LMFDB - Embedded newform 84.9.p.b.53.19

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			53.19
Character	$\chi$	$=$	84.53
Dual form			84.9.p.b.65.19

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(77.2965 - 24.2126i) q^{3} +(-128.664 + 74.2840i) q^{5} +(2032.57 + 1278.06i) q^{7} +(5388.50 - 3743.10i) q^{9} +O(q^{10})$	\(q+(77.2965 - 24.2126i) q^{3} +(-128.664 + 74.2840i) q^{5} +(2032.57 + 1278.06i) q^{7} +(5388.50 - 3743.10i) q^{9} +(1287.13 + 743.127i) q^{11} -19136.1 q^{13} +(-8146.65 + 8857.18i) q^{15} +(107218. + 61902.1i) q^{17} +(24996.8 + 43295.7i) q^{19} +(188056. + 49576.0i) q^{21} +(163899. - 94626.9i) q^{23} +(-184276. + 319176. i) q^{25} +(325882. - 419798. i) q^{27} -134496. i q^{29} +(3894.88 - 6746.14i) q^{31} +(117484. + 26276.3i) q^{33} +(-356458. - 13452.7i) q^{35} +(557357. + 965370. i) q^{37} +(-1.47915e6 + 463335. i) q^{39} -1.97847e6i q^{41} +3.88471e6 q^{43} +(-415252. + 881880. i) q^{45} +(6.93068e6 - 4.00143e6i) q^{47} +(2.49791e6 + 5.19552e6i) q^{49} +(9.78635e6 + 2.18880e6i) q^{51} +(151016. + 87189.3i) q^{53} -220810. q^{55} +(2.98046e6 + 2.74137e6i) q^{57} +(1.40220e7 + 8.09562e6i) q^{59} +(5.04668e6 + 8.74110e6i) q^{61} +(1.57364e7 - 721278. i) q^{63} +(2.46212e6 - 1.42151e6i) q^{65} +(-8.47085e6 + 1.46719e7i) q^{67} +(1.03776e7 - 1.12827e7i) q^{69} -4.07358e7i q^{71} +(-160357. + 277747. i) q^{73} +(-6.51583e6 + 2.91330e7i) q^{75} +(1.66643e6 + 3.15550e6i) q^{77} +(3.86159e6 + 6.68846e6i) q^{79} +(1.50251e7 - 4.03394e7i) q^{81} +966315. i q^{83} -1.83933e7 q^{85} +(-3.25650e6 - 1.03961e7i) q^{87} +(-9.66146e7 + 5.57805e7i) q^{89} +(-3.88955e7 - 2.44572e7i) q^{91} +(137719. - 615758. i) q^{93} +(-6.43235e6 - 3.71372e6i) q^{95} -4.98046e7 q^{97} +(9.71732e6 - 813530. i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/84\mathbb{Z}\right)^\times$.

$n$	$29$	$43$	$73$
$\chi(n)$	$-1$	$1$	$e\left(\frac{2}{3}\right)$

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
$3$	77.2965	−	24.2126i	0.954278	−	0.298921i
$3$	77.2965	−	24.2126i	0.954278	−	0.298921i
$4$		0			0
$5$	−128.664	+	74.2840i	−0.205862	+	0.118854i	−0.599387	−	0.800460i	$-0.704589\pi$
$5$	−128.664	+	74.2840i	−0.205862	+	0.118854i	0.393525	+	0.919314i	$0.371256\pi$
$6$		0			0
$7$	2032.57	+	1278.06i	0.846553	+	0.532305i
$7$	2032.57	+	1278.06i	0.846553	+	0.532305i
$8$		0			0
$9$	5388.50	−	3743.10i	0.821292	−	0.570507i
$10$		0			0
$11$	1287.13	+	743.127i	0.0879130	+	0.0507566i	0.543312	−	0.839531i	$-0.317170\pi$
$11$	1287.13	+	743.127i	0.0879130	+	0.0507566i	−0.455399	+	0.890287i	$0.650503\pi$
$12$		0			0
$13$	−19136.1			−0.670008			−0.335004	−	0.942217i	$-0.608738\pi$
$13$	−19136.1			−0.670008			−0.335004	+	0.942217i	$0.608738\pi$
$14$		0			0
$15$	−8146.65	+	8857.18i	−0.160921	+	0.174957i
$16$		0			0
$17$	107218.	+	61902.1i	1.28372	+	0.741156i	0.977526	−	0.210814i	$-0.0676114\pi$
$17$	107218.	+	61902.1i	1.28372	+	0.741156i	0.306193	+	0.951969i	$0.400945\pi$
$18$		0			0
$19$	24996.8	+	43295.7i	0.191809	+	0.332223i	0.945850	−	0.324604i	$-0.105231\pi$
$19$	24996.8	+	43295.7i	0.191809	+	0.332223i	−0.754041	+	0.656828i	$0.771898\pi$
$20$		0			0
$21$	188056.	+	49576.0i	0.966964	+	0.254914i
$22$		0			0
$23$	163899.	−	94626.9i	0.585685	−	0.338145i	−0.177705	−	0.984084i	$-0.556867\pi$
$23$	163899.	−	94626.9i	0.585685	−	0.338145i	0.763389	+	0.645939i	$0.223534\pi$
$24$		0			0
$25$	−184276.	+	319176.i	−0.471747	+	0.817090i
$26$		0			0
$27$	325882.	−	419798.i	0.613204	−	0.789924i
$28$		0			0
$29$		−	134496.i		−	0.190159i	−0.995470	−	0.0950797i	$-0.969689\pi$
$29$		−	134496.i		−	0.190159i	0.995470	−	0.0950797i	$-0.0303106\pi$
$30$		0			0
$31$	3894.88	−	6746.14i	0.00421743	−	0.00730480i	−0.863909	−	0.503648i	$-0.831991\pi$
$31$	3894.88	−	6746.14i	0.00421743	−	0.00730480i	0.868126	+	0.496343i	$0.165324\pi$
$32$		0			0
$33$	117484.	+	26276.3i	0.0990657	+	0.0221569i
$34$		0			0
$35$	−356458.	−	13452.7i	−0.237540	−	0.00896472i
$36$		0			0
$37$	557357.	+	965370.i	0.297390	+	0.515095i	0.975538	−	0.219831i	$-0.0705505\pi$
$37$	557357.	+	965370.i	0.297390	+	0.515095i	−0.678148	+	0.734925i	$0.737217\pi$
$38$		0			0
$39$	−1.47915e6	+	463335.i	−0.639374	+	0.200280i
$40$		0			0
$41$		−	1.97847e6i		−	0.700156i	−0.936720	−	0.350078i	$-0.886155\pi$
$41$		−	1.97847e6i		−	0.700156i	0.936720	−	0.350078i	$-0.113845\pi$
$42$		0			0
$43$	3.88471e6			1.13628			0.568139	−	0.822933i	$-0.307664\pi$
$43$	3.88471e6			1.13628			0.568139	+	0.822933i	$0.307664\pi$
$44$		0			0
$45$	−415252.	+	881880.i	−0.101265	+	0.215060i
$46$		0			0
$47$	6.93068e6	−	4.00143e6i	1.42031	−	0.820018i	0.423988	−	0.905668i	$-0.360630\pi$
$47$	6.93068e6	−	4.00143e6i	1.42031	−	0.820018i	0.996325	+	0.0856494i	$0.0272965\pi$
$48$		0			0
$49$	2.49791e6	+	5.19552e6i	0.433303	+	0.901248i
$50$		0			0
$51$	9.78635e6	+	2.18880e6i	1.44657	+	0.323538i
$52$		0			0
$53$	151016.	+	87189.3i	0.0191391	+	0.0110499i	0.509539	−	0.860448i	$-0.329816\pi$
$53$	151016.	+	87189.3i	0.0191391	+	0.0110499i	−0.490400	+	0.871497i	$0.663149\pi$
$54$		0			0
$55$	−220810.			−0.0241306
$56$		0			0
$57$	2.98046e6	+	2.74137e6i	0.282348	+	0.259697i
$58$		0			0
$59$	1.40220e7	+	8.09562e6i	1.15718	+	0.668101i	0.950628	−	0.310334i	$-0.100441\pi$
$59$	1.40220e7	+	8.09562e6i	1.15718	+	0.668101i	0.206557	+	0.978435i	$0.433774\pi$
$60$		0			0
$61$	5.04668e6	+	8.74110e6i	0.364491	+	0.631316i	0.988694	−	0.149945i	$-0.0479098\pi$
$61$	5.04668e6	+	8.74110e6i	0.364491	+	0.631316i	−0.624204	+	0.781262i	$0.714576\pi$
$62$		0			0
$63$	1.57364e7	−	721278.i	0.998951	−	0.0457868i
$64$		0			0
$65$	2.46212e6	−	1.42151e6i	0.137929	−	0.0796335i
$66$		0			0
$67$	−8.47085e6	+	1.46719e7i	−0.420366	+	0.728095i	−0.995975	−	0.0896297i	$-0.971432\pi$
$67$	−8.47085e6	+	1.46719e7i	−0.420366	+	0.728095i	0.575609	+	0.817725i	$0.304765\pi$
$68$		0			0
$69$	1.03776e7	−	1.12827e7i	0.457827	−	0.497758i
$70$		0			0
$71$		−	4.07358e7i		−	1.60303i	−0.597973	−	0.801516i	$-0.704027\pi$
$71$		−	4.07358e7i		−	1.60303i	0.597973	−	0.801516i	$-0.295973\pi$
$72$		0			0
$73$	−160357.	+	277747.i	−0.00564673	+	0.00978042i	−0.868835	−	0.495102i	$-0.835131\pi$
$73$	−160357.	+	277747.i	−0.00564673	+	0.00978042i	0.863188	+	0.504882i	$0.168464\pi$
$74$		0			0
$75$	−6.51583e6	+	2.91330e7i	−0.205932	+	0.920746i
$76$		0			0
$77$	1.66643e6	+	3.15550e6i	0.0474050	+	0.0897647i
$78$		0			0
$79$	3.86159e6	+	6.68846e6i	0.0991419	+	0.171719i	0.911330	−	0.411677i	$-0.135057\pi$
$79$	3.86159e6	+	6.68846e6i	0.0991419	+	0.171719i	−0.812188	+	0.583396i	$0.801724\pi$
$80$		0			0
$81$	1.50251e7	−	4.03394e7i	0.349042	−	0.937107i
$82$		0			0
$83$			966315.i			0.0203613i	0.999948	+	0.0101807i	$0.00324066\pi$
$83$			966315.i			0.0203613i	−0.999948	+	0.0101807i	$0.996759\pi$
$84$		0			0
$85$	−1.83933e7			−0.352359
$86$		0			0
$87$	−3.25650e6	−	1.03961e7i	−0.0568427	−	0.181465i
$88$		0			0
$89$	−9.66146e7	+	5.57805e7i	−1.53987	+	0.889042i	−0.541020	+	0.841010i	$0.681962\pi$
$89$	−9.66146e7	+	5.57805e7i	−1.53987	+	0.889042i	−0.998846	+	0.0480325i	$0.984705\pi$
$90$		0			0
$91$	−3.88955e7	−	2.44572e7i	−0.567197	−	0.356649i
$92$		0			0
$93$	137719.	−	615758.i	0.00184104	−	0.00823149i
$94$		0			0
$95$	−6.43235e6	−	3.71372e6i	−0.0789724	−	0.0455947i
$96$		0			0
$97$	−4.98046e7			−0.562578			−0.281289	−	0.959623i	$-0.590762\pi$
$97$	−4.98046e7			−0.562578			−0.281289	+	0.959623i	$0.590762\pi$
$98$		0			0
$99$	9.71732e6	−	813530.i	0.101159	−	0.00846902i
$100$		0			0
$101$	5.23023e7	+	3.01967e7i	0.502615	+	0.290185i	0.729793	−	0.683669i	$-0.239617\pi$
$101$	5.23023e7	+	3.01967e7i	0.502615	+	0.290185i	−0.227178	+	0.973853i	$0.572950\pi$
$102$		0			0
$103$	−9.42065e7	−	1.63170e8i	−0.837013	−	1.44975i	−0.892381	−	0.451283i	$-0.850967\pi$
$103$	−9.42065e7	−	1.63170e8i	−0.837013	−	1.44975i	0.0553684	−	0.998466i	$-0.482367\pi$
$104$		0			0
$105$	−2.78787e7	+	7.59093e6i	−0.229359	+	0.0624508i
$106$		0			0
$107$	−2.69629e7	+	1.55671e7i	−0.205699	+	0.118760i	−0.599311	−	0.800516i	$-0.704559\pi$
$107$	−2.69629e7	+	1.55671e7i	−0.205699	+	0.118760i	0.393612	+	0.919277i	$0.371225\pi$
$108$		0			0
$109$	−1.10190e7	+	1.90855e7i	−0.0780614	+	0.135206i	−0.902413	−	0.430871i	$-0.858206\pi$
$109$	−1.10190e7	+	1.90855e7i	−0.0780614	+	0.135206i	0.824352	+	0.566077i	$0.191540\pi$
$110$		0			0
$111$	6.64558e7	+	6.11247e7i	0.437765	+	0.402647i
$112$		0			0
$113$		−	3.15738e7i		−	0.193648i	−0.995302	−	0.0968241i	$-0.969132\pi$
$113$		−	3.15738e7i		−	0.193648i	0.995302	−	0.0968241i	$-0.0308684\pi$
$114$		0			0
$115$	−1.40585e7	+	2.43501e7i	−0.0803801	+	0.139222i
$116$		0			0
$117$	−1.03115e8	+	7.16284e7i	−0.550273	+	0.382245i
$118$		0			0
$119$	1.38813e8	+	2.62851e8i	0.692215	+	1.31076i
$120$		0			0
$121$	−1.06075e8	−	1.83727e8i	−0.494848	−	0.857101i
$122$		0			0
$123$	−4.79040e7	−	1.52929e8i	−0.209291	−	0.668144i
$124$		0			0
$125$		−	1.12790e8i		−	0.461986i
$126$		0			0
$127$	−3.07954e8			−1.18378			−0.591890	−	0.806019i	$-0.701618\pi$
$127$	−3.07954e8			−1.18378			−0.591890	+	0.806019i	$0.701618\pi$
$128$		0			0
$129$	3.00274e8	−	9.40589e7i	1.08432	−	0.339657i
$130$		0			0
$131$	−3.92963e8	+	2.26877e8i	−1.33434	+	0.770381i	−0.985962	−	0.166973i	$-0.946601\pi$
$131$	−3.92963e8	+	2.26877e8i	−1.33434	+	0.770381i	−0.348378	+	0.937354i	$0.613267\pi$
$132$		0			0
$133$	−4.52686e6	+	1.19949e8i	−0.0144674	+	0.383345i
$134$		0			0
$135$	−1.07449e7	+	7.82206e7i	−0.0323495	+	0.235497i
$136$		0			0
$137$	−4.75361e8	−	2.74450e8i	−1.34940	−	0.779078i	−0.361238	−	0.932474i	$-0.617646\pi$
$137$	−4.75361e8	−	2.74450e8i	−1.34940	−	0.779078i	−0.988165	+	0.153396i	$0.950979\pi$
$138$		0			0
$139$	−5.88028e8			−1.57521			−0.787605	−	0.616180i	$-0.788679\pi$
$139$	−5.88028e8			−1.57521			−0.787605	+	0.616180i	$0.788679\pi$
$140$		0			0
$141$	4.38832e8	−	4.77106e8i	1.11025	−	1.20709i
$142$		0			0
$143$	−2.46307e7	−	1.42206e7i	−0.0589025	−	0.0340073i
$144$		0			0
$145$	9.99092e6	+	1.73048e7i	0.0226013	+	0.0391466i
$146$		0			0
$147$	3.18876e8	+	3.41114e8i	0.682894	+	0.730518i
$148$		0			0
$149$	5.74757e8	−	3.31836e8i	1.16611	−	0.673254i	0.213349	−	0.976976i	$-0.431563\pi$
$149$	5.74757e8	−	3.31836e8i	1.16611	−	0.673254i	0.952761	+	0.303722i	$0.0982294\pi$
$150$		0			0
$151$	−1.33095e8	+	2.30528e8i	−0.256009	+	0.443420i	−0.965169	−	0.261627i	$-0.915741\pi$
$151$	−1.33095e8	+	2.30528e8i	−0.256009	+	0.443420i	0.709160	+	0.705047i	$0.249074\pi$
$152$		0			0
$153$	8.09447e8	−	6.77666e7i	1.47714	−	0.123666i
$154$		0			0
$155$			1.15731e6i			0.00200504i
$156$		0			0
$157$	−1.82010e8	+	3.15250e8i	−0.299568	+	0.518868i	−0.976037	−	0.217604i	$-0.930176\pi$
$157$	−1.82010e8	+	3.15250e8i	−0.299568	+	0.518868i	0.676469	+	0.736471i	$0.263509\pi$
$158$		0			0
$159$	1.37841e7	+	3.08293e6i	0.0215670	+	0.00482364i
$160$		0			0
$161$	4.54075e8	+	1.71367e7i	0.675809	+	0.0255050i
$162$		0			0
$163$	2.03282e8	+	3.52095e8i	0.287971	+	0.498780i	0.973325	−	0.229430i	$-0.0736861\pi$
$163$	2.03282e8	+	3.52095e8i	0.287971	+	0.498780i	−0.685355	+	0.728210i	$0.740353\pi$
$164$		0			0
$165$	−1.70678e7	+	5.34638e6i	−0.0230273	+	0.00721314i
$166$		0			0
$167$		−	3.84530e8i		−	0.494385i	−0.968966	−	0.247192i	$-0.920492\pi$
$167$		−	3.84530e8i		−	0.494385i	0.968966	−	0.247192i	$-0.0795079\pi$
$168$		0			0
$169$	−4.49540e8			−0.551089
$170$		0			0
$171$	2.96755e8	+	1.39733e8i	0.347067	+	0.163424i
$172$		0			0
$173$	4.25512e8	−	2.45670e8i	0.475037	−	0.274263i	−0.243309	−	0.969949i	$-0.578233\pi$
$173$	4.25512e8	−	2.45670e8i	0.475037	−	0.274263i	0.718346	+	0.695686i	$0.244899\pi$
$174$		0			0
$175$	−7.82482e8	+	4.13232e8i	−0.834300	+	0.440597i
$176$		0			0
$177$	1.27987e9	+	2.86253e8i	1.30399	+	0.291647i
$178$		0			0
$179$	1.40579e9	+	8.11634e8i	1.36933	+	0.790584i	0.990843	−	0.135021i	$-0.0431103\pi$
$179$	1.40579e9	+	8.11634e8i	1.36933	+	0.790584i	0.378490	+	0.925606i	$0.376444\pi$
$180$		0			0
$181$	9.81536e8			0.914517			0.457259	−	0.889334i	$-0.348831\pi$
$181$	9.81536e8			0.914517			0.457259	+	0.889334i	$0.348831\pi$
$182$		0			0
$183$	6.01736e8	+	5.53464e8i	0.536539	+	0.493497i
$184$		0			0
$185$	−1.43423e8	−	8.28054e7i	−0.122443	−	0.0706922i
$186$		0			0
$187$	9.20022e7	+	1.59353e8i	0.0752371	+	0.130314i
$188$		0			0
$189$	1.19891e9	−	4.36772e8i	0.939590	−	0.342301i
$190$		0			0
$191$	−2.09074e8	+	1.20709e8i	−0.157097	+	0.0906998i	−0.576488	−	0.817106i	$-0.695577\pi$
$191$	−2.09074e8	+	1.20709e8i	−0.157097	+	0.0906998i	0.419391	+	0.907806i	$0.362244\pi$
$192$		0			0
$193$	8.90212e7	−	1.54189e8i	0.0641600	−	0.111128i	−0.832161	−	0.554534i	$-0.812897\pi$
$193$	8.90212e7	−	1.54189e8i	0.0641600	−	0.111128i	0.896321	+	0.443406i	$0.146230\pi$
$194$		0			0
$195$	1.55895e8	−	1.69492e8i	0.107819	−	0.117222i
$196$		0			0
$197$		−	1.01481e9i		−	0.673784i	−0.941543	−	0.336892i	$-0.890624\pi$
$197$		−	1.01481e9i		−	0.673784i	0.941543	−	0.336892i	$-0.109376\pi$
$198$		0			0
$199$	−3.11596e8	+	5.39701e8i	−0.198692	+	0.344144i	−0.948105	−	0.317959i	$-0.897003\pi$
$199$	−3.11596e8	+	5.39701e8i	−0.198692	+	0.344144i	0.749413	+	0.662103i	$0.230336\pi$
$200$		0			0
$201$	−2.99521e8	+	1.33919e9i	−0.183503	+	0.820461i
$202$		0			0
$203$	1.71895e8	−	2.73373e8i	0.101223	−	0.160980i
$204$		0			0
$205$	1.46969e8	+	2.54558e8i	0.0832167	+	0.144136i
$206$		0			0
$207$	5.28970e8	−	1.12339e9i	0.288104	−	0.611853i
$208$		0			0
$209$			7.43031e7i			0.0389423i
$210$		0			0
$211$	−3.56230e9			−1.79722			−0.898609	−	0.438750i	$-0.855421\pi$
$211$	−3.56230e9			−1.79722			−0.898609	+	0.438750i	$0.855421\pi$
$212$		0			0
$213$	−9.86319e8	−	3.14873e9i	−0.479180	−	1.52974i
$214$		0			0
$215$	−4.99821e8	+	2.88572e8i	−0.233916	+	0.135052i
$216$		0			0
$217$	1.65386e7	−	8.73411e6i	0.00745866	−	0.00393894i
$218$		0			0
$219$	−5.67007e6	+	2.53515e7i	−0.00246497	+	0.0110212i
$220$		0			0
$221$	−2.05173e9	−	1.18456e9i	−0.860103	−	0.496580i
$222$		0			0
$223$	−2.11614e9			−0.855708			−0.427854	−	0.903848i	$-0.640730\pi$
$223$	−2.11614e9			−0.855708			−0.427854	+	0.903848i	$0.640730\pi$
$224$		0			0
$225$	2.01734e8	+	2.40964e9i	0.0787136	+	0.940205i
$226$		0			0
$227$	−9.17729e8	−	5.29851e8i	−0.345630	−	0.199549i	0.317129	−	0.948382i	$-0.397281\pi$
$227$	−9.17729e8	−	5.29851e8i	−0.345630	−	0.199549i	−0.662759	+	0.748833i	$0.730615\pi$
$228$		0			0
$229$	−8.91333e8	−	1.54383e9i	−0.324114	−	0.561382i	0.657218	−	0.753700i	$-0.271733\pi$
$229$	−8.91333e8	−	1.54383e9i	−0.324114	−	0.561382i	−0.981333	+	0.192318i	$0.938400\pi$
$230$		0			0
$231$	2.05212e8	+	2.03561e8i	0.0720701	+	0.0714901i
$232$		0			0
$233$	−1.70922e9	+	9.86817e8i	−0.579927	+	0.334821i	−0.761105	−	0.648629i	$-0.775343\pi$
$233$	−1.70922e9	+	9.86817e8i	−0.579927	+	0.334821i	0.181177	+	0.983450i	$0.442009\pi$
$234$		0			0
$235$	−5.94484e8	+	1.02968e9i	−0.194926	+	0.337621i
$236$		0			0
$237$	4.60432e8	+	4.23496e8i	0.145939	+	0.134232i
$238$		0			0
$239$			6.12916e9i			1.87849i	0.343244	+	0.939246i	$0.388474\pi$
$239$			6.12916e9i			1.87849i	−0.343244	+	0.939246i	$0.611526\pi$
$240$		0			0
$241$	2.44906e9	−	4.24190e9i	0.725992	−	1.25746i	−0.232572	−	0.972579i	$-0.574714\pi$
$241$	2.44906e9	−	4.24190e9i	0.725992	−	1.25746i	0.958564	−	0.284876i	$-0.0919526\pi$
$242$		0			0
$243$	1.84669e8	−	3.48189e9i	0.0529626	−	0.998596i
$244$		0			0
$245$	−7.07334e8	−	4.82920e8i	−0.196318	−	0.134033i
$246$		0			0
$247$	−4.78341e8	−	8.28511e8i	−0.128514	−	0.222592i
$248$		0			0
$249$	2.33970e7	+	7.46928e7i	0.00608643	+	0.0194304i
$250$		0			0
$251$			9.39382e8i			0.236672i	0.992974	+	0.118336i	$0.0377560\pi$
$251$			9.39382e8i			0.236672i	−0.992974	+	0.118336i	$0.962244\pi$
$252$		0			0
$253$	2.81279e8			0.0686524
$254$		0			0
$255$	−1.42174e9	+	4.45351e8i	−0.336248	+	0.105327i
$256$		0			0
$257$	3.62926e9	−	2.09536e9i	0.831929	−	0.480314i	−0.0225839	−	0.999745i	$-0.507189\pi$
$257$	3.62926e9	−	2.09536e9i	0.831929	−	0.480314i	0.854513	+	0.519431i	$0.173856\pi$
$258$		0			0
$259$	−1.00936e8	+	2.67452e9i	−0.0224309	+	0.594357i
$260$		0			0
$261$	−5.03432e8	−	7.24732e8i	−0.108487	−	0.156176i
$262$		0			0
$263$	2.72986e9	+	1.57608e9i	0.570580	+	0.329425i	0.757381	−	0.652973i	$-0.226479\pi$
$263$	2.72986e9	+	1.57608e9i	0.570580	+	0.329425i	−0.186801	+	0.982398i	$0.559812\pi$
$264$		0			0
$265$	−2.59071e7			−0.00525334
$266$		0			0
$267$	−6.11738e9	+	6.65093e9i	−1.20371	+	1.30869i
$268$		0			0
$269$	−7.16947e9	−	4.13929e9i	−1.36923	−	0.790528i	−0.378404	−	0.925640i	$-0.623527\pi$
$269$	−7.16947e9	−	4.13929e9i	−1.36923	−	0.790528i	−0.990830	+	0.135112i	$0.956860\pi$
$270$		0			0
$271$	−2.89546e9	−	5.01508e9i	−0.536834	−	0.929824i	−0.999072	−	0.0430684i	$-0.986287\pi$
$271$	−2.89546e9	−	5.01508e9i	−0.536834	−	0.929824i	0.462238	−	0.886756i	$-0.347047\pi$
$272$		0			0
$273$	−3.59866e9	−	9.48691e8i	−0.647874	−	0.170795i
$274$		0			0
$275$	−4.74377e8	+	2.73881e8i	−0.0829454	+	0.0478886i
$276$		0			0
$277$	2.03945e9	−	3.53243e9i	0.346413	−	0.600004i	−0.639197	−	0.769043i	$-0.720733\pi$
$277$	2.03945e9	−	3.53243e9i	0.346413	−	0.600004i	0.985609	+	0.169039i	$0.0540663\pi$
$278$		0			0
$279$	−4.26388e6	−	5.09305e7i	−0.000703701	−	0.00840545i
$280$		0			0
$281$			1.10129e10i			1.76635i	0.469042	+	0.883176i	$0.344599\pi$
$281$			1.10129e10i			1.76635i	−0.469042	+	0.883176i	$0.655401\pi$
$282$		0			0
$283$	3.16645e9	−	5.48445e9i	0.493659	−	0.855043i	−0.506314	−	0.862349i	$-0.668992\pi$
$283$	3.16645e9	−	5.48445e9i	0.493659	−	0.855043i	0.999973	+	0.00730649i	$0.00232575\pi$
$284$		0			0
$285$	−5.87117e8	−	1.31314e8i	−0.0889909	−	0.0199035i
$286$		0			0
$287$	2.52862e9	−	4.02139e9i	0.372697	−	0.592719i
$288$		0			0
$289$	4.17585e9	+	7.23279e9i	0.598623	+	1.03685i
$290$		0			0
$291$	−3.84972e9	+	1.20590e9i	−0.536856	+	0.168166i
$292$		0			0
$293$		−	7.31235e9i		−	0.992170i	−0.868274	−	0.496085i	$-0.834770\pi$
$293$		−	7.31235e9i		−	0.992170i	0.868274	−	0.496085i	$-0.165230\pi$
$294$		0			0
$295$	−2.40550e9			−0.317627
$296$		0			0
$297$	7.31417e8	−	2.98165e8i	0.0940025	−	0.0383204i
$298$		0			0
$299$	−3.13638e9	+	1.81079e9i	−0.392414	+	0.226560i
$300$		0			0
$301$	7.89595e9	+	4.96490e9i	0.961919	+	0.604846i
$302$		0			0
$303$	4.77392e9	+	1.06773e9i	0.566376	+	0.126675i
$304$		0			0
$305$	−1.29865e9	−	7.49775e8i	−0.150069	−	0.0866426i
$306$		0			0
$307$	1.30162e10			1.46531			0.732657	−	0.680598i	$-0.238280\pi$
$307$	1.30162e10			1.46531			0.732657	+	0.680598i	$0.238280\pi$
$308$		0			0
$309$	−1.12326e10	−	1.03315e10i	−1.23210	−	1.13326i
$310$		0			0
$311$	−9.68235e9	−	5.59010e9i	−1.03500	−	0.597556i	−0.116585	−	0.993181i	$-0.537195\pi$
$311$	−9.68235e9	−	5.59010e9i	−1.03500	−	0.597556i	−0.918412	+	0.395625i	$0.870528\pi$
$312$		0			0
$313$	3.89082e9	+	6.73910e9i	0.405381	+	0.702141i	0.994366	−	0.106003i	$-0.0338054\pi$
$313$	3.89082e9	+	6.73910e9i	0.405381	+	0.702141i	−0.588984	+	0.808144i	$0.700472\pi$
$314$		0			0
$315$	−1.97113e9	+	1.26177e9i	−0.200204	+	0.128156i
$316$		0			0
$317$	1.30191e10	−	7.51657e9i	1.28927	−	0.744360i	0.310745	−	0.950493i	$-0.399422\pi$
$317$	1.30191e10	−	7.51657e9i	1.28927	−	0.744360i	0.978524	+	0.206134i	$0.0660882\pi$
$318$		0			0
$319$	9.99478e7	−	1.73115e8i	0.00965185	−	0.0167175i
$320$		0			0
$321$	−1.70722e9	+	1.85612e9i	−0.160794	+	0.174818i
$322$		0			0
$323$			6.18940e9i			0.568642i
$324$		0			0
$325$	3.52633e9	−	6.10778e9i	0.316075	−	0.547457i
$326$		0			0
$327$	−3.89622e8	+	1.74204e9i	−0.0340763	+	0.152359i
$328$		0			0
$329$	1.92012e10	+	7.24650e8i	1.63887	+	0.0618507i
$330$		0			0
$331$	7.08901e9	+	1.22785e10i	0.590573	+	1.02290i	0.994155	+	0.107959i	$0.0344314\pi$
$331$	7.08901e9	+	1.22785e10i	0.590573	+	1.02290i	−0.403583	+	0.914943i	$0.632235\pi$
$332$		0			0
$333$	6.61679e9	+	3.11565e9i	0.538109	+	0.253380i
$334$		0			0
$335$		−	2.51699e9i		−	0.199849i
$336$		0			0
$337$	−5.70944e9			−0.442664			−0.221332	−	0.975199i	$-0.571040\pi$
$337$	−5.70944e9			−0.442664			−0.221332	+	0.975199i	$0.571040\pi$
$338$		0			0
$339$	−7.64485e8	−	2.44055e9i	−0.0578855	−	0.184794i
$340$		0			0
$341$	1.00265e7	−	5.78879e6i	0.000741534	−	0.000428125i
$342$		0			0
$343$	−1.56302e9	+	1.37528e10i	−0.112925	+	0.993604i
$344$		0			0
$345$	−4.97096e8	+	2.22257e9i	−0.0350884	+	0.156884i
$346$		0			0
$347$	6.65061e9	+	3.83973e9i	0.458715	+	0.264839i	0.711504	−	0.702682i	$-0.248014\pi$
$347$	6.65061e9	+	3.83973e9i	0.458715	+	0.264839i	−0.252789	+	0.967522i	$0.581348\pi$
$348$		0			0
$349$	1.45874e10			0.983275			0.491638	−	0.870800i	$-0.336398\pi$
$349$	1.45874e10			0.983275			0.491638	+	0.870800i	$0.336398\pi$
$350$		0			0
$351$	−6.23611e9	+	8.03330e9i	−0.410852	+	0.529256i
$352$		0			0
$353$	−3.81550e8	−	2.20288e8i	−0.0245727	−	0.0141871i	0.487663	−	0.873032i	$-0.337849\pi$
$353$	−3.81550e8	−	2.20288e8i	−0.0245727	−	0.0141871i	−0.512236	+	0.858845i	$0.671183\pi$
$354$		0			0
$355$	3.02602e9	+	5.24121e9i	0.190528	+	0.330003i
$356$		0			0
$357$	1.70940e10	+	1.69565e10i	1.05238	+	1.04391i
$358$		0			0
$359$	1.90582e10	−	1.10033e10i	1.14737	−	0.662435i	0.199126	−	0.979974i	$-0.436190\pi$
$359$	1.90582e10	−	1.10033e10i	1.14737	−	0.662435i	0.948245	+	0.317539i	$0.102856\pi$
$360$		0			0
$361$	7.24211e9	−	1.25437e10i	0.426418	−	0.738578i
$362$		0			0
$363$	−1.26477e10	−	1.16331e10i	−0.728428	−	0.669992i
$364$		0			0
$365$		−	4.76479e7i		−	0.00268455i
$366$		0			0
$367$	8.47008e9	−	1.46706e10i	0.466899	−	0.808693i	−0.532386	−	0.846502i	$-0.678704\pi$
$367$	8.47008e9	−	1.46706e10i	0.466899	−	0.808693i	0.999285	+	0.0378087i	$0.0120377\pi$
$368$		0			0
$369$	−7.40563e9	−	1.06610e10i	−0.399444	−	0.575033i
$370$		0			0
$371$	1.95518e8	+	3.70227e8i	0.0103203	+	0.0195422i
$372$		0			0
$373$	8.92192e9	+	1.54532e10i	0.460917	+	0.798332i	0.999007	−	0.0445556i	$-0.0141872\pi$
$373$	8.92192e9	+	1.54532e10i	0.460917	+	0.798332i	−0.538090	+	0.842888i	$0.680854\pi$
$374$		0			0
$375$	−2.73093e9	−	8.71824e9i	−0.138097	−	0.440863i
$376$		0			0
$377$			2.57373e9i			0.127408i
$378$		0			0
$379$	−2.52632e9			−0.122442			−0.0612210	−	0.998124i	$-0.519499\pi$
$379$	−2.52632e9			−0.122442			−0.0612210	+	0.998124i	$0.519499\pi$
$380$		0			0
$381$	−2.38038e10	+	7.45637e9i	−1.12966	+	0.353857i
$382$		0			0
$383$	−2.72614e10	+	1.57394e10i	−1.26693	+	0.731463i	−0.974406	−	0.224795i	$-0.927829\pi$
$383$	−2.72614e10	+	1.57394e10i	−1.26693	+	0.731463i	−0.292525	+	0.956258i	$0.594496\pi$
$384$		0			0
$385$	−4.48812e8	−	2.82209e8i	−0.0204278	−	0.0128448i
$386$		0			0
$387$	2.09327e10	−	1.45408e10i	0.933216	−	0.648255i
$388$		0			0
$389$	−2.78079e10	−	1.60549e10i	−1.21442	−	0.701148i	−0.250704	−	0.968064i	$-0.580662\pi$
$389$	−2.78079e10	−	1.60549e10i	−1.21442	−	0.701148i	−0.963720	+	0.266916i	$0.913995\pi$
$390$		0			0
$391$	2.34304e10			1.00247
$392$		0			0
$393$	−2.48814e10	+	2.70515e10i	−1.04305	+	1.13402i
$394$		0			0
$395$	−9.93692e8	−	5.73708e8i	−0.0408191	−	0.0235669i
$396$		0			0
$397$	−1.96012e10	−	3.39503e10i	−0.789080	−	1.36673i	−0.926531	−	0.376218i	$-0.877224\pi$
$397$	−1.96012e10	−	3.39503e10i	−0.789080	−	1.36673i	0.137451	−	0.990509i	$-0.456109\pi$
$398$		0			0
$399$	2.55437e9	+	9.38125e9i	0.100784	+	0.370143i
$400$		0			0
$401$	−2.01562e10	+	1.16372e10i	−0.779528	+	0.450061i	−0.836263	−	0.548329i	$-0.815264\pi$
$401$	−2.01562e10	+	1.16372e10i	−0.779528	+	0.450061i	0.0567349	+	0.998389i	$0.481931\pi$
$402$		0			0
$403$	−7.45329e7	+	1.29095e8i	−0.00282571	+	0.00489428i
$404$		0			0
$405$	1.06338e9	+	6.30634e9i	0.0395248	+	0.234400i
$406$		0			0
$407$			1.65675e9i			0.0603780i
$408$		0			0
$409$	1.99570e10	−	3.45665e10i	0.713184	−	1.23527i	−0.250472	−	0.968124i	$-0.580586\pi$
$409$	1.99570e10	−	3.45665e10i	0.713184	−	1.23527i	0.963656	−	0.267147i	$-0.0860808\pi$
$410$		0			0
$411$	−4.33889e10	−	9.70429e9i	−1.52059	−	0.340092i
$412$		0			0
$413$	1.81541e10	+	3.43760e10i	0.623985	+	1.18156i
$414$		0			0
$415$	−7.17818e7	−	1.24330e8i	−0.00242004	−	0.00419163i
$416$		0			0
$417$	−4.54525e10	+	1.42377e10i	−1.50319	+	0.470864i
$418$		0			0
$419$		−	4.37620e10i		−	1.41985i	−0.704280	−	0.709923i	$-0.748730\pi$
$419$		−	4.37620e10i		−	1.41985i	0.704280	−	0.709923i	$-0.251270\pi$
$420$		0			0
$421$	−5.09222e10			−1.62099			−0.810493	−	0.585749i	$-0.800800\pi$
$421$	−5.09222e10			−1.62099			−0.810493	+	0.585749i	$0.800800\pi$
$422$		0			0
$423$	2.23682e10	−	4.75039e10i	0.698666	−	1.48377i
$424$		0			0
$425$	−3.95153e10	+	2.28142e10i	−1.21118	+	0.699276i
$426$		0			0
$427$	−9.13943e8	+	2.42169e10i	−0.0274921	+	0.728463i
$428$		0			0
$429$	−2.24819e9	−	5.02826e8i	−0.0663748	−	0.0148453i
$430$		0			0
$431$	−1.86657e10	−	1.07767e10i	−0.540924	−	0.312302i	0.204530	−	0.978860i	$-0.434434\pi$
$431$	−1.86657e10	−	1.07767e10i	−0.540924	−	0.312302i	−0.745453	+	0.666558i	$0.767767\pi$
$432$		0			0
$433$	−4.30006e10			−1.22327			−0.611636	−	0.791139i	$-0.709488\pi$
$433$	−4.30006e10			−1.22327			−0.611636	+	0.791139i	$0.709488\pi$
$434$		0			0
$435$	1.19126e9	+	1.09569e9i	0.0332697	+	0.0306007i
$436$		0			0
$437$	8.19387e9	+	4.73073e9i	0.224679	+	0.129719i
$438$		0			0
$439$	−1.32648e9	−	2.29753e9i	−0.0357143	−	0.0618590i	0.847616	−	0.530611i	$-0.178037\pi$
$439$	−1.32648e9	−	2.29753e9i	−0.0357143	−	0.0618590i	−0.883330	+	0.468752i	$0.844704\pi$
$440$		0			0
$441$	3.29073e10	+	1.86461e10i	0.870038	+	0.492985i
$442$		0			0
$443$	4.65000e10	−	2.68468e10i	1.20736	−	0.697071i	0.245180	−	0.969478i	$-0.421153\pi$
$443$	4.65000e10	−	2.68468e10i	1.20736	−	0.697071i	0.962182	+	0.272407i	$0.0878196\pi$
$444$		0			0
$445$	8.28720e9	−	1.43538e10i	0.211333	−	0.366040i
$446$		0			0
$447$	3.63921e10	−	3.95662e10i	0.911543	−	0.991046i
$448$		0			0
$449$			2.86415e10i			0.704710i	0.935866	+	0.352355i	$0.114619\pi$
$449$			2.86415e10i			0.704710i	−0.935866	+	0.352355i	$0.885381\pi$
$450$		0			0
$451$	1.47026e9	−	2.54656e9i	0.0355376	−	0.0615529i
$452$		0			0
$453$	−4.70612e9	+	2.10416e10i	−0.111756	+	0.499672i
$454$		0			0
$455$	6.82122e9	+	2.57432e8i	0.159154	+	0.00600644i
$456$		0			0
$457$	−9.61706e8	−	1.66572e9i	−0.0220484	−	0.0381890i	0.854791	−	0.518973i	$-0.173685\pi$
$457$	−9.61706e8	−	1.66572e9i	−0.0220484	−	0.0381890i	−0.876839	+	0.480784i	$0.840352\pi$
$458$		0			0
$459$	6.09266e10	−	2.48369e10i	1.37264	−	0.559561i
$460$		0			0
$461$			7.86043e10i			1.74037i	0.492722	+	0.870187i	$0.336002\pi$
$461$			7.86043e10i			1.74037i	−0.492722	+	0.870187i	$0.663998\pi$
$462$		0			0
$463$	−3.69107e10			−0.803210			−0.401605	−	0.915813i	$-0.631547\pi$
$463$	−3.69107e10			−0.803210			−0.401605	+	0.915813i	$0.631547\pi$
$464$		0			0
$465$	2.80215e7	+	8.94561e7i	0.000599349	+	0.00191337i
$466$		0			0
$467$	−4.06680e9	+	2.34797e9i	−0.0855039	+	0.0493657i	−0.542142	−	0.840287i	$-0.682387\pi$
$467$	−4.06680e9	+	2.34797e9i	−0.0855039	+	0.0493657i	0.456638	+	0.889652i	$0.349053\pi$
$468$		0			0
$469$	−3.59693e10	+	1.89955e10i	−0.743431	+	0.392608i
$470$		0			0
$471$	−6.43569e9	+	2.87747e10i	−0.130771	+	0.584691i
$472$		0			0
$473$	5.00014e9	+	2.88683e9i	0.0998936	+	0.0576736i
$474$		0			0
$475$	−1.84252e10			−0.361942
$476$		0			0
$477$	1.14011e9	−	9.54495e7i	0.0220228	−	0.00184374i
$478$		0			0
$479$	−2.34637e10	−	1.35468e10i	−0.445712	−	0.257332i	0.260305	−	0.965526i	$-0.416177\pi$
$479$	−2.34637e10	−	1.35468e10i	−0.445712	−	0.257332i	−0.706018	+	0.708194i	$0.749510\pi$
$480$		0			0
$481$	−1.06656e10	−	1.84734e10i	−0.199254	−	0.345118i
$482$		0			0
$483$	3.55133e10	−	9.66973e9i	0.652534	−	0.177675i
$484$		0			0
$485$	6.40805e9	−	3.69969e9i	0.115813	−	0.0668649i
$486$		0			0
$487$	−7.81188e9	+	1.35306e10i	−0.138880	+	0.240547i	−0.927073	−	0.374881i	$-0.877684\pi$
$487$	−7.81188e9	+	1.35306e10i	−0.138880	+	0.240547i	0.788193	+	0.615428i	$0.211017\pi$
$488$		0			0
$489$	2.42381e10	+	2.22937e10i	0.423900	+	0.389894i
$490$		0			0
$491$			3.79528e10i			0.653008i	0.945196	+	0.326504i	$0.105871\pi$
$491$			3.79528e10i			0.653008i	−0.945196	+	0.326504i	$0.894129\pi$
$492$		0			0
$493$	8.32559e9	−	1.44203e10i	0.140938	−	0.244111i
$494$		0			0
$495$	−1.18983e9	+	8.26514e8i	−0.0198183	+	0.0137667i
$496$		0			0
$497$	5.20629e10	−	8.27984e10i	0.853302	−	1.35705i
$498$		0			0
$499$	4.57830e10	+	7.92984e10i	0.738417	+	1.27898i	0.953208	+	0.302316i	$0.0977597\pi$
$499$	4.57830e10	+	7.92984e10i	0.738417	+	1.27898i	−0.214791	+	0.976660i	$0.568907\pi$
$500$		0			0
$501$	−9.31048e9	−	2.97229e10i	−0.147782	−	0.471780i
$502$		0			0
$503$		−	8.73647e10i		−	1.36478i	−0.730986	−	0.682392i	$-0.760940\pi$
$503$		−	8.73647e10i		−	1.36478i	0.730986	−	0.682392i	$-0.239060\pi$
$504$		0			0
$505$	−8.97254e9			−0.137959
$506$		0			0
$507$	−3.47479e10	+	1.08845e10i	−0.525892	+	0.164732i
$508$		0			0
$509$	4.07389e10	−	2.35206e10i	0.606930	−	0.350411i	−0.164833	−	0.986321i	$-0.552709\pi$
$509$	4.07389e10	−	2.35206e10i	0.606930	−	0.350411i	0.771763	+	0.635910i	$0.219375\pi$
$510$		0			0
$511$	−6.80915e8	+	3.59594e8i	−0.00998641	+	0.00527386i
$512$		0			0
$513$	2.63214e10	+	3.61568e9i	0.380049	+	0.0522061i
$514$		0			0
$515$	2.42419e10	+	1.39961e10i	0.344618	+	0.198965i
$516$		0			0
$517$	1.18943e10			0.166485
$518$		0			0
$519$	2.69423e10	−	2.92922e10i	0.371335	−	0.403722i
$520$		0			0
$521$	−8.81832e10	−	5.09126e10i	−1.19684	−	0.690994i	−0.236989	−	0.971512i	$-0.576160\pi$
$521$	−8.81832e10	−	5.09126e10i	−1.19684	−	0.690994i	−0.959849	+	0.280518i	$0.909494\pi$
$522$		0			0
$523$	9.34191e9	+	1.61807e10i	0.124862	+	0.216267i	0.921679	−	0.387954i	$-0.126818\pi$
$523$	9.34191e9	+	1.61807e10i	0.124862	+	0.216267i	−0.796817	+	0.604220i	$0.793485\pi$
$524$		0			0
$525$	−5.04777e10	+	5.08873e10i	−0.664450	+	0.669841i
$526$		0			0
$527$	8.35200e8	−	4.82203e8i	0.0108280	−	0.00625154i
$528$		0			0
$529$	−2.12470e10	+	3.68009e10i	−0.271316	+	0.469933i
$530$		0			0
$531$	1.05860e11	−	8.86259e9i	1.33154	−	0.111476i
$532$		0			0
$533$			3.78603e10i			0.469111i
$534$		0			0
$535$	2.31277e9	−	4.00583e9i	0.0282304	−	0.0488965i
$536$		0			0
$537$	1.28315e11	+	2.86986e10i	1.54305	+	0.345115i
$538$		0			0
$539$	−6.45788e8	+	8.54359e9i	−0.00765129	+	0.101224i
$540$		0			0
$541$	−7.25970e9	−	1.25742e10i	−0.0847480	−	0.146788i	0.820536	−	0.571595i	$-0.193675\pi$
$541$	−7.25970e9	−	1.25742e10i	−0.0847480	−	0.146788i	−0.905284	+	0.424807i	$0.860342\pi$
$542$		0			0
$543$	7.58693e10	−	2.37655e10i	0.872704	−	0.273369i
$544$		0			0
$545$		−	3.27415e9i		−	0.0371118i
$546$		0			0
$547$	−4.95868e10			−0.553881			−0.276941	−	0.960887i	$-0.589321\pi$
$547$	−4.95868e10			−0.553881			−0.276941	+	0.960887i	$0.589321\pi$
$548$		0			0
$549$	5.99128e10	+	2.82112e10i	0.659524	+	0.310551i
$550$		0			0
$551$	5.82310e9	−	3.36197e9i	0.0631754	−	0.0364743i
$552$		0			0
$553$	−6.99325e8	+	1.85301e10i	−0.00747788	+	0.198143i
$554$		0			0
$555$	−1.30910e10	−	2.92792e9i	−0.137976	−	0.0308594i
$556$		0			0
$557$	2.02239e10	+	1.16763e10i	0.210108	+	0.121306i	0.601362	−	0.798977i	$-0.294625\pi$
$557$	2.02239e10	+	1.16763e10i	0.210108	+	0.121306i	−0.391253	+	0.920283i	$0.627958\pi$
$558$		0			0
$559$	−7.43382e10			−0.761316
$560$		0			0
$561$	1.09698e10	+	1.00898e10i	0.110751	+	0.101866i
$562$		0			0
$563$	1.24693e10	+	7.19918e9i	0.124111	+	0.0716554i	0.560770	−	0.827972i	$-0.310505\pi$
$563$	1.24693e10	+	7.19918e9i	0.124111	+	0.0716554i	−0.436659	+	0.899627i	$0.643838\pi$
$564$		0			0
$565$	2.34543e9	+	4.06241e9i	0.0230160	+	0.0398648i
$566$		0			0
$567$	8.20960e10	−	6.27897e10i	0.794309	−	0.607514i
$568$		0			0
$569$	1.01083e11	−	5.83606e10i	0.964342	−	0.556763i	0.0668352	−	0.997764i	$-0.478710\pi$
$569$	1.01083e11	−	5.83606e10i	0.964342	−	0.556763i	0.897507	+	0.441001i	$0.145376\pi$
$570$		0			0
$571$	−9.09590e10	+	1.57546e11i	−0.855660	+	1.48205i	0.0203721	+	0.999792i	$0.493515\pi$
$571$	−9.09590e10	+	1.57546e11i	−0.855660	+	1.48205i	−0.876032	+	0.482253i	$0.839818\pi$
$572$		0			0
$573$	−1.32380e10	+	1.43926e10i	−0.122802	+	0.133512i
$574$		0			0
$575$			6.97500e10i			0.638076i
$576$		0			0
$577$	1.37201e10	−	2.37638e10i	0.123781	−	0.214394i	−0.797475	−	0.603352i	$-0.793831\pi$
$577$	1.37201e10	−	2.37638e10i	0.123781	−	0.214394i	0.921256	+	0.388958i	$0.127165\pi$
$578$		0			0
$579$	3.14770e9	−	1.40737e10i	0.0280078	−	0.125226i
$580$		0			0
$581$	−1.23501e9	+	1.96411e9i	−0.0108384	+	0.0172370i
$582$		0			0
$583$	1.29586e8	+	2.24449e8i	0.00112171	+	0.00194287i
$584$		0			0
$585$	7.94630e9	−	1.68758e10i	0.0678487	−	0.144092i
$586$		0			0
$587$			2.45799e10i			0.207027i	0.994628	+	0.103513i	$0.0330085\pi$
$587$			2.45799e10i			0.207027i	−0.994628	+	0.103513i	$0.966992\pi$
$588$		0			0
$589$	3.89438e8			0.00323577
$590$		0			0
$591$	−2.45712e10	−	7.84414e10i	−0.201408	−	0.642977i
$592$		0			0
$593$	1.13374e11	−	6.54565e10i	0.916843	−	0.529339i	0.0342164	−	0.999414i	$-0.489106\pi$
$593$	1.13374e11	−	6.54565e10i	0.916843	−	0.529339i	0.882626	+	0.470075i	$0.155773\pi$
$594$		0			0
$595$	−3.73858e10	−	2.35079e10i	−0.298290	−	0.187562i
$596$		0			0
$597$	−1.10178e10	+	4.92615e10i	−0.0867352	+	0.387803i
$598$		0			0
$599$	3.88976e10	+	2.24575e10i	0.302145	+	0.174443i	0.643406	−	0.765525i	$-0.277521\pi$
$599$	3.88976e10	+	2.24575e10i	0.302145	+	0.174443i	−0.341261	+	0.939968i	$0.610854\pi$
$600$		0			0
$601$	2.63320e10			0.201830			0.100915	−	0.994895i	$-0.467823\pi$
$601$	2.63320e10			0.201830			0.100915	+	0.994895i	$0.467823\pi$
$602$		0			0
$603$	9.27336e9	+	1.10767e11i	0.0701404	+	0.837801i
$604$		0			0
$605$	2.72960e10	+	1.57594e10i	0.203741	+	0.117630i
$606$		0			0
$607$	−4.40312e10	−	7.62643e10i	−0.324344	−	0.561781i	0.657035	−	0.753860i	$-0.271810\pi$
$607$	−4.40312e10	−	7.62643e10i	−0.324344	−	0.561781i	−0.981379	+	0.192079i	$0.938477\pi$
$608$		0			0
$609$	6.66778e9	−	2.52928e10i	0.0484743	−	0.183877i
$610$		0			0
$611$	−1.32626e11	+	7.65718e10i	−0.951622	+	0.549419i
$612$		0			0
$613$	5.86210e10	−	1.01535e11i	0.415156	−	0.719072i	−0.580289	−	0.814411i	$-0.697060\pi$
$613$	5.86210e10	−	1.01535e11i	0.415156	−	0.719072i	0.995445	+	0.0953392i	$0.0303936\pi$
$614$		0			0
$615$	1.75237e10	+	1.61179e10i	0.122497	+	0.112670i
$616$		0			0
$617$		−	2.60784e10i		−	0.179945i	−0.995944	−	0.0899725i	$-0.971322\pi$
$617$		−	2.60784e10i		−	0.179945i	0.995944	−	0.0899725i	$-0.0286779\pi$
$618$		0			0
$619$	−4.62386e10	+	8.00875e10i	−0.314950	+	0.545510i	−0.979427	−	0.201799i	$-0.935321\pi$
$619$	−4.62386e10	+	8.00875e10i	−0.314950	+	0.545510i	0.664477	+	0.747309i	$0.268654\pi$
$620$		0			0
$621$	1.36874e10	−	9.96415e10i	0.0920354	−	0.669999i
$622$		0			0
$623$	−2.67667e11	−	1.01017e10i	−1.77682	−	0.0670569i
$624$		0			0
$625$	−6.36045e10	−	1.10166e11i	−0.416838	−	0.721985i
$626$		0			0
$627$	1.79907e9	+	5.74337e9i	0.0116407	+	0.0371618i
$628$		0			0
$629$			1.38006e11i			0.881649i
$630$		0			0
$631$	9.74236e10			0.614535			0.307268	−	0.951623i	$-0.400585\pi$
$631$	9.74236e10			0.614535			0.307268	+	0.951623i	$0.400585\pi$
$632$		0			0
$633$	−2.75353e11	+	8.62526e10i	−1.71505	+	0.537226i
$634$		0			0
$635$	3.96225e10	−	2.28761e10i	0.243695	−	0.140698i
$636$		0			0
$637$	−4.78002e10	−	9.94220e10i	−0.290317	−	0.603844i
$638$		0			0
$639$	−1.52478e11	−	2.19505e11i	−0.914542	−	1.31656i
$640$		0			0
$641$	−1.81368e11	−	1.04713e11i	−1.07431	−	0.620251i	−0.144952	−	0.989439i	$-0.546303\pi$
$641$	−1.81368e11	−	1.04713e11i	−1.07431	−	0.620251i	−0.929355	+	0.369187i	$0.879636\pi$
$642$		0			0
$643$	−2.72807e11			−1.59592			−0.797960	−	0.602710i	$-0.794087\pi$
$643$	−2.72807e11			−1.59592			−0.797960	+	0.602710i	$0.794087\pi$
$644$		0			0
$645$	−3.16473e10	+	3.44076e10i	−0.182851	+	0.198799i
$646$		0			0
$647$	2.34021e10	+	1.35112e10i	0.133548	+	0.0771040i	0.565286	−	0.824895i	$-0.308766\pi$
$647$	2.34021e10	+	1.35112e10i	0.133548	+	0.0771040i	−0.431737	+	0.901999i	$0.642099\pi$
$648$		0			0
$649$	1.20322e10	+	2.08403e10i	0.0678211	+	0.117470i
$650$		0			0
$651$	1.06690e9	−	1.07556e9i	0.00594020	−	0.00598840i
$652$		0			0
$653$	−3.03406e11	+	1.75172e11i	−1.66868	+	0.963410i	−0.700323	+	0.713827i	$0.746960\pi$
$653$	−3.03406e11	+	1.75172e11i	−1.66868	+	0.963410i	−0.968353	+	0.249584i	$0.919706\pi$
$654$		0			0
$655$	3.37067e10	−	5.83817e10i	0.183126	−	0.317184i
$656$		0			0
$657$	1.75549e8	+	2.09687e9i	0.000942187	+	0.0112541i
$658$		0			0
$659$		−	4.90923e10i		−	0.260299i	−0.991494	−	0.130149i	$-0.958454\pi$
$659$		−	4.90923e10i		−	0.260299i	0.991494	−	0.130149i	$-0.0415457\pi$
$660$		0			0
$661$	1.27841e11	−	2.21426e11i	0.669673	−	1.15991i	−0.308323	−	0.951282i	$-0.599768\pi$
$661$	1.27841e11	−	2.21426e11i	0.669673	−	1.15991i	0.977996	−	0.208626i	$-0.0668990\pi$
$662$		0			0
$663$	−1.87273e11	−	4.18851e10i	−0.969215	−	0.216773i
$664$		0			0
$665$	−8.32786e9	−	1.57694e10i	−0.0425840	−	0.0806357i
$666$		0			0
$667$	−1.27270e10	−	2.20437e10i	−0.0643015	−	0.111373i
$668$		0			0
$669$	−1.63571e11	+	5.12374e10i	−0.816583	+	0.255789i
$670$		0			0
$671$			1.50013e10i			0.0740012i
$672$		0			0
$673$	3.21465e11			1.56702			0.783509	−	0.621380i	$-0.213428\pi$
$673$	3.21465e11			1.56702			0.783509	+	0.621380i	$0.213428\pi$
$674$		0			0
$675$	7.39371e10	+	1.81372e11i	0.356162	+	0.873688i
$676$		0			0
$677$	1.52217e10	−	8.78827e9i	0.0724618	−	0.0418359i	−0.463331	−	0.886185i	$-0.653346\pi$
$677$	1.52217e10	−	8.78827e9i	0.0724618	−	0.0418359i	0.535793	+	0.844349i	$0.320013\pi$
$678$		0			0
$679$	−1.01232e11	−	6.36535e10i	−0.476252	−	0.299463i
$680$		0			0
$681$	−8.37663e10	−	1.87350e10i	−0.389476	−	0.0871096i
$682$		0			0
$683$	3.36663e11	+	1.94373e11i	1.54708	+	0.893207i	0.998363	+	0.0571993i	$0.0182170\pi$
$683$	3.36663e11	+	1.94373e11i	1.54708	+	0.893207i	0.548717	+	0.836008i	$0.315116\pi$
$684$		0			0
$685$	8.15490e10			0.370387
$686$		0			0
$687$	−1.06277e11	−	9.77515e10i	−0.477104	−	0.438830i
$688$		0			0
$689$	−2.88986e9	−	1.66846e9i	−0.0128233	−	0.00740355i
$690$		0			0
$691$	1.78274e11	+	3.08779e11i	0.781943	+	1.35437i	0.930808	+	0.365508i	$0.119105\pi$
$691$	1.78274e11	+	3.08779e11i	0.781943	+	1.35437i	−0.148865	+	0.988857i	$0.547562\pi$
$692$		0			0
$693$	2.07909e10	+	1.07658e10i	0.0901448	+	0.0466781i
$694$		0			0
$695$	7.56578e10	−	4.36811e10i	0.324276	−	0.187221i
$696$		0			0
$697$	1.22472e11	−	2.12127e11i	0.518925	−	0.898804i
$698$		0			0
$699$	−1.08223e11	+	1.17662e11i	−0.453327	+	0.492865i
$700$		0			0
$701$		−	4.31505e11i		−	1.78696i	−0.449106	−	0.893479i	$-0.648257\pi$
$701$		−	4.31505e11i		−	1.78696i	0.449106	−	0.893479i	$-0.351743\pi$
$702$		0			0
$703$	−2.78642e10	+	4.82623e10i	−0.114084	+	0.197600i
$704$		0			0
$705$	−2.10204e10	+	9.39845e10i	−0.0850912	+	0.380452i
$706$		0			0
$707$	6.77149e10	+	1.28223e11i	0.271023	+	0.513201i
$708$		0			0
$709$	1.00232e11	+	1.73606e11i	0.396661	+	0.687038i	0.993312	−	0.115464i	$-0.0368354\pi$
$709$	1.00232e11	+	1.73606e11i	0.396661	+	0.687038i	−0.596650	+	0.802501i	$0.703502\pi$
$710$		0			0
$711$	4.58437e10	+	2.15865e10i	0.179391	+	0.0844702i
$712$		0			0
$713$		−	1.47424e9i		−	0.00570441i
$714$		0			0
$715$	4.22544e9			0.0161677
$716$		0			0
$717$	1.48403e11	+	4.73763e11i	0.561521	+	1.79260i
$718$		0			0
$719$	−2.35196e11	+	1.35791e11i	−0.880065	+	0.508106i	−0.870680	−	0.491850i	$-0.836321\pi$
$719$	−2.35196e11	+	1.35791e11i	−0.880065	+	0.508106i	−0.00938510	+	0.999956i	$0.502987\pi$
$720$		0			0
$721$	1.70606e10	−	4.52058e11i	0.0631326	−	1.67283i
$722$		0			0
$723$	8.65966e10	−	3.87183e11i	0.316918	−	1.41698i
$724$		0			0
$725$	4.29279e10	+	2.47844e10i	0.155377	+	0.0897072i
$726$		0			0
$727$	6.54370e10			0.234253			0.117127	−	0.993117i	$-0.462632\pi$
$727$	6.54370e10			0.234253			0.117127	+	0.993117i	$0.462632\pi$
$728$		0			0
$729$	−7.00314e10	−	2.73609e11i	−0.247961	−	0.968770i
$730$		0			0
$731$	4.16509e11	+	2.40471e11i	1.45866	+	0.842159i
$732$		0			0
$733$	1.13417e11	+	1.96443e11i	0.392880	+	0.680489i	0.992828	−	0.119550i	$-0.0381453\pi$
$733$	1.13417e11	+	1.96443e11i	0.392880	+	0.680489i	−0.599948	+	0.800039i	$0.704812\pi$
$734$		0			0
$735$	−6.63672e10	−	2.02016e10i	−0.227407	−	0.0692208i
$736$		0			0
$737$	−2.18062e10	+	1.25898e10i	−0.0739113	+	0.0426727i
$738$		0			0
$739$	−2.39672e11	+	4.15125e11i	−0.803601	+	1.39188i	0.113631	+	0.993523i	$0.463752\pi$
$739$	−2.39672e11	+	4.15125e11i	−0.803601	+	1.39188i	−0.917232	+	0.398354i	$0.869582\pi$
$740$		0			0
$741$	−5.70345e10	−	5.24591e10i	−0.189175	−	0.173999i
$742$		0			0
$743$		−	3.45306e11i		−	1.13305i	−0.824045	−	0.566525i	$-0.808288\pi$
$743$		−	3.45306e11i		−	1.13305i	0.824045	−	0.566525i	$-0.191712\pi$
$744$		0			0
$745$	−4.93003e10	+	8.53906e10i	−0.160038	+	0.277195i
$746$		0			0
$747$	3.61701e9	+	5.20699e9i	0.0116163	+	0.0167226i
$748$		0			0
$749$	−7.46999e10	−	2.81916e9i	−0.237352	−	0.00895763i
$750$		0			0
$751$	1.90819e11	+	3.30509e11i	0.599877	+	1.03902i	0.992839	+	0.119463i	$0.0381173\pi$
$751$	1.90819e11	+	3.30509e11i	0.599877	+	1.03902i	−0.392961	+	0.919555i	$0.628549\pi$
$752$		0			0
$753$	2.27449e10	+	7.26110e10i	0.0707463	+	0.225851i
$754$		0			0
$755$		−	3.95474e10i		−	0.121711i
$756$		0			0
$757$	5.83016e11			1.77540			0.887701	−	0.460420i	$-0.152301\pi$
$757$	5.83016e11			1.77540			0.887701	+	0.460420i	$0.152301\pi$
$758$		0			0
$759$	2.17419e10	−	6.81050e9i	0.0655135	−	0.0205216i
$760$		0			0
$761$	1.74586e11	−	1.00798e11i	0.520561	−	0.300546i	−0.216603	−	0.976260i	$-0.569498\pi$
$761$	1.74586e11	−	1.00798e11i	0.520561	−	0.300546i	0.737164	+	0.675714i	$0.236164\pi$
$762$		0			0
$763$	−4.67894e10	+	2.47096e10i	−0.138054	+	0.0729069i
$764$		0			0
$765$	−9.91125e10	+	6.88481e10i	−0.289389	+	0.201023i
$766$		0			0
$767$	−2.68327e11	−	1.54919e11i	−0.775324	−	0.447633i
$768$		0			0
$769$	2.93939e11			0.840526			0.420263	−	0.907402i	$-0.361938\pi$
$769$	2.93939e11			0.840526			0.420263	+	0.907402i	$0.361938\pi$
$770$		0			0
$771$	2.29795e11	−	2.49838e11i	0.650315	−	0.707034i
$772$		0			0
$773$	2.99140e11	+	1.72709e11i	0.837832	+	0.483722i	0.856527	−	0.516103i	$-0.172618\pi$
$773$	2.99140e11	+	1.72709e11i	0.837832	+	0.483722i	−0.0186949	+	0.999825i	$0.505951\pi$
$774$		0			0
$775$	1.43547e9	+	2.48631e9i	0.00397912	+	0.00689204i
$776$		0			0
$777$	5.69552e10	+	2.09175e11i	0.156260	+	0.573887i
$778$		0			0
$779$	8.56594e10	−	4.94555e10i	0.232608	−	0.134296i
$780$		0			0
$781$	3.02719e10	−	5.24324e10i	0.0813645	−	0.140927i
$782$		0			0
$783$	−5.64612e10	−	4.38299e10i	−0.150212	−	0.116607i
$784$		0			0
$785$		−	5.40817e10i		−	0.142420i
$786$		0			0
$787$	−5.87048e10	+	1.01680e11i	−0.153029	+	0.265054i	−0.932340	−	0.361584i	$-0.882236\pi$
$787$	−5.87048e10	+	1.01680e11i	−0.153029	+	0.265054i	0.779310	+	0.626638i	$0.215570\pi$
$788$		0			0
$789$	2.49169e11	+	5.57288e10i	0.642964	+	0.143804i
$790$		0			0
$791$	4.03534e10	−	6.41761e10i	0.103080	−	0.163933i
$792$		0			0
$793$	−9.65738e10	−	1.67271e11i	−0.244212	−	0.422987i
$794$		0			0
$795$	−2.00253e9	+	6.27278e8i	−0.00501314	+	0.00157033i
$796$		0			0
$797$		−	4.99180e11i		−	1.23715i	−0.785724	−	0.618577i	$-0.787709\pi$
$797$		−	4.99180e11i		−	1.23715i	0.785724	−	0.618577i	$-0.212291\pi$
$798$		0			0
$799$	9.90787e11			2.43104
$800$		0			0
$801$	−3.11816e11	+	6.62211e11i	−0.757475	+	1.60867i
$802$		0			0
$803$	−4.12802e8	+	2.38331e8i	−0.000992841	+	0.000573217i
$804$		0			0
$805$	−5.96960e10	+	3.15256e10i	−0.142155	+	0.0750724i
$806$		0			0
$807$	−6.54398e11	−	1.46362e11i	−1.54294	−	0.345090i
$808$		0			0
$809$	−2.51785e11	−	1.45368e11i	−0.587809	−	0.339372i	0.176422	−	0.984315i	$-0.443548\pi$
$809$	−2.51785e11	−	1.45368e11i	−0.587809	−	0.339372i	−0.764231	+	0.644943i	$0.776881\pi$
$810$		0			0
$811$	−4.42552e11			−1.02301			−0.511506	−	0.859280i	$-0.670912\pi$
$811$	−4.42552e11			−1.02301			−0.511506	+	0.859280i	$0.670912\pi$
$812$		0			0
$813$	−3.45237e11	−	3.17542e11i	−0.790233	−	0.726840i
$814$		0			0
$815$	−5.23100e10	−	3.02012e10i	−0.118564	−	0.0684532i
$816$		0			0
$817$	9.71051e10	+	1.68191e11i	0.217949	+	0.377498i
$818$		0			0
$819$	−3.01134e11	+	1.38025e10i	−0.669306	+	0.0306776i
$820$		0			0
$821$	1.77632e11	−	1.02556e11i	0.390974	−	0.225729i	−0.291608	−	0.956538i	$-0.594190\pi$
$821$	1.77632e11	−	1.02556e11i	0.390974	−	0.225729i	0.682582	+	0.730809i	$0.260857\pi$
$822$		0			0
$823$	−1.08302e11	+	1.87584e11i	−0.236068	+	0.408882i	−0.959582	−	0.281427i	$-0.909192\pi$
$823$	−1.08302e11	+	1.87584e11i	−0.236068	+	0.408882i	0.723515	+	0.690309i	$0.242525\pi$
$824$		0			0
$825$	−3.00363e10	+	3.26560e10i	−0.0648381	+	0.0704931i
$826$		0			0
$827$		−	3.39320e11i		−	0.725417i	−0.931903	−	0.362708i	$-0.881852\pi$
$827$		−	3.39320e11i		−	0.725417i	0.931903	−	0.362708i	$-0.118148\pi$
$828$		0			0
$829$	2.50521e11	−	4.33915e11i	0.530427	−	0.918726i	−0.468943	−	0.883229i	$-0.655365\pi$
$829$	2.50521e11	−	4.33915e11i	0.530427	−	0.918726i	0.999370	−	0.0354979i	$-0.0113017\pi$
$830$		0			0
$831$	7.21130e10	−	3.22425e11i	0.151220	−	0.676121i
$832$		0			0
$833$	−5.37937e10	+	7.11676e11i	−0.111725	+	1.47809i
$834$		0			0
$835$	2.85645e10	+	4.94751e10i	0.0587598	+	0.101775i
$836$		0			0
$837$	−1.56274e9	−	3.83351e9i	−0.00318409	−	0.00781079i
$838$		0			0
$839$			7.31744e11i			1.47676i	0.674383	+	0.738382i	$0.264410\pi$
$839$			7.31744e11i			1.47676i	−0.674383	+	0.738382i	$0.735590\pi$
$840$		0			0
$841$	4.82157e11			0.963839
$842$		0			0
$843$	2.66651e11	+	8.51260e11i	0.528000	+	1.68559i
$844$		0			0
$845$	5.78395e10	−	3.33936e10i	0.113448	−	0.0654994i
$846$		0			0
$847$	1.92100e10	−	5.09010e11i	0.0373244	−	0.988991i
$848$		0			0
$849$	1.11963e11	−	5.00597e11i	0.215498	−	0.963513i
$850$		0			0
$851$	1.82700e11	+	1.05482e11i	0.348353	+	0.201122i
$852$		0			0
$853$	−7.25697e11			−1.37075			−0.685377	−	0.728189i	$-0.740362\pi$
$853$	−7.25697e11			−1.37075			−0.685377	+	0.728189i	$0.740362\pi$
$854$		0			0
$855$	−4.85616e10	+	4.06555e9i	−0.0908716	+	0.00760773i
$856$		0			0
$857$	2.73312e11	+	1.57797e11i	0.506682	+	0.292533i	0.731469	−	0.681875i	$-0.238835\pi$
$857$	2.73312e11	+	1.57797e11i	0.506682	+	0.292533i	−0.224787	+	0.974408i	$0.572169\pi$
$858$		0			0
$859$	3.83337e10	+	6.63959e10i	0.0704057	+	0.121946i	0.899079	−	0.437786i	$-0.144237\pi$
$859$	3.83337e10	+	6.63959e10i	0.0704057	+	0.121946i	−0.828673	+	0.559732i	$0.810904\pi$
$860$		0			0
$861$	9.80848e10	−	3.72064e11i	0.178480	−	0.677026i
$862$		0			0
$863$	−5.52823e11	+	3.19173e11i	−0.996651	+	0.575417i	−0.907256	−	0.420579i	$-0.861827\pi$
$863$	−5.52823e11	+	3.19173e11i	−0.996651	+	0.575417i	−0.0893954	+	0.995996i	$0.528493\pi$
$864$		0			0
$865$	−3.64987e10	+	6.32176e10i	−0.0651948	+	0.112921i
$866$		0			0
$867$	4.97903e11	+	4.57961e11i	0.881188	+	0.810498i
$868$		0			0
$869$			1.14786e10i			0.0201284i
$870$		0			0
$871$	1.62099e11	−	2.80764e11i	0.281649	−	0.487830i
$872$		0			0
$873$	−2.68372e11	+	1.86424e11i	−0.462041	+	0.320955i
$874$		0			0
$875$	1.44152e11	−	2.29253e11i	0.245917	−	0.391095i
$876$		0			0
$877$	4.13719e10	+	7.16582e10i	0.0699370	+	0.121134i	0.898873	−	0.438208i	$-0.144387\pi$
$877$	4.13719e10	+	7.16582e10i	0.0699370	+	0.121134i	−0.828936	+	0.559343i	$0.811053\pi$
$878$		0			0
$879$	−1.77051e11	−	5.65219e11i	−0.296581	−	0.946806i
$880$		0			0
$881$		−	2.06167e11i		−	0.342228i	−0.985251	−	0.171114i	$-0.945263\pi$
$881$		−	2.06167e11i		−	0.342228i	0.985251	−	0.171114i	$-0.0547366\pi$
$882$		0			0
$883$	−5.31453e11			−0.874223			−0.437111	−	0.899407i	$-0.643998\pi$
$883$	−5.31453e11			−0.874223			−0.437111	+	0.899407i	$0.643998\pi$
$884$		0			0
$885$	−1.85937e11	+	5.82434e10i	−0.303104	+	0.0949454i
$886$		0			0
$887$	−2.04096e11	+	1.17835e11i	−0.329716	+	0.190361i	−0.655715	−	0.755009i	$-0.727633\pi$
$887$	−2.04096e11	+	1.17835e11i	−0.329716	+	0.190361i	0.325999	+	0.945370i	$0.394299\pi$
$888$		0			0
$889$	−6.25939e11	−	3.93585e11i	−1.00213	−	0.630132i
$890$		0			0
$891$	4.93167e10	−	4.07566e10i	0.0782497	−	0.0646677i
$892$		0			0
$893$	3.46489e11	+	2.00045e11i	0.544858	+	0.314574i
$894$		0			0
$895$	−2.41166e11			−0.375858
$896$		0			0
$897$	−1.98587e11	+	2.15908e11i	−0.306748	+	0.333502i
$898$		0			0
$899$	−9.07330e8	−	5.23847e8i	−0.00138908	−	0.000801984i
$900$		0			0
$901$	1.07944e10	+	1.86964e10i	0.0163794	+	0.0283700i
$902$		0			0
$903$	7.30543e11	+	1.92588e11i	1.09874	+	0.289653i
$904$		0			0
$905$	−1.26288e11	+	7.29125e10i	−0.188264	+	0.108694i
$906$		0			0
$907$	−2.76030e11	+	4.78098e11i	−0.407875	+	0.706461i	−0.994651	−	0.103288i	$-0.967064\pi$
$907$	−2.76030e11	+	4.78098e11i	−0.407875	+	0.706461i	0.586776	+	0.809749i	$0.300397\pi$
$908$		0			0
$909$	3.94860e11	−	3.30575e10i	0.578346	−	0.0484189i
$910$		0			0
$911$		−	9.26375e11i		−	1.34497i	−0.740110	−	0.672486i	$-0.765226\pi$
$911$		−	9.26375e11i		−	1.34497i	0.740110	−	0.672486i	$-0.234774\pi$
$912$		0			0
$913$	−7.18095e8	+	1.24378e9i	−0.00103347	+	0.00179003i
$914$		0			0
$915$	−1.18535e11	−	2.65113e10i	−0.169107	−	0.0378222i
$916$		0			0
$917$	−1.08869e12	−	4.10870e10i	−1.53967	−	0.0581068i
$918$		0			0
$919$	−5.33790e11	−	9.24552e11i	−0.748357	−	1.29619i	−0.948610	−	0.316447i	$-0.897510\pi$
$919$	−5.33790e11	−	9.24552e11i	−0.748357	−	1.29619i	0.200253	−	0.979744i	$-0.435823\pi$
$920$		0			0
$921$	1.00611e12	−	3.15156e11i	1.39832	−	0.438013i
$922$		0			0
$923$			7.79524e11i			1.07405i
$924$		0			0
$925$	−4.10830e11			−0.561172
$926$		0			0
$927$	−1.11840e12	−	5.26620e11i	−1.51452	−	0.713146i
$928$		0			0
$929$	−1.90991e11	+	1.10269e11i	−0.256420	+	0.148044i	−0.622700	−	0.782461i	$-0.713964\pi$
$929$	−1.90991e11	+	1.10269e11i	−0.256420	+	0.148044i	0.366281	+	0.930504i	$0.380631\pi$
$930$		0			0
$931$	−1.62504e11	+	2.38020e11i	−0.216304	+	0.316821i
$932$		0			0
$933$	−8.83762e11	−	1.97661e11i	−1.16630	−	0.260852i
$934$		0			0
$935$	−2.36747e10	−	1.36686e10i	−0.0309769	−	0.0178845i
$936$		0			0
$937$	−1.07968e12			−1.40068			−0.700339	−	0.713810i	$-0.746968\pi$
$937$	−1.07968e12			−1.40068			−0.700339	+	0.713810i	$0.746968\pi$
$938$		0			0
$939$	4.63918e11	+	4.26702e11i	0.596731	+	0.548861i
$940$		0			0
$941$	−2.28864e11	−	1.32134e11i	−0.291889	−	0.168522i	0.346904	−	0.937901i	$-0.387233\pi$
$941$	−2.28864e11	−	1.32134e11i	−0.291889	−	0.168522i	−0.638794	+	0.769378i	$0.720566\pi$
$942$		0			0
$943$	−1.87217e11	−	3.24269e11i	−0.236754	−	0.410071i
$944$		0			0
$945$	−1.21811e11	+	1.45256e11i	−0.152742	+	0.182141i
$946$		0			0
$947$	−1.06831e12	+	6.16789e11i	−1.32830	+	0.766897i	−0.985037	−	0.172341i	$-0.944867\pi$
$947$	−1.06831e12	+	6.16789e11i	−1.32830	+	0.766897i	−0.343267	+	0.939238i	$0.611534\pi$
$948$		0			0
$949$	3.06861e9	−	5.31499e9i	0.00378335	−	0.00655296i
$950$		0			0
$951$	8.24334e11	−	8.96231e11i	1.00782	−	1.09572i
$952$		0			0
$953$			1.30737e12i			1.58499i	0.609879	+	0.792495i	$0.291218\pi$
$953$			1.30737e12i			1.58499i	−0.609879	+	0.792495i	$0.708782\pi$
$954$		0			0
$955$	1.79335e10	−	3.10618e10i	0.0215602	−	0.0373433i
$956$		0			0
$957$	3.53406e9	−	1.58012e10i	0.00421333	−	0.0188383i
$958$		0			0
$959$	−6.15442e11	−	1.16538e12i	−0.727634	−	1.37782i
$960$		0			0
$961$	4.26415e11	+	7.38573e11i	0.499964	+	0.865964i
$962$		0			0
$963$	−8.70208e10	+	1.84808e11i	−0.101185	+	0.214890i
$964$		0			0
$965$			2.64514e10i			0.0305028i
$966$		0			0
$967$	6.86530e11			0.785152			0.392576	−	0.919720i	$-0.371584\pi$
$967$	6.86530e11			0.785152			0.392576	+	0.919720i	$0.371584\pi$
$968$		0			0
$969$	1.49862e11	+	4.78419e11i	0.169979	+	0.542642i
$970$		0			0
$971$	1.27451e12	−	7.35841e11i	1.43373	−	0.827765i	0.436327	−	0.899788i	$-0.356279\pi$
$971$	1.27451e12	−	7.35841e11i	1.43373	−	0.827765i	0.997403	+	0.0720235i	$0.0229457\pi$
$972$		0			0
$973$	−1.19521e12	−	7.51537e11i	−1.33350	−	0.838492i
$974$		0			0
$975$	1.24688e11	−	5.57492e11i	0.137976	−	0.616908i
$976$		0			0
$977$	1.63241e11	+	9.42470e10i	0.179164	+	0.103440i	0.586900	−	0.809660i	$-0.300348\pi$
$977$	1.63241e11	+	9.42470e10i	0.179164	+	0.103440i	−0.407736	+	0.913100i	$0.633682\pi$
$978$		0			0
$979$	−1.65808e11			−0.180499
$980$		0			0
$981$	1.20629e10	+	1.44087e11i	0.0130250	+	0.155579i
$982$		0			0
$983$	−7.58476e11	−	4.37906e11i	−0.812321	−	0.468994i	0.0354401	−	0.999372i	$-0.488717\pi$
$983$	−7.58476e11	−	4.37906e11i	−0.812321	−	0.468994i	−0.847761	+	0.530378i	$0.822050\pi$
$984$		0			0
$985$	7.53843e10	+	1.30569e11i	0.0800822	+	0.138706i
$986$		0			0
$987$	1.50173e12	−	4.08898e11i	1.58243	−	0.430870i
$988$		0			0
$989$	6.36698e11	−	3.67598e11i	0.665500	−	0.384227i
$990$		0			0
$991$	2.13103e11	−	3.69104e11i	0.220950	−	0.382697i	−0.734147	−	0.678991i	$-0.762418\pi$
$991$	2.13103e11	−	3.69104e11i	0.220950	−	0.382697i	0.955097	+	0.296294i	$0.0957509\pi$
$992$		0			0
$993$	8.45250e11	+	7.77443e11i	0.869337	+	0.799598i
$994$		0			0
$995$		−	9.25865e10i		−	0.0944616i
$996$		0			0
$997$	−1.09201e11	+	1.89142e11i	−0.110521	+	0.191429i	−0.915981	−	0.401223i	$-0.868585\pi$
$997$	−1.09201e11	+	1.89142e11i	−0.110521	+	0.191429i	0.805459	+	0.592651i	$0.201919\pi$
$998$		0			0
$999$	5.86893e11	+	8.06195e10i	0.589247	+	0.0809427i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	84.9.p.b.53.19	yes	40
3.2	odd	2	inner	84.9.p.b.53.8	✓	40
7.2	even	3	inner	84.9.p.b.65.8	yes	40
21.2	odd	6	inner	84.9.p.b.65.19	yes	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.9.p.b.53.8	✓	40	3.2	odd	2	inner
84.9.p.b.53.19	yes	40	1.1	even	1	trivial
84.9.p.b.65.8	yes	40	7.2	even	3	inner
84.9.p.b.65.19	yes	40	21.2	odd	6	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)

\(f(q)\)	\(=\)	\(q+(77.2965 - 24.2126i) q^{3} +(-128.664 + 74.2840i) q^{5} +(2032.57 + 1278.06i) q^{7} +(5388.50 - 3743.10i) q^{9} +O(q^{10})\)	\(q+(77.2965 - 24.2126i) q^{3} +(-128.664 + 74.2840i) q^{5} +(2032.57 + 1278.06i) q^{7} +(5388.50 - 3743.10i) q^{9} +(1287.13 + 743.127i) q^{11} -19136.1 q^{13} +(-8146.65 + 8857.18i) q^{15} +(107218. + 61902.1i) q^{17} +(24996.8 + 43295.7i) q^{19} +(188056. + 49576.0i) q^{21} +(163899. - 94626.9i) q^{23} +(-184276. + 319176. i) q^{25} +(325882. - 419798. i) q^{27} -134496. i q^{29} +(3894.88 - 6746.14i) q^{31} +(117484. + 26276.3i) q^{33} +(-356458. - 13452.7i) q^{35} +(557357. + 965370. i) q^{37} +(-1.47915e6 + 463335. i) q^{39} -1.97847e6i q^{41} +3.88471e6 q^{43} +(-415252. + 881880. i) q^{45} +(6.93068e6 - 4.00143e6i) q^{47} +(2.49791e6 + 5.19552e6i) q^{49} +(9.78635e6 + 2.18880e6i) q^{51} +(151016. + 87189.3i) q^{53} -220810. q^{55} +(2.98046e6 + 2.74137e6i) q^{57} +(1.40220e7 + 8.09562e6i) q^{59} +(5.04668e6 + 8.74110e6i) q^{61} +(1.57364e7 - 721278. i) q^{63} +(2.46212e6 - 1.42151e6i) q^{65} +(-8.47085e6 + 1.46719e7i) q^{67} +(1.03776e7 - 1.12827e7i) q^{69} -4.07358e7i q^{71} +(-160357. + 277747. i) q^{73} +(-6.51583e6 + 2.91330e7i) q^{75} +(1.66643e6 + 3.15550e6i) q^{77} +(3.86159e6 + 6.68846e6i) q^{79} +(1.50251e7 - 4.03394e7i) q^{81} +966315. i q^{83} -1.83933e7 q^{85} +(-3.25650e6 - 1.03961e7i) q^{87} +(-9.66146e7 + 5.57805e7i) q^{89} +(-3.88955e7 - 2.44572e7i) q^{91} +(137719. - 615758. i) q^{93} +(-6.43235e6 - 3.71372e6i) q^{95} -4.98046e7 q^{97} +(9.71732e6 - 813530. i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 81 q^{3} - 34 q^{7} + 4771 q^{9}+O(q^{10}) \) 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

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