LMFDB - Embedded newform 378.4.g.c.163.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [378,4,Mod(109,378)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(378, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 4]))

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

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

chi := DirichletCharacter("378.109");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 378 = 2 \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	378.g (of order $3$, degree $2$, 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:	$22.3027219822$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} + 18x^{6} + 9x^{5} + 283x^{4} - 48x^{3} + 186x^{2} + 40x + 100 $ x^8 - x^7 + 18x^6 + 9x^5 + 283x^4 - 48x^3 + 186x^2 + 40x + 100
Coefficient ring:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{3} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			163.4
Root			$-0.338925 - 0.587036i$ of defining polynomial
Character	$\chi$	$=$	378.163
Dual form			378.4.g.c.109.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.00000 + 1.73205i) q^{2} +(-2.00000 - 3.46410i) q^{4} +(8.05411 - 13.9501i) q^{5} +(6.77345 + 17.2372i) q^{7} +8.00000 q^{8} +O(q^{10})$	\(q+(-1.00000 + 1.73205i) q^{2} +(-2.00000 - 3.46410i) q^{4} +(8.05411 - 13.9501i) q^{5} +(6.77345 + 17.2372i) q^{7} +8.00000 q^{8} +(16.1082 + 27.9003i) q^{10} +(6.04111 + 10.4635i) q^{11} -39.7518 q^{13} +(-36.6291 - 5.50523i) q^{14} +(-8.00000 + 13.8564i) q^{16} +(62.7914 + 108.758i) q^{17} +(62.5108 - 108.272i) q^{19} -64.4329 q^{20} -24.1645 q^{22} +(-24.1212 + 41.7792i) q^{23} +(-67.2373 - 116.458i) q^{25} +(39.7518 - 68.8521i) q^{26} +(46.1645 - 57.9383i) q^{28} +93.0461 q^{29} +(-11.3809 - 19.7123i) q^{31} +(-16.0000 - 27.7128i) q^{32} -251.166 q^{34} +(295.015 + 44.3397i) q^{35} +(200.361 - 347.035i) q^{37} +(125.022 + 216.544i) q^{38} +(64.4329 - 111.601i) q^{40} +354.070 q^{41} +225.918 q^{43} +(24.1645 - 41.8541i) q^{44} +(-48.2424 - 83.5583i) q^{46} +(124.107 - 214.960i) q^{47} +(-251.241 + 233.510i) q^{49} +268.949 q^{50} +(79.5036 + 137.704i) q^{52} +(167.460 + 290.050i) q^{53} +194.623 q^{55} +(54.1876 + 137.897i) q^{56} +(-93.0461 + 161.161i) q^{58} +(-91.4364 - 158.373i) q^{59} +(-304.912 + 528.123i) q^{61} +45.5237 q^{62} +64.0000 q^{64} +(-320.165 + 554.542i) q^{65} +(-59.1898 - 102.520i) q^{67} +(251.166 - 435.032i) q^{68} +(-371.814 + 466.641i) q^{70} +81.4905 q^{71} +(489.415 + 847.692i) q^{73} +(400.721 + 694.069i) q^{74} -500.086 q^{76} +(-139.442 + 175.006i) q^{77} +(346.714 - 600.527i) q^{79} +(128.866 + 223.202i) q^{80} +(-354.070 + 613.268i) q^{82} +1409.05 q^{83} +2022.92 q^{85} +(-225.918 + 391.301i) q^{86} +(48.3289 + 83.7081i) q^{88} +(-605.706 + 1049.11i) q^{89} +(-269.256 - 685.209i) q^{91} +192.970 q^{92} +(248.215 + 429.921i) q^{94} +(-1006.94 - 1744.07i) q^{95} -1336.50 q^{97} +(-153.211 - 668.672i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{2} - 16 q^{4} + 2 q^{5} + 6 q^{7} + 64 q^{8}+O(q^{10}) $ 8 * q - 8 * q^2 - 16 * q^4 + 2 * q^5 + 6 * q^7 + 64 * q^8	$ 8 q - 8 q^{2} - 16 q^{4} + 2 q^{5} + 6 q^{7} + 64 q^{8} + 4 q^{10} - 32 q^{11} - 4 q^{13} - 36 q^{14} - 64 q^{16} + 58 q^{17} + 70 q^{19} - 16 q^{20} + 128 q^{22} - 86 q^{23} - 156 q^{25} + 4 q^{26} + 48 q^{28} + 212 q^{29} - 64 q^{31} - 128 q^{32} - 232 q^{34} - 8 q^{35} - 146 q^{37} + 140 q^{38} + 16 q^{40} + 780 q^{41} + 880 q^{43} - 128 q^{44} - 172 q^{46} + 306 q^{47} + 50 q^{49} + 624 q^{50} + 8 q^{52} + 90 q^{53} - 64 q^{55} + 48 q^{56} - 212 q^{58} - 148 q^{59} - 364 q^{61} + 256 q^{62} + 512 q^{64} - 1296 q^{65} - 954 q^{67} + 232 q^{68} + 20 q^{70} + 1360 q^{71} - 54 q^{73} - 292 q^{74} - 560 q^{76} - 2224 q^{77} - 226 q^{79} + 32 q^{80} - 780 q^{82} + 3136 q^{83} + 3920 q^{85} - 880 q^{86} - 256 q^{88} + 1458 q^{89} + 3836 q^{91} + 688 q^{92} + 612 q^{94} - 1310 q^{95} - 4344 q^{97} - 776 q^{98}+O(q^{100}) $ 8 * q - 8 * q^2 - 16 * q^4 + 2 * q^5 + 6 * q^7 + 64 * q^8 + 4 * q^10 - 32 * q^11 - 4 * q^13 - 36 * q^14 - 64 * q^16 + 58 * q^17 + 70 * q^19 - 16 * q^20 + 128 * q^22 - 86 * q^23 - 156 * q^25 + 4 * q^26 + 48 * q^28 + 212 * q^29 - 64 * q^31 - 128 * q^32 - 232 * q^34 - 8 * q^35 - 146 * q^37 + 140 * q^38 + 16 * q^40 + 780 * q^41 + 880 * q^43 - 128 * q^44 - 172 * q^46 + 306 * q^47 + 50 * q^49 + 624 * q^50 + 8 * q^52 + 90 * q^53 - 64 * q^55 + 48 * q^56 - 212 * q^58 - 148 * q^59 - 364 * q^61 + 256 * q^62 + 512 * q^64 - 1296 * q^65 - 954 * q^67 + 232 * q^68 + 20 * q^70 + 1360 * q^71 - 54 * q^73 - 292 * q^74 - 560 * q^76 - 2224 * q^77 - 226 * q^79 + 32 * q^80 - 780 * q^82 + 3136 * q^83 + 3920 * q^85 - 880 * q^86 - 256 * q^88 + 1458 * q^89 + 3836 * q^91 + 688 * q^92 + 612 * q^94 - 1310 * q^95 - 4344 * q^97 - 776 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/378\mathbb{Z}\right)^\times$.

$n$	$29$	$325$
$\chi(n)$	$1$	$e\left(\frac{1}{3}\right)$

Coefficient data

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

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

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

$2$	−1.00000	+	1.73205i	−0.353553	+	0.612372i
$2$	−1.00000	+	1.73205i	−0.353553	+	0.612372i
$3$		0			0
$3$		0			0
$4$	−2.00000	−	3.46410i	−0.250000	−	0.433013i
$5$	8.05411	−	13.9501i	0.720381	−	1.24774i	−0.240466	−	0.970658i	$-0.577300\pi$
$5$	8.05411	−	13.9501i	0.720381	−	1.24774i	0.960847	−	0.277080i	$-0.0893665\pi$
$6$		0			0
$7$	6.77345	+	17.2372i	0.365732	+	0.930720i
$7$	6.77345	+	17.2372i	0.365732	+	0.930720i
$8$	8.00000			0.353553
$9$		0			0
$10$	16.1082	+	27.9003i	0.509387	+	0.882283i
$11$	6.04111	+	10.4635i	0.165588	+	0.286806i	0.936864	−	0.349694i	$-0.113715\pi$
$11$	6.04111	+	10.4635i	0.165588	+	0.286806i	−0.771276	+	0.636501i	$0.780381\pi$
$12$		0			0
$13$	−39.7518			−0.848089			−0.424045	−	0.905641i	$-0.639390\pi$
$13$	−39.7518			−0.848089			−0.424045	+	0.905641i	$0.639390\pi$
$14$	−36.6291	−	5.50523i	−0.699253	−	0.105095i
$15$		0			0
$16$	−8.00000	+	13.8564i	−0.125000	+	0.216506i
$17$	62.7914	+	108.758i	0.895833	+	1.55163i	0.832770	+	0.553619i	$0.186754\pi$
$17$	62.7914	+	108.758i	0.895833	+	1.55163i	0.0630630	+	0.998010i	$0.479913\pi$
$18$		0			0
$19$	62.5108	−	108.272i	0.754787	−	1.30733i	−0.190693	−	0.981650i	$-0.561073\pi$
$19$	62.5108	−	108.272i	0.754787	−	1.30733i	0.945480	−	0.325680i	$-0.105593\pi$
$20$	−64.4329			−0.720381
$21$		0			0
$22$	−24.1645			−0.234176
$23$	−24.1212	+	41.7792i	−0.218679	+	0.378763i	−0.954404	−	0.298517i	$-0.903508\pi$
$23$	−24.1212	+	41.7792i	−0.218679	+	0.378763i	0.735725	+	0.677280i	$0.236841\pi$
$24$		0			0
$25$	−67.2373	−	116.458i	−0.537899	−	0.931668i
$26$	39.7518	−	68.8521i	0.299845	−	0.519346i
$27$		0			0
$28$	46.1645	−	57.9383i	0.311581	−	0.391047i
$29$	93.0461			0.595801			0.297900	−	0.954597i	$-0.403714\pi$
$29$	93.0461			0.595801			0.297900	+	0.954597i	$0.403714\pi$
$30$		0			0
$31$	−11.3809	−	19.7123i	−0.0659379	−	0.114208i	0.831172	−	0.556016i	$-0.187671\pi$
$31$	−11.3809	−	19.7123i	−0.0659379	−	0.114208i	−0.897110	+	0.441808i	$0.854337\pi$
$32$	−16.0000	−	27.7128i	−0.0883883	−	0.153093i
$33$		0			0
$34$	−251.166			−1.26690
$35$	295.015	+	44.3397i	1.42476	+	0.214137i
$36$		0			0
$37$	200.361	−	347.035i	0.890245	−	1.54195i	0.0506642	−	0.998716i	$-0.483866\pi$
$37$	200.361	−	347.035i	0.890245	−	1.54195i	0.839581	−	0.543234i	$-0.182800\pi$
$38$	125.022	+	216.544i	0.533715	+	0.924422i
$39$		0			0
$40$	64.4329	−	111.601i	0.254693	−	0.441142i
$41$	354.070			1.34870			0.674348	−	0.738414i	$-0.264425\pi$
$41$	354.070			1.34870			0.674348	+	0.738414i	$0.264425\pi$
$42$		0			0
$43$	225.918			0.801212			0.400606	−	0.916250i	$-0.368800\pi$
$43$	225.918			0.801212			0.400606	+	0.916250i	$0.368800\pi$
$44$	24.1645	−	41.8541i	0.0827938	−	0.143403i
$45$		0			0
$46$	−48.2424	−	83.5583i	−0.154630	−	0.267826i
$47$	124.107	−	214.960i	0.385169	−	0.667132i	−0.606624	−	0.794989i	$-0.707477\pi$
$47$	124.107	−	214.960i	0.385169	−	0.667132i	0.991793	+	0.127857i	$0.0408099\pi$
$48$		0			0
$49$	−251.241	+	233.510i	−0.732481	+	0.680788i
$50$	268.949			0.760704
$51$		0			0
$52$	79.5036	+	137.704i	0.212022	+	0.367233i
$53$	167.460	+	290.050i	0.434008	+	0.751724i	0.997214	−	0.0745926i	$-0.0237656\pi$
$53$	167.460	+	290.050i	0.434008	+	0.751724i	−0.563206	+	0.826316i	$0.690432\pi$
$54$		0			0
$55$	194.623			0.477145
$56$	54.1876	+	137.897i	0.129306	+	0.329059i
$57$		0			0
$58$	−93.0461	+	161.161i	−0.210647	+	0.364852i
$59$	−91.4364	−	158.373i	−0.201763	−	0.349464i	0.747334	−	0.664449i	$-0.231334\pi$
$59$	−91.4364	−	158.373i	−0.201763	−	0.349464i	−0.949097	+	0.314985i	$0.898000\pi$
$60$		0			0
$61$	−304.912	+	528.123i	−0.639999	+	1.10851i	0.345433	+	0.938443i	$0.387732\pi$
$61$	−304.912	+	528.123i	−0.639999	+	1.10851i	−0.985432	+	0.170068i	$0.945601\pi$
$62$	45.5237			0.0932502
$63$		0			0
$64$	64.0000			0.125000
$65$	−320.165	+	554.542i	−0.610948	+	1.05819i
$66$		0			0
$67$	−59.1898	−	102.520i	−0.107928	−	0.186937i	0.807003	−	0.590548i	$-0.201088\pi$
$67$	−59.1898	−	102.520i	−0.107928	−	0.186937i	−0.914931	+	0.403611i	$0.867755\pi$
$68$	251.166	−	435.032i	0.447917	−	0.775814i
$69$		0			0
$70$	−371.814	+	466.641i	−0.634860	+	0.796775i
$71$	81.4905			0.136213			0.0681066	−	0.997678i	$-0.478304\pi$
$71$	81.4905			0.136213			0.0681066	+	0.997678i	$0.478304\pi$
$72$		0			0
$73$	489.415	+	847.692i	0.784681	+	1.35911i	0.929189	+	0.369604i	$0.120507\pi$
$73$	489.415	+	847.692i	0.784681	+	1.35911i	−0.144508	+	0.989504i	$0.546160\pi$
$74$	400.721	+	694.069i	0.629498	+	1.09032i
$75$		0			0
$76$	−500.086			−0.754787
$77$	−139.442	+	175.006i	−0.206376	+	0.259010i
$78$		0			0
$79$	346.714	−	600.527i	0.493777	−	0.855247i	−0.506197	−	0.862418i	$-0.668949\pi$
$79$	346.714	−	600.527i	0.493777	−	0.855247i	0.999974	+	0.00717076i	$0.00228254\pi$
$80$	128.866	+	223.202i	0.180095	+	0.311934i
$81$		0			0
$82$	−354.070	+	613.268i	−0.476836	+	0.825904i
$83$	1409.05			1.86341			0.931707	−	0.363212i	$-0.118320\pi$
$83$	1409.05			1.86341			0.931707	+	0.363212i	$0.118320\pi$
$84$		0			0
$85$	2022.92			2.58137
$86$	−225.918	+	391.301i	−0.283271	+	0.490640i
$87$		0			0
$88$	48.3289	+	83.7081i	0.0585441	+	0.101401i
$89$	−605.706	+	1049.11i	−0.721401	+	1.24950i	0.239037	+	0.971010i	$0.423168\pi$
$89$	−605.706	+	1049.11i	−0.721401	+	1.24950i	−0.960438	+	0.278493i	$0.910165\pi$
$90$		0			0
$91$	−269.256	−	685.209i	−0.310173	−	0.789334i
$92$	192.970			0.218679
$93$		0			0
$94$	248.215	+	429.921i	0.272355	+	0.471733i
$95$	−1006.94	−	1744.07i	−1.08747	−	1.88355i
$96$		0			0
$97$	−1336.50			−1.39898			−0.699492	−	0.714641i	$-0.746590\pi$
$97$	−1336.50			−1.39898			−0.699492	+	0.714641i	$0.746590\pi$
$98$	−153.211	−	668.672i	−0.157925	−	0.689246i
$99$		0			0
$100$	−268.949	+	465.834i	−0.268949	+	0.465834i
$101$	103.208	+	178.762i	0.101679	+	0.176114i	0.912377	−	0.409352i	$-0.134245\pi$
$101$	103.208	+	178.762i	0.101679	+	0.176114i	−0.810697	+	0.585465i	$0.800912\pi$
$102$		0			0
$103$	344.223	−	596.212i	0.329294	−	0.570354i	−0.653078	−	0.757291i	$-0.726523\pi$
$103$	344.223	−	596.212i	0.329294	−	0.570354i	0.982372	+	0.186936i	$0.0598559\pi$
$104$	−318.014			−0.299845
$105$		0			0
$106$	−669.841			−0.613780
$107$	226.713	−	392.678i	0.204833	−	0.354781i	−0.745247	−	0.666789i	$-0.767668\pi$
$107$	226.713	−	392.678i	0.204833	−	0.354781i	0.950079	+	0.312008i	$0.101002\pi$
$108$		0			0
$109$	−685.492	−	1187.31i	−0.602369	−	1.04333i	−0.992461	−	0.122558i	$-0.960890\pi$
$109$	−685.492	−	1187.31i	−0.602369	−	1.04333i	0.390093	−	0.920776i	$-0.372443\pi$
$110$	−194.623	+	337.097i	−0.168696	+	0.292191i
$111$		0			0
$112$	−293.033	−	44.0418i	−0.247223	−	0.0371568i
$113$	1807.69			1.50489			0.752446	−	0.658654i	$-0.228874\pi$
$113$	1807.69			1.50489			0.752446	+	0.658654i	$0.228874\pi$
$114$		0			0
$115$	388.550	+	672.988i	0.315065	+	0.545708i
$116$	−186.092	−	322.321i	−0.148950	−	0.257989i
$117$		0			0
$118$	365.746			0.285336
$119$	−1449.37	+	1819.01i	−1.11650	+	1.40125i
$120$		0			0
$121$	592.510	−	1026.26i	0.445161	−	0.771042i
$122$	−609.824	−	1056.25i	−0.452548	−	0.783836i
$123$		0			0
$124$	−45.5237	+	78.8493i	−0.0329689	+	0.0571039i
$125$	−152.620			−0.109206
$126$		0			0
$127$	−196.476			−0.137279			−0.0686394	−	0.997642i	$-0.521866\pi$
$127$	−196.476			−0.137279			−0.0686394	+	0.997642i	$0.521866\pi$
$128$	−64.0000	+	110.851i	−0.0441942	+	0.0765466i
$129$		0			0
$130$	−640.330	−	1109.08i	−0.432005	−	0.748255i
$131$	753.242	−	1304.65i	0.502374	−	0.870138i	−0.497622	−	0.867394i	$-0.665793\pi$
$131$	753.242	−	1304.65i	0.502374	−	0.870138i	0.999996	−	0.00274396i	$-0.000873432\pi$
$132$		0			0
$133$	2289.71	+	344.136i	1.49281	+	0.224364i
$134$	236.759			0.152634
$135$		0			0
$136$	502.332	+	870.064i	0.316725	+	0.548583i
$137$	−157.673	−	273.097i	−0.0983277	−	0.170309i	0.812665	−	0.582731i	$-0.198016\pi$
$137$	−157.673	−	273.097i	−0.0983277	−	0.170309i	−0.910993	+	0.412423i	$0.864683\pi$
$138$		0			0
$139$	−324.501			−0.198013			−0.0990064	−	0.995087i	$-0.531566\pi$
$139$	−324.501			−0.198013			−0.0990064	+	0.995087i	$0.531566\pi$
$140$	−436.433	−	1110.64i	−0.263466	−	0.670474i
$141$		0			0
$142$	−81.4905	+	141.146i	−0.0481587	+	0.0834132i
$143$	−240.145	−	415.943i	−0.140433	−	0.243237i
$144$		0			0
$145$	749.403	−	1298.00i	0.429204	−	0.743403i
$146$	−1957.66			−1.10971
$147$		0			0
$148$	−1602.88			−0.890245
$149$	638.745	−	1106.34i	0.351195	−	0.608287i	−0.635264	−	0.772295i	$-0.719109\pi$
$149$	638.745	−	1106.34i	0.351195	−	0.608287i	0.986459	+	0.164008i	$0.0524422\pi$
$150$		0			0
$151$	−258.424	−	447.604i	−0.139273	−	0.241228i	0.787948	−	0.615741i	$-0.211143\pi$
$151$	−258.424	−	447.604i	−0.139273	−	0.241228i	−0.927222	+	0.374513i	$0.877810\pi$
$152$	500.086	−	866.175i	0.266858	−	0.462211i
$153$		0			0
$154$	−163.677	−	416.527i	−0.0856457	−	0.217953i
$155$	−366.653			−0.190002
$156$		0			0
$157$	649.320	+	1124.66i	0.330073	+	0.571702i	0.982526	−	0.186127i	$-0.0595935\pi$
$157$	649.320	+	1124.66i	0.330073	+	0.571702i	−0.652453	+	0.757829i	$0.726260\pi$
$158$	693.428	+	1201.05i	0.349153	+	0.604751i
$159$		0			0
$160$	−515.463			−0.254693
$161$	−883.539	−	132.793i	−0.432501	−	0.0650034i
$162$		0			0
$163$	−1487.02	+	2575.60i	−0.714556	+	1.23765i	0.248574	+	0.968613i	$0.420038\pi$
$163$	−1487.02	+	2575.60i	−0.714556	+	1.23765i	−0.963130	+	0.269035i	$0.913295\pi$
$164$	−708.141	−	1226.54i	−0.337174	−	0.584002i
$165$		0			0
$166$	−1409.05	+	2440.55i	−0.658816	+	1.14110i
$167$	−3607.22			−1.67147			−0.835733	−	0.549135i	$-0.814957\pi$
$167$	−3607.22			−1.67147			−0.835733	+	0.549135i	$0.814957\pi$
$168$		0			0
$169$	−616.796			−0.280745
$170$	−2022.92	+	3503.79i	−0.912651	+	1.58076i
$171$		0			0
$172$	−451.835	−	782.602i	−0.200303	−	0.346935i
$173$	−1263.21	+	2187.94i	−0.555143	+	0.961536i	0.442749	+	0.896645i	$0.354003\pi$
$173$	−1263.21	+	2187.94i	−0.555143	+	0.961536i	−0.997892	+	0.0648904i	$0.979330\pi$
$174$		0			0
$175$	1551.99	−	1947.81i	0.670396	−	0.841374i
$176$	−193.316			−0.0827938
$177$		0			0
$178$	−1211.41	−	2098.23i	−0.510108	−	0.883532i
$179$	−711.221	−	1231.87i	−0.296979	−	0.514382i	0.678465	−	0.734633i	$-0.262646\pi$
$179$	−711.221	−	1231.87i	−0.296979	−	0.514382i	−0.975443	+	0.220251i	$0.929312\pi$
$180$		0			0
$181$	2494.29			1.02431			0.512153	−	0.858894i	$-0.328848\pi$
$181$	2494.29			1.02431			0.512153	+	0.858894i	$0.328848\pi$
$182$	1456.07	+	218.843i	0.593029	+	0.0891302i
$183$		0			0
$184$	−192.970	+	334.233i	−0.0773148	+	0.133913i
$185$	−3227.45	−	5590.11i	−1.28263	−	2.22158i
$186$		0			0
$187$	−758.661	+	1314.04i	−0.296678	+	0.513861i
$188$	−992.859			−0.385169
$189$		0			0
$190$	4027.75			1.53791
$191$	−1846.91	+	3198.94i	−0.699674	+	1.21187i	0.268906	+	0.963167i	$0.413338\pi$
$191$	−1846.91	+	3198.94i	−0.699674	+	1.21187i	−0.968580	+	0.248704i	$0.919995\pi$
$192$		0			0
$193$	38.5601	+	66.7881i	0.0143815	+	0.0249094i	0.873127	−	0.487494i	$-0.162089\pi$
$193$	38.5601	+	66.7881i	0.0143815	+	0.0249094i	−0.858745	+	0.512403i	$0.828755\pi$
$194$	1336.50	−	2314.89i	0.494615	−	0.856699i
$195$		0			0
$196$	1311.38	+	403.304i	0.477910	+	0.146977i
$197$	−1259.96			−0.455679			−0.227839	−	0.973699i	$-0.573166\pi$
$197$	−1259.96			−0.455679			−0.227839	+	0.973699i	$0.573166\pi$
$198$		0			0
$199$	−2725.37	−	4720.48i	−0.970837	−	1.68154i	−0.693041	−	0.720898i	$-0.743729\pi$
$199$	−2725.37	−	4720.48i	−0.970837	−	1.68154i	−0.277796	−	0.960640i	$-0.589604\pi$
$200$	−537.899	−	931.668i	−0.190176	−	0.329394i
$201$		0			0
$202$	−412.833			−0.143796
$203$	630.242	+	1603.85i	0.217903	+	0.554524i
$204$		0			0
$205$	2851.72	−	4939.33i	0.971575	−	1.68282i
$206$	688.446	+	1192.42i	0.232846	+	0.403301i
$207$		0			0
$208$	318.014	−	550.817i	0.106011	−	0.183617i
$209$	1510.54			0.499934
$210$		0			0
$211$	326.412			0.106498			0.0532492	−	0.998581i	$-0.483042\pi$
$211$	326.412			0.106498			0.0532492	+	0.998581i	$0.483042\pi$
$212$	669.841	−	1160.20i	0.217004	−	0.375862i
$213$		0			0
$214$	453.425	+	785.355i	0.144839	+	0.250868i
$215$	1819.57	−	3151.58i	0.577178	−	0.999702i
$216$		0			0
$217$	262.697	−	329.695i	0.0821799	−	0.103139i
$218$	2741.97			0.851878
$219$		0			0
$220$	−389.246	−	674.195i	−0.119286	−	0.206610i
$221$	−2496.07	−	4323.32i	−0.759746	−	1.31592i
$222$		0			0
$223$	−3095.25			−0.929477			−0.464738	−	0.885448i	$-0.653852\pi$
$223$	−3095.25			−0.929477			−0.464738	+	0.885448i	$0.653852\pi$
$224$	369.316	−	463.506i	0.110160	−	0.138256i
$225$		0			0
$226$	−1807.69	+	3131.01i	−0.532060	+	0.921555i
$227$	313.357	+	542.750i	0.0916221	+	0.158694i	0.908194	−	0.418550i	$-0.137461\pi$
$227$	313.357	+	542.750i	0.0916221	+	0.158694i	−0.816572	+	0.577244i	$0.804128\pi$
$228$		0			0
$229$	−3333.25	+	5773.36i	−0.961867	+	1.66600i	−0.244059	+	0.969760i	$0.578479\pi$
$229$	−3333.25	+	5773.36i	−0.961867	+	1.66600i	−0.717808	+	0.696241i	$0.754854\pi$
$230$	−1554.20			−0.445569
$231$		0			0
$232$	744.369			0.210647
$233$	−1750.26	+	3031.53i	−0.492117	+	0.852371i	−0.999959	−	0.00907927i	$-0.997110\pi$
$233$	−1750.26	+	3031.53i	−0.492117	+	0.852371i	0.507842	+	0.861450i	$0.330443\pi$
$234$		0			0
$235$	−1999.15	−	3462.63i	−0.554937	−	0.961179i
$236$	−365.746	+	633.490i	−0.100881	+	0.174732i
$237$		0			0
$238$	−1701.26	−	4329.39i	−0.463345	−	1.17913i
$239$	−119.098			−0.0322335			−0.0161168	−	0.999870i	$-0.505130\pi$
$239$	−119.098			−0.0322335			−0.0161168	+	0.999870i	$0.505130\pi$
$240$		0			0
$241$	−635.903	−	1101.42i	−0.169967	−	0.294392i	0.768441	−	0.639921i	$-0.221033\pi$
$241$	−635.903	−	1101.42i	−0.169967	−	0.294392i	−0.938408	+	0.345529i	$0.887699\pi$
$242$	1185.02	+	2052.51i	0.314777	+	0.545209i
$243$		0			0
$244$	2439.29			0.639999
$245$	1233.98	+	5385.56i	0.321779	+	1.40437i
$246$		0			0
$247$	−2484.91	+	4304.00i	−0.640127	+	1.10873i
$248$	−91.0474	−	157.699i	−0.0233126	−	0.0403785i
$249$		0			0
$250$	152.620	−	264.346i	0.0386102	−	0.0668748i
$251$	330.354			0.0830749			0.0415374	−	0.999137i	$-0.486774\pi$
$251$	330.354			0.0830749			0.0415374	+	0.999137i	$0.486774\pi$
$252$		0			0
$253$	−582.876			−0.144842
$254$	196.476	−	340.306i	0.0485354	−	0.0840658i
$255$		0			0
$256$	−128.000	−	221.703i	−0.0312500	−	0.0541266i
$257$	1260.40	−	2183.08i	0.305921	−	0.529870i	−0.671545	−	0.740964i	$-0.734369\pi$
$257$	1260.40	−	2183.08i	0.305921	−	0.529870i	0.977466	+	0.211093i	$0.0677024\pi$
$258$		0			0
$259$	7339.03	+	1103.03i	1.76072	+	0.264629i
$260$	2561.32			0.610948
$261$		0			0
$262$	1506.48	+	2609.31i	0.355232	+	0.615281i
$263$	713.729	+	1236.21i	0.167340	+	0.289841i	0.937484	−	0.348029i	$-0.113149\pi$
$263$	713.729	+	1236.21i	0.167340	+	0.289841i	−0.770144	+	0.637870i	$0.779816\pi$
$264$		0			0
$265$	5394.97			1.25061
$266$	−2885.78	+	3621.77i	−0.665182	+	0.834830i
$267$		0			0
$268$	−236.759	+	410.079i	−0.0539641	+	0.0934686i
$269$	−4094.93	−	7092.63i	−0.928151	−	1.60760i	−0.786414	−	0.617700i	$-0.788065\pi$
$269$	−4094.93	−	7092.63i	−0.928151	−	1.60760i	−0.141737	−	0.989904i	$-0.545269\pi$
$270$		0			0
$271$	−3197.35	+	5537.98i	−0.716699	+	1.24136i	0.245602	+	0.969371i	$0.421014\pi$
$271$	−3197.35	+	5537.98i	−0.716699	+	1.24136i	−0.962301	+	0.271988i	$0.912319\pi$
$272$	−2009.33			−0.447917
$273$		0			0
$274$	630.691			0.139056
$275$	812.377	−	1407.08i	0.178139	−	0.308545i
$276$		0			0
$277$	3319.70	+	5749.90i	0.720078	+	1.24721i	0.960968	+	0.276660i	$0.0892275\pi$
$277$	3319.70	+	5749.90i	0.720078	+	1.24721i	−0.240890	+	0.970553i	$0.577439\pi$
$278$	324.501	−	562.052i	0.0700081	−	0.121258i
$279$		0			0
$280$	2360.12	+	354.718i	0.503729	+	0.0757087i
$281$	4929.95			1.04661			0.523303	−	0.852147i	$-0.324700\pi$
$281$	4929.95			1.04661			0.523303	+	0.852147i	$0.324700\pi$
$282$		0			0
$283$	−1391.39	−	2409.96i	−0.292261	−	0.506210i	0.682083	−	0.731274i	$-0.261074\pi$
$283$	−1391.39	−	2409.96i	−0.292261	−	0.506210i	−0.974344	+	0.225064i	$0.927741\pi$
$284$	−162.981	−	282.291i	−0.0340533	−	0.0589821i
$285$		0			0
$286$	960.580			0.198602
$287$	2398.28	+	6103.18i	0.493261	+	1.25526i
$288$		0			0
$289$	−5429.03	+	9403.36i	−1.10503	+	1.91397i
$290$	1498.81	+	2596.01i	0.303493	+	0.525665i
$291$		0			0
$292$	1957.66	−	3390.77i	0.392341	−	0.679554i
$293$	1639.57			0.326910			0.163455	−	0.986551i	$-0.447736\pi$
$293$	1639.57			0.326910			0.163455	+	0.986551i	$0.447736\pi$
$294$		0			0
$295$	−2945.76			−0.581385
$296$	1602.88	−	2776.28i	0.314749	−	0.545162i
$297$		0			0
$298$	1277.49	+	2212.68i	0.248332	+	0.430124i
$299$	958.861	−	1660.80i	0.185459	−	0.321225i
$300$		0			0
$301$	1530.24	+	3894.18i	0.293029	+	0.745705i
$302$	1033.70			0.196962
$303$		0			0
$304$	1000.17	+	1732.35i	0.188697	+	0.326832i
$305$	4911.59	+	8507.12i	0.922087	+	1.59710i
$306$		0			0
$307$	−10332.5			−1.92087			−0.960436	−	0.278502i	$-0.910162\pi$
$307$	−10332.5			−1.92087			−0.960436	+	0.278502i	$0.910162\pi$
$308$	885.123	+	133.031i	0.163749	+	0.0246108i
$309$		0			0
$310$	366.653	−	635.061i	0.0671757	−	0.116352i
$311$	4626.39	+	8013.14i	0.843532	+	1.46104i	0.886890	+	0.461980i	$0.152861\pi$
$311$	4626.39	+	8013.14i	0.843532	+	1.46104i	−0.0433584	+	0.999060i	$0.513806\pi$
$312$		0			0
$313$	1238.89	−	2145.82i	0.223726	−	0.387504i	−0.732211	−	0.681078i	$-0.761511\pi$
$313$	1238.89	−	2145.82i	0.223726	−	0.387504i	0.955936	+	0.293574i	$0.0948447\pi$
$314$	−2597.28			−0.466793
$315$		0			0
$316$	−2773.71			−0.493777
$317$	1441.61	−	2496.94i	0.255422	−	0.442404i	−0.709588	−	0.704617i	$-0.751119\pi$
$317$	1441.61	−	2496.94i	0.255422	−	0.442404i	0.965010	+	0.262213i	$0.0844523\pi$
$318$		0			0
$319$	562.102	+	973.589i	0.0986573	+	0.170879i
$320$	515.463	−	892.808i	0.0900477	−	0.155967i
$321$		0			0
$322$	1113.54	−	1397.54i	0.192718	−	0.241869i
$323$	15700.6			2.70465
$324$		0			0
$325$	2672.80	+	4629.43i	0.456186	+	0.790137i
$326$	−2974.05	−	5151.20i	−0.505267	−	0.875149i
$327$		0			0
$328$	2832.56			0.476836
$329$	4545.95	+	683.240i	0.761782	+	0.114493i
$330$		0			0
$331$	248.565	−	430.528i	0.0412761	−	0.0714923i	−0.844649	−	0.535320i	$-0.820191\pi$
$331$	248.565	−	430.528i	0.0412761	−	0.0714923i	0.885925	+	0.463828i	$0.153524\pi$
$332$	−2818.10	−	4881.09i	−0.465853	−	0.806882i
$333$		0			0
$334$	3607.22	−	6247.89i	0.590953	−	1.02356i
$335$	−1906.89			−0.310998
$336$		0			0
$337$	6711.07			1.08479			0.542397	−	0.840123i	$-0.317517\pi$
$337$	6711.07			1.08479			0.542397	+	0.840123i	$0.317517\pi$
$338$	616.796	−	1068.32i	0.0992583	−	0.171920i
$339$		0			0
$340$	−4045.83	−	7007.59i	−0.645341	−	1.11776i
$341$	137.507	−	238.169i	0.0218370	−	0.0378228i
$342$		0			0
$343$	−5726.82	−	2749.02i	−0.901514	−	0.432749i
$344$	1807.34			0.283271
$345$		0			0
$346$	−2526.41	−	4375.87i	−0.392545	−	0.679908i
$347$	−80.8131	−	139.972i	−0.0125022	−	0.0216545i	0.859707	−	0.510788i	$-0.170646\pi$
$347$	−80.8131	−	139.972i	−0.0125022	−	0.0216545i	−0.872209	+	0.489134i	$0.837313\pi$
$348$		0			0
$349$	−3970.85			−0.609040			−0.304520	−	0.952506i	$-0.598496\pi$
$349$	−3970.85			−0.609040			−0.304520	+	0.952506i	$0.598496\pi$
$350$	1821.71	+	4635.93i	0.278213	+	0.708002i
$351$		0			0
$352$	193.316	−	334.833i	0.0292720	−	0.0507007i
$353$	−5777.83	−	10007.5i	−0.871169	−	1.50891i	−0.860788	−	0.508964i	$-0.830029\pi$
$353$	−5777.83	−	10007.5i	−0.871169	−	1.50891i	−0.0103815	−	0.999946i	$-0.503305\pi$
$354$		0			0
$355$	656.333	−	1136.80i	0.0981255	−	0.169958i
$356$	4845.65			0.721401
$357$		0			0
$358$	2844.88			0.419991
$359$	−4270.56	+	7396.82i	−0.627831	+	1.08744i	0.360155	+	0.932892i	$0.382724\pi$
$359$	−4270.56	+	7396.82i	−0.627831	+	1.08744i	−0.987986	+	0.154543i	$0.950610\pi$
$360$		0			0
$361$	−4385.70	−	7596.25i	−0.639407	−	1.10749i
$362$	−2494.29	+	4320.24i	−0.362147	+	0.627257i
$363$		0			0
$364$	−1835.12	+	2303.15i	−0.264248	+	0.331642i
$365$	15767.2			2.26108
$366$		0			0
$367$	−1901.32	−	3293.18i	−0.270431	−	0.468400i	0.698541	−	0.715570i	$-0.253833\pi$
$367$	−1901.32	−	3293.18i	−0.270431	−	0.468400i	−0.968972	+	0.247170i	$0.920499\pi$
$368$	−385.939	−	668.467i	−0.0546698	−	0.0946908i
$369$		0			0
$370$	12909.8			1.81392
$371$	−3865.35	+	4851.18i	−0.540914	+	0.678869i
$372$		0			0
$373$	4296.45	−	7441.66i	0.596412	−	1.03302i	−0.396934	−	0.917847i	$-0.629926\pi$
$373$	4296.45	−	7441.66i	0.596412	−	1.03302i	0.993346	−	0.115168i	$-0.0367407\pi$
$374$	−1517.32	−	2628.08i	−0.209783	−	0.363355i
$375$		0			0
$376$	992.859	−	1719.68i	0.136178	−	0.235867i
$377$	−3698.75			−0.505292
$378$		0			0
$379$	−9415.95			−1.27616			−0.638080	−	0.769970i	$-0.720271\pi$
$379$	−9415.95			−1.27616			−0.638080	+	0.769970i	$0.720271\pi$
$380$	−4027.75	+	6976.27i	−0.543735	+	0.941776i
$381$		0			0
$382$	−3693.82	−	6397.88i	−0.494744	−	0.856922i
$383$	−4618.37	+	7999.25i	−0.616156	+	1.06721i	0.374025	+	0.927419i	$0.377978\pi$
$383$	−4618.37	+	7999.25i	−0.616156	+	1.06721i	−0.990181	+	0.139795i	$0.955356\pi$
$384$		0			0
$385$	1318.27	+	3354.76i	0.174507	+	0.444089i
$386$	−154.241			−0.0203384
$387$		0			0
$388$	2673.01	+	4629.78i	0.349746	+	0.605777i
$389$	1003.96	+	1738.90i	0.130855	+	0.226648i	0.924006	−	0.382377i	$-0.124894\pi$
$389$	1003.96	+	1738.90i	0.130855	+	0.226648i	−0.793151	+	0.609025i	$0.791561\pi$
$390$		0			0
$391$	−6058.42			−0.783600
$392$	−2009.93	+	1868.08i	−0.258971	+	0.240695i
$393$		0			0
$394$	1259.96	−	2182.32i	0.161107	−	0.279045i
$395$	−5584.95	−	9673.41i	−0.711416	−	1.23221i
$396$		0			0
$397$	−635.532	+	1100.77i	−0.0803437	+	0.139159i	−0.903398	−	0.428804i	$-0.858935\pi$
$397$	−635.532	+	1100.77i	−0.0803437	+	0.139159i	0.823054	+	0.567963i	$0.192268\pi$
$398$	10901.5			1.37297
$399$		0			0
$400$	2151.59			0.268949
$401$	−7512.71	+	13012.4i	−0.935578	+	1.62047i	−0.161978	+	0.986794i	$0.551787\pi$
$401$	−7512.71	+	13012.4i	−0.935578	+	1.62047i	−0.773600	+	0.633674i	$0.781546\pi$
$402$		0			0
$403$	452.412	+	783.600i	0.0559212	+	0.0968583i
$404$	412.833	−	715.048i	0.0508396	−	0.0880568i
$405$		0			0
$406$	−3408.20	−	512.240i	−0.416616	−	0.0626159i
$407$	4841.61			0.589655
$408$		0			0
$409$	−239.226	−	414.352i	−0.0289217	−	0.0500939i	0.851202	−	0.524838i	$-0.175874\pi$
$409$	−239.226	−	414.352i	−0.0289217	−	0.0500939i	−0.880124	+	0.474744i	$0.842541\pi$
$410$	5703.44	+	9878.66i	0.687007	+	1.18993i
$411$		0			0
$412$	−2753.78			−0.329294
$413$	2110.56	−	2648.83i	0.251462	−	0.315595i
$414$		0			0
$415$	11348.6	−	19656.4i	1.34237	−	2.32505i
$416$	636.028	+	1101.63i	0.0749612	+	0.129837i
$417$		0			0
$418$	−1510.54	+	2616.33i	−0.176753	+	0.306146i
$419$	−11608.0			−1.35343			−0.676715	−	0.736245i	$-0.736597\pi$
$419$	−11608.0			−1.35343			−0.676715	+	0.736245i	$0.736597\pi$
$420$		0			0
$421$	−3833.92			−0.443833			−0.221917	−	0.975066i	$-0.571231\pi$
$421$	−3833.92			−0.443833			−0.221917	+	0.975066i	$0.571231\pi$
$422$	−326.412	+	565.363i	−0.0376529	+	0.0652167i
$423$		0			0
$424$	1339.68	+	2320.40i	0.153445	+	0.265775i
$425$	8443.86	−	14625.2i	0.963735	−	1.66924i
$426$		0			0
$427$	−11168.7	−	1678.61i	−1.26578	−	0.190243i
$428$	−1813.70			−0.204833
$429$		0			0
$430$	3639.13	+	6303.16i	0.408127	+	0.706896i
$431$	−499.024	−	864.334i	−0.0557706	−	0.0965975i	0.836792	−	0.547521i	$-0.184428\pi$
$431$	−499.024	−	864.334i	−0.0557706	−	0.0965975i	−0.892563	+	0.450923i	$0.851095\pi$
$432$		0			0
$433$	1298.17			0.144079			0.0720396	−	0.997402i	$-0.477049\pi$
$433$	1298.17			0.144079			0.0720396	+	0.997402i	$0.477049\pi$
$434$	308.352	+	784.700i	0.0341046	+	0.0867899i
$435$		0			0
$436$	−2741.97	+	4749.23i	−0.301184	+	0.521667i
$437$	3015.67	+	5223.30i	0.330112	+	0.571772i
$438$		0			0
$439$	966.704	−	1674.38i	0.105099	−	0.182036i	−0.808680	−	0.588249i	$-0.799818\pi$
$439$	966.704	−	1674.38i	0.105099	−	0.182036i	0.913778	+	0.406213i	$0.133151\pi$
$440$	1556.99			0.168696
$441$		0			0
$442$	9984.29			1.07444
$443$	7267.01	−	12586.8i	0.779382	−	1.34993i	−0.152917	−	0.988239i	$-0.548867\pi$
$443$	7267.01	−	12586.8i	0.779382	−	1.34993i	0.932299	−	0.361690i	$-0.117800\pi$
$444$		0			0
$445$	9756.84	+	16899.3i	1.03937	+	1.80024i
$446$	3095.25	−	5361.13i	0.328620	−	0.569186i
$447$		0			0
$448$	433.500	+	1103.18i	0.0457165	+	0.116340i
$449$	−4799.65			−0.504476			−0.252238	−	0.967665i	$-0.581167\pi$
$449$	−4799.65			−0.504476			−0.252238	+	0.967665i	$0.581167\pi$
$450$		0			0
$451$	2138.98	+	3704.82i	0.223327	+	0.386814i
$452$	−3615.37	−	6262.01i	−0.376223	−	0.651638i
$453$		0			0
$454$	−1253.43			−0.129573
$455$	−11727.4	−	1762.58i	−1.20832	−	0.181607i
$456$		0			0
$457$	−2367.14	+	4100.00i	−0.242297	+	0.419671i	−0.961368	−	0.275265i	$-0.911234\pi$
$457$	−2367.14	+	4100.00i	−0.242297	+	0.419671i	0.719071	+	0.694937i	$0.244568\pi$
$458$	−6666.50	−	11546.7i	−0.680142	−	1.17804i
$459$		0			0
$460$	1554.20	−	2691.95i	0.157532	−	0.272854i
$461$	−8137.41			−0.822119			−0.411060	−	0.911608i	$-0.634841\pi$
$461$	−8137.41			−0.822119			−0.411060	+	0.911608i	$0.634841\pi$
$462$		0			0
$463$	8671.77			0.870435			0.435218	−	0.900325i	$-0.356671\pi$
$463$	8671.77			0.870435			0.435218	+	0.900325i	$0.356671\pi$
$464$	−744.369	+	1289.28i	−0.0744751	+	0.128995i
$465$		0			0
$466$	−3500.51	−	6063.07i	−0.347979	−	0.602717i
$467$	7794.04	−	13499.7i	0.772303	−	1.33767i	−0.163995	−	0.986461i	$-0.552438\pi$
$467$	7794.04	−	13499.7i	0.772303	−	1.33767i	0.936298	−	0.351206i	$-0.114228\pi$
$468$		0			0
$469$	1366.23	−	1714.68i	0.134513	−	0.168820i
$470$	7996.60			0.784799
$471$		0			0
$472$	−731.492	−	1266.98i	−0.0713340	−	0.123554i
$473$	1364.79	+	2363.89i	0.132671	+	0.229793i
$474$		0			0
$475$	−16812.2			−1.62400
$476$	9199.98	+	1382.73i	0.885883	+	0.133145i
$477$		0			0
$478$	119.098	−	206.284i	0.0113963	−	0.0197389i
$479$	−4767.04	−	8256.76i	−0.454722	−	0.787601i	0.543951	−	0.839117i	$-0.316928\pi$
$479$	−4767.04	−	8256.76i	−0.454722	−	0.787601i	−0.998672	+	0.0515163i	$0.983595\pi$
$480$		0			0
$481$	−7964.69	+	13795.2i	−0.755007	+	1.30771i
$482$	2543.61			0.240370
$483$		0			0
$484$	−4740.08			−0.445161
$485$	−10764.3	+	18644.4i	−1.00780	+	1.74556i
$486$		0			0
$487$	4398.55	+	7618.51i	0.409276	+	0.708887i	0.994809	−	0.101762i	$-0.0324480\pi$
$487$	4398.55	+	7618.51i	0.409276	+	0.708887i	−0.585533	+	0.810649i	$0.699115\pi$
$488$	−2439.29	+	4224.98i	−0.226274	+	0.391918i
$489$		0			0
$490$	−10562.0	−	3248.25i	−0.973764	−	0.299471i
$491$	15749.6			1.44760			0.723799	−	0.690010i	$-0.242394\pi$
$491$	15749.6			1.44760			0.723799	+	0.690010i	$0.242394\pi$
$492$		0			0
$493$	5842.50	+	10119.5i	0.533738	+	0.924461i
$494$	−4969.83	−	8608.00i	−0.452638	−	0.783992i
$495$		0			0
$496$	364.190			0.0329689
$497$	551.971	+	1404.67i	0.0498175	+	0.126776i
$498$		0			0
$499$	6490.60	−	11242.0i	0.582283	−	1.00854i	−0.412926	−	0.910765i	$-0.635493\pi$
$499$	6490.60	−	11242.0i	0.582283	−	1.00854i	0.995208	−	0.0977783i	$-0.0311736\pi$
$500$	305.240	+	528.692i	0.0273015	+	0.0472876i
$501$		0			0
$502$	−330.354	+	572.191i	−0.0293714	+	0.0508727i
$503$	−13047.4			−1.15657			−0.578284	−	0.815836i	$-0.696277\pi$
$503$	−13047.4			−1.15657			−0.578284	+	0.815836i	$0.696277\pi$
$504$		0			0
$505$	3325.00			0.292991
$506$	582.876	−	1009.57i	0.0512095	−	0.0886974i
$507$		0			0
$508$	392.952	+	680.612i	0.0343197	+	0.0594435i
$509$	−381.720	+	661.159i	−0.0332406	+	0.0575744i	−0.882167	−	0.470937i	$-0.843916\pi$
$509$	−381.720	+	661.159i	−0.0332406	+	0.0575744i	0.848927	+	0.528511i	$0.177249\pi$
$510$		0			0
$511$	−11296.8	+	14177.9i	−0.977967	+	1.22739i
$512$	512.000			0.0441942
$513$		0			0
$514$	2520.80	+	4366.15i	0.216319	+	0.374675i
$515$	−5544.82	−	9603.91i	−0.474435	−	0.821745i
$516$		0			0
$517$	2998.99			0.255117
$518$	−9249.54	+	11608.5i	−0.784559	+	0.984653i
$519$		0			0
$520$	−2561.32	+	4436.34i	−0.216003	+	0.374127i
$521$	−1015.38	−	1758.68i	−0.0853829	−	0.147887i	0.820171	−	0.572118i	$-0.193878\pi$
$521$	−1015.38	−	1758.68i	−0.0853829	−	0.147887i	−0.905554	+	0.424230i	$0.860545\pi$
$522$		0			0
$523$	6482.87	−	11228.7i	0.542019	−	0.938805i	−0.456769	−	0.889585i	$-0.650993\pi$
$523$	6482.87	−	11228.7i	0.542019	−	0.938805i	0.998788	−	0.0492192i	$-0.0156733\pi$
$524$	−6025.94			−0.502374
$525$		0			0
$526$	−2854.91			−0.236654
$527$	1429.25	−	2475.53i	0.118139	−	0.204622i
$528$		0			0
$529$	4919.83	+	8521.40i	0.404359	+	0.700370i
$530$	−5394.97	+	9344.36i	−0.442156	+	0.765836i
$531$		0			0
$532$	−3387.31	−	8620.08i	−0.276050	−	0.702496i
$533$	−14074.9			−1.14381
$534$		0			0
$535$	−3651.93	−	6325.34i	−0.295116	−	0.511155i
$536$	−473.519	−	820.158i	−0.0381584	−	0.0660923i
$537$		0			0
$538$	16379.7			1.31260
$539$	−3961.11	−	1218.20i	−0.316544	−	0.0973500i
$540$		0			0
$541$	7812.43	−	13531.5i	0.620855	−	1.07535i	−0.368471	−	0.929639i	$-0.620119\pi$
$541$	7812.43	−	13531.5i	0.620855	−	1.07535i	0.989327	−	0.145714i	$-0.0465479\pi$
$542$	−6394.70	−	11076.0i	−0.506782	−	0.877773i
$543$		0			0
$544$	2009.33	−	3480.26i	0.158362	−	0.274292i
$545$	−22084.1			−1.73574
$546$		0			0
$547$	−14437.8			−1.12855			−0.564273	−	0.825588i	$-0.690843\pi$
$547$	−14437.8			−1.12855			−0.564273	+	0.825588i	$0.690843\pi$
$548$	−630.691	+	1092.39i	−0.0491639	+	0.0851543i
$549$		0			0
$550$	1624.75	+	2814.16i	0.125963	+	0.218175i
$551$	5816.38	−	10074.3i	0.449703	−	0.778908i
$552$		0			0
$553$	12699.8	+	1908.74i	0.976586	+	0.146777i
$554$	−13278.8			−1.01834
$555$		0			0
$556$	649.001	+	1124.10i	0.0495032	+	0.0857421i
$557$	−8242.53	−	14276.5i	−0.627015	−	1.08602i	−0.988148	−	0.153507i	$-0.950943\pi$
$557$	−8242.53	−	14276.5i	−0.627015	−	1.08602i	0.361133	−	0.932514i	$-0.382390\pi$
$558$		0			0
$559$	−8980.63			−0.679499
$560$	−2974.51	+	3733.13i	−0.224457	+	0.281703i
$561$		0			0
$562$	−4929.95	+	8538.92i	−0.370031	+	0.640912i
$563$	8366.91	+	14491.9i	0.626329	+	1.08483i	0.988282	+	0.152637i	$0.0487766\pi$
$563$	8366.91	+	14491.9i	0.626329	+	1.08483i	−0.361954	+	0.932196i	$0.617890\pi$
$564$		0			0
$565$	14559.3	−	25217.5i	1.08410	−	1.87771i
$566$	5565.57			0.413319
$567$		0			0
$568$	651.924			0.0481587
$569$	−7508.37	+	13004.9i	−0.553194	+	0.958160i	0.444848	+	0.895606i	$0.353258\pi$
$569$	−7508.37	+	13004.9i	−0.553194	+	0.958160i	−0.998042	+	0.0625539i	$0.980075\pi$
$570$		0			0
$571$	−3236.25	−	5605.35i	−0.237186	−	0.410817i	0.722720	−	0.691141i	$-0.242892\pi$
$571$	−3236.25	−	5605.35i	−0.237186	−	0.410817i	−0.959906	+	0.280323i	$0.909558\pi$
$572$	−960.580	+	1663.77i	−0.0702166	+	0.121619i
$573$		0			0
$574$	−12969.3	−	1949.24i	−0.943080	−	0.141742i
$575$	6487.38			0.470509
$576$		0			0
$577$	−4955.10	−	8582.48i	−0.357510	−	0.619226i	0.630034	−	0.776567i	$-0.283041\pi$
$577$	−4955.10	−	8582.48i	−0.357510	−	0.619226i	−0.987544	+	0.157342i	$0.949708\pi$
$578$	−10858.1	−	18806.7i	−0.781377	−	1.35338i
$579$		0			0
$580$	−5995.23			−0.429204
$581$	9544.12	+	24288.0i	0.681509	+	1.73432i
$582$		0			0
$583$	−2023.29	+	3504.45i	−0.143733	+	0.248952i
$584$	3915.32	+	6781.54i	0.277427	+	0.480517i
$585$		0			0
$586$	−1639.57	+	2839.81i	−0.115580	+	0.200191i
$587$	1112.26			0.0782079			0.0391039	−	0.999235i	$-0.487550\pi$
$587$	1112.26			0.0782079			0.0391039	+	0.999235i	$0.487550\pi$
$588$		0			0
$589$	−2845.72			−0.199076
$590$	2945.76	−	5102.20i	0.205551	−	0.356024i
$591$		0			0
$592$	3205.77	+	5552.56i	0.222561	+	0.385488i
$593$	−379.191	+	656.778i	−0.0262589	+	0.0454817i	−0.878856	−	0.477087i	$-0.841693\pi$
$593$	−379.191	+	656.778i	−0.0262589	+	0.0454817i	0.852597	+	0.522568i	$0.175026\pi$
$594$		0			0
$595$	13702.1	+	34869.4i	0.944087	+	2.40253i
$596$	−5109.96			−0.351195
$597$		0			0
$598$	1917.72	+	3321.59i	0.131140	+	0.227140i
$599$	−8957.35	−	15514.6i	−0.610997	−	1.05828i	−0.991073	−	0.133323i	$-0.957435\pi$
$599$	−8957.35	−	15514.6i	−0.610997	−	1.05828i	0.380076	−	0.924955i	$-0.375898\pi$
$600$		0			0
$601$	19337.1			1.31244			0.656221	−	0.754569i	$-0.272154\pi$
$601$	19337.1			1.31244			0.656221	+	0.754569i	$0.272154\pi$
$602$	−8275.17	−	1243.73i	−0.560250	−	0.0842037i
$603$		0			0
$604$	−1033.70	+	1790.42i	−0.0696366	+	0.120614i
$605$	−9544.28	−	16531.2i	−0.641372	−	1.11089i
$606$		0			0
$607$	−787.211	+	1363.49i	−0.0526391	+	0.0911736i	−0.891144	−	0.453720i	$-0.850097\pi$
$607$	−787.211	+	1363.49i	−0.0526391	+	0.0911736i	0.838505	+	0.544894i	$0.183430\pi$
$608$	−4000.69			−0.266858
$609$		0			0
$610$	−19646.3			−1.30403
$611$	−4933.49	+	8545.06i	−0.326657	+	0.565787i
$612$		0			0
$613$	2597.21	+	4498.49i	0.171126	+	0.296399i	0.938814	−	0.344425i	$-0.111926\pi$
$613$	2597.21	+	4498.49i	0.171126	+	0.296399i	−0.767688	+	0.640824i	$0.778593\pi$
$614$	10332.5	−	17896.4i	0.679131	−	1.17629i
$615$		0			0
$616$	−1115.54	+	1400.05i	−0.0729649	+	0.0915739i
$617$	1699.80			0.110910			0.0554550	−	0.998461i	$-0.482339\pi$
$617$	1699.80			0.110910			0.0554550	+	0.998461i	$0.482339\pi$
$618$		0			0
$619$	6362.04	+	11019.4i	0.413105	+	0.715518i	0.995227	−	0.0975825i	$-0.0311110\pi$
$619$	6362.04	+	11019.4i	0.413105	+	0.715518i	−0.582123	+	0.813101i	$0.697778\pi$
$620$	733.305	+	1270.12i	0.0475004	+	0.0822731i
$621$		0			0
$622$	−18505.6			−1.19293
$623$	−22186.5	−	3334.55i	−1.42678	−	0.214440i
$624$		0			0
$625$	7175.45	−	12428.2i	0.459229	−	0.795407i
$626$	2477.78	+	4291.63i	0.158198	+	0.274007i
$627$		0			0
$628$	2597.28	−	4498.62i	0.165036	−	0.285851i
$629$	50323.7			3.19004
$630$		0			0
$631$	−14921.6			−0.941392			−0.470696	−	0.882295i	$-0.655997\pi$
$631$	−14921.6			−0.941392			−0.470696	+	0.882295i	$0.655997\pi$
$632$	2773.71	−	4804.21i	0.174577	−	0.302375i
$633$		0			0
$634$	2883.22	+	4993.88i	0.180611	+	0.312827i
$635$	−1582.44	+	2740.86i	−0.0988931	+	0.171288i
$636$		0			0
$637$	9987.27	−	9282.45i	0.621209	−	0.577369i
$638$	−2248.41			−0.139522
$639$		0			0
$640$	1030.93	+	1785.62i	0.0636733	+	0.110285i
$641$	13483.7	+	23354.5i	0.830850	+	1.43907i	0.897365	+	0.441289i	$0.145478\pi$
$641$	13483.7	+	23354.5i	0.830850	+	1.43907i	−0.0665155	+	0.997785i	$0.521188\pi$
$642$		0			0
$643$	12146.2			0.744947			0.372473	−	0.928043i	$-0.378510\pi$
$643$	12146.2			0.744947			0.372473	+	0.928043i	$0.378510\pi$
$644$	1307.07	+	3326.25i	0.0799779	+	0.203529i
$645$		0			0
$646$	−15700.6	+	27194.2i	−0.956239	+	1.65626i
$647$	−13575.9	−	23514.1i	−0.824919	−	1.42880i	−0.901981	−	0.431775i	$-0.857887\pi$
$647$	−13575.9	−	23514.1i	−0.824919	−	1.42880i	0.0770625	−	0.997026i	$-0.475446\pi$
$648$		0			0
$649$	1104.76	−	1913.49i	0.0668189	−	0.115734i
$650$	−10691.2			−0.645144
$651$		0			0
$652$	11896.2			0.714556
$653$	−6320.52	+	10947.5i	−0.378776	+	0.656060i	−0.990884	−	0.134714i	$-0.956988\pi$
$653$	−6320.52	+	10947.5i	−0.378776	+	0.656060i	0.612108	+	0.790774i	$0.290322\pi$
$654$		0			0
$655$	−12133.4	−	21015.6i	−0.723802	−	1.25366i
$656$	−2832.56	+	4906.14i	−0.168587	+	0.292001i
$657$		0			0
$658$	−5729.35	+	7190.57i	−0.339443	+	0.426015i
$659$	11360.0			0.671506			0.335753	−	0.941950i	$-0.391009\pi$
$659$	11360.0			0.671506			0.335753	+	0.941950i	$0.391009\pi$
$660$		0			0
$661$	−2078.99	−	3600.91i	−0.122335	−	0.211890i	0.798353	−	0.602189i	$-0.205705\pi$
$661$	−2078.99	−	3600.91i	−0.122335	−	0.211890i	−0.920688	+	0.390300i	$0.872371\pi$
$662$	497.131	+	861.056i	0.0291866	+	0.0505527i
$663$		0			0
$664$	11272.4			0.658816
$665$	23242.4	−	29170.1i	1.35534	−	1.70100i
$666$		0			0
$667$	−2244.38	+	3887.39i	−0.130289	+	0.225668i
$668$	7214.44	+	12495.8i	0.417867	+	0.723766i
$669$		0			0
$670$	1906.89	−	3302.82i	0.109954	−	0.190447i
$671$	−7368.03			−0.423904
$672$		0			0
$673$	8753.81			0.501389			0.250694	−	0.968066i	$-0.419341\pi$
$673$	8753.81			0.501389			0.250694	+	0.968066i	$0.419341\pi$
$674$	−6711.07	+	11623.9i	−0.383532	+	0.664297i
$675$		0			0
$676$	1233.59	+	2136.65i	0.0701862	+	0.121566i
$677$	5802.61	−	10050.4i	0.329413	−	0.570560i	−0.652983	−	0.757373i	$-0.726483\pi$
$677$	5802.61	−	10050.4i	0.329413	−	0.570560i	0.982395	+	0.186813i	$0.0598158\pi$
$678$		0			0
$679$	−9052.73	−	23037.6i	−0.511652	−	1.30206i
$680$	16183.3			0.912651
$681$		0			0
$682$	275.014	+	476.338i	0.0154411	+	0.0267448i
$683$	9199.15	+	15933.4i	0.515367	+	0.892642i	0.999841	+	0.0178364i	$0.00567781\pi$
$683$	9199.15	+	15933.4i	0.515367	+	0.892642i	−0.484474	+	0.874806i	$0.660989\pi$
$684$		0			0
$685$	−5079.66			−0.283334
$686$	10488.3	−	7170.13i	0.583737	−	0.399063i
$687$		0			0
$688$	−1807.34	+	3130.41i	−0.100152	+	0.173468i
$689$	−6656.84	−	11530.0i	−0.368077	−	0.637529i
$690$		0			0
$691$	20.7919	−	36.0127i	0.00114466	−	0.00198261i	−0.865453	−	0.500991i	$-0.832969\pi$
$691$	20.7919	−	36.0127i	0.00114466	−	0.00198261i	0.866597	+	0.499008i	$0.166302\pi$
$692$	10105.6			0.555143
$693$		0			0
$694$	323.252			0.0176808
$695$	−2613.56	+	4526.83i	−0.142645	+	0.247068i
$696$		0			0
$697$	22232.6	+	38508.0i	1.20821	+	2.09267i
$698$	3970.85	−	6877.72i	0.215328	−	0.372959i
$699$		0			0
$700$	−9851.38	−	1480.63i	−0.531924	−	0.0799464i
$701$	−29655.0			−1.59779			−0.798896	−	0.601469i	$-0.794582\pi$
$701$	−29655.0			−1.59779			−0.798896	+	0.601469i	$0.794582\pi$
$702$		0			0
$703$	−25049.4	−	43386.8i	−1.34389	−	2.32769i
$704$	386.631	+	669.665i	0.0206985	+	0.0358508i
$705$		0			0
$706$	23111.3			1.23202
$707$	−2382.28	+	2989.85i	−0.126725	+	0.159045i
$708$		0			0
$709$	4375.23	−	7578.13i	0.231757	−	0.401414i	−0.726569	−	0.687094i	$-0.758886\pi$
$709$	4375.23	−	7578.13i	0.231757	−	0.401414i	0.958325	+	0.285680i	$0.0922194\pi$
$710$	1312.67	+	2273.60i	0.0693852	+	0.120179i
$711$		0			0
$712$	−4845.65	+	8392.91i	−0.255054	+	0.441766i
$713$	1098.09			0.0576769
$714$		0			0
$715$	−7736.62			−0.404662
$716$	−2844.88	+	4927.48i	−0.148489	+	0.257191i
$717$		0			0
$718$	−8541.11	−	14793.6i	−0.443944	−	0.768933i
$719$	−9541.72	+	16526.8i	−0.494918	+	0.857224i	−0.999983	−	0.00585789i	$-0.998135\pi$
$719$	−9541.72	+	16526.8i	−0.494918	+	0.857224i	0.505065	+	0.863082i	$0.331469\pi$
$720$		0			0
$721$	12608.6	+	1895.03i	0.651274	+	0.0978842i
$722$	17542.8			0.904259
$723$		0			0
$724$	−4988.59	−	8640.49i	−0.256076	−	0.443537i
$725$	−6256.17	−	10836.0i	−0.320480	−	0.555088i
$726$		0			0
$727$	15050.9			0.767821			0.383910	−	0.923370i	$-0.374577\pi$
$727$	15050.9			0.767821			0.383910	+	0.923370i	$0.374577\pi$
$728$	−2154.05	−	5481.67i	−0.109663	−	0.279072i
$729$		0			0
$730$	−15767.2	+	27309.6i	−0.799412	+	1.38462i
$731$	14185.7	+	24570.4i	0.717752	+	1.24318i
$732$		0			0
$733$	−5279.02	+	9143.53i	−0.266010	+	0.460742i	−0.967828	−	0.251614i	$-0.919039\pi$
$733$	−5279.02	+	9143.53i	−0.266010	+	0.460742i	0.701818	+	0.712356i	$0.252372\pi$
$734$	7605.28			0.382447
$735$		0			0
$736$	1543.76			0.0773148
$737$	715.145	−	1238.67i	0.0357432	−	0.0619090i
$738$		0			0
$739$	16623.4	+	28792.5i	0.827470	+	1.43322i	0.900017	+	0.435856i	$0.143554\pi$
$739$	16623.4	+	28792.5i	0.827470	+	1.43322i	−0.0725462	+	0.997365i	$0.523112\pi$
$740$	−12909.8	+	22360.4i	−0.641316	+	1.11079i
$741$		0			0
$742$	−4537.13	−	11546.2i	−0.224479	−	0.571258i
$743$	32799.7			1.61952			0.809761	−	0.586760i	$-0.199597\pi$
$743$	32799.7			1.61952			0.809761	+	0.586760i	$0.199597\pi$
$744$		0			0
$745$	−10289.0	−	17821.1i	−0.505988	−	0.876397i
$746$	8592.89	+	14883.3i	0.421727	+	0.730452i
$747$		0			0
$748$	6069.29			0.296678
$749$	8304.28	+	1248.10i	0.405116	+	0.0608875i
$750$		0			0
$751$	12113.3	−	20980.8i	0.588574	−	1.01944i	−0.405845	−	0.913942i	$-0.633023\pi$
$751$	12113.3	−	20980.8i	0.588574	−	1.01944i	0.994419	−	0.105499i	$-0.0336439\pi$
$752$	1985.72	+	3439.37i	0.0962922	+	0.166783i
$753$		0			0
$754$	3698.75	−	6406.42i	0.178648	−	0.309427i
$755$	−8325.51			−0.401319
$756$		0			0
$757$	−3116.97			−0.149654			−0.0748271	−	0.997197i	$-0.523840\pi$
$757$	−3116.97			−0.149654			−0.0748271	+	0.997197i	$0.523840\pi$
$758$	9415.95	−	16308.9i	0.451191	−	0.781485i
$759$		0			0
$760$	−8055.50	−	13952.5i	−0.384478	−	0.665936i
$761$	−4355.33	+	7543.65i	−0.207464	+	0.359339i	−0.950915	−	0.309452i	$-0.899854\pi$
$761$	−4355.33	+	7543.65i	−0.207464	+	0.359339i	0.743451	+	0.668791i	$0.233188\pi$
$762$		0			0
$763$	15822.7	−	19858.1i	0.750746	−	0.942217i
$764$	14775.3			0.699674
$765$		0			0
$766$	−9236.74	−	15998.5i	−0.435688	−	0.754634i
$767$	3634.76	+	6295.59i	0.171113	+	0.296376i
$768$		0			0
$769$	−14498.2			−0.679868			−0.339934	−	0.940449i	$-0.610405\pi$
$769$	−14498.2			−0.679868			−0.339934	+	0.940449i	$0.610405\pi$
$770$	−7128.88	−	1071.45i	−0.333645	−	0.0501457i
$771$		0			0
$772$	154.241	−	267.153i	0.00719073	−	0.0124547i
$773$	−7418.91	−	12849.9i	−0.345200	−	0.597904i	0.640190	−	0.768217i	$-0.278856\pi$
$773$	−7418.91	−	12849.9i	−0.345200	−	0.597904i	−0.985390	+	0.170312i	$0.945522\pi$
$774$		0			0
$775$	−1530.45	+	2650.81i	−0.0709358	+	0.122864i
$776$	−10692.0			−0.494615
$777$		0			0
$778$	−4015.82			−0.185057
$779$	22133.2	−	38335.9i	1.01798	−	1.76319i
$780$		0			0
$781$	492.293	+	852.677i	0.0225552	+	0.0390668i
$782$	6058.42	−	10493.5i	0.277044	−	0.479855i
$783$		0			0
$784$	−1225.69	−	5349.38i	−0.0558348	−	0.243685i
$785$	20918.8			0.951112
$786$		0			0
$787$	−22013.0	−	38127.6i	−0.997050	−	1.72694i	−0.564991	−	0.825097i	$-0.691120\pi$
$787$	−22013.0	−	38127.6i	−0.997050	−	1.72694i	−0.432059	−	0.901845i	$-0.642213\pi$
$788$	2519.93	+	4364.64i	0.113920	+	0.197315i
$789$		0			0
$790$	22339.8			1.00609
$791$	12244.3	+	31159.4i	0.550387	+	1.40063i
$792$		0			0
$793$	12120.8	−	20993.8i	0.542776	−	0.940116i
$794$	−1271.06	−	2201.55i	−0.0568116	−	0.0984005i
$795$		0			0
$796$	−10901.5	+	18881.9i	−0.485418	+	0.840769i
$797$	14499.6			0.644419			0.322209	−	0.946668i	$-0.395574\pi$
$797$	14499.6			0.644419			0.322209	+	0.946668i	$0.395574\pi$
$798$		0			0
$799$	31171.5			1.38019
$800$	−2151.59	+	3726.67i	−0.0950880	+	0.164697i
$801$		0			0
$802$	−15025.4	−	26024.8i	−0.661553	−	1.14584i
$803$	−5913.23	+	10242.0i	−0.259867	+	0.450103i
$804$		0			0
$805$	−8968.59	+	11255.9i	−0.392673	+	0.492820i
$806$	−1809.65			−0.0790845
$807$		0			0
$808$	825.666	+	1430.10i	0.0359490	+	0.0622656i
$809$	8965.81	+	15529.2i	0.389643	+	0.674881i	0.992401	−	0.123042i	$-0.0392651\pi$
$809$	8965.81	+	15529.2i	0.389643	+	0.674881i	−0.602758	+	0.797924i	$0.705932\pi$
$810$		0			0
$811$	7961.18			0.344704			0.172352	−	0.985035i	$-0.444863\pi$
$811$	7961.18			0.344704			0.172352	+	0.985035i	$0.444863\pi$
$812$	4295.42	−	5390.93i	0.185640	−	0.232986i
$813$		0			0
$814$	−4841.61	+	8385.91i	−0.208474	+	0.361088i
$815$	23953.3	+	41488.3i	1.02951	+	1.78316i
$816$		0			0
$817$	14122.3	−	24460.5i	0.604745	−	1.04745i
$818$	956.906			0.0409015
$819$		0			0
$820$	−22813.8			−0.971575
$821$	3354.32	−	5809.86i	0.142590	−	0.246974i	−0.785881	−	0.618378i	$-0.787790\pi$
$821$	3354.32	−	5809.86i	0.142590	−	0.246974i	0.928471	+	0.371404i	$0.121123\pi$
$822$		0			0
$823$	279.982	+	484.944i	0.0118585	+	0.0205396i	0.871894	−	0.489695i	$-0.162892\pi$
$823$	279.982	+	484.944i	0.0118585	+	0.0205396i	−0.860035	+	0.510235i	$0.829559\pi$
$824$	2753.78	−	4769.70i	0.116423	−	0.201651i
$825$		0			0
$826$	2477.36	+	6304.43i	0.104356	+	0.265568i
$827$	−41169.0			−1.73106			−0.865530	−	0.500858i	$-0.833018\pi$
$827$	−41169.0			−1.73106			−0.865530	+	0.500858i	$0.833018\pi$
$828$		0			0
$829$	2086.87	+	3614.57i	0.0874308	+	0.151435i	0.906424	−	0.422368i	$-0.138801\pi$
$829$	2086.87	+	3614.57i	0.0874308	+	0.151435i	−0.818994	+	0.573803i	$0.805468\pi$
$830$	22697.3	+	39312.8i	0.949198	+	1.64406i
$831$		0			0
$832$	−2544.11			−0.106011
$833$	−41171.9	−	12662.0i	−1.71251	−	0.526666i
$834$		0			0
$835$	−29052.9	+	50321.2i	−1.20409	+	2.08555i
$836$	−3021.08	−	5232.66i	−0.124983	−	0.216478i
$837$		0			0
$838$	11608.0	−	20105.6i	0.478510	−	0.828803i
$839$	9010.93			0.370789			0.185394	−	0.982664i	$-0.440644\pi$
$839$	9010.93			0.370789			0.185394	+	0.982664i	$0.440644\pi$
$840$		0			0
$841$	−15731.4			−0.645021
$842$	3833.92	−	6640.55i	0.156919	−	0.271791i
$843$		0			0
$844$	−652.825	−	1130.73i	−0.0266246	−	0.0461152i
$845$	−4967.75	+	8604.39i	−0.202243	+	0.350296i
$846$		0			0
$847$	21703.1	+	3261.90i	0.880434	+	0.132326i
$848$	−5358.73			−0.217004
$849$		0			0
$850$	16887.7	+	29250.4i	0.681463	+	1.18033i
$851$	9665.88	+	16741.8i	0.389356	+	0.674385i
$852$		0			0
$853$	−24474.7			−0.982411			−0.491205	−	0.871044i	$-0.663444\pi$
$853$	−24474.7			−0.982411			−0.491205	+	0.871044i	$0.663444\pi$
$854$	14076.1	−	17666.1i	0.564021	−	0.707869i
$855$		0			0
$856$	1813.70	−	3141.42i	0.0724194	−	0.125434i
$857$	14742.4	+	25534.5i	0.587619	+	1.01779i	0.994543	+	0.104324i	$0.0332678\pi$
$857$	14742.4	+	25534.5i	0.587619	+	1.01779i	−0.406925	+	0.913462i	$0.633399\pi$
$858$		0			0
$859$	−3986.84	+	6905.41i	−0.158358	+	0.274284i	−0.934277	−	0.356549i	$-0.883953\pi$
$859$	−3986.84	+	6905.41i	−0.158358	+	0.274284i	0.775919	+	0.630833i	$0.217287\pi$
$860$	−14556.5			−0.577178
$861$		0			0
$862$	1996.09			0.0788715
$863$	13792.0	−	23888.4i	0.544015	−	0.942261i	−0.454653	−	0.890668i	$-0.650237\pi$
$863$	13792.0	−	23888.4i	0.544015	−	0.942261i	0.998668	−	0.0515928i	$-0.0164298\pi$
$864$		0			0
$865$	20348.0	+	35243.7i	0.799829	+	1.38535i
$866$	−1298.17	+	2248.50i	−0.0509397	+	0.0882301i
$867$		0			0
$868$	−1667.49	−	250.618i	−0.0652055	−	0.00980017i
$869$	8378.16			0.327054
$870$		0			0
$871$	2352.90	+	4075.34i	0.0915327	+	0.158539i
$872$	−5483.93	−	9498.45i	−0.212970	−	0.368874i
$873$		0			0
$874$	−12062.7			−0.466849
$875$	−1033.76	−	2630.74i	−0.0399401	−	0.101640i
$876$		0			0
$877$	−7078.94	+	12261.1i	−0.272564	+	0.472095i	−0.969518	−	0.245021i	$-0.921205\pi$
$877$	−7078.94	+	12261.1i	−0.272564	+	0.472095i	0.696954	+	0.717116i	$0.254538\pi$
$878$	1933.41	+	3348.76i	0.0743159	+	0.128719i
$879$		0			0
$880$	−1556.99	+	2696.78i	−0.0596431	+	0.103305i
$881$	31107.6			1.18960			0.594802	−	0.803873i	$-0.297231\pi$
$881$	31107.6			1.18960			0.594802	+	0.803873i	$0.297231\pi$
$882$		0			0
$883$	9141.04			0.348381			0.174190	−	0.984712i	$-0.444269\pi$
$883$	9141.04			0.348381			0.174190	+	0.984712i	$0.444269\pi$
$884$	−9984.29	+	17293.3i	−0.379873	+	0.657960i
$885$		0			0
$886$	14534.0	+	25173.7i	0.551106	+	0.954544i
$887$	17902.3	−	31007.8i	0.677680	−	1.17378i	−0.297998	−	0.954567i	$-0.596319\pi$
$887$	17902.3	−	31007.8i	0.677680	−	1.17378i	0.975678	−	0.219209i	$-0.0703478\pi$
$888$		0			0
$889$	−1330.82	−	3386.69i	−0.0502072	−	0.127768i
$890$	−39027.4			−1.46989
$891$		0			0
$892$	6190.50	+	10722.3i	0.232369	+	0.402475i
$893$	−15516.1	−	26874.7i	−0.581441	−	1.00709i
$894$		0			0
$895$	−22913.0			−0.855751
$896$	−2344.26	−	352.335i	−0.0874066	−	0.0131369i
$897$		0			0
$898$	4799.65	−	8313.24i	0.178359	−	0.308927i
$899$	−1058.95	−	1834.16i	−0.0392858	−	0.0680451i
$900$		0			0
$901$	−21030.1	+	36425.3i	−0.777597	+	1.34684i
$902$	−8555.92			−0.315833
$903$		0			0
$904$	14461.5			0.532060
$905$	20089.3	−	34795.7i	0.737891	−	1.27806i
$906$		0			0
$907$	11155.9	+	19322.6i	0.408407	+	0.707382i	0.994711	−	0.102709i	$-0.0327511\pi$
$907$	11155.9	+	19322.6i	0.408407	+	0.707382i	−0.586304	+	0.810091i	$0.699418\pi$
$908$	1253.43	−	2171.00i	0.0458110	−	0.0793470i
$909$		0			0
$910$	14780.2	−	18549.8i	0.538418	−	0.675737i
$911$	11553.9			0.420195			0.210097	−	0.977680i	$-0.432622\pi$
$911$	11553.9			0.420195			0.210097	+	0.977680i	$0.432622\pi$
$912$		0			0
$913$	8512.23	+	14743.6i	0.308558	+	0.534439i
$914$	−4734.27	−	8200.00i	−0.171330	−	0.296753i
$915$		0			0
$916$	26666.0			0.961867
$917$	27590.6	+	4146.77i	0.993589	+	0.149333i
$918$		0			0
$919$	13687.1	−	23706.8i	0.491291	−	0.850942i	−0.508658	−	0.860968i	$-0.669858\pi$
$919$	13687.1	−	23706.8i	0.491291	−	0.850942i	0.999950	+	0.0100269i	$0.00319171\pi$
$920$	3108.40	+	5383.90i	0.111392	+	0.192937i
$921$		0			0
$922$	8137.41	−	14094.4i	0.290663	−	0.503443i
$923$	−3239.39			−0.115521
$924$		0			0
$925$	−53886.8			−1.91545
$926$	−8671.77	+	15020.0i	−0.307745	+	0.533031i
$927$		0			0
$928$	−1488.74	−	2578.57i	−0.0526618	−	0.0912130i
$929$	−18587.1	+	32193.7i	−0.656428	+	1.13697i	0.325105	+	0.945678i	$0.394600\pi$
$929$	−18587.1	+	32193.7i	−0.656428	+	1.13697i	−0.981534	+	0.191289i	$0.938733\pi$
$930$		0			0
$931$	9577.32	+	41799.2i	0.337147	+	1.47144i
$932$	14002.1			0.492117
$933$		0			0
$934$	15588.1	+	26999.4i	0.546100	+	0.945874i
$935$	12220.7	+	21166.8i	0.427442	+	0.740352i
$936$		0			0
$937$	603.266			0.0210329			0.0105165	−	0.999945i	$-0.496652\pi$
$937$	603.266			0.0210329			0.0105165	+	0.999945i	$0.496652\pi$
$938$	1603.68	+	4081.06i	0.0558229	+	0.142059i
$939$		0			0
$940$	−7996.60	+	13850.5i	−0.277468	+	0.480589i
$941$	8075.89	+	13987.9i	0.279773	+	0.484581i	0.971328	−	0.237742i	$-0.0764074\pi$
$941$	8075.89	+	13987.9i	0.279773	+	0.484581i	−0.691555	+	0.722324i	$0.743074\pi$
$942$		0			0
$943$	−8540.61	+	14792.8i	−0.294932	+	0.510837i
$944$	2925.97			0.100881
$945$		0			0
$946$	−5459.18			−0.187625
$947$	−8506.64	+	14733.9i	−0.291899	+	0.505585i	−0.974259	−	0.225432i	$-0.927621\pi$
$947$	−8506.64	+	14733.9i	−0.291899	+	0.505585i	0.682359	+	0.731017i	$0.260954\pi$
$948$		0			0
$949$	−19455.1	−	33697.3i	−0.665480	−	1.15264i
$950$	16812.2	−	29119.6i	0.574169	−	0.994491i
$951$		0			0
$952$	−11594.9	+	14552.1i	−0.394742	+	0.495417i
$953$	5233.67			0.177896			0.0889482	−	0.996036i	$-0.471649\pi$
$953$	5233.67			0.177896			0.0889482	+	0.996036i	$0.471649\pi$
$954$		0			0
$955$	29750.4	+	51529.2i	1.00806	+	1.74602i
$956$	238.196	+	412.568i	0.00805838	+	0.0139575i
$957$		0			0
$958$	19068.2			0.643074
$959$	3639.44	−	4567.65i	0.122548	−	0.153803i
$960$		0			0
$961$	14636.4	−	25351.1i	0.491304	−	0.850964i
$962$	−15929.4	−	27590.5i	−0.533871	−	0.924691i
$963$		0			0
$964$	−2543.61	+	4405.66i	−0.0849836	+	0.147196i
$965$	1242.27			0.0414405
$966$		0			0
$967$	16834.4			0.559832			0.279916	−	0.960024i	$-0.409693\pi$
$967$	16834.4			0.559832			0.279916	+	0.960024i	$0.409693\pi$
$968$	4740.08	−	8210.06i	0.157388	−	0.272605i
$969$		0			0
$970$	−21528.7	−	37288.8i	−0.712623	−	1.23430i
$971$	−26609.2	+	46088.5i	−0.879434	+	1.52322i	−0.0274708	+	0.999623i	$0.508745\pi$
$971$	−26609.2	+	46088.5i	−0.879434	+	1.52322i	−0.851963	+	0.523602i	$0.824588\pi$
$972$		0			0
$973$	−2197.99	−	5593.48i	−0.0724196	−	0.184295i
$974$	−17594.2			−0.578803
$975$		0			0
$976$	−4878.59	−	8449.96i	−0.160000	−	0.277128i
$977$	−27579.9	−	47769.9i	−0.903133	−	1.56427i	−0.823405	−	0.567455i	$-0.807928\pi$
$977$	−27579.9	−	47769.9i	−0.903133	−	1.56427i	−0.0797279	−	0.996817i	$-0.525405\pi$
$978$		0			0
$979$	−14636.6			−0.477821
$980$	16188.2	−	15045.7i	0.527665	−	0.490427i
$981$		0			0
$982$	−15749.6	+	27279.2i	−0.511804	+	0.886470i
$983$	25566.4	+	44282.2i	0.829542	+	1.43681i	0.898398	+	0.439183i	$0.144732\pi$
$983$	25566.4	+	44282.2i	0.829542	+	1.43681i	−0.0688555	+	0.997627i	$0.521935\pi$
$984$		0			0
$985$	−10147.9	+	17576.6i	−0.328262	+	0.568567i
$986$	−23370.0			−0.754820
$987$		0			0
$988$	19879.3			0.640127
$989$	−5449.41	+	9438.65i	−0.175208	+	0.303470i
$990$		0			0
$991$	−18879.1	−	32699.6i	−0.605161	−	1.04817i	−0.992026	−	0.126033i	$-0.959776\pi$
$991$	−18879.1	−	32699.6i	−0.605161	−	1.04817i	0.386865	−	0.922136i	$-0.373558\pi$
$992$	−364.190	+	630.795i	−0.0116563	+	0.0201893i
$993$		0			0
$994$	−2984.92	−	448.624i	−0.0952475	−	0.0143154i
$995$	−87801.8			−2.79749
$996$		0			0
$997$	12028.9	+	20834.6i	0.382105	+	0.661825i	0.991363	−	0.131147i	$-0.0418660\pi$
$997$	12028.9	+	20834.6i	0.382105	+	0.661825i	−0.609258	+	0.792972i	$0.708533\pi$
$998$	12981.2	+	22484.1i	0.411736	+	0.713148i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	378.4.g.c.163.4	yes	8
3.2	odd	2		378.4.g.f.163.1	yes	8
7.4	even	3	inner	378.4.g.c.109.4	✓	8
21.11	odd	6		378.4.g.f.109.1	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
378.4.g.c.109.4	✓	8	7.4	even	3	inner
378.4.g.c.163.4	yes	8	1.1	even	1	trivial
378.4.g.f.109.1	yes	8	21.11	odd	6
378.4.g.f.163.1	yes	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:	\( N \)	\(=\)	\( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	378.g (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.00000 + 1.73205i) q^{2} +(-2.00000 - 3.46410i) q^{4} +(8.05411 - 13.9501i) q^{5} +(6.77345 + 17.2372i) q^{7} +8.00000 q^{8} +O(q^{10})\)	\(q+(-1.00000 + 1.73205i) q^{2} +(-2.00000 - 3.46410i) q^{4} +(8.05411 - 13.9501i) q^{5} +(6.77345 + 17.2372i) q^{7} +8.00000 q^{8} +(16.1082 + 27.9003i) q^{10} +(6.04111 + 10.4635i) q^{11} -39.7518 q^{13} +(-36.6291 - 5.50523i) q^{14} +(-8.00000 + 13.8564i) q^{16} +(62.7914 + 108.758i) q^{17} +(62.5108 - 108.272i) q^{19} -64.4329 q^{20} -24.1645 q^{22} +(-24.1212 + 41.7792i) q^{23} +(-67.2373 - 116.458i) q^{25} +(39.7518 - 68.8521i) q^{26} +(46.1645 - 57.9383i) q^{28} +93.0461 q^{29} +(-11.3809 - 19.7123i) q^{31} +(-16.0000 - 27.7128i) q^{32} -251.166 q^{34} +(295.015 + 44.3397i) q^{35} +(200.361 - 347.035i) q^{37} +(125.022 + 216.544i) q^{38} +(64.4329 - 111.601i) q^{40} +354.070 q^{41} +225.918 q^{43} +(24.1645 - 41.8541i) q^{44} +(-48.2424 - 83.5583i) q^{46} +(124.107 - 214.960i) q^{47} +(-251.241 + 233.510i) q^{49} +268.949 q^{50} +(79.5036 + 137.704i) q^{52} +(167.460 + 290.050i) q^{53} +194.623 q^{55} +(54.1876 + 137.897i) q^{56} +(-93.0461 + 161.161i) q^{58} +(-91.4364 - 158.373i) q^{59} +(-304.912 + 528.123i) q^{61} +45.5237 q^{62} +64.0000 q^{64} +(-320.165 + 554.542i) q^{65} +(-59.1898 - 102.520i) q^{67} +(251.166 - 435.032i) q^{68} +(-371.814 + 466.641i) q^{70} +81.4905 q^{71} +(489.415 + 847.692i) q^{73} +(400.721 + 694.069i) q^{74} -500.086 q^{76} +(-139.442 + 175.006i) q^{77} +(346.714 - 600.527i) q^{79} +(128.866 + 223.202i) q^{80} +(-354.070 + 613.268i) q^{82} +1409.05 q^{83} +2022.92 q^{85} +(-225.918 + 391.301i) q^{86} +(48.3289 + 83.7081i) q^{88} +(-605.706 + 1049.11i) q^{89} +(-269.256 - 685.209i) q^{91} +192.970 q^{92} +(248.215 + 429.921i) q^{94} +(-1006.94 - 1744.07i) q^{95} -1336.50 q^{97} +(-153.211 - 668.672i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{2} - 16 q^{4} + 2 q^{5} + 6 q^{7} + 64 q^{8}+O(q^{10}) \) 8 * q - 8 * q^2 - 16 * q^4 + 2 * q^5 + 6 * q^7 + 64 * q^8	\( 8 q - 8 q^{2} - 16 q^{4} + 2 q^{5} + 6 q^{7} + 64 q^{8} + 4 q^{10} - 32 q^{11} - 4 q^{13} - 36 q^{14} - 64 q^{16} + 58 q^{17} + 70 q^{19} - 16 q^{20} + 128 q^{22} - 86 q^{23} - 156 q^{25} + 4 q^{26} + 48 q^{28} + 212 q^{29} - 64 q^{31} - 128 q^{32} - 232 q^{34} - 8 q^{35} - 146 q^{37} + 140 q^{38} + 16 q^{40} + 780 q^{41} + 880 q^{43} - 128 q^{44} - 172 q^{46} + 306 q^{47} + 50 q^{49} + 624 q^{50} + 8 q^{52} + 90 q^{53} - 64 q^{55} + 48 q^{56} - 212 q^{58} - 148 q^{59} - 364 q^{61} + 256 q^{62} + 512 q^{64} - 1296 q^{65} - 954 q^{67} + 232 q^{68} + 20 q^{70} + 1360 q^{71} - 54 q^{73} - 292 q^{74} - 560 q^{76} - 2224 q^{77} - 226 q^{79} + 32 q^{80} - 780 q^{82} + 3136 q^{83} + 3920 q^{85} - 880 q^{86} - 256 q^{88} + 1458 q^{89} + 3836 q^{91} + 688 q^{92} + 612 q^{94} - 1310 q^{95} - 4344 q^{97} - 776 q^{98}+O(q^{100}) \) 8 * q - 8 * q^2 - 16 * q^4 + 2 * q^5 + 6 * q^7 + 64 * q^8 + 4 * q^10 - 32 * q^11 - 4 * q^13 - 36 * q^14 - 64 * q^16 + 58 * q^17 + 70 * q^19 - 16 * q^20 + 128 * q^22 - 86 * q^23 - 156 * q^25 + 4 * q^26 + 48 * q^28 + 212 * q^29 - 64 * q^31 - 128 * q^32 - 232 * q^34 - 8 * q^35 - 146 * q^37 + 140 * q^38 + 16 * q^40 + 780 * q^41 + 880 * q^43 - 128 * q^44 - 172 * q^46 + 306 * q^47 + 50 * q^49 + 624 * q^50 + 8 * q^52 + 90 * q^53 - 64 * q^55 + 48 * q^56 - 212 * q^58 - 148 * q^59 - 364 * q^61 + 256 * q^62 + 512 * q^64 - 1296 * q^65 - 954 * q^67 + 232 * q^68 + 20 * q^70 + 1360 * q^71 - 54 * q^73 - 292 * q^74 - 560 * q^76 - 2224 * q^77 - 226 * q^79 + 32 * q^80 - 780 * q^82 + 3136 * q^83 + 3920 * q^85 - 880 * q^86 - 256 * q^88 + 1458 * q^89 + 3836 * q^91 + 688 * q^92 + 612 * q^94 - 1310 * q^95 - 4344 * q^97 - 776 * q^98

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