LMFDB - Embedded newform 20.9.f.a.13.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [20,9,Mod(13,20)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(20, base_ring=CyclotomicField(4))

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

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

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

chi := DirichletCharacter("20.13");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 20 = 2^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	20.f (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:	$8.14757220122$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4 x^{7} + 8 x^{6} + 22254 x^{5} + 4820745 x^{4} + 50131374 x^{3} + 307615702 x^{2} + \cdots + 2405464244 $ x^8 - 4x^7 + 8x^6 + 22254x^5 + 4820745x^4 + 50131374x^3 + 307615702x^2 - 1770757924*x + 2405464244
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 2^{13}\cdot 5^{3} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			13.1
Root			$-7.21849 - 8.21849i$ of defining polynomial
Character	$\chi$	$=$	20.13
Dual form			20.9.f.a.17.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-103.804 - 103.804i) q^{3} +(304.194 - 545.977i) q^{5} +(-1625.71 + 1625.71i) q^{7} +14989.6i q^{9} +O(q^{10})$	\(q+(-103.804 - 103.804i) q^{3} +(304.194 - 545.977i) q^{5} +(-1625.71 + 1625.71i) q^{7} +14989.6i q^{9} -7356.04 q^{11} +(4128.03 + 4128.03i) q^{13} +(-88251.3 + 25098.1i) q^{15} +(36330.8 - 36330.8i) q^{17} +246578. i q^{19} +337512. q^{21} +(-351384. - 351384. i) q^{23} +(-205557. - 332166. i) q^{25} +(874925. - 874925. i) q^{27} +124160. i q^{29} -520399. q^{31} +(763588. + 763588. i) q^{33} +(393070. + 1.38213e6i) q^{35} +(-1.16648e6 + 1.16648e6i) q^{37} -857012. i q^{39} -3.72458e6 q^{41} +(-2.86749e6 - 2.86749e6i) q^{43} +(8.18398e6 + 4.55975e6i) q^{45} +(-2.45745e6 + 2.45745e6i) q^{47} +478916. i q^{49} -7.54258e6 q^{51} +(4.77983e6 + 4.77983e6i) q^{53} +(-2.23766e6 + 4.01623e6i) q^{55} +(2.55958e7 - 2.55958e7i) q^{57} -7.42325e6i q^{59} -5.46517e6 q^{61} +(-2.43688e7 - 2.43688e7i) q^{63} +(3.50953e6 - 998087. i) q^{65} +(1.14987e7 - 1.14987e7i) q^{67} +7.29503e7i q^{69} +2.65449e7 q^{71} +(-1.87105e7 - 1.87105e7i) q^{73} +(-1.31425e7 + 5.58179e7i) q^{75} +(1.19588e7 - 1.19588e7i) q^{77} +1.79128e6i q^{79} -8.32948e7 q^{81} +(1.15125e7 + 1.15125e7i) q^{83} +(-8.78418e6 - 3.08874e7i) q^{85} +(1.28883e7 - 1.28883e7i) q^{87} -3.57299e7i q^{89} -1.34220e7 q^{91} +(5.40195e7 + 5.40195e7i) q^{93} +(1.34626e8 + 7.50076e7i) q^{95} +(-3.39701e7 + 3.39701e7i) q^{97} -1.10264e8i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 70 q^{3} + 894 q^{5} - 2030 q^{7}+O(q^{10}) $ 8 * q - 70 * q^3 + 894 * q^5 - 2030 * q^7	$ 8 q - 70 q^{3} + 894 q^{5} - 2030 q^{7} - 420 q^{11} + 33180 q^{13} - 48478 q^{15} + 43620 q^{17} + 108668 q^{21} - 663270 q^{23} + 163396 q^{25} + 1576040 q^{27} - 3178492 q^{31} - 944020 q^{33} + 2571618 q^{35} + 5344080 q^{37} - 10185252 q^{41} - 10342710 q^{43} + 20284834 q^{45} + 19232250 q^{47} - 47126684 q^{51} - 24320640 q^{53} + 21483180 q^{55} + 88218320 q^{57} - 82515684 q^{61} - 77441350 q^{63} + 72045768 q^{65} + 100675930 q^{67} - 99290076 q^{71} - 93528520 q^{73} + 76524178 q^{75} + 134199660 q^{77} - 161920268 q^{81} - 10450350 q^{83} + 51676156 q^{85} + 164801600 q^{87} - 130681068 q^{91} - 50183620 q^{93} + 84367944 q^{95} - 179570760 q^{97}+O(q^{100}) $ 8 * q - 70 * q^3 + 894 * q^5 - 2030 * q^7 - 420 * q^11 + 33180 * q^13 - 48478 * q^15 + 43620 * q^17 + 108668 * q^21 - 663270 * q^23 + 163396 * q^25 + 1576040 * q^27 - 3178492 * q^31 - 944020 * q^33 + 2571618 * q^35 + 5344080 * q^37 - 10185252 * q^41 - 10342710 * q^43 + 20284834 * q^45 + 19232250 * q^47 - 47126684 * q^51 - 24320640 * q^53 + 21483180 * q^55 + 88218320 * q^57 - 82515684 * q^61 - 77441350 * q^63 + 72045768 * q^65 + 100675930 * q^67 - 99290076 * q^71 - 93528520 * q^73 + 76524178 * q^75 + 134199660 * q^77 - 161920268 * q^81 - 10450350 * q^83 + 51676156 * q^85 + 164801600 * q^87 - 130681068 * q^91 - 50183620 * q^93 + 84367944 * q^95 - 179570760 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/20\mathbb{Z}\right)^\times$.

$n$	$11$	$17$
$\chi(n)$	$1$	$e\left(\frac{3}{4}\right)$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	−103.804	−	103.804i	−1.28153	−	1.28153i	−0.939795	−	0.341737i	$-0.888985\pi$
$3$	−103.804	−	103.804i	−1.28153	−	1.28153i	−0.341737	−	0.939795i	$-0.611015\pi$
$4$		0			0
$5$	304.194	−	545.977i	0.486710	−	0.873563i
$5$	304.194	−	545.977i	0.486710	−	0.873563i
$6$		0			0
$7$	−1625.71	+	1625.71i	−0.677098	+	0.677098i	−0.959343	−	0.282244i	$-0.908921\pi$
$7$	−1625.71	+	1625.71i	−0.677098	+	0.677098i	0.282244	+	0.959343i	$0.408921\pi$
$8$		0			0
$9$			14989.6i			2.28465i
$10$		0			0
$11$	−7356.04			−0.502427			−0.251214	−	0.967932i	$-0.580830\pi$
$11$	−7356.04			−0.502427			−0.251214	+	0.967932i	$0.580830\pi$
$12$		0			0
$13$	4128.03	+	4128.03i	0.144534	+	0.144534i	0.775671	−	0.631137i	$-0.217412\pi$
$13$	4128.03	+	4128.03i	0.144534	+	0.144534i	−0.631137	+	0.775671i	$0.717412\pi$
$14$		0			0
$15$	−88251.3	+	25098.1i	−1.74324	+	0.495765i
$16$		0			0
$17$	36330.8	−	36330.8i	0.434990	−	0.434990i	−0.455332	−	0.890322i	$-0.650479\pi$
$17$	36330.8	−	36330.8i	0.434990	−	0.434990i	0.890322	+	0.455332i	$0.150479\pi$
$18$		0			0
$19$			246578.i			1.89208i	0.324044	+	0.946042i	$0.394957\pi$
$19$			246578.i			1.89208i	−0.324044	+	0.946042i	$0.605043\pi$
$20$		0			0
$21$	337512.			1.73545
$22$		0			0
$23$	−351384.	−	351384.i	−1.25566	−	1.25566i	−0.953146	−	0.302512i	$-0.902175\pi$
$23$	−351384.	−	351384.i	−1.25566	−	1.25566i	−0.302512	−	0.953146i	$-0.597825\pi$
$24$		0			0
$25$	−205557.	−	332166.i	−0.526226	−	0.850345i
$26$		0			0
$27$	874925.	−	874925.i	1.64632	−	1.64632i
$28$		0			0
$29$			124160.i			0.175545i	0.996141	+	0.0877725i	$0.0279748\pi$
$29$			124160.i			0.175545i	−0.996141	+	0.0877725i	$0.972025\pi$
$30$		0			0
$31$	−520399.			−0.563494			−0.281747	−	0.959489i	$-0.590914\pi$
$31$	−520399.			−0.563494			−0.281747	+	0.959489i	$0.590914\pi$
$32$		0			0
$33$	763588.	+	763588.i	0.643877	+	0.643877i
$34$		0			0
$35$	393070.	+	1.38213e6i	0.261937	+	0.921039i
$36$		0			0
$37$	−1.16648e6	+	1.16648e6i	−0.622403	+	0.622403i	−0.946145	−	0.323742i	$-0.895059\pi$
$37$	−1.16648e6	+	1.16648e6i	−0.622403	+	0.622403i	0.323742	+	0.946145i	$0.395059\pi$
$38$		0			0
$39$		−	857012.i		−	0.370449i
$40$		0			0
$41$	−3.72458e6			−1.31808			−0.659040	−	0.752108i	$-0.729037\pi$
$41$	−3.72458e6			−1.31808			−0.659040	+	0.752108i	$0.729037\pi$
$42$		0			0
$43$	−2.86749e6	−	2.86749e6i	−0.838740	−	0.838740i	0.149953	−	0.988693i	$-0.452088\pi$
$43$	−2.86749e6	−	2.86749e6i	−0.838740	−	0.838740i	−0.988693	+	0.149953i	$0.952088\pi$
$44$		0			0
$45$	8.18398e6	+	4.55975e6i	1.99579	+	1.11196i
$46$		0			0
$47$	−2.45745e6	+	2.45745e6i	−0.503608	+	0.503608i	−0.912557	−	0.408949i	$-0.865895\pi$
$47$	−2.45745e6	+	2.45745e6i	−0.503608	+	0.503608i	0.408949	+	0.912557i	$0.365895\pi$
$48$		0			0
$49$			478916.i			0.0830760i
$50$		0			0
$51$	−7.54258e6			−1.11491
$52$		0			0
$53$	4.77983e6	+	4.77983e6i	0.605772	+	0.605772i	0.941838	−	0.336066i	$-0.109097\pi$
$53$	4.77983e6	+	4.77983e6i	0.605772	+	0.605772i	−0.336066	+	0.941838i	$0.609097\pi$
$54$		0			0
$55$	−2.23766e6	+	4.01623e6i	−0.244537	+	0.438902i
$56$		0			0
$57$	2.55958e7	−	2.55958e7i	2.42477	−	2.42477i
$58$		0			0
$59$		−	7.42325e6i		−	0.612613i	−0.951933	−	0.306307i	$-0.900907\pi$
$59$		−	7.42325e6i		−	0.612613i	0.951933	−	0.306307i	$-0.0990933\pi$
$60$		0			0
$61$	−5.46517e6			−0.394715			−0.197358	−	0.980332i	$-0.563236\pi$
$61$	−5.46517e6			−0.394715			−0.197358	+	0.980332i	$0.563236\pi$
$62$		0			0
$63$	−2.43688e7	−	2.43688e7i	−1.54693	−	1.54693i
$64$		0			0
$65$	3.50953e6	−	998087.i	0.196605	−	0.0559133i
$66$		0			0
$67$	1.14987e7	−	1.14987e7i	0.570622	−	0.570622i	−0.361680	−	0.932302i	$-0.617797\pi$
$67$	1.14987e7	−	1.14987e7i	0.570622	−	0.570622i	0.932302	+	0.361680i	$0.117797\pi$
$68$		0			0
$69$			7.29503e7i			3.21833i
$70$		0			0
$71$	2.65449e7			1.04459			0.522297	−	0.852764i	$-0.325075\pi$
$71$	2.65449e7			1.04459			0.522297	+	0.852764i	$0.325075\pi$
$72$		0			0
$73$	−1.87105e7	−	1.87105e7i	−0.658862	−	0.658862i	0.296249	−	0.955111i	$-0.404264\pi$
$73$	−1.87105e7	−	1.87105e7i	−0.658862	−	0.658862i	−0.955111	+	0.296249i	$0.904264\pi$
$74$		0			0
$75$	−1.31425e7	+	5.58179e7i	−0.415369	+	1.76412i
$76$		0			0
$77$	1.19588e7	−	1.19588e7i	0.340193	−	0.340193i
$78$		0			0
$79$			1.79128e6i			0.0459892i	0.999736	+	0.0229946i	$0.00732005\pi$
$79$			1.79128e6i			0.0459892i	−0.999736	+	0.0229946i	$0.992680\pi$
$80$		0			0
$81$	−8.32948e7			−1.93499
$82$		0			0
$83$	1.15125e7	+	1.15125e7i	0.242581	+	0.242581i	0.817917	−	0.575336i	$-0.195129\pi$
$83$	1.15125e7	+	1.15125e7i	0.242581	+	0.242581i	−0.575336	+	0.817917i	$0.695129\pi$
$84$		0			0
$85$	−8.78418e6	−	3.08874e7i	−0.168277	−	0.591706i
$86$		0			0
$87$	1.28883e7	−	1.28883e7i	0.224967	−	0.224967i
$88$		0			0
$89$		−	3.57299e7i		−	0.569472i	−0.958606	−	0.284736i	$-0.908094\pi$
$89$		−	3.57299e7i		−	0.569472i	0.958606	−	0.284736i	$-0.0919059\pi$
$90$		0			0
$91$	−1.34220e7			−0.195727
$92$		0			0
$93$	5.40195e7	+	5.40195e7i	0.722136	+	0.722136i
$94$		0			0
$95$	1.34626e8	+	7.50076e7i	1.65286	+	0.920897i
$96$		0			0
$97$	−3.39701e7	+	3.39701e7i	−0.383716	+	0.383716i	−0.872439	−	0.488723i	$-0.837463\pi$
$97$	−3.39701e7	+	3.39701e7i	−0.383716	+	0.383716i	0.488723	+	0.872439i	$0.337463\pi$
$98$		0			0
$99$		−	1.10264e8i		−	1.14787i
$100$		0			0
$101$	−2.80333e7			−0.269395			−0.134697	−	0.990887i	$-0.543006\pi$
$101$	−2.80333e7			−0.269395			−0.134697	+	0.990887i	$0.543006\pi$
$102$		0			0
$103$	−9.32395e7	−	9.32395e7i	−0.828421	−	0.828421i	0.158877	−	0.987298i	$-0.449213\pi$
$103$	−9.32395e7	−	9.32395e7i	−0.828421	−	0.828421i	−0.987298	+	0.158877i	$0.949213\pi$
$104$		0			0
$105$	1.02669e8	−	1.84274e8i	0.844660	−	1.51602i
$106$		0			0
$107$	−5.95511e7	+	5.95511e7i	−0.454312	+	0.454312i	−0.896783	−	0.442471i	$-0.854102\pi$
$107$	−5.95511e7	+	5.95511e7i	−0.454312	+	0.454312i	0.442471	+	0.896783i	$0.354102\pi$
$108$		0			0
$109$		−	1.86471e8i		−	1.32101i	−0.750821	−	0.660505i	$-0.770342\pi$
$109$		−	1.86471e8i		−	1.32101i	0.750821	−	0.660505i	$-0.229658\pi$
$110$		0			0
$111$	2.42172e8			1.59526
$112$		0			0
$113$	5.50726e7	+	5.50726e7i	0.337771	+	0.337771i	0.855528	−	0.517757i	$-0.173233\pi$
$113$	5.50726e7	+	5.50726e7i	0.337771	+	0.337771i	−0.517757	+	0.855528i	$0.673233\pi$
$114$		0			0
$115$	−2.98737e8	+	8.49588e7i	−1.70804	+	0.485755i
$116$		0			0
$117$	−6.18775e7	+	6.18775e7i	−0.330209	+	0.330209i
$118$		0			0
$119$			1.18127e8i			0.589063i
$120$		0			0
$121$	−1.60248e8			−0.747567
$122$		0			0
$123$	3.86627e8	+	3.86627e8i	1.68916	+	1.68916i
$124$		0			0
$125$	−2.43884e8	+	1.11865e7i	−0.998950	+	0.0458201i
$126$		0			0
$127$	−1.21468e8	+	1.21468e8i	−0.466927	+	0.466927i	−0.900917	−	0.433991i	$-0.857105\pi$
$127$	−1.21468e8	+	1.21468e8i	−0.466927	+	0.466927i	0.433991	+	0.900917i	$0.357105\pi$
$128$		0			0
$129$			5.95314e8i			2.14975i
$130$		0			0
$131$	3.77572e8			1.28208			0.641039	−	0.767509i	$-0.278504\pi$
$131$	3.77572e8			1.28208			0.641039	+	0.767509i	$0.278504\pi$
$132$		0			0
$133$	−4.00865e8	−	4.00865e8i	−1.28113	−	1.28113i
$134$		0			0
$135$	−2.11542e8	−	7.43836e8i	−0.636886	−	2.23945i
$136$		0			0
$137$	−1.06114e8	+	1.06114e8i	−0.301225	+	0.301225i	−0.841493	−	0.540268i	$-0.818323\pi$
$137$	−1.06114e8	+	1.06114e8i	−0.301225	+	0.301225i	0.540268	+	0.841493i	$0.318323\pi$
$138$		0			0
$139$		−	8.27359e7i		−	0.221633i	−0.993841	−	0.110817i	$-0.964653\pi$
$139$		−	8.27359e7i		−	0.221633i	0.993841	−	0.110817i	$-0.0353466\pi$
$140$		0			0
$141$	5.10187e8			1.29078
$142$		0			0
$143$	−3.03659e7	−	3.03659e7i	−0.0726177	−	0.0726177i
$144$		0			0
$145$	6.77883e7	+	3.77686e7i	0.153350	+	0.0854396i
$146$		0			0
$147$	4.97135e7	−	4.97135e7i	0.106465	−	0.106465i
$148$		0			0
$149$		−	5.63570e8i		−	1.14341i	−0.820458	−	0.571706i	$-0.806282\pi$
$149$		−	5.63570e8i		−	1.14341i	0.820458	−	0.571706i	$-0.193718\pi$
$150$		0			0
$151$	−2.04916e8			−0.394156			−0.197078	−	0.980388i	$-0.563145\pi$
$151$	−2.04916e8			−0.394156			−0.197078	+	0.980388i	$0.563145\pi$
$152$		0			0
$153$	5.44585e8	+	5.44585e8i	0.993802	+	0.993802i
$154$		0			0
$155$	−1.58302e8	+	2.84126e8i	−0.274258	+	0.492248i
$156$		0			0
$157$	−2.73948e8	+	2.73948e8i	−0.450888	+	0.450888i	−0.895649	−	0.444761i	$-0.853289\pi$
$157$	−2.73948e8	+	2.73948e8i	−0.450888	+	0.450888i	0.444761	+	0.895649i	$0.353289\pi$
$158$		0			0
$159$		−	9.92333e8i		−	1.55263i
$160$		0			0
$161$	1.14250e9			1.70041
$162$		0			0
$163$	1.98021e8	+	1.98021e8i	0.280518	+	0.280518i	0.833315	−	0.552798i	$-0.186440\pi$
$163$	1.98021e8	+	1.98021e8i	0.280518	+	0.280518i	−0.552798	+	0.833315i	$0.686440\pi$
$164$		0			0
$165$	6.49180e8	−	1.84623e8i	0.875849	−	0.249086i
$166$		0			0
$167$	2.15477e8	−	2.15477e8i	0.277035	−	0.277035i	−0.554889	−	0.831924i	$-0.687239\pi$
$167$	2.15477e8	−	2.15477e8i	0.277035	−	0.277035i	0.831924	+	0.554889i	$0.187239\pi$
$168$		0			0
$169$		−	7.81650e8i		−	0.958220i
$170$		0			0
$171$	−3.69611e9			−4.32275
$172$		0			0
$173$	−4.80990e8	−	4.80990e8i	−0.536972	−	0.536972i	0.385666	−	0.922638i	$-0.373972\pi$
$173$	−4.80990e8	−	4.80990e8i	−0.536972	−	0.536972i	−0.922638	+	0.385666i	$0.873972\pi$
$174$		0			0
$175$	8.74183e8	+	2.05830e8i	0.932074	+	0.219460i
$176$		0			0
$177$	−7.70565e8	+	7.70565e8i	−0.785084	+	0.785084i
$178$		0			0
$179$			1.16291e9i			1.13275i	0.824148	+	0.566374i	$0.191654\pi$
$179$			1.16291e9i			1.13275i	−0.824148	+	0.566374i	$0.808346\pi$
$180$		0			0
$181$	1.73333e9			1.61498			0.807488	−	0.589884i	$-0.200826\pi$
$181$	1.73333e9			1.61498			0.807488	+	0.589884i	$0.200826\pi$
$182$		0			0
$183$	5.67307e8	+	5.67307e8i	0.505841	+	0.505841i
$184$		0			0
$185$	2.82036e8	+	9.91711e8i	0.240779	+	0.846639i
$186$		0			0
$187$	−2.67251e8	+	2.67251e8i	−0.218551	+	0.218551i
$188$		0			0
$189$			2.84475e9i			2.22945i
$190$		0			0
$191$	−2.95875e8			−0.222318			−0.111159	−	0.993803i	$-0.535456\pi$
$191$	−2.95875e8			−0.222318			−0.111159	+	0.993803i	$0.535456\pi$
$192$		0			0
$193$	−1.09525e9	−	1.09525e9i	−0.789376	−	0.789376i	0.192016	−	0.981392i	$-0.438497\pi$
$193$	−1.09525e9	−	1.09525e9i	−0.789376	−	0.789376i	−0.981392	+	0.192016i	$0.938497\pi$
$194$		0			0
$195$	−4.67909e8	−	2.60698e8i	−0.323611	−	0.180301i
$196$		0			0
$197$	8.46906e8	−	8.46906e8i	0.562303	−	0.562303i	−0.367658	−	0.929961i	$-0.619840\pi$
$197$	8.46906e8	−	8.46906e8i	0.562303	−	0.562303i	0.929961	+	0.367658i	$0.119840\pi$
$198$		0			0
$199$		−	8.64462e8i		−	0.551231i	−0.961268	−	0.275616i	$-0.911118\pi$
$199$		−	8.64462e8i		−	0.551231i	0.961268	−	0.275616i	$-0.0888817\pi$
$200$		0			0
$201$	−2.38722e9			−1.46254
$202$		0			0
$203$	−2.01848e8	−	2.01848e8i	−0.118861	−	0.118861i
$204$		0			0
$205$	−1.13299e9	+	2.03353e9i	−0.641523	+	1.15143i
$206$		0			0
$207$	5.26712e9	−	5.26712e9i	2.86874	−	2.86874i
$208$		0			0
$209$		−	1.81384e9i		−	0.950635i
$210$		0			0
$211$	3.50884e9			1.77025			0.885124	−	0.465355i	$-0.154073\pi$
$211$	3.50884e9			1.77025			0.885124	+	0.465355i	$0.154073\pi$
$212$		0			0
$213$	−2.75547e9	−	2.75547e9i	−1.33868	−	1.33868i
$214$		0			0
$215$	−2.43785e9	+	6.93310e8i	−1.14092	+	0.324469i
$216$		0			0
$217$	8.46019e8	−	8.46019e8i	0.381541	−	0.381541i
$218$		0			0
$219$			3.88446e9i			1.68871i
$220$		0			0
$221$	2.99949e8			0.125742
$222$		0			0
$223$	1.43975e9	+	1.43975e9i	0.582195	+	0.582195i	0.935506	−	0.353311i	$-0.114944\pi$
$223$	1.43975e9	+	1.43975e9i	0.582195	+	0.582195i	−0.353311	+	0.935506i	$0.614944\pi$
$224$		0			0
$225$	4.97904e9	−	3.08122e9i	1.94274	−	1.20224i
$226$		0			0
$227$	−1.74616e9	+	1.74616e9i	−0.657628	+	0.657628i	−0.954818	−	0.297190i	$-0.903950\pi$
$227$	−1.74616e9	+	1.74616e9i	−0.657628	+	0.657628i	0.297190	+	0.954818i	$0.403950\pi$
$228$		0			0
$229$			4.57752e9i			1.66452i	0.554388	+	0.832259i	$0.312953\pi$
$229$			4.57752e9i			1.66452i	−0.554388	+	0.832259i	$0.687047\pi$
$230$		0			0
$231$	−2.48275e9			−0.871936
$232$		0			0
$233$	−2.88573e9	−	2.88573e9i	−0.979113	−	0.979113i	0.0206737	−	0.999786i	$-0.493419\pi$
$233$	−2.88573e9	−	2.88573e9i	−0.979113	−	0.979113i	−0.999786	+	0.0206737i	$0.993419\pi$
$234$		0			0
$235$	5.94170e8	+	2.08925e9i	0.194822	+	0.685045i
$236$		0			0
$237$	1.85943e8	−	1.85943e8i	0.0589367	−	0.0589367i
$238$		0			0
$239$		−	5.39020e9i		−	1.65201i	−0.563661	−	0.826007i	$-0.690607\pi$
$239$		−	5.39020e9i		−	1.65201i	0.563661	−	0.826007i	$-0.309393\pi$
$240$		0			0
$241$	−2.90282e9			−0.860501			−0.430250	−	0.902710i	$-0.641575\pi$
$241$	−2.90282e9			−0.860501			−0.430250	+	0.902710i	$0.641575\pi$
$242$		0			0
$243$	2.90597e9	+	2.90597e9i	0.833423	+	0.833423i
$244$		0			0
$245$	2.61477e8	+	1.45683e8i	0.0725721	+	0.0404339i
$246$		0			0
$247$	−1.01788e9	+	1.01788e9i	−0.273470	+	0.273470i
$248$		0			0
$249$		−	2.39009e9i		−	0.621752i
$250$		0			0
$251$	−2.57470e9			−0.648681			−0.324340	−	0.945940i	$-0.605142\pi$
$251$	−2.57470e9			−0.648681			−0.324340	+	0.945940i	$0.605142\pi$
$252$		0			0
$253$	2.58480e9	+	2.58480e9i	0.630877	+	0.630877i
$254$		0			0
$255$	−2.29441e9	+	4.11808e9i	−0.542638	+	0.973944i
$256$		0			0
$257$	−5.21766e9	+	5.21766e9i	−1.19603	+	1.19603i	−0.220689	+	0.975344i	$0.570831\pi$
$257$	−5.21766e9	+	5.21766e9i	−1.19603	+	1.19603i	−0.975344	+	0.220689i	$0.929169\pi$
$258$		0			0
$259$		−	3.79274e9i		−	0.842856i
$260$		0			0
$261$	−1.86110e9			−0.401059
$262$		0			0
$263$	2.52818e9	+	2.52818e9i	0.528426	+	0.528426i	0.920103	−	0.391677i	$-0.128105\pi$
$263$	2.52818e9	+	2.52818e9i	0.528426	+	0.528426i	−0.391677	+	0.920103i	$0.628105\pi$
$264$		0			0
$265$	4.06368e9	−	1.15568e9i	0.824016	−	0.234345i
$266$		0			0
$267$	−3.70891e9	+	3.70891e9i	−0.729797	+	0.729797i
$268$		0			0
$269$			1.23618e9i			0.236087i	0.993008	+	0.118044i	$0.0376623\pi$
$269$			1.23618e9i			0.236087i	−0.993008	+	0.118044i	$0.962338\pi$
$270$		0			0
$271$	8.22005e9			1.52404			0.762022	−	0.647552i	$-0.224207\pi$
$271$	8.22005e9			1.52404			0.762022	+	0.647552i	$0.224207\pi$
$272$		0			0
$273$	1.39326e9	+	1.39326e9i	0.250831	+	0.250831i
$274$		0			0
$275$	1.51209e9	+	2.44343e9i	0.264390	+	0.427237i
$276$		0			0
$277$	−6.42663e7	+	6.42663e7i	−0.0109160	+	0.0109160i	−0.712544	−	0.701628i	$-0.752457\pi$
$277$	−6.42663e7	+	6.42663e7i	−0.0109160	+	0.0109160i	0.701628	+	0.712544i	$0.252457\pi$
$278$		0			0
$279$		−	7.80057e9i		−	1.28739i
$280$		0			0
$281$	2.39407e9			0.383982			0.191991	−	0.981397i	$-0.438506\pi$
$281$	2.39407e9			0.383982			0.191991	+	0.981397i	$0.438506\pi$
$282$		0			0
$283$	−5.94041e9	−	5.94041e9i	−0.926127	−	0.926127i	0.0713260	−	0.997453i	$-0.477277\pi$
$283$	−5.94041e9	−	5.94041e9i	−0.926127	−	0.926127i	−0.997453	+	0.0713260i	$0.977277\pi$
$284$		0			0
$285$	−6.18864e9	−	2.17609e10i	−0.938029	−	3.29835i
$286$		0			0
$287$	6.05509e9	−	6.05509e9i	0.892469	−	0.892469i
$288$		0			0
$289$			4.33590e9i			0.621567i
$290$		0			0
$291$	7.05248e9			0.983490
$292$		0			0
$293$	3.10983e8	+	3.10983e8i	0.0421955	+	0.0421955i	0.727890	−	0.685694i	$-0.240501\pi$
$293$	3.10983e8	+	3.10983e8i	0.0421955	+	0.0421955i	−0.685694	+	0.727890i	$0.740501\pi$
$294$		0			0
$295$	−4.05293e9	−	2.25811e9i	−0.535156	−	0.298165i
$296$		0			0
$297$	−6.43598e9	+	6.43598e9i	−0.827159	+	0.827159i
$298$		0			0
$299$		−	2.90105e9i		−	0.362969i
$300$		0			0
$301$	9.32342e9			1.13582
$302$		0			0
$303$	2.90998e9	+	2.90998e9i	0.345238	+	0.345238i
$304$		0			0
$305$	−1.66247e9	+	2.98386e9i	−0.192112	+	0.344809i
$306$		0			0
$307$	3.04928e9	−	3.04928e9i	0.343276	−	0.343276i	−0.514321	−	0.857597i	$-0.671956\pi$
$307$	3.04928e9	−	3.04928e9i	0.343276	−	0.343276i	0.857597	+	0.514321i	$0.171956\pi$
$308$		0			0
$309$			1.93573e10i			2.12330i
$310$		0			0
$311$	−7.16164e9			−0.765546			−0.382773	−	0.923843i	$-0.625031\pi$
$311$	−7.16164e9			−0.765546			−0.382773	+	0.923843i	$0.625031\pi$
$312$		0			0
$313$	7.06143e9	+	7.06143e9i	0.735725	+	0.735725i	0.971748	−	0.236023i	$-0.0758440\pi$
$313$	7.06143e9	+	7.06143e9i	0.735725	+	0.735725i	−0.236023	+	0.971748i	$0.575844\pi$
$314$		0			0
$315$	−2.07176e10	+	5.89196e9i	−2.10425	+	0.598436i
$316$		0			0
$317$	8.58096e9	−	8.58096e9i	0.849765	−	0.849765i	−0.140339	−	0.990104i	$-0.544819\pi$
$317$	8.58096e9	−	8.58096e9i	0.849765	−	0.849765i	0.990104	+	0.140339i	$0.0448192\pi$
$318$		0			0
$319$		−	9.13323e8i		−	0.0881986i
$320$		0			0
$321$	1.23633e10			1.16443
$322$		0			0
$323$	8.95839e9	+	8.95839e9i	0.823038	+	0.823038i
$324$		0			0
$325$	5.22645e8	−	2.21973e9i	0.0468461	−	0.198961i
$326$		0			0
$327$	−1.93565e10	+	1.93565e10i	−1.69292	+	1.69292i
$328$		0			0
$329$		−	7.99021e9i		−	0.681985i
$330$		0			0
$331$	−1.09080e10			−0.908728			−0.454364	−	0.890816i	$-0.650133\pi$
$331$	−1.09080e10			−0.908728			−0.454364	+	0.890816i	$0.650133\pi$
$332$		0			0
$333$	−1.74851e10	−	1.74851e10i	−1.42198	−	1.42198i
$334$		0			0
$335$	−2.78019e9	−	9.77585e9i	−0.220747	−	0.776203i
$336$		0			0
$337$	−1.10678e10	+	1.10678e10i	−0.858105	+	0.858105i	−0.991115	−	0.133010i	$-0.957536\pi$
$337$	−1.10678e10	+	1.10678e10i	−0.858105	+	0.858105i	0.133010	+	0.991115i	$0.457536\pi$
$338$		0			0
$339$		−	1.14335e10i		−	0.865728i
$340$		0			0
$341$	3.82807e9			0.283115
$342$		0			0
$343$	−1.01505e10	−	1.01505e10i	−0.733349	−	0.733349i
$344$		0			0
$345$	3.98292e10	+	2.21911e10i	2.81142	+	1.56640i
$346$		0			0
$347$	1.61976e8	−	1.61976e8i	0.0111721	−	0.0111721i	−0.701499	−	0.712671i	$-0.747485\pi$
$347$	1.61976e8	−	1.61976e8i	0.0111721	−	0.0111721i	0.712671	+	0.701499i	$0.247485\pi$
$348$		0			0
$349$			1.18081e10i			0.795937i	0.917399	+	0.397968i	$0.130285\pi$
$349$			1.18081e10i			0.795937i	−0.917399	+	0.397968i	$0.869715\pi$
$350$		0			0
$351$	7.22342e9			0.475899
$352$		0			0
$353$	1.52366e10	+	1.52366e10i	0.981271	+	0.981271i	0.999828	−	0.0185564i	$-0.00590703\pi$
$353$	1.52366e10	+	1.52366e10i	0.981271	+	0.981271i	−0.0185564	+	0.999828i	$0.505907\pi$
$354$		0			0
$355$	8.07480e9	−	1.44929e10i	0.508415	−	0.912519i
$356$		0			0
$357$	1.22621e10	−	1.22621e10i	0.754903	−	0.754903i
$358$		0			0
$359$			1.16028e10i			0.698529i	0.937024	+	0.349265i	$0.113569\pi$
$359$			1.16028e10i			0.698529i	−0.937024	+	0.349265i	$0.886431\pi$
$360$		0			0
$361$	−4.38173e10			−2.57998
$362$		0			0
$363$	1.66344e10	+	1.66344e10i	0.958031	+	0.958031i
$364$		0			0
$365$	−1.59071e10	+	4.52389e9i	−0.896233	+	0.254883i
$366$		0			0
$367$	1.37224e10	−	1.37224e10i	0.756422	−	0.756422i	−0.219247	−	0.975669i	$-0.570360\pi$
$367$	1.37224e10	−	1.37224e10i	0.756422	−	0.756422i	0.975669	+	0.219247i	$0.0703601\pi$
$368$		0			0
$369$		−	5.58299e10i		−	3.01135i
$370$		0			0
$371$	−1.55413e10			−0.820334
$372$		0			0
$373$	−1.59252e10	−	1.59252e10i	−0.822715	−	0.822715i	0.163782	−	0.986497i	$-0.447631\pi$
$373$	−1.59252e10	−	1.59252e10i	−0.822715	−	0.822715i	−0.986497	+	0.163782i	$0.947631\pi$
$374$		0			0
$375$	2.64774e10	+	2.41550e10i	1.33891	+	1.22147i
$376$		0			0
$377$	−5.12534e8	+	5.12534e8i	−0.0253722	+	0.0253722i
$378$		0			0
$379$		−	2.13124e10i		−	1.03294i	−0.856305	−	0.516470i	$-0.827246\pi$
$379$		−	2.13124e10i		−	1.03294i	0.856305	−	0.516470i	$-0.172754\pi$
$380$		0			0
$381$	2.52179e10			1.19676
$382$		0			0
$383$	6.73538e9	+	6.73538e9i	0.313016	+	0.313016i	0.846077	−	0.533061i	$-0.178958\pi$
$383$	6.73538e9	+	6.73538e9i	0.313016	+	0.313016i	−0.533061	+	0.846077i	$0.678958\pi$
$384$		0			0
$385$	−2.89144e9	−	1.01670e10i	−0.131605	−	0.462755i
$386$		0			0
$387$	4.29825e10	−	4.29825e10i	1.91623	−	1.91623i
$388$		0			0
$389$			1.58368e10i			0.691622i	0.938304	+	0.345811i	$0.112396\pi$
$389$			1.58368e10i			0.691622i	−0.938304	+	0.345811i	$0.887604\pi$
$390$		0			0
$391$	−2.55322e10			−1.09240
$392$		0			0
$393$	−3.91935e10	−	3.91935e10i	−1.64302	−	1.64302i
$394$		0			0
$395$	9.78000e8	+	5.44898e8i	0.0401745	+	0.0223834i
$396$		0			0
$397$	−2.65538e10	+	2.65538e10i	−1.06897	+	1.06897i	−0.0715274	+	0.997439i	$0.522787\pi$
$397$	−2.65538e10	+	2.65538e10i	−1.06897	+	1.06897i	−0.997439	+	0.0715274i	$0.977213\pi$
$398$		0			0
$399$			8.32230e10i			3.28361i
$400$		0			0
$401$	8.55208e9			0.330746			0.165373	−	0.986231i	$-0.447117\pi$
$401$	8.55208e9			0.330746			0.165373	+	0.986231i	$0.447117\pi$
$402$		0			0
$403$	−2.14822e9	−	2.14822e9i	−0.0814439	−	0.0814439i
$404$		0			0
$405$	−2.53378e10	+	4.54771e10i	−0.941778	+	1.69033i
$406$		0			0
$407$	8.58070e9	−	8.58070e9i	0.312712	−	0.312712i
$408$		0			0
$409$			2.58435e10i			0.923544i	0.886999	+	0.461772i	$0.152786\pi$
$409$			2.58435e10i			0.923544i	−0.886999	+	0.461772i	$0.847214\pi$
$410$		0			0
$411$	2.20302e10			0.772060
$412$		0			0
$413$	1.20681e10	+	1.20681e10i	0.414799	+	0.414799i
$414$		0			0
$415$	9.78760e9	−	2.78353e9i	0.329977	−	0.0938433i
$416$		0			0
$417$	−8.58833e9	+	8.58833e9i	−0.284030	+	0.284030i
$418$		0			0
$419$			3.27801e10i			1.06354i	0.846889	+	0.531770i	$0.178473\pi$
$419$			3.27801e10i			1.06354i	−0.846889	+	0.531770i	$0.821527\pi$
$420$		0			0
$421$	1.44471e10			0.459888			0.229944	−	0.973204i	$-0.426146\pi$
$421$	1.44471e10			0.459888			0.229944	+	0.973204i	$0.426146\pi$
$422$		0			0
$423$	−3.68362e10	−	3.68362e10i	−1.15057	−	1.15057i
$424$		0			0
$425$	−1.95359e10	−	4.59981e9i	−0.598795	−	0.140989i
$426$		0			0
$427$	8.88479e9	−	8.88479e9i	0.267261	−	0.267261i
$428$		0			0
$429$			6.30422e9i			0.186124i
$430$		0			0
$431$	−5.66622e10			−1.64204			−0.821021	−	0.570897i	$-0.806595\pi$
$431$	−5.66622e10			−1.64204			−0.821021	+	0.570897i	$0.806595\pi$
$432$		0			0
$433$	−6.19308e9	−	6.19308e9i	−0.176179	−	0.176179i	0.613509	−	0.789688i	$-0.289758\pi$
$433$	−6.19308e9	−	6.19308e9i	−0.176179	−	0.176179i	−0.789688	+	0.613509i	$0.789758\pi$
$434$		0			0
$435$	−3.11617e9	−	1.09572e10i	−0.0870290	−	0.306016i
$436$		0			0
$437$	8.66438e10	−	8.66438e10i	2.37581	−	2.37581i
$438$		0			0
$439$		−	3.41996e9i		−	0.0920796i	−0.998940	−	0.0460398i	$-0.985340\pi$
$439$		−	3.41996e9i		−	0.0920796i	0.998940	−	0.0460398i	$-0.0146601\pi$
$440$		0			0
$441$	−7.17877e9			−0.189800
$442$		0			0
$443$	3.03768e10	+	3.03768e10i	0.788728	+	0.788728i	0.981286	−	0.192558i	$-0.0616783\pi$
$443$	3.03768e10	+	3.03768e10i	0.788728	+	0.788728i	−0.192558	+	0.981286i	$0.561678\pi$
$444$		0			0
$445$	−1.95077e10	−	1.08688e10i	−0.497470	−	0.277168i
$446$		0			0
$447$	−5.85009e10	+	5.85009e10i	−1.46532	+	1.46532i
$448$		0			0
$449$		−	6.57626e10i		−	1.61806i	−0.587769	−	0.809029i	$-0.699994\pi$
$449$		−	6.57626e10i		−	1.61806i	0.587769	−	0.809029i	$-0.300006\pi$
$450$		0			0
$451$	2.73981e10			0.662239
$452$		0			0
$453$	2.12711e10	+	2.12711e10i	0.505124	+	0.505124i
$454$		0			0
$455$	−4.08288e9	+	7.32809e9i	−0.0952623	+	0.170980i
$456$		0			0
$457$	1.29272e10	−	1.29272e10i	0.296374	−	0.296374i	−0.543218	−	0.839592i	$-0.682794\pi$
$457$	1.29272e10	−	1.29272e10i	0.296374	−	0.296374i	0.839592	+	0.543218i	$0.182794\pi$
$458$		0			0
$459$		−	6.35735e10i		−	1.43227i
$460$		0			0
$461$	−7.97449e9			−0.176563			−0.0882814	−	0.996096i	$-0.528137\pi$
$461$	−7.97449e9			−0.176563			−0.0882814	+	0.996096i	$0.528137\pi$
$462$		0			0
$463$	3.63599e10	+	3.63599e10i	0.791222	+	0.791222i	0.981693	−	0.190471i	$-0.0610015\pi$
$463$	3.63599e10	+	3.63599e10i	0.791222	+	0.791222i	−0.190471	+	0.981693i	$0.561001\pi$
$464$		0			0
$465$	4.59259e10	−	1.30610e10i	0.982303	−	0.279361i
$466$		0			0
$467$	−3.11324e10	+	3.11324e10i	−0.654554	+	0.654554i	−0.954086	−	0.299532i	$-0.903169\pi$
$467$	−3.11324e10	+	3.11324e10i	−0.654554	+	0.654554i	0.299532	+	0.954086i	$0.403169\pi$
$468$		0			0
$469$			3.73871e10i			0.772735i
$470$		0			0
$471$	5.68738e10			1.15566
$472$		0			0
$473$	2.10933e10	+	2.10933e10i	0.421406	+	0.421406i
$474$		0			0
$475$	8.19049e10	−	5.06859e10i	1.60892	−	0.995664i
$476$		0			0
$477$	−7.16478e10	+	7.16478e10i	−1.38398	+	1.38398i
$478$		0			0
$479$		−	8.37952e10i		−	1.59176i	−0.605455	−	0.795879i	$-0.707009\pi$
$479$		−	8.37952e10i		−	1.59176i	0.605455	−	0.795879i	$-0.292991\pi$
$480$		0			0
$481$	−9.63055e9			−0.179916
$482$		0			0
$483$	−1.18596e11	−	1.18596e11i	−2.17913	−	2.17913i
$484$		0			0
$485$	8.21340e9	+	2.88804e10i	0.148442	+	0.521959i
$486$		0			0
$487$	2.19566e9	−	2.19566e9i	0.0390346	−	0.0390346i	−0.687320	−	0.726355i	$-0.741213\pi$
$487$	2.19566e9	−	2.19566e9i	0.0390346	−	0.0390346i	0.726355	+	0.687320i	$0.241213\pi$
$488$		0			0
$489$		−	4.11108e10i		−	0.718985i
$490$		0			0
$491$	1.83225e10			0.315253			0.157627	−	0.987499i	$-0.449616\pi$
$491$	1.83225e10			0.315253			0.157627	+	0.987499i	$0.449616\pi$
$492$		0			0
$493$	4.51082e9	+	4.51082e9i	0.0763604	+	0.0763604i
$494$		0			0
$495$	−6.02017e10	−	3.35417e10i	−1.00274	−	0.558681i
$496$		0			0
$497$	−4.31544e10	+	4.31544e10i	−0.707293	+	0.707293i
$498$		0			0
$499$			1.18277e10i			0.190765i	0.995441	+	0.0953827i	$0.0304075\pi$
$499$			1.18277e10i			0.190765i	−0.995441	+	0.0953827i	$0.969593\pi$
$500$		0			0
$501$	−4.47348e10			−0.710060
$502$		0			0
$503$	7.44684e10	+	7.44684e10i	1.16332	+	1.16332i	0.983744	+	0.179578i	$0.0574732\pi$
$503$	7.44684e10	+	7.44684e10i	1.16332	+	1.16332i	0.179578	+	0.983744i	$0.442527\pi$
$504$		0			0
$505$	−8.52757e9	+	1.53056e10i	−0.131117	+	0.235333i
$506$		0			0
$507$	−8.11385e10	+	8.11385e10i	−1.22799	+	1.22799i
$508$		0			0
$509$		−	6.89795e10i		−	1.02766i	−0.857892	−	0.513829i	$-0.828226\pi$
$509$		−	6.89795e10i		−	1.02766i	0.857892	−	0.513829i	$-0.171774\pi$
$510$		0			0
$511$	6.08359e10			0.892228
$512$		0			0
$513$	2.15737e11	+	2.15737e11i	3.11498	+	3.11498i
$514$		0			0
$515$	−7.92696e10	+	2.25437e10i	−1.12688	+	0.320477i
$516$		0			0
$517$	1.80771e10	−	1.80771e10i	0.253027	−	0.253027i
$518$		0			0
$519$			9.98575e10i			1.37629i
$520$		0			0
$521$	−1.11354e11			−1.51131			−0.755657	−	0.654968i	$-0.772682\pi$
$521$	−1.11354e11			−1.51131			−0.755657	+	0.654968i	$0.772682\pi$
$522$		0			0
$523$	−6.63460e10	−	6.63460e10i	−0.886764	−	0.886764i	0.107447	−	0.994211i	$-0.465733\pi$
$523$	−6.63460e10	−	6.63460e10i	−0.886764	−	0.886764i	−0.994211	+	0.107447i	$0.965733\pi$
$524$		0			0
$525$	−6.93779e10	−	1.12110e11i	−0.913237	−	1.47573i
$526$		0			0
$527$	−1.89065e10	+	1.89065e10i	−0.245115	+	0.245115i
$528$		0			0
$529$			1.68631e11i			2.15335i
$530$		0			0
$531$	1.11272e11			1.39961
$532$		0			0
$533$	−1.53751e10	−	1.53751e10i	−0.190507	−	0.190507i
$534$		0			0
$535$	1.43984e10	+	5.06286e10i	0.175752	+	0.617989i
$536$		0			0
$537$	1.20715e11	−	1.20715e11i	1.45165	−	1.45165i
$538$		0			0
$539$		−	3.52293e9i		−	0.0417396i
$540$		0			0
$541$	1.47989e11			1.72759			0.863796	−	0.503843i	$-0.168081\pi$
$541$	1.47989e11			1.72759			0.863796	+	0.503843i	$0.168081\pi$
$542$		0			0
$543$	−1.79927e11	−	1.79927e11i	−2.06965	−	2.06965i
$544$		0			0
$545$	−1.01809e11	−	5.67235e10i	−1.15399	−	0.642950i
$546$		0			0
$547$	−8.29905e10	+	8.29905e10i	−0.926999	+	0.926999i	−0.997511	−	0.0705123i	$-0.977537\pi$
$547$	−8.29905e10	+	8.29905e10i	−0.926999	+	0.926999i	0.0705123	+	0.997511i	$0.477537\pi$
$548$		0			0
$549$		−	8.19207e10i		−	0.901788i
$550$		0			0
$551$	−3.06151e10			−0.332146
$552$		0			0
$553$	−2.91211e9	−	2.91211e9i	−0.0311392	−	0.0311392i
$554$		0			0
$555$	7.36672e10	−	1.32220e11i	0.776430	−	1.39356i
$556$		0			0
$557$	6.49903e10	−	6.49903e10i	0.675193	−	0.675193i	−0.283716	−	0.958908i	$-0.591567\pi$
$557$	6.49903e10	−	6.49903e10i	0.675193	−	0.675193i	0.958908	+	0.283716i	$0.0915672\pi$
$558$		0			0
$559$		−	2.36741e10i		−	0.242452i
$560$		0			0
$561$	5.54835e10			0.560161
$562$		0			0
$563$	1.09888e10	+	1.09888e10i	0.109374	+	0.109374i	0.759676	−	0.650302i	$-0.225358\pi$
$563$	1.09888e10	+	1.09888e10i	0.109374	+	0.109374i	−0.650302	+	0.759676i	$0.725358\pi$
$564$		0			0
$565$	4.68211e10	−	1.33156e10i	0.459460	−	0.130668i
$566$		0			0
$567$	1.35413e11	−	1.35413e11i	1.31018	−	1.31018i
$568$		0			0
$569$		−	3.50530e10i		−	0.334408i	−0.985922	−	0.167204i	$-0.946526\pi$
$569$		−	3.50530e10i		−	0.334408i	0.985922	−	0.167204i	$-0.0534738\pi$
$570$		0			0
$571$	9.68845e10			0.911401			0.455701	−	0.890133i	$-0.349389\pi$
$571$	9.68845e10			0.911401			0.455701	+	0.890133i	$0.349389\pi$
$572$		0			0
$573$	3.07130e10	+	3.07130e10i	0.284908	+	0.284908i
$574$		0			0
$575$	−4.44884e10	+	1.88947e11i	−0.406982	+	1.72850i
$576$		0			0
$577$	−8.47132e10	+	8.47132e10i	−0.764272	+	0.764272i	−0.977091	−	0.212820i	$-0.931735\pi$
$577$	−8.47132e10	+	8.47132e10i	−0.764272	+	0.764272i	0.212820	+	0.977091i	$0.431735\pi$
$578$		0			0
$579$			2.27383e11i			2.02322i
$580$		0			0
$581$	−3.74320e10			−0.328503
$582$		0			0
$583$	−3.51606e10	−	3.51606e10i	−0.304356	−	0.304356i
$584$		0			0
$585$	1.49609e10	+	5.26064e10i	0.127742	+	0.449175i
$586$		0			0
$587$	6.08713e10	−	6.08713e10i	0.512696	−	0.512696i	−0.402656	−	0.915352i	$-0.631913\pi$
$587$	6.08713e10	−	6.08713e10i	0.512696	−	0.512696i	0.915352	+	0.402656i	$0.131913\pi$
$588$		0			0
$589$		−	1.28319e11i		−	1.06618i
$590$		0			0
$591$	−1.75825e11			−1.44122
$592$		0			0
$593$	−4.05829e10	−	4.05829e10i	−0.328189	−	0.328189i	0.523709	−	0.851897i	$-0.324548\pi$
$593$	−4.05829e10	−	4.05829e10i	−0.328189	−	0.328189i	−0.851897	+	0.523709i	$0.824548\pi$
$594$		0			0
$595$	6.44946e10	+	3.59335e10i	0.514583	+	0.286703i
$596$		0			0
$597$	−8.97348e10	+	8.97348e10i	−0.706421	+	0.706421i
$598$		0			0
$599$		−	2.32471e11i		−	1.80576i	−0.429889	−	0.902882i	$-0.641447\pi$
$599$		−	2.32471e11i		−	1.80576i	0.429889	−	0.902882i	$-0.358553\pi$
$600$		0			0
$601$	−1.18707e11			−0.909871			−0.454936	−	0.890524i	$-0.650338\pi$
$601$	−1.18707e11			−0.909871			−0.454936	+	0.890524i	$0.650338\pi$
$602$		0			0
$603$	1.72361e11	+	1.72361e11i	1.30367	+	1.30367i
$604$		0			0
$605$	−4.87463e10	+	8.74915e10i	−0.363848	+	0.653047i
$606$		0			0
$607$	4.73692e10	−	4.73692e10i	0.348932	−	0.348932i	−0.510779	−	0.859712i	$-0.670643\pi$
$607$	4.73692e10	−	4.73692e10i	0.348932	−	0.348932i	0.859712	+	0.510779i	$0.170643\pi$
$608$		0			0
$609$			4.19053e10i			0.304649i
$610$		0			0
$611$	−2.02888e10			−0.145577
$612$		0			0
$613$	2.10662e10	+	2.10662e10i	0.149191	+	0.149191i	0.777757	−	0.628565i	$-0.216358\pi$
$613$	2.10662e10	+	2.10662e10i	0.149191	+	0.149191i	−0.628565	+	0.777757i	$0.716358\pi$
$614$		0			0
$615$	3.28699e11	−	9.34798e10i	2.29772	−	0.653457i
$616$		0			0
$617$	1.01733e11	−	1.01733e11i	0.701972	−	0.701972i	−0.262861	−	0.964834i	$-0.584666\pi$
$617$	1.01733e11	−	1.01733e11i	0.701972	−	0.701972i	0.964834	+	0.262861i	$0.0846662\pi$
$618$		0			0
$619$		−	3.53524e10i		−	0.240800i	−0.992725	−	0.120400i	$-0.961582\pi$
$619$		−	3.53524e10i		−	0.240800i	0.992725	−	0.120400i	$-0.0384177\pi$
$620$		0			0
$621$	−6.14870e11			−4.13444
$622$		0			0
$623$	5.80866e10	+	5.80866e10i	0.385588	+	0.385588i
$624$		0			0
$625$	−6.80805e10	+	1.36558e11i	−0.446172	+	0.894947i
$626$		0			0
$627$	−1.88284e11	+	1.88284e11i	−1.21827	+	1.21827i
$628$		0			0
$629$			8.47587e10i			0.541479i
$630$		0			0
$631$	1.25311e11			0.790444			0.395222	−	0.918586i	$-0.370668\pi$
$631$	1.25311e11			0.790444			0.395222	+	0.918586i	$0.370668\pi$
$632$		0			0
$633$	−3.64233e11	−	3.64233e11i	−2.26863	−	2.26863i
$634$		0			0
$635$	2.93690e10	+	1.03269e11i	0.180632	+	0.635148i
$636$		0			0
$637$	−1.97698e9	+	1.97698e9i	−0.0120073	+	0.0120073i
$638$		0			0
$639$			3.97898e11i			2.38654i
$640$		0			0
$641$	3.11520e11			1.84525			0.922623	−	0.385702i	$-0.126041\pi$
$641$	3.11520e11			1.84525			0.922623	+	0.385702i	$0.126041\pi$
$642$		0			0
$643$	−1.50924e10	−	1.50924e10i	−0.0882904	−	0.0882904i	0.661582	−	0.749873i	$-0.269885\pi$
$643$	−1.50924e10	−	1.50924e10i	−0.0882904	−	0.0882904i	−0.749873	+	0.661582i	$0.769885\pi$
$644$		0			0
$645$	3.25028e11	+	1.81091e11i	1.87794	+	1.04630i
$646$		0			0
$647$	−3.79847e10	+	3.79847e10i	−0.216766	+	0.216766i	−0.807134	−	0.590368i	$-0.798983\pi$
$647$	−3.79847e10	+	3.79847e10i	−0.216766	+	0.216766i	0.590368	+	0.807134i	$0.298983\pi$
$648$		0			0
$649$			5.46058e10i			0.307794i
$650$		0			0
$651$	−1.75641e11			−0.977914
$652$		0			0
$653$	1.69412e11	+	1.69412e11i	0.931732	+	0.931732i	0.997814	−	0.0660822i	$-0.0210499\pi$
$653$	1.69412e11	+	1.69412e11i	0.931732	+	0.931732i	−0.0660822	+	0.997814i	$0.521050\pi$
$654$		0			0
$655$	1.14855e11	−	2.06145e11i	0.624000	−	1.11998i
$656$		0			0
$657$	2.80463e11	−	2.80463e11i	1.50527	−	1.50527i
$658$		0			0
$659$		−	2.02230e11i		−	1.07227i	−0.844132	−	0.536135i	$-0.819884\pi$
$659$		−	2.02230e11i		−	1.07227i	0.844132	−	0.536135i	$-0.180116\pi$
$660$		0			0
$661$	1.19184e11			0.624326			0.312163	−	0.950029i	$-0.398947\pi$
$661$	1.19184e11			0.624326			0.312163	+	0.950029i	$0.398947\pi$
$662$		0			0
$663$	−3.11360e10	−	3.11360e10i	−0.161142	−	0.161142i
$664$		0			0
$665$	−3.40804e11	+	9.69225e10i	−1.74268	+	0.495608i
$666$		0			0
$667$	4.36278e10	−	4.36278e10i	0.220424	−	0.220424i
$668$		0			0
$669$		−	2.98905e11i		−	1.49220i
$670$		0			0
$671$	4.02020e10			0.198316
$672$		0			0
$673$	−2.67365e11	−	2.67365e11i	−1.30330	−	1.30330i	−0.926154	−	0.377145i	$-0.876906\pi$
$673$	−2.67365e11	−	2.67365e11i	−1.30330	−	1.30330i	−0.377145	−	0.926154i	$-0.623094\pi$
$674$		0			0
$675$	−4.70467e11	−	1.10773e11i	−2.26628	−	0.533605i
$676$		0			0
$677$	1.16174e11	−	1.16174e11i	0.553037	−	0.553037i	−0.374279	−	0.927316i	$-0.622110\pi$
$677$	1.16174e11	−	1.16174e11i	0.553037	−	0.553037i	0.927316	+	0.374279i	$0.122110\pi$
$678$		0			0
$679$		−	1.10451e11i		−	0.519627i
$680$		0			0
$681$	3.62517e11			1.68554
$682$		0			0
$683$	5.01803e10	+	5.01803e10i	0.230595	+	0.230595i	0.812941	−	0.582346i	$-0.197865\pi$
$683$	5.01803e10	+	5.01803e10i	0.230595	+	0.230595i	−0.582346	+	0.812941i	$0.697865\pi$
$684$		0			0
$685$	2.56566e10	+	9.02152e10i	0.116530	+	0.409749i
$686$		0			0
$687$	4.75166e11	−	4.75166e11i	2.13313	−	2.13313i
$688$		0			0
$689$			3.94625e10i			0.175109i
$690$		0			0
$691$	−3.63223e11			−1.59317			−0.796583	−	0.604530i	$-0.793361\pi$
$691$	−3.63223e11			−1.59317			−0.796583	+	0.604530i	$0.793361\pi$
$692$		0			0
$693$	1.79258e11	+	1.79258e11i	0.777222	+	0.777222i
$694$		0			0
$695$	−4.51719e10	−	2.51678e10i	−0.193611	−	0.107871i
$696$		0			0
$697$	−1.35317e11	+	1.35317e11i	−0.573352	+	0.573352i
$698$		0			0
$699$			5.99102e11i			2.50953i
$700$		0			0
$701$	2.17997e11			0.902774			0.451387	−	0.892328i	$-0.350929\pi$
$701$	2.17997e11			0.902774			0.451387	+	0.892328i	$0.350929\pi$
$702$		0			0
$703$	−2.87630e11	−	2.87630e11i	−1.17764	−	1.17764i
$704$		0			0
$705$	1.55196e11	−	2.78550e11i	0.628237	−	1.12758i
$706$		0			0
$707$	4.55742e10	−	4.55742e10i	0.182407	−	0.182407i
$708$		0			0
$709$			2.27164e11i			0.898991i	0.893283	+	0.449495i	$0.148396\pi$
$709$			2.27164e11i			0.898991i	−0.893283	+	0.449495i	$0.851604\pi$
$710$		0			0
$711$	−2.68506e10			−0.105069
$712$		0			0
$713$	1.82860e11	+	1.82860e11i	0.707556	+	0.707556i
$714$		0			0
$715$	−2.58162e10	+	7.34197e9i	−0.0987799	+	0.0280924i
$716$		0			0
$717$	−5.59525e11	+	5.59525e11i	−2.11711	+	2.11711i
$718$		0			0
$719$			2.40489e11i			0.899869i	0.893062	+	0.449934i	$0.148553\pi$
$719$			2.40489e11i			0.899869i	−0.893062	+	0.449934i	$0.851447\pi$
$720$		0			0
$721$	3.03161e11			1.12185
$722$		0			0
$723$	3.01324e11	+	3.01324e11i	1.10276	+	1.10276i
$724$		0			0
$725$	4.12416e10	−	2.55219e10i	0.149274	−	0.0923763i
$726$		0			0
$727$	−1.86233e11	+	1.86233e11i	−0.666682	+	0.666682i	−0.956946	−	0.290265i	$-0.906257\pi$
$727$	−1.86233e11	+	1.86233e11i	−0.666682	+	0.666682i	0.290265	+	0.956946i	$0.406257\pi$
$728$		0			0
$729$		−	5.68058e10i		−	0.201133i
$730$		0			0
$731$	−2.08356e11			−0.729688
$732$		0			0
$733$	−2.62033e11	−	2.62033e11i	−0.907694	−	0.907694i	0.0883914	−	0.996086i	$-0.471827\pi$
$733$	−2.62033e11	−	2.62033e11i	−0.907694	−	0.907694i	−0.996086	+	0.0883914i	$0.971827\pi$
$734$		0			0
$735$	−1.20199e10	−	4.22650e10i	−0.0411861	−	0.144821i
$736$		0			0
$737$	−8.45848e10	+	8.45848e10i	−0.286696	+	0.286696i
$738$		0			0
$739$		−	4.62245e11i		−	1.54987i	−0.632043	−	0.774933i	$-0.717783\pi$
$739$		−	4.62245e11i		−	1.54987i	0.632043	−	0.774933i	$-0.282217\pi$
$740$		0			0
$741$	2.11321e11			0.700921
$742$		0			0
$743$	−8.72415e10	−	8.72415e10i	−0.286265	−	0.286265i	0.549336	−	0.835601i	$-0.314881\pi$
$743$	−8.72415e10	−	8.72415e10i	−0.286265	−	0.286265i	−0.835601	+	0.549336i	$0.814881\pi$
$744$		0			0
$745$	−3.07696e11	−	1.71435e11i	−0.998843	−	0.556511i
$746$		0			0
$747$	−1.72568e11	+	1.72568e11i	−0.554214	+	0.554214i
$748$		0			0
$749$		−	1.93626e11i		−	0.615228i
$750$		0			0
$751$	−2.78859e11			−0.876648			−0.438324	−	0.898817i	$-0.644428\pi$
$751$	−2.78859e11			−0.876648			−0.438324	+	0.898817i	$0.644428\pi$
$752$		0			0
$753$	2.67264e11	+	2.67264e11i	0.831306	+	0.831306i
$754$		0			0
$755$	−6.23343e10	+	1.11880e11i	−0.191840	+	0.344320i
$756$		0			0
$757$	−1.49667e11	+	1.49667e11i	−0.455768	+	0.455768i	−0.897263	−	0.441496i	$-0.854448\pi$
$757$	−1.49667e11	+	1.49667e11i	−0.455768	+	0.455768i	0.441496	+	0.897263i	$0.354448\pi$
$758$		0			0
$759$		−	5.36626e11i		−	1.61698i
$760$		0			0
$761$	−5.68435e11			−1.69489			−0.847447	−	0.530880i	$-0.821861\pi$
$761$	−5.68435e11			−1.69489			−0.847447	+	0.530880i	$0.821861\pi$
$762$		0			0
$763$	3.03149e11	+	3.03149e11i	0.894454	+	0.894454i
$764$		0			0
$765$	4.62990e11	−	1.31671e11i	1.35184	−	0.384455i
$766$		0			0
$767$	3.06434e10	−	3.06434e10i	0.0885432	−	0.0885432i
$768$		0			0
$769$			2.23982e11i			0.640483i	0.947336	+	0.320241i	$0.103764\pi$
$769$			2.23982e11i			0.640483i	−0.947336	+	0.320241i	$0.896236\pi$
$770$		0			0
$771$	1.08323e12			3.06551
$772$		0			0
$773$	3.00747e11	+	3.00747e11i	0.842333	+	0.842333i	0.989162	−	0.146829i	$-0.0469067\pi$
$773$	3.00747e11	+	3.00747e11i	0.842333	+	0.842333i	−0.146829	+	0.989162i	$0.546907\pi$
$774$		0			0
$775$	1.06972e11	+	1.72859e11i	0.296525	+	0.479164i
$776$		0			0
$777$	−3.93702e11	+	3.93702e11i	−1.08015	+	1.08015i
$778$		0			0
$779$		−	9.18400e11i		−	2.49392i
$780$		0			0
$781$	−1.95265e11			−0.524833
$782$		0			0
$783$	1.08630e11	+	1.08630e11i	0.289004	+	0.289004i
$784$		0			0
$785$	6.62359e10	+	2.32902e11i	0.174427	+	0.613331i
$786$		0			0
$787$	−3.58727e11	+	3.58727e11i	−0.935114	+	0.935114i	−0.998019	−	0.0629054i	$-0.979963\pi$
$787$	−3.58727e11	+	3.58727e11i	−0.935114	+	0.935114i	0.0629054	+	0.998019i	$0.479963\pi$
$788$		0			0
$789$		−	5.24870e11i		−	1.35439i
$790$		0			0
$791$	−1.79064e11			−0.457408
$792$		0			0
$793$	−2.25603e10	−	2.25603e10i	−0.0570497	−	0.0570497i
$794$		0			0
$795$	−5.41791e11	−	3.01862e11i	−1.35632	−	0.755683i
$796$		0			0
$797$	3.93532e11	−	3.93532e11i	0.975319	−	0.975319i	−0.0243841	−	0.999703i	$-0.507762\pi$
$797$	3.93532e11	−	3.93532e11i	0.975319	−	0.975319i	0.999703	+	0.0243841i	$0.00776245\pi$
$798$		0			0
$799$			1.78562e11i			0.438130i
$800$		0			0
$801$	5.35578e11			1.30104
$802$		0			0
$803$	1.37635e11	+	1.37635e11i	0.331030	+	0.331030i
$804$		0			0
$805$	3.47542e11	−	6.23779e11i	0.827606	−	1.48541i
$806$		0			0
$807$	1.28321e11	−	1.28321e11i	0.302553	−	0.302553i
$808$		0			0
$809$			6.87091e11i			1.60406i	0.597285	+	0.802029i	$0.296246\pi$
$809$			6.87091e11i			1.60406i	−0.597285	+	0.802029i	$0.703754\pi$
$810$		0			0
$811$	−1.86860e11			−0.431949			−0.215974	−	0.976399i	$-0.569293\pi$
$811$	−1.86860e11			−0.431949			−0.215974	+	0.976399i	$0.569293\pi$
$812$		0			0
$813$	−8.53275e11	−	8.53275e11i	−1.95311	−	1.95311i
$814$		0			0
$815$	1.68352e11	−	4.78781e10i	0.381581	−	0.108519i
$816$		0			0
$817$	7.07060e11	−	7.07060e11i	1.58697	−	1.58697i
$818$		0			0
$819$		−	2.01190e11i		−	0.447168i
$820$		0			0
$821$	−1.49846e11			−0.329816			−0.164908	−	0.986309i	$-0.552733\pi$
$821$	−1.49846e11			−0.329816			−0.164908	+	0.986309i	$0.552733\pi$
$822$		0			0
$823$	1.02577e11	+	1.02577e11i	0.223588	+	0.223588i	0.810008	−	0.586419i	$-0.199463\pi$
$823$	1.02577e11	+	1.02577e11i	0.223588	+	0.223588i	−0.586419	+	0.810008i	$0.699463\pi$
$824$		0			0
$825$	9.66770e10	−	4.10599e11i	0.208693	−	0.886343i
$826$		0			0
$827$	−4.29025e11	+	4.29025e11i	−0.917193	+	0.917193i	−0.996824	−	0.0796316i	$-0.974626\pi$
$827$	−4.29025e11	+	4.29025e11i	−0.917193	+	0.917193i	0.0796316	+	0.996824i	$0.474626\pi$
$828$		0			0
$829$			3.52109e11i			0.745518i	0.927928	+	0.372759i	$0.121588\pi$
$829$			3.52109e11i			0.745518i	−0.927928	+	0.372759i	$0.878412\pi$
$830$		0			0
$831$	1.33422e10			0.0279785
$832$		0			0
$833$	1.73994e10	+	1.73994e10i	0.0361372	+	0.0361372i
$834$		0			0
$835$	−5.20987e10	−	1.83192e11i	−0.107172	−	0.376844i
$836$		0			0
$837$	−4.55310e11	+	4.55310e11i	−0.927694	+	0.927694i
$838$		0			0
$839$		−	1.48205e9i		−	0.00299100i	−0.999999	−	0.00149550i	$-0.999524\pi$
$839$		−	1.48205e9i		−	0.00299100i	0.999999	−	0.00149550i	$-0.000476032\pi$
$840$		0			0
$841$	4.84831e11			0.969184
$842$		0			0
$843$	−2.48514e11	−	2.48514e11i	−0.492085	−	0.492085i
$844$		0			0
$845$	−4.26763e11	−	2.37773e11i	−0.837066	−	0.466376i
$846$		0			0
$847$	2.60517e11	−	2.60517e11i	0.506176	−	0.506176i
$848$		0			0
$849$			1.23328e12i			2.37372i
$850$		0			0
$851$	8.19769e11			1.56305
$852$		0			0
$853$	4.15266e11	+	4.15266e11i	0.784388	+	0.784388i	0.980568	−	0.196180i	$-0.0628538\pi$
$853$	4.15266e11	+	4.15266e11i	0.784388	+	0.784388i	−0.196180	+	0.980568i	$0.562854\pi$
$854$		0			0
$855$	−1.12433e12	+	2.01799e12i	−2.10393	+	3.77620i
$856$		0			0
$857$	4.48030e10	−	4.48030e10i	0.0830585	−	0.0830585i	−0.664357	−	0.747415i	$-0.731295\pi$
$857$	4.48030e10	−	4.48030e10i	0.0830585	−	0.0830585i	0.747415	+	0.664357i	$0.231295\pi$
$858$		0			0
$859$			6.15779e11i			1.13097i	0.824757	+	0.565487i	$0.191312\pi$
$859$			6.15779e11i			1.13097i	−0.824757	+	0.565487i	$0.808688\pi$
$860$		0			0
$861$	−1.25709e12			−2.28746
$862$		0			0
$863$	4.13379e11	+	4.13379e11i	0.745256	+	0.745256i	0.973584	−	0.228328i	$-0.0733259\pi$
$863$	4.13379e11	+	4.13379e11i	0.745256	+	0.745256i	−0.228328	+	0.973584i	$0.573326\pi$
$864$		0			0
$865$	−4.08924e11	+	1.16295e11i	−0.730429	+	0.207729i
$866$		0			0
$867$	4.50084e11	−	4.50084e11i	0.796558	−	0.796558i
$868$		0			0
$869$		−	1.31768e10i		−	0.0231062i
$870$		0			0
$871$	9.49337e10			0.164948
$872$		0			0
$873$	−5.09199e11	−	5.09199e11i	−0.876659	−	0.876659i
$874$		0			0
$875$	3.78300e11	−	4.14672e11i	0.645362	−	0.707412i
$876$		0			0
$877$	5.55718e11	−	5.55718e11i	0.939412	−	0.939412i	−0.0588548	−	0.998267i	$-0.518745\pi$
$877$	5.55718e11	−	5.55718e11i	0.939412	−	0.939412i	0.998267	+	0.0588548i	$0.0187449\pi$
$878$		0			0
$879$		−	6.45626e10i		−	0.108150i
$880$		0			0
$881$	−9.08891e11			−1.50872			−0.754359	−	0.656462i	$-0.772052\pi$
$881$	−9.08891e11			−1.50872			−0.754359	+	0.656462i	$0.772052\pi$
$882$		0			0
$883$	1.59884e11	+	1.59884e11i	0.263005	+	0.263005i	0.826274	−	0.563269i	$-0.190457\pi$
$883$	1.59884e11	+	1.59884e11i	0.263005	+	0.263005i	−0.563269	+	0.826274i	$0.690457\pi$
$884$		0			0
$885$	1.86310e11	+	6.55112e11i	0.303712	+	1.06793i
$886$		0			0
$887$	3.25058e11	−	3.25058e11i	0.525130	−	0.525130i	−0.393987	−	0.919116i	$-0.628904\pi$
$887$	3.25058e11	−	3.25058e11i	0.525130	−	0.525130i	0.919116	+	0.393987i	$0.128904\pi$
$888$		0			0
$889$		−	3.94946e11i		−	0.632310i
$890$		0			0
$891$	6.12720e11			0.972190
$892$		0			0
$893$	−6.05953e11	−	6.05953e11i	−0.952869	−	0.952869i
$894$		0			0
$895$	6.34922e11	+	3.53750e11i	0.989528	+	0.551321i
$896$		0			0
$897$	−3.01141e11	+	3.01141e11i	−0.465157	+	0.465157i
$898$		0			0
$899$		−	6.46125e10i		−	0.0989186i
$900$		0			0
$901$	3.47311e11			0.527010
$902$		0			0
$903$	−9.67810e11	−	9.67810e11i	−1.45559	−	1.45559i
$904$		0			0
$905$	5.27268e11	−	9.46357e11i	0.786026	−	1.41078i
$906$		0			0
$907$	−9.43868e10	+	9.43868e10i	−0.139470	+	0.139470i	−0.773395	−	0.633925i	$-0.781443\pi$
$907$	−9.43868e10	+	9.43868e10i	−0.139470	+	0.139470i	0.633925	+	0.773395i	$0.281443\pi$
$908$		0			0
$909$		−	4.20209e11i		−	0.615474i
$910$		0			0
$911$	−8.68878e11			−1.26149			−0.630747	−	0.775988i	$-0.717251\pi$
$911$	−8.68878e11			−1.26149			−0.630747	+	0.775988i	$0.717251\pi$
$912$		0			0
$913$	−8.46864e10	−	8.46864e10i	−0.121879	−	0.121879i
$914$		0			0
$915$	4.82308e11	−	1.37165e11i	0.688082	−	0.195686i
$916$		0			0
$917$	−6.13823e11	+	6.13823e11i	−0.868092	+	0.868092i
$918$		0			0
$919$			5.29410e11i			0.742215i	0.928590	+	0.371108i	$0.121022\pi$
$919$			5.29410e11i			0.742215i	−0.928590	+	0.371108i	$0.878978\pi$
$920$		0			0
$921$	−6.33055e11			−0.879839
$922$		0			0
$923$	1.09578e11	+	1.09578e11i	0.150979	+	0.150979i
$924$		0			0
$925$	6.27245e11	+	1.47687e11i	0.856782	+	0.201733i
$926$		0			0
$927$	1.39762e12	−	1.39762e12i	1.89266	−	1.89266i
$928$		0			0
$929$			3.26350e11i			0.438147i	0.975708	+	0.219074i	$0.0703035\pi$
$929$			3.26350e11i			0.438147i	−0.975708	+	0.219074i	$0.929696\pi$
$930$		0			0
$931$	−1.18090e11			−0.157187
$932$		0			0
$933$	7.43408e11	+	7.43408e11i	0.981072	+	0.981072i
$934$		0			0
$935$	6.46168e10	+	2.27209e11i	0.0845472	+	0.297289i
$936$		0			0
$937$	−2.74343e11	+	2.74343e11i	−0.355906	+	0.355906i	−0.862301	−	0.506396i	$-0.830978\pi$
$937$	−2.74343e11	+	2.74343e11i	−0.355906	+	0.355906i	0.506396	+	0.862301i	$0.330978\pi$
$938$		0			0
$939$		−	1.46601e12i		−	1.88571i
$940$		0			0
$941$	8.80306e11			1.12273			0.561365	−	0.827568i	$-0.310276\pi$
$941$	8.80306e11			1.12273			0.561365	+	0.827568i	$0.310276\pi$
$942$		0			0
$943$	1.30876e12	+	1.30876e12i	1.65506	+	1.65506i
$944$		0			0
$945$	1.55317e12	+	8.65357e11i	1.94756	+	1.08510i
$946$		0			0
$947$	5.38215e11	−	5.38215e11i	0.669200	−	0.669200i	−0.288331	−	0.957531i	$-0.593100\pi$
$947$	5.38215e11	−	5.38215e11i	0.669200	−	0.669200i	0.957531	+	0.288331i	$0.0931003\pi$
$948$		0			0
$949$		−	1.54475e11i		−	0.190455i
$950$		0			0
$951$	−1.78148e12			−2.17800
$952$		0			0
$953$	−7.87404e11	−	7.87404e11i	−0.954610	−	0.954610i	0.0444032	−	0.999014i	$-0.485861\pi$
$953$	−7.87404e11	−	7.87404e11i	−0.954610	−	0.954610i	−0.999014	+	0.0444032i	$0.985861\pi$
$954$		0			0
$955$	−9.00033e10	+	1.61541e11i	−0.108204	+	0.194209i
$956$		0			0
$957$	−9.48067e10	+	9.48067e10i	−0.113029	+	0.113029i
$958$		0			0
$959$		−	3.45022e11i		−	0.407918i
$960$		0			0
$961$	−5.82076e11			−0.682474
$962$		0			0
$963$	−8.92647e11	−	8.92647e11i	−1.03795	−	1.03795i
$964$		0			0
$965$	−9.31150e11	+	2.64813e11i	−1.07377	+	0.305372i
$966$		0			0
$967$	−6.84900e11	+	6.84900e11i	−0.783288	+	0.783288i	−0.980384	−	0.197096i	$-0.936849\pi$
$967$	−6.84900e11	+	6.84900e11i	−0.783288	+	0.783288i	0.197096	+	0.980384i	$0.436849\pi$
$968$		0			0
$969$		−	1.85984e12i		−	2.10950i
$970$		0			0
$971$	−1.46919e12			−1.65273			−0.826366	−	0.563134i	$-0.809596\pi$
$971$	−1.46919e12			−1.65273			−0.826366	+	0.563134i	$0.809596\pi$
$972$		0			0
$973$	1.34505e11	+	1.34505e11i	0.150067	+	0.150067i
$974$		0			0
$975$	−2.84670e11	+	1.76165e11i	−0.315010	+	0.194940i
$976$		0			0
$977$	8.05637e11	−	8.05637e11i	0.884222	−	0.884222i	−0.109738	−	0.993961i	$-0.535001\pi$
$977$	8.05637e11	−	8.05637e11i	0.884222	−	0.884222i	0.993961	+	0.109738i	$0.0350013\pi$
$978$		0			0
$979$			2.62831e11i			0.286118i
$980$		0			0
$981$	2.79513e12			3.01805
$982$		0			0
$983$	−7.89432e10	−	7.89432e10i	−0.0845474	−	0.0845474i	0.663568	−	0.748116i	$-0.269041\pi$
$983$	−7.89432e10	−	7.89432e10i	−0.0845474	−	0.0845474i	−0.748116	+	0.663568i	$0.769041\pi$
$984$		0			0
$985$	−2.04768e11	−	7.20015e11i	−0.217529	−	0.764886i
$986$		0			0
$987$	−8.29417e11	+	8.29417e11i	−0.873986	+	0.873986i
$988$		0			0
$989$			2.01518e12i			2.10634i
$990$		0			0
$991$	−9.14136e11			−0.947799			−0.473899	−	0.880579i	$-0.657154\pi$
$991$	−9.14136e11			−0.947799			−0.473899	+	0.880579i	$0.657154\pi$
$992$		0			0
$993$	1.13230e12	+	1.13230e12i	1.16456	+	1.16456i
$994$		0			0
$995$	−4.71977e11	−	2.62964e11i	−0.481535	−	0.268290i
$996$		0			0
$997$	6.87339e11	−	6.87339e11i	0.695650	−	0.695650i	−0.267819	−	0.963469i	$-0.586303\pi$
$997$	6.87339e11	−	6.87339e11i	0.695650	−	0.695650i	0.963469	+	0.267819i	$0.0863031\pi$
$998$		0			0
$999$			2.04117e12i			2.04936i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	20.9.f.a.13.1	✓	8
3.2	odd	2		180.9.l.a.73.2		8
4.3	odd	2		80.9.p.d.33.4		8
5.2	odd	4	inner	20.9.f.a.17.1	yes	8
5.3	odd	4		100.9.f.b.57.4		8
5.4	even	2		100.9.f.b.93.4		8
15.2	even	4		180.9.l.a.37.2		8
20.7	even	4		80.9.p.d.17.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
20.9.f.a.13.1	✓	8	1.1	even	1	trivial
20.9.f.a.17.1	yes	8	5.2	odd	4	inner
80.9.p.d.17.4		8	20.7	even	4
80.9.p.d.33.4		8	4.3	odd	2
100.9.f.b.57.4		8	5.3	odd	4
100.9.f.b.93.4		8	5.4	even	2
180.9.l.a.37.2		8	15.2	even	4
180.9.l.a.73.2		8	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 20 = 2^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	20.f (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-103.804 - 103.804i) q^{3} +(304.194 - 545.977i) q^{5} +(-1625.71 + 1625.71i) q^{7} +14989.6i q^{9} +O(q^{10})\)	\(q+(-103.804 - 103.804i) q^{3} +(304.194 - 545.977i) q^{5} +(-1625.71 + 1625.71i) q^{7} +14989.6i q^{9} -7356.04 q^{11} +(4128.03 + 4128.03i) q^{13} +(-88251.3 + 25098.1i) q^{15} +(36330.8 - 36330.8i) q^{17} +246578. i q^{19} +337512. q^{21} +(-351384. - 351384. i) q^{23} +(-205557. - 332166. i) q^{25} +(874925. - 874925. i) q^{27} +124160. i q^{29} -520399. q^{31} +(763588. + 763588. i) q^{33} +(393070. + 1.38213e6i) q^{35} +(-1.16648e6 + 1.16648e6i) q^{37} -857012. i q^{39} -3.72458e6 q^{41} +(-2.86749e6 - 2.86749e6i) q^{43} +(8.18398e6 + 4.55975e6i) q^{45} +(-2.45745e6 + 2.45745e6i) q^{47} +478916. i q^{49} -7.54258e6 q^{51} +(4.77983e6 + 4.77983e6i) q^{53} +(-2.23766e6 + 4.01623e6i) q^{55} +(2.55958e7 - 2.55958e7i) q^{57} -7.42325e6i q^{59} -5.46517e6 q^{61} +(-2.43688e7 - 2.43688e7i) q^{63} +(3.50953e6 - 998087. i) q^{65} +(1.14987e7 - 1.14987e7i) q^{67} +7.29503e7i q^{69} +2.65449e7 q^{71} +(-1.87105e7 - 1.87105e7i) q^{73} +(-1.31425e7 + 5.58179e7i) q^{75} +(1.19588e7 - 1.19588e7i) q^{77} +1.79128e6i q^{79} -8.32948e7 q^{81} +(1.15125e7 + 1.15125e7i) q^{83} +(-8.78418e6 - 3.08874e7i) q^{85} +(1.28883e7 - 1.28883e7i) q^{87} -3.57299e7i q^{89} -1.34220e7 q^{91} +(5.40195e7 + 5.40195e7i) q^{93} +(1.34626e8 + 7.50076e7i) q^{95} +(-3.39701e7 + 3.39701e7i) q^{97} -1.10264e8i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 70 q^{3} + 894 q^{5} - 2030 q^{7}+O(q^{10}) \) 8 * q - 70 * q^3 + 894 * q^5 - 2030 * q^7	\( 8 q - 70 q^{3} + 894 q^{5} - 2030 q^{7} - 420 q^{11} + 33180 q^{13} - 48478 q^{15} + 43620 q^{17} + 108668 q^{21} - 663270 q^{23} + 163396 q^{25} + 1576040 q^{27} - 3178492 q^{31} - 944020 q^{33} + 2571618 q^{35} + 5344080 q^{37} - 10185252 q^{41} - 10342710 q^{43} + 20284834 q^{45} + 19232250 q^{47} - 47126684 q^{51} - 24320640 q^{53} + 21483180 q^{55} + 88218320 q^{57} - 82515684 q^{61} - 77441350 q^{63} + 72045768 q^{65} + 100675930 q^{67} - 99290076 q^{71} - 93528520 q^{73} + 76524178 q^{75} + 134199660 q^{77} - 161920268 q^{81} - 10450350 q^{83} + 51676156 q^{85} + 164801600 q^{87} - 130681068 q^{91} - 50183620 q^{93} + 84367944 q^{95} - 179570760 q^{97}+O(q^{100}) \) 8 * q - 70 * q^3 + 894 * q^5 - 2030 * q^7 - 420 * q^11 + 33180 * q^13 - 48478 * q^15 + 43620 * q^17 + 108668 * q^21 - 663270 * q^23 + 163396 * q^25 + 1576040 * q^27 - 3178492 * q^31 - 944020 * q^33 + 2571618 * q^35 + 5344080 * q^37 - 10185252 * q^41 - 10342710 * q^43 + 20284834 * q^45 + 19232250 * q^47 - 47126684 * q^51 - 24320640 * q^53 + 21483180 * q^55 + 88218320 * q^57 - 82515684 * q^61 - 77441350 * q^63 + 72045768 * q^65 + 100675930 * q^67 - 99290076 * q^71 - 93528520 * q^73 + 76524178 * q^75 + 134199660 * q^77 - 161920268 * q^81 - 10450350 * q^83 + 51676156 * q^85 + 164801600 * q^87 - 130681068 * q^91 - 50183620 * q^93 + 84367944 * q^95 - 179570760 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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