6i) q^{69} +32493.9i q^{71} +(-29301.5 - 9522.18i) q^{72} -12737.8i q^{73} -9260.03i q^{74} +(-54149.3 - 74533.9i) q^{75} +14897.5i q^{76} +(286.672 + 394.590i) q^{78} +8597.89 q^{79} -57797.8 q^{80} +(-47768.3 - 34712.7i) q^{81} +23342.3i q^{82} +31669.6 q^{83} -60880.4 q^{85} +43807.8i q^{86} +(-45334.8 + 32936.0i) q^{87} -16568.7 q^{88} -31945.9 q^{89} +(46847.6 + 15224.2i) q^{90} -112396. i q^{92} +(-36585.3 + 26579.4i) q^{93} +18801.4i q^{94} -51582.8i q^{95} +(67522.8 - 49055.7i) q^{96} -143107. i q^{97} +(-30200.3 - 9814.28i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 172 q^{4} + 1212 q^{9}+O(q^{10}) \) 16 * q - 172 * q^4 + 1212 * q^9	\( 16 q - 172 q^{4} + 1212 q^{9} + 1188 q^{15} + 5716 q^{16} + 876 q^{18} - 21900 q^{22} + 13156 q^{25} - 900 q^{30} - 15132 q^{36} + 20932 q^{37} + 34836 q^{39} + 111052 q^{43} - 163392 q^{46} - 63192 q^{51} - 31368 q^{57} + 83412 q^{58} - 120132 q^{60} - 158884 q^{64} + 204404 q^{67} - 661728 q^{72} - 277512 q^{78} + 502616 q^{79} - 358524 q^{81} + 205152 q^{85} + 719028 q^{88} - 35352 q^{93} + 215472 q^{99}+O(q^{100}) \) 16 * q - 172 * q^4 + 1212 * q^9 + 1188 * q^15 + 5716 * q^16 + 876 * q^18 - 21900 * q^22 + 13156 * q^25 - 900 * q^30 - 15132 * q^36 + 20932 * q^37 + 34836 * q^39 + 111052 * q^43 - 163392 * q^46 - 63192 * q^51 - 31368 * q^57 + 83412 * q^58 - 120132 * q^60 - 158884 * q^64 + 204404 * q^67 - 661728 * q^72 - 277512 * q^78 + 502616 * q^79 - 358524 * q^81 + 205152 * q^85 + 719028 * q^88 - 35352 * q^93 + 215472 * q^99

Introduction

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