LMFDB - Embedded newform 342.4.g.f.163.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [342,4,Mod(163,342)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("342.163"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(342, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 4])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 342 = 2 \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	342.g (of order $3$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [6,-6,0,-12,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$20.1786532220$
Analytic rank:	$0$
Dimension:	$6$
Relative dimension:	$3$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} + 64x^{4} + 33x^{3} + 3984x^{2} - 945x + 225 $ x^6 - x^5 + 64x^4 + 33x^3 + 3984x^2 - 945x + 225
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 38)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			163.1
Root			$0.118706 - 0.205606i$ of defining polynomial
Character	$\chi$	$=$	342.163
Dual form			342.4.g.f.235.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-1.00000 + 1.73205i) q^{2} +(-2.00000 - 3.46410i) q^{4} +(-10.3546 + 17.9347i) q^{5} +8.76259 q^{7} +8.00000 q^{8} +(-20.7092 - 35.8694i) q^{10} +62.1780 q^{11} +(32.2611 + 55.8778i) q^{13} +(-8.76259 + 15.1772i) q^{14} +(-8.00000 + 13.8564i) q^{16} +(-23.2077 + 40.1970i) q^{17} +(17.3798 + 80.9750i) q^{19} +82.8369 q^{20} +(-62.1780 + 107.696i) q^{22} +(-18.6424 - 32.2895i) q^{23} +(-151.936 - 263.161i) q^{25} -129.044 q^{26} +(-17.5252 - 30.3545i) q^{28} +(-33.2119 - 57.5247i) q^{29} +112.370 q^{31} +(-16.0000 - 27.7128i) q^{32} +(-46.4155 - 80.3939i) q^{34} +(-90.7332 + 157.155i) q^{35} -189.018 q^{37} +(-157.633 - 50.8723i) q^{38} +(-82.8369 + 143.478i) q^{40} +(-120.005 + 207.854i) q^{41} +(-84.1370 + 145.730i) q^{43} +(-124.356 - 215.391i) q^{44} +74.5695 q^{46} +(93.9241 + 162.681i) q^{47} -266.217 q^{49} +607.744 q^{50} +(129.044 - 223.511i) q^{52} +(-56.5285 - 97.9102i) q^{53} +(-643.830 + 1115.15i) q^{55} +70.1007 q^{56} +132.848 q^{58} +(92.9940 - 161.070i) q^{59} +(-154.322 - 267.293i) q^{61} +(-112.370 + 194.631i) q^{62} +64.0000 q^{64} -1336.20 q^{65} +(-19.7279 - 34.1697i) q^{67} +185.662 q^{68} +(-181.466 - 314.309i) q^{70} +(-175.101 + 303.284i) q^{71} +(4.80280 - 8.31870i) q^{73} +(189.018 - 327.389i) q^{74} +(245.746 - 222.155i) q^{76} +544.840 q^{77} +(588.269 - 1018.91i) q^{79} +(-165.674 - 286.956i) q^{80} +(-240.010 - 415.709i) q^{82} +257.980 q^{83} +(-480.614 - 832.448i) q^{85} +(-168.274 - 291.459i) q^{86} +497.424 q^{88} +(-66.8743 - 115.830i) q^{89} +(282.691 + 489.634i) q^{91} +(-74.5695 + 129.158i) q^{92} -375.696 q^{94} +(-1632.22 - 526.763i) q^{95} +(-598.387 + 1036.44i) q^{97} +(266.217 - 461.101i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{2} - 12 q^{4} + q^{5} + 52 q^{7} + 48 q^{8} + 2 q^{10} - 8 q^{11} + 129 q^{13} - 52 q^{14} - 48 q^{16} + 51 q^{17} + 40 q^{19} - 8 q^{20} + 8 q^{22} - 47 q^{23} - 338 q^{25} - 516 q^{26} - 104 q^{28}+ \cdots + 1354 q^{98}+O(q^{100}) $ 6 * q - 6 * q^2 - 12 * q^4 + q^5 + 52 * q^7 + 48 * q^8 + 2 * q^10 - 8 * q^11 + 129 * q^13 - 52 * q^14 - 48 * q^16 + 51 * q^17 + 40 * q^19 - 8 * q^20 + 8 * q^22 - 47 * q^23 - 338 * q^25 - 516 * q^26 - 104 * q^28 + 125 * q^29 - 100 * q^31 - 96 * q^32 + 102 * q^34 + 84 * q^35 - 376 * q^37 - 322 * q^38 + 8 * q^40 - 475 * q^41 - 73 * q^43 + 16 * q^44 + 188 * q^46 + 241 * q^47 - 1354 * q^49 + 1352 * q^50 + 516 * q^52 - 29 * q^53 - 1838 * q^55 + 416 * q^56 - 500 * q^58 + 1065 * q^59 - 981 * q^61 + 100 * q^62 + 384 * q^64 - 586 * q^65 + 877 * q^67 - 408 * q^68 + 168 * q^70 - 2135 * q^71 + 667 * q^73 + 376 * q^74 + 484 * q^76 + 492 * q^77 + 1671 * q^79 + 16 * q^80 - 950 * q^82 - 1176 * q^83 - 1929 * q^85 - 146 * q^86 - 64 * q^88 - 693 * q^89 + 1676 * q^91 - 188 * q^92 - 964 * q^94 - 4489 * q^95 - 985 * q^97 + 1354 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/342\mathbb{Z}\right)^\times$.

$n$	$191$	$325$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$

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
$5$	−10.3546	+	17.9347i	−0.926145	+	1.60413i	−0.136436	+	0.990649i	$0.543565\pi$
$5$	−10.3546	+	17.9347i	−0.926145	+	1.60413i	−0.789709	+	0.613481i	$0.789769\pi$
$6$		0			0
$7$	8.76259			0.473135			0.236568	−	0.971615i	$-0.423978\pi$
$7$	8.76259			0.473135			0.236568	+	0.971615i	$0.423978\pi$
$8$	8.00000			0.353553
$9$		0			0
$10$	−20.7092	−	35.8694i	−0.654883	−	1.13429i
$11$	62.1780			1.70431			0.852154	−	0.523291i	$-0.175296\pi$
$11$	62.1780			1.70431			0.852154	+	0.523291i	$0.175296\pi$
$12$		0			0
$13$	32.2611	+	55.8778i	0.688278	+	1.19213i	0.972395	+	0.233343i	$0.0749664\pi$
$13$	32.2611	+	55.8778i	0.688278	+	1.19213i	−0.284117	+	0.958790i	$0.591700\pi$
$14$	−8.76259	+	15.1772i	−0.167279	+	0.289735i
$15$		0			0
$16$	−8.00000	+	13.8564i	−0.125000	+	0.216506i
$17$	−23.2077	+	40.1970i	−0.331100	+	0.573482i	−0.982728	−	0.185057i	$-0.940753\pi$
$17$	−23.2077	+	40.1970i	−0.331100	+	0.573482i	0.651628	+	0.758539i	$0.274086\pi$
$18$		0			0
$19$	17.3798	+	80.9750i	0.209852	+	0.977733i
$19$	17.3798	+	80.9750i	0.209852	+	0.977733i
$20$	82.8369			0.926145
$21$		0			0
$22$	−62.1780	+	107.696i	−0.602564	+	1.04367i
$23$	−18.6424	−	32.2895i	−0.169009	−	0.292732i	0.769063	−	0.639173i	$-0.220723\pi$
$23$	−18.6424	−	32.2895i	−0.169009	−	0.292732i	−0.938072	+	0.346441i	$0.887390\pi$
$24$		0			0
$25$	−151.936	−	263.161i	−1.21549	−	2.10529i
$26$	−129.044			−0.973372
$27$		0			0
$28$	−17.5252	−	30.3545i	−0.118284	−	0.204874i
$29$	−33.2119	−	57.5247i	−0.212665	−	0.368347i	0.739883	−	0.672736i	$-0.234881\pi$
$29$	−33.2119	−	57.5247i	−0.212665	−	0.368347i	−0.952548	+	0.304389i	$0.901548\pi$
$30$		0			0
$31$	112.370			0.651043			0.325521	−	0.945535i	$-0.394460\pi$
$31$	112.370			0.651043			0.325521	+	0.945535i	$0.394460\pi$
$32$	−16.0000	−	27.7128i	−0.0883883	−	0.153093i
$33$		0			0
$34$	−46.4155	−	80.3939i	−0.234123	−	0.405513i
$35$	−90.7332	+	157.155i	−0.438192	+	0.758970i
$36$		0			0
$37$	−189.018			−0.839848			−0.419924	−	0.907559i	$-0.637943\pi$
$37$	−189.018			−0.839848			−0.419924	+	0.907559i	$0.637943\pi$
$38$	−157.633	−	50.8723i	−0.672931	−	0.217173i
$39$		0			0
$40$	−82.8369	+	143.478i	−0.327442	+	0.567146i
$41$	−120.005	+	207.854i	−0.457112	+	0.791742i	−0.998807	−	0.0488336i	$-0.984450\pi$
$41$	−120.005	+	207.854i	−0.457112	+	0.791742i	0.541695	+	0.840575i	$0.317783\pi$
$42$		0			0
$43$	−84.1370	+	145.730i	−0.298390	+	0.516827i	−0.975768	−	0.218809i	$-0.929783\pi$
$43$	−84.1370	+	145.730i	−0.298390	+	0.516827i	0.677378	+	0.735635i	$0.263116\pi$
$44$	−124.356	−	215.391i	−0.426077	−	0.737987i
$45$		0			0
$46$	74.5695			0.239015
$47$	93.9241	+	162.681i	0.291494	+	0.504883i	0.974163	−	0.225845i	$-0.0725142\pi$
$47$	93.9241	+	162.681i	0.291494	+	0.504883i	−0.682669	+	0.730728i	$0.739181\pi$
$48$		0			0
$49$	−266.217			−0.776143
$50$	607.744			1.71896
$51$		0			0
$52$	129.044	−	223.511i	0.344139	−	0.596066i
$53$	−56.5285	−	97.9102i	−0.146505	−	0.253755i	0.783428	−	0.621482i	$-0.213469\pi$
$53$	−56.5285	−	97.9102i	−0.146505	−	0.253755i	−0.929934	+	0.367728i	$0.880136\pi$
$54$		0			0
$55$	−643.830	+	1115.15i	−1.57844	+	2.73393i
$56$	70.1007			0.167279
$57$		0			0
$58$	132.848			0.300754
$59$	92.9940	−	161.070i	0.205200	−	0.355417i	−0.744997	−	0.667068i	$-0.767549\pi$
$59$	92.9940	−	161.070i	0.205200	−	0.355417i	0.950196	+	0.311652i	$0.100882\pi$
$60$		0			0
$61$	−154.322	−	267.293i	−0.323916	−	0.561038i	0.657377	−	0.753562i	$-0.271666\pi$
$61$	−154.322	−	267.293i	−0.323916	−	0.561038i	−0.981292	+	0.192524i	$0.938333\pi$
$62$	−112.370	+	194.631i	−0.230178	+	0.398681i
$63$		0			0
$64$	64.0000			0.125000
$65$	−1336.20			−2.54978
$66$		0			0
$67$	−19.7279	−	34.1697i	−0.0359723	−	0.0623058i	0.847479	−	0.530829i	$-0.178119\pi$
$67$	−19.7279	−	34.1697i	−0.0359723	−	0.0623058i	−0.883451	+	0.468523i	$0.844786\pi$
$68$	185.662			0.331100
$69$		0			0
$70$	−181.466	−	314.309i	−0.309848	−	0.536673i
$71$	−175.101	+	303.284i	−0.292686	+	0.506946i	−0.974444	−	0.224631i	$-0.927882\pi$
$71$	−175.101	+	303.284i	−0.292686	+	0.506946i	0.681758	+	0.731578i	$0.261216\pi$
$72$		0			0
$73$	4.80280	−	8.31870i	0.00770035	−	0.0133374i	−0.862150	−	0.506654i	$-0.830882\pi$
$73$	4.80280	−	8.31870i	0.00770035	−	0.0133374i	0.869850	+	0.493316i	$0.164216\pi$
$74$	189.018	−	327.389i	0.296931	−	0.514300i
$75$		0			0
$76$	245.746	−	222.155i	0.370908	−	0.335302i
$77$	544.840			0.806368
$78$		0			0
$79$	588.269	−	1018.91i	0.837791	−	1.45110i	−0.0539471	−	0.998544i	$-0.517180\pi$
$79$	588.269	−	1018.91i	0.837791	−	1.45110i	0.891738	−	0.452552i	$-0.149486\pi$
$80$	−165.674	−	286.956i	−0.231536	−	0.401033i
$81$		0			0
$82$	−240.010	−	415.709i	−0.323227	−	0.559846i
$83$	257.980			0.341168			0.170584	−	0.985343i	$-0.445435\pi$
$83$	257.980			0.341168			0.170584	+	0.985343i	$0.445435\pi$
$84$		0			0
$85$	−480.614	−	832.448i	−0.613293	−	1.06226i
$86$	−168.274	−	291.459i	−0.210994	−	0.365452i
$87$		0			0
$88$	497.424			0.602564
$89$	−66.8743	−	115.830i	−0.0796479	−	0.137954i	0.823450	−	0.567389i	$-0.192046\pi$
$89$	−66.8743	−	115.830i	−0.0796479	−	0.137954i	−0.903098	+	0.429434i	$0.858713\pi$
$90$		0			0
$91$	282.691	+	489.634i	0.325649	+	0.564040i
$92$	−74.5695	+	129.158i	−0.0845044	+	0.146366i
$93$		0			0
$94$	−375.696			−0.412235
$95$	−1632.22	−	526.763i	−1.76276	−	0.568892i
$96$		0			0
$97$	−598.387	+	1036.44i	−0.626361	+	1.08489i	0.361915	+	0.932211i	$0.382123\pi$
$97$	−598.387	+	1036.44i	−0.626361	+	1.08489i	−0.988276	+	0.152678i	$0.951210\pi$
$98$	266.217	−	461.101i	0.274408	−	0.475289i
$99$		0			0
$100$	−607.744	+	1052.64i	−0.607744	+	1.05264i
$101$	434.472	+	752.527i	0.428035	+	0.741378i	0.996698	−	0.0811919i	$-0.0258727\pi$
$101$	434.472	+	752.527i	0.428035	+	0.741378i	−0.568663	+	0.822570i	$0.692539\pi$
$102$		0			0
$103$	−1491.56			−1.42687			−0.713437	−	0.700720i	$-0.752862\pi$
$103$	−1491.56			−1.42687			−0.713437	+	0.700720i	$0.752862\pi$
$104$	258.089	+	447.023i	0.243343	+	0.421482i
$105$		0			0
$106$	226.114			0.207190
$107$	1610.18			1.45478			0.727392	−	0.686222i	$-0.240732\pi$
$107$	1610.18			1.45478			0.727392	+	0.686222i	$0.240732\pi$
$108$		0			0
$109$	25.5111	−	44.1865i	0.0224176	−	0.0388284i	−0.854599	−	0.519289i	$-0.826197\pi$
$109$	25.5111	−	44.1865i	0.0224176	−	0.0388284i	0.877017	+	0.480460i	$0.159530\pi$
$110$	−1287.66	−	2230.29i	−1.11612	−	1.93318i
$111$		0			0
$112$	−70.1007	+	121.418i	−0.0591419	+	0.102437i
$113$	−1789.55			−1.48979			−0.744895	−	0.667181i	$-0.767501\pi$
$113$	−1789.55			−1.48979			−0.744895	+	0.667181i	$0.767501\pi$
$114$		0			0
$115$	772.139			0.626107
$116$	−132.848	+	230.099i	−0.106333	+	0.184173i
$117$		0			0
$118$	185.988	+	322.141i	0.145098	+	0.251317i
$119$	−203.360	+	352.229i	−0.156655	+	0.271335i
$120$		0			0
$121$	2535.11			1.90466
$122$	617.286			0.458086
$123$		0			0
$124$	−224.741	−	389.263i	−0.162761	−	0.281910i
$125$	3704.31			2.65059
$126$		0			0
$127$	−138.888	−	240.560i	−0.0970416	−	0.168081i	0.813417	−	0.581681i	$-0.197605\pi$
$127$	−138.888	−	240.560i	−0.0970416	−	0.168081i	−0.910459	+	0.413600i	$0.864271\pi$
$128$	−64.0000	+	110.851i	−0.0441942	+	0.0765466i
$129$		0			0
$130$	1336.20	−	2314.37i	0.901484	−	1.56142i
$131$	−911.342	+	1578.49i	−0.607819	+	1.05277i	0.383780	+	0.923424i	$0.374622\pi$
$131$	−911.342	+	1578.49i	−0.607819	+	1.05277i	−0.991599	+	0.129349i	$0.958711\pi$
$132$		0			0
$133$	152.292	+	709.550i	0.0992886	+	0.462600i
$134$	78.9115			0.0508725
$135$		0			0
$136$	−185.662	+	321.576i	−0.117062	+	0.202757i
$137$	330.481	+	572.411i	0.206094	+	0.356966i	0.950481	−	0.310783i	$-0.100591\pi$
$137$	330.481	+	572.411i	0.206094	+	0.356966i	−0.744386	+	0.667749i	$0.767258\pi$
$138$		0			0
$139$	622.975	+	1079.02i	0.380144	+	0.658429i	0.991083	−	0.133249i	$-0.0425410\pi$
$139$	622.975	+	1079.02i	0.380144	+	0.658429i	−0.610938	+	0.791678i	$0.709208\pi$
$140$	725.866			0.438192
$141$		0			0
$142$	−350.202	−	606.568i	−0.206960	−	0.358465i
$143$	2005.93	+	3474.37i	1.17304	+	2.03176i
$144$		0			0
$145$	1375.59			0.787835
$146$	9.60561	+	16.6374i	0.00544497	+	0.00943096i
$147$		0			0
$148$	378.036	+	654.777i	0.209962	+	0.363665i
$149$	1136.01	−	1967.62i	0.624599	−	1.08184i	−0.364019	−	0.931392i	$-0.618596\pi$
$149$	1136.01	−	1967.62i	0.624599	−	1.08184i	0.988618	−	0.150446i	$-0.0480710\pi$
$150$		0			0
$151$	3637.77			1.96051			0.980256	−	0.197734i	$-0.0633581\pi$
$151$	3637.77			1.96051			0.980256	+	0.197734i	$0.0633581\pi$
$152$	139.038	+	647.800i	0.0741941	+	0.345681i
$153$		0			0
$154$	−544.840	+	943.691i	−0.285094	+	0.493798i
$155$	−1163.55	+	2015.33i	−0.602960	+	1.04436i
$156$		0			0
$157$	225.885	−	391.244i	0.114825	−	0.198883i	−0.802885	−	0.596135i	$-0.796702\pi$
$157$	225.885	−	391.244i	0.114825	−	0.198883i	0.917710	+	0.397251i	$0.130036\pi$
$158$	1176.54	+	2037.82i	0.592408	+	1.02608i
$159$		0			0
$160$	662.695			0.327442
$161$	−163.355	−	282.940i	−0.0799641	−	0.138502i
$162$		0			0
$163$	215.631			0.103617			0.0518084	−	0.998657i	$-0.483502\pi$
$163$	215.631			0.103617			0.0518084	+	0.998657i	$0.483502\pi$
$164$	960.039			0.457112
$165$		0			0
$166$	−257.980	+	446.834i	−0.120621	+	0.208922i
$167$	−1660.74	−	2876.49i	−0.769533	−	1.33287i	−0.937816	−	0.347132i	$-0.887156\pi$
$167$	−1660.74	−	2876.49i	−0.769533	−	1.33287i	0.168283	−	0.985739i	$-0.446178\pi$
$168$		0			0
$169$	−983.055	+	1702.70i	−0.447453	+	0.775012i
$170$	1922.46			0.867328
$171$		0			0
$172$	673.096			0.298390
$173$	736.228	−	1275.19i	0.323552	−	0.560408i	−0.657667	−	0.753309i	$-0.728456\pi$
$173$	736.228	−	1275.19i	0.323552	−	0.560408i	0.981218	+	0.192901i	$0.0617898\pi$
$174$		0			0
$175$	−1331.35	−	2305.97i	−0.575091	−	0.996086i
$176$	−497.424	+	861.564i	−0.213038	+	0.368993i
$177$		0			0
$178$	267.497			0.112639
$179$	−3573.98			−1.49236			−0.746179	−	0.665745i	$-0.768114\pi$
$179$	−3573.98			−1.49236			−0.746179	+	0.665745i	$0.768114\pi$
$180$		0			0
$181$	1471.86	+	2549.34i	0.604436	+	1.04691i	0.992140	+	0.125130i	$0.0399347\pi$
$181$	1471.86	+	2549.34i	0.604436	+	1.04691i	−0.387705	+	0.921784i	$0.626732\pi$
$182$	−1130.76			−0.460537
$183$		0			0
$184$	−149.139	−	258.316i	−0.0597537	−	0.103496i
$185$	1957.21	−	3389.98i	0.777821	−	1.34722i
$186$		0			0
$187$	−1443.01	+	2499.37i	−0.564296	+	0.977390i
$188$	375.696	−	650.725i	0.145747	−	0.252442i
$189$		0			0
$190$	2544.60	−	2300.33i	0.971605	−	0.878335i
$191$	700.359			0.265321			0.132660	−	0.991162i	$-0.457648\pi$
$191$	700.359			0.265321			0.132660	+	0.991162i	$0.457648\pi$
$192$		0			0
$193$	29.9508	−	51.8763i	0.0111705	−	0.0193478i	−0.860386	−	0.509643i	$-0.829778\pi$
$193$	29.9508	−	51.8763i	0.0111705	−	0.0193478i	0.871557	+	0.490295i	$0.163111\pi$
$194$	−1196.77	−	2072.87i	−0.442904	−	0.767132i
$195$		0			0
$196$	532.434	+	922.203i	0.194036	+	0.336080i
$197$	374.382			0.135399			0.0676994	−	0.997706i	$-0.478434\pi$
$197$	374.382			0.135399			0.0676994	+	0.997706i	$0.478434\pi$
$198$		0			0
$199$	−2301.88	−	3986.98i	−0.819981	−	1.42025i	−0.905696	−	0.423929i	$-0.860651\pi$
$199$	−2301.88	−	3986.98i	−0.819981	−	1.42025i	0.0857147	−	0.996320i	$-0.472683\pi$
$200$	−1215.49	−	2105.29i	−0.429740	−	0.744332i
$201$		0			0
$202$	−1737.89			−0.605333
$203$	−291.022	−	504.065i	−0.100619	−	0.174278i
$204$		0			0
$205$	−2485.21	−	4304.51i	−0.846705	−	1.46654i
$206$	1491.56	−	2583.46i	0.504476	−	0.873778i
$207$		0			0
$208$	−1032.35			−0.344139
$209$	1080.64	+	5034.86i	0.357653	+	1.66636i
$210$		0			0
$211$	1478.36	−	2560.60i	0.482344	−	0.835444i	−0.517451	−	0.855713i	$-0.673119\pi$
$211$	1478.36	−	2560.60i	0.482344	−	0.835444i	0.999795	+	0.0202687i	$0.00645218\pi$
$212$	−226.114	+	391.641i	−0.0732527	+	0.126877i
$213$		0			0
$214$	−1610.18	+	2788.91i	−0.514344	+	0.890870i
$215$	−1742.41	−	3017.95i	−0.552705	−	0.957313i
$216$		0			0
$217$	984.655			0.308031
$218$	51.0222	+	88.3730i	0.0158516	+	0.0274558i
$219$		0			0
$220$	5150.64			1.57844
$221$	−2994.83			−0.911555
$222$		0			0
$223$	−265.320	+	459.548i	−0.0796733	+	0.137998i	−0.903109	−	0.429411i	$-0.858721\pi$
$223$	−265.320	+	459.548i	−0.0796733	+	0.137998i	0.823436	+	0.567410i	$0.192054\pi$
$224$	−140.201	−	242.836i	−0.0418196	−	0.0724337i
$225$		0			0
$226$	1789.55	−	3099.58i	0.526720	−	0.912307i
$227$	3479.72			1.01743			0.508716	−	0.860934i	$-0.330120\pi$
$227$	3479.72			1.01743			0.508716	+	0.860934i	$0.330120\pi$
$228$		0			0
$229$	3566.48			1.02917			0.514584	−	0.857440i	$-0.327946\pi$
$229$	3566.48			1.02917			0.514584	+	0.857440i	$0.327946\pi$
$230$	−772.139	+	1337.38i	−0.221362	+	0.383411i
$231$		0			0
$232$	−265.695	−	460.197i	−0.0751885	−	0.130230i
$233$	−2995.60	+	5188.53i	−0.842267	+	1.45885i	0.0457068	+	0.998955i	$0.485446\pi$
$233$	−2995.60	+	5188.53i	−0.842267	+	1.45885i	−0.887974	+	0.459894i	$0.847887\pi$
$234$		0			0
$235$	−3890.19			−1.07986
$236$	−743.952			−0.205200
$237$		0			0
$238$	−406.719	−	704.459i	−0.110772	−	0.191863i
$239$	606.478			0.164141			0.0820707	−	0.996627i	$-0.473847\pi$
$239$	606.478			0.164141			0.0820707	+	0.996627i	$0.473847\pi$
$240$		0			0
$241$	1722.58	+	2983.60i	0.460421	+	0.797472i	0.998982	−	0.0451143i	$-0.0143652\pi$
$241$	1722.58	+	2983.60i	0.460421	+	0.797472i	−0.538561	+	0.842586i	$0.681032\pi$
$242$	−2535.11	+	4390.94i	−0.673401	+	1.16636i
$243$		0			0
$244$	−617.286	+	1069.17i	−0.161958	+	0.280519i
$245$	2756.58	−	4774.53i	0.718821	−	1.24503i
$246$		0			0
$247$	−3964.01	+	3583.48i	−1.02115	+	0.923124i
$248$	898.963			0.230178
$249$		0			0
$250$	−3704.31	+	6416.05i	−0.937124	+	1.62315i
$251$	902.687	+	1563.50i	0.227000	+	0.393176i	0.956918	−	0.290359i	$-0.0937748\pi$
$251$	902.687	+	1563.50i	0.227000	+	0.393176i	−0.729917	+	0.683535i	$0.760441\pi$
$252$		0			0
$253$	−1159.15	−	2007.70i	−0.288043	−	0.498905i
$254$	555.551			0.137238
$255$		0			0
$256$	−128.000	−	221.703i	−0.0312500	−	0.0541266i
$257$	1118.48	+	1937.27i	0.271475	+	0.470208i	0.969240	−	0.246119i	$-0.0791553\pi$
$257$	1118.48	+	1937.27i	0.271475	+	0.470208i	−0.697765	+	0.716327i	$0.745822\pi$
$258$		0			0
$259$	−1656.29			−0.397362
$260$	2672.41	+	4628.75i	0.637445	+	1.10409i
$261$		0			0
$262$	−1822.68	−	3156.98i	−0.429793	−	0.744423i
$263$	416.129	−	720.757i	0.0975651	−	0.168988i	−0.813111	−	0.582108i	$-0.802228\pi$
$263$	416.129	−	720.757i	0.0975651	−	0.168988i	0.910676	+	0.413121i	$0.135561\pi$
$264$		0			0
$265$	2341.32			0.542741
$266$	−1381.27	−	445.773i	−0.318387	−	0.102752i
$267$		0			0
$268$	−78.9115	+	136.679i	−0.0179861	+	0.0311529i
$269$	−1741.94	+	3017.13i	−0.394826	+	0.683858i	−0.993079	−	0.117449i	$-0.962528\pi$
$269$	−1741.94	+	3017.13i	−0.394826	+	0.683858i	0.598253	+	0.801307i	$0.295862\pi$
$270$		0			0
$271$	−894.554	+	1549.41i	−0.200518	+	0.347307i	−0.948695	−	0.316192i	$-0.897596\pi$
$271$	−894.554	+	1549.41i	−0.200518	+	0.347307i	0.748178	+	0.663498i	$0.230929\pi$
$272$	−371.324	−	643.151i	−0.0827750	−	0.143371i
$273$		0			0
$274$	−1321.93			−0.291462
$275$	−9447.09	−	16362.8i	−2.07157	−	3.58806i
$276$		0			0
$277$	−5548.66			−1.20356			−0.601781	−	0.798661i	$-0.705542\pi$
$277$	−5548.66			−1.20356			−0.601781	+	0.798661i	$0.705542\pi$
$278$	−2491.90			−0.537605
$279$		0			0
$280$	−725.866	+	1257.24i	−0.154924	+	0.268337i
$281$	−22.8948	−	39.6549i	−0.00486046	−	0.00841856i	0.863585	−	0.504203i	$-0.168214\pi$
$281$	−22.8948	−	39.6549i	−0.00486046	−	0.00841856i	−0.868445	+	0.495785i	$0.834880\pi$
$282$		0			0
$283$	3060.37	−	5300.72i	0.642827	−	1.11341i	−0.341971	−	0.939710i	$-0.611095\pi$
$283$	3060.37	−	5300.72i	0.642827	−	1.11341i	0.984799	−	0.173699i	$-0.0555721\pi$
$284$	1400.81			0.292686
$285$		0			0
$286$	−8023.72			−1.65893
$287$	−1051.55	+	1821.34i	−0.216276	+	0.374601i
$288$		0			0
$289$	1379.30	+	2389.02i	0.280746	+	0.486266i
$290$	−1375.59	+	2382.58i	−0.278542	+	0.482449i
$291$		0			0
$292$	−38.4224			−0.00770035
$293$	7395.39			1.47455			0.737276	−	0.675592i	$-0.236112\pi$
$293$	7395.39			1.47455			0.737276	+	0.675592i	$0.236112\pi$
$294$		0			0
$295$	1925.83	+	3335.64i	0.380090	+	0.658334i
$296$	−1512.14			−0.296931
$297$		0			0
$298$	2272.01	+	3935.24i	0.441658	+	0.764975i
$299$	1202.85	−	2083.39i	0.232650	−	0.402962i
$300$		0			0
$301$	−737.258	+	1276.97i	−0.141179	+	0.244529i
$302$	−3637.77	+	6300.80i	−0.693146	+	1.20056i
$303$		0			0
$304$	−1261.06	−	406.978i	−0.237917	−	0.0767822i
$305$	6391.76			1.19997
$306$		0			0
$307$	287.534	−	498.024i	0.0534542	−	0.0925854i	−0.838060	−	0.545578i	$-0.816310\pi$
$307$	287.534	−	498.024i	0.0534542	−	0.0925854i	0.891514	+	0.452992i	$0.149644\pi$
$308$	−1089.68	−	1887.38i	−0.201592	−	0.349168i
$309$		0			0
$310$	−2327.10	−	4030.66i	−0.426357	−	0.738472i
$311$	36.6434			0.00668120			0.00334060	−	0.999994i	$-0.498937\pi$
$311$	36.6434			0.00668120			0.00334060	+	0.999994i	$0.498937\pi$
$312$		0			0
$313$	2441.86	+	4229.43i	0.440966	+	0.763775i	0.997761	−	0.0668742i	$-0.0213026\pi$
$313$	2441.86	+	4229.43i	0.440966	+	0.763775i	−0.556795	+	0.830650i	$0.687969\pi$
$314$	451.770	+	782.489i	0.0811938	+	0.140632i
$315$		0			0
$316$	−4706.15			−0.837791
$317$	−2933.65	−	5081.23i	−0.519779	−	0.900284i	−0.999736	−	0.0229917i	$-0.992681\pi$
$317$	−2933.65	−	5081.23i	−0.519779	−	0.900284i	0.479956	−	0.877292i	$-0.340652\pi$
$318$		0			0
$319$	−2065.05	−	3576.77i	−0.362447	−	0.627777i
$320$	−662.695	+	1147.82i	−0.115768	+	0.200516i
$321$		0			0
$322$	653.422			0.113086
$323$	−3658.29	−	1180.63i	−0.630195	−	0.203381i
$324$		0			0
$325$	9803.25	−	16979.7i	1.67319	−	2.89805i
$326$	−215.631	+	373.484i	−0.0366340	+	0.0634520i
$327$		0			0
$328$	−960.039	+	1662.84i	−0.161614	+	0.279923i
$329$	823.018	+	1425.51i	0.137916	+	0.238878i
$330$		0			0
$331$	−8866.06			−1.47227			−0.736137	−	0.676833i	$-0.763352\pi$
$331$	−8866.06			−1.47227			−0.736137	+	0.676833i	$0.763352\pi$
$332$	−515.960	−	893.668i	−0.0852921	−	0.147730i
$333$		0			0
$334$	6642.96			1.08828
$335$	817.098			0.133262
$336$		0			0
$337$	−1513.60	+	2621.63i	−0.244662	+	0.423767i	−0.962037	−	0.272921i	$-0.912010\pi$
$337$	−1513.60	+	2621.63i	−0.244662	+	0.423767i	0.717375	+	0.696688i	$0.245344\pi$
$338$	−1966.11	−	3405.40i	−0.316397	−	0.548016i
$339$		0			0
$340$	−1922.46	+	3329.79i	−0.306647	+	0.531128i
$341$	6986.97			1.10958
$342$		0			0
$343$	−5338.32			−0.840356
$344$	−673.096	+	1165.84i	−0.105497	+	0.182726i
$345$		0			0
$346$	1472.46	+	2550.37i	0.228785	+	0.396268i
$347$	−4437.88	+	7686.63i	−0.686564	+	1.18916i	0.286378	+	0.958117i	$0.407549\pi$
$347$	−4437.88	+	7686.63i	−0.686564	+	1.18916i	−0.972942	+	0.231048i	$0.925785\pi$
$348$		0			0
$349$	2808.70			0.430791			0.215395	−	0.976527i	$-0.430896\pi$
$349$	2808.70			0.430791			0.215395	+	0.976527i	$0.430896\pi$
$350$	5325.41			0.813301
$351$		0			0
$352$	−994.849	−	1723.13i	−0.150641	−	0.260918i
$353$	−11702.4			−1.76446			−0.882230	−	0.470819i	$-0.843959\pi$
$353$	−11702.4			−1.76446			−0.882230	+	0.470819i	$0.843959\pi$
$354$		0			0
$355$	−3626.21	−	6280.78i	−0.542139	−	0.939012i
$356$	−267.497	+	463.319i	−0.0398239	+	0.0689771i
$357$		0			0
$358$	3573.98	−	6190.32i	0.527628	−	0.913879i
$359$	−843.014	+	1460.14i	−0.123935	+	0.214661i	−0.921316	−	0.388815i	$-0.872885\pi$
$359$	−843.014	+	1460.14i	−0.123935	+	0.214661i	0.797381	+	0.603476i	$0.206218\pi$
$360$		0			0
$361$	−6254.89	+	2814.66i	−0.911924	+	0.410359i
$362$	−5887.46			−0.854801
$363$		0			0
$364$	1130.76	−	1958.54i	0.162824	−	0.282020i
$365$	99.4624	+	172.274i	0.0142633	+	0.0247047i
$366$		0			0
$367$	4218.01	+	7305.81i	0.599941	+	1.03913i	0.992829	+	0.119542i	$0.0381428\pi$
$367$	4218.01	+	7305.81i	0.599941	+	1.03913i	−0.392888	+	0.919586i	$0.628524\pi$
$368$	596.556			0.0845044
$369$		0			0
$370$	3914.42	+	6779.97i	0.550002	+	0.952632i
$371$	−495.336	−	857.947i	−0.0693169	−	0.120060i
$372$		0			0
$373$	−1743.67			−0.242048			−0.121024	−	0.992650i	$-0.538618\pi$
$373$	−1743.67			−0.242048			−0.121024	+	0.992650i	$0.538618\pi$
$374$	−2886.02	−	4998.74i	−0.399018	−	0.691119i
$375$		0			0
$376$	751.393	+	1301.45i	0.103059	+	0.178503i
$377$	2142.90	−	3711.62i	0.292746	−	0.507050i
$378$		0			0
$379$	−1591.45			−0.215693			−0.107846	−	0.994168i	$-0.534395\pi$
$379$	−1591.45			−0.215693			−0.107846	+	0.994168i	$0.534395\pi$
$380$	1439.69	+	6707.72i	0.194354	+	0.905523i
$381$		0			0
$382$	−700.359	+	1213.06i	−0.0938050	+	0.162475i
$383$	4502.45	−	7798.48i	0.600691	−	1.04043i	−0.392026	−	0.919954i	$-0.628226\pi$
$383$	4502.45	−	7798.48i	0.600691	−	1.04043i	0.992717	−	0.120473i	$-0.0384410\pi$
$384$		0			0
$385$	−5641.61	+	9771.56i	−0.746814	+	1.29352i
$386$	59.9015	+	103.753i	0.00789873	+	0.0136810i
$387$		0			0
$388$	4787.10			0.626361
$389$	2034.97	+	3524.66i	0.265236	+	0.459402i	0.967625	−	0.252391i	$-0.0812168\pi$
$389$	2034.97	+	3524.66i	0.265236	+	0.459402i	−0.702389	+	0.711793i	$0.747883\pi$
$390$		0			0
$391$	1730.59			0.223835
$392$	−2129.74			−0.274408
$393$		0			0
$394$	−374.382	+	648.448i	−0.0478707	+	0.0829145i
$395$	12182.6	+	21100.9i	1.55183	+	2.68785i
$396$		0			0
$397$	−164.785	+	285.416i	−0.0208321	+	0.0360822i	−0.876253	−	0.481850i	$-0.839965\pi$
$397$	−164.785	+	285.416i	−0.0208321	+	0.0360822i	0.855421	+	0.517933i	$0.173298\pi$
$398$	9207.53			1.15963
$399$		0			0
$400$	4861.96			0.607744
$401$	4093.53	−	7090.20i	0.509778	−	0.882962i	−0.490158	−	0.871634i	$-0.663061\pi$
$401$	4093.53	−	7090.20i	0.509778	−	0.882962i	0.999936	−	0.0113281i	$-0.00360591\pi$
$402$		0			0
$403$	3625.19	+	6279.02i	0.448098	+	0.776129i
$404$	1737.89	−	3010.11i	0.214017	−	0.370689i
$405$		0			0
$406$	1164.09			0.142297
$407$	−11752.8			−1.43136
$408$		0			0
$409$	−2499.25	−	4328.83i	−0.302152	−	0.523342i	0.674472	−	0.738301i	$-0.264372\pi$
$409$	−2499.25	−	4328.83i	−0.302152	−	0.523342i	−0.976623	+	0.214959i	$0.931038\pi$
$410$	9940.83			1.19742
$411$		0			0
$412$	2983.12	+	5166.92i	0.356718	+	0.617854i
$413$	814.868	−	1411.39i	0.0970873	−	0.168160i
$414$		0			0
$415$	−2671.28	+	4626.80i	−0.315971	+	0.547278i
$416$	1032.35	−	1788.09i	0.121672	−	0.210741i
$417$		0			0
$418$	−9801.28	−	3163.14i	−1.14688	−	0.370130i
$419$	−11941.1			−1.39227			−0.696134	−	0.717912i	$-0.745098\pi$
$419$	−11941.1			−1.39227			−0.696134	+	0.717912i	$0.745098\pi$
$420$		0			0
$421$	5991.40	−	10377.4i	0.693594	−	1.20134i	−0.277059	−	0.960853i	$-0.589360\pi$
$421$	5991.40	−	10377.4i	0.693594	−	1.20134i	0.970653	−	0.240486i	$-0.0773069\pi$
$422$	2956.72	+	5121.19i	0.341069	+	0.590748i
$423$		0			0
$424$	−452.228	−	783.282i	−0.0517975	−	0.0897159i
$425$	14104.4			1.60979
$426$		0			0
$427$	−1352.26	−	2342.18i	−0.153256	−	0.265447i
$428$	−3220.36	−	5577.83i	−0.363696	−	0.629940i
$429$		0			0
$430$	6969.65			0.781643
$431$	3541.49	+	6134.04i	0.395795	+	0.685537i	0.993202	−	0.116401i	$-0.0371358\pi$
$431$	3541.49	+	6134.04i	0.395795	+	0.685537i	−0.597407	+	0.801938i	$0.703802\pi$
$432$		0			0
$433$	3701.50	+	6411.19i	0.410815	+	0.711552i	0.994979	−	0.100084i	$-0.0319110\pi$
$433$	3701.50	+	6411.19i	0.410815	+	0.711552i	−0.584164	+	0.811635i	$0.698578\pi$
$434$	−984.655	+	1705.47i	−0.108905	+	0.188630i
$435$		0			0
$436$	−204.089			−0.0224176
$437$	2290.64	−	2070.75i	0.250747	−	0.226676i
$438$		0			0
$439$	3837.72	−	6647.12i	0.417231	−	0.722665i	−0.578429	−	0.815733i	$-0.696334\pi$
$439$	3837.72	−	6647.12i	0.417231	−	0.722665i	0.995660	+	0.0930680i	$0.0296674\pi$
$440$	−5150.64	+	8921.17i	−0.558061	+	0.966591i
$441$		0			0
$442$	2994.83	−	5187.19i	0.322284	−	0.558211i
$443$	−1341.94	−	2324.32i	−0.143923	−	0.249281i	0.785048	−	0.619435i	$-0.212638\pi$
$443$	−1341.94	−	2324.32i	−0.143923	−	0.249281i	−0.928970	+	0.370154i	$0.879305\pi$
$444$		0			0
$445$	2769.83			0.295062
$446$	−530.640	−	919.095i	−0.0563375	−	0.0975794i
$447$		0			0
$448$	560.806			0.0591419
$449$	14097.6			1.48175			0.740874	−	0.671644i	$-0.234411\pi$
$449$	14097.6			1.48175			0.740874	+	0.671644i	$0.234411\pi$
$450$		0			0
$451$	−7461.67	+	12924.0i	−0.779060	+	1.34937i
$452$	3579.09	+	6199.17i	0.372448	+	0.645098i
$453$		0			0
$454$	−3479.72	+	6027.05i	−0.359717	+	0.623048i
$455$	−11708.6			−1.20639
$456$		0			0
$457$	7040.20			0.720627			0.360314	−	0.932831i	$-0.382670\pi$
$457$	7040.20			0.720627			0.360314	+	0.932831i	$0.382670\pi$
$458$	−3566.48	+	6177.32i	−0.363866	+	0.630234i
$459$		0			0
$460$	−1544.28	−	2674.77i	−0.156527	−	0.271112i
$461$	3676.54	−	6367.96i	0.371440	−	0.643352i	−0.618348	−	0.785905i	$-0.712198\pi$
$461$	3676.54	−	6367.96i	0.371440	−	0.643352i	0.989787	+	0.142552i	$0.0455310\pi$
$462$		0			0
$463$	14054.7			1.41075			0.705377	−	0.708832i	$-0.250778\pi$
$463$	14054.7			1.41075			0.705377	+	0.708832i	$0.250778\pi$
$464$	1062.78			0.106333
$465$		0			0
$466$	−5991.20	−	10377.1i	−0.595573	−	1.03156i
$467$	−5875.16			−0.582163			−0.291081	−	0.956698i	$-0.594015\pi$
$467$	−5875.16			−0.582163			−0.291081	+	0.956698i	$0.594015\pi$
$468$		0			0
$469$	−172.867	−	299.415i	−0.0170198	−	0.0294791i
$470$	3890.19	−	6738.01i	0.381790	−	0.661279i
$471$		0			0
$472$	743.952	−	1288.56i	0.0725491	−	0.125659i
$473$	−5231.48	+	9061.18i	−0.508549	+	0.880832i
$474$		0			0
$475$	18668.8	−	16876.7i	1.80334	−	1.63022i
$476$	1626.88			0.156655
$477$		0			0
$478$	−606.478	+	1050.45i	−0.0580328	+	0.100516i
$479$	−88.8904	−	153.963i	−0.00847914	−	0.0146863i	0.861755	−	0.507325i	$-0.169366\pi$
$479$	−88.8904	−	153.963i	−0.00847914	−	0.0146863i	−0.870234	+	0.492639i	$0.836032\pi$
$480$		0			0
$481$	−6097.92	−	10561.9i	−0.578049	−	1.00121i
$482$	−6890.34			−0.651133
$483$		0			0
$484$	−5070.22	−	8781.87i	−0.476166	−	0.824744i
$485$	−12392.1	−	21463.8i	−1.16020	−	2.00953i
$486$		0			0
$487$	12319.9			1.14634			0.573172	−	0.819435i	$-0.305713\pi$
$487$	12319.9			1.14634			0.573172	+	0.819435i	$0.305713\pi$
$488$	−1234.57	−	2138.34i	−0.114521	−	0.198357i
$489$		0			0
$490$	5513.15	+	9549.06i	0.508283	+	0.880372i
$491$	−4220.92	+	7310.84i	−0.387958	+	0.671963i	−0.992175	−	0.124857i	$-0.960153\pi$
$491$	−4220.92	+	7310.84i	−0.387958	+	0.671963i	0.604217	+	0.796820i	$0.293486\pi$
$492$		0			0
$493$	3083.09			0.281654
$494$	−2242.76	−	10449.4i	−0.204265	−	0.951698i
$495$		0			0
$496$	−898.963	+	1557.05i	−0.0813803	+	0.140955i
$497$	−1534.34	+	2657.55i	−0.138480	+	0.239854i
$498$		0			0
$499$	3105.62	−	5379.09i	0.278610	−	0.482567i	−0.692429	−	0.721486i	$-0.743460\pi$
$499$	3105.62	−	5379.09i	0.278610	−	0.482567i	0.971040	+	0.238918i	$0.0767929\pi$
$500$	−7408.61	−	12832.1i	−0.662646	−	1.14774i
$501$		0			0
$502$	−3610.75			−0.321027
$503$	7836.70	+	13573.6i	0.694675	+	1.20321i	0.970290	+	0.241944i	$0.0777850\pi$
$503$	7836.70	+	13573.6i	0.694675	+	1.20321i	−0.275616	+	0.961268i	$0.588882\pi$
$504$		0			0
$505$	−17995.1			−1.58569
$506$	4636.59			0.407354
$507$		0			0
$508$	−555.551	+	962.242i	−0.0485208	+	0.0840405i
$509$	−9897.72	−	17143.3i	−0.861903	−	1.49286i	−0.870090	−	0.492893i	$-0.835939\pi$
$509$	−9897.72	−	17143.3i	−0.861903	−	1.49286i	0.00818701	−	0.999966i	$-0.497394\pi$
$510$		0			0
$511$	42.0850	−	72.8933i	0.00364331	−	0.00631039i
$512$	512.000			0.0441942
$513$		0			0
$514$	−4473.93			−0.383923
$515$	15444.5	−	26750.7i	1.32149	−	2.28889i
$516$		0			0
$517$	5840.02	+	10115.2i	0.496796	+	0.860476i
$518$	1656.29	−	2868.77i	0.140489	−	0.243333i
$519$		0			0
$520$	−10689.6			−0.901484
$521$	21391.1			1.79877			0.899385	−	0.437157i	$-0.144014\pi$
$521$	21391.1			1.79877			0.899385	+	0.437157i	$0.144014\pi$
$522$		0			0
$523$	−1601.07	−	2773.13i	−0.133862	−	0.231855i	0.791300	−	0.611428i	$-0.209404\pi$
$523$	−1601.07	−	2773.13i	−0.133862	−	0.231855i	−0.925162	+	0.379572i	$0.876071\pi$
$524$	7290.73			0.607819
$525$		0			0
$526$	832.258	+	1441.51i	0.0689890	+	0.119492i
$527$	−2607.86	+	4516.95i	−0.215560	+	0.373361i
$528$		0			0
$529$	5388.42	−	9333.02i	0.442872	−	0.767077i
$530$	−2341.32	+	4055.29i	−0.191888	+	0.332360i
$531$		0			0
$532$	2153.37	−	1946.65i	0.175490	−	0.158643i
$533$	−15485.9			−1.25848
$534$		0			0
$535$	−16672.8	+	28878.1i	−1.34734	+	2.33366i
$536$	−157.823	−	273.357i	−0.0127181	−	0.0220284i
$537$		0			0
$538$	−3483.89	−	6034.27i	−0.279184	−	0.483561i
$539$	−16552.9			−1.32279
$540$		0			0
$541$	3681.50	+	6376.55i	0.292569	+	0.506745i	0.974417	−	0.224750i	$-0.0721564\pi$
$541$	3681.50	+	6376.55i	0.292569	+	0.506745i	−0.681847	+	0.731495i	$0.738823\pi$
$542$	−1789.11	−	3098.82i	−0.141787	−	0.245583i
$543$		0			0
$544$	1485.29			0.117062
$545$	528.315	+	915.068i	0.0415239	+	0.0719215i
$546$		0			0
$547$	6484.65	+	11231.7i	0.506880	+	0.877942i	0.999968	+	0.00796294i	$0.00253471\pi$
$547$	6484.65	+	11231.7i	0.506880	+	0.877942i	−0.493088	+	0.869979i	$0.664132\pi$
$548$	1321.93	−	2289.64i	0.103047	−	0.178483i
$549$		0			0
$550$	37788.4			2.92964
$551$	4080.84	−	3689.10i	0.315517	−	0.285228i
$552$		0			0
$553$	5154.76	−	8928.31i	0.396388	−	0.686565i
$554$	5548.66	−	9610.56i	0.425523	−	0.737028i
$555$		0			0
$556$	2491.90	−	4316.10i	0.190072	−	0.329215i
$557$	9997.27	+	17315.8i	0.760499	+	1.31722i	0.942594	+	0.333942i	$0.108379\pi$
$557$	9997.27	+	17315.8i	0.760499	+	1.31722i	−0.182095	+	0.983281i	$0.558288\pi$
$558$		0			0
$559$	−10857.4			−0.821502
$560$	−1451.73	−	2514.47i	−0.109548	−	0.189743i
$561$		0			0
$562$	91.5792			0.00687372
$563$	15654.6			1.17187			0.585936	−	0.810358i	$-0.300727\pi$
$563$	15654.6			1.17187			0.585936	+	0.810358i	$0.300727\pi$
$564$		0			0
$565$	18530.1	−	32095.0i	1.37976	−	2.38982i
$566$	6120.74	+	10601.4i	0.454548	+	0.787300i
$567$		0			0
$568$	−1400.81	+	2426.27i	−0.103480	+	0.179233i
$569$	6450.45			0.475249			0.237625	−	0.971357i	$-0.423631\pi$
$569$	6450.45			0.475249			0.237625	+	0.971357i	$0.423631\pi$
$570$		0			0
$571$	18704.6			1.37086			0.685431	−	0.728138i	$-0.259614\pi$
$571$	18704.6			1.37086			0.685431	+	0.728138i	$0.259614\pi$
$572$	8023.72	−	13897.5i	0.586519	−	1.01588i
$573$		0			0
$574$	−2103.11	−	3642.69i	−0.152930	−	0.264883i
$575$	−5664.90	+	9811.90i	−0.410857	+	0.711625i
$576$		0			0
$577$	−12394.4			−0.894257			−0.447128	−	0.894470i	$-0.647553\pi$
$577$	−12394.4			−0.894257			−0.447128	+	0.894470i	$0.647553\pi$
$578$	−5517.21			−0.397034
$579$		0			0
$580$	−2751.17	−	4765.17i	−0.196959	−	0.341143i
$581$	2260.57			0.161419
$582$		0			0
$583$	−3514.83	−	6087.87i	−0.249690	−	0.432476i
$584$	38.4224	−	66.5496i	0.00272248	−	0.00471548i
$585$		0			0
$586$	−7395.39	+	12809.2i	−0.521333	+	0.902974i
$587$	−3212.31	+	5563.88i	−0.225871	+	0.391220i	−0.956580	−	0.291469i	$-0.905856\pi$
$587$	−3212.31	+	5563.88i	−0.225871	+	0.391220i	0.730710	+	0.682688i	$0.239189\pi$
$588$		0			0
$589$	1952.97	+	9099.19i	0.136623	+	0.636546i
$590$	−7703.34			−0.537528
$591$		0			0
$592$	1512.14	−	2619.11i	0.104981	−	0.181832i
$593$	9050.05	+	15675.2i	0.626714	+	1.08550i	0.988207	+	0.153125i	$0.0489337\pi$
$593$	9050.05	+	15675.2i	0.626714	+	1.08550i	−0.361493	+	0.932375i	$0.617733\pi$
$594$		0			0
$595$	−4211.42	−	7294.40i	−0.290171	−	0.502590i
$596$	−9088.05			−0.624599
$597$		0			0
$598$	2405.69	+	4166.78i	0.164509	+	0.284937i
$599$	−1423.60	−	2465.74i	−0.0971061	−	0.168193i	0.813380	−	0.581733i	$-0.197625\pi$
$599$	−1423.60	−	2465.74i	−0.0971061	−	0.168193i	−0.910486	+	0.413541i	$0.864292\pi$
$600$		0			0
$601$	6934.19			0.470635			0.235317	−	0.971919i	$-0.424387\pi$
$601$	6934.19			0.470635			0.235317	+	0.971919i	$0.424387\pi$
$602$	−1474.52	−	2553.94i	−0.0998286	−	0.172908i
$603$		0			0
$604$	−7275.53	−	12601.6i	−0.490128	−	0.848926i
$605$	−26250.1	+	45466.5i	−1.76400	+	3.05533i
$606$		0			0
$607$	9089.78			0.607813			0.303907	−	0.952702i	$-0.401709\pi$
$607$	9089.78			0.607813			0.303907	+	0.952702i	$0.401709\pi$
$608$	1965.97	−	1777.24i	0.131136	−	0.118547i
$609$		0			0
$610$	−6391.76	+	11070.9i	−0.424254	+	0.734829i
$611$	−6060.19	+	10496.6i	−0.401259	+	0.695000i
$612$		0			0
$613$	−12818.6	+	22202.5i	−0.844598	+	1.46289i	0.0413712	+	0.999144i	$0.486827\pi$
$613$	−12818.6	+	22202.5i	−0.844598	+	1.46289i	−0.885970	+	0.463743i	$0.846506\pi$
$614$	575.069	+	996.048i	0.0377979	+	0.0654678i
$615$		0			0
$616$	4358.72			0.285094
$617$	10891.1	+	18863.9i	0.710630	+	1.23085i	0.964621	+	0.263641i	$0.0849234\pi$
$617$	10891.1	+	18863.9i	0.710630	+	1.23085i	−0.253991	+	0.967207i	$0.581743\pi$
$618$		0			0
$619$	−23717.7			−1.54006			−0.770029	−	0.638008i	$-0.779758\pi$
$619$	−23717.7			−1.54006			−0.770029	+	0.638008i	$0.779758\pi$
$620$	9308.42			0.602960
$621$		0			0
$622$	−36.6434	+	63.4682i	−0.00236216	+	0.00409138i
$623$	−585.992	−	1014.97i	−0.0376842	−	0.0652710i
$624$		0			0
$625$	−19364.7	+	33540.6i	−1.23934	+	2.14660i
$626$	−9767.46			−0.623620
$627$		0			0
$628$	−1807.08			−0.114825
$629$	4386.68	−	7597.95i	0.278074	−	0.481638i
$630$		0			0
$631$	13101.6	+	22692.6i	0.826571	+	1.43166i	0.900712	+	0.434416i	$0.143045\pi$
$631$	13101.6	+	22692.6i	0.826571	+	1.43166i	−0.0741409	+	0.997248i	$0.523621\pi$
$632$	4706.15	−	8151.30i	0.296204	−	0.513040i
$633$		0			0
$634$	11734.6			0.735079
$635$	5752.51			0.359498
$636$		0			0
$637$	−8588.45	−	14875.6i	−0.534202	−	0.925265i
$638$	8260.20			0.512577
$639$		0			0
$640$	−1325.39	−	2295.64i	−0.0818604	−	0.141786i
$641$	2092.66	−	3624.59i	0.128947	−	0.223343i	−0.794322	−	0.607497i	$-0.792174\pi$
$641$	2092.66	−	3624.59i	0.128947	−	0.223343i	0.923269	+	0.384154i	$0.125507\pi$
$642$		0			0
$643$	−16099.1	+	27884.5i	−0.987383	+	1.71020i	−0.356556	+	0.934274i	$0.616049\pi$
$643$	−16099.1	+	27884.5i	−0.987383	+	1.71020i	−0.630827	+	0.775924i	$0.717284\pi$
$644$	−653.422	+	1131.76i	−0.0399820	+	0.0692509i
$645$		0			0
$646$	5703.20	−	5155.72i	0.347352	−	0.314008i
$647$	−371.159			−0.0225530			−0.0112765	−	0.999936i	$-0.503589\pi$
$647$	−371.159			−0.0225530			−0.0112765	+	0.999936i	$0.503589\pi$
$648$		0			0
$649$	5782.19	−	10015.0i	0.349724	−	0.605739i
$650$	19606.5	+	33959.4i	1.18312	+	2.04923i
$651$		0			0
$652$	−431.262	−	746.968i	−0.0259042	−	0.0448674i
$653$	−1659.89			−0.0994741			−0.0497370	−	0.998762i	$-0.515838\pi$
$653$	−1659.89			−0.0994741			−0.0497370	+	0.998762i	$0.515838\pi$
$654$		0			0
$655$	−18873.2	−	32689.3i	−1.12586	−	1.95004i
$656$	−1920.08	−	3325.67i	−0.114278	−	0.197935i
$657$		0			0
$658$	−3292.07			−0.195043
$659$	−12365.0	−	21416.8i	−0.730914	−	1.26598i	−0.956493	−	0.291756i	$-0.905760\pi$
$659$	−12365.0	−	21416.8i	−0.730914	−	1.26598i	0.225578	−	0.974225i	$-0.427573\pi$
$660$		0			0
$661$	4199.93	+	7274.48i	0.247138	+	0.428055i	0.962731	−	0.270462i	$-0.0871766\pi$
$661$	4199.93	+	7274.48i	0.247138	+	0.428055i	−0.715593	+	0.698518i	$0.753843\pi$
$662$	8866.06	−	15356.5i	0.520527	−	0.901580i
$663$		0			0
$664$	2063.84			0.120621
$665$	−14302.5	−	4615.81i	−0.834026	−	0.269163i
$666$		0			0
$667$	−1238.30	+	2144.79i	−0.0718846	+	0.124508i
$668$	−6642.96	+	11506.0i	−0.384766	+	0.666435i
$669$		0			0
$670$	−817.098	+	1415.26i	−0.0471153	+	0.0816061i
$671$	−9595.41	−	16619.7i	−0.552052	−	0.956182i
$672$		0			0
$673$	27850.0			1.59516			0.797578	−	0.603216i	$-0.206114\pi$
$673$	27850.0			1.59516			0.797578	+	0.603216i	$0.206114\pi$
$674$	−3027.20	−	5243.27i	−0.173002	−	0.299649i
$675$		0			0
$676$	7864.44			0.447453
$677$	−20343.1			−1.15487			−0.577437	−	0.816435i	$-0.695947\pi$
$677$	−20343.1			−1.15487			−0.577437	+	0.816435i	$0.695947\pi$
$678$		0			0
$679$	−5243.42	+	9081.86i	−0.296353	+	0.513299i
$680$	−3844.91	−	6659.59i	−0.216832	−	0.375564i
$681$		0			0
$682$	−6986.97	+	12101.8i	−0.392295	+	0.679474i
$683$	26794.4			1.50111			0.750556	−	0.660807i	$-0.229786\pi$
$683$	26794.4			1.50111			0.750556	+	0.660807i	$0.229786\pi$
$684$		0			0
$685$	−13688.0			−0.763493
$686$	5338.32	−	9246.24i	0.297111	−	0.514611i
$687$		0			0
$688$	−1346.19	−	2331.67i	−0.0745975	−	0.129207i
$689$	3647.34	−	6317.38i	0.201673	−	0.349308i
$690$		0			0
$691$	29092.3			1.60162			0.800812	−	0.598916i	$-0.204402\pi$
$691$	29092.3			1.60162			0.800812	+	0.598916i	$0.204402\pi$
$692$	−5889.83			−0.323552
$693$		0			0
$694$	−8875.76	−	15373.3i	−0.485474	−	0.840866i
$695$	−25802.7			−1.40827
$696$		0			0
$697$	−5570.08	−	9647.66i	−0.302700	−	0.524291i
$698$	−2808.70	+	4864.80i	−0.152308	+	0.263804i
$699$		0			0
$700$	−5325.41	+	9223.89i	−0.287545	+	0.498043i
$701$	10761.5	−	18639.5i	0.579824	−	1.00428i	−0.415675	−	0.909513i	$-0.636455\pi$
$701$	10761.5	−	18639.5i	0.579824	−	1.00428i	0.995499	−	0.0947713i	$-0.0302120\pi$
$702$		0			0
$703$	−3285.09	−	15305.7i	−0.176244	−	0.821147i
$704$	3979.39			0.213038
$705$		0			0
$706$	11702.4	−	20269.1i	0.623831	−	1.08051i
$707$	3807.09	+	6594.08i	0.202518	+	0.350772i
$708$		0			0
$709$	−7482.91	−	12960.8i	−0.396370	−	0.686533i	0.596905	−	0.802312i	$-0.296397\pi$
$709$	−7482.91	−	12960.8i	−0.396370	−	0.686533i	−0.993275	+	0.115779i	$0.963064\pi$
$710$	14504.8			0.766700
$711$		0			0
$712$	−534.994	−	926.637i	−0.0281598	−	0.0487742i
$713$	−2094.85	−	3628.39i	−0.110032	−	0.190581i
$714$		0			0
$715$	−83082.6			−4.34561
$716$	7147.97	+	12380.6i	0.373090	+	0.646210i
$717$		0			0
$718$	−1686.03	−	2920.29i	−0.0876351	−	0.151788i
$719$	3678.90	−	6372.03i	0.190820	−	0.330510i	−0.754702	−	0.656068i	$-0.772219\pi$
$719$	3678.90	−	6372.03i	0.190820	−	0.330510i	0.945522	+	0.325557i	$0.105552\pi$
$720$		0			0
$721$	−13069.9			−0.675104
$722$	1379.76	−	13648.4i	0.0711210	−	0.703521i
$723$		0			0
$724$	5887.46	−	10197.4i	0.302218	−	0.523457i
$725$	−10092.2	+	17480.1i	−0.516984	+	0.895443i
$726$		0			0
$727$	−14574.2	+	25243.2i	−0.743503	+	1.28779i	0.207387	+	0.978259i	$0.433504\pi$
$727$	−14574.2	+	25243.2i	−0.743503	+	1.28779i	−0.950891	+	0.309527i	$0.899829\pi$
$728$	2261.52	+	3917.08i	0.115134	+	0.199418i
$729$		0			0
$730$	−397.849			−0.0201713
$731$	−3905.26	−	6764.11i	−0.197594	−	0.342243i
$732$		0			0
$733$	21514.5			1.08412			0.542059	−	0.840341i	$-0.317645\pi$
$733$	21514.5			1.08412			0.542059	+	0.840341i	$0.317645\pi$
$734$	−16872.1			−0.848445
$735$		0			0
$736$	−596.556	+	1033.27i	−0.0298768	+	0.0517482i
$737$	−1226.64	−	2124.60i	−0.0613079	−	0.106188i
$738$		0			0
$739$	11558.0	−	20019.0i	0.575326	−	0.996495i	−0.420680	−	0.907209i	$-0.638208\pi$
$739$	11558.0	−	20019.0i	0.575326	−	0.996495i	0.996006	−	0.0892854i	$-0.0284583\pi$
$740$	−15657.7			−0.777821
$741$		0			0
$742$	1981.34			0.0980289
$743$	7551.47	−	13079.5i	0.372862	−	0.645816i	−0.617142	−	0.786852i	$-0.711710\pi$
$743$	7551.47	−	13079.5i	0.372862	−	0.645816i	0.990005	+	0.141035i	$0.0450430\pi$
$744$		0			0
$745$	23525.8	+	40747.9i	1.15694	+	2.00388i
$746$	1743.67	−	3020.13i	0.0855770	−	0.148224i
$747$		0			0
$748$	11544.1			0.564296
$749$	14109.3			0.688310
$750$		0			0
$751$	−14297.9	−	24764.7i	−0.694726	−	1.20330i	−0.970273	−	0.242013i	$-0.922192\pi$
$751$	−14297.9	−	24764.7i	−0.694726	−	1.20330i	0.275547	−	0.961288i	$-0.411141\pi$
$752$	−3005.57			−0.145747
$753$		0			0
$754$	4285.81	+	7423.23i	0.207002	+	0.358539i
$755$	−37667.7	+	65242.3i	−1.81572	+	3.14492i
$756$		0			0
$757$	3822.82	−	6621.32i	0.183544	−	0.317908i	−0.759541	−	0.650460i	$-0.774576\pi$
$757$	3822.82	−	6621.32i	0.183544	−	0.317908i	0.943085	+	0.332552i	$0.107910\pi$
$758$	1591.45	−	2756.48i	0.0762589	−	0.132084i
$759$		0			0
$760$	−13057.8	−	4214.10i	−0.623231	−	0.201134i
$761$	−10724.9			−0.510876			−0.255438	−	0.966825i	$-0.582220\pi$
$761$	−10724.9			−0.510876			−0.255438	+	0.966825i	$0.582220\pi$
$762$		0			0
$763$	223.543	−	387.188i	0.0106066	−	0.0183711i
$764$	−1400.72	−	2426.12i	−0.0663301	−	0.114887i
$765$		0			0
$766$	9004.90	+	15597.0i	0.424752	+	0.735693i
$767$	12000.4			0.564938
$768$		0			0
$769$	1981.77	+	3432.52i	0.0929315	+	0.160962i	0.908743	−	0.417355i	$-0.137043\pi$
$769$	1981.77	+	3432.52i	0.0929315	+	0.160962i	−0.815812	+	0.578317i	$0.803710\pi$
$770$	−11283.2	−	19543.1i	−0.528077	−	0.914656i
$771$		0			0
$772$	−239.606			−0.0111705
$773$	−2222.65	−	3849.74i	−0.103419	−	0.179128i	0.809672	−	0.586883i	$-0.199645\pi$
$773$	−2222.65	−	3849.74i	−0.103419	−	0.179128i	−0.913091	+	0.407755i	$0.866312\pi$
$774$		0			0
$775$	−17073.1	−	29571.5i	−0.791335	−	1.37063i
$776$	−4787.10	+	8291.49i	−0.221452	+	0.383566i
$777$		0			0
$778$	−8139.86			−0.375100
$779$	−18916.7	−	6104.92i	−0.870038	−	0.280785i
$780$		0			0
$781$	−10887.4	+	18857.6i	−0.498826	+	0.863992i
$782$	−1730.59	+	2997.47i	−0.0791378	+	0.137071i
$783$		0			0
$784$	2129.74	−	3688.81i	0.0970179	−	0.168040i
$785$	4677.91	+	8102.37i	0.212690	+	0.368390i
$786$		0			0
$787$	3452.08			0.156357			0.0781787	−	0.996939i	$-0.475090\pi$
$787$	3452.08			0.156357			0.0781787	+	0.996939i	$0.475090\pi$
$788$	−748.763	−	1296.90i	−0.0338497	−	0.0586294i
$789$		0			0
$790$	−48730.4			−2.19462
$791$	−15681.1			−0.704872
$792$		0			0
$793$	9957.16	−	17246.3i	0.445888	−	0.772301i
$794$	−329.570	−	570.832i	−0.0147305	−	0.0255140i
$795$		0			0
$796$	−9207.53	+	15947.9i	−0.409990	+	0.710124i
$797$	25298.0			1.12434			0.562171	−	0.827021i	$-0.309966\pi$
$797$	25298.0			1.12434			0.562171	+	0.827021i	$0.309966\pi$
$798$		0			0
$799$	−8719.06			−0.386055
$800$	−4861.96	+	8421.15i	−0.214870	+	0.372166i
$801$		0			0
$802$	8187.06	+	14180.4i	0.360468	+	0.624348i
$803$	298.629	−	517.240i	0.0131238	−	0.0227310i
$804$		0			0
$805$	6765.93			0.296233
$806$	−14500.8			−0.633707
$807$		0			0
$808$	3475.77	+	6020.21i	0.151333	+	0.262117i
$809$	23140.8			1.00567			0.502834	−	0.864383i	$-0.332291\pi$
$809$	23140.8			1.00567			0.502834	+	0.864383i	$0.332291\pi$
$810$		0			0
$811$	5554.38	+	9620.46i	0.240494	+	0.416548i	0.960855	−	0.277051i	$-0.0893572\pi$
$811$	5554.38	+	9620.46i	0.240494	+	0.416548i	−0.720361	+	0.693599i	$0.756024\pi$
$812$	−1164.09	+	2016.26i	−0.0503097	+	0.0871390i
$813$		0			0
$814$	11752.8	−	20356.4i	0.506062	−	0.876525i
$815$	−2232.78	+	3867.28i	−0.0959641	+	0.166215i
$816$		0			0
$817$	−13262.7	−	4280.24i	−0.567937	−	0.183289i
$818$	9997.00			0.427307
$819$		0			0
$820$	−9940.83	+	17218.0i	−0.423352	+	0.733268i
$821$	6204.51	+	10746.5i	0.263750	+	0.456829i	0.967235	−	0.253881i	$-0.0817072\pi$
$821$	6204.51	+	10746.5i	0.263750	+	0.456829i	−0.703485	+	0.710710i	$0.748374\pi$
$822$		0			0
$823$	9536.05	+	16516.9i	0.403895	+	0.699567i	0.994192	−	0.107619i	$-0.0343226\pi$
$823$	9536.05	+	16516.9i	0.403895	+	0.699567i	−0.590297	+	0.807186i	$0.700989\pi$
$824$	−11932.5			−0.504476
$825$		0			0
$826$	1629.74	+	2822.79i	0.0686511	+	0.118907i
$827$	2143.56	+	3712.76i	0.0901317	+	0.156113i	0.907566	−	0.419909i	$-0.137938\pi$
$827$	2143.56	+	3712.76i	0.0901317	+	0.156113i	−0.817435	+	0.576021i	$0.804605\pi$
$828$		0			0
$829$	−22544.9			−0.944533			−0.472266	−	0.881456i	$-0.656564\pi$
$829$	−22544.9			−0.944533			−0.472266	+	0.881456i	$0.656564\pi$
$830$	−5342.56	−	9253.59i	−0.223425	−	0.386984i
$831$		0			0
$832$	2064.71	+	3576.18i	0.0860348	+	0.149017i
$833$	6178.29	−	10701.1i	0.256981	−	0.445104i
$834$		0			0
$835$	68785.3			2.85080
$836$	15280.0	−	13813.2i	0.632141	−	0.571458i
$837$		0			0
$838$	11941.1	−	20682.6i	0.492241	−	0.852587i
$839$	19298.9	−	33426.7i	0.794127	−	1.37547i	−0.129266	−	0.991610i	$-0.541262\pi$
$839$	19298.9	−	33426.7i	0.794127	−	1.37547i	0.923392	−	0.383858i	$-0.125405\pi$
$840$		0			0
$841$	9988.44	−	17300.5i	0.409547	−	0.709356i
$842$	11982.8	+	20754.8i	0.490445	+	0.849475i
$843$		0			0
$844$	−11826.9			−0.482344
$845$	−20358.3	−	35261.6i	−0.828813	−	1.43555i
$846$		0			0
$847$	22214.1			0.901164
$848$	1808.91			0.0732527
$849$		0			0
$850$	−14104.4	+	24429.5i	−0.569148	+	0.985793i
$851$	3523.74	+	6103.30i	0.141942	+	0.245850i
$852$		0			0
$853$	−1804.94	+	3126.25i	−0.0724502	+	0.125488i	−0.899975	−	0.435942i	$-0.856415\pi$
$853$	−1804.94	+	3126.25i	−0.0724502	+	0.125488i	0.827524	+	0.561430i	$0.189749\pi$
$854$	5409.03			0.216737
$855$		0			0
$856$	12881.4			0.514344
$857$	−1209.06	+	2094.15i	−0.0481921	+	0.0834711i	−0.889115	−	0.457683i	$-0.848679\pi$
$857$	−1209.06	+	2094.15i	−0.0481921	+	0.0834711i	0.840923	+	0.541155i	$0.182013\pi$
$858$		0			0
$859$	−17426.1	−	30183.0i	−0.692168	−	1.19887i	−0.971126	−	0.238567i	$-0.923322\pi$
$859$	−17426.1	−	30183.0i	−0.692168	−	1.19887i	0.278958	−	0.960303i	$-0.410011\pi$
$860$	−6969.65	+	12071.8i	−0.276353	+	0.478657i
$861$		0			0
$862$	−14166.0			−0.559739
$863$	32080.0			1.26537			0.632685	−	0.774409i	$-0.281953\pi$
$863$	32080.0			1.26537			0.632685	+	0.774409i	$0.281953\pi$
$864$		0			0
$865$	15246.7	+	26408.1i	0.599311	+	1.03804i
$866$	−14806.0			−0.580980
$867$		0			0
$868$	−1969.31	−	3410.95i	−0.0770078	−	0.133381i
$869$	36577.4	−	63354.0i	1.42785	−	2.47311i
$870$		0			0
$871$	1272.89	−	2204.70i	0.0495179	−	0.0857675i
$872$	204.089	−	353.492i	0.00792582	−	0.0137279i
$873$		0			0
$874$	1296.00	+	6038.26i	0.0501578	+	0.233693i
$875$	32459.3			1.25409
$876$		0			0
$877$	−21507.3	+	37251.7i	−0.828107	+	1.43432i	0.0714140	+	0.997447i	$0.477249\pi$
$877$	−21507.3	+	37251.7i	−0.828107	+	1.43432i	−0.899521	+	0.436877i	$0.856084\pi$
$878$	7675.43	+	13294.2i	0.295027	+	0.511001i
$879$		0			0
$880$	−10301.3	−	17842.3i	−0.394609	−	0.683483i
$881$	−20270.3			−0.775170			−0.387585	−	0.921834i	$-0.626691\pi$
$881$	−20270.3			−0.775170			−0.387585	+	0.921834i	$0.626691\pi$
$882$		0			0
$883$	9829.17	+	17024.6i	0.374607	+	0.648838i	0.990268	−	0.139173i	$-0.0444443\pi$
$883$	9829.17	+	17024.6i	0.374607	+	0.648838i	−0.615661	+	0.788011i	$0.711111\pi$
$884$	5989.65	+	10374.4i	0.227889	+	0.394715i
$885$		0			0
$886$	5367.78			0.203537
$887$	22422.7	+	38837.3i	0.848795	+	1.47016i	0.882284	+	0.470717i	$0.156005\pi$
$887$	22422.7	+	38837.3i	0.848795	+	1.47016i	−0.0334891	+	0.999439i	$0.510662\pi$
$888$		0			0
$889$	−1217.02	−	2107.93i	−0.0459138	−	0.0795251i
$890$	−2769.83	+	4797.49i	−0.104320	+	0.180688i
$891$		0			0
$892$	2122.56			0.0796733
$893$	−11540.7	+	10432.9i	−0.432470	+	0.390955i
$894$		0			0
$895$	37007.2	−	64098.4i	1.38214	−	2.39394i
$896$	−560.806	+	971.344i	−0.0209098	+	0.0362169i
$897$		0			0
$898$	−14097.6	+	24417.7i	−0.523877	+	0.907382i
$899$	−3732.03	−	6464.07i	−0.138454	−	0.239810i
$900$		0			0
$901$	5247.59			0.194032
$902$	−14923.3	−	25848.0i	−0.550879	−	0.954150i
$903$		0			0
$904$	−14316.4			−0.526720
$905$	−60962.4			−2.23918
$906$		0			0
$907$	−3605.66	+	6245.18i	−0.132000	+	0.228630i	−0.924447	−	0.381310i	$-0.875473\pi$
$907$	−3605.66	+	6245.18i	−0.132000	+	0.228630i	0.792448	+	0.609940i	$0.208806\pi$
$908$	−6959.44	−	12054.1i	−0.254358	−	0.440561i
$909$		0			0
$910$	11708.6	−	20279.9i	0.426524	−	0.738761i
$911$	−29098.8			−1.05827			−0.529136	−	0.848537i	$-0.677484\pi$
$911$	−29098.8			−1.05827			−0.529136	+	0.848537i	$0.677484\pi$
$912$		0			0
$913$	16040.7			0.581456
$914$	−7040.20	+	12194.0i	−0.254780	+	0.441292i
$915$		0			0
$916$	−7132.95	−	12354.6i	−0.257292	−	0.445643i
$917$	−7985.71	+	13831.7i	−0.287581	+	0.498104i
$918$		0			0
$919$	−15321.9			−0.549972			−0.274986	−	0.961448i	$-0.588673\pi$
$919$	−15321.9			−0.549972			−0.274986	+	0.961448i	$0.588673\pi$
$920$	6177.11			0.221362
$921$		0			0
$922$	7353.08	+	12735.9i	0.262647	+	0.454919i
$923$	−22595.8			−0.805796
$924$		0			0
$925$	28718.7	+	49742.2i	1.02083	+	1.76812i
$926$	−14054.7	+	24343.5i	−0.498777	+	0.863907i
$927$		0			0
$928$	−1062.78	+	1840.79i	−0.0375943	+	0.0651152i
$929$	−10272.5	+	17792.4i	−0.362787	+	0.628365i	−0.988418	−	0.151754i	$-0.951508\pi$
$929$	−10272.5	+	17792.4i	−0.362787	+	0.628365i	0.625632	+	0.780118i	$0.284841\pi$
$930$		0			0
$931$	−4626.80	−	21556.9i	−0.162876	−	0.758861i
$932$	23964.8			0.842267
$933$		0			0
$934$	5875.16	−	10176.1i	0.205826	−	0.356501i
$935$	−29883.6	−	51760.0i	−1.04524	−	1.81041i
$936$		0			0
$937$	−23917.7	−	41426.6i	−0.833891	−	1.44434i	−0.894930	−	0.446206i	$-0.852775\pi$
$937$	−23917.7	−	41426.6i	−0.833891	−	1.44434i	0.0610397	−	0.998135i	$-0.480558\pi$
$938$	691.469			0.0240696
$939$		0			0
$940$	7780.39	+	13476.0i	0.269966	+	0.467595i
$941$	−222.655	−	385.650i	−0.00771344	−	0.0133601i	0.862143	−	0.506665i	$-0.169122\pi$
$941$	−222.655	−	385.650i	−0.00771344	−	0.0133601i	−0.869856	+	0.493305i	$0.835789\pi$
$942$		0			0
$943$	8948.70			0.309024
$944$	1487.90	+	2577.13i	0.0513000	+	0.0888541i
$945$		0			0
$946$	−10463.0	−	18122.4i	−0.359598	−	0.622842i
$947$	22023.0	−	38145.0i	0.755705	−	1.30892i	−0.189318	−	0.981916i	$-0.560628\pi$
$947$	22023.0	−	38145.0i	0.755705	−	1.30892i	0.945023	−	0.327004i	$-0.106039\pi$
$948$		0			0
$949$	619.775			0.0211999
$950$	10562.5	+	49212.1i	0.360728	+	1.68068i
$951$		0			0
$952$	−1626.88	+	2817.83i	−0.0553859	+	0.0959313i
$953$	1758.66	−	3046.10i	0.0597783	−	0.103539i	−0.834588	−	0.550875i	$-0.814294\pi$
$953$	1758.66	−	3046.10i	0.0597783	−	0.103539i	0.894366	+	0.447336i	$0.147627\pi$
$954$		0			0
$955$	−7251.95	+	12560.7i	−0.245725	+	0.425609i
$956$	−1212.96	−	2100.90i	−0.0410354	−	0.0710753i
$957$		0			0
$958$	355.562			0.0119913
$959$	2895.87	+	5015.80i	0.0975105	+	0.168893i
$960$		0			0
$961$	−17163.9			−0.576144
$962$	24391.7			0.817484
$963$		0			0
$964$	6890.34	−	11934.4i	0.230210	−	0.398736i
$965$	620.257	+	1074.32i	0.0206910	+	0.0358378i
$966$		0			0
$967$	7621.77	−	13201.3i	0.253464	−	0.439012i	−0.711013	−	0.703179i	$-0.751763\pi$
$967$	7621.77	−	13201.3i	0.253464	−	0.439012i	0.964477	+	0.264166i	$0.0850967\pi$
$968$	20280.9			0.673401
$969$		0			0
$970$	49568.5			1.64077
$971$	1633.51	−	2829.33i	0.0539876	−	0.0935092i	−0.837769	−	0.546025i	$-0.816140\pi$
$971$	1633.51	−	2829.33i	0.0539876	−	0.0935092i	0.891756	+	0.452516i	$0.149474\pi$
$972$		0			0
$973$	5458.87	+	9455.05i	0.179860	+	0.311526i
$974$	−12319.9	+	21338.7i	−0.405293	+	0.701989i
$975$		0			0
$976$	4938.29			0.161958
$977$	−40038.2			−1.31109			−0.655545	−	0.755156i	$-0.727561\pi$
$977$	−40038.2			−1.31109			−0.655545	+	0.755156i	$0.727561\pi$
$978$		0			0
$979$	−4158.11	−	7202.06i	−0.135744	−	0.235116i
$980$	−22052.6			−0.718821
$981$		0			0
$982$	−8441.83	−	14621.7i	−0.274328	−	0.475149i
$983$	11127.2	−	19272.9i	0.361040	−	0.625340i	−0.627092	−	0.778945i	$-0.715755\pi$
$983$	11127.2	−	19272.9i	0.361040	−	0.625340i	0.988132	+	0.153605i	$0.0490885\pi$
$984$		0			0
$985$	−3876.58	+	6714.43i	−0.125399	+	0.217197i
$986$	−3083.09	+	5340.07i	−0.0995797	+	0.172477i
$987$		0			0
$988$	20341.6	+	6564.78i	0.655012	+	0.211390i
$989$	6274.06			0.201722
$990$		0			0
$991$	−7389.74	+	12799.4i	−0.236875	+	0.410279i	−0.959816	−	0.280630i	$-0.909456\pi$
$991$	−7389.74	+	12799.4i	−0.236875	+	0.410279i	0.722941	+	0.690910i	$0.242790\pi$
$992$	−1797.93	−	3114.10i	−0.0575446	−	0.0996701i
$993$		0			0
$994$	−3068.68	−	5315.10i	−0.0979200	−	0.169602i
$995$	95340.5			3.03768
$996$		0			0
$997$	−14444.2	−	25018.1i	−0.458829	−	0.794716i	0.540070	−	0.841620i	$-0.318398\pi$
$997$	−14444.2	−	25018.1i	−0.458829	−	0.794716i	−0.998899	+	0.0469043i	$0.985064\pi$
$998$	6211.24	+	10758.2i	0.197007	+	0.341227i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	342.4.g.f.163.1		6
3.2	odd	2		38.4.c.c.11.2	yes	6
12.11	even	2		304.4.i.e.49.2		6
19.7	even	3	inner	342.4.g.f.235.1		6
57.8	even	6		722.4.a.k.1.2		3
57.11	odd	6		722.4.a.j.1.2		3
57.26	odd	6		38.4.c.c.7.2	✓	6
228.83	even	6		304.4.i.e.273.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.4.c.c.7.2	✓	6	57.26	odd	6
38.4.c.c.11.2	yes	6	3.2	odd	2
304.4.i.e.49.2		6	12.11	even	2
304.4.i.e.273.2		6	228.83	even	6
342.4.g.f.163.1		6	1.1	even	1	trivial
342.4.g.f.235.1		6	19.7	even	3	inner
722.4.a.j.1.2		3	57.11	odd	6
722.4.a.k.1.2		3	57.8	even	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 342 = 2 \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	342.g (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.00000 + 1.73205i) q^{2} +(-2.00000 - 3.46410i) q^{4} +(-10.3546 + 17.9347i) q^{5} +8.76259 q^{7} +8.00000 q^{8} +(-20.7092 - 35.8694i) q^{10} +62.1780 q^{11} +(32.2611 + 55.8778i) q^{13} +(-8.76259 + 15.1772i) q^{14} +(-8.00000 + 13.8564i) q^{16} +(-23.2077 + 40.1970i) q^{17} +(17.3798 + 80.9750i) q^{19} +82.8369 q^{20} +(-62.1780 + 107.696i) q^{22} +(-18.6424 - 32.2895i) q^{23} +(-151.936 - 263.161i) q^{25} -129.044 q^{26} +(-17.5252 - 30.3545i) q^{28} +(-33.2119 - 57.5247i) q^{29} +112.370 q^{31} +(-16.0000 - 27.7128i) q^{32} +(-46.4155 - 80.3939i) q^{34} +(-90.7332 + 157.155i) q^{35} -189.018 q^{37} +(-157.633 - 50.8723i) q^{38} +(-82.8369 + 143.478i) q^{40} +(-120.005 + 207.854i) q^{41} +(-84.1370 + 145.730i) q^{43} +(-124.356 - 215.391i) q^{44} +74.5695 q^{46} +(93.9241 + 162.681i) q^{47} -266.217 q^{49} +607.744 q^{50} +(129.044 - 223.511i) q^{52} +(-56.5285 - 97.9102i) q^{53} +(-643.830 + 1115.15i) q^{55} +70.1007 q^{56} +132.848 q^{58} +(92.9940 - 161.070i) q^{59} +(-154.322 - 267.293i) q^{61} +(-112.370 + 194.631i) q^{62} +64.0000 q^{64} -1336.20 q^{65} +(-19.7279 - 34.1697i) q^{67} +185.662 q^{68} +(-181.466 - 314.309i) q^{70} +(-175.101 + 303.284i) q^{71} +(4.80280 - 8.31870i) q^{73} +(189.018 - 327.389i) q^{74} +(245.746 - 222.155i) q^{76} +544.840 q^{77} +(588.269 - 1018.91i) q^{79} +(-165.674 - 286.956i) q^{80} +(-240.010 - 415.709i) q^{82} +257.980 q^{83} +(-480.614 - 832.448i) q^{85} +(-168.274 - 291.459i) q^{86} +497.424 q^{88} +(-66.8743 - 115.830i) q^{89} +(282.691 + 489.634i) q^{91} +(-74.5695 + 129.158i) q^{92} -375.696 q^{94} +(-1632.22 - 526.763i) q^{95} +(-598.387 + 1036.44i) q^{97} +(266.217 - 461.101i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{2} - 12 q^{4} + q^{5} + 52 q^{7} + 48 q^{8} + 2 q^{10} - 8 q^{11} + 129 q^{13} - 52 q^{14} - 48 q^{16} + 51 q^{17} + 40 q^{19} - 8 q^{20} + 8 q^{22} - 47 q^{23} - 338 q^{25} - 516 q^{26} - 104 q^{28}+ \cdots + 1354 q^{98}+O(q^{100}) \) 6 * q - 6 * q^2 - 12 * q^4 + q^5 + 52 * q^7 + 48 * q^8 + 2 * q^10 - 8 * q^11 + 129 * q^13 - 52 * q^14 - 48 * q^16 + 51 * q^17 + 40 * q^19 - 8 * q^20 + 8 * q^22 - 47 * q^23 - 338 * q^25 - 516 * q^26 - 104 * q^28 + 125 * q^29 - 100 * q^31 - 96 * q^32 + 102 * q^34 + 84 * q^35 - 376 * q^37 - 322 * q^38 + 8 * q^40 - 475 * q^41 - 73 * q^43 + 16 * q^44 + 188 * q^46 + 241 * q^47 - 1354 * q^49 + 1352 * q^50 + 516 * q^52 - 29 * q^53 - 1838 * q^55 + 416 * q^56 - 500 * q^58 + 1065 * q^59 - 981 * q^61 + 100 * q^62 + 384 * q^64 - 586 * q^65 + 877 * q^67 - 408 * q^68 + 168 * q^70 - 2135 * q^71 + 667 * q^73 + 376 * q^74 + 484 * q^76 + 492 * q^77 + 1671 * q^79 + 16 * q^80 - 950 * q^82 - 1176 * q^83 - 1929 * q^85 - 146 * q^86 - 64 * q^88 - 693 * q^89 + 1676 * q^91 - 188 * q^92 - 964 * q^94 - 4489 * q^95 - 985 * q^97 + 1354 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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