LMFDB - Embedded newform 162.8.c.p.55.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [162,8,Mod(55,162)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(162, base_ring=CyclotomicField(6))

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

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("162.55");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 162 = 2 \cdot 3^{4} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	162.c (of order $3$, degree $2$, not 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:	$50.6063741284$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{329})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} + 83x^{2} + 82x + 6724 $ x^4 - x^3 + 83x^2 + 82x + 6724
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{6} $
Twist minimal:	no (minimal twist has level 54)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			55.1
Root			$4.78459 - 8.28715i$ of defining polynomial
Character	$\chi$	$=$	162.55
Dual form			162.8.c.p.109.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(4.00000 - 6.92820i) q^{2} +(-32.0000 - 55.4256i) q^{4} +(-256.868 - 444.908i) q^{5} +(-464.868 + 805.175i) q^{7} -512.000 q^{8} -4109.89 q^{10} +(3766.44 - 6523.67i) q^{11} +(-7037.36 - 12189.1i) q^{13} +(3718.94 + 6441.40i) q^{14} +(-2048.00 + 3547.24i) q^{16} -3057.29 q^{17} -15757.5 q^{19} +(-16439.5 + 28474.1i) q^{20} +(-30131.5 - 52189.4i) q^{22} +(-19804.4 - 34302.2i) q^{23} +(-92899.7 + 160907. i) q^{25} -112598. q^{26} +59503.1 q^{28} +(-12500.4 + 21651.3i) q^{29} +(8391.33 + 14534.2i) q^{31} +(16384.0 + 28377.9i) q^{32} +(-12229.1 + 21181.5i) q^{34} +477638. q^{35} +389561. q^{37} +(-63030.0 + 109171. i) q^{38} +(131516. + 227793. i) q^{40} +(127058. + 220071. i) q^{41} +(-128452. + 222485. i) q^{43} -482105. q^{44} -316870. q^{46} +(-26238.0 + 45445.6i) q^{47} +(-20432.7 - 35390.4i) q^{49} +(743197. + 1.28726e6i) q^{50} +(-450391. + 780100. i) q^{52} +571071. q^{53} -3.86991e6 q^{55} +(238012. - 412249. i) q^{56} +(100003. + 173211. i) q^{58} +(148444. + 257113. i) q^{59} +(696724. - 1.20676e6i) q^{61} +134261. q^{62} +262144. q^{64} +(-3.61534e6 + 6.26195e6i) q^{65} +(19493.5 + 33763.7i) q^{67} +(97833.2 + 169452. i) q^{68} +(1.91055e6 - 3.30918e6i) q^{70} -1.17263e6 q^{71} +2.81003e6 q^{73} +(1.55824e6 - 2.69895e6i) q^{74} +(504240. + 873369. i) q^{76} +(3.50180e6 + 6.06529e6i) q^{77} +(-720481. + 1.24791e6i) q^{79} +2.10426e6 q^{80} +2.03293e6 q^{82} +(-3.13138e6 + 5.42372e6i) q^{83} +(785319. + 1.36021e6i) q^{85} +(1.02762e6 + 1.77988e6i) q^{86} +(-1.92842e6 + 3.34012e6i) q^{88} +5.01788e6 q^{89} +1.30858e7 q^{91} +(-1.26748e6 + 2.19534e6i) q^{92} +(209904. + 363565. i) q^{94} +(4.04759e6 + 7.01064e6i) q^{95} +(3.27822e6 - 5.67804e6i) q^{97} -326923. q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 16 q^{2} - 128 q^{4} - 48 q^{5} - 880 q^{7} - 2048 q^{8} - 768 q^{10} + 7230 q^{11} - 8560 q^{13} + 7040 q^{14} - 8192 q^{16} - 51408 q^{17} + 74096 q^{19} - 3072 q^{20} - 57840 q^{22} - 59628 q^{23}+ \cdots + 12483264 q^{98}+O(q^{100}) $ 4 * q + 16 * q^2 - 128 * q^4 - 48 * q^5 - 880 * q^7 - 2048 * q^8 - 768 * q^10 + 7230 * q^11 - 8560 * q^13 + 7040 * q^14 - 8192 * q^16 - 51408 * q^17 + 74096 * q^19 - 3072 * q^20 - 57840 * q^22 - 59628 * q^23 - 324584 * q^25 - 136960 * q^26 + 112640 * q^28 - 275280 * q^29 - 245584 * q^31 + 65536 * q^32 - 205632 * q^34 + 1001604 * q^35 - 142120 * q^37 + 296384 * q^38 + 24576 * q^40 + 494520 * q^41 - 825280 * q^43 - 925440 * q^44 - 954048 * q^46 - 927708 * q^47 + 780204 * q^49 + 2596672 * q^50 - 547840 * q^52 + 2764224 * q^53 - 8021952 * q^55 + 450560 * q^56 + 2202240 * q^58 + 1588920 * q^59 + 3021968 * q^61 - 3929344 * q^62 + 1048576 * q^64 - 9799080 * q^65 + 1882160 * q^67 + 1645056 * q^68 + 4006416 * q^70 + 6788880 * q^71 + 1790180 * q^73 - 568480 * q^74 - 2371072 * q^76 + 7018656 * q^77 + 369920 * q^79 + 393216 * q^80 + 7912320 * q^82 - 9352050 * q^83 - 8976744 * q^85 + 6602240 * q^86 - 3701760 * q^88 + 1825920 * q^89 + 26720080 * q^91 - 3816192 * q^92 + 7421664 * q^94 + 32688588 * q^95 - 1359790 * q^97 + 12483264 * 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 $

$2$	4.00000	−	6.92820i	0.353553	−	0.612372i
$2$	4.00000	−	6.92820i	0.353553	−	0.612372i
$3$		0			0
$3$		0			0
$4$	−32.0000	−	55.4256i	−0.250000	−	0.433013i
$5$	−256.868	−	444.908i	−0.918998	−	1.59175i	−0.800940	−	0.598745i	$-0.795666\pi$
$5$	−256.868	−	444.908i	−0.918998	−	1.59175i	−0.118059	−	0.993007i	$-0.537667\pi$
$6$		0			0
$7$	−464.868	+	805.175i	−0.512255	+	0.887252i	0.487644	+	0.873043i	$0.337856\pi$
$7$	−464.868	+	805.175i	−0.512255	+	0.887252i	−0.999899	+	0.0142093i	$0.995477\pi$
$8$	−512.000			−0.353553
$9$		0			0
$10$	−4109.89			−1.29966
$11$	3766.44	−	6523.67i	0.853212	−	1.47781i	−0.0250819	−	0.999685i	$-0.507985\pi$
$11$	3766.44	−	6523.67i	0.853212	−	1.47781i	0.878294	−	0.478121i	$-0.158682\pi$
$12$		0			0
$13$	−7037.36	−	12189.1i	−0.888398	−	1.53875i	−0.841768	−	0.539839i	$-0.818485\pi$
$13$	−7037.36	−	12189.1i	−0.888398	−	1.53875i	−0.0466303	−	0.998912i	$-0.514848\pi$
$14$	3718.94	+	6441.40i	0.362219	+	0.627382i
$15$		0			0
$16$	−2048.00	+	3547.24i	−0.125000	+	0.216506i
$17$	−3057.29			−0.150926			−0.0754632	−	0.997149i	$-0.524044\pi$
$17$	−3057.29			−0.150926			−0.0754632	+	0.997149i	$0.524044\pi$
$18$		0			0
$19$	−15757.5			−0.527047			−0.263524	−	0.964653i	$-0.584885\pi$
$19$	−15757.5			−0.527047			−0.263524	+	0.964653i	$0.584885\pi$
$20$	−16439.5	+	28474.1i	−0.459499	+	0.795876i
$21$		0			0
$22$	−30131.5	−	52189.4i	−0.603312	−	1.04497i
$23$	−19804.4	−	34302.2i	−0.339401	−	0.587860i	0.644919	−	0.764251i	$-0.276891\pi$
$23$	−19804.4	−	34302.2i	−0.339401	−	0.587860i	−0.984320	+	0.176391i	$0.943558\pi$
$24$		0			0
$25$	−92899.7	+	160907.i	−1.18912	+	2.05961i
$26$	−112598.			−1.25639
$27$		0			0
$28$	59503.1			0.512255
$29$	−12500.4	+	21651.3i	−0.0951768	+	0.164851i	−0.909682	−	0.415305i	$-0.863675\pi$
$29$	−12500.4	+	21651.3i	−0.0951768	+	0.164851i	0.814506	+	0.580156i	$0.197008\pi$
$30$		0			0
$31$	8391.33	+	14534.2i	0.0505900	+	0.0876245i	0.890211	−	0.455548i	$-0.150556\pi$
$31$	8391.33	+	14534.2i	0.0505900	+	0.0876245i	−0.839621	+	0.543172i	$0.817223\pi$
$32$	16384.0	+	28377.9i	0.0883883	+	0.153093i
$33$		0			0
$34$	−12229.1	+	21181.5i	−0.0533605	+	0.0924231i
$35$	477638.			1.88305
$36$		0			0
$37$	389561.			1.26435			0.632177	−	0.774824i	$-0.282161\pi$
$37$	389561.			1.26435			0.632177	+	0.774824i	$0.282161\pi$
$38$	−63030.0	+	109171.i	−0.186339	+	0.322749i
$39$		0			0
$40$	131516.	+	227793.i	0.324915	+	0.562769i
$41$	127058.	+	220071.i	0.287912	+	0.498677i	0.973311	−	0.229490i	$-0.0737058\pi$
$41$	127058.	+	220071.i	0.287912	+	0.498677i	−0.685400	+	0.728167i	$0.740372\pi$
$42$		0			0
$43$	−128452.	+	222485.i	−0.246378	+	0.426739i	−0.962518	−	0.271218i	$-0.912574\pi$
$43$	−128452.	+	222485.i	−0.246378	+	0.426739i	0.716140	+	0.697956i	$0.245907\pi$
$44$	−482105.			−0.853212
$45$		0			0
$46$	−316870.			−0.479986
$47$	−26238.0	+	45445.6i	−0.0368628	+	0.0638483i	−0.883868	−	0.467736i	$-0.845070\pi$
$47$	−26238.0	+	45445.6i	−0.0368628	+	0.0638483i	0.847005	+	0.531584i	$0.178403\pi$
$48$		0			0
$49$	−20432.7	−	35390.4i	−0.0248107	−	0.0429734i
$50$	743197.	+	1.28726e6i	0.840832	+	1.45636i
$51$		0			0
$52$	−450391.	+	780100.i	−0.444199	+	0.769376i
$53$	571071.			0.526895			0.263448	−	0.964674i	$-0.415140\pi$
$53$	571071.			0.526895			0.263448	+	0.964674i	$0.415140\pi$
$54$		0			0
$55$	−3.86991e6			−3.13640
$56$	238012.	−	412249.i	0.181110	−	0.313691i
$57$		0			0
$58$	100003.	+	173211.i	0.0673001	+	0.116567i
$59$	148444.	+	257113.i	0.0940982	+	0.162983i	0.909232	−	0.416290i	$-0.136670\pi$
$59$	148444.	+	257113.i	0.0940982	+	0.162983i	−0.815134	+	0.579273i	$0.803337\pi$
$60$		0			0
$61$	696724.	−	1.20676e6i	0.393012	−	0.680717i	−0.599833	−	0.800125i	$-0.704766\pi$
$61$	696724.	−	1.20676e6i	0.393012	−	0.680717i	0.992845	+	0.119408i	$0.0380996\pi$
$62$	134261.			0.0715451
$63$		0			0
$64$	262144.			0.125000
$65$	−3.61534e6	+	6.26195e6i	−1.63287	+	2.82822i
$66$		0			0
$67$	19493.5	+	33763.7i	0.00791822	+	0.0137148i	0.869957	−	0.493127i	$-0.164146\pi$
$67$	19493.5	+	33763.7i	0.00791822	+	0.0137148i	−0.862039	+	0.506842i	$0.830813\pi$
$68$	97833.2	+	169452.i	0.0377316	+	0.0653530i
$69$		0			0
$70$	1.91055e6	−	3.30918e6i	0.665757	−	1.15313i
$71$	−1.17263e6			−0.388828			−0.194414	−	0.980920i	$-0.562280\pi$
$71$	−1.17263e6			−0.388828			−0.194414	+	0.980920i	$0.562280\pi$
$72$		0			0
$73$	2.81003e6			0.845436			0.422718	−	0.906261i	$-0.361076\pi$
$73$	2.81003e6			0.845436			0.422718	+	0.906261i	$0.361076\pi$
$74$	1.55824e6	−	2.69895e6i	0.447017	−	0.774256i
$75$		0			0
$76$	504240.	+	873369.i	0.131762	+	0.228218i
$77$	3.50180e6	+	6.06529e6i	0.874124	+	1.51403i
$78$		0			0
$79$	−720481.	+	1.24791e6i	−0.164410	+	0.284766i	−0.936446	−	0.350813i	$-0.885905\pi$
$79$	−720481.	+	1.24791e6i	−0.164410	+	0.284766i	0.772036	+	0.635579i	$0.219239\pi$
$80$	2.10426e6			0.459499
$81$		0			0
$82$	2.03293e6			0.407168
$83$	−3.13138e6	+	5.42372e6i	−0.601123	+	1.04117i	0.391529	+	0.920166i	$0.371946\pi$
$83$	−3.13138e6	+	5.42372e6i	−0.601123	+	1.04117i	−0.992651	+	0.121009i	$0.961387\pi$
$84$		0			0
$85$	785319.	+	1.36021e6i	0.138701	+	0.240237i
$86$	1.02762e6	+	1.77988e6i	0.174215	+	0.301750i
$87$		0			0
$88$	−1.92842e6	+	3.34012e6i	−0.301656	+	0.522484i
$89$	5.01788e6			0.754493			0.377246	−	0.926113i	$-0.376871\pi$
$89$	5.01788e6			0.754493			0.377246	+	0.926113i	$0.376871\pi$
$90$		0			0
$91$	1.30858e7			1.82035
$92$	−1.26748e6	+	2.19534e6i	−0.169701	+	0.293930i
$93$		0			0
$94$	209904.	+	363565.i	0.0260660	+	0.0451476i
$95$	4.04759e6	+	7.01064e6i	0.484356	+	0.838928i
$96$		0			0
$97$	3.27822e6	−	5.67804e6i	0.364701	−	0.631681i	−0.624027	−	0.781403i	$-0.714505\pi$
$97$	3.27822e6	−	5.67804e6i	0.364701	−	0.631681i	0.988728	+	0.149722i	$0.0478379\pi$
$98$	−326923.			−0.0350876
$99$		0			0
$100$	1.18912e7			1.18912
$101$	2.63601e6	−	4.56571e6i	0.254579	−	0.440944i	−0.710202	−	0.703998i	$-0.751396\pi$
$101$	2.63601e6	−	4.56571e6i	0.254579	−	0.440944i	0.964781	+	0.263054i	$0.0847297\pi$
$102$		0			0
$103$	−7.17000e6	−	1.24188e7i	−0.646531	−	1.11982i	−0.983946	−	0.178468i	$-0.942886\pi$
$103$	−7.17000e6	−	1.24188e7i	−0.646531	−	1.11982i	0.337415	−	0.941356i	$-0.390447\pi$
$104$	3.60313e6	+	6.24080e6i	0.314096	+	0.544031i
$105$		0			0
$106$	2.28428e6	−	3.95649e6i	0.186286	−	0.322656i
$107$	−1.44748e7			−1.14227			−0.571135	−	0.820856i	$-0.693497\pi$
$107$	−1.44748e7			−1.14227			−0.571135	+	0.820856i	$0.693497\pi$
$108$		0			0
$109$	−2.18620e7			−1.61695			−0.808474	−	0.588531i	$-0.799706\pi$
$109$	−2.18620e7			−1.61695			−0.808474	+	0.588531i	$0.799706\pi$
$110$	−1.54796e7	+	2.68115e7i	−1.10889	+	1.92065i
$111$		0			0
$112$	−1.90410e6	−	3.29800e6i	−0.128064	−	0.221813i
$113$	1.14181e7	+	1.97768e7i	0.744425	+	1.28938i	0.950463	+	0.310838i	$0.100609\pi$
$113$	1.14181e7	+	1.97768e7i	0.744425	+	1.28938i	−0.206038	+	0.978544i	$0.566057\pi$
$114$		0			0
$115$	−1.01742e7	+	1.76222e7i	−0.623818	+	1.08048i
$116$	1.60005e6			0.0951768
$117$		0			0
$118$	2.37511e6			0.133075
$119$	1.42123e6	−	2.46165e6i	0.0773128	−	0.133910i
$120$		0			0
$121$	−1.86286e7	−	3.22657e7i	−0.955941	−	1.65574i
$122$	−5.57379e6	−	9.65409e6i	−0.277902	−	0.481340i
$123$		0			0
$124$	537045.	−	930189.i	0.0252950	−	0.0438122i
$125$	5.53161e7			2.53318
$126$		0			0
$127$	−4.32493e7			−1.87355			−0.936777	−	0.349926i	$-0.886207\pi$
$127$	−4.32493e7			−1.87355			−0.936777	+	0.349926i	$0.886207\pi$
$128$	1.04858e6	−	1.81619e6i	0.0441942	−	0.0765466i
$129$		0			0
$130$	2.89227e7	+	5.00956e7i	1.15462	+	1.99985i
$131$	−1.01142e7	−	1.75183e7i	−0.393082	−	0.680838i	0.599773	−	0.800171i	$-0.295258\pi$
$131$	−1.01142e7	−	1.75183e7i	−0.393082	−	0.680838i	−0.992854	+	0.119333i	$0.961924\pi$
$132$		0			0
$133$	7.32515e6	−	1.26875e7i	0.269983	−	0.467624i
$134$	311896.			0.0111981
$135$		0			0
$136$	1.56533e6			0.0533605
$137$	1.38583e7	−	2.40033e7i	0.460456	−	0.797533i	−0.538528	−	0.842608i	$-0.681019\pi$
$137$	1.38583e7	−	2.40033e7i	0.460456	−	0.797533i	0.998984	+	0.0450750i	$0.0143527\pi$
$138$		0			0
$139$	2.76358e7	+	4.78666e7i	0.872810	+	1.51175i	0.859077	+	0.511846i	$0.171038\pi$
$139$	2.76358e7	+	4.78666e7i	0.872810	+	1.51175i	0.0137331	+	0.999906i	$0.495628\pi$
$140$	−1.52844e7	−	2.64734e7i	−0.470762	−	0.815383i
$141$		0			0
$142$	−4.69052e6	+	8.12422e6i	−0.137471	+	0.238107i
$143$	−1.06023e8			−3.03197
$144$		0			0
$145$	1.28438e7			0.349869
$146$	1.12401e7	−	1.94685e7i	0.298907	−	0.517722i
$147$		0			0
$148$	−1.24659e7	−	2.15916e7i	−0.316089	−	0.547482i
$149$	−2.77455e6	−	4.80566e6i	−0.0687133	−	0.119015i	0.829622	−	0.558326i	$-0.188556\pi$
$149$	−2.77455e6	−	4.80566e6i	−0.0687133	−	0.119015i	−0.898335	+	0.439311i	$0.855223\pi$
$150$		0			0
$151$	1.45783e7	−	2.52503e7i	0.344578	−	0.596826i	−0.640699	−	0.767792i	$-0.721355\pi$
$151$	1.45783e7	−	2.52503e7i	0.344578	−	0.596826i	0.985277	+	0.170966i	$0.0546887\pi$
$152$	8.06784e6			0.186339
$153$		0			0
$154$	5.60287e7			1.23620
$155$	4.31092e6	−	7.46674e6i	0.0929843	−	0.161053i
$156$		0			0
$157$	1.69551e7	+	2.93672e7i	0.349665	+	0.605638i	0.986190	−	0.165618i	$-0.0529620\pi$
$157$	1.69551e7	+	2.93672e7i	0.349665	+	0.605638i	−0.636525	+	0.771256i	$0.719629\pi$
$158$	5.76385e6	+	9.98328e6i	0.116255	+	0.201360i
$159$		0			0
$160$	8.41704e6	−	1.45787e7i	0.162457	−	0.281385i
$161$	3.68256e7			0.695440
$162$		0			0
$163$	−8.50849e7			−1.53885			−0.769425	−	0.638738i	$-0.779457\pi$
$163$	−8.50849e7			−1.53885			−0.769425	+	0.638738i	$0.779457\pi$
$164$	8.13172e6	−	1.40846e7i	0.143956	−	0.249339i
$165$		0			0
$166$	2.50511e7	+	4.33897e7i	0.425058	+	0.736222i
$167$	−2.43589e7	−	4.21908e7i	−0.404715	−	0.700987i	0.589573	−	0.807715i	$-0.299296\pi$
$167$	−2.43589e7	−	4.21908e7i	−0.404715	−	0.700987i	−0.994288	+	0.106728i	$0.965963\pi$
$168$		0			0
$169$	−6.76745e7	+	1.17216e8i	−1.07850	+	1.86802i
$170$	1.25651e7			0.196153
$171$		0			0
$172$	1.64419e7			0.246378
$173$	−2.46741e7	+	4.27368e7i	−0.362310	+	0.627539i	−0.988341	−	0.152259i	$-0.951345\pi$
$173$	−2.46741e7	+	4.27368e7i	−0.362310	+	0.627539i	0.626031	+	0.779798i	$0.284678\pi$
$174$		0			0
$175$	−8.63721e7	−	1.49601e8i	−1.21826	−	2.11009i
$176$	1.54273e7	+	2.67210e7i	0.213303	+	0.369452i
$177$		0			0
$178$	2.00715e7	−	3.47649e7i	0.266753	−	0.462031i
$179$	−4.83660e7			−0.630310			−0.315155	−	0.949040i	$-0.602057\pi$
$179$	−4.83660e7			−0.630310			−0.315155	+	0.949040i	$0.602057\pi$
$180$		0			0
$181$	−7.66634e7			−0.960978			−0.480489	−	0.877001i	$-0.659541\pi$
$181$	−7.66634e7			−0.960978			−0.480489	+	0.877001i	$0.659541\pi$
$182$	5.23430e7	−	9.06608e7i	0.643590	−	1.11473i
$183$		0			0
$184$	1.01398e7	+	1.75627e7i	0.119996	+	0.207840i
$185$	−1.00066e8	−	1.73319e8i	−1.16194	−	2.01254i
$186$		0			0
$187$	−1.15151e7	+	1.99447e7i	−0.128772	+	0.223040i
$188$	3.35847e6			0.0368628
$189$		0			0
$190$	6.47615e7			0.684982
$191$	7.71024e7	−	1.33545e8i	0.800665	−	1.38679i	−0.118514	−	0.992952i	$-0.537813\pi$
$191$	7.71024e7	−	1.33545e8i	0.800665	−	1.38679i	0.919179	−	0.393840i	$-0.128854\pi$
$192$		0			0
$193$	−6.72489e7	−	1.16478e8i	−0.673340	−	1.16626i	−0.976951	−	0.213463i	$-0.931526\pi$
$193$	−6.72489e7	−	1.16478e8i	−0.673340	−	1.16626i	0.303611	−	0.952796i	$-0.401808\pi$
$194$	−2.62258e7	−	4.54243e7i	−0.257883	−	0.446666i
$195$		0			0
$196$	−1.30769e6	+	2.26499e6i	−0.0124054	+	0.0214867i
$197$	1.23329e8			1.14930			0.574651	−	0.818398i	$-0.305138\pi$
$197$	1.23329e8			1.14930			0.574651	+	0.818398i	$0.305138\pi$
$198$		0			0
$199$	1.26860e8			1.14114			0.570571	−	0.821248i	$-0.306722\pi$
$199$	1.26860e8			1.14114			0.570571	+	0.821248i	$0.306722\pi$
$200$	4.75646e7	−	8.23843e7i	0.420416	−	0.728182i
$201$		0			0
$202$	−2.10881e7	−	3.65257e7i	−0.180015	−	0.311795i
$203$	−1.16221e7	−	2.01300e7i	−0.0975096	−	0.168892i
$204$		0			0
$205$	6.52743e7	−	1.13058e8i	0.529180	−	0.916567i
$206$	−1.14720e8			−0.914332
$207$		0			0
$208$	5.76500e7			0.444199
$209$	−5.93497e7	+	1.02797e8i	−0.449683	+	0.778874i
$210$		0			0
$211$	−2.04917e7	−	3.54927e7i	−0.150172	−	0.260106i	0.781118	−	0.624383i	$-0.214649\pi$
$211$	−2.04917e7	−	3.54927e7i	−0.150172	−	0.260106i	−0.931291	+	0.364277i	$0.881316\pi$
$212$	−1.82743e7	−	3.16520e7i	−0.131724	−	0.228152i
$213$		0			0
$214$	−5.78991e7	+	1.00284e8i	−0.403853	+	0.699494i
$215$	1.31981e8			0.905683
$216$		0			0
$217$	−1.56034e7			−0.103660
$218$	−8.74478e7	+	1.51464e8i	−0.571678	+	0.990175i
$219$		0			0
$220$	1.23837e8	+	2.14492e8i	0.784100	+	1.35810i
$221$	2.15152e7	+	3.72655e7i	0.134083	+	0.232238i
$222$		0			0
$223$	7.06819e7	−	1.22425e8i	0.426816	−	0.739268i	−0.569772	−	0.821803i	$-0.692968\pi$
$223$	7.06819e7	−	1.22425e8i	0.426816	−	0.739268i	0.996588	+	0.0825352i	$0.0263017\pi$
$224$	−3.04656e7			−0.181110
$225$		0			0
$226$	1.82690e8			1.05278
$227$	1.04100e8	−	1.80307e8i	0.590693	−	1.02311i	−0.403446	−	0.915004i	$-0.632188\pi$
$227$	1.04100e8	−	1.80307e8i	0.590693	−	1.02311i	0.994139	−	0.108108i	$-0.0344791\pi$
$228$		0			0
$229$	8.31954e7	+	1.44099e8i	0.457799	+	0.792931i	0.998844	−	0.0480623i	$-0.0153046\pi$
$229$	8.31954e7	+	1.44099e8i	0.457799	+	0.792931i	−0.541045	+	0.840993i	$0.681971\pi$
$230$	8.13936e7	+	1.40978e8i	0.441106	+	0.764018i
$231$		0			0
$232$	6.40021e6	−	1.10855e7i	0.0336501	−	0.0582836i
$233$	−2.64825e8			−1.37156			−0.685778	−	0.727810i	$-0.740538\pi$
$233$	−2.64825e8			−1.37156			−0.685778	+	0.727810i	$0.740538\pi$
$234$		0			0
$235$	2.69588e7			0.135508
$236$	9.50043e6	−	1.64552e7i	0.0470491	−	0.0814915i
$237$		0			0
$238$	−1.13699e7	−	1.96932e7i	−0.0546684	−	0.0946884i
$239$	2.57636e7	+	4.46239e7i	0.122071	+	0.211434i	0.920584	−	0.390544i	$-0.127713\pi$
$239$	2.57636e7	+	4.46239e7i	0.122071	+	0.211434i	−0.798513	+	0.601978i	$0.794380\pi$
$240$		0			0
$241$	6.70898e7	−	1.16203e8i	0.308743	−	0.534758i	−0.669345	−	0.742952i	$-0.733425\pi$
$241$	6.70898e7	−	1.16203e8i	0.308743	−	0.534758i	0.978088	+	0.208194i	$0.0667585\pi$
$242$	−2.98058e8			−1.35191
$243$		0			0
$244$	−8.91806e7			−0.393012
$245$	−1.04970e7	+	1.81813e7i	−0.0456020	+	0.0789850i
$246$		0			0
$247$	1.10891e8	+	1.92069e8i	0.468228	+	0.810995i
$248$	−4.29636e6	−	7.44151e6i	−0.0178863	−	0.0309799i
$249$		0			0
$250$	2.21265e8	−	3.83241e8i	0.895616	−	1.55125i
$251$	−3.18238e6			−0.0127027			−0.00635133	−	0.999980i	$-0.502022\pi$
$251$	−3.18238e6			−0.0127027			−0.00635133	+	0.999980i	$0.502022\pi$
$252$		0			0
$253$	−2.98368e8			−1.15832
$254$	−1.72997e8	+	2.99640e8i	−0.662402	+	1.14731i
$255$		0			0
$256$	−8.38861e6	−	1.45295e7i	−0.0312500	−	0.0541266i
$257$	−2.10138e8	−	3.63970e8i	−0.772216	−	1.33752i	−0.936346	−	0.351079i	$-0.885815\pi$
$257$	−2.10138e8	−	3.63970e8i	−0.772216	−	1.33752i	0.164129	−	0.986439i	$-0.447519\pi$
$258$		0			0
$259$	−1.81094e8	+	3.13664e8i	−0.647672	+	1.12180i
$260$	4.62764e8			1.63287
$261$		0			0
$262$	−1.61828e8			−0.555902
$263$	3.66087e7	−	6.34081e7i	0.124091	−	0.214931i	−0.797287	−	0.603601i	$-0.793732\pi$
$263$	3.66087e7	−	6.34081e7i	0.124091	−	0.214931i	0.921377	+	0.388670i	$0.127065\pi$
$264$		0			0
$265$	−1.46690e8	−	2.54074e8i	−0.484216	−	0.838687i
$266$	−5.86012e7	−	1.01500e8i	−0.190907	−	0.330660i
$267$		0			0
$268$	1.24758e6	−	2.16088e6i	0.00395911	−	0.00685738i
$269$	4.03323e8			1.26334			0.631670	−	0.775237i	$-0.282370\pi$
$269$	4.03323e8			1.26334			0.631670	+	0.775237i	$0.282370\pi$
$270$		0			0
$271$	−7.59082e7			−0.231684			−0.115842	−	0.993268i	$-0.536957\pi$
$271$	−7.59082e7			−0.231684			−0.115842	+	0.993268i	$0.536957\pi$
$272$	6.26132e6	−	1.08449e7i	0.0188658	−	0.0326765i
$273$		0			0
$274$	−1.10866e8	−	1.92026e8i	−0.325591	−	0.563941i
$275$	6.99802e8	+	1.21209e9i	2.02914	+	3.51457i
$276$		0			0
$277$	−8.90979e7	+	1.54322e8i	−0.251877	+	0.436263i	−0.964043	−	0.265748i	$-0.914381\pi$
$277$	−8.90979e7	+	1.54322e8i	−0.251877	+	0.436263i	0.712166	+	0.702011i	$0.247714\pi$
$278$	4.42172e8			1.23434
$279$		0			0
$280$	−2.44551e8			−0.665757
$281$	−2.93797e8	+	5.08872e8i	−0.789907	+	1.36816i	0.136117	+	0.990693i	$0.456538\pi$
$281$	−2.93797e8	+	5.08872e8i	−0.789907	+	1.36816i	−0.926024	+	0.377466i	$0.876796\pi$
$282$		0			0
$283$	1.42459e8	+	2.46746e8i	0.373626	+	0.647138i	0.990120	−	0.140221i	$-0.0447812\pi$
$283$	1.42459e8	+	2.46746e8i	0.373626	+	0.647138i	−0.616495	+	0.787359i	$0.711448\pi$
$284$	3.75242e7	+	6.49938e7i	0.0972069	+	0.168367i
$285$		0			0
$286$	−4.24093e8	+	7.34550e8i	−1.07196	+	1.85669i
$287$	−2.36261e8			−0.589937
$288$		0			0
$289$	−4.00992e8			−0.977221
$290$	5.13752e7	−	8.89845e7i	0.123697	−	0.214250i
$291$		0			0
$292$	−8.99210e7	−	1.55748e8i	−0.211359	−	0.366085i
$293$	1.75728e8	+	3.04371e8i	0.408136	+	0.706913i	0.994681	−	0.103004i	$-0.0328454\pi$
$293$	1.75728e8	+	3.04371e8i	0.408136	+	0.706913i	−0.586545	+	0.809917i	$0.699512\pi$
$294$		0			0
$295$	7.62611e7	−	1.32088e8i	0.172952	−	0.299562i
$296$	−1.99455e8			−0.447017
$297$		0			0
$298$	−4.43928e7			−0.0971753
$299$	−2.78741e8	+	4.82793e8i	−0.603047	+	1.04451i
$300$		0			0
$301$	−1.19426e8	−	2.06853e8i	−0.252416	−	0.437198i
$302$	−1.16626e8	−	2.02003e8i	−0.243653	−	0.422020i
$303$		0			0
$304$	3.22713e7	−	5.58956e7i	0.0658809	−	0.114109i
$305$	−7.15864e8			−1.44471
$306$		0			0
$307$	3.58206e8			0.706560			0.353280	−	0.935518i	$-0.385066\pi$
$307$	3.58206e8			0.706560			0.353280	+	0.935518i	$0.385066\pi$
$308$	2.24115e8	−	3.88178e8i	0.437062	−	0.757014i
$309$		0			0
$310$	−3.44874e7	−	5.97339e7i	−0.0657498	−	0.113882i
$311$	1.48496e8	+	2.57203e8i	0.279932	+	0.484857i	0.971368	−	0.237581i	$-0.0763546\pi$
$311$	1.48496e8	+	2.57203e8i	0.279932	+	0.484857i	−0.691435	+	0.722438i	$0.743021\pi$
$312$		0			0
$313$	−2.66043e8	+	4.60799e8i	−0.490395	+	0.849389i	−0.999939	−	0.0110555i	$-0.996481\pi$
$313$	−2.66043e8	+	4.60799e8i	−0.490395	+	0.849389i	0.509544	+	0.860445i	$0.329814\pi$
$314$	2.71282e8			0.494501
$315$		0			0
$316$	9.22216e7			0.164410
$317$	−1.39087e8	+	2.40906e8i	−0.245233	+	0.424757i	−0.962197	−	0.272354i	$-0.912198\pi$
$317$	−1.39087e8	+	2.40906e8i	−0.245233	+	0.424757i	0.716964	+	0.697110i	$0.245531\pi$
$318$		0			0
$319$	9.41641e7	+	1.63097e8i	0.162412	+	0.281306i
$320$	−6.73364e7	−	1.16630e8i	−0.114875	−	0.198969i
$321$		0			0
$322$	1.47303e8	−	2.55135e8i	0.245875	−	0.425868i
$323$	4.81752e7			0.0795453
$324$		0			0
$325$	2.61507e9			4.22563
$326$	−3.40340e8	+	5.89486e8i	−0.544065	+	0.942349i
$327$		0			0
$328$	−6.50538e7	−	1.12676e8i	−0.101792	−	0.176309i
$329$	−2.43944e7	−	4.22524e7i	−0.0377664	−	0.0654132i
$330$		0			0
$331$	−1.79071e7	+	3.10160e7i	−0.0271411	+	0.0470097i	−0.879277	−	0.476311i	$-0.841974\pi$
$331$	−1.79071e7	+	3.10160e7i	−0.0271411	+	0.0470097i	0.852136	+	0.523321i	$0.175307\pi$
$332$	4.00817e8			0.601123
$333$		0			0
$334$	−3.89742e8			−0.572353
$335$	1.00145e7	−	1.73456e7i	0.0145537	−	0.0252077i
$336$		0			0
$337$	7.44599e7	+	1.28968e8i	0.105979	+	0.183560i	0.914138	−	0.405404i	$-0.132869\pi$
$337$	7.44599e7	+	1.28968e8i	0.105979	+	0.183560i	−0.808159	+	0.588964i	$0.799536\pi$
$338$	5.41396e8	+	9.37726e8i	0.762617	+	1.32089i
$339$		0			0
$340$	5.02604e7	−	8.70536e7i	0.0693505	−	0.120119i
$341$	1.26422e8			0.172656
$342$		0			0
$343$	−7.27683e8			−0.973673
$344$	6.57674e7	−	1.13913e8i	0.0871077	−	0.150875i
$345$		0			0
$346$	1.97393e8	+	3.41894e8i	0.256192	+	0.443737i
$347$	−5.99635e7	−	1.03860e8i	−0.0770431	−	0.133443i	0.824930	−	0.565235i	$-0.191215\pi$
$347$	−5.99635e7	−	1.03860e8i	−0.0770431	−	0.133443i	−0.901973	+	0.431793i	$0.857881\pi$
$348$		0			0
$349$	7.04809e8	−	1.22076e9i	0.887528	−	1.53724i	0.0447402	−	0.998999i	$-0.485754\pi$
$349$	7.04809e8	−	1.22076e9i	0.887528	−	1.53724i	0.842788	−	0.538245i	$-0.180913\pi$
$350$	−1.38195e9			−1.72288
$351$		0			0
$352$	2.46838e8			0.301656
$353$	6.73687e8	−	1.16686e9i	0.815167	−	1.41191i	−0.0940405	−	0.995568i	$-0.529978\pi$
$353$	6.73687e8	−	1.16686e9i	0.815167	−	1.41191i	0.909208	−	0.416343i	$-0.136688\pi$
$354$		0			0
$355$	3.01211e8	+	5.21713e8i	0.357332	+	0.618917i
$356$	−1.60572e8	−	2.78119e8i	−0.188623	−	0.326705i
$357$		0			0
$358$	−1.93464e8	+	3.35089e8i	−0.222848	+	0.385984i
$359$	1.20973e9			1.37993			0.689967	−	0.723841i	$-0.257625\pi$
$359$	1.20973e9			1.37993			0.689967	+	0.723841i	$0.257625\pi$
$360$		0			0
$361$	−6.45573e8			−0.722221
$362$	−3.06654e8	+	5.31140e8i	−0.339757	+	0.588476i
$363$		0			0
$364$	−4.18744e8	−	7.25287e8i	−0.455087	−	0.788233i
$365$	−7.21806e8	−	1.25021e9i	−0.776955	−	1.34572i
$366$		0			0
$367$	−6.77249e8	+	1.17303e9i	−0.715183	+	1.23873i	0.247706	+	0.968835i	$0.420323\pi$
$367$	−6.77249e8	+	1.17303e9i	−0.715183	+	1.23873i	−0.962889	+	0.269898i	$0.913010\pi$
$368$	1.62237e8			0.169701
$369$		0			0
$370$	−1.60105e9			−1.64323
$371$	−2.65472e8	+	4.59812e8i	−0.269905	+	0.467489i
$372$		0			0
$373$	3.04972e8	+	5.28227e8i	0.304284	+	0.527035i	0.977102	−	0.212773i	$-0.0682495\pi$
$373$	3.04972e8	+	5.28227e8i	0.304284	+	0.527035i	−0.672818	+	0.739808i	$0.734916\pi$
$374$	9.21208e7	+	1.59558e8i	0.0910557	+	0.157713i
$375$		0			0
$376$	1.34339e7	−	2.32681e7i	0.0130330	−	0.0225738i
$377$	3.51879e8			0.338220
$378$		0			0
$379$	−2.77216e7			−0.0261565			−0.0130783	−	0.999914i	$-0.504163\pi$
$379$	−2.77216e7			−0.0261565			−0.0130783	+	0.999914i	$0.504163\pi$
$380$	2.59046e8	−	4.48681e8i	0.242178	−	0.419464i
$381$		0			0
$382$	−6.16819e8	−	1.06836e9i	−0.566156	−	0.980611i
$383$	−9.52673e8	−	1.65008e9i	−0.866459	−	1.50075i	−0.865591	−	0.500752i	$-0.833057\pi$
$383$	−9.52673e8	−	1.65008e9i	−0.866459	−	1.50075i	−0.000868235	−	1.00000i	$-0.500276\pi$
$384$		0			0
$385$	1.79900e9	−	3.11595e9i	1.60664	−	2.78278i
$386$	−1.07598e9			−0.952246
$387$		0			0
$388$	−4.19612e8			−0.364701
$389$	−3.65276e8	+	6.32676e8i	−0.314628	+	0.544952i	−0.979358	−	0.202132i	$-0.935213\pi$
$389$	−3.65276e8	+	6.32676e8i	−0.314628	+	0.544952i	0.664730	+	0.747083i	$0.268546\pi$
$390$		0			0
$391$	6.05476e7	+	1.04872e8i	0.0512246	+	0.0887236i
$392$	1.04615e7	+	1.81199e7i	0.00877191	+	0.0151934i
$393$		0			0
$394$	4.93317e8	−	8.54450e8i	0.406340	−	0.703801i
$395$	7.40274e8			0.604369
$396$		0			0
$397$	−7.11145e7			−0.0570415			−0.0285208	−	0.999593i	$-0.509080\pi$
$397$	−7.11145e7			−0.0570415			−0.0285208	+	0.999593i	$0.509080\pi$
$398$	5.07441e8	−	8.78914e8i	0.403455	−	0.698804i
$399$		0			0
$400$	−3.80517e8	−	6.59075e8i	−0.297279	−	0.514902i
$401$	6.71791e8	+	1.16358e9i	0.520270	+	0.901134i	0.999722	+	0.0235661i	$0.00750200\pi$
$401$	6.71791e8	+	1.16358e9i	0.520270	+	0.901134i	−0.479452	+	0.877568i	$0.659165\pi$
$402$		0			0
$403$	1.18106e8	−	2.04565e8i	0.0898882	−	0.155691i
$404$	−3.37410e8			−0.254579
$405$		0			0
$406$	−1.85953e8			−0.137899
$407$	1.46726e9	−	2.54136e9i	1.07876	−	1.86847i
$408$		0			0
$409$	3.56823e8	+	6.18035e8i	0.257882	+	0.446665i	0.965674	−	0.259756i	$-0.0836421\pi$
$409$	3.56823e8	+	6.18035e8i	0.257882	+	0.446665i	−0.707792	+	0.706421i	$0.750309\pi$
$410$	−5.22194e8	−	9.04467e8i	−0.374187	−	0.648111i
$411$		0			0
$412$	−4.58880e8	+	7.94804e8i	−0.323265	+	0.559912i
$413$	−2.76028e8			−0.192809
$414$		0			0
$415$	3.21741e9			2.20972
$416$	2.30600e8	−	3.99411e8i	0.157048	−	0.272015i
$417$		0			0
$418$	4.74798e8	+	8.22374e8i	0.317974	+	0.550747i
$419$	−4.62128e8	−	8.00430e8i	−0.306912	−	0.531587i	0.670773	−	0.741662i	$-0.265962\pi$
$419$	−4.62128e8	−	8.00430e8i	−0.306912	−	0.531587i	−0.977685	+	0.210076i	$0.932629\pi$
$420$		0			0
$421$	−1.27849e9	+	2.21441e9i	−0.835045	+	1.44634i	0.0589493	+	0.998261i	$0.481225\pi$
$421$	−1.27849e9	+	2.21441e9i	−0.835045	+	1.44634i	−0.893994	+	0.448079i	$0.852108\pi$
$422$	−3.27867e8			−0.212376
$423$		0			0
$424$	−2.92388e8			−0.186286
$425$	2.84021e8	−	4.91939e8i	0.179469	−	0.310849i
$426$		0			0
$427$	6.47769e8	+	1.12197e9i	0.402645	+	0.697402i
$428$	4.63193e8	+	8.02274e8i	0.285567	+	0.494617i
$429$		0			0
$430$	5.27923e8	−	9.14390e8i	0.320207	−	0.554615i
$431$	−2.72540e9			−1.63968			−0.819840	−	0.572593i	$-0.805938\pi$
$431$	−2.72540e9			−1.63968			−0.819840	+	0.572593i	$0.805938\pi$
$432$		0			0
$433$	−1.62728e9			−0.963284			−0.481642	−	0.876368i	$-0.659959\pi$
$433$	−1.62728e9			−0.963284			−0.481642	+	0.876368i	$0.659959\pi$
$434$	−6.24137e7	+	1.08104e8i	−0.0366493	+	0.0634785i
$435$		0			0
$436$	6.99583e8	+	1.21171e9i	0.404237	+	0.700159i
$437$	3.12067e8	+	5.40516e8i	0.178880	+	0.309830i
$438$		0			0
$439$	−8.12016e8	+	1.40645e9i	−0.458077	+	0.793413i	−0.998859	−	0.0477496i	$-0.984795\pi$
$439$	−8.12016e8	+	1.40645e9i	−0.458077	+	0.793413i	0.540782	+	0.841163i	$0.318128\pi$
$440$	1.98139e9			1.10889
$441$		0			0
$442$	3.44244e8			0.189622
$443$	−1.15001e9	+	1.99187e9i	−0.628475	+	1.08855i	0.359383	+	0.933190i	$0.382987\pi$
$443$	−1.15001e9	+	1.99187e9i	−0.628475	+	1.08855i	−0.987858	+	0.155360i	$0.950346\pi$
$444$		0			0
$445$	−1.28893e9	−	2.23249e9i	−0.693377	−	1.20097i
$446$	−5.65455e8	−	9.79397e8i	−0.301805	−	0.522741i
$447$		0			0
$448$	−1.21862e8	+	2.11072e8i	−0.0640319	+	0.110906i
$449$	2.21255e8			0.115353			0.0576767	−	0.998335i	$-0.481631\pi$
$449$	2.21255e8			0.115353			0.0576767	+	0.998335i	$0.481631\pi$
$450$		0			0
$451$	1.91423e9			0.982598
$452$	7.30761e8	−	1.26572e9i	0.372212	−	0.644691i
$453$		0			0
$454$	−8.32804e8	−	1.44246e9i	−0.417683	−	0.723449i
$455$	−3.36131e9	−	5.82196e9i	−1.67290	−	2.89754i
$456$		0			0
$457$	−1.07774e9	+	1.86670e9i	−0.528211	+	0.914889i	0.471248	+	0.882001i	$0.343804\pi$
$457$	−1.07774e9	+	1.86670e9i	−0.528211	+	0.914889i	−0.999459	+	0.0328880i	$0.989530\pi$
$458$	1.33113e9			0.647426
$459$		0			0
$460$	1.30230e9			0.623818
$461$	1.12474e9	−	1.94811e9i	0.534686	−	0.926104i	−0.464492	−	0.885577i	$-0.653763\pi$
$461$	1.12474e9	−	1.94811e9i	0.534686	−	0.926104i	0.999178	−	0.0405266i	$-0.0129036\pi$
$462$		0			0
$463$	1.88439e9	+	3.26386e9i	0.882344	+	1.52826i	0.848728	+	0.528829i	$0.177369\pi$
$463$	1.88439e9	+	3.26386e9i	0.882344	+	1.52826i	0.0336156	+	0.999435i	$0.489298\pi$
$464$	−5.12016e7	−	8.86838e7i	−0.0237942	−	0.0412127i
$465$		0			0
$466$	−1.05930e9	+	1.83476e9i	−0.484919	+	0.839904i
$467$	1.00692e9			0.457497			0.228748	−	0.973486i	$-0.426537\pi$
$467$	1.00692e9			0.457497			0.228748	+	0.973486i	$0.426537\pi$
$468$		0			0
$469$	−3.62476e7			−0.0162246
$470$	1.07835e8	−	1.86776e8i	0.0479091	−	0.0829811i
$471$		0			0
$472$	−7.60035e7	−	1.31642e8i	−0.0332687	−	0.0576232i
$473$	9.67614e8	+	1.67596e9i	0.420425	+	0.728197i
$474$		0			0
$475$	1.46387e9	−	2.53549e9i	0.626720	−	1.08551i
$476$	−1.81918e8			−0.0773128
$477$		0			0
$478$	4.12218e8			0.172635
$479$	5.42944e8	−	9.40407e8i	0.225726	−	0.390968i	−0.730811	−	0.682580i	$-0.760858\pi$
$479$	5.42944e8	−	9.40407e8i	0.225726	−	0.390968i	0.956537	+	0.291611i	$0.0941914\pi$
$480$		0			0
$481$	−2.74148e9	−	4.74838e9i	−1.12325	−	1.94553i
$482$	−5.36718e8	−	9.29623e8i	−0.218314	−	0.378131i
$483$		0			0
$484$	−1.19223e9	+	2.06500e9i	−0.477971	+	0.827870i
$485$	−3.36828e9			−1.34064
$486$		0			0
$487$	−2.90861e9			−1.14113			−0.570564	−	0.821253i	$-0.693275\pi$
$487$	−2.90861e9			−1.14113			−0.570564	+	0.821253i	$0.693275\pi$
$488$	−3.56723e8	+	6.17862e8i	−0.138951	+	0.240670i
$489$		0			0
$490$	8.39760e7	+	1.45451e8i	0.0322455	+	0.0558508i
$491$	2.09312e9	+	3.62538e9i	0.798010	+	1.38219i	0.920910	+	0.389774i	$0.127447\pi$
$491$	2.09312e9	+	3.62538e9i	0.798010	+	1.38219i	−0.122901	+	0.992419i	$0.539220\pi$
$492$		0			0
$493$	3.82173e7	−	6.61943e7i	0.0143647	−	0.0248804i
$494$	1.77426e9			0.662174
$495$		0			0
$496$	−6.87418e7			−0.0252950
$497$	5.45118e8	−	9.44173e8i	0.199179	−	0.344988i
$498$		0			0
$499$	−1.87965e9	−	3.25565e9i	−0.677213	−	1.17297i	−0.975817	−	0.218591i	$-0.929854\pi$
$499$	−1.87965e9	−	3.25565e9i	−0.677213	−	1.17297i	0.298603	−	0.954377i	$-0.403479\pi$
$500$	−1.77012e9	−	3.06593e9i	−0.633296	−	1.09690i
$501$		0			0
$502$	−1.27295e7	+	2.20482e7i	−0.00449107	+	0.00777875i
$503$	−2.94320e9			−1.03117			−0.515587	−	0.856837i	$-0.672426\pi$
$503$	−2.94320e9			−1.03117			−0.515587	+	0.856837i	$0.672426\pi$
$504$		0			0
$505$	−2.70843e9			−0.935831
$506$	−1.19347e9	+	2.06715e9i	−0.409530	+	0.709326i
$507$		0			0
$508$	1.38398e9	+	2.39712e9i	0.468389	+	0.811273i
$509$	5.95726e8	+	1.03183e9i	0.200233	+	0.346813i	0.948603	−	0.316468i	$-0.102497\pi$
$509$	5.95726e8	+	1.03183e9i	0.200233	+	0.346813i	−0.748371	+	0.663281i	$0.769164\pi$
$510$		0			0
$511$	−1.30629e9	+	2.26256e9i	−0.433079	+	0.750115i
$512$	−1.34218e8			−0.0441942
$513$		0			0
$514$	−3.36221e9			−1.09208
$515$	−3.68349e9	+	6.37999e9i	−1.18832	+	2.05823i
$516$		0			0
$517$	1.97648e8	+	3.42336e8i	0.0629036	+	0.108952i
$518$	1.44875e9	+	2.50931e9i	0.457973	+	0.793233i
$519$		0			0
$520$	1.85105e9	−	3.20612e9i	0.577308	−	0.999927i
$521$	3.63894e9			1.12731			0.563655	−	0.826010i	$-0.309395\pi$
$521$	3.63894e9			1.12731			0.563655	+	0.826010i	$0.309395\pi$
$522$		0			0
$523$	−4.37296e9			−1.33666			−0.668328	−	0.743867i	$-0.732990\pi$
$523$	−4.37296e9			−1.33666			−0.668328	+	0.743867i	$0.732990\pi$
$524$	−6.47310e8	+	1.12117e9i	−0.196541	+	0.340419i
$525$		0			0
$526$	−2.92869e8	−	5.07265e8i	−0.0877453	−	0.151979i
$527$	−2.56547e7	−	4.44352e7i	−0.00763537	−	0.0132248i
$528$		0			0
$529$	9.17988e8	−	1.59000e9i	0.269614	−	0.466985i
$530$	−2.34704e9			−0.684785
$531$		0			0
$532$	−9.37620e8			−0.269983
$533$	1.78831e9	−	3.09744e9i	0.511560	−	0.886048i
$534$		0			0
$535$	3.71810e9	+	6.43995e9i	1.04974	+	1.81821i
$536$	−9.98066e6	−	1.72870e7i	−0.00279951	−	0.00484890i
$537$		0			0
$538$	1.61329e9	−	2.79431e9i	0.446658	−	0.773635i
$539$	−3.07834e8			−0.0846752
$540$		0			0
$541$	−3.72635e7			−0.0101180			−0.00505898	−	0.999987i	$-0.501610\pi$
$541$	−3.72635e7			−0.0101180			−0.00505898	+	0.999987i	$0.501610\pi$
$542$	−3.03633e8	+	5.25907e8i	−0.0819127	+	0.141877i
$543$		0			0
$544$	−5.00906e7	−	8.67595e7i	−0.0133401	−	0.0231058i
$545$	5.61563e9	+	9.72656e9i	1.48597	+	2.57378i
$546$		0			0
$547$	−2.66081e9	+	4.60866e9i	−0.695118	+	1.20398i	0.275022	+	0.961438i	$0.411315\pi$
$547$	−2.66081e9	+	4.60866e9i	−0.695118	+	1.20398i	−0.970141	+	0.242543i	$0.922019\pi$
$548$	−1.77386e9			−0.460456
$549$		0			0
$550$	1.11968e10			2.86963
$551$	1.96975e8	−	3.41171e8i	0.0501627	−	0.0868843i
$552$		0			0
$553$	−6.69857e8	−	1.16023e9i	−0.168440	−	0.291746i
$554$	7.12783e8	+	1.23458e9i	0.178104	+	0.308485i
$555$		0			0
$556$	1.76869e9	−	3.06346e9i	0.436405	−	0.755876i
$557$	6.10360e9			1.49655			0.748277	−	0.663386i	$-0.230881\pi$
$557$	6.10360e9			1.49655			0.748277	+	0.663386i	$0.230881\pi$
$558$		0			0
$559$	3.61585e9			0.875526
$560$	−9.78203e8	+	1.69430e9i	−0.235381	+	0.407692i
$561$		0			0
$562$	2.35038e9	+	4.07098e9i	0.558548	+	0.967434i
$563$	−4.53011e8	−	7.84638e8i	−0.106987	−	0.185306i	0.807562	−	0.589783i	$-0.200787\pi$
$563$	−4.53011e8	−	7.84638e8i	−0.106987	−	0.185306i	−0.914548	+	0.404477i	$0.867454\pi$
$564$		0			0
$565$	5.86591e9	−	1.01600e10i	1.36825	−	2.36988i
$566$	2.27934e9			0.528386
$567$		0			0
$568$	6.00387e8			0.137471
$569$	3.01597e9	−	5.22381e9i	0.686331	−	1.18876i	−0.286685	−	0.958025i	$-0.592553\pi$
$569$	3.01597e9	−	5.22381e9i	0.686331	−	1.18876i	0.973016	−	0.230736i	$-0.0741133\pi$
$570$		0			0
$571$	5.83004e8	+	1.00979e9i	0.131052	+	0.226989i	0.924083	−	0.382193i	$-0.124831\pi$
$571$	5.83004e8	+	1.00979e9i	0.131052	+	0.226989i	−0.793030	+	0.609182i	$0.791498\pi$
$572$	3.39274e9	+	5.87640e9i	0.757992	+	1.31288i
$573$		0			0
$574$	−9.45044e8	+	1.63686e9i	−0.208574	+	0.361261i
$575$	7.35927e9			1.61435
$576$		0			0
$577$	2.91738e9			0.632233			0.316117	−	0.948720i	$-0.397621\pi$
$577$	2.91738e9			0.632233			0.316117	+	0.948720i	$0.397621\pi$
$578$	−1.60397e9	+	2.77815e9i	−0.345500	+	0.598423i
$579$		0			0
$580$	−4.11002e8	−	7.11876e8i	−0.0874673	−	0.151498i
$581$	−2.91136e9	−	5.04262e9i	−0.615856	−	1.06669i
$582$		0			0
$583$	2.15091e9	−	3.72548e9i	0.449553	−	0.778649i
$584$	−1.43874e9			−0.298907
$585$		0			0
$586$	2.81165e9			0.577192
$587$	−1.25480e9	+	2.17338e9i	−0.256060	+	0.443509i	−0.965183	−	0.261576i	$-0.915758\pi$
$587$	−1.25480e9	+	2.17338e9i	−0.256060	+	0.443509i	0.709123	+	0.705085i	$0.249091\pi$
$588$		0			0
$589$	−1.32226e8	−	2.29023e8i	−0.0266633	−	0.0461822i
$590$	−6.10089e8	−	1.05671e9i	−0.122296	−	0.211822i
$591$		0			0
$592$	−7.97820e8	+	1.38186e9i	−0.158044	+	0.273741i
$593$	−6.93736e9			−1.36616			−0.683082	−	0.730341i	$-0.739361\pi$
$593$	−6.93736e9			−1.36616			−0.683082	+	0.730341i	$0.739361\pi$
$594$		0			0
$595$	−1.46028e9			−0.284201
$596$	−1.77571e8	+	3.07562e8i	−0.0343566	+	0.0595075i
$597$		0			0
$598$	2.22993e9	+	3.86234e9i	0.426419	+	0.738579i
$599$	−1.27967e9	−	2.21646e9i	−0.243280	−	0.421373i	0.718367	−	0.695664i	$-0.244890\pi$
$599$	−1.27967e9	−	2.21646e9i	−0.243280	−	0.421373i	−0.961647	+	0.274292i	$0.911557\pi$
$600$		0			0
$601$	−5.69651e8	+	9.86665e8i	−0.107041	+	0.185400i	−0.914570	−	0.404427i	$-0.867471\pi$
$601$	−5.69651e8	+	9.86665e8i	−0.107041	+	0.185400i	0.807530	+	0.589827i	$0.200804\pi$
$602$	−1.91082e9			−0.356971
$603$		0			0
$604$	−1.86602e9			−0.344578
$605$	−9.57017e9	+	1.65760e10i	−1.75702	+	3.04324i
$606$		0			0
$607$	−3.64120e9	−	6.30675e9i	−0.660822	−	1.14458i	−0.980400	−	0.197018i	$-0.936874\pi$
$607$	−3.64120e9	−	6.30675e9i	−0.660822	−	1.14458i	0.319578	−	0.947560i	$-0.396459\pi$
$608$	−2.58171e8	−	4.47165e8i	−0.0465848	−	0.0806873i
$609$		0			0
$610$	−2.86345e9	+	4.95965e9i	−0.510782	+	0.884701i
$611$	7.38585e8			0.130996
$612$		0			0
$613$	−6.91313e9			−1.21217			−0.606084	−	0.795400i	$-0.707261\pi$
$613$	−6.91313e9			−1.21217			−0.606084	+	0.795400i	$0.707261\pi$
$614$	1.43283e9	−	2.48173e9i	0.249807	−	0.432678i
$615$		0			0
$616$	−1.79292e9	−	3.10543e9i	−0.309050	−	0.535290i
$617$	−7.01580e8	−	1.21517e9i	−0.120248	−	0.208276i	0.799617	−	0.600510i	$-0.205036\pi$
$617$	−7.01580e8	−	1.21517e9i	−0.120248	−	0.208276i	−0.919866	+	0.392234i	$0.871702\pi$
$618$		0			0
$619$	−5.09778e9	+	8.82962e9i	−0.863901	+	1.49632i	0.00423255	+	0.999991i	$0.498653\pi$
$619$	−5.09778e9	+	8.82962e9i	−0.863901	+	1.49632i	−0.868134	+	0.496330i	$0.834681\pi$
$620$	−5.51798e8			−0.0929843
$621$		0			0
$622$	2.37594e9			0.395884
$623$	−2.33265e9	+	4.04027e9i	−0.386493	+	0.669425i
$624$		0			0
$625$	−6.95115e9	−	1.20397e10i	−1.13888	−	1.97259i
$626$	2.12834e9	+	3.68640e9i	0.346762	+	0.600609i
$627$		0			0
$628$	1.08513e9	−	1.87950e9i	0.174833	−	0.302819i
$629$	−1.19100e9			−0.190824
$630$		0			0
$631$	−1.63463e9			−0.259010			−0.129505	−	0.991579i	$-0.541339\pi$
$631$	−1.63463e9			−0.259010			−0.129505	+	0.991579i	$0.541339\pi$
$632$	3.68886e8	−	6.38930e8i	0.0581277	−	0.100680i
$633$		0			0
$634$	1.11270e9	+	1.92725e9i	0.173406	+	0.300348i
$635$	1.11094e10	+	1.92420e10i	1.72179	+	2.98223i
$636$		0			0
$637$	−2.87584e8	+	4.98110e8i	−0.0440836	+	0.0763550i
$638$	1.50663e9			0.229685
$639$		0			0
$640$	−1.07738e9			−0.162457
$641$	8.20607e7	−	1.42133e8i	0.0123064	−	0.0213154i	−0.859807	−	0.510620i	$-0.829416\pi$
$641$	8.20607e7	−	1.42133e8i	0.0123064	−	0.0213154i	0.872113	+	0.489304i	$0.162749\pi$
$642$		0			0
$643$	−1.24387e9	−	2.15444e9i	−0.184516	−	0.319592i	0.758897	−	0.651211i	$-0.225739\pi$
$643$	−1.24387e9	−	2.15444e9i	−0.184516	−	0.319592i	−0.943413	+	0.331619i	$0.892405\pi$
$644$	−1.17842e9	−	2.04108e9i	−0.173860	−	0.301134i
$645$		0			0
$646$	1.92701e8	−	3.33767e8i	0.0281235	−	0.0487114i
$647$	−2.24150e9			−0.325368			−0.162684	−	0.986678i	$-0.552015\pi$
$647$	−2.24150e9			−0.325368			−0.162684	+	0.986678i	$0.552015\pi$
$648$		0			0
$649$	2.23643e9			0.321143
$650$	1.04603e10	−	1.81177e10i	1.49399	−	2.58766i
$651$		0			0
$652$	2.72272e9	+	4.71589e9i	0.384712	+	0.666341i
$653$	−5.04729e9	−	8.74216e9i	−0.709353	−	1.22863i	−0.965098	−	0.261890i	$-0.915654\pi$
$653$	−5.04729e9	−	8.74216e9i	−0.709353	−	1.22863i	0.255745	−	0.966744i	$-0.417679\pi$
$654$		0			0
$655$	−5.19604e9	+	8.99980e9i	−0.722483	+	1.25138i
$656$	−1.04086e9			−0.143956
$657$		0			0
$658$	−3.90311e8			−0.0534097
$659$	7.15278e8	−	1.23890e9i	0.0973590	−	0.168631i	−0.813232	−	0.581940i	$-0.802294\pi$
$659$	7.15278e8	−	1.23890e9i	0.0973590	−	0.168631i	0.910591	+	0.413309i	$0.135627\pi$
$660$		0			0
$661$	−2.72158e9	−	4.71392e9i	−0.366536	−	0.634859i	0.622485	−	0.782631i	$-0.286123\pi$
$661$	−2.72158e9	−	4.71392e9i	−0.366536	−	0.634859i	−0.989021	+	0.147772i	$0.952790\pi$
$662$	1.43257e8	+	2.48128e8i	0.0191916	+	0.0332409i
$663$		0			0
$664$	1.60327e9	−	2.77694e9i	0.212529	−	0.368111i
$665$	−7.52638e9			−0.992454
$666$		0			0
$667$	9.90250e8			0.129212
$668$	−1.55897e9	+	2.70021e9i	−0.202357	+	0.350493i
$669$		0			0
$670$	−8.01159e7	−	1.38765e8i	−0.0102910	−	0.0178245i
$671$	−5.24834e9	−	9.09039e9i	−0.670646	−	1.16159i
$672$		0			0
$673$	3.84720e9	−	6.66354e9i	0.486510	−	0.842660i	−0.513370	−	0.858168i	$-0.671603\pi$
$673$	3.84720e9	−	6.66354e9i	0.486510	−	0.842660i	0.999880	+	0.0155073i	$0.00493633\pi$
$674$	1.19136e9			0.149876
$675$		0			0
$676$	8.66234e9			1.07850
$677$	−3.31338e9	+	5.73895e9i	−0.410404	+	0.710840i	−0.994934	−	0.100532i	$-0.967946\pi$
$677$	−3.31338e9	+	5.73895e9i	−0.410404	+	0.710840i	0.584530	+	0.811372i	$0.301279\pi$
$678$		0			0
$679$	3.04788e9	+	5.27908e9i	0.373640	+	0.647163i
$680$	−4.02083e8	−	6.96428e8i	−0.0490382	−	0.0849367i
$681$		0			0
$682$	5.05687e8	−	8.75876e8i	0.0610431	−	0.105730i
$683$	1.25637e10			1.50885			0.754426	−	0.656386i	$-0.227916\pi$
$683$	1.25637e10			1.50885			0.754426	+	0.656386i	$0.227916\pi$
$684$		0			0
$685$	−1.42390e10			−1.69263
$686$	−2.91073e9	+	5.04154e9i	−0.344245	+	0.596250i
$687$		0			0
$688$	−5.26140e8	−	9.11300e8i	−0.0615944	−	0.106685i
$689$	−4.01883e9	−	6.96082e9i	−0.468093	−	0.810761i
$690$		0			0
$691$	−6.86500e9	+	1.18905e10i	−0.791530	+	1.37097i	0.133489	+	0.991050i	$0.457382\pi$
$691$	−6.86500e9	+	1.18905e10i	−0.791530	+	1.37097i	−0.925019	+	0.379921i	$0.875951\pi$
$692$	3.15829e9			0.362310
$693$		0			0
$694$	−9.59416e8			−0.108955
$695$	1.41975e10	−	2.45908e10i	1.60422	−	2.77859i
$696$		0			0
$697$	−3.88453e8	−	6.72821e8i	−0.0434534	−	0.0752635i
$698$	−5.63847e9	−	9.76611e9i	−0.627577	−	1.08700i
$699$		0			0
$700$	−5.52782e9	+	9.57446e9i	−0.609131	+	1.05505i
$701$	−1.06175e10			−1.16415			−0.582076	−	0.813134i	$-0.697759\pi$
$701$	−1.06175e10			−1.16415			−0.582076	+	0.813134i	$0.697759\pi$
$702$		0			0
$703$	−6.13850e9			−0.666375
$704$	9.87350e8	−	1.71014e9i	0.106651	−	0.184726i
$705$		0			0
$706$	−5.38950e9	−	9.33488e9i	−0.576410	−	0.998372i
$707$	2.45080e9	+	4.24490e9i	0.260819	+	0.451752i
$708$		0			0
$709$	6.45277e9	−	1.11765e10i	0.679962	−	1.17773i	−0.295030	−	0.955488i	$-0.595330\pi$
$709$	6.45277e9	−	1.11765e10i	0.679962	−	1.17773i	0.974992	−	0.222240i	$-0.0713369\pi$
$710$	4.81938e9			0.505344
$711$		0			0
$712$	−2.56915e9			−0.266753
$713$	3.32370e8	−	5.75681e8i	0.0343406	−	0.0594797i
$714$		0			0
$715$	2.72339e10	+	4.71706e10i	2.78637	+	4.82614i
$716$	1.54771e9	+	2.68071e9i	0.157577	+	0.272932i
$717$		0			0
$718$	4.83893e9	−	8.38126e9i	0.487880	−	0.845034i
$719$	9.38527e9			0.941664			0.470832	−	0.882223i	$-0.343954\pi$
$719$	9.38527e9			0.941664			0.470832	+	0.882223i	$0.343954\pi$
$720$		0			0
$721$	1.33324e10			1.32475
$722$	−2.58229e9	+	4.47266e9i	−0.255344	+	0.442268i
$723$		0			0
$724$	2.45323e9	+	4.24912e9i	0.240244	+	0.416116i
$725$	−2.32257e9	−	4.02280e9i	−0.226352	−	0.392054i
$726$		0			0
$727$	1.27432e9	−	2.20719e9i	0.123001	−	0.213044i	−0.797949	−	0.602725i	$-0.794081\pi$
$727$	1.27432e9	−	2.20719e9i	0.123001	−	0.213044i	0.920950	+	0.389681i	$0.127415\pi$
$728$	−6.69991e9			−0.643590
$729$		0			0
$730$	−1.15489e10			−1.09878
$731$	3.92715e8	−	6.80202e8i	0.0371849	−	0.0644061i
$732$		0			0
$733$	−6.37526e9	−	1.10423e10i	−0.597908	−	1.03561i	−0.993129	−	0.117021i	$-0.962666\pi$
$733$	−6.37526e9	−	1.10423e10i	−0.597908	−	1.03561i	0.395222	−	0.918586i	$-0.370668\pi$
$734$	5.41799e9	+	9.38424e9i	0.505711	+	0.875916i
$735$		0			0
$736$	6.48949e8	−	1.12401e9i	0.0599982	−	0.103920i
$737$	2.93684e8			0.0270237
$738$		0			0
$739$	5.86108e9			0.534222			0.267111	−	0.963666i	$-0.413931\pi$
$739$	5.86108e9			0.534222			0.267111	+	0.963666i	$0.413931\pi$
$740$	−6.40420e9	+	1.10924e10i	−0.580970	+	1.00627i
$741$		0			0
$742$	2.12378e9	+	3.67849e9i	0.190852	+	0.330565i
$743$	−4.54298e9	−	7.86867e9i	−0.406331	−	0.703785i	0.588145	−	0.808756i	$-0.299859\pi$
$743$	−4.54298e9	−	7.86867e9i	−0.406331	−	0.703785i	−0.994475	+	0.104970i	$0.966525\pi$
$744$		0			0
$745$	−1.42539e9	+	2.46884e9i	−0.126295	+	0.218749i
$746$	4.87955e9			0.430323
$747$		0			0
$748$	1.47393e9			0.128772
$749$	6.72886e9	−	1.16547e10i	0.585134	−	1.01348i
$750$		0			0
$751$	−9.81435e9	−	1.69989e10i	−0.845516	−	1.46448i	−0.885173	−	0.465263i	$-0.845960\pi$
$751$	−9.81435e9	−	1.69989e10i	−0.845516	−	1.46448i	0.0396571	−	0.999213i	$-0.487373\pi$
$752$	−1.07471e8	−	1.86145e8i	−0.00921571	−	0.0159621i
$753$		0			0
$754$	1.40752e9	−	2.43789e9i	0.119579	−	0.207116i
$755$	−1.49788e10			−1.26667
$756$		0			0
$757$	9.09151e9			0.761729			0.380864	−	0.924631i	$-0.375627\pi$
$757$	9.09151e9			0.761729			0.380864	+	0.924631i	$0.375627\pi$
$758$	−1.10886e8	+	1.92061e8i	−0.00924773	+	0.0160175i
$759$		0			0
$760$	−2.07237e9	−	3.58945e9i	−0.171246	−	0.296606i
$761$	−2.33119e9	−	4.03775e9i	−0.191749	−	0.332118i	0.754081	−	0.656781i	$-0.228082\pi$
$761$	−2.33119e9	−	4.03775e9i	−0.191749	−	0.332118i	−0.945830	+	0.324663i	$0.894749\pi$
$762$		0			0
$763$	1.01629e10	−	1.76027e10i	0.828290	−	1.43464i
$764$	−9.86910e9			−0.800665
$765$		0			0
$766$	−1.52428e10			−1.22536
$767$	2.08931e9	−	3.61879e9i	0.167193	−	0.289588i
$768$		0			0
$769$	1.57011e8	+	2.71950e8i	0.0124505	+	0.0215649i	0.872184	−	0.489179i	$-0.162703\pi$
$769$	1.57011e8	+	2.71950e8i	0.0124505	+	0.0215649i	−0.859733	+	0.510744i	$0.829370\pi$
$770$	−1.43920e10	−	2.49276e10i	−1.13606	−	1.96772i
$771$		0			0
$772$	−4.30393e9	+	7.45462e9i	−0.336670	+	0.583129i
$773$	2.16691e10			1.68738			0.843688	−	0.536834i	$-0.180380\pi$
$773$	2.16691e10			1.68738			0.843688	+	0.536834i	$0.180380\pi$
$774$		0			0
$775$	−3.11821e9			−0.240630
$776$	−1.67845e9	+	2.90716e9i	−0.128941	+	0.223333i
$777$		0			0
$778$	2.92221e9	+	5.06141e9i	0.222476	+	0.385339i
$779$	−2.00212e9	−	3.46777e9i	−0.151743	−	0.262827i
$780$		0			0
$781$	−4.41665e9	+	7.64986e9i	−0.331752	+	0.574612i
$782$	9.68762e8			0.0724425
$783$		0			0
$784$	1.67385e8			0.0124054
$785$	8.71046e9	−	1.50870e10i	0.642684	−	1.11316i
$786$		0			0
$787$	−6.63078e8	−	1.14848e9i	−0.0484901	−	0.0839873i	0.840762	−	0.541405i	$-0.182108\pi$
$787$	−6.63078e8	−	1.14848e9i	−0.0484901	−	0.0839873i	−0.889252	+	0.457418i	$0.848774\pi$
$788$	−3.94654e9	−	6.83560e9i	−0.287326	−	0.497663i
$789$		0			0
$790$	2.96109e9	−	5.12877e9i	0.213677	−	0.370099i
$791$	−2.12317e10			−1.52534
$792$		0			0
$793$	−1.96124e10			−1.39661
$794$	−2.84458e8	+	4.92695e8i	−0.0201672	+	0.0349307i
$795$		0			0
$796$	−4.05953e9	−	7.03131e9i	−0.285286	−	0.494129i
$797$	−1.01981e10	−	1.76636e10i	−0.713534	−	1.23588i	−0.963522	−	0.267628i	$-0.913760\pi$
$797$	−1.01981e10	−	1.76636e10i	−0.713534	−	1.23588i	0.249988	−	0.968249i	$-0.419573\pi$
$798$		0			0
$799$	8.02172e7	−	1.38940e8i	0.00556357	−	0.00963639i
$800$	−6.08827e9			−0.420416
$801$		0			0
$802$	1.07487e10			0.735773
$803$	1.05838e10	−	1.83317e10i	0.721336	−	1.24939i
$804$		0			0
$805$	−9.45932e9	−	1.63840e10i	−0.639108	−	1.10697i
$806$	−9.44844e8	−	1.63652e9i	−0.0635605	−	0.110090i
$807$		0			0
$808$	−1.34964e9	+	2.33764e9i	−0.0900073	+	0.155897i
$809$	−1.90447e10			−1.26461			−0.632303	−	0.774721i	$-0.717890\pi$
$809$	−1.90447e10			−1.26461			−0.632303	+	0.774721i	$0.717890\pi$
$810$		0			0
$811$	1.63087e10			1.07361			0.536806	−	0.843706i	$-0.319631\pi$
$811$	1.63087e10			1.07361			0.536806	+	0.843706i	$0.319631\pi$
$812$	−7.43812e8	+	1.28832e9i	−0.0487548	+	0.0844458i
$813$		0			0
$814$	−1.17381e10	−	2.03309e10i	−0.762800	−	1.32121i
$815$	2.18556e10	+	3.78550e10i	1.41420	+	2.44947i
$816$		0			0
$817$	2.02408e9	−	3.50581e9i	0.129853	−	0.224911i
$818$	5.70916e9			0.364700
$819$		0			0
$820$	−8.35511e9			−0.529180
$821$	−7.84871e9	+	1.35944e10i	−0.494991	+	0.857350i	−0.999983	−	0.00577401i	$-0.998162\pi$
$821$	−7.84871e9	+	1.35944e10i	−0.494991	+	0.857350i	0.504992	+	0.863124i	$0.331495\pi$
$822$		0			0
$823$	9.11320e9	+	1.57845e10i	0.569864	+	0.987034i	0.996579	+	0.0826475i	$0.0263375\pi$
$823$	9.11320e9	+	1.57845e10i	0.569864	+	0.987034i	−0.426715	+	0.904386i	$0.640329\pi$
$824$	3.67104e9	+	6.35843e9i	0.228583	+	0.395918i
$825$		0			0
$826$	−1.10411e9	+	1.91238e9i	−0.0681684	+	0.118071i
$827$	−1.29185e10			−0.794222			−0.397111	−	0.917771i	$-0.629987\pi$
$827$	−1.29185e10			−0.794222			−0.397111	+	0.917771i	$0.629987\pi$
$828$		0			0
$829$	1.25055e10			0.762357			0.381179	−	0.924501i	$-0.375518\pi$
$829$	1.25055e10			0.762357			0.381179	+	0.924501i	$0.375518\pi$
$830$	1.28696e10	−	2.22909e10i	0.781255	−	1.35317i
$831$		0			0
$832$	−1.84480e9	−	3.19529e9i	−0.111050	−	0.192344i
$833$	6.24686e7	+	1.08199e8i	0.00374459	+	0.00648582i
$834$		0			0
$835$	−1.25140e10	+	2.16749e10i	−0.743865	+	1.28841i
$836$	7.59676e9			0.449683
$837$		0			0
$838$	−7.39405e9			−0.434039
$839$	−3.49281e9	+	6.04972e9i	−0.204177	+	0.353646i	−0.949870	−	0.312644i	$-0.898785\pi$
$839$	−3.49281e9	+	6.04972e9i	−0.204177	+	0.353646i	0.745693	+	0.666290i	$0.232119\pi$
$840$		0			0
$841$	8.31242e9	+	1.43975e10i	0.481883	+	0.834645i
$842$	1.02279e10	+	1.77153e10i	0.590466	+	1.02272i
$843$		0			0
$844$	−1.31147e9	+	2.27153e9i	−0.0750861	+	0.130053i
$845$	6.95336e10			3.96457
$846$		0			0
$847$	3.46393e10			1.95874
$848$	−1.16955e9	+	2.02573e9i	−0.0658619	+	0.114076i
$849$		0			0
$850$	−2.27217e9	−	3.93551e9i	−0.126904	−	0.219804i
$851$	−7.71500e9	−	1.33628e10i	−0.429123	−	0.743264i
$852$		0			0
$853$	1.44807e10	−	2.50814e10i	0.798858	−	1.38366i	−0.121503	−	0.992591i	$-0.538771\pi$
$853$	1.44807e10	−	2.50814e10i	0.798858	−	1.38366i	0.920361	−	0.391071i	$-0.127895\pi$
$854$	1.03643e10			0.569426
$855$		0			0
$856$	7.41109e9			0.403853
$857$	−9.43344e9	+	1.63392e10i	−0.511962	+	0.886743i	0.487942	+	0.872876i	$0.337748\pi$
$857$	−9.43344e9	+	1.63392e10i	−0.511962	+	0.886743i	−0.999904	+	0.0138675i	$0.995586\pi$
$858$		0			0
$859$	−1.28385e10	−	2.22369e10i	−0.691095	−	1.19701i	−0.971480	−	0.237123i	$-0.923795\pi$
$859$	−1.28385e10	−	2.22369e10i	−0.691095	−	1.19701i	0.280385	−	0.959888i	$-0.409538\pi$
$860$	−4.22338e9	−	7.31512e9i	−0.226421	−	0.392172i
$861$		0			0
$862$	−1.09016e10	+	1.88821e10i	−0.579714	+	1.00409i
$863$	−2.32394e10			−1.23080			−0.615400	−	0.788215i	$-0.711005\pi$
$863$	−2.32394e10			−1.23080			−0.615400	+	0.788215i	$0.711005\pi$
$864$		0			0
$865$	2.53519e10			1.33185
$866$	−6.50911e9	+	1.12741e10i	−0.340572	+	0.589888i
$867$		0			0
$868$	4.99310e8	+	8.64830e8i	0.0259150	+	0.0448861i
$869$	5.42730e9	+	9.40036e9i	0.280553	+	0.485932i
$870$		0			0
$871$	2.74365e8	−	4.75214e8i	0.0140691	−	0.0243683i
$872$	1.11933e10			0.571678
$873$		0			0
$874$	4.99307e9			0.252975
$875$	−2.57147e10	+	4.45391e10i	−1.29764	+	2.24757i
$876$		0			0
$877$	4.88371e9	+	8.45883e9i	0.244484	+	0.423459i	0.961986	−	0.273097i	$-0.0880480\pi$
$877$	4.88371e9	+	8.45883e9i	0.244484	+	0.423459i	−0.717502	+	0.696556i	$0.754715\pi$
$878$	6.49613e9	+	1.12516e10i	0.323910	+	0.561028i
$879$		0			0
$880$	7.92558e9	−	1.37275e10i	0.392050	−	0.679051i
$881$	−2.48003e10			−1.22192			−0.610959	−	0.791662i	$-0.709216\pi$
$881$	−2.48003e10			−1.22192			−0.610959	+	0.791662i	$0.709216\pi$
$882$		0			0
$883$	2.25403e10			1.10178			0.550892	−	0.834576i	$-0.314288\pi$
$883$	2.25403e10			1.10178			0.550892	+	0.834576i	$0.314288\pi$
$884$	1.37697e9	−	2.38499e9i	0.0670414	−	0.116119i
$885$		0			0
$886$	9.20007e9	+	1.59350e10i	0.444399	+	0.769722i
$887$	−1.72287e10	−	2.98409e10i	−0.828932	−	1.43575i	−0.898877	−	0.438202i	$-0.855616\pi$
$887$	−1.72287e10	−	2.98409e10i	−0.828932	−	1.43575i	0.0699446	−	0.997551i	$-0.477718\pi$
$888$		0			0
$889$	2.01052e10	−	3.48233e10i	0.959738	−	1.66231i
$890$	−2.06229e10			−0.980584
$891$		0			0
$892$	−9.04728e9			−0.426816
$893$	4.13446e8	−	7.16109e8i	0.0194285	−	0.0336511i
$894$		0			0
$895$	1.24237e10	+	2.15184e10i	0.579254	+	1.00330i
$896$	9.74898e8	+	1.68857e9i	0.0452774	+	0.0784227i
$897$		0			0
$898$	8.85019e8	−	1.53290e9i	0.0407836	−	0.0706392i
$899$	−4.19580e8			−0.0192600
$900$		0			0
$901$	−1.74593e9			−0.0795224
$902$	7.65692e9	−	1.32622e10i	0.347401	−	0.601716i
$903$		0			0
$904$	−5.84609e9	−	1.01257e10i	−0.263194	−	0.455865i
$905$	1.96924e10	+	3.41082e10i	0.883137	+	1.52964i
$906$		0			0
$907$	−1.46001e10	+	2.52881e10i	−0.649726	+	1.12536i	0.333463	+	0.942763i	$0.391783\pi$
$907$	−1.46001e10	+	2.52881e10i	−0.649726	+	1.12536i	−0.983188	+	0.182594i	$0.941550\pi$
$908$	−1.33249e10			−0.590693
$909$		0			0
$910$	−5.37810e10			−2.36583
$911$	−6.84514e9	+	1.18561e10i	−0.299963	+	0.519551i	−0.976127	−	0.217200i	$-0.930308\pi$
$911$	−6.84514e9	+	1.18561e10i	−0.299963	+	0.519551i	0.676164	+	0.736751i	$0.263641\pi$
$912$		0			0
$913$	2.35884e10	+	4.08562e10i	1.02577	+	1.77669i
$914$	8.62193e9	+	1.49336e10i	0.373502	+	0.646924i
$915$		0			0
$916$	5.32450e9	−	9.22231e9i	0.228900	−	0.396466i
$917$	1.88071e10			0.805433
$918$		0			0
$919$	−2.26823e10			−0.964014			−0.482007	−	0.876167i	$-0.660092\pi$
$919$	−2.26823e10			−0.964014			−0.482007	+	0.876167i	$0.660092\pi$
$920$	5.20919e9	−	9.02259e9i	0.220553	−	0.382009i
$921$		0			0
$922$	−8.99792e9	−	1.55849e10i	−0.378080	−	0.654854i
$923$	8.25222e9	+	1.42933e10i	0.345434	+	0.598309i
$924$		0			0
$925$	−3.61900e10	+	6.26830e10i	−1.50346	+	2.60408i
$926$	3.01503e10			1.24782
$927$		0			0
$928$	−8.19226e8			−0.0336501
$929$	−5.42790e9	+	9.40139e9i	−0.222114	+	0.384713i	−0.955450	−	0.295154i	$-0.904629\pi$
$929$	−5.42790e9	+	9.40139e9i	−0.222114	+	0.384713i	0.733335	+	0.679867i	$0.237962\pi$
$930$		0			0
$931$	3.21968e8	+	5.57665e8i	0.0130764	+	0.0226490i
$932$	8.47441e9	+	1.46781e10i	0.342889	+	0.593902i
$933$		0			0
$934$	4.02770e9	−	6.97618e9i	0.161750	−	0.280158i
$935$	1.18314e10			0.473366
$936$		0			0
$937$	−2.04665e10			−0.812744			−0.406372	−	0.913708i	$-0.633206\pi$
$937$	−2.04665e10			−0.812744			−0.406372	+	0.913708i	$0.633206\pi$
$938$	−1.44990e8	+	2.51130e8i	−0.00573626	+	0.00993549i
$939$		0			0
$940$	−8.62682e8	−	1.49421e9i	−0.0338769	−	0.0586765i
$941$	1.49376e10	+	2.58727e10i	0.584409	+	1.01223i	0.994949	+	0.100383i	$0.0320069\pi$
$941$	1.49376e10	+	2.58727e10i	0.584409	+	1.01223i	−0.410540	+	0.911843i	$0.634660\pi$
$942$		0			0
$943$	5.03261e9	−	8.71674e9i	0.195435	−	0.338503i
$944$	−1.21606e9			−0.0470491
$945$		0			0
$946$	1.54818e10			0.594570
$947$	1.51587e10	−	2.62557e10i	0.580013	−	1.00461i	−0.415464	−	0.909610i	$-0.636381\pi$
$947$	1.51587e10	−	2.62557e10i	0.580013	−	1.00461i	0.995477	−	0.0950029i	$-0.0302860\pi$
$948$		0			0
$949$	−1.97752e10	−	3.42516e10i	−0.751084	−	1.30092i
$950$	−1.17109e10	−	2.02839e10i	−0.443158	−	0.767572i
$951$		0			0
$952$	−7.27672e8	+	1.26036e9i	−0.0273342	+	0.0473442i
$953$	1.84836e10			0.691770			0.345885	−	0.938277i	$-0.387579\pi$
$953$	1.84836e10			0.691770			0.345885	+	0.938277i	$0.387579\pi$
$954$		0			0
$955$	−7.92205e10			−2.94324
$956$	1.64887e9	−	2.85593e9i	0.0610357	−	0.105717i
$957$		0			0
$958$	−4.34355e9	−	7.52326e9i	−0.159612	−	0.276456i
$959$	1.28846e10	+	2.23167e10i	0.471742	+	0.817081i
$960$		0			0
$961$	1.36155e10	−	2.35827e10i	0.494881	−	0.857160i
$962$	−4.38636e10			−1.58852
$963$		0			0
$964$	−8.58749e9			−0.308743
$965$	−3.45481e10	+	5.98391e10i	−1.23760	+	2.14358i
$966$		0			0
$967$	2.68223e10	+	4.64576e10i	0.953901	+	1.65221i	0.736863	+	0.676042i	$0.236306\pi$
$967$	2.68223e10	+	4.64576e10i	0.953901	+	1.65221i	0.217038	+	0.976163i	$0.430360\pi$
$968$	9.53784e9	+	1.65200e10i	0.337976	+	0.585392i
$969$		0			0
$970$	−1.34731e10	+	2.33361e10i	−0.473987	+	0.820970i
$971$	1.49741e10			0.524895			0.262447	−	0.964946i	$-0.415470\pi$
$971$	1.49741e10			0.524895			0.262447	+	0.964946i	$0.415470\pi$
$972$		0			0
$973$	−5.13879e10			−1.78841
$974$	−1.16344e10	+	2.01514e10i	−0.403449	+	0.698795i
$975$		0			0
$976$	2.85378e9	+	4.94289e9i	0.0982531	+	0.170179i
$977$	1.22943e10	+	2.12944e10i	0.421769	+	0.730525i	0.996113	−	0.0880893i	$-0.0280761\pi$
$977$	1.22943e10	+	2.12944e10i	0.421769	+	0.730525i	−0.574344	+	0.818614i	$0.694743\pi$
$978$		0			0
$979$	1.88995e10	−	3.27350e10i	0.643742	−	1.11499i
$980$	1.34362e9			0.0456020
$981$		0			0
$982$	3.34899e10			1.12856
$983$	−1.52926e10	+	2.64875e10i	−0.513503	+	0.889413i	0.486375	+	0.873750i	$0.338319\pi$
$983$	−1.52926e10	+	2.64875e10i	−0.513503	+	0.889413i	−0.999877	+	0.0156623i	$0.995014\pi$
$984$		0			0
$985$	−3.16793e10	−	5.48702e10i	−1.05621	−	1.82940i
$986$	−3.05739e8	−	5.29555e8i	−0.0101574	−	0.0175931i
$987$		0			0
$988$	7.09703e9	−	1.22924e10i	0.234114	−	0.405497i
$989$	1.01756e10			0.334483
$990$		0			0
$991$	5.03888e10			1.64466			0.822330	−	0.569010i	$-0.192674\pi$
$991$	5.03888e10			1.64466			0.822330	+	0.569010i	$0.192674\pi$
$992$	−2.74967e8	+	4.76257e8i	−0.00894314	+	0.0154900i
$993$		0			0
$994$	−4.36095e9	−	7.55338e9i	−0.140841	−	0.243943i
$995$	−3.25863e10	−	5.64412e10i	−1.04871	−	1.81642i
$996$		0			0
$997$	1.54996e8	−	2.68461e8i	0.00495322	−	0.00857923i	−0.863538	−	0.504283i	$-0.831757\pi$
$997$	1.54996e8	−	2.68461e8i	0.00495322	−	0.00857923i	0.868491	+	0.495704i	$0.165090\pi$
$998$	−3.00744e10			−0.957724
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	162.8.c.p.55.1		4
3.2	odd	2		162.8.c.m.55.2		4
9.2	odd	6		54.8.a.h.1.1	yes	2
9.4	even	3	inner	162.8.c.p.109.1		4
9.5	odd	6		162.8.c.m.109.2		4
9.7	even	3		54.8.a.g.1.2	✓	2
36.7	odd	6		432.8.a.p.1.2		2
36.11	even	6		432.8.a.k.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
54.8.a.g.1.2	✓	2	9.7	even	3
54.8.a.h.1.1	yes	2	9.2	odd	6
162.8.c.m.55.2		4	3.2	odd	2
162.8.c.m.109.2		4	9.5	odd	6
162.8.c.p.55.1		4	1.1	even	1	trivial
162.8.c.p.109.1		4	9.4	even	3	inner
432.8.a.k.1.1		2	36.11	even	6
432.8.a.p.1.2		2	36.7	odd	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 162 = 2 \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	162.c (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(4.00000 - 6.92820i) q^{2} +(-32.0000 - 55.4256i) q^{4} +(-256.868 - 444.908i) q^{5} +(-464.868 + 805.175i) q^{7} -512.000 q^{8} -4109.89 q^{10} +(3766.44 - 6523.67i) q^{11} +(-7037.36 - 12189.1i) q^{13} +(3718.94 + 6441.40i) q^{14} +(-2048.00 + 3547.24i) q^{16} -3057.29 q^{17} -15757.5 q^{19} +(-16439.5 + 28474.1i) q^{20} +(-30131.5 - 52189.4i) q^{22} +(-19804.4 - 34302.2i) q^{23} +(-92899.7 + 160907. i) q^{25} -112598. q^{26} +59503.1 q^{28} +(-12500.4 + 21651.3i) q^{29} +(8391.33 + 14534.2i) q^{31} +(16384.0 + 28377.9i) q^{32} +(-12229.1 + 21181.5i) q^{34} +477638. q^{35} +389561. q^{37} +(-63030.0 + 109171. i) q^{38} +(131516. + 227793. i) q^{40} +(127058. + 220071. i) q^{41} +(-128452. + 222485. i) q^{43} -482105. q^{44} -316870. q^{46} +(-26238.0 + 45445.6i) q^{47} +(-20432.7 - 35390.4i) q^{49} +(743197. + 1.28726e6i) q^{50} +(-450391. + 780100. i) q^{52} +571071. q^{53} -3.86991e6 q^{55} +(238012. - 412249. i) q^{56} +(100003. + 173211. i) q^{58} +(148444. + 257113. i) q^{59} +(696724. - 1.20676e6i) q^{61} +134261. q^{62} +262144. q^{64} +(-3.61534e6 + 6.26195e6i) q^{65} +(19493.5 + 33763.7i) q^{67} +(97833.2 + 169452. i) q^{68} +(1.91055e6 - 3.30918e6i) q^{70} -1.17263e6 q^{71} +2.81003e6 q^{73} +(1.55824e6 - 2.69895e6i) q^{74} +(504240. + 873369. i) q^{76} +(3.50180e6 + 6.06529e6i) q^{77} +(-720481. + 1.24791e6i) q^{79} +2.10426e6 q^{80} +2.03293e6 q^{82} +(-3.13138e6 + 5.42372e6i) q^{83} +(785319. + 1.36021e6i) q^{85} +(1.02762e6 + 1.77988e6i) q^{86} +(-1.92842e6 + 3.34012e6i) q^{88} +5.01788e6 q^{89} +1.30858e7 q^{91} +(-1.26748e6 + 2.19534e6i) q^{92} +(209904. + 363565. i) q^{94} +(4.04759e6 + 7.01064e6i) q^{95} +(3.27822e6 - 5.67804e6i) q^{97} -326923. q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 16 q^{2} - 128 q^{4} - 48 q^{5} - 880 q^{7} - 2048 q^{8} - 768 q^{10} + 7230 q^{11} - 8560 q^{13} + 7040 q^{14} - 8192 q^{16} - 51408 q^{17} + 74096 q^{19} - 3072 q^{20} - 57840 q^{22} - 59628 q^{23}+ \cdots + 12483264 q^{98}+O(q^{100}) \) 4 * q + 16 * q^2 - 128 * q^4 - 48 * q^5 - 880 * q^7 - 2048 * q^8 - 768 * q^10 + 7230 * q^11 - 8560 * q^13 + 7040 * q^14 - 8192 * q^16 - 51408 * q^17 + 74096 * q^19 - 3072 * q^20 - 57840 * q^22 - 59628 * q^23 - 324584 * q^25 - 136960 * q^26 + 112640 * q^28 - 275280 * q^29 - 245584 * q^31 + 65536 * q^32 - 205632 * q^34 + 1001604 * q^35 - 142120 * q^37 + 296384 * q^38 + 24576 * q^40 + 494520 * q^41 - 825280 * q^43 - 925440 * q^44 - 954048 * q^46 - 927708 * q^47 + 780204 * q^49 + 2596672 * q^50 - 547840 * q^52 + 2764224 * q^53 - 8021952 * q^55 + 450560 * q^56 + 2202240 * q^58 + 1588920 * q^59 + 3021968 * q^61 - 3929344 * q^62 + 1048576 * q^64 - 9799080 * q^65 + 1882160 * q^67 + 1645056 * q^68 + 4006416 * q^70 + 6788880 * q^71 + 1790180 * q^73 - 568480 * q^74 - 2371072 * q^76 + 7018656 * q^77 + 369920 * q^79 + 393216 * q^80 + 7912320 * q^82 - 9352050 * q^83 - 8976744 * q^85 + 6602240 * q^86 - 3701760 * q^88 + 1825920 * q^89 + 26720080 * q^91 - 3816192 * q^92 + 7421664 * q^94 + 32688588 * q^95 - 1359790 * q^97 + 12483264 * q^98

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