LMFDB - Embedded newform 84.9.p.b.53.4

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			53.4
Character	$\chi$	$=$	84.53
Dual form			84.9.p.b.65.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-71.6341 + 37.8094i) q^{3} +(-304.163 + 175.609i) q^{5} +(517.537 + 2344.56i) q^{7} +(3701.90 - 5416.89i) q^{9} +O(q^{10})$	\(q+(-71.6341 + 37.8094i) q^{3} +(-304.163 + 175.609i) q^{5} +(517.537 + 2344.56i) q^{7} +(3701.90 - 5416.89i) q^{9} +(18965.6 + 10949.8i) q^{11} +5256.11 q^{13} +(15148.8 - 24079.8i) q^{15} +(5364.57 + 3097.23i) q^{17} +(51832.6 + 89776.8i) q^{19} +(-125720. - 148383. i) q^{21} +(237751. - 137265. i) q^{23} +(-133636. + 231464. i) q^{25} +(-60373.3 + 528001. i) q^{27} +272128. i q^{29} +(-53915.5 + 93384.4i) q^{31} +(-1.77259e6 - 67301.5i) q^{33} +(-569140. - 622244. i) q^{35} +(-1.46687e6 - 2.54070e6i) q^{37} +(-376517. + 198730. i) q^{39} +3.88684e6i q^{41} -3.82188e6 q^{43} +(-174729. + 2.29770e6i) q^{45} +(-3.87081e6 + 2.23481e6i) q^{47} +(-5.22911e6 + 2.42679e6i) q^{49} +(-501391. - 19036.7i) q^{51} +(8.81015e6 + 5.08654e6i) q^{53} -7.69153e6 q^{55} +(-7.10739e6 - 4.47132e6i) q^{57} +(-3.48596e6 - 2.01262e6i) q^{59} +(-4.96449e6 - 8.59875e6i) q^{61} +(1.46161e7 + 5.87589e6i) q^{63} +(-1.59871e6 + 923018. i) q^{65} +(-1.65440e6 + 2.86550e6i) q^{67} +(-1.18411e7 + 1.88221e7i) q^{69} +1.30051e6i q^{71} +(-1.72530e7 + 2.98831e7i) q^{73} +(821373. - 2.16334e7i) q^{75} +(-1.58571e7 + 5.01330e7i) q^{77} +(-2.99380e6 - 5.18541e6i) q^{79} +(-1.56386e7 - 4.01055e7i) q^{81} -6.41735e7i q^{83} -2.17560e6 q^{85} +(-1.02890e7 - 1.94936e7i) q^{87} +(-5.01926e7 + 2.89787e7i) q^{89} +(2.72023e6 + 1.23233e7i) q^{91} +(331384. - 8.72802e6i) q^{93} +(-3.15311e7 - 1.82045e7i) q^{95} +6.99710e7 q^{97} +(1.29523e8 - 6.21996e7i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−71.6341	+	37.8094i	−0.884372	+	0.466783i
$3$	−71.6341	+	37.8094i	−0.884372	+	0.466783i
$4$		0			0
$5$	−304.163	+	175.609i	−0.486661	+	0.280974i	−0.723188	−	0.690651i	$-0.757324\pi$
$5$	−304.163	+	175.609i	−0.486661	+	0.280974i	0.236527	+	0.971625i	$0.423991\pi$
$6$		0			0
$7$	517.537	+	2344.56i	0.215550	+	0.976493i
$7$	517.537	+	2344.56i	0.215550	+	0.976493i
$8$		0			0
$9$	3701.90	−	5416.89i	0.564228	−	0.825619i
$10$		0			0
$11$	18965.6	+	10949.8i	1.29538	+	0.747888i	0.979602	−	0.200946i	$-0.0644015\pi$
$11$	18965.6	+	10949.8i	1.29538	+	0.747888i	0.315777	+	0.948833i	$0.397735\pi$
$12$		0			0
$13$	5256.11			0.184031			0.0920156	−	0.995758i	$-0.470669\pi$
$13$	5256.11			0.184031			0.0920156	+	0.995758i	$0.470669\pi$
$14$		0			0
$15$	15148.8	−	24079.8i	0.299236	−	0.475650i
$16$		0			0
$17$	5364.57	+	3097.23i	0.0642302	+	0.0370833i	0.531771	−	0.846888i	$-0.321527\pi$
$17$	5364.57	+	3097.23i	0.0642302	+	0.0370833i	−0.467541	+	0.883971i	$0.654860\pi$
$18$		0			0
$19$	51832.6	+	89776.8i	0.397731	+	0.688889i	0.993446	−	0.114306i	$-0.0364646\pi$
$19$	51832.6	+	89776.8i	0.397731	+	0.688889i	−0.595715	+	0.803196i	$0.703131\pi$
$20$		0			0
$21$	−125720.	−	148383.i	−0.646437	−	0.762968i
$22$		0			0
$23$	237751.	−	137265.i	0.849592	−	0.490512i	−0.0109212	−	0.999940i	$-0.503476\pi$
$23$	237751.	−	137265.i	0.849592	−	0.490512i	0.860513	+	0.509428i	$0.170143\pi$
$24$		0			0
$25$	−133636.	+	231464.i	−0.342108	+	0.592548i
$26$		0			0
$27$	−60373.3	+	528001.i	−0.113603	+	0.993526i
$28$		0			0
$29$			272128.i			0.384752i	0.981321	+	0.192376i	$0.0616193\pi$
$29$			272128.i			0.384752i	−0.981321	+	0.192376i	$0.938381\pi$
$30$		0			0
$31$	−53915.5	+	93384.4i	−0.0583804	+	0.101118i	−0.893739	−	0.448588i	$-0.851927\pi$
$31$	−53915.5	+	93384.4i	−0.0583804	+	0.101118i	0.835358	+	0.549706i	$0.185260\pi$
$32$		0			0
$33$	−1.77259e6	−	67301.5i	−1.49470	−	0.0567504i
$34$		0			0
$35$	−569140.	−	622244.i	−0.379269	−	0.414657i
$36$		0			0
$37$	−1.46687e6	−	2.54070e6i	−0.782683	−	1.35565i	−0.930374	−	0.366613i	$-0.880517\pi$
$37$	−1.46687e6	−	2.54070e6i	−0.782683	−	1.35565i	0.147691	−	0.989034i	$-0.452816\pi$
$38$		0			0
$39$	−376517.	+	198730.i	−0.162752	+	0.0859025i
$40$		0			0
$41$			3.88684e6i			1.37550i	0.725946	+	0.687752i	$0.241402\pi$
$41$			3.88684e6i			1.37550i	−0.725946	+	0.687752i	$0.758598\pi$
$42$		0			0
$43$	−3.82188e6			−1.11790			−0.558950	−	0.829201i	$-0.688796\pi$
$43$	−3.82188e6			−1.11790			−0.558950	+	0.829201i	$0.688796\pi$
$44$		0			0
$45$	−174729.	+	2.29770e6i	−0.0426104	+	0.560330i
$46$		0			0
$47$	−3.87081e6	+	2.23481e6i	−0.793251	+	0.457984i	−0.841106	−	0.540870i	$-0.818095\pi$
$47$	−3.87081e6	+	2.23481e6i	−0.793251	+	0.457984i	0.0478546	+	0.998854i	$0.484762\pi$
$48$		0			0
$49$	−5.22911e6	+	2.42679e6i	−0.907076	+	0.420967i
$50$		0			0
$51$	−501391.	−	19036.7i	−0.0741132	−	0.00281392i
$52$		0			0
$53$	8.81015e6	+	5.08654e6i	1.11655	+	0.644643i	0.940519	−	0.339741i	$-0.110339\pi$
$53$	8.81015e6	+	5.08654e6i	1.11655	+	0.644643i	0.176035	+	0.984384i	$0.443673\pi$
$54$		0			0
$55$	−7.69153e6			−0.840547
$56$		0			0
$57$	−7.10739e6	−	4.47132e6i	−0.673303	−	0.423581i
$58$		0			0
$59$	−3.48596e6	−	2.01262e6i	−0.287683	−	0.166094i	0.349213	−	0.937043i	$-0.386449\pi$
$59$	−3.48596e6	−	2.01262e6i	−0.287683	−	0.166094i	−0.636896	+	0.770949i	$0.719782\pi$
$60$		0			0
$61$	−4.96449e6	−	8.59875e6i	−0.358555	−	0.621035i	0.629165	−	0.777272i	$-0.283397\pi$
$61$	−4.96449e6	−	8.59875e6i	−0.358555	−	0.621035i	−0.987720	+	0.156237i	$0.950064\pi$
$62$		0			0
$63$	1.46161e7	+	5.87589e6i	0.927831	+	0.373002i
$64$		0			0
$65$	−1.59871e6	+	923018.i	−0.0895607	+	0.0517079i
$66$		0			0
$67$	−1.65440e6	+	2.86550e6i	−0.0820995	+	0.142200i	−0.904152	−	0.427212i	$-0.859496\pi$
$67$	−1.65440e6	+	2.86550e6i	−0.0820995	+	0.142200i	0.822052	+	0.569412i	$0.192829\pi$
$68$		0			0
$69$	−1.18411e7	+	1.88221e7i	−0.522393	+	0.830370i
$70$		0			0
$71$			1.30051e6i			0.0511776i	0.999673	+	0.0255888i	$0.00814606\pi$
$71$			1.30051e6i			0.0511776i	−0.999673	+	0.0255888i	$0.991854\pi$
$72$		0			0
$73$	−1.72530e7	+	2.98831e7i	−0.607538	+	1.05229i	0.384107	+	0.923289i	$0.374509\pi$
$73$	−1.72530e7	+	2.98831e7i	−0.607538	+	1.05229i	−0.991645	+	0.128998i	$0.958824\pi$
$74$		0			0
$75$	821373.	−	2.16334e7i	0.0259595	−	0.683723i
$76$		0			0
$77$	−1.58571e7	+	5.01330e7i	−0.451087	+	1.42614i
$78$		0			0
$79$	−2.99380e6	−	5.18541e6i	−0.0768625	−	0.133130i	0.825032	−	0.565086i	$-0.191157\pi$
$79$	−2.99380e6	−	5.18541e6i	−0.0768625	−	0.133130i	−0.901895	+	0.431956i	$0.857824\pi$
$80$		0			0
$81$	−1.56386e7	−	4.01055e7i	−0.363293	−	0.931675i
$82$		0			0
$83$		−	6.41735e7i		−	1.35221i	−0.736806	−	0.676104i	$-0.763667\pi$
$83$		−	6.41735e7i		−	1.35221i	0.736806	−	0.676104i	$-0.236333\pi$
$84$		0			0
$85$	−2.17560e6			−0.0416777
$86$		0			0
$87$	−1.02890e7	−	1.94936e7i	−0.179596	−	0.340264i
$88$		0			0
$89$	−5.01926e7	+	2.89787e7i	−0.799982	+	0.461870i	−0.843465	−	0.537184i	$-0.819488\pi$
$89$	−5.01926e7	+	2.89787e7i	−0.799982	+	0.461870i	0.0434830	+	0.999054i	$0.486155\pi$
$90$		0			0
$91$	2.72023e6	+	1.23233e7i	0.0396680	+	0.179705i
$92$		0			0
$93$	331384.	−	8.72802e6i	0.00442996	−	0.116677i
$94$		0			0
$95$	−3.15311e7	−	1.82045e7i	−0.387120	−	0.223504i
$96$		0			0
$97$	6.99710e7			0.790371			0.395185	−	0.918601i	$-0.370680\pi$
$97$	6.99710e7			0.790371			0.395185	+	0.918601i	$0.370680\pi$
$98$		0			0
$99$	1.29523e8	−	6.21996e7i	1.34836	−	0.647511i
$100$		0			0
$101$	9.90333e7	+	5.71769e7i	0.951690	+	0.549459i	0.893606	−	0.448853i	$-0.148167\pi$
$101$	9.90333e7	+	5.71769e7i	0.951690	+	0.549459i	0.0580846	+	0.998312i	$0.481501\pi$
$102$		0			0
$103$	1.76954e7	+	3.06494e7i	0.157222	+	0.272316i	0.933866	−	0.357624i	$-0.116413\pi$
$103$	1.76954e7	+	3.06494e7i	0.157222	+	0.272316i	−0.776644	+	0.629940i	$0.783080\pi$
$104$		0			0
$105$	6.42965e7	+	2.30551e7i	0.528969	+	0.189675i
$106$		0			0
$107$	2.24998e8	−	1.29903e8i	1.71650	−	0.991023i	0.791405	−	0.611293i	$-0.209350\pi$
$107$	2.24998e8	−	1.29903e8i	1.71650	−	0.991023i	0.925097	−	0.379730i	$-0.123983\pi$
$108$		0			0
$109$	−1.69275e7	+	2.93193e7i	−0.119919	+	0.207705i	−0.919735	−	0.392539i	$-0.871597\pi$
$109$	−1.69275e7	+	2.93193e7i	−0.119919	+	0.207705i	0.799817	+	0.600244i	$0.204930\pi$
$110$		0			0
$111$	2.01141e8	+	1.26539e8i	1.32497	+	0.833553i
$112$		0			0
$113$		−	2.64579e8i		−	1.62271i	−0.584553	−	0.811355i	$-0.698730\pi$
$113$		−	2.64579e8i		−	1.62271i	0.584553	−	0.811355i	$-0.301270\pi$
$114$		0			0
$115$	−4.82100e7	+	8.35021e7i	−0.275642	+	0.477426i
$116$		0			0
$117$	1.94576e7	−	2.84718e7i	0.103836	−	0.151940i
$118$		0			0
$119$	−4.48529e6	+	1.41805e7i	−0.0223667	+	0.0707136i
$120$		0			0
$121$	1.32618e8	+	2.29701e8i	0.618672	+	1.07157i
$122$		0			0
$123$	−1.46959e8	−	2.78431e8i	−0.642061	−	1.21646i
$124$		0			0
$125$		−	2.31065e8i		−	0.946440i
$126$		0			0
$127$	2.30365e8			0.885526			0.442763	−	0.896639i	$-0.353998\pi$
$127$	2.30365e8			0.885526			0.442763	+	0.896639i	$0.353998\pi$
$128$		0			0
$129$	2.73777e8	−	1.44503e8i	0.988640	−	0.521816i
$130$		0			0
$131$	−4.91276e8	+	2.83638e8i	−1.66817	+	0.963119i	−0.699547	+	0.714586i	$0.746615\pi$
$131$	−4.91276e8	+	2.83638e8i	−1.66817	+	0.963119i	−0.968624	+	0.248533i	$0.920052\pi$
$132$		0			0
$133$	−1.83662e8	+	1.67987e8i	−0.586965	+	0.536871i
$134$		0			0
$135$	−7.43581e7	−	1.71200e8i	−0.223869	−	0.515430i
$136$		0			0
$137$	2.33556e8	+	1.34844e8i	0.662993	+	0.382779i	0.793416	−	0.608679i	$-0.208300\pi$
$137$	2.33556e8	+	1.34844e8i	0.662993	+	0.382779i	−0.130423	+	0.991458i	$0.541634\pi$
$138$		0			0
$139$	−3.84729e8			−1.03061			−0.515307	−	0.857006i	$-0.672322\pi$
$139$	−3.84729e8			−1.03061			−0.515307	+	0.857006i	$0.672322\pi$
$140$		0			0
$141$	1.92785e8	−	3.06442e8i	0.487750	−	0.775304i
$142$		0			0
$143$	9.96856e7	+	5.75535e7i	0.238390	+	0.137635i
$144$		0			0
$145$	−4.77880e7	−	8.27712e7i	−0.108105	−	0.187244i
$146$		0			0
$147$	2.82828e8	−	3.71551e8i	0.605693	−	0.795699i
$148$		0			0
$149$	−7.04874e8	+	4.06959e8i	−1.43010	+	0.825669i	−0.997128	−	0.0757387i	$-0.975868\pi$
$149$	−7.04874e8	+	4.06959e8i	−1.43010	+	0.825669i	−0.432972	+	0.901407i	$0.642535\pi$
$150$		0			0
$151$	−1.11416e8	+	1.92977e8i	−0.214308	+	0.371192i	−0.953058	−	0.302787i	$-0.902083\pi$
$151$	−1.11416e8	+	1.92977e8i	−0.214308	+	0.371192i	0.738750	+	0.673979i	$0.235416\pi$
$152$		0			0
$153$	3.66365e7	−	1.75936e7i	0.0668571	−	0.0321062i
$154$		0			0
$155$		−	3.78721e7i		−	0.0656134i
$156$		0			0
$157$	−5.91423e8	+	1.02437e9i	−0.973419	+	1.68601i	−0.288360	+	0.957522i	$0.593110\pi$
$157$	−5.91423e8	+	1.02437e9i	−0.973419	+	1.68601i	−0.685058	+	0.728488i	$0.740223\pi$
$158$		0			0
$159$	−8.23426e8	−	3.12637e7i	−1.28836	−	0.0489161i
$160$		0			0
$161$	4.44871e8	+	4.86381e8i	0.662111	+	0.723890i
$162$		0			0
$163$	−7.20435e7	−	1.24783e8i	−0.102057	−	0.176769i	0.810475	−	0.585774i	$-0.199209\pi$
$163$	−7.20435e7	−	1.24783e8i	−0.102057	−	0.176769i	−0.912532	+	0.409005i	$0.865876\pi$
$164$		0			0
$165$	5.50976e8	−	2.90812e8i	0.743356	−	0.392353i
$166$		0			0
$167$		−	1.62019e8i		−	0.208305i	−0.994561	−	0.104152i	$-0.966787\pi$
$167$		−	1.62019e8i		−	0.208305i	0.994561	−	0.104152i	$-0.0332130\pi$
$168$		0			0
$169$	−7.88104e8			−0.966133
$170$		0			0
$171$	6.78190e8	+	5.15731e7i	0.793171	+	0.0603169i
$172$		0			0
$173$	6.69805e8	−	3.86712e8i	0.747763	−	0.431721i	−0.0771221	−	0.997022i	$-0.524573\pi$
$173$	6.69805e8	−	3.86712e8i	0.747763	−	0.431721i	0.824885	+	0.565301i	$0.191240\pi$
$174$		0			0
$175$	−6.11842e8	−	1.93526e8i	−0.652360	−	0.206342i
$176$		0			0
$177$	3.25810e8	+	1.23703e7i	0.331949	+	0.0126034i
$178$		0			0
$179$	−6.62081e8	−	3.82253e8i	−0.644910	−	0.372339i	0.141593	−	0.989925i	$-0.454777\pi$
$179$	−6.62081e8	−	3.82253e8i	−0.644910	−	0.372339i	−0.786504	+	0.617586i	$0.788111\pi$
$180$		0			0
$181$	−1.58706e9			−1.47870			−0.739350	−	0.673321i	$-0.764867\pi$
$181$	−1.58706e9			−1.47870			−0.739350	+	0.673321i	$0.764867\pi$
$182$		0			0
$183$	6.80741e8	+	4.28260e8i	0.606984	+	0.381859i
$184$		0			0
$185$	8.92337e8	+	5.15191e8i	0.761802	+	0.439826i
$186$		0			0
$187$	6.78283e7	+	1.17482e8i	0.0554683	+	0.0960739i
$188$		0			0
$189$	−1.26917e9	+	1.31711e8i	−0.994658	+	0.103223i
$190$		0			0
$191$	−2.29634e8	+	1.32579e8i	−0.172545	+	0.0996192i	−0.583786	−	0.811908i	$-0.698429\pi$
$191$	−2.29634e8	+	1.32579e8i	−0.172545	+	0.0996192i	0.411240	+	0.911527i	$0.365096\pi$
$192$		0			0
$193$	−1.33759e9	+	2.31678e9i	−0.964040	+	1.66977i	−0.251868	+	0.967762i	$0.581045\pi$
$193$	−1.33759e9	+	2.31678e9i	−0.964040	+	1.66977i	−0.712172	+	0.702005i	$0.752288\pi$
$194$		0			0
$195$	7.96238e7	−	1.26566e8i	0.0550686	−	0.0875344i
$196$		0			0
$197$			1.68956e8i			0.112178i	0.998426	+	0.0560891i	$0.0178631\pi$
$197$			1.68956e8i			0.112178i	−0.998426	+	0.0560891i	$0.982137\pi$
$198$		0			0
$199$	1.00634e8	−	1.74303e8i	0.0641699	−	0.111145i	−0.832156	−	0.554542i	$-0.812893\pi$
$199$	1.00634e8	−	1.74303e8i	0.0641699	−	0.111145i	0.896325	+	0.443397i	$0.146227\pi$
$200$		0			0
$201$	1.01685e7	−	2.67819e8i	0.00622979	−	0.164081i
$202$		0			0
$203$	−6.38020e8	+	1.40836e8i	−0.375708	+	0.0829335i
$204$		0			0
$205$	−6.82563e8	−	1.18223e9i	−0.386480	−	0.669404i
$206$		0			0
$207$	1.36578e8	−	1.79601e9i	0.0743875	−	0.978200i
$208$		0			0
$209$			2.27023e9i			1.18983i
$210$		0			0
$211$	8.43945e8			0.425779			0.212889	−	0.977076i	$-0.431713\pi$
$211$	8.43945e8			0.425779			0.212889	+	0.977076i	$0.431713\pi$
$212$		0			0
$213$	−4.91715e7	−	9.31609e7i	−0.0238888	−	0.0452601i
$214$		0			0
$215$	1.16247e9	−	6.71155e8i	0.544038	−	0.314101i
$216$		0			0
$217$	−2.46849e8	−	7.80783e7i	−0.111325	−	0.0352120i
$218$		0			0
$219$	1.06043e8	−	2.79298e9i	0.0461006	−	1.21420i
$220$		0			0
$221$	2.81968e7	+	1.62794e7i	0.0118203	+	0.00682448i
$222$		0			0
$223$	2.64503e9			1.06958			0.534788	−	0.844986i	$-0.320391\pi$
$223$	2.64503e9			1.06958			0.534788	+	0.844986i	$0.320391\pi$
$224$		0			0
$225$	7.59108e8	+	1.58075e9i	0.296192	+	0.616783i
$226$		0			0
$227$	−2.24625e9	−	1.29687e9i	−0.845968	−	0.488420i	0.0133202	−	0.999911i	$-0.495760\pi$
$227$	−2.24625e9	−	1.29687e9i	−0.845968	−	0.488420i	−0.859288	+	0.511491i	$0.829093\pi$
$228$		0			0
$229$	1.63276e9	+	2.82802e9i	0.593717	+	1.02835i	0.993727	+	0.111837i	$0.0356736\pi$
$229$	1.63276e9	+	2.82802e9i	0.593717	+	1.02835i	−0.400009	+	0.916511i	$0.630993\pi$
$230$		0			0
$231$	−7.59590e8	−	4.19078e9i	−0.266766	−	1.47179i
$232$		0			0
$233$	−1.83584e9	+	1.05993e9i	−0.622891	+	0.359626i	−0.777994	−	0.628272i	$-0.783763\pi$
$233$	−1.83584e9	+	1.05993e9i	−0.622891	+	0.359626i	0.155103	+	0.987898i	$0.450429\pi$
$234$		0			0
$235$	7.84905e8	−	1.35950e9i	0.257363	−	0.445765i
$236$		0			0
$237$	4.10516e8	+	2.58259e8i	0.130118	+	0.0818581i
$238$		0			0
$239$		−	5.14130e9i		−	1.57573i	−0.615849	−	0.787864i	$-0.711187\pi$
$239$		−	5.14130e9i		−	1.57573i	0.615849	−	0.787864i	$-0.288813\pi$
$240$		0			0
$241$	−2.34527e9	+	4.06212e9i	−0.695223	+	1.20416i	0.274882	+	0.961478i	$0.411361\pi$
$241$	−2.34527e9	+	4.06212e9i	−0.695223	+	1.20416i	−0.970105	+	0.242684i	$0.921972\pi$
$242$		0			0
$243$	2.63662e9	+	2.28164e9i	0.756176	+	0.654368i
$244$		0			0
$245$	1.16434e9	−	1.65642e9i	0.323158	−	0.459732i
$246$		0			0
$247$	2.72438e8	+	4.71877e8i	0.0731948	+	0.126777i
$248$		0			0
$249$	2.42636e9	+	4.59701e9i	0.631187	+	1.19585i
$250$		0			0
$251$		−	1.01750e9i		−	0.256353i	−0.991751	−	0.128176i	$-0.959088\pi$
$251$		−	1.01750e9i		−	0.256353i	0.991751	−	0.128176i	$-0.0409124\pi$
$252$		0			0
$253$	6.01213e9			1.46739
$254$		0			0
$255$	1.55847e8	−	8.22582e7i	0.0368586	−	0.0194544i
$256$		0			0
$257$	6.47872e9	−	3.74049e9i	1.48510	−	0.857425i	0.485248	−	0.874377i	$-0.338729\pi$
$257$	6.47872e9	−	3.74049e9i	1.48510	−	0.857425i	0.999856	+	0.0169511i	$0.00539597\pi$
$258$		0			0
$259$	5.19766e9	−	4.75408e9i	1.15507	−	1.05649i
$260$		0			0
$261$	1.47409e9	+	1.00739e9i	0.317659	+	0.217088i
$262$		0			0
$263$	5.43347e9	+	3.13702e9i	1.13568	+	0.655683i	0.945356	−	0.326039i	$-0.105714\pi$
$263$	5.43347e9	+	3.13702e9i	1.13568	+	0.655683i	0.190320	+	0.981722i	$0.439048\pi$
$264$		0			0
$265$	−3.57296e9			−0.724511
$266$		0			0
$267$	2.49984e9	−	3.97362e9i	0.491889	−	0.781882i
$268$		0			0
$269$	−3.56865e9	−	2.06036e9i	−0.681546	−	0.393491i	0.118891	−	0.992907i	$-0.462066\pi$
$269$	−3.56865e9	−	2.06036e9i	−0.681546	−	0.393491i	−0.800437	+	0.599416i	$0.795399\pi$
$270$		0			0
$271$	1.46426e9	+	2.53617e9i	0.271482	+	0.470221i	0.969242	−	0.246111i	$-0.0791529\pi$
$271$	1.46426e9	+	2.53617e9i	0.271482	+	0.470221i	−0.697759	+	0.716332i	$0.745820\pi$
$272$		0			0
$273$	−6.60797e8	−	7.79916e8i	−0.118964	−	0.140410i
$274$		0			0
$275$	−5.06898e9	+	2.92658e9i	−0.886318	+	0.511716i
$276$		0			0
$277$	1.98345e9	−	3.43544e9i	0.336901	−	0.583531i	−0.646947	−	0.762535i	$-0.723954\pi$
$277$	1.98345e9	−	3.43544e9i	0.336901	−	0.583531i	0.983848	+	0.179005i	$0.0572877\pi$
$278$		0			0
$279$	3.06263e8	+	6.37754e8i	0.0505449	+	0.105253i
$280$		0			0
$281$			3.58332e9i			0.574725i	0.957822	+	0.287362i	$0.0927784\pi$
$281$			3.58332e9i			0.574725i	−0.957822	+	0.287362i	$0.907222\pi$
$282$		0			0
$283$	1.54115e9	−	2.66935e9i	0.240270	−	0.416160i	−0.720521	−	0.693433i	$-0.756097\pi$
$283$	1.54115e9	−	2.66935e9i	0.240270	−	0.416160i	0.960791	+	0.277273i	$0.0894307\pi$
$284$		0			0
$285$	2.94701e9	+	1.11891e8i	0.446685	+	0.0169597i
$286$		0			0
$287$	−9.11294e9	+	2.01158e9i	−1.34317	+	0.296490i
$288$		0			0
$289$	−3.46869e9	−	6.00795e9i	−0.497250	−	0.861262i
$290$		0			0
$291$	−5.01231e9	+	2.64556e9i	−0.698982	+	0.368931i
$292$		0			0
$293$		−	9.08460e8i		−	0.123264i	−0.998099	−	0.0616319i	$-0.980370\pi$
$293$		−	9.08460e8i		−	0.123264i	0.998099	−	0.0616319i	$-0.0196305\pi$
$294$		0			0
$295$	1.41373e9			0.186672
$296$		0			0
$297$	−6.92653e9	+	9.35280e9i	−0.890205	+	1.20203i
$298$		0			0
$299$	1.24964e9	−	7.21482e8i	0.156351	−	0.0902695i
$300$		0			0
$301$	−1.97796e9	−	8.96062e9i	−0.240964	−	1.09162i
$302$		0			0
$303$	−9.25599e9	−	3.51430e8i	−1.09813	−	0.0416934i
$304$		0			0
$305$	3.02003e9	+	1.74361e9i	0.348989	+	0.201489i
$306$		0			0
$307$	−3.97074e9			−0.447011			−0.223505	−	0.974703i	$-0.571750\pi$
$307$	−3.97074e9			−0.447011			−0.223505	+	0.974703i	$0.571750\pi$
$308$		0			0
$309$	−2.42643e9	−	1.52649e9i	−0.266155	−	0.167440i
$310$		0			0
$311$	9.43187e9	+	5.44550e9i	1.00822	+	0.582098i	0.910671	−	0.413133i	$-0.135565\pi$
$311$	9.43187e9	+	5.44550e9i	1.00822	+	0.582098i	0.0975519	+	0.995230i	$0.468899\pi$
$312$		0			0
$313$	1.16812e9	+	2.02324e9i	0.121706	+	0.210800i	0.920440	−	0.390883i	$-0.127830\pi$
$313$	1.16812e9	+	2.02324e9i	0.121706	+	0.210800i	−0.798735	+	0.601683i	$0.794497\pi$
$314$		0			0
$315$	−5.47753e9	+	7.79481e8i	−0.556342	+	0.0791705i
$316$		0			0
$317$	−4.60582e9	+	2.65917e9i	−0.456110	+	0.263335i	−0.710407	−	0.703791i	$-0.751489\pi$
$317$	−4.60582e9	+	2.65917e9i	−0.456110	+	0.263335i	0.254297	+	0.967126i	$0.418156\pi$
$318$		0			0
$319$	−2.97975e9	+	5.16108e9i	−0.287751	+	0.498400i
$320$		0			0
$321$	−1.12060e10	+	1.78125e10i	−1.05543	+	1.67767i
$322$		0			0
$323$			6.42151e8i			0.0589966i
$324$		0			0
$325$	−7.02405e8	+	1.21660e9i	−0.0629584	+	0.109047i
$326$		0			0
$327$	1.04042e8	−	2.74028e9i	0.00909954	−	0.239665i
$328$		0			0
$329$	−7.24294e9	−	7.91875e9i	−0.618204	−	0.675885i
$330$		0			0
$331$	−7.11009e9	−	1.23150e10i	−0.592329	−	1.02594i	−0.993918	−	0.110124i	$-0.964875\pi$
$331$	−7.11009e9	−	1.23150e10i	−0.592329	−	1.02594i	0.401589	−	0.915820i	$-0.368458\pi$
$332$		0			0
$333$	−1.91929e10	−	1.45953e9i	−1.56086	−	0.118696i
$334$		0			0
$335$		−	1.16210e9i		−	0.0922712i
$336$		0			0
$337$	2.20560e10			1.71004			0.855022	−	0.518592i	$-0.173544\pi$
$337$	2.20560e10			1.71004			0.855022	+	0.518592i	$0.173544\pi$
$338$		0			0
$339$	1.00036e10	+	1.89529e10i	0.757453	+	1.43508i
$340$		0			0
$341$	−2.04509e9	+	1.18073e9i	−0.151249	+	0.0873239i
$342$		0			0
$343$	−8.39601e9	−	1.10040e10i	−0.606592	−	0.795014i
$344$		0			0
$345$	2.96316e8	−	7.80439e9i	0.0209160	−	0.550887i
$346$		0			0
$347$	1.50837e9	+	8.70860e8i	0.104038	+	0.0600662i	0.551116	−	0.834429i	$-0.314202\pi$
$347$	1.50837e9	+	8.70860e8i	0.104038	+	0.0600662i	−0.447078	+	0.894495i	$0.647535\pi$
$348$		0			0
$349$	−1.17299e9			−0.0790664			−0.0395332	−	0.999218i	$-0.512587\pi$
$349$	−1.17299e9			−0.0790664			−0.0395332	+	0.999218i	$0.512587\pi$
$350$		0			0
$351$	−3.17329e8	+	2.77523e9i	−0.0209065	+	0.182840i
$352$		0			0
$353$	2.39916e10	+	1.38515e10i	1.54511	+	0.892070i	0.998504	+	0.0546809i	$0.0174141\pi$
$353$	2.39916e10	+	1.38515e10i	1.54511	+	0.892070i	0.546607	+	0.837389i	$0.315919\pi$
$354$		0			0
$355$	−2.28381e8	−	3.95567e8i	−0.0143796	−	0.0249061i
$356$		0			0
$357$	−2.14855e8	−	1.18539e9i	−0.0132274	−	0.0729775i
$358$		0			0
$359$	7.21825e9	−	4.16746e9i	0.434565	−	0.250896i	−0.266725	−	0.963773i	$-0.585941\pi$
$359$	7.21825e9	−	4.16746e9i	0.434565	−	0.250896i	0.701289	+	0.712877i	$0.252608\pi$
$360$		0			0
$361$	3.11854e9	−	5.40146e9i	0.183621	−	0.318041i
$362$		0			0
$363$	−1.81848e10	−	1.14402e10i	−1.04733	−	0.658882i
$364$		0			0
$365$		−	1.21191e10i		−	0.682809i
$366$		0			0
$367$	−1.00980e10	+	1.74902e10i	−0.556633	+	0.964117i	0.441141	+	0.897438i	$0.354574\pi$
$367$	−1.00980e10	+	1.74902e10i	−0.556633	+	0.964117i	−0.997774	+	0.0666792i	$0.978760\pi$
$368$		0			0
$369$	2.10546e10	+	1.43887e10i	1.13564	+	0.776098i
$370$		0			0
$371$	−7.36612e9	+	2.32884e10i	−0.388815	+	1.22926i
$372$		0			0
$373$	−1.01297e10	−	1.75452e10i	−0.523313	−	0.906404i	−0.999632	−	0.0271316i	$-0.991363\pi$
$373$	−1.01297e10	−	1.75452e10i	−0.523313	−	0.906404i	0.476319	−	0.879272i	$-0.341971\pi$
$374$		0			0
$375$	8.73641e9	+	1.65521e10i	0.441782	+	0.837005i
$376$		0			0
$377$			1.43033e9i			0.0708064i
$378$		0			0
$379$	3.63264e10			1.76062			0.880310	−	0.474399i	$-0.157335\pi$
$379$	3.63264e10			1.76062			0.880310	+	0.474399i	$0.157335\pi$
$380$		0			0
$381$	−1.65020e10	+	8.70995e9i	−0.783134	+	0.413348i
$382$		0			0
$383$	−2.06374e10	+	1.19150e10i	−0.959092	+	0.553732i	−0.895893	−	0.444269i	$-0.853463\pi$
$383$	−2.06374e10	+	1.19150e10i	−0.959092	+	0.553732i	−0.0631983	+	0.998001i	$0.520130\pi$
$384$		0			0
$385$	−3.98065e9	−	1.80332e10i	−0.181180	−	0.820788i
$386$		0			0
$387$	−1.41482e10	+	2.07027e10i	−0.630751	+	0.922960i
$388$		0			0
$389$	−1.58409e10	−	9.14576e9i	−0.691803	−	0.399412i	0.112484	−	0.993653i	$-0.464119\pi$
$389$	−1.58409e10	−	9.14576e9i	−0.691803	−	0.399412i	−0.804287	+	0.594241i	$0.797453\pi$
$390$		0			0
$391$	1.70057e9			0.0727592
$392$		0			0
$393$	2.44680e10	−	3.88930e10i	1.02572	−	1.63043i
$394$		0			0
$395$	1.82121e9	+	1.05147e9i	0.0748119	+	0.0431927i
$396$		0			0
$397$	−6.71463e9	−	1.16301e10i	−0.270309	−	0.468188i	0.698632	−	0.715481i	$-0.253792\pi$
$397$	−6.71463e9	−	1.16301e10i	−0.270309	−	0.468188i	−0.968941	+	0.247293i	$0.920459\pi$
$398$		0			0
$399$	6.80494e9	−	1.89778e10i	0.268493	−	0.748779i
$400$		0			0
$401$	−2.46355e10	+	1.42233e10i	−0.952761	+	0.550077i	−0.893937	−	0.448192i	$-0.852068\pi$
$401$	−2.46355e10	+	1.42233e10i	−0.952761	+	0.550077i	−0.0588233	+	0.998268i	$0.518735\pi$
$402$		0			0
$403$	−2.83386e8	+	4.90839e8i	−0.0107438	+	0.0186088i
$404$		0			0
$405$	1.17996e10	+	9.45235e9i	0.438577	+	0.351334i
$406$		0			0
$407$		−	6.42480e10i		−	2.34143i
$408$		0			0
$409$	5.37779e9	−	9.31460e9i	0.192181	−	0.332867i	−0.753792	−	0.657113i	$-0.771777\pi$
$409$	5.37779e9	−	9.31460e9i	0.192181	−	0.332867i	0.945973	+	0.324246i	$0.105111\pi$
$410$		0			0
$411$	−2.18289e10	−	8.28797e8i	−0.765007	−	0.0290456i
$412$		0			0
$413$	2.91459e9	−	9.21464e9i	0.100179	−	0.316722i
$414$		0			0
$415$	1.12694e10	+	1.95192e10i	0.379935	+	0.658066i
$416$		0			0
$417$	2.75597e10	−	1.45464e10i	0.911446	−	0.481072i
$418$		0			0
$419$		−	6.21950e9i		−	0.201790i	−0.994897	−	0.100895i	$-0.967829\pi$
$419$		−	6.21950e9i		−	0.201790i	0.994897	−	0.100895i	$-0.0321706\pi$
$420$		0			0
$421$	4.21071e10			1.34038			0.670189	−	0.742191i	$-0.266213\pi$
$421$	4.21071e10			1.34038			0.670189	+	0.742191i	$0.266213\pi$
$422$		0			0
$423$	−2.22363e9	+	2.92408e10i	−0.0694545	+	0.913331i
$424$		0			0
$425$	−1.43380e9	+	8.27803e8i	−0.0439472	+	0.0253730i
$426$		0			0
$427$	1.75910e10	−	1.60897e10i	0.529150	−	0.483991i
$428$		0			0
$429$	−9.31695e9	−	3.53744e8i	−0.275071	−	0.0104438i
$430$		0			0
$431$	2.76898e10	+	1.59867e10i	0.802437	+	0.463287i	0.844323	−	0.535835i	$-0.180003\pi$
$431$	2.76898e10	+	1.59867e10i	0.802437	+	0.463287i	−0.0418853	+	0.999122i	$0.513336\pi$
$432$		0			0
$433$	−8.52497e9			−0.242517			−0.121258	−	0.992621i	$-0.538693\pi$
$433$	−8.52497e9			−0.242517			−0.121258	+	0.992621i	$0.538693\pi$
$434$		0			0
$435$	6.55278e9	+	4.12241e9i	0.183007	+	0.115131i
$436$		0			0
$437$	2.46465e10	+	1.42297e10i	0.675817	+	0.390183i
$438$		0			0
$439$	3.60666e10	+	6.24691e10i	0.971062	+	1.68193i	0.692362	+	0.721550i	$0.256570\pi$
$439$	3.60666e10	+	6.24691e10i	0.971062	+	1.68193i	0.278699	+	0.960378i	$0.410097\pi$
$440$		0			0
$441$	−6.21201e9	+	3.73092e10i	−0.164240	+	0.986420i
$442$		0			0
$443$	3.78514e10	−	2.18535e10i	0.982803	−	0.567422i	0.0796879	−	0.996820i	$-0.474608\pi$
$443$	3.78514e10	−	2.18535e10i	0.982803	−	0.567422i	0.903115	+	0.429398i	$0.141274\pi$
$444$		0			0
$445$	1.01778e10	−	1.76285e10i	0.259546	−	0.449548i
$446$		0			0
$447$	3.51062e10	−	5.58030e10i	0.879333	−	1.39774i
$448$		0			0
$449$		−	5.55024e10i		−	1.36561i	−0.730601	−	0.682805i	$-0.760760\pi$
$449$		−	5.55024e10i		−	1.36561i	0.730601	−	0.682805i	$-0.239240\pi$
$450$		0			0
$451$	−4.25603e10	+	7.37165e10i	−1.02872	+	1.78180i
$452$		0			0
$453$	6.84800e8	−	1.80363e10i	0.0162619	−	0.428307i
$454$		0			0
$455$	−2.99146e9	−	3.27059e9i	−0.0697972	−	0.0763097i
$456$		0			0
$457$	−1.02780e10	−	1.78020e10i	−0.235636	−	0.408134i	0.723821	−	0.689988i	$-0.242384\pi$
$457$	−1.02780e10	−	1.78020e10i	−0.235636	−	0.408134i	−0.959457	+	0.281854i	$0.909051\pi$
$458$		0			0
$459$	−1.95922e9	+	2.64550e9i	−0.0441400	+	0.0596016i
$460$		0			0
$461$			6.02694e10i			1.33442i	0.744869	+	0.667211i	$0.232512\pi$
$461$			6.02694e10i			1.33442i	−0.744869	+	0.667211i	$0.767488\pi$
$462$		0			0
$463$	4.14183e10			0.901299			0.450649	−	0.892701i	$-0.351193\pi$
$463$	4.14183e10			0.901299			0.450649	+	0.892701i	$0.351193\pi$
$464$		0			0
$465$	1.43192e9	+	2.71294e9i	0.0306272	+	0.0580267i
$466$		0			0
$467$	4.43515e10	−	2.56064e10i	0.932484	−	0.538370i	0.0448874	−	0.998992i	$-0.485707\pi$
$467$	4.43515e10	−	2.56064e10i	0.932484	−	0.538370i	0.887596	+	0.460622i	$0.152374\pi$
$468$		0			0
$469$	−7.57454e9	−	2.39583e9i	−0.156554	−	0.0495182i
$470$		0			0
$471$	3.63510e9	−	9.57415e10i	0.0738639	−	1.94544i
$472$		0			0
$473$	−7.24844e10	−	4.18489e10i	−1.44810	−	0.836064i
$474$		0			0
$475$	−2.77068e10			−0.544266
$476$		0			0
$477$	6.01675e10	−	2.88937e10i	1.16222	−	0.558123i
$478$		0			0
$479$	9.07271e9	+	5.23813e9i	0.172344	+	0.0995026i	0.583690	−	0.811976i	$-0.301608\pi$
$479$	9.07271e9	+	5.23813e9i	0.172344	+	0.0995026i	−0.411347	+	0.911479i	$0.634942\pi$
$480$		0			0
$481$	−7.71005e9	−	1.33542e10i	−0.144038	−	0.249481i
$482$		0			0
$483$	−5.02577e10	−	1.80211e10i	−0.923452	−	0.331126i
$484$		0			0
$485$	−2.12826e10	+	1.22875e10i	−0.384642	+	0.222073i
$486$		0			0
$487$	−4.86999e10	+	8.43507e10i	−0.865789	+	1.49959i	0.000472652		1.00000i	$0.499850\pi$
$487$	−4.86999e10	+	8.43507e10i	−0.865789	+	1.49959i	−0.866262	+	0.499591i	$0.833484\pi$
$488$		0			0
$489$	9.87875e9	+	6.21480e9i	0.172769	+	0.108691i
$490$		0			0
$491$			7.00189e10i			1.20473i	0.798221	+	0.602364i	$0.205775\pi$
$491$			7.00189e10i			1.20473i	−0.798221	+	0.602364i	$0.794225\pi$
$492$		0			0
$493$	−8.42844e8	+	1.45985e9i	−0.0142679	+	0.0247127i
$494$		0			0
$495$	−2.84733e10	+	4.16641e10i	−0.474260	+	0.693971i
$496$		0			0
$497$	−3.04912e9	+	6.73061e8i	−0.0499746	+	0.0110314i
$498$		0			0
$499$	−9.68021e9	−	1.67666e10i	−0.156129	−	0.270423i	0.777341	−	0.629080i	$-0.216568\pi$
$499$	−9.68021e9	−	1.67666e10i	−0.156129	−	0.270423i	−0.933469	+	0.358657i	$0.883235\pi$
$500$		0			0
$501$	6.12583e9	+	1.16061e10i	0.0972330	+	0.184219i
$502$		0			0
$503$			8.67260e10i			1.35481i	0.735612	+	0.677404i	$0.236895\pi$
$503$			8.67260e10i			1.35481i	−0.735612	+	0.677404i	$0.763105\pi$
$504$		0			0
$505$	−4.01630e10			−0.617534
$506$		0			0
$507$	5.64552e10	−	2.97977e10i	0.854421	−	0.450974i
$508$		0			0
$509$	2.99763e10	−	1.73068e10i	0.446588	−	0.257838i	−0.259800	−	0.965662i	$-0.583657\pi$
$509$	2.99763e10	−	1.73068e10i	0.446588	−	0.257838i	0.706388	+	0.707825i	$0.250323\pi$
$510$		0			0
$511$	−7.89917e10	−	2.49851e10i	−1.15851	−	0.366436i
$512$		0			0
$513$	−5.05315e10	+	2.19475e10i	−0.729613	+	0.316896i
$514$		0			0
$515$	−1.07646e10	−	6.21493e9i	−0.153027	−	0.0883502i
$516$		0			0
$517$	−9.78833e10			−1.37008
$518$		0			0
$519$	−3.33596e10	+	5.30267e10i	−0.459781	+	0.730845i
$520$		0			0
$521$	−2.05021e10	−	1.18369e10i	−0.278258	−	0.160652i	0.354377	−	0.935103i	$-0.384693\pi$
$521$	−2.05021e10	−	1.18369e10i	−0.278258	−	0.160652i	−0.632634	+	0.774451i	$0.718026\pi$
$522$		0			0
$523$	−1.66720e10	−	2.88768e10i	−0.222834	−	0.385960i	0.732833	−	0.680408i	$-0.238197\pi$
$523$	−1.66720e10	−	2.88768e10i	−0.222834	−	0.385960i	−0.955667	+	0.294448i	$0.904864\pi$
$524$		0			0
$525$	5.11459e10	−	9.27032e9i	0.673246	−	0.122027i
$526$		0			0
$527$	−5.78467e8	+	3.33978e8i	−0.00749956	+	0.00432987i
$528$		0			0
$529$	−1.47191e9	+	2.54943e9i	−0.0187957	+	0.0325552i
$530$		0			0
$531$	−2.38068e10	+	1.14325e10i	−0.299449	+	0.143802i
$532$		0			0
$533$			2.04297e10i			0.253135i
$534$		0			0
$535$	−4.56241e10	+	7.90233e10i	−0.556903	+	0.964584i
$536$		0			0
$537$	6.18804e10	+	2.34946e9i	0.744142	+	0.0282534i
$538$		0			0
$539$	−1.25746e11	−	1.12322e10i	−1.48984	−	0.133079i
$540$		0			0
$541$	−2.69141e10	−	4.66165e10i	−0.314188	−	0.544190i	0.665076	−	0.746775i	$-0.268399\pi$
$541$	−2.69141e10	−	4.66165e10i	−0.314188	−	0.544190i	−0.979265	+	0.202585i	$0.935066\pi$
$542$		0			0
$543$	1.13688e11	−	6.00059e10i	1.30772	−	0.690231i
$544$		0			0
$545$		−	1.18905e10i		−	0.134776i
$546$		0			0
$547$	−9.02277e10			−1.00784			−0.503919	−	0.863751i	$-0.668109\pi$
$547$	−9.02277e10			−1.00784			−0.503919	+	0.863751i	$0.668109\pi$
$548$		0			0
$549$	−6.49565e10	−	4.93964e9i	−0.715045	−	0.0543758i
$550$		0			0
$551$	−2.44308e10	+	1.41051e10i	−0.265052	+	0.153028i
$552$		0			0
$553$	1.06081e10	−	9.70278e9i	0.113432	−	0.103752i
$554$		0			0
$555$	−8.34009e10	−	3.16655e9i	−0.879020	−	0.0333745i
$556$		0			0
$557$	5.83460e10	+	3.36861e10i	0.606165	+	0.349969i	0.771463	−	0.636274i	$-0.219525\pi$
$557$	5.83460e10	+	3.36861e10i	0.606165	+	0.349969i	−0.165298	+	0.986244i	$0.552859\pi$
$558$		0			0
$559$	−2.00882e10			−0.205728
$560$		0			0
$561$	−9.30075e9	−	5.85118e9i	−0.0939002	−	0.0590734i
$562$		0			0
$563$	3.24191e10	+	1.87172e10i	0.322676	+	0.186297i	0.652585	−	0.757716i	$-0.273685\pi$
$563$	3.24191e10	+	1.87172e10i	0.322676	+	0.186297i	−0.329909	+	0.944013i	$0.607018\pi$
$564$		0			0
$565$	4.64623e10	+	8.04750e10i	0.455939	+	0.789710i
$566$		0			0
$567$	8.59363e10	−	5.74217e10i	0.831466	−	0.555576i
$568$		0			0
$569$	−8.42286e10	+	4.86294e10i	−0.803546	+	0.463927i	−0.844710	−	0.535225i	$-0.820227\pi$
$569$	−8.42286e10	+	4.86294e10i	−0.803546	+	0.463927i	0.0411637	+	0.999152i	$0.486893\pi$
$570$		0			0
$571$	−6.88613e10	+	1.19271e11i	−0.647785	+	1.12200i	0.335865	+	0.941910i	$0.390971\pi$
$571$	−6.88613e10	+	1.19271e11i	−0.647785	+	1.12200i	−0.983651	+	0.180087i	$0.942362\pi$
$572$		0			0
$573$	1.14369e10	−	1.81796e10i	0.106094	−	0.168642i
$574$		0			0
$575$			7.33743e10i			0.671232i
$576$		0			0
$577$	4.68159e10	−	8.10876e10i	0.422367	−	0.731562i	−0.573803	−	0.818993i	$-0.694533\pi$
$577$	4.68159e10	−	8.10876e10i	0.422367	−	0.731562i	0.996171	+	0.0874315i	$0.0278659\pi$
$578$		0			0
$579$	8.22133e9	−	2.16534e11i	0.0731523	−	1.92669i
$580$		0			0
$581$	1.50459e11	−	3.32121e10i	1.32042	−	0.291469i
$582$		0			0
$583$	1.11393e11	+	1.92939e11i	0.964241	+	1.67011i
$584$		0			0
$585$	−9.18397e8	+	1.20770e10i	−0.00784164	+	0.103118i
$586$		0			0
$587$		−	1.43541e11i		−	1.20899i	−0.796609	−	0.604495i	$-0.793375\pi$
$587$		−	1.43541e11i		−	1.20899i	0.796609	−	0.604495i	$-0.206625\pi$
$588$		0			0
$589$	−1.11783e10			−0.0928786
$590$		0			0
$591$	−6.38812e9	−	1.21030e10i	−0.0523628	−	0.0992072i
$592$		0			0
$593$	6.33451e10	−	3.65723e10i	0.512264	−	0.295756i	−0.221500	−	0.975160i	$-0.571095\pi$
$593$	6.33451e10	−	3.65723e10i	0.512264	−	0.295756i	0.733764	+	0.679404i	$0.237762\pi$
$594$		0			0
$595$	−1.12595e9	−	5.10083e9i	−0.00898365	−	0.0406980i
$596$		0			0
$597$	−6.18531e8	+	1.62909e10i	−0.00486927	+	0.128247i
$598$		0			0
$599$	1.57500e11	+	9.09327e10i	1.22341	+	0.706338i	0.965644	−	0.259868i	$-0.0836792\pi$
$599$	1.57500e11	+	9.09327e10i	1.22341	+	0.706338i	0.257769	+	0.966206i	$0.417013\pi$
$600$		0			0
$601$	−5.85889e10			−0.449073			−0.224537	−	0.974466i	$-0.572087\pi$
$601$	−5.85889e10			−0.449073			−0.224537	+	0.974466i	$0.572087\pi$
$602$		0			0
$603$	9.39767e9	+	1.95695e10i	0.0710806	+	0.148016i
$604$		0			0
$605$	−8.06748e10	−	4.65776e10i	−0.602166	−	0.347661i
$606$		0			0
$607$	7.51158e10	+	1.30104e11i	0.553320	+	0.958378i	0.998032	+	0.0627046i	$0.0199726\pi$
$607$	7.51158e10	+	1.30104e11i	0.553320	+	0.958378i	−0.444712	+	0.895673i	$0.646694\pi$
$608$		0			0
$609$	4.03791e10	−	3.42118e10i	0.293553	−	0.248718i
$610$		0			0
$611$	−2.03454e10	+	1.17464e10i	−0.145983	+	0.0842833i
$612$		0			0
$613$	2.93318e9	−	5.08041e9i	0.0207729	−	0.0359797i	−0.855452	−	0.517882i	$-0.826721\pi$
$613$	2.93318e9	−	5.08041e9i	0.0207729	−	0.0359797i	0.876225	+	0.481902i	$0.160054\pi$
$614$		0			0
$615$	9.35944e10	+	5.88810e10i	0.654258	+	0.411600i
$616$		0			0
$617$		−	2.93100e10i		−	0.202244i	−0.994874	−	0.101122i	$-0.967757\pi$
$617$		−	2.93100e10i		−	0.202244i	0.994874	−	0.101122i	$-0.0322432\pi$
$618$		0			0
$619$	1.26735e11	−	2.19512e11i	0.863247	−	1.49519i	−0.00553014	−	0.999985i	$-0.501760\pi$
$619$	1.26735e11	−	2.19512e11i	0.863247	−	1.49519i	0.868777	−	0.495203i	$-0.164906\pi$
$620$		0			0
$621$	5.81224e10	+	1.33820e11i	0.390820	+	0.899815i
$622$		0			0
$623$	−9.39189e10	−	1.02682e11i	−0.623449	−	0.681620i
$624$		0			0
$625$	−1.16246e10	−	2.01343e10i	−0.0761828	−	0.131952i
$626$		0			0
$627$	−8.58361e10	−	1.62626e11i	−0.555392	−	1.05225i
$628$		0			0
$629$		−	1.81730e10i		−	0.116098i
$630$		0			0
$631$	8.13656e10			0.513244			0.256622	−	0.966512i	$-0.417391\pi$
$631$	8.13656e10			0.513244			0.256622	+	0.966512i	$0.417391\pi$
$632$		0			0
$633$	−6.04553e10	+	3.19090e10i	−0.376547	+	0.198746i
$634$		0			0
$635$	−7.00684e10	+	4.04540e10i	−0.430951	+	0.248809i
$636$		0			0
$637$	−2.74848e10	+	1.27555e10i	−0.166930	+	0.0774710i
$638$		0			0
$639$	7.04471e9	+	4.81436e9i	0.0422532	+	0.0288759i
$640$		0			0
$641$	−3.24373e10	−	1.87277e10i	−0.192137	−	0.110931i	0.400845	−	0.916146i	$-0.368716\pi$
$641$	−3.24373e10	−	1.87277e10i	−0.192137	−	0.110931i	−0.592983	+	0.805215i	$0.702050\pi$
$642$		0			0
$643$	2.00046e11			1.17027			0.585134	−	0.810937i	$-0.301042\pi$
$643$	2.00046e11			1.17027			0.585134	+	0.810937i	$0.301042\pi$
$644$		0			0
$645$	−5.78969e10	+	9.20300e10i	−0.334516	+	0.531729i
$646$		0			0
$647$	−7.77819e10	−	4.49074e10i	−0.443876	−	0.256272i	0.261365	−	0.965240i	$-0.415828\pi$
$647$	−7.77819e10	−	4.49074e10i	−0.443876	−	0.256272i	−0.705240	+	0.708968i	$0.749161\pi$
$648$		0			0
$649$	−4.40756e10	−	7.63413e10i	−0.248439	−	0.430309i
$650$		0			0
$651$	2.06349e10	−	3.74012e9i	0.114889	−	0.0208239i
$652$		0			0
$653$	2.01888e11	−	1.16560e11i	1.11034	−	0.641057i	0.171425	−	0.985197i	$-0.445163\pi$
$653$	2.01888e11	−	1.16560e11i	1.11034	−	0.641057i	0.938918	+	0.344140i	$0.111830\pi$
$654$		0			0
$655$	9.96187e10	−	1.72545e11i	0.541222	−	0.937424i
$656$		0			0
$657$	9.80044e10	+	2.04082e11i	0.525998	+	1.09532i
$658$		0			0
$659$			9.81940e9i			0.0520647i	0.999661	+	0.0260323i	$0.00828729\pi$
$659$			9.81940e9i			0.0520647i	−0.999661	+	0.0260323i	$0.991713\pi$
$660$		0			0
$661$	6.56638e10	−	1.13733e11i	0.343970	−	0.595773i	−0.641196	−	0.767377i	$-0.721562\pi$
$661$	6.56638e10	−	1.13733e11i	0.343970	−	0.595773i	0.985166	+	0.171604i	$0.0548949\pi$
$662$		0			0
$663$	−2.63537e9	−	1.00059e8i	−0.0136391	−	0.000517848i
$664$		0			0
$665$	2.63630e10	−	8.33481e10i	0.134806	−	0.426196i
$666$		0			0
$667$	3.73537e10	+	6.46986e10i	0.188726	+	0.326882i
$668$		0			0
$669$	−1.89475e11	+	1.00007e11i	−0.945904	+	0.499260i
$670$		0			0
$671$		−	2.17441e11i		−	1.07263i
$672$		0			0
$673$	4.64241e10			0.226299			0.113150	−	0.993578i	$-0.463906\pi$
$673$	4.64241e10			0.226299			0.113150	+	0.993578i	$0.463906\pi$
$674$		0			0
$675$	−1.14145e11	−	8.45340e10i	−0.549847	−	0.407208i
$676$		0			0
$677$	−9.80170e10	+	5.65901e10i	−0.466602	+	0.269393i	−0.714816	−	0.699312i	$-0.753490\pi$
$677$	−9.80170e10	+	5.65901e10i	−0.466602	+	0.269393i	0.248214	+	0.968705i	$0.420156\pi$
$678$		0			0
$679$	3.62125e10	+	1.64051e11i	0.170365	+	0.771791i
$680$		0			0
$681$	2.09942e11	+	7.97103e9i	0.976137	+	0.0370618i
$682$		0			0
$683$	−1.14701e11	−	6.62226e10i	−0.527089	−	0.304315i	0.212741	−	0.977109i	$-0.431761\pi$
$683$	−1.14701e11	−	6.62226e10i	−0.527089	−	0.304315i	−0.739830	+	0.672793i	$0.765094\pi$
$684$		0			0
$685$	−9.47188e10			−0.430203
$686$		0			0
$687$	−2.23887e11	−	1.40849e11i	−1.00508	−	0.632306i
$688$		0			0
$689$	4.63071e10	+	2.67354e10i	0.205481	+	0.118634i
$690$		0			0
$691$	−7.21691e10	−	1.25001e11i	−0.316548	−	0.548277i	0.663218	−	0.748427i	$-0.269190\pi$
$691$	−7.21691e10	−	1.25001e11i	−0.316548	−	0.548277i	−0.979765	+	0.200150i	$0.935857\pi$
$692$		0			0
$693$	2.12863e11	+	2.71483e11i	0.922929	+	1.17709i
$694$		0			0
$695$	1.17020e11	−	6.75617e10i	0.501559	−	0.289575i
$696$		0			0
$697$	−1.20385e10	+	2.08512e10i	−0.0510082	+	0.0883488i
$698$		0			0
$699$	9.14340e10	−	1.45339e11i	0.383000	−	0.608798i
$700$		0			0
$701$			4.52990e11i			1.87593i	0.346728	+	0.937966i	$0.387293\pi$
$701$			4.52990e11i			1.87593i	−0.346728	+	0.937966i	$0.612707\pi$
$702$		0			0
$703$	1.52064e11	−	2.63382e11i	0.622594	−	1.07836i
$704$		0			0
$705$	−4.82431e9	+	1.27063e11i	−0.0195289	+	0.514355i
$706$		0			0
$707$	−8.28012e10	+	2.61780e11i	−0.331405	+	1.04775i
$708$		0			0
$709$	1.02894e11	+	1.78218e11i	0.407198	+	0.705288i	0.994575	−	0.104026i	$-0.0331724\pi$
$709$	1.02894e11	+	1.78218e11i	0.407198	+	0.705288i	−0.587376	+	0.809314i	$0.699839\pi$
$710$		0			0
$711$	−3.91715e10	−	2.97881e9i	−0.153282	−	0.0116564i
$712$		0			0
$713$			2.96029e10i			0.114545i
$714$		0			0
$715$	−4.04275e10			−0.154687
$716$		0			0
$717$	1.94389e11	+	3.68292e11i	0.735522	+	1.39353i
$718$		0			0
$719$	2.56726e11	−	1.48221e11i	0.960626	−	0.554618i	0.0642602	−	0.997933i	$-0.479531\pi$
$719$	2.56726e11	−	1.48221e11i	0.960626	−	0.554618i	0.896366	+	0.443316i	$0.146198\pi$
$720$		0			0
$721$	−6.27012e10	+	5.73501e10i	−0.232025	+	0.212223i
$722$		0			0
$723$	1.44149e10	−	3.79660e11i	0.0527542	−	1.38944i
$724$		0			0
$725$	−6.29878e10	−	3.63660e10i	−0.227984	−	0.131627i
$726$		0			0
$727$	2.91589e11			1.04384			0.521920	−	0.852995i	$-0.325216\pi$
$727$	2.91589e11			1.04384			0.521920	+	0.852995i	$0.325216\pi$
$728$		0			0
$729$	−2.75140e11	−	6.37543e10i	−0.974189	−	0.225735i
$730$		0			0
$731$	−2.05027e10	−	1.18373e10i	−0.0718029	−	0.0414554i
$732$		0			0
$733$	−2.29401e11	−	3.97334e11i	−0.794655	−	1.37638i	−0.923058	−	0.384661i	$-0.874318\pi$
$733$	−2.29401e11	−	3.97334e11i	−0.794655	−	1.37638i	0.128403	−	0.991722i	$-0.459015\pi$
$734$		0			0
$735$	−2.07782e10	+	1.62679e11i	−0.0711965	+	0.557419i
$736$		0			0
$737$	−6.27534e10	+	3.62307e10i	−0.212700	+	0.122802i
$738$		0			0
$739$	1.35783e11	−	2.35184e11i	0.455270	−	0.788551i	−0.543434	−	0.839452i	$-0.682876\pi$
$739$	1.35783e11	−	2.35184e11i	0.455270	−	0.788551i	0.998704	+	0.0509014i	$0.0162094\pi$
$740$		0			0
$741$	−3.73572e10	−	2.35018e10i	−0.123909	−	0.0779521i
$742$		0			0
$743$		−	1.93861e11i		−	0.636115i	−0.948072	−	0.318057i	$-0.896970\pi$
$743$		−	1.93861e11i		−	0.636115i	0.948072	−	0.318057i	$-0.103030\pi$
$744$		0			0
$745$	1.42931e11	−	2.47564e11i	0.463982	−	0.803641i
$746$		0			0
$747$	−3.47621e11	−	2.37564e11i	−1.11641	−	0.762953i
$748$		0			0
$749$	4.21010e11	+	4.60292e11i	1.33772	+	1.46254i
$750$		0			0
$751$	2.69574e11	+	4.66915e11i	0.847457	+	1.46784i	0.883471	+	0.468487i	$0.155201\pi$
$751$	2.69574e11	+	4.66915e11i	0.847457	+	1.46784i	−0.0360138	+	0.999351i	$0.511466\pi$
$752$		0			0
$753$	3.84709e10	+	7.28875e10i	0.119661	+	0.226711i
$754$		0			0
$755$		−	7.82621e10i		−	0.240859i
$756$		0			0
$757$	−5.76575e11			−1.75579			−0.877895	−	0.478854i	$-0.841052\pi$
$757$	−5.76575e11			−1.75579			−0.877895	+	0.478854i	$0.841052\pi$
$758$		0			0
$759$	−4.30674e11	+	2.27315e11i	−1.29772	+	0.684953i
$760$		0			0
$761$	−3.72013e11	+	2.14782e11i	−1.10922	+	0.640411i	−0.938628	−	0.344930i	$-0.887903\pi$
$761$	−3.72013e11	+	2.14782e11i	−1.10922	+	0.640411i	−0.170596	+	0.985341i	$0.554569\pi$
$762$		0			0
$763$	−7.75014e10	−	2.45137e10i	−0.228671	−	0.0723287i
$764$		0			0
$765$	−8.05387e9	+	1.17850e10i	−0.0235157	+	0.0344099i
$766$		0			0
$767$	−1.83226e10	−	1.05786e10i	−0.0529426	−	0.0305664i
$768$		0			0
$769$	−3.02231e11			−0.864240			−0.432120	−	0.901816i	$-0.642234\pi$
$769$	−3.02231e11			−0.864240			−0.432120	+	0.901816i	$0.642234\pi$
$770$		0			0
$771$	−3.22672e11	+	5.12904e11i	−0.913154	+	1.45150i
$772$		0			0
$773$	−5.31443e11	−	3.06829e11i	−1.48847	−	0.859366i	−0.488553	−	0.872534i	$-0.662475\pi$
$773$	−5.31443e11	−	3.06829e11i	−1.48847	−	0.859366i	−0.999913	+	0.0131677i	$0.995808\pi$
$774$		0			0
$775$	−1.44101e10	−	2.49590e10i	−0.0399447	−	0.0691863i
$776$		0			0
$777$	−1.92581e11	+	5.37074e11i	−0.528360	+	1.47350i
$778$		0			0
$779$	−3.48948e11	+	2.01465e11i	−0.947570	+	0.547080i
$780$		0			0
$781$	−1.42404e10	+	2.46650e10i	−0.0382751	+	0.0662944i
$782$		0			0
$783$	−1.43684e11	−	1.64293e10i	−0.382261	−	0.0437090i
$784$		0			0
$785$		−	4.15436e11i		−	1.09402i
$786$		0			0
$787$	2.22703e11	−	3.85734e11i	0.580534	−	1.00552i	−0.414882	−	0.909875i	$-0.636177\pi$
$787$	2.22703e11	−	3.85734e11i	0.580534	−	1.00552i	0.995416	−	0.0956397i	$-0.0304897\pi$
$788$		0			0
$789$	−5.07831e11	−	1.92812e10i	−1.31042	−	0.0497538i
$790$		0			0
$791$	6.20320e11	−	1.36929e11i	1.58457	−	0.349776i
$792$		0			0
$793$	−2.60939e10	−	4.51960e10i	−0.0659852	−	0.114290i
$794$		0			0
$795$	2.55946e11	−	1.35091e11i	0.640737	−	0.338189i
$796$		0			0
$797$			4.19587e11i			1.03989i	0.854199	+	0.519947i	$0.174048\pi$
$797$			4.19587e11i			1.03989i	−0.854199	+	0.519947i	$0.825952\pi$
$798$		0			0
$799$	−2.76870e10			−0.0679342
$800$		0			0
$801$	−2.88336e10	+	3.79164e11i	−0.0700438	+	0.921080i
$802$		0			0
$803$	−6.54429e11	+	3.77835e11i	−1.57398	+	0.908740i
$804$		0			0
$805$	−2.20726e11	−	6.98157e10i	−0.525618	−	0.166253i
$806$		0			0
$807$	3.33538e11	+	1.26637e10i	0.786415	+	0.0298585i
$808$		0			0
$809$	3.33490e11	+	1.92541e11i	0.778554	+	0.449498i	0.835918	−	0.548855i	$-0.184936\pi$
$809$	3.33490e11	+	1.92541e11i	0.778554	+	0.449498i	−0.0573635	+	0.998353i	$0.518269\pi$
$810$		0			0
$811$	−3.48989e11			−0.806729			−0.403365	−	0.915039i	$-0.632159\pi$
$811$	−3.48989e11			−0.806729			−0.403365	+	0.915039i	$0.632159\pi$
$812$		0			0
$813$	−2.00782e11	−	1.26314e11i	−0.459582	−	0.289127i
$814$		0			0
$815$	4.38259e10	+	2.53029e10i	0.0993347	+	0.0573509i
$816$		0			0
$817$	−1.98098e11	−	3.43116e11i	−0.444623	−	0.770110i
$818$		0			0
$819$	7.68238e10	+	3.08843e10i	0.170750	+	0.0686440i
$820$		0			0
$821$	5.42071e11	−	3.12965e11i	1.19312	−	0.688847i	0.234106	−	0.972211i	$-0.424784\pi$
$821$	5.42071e11	−	3.12965e11i	1.19312	−	0.688847i	0.959012	+	0.283364i	$0.0914504\pi$
$822$		0			0
$823$	8.27256e10	−	1.43285e11i	0.180319	−	0.312321i	−0.761670	−	0.647965i	$-0.775620\pi$
$823$	8.27256e10	−	1.43285e11i	0.180319	−	0.312321i	0.941989	+	0.335644i	$0.108954\pi$
$824$		0			0
$825$	2.52460e11	−	4.01298e11i	0.544975	−	0.866265i
$826$		0			0
$827$			4.00173e11i			0.855511i	0.903894	+	0.427756i	$0.140696\pi$
$827$			4.00173e11i			0.855511i	−0.903894	+	0.427756i	$0.859304\pi$
$828$		0			0
$829$	3.84418e11	−	6.65832e11i	0.813927	−	1.40976i	−0.0961681	−	0.995365i	$-0.530659\pi$
$829$	3.84418e11	−	6.65832e11i	0.813927	−	1.40976i	0.910096	−	0.414399i	$-0.136008\pi$
$830$		0			0
$831$	−1.21910e10	+	3.21088e11i	−0.0255644	+	0.673318i
$832$		0			0
$833$	−3.55683e10	−	3.17711e9i	−0.0738725	−	0.00659860i
$834$		0			0
$835$	2.84519e10	+	4.92801e10i	0.0585281	+	0.101374i
$836$		0			0
$837$	−4.60520e10	−	3.41053e10i	−0.0938310	−	0.0694897i
$838$		0			0
$839$		−	1.62737e11i		−	0.328427i	−0.986425	−	0.164214i	$-0.947491\pi$
$839$		−	1.62737e11i		−	0.328427i	0.986425	−	0.164214i	$-0.0525087\pi$
$840$		0			0
$841$	4.26193e11			0.851966
$842$		0			0
$843$	−1.35483e11	−	2.56688e11i	−0.268271	−	0.508270i
$844$		0			0
$845$	2.39712e11	−	1.38398e11i	0.470179	−	0.271458i
$846$		0			0
$847$	−4.69912e11	+	4.29809e11i	−0.913026	+	0.835106i
$848$		0			0
$849$	−9.47247e9	+	2.49487e11i	−0.0182319	+	0.480194i
$850$		0			0
$851$	−6.97500e11	−	4.02702e11i	−1.32992	−	0.767831i
$852$		0			0
$853$	7.31968e11			1.38260			0.691299	−	0.722569i	$-0.257039\pi$
$853$	7.31968e11			1.38260			0.691299	+	0.722569i	$0.257039\pi$
$854$		0			0
$855$	−2.15337e11	+	1.03409e11i	−0.402953	+	0.193506i
$856$		0			0
$857$	5.35957e11	+	3.09435e11i	0.993589	+	0.573649i	0.906345	−	0.422538i	$-0.138861\pi$
$857$	5.35957e11	+	3.09435e11i	0.993589	+	0.573649i	0.0872440	+	0.996187i	$0.472194\pi$
$858$		0			0
$859$	1.98464e11	+	3.43750e11i	0.364510	+	0.631350i	0.988697	−	0.149925i	$-0.0479031\pi$
$859$	1.98464e11	+	3.43750e11i	0.364510	+	0.631350i	−0.624187	+	0.781275i	$0.714570\pi$
$860$		0			0
$861$	5.76741e11	−	4.88653e11i	1.04946	−	0.889176i
$862$		0			0
$863$	−1.17285e10	+	6.77144e9i	−0.0211445	+	0.0122078i	−0.510535	−	0.859857i	$-0.670553\pi$
$863$	−1.17285e10	+	6.77144e9i	−0.0211445	+	0.0122078i	0.489390	+	0.872065i	$0.337219\pi$
$864$		0			0
$865$	−1.35820e11	+	2.35247e11i	−0.242605	+	0.420203i
$866$		0			0
$867$	4.75634e11	+	2.99225e11i	0.841776	+	0.529568i
$868$		0			0
$869$		−	1.31126e11i		−	0.229938i
$870$		0			0
$871$	−8.69570e9	+	1.50614e10i	−0.0151089	+	0.0261693i
$872$		0			0
$873$	2.59026e11	−	3.79025e11i	0.445949	−	0.652545i
$874$		0			0
$875$	5.41744e11	−	1.19584e11i	0.924192	−	0.204006i
$876$		0			0
$877$	3.82473e11	+	6.62463e11i	0.646551	+	1.11986i	0.983941	+	0.178494i	$0.0571224\pi$
$877$	3.82473e11	+	6.62463e11i	0.646551	+	1.11986i	−0.337390	+	0.941365i	$0.609544\pi$
$878$		0			0
$879$	3.43483e10	+	6.50767e10i	0.0575373	+	0.109011i
$880$		0			0
$881$		−	2.96363e11i		−	0.491949i	−0.969276	−	0.245974i	$-0.920892\pi$
$881$		−	2.96363e11i		−	0.491949i	0.969276	−	0.245974i	$-0.0791079\pi$
$882$		0			0
$883$	−5.25932e11			−0.865141			−0.432571	−	0.901600i	$-0.642393\pi$
$883$	−5.25932e11			−0.865141			−0.432571	+	0.901600i	$0.642393\pi$
$884$		0			0
$885$	−1.01272e11	+	5.34524e10i	−0.165088	+	0.0871353i
$886$		0			0
$887$	5.27686e11	−	3.04660e11i	0.852474	−	0.492176i	−0.00901088	−	0.999959i	$-0.502868\pi$
$887$	5.27686e11	−	3.04660e11i	0.852474	−	0.492176i	0.861485	+	0.507783i	$0.169535\pi$
$888$		0			0
$889$	1.19222e11	+	5.40104e11i	0.190875	+	0.864709i
$890$		0			0
$891$	1.42553e11	−	9.31867e11i	0.226185	−	1.47857i
$892$		0			0
$893$	−4.01269e11	−	2.31673e11i	−0.631000	−	0.364308i
$894$		0			0
$895$	2.68507e11			0.418470
$896$		0			0
$897$	−6.22384e10	+	9.89311e10i	−0.0961365	+	0.152814i
$898$		0			0
$899$	−2.54125e10	−	1.46719e10i	−0.0389053	−	0.0224620i
$900$		0			0
$901$	3.15084e10	+	5.45742e10i	0.0478110	+	0.0828110i
$902$		0			0
$903$	4.80485e11	+	5.67101e11i	0.722652	+	0.852922i
$904$		0			0
$905$	4.82726e11	−	2.78702e11i	0.719625	−	0.415476i
$906$		0			0
$907$	−8.06279e10	+	1.39652e11i	−0.119140	+	0.206356i	−0.919427	−	0.393261i	$-0.871347\pi$
$907$	−8.06279e10	+	1.39652e11i	−0.119140	+	0.206356i	0.800287	+	0.599617i	$0.204680\pi$
$908$		0			0
$909$	6.76332e11	−	3.24789e11i	0.990614	−	0.475714i
$910$		0			0
$911$			3.61832e11i			0.525332i	0.964887	+	0.262666i	$0.0846017\pi$
$911$			3.61832e11i			0.525332i	−0.964887	+	0.262666i	$0.915398\pi$
$912$		0			0
$913$	7.02688e11	−	1.21709e12i	1.01130	−	1.75162i
$914$		0			0
$915$	−2.82262e11	−	1.07169e10i	−0.402688	−	0.0152892i
$916$		0			0
$917$	−9.19260e11	−	1.00503e12i	−1.30005	−	1.42136i
$918$		0			0
$919$	5.60480e11	+	9.70779e11i	0.785774	+	1.36100i	0.928535	+	0.371244i	$0.121069\pi$
$919$	5.60480e11	+	9.70779e11i	0.785774	+	1.36100i	−0.142761	+	0.989757i	$0.545598\pi$
$920$		0			0
$921$	2.84441e11	−	1.50131e11i	0.395324	−	0.208657i
$922$		0			0
$923$			6.83563e9i			0.00941828i
$924$		0			0
$925$	7.84107e11			1.07105
$926$		0			0
$927$	2.31531e11	+	1.76068e10i	0.313538	+	0.0238431i
$928$		0			0
$929$	−3.60741e11	+	2.08274e11i	−0.484320	+	0.279622i	−0.722215	−	0.691669i	$-0.756876\pi$
$929$	−3.60741e11	+	2.08274e11i	−0.484320	+	0.279622i	0.237895	+	0.971291i	$0.423543\pi$
$930$		0			0
$931$	−4.88908e11	−	3.43666e11i	−0.650771	−	0.457444i
$932$		0			0
$933$	−8.81535e11	−	3.34700e10i	−1.16336	−	0.0441701i
$934$		0			0
$935$	−4.12617e10	−	2.38225e10i	−0.0539885	−	0.0311702i
$936$		0			0
$937$	−1.20353e12			−1.56134			−0.780670	−	0.624944i	$-0.785122\pi$
$937$	−1.20353e12			−1.56134			−0.780670	+	0.624944i	$0.785122\pi$
$938$		0			0
$939$	−1.60175e11	−	1.00767e11i	−0.206031	−	0.129616i
$940$		0			0
$941$	−1.24560e12	−	7.19150e11i	−1.58863	−	0.917194i	−0.993534	−	0.113536i	$-0.963782\pi$
$941$	−1.24560e12	−	7.19150e11i	−1.58863	−	0.917194i	−0.595092	−	0.803657i	$-0.702885\pi$
$942$		0			0
$943$	5.33529e11	+	9.24100e11i	0.674701	+	1.16862i
$944$		0			0
$945$	3.62906e11	−	2.62939e11i	0.455058	−	0.329707i
$946$		0			0
$947$	−5.12016e11	+	2.95613e11i	−0.636625	+	0.367556i	−0.783313	−	0.621627i	$-0.786472\pi$
$947$	−5.12016e11	+	2.95613e11i	−0.636625	+	0.367556i	0.146688	+	0.989183i	$0.453139\pi$
$948$		0			0
$949$	−9.06838e10	+	1.57069e11i	−0.111806	+	0.193653i
$950$		0			0
$951$	2.29392e11	−	3.64630e11i	0.280451	−	0.445790i
$952$		0			0
$953$			1.23642e12i			1.49897i	0.662020	+	0.749486i	$0.269699\pi$
$953$			1.23642e12i			1.49897i	−0.662020	+	0.749486i	$0.730301\pi$
$954$		0			0
$955$	4.65642e10	−	8.06515e10i	0.0559807	−	0.0969615i
$956$		0			0
$957$	1.83146e10	−	4.82372e11i	0.0218348	−	0.575088i
$958$		0			0
$959$	−1.95275e11	+	6.17372e11i	−0.230873	+	0.729916i
$960$		0			0
$961$	4.20632e11	+	7.28556e11i	0.493183	+	0.854219i
$962$		0			0
$963$	1.29252e11	−	1.69968e12i	0.150291	−	1.97634i
$964$		0			0
$965$		−	9.39572e11i		−	1.08348i
$966$		0			0
$967$	−3.68784e11			−0.421761			−0.210880	−	0.977512i	$-0.567633\pi$
$967$	−3.68784e11			−0.421761			−0.210880	+	0.977512i	$0.567633\pi$
$968$		0			0
$969$	−2.42793e10	−	4.60000e10i	−0.0275386	−	0.0521750i
$970$		0			0
$971$	1.16250e12	−	6.71168e11i	1.30772	−	0.755012i	0.326005	−	0.945368i	$-0.394297\pi$
$971$	1.16250e12	−	6.71168e11i	1.30772	−	0.755012i	0.981715	+	0.190356i	$0.0609642\pi$
$972$		0			0
$973$	−1.99111e11	−	9.02020e11i	−0.222149	−	1.00639i
$974$		0			0
$975$	4.31723e9	−	1.13708e11i	0.00477735	−	0.125826i
$976$		0			0
$977$	1.23413e12	+	7.12528e11i	1.35452	+	0.782030i	0.988878	−	0.148727i	$-0.0475176\pi$
$977$	1.23413e12	+	7.12528e11i	1.35452	+	0.782030i	0.365638	+	0.930757i	$0.380851\pi$
$978$		0			0
$979$	−1.26925e12			−1.38171
$980$		0			0
$981$	9.61553e10	+	2.00231e11i	0.103824	+	0.216200i
$982$		0			0
$983$	7.46080e11	+	4.30749e11i	0.799045	+	0.461329i	0.843137	−	0.537699i	$-0.180706\pi$
$983$	7.46080e11	+	4.30749e11i	0.799045	+	0.461329i	−0.0440921	+	0.999027i	$0.514039\pi$
$984$		0			0
$985$	−2.96701e10	−	5.13901e10i	−0.0315191	−	0.0545927i
$986$		0			0
$987$	8.18245e11	+	2.93402e11i	0.862214	+	0.309168i
$988$		0			0
$989$	−9.08654e11	+	5.24612e11i	−0.949759	+	0.548344i
$990$		0			0
$991$	−8.80347e11	+	1.52481e12i	−0.912765	+	1.58096i	−0.102625	+	0.994720i	$0.532724\pi$
$991$	−8.80347e11	+	1.52481e12i	−0.912765	+	1.58096i	−0.810141	+	0.586236i	$0.800609\pi$
$992$		0			0
$993$	9.74949e11	+	6.13349e11i	1.00273	+	0.630827i
$994$		0			0
$995$			7.06886e10i			0.0721202i
$996$		0			0
$997$	2.26364e10	−	3.92074e10i	0.0229101	−	0.0396815i	−0.854343	−	0.519710i	$-0.826040\pi$
$997$	2.26364e10	−	3.92074e10i	0.0229101	−	0.0396815i	0.877253	+	0.480028i	$0.159374\pi$
$998$		0			0
$999$	1.43005e12	−	6.21120e11i	1.43579	−	0.623610i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	84.9.p.b.53.4	✓	40
3.2	odd	2	inner	84.9.p.b.53.10	yes	40
7.2	even	3	inner	84.9.p.b.65.10	yes	40
21.2	odd	6	inner	84.9.p.b.65.4	yes	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.9.p.b.53.4	✓	40	1.1	even	1	trivial
84.9.p.b.53.10	yes	40	3.2	odd	2	inner
84.9.p.b.65.4	yes	40	21.2	odd	6	inner
84.9.p.b.65.10	yes	40	7.2	even	3	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)

\(f(q)\)	\(=\)	\(q+(-71.6341 + 37.8094i) q^{3} +(-304.163 + 175.609i) q^{5} +(517.537 + 2344.56i) q^{7} +(3701.90 - 5416.89i) q^{9} +O(q^{10})\)	\(q+(-71.6341 + 37.8094i) q^{3} +(-304.163 + 175.609i) q^{5} +(517.537 + 2344.56i) q^{7} +(3701.90 - 5416.89i) q^{9} +(18965.6 + 10949.8i) q^{11} +5256.11 q^{13} +(15148.8 - 24079.8i) q^{15} +(5364.57 + 3097.23i) q^{17} +(51832.6 + 89776.8i) q^{19} +(-125720. - 148383. i) q^{21} +(237751. - 137265. i) q^{23} +(-133636. + 231464. i) q^{25} +(-60373.3 + 528001. i) q^{27} +272128. i q^{29} +(-53915.5 + 93384.4i) q^{31} +(-1.77259e6 - 67301.5i) q^{33} +(-569140. - 622244. i) q^{35} +(-1.46687e6 - 2.54070e6i) q^{37} +(-376517. + 198730. i) q^{39} +3.88684e6i q^{41} -3.82188e6 q^{43} +(-174729. + 2.29770e6i) q^{45} +(-3.87081e6 + 2.23481e6i) q^{47} +(-5.22911e6 + 2.42679e6i) q^{49} +(-501391. - 19036.7i) q^{51} +(8.81015e6 + 5.08654e6i) q^{53} -7.69153e6 q^{55} +(-7.10739e6 - 4.47132e6i) q^{57} +(-3.48596e6 - 2.01262e6i) q^{59} +(-4.96449e6 - 8.59875e6i) q^{61} +(1.46161e7 + 5.87589e6i) q^{63} +(-1.59871e6 + 923018. i) q^{65} +(-1.65440e6 + 2.86550e6i) q^{67} +(-1.18411e7 + 1.88221e7i) q^{69} +1.30051e6i q^{71} +(-1.72530e7 + 2.98831e7i) q^{73} +(821373. - 2.16334e7i) q^{75} +(-1.58571e7 + 5.01330e7i) q^{77} +(-2.99380e6 - 5.18541e6i) q^{79} +(-1.56386e7 - 4.01055e7i) q^{81} -6.41735e7i q^{83} -2.17560e6 q^{85} +(-1.02890e7 - 1.94936e7i) q^{87} +(-5.01926e7 + 2.89787e7i) q^{89} +(2.72023e6 + 1.23233e7i) q^{91} +(331384. - 8.72802e6i) q^{93} +(-3.15311e7 - 1.82045e7i) q^{95} +6.99710e7 q^{97} +(1.29523e8 - 6.21996e7i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 81 q^{3} - 34 q^{7} + 4771 q^{9}+O(q^{10}) \) 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

Related objects

Downloads

Learn more