LMFDB - Embedded newform 162.6.c.p.55.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [162,6,Mod(55,162)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(162, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4])) N = Newforms(chi, 6, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("162.55"); S:= CuspForms(chi, 6); N := Newforms(S);

Level:	$ N $	$=$	$ 162 = 2 \cdot 3^{4} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	162.c (of order $3$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,8,0,-32,-12,0,176] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$25.9821788097$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\zeta_{12})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 3^{3} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			55.2
Root			$-0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	162.55
Dual form			162.6.c.p.109.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(2.00000 - 3.46410i) q^{2} +(-8.00000 - 13.8564i) q^{4} +(9.99038 + 17.3038i) q^{5} +(101.158 - 175.210i) q^{7} -64.0000 q^{8} +79.9230 q^{10} +(10.4923 - 18.1731i) q^{11} +(116.696 + 202.124i) q^{13} +(-404.631 - 700.841i) q^{14} +(-128.000 + 221.703i) q^{16} +1304.81 q^{17} -1429.93 q^{19} +(159.846 - 276.862i) q^{20} +(-41.9691 - 72.6926i) q^{22} +(-964.146 - 1669.95i) q^{23} +(1362.88 - 2360.59i) q^{25} +933.569 q^{26} -3237.05 q^{28} +(2187.15 - 3788.25i) q^{29} +(-4576.07 - 7925.98i) q^{31} +(512.000 + 886.810i) q^{32} +(2609.62 - 4520.00i) q^{34} +4042.41 q^{35} -8889.11 q^{37} +(-2859.86 + 4953.42i) q^{38} +(-639.384 - 1107.45i) q^{40} +(-7905.70 - 13693.1i) q^{41} +(-2019.18 + 3497.33i) q^{43} -335.753 q^{44} -7713.17 q^{46} +(-13343.7 + 23112.0i) q^{47} +(-12062.3 - 20892.4i) q^{49} +(-5451.54 - 9442.34i) q^{50} +(1867.14 - 3233.98i) q^{52} +37821.7 q^{53} +419.287 q^{55} +(-6474.09 + 11213.5i) q^{56} +(-8748.59 - 15153.0i) q^{58} +(14840.6 + 25704.7i) q^{59} +(8559.30 - 14825.1i) q^{61} -36608.5 q^{62} +4096.00 q^{64} +(-2331.68 + 4038.58i) q^{65} +(-34359.8 - 59513.0i) q^{67} +(-10438.5 - 18080.0i) q^{68} +(8084.83 - 14003.3i) q^{70} +22122.8 q^{71} -3260.65 q^{73} +(-17778.2 + 30792.8i) q^{74} +(11439.4 + 19813.7i) q^{76} +(-2122.75 - 3676.71i) q^{77} +(-33869.5 + 58663.7i) q^{79} -5115.08 q^{80} -63245.6 q^{82} +(23085.8 - 39985.9i) q^{83} +(13035.6 + 22578.3i) q^{85} +(8076.73 + 13989.3i) q^{86} +(-671.505 + 1163.08i) q^{88} +26065.6 q^{89} +47218.8 q^{91} +(-15426.3 + 26719.2i) q^{92} +(53374.8 + 92447.9i) q^{94} +(-14285.6 - 24743.3i) q^{95} +(-8814.24 + 15266.7i) q^{97} -96498.0 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{2} - 32 q^{4} - 12 q^{5} + 176 q^{7} - 256 q^{8} - 96 q^{10} - 540 q^{11} + 446 q^{13} - 704 q^{14} - 512 q^{16} + 3120 q^{17} - 3184 q^{19} - 192 q^{20} + 2160 q^{22} - 1404 q^{23} + 4828 q^{25}+ \cdots - 64080 q^{98}+O(q^{100}) $ 4 * q + 8 * q^2 - 32 * q^4 - 12 * q^5 + 176 * q^7 - 256 * q^8 - 96 * q^10 - 540 * q^11 + 446 * q^13 - 704 * q^14 - 512 * q^16 + 3120 * q^17 - 3184 * q^19 - 192 * q^20 + 2160 * q^22 - 1404 * q^23 + 4828 * q^25 + 3568 * q^26 - 5632 * q^28 - 2340 * q^29 + 776 * q^31 + 2048 * q^32 + 6240 * q^34 + 9768 * q^35 + 7052 * q^37 - 6368 * q^38 + 768 * q^40 - 27840 * q^41 - 23104 * q^43 + 17280 * q^44 - 11232 * q^46 - 38784 * q^47 - 8010 * q^49 - 19312 * q^50 + 7136 * q^52 + 91344 * q^53 + 36720 * q^55 - 11264 * q^56 + 9360 * q^58 - 2160 * q^59 - 8974 * q^61 + 6208 * q^62 + 16384 * q^64 + 2136 * q^65 - 71656 * q^67 - 24960 * q^68 + 19536 * q^70 + 49416 * q^71 + 120644 * q^73 + 14104 * q^74 + 25472 * q^76 - 19008 * q^77 - 74392 * q^79 + 6144 * q^80 - 222720 * q^82 - 29496 * q^83 + 17910 * q^85 + 92416 * q^86 + 34560 * q^88 - 221952 * q^89 + 83248 * q^91 - 22464 * q^92 + 155136 * q^94 - 23388 * q^95 + 5564 * q^97 - 64080 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/162\mathbb{Z}\right)^\times$.

$n$	$83$
$\chi(n)$	$e\left(\frac{2}{3}\right)$

Coefficient data

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

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

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

$2$	2.00000	−	3.46410i	0.353553	−	0.612372i
$2$	2.00000	−	3.46410i	0.353553	−	0.612372i
$3$		0			0
$3$		0			0
$4$	−8.00000	−	13.8564i	−0.250000	−	0.433013i
$5$	9.99038	+	17.3038i	0.178713	+	0.309541i	0.941440	−	0.337180i	$-0.109473\pi$
$5$	9.99038	+	17.3038i	0.178713	+	0.309541i	−0.762727	+	0.646721i	$0.776140\pi$
$6$		0			0
$7$	101.158	−	175.210i	0.780286	−	1.35149i	−0.151489	−	0.988459i	$-0.548407\pi$
$7$	101.158	−	175.210i	0.780286	−	1.35149i	0.931775	−	0.363036i	$-0.118260\pi$
$8$	−64.0000			−0.353553
$9$		0			0
$10$	79.9230			0.252739
$11$	10.4923	−	18.1731i	0.0261449	−	0.0452844i	−0.852657	−	0.522471i	$-0.825010\pi$
$11$	10.4923	−	18.1731i	0.0261449	−	0.0452844i	0.878802	+	0.477187i	$0.158344\pi$
$12$		0			0
$13$	116.696	+	202.124i	0.191513	+	0.331710i	0.945752	−	0.324890i	$-0.105327\pi$
$13$	116.696	+	202.124i	0.191513	+	0.331710i	−0.754239	+	0.656600i	$0.771994\pi$
$14$	−404.631	−	700.841i	−0.551745	−	0.955651i
$15$		0			0
$16$	−128.000	+	221.703i	−0.125000	+	0.216506i
$17$	1304.81			1.09503			0.547514	−	0.836796i	$-0.315574\pi$
$17$	1304.81			1.09503			0.547514	+	0.836796i	$0.315574\pi$
$18$		0			0
$19$	−1429.93			−0.908722			−0.454361	−	0.890818i	$-0.650132\pi$
$19$	−1429.93			−0.908722			−0.454361	+	0.890818i	$0.650132\pi$
$20$	159.846	−	276.862i	0.0893567	−	0.154770i
$21$		0			0
$22$	−41.9691	−	72.6926i	−0.0184873	−	0.0320209i
$23$	−964.146	−	1669.95i	−0.380035	−	0.658239i	0.611032	−	0.791606i	$-0.290755\pi$
$23$	−964.146	−	1669.95i	−0.380035	−	0.658239i	−0.991067	+	0.133366i	$0.957421\pi$
$24$		0			0
$25$	1362.88	−	2360.59i	0.436123	−	0.755387i
$26$	933.569			0.270840
$27$		0			0
$28$	−3237.05			−0.780286
$29$	2187.15	−	3788.25i	0.482929	−	0.836457i	−0.516879	−	0.856058i	$-0.672906\pi$
$29$	2187.15	−	3788.25i	0.482929	−	0.836457i	0.999808	+	0.0196013i	$0.00623968\pi$
$30$		0			0
$31$	−4576.07	−	7925.98i	−0.855240	−	1.48132i	−0.876422	−	0.481544i	$-0.840076\pi$
$31$	−4576.07	−	7925.98i	−0.855240	−	1.48132i	0.0211813	−	0.999776i	$-0.493257\pi$
$32$	512.000	+	886.810i	0.0883883	+	0.153093i
$33$		0			0
$34$	2609.62	−	4520.00i	0.387151	−	0.670565i
$35$	4042.41			0.557790
$36$		0			0
$37$	−8889.11			−1.06747			−0.533733	−	0.845653i	$-0.679211\pi$
$37$	−8889.11			−1.06747			−0.533733	+	0.845653i	$0.679211\pi$
$38$	−2859.86	+	4953.42i	−0.321282	+	0.556476i
$39$		0			0
$40$	−639.384	−	1107.45i	−0.0631847	−	0.109439i
$41$	−7905.70	−	13693.1i	−0.734481	−	1.27216i	−0.954951	−	0.296765i	$-0.904092\pi$
$41$	−7905.70	−	13693.1i	−0.734481	−	1.27216i	0.220469	−	0.975394i	$-0.429241\pi$
$42$		0			0
$43$	−2019.18	+	3497.33i	−0.166535	+	0.288446i	−0.937199	−	0.348795i	$-0.886591\pi$
$43$	−2019.18	+	3497.33i	−0.166535	+	0.288446i	0.770665	+	0.637241i	$0.219924\pi$
$44$	−335.753			−0.0261449
$45$		0			0
$46$	−7713.17			−0.537450
$47$	−13343.7	+	23112.0i	−0.881113	+	1.52613i	−0.0310080	+	0.999519i	$0.509872\pi$
$47$	−13343.7	+	23112.0i	−0.881113	+	1.52613i	−0.850105	+	0.526613i	$0.823462\pi$
$48$		0			0
$49$	−12062.3	−	20892.4i	−0.717692	−	1.24308i
$50$	−5451.54	−	9442.34i	−0.308386	−	0.534139i
$51$		0			0
$52$	1867.14	−	3233.98i	0.0957565	−	0.165855i
$53$	37821.7			1.84949			0.924744	−	0.380590i	$-0.124279\pi$
$53$	37821.7			1.84949			0.924744	+	0.380590i	$0.124279\pi$
$54$		0			0
$55$	419.287			0.0186898
$56$	−6474.09	+	11213.5i	−0.275873	+	0.477826i
$57$		0			0
$58$	−8748.59	−	15153.0i	−0.341482	−	0.591464i
$59$	14840.6	+	25704.7i	0.555037	+	0.961352i	0.997901	+	0.0647630i	$0.0206292\pi$
$59$	14840.6	+	25704.7i	0.555037	+	0.961352i	−0.442864	+	0.896589i	$0.646038\pi$
$60$		0			0
$61$	8559.30	−	14825.1i	0.294519	−	0.510122i	−0.680354	−	0.732884i	$-0.738174\pi$
$61$	8559.30	−	14825.1i	0.294519	−	0.510122i	0.974873	+	0.222762i	$0.0715072\pi$
$62$	−36608.5			−1.20949
$63$		0			0
$64$	4096.00			0.125000
$65$	−2331.68	+	4038.58i	−0.0684519	+	0.118562i
$66$		0			0
$67$	−34359.8	−	59513.0i	−0.935113	−	1.61966i	−0.774433	−	0.632656i	$-0.781965\pi$
$67$	−34359.8	−	59513.0i	−0.935113	−	1.61966i	−0.160680	−	0.987007i	$-0.551369\pi$
$68$	−10438.5	−	18080.0i	−0.273757	−	0.474161i
$69$		0			0
$70$	8084.83	−	14003.3i	0.197209	−	0.341575i
$71$	22122.8			0.520827			0.260414	−	0.965497i	$-0.416141\pi$
$71$	22122.8			0.520827			0.260414	+	0.965497i	$0.416141\pi$
$72$		0			0
$73$	−3260.65			−0.0716139			−0.0358070	−	0.999359i	$-0.511400\pi$
$73$	−3260.65			−0.0716139			−0.0358070	+	0.999359i	$0.511400\pi$
$74$	−17778.2	+	30792.8i	−0.377406	+	0.653687i
$75$		0			0
$76$	11439.4	+	19813.7i	0.227180	+	0.393488i
$77$	−2122.75	−	3676.71i	−0.0408011	−	0.0706695i
$78$		0			0
$79$	−33869.5	+	58663.7i	−0.610578	+	1.05755i	0.380565	+	0.924754i	$0.375729\pi$
$79$	−33869.5	+	58663.7i	−0.610578	+	1.05755i	−0.991143	+	0.132798i	$0.957604\pi$
$80$	−5115.08			−0.0893567
$81$		0			0
$82$	−63245.6			−1.03871
$83$	23085.8	−	39985.9i	0.367833	−	0.637105i	−0.621394	−	0.783499i	$-0.713433\pi$
$83$	23085.8	−	39985.9i	0.367833	−	0.637105i	0.989226	+	0.146393i	$0.0467665\pi$
$84$		0			0
$85$	13035.6	+	22578.3i	0.195696	+	0.338956i
$86$	8076.73	+	13989.3i	0.117758	+	0.203962i
$87$		0			0
$88$	−671.505	+	1163.08i	−0.00924363	+	0.0160104i
$89$	26065.6			0.348813			0.174407	−	0.984674i	$-0.444199\pi$
$89$	26065.6			0.348813			0.174407	+	0.984674i	$0.444199\pi$
$90$		0			0
$91$	47218.8			0.597739
$92$	−15426.3	+	26719.2i	−0.190017	+	0.329120i
$93$		0			0
$94$	53374.8	+	92447.9i	0.623041	+	1.07914i
$95$	−14285.6	−	24743.3i	−0.162401	−	0.281286i
$96$		0			0
$97$	−8814.24	+	15266.7i	−0.0951165	+	0.164747i	−0.909657	−	0.415360i	$-0.863656\pi$
$97$	−8814.24	+	15266.7i	−0.0951165	+	0.164747i	0.814541	+	0.580106i	$0.196989\pi$
$98$	−96498.0			−1.01497
$99$		0			0
$100$	−43612.3			−0.436123
$101$	82822.6	−	143453.i	0.807878	−	1.39929i	−0.106453	−	0.994318i	$-0.533950\pi$
$101$	82822.6	−	143453.i	0.807878	−	1.39929i	0.914331	−	0.404967i	$-0.132717\pi$
$102$		0			0
$103$	32320.9	+	55981.5i	0.300186	+	0.519938i	0.976178	−	0.216972i	$-0.0696179\pi$
$103$	32320.9	+	55981.5i	0.300186	+	0.519938i	−0.675992	+	0.736909i	$0.736285\pi$
$104$	−7468.55	−	12935.9i	−0.0677101	−	0.117277i
$105$		0			0
$106$	75643.4	−	131018.i	0.653893	−	1.13258i
$107$	28673.3			0.242113			0.121057	−	0.992646i	$-0.461372\pi$
$107$	28673.3			0.242113			0.121057	+	0.992646i	$0.461372\pi$
$108$		0			0
$109$	173801.			1.40115			0.700575	−	0.713578i	$-0.252927\pi$
$109$	173801.			1.40115			0.700575	+	0.713578i	$0.252927\pi$
$110$	838.574	−	1452.45i	0.00660784	−	0.0114451i
$111$		0			0
$112$	25896.4	+	44853.8i	0.195071	+	0.337874i
$113$	79890.0	+	138374.i	0.588567	+	1.01943i	0.994420	+	0.105490i	$0.0336412\pi$
$113$	79890.0	+	138374.i	0.588567	+	1.01943i	−0.405853	+	0.913938i	$0.633025\pi$
$114$		0			0
$115$	19264.4	−	33366.9i	0.135835	−	0.235272i
$116$	−69988.7			−0.482929
$117$		0			0
$118$	118725.			0.784941
$119$	131992.	−	228616.i	0.854435	−	1.47993i
$120$		0			0
$121$	80305.3	+	139093.i	0.498633	+	0.863657i
$122$	−34237.2	−	59300.6i	−0.208257	−	0.360711i
$123$		0			0
$124$	−73217.1	+	126816.i	−0.427620	+	0.740660i
$125$	116903.			0.669191
$126$		0			0
$127$	48402.8			0.266294			0.133147	−	0.991096i	$-0.457492\pi$
$127$	48402.8			0.266294			0.133147	+	0.991096i	$0.457492\pi$
$128$	8192.00	−	14189.0i	0.0441942	−	0.0765466i
$129$		0			0
$130$	9326.71	+	16154.3i	0.0484028	+	0.0838361i
$131$	30009.1	+	51977.3i	0.152783	+	0.264628i	0.932250	−	0.361816i	$-0.117843\pi$
$131$	30009.1	+	51977.3i	0.152783	+	0.264628i	−0.779467	+	0.626444i	$0.784510\pi$
$132$		0			0
$133$	−144648.	+	250538.i	−0.709063	+	1.22813i
$134$	−274879.			−1.32245
$135$		0			0
$136$	−83507.9			−0.387151
$137$	−129008.	+	223449.i	−0.587240	+	1.01713i	0.407352	+	0.913271i	$0.366452\pi$
$137$	−129008.	+	223449.i	−0.587240	+	1.01713i	−0.994592	+	0.103859i	$0.966881\pi$
$138$		0			0
$139$	163161.	+	282603.i	0.716275	+	1.24062i	0.962466	+	0.271403i	$0.0874876\pi$
$139$	163161.	+	282603.i	0.716275	+	1.24062i	−0.246191	+	0.969221i	$0.579179\pi$
$140$	−32339.3	−	56013.3i	−0.139448	−	0.241530i
$141$		0			0
$142$	44245.5	−	76635.5i	0.184140	−	0.318940i
$143$	4897.63			0.0200284
$144$		0			0
$145$	87401.7			0.345223
$146$	−6521.30	+	11295.2i	−0.0253193	+	0.0438544i
$147$		0			0
$148$	71112.9	+	123171.i	0.266867	+	0.462226i
$149$	172196.	+	298253.i	0.635416	+	1.10057i	0.986427	+	0.164202i	$0.0525049\pi$
$149$	172196.	+	298253.i	0.635416	+	1.10057i	−0.351010	+	0.936372i	$0.614162\pi$
$150$		0			0
$151$	−80917.2	+	140153.i	−0.288801	+	0.500218i	−0.973524	−	0.228586i	$-0.926590\pi$
$151$	−80917.2	+	140153.i	−0.288801	+	0.500218i	0.684723	+	0.728804i	$0.259923\pi$
$152$	91515.6			0.321282
$153$		0			0
$154$	−16982.0			−0.0577014
$155$	91433.3	−	158367.i	0.305686	−	0.529463i
$156$		0			0
$157$	−219924.	−	380919.i	−0.712071	−	1.23334i	−0.964078	−	0.265618i	$-0.914424\pi$
$157$	−219924.	−	380919.i	−0.712071	−	1.23334i	0.252008	−	0.967725i	$-0.418909\pi$
$158$	135478.	+	234655.i	0.431744	+	0.747802i
$159$		0			0
$160$	−10230.2	+	17719.1i	−0.0315924	+	0.0547196i
$161$	−390123.			−1.18614
$162$		0			0
$163$	133043.			0.392213			0.196107	−	0.980583i	$-0.437170\pi$
$163$	133043.			0.392213			0.196107	+	0.980583i	$0.437170\pi$
$164$	−126491.	+	219089.i	−0.367241	+	0.636079i
$165$		0			0
$166$	−92343.4	−	159943.i	−0.260097	−	0.450501i
$167$	−138865.	−	240521.i	−0.385302	−	0.667363i	0.606509	−	0.795077i	$-0.292569\pi$
$167$	−138865.	−	240521.i	−0.385302	−	0.667363i	−0.991811	+	0.127714i	$0.959236\pi$
$168$		0			0
$169$	158411.	−	274375.i	0.426646	−	0.738972i
$170$	104285.			0.276756
$171$		0			0
$172$	64613.8			0.166535
$173$	−115075.	+	199316.i	−0.292325	+	0.506322i	−0.974359	−	0.224999i	$-0.927762\pi$
$173$	−115075.	+	199316.i	−0.292325	+	0.506322i	0.682034	+	0.731320i	$0.261096\pi$
$174$		0			0
$175$	−275732.	−	477583.i	−0.680601	−	1.17884i
$176$	2686.02	+	4652.32i	0.00653624	+	0.0113211i
$177$		0			0
$178$	52131.2	−	90293.9i	0.123324	−	0.213604i
$179$	−211343.			−0.493011			−0.246505	−	0.969141i	$-0.579282\pi$
$179$	−211343.			−0.493011			−0.246505	+	0.969141i	$0.579282\pi$
$180$		0			0
$181$	−215516.			−0.488971			−0.244485	−	0.969653i	$-0.578619\pi$
$181$	−215516.			−0.488971			−0.244485	+	0.969653i	$0.578619\pi$
$182$	94437.7	−	163571.i	0.211333	−	0.366039i
$183$		0			0
$184$	61705.3	+	106877.i	0.134363	+	0.232723i
$185$	−88805.6	−	153816.i	−0.190770	−	0.330424i
$186$		0			0
$187$	13690.4	−	23712.5i	0.0286295	−	0.0495877i
$188$	426998.			0.881113
$189$		0			0
$190$	−114284.			−0.229669
$191$	−281147.	+	486961.i	−0.557635	+	0.965853i	0.440058	+	0.897969i	$0.354958\pi$
$191$	−281147.	+	486961.i	−0.557635	+	0.965853i	−0.997693	+	0.0678833i	$0.978375\pi$
$192$		0			0
$193$	202689.	+	351068.i	0.391685	+	0.678419i	0.992672	−	0.120840i	$-0.0385589\pi$
$193$	202689.	+	351068.i	0.391685	+	0.678419i	−0.600987	+	0.799259i	$0.705226\pi$
$194$	35257.0	+	61066.9i	0.0672575	+	0.116493i
$195$		0			0
$196$	−192996.	+	334279.i	−0.358846	+	0.621540i
$197$	1.02252e6			1.87718			0.938589	−	0.345036i	$-0.112133\pi$
$197$	1.02252e6			1.87718			0.938589	+	0.345036i	$0.112133\pi$
$198$		0			0
$199$	−11535.1			−0.0206485			−0.0103243	−	0.999947i	$-0.503286\pi$
$199$	−11535.1			−0.0206485			−0.0103243	+	0.999947i	$0.503286\pi$
$200$	−87224.6	+	151077.i	−0.154193	+	0.267070i
$201$		0			0
$202$	−331291.	−	573812.i	−0.571256	−	0.989444i
$203$	−442493.	−	766421.i	−0.753645	−	1.30535i
$204$		0			0
$205$	157962.	−	273598.i	0.262523	−	0.454704i
$206$	258567.			0.424527
$207$		0			0
$208$	−59748.4			−0.0957565
$209$	−15003.2	+	25986.3i	−0.0237585	+	0.0411509i
$210$		0			0
$211$	−327550.	−	567334.i	−0.506491	−	0.877269i	−0.999972	−	0.00751170i	$-0.997609\pi$
$211$	−327550.	−	567334.i	−0.506491	−	0.877269i	0.493481	−	0.869757i	$-0.335724\pi$
$212$	−302574.	−	524073.i	−0.462372	−	0.800852i
$213$		0			0
$214$	57346.7	−	99327.4i	0.0856000	−	0.148264i
$215$	−80689.6			−0.119048
$216$		0			0
$217$	−1.85162e6			−2.66933
$218$	347601.	−	602063.i	0.495382	−	0.858026i
$219$		0			0
$220$	−3354.30	−	5809.81i	−0.00467245	−	0.00809292i
$221$	152266.	+	263733.i	0.209712	+	0.363232i
$222$		0			0
$223$	435975.	−	755130.i	0.587082	−	1.01686i	−0.407530	−	0.913192i	$-0.633610\pi$
$223$	435975.	−	755130.i	0.587082	−	1.01686i	0.994612	−	0.103665i	$-0.0330568\pi$
$224$	207171.			0.275873
$225$		0			0
$226$	639120.			0.832360
$227$	−323662.	+	560600.i	−0.416896	+	0.722085i	−0.995625	−	0.0934346i	$-0.970215\pi$
$227$	−323662.	+	560600.i	−0.416896	+	0.722085i	0.578729	+	0.815520i	$0.303549\pi$
$228$		0			0
$229$	132358.	+	229251.i	0.166787	+	0.288883i	0.937288	−	0.348555i	$-0.113327\pi$
$229$	132358.	+	229251.i	0.166787	+	0.288883i	−0.770501	+	0.637438i	$0.779994\pi$
$230$	−77057.5	−	133467.i	−0.0960495	−	0.166363i
$231$		0			0
$232$	−139977.	+	242448.i	−0.170741	+	0.295732i
$233$	−859254.			−1.03689			−0.518444	−	0.855112i	$-0.673488\pi$
$233$	−859254.			−1.03689			−0.518444	+	0.855112i	$0.673488\pi$
$234$		0			0
$235$	−533235.			−0.629867
$236$	237450.	−	411275.i	0.277518	−	0.480676i
$237$		0			0
$238$	−527967.	−	914465.i	−0.604177	−	1.04647i
$239$	−181801.	−	314888.i	−0.205873	−	0.356583i	0.744537	−	0.667581i	$-0.232670\pi$
$239$	−181801.	−	314888.i	−0.205873	−	0.356583i	−0.950411	+	0.310998i	$0.899337\pi$
$240$		0			0
$241$	665386.	−	1.15248e6i	0.737957	−	1.27818i	−0.215456	−	0.976514i	$-0.569124\pi$
$241$	665386.	−	1.15248e6i	0.737957	−	1.27818i	0.953414	−	0.301666i	$-0.0975428\pi$
$242$	642443.			0.705173
$243$		0			0
$244$	−273898.			−0.294519
$245$	241013.	−	417447.i	0.256522	−	0.444310i
$246$		0			0
$247$	−166867.	−	289023.i	−0.174032	−	0.301432i
$248$	292868.	+	507263.i	0.302373	+	0.523726i
$249$		0			0
$250$	233806.	−	404963.i	0.236595	−	0.409794i
$251$	−615398.			−0.616555			−0.308278	−	0.951296i	$-0.599753\pi$
$251$	−615398.			−0.616555			−0.308278	+	0.951296i	$0.599753\pi$
$252$		0			0
$253$	−40464.3			−0.0397439
$254$	96805.7	−	167672.i	0.0941492	−	0.163071i
$255$		0			0
$256$	−32768.0	−	56755.8i	−0.0312500	−	0.0541266i
$257$	−588685.	−	1.01963e6i	−0.555968	−	0.962965i	−0.997828	−	0.0658808i	$-0.979014\pi$
$257$	−588685.	−	1.01963e6i	−0.555968	−	0.962965i	0.441859	−	0.897084i	$-0.354319\pi$
$258$		0			0
$259$	−899202.	+	1.55746e6i	−0.832929	+	1.44267i
$260$	74613.7			0.0684519
$261$		0			0
$262$	240073.			0.216068
$263$	−69008.5	+	119526.i	−0.0615195	+	0.106555i	−0.895145	−	0.445775i	$-0.852928\pi$
$263$	−69008.5	+	119526.i	−0.0615195	+	0.106555i	0.833625	+	0.552330i	$0.186261\pi$
$264$		0			0
$265$	377853.	+	654461.i	0.330528	+	0.572492i
$266$	578594.	+	1.00215e6i	0.501383	+	0.868421i
$267$		0			0
$268$	−549757.	+	952207.i	−0.467556	+	0.809831i
$269$	985165.			0.830096			0.415048	−	0.909800i	$-0.363765\pi$
$269$	985165.			0.830096			0.415048	+	0.909800i	$0.363765\pi$
$270$		0			0
$271$	841010.			0.695629			0.347815	−	0.937563i	$-0.386924\pi$
$271$	841010.			0.695629			0.347815	+	0.937563i	$0.386924\pi$
$272$	−167016.	+	289280.i	−0.136879	+	0.237081i
$273$		0			0
$274$	516033.	+	893795.i	0.415242	+	0.719220i
$275$	−28599.5	−	49535.8i	−0.0228048	−	0.0394991i
$276$		0			0
$277$	−10653.6	+	18452.5i	−0.00834248	+	0.0144496i	−0.870167	−	0.492758i	$-0.835989\pi$
$277$	−10653.6	+	18452.5i	−0.00834248	+	0.0144496i	0.861824	+	0.507207i	$0.169322\pi$
$278$	1.30529e6			1.01297
$279$		0			0
$280$	−258715.			−0.197209
$281$	−841126.	+	1.45687e6i	−0.635470	+	1.10067i	0.350945	+	0.936396i	$0.385860\pi$
$281$	−841126.	+	1.45687e6i	−0.635470	+	1.10067i	−0.986415	+	0.164271i	$0.947473\pi$
$282$		0			0
$283$	395785.	+	685520.i	0.293761	+	0.508808i	0.974696	−	0.223535i	$-0.0717598\pi$
$283$	395785.	+	685520.i	0.293761	+	0.508808i	−0.680935	+	0.732344i	$0.738427\pi$
$284$	−176982.	−	306542.i	−0.130207	−	0.225525i
$285$		0			0
$286$	9795.26	−	16965.9i	0.00708110	−	0.0122648i
$287$	−3.19889e6			−2.29242
$288$		0			0
$289$	282676.			0.199087
$290$	174803.	−	302769.i	0.122055	−	0.211405i
$291$		0			0
$292$	26085.2	+	45180.9i	0.0179035	+	0.0310097i
$293$	1.15302e6	+	1.99710e6i	0.784638	+	1.35903i	0.929215	+	0.369539i	$0.120485\pi$
$293$	1.15302e6	+	1.99710e6i	0.784638	+	1.35903i	−0.144577	+	0.989494i	$0.546182\pi$
$294$		0			0
$295$	−296527.	+	513599.i	−0.198385	+	0.343613i
$296$	568903.			0.377406
$297$		0			0
$298$	1.37757e6			0.898615
$299$	225024.	−	389753.i	0.145563	−	0.252123i
$300$		0			0
$301$	408511.	+	707563.i	0.259889	+	0.450141i
$302$	323669.	+	560611.i	0.204213	+	0.353707i
$303$		0			0
$304$	183031.	−	317019.i	0.113590	−	0.196744i
$305$	342043.			0.210538
$306$		0			0
$307$	2.92870e6			1.77349			0.886745	−	0.462260i	$-0.152961\pi$
$307$	2.92870e6			1.77349			0.886745	+	0.462260i	$0.152961\pi$
$308$	−33963.9	+	58827.3i	−0.0204005	+	0.0353348i
$309$		0			0
$310$	−365733.	−	633469.i	−0.216153	−	0.374387i
$311$	−540095.	−	935473.i	−0.316643	−	0.548441i	0.663143	−	0.748493i	$-0.269222\pi$
$311$	−540095.	−	935473.i	−0.316643	−	0.548441i	−0.979785	+	0.200052i	$0.935889\pi$
$312$		0			0
$313$	321741.	−	557272.i	0.185629	−	0.321519i	−0.758159	−	0.652069i	$-0.773901\pi$
$313$	321741.	−	557272.i	0.185629	−	0.321519i	0.943788	+	0.330550i	$0.107234\pi$
$314$	−1.75939e6			−1.00702
$315$		0			0
$316$	1.08382e6			0.610578
$317$	577091.	−	999552.i	0.322550	−	0.558672i	−0.658464	−	0.752612i	$-0.728793\pi$
$317$	577091.	−	999552.i	0.322550	−	0.558672i	0.981013	+	0.193940i	$0.0621268\pi$
$318$		0			0
$319$	−45896.3	−	79494.7i	−0.0252523	−	0.0437382i
$320$	40920.6	+	70876.6i	0.0223392	+	0.0386926i
$321$		0			0
$322$	−780246.	+	1.35143e6i	−0.419365	+	0.726361i
$323$	−1.86579e6			−0.995076
$324$		0			0
$325$	636174.			0.334093
$326$	266085.	−	460874.i	0.138668	−	0.240180i
$327$		0			0
$328$	505965.	+	876357.i	0.259678	+	0.449776i
$329$	2.69964e6	+	4.67591e6i	1.37504	+	2.38164i
$330$		0			0
$331$	107519.	−	186228.i	0.0539404	−	0.0934274i	−0.837794	−	0.545986i	$-0.816155\pi$
$331$	107519.	−	186228.i	0.0539404	−	0.0934274i	0.891735	+	0.452558i	$0.149489\pi$
$332$	−738747.			−0.367833
$333$		0			0
$334$	−1.11092e6			−0.544900
$335$	686535.	−	1.18911e6i	0.334234	−	0.578911i
$336$		0			0
$337$	1.71484e6	+	2.97019e6i	0.822526	+	1.42466i	0.903796	+	0.427964i	$0.140769\pi$
$337$	1.71484e6	+	2.97019e6i	0.822526	+	1.42466i	−0.0812700	+	0.996692i	$0.525898\pi$
$338$	−633642.	−	1.09750e6i	−0.301684	−	0.522532i
$339$		0			0
$340$	208569.	−	361252.i	0.0978481	−	0.169478i
$341$	−192053.			−0.0894408
$342$		0			0
$343$	−1.48044e6			−0.679448
$344$	129228.	−	223829.i	0.0588788	−	0.101981i
$345$		0			0
$346$	460300.	+	797263.i	0.206705	+	0.358024i
$347$	−802264.	−	1.38956e6i	−0.357679	−	0.619518i	0.629894	−	0.776681i	$-0.283098\pi$
$347$	−802264.	−	1.38956e6i	−0.357679	−	0.619518i	−0.987573	+	0.157163i	$0.949765\pi$
$348$		0			0
$349$	−356261.	+	617062.i	−0.156569	+	0.271185i	−0.933629	−	0.358241i	$-0.883377\pi$
$349$	−356261.	+	617062.i	−0.156569	+	0.271185i	0.777060	+	0.629426i	$0.216710\pi$
$350$	−2.20586e6			−0.962516
$351$		0			0
$352$	21488.2			0.00924363
$353$	−1.43528e6	+	2.48597e6i	−0.613054	+	1.06184i	0.377668	+	0.925941i	$0.376726\pi$
$353$	−1.43528e6	+	2.48597e6i	−0.613054	+	1.06184i	−0.990723	+	0.135900i	$0.956607\pi$
$354$		0			0
$355$	221015.	+	382809.i	0.0930788	+	0.161217i
$356$	−208525.	−	361176.i	−0.0872033	−	0.151041i
$357$		0			0
$358$	−422687.	+	732115.i	−0.174306	+	0.301906i
$359$	3.96829e6			1.62505			0.812525	−	0.582926i	$-0.198092\pi$
$359$	3.96829e6			1.62505			0.812525	+	0.582926i	$0.198092\pi$
$360$		0			0
$361$	−431397.			−0.174225
$362$	−431032.	+	746569.i	−0.172877	+	0.299432i
$363$		0			0
$364$	−377751.	−	654284.i	−0.149435	−	0.258829i
$365$	−32575.2	−	56421.8i	−0.0127984	−	0.0221674i
$366$		0			0
$367$	−1.79190e6	+	3.10365e6i	−0.694461	+	1.20284i	0.275902	+	0.961186i	$0.411024\pi$
$367$	−1.79190e6	+	3.10365e6i	−0.694461	+	1.20284i	−0.970362	+	0.241655i	$0.922310\pi$
$368$	493643.			0.190017
$369$		0			0
$370$	−710445.			−0.269790
$371$	3.82596e6	−	6.62675e6i	1.44313	−	2.49957i
$372$		0			0
$373$	317676.	+	550230.i	0.118226	+	0.204773i	0.919065	−	0.394107i	$-0.128946\pi$
$373$	317676.	+	550230.i	0.118226	+	0.204773i	−0.800839	+	0.598880i	$0.795613\pi$
$374$	−54761.7	−	94850.1i	−0.0202441	−	0.0350638i
$375$		0			0
$376$	853997.	−	1.47917e6i	0.311520	−	0.539569i
$377$	1.02093e6			0.369948
$378$		0			0
$379$	399337.			0.142804			0.0714022	−	0.997448i	$-0.477253\pi$
$379$	399337.			0.142804			0.0714022	+	0.997448i	$0.477253\pi$
$380$	−228569.	+	395893.i	−0.0812004	+	0.140643i
$381$		0			0
$382$	1.12459e6	+	1.94784e6i	0.394308	+	0.682961i
$383$	−1.63313e6	−	2.82866e6i	−0.568883	−	0.985334i	−0.996677	−	0.0814575i	$-0.974043\pi$
$383$	−1.63313e6	−	2.82866e6i	−0.568883	−	0.985334i	0.427794	−	0.903876i	$-0.359291\pi$
$384$		0			0
$385$	42414.1	−	73463.4i	0.0145834	−	0.0252592i
$386$	1.62151e6			0.553927
$387$		0			0
$388$	282056.			0.0951165
$389$	−526396.	+	911744.i	−0.176376	+	0.305491i	−0.940636	−	0.339416i	$-0.889771\pi$
$389$	−526396.	+	911744.i	−0.176376	+	0.305491i	0.764261	+	0.644907i	$0.223104\pi$
$390$		0			0
$391$	−1.25803e6	−	2.17897e6i	−0.416149	−	0.720791i
$392$	771984.	+	1.33712e6i	0.253742	+	0.439495i
$393$		0			0
$394$	2.04504e6	−	3.54211e6i	0.663683	−	1.14953i
$395$	−1.35348e6			−0.436474
$396$		0			0
$397$	1.37875e6			0.439045			0.219522	−	0.975607i	$-0.429550\pi$
$397$	1.37875e6			0.439045			0.219522	+	0.975607i	$0.429550\pi$
$398$	−23070.2	+	39958.8i	−0.00730036	+	0.0126446i
$399$		0			0
$400$	348898.	+	604310.i	0.109031	+	0.188847i
$401$	830386.	+	1.43827e6i	0.257881	+	0.446663i	0.965674	−	0.259757i	$-0.0836424\pi$
$401$	830386.	+	1.43827e6i	0.257881	+	0.446663i	−0.707793	+	0.706420i	$0.750309\pi$
$402$		0			0
$403$	1.06802e6	−	1.84986e6i	0.327579	−	0.567384i
$404$	−2.65032e6			−0.807878
$405$		0			0
$406$	−3.53995e6			−1.06581
$407$	−93266.9	+	161543.i	−0.0279088	+	0.0483395i
$408$		0			0
$409$	−2.70023e6	−	4.67693e6i	−0.798164	−	1.38246i	−0.920811	−	0.390010i	$-0.872472\pi$
$409$	−2.70023e6	−	4.67693e6i	−0.798164	−	1.38246i	0.122647	−	0.992450i	$-0.460862\pi$
$410$	−631848.	−	1.09439e6i	−0.185632	−	0.321524i
$411$		0			0
$412$	517135.	−	895703.i	0.150093	−	0.259969i
$413$	6.00497e6			1.73235
$414$		0			0
$415$	922546.			0.262947
$416$	−119497.	+	206975.i	−0.0338550	+	0.0586386i
$417$		0			0
$418$	60012.9	+	103945.i	0.0167998	+	0.0290981i
$419$	−3.00517e6	−	5.20510e6i	−0.836245	−	1.44842i	−0.893013	−	0.450032i	$-0.851413\pi$
$419$	−3.00517e6	−	5.20510e6i	−0.836245	−	1.44842i	0.0567675	−	0.998387i	$-0.481921\pi$
$420$		0			0
$421$	1.86568e6	−	3.23146e6i	0.513018	−	0.888573i	−0.486868	−	0.873476i	$-0.661861\pi$
$421$	1.86568e6	−	3.23146e6i	0.513018	−	0.888573i	0.999886	−	0.0150977i	$-0.00480593\pi$
$422$	−2.62040e6			−0.716287
$423$		0			0
$424$	−2.42059e6			−0.653893
$425$	1.77831e6	−	3.08012e6i	0.477567	−	0.827171i
$426$		0			0
$427$	−1.73168e6	−	2.99935e6i	−0.459618	−	0.796082i
$428$	−229387.	−	397310.i	−0.0605284	−	0.104838i
$429$		0			0
$430$	−161379.	+	279517.i	−0.0420897	+	0.0729016i
$431$	−293652.			−0.0761448			−0.0380724	−	0.999275i	$-0.512122\pi$
$431$	−293652.			−0.0761448			−0.0380724	+	0.999275i	$0.512122\pi$
$432$		0			0
$433$	−731246.			−0.187432			−0.0937160	−	0.995599i	$-0.529875\pi$
$433$	−731246.			−0.187432			−0.0937160	+	0.995599i	$0.529875\pi$
$434$	−3.70324e6	+	6.41419e6i	−0.943750	+	1.63462i
$435$		0			0
$436$	−1.39040e6	−	2.40825e6i	−0.350288	−	0.606716i
$437$	1.37866e6	+	2.38791e6i	0.345346	+	0.598156i
$438$		0			0
$439$	−970387.	+	1.68076e6i	−0.240316	+	0.416240i	−0.960804	−	0.277227i	$-0.910585\pi$
$439$	−970387.	+	1.68076e6i	−0.240316	+	0.416240i	0.720488	+	0.693467i	$0.243918\pi$
$440$	−26834.4			−0.00660784
$441$		0			0
$442$	1.21813e6			0.296578
$443$	−2.27045e6	+	3.93254e6i	−0.549672	+	0.952059i	0.448625	+	0.893720i	$0.351914\pi$
$443$	−2.27045e6	+	3.93254e6i	−0.549672	+	0.952059i	−0.998297	+	0.0583394i	$0.981419\pi$
$444$		0			0
$445$	260405.	+	451035.i	0.0623376	+	0.107972i
$446$	−1.74390e6	−	3.02052e6i	−0.415130	−	0.719026i
$447$		0			0
$448$	414342.	−	717661.i	0.0975357	−	0.168937i
$449$	6.94484e6			1.62572			0.812861	−	0.582457i	$-0.197909\pi$
$449$	6.94484e6			1.62572			0.812861	+	0.582457i	$0.197909\pi$
$450$		0			0
$451$	−331795.			−0.0768119
$452$	1.27824e6	−	2.21398e6i	0.294284	−	0.509714i
$453$		0			0
$454$	1.29465e6	+	2.24240e6i	0.294790	+	0.510591i
$455$	471734.	+	817068.i	0.106824	+	0.185025i
$456$		0			0
$457$	−1.27256e6	+	2.20414e6i	−0.285028	+	0.493684i	−0.972616	−	0.232418i	$-0.925336\pi$
$457$	−1.27256e6	+	2.20414e6i	−0.285028	+	0.493684i	0.687588	+	0.726101i	$0.258670\pi$
$458$	1.05887e6			0.235872
$459$		0			0
$460$	−616460.			−0.135835
$461$	−3.93640e6	+	6.81805e6i	−0.862675	+	1.49420i	0.00666268	+	0.999978i	$0.497879\pi$
$461$	−3.93640e6	+	6.81805e6i	−0.862675	+	1.49420i	−0.869338	+	0.494219i	$0.835454\pi$
$462$		0			0
$463$	−2.98583e6	−	5.17160e6i	−0.647309	−	1.12117i	−0.983763	−	0.179473i	$-0.942561\pi$
$463$	−2.98583e6	−	5.17160e6i	−0.647309	−	1.12117i	0.336453	−	0.941700i	$-0.390773\pi$
$464$	559910.	+	969792.i	0.120732	+	0.209114i
$465$		0			0
$466$	−1.71851e6	+	2.97654e6i	−0.366595	+	0.634961i
$467$	1.75432e6			0.372235			0.186118	−	0.982527i	$-0.440409\pi$
$467$	1.75432e6			0.372235			0.186118	+	0.982527i	$0.440409\pi$
$468$		0			0
$469$	−1.39030e7			−2.91862
$470$	−1.06647e6	+	1.84718e6i	−0.222691	+	0.385713i
$471$		0			0
$472$	−949799.	−	1.64510e6i	−0.196235	−	0.339889i
$473$	42371.6	+	73389.8i	0.00870807	+	0.0150828i
$474$		0			0
$475$	−1.94883e6	+	3.37547e6i	−0.396315	+	0.686437i
$476$	−4.22373e6			−0.854435
$477$		0			0
$478$	−1.45440e6			−0.291149
$479$	−222551.	+	385470.i	−0.0443191	+	0.0767630i	−0.887334	−	0.461127i	$-0.847445\pi$
$479$	−222551.	+	385470.i	−0.0443191	+	0.0767630i	0.843015	+	0.537890i	$0.180779\pi$
$480$		0			0
$481$	−1.03733e6	−	1.79670e6i	−0.204434	−	0.354089i
$482$	−2.66155e6	−	4.60993e6i	−0.521815	−	0.903810i
$483$		0			0
$484$	1.28489e6	−	2.22549e6i	0.249316	−	0.431829i
$485$	−352231.			−0.0679943
$486$		0			0
$487$	9.62628e6			1.83923			0.919615	−	0.392821i	$-0.128501\pi$
$487$	9.62628e6			1.83923			0.919615	+	0.392821i	$0.128501\pi$
$488$	−547795.	+	948809.i	−0.104128	+	0.180355i
$489$		0			0
$490$	−964052.	−	1.66979e6i	−0.181389	−	0.314174i
$491$	2.67834e6	+	4.63902e6i	0.501374	+	0.868405i	0.999999	+	0.00158682i	$0.000505100\pi$
$491$	2.67834e6	+	4.63902e6i	0.501374	+	0.868405i	−0.498625	+	0.866818i	$0.666162\pi$
$492$		0			0
$493$	2.85381e6	−	4.94295e6i	0.528821	−	0.915944i
$494$	−1.33494e6			−0.246118
$495$		0			0
$496$	2.34295e6			0.427620
$497$	2.23789e6	−	3.87614e6i	0.406394	−	0.703895i
$498$		0			0
$499$	−3.06721e6	−	5.31256e6i	−0.551432	−	0.955108i	−0.998172	−	0.0604440i	$-0.980748\pi$
$499$	−3.06721e6	−	5.31256e6i	−0.551432	−	0.955108i	0.446740	−	0.894664i	$-0.352585\pi$
$500$	−935223.	−	1.61985e6i	−0.167298	−	0.289768i
$501$		0			0
$502$	−1.23080e6	+	2.13180e6i	−0.217985	+	0.377562i
$503$	−5.69058e6			−1.00285			−0.501425	−	0.865201i	$-0.667191\pi$
$503$	−5.69058e6			−1.00285			−0.501425	+	0.865201i	$0.667191\pi$
$504$		0			0
$505$	3.30972e6			0.577514
$506$	−80928.6	+	140172.i	−0.0140516	+	0.0243381i
$507$		0			0
$508$	−387223.	−	670690.i	−0.0665735	−	0.115309i
$509$	2.87553e6	+	4.98057e6i	0.491953	+	0.852088i	0.999957	−	0.00926687i	$-0.00294978\pi$
$509$	2.87553e6	+	4.98057e6i	0.491953	+	0.852088i	−0.508004	+	0.861355i	$0.669616\pi$
$510$		0			0
$511$	−329840.	+	571300.i	−0.0558793	+	0.0967858i
$512$	−262144.			−0.0441942
$513$		0			0
$514$	−4.70948e6			−0.786258
$515$	−645796.	+	1.11855e6i	−0.107295	+	0.185840i
$516$		0			0
$517$	280011.	+	484994.i	0.0460733	+	0.0798013i
$518$	3.59681e6	+	6.22985e6i	0.588970	+	1.02013i
$519$		0			0
$520$	149227.	−	258469.i	0.0242014	−	0.0419180i
$521$	6.91604e6			1.11625			0.558127	−	0.829756i	$-0.311520\pi$
$521$	6.91604e6			1.11625			0.558127	+	0.829756i	$0.311520\pi$
$522$		0			0
$523$	175282.			0.0280209			0.0140105	−	0.999902i	$-0.495540\pi$
$523$	175282.			0.0280209			0.0140105	+	0.999902i	$0.495540\pi$
$524$	480146.	−	831638.i	0.0763915	−	0.132314i
$525$		0			0
$526$	276034.	+	478105.i	0.0435009	+	0.0753457i
$527$	−5.97091e6	−	1.03419e7i	−0.936513	−	1.62209i
$528$		0			0
$529$	1.35902e6	−	2.35389e6i	0.211147	−	0.365718i
$530$	3.02283e6			0.467437
$531$		0			0
$532$	4.62875e6			0.709063
$533$	1.84513e6	−	3.19586e6i	0.281325	−	0.487270i
$534$		0			0
$535$	286458.	+	496159.i	0.0432689	+	0.0749440i
$536$	2.19903e6	+	3.80883e6i	0.330612	+	0.572637i
$537$		0			0
$538$	1.97033e6	−	3.41271e6i	0.293483	−	0.508328i
$539$	−506241.			−0.0750561
$540$		0			0
$541$	3.11921e6			0.458196			0.229098	−	0.973403i	$-0.426422\pi$
$541$	3.11921e6			0.458196			0.229098	+	0.973403i	$0.426422\pi$
$542$	1.68202e6	−	2.91334e6i	0.245942	−	0.425984i
$543$		0			0
$544$	668063.	+	1.15712e6i	0.0967878	+	0.167641i
$545$	1.73633e6	+	3.00742e6i	0.250404	+	0.433713i
$546$		0			0
$547$	−2.45367e6	+	4.24987e6i	−0.350628	+	0.607306i	−0.986360	−	0.164604i	$-0.947365\pi$
$547$	−2.45367e6	+	4.24987e6i	−0.350628	+	0.607306i	0.635731	+	0.771910i	$0.280699\pi$
$548$	4.12826e6			0.587240
$549$		0			0
$550$	−228796.			−0.0322509
$551$	−3.12747e6	+	5.41693e6i	−0.438848	+	0.760107i
$552$		0			0
$553$	6.85232e6	+	1.18686e7i	0.952850	+	1.65039i
$554$	42614.2	+	73810.0i	0.00589903	+	0.0102174i
$555$		0			0
$556$	2.61058e6	−	4.52165e6i	0.358137	−	0.620312i
$557$	−1.41122e7			−1.92734			−0.963669	−	0.267099i	$-0.913935\pi$
$557$	−1.41122e7			−1.92734			−0.963669	+	0.267099i	$0.913935\pi$
$558$		0			0
$559$	−942523.			−0.127574
$560$	−517429.	+	896214.i	−0.0697238	+	0.120765i
$561$		0			0
$562$	3.36450e6	+	5.82749e6i	0.449345	+	0.778289i
$563$	−3.72663e6	−	6.45472e6i	−0.495502	−	0.858235i	0.504484	−	0.863421i	$-0.331683\pi$
$563$	−3.72663e6	−	6.45472e6i	−0.495502	−	0.858235i	−0.999987	+	0.00518570i	$0.998349\pi$
$564$		0			0
$565$	−1.59626e6	+	2.76481e6i	−0.210370	+	0.364371i
$566$	3.16628e6			0.415440
$567$		0			0
$568$	−1.41586e6			−0.184140
$569$	3.24943e6	−	5.62817e6i	0.420752	−	0.728763i	−0.575261	−	0.817970i	$-0.695100\pi$
$569$	3.24943e6	−	5.62817e6i	0.420752	−	0.728763i	0.996013	+	0.0892062i	$0.0284330\pi$
$570$		0			0
$571$	5.56754e6	+	9.64326e6i	0.714617	+	1.23775i	0.963107	+	0.269118i	$0.0867321\pi$
$571$	5.56754e6	+	9.64326e6i	0.714617	+	1.23775i	−0.248491	+	0.968634i	$0.579935\pi$
$572$	−39181.0	−	67863.5i	−0.00500709	−	0.00867254i
$573$		0			0
$574$	−6.39778e6	+	1.10813e7i	−0.810493	+	1.40382i
$575$	−5.25608e6			−0.662968
$576$		0			0
$577$	−5.28971e6			−0.661443			−0.330721	−	0.943728i	$-0.607292\pi$
$577$	−5.28971e6			−0.661443			−0.330721	+	0.943728i	$0.607292\pi$
$578$	565352.	−	979218.i	0.0703881	−	0.121916i
$579$		0			0
$580$	−699214.	−	1.21107e6i	−0.0863058	−	0.149486i
$581$	−4.67062e6	−	8.08975e6i	−0.574030	−	0.994248i
$582$		0			0
$583$	396835.	−	687339.i	0.0483547	−	0.0837529i
$584$	208682.			0.0253193
$585$		0			0
$586$	9.22419e6			1.10965
$587$	4.68325e6	−	8.11163e6i	0.560986	−	0.971657i	−0.436424	−	0.899741i	$-0.643755\pi$
$587$	4.68325e6	−	8.11163e6i	0.560986	−	0.971657i	0.997411	−	0.0719160i	$-0.0229114\pi$
$588$		0			0
$589$	6.54346e6	+	1.13336e7i	0.777176	+	1.34611i
$590$	1.18611e6	+	2.05440e6i	0.140279	+	0.242971i
$591$		0			0
$592$	1.13781e6	−	1.97074e6i	0.133433	−	0.231113i
$593$	5.82148e6			0.679825			0.339912	−	0.940457i	$-0.389603\pi$
$593$	5.82148e6			0.679825			0.339912	+	0.940457i	$0.389603\pi$
$594$		0			0
$595$	5.27459e6			0.610796
$596$	2.75514e6	−	4.77205e6i	0.317708	−	0.550287i
$597$		0			0
$598$	−900097.	−	1.55901e6i	−0.102929	−	0.178278i
$599$	7.98480e6	+	1.38301e7i	0.909279	+	1.57492i	0.815068	+	0.579365i	$0.196699\pi$
$599$	7.98480e6	+	1.38301e7i	0.909279	+	1.57492i	0.0942102	+	0.995552i	$0.469967\pi$
$600$		0			0
$601$	334611.	−	579563.i	0.0377880	−	0.0654508i	−0.846513	−	0.532368i	$-0.821302\pi$
$601$	334611.	−	579563.i	0.0377880	−	0.0654508i	0.884301	+	0.466918i	$0.154636\pi$
$602$	3.26809e6			0.367539
$603$		0			0
$604$	2.58935e6			0.288801
$605$	−1.60456e6	+	2.77918e6i	−0.178225	+	0.308694i
$606$		0			0
$607$	1.30270e6	+	2.25634e6i	0.143507	+	0.248561i	0.928815	−	0.370544i	$-0.120829\pi$
$607$	1.30270e6	+	2.25634e6i	0.143507	+	0.248561i	−0.785308	+	0.619105i	$0.787495\pi$
$608$	−732124.	−	1.26808e6i	−0.0803204	−	0.139119i
$609$		0			0
$610$	684085.	−	1.18487e6i	0.0744365	−	0.128928i
$611$	−6.22863e6			−0.674978
$612$		0			0
$613$	4.33667e6			0.466128			0.233064	−	0.972461i	$-0.425125\pi$
$613$	4.33667e6			0.466128			0.233064	+	0.972461i	$0.425125\pi$
$614$	5.85739e6	−	1.01453e7i	0.627023	−	1.08604i
$615$		0			0
$616$	135856.	+	235309.i	0.0144254	+	0.0249854i
$617$	2.45487e6	+	4.25195e6i	0.259606	+	0.449651i	0.966136	−	0.258032i	$-0.0830739\pi$
$617$	2.45487e6	+	4.25195e6i	0.259606	+	0.449651i	−0.706530	+	0.707683i	$0.749741\pi$
$618$		0			0
$619$	4.49719e6	−	7.78936e6i	0.471753	−	0.817100i	−0.527725	−	0.849415i	$-0.676955\pi$
$619$	4.49719e6	−	7.78936e6i	0.471753	−	0.817100i	0.999478	+	0.0323155i	$0.0102881\pi$
$620$	−2.92587e6			−0.305686
$621$		0			0
$622$	−4.32076e6			−0.447800
$623$	2.63674e6	−	4.56696e6i	0.272174	−	0.471419i
$624$		0			0
$625$	−3.09111e6	−	5.35396e6i	−0.316530	−	0.548246i
$626$	−1.28696e6	−	2.22909e6i	−0.131260	−	0.227348i
$627$		0			0
$628$	−3.51878e6	+	6.09471e6i	−0.356035	+	0.616671i
$629$	−1.15986e7			−1.16891
$630$		0			0
$631$	−4.33057e6			−0.432984			−0.216492	−	0.976284i	$-0.569461\pi$
$631$	−4.33057e6			−0.432984			−0.216492	+	0.976284i	$0.569461\pi$
$632$	2.16765e6	−	3.75448e6i	0.215872	−	0.373901i
$633$		0			0
$634$	−2.30837e6	−	3.99821e6i	−0.228077	−	0.395041i
$635$	483563.	+	837556.i	0.0475903	+	0.0824289i
$636$		0			0
$637$	2.81524e6	−	4.87613e6i	0.274895	−	0.476132i
$638$	−367170.			−0.0357121
$639$		0			0
$640$	327365.			0.0315924
$641$	−2.13709e6	+	3.70154e6i	−0.205436	+	0.355826i	−0.950272	−	0.311422i	$-0.899195\pi$
$641$	−2.13709e6	+	3.70154e6i	−0.205436	+	0.355826i	0.744835	+	0.667248i	$0.232528\pi$
$642$		0			0
$643$	5.50562e6	+	9.53602e6i	0.525145	+	0.909578i	0.999571	+	0.0292823i	$0.00932219\pi$
$643$	5.50562e6	+	9.53602e6i	0.525145	+	0.909578i	−0.474426	+	0.880295i	$0.657344\pi$
$644$	3.12098e6	+	5.40570e6i	0.296536	+	0.513615i
$645$		0			0
$646$	−3.73158e6	+	6.46329e6i	−0.351813	+	0.609357i
$647$	−8.39293e6			−0.788230			−0.394115	−	0.919061i	$-0.628949\pi$
$647$	−8.39293e6			−0.788230			−0.394115	+	0.919061i	$0.628949\pi$
$648$		0			0
$649$	622847.			0.0580456
$650$	1.27235e6	−	2.20377e6i	0.118120	−	0.204589i
$651$		0			0
$652$	−1.06434e6	−	1.84349e6i	−0.0980533	−	0.169833i
$653$	3.53386e6	+	6.12082e6i	0.324314	+	0.561729i	0.981373	−	0.192110i	$-0.0615331\pi$
$653$	3.53386e6	+	6.12082e6i	0.324314	+	0.561729i	−0.657059	+	0.753839i	$0.728200\pi$
$654$		0			0
$655$	−599605.	+	1.03855e6i	−0.0546088	+	0.0945851i
$656$	4.04772e6			0.367241
$657$		0			0
$658$	2.15971e7			1.94460
$659$	2.37353e6	−	4.11108e6i	0.212903	−	0.368758i	−0.739719	−	0.672916i	$-0.765042\pi$
$659$	2.37353e6	−	4.11108e6i	0.212903	−	0.368758i	0.952622	+	0.304157i	$0.0983749\pi$
$660$		0			0
$661$	3.96475e6	+	6.86716e6i	0.352950	+	0.611327i	0.986765	−	0.162158i	$-0.0518454\pi$
$661$	3.96475e6	+	6.86716e6i	0.352950	+	0.611327i	−0.633815	+	0.773484i	$0.718512\pi$
$662$	−430075.	−	744911.i	−0.0381416	−	0.0660632i
$663$		0			0
$664$	−1.47749e6	+	2.55909e6i	−0.130049	+	0.225251i
$665$	−5.78037e6			−0.506876
$666$		0			0
$667$	−8.43492e6			−0.734119
$668$	−2.22184e6	+	3.84834e6i	−0.192651	+	0.333682i
$669$		0			0
$670$	−2.74614e6	−	4.75646e6i	−0.236339	−	0.409352i
$671$	−179613.	−	311099.i	−0.0154004	−	0.0266742i
$672$		0			0
$673$	−2.29624e6	+	3.97721e6i	−0.195425	+	0.338486i	−0.947040	−	0.321116i	$-0.895942\pi$
$673$	−2.29624e6	+	3.97721e6i	−0.195425	+	0.338486i	0.751615	+	0.659602i	$0.229275\pi$
$674$	1.37187e7			1.16323
$675$		0			0
$676$	−5.06914e6			−0.426646
$677$	−9.26736e6	+	1.60515e7i	−0.777113	+	1.34600i	0.156486	+	0.987680i	$0.449983\pi$
$677$	−9.26736e6	+	1.60515e7i	−0.777113	+	1.34600i	−0.933599	+	0.358319i	$0.883350\pi$
$678$		0			0
$679$	1.78326e6	+	3.08869e6i	0.148436	+	0.257099i
$680$	−834276.	−	1.44501e6i	−0.0691891	−	0.119839i
$681$		0			0
$682$	−384107.	+	665292.i	−0.0316221	+	0.0547711i
$683$	1.28540e7			1.05436			0.527179	−	0.849754i	$-0.323250\pi$
$683$	1.28540e7			1.05436			0.527179	+	0.849754i	$0.323250\pi$
$684$		0			0
$685$	−5.15536e6			−0.419791
$686$	−2.96089e6	+	5.12841e6i	−0.240221	+	0.416075i
$687$		0			0
$688$	−516911.	−	895315.i	−0.0416336	−	0.0721116i
$689$	4.41365e6	+	7.64466e6i	0.354201	+	0.613494i
$690$		0			0
$691$	−3.74818e6	+	6.49204e6i	−0.298624	+	0.517232i	−0.975821	−	0.218569i	$-0.929861\pi$
$691$	−3.74818e6	+	6.49204e6i	−0.298624	+	0.517232i	0.677197	+	0.735802i	$0.263194\pi$
$692$	3.68240e6			0.292325
$693$		0			0
$694$	−6.41811e6			−0.505835
$695$	−3.26008e6	+	5.64663e6i	−0.256016	+	0.443432i
$696$		0			0
$697$	−1.03154e7	−	1.78669e7i	−0.804278	−	1.39305i
$698$	1.42504e6	+	2.46825e6i	0.110711	+	0.191757i
$699$		0			0
$700$	−4.41172e6	+	7.64132e6i	−0.340301	+	0.589418i
$701$	−4.13449e6			−0.317780			−0.158890	−	0.987296i	$-0.550792\pi$
$701$	−4.13449e6			−0.317780			−0.158890	+	0.987296i	$0.550792\pi$
$702$		0			0
$703$	1.27108e7			0.970030
$704$	42976.3	−	74437.2i	0.00326812	−	0.00566055i
$705$		0			0
$706$	5.74111e6	+	9.94389e6i	0.433495	+	0.750835i
$707$	−1.67563e7	−	2.90227e7i	−1.26075	−	2.18368i
$708$		0			0
$709$	−557752.	+	966054.i	−0.0416702	+	0.0721749i	−0.886108	−	0.463478i	$-0.846601\pi$
$709$	−557752.	+	966054.i	−0.0416702	+	0.0721749i	0.844438	+	0.535653i	$0.179935\pi$
$710$	1.76812e6			0.131633
$711$		0			0
$712$	−1.66820e6			−0.123324
$713$	−8.82400e6	+	1.52836e7i	−0.650042	+	1.12591i
$714$		0			0
$715$	48929.2	+	84747.8i	0.00357934	+	0.00619960i
$716$	1.69075e6	+	2.92846e6i	0.123253	+	0.213480i
$717$		0			0
$718$	7.93657e6	−	1.37465e7i	0.574542	−	0.995136i
$719$	−1.06330e7			−0.767068			−0.383534	−	0.923527i	$-0.625293\pi$
$719$	−1.06330e7			−0.767068			−0.383534	+	0.923527i	$0.625293\pi$
$720$		0			0
$721$	1.30780e7			0.936924
$722$	−862795.	+	1.49440e6i	−0.0615977	+	0.106690i
$723$		0			0
$724$	1.72413e6	+	2.98628e6i	0.122243	+	0.211731i
$725$	−5.96166e6	−	1.03259e7i	−0.421233	−	0.729596i
$726$		0			0
$727$	5.37428e6	−	9.30853e6i	0.377124	−	0.653198i	−0.613518	−	0.789680i	$-0.710246\pi$
$727$	5.37428e6	−	9.30853e6i	0.377124	−	0.653198i	0.990643	+	0.136482i	$0.0435797\pi$
$728$	−3.02201e6			−0.211333
$729$		0			0
$730$	−260601.			−0.0180996
$731$	−2.63465e6	+	4.56335e6i	−0.182360	+	0.315857i
$732$		0			0
$733$	−9.70537e6	−	1.68102e7i	−0.667194	−	1.15561i	−0.978685	−	0.205365i	$-0.934162\pi$
$733$	−9.70537e6	−	1.68102e7i	−0.667194	−	1.15561i	0.311492	−	0.950249i	$-0.399171\pi$
$734$	7.16758e6	+	1.24146e7i	0.491058	+	0.850537i
$735$		0			0
$736$	987285.	−	1.71003e6i	0.0671813	−	0.116361i
$737$	−1.44205e6			−0.0977939
$738$		0			0
$739$	−2.50920e7			−1.69015			−0.845074	−	0.534649i	$-0.820444\pi$
$739$	−2.50920e7			−1.69015			−0.845074	+	0.534649i	$0.820444\pi$
$740$	−1.42089e6	+	2.46105e6i	−0.0953852	+	0.165212i
$741$		0			0
$742$	−1.53038e7	−	2.65070e7i	−1.02045	−	1.76746i
$743$	7.40755e6	+	1.28303e7i	0.492269	+	0.852635i	0.999960	−	0.00890388i	$-0.00283423\pi$
$743$	7.40755e6	+	1.28303e7i	0.492269	+	0.852635i	−0.507691	+	0.861539i	$0.669501\pi$
$744$		0			0
$745$	−3.44062e6	+	5.95932e6i	−0.227115	+	0.393374i
$746$	2.54140e6			0.167196
$747$		0			0
$748$	−438094.			−0.0286295
$749$	2.90053e6	−	5.02386e6i	0.188918	−	0.327215i
$750$		0			0
$751$	−8.52016e6	−	1.47574e7i	−0.551249	−	0.954792i	−0.998185	−	0.0602258i	$-0.980818\pi$
$751$	−8.52016e6	−	1.47574e7i	−0.551249	−	0.954792i	0.446935	−	0.894566i	$-0.352515\pi$
$752$	−3.41599e6	−	5.91666e6i	−0.220278	−	0.381533i
$753$		0			0
$754$	2.04185e6	−	3.53659e6i	0.130797	−	0.226546i
$755$	−3.23358e6			−0.206450
$756$		0			0
$757$	7.65160e6			0.485303			0.242651	−	0.970114i	$-0.421983\pi$
$757$	7.65160e6			0.485303			0.242651	+	0.970114i	$0.421983\pi$
$758$	798674.	−	1.38334e6i	0.0504890	−	0.0874495i
$759$		0			0
$760$	914275.	+	1.58357e6i	0.0574173	+	0.0994497i
$761$	−9.27073e6	−	1.60574e7i	−0.580300	−	1.00511i	−0.995444	−	0.0953528i	$-0.969602\pi$
$761$	−9.27073e6	−	1.60574e7i	−0.580300	−	1.00511i	0.415144	−	0.909756i	$-0.363731\pi$
$762$		0			0
$763$	1.75813e7	−	3.04516e7i	1.09330	−	1.89365i
$764$	8.99671e6			0.557635
$765$		0			0
$766$	−1.30650e7			−0.804522
$767$	−3.46368e6	+	5.99928e6i	−0.212593	+	0.368223i
$768$		0			0
$769$	4.19019e6	+	7.25762e6i	0.255516	+	0.442567i	0.965036	−	0.262119i	$-0.0844213\pi$
$769$	4.19019e6	+	7.25762e6i	0.255516	+	0.442567i	−0.709520	+	0.704686i	$0.751088\pi$
$770$	−169656.	−	293854.i	−0.0103120	−	0.0178609i
$771$		0			0
$772$	3.24303e6	−	5.61709e6i	0.195843	−	0.339209i
$773$	2.44216e7			1.47003			0.735015	−	0.678051i	$-0.237175\pi$
$773$	2.44216e7			1.47003			0.735015	+	0.678051i	$0.237175\pi$
$774$		0			0
$775$	−2.49466e7			−1.49196
$776$	564112.	−	977070.i	0.0336287	−	0.0582467i
$777$		0			0
$778$	2.10558e6	+	3.64698e6i	0.124716	+	0.216015i
$779$	1.13046e7	+	1.95801e7i	0.667439	+	1.15604i
$780$		0			0
$781$	232118.	−	402040.i	0.0136170	−	0.0235853i
$782$	−1.00642e7			−0.588523
$783$		0			0
$784$	6.17587e6			0.358846
$785$	4.39425e6	−	7.61106e6i	0.254513	−	0.440830i
$786$		0			0
$787$	−1.25097e7	−	2.16674e7i	−0.719960	−	1.24701i	−0.961015	−	0.276497i	$-0.910827\pi$
$787$	−1.25097e7	−	2.16674e7i	−0.719960	−	1.24701i	0.241054	−	0.970512i	$-0.422507\pi$
$788$	−8.18015e6	−	1.41684e7i	−0.469295	−	0.812842i
$789$		0			0
$790$	−2.70695e6	+	4.68858e6i	−0.154317	+	0.267284i
$791$	3.23259e7			1.83700
$792$		0			0
$793$	3.99535e6			0.225617
$794$	2.75750e6	−	4.77612e6i	0.155226	−	0.268859i
$795$		0			0
$796$	92280.9	+	159835.i	0.00516213	+	0.00894108i
$797$	−5.05956e6	−	8.76342e6i	−0.282142	−	0.488684i	0.689770	−	0.724028i	$-0.257711\pi$
$797$	−5.05956e6	−	8.76342e6i	−0.282142	−	0.488684i	−0.971912	+	0.235345i	$0.924378\pi$
$798$		0			0
$799$	−1.74110e7	+	3.01568e7i	−0.964844	+	1.67116i
$800$	2.79119e6			0.154193
$801$		0			0
$802$	6.64309e6			0.364699
$803$	−34211.6	+	59256.3i	−0.00187234	+	0.00324299i
$804$		0			0
$805$	−3.89748e6	−	6.75063e6i	−0.211980	−	0.367159i
$806$	−4.27208e6	−	7.39945e6i	−0.231634	−	0.401201i
$807$		0			0
$808$	−5.30065e6	+	9.18099e6i	−0.285628	+	0.494722i
$809$	−1.49150e7			−0.801218			−0.400609	−	0.916249i	$-0.631201\pi$
$809$	−1.49150e7			−0.801218			−0.400609	+	0.916249i	$0.631201\pi$
$810$		0			0
$811$	−1.89183e7			−1.01002			−0.505011	−	0.863113i	$-0.668512\pi$
$811$	−1.89183e7			−1.01002			−0.505011	+	0.863113i	$0.668512\pi$
$812$	−7.07990e6	+	1.22627e7i	−0.376822	+	0.652676i
$813$		0			0
$814$	373068.	+	646172.i	0.0197345	+	0.0341812i
$815$	1.32915e6	+	2.30215e6i	0.0700937	+	0.121406i
$816$		0			0
$817$	2.88729e6	−	5.00093e6i	0.151334	−	0.262117i
$818$	−2.16018e7			−1.12877
$819$		0			0
$820$	−5.05478e6			−0.262523
$821$	1.26791e7	−	2.19608e7i	0.656492	−	1.13708i	−0.325026	−	0.945705i	$-0.605373\pi$
$821$	1.26791e7	−	2.19608e7i	0.656492	−	1.13708i	0.981518	−	0.191372i	$-0.0612937\pi$
$822$		0			0
$823$	1.22397e7	+	2.11998e7i	0.629900	+	1.09102i	0.987571	+	0.157172i	$0.0502376\pi$
$823$	1.22397e7	+	2.11998e7i	0.629900	+	1.09102i	−0.357671	+	0.933848i	$0.616429\pi$
$824$	−2.06854e6	−	3.58281e6i	−0.106132	−	0.183826i
$825$		0			0
$826$	1.20099e7	−	2.08018e7i	0.612478	−	1.06084i
$827$	3.15281e7			1.60300			0.801502	−	0.597993i	$-0.204035\pi$
$827$	3.15281e7			1.60300			0.801502	+	0.597993i	$0.204035\pi$
$828$		0			0
$829$	−3.30952e7			−1.67255			−0.836274	−	0.548312i	$-0.815271\pi$
$829$	−3.30952e7			−1.67255			−0.836274	+	0.548312i	$0.815271\pi$
$830$	1.84509e6	−	3.19579e6i	0.0929657	−	0.161021i
$831$		0			0
$832$	477987.	+	827899.i	0.0239391	+	0.0414638i
$833$	−1.57390e7	−	2.72607e7i	−0.785893	−	1.36121i
$834$		0			0
$835$	2.77463e6	−	4.80580e6i	0.137717	−	0.238533i
$836$	480103.			0.0237585
$837$		0			0
$838$	−2.40413e7			−1.18263
$839$	−8.03959e6	+	1.39250e7i	−0.394302	+	0.682951i	−0.993012	−	0.118015i	$-0.962347\pi$
$839$	−8.03959e6	+	1.39250e7i	−0.394302	+	0.682951i	0.598710	+	0.800966i	$0.295680\pi$
$840$		0			0
$841$	688348.	+	1.19225e6i	0.0335597	+	0.0581271i
$842$	−7.46273e6	−	1.29258e7i	−0.362759	−	0.628316i
$843$		0			0
$844$	−5.24081e6	+	9.07734e6i	−0.253246	+	0.438634i
$845$	6.33033e6			0.304989
$846$		0			0
$847$	3.24940e7			1.55630
$848$	−4.84118e6	+	8.38517e6i	−0.231186	+	0.400426i
$849$		0			0
$850$	−7.11323e6	−	1.23205e7i	−0.337691	−	0.584898i
$851$	8.57040e6	+	1.48444e7i	0.405674	+	0.702648i
$852$		0			0
$853$	−2.84195e6	+	4.92240e6i	−0.133735	+	0.231635i	−0.925113	−	0.379691i	$-0.876030\pi$
$853$	−2.84195e6	+	4.92240e6i	−0.133735	+	0.231635i	0.791379	+	0.611326i	$0.209364\pi$
$854$	−1.38534e7			−0.649999
$855$		0			0
$856$	−1.83509e6			−0.0856000
$857$	2.25933e6	−	3.91328e6i	0.105082	−	0.182007i	−0.808690	−	0.588235i	$-0.799823\pi$
$857$	2.25933e6	−	3.91328e6i	0.105082	−	0.182007i	0.913772	+	0.406228i	$0.133156\pi$
$858$		0			0
$859$	−148468.	−	257154.i	−0.00686515	−	0.0118908i	0.862572	−	0.505934i	$-0.168852\pi$
$859$	−148468.	−	257154.i	−0.00686515	−	0.0118908i	−0.869438	+	0.494043i	$0.835519\pi$
$860$	645517.	+	1.11807e6i	0.0297619	+	0.0515492i
$861$		0			0
$862$	−587304.	+	1.01724e6i	−0.0269212	+	0.0466289i
$863$	1.89964e7			0.868251			0.434125	−	0.900852i	$-0.357057\pi$
$863$	1.89964e7			0.868251			0.434125	+	0.900852i	$0.357057\pi$
$864$		0			0
$865$	−4.59857e6			−0.208970
$866$	−1.46249e6	+	2.53311e6i	−0.0662672	+	0.114778i
$867$		0			0
$868$	1.48129e7	+	2.56568e7i	0.667332	+	1.15585i
$869$	710736.	+	1.23103e6i	0.0319270	+	0.0552992i
$870$		0			0
$871$	8.01932e6	−	1.38899e7i	0.358172	−	0.620373i
$872$	−1.11232e7			−0.495382
$873$		0			0
$874$	1.10293e7			0.488393
$875$	1.18256e7	−	2.04826e7i	0.522160	−	0.904408i
$876$		0			0
$877$	−7.85608e6	−	1.36071e7i	−0.344911	−	0.597403i	0.640427	−	0.768019i	$-0.278757\pi$
$877$	−7.85608e6	−	1.36071e7i	−0.344911	−	0.597403i	−0.985338	+	0.170616i	$0.945424\pi$
$878$	3.88155e6	+	6.72304e6i	0.169929	+	0.294326i
$879$		0			0
$880$	−53668.7	+	92957.0i	−0.00233623	+	0.00404646i
$881$	−1.34864e7			−0.585404			−0.292702	−	0.956204i	$-0.594554\pi$
$881$	−1.34864e7			−0.585404			−0.292702	+	0.956204i	$0.594554\pi$
$882$		0			0
$883$	3.54880e7			1.53172			0.765861	−	0.643006i	$-0.222313\pi$
$883$	3.54880e7			1.53172			0.765861	+	0.643006i	$0.222313\pi$
$884$	2.43626e6	−	4.21973e6i	0.104856	−	0.181616i
$885$		0			0
$886$	9.08182e6	+	1.57302e7i	0.388677	+	0.673208i
$887$	1.44698e7	+	2.50624e7i	0.617522	+	1.06958i	0.989936	+	0.141513i	$0.0451967\pi$
$887$	1.44698e7	+	2.50624e7i	0.617522	+	1.06958i	−0.372414	+	0.928067i	$0.621470\pi$
$888$		0			0
$889$	4.89632e6	−	8.48067e6i	0.207786	−	0.359895i
$890$	2.08324e6			0.0881587
$891$		0			0
$892$	−1.39512e7			−0.587082
$893$	1.90806e7	−	3.30485e7i	0.800687	−	1.38683i
$894$		0			0
$895$	−2.11140e6	−	3.65705e6i	−0.0881076	−	0.152607i
$896$	−1.65737e6	−	2.87064e6i	−0.0689682	−	0.119456i
$897$		0			0
$898$	1.38897e7	−	2.40576e7i	0.574780	−	0.995548i
$899$	−4.00341e7			−1.65208
$900$		0			0
$901$	4.93502e7			2.02524
$902$	−663590.	+	1.14937e6i	−0.0271571	+	0.0470375i
$903$		0			0
$904$	−5.11296e6	−	8.85591e6i	−0.208090	−	0.360423i
$905$	−2.15309e6	−	3.72926e6i	−0.0873856	−	0.151356i
$906$		0			0
$907$	−3.41845e6	+	5.92094e6i	−0.137979	+	0.238986i	−0.926731	−	0.375725i	$-0.877394\pi$
$907$	−3.41845e6	+	5.92094e6i	−0.137979	+	0.238986i	0.788753	+	0.614711i	$0.210727\pi$
$908$	1.03572e7			0.416896
$909$		0			0
$910$	3.77387e6			0.151072
$911$	−2.06379e7	+	3.57459e7i	−0.823891	+	1.42702i	0.0788725	+	0.996885i	$0.474868\pi$
$911$	−2.06379e7	+	3.57459e7i	−0.823891	+	1.42702i	−0.902764	+	0.430137i	$0.858465\pi$
$912$		0			0
$913$	−484446.	−	839085.i	−0.0192339	−	0.0333142i
$914$	5.09024e6	+	8.81656e6i	0.201545	+	0.349087i
$915$		0			0
$916$	2.11773e6	−	3.66802e6i	0.0833935	−	0.144442i
$917$	1.21426e7			0.476858
$918$		0			0
$919$	324016.			0.0126555			0.00632773	−	0.999980i	$-0.497986\pi$
$919$	324016.			0.0126555			0.00632773	+	0.999980i	$0.497986\pi$
$920$	−1.23292e6	+	2.13548e6i	−0.0480248	+	0.0831813i
$921$		0			0
$922$	1.57456e7	+	2.72722e7i	0.610003	+	1.05656i
$923$	2.58164e6	+	4.47153e6i	0.0997451	+	0.172764i
$924$		0			0
$925$	−1.21148e7	+	2.09835e7i	−0.465547	+	0.806350i
$926$	−2.38866e7			−0.915434
$927$		0			0
$928$	4.47928e6			0.170741
$929$	1.02225e7	−	1.77059e7i	0.388614	−	0.673099i	−0.603650	−	0.797250i	$-0.706287\pi$
$929$	1.02225e7	−	1.77059e7i	0.388614	−	0.673099i	0.992263	+	0.124151i	$0.0396208\pi$
$930$		0			0
$931$	1.72482e7	+	2.98747e7i	0.652182	+	1.12961i
$932$	6.87403e6	+	1.19062e7i	0.259222	+	0.448985i
$933$		0			0
$934$	3.50865e6	−	6.07715e6i	0.131605	−	0.227947i
$935$	547090.			0.0204659
$936$		0			0
$937$	661898.			0.0246287			0.0123144	−	0.999924i	$-0.496080\pi$
$937$	661898.			0.0246287			0.0123144	+	0.999924i	$0.496080\pi$
$938$	−2.78061e7	+	4.81615e7i	−1.03189	+	1.78728i
$939$		0			0
$940$	4.26588e6	+	7.38871e6i	0.157467	+	0.272740i
$941$	−4.39669e6	−	7.61529e6i	−0.161865	−	0.280358i	0.773673	−	0.633585i	$-0.218417\pi$
$941$	−4.39669e6	−	7.61529e6i	−0.161865	−	0.280358i	−0.935537	+	0.353228i	$0.885084\pi$
$942$		0			0
$943$	−1.52445e7	+	2.64042e7i	−0.558257	+	0.966929i
$944$	−7.59839e6			−0.277518
$945$		0			0
$946$	338973.			0.0123151
$947$	1.67722e7	−	2.90503e7i	0.607736	−	1.05263i	−0.383876	−	0.923385i	$-0.625411\pi$
$947$	1.67722e7	−	2.90503e7i	0.607736	−	1.05263i	0.991613	−	0.129246i	$-0.0412556\pi$
$948$		0			0
$949$	−380506.	−	659055.i	−0.0137150	−	0.0237551i
$950$	7.79532e6	+	1.35019e7i	0.280237	+	0.485384i
$951$		0			0
$952$	−8.44747e6	+	1.46314e7i	−0.302088	+	0.523233i
$953$	−2.34073e7			−0.834872			−0.417436	−	0.908706i	$-0.637071\pi$
$953$	−2.34073e7			−0.834872			−0.417436	+	0.908706i	$0.637071\pi$
$954$		0			0
$955$	−1.12351e7			−0.398628
$956$	−2.90881e6	+	5.03820e6i	−0.102937	+	0.178292i
$957$		0			0
$958$	890205.	+	1.54188e6i	0.0313384	+	0.0542796i
$959$	2.61003e7	+	4.52071e7i	0.916431	+	1.58730i
$960$		0			0
$961$	−2.75662e7	+	4.77461e7i	−0.962872	+	1.66774i
$962$	−8.29860e6			−0.289113
$963$		0			0
$964$	−2.12924e7			−0.737957
$965$	−4.04988e6	+	7.01461e6i	−0.139999	+	0.242485i
$966$		0			0
$967$	971981.	+	1.68352e6i	0.0334266	+	0.0578965i	0.882255	−	0.470772i	$-0.156025\pi$
$967$	971981.	+	1.68352e6i	0.0334266	+	0.0578965i	−0.848828	+	0.528669i	$0.822691\pi$
$968$	−5.13954e6	−	8.90195e6i	−0.176293	−	0.305349i
$969$		0			0
$970$	−704461.	+	1.22016e6i	−0.0240396	+	0.0416379i
$971$	5.80784e7			1.97682			0.988409	−	0.151811i	$-0.0485106\pi$
$971$	5.80784e7			1.97682			0.988409	+	0.151811i	$0.0485106\pi$
$972$		0			0
$973$	6.60200e7			2.23560
$974$	1.92526e7	−	3.33464e7i	0.650266	−	1.12629i
$975$		0			0
$976$	2.19118e6	+	3.79524e6i	0.0736298	+	0.127531i
$977$	2.29782e6	+	3.97995e6i	0.0770158	+	0.133395i	0.901961	−	0.431817i	$-0.142127\pi$
$977$	2.29782e6	+	3.97995e6i	0.0770158	+	0.133395i	−0.824945	+	0.565213i	$0.808794\pi$
$978$		0			0
$979$	273487.	−	473694.i	0.00911970	−	0.0157958i
$980$	−7.71242e6			−0.256522
$981$		0			0
$982$	2.14267e7			0.709049
$983$	−1.05477e7	+	1.82691e7i	−0.348155	+	0.603022i	−0.985922	−	0.167207i	$-0.946525\pi$
$983$	−1.05477e7	+	1.82691e7i	−0.348155	+	0.603022i	0.637767	+	0.770230i	$0.279858\pi$
$984$		0			0
$985$	1.02154e7	+	1.76935e7i	0.335477	+	0.581063i
$986$	−1.14153e7	−	1.97718e7i	−0.373933	−	0.647670i
$987$		0			0
$988$	−2.66988e6	+	4.62437e6i	−0.0870160	+	0.150716i
$989$	7.78714e6			0.253156
$990$		0			0
$991$	2.32369e7			0.751614			0.375807	−	0.926698i	$-0.377366\pi$
$991$	2.32369e7			0.751614			0.375807	+	0.926698i	$0.377366\pi$
$992$	4.68589e6	−	8.11621e6i	0.151187	−	0.261863i
$993$		0			0
$994$	−8.95155e6	−	1.55045e7i	−0.287364	−	0.497729i
$995$	−115240.	−	199602.i	−0.00369017	−	0.00639156i
$996$		0			0
$997$	−1.57477e7	+	2.72758e7i	−0.501740	+	0.869039i	0.498258	+	0.867029i	$0.333973\pi$
$997$	−1.57477e7	+	2.72758e7i	−0.501740	+	0.869039i	−0.999998	+	0.00201040i	$0.999360\pi$
$998$	−2.45377e7			−0.779842
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	162.6.c.p.55.2		4
3.2	odd	2		162.6.c.n.55.1		4
9.2	odd	6		162.6.a.g.1.2	yes	2
9.4	even	3	inner	162.6.c.p.109.2		4
9.5	odd	6		162.6.c.n.109.1		4
9.7	even	3		162.6.a.c.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
162.6.a.c.1.1	✓	2	9.7	even	3
162.6.a.g.1.2	yes	2	9.2	odd	6
162.6.c.n.55.1		4	3.2	odd	2
162.6.c.n.109.1		4	9.5	odd	6
162.6.c.p.55.2		4	1.1	even	1	trivial
162.6.c.p.109.2		4	9.4	even	3	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 162 = 2 \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	162.c (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(2.00000 - 3.46410i) q^{2} +(-8.00000 - 13.8564i) q^{4} +(9.99038 + 17.3038i) q^{5} +(101.158 - 175.210i) q^{7} -64.0000 q^{8} +79.9230 q^{10} +(10.4923 - 18.1731i) q^{11} +(116.696 + 202.124i) q^{13} +(-404.631 - 700.841i) q^{14} +(-128.000 + 221.703i) q^{16} +1304.81 q^{17} -1429.93 q^{19} +(159.846 - 276.862i) q^{20} +(-41.9691 - 72.6926i) q^{22} +(-964.146 - 1669.95i) q^{23} +(1362.88 - 2360.59i) q^{25} +933.569 q^{26} -3237.05 q^{28} +(2187.15 - 3788.25i) q^{29} +(-4576.07 - 7925.98i) q^{31} +(512.000 + 886.810i) q^{32} +(2609.62 - 4520.00i) q^{34} +4042.41 q^{35} -8889.11 q^{37} +(-2859.86 + 4953.42i) q^{38} +(-639.384 - 1107.45i) q^{40} +(-7905.70 - 13693.1i) q^{41} +(-2019.18 + 3497.33i) q^{43} -335.753 q^{44} -7713.17 q^{46} +(-13343.7 + 23112.0i) q^{47} +(-12062.3 - 20892.4i) q^{49} +(-5451.54 - 9442.34i) q^{50} +(1867.14 - 3233.98i) q^{52} +37821.7 q^{53} +419.287 q^{55} +(-6474.09 + 11213.5i) q^{56} +(-8748.59 - 15153.0i) q^{58} +(14840.6 + 25704.7i) q^{59} +(8559.30 - 14825.1i) q^{61} -36608.5 q^{62} +4096.00 q^{64} +(-2331.68 + 4038.58i) q^{65} +(-34359.8 - 59513.0i) q^{67} +(-10438.5 - 18080.0i) q^{68} +(8084.83 - 14003.3i) q^{70} +22122.8 q^{71} -3260.65 q^{73} +(-17778.2 + 30792.8i) q^{74} +(11439.4 + 19813.7i) q^{76} +(-2122.75 - 3676.71i) q^{77} +(-33869.5 + 58663.7i) q^{79} -5115.08 q^{80} -63245.6 q^{82} +(23085.8 - 39985.9i) q^{83} +(13035.6 + 22578.3i) q^{85} +(8076.73 + 13989.3i) q^{86} +(-671.505 + 1163.08i) q^{88} +26065.6 q^{89} +47218.8 q^{91} +(-15426.3 + 26719.2i) q^{92} +(53374.8 + 92447.9i) q^{94} +(-14285.6 - 24743.3i) q^{95} +(-8814.24 + 15266.7i) q^{97} -96498.0 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{2} - 32 q^{4} - 12 q^{5} + 176 q^{7} - 256 q^{8} - 96 q^{10} - 540 q^{11} + 446 q^{13} - 704 q^{14} - 512 q^{16} + 3120 q^{17} - 3184 q^{19} - 192 q^{20} + 2160 q^{22} - 1404 q^{23} + 4828 q^{25}+ \cdots - 64080 q^{98}+O(q^{100}) \) 4 * q + 8 * q^2 - 32 * q^4 - 12 * q^5 + 176 * q^7 - 256 * q^8 - 96 * q^10 - 540 * q^11 + 446 * q^13 - 704 * q^14 - 512 * q^16 + 3120 * q^17 - 3184 * q^19 - 192 * q^20 + 2160 * q^22 - 1404 * q^23 + 4828 * q^25 + 3568 * q^26 - 5632 * q^28 - 2340 * q^29 + 776 * q^31 + 2048 * q^32 + 6240 * q^34 + 9768 * q^35 + 7052 * q^37 - 6368 * q^38 + 768 * q^40 - 27840 * q^41 - 23104 * q^43 + 17280 * q^44 - 11232 * q^46 - 38784 * q^47 - 8010 * q^49 - 19312 * q^50 + 7136 * q^52 + 91344 * q^53 + 36720 * q^55 - 11264 * q^56 + 9360 * q^58 - 2160 * q^59 - 8974 * q^61 + 6208 * q^62 + 16384 * q^64 + 2136 * q^65 - 71656 * q^67 - 24960 * q^68 + 19536 * q^70 + 49416 * q^71 + 120644 * q^73 + 14104 * q^74 + 25472 * q^76 - 19008 * q^77 - 74392 * q^79 + 6144 * q^80 - 222720 * q^82 - 29496 * q^83 + 17910 * q^85 + 92416 * q^86 + 34560 * q^88 - 221952 * q^89 + 83248 * q^91 - 22464 * q^92 + 155136 * q^94 - 23388 * q^95 + 5564 * q^97 - 64080 * q^98

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