LMFDB - Embedded newform 20.9.f.a.17.4

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			17.4
Root			$1.89682 - 0.896816i$ of defining polynomial
Character	$\chi$	$=$	20.17
Dual form			20.9.f.a.13.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(80.1275 - 80.1275i) q^{3} +(621.850 + 62.6699i) q^{5} +(-1045.24 - 1045.24i) q^{7} -6279.84i q^{9} +O(q^{10})$	\(q+(80.1275 - 80.1275i) q^{3} +(621.850 + 62.6699i) q^{5} +(-1045.24 - 1045.24i) q^{7} -6279.84i q^{9} -1097.03 q^{11} +(36593.4 - 36593.4i) q^{13} +(54848.9 - 44805.7i) q^{15} +(-67233.5 - 67233.5i) q^{17} +177336. i q^{19} -167505. q^{21} +(-110984. + 110984. i) q^{23} +(382770. + 77942.6i) q^{25} +(22528.8 + 22528.8i) q^{27} +903078. i q^{29} -1.11746e6 q^{31} +(-87902.5 + 87902.5i) q^{33} +(-584479. - 715489. i) q^{35} +(1.48912e6 + 1.48912e6i) q^{37} -5.86427e6i q^{39} +4.18381e6 q^{41} +(-1.23717e6 + 1.23717e6i) q^{43} +(393557. - 3.90512e6i) q^{45} +(3.39500e6 + 3.39500e6i) q^{47} -3.57974e6i q^{49} -1.07745e7 q^{51} +(-6.26950e6 + 6.26950e6i) q^{53} +(-682190. - 68750.9i) q^{55} +(1.42095e7 + 1.42095e7i) q^{57} +4.61361e6i q^{59} -1.06398e7 q^{61} +(-6.56395e6 + 6.56395e6i) q^{63} +(2.50489e7 - 2.04623e7i) q^{65} +(-3.39626e6 - 3.39626e6i) q^{67} +1.77857e7i q^{69} -4.31250e7 q^{71} +(-9.30749e6 + 9.30749e6i) q^{73} +(3.69158e7 - 2.44251e7i) q^{75} +(1.14666e6 + 1.14666e6i) q^{77} -5.42990e7i q^{79} +4.48124e7 q^{81} +(4.43294e7 - 4.43294e7i) q^{83} +(-3.75956e7 - 4.60227e7i) q^{85} +(7.23614e7 + 7.23614e7i) q^{87} -2.12038e7i q^{89} -7.64978e7 q^{91} +(-8.95395e7 + 8.95395e7i) q^{93} +(-1.11136e7 + 1.10276e8i) q^{95} +(-4.71632e7 - 4.71632e7i) q^{97} +6.88919e6i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/20\mathbb{Z}\right)^\times$.

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

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
$3$	80.1275	−	80.1275i	0.989229	−	0.989229i	−0.0107140	−	0.999943i	$-0.503410\pi$
$3$	80.1275	−	80.1275i	0.989229	−	0.989229i	0.999943	+	0.0107140i	$0.00341043\pi$
$4$		0			0
$5$	621.850	+	62.6699i	0.994960	+	0.100272i
$5$	621.850	+	62.6699i	0.994960	+	0.100272i
$6$		0			0
$7$	−1045.24	−	1045.24i	−0.435336	−	0.435336i	0.455103	−	0.890439i	$-0.349603\pi$
$7$	−1045.24	−	1045.24i	−0.435336	−	0.435336i	−0.890439	+	0.455103i	$0.849603\pi$
$8$		0			0
$9$		−	6279.84i		−	0.957147i
$10$		0			0
$11$	−1097.03			−0.0749288			−0.0374644	−	0.999298i	$-0.511928\pi$
$11$	−1097.03			−0.0749288			−0.0374644	+	0.999298i	$0.511928\pi$
$12$		0			0
$13$	36593.4	−	36593.4i	1.28124	−	1.28124i	0.341270	−	0.939965i	$-0.389143\pi$
$13$	36593.4	−	36593.4i	1.28124	−	1.28124i	0.939965	−	0.341270i	$-0.110857\pi$
$14$		0			0
$15$	54848.9	−	44805.7i	1.08343	−	0.885051i
$16$		0			0
$17$	−67233.5	−	67233.5i	−0.804989	−	0.804989i	0.178881	−	0.983871i	$-0.442752\pi$
$17$	−67233.5	−	67233.5i	−0.804989	−	0.804989i	−0.983871	+	0.178881i	$0.942752\pi$
$18$		0			0
$19$			177336.i			1.36076i	0.732859	+	0.680380i	$0.238185\pi$
$19$			177336.i			1.36076i	−0.732859	+	0.680380i	$0.761815\pi$
$20$		0			0
$21$	−167505.			−0.861294
$22$		0			0
$23$	−110984.	+	110984.i	−0.396595	+	0.396595i	−0.877030	−	0.480435i	$-0.840479\pi$
$23$	−110984.	+	110984.i	−0.396595	+	0.396595i	0.480435	+	0.877030i	$0.340479\pi$
$24$		0			0
$25$	382770.	+	77942.6i	0.979891	+	0.199533i
$26$		0			0
$27$	22528.8	+	22528.8i	0.0423918	+	0.0423918i
$28$		0			0
$29$			903078.i			1.27683i	0.769692	+	0.638415i	$0.220410\pi$
$29$			903078.i			1.27683i	−0.769692	+	0.638415i	$0.779590\pi$
$30$		0			0
$31$	−1.11746e6			−1.21000			−0.605001	−	0.796224i	$-0.706827\pi$
$31$	−1.11746e6			−1.21000			−0.605001	+	0.796224i	$0.706827\pi$
$32$		0			0
$33$	−87902.5	+	87902.5i	−0.0741217	+	0.0741217i
$34$		0			0
$35$	−584479.	−	715489.i	−0.389490	−	0.476794i
$36$		0			0
$37$	1.48912e6	+	1.48912e6i	0.794553	+	0.794553i	0.982231	−	0.187678i	$-0.0600961\pi$
$37$	1.48912e6	+	1.48912e6i	0.794553	+	0.794553i	−0.187678	+	0.982231i	$0.560096\pi$
$38$		0			0
$39$		−	5.86427e6i		−	2.53487i
$40$		0			0
$41$	4.18381e6			1.48060			0.740298	−	0.672279i	$-0.234684\pi$
$41$	4.18381e6			1.48060			0.740298	+	0.672279i	$0.234684\pi$
$42$		0			0
$43$	−1.23717e6	+	1.23717e6i	−0.361872	+	0.361872i	−0.864502	−	0.502630i	$-0.832366\pi$
$43$	−1.23717e6	+	1.23717e6i	−0.361872	+	0.361872i	0.502630	+	0.864502i	$0.332366\pi$
$44$		0			0
$45$	393557.	−	3.90512e6i	0.0959748	−	0.952323i
$46$		0			0
$47$	3.39500e6	+	3.39500e6i	0.695741	+	0.695741i	0.963489	−	0.267748i	$-0.0862793\pi$
$47$	3.39500e6	+	3.39500e6i	0.695741	+	0.695741i	−0.267748	+	0.963489i	$0.586279\pi$
$48$		0			0
$49$		−	3.57974e6i		−	0.620965i
$50$		0			0
$51$	−1.07745e7			−1.59264
$52$		0			0
$53$	−6.26950e6	+	6.26950e6i	−0.794565	+	0.794565i	−0.982233	−	0.187668i	$-0.939907\pi$
$53$	−6.26950e6	+	6.26950e6i	−0.794565	+	0.794565i	0.187668	+	0.982233i	$0.439907\pi$
$54$		0			0
$55$	−682190.	−	68750.9i	−0.0745511	−	0.00751325i
$56$		0			0
$57$	1.42095e7	+	1.42095e7i	1.34610	+	1.34610i
$58$		0			0
$59$			4.61361e6i			0.380744i	0.981712	+	0.190372i	$0.0609694\pi$
$59$			4.61361e6i			0.380744i	−0.981712	+	0.190372i	$0.939031\pi$
$60$		0			0
$61$	−1.06398e7			−0.768446			−0.384223	−	0.923240i	$-0.625531\pi$
$61$	−1.06398e7			−0.768446			−0.384223	+	0.923240i	$0.625531\pi$
$62$		0			0
$63$	−6.56395e6	+	6.56395e6i	−0.416681	+	0.416681i
$64$		0			0
$65$	2.50489e7	−	2.04623e7i	1.40325	−	1.14631i
$66$		0			0
$67$	−3.39626e6	−	3.39626e6i	−0.168540	−	0.168540i	0.617798	−	0.786337i	$-0.288025\pi$
$67$	−3.39626e6	−	3.39626e6i	−0.168540	−	0.168540i	−0.786337	+	0.617798i	$0.788025\pi$
$68$		0			0
$69$			1.77857e7i			0.784646i
$70$		0			0
$71$	−4.31250e7			−1.69705			−0.848526	−	0.529153i	$-0.822510\pi$
$71$	−4.31250e7			−1.69705			−0.848526	+	0.529153i	$0.822510\pi$
$72$		0			0
$73$	−9.30749e6	+	9.30749e6i	−0.327749	+	0.327749i	−0.851730	−	0.523981i	$-0.824446\pi$
$73$	−9.30749e6	+	9.30749e6i	−0.327749	+	0.327749i	0.523981	+	0.851730i	$0.324446\pi$
$74$		0			0
$75$	3.69158e7	−	2.44251e7i	1.16672	−	0.771953i
$76$		0			0
$77$	1.14666e6	+	1.14666e6i	0.0326192	+	0.0326192i
$78$		0			0
$79$		−	5.42990e7i		−	1.39407i	−0.717039	−	0.697033i	$-0.754503\pi$
$79$		−	5.42990e7i		−	1.39407i	0.717039	−	0.697033i	$-0.245497\pi$
$80$		0			0
$81$	4.48124e7			1.04102
$82$		0			0
$83$	4.43294e7	−	4.43294e7i	0.934070	−	0.934070i	−0.0638872	−	0.997957i	$-0.520350\pi$
$83$	4.43294e7	−	4.43294e7i	0.934070	−	0.934070i	0.997957	+	0.0638872i	$0.0203498\pi$
$84$		0			0
$85$	−3.75956e7	−	4.60227e7i	−0.720214	−	0.881650i
$86$		0			0
$87$	7.23614e7	+	7.23614e7i	1.26308	+	1.26308i
$88$		0			0
$89$		−	2.12038e7i		−	0.337951i	−0.985620	−	0.168975i	$-0.945954\pi$
$89$		−	2.12038e7i		−	0.337951i	0.985620	−	0.168975i	$-0.0540458\pi$
$90$		0			0
$91$	−7.64978e7			−1.11554
$92$		0			0
$93$	−8.95395e7	+	8.95395e7i	−1.19697	+	1.19697i
$94$		0			0
$95$	−1.11136e7	+	1.10276e8i	−0.136446	+	1.35390i
$96$		0			0
$97$	−4.71632e7	−	4.71632e7i	−0.532741	−	0.532741i	0.388646	−	0.921387i	$-0.372943\pi$
$97$	−4.71632e7	−	4.71632e7i	−0.532741	−	0.532741i	−0.921387	+	0.388646i	$0.872943\pi$
$98$		0			0
$99$			6.88919e6i			0.0717178i
$100$		0			0
$101$	6.92150e7			0.665143			0.332571	−	0.943078i	$-0.392084\pi$
$101$	6.92150e7			0.665143			0.332571	+	0.943078i	$0.392084\pi$
$102$		0			0
$103$	−4.82442e7	+	4.82442e7i	−0.428643	+	0.428643i	−0.888166	−	0.459523i	$-0.848020\pi$
$103$	−4.82442e7	+	4.82442e7i	−0.428643	+	0.428643i	0.459523	+	0.888166i	$0.348020\pi$
$104$		0			0
$105$	−1.04163e8	−	1.04975e7i	−0.856953	−	0.0863635i
$106$		0			0
$107$	1.00650e8	+	1.00650e8i	0.767853	+	0.767853i	0.977728	−	0.209876i	$-0.0673059\pi$
$107$	1.00650e8	+	1.00650e8i	0.767853	+	0.767853i	−0.209876	+	0.977728i	$0.567306\pi$
$108$		0			0
$109$		−	1.97253e7i		−	0.139739i	−0.997556	−	0.0698696i	$-0.977742\pi$
$109$		−	1.97253e7i		−	0.139739i	0.997556	−	0.0698696i	$-0.0222583\pi$
$110$		0			0
$111$	2.38639e8			1.57199
$112$		0			0
$113$	5.38617e7	−	5.38617e7i	0.330344	−	0.330344i	−0.522373	−	0.852717i	$-0.674953\pi$
$113$	5.38617e7	−	5.38617e7i	0.330344	−	0.330344i	0.852717	+	0.522373i	$0.174953\pi$
$114$		0			0
$115$	−7.59704e7	+	6.20598e7i	−0.434363	+	0.354829i
$116$		0			0
$117$	−2.29800e8	−	2.29800e8i	−1.22633	−	1.22633i
$118$		0			0
$119$			1.40551e8i			0.700882i
$120$		0			0
$121$	−2.13155e8			−0.994386
$122$		0			0
$123$	3.35238e8	−	3.35238e8i	1.46465	−	1.46465i
$124$		0			0
$125$	2.33141e8	+	7.24567e7i	0.954945	+	0.296783i
$126$		0			0
$127$	−3.23608e8	−	3.23608e8i	−1.24395	−	1.24395i	−0.958347	−	0.285608i	$-0.907805\pi$
$127$	−3.23608e8	−	3.23608e8i	−1.24395	−	1.24395i	−0.285608	−	0.958347i	$-0.592195\pi$
$128$		0			0
$129$			1.98262e8i			0.715948i
$130$		0			0
$131$	1.08748e8			0.369263			0.184632	−	0.982808i	$-0.440891\pi$
$131$	1.08748e8			0.369263			0.184632	+	0.982808i	$0.440891\pi$
$132$		0			0
$133$	1.85359e8	−	1.85359e8i	0.592389	−	0.592389i
$134$		0			0
$135$	1.25976e7	+	1.54214e7i	0.0379275	+	0.0464289i
$136$		0			0
$137$	−8.11722e7	−	8.11722e7i	−0.230423	−	0.230423i	0.582447	−	0.812869i	$-0.302096\pi$
$137$	−8.11722e7	−	8.11722e7i	−0.230423	−	0.230423i	−0.812869	+	0.582447i	$0.802096\pi$
$138$		0			0
$139$			7.11069e7i			0.190481i	0.995454	+	0.0952407i	$0.0303621\pi$
$139$			7.11069e7i			0.190481i	−0.995454	+	0.0952407i	$0.969638\pi$
$140$		0			0
$141$	5.44065e8			1.37649
$142$		0			0
$143$	−4.01441e7	+	4.01441e7i	−0.0960014	+	0.0960014i
$144$		0			0
$145$	−5.65958e7	+	5.61579e8i	−0.128030	+	1.27039i
$146$		0			0
$147$	−2.86836e8	−	2.86836e8i	−0.614276	−	0.614276i
$148$		0			0
$149$			2.09227e8i			0.424494i	0.977216	+	0.212247i	$0.0680782\pi$
$149$			2.09227e8i			0.424494i	−0.977216	+	0.212247i	$0.931922\pi$
$150$		0			0
$151$	−5.31452e8			−1.02225			−0.511124	−	0.859507i	$-0.670771\pi$
$151$	−5.31452e8			−1.02225			−0.511124	+	0.859507i	$0.670771\pi$
$152$		0			0
$153$	−4.22216e8	+	4.22216e8i	−0.770493	+	0.770493i
$154$		0			0
$155$	−6.94894e8	−	7.00313e7i	−1.20390	−	0.121329i
$156$		0			0
$157$	2.19750e8	+	2.19750e8i	0.361684	+	0.361684i	0.864433	−	0.502748i	$-0.167678\pi$
$157$	2.19750e8	+	2.19750e8i	0.361684	+	0.361684i	−0.502748	+	0.864433i	$0.667678\pi$
$158$		0			0
$159$			1.00472e9i			1.57201i
$160$		0			0
$161$	2.32009e8			0.345304
$162$		0			0
$163$	3.89051e7	−	3.89051e7i	0.0551133	−	0.0551133i	−0.679013	−	0.734126i	$-0.737592\pi$
$163$	3.89051e7	−	3.89051e7i	0.0551133	−	0.0551133i	0.734126	+	0.679013i	$0.237592\pi$
$164$		0			0
$165$	−6.01710e7	+	4.91533e7i	−0.0811804	+	0.0663158i
$166$		0			0
$167$	4.40951e8	+	4.40951e8i	0.566923	+	0.566923i	0.931265	−	0.364342i	$-0.118706\pi$
$167$	4.40951e8	+	4.40951e8i	0.566923	+	0.566923i	−0.364342	+	0.931265i	$0.618706\pi$
$168$		0			0
$169$		−	1.86242e9i		−	2.28313i
$170$		0			0
$171$	1.11364e9			1.30245
$172$		0			0
$173$	4.44877e8	−	4.44877e8i	0.496655	−	0.496655i	−0.413740	−	0.910395i	$-0.635778\pi$
$173$	4.44877e8	−	4.44877e8i	0.496655	−	0.496655i	0.910395	+	0.413740i	$0.135778\pi$
$174$		0			0
$175$	−3.18619e8	−	4.81556e8i	−0.339718	−	0.513446i
$176$		0			0
$177$	3.69677e8	+	3.69677e8i	0.376643	+	0.376643i
$178$		0			0
$179$		−	1.83880e9i		−	1.79111i	−0.444951	−	0.895555i	$-0.646779\pi$
$179$		−	1.83880e9i		−	1.79111i	0.444951	−	0.895555i	$-0.353221\pi$
$180$		0			0
$181$	−4.39801e8			−0.409771			−0.204886	−	0.978786i	$-0.565682\pi$
$181$	−4.39801e8			−0.409771			−0.204886	+	0.978786i	$0.565682\pi$
$182$		0			0
$183$	−8.52540e8	+	8.52540e8i	−0.760169	+	0.760169i
$184$		0			0
$185$	8.32686e8	+	1.01933e9i	0.710877	+	0.870220i
$186$		0			0
$187$	7.37573e7	+	7.37573e7i	0.0603168	+	0.0603168i
$188$		0			0
$189$		−	4.70960e7i		−	0.0369094i
$190$		0			0
$191$	−1.69366e9			−1.27260			−0.636301	−	0.771441i	$-0.719536\pi$
$191$	−1.69366e9			−1.27260			−0.636301	+	0.771441i	$0.719536\pi$
$192$		0			0
$193$	2.60856e8	−	2.60856e8i	0.188006	−	0.188006i	−0.606828	−	0.794833i	$-0.707558\pi$
$193$	2.60856e8	−	2.60856e8i	0.188006	−	0.188006i	0.794833	+	0.606828i	$0.207558\pi$
$194$		0			0
$195$	3.67513e8	−	3.64670e9i	0.254176	−	2.52209i
$196$		0			0
$197$	1.19365e8	+	1.19365e8i	0.0792527	+	0.0792527i	0.745622	−	0.666369i	$-0.232153\pi$
$197$	1.19365e8	+	1.19365e8i	0.0792527	+	0.0792527i	−0.666369	+	0.745622i	$0.732153\pi$
$198$		0			0
$199$			2.90451e9i			1.85208i	0.377424	+	0.926040i	$0.376810\pi$
$199$			2.90451e9i			1.85208i	−0.377424	+	0.926040i	$0.623190\pi$
$200$		0			0
$201$	−5.44268e8			−0.333449
$202$		0			0
$203$	9.43935e8	−	9.43935e8i	0.555850	−	0.555850i
$204$		0			0
$205$	2.60170e9	+	2.62199e8i	1.47313	+	0.148462i
$206$		0			0
$207$	6.96959e8	+	6.96959e8i	0.379600	+	0.379600i
$208$		0			0
$209$		−	1.94543e8i		−	0.101960i
$210$		0			0
$211$	−4.74631e8			−0.239456			−0.119728	−	0.992807i	$-0.538202\pi$
$211$	−4.74631e8			−0.239456			−0.119728	+	0.992807i	$0.538202\pi$
$212$		0			0
$213$	−3.45550e9	+	3.45550e9i	−1.67877	+	1.67877i
$214$		0			0
$215$	−8.46866e8	+	6.91800e8i	−0.396334	+	0.323762i
$216$		0			0
$217$	1.16802e9	+	1.16802e9i	0.526758	+	0.526758i
$218$		0			0
$219$			1.49157e9i			0.648437i
$220$		0			0
$221$	−4.92060e9			−2.06276
$222$		0			0
$223$	3.99670e8	−	3.99670e8i	0.161615	−	0.161615i	−0.621667	−	0.783282i	$-0.713544\pi$
$223$	3.99670e8	−	3.99670e8i	0.161615	−	0.161615i	0.783282	+	0.621667i	$0.213544\pi$
$224$		0			0
$225$	4.89467e8	−	2.40373e9i	0.190982	−	0.937899i
$226$		0			0
$227$	1.47383e9	+	1.47383e9i	0.555066	+	0.555066i	0.927899	−	0.372833i	$-0.121613\pi$
$227$	1.47383e9	+	1.47383e9i	0.555066	+	0.555066i	−0.372833	+	0.927899i	$0.621613\pi$
$228$		0			0
$229$		−	1.10445e9i		−	0.401609i	−0.979631	−	0.200804i	$-0.935644\pi$
$229$		−	1.10445e9i		−	0.401609i	0.979631	−	0.200804i	$-0.0643556\pi$
$230$		0			0
$231$	1.83759e8			0.0645357
$232$		0			0
$233$	−1.40181e9	+	1.40181e9i	−0.475625	+	0.475625i	−0.903729	−	0.428104i	$-0.859182\pi$
$233$	−1.40181e9	+	1.40181e9i	−0.475625	+	0.475625i	0.428104	+	0.903729i	$0.359182\pi$
$234$		0			0
$235$	1.89841e9	+	2.32394e9i	0.622472	+	0.761998i
$236$		0			0
$237$	−4.35084e9	−	4.35084e9i	−1.37905	−	1.37905i
$238$		0			0
$239$		−	1.12813e9i		−	0.345753i	−0.984944	−	0.172876i	$-0.944694\pi$
$239$		−	1.12813e9i		−	0.345753i	0.984944	−	0.172876i	$-0.0553061\pi$
$240$		0			0
$241$	4.79763e9			1.42219			0.711096	−	0.703095i	$-0.248199\pi$
$241$	4.79763e9			1.42219			0.711096	+	0.703095i	$0.248199\pi$
$242$		0			0
$243$	3.44289e9	−	3.44289e9i	0.987412	−	0.987412i
$244$		0			0
$245$	2.24342e8	−	2.22606e9i	0.0622653	−	0.617835i
$246$		0			0
$247$	6.48931e9	+	6.48931e9i	1.74345	+	1.74345i
$248$		0			0
$249$		−	7.10401e9i		−	1.84802i
$250$		0			0
$251$	4.36253e7			0.0109912			0.00549558	−	0.999985i	$-0.498251\pi$
$251$	4.36253e7			0.0109912			0.00549558	+	0.999985i	$0.498251\pi$
$252$		0			0
$253$	1.21753e8	−	1.21753e8i	0.0297164	−	0.0297164i
$254$		0			0
$255$	−6.70013e9	−	6.75237e8i	−1.58461	−	0.159697i
$256$		0			0
$257$	−1.61519e9	−	1.61519e9i	−0.370246	−	0.370246i	0.497321	−	0.867567i	$-0.334317\pi$
$257$	−1.61519e9	−	1.61519e9i	−0.370246	−	0.370246i	−0.867567	+	0.497321i	$0.834317\pi$
$258$		0			0
$259$		−	3.11298e9i		−	0.691795i
$260$		0			0
$261$	5.67118e9			1.22211
$262$		0			0
$263$	1.63194e9	−	1.63194e9i	0.341099	−	0.341099i	−0.515681	−	0.856781i	$-0.672461\pi$
$263$	1.63194e9	−	1.63194e9i	0.341099	−	0.341099i	0.856781	+	0.515681i	$0.172461\pi$
$264$		0			0
$265$	−4.29160e9	+	3.50578e9i	−0.870233	+	0.710888i
$266$		0			0
$267$	−1.69901e9	−	1.69901e9i	−0.334310	−	0.334310i
$268$		0			0
$269$			5.35417e9i			1.02255i	0.859418	+	0.511273i	$0.170826\pi$
$269$			5.35417e9i			1.02255i	−0.859418	+	0.511273i	$0.829174\pi$
$270$		0			0
$271$	−1.55418e9			−0.288154			−0.144077	−	0.989567i	$-0.546021\pi$
$271$	−1.55418e9			−0.288154			−0.144077	+	0.989567i	$0.546021\pi$
$272$		0			0
$273$	−6.12958e9	+	6.12958e9i	−1.10352	+	1.10352i
$274$		0			0
$275$	−4.19911e8	−	8.55055e7i	−0.0734220	−	0.0149508i
$276$		0			0
$277$	−3.60456e9	−	3.60456e9i	−0.612257	−	0.612257i	0.331277	−	0.943534i	$-0.392521\pi$
$277$	−3.60456e9	−	3.60456e9i	−0.612257	−	0.612257i	−0.943534	+	0.331277i	$0.892521\pi$
$278$		0			0
$279$			7.01749e9i			1.15815i
$280$		0			0
$281$	6.33823e9			1.01658			0.508292	−	0.861185i	$-0.330277\pi$
$281$	6.33823e9			1.01658			0.508292	+	0.861185i	$0.330277\pi$
$282$		0			0
$283$	3.80086e9	−	3.80086e9i	0.592565	−	0.592565i	−0.345758	−	0.938324i	$-0.612378\pi$
$283$	3.80086e9	−	3.80086e9i	0.592565	−	0.592565i	0.938324	+	0.345758i	$0.112378\pi$
$284$		0			0
$285$	7.94565e9	+	9.72667e9i	1.20434	+	1.47430i
$286$		0			0
$287$	−4.37310e9	−	4.37310e9i	−0.644557	−	0.644557i
$288$		0			0
$289$			2.06493e9i			0.296015i
$290$		0			0
$291$	−7.55814e9			−1.05401
$292$		0			0
$293$	3.07238e9	−	3.07238e9i	0.416874	−	0.416874i	−0.467251	−	0.884125i	$-0.654755\pi$
$293$	3.07238e9	−	3.07238e9i	0.416874	−	0.416874i	0.884125	+	0.467251i	$0.154755\pi$
$294$		0			0
$295$	−2.89134e8	+	2.86897e9i	−0.0381779	+	0.378825i
$296$		0			0
$297$	−2.47148e7	−	2.47148e7i	−0.00317637	−	0.00317637i
$298$		0			0
$299$			8.12252e9i			1.01626i
$300$		0			0
$301$	2.58628e9			0.315072
$302$		0			0
$303$	5.54603e9	−	5.54603e9i	0.657978	−	0.657978i
$304$		0			0
$305$	−6.61635e9	−	6.66794e8i	−0.764573	−	0.0770535i
$306$		0			0
$307$	−7.52965e9	−	7.52965e9i	−0.847660	−	0.847660i	0.142181	−	0.989841i	$-0.454588\pi$
$307$	−7.52965e9	−	7.52965e9i	−0.847660	−	0.847660i	−0.989841	+	0.142181i	$0.954588\pi$
$308$		0			0
$309$			7.73137e9i			0.848052i
$310$		0			0
$311$	−7.43985e9			−0.795285			−0.397643	−	0.917540i	$-0.630172\pi$
$311$	−7.43985e9			−0.795285			−0.397643	+	0.917540i	$0.630172\pi$
$312$		0			0
$313$	−4.32589e9	+	4.32589e9i	−0.450711	+	0.450711i	−0.895591	−	0.444879i	$-0.853247\pi$
$313$	−4.32589e9	+	4.32589e9i	−0.450711	+	0.450711i	0.444879	+	0.895591i	$0.353247\pi$
$314$		0			0
$315$	−4.49316e9	+	3.67043e9i	−0.456362	+	0.372799i
$316$		0			0
$317$	−4.76363e9	−	4.76363e9i	−0.471738	−	0.471738i	0.430738	−	0.902477i	$-0.358253\pi$
$317$	−4.76363e9	−	4.76363e9i	−0.471738	−	0.471738i	−0.902477	+	0.430738i	$0.858253\pi$
$318$		0			0
$319$		−	9.90705e8i		−	0.0956713i
$320$		0			0
$321$	1.61296e10			1.51916
$322$		0			0
$323$	1.19229e10	−	1.19229e10i	1.09540	−	1.09540i
$324$		0			0
$325$	1.68590e10	−	1.11547e10i	1.51112	−	0.999822i
$326$		0			0
$327$	−1.58054e9	−	1.58054e9i	−0.138234	−	0.138234i
$328$		0			0
$329$		−	7.09719e9i		−	0.605763i
$330$		0			0
$331$	3.57347e9			0.297700			0.148850	−	0.988860i	$-0.452443\pi$
$331$	3.57347e9			0.297700			0.148850	+	0.988860i	$0.452443\pi$
$332$		0			0
$333$	9.35143e9	−	9.35143e9i	0.760504	−	0.760504i
$334$		0			0
$335$	−1.89912e9	−	2.32481e9i	−0.150790	−	0.184590i
$336$		0			0
$337$	−6.87579e9	−	6.87579e9i	−0.533093	−	0.533093i	0.388398	−	0.921492i	$-0.373028\pi$
$337$	−6.87579e9	−	6.87579e9i	−0.533093	−	0.533093i	−0.921492	+	0.388398i	$0.873028\pi$
$338$		0			0
$339$		−	8.63162e9i		−	0.653572i
$340$		0			0
$341$	1.22589e9			0.0906640
$342$		0			0
$343$	−9.76731e9	+	9.76731e9i	−0.705665	+	0.705665i
$344$		0			0
$345$	−1.11463e9	+	1.10600e10i	−0.0786779	+	0.780692i
$346$		0			0
$347$	1.78592e10	+	1.78592e10i	1.23181	+	1.23181i	0.963268	+	0.268541i	$0.0865414\pi$
$347$	1.78592e10	+	1.78592e10i	1.23181	+	1.23181i	0.268541	+	0.963268i	$0.413459\pi$
$348$		0			0
$349$		−	7.77066e9i		−	0.523789i	−0.965097	−	0.261894i	$-0.915653\pi$
$349$		−	7.77066e9i		−	0.523789i	0.965097	−	0.261894i	$-0.0843472\pi$
$350$		0			0
$351$	1.64881e9			0.108628
$352$		0			0
$353$	−1.58888e10	+	1.58888e10i	−1.02328	+	1.02328i	−0.0235547	+	0.999723i	$0.507498\pi$
$353$	−1.58888e10	+	1.58888e10i	−1.02328	+	1.02328i	−0.999723	+	0.0235547i	$0.992502\pi$
$354$		0			0
$355$	−2.68173e10	−	2.70264e9i	−1.68850	−	0.170167i
$356$		0			0
$357$	1.12620e10	+	1.12620e10i	0.693332	+	0.693332i
$358$		0			0
$359$		−	1.67233e9i		−	0.100680i	−0.998732	−	0.0503402i	$-0.983969\pi$
$359$		−	1.67233e9i		−	0.100680i	0.998732	−	0.0503402i	$-0.0160305\pi$
$360$		0			0
$361$	−1.44644e10			−0.851670
$362$		0			0
$363$	−1.70796e10	+	1.70796e10i	−0.983675	+	0.983675i
$364$		0			0
$365$	−6.37116e9	+	5.20456e9i	−0.358961	+	0.293233i
$366$		0			0
$367$	−1.52209e9	−	1.52209e9i	−0.0839030	−	0.0839030i	0.663910	−	0.747813i	$-0.268896\pi$
$367$	−1.52209e9	−	1.52209e9i	−0.0839030	−	0.0839030i	−0.747813	+	0.663910i	$0.768896\pi$
$368$		0			0
$369$		−	2.62737e10i		−	1.41715i
$370$		0			0
$371$	1.31063e10			0.691806
$372$		0			0
$373$	−4.21097e9	+	4.21097e9i	−0.217544	+	0.217544i	−0.807463	−	0.589919i	$-0.799160\pi$
$373$	−4.21097e9	+	4.21097e9i	−0.217544	+	0.217544i	0.589919	+	0.807463i	$0.299160\pi$
$374$		0			0
$375$	2.44868e10	−	1.28752e10i	1.23824	−	0.651073i
$376$		0			0
$377$	3.30466e10	+	3.30466e10i	1.63592	+	1.63592i
$378$		0			0
$379$			2.36508e10i			1.14627i	0.819459	+	0.573137i	$0.194274\pi$
$379$			2.36508e10i			1.14627i	−0.819459	+	0.573137i	$0.805726\pi$
$380$		0			0
$381$	−5.18598e10			−2.46111
$382$		0			0
$383$	1.50859e10	−	1.50859e10i	0.701093	−	0.701093i	−0.263553	−	0.964645i	$-0.584894\pi$
$383$	1.50859e10	−	1.50859e10i	0.701093	−	0.701093i	0.964645	+	0.263553i	$0.0848942\pi$
$384$		0			0
$385$	6.41192e8	+	7.84915e8i	0.0291840	+	0.0357256i
$386$		0			0
$387$	7.76921e9	+	7.76921e9i	0.346364	+	0.346364i
$388$		0			0
$389$			2.81919e10i			1.23119i	0.788061	+	0.615597i	$0.211085\pi$
$389$			2.81919e10i			1.23119i	−0.788061	+	0.615597i	$0.788915\pi$
$390$		0			0
$391$	1.49236e10			0.638509
$392$		0			0
$393$	8.71371e9	−	8.71371e9i	0.365286	−	0.365286i
$394$		0			0
$395$	3.40291e9	−	3.37658e10i	0.139786	−	1.38704i
$396$		0			0
$397$	−2.57578e10	−	2.57578e10i	−1.03693	−	1.03693i	−0.999292	−	0.0376335i	$-0.988018\pi$
$397$	−2.57578e10	−	2.57578e10i	−1.03693	−	1.03693i	−0.0376335	−	0.999292i	$-0.511982\pi$
$398$		0			0
$399$		−	2.97047e10i		−	1.17202i
$400$		0			0
$401$	4.54412e9			0.175741			0.0878703	−	0.996132i	$-0.471994\pi$
$401$	4.54412e9			0.175741			0.0878703	+	0.996132i	$0.471994\pi$
$402$		0			0
$403$	−4.08917e10	+	4.08917e10i	−1.55030	+	1.55030i
$404$		0			0
$405$	2.78666e10	+	2.80839e9i	1.03577	+	0.104385i
$406$		0			0
$407$	−1.63361e9	−	1.63361e9i	−0.0595349	−	0.0595349i
$408$		0			0
$409$		−	4.47521e10i		−	1.59926i	−0.600491	−	0.799632i	$-0.705028\pi$
$409$		−	4.47521e10i		−	1.59926i	0.600491	−	0.799632i	$-0.294972\pi$
$410$		0			0
$411$	−1.30082e10			−0.455881
$412$		0			0
$413$	4.82234e9	−	4.82234e9i	0.165752	−	0.165752i
$414$		0			0
$415$	3.03444e10	−	2.47881e10i	1.02302	−	0.835701i
$416$		0			0
$417$	5.69762e9	+	5.69762e9i	0.188430	+	0.188430i
$418$		0			0
$419$			3.86430e10i			1.25376i	0.779115	+	0.626881i	$0.215669\pi$
$419$			3.86430e10i			1.25376i	−0.779115	+	0.626881i	$0.784331\pi$
$420$		0			0
$421$	−2.31242e10			−0.736101			−0.368051	−	0.929806i	$-0.619975\pi$
$421$	−2.31242e10			−0.736101			−0.368051	+	0.929806i	$0.619975\pi$
$422$		0			0
$423$	2.13200e10	−	2.13200e10i	0.665927	−	0.665927i
$424$		0			0
$425$	−2.04946e10	−	3.09753e10i	−0.628180	−	0.949424i
$426$		0			0
$427$	1.11212e10	+	1.11212e10i	0.334533	+	0.334533i
$428$		0			0
$429$			6.43329e9i			0.189935i
$430$		0			0
$431$	5.28892e9			0.153270			0.0766351	−	0.997059i	$-0.475582\pi$
$431$	5.28892e9			0.153270			0.0766351	+	0.997059i	$0.475582\pi$
$432$		0			0
$433$	3.63085e10	−	3.63085e10i	1.03290	−	1.03290i	0.0334550	−	0.999440i	$-0.489349\pi$
$433$	3.63085e10	−	3.63085e10i	1.03290	−	1.03290i	0.999440	−	0.0334550i	$-0.0106510\pi$
$434$		0			0
$435$	4.04630e10	+	4.95328e10i	1.13006	+	1.38336i
$436$		0			0
$437$	−1.96813e10	−	1.96813e10i	−0.539671	−	0.539671i
$438$		0			0
$439$		−	1.91503e9i		−	0.0515605i	−0.999668	−	0.0257802i	$-0.991793\pi$
$439$		−	1.91503e9i		−	0.0515605i	0.999668	−	0.0257802i	$-0.00820702\pi$
$440$		0			0
$441$	−2.24802e10			−0.594354
$442$		0			0
$443$	1.28611e10	−	1.28611e10i	0.333935	−	0.333935i	−0.520143	−	0.854079i	$-0.674122\pi$
$443$	1.28611e10	−	1.28611e10i	0.333935	−	0.333935i	0.854079	+	0.520143i	$0.174122\pi$
$444$		0			0
$445$	1.32884e9	−	1.31856e10i	0.0338869	−	0.336247i
$446$		0			0
$447$	1.67648e10	+	1.67648e10i	0.419922	+	0.419922i
$448$		0			0
$449$			3.57906e10i			0.880611i	0.897848	+	0.440305i	$0.145130\pi$
$449$			3.57906e10i			0.880611i	−0.897848	+	0.440305i	$0.854870\pi$
$450$		0			0
$451$	−4.58977e9			−0.110939
$452$		0			0
$453$	−4.25839e10	+	4.25839e10i	−1.01124	+	1.01124i
$454$		0			0
$455$	−4.75702e10	−	4.79411e9i	−1.10991	−	0.111857i
$456$		0			0
$457$	1.19204e10	+	1.19204e10i	0.273291	+	0.273291i	0.830423	−	0.557133i	$-0.188099\pi$
$457$	1.19204e10	+	1.19204e10i	0.273291	+	0.273291i	−0.557133	+	0.830423i	$0.688099\pi$
$458$		0			0
$459$		−	3.02937e9i		−	0.0682499i
$460$		0			0
$461$	6.25621e9			0.138518			0.0692592	−	0.997599i	$-0.477936\pi$
$461$	6.25621e9			0.138518			0.0692592	+	0.997599i	$0.477936\pi$
$462$		0			0
$463$	−6.43592e10	+	6.43592e10i	−1.40051	+	1.40051i	−0.602066	+	0.798446i	$0.705656\pi$
$463$	−6.43592e10	+	6.43592e10i	−1.40051	+	1.40051i	−0.798446	+	0.602066i	$0.794344\pi$
$464$		0			0
$465$	−6.12916e10	+	5.00687e10i	−1.31096	+	1.07091i
$466$		0			0
$467$	−3.10806e10	−	3.10806e10i	−0.653465	−	0.653465i	0.300361	−	0.953826i	$-0.402893\pi$
$467$	−3.10806e10	−	3.10806e10i	−0.653465	−	0.653465i	−0.953826	+	0.300361i	$0.902893\pi$
$468$		0			0
$469$			7.09984e9i			0.146743i
$470$		0			0
$471$	3.52160e10			0.715577
$472$		0			0
$473$	1.35721e9	−	1.35721e9i	0.0271146	−	0.0271146i
$474$		0			0
$475$	−1.38220e10	+	6.78788e10i	−0.271517	+	1.33340i
$476$		0			0
$477$	3.93714e10	+	3.93714e10i	0.760515	+	0.760515i
$478$		0			0
$479$		−	6.65982e10i		−	1.26509i	−0.774525	−	0.632543i	$-0.782011\pi$
$479$		−	6.65982e10i		−	1.26509i	0.774525	−	0.632543i	$-0.217989\pi$
$480$		0			0
$481$	1.08984e11			2.03602
$482$		0			0
$483$	1.85903e10	−	1.85903e10i	0.341585	−	0.341585i
$484$		0			0
$485$	−2.63727e10	−	3.22842e10i	−0.476637	−	0.583475i
$486$		0			0
$487$	−5.20020e10	−	5.20020e10i	−0.924494	−	0.924494i	0.0728488	−	0.997343i	$-0.476791\pi$
$487$	−5.20020e10	−	5.20020e10i	−0.924494	−	0.924494i	−0.997343	+	0.0728488i	$0.976791\pi$
$488$		0			0
$489$		−	6.23474e9i		−	0.109039i
$490$		0			0
$491$	−1.21434e10			−0.208937			−0.104468	−	0.994528i	$-0.533314\pi$
$491$	−1.21434e10			−0.208937			−0.104468	+	0.994528i	$0.533314\pi$
$492$		0			0
$493$	6.07171e10	−	6.07171e10i	1.02783	−	1.02783i
$494$		0			0
$495$	−4.31745e8	+	4.28404e9i	−0.00719128	+	0.0713564i
$496$		0			0
$497$	4.50760e10	+	4.50760e10i	0.738789	+	0.738789i
$498$		0			0
$499$		−	2.78222e10i		−	0.448735i	−0.974505	−	0.224367i	$-0.927968\pi$
$499$		−	2.78222e10i		−	0.448735i	0.974505	−	0.224367i	$-0.0720316\pi$
$500$		0			0
$501$	7.06646e10			1.12163
$502$		0			0
$503$	4.71112e10	−	4.71112e10i	0.735957	−	0.735957i	−0.235836	−	0.971793i	$-0.575783\pi$
$503$	4.71112e10	−	4.71112e10i	0.735957	−	0.735957i	0.971793	+	0.235836i	$0.0757828\pi$
$504$		0			0
$505$	4.30414e10	+	4.33770e9i	0.661790	+	0.0666951i
$506$		0			0
$507$	−1.49231e11	−	1.49231e11i	−2.25853	−	2.25853i
$508$		0			0
$509$			1.11980e11i			1.66828i	0.551555	+	0.834139i	$0.314035\pi$
$509$			1.11980e11i			1.66828i	−0.551555	+	0.834139i	$0.685965\pi$
$510$		0			0
$511$	1.94572e10			0.285362
$512$		0			0
$513$	−3.99515e9	+	3.99515e9i	−0.0576852	+	0.0576852i
$514$		0			0
$515$	−3.30241e10	+	2.69772e10i	−0.469464	+	0.383502i
$516$		0			0
$517$	−3.72442e9	−	3.72442e9i	−0.0521311	−	0.0521311i
$518$		0			0
$519$		−	7.12937e10i		−	0.982611i
$520$		0			0
$521$	−1.16618e11			−1.58276			−0.791381	−	0.611324i	$-0.790637\pi$
$521$	−1.16618e11			−1.58276			−0.791381	+	0.611324i	$0.790637\pi$
$522$		0			0
$523$	−1.71807e10	+	1.71807e10i	−0.229633	+	0.229633i	−0.812539	−	0.582906i	$-0.801915\pi$
$523$	−1.71807e10	+	1.71807e10i	−0.229633	+	0.229633i	0.582906	+	0.812539i	$0.301915\pi$
$524$		0			0
$525$	−6.41160e10	−	1.30558e10i	−0.843974	−	0.171857i
$526$		0			0
$527$	7.51309e10	+	7.51309e10i	0.974039	+	0.974039i
$528$		0			0
$529$			5.36763e10i			0.685425i
$530$		0			0
$531$	2.89727e10			0.364428
$532$		0			0
$533$	1.53100e11	−	1.53100e11i	1.89699	−	1.89699i
$534$		0			0
$535$	5.62814e10	+	6.88968e10i	0.686989	+	0.840977i
$536$		0			0
$537$	−1.47338e11	−	1.47338e11i	−1.77182	−	1.77182i
$538$		0			0
$539$			3.92709e9i			0.0465281i
$540$		0			0
$541$	1.19657e11			1.39685			0.698425	−	0.715684i	$-0.253885\pi$
$541$	1.19657e11			1.39685			0.698425	+	0.715684i	$0.253885\pi$
$542$		0			0
$543$	−3.52401e10	+	3.52401e10i	−0.405358	+	0.405358i
$544$		0			0
$545$	1.23618e9	−	1.22662e10i	0.0140119	−	0.139035i
$546$		0			0
$547$	8.30421e10	+	8.30421e10i	0.927575	+	0.927575i	0.997549	−	0.0699737i	$-0.0222915\pi$
$547$	8.30421e10	+	8.30421e10i	0.927575	+	0.927575i	−0.0699737	+	0.997549i	$0.522292\pi$
$548$		0			0
$549$			6.68161e10i			0.735516i
$550$		0			0
$551$	−1.60148e11			−1.73746
$552$		0			0
$553$	−5.67556e10	+	5.67556e10i	−0.606887	+	0.606887i
$554$		0			0
$555$	1.48398e11	+	1.49555e10i	1.56407	+	0.157626i
$556$		0			0
$557$	−5.55977e10	−	5.55977e10i	−0.577611	−	0.577611i	0.356633	−	0.934245i	$-0.383925\pi$
$557$	−5.55977e10	−	5.55977e10i	−0.577611	−	0.577611i	−0.934245	+	0.356633i	$0.883925\pi$
$558$		0			0
$559$			9.05442e10i			0.927286i
$560$		0			0
$561$	1.18200e10			0.119334
$562$		0			0
$563$	−3.87578e10	+	3.87578e10i	−0.385767	+	0.385767i	−0.873175	−	0.487407i	$-0.837943\pi$
$563$	−3.87578e10	+	3.87578e10i	−0.385767	+	0.385767i	0.487407	+	0.873175i	$0.337943\pi$
$564$		0			0
$565$	3.68694e10	−	3.01184e10i	0.361803	−	0.295555i
$566$		0			0
$567$	−4.68398e10	−	4.68398e10i	−0.453192	−	0.453192i
$568$		0			0
$569$			1.87111e11i			1.78505i	0.451000	+	0.892524i	$0.351067\pi$
$569$			1.87111e11i			1.78505i	−0.451000	+	0.892524i	$0.648933\pi$
$570$		0			0
$571$	5.17121e7			0.000486461			0.000243231	−	1.00000i	$-0.499923\pi$
$571$	5.17121e7			0.000486461			0.000243231		1.00000i	$0.499923\pi$
$572$		0			0
$573$	−1.35709e11	+	1.35709e11i	−1.25889	+	1.25889i
$574$		0			0
$575$	−5.11315e10	+	3.38308e10i	−0.467754	+	0.309486i
$576$		0			0
$577$	5.02658e10	+	5.02658e10i	0.453491	+	0.453491i	0.896512	−	0.443020i	$-0.146093\pi$
$577$	5.02658e10	+	5.02658e10i	0.453491	+	0.453491i	−0.443020	+	0.896512i	$0.646093\pi$
$578$		0			0
$579$		−	4.18034e10i		−	0.371961i
$580$		0			0
$581$	−9.26699e10			−0.813269
$582$		0			0
$583$	6.87784e9	−	6.87784e9i	0.0595358	−	0.0595358i
$584$		0			0
$585$	−1.28500e11	−	1.57303e11i	−1.09718	−	1.34312i
$586$		0			0
$587$	2.81940e9	+	2.81940e9i	0.0237468	+	0.0237468i	0.718880	−	0.695134i	$-0.244655\pi$
$587$	2.81940e9	+	2.81940e9i	0.0237468	+	0.0237468i	−0.695134	+	0.718880i	$0.744655\pi$
$588$		0			0
$589$		−	1.98166e11i		−	1.64652i
$590$		0			0
$591$	1.91289e10			0.156798
$592$		0			0
$593$	1.27889e11	−	1.27889e11i	1.03423	−	1.03423i	0.0348321	−	0.999393i	$-0.488910\pi$
$593$	1.27889e11	−	1.27889e11i	1.03423	−	1.03423i	0.999393	−	0.0348321i	$-0.0110896\pi$
$594$		0			0
$595$	−8.80829e9	+	8.74014e10i	−0.0702787	+	0.697349i
$596$		0			0
$597$	2.32731e11	+	2.32731e11i	1.83213	+	1.83213i
$598$		0			0
$599$		−	1.00042e11i		−	0.777095i	−0.921429	−	0.388547i	$-0.872977\pi$
$599$		−	1.00042e11i		−	0.777095i	0.921429	−	0.388547i	$-0.127023\pi$
$600$		0			0
$601$	−9.00270e10			−0.690041			−0.345020	−	0.938595i	$-0.612128\pi$
$601$	−9.00270e10			−0.690041			−0.345020	+	0.938595i	$0.612128\pi$
$602$		0			0
$603$	−2.13280e10	+	2.13280e10i	−0.161317	+	0.161317i
$604$		0			0
$605$	−1.32551e11	−	1.33584e10i	−0.989374	−	0.0997089i
$606$		0			0
$607$	5.15328e10	+	5.15328e10i	0.379602	+	0.379602i	0.870959	−	0.491356i	$-0.163499\pi$
$607$	5.15328e10	+	5.15328e10i	0.379602	+	0.379602i	−0.491356	+	0.870959i	$0.663499\pi$
$608$		0			0
$609$		−	1.51270e11i		−	1.09973i
$610$		0			0
$611$	2.48469e11			1.78282
$612$		0			0
$613$	−4.94140e10	+	4.94140e10i	−0.349952	+	0.349952i	−0.860091	−	0.510140i	$-0.829594\pi$
$613$	−4.94140e10	+	4.94140e10i	−0.349952	+	0.349952i	0.510140	+	0.860091i	$0.329594\pi$
$614$		0			0
$615$	2.29477e11	−	1.87459e11i	1.60413	−	1.31040i
$616$		0			0
$617$	8.55221e10	+	8.55221e10i	0.590117	+	0.590117i	0.937663	−	0.347546i	$-0.112985\pi$
$617$	8.55221e10	+	8.55221e10i	0.590117	+	0.590117i	−0.347546	+	0.937663i	$0.612985\pi$
$618$		0			0
$619$		−	2.66904e11i		−	1.81799i	−0.416803	−	0.908997i	$-0.636850\pi$
$619$		−	2.66904e11i		−	1.81799i	0.416803	−	0.908997i	$-0.363150\pi$
$620$		0			0
$621$	−5.00064e9			−0.0336248
$622$		0			0
$623$	−2.21631e10	+	2.21631e10i	−0.147122	+	0.147122i
$624$		0			0
$625$	1.40438e11	+	5.96681e10i	0.920373	+	0.391041i
$626$		0			0
$627$	−1.55882e10	−	1.55882e10i	−0.100862	−	0.100862i
$628$		0			0
$629$		−	2.00237e11i		−	1.27921i
$630$		0			0
$631$	−9.16102e10			−0.577865			−0.288932	−	0.957349i	$-0.593300\pi$
$631$	−9.16102e10			−0.577865			−0.288932	+	0.957349i	$0.593300\pi$
$632$		0			0
$633$	−3.80310e10	+	3.80310e10i	−0.236877	+	0.236877i
$634$		0			0
$635$	−1.80955e11	−	2.21516e11i	−1.11295	−	1.36242i
$636$		0			0
$637$	−1.30995e11	−	1.30995e11i	−0.795602	−	0.795602i
$638$		0			0
$639$			2.70818e11i			1.62433i
$640$		0			0
$641$	−2.41221e11			−1.42884			−0.714420	−	0.699717i	$-0.753309\pi$
$641$	−2.41221e11			−1.42884			−0.714420	+	0.699717i	$0.753309\pi$
$642$		0			0
$643$	1.22336e10	−	1.22336e10i	0.0715668	−	0.0715668i	−0.670417	−	0.741984i	$-0.733885\pi$
$643$	1.22336e10	−	1.22336e10i	0.0715668	−	0.0715668i	0.741984	+	0.670417i	$0.233885\pi$
$644$		0			0
$645$	−1.24251e10	+	1.23289e11i	−0.0717894	+	0.712340i
$646$		0			0
$647$	2.09488e11	+	2.09488e11i	1.19548	+	1.19548i	0.975506	+	0.219975i	$0.0705974\pi$
$647$	2.09488e11	+	2.09488e11i	1.19548	+	1.19548i	0.219975	+	0.975506i	$0.429403\pi$
$648$		0			0
$649$		−	5.06128e9i		−	0.0285287i
$650$		0			0
$651$	1.87181e11			1.04217
$652$		0			0
$653$	−3.24462e10	+	3.24462e10i	−0.178448	+	0.178448i	−0.790679	−	0.612231i	$-0.790272\pi$
$653$	−3.24462e10	+	3.24462e10i	−0.178448	+	0.178448i	0.612231	+	0.790679i	$0.290272\pi$
$654$		0			0
$655$	6.76250e10	+	6.81523e9i	0.367402	+	0.0370267i
$656$		0			0
$657$	5.84495e10	+	5.84495e10i	0.313704	+	0.313704i
$658$		0			0
$659$			1.16092e11i			0.615545i	0.951460	+	0.307772i	$0.0995836\pi$
$659$			1.16092e11i			0.615545i	−0.951460	+	0.307772i	$0.900416\pi$
$660$		0			0
$661$	−3.38332e11			−1.77230			−0.886150	−	0.463399i	$-0.846630\pi$
$661$	−3.38332e11			−1.77230			−0.886150	+	0.463399i	$0.846630\pi$
$662$		0			0
$663$	−3.94275e11	+	3.94275e11i	−2.04054	+	2.04054i
$664$		0			0
$665$	1.26882e11	−	1.03649e11i	0.648803	−	0.530003i
$666$		0			0
$667$	−1.00227e11	−	1.00227e11i	−0.506384	−	0.506384i
$668$		0			0
$669$		−	6.40491e10i		−	0.319748i
$670$		0			0
$671$	1.16722e10			0.0575787
$672$		0			0
$673$	1.32298e11	−	1.32298e11i	0.644902	−	0.644902i	−0.306854	−	0.951757i	$-0.599276\pi$
$673$	1.32298e11	−	1.32298e11i	0.644902	−	0.644902i	0.951757	+	0.306854i	$0.0992764\pi$
$674$		0			0
$675$	6.86738e9	+	1.03793e10i	0.0330808	+	0.0499979i
$676$		0			0
$677$	2.19980e11	+	2.19980e11i	1.04720	+	1.04720i	0.998830	+	0.0483685i	$0.0154022\pi$
$677$	2.19980e11	+	2.19980e11i	1.04720	+	1.04720i	0.0483685	+	0.998830i	$0.484598\pi$
$678$		0			0
$679$			9.85940e10i			0.463843i
$680$		0			0
$681$	2.36189e11			1.09817
$682$		0			0
$683$	3.02709e11	−	3.02709e11i	1.39105	−	1.39105i	0.568060	−	0.822987i	$-0.307694\pi$
$683$	3.02709e11	−	3.02709e11i	1.39105	−	1.39105i	0.822987	−	0.568060i	$-0.192306\pi$
$684$		0			0
$685$	−4.53899e10	−	5.55640e10i	−0.206156	−	0.252366i
$686$		0			0
$687$	−8.84966e10	−	8.84966e10i	−0.397283	−	0.397283i
$688$		0			0
$689$			4.58844e11i			2.03605i
$690$		0			0
$691$	−1.31329e11			−0.576034			−0.288017	−	0.957625i	$-0.592996\pi$
$691$	−1.31329e11			−0.576034			−0.288017	+	0.957625i	$0.592996\pi$
$692$		0			0
$693$	7.20087e9	−	7.20087e9i	0.0312214	−	0.0312214i
$694$		0			0
$695$	−4.45626e9	+	4.42178e10i	−0.0190999	+	0.189521i
$696$		0			0
$697$	−2.81292e11	−	2.81292e11i	−1.19186	−	1.19186i
$698$		0			0
$699$			2.24647e11i			0.941004i
$700$		0			0
$701$	−1.97556e11			−0.818124			−0.409062	−	0.912507i	$-0.634144\pi$
$701$	−1.97556e11			−0.818124			−0.409062	+	0.912507i	$0.634144\pi$
$702$		0			0
$703$	−2.64074e11	+	2.64074e11i	−1.08120	+	1.08120i
$704$		0			0
$705$	3.38327e11	+	3.40965e10i	1.36956	+	0.138024i
$706$		0			0
$707$	−7.23465e10	−	7.23465e10i	−0.289561	−	0.289561i
$708$		0			0
$709$		−	2.71460e10i		−	0.107429i	−0.998556	−	0.0537144i	$-0.982894\pi$
$709$		−	2.71460e10i		−	0.107429i	0.998556	−	0.0537144i	$-0.0171061\pi$
$710$		0			0
$711$	−3.40989e11			−1.33433
$712$		0			0
$713$	1.24020e11	−	1.24020e11i	0.479881	−	0.479881i
$714$		0			0
$715$	−2.74794e10	+	2.24478e10i	−0.105144	+	0.0858913i
$716$		0			0
$717$	−9.03939e10	−	9.03939e10i	−0.342029	−	0.342029i
$718$		0			0
$719$		−	1.47425e11i		−	0.551640i	−0.961209	−	0.275820i	$-0.911051\pi$
$719$		−	1.47425e11i		−	0.551640i	0.961209	−	0.275820i	$-0.0889493\pi$
$720$		0			0
$721$	1.00854e11			0.373208
$722$		0			0
$723$	3.84422e11	−	3.84422e11i	1.40687	−	1.40687i
$724$		0			0
$725$	−7.03882e10	+	3.45671e11i	−0.254770	+	1.25115i
$726$		0			0
$727$	−1.41191e10	−	1.41191e10i	−0.0505438	−	0.0505438i	0.681383	−	0.731927i	$-0.261379\pi$
$727$	−1.41191e10	−	1.41191e10i	−0.0505438	−	0.0505438i	−0.731927	+	0.681383i	$0.761379\pi$
$728$		0			0
$729$		−	2.57727e11i		−	0.912535i
$730$		0			0
$731$	1.66358e11			0.582606
$732$		0			0
$733$	−8.48268e10	+	8.48268e10i	−0.293844	+	0.293844i	−0.838597	−	0.544753i	$-0.816624\pi$
$733$	−8.48268e10	+	8.48268e10i	−0.293844	+	0.293844i	0.544753	+	0.838597i	$0.316624\pi$
$734$		0			0
$735$	−1.60393e11	−	1.96345e11i	−0.549586	−	0.672775i
$736$		0			0
$737$	3.72581e9	+	3.72581e9i	0.0126285	+	0.0126285i
$738$		0			0
$739$		−	4.10110e11i		−	1.37506i	−0.726155	−	0.687532i	$-0.758694\pi$
$739$		−	4.10110e11i		−	1.37506i	0.726155	−	0.687532i	$-0.241306\pi$
$740$		0			0
$741$	1.03994e12			3.44935
$742$		0			0
$743$	1.64373e11	−	1.64373e11i	0.539355	−	0.539355i	−0.383984	−	0.923340i	$-0.625448\pi$
$743$	1.64373e11	−	1.64373e11i	0.539355	−	0.539355i	0.923340	+	0.383984i	$0.125448\pi$
$744$		0			0
$745$	−1.31122e10	+	1.30108e11i	−0.0425648	+	0.422355i
$746$		0			0
$747$	−2.78381e11	−	2.78381e11i	−0.894042	−	0.894042i
$748$		0			0
$749$		−	2.10407e11i		−	0.668548i
$750$		0			0
$751$	3.54155e11			1.11336			0.556678	−	0.830728i	$-0.312076\pi$
$751$	3.54155e11			1.11336			0.556678	+	0.830728i	$0.312076\pi$
$752$		0			0
$753$	3.49559e9	−	3.49559e9i	0.0108728	−	0.0108728i
$754$		0			0
$755$	−3.30484e11	−	3.33061e10i	−1.01710	−	0.102503i
$756$		0			0
$757$	1.53872e11	+	1.53872e11i	0.468570	+	0.468570i	0.901451	−	0.432881i	$-0.142503\pi$
$757$	1.53872e11	+	1.53872e11i	0.468570	+	0.468570i	−0.432881	+	0.901451i	$0.642503\pi$
$758$		0			0
$759$		−	1.95115e10i		−	0.0587926i
$760$		0			0
$761$	−8.09957e10			−0.241503			−0.120752	−	0.992683i	$-0.538530\pi$
$761$	−8.09957e10			−0.241503			−0.120752	+	0.992683i	$0.538530\pi$
$762$		0			0
$763$	−2.06177e10	+	2.06177e10i	−0.0608335	+	0.0608335i
$764$		0			0
$765$	−2.89015e11	+	2.36095e11i	−0.843868	+	0.689351i
$766$		0			0
$767$	1.68827e11	+	1.68827e11i	0.487822	+	0.487822i
$768$		0			0
$769$			4.78764e11i			1.36904i	0.728994	+	0.684520i	$0.239988\pi$
$769$			4.78764e11i			1.36904i	−0.728994	+	0.684520i	$0.760012\pi$
$770$		0			0
$771$	−2.58842e11			−0.732515
$772$		0			0
$773$	8.24114e10	−	8.24114e10i	0.230818	−	0.230818i	−0.582216	−	0.813034i	$-0.697814\pi$
$773$	8.24114e10	−	8.24114e10i	0.230818	−	0.230818i	0.813034	+	0.582216i	$0.197814\pi$
$774$		0			0
$775$	−4.27731e11	−	8.70979e10i	−1.18567	−	0.241435i
$776$		0			0
$777$	−2.49436e11	−	2.49436e11i	−0.684344	−	0.684344i
$778$		0			0
$779$			7.41939e11i			2.01474i
$780$		0			0
$781$	4.73095e10			0.127158
$782$		0			0
$783$	−2.03452e10	+	2.03452e10i	−0.0541272	+	0.0541272i
$784$		0			0
$785$	1.22880e11	+	1.50423e11i	0.323595	+	0.396128i
$786$		0			0
$787$	−1.42765e11	−	1.42765e11i	−0.372153	−	0.372153i	0.496108	−	0.868261i	$-0.334762\pi$
$787$	−1.42765e11	−	1.42765e11i	−0.372153	−	0.372153i	−0.868261	+	0.496108i	$0.834762\pi$
$788$		0			0
$789$		−	2.61526e11i		−	0.674851i
$790$		0			0
$791$	−1.12597e11			−0.287622
$792$		0			0
$793$	−3.89345e11	+	3.89345e11i	−0.984560	+	0.984560i
$794$		0			0
$795$	−6.29656e10	+	6.24784e11i	−0.157629	+	1.56409i
$796$		0			0
$797$	−1.16899e11	−	1.16899e11i	−0.289720	−	0.289720i	0.547249	−	0.836970i	$-0.315675\pi$
$797$	−1.16899e11	−	1.16899e11i	−0.289720	−	0.289720i	−0.836970	+	0.547249i	$0.815675\pi$
$798$		0			0
$799$		−	4.56515e11i		−	1.12013i
$800$		0			0
$801$	−1.33156e11			−0.323468
$802$		0			0
$803$	1.02106e10	−	1.02106e10i	0.0245578	−	0.0245578i
$804$		0			0
$805$	1.44275e11	+	1.45400e10i	0.343564	+	0.0346243i
$806$		0			0
$807$	4.29017e11	+	4.29017e11i	1.01153	+	1.01153i
$808$		0			0
$809$		−	1.45998e10i		−	0.0340841i	−0.999855	−	0.0170421i	$-0.994575\pi$
$809$		−	1.45998e10i		−	0.0340841i	0.999855	−	0.0170421i	$-0.00542492\pi$
$810$		0			0
$811$	4.38468e11			1.01357			0.506786	−	0.862072i	$-0.330833\pi$
$811$	4.38468e11			1.01357			0.506786	+	0.862072i	$0.330833\pi$
$812$		0			0
$813$	−1.24533e11	+	1.24533e11i	−0.285050	+	0.285050i
$814$		0			0
$815$	2.66313e10	−	2.17550e10i	0.0603619	−	0.0493092i
$816$		0			0
$817$	−2.19394e11	−	2.19394e11i	−0.492421	−	0.492421i
$818$		0			0
$819$			4.80394e11i			1.06773i
$820$		0			0
$821$	5.60669e11			1.23405			0.617027	−	0.786942i	$-0.288337\pi$
$821$	5.60669e11			1.23405			0.617027	+	0.786942i	$0.288337\pi$
$822$		0			0
$823$	4.33738e11	−	4.33738e11i	0.945427	−	0.945427i	−0.0531589	−	0.998586i	$-0.516929\pi$
$823$	4.33738e11	−	4.33738e11i	0.945427	−	0.945427i	0.998586	+	0.0531589i	$0.0169290\pi$
$824$		0			0
$825$	−4.04978e10	+	2.67951e10i	−0.0874209	+	0.0578415i
$826$		0			0
$827$	−1.80033e11	−	1.80033e11i	−0.384885	−	0.384885i	0.487974	−	0.872858i	$-0.337736\pi$
$827$	−1.80033e11	−	1.80033e11i	−0.384885	−	0.384885i	−0.872858	+	0.487974i	$0.837736\pi$
$828$		0			0
$829$		−	8.74273e10i		−	0.185110i	−0.995708	−	0.0925548i	$-0.970497\pi$
$829$		−	8.74273e10i		−	0.185110i	0.995708	−	0.0925548i	$-0.0295033\pi$
$830$		0			0
$831$	−5.77650e11			−1.21132
$832$		0			0
$833$	−2.40678e11	+	2.40678e11i	−0.499870	+	0.499870i
$834$		0			0
$835$	2.46571e11	+	3.01840e11i	0.507220	+	0.620913i
$836$		0			0
$837$	−2.51751e10	−	2.51751e10i	−0.0512942	−	0.0512942i
$838$		0			0
$839$			4.54007e11i			0.916252i	0.888887	+	0.458126i	$0.151479\pi$
$839$			4.54007e11i			0.916252i	−0.888887	+	0.458126i	$0.848521\pi$
$840$		0			0
$841$	−3.15303e11			−0.630295
$842$		0			0
$843$	5.07867e11	−	5.07867e11i	1.00563	−	1.00563i
$844$		0			0
$845$	1.16717e11	−	1.15814e12i	0.228933	−	2.27162i
$846$		0			0
$847$	2.22799e11	+	2.22799e11i	0.432892	+	0.432892i
$848$		0			0
$849$		−	6.09107e11i		−	1.17236i
$850$		0			0
$851$	−3.30536e11			−0.630231
$852$		0			0
$853$	−5.31183e10	+	5.31183e10i	−0.100334	+	0.100334i	−0.755492	−	0.655158i	$-0.772602\pi$
$853$	−5.31183e10	+	5.31183e10i	−0.100334	+	0.100334i	0.655158	+	0.755492i	$0.272602\pi$
$854$		0			0
$855$	6.92517e11	+	6.97917e10i	1.29588	+	0.130599i
$856$		0			0
$857$	−6.47292e11	−	6.47292e11i	−1.19999	−	1.19999i	−0.974170	−	0.225818i	$-0.927495\pi$
$857$	−6.47292e11	−	6.47292e11i	−1.19999	−	1.19999i	−0.225818	−	0.974170i	$-0.572505\pi$
$858$		0			0
$859$			5.81662e11i			1.06831i	0.845386	+	0.534156i	$0.179371\pi$
$859$			5.81662e11i			1.06831i	−0.845386	+	0.534156i	$0.820629\pi$
$860$		0			0
$861$	−7.00811e11			−1.27523
$862$		0			0
$863$	3.75645e11	−	3.75645e11i	0.677228	−	0.677228i	−0.282144	−	0.959372i	$-0.591046\pi$
$863$	3.75645e11	−	3.75645e11i	0.677228	−	0.677228i	0.959372	+	0.282144i	$0.0910457\pi$
$864$		0			0
$865$	3.04527e11	−	2.48766e11i	0.543953	−	0.444352i
$866$		0			0
$867$	1.65458e11	+	1.65458e11i	0.292827	+	0.292827i
$868$		0			0
$869$			5.95677e10i			0.104456i
$870$		0			0
$871$	−2.48561e11			−0.431878
$872$		0			0
$873$	−2.96177e11	+	2.96177e11i	−0.509912	+	0.509912i
$874$		0			0
$875$	−1.67954e11	−	3.19424e11i	−0.286522	−	0.544922i
$876$		0			0
$877$	−2.03335e11	−	2.03335e11i	−0.343727	−	0.343727i	0.514039	−	0.857767i	$-0.328148\pi$
$877$	−2.03335e11	−	2.03335e11i	−0.343727	−	0.343727i	−0.857767	+	0.514039i	$0.828148\pi$
$878$		0			0
$879$		−	4.92365e11i		−	0.824767i
$880$		0			0
$881$	7.98932e11			1.32619			0.663096	−	0.748535i	$-0.269242\pi$
$881$	7.98932e11			1.32619			0.663096	+	0.748535i	$0.269242\pi$
$882$		0			0
$883$	2.06797e11	−	2.06797e11i	0.340174	−	0.340174i	−0.516259	−	0.856433i	$-0.672676\pi$
$883$	2.06797e11	−	2.06797e11i	0.340174	−	0.340174i	0.856433	+	0.516259i	$0.172676\pi$
$884$		0			0
$885$	2.06716e11	+	2.53051e11i	0.336978	+	0.412511i
$886$		0			0
$887$	5.47416e11	+	5.47416e11i	0.884348	+	0.884348i	0.993973	−	0.109625i	$-0.0349651\pi$
$887$	5.47416e11	+	5.47416e11i	0.884348	+	0.884348i	−0.109625	+	0.993973i	$0.534965\pi$
$888$		0			0
$889$			6.76498e11i			1.08308i
$890$		0			0
$891$	−4.91606e10			−0.0780021
$892$		0			0
$893$	−6.02054e11	+	6.02054e11i	−0.946738	+	0.946738i
$894$		0			0
$895$	1.15237e11	−	1.14346e12i	0.179598	−	1.78208i
$896$		0			0
$897$	6.50837e11	+	6.50837e11i	1.00532	+	1.00532i
$898$		0			0
$899$		−	1.00916e12i		−	1.54497i
$900$		0			0
$901$	8.43041e11			1.27923
$902$		0			0
$903$	2.07232e11	−	2.07232e11i	0.311678	−	0.311678i
$904$		0			0
$905$	−2.73490e11	−	2.75623e10i	−0.407706	−	0.0410885i
$906$		0			0
$907$	5.97670e11	+	5.97670e11i	0.883146	+	0.883146i	0.993853	−	0.110707i	$-0.0353115\pi$
$907$	5.97670e11	+	5.97670e11i	0.883146	+	0.883146i	−0.110707	+	0.993853i	$0.535311\pi$
$908$		0			0
$909$		−	4.34659e11i		−	0.636639i
$910$		0			0
$911$	−9.81526e11			−1.42504			−0.712522	−	0.701650i	$-0.752447\pi$
$911$	−9.81526e11			−1.42504			−0.712522	+	0.701650i	$0.752447\pi$
$912$		0			0
$913$	−4.86308e10	+	4.86308e10i	−0.0699887	+	0.0699887i
$914$		0			0
$915$	−5.83580e11	+	4.76723e11i	−0.832561	+	0.680114i
$916$		0			0
$917$	−1.13668e11	−	1.13668e11i	−0.160754	−	0.160754i
$918$		0			0
$919$		−	6.40387e11i		−	0.897802i	−0.893582	−	0.448901i	$-0.851816\pi$
$919$		−	6.40387e11i		−	0.897802i	0.893582	−	0.448901i	$-0.148184\pi$
$920$		0			0
$921$	−1.20666e12			−1.67706
$922$		0			0
$923$	−1.57809e12	+	1.57809e12i	−2.17432	+	2.17432i
$924$		0			0
$925$	4.53925e11	+	6.86056e11i	0.620036	+	0.937115i
$926$		0			0
$927$	3.02966e11	+	3.02966e11i	0.410274	+	0.410274i
$928$		0			0
$929$		−	4.07354e11i		−	0.546901i	−0.961886	−	0.273451i	$-0.911835\pi$
$929$		−	4.07354e11i		−	0.546901i	0.961886	−	0.273451i	$-0.0881650\pi$
$930$		0			0
$931$	6.34815e11			0.844985
$932$		0			0
$933$	−5.96137e11	+	5.96137e11i	−0.786719	+	0.786719i
$934$		0			0
$935$	4.12436e10	+	5.04883e10i	0.0539648	+	0.0660609i
$936$		0			0
$937$	7.94225e11	+	7.94225e11i	1.03035	+	1.03035i	0.999525	+	0.0308260i	$0.00981379\pi$
$937$	7.94225e11	+	7.94225e11i	1.03035	+	1.03035i	0.0308260	+	0.999525i	$0.490186\pi$
$938$		0			0
$939$			6.93246e11i			0.891713i
$940$		0			0
$941$	−6.85413e11			−0.874166			−0.437083	−	0.899421i	$-0.643988\pi$
$941$	−6.85413e11			−0.874166			−0.437083	+	0.899421i	$0.643988\pi$
$942$		0			0
$943$	−4.64334e11	+	4.64334e11i	−0.587197	+	0.587197i
$944$		0			0
$945$	2.95150e9	−	2.92867e10i	0.00370097	−	0.0367234i
$946$		0			0
$947$	−3.45694e11	−	3.45694e11i	−0.429825	−	0.429825i	0.458744	−	0.888569i	$-0.348300\pi$
$947$	−3.45694e11	−	3.45694e11i	−0.429825	−	0.429825i	−0.888569	+	0.458744i	$0.848300\pi$
$948$		0			0
$949$			6.81184e11i			0.839846i
$950$		0			0
$951$	−7.63396e11			−0.933314
$952$		0			0
$953$	9.65719e11	−	9.65719e11i	1.17079	−	1.17079i	0.188769	−	0.982022i	$-0.439550\pi$
$953$	9.65719e11	−	9.65719e11i	1.17079	−	1.17079i	0.982022	−	0.188769i	$-0.0604498\pi$
$954$		0			0
$955$	−1.05320e12	−	1.06141e11i	−1.26619	−	0.127606i
$956$		0			0
$957$	−7.93827e10	−	7.93827e10i	−0.0946408	−	0.0946408i
$958$		0			0
$959$			1.69689e11i			0.200623i
$960$		0			0
$961$	3.95832e11			0.464107
$962$		0			0
$963$	6.32065e11	−	6.32065e11i	0.734947	−	0.734947i
$964$		0			0
$965$	1.78561e11	−	1.45865e11i	0.205910	−	0.168207i
$966$		0			0
$967$	8.09654e11	+	8.09654e11i	0.925962	+	0.925962i	0.997442	−	0.0714797i	$-0.0227721\pi$
$967$	8.09654e11	+	8.09654e11i	0.925962	+	0.925962i	−0.0714797	+	0.997442i	$0.522772\pi$
$968$		0			0
$969$		−	1.91071e12i		−	2.16720i
$970$		0			0
$971$	1.55047e12			1.74416			0.872082	−	0.489361i	$-0.162770\pi$
$971$	1.55047e12			1.74416			0.872082	+	0.489361i	$0.162770\pi$
$972$		0			0
$973$	7.43239e10	−	7.43239e10i	0.0829234	−	0.0829234i
$974$		0			0
$975$	4.57076e11	−	2.24467e12i	0.505790	−	2.48390i
$976$		0			0
$977$	5.55158e11	+	5.55158e11i	0.609310	+	0.609310i	0.942766	−	0.333456i	$-0.108215\pi$
$977$	5.55158e11	+	5.55158e11i	0.609310	+	0.609310i	−0.333456	+	0.942766i	$0.608215\pi$
$978$		0			0
$979$			2.32612e10i			0.0253222i
$980$		0			0
$981$	−1.23872e11			−0.133751
$982$		0			0
$983$	−4.48883e11	+	4.48883e11i	−0.480750	+	0.480750i	−0.905371	−	0.424621i	$-0.860407\pi$
$983$	−4.48883e11	+	4.48883e11i	−0.480750	+	0.480750i	0.424621	+	0.905371i	$0.360407\pi$
$984$		0			0
$985$	6.67468e10	+	8.17081e10i	0.0709064	+	0.0868000i
$986$		0			0
$987$	−5.68680e11	−	5.68680e11i	−0.599238	−	0.599238i
$988$		0			0
$989$		−	2.74611e11i		−	0.287033i
$990$		0			0
$991$	1.03897e12			1.07723			0.538616	−	0.842551i	$-0.318947\pi$
$991$	1.03897e12			1.07723			0.538616	+	0.842551i	$0.318947\pi$
$992$		0			0
$993$	2.86334e11	−	2.86334e11i	0.294493	−	0.294493i
$994$		0			0
$995$	−1.82025e11	+	1.80617e12i	−0.185712	+	1.84275i
$996$		0			0
$997$	−4.06518e11	−	4.06518e11i	−0.411433	−	0.411433i	0.470804	−	0.882238i	$-0.343964\pi$
$997$	−4.06518e11	−	4.06518e11i	−0.411433	−	0.411433i	−0.882238	+	0.470804i	$0.843964\pi$
$998$		0			0
$999$			6.70961e10i			0.0673651i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	20.9.f.a.17.4	yes	8
3.2	odd	2		180.9.l.a.37.1		8
4.3	odd	2		80.9.p.d.17.1		8
5.2	odd	4		100.9.f.b.93.1		8
5.3	odd	4	inner	20.9.f.a.13.4	✓	8
5.4	even	2		100.9.f.b.57.1		8
15.8	even	4		180.9.l.a.73.1		8
20.3	even	4		80.9.p.d.33.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
20.9.f.a.13.4	✓	8	5.3	odd	4	inner
20.9.f.a.17.4	yes	8	1.1	even	1	trivial
80.9.p.d.17.1		8	4.3	odd	2
80.9.p.d.33.1		8	20.3	even	4
100.9.f.b.57.1		8	5.4	even	2
100.9.f.b.93.1		8	5.2	odd	4
180.9.l.a.37.1		8	3.2	odd	2
180.9.l.a.73.1		8	15.8	even	4

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)

\(f(q)\)	\(=\)	\(q+(80.1275 - 80.1275i) q^{3} +(621.850 + 62.6699i) q^{5} +(-1045.24 - 1045.24i) q^{7} -6279.84i q^{9} +O(q^{10})\)	\(q+(80.1275 - 80.1275i) q^{3} +(621.850 + 62.6699i) q^{5} +(-1045.24 - 1045.24i) q^{7} -6279.84i q^{9} -1097.03 q^{11} +(36593.4 - 36593.4i) q^{13} +(54848.9 - 44805.7i) q^{15} +(-67233.5 - 67233.5i) q^{17} +177336. i q^{19} -167505. q^{21} +(-110984. + 110984. i) q^{23} +(382770. + 77942.6i) q^{25} +(22528.8 + 22528.8i) q^{27} +903078. i q^{29} -1.11746e6 q^{31} +(-87902.5 + 87902.5i) q^{33} +(-584479. - 715489. i) q^{35} +(1.48912e6 + 1.48912e6i) q^{37} -5.86427e6i q^{39} +4.18381e6 q^{41} +(-1.23717e6 + 1.23717e6i) q^{43} +(393557. - 3.90512e6i) q^{45} +(3.39500e6 + 3.39500e6i) q^{47} -3.57974e6i q^{49} -1.07745e7 q^{51} +(-6.26950e6 + 6.26950e6i) q^{53} +(-682190. - 68750.9i) q^{55} +(1.42095e7 + 1.42095e7i) q^{57} +4.61361e6i q^{59} -1.06398e7 q^{61} +(-6.56395e6 + 6.56395e6i) q^{63} +(2.50489e7 - 2.04623e7i) q^{65} +(-3.39626e6 - 3.39626e6i) q^{67} +1.77857e7i q^{69} -4.31250e7 q^{71} +(-9.30749e6 + 9.30749e6i) q^{73} +(3.69158e7 - 2.44251e7i) q^{75} +(1.14666e6 + 1.14666e6i) q^{77} -5.42990e7i q^{79} +4.48124e7 q^{81} +(4.43294e7 - 4.43294e7i) q^{83} +(-3.75956e7 - 4.60227e7i) q^{85} +(7.23614e7 + 7.23614e7i) q^{87} -2.12038e7i q^{89} -7.64978e7 q^{91} +(-8.95395e7 + 8.95395e7i) q^{93} +(-1.11136e7 + 1.10276e8i) q^{95} +(-4.71632e7 - 4.71632e7i) q^{97} +6.88919e6i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 70 q^{3} + 894 q^{5} - 2030 q^{7}+O(q^{10}) \) 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

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Database

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