LMFDB - Embedded newform 147.6.c.c.146.16

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			146.16
Root			$-8.95007 - 5.16733i$ of defining polynomial
Character	$\chi$	$=$	147.146
Dual form			147.6.c.c.146.2

$q$-expansion

$f(q)$	$=$	$q+10.3347i q^{2} +(12.5210 - 9.28579i) q^{3} -74.8050 q^{4} +31.7408 q^{5} +(95.9654 + 129.400i) q^{6} -442.375i q^{8} +(70.5484 - 232.534i) q^{9} +O(q^{10})$	\(q+10.3347i q^{2} +(12.5210 - 9.28579i) q^{3} -74.8050 q^{4} +31.7408 q^{5} +(95.9654 + 129.400i) q^{6} -442.375i q^{8} +(70.5484 - 232.534i) q^{9} +328.031i q^{10} +453.489i q^{11} +(-936.630 + 694.623i) q^{12} +551.786i q^{13} +(397.426 - 294.739i) q^{15} +2178.03 q^{16} +538.161 q^{17} +(2403.15 + 729.093i) q^{18} +1366.77i q^{19} -2374.38 q^{20} -4686.65 q^{22} +3237.97i q^{23} +(-4107.80 - 5538.96i) q^{24} -2117.52 q^{25} -5702.52 q^{26} +(-1275.93 - 3566.64i) q^{27} +1604.88i q^{29} +(3046.02 + 4107.25i) q^{30} +7069.92i q^{31} +8353.19i q^{32} +(4211.00 + 5678.11i) q^{33} +5561.71i q^{34} +(-5277.37 + 17394.7i) q^{36} -1951.06 q^{37} -14125.1 q^{38} +(5123.77 + 6908.89i) q^{39} -14041.4i q^{40} +1800.47 q^{41} +7882.36 q^{43} -33923.2i q^{44} +(2239.27 - 7380.82i) q^{45} -33463.3 q^{46} -6090.22 q^{47} +(27271.0 - 20224.7i) q^{48} -21883.8i q^{50} +(6738.29 - 4997.25i) q^{51} -41276.4i q^{52} -14177.9i q^{53} +(36860.0 - 13186.2i) q^{54} +14394.1i q^{55} +(12691.6 + 17113.3i) q^{57} -16585.9 q^{58} +16918.9 q^{59} +(-29729.4 + 22047.9i) q^{60} -29718.9i q^{61} -73065.1 q^{62} -16630.3 q^{64} +17514.2i q^{65} +(-58681.3 + 43519.2i) q^{66} +13614.3 q^{67} -40257.1 q^{68} +(30067.1 + 40542.4i) q^{69} +31338.5i q^{71} +(-102867. - 31208.8i) q^{72} -9821.33i q^{73} -20163.6i q^{74} +(-26513.3 + 19662.8i) q^{75} -102242. i q^{76} +(-71401.0 + 52952.4i) q^{78} +99798.2 q^{79} +69132.6 q^{80} +(-49094.8 - 32809.8i) q^{81} +18607.2i q^{82} -79557.4 q^{83} +17081.7 q^{85} +81461.4i q^{86} +(14902.6 + 20094.6i) q^{87} +200612. q^{88} +953.735 q^{89} +(76278.2 + 23142.0i) q^{90} -242216. i q^{92} +(65649.7 + 88522.1i) q^{93} -62940.3i q^{94} +43382.6i q^{95} +(77566.0 + 104590. i) q^{96} +115552. i q^{97} +(105451. + 31992.9i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$			10.3347i			1.82693i	0.406922	+	0.913463i	$0.366602\pi$
$2$			10.3347i			1.82693i	−0.406922	+	0.913463i	$0.633398\pi$
$3$	12.5210	−	9.28579i	0.803219	−	0.595683i
$3$	12.5210	−	9.28579i	0.803219	−	0.595683i
$4$	−74.8050			−2.33766
$5$	31.7408			0.567798			0.283899	−	0.958854i	$-0.408372\pi$
$5$	31.7408			0.567798			0.283899	+	0.958854i	$0.408372\pi$
$6$	95.9654	+	129.400i	1.08827	+	1.46742i
$7$		0			0
$7$		0			0
$8$		−	442.375i		−	2.44380i
$9$	70.5484	−	232.534i	0.290323	−	0.956929i
$10$			328.031i			1.03732i
$11$			453.489i			1.13002i	0.825085	+	0.565008i	$0.191127\pi$
$11$			453.489i			1.13002i	−0.825085	+	0.565008i	$0.808873\pi$
$12$	−936.630	+	694.623i	−1.87765	+	1.39250i
$13$			551.786i			0.905550i	0.891625	+	0.452775i	$0.149566\pi$
$13$			551.786i			0.905550i	−0.891625	+	0.452775i	$0.850434\pi$
$14$		0			0
$15$	397.426	−	294.739i	0.456066	−	0.338228i
$16$	2178.03			2.12698
$17$	538.161			0.451637			0.225819	−	0.974169i	$-0.427494\pi$
$17$	538.161			0.451637			0.225819	+	0.974169i	$0.427494\pi$
$18$	2403.15	+	729.093i	1.74824	+	0.530398i
$19$			1366.77i			0.868586i	0.900772	+	0.434293i	$0.143002\pi$
$19$			1366.77i			0.868586i	−0.900772	+	0.434293i	$0.856998\pi$
$20$	−2374.38			−1.32732
$21$		0			0
$22$	−4686.65			−2.06446
$23$			3237.97i			1.27630i	0.769912	+	0.638150i	$0.220300\pi$
$23$			3237.97i			1.27630i	−0.769912	+	0.638150i	$0.779700\pi$
$24$	−4107.80	−	5538.96i	−1.45573	−	1.96291i
$25$	−2117.52			−0.677606
$26$	−5702.52			−1.65437
$27$	−1275.93	−	3566.64i	−0.336834	−	0.941564i
$28$		0			0
$29$			1604.88i			0.354363i	0.984178	+	0.177181i	$0.0566979\pi$
$29$			1604.88i			0.354363i	−0.984178	+	0.177181i	$0.943302\pi$
$30$	3046.02	+	4107.25i	0.617917	+	0.833199i
$31$			7069.92i			1.32133i	0.750683	+	0.660663i	$0.229725\pi$
$31$			7069.92i			1.32133i	−0.750683	+	0.660663i	$0.770275\pi$
$32$			8353.19i			1.44204i
$33$	4211.00	+	5678.11i	0.673132	+	0.907651i
$34$			5561.71i			0.825108i
$35$		0			0
$36$	−5277.37	+	17394.7i	−0.678675	+	2.23697i
$37$	−1951.06			−0.234297			−0.117149	−	0.993114i	$-0.537375\pi$
$37$	−1951.06			−0.234297			−0.117149	+	0.993114i	$0.537375\pi$
$38$	−14125.1			−1.58684
$39$	5123.77	+	6908.89i	0.539421	+	0.727355i
$40$		−	14041.4i		−	1.38758i
$41$	1800.47			0.167273			0.0836365	−	0.996496i	$-0.473347\pi$
$41$	1800.47			0.167273			0.0836365	+	0.996496i	$0.473347\pi$
$42$		0			0
$43$	7882.36			0.650107			0.325054	−	0.945696i	$-0.394618\pi$
$43$	7882.36			0.650107			0.325054	+	0.945696i	$0.394618\pi$
$44$		−	33923.2i		−	2.64159i
$45$	2239.27	−	7380.82i	0.164844	−	0.543342i
$46$	−33463.3			−2.33170
$47$	−6090.22			−0.402150			−0.201075	−	0.979576i	$-0.564444\pi$
$47$	−6090.22			−0.402150			−0.201075	+	0.979576i	$0.564444\pi$
$48$	27271.0	−	20224.7i	1.70843	−	1.26701i
$49$		0			0
$50$		−	21883.8i		−	1.23794i
$51$	6738.29	−	4997.25i	0.362764	−	0.269033i
$52$		−	41276.4i		−	2.11687i
$53$		−	14177.9i		−	0.693302i	−0.937994	−	0.346651i	$-0.887319\pi$
$53$		−	14177.9i		−	0.693302i	0.937994	−	0.346651i	$-0.112681\pi$
$54$	36860.0	−	13186.2i	1.72017	−	0.615370i
$55$			14394.1i			0.641621i
$56$		0			0
$57$	12691.6	+	17113.3i	0.517402	+	0.697665i
$58$	−16585.9			−0.647394
$59$	16918.9			0.632764			0.316382	−	0.948632i	$-0.397532\pi$
$59$	16918.9			0.632764			0.316382	+	0.948632i	$0.397532\pi$
$60$	−29729.4	+	22047.9i	−1.06613	+	0.790660i
$61$		−	29718.9i		−	1.02261i	−0.859401	−	0.511303i	$-0.829163\pi$
$61$		−	29718.9i		−	1.02261i	0.859401	−	0.511303i	$-0.170837\pi$
$62$	−73065.1			−2.41396
$63$		0			0
$64$	−16630.3			−0.507518
$65$			17514.2i			0.514169i
$66$	−58681.3	+	43519.2i	−1.65821	+	1.22976i
$67$	13614.3			0.370516			0.185258	−	0.982690i	$-0.440688\pi$
$67$	13614.3			0.370516			0.185258	+	0.982690i	$0.440688\pi$
$68$	−40257.1			−1.05577
$69$	30067.1	+	40542.4i	0.760271	+	1.02515i
$70$		0			0
$71$			31338.5i			0.737790i	0.929471	+	0.368895i	$0.120264\pi$
$71$			31338.5i			0.737790i	−0.929471	+	0.368895i	$0.879736\pi$
$72$	−102867.	−	31208.8i	−2.33854	−	0.709490i
$73$		−	9821.33i		−	0.215707i	−0.994167	−	0.107853i	$-0.965602\pi$
$73$		−	9821.33i		−	0.215707i	0.994167	−	0.107853i	$-0.0343977\pi$
$74$		−	20163.6i		−	0.428044i
$75$	−26513.3	+	19662.8i	−0.544266	+	0.403639i
$76$		−	102242.i		−	2.03046i
$77$		0			0
$78$	−71401.0	+	52952.4i	−1.32882	+	0.985482i
$79$	99798.2			1.79910			0.899550	−	0.436818i	$-0.143895\pi$
$79$	99798.2			1.79910			0.899550	+	0.436818i	$0.143895\pi$
$80$	69132.6			1.20770
$81$	−49094.8	−	32809.8i	−0.831426	−	0.555636i
$82$			18607.2i			0.305595i
$83$	−79557.4			−1.26761			−0.633804	−	0.773493i	$-0.718508\pi$
$83$	−79557.4			−1.26761			−0.633804	+	0.773493i	$0.718508\pi$
$84$		0			0
$85$	17081.7			0.256439
$86$			81461.4i			1.18770i
$87$	14902.6	+	20094.6i	0.211088	+	0.284631i
$88$	200612.			2.76154
$89$	953.735			0.0127630			0.00638150	−	0.999980i	$-0.497969\pi$
$89$	953.735			0.0127630			0.00638150	+	0.999980i	$0.497969\pi$
$90$	76278.2	+	23142.0i	0.992645	+	0.301159i
$91$		0			0
$92$		−	242216.i		−	2.98355i
$93$	65649.7	+	88522.1i	0.787092	+	1.06131i
$94$		−	62940.3i		−	0.734699i
$95$			43382.6i			0.493181i
$96$	77566.0	+	104590.i	0.859000	+	1.15828i
$97$			115552.i			1.24695i	0.781842	+	0.623476i	$0.214280\pi$
$97$			115552.i			1.24695i	−0.781842	+	0.623476i	$0.785720\pi$
$98$		0			0
$99$	105451.	+	31992.9i	1.08135	+	0.328069i
$100$	158401.			1.58401
$101$	−89983.4			−0.877726			−0.438863	−	0.898554i	$-0.644619\pi$
$101$	−89983.4			−0.877726			−0.438863	+	0.898554i	$0.644619\pi$
$102$	51644.8	+	69637.9i	0.491503	+	0.662743i
$103$			156663.i			1.45503i	0.686089	+	0.727517i	$0.259326\pi$
$103$			156663.i			1.45503i	−0.686089	+	0.727517i	$0.740674\pi$
$104$	244096.			2.21298
$105$		0			0
$106$	146524.			1.26661
$107$		−	139107.i		−	1.17460i	−0.809370	−	0.587299i	$-0.800191\pi$
$107$		−	139107.i		−	1.17460i	0.809370	−	0.587299i	$-0.199809\pi$
$108$	95445.6	+	266803.i	0.787402	+	2.20105i
$109$	24226.4			0.195309			0.0976547	−	0.995220i	$-0.468866\pi$
$109$	24226.4			0.195309			0.0976547	+	0.995220i	$0.468866\pi$
$110$	−148758.			−1.17219
$111$	−24429.2	+	18117.2i	−0.188192	+	0.139567i
$112$		0			0
$113$			38230.9i			0.281655i	0.990034	+	0.140828i	$0.0449763\pi$
$113$			38230.9i			0.281655i	−0.990034	+	0.140828i	$0.955024\pi$
$114$	−176860.	+	131163.i	−1.27458	+	0.945255i
$115$			102776.i			0.724680i
$116$		−	120053.i		−	0.828378i
$117$	128309.	+	38927.6i	0.866547	+	0.262902i
$118$			174851.i			1.15601i
$119$		0			0
$120$	−130385.	−	175811.i	−0.826561	−	1.11453i
$121$	−44601.2			−0.276938
$122$	307135.			1.86822
$123$	22543.6	−	16718.8i	0.134357	−	0.0996417i
$124$		−	528865.i		−	3.08881i
$125$	−166402.			−0.952541
$126$		0			0
$127$	−65650.6			−0.361184			−0.180592	−	0.983558i	$-0.557801\pi$
$127$	−65650.6			−0.361184			−0.180592	+	0.983558i	$0.557801\pi$
$128$			95433.4i			0.514844i
$129$	98694.6	−	73193.9i	0.522179	−	0.387258i
$130$	−181003.			−0.939349
$131$	205595.			1.04673			0.523364	−	0.852109i	$-0.324677\pi$
$131$	205595.			1.04673			0.523364	+	0.852109i	$0.324677\pi$
$132$	−315004.	−	424751.i	−1.57355	−	2.12178i
$133$		0			0
$134$			140699.i			0.676905i
$135$	−40498.9	−	113208.i	−0.191253	−	0.534618i
$136$		−	238069.i		−	1.10371i
$137$		−	410472.i		−	1.86845i	−0.356683	−	0.934225i	$-0.616092\pi$
$137$		−	410472.i		−	1.86845i	0.356683	−	0.934225i	$-0.383908\pi$
$138$	−418992.	+	310733.i	−1.87287	+	1.38896i
$139$		−	73875.8i		−	0.324314i	−0.986765	−	0.162157i	$-0.948155\pi$
$139$		−	73875.8i		−	0.324314i	0.986765	−	0.162157i	$-0.0518450\pi$
$140$		0			0
$141$	−76255.4	+	56552.5i	−0.323015	+	0.239554i
$142$	−323873.			−1.34789
$143$	−250229.			−1.02329
$144$	153657.	−	506466.i	0.617511	−	2.03537i
$145$			50940.3i			0.201206i
$146$	101500.			0.394080
$147$		0			0
$148$	145949.			0.547707
$149$			57972.3i			0.213922i	0.994263	+	0.106961i	$0.0341119\pi$
$149$			57972.3i			0.213922i	−0.994263	+	0.106961i	$0.965888\pi$
$150$	−203208.	−	274006.i	−0.737418	−	0.994334i
$151$	−172435.			−0.615435			−0.307718	−	0.951478i	$-0.599565\pi$
$151$	−172435.			−0.615435			−0.307718	+	0.951478i	$0.599565\pi$
$152$	604627.			2.12265
$153$	37966.4	−	125141.i	0.131121	−	0.432185i
$154$		0			0
$155$			224405.i			0.750246i
$156$	−383284.	−	516820.i	−1.26098	−	1.70031i
$157$			239287.i			0.774764i	0.921919	+	0.387382i	$0.126620\pi$
$157$			239287.i			0.774764i	−0.921919	+	0.387382i	$0.873380\pi$
$158$			1.03138e6i			3.28682i
$159$	−131653.	−	177521.i	−0.412989	−	0.556874i
$160$			265137.i			0.818787i
$161$		0			0
$162$	339077.	−	507378.i	1.01511	−	1.51895i
$163$	148378.			0.437422			0.218711	−	0.975790i	$-0.429815\pi$
$163$	148378.			0.437422			0.218711	+	0.975790i	$0.429815\pi$
$164$	−134684.			−0.391027
$165$	133661.	+	180228.i	0.382203	+	0.515362i
$166$		−	822198.i		−	2.31583i
$167$	549931.			1.52587			0.762934	−	0.646476i	$-0.223758\pi$
$167$	549931.			1.52587			0.762934	+	0.646476i	$0.223758\pi$
$168$		0			0
$169$	66825.0			0.179979
$170$			176533.i			0.468494i
$171$	317821.	+	96423.7i	0.831175	+	0.252170i
$172$	−589640.			−1.51973
$173$	316273.			0.803428			0.401714	−	0.915765i	$-0.368415\pi$
$173$	316273.			0.803428			0.401714	+	0.915765i	$0.368415\pi$
$174$	−207671.	+	154013.i	−0.520000	+	0.385642i
$175$		0			0
$176$			987713.i			2.40353i
$177$	211840.	−	157105.i	0.508248	−	0.376927i
$178$			9856.52i			0.0233170i
$179$		−	563606.i		−	1.31475i	−0.753564	−	0.657374i	$-0.771667\pi$
$179$		−	563606.i		−	1.31475i	0.753564	−	0.657374i	$-0.228333\pi$
$180$	−167508.	+	552122.i	−0.385350	+	1.27015i
$181$			329274.i			0.747069i	0.927616	+	0.373535i	$0.121854\pi$
$181$			329274.i			0.747069i	−0.927616	+	0.373535i	$0.878146\pi$
$182$		0			0
$183$	−275963.	−	372109.i	−0.609149	−	0.821377i
$184$	1.43240e6			3.11902
$185$	−61928.4			−0.133033
$186$	−914845.	+	678467.i	−1.93894	+	1.43796i
$187$			244050.i			0.510358i
$188$	455579.			0.940090
$189$		0			0
$190$	−448344.			−0.901004
$191$		−	805475.i		−	1.59760i	−0.601595	−	0.798801i	$-0.705468\pi$
$191$		−	805475.i		−	1.59760i	0.601595	−	0.798801i	$-0.294532\pi$
$192$	−208228.	+	154426.i	−0.407648	+	0.302320i
$193$	635701.			1.22846			0.614228	−	0.789128i	$-0.289467\pi$
$193$	635701.			1.22846			0.614228	+	0.789128i	$0.289467\pi$
$194$	−1.19419e6			−2.27809
$195$	162633.	+	219294.i	0.306282	+	0.412991i
$196$		0			0
$197$		−	137701.i		−	0.252796i	−0.991980	−	0.126398i	$-0.959658\pi$
$197$		−	137701.i		−	0.252796i	0.991980	−	0.126398i	$-0.0403417\pi$
$198$	−330636.	+	1.08980e6i	−0.599358	+	1.97554i
$199$		−	895559.i		−	1.60310i	−0.597926	−	0.801552i	$-0.704008\pi$
$199$		−	895559.i		−	1.60310i	0.597926	−	0.801552i	$-0.295992\pi$
$200$			936737.i			1.65593i
$201$	170463.	−	126419.i	0.297606	−	0.220710i
$202$		−	929947.i		−	1.60354i
$203$		0			0
$204$	−504058.	+	373819.i	−0.848018	+	0.628907i
$205$	57148.4			0.0949772
$206$	−1.61906e6			−2.65824
$207$	752936.	+	228433.i	1.22133	+	0.370539i
$208$			1.20181e6i			1.92609i
$209$	−619817.			−0.981516
$210$		0			0
$211$	−392533.			−0.606974			−0.303487	−	0.952836i	$-0.598151\pi$
$211$	−392533.			−0.606974			−0.303487	+	0.952836i	$0.598151\pi$
$212$			1.06058e6i			1.62070i
$213$	291003.	+	392388.i	0.439489	+	0.592607i
$214$	1.43762e6			2.14590
$215$	250193.			0.369129
$216$	−1.57779e6	+	564437.i	−2.30099	+	0.823155i
$217$		0			0
$218$			250372.i			0.356816i
$219$	−91198.8	−	122972.i	−0.128493	−	0.173260i
$220$		−	1.07675e6i		−	1.49989i
$221$			296950.i			0.408980i
$222$	−187235.	−	252467.i	−0.254979	−	0.343813i
$223$		−	1.01457e6i		−	1.36622i	−0.730317	−	0.683109i	$-0.760628\pi$
$223$		−	1.01457e6i		−	1.36622i	0.730317	−	0.683109i	$-0.239372\pi$
$224$		0			0
$225$	−149388.	+	492394.i	−0.196724	+	0.648421i
$226$	−395103.			−0.514563
$227$	−313370.			−0.403639			−0.201820	−	0.979423i	$-0.564685\pi$
$227$	−313370.			−0.403639			−0.201820	+	0.979423i	$0.564685\pi$
$228$	−949393.	−	1.28016e6i	−1.20951	−	1.63090i
$229$			450743.i			0.567990i	0.958826	+	0.283995i	$0.0916599\pi$
$229$			450743.i			0.567990i	−0.958826	+	0.283995i	$0.908340\pi$
$230$	−1.06215e6			−1.32394
$231$		0			0
$232$	709960.			0.865992
$233$		−	134459.i		−	0.162255i	−0.996704	−	0.0811276i	$-0.974148\pi$
$233$		−	134459.i		−	0.162255i	0.996704	−	0.0811276i	$-0.0258521\pi$
$234$	−402303.	+	1.32603e6i	−0.480302	+	1.58312i
$235$	−193309.			−0.228340
$236$	−1.26562e6			−1.47918
$237$	1.24957e6	−	926705.i	1.44507	−	1.07169i
$238$		0			0
$239$		−	1.26368e6i		−	1.43101i	−0.698605	−	0.715507i	$-0.746196\pi$
$239$		−	1.26368e6i		−	1.43101i	0.698605	−	0.715507i	$-0.253804\pi$
$240$	865605.	−	641950.i	0.970045	−	0.719405i
$241$		−	486586.i		−	0.539656i	−0.962909	−	0.269828i	$-0.913033\pi$
$241$		−	486586.i		−	0.539656i	0.962909	−	0.269828i	$-0.0869668\pi$
$242$		−	460938.i		−	0.505945i
$243$	−919379.	+	45074.9i	−0.998800	+	0.0489687i
$244$			2.22312e6i			2.39050i
$245$		0			0
$246$	172783.	+	232980.i	0.182038	+	0.245460i
$247$	−754167.			−0.786548
$248$	3.12755e6			3.22906
$249$	−996134.	+	738753.i	−1.01817	+	0.755093i
$250$		−	1.71971e6i		−	1.74022i
$251$	453648.			0.454501			0.227250	−	0.973836i	$-0.427026\pi$
$251$	453648.			0.454501			0.227250	+	0.973836i	$0.427026\pi$
$252$		0			0
$253$	−1.46838e6			−1.44224
$254$		−	678476.i		−	0.659857i
$255$	213879.	−	158617.i	0.205976	−	0.152756i
$256$	−1.51844e6			−1.44810
$257$	1.98345e6			1.87322			0.936610	−	0.350373i	$-0.113945\pi$
$257$	1.98345e6			1.87322			0.936610	+	0.350373i	$0.113945\pi$
$258$	756433.	+	1.01997e6i	0.707492	+	0.953982i
$259$		0			0
$260$		−	1.31015e6i		−	1.20195i
$261$	373189.	+	113222.i	0.339100	+	0.102879i
$262$			2.12475e6i			1.91229i
$263$		−	782254.i		−	0.697363i	−0.937241	−	0.348681i	$-0.886630\pi$
$263$		−	782254.i		−	0.697363i	0.937241	−	0.348681i	$-0.113370\pi$
$264$	2.51186e6	−	1.86284e6i	2.21812	−	1.64500i
$265$		−	450019.i		−	0.393655i
$266$		0			0
$267$	11941.7	−	8856.17i	0.0102515	−	0.00760270i
$268$	−1.01841e6			−0.866139
$269$	1.68301e6			1.41809			0.709047	−	0.705161i	$-0.249125\pi$
$269$	1.68301e6			1.41809			0.709047	+	0.705161i	$0.249125\pi$
$270$	1.16997e6	−	418543.i	0.976707	−	0.349406i
$271$			813635.i			0.672986i	0.941686	+	0.336493i	$0.109241\pi$
$271$			813635.i			0.672986i	−0.941686	+	0.336493i	$0.890759\pi$
$272$	1.17213e6			0.960625
$273$		0			0
$274$	4.24208e6			3.41352
$275$		−	960271.i		−	0.765706i
$276$	−2.24917e6	−	3.03278e6i	−1.77725	−	2.39645i
$277$	375524.			0.294062			0.147031	−	0.989132i	$-0.453028\pi$
$277$	375524.			0.294062			0.147031	+	0.989132i	$0.453028\pi$
$278$	763481.			0.592497
$279$	1.64399e6	+	498771.i	1.26441	+	0.383611i
$280$		0			0
$281$			264282.i			0.199665i	0.995004	+	0.0998326i	$0.0318307\pi$
$281$			264282.i			0.199665i	−0.995004	+	0.0998326i	$0.968169\pi$
$282$	−584450.	−	788073.i	−0.437648	−	0.590124i
$283$			1.84525e6i			1.36959i	0.728738	+	0.684793i	$0.240107\pi$
$283$			1.84525e6i			1.36959i	−0.728738	+	0.684793i	$0.759893\pi$
$284$		−	2.34428e6i		−	1.72470i
$285$	402841.	+	543191.i	0.293780	+	0.396132i
$286$		−	2.58603e6i		−	1.86947i
$287$		0			0
$288$	1.94240e6	+	589304.i	1.37993	+	0.418657i
$289$	−1.13024e6			−0.796024
$290$	−526450.			−0.367589
$291$	1.07300e6	+	1.44683e6i	0.742789	+	1.00158i
$292$			734685.i			0.504248i
$293$	−1.96922e6			−1.34006			−0.670032	−	0.742332i	$-0.733720\pi$
$293$	−1.96922e6			−1.34006			−0.670032	+	0.742332i	$0.733720\pi$
$294$		0			0
$295$	537020.			0.359282
$296$			863102.i			0.572576i
$297$	1.61743e6	−	578618.i	1.06398	−	0.380628i
$298$	−599123.			−0.390819
$299$	−1.78667e6			−1.15575
$300$	1.98333e6	−	1.47088e6i	1.27231	−	0.943569i
$301$		0			0
$302$		−	1.78205e6i		−	1.12435i
$303$	−1.12668e6	+	835566.i	−0.705006	+	0.522847i
$304$			2.97688e6i			1.84747i
$305$		−	943303.i		−	0.580633i
$306$	1.29328e6	+	392369.i	0.789570	+	0.239547i
$307$			2.38073e6i			1.44166i	0.693110	+	0.720832i	$0.256240\pi$
$307$			2.38073e6i			1.44166i	−0.693110	+	0.720832i	$0.743760\pi$
$308$		0			0
$309$	1.45474e6	+	1.96157e6i	0.866740	+	1.16871i
$310$	−2.31915e6			−1.37064
$311$	−3.37378e6			−1.97795			−0.988976	−	0.148078i	$-0.952691\pi$
$311$	−3.37378e6			−1.97795			−0.988976	+	0.148078i	$0.952691\pi$
$312$	3.05632e6	−	2.26663e6i	1.77751	−	1.31824i
$313$		−	2.68664e6i		−	1.55006i	−0.631923	−	0.775031i	$-0.717734\pi$
$313$		−	2.68664e6i		−	1.55006i	0.631923	−	0.775031i	$-0.282266\pi$
$314$	−2.47294e6			−1.41544
$315$		0			0
$316$	−7.46541e6			−4.20568
$317$			690592.i			0.385988i	0.981200	+	0.192994i	$0.0618197\pi$
$317$			690592.i			0.385988i	−0.981200	+	0.192994i	$0.938180\pi$
$318$	1.83462e6	−	1.36059e6i	1.01737	−	0.754500i
$319$	−727796.			−0.400436
$320$	−527861.			−0.288167
$321$	−1.29172e6	−	1.74175e6i	−0.699689	−	0.943460i
$322$		0			0
$323$			735544.i			0.392286i
$324$	3.67254e6	+	2.45433e6i	1.94359	+	1.29889i
$325$		−	1.16842e6i		−	0.613606i
$326$			1.53344e6i			0.799138i
$327$	303338.	−	224961.i	0.156876	−	0.116343i
$328$		−	796482.i		−	0.408782i
$329$		0			0
$330$	−1.86259e6	+	1.38134e6i	−0.941528	+	0.698256i
$331$	3.23377e6			1.62233			0.811164	−	0.584819i	$-0.198834\pi$
$331$	3.23377e6			1.62233			0.811164	+	0.584819i	$0.198834\pi$
$332$	5.95129e6			2.96323
$333$	−137644.	+	453688.i	−0.0680218	+	0.224206i
$334$			5.68335e6i			2.78765i
$335$	432128.			0.210378
$336$		0			0
$337$	−751164.			−0.360296			−0.180148	−	0.983639i	$-0.557658\pi$
$337$	−751164.			−0.360296			−0.180148	+	0.983639i	$0.557658\pi$
$338$			690613.i			0.328808i
$339$	355003.	+	478687.i	0.167777	+	0.226231i
$340$	−1.27780e6			−0.599466
$341$	−3.20613e6			−1.49312
$342$	−996505.	+	3.28457e6i	−0.460696	+	1.51849i
$343$		0			0
$344$		−	3.48696e6i		−	1.58873i
$345$	954354.	+	1.28685e6i	0.431680	+	0.582077i
$346$			3.26857e6i			1.46780i
$347$		−	2.12687e6i		−	0.948236i	−0.880461	−	0.474118i	$-0.842767\pi$
$347$		−	2.12687e6i		−	0.948236i	0.880461	−	0.474118i	$-0.157233\pi$
$348$	−1.11479e6	−	1.50318e6i	−0.493451	−	0.665370i
$349$			1.87399e6i			0.823575i	0.911280	+	0.411787i	$0.135095\pi$
$349$			1.87399e6i			0.823575i	−0.911280	+	0.411787i	$0.864905\pi$
$350$		0			0
$351$	1.96802e6	−	704038.i	0.852633	−	0.305020i
$352$	−3.78808e6			−1.62953
$353$	−3.19891e6			−1.36636			−0.683180	−	0.730250i	$-0.739403\pi$
$353$	−3.19891e6			−1.36636			−0.683180	+	0.730250i	$0.739403\pi$
$354$	1.62363e6	+	2.18930e6i	0.688617	+	0.928531i
$355$			994711.i			0.418915i
$356$	−71344.1			−0.0298355
$357$		0			0
$358$	5.82467e6			2.40195
$359$			246921.i			0.101117i	0.998721	+	0.0505583i	$0.0161001\pi$
$359$			246921.i			0.101117i	−0.998721	+	0.0505583i	$0.983900\pi$
$360$	−3.26509e6	−	990595.i	−1.32782	−	0.402847i
$361$	608029.			0.245559
$362$	−3.40293e6			−1.36484
$363$	−558449.	+	414157.i	−0.222442	+	0.164967i
$364$		0			0
$365$		−	311737.i		−	0.122478i
$366$	3.84562e6	−	2.85199e6i	1.50059	−	1.11287i
$367$			454308.i			0.176070i	0.996117	+	0.0880349i	$0.0280587\pi$
$367$			454308.i			0.176070i	−0.996117	+	0.0880349i	$0.971941\pi$
$368$			7.05239e6i			2.71467i
$369$	127020.	−	418670.i	0.0485631	−	0.160068i
$370$		−	640009.i		−	0.243042i
$371$		0			0
$372$	−4.91093e6	−	6.62189e6i	−1.83995	−	2.48099i
$373$	2.35605e6			0.876825			0.438412	−	0.898774i	$-0.355541\pi$
$373$	2.35605e6			0.876825			0.438412	+	0.898774i	$0.355541\pi$
$374$	−2.52217e6			−0.932386
$375$	−2.08351e6	+	1.54517e6i	−0.765099	+	0.567413i
$376$			2.69416e6i			0.982775i
$377$	−885551.			−0.320893
$378$		0			0
$379$	1.39242e6			0.497935			0.248968	−	0.968512i	$-0.419909\pi$
$379$	1.39242e6			0.497935			0.248968	+	0.968512i	$0.419909\pi$
$380$		−	3.24523e6i		−	1.15289i
$381$	−822007.	+	609617.i	−0.290110	+	0.215152i
$382$	8.32431e6			2.91870
$383$	4.55786e6			1.58768			0.793841	−	0.608125i	$-0.208078\pi$
$383$	4.55786e6			1.58768			0.793841	+	0.608125i	$0.208078\pi$
$384$	886174.	+	1.19492e6i	0.306684	+	0.413532i
$385$		0			0
$386$			6.56975e6i			2.24430i
$387$	556088.	−	1.83291e6i	0.188741	−	0.622106i
$388$		−	8.64391e6i		−	2.91495i
$389$		−	1.66786e6i		−	0.558839i	−0.960169	−	0.279420i	$-0.909858\pi$
$389$		−	1.66786e6i		−	0.558839i	0.960169	−	0.279420i	$-0.0901420\pi$
$390$	−2.26633e6	+	1.68075e6i	−0.754503	+	0.559554i
$391$			1.74255e6i			0.576425i
$392$		0			0
$393$	2.57424e6	−	1.90911e6i	0.840752	−	0.623518i
$394$	1.42309e6			0.461840
$395$	3.16768e6			1.02152
$396$	−7.88830e6	−	2.39323e6i	−2.52782	−	0.766914i
$397$			1.72340e6i			0.548793i	0.961617	+	0.274397i	$0.0884781\pi$
$397$			1.72340e6i			0.548793i	−0.961617	+	0.274397i	$0.911522\pi$
$398$	9.25529e6			2.92875
$399$		0			0
$400$	−4.61202e6			−1.44126
$401$			5.66032e6i			1.75784i	0.476966	+	0.878922i	$0.341737\pi$
$401$			5.66032e6i			1.75784i	−0.476966	+	0.878922i	$0.658263\pi$
$402$	1.30650e6	+	1.76168e6i	0.403221	+	0.543703i
$403$	−3.90108e6			−1.19653
$404$	6.73121e6			2.05182
$405$	−1.55831e6	−	1.04141e6i	−0.472081	−	0.315489i
$406$		0			0
$407$		−	884786.i		−	0.264760i
$408$	−2.21066e6	−	2.98085e6i	−0.657463	−	0.886522i
$409$		−	2.24354e6i		−	0.663170i	−0.943425	−	0.331585i	$-0.892417\pi$
$409$		−	2.24354e6i		−	0.663170i	0.943425	−	0.331585i	$-0.107583\pi$
$410$			590609.i			0.173516i
$411$	−3.81155e6	−	5.13949e6i	−1.11301	−	1.50078i
$412$		−	1.17192e7i		−	3.40137i
$413$		0			0
$414$	−2.36078e6	+	7.78133e6i	−0.676947	+	2.23128i
$415$	−2.52522e6			−0.719745
$416$	−4.60918e6			−1.30584
$417$	−685995.	−	924995.i	−0.193188	−	0.260495i
$418$		−	6.40559e6i		−	1.79316i
$419$	−6.62738e6			−1.84419			−0.922097	−	0.386959i	$-0.873526\pi$
$419$	−6.62738e6			−1.84419			−0.922097	+	0.386959i	$0.873526\pi$
$420$		0			0
$421$	−633684.			−0.174248			−0.0871240	−	0.996197i	$-0.527768\pi$
$421$	−633684.			−0.174248			−0.0871240	+	0.996197i	$0.527768\pi$
$422$		−	4.05669e6i		−	1.10890i
$423$	−429655.	+	1.41618e6i	−0.116753	+	0.384829i
$424$	−6.27196e6			−1.69429
$425$	−1.13957e6			−0.306032
$426$	−4.05519e6	+	3.00741e6i	−1.08265	+	0.802914i
$427$		0			0
$428$			1.04059e7i			2.74581i
$429$	−3.13310e6	+	2.32357e6i	−0.821924	+	0.609555i
$430$			2.58565e6i			0.674372i
$431$			3.73993e6i			0.969772i	0.874577	+	0.484886i	$0.161139\pi$
$431$			3.73993e6i			0.969772i	−0.874577	+	0.484886i	$0.838861\pi$
$432$	−2.77901e6	−	7.76825e6i	−0.716440	−	2.00269i
$433$			3.36951e6i			0.863668i	0.901953	+	0.431834i	$0.142133\pi$
$433$			3.36951e6i			0.863668i	−0.901953	+	0.431834i	$0.857867\pi$
$434$		0			0
$435$	473021.	+	637821.i	0.119855	+	0.161613i
$436$	−1.81226e6			−0.456566
$437$	−4.42557e6			−1.10858
$438$	1.27088e6	−	942508.i	0.316533	−	0.234747i
$439$			1.28182e6i			0.317443i	0.987323	+	0.158722i	$0.0507372\pi$
$439$			1.28182e6i			0.317443i	−0.987323	+	0.158722i	$0.949263\pi$
$440$	6.36760e6			1.56799
$441$		0			0
$442$	−3.06887e6			−0.747177
$443$		−	4.24150e6i		−	1.02686i	−0.858132	−	0.513428i	$-0.828375\pi$
$443$		−	4.24150e6i		−	1.02686i	0.858132	−	0.513428i	$-0.171625\pi$
$444$	1.82743e6	−	1.35525e6i	0.439929	−	0.326260i
$445$	30272.3			0.00724680
$446$	1.04852e7			2.49598
$447$	538318.	+	725868.i	0.127430	+	0.171826i
$448$		0			0
$449$			5.30045e6i			1.24079i	0.784291	+	0.620393i	$0.213027\pi$
$449$			5.30045e6i			1.24079i	−0.784291	+	0.620393i	$0.786973\pi$
$450$	−5.08873e6	−	1.54387e6i	−1.18462	−	0.359401i
$451$			816492.i			0.189021i
$452$		−	2.85986e6i		−	0.658413i
$453$	−2.15905e6	+	1.60119e6i	−0.494330	+	0.366605i
$454$		−	3.23857e6i		−	0.737419i
$455$		0			0
$456$	7.57050e6	−	5.61443e6i	1.70495	−	1.26443i
$457$	−7.00723e6			−1.56948			−0.784741	−	0.619824i	$-0.787204\pi$
$457$	−7.00723e6			−1.56948			−0.784741	+	0.619824i	$0.787204\pi$
$458$	−4.65827e6			−1.03768
$459$	−686653.	−	1.91943e6i	−0.152127	−	0.425246i
$460$		−	7.68815e6i		−	1.69405i
$461$	640032.			0.140265			0.0701325	−	0.997538i	$-0.477658\pi$
$461$	640032.			0.140265			0.0701325	+	0.997538i	$0.477658\pi$
$462$		0			0
$463$	3.47378e6			0.753096			0.376548	−	0.926397i	$-0.377111\pi$
$463$	3.47378e6			0.753096			0.376548	+	0.926397i	$0.377111\pi$
$464$			3.49548e6i			0.753724i
$465$	2.08378e6	+	2.80977e6i	0.446909	+	0.602612i
$466$	1.38958e6			0.296428
$467$	−1.23565e6			−0.262182			−0.131091	−	0.991370i	$-0.541848\pi$
$467$	−1.23565e6			−0.262182			−0.131091	+	0.991370i	$0.541848\pi$
$468$	−9.59815e6	−	2.91198e6i	−2.02569	−	0.614574i
$469$		0			0
$470$		−	1.99778e6i		−	0.417160i
$471$	2.22196e6	+	2.99610e6i	0.461514	+	0.622305i
$472$		−	7.48449e6i		−	1.54635i
$473$			3.57456e6i			0.734632i
$474$	9.57717e6	+	1.29139e7i	1.95790	+	2.64004i
$475$		−	2.89417e6i		−	0.588559i
$476$		0			0
$477$	−3.29684e6	−	1.00023e6i	−0.663441	−	0.201281i
$478$	1.30597e7			2.61436
$479$	1.56481e6			0.311618			0.155809	−	0.987787i	$-0.450201\pi$
$479$	1.56481e6			0.311618			0.155809	+	0.987787i	$0.450201\pi$
$480$	2.46201e6	+	3.31977e6i	0.487738	+	0.657666i
$481$		−	1.07657e6i		−	0.212168i
$482$	5.02870e6			0.985911
$483$		0			0
$484$	3.33639e6			0.647387
$485$			3.66773e6i			0.708017i
$486$	−465833.	−	9.50146e6i	−0.0894623	−	1.82473i
$487$	−2.89980e6			−0.554045			−0.277023	−	0.960863i	$-0.589348\pi$
$487$	−2.89980e6			−0.554045			−0.277023	+	0.960863i	$0.589348\pi$
$488$	−1.31469e7			−2.49904
$489$	1.85784e6	−	1.37781e6i	0.351346	−	0.260565i
$490$		0			0
$491$		−	4.46273e6i		−	0.835405i	−0.908584	−	0.417702i	$-0.862835\pi$
$491$		−	4.46273e6i		−	0.835405i	0.908584	−	0.417702i	$-0.137165\pi$
$492$	−1.68637e6	+	1.25065e6i	−0.314080	+	0.232928i
$493$			863685.i			0.160043i
$494$		−	7.79405e6i		−	1.43696i
$495$	3.34712e6	+	1.01548e6i	0.613985	+	0.186277i
$496$			1.53985e7i			2.81044i
$497$		0			0
$498$	−7.63475e6	−	1.02947e7i	−1.37950	−	1.86012i
$499$	2.66828e6			0.479711			0.239855	−	0.970809i	$-0.422900\pi$
$499$	2.66828e6			0.479711			0.239855	+	0.970809i	$0.422900\pi$
$500$	1.24477e7			2.22671
$501$	6.88566e6	−	5.10654e6i	1.22561	−	0.908935i
$502$			4.68830e6i			0.830339i
$503$	6.90952e6			1.21767			0.608833	−	0.793299i	$-0.291638\pi$
$503$	6.90952e6			1.21767			0.608833	+	0.793299i	$0.291638\pi$
$504$		0			0
$505$	−2.85615e6			−0.498370
$506$		−	1.51752e7i		−	2.63487i
$507$	836712.	−	620522.i	0.144563	−	0.107211i
$508$	4.91099e6			0.844326
$509$	−4.56189e6			−0.780460			−0.390230	−	0.920717i	$-0.627604\pi$
$509$	−4.56189e6			−0.780460			−0.390230	+	0.920717i	$0.627604\pi$
$510$	1.63925e6	+	2.21036e6i	0.279074	+	0.376304i
$511$		0			0
$512$		−	1.26387e7i		−	2.13073i
$513$	4.87479e6	−	1.74390e6i	0.817829	−	0.292569i
$514$			2.04983e7i			3.42224i
$515$			4.97262e6i			0.826165i
$516$	−7.38285e6	+	5.47527e6i	−1.22067	+	0.905277i
$517$		−	2.76185e6i		−	0.454437i
$518$		0			0
$519$	3.96004e6	−	2.93684e6i	0.645329	−	0.478589i
$520$	7.74783e6			1.25653
$521$	532535.			0.0859515			0.0429758	−	0.999076i	$-0.486316\pi$
$521$	532535.			0.0859515			0.0429758	+	0.999076i	$0.486316\pi$
$522$	−1.17011e6	+	3.85678e6i	−0.187953	+	0.619510i
$523$			1.32596e6i			0.211971i	0.994368	+	0.105985i	$0.0337996\pi$
$523$			1.32596e6i			0.211971i	−0.994368	+	0.105985i	$0.966200\pi$
$524$	−1.53795e7			−2.44689
$525$		0			0
$526$	8.08433e6			1.27403
$527$			3.80475e6i			0.596760i
$528$	9.17169e6	+	1.23671e7i	1.43174	+	1.93056i
$529$	−4.04808e6			−0.628941
$530$	4.65079e6			0.719179
$531$	1.19360e6	−	3.93421e6i	0.183706	−	0.605510i
$532$		0			0
$533$			993473.i			0.151474i
$534$	91525.5	+	123413.i	0.0138896	+	0.0187287i
$535$		−	4.41537e6i		−	0.666934i
$536$		−	6.02261e6i		−	0.905467i
$537$	−5.23352e6	−	7.05688e6i	−0.783174	−	1.05603i
$538$			1.73933e7i			2.59075i
$539$		0			0
$540$	3.02952e6	+	8.46854e6i	0.447085	+	1.24975i
$541$	−1.12979e7			−1.65961			−0.829804	−	0.558055i	$-0.811548\pi$
$541$	−1.12979e7			−1.65961			−0.829804	+	0.558055i	$0.811548\pi$
$542$	−8.40863e6			−1.22950
$543$	3.05757e6	+	4.12282e6i	0.445017	+	0.600060i
$544$			4.49536e6i			0.651280i
$545$	768967.			0.110896
$546$		0			0
$547$	−1.17217e7			−1.67503			−0.837516	−	0.546413i	$-0.815993\pi$
$547$	−1.17217e7			−1.67503			−0.837516	+	0.546413i	$0.815993\pi$
$548$			3.07053e7i			4.36780i
$549$	−6.91065e6	−	2.09662e6i	−0.978561	−	0.296886i
$550$	9.92407e6			1.39889
$551$	−2.19351e6			−0.307794
$552$	1.79350e7	−	1.33009e7i	2.50526	−	1.85795i
$553$		0			0
$554$			3.88091e6i			0.537229i
$555$	−775403.	+	575054.i	−0.106855	+	0.0792458i
$556$			5.52628e6i			0.758134i
$557$			5.47372e6i			0.747558i	0.927518	+	0.373779i	$0.121938\pi$
$557$			5.47372e6i			0.747558i	−0.927518	+	0.373779i	$0.878062\pi$
$558$	−5.15463e6	+	1.69901e7i	−0.700828	+	2.30999i
$559$			4.34938e6i			0.588705i
$560$		0			0
$561$	2.26620e6	+	3.05574e6i	0.304012	+	0.409929i
$562$	−2.73127e6			−0.364773
$563$	1.44586e7			1.92245			0.961224	−	0.275769i	$-0.0889326\pi$
$563$	1.44586e7			1.92245			0.961224	+	0.275769i	$0.0889326\pi$
$564$	5.70428e6	−	4.23041e6i	0.755098	−	0.559996i
$565$			1.21348e6i			0.159923i
$566$	−1.90700e7			−2.50213
$567$		0			0
$568$	1.38634e7			1.80301
$569$		−	4.69732e6i		−	0.608233i	−0.952635	−	0.304116i	$-0.901639\pi$
$569$		−	4.69732e6i		−	0.608233i	0.952635	−	0.304116i	$-0.0983612\pi$
$570$	−5.61369e6	+	4.16322e6i	−0.723704	+	0.536713i
$571$	5.07275e6			0.651108			0.325554	−	0.945523i	$-0.394449\pi$
$571$	5.07275e6			0.651108			0.325554	+	0.945523i	$0.394449\pi$
$572$	1.87184e7			2.39209
$573$	−7.47947e6	−	1.00853e7i	−0.951665	−	1.28323i
$574$		0			0
$575$		−	6.85645e6i		−	0.864828i
$576$	−1.17324e6	+	3.86712e6i	−0.147344	+	0.485659i
$577$		−	7.54076e6i		−	0.942922i	−0.881887	−	0.471461i	$-0.843727\pi$
$577$		−	7.54076e6i		−	0.942922i	0.881887	−	0.471461i	$-0.156273\pi$
$578$		−	1.16806e7i		−	1.45428i
$579$	7.95959e6	−	5.90299e6i	0.986720	−	0.731771i
$580$		−	3.81059e6i		−	0.470351i
$581$		0			0
$582$	−1.49525e7	+	1.10890e7i	−1.82981	+	1.35702i
$583$	6.42953e6			0.783443
$584$	−4.34471e6			−0.527144
$585$	4.07263e6	+	1.23560e6i	0.492023	+	0.149275i
$586$		−	2.03512e7i		−	2.44820i
$587$	1.49725e7			1.79349			0.896743	−	0.442551i	$-0.145926\pi$
$587$	1.49725e7			1.79349			0.896743	+	0.442551i	$0.145926\pi$
$588$		0			0
$589$	−9.66297e6			−1.14768
$590$			5.54991e6i			0.656381i
$591$	−1.27866e6	−	1.72414e6i	−0.150586	−	0.203051i
$592$	−4.24948e6			−0.498347
$593$	−3.02527e6			−0.353287			−0.176644	−	0.984275i	$-0.556524\pi$
$593$	−3.02527e6			−0.353287			−0.176644	+	0.984275i	$0.556524\pi$
$594$	5.97981e6	+	1.67156e7i	0.695379	+	1.94382i
$595$		0			0
$596$		−	4.33662e6i		−	0.500075i
$597$	−8.31597e6	−	1.12132e7i	−0.954942	−	1.28764i
$598$		−	1.84646e7i		−	2.11148i
$599$			5.80456e6i			0.661001i	0.943806	+	0.330501i	$0.107218\pi$
$599$			5.80456e6i			0.661001i	−0.943806	+	0.330501i	$0.892782\pi$
$600$	8.69834e6	+	1.17288e7i	0.986412	+	1.33008i
$601$		−	8.90296e6i		−	1.00542i	−0.864455	−	0.502711i	$-0.832336\pi$
$601$		−	8.90296e6i		−	1.00542i	0.864455	−	0.502711i	$-0.167664\pi$
$602$		0			0
$603$	960464.	−	3.16577e6i	0.107569	−	0.354557i
$604$	1.28990e7			1.43868
$605$	−1.41568e6			−0.157245
$606$	−8.63528e6	−	1.16438e7i	−0.955202	−	1.28799i
$607$		−	344344.i		−	0.0379333i	−0.999820	−	0.0189667i	$-0.993962\pi$
$607$		−	344344.i		−	0.0379333i	0.999820	−	0.0189667i	$-0.00603764\pi$
$608$	−1.14169e7			−1.25254
$609$		0			0
$610$	9.74871e6			1.06077
$611$		−	3.36050e6i		−	0.364167i
$612$	−2.84008e6	+	9.36114e6i	−0.306515	+	1.01030i
$613$	−40196.2			−0.00432050			−0.00216025	−	0.999998i	$-0.500688\pi$
$613$	−40196.2			−0.00432050			−0.00216025	+	0.999998i	$0.500688\pi$
$614$	−2.46040e7			−2.63381
$615$	715552.	−	530668.i	0.0762875	−	0.0565763i
$616$		0			0
$617$			4.04424e6i			0.427685i	0.976868	+	0.213843i	$0.0685980\pi$
$617$			4.04424e6i			0.427685i	−0.976868	+	0.213843i	$0.931402\pi$
$618$	−2.02721e7	+	1.50342e7i	−2.13515	+	1.58347i
$619$		−	7.68259e6i		−	0.805900i	−0.915222	−	0.402950i	$-0.867985\pi$
$619$		−	7.68259e6i		−	0.805900i	0.915222	−	0.402950i	$-0.132015\pi$
$620$		−	1.67866e7i		−	1.75382i
$621$	1.15487e7	−	4.13140e6i	1.20172	−	0.429901i
$622$		−	3.48668e7i		−	3.61357i
$623$		0			0
$624$	1.11597e7	+	1.50478e7i	1.14734	+	1.54707i
$625$	1.33551e6			0.136756
$626$	2.77655e7			2.83185
$627$	−7.76069e6	+	5.75548e6i	−0.788373	+	0.584673i
$628$		−	1.78998e7i		−	1.81113i
$629$	−1.04999e6			−0.105817
$630$		0			0
$631$	1.42740e7			1.42716			0.713582	−	0.700572i	$-0.247072\pi$
$631$	1.42740e7			1.42716			0.713582	+	0.700572i	$0.247072\pi$
$632$		−	4.41483e7i		−	4.39664i
$633$	−4.91488e6	+	3.64498e6i	−0.487533	+	0.361564i
$634$	−7.13703e6			−0.705171
$635$	−2.08380e6			−0.205080
$636$	9.84831e6	+	1.32795e7i	0.965426	+	1.30178i
$637$		0			0
$638$		−	7.52152e6i		−	0.731566i
$639$	7.28726e6	+	2.21088e6i	0.706012	+	0.214197i
$640$			3.02914e6i			0.292327i
$641$		−	7.73472e6i		−	0.743532i	−0.928326	−	0.371766i	$-0.878752\pi$
$641$		−	7.73472e6i		−	0.743532i	0.928326	−	0.371766i	$-0.121248\pi$
$642$	1.80004e7	−	1.33494e7i	1.72363	−	1.27828i
$643$			1.41911e7i			1.35360i	0.736168	+	0.676799i	$0.236633\pi$
$643$			1.41911e7i			1.35360i	−0.736168	+	0.676799i	$0.763367\pi$
$644$		0			0
$645$	3.13265e6	−	2.32324e6i	0.296492	−	0.219884i
$646$	−7.60159e6			−0.716677
$647$	−678952.			−0.0637644			−0.0318822	−	0.999492i	$-0.510150\pi$
$647$	−678952.			−0.0637644			−0.0318822	+	0.999492i	$0.510150\pi$
$648$	−1.45142e7	+	2.17183e7i	−1.35786	+	2.03184i
$649$			7.67252e6i			0.715034i
$650$	1.20752e7			1.12101
$651$		0			0
$652$	−1.10994e7			−1.02254
$653$		−	2.65332e6i		−	0.243505i	−0.992561	−	0.121752i	$-0.961149\pi$
$653$		−	2.65332e6i		−	0.243505i	0.992561	−	0.121752i	$-0.0388514\pi$
$654$	2.32490e6	+	3.13489e6i	0.212549	+	0.286601i
$655$	6.52575e6			0.594329
$656$	3.92148e6			0.355787
$657$	−2.28379e6	−	692879.i	−0.206416	−	0.0626245i
$658$		0			0
$659$		−	1.07933e7i		−	0.968149i	−0.875027	−	0.484074i	$-0.839156\pi$
$659$		−	1.07933e7i		−	0.968149i	0.875027	−	0.484074i	$-0.160844\pi$
$660$	−9.99849e6	−	1.34820e7i	−0.893459	−	1.20474i
$661$			2.05761e7i			1.83172i	0.401496	+	0.915861i	$0.368490\pi$
$661$			2.05761e7i			1.83172i	−0.401496	+	0.915861i	$0.631510\pi$
$662$			3.34198e7i			2.96387i
$663$	2.75741e6	+	3.71809e6i	0.243623	+	0.328501i
$664$			3.51942e7i			3.09778i
$665$		0			0
$666$	−4.68871e6	−	1.42251e6i	−0.409607	−	0.124271i
$667$	−5.19655e6			−0.452273
$668$	−4.11376e7			−3.56696
$669$	−9.42108e6	−	1.27034e7i	−0.813833	−	1.09737i
$670$			4.46589e6i			0.384345i
$671$	1.34772e7			1.15556
$672$		0			0
$673$	−1.73039e7			−1.47268			−0.736338	−	0.676614i	$-0.763447\pi$
$673$	−1.73039e7			−1.47268			−0.736338	+	0.676614i	$0.763447\pi$
$674$		−	7.76302e6i		−	0.658235i
$675$	2.70180e6	+	7.55243e6i	0.228241	+	0.638009i
$676$	−4.99884e6			−0.420729
$677$	−2.14664e6			−0.180006			−0.0900031	−	0.995941i	$-0.528688\pi$
$677$	−2.14664e6			−0.180006			−0.0900031	+	0.995941i	$0.528688\pi$
$678$	−4.94706e6	+	3.66884e6i	−0.413307	+	0.306517i
$679$		0			0
$680$		−	7.55651e6i		−	0.626685i
$681$	−3.92369e6	+	2.90989e6i	−0.324211	+	0.240441i
$682$		−	3.31342e7i		−	2.72782i
$683$		−	2.60907e6i		−	0.214010i	−0.994258	−	0.107005i	$-0.965874\pi$
$683$		−	2.60907e6i		−	0.214010i	0.994258	−	0.107005i	$-0.0341260\pi$
$684$	−2.37746e7	−	7.21298e6i	−1.94300	−	0.589487i
$685$		−	1.30287e7i		−	1.06090i
$686$		0			0
$687$	4.18550e6	+	5.64373e6i	0.338342	+	0.456220i
$688$	1.71680e7			1.38277
$689$	7.82318e6			0.627820
$690$	−1.32992e7	+	9.86292e6i	−1.06341	+	0.788647i
$691$			8.38905e6i			0.668371i	0.942507	+	0.334185i	$0.108461\pi$
$691$			8.38905e6i			0.668371i	−0.942507	+	0.334185i	$0.891539\pi$
$692$	−2.36588e7			−1.87814
$693$		0			0
$694$	2.19804e7			1.73236
$695$		−	2.34488e6i		−	0.184144i
$696$	8.88937e6	−	6.59253e6i	0.695581	−	0.515857i
$697$	968942.			0.0755467
$698$	−1.93670e7			−1.50461
$699$	−1.24855e6	−	1.68355e6i	−0.0966527	−	0.130327i
$700$		0			0
$701$			1.05701e6i			0.0812425i	0.999175	+	0.0406213i	$0.0129337\pi$
$701$			1.05701e6i			0.0812425i	−0.999175	+	0.0406213i	$0.987066\pi$
$702$	7.27599e6	+	2.03388e7i	0.557249	+	1.55770i
$703$		−	2.66666e6i		−	0.203507i
$704$		−	7.54168e6i		−	0.573504i
$705$	−2.42041e6	+	1.79502e6i	−0.183407	+	0.136018i
$706$		−	3.30596e7i		−	2.49624i
$707$		0			0
$708$	−1.58467e7	+	1.17523e7i	−1.18811	+	0.881126i
$709$	8.71222e6			0.650898			0.325449	−	0.945560i	$-0.394485\pi$
$709$	8.71222e6			0.650898			0.325449	+	0.945560i	$0.394485\pi$
$710$	−1.02800e7			−0.765327
$711$	7.04061e6	−	2.32065e7i	0.522319	−	1.72161i
$712$		−	421908.i		−	0.0311902i
$713$	−2.28921e7			−1.68641
$714$		0			0
$715$	−7.94248e6			−0.581020
$716$			4.21605e7i			3.07343i
$717$	−1.17343e7	−	1.58225e7i	−0.852431	−	1.14942i
$718$	−2.55185e6			−0.184732
$719$	2.34094e7			1.68876			0.844380	−	0.535744i	$-0.179969\pi$
$719$	2.34094e7			1.68876			0.844380	+	0.535744i	$0.179969\pi$
$720$	4.87719e6	−	1.60757e7i	0.350621	−	1.15568i
$721$		0			0
$722$			6.28377e6i			0.448618i
$723$	−4.51833e6	−	6.09252e6i	−0.321464	−	0.433462i
$724$		−	2.46313e7i		−	1.74639i
$725$		−	3.39837e6i		−	0.240118i
$726$	−4.28017e6	−	5.77138e6i	−0.301383	−	0.406385i
$727$			1.80153e7i			1.26417i	0.774899	+	0.632085i	$0.217801\pi$
$727$			1.80153e7i			1.26417i	−0.774899	+	0.632085i	$0.782199\pi$
$728$		0			0
$729$	−1.10929e7	+	9.10153e6i	−0.773086	+	0.634301i
$730$	3.22170e6			0.223758
$731$	4.24198e6			0.293613
$732$	2.06434e7	+	2.78356e7i	1.42398	+	1.92010i
$733$			7.02971e6i			0.483256i	0.970369	+	0.241628i	$0.0776814\pi$
$733$			7.02971e6i			0.483256i	−0.970369	+	0.241628i	$0.922319\pi$
$734$	−4.69511e6			−0.321667
$735$		0			0
$736$	−2.70474e7			−1.84048
$737$			6.17391e6i			0.418689i
$738$	4.32680e6	+	1.31271e6i	0.292433	+	0.0887212i
$739$	1.59687e7			1.07562			0.537810	−	0.843066i	$-0.319252\pi$
$739$	1.59687e7			1.07562			0.537810	+	0.843066i	$0.319252\pi$
$740$	4.63256e6			0.310987
$741$	−9.44289e6	+	7.00303e6i	−0.631770	+	0.468533i
$742$		0			0
$743$			2.50742e7i			1.66630i	0.553044	+	0.833152i	$0.313466\pi$
$743$			2.50742e7i			1.66630i	−0.553044	+	0.833152i	$0.686534\pi$
$744$	3.91600e7	−	2.90418e7i	2.59364	−	1.92350i
$745$			1.84009e6i			0.121464i
$746$			2.43490e7i			1.60189i
$747$	−5.61264e6	+	1.84998e7i	−0.368015	+	1.21301i
$748$		−	1.82562e7i		−	1.19304i
$749$		0			0
$750$	−1.59688e7	−	2.15324e7i	−1.03662	−	1.39778i
$751$	−2.18908e6			−0.141632			−0.0708159	−	0.997489i	$-0.522560\pi$
$751$	−2.18908e6			−0.141632			−0.0708159	+	0.997489i	$0.522560\pi$
$752$	−1.32647e7			−0.855367
$753$	5.68011e6	−	4.21248e6i	0.365064	−	0.270739i
$754$		−	9.15187e6i		−	0.586248i
$755$	−5.47323e6			−0.349443
$756$		0			0
$757$	−1.44147e6			−0.0914251			−0.0457126	−	0.998955i	$-0.514556\pi$
$757$	−1.44147e6			−0.0914251			−0.0457126	+	0.998955i	$0.514556\pi$
$758$			1.43902e7i			0.909691i
$759$	−1.83855e7	+	1.36351e7i	−1.15844	+	0.859119i
$760$	1.91914e7			1.20524
$761$	−1.36473e6			−0.0854253			−0.0427126	−	0.999087i	$-0.513600\pi$
$761$	−1.36473e6			−0.0854253			−0.0427126	+	0.999087i	$0.513600\pi$
$762$	−6.30018e6	−	8.49516e6i	−0.393066	−	0.530010i
$763$		0			0
$764$			6.02536e7i			3.73465i
$765$	1.20509e6	−	3.97207e6i	0.0744499	−	0.245394i
$766$			4.71038e7i			2.90058i
$767$			9.33561e6i			0.572999i
$768$	−1.90123e7	+	1.40999e7i	−1.16314	+	0.862608i
$769$			1.59541e7i			0.972875i	0.873715	+	0.486437i	$0.161704\pi$
$769$			1.59541e7i			0.972875i	−0.873715	+	0.486437i	$0.838296\pi$
$770$		0			0
$771$	2.48347e7	−	1.84179e7i	1.50461	−	1.11585i
$772$	−4.75537e7			−2.87171
$773$	1.72753e7			1.03986			0.519931	−	0.854208i	$-0.325958\pi$
$773$	1.72753e7			1.03986			0.519931	+	0.854208i	$0.325958\pi$
$774$	1.89425e7	+	5.74697e6i	1.13654	+	0.344815i
$775$		−	1.49707e7i		−	0.895338i
$776$	5.11175e7			3.04730
$777$		0			0
$778$	1.72368e7			1.02096
$779$			2.46083e6i			0.145291i
$780$	−1.21657e7	−

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)

\(f(q)\)	\(=\)	\(q+10.3347i q^{2} +(12.5210 - 9.28579i) q^{3} -74.8050 q^{4} +31.7408 q^{5} +(95.9654 + 129.400i) q^{6} -442.375i q^{8} +(70.5484 - 232.534i) q^{9} +O(q^{10})\)	\(q+10.3347i q^{2} +(12.5210 - 9.28579i) q^{3} -74.8050 q^{4} +31.7408 q^{5} +(95.9654 + 129.400i) q^{6} -442.375i q^{8} +(70.5484 - 232.534i) q^{9} +328.031i q^{10} +453.489i q^{11} +(-936.630 + 694.623i) q^{12} +551.786i q^{13} +(397.426 - 294.739i) q^{15} +2178.03 q^{16} +538.161 q^{17} +(2403.15 + 729.093i) q^{18} +1366.77i q^{19} -2374.38 q^{20} -4686.65 q^{22} +3237.97i q^{23} +(-4107.80 - 5538.96i) q^{24} -2117.52 q^{25} -5702.52 q^{26} +(-1275.93 - 3566.64i) q^{27} +1604.88i q^{29} +(3046.02 + 4107.25i) q^{30} +7069.92i q^{31} +8353.19i q^{32} +(4211.00 + 5678.11i) q^{33} +5561.71i q^{34} +(-5277.37 + 17394.7i) q^{36} -1951.06 q^{37} -14125.1 q^{38} +(5123.77 + 6908.89i) q^{39} -14041.4i q^{40} +1800.47 q^{41} +7882.36 q^{43} -33923.2i q^{44} +(2239.27 - 7380.82i) q^{45} -33463.3 q^{46} -6090.22 q^{47} +(27271.0 - 20224.7i) q^{48} -21883.8i q^{50} +(6738.29 - 4997.25i) q^{51} -41276.4i q^{52} -14177.9i q^{53} +(36860.0 - 13186.2i) q^{54} +14394.1i q^{55} +(12691.6 + 17113.3i) q^{57} -16585.9 q^{58} +16918.9 q^{59} +(-29729.4 + 22047.9i) q^{60} -29718.9i q^{61} -73065.1 q^{62} -16630.3 q^{64} +17514.2i q^{65} +(-58681.3 + 43519.2i) q^{66} +13614.3 q^{67} -40257.1 q^{68} +(30067.1 + 40542.4i) q^{69} +31338.5i q^{71} +(-102867. - 31208.8i) q^{72} -9821.33i q^{73} -20163.6i q^{74} +(-26513.3 + 19662.8i) q^{75} -102242. i q^{76} +(-71401.0 + 52952.4i) q^{78} +99798.2 q^{79} +69132.6 q^{80} +(-49094.8 - 32809.8i) q^{81} +18607.2i q^{82} -79557.4 q^{83} +17081.7 q^{85} +81461.4i q^{86} +(14902.6 + 20094.6i) q^{87} +200612. q^{88} +953.735 q^{89} +(76278.2 + 23142.0i) q^{90} -242216. i q^{92} +(65649.7 + 88522.1i) q^{93} -62940.3i q^{94} +43382.6i q^{95} +(77566.0 + 104590. i) q^{96} +115552. i q^{97} +(105451. + 31992.9i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 172 q^{4} + 1212 q^{9}+O(q^{10}) \) 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

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more