LMFDB - Embedded newform 108.5.k.a.65.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [108,5,Mod(5,108)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(108, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([0, 5]))

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

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

chi := DirichletCharacter("108.5");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 108 = 2^{2} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	108.k (of order $18$, degree $6$, 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:	$11.1639560131$
Analytic rank:	$0$
Dimension:	$72$
Relative dimension:	$12$ over $\Q(\zeta_{18})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			65.1
Character	$\chi$	$=$	108.65
Dual form			108.5.k.a.5.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-8.89497 + 1.37096i) q^{3} +(-1.59659 + 4.38660i) q^{5} +(12.9775 - 73.5989i) q^{7} +(77.2410 - 24.3892i) q^{9} +O(q^{10})$	\(q+(-8.89497 + 1.37096i) q^{3} +(-1.59659 + 4.38660i) q^{5} +(12.9775 - 73.5989i) q^{7} +(77.2410 - 24.3892i) q^{9} +(54.6352 + 150.109i) q^{11} +(-161.958 + 135.899i) q^{13} +(8.18781 - 41.2076i) q^{15} +(-295.081 + 170.365i) q^{17} +(-169.557 + 293.681i) q^{19} +(-14.5334 + 672.451i) q^{21} +(946.685 - 166.926i) q^{23} +(462.085 + 387.735i) q^{25} +(-653.619 + 322.835i) q^{27} +(-502.780 + 599.189i) q^{29} +(226.590 + 1285.06i) q^{31} +(-691.772 - 1260.31i) q^{33} +(302.129 + 174.434i) q^{35} +(-611.487 - 1059.13i) q^{37} +(1254.30 - 1430.85i) q^{39} +(-11.6088 - 13.8349i) q^{41} +(-1615.74 + 588.079i) q^{43} +(-16.3366 + 377.765i) q^{45} +(3753.45 + 661.834i) q^{47} +(-2992.18 - 1089.06i) q^{49} +(2391.18 - 1919.94i) q^{51} -850.088i q^{53} -745.699 q^{55} +(1105.58 - 2844.73i) q^{57} +(-1077.21 + 2959.61i) q^{59} +(-630.093 + 3573.43i) q^{61} +(-792.627 - 6001.36i) q^{63} +(-337.554 - 927.421i) q^{65} +(3089.43 - 2592.34i) q^{67} +(-8191.89 + 2782.67i) q^{69} +(3906.92 - 2255.66i) q^{71} +(-2203.63 + 3816.80i) q^{73} +(-4641.80 - 2815.39i) q^{75} +(11756.9 - 2073.06i) q^{77} +(-6846.72 - 5745.08i) q^{79} +(5371.33 - 3767.69i) q^{81} +(-715.401 + 852.581i) q^{83} +(-276.200 - 1566.41i) q^{85} +(3650.75 - 6019.06i) q^{87} +(4214.90 + 2433.47i) q^{89} +(7900.21 + 13683.6i) q^{91} +(-3777.27 - 11119.9i) q^{93} +(-1017.55 - 1212.67i) q^{95} +(-12919.2 + 4702.22i) q^{97} +(7881.12 + 10262.1i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 72 q + 9 q^{5} - 102 q^{9}+O(q^{10}) $ 72 * q + 9 * q^5 - 102 * q^9	$ 72 q + 9 q^{5} - 102 q^{9} + 18 q^{11} - 225 q^{15} - 282 q^{21} - 1278 q^{23} + 441 q^{25} + 54 q^{27} + 1854 q^{29} - 1665 q^{31} - 45 q^{33} - 2673 q^{35} + 6951 q^{39} - 5472 q^{41} + 1260 q^{43} + 5553 q^{45} + 5103 q^{47} - 5904 q^{49} + 1899 q^{51} + 1107 q^{57} - 10944 q^{59} + 8352 q^{61} - 11985 q^{63} + 8757 q^{65} + 378 q^{67} + 5607 q^{69} - 19764 q^{71} + 6111 q^{73} - 3453 q^{75} - 5679 q^{77} - 5652 q^{79} - 20466 q^{81} - 20061 q^{83} + 26100 q^{85} + 40545 q^{87} + 15633 q^{89} - 6039 q^{91} + 40179 q^{93} + 48024 q^{95} - 37530 q^{97} + 12177 q^{99}+O(q^{100}) $ 72 * q + 9 * q^5 - 102 * q^9 + 18 * q^11 - 225 * q^15 - 282 * q^21 - 1278 * q^23 + 441 * q^25 + 54 * q^27 + 1854 * q^29 - 1665 * q^31 - 45 * q^33 - 2673 * q^35 + 6951 * q^39 - 5472 * q^41 + 1260 * q^43 + 5553 * q^45 + 5103 * q^47 - 5904 * q^49 + 1899 * q^51 + 1107 * q^57 - 10944 * q^59 + 8352 * q^61 - 11985 * q^63 + 8757 * q^65 + 378 * q^67 + 5607 * q^69 - 19764 * q^71 + 6111 * q^73 - 3453 * q^75 - 5679 * q^77 - 5652 * q^79 - 20466 * q^81 - 20061 * q^83 + 26100 * q^85 + 40545 * q^87 + 15633 * q^89 - 6039 * q^91 + 40179 * q^93 + 48024 * q^95 - 37530 * q^97 + 12177 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/108\mathbb{Z}\right)^\times$.

$n$	$29$	$55$
$\chi(n)$	$e\left(\frac{13}{18}\right)$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	−8.89497	+	1.37096i	−0.988330	+	0.152328i
$3$	−8.89497	+	1.37096i	−0.988330	+	0.152328i
$4$		0			0
$5$	−1.59659	+	4.38660i	−0.0638637	+	0.175464i	−0.967520	−	0.252794i	$-0.918650\pi$
$5$	−1.59659	+	4.38660i	−0.0638637	+	0.175464i	0.903656	+	0.428258i	$0.140873\pi$
$6$		0			0
$7$	12.9775	−	73.5989i	0.264846	−	1.50202i	−0.504626	−	0.863338i	$-0.668370\pi$
$7$	12.9775	−	73.5989i	0.264846	−	1.50202i	0.769473	−	0.638680i	$-0.220519\pi$
$8$		0			0
$9$	77.2410	−	24.3892i	0.953592	−	0.301101i
$10$		0			0
$11$	54.6352	+	150.109i	0.451531	+	1.24057i	0.931647	+	0.363366i	$0.118372\pi$
$11$	54.6352	+	150.109i	0.451531	+	1.24057i	−0.480116	+	0.877205i	$0.659405\pi$
$12$		0			0
$13$	−161.958	+	135.899i	−0.958332	+	0.804136i	−0.980681	−	0.195614i	$-0.937330\pi$
$13$	−161.958	+	135.899i	−0.958332	+	0.804136i	0.0223489	+	0.999750i	$0.492886\pi$
$14$		0			0
$15$	8.18781	−	41.2076i	0.0363903	−	0.183145i
$16$		0			0
$17$	−295.081	+	170.365i	−1.02104	+	0.589499i	−0.914406	−	0.404799i	$-0.867342\pi$
$17$	−295.081	+	170.365i	−1.02104	+	0.589499i	−0.106637	+	0.994298i	$0.534008\pi$
$18$		0			0
$19$	−169.557	+	293.681i	−0.469686	+	0.813520i	−0.999399	−	0.0346569i	$-0.988966\pi$
$19$	−169.557	+	293.681i	−0.469686	+	0.813520i	0.529713	+	0.848177i	$0.322299\pi$
$20$		0			0
$21$	−14.5334	+	672.451i	−0.0329555	+	1.52483i
$22$		0			0
$23$	946.685	−	166.926i	1.78958	−	0.315550i	0.822257	−	0.569116i	$-0.192715\pi$
$23$	946.685	−	166.926i	1.78958	−	0.315550i	0.967318	+	0.253566i	$0.0816035\pi$
$24$		0			0
$25$	462.085	+	387.735i	0.739335	+	0.620376i
$26$		0			0
$27$	−653.619	+	322.835i	−0.896597	+	0.442847i
$28$		0			0
$29$	−502.780	+	599.189i	−0.597835	+	0.712472i	−0.977092	−	0.212819i	$-0.931735\pi$
$29$	−502.780	+	599.189i	−0.597835	+	0.712472i	0.379256	+	0.925292i	$0.376180\pi$
$30$		0			0
$31$	226.590	+	1285.06i	0.235786	+	1.33721i	0.840952	+	0.541110i	$0.181996\pi$
$31$	226.590	+	1285.06i	0.235786	+	1.33721i	−0.605166	+	0.796099i	$0.706893\pi$
$32$		0			0
$33$	−691.772	−	1260.31i	−0.635236	−	1.15731i
$34$		0			0
$35$	302.129	+	174.434i	0.246636	+	0.142395i
$36$		0			0
$37$	−611.487	−	1059.13i	−0.446667	−	0.773650i	0.551500	−	0.834175i	$-0.314056\pi$
$37$	−611.487	−	1059.13i	−0.446667	−	0.773650i	−0.998167	+	0.0605250i	$0.980723\pi$
$38$		0			0
$39$	1254.30	−	1430.85i	0.824655	−	0.940733i
$40$		0			0
$41$	−11.6088	−	13.8349i	−0.00690590	−	0.00823013i	0.762580	−	0.646893i	$-0.223932\pi$
$41$	−11.6088	−	13.8349i	−0.00690590	−	0.00823013i	−0.769486	+	0.638663i	$0.779488\pi$
$42$		0			0
$43$	−1615.74	+	588.079i	−0.873843	+	0.318053i	−0.739722	−	0.672912i	$-0.765043\pi$
$43$	−1615.74	+	588.079i	−0.873843	+	0.318053i	−0.134120	+	0.990965i	$0.542821\pi$
$44$		0			0
$45$	−16.3366	+	377.765i	−0.00806744	+	0.186551i
$46$		0			0
$47$	3753.45	+	661.834i	1.69916	+	0.299608i	0.937401	−	0.348252i	$-0.113225\pi$
$47$	3753.45	+	661.834i	1.69916	+	0.299608i	0.761760	+	0.647860i	$0.224336\pi$
$48$		0			0
$49$	−2992.18	−	1089.06i	−1.24622	−	0.453587i
$50$		0			0
$51$	2391.18	−	1919.94i	0.919330	−	0.738154i
$52$		0			0
$53$		−	850.088i		−	0.302630i	−0.988486	−	0.151315i	$-0.951649\pi$
$53$		−	850.088i		−	0.302630i	0.988486	−	0.151315i	$-0.0483508\pi$
$54$		0			0
$55$	−745.699			−0.246512
$56$		0			0
$57$	1105.58	−	2844.73i	0.340282	−	0.875572i
$58$		0			0
$59$	−1077.21	+	2959.61i	−0.309455	+	0.850220i	0.683308	+	0.730130i	$0.260540\pi$
$59$	−1077.21	+	2959.61i	−0.309455	+	0.850220i	−0.992763	+	0.120090i	$0.961682\pi$
$60$		0			0
$61$	−630.093	+	3573.43i	−0.169334	+	0.960342i	0.775148	+	0.631780i	$0.217675\pi$
$61$	−630.093	+	3573.43i	−0.169334	+	0.960342i	−0.944482	+	0.328562i	$0.893436\pi$
$62$		0			0
$63$	−792.627	−	6001.36i	−0.199704	−	1.51206i
$64$		0			0
$65$	−337.554	−	927.421i	−0.0798944	−	0.219508i
$66$		0			0
$67$	3089.43	−	2592.34i	0.688223	−	0.577488i	−0.230173	−	0.973150i	$-0.573929\pi$
$67$	3089.43	−	2592.34i	0.688223	−	0.577488i	0.918396	+	0.395662i	$0.129485\pi$
$68$		0			0
$69$	−8191.89	+	2782.67i	−1.72062	+	0.584471i
$70$		0			0
$71$	3906.92	−	2255.66i	0.775029	−	0.447463i	−0.0596365	−	0.998220i	$-0.518994\pi$
$71$	3906.92	−	2255.66i	0.775029	−	0.447463i	0.834666	+	0.550757i	$0.185661\pi$
$72$		0			0
$73$	−2203.63	+	3816.80i	−0.413517	+	0.716232i	−0.995271	−	0.0971321i	$-0.969033\pi$
$73$	−2203.63	+	3816.80i	−0.413517	+	0.716232i	0.581755	+	0.813364i	$0.302366\pi$
$74$		0			0
$75$	−4641.80	−	2815.39i	−0.825208	−	0.500514i
$76$		0			0
$77$	11756.9	−	2073.06i	1.98295	−	0.349647i
$78$		0			0
$79$	−6846.72	−	5745.08i	−1.09706	−	0.920538i	−0.0998315	−	0.995004i	$-0.531830\pi$
$79$	−6846.72	−	5745.08i	−1.09706	−	0.920538i	−0.997224	+	0.0744659i	$0.976275\pi$
$80$		0			0
$81$	5371.33	−	3767.69i	0.818676	−	0.574256i
$82$		0			0
$83$	−715.401	+	852.581i	−0.103847	+	0.123760i	−0.815464	−	0.578807i	$-0.803518\pi$
$83$	−715.401	+	852.581i	−0.103847	+	0.123760i	0.711618	+	0.702567i	$0.247963\pi$
$84$		0			0
$85$	−276.200	−	1566.41i	−0.0382284	−	0.216804i
$86$		0			0
$87$	3650.75	−	6019.06i	0.482329	−	0.795225i
$88$		0			0
$89$	4214.90	+	2433.47i	0.532117	+	0.307218i	0.741878	−	0.670535i	$-0.233935\pi$
$89$	4214.90	+	2433.47i	0.532117	+	0.307218i	−0.209761	+	0.977753i	$0.567269\pi$
$90$		0			0
$91$	7900.21	+	13683.6i	0.954016	+	1.65240i
$92$		0			0
$93$	−3777.27	−	11119.9i	−0.436729	−	1.28569i
$94$		0			0
$95$	−1017.55	−	1212.67i	−0.112748	−	0.134367i
$96$		0			0
$97$	−12919.2	+	4702.22i	−1.37307	+	0.499757i	−0.920071	−	0.391753i	$-0.871869\pi$
$97$	−12919.2	+	4702.22i	−1.37307	+	0.499757i	−0.453001	+	0.891510i	$0.649647\pi$
$98$		0			0
$99$	7881.12	+	10262.1i	0.804114	+	1.04704i
$100$		0			0
$101$	−7328.44	−	1292.20i	−0.718404	−	0.126674i	−0.197517	−	0.980299i	$-0.563288\pi$
$101$	−7328.44	−	1292.20i	−0.718404	−	0.126674i	−0.520887	+	0.853625i	$0.674399\pi$
$102$		0			0
$103$	−4986.77	−	1815.04i	−0.470051	−	0.171085i	0.0961244	−	0.995369i	$-0.469355\pi$
$103$	−4986.77	−	1815.04i	−0.470051	−	0.171085i	−0.566176	+	0.824285i	$0.691578\pi$
$104$		0			0
$105$	−2926.57	−	1137.38i	−0.265449	−	0.103164i
$106$		0			0
$107$			3224.64i			0.281652i	0.990034	+	0.140826i	$0.0449759\pi$
$107$			3224.64i			0.281652i	−0.990034	+	0.140826i	$0.955024\pi$
$108$		0			0
$109$	2220.09			0.186861			0.0934304	−	0.995626i	$-0.470217\pi$
$109$	2220.09			0.186861			0.0934304	+	0.995626i	$0.470217\pi$
$110$		0			0
$111$	6891.18	+	8582.58i	0.559303	+	0.696582i
$112$		0			0
$113$	−4324.96	+	11882.7i	−0.338708	+	0.930593i	0.647054	+	0.762444i	$0.276001\pi$
$113$	−4324.96	+	11882.7i	−0.338708	+	0.930593i	−0.985762	+	0.168148i	$0.946221\pi$
$114$		0			0
$115$	−779.232	+	4419.25i	−0.0589212	+	0.334158i
$116$		0			0
$117$	−9195.33	+	14447.0i	−0.671731	+	1.05537i
$118$		0			0
$119$	8709.29	+	23928.6i	0.615019	+	1.68975i
$120$		0			0
$121$	−8332.07	+	6991.44i	−0.569092	+	0.477525i
$122$		0			0
$123$	122.227	+	107.145i	0.00807899	+	0.00708212i
$124$		0			0
$125$	−4965.30	+	2866.72i	−0.317779	+	0.183470i
$126$		0			0
$127$	2750.24	−	4763.56i	0.170516	−	0.295342i	−0.768085	−	0.640348i	$-0.778790\pi$
$127$	2750.24	−	4763.56i	0.170516	−	0.295342i	0.938600	+	0.345007i	$0.112123\pi$
$128$		0			0
$129$	13565.7	−	7446.05i	0.815196	−	0.447452i
$130$		0			0
$131$	23651.6	−	4170.42i	1.37822	−	0.243017i	0.565056	−	0.825053i	$-0.308855\pi$
$131$	23651.6	−	4170.42i	1.37822	−	0.243017i	0.813163	+	0.582035i	$0.197744\pi$
$132$		0			0
$133$	19414.1	+	16290.4i	1.09753	+	0.920934i
$134$		0			0
$135$	−372.586	−	3382.61i	−0.0204437	−	0.185603i
$136$		0			0
$137$	−6566.41	+	7825.54i	−0.349854	+	0.416940i	−0.912060	−	0.410058i	$-0.865509\pi$
$137$	−6566.41	+	7825.54i	−0.349854	+	0.416940i	0.562205	+	0.826998i	$0.309953\pi$
$138$		0			0
$139$	−4736.20	−	26860.3i	−0.245132	−	1.39021i	−0.820185	−	0.572098i	$-0.806130\pi$
$139$	−4736.20	−	26860.3i	−0.245132	−	1.39021i	0.575053	−	0.818116i	$-0.304981\pi$
$140$		0			0
$141$	−34294.1	−	741.184i	−1.72497	−	0.0372810i
$142$		0			0
$143$	−29248.3	−	16886.5i	−1.43030	−	0.825787i
$144$		0			0
$145$	−1825.67	−	3162.16i	−0.0868334	−	0.150400i
$146$		0			0
$147$	28108.4	+	5585.04i	1.30077	+	0.258459i
$148$		0			0
$149$	−5860.70	−	6984.51i	−0.263984	−	0.314603i	0.617728	−	0.786392i	$-0.288053\pi$
$149$	−5860.70	−	6984.51i	−0.263984	−	0.314603i	−0.881712	+	0.471788i	$0.843609\pi$
$150$		0			0
$151$	−10348.1	+	3766.40i	−0.453844	+	0.165186i	−0.558820	−	0.829289i	$-0.688746\pi$
$151$	−10348.1	+	3766.40i	−0.453844	+	0.165186i	0.104976	+	0.994475i	$0.466523\pi$
$152$		0			0
$153$	−18637.3	+	20356.0i	−0.796159	+	0.869579i
$154$		0			0
$155$	−5998.81	−	1057.75i	−0.249690	−	0.0440272i
$156$		0			0
$157$	27245.9	+	9916.68i	1.10535	+	0.402316i	0.829287	−	0.558823i	$-0.188747\pi$
$157$	27245.9	+	9916.68i	1.10535	+	0.402316i	0.276066	+	0.961139i	$0.410969\pi$
$158$		0			0
$159$	1165.43	+	7561.50i	0.0460991	+	0.299098i
$160$		0			0
$161$		−	71841.2i		−	2.77155i
$162$		0			0
$163$	28656.2			1.07856			0.539279	−	0.842127i	$-0.318697\pi$
$163$	28656.2			1.07856			0.539279	+	0.842127i	$0.318697\pi$
$164$		0			0
$165$	6632.97	−	1022.32i	0.243635	−	0.0375508i
$166$		0			0
$167$	13011.4	−	35748.5i	0.466542	−	1.28181i	−0.453941	−	0.891032i	$-0.649982\pi$
$167$	13011.4	−	35748.5i	0.466542	−	1.28181i	0.920483	−	0.390782i	$-0.127795\pi$
$168$		0			0
$169$	2802.33	−	15892.8i	0.0981173	−	0.556451i
$170$		0			0
$171$	−5934.07	+	26819.5i	−0.202937	+	0.917189i
$172$		0			0
$173$	3685.17	+	10124.9i	0.123130	+	0.338298i	0.985909	−	0.167285i	$-0.0534999\pi$
$173$	3685.17	+	10124.9i	0.123130	+	0.338298i	−0.862778	+	0.505582i	$0.831278\pi$
$174$		0			0
$175$	34533.5	−	28977.1i	1.12763	−	0.946191i
$176$		0			0
$177$	5524.26	−	27802.5i	0.176331	−	0.887436i
$178$		0			0
$179$	−40319.4	+	23278.4i	−1.25837	+	0.726519i	−0.972757	−	0.231827i	$-0.925530\pi$
$179$	−40319.4	+	23278.4i	−1.25837	+	0.726519i	−0.285611	+	0.958346i	$0.592196\pi$
$180$		0			0
$181$	4651.64	−	8056.88i	0.141987	−	0.245929i	−0.786258	−	0.617899i	$-0.787984\pi$
$181$	4651.64	−	8056.88i	0.141987	−	0.245929i	0.928245	+	0.371970i	$0.121317\pi$
$182$		0			0
$183$	705.637	−	32649.4i	0.0210707	−	0.974929i
$184$		0			0
$185$	5622.27	−	991.357i	0.164274	−	0.0289659i
$186$		0			0
$187$	−41695.2	−	34986.4i	−1.19235	−	1.00050i
$188$		0			0
$189$	15278.0	+	52295.2i	0.427703	+	1.46399i
$190$		0			0
$191$	−40513.7	+	48282.4i	−1.11054	+	1.32349i	−0.169377	+	0.985551i	$0.554175\pi$
$191$	−40513.7	+	48282.4i	−1.11054	+	1.32349i	−0.941167	+	0.337943i	$0.890269\pi$
$192$		0			0
$193$	−8351.64	−	47364.5i	−0.224211	−	1.27156i	−0.864187	−	0.503170i	$-0.832167\pi$
$193$	−8351.64	−	47364.5i	−0.224211	−	1.27156i	0.639976	−	0.768395i	$-0.278944\pi$
$194$		0			0
$195$	4273.98	+	7786.61i	0.112399	+	0.204776i
$196$		0			0
$197$	−21887.7	−	12636.9i	−0.563985	−	0.325617i	0.190758	−	0.981637i	$-0.438905\pi$
$197$	−21887.7	−	12636.9i	−0.563985	−	0.325617i	−0.754743	+	0.656020i	$0.772239\pi$
$198$		0			0
$199$	6646.68	+	11512.4i	0.167841	+	0.290709i	0.937661	−	0.347552i	$-0.112987\pi$
$199$	6646.68	+	11512.4i	0.167841	+	0.290709i	−0.769819	+	0.638262i	$0.779654\pi$
$200$		0			0
$201$	−23926.4	+	27294.3i	−0.592224	+	0.675585i
$202$		0			0
$203$	37574.9	+	44780.0i	0.911812	+	1.08666i
$204$		0			0
$205$	79.2226	−	28.8347i	0.00188513	−	0.000686131i
$206$		0			0
$207$	69051.7	−	35982.4i	1.61151	−	0.839750i
$208$		0			0
$209$	−53347.9	−	9406.67i	−1.22131	−	0.215349i
$210$		0			0
$211$	−28402.4	−	10337.6i	−0.637954	−	0.232196i	0.00273545	−	0.999996i	$-0.499129\pi$
$211$	−28402.4	−	10337.6i	−0.637954	−	0.232196i	−0.640690	+	0.767800i	$0.721352\pi$
$212$		0			0
$213$	−31659.5	+	25420.3i	−0.697823	+	0.560300i
$214$		0			0
$215$		−	8026.51i		−	0.173640i
$216$		0			0
$217$	97519.4			2.07096
$218$		0			0
$219$	14368.6	−	36971.4i	0.299589	−	0.770864i
$220$		0			0
$221$	24638.4	−	67693.3i	0.504460	−	1.38599i
$222$		0			0
$223$	−9815.65	+	55667.3i	−0.197383	+	1.11941i	0.711602	+	0.702583i	$0.247970\pi$
$223$	−9815.65	+	55667.3i	−0.197383	+	1.11941i	−0.908984	+	0.416830i	$0.863141\pi$
$224$		0			0
$225$	45148.4	+	18679.1i	0.891820	+	0.368971i
$226$		0			0
$227$	−22493.6	−	61800.7i	−0.436523	−	1.19934i	−0.941739	−	0.336344i	$-0.890809\pi$
$227$	−22493.6	−	61800.7i	−0.436523	−	1.19934i	0.505216	−	0.862993i	$-0.331413\pi$
$228$		0			0
$229$	42755.1	−	35875.8i	0.815299	−	0.684117i	−0.136567	−	0.990631i	$-0.543607\pi$
$229$	42755.1	−	35875.8i	0.815299	−	0.684117i	0.951866	+	0.306514i	$0.0991625\pi$
$230$		0			0
$231$	−101735.	+	34557.9i	−1.90654	+	0.647625i
$232$		0			0
$233$	6779.35	−	3914.06i	0.124875	−	0.0720968i	−0.436261	−	0.899820i	$-0.643698\pi$
$233$	6779.35	−	3914.06i	0.124875	−	0.0720968i	0.561136	+	0.827723i	$0.310364\pi$
$234$		0			0
$235$	−8895.93	+	15408.2i	−0.161085	+	0.279008i
$236$		0			0
$237$	68777.6	+	41715.8i	1.22448	+	0.742683i
$238$		0			0
$239$	−71682.9	+	12639.6i	−1.25493	+	0.221278i	−0.761303	−	0.648396i	$-0.775440\pi$
$239$	−71682.9	+	12639.6i	−1.25493	+	0.221278i	−0.493627	+	0.869674i	$0.664329\pi$
$240$		0			0
$241$	−4923.27	−	4131.12i	−0.0847656	−	0.0711268i	0.599420	−	0.800435i	$-0.295398\pi$
$241$	−4923.27	−	4131.12i	−0.0847656	−	0.0711268i	−0.684186	+	0.729308i	$0.739842\pi$
$242$		0			0
$243$	−42612.5	+	40877.4i	−0.721646	+	0.692262i
$244$		0			0
$245$	9554.58	−	11386.7i	0.159177	−	0.189699i
$246$		0			0
$247$	−12449.8	−	70606.5i	−0.204066	−	1.15731i
$248$		0			0
$249$	5194.62	−	8564.47i	0.0837828	−	0.138134i
$250$		0			0
$251$	8404.40	+	4852.28i	0.133401	+	0.0770191i	0.565215	−	0.824943i	$-0.308793\pi$
$251$	8404.40	+	4852.28i	0.133401	+	0.0770191i	−0.431814	+	0.901963i	$0.642126\pi$
$252$		0			0
$253$	76779.5	+	132986.i	1.19951	+	2.07761i
$254$		0			0
$255$	4604.27	+	13554.5i	0.0708077	+	0.208451i
$256$		0			0
$257$	41364.9	+	49296.7i	0.626276	+	0.746366i	0.982136	−	0.188173i	$-0.0602565\pi$
$257$	41364.9	+	49296.7i	0.626276	+	0.746366i	−0.355860	+	0.934539i	$0.615812\pi$
$258$		0			0
$259$	−85886.1	+	31260.0i	−1.28033	+	0.466004i
$260$		0			0
$261$	−24221.4	+	58544.4i	−0.355565	+	0.859417i
$262$		0			0
$263$	43081.5	+	7596.43i	0.622844	+	0.109824i	0.476159	−	0.879359i	$-0.342029\pi$
$263$	43081.5	+	7596.43i	0.622844	+	0.109824i	0.146685	+	0.989183i	$0.453140\pi$
$264$		0			0
$265$	3729.00	+	1357.24i	0.0531007	+	0.0193271i
$266$		0			0
$267$	−40827.6	−	15867.2i	−0.572706	−	0.222576i
$268$		0			0
$269$		−	60652.2i		−	0.838189i	−0.907943	−	0.419094i	$-0.862348\pi$
$269$		−	60652.2i		−	0.838189i	0.907943	−	0.419094i	$-0.137652\pi$
$270$		0			0
$271$	−31866.2			−0.433902			−0.216951	−	0.976183i	$-0.569611\pi$
$271$	−31866.2			−0.433902			−0.216951	+	0.976183i	$0.569611\pi$
$272$		0			0
$273$	−89031.6	−	110884.i	−1.19459	−	1.48780i
$274$		0			0
$275$	−32956.4	+	90547.1i	−0.435788	+	1.19732i
$276$		0			0
$277$	−8963.47	+	50834.4i	−0.116820	+	0.662518i	0.869013	+	0.494789i	$0.164755\pi$
$277$	−8963.47	+	50834.4i	−0.116820	+	0.662518i	−0.985833	+	0.167729i	$0.946357\pi$
$278$		0			0
$279$	48843.6	+	93732.8i	0.627479	+	1.20416i
$280$		0			0
$281$	45325.4	+	124530.i	0.574022	+	1.57711i	0.798089	+	0.602539i	$0.205844\pi$
$281$	45325.4	+	124530.i	0.574022	+	1.57711i	−0.224067	+	0.974574i	$0.571934\pi$
$282$		0			0
$283$	30812.7	−	25854.9i	0.384730	−	0.322827i	−0.429826	−	0.902912i	$-0.641425\pi$
$283$	30812.7	−	25854.9i	0.384730	−	0.322827i	0.814556	+	0.580085i	$0.196981\pi$
$284$		0			0
$285$	10713.6	+	9391.61i	0.131900	+	0.115625i
$286$		0			0
$287$	−1168.88	+	674.855i	−0.0141908	+	0.00819307i
$288$		0			0
$289$	16288.2	−	28212.0i	0.195019	−	0.337783i
$290$		0			0
$291$	108470.	−	59537.8i	1.28092	−	0.703083i
$292$		0			0
$293$	109933.	−	19384.2i	1.28054	−	0.225794i	0.508331	−	0.861162i	$-0.330263\pi$
$293$	109933.	−	19384.2i	1.28054	−	0.225794i	0.772210	+	0.635368i	$0.219152\pi$
$294$		0			0
$295$	−11262.8	−	9450.60i	−0.129420	−	0.108596i
$296$		0			0
$297$	−84171.2	−	80476.0i	−0.954224	−	0.912334i
$298$		0			0
$299$	−130638.	+	155689.i	−1.46126	+	1.74146i
$300$		0			0
$301$	22313.8	+	126548.i	0.246287	+	1.39676i
$302$		0			0
$303$	66957.8	+	1447.13i	0.729316	+	0.0157624i
$304$		0			0
$305$	−14669.2	−	8469.28i	−0.157691	−	0.0910431i
$306$		0			0
$307$	−45164.5	−	78227.2i	−0.479204	−	0.830006i	0.520511	−	0.853855i	$-0.325741\pi$
$307$	−45164.5	−	78227.2i	−0.479204	−	0.830006i	−0.999716	+	0.0238487i	$0.992408\pi$
$308$		0			0
$309$	46845.5	+	9308.05i	0.490627	+	0.0974859i
$310$		0			0
$311$	38249.6	+	45584.1i	0.395463	+	0.471295i	0.926631	−	0.375972i	$-0.122691\pi$
$311$	38249.6	+	45584.1i	0.395463	+	0.471295i	−0.531168	+	0.847267i	$0.678247\pi$
$312$		0			0
$313$	−90164.7	+	32817.3i	−0.920339	+	0.334976i	−0.758373	−	0.651820i	$-0.774006\pi$
$313$	−90164.7	+	32817.3i	−0.920339	+	0.334976i	−0.161966	+	0.986796i	$0.551783\pi$
$314$		0			0
$315$	27591.1	+	6104.79i	0.278066	+	0.0615247i
$316$		0			0
$317$	84008.5	+	14813.0i	0.835997	+	0.147409i	0.575229	−	0.817993i	$-0.304913\pi$
$317$	84008.5	+	14813.0i	0.835997	+	0.147409i	0.260768	+	0.965401i	$0.416024\pi$
$318$		0			0
$319$	−117413.	−	42734.9i	−1.15381	−	0.419954i
$320$		0			0
$321$	−4420.84	−	28683.1i	−0.0429037	−	0.278366i
$322$		0			0
$323$		−	115546.i		−	1.10752i
$324$		0			0
$325$	−127531.			−1.20740
$326$		0			0
$327$	−19747.7	+	3043.65i	−0.184680	+	0.0284642i
$328$		0			0
$329$	97420.4	−	267660.i	0.900033	−	2.47282i
$330$		0			0
$331$	11786.7	−	66845.7i	0.107581	−	0.610123i	−0.882577	−	0.470168i	$-0.844193\pi$
$331$	11786.7	−	66845.7i	0.107581	−	0.610123i	0.990158	−	0.139955i	$-0.0446957\pi$
$332$		0			0
$333$	−73063.1	−	66894.3i	−0.658885	−	0.603255i
$334$		0			0
$335$	6439.01	+	17691.0i	0.0573759	+	0.157639i
$336$		0			0
$337$	15610.7	−	13098.9i	0.137456	−	0.115339i	−0.571468	−	0.820625i	$-0.693626\pi$
$337$	15610.7	−	13098.9i	0.137456	−	0.115339i	0.708923	+	0.705286i	$0.249181\pi$
$338$		0			0
$339$	22179.7	−	111626.i	0.193000	−	0.971328i
$340$		0			0
$341$	−180519.	+	104223.i	−1.55244	+	0.896301i
$342$		0			0
$343$	−29266.2	+	50690.6i	−0.248759	+	0.430863i
$344$		0			0
$345$	872.657	−	40377.3i	0.00733172	−	0.339234i
$346$		0			0
$347$	117734.	−	20759.6i	0.977781	−	0.172409i	0.338151	−	0.941092i	$-0.390199\pi$
$347$	117734.	−	20759.6i	0.977781	−	0.172409i	0.639630	+	0.768683i	$0.279088\pi$
$348$		0			0
$349$	69392.2	+	58227.0i	0.569718	+	0.478050i	0.881552	−	0.472087i	$-0.156499\pi$
$349$	69392.2	+	58227.0i	0.569718	+	0.478050i	−0.311835	+	0.950136i	$0.600943\pi$
$350$		0			0
$351$	61986.0	−	141112.i	0.503129	−	1.14538i
$352$		0			0
$353$	126461.	−	150710.i	1.01486	−	1.20947i	0.0371955	−	0.999308i	$-0.488158\pi$
$353$	126461.	−	150710.i	1.01486	−	1.20947i	0.977667	−	0.210158i	$-0.0673980\pi$
$354$		0			0
$355$	3656.93	+	20739.5i	0.0290175	+	0.164567i
$356$		0			0
$357$	−110274.	−	200904.i	−0.865239	−	1.57635i
$358$		0			0
$359$	188927.	+	109077.i	1.46590	+	0.846338i	0.999273	−	0.0381164i	$-0.0121358\pi$
$359$	188927.	+	109077.i	1.46590	+	0.846338i	0.466627	+	0.884454i	$0.345469\pi$
$360$		0			0
$361$	7661.62	+	13270.3i	0.0587904	+	0.101828i
$362$		0			0
$363$	64528.6	−	73611.5i	0.489710	−	0.558641i
$364$		0			0
$365$	−13224.5	−	15760.3i	−0.0992643	−	0.118299i
$366$		0			0
$367$	138372.	−	50363.4i	1.02735	−	0.373924i	0.227277	−	0.973830i	$-0.427018\pi$
$367$	138372.	−	50363.4i	1.02735	−	0.373924i	0.800070	+	0.599906i	$0.204795\pi$
$368$		0			0
$369$	−1234.10	−	785.487i	−0.00906352	−	0.00576881i
$370$		0			0
$371$	−62565.5	−	11032.0i	−0.454556	−	0.0801504i
$372$		0			0
$373$	−71253.4	−	25934.1i	−0.512139	−	0.186403i	0.0730067	−	0.997331i	$-0.476741\pi$
$373$	−71253.4	−	25934.1i	−0.512139	−	0.186403i	−0.585146	+	0.810928i	$0.698963\pi$
$374$		0			0
$375$	40236.0	−	32306.6i	0.286123	−	0.229736i
$376$		0			0
$377$		−	165371.i		−	1.16353i
$378$		0			0
$379$	138702.			0.965616			0.482808	−	0.875726i	$-0.339617\pi$
$379$	138702.			0.965616			0.482808	+	0.875726i	$0.339617\pi$
$380$		0			0
$381$	−17932.7	+	46142.2i	−0.123537	+	0.317869i
$382$		0			0
$383$	−65429.9	+	179767.i	−0.446045	+	1.22550i	0.489411	+	0.872053i	$0.337212\pi$
$383$	−65429.9	+	179767.i	−0.446045	+	1.22550i	−0.935455	+	0.353445i	$0.885010\pi$
$384$		0			0
$385$	−9677.29	+	54882.6i	−0.0652878	+	0.370266i
$386$		0			0
$387$	−110458.	+	84830.3i	−0.737523	+	0.566408i
$388$		0			0
$389$	64370.0	+	176855.i	0.425387	+	1.16874i	0.948583	+	0.316529i	$0.102517\pi$
$389$	64370.0	+	176855.i	0.425387	+	1.16874i	−0.523196	+	0.852212i	$0.675260\pi$
$390$		0			0
$391$	−250911.	+	210539.i	−1.64122	+	1.37714i
$392$		0			0
$393$	−204663.	+	69521.1i	−1.32512	+	0.450123i
$394$		0			0
$395$	36132.8	−	20861.3i	0.231584	−	0.133705i
$396$		0			0
$397$	−46197.3	+	80016.0i	−0.293113	+	0.507687i	−0.974544	−	0.224195i	$-0.928025\pi$
$397$	−46197.3	+	80016.0i	−0.293113	+	0.507687i	0.681431	+	0.731882i	$0.261358\pi$
$398$		0			0
$399$	−195022.	−	118287.i	−1.22500	−	0.743002i
$400$		0			0
$401$	692.633	−	122.130i	0.00430739	−	0.000759510i	−0.171494	−	0.985185i	$-0.554859\pi$
$401$	692.633	−	122.130i	0.00430739	−	0.000759510i	0.175801	+	0.984426i	$0.443748\pi$
$402$		0			0
$403$	−211336.	−	177332.i	−1.30126	−	1.09189i
$404$		0			0
$405$	7951.54	+	29577.4i	0.0484776	+	0.180322i
$406$		0			0
$407$	125576.	−	149655.i	0.758084	−	0.903449i
$408$		0			0
$409$	−22043.3	−	125014.i	−0.131774	−	0.747327i	−0.977052	−	0.213000i	$-0.931676\pi$
$409$	−22043.3	−	125014.i	−0.131774	−	0.747327i	0.845278	−	0.534326i	$-0.179435\pi$
$410$		0			0
$411$	47679.6	−	78610.2i	0.282260	−	0.465367i
$412$		0			0
$413$	203845.	+	117690.i	1.19509	+	0.689984i
$414$		0			0
$415$	−2597.73	−	4499.40i	−0.0150834	−	0.0261251i
$416$		0			0
$417$	78952.7	+	232429.i	0.454041	+	1.33665i
$418$		0			0
$419$	−55961.7	−	66692.5i	−0.318759	−	0.379882i	0.582743	−	0.812656i	$-0.301979\pi$
$419$	−55961.7	−	66692.5i	−0.318759	−	0.379882i	−0.901503	+	0.432774i	$0.857535\pi$
$420$		0			0
$421$	49871.4	−	18151.7i	0.281376	−	0.102413i	−0.197477	−	0.980308i	$-0.563275\pi$
$421$	49871.4	−	18151.7i	0.281376	−	0.102413i	0.478853	+	0.877895i	$0.341053\pi$
$422$		0			0
$423$	306061.	−	40422.9i	1.71052	−	0.225916i
$424$		0			0
$425$	−202409.	−	35690.2i	−1.12060	−	0.197593i
$426$		0			0
$427$	254824.	+	92748.2i	1.39760	+	0.508686i
$428$		0			0
$429$	283313.	+	110107.i	1.53940	+	0.598274i
$430$		0			0
$431$			133930.i			0.720980i	0.932763	+	0.360490i	$0.117391\pi$
$431$			133930.i			0.720980i	−0.932763	+	0.360490i	$0.882609\pi$
$432$		0			0
$433$	−42018.9			−0.224114			−0.112057	−	0.993702i	$-0.535744\pi$
$433$	−42018.9			−0.224114			−0.112057	+	0.993702i	$0.535744\pi$
$434$		0			0
$435$	20574.5	+	25624.4i	0.108730	+	0.135417i
$436$		0			0
$437$	−111494.	+	306327.i	−0.583832	+	1.60406i
$438$		0			0
$439$	−39764.9	+	225518.i	−0.206334	+	1.17018i	0.688993	+	0.724768i	$0.258053\pi$
$439$	−39764.9	+	225518.i	−0.206334	+	1.17018i	−0.895327	+	0.445410i	$0.853058\pi$
$440$		0			0
$441$	−257680.	−	11143.4i	−1.32496	−	0.0572984i
$442$		0			0
$443$	−80393.0	−	220878.i	−0.409648	−	1.12550i	−0.957377	−	0.288843i	$-0.906730\pi$
$443$	−80393.0	−	220878.i	−0.409648	−	1.12550i	0.547728	−	0.836656i	$-0.315493\pi$
$444$		0			0
$445$	−17404.2	+	14603.8i	−0.0878888	+	0.0737474i
$446$		0			0
$447$	61706.2	+	54092.3i	0.308826	+	0.270720i
$448$		0			0
$449$	−143947.	+	83107.7i	−0.714018	+	0.412239i	−0.812547	−	0.582895i	$-0.801920\pi$
$449$	−143947.	+	83107.7i	−0.714018	+	0.412239i	0.0985287	+	0.995134i	$0.468586\pi$
$450$		0			0
$451$	1442.49	−	2498.46i	0.00709184	−	0.0122834i
$452$		0			0
$453$	86882.5	−	47688.8i	0.423385	−	0.232391i
$454$		0			0
$455$	−72637.8	+	12808.0i	−0.350865	+	0.0618669i
$456$		0			0
$457$	101420.	+	85101.3i	0.485613	+	0.407478i	0.852451	−	0.522807i	$-0.175115\pi$
$457$	101420.	+	85101.3i	0.485613	+	0.407478i	−0.366838	+	0.930285i	$0.619560\pi$
$458$		0			0
$459$	137871.	−	206617.i	0.654406	−	0.980709i
$460$		0			0
$461$	−42494.6	+	50643.1i	−0.199955	+	0.238297i	−0.856699	−	0.515816i	$-0.827489\pi$
$461$	−42494.6	+	50643.1i	−0.199955	+	0.238297i	0.656744	+	0.754113i	$0.271933\pi$
$462$		0			0
$463$	−25053.8	−	142087.i	−0.116872	−	0.662817i	−0.985806	−	0.167888i	$-0.946305\pi$
$463$	−25053.8	−	142087.i	−0.116872	−	0.662817i	0.868934	−	0.494929i	$-0.164806\pi$
$464$		0			0
$465$	54809.4	+	1184.57i	0.253483	+	0.00547842i
$466$		0			0
$467$	324271.	+	187218.i	1.48687	+	0.858446i	0.999888	−	0.0149642i	$-0.00476345\pi$
$467$	324271.	+	187218.i	1.48687	+	0.858446i	0.486985	+	0.873411i	$0.338097\pi$
$468$		0			0
$469$	−150701.	−	261021.i	−0.685124	−	1.18667i
$470$		0			0
$471$	−255946.	−	50855.7i	−1.15374	−	0.229244i
$472$		0			0
$473$	−176552.	−	210407.i	−0.789134	−	0.940453i
$474$		0			0
$475$	−192220.	+	69962.3i	−0.851944	+	0.310082i
$476$		0			0
$477$	−20733.0	−	65661.6i	−0.0911223	−	0.288586i
$478$		0			0
$479$	376454.	+	66379.1i	1.64075	+	0.289308i	0.916440	−	0.400171i	$-0.131049\pi$
$479$	376454.	+	66379.1i	1.64075	+	0.289308i	0.724306	+	0.689479i	$0.242160\pi$
$480$		0			0
$481$	242970.	+	88433.7i	1.05018	+	0.382233i
$482$		0			0
$483$	98491.1	+	639026.i	0.422185	+	2.73920i
$484$		0			0
$485$		−	64179.1i		−	0.272841i
$486$		0			0
$487$	−170794.			−0.720135			−0.360068	−	0.932926i	$-0.617246\pi$
$487$	−170794.			−0.720135			−0.360068	+	0.932926i	$0.617246\pi$
$488$		0			0
$489$	−254896.	+	39286.4i	−1.06597	+	0.164295i
$490$		0			0
$491$	−93039.5	+	255624.i	−0.385926	+	1.06032i	0.582892	+	0.812550i	$0.301921\pi$
$491$	−93039.5	+	255624.i	−0.385926	+	1.06032i	−0.968818	+	0.247774i	$0.920301\pi$
$492$		0			0
$493$	46279.8	−	262466.i	0.190413	−	1.07989i
$494$		0			0
$495$	−57598.5	+	18187.0i	−0.235072	+	0.0742252i
$496$		0			0
$497$	−115312.	−	316818.i	−0.466834	−	1.28262i
$498$		0			0
$499$	310696.	−	260705.i	1.24777	−	1.04700i	0.250896	−	0.968014i	$-0.419275\pi$
$499$	310696.	−	260705.i	1.24777	−	1.04700i	0.996875	−	0.0789903i	$-0.0251696\pi$
$500$		0			0
$501$	−66726.3	+	335820.i	−0.265841	+	1.33792i
$502$		0			0
$503$	−258998.	+	149532.i	−1.02367	+	0.591016i	−0.915165	−	0.403080i	$-0.867940\pi$
$503$	−258998.	+	149532.i	−1.02367	+	0.591016i	−0.108505	+	0.994096i	$0.534606\pi$
$504$		0			0
$505$	17368.9	−	30083.8i	0.0681067	−	0.117964i
$506$		0			0
$507$	−3138.31	+	145208.i	−0.0122090	+	0.564903i
$508$		0			0
$509$	−75484.0	+	13309.9i	−0.291353	+	0.0513734i	−0.317414	−	0.948287i	$-0.602815\pi$
$509$	−75484.0	+	13309.9i	−0.291353	+	0.0513734i	0.0260613	+	0.999660i	$0.491703\pi$
$510$		0			0
$511$	252315.	+	211717.i	0.966275	+	0.810801i
$512$		0			0
$513$	16015.0	−	246694.i	0.0608545	−	0.937398i
$514$		0			0
$515$	15923.7	−	18977.1i	0.0600384	−	0.0715510i
$516$		0			0
$517$	105723.	+	599586.i	0.395539	+	2.24321i
$518$		0			0
$519$	−46660.2	−	85008.6i	−0.173226	−	0.315593i
$520$		0			0
$521$	−147819.	−	85343.1i	−0.544570	−	0.314408i	0.202359	−	0.979311i	$-0.435139\pi$
$521$	−147819.	−	85343.1i	−0.544570	−	0.314408i	−0.746929	+	0.664904i	$0.768473\pi$
$522$		0			0
$523$	150342.	+	260400.i	0.549639	+	0.952003i	0.998299	+	0.0583002i	$0.0185680\pi$
$523$	150342.	+	260400.i	0.549639	+	0.952003i	−0.448660	+	0.893702i	$0.648099\pi$
$524$		0			0
$525$	−267449.	+	305094.i	−0.970335	+	1.10692i
$526$		0			0
$527$	−285792.	−	340594.i	−1.02903	−	1.22635i
$528$		0			0
$529$	605384.	−	220342.i	2.16331	−	0.787382i
$530$		0			0
$531$	−11022.2	+	254876.i	−0.0390912	+	0.903940i
$532$		0			0
$533$	3760.29	+	663.040i	0.0132363	+	0.00233392i
$534$		0			0
$535$	−14145.2	−	5148.44i	−0.0494199	−	0.0179874i
$536$		0			0
$537$	326726.	−	262337.i	1.13301	−	0.909726i
$538$		0			0
$539$		−	508654.i		−	1.75083i
$540$		0			0
$541$	−45127.6			−0.154187			−0.0770935	−	0.997024i	$-0.524564\pi$
$541$	−45127.6			−0.154187			−0.0770935	+	0.997024i	$0.524564\pi$
$542$		0			0
$543$	−30330.6	+	78042.9i	−0.102868	+	0.264688i
$544$		0			0
$545$	−3544.59	+	9738.67i	−0.0119336	+	0.0327874i
$546$		0			0
$547$	101751.	−	577057.i	0.340066	−	1.92861i	−0.0298729	−	0.999554i	$-0.509510\pi$
$547$	101751.	−	577057.i	0.340066	−	1.92861i	0.369939	−	0.929056i	$-0.379379\pi$
$548$		0			0
$549$	38484.3	+	291383.i	0.127685	+	0.966761i
$550$		0			0
$551$	−90720.7	−	249253.i	−0.298816	−	0.820989i
$552$		0			0
$553$	−511685.	+	429354.i	−1.67322	+	1.40399i
$554$		0			0
$555$	−48650.8	+	16526.0i	−0.157944	+	0.0536514i
$556$		0			0
$557$	−102333.	+	59082.0i	−0.329841	+	0.190434i	−0.655771	−	0.754960i	$-0.727656\pi$
$557$	−102333.	+	59082.0i	−0.329841	+	0.190434i	0.325929	+	0.945394i	$0.394323\pi$
$558$		0			0
$559$	181762.	−	314821.i	0.581674	−	1.00749i
$560$		0			0
$561$	418843.	+	254041.i	1.33084	+	0.807195i
$562$		0			0
$563$	−364046.	+	64191.1i	−1.14852	+	0.202515i	−0.715330	−	0.698787i	$-0.753724\pi$
$563$	−364046.	+	64191.1i	−1.14852	+	0.202515i	−0.433191	+	0.901302i	$0.642613\pi$
$564$		0			0
$565$	−45219.7	−	37943.8i	−0.141655	−	0.118862i
$566$		0			0
$567$	−207592.	−	444219.i	−0.645719	−	1.38176i
$568$		0			0
$569$	34692.3	−	41344.7i	0.107154	−	0.127701i	−0.709802	−	0.704402i	$-0.751215\pi$
$569$	34692.3	−	41344.7i	0.107154	−	0.127701i	0.816956	+	0.576700i	$0.195660\pi$
$570$		0			0
$571$	−36084.4	−	204645.i	−0.110674	−	0.627665i	−0.988801	−	0.149238i	$-0.952318\pi$
$571$	−36084.4	−	204645.i	−0.110674	−	0.627665i	0.878127	−	0.478428i	$-0.158793\pi$
$572$		0			0
$573$	294175.	−	485013.i	0.895977	−	1.47722i
$574$		0			0
$575$	502172.	+	289929.i	1.51886	+	0.876912i
$576$		0			0
$577$	−328925.	−	569715.i	−0.987974	−	1.71122i	−0.627893	−	0.778300i	$-0.716083\pi$
$577$	−328925.	−	569715.i	−0.987974	−	1.71122i	−0.360081	−	0.932921i	$-0.617251\pi$
$578$		0			0
$579$	139222.	+	409856.i	0.415290	+	1.22257i
$580$		0			0
$581$	53464.9	+	63717.0i	0.158386	+	0.188757i
$582$		0			0
$583$	127606.	−	46444.7i	0.375434	−	0.136647i
$584$		0			0
$585$	−48692.1	−	63402.3i	−0.142281	−	0.185265i
$586$		0			0
$587$	−339150.	−	59801.2i	−0.984272	−	0.173554i	−0.341725	−	0.939800i	$-0.611011\pi$
$587$	−339150.	−	59801.2i	−0.984272	−	0.173554i	−0.642547	+	0.766246i	$0.722122\pi$
$588$		0			0
$589$	−415817.	−	151345.i	−1.19859	−	0.436252i
$590$		0			0
$591$	212015.	+	82397.5i	0.607004	+	0.235906i
$592$		0			0
$593$			222202.i			0.631886i	0.948778	+	0.315943i	$0.102321\pi$
$593$			222202.i			0.631886i	−0.948778	+	0.315943i	$0.897679\pi$
$594$		0			0
$595$	−118870.			−0.335768
$596$		0			0
$597$	−74904.9	−	93290.0i	−0.210166	−	0.261750i
$598$		0			0
$599$	215568.	−	592269.i	0.600801	−	1.65069i	−0.148853	−	0.988859i	$-0.547558\pi$
$599$	215568.	−	592269.i	0.600801	−	1.65069i	0.749655	−	0.661829i	$-0.230220\pi$
$600$		0			0
$601$	50236.9	−	284908.i	0.139083	−	0.788778i	−0.832846	−	0.553505i	$-0.813290\pi$
$601$	50236.9	−	284908.i	0.139083	−	0.788778i	0.971929	−	0.235274i	$-0.0755987\pi$
$602$		0			0
$603$	175406.	−	275584.i	0.482402	−	0.757913i
$604$		0			0
$605$	−17365.7	−	47712.0i	−0.0474441	−	0.130352i
$606$		0			0
$607$	319799.	−	268343.i	0.867959	−	0.728304i	−0.0957082	−	0.995409i	$-0.530512\pi$
$607$	319799.	−	268343.i	0.867959	−	0.728304i	0.963667	+	0.267105i	$0.0860671\pi$
$608$		0			0
$609$	−395619.	−	346803.i	−1.06670	−	0.935079i
$610$		0			0
$611$	−697844.	+	402900.i	−1.86929	+	1.07923i
$612$		0			0
$613$	309732.	−	536471.i	0.824261	−	1.42766i	−0.0782216	−	0.996936i	$-0.524924\pi$
$613$	309732.	−	536471.i	0.824261	−	1.42766i	0.902483	−	0.430726i	$-0.141742\pi$
$614$		0			0
$615$	−665.151	+	365.094i	−0.00175861	+	0.000965283i
$616$		0			0
$617$	−209190.	+	36885.8i	−0.549502	+	0.0968921i	−0.441505	−	0.897259i	$-0.645555\pi$
$617$	−209190.	+	36885.8i	−0.549502	+	0.0968921i	−0.107998	+	0.994151i	$0.534444\pi$
$618$		0			0
$619$	423509.	+	355367.i	1.10530	+	0.927460i	0.997770	−	0.0667411i	$-0.0212601\pi$
$619$	423509.	+	355367.i	1.10530	+	0.927460i	0.107534	+	0.994201i	$0.465705\pi$
$620$		0			0
$621$	−564882.	+	414730.i	−1.46479	+	1.07543i
$622$		0			0
$623$	233800.	−	278632.i	0.602376	−	0.717884i
$624$		0			0
$625$	60818.7	+	344920.i	0.155696	+	0.882995i
$626$		0			0
$627$	487424.	+	10534.5i	1.23986	+	0.0267965i
$628$		0			0
$629$	360877.	+	208352.i	0.912133	+	0.526620i
$630$		0			0
$631$	−86024.6	−	148999.i	−0.216055	−	0.374218i	0.737544	−	0.675300i	$-0.235986\pi$
$631$	−86024.6	−	148999.i	−0.216055	−	0.374218i	−0.953598	+	0.301082i	$0.902652\pi$
$632$		0			0
$633$	266811.	+	53014.4i	0.665879	+	0.132308i
$634$		0			0
$635$	16504.8	+	19669.7i	0.0409321	+	0.0487810i
$636$		0			0
$637$	632610.	−	230251.i	1.55904	−	0.567444i
$638$		0			0
$639$	246761.	−	269516.i	0.604330	−	0.660060i
$640$		0			0
$641$	220023.	+	38796.0i	0.535491	+	0.0944216i	0.434853	−	0.900502i	$-0.356800\pi$
$641$	220023.	+	38796.0i	0.535491	+	0.0944216i	0.100638	+	0.994923i	$0.467911\pi$
$642$		0			0
$643$	249787.	+	90915.0i	0.604154	+	0.219894i	0.625943	−	0.779868i	$-0.284714\pi$
$643$	249787.	+	90915.0i	0.604154	+	0.219894i	−0.0217893	+	0.999763i	$0.506936\pi$
$644$		0			0
$645$	11004.0	+	71395.6i	0.0264503	+	0.171614i
$646$		0			0
$647$		−	182160.i		−	0.435155i	−0.976043	−	0.217577i	$-0.930185\pi$
$647$		−	182160.i		−	0.435155i	0.976043	−	0.217577i	$-0.0698155\pi$
$648$		0			0
$649$	−503119.			−1.19449
$650$		0			0
$651$	−867432.	+	133695.i	−2.04679	+	0.315466i
$652$		0			0
$653$	65152.7	−	179006.i	0.152794	−	0.419798i	−0.839553	−	0.543278i	$-0.817183\pi$
$653$	65152.7	−	179006.i	0.152794	−	0.419798i	0.992347	+	0.123480i	$0.0394054\pi$
$654$		0			0
$655$	−19468.0	+	110409.i	−0.0453774	+	0.257348i
$656$		0			0
$657$	−77121.8	+	348558.i	−0.178668	+	0.807504i
$658$		0			0
$659$	−116401.	−	319808.i	−0.268030	−	0.736407i	−0.998566	−	0.0535329i	$-0.982952\pi$
$659$	−116401.	−	319808.i	−0.268030	−	0.736407i	0.730536	−	0.682875i	$-0.239270\pi$
$660$		0			0
$661$	−252525.	+	211893.i	−0.577964	+	0.484969i	−0.884278	−	0.466961i	$-0.845349\pi$
$661$	−252525.	+	211893.i	−0.577964	+	0.484969i	0.306314	+	0.951931i	$0.400904\pi$
$662$		0			0
$663$	−126353.	+	635908.i	−0.287447	+	1.44666i
$664$		0			0
$665$	−102456.	+	59153.0i	−0.231683	+	0.133762i
$666$		0			0
$667$	−375954.	+	651171.i	−0.845050	+	1.46367i
$668$		0			0
$669$	10992.5	−	508616.i	0.0245609	−	1.13642i
$670$		0			0
$671$	−570830.	+	100653.i	−1.26783	+	0.223553i
$672$		0			0
$673$	−139865.	−	117361.i	−0.308801	−	0.259115i	0.475195	−	0.879880i	$-0.342378\pi$
$673$	−139865.	−	117361.i	−0.308801	−	0.259115i	−0.783996	+	0.620765i	$0.786822\pi$
$674$		0			0
$675$	−427202.	−	104254.i	−0.937618	−	0.228815i
$676$		0			0
$677$	−15878.7	+	18923.4i	−0.0346447	+	0.0412879i	−0.783089	−	0.621910i	$-0.786357\pi$
$677$	−15878.7	+	18923.4i	−0.0346447	+	0.0412879i	0.748444	+	0.663198i	$0.230801\pi$
$678$		0			0
$679$	178419.	+	1.01186e6i	0.386991	+	2.19474i
$680$		0			0
$681$	284806.	+	518877.i	0.614122	+	1.11885i
$682$		0			0
$683$	582433.	+	336268.i	1.24855	+	0.720849i	0.970819	−	0.239811i	$-0.0770856\pi$
$683$	582433.	+	336268.i	1.24855	+	0.720849i	0.277727	+	0.960660i	$0.410419\pi$
$684$		0			0
$685$	−23843.7	−	41298.5i	−0.0508150	−	0.0880142i
$686$		0			0
$687$	−331121.	+	377729.i	−0.701574	+	0.800327i
$688$		0			0
$689$	115526.	+	137679.i	0.243356	+	0.290020i
$690$		0			0
$691$	−422287.	+	153700.i	−0.884406	+	0.321897i	−0.743986	−	0.668195i	$-0.767067\pi$
$691$	−422287.	+	153700.i	−0.884406	+	0.321897i	−0.140419	+	0.990092i	$0.544845\pi$
$692$		0			0
$693$	857553.	−	446866.i	1.78564	−	0.930488i
$694$		0			0
$695$	125387.	+	22109.2i	0.259588	+	0.0457723i
$696$		0			0
$697$	5782.53	+	2104.67i	0.0119029	+	0.00433229i
$698$		0			0
$699$	−54936.1	+	44109.7i	−0.112436	+	0.0902775i
$700$		0			0
$701$		−	819779.i		−	1.66825i	−0.551577	−	0.834124i	$-0.685974\pi$
$701$		−	819779.i		−	1.66825i	0.551577	−	0.834124i	$-0.314026\pi$
$702$		0			0
$703$	414727.			0.839173
$704$		0			0
$705$	58005.1	−	149251.i	0.116705	−	0.300290i
$706$		0			0
$707$	−190209.	+	522595.i	−0.380533	+	1.04551i
$708$		0			0
$709$	−144814.	+	821282.i	−0.288084	+	1.63380i	0.405975	+	0.913884i	$0.366932\pi$
$709$	−144814.	+	821282.i	−0.288084	+	1.63380i	−0.694058	+	0.719919i	$0.744179\pi$
$710$		0			0
$711$	−668965.	−	276769.i	−1.32332	−	0.547493i
$712$		0			0
$713$	429020.	+	1.17872e6i	0.843914	+	2.31863i
$714$		0			0
$715$	120772.	−	101340.i	0.236240	−	0.198229i
$716$		0			0
$717$	620289.	−	210703.i	1.20658	−	0.409857i
$718$		0			0
$719$	149544.	−	86339.5i	0.289276	−	0.167014i	−0.348339	−	0.937369i	$-0.613254\pi$
$719$	149544.	−	86339.5i	0.289276	−	0.167014i	0.637615	+	0.770355i	$0.279921\pi$
$720$		0			0
$721$	−198300.	+	343466.i	−0.381463	+	0.660714i
$722$		0			0
$723$	49455.9	+	29996.6i	0.0946111	+	0.0573846i
$724$		0			0
$725$	−464653.	+	81930.9i	−0.884002	+	0.155873i
$726$		0			0
$727$	−692985.	−	581483.i	−1.31116	−	1.10019i	−0.988100	−	0.153813i	$-0.950845\pi$
$727$	−692985.	−	581483.i	−1.31116	−	1.10019i	−0.323058	−	0.946379i	$-0.604711\pi$
$728$		0			0
$729$	322996.	−	422023.i	0.607774	−	0.794110i
$730$		0			0
$731$	376585.	−	448797.i	0.704739	−	0.839875i
$732$		0			0
$733$	−89146.5	−	505575.i	−0.165919	−	0.940974i	−0.948112	−	0.317937i	$-0.897010\pi$
$733$	−89146.5	−	505575.i	−0.165919	−	0.940974i	0.782193	−	0.623037i	$-0.214101\pi$
$734$		0			0
$735$	−69377.0	+	114383.i	−0.128422	+	0.211733i
$736$		0			0
$737$	557926.	+	322119.i	1.02717	+	0.593036i
$738$		0			0
$739$	258778.	+	448217.i	0.473848	+	0.820729i	0.999552	−	0.0299391i	$-0.00953133\pi$
$739$	258778.	+	448217.i	0.473848	+	0.820729i	−0.525704	+	0.850668i	$0.676198\pi$
$740$		0			0
$741$	207539.	+	610975.i	0.377976	+	1.11272i
$742$		0			0
$743$	507847.	+	605228.i	0.919930	+	1.09633i	0.995072	+	0.0991572i	$0.0316147\pi$
$743$	507847.	+	605228.i	0.919930	+	1.09633i	−0.0751416	+	0.997173i	$0.523941\pi$
$744$		0			0
$745$	39995.4	−	14557.1i	0.0720606	−	0.0262279i
$746$		0			0
$747$	−34464.4	+	83302.3i	−0.0617632	+	0.149285i
$748$		0			0
$749$	237330.	+	41847.6i	0.423047	+	0.0745946i
$750$		0			0
$751$	−299004.	−	108829.i	−0.530148	−	0.192958i	0.0630562	−	0.998010i	$-0.479915\pi$
$751$	−299004.	−	108829.i	−0.530148	−	0.192958i	−0.593205	+	0.805052i	$0.702137\pi$
$752$		0			0
$753$	−81409.1	−	31638.8i	−0.143576	−	0.0557995i
$754$		0			0
$755$		−	51406.4i		−	0.0901827i
$756$		0			0
$757$	−265313.			−0.462985			−0.231493	−	0.972837i	$-0.574361\pi$
$757$	−265313.			−0.462985			−0.231493	+	0.972837i	$0.574361\pi$
$758$		0			0
$759$	−865269.	−	1.07765e6i	−1.50199	−	1.87065i
$760$		0			0
$761$	40717.9	−	111871.i	0.0703098	−	0.193175i	−0.899561	−	0.436796i	$-0.856113\pi$
$761$	40717.9	−	111871.i	0.0703098	−	0.193175i	0.969870	+	0.243621i	$0.0783354\pi$
$762$		0			0
$763$	28811.2	−	163396.i	0.0494894	−	0.280668i
$764$		0			0
$765$	−59537.5	−	114255.i	−0.101734	−	0.195232i
$766$		0			0
$767$	−227745.	−	625726.i	−0.387132	−	1.06364i
$768$		0			0
$769$	155869.	−	130789.i	0.263576	−	0.221167i	−0.501416	−	0.865206i	$-0.667187\pi$
$769$	155869.	−	130789.i	0.263576	−	0.221167i	0.764992	+	0.644040i	$0.222743\pi$
$770$		0			0
$771$	−435523.	−	381784.i	−0.732660	−	0.642257i
$772$		0			0
$773$	−861696.	+	497500.i	−1.44210	+	0.832596i	−0.997990	−	0.0633765i	$-0.979813\pi$
$773$	−861696.	+	497500.i	−1.44210	+	0.832596i	−0.444109	+	0.895973i	$0.646480\pi$
$774$		0			0
$775$	−393558.	+	681663.i	−0.655248	+	1.13492i
$776$		0			0
$777$	721098.	−	395803.i	1.19441	−	0.655597i
$778$		0			0
$779$	6031.38	−	1063.50i	0.00993898	−	0.00175251i
$780$		0			0
$781$	552051.	+	463226.i	0.905060	+	0.759435i
$782$		0			0
$783$	135187.	−	553957.i	0.220502	−	0.903550i
$784$		0			0
$785$	−87001.1	+	103684.i	−0.141184	+	0.168256i
$786$		0			0
$787$	−108800.	−	617037.i	−0.175663	−	0.996236i	−0.937376	−	0.348320i	$-0.886752\pi$
$787$	−108800.	−	617037.i	−0.175663	−	0.996236i	0.761712	−	0.647915i	$-0.224359\pi$
$788$		0			0
$789$	−393623.	−	8507.20i	−0.632305	−	0.0136657i
$790$		0			0
$791$	818429.	+	472520.i	1.30806	+	0.755210i
$792$		0			0
$793$	−383577.	−	664375.i	−0.609967	−	1.05649i
$794$		0			0
$795$	−35030.0	−	6960.35i	−0.0554251	−	0.0110128i
$796$		0			0
$797$	495189.	+	590143.i	0.779569	+	0.929054i	0.998914	−	0.0465936i	$-0.0148366\pi$
$797$	495189.	+	590143.i	0.779569	+	0.929054i	−0.219345	+	0.975647i	$0.570392\pi$
$798$		0			0
$799$	−1.22033e6	+	444162.i	−1.91153	+	0.695742i
$800$		0			0
$801$	384914.	+	85165.8i	0.599927	+	0.132739i
$802$		0			0
$803$	−693332.	−	122253.i	−1.07525	−	0.189596i
$804$		0			0
$805$	315139.	+	114701.i	0.486307	+	0.177001i
$806$		0			0
$807$	83151.4	+	539499.i	0.127680	+	0.828407i
$808$		0			0
$809$			592350.i			0.905069i	0.891747	+	0.452534i	$0.149480\pi$
$809$			592350.i			0.905069i	−0.891747	+	0.452534i	$0.850520\pi$
$810$		0			0
$811$	123096.			0.187156			0.0935780	−	0.995612i	$-0.470170\pi$
$811$	123096.			0.187156			0.0935780	+	0.995612i	$0.470170\pi$
$812$		0			0
$813$	283449.	−	43687.1i	0.428838	−	0.0660956i
$814$		0			0
$815$	−45752.3	+	125704.i	−0.0688808	+	0.189248i
$816$		0			0
$817$	101251.	−	574223.i	0.151689	−	0.860273i
$818$		0			0
$819$	943951.	+	864251.i	1.40728	+	1.28846i
$820$		0			0
$821$	−85712.1	−	235492.i	−0.127162	−	0.349374i	0.859732	−	0.510745i	$-0.170630\pi$
$821$	−85712.1	−	235492.i	−0.127162	−	0.349374i	−0.986894	+	0.161372i	$0.948408\pi$
$822$		0			0
$823$	−735747.	+	617365.i	−1.08625	+	0.911470i	−0.996425	−	0.0844878i	$-0.973075\pi$
$823$	−735747.	+	617365.i	−1.08625	+	0.911470i	−0.0898230	+	0.995958i	$0.528630\pi$
$824$		0			0
$825$	169011.	−	850595.i	0.248317	−	1.24973i
$826$		0			0
$827$	1.04316e6	−	602271.i	1.52525	−	0.880604i	0.525699	−	0.850670i	$-0.323804\pi$
$827$	1.04316e6	−	602271.i	1.52525	−	0.880604i	0.999552	−	0.0299338i	$-0.00952965\pi$
$828$		0			0
$829$	290735.	−	503567.i	0.423046	−	0.732738i	−0.573189	−	0.819423i	$-0.694294\pi$
$829$	290735.	−	503567.i	0.423046	−	0.732738i	0.996236	+	0.0866852i	$0.0276274\pi$
$830$		0			0
$831$	10038.1	−	464459.i	0.0145362	−	0.672582i
$832$		0			0
$833$	1.06847e6	−	188401.i	1.53983	−	0.271514i
$834$		0			0
$835$	136041.	+	114152.i	0.195117	+	0.163723i
$836$		0			0
$837$	−562966.	−	766788.i	−0.803584	−	1.09452i
$838$		0			0
$839$	366835.	−	437177.i	0.521131	−	0.621060i	−0.439717	−	0.898137i	$-0.644921\pi$
$839$	366835.	−	437177.i	0.521131	−	0.621060i	0.960848	+	0.277077i	$0.0893656\pi$
$840$		0			0
$841$	16577.5	+	94015.5i	0.0234383	+	0.132925i
$842$		0			0
$843$	−573893.	−	1.04556e6i	−0.807562	−	1.47127i
$844$		0			0
$845$	65241.2	+	37667.0i	0.0913711	+	0.0527531i
$846$		0			0
$847$	406433.	+	703962.i	0.566529	+	0.981256i
$848$		0			0
$849$	−238632.	+	272221.i	−0.331065	+	0.377665i
$850$		0			0
$851$	−755682.	−	900587.i	−1.04347	−	1.24356i
$852$		0			0
$853$	−412293.	+	150062.i	−0.566640	+	0.206240i	−0.609424	−	0.792844i	$-0.708599\pi$
$853$	−412293.	+	150062.i	−0.566640	+	0.206240i	0.0427842	+	0.999084i	$0.486377\pi$
$854$		0			0
$855$	−108172.	−	68850.3i	−0.147973	−	0.0941832i
$856$		0			0
$857$	576593.	+	101669.i	0.785068	+	0.138429i	0.551792	−	0.833982i	$-0.313944\pi$
$857$	576593.	+	101669.i	0.785068	+	0.138429i	0.233276	+	0.972410i	$0.425055\pi$
$858$		0			0
$859$	−321196.	−	116906.i	−0.435296	−	0.158435i	0.115072	−	0.993357i	$-0.463290\pi$
$859$	−321196.	−	116906.i	−0.435296	−	0.158435i	−0.550368	+	0.834923i	$0.685512\pi$
$860$		0			0
$861$	9471.98	−	7605.30i	0.0127772	−	0.0102591i
$862$		0			0
$863$		−	385745.i		−	0.517940i	−0.965885	−	0.258970i	$-0.916617\pi$
$863$		−	385745.i		−	0.517940i	0.965885	−	0.258970i	$-0.0833830\pi$
$864$		0			0
$865$	−50297.7			−0.0672227
$866$		0			0
$867$	−106206.	+	273275.i	−0.141289	+	0.363548i
$868$		0			0
$869$	488317.	−	1.34164e6i	0.646639	−	1.77663i
$870$		0			0
$871$	−148062.	+	839702.i	−0.195168	+	1.10685i
$872$		0			0
$873$	−883211.	+	678294.i	−1.15887	+	0.889999i
$874$		0			0
$875$	146550.	+	402643.i	0.191412	+	0.525901i
$876$		0			0
$877$	−508664.	+	426820.i	−0.661351	+	0.554939i	−0.910491	−	0.413528i	$-0.864296\pi$
$877$	−508664.	+	426820.i	−0.661351	+	0.554939i	0.249141	+	0.968467i	$0.419852\pi$
$878$		0			0
$879$	−951277.	+	323135.i	−1.23120	+	0.418221i
$880$		0			0
$881$	−1.23270e6	+	711698.i	−1.58820	+	0.916946i	−0.594593	+	0.804026i	$0.702687\pi$
$881$	−1.23270e6	+	711698.i	−1.58820	+	0.916946i	−0.993604	+	0.112920i	$0.963980\pi$
$882$		0			0
$883$	445152.	−	771025.i	0.570935	−	0.988888i	−0.425535	−	0.904942i	$-0.639914\pi$
$883$	445152.	−	771025.i	0.570935	−	0.988888i	0.996470	−	0.0839465i	$-0.0267525\pi$
$884$		0			0
$885$	113138.	+	68622.0i	0.144452	+	0.0876147i
$886$		0			0
$887$	703483.	−	124043.i	0.894142	−	0.157661i	0.292347	−	0.956312i	$-0.405564\pi$
$887$	703483.	−	124043.i	0.894142	−	0.157661i	0.601795	+	0.798651i	$0.294453\pi$
$888$		0			0
$889$	−314902.	−	264234.i	−0.398448	−	0.334337i
$890$		0			0
$891$	859029.	+	600437.i	1.08206	+	0.756331i
$892$		0			0
$893$	−830789.	+	990096.i	−1.04181	+	1.24158i
$894$		0			0
$895$	−37739.5	−	214031.i	−0.0471140	−	0.267197i
$896$		0			0
$897$	948581.	−	1.56394e6i	1.17893	−	1.94373i
$898$		0			0
$899$	−883918.	−	510330.i	−1.09369	−	0.631440i
$900$		0			0
$901$	144825.	+	250845.i	0.178400	+	0.308998i
$902$		0			0
$903$	−371973.	−	1.09505e6i	−0.456179	−	1.34295i
$904$		0			0
$905$	27915.6	+	33268.5i	0.0340839	+	0.0406196i
$906$		0			0
$907$	1.32996e6	−	484066.i	1.61668	−	0.588424i	0.633935	−	0.773386i	$-0.281439\pi$
$907$	1.32996e6	−	484066.i	1.61668	−	0.588424i	0.982746	+	0.184963i	$0.0592165\pi$
$908$		0			0
$909$	−597572.	+	78924.0i	−0.723206	+	0.0955172i
$910$		0			0
$911$	217051.	+	38272.0i	0.261532	+	0.0461152i	0.302877	−	0.953030i	$-0.402053\pi$
$911$	217051.	+	38272.0i	0.261532	+	0.0461152i	−0.0413443	+	0.999145i	$0.513164\pi$
$912$		0			0
$913$	−167066.	−	60807.1i	−0.200423	−	0.0729479i
$914$		0			0
$915$	142093.	+	55223.2i	0.169719	+	0.0659598i
$916$		0			0
$917$		−	1.79485e6i		−	2.13447i
$918$		0			0
$919$	743610.			0.880470			0.440235	−	0.897883i	$-0.354895\pi$
$919$	743610.			0.880470			0.440235	+	0.897883i	$0.354895\pi$
$920$		0			0
$921$	508983.	+	633910.i	0.600045	+	0.747323i
$922$		0			0
$923$	−326216.	+	896270.i	−0.382914	+	1.05205i
$924$		0			0
$925$	128102.	−	726501.i	0.149717	−	0.849089i
$926$		0			0
$927$	−429450.	−	18571.7i	−0.499751	−	0.0216119i
$928$		0			0
$929$	287994.	+	791258.i	0.333697	+	0.916826i	0.987141	+	0.159851i	$0.0511013\pi$
$929$	287994.	+	791258.i	0.333697	+	0.916826i	−0.653444	+	0.756975i	$0.726676\pi$
$930$		0			0
$931$	827180.	−	694087.i	0.954335	−	0.800782i
$932$		0			0
$933$	−402723.	−	353031.i	−0.462640	−	0.405555i
$934$		0			0
$935$	220042.	−	127041.i	0.251699	−	0.145319i
$936$		0			0
$937$	30873.8	−	53475.0i	0.0351650	−	0.0609076i	−0.847907	−	0.530145i	$-0.822138\pi$
$937$	30873.8	−	53475.0i	0.0351650	−	0.0609076i	0.883072	+	0.469237i	$0.155471\pi$
$938$		0			0
$939$	757021.	−	415520.i	0.858572	−	0.471261i
$940$		0			0
$941$	1.03510e6	−	182516.i	1.16897	−	0.206120i	0.444725	−	0.895667i	$-0.353301\pi$
$941$	1.03510e6	−	182516.i	1.16897	−	0.206120i	0.724241	+	0.689547i	$0.242190\pi$
$942$		0			0
$943$	−13299.3	−	11159.4i	−0.0149557	−	0.0125493i
$944$		0			0
$945$	−253791.	−	16475.7i	−0.284193	−	0.0184494i
$946$		0			0
$947$	−193091.	+	230117.i	−0.215309	+	0.256595i	−0.862879	−	0.505411i	$-0.831341\pi$
$947$	−193091.	+	230117.i	−0.215309	+	0.256595i	0.647570	+	0.762006i	$0.275785\pi$
$948$		0			0
$949$	−161803.	−	917633.i	−0.179662	−	1.01891i
$950$		0			0
$951$	−767561.	−	16588.9i	−0.848695	−	0.0183425i
$952$		0			0
$953$	−1.08256e6	−	625018.i	−1.19198	−	0.688188i	−0.233222	−	0.972424i	$-0.574927\pi$
$953$	−1.08256e6	−	625018.i	−1.19198	−	0.688188i	−0.958754	+	0.284236i	$0.908260\pi$
$954$		0			0
$955$	−147112.	−	254805.i	−0.161302	−	0.279384i
$956$		0			0
$957$	1.10297e6	+	219158.i	1.20432	+	0.239294i
$958$		0			0
$959$	490736.	+	584836.i	0.533594	+	0.635912i
$960$		0			0
$961$	−732205.	+	266501.i	−0.792841	+	0.288570i
$962$		0			0
$963$	78646.4	+	249074.i	0.0848060	+	0.268582i
$964$		0			0
$965$	221103.	+	38986.5i	0.237433	+	0.0418658i
$966$		0			0
$967$	1.40955e6	+	513035.i	1.50740	+	0.548648i	0.957964	−	0.286889i	$-0.0926212\pi$
$967$	1.40955e6	+	513035.i	1.50740	+	0.548648i	0.549434	+	0.835537i	$0.314843\pi$
$968$		0			0
$969$	158409.	+	1.02778e6i	0.168706	+	1.09459i
$970$		0			0
$971$		−	1.27140e6i		−	1.34848i	−0.738513	−	0.674239i	$-0.764472\pi$
$971$		−	1.27140e6i		−	1.34848i	0.738513	−	0.674239i	$-0.235528\pi$
$972$		0			0
$973$	−2.03835e6			−2.15305
$974$		0			0
$975$	1.13439e6	−	174840.i	1.19331	−	0.183921i
$976$		0			0
$977$	44349.1	−	121848.i	0.0464617	−	0.127653i	−0.914292	−	0.405057i	$-0.867252\pi$
$977$	44349.1	−	121848.i	0.0464617	−	0.127653i	0.960753	+	0.277404i	$0.0894740\pi$
$978$		0			0
$979$	−135005.	+	765649.i	−0.140858	+	0.798848i
$980$		0			0
$981$	171482.	−	54146.3i	0.178189	−	0.0562641i
$982$		0			0
$983$	276289.	+	759098.i	0.285928	+	0.785581i	0.996625	+	0.0820832i	$0.0261573\pi$
$983$	276289.	+	759098.i	0.285928	+	0.785581i	−0.710697	+	0.703498i	$0.751620\pi$
$984$		0			0
$985$	90378.6	−	75836.7i	0.0931523	−	0.0781640i
$986$		0			0
$987$	−499601.	+	2.51439e6i	−0.512849	+	2.58106i
$988$		0			0
$989$	−1.43143e6	+	826435.i	−1.46345	+	0.844921i
$990$		0			0
$991$	−469573.	+	813325.i	−0.478141	+	0.828164i	−0.999686	−	0.0250593i	$-0.992023\pi$
$991$	−469573.	+	813325.i	−0.478141	+	0.828164i	0.521545	+	0.853224i	$0.325356\pi$
$992$		0			0
$993$	−13199.9	+	610749.i	−0.0133866	+	0.619391i
$994$		0			0
$995$	−61112.3	+	10775.7i	−0.0617280	+	0.0108843i
$996$		0			0
$997$	−448969.	−	376730.i	−0.451675	−	0.379000i	0.388382	−	0.921498i	$-0.373034\pi$
$997$	−448969.	−	376730.i	−0.451675	−	0.379000i	−0.840057	+	0.542498i	$0.817479\pi$
$998$		0			0
$999$	741604.	+	494856.i	0.743089	+	0.495848i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	108.5.k.a.65.1	yes	72
3.2	odd	2		324.5.k.a.197.6		72
27.5	odd	18	inner	108.5.k.a.5.1	✓	72
27.22	even	9		324.5.k.a.125.6		72

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.5.k.a.5.1	✓	72	27.5	odd	18	inner
108.5.k.a.65.1	yes	72	1.1	even	1	trivial
324.5.k.a.125.6		72	27.22	even	9
324.5.k.a.197.6		72	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:	\( N \)	\(=\)	\( 108 = 2^{2} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	108.k (of order \(18\), degree \(6\), minimal)

\(f(q)\)	\(=\)	\(q+(-8.89497 + 1.37096i) q^{3} +(-1.59659 + 4.38660i) q^{5} +(12.9775 - 73.5989i) q^{7} +(77.2410 - 24.3892i) q^{9} +O(q^{10})\)	\(q+(-8.89497 + 1.37096i) q^{3} +(-1.59659 + 4.38660i) q^{5} +(12.9775 - 73.5989i) q^{7} +(77.2410 - 24.3892i) q^{9} +(54.6352 + 150.109i) q^{11} +(-161.958 + 135.899i) q^{13} +(8.18781 - 41.2076i) q^{15} +(-295.081 + 170.365i) q^{17} +(-169.557 + 293.681i) q^{19} +(-14.5334 + 672.451i) q^{21} +(946.685 - 166.926i) q^{23} +(462.085 + 387.735i) q^{25} +(-653.619 + 322.835i) q^{27} +(-502.780 + 599.189i) q^{29} +(226.590 + 1285.06i) q^{31} +(-691.772 - 1260.31i) q^{33} +(302.129 + 174.434i) q^{35} +(-611.487 - 1059.13i) q^{37} +(1254.30 - 1430.85i) q^{39} +(-11.6088 - 13.8349i) q^{41} +(-1615.74 + 588.079i) q^{43} +(-16.3366 + 377.765i) q^{45} +(3753.45 + 661.834i) q^{47} +(-2992.18 - 1089.06i) q^{49} +(2391.18 - 1919.94i) q^{51} -850.088i q^{53} -745.699 q^{55} +(1105.58 - 2844.73i) q^{57} +(-1077.21 + 2959.61i) q^{59} +(-630.093 + 3573.43i) q^{61} +(-792.627 - 6001.36i) q^{63} +(-337.554 - 927.421i) q^{65} +(3089.43 - 2592.34i) q^{67} +(-8191.89 + 2782.67i) q^{69} +(3906.92 - 2255.66i) q^{71} +(-2203.63 + 3816.80i) q^{73} +(-4641.80 - 2815.39i) q^{75} +(11756.9 - 2073.06i) q^{77} +(-6846.72 - 5745.08i) q^{79} +(5371.33 - 3767.69i) q^{81} +(-715.401 + 852.581i) q^{83} +(-276.200 - 1566.41i) q^{85} +(3650.75 - 6019.06i) q^{87} +(4214.90 + 2433.47i) q^{89} +(7900.21 + 13683.6i) q^{91} +(-3777.27 - 11119.9i) q^{93} +(-1017.55 - 1212.67i) q^{95} +(-12919.2 + 4702.22i) q^{97} +(7881.12 + 10262.1i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q + 9 q^{5} - 102 q^{9}+O(q^{10}) \) 72 * q + 9 * q^5 - 102 * q^9	\( 72 q + 9 q^{5} - 102 q^{9} + 18 q^{11} - 225 q^{15} - 282 q^{21} - 1278 q^{23} + 441 q^{25} + 54 q^{27} + 1854 q^{29} - 1665 q^{31} - 45 q^{33} - 2673 q^{35} + 6951 q^{39} - 5472 q^{41} + 1260 q^{43} + 5553 q^{45} + 5103 q^{47} - 5904 q^{49} + 1899 q^{51} + 1107 q^{57} - 10944 q^{59} + 8352 q^{61} - 11985 q^{63} + 8757 q^{65} + 378 q^{67} + 5607 q^{69} - 19764 q^{71} + 6111 q^{73} - 3453 q^{75} - 5679 q^{77} - 5652 q^{79} - 20466 q^{81} - 20061 q^{83} + 26100 q^{85} + 40545 q^{87} + 15633 q^{89} - 6039 q^{91} + 40179 q^{93} + 48024 q^{95} - 37530 q^{97} + 12177 q^{99}+O(q^{100}) \) 72 * q + 9 * q^5 - 102 * q^9 + 18 * q^11 - 225 * q^15 - 282 * q^21 - 1278 * q^23 + 441 * q^25 + 54 * q^27 + 1854 * q^29 - 1665 * q^31 - 45 * q^33 - 2673 * q^35 + 6951 * q^39 - 5472 * q^41 + 1260 * q^43 + 5553 * q^45 + 5103 * q^47 - 5904 * q^49 + 1899 * q^51 + 1107 * q^57 - 10944 * q^59 + 8352 * q^61 - 11985 * q^63 + 8757 * q^65 + 378 * q^67 + 5607 * q^69 - 19764 * q^71 + 6111 * q^73 - 3453 * q^75 - 5679 * q^77 - 5652 * q^79 - 20466 * q^81 - 20061 * q^83 + 26100 * q^85 + 40545 * q^87 + 15633 * q^89 - 6039 * q^91 + 40179 * q^93 + 48024 * q^95 - 37530 * q^97 + 12177 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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