LMFDB - Embedded newform 38.5.b.a.37.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [38,5,Mod(37,38)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(38, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1]))

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("38.37");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 38 = 2 \cdot 19 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	38.b (of order $2$, degree $1$, 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:	$3.92805859719$
Analytic rank:	$0$
Dimension:	$8$
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} + 450x^{6} + 68229x^{4} + 4001228x^{2} + 77475204 $ x^8 + 450x^6 + 68229x^4 + 4001228*x^2 + 77475204
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			37.3
Root			$8.07810i$ of defining polynomial
Character	$\chi$	$=$	38.37
Dual form			38.5.b.a.37.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.82843i q^{2} +6.66389i q^{3} -8.00000 q^{4} +26.7027 q^{5} +18.8483 q^{6} +51.4469 q^{7} +22.6274i q^{8} +36.5926 q^{9} +O(q^{10})$	\(q-2.82843i q^{2} +6.66389i q^{3} -8.00000 q^{4} +26.7027 q^{5} +18.8483 q^{6} +51.4469 q^{7} +22.6274i q^{8} +36.5926 q^{9} -75.5266i q^{10} -25.0891 q^{11} -53.3111i q^{12} -53.2531i q^{13} -145.514i q^{14} +177.944i q^{15} +64.0000 q^{16} -24.4913 q^{17} -103.499i q^{18} +(-78.0101 + 352.470i) q^{19} -213.622 q^{20} +342.837i q^{21} +70.9628i q^{22} -612.242 q^{23} -150.787 q^{24} +88.0343 q^{25} -150.622 q^{26} +783.624i q^{27} -411.576 q^{28} -1345.96i q^{29} +503.301 q^{30} -912.680i q^{31} -181.019i q^{32} -167.191i q^{33} +69.2720i q^{34} +1373.77 q^{35} -292.741 q^{36} +325.059i q^{37} +(996.937 + 220.646i) q^{38} +354.872 q^{39} +604.213i q^{40} -51.7494i q^{41} +969.689 q^{42} -2545.26 q^{43} +200.713 q^{44} +977.121 q^{45} +1731.68i q^{46} +2998.19 q^{47} +426.489i q^{48} +245.788 q^{49} -248.999i q^{50} -163.208i q^{51} +426.024i q^{52} -4067.86i q^{53} +2216.42 q^{54} -669.948 q^{55} +1164.11i q^{56} +(-2348.82 - 519.851i) q^{57} -3806.96 q^{58} +1078.88i q^{59} -1423.55i q^{60} -6530.89 q^{61} -2581.45 q^{62} +1882.58 q^{63} -512.000 q^{64} -1422.00i q^{65} -472.888 q^{66} +7386.18i q^{67} +195.931 q^{68} -4079.92i q^{69} -3885.62i q^{70} +5806.02i q^{71} +827.995i q^{72} +3629.56 q^{73} +919.405 q^{74} +586.651i q^{75} +(624.081 - 2819.76i) q^{76} -1290.76 q^{77} -1003.73i q^{78} +3903.16i q^{79} +1708.97 q^{80} -2257.99 q^{81} -146.369 q^{82} -997.967 q^{83} -2742.69i q^{84} -653.985 q^{85} +7199.08i q^{86} +8969.34 q^{87} -567.702i q^{88} -6152.85i q^{89} -2763.71i q^{90} -2739.71i q^{91} +4897.94 q^{92} +6082.00 q^{93} -8480.16i q^{94} +(-2083.08 + 9411.91i) q^{95} +1206.29 q^{96} +5023.58i q^{97} -695.195i q^{98} -918.076 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 64 q^{4} + 18 q^{5} - 32 q^{6} - 162 q^{7} - 268 q^{9}+O(q^{10}) $ 8 * q - 64 * q^4 + 18 * q^5 - 32 * q^6 - 162 * q^7 - 268 * q^9	$ 8 q - 64 q^{4} + 18 q^{5} - 32 q^{6} - 162 q^{7} - 268 q^{9} - 6 q^{11} + 512 q^{16} + 510 q^{17} - 12 q^{19} - 144 q^{20} - 396 q^{23} + 256 q^{24} + 3458 q^{25} - 192 q^{26} + 1296 q^{28} - 2752 q^{30} + 1002 q^{35} + 2144 q^{36} - 3216 q^{38} - 6588 q^{39} + 1376 q^{42} - 8654 q^{43} + 48 q^{44} - 10334 q^{45} + 3210 q^{47} + 9222 q^{49} + 9088 q^{54} + 17146 q^{55} - 14076 q^{57} - 960 q^{58} + 1314 q^{61} - 15168 q^{62} + 29938 q^{63} - 4096 q^{64} + 4928 q^{66} - 4080 q^{68} + 23398 q^{73} + 13152 q^{74} + 96 q^{76} - 44622 q^{77} + 1152 q^{80} - 20368 q^{81} + 16512 q^{82} - 10440 q^{83} + 21274 q^{85} - 14316 q^{87} + 3168 q^{92} + 19416 q^{93} - 34686 q^{95} - 2048 q^{96} - 56798 q^{99}+O(q^{100}) $ 8 * q - 64 * q^4 + 18 * q^5 - 32 * q^6 - 162 * q^7 - 268 * q^9 - 6 * q^11 + 512 * q^16 + 510 * q^17 - 12 * q^19 - 144 * q^20 - 396 * q^23 + 256 * q^24 + 3458 * q^25 - 192 * q^26 + 1296 * q^28 - 2752 * q^30 + 1002 * q^35 + 2144 * q^36 - 3216 * q^38 - 6588 * q^39 + 1376 * q^42 - 8654 * q^43 + 48 * q^44 - 10334 * q^45 + 3210 * q^47 + 9222 * q^49 + 9088 * q^54 + 17146 * q^55 - 14076 * q^57 - 960 * q^58 + 1314 * q^61 - 15168 * q^62 + 29938 * q^63 - 4096 * q^64 + 4928 * q^66 - 4080 * q^68 + 23398 * q^73 + 13152 * q^74 + 96 * q^76 - 44622 * q^77 + 1152 * q^80 - 20368 * q^81 + 16512 * q^82 - 10440 * q^83 + 21274 * q^85 - 14316 * q^87 + 3168 * q^92 + 19416 * q^93 - 34686 * q^95 - 2048 * q^96 - 56798 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/38\mathbb{Z}\right)^\times$.

$n$	$21$
$\chi(n)$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		−	2.82843i		−	0.707107i
$2$		−	2.82843i		−	0.707107i
$3$			6.66389i			0.740432i	0.928946	+	0.370216i	$0.120716\pi$
$3$			6.66389i			0.740432i	−0.928946	+	0.370216i	$0.879284\pi$
$4$	−8.00000			−0.500000
$5$	26.7027			1.06811			0.534054	−	0.845450i	$-0.320668\pi$
$5$	26.7027			1.06811			0.534054	+	0.845450i	$0.320668\pi$
$6$	18.8483			0.523565
$7$	51.4469			1.04994			0.524969	−	0.851121i	$-0.324077\pi$
$7$	51.4469			1.04994			0.524969	+	0.851121i	$0.324077\pi$
$8$			22.6274i			0.353553i
$9$	36.5926			0.451760
$10$		−	75.5266i		−	0.755266i
$11$	−25.0891			−0.207348			−0.103674	−	0.994611i	$-0.533060\pi$
$11$	−25.0891			−0.207348			−0.103674	+	0.994611i	$0.533060\pi$
$12$		−	53.3111i		−	0.370216i
$13$		−	53.2531i		−	0.315107i	−0.987510	−	0.157553i	$-0.949639\pi$
$13$		−	53.2531i		−	0.315107i	0.987510	−	0.157553i	$-0.0503606\pi$
$14$		−	145.514i		−	0.742418i
$15$			177.944i			0.790862i
$16$	64.0000			0.250000
$17$	−24.4913			−0.0847451			−0.0423726	−	0.999102i	$-0.513492\pi$
$17$	−24.4913			−0.0847451			−0.0423726	+	0.999102i	$0.513492\pi$
$18$		−	103.499i		−	0.319443i
$19$	−78.0101	+	352.470i	−0.216094	+	0.976372i
$19$	−78.0101	+	352.470i	−0.216094	+	0.976372i
$20$	−213.622			−0.534054
$21$			342.837i			0.777408i
$22$			70.9628i			0.146617i
$23$	−612.242			−1.15736			−0.578679	−	0.815555i	$-0.696432\pi$
$23$	−612.242			−1.15736			−0.578679	+	0.815555i	$0.696432\pi$
$24$	−150.787			−0.261782
$25$	88.0343			0.140855
$26$	−150.622			−0.222814
$27$			783.624i			1.07493i
$28$	−411.576			−0.524969
$29$		−	1345.96i		−	1.60043i	−0.599713	−	0.800215i	$-0.704718\pi$
$29$		−	1345.96i		−	1.60043i	0.599713	−	0.800215i	$-0.295282\pi$
$30$	503.301			0.559224
$31$		−	912.680i		−	0.949719i	−0.880062	−	0.474860i	$-0.842499\pi$
$31$		−	912.680i		−	0.949719i	0.880062	−	0.474860i	$-0.157501\pi$
$32$		−	181.019i		−	0.176777i
$33$		−	167.191i		−	0.153527i
$34$			69.2720i			0.0599238i
$35$	1373.77			1.12145
$36$	−292.741			−0.225880
$37$			325.059i			0.237443i	0.992928	+	0.118721i	$0.0378795\pi$
$37$			325.059i			0.237443i	−0.992928	+	0.118721i	$0.962120\pi$
$38$	996.937	+	220.646i	0.690400	+	0.152802i
$39$	354.872			0.233315
$40$			604.213i			0.377633i
$41$		−	51.7494i		−	0.0307849i	−0.999882	−	0.0153924i	$-0.995100\pi$
$41$		−	51.7494i		−	0.0307849i	0.999882	−	0.0153924i	$-0.00489976\pi$
$42$	969.689			0.549710
$43$	−2545.26			−1.37656			−0.688280	−	0.725445i	$-0.741634\pi$
$43$	−2545.26			−1.37656			−0.688280	+	0.725445i	$0.741634\pi$
$44$	200.713			0.103674
$45$	977.121			0.482529
$46$			1731.68i			0.818376i
$47$	2998.19			1.35726			0.678630	−	0.734480i	$-0.262574\pi$
$47$	2998.19			1.35726			0.678630	+	0.734480i	$0.262574\pi$
$48$			426.489i			0.185108i
$49$	245.788			0.102369
$50$		−	248.999i		−	0.0995994i
$51$		−	163.208i		−	0.0627480i
$52$			426.024i			0.157553i
$53$		−	4067.86i		−	1.44815i	−0.689720	−	0.724077i	$-0.742266\pi$
$53$		−	4067.86i		−	1.44815i	0.689720	−	0.724077i	$-0.257734\pi$
$54$	2216.42			0.760090
$55$	−669.948			−0.221470
$56$			1164.11i			0.371209i
$57$	−2348.82	−	519.851i	−0.722938	−	0.160003i
$58$	−3806.96			−1.13168
$59$			1078.88i			0.309933i	0.987920	+	0.154966i	$0.0495269\pi$
$59$			1078.88i			0.309933i	−0.987920	+	0.154966i	$0.950473\pi$
$60$		−	1423.55i		−	0.395431i
$61$	−6530.89			−1.75514			−0.877572	−	0.479445i	$-0.840838\pi$
$61$	−6530.89			−1.75514			−0.877572	+	0.479445i	$0.840838\pi$
$62$	−2581.45			−0.671553
$63$	1882.58			0.474320
$64$	−512.000			−0.125000
$65$		−	1422.00i		−	0.336568i
$66$	−472.888			−0.108560
$67$			7386.18i			1.64540i	0.568478	+	0.822698i	$0.307532\pi$
$67$			7386.18i			1.64540i	−0.568478	+	0.822698i	$0.692468\pi$
$68$	195.931			0.0423726
$69$		−	4079.92i		−	0.856945i
$70$		−	3885.62i		−	0.792983i
$71$			5806.02i			1.15176i	0.817535	+	0.575879i	$0.195340\pi$
$71$			5806.02i			1.15176i	−0.817535	+	0.575879i	$0.804660\pi$
$72$			827.995i			0.159721i
$73$	3629.56			0.681096			0.340548	−	0.940227i	$-0.389387\pi$
$73$	3629.56			0.681096			0.340548	+	0.940227i	$0.389387\pi$
$74$	919.405			0.167897
$75$			586.651i			0.104293i
$76$	624.081	−	2819.76i	0.108047	−	0.488186i
$77$	−1290.76			−0.217703
$78$		−	1003.73i		−	0.164979i
$79$			3903.16i			0.625407i	0.949851	+	0.312703i	$0.101235\pi$
$79$			3903.16i			0.625407i	−0.949851	+	0.312703i	$0.898765\pi$
$80$	1708.97			0.267027
$81$	−2257.99			−0.344153
$82$	−146.369			−0.0217682
$83$	−997.967			−0.144864			−0.0724319	−	0.997373i	$-0.523076\pi$
$83$	−997.967			−0.144864			−0.0724319	+	0.997373i	$0.523076\pi$
$84$		−	2742.69i		−	0.388704i
$85$	−653.985			−0.0905169
$86$			7199.08i			0.973375i
$87$	8969.34			1.18501
$88$		−	567.702i		−	0.0733087i
$89$		−	6152.85i		−	0.776777i	−0.921496	−	0.388389i	$-0.873032\pi$
$89$		−	6152.85i		−	0.776777i	0.921496	−	0.388389i	$-0.126968\pi$
$90$		−	2763.71i		−	0.341199i
$91$		−	2739.71i		−	0.330843i
$92$	4897.94			0.578679
$93$	6082.00			0.703203
$94$		−	8480.16i		−	0.959728i
$95$	−2083.08	+	9411.91i	−0.230812	+	1.04287i
$96$	1206.29			0.130891
$97$			5023.58i			0.533912i	0.963709	+	0.266956i	$0.0860178\pi$
$97$			5023.58i			0.533912i	−0.963709	+	0.266956i	$0.913982\pi$
$98$		−	695.195i		−	0.0723860i
$99$	−918.076			−0.0936717
$100$	−704.274			−0.0704274
$101$	15793.7			1.54825			0.774123	−	0.633035i	$-0.218191\pi$
$101$	15793.7			1.54825			0.774123	+	0.633035i	$0.218191\pi$
$102$	−461.621			−0.0443695
$103$		−	13068.7i		−	1.23185i	−0.787806	−	0.615924i	$-0.788783\pi$
$103$		−	13068.7i		−	1.23185i	0.787806	−	0.615924i	$-0.211217\pi$
$104$	1204.98			0.111407
$105$			9154.67i			0.830355i
$106$	−11505.7			−1.02400
$107$		−	8273.09i		−	0.722604i	−0.932449	−	0.361302i	$-0.882332\pi$
$107$		−	8273.09i		−	0.722604i	0.932449	−	0.361302i	$-0.117668\pi$
$108$		−	6268.99i		−	0.537465i
$109$			14219.6i			1.19683i	0.801186	+	0.598416i	$0.204203\pi$
$109$			14219.6i			1.19683i	−0.801186	+	0.598416i	$0.795797\pi$
$110$			1894.90i			0.156603i
$111$	−2166.16			−0.175810
$112$	3292.60			0.262484
$113$		−	20747.4i		−	1.62483i	−0.583080	−	0.812414i	$-0.698153\pi$
$113$		−	20747.4i		−	1.62483i	0.583080	−	0.812414i	$-0.301847\pi$
$114$	−1470.36	+	6643.48i	−0.113139	+	0.511194i
$115$	−16348.5			−1.23618
$116$			10767.7i			0.800215i
$117$		−	1948.67i		−	0.142353i
$118$	3051.52			0.219155
$119$	−1260.00			−0.0889771
$120$	−4026.41			−0.279612
$121$	−14011.5			−0.957007
$122$			18472.1i			1.24107i
$123$	344.852			0.0227941
$124$			7301.44i			0.474860i
$125$	−14338.4			−0.917660
$126$		−	5324.73i		−	0.335395i
$127$		−	11117.8i		−	0.689305i	−0.938730	−	0.344653i	$-0.887997\pi$
$127$		−	11117.8i		−	0.689305i	0.938730	−	0.344653i	$-0.112003\pi$
$128$			1448.15i			0.0883883i
$129$		−	16961.3i		−	1.01925i
$130$	−4022.02			−0.237990
$131$	223.317			0.0130130			0.00650652	−	0.999979i	$-0.497929\pi$
$131$	223.317			0.0130130			0.00650652	+	0.999979i	$0.497929\pi$
$132$			1337.53i			0.0767637i
$133$	−4013.38	+	18133.5i	−0.226886	+	1.02513i
$134$	20891.3			1.16347
$135$			20924.9i			1.14814i
$136$		−	554.176i		−	0.0299619i
$137$	18953.1			1.00981			0.504903	−	0.863176i	$-0.331528\pi$
$137$	18953.1			1.00981			0.504903	+	0.863176i	$0.331528\pi$
$138$	−11539.7			−0.605952
$139$	−11115.5			−0.575309			−0.287655	−	0.957734i	$-0.592875\pi$
$139$	−11115.5			−0.575309			−0.287655	+	0.957734i	$0.592875\pi$
$140$	−10990.2			−0.560723
$141$			19979.6i			1.00496i
$142$	16421.9			0.814416
$143$			1336.07i			0.0653368i
$144$	2341.92			0.112940
$145$		−	35940.8i		−	1.70943i
$146$		−	10265.9i		−	0.481608i
$147$			1637.91i			0.0757975i
$148$		−	2600.47i		−	0.118721i
$149$	−19727.7			−0.888597			−0.444299	−	0.895879i	$-0.646547\pi$
$149$	−19727.7			−0.888597			−0.444299	+	0.895879i	$0.646547\pi$
$150$	1659.30			0.0737466
$151$			770.565i			0.0337952i	0.999857	+	0.0168976i	$0.00537894\pi$
$151$			770.565i			0.0337952i	−0.999857	+	0.0168976i	$0.994621\pi$
$152$	−7975.50	−	1765.17i	−0.345200	−	0.0764009i
$153$	−896.201			−0.0382845
$154$			3650.82i			0.153939i
$155$		−	24371.0i		−	1.01440i
$156$	−2838.98			−0.116658
$157$	46244.5			1.87612			0.938061	−	0.346470i	$-0.112620\pi$
$157$	46244.5			1.87612			0.938061	+	0.346470i	$0.112620\pi$
$158$	11039.8			0.442229
$159$	27107.8			1.07226
$160$		−	4833.71i		−	0.188817i
$161$	−31498.0			−1.21515
$162$			6386.55i			0.243353i
$163$	32951.9			1.24024			0.620119	−	0.784507i	$-0.287084\pi$
$163$	32951.9			1.24024			0.620119	+	0.784507i	$0.287084\pi$
$164$			413.995i			0.0153924i
$165$		−	4464.46i		−	0.163984i
$166$			2822.68i			0.102434i
$167$		−	48049.1i		−	1.72287i	−0.507870	−	0.861434i	$-0.669567\pi$
$167$		−	48049.1i		−	1.72287i	0.507870	−	0.861434i	$-0.330433\pi$
$168$	−7757.51			−0.274855
$169$	25725.1			0.900708
$170$			1849.75i			0.0640051i
$171$	−2854.59	+	12897.8i	−0.0976228	+	0.441086i
$172$	20362.1			0.688280
$173$			46219.6i			1.54431i	0.635436	+	0.772153i	$0.280820\pi$
$173$			46219.6i			1.54431i	−0.635436	+	0.772153i	$0.719180\pi$
$174$		−	25369.1i		−	0.837929i
$175$	4529.10			0.147889
$176$	−1605.70			−0.0518371
$177$	−7189.51			−0.229484
$178$	−17402.9			−0.549264
$179$			39826.5i			1.24299i	0.783420	+	0.621493i	$0.213473\pi$
$179$			39826.5i			1.24299i	−0.783420	+	0.621493i	$0.786527\pi$
$180$	−7816.96			−0.241264
$181$			51844.5i			1.58251i	0.611488	+	0.791254i	$0.290571\pi$
$181$			51844.5i			1.58251i	−0.611488	+	0.791254i	$0.709429\pi$
$182$	−7749.06			−0.233941
$183$		−	43521.1i		−	1.29957i
$184$		−	13853.5i		−	0.409188i
$185$			8679.95i			0.253614i
$186$		−	17202.5i		−	0.497239i
$187$	614.467			0.0175718
$188$	−23985.5			−0.678630
$189$			40315.1i			1.12861i
$190$	26620.9	+	5891.84i	0.737421	+	0.163209i
$191$	−38958.8			−1.06792			−0.533960	−	0.845509i	$-0.679297\pi$
$191$	−38958.8			−1.06792			−0.533960	+	0.845509i	$0.679297\pi$
$192$		−	3411.91i		−	0.0925540i
$193$			43058.1i			1.15595i	0.816053	+	0.577977i	$0.196158\pi$
$193$			43058.1i			1.15595i	−0.816053	+	0.577977i	$0.803842\pi$
$194$	14208.8			0.377533
$195$	9476.05			0.249206
$196$	−1966.31			−0.0511846
$197$	26311.0			0.677962			0.338981	−	0.940793i	$-0.389918\pi$
$197$	26311.0			0.677962			0.338981	+	0.940793i	$0.389918\pi$
$198$			2596.71i			0.0662359i
$199$	−11773.1			−0.297293			−0.148647	−	0.988890i	$-0.547492\pi$
$199$	−11773.1			−0.297293			−0.148647	+	0.988890i	$0.547492\pi$
$200$			1991.99i			0.0497997i
$201$	−49220.7			−1.21830
$202$		−	44671.2i		−	1.09478i
$203$		−	69245.7i		−	1.68035i
$204$			1305.66i			0.0313740i
$205$		−	1381.85i		−	0.0328816i
$206$	−36963.8			−0.871048
$207$	−22403.5			−0.522848
$208$		−	3408.20i		−	0.0787767i
$209$	1957.21	−	8843.18i	0.0448068	−	0.202449i
$210$	25893.3			0.587150
$211$		−	12495.7i		−	0.280670i	−0.990104	−	0.140335i	$-0.955182\pi$
$211$		−	12495.7i		−	0.280670i	0.990104	−	0.140335i	$-0.0448179\pi$
$212$			32542.9i			0.724077i
$213$	−38690.7			−0.852799
$214$	−23399.8			−0.510958
$215$	−67965.3			−1.47032
$216$	−17731.4			−0.380045
$217$		−	46954.6i		−	0.997146i
$218$	40219.0			0.846288
$219$			24187.0i			0.504305i
$220$	5359.58			0.110735
$221$			1304.24i			0.0267038i
$222$			6126.82i			0.124317i
$223$			59706.5i			1.20064i	0.799761	+	0.600319i	$0.204960\pi$
$223$			59706.5i			1.20064i	−0.799761	+	0.600319i	$0.795040\pi$
$224$		−	9312.89i		−	0.185605i
$225$	3221.40			0.0636326
$226$	−58682.6			−1.14893
$227$		−	57964.2i		−	1.12489i	−0.826836	−	0.562443i	$-0.809862\pi$
$227$		−	57964.2i		−	1.12489i	0.826836	−	0.562443i	$-0.190138\pi$
$228$	18790.6	+	4158.80i	0.361469	+	0.0800016i
$229$	40157.9			0.765773			0.382886	−	0.923795i	$-0.374930\pi$
$229$	40157.9			0.765773			0.382886	+	0.923795i	$0.374930\pi$
$230$			46240.6i			0.874114i
$231$		−	8601.48i		−	0.161194i
$232$	30455.6			0.565838
$233$	71740.1			1.32145			0.660724	−	0.750629i	$-0.270249\pi$
$233$	71740.1			1.32145			0.660724	+	0.750629i	$0.270249\pi$
$234$	−5511.66			−0.100659
$235$	80059.7			1.44970
$236$		−	8631.00i		−	0.154966i
$237$	−26010.2			−0.463071
$238$			3563.83i			0.0629163i
$239$	44178.4			0.773417			0.386709	−	0.922202i	$-0.373612\pi$
$239$	44178.4			0.773417			0.386709	+	0.922202i	$0.373612\pi$
$240$			11388.4i			0.197715i
$241$		−	33823.7i		−	0.582353i	−0.956669	−	0.291177i	$-0.905953\pi$
$241$		−	33823.7i		−	0.582353i	0.956669	−	0.291177i	$-0.0940467\pi$
$242$			39630.6i			0.676706i
$243$			48426.6i			0.820108i
$244$	52247.1			0.877572
$245$	6563.22			0.109341
$246$		−	975.390i		−	0.0161179i
$247$	18770.1	+	4154.27i	0.307662	+	0.0680928i
$248$	20651.6			0.335777
$249$		−	6650.34i		−	0.107262i
$250$			40555.2i			0.648884i
$251$	−64952.8			−1.03098			−0.515490	−	0.856896i	$-0.672390\pi$
$251$	−64952.8			−1.03098			−0.515490	+	0.856896i	$0.672390\pi$
$252$	−15060.6			−0.237160
$253$	15360.6			0.239976
$254$	−31445.9			−0.487412
$255$		−	4358.08i		−	0.0670217i
$256$	4096.00			0.0625000
$257$			1155.59i			0.0174959i	0.999962	+	0.00874796i	$0.00278460\pi$
$257$			1155.59i			0.0174959i	−0.999962	+	0.00874796i	$0.997215\pi$
$258$	−47973.9			−0.720718
$259$			16723.3i			0.249300i
$260$			11376.0i			0.168284i
$261$		−	49252.2i		−	0.723011i
$262$		−	631.635i		−	0.00920161i
$263$	6732.94			0.0973404			0.0486702	−	0.998815i	$-0.484502\pi$
$263$	6732.94			0.0973404			0.0486702	+	0.998815i	$0.484502\pi$
$264$	3783.11			0.0542801
$265$		−	108623.i		−	1.54678i
$266$	51289.4	+	11351.6i	0.724877	+	0.160432i
$267$	41001.9			0.575151
$268$		−	59089.5i		−	0.822698i
$269$		−	22231.5i		−	0.307230i	−0.988131	−	0.153615i	$-0.950908\pi$
$269$		−	22231.5i		−	0.307230i	0.988131	−	0.153615i	$-0.0490915\pi$
$270$	59184.5			0.811859
$271$	−46597.6			−0.634490			−0.317245	−	0.948344i	$-0.602758\pi$
$271$	−46597.6			−0.634490			−0.317245	+	0.948344i	$0.602758\pi$
$272$	−1567.45			−0.0211863
$273$	18257.1			0.244966
$274$		−	53607.3i		−	0.714041i
$275$	−2208.70			−0.0292060
$276$			32639.3i			0.428473i
$277$	105817.			1.37910			0.689548	−	0.724240i	$-0.257809\pi$
$277$	105817.			1.37910			0.689548	+	0.724240i	$0.257809\pi$
$278$			31439.5i			0.406805i
$279$		−	33397.3i		−	0.429045i
$280$			31084.9i			0.396491i
$281$			20298.2i			0.257067i	0.991705	+	0.128533i	$0.0410269\pi$
$281$			20298.2i			0.257067i	−0.991705	+	0.128533i	$0.958973\pi$
$282$	56510.8			0.710614
$283$	−55710.6			−0.695609			−0.347804	−	0.937567i	$-0.613073\pi$
$283$	−55710.6			−0.695609			−0.347804	+	0.937567i	$0.613073\pi$
$284$		−	46448.1i		−	0.575879i
$285$	−62720.0	−	13881.4i	−0.772176	−	0.170901i
$286$	3778.99			0.0462001
$287$		−	2662.35i		−	0.0323222i
$288$		−	6623.96i		−	0.0798607i
$289$	−82921.2			−0.992818
$290$	−101656.			−1.20875
$291$	−33476.6			−0.395326
$292$	−29036.5			−0.340548
$293$			84053.1i			0.979081i	0.871980	+	0.489541i	$0.162836\pi$
$293$			84053.1i			0.979081i	−0.871980	+	0.489541i	$0.837164\pi$
$294$	4632.70			0.0535969
$295$			28808.9i			0.331042i
$296$	−7355.24			−0.0839486
$297$		−	19660.4i		−	0.222885i
$298$			55798.5i			0.628333i
$299$			32603.8i			0.364691i
$300$		−	4693.21i		−	0.0521467i
$301$	−130946.			−1.44530
$302$	2179.49			0.0238968
$303$			105247.i			1.14637i
$304$	−4992.64	+	22558.1i	−0.0540236	+	0.244093i
$305$	−174392.			−1.87468
$306$			2534.84i			0.0270712i
$307$		−	10396.7i		−	0.110311i	−0.998478	−	0.0551555i	$-0.982435\pi$
$307$		−	10396.7i		−	0.110311i	0.998478	−	0.0551555i	$-0.0175655\pi$
$308$	10326.1			0.108851
$309$	87088.2			0.912100
$310$	−68931.7			−0.717291
$311$	118107.			1.22111			0.610557	−	0.791973i	$-0.290946\pi$
$311$	118107.			1.22111			0.610557	+	0.791973i	$0.290946\pi$
$312$			8029.85i			0.0824894i
$313$	−8391.17			−0.0856513			−0.0428257	−	0.999083i	$-0.513636\pi$
$313$	−8391.17			−0.0856513			−0.0428257	+	0.999083i	$0.513636\pi$
$314$		−	130799.i		−	1.32662i
$315$	50269.9			0.506625
$316$		−	31225.3i		−	0.312703i
$317$			143474.i			1.42776i	0.700270	+	0.713878i	$0.253063\pi$
$317$			143474.i			1.42776i	−0.700270	+	0.713878i	$0.746937\pi$
$318$		−	76672.4i		−	0.758202i
$319$			33769.0i			0.331847i
$320$	−13671.8			−0.133514
$321$	55131.0			0.535039
$322$			89089.8i			0.859244i
$323$	1910.57	−	8632.47i	0.0183129	−	0.0827428i
$324$	18063.9			0.172076
$325$		−	4688.09i		−	0.0443843i
$326$		−	93202.1i		−	0.876981i
$327$	−94757.6			−0.886173
$328$	1170.96			0.0108841
$329$	154248.			1.42504
$330$	−12627.4			−0.115954
$331$		−	16943.3i		−	0.154647i	−0.997006	−	0.0773237i	$-0.975363\pi$
$331$		−	16943.3i		−	0.154647i	0.997006	−	0.0773237i	$-0.0246375\pi$
$332$	7983.73			0.0724319
$333$			11894.7i			0.107267i
$334$	−135903.			−1.21825
$335$			197231.i			1.75746i
$336$			21941.6i			0.194352i
$337$		−	217168.i		−	1.91221i	−0.293021	−	0.956106i	$-0.594661\pi$
$337$		−	217168.i		−	1.91221i	0.293021	−	0.956106i	$-0.405339\pi$
$338$		−	72761.6i		−	0.636897i
$339$	138259.			1.20308
$340$	5231.88			0.0452585
$341$			22898.4i			0.196923i
$342$	36480.5	+	8074.00i	0.311895	+	0.0690298i
$343$	−110879.			−0.942456
$344$		−	57592.7i		−	0.486688i
$345$		−	108945.i		−	0.915310i
$346$	130729.			1.09199
$347$	64555.0			0.536131			0.268066	−	0.963401i	$-0.413616\pi$
$347$	64555.0			0.536131			0.268066	+	0.963401i	$0.413616\pi$
$348$	−71754.8			−0.592505
$349$	−113275.			−0.930001			−0.465001	−	0.885310i	$-0.653946\pi$
$349$	−113275.			−0.930001			−0.465001	+	0.885310i	$0.653946\pi$
$350$		−	12810.2i		−	0.104573i
$351$	41730.4			0.338718
$352$			4541.62i			0.0366543i
$353$	27685.3			0.222178			0.111089	−	0.993810i	$-0.464566\pi$
$353$	27685.3			0.222178			0.111089	+	0.993810i	$0.464566\pi$
$354$			20335.0i			0.162270i
$355$			155036.i			1.23020i
$356$			49222.8i			0.388389i
$357$		−	8396.53i		−	0.0658815i
$358$	112646.			0.878923
$359$	65336.2			0.506950			0.253475	−	0.967342i	$-0.418426\pi$
$359$	65336.2			0.506950			0.253475	+	0.967342i	$0.418426\pi$
$360$			22109.7i			0.170600i
$361$	−118150.	−	54992.5i	−0.906606	−	0.421977i
$362$	146638.			1.11900
$363$		−	93371.3i		−	0.708599i
$364$			21917.7i			0.165421i
$365$	96919.1			0.727484
$366$	−123096.			−0.918931
$367$	177298.			1.31635			0.658176	−	0.752864i	$-0.271328\pi$
$367$	177298.			1.31635			0.658176	+	0.752864i	$0.271328\pi$
$368$	−39183.5			−0.289340
$369$		−	1893.64i		−	0.0139074i
$370$	24550.6			0.179332
$371$		−	209279.i		−	1.52047i
$372$	−48656.0			−0.351601
$373$			15381.6i			0.110557i	0.998471	+	0.0552783i	$0.0176046\pi$
$373$			15381.6i			0.110557i	−0.998471	+	0.0552783i	$0.982395\pi$
$374$		−	1737.97i		−	0.0124251i
$375$		−	95549.8i		−	0.679465i
$376$			67841.3i			0.479864i
$377$	−71676.6			−0.504307
$378$	114028.			0.798047
$379$			232626.i			1.61949i	0.586780	+	0.809746i	$0.300395\pi$
$379$			232626.i			1.61949i	−0.586780	+	0.809746i	$0.699605\pi$
$380$	16664.6	−	75295.3i	0.115406	−	0.521436i
$381$	74087.8			0.510384
$382$			110192.i			0.755134i
$383$		−	52033.6i		−	0.354721i	−0.984146	−	0.177360i	$-0.943244\pi$
$383$		−	52033.6i		−	0.354721i	0.984146	−	0.177360i	$-0.0567558\pi$
$384$	−9650.34			−0.0654456
$385$	−34466.8			−0.232530
$386$	121787.			0.817383
$387$	−93137.6			−0.621875
$388$		−	40188.6i		−	0.266956i
$389$	57687.6			0.381227			0.190613	−	0.981665i	$-0.438952\pi$
$389$	57687.6			0.381227			0.190613	+	0.981665i	$0.438952\pi$
$390$		−	26802.3i		−	0.176215i
$391$	14994.6			0.0980804
$392$			5561.56i			0.0361930i
$393$			1488.16i			0.00963527i
$394$		−	74418.9i		−	0.479392i
$395$			104225.i			0.668002i
$396$	7344.61			0.0468358
$397$	16914.2			0.107317			0.0536586	−	0.998559i	$-0.482912\pi$
$397$	16914.2			0.107317			0.0536586	+	0.998559i	$0.482912\pi$
$398$			33299.4i			0.210218i
$399$	−120840.	−	26744.7i	−0.759039	−	0.167993i
$400$	5634.19			0.0352137
$401$		−	23481.6i		−	0.146029i	−0.997331	−	0.0730144i	$-0.976738\pi$
$401$		−	23481.6i		−	0.146029i	0.997331	−	0.0730144i	$-0.0232619\pi$
$402$			139217.i			0.861471i
$403$	−48603.0			−0.299263
$404$	−126349.			−0.774123
$405$	−60294.3			−0.367592
$406$	−195856.			−1.18819
$407$		−	8155.45i		−	0.0492333i
$408$	3692.97			0.0221848
$409$		−	310776.i		−	1.85781i	−0.370321	−	0.928904i	$-0.620752\pi$
$409$		−	310776.i		−	1.85781i	0.370321	−	0.928904i	$-0.379248\pi$
$410$	−3908.46			−0.0232508
$411$			126301.i			0.747693i
$412$			104549.i			0.615924i
$413$			55504.9i			0.325410i
$414$			63366.7i			0.369710i
$415$	−26648.4			−0.154730
$416$	−9639.83			−0.0557035
$417$		−	74072.8i		−	0.425977i
$418$	−25012.3	−	5535.81i	−0.143153	−	0.0316832i
$419$	−271562.			−1.54682			−0.773411	−	0.633905i	$-0.781451\pi$
$419$	−271562.			−1.54682			−0.773411	+	0.633905i	$0.781451\pi$
$420$		−	73237.3i		−	0.415178i
$421$			322467.i			1.81937i	0.415299	+	0.909685i	$0.363677\pi$
$421$			322467.i			1.81937i	−0.415299	+	0.909685i	$0.636323\pi$
$422$	−35343.2			−0.198464
$423$	109711.			0.613156
$424$	92045.2			0.512000
$425$	−2156.08			−0.0119368
$426$			109434.i			0.603020i
$427$	−335994.			−1.84279
$428$			66184.7i			0.361302i
$429$	−8903.44			−0.0483775
$430$			192235.i			1.03967i
$431$		−	157951.i		−	0.850290i	−0.905125	−	0.425145i	$-0.860223\pi$
$431$		−	157951.i		−	0.850290i	0.905125	−	0.425145i	$-0.139777\pi$
$432$			50151.9i			0.268732i
$433$			36946.7i			0.197061i	0.995134	+	0.0985304i	$0.0314142\pi$
$433$			36946.7i			0.197061i	−0.995134	+	0.0985304i	$0.968586\pi$
$434$	−132808.			−0.705089
$435$	239506.			1.26572
$436$		−	113756.i		−	0.598416i
$437$	47761.1	−	215797.i	0.250099	−	1.13001i
$438$	68411.1			0.356598
$439$		−	194377.i		−	1.00859i	−0.863531	−	0.504296i	$-0.831752\pi$
$439$		−	194377.i		−	1.00859i	0.863531	−	0.504296i	$-0.168248\pi$
$440$		−	15159.2i		−	0.0783016i
$441$	8994.03			0.0462463
$442$	3688.94			0.0188824
$443$	30671.7			0.156290			0.0781449	−	0.996942i	$-0.475100\pi$
$443$	30671.7			0.156290			0.0781449	+	0.996942i	$0.475100\pi$
$444$	17329.3			0.0879051
$445$		−	164298.i		−	0.829682i
$446$	168875.			0.848979
$447$		−	131464.i		−	0.657946i
$448$	−26340.8			−0.131242
$449$			236846.i			1.17483i	0.809287	+	0.587413i	$0.199854\pi$
$449$			236846.i			1.17483i	−0.809287	+	0.587413i	$0.800146\pi$
$450$		−	9111.50i		−	0.0449951i
$451$			1298.35i			0.00638319i
$452$			165980.i			0.812414i
$453$	−5134.96			−0.0250231
$454$	−163948.			−0.795414
$455$		−	73157.6i		−	0.353376i
$456$	11762.9	−	53147.8i	0.0565697	−	0.255597i
$457$	−82060.7			−0.392919			−0.196459	−	0.980512i	$-0.562944\pi$
$457$	−82060.7			−0.392919			−0.196459	+	0.980512i	$0.562944\pi$
$458$		−	113584.i		−	0.541483i
$459$		−	19192.0i		−	0.0910951i
$460$	130788.			0.618092
$461$	−16781.0			−0.0789614			−0.0394807	−	0.999220i	$-0.512570\pi$
$461$	−16781.0			−0.0789614			−0.0394807	+	0.999220i	$0.512570\pi$
$462$	−24328.7			−0.113981
$463$	100169.			0.467274			0.233637	−	0.972324i	$-0.424937\pi$
$463$	100169.			0.467274			0.233637	+	0.972324i	$0.424937\pi$
$464$		−	86141.6i		−	0.400108i
$465$	162406.			0.751097
$466$		−	202912.i		−	0.934405i
$467$	−216772.			−0.993960			−0.496980	−	0.867762i	$-0.665558\pi$
$467$	−216772.			−0.993960			−0.496980	+	0.867762i	$0.665558\pi$
$468$			15589.3i			0.0711764i
$469$			379997.i			1.72756i
$470$		−	226443.i		−	1.02509i
$471$			308168.i			1.38914i
$472$	−24412.2			−0.109578
$473$	63858.4			0.285427
$474$			73568.1i			0.327441i
$475$	−6867.56	+	31029.5i	−0.0304379	+	0.137527i
$476$	10080.0			0.0444885
$477$		−	148854.i		−	0.654218i
$478$		−	124955.i		−	0.546889i
$479$	91676.2			0.399563			0.199782	−	0.979840i	$-0.435977\pi$
$479$	91676.2			0.399563			0.199782	+	0.979840i	$0.435977\pi$
$480$	32211.3			0.139806
$481$	17310.4			0.0748198
$482$	−95667.8			−0.411786
$483$		−	209899.i		−	0.899739i
$484$	112092.			0.478503
$485$			134143.i			0.570276i
$486$	136971.			0.579904
$487$		−	107326.i		−	0.452528i	−0.974066	−	0.226264i	$-0.927349\pi$
$487$		−	107326.i		−	0.452528i	0.974066	−	0.226264i	$-0.0726512\pi$
$488$		−	147777.i		−	0.620537i
$489$			219588.i			0.918313i
$490$		−	18563.6i		−	0.0773160i
$491$	211818.			0.878619			0.439310	−	0.898336i	$-0.355223\pi$
$491$	211818.			0.878619			0.439310	+	0.898336i	$0.355223\pi$
$492$	−2758.82			−0.0113971
$493$			32964.4i			0.135629i
$494$	11750.1	−	53089.9i	0.0481489	−	0.217550i
$495$	−24515.1			−0.100051
$496$		−	58411.5i		−	0.237430i
$497$			298702.i			1.20928i
$498$	−18810.0			−0.0758456
$499$	145489.			0.584291			0.292146	−	0.956374i	$-0.405631\pi$
$499$	145489.			0.584291			0.292146	+	0.956374i	$0.405631\pi$
$500$	114707.			0.458830
$501$	320194.			1.27567
$502$			183714.i			0.729013i
$503$	−33735.6			−0.133338			−0.0666689	−	0.997775i	$-0.521237\pi$
$503$	−33735.6			−0.133338			−0.0666689	+	0.997775i	$0.521237\pi$
$504$			42597.8i			0.167697i
$505$	421733.			1.65369
$506$		−	43446.4i		−	0.169689i
$507$			171429.i			0.666913i
$508$			88942.4i			0.344653i
$509$			128247.i			0.495009i	0.968887	+	0.247505i	$0.0796105\pi$
$509$			128247.i			0.495009i	−0.968887	+	0.247505i	$0.920389\pi$
$510$	−12326.5			−0.0473915
$511$	186730.			0.715108
$512$		−	11585.2i		−	0.0441942i
$513$	−276204.	−	61130.6i	−1.04953	−	0.232286i
$514$	3268.50			0.0123715
$515$		−	348969.i		−	1.31575i
$516$			135691.i			0.509625i
$517$	−75222.0			−0.281426
$518$	47300.6			0.176282
$519$	−308002.			−1.14345
$520$	32176.2			0.118995
$521$			38092.9i			0.140336i	0.997535	+	0.0701680i	$0.0223535\pi$
$521$			38092.9i			0.140336i	−0.997535	+	0.0701680i	$0.977646\pi$
$522$	−139306.			−0.511246
$523$		−	96329.4i		−	0.352173i	−0.984375	−	0.176086i	$-0.943656\pi$
$523$		−	96329.4i		−	0.352173i	0.984375	−	0.176086i	$-0.0563438\pi$
$524$	−1786.53			−0.00650652
$525$			30181.4i			0.109502i
$526$		−	19043.6i		−	0.0688301i
$527$			22352.8i			0.0804841i
$528$		−	10700.2i		−	0.0383818i
$529$	94999.8			0.339478
$530$	−307232.			−1.09374
$531$			39478.8i			0.140015i
$532$	32107.0	−	145068.i	0.113443	−	0.512565i
$533$	−2755.81			−0.00970053
$534$		−	115971.i		−	0.406693i
$535$		−	220914.i		−	0.771819i
$536$	−167130.			−0.581735
$537$	−265399.			−0.920346
$538$	−62880.1			−0.217244
$539$	−6166.62			−0.0212261
$540$		−	167399.i		−	0.574071i
$541$	−163021.			−0.556993			−0.278496	−	0.960437i	$-0.589836\pi$
$541$	−163021.			−0.556993			−0.278496	+	0.960437i	$0.589836\pi$
$542$			131798.i			0.448652i
$543$	−345486.			−1.17174
$544$			4433.41i			0.0149810i
$545$			379701.i			1.27835i
$546$		−	51638.9i		−	0.173217i
$547$		−	450817.i		−	1.50670i	−0.657621	−	0.753349i	$-0.728437\pi$
$547$		−	450817.i		−	1.50670i	0.657621	−	0.753349i	$-0.271563\pi$
$548$	−151624.			−0.504903
$549$	−238982.			−0.792904
$550$			6247.16i			0.0206518i
$551$	474412.	+	104999.i	1.56262	+	0.345844i
$552$	92318.0			0.302976
$553$			200806.i			0.656638i
$554$		−	299295.i		−	0.975168i
$555$	−57842.2			−0.187784
$556$	88924.4			0.287655
$557$	−270930.			−0.873268			−0.436634	−	0.899639i	$-0.643829\pi$
$557$	−270930.			−0.873268			−0.436634	+	0.899639i	$0.643829\pi$
$558$	−94461.9			−0.303381
$559$			135543.i			0.433764i
$560$	87921.4			0.280362
$561$			4094.74i			0.0130107i
$562$	57412.1			0.181774
$563$			59429.1i			0.187492i	0.995596	+	0.0937459i	$0.0298841\pi$
$563$			59429.1i			0.187492i	−0.995596	+	0.0937459i	$0.970116\pi$
$564$		−	159837.i		−	0.502480i
$565$		−	554013.i		−	1.73549i
$566$			157573.i			0.491870i
$567$	−116166.			−0.361339
$568$	−131375.			−0.407208
$569$			364145.i			1.12473i	0.826888	+	0.562367i	$0.190109\pi$
$569$			364145.i			1.12473i	−0.826888	+	0.562367i	$0.809891\pi$
$570$	−39262.6	+	177399.i	−0.120845	+	0.546011i
$571$	410057.			1.25769			0.628843	−	0.777532i	$-0.283529\pi$
$571$	410057.			1.25769			0.628843	+	0.777532i	$0.283529\pi$
$572$		−	10688.6i		−	0.0326684i
$573$		−	259617.i		−	0.790723i
$574$	−7530.26			−0.0228553
$575$	−53898.3			−0.163020
$576$	−18735.4			−0.0564700
$577$	−44617.5			−0.134015			−0.0670075	−	0.997752i	$-0.521345\pi$
$577$	−44617.5			−0.134015			−0.0670075	+	0.997752i	$0.521345\pi$
$578$			234536.i			0.702029i
$579$	−286935.			−0.855906
$580$			287527.i			0.854717i
$581$	−51342.3			−0.152098
$582$			94686.0i			0.279537i
$583$			102059.i			0.300272i
$584$			82127.6i			0.240804i
$585$		−	52034.7i		−	0.152048i
$586$	237738.			0.692315
$587$	−371422.			−1.07793			−0.538966	−	0.842328i	$-0.681185\pi$
$587$	−371422.			−1.07793			−0.538966	+	0.842328i	$0.681185\pi$
$588$		−	13103.3i		−	0.0378987i
$589$	321693.	+	71198.3i	0.927280	+	0.205229i
$590$	81483.9			0.234082
$591$			175334.i			0.501985i
$592$			20803.8i			0.0593606i
$593$	−327963.			−0.932644			−0.466322	−	0.884615i	$-0.654421\pi$
$593$	−327963.			−0.932644			−0.466322	+	0.884615i	$0.654421\pi$
$594$	−55608.1			−0.157603
$595$	−33645.5			−0.0950372
$596$	157822.			0.444299
$597$		−	78454.7i		−	0.220125i
$598$	92217.4			0.257876
$599$		−	81737.2i		−	0.227806i	−0.993492	−	0.113903i	$-0.963665\pi$
$599$		−	81737.2i		−	0.227806i	0.993492	−	0.113903i	$-0.0363354\pi$
$600$	−13274.4			−0.0368733
$601$		−	75524.8i		−	0.209093i	−0.994520	−	0.104547i	$-0.966661\pi$
$601$		−	75524.8i		−	0.209093i	0.994520	−	0.104547i	$-0.0333392\pi$
$602$			370371.i			1.02198i
$603$			270279.i			0.743325i
$604$		−	6164.52i		−	0.0168976i
$605$	−374146.			−1.02219
$606$	297684.			0.810607
$607$		−	97397.5i		−	0.264345i	−0.991227	−	0.132172i	$-0.957805\pi$
$607$		−	97397.5i		−	0.264345i	0.991227	−	0.132172i	$-0.0421952\pi$
$608$	63804.0	+	14121.3i	0.172600	+	0.0382005i
$609$	461445.			1.24419
$610$			493256.i			1.32560i
$611$		−	159663.i		−	0.427682i
$612$	7169.61			0.0191422
$613$	185283.			0.493078			0.246539	−	0.969133i	$-0.420707\pi$
$613$	185283.			0.493078			0.246539	+	0.969133i	$0.420707\pi$
$614$	−29406.3			−0.0780017
$615$	9208.49			0.0243466
$616$		−	29206.6i		−	0.0769695i
$617$	628408.			1.65071			0.825357	−	0.564612i	$-0.190974\pi$
$617$	628408.			1.65071			0.825357	+	0.564612i	$0.190974\pi$
$618$		−	246323.i		−	0.644952i
$619$	203791.			0.531867			0.265933	−	0.963991i	$-0.414320\pi$
$619$	203791.			0.531867			0.265933	+	0.963991i	$0.414320\pi$
$620$			194968.i			0.507201i
$621$		−	479768.i		−	1.24408i
$622$		−	334058.i		−	0.863458i
$623$		−	316545.i		−	0.815568i
$624$	22711.8			0.0583288
$625$	−437896.			−1.12101
$626$			23733.8i			0.0605646i
$627$	58930.0	+	13042.6i	0.149900	+	0.0331764i
$628$	−369956.			−0.938061
$629$		−	7961.13i		−	0.0201221i
$630$		−	142185.i		−	0.358238i
$631$	496722.			1.24754			0.623771	−	0.781607i	$-0.285600\pi$
$631$	496722.			1.24754			0.623771	+	0.781607i	$0.285600\pi$
$632$	−88318.5			−0.221115
$633$	83270.0			0.207817
$634$	405805.			1.00958
$635$		−	296875.i		−	0.736252i
$636$	−216862.			−0.536130
$637$		−	13089.0i		−	0.0322572i
$638$	95513.2			0.234651
$639$			212457.i			0.520319i
$640$			38669.6i			0.0944083i
$641$		−	44345.5i		−	0.107928i	−0.998543	−	0.0539639i	$-0.982814\pi$
$641$		−	44345.5i		−	0.107928i	0.998543	−	0.0539639i	$-0.0171856\pi$
$642$		−	155934.i		−	0.378330i
$643$	139327.			0.336987			0.168494	−	0.985703i	$-0.446110\pi$
$643$	139327.			0.336987			0.168494	+	0.985703i	$0.446110\pi$
$644$	251984.			0.607577
$645$		−	452913.i		−	1.08867i
$646$	−24416.3	−	5403.91i	−0.0585080	−	0.0129492i
$647$	−617612.			−1.47539			−0.737696	−	0.675133i	$-0.764086\pi$
$647$	−617612.			−1.47539			−0.737696	+	0.675133i	$0.764086\pi$
$648$		−	51092.4i		−	0.121676i
$649$		−	27068.1i		−	0.0642640i
$650$	−13259.9			−0.0313845
$651$	312900.			0.738319
$652$	−263615.			−0.620119
$653$	570062.			1.33689			0.668446	−	0.743761i	$-0.266960\pi$
$653$	570062.			1.33689			0.668446	+	0.743761i	$0.266960\pi$
$654$			268015.i			0.626619i
$655$	5963.16			0.0138993
$656$		−	3311.96i		−	0.00769622i
$657$	132815.			0.307692
$658$		−	436278.i		−	1.00765i
$659$		−	716546.i		−	1.64996i	−0.565162	−	0.824980i	$-0.691186\pi$
$659$		−	716546.i		−	1.64996i	0.565162	−	0.824980i	$-0.308814\pi$
$660$			35715.7i			0.0819919i
$661$		−	187692.i		−	0.429579i	−0.976660	−	0.214790i	$-0.931093\pi$
$661$		−	187692.i		−	0.429579i	0.976660	−	0.214790i	$-0.0689066\pi$
$662$	−47923.0			−0.109352
$663$	−8691.30			−0.0197723
$664$		−	22581.4i		−	0.0512171i
$665$	−107168.	+	484214.i	−0.242338	+	1.09495i
$666$	33643.4			0.0758493
$667$			824055.i			1.85227i
$668$			384393.i			0.861434i
$669$	−397877.			−0.888990
$670$	557854.			1.24271
$671$	163854.			0.363926
$672$	62060.1			0.137428
$673$			504263.i			1.11334i	0.830734	+	0.556669i	$0.187921\pi$
$673$			504263.i			1.11334i	−0.830734	+	0.556669i	$0.812079\pi$
$674$	−614244.			−1.35214
$675$			68985.8i			0.151409i
$676$	−205801.			−0.450354
$677$		−	689926.i		−	1.50531i	−0.658417	−	0.752653i	$-0.728774\pi$
$677$		−	689926.i		−	1.50531i	0.658417	−	0.752653i	$-0.271226\pi$
$678$		−	391055.i		−	0.850703i
$679$			258448.i			0.560574i
$680$		−	14798.0i		−	0.0320026i
$681$	386267.			0.832901
$682$	64766.3			0.139245
$683$			226300.i			0.485113i	0.970137	+	0.242556i	$0.0779859\pi$
$683$			226300.i			0.485113i	−0.970137	+	0.242556i	$0.922014\pi$
$684$	22836.7	−	103182.i	0.0488114	−	0.220543i
$685$	506098.			1.07858
$686$			313613.i			0.666417i
$687$			267608.i			0.567003i
$688$	−162897.			−0.344140
$689$	−216626.			−0.456323
$690$	−308142.			−0.647222
$691$	−921891.			−1.93074			−0.965370	−	0.260886i	$-0.915985\pi$
$691$	−921891.			−1.93074			−0.965370	+	0.260886i	$0.915985\pi$
$692$		−	369756.i		−	0.772153i
$693$	−47232.2			−0.0983494
$694$		−	182589.i		−	0.379102i
$695$	−296815.			−0.614492
$696$			202953.i			0.418964i
$697$			1267.41i			0.00260887i
$698$			320390.i			0.657610i
$699$			478068.i			0.978443i
$700$	−36232.8			−0.0739444
$701$	−561368.			−1.14238			−0.571191	−	0.820817i	$-0.693519\pi$
$701$	−561368.			−1.14238			−0.571191	+	0.820817i	$0.693519\pi$
$702$		−	118031.i		−	0.239510i
$703$	−114574.	−	25357.9i	−0.231832	−	0.0513100i
$704$	12845.6			0.0259185
$705$			533509.i			1.07341i
$706$		−	78306.0i		−	0.157103i
$707$	812536.			1.62556
$708$	57516.1			0.114742
$709$	−287728.			−0.572386			−0.286193	−	0.958172i	$-0.592390\pi$
$709$	−287728.			−0.572386			−0.286193	+	0.958172i	$0.592390\pi$
$710$	438509.			0.869885
$711$			142827.i			0.282534i
$712$	139223.			0.274632
$713$			558782.i			1.09917i
$714$	−23749.0			−0.0465853
$715$			35676.8i			0.0697868i
$716$		−	318612.i		−	0.621493i
$717$			294400.i			0.572663i
$718$		−	184799.i		−	0.358468i
$719$	−805860.			−1.55884			−0.779420	−	0.626502i	$-0.784486\pi$
$719$	−805860.			−1.55884			−0.779420	+	0.626502i	$0.784486\pi$
$720$	62535.7			0.120632
$721$		−	672344.i		−	1.29336i
$722$	−155542.	+	334178.i	−0.298383	+	0.641068i
$723$	225397.			0.431193
$724$		−	414756.i		−	0.791254i
$725$		−	118491.i		−	0.225428i
$726$	−264094.			−0.501055
$727$	−348604.			−0.659575			−0.329787	−	0.944055i	$-0.606977\pi$
$727$	−348604.			−0.659575			−0.329787	+	0.944055i	$0.606977\pi$
$728$	61992.5			0.116970
$729$	−505606.			−0.951387
$730$		−	274129.i		−	0.514409i
$731$	62336.8			0.116657
$732$			348169.i			0.649783i
$733$	238795.			0.444443			0.222222	−	0.974996i	$-0.428669\pi$
$733$	238795.			0.444443			0.222222	+	0.974996i	$0.428669\pi$
$734$		−	501475.i		−	0.930801i
$735$			43736.6i			0.0809599i
$736$			110828.i			0.204594i
$737$		−	185313.i		−	0.341170i
$738$	−5356.03			−0.00983401
$739$	502122.			0.919433			0.459717	−	0.888066i	$-0.347951\pi$
$739$	502122.			0.919433			0.459717	+	0.888066i	$0.347951\pi$
$740$		−	69439.6i		−	0.126807i
$741$	−27683.6	+	125082.i	−0.0504181	+	0.227803i
$742$	−591931.			−1.07514
$743$		−	411861.i		−	0.746058i	−0.927820	−	0.373029i	$-0.878319\pi$
$743$		−	411861.i		−	0.746058i	0.927820	−	0.373029i	$-0.121681\pi$
$744$			137620.i			0.248620i
$745$	−526784.			−0.949118
$746$	43505.8			0.0781753
$747$	−36518.2			−0.0654437
$748$	−4915.73			−0.00878588
$749$		−	425625.i		−	0.758689i
$750$	−270256.			−0.480454
$751$			876407.i			1.55391i	0.629556	+	0.776955i	$0.283237\pi$
$751$			876407.i			1.55391i	−0.629556	+	0.776955i	$0.716763\pi$
$752$	191884.			0.339315
$753$		−	432838.i		−	0.763371i
$754$			202732.i			0.356599i
$755$			20576.2i			0.0360970i
$756$		−	322520.i		−	0.564305i
$757$	−447975.			−0.781739			−0.390870	−	0.920446i	$-0.627826\pi$
$757$	−447975.			−0.781739			−0.390870	+	0.920446i	$0.627826\pi$
$758$	657964.			1.14515
$759$			102362.i			0.177686i
$760$	−212967.	−	47134.7i	−0.368711	−	0.0816044i
$761$	527400.			0.910690			0.455345	−	0.890315i	$-0.349516\pi$
$761$	527400.			0.910690			0.455345	+	0.890315i	$0.349516\pi$
$762$		−	209552.i		−	0.360896i
$763$			731553.i			1.25660i
$764$	311671.			0.533960
$765$	−23931.0			−0.0408919
$766$	−147173.			−0.250825
$767$	57453.4			0.0976619
$768$			27295.3i			0.0462770i
$769$	549287.			0.928852			0.464426	−	0.885612i	$-0.346261\pi$
$769$	549287.			0.928852			0.464426	+	0.885612i	$0.346261\pi$
$770$			97486.7i			0.164424i
$771$	−7700.71			−0.0129545
$772$		−	344465.i		−	0.577977i
$773$		−	1.03569e6i		−	1.73329i	−0.498927	−	0.866644i	$-0.666272\pi$
$773$		−	1.03569e6i		−	1.73329i	0.498927	−	0.866644i	$-0.333728\pi$
$774$			263433.i			0.439732i
$775$		−	80347.2i		−	0.133773i
$776$	−113671.			−0.188766
$777$	−111442.			−0.184590
$778$		−	163165.i		−	0.269568i
$779$	18240.1	+	4036.98i	0.0300575	+	0.00665244i
$780$	−75808.4			−0.124603
$781$		−	145668.i		−	0.238815i
$782$		−	42411.2i		−	0.0693533i
$783$	1.05473e6			1.72035
$784$	15730.5			0.0255923
$785$	1.23485e6			2.00390
$786$	4209.15			0.00681317
$787$			1.04115e6i			1.68098i	0.541824	+	0.840492i	$0.317734\pi$
$787$			1.04115e6i			1.68098i	−0.541824	+	0.840492i	$0.682266\pi$
$788$	−210488.			−0.338981
$789$			44867.6i			0.0720740i
$790$	294793.			0.472349
$791$		−	1.06739e6i		−	1.70597i
$792$		−	20773.7i		−	0.0331179i
$793$			347790.i			0.553058i
$794$		−	47840.5i		−	0.0758847i
$795$	723851.			1.14529
$796$	94184.8			0.148647
$797$			476576.i			0.750266i	0.926971	+	0.375133i	$0.122403\pi$
$797$			476576.i			0.750266i	−0.926971	+	0.375133i	$0.877597\pi$
$798$	−75645.5	+	341787.i	−0.118789	+	0.536722i
$799$	−73429.7			−0.115021
$800$		−	15935.9i		−	0.0248999i
$801$		−	225149.i		−	0.350917i
$802$	−66415.9			−0.103258
$803$	−91062.5			−0.141224
$804$	393766.			0.609152
$805$	−841082.			−1.29792
$806$			137470.i			0.211611i
$807$	148148.			0.227483
$808$			357370.i			0.547388i
$809$	−1.00569e6			−1.53663			−0.768314	−	0.640074i	$-0.778904\pi$
$809$	−1.00569e6			−1.53663			−0.768314	+	0.640074i	$0.778904\pi$
$810$			170538.i			0.259927i
$811$		−	257961.i		−	0.392205i	−0.980583	−	0.196102i	$-0.937171\pi$
$811$		−	257961.i		−	0.392205i	0.980583	−	0.196102i	$-0.0628285\pi$
$812$			553965.i			0.840176i
$813$		−	310521.i		−	0.469797i
$814$	−23067.1			−0.0348132
$815$	879905.			1.32471
$816$		−	10445.3i		−	0.0156870i
$817$	198556.	−	897129.i	0.297467	−	1.34404i
$818$	−879007.			−1.31367
$819$		−	100253.i		−	0.149461i
$820$			11054.8i			0.0164408i
$821$	−65320.9			−0.0969094			−0.0484547	−	0.998825i	$-0.515430\pi$
$821$	−65320.9			−0.0969094			−0.0484547	+	0.998825i	$0.515430\pi$
$822$	357233.			0.528699
$823$	−109324.			−0.161404			−0.0807020	−	0.996738i	$-0.525716\pi$
$823$	−109324.			−0.161404			−0.0807020	+	0.996738i	$0.525716\pi$
$824$	295710.			0.435524
$825$		−	14718.6i		−	0.0216251i
$826$	156991.			0.230100
$827$			988603.i			1.44548i	0.691122	+	0.722738i	$0.257117\pi$
$827$			988603.i			1.44548i	−0.691122	+	0.722738i	$0.742883\pi$
$828$	179228.			0.261424
$829$		−	467829.i		−	0.680735i	−0.940292	−	0.340368i	$-0.889448\pi$
$829$		−	467829.i		−	0.680735i	0.940292	−	0.340368i	$-0.110552\pi$
$830$			75373.1i			0.109411i
$831$			705151.i			1.02113i
$832$			27265.6i			0.0393884i
$833$	−6019.69			−0.00867529
$834$	−209509.			−0.301211
$835$		−	1.28304e6i		−	1.84021i
$836$	−15657.6	+	70745.4i	−0.0224034	+	0.101225i
$837$	715198.			1.02088
$838$			768092.i			1.09377i
$839$		−	657894.i		−	0.934613i	−0.884095	−	0.467306i	$-0.845224\pi$
$839$		−	657894.i		−	0.934613i	0.884095	−	0.467306i	$-0.154776\pi$
$840$	−207147.			−0.293575
$841$	−1.10433e6			−1.56138
$842$	912074.			1.28649
$843$	−135265.			−0.190340
$844$			99965.6i			0.140335i
$845$	686930.			0.962053
$846$		−	310311.i		−	0.433567i
$847$	−720851.			−1.00480
$848$		−	260343.i		−	0.362038i
$849$		−	371249.i		−	0.515051i
$850$			6098.31i			0.00844057i
$851$		−	199015.i		−	0.274806i
$852$	309525.			0.426400
$853$	−226576.			−0.311398			−0.155699	−	0.987805i	$-0.549763\pi$
$853$	−226576.			−0.311398			−0.155699	+	0.987805i	$0.549763\pi$
$854$			950336.i			1.30305i
$855$	−76225.2	+	344406.i	−0.104272	+	0.471128i
$856$	187199.			0.255479
$857$		−	183273.i		−	0.249539i	−0.992186	−	0.124769i	$-0.960181\pi$
$857$		−	183273.i		−	0.249539i	0.992186	−	0.124769i	$-0.0398191\pi$
$858$			25182.7i			0.0342081i
$859$	565839.			0.766843			0.383421	−	0.923574i	$-0.374746\pi$
$859$	565839.			0.766843			0.383421	+	0.923574i	$0.374746\pi$
$860$	543723.			0.735158
$861$	17741.6			0.0239324
$862$	−446752.			−0.601246
$863$			247431.i			0.332225i	0.986107	+	0.166113i	$0.0531216\pi$
$863$			247431.i			0.332225i	−0.986107	+	0.166113i	$0.946878\pi$
$864$	141851.			0.190023
$865$			1.23419e6i			1.64949i
$866$	104501.			0.139343
$867$		−	552578.i		−	0.735115i
$868$			375637.i			0.498573i
$869$		−	97927.0i		−	0.129677i
$870$		−	677424.i		−	0.894999i
$871$	393337.			0.518476
$872$	−321752.			−0.423144
$873$			183826.i			0.241200i
$874$	−610367.	−	135089.i	−0.799040	−	0.176846i
$875$	−737669.			−0.963486
$876$		−	193496.i		−	0.252153i
$877$			58891.6i			0.0765692i	0.999267	+	0.0382846i	$0.0121893\pi$
$877$			58891.6i			0.0765692i	−0.999267	+	0.0382846i	$0.987811\pi$
$878$	−549780.			−0.713182
$879$	−560121.			−0.724943
$880$	−42876.7			−0.0553676
$881$	−844161.			−1.08761			−0.543806	−	0.839211i	$-0.683017\pi$
$881$	−844161.			−1.08761			−0.543806	+	0.839211i	$0.683017\pi$
$882$		−	25439.0i		−	0.0327011i
$883$	−1.34541e6			−1.72558			−0.862788	−	0.505565i	$-0.831284\pi$
$883$	−1.34541e6			−1.72558			−0.862788	+	0.505565i	$0.831284\pi$
$884$		−	10433.9i		−	0.0133519i
$885$	−191979.			−0.245114
$886$		−	86752.7i		−	0.110514i
$887$		−	268542.i		−	0.341323i	−0.985330	−	0.170661i	$-0.945410\pi$
$887$		−	268542.i		−	0.341323i	0.985330	−	0.170661i	$-0.0545904\pi$
$888$		−	49014.5i		−	0.0621583i
$889$		−	571977.i		−	0.723727i
$890$	−464704.			−0.586674
$891$	56650.9			0.0713594
$892$		−	477652.i		−	0.600319i
$893$	−233889.	+	1.05677e6i	−0.293296	+	1.32519i
$894$	−371835.			−0.465238
$895$			1.06348e6i			1.32764i
$896$			74503.1i			0.0928023i
$897$	−217268.			−0.270029
$898$	669902.			0.830728
$899$	−1.22843e6			−1.51996
$900$	−25771.2			−0.0318163
$901$			99627.4i			0.122724i
$902$	3672.28			0.00451360
$903$		−	872609.i		−	1.07015i
$904$	469461.			0.574464
$905$			1.38439e6i			1.69029i
$906$			14523.9i			0.0176940i
$907$		−	1.24035e6i		−	1.50775i	−0.657017	−	0.753876i	$-0.728182\pi$
$907$		−	1.24035e6i		−	1.50775i	0.657017	−	0.753876i	$-0.271818\pi$
$908$			463714.i			0.562443i
$909$	577931.			0.699436
$910$	−206921.			−0.249874
$911$			191069.i			0.230226i	0.993352	+	0.115113i	$0.0367229\pi$
$911$			191069.i			0.230226i	−0.993352	+	0.115113i	$0.963277\pi$
$912$	−150325.	−	33270.4i	−0.180734	−	0.0400008i
$913$	25038.1			0.0300372
$914$			232103.i			0.277836i
$915$		−	1.16213e6i		−	1.38808i
$916$	−321263.			−0.382886
$917$	11489.0			0.0136629
$918$	−54283.2			−0.0644139
$919$	830092.			0.982868			0.491434	−	0.870915i	$-0.336473\pi$
$919$	830092.			0.982868			0.491434	+	0.870915i	$0.336473\pi$
$920$		−	369925.i		−	0.437057i
$921$	69282.5			0.0816779
$922$			47463.7i			0.0558342i
$923$	309188.			0.362927
$924$			68811.8i			0.0805971i
$925$			28616.3i			0.0334449i
$926$		−	283321.i		−	0.330413i
$927$		−	478216.i		−	0.556500i
$928$	−243645.			−0.282919
$929$	1.18305e6			1.37080			0.685399	−	0.728168i	$-0.259628\pi$
$929$	1.18305e6			1.37080			0.685399	+	0.728168i	$0.259628\pi$
$930$		−	459353.i		−	0.531106i
$931$	−19174.0	+	86633.2i	−0.0221214	+	0.0999505i
$932$	−573921.			−0.660724
$933$			787054.i			0.904152i
$934$			613123.i			0.702836i
$935$	16407.9			0.0187685
$936$	44093.3			0.0503293
$937$	1.14337e6			1.30229			0.651144	−	0.758954i	$-0.274289\pi$
$937$	1.14337e6			1.30229			0.651144	+	0.758954i	$0.274289\pi$
$938$	1.07479e6			1.22157
$939$		−	55917.9i		−	0.0634190i
$940$	−640478.			−0.724851
$941$		−	1.12716e6i		−	1.27294i	−0.771302	−	0.636469i	$-0.780394\pi$
$941$		−	1.12716e6i		−	1.27294i	0.771302	−	0.636469i	$-0.219606\pi$
$942$	871632.			0.982271
$943$			31683.2i			0.0356291i
$944$			69048.0i			0.0774832i
$945$			1.07652e6i			1.20548i
$946$		−	180619.i		−	0.201828i
$947$	1.01859e6			1.13579			0.567897	−	0.823099i	$-0.307757\pi$
$947$	1.01859e6			1.13579			0.567897	+	0.823099i	$0.307757\pi$
$948$	208082.			0.231536
$949$		−	193285.i		−	0.214618i
$950$	87764.6	+	19424.4i	0.0972461	+	0.0215229i
$951$	−956093.			−1.05716
$952$		−	28510.6i		−	0.0314582i
$953$		−	18352.7i		−	0.0202076i	−0.999949	−	0.0101038i	$-0.996784\pi$
$953$		−	18352.7i		−	0.0202076i	0.999949	−	0.0101038i	$-0.00321619\pi$
$954$	−421021.			−0.462602
$955$	−1.04031e6			−1.14066
$956$	−353427.			−0.386709
$957$	−225033.			−0.245710
$958$		−	259299.i		−	0.282534i
$959$	975077.			1.06023
$960$		−	91107.3i		−	0.0988577i
$961$	90535.6			0.0980331
$962$		−	48961.1i		−	0.0529056i
$963$		−	302734.i		−	0.326444i
$964$			270589.i			0.291177i
$965$			1.14977e6i			1.23468i
$966$	−593685.			−0.636212
$967$	301406.			0.322329			0.161164	−	0.986928i	$-0.448475\pi$
$967$	301406.			0.322329			0.161164	+	0.986928i	$0.448475\pi$
$968$		−	317045.i		−	0.338353i
$969$	57525.9	+	12731.8i	0.0612654	+	0.0135595i
$970$	379414.			0.403246
$971$			796081.i			0.844342i	0.906516	+	0.422171i	$0.138732\pi$
$971$			796081.i			0.844342i	−0.906516	+	0.422171i	$0.861268\pi$
$972$		−	387413.i		−	0.410054i
$973$	−571861.			−0.604039
$974$	−303562.			−0.319985
$975$	31240.9			0.0328636
$976$	−417977.			−0.438786
$977$			664866.i			0.696538i	0.937395	+	0.348269i	$0.113230\pi$
$977$			664866.i			0.696538i	−0.937395	+	0.348269i	$0.886770\pi$
$978$	621088.			0.649345
$979$			154370.i			0.161063i
$980$	−52505.7			−0.0546707
$981$			520330.i			0.540681i
$982$		−	599113.i		−	0.621278i
$983$		−	1.05287e6i		−	1.08960i	−0.838566	−	0.544801i	$-0.816605\pi$
$983$		−	1.05287e6i		−	1.08960i	0.838566	−	0.544801i	$-0.183395\pi$
$984$			7803.12i			0.00805894i
$985$	702576.			0.724137
$986$	93237.5			0.0959040
$987$			1.02789e6i			1.05514i
$988$	−150161.	−	33234.2i	−0.153831	−	0.0340464i
$989$	1.55832e6			1.59317
$990$			69339.2i			0.0707471i
$991$			1.85377e6i			1.88759i	0.330529	+	0.943796i	$0.392773\pi$
$991$			1.85377e6i			1.88759i	−0.330529	+	0.943796i	$0.607227\pi$
$992$	−165213.			−0.167888
$993$	112908.			0.114506
$994$	844856.			0.855087
$995$	−314374.			−0.317541
$996$			53202.7i			0.0536309i
$997$	494751.			0.497733			0.248867	−	0.968538i	$-0.419942\pi$
$997$	494751.			0.497733			0.248867	+	0.968538i	$0.419942\pi$
$998$		−	411505.i		−	0.413156i
$999$	−254724.			−0.255234

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	38.5.b.a.37.3	✓	8
3.2	odd	2		342.5.d.a.37.6		8
4.3	odd	2		304.5.e.e.113.4		8
19.18	odd	2	inner	38.5.b.a.37.6	yes	8
57.56	even	2		342.5.d.a.37.2		8
76.75	even	2		304.5.e.e.113.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.5.b.a.37.3	✓	8	1.1	even	1	trivial
38.5.b.a.37.6	yes	8	19.18	odd	2	inner
304.5.e.e.113.4		8	4.3	odd	2
304.5.e.e.113.5		8	76.75	even	2
342.5.d.a.37.2		8	57.56	even	2
342.5.d.a.37.6		8	3.2	odd	2

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:	\( N \)	\(=\)	\( 38 = 2 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	38.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.82843i q^{2} +6.66389i q^{3} -8.00000 q^{4} +26.7027 q^{5} +18.8483 q^{6} +51.4469 q^{7} +22.6274i q^{8} +36.5926 q^{9} +O(q^{10})\)	\(q-2.82843i q^{2} +6.66389i q^{3} -8.00000 q^{4} +26.7027 q^{5} +18.8483 q^{6} +51.4469 q^{7} +22.6274i q^{8} +36.5926 q^{9} -75.5266i q^{10} -25.0891 q^{11} -53.3111i q^{12} -53.2531i q^{13} -145.514i q^{14} +177.944i q^{15} +64.0000 q^{16} -24.4913 q^{17} -103.499i q^{18} +(-78.0101 + 352.470i) q^{19} -213.622 q^{20} +342.837i q^{21} +70.9628i q^{22} -612.242 q^{23} -150.787 q^{24} +88.0343 q^{25} -150.622 q^{26} +783.624i q^{27} -411.576 q^{28} -1345.96i q^{29} +503.301 q^{30} -912.680i q^{31} -181.019i q^{32} -167.191i q^{33} +69.2720i q^{34} +1373.77 q^{35} -292.741 q^{36} +325.059i q^{37} +(996.937 + 220.646i) q^{38} +354.872 q^{39} +604.213i q^{40} -51.7494i q^{41} +969.689 q^{42} -2545.26 q^{43} +200.713 q^{44} +977.121 q^{45} +1731.68i q^{46} +2998.19 q^{47} +426.489i q^{48} +245.788 q^{49} -248.999i q^{50} -163.208i q^{51} +426.024i q^{52} -4067.86i q^{53} +2216.42 q^{54} -669.948 q^{55} +1164.11i q^{56} +(-2348.82 - 519.851i) q^{57} -3806.96 q^{58} +1078.88i q^{59} -1423.55i q^{60} -6530.89 q^{61} -2581.45 q^{62} +1882.58 q^{63} -512.000 q^{64} -1422.00i q^{65} -472.888 q^{66} +7386.18i q^{67} +195.931 q^{68} -4079.92i q^{69} -3885.62i q^{70} +5806.02i q^{71} +827.995i q^{72} +3629.56 q^{73} +919.405 q^{74} +586.651i q^{75} +(624.081 - 2819.76i) q^{76} -1290.76 q^{77} -1003.73i q^{78} +3903.16i q^{79} +1708.97 q^{80} -2257.99 q^{81} -146.369 q^{82} -997.967 q^{83} -2742.69i q^{84} -653.985 q^{85} +7199.08i q^{86} +8969.34 q^{87} -567.702i q^{88} -6152.85i q^{89} -2763.71i q^{90} -2739.71i q^{91} +4897.94 q^{92} +6082.00 q^{93} -8480.16i q^{94} +(-2083.08 + 9411.91i) q^{95} +1206.29 q^{96} +5023.58i q^{97} -695.195i q^{98} -918.076 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 64 q^{4} + 18 q^{5} - 32 q^{6} - 162 q^{7} - 268 q^{9}+O(q^{10}) \) 8 * q - 64 * q^4 + 18 * q^5 - 32 * q^6 - 162 * q^7 - 268 * q^9	\( 8 q - 64 q^{4} + 18 q^{5} - 32 q^{6} - 162 q^{7} - 268 q^{9} - 6 q^{11} + 512 q^{16} + 510 q^{17} - 12 q^{19} - 144 q^{20} - 396 q^{23} + 256 q^{24} + 3458 q^{25} - 192 q^{26} + 1296 q^{28} - 2752 q^{30} + 1002 q^{35} + 2144 q^{36} - 3216 q^{38} - 6588 q^{39} + 1376 q^{42} - 8654 q^{43} + 48 q^{44} - 10334 q^{45} + 3210 q^{47} + 9222 q^{49} + 9088 q^{54} + 17146 q^{55} - 14076 q^{57} - 960 q^{58} + 1314 q^{61} - 15168 q^{62} + 29938 q^{63} - 4096 q^{64} + 4928 q^{66} - 4080 q^{68} + 23398 q^{73} + 13152 q^{74} + 96 q^{76} - 44622 q^{77} + 1152 q^{80} - 20368 q^{81} + 16512 q^{82} - 10440 q^{83} + 21274 q^{85} - 14316 q^{87} + 3168 q^{92} + 19416 q^{93} - 34686 q^{95} - 2048 q^{96} - 56798 q^{99}+O(q^{100}) \) 8 * q - 64 * q^4 + 18 * q^5 - 32 * q^6 - 162 * q^7 - 268 * q^9 - 6 * q^11 + 512 * q^16 + 510 * q^17 - 12 * q^19 - 144 * q^20 - 396 * q^23 + 256 * q^24 + 3458 * q^25 - 192 * q^26 + 1296 * q^28 - 2752 * q^30 + 1002 * q^35 + 2144 * q^36 - 3216 * q^38 - 6588 * q^39 + 1376 * q^42 - 8654 * q^43 + 48 * q^44 - 10334 * q^45 + 3210 * q^47 + 9222 * q^49 + 9088 * q^54 + 17146 * q^55 - 14076 * q^57 - 960 * q^58 + 1314 * q^61 - 15168 * q^62 + 29938 * q^63 - 4096 * q^64 + 4928 * q^66 - 4080 * q^68 + 23398 * q^73 + 13152 * q^74 + 96 * q^76 - 44622 * q^77 + 1152 * q^80 - 20368 * q^81 + 16512 * q^82 - 10440 * q^83 + 21274 * q^85 - 14316 * q^87 + 3168 * q^92 + 19416 * q^93 - 34686 * q^95 - 2048 * q^96 - 56798 * q^99

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