LMFDB - Embedded newform 80.9.p.a.33.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [80,9,Mod(17,80)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(80, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

N = Newforms(chi, 9, names="a")

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

chi := DirichletCharacter("80.17");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 80 = 2^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	80.p (of order $4$, 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:	$32.5902888049$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(i, \sqrt{601})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 301x^{2} + 22500 $ x^4 + 301*x^2 + 22500
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 10)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			33.2
Root			$11.7577i$ of defining polynomial
Character	$\chi$	$=$	80.33
Dual form			80.9.p.a.17.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(39.7883 + 39.7883i) q^{3} +(-401.365 - 479.094i) q^{5} +(-144.447 + 144.447i) q^{7} -3394.79i q^{9} +O(q^{10})$	\(q+(39.7883 + 39.7883i) q^{3} +(-401.365 - 479.094i) q^{5} +(-144.447 + 144.447i) q^{7} -3394.79i q^{9} -13600.3 q^{11} +(30883.7 + 30883.7i) q^{13} +(3092.72 - 35031.9i) q^{15} +(4949.02 - 4949.02i) q^{17} +176579. i q^{19} -11494.6 q^{21} +(229482. + 229482. i) q^{23} +(-68437.7 + 384583. i) q^{25} +(396123. - 396123. i) q^{27} -144167. i q^{29} -1.50971e6 q^{31} +(-541132. - 541132. i) q^{33} +(127179. + 11227.8i) q^{35} +(-1.60715e6 + 1.60715e6i) q^{37} +2.45761e6i q^{39} +3.62276e6 q^{41} +(575576. + 575576. i) q^{43} +(-1.62642e6 + 1.36255e6i) q^{45} +(-3.11186e6 + 3.11186e6i) q^{47} +5.72307e6i q^{49} +393826. q^{51} +(8.16813e6 + 8.16813e6i) q^{53} +(5.45868e6 + 6.51582e6i) q^{55} +(-7.02575e6 + 7.02575e6i) q^{57} +1.54905e7i q^{59} -1.92688e7 q^{61} +(490366. + 490366. i) q^{63} +(2.40057e6 - 2.71918e7i) q^{65} +(1.00075e7 - 1.00075e7i) q^{67} +1.82614e7i q^{69} -1.84257e7 q^{71} +(2.45702e7 + 2.45702e7i) q^{73} +(-1.80249e7 + 1.25789e7i) q^{75} +(1.96452e6 - 1.96452e6i) q^{77} -2.17799e7i q^{79} +9.24891e6 q^{81} +(-4.00189e6 - 4.00189e6i) q^{83} +(-4.35741e6 - 384685. i) q^{85} +(5.73617e6 - 5.73617e6i) q^{87} -6.14525e7i q^{89} -8.92209e6 q^{91} +(-6.00686e7 - 6.00686e7i) q^{93} +(8.45978e7 - 7.08724e7i) q^{95} +(1.55814e7 - 1.55814e7i) q^{97} +4.61701e7i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 86 q^{3} - 870 q^{5} - 5726 q^{7}+O(q^{10}) $ 4 * q - 86 * q^3 - 870 * q^5 - 5726 * q^7	$ 4 q - 86 q^{3} - 870 q^{5} - 5726 q^{7} + 14732 q^{11} + 45576 q^{13} + 115090 q^{15} + 3616 q^{17} + 877268 q^{21} + 456794 q^{23} - 913600 q^{25} + 889240 q^{27} - 4672348 q^{31} - 4553788 q^{33} - 2956030 q^{35} - 5554884 q^{37} + 10738708 q^{41} - 4913286 q^{43} - 12173390 q^{45} + 5448474 q^{47} + 1827812 q^{51} + 20290316 q^{53} + 9506940 q^{55} - 2593360 q^{57} - 43572012 q^{61} - 37877126 q^{63} + 15450660 q^{65} - 9518486 q^{67} - 20406908 q^{71} - 11608364 q^{73} + 21302450 q^{75} - 110066908 q^{77} + 96218224 q^{81} - 64264686 q^{83} - 12424120 q^{85} - 8501600 q^{87} + 70189812 q^{91} + 16706132 q^{93} + 82819000 q^{95} + 34113396 q^{97}+O(q^{100}) $ 4 * q - 86 * q^3 - 870 * q^5 - 5726 * q^7 + 14732 * q^11 + 45576 * q^13 + 115090 * q^15 + 3616 * q^17 + 877268 * q^21 + 456794 * q^23 - 913600 * q^25 + 889240 * q^27 - 4672348 * q^31 - 4553788 * q^33 - 2956030 * q^35 - 5554884 * q^37 + 10738708 * q^41 - 4913286 * q^43 - 12173390 * q^45 + 5448474 * q^47 + 1827812 * q^51 + 20290316 * q^53 + 9506940 * q^55 - 2593360 * q^57 - 43572012 * q^61 - 37877126 * q^63 + 15450660 * q^65 - 9518486 * q^67 - 20406908 * q^71 - 11608364 * q^73 + 21302450 * q^75 - 110066908 * q^77 + 96218224 * q^81 - 64264686 * q^83 - 12424120 * q^85 - 8501600 * q^87 + 70189812 * q^91 + 16706132 * q^93 + 82819000 * q^95 + 34113396 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/80\mathbb{Z}\right)^\times$.

$n$	$17$	$21$	$31$
$\chi(n)$	$e\left(\frac{3}{4}\right)$	$1$	$1$

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
$3$	39.7883	+	39.7883i	0.491213	+	0.491213i	0.908688	−	0.417475i	$-0.137085\pi$
$3$	39.7883	+	39.7883i	0.491213	+	0.491213i	−0.417475	+	0.908688i	$0.637085\pi$
$4$		0			0
$5$	−401.365	−	479.094i	−0.642184	−	0.766551i
$5$	−401.365	−	479.094i	−0.642184	−	0.766551i
$6$		0			0
$7$	−144.447	+	144.447i	−0.0601610	+	0.0601610i	−0.736547	−	0.676386i	$-0.763545\pi$
$7$	−144.447	+	144.447i	−0.0601610	+	0.0601610i	0.676386	+	0.736547i	$0.263545\pi$
$8$		0			0
$9$		−	3394.79i		−	0.517420i
$10$		0			0
$11$	−13600.3			−0.928918			−0.464459	−	0.885595i	$-0.653751\pi$
$11$	−13600.3			−0.928918			−0.464459	+	0.885595i	$0.653751\pi$
$12$		0			0
$13$	30883.7	+	30883.7i	1.08132	+	1.08132i	0.996386	+	0.0849366i	$0.0270688\pi$
$13$	30883.7	+	30883.7i	1.08132	+	1.08132i	0.0849366	+	0.996386i	$0.472931\pi$
$14$		0			0
$15$	3092.72	−	35031.9i	0.0610908	−	0.691989i
$16$		0			0
$17$	4949.02	−	4949.02i	0.0592549	−	0.0592549i	−0.676858	−	0.736113i	$-0.736659\pi$
$17$	4949.02	−	4949.02i	0.0592549	−	0.0592549i	0.736113	+	0.676858i	$0.236659\pi$
$18$		0			0
$19$			176579.i			1.35495i	0.735546	+	0.677475i	$0.236926\pi$
$19$			176579.i			1.35495i	−0.735546	+	0.677475i	$0.763074\pi$
$20$		0			0
$21$	−11494.6			−0.0591038
$22$		0			0
$23$	229482.	+	229482.i	0.820043	+	0.820043i	0.986114	−	0.166071i	$-0.0531080\pi$
$23$	229482.	+	229482.i	0.820043	+	0.820043i	−0.166071	+	0.986114i	$0.553108\pi$
$24$		0			0
$25$	−68437.7	+	384583.i	−0.175200	+	0.984533i
$26$		0			0
$27$	396123.	−	396123.i	0.745376	−	0.745376i
$28$		0			0
$29$		−	144167.i		−	0.203833i	−0.994793	−	0.101917i	$-0.967503\pi$
$29$		−	144167.i		−	0.203833i	0.994793	−	0.101917i	$-0.0324975\pi$
$30$		0			0
$31$	−1.50971e6			−1.63473			−0.817365	−	0.576120i	$-0.804566\pi$
$31$	−1.50971e6			−1.63473			−0.817365	+	0.576120i	$0.804566\pi$
$32$		0			0
$33$	−541132.	−	541132.i	−0.456297	−	0.456297i
$34$		0			0
$35$	127179.	+	11227.8i	0.0847509	+	0.00748206i
$36$		0			0
$37$	−1.60715e6	+	1.60715e6i	−0.857532	+	0.857532i	−0.991047	−	0.133515i	$-0.957374\pi$
$37$	−1.60715e6	+	1.60715e6i	−0.857532	+	0.857532i	0.133515	+	0.991047i	$0.457374\pi$
$38$		0			0
$39$			2.45761e6i			1.06232i
$40$		0			0
$41$	3.62276e6			1.28205			0.641023	−	0.767522i	$-0.278510\pi$
$41$	3.62276e6			1.28205			0.641023	+	0.767522i	$0.278510\pi$
$42$		0			0
$43$	575576.	+	575576.i	0.168356	+	0.168356i	0.786256	−	0.617900i	$-0.212017\pi$
$43$	575576.	+	575576.i	0.168356	+	0.168356i	−0.617900	+	0.786256i	$0.712017\pi$
$44$		0			0
$45$	−1.62642e6	+	1.36255e6i	−0.396628	+	0.332278i
$46$		0			0
$47$	−3.11186e6	+	3.11186e6i	−0.637718	+	0.637718i	−0.949992	−	0.312274i	$-0.898909\pi$
$47$	−3.11186e6	+	3.11186e6i	−0.637718	+	0.637718i	0.312274	+	0.949992i	$0.398909\pi$
$48$		0			0
$49$			5.72307e6i			0.992761i
$50$		0			0
$51$	393826.			0.0582135
$52$		0			0
$53$	8.16813e6	+	8.16813e6i	1.03519	+	1.03519i	0.999358	+	0.0358294i	$0.0114073\pi$
$53$	8.16813e6	+	8.16813e6i	1.03519	+	1.03519i	0.0358294	+	0.999358i	$0.488593\pi$
$54$		0			0
$55$	5.45868e6	+	6.51582e6i	0.596536	+	0.712063i
$56$		0			0
$57$	−7.02575e6	+	7.02575e6i	−0.665569	+	0.665569i
$58$		0			0
$59$			1.54905e7i			1.27837i	0.769051	+	0.639187i	$0.220729\pi$
$59$			1.54905e7i			1.27837i	−0.769051	+	0.639187i	$0.779271\pi$
$60$		0			0
$61$	−1.92688e7			−1.39167			−0.695833	−	0.718204i	$-0.744965\pi$
$61$	−1.92688e7			−1.39167			−0.695833	+	0.718204i	$0.744965\pi$
$62$		0			0
$63$	490366.	+	490366.i	0.0311285	+	0.0311285i
$64$		0			0
$65$	2.40057e6	−	2.71918e7i	0.134481	−	1.52330i
$66$		0			0
$67$	1.00075e7	−	1.00075e7i	0.496624	−	0.496624i	−0.413761	−	0.910385i	$-0.635785\pi$
$67$	1.00075e7	−	1.00075e7i	0.496624	−	0.496624i	0.910385	+	0.413761i	$0.135785\pi$
$68$		0			0
$69$			1.82614e7i			0.805632i
$70$		0			0
$71$	−1.84257e7			−0.725087			−0.362543	−	0.931967i	$-0.618092\pi$
$71$	−1.84257e7			−0.725087			−0.362543	+	0.931967i	$0.618092\pi$
$72$		0			0
$73$	2.45702e7	+	2.45702e7i	0.865203	+	0.865203i	0.991937	−	0.126734i	$-0.0404494\pi$
$73$	2.45702e7	+	2.45702e7i	0.865203	+	0.865203i	−0.126734	+	0.991937i	$0.540449\pi$
$74$		0			0
$75$	−1.80249e7	+	1.25789e7i	−0.569676	+	0.397555i
$76$		0			0
$77$	1.96452e6	−	1.96452e6i	0.0558847	−	0.0558847i
$78$		0			0
$79$		−	2.17799e7i		−	0.559176i	−0.960120	−	0.279588i	$-0.909802\pi$
$79$		−	2.17799e7i		−	0.559176i	0.960120	−	0.279588i	$-0.0901978\pi$
$80$		0			0
$81$	9.24891e6			0.214857
$82$		0			0
$83$	−4.00189e6	−	4.00189e6i	−0.0843242	−	0.0843242i	0.663687	−	0.748011i	$-0.268991\pi$
$83$	−4.00189e6	−	4.00189e6i	−0.0843242	−	0.0843242i	−0.748011	+	0.663687i	$0.768991\pi$
$84$		0			0
$85$	−4.35741e6	−	384685.i	−0.0834744	−	0.00736936i
$86$		0			0
$87$	5.73617e6	−	5.73617e6i	0.100126	−	0.100126i
$88$		0			0
$89$		−	6.14525e7i		−	0.979444i	−0.871879	−	0.489722i	$-0.837098\pi$
$89$		−	6.14525e7i		−	0.979444i	0.871879	−	0.489722i	$-0.162902\pi$
$90$		0			0
$91$	−8.92209e6			−0.130107
$92$		0			0
$93$	−6.00686e7	−	6.00686e7i	−0.803001	−	0.803001i
$94$		0			0
$95$	8.45978e7	−	7.08724e7i	1.03864	−	0.870127i
$96$		0			0
$97$	1.55814e7	−	1.55814e7i	0.176003	−	0.176003i	−0.613608	−	0.789611i	$-0.710282\pi$
$97$	1.55814e7	−	1.55814e7i	0.176003	−	0.176003i	0.789611	+	0.613608i	$0.210282\pi$
$98$		0			0
$99$			4.61701e7i			0.480640i
$100$		0			0
$101$	3.55113e7			0.341257			0.170629	−	0.985335i	$-0.445420\pi$
$101$	3.55113e7			0.341257			0.170629	+	0.985335i	$0.445420\pi$
$102$		0			0
$103$	−2.91511e7	−	2.91511e7i	−0.259004	−	0.259004i	0.565645	−	0.824649i	$-0.308627\pi$
$103$	−2.91511e7	−	2.91511e7i	−0.259004	−	0.259004i	−0.824649	+	0.565645i	$0.808627\pi$
$104$		0			0
$105$	4.61351e6	+	5.50698e6i	0.0379555	+	0.0453061i
$106$		0			0
$107$	9.89754e7	−	9.89754e7i	0.755079	−	0.755079i	−0.220344	−	0.975422i	$-0.570718\pi$
$107$	9.89754e7	−	9.89754e7i	0.755079	−	0.755079i	0.975422	+	0.220344i	$0.0707179\pi$
$108$		0			0
$109$			3.08220e7i			0.218351i	0.994022	+	0.109176i	$0.0348211\pi$
$109$			3.08220e7i			0.218351i	−0.994022	+	0.109176i	$0.965179\pi$
$110$		0			0
$111$	−1.27892e8			−0.842461
$112$		0			0
$113$	−6.07746e7	−	6.07746e7i	−0.372742	−	0.372742i	0.495733	−	0.868475i	$-0.334900\pi$
$113$	−6.07746e7	−	6.07746e7i	−0.372742	−	0.372742i	−0.868475	+	0.495733i	$0.834900\pi$
$114$		0			0
$115$	1.78375e7	−	2.02049e8i	0.101987	−	1.15522i
$116$		0			0
$117$	1.04844e8	−	1.04844e8i	0.559498	−	0.559498i
$118$		0			0
$119$			1.42974e6i			0.00712967i
$120$		0			0
$121$	−2.93911e7			−0.137111
$122$		0			0
$123$	1.44143e8	+	1.44143e8i	0.629758	+	0.629758i
$124$		0			0
$125$	2.11720e8	−	1.21570e8i	0.867205	−	0.497951i
$126$		0			0
$127$	−1.95509e8	+	1.95509e8i	−0.751539	+	0.751539i	−0.974766	−	0.223228i	$-0.928341\pi$
$127$	−1.95509e8	+	1.95509e8i	−0.751539	+	0.751539i	0.223228	+	0.974766i	$0.428341\pi$
$128$		0			0
$129$			4.58023e7i			0.165397i
$130$		0			0
$131$	3.40141e8			1.15498			0.577489	−	0.816399i	$-0.304033\pi$
$131$	3.40141e8			1.15498			0.577489	+	0.816399i	$0.304033\pi$
$132$		0			0
$133$	−2.55062e7	−	2.55062e7i	−0.0815152	−	0.0815152i
$134$		0			0
$135$	−3.48771e8	−	3.07905e7i	−1.05004	−	0.0927004i
$136$		0			0
$137$	−9.80014e7	+	9.80014e7i	−0.278196	+	0.278196i	−0.832388	−	0.554193i	$-0.813027\pi$
$137$	−9.80014e7	+	9.80014e7i	−0.278196	+	0.278196i	0.554193	+	0.832388i	$0.313027\pi$
$138$		0			0
$139$		−	3.51491e8i		−	0.941576i	−0.882246	−	0.470788i	$-0.843970\pi$
$139$		−	3.51491e8i		−	0.941576i	0.882246	−	0.470788i	$-0.156030\pi$
$140$		0			0
$141$	−2.47631e8			−0.626511
$142$		0			0
$143$	−4.20027e8	−	4.20027e8i	−1.00446	−	1.00446i
$144$		0			0
$145$	−6.90698e7	+	5.78637e7i	−0.156249	+	0.130898i
$146$		0			0
$147$	−2.27711e8	+	2.27711e8i	−0.487657	+	0.487657i
$148$		0			0
$149$			1.71452e8i			0.347855i	0.984758	+	0.173927i	$0.0556458\pi$
$149$			1.71452e8i			0.347855i	−0.984758	+	0.173927i	$0.944354\pi$
$150$		0			0
$151$	4.23728e8			0.815042			0.407521	−	0.913196i	$-0.366393\pi$
$151$	4.23728e8			0.815042			0.407521	+	0.913196i	$0.366393\pi$
$152$		0			0
$153$	−1.68009e7	−	1.68009e7i	−0.0306596	−	0.0306596i
$154$		0			0
$155$	6.05943e8	+	7.23292e8i	1.04980	+	1.25310i
$156$		0			0
$157$	4.05135e6	−	4.05135e6i	0.00666808	−	0.00666808i	−0.703765	−	0.710433i	$-0.748499\pi$
$157$	4.05135e6	−	4.05135e6i	0.00666808	−	0.00666808i	0.710433	+	0.703765i	$0.248499\pi$
$158$		0			0
$159$			6.49991e8i			1.01699i
$160$		0			0
$161$	−6.62957e7			−0.0986693
$162$		0			0
$163$	1.89503e8	+	1.89503e8i	0.268451	+	0.268451i	0.828476	−	0.560025i	$-0.189208\pi$
$163$	1.89503e8	+	1.89503e8i	0.268451	+	0.268451i	−0.560025	+	0.828476i	$0.689208\pi$
$164$		0			0
$165$	−4.20619e7	+	4.76444e8i	−0.0567483	+	0.642801i
$166$		0			0
$167$	−5.58235e8	+	5.58235e8i	−0.717714	+	0.717714i	−0.968137	−	0.250423i	$-0.919430\pi$
$167$	−5.58235e8	+	5.58235e8i	−0.717714	+	0.717714i	0.250423	+	0.968137i	$0.419430\pi$
$168$		0			0
$169$			1.09187e9i			1.33852i
$170$		0			0
$171$	5.99447e8			0.701078
$172$		0			0
$173$	−1.00283e8	−	1.00283e8i	−0.111955	−	0.111955i	0.648910	−	0.760865i	$-0.275225\pi$
$173$	−1.00283e8	−	1.00283e8i	−0.111955	−	0.111955i	−0.760865	+	0.648910i	$0.775225\pi$
$174$		0			0
$175$	−4.56662e7	−	6.54373e7i	−0.0486903	−	0.0697708i
$176$		0			0
$177$	−6.16341e8	+	6.16341e8i	−0.627954	+	0.627954i
$178$		0			0
$179$			8.62971e8i			0.840590i	0.907388	+	0.420295i	$0.138073\pi$
$179$			8.62971e8i			0.840590i	−0.907388	+	0.420295i	$0.861927\pi$
$180$		0			0
$181$	1.76106e9			1.64081			0.820407	−	0.571780i	$-0.193747\pi$
$181$	1.76106e9			1.64081			0.820407	+	0.571780i	$0.193747\pi$
$182$		0			0
$183$	−7.66671e8	−	7.66671e8i	−0.683604	−	0.683604i
$184$		0			0
$185$	1.41503e9	+	1.24923e8i	1.20803	+	0.106649i
$186$		0			0
$187$	−6.73082e7	+	6.73082e7i	−0.0550429	+	0.0550429i
$188$		0			0
$189$			1.14437e8i			0.0896852i
$190$		0			0
$191$	−2.10501e9			−1.58169			−0.790843	−	0.612019i	$-0.790358\pi$
$191$	−2.10501e9			−1.58169			−0.790843	+	0.612019i	$0.790358\pi$
$192$		0			0
$193$	−1.38679e9	−	1.38679e9i	−0.999496	−	0.999496i	0.000504122	−	1.00000i	$-0.499840\pi$
$193$	−1.38679e9	−	1.38679e9i	−0.999496	−	0.999496i	−1.00000		0.000504122i	$0.999840\pi$
$194$		0			0
$195$	1.17743e9	−	9.86400e8i	0.814322	−	0.682204i
$196$		0			0
$197$	−1.65638e9	+	1.65638e9i	−1.09976	+	1.09976i	−0.105317	+	0.994439i	$0.533586\pi$
$197$	−1.65638e9	+	1.65638e9i	−1.09976	+	1.09976i	−0.994439	+	0.105317i	$0.966414\pi$
$198$		0			0
$199$			2.74335e8i			0.174932i	0.996168	+	0.0874659i	$0.0278769\pi$
$199$			2.74335e8i			0.174932i	−0.996168	+	0.0874659i	$0.972123\pi$
$200$		0			0
$201$	7.96364e8			0.487896
$202$		0			0
$203$	2.08245e7	+	2.08245e7i	0.0122628	+	0.0122628i
$204$		0			0
$205$	−1.45405e9	−	1.73564e9i	−0.823309	−	0.982753i
$206$		0			0
$207$	7.79042e8	−	7.79042e8i	0.424306	−	0.424306i
$208$		0			0
$209$		−	2.40152e9i		−	1.25864i
$210$		0			0
$211$	1.58670e9			0.800506			0.400253	−	0.916405i	$-0.368922\pi$
$211$	1.58670e9			0.800506			0.400253	+	0.916405i	$0.368922\pi$
$212$		0			0
$213$	−7.33125e8	−	7.33125e8i	−0.356172	−	0.356172i
$214$		0			0
$215$	4.47392e7	−	5.06771e8i	0.0209380	−	0.237169i
$216$		0			0
$217$	2.18072e8	−	2.18072e8i	0.0983471	−	0.0983471i
$218$		0			0
$219$			1.95521e9i			0.849998i
$220$		0			0
$221$	3.05688e8			0.128147
$222$		0			0
$223$	−2.50984e9	−	2.50984e9i	−1.01491	−	1.01491i	−0.999887	−	0.0150215i	$-0.995218\pi$
$223$	−2.50984e9	−	2.50984e9i	−1.01491	−	1.01491i	−0.0150215	−	0.999887i	$-0.504782\pi$
$224$		0			0
$225$	1.30558e9	+	2.32331e8i	0.509417	+	0.0906521i
$226$		0			0
$227$	1.53379e9	−	1.53379e9i	0.577645	−	0.577645i	−0.356609	−	0.934254i	$-0.616067\pi$
$227$	1.53379e9	−	1.53379e9i	0.577645	−	0.577645i	0.934254	+	0.356609i	$0.116067\pi$
$228$		0			0
$229$			1.85362e9i			0.674031i	0.941499	+	0.337015i	$0.109417\pi$
$229$			1.85362e9i			0.674031i	−0.941499	+	0.337015i	$0.890583\pi$
$230$		0			0
$231$	1.56329e8			0.0549026
$232$		0			0
$233$	7.16513e8	+	7.16513e8i	0.243109	+	0.243109i	0.818135	−	0.575026i	$-0.195008\pi$
$233$	7.16513e8	+	7.16513e8i	0.243109	+	0.243109i	−0.575026	+	0.818135i	$0.695008\pi$
$234$		0			0
$235$	2.73987e9	+	2.41884e8i	0.898376	+	0.0793113i
$236$		0			0
$237$	8.66586e8	−	8.66586e8i	0.274674	−	0.274674i
$238$		0			0
$239$		−	3.15340e9i		−	0.966467i	−0.875492	−	0.483233i	$-0.839462\pi$
$239$		−	3.15340e9i		−	0.966467i	0.875492	−	0.483233i	$-0.160538\pi$
$240$		0			0
$241$	1.08337e9			0.321152			0.160576	−	0.987023i	$-0.448665\pi$
$241$	1.08337e9			0.321152			0.160576	+	0.987023i	$0.448665\pi$
$242$		0			0
$243$	−2.23097e9	−	2.23097e9i	−0.639835	−	0.639835i
$244$		0			0
$245$	2.74189e9	−	2.29704e9i	0.761002	−	0.637535i
$246$		0			0
$247$	−5.45339e9	+	5.45339e9i	−1.46514	+	1.46514i
$248$		0			0
$249$		−	3.18456e8i		−	0.0828423i
$250$		0			0
$251$	6.99442e9			1.76221			0.881103	−	0.472925i	$-0.156802\pi$
$251$	6.99442e9			1.76221			0.881103	+	0.472925i	$0.156802\pi$
$252$		0			0
$253$	−3.12102e9	−	3.12102e9i	−0.761753	−	0.761753i
$254$		0			0
$255$	−1.58068e8	−	1.88680e8i	−0.0373838	−	0.0446236i
$256$		0			0
$257$	−2.42952e8	+	2.42952e8i	−0.0556914	+	0.0556914i	−0.734404	−	0.678713i	$-0.762538\pi$
$257$	−2.42952e8	+	2.42952e8i	−0.0556914	+	0.0556914i	0.678713	+	0.734404i	$0.262538\pi$
$258$		0			0
$259$		−	4.64296e8i		−	0.103180i
$260$		0			0
$261$	−4.89418e8			−0.105467
$262$		0			0
$263$	−1.31782e9	−	1.31782e9i	−0.275445	−	0.275445i	0.555843	−	0.831287i	$-0.312396\pi$
$263$	−1.31782e9	−	1.31782e9i	−0.275445	−	0.275445i	−0.831287	+	0.555843i	$0.812396\pi$
$264$		0			0
$265$	6.34905e8	−	7.19170e9i	0.128743	−	1.45830i
$266$		0			0
$267$	2.44509e9	−	2.44509e9i	0.481115	−	0.481115i
$268$		0			0
$269$			8.21589e9i			1.56908i	0.620077	+	0.784541i	$0.287101\pi$
$269$			8.21589e9i			1.56908i	−0.620077	+	0.784541i	$0.712899\pi$
$270$		0			0
$271$	−7.16847e9			−1.32908			−0.664538	−	0.747255i	$-0.731371\pi$
$271$	−7.16847e9			−1.32908			−0.664538	+	0.747255i	$0.731371\pi$
$272$		0			0
$273$	−3.54994e8	−	3.54994e8i	−0.0639103	−	0.0639103i
$274$		0			0
$275$	9.30772e8	−	5.23044e9i	0.162747	−	0.914550i
$276$		0			0
$277$	1.02569e9	−	1.02569e9i	0.174219	−	0.174219i	−0.614611	−	0.788830i	$-0.710687\pi$
$277$	1.02569e9	−	1.02569e9i	0.174219	−	0.174219i	0.788830	+	0.614611i	$0.210687\pi$
$278$		0			0
$279$			5.12514e9i			0.845841i
$280$		0			0
$281$	−6.99805e9			−1.12241			−0.561205	−	0.827677i	$-0.689662\pi$
$281$	−6.99805e9			−1.12241			−0.561205	+	0.827677i	$0.689662\pi$
$282$		0			0
$283$	5.65931e8	+	5.65931e8i	0.0882302	+	0.0882302i	0.749844	−	0.661614i	$-0.230128\pi$
$283$	5.65931e8	+	5.65931e8i	0.0882302	+	0.0882302i	−0.661614	+	0.749844i	$0.730128\pi$
$284$		0			0
$285$	6.18589e9	+	5.46108e8i	0.937610	+	0.0827750i
$286$		0			0
$287$	−5.23295e8	+	5.23295e8i	−0.0771292	+	0.0771292i
$288$		0			0
$289$			6.92677e9i			0.992978i
$290$		0			0
$291$	1.23991e9			0.172910
$292$		0			0
$293$	−4.98809e9	−	4.98809e9i	−0.676805	−	0.676805i	0.282471	−	0.959276i	$-0.408846\pi$
$293$	−4.98809e9	−	4.98809e9i	−0.676805	−	0.676805i	−0.959276	+	0.282471i	$0.908846\pi$
$294$		0			0
$295$	7.42142e9	−	6.21735e9i	0.979939	−	0.820951i
$296$		0			0
$297$	−5.38739e9	+	5.38739e9i	−0.692393	+	0.692393i
$298$		0			0
$299$			1.41745e10i			1.77346i
$300$		0			0
$301$	−1.66280e8			−0.0202569
$302$		0			0
$303$	1.41293e9	+	1.41293e9i	0.167630	+	0.167630i
$304$		0			0
$305$	7.73381e9	+	9.23156e9i	0.893705	+	1.06678i
$306$		0			0
$307$	−1.00289e10	+	1.00289e10i	−1.12901	+	1.12901i	−0.138672	+	0.990338i	$0.544283\pi$
$307$	−1.00289e10	+	1.00289e10i	−1.12901	+	1.12901i	−0.990338	+	0.138672i	$0.955717\pi$
$308$		0			0
$309$		−	2.31974e9i		−	0.254452i
$310$		0			0
$311$	6.55125e9			0.700298			0.350149	−	0.936694i	$-0.386131\pi$
$311$	6.55125e9			0.700298			0.350149	+	0.936694i	$0.386131\pi$
$312$		0			0
$313$	−2.50993e9	−	2.50993e9i	−0.261508	−	0.261508i	0.564159	−	0.825666i	$-0.309201\pi$
$313$	−2.50993e9	−	2.50993e9i	−0.261508	−	0.261508i	−0.825666	+	0.564159i	$0.809201\pi$
$314$		0			0
$315$	3.81159e7	−	4.31747e8i	0.00387137	−	0.0438518i
$316$		0			0
$317$	4.00611e9	−	4.00611e9i	0.396721	−	0.396721i	−0.480354	−	0.877075i	$-0.659492\pi$
$317$	4.00611e9	−	4.00611e9i	0.396721	−	0.396721i	0.877075	+	0.480354i	$0.159492\pi$
$318$		0			0
$319$			1.96072e9i			0.189344i
$320$		0			0
$321$	7.87612e9			0.741809
$322$		0			0
$323$	8.73891e8	+	8.73891e8i	0.0802874	+	0.0802874i
$324$		0			0
$325$	−1.39909e10	+	9.76373e9i	−1.25405	+	0.875150i
$326$		0			0
$327$	−1.22636e9	+	1.22636e9i	−0.107257	+	0.107257i
$328$		0			0
$329$		−	8.98996e8i		−	0.0767316i
$330$		0			0
$331$	−1.60673e10			−1.33854			−0.669271	−	0.743018i	$-0.733394\pi$
$331$	−1.60673e10			−1.33854			−0.669271	+	0.743018i	$0.733394\pi$
$332$		0			0
$333$	5.45594e9	+	5.45594e9i	0.443704	+	0.443704i
$334$		0			0
$335$	−8.81122e9	−	7.77881e8i	−0.699611	−	0.0617638i
$336$		0			0
$337$	−1.41455e10	+	1.41455e10i	−1.09673	+	1.09673i	−0.101938	+	0.994791i	$0.532504\pi$
$337$	−1.41455e10	+	1.41455e10i	−1.09673	+	1.09673i	−0.994791	+	0.101938i	$0.967496\pi$
$338$		0			0
$339$		−	4.83623e9i		−	0.366192i
$340$		0			0
$341$	2.05325e10			1.51853
$342$		0			0
$343$	−1.65939e9	−	1.65939e9i	−0.119887	−	0.119887i
$344$		0			0
$345$	8.74891e9	−	7.32946e9i	0.617558	−	0.517364i
$346$		0			0
$347$	−6.70725e8	+	6.70725e8i	−0.0462622	+	0.0462622i	−0.729859	−	0.683597i	$-0.760415\pi$
$347$	−6.70725e8	+	6.70725e8i	−0.0462622	+	0.0462622i	0.683597	+	0.729859i	$0.260415\pi$
$348$		0			0
$349$			1.51291e10i			1.01979i	0.860236	+	0.509896i	$0.170316\pi$
$349$			1.51291e10i			1.01979i	−0.860236	+	0.509896i	$0.829684\pi$
$350$		0			0
$351$	2.44675e10			1.61198
$352$		0			0
$353$	−3.57220e9	−	3.57220e9i	−0.230058	−	0.230058i	0.582659	−	0.812717i	$-0.302012\pi$
$353$	−3.57220e9	−	3.57220e9i	−0.230058	−	0.230058i	−0.812717	+	0.582659i	$0.802012\pi$
$354$		0			0
$355$	7.39541e9	+	8.82763e9i	0.465639	+	0.555816i
$356$		0			0
$357$	−5.68869e7	+	5.68869e7i	−0.00350219	+	0.00350219i
$358$		0			0
$359$			1.30385e10i			0.784963i	0.919760	+	0.392482i	$0.128383\pi$
$359$			1.30385e10i			0.784963i	−0.919760	+	0.392482i	$0.871617\pi$
$360$		0			0
$361$	−1.41964e10			−0.835891
$362$		0			0
$363$	−1.16942e9	−	1.16942e9i	−0.0673509	−	0.0673509i
$364$		0			0
$365$	1.90983e9	−	2.16331e10i	0.107603	−	1.21884i
$366$		0			0
$367$	2.24627e9	−	2.24627e9i	0.123822	−	0.123822i	−0.642480	−	0.766302i	$-0.722094\pi$
$367$	2.24627e9	−	2.24627e9i	0.123822	−	0.123822i	0.766302	+	0.642480i	$0.222094\pi$
$368$		0			0
$369$		−	1.22985e10i		−	0.663356i
$370$		0			0
$371$	−2.35972e9			−0.124556
$372$		0			0
$373$	−1.21704e10	−	1.21704e10i	−0.628736	−	0.628736i	0.319014	−	0.947750i	$-0.396648\pi$
$373$	−1.21704e10	−	1.21704e10i	−0.628736	−	0.628736i	−0.947750	+	0.319014i	$0.896648\pi$
$374$		0			0
$375$	1.32610e10	+	3.58691e9i	0.670582	+	0.181383i
$376$		0			0
$377$	4.45242e9	−	4.45242e9i	0.220410	−	0.220410i
$378$		0			0
$379$		−	3.40471e10i		−	1.65015i	−0.565024	−	0.825075i	$-0.691133\pi$
$379$		−	3.40471e10i		−	1.65015i	0.565024	−	0.825075i	$-0.308867\pi$
$380$		0			0
$381$	−1.55579e10			−0.738331
$382$		0			0
$383$	−7.51136e9	−	7.51136e9i	−0.349079	−	0.349079i	0.510687	−	0.859766i	$-0.329391\pi$
$383$	−7.51136e9	−	7.51136e9i	−0.349079	−	0.349079i	−0.859766	+	0.510687i	$0.829391\pi$
$384$		0			0
$385$	−1.72968e9	−	1.52701e8i	−0.0787267	−	0.00695022i
$386$		0			0
$387$	1.95396e9	−	1.95396e9i	0.0871107	−	0.0871107i
$388$		0			0
$389$			9.40651e9i			0.410800i	0.978678	+	0.205400i	$0.0658495\pi$
$389$			9.40651e9i			0.410800i	−0.978678	+	0.205400i	$0.934150\pi$
$390$		0			0
$391$	2.27142e9			0.0971831
$392$		0			0
$393$	1.35336e10	+	1.35336e10i	0.567340	+	0.567340i
$394$		0			0
$395$	−1.04346e10	+	8.74170e9i	−0.428637	+	0.359093i
$396$		0			0
$397$	9.53889e9	−	9.53889e9i	0.384004	−	0.384004i	−0.488538	−	0.872542i	$-0.662470\pi$
$397$	9.53889e9	−	9.53889e9i	0.384004	−	0.384004i	0.872542	+	0.488538i	$0.162470\pi$
$398$		0			0
$399$		−	2.02969e9i		−	0.0800827i
$400$		0			0
$401$	1.29102e10			0.499293			0.249646	−	0.968337i	$-0.419686\pi$
$401$	1.29102e10			0.499293			0.249646	+	0.968337i	$0.419686\pi$
$402$		0			0
$403$	−4.66253e10	−	4.66253e10i	−1.76767	−	1.76767i
$404$		0			0
$405$	−3.71219e9	−	4.43110e9i	−0.137978	−	0.164699i
$406$		0			0
$407$	2.18577e10	−	2.18577e10i	0.796577	−	0.796577i
$408$		0			0
$409$		−	3.89504e10i		−	1.39194i	−0.718073	−	0.695968i	$-0.754976\pi$
$409$		−	3.89504e10i		−	1.39194i	0.718073	−	0.695968i	$-0.245024\pi$
$410$		0			0
$411$	−7.79861e9			−0.273307
$412$		0			0
$413$	−2.23755e9	−	2.23755e9i	−0.0769083	−	0.0769083i
$414$		0			0
$415$	−3.11065e8	+	3.52350e9i	−0.0104872	+	0.118790i
$416$		0			0
$417$	1.39852e10	−	1.39852e10i	0.462514	−	0.462514i
$418$		0			0
$419$			6.53853e9i			0.212141i	0.994359	+	0.106070i	$0.0338269\pi$
$419$			6.53853e9i			0.212141i	−0.994359	+	0.106070i	$0.966173\pi$
$420$		0			0
$421$	1.36111e10			0.433277			0.216638	−	0.976252i	$-0.430491\pi$
$421$	1.36111e10			0.433277			0.216638	+	0.976252i	$0.430491\pi$
$422$		0			0
$423$	1.05641e10	+	1.05641e10i	0.329968	+	0.329968i
$424$		0			0
$425$	1.56461e9	+	2.24201e9i	0.0479569	+	0.0687198i
$426$		0			0
$427$	2.78331e9	−	2.78331e9i	0.0837241	−	0.0837241i
$428$		0			0
$429$		−	3.34243e10i		−	0.986808i
$430$		0			0
$431$	2.81277e10			0.815126			0.407563	−	0.913177i	$-0.366379\pi$
$431$	2.81277e10			0.815126			0.407563	+	0.913177i	$0.366379\pi$
$432$		0			0
$433$	−2.09524e9	−	2.09524e9i	−0.0596051	−	0.0596051i	0.676676	−	0.736281i	$-0.263420\pi$
$433$	−2.09524e9	−	2.09524e9i	−0.0596051	−	0.0596051i	−0.736281	+	0.676676i	$0.763420\pi$
$434$		0			0
$435$	−5.05046e9	−	4.45869e8i	−0.141050	−	0.0124523i
$436$		0			0
$437$	−4.05215e10	+	4.05215e10i	−1.11112	+	1.11112i
$438$		0			0
$439$		−	1.07942e10i		−	0.290623i	−0.989386	−	0.145312i	$-0.953581\pi$
$439$		−	1.07942e10i		−	0.290623i	0.989386	−	0.145312i	$-0.0464185\pi$
$440$		0			0
$441$	1.94286e10			0.513674
$442$		0			0
$443$	3.89099e10	+	3.89099e10i	1.01029	+	1.01029i	0.999947	+	0.0103426i	$0.00329222\pi$
$443$	3.89099e10	+	3.89099e10i	1.01029	+	1.01029i	0.0103426	+	0.999947i	$0.496708\pi$
$444$		0			0
$445$	−2.94415e10	+	2.46649e10i	−0.750793	+	0.628983i
$446$		0			0
$447$	−6.82178e9	+	6.82178e9i	−0.170871	+	0.170871i
$448$		0			0
$449$			4.43789e9i			0.109192i	0.998509	+	0.0545961i	$0.0173871\pi$
$449$			4.43789e9i			0.109192i	−0.998509	+	0.0545961i	$0.982613\pi$
$450$		0			0
$451$	−4.92705e10			−1.19092
$452$		0			0
$453$	1.68594e10	+	1.68594e10i	0.400359	+	0.400359i
$454$		0			0
$455$	3.58101e9	+	4.27452e9i	0.0835526	+	0.0997337i
$456$		0			0
$457$	1.68505e10	−	1.68505e10i	0.386321	−	0.386321i	−0.487052	−	0.873373i	$-0.661928\pi$
$457$	1.68505e10	−	1.68505e10i	0.386321	−	0.386321i	0.873373	+	0.487052i	$0.161928\pi$
$458$		0			0
$459$		−	3.92085e9i		−	0.0883343i
$460$		0			0
$461$	3.57328e10			0.791158			0.395579	−	0.918432i	$-0.370544\pi$
$461$	3.57328e10			0.791158			0.395579	+	0.918432i	$0.370544\pi$
$462$		0			0
$463$	6.28237e10	+	6.28237e10i	1.36710	+	1.36710i	0.864550	+	0.502547i	$0.167603\pi$
$463$	6.28237e10	+	6.28237e10i	1.36710	+	1.36710i	0.502547	+	0.864550i	$0.332397\pi$
$464$		0			0
$465$	−4.66911e9	+	5.28880e10i	−0.0998670	+	1.13121i
$466$		0			0
$467$	2.40702e10	−	2.40702e10i	0.506071	−	0.506071i	−0.407247	−	0.913318i	$-0.633511\pi$
$467$	2.40702e10	−	2.40702e10i	0.506071	−	0.506071i	0.913318	+	0.407247i	$0.133511\pi$
$468$		0			0
$469$			2.89111e9i			0.0597548i
$470$		0			0
$471$	3.22392e8			0.00655089
$472$		0			0
$473$	−7.82799e9	−	7.82799e9i	−0.156389	−	0.156389i
$474$		0			0
$475$	−6.79091e10	−	1.20846e10i	−1.33399	−	0.237388i
$476$		0			0
$477$	2.77291e10	−	2.77291e10i	0.535626	−	0.535626i
$478$		0			0
$479$		−	6.79331e10i		−	1.29044i	−0.763995	−	0.645222i	$-0.776765\pi$
$479$		−	6.79331e10i		−	1.29044i	0.763995	−	0.645222i	$-0.223235\pi$
$480$		0			0
$481$	−9.92695e10			−1.85454
$482$		0			0
$483$	−2.63779e9	−	2.63779e9i	−0.0484677	−	0.0484677i
$484$		0			0
$485$	−1.37188e10	−	1.21113e9i	−0.247941	−	0.0218890i
$486$		0			0
$487$	−9.43406e9	+	9.43406e9i	−0.167719	+	0.167719i	−0.785976	−	0.618257i	$-0.787839\pi$
$487$	−9.43406e9	+	9.43406e9i	−0.167719	+	0.167719i	0.618257	+	0.785976i	$0.287839\pi$
$488$		0			0
$489$			1.50800e10i			0.263733i
$490$		0			0
$491$	−5.75323e10			−0.989887			−0.494943	−	0.868925i	$-0.664811\pi$
$491$	−5.75323e10			−0.989887			−0.494943	+	0.868925i	$0.664811\pi$
$492$		0			0
$493$	−7.13488e8	−	7.13488e8i	−0.0120781	−	0.0120781i
$494$		0			0
$495$	2.21198e10	−	1.85311e10i	0.368435	−	0.308659i
$496$		0			0
$497$	2.66153e9	−	2.66153e9i	0.0436220	−	0.0436220i
$498$		0			0
$499$			1.11776e11i			1.80280i	0.432988	+	0.901400i	$0.357459\pi$
$499$			1.11776e11i			1.80280i	−0.432988	+	0.901400i	$0.642541\pi$
$500$		0			0
$501$	−4.44224e10			−0.705101
$502$		0			0
$503$	4.45397e10	+	4.45397e10i	0.695785	+	0.695785i	0.963499	−	0.267714i	$-0.0862681\pi$
$503$	4.45397e10	+	4.45397e10i	0.695785	+	0.695785i	−0.267714	+	0.963499i	$0.586268\pi$
$504$		0			0
$505$	−1.42530e10	−	1.70133e10i	−0.219150	−	0.261591i
$506$		0			0
$507$	−4.34436e10	+	4.34436e10i	−0.657498	+	0.657498i
$508$		0			0
$509$			2.54580e10i			0.379275i	0.981854	+	0.189637i	$0.0607312\pi$
$509$			2.54580e10i			0.379275i	−0.981854	+	0.189637i	$0.939269\pi$
$510$		0			0
$511$	−7.09818e9			−0.104103
$512$		0			0
$513$	6.99469e10	+	6.99469e10i	1.00995	+	1.00995i
$514$		0			0
$515$	−2.26590e9	+	2.56663e10i	−0.0322116	+	0.364867i
$516$		0			0
$517$	4.23222e10	−	4.23222e10i	0.592388	−	0.592388i
$518$		0			0
$519$		−	7.98017e9i		−	0.109987i
$520$		0			0
$521$	7.42766e9			0.100809			0.0504047	−	0.998729i	$-0.483949\pi$
$521$	7.42766e9			0.100809			0.0504047	+	0.998729i	$0.483949\pi$
$522$		0			0
$523$	−5.49185e10	−	5.49185e10i	−0.734027	−	0.734027i	0.237388	−	0.971415i	$-0.423709\pi$
$523$	−5.49185e10	−	5.49185e10i	−0.734027	−	0.734027i	−0.971415	+	0.237388i	$0.923709\pi$
$524$		0			0
$525$	7.86661e8	−	4.42061e9i	0.0103550	−	0.0581896i
$526$		0			0
$527$	−7.47158e9	+	7.47158e9i	−0.0968657	+	0.0968657i
$528$		0			0
$529$			2.70127e10i			0.344942i
$530$		0			0
$531$	5.25871e10			0.661456
$532$		0			0
$533$	1.11884e11	+	1.11884e11i	1.38631	+	1.38631i
$534$		0			0
$535$	−8.71438e10	−	7.69331e9i	−1.06371	−	0.0939071i
$536$		0			0
$537$	−3.43361e10	+	3.43361e10i	−0.412909	+	0.412909i
$538$		0			0
$539$		−	7.78354e10i		−	0.922194i
$540$		0			0
$541$	−1.34348e11			−1.56835			−0.784174	−	0.620540i	$-0.786913\pi$
$541$	−1.34348e11			−1.56835			−0.784174	+	0.620540i	$0.786913\pi$
$542$		0			0
$543$	7.00694e10	+	7.00694e10i	0.805989	+	0.805989i
$544$		0			0
$545$	1.47667e10	−	1.23709e10i	0.167377	−	0.140222i
$546$		0			0
$547$	2.62302e10	−	2.62302e10i	0.292990	−	0.292990i	−0.545271	−	0.838260i	$-0.683573\pi$
$547$	2.62302e10	−	2.62302e10i	0.292990	−	0.292990i	0.838260	+	0.545271i	$0.183573\pi$
$548$		0			0
$549$			6.54135e10i			0.720075i
$550$		0			0
$551$	2.54569e10			0.276184
$552$		0			0
$553$	3.14604e9	+	3.14604e9i	0.0336406	+	0.0336406i
$554$		0			0
$555$	5.13312e10	+	6.12721e10i	0.541015	+	0.645789i
$556$		0			0
$557$	1.10867e11	−	1.10867e11i	1.15181	−	1.15181i	0.165621	−	0.986189i	$-0.447037\pi$
$557$	1.10867e11	−	1.10867e11i	1.15181	−	1.15181i	0.986189	−	0.165621i	$-0.0529629\pi$
$558$		0			0
$559$			3.55518e10i			0.364094i
$560$		0			0
$561$	−5.35615e9			−0.0540756
$562$		0			0
$563$	1.20925e11	+	1.20925e11i	1.20361	+	1.20361i	0.973062	+	0.230543i	$0.0740502\pi$
$563$	1.20925e11	+	1.20925e11i	1.20361	+	1.20361i	0.230543	+	0.973062i	$0.425950\pi$
$564$		0			0
$565$	−4.72398e9	+	5.35096e10i	−0.0463569	+	0.525095i
$566$		0			0
$567$	−1.33597e9	+	1.33597e9i	−0.0129260	+	0.0129260i
$568$		0			0
$569$		−	6.53746e10i		−	0.623677i	−0.950135	−	0.311839i	$-0.899055\pi$
$569$		−	6.53746e10i		−	0.623677i	0.950135	−	0.311839i	$-0.100945\pi$
$570$		0			0
$571$	5.53735e10			0.520904			0.260452	−	0.965487i	$-0.416128\pi$
$571$	5.53735e10			0.520904			0.260452	+	0.965487i	$0.416128\pi$
$572$		0			0
$573$	−8.37546e10	−	8.37546e10i	−0.776945	−	0.776945i
$574$		0			0
$575$	−1.03960e11	+	7.25496e10i	−0.951031	+	0.663688i
$576$		0			0
$577$	−5.29155e10	+	5.29155e10i	−0.477397	+	0.477397i	−0.904298	−	0.426901i	$-0.859605\pi$
$577$	−5.29155e10	+	5.29155e10i	−0.477397	+	0.477397i	0.426901	+	0.904298i	$0.359605\pi$
$578$		0			0
$579$		−	1.10356e11i		−	0.981931i
$580$		0			0
$581$	1.15612e9			0.0101461
$582$		0			0
$583$	−1.11089e11	−	1.11089e11i	−0.961604	−	0.961604i
$584$		0			0
$585$	−9.23105e10	−	8.14944e9i	−0.788184	−	0.0695832i
$586$		0			0
$587$	1.57769e11	−	1.57769e11i	1.32883	−	1.32883i	0.422445	−	0.906389i	$-0.361172\pi$
$587$	1.57769e11	−	1.57769e11i	1.32883	−	1.32883i	0.906389	−	0.422445i	$-0.138828\pi$
$588$		0			0
$589$		−	2.66582e11i		−	2.21498i
$590$		0			0
$591$	−1.31809e11			−1.08043
$592$		0			0
$593$	−3.09235e10	−	3.09235e10i	−0.250075	−	0.250075i	0.570927	−	0.821001i	$-0.306584\pi$
$593$	−3.09235e10	−	3.09235e10i	−0.250075	−	0.250075i	−0.821001	+	0.570927i	$0.806584\pi$
$594$		0			0
$595$	6.84980e8	−	5.73847e8i	0.00546525	−	0.00457856i
$596$		0			0
$597$	−1.09153e10	+	1.09153e10i	−0.0859288	+	0.0859288i
$598$		0			0
$599$		−	1.42432e11i		−	1.10637i	−0.833059	−	0.553184i	$-0.813413\pi$
$599$		−	1.42432e11i		−	1.10637i	0.833059	−	0.553184i	$-0.186587\pi$
$600$		0			0
$601$	9.37468e10			0.718553			0.359276	−	0.933231i	$-0.383024\pi$
$601$	9.37468e10			0.718553			0.359276	+	0.933231i	$0.383024\pi$
$602$		0			0
$603$	−3.39735e10	−	3.39735e10i	−0.256963	−	0.256963i
$604$		0			0
$605$	1.17965e10	+	1.40811e10i	0.0880507	+	0.105103i
$606$		0			0
$607$	1.60965e10	−	1.60965e10i	0.118571	−	0.118571i	−0.645332	−	0.763902i	$-0.723281\pi$
$607$	1.60965e10	−	1.60965e10i	0.118571	−	0.118571i	0.763902	+	0.645332i	$0.223281\pi$
$608$		0			0
$609$			1.65714e9i			0.0120473i
$610$		0			0
$611$	−1.92211e11			−1.37916
$612$		0			0
$613$	8.07997e10	+	8.07997e10i	0.572226	+	0.572226i	0.932750	−	0.360524i	$-0.117402\pi$
$613$	8.07997e10	+	8.07997e10i	0.572226	+	0.572226i	−0.360524	+	0.932750i	$0.617402\pi$
$614$		0			0
$615$	1.12042e10	−	1.26912e11i	0.0783212	−	0.887161i
$616$		0			0
$617$	1.53536e11	−	1.53536e11i	1.05942	−	1.05942i	0.0613041	−	0.998119i	$-0.480474\pi$
$617$	1.53536e11	−	1.53536e11i	1.05942	−	1.05942i	0.998119	−	0.0613041i	$-0.0195260\pi$
$618$		0			0
$619$			4.56835e10i			0.311169i	0.987823	+	0.155585i	$0.0497261\pi$
$619$			4.56835e10i			0.311169i	−0.987823	+	0.155585i	$0.950274\pi$
$620$		0			0
$621$	1.81806e11			1.22248
$622$		0			0
$623$	8.87661e9	+	8.87661e9i	0.0589244	+	0.0589244i
$624$		0			0
$625$	−1.43220e11	−	5.26399e10i	−0.938610	−	0.344981i
$626$		0			0
$627$	9.55522e10	−	9.55522e10i	0.618259	−	0.618259i
$628$		0			0
$629$			1.59077e10i			0.101626i
$630$		0			0
$631$	−6.87475e10			−0.433650			−0.216825	−	0.976210i	$-0.569570\pi$
$631$	−6.87475e10			−0.433650			−0.216825	+	0.976210i	$0.569570\pi$
$632$		0			0
$633$	6.31320e10	+	6.31320e10i	0.393219	+	0.393219i
$634$		0			0
$635$	1.72137e11	+	1.51968e10i	1.05872	+	0.0934668i
$636$		0			0
$637$	−1.76749e11	+	1.76749e11i	−1.07350	+	1.07350i
$638$		0			0
$639$			6.25513e10i			0.375174i
$640$		0			0
$641$	9.51984e9			0.0563894			0.0281947	−	0.999602i	$-0.491024\pi$
$641$	9.51984e9			0.0563894			0.0281947	+	0.999602i	$0.491024\pi$
$642$		0			0
$643$	9.40725e10	+	9.40725e10i	0.550325	+	0.550325i	0.926534	−	0.376210i	$-0.122773\pi$
$643$	9.40725e10	+	9.40725e10i	0.550325	+	0.550325i	−0.376210	+	0.926534i	$0.622773\pi$
$644$		0			0
$645$	2.19436e10	−	1.83834e10i	0.126785	−	0.106215i
$646$		0			0
$647$	−1.03789e11	+	1.03789e11i	−0.592289	+	0.592289i	−0.938249	−	0.345960i	$-0.887553\pi$
$647$	−1.03789e11	+	1.03789e11i	−0.592289	+	0.592289i	0.345960	+	0.938249i	$0.387553\pi$
$648$		0			0
$649$		−	2.10676e11i		−	1.18750i
$650$		0			0
$651$	1.73534e10			0.0966187
$652$		0			0
$653$	−1.14827e11	−	1.14827e11i	−0.631527	−	0.631527i	0.316924	−	0.948451i	$-0.397350\pi$
$653$	−1.14827e11	−	1.14827e11i	−0.631527	−	0.631527i	−0.948451	+	0.316924i	$0.897350\pi$
$654$		0			0
$655$	−1.36521e11	−	1.62959e11i	−0.741708	−	0.885349i
$656$		0			0
$657$	8.34108e10	−	8.34108e10i	0.447673	−	0.447673i
$658$		0			0
$659$			1.16978e10i			0.0620246i	0.999519	+	0.0310123i	$0.00987310\pi$
$659$			1.16978e10i			0.0620246i	−0.999519	+	0.0310123i	$0.990127\pi$
$660$		0			0
$661$	−1.06677e11			−0.558809			−0.279404	−	0.960174i	$-0.590137\pi$
$661$	−1.06677e11			−0.558809			−0.279404	+	0.960174i	$0.590137\pi$
$662$		0			0
$663$	1.21628e10	+	1.21628e10i	0.0629476	+	0.0629476i
$664$		0			0
$665$	−1.98258e9	+	2.24571e10i	−0.0101378	+	0.114833i
$666$		0			0
$667$	3.30838e10	−	3.30838e10i	0.167152	−	0.167152i
$668$		0			0
$669$		−	1.99724e11i		−	0.997073i
$670$		0			0
$671$	2.62061e11			1.29274
$672$		0			0
$673$	5.13966e10	+	5.13966e10i	0.250539	+	0.250539i	0.821191	−	0.570653i	$-0.193310\pi$
$673$	5.13966e10	+	5.13966e10i	0.250539	+	0.250539i	−0.570653	+	0.821191i	$0.693310\pi$
$674$		0			0
$675$	1.25233e11	+	1.79452e11i	0.603257	+	0.864438i
$676$		0			0
$677$	−2.46063e11	+	2.46063e11i	−1.17136	+	1.17136i	−0.189479	+	0.981885i	$0.560680\pi$
$677$	−2.46063e11	+	2.46063e11i	−1.17136	+	1.17136i	−0.981885	+	0.189479i	$0.939320\pi$
$678$		0			0
$679$			4.50136e9i			0.0211770i
$680$		0			0
$681$	1.22053e11			0.567494
$682$		0			0
$683$	−1.11448e11	−	1.11448e11i	−0.512142	−	0.512142i	0.403040	−	0.915182i	$-0.367953\pi$
$683$	−1.11448e11	−	1.11448e11i	−0.512142	−	0.512142i	−0.915182	+	0.403040i	$0.867953\pi$
$684$		0			0
$685$	8.62863e10	+	7.61760e9i	0.391904	+	0.0345984i
$686$		0			0
$687$	−7.37525e10	+	7.37525e10i	−0.331093	+	0.331093i
$688$		0			0
$689$			5.04523e11i			2.23874i
$690$		0			0
$691$	2.73439e11			1.19936			0.599678	−	0.800241i	$-0.295295\pi$
$691$	2.73439e11			1.19936			0.599678	+	0.800241i	$0.295295\pi$
$692$		0			0
$693$	−6.66912e9	−	6.66912e9i	−0.0289158	−	0.0289158i
$694$		0			0
$695$	−1.68397e11	+	1.41076e11i	−0.721766	+	0.604664i
$696$		0			0
$697$	1.79291e10	−	1.79291e10i	0.0759674	−	0.0759674i
$698$		0			0
$699$			5.70176e10i			0.238836i
$700$		0			0
$701$	7.12259e10			0.294962			0.147481	−	0.989065i	$-0.452883\pi$
$701$	7.12259e10			0.294962			0.147481	+	0.989065i	$0.452883\pi$
$702$		0			0
$703$	−2.83789e11	−	2.83789e11i	−1.16191	−	1.16191i
$704$		0			0
$705$	9.93904e10	+	1.18639e11i	0.402335	+	0.480253i
$706$		0			0
$707$	−5.12950e9	+	5.12950e9i	−0.0205304	+	0.0205304i
$708$		0			0
$709$			3.40136e11i			1.34607i	0.739610	+	0.673036i	$0.235010\pi$
$709$			3.40136e11i			1.34607i	−0.739610	+	0.673036i	$0.764990\pi$
$710$		0			0
$711$	−7.39383e10			−0.289328
$712$		0			0
$713$	−3.46450e11	−	3.46450e11i	−1.34055	−	1.34055i
$714$		0			0
$715$	−3.26485e10	+	3.69816e11i	−0.124922	+	1.41502i
$716$		0			0
$717$	1.25468e11	−	1.25468e11i	0.474741	−	0.474741i
$718$		0			0
$719$		−	3.73670e10i		−	0.139821i	−0.997553	−	0.0699105i	$-0.977729\pi$
$719$		−	3.73670e10i		−	0.139821i	0.997553	−	0.0699105i	$-0.0222714\pi$
$720$		0			0
$721$	8.42155e9			0.0311639
$722$		0			0
$723$	4.31056e10	+	4.31056e10i	0.157754	+	0.157754i
$724$		0			0
$725$	5.54443e10	+	9.86648e9i	0.200680	+	0.0357117i
$726$		0			0
$727$	−2.20574e11	+	2.20574e11i	−0.789618	+	0.789618i	−0.981431	−	0.191814i	$-0.938563\pi$
$727$	−2.20574e11	+	2.20574e11i	−0.789618	+	0.789618i	0.191814	+	0.981431i	$0.438563\pi$
$728$		0			0
$729$		−	2.38215e11i		−	0.843448i
$730$		0			0
$731$	5.69708e9			0.0199518
$732$		0			0
$733$	2.40945e11	+	2.40945e11i	0.834646	+	0.834646i	0.988148	−	0.153502i	$-0.0490552\pi$
$733$	2.40945e11	+	2.40945e11i	0.834646	+	0.834646i	−0.153502	+	0.988148i	$0.549055\pi$
$734$		0			0
$735$	2.00490e11	+	1.76999e10i	0.686980	+	0.0606486i
$736$		0			0
$737$	−1.36105e11	+	1.36105e11i	−0.461323	+	0.461323i
$738$		0			0
$739$			1.27384e11i			0.427106i	0.976931	+	0.213553i	$0.0685037\pi$
$739$			1.27384e11i			0.427106i	−0.976931	+	0.213553i	$0.931496\pi$
$740$		0			0
$741$	−4.33962e11			−1.43939
$742$		0			0
$743$	2.08172e11	+	2.08172e11i	0.683072	+	0.683072i	0.960691	−	0.277619i	$-0.0895452\pi$
$743$	2.08172e11	+	2.08172e11i	0.683072	+	0.683072i	−0.277619	+	0.960691i	$0.589545\pi$
$744$		0			0
$745$	8.21417e10	−	6.88148e10i	0.266648	−	0.223387i
$746$		0			0
$747$	−1.35856e10	+	1.35856e10i	−0.0436310	+	0.0436310i
$748$		0			0
$749$			2.85933e10i			0.0908527i
$750$		0			0
$751$	8.78634e10			0.276216			0.138108	−	0.990417i	$-0.455898\pi$
$751$	8.78634e10			0.276216			0.138108	+	0.990417i	$0.455898\pi$
$752$		0			0
$753$	2.78296e11	+	2.78296e11i	0.865618	+	0.865618i
$754$		0			0
$755$	−1.70070e11	−	2.03006e11i	−0.523406	−	0.624771i
$756$		0			0
$757$	1.18459e11	−	1.18459e11i	0.360733	−	0.360733i	−0.503350	−	0.864083i	$-0.667899\pi$
$757$	1.18459e11	−	1.18459e11i	0.360733	−	0.360733i	0.864083	+	0.503350i	$0.167899\pi$
$758$		0			0
$759$		−	2.48360e11i		−	0.748366i
$760$		0			0
$761$	−5.02471e10			−0.149821			−0.0749105	−	0.997190i	$-0.523867\pi$
$761$	−5.02471e10			−0.149821			−0.0749105	+	0.997190i	$0.523867\pi$
$762$		0			0
$763$	−4.45214e9	−	4.45214e9i	−0.0131362	−	0.0131362i
$764$		0			0
$765$	−1.30593e9	+	1.47925e10i	−0.00381305	+	0.0431913i
$766$		0			0
$767$	−4.78404e11	+	4.78404e11i	−1.38234	+	1.38234i
$768$		0			0
$769$			6.28311e11i			1.79668i	0.439306	+	0.898338i	$0.355224\pi$
$769$			6.28311e11i			1.79668i	−0.439306	+	0.898338i	$0.644776\pi$
$770$		0			0
$771$	−1.93333e10			−0.0547126
$772$		0			0
$773$	−7.85266e10	−	7.85266e10i	−0.219937	−	0.219937i	0.588535	−	0.808472i	$-0.299705\pi$
$773$	−7.85266e10	−	7.85266e10i	−0.219937	−	0.219937i	−0.808472	+	0.588535i	$0.799705\pi$
$774$		0			0
$775$	1.03321e11	−	5.80608e11i	0.286405	−	1.60945i
$776$		0			0
$777$	1.84735e10	−	1.84735e10i	0.0506834	−	0.0506834i
$778$		0			0
$779$			6.39701e11i			1.73711i
$780$		0			0
$781$	2.50594e11			0.673546
$782$		0			0
$783$	−5.71081e10	−	5.71081e10i	−0.151932	−	0.151932i
$784$		0			0
$785$	−3.56704e9	−	3.14909e8i	−0.00939355	−	0.000829291i
$786$		0			0
$787$	3.23753e11	−	3.23753e11i	0.843946	−	0.843946i	−0.145423	−	0.989370i	$-0.546454\pi$
$787$	3.23753e11	−	3.23753e11i	0.843946	−	0.843946i	0.989370	+	0.145423i	$0.0464544\pi$
$788$		0			0
$789$		−	1.04868e11i		−	0.270604i
$790$		0			0
$791$	1.75574e10			0.0448491
$792$		0			0
$793$	−5.95090e11	−	5.95090e11i	−1.50484	−	1.50484i
$794$		0			0
$795$	3.11407e11	−	2.60883e11i	0.779578	−	0.653098i
$796$		0			0
$797$	2.62197e10	−	2.62197e10i	0.0649821	−	0.0649821i	−0.673869	−	0.738851i	$-0.735369\pi$
$797$	2.62197e10	−	2.62197e10i	0.0649821	−	0.0649821i	0.738851	+	0.673869i	$0.235369\pi$
$798$		0			0
$799$			3.08014e10i			0.0755758i
$800$		0			0
$801$	−2.08618e11			−0.506783
$802$		0			0
$803$	−3.34162e11	−	3.34162e11i	−0.803703	−	0.803703i
$804$		0			0
$805$	2.66088e10	+	3.17619e10i	0.0633638	+	0.0756350i
$806$		0			0
$807$	−3.26896e11	+	3.26896e11i	−0.770753	+	0.770753i
$808$		0			0
$809$			1.16592e11i			0.272193i	0.990696	+	0.136096i	$0.0434557\pi$
$809$			1.16592e11i			0.272193i	−0.990696	+	0.136096i	$0.956544\pi$
$810$		0			0
$811$	−7.33939e10			−0.169659			−0.0848294	−	0.996395i	$-0.527035\pi$
$811$	−7.33939e10			−0.169659			−0.0848294	+	0.996395i	$0.527035\pi$
$812$		0			0
$813$	−2.85221e11	−	2.85221e11i	−0.652859	−	0.652859i
$814$		0			0
$815$	1.47300e10	−	1.66849e11i	0.0333865	−	0.378176i
$816$		0			0
$817$	−1.01634e11	+	1.01634e11i	−0.228114	+	0.228114i
$818$		0			0
$819$			3.02886e10i			0.0673199i
$820$		0			0
$821$	3.89632e11			0.857594			0.428797	−	0.903401i	$-0.358938\pi$
$821$	3.89632e11			0.857594			0.428797	+	0.903401i	$0.358938\pi$
$822$		0			0
$823$	6.47224e11	+	6.47224e11i	1.41077	+	1.41077i	0.754714	+	0.656054i	$0.227776\pi$
$823$	6.47224e11	+	6.47224e11i	1.41077	+	1.41077i	0.656054	+	0.754714i	$0.272224\pi$
$824$		0			0
$825$	2.45144e11	−	1.71076e11i	0.529182	−	0.369296i
$826$		0			0
$827$	5.28597e11	−	5.28597e11i	1.13006	−	1.13006i	0.139896	−	0.990166i	$-0.455323\pi$
$827$	5.28597e11	−	5.28597e11i	1.13006	−	1.13006i	0.990166	−	0.139896i	$-0.0446770\pi$
$828$		0			0
$829$		−	4.48986e11i		−	0.950637i	−0.879814	−	0.475319i	$-0.842333\pi$
$829$		−	4.48986e11i		−	0.950637i	0.879814	−	0.475319i	$-0.157667\pi$
$830$		0			0
$831$	8.16205e10			0.171157
$832$		0			0
$833$	2.83236e10	+	2.83236e10i	0.0588259	+	0.0588259i
$834$		0			0
$835$	4.91503e11	+	4.33914e10i	1.01107	+	0.0892601i
$836$		0			0
$837$	−5.98031e11	+	5.98031e11i	−1.21849	+	1.21849i
$838$		0			0
$839$			3.12579e11i			0.630829i	0.948954	+	0.315415i	$0.102144\pi$
$839$			3.12579e11i			0.630829i	−0.948954	+	0.315415i	$0.897856\pi$
$840$		0			0
$841$	4.79462e11			0.958452
$842$		0			0
$843$	−2.78440e11	−	2.78440e11i	−0.551343	−	0.551343i
$844$		0			0
$845$	5.23109e11	−	4.38238e11i	1.02604	−	0.859575i
$846$		0			0
$847$	4.24544e9	−	4.24544e9i	0.00824877	−	0.00824877i
$848$		0			0
$849$			4.50348e10i			0.0866797i
$850$		0			0
$851$	−7.37624e11			−1.40643
$852$		0			0
$853$	9.76699e10	+	9.76699e10i	0.184487	+	0.184487i	0.793308	−	0.608821i	$-0.208357\pi$
$853$	9.76699e10	+	9.76699e10i	0.184487	+	0.184487i	−0.608821	+	0.793308i	$0.708357\pi$
$854$		0			0
$855$	−2.40597e11	−	2.87192e11i	−0.450221	−	0.537412i
$856$		0			0
$857$	3.15076e11	−	3.15076e11i	0.584107	−	0.584107i	−0.351923	−	0.936029i	$-0.614472\pi$
$857$	3.15076e11	−	3.15076e11i	0.584107	−	0.584107i	0.936029	+	0.351923i	$0.114472\pi$
$858$		0			0
$859$		−	5.11155e10i		−	0.0938815i	−0.998898	−	0.0469408i	$-0.985053\pi$
$859$		−	5.11155e10i		−	0.0938815i	0.998898	−	0.0469408i	$-0.0149472\pi$
$860$		0			0
$861$	−4.16420e10			−0.0757738
$862$		0			0
$863$	−4.56565e11	−	4.56565e11i	−0.823113	−	0.823113i	0.163440	−	0.986553i	$-0.447741\pi$
$863$	−4.56565e11	−	4.56565e11i	−0.823113	−	0.823113i	−0.986553	+	0.163440i	$0.947741\pi$
$864$		0			0
$865$	−7.79495e9	+	8.82951e10i	−0.0139235	+	0.157715i
$866$		0			0
$867$	−2.75604e11	+	2.75604e11i	−0.487764	+	0.487764i
$868$		0			0
$869$			2.96213e11i			0.519428i
$870$		0			0
$871$	6.18138e11			1.07402
$872$		0			0
$873$	−5.28956e10	−	5.28956e10i	−0.0910673	−	0.0910673i
$874$		0			0
$875$	−1.30219e10	+	4.81426e10i	−0.0222147	+	0.0821292i
$876$		0			0
$877$	4.67465e11	−	4.67465e11i	0.790225	−	0.790225i	−0.191306	−	0.981530i	$-0.561272\pi$
$877$	4.67465e11	−	4.67465e11i	0.790225	−	0.790225i	0.981530	+	0.191306i	$0.0612722\pi$
$878$		0			0
$879$		−	3.96935e11i		−	0.664911i
$880$		0			0
$881$	8.25600e10			0.137046			0.0685229	−	0.997650i	$-0.478171\pi$
$881$	8.25600e10			0.137046			0.0685229	+	0.997650i	$0.478171\pi$
$882$		0			0
$883$	−6.74600e11	−	6.74600e11i	−1.10969	−	1.10969i	−0.993190	−	0.116504i	$-0.962831\pi$
$883$	−6.74600e11	−	6.74600e11i	−1.10969	−	1.10969i	−0.116504	−	0.993190i	$-0.537169\pi$
$884$		0			0
$885$	5.42663e11	+	4.79079e10i	0.884620	+	0.0780969i
$886$		0			0
$887$	6.85055e11	−	6.85055e11i	1.10670	−	1.10670i	0.113122	−	0.993581i	$-0.463915\pi$
$887$	6.85055e11	−	6.85055e11i	1.10670	−	1.10670i	0.993581	−	0.113122i	$-0.0360850\pi$
$888$		0			0
$889$		−	5.64812e10i		−	0.0904267i
$890$		0			0
$891$	−1.25788e11			−0.199585
$892$		0			0
$893$	−5.49488e11	−	5.49488e11i	−0.864077	−	0.864077i
$894$		0			0
$895$	4.13445e11	−	3.46366e11i	0.644355	−	0.539813i
$896$		0			0
$897$	−5.63977e11	+	5.63977e11i	−0.871148	+	0.871148i
$898$		0			0
$899$			2.17651e11i			0.333212i
$900$		0			0
$901$	8.08485e10			0.122680
$902$		0			0
$903$	−6.61599e9	−	6.61599e9i	−0.00995048	−	0.00995048i
$904$		0			0
$905$	−7.06827e11	−	8.43713e11i	−1.05370	−	1.25777i
$906$		0			0
$907$	−7.09456e11	+	7.09456e11i	−1.04833	+	1.04833i	−0.0495540	+	0.998771i	$0.515780\pi$
$907$	−7.09456e11	+	7.09456e11i	−1.04833	+	1.04833i	−0.998771	+	0.0495540i	$0.984220\pi$
$908$		0			0
$909$		−	1.20554e11i		−	0.176573i
$910$		0			0
$911$	1.23544e12			1.79370			0.896849	−	0.442338i	$-0.145851\pi$
$911$	1.23544e12			1.79370			0.896849	+	0.442338i	$0.145851\pi$
$912$		0			0
$913$	5.44268e10	+	5.44268e10i	0.0783303	+	0.0783303i
$914$		0			0
$915$	−5.95930e10	+	6.75022e11i	−0.0850180	+	0.963017i
$916$		0			0
$917$	−4.91322e10	+	4.91322e10i	−0.0694846	+	0.0694846i
$918$		0			0
$919$		−	2.44254e10i		−	0.0342437i	−0.999853	−	0.0171218i	$-0.994550\pi$
$919$		−	2.44254e10i		−	0.0342437i	0.999853	−	0.0171218i	$-0.00545031\pi$
$920$		0			0
$921$	−7.98061e11			−1.10917
$922$		0			0
$923$	−5.69052e11	−	5.69052e11i	−0.784053	−	0.784053i
$924$		0			0
$925$	−5.08094e11	−	7.28073e11i	−0.694028	−	0.994508i
$926$		0			0
$927$	−9.89618e10	+	9.89618e10i	−0.134014	+	0.134014i
$928$		0			0
$929$		−	6.56877e11i		−	0.881904i	−0.897531	−	0.440952i	$-0.854641\pi$
$929$		−	6.56877e11i		−	0.881904i	0.897531	−	0.440952i	$-0.145359\pi$
$930$		0			0
$931$	−1.01057e12			−1.34514
$932$		0			0
$933$	2.60663e11	+	2.60663e11i	0.343995	+	0.343995i
$934$		0			0
$935$	5.92621e10	+	5.23183e9i	0.0775408	+	0.00684553i
$936$		0			0
$937$	5.13469e11	−	5.13469e11i	0.666126	−	0.666126i	−0.290691	−	0.956817i	$-0.593885\pi$
$937$	5.13469e11	−	5.13469e11i	0.666126	−	0.666126i	0.956817	+	0.290691i	$0.0938852\pi$
$938$		0			0
$939$		−	1.99732e11i		−	0.256912i
$940$		0			0
$941$	−9.79703e11			−1.24950			−0.624750	−	0.780825i	$-0.714799\pi$
$941$	−9.79703e11			−1.24950			−0.624750	+	0.780825i	$0.714799\pi$
$942$		0			0
$943$	8.31356e11	+	8.31356e11i	1.05133	+	1.05133i
$944$		0			0
$945$	5.48263e10	−	4.59312e10i	0.0687483	−	0.0575944i
$946$		0			0
$947$	6.56873e11	−	6.56873e11i	0.816736	−	0.816736i	−0.168898	−	0.985634i	$-0.554021\pi$
$947$	6.56873e11	−	6.56873e11i	0.816736	−	0.816736i	0.985634	+	0.168898i	$0.0540208\pi$
$948$		0			0
$949$			1.51764e12i			1.87113i
$950$		0			0
$951$	3.18792e11			0.389749
$952$		0			0
$953$	1.07764e12	+	1.07764e12i	1.30647	+	1.30647i	0.923942	+	0.382532i	$0.124948\pi$
$953$	1.07764e12	+	1.07764e12i	1.30647	+	1.30647i	0.382532	+	0.923942i	$0.375052\pi$
$954$		0			0
$955$	8.44876e11	+	1.00850e12i	1.01573	+	1.21244i
$956$		0			0
$957$	−7.80135e10	+	7.80135e10i	−0.0930084	+	0.0930084i
$958$		0			0
$959$		−	2.83120e10i		−	0.0334731i
$960$		0			0
$961$	1.42633e12			1.67234
$962$		0			0
$963$	−3.36001e11	−	3.36001e11i	−0.390693	−	0.390693i
$964$		0			0
$965$	−1.07794e11	+	1.22101e12i	−0.124305	+	1.40802i
$966$		0			0
$967$	−1.50501e11	+	1.50501e11i	−0.172121	+	0.172121i	−0.787910	−	0.615790i	$-0.788837\pi$
$967$	−1.50501e11	+	1.50501e11i	−0.172121	+	0.172121i	0.615790	+	0.787910i	$0.288837\pi$
$968$		0			0
$969$			6.95412e10i			0.0788764i
$970$		0			0
$971$	−9.35157e11			−1.05198			−0.525990	−	0.850491i	$-0.676305\pi$
$971$	−9.35157e11			−1.05198			−0.525990	+	0.850491i	$0.676305\pi$
$972$		0			0
$973$	5.07717e10	+	5.07717e10i	0.0566462	+	0.0566462i
$974$		0			0
$975$	−9.45157e11	−	1.68193e11i	−1.04589	−	0.186119i
$976$		0			0
$977$	−1.02379e12	+	1.02379e12i	−1.12365	+	1.12365i	−0.132465	+	0.991188i	$0.542289\pi$
$977$	−1.02379e12	+	1.02379e12i	−1.12365	+	1.12365i	−0.991188	+	0.132465i	$0.957711\pi$
$978$		0			0
$979$			8.35771e11i			0.909823i
$980$		0			0
$981$	1.04634e11			0.112979
$982$		0			0
$983$	2.56277e11	+	2.56277e11i	0.274470	+	0.274470i	0.830897	−	0.556426i	$-0.187828\pi$
$983$	2.56277e11	+	2.56277e11i	0.274470	+	0.274470i	−0.556426	+	0.830897i	$0.687828\pi$
$984$		0			0
$985$	1.45838e12	+	1.28750e11i	1.54926	+	0.136774i
$986$		0			0
$987$	3.57695e10	−	3.57695e10i	0.0376916	−	0.0376916i
$988$		0			0
$989$			2.64168e11i			0.276118i
$990$		0			0
$991$	1.19873e12			1.24288			0.621439	−	0.783463i	$-0.286548\pi$
$991$	1.19873e12			1.24288			0.621439	+	0.783463i	$0.286548\pi$
$992$		0			0
$993$	−6.39292e11	−	6.39292e11i	−0.657509	−	0.657509i
$994$		0			0
$995$	1.31432e11	−	1.10108e11i	0.134094	−	0.112338i
$996$		0			0
$997$	7.46399e10	−	7.46399e10i	0.0755424	−	0.0755424i	−0.668326	−	0.743868i	$-0.732989\pi$
$997$	7.46399e10	−	7.46399e10i	0.0755424	−	0.0755424i	0.743868	+	0.668326i	$0.232989\pi$
$998$		0			0
$999$			1.27326e12i			1.27837i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	80.9.p.a.33.2		4
4.3	odd	2		10.9.c.b.3.1	✓	4
5.2	odd	4	inner	80.9.p.a.17.2		4
12.11	even	2		90.9.g.a.73.2		4
20.3	even	4		50.9.c.b.7.2		4
20.7	even	4		10.9.c.b.7.1	yes	4
20.19	odd	2		50.9.c.b.43.2		4
60.47	odd	4		90.9.g.a.37.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
10.9.c.b.3.1	✓	4	4.3	odd	2
10.9.c.b.7.1	yes	4	20.7	even	4
50.9.c.b.7.2		4	20.3	even	4
50.9.c.b.43.2		4	20.19	odd	2
80.9.p.a.17.2		4	5.2	odd	4	inner
80.9.p.a.33.2		4	1.1	even	1	trivial
90.9.g.a.37.2		4	60.47	odd	4
90.9.g.a.73.2		4	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 80 = 2^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	80.p (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(39.7883 + 39.7883i) q^{3} +(-401.365 - 479.094i) q^{5} +(-144.447 + 144.447i) q^{7} -3394.79i q^{9} +O(q^{10})\)	\(q+(39.7883 + 39.7883i) q^{3} +(-401.365 - 479.094i) q^{5} +(-144.447 + 144.447i) q^{7} -3394.79i q^{9} -13600.3 q^{11} +(30883.7 + 30883.7i) q^{13} +(3092.72 - 35031.9i) q^{15} +(4949.02 - 4949.02i) q^{17} +176579. i q^{19} -11494.6 q^{21} +(229482. + 229482. i) q^{23} +(-68437.7 + 384583. i) q^{25} +(396123. - 396123. i) q^{27} -144167. i q^{29} -1.50971e6 q^{31} +(-541132. - 541132. i) q^{33} +(127179. + 11227.8i) q^{35} +(-1.60715e6 + 1.60715e6i) q^{37} +2.45761e6i q^{39} +3.62276e6 q^{41} +(575576. + 575576. i) q^{43} +(-1.62642e6 + 1.36255e6i) q^{45} +(-3.11186e6 + 3.11186e6i) q^{47} +5.72307e6i q^{49} +393826. q^{51} +(8.16813e6 + 8.16813e6i) q^{53} +(5.45868e6 + 6.51582e6i) q^{55} +(-7.02575e6 + 7.02575e6i) q^{57} +1.54905e7i q^{59} -1.92688e7 q^{61} +(490366. + 490366. i) q^{63} +(2.40057e6 - 2.71918e7i) q^{65} +(1.00075e7 - 1.00075e7i) q^{67} +1.82614e7i q^{69} -1.84257e7 q^{71} +(2.45702e7 + 2.45702e7i) q^{73} +(-1.80249e7 + 1.25789e7i) q^{75} +(1.96452e6 - 1.96452e6i) q^{77} -2.17799e7i q^{79} +9.24891e6 q^{81} +(-4.00189e6 - 4.00189e6i) q^{83} +(-4.35741e6 - 384685. i) q^{85} +(5.73617e6 - 5.73617e6i) q^{87} -6.14525e7i q^{89} -8.92209e6 q^{91} +(-6.00686e7 - 6.00686e7i) q^{93} +(8.45978e7 - 7.08724e7i) q^{95} +(1.55814e7 - 1.55814e7i) q^{97} +4.61701e7i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 86 q^{3} - 870 q^{5} - 5726 q^{7}+O(q^{10}) \) 4 * q - 86 * q^3 - 870 * q^5 - 5726 * q^7	\( 4 q - 86 q^{3} - 870 q^{5} - 5726 q^{7} + 14732 q^{11} + 45576 q^{13} + 115090 q^{15} + 3616 q^{17} + 877268 q^{21} + 456794 q^{23} - 913600 q^{25} + 889240 q^{27} - 4672348 q^{31} - 4553788 q^{33} - 2956030 q^{35} - 5554884 q^{37} + 10738708 q^{41} - 4913286 q^{43} - 12173390 q^{45} + 5448474 q^{47} + 1827812 q^{51} + 20290316 q^{53} + 9506940 q^{55} - 2593360 q^{57} - 43572012 q^{61} - 37877126 q^{63} + 15450660 q^{65} - 9518486 q^{67} - 20406908 q^{71} - 11608364 q^{73} + 21302450 q^{75} - 110066908 q^{77} + 96218224 q^{81} - 64264686 q^{83} - 12424120 q^{85} - 8501600 q^{87} + 70189812 q^{91} + 16706132 q^{93} + 82819000 q^{95} + 34113396 q^{97}+O(q^{100}) \) 4 * q - 86 * q^3 - 870 * q^5 - 5726 * q^7 + 14732 * q^11 + 45576 * q^13 + 115090 * q^15 + 3616 * q^17 + 877268 * q^21 + 456794 * q^23 - 913600 * q^25 + 889240 * q^27 - 4672348 * q^31 - 4553788 * q^33 - 2956030 * q^35 - 5554884 * q^37 + 10738708 * q^41 - 4913286 * q^43 - 12173390 * q^45 + 5448474 * q^47 + 1827812 * q^51 + 20290316 * q^53 + 9506940 * q^55 - 2593360 * q^57 - 43572012 * q^61 - 37877126 * q^63 + 15450660 * q^65 - 9518486 * q^67 - 20406908 * q^71 - 11608364 * q^73 + 21302450 * q^75 - 110066908 * q^77 + 96218224 * q^81 - 64264686 * q^83 - 12424120 * q^85 - 8501600 * q^87 + 70189812 * q^91 + 16706132 * q^93 + 82819000 * q^95 + 34113396 * q^97

Introduction

L-functions

Modular forms

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Database

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