LMFDB - Embedded newform 810.4.e.bg.271.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [810,4,Mod(271,810)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(810, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([2, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("810.271");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 810 = 2 \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	810.e (of order $3$, degree $2$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$47.7915471046$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 254x^{6} + 23581x^{4} + 947376x^{2} + 13883076 $ x^8 + 254x^6 + 23581x^4 + 947376*x^2 + 13883076
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 3^{10} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			271.3
Root			$6.40373i$ of defining polynomial
Character	$\chi$	$=$	810.271
Dual form			810.4.e.bg.541.3

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.00000 - 1.73205i) q^{2} +(-2.00000 + 3.46410i) q^{4} +(2.50000 - 4.33013i) q^{5} +(4.90945 + 8.50341i) q^{7} +8.00000 q^{8} +O(q^{10})$	\(q+(-1.00000 - 1.73205i) q^{2} +(-2.00000 + 3.46410i) q^{4} +(2.50000 - 4.33013i) q^{5} +(4.90945 + 8.50341i) q^{7} +8.00000 q^{8} -10.0000 q^{10} +(-9.53931 - 16.5226i) q^{11} +(43.7409 - 75.7615i) q^{13} +(9.81890 - 17.0068i) q^{14} +(-8.00000 - 13.8564i) q^{16} -10.8482 q^{17} +103.354 q^{19} +(10.0000 + 17.3205i) q^{20} +(-19.0786 + 33.0451i) q^{22} +(-98.0970 + 169.909i) q^{23} +(-12.5000 - 21.6506i) q^{25} -174.964 q^{26} -39.2756 q^{28} +(-6.04914 - 10.4774i) q^{29} +(-117.420 + 203.378i) q^{31} +(-16.0000 + 27.7128i) q^{32} +(10.8482 + 18.7896i) q^{34} +49.0945 q^{35} +304.576 q^{37} +(-103.354 - 179.014i) q^{38} +(20.0000 - 34.6410i) q^{40} +(182.380 - 315.891i) q^{41} +(-169.759 - 294.031i) q^{43} +76.3145 q^{44} +392.388 q^{46} +(-118.565 - 205.360i) q^{47} +(123.295 - 213.553i) q^{49} +(-25.0000 + 43.3013i) q^{50} +(174.964 + 303.046i) q^{52} +31.2915 q^{53} -95.3931 q^{55} +(39.2756 + 68.0273i) q^{56} +(-12.0983 + 20.9549i) q^{58} +(64.8469 - 112.318i) q^{59} +(269.470 + 466.736i) q^{61} +469.680 q^{62} +64.0000 q^{64} +(-218.704 - 378.807i) q^{65} +(-168.531 + 291.904i) q^{67} +(21.6964 - 37.5792i) q^{68} +(-49.0945 - 85.0341i) q^{70} +264.767 q^{71} +950.225 q^{73} +(-304.576 - 527.540i) q^{74} +(-206.708 + 358.028i) q^{76} +(93.6655 - 162.233i) q^{77} +(-187.548 - 324.843i) q^{79} -80.0000 q^{80} -729.519 q^{82} +(-538.552 - 932.800i) q^{83} +(-27.1205 + 46.9740i) q^{85} +(-339.517 + 588.061i) q^{86} +(-76.3145 - 132.181i) q^{88} -909.671 q^{89} +858.975 q^{91} +(-392.388 - 679.636i) q^{92} +(-237.129 + 410.720i) q^{94} +(258.385 - 447.535i) q^{95} +(-658.853 - 1141.17i) q^{97} -493.179 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{2} - 16 q^{4} + 20 q^{5} - 2 q^{7} + 64 q^{8}+O(q^{10}) $ 8 * q - 8 * q^2 - 16 * q^4 + 20 * q^5 - 2 * q^7 + 64 * q^8	$ 8 q - 8 q^{2} - 16 q^{4} + 20 q^{5} - 2 q^{7} + 64 q^{8} - 80 q^{10} + 36 q^{11} - 32 q^{13} - 4 q^{14} - 64 q^{16} + 180 q^{17} + 328 q^{19} + 80 q^{20} + 72 q^{22} - 42 q^{23} - 100 q^{25} + 128 q^{26} + 16 q^{28} - 36 q^{29} - 446 q^{31} - 128 q^{32} - 180 q^{34} - 20 q^{35} + 1372 q^{37} - 328 q^{38} + 160 q^{40} - 108 q^{41} - 470 q^{43} - 288 q^{44} + 168 q^{46} - 804 q^{47} - 1338 q^{49} - 200 q^{50} - 128 q^{52} + 336 q^{53} + 360 q^{55} - 16 q^{56} - 72 q^{58} - 768 q^{59} - 596 q^{61} + 1784 q^{62} + 512 q^{64} + 160 q^{65} + 424 q^{67} - 360 q^{68} + 20 q^{70} + 696 q^{71} - 188 q^{73} - 1372 q^{74} - 656 q^{76} - 630 q^{77} - 1088 q^{79} - 640 q^{80} + 432 q^{82} - 522 q^{83} + 450 q^{85} - 940 q^{86} + 288 q^{88} + 2256 q^{89} + 5684 q^{91} - 168 q^{92} - 1608 q^{94} + 820 q^{95} - 2486 q^{97} + 5352 q^{98}+O(q^{100}) $ 8 * q - 8 * q^2 - 16 * q^4 + 20 * q^5 - 2 * q^7 + 64 * q^8 - 80 * q^10 + 36 * q^11 - 32 * q^13 - 4 * q^14 - 64 * q^16 + 180 * q^17 + 328 * q^19 + 80 * q^20 + 72 * q^22 - 42 * q^23 - 100 * q^25 + 128 * q^26 + 16 * q^28 - 36 * q^29 - 446 * q^31 - 128 * q^32 - 180 * q^34 - 20 * q^35 + 1372 * q^37 - 328 * q^38 + 160 * q^40 - 108 * q^41 - 470 * q^43 - 288 * q^44 + 168 * q^46 - 804 * q^47 - 1338 * q^49 - 200 * q^50 - 128 * q^52 + 336 * q^53 + 360 * q^55 - 16 * q^56 - 72 * q^58 - 768 * q^59 - 596 * q^61 + 1784 * q^62 + 512 * q^64 + 160 * q^65 + 424 * q^67 - 360 * q^68 + 20 * q^70 + 696 * q^71 - 188 * q^73 - 1372 * q^74 - 656 * q^76 - 630 * q^77 - 1088 * q^79 - 640 * q^80 + 432 * q^82 - 522 * q^83 + 450 * q^85 - 940 * q^86 + 288 * q^88 + 2256 * q^89 + 5684 * q^91 - 168 * q^92 - 1608 * q^94 + 820 * q^95 - 2486 * q^97 + 5352 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/810\mathbb{Z}\right)^\times$.

$n$	$487$	$731$
$\chi(n)$	$1$	$e\left(\frac{1}{3}\right)$

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 $

$2$	−1.00000	−	1.73205i	−0.353553	−	0.612372i
$2$	−1.00000	−	1.73205i	−0.353553	−	0.612372i
$3$		0			0
$3$		0			0
$4$	−2.00000	+	3.46410i	−0.250000	+	0.433013i
$5$	2.50000	−	4.33013i	0.223607	−	0.387298i
$5$	2.50000	−	4.33013i	0.223607	−	0.387298i
$6$		0			0
$7$	4.90945	+	8.50341i	0.265085	+	0.459141i	0.967586	−	0.252542i	$-0.0812665\pi$
$7$	4.90945	+	8.50341i	0.265085	+	0.459141i	−0.702501	+	0.711683i	$0.747933\pi$
$8$	8.00000			0.353553
$9$		0			0
$10$	−10.0000			−0.316228
$11$	−9.53931	−	16.5226i	−0.261474	−	0.452886i	0.705160	−	0.709048i	$-0.250875\pi$
$11$	−9.53931	−	16.5226i	−0.261474	−	0.452886i	−0.966634	+	0.256162i	$0.917542\pi$
$12$		0			0
$13$	43.7409	−	75.7615i	0.933196	−	1.61634i	0.155375	−	0.987856i	$-0.450341\pi$
$13$	43.7409	−	75.7615i	0.933196	−	1.61634i	0.777821	−	0.628486i	$-0.216325\pi$
$14$	9.81890	−	17.0068i	0.187444	−	0.324662i
$15$		0			0
$16$	−8.00000	−	13.8564i	−0.125000	−	0.216506i
$17$	−10.8482			−0.154769			−0.0773845	−	0.997001i	$-0.524657\pi$
$17$	−10.8482			−0.154769			−0.0773845	+	0.997001i	$0.524657\pi$
$18$		0			0
$19$	103.354			1.24795			0.623974	−	0.781445i	$-0.285517\pi$
$19$	103.354			1.24795			0.623974	+	0.781445i	$0.285517\pi$
$20$	10.0000	+	17.3205i	0.111803	+	0.193649i
$21$		0			0
$22$	−19.0786	+	33.0451i	−0.184890	+	0.320239i
$23$	−98.0970	+	169.909i	−0.889332	+	1.54037i	−0.0486659	+	0.998815i	$0.515497\pi$
$23$	−98.0970	+	169.909i	−0.889332	+	1.54037i	−0.840666	+	0.541553i	$0.817836\pi$
$24$		0			0
$25$	−12.5000	−	21.6506i	−0.100000	−	0.173205i
$26$	−174.964			−1.31974
$27$		0			0
$28$	−39.2756			−0.265085
$29$	−6.04914	−	10.4774i	−0.0387344	−	0.0670900i	0.846008	−	0.533170i	$-0.178999\pi$
$29$	−6.04914	−	10.4774i	−0.0387344	−	0.0670900i	−0.884743	+	0.466080i	$0.845666\pi$
$30$		0			0
$31$	−117.420	+	203.378i	−0.680299	+	1.17831i	0.294590	+	0.955624i	$0.404817\pi$
$31$	−117.420	+	203.378i	−0.680299	+	1.17831i	−0.974890	+	0.222689i	$0.928517\pi$
$32$	−16.0000	+	27.7128i	−0.0883883	+	0.153093i
$33$		0			0
$34$	10.8482	+	18.7896i	0.0547191	+	0.0947762i
$35$	49.0945			0.237100
$36$		0			0
$37$	304.576			1.35329			0.676647	−	0.736307i	$-0.263432\pi$
$37$	304.576			1.35329			0.676647	+	0.736307i	$0.263432\pi$
$38$	−103.354	−	179.014i	−0.441216	−	0.764208i
$39$		0			0
$40$	20.0000	−	34.6410i	0.0790569	−	0.136931i
$41$	182.380	−	315.891i	0.694706	−	1.20327i	−0.275574	−	0.961280i	$-0.588868\pi$
$41$	182.380	−	315.891i	0.694706	−	1.20327i	0.970280	−	0.241986i	$-0.0777987\pi$
$42$		0			0
$43$	−169.759	−	294.031i	−0.602045	−	1.04277i	−0.992511	−	0.122156i	$-0.961019\pi$
$43$	−169.759	−	294.031i	−0.602045	−	1.04277i	0.390466	−	0.920618i	$-0.372314\pi$
$44$	76.3145			0.261474
$45$		0			0
$46$	392.388			1.25771
$47$	−118.565	−	205.360i	−0.367967	−	0.637337i	0.621281	−	0.783588i	$-0.286613\pi$
$47$	−118.565	−	205.360i	−0.367967	−	0.637337i	−0.989247	+	0.146251i	$0.953279\pi$
$48$		0			0
$49$	123.295	−	213.553i	0.359460	−	0.622602i
$50$	−25.0000	+	43.3013i	−0.0707107	+	0.122474i
$51$		0			0
$52$	174.964	+	303.046i	0.466598	+	0.808171i
$53$	31.2915			0.0810986			0.0405493	−	0.999178i	$-0.487089\pi$
$53$	31.2915			0.0810986			0.0405493	+	0.999178i	$0.487089\pi$
$54$		0			0
$55$	−95.3931			−0.233869
$56$	39.2756	+	68.0273i	0.0937218	+	0.162331i
$57$		0			0
$58$	−12.0983	+	20.9549i	−0.0273894	+	0.0474398i
$59$	64.8469	−	112.318i	0.143091	−	0.247840i	−0.785568	−	0.618775i	$-0.787629\pi$
$59$	64.8469	−	112.318i	0.143091	−	0.247840i	0.928659	+	0.370935i	$0.120963\pi$
$60$		0			0
$61$	269.470	+	466.736i	0.565608	+	0.979662i	0.996993	+	0.0774942i	$0.0246919\pi$
$61$	269.470	+	466.736i	0.565608	+	0.979662i	−0.431384	+	0.902168i	$0.641975\pi$
$62$	469.680			0.962088
$63$		0			0
$64$	64.0000			0.125000
$65$	−218.704	−	378.807i	−0.417338	−	0.722850i
$66$		0			0
$67$	−168.531	+	291.904i	−0.307303	+	0.532264i	−0.977771	−	0.209674i	$-0.932760\pi$
$67$	−168.531	+	291.904i	−0.307303	+	0.532264i	0.670468	+	0.741938i	$0.266093\pi$
$68$	21.6964	−	37.5792i	0.0386922	−	0.0670169i
$69$		0			0
$70$	−49.0945	−	85.0341i	−0.0838273	−	0.145193i
$71$	264.767			0.442564			0.221282	−	0.975210i	$-0.428976\pi$
$71$	264.767			0.442564			0.221282	+	0.975210i	$0.428976\pi$
$72$		0			0
$73$	950.225			1.52350			0.761749	−	0.647872i	$-0.224341\pi$
$73$	950.225			1.52350			0.761749	+	0.647872i	$0.224341\pi$
$74$	−304.576	−	527.540i	−0.478462	−	0.828720i
$75$		0			0
$76$	−206.708	+	358.028i	−0.311987	+	0.540377i
$77$	93.6655	−	162.233i	0.138626	−	0.240107i
$78$		0			0
$79$	−187.548	−	324.843i	−0.267099	−	0.462629i	0.701012	−	0.713149i	$-0.252732\pi$
$79$	−187.548	−	324.843i	−0.267099	−	0.462629i	−0.968111	+	0.250520i	$0.919398\pi$
$80$	−80.0000			−0.111803
$81$		0			0
$82$	−729.519			−0.982462
$83$	−538.552	−	932.800i	−0.712214	−	1.23359i	−0.964024	−	0.265814i	$-0.914359\pi$
$83$	−538.552	−	932.800i	−0.712214	−	1.23359i	0.251810	−	0.967777i	$-0.418974\pi$
$84$		0			0
$85$	−27.1205	+	46.9740i	−0.0346074	+	0.0599417i
$86$	−339.517	+	588.061i	−0.425710	+	0.737352i
$87$		0			0
$88$	−76.3145	−	132.181i	−0.0924449	−	0.160119i
$89$	−909.671			−1.08343			−0.541713	−	0.840564i	$-0.682224\pi$
$89$	−909.671			−1.08343			−0.541713	+	0.840564i	$0.682224\pi$
$90$		0			0
$91$	858.975			0.989506
$92$	−392.388	−	679.636i	−0.444666	−	0.770184i
$93$		0			0
$94$	−237.129	+	410.720i	−0.260192	+	0.450665i
$95$	258.385	−	447.535i	0.279049	−	0.483328i
$96$		0			0
$97$	−658.853	−	1141.17i	−0.689654	−	1.19452i	−0.971950	−	0.235188i	$-0.924429\pi$
$97$	−658.853	−	1141.17i	−0.689654	−	1.19452i	0.282296	−	0.959327i	$-0.408904\pi$
$98$	−493.179			−0.508353
$99$		0			0
$100$	100.000			0.100000
$101$	−321.491	−	556.838i	−0.316728	−	0.548589i	0.663075	−	0.748553i	$-0.269251\pi$
$101$	−321.491	−	556.838i	−0.316728	−	0.548589i	−0.979803	+	0.199964i	$0.935918\pi$
$102$		0			0
$103$	−27.6827	+	47.9479i	−0.0264821	+	0.0458684i	−0.878963	−	0.476890i	$-0.841764\pi$
$103$	−27.6827	+	47.9479i	−0.0264821	+	0.0458684i	0.852481	+	0.522759i	$0.175097\pi$
$104$	349.927	−	606.092i	0.329934	−	0.571463i
$105$		0			0
$106$	−31.2915	−	54.1985i	−0.0286727	−	0.0496625i
$107$	−178.586			−0.161351			−0.0806757	−	0.996740i	$-0.525708\pi$
$107$	−178.586			−0.161351			−0.0806757	+	0.996740i	$0.525708\pi$
$108$		0			0
$109$	723.780			0.636014			0.318007	−	0.948088i	$-0.396986\pi$
$109$	723.780			0.636014			0.318007	+	0.948088i	$0.396986\pi$
$110$	95.3931	+	165.226i	0.0826853	+	0.143215i
$111$		0			0
$112$	78.5512	−	136.055i	0.0662713	−	0.114785i
$113$	591.131	−	1023.87i	0.492114	−	0.852367i	−0.507845	−	0.861449i	$-0.669558\pi$
$113$	591.131	−	1023.87i	0.492114	−	0.852367i	0.999959	+	0.00908187i	$0.00289089\pi$
$114$		0			0
$115$	490.485	+	849.545i	0.397721	+	0.688874i
$116$	48.3932			0.0387344
$117$		0			0
$118$	−259.387			−0.202361
$119$	−53.2586	−	92.2466i	−0.0410270	−	0.0710608i
$120$		0			0
$121$	483.503	−	837.452i	0.363263	−	0.629190i
$122$	538.940	−	933.472i	0.399946	−	0.692726i
$123$		0			0
$124$	−469.680	−	813.510i	−0.340150	−	0.589156i
$125$	−125.000			−0.0894427
$126$		0			0
$127$	−232.957			−0.162769			−0.0813844	−	0.996683i	$-0.525934\pi$
$127$	−232.957			−0.162769			−0.0813844	+	0.996683i	$0.525934\pi$
$128$	−64.0000	−	110.851i	−0.0441942	−	0.0765466i
$129$		0			0
$130$	−437.409	+	757.615i	−0.295102	+	0.511132i
$131$	1075.42	−	1862.69i	0.717254	−	1.24232i	−0.244830	−	0.969566i	$-0.578732\pi$
$131$	1075.42	−	1862.69i	0.717254	−	1.24232i	0.962084	−	0.272754i	$-0.0879345\pi$
$132$		0			0
$133$	507.410	+	878.860i	0.330812	+	0.572984i
$134$	674.123			0.434592
$135$		0			0
$136$	−86.7855			−0.0547191
$137$	−1175.87	−	2036.66i	−0.733294	−	1.27010i	−0.955468	−	0.295095i	$-0.904649\pi$
$137$	−1175.87	−	2036.66i	−0.733294	−	1.27010i	0.222174	−	0.975007i	$-0.428685\pi$
$138$		0			0
$139$	−534.601	+	925.956i	−0.326218	+	0.565025i	−0.981758	−	0.190135i	$-0.939108\pi$
$139$	−534.601	+	925.956i	−0.326218	+	0.565025i	0.655540	+	0.755160i	$0.272441\pi$
$140$	−98.1890	+	170.068i	−0.0592749	+	0.102667i
$141$		0			0
$142$	−264.767	−	458.589i	−0.156470	−	0.271014i
$143$	−1669.03			−0.976024
$144$		0			0
$145$	−60.4914			−0.0346451
$146$	−950.225	−	1645.84i	−0.538638	−	0.932949i
$147$		0			0
$148$	−609.151	+	1055.08i	−0.338324	+	0.585994i
$149$	−155.923	+	270.066i	−0.0857295	+	0.148488i	−0.905702	−	0.423915i	$-0.860655\pi$
$149$	−155.923	+	270.066i	−0.0857295	+	0.148488i	0.819972	+	0.572403i	$0.193989\pi$
$150$		0			0
$151$	−1267.21	−	2194.88i	−0.682943	−	1.18289i	−0.974078	−	0.226210i	$-0.927366\pi$
$151$	−1267.21	−	2194.88i	−0.682943	−	1.18289i	0.291135	−	0.956682i	$-0.405967\pi$
$152$	826.831			0.441216
$153$		0			0
$154$	−374.662			−0.196046
$155$	587.100	+	1016.89i	0.304239	+	0.526957i
$156$		0			0
$157$	528.670	−	915.683i	0.268742	−	0.465474i	−0.699795	−	0.714343i	$-0.746725\pi$
$157$	528.670	−	915.683i	0.268742	−	0.465474i	0.968537	+	0.248869i	$0.0800588\pi$
$158$	−375.097	+	649.687i	−0.188868	+	0.327128i
$159$		0			0
$160$	80.0000	+	138.564i	0.0395285	+	0.0684653i
$161$	−1926.41			−0.942996
$162$		0			0
$163$	2665.14			1.28067			0.640337	−	0.768094i	$-0.278795\pi$
$163$	2665.14			1.28067			0.640337	+	0.768094i	$0.278795\pi$
$164$	729.519	+	1263.56i	0.347353	+	0.601633i
$165$		0			0
$166$	−1077.10	+	1865.60i	−0.503611	+	0.872281i
$167$	767.253	−	1328.92i	0.355520	−	0.615779i	−0.631687	−	0.775224i	$-0.717637\pi$
$167$	767.253	−	1328.92i	0.355520	−	0.615779i	0.987207	+	0.159445i	$0.0509705\pi$
$168$		0			0
$169$	−2728.03	−	4725.09i	−1.24171	−	2.15070i
$170$	108.482			0.0489422
$171$		0			0
$172$	1358.07			0.602045
$173$	1003.92	+	1738.84i	0.441195	+	0.764173i	0.997778	−	0.0666192i	$-0.0212213\pi$
$173$	1003.92	+	1738.84i	0.441195	+	0.764173i	−0.556583	+	0.830792i	$0.687888\pi$
$174$		0			0
$175$	122.736	−	212.585i	0.0530171	−	0.0918282i
$176$	−152.629	+	264.361i	−0.0653684	+	0.113221i
$177$		0			0
$178$	909.671	+	1575.60i	0.383049	+	0.663460i
$179$	−4588.33			−1.91591			−0.957955	−	0.286917i	$-0.907370\pi$
$179$	−4588.33			−1.91591			−0.957955	+	0.286917i	$0.907370\pi$
$180$		0			0
$181$	15.7678			0.00647519			0.00323759	−	0.999995i	$-0.498969\pi$
$181$	15.7678			0.00647519			0.00323759	+	0.999995i	$0.498969\pi$
$182$	−858.975	−	1487.79i	−0.349843	−	0.605946i
$183$		0			0
$184$	−784.776	+	1359.27i	−0.314426	+	0.544603i
$185$	761.439	−	1318.85i	0.302606	−	0.524129i
$186$		0			0
$187$	103.484	+	179.240i	0.0404680	+	0.0700926i
$188$	948.517			0.367967
$189$		0			0
$190$	−1033.54			−0.394636
$191$	798.864	+	1383.67i	0.302637	+	0.524183i	0.976733	−	0.214461i	$-0.0687996\pi$
$191$	798.864	+	1383.67i	0.302637	+	0.524183i	−0.674095	+	0.738645i	$0.735466\pi$
$192$		0			0
$193$	−289.934	+	502.180i	−0.108134	+	0.187294i	−0.915014	−	0.403421i	$-0.867821\pi$
$193$	−289.934	+	502.180i	−0.108134	+	0.187294i	0.806880	+	0.590715i	$0.201154\pi$
$194$	−1317.71	+	2282.33i	−0.487659	+	0.844650i
$195$		0			0
$196$	493.179	+	854.210i	0.179730	+	0.311301i
$197$	2095.96			0.758026			0.379013	−	0.925391i	$-0.376264\pi$
$197$	2095.96			0.758026			0.379013	+	0.925391i	$0.376264\pi$
$198$		0			0
$199$	−1748.58			−0.622881			−0.311440	−	0.950266i	$-0.600811\pi$
$199$	−1748.58			−0.622881			−0.311440	+	0.950266i	$0.600811\pi$
$200$	−100.000	−	173.205i	−0.0353553	−	0.0612372i
$201$		0			0
$202$	−642.982	+	1113.68i	−0.223961	+	0.387911i
$203$	59.3959	−	102.877i	0.0205358	−	0.0355691i
$204$		0			0
$205$	−911.899	−	1579.45i	−0.310682	−	0.538117i
$206$	110.731			0.0374514
$207$		0			0
$208$	−1399.71			−0.466598
$209$	−985.924	−	1707.67i	−0.326305	−	0.565177i
$210$		0			0
$211$	−561.332	+	972.255i	−0.183145	+	0.317217i	−0.942950	−	0.332934i	$-0.891961\pi$
$211$	−561.332	+	972.255i	−0.183145	+	0.317217i	0.759805	+	0.650151i	$0.225295\pi$
$212$	−62.5831	+	108.397i	−0.0202746	+	0.0351167i
$213$		0			0
$214$	178.586	+	309.321i	0.0570463	+	0.0988072i
$215$	−1697.59			−0.538486
$216$		0			0
$217$	−2305.87			−0.721349
$218$	−723.780	−	1253.62i	−0.224865	−	0.389477i
$219$		0			0
$220$	190.786	−	330.451i	0.0584673	−	0.101268i
$221$	−474.509	+	821.874i	−0.144430	+	0.250159i
$222$		0			0
$223$	−2226.32	−	3856.09i	−0.668543	−	1.15795i	−0.978312	−	0.207139i	$-0.933585\pi$
$223$	−2226.32	−	3856.09i	−0.668543	−	1.15795i	0.309768	−	0.950812i	$-0.399749\pi$
$224$	−314.205			−0.0937218
$225$		0			0
$226$	−2364.52			−0.695955
$227$	1635.80	+	2833.29i	0.478290	+	0.828423i	0.999690	−	0.0248892i	$-0.00792330\pi$
$227$	1635.80	+	2833.29i	0.478290	+	0.828423i	−0.521400	+	0.853313i	$0.674590\pi$
$228$		0			0
$229$	−38.6644	+	66.9688i	−0.0111573	+	0.0193250i	−0.871550	−	0.490306i	$-0.836885\pi$
$229$	−38.6644	+	66.9688i	−0.0111573	+	0.0193250i	0.860393	+	0.509631i	$0.170218\pi$
$230$	980.970	−	1699.09i	0.281232	−	0.487107i
$231$		0			0
$232$	−48.3932	−	83.8194i	−0.0136947	−	0.0237199i
$233$	4243.59			1.19316			0.596581	−	0.802553i	$-0.296525\pi$
$233$	4243.59			1.19316			0.596581	+	0.802553i	$0.296525\pi$
$234$		0			0
$235$	−1185.65			−0.329119
$236$	259.387	+	449.272i	0.0715453	+	0.123920i
$237$		0			0
$238$	−106.517	+	184.493i	−0.0290104	+	0.0502476i
$239$	39.2473	−	67.9783i	0.0106222	−	0.0183981i	−0.860665	−	0.509171i	$-0.829952\pi$
$239$	39.2473	−	67.9783i	0.0106222	−	0.0183981i	0.871288	+	0.490773i	$0.163285\pi$
$240$		0			0
$241$	2698.85	+	4674.54i	0.721361	+	1.24943i	0.960454	+	0.278438i	$0.0898167\pi$
$241$	2698.85	+	4674.54i	0.721361	+	1.24943i	−0.239093	+	0.970997i	$0.576850\pi$
$242$	−1934.01			−0.513731
$243$		0			0
$244$	−2155.76			−0.565608
$245$	−616.473	−	1067.76i	−0.160755	−	0.278436i
$246$		0			0
$247$	4520.79	−	7830.24i	1.16458	−	2.01711i
$248$	−939.361	+	1627.02i	−0.240522	+	0.416596i
$249$		0			0
$250$	125.000	+	216.506i	0.0316228	+	0.0547723i
$251$	−1627.51			−0.409274			−0.204637	−	0.978838i	$-0.565601\pi$
$251$	−1627.51			−0.409274			−0.204637	+	0.978838i	$0.565601\pi$
$252$		0			0
$253$	3743.11			0.930148
$254$	232.957	+	403.494i	0.0575474	+	0.0996751i
$255$		0			0
$256$	−128.000	+	221.703i	−0.0312500	+	0.0541266i
$257$	3091.57	−	5354.76i	0.750378	−	1.29969i	−0.197262	−	0.980351i	$-0.563205\pi$
$257$	3091.57	−	5354.76i	0.750378	−	1.29969i	0.947640	−	0.319341i	$-0.103462\pi$
$258$		0			0
$259$	1495.30	+	2589.93i	0.358739	+	0.621353i
$260$	1749.64			0.417338
$261$		0			0
$262$	−4301.70			−1.01435
$263$	−3644.64	−	6312.70i	−0.854517	−	1.48007i	−0.877093	−	0.480321i	$-0.840520\pi$
$263$	−3644.64	−	6312.70i	−0.854517	−	1.48007i	0.0225757	−	0.999745i	$-0.492813\pi$
$264$		0			0
$265$	78.2289	−	135.496i	0.0181342	−	0.0314093i
$266$	1014.82	−	1757.72i	0.233920	−	0.405161i
$267$		0			0
$268$	−674.123	−	1167.62i	−0.153652	−	0.266132i
$269$	−1903.32			−0.431404			−0.215702	−	0.976459i	$-0.569204\pi$
$269$	−1903.32			−0.431404			−0.215702	+	0.976459i	$0.569204\pi$
$270$		0			0
$271$	6485.88			1.45383			0.726917	−	0.686725i	$-0.240952\pi$
$271$	6485.88			1.45383			0.726917	+	0.686725i	$0.240952\pi$
$272$	86.7855	+	150.317i	0.0193461	+	0.0335085i
$273$		0			0
$274$	−2351.74	+	4073.33i	−0.518517	+	0.898098i
$275$	−238.483	+	413.064i	−0.0522947	+	0.0905772i
$276$		0			0
$277$	2923.89	+	5064.32i	0.634222	+	1.09850i	0.986679	+	0.162677i	$0.0520127\pi$
$277$	2923.89	+	5064.32i	0.634222	+	1.09850i	−0.352458	+	0.935828i	$0.614654\pi$
$278$	2138.40			0.461341
$279$		0			0
$280$	392.756			0.0838273
$281$	1868.06	+	3235.58i	0.396581	+	0.686899i	0.993302	−	0.115550i	$-0.0368631\pi$
$281$	1868.06	+	3235.58i	0.396581	+	0.686899i	−0.596720	+	0.802449i	$0.703530\pi$
$282$		0			0
$283$	1430.05	−	2476.93i	0.300381	−	0.520275i	−0.675841	−	0.737047i	$-0.736219\pi$
$283$	1430.05	−	2476.93i	0.300381	−	0.520275i	0.976222	+	0.216772i	$0.0695528\pi$
$284$	−529.533	+	917.178i	−0.110641	+	0.191636i
$285$		0			0
$286$	1669.03	+	2890.85i	0.345077	+	0.597690i
$287$	3581.54			0.736625
$288$		0			0
$289$	−4795.32			−0.976047
$290$	60.4914	+	104.774i	0.0122489	+	0.0212157i
$291$		0			0
$292$	−1900.45	+	3291.68i	−0.380875	+	0.659694i
$293$	−4019.37	+	6961.75i	−0.801413	+	1.38809i	0.117273	+	0.993100i	$0.462585\pi$
$293$	−4019.37	+	6961.75i	−0.801413	+	1.38809i	−0.918686	+	0.394989i	$0.870749\pi$
$294$		0			0
$295$	−324.234	−	561.590i	−0.0639920	−	0.110837i
$296$	2436.60			0.478462
$297$		0			0
$298$	623.691			0.121240
$299$	8581.70	+	14863.9i	1.65984	+	2.87493i
$300$		0			0
$301$	1666.84	−	2887.06i	0.319187	−	0.552848i
$302$	−2534.43	+	4389.76i	−0.482914	+	0.836431i
$303$		0			0
$304$	−826.831	−	1432.11i	−0.155993	−	0.270188i
$305$	2694.70			0.505895
$306$		0			0
$307$	3066.57			0.570093			0.285046	−	0.958514i	$-0.407991\pi$
$307$	3066.57			0.570093			0.285046	+	0.958514i	$0.407991\pi$
$308$	374.662	+	648.934i	0.0693128	+	0.120053i
$309$		0			0
$310$	1174.20	−	2033.78i	0.215129	−	0.372615i
$311$	−4864.61	+	8425.75i	−0.886967	+	1.53627i	−0.0435236	+	0.999052i	$0.513858\pi$
$311$	−4864.61	+	8425.75i	−0.886967	+	1.53627i	−0.843443	+	0.537219i	$0.819475\pi$
$312$		0			0
$313$	358.564	+	621.051i	0.0647515	+	0.112153i	0.896584	−	0.442874i	$-0.146041\pi$
$313$	358.564	+	621.051i	0.0647515	+	0.112153i	−0.831832	+	0.555027i	$0.812708\pi$
$314$	−2114.68			−0.380058
$315$		0			0
$316$	1500.39			0.267099
$317$	1566.03	+	2712.44i	0.277466	+	0.480585i	0.970754	−	0.240075i	$-0.0771720\pi$
$317$	1566.03	+	2712.44i	0.277466	+	0.480585i	−0.693288	+	0.720660i	$0.743839\pi$
$318$		0			0
$319$	−115.409	+	199.895i	−0.0202561	+	0.0350845i
$320$	160.000	−	277.128i	0.0279508	−	0.0484123i
$321$		0			0
$322$	1926.41	+	3336.64i	0.333399	+	0.577465i
$323$	−1121.20			−0.193143
$324$		0			0
$325$	−2187.04			−0.373278
$326$	−2665.14	−	4616.16i	−0.452787	−	0.784250i
$327$		0			0
$328$	1459.04	−	2527.13i	0.245616	−	0.425419i
$329$	1164.17	−	2016.41i	0.195085	−	0.337897i
$330$		0			0
$331$	2411.07	+	4176.10i	0.400376	+	0.693471i	0.993771	−	0.111440i	$-0.0355463\pi$
$331$	2411.07	+	4176.10i	0.400376	+	0.693471i	−0.593395	+	0.804911i	$0.702213\pi$
$332$	4308.42			0.712214
$333$		0			0
$334$	−3069.01			−0.502781
$335$	842.654	+	1459.52i	0.137430	+	0.238036i
$336$		0			0
$337$	3180.39	−	5508.60i	0.514086	−	0.890423i	−0.485780	−	0.874081i	$-0.661465\pi$
$337$	3180.39	−	5508.60i	0.514086	−	0.890423i	0.999866	−	0.0163423i	$-0.00520214\pi$
$338$	−5456.06	+	9450.18i	−0.878020	+	1.52078i
$339$		0			0
$340$	−108.482	−	187.896i	−0.0173037	−	0.0299709i
$341$	4480.43			0.711521
$342$		0			0
$343$	5789.12			0.911320
$344$	−1358.07	−	2352.24i	−0.212855	−	0.368676i
$345$		0			0
$346$	2007.84	−	3477.69i	0.311972	−	0.540352i
$347$	5165.33	−	8946.61i	0.799104	−	1.38409i	−0.121096	−	0.992641i	$-0.538641\pi$
$347$	5165.33	−	8946.61i	0.799104	−	1.38409i	0.920200	−	0.391449i	$-0.128026\pi$
$348$		0			0
$349$	−4815.36	−	8340.44i	−0.738567	−	1.27924i	−0.953140	−	0.302528i	$-0.902169\pi$
$349$	−4815.36	−	8340.44i	−0.738567	−	1.27924i	0.214573	−	0.976708i	$-0.431164\pi$
$350$	−490.945			−0.0749774
$351$		0			0
$352$	610.516			0.0924449
$353$	5697.58	+	9868.50i	0.859070	+	1.48795i	0.872817	+	0.488047i	$0.162291\pi$
$353$	5697.58	+	9868.50i	0.859070	+	1.48795i	−0.0137475	+	0.999905i	$0.504376\pi$
$354$		0			0
$355$	661.917	−	1146.47i	0.0989602	−	0.171404i
$356$	1819.34	−	3151.19i	0.270856	−	0.469137i
$357$		0			0
$358$	4588.33	+	7947.22i	0.677377	+	1.17325i
$359$	1549.64			0.227819			0.113909	−	0.993491i	$-0.463663\pi$
$359$	1549.64			0.227819			0.113909	+	0.993491i	$0.463663\pi$
$360$		0			0
$361$	3823.01			0.557372
$362$	−15.7678	−	27.3106i	−0.00228933	−	0.00396523i
$363$		0			0
$364$	−1717.95	+	2975.58i	−0.247376	+	0.428469i
$365$	2375.56	−	4114.59i	0.340665	−	0.590049i
$366$		0			0
$367$	2523.39	+	4370.64i	0.358910	+	0.621650i	0.987779	−	0.155861i	$-0.0498152\pi$
$367$	2523.39	+	4370.64i	0.358910	+	0.621650i	−0.628869	+	0.777511i	$0.716482\pi$
$368$	3139.10			0.444666
$369$		0			0
$370$	−3045.76			−0.427949
$371$	153.624	+	266.085i	0.0214980	+	0.0372357i
$372$		0			0
$373$	5092.02	−	8819.63i	0.706849	−	1.22430i	−0.259171	−	0.965831i	$-0.583449\pi$
$373$	5092.02	−	8819.63i	0.706849	−	1.22430i	0.966020	−	0.258467i	$-0.0832172\pi$
$374$	206.968	−	358.480i	0.0286152	−	0.0495630i
$375$		0			0
$376$	−948.517	−	1642.88i	−0.130096	−	0.225333i
$377$	−1058.38			−0.144587
$378$		0			0
$379$	−7332.90			−0.993841			−0.496921	−	0.867796i	$-0.665536\pi$
$379$	−7332.90			−0.993841			−0.496921	+	0.867796i	$0.665536\pi$
$380$	1033.54	+	1790.14i	0.139525	+	0.241664i
$381$		0			0
$382$	1597.73	−	2767.34i	0.213997	−	0.370654i
$383$	−542.748	+	940.067i	−0.0724103	+	0.125418i	−0.899957	−	0.435978i	$-0.856402\pi$
$383$	−542.748	+	940.067i	−0.0724103	+	0.125418i	0.827547	+	0.561397i	$0.189736\pi$
$384$		0			0
$385$	−468.328	−	811.167i	−0.0619953	−	0.107379i
$386$	1159.74			0.152925
$387$		0			0
$388$	5270.83			0.689654
$389$	−213.944	−	370.562i	−0.0278853	−	0.0482988i	0.851746	−	0.523955i	$-0.175544\pi$
$389$	−213.944	−	370.562i	−0.0278853	−	0.0482988i	−0.879631	+	0.475656i	$0.842211\pi$
$390$		0			0
$391$	1064.17	−	1843.20i	0.137641	−	0.238401i
$392$	986.357	−	1708.42i	0.127088	−	0.220123i
$393$		0			0
$394$	−2095.96	−	3630.31i	−0.268003	−	0.464194i
$395$	−1875.48			−0.238901
$396$		0			0
$397$	−2668.85			−0.337394			−0.168697	−	0.985668i	$-0.553956\pi$
$397$	−2668.85			−0.337394			−0.168697	+	0.985668i	$0.553956\pi$
$398$	1748.58	+	3028.62i	0.220222	+	0.381435i
$399$		0			0
$400$	−200.000	+	346.410i	−0.0250000	+	0.0433013i
$401$	−5967.59	+	10336.2i	−0.743160	+	1.28719i	0.207889	+	0.978152i	$0.433341\pi$
$401$	−5967.59	+	10336.2i	−0.743160	+	1.28719i	−0.951049	+	0.309039i	$0.899993\pi$
$402$		0			0
$403$	10272.1	+	17791.8i	1.26970	+	2.19919i
$404$	2571.93			0.316728
$405$		0			0
$406$	−237.584			−0.0290421
$407$	−2905.44	−	5032.37i	−0.353851	−	0.612888i
$408$		0			0
$409$	383.212	−	663.743i	0.0463291	−	0.0802444i	−0.841931	−	0.539585i	$-0.818581\pi$
$409$	383.212	−	663.743i	0.0463291	−	0.0802444i	0.888260	+	0.459341i	$0.151914\pi$
$410$	−1823.80	+	3158.91i	−0.219685	+	0.380506i
$411$		0			0
$412$	−110.731	−	191.792i	−0.0132411	−	0.0229342i
$413$	1273.45			0.151725
$414$		0			0
$415$	−5385.52			−0.637024
$416$	1399.71	+	2424.37i	0.164967	+	0.285732i
$417$		0			0
$418$	−1971.85	+	3415.34i	−0.230733	+	0.399641i
$419$	−5817.04	+	10075.4i	−0.678237	+	1.17474i	0.297275	+	0.954792i	$0.403922\pi$
$419$	−5817.04	+	10075.4i	−0.678237	+	1.17474i	−0.975511	+	0.219948i	$0.929411\pi$
$420$		0			0
$421$	−5819.41	−	10079.5i	−0.673683	−	1.16685i	−0.976852	−	0.213916i	$-0.931378\pi$
$421$	−5819.41	−	10079.5i	−0.673683	−	1.16685i	0.303169	−	0.952937i	$-0.401955\pi$
$422$	2245.33			0.259007
$423$		0			0
$424$	250.332			0.0286727
$425$	135.602	+	234.870i	0.0154769	+	0.0268068i
$426$		0			0
$427$	−2645.90	+	4582.83i	−0.299869	+	0.519388i
$428$	357.173	−	618.641i	0.0403379	−	0.0698672i
$429$		0			0
$430$	1697.59	+	2940.31i	0.190383	+	0.329754i
$431$	7408.63			0.827984			0.413992	−	0.910280i	$-0.364134\pi$
$431$	7408.63			0.827984			0.413992	+	0.910280i	$0.364134\pi$
$432$		0			0
$433$	9830.81			1.09108			0.545541	−	0.838084i	$-0.316324\pi$
$433$	9830.81			1.09108			0.545541	+	0.838084i	$0.316324\pi$
$434$	2305.87	+	3993.89i	0.255035	+	0.441734i
$435$		0			0
$436$	−1447.56	+	2507.25i	−0.159003	+	0.275402i
$437$	−10138.7	+	17560.7i	−1.10984	+	1.92230i
$438$		0			0
$439$	1865.84	+	3231.73i	0.202851	+	0.351349i	0.949446	−	0.313930i	$-0.101646\pi$
$439$	1865.84	+	3231.73i	0.202851	+	0.351349i	−0.746595	+	0.665279i	$0.768313\pi$
$440$	−763.145			−0.0826853
$441$		0			0
$442$	1898.04			0.204254
$443$	−4067.44	−	7045.01i	−0.436230	−	0.755572i	0.561165	−	0.827704i	$-0.310353\pi$
$443$	−4067.44	−	7045.01i	−0.436230	−	0.755572i	−0.997395	+	0.0721315i	$0.977020\pi$
$444$		0			0
$445$	−2274.18	+	3938.99i	−0.242261	+	0.419609i
$446$	−4452.63	+	7712.19i	−0.472732	+	0.818795i
$447$		0			0
$448$	314.205	+	544.219i	0.0331357	+	0.0573927i
$449$	−17834.3			−1.87450			−0.937250	−	0.348658i	$-0.886637\pi$
$449$	−17834.3			−1.87450			−0.937250	+	0.348658i	$0.886637\pi$
$450$		0			0
$451$	−6959.11			−0.726589
$452$	2364.52	+	4095.47i	0.246057	+	0.426183i
$453$		0			0
$454$	3271.60	−	5666.58i	0.338202	−	0.585784i
$455$	2147.44	−	3719.47i	0.221260	−	0.383234i
$456$		0			0
$457$	2478.35	+	4292.62i	0.253681	+	0.439388i	0.964536	−	0.263950i	$-0.0850254\pi$
$457$	2478.35	+	4292.62i	0.253681	+	0.439388i	−0.710856	+	0.703338i	$0.751692\pi$
$458$	154.658			0.0157788
$459$		0			0
$460$	−3923.88			−0.397721
$461$	1794.26	+	3107.76i	0.181274	+	0.313975i	0.942315	−	0.334729i	$-0.108645\pi$
$461$	1794.26	+	3107.76i	0.181274	+	0.313975i	−0.761041	+	0.648704i	$0.775311\pi$
$462$		0			0
$463$	−5250.52	+	9094.16i	−0.527024	+	0.912832i	0.472480	+	0.881341i	$0.343359\pi$
$463$	−5250.52	+	9094.16i	−0.527024	+	0.912832i	−0.999504	+	0.0314911i	$0.989974\pi$
$464$	−96.7863	+	167.639i	−0.00968360	+	0.0167725i
$465$		0			0
$466$	−4243.59	−	7350.11i	−0.421847	−	0.730660i
$467$	11303.2			1.12002			0.560009	−	0.828486i	$-0.310798\pi$
$467$	11303.2			1.12002			0.560009	+	0.828486i	$0.310798\pi$
$468$		0			0
$469$	−3309.57			−0.325846
$470$	1185.65	+	2053.60i	0.116361	+	0.201544i
$471$		0			0
$472$	518.775	−	898.544i	0.0505901	−	0.0876247i
$473$	−3238.76	+	5609.70i	−0.314838	+	0.545316i
$474$		0			0
$475$	−1291.92	−	2237.68i	−0.124795	−	0.216151i
$476$	426.069			0.0410270
$477$		0			0
$478$	−156.989			−0.0150220
$479$	8646.24	+	14975.7i	0.824753	+	1.42851i	0.902108	+	0.431511i	$0.142019\pi$
$479$	8646.24	+	14975.7i	0.824753	+	1.42851i	−0.0773548	+	0.997004i	$0.524647\pi$
$480$		0			0
$481$	13322.4	−	23075.1i	1.26289	−	2.18739i
$482$	5397.70	−	9349.09i	0.510080	−	0.883484i
$483$		0			0
$484$	1934.01	+	3349.81i	0.181631	+	0.314595i
$485$	−6588.53			−0.616845
$486$		0			0
$487$	1617.93			0.150545			0.0752724	−	0.997163i	$-0.476017\pi$
$487$	1617.93			0.150545			0.0752724	+	0.997163i	$0.476017\pi$
$488$	2155.76	+	3733.89i	0.199973	+	0.346363i
$489$		0			0
$490$	−1232.95	+	2135.53i	−0.113671	+	0.196884i
$491$	1370.85	−	2374.38i	0.125999	−	0.218236i	−0.796124	−	0.605133i	$-0.793120\pi$
$491$	1370.85	−	2374.38i	0.125999	−	0.218236i	0.922123	+	0.386897i	$0.126453\pi$
$492$		0			0
$493$	65.6222	+	113.661i	0.00599488	+	0.0103834i
$494$	−18083.2			−1.64696
$495$		0			0
$496$	3757.44			0.340150
$497$	1299.86	+	2251.42i	0.117317	+	0.203199i
$498$		0			0
$499$	−6878.59	+	11914.1i	−0.617090	+	1.06883i	0.372924	+	0.927862i	$0.378355\pi$
$499$	−6878.59	+	11914.1i	−0.617090	+	1.06883i	−0.990014	+	0.140969i	$0.954978\pi$
$500$	250.000	−	433.013i	0.0223607	−	0.0387298i
$501$		0			0
$502$	1627.51	+	2818.94i	0.144700	+	0.250628i
$503$	−5929.32			−0.525597			−0.262798	−	0.964851i	$-0.584645\pi$
$503$	−5929.32			−0.525597			−0.262798	+	0.964851i	$0.584645\pi$
$504$		0			0
$505$	−3214.91			−0.283290
$506$	−3743.11	−	6483.26i	−0.328857	−	0.569597i
$507$		0			0
$508$	465.915	−	806.988i	0.0406922	−	0.0704809i
$509$	8275.81	−	14334.1i	0.720666	−	1.24823i	−0.240067	−	0.970756i	$-0.577169\pi$
$509$	8275.81	−	14334.1i	0.720666	−	1.24823i	0.960733	−	0.277474i	$-0.0894972\pi$
$510$		0			0
$511$	4665.08	+	8080.16i	0.403857	+	0.699501i
$512$	512.000			0.0441942
$513$		0			0
$514$	−12366.3			−1.06119
$515$	138.414	+	239.740i	0.0118432	+	0.0205130i
$516$		0			0
$517$	−2262.05	+	3917.99i	−0.192427	+	0.333294i
$518$	2990.60	−	5179.86i	0.253666	−	0.439363i
$519$		0			0
$520$	−1749.64	−	3030.46i	−0.147551	−	0.255566i
$521$	20294.3			1.70654			0.853271	−	0.521468i	$-0.174616\pi$
$521$	20294.3			1.70654			0.853271	+	0.521468i	$0.174616\pi$
$522$		0			0
$523$	21068.6			1.76150			0.880752	−	0.473578i	$-0.157038\pi$
$523$	21068.6			1.76150			0.880752	+	0.473578i	$0.157038\pi$
$524$	4301.70	+	7450.76i	0.358627	+	0.621160i
$525$		0			0
$526$	−7289.27	+	12625.4i	−0.604235	+	1.04657i
$527$	1273.79	−	2206.28i	0.105289	−	0.182366i
$528$		0			0
$529$	−13162.5	−	22798.2i	−1.08182	−	1.87377i
$530$	−312.915			−0.0256456
$531$		0			0
$532$	−4059.28			−0.330812
$533$	−15954.9	−	27634.7i	−1.29659	−	2.24576i
$534$		0			0
$535$	−446.466	+	773.302i	−0.0360793	+	0.0624911i
$536$	−1348.25	+	2335.23i	−0.108648	+	0.188184i
$537$		0			0
$538$	1903.32	+	3296.65i	0.152524	+	0.264180i
$539$	−4704.58			−0.375957
$540$		0			0
$541$	−14520.4			−1.15394			−0.576968	−	0.816767i	$-0.695764\pi$
$541$	−14520.4			−1.15394			−0.576968	+	0.816767i	$0.695764\pi$
$542$	−6485.88	−	11233.9i	−0.514008	−	0.890288i
$543$		0			0
$544$	173.571	−	300.634i	0.0136798	−	0.0236941i
$545$	1809.45	−	3134.06i	0.142217	−	0.246327i
$546$		0			0
$547$	3846.59	+	6662.49i	0.300673	+	0.520782i	0.976289	−	0.216473i	$-0.0694553\pi$
$547$	3846.59	+	6662.49i	0.300673	+	0.520782i	−0.675615	+	0.737254i	$0.736122\pi$
$548$	9406.95			0.733294
$549$		0			0
$550$	953.931			0.0739559
$551$	−625.202	−	1082.88i	−0.0483385	−	0.0837247i
$552$		0			0
$553$	1841.52	−	3189.60i	0.141608	−	0.245273i
$554$	5847.78	−	10128.6i	0.448463	−	0.776760i
$555$		0			0
$556$	−2138.40	−	3703.82i	−0.163109	−	0.282513i
$557$	3868.38			0.294270			0.147135	−	0.989116i	$-0.452995\pi$
$557$	3868.38			0.294270			0.147135	+	0.989116i	$0.452995\pi$
$558$		0			0
$559$	−29701.6			−2.24730
$560$	−392.756	−	680.273i	−0.0296374	−	0.0513335i
$561$		0			0
$562$	3736.13	−	6471.16i	0.280425	−	0.485711i
$563$	2571.34	−	4453.69i	0.192485	−	0.333394i	−0.753588	−	0.657347i	$-0.771679\pi$
$563$	2571.34	−	4453.69i	0.192485	−	0.333394i	0.946073	+	0.323953i	$0.105012\pi$
$564$		0			0
$565$	−2955.65	−	5119.34i	−0.220080	−	0.381190i
$566$	−5720.22			−0.424803
$567$		0			0
$568$	2118.13			0.156470
$569$	6020.87	+	10428.4i	0.443599	+	0.768336i	0.997953	−	0.0639446i	$-0.0203681\pi$
$569$	6020.87	+	10428.4i	0.443599	+	0.768336i	−0.554354	+	0.832281i	$0.687035\pi$
$570$		0			0
$571$	686.300	−	1188.71i	0.0502990	−	0.0871205i	−0.839780	−	0.542927i	$-0.817316\pi$
$571$	686.300	−	1188.71i	0.0502990	−	0.0871205i	0.890079	+	0.455807i	$0.150649\pi$
$572$	3338.06	−	5781.70i	0.244006	−	0.422631i
$573$		0			0
$574$	−3581.54	−	6203.40i	−0.260436	−	0.451089i
$575$	4904.85			0.355733
$576$		0			0
$577$	−16953.7			−1.22321			−0.611605	−	0.791164i	$-0.709476\pi$
$577$	−16953.7			−1.22321			−0.611605	+	0.791164i	$0.709476\pi$
$578$	4795.32	+	8305.73i	0.345085	+	0.597704i
$579$		0			0
$580$	120.983	−	209.549i	0.00866128	−	0.0150018i
$581$	5287.99	−	9159.06i	0.377595	−	0.654014i
$582$		0			0
$583$	−298.500	−	517.017i	−0.0212051	−	0.0367284i
$584$	7601.80			0.538638
$585$		0			0
$586$	16077.5			1.13337
$587$	6767.07	+	11720.9i	0.475821	+	0.824146i	0.999616	−	0.0276983i	$-0.00881777\pi$
$587$	6767.07	+	11720.9i	0.475821	+	0.824146i	−0.523796	+	0.851844i	$0.675484\pi$
$588$		0			0
$589$	−12135.8	+	21019.9i	−0.848977	+	1.47047i
$590$	−648.469	+	1123.18i	−0.0452492	+	0.0783739i
$591$		0			0
$592$	−2436.60	−	4220.32i	−0.169162	−	0.292997i
$593$	6652.32			0.460671			0.230336	−	0.973111i	$-0.426018\pi$
$593$	6652.32			0.460671			0.230336	+	0.973111i	$0.426018\pi$
$594$		0			0
$595$	−532.586			−0.0366956
$596$	−623.691	−	1080.26i	−0.0428647	−	0.0742439i
$597$		0			0
$598$	17163.4	−	29727.9i	1.17369	−	2.03288i
$599$	9521.24	−	16491.3i	0.649461	−	1.12490i	−0.333791	−	0.942647i	$-0.608328\pi$
$599$	9521.24	−	16491.3i	0.649461	−	1.12490i	0.983252	−	0.182253i	$-0.0583389\pi$
$600$		0			0
$601$	8537.22	+	14786.9i	0.579435	+	1.00361i	0.995544	+	0.0942963i	$0.0300601\pi$
$601$	8537.22	+	14786.9i	0.579435	+	1.00361i	−0.416109	+	0.909315i	$0.636607\pi$
$602$	−6667.37			−0.451398
$603$		0			0
$604$	10137.7			0.682943
$605$	−2417.52	−	4187.26i	−0.162456	−	0.281382i
$606$		0			0
$607$	−9180.13	+	15900.5i	−0.613855	+	1.06323i	0.376729	+	0.926323i	$0.377049\pi$
$607$	−9180.13	+	15900.5i	−0.613855	+	1.06323i	−0.990584	+	0.136905i	$0.956285\pi$
$608$	−1653.66	+	2864.23i	−0.110304	+	0.191052i
$609$		0			0
$610$	−2694.70	−	4667.36i	−0.178861	−	0.309796i
$611$	−20744.5			−1.37354
$612$		0			0
$613$	−6500.25			−0.428291			−0.214146	−	0.976802i	$-0.568697\pi$
$613$	−6500.25			−0.428291			−0.214146	+	0.976802i	$0.568697\pi$
$614$	−3066.57	−	5311.46i	−0.201558	−	0.349109i
$615$		0			0
$616$	749.324	−	1297.87i	0.0490116	−	0.0848905i
$617$	−3553.17	+	6154.26i	−0.231840	+	0.401558i	−0.958350	−	0.285598i	$-0.907808\pi$
$617$	−3553.17	+	6154.26i	−0.231840	+	0.401558i	0.726510	+	0.687156i	$0.241141\pi$
$618$		0			0
$619$	−9784.01	−	16946.4i	−0.635303	−	1.10038i	−0.986451	−	0.164057i	$-0.947542\pi$
$619$	−9784.01	−	16946.4i	−0.635303	−	1.10038i	0.351148	−	0.936320i	$-0.385791\pi$
$620$	−4696.80			−0.304239
$621$		0			0
$622$	19458.4			1.25436
$623$	−4465.98	−	7735.31i	−0.287200	−	0.497445i
$624$		0			0
$625$	−312.500	+	541.266i	−0.0200000	+	0.0346410i
$626$	717.127	−	1242.10i	0.0457862	−	0.0793041i
$627$		0			0
$628$	2114.68	+	3662.73i	0.134371	+	0.232737i
$629$	−3304.09			−0.209448
$630$		0			0
$631$	26180.9			1.65174			0.825869	−	0.563862i	$-0.190685\pi$
$631$	26180.9			1.65174			0.825869	+	0.563862i	$0.190685\pi$
$632$	−1500.39	−	2598.75i	−0.0944338	−	0.163564i
$633$		0			0
$634$	3132.05	−	5424.87i	0.196198	−	0.339825i
$635$	−582.393	+	1008.74i	−0.0363962	+	0.0630400i
$636$		0			0
$637$	−10786.0	−	18682.0i	−0.670892	−	1.16202i
$638$	461.637			0.0286464
$639$		0			0
$640$	−640.000			−0.0395285
$641$	−12003.1	−	20790.0i	−0.739618	−	1.28106i	−0.952667	−	0.304014i	$-0.901673\pi$
$641$	−12003.1	−	20790.0i	−0.739618	−	1.28106i	0.213050	−	0.977041i	$-0.431660\pi$
$642$		0			0
$643$	−4655.69	+	8063.89i	−0.285540	+	0.494570i	−0.972740	−	0.231898i	$-0.925506\pi$
$643$	−4655.69	+	8063.89i	−0.285540	+	0.494570i	0.687200	+	0.726468i	$0.258840\pi$
$644$	3852.82	−	6673.28i	0.235749	−	0.408329i
$645$		0			0
$646$	1121.20	+	1941.98i	0.0682865	+	0.118276i
$647$	−31854.2			−1.93558			−0.967789	−	0.251764i	$-0.918989\pi$
$647$	−31854.2			−1.93558			−0.967789	+	0.251764i	$0.918989\pi$
$648$		0			0
$649$	−2474.38			−0.149658
$650$	2187.04	+	3788.07i	0.131974	+	0.228585i
$651$		0			0
$652$	−5330.28	+	9232.32i	−0.320169	+	0.554548i
$653$	−4856.41	+	8411.56i	−0.291035	+	0.504088i	−0.974055	−	0.226312i	$-0.927333\pi$
$653$	−4856.41	+	8411.56i	−0.291035	+	0.504088i	0.683019	+	0.730400i	$0.260666\pi$
$654$		0			0
$655$	−5377.12	−	9313.45i	−0.320766	−	0.555582i
$656$	−5836.15			−0.347353
$657$		0			0
$658$	−4656.70			−0.275892
$659$	−6998.63	−	12122.0i	−0.413699	−	0.716548i	0.581592	−	0.813481i	$-0.302430\pi$
$659$	−6998.63	−	12122.0i	−0.413699	−	0.716548i	−0.995291	+	0.0969327i	$0.969097\pi$
$660$		0			0
$661$	−13935.0	+	24136.1i	−0.819982	+	1.42025i	0.0857125	+	0.996320i	$0.472683\pi$
$661$	−13935.0	+	24136.1i	−0.819982	+	1.42025i	−0.905695	+	0.423931i	$0.860650\pi$
$662$	4822.14	−	8352.19i	0.283108	−	0.490358i
$663$		0			0
$664$	−4308.42	−	7462.40i	−0.251806	−	0.436140i
$665$	5074.10			0.295888
$666$		0			0
$667$	2373.61			0.137791
$668$	3069.01	+	5315.68i	0.177760	+	0.307889i
$669$		0			0
$670$	1685.31	−	2919.04i	0.0971778	−	0.168317i
$671$	5141.12	−	8904.68i	0.295783	−	0.512312i
$672$		0			0
$673$	−6072.89	−	10518.5i	−0.347834	−	0.602467i	0.638030	−	0.770011i	$-0.279750\pi$
$673$	−6072.89	−	10518.5i	−0.347834	−	0.602467i	−0.985865	+	0.167545i	$0.946416\pi$
$674$	−12721.6			−0.727027
$675$		0			0
$676$	21824.3			1.24171
$677$	−3289.74	−	5697.99i	−0.186758	−	0.323474i	0.757410	−	0.652940i	$-0.226465\pi$
$677$	−3289.74	−	5697.99i	−0.186758	−	0.323474i	−0.944167	+	0.329466i	$0.893131\pi$
$678$		0			0
$679$	6469.21	−	11205.0i	0.365634	−	0.633297i
$680$	−216.964	+	375.792i	−0.0122356	+	0.0211926i
$681$		0			0
$682$	−4480.43	−	7760.33i	−0.251561	−	0.435716i
$683$	21314.6			1.19412			0.597058	−	0.802198i	$-0.296336\pi$
$683$	21314.6			1.19412			0.597058	+	0.802198i	$0.296336\pi$
$684$		0			0
$685$	−11758.7			−0.655878
$686$	−5789.12	−	10027.0i	−0.322200	−	0.558067i
$687$		0			0
$688$	−2716.14	+	4704.49i	−0.150511	+	0.260693i
$689$	1368.72	−	2370.69i	0.0756808	−	0.131083i
$690$		0			0
$691$	7169.95	+	12418.7i	0.394729	+	0.683690i	0.993067	−	0.117554i	$-0.0375052\pi$
$691$	7169.95	+	12418.7i	0.394729	+	0.683690i	−0.598338	+	0.801244i	$0.704172\pi$
$692$	−8031.38			−0.441195
$693$		0			0
$694$	−20661.3			−1.13010
$695$	2673.00	+	4629.78i	0.145889	+	0.252687i
$696$		0			0
$697$	−1978.49	+	3426.84i	−0.107519	+	0.186228i
$698$	−9630.71	+	16680.9i	−0.522246	+	0.904557i
$699$		0			0
$700$	490.945	+	850.341i	0.0265085	+	0.0459141i
$701$	8307.83			0.447621			0.223811	−	0.974633i	$-0.428150\pi$
$701$	8307.83			0.447621			0.223811	+	0.974633i	$0.428150\pi$
$702$		0			0
$703$	31479.0			1.68884
$704$	−610.516	−	1057.44i	−0.0326842	−	0.0566107i
$705$		0			0
$706$	11395.2	−	19737.0i	0.607454	−	1.05214i
$707$	3156.69	−	5467.54i	0.167920	−	0.290846i
$708$		0			0
$709$	3598.27	+	6232.38i	0.190600	+	0.330130i	0.945449	−	0.325769i	$-0.105623\pi$
$709$	3598.27	+	6232.38i	0.190600	+	0.330130i	−0.754849	+	0.655899i	$0.772290\pi$
$710$	−2647.67			−0.139951
$711$		0			0
$712$	−7277.37			−0.383049
$713$	−23037.1	−	39901.5i	−1.21002	−	2.09582i
$714$		0			0
$715$	−4172.58	+	7227.12i	−0.218246	+	0.378013i
$716$	9176.66	−	15894.4i	0.478978	−	0.829614i
$717$		0			0
$718$	−1549.64	−	2684.06i	−0.0805461	−	0.139510i
$719$	−4830.88			−0.250572			−0.125286	−	0.992121i	$-0.539985\pi$
$719$	−4830.88			−0.250572			−0.125286	+	0.992121i	$0.539985\pi$
$720$		0			0
$721$	−543.628			−0.0280801
$722$	−3823.01	−	6621.65i	−0.197061	−	0.341319i
$723$		0			0
$724$	−31.5355	+	54.6212i	−0.00161880	+	0.00280384i
$725$	−151.229	+	261.936i	−0.00774688	+	0.0134180i
$726$		0			0
$727$	14009.6	+	24265.3i	0.714699	+	1.23790i	0.963075	+	0.269232i	$0.0867698\pi$
$727$	14009.6	+	24265.3i	0.714699	+	1.23790i	−0.248376	+	0.968664i	$0.579897\pi$
$728$	6871.80			0.349843
$729$		0			0
$730$	−9502.25			−0.481773
$731$	1841.57	+	3189.70i	0.0931779	+	0.161389i
$732$		0			0
$733$	4462.60	−	7729.45i	0.224870	−	0.389487i	−0.731410	−	0.681938i	$-0.761138\pi$
$733$	4462.60	−	7729.45i	0.224870	−	0.389487i	0.956281	+	0.292451i	$0.0944708\pi$
$734$	5046.78	−	8741.28i	0.253788	−	0.439573i
$735$		0			0
$736$	−3139.10	−	5437.09i	−0.157213	−	0.272301i
$737$	6430.67			0.321407
$738$		0			0
$739$	−30797.6			−1.53303			−0.766515	−	0.642226i	$-0.778011\pi$
$739$	−30797.6			−1.53303			−0.766515	+	0.642226i	$0.778011\pi$
$740$	3045.76	+	5275.40i	0.151303	+	0.262064i
$741$		0			0
$742$	307.248	−	532.170i	0.0152014	−	0.0263296i
$743$	8559.00	−	14824.6i	0.422610	−	0.731982i	−0.573584	−	0.819147i	$-0.694447\pi$
$743$	8559.00	−	14824.6i	0.422610	−	0.731982i	0.996194	+	0.0871645i	$0.0277806\pi$
$744$		0			0
$745$	779.614	+	1350.33i	0.0383394	+	0.0664058i
$746$	−20368.1			−0.999635
$747$		0			0
$748$	−827.874			−0.0404680
$749$	−876.761	−	1518.59i	−0.0427719	−	0.0740831i
$750$		0			0
$751$	−16623.2	+	28792.3i	−0.807711	+	1.39900i	0.106735	+	0.994288i	$0.465960\pi$
$751$	−16623.2	+	28792.3i	−0.807711	+	1.39900i	−0.914446	+	0.404709i	$0.867373\pi$
$752$	−1897.03	+	3285.76i	−0.0919916	+	0.159334i
$753$		0			0
$754$	1058.38	+	1833.17i	0.0511193	+	0.0885412i
$755$	−12672.1			−0.610843
$756$		0			0
$757$	41146.4			1.97555			0.987775	−	0.155888i	$-0.0498239\pi$
$757$	41146.4			1.97555			0.987775	+	0.155888i	$0.0498239\pi$
$758$	7332.90	+	12701.0i	0.351376	+	0.608601i
$759$		0			0
$760$	2067.08	−	3580.28i	0.0986589	−	0.170882i
$761$	−16150.0	+	27972.7i	−0.769301	+	1.33247i	0.168642	+	0.985677i	$0.446062\pi$
$761$	−16150.0	+	27972.7i	−0.769301	+	1.33247i	−0.937943	+	0.346790i	$0.887272\pi$
$762$		0			0
$763$	3553.36	+	6154.60i	0.168598	+	0.292020i
$764$	−6390.91			−0.302637
$765$		0			0
$766$	2170.99			0.102404
$767$	−5672.92	−	9825.78i	−0.267063	−	0.462567i
$768$		0			0
$769$	−14965.2	+	25920.6i	−0.701769	+	1.21550i	0.266076	+	0.963952i	$0.414273\pi$
$769$	−14965.2	+	25920.6i	−0.701769	+	1.21550i	−0.967845	+	0.251548i	$0.919060\pi$
$770$	−936.655	+	1622.33i	−0.0438373	+	0.0759284i
$771$		0			0
$772$	−1159.74	−	2008.72i	−0.0540671	−	0.0936469i
$773$	15385.0			0.715860			0.357930	−	0.933748i	$-0.383483\pi$
$773$	15385.0			0.715860			0.357930	+	0.933748i	$0.383483\pi$
$774$		0			0
$775$	5871.00			0.272120
$776$	−5270.83	−	9129.34i	−0.243829	−	0.422325i
$777$		0			0
$778$	−427.888	+	741.124i	−0.0197179	+	0.0341524i
$779$	18849.6	−	32648.5i	0.866956	−	1.50161i
$780$		0			0
$781$	−2525.69	−	4374.63i	−0.115719	−	0.200431i
$782$	−4256.70			−0.194654
$783$		0			0
$784$	−3945.43			−0.179730
$785$	−2643.35	−	4578.42i	−0.120185	−	0.208167i
$786$		0			0
$787$	7942.82	−	13757.4i	0.359760	−	0.623122i	−0.628161	−	0.778084i	$-0.716192\pi$
$787$	7942.82	−	13757.4i	0.359760	−	0.623122i	0.987921	+	0.154961i	$0.0495253\pi$
$788$	−4191.92	+	7260.63i	−0.189507	+	0.328235i
$789$		0			0
$790$	1875.48	+	3248.43i	0.0844642	+	0.146296i
$791$	11608.5			0.521809
$792$		0			0
$793$	47147.5			2.11129
$794$	2668.85	+	4622.58i	0.119287	+	0.206611i
$795$		0			0
$796$	3497.15	−	6057.25i	0.155720	−	0.269715i
$797$	−16511.2	+	28598.2i	−0.733821	+	1.27102i	0.221418	+	0.975179i	$0.428932\pi$
$797$	−16511.2	+	28598.2i	−0.733821	+	1.27102i	−0.955239	+	0.295836i	$0.904402\pi$
$798$		0			0
$799$	1286.21	+	2227.78i	0.0569498	+	0.0986399i
$800$	800.000			0.0353553
$801$		0			0
$802$	23870.4			1.05099
$803$	−9064.49	−	15700.2i	−0.398355	−	0.689971i
$804$		0			0
$805$	−4816.02	+	8341.60i	−0.210860	+	0.365221i
$806$	20544.2	−	35583.7i	0.897816	−	1.55506i
$807$		0			0
$808$	−2571.93	−	4454.71i	−0.111980	−	0.193956i
$809$	11391.7			0.495070			0.247535	−	0.968879i	$-0.420379\pi$
$809$	11391.7			0.495070			0.247535	+	0.968879i	$0.420379\pi$
$810$		0			0
$811$	−14491.3			−0.627446			−0.313723	−	0.949515i	$-0.601576\pi$
$811$	−14491.3			−0.627446			−0.313723	+	0.949515i	$0.601576\pi$
$812$	237.584	+	411.507i	0.0102679	+	0.0177846i
$813$		0			0
$814$	−5810.88	+	10064.7i	−0.250210	+	0.433377i
$815$	6662.85	−	11540.4i	0.286368	−	0.496003i
$816$		0			0
$817$	−17545.2	−	30389.2i	−0.751321	−	1.30133i
$818$	−1532.85			−0.0655193
$819$		0			0
$820$	7295.19			0.310682
$821$	8212.54	+	14224.5i	0.349110	+	0.604677i	0.986092	−	0.166202i	$-0.0531505\pi$
$821$	8212.54	+	14224.5i	0.349110	+	0.604677i	−0.636981	+	0.770879i	$0.719817\pi$
$822$		0			0
$823$	−14450.3	+	25028.6i	−0.612035	+	1.06008i	0.378862	+	0.925453i	$0.376316\pi$
$823$	−14450.3	+	25028.6i	−0.612035	+	1.06008i	−0.990897	+	0.134623i	$0.957018\pi$
$824$	−221.462	+	383.583i	−0.00936285	+	0.0162169i
$825$		0			0
$826$	−1273.45	−	2205.68i	−0.0536428	−	0.0929121i
$827$	7264.81			0.305468			0.152734	−	0.988267i	$-0.451192\pi$
$827$	7264.81			0.305468			0.152734	+	0.988267i	$0.451192\pi$
$828$		0			0
$829$	−40403.8			−1.69274			−0.846370	−	0.532595i	$-0.821217\pi$
$829$	−40403.8			−1.69274			−0.846370	+	0.532595i	$0.821217\pi$
$830$	5385.52	+	9328.00i	0.225222	+	0.390096i
$831$		0			0
$832$	2799.42	−	4848.73i	0.116649	−	0.202043i
$833$	−1337.52	+	2316.66i	−0.0556332	+	0.0963595i
$834$		0			0
$835$	−3836.27	−	6644.61i	−0.158993	−	0.275385i
$836$	7887.40			0.326305
$837$		0			0
$838$	23268.2			0.959172
$839$	−20453.7	−	35426.8i	−0.841645	−	1.45777i	−0.888503	−	0.458870i	$-0.848254\pi$
$839$	−20453.7	−	35426.8i	−0.841645	−	1.45777i	0.0468588	−	0.998902i	$-0.485079\pi$
$840$		0			0
$841$	12121.3	−	20994.7i	0.496999	−	0.860828i
$842$	−11638.8	+	20159.0i	−0.476366	+	0.825090i
$843$		0			0
$844$	−2245.33	−	3889.02i	−0.0915727	−	0.158608i
$845$	−27280.3			−1.11062
$846$		0			0
$847$	9494.93			0.385183
$848$	−250.332	−	433.588i	−0.0101373	−	0.0175584i
$849$		0			0
$850$	271.205	−	469.740i	0.0109438	−	0.0189552i
$851$	−29877.9	+	51750.1i	−1.20353	+	2.08457i
$852$		0			0
$853$	2393.17	+	4145.09i	0.0960615	+	0.166383i	0.910051	−	0.414496i	$-0.136042\pi$
$853$	2393.17	+	4145.09i	0.0960615	+	0.166383i	−0.813990	+	0.580879i	$0.802709\pi$
$854$	10583.6			0.424079
$855$		0			0
$856$	−1428.69			−0.0570463
$857$	−13891.0	−	24059.9i	−0.553683	−	0.959007i	−0.998005	−	0.0631400i	$-0.979889\pi$
$857$	−13891.0	−	24059.9i	−0.553683	−	0.959007i	0.444321	−	0.895867i	$-0.353445\pi$
$858$		0			0
$859$	−4523.68	+	7835.25i	−0.179681	+	0.311217i	−0.941771	−	0.336254i	$-0.890840\pi$
$859$	−4523.68	+	7835.25i	−0.179681	+	0.311217i	0.762090	+	0.647471i	$0.224173\pi$
$860$	3395.17	−	5880.61i	0.134621	−	0.233171i
$861$		0			0
$862$	−7408.63	−	12832.1i	−0.292737	−	0.507035i
$863$	−9705.84			−0.382840			−0.191420	−	0.981508i	$-0.561309\pi$
$863$	−9705.84			−0.382840			−0.191420	+	0.981508i	$0.561309\pi$
$864$		0			0
$865$	10039.2			0.394617
$866$	−9830.81	−	17027.5i	−0.385756	−	0.668149i
$867$		0			0
$868$	4611.74	−	7987.77i	0.180337	−	0.312353i
$869$	−3578.16	+	6197.56i	−0.139679	+	0.241931i
$870$		0			0
$871$	14743.4	+	25536.3i	0.573548	+	0.993414i
$872$	5790.24			0.224865
$873$		0			0
$874$	40554.8			1.56955
$875$	−613.681	−	1062.93i	−0.0237100	−	0.0410668i
$876$		0			0
$877$	675.238	−	1169.55i	0.0259990	−	0.0450317i	−0.852733	−	0.522347i	$-0.825057\pi$
$877$	675.238	−	1169.55i	0.0259990	−	0.0450317i	0.878732	+	0.477315i	$0.158390\pi$
$878$	3731.68	−	6463.46i	0.143438	−	0.248441i
$879$		0			0
$880$	763.145	+	1321.81i	0.0292337	+	0.0506342i
$881$	15199.2			0.581243			0.290622	−	0.956838i	$-0.406138\pi$
$881$	15199.2			0.581243			0.290622	+	0.956838i	$0.406138\pi$
$882$		0			0
$883$	−5061.18			−0.192890			−0.0964452	−	0.995338i	$-0.530747\pi$
$883$	−5061.18			−0.192890			−0.0964452	+	0.995338i	$0.530747\pi$
$884$	−1898.04	−	3287.50i	−0.0722148	−	0.125080i
$885$		0			0
$886$	−8134.88	+	14090.0i	−0.308461	+	0.534270i
$887$	−12071.9	+	20909.2i	−0.456974	+	0.791502i	−0.998799	−	0.0489889i	$-0.984400\pi$
$887$	−12071.9	+	20909.2i	−0.456974	+	0.791502i	0.541825	+	0.840491i	$0.317733\pi$
$888$		0			0
$889$	−1143.69	−	1980.93i	−0.0431476	−	0.0747338i
$890$	9096.71			0.342609
$891$		0			0
$892$	17810.5			0.668543
$893$	−12254.1	−	21224.7i	−0.459203	−	0.795363i
$894$		0			0
$895$	−11470.8	+	19868.1i	−0.428411	+	0.742029i
$896$	628.409	−	1088.44i	0.0234305	−	0.0405827i
$897$		0			0
$898$	17834.3	+	30889.8i	0.662736	+	1.14789i
$899$	2841.16			0.105404
$900$		0			0
$901$	−339.456			−0.0125515
$902$	6959.11	+	12053.5i	0.256888	+	0.444943i
$903$		0			0
$904$	4729.05	−	8190.95i	0.173989	−	0.301357i
$905$	39.4194	−	68.2765i	0.00144790	−	0.00250783i
$906$		0			0
$907$	−10600.0	−	18359.8i	−0.388058	−	0.672136i	0.604130	−	0.796885i	$-0.293521\pi$
$907$	−10600.0	−	18359.8i	−0.388058	−	0.672136i	−0.992188	+	0.124750i	$0.960187\pi$
$908$	−13086.4			−0.478290
$909$		0			0
$910$	−8589.75			−0.312909
$911$	15948.8	+	27624.1i	0.580030	+	1.00464i	0.995475	+	0.0950229i	$0.0302924\pi$
$911$	15948.8	+	27624.1i	0.580030	+	1.00464i	−0.415445	+	0.909618i	$0.636374\pi$
$912$		0			0
$913$	−10274.8	+	17796.5i	−0.372451	+	0.645103i
$914$	4956.69	−	8585.24i	0.179379	−	0.310694i
$915$		0			0
$916$	−154.658	−	267.875i	−0.00557864	−	0.00966249i
$917$	21119.0			0.760534
$918$		0			0
$919$	11183.2			0.401416			0.200708	−	0.979651i	$-0.435676\pi$
$919$	11183.2			0.401416			0.200708	+	0.979651i	$0.435676\pi$
$920$	3923.88	+	6796.36i	0.140616	+	0.243554i
$921$		0			0
$922$	3588.53	−	6215.52i	0.128180	−	0.222014i
$923$	11581.1	−	20059.1i	0.412998	−	0.715334i
$924$		0			0
$925$	−3807.19	−	6594.25i	−0.135329	−	0.234398i
$926$	21002.1			0.745325
$927$		0			0
$928$	387.145			0.0136947
$929$	12485.5	+	21625.5i	0.440943	+	0.763737i	0.997760	−	0.0668992i	$-0.0213106\pi$
$929$	12485.5	+	21625.5i	0.440943	+	0.763737i	−0.556816	+	0.830636i	$0.687977\pi$
$930$		0			0
$931$	12743.0	−	22071.5i	0.448587	−	0.776975i
$932$	−8487.18	+	14700.2i	−0.298291	+	0.516654i
$933$		0			0
$934$	−11303.2	−	19577.7i	−0.395986	−	0.685869i
$935$	1034.84			0.0361957
$936$		0			0
$937$	−26746.4			−0.932514			−0.466257	−	0.884649i	$-0.654398\pi$
$937$	−26746.4			−0.932514			−0.466257	+	0.884649i	$0.654398\pi$
$938$	3309.57	+	5732.35i	0.115204	+	0.199539i
$939$		0			0
$940$	2371.29	−	4107.20i	0.0822798	−	0.142513i
$941$	3071.90	−	5320.68i	0.106420	−	0.184324i	−0.807898	−	0.589323i	$-0.799395\pi$
$941$	3071.90	−	5320.68i	0.106420	−	0.184324i	0.914317	+	0.404998i	$0.132728\pi$
$942$		0			0
$943$	35781.8	+	61975.9i	1.23565	+	2.14021i
$944$	−2075.10			−0.0715453
$945$		0			0
$946$	12955.0			0.445248
$947$	18389.2	+	31851.1i	0.631013	+	1.09295i	0.987345	+	0.158588i	$0.0506940\pi$
$947$	18389.2	+	31851.1i	0.631013	+	1.09295i	−0.356332	+	0.934360i	$0.615973\pi$
$948$		0			0
$949$	41563.7	−	71990.4i	1.42172	−	2.46250i
$950$	−2583.85	+	4475.35i	−0.0882432	+	0.152842i
$951$		0			0
$952$	−426.069	−	737.973i	−0.0145052	−	0.0251238i
$953$	−617.414			−0.0209863			−0.0104932	−	0.999945i	$-0.503340\pi$
$953$	−617.414			−0.0209863			−0.0104932	+	0.999945i	$0.503340\pi$
$954$		0			0
$955$	7988.64			0.270687
$956$	156.989	+	271.913i	0.00531108	+	0.00919906i
$957$		0			0
$958$	17292.5	−	29951.4i	0.583188	−	1.01011i
$959$	11545.7	−	19997.8i	0.388771	−	0.673371i
$960$		0			0
$961$	−12679.5	−	21961.5i	−0.425614	−	0.737185i
$962$	−53289.6			−1.78599
$963$		0			0
$964$	−21590.8			−0.721361
$965$	1449.67	+	2510.90i	0.0483591	+	0.0837603i
$966$		0			0
$967$	713.826	−	1236.38i	0.0237385	−	0.0411162i	−0.853912	−	0.520417i	$-0.825776\pi$
$967$	713.826	−	1236.38i	0.0237385	−	0.0411162i	0.877651	+	0.479301i	$0.159110\pi$
$968$	3868.02	−	6699.61i	0.128433	−	0.222452i
$969$		0			0
$970$	6588.53	+	11411.7i	0.218088	+	0.377739i
$971$	−26134.6			−0.863748			−0.431874	−	0.901934i	$-0.642147\pi$
$971$	−26134.6			−0.863748			−0.431874	+	0.901934i	$0.642147\pi$
$972$		0			0
$973$	−10498.4			−0.345902
$974$	−1617.93	−	2802.34i	−0.0532257	−	0.0921895i
$975$		0			0
$976$	4311.52	−	7467.77i	0.141402	−	0.244916i
$977$	10006.5	−	17331.7i	0.327672	−	0.567545i	−0.654377	−	0.756168i	$-0.727069\pi$
$977$	10006.5	−	17331.7i	0.327672	−	0.567545i	0.982049	+	0.188623i	$0.0604026\pi$
$978$		0			0
$979$	8677.63	+	15030.1i	0.283287	+	0.490668i
$980$	4931.79			0.160755
$981$		0			0
$982$	−5483.39			−0.178189
$983$	−1056.68	−	1830.23i	−0.0342858	−	0.0593848i	0.848373	−	0.529398i	$-0.177582\pi$
$983$	−1056.68	−	1830.23i	−0.0342858	−	0.0593848i	−0.882659	+	0.470014i	$0.844249\pi$
$984$		0			0
$985$	5239.91	−	9075.78i	0.169500	−	0.293582i
$986$	131.244	−	227.322i	0.00423902	−	0.00734220i
$987$		0			0
$988$	18083.2	+	31320.9i	0.582289	+	1.00855i
$989$	66611.3			2.14167
$990$		0			0
$991$	11029.8			0.353554			0.176777	−	0.984251i	$-0.443433\pi$
$991$	11029.8			0.353554			0.176777	+	0.984251i	$0.443433\pi$
$992$	−3757.44	−	6508.08i	−0.120261	−	0.208298i
$993$		0			0
$994$	2599.72	−	4502.84i	0.0829557	−	0.143684i
$995$	−4371.44	+	7571.56i	−0.139280	+	0.241241i
$996$		0			0
$997$	5027.99	+	8708.74i	0.159717	+	0.276639i	0.934767	−	0.355262i	$-0.115608\pi$
$997$	5027.99	+	8708.74i	0.159717	+	0.276639i	−0.775049	+	0.631901i	$0.782275\pi$
$998$	27514.4			0.872697
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	810.4.e.bg.271.3		8
3.2	odd	2		810.4.e.bh.271.3		8
9.2	odd	6		810.4.e.bh.541.3		8
9.4	even	3		810.4.a.u.1.2	yes	4
9.5	odd	6		810.4.a.t.1.2	✓	4
9.7	even	3	inner	810.4.e.bg.541.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
810.4.a.t.1.2	✓	4	9.5	odd	6
810.4.a.u.1.2	yes	4	9.4	even	3
810.4.e.bg.271.3		8	1.1	even	1	trivial
810.4.e.bg.541.3		8	9.7	even	3	inner
810.4.e.bh.271.3		8	3.2	odd	2
810.4.e.bh.541.3		8	9.2	odd	6

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:	\( N \)	\(=\)	\( 810 = 2 \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	810.e (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.00000 - 1.73205i) q^{2} +(-2.00000 + 3.46410i) q^{4} +(2.50000 - 4.33013i) q^{5} +(4.90945 + 8.50341i) q^{7} +8.00000 q^{8} +O(q^{10})\)	\(q+(-1.00000 - 1.73205i) q^{2} +(-2.00000 + 3.46410i) q^{4} +(2.50000 - 4.33013i) q^{5} +(4.90945 + 8.50341i) q^{7} +8.00000 q^{8} -10.0000 q^{10} +(-9.53931 - 16.5226i) q^{11} +(43.7409 - 75.7615i) q^{13} +(9.81890 - 17.0068i) q^{14} +(-8.00000 - 13.8564i) q^{16} -10.8482 q^{17} +103.354 q^{19} +(10.0000 + 17.3205i) q^{20} +(-19.0786 + 33.0451i) q^{22} +(-98.0970 + 169.909i) q^{23} +(-12.5000 - 21.6506i) q^{25} -174.964 q^{26} -39.2756 q^{28} +(-6.04914 - 10.4774i) q^{29} +(-117.420 + 203.378i) q^{31} +(-16.0000 + 27.7128i) q^{32} +(10.8482 + 18.7896i) q^{34} +49.0945 q^{35} +304.576 q^{37} +(-103.354 - 179.014i) q^{38} +(20.0000 - 34.6410i) q^{40} +(182.380 - 315.891i) q^{41} +(-169.759 - 294.031i) q^{43} +76.3145 q^{44} +392.388 q^{46} +(-118.565 - 205.360i) q^{47} +(123.295 - 213.553i) q^{49} +(-25.0000 + 43.3013i) q^{50} +(174.964 + 303.046i) q^{52} +31.2915 q^{53} -95.3931 q^{55} +(39.2756 + 68.0273i) q^{56} +(-12.0983 + 20.9549i) q^{58} +(64.8469 - 112.318i) q^{59} +(269.470 + 466.736i) q^{61} +469.680 q^{62} +64.0000 q^{64} +(-218.704 - 378.807i) q^{65} +(-168.531 + 291.904i) q^{67} +(21.6964 - 37.5792i) q^{68} +(-49.0945 - 85.0341i) q^{70} +264.767 q^{71} +950.225 q^{73} +(-304.576 - 527.540i) q^{74} +(-206.708 + 358.028i) q^{76} +(93.6655 - 162.233i) q^{77} +(-187.548 - 324.843i) q^{79} -80.0000 q^{80} -729.519 q^{82} +(-538.552 - 932.800i) q^{83} +(-27.1205 + 46.9740i) q^{85} +(-339.517 + 588.061i) q^{86} +(-76.3145 - 132.181i) q^{88} -909.671 q^{89} +858.975 q^{91} +(-392.388 - 679.636i) q^{92} +(-237.129 + 410.720i) q^{94} +(258.385 - 447.535i) q^{95} +(-658.853 - 1141.17i) q^{97} -493.179 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{2} - 16 q^{4} + 20 q^{5} - 2 q^{7} + 64 q^{8}+O(q^{10}) \) 8 * q - 8 * q^2 - 16 * q^4 + 20 * q^5 - 2 * q^7 + 64 * q^8	\( 8 q - 8 q^{2} - 16 q^{4} + 20 q^{5} - 2 q^{7} + 64 q^{8} - 80 q^{10} + 36 q^{11} - 32 q^{13} - 4 q^{14} - 64 q^{16} + 180 q^{17} + 328 q^{19} + 80 q^{20} + 72 q^{22} - 42 q^{23} - 100 q^{25} + 128 q^{26} + 16 q^{28} - 36 q^{29} - 446 q^{31} - 128 q^{32} - 180 q^{34} - 20 q^{35} + 1372 q^{37} - 328 q^{38} + 160 q^{40} - 108 q^{41} - 470 q^{43} - 288 q^{44} + 168 q^{46} - 804 q^{47} - 1338 q^{49} - 200 q^{50} - 128 q^{52} + 336 q^{53} + 360 q^{55} - 16 q^{56} - 72 q^{58} - 768 q^{59} - 596 q^{61} + 1784 q^{62} + 512 q^{64} + 160 q^{65} + 424 q^{67} - 360 q^{68} + 20 q^{70} + 696 q^{71} - 188 q^{73} - 1372 q^{74} - 656 q^{76} - 630 q^{77} - 1088 q^{79} - 640 q^{80} + 432 q^{82} - 522 q^{83} + 450 q^{85} - 940 q^{86} + 288 q^{88} + 2256 q^{89} + 5684 q^{91} - 168 q^{92} - 1608 q^{94} + 820 q^{95} - 2486 q^{97} + 5352 q^{98}+O(q^{100}) \) 8 * q - 8 * q^2 - 16 * q^4 + 20 * q^5 - 2 * q^7 + 64 * q^8 - 80 * q^10 + 36 * q^11 - 32 * q^13 - 4 * q^14 - 64 * q^16 + 180 * q^17 + 328 * q^19 + 80 * q^20 + 72 * q^22 - 42 * q^23 - 100 * q^25 + 128 * q^26 + 16 * q^28 - 36 * q^29 - 446 * q^31 - 128 * q^32 - 180 * q^34 - 20 * q^35 + 1372 * q^37 - 328 * q^38 + 160 * q^40 - 108 * q^41 - 470 * q^43 - 288 * q^44 + 168 * q^46 - 804 * q^47 - 1338 * q^49 - 200 * q^50 - 128 * q^52 + 336 * q^53 + 360 * q^55 - 16 * q^56 - 72 * q^58 - 768 * q^59 - 596 * q^61 + 1784 * q^62 + 512 * q^64 + 160 * q^65 + 424 * q^67 - 360 * q^68 + 20 * q^70 + 696 * q^71 - 188 * q^73 - 1372 * q^74 - 656 * q^76 - 630 * q^77 - 1088 * q^79 - 640 * q^80 + 432 * q^82 - 522 * q^83 + 450 * q^85 - 940 * q^86 + 288 * q^88 + 2256 * q^89 + 5684 * q^91 - 168 * q^92 - 1608 * q^94 + 820 * q^95 - 2486 * q^97 + 5352 * q^98

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