LMFDB - Embedded newform 147.6.c.c.146.5

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			146.5
Root			$3.16536 + 1.82752i$ of defining polynomial
Character	$\chi$	$=$	147.146
Dual form			147.6.c.c.146.11

$q$-expansion

$f(q)$	$=$	$q-3.65505i q^{2} +(-15.5635 + 0.882561i) q^{3} +18.6406 q^{4} +74.3244 q^{5} +(3.22580 + 56.8851i) q^{6} -185.094i q^{8} +(241.442 - 27.4714i) q^{9} +O(q^{10})$	\(q-3.65505i q^{2} +(-15.5635 + 0.882561i) q^{3} +18.6406 q^{4} +74.3244 q^{5} +(3.22580 + 56.8851i) q^{6} -185.094i q^{8} +(241.442 - 27.4714i) q^{9} -271.659i q^{10} +39.3914i q^{11} +(-290.113 + 16.4515i) q^{12} +589.288i q^{13} +(-1156.74 + 65.5958i) q^{15} -80.0263 q^{16} +1237.51 q^{17} +(-100.409 - 882.482i) q^{18} +2767.79i q^{19} +1385.45 q^{20} +143.977 q^{22} +463.253i q^{23} +(163.357 + 2880.70i) q^{24} +2399.11 q^{25} +2153.88 q^{26} +(-3733.43 + 640.637i) q^{27} -5294.57i q^{29} +(239.756 + 4227.95i) q^{30} -2846.74i q^{31} -5630.50i q^{32} +(-34.7653 - 613.066i) q^{33} -4523.16i q^{34} +(4500.64 - 512.084i) q^{36} +7847.11 q^{37} +10116.4 q^{38} +(-520.083 - 9171.36i) q^{39} -13757.0i q^{40} +12162.5 q^{41} +5350.18 q^{43} +734.281i q^{44} +(17945.0 - 2041.79i) q^{45} +1693.21 q^{46} +6755.99 q^{47} +(1245.49 - 70.6281i) q^{48} -8768.86i q^{50} +(-19259.9 + 1092.18i) q^{51} +10984.7i q^{52} -16011.8i q^{53} +(2341.56 + 13645.9i) q^{54} +2927.74i q^{55} +(-2442.74 - 43076.3i) q^{57} -19351.9 q^{58} +7746.64 q^{59} +(-21562.4 + 1222.75i) q^{60} +5617.58i q^{61} -10405.0 q^{62} -23140.6 q^{64} +43798.5i q^{65} +(-2240.79 + 127.069i) q^{66} -12887.4 q^{67} +23068.0 q^{68} +(-408.849 - 7209.82i) q^{69} +60637.7i q^{71} +(-5084.79 - 44689.5i) q^{72} -51641.4i q^{73} -28681.6i q^{74} +(-37338.5 + 2117.36i) q^{75} +51593.3i q^{76} +(-33521.7 + 1900.93i) q^{78} +67372.2 q^{79} -5947.90 q^{80} +(57539.6 - 13265.5i) q^{81} -44454.5i q^{82} -8712.06 q^{83} +91977.1 q^{85} -19555.1i q^{86} +(4672.78 + 82401.9i) q^{87} +7291.11 q^{88} -18112.0 q^{89} +(-7462.85 - 65589.9i) q^{90} +8635.34i q^{92} +(2512.42 + 44305.1i) q^{93} -24693.4i q^{94} +205714. i q^{95} +(4969.26 + 87630.1i) q^{96} -18641.1i q^{97} +(1082.14 + 9510.75i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$		−	3.65505i		−	0.646127i	−0.946377	−	0.323064i	$-0.895287\pi$
$2$		−	3.65505i		−	0.646127i	0.946377	−	0.323064i	$-0.104713\pi$
$3$	−15.5635	+	0.882561i	−0.998396	+	0.0566163i
$3$	−15.5635	+	0.882561i	−0.998396	+	0.0566163i
$4$	18.6406			0.582520
$5$	74.3244			1.32955			0.664777	−	0.747042i	$-0.268526\pi$
$5$	74.3244			1.32955			0.664777	+	0.747042i	$0.268526\pi$
$6$	3.22580	+	56.8851i	0.0365813	+	0.645091i
$7$		0			0
$7$		0			0
$8$		−	185.094i		−	1.02251i
$9$	241.442	−	27.4714i	0.993589	−	0.113051i
$10$		−	271.659i		−	0.859061i
$11$			39.3914i			0.0981567i	0.998795	+	0.0490783i	$0.0156284\pi$
$11$			39.3914i			0.0981567i	−0.998795	+	0.0490783i	$0.984372\pi$
$12$	−290.113	+	16.4515i	−0.581586	+	0.0329801i
$13$			589.288i			0.967096i	0.875318	+	0.483548i	$0.160652\pi$
$13$			589.288i			0.967096i	−0.875318	+	0.483548i	$0.839348\pi$
$14$		0			0
$15$	−1156.74	+	65.5958i	−1.32742	+	0.0752745i
$16$	−80.0263			−0.0781507
$17$	1237.51			1.03855			0.519274	−	0.854608i	$-0.326203\pi$
$17$	1237.51			1.03855			0.519274	+	0.854608i	$0.326203\pi$
$18$	−100.409	−	882.482i	−0.0730453	−	0.641985i
$19$			2767.79i			1.75893i	0.475964	+	0.879465i	$0.342099\pi$
$19$			2767.79i			1.75893i	−0.475964	+	0.879465i	$0.657901\pi$
$20$	1385.45			0.774492
$21$		0			0
$22$	143.977			0.0634217
$23$			463.253i			0.182599i	0.995823	+	0.0912997i	$0.0291021\pi$
$23$			463.253i			0.182599i	−0.995823	+	0.0912997i	$0.970898\pi$
$24$	163.357	+	2880.70i	0.0578907	+	1.02087i
$25$	2399.11			0.767716
$26$	2153.88			0.624867
$27$	−3733.43	+	640.637i	−0.985595	+	0.169123i
$28$		0			0
$29$		−	5294.57i		−	1.16906i	−0.811373	−	0.584529i	$-0.801279\pi$
$29$		−	5294.57i		−	1.16906i	0.811373	−	0.584529i	$-0.198721\pi$
$30$	239.756	+	4227.95i	0.0486369	+	0.857683i
$31$		−	2846.74i		−	0.532039i	−0.963968	−	0.266019i	$-0.914291\pi$
$31$		−	2846.74i		−	0.532039i	0.963968	−	0.266019i	$-0.0857085\pi$
$32$		−	5630.50i		−	0.972014i
$33$	−34.7653	−	613.066i	−0.00555727	−	0.0979992i
$34$		−	4523.16i		−	0.671034i
$35$		0			0
$36$	4500.64	−	512.084i	0.578785	−	0.0658545i
$37$	7847.11			0.942336			0.471168	−	0.882044i	$-0.343833\pi$
$37$	7847.11			0.942336			0.471168	+	0.882044i	$0.343833\pi$
$38$	10116.4			1.13649
$39$	−520.083	−	9171.36i	−0.0547534	−	0.965544i
$40$		−	13757.0i		−	1.35948i
$41$	12162.5			1.12996			0.564980	−	0.825105i	$-0.308884\pi$
$41$	12162.5			1.12996			0.564980	+	0.825105i	$0.308884\pi$
$42$		0			0
$43$	5350.18			0.441262			0.220631	−	0.975357i	$-0.429188\pi$
$43$	5350.18			0.441262			0.220631	+	0.975357i	$0.429188\pi$
$44$			734.281i			0.0571782i
$45$	17945.0	−	2041.79i	1.32103	−	0.150307i
$46$	1693.21			0.117982
$47$	6755.99			0.446112			0.223056	−	0.974806i	$-0.428397\pi$
$47$	6755.99			0.446112			0.223056	+	0.974806i	$0.428397\pi$
$48$	1245.49	−	70.6281i	0.0780253	−	0.00442460i
$49$		0			0
$50$		−	8768.86i		−	0.496042i
$51$	−19259.9	+	1092.18i	−1.03688	+	0.0587987i
$52$			10984.7i			0.563352i
$53$		−	16011.8i		−	0.782977i	−0.920183	−	0.391489i	$-0.871960\pi$
$53$		−	16011.8i		−	0.782977i	0.920183	−	0.391489i	$-0.128040\pi$
$54$	2341.56	+	13645.9i	0.109275	+	0.636820i
$55$			2927.74i			0.130505i
$56$		0			0
$57$	−2442.74	−	43076.3i	−0.0995841	−	1.75611i
$58$	−19351.9			−0.755360
$59$	7746.64			0.289723			0.144862	−	0.989452i	$-0.453726\pi$
$59$	7746.64			0.289723			0.144862	+	0.989452i	$0.453726\pi$
$60$	−21562.4	+	1222.75i	−0.773250	+	0.0438489i
$61$			5617.58i			0.193297i	0.995319	+	0.0966484i	$0.0308122\pi$
$61$			5617.58i			0.193297i	−0.995319	+	0.0966484i	$0.969188\pi$
$62$	−10405.0			−0.343765
$63$		0			0
$64$	−23140.6			−0.706195
$65$			43798.5i			1.28581i
$66$	−2240.79	+	127.069i	−0.0633199	+	0.00359070i
$67$	−12887.4			−0.350734			−0.175367	−	0.984503i	$-0.556111\pi$
$67$	−12887.4			−0.350734			−0.175367	+	0.984503i	$0.556111\pi$
$68$	23068.0			0.604975
$69$	−408.849	−	7209.82i	−0.0103381	−	0.182306i
$70$		0			0
$71$			60637.7i			1.42757i	0.700366	+	0.713784i	$0.253020\pi$
$71$			60637.7i			1.42757i	−0.700366	+	0.713784i	$0.746980\pi$
$72$	−5084.79	−	44689.5i	−0.115596	−	1.01595i
$73$		−	51641.4i		−	1.13420i	−0.823648	−	0.567101i	$-0.808065\pi$
$73$		−	51641.4i		−	1.13420i	0.823648	−	0.567101i	$-0.191935\pi$
$74$		−	28681.6i		−	0.608869i
$75$	−37338.5	+	2117.36i	−0.766484	+	0.0434652i
$76$			51593.3i			1.02461i
$77$		0			0
$78$	−33521.7	+	1900.93i	−0.623864	+	0.0353776i
$79$	67372.2			1.21454			0.607272	−	0.794494i	$-0.292264\pi$
$79$	67372.2			1.21454			0.607272	+	0.794494i	$0.292264\pi$
$80$	−5947.90			−0.103906
$81$	57539.6	−	13265.5i	0.974439	−	0.224652i
$82$		−	44454.5i		−	0.730097i
$83$	−8712.06			−0.138812			−0.0694058	−	0.997589i	$-0.522110\pi$
$83$	−8712.06			−0.138812			−0.0694058	+	0.997589i	$0.522110\pi$
$84$		0			0
$85$	91977.1			1.38081
$86$		−	19555.1i		−	0.285112i
$87$	4672.78	+	82401.9i	0.0661877	+	1.16718i
$88$	7291.11			0.100366
$89$	−18112.0			−0.242378			−0.121189	−	0.992629i	$-0.538671\pi$
$89$	−18112.0			−0.242378			−0.121189	+	0.992629i	$0.538671\pi$
$90$	−7462.85	−	65589.9i	−0.0971177	−	0.853554i
$91$		0			0
$92$			8635.34i			0.106368i
$93$	2512.42	+	44305.1i	0.0301221	+	0.531185i
$94$		−	24693.4i		−	0.288245i
$95$			205714.i			2.33859i
$96$	4969.26	+	87630.1i	0.0550318	+	0.970454i
$97$		−	18641.1i		−	0.201160i	−0.994929	−	0.100580i	$-0.967930\pi$
$97$		−	18641.1i		−	0.201160i	0.994929	−	0.100580i	$-0.0320698\pi$
$98$		0			0
$99$	1082.14	+	9510.75i	0.0110967	+	0.0975274i
$100$	44721.0			0.447210
$101$	−170598.			−1.66407			−0.832034	−	0.554725i	$-0.812824\pi$
$101$	−170598.			−1.66407			−0.832034	+	0.554725i	$0.812824\pi$
$102$	3991.96	+	70395.9i	0.0379914	+	0.669957i
$103$		−	58887.8i		−	0.546931i	−0.961882	−	0.273465i	$-0.911830\pi$
$103$		−	58887.8i		−	0.546931i	0.961882	−	0.273465i	$-0.0881699\pi$
$104$	109074.			0.988864
$105$		0			0
$106$	−58523.7			−0.505903
$107$		−	147582.i		−	1.24616i	−0.782157	−	0.623082i	$-0.785880\pi$
$107$		−	147582.i		−	1.24616i	0.782157	−	0.623082i	$-0.214120\pi$
$108$	−69593.5	+	11941.9i	−0.574129	+	0.0985175i
$109$	26156.0			0.210865			0.105433	−	0.994426i	$-0.466377\pi$
$109$	26156.0			0.210865			0.105433	+	0.994426i	$0.466377\pi$
$110$	10701.0			0.0843226
$111$	−122128.	+	6925.56i	−0.940824	+	0.0533516i
$112$		0			0
$113$			19094.4i			0.140672i	0.997523	+	0.0703362i	$0.0224072\pi$
$113$			19094.4i			0.140672i	−0.997523	+	0.0703362i	$0.977593\pi$
$114$	−157446.	+	8928.33i	−1.13467	+	0.0643440i
$115$			34431.0i			0.242776i
$116$		−	98694.2i		−	0.681000i
$117$	16188.6	+	142279.i	0.109331	+	0.960896i
$118$		−	28314.3i		−	0.187198i
$119$		0			0
$120$	12141.4	+	214106.i	0.0769688	+	1.35730i
$121$	159499.			0.990365
$122$	20532.5			0.124894
$123$	−189290.	+	10734.1i	−1.12815	+	0.0639741i
$124$		−	53065.0i		−	0.309923i
$125$	−53951.2			−0.308835
$126$		0			0
$127$	−42178.1			−0.232048			−0.116024	−	0.993246i	$-0.537015\pi$
$127$	−42178.1			−0.232048			−0.116024	+	0.993246i	$0.537015\pi$
$128$		−	95596.2i		−	0.515722i
$129$	−83267.2	+	4721.86i	−0.440555	+	0.0249827i
$130$	160085.			0.830794
$131$	−283072.			−1.44118			−0.720590	−	0.693361i	$-0.756129\pi$
$131$	−283072.			−1.44118			−0.720590	+	0.693361i	$0.756129\pi$
$132$	−648.048	−	11427.9i	−0.00323722	−	0.0570865i
$133$		0			0
$134$			47104.0i			0.226619i
$135$	−277485.	+	47615.0i	−1.31040	+	0.224858i
$136$		−	229055.i		−	1.06192i
$137$			263686.i			1.20029i	0.799891	+	0.600145i	$0.204890\pi$
$137$			263686.i			1.20029i	−0.799891	+	0.600145i	$0.795110\pi$
$138$	−26352.2	+	1494.36i	−0.117793	+	0.00667972i
$139$			363744.i			1.59683i	0.602108	+	0.798415i	$0.294328\pi$
$139$			363744.i			1.59683i	−0.602108	+	0.798415i	$0.705672\pi$
$140$		0			0
$141$	−105147.	+	5962.57i	−0.445397	+	0.0252572i
$142$	221634.			0.922390
$143$	−23212.9			−0.0949269
$144$	−19321.7	+	2198.43i	−0.0776497	+	0.00883501i
$145$		−	393516.i		−	1.55433i
$146$	−188752.			−0.732839
$147$		0			0
$148$	146275.			0.548929
$149$			385877.i			1.42391i	0.702224	+	0.711957i	$0.252191\pi$
$149$			385877.i			1.42391i	−0.702224	+	0.711957i	$0.747809\pi$
$150$	7739.06	+	136474.i	0.0280841	+	0.495246i
$151$	−276533.			−0.986970			−0.493485	−	0.869754i	$-0.664277\pi$
$151$	−276533.			−0.986970			−0.493485	+	0.869754i	$0.664277\pi$
$152$	512300.			1.79852
$153$	298787.	−	33996.1i	1.03189	−	0.117409i
$154$		0			0
$155$		−	211582.i		−	0.707375i
$156$	−9694.67	−	170960.i	−0.0318949	−	0.562449i
$157$		−	378991.i		−	1.22710i	−0.789656	−	0.613550i	$-0.789741\pi$
$157$		−	378991.i		−	1.22710i	0.789656	−	0.613550i	$-0.210259\pi$
$158$		−	246248.i		−	0.784749i
$159$	14131.3	+	249198.i	0.0443293	+	0.781722i
$160$		−	418484.i		−	1.29235i
$161$		0			0
$162$	−48486.0	−	210310.i	−0.145154	−	0.629611i
$163$	−231513.			−0.682505			−0.341252	−	0.939972i	$-0.610851\pi$
$163$	−231513.			−0.682505			−0.341252	+	0.939972i	$0.610851\pi$
$164$	226717.			0.658224
$165$	−2583.91	−	45565.8i	−0.00738869	−	0.130295i
$166$			31843.0i			0.0896899i
$167$	−173708.			−0.481981			−0.240990	−	0.970527i	$-0.577472\pi$
$167$	−173708.			−0.481981			−0.240990	+	0.970527i	$0.577472\pi$
$168$		0			0
$169$	24032.3			0.0647260
$170$		−	336181.i		−	0.892176i
$171$	76034.9	+	668260.i	0.198849	+	1.74765i
$172$	99730.7			0.257044
$173$	−490510.			−1.24604			−0.623021	−	0.782205i	$-0.714095\pi$
$173$	−490510.			−1.24604			−0.623021	+	0.782205i	$0.714095\pi$
$174$	301183.	−	17079.2i	0.754148	−	0.0427657i
$175$		0			0
$176$		−	3152.35i		−	0.00767101i
$177$	−120564.	+	6836.88i	−0.289259	+	0.0164031i
$178$			66200.4i			0.156607i
$179$		−	641321.i		−	1.49604i	−0.663677	−	0.748019i	$-0.731005\pi$
$179$		−	641321.i		−	1.49604i	0.663677	−	0.748019i	$-0.268995\pi$
$180$	334507.	−	38060.3i	0.769527	−	0.0875571i
$181$			46939.3i			0.106498i	0.998581	+	0.0532489i	$0.0169577\pi$
$181$			46939.3i			0.106498i	−0.998581	+	0.0532489i	$0.983042\pi$
$182$		0			0
$183$	−4957.85	−	87428.9i	−0.0109437	−	0.192987i
$184$	85745.4			0.186709
$185$	583232.			1.25289
$186$	161937.	−	9183.01i	0.343213	−	0.0194627i
$187$			48747.3i			0.101940i
$188$	125936.			0.259869
$189$		0			0
$190$	751894.			1.51103
$191$			761400.i			1.51018i	0.655620	+	0.755091i	$0.272407\pi$
$191$			761400.i			1.51018i	−0.655620	+	0.755091i	$0.727593\pi$
$192$	360148.	−	20423.0i	0.705062	−	0.0399821i
$193$	556313.			1.07504			0.537522	−	0.843250i	$-0.319361\pi$
$193$	556313.			1.07504			0.537522	+	0.843250i	$0.319361\pi$
$194$	−68133.9			−0.129975
$195$	−38654.8	−	681656.i	−0.0727976	−	1.28374i
$196$		0			0
$197$		−	545896.i		−	1.00218i	−0.865396	−	0.501088i	$-0.832933\pi$
$197$		−	545896.i		−	1.00218i	0.865396	−	0.501088i	$-0.167067\pi$
$198$	34762.2	−	3955.26i	0.0630151	−	0.00716988i
$199$			211201.i			0.378062i	0.981971	+	0.189031i	$0.0605346\pi$
$199$			211201.i			0.378062i	−0.981971	+	0.189031i	$0.939465\pi$
$200$		−	444061.i		−	0.784996i
$201$	200572.	−	11373.9i	0.350171	−	0.0198573i
$202$			623544.i			1.07520i
$203$		0			0
$204$	−359017.	+	20358.9i	−0.604004	+	0.0342514i
$205$	903969.			1.50234
$206$	−215238.			−0.353387
$207$	12726.2	+	111849.i	0.0206430	+	0.181429i
$208$		−	47158.6i		−	0.0755792i
$209$	−109027.			−0.172651
$210$		0			0
$211$	−670640.			−1.03701			−0.518506	−	0.855074i	$-0.673512\pi$
$211$	−670640.			−1.03701			−0.518506	+	0.855074i	$0.673512\pi$
$212$		−	298469.i		−	0.456100i
$213$	−53516.5	−	943732.i	−0.0808236	−	1.42528i
$214$	−539420.			−0.805180
$215$	397649.			0.586683
$216$	118578.	+	691035.i	0.172930	+	1.00778i
$217$		0			0
$218$		−	95601.5i		−	0.136246i
$219$	45576.7	+	803718.i	0.0642144	+	1.13238i
$220$			54575.0i			0.0760216i
$221$			729250.i			1.00437i
$222$	25313.2	+	446384.i	0.0344719	+	0.607892i
$223$		−	18729.0i		−	0.0252204i	−0.999920	−	0.0126102i	$-0.995986\pi$
$223$		−	18729.0i		−	0.0252204i	0.999920	−	0.0126102i	$-0.00401406\pi$
$224$		0			0
$225$	579247.	−	65906.9i	0.762794	−	0.0867910i
$226$	69790.8			0.0908923
$227$	447047.			0.575822			0.287911	−	0.957657i	$-0.407039\pi$
$227$	447047.			0.575822			0.287911	+	0.957657i	$0.407039\pi$
$228$	−45534.2	−	802970.i	−0.0580097	−	1.02297i
$229$		−	22000.4i		−	0.0277231i	−0.999904	−	0.0138616i	$-0.995588\pi$
$229$		−	22000.4i		−	0.0277231i	0.999904	−	0.0138616i	$-0.00441241\pi$
$230$	125847.			0.156864
$231$		0			0
$232$	−979993.			−1.19537
$233$		−	533090.i		−	0.643296i	−0.946859	−	0.321648i	$-0.895763\pi$
$233$		−	533090.i		−	0.643296i	0.946859	−	0.321648i	$-0.104237\pi$
$234$	520036.	−	59170.0i	0.620861	−	0.0706418i
$235$	502135.			0.593131
$236$	144402.			0.168770
$237$	−1.04854e6	+	59460.1i	−1.21259	+	0.0687629i
$238$		0			0
$239$		−	269068.i		−	0.304697i	−0.988327	−	0.152348i	$-0.951316\pi$
$239$		−	269068.i		−	0.304697i	0.988327	−	0.152348i	$-0.0486836\pi$
$240$	92569.9	−	5249.39i	0.103739	−	0.00588275i
$241$			266824.i			0.295926i	0.988993	+	0.147963i	$0.0472716\pi$
$241$			266824.i			0.295926i	−0.988993	+	0.147963i	$0.952728\pi$
$242$		−	582977.i		−	0.639902i
$243$	−883808.	+	257239.i	−0.960157	+	0.279461i
$244$			104715.i			0.112599i
$245$		0			0
$246$	39233.8	+	691865.i	0.0413354	+	0.728926i
$247$	−1.63102e6			−1.70105
$248$	−526914.			−0.544014
$249$	135590.	−	7688.92i	0.138589	−	0.00785899i
$250$			197194.i			0.199546i
$251$	947580.			0.949361			0.474681	−	0.880158i	$-0.342564\pi$
$251$	947580.			0.949361			0.474681	+	0.880158i	$0.342564\pi$
$252$		0			0
$253$	−18248.2			−0.0179233
$254$			154163.i			0.149932i
$255$	−1.43148e6	+	81175.4i	−1.37859	+	0.0781761i
$256$	−1.08991e6			−1.03942
$257$	−526317.			−0.497066			−0.248533	−	0.968623i	$-0.579948\pi$
$257$	−526317.			−0.497066			−0.248533	+	0.968623i	$0.579948\pi$
$258$	17258.6	+	304346.i	0.0161420	+	0.284654i
$259$		0			0
$260$			816432.i			0.749008i
$261$	−145449.	−	1.27833e6i	−0.132163	−	1.16156i
$262$			1.03464e6i			0.931186i
$263$			1.36537e6i			1.21720i	0.793477	+	0.608601i	$0.208269\pi$
$263$			1.36537e6i			1.21720i	−0.793477	+	0.608601i	$0.791731\pi$
$264$	−113475.	+	6434.85i	−0.100205	+	0.00568236i
$265$		−	1.19006e6i		−	1.04101i
$266$		0			0
$267$	281886.	−	15985.0i	0.241989	−	0.0137225i
$268$	−240229.			−0.204310
$269$	−29229.0			−0.0246283			−0.0123141	−	0.999924i	$-0.503920\pi$
$269$	−29229.0			−0.0246283			−0.0123141	+	0.999924i	$0.503920\pi$
$270$	174035.	+	1.01422e6i	0.145287	+	0.846686i
$271$			423724.i			0.350477i	0.984526	+	0.175239i	$0.0560697\pi$
$271$			423724.i			0.350477i	−0.984526	+	0.175239i	$0.943930\pi$
$272$	−99033.3			−0.0811632
$273$		0			0
$274$	963785.			0.775540
$275$			94504.4i			0.0753564i
$276$	−7621.21	−	134396.i	−0.00602215	−	0.106197i
$277$	−746981.			−0.584938			−0.292469	−	0.956275i	$-0.594477\pi$
$277$	−746981.			−0.584938			−0.292469	+	0.956275i	$0.594477\pi$
$278$	1.32950e6			1.03175
$279$	−78203.9	−	687323.i	−0.0601475	−	0.528628i
$280$		0			0
$281$			841921.i			0.636071i	0.948079	+	0.318035i	$0.103023\pi$
$281$			841921.i			0.636071i	−0.948079	+	0.318035i	$0.896977\pi$
$282$	21793.5	+	384315.i	0.0163194	+	0.287783i
$283$		−	1.39274e6i		−	1.03373i	−0.856068	−	0.516863i	$-0.827100\pi$
$283$		−	1.39274e6i		−	1.03373i	0.856068	−	0.516863i	$-0.172900\pi$
$284$			1.13033e6i			0.831587i
$285$	−181555.	−	3.20162e6i	−0.132403	−	2.33484i
$286$			84844.2i			0.0613348i
$287$		0			0
$288$	−154678.	−	1.35944e6i	−0.109887	−	0.965782i
$289$	111574.			0.0785809
$290$	−1.43832e6			−1.00429
$291$	16451.9	+	290119.i	0.0113889	+	0.200837i
$292$		−	962628.i		−	0.660696i
$293$	537097.			0.365497			0.182748	−	0.983160i	$-0.441501\pi$
$293$	537097.			0.365497			0.182748	+	0.983160i	$0.441501\pi$
$294$		0			0
$295$	575764.			0.385203
$296$		−	1.45245e6i		−	0.963547i
$297$	−25235.6	−	147065.i	−0.0166006	−	0.0967427i
$298$	1.41040e6			0.920029
$299$	−272990.			−0.176591
$300$	−696013.	+	39469.0i	−0.446492	+	0.0253194i
$301$		0			0
$302$			1.01074e6i			0.637708i
$303$	2.65510e6	−	150563.i	1.66140	−	0.0942134i
$304$		−	221496.i		−	0.137462i
$305$			417523.i			0.256999i
$306$	−124257.	−	1.09208e6i	−0.0758610	−	0.666732i
$307$		−	511397.i		−	0.309679i	−0.987940	−	0.154840i	$-0.950514\pi$
$307$		−	511397.i		−	0.309679i	0.987940	−	0.154840i	$-0.0494861\pi$
$308$		0			0
$309$	51972.1	+	916497.i	0.0309652	+	0.546053i
$310$	−773342.			−0.457054
$311$	−2.41299e6			−1.41467			−0.707335	−	0.706878i	$-0.750103\pi$
$311$	−2.41299e6			−1.41467			−0.707335	+	0.706878i	$0.750103\pi$
$312$	−1.69756e6	+	96264.1i	−0.987278	+	0.0559858i
$313$			2.72163e6i			1.57025i	0.619339	+	0.785124i	$0.287401\pi$
$313$			2.72163e6i			1.57025i	−0.619339	+	0.785124i	$0.712599\pi$
$314$	−1.38523e6			−0.792862
$315$		0			0
$316$	1.25586e6			0.707495
$317$			1.65080e6i			0.922672i	0.887226	+	0.461336i	$0.152630\pi$
$317$			1.65080e6i			0.922672i	−0.887226	+	0.461336i	$0.847370\pi$
$318$	910831.	−	51650.7i	0.505091	−	0.0286424i
$319$	208561.			0.114751
$320$	−1.71991e6			−0.938925
$321$	130250.	+	2.29689e6i	0.0705532	+	1.24416i
$322$		0			0
$323$			3.42516e6i			1.82673i
$324$	1.07258e6	−	247277.i	0.567630	−	0.130865i
$325$			1.41377e6i			0.742455i
$326$			846190.i			0.440985i
$327$	−407078.	+	23084.3i	−0.210527	+	0.0119384i
$328$		−	2.25120e6i		−	1.15539i
$329$		0			0
$330$	−166545.	+	9444.31i	−0.0841873	+	0.00477403i
$331$	−2.62569e6			−1.31727			−0.658633	−	0.752464i	$-0.728865\pi$
$331$	−2.62569e6			−1.31727			−0.658633	+	0.752464i	$0.728865\pi$
$332$	−162398.			−0.0808605
$333$	1.89462e6	−	215571.i	0.936295	−	0.106532i
$334$			634912.i			0.311421i
$335$	−957847.			−0.466320
$336$		0			0
$337$	−3.01254e6			−1.44497			−0.722483	−	0.691388i	$-0.756999\pi$
$337$	−3.01254e6			−1.44497			−0.722483	+	0.691388i	$0.756999\pi$
$338$		−	87839.2i		−	0.0418212i
$339$	−16851.9	−	297174.i	−0.00796436	−	0.140447i
$340$	1.71451e6			0.804347
$341$	112137.			0.0522232
$342$	2.44252e6	−	277911.i	1.12921	−	0.128482i
$343$		0			0
$344$		−	990285.i		−	0.451195i
$345$	−30387.5	−	535866.i	−0.0137451	−	0.242386i
$346$			1.79284e6i			0.805101i
$347$		−	2.90956e6i		−	1.29719i	−0.761134	−	0.648595i	$-0.775357\pi$
$347$		−	2.90956e6i		−	1.29719i	0.761134	−	0.648595i	$-0.224643\pi$
$348$	87103.7	+	1.53602e6i	0.0385557	+	0.679907i
$349$			3.93286e6i			1.72840i	0.503147	+	0.864201i	$0.332176\pi$
$349$			3.93286e6i			1.72840i	−0.503147	+	0.864201i	$0.667824\pi$
$350$		0			0
$351$	−377520.	−	2.20007e6i	−0.163558	−	0.953165i
$352$	221793.			0.0954096
$353$	1.93598e6			0.826922			0.413461	−	0.910522i	$-0.364320\pi$
$353$	1.93598e6			0.826922			0.413461	+	0.910522i	$0.364320\pi$
$354$	24989.1	+	440669.i	0.0105985	+	0.186898i
$355$			4.50686e6i			1.89803i
$356$	−337620.			−0.141190
$357$		0			0
$358$	−2.34406e6			−0.966631
$359$		−	1.39251e6i		−	0.570247i	−0.958491	−	0.285123i	$-0.907965\pi$
$359$		−	1.39251e6i		−	0.570247i	0.958491	−	0.285123i	$-0.0920346\pi$
$360$	−377924.	−	3.32152e6i	−0.153691	−	1.35077i
$361$	−5.18454e6			−2.09383
$362$	171565.			0.0688111
$363$	−2.48236e6	+	140768.i	−0.988777	+	0.0560708i
$364$		0			0
$365$		−	3.83821e6i		−	1.50798i
$366$	−319557.	+	18121.2i	−0.124694	+	0.00707105i
$367$			985240.i			0.381836i	0.981606	+	0.190918i	$0.0611465\pi$
$367$			985240.i			0.381836i	−0.981606	+	0.190918i	$0.938854\pi$
$368$		−	37072.5i		−	0.0142703i
$369$	2.93654e6	−	334120.i	1.12272	−	0.127743i
$370$		−	2.13174e6i		−	0.809524i
$371$		0			0
$372$	46833.1	+	825875.i	0.0175467	+	0.309426i
$373$	2.51585e6			0.936296			0.468148	−	0.883650i	$-0.344921\pi$
$373$	2.51585e6			0.936296			0.468148	+	0.883650i	$0.344921\pi$
$374$	178173.			0.0658664
$375$	839667.	−	47615.2i	0.308339	−	0.0174851i
$376$		−	1.25049e6i		−	0.456154i
$377$	3.12003e6			1.13059
$378$		0			0
$379$	−627888.			−0.224535			−0.112268	−	0.993678i	$-0.535811\pi$
$379$	−627888.			−0.224535			−0.112268	+	0.993678i	$0.535811\pi$
$380$			3.83464e6i			1.36228i
$381$	656437.	−	37224.8i	0.231676	−	0.0131377i
$382$	2.78295e6			0.975770
$383$	3.10523e6			1.08167			0.540837	−	0.841127i	$-0.318107\pi$
$383$	3.10523e6			1.08167			0.540837	+	0.841127i	$0.318107\pi$
$384$	84369.5	+	1.48781e6i	0.0291983	+	0.514895i
$385$		0			0
$386$		−	2.03335e6i		−	0.694615i
$387$	1.29176e6	−	146977.i	0.438434	−	0.0498852i
$388$		−	347481.i		−	0.117180i
$389$			669791.i			0.224422i	0.993684	+	0.112211i	$0.0357932\pi$
$389$			669791.i			0.224422i	−0.993684	+	0.112211i	$0.964207\pi$
$390$	−2.49148e6	+	141285.i	−0.829462	+	0.0470365i
$391$			573281.i			0.189638i
$392$		0			0
$393$	4.40558e6	−	249828.i	1.43887	−	0.0815943i
$394$	−1.99527e6			−0.647533
$395$	5.00740e6			1.61480
$396$	20171.7	+	177286.i	0.00646405	+	0.0568117i
$397$		−	3.49472e6i		−	1.11285i	−0.830898	−	0.556425i	$-0.812173\pi$
$397$		−	3.49472e6i		−	1.11285i	0.830898	−	0.556425i	$-0.187827\pi$
$398$	771949.			0.244276
$399$		0			0
$400$	−191992.			−0.0599975
$401$		−	2.81146e6i		−	0.873115i	−0.899676	−	0.436558i	$-0.856198\pi$
$401$		−	2.81146e6i		−	0.873115i	0.899676	−	0.436558i	$-0.143802\pi$
$402$	−41572.1	−	733101.i	−0.0128303	−	0.226255i
$403$	1.67755e6			0.514532
$404$	−3.18006e6			−0.969353
$405$	4.27660e6	−	985950.i	1.29557	−	0.298688i
$406$		0			0
$407$			309109.i			0.0924965i
$408$	202155.	+	3.56489e6i	0.0601222	+	1.06022i
$409$			2.79035e6i			0.824803i	0.911002	+	0.412402i	$0.135310\pi$
$409$			2.79035e6i			0.824803i	−0.911002	+	0.412402i	$0.864690\pi$
$410$		−	3.30405e6i		−	0.970704i
$411$	−232719.	−	4.10387e6i	−0.0679560	−	1.19836i
$412$		−	1.09771e6i		−	0.318598i
$413$		0			0
$414$	408813.	−	46514.9i	0.117226	−	0.0133380i
$415$	−647518.			−0.184557
$416$	3.31799e6			0.940030
$417$	−321026.	−	5.66111e6i	−0.0904066	−	1.59427i
$418$			398499.i			0.111554i
$419$	−3.40729e6			−0.948144			−0.474072	−	0.880486i	$-0.657216\pi$
$419$	−3.40729e6			−0.948144			−0.474072	+	0.880486i	$0.657216\pi$
$420$		0			0
$421$	3.77099e6			1.03693			0.518465	−	0.855099i	$-0.326504\pi$
$421$	3.77099e6			1.03693			0.518465	+	0.855099i	$0.326504\pi$
$422$			2.45122e6i			0.670041i
$423$	1.63118e6	−	185596.i	0.443252	−	0.0504334i
$424$	−2.96368e6			−0.800601
$425$	2.96892e6			0.797309
$426$	−3.44938e6	+	195605.i	−0.920911	+	0.0522223i
$427$		0			0
$428$		−	2.75103e6i		−	0.725915i
$429$	361273.	−	20486.8i	0.0947746	−	0.00537441i
$430$		−	1.45342e6i		−	0.379071i
$431$		−	3.00701e6i		−	0.779726i	−0.920873	−	0.389863i	$-0.872522\pi$
$431$		−	3.00701e6i		−	0.779726i	0.920873	−	0.389863i	$-0.127478\pi$
$432$	298773.	−	51267.8i	0.0770249	−	0.0132171i
$433$		−	1.50880e6i		−	0.386734i	−0.981126	−	0.193367i	$-0.938059\pi$
$433$		−	1.50880e6i		−	0.386734i	0.981126	−	0.193367i	$-0.0619408\pi$
$434$		0			0
$435$	347302.	+	6.12447e6i	0.0880002	+	1.55183i
$436$	487565.			0.122833
$437$	−1.28219e6			−0.321179
$438$	2.93763e6	−	166585.i	0.731664	−	0.0414906i
$439$		−	1.97163e6i		−	0.488275i	−0.969741	−	0.244138i	$-0.921495\pi$
$439$		−	1.97163e6i		−	0.488275i	0.969741	−	0.244138i	$-0.0785049\pi$
$440$	541907.			0.133442
$441$		0			0
$442$	2.66544e6			0.648954
$443$		−	2.13681e6i		−	0.517316i	−0.965969	−	0.258658i	$-0.916720\pi$
$443$		−	2.13681e6i		−	0.517316i	0.965969	−	0.258658i	$-0.0832803\pi$
$444$	−2.27655e6	+	129097.i	−0.548049	+	0.0310784i
$445$	−1.34617e6			−0.322254
$446$	−68455.3			−0.0162956
$447$	−340560.	−	6.00558e6i	−0.0806167	−	1.42163i
$448$		0			0
$449$		−	5.91347e6i		−	1.38429i	−0.721759	−	0.692144i	$-0.756666\pi$
$449$		−	5.91347e6i		−	1.38429i	0.721759	−	0.692144i	$-0.243334\pi$
$450$	−240893.	−	2.11717e6i	−0.0560780	−	0.492862i
$451$			479098.i			0.110913i
$452$			355931.i			0.0819445i
$453$	4.30380e6	−	244057.i	0.985387	−	0.0558786i
$454$		−	1.63398e6i		−	0.372054i
$455$		0			0
$456$	−7.97316e6	+	452136.i	−1.79564	+	0.101826i
$457$	−7.51891e6			−1.68409			−0.842044	−	0.539410i	$-0.818647\pi$
$457$	−7.51891e6			−1.68409			−0.842044	+	0.539410i	$0.818647\pi$
$458$	−80412.5			−0.0179126
$459$	−4.62016e6	+	792795.i	−1.02359	+	0.175642i
$460$			641816.i			0.141422i
$461$	929187.			0.203634			0.101817	−	0.994803i	$-0.467534\pi$
$461$	929187.			0.203634			0.101817	+	0.994803i	$0.467534\pi$
$462$		0			0
$463$	1.21496e6			0.263396			0.131698	−	0.991290i	$-0.457957\pi$
$463$	1.21496e6			0.263396			0.131698	+	0.991290i	$0.457957\pi$
$464$			423705.i			0.0913627i
$465$	186734.	+	3.29295e6i	0.0400489	+	0.706240i
$466$	−1.94847e6			−0.415651
$467$	−3.03117e6			−0.643159			−0.321580	−	0.946883i	$-0.604214\pi$
$467$	−3.03117e6			−0.643159			−0.321580	+	0.946883i	$0.604214\pi$
$468$	301765.	+	2.65217e6i	0.0636876	+	0.559741i
$469$		0			0
$470$		−	1.83532e6i		−	0.383238i
$471$	334483.	+	5.89841e6i	0.0694739	+	1.22513i
$472$		−	1.43386e6i		−	0.296245i
$473$			210751.i			0.0433129i
$474$	217329.	+	3.83248e6i	0.0444296	+	0.783490i
$475$			6.64023e6i			1.35036i
$476$		0			0
$477$	−439865.	−	3.86591e6i	−0.0885164	−	0.777958i
$478$	−983457.			−0.196873
$479$	−81905.2			−0.0163107			−0.00815535	−	0.999967i	$-0.502596\pi$
$479$	−81905.2			−0.0163107			−0.00815535	+	0.999967i	$0.502596\pi$
$480$	369337.	+	6.51305e6i	0.0731678	+	1.29027i
$481$			4.62421e6i			0.911329i
$482$	975255.			0.191206
$483$		0			0
$484$	2.97317e6			0.576907
$485$		−	1.38549e6i		−	0.267453i
$486$	940222.	+	3.23036e6i	0.180567	+	0.620383i
$487$	4.41516e6			0.843575			0.421787	−	0.906695i	$-0.361403\pi$
$487$	4.41516e6			0.843575			0.421787	+	0.906695i	$0.361403\pi$
$488$	1.03978e6			0.197648
$489$	3.60314e6	−	204324.i	0.681410	−	0.0386409i
$490$		0			0
$491$			9.60786e6i			1.79855i	0.437381	+	0.899276i	$0.355906\pi$
$491$			9.60786e6i			1.79855i	−0.437381	+	0.899276i	$0.644094\pi$
$492$	−3.52849e6	+	200091.i	−0.657168	+	0.0372662i
$493$		−	6.55209e6i		−	1.21412i
$494$			5.96147e6i			1.09910i
$495$	80429.1	+	706880.i	0.0147537	+	0.129668i
$496$			227814.i			0.0415792i
$497$		0			0
$498$	−28103.4	−	495587.i	−0.00507791	−	0.0895460i
$499$	1.00695e7			1.81032			0.905160	−	0.425071i	$-0.139751\pi$
$499$	1.00695e7			1.81032			0.905160	+	0.425071i	$0.139751\pi$
$500$	−1.00568e6			−0.179902
$501$	2.70350e6	−	153308.i	0.481208	−	0.0272880i
$502$		−	3.46345e6i		−	0.613408i
$503$	−2.68345e6			−0.472904			−0.236452	−	0.971643i	$-0.575985\pi$
$503$	−2.68345e6			−0.472904			−0.236452	+	0.971643i	$0.575985\pi$
$504$		0			0
$505$	−1.26796e7			−2.21247
$506$			66698.0i			0.0115808i
$507$	−374026.	+	21210.0i	−0.0646222	+	0.00366455i
$508$	−786227.			−0.135173
$509$	−7.51287e6			−1.28532			−0.642661	−	0.766151i	$-0.722169\pi$
$509$	−7.51287e6			−1.28532			−0.642661	+	0.766151i	$0.722169\pi$
$510$	296700.	+	5.23213e6i	0.0505117	+	0.890745i
$511$		0			0
$512$			924585.i			0.155873i
$513$	−1.77315e6	−	1.03333e7i	−0.297475	−	1.73359i
$514$			1.92371e6i			0.321168i
$515$		−	4.37680e6i		−	0.727174i
$516$	−1.55215e6	+	88018.4i	−0.256632	+	0.0145529i
$517$			266128.i			0.0437889i
$518$		0			0
$519$	7.63403e6	−	432905.i	1.24404	−	0.0705463i
$520$	8.10683e6			1.31475
$521$	−9.46018e6			−1.52688			−0.763440	−	0.645878i	$-0.776491\pi$
$521$	−9.46018e6			−1.52688			−0.763440	+	0.645878i	$0.776491\pi$
$522$	−4.67237e6	+	531624.i	−0.750517	+	0.0853942i
$523$		−	1.32587e6i		−	0.211957i	−0.994368	−	0.105979i	$-0.966202\pi$
$523$		−	1.32587e6i		−	0.211957i	0.994368	−	0.105979i	$-0.0337975\pi$
$524$	−5.27664e6			−0.839517
$525$		0			0
$526$	4.99051e6			0.786466
$527$		−	3.52287e6i		−	0.552548i
$528$	2782.14	+	49061.4i	0.000434304	+	0.00765871i
$529$	6.22174e6			0.966657
$530$	−4.34974e6			−0.672626
$531$	1.87037e6	−	212811.i	0.287866	−	0.0327535i
$532$		0			0
$533$			7.16721e6i			1.09278i
$534$	−58425.9	−	1.03031e6i	−0.00886650	−	0.156356i
$535$		−	1.09690e7i		−	1.65684i
$536$			2.38538e6i			0.358629i
$537$	566005.	+	9.98116e6i	0.0847002	+	1.49364i
$538$			106833.i			0.0159130i
$539$		0			0
$540$	−5.17249e6	+	887573.i	−0.763335	+	0.130984i
$541$	6.92295e6			1.01695			0.508473	−	0.861078i	$-0.330210\pi$
$541$	6.92295e6			1.01695			0.508473	+	0.861078i	$0.330210\pi$
$542$	1.54873e6			0.226453
$543$	−41426.8	−	730538.i	−0.00602951	−	0.106327i
$544$		−	6.96780e6i		−	1.00948i
$545$	1.94403e6			0.280357
$546$		0			0
$547$	−9.97756e6			−1.42579			−0.712896	−	0.701270i	$-0.752617\pi$
$547$	−9.97756e6			−1.42579			−0.712896	+	0.701270i	$0.752617\pi$
$548$			4.91528e6i			0.699193i
$549$	154323.	+	1.35632e6i	0.0218524	+	0.192058i
$550$	345418.			0.0486898
$551$	1.46542e7			2.05629
$552$	−1.33449e6	+	75675.5i	−0.186410	+	0.0105708i
$553$		0			0
$554$			2.73025e6i			0.377944i
$555$	−9.07710e6	+	514738.i	−1.25088	+	0.0709338i
$556$			6.78041e6i			0.930185i
$557$			1.25386e7i			1.71242i	0.516628	+	0.856210i	$0.327187\pi$
$557$			1.25386e7i			1.71242i	−0.516628	+	0.856210i	$0.672813\pi$
$558$	−2.51220e6	+	285839.i	−0.341561	+	0.0388629i
$559$			3.15280e6i			0.426743i
$560$		0			0
$561$	−43022.4	−	758676.i	−0.00577149	−	0.101777i
$562$	3.07726e6			0.410982
$563$	−6.87659e6			−0.914329			−0.457164	−	0.889382i	$-0.651135\pi$
$563$	−6.87659e6			−0.914329			−0.457164	+	0.889382i	$0.651135\pi$
$564$	−1.96000e6	+	111146.i	−0.259452	+	0.0147128i
$565$			1.41918e6i			0.187032i
$566$	−5.09055e6			−0.667918
$567$		0			0
$568$	1.12237e7			1.45970
$569$			2.64936e6i			0.343053i	0.985180	+	0.171526i	$0.0548699\pi$
$569$			2.64936e6i			0.343053i	−0.985180	+	0.171526i	$0.945130\pi$
$570$	−1.17021e7	+	663592.i	−1.50860	+	0.0855488i
$571$	1.09557e6			0.140621			0.0703105	−	0.997525i	$-0.477601\pi$
$571$	1.09557e6			0.140621			0.0703105	+	0.997525i	$0.477601\pi$
$572$	−432703.			−0.0552968
$573$	−671982.	−	1.18500e7i	−0.0855009	−	1.50776i
$574$		0			0
$575$			1.11140e6i			0.140184i
$576$	−5.58712e6	+	635704.i	−0.701668	+	0.0798360i
$577$			2.90428e6i			0.363161i	0.983376	+	0.181581i	$0.0581214\pi$
$577$			2.90428e6i			0.363161i	−0.983376	+	0.181581i	$0.941879\pi$
$578$		−	407807.i		−	0.0507733i
$579$	−8.65816e6	+	490980.i	−1.07332	+	0.0608650i
$580$		−	7.33539e6i		−	0.905426i
$581$		0			0
$582$	1.06040e6	−	60132.3i	0.129766	−	0.00735869i
$583$	630726.			0.0768545
$584$	−9.55850e6			−1.15973
$585$	1.20321e6	+	1.05748e7i	0.145362	+	1.27756i
$586$		−	1.96311e6i		−	0.236157i
$587$	−2.80932e6			−0.336516			−0.168258	−	0.985743i	$-0.553814\pi$
$587$	−2.80932e6			−0.336516			−0.168258	+	0.985743i	$0.553814\pi$
$588$		0			0
$589$	7.87916e6			0.935819
$590$		−	2.10444e6i		−	0.248890i
$591$	481786.	+	8.49602e6i	0.0567395	+	1.00057i
$592$	−627976.			−0.0736442
$593$	−471223.			−0.0550288			−0.0275144	−	0.999621i	$-0.508759\pi$
$593$	−471223.			−0.0550288			−0.0275144	+	0.999621i	$0.508759\pi$
$594$	−537529.	+	92237.3i	−0.0625081	+	0.0107261i
$595$		0			0
$596$			7.19300e6i			0.829458i
$597$	−186398.	−	3.28701e6i	−0.0214045	−	0.377455i
$598$			997791.i			0.114100i
$599$			1.73472e6i			0.197544i	0.995110	+	0.0987719i	$0.0314914\pi$
$599$			1.73472e6i			0.197544i	−0.995110	+	0.0987719i	$0.968509\pi$
$600$	391911.	+	6.91112e6i	0.0444436	+	0.783737i
$601$		−	3.42903e6i		−	0.387244i	−0.981076	−	0.193622i	$-0.937976\pi$
$601$		−	3.42903e6i		−	0.387244i	0.981076	−	0.193622i	$-0.0620235\pi$
$602$		0			0
$603$	−3.11156e6	+	354034.i	−0.348486	+	0.0396508i
$604$	−5.15475e6			−0.574930
$605$	1.18547e7			1.31674
$606$	−550316.	−	9.70451e6i	−0.0608738	−	1.07347i
$607$		−	9.81466e6i		−	1.08119i	−0.841282	−	0.540597i	$-0.818198\pi$
$607$		−	9.81466e6i		−	1.08119i	0.841282	−	0.540597i	$-0.181802\pi$
$608$	1.55840e7			1.70970
$609$		0			0
$610$	1.52607e6			0.166054
$611$			3.98122e6i			0.431433i
$612$	5.56958e6	−	633709.i	0.601096	−	0.0683930i
$613$	−2.75780e6			−0.296422			−0.148211	−	0.988956i	$-0.547352\pi$
$613$	−2.75780e6			−0.296422			−0.148211	+	0.988956i	$0.547352\pi$
$614$	−1.86918e6			−0.200092
$615$	−1.40689e7	+	797808.i	−1.49993	+	0.0850571i
$616$		0			0
$617$		−	5.46419e6i		−	0.577847i	−0.957352	−	0.288923i	$-0.906703\pi$
$617$		−	5.46419e6i		−	0.577847i	0.957352	−	0.288923i	$-0.0932973\pi$
$618$	3.34984e6	−	189960.i	0.352820	−	0.0200074i
$619$		−	4.24390e6i		−	0.445183i	−0.974912	−	0.222591i	$-0.928548\pi$
$619$		−	4.24390e6i		−	0.445183i	0.974912	−	0.222591i	$-0.0714516\pi$
$620$		−	3.94402e6i		−	0.412060i
$621$	−296777.	−	1.72952e6i	−0.0308817	−	0.179969i
$622$			8.81960e6i			0.914057i
$623$		0			0
$624$	41620.3	+	733950.i	0.00427901	+	0.0754580i
$625$	−1.15071e7			−1.17833
$626$	9.94768e6			1.01458
$627$	1.69684e6	−	96222.9i	0.172374	−	0.00977484i
$628$		−	7.06464e6i		−	0.714810i
$629$	9.71088e6			0.978660
$630$		0			0
$631$	1.97794e6			0.197761			0.0988804	−	0.995099i	$-0.468474\pi$
$631$	1.97794e6			0.197761			0.0988804	+	0.995099i	$0.468474\pi$
$632$		−	1.24702e7i		−	1.24188i
$633$	1.04375e7	−	591881.i	1.03535	−	0.0587118i
$634$	6.03376e6			0.596163
$635$	−3.13486e6			−0.308521
$636$	263417.	+	4.64521e6i	0.0258227	+	0.455368i
$637$		0			0
$638$		−	762299.i		−	0.0741436i
$639$	1.66580e6	+	1.46405e7i	0.161388	+	1.41842i
$640$		−	7.10513e6i		−	0.685681i
$641$		−	6.87312e6i		−	0.660707i	−0.943857	−	0.330353i	$-0.892832\pi$
$641$		−	6.87312e6i		−	0.660707i	0.943857	−	0.330353i	$-0.107168\pi$
$642$	8.39524e6	−	476071.i	0.803888	−	0.0455863i
$643$			4.81869e6i			0.459623i	0.973235	+	0.229811i	$0.0738109\pi$
$643$			4.81869e6i			0.459623i	−0.973235	+	0.229811i	$0.926189\pi$
$644$		0			0
$645$	−6.18878e6	+	350949.i	−0.585742	+	0.0332158i
$646$	1.25191e7			1.18030
$647$	−7.50407e6			−0.704752			−0.352376	−	0.935859i	$-0.614626\pi$
$647$	−7.50407e6			−0.704752			−0.352376	+	0.935859i	$0.614626\pi$
$648$	−2.45536e6	−	1.06502e7i	−0.229709	−	0.996372i
$649$			305151.i			0.0284383i
$650$	5.16739e6			0.479720
$651$		0			0
$652$	−4.31554e6			−0.397573
$653$		−	1.40288e7i		−	1.28747i	−0.765248	−	0.643735i	$-0.777384\pi$
$653$		−	1.40288e7i		−	1.28747i	0.765248	−	0.643735i	$-0.222616\pi$
$654$	84374.1	+	1.48789e6i	0.00771374	+	0.136027i
$655$	−2.10391e7			−1.91613
$656$	−973319.			−0.0883071
$657$	−1.41866e6	−	1.24684e7i	−0.128223	−	1.12693i
$658$		0			0
$659$		−	4.93455e6i		−	0.442623i	−0.975203	−	0.221311i	$-0.928966\pi$
$659$		−	4.93455e6i		−	0.442623i	0.975203	−	0.221311i	$-0.0710337\pi$
$660$	−48165.7	−	849375.i	−0.00430406	−	0.0758996i
$661$		−	1.83713e7i		−	1.63544i	−0.575615	−	0.817721i	$-0.695237\pi$
$661$		−	1.83713e7i		−	1.63544i	0.575615	−	0.817721i	$-0.304763\pi$
$662$			9.59702e6i			0.851121i
$663$	−643608.	−	1.13496e7i	−0.0568640	−	1.00276i
$664$			1.61255e6i			0.141936i
$665$		0			0
$666$	−787923.	−	6.92494e6i	−0.0688332	−	0.604965i
$667$	2.45273e6			0.213469
$668$	−3.23804e6			−0.280763
$669$	16529.5	+	291488.i	0.00142789	+	0.0251800i
$670$			3.50097e6i			0.301302i
$671$	−221284.			−0.0189734
$672$		0			0
$673$	5.27763e6			0.449160			0.224580	−	0.974456i	$-0.427899\pi$
$673$	5.27763e6			0.449160			0.224580	+	0.974456i	$0.427899\pi$
$674$			1.10110e7i			0.933632i
$675$	−8.95691e6	+	1.53696e6i	−0.756657	+	0.129838i
$676$	447978.			0.0377042
$677$	1.37062e7			1.14933			0.574666	−	0.818388i	$-0.305132\pi$
$677$	1.37062e7			1.14933			0.574666	+	0.818388i	$0.305132\pi$
$678$	−1.08619e6	+	61594.6i	−0.0907465	+	0.00514599i
$679$		0			0
$680$		−	1.70244e7i		−	1.41189i
$681$	−6.95759e6	+	394546.i	−0.574899	+	0.0326009i
$682$		−	409866.i		−	0.0337428i
$683$		−	1.56582e7i		−	1.28437i	−0.766549	−	0.642186i	$-0.778028\pi$
$683$		−	1.56582e7i		−	1.28437i	0.766549	−	0.642186i	$-0.221972\pi$
$684$	1.41734e6	+	1.24568e7i	0.115833	+	1.01804i
$685$			1.95983e7i			1.59585i
$686$		0			0
$687$	19416.7	+	342402.i	0.00156958	+	0.0276786i
$688$	−428155.			−0.0344850
$689$	9.43554e6			0.757214
$690$	−1.95861e6	+	111068.i	−0.156612	+	0.00888106i
$691$			4.22151e6i			0.336336i	0.985758	+	0.168168i	$0.0537850\pi$
$691$			4.22151e6i			0.336336i	−0.985758	+	0.168168i	$0.946215\pi$
$692$	−9.14342e6			−0.725844
$693$		0			0
$694$	−1.06346e7			−0.838150
$695$			2.70350e7i			2.12307i
$696$	1.52521e7	−	864904.i	1.19345	−	0.0676776i
$697$	1.50512e7			1.17352
$698$	1.43748e7			1.11677
$699$	470485.	+	8.29673e6i	0.0364211	+	0.642264i
$700$		0			0
$701$		−	2.08992e7i		−	1.60633i	−0.595758	−	0.803164i	$-0.703148\pi$
$701$		−	2.08992e7i		−	1.60633i	0.595758	−	0.803164i	$-0.296852\pi$
$702$	−8.04134e6	+	1.37985e6i	−0.615865	+	0.105679i
$703$			2.17191e7i			1.65750i
$704$		−	911541.i		−	0.0693177i
$705$	−7.81495e6	+	443164.i	−0.592179	+	0.0335809i
$706$		−	7.07610e6i		−	0.534297i
$707$		0			0
$708$	−2.24740e6	+	127444.i	−0.168499	+	0.00955511i
$709$	2.06100e6			0.153979			0.0769895	−	0.997032i	$-0.475469\pi$
$709$	2.06100e6			0.153979			0.0769895	+	0.997032i	$0.475469\pi$
$710$	1.64728e7			1.22637
$711$	1.62665e7	−	1.85081e6i	1.20676	−	0.137305i
$712$			3.35243e6i			0.247833i
$713$	1.31876e6			0.0971499
$714$		0			0
$715$	−1.72528e6			−0.126210
$716$		−	1.19546e7i		−	0.871472i
$717$	237469.	+	4.18763e6i	0.0172508	+	0.304208i
$718$	−5.08970e6			−0.368452
$719$	−2.24934e7			−1.62268			−0.811339	−	0.584575i	$-0.801261\pi$
$719$	−2.24934e7			−1.62268			−0.811339	+	0.584575i	$0.801261\pi$
$720$	−1.43607e6	+	163397.i	−0.103239	+	0.0117466i
$721$		0			0
$722$			1.89497e7i			1.35288i
$723$	−235489.	−	4.15271e6i	−0.0167542	−	0.295451i
$724$			874979.i			0.0620370i
$725$		−	1.27023e7i		−	0.897504i
$726$	514513.	+	9.07314e6i	0.0362289	+	0.638875i
$727$		−	2.23831e7i		−	1.57067i	−0.619073	−	0.785334i	$-0.712491\pi$
$727$		−	2.23831e7i		−	1.57067i	0.619073	−	0.785334i	$-0.287509\pi$
$728$		0			0
$729$	1.35281e7	−	4.78355e6i	0.942795	−	0.333374i
$730$	−1.40288e7			−0.974350
$731$	6.62090e6			0.458272
$732$	−92417.6	−	1.62973e6i	−0.00637495	−	0.112419i
$733$			1.24379e7i			0.855040i	0.904006	+	0.427520i	$0.140613\pi$
$733$			1.24379e7i			0.855040i	−0.904006	+	0.427520i	$0.859387\pi$
$734$	3.60110e6			0.246715
$735$		0			0
$736$	2.60835e6			0.177489
$737$		−	507652.i		−	0.0344269i
$738$	−1.22123e6	−	1.07332e7i	−0.0825382	−	0.725417i
$739$	−1.34548e7			−0.906291			−0.453145	−	0.891437i	$-0.649698\pi$
$739$	−1.34548e7			−0.906291			−0.453145	+	0.891437i	$0.649698\pi$
$740$	1.08718e7			0.729832
$741$	2.53844e7	−	1.43948e6i	1.69832	−	0.0963073i
$742$		0			0
$743$			1.80287e7i			1.19810i	0.800713	+	0.599049i	$0.204454\pi$
$743$			1.80287e7i			1.19810i	−0.800713	+	0.599049i	$0.795546\pi$
$744$	8.20060e6	−	465034.i	0.543142	−	0.0308001i
$745$			2.86801e7i			1.89317i
$746$		−	9.19556e6i		−	0.604966i
$747$	−2.10346e6	+	239332.i	−0.137922	+	0.0156928i
$748$			908680.i			0.0593823i
$749$		0			0
$750$	−174036.	−	3.06902e6i	−0.0112976	−	0.199226i
$751$	8.24075e6			0.533171			0.266586	−	0.963811i	$-0.414105\pi$
$751$	8.24075e6			0.533171			0.266586	+	0.963811i	$0.414105\pi$
$752$	−540657.			−0.0348640
$753$	−1.47476e7	+	836297.i	−0.947838	+	0.0537493i
$754$		−	1.14039e7i		−	0.730505i
$755$	−2.05531e7			−1.31223
$756$		0			0
$757$	−1.17983e7			−0.748308			−0.374154	−	0.927367i	$-0.622067\pi$
$757$	−1.17983e7			−0.748308			−0.374154	+	0.927367i	$0.622067\pi$
$758$			2.29496e6i			0.145078i
$759$	284005.	−	16105.2i	0.0178946	−	0.00101475i
$760$	3.80764e7			2.39123
$761$	−1.22420e7			−0.766286			−0.383143	−	0.923689i	$-0.625158\pi$
$761$	−1.22420e7			−0.766286			−0.383143	+	0.923689i	$0.625158\pi$
$762$	−136058.	−	2.39931e6i	−0.00848862	−	0.149692i
$763$		0			0
$764$			1.41930e7i			0.879711i
$765$	2.22072e7	−	2.52674e6i	1.37195	−	0.156101i
$766$		−	1.13498e7i		−	0.698899i
$767$			4.56500e6i			0.280190i
$768$	1.69627e7	−	961910.i	1.03775	−	0.0588479i
$769$		−	2.89879e6i		−	0.176767i	−0.996087	−	0.0883834i	$-0.971830\pi$
$769$		−	2.89879e6i		−	0.176767i	0.996087	−	0.0883834i	$-0.0281701\pi$
$770$		0			0
$771$	8.19131e6	−	464507.i	0.496269	−	0.0281421i
$772$	1.03700e7			0.626234
$773$	3.49988e6			0.210671			0.105336	−	0.994437i	$-0.466408\pi$
$773$	3.49988e6			0.210671			0.105336	+	0.994437i	$0.466408\pi$
$774$	−537207.	−	4.72144e6i	−0.0322322	−	0.283284i
$775$		−	6.82964e6i		−	0.408455i
$776$	−3.45035e6			−0.205688
$777$		0			0
$778$	2.44812e6			0.145005
$779$			3.36632e7i			1.98752i
$780$

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)

\(f(q)\)	\(=\)	\(q-3.65505i q^{2} +(-15.5635 + 0.882561i) q^{3} +18.6406 q^{4} +74.3244 q^{5} +(3.22580 + 56.8851i) q^{6} -185.094i q^{8} +(241.442 - 27.4714i) q^{9} +O(q^{10})\)	\(q-3.65505i q^{2} +(-15.5635 + 0.882561i) q^{3} +18.6406 q^{4} +74.3244 q^{5} +(3.22580 + 56.8851i) q^{6} -185.094i q^{8} +(241.442 - 27.4714i) q^{9} -271.659i q^{10} +39.3914i q^{11} +(-290.113 + 16.4515i) q^{12} +589.288i q^{13} +(-1156.74 + 65.5958i) q^{15} -80.0263 q^{16} +1237.51 q^{17} +(-100.409 - 882.482i) q^{18} +2767.79i q^{19} +1385.45 q^{20} +143.977 q^{22} +463.253i q^{23} +(163.357 + 2880.70i) q^{24} +2399.11 q^{25} +2153.88 q^{26} +(-3733.43 + 640.637i) q^{27} -5294.57i q^{29} +(239.756 + 4227.95i) q^{30} -2846.74i q^{31} -5630.50i q^{32} +(-34.7653 - 613.066i) q^{33} -4523.16i q^{34} +(4500.64 - 512.084i) q^{36} +7847.11 q^{37} +10116.4 q^{38} +(-520.083 - 9171.36i) q^{39} -13757.0i q^{40} +12162.5 q^{41} +5350.18 q^{43} +734.281i q^{44} +(17945.0 - 2041.79i) q^{45} +1693.21 q^{46} +6755.99 q^{47} +(1245.49 - 70.6281i) q^{48} -8768.86i q^{50} +(-19259.9 + 1092.18i) q^{51} +10984.7i q^{52} -16011.8i q^{53} +(2341.56 + 13645.9i) q^{54} +2927.74i q^{55} +(-2442.74 - 43076.3i) q^{57} -19351.9 q^{58} +7746.64 q^{59} +(-21562.4 + 1222.75i) q^{60} +5617.58i q^{61} -10405.0 q^{62} -23140.6 q^{64} +43798.5i q^{65} +(-2240.79 + 127.069i) q^{66} -12887.4 q^{67} +23068.0 q^{68} +(-408.849 - 7209.82i) q^{69} +60637.7i q^{71} +(-5084.79 - 44689.5i) q^{72} -51641.4i q^{73} -28681.6i q^{74} +(-37338.5 + 2117.36i) q^{75} +51593.3i q^{76} +(-33521.7 + 1900.93i) q^{78} +67372.2 q^{79} -5947.90 q^{80} +(57539.6 - 13265.5i) q^{81} -44454.5i q^{82} -8712.06 q^{83} +91977.1 q^{85} -19555.1i q^{86} +(4672.78 + 82401.9i) q^{87} +7291.11 q^{88} -18112.0 q^{89} +(-7462.85 - 65589.9i) q^{90} +8635.34i q^{92} +(2512.42 + 44305.1i) q^{93} -24693.4i q^{94} +205714. i q^{95} +(4969.26 + 87630.1i) q^{96} -18641.1i q^{97} +(1082.14 + 9510.75i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 172 q^{4} + 1212 q^{9}+O(q^{10}) \) 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

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