LMFDB - Embedded newform 160.8.n.b.127.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [160,8,Mod(63,160)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(160, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 0, 3]))

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

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

chi := DirichletCharacter("160.63");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 160 = 2^{5} \cdot 5 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	160.n (of order $4$, degree $2$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$49.9816040775$
Analytic rank:	$0$
Dimension:	$20$
Relative dimension:	$10$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 2 x^{19} - 13755 x^{18} - 18266 x^{17} + 77176511 x^{16} + 352443750 x^{15} + \cdots + 51\!\cdots\!88 $ x^20 - 2x^19 - 13755x^18 - 18266x^17 + 77176511x^16 + 352443750x^15 - 226611924763x^14 - 1758434718274x^13 + 370823584610133x^12 + 4065620251006082x^11 - 330927571498916861x^10 - 4694431557626426502x^9 + 143801066495201741005x^8 + 2480516059849648878514x^7 - 22890674373121550880249x^6 - 443749301321387555132390x^5 + 2167585477982423920911062x^4 + 35131537735851817170161256x^3 - 147219457505527453660598772x^2 - 1084930630243156967937214968*x + 5114948838423692248181226888
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{69}\cdot 5^{10} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			127.2
Root			$47.4084 - 1.00000i$ of defining polynomial
Character	$\chi$	$=$	160.127
Dual form			160.8.n.b.63.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-44.4084 - 44.4084i) q^{3} +(-259.462 + 103.945i) q^{5} +(-261.567 + 261.567i) q^{7} +1757.21i q^{9} +4843.71i q^{11} +(1479.02 - 1479.02i) q^{13} +(16138.3 + 6906.24i) q^{15} +(907.816 + 907.816i) q^{17} +13306.0 q^{19} +23231.6 q^{21} +(31405.7 + 31405.7i) q^{23} +(56515.8 - 53939.6i) q^{25} +(-19086.2 + 19086.2i) q^{27} -31344.6i q^{29} +216018. i q^{31} +(215101. - 215101. i) q^{33} +(40678.0 - 95055.4i) q^{35} +(-231635. - 231635. i) q^{37} -131362. q^{39} +565257. q^{41} +(-393861. - 393861. i) q^{43} +(-182654. - 455929. i) q^{45} +(-315328. + 315328. i) q^{47} +686708. i q^{49} -80629.3i q^{51} +(-292177. + 292177. i) q^{53} +(-503481. - 1.25676e6i) q^{55} +(-590897. - 590897. i) q^{57} -1.72159e6 q^{59} -3.08385e6 q^{61} +(-459629. - 459629. i) q^{63} +(-230011. + 537485. i) q^{65} +(-45437.5 + 45437.5i) q^{67} -2.78935e6i q^{69} -1.19240e6i q^{71} +(4.18729e6 - 4.18729e6i) q^{73} +(-4.90515e6 - 114401. i) q^{75} +(-1.26696e6 - 1.26696e6i) q^{77} -5.76074e6 q^{79} +5.53820e6 q^{81} +(-1.79184e6 - 1.79184e6i) q^{83} +(-329907. - 141180. i) q^{85} +(-1.39196e6 + 1.39196e6i) q^{87} -7.42470e6i q^{89} +773724. i q^{91} +(9.59302e6 - 9.59302e6i) q^{93} +(-3.45239e6 + 1.38309e6i) q^{95} +(-1.02064e7 - 1.02064e7i) q^{97} -8.51143e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 58 q^{3} - 54 q^{5} + 2466 q^{7} - 1172 q^{13} - 11138 q^{15} - 25136 q^{17} + 64784 q^{19} - 71268 q^{21} - 39922 q^{23} + 118056 q^{25} + 31792 q^{27} + 59756 q^{33} - 378426 q^{35} - 647408 q^{37}+ \cdots - 7823900 q^{99}+O(q^{100}) $ 20 * q + 58 * q^3 - 54 * q^5 + 2466 * q^7 - 1172 * q^13 - 11138 * q^15 - 25136 * q^17 + 64784 * q^19 - 71268 * q^21 - 39922 * q^23 + 118056 * q^25 + 31792 * q^27 + 59756 * q^33 - 378426 * q^35 - 647408 * q^37 + 1415100 * q^39 - 1151884 * q^41 - 1589406 * q^43 + 2340830 * q^45 - 1417862 * q^47 - 2115336 * q^53 - 397400 * q^55 - 2093064 * q^57 + 6498576 * q^59 + 1936564 * q^61 + 651590 * q^63 + 4800532 * q^65 + 1634150 * q^67 - 2406004 * q^73 - 2894398 * q^75 - 2176156 * q^77 - 11749104 * q^79 - 12415936 * q^81 - 13229814 * q^83 - 10242140 * q^85 - 37792 * q^87 - 9544900 * q^93 + 30202616 * q^95 + 24630564 * q^97 - 7823900 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/160\mathbb{Z}\right)^\times$.

$n$	$31$	$97$	$101$
$\chi(n)$	$-1$	$e\left(\frac{1}{4}\right)$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	−44.4084	−	44.4084i	−0.949600	−	0.949600i	0.0491894	−	0.998789i	$-0.484336\pi$
$3$	−44.4084	−	44.4084i	−0.949600	−	0.949600i	−0.998789	+	0.0491894i	$0.984336\pi$
$4$		0			0
$5$	−259.462	+	103.945i	−0.928278	+	0.371886i
$5$	−259.462	+	103.945i	−0.928278	+	0.371886i
$6$		0			0
$7$	−261.567	+	261.567i	−0.288231	+	0.288231i	−0.836380	−	0.548150i	$-0.815332\pi$
$7$	−261.567	+	261.567i	−0.288231	+	0.288231i	0.548150	+	0.836380i	$0.315332\pi$
$8$		0			0
$9$			1757.21i			0.803481i
$10$		0			0
$11$			4843.71i			1.09725i	0.836070	+	0.548623i	$0.184848\pi$
$11$			4843.71i			1.09725i	−0.836070	+	0.548623i	$0.815152\pi$
$12$		0			0
$13$	1479.02	−	1479.02i	0.186712	−	0.186712i	−0.607561	−	0.794273i	$-0.707852\pi$
$13$	1479.02	−	1479.02i	0.186712	−	0.186712i	0.794273	+	0.607561i	$0.207852\pi$
$14$		0			0
$15$	16138.3	+	6906.24i	1.23464	+	0.528350i
$16$		0			0
$17$	907.816	+	907.816i	0.0448153	+	0.0448153i	0.729159	−	0.684344i	$-0.239911\pi$
$17$	907.816	+	907.816i	0.0448153	+	0.0448153i	−0.684344	+	0.729159i	$0.739911\pi$
$18$		0			0
$19$	13306.0			0.445050			0.222525	−	0.974927i	$-0.428570\pi$
$19$	13306.0			0.445050			0.222525	+	0.974927i	$0.428570\pi$
$20$		0			0
$21$	23231.6			0.547408
$22$		0			0
$23$	31405.7	+	31405.7i	0.538221	+	0.538221i	0.923006	−	0.384785i	$-0.125724\pi$
$23$	31405.7	+	31405.7i	0.538221	+	0.538221i	−0.384785	+	0.923006i	$0.625724\pi$
$24$		0			0
$25$	56515.8	−	53939.6i	0.723402	−	0.690427i
$26$		0			0
$27$	−19086.2	+	19086.2i	−0.186615	+	0.186615i
$28$		0			0
$29$		−	31344.6i		−	0.238655i	−0.992855	−	0.119327i	$-0.961926\pi$
$29$		−	31344.6i		−	0.238655i	0.992855	−	0.119327i	$-0.0380738\pi$
$30$		0			0
$31$			216018.i			1.30234i	0.758932	+	0.651170i	$0.225721\pi$
$31$			216018.i			1.30234i	−0.758932	+	0.651170i	$0.774279\pi$
$32$		0			0
$33$	215101.	−	215101.i	1.04194	−	1.04194i
$34$		0			0
$35$	40678.0	−	95055.4i	0.160369	−	0.374747i
$36$		0			0
$37$	−231635.	−	231635.i	−0.751792	−	0.751792i	0.223022	−	0.974813i	$-0.428408\pi$
$37$	−231635.	−	231635.i	−0.751792	−	0.751792i	−0.974813	+	0.223022i	$0.928408\pi$
$38$		0			0
$39$	−131362.			−0.354603
$40$		0			0
$41$	565257.			1.28086			0.640431	−	0.768016i	$-0.278756\pi$
$41$	565257.			1.28086			0.640431	+	0.768016i	$0.278756\pi$
$42$		0			0
$43$	−393861.	−	393861.i	−0.755446	−	0.755446i	0.220044	−	0.975490i	$-0.429380\pi$
$43$	−393861.	−	393861.i	−0.755446	−	0.755446i	−0.975490	+	0.220044i	$0.929380\pi$
$44$		0			0
$45$	−182654.	−	455929.i	−0.298803	−	0.745854i
$46$		0			0
$47$	−315328.	+	315328.i	−0.443016	+	0.443016i	−0.893024	−	0.450008i	$-0.851421\pi$
$47$	−315328.	+	315328.i	−0.443016	+	0.443016i	0.450008	+	0.893024i	$0.351421\pi$
$48$		0			0
$49$			686708.i			0.833846i
$50$		0			0
$51$		−	80629.3i		−	0.0851133i
$52$		0			0
$53$	−292177.	+	292177.i	−0.269576	+	0.269576i	−0.828929	−	0.559354i	$-0.811049\pi$
$53$	−292177.	+	292177.i	−0.269576	+	0.269576i	0.559354	+	0.828929i	$0.311049\pi$
$54$		0			0
$55$	−503481.	−	1.25676e6i	−0.408050	−	1.01855i
$56$		0			0
$57$	−590897.	−	590897.i	−0.422619	−	0.422619i
$58$		0			0
$59$	−1.72159e6			−1.09131			−0.545655	−	0.838010i	$-0.683719\pi$
$59$	−1.72159e6			−1.09131			−0.545655	+	0.838010i	$0.683719\pi$
$60$		0			0
$61$	−3.08385e6			−1.73955			−0.869777	−	0.493444i	$-0.835738\pi$
$61$	−3.08385e6			−1.73955			−0.869777	+	0.493444i	$0.835738\pi$
$62$		0			0
$63$	−459629.	−	459629.i	−0.231588	−	0.231588i
$64$		0			0
$65$	−230011.	+	537485.i	−0.103885	+	0.242756i
$66$		0			0
$67$	−45437.5	+	45437.5i	−0.0184566	+	0.0184566i	−0.716275	−	0.697818i	$-0.754154\pi$
$67$	−45437.5	+	45437.5i	−0.0184566	+	0.0184566i	0.697818	+	0.716275i	$0.254154\pi$
$68$		0			0
$69$		−	2.78935e6i		−	1.02219i
$70$		0			0
$71$		−	1.19240e6i		−	0.395383i	−0.980264	−	0.197692i	$-0.936656\pi$
$71$		−	1.19240e6i		−	0.395383i	0.980264	−	0.197692i	$-0.0633445\pi$
$72$		0			0
$73$	4.18729e6	−	4.18729e6i	1.25981	−	1.25981i	0.308620	−	0.951185i	$-0.400133\pi$
$73$	4.18729e6	−	4.18729e6i	1.25981	−	1.25981i	0.951185	−	0.308620i	$-0.0998671\pi$
$74$		0			0
$75$	−4.90515e6	−	114401.i	−1.34257	−	0.0313123i
$76$		0			0
$77$	−1.26696e6	−	1.26696e6i	−0.316260	−	0.316260i
$78$		0			0
$79$	−5.76074e6			−1.31457			−0.657285	−	0.753642i	$-0.728295\pi$
$79$	−5.76074e6			−1.31457			−0.657285	+	0.753642i	$0.728295\pi$
$80$		0			0
$81$	5.53820e6			1.15790
$82$		0			0
$83$	−1.79184e6	−	1.79184e6i	−0.343974	−	0.343974i	0.513885	−	0.857859i	$-0.328206\pi$
$83$	−1.79184e6	−	1.79184e6i	−0.343974	−	0.343974i	−0.857859	+	0.513885i	$0.828206\pi$
$84$		0			0
$85$	−329907.	−	141180.i	−0.0582673	−	0.0249349i
$86$		0			0
$87$	−1.39196e6	+	1.39196e6i	−0.226626	+	0.226626i
$88$		0			0
$89$		−	7.42470e6i		−	1.11638i	−0.829712	−	0.558192i	$-0.811495\pi$
$89$		−	7.42470e6i		−	1.11638i	0.829712	−	0.558192i	$-0.188505\pi$
$90$		0			0
$91$			773724.i			0.107632i
$92$		0			0
$93$	9.59302e6	−	9.59302e6i	1.23670	−	1.23670i
$94$		0			0
$95$	−3.45239e6	+	1.38309e6i	−0.413130	+	0.165508i
$96$		0			0
$97$	−1.02064e7	−	1.02064e7i	−1.13546	−	1.13546i	−0.989255	−	0.146202i	$-0.953295\pi$
$97$	−1.02064e7	−	1.02064e7i	−1.13546	−	1.13546i	−0.146202	−	0.989255i	$-0.546705\pi$
$98$		0			0
$99$	−8.51143e6			−0.881615
$100$		0			0
$101$	1.44853e7			1.39895			0.699475	−	0.714657i	$-0.253417\pi$
$101$	1.44853e7			1.39895			0.699475	+	0.714657i	$0.253417\pi$
$102$		0			0
$103$	1.42361e7	+	1.42361e7i	1.28370	+	1.28370i	0.938549	+	0.345147i	$0.112171\pi$
$103$	1.42361e7	+	1.42361e7i	1.28370	+	1.28370i	0.345147	+	0.938549i	$0.387829\pi$
$104$		0			0
$105$	−6.02770e6	+	2.41481e6i	−0.508147	+	0.203573i
$106$		0			0
$107$	9.64958e6	−	9.64958e6i	0.761491	−	0.761491i	−0.215100	−	0.976592i	$-0.569008\pi$
$107$	9.64958e6	−	9.64958e6i	0.761491	−	0.761491i	0.976592	+	0.215100i	$0.0690079\pi$
$108$		0			0
$109$		−	1.17722e7i		−	0.870695i	−0.900262	−	0.435348i	$-0.856625\pi$
$109$		−	1.17722e7i		−	0.870695i	0.900262	−	0.435348i	$-0.143375\pi$
$110$		0			0
$111$			2.05730e7i			1.42780i
$112$		0			0
$113$	−870440.	+	870440.i	−0.0567498	+	0.0567498i	−0.734912	−	0.678162i	$-0.762777\pi$
$113$	−870440.	+	870440.i	−0.0567498	+	0.0567498i	0.678162	+	0.734912i	$0.262777\pi$
$114$		0			0
$115$	−1.14131e7	−	4.88410e6i	−0.699776	−	0.299462i
$116$		0			0
$117$	2.59895e6	+	2.59895e6i	0.150019	+	0.150019i
$118$		0			0
$119$	−474910.			−0.0258343
$120$		0			0
$121$	−3.97436e6			−0.203947
$122$		0			0
$123$	−2.51021e7	−	2.51021e7i	−1.21631	−	1.21631i
$124$		0			0
$125$	−9.05690e6	+	1.98698e7i	−0.414758	+	0.909932i
$126$		0			0
$127$	1.13675e7	−	1.13675e7i	0.492439	−	0.492439i	−0.416635	−	0.909074i	$-0.636791\pi$
$127$	1.13675e7	−	1.13675e7i	0.492439	−	0.492439i	0.909074	+	0.416635i	$0.136791\pi$
$128$		0			0
$129$			3.49815e7i			1.43474i
$130$		0			0
$131$		−	5.02306e6i		−	0.195218i	−0.995225	−	0.0976088i	$-0.968881\pi$
$131$		−	5.02306e6i		−	0.195218i	0.995225	−	0.0976088i	$-0.0311194\pi$
$132$		0			0
$133$	−3.48040e6	+	3.48040e6i	−0.128277	+	0.128277i
$134$		0			0
$135$	2.96822e6	−	6.93606e6i	0.103831	−	0.242630i
$136$		0			0
$137$	−3.61428e7	−	3.61428e7i	−1.20088	−	1.20088i	−0.973899	−	0.226981i	$-0.927115\pi$
$137$	−3.61428e7	−	3.61428e7i	−1.20088	−	1.20088i	−0.226981	−	0.973899i	$-0.572885\pi$
$138$		0			0
$139$	−3.63256e7			−1.14726			−0.573629	−	0.819115i	$-0.694465\pi$
$139$	−3.63256e7			−1.14726			−0.573629	+	0.819115i	$0.694465\pi$
$140$		0			0
$141$	2.80064e7			0.841377
$142$		0			0
$143$	7.16393e6	+	7.16393e6i	0.204868	+	0.204868i
$144$		0			0
$145$	3.25812e6	+	8.13272e6i	0.0887523	+	0.221538i
$146$		0			0
$147$	3.04956e7	−	3.04956e7i	0.791820	−	0.791820i
$148$		0			0
$149$			3.03216e7i			0.750931i	0.926836	+	0.375466i	$0.122517\pi$
$149$			3.03216e7i			0.750931i	−0.926836	+	0.375466i	$0.877483\pi$
$150$		0			0
$151$			6.16736e7i			1.45774i	0.684652	+	0.728870i	$0.259954\pi$
$151$			6.16736e7i			1.45774i	−0.684652	+	0.728870i	$0.740046\pi$
$152$		0			0
$153$	−1.59523e6	+	1.59523e6i	−0.0360083	+	0.0360083i
$154$		0			0
$155$	−2.24541e7	−	5.60484e7i	−0.484322	−	1.20893i
$156$		0			0
$157$	−2.91931e7	−	2.91931e7i	−0.602049	−	0.602049i	0.338807	−	0.940856i	$-0.389977\pi$
$157$	−2.91931e7	−	2.91931e7i	−0.602049	−	0.602049i	−0.940856	+	0.338807i	$0.889977\pi$
$158$		0			0
$159$	2.59502e7			0.511978
$160$		0			0
$161$	−1.64294e7			−0.310264
$162$		0			0
$163$	2.72546e7	+	2.72546e7i	0.492928	+	0.492928i	0.909227	−	0.416300i	$-0.136673\pi$
$163$	2.72546e7	+	2.72546e7i	0.492928	+	0.492928i	−0.416300	+	0.909227i	$0.636673\pi$
$164$		0			0
$165$	−3.34518e7	+	7.81694e7i	−0.579730	+	1.35470i
$166$		0			0
$167$	5.82194e7	−	5.82194e7i	0.967298	−	0.967298i	−0.0321843	−	0.999482i	$-0.510246\pi$
$167$	5.82194e7	−	5.82194e7i	0.967298	−	0.967298i	0.999482	+	0.0321843i	$0.0102463\pi$
$168$		0			0
$169$			5.83735e7i			0.930278i
$170$		0			0
$171$			2.33814e7i			0.357589i
$172$		0			0
$173$	1.51226e7	−	1.51226e7i	0.222057	−	0.222057i	−0.587307	−	0.809364i	$-0.699812\pi$
$173$	1.51226e7	−	1.51226e7i	0.222057	−	0.222057i	0.809364	+	0.587307i	$0.199812\pi$
$174$		0			0
$175$	−673825.	+	2.88915e7i	−0.00950417	+	0.407509i
$176$		0			0
$177$	7.64532e7	+	7.64532e7i	1.03631	+	1.03631i
$178$		0			0
$179$	8.05122e7			1.04924			0.524621	−	0.851336i	$-0.324207\pi$
$179$	8.05122e7			1.04924			0.524621	+	0.851336i	$0.324207\pi$
$180$		0			0
$181$	−3.14755e7			−0.394546			−0.197273	−	0.980349i	$-0.563209\pi$
$181$	−3.14755e7			−0.394546			−0.197273	+	0.980349i	$0.563209\pi$
$182$		0			0
$183$	1.36949e8	+	1.36949e8i	1.65188	+	1.65188i
$184$		0			0
$185$	8.41776e7	+	3.60230e7i	0.977453	+	0.418291i
$186$		0			0
$187$	−4.39720e6	+	4.39720e6i	−0.0491734	+	0.0491734i
$188$		0			0
$189$		−	9.98465e6i		−	0.107576i
$190$		0			0
$191$			8.36266e7i			0.868416i	0.900813	+	0.434208i	$0.142972\pi$
$191$			8.36266e7i			0.868416i	−0.900813	+	0.434208i	$0.857028\pi$
$192$		0			0
$193$	7.01579e7	−	7.01579e7i	0.702467	−	0.702467i	−0.262472	−	0.964939i	$-0.584538\pi$
$193$	7.01579e7	−	7.01579e7i	0.702467	−	0.702467i	0.964939	+	0.262472i	$0.0845379\pi$
$194$		0			0
$195$	3.40833e7	−	1.36544e7i	0.329170	−	0.131872i
$196$		0			0
$197$	1.92214e7	+	1.92214e7i	0.179124	+	0.179124i	0.790974	−	0.611850i	$-0.209574\pi$
$197$	1.92214e7	+	1.92214e7i	0.179124	+	0.179124i	−0.611850	+	0.790974i	$0.709574\pi$
$198$		0			0
$199$	1.93689e8			1.74229			0.871143	−	0.491029i	$-0.163379\pi$
$199$	1.93689e8			1.74229			0.871143	+	0.491029i	$0.163379\pi$
$200$		0			0
$201$	4.03561e6			0.0350528
$202$		0			0
$203$	8.19872e6	+	8.19872e6i	0.0687876	+	0.0687876i
$204$		0			0
$205$	−1.46662e8	+	5.87558e7i	−1.18900	+	0.476334i
$206$		0			0
$207$	−5.51865e7	+	5.51865e7i	−0.432451	+	0.432451i
$208$		0			0
$209$			6.44502e7i			0.488329i
$210$		0			0
$211$		−	8.05752e7i		−	0.590490i	−0.955422	−	0.295245i	$-0.904599\pi$
$211$		−	8.05752e7i		−	0.590490i	0.955422	−	0.295245i	$-0.0954013\pi$
$212$		0			0
$213$	−5.29526e7	+	5.29526e7i	−0.375456	+	0.375456i
$214$		0			0
$215$	1.43132e8	+	6.12518e7i	0.982204	+	0.420324i
$216$		0			0
$217$	−5.65032e7	−	5.65032e7i	−0.375374	−	0.375374i
$218$		0			0
$219$	−3.71902e8			−2.39262
$220$		0			0
$221$	2.68535e6			0.0167351
$222$		0			0
$223$	−1.31377e8	−	1.31377e8i	−0.793326	−	0.793326i	0.188708	−	0.982033i	$-0.439570\pi$
$223$	−1.31377e8	−	1.31377e8i	−0.793326	−	0.793326i	−0.982033	+	0.188708i	$0.939570\pi$
$224$		0			0
$225$	9.47834e7	+	9.93102e7i	0.554745	+	0.581239i
$226$		0			0
$227$	1.53191e8	−	1.53191e8i	0.869247	−	0.869247i	−0.123142	−	0.992389i	$-0.539297\pi$
$227$	1.53191e8	−	1.53191e8i	0.869247	−	0.869247i	0.992389	+	0.123142i	$0.0392971\pi$
$228$		0			0
$229$		−	2.99759e8i		−	1.64949i	−0.565508	−	0.824743i	$-0.691320\pi$
$229$		−	2.99759e8i		−	1.64949i	0.565508	−	0.824743i	$-0.308680\pi$
$230$		0			0
$231$			1.12527e8i			0.600641i
$232$		0			0
$233$	7.06102e7	−	7.06102e7i	0.365697	−	0.365697i	−0.500208	−	0.865905i	$-0.666743\pi$
$233$	7.06102e7	−	7.06102e7i	0.365697	−	0.365697i	0.865905	+	0.500208i	$0.166743\pi$
$234$		0			0
$235$	4.90387e7	−	1.14592e8i	0.246491	−	0.575994i
$236$		0			0
$237$	2.55825e8	+	2.55825e8i	1.24832	+	1.24832i
$238$		0			0
$239$	−2.24938e8			−1.06579			−0.532894	−	0.846182i	$-0.678896\pi$
$239$	−2.24938e8			−1.06579			−0.532894	+	0.846182i	$0.678896\pi$
$240$		0			0
$241$	−3.36645e8			−1.54922			−0.774610	−	0.632440i	$-0.782054\pi$
$241$	−3.36645e8			−1.54922			−0.774610	+	0.632440i	$0.782054\pi$
$242$		0			0
$243$	−2.04201e8	−	2.04201e8i	−0.912927	−	0.912927i
$244$		0			0
$245$	−7.13801e7	−	1.78174e8i	−0.310096	−	0.774041i
$246$		0			0
$247$	1.96797e7	−	1.96797e7i	0.0830960	−	0.0830960i
$248$		0			0
$249$			1.59146e8i			0.653276i
$250$		0			0
$251$		−	1.64376e8i		−	0.656115i	−0.944658	−	0.328057i	$-0.893606\pi$
$251$		−	1.64376e8i		−	0.656115i	0.944658	−	0.328057i	$-0.106394\pi$
$252$		0			0
$253$	−1.52120e8	+	1.52120e8i	−0.590561	+	0.590561i
$254$		0			0
$255$	8.38104e6	+	2.09202e7i	0.0316524	+	0.0790089i
$256$		0			0
$257$	−2.11466e8	−	2.11466e8i	−0.777095	−	0.777095i	0.202241	−	0.979336i	$-0.435178\pi$
$257$	−2.11466e8	−	2.11466e8i	−0.777095	−	0.777095i	−0.979336	+	0.202241i	$0.935178\pi$
$258$		0			0
$259$	1.21176e8			0.433379
$260$		0			0
$261$	5.50791e7			0.191754
$262$		0			0
$263$	−1.40979e8	−	1.40979e8i	−0.477870	−	0.477870i	0.426580	−	0.904450i	$-0.359718\pi$
$263$	−1.40979e8	−	1.40979e8i	−0.477870	−	0.477870i	−0.904450	+	0.426580i	$0.859718\pi$
$264$		0			0
$265$	4.54383e7	−	1.06179e8i	0.149990	−	0.350493i
$266$		0			0
$267$	−3.29719e8	+	3.29719e8i	−1.06012	+	1.06012i
$268$		0			0
$269$		−	3.21309e8i		−	1.00644i	−0.864157	−	0.503222i	$-0.832148\pi$
$269$		−	3.21309e8i		−	1.00644i	0.864157	−	0.503222i	$-0.167852\pi$
$270$		0			0
$271$		−	8.11990e7i		−	0.247832i	−0.992293	−	0.123916i	$-0.960455\pi$
$271$		−	8.11990e7i		−	0.247832i	0.992293	−	0.123916i	$-0.0395454\pi$
$272$		0			0
$273$	3.43599e7	−	3.43599e7i	0.102207	−	0.102207i
$274$		0			0
$275$	2.61268e8	+	2.73746e8i	0.757568	+	0.793749i
$276$		0			0
$277$	3.17066e8	+	3.17066e8i	0.896334	+	0.896334i	0.995110	−	0.0987754i	$-0.0314926\pi$
$277$	3.17066e8	+	3.17066e8i	0.896334	+	0.896334i	−0.0987754	+	0.995110i	$0.531493\pi$
$278$		0			0
$279$	−3.79589e8			−1.04640
$280$		0			0
$281$	−5.04472e8			−1.35633			−0.678164	−	0.734910i	$-0.737224\pi$
$281$	−5.04472e8			−1.35633			−0.678164	+	0.734910i	$0.737224\pi$
$282$		0			0
$283$	−2.19661e8	−	2.19661e8i	−0.576104	−	0.576104i	0.357724	−	0.933828i	$-0.383553\pi$
$283$	−2.19661e8	−	2.19661e8i	−0.576104	−	0.576104i	−0.933828	+	0.357724i	$0.883553\pi$
$284$		0			0
$285$	2.14736e8	+	9.18941e7i	0.549475	+	0.235142i
$286$		0			0
$287$	−1.47853e8	+	1.47853e8i	−0.369183	+	0.369183i
$288$		0			0
$289$		−	4.08690e8i		−	0.995983i
$290$		0			0
$291$			9.06498e8i			2.15646i
$292$		0			0
$293$	1.84660e8	−	1.84660e8i	0.428881	−	0.428881i	−0.459366	−	0.888247i	$-0.651923\pi$
$293$	1.84660e8	−	1.84660e8i	0.428881	−	0.428881i	0.888247	+	0.459366i	$0.151923\pi$
$294$		0			0
$295$	4.46688e8	−	1.78952e8i	1.01304	−	0.405843i
$296$		0			0
$297$	−9.24480e7	−	9.24480e7i	−0.204762	−	0.204762i
$298$		0			0
$299$	9.28991e7			0.200984
$300$		0			0
$301$	2.06042e8			0.435485
$302$		0			0
$303$	−6.43268e8	−	6.43268e8i	−1.32844	−	1.32844i
$304$		0			0
$305$	8.00140e8	−	3.20551e8i	1.61479	−	0.646916i
$306$		0			0
$307$	−7.33835e7	+	7.33835e7i	−0.144749	+	0.144749i	−0.775767	−	0.631019i	$-0.782637\pi$
$307$	−7.33835e7	+	7.33835e7i	−0.144749	+	0.144749i	0.631019	+	0.775767i	$0.282637\pi$
$308$		0			0
$309$		−	1.26441e9i		−	2.43800i
$310$		0			0
$311$		−	5.13493e8i		−	0.967995i	−0.875069	−	0.483997i	$-0.839184\pi$
$311$		−	5.13493e8i		−	0.967995i	0.875069	−	0.483997i	$-0.160816\pi$
$312$		0			0
$313$	5.27608e8	−	5.27608e8i	0.972537	−	0.972537i	−0.0270954	−	0.999633i	$-0.508626\pi$
$313$	5.27608e8	−	5.27608e8i	0.972537	−	0.972537i	0.999633	+	0.0270954i	$0.00862580\pi$
$314$		0			0
$315$	1.67032e8	+	7.14799e7i	0.301102	+	0.128854i
$316$		0			0
$317$	2.79666e8	+	2.79666e8i	0.493097	+	0.493097i	0.909281	−	0.416184i	$-0.136633\pi$
$317$	2.79666e8	+	2.79666e8i	0.493097	+	0.493097i	−0.416184	+	0.909281i	$0.636633\pi$
$318$		0			0
$319$	1.51824e8			0.261863
$320$		0			0
$321$	−8.57045e8			−1.44622
$322$		0			0
$323$	1.20794e7	+	1.20794e7i	0.0199451	+	0.0199451i
$324$		0			0
$325$	3.81011e6	−	1.63365e8i	0.00615666	−	0.263978i
$326$		0			0
$327$	−5.22786e8	+	5.22786e8i	−0.826812	+	0.826812i
$328$		0			0
$329$		−	1.64959e8i		−	0.255382i
$330$		0			0
$331$			5.02158e8i			0.761101i	0.924760	+	0.380550i	$0.124265\pi$
$331$			5.02158e8i			0.761101i	−0.924760	+	0.380550i	$0.875735\pi$
$332$		0			0
$333$	4.07031e8	−	4.07031e8i	0.604050	−	0.604050i
$334$		0			0
$335$	7.06627e6	−	1.65123e7i	0.0102691	−	0.0239966i
$336$		0			0
$337$	5.37281e8	+	5.37281e8i	0.764711	+	0.764711i	0.977170	−	0.212459i	$-0.0681473\pi$
$337$	5.37281e8	+	5.37281e8i	0.764711	+	0.764711i	−0.212459	+	0.977170i	$0.568147\pi$
$338$		0			0
$339$	7.73097e7			0.107779
$340$		0			0
$341$	−1.04633e9			−1.42899
$342$		0			0
$343$	−3.95032e8	−	3.95032e8i	−0.528571	−	0.528571i
$344$		0			0
$345$	2.89940e8	+	7.23730e8i	0.380138	+	0.948877i
$346$		0			0
$347$	3.25029e8	−	3.25029e8i	0.417609	−	0.417609i	−0.466770	−	0.884379i	$-0.654582\pi$
$347$	3.25029e8	−	3.25029e8i	0.417609	−	0.417609i	0.884379	+	0.466770i	$0.154582\pi$
$348$		0			0
$349$		−	1.22607e8i		−	0.154392i	−0.997016	−	0.0771962i	$-0.975403\pi$
$349$		−	1.22607e8i		−	0.154392i	0.997016	−	0.0771962i	$-0.0245968\pi$
$350$		0			0
$351$			5.64576e7i			0.0696863i
$352$		0			0
$353$	1.43557e8	−	1.43557e8i	0.173705	−	0.173705i	−0.614900	−	0.788605i	$-0.710804\pi$
$353$	1.43557e8	−	1.43557e8i	0.173705	−	0.173705i	0.788605	+	0.614900i	$0.210804\pi$
$354$		0			0
$355$	1.23944e8	+	3.09382e8i	0.147037	+	0.367026i
$356$		0			0
$357$	2.10900e7	+	2.10900e7i	0.0245323	+	0.0245323i
$358$		0			0
$359$	−1.16713e9			−1.33134			−0.665672	−	0.746245i	$-0.731855\pi$
$359$	−1.16713e9			−1.33134			−0.665672	+	0.746245i	$0.731855\pi$
$360$		0			0
$361$	−7.16823e8			−0.801931
$362$		0			0
$363$	1.76495e8	+	1.76495e8i	0.193668	+	0.193668i
$364$		0			0
$365$	−6.51193e8	+	1.52169e9i	−0.700946	+	1.63795i
$366$		0			0
$367$	1.34907e8	−	1.34907e8i	0.142464	−	0.142464i	−0.632278	−	0.774742i	$-0.717880\pi$
$367$	1.34907e8	−	1.34907e8i	0.142464	−	0.142464i	0.774742	+	0.632278i	$0.217880\pi$
$368$		0			0
$369$			9.93276e8i			1.02915i
$370$		0			0
$371$		−	1.52848e8i		−	0.155400i
$372$		0			0
$373$	−9.25897e8	+	9.25897e8i	−0.923808	+	0.923808i	−0.997296	−	0.0734878i	$-0.976587\pi$
$373$	−9.25897e8	+	9.25897e8i	−0.923808	+	0.923808i	0.0734878	+	0.997296i	$0.476587\pi$
$374$		0			0
$375$	1.28459e9	−	4.80184e8i	1.25793	−	0.470217i
$376$		0			0
$377$	−4.63592e7	−	4.63592e7i	−0.0445596	−	0.0445596i
$378$		0			0
$379$	3.19599e8			0.301556			0.150778	−	0.988568i	$-0.451822\pi$
$379$	3.19599e8			0.301556			0.150778	+	0.988568i	$0.451822\pi$
$380$		0			0
$381$	−1.00963e9			−0.935240
$382$		0			0
$383$	−5.78279e8	−	5.78279e8i	−0.525946	−	0.525946i	0.393415	−	0.919361i	$-0.371294\pi$
$383$	−5.78279e8	−	5.78279e8i	−0.525946	−	0.525946i	−0.919361	+	0.393415i	$0.871294\pi$
$384$		0			0
$385$	4.60421e8	+	1.97032e8i	0.411190	+	0.175965i
$386$		0			0
$387$	6.92097e8	−	6.92097e8i	0.606986	−	0.606986i
$388$		0			0
$389$			3.77486e8i			0.325145i	0.986697	+	0.162573i	$0.0519792\pi$
$389$			3.77486e8i			0.325145i	−0.986697	+	0.162573i	$0.948021\pi$
$390$		0			0
$391$			5.70212e7i			0.0482412i
$392$		0			0
$393$	−2.23066e8	+	2.23066e8i	−0.185379	+	0.185379i
$394$		0			0
$395$	1.49469e9	−	5.98802e8i	1.22029	−	0.488870i
$396$		0			0
$397$	3.07795e8	+	3.07795e8i	0.246885	+	0.246885i	0.819691	−	0.572806i	$-0.194145\pi$
$397$	3.07795e8	+	3.07795e8i	0.246885	+	0.246885i	−0.572806	+	0.819691i	$0.694145\pi$
$398$		0			0
$399$	3.09118e8			0.243624
$400$		0			0
$401$	7.45129e8			0.577067			0.288533	−	0.957470i	$-0.406832\pi$
$401$	7.45129e8			0.577067			0.288533	+	0.957470i	$0.406832\pi$
$402$		0			0
$403$	3.19494e8	+	3.19494e8i	0.243162	+	0.243162i
$404$		0			0
$405$	−1.43695e9	+	5.75670e8i	−1.07485	+	0.430607i
$406$		0			0
$407$	1.12197e9	−	1.12197e9i	0.824900	−	0.824900i
$408$		0			0
$409$		−	4.05732e8i		−	0.293230i	−0.989194	−	0.146615i	$-0.953162\pi$
$409$		−	4.05732e8i		−	0.293230i	0.989194	−	0.146615i	$-0.0468378\pi$
$410$		0			0
$411$			3.21009e9i			2.28071i
$412$		0			0
$413$	4.50312e8	−	4.50312e8i	0.314549	−	0.314549i
$414$		0			0
$415$	6.51167e8	+	2.78661e8i	0.447223	+	0.191385i
$416$		0			0
$417$	1.61316e9	+	1.61316e9i	1.08944	+	1.08944i
$418$		0			0
$419$	−1.51364e9			−1.00525			−0.502623	−	0.864506i	$-0.667632\pi$
$419$	−1.51364e9			−1.00525			−0.502623	+	0.864506i	$0.667632\pi$
$420$		0			0
$421$	−4.79908e8			−0.313452			−0.156726	−	0.987642i	$-0.550094\pi$
$421$	−4.79908e8			−0.313452			−0.156726	+	0.987642i	$0.550094\pi$
$422$		0			0
$423$	−5.54098e8	−	5.54098e8i	−0.355955	−	0.355955i
$424$		0			0
$425$	1.00273e8	+	2.33863e6i	0.0633612	+	0.00147775i
$426$		0			0
$427$	8.06633e8	−	8.06633e8i	0.501393	−	0.501393i
$428$		0			0
$429$		−	6.36277e8i		−	0.389086i
$430$		0			0
$431$			8.83724e8i			0.531675i	0.964018	+	0.265837i	$0.0856484\pi$
$431$			8.83724e8i			0.531675i	−0.964018	+	0.265837i	$0.914352\pi$
$432$		0			0
$433$	−3.12796e7	+	3.12796e7i	−0.0185163	+	0.0185163i	−0.716304	−	0.697788i	$-0.754168\pi$
$433$	−3.12796e7	+	3.12796e7i	−0.0185163	+	0.0185163i	0.697788	+	0.716304i	$0.254168\pi$
$434$		0			0
$435$	2.16473e8	−	5.05849e8i	0.126093	−	0.294652i
$436$		0			0
$437$	4.17883e8	+	4.17883e8i	0.239535	+	0.239535i
$438$		0			0
$439$	−1.14996e9			−0.648720			−0.324360	−	0.945934i	$-0.605149\pi$
$439$	−1.14996e9			−0.648720			−0.324360	+	0.945934i	$0.605149\pi$
$440$		0			0
$441$	−1.20669e9			−0.669979
$442$		0			0
$443$	1.77500e9	+	1.77500e9i	0.970028	+	0.970028i	0.999564	−	0.0295356i	$-0.00940284\pi$
$443$	1.77500e9	+	1.77500e9i	0.970028	+	0.970028i	−0.0295356	+	0.999564i	$0.509403\pi$
$444$		0			0
$445$	7.71763e8	+	1.92643e9i	0.415168	+	1.03632i
$446$		0			0
$447$	1.34653e9	−	1.34653e9i	0.713085	−	0.713085i
$448$		0			0
$449$			7.01757e8i			0.365868i	0.983125	+	0.182934i	$0.0585595\pi$
$449$			7.01757e8i			0.365868i	−0.983125	+	0.182934i	$0.941441\pi$
$450$		0			0
$451$			2.73794e9i			1.40542i
$452$		0			0
$453$	2.73883e9	−	2.73883e9i	1.38427	−	1.38427i
$454$		0			0
$455$	−8.04250e7	−	2.00752e8i	−0.0400268	−	0.0999125i
$456$		0			0
$457$	−2.00280e9	−	2.00280e9i	−0.981592	−	0.981592i	0.0182412	−	0.999834i	$-0.494193\pi$
$457$	−2.00280e9	−	2.00280e9i	−0.981592	−	0.981592i	−0.999834	+	0.0182412i	$0.994193\pi$
$458$		0			0
$459$	−3.46535e7			−0.0167264
$460$		0			0
$461$	2.77556e9			1.31946			0.659731	−	0.751502i	$-0.270670\pi$
$461$	2.77556e9			1.31946			0.659731	+	0.751502i	$0.270670\pi$
$462$		0			0
$463$	5.66758e8	+	5.66758e8i	0.265377	+	0.265377i	0.827234	−	0.561857i	$-0.189913\pi$
$463$	5.66758e8	+	5.66758e8i	0.265377	+	0.265377i	−0.561857	+	0.827234i	$0.689913\pi$
$464$		0			0
$465$	−1.49187e9	+	3.48617e9i	−0.688091	+	1.60792i
$466$		0			0
$467$	1.67690e9	−	1.67690e9i	0.761902	−	0.761902i	−0.214764	−	0.976666i	$-0.568898\pi$
$467$	1.67690e9	−	1.67690e9i	0.761902	−	0.761902i	0.976666	+	0.214764i	$0.0688983\pi$
$468$		0			0
$469$		−	2.37699e7i		−	0.0106395i
$470$		0			0
$471$			2.59284e9i			1.14341i
$472$		0			0
$473$	1.90775e9	−	1.90775e9i	0.828909	−	0.828909i
$474$		0			0
$475$	7.51997e8	−	7.17719e8i	0.321950	−	0.307275i
$476$		0			0
$477$	−5.13417e8	−	5.13417e8i	−0.216599	−	0.216599i
$478$		0			0
$479$	2.74743e9			1.14223			0.571114	−	0.820871i	$-0.306511\pi$
$479$	2.74743e9			1.14223			0.571114	+	0.820871i	$0.306511\pi$
$480$		0			0
$481$	−6.85183e8			−0.280736
$482$		0			0
$483$	7.29604e8	+	7.29604e8i	0.294627	+	0.294627i
$484$		0			0
$485$	3.70907e9	+	1.58726e9i	1.47628	+	0.631760i
$486$		0			0
$487$	−1.32008e9	+	1.32008e9i	−0.517902	+	0.517902i	−0.916936	−	0.399034i	$-0.869346\pi$
$487$	−1.32008e9	+	1.32008e9i	−0.517902	+	0.517902i	0.399034	+	0.916936i	$0.369346\pi$
$488$		0			0
$489$		−	2.42067e9i		−	0.936168i
$490$		0			0
$491$			7.59558e8i			0.289585i	0.989462	+	0.144792i	$0.0462514\pi$
$491$			7.59558e8i			0.289585i	−0.989462	+	0.144792i	$0.953749\pi$
$492$		0			0
$493$	2.84551e7	−	2.84551e7i	0.0106954	−	0.0106954i
$494$		0			0
$495$	2.20839e9	−	8.84723e8i	0.818385	−	0.327860i
$496$		0			0
$497$	3.11893e8	+	3.11893e8i	0.113962	+	0.113962i
$498$		0			0
$499$	−3.16728e9			−1.14113			−0.570565	−	0.821253i	$-0.693276\pi$
$499$	−3.16728e9			−1.14113			−0.570565	+	0.821253i	$0.693276\pi$
$500$		0			0
$501$	−5.17086e9			−1.83709
$502$		0			0
$503$	−2.45155e9	−	2.45155e9i	−0.858920	−	0.858920i	0.132291	−	0.991211i	$-0.457767\pi$
$503$	−2.45155e9	−	2.45155e9i	−0.858920	−	0.858920i	−0.991211	+	0.132291i	$0.957767\pi$
$504$		0			0
$505$	−3.75838e9	+	1.50568e9i	−1.29862	+	0.520250i
$506$		0			0
$507$	2.59228e9	−	2.59228e9i	0.883392	−	0.883392i
$508$		0			0
$509$			4.49129e9i			1.50959i	0.655961	+	0.754794i	$0.272263\pi$
$509$			4.49129e9i			1.50959i	−0.655961	+	0.754794i	$0.727737\pi$
$510$		0			0
$511$			2.19052e9i			0.726229i
$512$		0			0
$513$	−2.53960e8	+	2.53960e8i	−0.0830529	+	0.0830529i
$514$		0			0
$515$	−5.17351e9	−	2.21395e9i	−1.66902	−	0.714239i
$516$		0			0
$517$	−1.52736e9	−	1.52736e9i	−0.486098	−	0.486098i
$518$		0			0
$519$	−1.34314e9			−0.421731
$520$		0			0
$521$	6.44243e8			0.199580			0.0997902	−	0.995009i	$-0.468183\pi$
$521$	6.44243e8			0.199580			0.0997902	+	0.995009i	$0.468183\pi$
$522$		0			0
$523$	−2.95865e9	−	2.95865e9i	−0.904353	−	0.904353i	0.0914562	−	0.995809i	$-0.470848\pi$
$523$	−2.95865e9	−	2.95865e9i	−0.904353	−	0.904353i	−0.995809	+	0.0914562i	$0.970848\pi$
$524$		0			0
$525$	1.31295e9	−	1.25310e9i	0.395996	−	0.377945i
$526$		0			0
$527$	−1.96105e8	+	1.96105e8i	−0.0583648	+	0.0583648i
$528$		0			0
$529$		−	1.43219e9i		−	0.420635i
$530$		0			0
$531$		−	3.02520e9i		−	0.876847i
$532$		0			0
$533$	8.36024e8	−	8.36024e8i	0.239152	−	0.239152i
$534$		0			0
$535$	−1.50067e9	+	3.50672e9i	−0.423688	+	0.990064i
$536$		0			0
$537$	−3.57542e9	−	3.57542e9i	−0.996361	−	0.996361i
$538$		0			0
$539$	−3.32622e9			−0.914934
$540$		0			0
$541$	−2.47049e9			−0.670800			−0.335400	−	0.942076i	$-0.608871\pi$
$541$	−2.47049e9			−0.670800			−0.335400	+	0.942076i	$0.608871\pi$
$542$		0			0
$543$	1.39778e9	+	1.39778e9i	0.374661	+	0.374661i
$544$		0			0
$545$	1.22367e9	+	3.05444e9i	0.323799	+	0.808248i
$546$		0			0
$547$	−1.45879e9	+	1.45879e9i	−0.381100	+	0.381100i	−0.871498	−	0.490399i	$-0.836851\pi$
$547$	−1.45879e9	+	1.45879e9i	−0.381100	+	0.381100i	0.490399	+	0.871498i	$0.336851\pi$
$548$		0			0
$549$		−	5.41897e9i		−	1.39770i
$550$		0			0
$551$		−	4.17070e8i		−	0.106213i
$552$		0			0
$553$	1.50682e9	−	1.50682e9i	0.378899	−	0.378899i
$554$		0			0
$555$	−2.13847e9	−	5.33792e9i	−0.530980	−	1.32540i
$556$		0			0
$557$	1.88379e9	+	1.88379e9i	0.461890	+	0.461890i	0.899274	−	0.437385i	$-0.144095\pi$
$557$	1.88379e9	+	1.88379e9i	0.461890	+	0.461890i	−0.437385	+	0.899274i	$0.644095\pi$
$558$		0			0
$559$	−1.16505e9			−0.282101
$560$		0			0
$561$	3.90545e8			0.0933902
$562$		0			0
$563$	−5.79162e9	−	5.79162e9i	−1.36779	−	1.36779i	−0.863576	−	0.504219i	$-0.831780\pi$
$563$	−5.79162e9	−	5.79162e9i	−1.36779	−	1.36779i	−0.504219	−	0.863576i	$-0.668220\pi$
$564$		0			0
$565$	1.35368e8	−	3.16324e8i	0.0315751	−	0.0737840i
$566$		0			0
$567$	−1.44861e9	+	1.44861e9i	−0.333742	+	0.333742i
$568$		0			0
$569$		−	7.90563e9i		−	1.79905i	−0.436867	−	0.899526i	$-0.643912\pi$
$569$		−	7.90563e9i		−	1.79905i	0.436867	−	0.899526i	$-0.356088\pi$
$570$		0			0
$571$			4.35415e9i			0.978762i	0.872070	+	0.489381i	$0.162777\pi$
$571$			4.35415e9i			0.978762i	−0.872070	+	0.489381i	$0.837223\pi$
$572$		0			0
$573$	3.71372e9	−	3.71372e9i	0.824648	−	0.824648i
$574$		0			0
$575$	3.46893e9	+	8.09045e7i	0.760953	+	0.0177474i
$576$		0			0
$577$	3.90259e9	+	3.90259e9i	0.845741	+	0.845741i	0.989598	−	0.143857i	$-0.0459507\pi$
$577$	3.90259e9	+	3.90259e9i	0.845741	+	0.845741i	−0.143857	+	0.989598i	$0.545951\pi$
$578$		0			0
$579$	−6.23120e9			−1.33413
$580$		0			0
$581$	9.37373e8			0.198288
$582$		0			0
$583$	−1.41522e9	−	1.41522e9i	−0.295791	−	0.295791i
$584$		0			0
$585$	−9.44475e8	−	4.04179e8i	−0.195050	−	0.0834695i
$586$		0			0
$587$	3.86538e9	−	3.86538e9i	0.788786	−	0.788786i	−0.192509	−	0.981295i	$-0.561663\pi$
$587$	3.86538e9	−	3.86538e9i	0.788786	−	0.788786i	0.981295	+	0.192509i	$0.0616625\pi$
$588$		0			0
$589$			2.87433e9i			0.579606i
$590$		0			0
$591$		−	1.70719e9i		−	0.340193i
$592$		0			0
$593$	2.19980e9	−	2.19980e9i	0.433203	−	0.433203i	−0.456513	−	0.889717i	$-0.650902\pi$
$593$	2.19980e9	−	2.19980e9i	0.433203	−	0.433203i	0.889717	+	0.456513i	$0.150902\pi$
$594$		0			0
$595$	1.23221e8	−	4.93647e7i	0.0239814	−	0.00960742i
$596$		0			0
$597$	−8.60142e9	−	8.60142e9i	−1.65448	−	1.65448i
$598$		0			0
$599$	−2.97273e7			−0.00565148			−0.00282574	−	0.999996i	$-0.500899\pi$
$599$	−2.97273e7			−0.00565148			−0.00282574	+	0.999996i	$0.500899\pi$
$600$		0			0
$601$	8.86874e8			0.166648			0.0833242	−	0.996522i	$-0.473446\pi$
$601$	8.86874e8			0.166648			0.0833242	+	0.996522i	$0.473446\pi$
$602$		0			0
$603$	−7.98433e7	−	7.98433e7i	−0.0148295	−	0.0148295i
$604$		0			0
$605$	1.03119e9	−	4.13116e8i	0.189320	−	0.0758452i
$606$		0			0
$607$	−7.56606e9	+	7.56606e9i	−1.37312	+	1.37312i	−0.517348	+	0.855775i	$0.673081\pi$
$607$	−7.56606e9	+	7.56606e9i	−1.37312	+	1.37312i	−0.855775	+	0.517348i	$0.826919\pi$
$608$		0			0
$609$		−	7.28184e8i		−	0.130641i
$610$		0			0
$611$			9.32750e8i			0.165433i
$612$		0			0
$613$	−1.67645e9	+	1.67645e9i	−0.293954	+	0.293954i	−0.838640	−	0.544686i	$-0.816649\pi$
$613$	−1.67645e9	+	1.67645e9i	−0.293954	+	0.293954i	0.544686	+	0.838640i	$0.316649\pi$
$614$		0			0
$615$	9.12229e9	+	3.90379e9i	1.58140	+	0.676743i
$616$		0			0
$617$	6.41892e9	+	6.41892e9i	1.10018	+	1.10018i	0.994388	+	0.105793i	$0.0337380\pi$
$617$	6.41892e9	+	6.41892e9i	1.10018	+	1.10018i	0.105793	+	0.994388i	$0.466262\pi$
$618$		0			0
$619$	−5.97506e8			−0.101257			−0.0506285	−	0.998718i	$-0.516122\pi$
$619$	−5.97506e8			−0.101257			−0.0506285	+	0.998718i	$0.516122\pi$
$620$		0			0
$621$	−1.19883e9			−0.200880
$622$		0			0
$623$	1.94206e9	+	1.94206e9i	0.321776	+	0.321776i
$624$		0			0
$625$	2.84545e8	−	6.09688e9i	0.0466198	−	0.998913i
$626$		0			0
$627$	2.86213e9	−	2.86213e9i	0.463717	−	0.463717i
$628$		0			0
$629$		−	4.20563e8i		−	0.0673836i
$630$		0			0
$631$			5.29242e9i			0.838593i	0.907849	+	0.419297i	$0.137723\pi$
$631$			5.29242e9i			0.838593i	−0.907849	+	0.419297i	$0.862277\pi$
$632$		0			0
$633$	−3.57822e9	+	3.57822e9i	−0.560730	+	0.560730i
$634$		0			0
$635$	−1.76783e9	+	4.13103e9i	−0.273989	+	0.640252i
$636$		0			0
$637$	1.01565e9	+	1.01565e9i	0.155689	+	0.155689i
$638$		0			0
$639$	2.09530e9			0.317683
$640$		0			0
$641$	1.28072e10			1.92066			0.960329	−	0.278869i	$-0.0899596\pi$
$641$	1.28072e10			1.92066			0.960329	+	0.278869i	$0.0899596\pi$
$642$		0			0
$643$	7.57296e9	+	7.57296e9i	1.12338	+	1.12338i	0.991229	+	0.132153i	$0.0421889\pi$
$643$	7.57296e9	+	7.57296e9i	1.12338	+	1.12338i	0.132153	+	0.991229i	$0.457811\pi$
$644$		0			0
$645$	−3.63616e9	−	9.07635e9i	−0.533561	−	1.33184i
$646$		0			0
$647$	3.43154e9	−	3.43154e9i	0.498108	−	0.498108i	−0.412740	−	0.910849i	$-0.635428\pi$
$647$	3.43154e9	−	3.43154e9i	0.498108	−	0.498108i	0.910849	+	0.412740i	$0.135428\pi$
$648$		0			0
$649$		−	8.33890e9i		−	1.19744i
$650$		0			0
$651$			5.01844e9i			0.712911i
$652$		0			0
$653$	−1.60565e9	+	1.60565e9i	−0.225660	+	0.225660i	−0.810877	−	0.585217i	$-0.801009\pi$
$653$	−1.60565e9	+	1.60565e9i	−0.225660	+	0.225660i	0.585217	+	0.810877i	$0.301009\pi$
$654$		0			0
$655$	5.22123e8	+	1.30329e9i	0.0725987	+	0.181216i
$656$		0			0
$657$	7.35796e9	+	7.35796e9i	1.01223	+	1.01223i
$658$		0			0
$659$	4.43687e9			0.603917			0.301959	−	0.953321i	$-0.402360\pi$
$659$	4.43687e9			0.603917			0.301959	+	0.953321i	$0.402360\pi$
$660$		0			0
$661$	−8.67113e9			−1.16781			−0.583903	−	0.811823i	$-0.698475\pi$
$661$	−8.67113e9			−1.16781			−0.583903	+	0.811823i	$0.698475\pi$
$662$		0			0
$663$	−1.19252e8	−	1.19252e8i	−0.0158916	−	0.0158916i
$664$		0			0
$665$	5.41260e8	−	1.26480e9i	0.0713724	−	0.166781i
$666$		0			0
$667$	9.84399e8	−	9.84399e8i	0.128449	−	0.128449i
$668$		0			0
$669$			1.16685e10i			1.50668i
$670$		0			0
$671$		−	1.49373e10i		−	1.90872i
$672$		0			0
$673$	−3.38498e9	+	3.38498e9i	−0.428059	+	0.428059i	−0.887967	−	0.459908i	$-0.847882\pi$
$673$	−3.38498e9	+	3.38498e9i	−0.428059	+	0.428059i	0.459908	+	0.887967i	$0.347882\pi$
$674$		0			0
$675$	−4.91681e7	+	2.10817e9i	−0.00615347	+	0.263842i
$676$		0			0
$677$	6.44910e8	+	6.44910e8i	0.0798802	+	0.0798802i	0.745918	−	0.666038i	$-0.232011\pi$
$677$	6.44910e8	+	6.44910e8i	0.0798802	+	0.0798802i	−0.666038	+	0.745918i	$0.732011\pi$
$678$		0			0
$679$	5.33931e9			0.654547
$680$		0			0
$681$	−1.36059e10			−1.65087
$682$		0			0
$683$	−9.58841e9	−	9.58841e9i	−1.15153	−	1.15153i	−0.986247	−	0.165280i	$-0.947147\pi$
$683$	−9.58841e9	−	9.58841e9i	−1.15153	−	1.15153i	−0.165280	−	0.986247i	$-0.552853\pi$
$684$		0			0
$685$	1.31345e10	+	5.62080e9i	1.56134	+	0.668161i
$686$		0			0
$687$	−1.33118e10	+	1.33118e10i	−1.56635	+	1.56635i
$688$		0			0
$689$			8.64270e8i			0.100666i
$690$		0			0
$691$		−	1.41809e10i		−	1.63504i	−0.575897	−	0.817522i	$-0.695347\pi$
$691$		−	1.41809e10i		−	1.63504i	0.575897	−	0.817522i	$-0.304653\pi$
$692$		0			0
$693$	2.22631e9	−	2.22631e9i	0.254109	−	0.254109i
$694$		0			0
$695$	9.42510e9	−	3.77587e9i	1.06497	−	0.426649i
$696$		0			0
$697$	5.13149e8	+	5.13149e8i	0.0574022	+	0.0574022i
$698$		0			0
$699$	−6.27137e9			−0.694532
$700$		0			0
$701$	4.28213e9			0.469512			0.234756	−	0.972054i	$-0.424571\pi$
$701$	4.28213e9			0.469512			0.234756	+	0.972054i	$0.424571\pi$
$702$		0			0
$703$	−3.08212e9	−	3.08212e9i	−0.334585	−	0.334585i
$704$		0			0
$705$	−7.26659e9	+	2.91113e9i	−0.781032	+	0.312896i
$706$		0			0
$707$	−3.78888e9	+	3.78888e9i	−0.403220	+	0.403220i
$708$		0			0
$709$			3.43170e8i			0.0361615i	0.999837	+	0.0180808i	$0.00575560\pi$
$709$			3.43170e8i			0.0361615i	−0.999837	+	0.0180808i	$0.994244\pi$
$710$		0			0
$711$		−	1.01228e10i		−	1.05623i
$712$		0			0
$713$	−6.78420e9	+	6.78420e9i	−0.700947	+	0.700947i
$714$		0			0
$715$	−2.60342e9	−	1.11411e9i	−0.266363	−	0.113987i
$716$		0			0
$717$	9.98915e9	+	9.98915e9i	1.01207	+	1.01207i
$718$		0			0
$719$	1.29027e10			1.29458			0.647289	−	0.762245i	$-0.275903\pi$
$719$	1.29027e10			1.29458			0.647289	+	0.762245i	$0.275903\pi$
$720$		0			0
$721$	−7.44742e9			−0.740001
$722$		0			0
$723$	1.49499e10	+	1.49499e10i	1.47114	+	1.47114i
$724$		0			0
$725$	−1.69072e9	−	1.77146e9i	−0.164774	−	0.172643i
$726$		0			0
$727$	−3.85294e9	+	3.85294e9i	−0.371896	+	0.371896i	−0.868168	−	0.496271i	$-0.834702\pi$
$727$	−3.85294e9	+	3.85294e9i	−0.371896	+	0.371896i	0.496271	+	0.868168i	$0.334702\pi$
$728$		0			0
$729$			6.02444e9i			0.575931i
$730$		0			0
$731$		−	7.15107e8i		−	0.0677111i
$732$		0			0
$733$	−1.24419e10	+	1.24419e10i	−1.16687	+	1.16687i	−0.183933	+	0.982939i	$0.558883\pi$
$733$	−1.24419e10	+	1.24419e10i	−1.16687	+	1.16687i	−0.982939	+	0.183933i	$0.941117\pi$
$734$		0			0
$735$	−4.74257e9	+	1.10823e10i	−0.440563	+	1.02950i
$736$		0			0
$737$	−2.20086e8	−	2.20086e8i	−0.0202514	−	0.0202514i
$738$		0			0
$739$	1.74752e10			1.59282			0.796409	−	0.604759i	$-0.206730\pi$
$739$	1.74752e10			1.59282			0.796409	+	0.604759i	$0.206730\pi$
$740$		0			0
$741$	−1.74789e9			−0.157816
$742$		0			0
$743$	−7.57887e9	−	7.57887e9i	−0.677866	−	0.677866i	0.281651	−	0.959517i	$-0.409118\pi$
$743$	−7.57887e9	−	7.57887e9i	−0.677866	−	0.677866i	−0.959517	+	0.281651i	$0.909118\pi$
$744$		0			0
$745$	−3.15179e9	−	7.86730e9i	−0.279261	−	0.697073i
$746$		0			0
$747$	3.14864e9	−	3.14864e9i	0.276377	−	0.276377i
$748$		0			0
$749$			5.04803e9i			0.438970i
$750$		0			0
$751$		−	1.44627e10i		−	1.24597i	−0.782232	−	0.622987i	$-0.785919\pi$
$751$		−	1.44627e10i		−	1.24597i	0.782232	−	0.622987i	$-0.214081\pi$
$752$		0			0
$753$	−7.29966e9	+	7.29966e9i	−0.623046	+	0.623046i
$754$		0			0
$755$	−6.41068e9	−	1.60019e10i	−0.542113	−	1.35319i
$756$		0			0
$757$	−7.62134e9	−	7.62134e9i	−0.638551	−	0.638551i	0.311647	−	0.950198i	$-0.399119\pi$
$757$	−7.62134e9	−	7.62134e9i	−0.638551	−	0.638551i	−0.950198	+	0.311647i	$0.899119\pi$
$758$		0			0
$759$	1.35108e10			1.12159
$760$		0			0
$761$	−2.68630e9			−0.220957			−0.110479	−	0.993879i	$-0.535238\pi$
$761$	−2.68630e9			−0.220957			−0.110479	+	0.993879i	$0.535238\pi$
$762$		0			0
$763$	3.07923e9	+	3.07923e9i	0.250961	+	0.250961i
$764$		0			0
$765$	2.48084e8	−	5.79716e8i	0.0200347	−	0.0468167i
$766$		0			0
$767$	−2.54626e9	+	2.54626e9i	−0.203760	+	0.203760i
$768$		0			0
$769$			1.53439e10i			1.21673i	0.793658	+	0.608364i	$0.208174\pi$
$769$			1.53439e10i			1.21673i	−0.793658	+	0.608364i	$0.791826\pi$
$770$		0			0
$771$			1.87817e10i			1.47586i
$772$		0			0
$773$	4.80566e9	−	4.80566e9i	0.374218	−	0.374218i	−0.494793	−	0.869011i	$-0.664756\pi$
$773$	4.80566e9	−	4.80566e9i	0.374218	−	0.374218i	0.869011	+	0.494793i	$0.164756\pi$
$774$		0			0
$775$	1.16519e10	+	1.22084e10i	0.899171	+	0.942114i
$776$		0			0
$777$	−5.38123e9	−	5.38123e9i	−0.411537	−	0.411537i
$778$		0			0
$779$	7.52128e9			0.570047
$780$		0			0
$781$	5.77565e9			0.433832
$782$		0			0
$783$	5.98249e8	+	5.98249e8i	0.0445365	+	0.0445365i
$784$		0			0
$785$	1.06090e10	+	4.54001e9i	0.782763	+	0.334976i
$786$		0			0
$787$	1.65336e10	−	1.65336e10i	1.20908	−	1.20908i	0.237757	−	0.971325i	$-0.423588\pi$
$787$	1.65336e10	−	1.65336e10i	1.20908	−	1.20908i	0.971325	−	0.237757i	$-0.0764121\pi$
$788$		0			0
$789$			1.25213e10i			0.907571i
$790$		0			0
$791$		−	4.55357e8i		−	0.0327141i
$792$		0			0
$793$	−4.56106e9	+	4.56106e9i	−0.324795	+	0.324795i
$794$		0			0
$795$	−6.73309e9	+	2.69741e9i	−0.475258	+	0.190397i
$796$		0			0
$797$	1.61801e10	+	1.61801e10i	1.13208	+	1.13208i	0.989831	+	0.142247i	$0.0454327\pi$
$797$	1.61801e10	+	1.61801e10i	1.13208	+	1.13208i	0.142247	+	0.989831i	$0.454567\pi$
$798$		0			0
$799$	−5.72520e8			−0.0397079
$800$		0			0
$801$	1.30468e10			0.896993
$802$		0			0
$803$	2.02820e10	+	2.02820e10i	1.38232	+	1.38232i
$804$		0			0
$805$	4.26280e9	−	1.70776e9i	0.288011	−	0.115383i
$806$		0			0
$807$	−1.42688e10	+	1.42688e10i	−0.955720	+	0.955720i
$808$		0			0
$809$		−	3.48033e9i		−	0.231100i	−0.993302	−	0.115550i	$-0.963137\pi$
$809$		−	3.48033e9i		−	0.231100i	0.993302	−	0.115550i	$-0.0368631\pi$
$810$		0			0
$811$			1.34188e10i			0.883367i	0.897171	+	0.441683i	$0.145618\pi$
$811$			1.34188e10i			0.883367i	−0.897171	+	0.441683i	$0.854382\pi$
$812$		0			0
$813$	−3.60592e9	+	3.60592e9i	−0.235342	+	0.235342i
$814$		0			0
$815$	−9.90451e9	−	4.23854e9i	−0.640887	−	0.274261i
$816$		0			0
$817$	−5.24070e9	−	5.24070e9i	−0.336211	−	0.336211i
$818$		0			0
$819$	−1.35960e9			−0.0864802
$820$		0			0
$821$	1.77890e10			1.12189			0.560947	−	0.827852i	$-0.310437\pi$
$821$	1.77890e10			1.12189			0.560947	+	0.827852i	$0.310437\pi$
$822$		0			0
$823$	−6.48743e9	−	6.48743e9i	−0.405670	−	0.405670i	0.474555	−	0.880226i	$-0.342609\pi$
$823$	−6.48743e9	−	6.48743e9i	−0.405670	−	0.405670i	−0.880226	+	0.474555i	$0.842609\pi$
$824$		0			0
$825$	5.54124e8	−	2.37591e10i	0.0343573	−	1.47313i
$826$		0			0
$827$	−8.87561e8	+	8.87561e8i	−0.0545668	+	0.0545668i	−0.733864	−	0.679297i	$-0.762285\pi$
$827$	−8.87561e8	+	8.87561e8i	−0.0545668	+	0.0545668i	0.679297	+	0.733864i	$0.262285\pi$
$828$		0			0
$829$		−	2.15941e10i		−	1.31642i	−0.752834	−	0.658211i	$-0.771314\pi$
$829$		−	2.15941e10i		−	1.31642i	0.752834	−	0.658211i	$-0.228686\pi$
$830$		0			0
$831$		−	2.81608e10i		−	1.70232i
$832$		0			0
$833$	−6.23405e8	+	6.23405e8i	−0.0373691	+	0.0373691i
$834$		0			0
$835$	−9.05407e9	+	2.11573e10i	−0.538197	+	1.25765i
$836$		0			0
$837$	−4.12296e9	−	4.12296e9i	−0.243036	−	0.243036i
$838$		0			0
$839$	9.54702e9			0.558086			0.279043	−	0.960279i	$-0.409983\pi$
$839$	9.54702e9			0.558086			0.279043	+	0.960279i	$0.409983\pi$
$840$		0			0
$841$	1.62674e10			0.943044
$842$		0			0
$843$	2.24028e10	+	2.24028e10i	1.28797	+	1.28797i
$844$		0			0
$845$	−6.06765e9	−	1.51457e10i	−0.345957	−	0.863557i
$846$		0			0
$847$	1.03956e9	−	1.03956e9i	0.0587839	−	0.0587839i
$848$		0			0
$849$			1.95096e10i			1.09414i
$850$		0			0
$851$		−	1.45493e10i		−	0.809261i
$852$		0			0
$853$	−4.50377e9	+	4.50377e9i	−0.248459	+	0.248459i	−0.820338	−	0.571879i	$-0.806215\pi$
$853$	−4.50377e9	+	4.50377e9i	−0.248459	+	0.248459i	0.571879	+	0.820338i	$0.306215\pi$
$854$		0			0
$855$	−2.43039e9	−	6.06658e9i	−0.132982	−	0.331942i
$856$		0			0
$857$	−2.16952e10	−	2.16952e10i	−1.17742	−	1.17742i	−0.980400	−	0.197020i	$-0.936874\pi$
$857$	−2.16952e10	−	2.16952e10i	−1.17742	−	1.17742i	−0.197020	−	0.980400i	$-0.563126\pi$
$858$		0			0
$859$	−1.35381e10			−0.728755			−0.364378	−	0.931251i	$-0.618718\pi$
$859$	−1.35381e10			−0.728755			−0.364378	+	0.931251i	$0.618718\pi$
$860$		0			0
$861$	1.31318e10			0.701153
$862$		0			0
$863$	−2.07408e10	−	2.07408e10i	−1.09847	−	1.09847i	−0.994590	−	0.103877i	$-0.966875\pi$
$863$	−2.07408e10	−	2.07408e10i	−1.09847	−	1.09847i	−0.103877	−	0.994590i	$-0.533125\pi$
$864$		0			0
$865$	−2.35181e9	+	5.49566e9i	−0.123551	+	0.288711i
$866$		0			0
$867$	−1.81493e10	+	1.81493e10i	−0.945786	+	0.945786i
$868$		0			0
$869$		−	2.79034e10i		−	1.44241i
$870$		0			0
$871$			1.34406e8i			0.00689213i
$872$		0			0
$873$	1.79348e10	−	1.79348e10i	0.912318	−	0.912318i
$874$		0			0
$875$	−2.82830e9	−	7.56628e9i	−0.142724	−	0.381816i
$876$		0			0
$877$	−1.69228e10	−	1.69228e10i	−0.847178	−	0.847178i	0.142602	−	0.989780i	$-0.454453\pi$
$877$	−1.69228e10	−	1.69228e10i	−0.847178	−	0.847178i	−0.989780	+	0.142602i	$0.954453\pi$
$878$		0			0
$879$	−1.64009e10			−0.814530
$880$		0			0
$881$	3.79762e10			1.87109			0.935547	−	0.353203i	$-0.114907\pi$
$881$	3.79762e10			1.87109			0.935547	+	0.353203i	$0.114907\pi$
$882$		0			0
$883$	5.98400e9	+	5.98400e9i	0.292502	+	0.292502i	0.838068	−	0.545566i	$-0.183685\pi$
$883$	5.98400e9	+	5.98400e9i	0.292502	+	0.292502i	−0.545566	+	0.838068i	$0.683685\pi$
$884$		0			0
$885$	−2.77836e10	−	1.18897e10i	−1.34737	−	0.576594i
$886$		0			0
$887$	1.83183e10	−	1.83183e10i	0.881360	−	0.881360i	−0.112313	−	0.993673i	$-0.535826\pi$
$887$	1.83183e10	−	1.83183e10i	0.881360	−	0.881360i	0.993673	+	0.112313i	$0.0358259\pi$
$888$		0			0
$889$			5.94674e9i			0.283872i
$890$		0			0
$891$			2.68254e10i			1.27050i
$892$		0			0
$893$	−4.19574e9	+	4.19574e9i	−0.197164	+	0.197164i
$894$		0			0
$895$	−2.08898e10	+	8.36886e9i	−0.973990	+	0.390199i
$896$		0			0
$897$	−4.12550e9	−	4.12550e9i	−0.190855	−	0.190855i
$898$		0			0
$899$	6.77100e9			0.310809
$900$		0			0
$901$	−5.30486e8			−0.0241623
$902$		0			0
$903$	−9.15000e9	−	9.15000e9i	−0.413537	−	0.413537i
$904$		0			0
$905$	8.16670e9	−	3.27173e9i	0.366249	−	0.146726i
$906$		0			0
$907$	2.05995e10	−	2.05995e10i	0.916708	−	0.916708i	−0.0800808	−	0.996788i	$-0.525518\pi$
$907$	2.05995e10	−	2.05995e10i	0.916708	−	0.916708i	0.996788	+	0.0800808i	$0.0255178\pi$
$908$		0			0
$909$			2.54537e10i			1.12403i
$910$		0			0
$911$		−	3.91426e10i		−	1.71528i	−0.514250	−	0.857640i	$-0.671930\pi$
$911$		−	3.91426e10i		−	1.71528i	0.514250	−	0.857640i	$-0.328070\pi$
$912$		0			0
$913$	8.67916e9	−	8.67916e9i	0.377424	−	0.377424i
$914$		0			0
$915$	−4.97681e10	−	2.12978e10i	−2.14772	−	0.919094i
$916$		0			0
$917$	1.31387e9	+	1.31387e9i	0.0562677	+	0.0562677i
$918$		0			0
$919$	−3.08816e10			−1.31249			−0.656245	−	0.754548i	$-0.727856\pi$
$919$	−3.08816e10			−1.31249			−0.656245	+	0.754548i	$0.727856\pi$
$920$		0			0
$921$	6.51769e9			0.274906
$922$		0			0
$923$	−1.76358e9	−	1.76358e9i	−0.0738226	−	0.0738226i
$924$		0			0
$925$	−2.55853e10	−	5.96716e8i	−1.06290	−	0.0247897i
$926$		0			0
$927$	−2.50159e10	+	2.50159e10i	−1.03142	+	1.03142i
$928$		0			0
$929$			1.91058e10i			0.781825i	0.920428	+	0.390912i	$0.127840\pi$
$929$			1.91058e10i			0.781825i	−0.920428	+	0.390912i	$0.872160\pi$
$930$		0			0
$931$			9.13731e9i			0.371103i
$932$		0			0
$933$	−2.28034e10	+	2.28034e10i	−0.919208	+	0.919208i
$934$		0			0
$935$	6.83837e8	−	1.59797e9i	0.0273597	−	0.0639335i
$936$		0			0
$937$	−1.81182e10	−	1.81182e10i	−0.719493	−	0.719493i	0.249009	−	0.968501i	$-0.419895\pi$
$937$	−1.81182e10	−	1.81182e10i	−0.719493	−	0.719493i	−0.968501	+	0.249009i	$0.919895\pi$
$938$		0			0
$939$	−4.68605e10			−1.84704
$940$		0			0
$941$	1.15031e10			0.450039			0.225020	−	0.974354i	$-0.427755\pi$
$941$	1.15031e10			0.450039			0.225020	+	0.974354i	$0.427755\pi$
$942$		0			0
$943$	1.77523e10	+	1.77523e10i	0.689387	+	0.689387i
$944$		0			0
$945$	1.03786e9	+	2.59063e9i	0.0400061	+	0.0998607i
$946$		0			0
$947$	3.38548e10	−	3.38548e10i	1.29537	−	1.29537i	0.363960	−	0.931415i	$-0.381424\pi$
$947$	3.38548e10	−	3.38548e10i	1.29537	−	1.29537i	0.931415	−	0.363960i	$-0.118576\pi$
$948$		0			0
$949$		−	1.23862e10i		−	0.470441i
$950$		0			0
$951$		−	2.48390e10i		−	0.936490i
$952$		0			0
$953$	−3.05822e9	+	3.05822e9i	−0.114457	+	0.114457i	−0.762016	−	0.647559i	$-0.775790\pi$
$953$	−3.05822e9	+	3.05822e9i	−0.114457	+	0.114457i	0.647559	+	0.762016i	$0.275790\pi$
$954$		0			0
$955$	−8.69259e9	−	2.16979e10i	−0.322952	−	0.806132i
$956$		0			0
$957$	−6.74227e9	−	6.74227e9i	−0.248665	−	0.248665i
$958$		0			0
$959$	1.89075e10			0.692261
$960$		0			0
$961$	−1.91512e10			−0.696087
$962$		0			0
$963$	1.69564e10	+	1.69564e10i	0.611844	+	0.611844i
$964$		0			0
$965$	−1.09107e10	+	2.54959e10i	−0.390847	+	0.913323i
$966$		0			0
$967$	−3.23554e10	+	3.23554e10i	−1.15068	+	1.15068i	−0.164262	+	0.986417i	$0.552524\pi$
$967$	−3.23554e10	+	3.23554e10i	−1.15068	+	1.15068i	−0.986417	+	0.164262i	$0.947476\pi$
$968$		0			0
$969$		−	1.07285e9i		−	0.0378797i
$970$		0			0
$971$		−	4.12508e9i		−	0.144599i	−0.997383	−	0.0722994i	$-0.976966\pi$
$971$		−	4.12508e9i		−	0.144599i	0.997383	−	0.0722994i	$-0.0230337\pi$
$972$		0			0
$973$	9.50159e9	−	9.50159e9i	0.330675	−	0.330675i
$974$		0			0
$975$	−7.42399e9	+	7.08559e9i	−0.256520	+	0.244827i
$976$		0			0
$977$	−3.92345e8	−	3.92345e8i	−0.0134598	−	0.0134598i	0.700345	−	0.713805i	$-0.253030\pi$
$977$	−3.92345e8	−	3.92345e8i	−0.0134598	−	0.0134598i	−0.713805	+	0.700345i	$0.753030\pi$
$978$		0			0
$979$	3.59631e10			1.22495
$980$		0			0
$981$	2.06863e10			0.699587
$982$		0			0
$983$	1.27089e9	+	1.27089e9i	0.0426747	+	0.0426747i	0.728122	−	0.685447i	$-0.240393\pi$
$983$	1.27089e9	+	1.27089e9i	0.0426747	+	0.0426747i	−0.685447	+	0.728122i	$0.740393\pi$
$984$		0			0
$985$	−6.98520e9	−	2.98925e9i	−0.232891	−	0.0996633i
$986$		0			0
$987$	−7.32556e9	+	7.32556e9i	−0.242511	+	0.242511i
$988$		0			0
$989$		−	2.47390e10i		−	0.813194i
$990$		0			0
$991$			4.10559e9i			0.134004i	0.997753	+	0.0670020i	$0.0213434\pi$
$991$			4.10559e9i			0.134004i	−0.997753	+	0.0670020i	$0.978657\pi$
$992$		0			0
$993$	2.23000e10	−	2.23000e10i	0.722741	−	0.722741i
$994$		0			0
$995$	−5.02549e10	+	2.01331e10i	−1.61733	+	0.647932i
$996$		0			0
$997$	−2.89545e10	−	2.89545e10i	−0.925300	−	0.925300i	0.0720978	−	0.997398i	$-0.477031\pi$
$997$	−2.89545e10	−	2.89545e10i	−0.925300	−	0.925300i	−0.997398	+	0.0720978i	$0.977031\pi$
$998$		0			0
$999$	8.84205e9			0.280591

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	160.8.n.b.127.2	yes	20
4.3	odd	2		160.8.n.a.127.9	yes	20
5.3	odd	4		160.8.n.a.63.9	✓	20
20.3	even	4	inner	160.8.n.b.63.2	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
160.8.n.a.63.9	✓	20	5.3	odd	4
160.8.n.a.127.9	yes	20	4.3	odd	2
160.8.n.b.63.2	yes	20	20.3	even	4	inner
160.8.n.b.127.2	yes	20	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 160 = 2^{5} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	160.n (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-44.4084 - 44.4084i) q^{3} +(-259.462 + 103.945i) q^{5} +(-261.567 + 261.567i) q^{7} +1757.21i q^{9} +4843.71i q^{11} +(1479.02 - 1479.02i) q^{13} +(16138.3 + 6906.24i) q^{15} +(907.816 + 907.816i) q^{17} +13306.0 q^{19} +23231.6 q^{21} +(31405.7 + 31405.7i) q^{23} +(56515.8 - 53939.6i) q^{25} +(-19086.2 + 19086.2i) q^{27} -31344.6i q^{29} +216018. i q^{31} +(215101. - 215101. i) q^{33} +(40678.0 - 95055.4i) q^{35} +(-231635. - 231635. i) q^{37} -131362. q^{39} +565257. q^{41} +(-393861. - 393861. i) q^{43} +(-182654. - 455929. i) q^{45} +(-315328. + 315328. i) q^{47} +686708. i q^{49} -80629.3i q^{51} +(-292177. + 292177. i) q^{53} +(-503481. - 1.25676e6i) q^{55} +(-590897. - 590897. i) q^{57} -1.72159e6 q^{59} -3.08385e6 q^{61} +(-459629. - 459629. i) q^{63} +(-230011. + 537485. i) q^{65} +(-45437.5 + 45437.5i) q^{67} -2.78935e6i q^{69} -1.19240e6i q^{71} +(4.18729e6 - 4.18729e6i) q^{73} +(-4.90515e6 - 114401. i) q^{75} +(-1.26696e6 - 1.26696e6i) q^{77} -5.76074e6 q^{79} +5.53820e6 q^{81} +(-1.79184e6 - 1.79184e6i) q^{83} +(-329907. - 141180. i) q^{85} +(-1.39196e6 + 1.39196e6i) q^{87} -7.42470e6i q^{89} +773724. i q^{91} +(9.59302e6 - 9.59302e6i) q^{93} +(-3.45239e6 + 1.38309e6i) q^{95} +(-1.02064e7 - 1.02064e7i) q^{97} -8.51143e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 58 q^{3} - 54 q^{5} + 2466 q^{7} - 1172 q^{13} - 11138 q^{15} - 25136 q^{17} + 64784 q^{19} - 71268 q^{21} - 39922 q^{23} + 118056 q^{25} + 31792 q^{27} + 59756 q^{33} - 378426 q^{35} - 647408 q^{37}+ \cdots - 7823900 q^{99}+O(q^{100}) \) 20 * q + 58 * q^3 - 54 * q^5 + 2466 * q^7 - 1172 * q^13 - 11138 * q^15 - 25136 * q^17 + 64784 * q^19 - 71268 * q^21 - 39922 * q^23 + 118056 * q^25 + 31792 * q^27 + 59756 * q^33 - 378426 * q^35 - 647408 * q^37 + 1415100 * q^39 - 1151884 * q^41 - 1589406 * q^43 + 2340830 * q^45 - 1417862 * q^47 - 2115336 * q^53 - 397400 * q^55 - 2093064 * q^57 + 6498576 * q^59 + 1936564 * q^61 + 651590 * q^63 + 4800532 * q^65 + 1634150 * q^67 - 2406004 * q^73 - 2894398 * q^75 - 2176156 * q^77 - 11749104 * q^79 - 12415936 * q^81 - 13229814 * q^83 - 10242140 * q^85 - 37792 * q^87 - 9544900 * q^93 + 30202616 * q^95 + 24630564 * q^97 - 7823900 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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