LMFDB - Embedded newform 84.9.p.b.65.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [84,9,Mod(53,84)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(84, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 3, 4]))

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

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

chi := DirichletCharacter("84.53");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 84 = 2^{2} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	84.p (of order $6$, 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:	$34.2198032451$
Analytic rank:	$0$
Dimension:	$40$
Relative dimension:	$20$ over $\Q(\zeta_{6})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			65.5
Character	$\chi$	$=$	84.65
Dual form			84.9.p.b.53.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-71.4876 + 38.0858i) q^{3} +(357.784 + 206.567i) q^{5} +(-24.5961 + 2400.87i) q^{7} +(3659.95 - 5445.32i) q^{9} +O(q^{10})$	\(q+(-71.4876 + 38.0858i) q^{3} +(357.784 + 206.567i) q^{5} +(-24.5961 + 2400.87i) q^{7} +(3659.95 - 5445.32i) q^{9} +(7844.81 - 4529.20i) q^{11} +31045.1 q^{13} +(-33444.3 - 1140.48i) q^{15} +(-90466.0 + 52230.6i) q^{17} +(122784. - 212669. i) q^{19} +(-89680.8 - 172569. i) q^{21} +(364494. + 210441. i) q^{23} +(-109973. - 190479. i) q^{25} +(-54252.1 + 528665. i) q^{27} -162607. i q^{29} +(358877. + 621593. i) q^{31} +(-388308. + 622557. i) q^{33} +(-504740. + 853913. i) q^{35} +(-938176. + 1.62497e6i) q^{37} +(-2.21934e6 + 1.18238e6i) q^{39} +5.16504e6i q^{41} +982646. q^{43} +(2.43429e6 - 1.19222e6i) q^{45} +(6.43455e6 + 3.71499e6i) q^{47} +(-5.76359e6 - 118104. i) q^{49} +(4.47796e6 - 7.17931e6i) q^{51} +(-7.45705e6 + 4.30533e6i) q^{53} +3.74233e6 q^{55} +(-677909. + 1.98795e7i) q^{57} +(-1.43142e7 + 8.26429e6i) q^{59} +(8.43266e6 - 1.46058e7i) q^{61} +(1.29835e7 + 8.92102e6i) q^{63} +(1.11074e7 + 6.41288e6i) q^{65} +(-574689. - 995390. i) q^{67} +(-3.40716e7 - 1.16187e6i) q^{69} -1.64603e6i q^{71} +(2.34398e7 + 4.05990e7i) q^{73} +(1.51162e7 + 9.42847e6i) q^{75} +(1.06811e7 + 1.89458e7i) q^{77} +(1.49381e6 - 2.58735e6i) q^{79} +(-1.62562e7 - 3.98592e7i) q^{81} +1.80139e7i q^{83} -4.31564e7 q^{85} +(6.19303e6 + 1.16244e7i) q^{87} +(-9.59768e7 - 5.54122e7i) q^{89} +(-763589. + 7.45354e7i) q^{91} +(-4.93291e7 - 3.07681e7i) q^{93} +(8.78605e7 - 5.07263e7i) q^{95} -2.73581e6 q^{97} +(4.04867e6 - 5.92941e7i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q + 81 q^{3} - 34 q^{7} + 4771 q^{9}+O(q^{10}) $ 40 * q + 81 * q^3 - 34 * q^7 + 4771 * q^9	$ 40 q + 81 q^{3} - 34 q^{7} + 4771 q^{9} - 55464 q^{13} + 68482 q^{15} + 311690 q^{19} - 172343 q^{21} + 1766792 q^{25} - 3451932 q^{27} + 31596 q^{31} + 1874885 q^{33} - 1853482 q^{37} + 11217526 q^{39} - 13372600 q^{43} - 527785 q^{45} - 12653462 q^{49} - 1103461 q^{51} + 71577224 q^{55} - 17195214 q^{57} - 21761970 q^{61} + 21945045 q^{63} - 26337350 q^{67} - 5588722 q^{69} + 41115682 q^{73} - 17971730 q^{75} - 120916932 q^{79} - 24550133 q^{81} + 139250060 q^{85} - 16321046 q^{87} + 345074940 q^{91} + 25774675 q^{93} - 707216948 q^{97} - 94510994 q^{99}+O(q^{100}) $ 40 * q + 81 * q^3 - 34 * q^7 + 4771 * q^9 - 55464 * q^13 + 68482 * q^15 + 311690 * q^19 - 172343 * q^21 + 1766792 * q^25 - 3451932 * q^27 + 31596 * q^31 + 1874885 * q^33 - 1853482 * q^37 + 11217526 * q^39 - 13372600 * q^43 - 527785 * q^45 - 12653462 * q^49 - 1103461 * q^51 + 71577224 * q^55 - 17195214 * q^57 - 21761970 * q^61 + 21945045 * q^63 - 26337350 * q^67 - 5588722 * q^69 + 41115682 * q^73 - 17971730 * q^75 - 120916932 * q^79 - 24550133 * q^81 + 139250060 * q^85 - 16321046 * q^87 + 345074940 * q^91 + 25774675 * q^93 - 707216948 * q^97 - 94510994 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/84\mathbb{Z}\right)^\times$.

$n$	$29$	$43$	$73$
$\chi(n)$	$-1$	$1$	$e\left(\frac{1}{3}\right)$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	−71.4876	+	38.0858i	−0.882563	+	0.470194i
$3$	−71.4876	+	38.0858i	−0.882563	+	0.470194i
$4$		0			0
$5$	357.784	+	206.567i	0.572454	+	0.330506i	0.758129	−	0.652105i	$-0.226114\pi$
$5$	357.784	+	206.567i	0.572454	+	0.330506i	−0.185675	+	0.982611i	$0.559447\pi$
$6$		0			0
$7$	−24.5961	+	2400.87i	−0.0102441	+	0.999948i
$7$	−24.5961	+	2400.87i	−0.0102441	+	0.999948i
$8$		0			0
$9$	3659.95	−	5445.32i	0.557834	−	0.829952i
$10$		0			0
$11$	7844.81	−	4529.20i	0.535811	−	0.309351i	−0.207569	−	0.978220i	$-0.566555\pi$
$11$	7844.81	−	4529.20i	0.535811	−	0.309351i	0.743379	+	0.668870i	$0.233222\pi$
$12$		0			0
$13$	31045.1			1.08698			0.543488	−	0.839417i	$-0.317103\pi$
$13$	31045.1			1.08698			0.543488	+	0.839417i	$0.317103\pi$
$14$		0			0
$15$	−33444.3	−	1140.48i	−0.660629	−	0.0225280i
$16$		0			0
$17$	−90466.0	+	52230.6i	−1.08315	+	0.625359i	−0.931745	−	0.363113i	$-0.881714\pi$
$17$	−90466.0	+	52230.6i	−1.08315	+	0.625359i	−0.151408	+	0.988471i	$0.548381\pi$
$18$		0			0
$19$	122784.	−	212669.i	0.942169	−	1.63188i	0.180847	−	0.983511i	$-0.442116\pi$
$19$	122784.	−	212669.i	0.942169	−	1.63188i	0.761322	−	0.648373i	$-0.224550\pi$
$20$		0			0
$21$	−89680.8	−	172569.i	−0.461129	−	0.887333i
$22$		0			0
$23$	364494.	+	210441.i	1.30250	+	0.752002i	0.980833	−	0.194850i	$-0.0624219\pi$
$23$	364494.	+	210441.i	1.30250	+	0.752002i	0.321672	+	0.946851i	$0.395755\pi$
$24$		0			0
$25$	−109973.	−	190479.i	−0.281531	−	0.487626i
$26$		0			0
$27$	−54252.1	+	528665.i	−0.102085	+	0.994776i
$28$		0			0
$29$		−	162607.i		−	0.229905i	−0.993371	−	0.114952i	$-0.963328\pi$
$29$		−	162607.i		−	0.229905i	0.993371	−	0.114952i	$-0.0366716\pi$
$30$		0			0
$31$	358877.	+	621593.i	0.388596	+	0.673069i	0.992261	−	0.124170i	$-0.0396267\pi$
$31$	358877.	+	621593.i	0.388596	+	0.673069i	−0.603665	+	0.797238i	$0.706293\pi$
$32$		0			0
$33$	−388308.	+	622557.i	−0.327432	+	0.524957i
$34$		0			0
$35$	−504740.	+	853913.i	−0.336353	+	0.569038i
$36$		0			0
$37$	−938176.	+	1.62497e6i	−0.500585	+	0.867038i	0.499415	+	0.866363i	$0.333548\pi$
$37$	−938176.	+	1.62497e6i	−0.500585	+	0.867038i	−1.00000		0.000675125i	$0.999785\pi$
$38$		0			0
$39$	−2.21934e6	+	1.18238e6i	−0.959324	+	0.511090i
$40$		0			0
$41$			5.16504e6i			1.82784i	0.405892	+	0.913921i	$0.366961\pi$
$41$			5.16504e6i			1.82784i	−0.405892	+	0.913921i	$0.633039\pi$
$42$		0			0
$43$	982646.			0.287424			0.143712	−	0.989620i	$-0.454096\pi$
$43$	982646.			0.287424			0.143712	+	0.989620i	$0.454096\pi$
$44$		0			0
$45$	2.43429e6	−	1.19222e6i	0.593639	−	0.290742i
$46$		0			0
$47$	6.43455e6	+	3.71499e6i	1.31864	+	0.761318i	0.983510	−	0.180853i	$-0.0578859\pi$
$47$	6.43455e6	+	3.71499e6i	1.31864	+	0.761318i	0.335132	+	0.942171i	$0.391219\pi$
$48$		0			0
$49$	−5.76359e6	−	118104.i	−0.999790	−	0.0204871i
$50$		0			0
$51$	4.47796e6	−	7.17931e6i	0.661911	−	1.06121i
$52$		0			0
$53$	−7.45705e6	+	4.30533e6i	−0.945069	+	0.545636i	−0.891546	−	0.452931i	$-0.850379\pi$
$53$	−7.45705e6	+	4.30533e6i	−0.945069	+	0.545636i	−0.0535234	+	0.998567i	$0.517045\pi$
$54$		0			0
$55$	3.74233e6			0.408969
$56$		0			0
$57$	−677909.	+	1.98795e7i	−0.0642203	+	1.88324i
$58$		0			0
$59$	−1.43142e7	+	8.26429e6i	−1.18129	+	0.682020i	−0.956314	−	0.292343i	$-0.905565\pi$
$59$	−1.43142e7	+	8.26429e6i	−1.18129	+	0.682020i	−0.224980	+	0.974363i	$0.572232\pi$
$60$		0			0
$61$	8.43266e6	−	1.46058e7i	0.609039	−	1.05489i	−0.382360	−	0.924013i	$-0.624889\pi$
$61$	8.43266e6	−	1.46058e7i	0.609039	−	1.05489i	0.991399	−	0.130873i	$-0.0417781\pi$
$62$		0			0
$63$	1.29835e7	+	8.92102e6i	0.824194	+	0.566307i
$64$		0			0
$65$	1.11074e7	+	6.41288e6i	0.622244	+	0.359252i
$66$		0			0
$67$	−574689.	−	995390.i	−0.0285190	−	0.0493963i	0.851414	−	0.524495i	$-0.175746\pi$
$67$	−574689.	−	995390.i	−0.0285190	−	0.0493963i	−0.879933	+	0.475098i	$0.842412\pi$
$68$		0			0
$69$	−3.40716e7	−	1.16187e6i	−1.50313	−	0.0512581i
$70$		0			0
$71$		−	1.64603e6i		−	0.0647745i	−0.999475	−	0.0323873i	$-0.989689\pi$
$71$		−	1.64603e6i		−	0.0647745i	0.999475	−	0.0323873i	$-0.0103110\pi$
$72$		0			0
$73$	2.34398e7	+	4.05990e7i	0.825397	+	1.42963i	0.901615	+	0.432539i	$0.142382\pi$
$73$	2.34398e7	+	4.05990e7i	0.825397	+	1.42963i	−0.0762180	+	0.997091i	$0.524284\pi$
$74$		0			0
$75$	1.51162e7	+	9.42847e6i	0.477748	+	0.297986i
$76$		0			0
$77$	1.06811e7	+	1.89458e7i	0.303845	+	0.538952i
$78$		0			0
$79$	1.49381e6	−	2.58735e6i	0.0383518	−	0.0664273i	−0.846212	−	0.532846i	$-0.821123\pi$
$79$	1.49381e6	−	2.58735e6i	0.0383518	−	0.0664273i	0.884564	+	0.466419i	$0.154456\pi$
$80$		0			0
$81$	−1.62562e7	−	3.98592e7i	−0.377642	−	0.925952i
$82$		0			0
$83$			1.80139e7i			0.379574i	0.981825	+	0.189787i	$0.0607797\pi$
$83$			1.80139e7i			0.379574i	−0.981825	+	0.189787i	$0.939220\pi$
$84$		0			0
$85$	−4.31564e7			−0.826741
$86$		0			0
$87$	6.19303e6	+	1.16244e7i	0.108100	+	0.202906i
$88$		0			0
$89$	−9.59768e7	−	5.54122e7i	−1.52970	−	0.883173i	−0.999374	−	0.0353792i	$-0.988736\pi$
$89$	−9.59768e7	−	5.54122e7i	−1.52970	−	0.883173i	−0.530326	−	0.847794i	$-0.677931\pi$
$90$		0			0
$91$	−763589.	+	7.45354e7i	−0.0111351	+	1.08692i
$92$		0			0
$93$	−4.93291e7	−	3.07681e7i	−0.659434	−	0.411310i
$94$		0			0
$95$	8.78605e7	−	5.07263e7i	1.07870	−	0.622786i
$96$		0			0
$97$	−2.73581e6			−0.0309029			−0.0154515	−	0.999881i	$-0.504919\pi$
$97$	−2.73581e6			−0.0309029			−0.0154515	+	0.999881i	$0.504919\pi$
$98$		0			0
$99$	4.04867e6	−	5.92941e7i	0.0421475	−	0.617264i
$100$		0			0
$101$	9.95515e7	−	5.74761e7i	0.956670	−	0.552334i	0.0615235	−	0.998106i	$-0.480404\pi$
$101$	9.95515e7	−	5.74761e7i	0.956670	−	0.552334i	0.895147	+	0.445772i	$0.147071\pi$
$102$		0			0
$103$	6.51787e7	−	1.12893e8i	0.579104	−	1.00304i	−0.416478	−	0.909146i	$-0.636736\pi$
$103$	6.51787e7	−	1.12893e8i	0.579104	−	1.00304i	0.995582	−	0.0938925i	$-0.0299310\pi$
$104$		0			0
$105$	3.56075e6	−	8.02676e7i	0.0292944	−	0.660364i
$106$		0			0
$107$	−2.58935e6	−	1.49496e6i	−0.0197540	−	0.0114050i	0.490090	−	0.871672i	$-0.336964\pi$
$107$	−2.58935e6	−	1.49496e6i	−0.0197540	−	0.0114050i	−0.509844	+	0.860267i	$0.670297\pi$
$108$		0			0
$109$	3.97892e7	+	6.89170e7i	0.281877	+	0.488225i	0.971847	−	0.235613i	$-0.0757097\pi$
$109$	3.97892e7	+	6.89170e7i	0.281877	+	0.488225i	−0.689970	+	0.723838i	$0.742376\pi$
$110$		0			0
$111$	5.17980e6	−	1.51896e8i	0.0341209	−	1.00059i
$112$		0			0
$113$		−	4.55419e7i		−	0.279317i	−0.990200	−	0.139658i	$-0.955400\pi$
$113$		−	4.55419e7i		−	0.279317i	0.990200	−	0.139658i	$-0.0446004\pi$
$114$		0			0
$115$	8.69401e7	+	1.50585e8i	0.497083	+	0.860973i
$116$		0			0
$117$	1.13624e8	−	1.69050e8i	0.606352	−	0.902138i
$118$		0			0
$119$	−1.23174e8	−	2.18482e8i	−0.614230	−	1.08950i
$120$		0			0
$121$	−6.61521e7	+	1.14579e8i	−0.308604	+	0.534519i
$122$		0			0
$123$	−1.96715e8	−	3.69237e8i	−0.859441	−	1.61319i
$124$		0			0
$125$		−	2.52247e8i		−	1.03320i
$126$		0			0
$127$	−2.11231e8			−0.811977			−0.405988	−	0.913878i	$-0.633073\pi$
$127$	−2.11231e8			−0.811977			−0.405988	+	0.913878i	$0.633073\pi$
$128$		0			0
$129$	−7.02470e7	+	3.74248e7i	−0.253670	+	0.135145i
$130$		0			0
$131$	1.99192e8	+	1.15004e8i	0.676374	+	0.390505i	0.798488	−	0.602011i	$-0.205634\pi$
$131$	1.99192e8	+	1.15004e8i	0.676374	+	0.390505i	−0.122113	+	0.992516i	$0.538967\pi$
$132$		0			0
$133$	5.07571e8	+	3.00021e8i	1.62215	+	0.958837i
$134$		0			0
$135$	−1.28615e8	+	1.77941e8i	−0.387219	+	0.535724i
$136$		0			0
$137$	−3.06374e8	+	1.76885e8i	−0.869701	+	0.502122i	−0.867249	−	0.497875i	$-0.834114\pi$
$137$	−3.06374e8	+	1.76885e8i	−0.869701	+	0.502122i	−0.00245210	+	0.999997i	$0.500781\pi$
$138$		0			0
$139$	−7.32077e7			−0.196109			−0.0980545	−	0.995181i	$-0.531262\pi$
$139$	−7.32077e7			−0.196109			−0.0980545	+	0.995181i	$0.531262\pi$
$140$		0			0
$141$	−6.01479e8	−	2.05110e7i	−1.52175	−	0.0518931i
$142$		0			0
$143$	2.43543e8	−	1.40610e8i	0.582413	−	0.336256i
$144$		0			0
$145$	3.35893e7	−	5.81783e7i	0.0759851	−	0.131610i
$146$		0			0
$147$	4.16523e8	−	2.11068e8i	0.892011	−	0.452015i
$148$		0			0
$149$	−8.80690e7	−	5.08467e7i	−0.178681	−	0.103161i	0.407992	−	0.912986i	$-0.366229\pi$
$149$	−8.80690e7	−	5.08467e7i	−0.178681	−	0.103161i	−0.586673	+	0.809824i	$0.699563\pi$
$150$		0			0
$151$	2.50737e8	+	4.34290e8i	0.482293	+	0.835356i	0.999793	−	0.0203270i	$-0.00647073\pi$
$151$	2.50737e8	+	4.34290e8i	0.482293	+	0.835356i	−0.517500	+	0.855683i	$0.673137\pi$
$152$		0			0
$153$	−4.66891e7	+	6.83778e8i	−0.0852021	+	1.24781i
$154$		0			0
$155$			2.96528e8i			0.513735i
$156$		0			0
$157$	2.79449e8	+	4.84019e8i	0.459942	+	0.796643i	0.998957	−	0.0456528i	$-0.0145368\pi$
$157$	2.79449e8	+	4.84019e8i	0.459942	+	0.796643i	−0.539015	+	0.842296i	$0.681203\pi$
$158$		0			0
$159$	3.69115e8	−	5.91785e8i	0.577528	−	0.925924i
$160$		0			0
$161$	−5.14207e8	+	8.69929e8i	−0.765305	+	1.29473i
$162$		0			0
$163$	−3.44082e8	+	5.95967e8i	−0.487429	+	0.844252i	−0.999896	−	0.0144555i	$-0.995399\pi$
$163$	−3.44082e8	+	5.95967e8i	−0.487429	+	0.844252i	0.512467	+	0.858707i	$0.328732\pi$
$164$		0			0
$165$	−2.67530e8	+	1.42529e8i	−0.360941	+	0.192295i
$166$		0			0
$167$			5.28526e8i			0.679517i	0.940513	+	0.339758i	$0.110345\pi$
$167$			5.28526e8i			0.679517i	−0.940513	+	0.339758i	$0.889655\pi$
$168$		0			0
$169$	1.48068e8			0.181516
$170$		0			0
$171$	−7.08664e8	−	1.44696e9i	−0.828812	−	1.69228i
$172$		0			0
$173$	5.13678e7	+	2.96572e7i	0.0573464	+	0.0331090i	0.528399	−	0.848996i	$-0.322792\pi$
$173$	5.13678e7	+	2.96572e7i	0.0573464	+	0.0331090i	−0.471053	+	0.882105i	$0.656126\pi$
$174$		0			0
$175$	4.60021e8	−	2.59346e8i	0.490484	−	0.276521i
$176$		0			0
$177$	7.08534e8	−	1.13596e9i	0.721884	−	1.15736i
$178$		0			0
$179$	5.94393e8	−	3.43173e8i	0.578978	−	0.334273i	−0.181749	−	0.983345i	$-0.558176\pi$
$179$	5.94393e8	−	3.43173e8i	0.578978	−	0.334273i	0.760727	+	0.649072i	$0.224843\pi$
$180$		0			0
$181$	1.01672e9			0.947303			0.473651	−	0.880712i	$-0.342936\pi$
$181$	1.01672e9			0.947303			0.473651	+	0.880712i	$0.342936\pi$
$182$		0			0
$183$	−4.65578e7	+	1.36530e9i	−0.0415134	+	1.21737i
$184$		0			0
$185$	−6.71328e8	+	3.87592e8i	−0.573123	+	0.330893i
$186$		0			0
$187$	−4.73126e8	+	8.19478e8i	−0.386910	+	0.670148i
$188$		0			0
$189$	−1.26792e9	−	1.43256e8i	−0.993678	−	0.112270i
$190$		0			0
$191$	7.99680e8	+	4.61695e8i	0.600873	+	0.346914i	0.769385	−	0.638785i	$-0.220563\pi$
$191$	7.99680e8	+	4.61695e8i	0.600873	+	0.346914i	−0.168512	+	0.985700i	$0.553896\pi$
$192$		0			0
$193$	−5.58182e8	−	9.66799e8i	−0.402297	−	0.696798i	0.591706	−	0.806154i	$-0.298455\pi$
$193$	−5.58182e8	−	9.66799e8i	−0.402297	−	0.696798i	−0.994003	+	0.109356i	$0.965121\pi$
$194$		0			0
$195$	−1.03828e9	−	3.54064e7i	−0.718088	−	0.0244874i
$196$		0			0
$197$			2.41802e9i			1.60544i	0.596354	+	0.802722i	$0.296615\pi$
$197$			2.41802e9i			1.60544i	−0.596354	+	0.802722i	$0.703385\pi$
$198$		0			0
$199$	8.94414e8	+	1.54917e9i	0.570330	+	0.987841i	0.996532	+	0.0832126i	$0.0265181\pi$
$199$	8.94414e8	+	1.54917e9i	0.570330	+	0.987841i	−0.426202	+	0.904628i	$0.640149\pi$
$200$		0			0
$201$	7.89933e7	+	4.92706e7i	0.0483956	+	0.0301859i
$202$		0			0
$203$	3.90400e8	+	3.99951e6i	0.229893	+	0.00235517i
$204$		0			0
$205$	−1.06693e9	+	1.84797e9i	−0.604114	+	1.04636i
$206$		0			0
$207$	2.47995e9	−	1.21458e9i	1.35071	−	0.661525i
$208$		0			0
$209$		−	2.22446e9i		−	1.16584i
$210$		0			0
$211$	1.83517e9			0.925862			0.462931	−	0.886394i	$-0.346798\pi$
$211$	1.83517e9			0.925862			0.462931	+	0.886394i	$0.346798\pi$
$212$		0			0
$213$	6.26903e7	+	1.17671e8i	0.0304566	+	0.0571676i
$214$		0			0
$215$	3.51575e8	+	2.02982e8i	0.164537	+	0.0949955i
$216$		0			0
$217$	−1.50119e9	+	8.46330e8i	−0.677014	+	0.381681i
$218$		0			0
$219$	−3.22190e9	−	2.00960e9i	−1.40067	−	0.873641i
$220$		0			0
$221$	−2.80853e9	+	1.62150e9i	−1.17736	+	0.679750i
$222$		0			0
$223$	3.00307e8			0.121436			0.0607179	−	0.998155i	$-0.480661\pi$
$223$	3.00307e8			0.121436			0.0607179	+	0.998155i	$0.480661\pi$
$224$		0			0
$225$	−1.43971e9	−	9.83054e7i	−0.561754	−	0.0383572i
$226$		0			0
$227$	4.27881e9	−	2.47037e9i	1.61146	−	0.930376i	0.622426	−	0.782679i	$-0.286147\pi$
$227$	4.27881e9	−	2.47037e9i	1.61146	−	0.930376i	0.989033	−	0.147697i	$-0.0471861\pi$
$228$		0			0
$229$	−2.59755e8	+	4.49909e8i	−0.0944544	+	0.163600i	−0.909381	−	0.415965i	$-0.863444\pi$
$229$	−2.59755e8	+	4.49909e8i	−0.0944544	+	0.163600i	0.814926	+	0.579564i	$0.196777\pi$
$230$		0			0
$231$	−1.48513e9	−	9.47592e8i	−0.521575	−	0.332792i
$232$		0			0
$233$	−1.68419e9	−	9.72367e8i	−0.571435	−	0.329918i	0.186287	−	0.982495i	$-0.440355\pi$
$233$	−1.68419e9	−	9.72367e8i	−0.571435	−	0.329918i	−0.757722	+	0.652577i	$0.773688\pi$
$234$		0			0
$235$	1.53479e9	+	2.65833e9i	0.503241	+	0.871639i
$236$		0			0
$237$	−8.24750e6	+	2.41856e8i	−0.00261414	+	0.0766591i
$238$		0			0
$239$		−	3.56441e9i		−	1.09244i	−0.837643	−	0.546218i	$-0.816067\pi$
$239$		−	3.56441e9i		−	1.09244i	0.837643	−	0.546218i	$-0.183933\pi$
$240$		0			0
$241$	−1.33494e9	−	2.31219e9i	−0.395726	−	0.685417i	0.597468	−	0.801893i	$-0.296174\pi$
$241$	−1.33494e9	−	2.31219e9i	−0.395726	−	0.685417i	−0.993194	+	0.116476i	$0.962840\pi$
$242$		0			0
$243$	2.68019e9	+	2.23031e9i	0.768670	+	0.639646i
$244$		0			0
$245$	−2.03772e9	−	1.23282e9i	−0.565563	−	0.342165i
$246$		0			0
$247$	3.81186e9	−	6.60233e9i	1.02411	−	1.77382i
$248$		0			0
$249$	−6.86074e8	−	1.28777e9i	−0.178473	−	0.334998i
$250$		0			0
$251$		−	5.77566e9i		−	1.45515i	−0.686030	−	0.727573i	$-0.740648\pi$
$251$		−	5.77566e9i		−	1.45515i	0.686030	−	0.727573i	$-0.259352\pi$
$252$		0			0
$253$	3.81252e9			0.930528
$254$		0			0
$255$	3.08515e9	−	1.64364e9i	0.729650	−	0.388729i
$256$		0			0
$257$	−1.35968e9	−	7.85011e8i	−0.311677	−	0.179947i	0.336000	−	0.941862i	$-0.390926\pi$
$257$	−1.35968e9	−	7.85011e8i	−0.311677	−	0.179947i	−0.647676	+	0.761915i	$0.724259\pi$
$258$		0			0
$259$	−3.87827e9	−	2.29241e9i	−0.861864	−	0.509440i
$260$		0			0
$261$	−8.85449e8	−	5.95135e8i	−0.190810	−	0.128249i
$262$		0			0
$263$	2.75489e9	−	1.59053e9i	0.575812	−	0.332445i	−0.183655	−	0.982991i	$-0.558793\pi$
$263$	2.75489e9	−	1.59053e9i	0.575812	−	0.332445i	0.759467	+	0.650546i	$0.225460\pi$
$264$		0			0
$265$	−3.55735e9			−0.721345
$266$		0			0
$267$	8.97157e9	+	3.05938e8i	1.76532	+	0.0601990i
$268$		0			0
$269$	2.59695e9	−	1.49935e9i	0.495968	−	0.286347i	−0.231079	−	0.972935i	$-0.574226\pi$
$269$	2.59695e9	−	1.49935e9i	0.495968	−	0.286347i	0.727047	+	0.686588i	$0.240892\pi$
$270$		0			0
$271$	−3.53154e9	+	6.11680e9i	−0.654767	+	1.13409i	0.327185	+	0.944960i	$0.393900\pi$
$271$	−3.53154e9	+	6.11680e9i	−0.654767	+	1.13409i	−0.981952	+	0.189129i	$0.939434\pi$
$272$		0			0
$273$	−2.78415e9	−	5.35744e9i	−0.501236	−	0.964510i
$274$		0			0
$275$	−1.72543e9	−	9.96180e8i	−0.301695	−	0.174183i
$276$		0			0
$277$	1.39299e9	+	2.41273e9i	0.236607	+	0.409816i	0.959739	−	0.280895i	$-0.0906311\pi$
$277$	1.39299e9	+	2.41273e9i	0.236607	+	0.409816i	−0.723131	+	0.690711i	$0.757298\pi$
$278$		0			0
$279$	4.69824e9	+	3.20802e8i	0.775387	+	0.0529444i
$280$		0			0
$281$			9.55083e9i			1.53185i	0.642930	+	0.765925i	$0.277718\pi$
$281$			9.55083e9i			1.53185i	−0.642930	+	0.765925i	$0.722282\pi$
$282$		0			0
$283$	−3.25958e9	−	5.64576e9i	−0.508178	−	0.880191i	−0.999955	−	0.00946946i	$-0.996986\pi$
$283$	−3.25958e9	−	5.64576e9i	−0.508178	−	0.880191i	0.491777	−	0.870721i	$-0.336348\pi$
$284$		0			0
$285$	−4.34899e9	+	6.97254e9i	−0.659187	+	1.05685i
$286$		0			0
$287$	−1.24006e10	−	1.27040e8i	−1.82775	−	0.0187246i
$288$		0			0
$289$	1.96819e9	−	3.40901e9i	0.282147	−	0.488693i
$290$		0			0
$291$	1.95577e8	−	1.04196e8i	0.0272738	−	0.0145304i
$292$		0			0
$293$			1.12009e9i			0.151978i	0.997109	+	0.0759892i	$0.0242114\pi$
$293$			1.12009e9i			0.151978i	−0.997109	+	0.0759892i	$0.975789\pi$
$294$		0			0
$295$	−6.82850e9			−0.901649
$296$		0			0
$297$	1.96883e9	+	4.39299e9i	0.253036	+	0.564592i
$298$		0			0
$299$	1.13158e10	+	6.53316e9i	1.41579	+	0.817407i
$300$		0			0
$301$	−2.41693e7	+	2.35921e9i	−0.00294440	+	0.287409i
$302$		0			0
$303$	−4.92768e9	+	7.90032e9i	−0.584617	+	0.937290i
$304$		0			0
$305$	6.03414e9	−	3.48381e9i	0.697294	−	0.402583i
$306$		0			0
$307$	−4.29211e9			−0.483190			−0.241595	−	0.970377i	$-0.577671\pi$
$307$	−4.29211e9			−0.483190			−0.241595	+	0.970377i	$0.577671\pi$
$308$		0			0
$309$	−3.59860e8	+	1.05528e10i	−0.0394730	+	1.15754i
$310$		0			0
$311$	1.02202e10	−	5.90066e9i	1.09250	−	0.630753i	0.158256	−	0.987398i	$-0.449413\pi$
$311$	1.02202e10	−	5.90066e9i	1.09250	−	0.630753i	0.934240	+	0.356646i	$0.116080\pi$
$312$		0			0
$313$	7.22454e9	−	1.25133e10i	0.752719	−	1.30375i	−0.193781	−	0.981045i	$-0.562075\pi$
$313$	7.22454e9	−	1.25133e10i	0.752719	−	1.30375i	0.946500	−	0.322703i	$-0.104592\pi$
$314$		0			0
$315$	2.80250e9	+	5.87375e9i	0.284645	+	0.596586i
$316$		0			0
$317$	−9.91162e9	−	5.72247e9i	−0.981539	−	0.566692i	−0.0788043	−	0.996890i	$-0.525110\pi$
$317$	−9.91162e9	−	5.72247e9i	−0.981539	−	0.566692i	−0.902734	+	0.430198i	$0.858444\pi$
$318$		0			0
$319$	−7.36482e8	−	1.27562e9i	−0.0711212	−	0.123186i
$320$		0			0
$321$	2.42043e8	+	8.25387e6i	0.0227967	+	0.000777388i
$322$		0			0
$323$			2.56524e10i			2.35677i
$324$		0			0
$325$	−3.41412e9	−	5.91344e9i	−0.306017	−	0.530037i
$326$		0			0
$327$	−5.46919e9	−	3.41131e9i	−0.478335	−	0.298352i
$328$		0			0
$329$	−9.07749e9	+	1.53572e10i	−0.774786	+	1.31077i
$330$		0			0
$331$	−2.93617e9	+	5.08560e9i	−0.244607	+	0.423672i	−0.962021	−	0.272975i	$-0.911992\pi$
$331$	−2.93617e9	+	5.08560e9i	−0.244607	+	0.423672i	0.717414	+	0.696647i	$0.245326\pi$
$332$		0			0
$333$	5.41479e9	+	1.10560e10i	0.440357	+	0.899125i
$334$		0			0
$335$		−	4.74846e8i		−	0.0377028i
$336$		0			0
$337$	−9.13961e8			−0.0708611			−0.0354305	−	0.999372i	$-0.511280\pi$
$337$	−9.13961e8			−0.0708611			−0.0354305	+	0.999372i	$0.511280\pi$
$338$		0			0
$339$	1.73450e9	+	3.25568e9i	0.131333	+	0.246515i
$340$		0			0
$341$	5.63064e9	+	3.25085e9i	0.416428	+	0.240425i
$342$		0			0
$343$	4.25316e8	−	1.38348e10i	0.0307280	−	0.999528i
$344$		0			0
$345$	−1.19503e10	−	7.45376e9i	−0.843531	−	0.526137i
$346$		0			0
$347$	−1.04074e10	+	6.00873e9i	−0.717836	+	0.414443i	−0.813956	−	0.580927i	$-0.802690\pi$
$347$	−1.04074e10	+	6.00873e9i	−0.717836	+	0.414443i	0.0961196	+	0.995370i	$0.469357\pi$
$348$		0			0
$349$	2.38112e10			1.60502			0.802509	−	0.596639i	$-0.203498\pi$
$349$	2.38112e10			1.60502			0.802509	+	0.596639i	$0.203498\pi$
$350$		0			0
$351$	−1.68426e9	+	1.64124e10i	−0.110964	+	1.08130i
$352$		0			0
$353$	1.14264e10	−	6.59704e9i	0.735887	−	0.424864i	−0.0846850	−	0.996408i	$-0.526988\pi$
$353$	1.14264e10	−	6.59704e9i	0.735887	−	0.424864i	0.820572	+	0.571543i	$0.193655\pi$
$354$		0			0
$355$	3.40015e8	−	5.88923e8i	0.0214084	−	0.0370804i
$356$		0			0
$357$	1.71265e10	+	1.09276e10i	1.05437	+	0.672747i
$358$		0			0
$359$	6.65854e8	+	3.84431e8i	0.0400868	+	0.0231441i	0.519909	−	0.854221i	$-0.325966\pi$
$359$	6.65854e8	+	3.84431e8i	0.0400868	+	0.0231441i	−0.479823	+	0.877365i	$0.659299\pi$
$360$		0			0
$361$	−2.16602e10	−	3.75166e10i	−1.27537	−	2.20900i
$362$		0			0
$363$	3.65235e8	−	1.07104e10i	0.0210351	−	0.616850i
$364$		0			0
$365$			1.93675e10i			1.09120i
$366$		0			0
$367$	−3.71915e9	−	6.44175e9i	−0.205012	−	0.355091i	0.745125	−	0.666925i	$-0.232390\pi$
$367$	−3.71915e9	−	6.44175e9i	−0.205012	−	0.355091i	−0.950137	+	0.311834i	$0.899057\pi$
$368$		0			0
$369$	2.81253e10	+	1.89038e10i	1.51702	+	1.01963i
$370$		0			0
$371$	−1.01531e10	−	1.80093e10i	−0.535926	−	0.950609i
$372$		0			0
$373$	−5.94751e9	+	1.03014e10i	−0.307255	+	0.532182i	−0.977761	−	0.209723i	$-0.932744\pi$
$373$	−5.94751e9	+	1.03014e10i	−0.307255	+	0.532182i	0.670506	+	0.741905i	$0.266077\pi$
$374$		0			0
$375$	9.60702e9	+	1.80325e10i	0.485807	+	0.911868i
$376$		0			0
$377$		−	5.04817e9i		−	0.249901i
$378$		0			0
$379$	4.47934e9			0.217099			0.108549	−	0.994091i	$-0.465379\pi$
$379$	4.47934e9			0.217099			0.108549	+	0.994091i	$0.465379\pi$
$380$		0			0
$381$	1.51004e10	−	8.04491e9i	0.716621	−	0.381787i
$382$		0			0
$383$	−9.63688e9	−	5.56385e9i	−0.447859	−	0.258572i	0.259067	−	0.965859i	$-0.416585\pi$
$383$	−9.63688e9	−	5.56385e9i	−0.447859	−	0.258572i	−0.706926	+	0.707288i	$0.749918\pi$
$384$		0			0
$385$	−9.20467e7	+	8.98485e9i	−0.00418953	+	0.408948i
$386$		0			0
$387$	3.59643e9	−	5.35082e9i	0.160335	−	0.238548i
$388$		0			0
$389$	2.73476e10	−	1.57891e10i	1.19432	−	0.689541i	0.235037	−	0.971986i	$-0.424479\pi$
$389$	2.73476e10	−	1.57891e10i	1.19432	−	0.689541i	0.959283	+	0.282445i	$0.0911456\pi$
$390$		0			0
$391$	−4.39658e10			−1.88108
$392$		0			0
$393$	−1.86198e10	−	6.34951e8i	−0.780556	−	0.0266177i
$394$		0			0
$395$	1.06892e9	−	6.17141e8i	0.0439093	−	0.0253510i
$396$		0			0
$397$	4.10369e9	−	7.10780e9i	0.165201	−	0.286136i	−0.771526	−	0.636198i	$-0.780506\pi$
$397$	4.10369e9	−	7.10780e9i	0.165201	−	0.286136i	0.936727	+	0.350062i	$0.113839\pi$
$398$		0			0
$399$	−4.77115e10	−	2.11653e9i	−1.88249	−	0.0835091i
$400$		0			0
$401$	−2.82816e10	−	1.63284e10i	−1.09377	−	0.631488i	−0.159192	−	0.987248i	$-0.550889\pi$
$401$	−2.82816e10	−	1.63284e10i	−1.09377	−	0.631488i	−0.934577	+	0.355760i	$0.884222\pi$
$402$		0			0
$403$	1.11414e10	+	1.92974e10i	0.422395	+	0.731609i
$404$		0			0
$405$	2.41736e9	−	1.76190e10i	0.0898506	−	0.654878i
$406$		0			0
$407$			1.69968e10i			0.619424i
$408$		0			0
$409$	1.54794e10	+	2.68111e10i	0.553173	+	0.958123i	0.998043	+	0.0625284i	$0.0199164\pi$
$409$	1.54794e10	+	2.68111e10i	0.553173	+	0.958123i	−0.444870	+	0.895595i	$0.646750\pi$
$410$		0			0
$411$	1.51651e10	−	2.43136e10i	0.531471	−	0.852083i
$412$		0			0
$413$	−1.94894e10	−	3.45698e10i	−0.669883	−	1.18822i
$414$		0			0
$415$	−3.72107e9	+	6.44509e9i	−0.125452	+	0.217288i
$416$		0			0
$417$	5.23344e9	−	2.78817e9i	0.173079	−	0.0922094i
$418$		0			0
$419$		−	1.73085e10i		−	0.561569i	−0.959771	−	0.280784i	$-0.909405\pi$
$419$		−	1.73085e10i		−	0.561569i	0.959771	−	0.280784i	$-0.0905946\pi$
$420$		0			0
$421$	−5.23334e10			−1.66591			−0.832954	−	0.553342i	$-0.813352\pi$
$421$	−5.23334e10			−1.66591			−0.832954	+	0.553342i	$0.813352\pi$
$422$		0			0
$423$	4.37794e10	−	2.14415e10i	1.36744	−	0.669720i
$424$		0			0
$425$	1.98976e10	+	1.14879e10i	0.609882	+	0.352116i
$426$		0			0
$427$	3.48593e10	+	2.06050e10i	1.04859	+	0.619814i
$428$		0			0
$429$	−1.20551e10	+	1.93274e10i	−0.355910	+	0.570615i
$430$		0			0
$431$	−6.98669e9	+	4.03377e9i	−0.202471	+	0.116897i	−0.597807	−	0.801640i	$-0.703961\pi$
$431$	−6.98669e9	+	4.03377e9i	−0.202471	+	0.116897i	0.395337	+	0.918536i	$0.370628\pi$
$432$		0			0
$433$	1.12651e10			0.320466			0.160233	−	0.987079i	$-0.448775\pi$
$433$	1.12651e10			0.320466			0.160233	+	0.987079i	$0.448775\pi$
$434$		0			0
$435$	−1.85451e8	+	5.43830e9i	−0.00517931	+	0.151882i
$436$		0			0
$437$	8.95084e10	−	5.16777e10i	2.45436	−	1.41703i
$438$		0			0
$439$	2.61490e10	−	4.52914e10i	0.704039	−	1.21943i	−0.262998	−	0.964796i	$-0.584711\pi$
$439$	2.61490e10	−	4.52914e10i	0.704039	−	1.21943i	0.967037	−	0.254635i	$-0.0819553\pi$
$440$		0			0
$441$	−2.17376e10	+	3.09523e10i	−0.574721	+	0.818350i
$442$		0			0
$443$	−4.97977e10	−	2.87507e10i	−1.29299	−	0.746507i	−0.313806	−	0.949487i	$-0.601604\pi$
$443$	−4.97977e10	−	2.87507e10i	−1.29299	−	0.746507i	−0.979183	+	0.202980i	$0.934937\pi$
$444$		0			0
$445$	−2.28926e10	−	3.96512e10i	−0.583789	−	1.01115i
$446$		0			0
$447$	8.23238e9	+	2.80731e8i	0.206203	+	0.00703171i
$448$		0			0
$449$		−	1.04552e10i		−	0.257244i	−0.991694	−	0.128622i	$-0.958945\pi$
$449$		−	1.04552e10i		−	0.257244i	0.991694	−	0.128622i	$-0.0410555\pi$
$450$		0			0
$451$	2.33935e10	+	4.05188e10i	0.565444	+	0.979378i
$452$		0			0
$453$	−3.44648e10	−	2.14968e10i	−0.818434	−	0.510483i
$454$		0			0
$455$	−1.56697e10	+	2.65098e10i	−0.365608	+	0.618531i
$456$		0			0
$457$	3.67160e10	−	6.35940e10i	0.841765	−	1.45798i	−0.0466366	−	0.998912i	$-0.514850\pi$
$457$	3.67160e10	−	6.35940e10i	0.841765	−	1.45798i	0.888401	−	0.459068i	$-0.151816\pi$
$458$		0			0
$459$	−2.27045e10	−	5.06598e10i	−0.511518	−	1.14133i
$460$		0			0
$461$		−	1.69739e10i		−	0.375819i	−0.982186	−	0.187909i	$-0.939829\pi$
$461$		−	1.69739e10i		−	0.375819i	0.982186	−	0.187909i	$-0.0601711\pi$
$462$		0			0
$463$	5.85022e10			1.27306			0.636529	−	0.771253i	$-0.280370\pi$
$463$	5.85022e10			1.27306			0.636529	+	0.771253i	$0.280370\pi$
$464$		0			0
$465$	−1.12935e10	−	2.11981e10i	−0.241555	−	0.453403i
$466$		0			0
$467$	4.27447e10	+	2.46786e10i	0.898699	+	0.518864i	0.876778	−	0.480895i	$-0.159688\pi$
$467$	4.27447e10	+	2.46786e10i	0.898699	+	0.518864i	0.0219213	+	0.999760i	$0.493022\pi$
$468$		0			0
$469$	2.40394e9	−	1.35527e9i	0.0496858	−	0.0280114i
$470$		0			0
$471$	−3.84113e10	−	2.39583e10i	−0.780505	−	0.486825i
$472$		0			0
$473$	7.70866e9	−	4.45060e9i	0.154005	−	0.0889148i
$474$		0			0
$475$	−5.40119e10			−1.06100
$476$		0			0
$477$	−3.84855e9	+	5.63633e10i	−0.0743402	+	1.08874i
$478$		0			0
$479$	−6.29816e10	+	3.63625e10i	−1.19639	+	0.690735i	−0.959748	−	0.280863i	$-0.909379\pi$
$479$	−6.29816e10	+	3.63625e10i	−1.19639	+	0.690735i	−0.236639	+	0.971598i	$0.576046\pi$
$480$		0			0
$481$	−2.91258e10	+	5.04473e10i	−0.544123	+	0.942449i
$482$		0			0
$483$	3.62754e9	−	8.17731e10i	0.0666536	−	1.50253i
$484$		0			0
$485$	−9.78830e8	−	5.65128e8i	−0.0176905	−	0.0102136i
$486$		0			0
$487$	3.97466e10	+	6.88431e10i	0.706617	+	1.22390i	0.966105	+	0.258150i	$0.0831130\pi$
$487$	3.97466e10	+	6.88431e10i	0.706617	+	1.22390i	−0.259488	+	0.965746i	$0.583554\pi$
$488$		0			0
$489$	1.89972e9	−	5.57089e10i	0.0332242	−	0.974292i
$490$		0			0
$491$			9.72112e10i			1.67259i	0.548277	+	0.836297i	$0.315284\pi$
$491$			9.72112e10i			1.67259i	−0.548277	+	0.836297i	$0.684716\pi$
$492$		0			0
$493$	8.49308e9	+	1.47105e10i	0.143773	+	0.249022i
$494$		0			0
$495$	1.36967e10	−	2.03782e10i	0.228137	−	0.339425i
$496$		0			0
$497$	3.95191e9	+	4.04859e7i	0.0647711	+	0.000663557i
$498$		0			0
$499$	−3.88834e10	+	6.73480e10i	−0.627136	+	1.08623i	0.360988	+	0.932571i	$0.382440\pi$
$499$	−3.88834e10	+	6.73480e10i	−0.627136	+	1.08623i	−0.988124	+	0.153661i	$0.950894\pi$
$500$		0			0
$501$	−2.01293e10	−	3.77830e10i	−0.319505	−	0.599716i
$502$		0			0
$503$		−	1.48483e10i		−	0.231955i	−0.993252	−	0.115978i	$-0.963000\pi$
$503$		−	1.48483e10i		−	0.231955i	0.993252	−	0.115978i	$-0.0370001\pi$
$504$		0			0
$505$	4.74905e10			0.730200
$506$		0			0
$507$	−1.05850e10	+	5.63928e9i	−0.160199	+	0.0853478i
$508$		0			0
$509$	4.32489e10	+	2.49698e10i	0.644324	+	0.372001i	0.786278	−	0.617873i	$-0.212005\pi$
$509$	4.32489e10	+	2.49698e10i	0.644324	+	0.372001i	−0.141954	+	0.989873i	$0.545339\pi$
$510$		0			0
$511$	−9.80496e10	+	5.52775e10i	−1.43801	+	0.810709i
$512$		0			0
$513$	1.05769e11	+	7.64495e10i	1.52718	+	1.10384i
$514$		0			0
$515$	4.66398e10	−	2.69275e10i	0.663021	−	0.382796i
$516$		0			0
$517$	6.73037e10			0.942057
$518$		0			0
$519$	−4.80167e9	−	1.63741e8i	−0.0661795	−	0.00225678i
$520$		0			0
$521$	−2.02087e10	+	1.16675e10i	−0.274276	+	0.158353i	−0.630829	−	0.775922i	$-0.717285\pi$
$521$	−2.02087e10	+	1.16675e10i	−0.274276	+	0.158353i	0.356553	+	0.934275i	$0.383952\pi$
$522$		0			0
$523$	−2.98702e10	+	5.17367e10i	−0.399237	+	0.691499i	−0.993632	−	0.112674i	$-0.964058\pi$
$523$	−2.98702e10	+	5.17367e10i	−0.399237	+	0.691499i	0.594395	+	0.804173i	$0.297392\pi$
$524$		0			0
$525$	−2.30084e10	+	3.60603e10i	−0.302865	+	0.474670i
$526$		0			0
$527$	−6.49324e10	−	3.74887e10i	−0.841819	−	0.486024i
$528$		0			0
$529$	4.94152e10	+	8.55897e10i	0.631013	+	1.09295i
$530$		0			0
$531$	−7.38748e9	+	1.08192e11i	−0.0929219	+	1.36087i
$532$		0			0
$533$			1.60349e11i			1.98682i
$534$		0			0
$535$	−6.17617e8	−	1.06974e9i	−0.00753884	−	0.0130576i
$536$		0			0
$537$	−2.94217e10	+	4.71705e10i	−0.353811	+	0.567249i
$538$		0			0
$539$	−4.57492e10	+	2.51780e10i	−0.542036	+	0.298308i
$540$		0			0
$541$	4.10426e10	−	7.10879e10i	0.479122	−	0.829864i	−0.520591	−	0.853806i	$-0.674289\pi$
$541$	4.10426e10	−	7.10879e10i	0.479122	−	0.829864i	0.999713	+	0.0239424i	$0.00762184\pi$
$542$		0			0
$543$	−7.26832e10	+	3.87227e10i	−0.836054	+	0.445417i
$544$		0			0
$545$			3.28765e10i			0.372649i
$546$		0			0
$547$	−3.41404e10			−0.381346			−0.190673	−	0.981654i	$-0.561067\pi$
$547$	−3.41404e10			−0.381346			−0.190673	+	0.981654i	$0.561067\pi$
$548$		0			0
$549$	−4.86701e10	−	9.93750e10i	−0.535763	−	1.09393i
$550$		0			0
$551$	−3.45815e10	−	1.99657e10i	−0.375178	−	0.216609i
$552$		0			0
$553$	6.17515e9	+	3.65008e9i	0.0660309	+	0.0390303i
$554$		0			0
$555$	3.32299e10	−	5.32760e10i	0.350233	−	0.561513i
$556$		0			0
$557$	−3.24544e10	+	1.87376e10i	−0.337173	+	0.194667i	−0.659021	−	0.752124i	$-0.729029\pi$
$557$	−3.24544e10	+	1.87376e10i	−0.337173	+	0.194667i	0.321848	+	0.946791i	$0.395696\pi$
$558$		0			0
$559$	3.05063e10			0.312423
$560$		0			0
$561$	2.61219e9	−	7.66018e10i	0.0263726	−	0.773371i
$562$		0			0
$563$	1.11108e11	−	6.41483e10i	1.10589	−	0.638486i	0.168128	−	0.985765i	$-0.446228\pi$
$563$	1.11108e11	−	6.41483e10i	1.10589	−	0.638486i	0.937762	+	0.347279i	$0.112894\pi$
$564$		0			0
$565$	9.40743e9	−	1.62941e10i	0.0923160	−	0.159896i
$566$		0			0
$567$	9.60967e10	−	3.80488e10i	0.929772	−	0.368136i
$568$		0			0
$569$	−1.88723e10	−	1.08960e10i	−0.180043	−	0.103948i	0.407270	−	0.913308i	$-0.366481\pi$
$569$	−1.88723e10	−	1.08960e10i	−0.180043	−	0.103948i	−0.587313	+	0.809360i	$0.699814\pi$
$570$		0			0
$571$	−1.27950e10	−	2.21615e10i	−0.120363	−	0.208476i	0.799548	−	0.600603i	$-0.205073\pi$
$571$	−1.27950e10	−	2.21615e10i	−0.120363	−	0.208476i	−0.919911	+	0.392127i	$0.871739\pi$
$572$		0			0
$573$	−7.47512e10	−	2.54908e9i	−0.693425	−	0.0236464i
$574$		0			0
$575$		−	9.25713e10i		−	0.846847i
$576$		0			0
$577$	−6.09724e9	−	1.05607e10i	−0.0550086	−	0.0952776i	0.837210	−	0.546882i	$-0.184185\pi$
$577$	−6.09724e9	−	1.05607e10i	−0.0550086	−	0.0952776i	−0.892218	+	0.451604i	$0.850852\pi$
$578$		0			0
$579$	7.67243e10	+	4.78554e10i	0.682683	+	0.425810i
$580$		0			0
$581$	−4.32492e10	−	4.43073e8i	−0.379554	−	0.00388839i
$582$		0			0
$583$	−3.89994e10	+	6.75490e10i	−0.337586	+	0.584715i
$584$		0			0
$585$	7.55728e10	−	3.70127e10i	0.645271	−	0.316029i
$586$		0			0
$587$			1.76523e11i			1.48679i	0.668854	+	0.743394i	$0.266785\pi$
$587$			1.76523e11i			1.48679i	−0.668854	+	0.743394i	$0.733215\pi$
$588$		0			0
$589$	1.76258e11			1.46449
$590$		0			0
$591$	−9.20921e10	−	1.72858e11i	−0.754870	−	1.41690i
$592$		0			0
$593$	−8.53263e9	−	4.92631e9i	−0.0690024	−	0.0398385i	0.465102	−	0.885257i	$-0.346018\pi$
$593$	−8.53263e9	−	4.92631e9i	−0.0690024	−	0.0398385i	−0.534104	+	0.845419i	$0.679351\pi$
$594$		0			0
$595$	1.06148e9	−	1.03613e11i	0.00846922	−	0.826697i
$596$		0			0
$597$	−1.22941e11	−	7.66820e10i	−0.967829	−	0.603666i
$598$		0			0
$599$	−3.04680e10	+	1.75907e10i	−0.236666	+	0.136639i	−0.613644	−	0.789583i	$-0.710297\pi$
$599$	−3.04680e10	+	1.75907e10i	−0.236666	+	0.136639i	0.376977	+	0.926223i	$0.376963\pi$
$600$		0			0
$601$	1.61991e11			1.24163			0.620815	−	0.783957i	$-0.286802\pi$
$601$	1.61991e11			1.24163			0.620815	+	0.783957i	$0.286802\pi$
$602$		0			0
$603$	−7.52355e9	−	5.13717e8i	−0.0569054	−	0.00388557i
$604$		0			0
$605$	−4.73363e10	+	2.73296e10i	−0.353324	+	0.203992i
$606$		0			0
$607$	−3.24177e10	+	5.61491e10i	−0.238796	+	0.413607i	−0.960369	−	0.278731i	$-0.910086\pi$
$607$	−3.24177e10	+	5.61491e10i	−0.238796	+	0.413607i	0.721573	+	0.692338i	$0.243419\pi$
$608$		0			0
$609$	−2.80611e10	+	1.45828e10i	−0.204002	+	0.106016i
$610$		0			0
$611$	1.99761e11	+	1.15332e11i	1.43333	+	0.827534i
$612$		0			0
$613$	1.03362e9	+	1.79027e9i	0.00732010	+	0.0126788i	0.869662	−	0.493647i	$-0.164337\pi$
$613$	1.03362e9	+	1.79027e9i	0.00732010	+	0.0126788i	−0.862342	+	0.506326i	$0.831003\pi$
$614$		0			0
$615$	5.89064e9	−	1.72741e11i	0.0411777	−	1.20753i
$616$		0			0
$617$		−	2.42496e11i		−	1.67326i	−0.547766	−	0.836632i	$-0.684521\pi$
$617$		−	2.42496e11i		−	1.67326i	0.547766	−	0.836632i	$-0.315479\pi$
$618$		0			0
$619$	5.06681e10	+	8.77597e10i	0.345121	+	0.597768i	0.985376	−	0.170395i	$-0.0545045\pi$
$619$	5.06681e10	+	8.77597e10i	0.345121	+	0.597768i	−0.640255	+	0.768163i	$0.721171\pi$
$620$		0			0
$621$	−1.31027e11	+	1.81278e11i	−0.881039	+	1.21893i
$622$		0			0
$623$	1.35398e11	−	2.29065e11i	0.898797	−	1.52057i
$624$		0			0
$625$	9.14760e9	−	1.58441e10i	0.0599497	−	0.103836i
$626$		0			0
$627$	8.47203e10	+	1.59021e11i	0.548172	+	1.02893i
$628$		0			0
$629$		−	1.96006e11i		−	1.25218i
$630$		0			0
$631$	−2.45898e10			−0.155109			−0.0775547	−	0.996988i	$-0.524711\pi$
$631$	−2.45898e10			−0.155109			−0.0775547	+	0.996988i	$0.524711\pi$
$632$		0			0
$633$	−1.31192e11	+	6.98938e10i	−0.817132	+	0.435335i
$634$		0			0
$635$	−7.55752e10	−	4.36334e10i	−0.464819	−	0.268364i
$636$		0			0
$637$	−1.78931e11	−	3.66656e9i	−1.08675	−	0.0222690i
$638$		0			0
$639$	−8.96315e9	−	6.02439e9i	−0.0537598	−	0.0361335i
$640$		0			0
$641$	−7.70167e10	+	4.44656e10i	−0.456197	+	0.263386i	−0.710444	−	0.703754i	$-0.751506\pi$
$641$	−7.70167e10	+	4.44656e10i	−0.456197	+	0.263386i	0.254247	+	0.967139i	$0.418173\pi$
$642$		0			0
$643$	−2.51164e11			−1.46931			−0.734654	−	0.678442i	$-0.762656\pi$
$643$	−2.51164e11			−1.46931			−0.734654	+	0.678442i	$0.762656\pi$
$644$		0			0
$645$	−3.28639e10	−	1.12069e9i	−0.189881	−	0.00647510i
$646$		0			0
$647$	−7.50709e10	+	4.33422e10i	−0.428405	+	0.247340i	−0.698667	−	0.715447i	$-0.746223\pi$
$647$	−7.50709e10	+	4.33422e10i	−0.428405	+	0.247340i	0.270262	+	0.962787i	$0.412890\pi$
$648$		0			0
$649$	−7.48612e10	+	1.29663e11i	−0.421967	+	0.730868i
$650$		0			0
$651$	7.50836e10	−	1.17676e11i	0.418043	−	0.655186i
$652$		0			0
$653$	8.86421e10	+	5.11776e10i	0.487514	+	0.281467i	0.723543	−	0.690280i	$-0.242512\pi$
$653$	8.86421e10	+	5.11776e10i	0.487514	+	0.281467i	−0.236028	+	0.971746i	$0.575846\pi$
$654$		0			0
$655$	4.75118e10	+	8.22929e10i	0.258129	+	0.447092i
$656$		0			0
$657$	3.06863e11	+	2.09530e10i	1.64696	+	0.112456i
$658$		0			0
$659$		−	3.22261e11i		−	1.70870i	−0.519696	−	0.854351i	$-0.673955\pi$
$659$		−	3.22261e11i		−	1.70870i	0.519696	−	0.854351i	$-0.326045\pi$
$660$		0			0
$661$	−1.11367e11	−	1.92894e11i	−0.583381	−	1.01045i	−0.995075	−	0.0991232i	$-0.968396\pi$
$661$	−1.11367e11	−	1.92894e11i	−0.583381	−	1.01045i	0.411694	−	0.911322i	$-0.364937\pi$
$662$		0			0
$663$	1.39019e11	−	2.22882e11i	0.719481	−	1.15351i
$664$		0			0
$665$	1.19626e11	+	2.12190e11i	0.611703	+	1.08502i
$666$		0			0
$667$	3.42193e10	−	5.92695e10i	0.172889	−	0.299452i
$668$		0			0
$669$	−2.14683e10	+	1.14374e10i	−0.107175	+	0.0570984i
$670$		0			0
$671$		−	1.52773e11i		−	0.753626i
$672$		0			0
$673$	−3.39213e10			−0.165353			−0.0826765	−	0.996576i	$-0.526347\pi$
$673$	−3.39213e10			−0.165353			−0.0826765	+	0.996576i	$0.526347\pi$
$674$		0			0
$675$	1.06666e11	−	4.78050e10i	0.513818	−	0.230281i
$676$		0			0
$677$	−1.51267e11	−	8.73339e10i	−0.720094	−	0.415746i	0.0946935	−	0.995506i	$-0.469813\pi$
$677$	−1.51267e11	−	8.73339e10i	−0.720094	−	0.415746i	−0.814787	+	0.579760i	$0.803146\pi$
$678$		0			0
$679$	6.72904e7	−	6.56835e9i	0.000316573	−	0.0309013i
$680$		0			0
$681$	−2.11796e11	+	3.39562e11i	−0.984756	+	1.57881i
$682$		0			0
$683$	2.12879e11	−	1.22906e11i	0.978251	−	0.564794i	0.0765096	−	0.997069i	$-0.475622\pi$
$683$	2.12879e11	−	1.22906e11i	0.978251	−	0.564794i	0.901742	+	0.432275i	$0.142289\pi$
$684$		0			0
$685$	−1.46154e11			−0.663818
$686$		0			0
$687$	1.43414e9	−	4.20559e10i	0.00643821	−	0.188799i
$688$		0			0
$689$	−2.31505e11	+	1.33659e11i	−1.02727	+	0.593093i
$690$		0			0
$691$	6.70229e10	−	1.16087e11i	0.293975	−	0.509180i	−0.680771	−	0.732497i	$-0.738355\pi$
$691$	6.70229e10	−	1.16087e11i	0.293975	−	0.509180i	0.974746	+	0.223316i	$0.0716883\pi$
$692$		0			0
$693$	1.42258e11	+	1.11788e10i	0.616800	+	0.0484686i
$694$		0			0
$695$	−2.61925e10	−	1.51223e10i	−0.112263	−	0.0648153i
$696$		0			0
$697$	−2.69773e11	−	4.67261e11i	−1.14306	−	1.97983i
$698$		0			0
$699$	1.57432e11	+	5.36857e9i	0.659453	+	0.0224879i
$700$		0			0
$701$		−	1.20616e10i		−	0.0499499i	−0.999688	−	0.0249749i	$-0.992049\pi$
$701$		−	1.20616e10i		−	0.0499499i	0.999688	−	0.0249749i	$-0.00795059\pi$
$702$		0			0
$703$	2.30387e11	+	3.99042e11i	0.943271	+	1.63379i
$704$		0			0
$705$	−2.10962e11	−	1.31584e11i	−0.853982	−	0.532655i
$706$		0			0
$707$	1.35544e11	+	2.40424e11i	0.542505	+	0.962278i
$708$		0			0
$709$	2.42877e11	−	4.20675e11i	0.961173	−	1.66480i	0.241609	−	0.970374i	$-0.422325\pi$
$709$	2.42877e11	−	4.20675e11i	0.961173	−	1.66480i	0.719564	−	0.694426i	$-0.244342\pi$
$710$		0			0
$711$	−8.62167e9	−	1.76038e10i	−0.0337375	−	0.0688856i
$712$		0			0
$713$			3.02090e11i			1.16890i
$714$		0			0
$715$	1.16181e11			0.444540
$716$		0			0
$717$	1.35753e11	+	2.54811e11i	0.513657	+	0.964144i
$718$		0			0
$719$	−6.29010e10	−	3.63159e10i	−0.235365	−	0.135888i	0.377680	−	0.925936i	$-0.376722\pi$
$719$	−6.29010e10	−	3.63159e10i	−0.235365	−	0.135888i	−0.613045	+	0.790048i	$0.710055\pi$
$720$		0			0
$721$	2.69438e11	+	1.59263e11i	0.997053	+	0.589349i
$722$		0			0
$723$	1.83493e11	+	1.14450e11i	0.671532	+	0.418856i
$724$		0			0
$725$	−3.09733e10	+	1.78824e10i	−0.112108	+	0.0647254i
$726$		0			0
$727$	−3.19868e11			−1.14507			−0.572537	−	0.819879i	$-0.694041\pi$
$727$	−3.19868e11			−1.14507			−0.572537	+	0.819879i	$0.694041\pi$
$728$		0			0
$729$	−2.76543e11	−	5.73623e10i	−0.979157	−	0.203103i
$730$		0			0
$731$	−8.88960e10	+	5.13242e10i	−0.311324	+	0.179743i
$732$		0			0
$733$	2.25805e11	−	3.91105e11i	0.782199	−	1.35481i	−0.148459	−	0.988919i	$-0.547431\pi$
$733$	2.25805e11	−	3.91105e11i	0.782199	−	1.35481i	0.930658	−	0.365890i	$-0.119235\pi$
$734$		0			0
$735$	1.92625e11	+	1.05232e10i	0.660029	+	0.0360577i
$736$		0			0
$737$	−9.01665e9	−	5.20576e9i	−0.0305615	−	0.0176447i
$738$		0			0
$739$	−1.22044e11	−	2.11387e11i	−0.409203	−	0.708761i	0.585597	−	0.810602i	$-0.300860\pi$
$739$	−1.22044e11	−	2.11387e11i	−0.409203	−	0.708761i	−0.994801	+	0.101841i	$0.967527\pi$
$740$		0			0
$741$	−2.10458e10	+	6.17162e11i	−0.0698059	+	2.04704i
$742$		0			0
$743$		−	2.81202e11i		−	0.922706i	−0.887217	−	0.461353i	$-0.847364\pi$
$743$		−	2.81202e11i		−	0.922706i	0.887217	−	0.461353i	$-0.152636\pi$
$744$		0			0
$745$	−2.10064e10	−	3.63842e10i	−0.0681911	−	0.118110i
$746$		0			0
$747$	9.80915e10	+	6.59301e10i	0.315028	+	0.211739i
$748$		0			0
$749$	3.65290e9	−	6.17992e9i	0.0116067	−	0.0196361i
$750$		0			0
$751$	−7.73501e9	+	1.33974e10i	−0.0243165	+	0.0421174i	−0.877928	−	0.478793i	$-0.841074\pi$
$751$	−7.73501e9	+	1.33974e10i	−0.0243165	+	0.0421174i	0.853611	+	0.520911i	$0.174408\pi$
$752$		0			0
$753$	2.19970e11	+	4.12888e11i	0.684202	+	1.28426i
$754$		0			0
$755$			2.07176e11i			0.637604i
$756$		0			0
$757$	−9.08011e10			−0.276508			−0.138254	−	0.990397i	$-0.544149\pi$
$757$	−9.08011e10			−0.276508			−0.138254	+	0.990397i	$0.544149\pi$
$758$		0			0
$759$	−2.72548e11	+	1.45203e11i	−0.821250	+	0.437529i
$760$		0			0
$761$	−2.59805e11	−	1.49999e11i	−0.774657	−	0.447248i	0.0598767	−	0.998206i	$-0.480929\pi$
$761$	−2.59805e11	−	1.49999e11i	−0.774657	−	0.447248i	−0.834533	+	0.550958i	$0.814263\pi$
$762$		0			0
$763$	−1.66440e11	+	9.38338e10i	−0.491087	+	0.276861i
$764$		0			0
$765$	−1.57950e11	+	2.35000e11i	−0.461184	+	0.686155i
$766$		0			0
$767$	−4.44385e11	+	2.56566e11i	−1.28404	+	0.741339i
$768$		0			0
$769$	−3.87255e11			−1.10737			−0.553684	−	0.832727i	$-0.686778\pi$
$769$	−3.87255e11			−1.10737			−0.553684	+	0.832727i	$0.686778\pi$
$770$		0			0
$771$	1.27098e11	+	4.33415e9i	0.359684	+	0.0122655i
$772$		0			0
$773$	−2.38795e11	+	1.37868e11i	−0.668816	+	0.386141i	−0.795628	−	0.605786i	$-0.792859\pi$
$773$	−2.38795e11	+	1.37868e11i	−0.668816	+	0.386141i	0.126812	+	0.991927i	$0.459526\pi$
$774$		0			0
$775$	7.89336e10	−	1.36717e11i	0.218804	−	0.378979i
$776$		0			0
$777$	3.64556e11	+	1.61721e10i	1.00019	+	0.0443693i
$778$		0			0
$779$	1.09844e12	+	6.34187e11i	2.98283	+	1.72214i
$780$		0			0
$781$	−7.45520e9	−	1.29128e10i	−0.0200380	−	0.0347069i
$782$		0			0
$783$	8.59648e10	+	8.82179e9i	0.228704	+	0.0234698i
$784$		0			0
$785$			2.30899e11i			0.608055i
$786$		0			0
$787$	8.60930e10	+	1.49117e11i	0.224424	+	0.388713i	0.956146	−	0.292889i	$-0.0946167\pi$
$787$	8.60930e10	+	1.49117e11i	0.224424	+	0.388713i	−0.731723	+	0.681603i	$0.761283\pi$
$788$		0			0
$789$	−1.36363e11	+	2.18625e11i	−0.351876	+	0.564147i
$790$		0			0
$791$	1.09340e11	+	1.12015e9i	0.279302	+	0.00286135i
$792$		0			0
$793$	2.61793e11	−	4.53438e11i	0.662011	−	1.14664i
$794$		0			0
$795$	2.54306e11	−	1.35484e11i	0.636632	−	0.339172i
$796$		0			0
$797$		−	4.22870e10i		−	0.104803i	−0.998626	−	0.0524015i	$-0.983312\pi$
$797$		−	4.22870e10i		−	0.104803i	0.998626	−	0.0524015i	$-0.0166876\pi$
$798$		0			0
$799$	−7.76144e11			−1.90439
$800$		0			0
$801$	−6.53008e11	+	3.19818e11i	−1.58631	+	0.776914i
$802$		0			0
$803$	3.67762e11	+	2.12327e11i	0.884514	+	0.510674i
$804$		0			0
$805$	−3.63673e11	+	2.05028e11i	−0.866020	+	0.488237i
$806$		0			0
$807$	−1.28546e11	+	2.06091e11i	−0.303084	+	0.485921i
$808$		0			0
$809$	−2.93443e11	+	1.69420e11i	−0.685063	+	0.395521i	−0.801760	−	0.597647i	$-0.796103\pi$
$809$	−2.93443e11	+	1.69420e11i	−0.685063	+	0.395521i	0.116697	+	0.993168i	$0.462769\pi$
$810$		0			0
$811$	3.33897e11			0.771843			0.385922	−	0.922532i	$-0.373884\pi$
$811$	3.33897e11			0.771843			0.385922	+	0.922532i	$0.373884\pi$
$812$		0			0
$813$	1.94981e10	−	5.71777e11i	0.0446303	−	1.30877i
$814$		0			0
$815$	−2.46214e11	+	1.42152e11i	−0.558061	+	0.322197i
$816$		0			0
$817$	1.20654e11	−	2.08978e11i	0.270802	−	0.469043i
$818$		0			0
$819$	4.03074e11	+	2.76954e11i	0.895879	+	0.615562i
$820$		0			0
$821$	−8.47558e10	−	4.89338e10i	−0.186551	−	0.107705i	0.403816	−	0.914840i	$-0.367684\pi$
$821$	−8.47558e10	−	4.89338e10i	−0.186551	−	0.107705i	−0.590367	+	0.807135i	$0.701017\pi$
$822$		0			0
$823$	−1.15554e11	−	2.00145e11i	−0.251875	−	0.436261i	0.712167	−	0.702010i	$-0.247714\pi$
$823$	−1.15554e11	−	2.00145e11i	−0.251875	−	0.436261i	−0.964042	+	0.265749i	$0.914381\pi$
$824$		0			0
$825$	1.61287e11	+	5.50004e9i	0.348165	+	0.0118727i
$826$		0			0
$827$		−	6.03031e11i		−	1.28919i	−0.764524	−	0.644596i	$-0.777026\pi$
$827$		−	6.03031e11i		−	1.28919i	0.764524	−	0.644596i	$-0.222974\pi$
$828$		0			0
$829$	3.63659e11	+	6.29876e11i	0.769975	+	1.33364i	0.937576	+	0.347781i	$0.113065\pi$
$829$	3.63659e11	+	6.29876e11i	0.769975	+	1.33364i	−0.167601	+	0.985855i	$0.553602\pi$
$830$		0			0
$831$	−1.91472e11	−	1.19427e11i	−0.401514	−	0.250437i
$832$		0			0
$833$	5.27578e11	−	2.90351e11i	1.09574	−	0.603037i
$834$		0			0
$835$	−1.09176e11	+	1.89098e11i	−0.224585	+	0.388992i
$836$		0			0
$837$	−3.48084e11	+	1.56003e11i	−0.709222	+	0.317856i
$838$		0			0
$839$			7.47869e10i			0.150931i	0.997148	+	0.0754654i	$0.0240442\pi$
$839$			7.47869e10i			0.150931i	−0.997148	+	0.0754654i	$0.975956\pi$
$840$		0			0
$841$	4.73805e11			0.947144
$842$		0			0
$843$	−3.63751e11	−	6.82766e11i	−0.720267	−	1.35195i
$844$		0			0
$845$	5.29764e10	+	3.05859e10i	0.103910	+	0.0599922i
$846$		0			0
$847$	−2.73462e11	−	1.61641e11i	−0.531329	−	0.314064i
$848$		0			0
$849$	4.48043e11	+	2.79458e11i	0.862360	+	0.537881i
$850$		0			0
$851$	−6.83920e11	+	3.94861e11i	−1.30403	+	0.752881i
$852$		0			0
$853$	−2.19915e11			−0.415392			−0.207696	−	0.978193i	$-0.566597\pi$
$853$	−2.19915e11			−0.415392			−0.207696	+	0.978193i	$0.566597\pi$
$854$		0			0
$855$	4.53445e10	−	6.64084e11i	0.0848516	−	1.24268i
$856$		0			0
$857$	2.33611e11	−	1.34876e11i	0.433082	−	0.250040i	−0.267576	−	0.963537i	$-0.586223\pi$
$857$	2.33611e11	−	1.34876e11i	0.433082	−	0.250040i	0.700659	+	0.713496i	$0.252889\pi$
$858$		0			0
$859$	9.09330e10	−	1.57501e11i	0.167012	−	0.289274i	−0.770356	−	0.637614i	$-0.779921\pi$
$859$	9.09330e10	−	1.57501e11i	0.167012	−	0.289274i	0.937368	+	0.348340i	$0.113255\pi$
$860$		0			0
$861$	8.91329e11	−	4.63205e11i	1.62190	−	0.842870i
$862$		0			0
$863$	−7.08745e11	−	4.09194e11i	−1.27775	−	0.737711i	−0.301318	−	0.953524i	$-0.597427\pi$
$863$	−7.08745e11	−	4.09194e11i	−1.27775	−	0.737711i	−0.976435	+	0.215813i	$0.930760\pi$
$864$		0			0
$865$	1.22524e10	+	2.12217e10i	0.0218855	+	0.0379067i
$866$		0			0
$867$	−1.08666e10	+	3.18662e11i	−0.0192318	+	0.563967i
$868$		0			0
$869$		−	2.70630e10i		−	0.0474566i
$870$		0			0
$871$	−1.78413e10	−	3.09020e10i	−0.0309994	−	0.0536926i
$872$		0			0
$873$	−1.00129e10	+	1.48974e10i	−0.0172387	+	0.0256480i
$874$		0			0
$875$	6.05614e11	+	6.20430e9i	1.03315	+	0.0105843i
$876$		0			0
$877$	−3.17221e11	+	5.49443e11i	−0.536245	+	0.928804i	0.462857	+	0.886433i	$0.346824\pi$
$877$	−3.17221e11	+	5.49443e11i	−0.536245	+	0.928804i	−0.999102	+	0.0423708i	$0.986509\pi$
$878$		0			0
$879$	−4.26594e10	−	8.00724e10i	−0.0714594	−	0.134130i
$880$		0			0
$881$		−	2.52476e11i		−	0.419099i	−0.977798	−	0.209549i	$-0.932800\pi$
$881$		−	2.52476e11i		−	0.419099i	0.977798	−	0.209549i	$-0.0671997\pi$
$882$		0			0
$883$	2.88115e11			0.473939			0.236970	−	0.971517i	$-0.423846\pi$
$883$	2.88115e11			0.473939			0.236970	+	0.971517i	$0.423846\pi$
$884$		0			0
$885$	4.88153e11	−	2.60069e11i	0.795761	−	0.423950i
$886$		0			0
$887$	−7.46455e11	−	4.30966e11i	−1.20589	−	0.696224i	−0.244035	−	0.969766i	$-0.578471\pi$
$887$	−7.46455e11	−	4.30966e11i	−1.20589	−	0.696224i	−0.961860	+	0.273543i	$0.911804\pi$
$888$		0			0
$889$	5.19547e9	−	5.07140e11i	0.00831798	−	0.811934i
$890$		0			0
$891$	−3.08057e11	−	2.39060e11i	−0.488788	−	0.379311i
$892$		0			0
$893$	1.58013e12	−	9.12286e11i	2.48477	−	1.43458i
$894$		0			0
$895$	2.83552e11			0.441917
$896$		0			0
$897$	−1.05776e12	−	3.60705e10i	−1.63386	−	0.0557162i
$898$		0			0
$899$	1.01076e11	−	5.83561e10i	0.154742	−	0.0893403i
$900$		0			0
$901$	4.49740e11	−	7.78972e11i	0.682436	−	1.18201i
$902$		0			0
$903$	−8.81244e10	−	1.69575e11i	−0.132539	−	0.255041i
$904$		0			0
$905$	3.63767e11	+	2.10021e11i	0.542287	+	0.313090i
$906$		0			0
$907$	−1.37499e11	−	2.38156e11i	−0.203176	−	0.351911i	0.746374	−	0.665526i	$-0.231793\pi$
$907$	−1.37499e11	−	2.38156e11i	−0.203176	−	0.351911i	−0.949550	+	0.313616i	$0.898460\pi$
$908$		0			0
$909$	5.13781e10	−	7.52449e11i	0.0752528	−	1.10210i
$910$		0			0
$911$		−	1.07639e12i		−	1.56278i	−0.624046	−	0.781388i	$-0.714512\pi$
$911$		−	1.07639e12i		−	1.56278i	0.624046	−	0.781388i	$-0.285488\pi$
$912$		0			0
$913$	8.15887e10	+	1.41316e11i	0.117421	+	0.203380i
$914$		0			0
$915$	−2.98682e11	+	4.78864e11i	−0.426113	+	0.683168i
$916$		0			0
$917$	−2.81009e11	+	4.75407e11i	−0.397413	+	0.672338i
$918$		0			0
$919$	4.03581e11	−	6.99023e11i	0.565808	−	0.980008i	−0.431166	−	0.902273i	$-0.641898\pi$
$919$	4.03581e11	−	6.99023e11i	0.565808	−	0.980008i	0.996974	−	0.0777355i	$-0.0247690\pi$
$920$		0			0
$921$	3.06833e11	−	1.63468e11i	0.426445	−	0.227193i
$922$		0			0
$923$		−	5.11012e10i		−	0.0704083i
$924$		0			0
$925$	4.12696e11			0.563720
$926$		0			0
$927$	−3.76186e11	−	7.68101e11i	−0.509430	−	1.04016i
$928$		0			0
$929$	−1.00425e12	−	5.79805e11i	−1.34828	−	0.778430i	−0.360274	−	0.932847i	$-0.617317\pi$
$929$	−1.00425e12	−	5.79805e11i	−1.34828	−	0.778430i	−0.988006	+	0.154417i	$0.950650\pi$
$930$		0			0
$931$	−7.32796e11	+	1.21123e12i	−0.975404	+	1.61224i
$932$		0			0
$933$	−5.05890e11	+	8.11070e11i	−0.667620	+	1.07036i
$934$		0			0
$935$	−3.38553e11	+	1.95464e11i	−0.442977	+	0.255753i
$936$		0			0
$937$	1.90181e11			0.246723			0.123362	−	0.992362i	$-0.460633\pi$
$937$	1.90181e11			0.246723			0.123362	+	0.992362i	$0.460633\pi$
$938$		0			0
$939$	−3.98877e10	+	1.16970e12i	−0.0513070	+	1.50456i
$940$		0			0
$941$	−1.27012e12	+	7.33304e11i	−1.61989	+	0.935245i	−0.632946	+	0.774196i	$0.718154\pi$
$941$	−1.27012e12	+	7.33304e11i	−1.61989	+	0.935245i	−0.986946	+	0.161049i	$0.948512\pi$
$942$		0			0
$943$	−1.08694e12	+	1.88263e12i	−1.37454	+	2.38077i
$944$		0			0
$945$	−4.24050e11	−	3.13165e11i	−0.531729	−	0.392686i
$946$		0			0
$947$	3.54955e11	+	2.04933e11i	0.441340	+	0.254808i	0.704166	−	0.710036i	$-0.251321\pi$
$947$	3.54955e11	+	2.04933e11i	0.441340	+	0.254808i	−0.262826	+	0.964843i	$0.584655\pi$
$948$		0			0
$949$	7.27692e11	+	1.26040e12i	0.897187	+	1.55397i
$950$		0			0
$951$	9.26502e11	+	3.15946e10i	1.13272	+	0.0386269i
$952$		0			0
$953$		−	9.32245e11i		−	1.13021i	−0.825020	−	0.565104i	$-0.808836\pi$
$953$		−	9.32245e11i		−	1.13021i	0.825020	−	0.565104i	$-0.191164\pi$
$954$		0			0
$955$	1.90742e11	+	3.30374e11i	0.229315	+	0.397185i
$956$		0			0
$957$	1.01232e11	+	6.31418e10i	0.120690	+	0.0752782i
$958$		0			0
$959$	−4.17144e11	−	7.39917e11i	−0.493186	−	0.874799i
$960$		0			0
$961$	1.68860e11	−	2.92474e11i	0.197986	−	0.342921i
$962$		0			0
$963$	−1.76174e10	+	8.62833e9i	−0.0204850	+	0.0100328i
$964$		0			0
$965$		−	4.61207e11i		−	0.531846i
$966$		0			0
$967$	8.17094e11			0.934472			0.467236	−	0.884133i	$-0.345250\pi$
$967$	8.17094e11			0.934472			0.467236	+	0.884133i	$0.345250\pi$
$968$		0			0
$969$	−9.76991e11	−	1.83383e12i	−1.10814	−	2.08000i
$970$		0			0
$971$	1.01292e12	+	5.84811e11i	1.13946	+	0.657867i	0.946297	−	0.323299i	$-0.104792\pi$
$971$	1.01292e12	+	5.84811e11i	1.13946	+	0.657867i	0.193163	+	0.981167i	$0.438125\pi$
$972$		0			0
$973$	1.80062e9	−	1.75762e11i	0.00200896	−	0.196099i
$974$		0			0
$975$	4.69285e11	+	2.92708e11i	0.519300	+	0.323904i
$976$		0			0
$977$	1.19410e12	−	6.89416e11i	1.31058	−	0.756665i	0.328389	−	0.944543i	$-0.393494\pi$
$977$	1.19410e12	−	6.89416e11i	1.31058	−	0.756665i	0.982192	+	0.187878i	$0.0601610\pi$
$978$		0			0
$979$	−1.00389e12			−1.09284
$980$		0			0
$981$	5.20901e11	+	3.55678e10i	0.562444	+	0.0384044i
$982$		0			0
$983$	−2.86042e11	+	1.65146e11i	−0.306348	+	0.176870i	−0.645291	−	0.763937i	$-0.723264\pi$
$983$	−2.86042e11	+	1.65146e11i	−0.306348	+	0.176870i	0.338943	+	0.940807i	$0.389931\pi$
$984$		0			0
$985$	−4.99482e11	+	8.65128e11i	−0.530609	+	0.919042i
$986$		0			0
$987$	6.40383e10	−	1.44357e12i	0.0674794	−	1.52114i
$988$		0			0
$989$	3.58169e11	+	2.06789e11i	0.374371	+	0.216143i
$990$		0			0
$991$	4.93244e11	+	8.54324e11i	0.511408	+	0.885785i	0.999913	+	0.0132234i	$0.00420927\pi$
$991$	4.93244e11	+	8.54324e11i	0.511408	+	0.885785i	−0.488504	+	0.872561i	$0.662457\pi$
$992$		0			0
$993$	1.62110e10	−	4.75383e11i	0.0166730	−	0.488930i
$994$		0			0
$995$			7.39024e11i			0.753991i
$996$		0			0
$997$	5.02553e11	+	8.70448e11i	0.508630	+	0.880972i	0.999950	+	0.00999334i	$0.00318103\pi$
$997$	5.02553e11	+	8.70448e11i	0.508630	+	0.880972i	−0.491321	+	0.870979i	$0.663486\pi$
$998$		0			0
$999$	−8.08165e11	−	5.84138e11i	−0.811406	−	0.586481i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	84.9.p.b.65.5	yes	40
3.2	odd	2	inner	84.9.p.b.65.18	yes	40
7.4	even	3	inner	84.9.p.b.53.18	yes	40
21.11	odd	6	inner	84.9.p.b.53.5	✓	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.9.p.b.53.5	✓	40	21.11	odd	6	inner
84.9.p.b.53.18	yes	40	7.4	even	3	inner
84.9.p.b.65.5	yes	40	1.1	even	1	trivial
84.9.p.b.65.18	yes	40	3.2	odd	2	inner

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:	\( N \)	\(=\)	\( 84 = 2^{2} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	84.p (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-71.4876 + 38.0858i) q^{3} +(357.784 + 206.567i) q^{5} +(-24.5961 + 2400.87i) q^{7} +(3659.95 - 5445.32i) q^{9} +O(q^{10})\)	\(q+(-71.4876 + 38.0858i) q^{3} +(357.784 + 206.567i) q^{5} +(-24.5961 + 2400.87i) q^{7} +(3659.95 - 5445.32i) q^{9} +(7844.81 - 4529.20i) q^{11} +31045.1 q^{13} +(-33444.3 - 1140.48i) q^{15} +(-90466.0 + 52230.6i) q^{17} +(122784. - 212669. i) q^{19} +(-89680.8 - 172569. i) q^{21} +(364494. + 210441. i) q^{23} +(-109973. - 190479. i) q^{25} +(-54252.1 + 528665. i) q^{27} -162607. i q^{29} +(358877. + 621593. i) q^{31} +(-388308. + 622557. i) q^{33} +(-504740. + 853913. i) q^{35} +(-938176. + 1.62497e6i) q^{37} +(-2.21934e6 + 1.18238e6i) q^{39} +5.16504e6i q^{41} +982646. q^{43} +(2.43429e6 - 1.19222e6i) q^{45} +(6.43455e6 + 3.71499e6i) q^{47} +(-5.76359e6 - 118104. i) q^{49} +(4.47796e6 - 7.17931e6i) q^{51} +(-7.45705e6 + 4.30533e6i) q^{53} +3.74233e6 q^{55} +(-677909. + 1.98795e7i) q^{57} +(-1.43142e7 + 8.26429e6i) q^{59} +(8.43266e6 - 1.46058e7i) q^{61} +(1.29835e7 + 8.92102e6i) q^{63} +(1.11074e7 + 6.41288e6i) q^{65} +(-574689. - 995390. i) q^{67} +(-3.40716e7 - 1.16187e6i) q^{69} -1.64603e6i q^{71} +(2.34398e7 + 4.05990e7i) q^{73} +(1.51162e7 + 9.42847e6i) q^{75} +(1.06811e7 + 1.89458e7i) q^{77} +(1.49381e6 - 2.58735e6i) q^{79} +(-1.62562e7 - 3.98592e7i) q^{81} +1.80139e7i q^{83} -4.31564e7 q^{85} +(6.19303e6 + 1.16244e7i) q^{87} +(-9.59768e7 - 5.54122e7i) q^{89} +(-763589. + 7.45354e7i) q^{91} +(-4.93291e7 - 3.07681e7i) q^{93} +(8.78605e7 - 5.07263e7i) q^{95} -2.73581e6 q^{97} +(4.04867e6 - 5.92941e7i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 81 q^{3} - 34 q^{7} + 4771 q^{9}+O(q^{10}) \) 40 * q + 81 * q^3 - 34 * q^7 + 4771 * q^9	\( 40 q + 81 q^{3} - 34 q^{7} + 4771 q^{9} - 55464 q^{13} + 68482 q^{15} + 311690 q^{19} - 172343 q^{21} + 1766792 q^{25} - 3451932 q^{27} + 31596 q^{31} + 1874885 q^{33} - 1853482 q^{37} + 11217526 q^{39} - 13372600 q^{43} - 527785 q^{45} - 12653462 q^{49} - 1103461 q^{51} + 71577224 q^{55} - 17195214 q^{57} - 21761970 q^{61} + 21945045 q^{63} - 26337350 q^{67} - 5588722 q^{69} + 41115682 q^{73} - 17971730 q^{75} - 120916932 q^{79} - 24550133 q^{81} + 139250060 q^{85} - 16321046 q^{87} + 345074940 q^{91} + 25774675 q^{93} - 707216948 q^{97} - 94510994 q^{99}+O(q^{100}) \) 40 * q + 81 * q^3 - 34 * q^7 + 4771 * q^9 - 55464 * q^13 + 68482 * q^15 + 311690 * q^19 - 172343 * q^21 + 1766792 * q^25 - 3451932 * q^27 + 31596 * q^31 + 1874885 * q^33 - 1853482 * q^37 + 11217526 * q^39 - 13372600 * q^43 - 527785 * q^45 - 12653462 * q^49 - 1103461 * q^51 + 71577224 * q^55 - 17195214 * q^57 - 21761970 * q^61 + 21945045 * q^63 - 26337350 * q^67 - 5588722 * q^69 + 41115682 * q^73 - 17971730 * q^75 - 120916932 * q^79 - 24550133 * q^81 + 139250060 * q^85 - 16321046 * q^87 + 345074940 * q^91 + 25774675 * q^93 - 707216948 * q^97 - 94510994 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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