LMFDB - Embedded newform 84.9.p.b.65.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("84.53");

S:= CuspForms(chi, 9);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$34.2198032451$
Analytic rank:	$0$
Dimension:	$40$
Relative dimension:	$20$ over $\Q(\zeta_{6})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			65.14
Character	$\chi$	$=$	84.65
Dual form			84.9.p.b.53.14

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(51.3357 + 62.6550i) q^{3} +(918.968 + 530.566i) q^{5} +(-1255.82 - 2046.39i) q^{7} +(-1290.30 + 6432.87i) q^{9} +O(q^{10})$	\(q+(51.3357 + 62.6550i) q^{3} +(918.968 + 530.566i) q^{5} +(-1255.82 - 2046.39i) q^{7} +(-1290.30 + 6432.87i) q^{9} +(22077.2 - 12746.3i) q^{11} +48286.6 q^{13} +(13933.2 + 84814.9i) q^{15} +(15116.9 - 8727.72i) q^{17} +(67655.0 - 117182. i) q^{19} +(63748.2 - 183736. i) q^{21} +(-428030. - 247123. i) q^{23} +(367689. + 636856. i) q^{25} +(-469290. + 249392. i) q^{27} +448191. i q^{29} +(192056. + 332652. i) q^{31} +(1.93196e6 + 728907. i) q^{33} +(-68313.3 - 2.54686e6i) q^{35} +(-40558.3 + 70249.1i) q^{37} +(2.47883e6 + 3.02540e6i) q^{39} +981836. i q^{41} +1.23519e6 q^{43} +(-4.59881e6 + 5.22701e6i) q^{45} +(-2.72478e6 - 1.57315e6i) q^{47} +(-2.61063e6 + 5.13980e6i) q^{49} +(1.32287e6 + 499103. i) q^{51} +(610035. - 352204. i) q^{53} +2.70509e7 q^{55} +(1.08151e7 - 1.77669e6i) q^{57} +(-1.15582e7 + 6.67310e6i) q^{59} +(542461. - 939570. i) q^{61} +(1.47846e7 - 5.43808e6i) q^{63} +(4.43739e7 + 2.56193e7i) q^{65} +(1.26543e7 + 2.19179e7i) q^{67} +(-6.48969e6 - 3.95044e7i) q^{69} +3.64547e7i q^{71} +(-4.68177e6 - 8.10906e6i) q^{73} +(-2.10267e7 + 5.57310e7i) q^{75} +(-5.38088e7 - 2.91715e7i) q^{77} +(825142. - 1.42919e6i) q^{79} +(-3.97170e7 - 1.66007e7i) q^{81} -3.97669e7i q^{83} +1.85225e7 q^{85} +(-2.80814e7 + 2.30082e7i) q^{87} +(-2.86471e7 - 1.65394e7i) q^{89} +(-6.06394e7 - 9.88133e7i) q^{91} +(-1.09829e7 + 2.91102e7i) q^{93} +(1.24346e8 - 7.17910e7i) q^{95} -1.05325e8 q^{97} +(5.35089e7 + 1.58466e8i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q + 81 q^{3} - 34 q^{7} + 4771 q^{9}+O(q^{10}) $ 40 * q + 81 * q^3 - 34 * q^7 + 4771 * q^9	$ 40 q + 81 q^{3} - 34 q^{7} + 4771 q^{9} - 55464 q^{13} + 68482 q^{15} + 311690 q^{19} - 172343 q^{21} + 1766792 q^{25} - 3451932 q^{27} + 31596 q^{31} + 1874885 q^{33} - 1853482 q^{37} + 11217526 q^{39} - 13372600 q^{43} - 527785 q^{45} - 12653462 q^{49} - 1103461 q^{51} + 71577224 q^{55} - 17195214 q^{57} - 21761970 q^{61} + 21945045 q^{63} - 26337350 q^{67} - 5588722 q^{69} + 41115682 q^{73} - 17971730 q^{75} - 120916932 q^{79} - 24550133 q^{81} + 139250060 q^{85} - 16321046 q^{87} + 345074940 q^{91} + 25774675 q^{93} - 707216948 q^{97} - 94510994 q^{99}+O(q^{100}) $ 40 * q + 81 * q^3 - 34 * q^7 + 4771 * q^9 - 55464 * q^13 + 68482 * q^15 + 311690 * q^19 - 172343 * q^21 + 1766792 * q^25 - 3451932 * q^27 + 31596 * q^31 + 1874885 * q^33 - 1853482 * q^37 + 11217526 * q^39 - 13372600 * q^43 - 527785 * q^45 - 12653462 * q^49 - 1103461 * q^51 + 71577224 * q^55 - 17195214 * q^57 - 21761970 * q^61 + 21945045 * q^63 - 26337350 * q^67 - 5588722 * q^69 + 41115682 * q^73 - 17971730 * q^75 - 120916932 * q^79 - 24550133 * q^81 + 139250060 * q^85 - 16321046 * q^87 + 345074940 * q^91 + 25774675 * q^93 - 707216948 * q^97 - 94510994 * q^99

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	51.3357	+	62.6550i	0.633774	+	0.773519i
$3$	51.3357	+	62.6550i	0.633774	+	0.773519i
$4$		0			0
$5$	918.968	+	530.566i	1.47035	+	0.848906i	0.999446	−	0.0332787i	$-0.0105949\pi$
$5$	918.968	+	530.566i	1.47035	+	0.848906i	0.470903	+	0.882185i	$0.343928\pi$
$6$		0			0
$7$	−1255.82	−	2046.39i	−0.523041	−	0.852308i
$7$	−1255.82	−	2046.39i	−0.523041	−	0.852308i
$8$		0			0
$9$	−1290.30	+	6432.87i	−0.196662	+	0.980471i
$10$		0			0
$11$	22077.2	−	12746.3i	1.50790	−	0.870586i	0.507942	−	0.861391i	$-0.330406\pi$
$11$	22077.2	−	12746.3i	1.50790	−	0.870586i	0.999958	−	0.00919529i	$-0.00292699\pi$
$12$		0			0
$13$	48286.6			1.69065			0.845324	−	0.534253i	$-0.179407\pi$
$13$	48286.6			1.69065			0.845324	+	0.534253i	$0.179407\pi$
$14$		0			0
$15$	13933.2	+	84814.9i	0.275223	+	1.67536i
$16$		0			0
$17$	15116.9	−	8727.72i	0.180995	−	0.104497i	−0.406765	−	0.913533i	$-0.633343\pi$
$17$	15116.9	−	8727.72i	0.180995	−	0.104497i	0.587760	+	0.809035i	$0.300010\pi$
$18$		0			0
$19$	67655.0	−	117182.i	0.519141	−	0.899179i	−0.480611	−	0.876934i	$-0.659585\pi$
$19$	67655.0	−	117182.i	0.519141	−	0.899179i	0.999753	−	0.0222452i	$-0.00708146\pi$
$20$		0			0
$21$	63748.2	−	183736.i	0.327786	−	0.944752i
$22$		0			0
$23$	−428030.	−	247123.i	−1.52955	−	0.883084i	−0.999381	−	0.0351895i	$-0.988797\pi$
$23$	−428030.	−	247123.i	−1.52955	−	0.883084i	−0.530165	−	0.847894i	$-0.677870\pi$
$24$		0			0
$25$	367689.	+	636856.i	0.941284	+	1.63035i
$26$		0			0
$27$	−469290.	+	249392.i	−0.883052	+	0.469275i
$28$		0			0
$29$			448191.i			0.633681i	0.948479	+	0.316841i	$0.102622\pi$
$29$			448191.i			0.633681i	−0.948479	+	0.316841i	$0.897378\pi$
$30$		0			0
$31$	192056.	+	332652.i	0.207961	+	0.360199i	0.951072	−	0.308969i	$-0.0999839\pi$
$31$	192056.	+	332652.i	0.207961	+	0.360199i	−0.743111	+	0.669168i	$0.766651\pi$
$32$		0			0
$33$	1.93196e6	+	728907.i	1.62908	+	0.614634i
$34$		0			0
$35$	−68313.3	−	2.54686e6i	−0.0455232	−	1.69720i
$36$		0			0
$37$	−40558.3	+	70249.1i	−0.0216408	+	0.0374830i	−0.876643	−	0.481141i	$-0.840222\pi$
$37$	−40558.3	+	70249.1i	−0.0216408	+	0.0374830i	0.855002	+	0.518624i	$0.173556\pi$
$38$		0			0
$39$	2.47883e6	+	3.02540e6i	1.07149	+	1.30775i
$40$		0			0
$41$			981836.i			0.347459i	0.984793	+	0.173729i	$0.0555818\pi$
$41$			981836.i			0.347459i	−0.984793	+	0.173729i	$0.944418\pi$
$42$		0			0
$43$	1.23519e6			0.361294			0.180647	−	0.983548i	$-0.442181\pi$
$43$	1.23519e6			0.361294			0.180647	+	0.983548i	$0.442181\pi$
$44$		0			0
$45$	−4.59881e6	+	5.22701e6i	−1.12149	+	1.27469i
$46$		0			0
$47$	−2.72478e6	−	1.57315e6i	−0.558394	−	0.322389i	0.194107	−	0.980980i	$-0.437819\pi$
$47$	−2.72478e6	−	1.57315e6i	−0.558394	−	0.322389i	−0.752501	+	0.658592i	$0.771153\pi$
$48$		0			0
$49$	−2.61063e6	+	5.13980e6i	−0.452857	+	0.891583i
$50$		0			0
$51$	1.32287e6	+	499103.i	0.195540	+	0.0737751i
$52$		0			0
$53$	610035.	−	352204.i	0.0773127	−	0.0446365i	−0.460845	−	0.887480i	$-0.652454\pi$
$53$	610035.	−	352204.i	0.0773127	−	0.0446365i	0.538158	+	0.842844i	$0.319120\pi$
$54$		0			0
$55$	2.70509e7			2.95619
$56$		0			0
$57$	1.08151e7	−	1.77669e6i	1.02455	−	0.168310i
$58$		0			0
$59$	−1.15582e7	+	6.67310e6i	−0.953851	+	0.550706i	−0.894275	−	0.447518i	$-0.852308\pi$
$59$	−1.15582e7	+	6.67310e6i	−0.953851	+	0.550706i	−0.0595758	+	0.998224i	$0.518975\pi$
$60$		0			0
$61$	542461.	−	939570.i	0.0391786	−	0.0678594i	−0.845771	−	0.533546i	$-0.820859\pi$
$61$	542461.	−	939570.i	0.0391786	−	0.0678594i	0.884950	+	0.465686i	$0.154193\pi$
$62$		0			0
$63$	1.47846e7	−	5.43808e6i	0.938525	−	0.345210i
$64$		0			0
$65$	4.43739e7	+	2.56193e7i	2.48584	+	1.43520i
$66$		0			0
$67$	1.26543e7	+	2.19179e7i	0.627971	+	1.08768i	0.987958	+	0.154721i	$0.0494479\pi$
$67$	1.26543e7	+	2.19179e7i	0.627971	+	1.08768i	−0.359987	+	0.932957i	$0.617219\pi$
$68$		0			0
$69$	−6.48969e6	−	3.95044e7i	−0.286304	−	1.74281i
$70$		0			0
$71$			3.64547e7i			1.43456i	0.696784	+	0.717281i	$0.254614\pi$
$71$			3.64547e7i			1.43456i	−0.696784	+	0.717281i	$0.745386\pi$
$72$		0			0
$73$	−4.68177e6	−	8.10906e6i	−0.164861	−	0.285548i	0.771745	−	0.635932i	$-0.219384\pi$
$73$	−4.68177e6	−	8.10906e6i	−0.164861	−	0.285548i	−0.936606	+	0.350384i	$0.886051\pi$
$74$		0			0
$75$	−2.10267e7	+	5.57310e7i	−0.664546	+	1.76137i
$76$		0			0
$77$	−5.38088e7	−	2.91715e7i	−1.53070	−	0.829842i
$78$		0			0
$79$	825142.	−	1.42919e6i	0.0211846	−	0.0366928i	−0.855239	−	0.518234i	$-0.826590\pi$
$79$	825142.	−	1.42919e6i	0.0211846	−	0.0366928i	0.876423	+	0.481541i	$0.159923\pi$
$80$		0			0
$81$	−3.97170e7	−	1.66007e7i	−0.922648	−	0.385643i
$82$		0			0
$83$		−	3.97669e7i		−	0.837934i	−0.908002	−	0.418967i	$-0.862392\pi$
$83$		−	3.97669e7i		−	0.837934i	0.908002	−	0.418967i	$-0.137608\pi$
$84$		0			0
$85$	1.85225e7			0.354834
$86$		0			0
$87$	−2.80814e7	+	2.30082e7i	−0.490164	+	0.401610i
$88$		0			0
$89$	−2.86471e7	−	1.65394e7i	−0.456584	−	0.263609i	0.254023	−	0.967198i	$-0.418246\pi$
$89$	−2.86471e7	−	1.65394e7i	−0.456584	−	0.263609i	−0.710607	+	0.703589i	$0.751580\pi$
$90$		0			0
$91$	−6.06394e7	−	9.88133e7i	−0.884278	−	1.44095i
$92$		0			0
$93$	−1.09829e7	+	2.91102e7i	−0.146821	+	0.389147i
$94$		0			0
$95$	1.24346e8	−	7.17910e7i	1.52664	−	0.881405i
$96$		0			0
$97$	−1.05325e8			−1.18972			−0.594861	−	0.803829i	$-0.702793\pi$
$97$	−1.05325e8			−1.18972			−0.594861	+	0.803829i	$0.702793\pi$
$98$		0			0
$99$	5.35089e7	+	1.58466e8i	0.557038	+	1.64966i
$100$		0			0
$101$	−8.31379e7	+	4.79997e7i	−0.798939	+	0.461268i	−0.843100	−	0.537757i	$-0.819272\pi$
$101$	−8.31379e7	+	4.79997e7i	−0.798939	+	0.461268i	0.0441610	+	0.999024i	$0.485939\pi$
$102$		0			0
$103$	1.80871e7	−	3.13279e7i	0.160702	−	0.278344i	−0.774419	−	0.632673i	$-0.781958\pi$
$103$	1.80871e7	−	3.13279e7i	0.160702	−	0.278344i	0.935121	+	0.354330i	$0.115291\pi$
$104$		0			0
$105$	1.56067e8	−	1.35025e8i	1.28397	−	1.11086i
$106$		0			0
$107$	4.04894e7	+	2.33766e7i	0.308892	+	0.178339i	0.646430	−	0.762973i	$-0.276261\pi$
$107$	4.04894e7	+	2.33766e7i	0.308892	+	0.178339i	−0.337539	+	0.941312i	$0.609594\pi$
$108$		0			0
$109$	2.05219e7	+	3.55449e7i	0.145382	+	0.251809i	0.929515	−	0.368783i	$-0.120226\pi$
$109$	2.05219e7	+	3.55449e7i	0.145382	+	0.251809i	−0.784133	+	0.620592i	$0.786892\pi$
$110$		0			0
$111$	−6.48355e6	+	1.06510e6i	−0.0427091	+	0.00701615i
$112$		0			0
$113$		−	2.16400e8i		−	1.32722i	−0.748078	−	0.663611i	$-0.769023\pi$
$113$		−	2.16400e8i		−	1.32722i	0.748078	−	0.663611i	$-0.230977\pi$
$114$		0			0
$115$	−2.62230e8	−	4.54196e8i	−1.49931	−	2.59688i
$116$		0			0
$117$	−6.23042e7	+	3.10622e8i	−0.332487	+	1.65763i
$118$		0			0
$119$	−3.68444e7	−	1.99745e7i	−0.183731	−	0.0996068i
$120$		0			0
$121$	2.17755e8	−	3.77162e8i	1.01584	−	1.75949i
$122$		0			0
$123$	−6.15169e7	+	5.04032e7i	−0.268766	+	0.220210i
$124$		0			0
$125$			3.65829e8i			1.49844i
$126$		0			0
$127$	−9.59175e7			−0.368708			−0.184354	−	0.982860i	$-0.559019\pi$
$127$	−9.59175e7			−0.368708			−0.184354	+	0.982860i	$0.559019\pi$
$128$		0			0
$129$	6.34095e7	+	7.73910e7i	0.228979	+	0.279468i
$130$		0			0
$131$	−2.83207e8	−	1.63510e8i	−0.961654	−	0.555211i	−0.0649724	−	0.997887i	$-0.520696\pi$
$131$	−2.83207e8	−	1.63510e8i	−0.961654	−	0.555211i	−0.896682	+	0.442676i	$0.854029\pi$
$132$		0			0
$133$	−3.24763e8	+	8.71094e6i	−1.03791	+	0.0278393i
$134$		0			0
$135$	−5.63582e8	−	1.98063e7i	−1.69677	−	0.0596304i
$136$		0			0
$137$	−3.94411e8	+	2.27713e8i	−1.11961	+	0.646407i	−0.941301	−	0.337568i	$-0.890396\pi$
$137$	−3.94411e8	+	2.27713e8i	−1.11961	+	0.646407i	−0.178308	+	0.983975i	$0.557062\pi$
$138$		0			0
$139$	4.86411e8			1.30300			0.651500	−	0.758649i	$-0.274140\pi$
$139$	4.86411e8			1.30300			0.651500	+	0.758649i	$0.274140\pi$
$140$		0			0
$141$	−4.13126e7	−	2.51480e8i	−0.104522	−	0.636250i
$142$		0			0
$143$	1.06603e9	−	6.15474e8i	2.54933	−	1.47186i
$144$		0			0
$145$	−2.37795e8	+	4.11873e8i	−0.537936	+	0.931733i
$146$		0			0
$147$	−4.56053e8	+	1.00286e8i	−0.976665	+	0.214769i
$148$		0			0
$149$	3.36769e8	+	1.94434e8i	0.683262	+	0.394482i	0.801083	−	0.598553i	$-0.204258\pi$
$149$	3.36769e8	+	1.94434e8i	0.683262	+	0.394482i	−0.117821	+	0.993035i	$0.537591\pi$
$150$		0			0
$151$	−2.33229e8	−	4.03964e8i	−0.448616	−	0.777026i	0.549680	−	0.835375i	$-0.314750\pi$
$151$	−2.33229e8	−	4.03964e8i	−0.448616	−	0.777026i	−0.998296	+	0.0583495i	$0.981416\pi$
$152$		0			0
$153$	3.66390e7	+	1.08506e8i	0.0668618	+	0.198011i
$154$		0			0
$155$			4.07595e8i			0.706158i
$156$		0			0
$157$	2.44394e8	+	4.23303e8i	0.402247	+	0.696712i	0.993997	−	0.109410i	$-0.0348961\pi$
$157$	2.44394e8	+	4.23303e8i	0.402247	+	0.696712i	−0.591750	+	0.806122i	$0.701563\pi$
$158$		0			0
$159$	5.33838e7	+	2.01411e7i	0.0835259	+	0.0315134i
$160$		0			0
$161$	3.18184e7	+	1.18626e9i	0.0473560	+	1.76553i
$162$		0			0
$163$	4.39751e8	−	7.61671e8i	0.622955	−	1.07899i	−0.365978	−	0.930624i	$-0.619265\pi$
$163$	4.39751e8	−	7.61671e8i	0.622955	−	1.07899i	0.988933	−	0.148366i	$-0.0474012\pi$
$164$		0			0
$165$	1.38868e9	+	1.69488e9i	1.87355	+	2.28666i
$166$		0			0
$167$		−	6.49558e8i		−	0.835126i	−0.908648	−	0.417563i	$-0.862884\pi$
$167$		−	6.49558e8i		−	0.835126i	0.908648	−	0.417563i	$-0.137116\pi$
$168$		0			0
$169$	1.51587e9			1.85829
$170$		0			0
$171$	6.66521e8	+	5.86416e8i	0.779524	+	0.685837i
$172$		0			0
$173$	−6.79031e8	−	3.92039e8i	−0.758063	−	0.437668i	0.0705369	−	0.997509i	$-0.477529\pi$
$173$	−6.79031e8	−	3.92039e8i	−0.758063	−	0.437668i	−0.828600	+	0.559841i	$0.810862\pi$
$174$		0			0
$175$	8.41505e8	−	1.55221e9i	0.897231	−	1.65500i
$176$		0			0
$177$	−1.01145e9	−	3.81608e8i	−1.03051	−	0.388798i
$178$		0			0
$179$	−1.48320e9	+	8.56326e8i	−1.44473	+	0.834117i	−0.998160	−	0.0606300i	$-0.980689\pi$
$179$	−1.48320e9	+	8.56326e8i	−1.44473	+	0.834117i	−0.446573	+	0.894747i	$0.647356\pi$
$180$		0			0
$181$	−2.97863e8			−0.277525			−0.138762	−	0.990326i	$-0.544312\pi$
$181$	−2.97863e8			−0.277525			−0.138762	+	0.990326i	$0.544312\pi$
$182$		0			0
$183$	8.67163e7	−	1.42455e7i	0.0773208	−	0.0127021i
$184$		0			0
$185$	−7.45437e7	+	4.30378e7i	−0.0636391	+	0.0367420i
$186$		0			0
$187$	2.22492e8	−	3.85367e8i	0.181948	−	0.315143i
$188$		0			0
$189$	1.09970e9	+	6.47159e8i	0.861839	+	0.507182i
$190$		0			0
$191$	3.34114e8	+	1.92901e8i	0.251051	+	0.144944i	0.620245	−	0.784408i	$-0.287033\pi$
$191$	3.34114e8	+	1.92901e8i	0.251051	+	0.144944i	−0.369195	+	0.929352i	$0.620366\pi$
$192$		0			0
$193$	−2.47406e8	−	4.28520e8i	−0.178312	−	0.308846i	0.762990	−	0.646410i	$-0.223730\pi$
$193$	−2.47406e8	−	4.28520e8i	−0.178312	−	0.308846i	−0.941303	+	0.337564i	$0.890397\pi$
$194$		0			0
$195$	6.72787e8	+	4.09543e9i	0.465306	+	2.83244i
$196$		0			0
$197$		−	1.64317e9i		−	1.09098i	−0.838117	−	0.545490i	$-0.816344\pi$
$197$		−	1.64317e9i		−	1.09098i	0.838117	−	0.545490i	$-0.183656\pi$
$198$		0			0
$199$	3.68613e8	+	6.38456e8i	0.235049	+	0.407117i	0.959287	−	0.282433i	$-0.0911416\pi$
$199$	3.68613e8	+	6.38456e8i	0.235049	+	0.407117i	−0.724238	+	0.689550i	$0.757808\pi$
$200$		0			0
$201$	−7.23650e8	+	1.91803e9i	−0.443348	+	1.17509i
$202$		0			0
$203$	9.17173e8	−	5.62847e8i	0.540091	−	0.331441i
$204$		0			0
$205$	−5.20929e8	+	9.02276e8i	−0.294960	+	0.510886i
$206$		0			0
$207$	2.14200e9	−	2.43460e9i	1.16664	−	1.32601i
$208$		0			0
$209$		−	3.44939e9i		−	1.80783i
$210$		0			0
$211$	−2.16570e9			−1.09262			−0.546309	−	0.837584i	$-0.683968\pi$
$211$	−2.16570e9			−1.09262			−0.546309	+	0.837584i	$0.683968\pi$
$212$		0			0
$213$	−2.28407e9	+	1.87142e9i	−1.10966	+	0.909188i
$214$		0			0
$215$	1.13510e9	+	6.55352e8i	0.531229	+	0.306705i
$216$		0			0
$217$	4.39546e8	−	8.10773e8i	0.198228	−	0.365646i
$218$		0			0
$219$	2.67732e8	−	7.09621e8i	0.116392	−	0.308496i
$220$		0			0
$221$	7.29942e8	−	4.21432e8i	0.305998	−	0.176668i
$222$		0			0
$223$	3.05532e8			0.123548			0.0617741	−	0.998090i	$-0.480324\pi$
$223$	3.05532e8			0.123548			0.0617741	+	0.998090i	$0.480324\pi$
$224$		0			0
$225$	−4.57124e9	+	1.54356e9i	−1.78363	+	0.602274i
$226$		0			0
$227$	−2.77545e9	+	1.60240e9i	−1.04527	+	0.603488i	−0.921322	−	0.388800i	$-0.872890\pi$
$227$	−2.77545e9	+	1.60240e9i	−1.04527	+	0.603488i	−0.123950	+	0.992288i	$0.539556\pi$
$228$		0			0
$229$	4.15474e8	−	7.19622e8i	0.151078	−	0.261675i	−0.780546	−	0.625098i	$-0.785059\pi$
$229$	4.15474e8	−	7.19622e8i	0.151078	−	0.261675i	0.931624	+	0.363423i	$0.118392\pi$
$230$		0			0
$231$	−9.34570e8	−	4.86893e9i	−0.328219	−	1.70996i
$232$		0			0
$233$	2.24927e9	+	1.29862e9i	0.763165	+	0.440614i	0.830431	−	0.557122i	$-0.188094\pi$
$233$	2.24927e9	+	1.29862e9i	0.763165	+	0.440614i	−0.0672659	+	0.997735i	$0.521428\pi$
$234$		0			0
$235$	−1.66933e9	−	2.89136e9i	−0.547356	−	0.948048i
$236$		0			0
$237$	1.31905e8	−	2.16690e7i	0.0418088	−	0.00686825i
$238$		0			0
$239$			2.89223e9i			0.886424i	0.896417	+	0.443212i	$0.146161\pi$
$239$			2.89223e9i			0.886424i	−0.896417	+	0.443212i	$0.853839\pi$
$240$		0			0
$241$	5.13572e8	+	8.89532e8i	0.152241	+	0.263690i	0.932051	−	0.362327i	$-0.118018\pi$
$241$	5.13572e8	+	8.89532e8i	0.152241	+	0.263690i	−0.779810	+	0.626017i	$0.784684\pi$
$242$		0			0
$243$	−9.98782e8	−	3.34067e9i	−0.286448	−	0.958096i
$244$		0			0
$245$	−5.12609e9	+	3.33820e9i	−1.42273	+	0.926506i
$246$		0			0
$247$	3.26683e9	−	5.65832e9i	0.877686	−	1.52020i
$248$		0			0
$249$	2.49160e9	−	2.04146e9i	0.648157	−	0.531060i
$250$		0			0
$251$		−	6.86268e9i		−	1.72902i	−0.502619	−	0.864508i	$-0.667630\pi$
$251$		−	6.86268e9i		−	1.72902i	0.502619	−	0.864508i	$-0.332370\pi$
$252$		0			0
$253$	−1.25996e10			−3.07520
$254$		0			0
$255$	9.50867e8	+	1.16053e9i	0.224884	+	0.274471i
$256$		0			0
$257$	5.48029e9	+	3.16405e9i	1.25624	+	0.725288i	0.972341	−	0.233568i	$-0.0750401\pi$
$257$	5.48029e9	+	3.16405e9i	1.25624	+	0.725288i	0.283895	+	0.958855i	$0.408373\pi$
$258$		0			0
$259$	1.94691e8	−	5.22210e6i	0.0432660	−	0.00116050i
$260$		0			0
$261$	−2.88315e9	−	5.78300e8i	−0.621306	−	0.124621i
$262$		0			0
$263$	−5.19074e9	+	2.99687e9i	−1.08494	+	0.626391i	−0.932225	−	0.361879i	$-0.882135\pi$
$263$	−5.19074e9	+	2.99687e9i	−1.08494	+	0.626391i	−0.152716	+	0.988270i	$0.548802\pi$
$264$		0			0
$265$	7.47470e8			0.151569
$266$		0			0
$267$	−4.34341e8	−	2.64395e9i	−0.0854645	−	0.520245i
$268$		0			0
$269$	8.94591e8	−	5.16492e8i	0.170850	−	0.0986404i	−0.412137	−	0.911122i	$-0.635217\pi$
$269$	8.94591e8	−	5.16492e8i	0.170850	−	0.0986404i	0.582987	+	0.812482i	$0.301884\pi$
$270$		0			0
$271$	2.31572e9	−	4.01095e9i	0.429348	−	0.743652i	−0.567468	−	0.823396i	$-0.692077\pi$
$271$	2.31572e9	−	4.01095e9i	0.429348	−	0.743652i	0.996815	+	0.0797434i	$0.0254101\pi$
$272$		0			0
$273$	3.07819e9	−	8.87201e9i	0.554172	−	1.59724i
$274$		0			0
$275$	1.62351e10	+	9.37332e9i	2.83872	+	1.63894i
$276$		0			0
$277$	4.68837e9	+	8.12050e9i	0.796348	+	1.37931i	0.921980	+	0.387238i	$0.126571\pi$
$277$	4.68837e9	+	8.12050e9i	0.796348	+	1.37931i	−0.125632	+	0.992077i	$0.540096\pi$
$278$		0			0
$279$	−2.38772e9	+	8.06255e8i	−0.394063	+	0.133062i
$280$		0			0
$281$		−	7.72737e9i		−	1.23939i	−0.784845	−	0.619693i	$-0.787257\pi$
$281$		−	7.72737e9i		−	1.23939i	0.784845	−	0.619693i	$-0.212743\pi$
$282$		0			0
$283$	2.76411e9	+	4.78757e9i	0.430932	+	0.746397i	0.996954	−	0.0779939i	$-0.0248515\pi$
$283$	2.76411e9	+	4.78757e9i	0.430932	+	0.746397i	−0.566022	+	0.824390i	$0.691518\pi$
$284$		0			0
$285$	1.08814e10	+	4.10544e9i	1.64933	+	0.622272i
$286$		0			0
$287$	2.00922e9	−	1.23301e9i	0.296142	−	0.181735i
$288$		0			0
$289$	−3.33553e9	+	5.77731e9i	−0.478161	+	0.828198i
$290$		0			0
$291$	−5.40694e9	−	6.59915e9i	−0.754014	−	0.920272i
$292$		0			0
$293$		−	7.46295e8i		−	0.101261i	−0.998717	−	0.0506303i	$-0.983877\pi$
$293$		−	7.46295e8i		−	0.101261i	0.998717	−	0.0506303i	$-0.0161230\pi$
$294$		0			0
$295$	−1.41621e10			−1.86999
$296$		0			0
$297$	−7.18178e9	+	1.14876e10i	−0.923010	+	1.47639i
$298$		0			0
$299$	−2.06681e10	−	1.19327e10i	−2.58593	−	1.49298i
$300$		0			0
$301$	−1.55118e9	−	2.52769e9i	−0.188972	−	0.307934i
$302$		0			0
$303$	−7.27536e9	−	2.74491e9i	−0.863146	−	0.325655i
$304$		0			0
$305$	9.97008e8	−	5.75623e8i	0.115212	−	0.0665180i
$306$		0			0
$307$	5.81698e9			0.654853			0.327427	−	0.944877i	$-0.393819\pi$
$307$	5.81698e9			0.654853			0.327427	+	0.944877i	$0.393819\pi$
$308$		0			0
$309$	2.89136e9	−	4.74986e8i	0.317153	−	0.0521011i
$310$		0			0
$311$	1.85960e9	−	1.07364e9i	0.198782	−	0.114767i	−0.397305	−	0.917687i	$-0.630055\pi$
$311$	1.85960e9	−	1.07364e9i	0.198782	−	0.114767i	0.596087	+	0.802920i	$0.296721\pi$
$312$		0			0
$313$	−5.44948e9	+	9.43878e9i	−0.567777	+	0.983419i	0.429008	+	0.903301i	$0.358863\pi$
$313$	−5.44948e9	+	9.43878e9i	−0.567777	+	0.983419i	−0.996785	+	0.0801184i	$0.974470\pi$
$314$		0			0
$315$	1.64718e10	+	2.84677e9i	1.67301	+	0.289141i
$316$		0			0
$317$	−1.69827e10	−	9.80498e9i	−1.68179	−	0.970979i	−0.960475	−	0.278367i	$-0.910207\pi$
$317$	−1.69827e10	−	9.80498e9i	−1.68179	−	0.970979i	−0.721310	−	0.692612i	$-0.756460\pi$
$318$		0			0
$319$	5.71275e9	+	9.89478e9i	0.551674	+	0.955528i
$320$		0			0
$321$	6.13891e8	+	3.73692e9i	0.0578191	+	0.351960i
$322$		0			0
$323$		−	2.36190e9i		−	0.216995i
$324$		0			0
$325$	1.77545e10	+	3.07516e10i	1.59138	+	2.75635i
$326$		0			0
$327$	−1.17356e9	+	3.11052e9i	−0.102640	+	0.272046i
$328$		0			0
$329$	2.02552e8	+	7.55157e9i	0.0172883	+	0.644546i
$330$		0			0
$331$	9.70186e9	−	1.68041e10i	0.808245	−	1.39992i	−0.105833	−	0.994384i	$-0.533751\pi$
$331$	9.70186e9	−	1.68041e10i	0.808245	−	1.39992i	0.914078	−	0.405538i	$-0.132916\pi$
$332$		0			0
$333$	−3.99571e8	−	3.51549e8i	−0.0324951	−	0.0285897i
$334$		0			0
$335$			2.68558e10i			2.13236i
$336$		0			0
$337$	1.05798e10			0.820269			0.410134	−	0.912025i	$-0.365482\pi$
$337$	1.05798e10			0.820269			0.410134	+	0.912025i	$0.365482\pi$
$338$		0			0
$339$	1.35585e10	−	1.11090e10i	1.02663	−	0.841158i
$340$		0			0
$341$	8.48012e9	+	4.89600e9i	0.627169	+	0.362096i
$342$		0			0
$343$	1.37965e10	−	1.11231e9i	0.996766	−	0.0803615i
$344$		0			0
$345$	1.49959e10	−	3.97465e10i	1.05851	−	2.80558i
$346$		0			0
$347$	1.26518e10	−	7.30451e9i	0.872638	−	0.503818i	0.00441390	−	0.999990i	$-0.498595\pi$
$347$	1.26518e10	−	7.30451e9i	0.872638	−	0.503818i	0.868224	+	0.496173i	$0.165262\pi$
$348$		0			0
$349$	1.30820e10			0.881806			0.440903	−	0.897555i	$-0.354658\pi$
$349$	1.30820e10			0.881806			0.440903	+	0.897555i	$0.354658\pi$
$350$		0			0
$351$	−2.26604e10	+	1.20423e10i	−1.49293	+	0.793379i
$352$		0			0
$353$	1.30913e10	−	7.55829e9i	0.843112	−	0.486771i	−0.0152086	−	0.999884i	$-0.504841\pi$
$353$	1.30913e10	−	7.55829e9i	0.843112	−	0.486771i	0.858321	+	0.513113i	$0.171508\pi$
$354$		0			0
$355$	−1.93416e10	+	3.35007e10i	−1.21781	+	2.10931i
$356$		0			0
$357$	−6.39927e8	−	3.33389e9i	−0.0393965	−	0.205248i
$358$		0			0
$359$	−5.65058e9	−	3.26236e9i	−0.340185	−	0.196406i	0.320169	−	0.947360i	$-0.396260\pi$
$359$	−5.65058e9	−	3.26236e9i	−0.340185	−	0.196406i	−0.660354	+	0.750955i	$0.729594\pi$
$360$		0			0
$361$	−6.62617e8	−	1.14769e9i	−0.0390152	−	0.0675763i
$362$		0			0
$363$	3.48097e10	−	5.71845e9i	2.00481	−	0.329345i
$364$		0			0
$365$		−	9.93596e9i		−	0.559807i
$366$		0			0
$367$	1.24009e10	+	2.14790e10i	0.683578	+	1.18399i	0.973881	+	0.227058i	$0.0729105\pi$
$367$	1.24009e10	+	2.14790e10i	0.683578	+	1.18399i	−0.290303	+	0.956935i	$0.593756\pi$
$368$		0			0
$369$	−6.31602e9	−	1.26686e9i	−0.340673	−	0.0683320i
$370$		0			0
$371$	−1.48684e9	−	8.06064e8i	−0.0784818	−	0.0425475i
$372$		0			0
$373$	−6.93425e9	+	1.20105e10i	−0.358232	+	0.620476i	−0.987666	−	0.156578i	$-0.949954\pi$
$373$	−6.93425e9	+	1.20105e10i	−0.358232	+	0.620476i	0.629434	+	0.777054i	$0.283287\pi$
$374$		0			0
$375$	−2.29210e10	+	1.87801e10i	−1.15907	+	0.949669i
$376$		0			0
$377$			2.16416e10i			1.07133i
$378$		0			0
$379$	−2.24421e10			−1.08770			−0.543848	−	0.839184i	$-0.683033\pi$
$379$	−2.24421e10			−1.08770			−0.543848	+	0.839184i	$0.683033\pi$
$380$		0			0
$381$	−4.92399e9	−	6.00971e9i	−0.233678	−	0.285203i
$382$		0			0
$383$	−4.79035e9	−	2.76571e9i	−0.222624	−	0.128532i	0.384541	−	0.923108i	$-0.374360\pi$
$383$	−4.79035e9	−	2.76571e9i	−0.222624	−	0.128532i	−0.607165	+	0.794576i	$0.707693\pi$
$384$		0			0
$385$	−3.39711e10	−	5.53568e10i	−1.54621	−	2.51958i
$386$		0			0
$387$	−1.59377e9	+	7.94584e9i	−0.0710529	+	0.354239i
$388$		0			0
$389$	−1.66098e10	+	9.58967e9i	−0.725381	+	0.418799i	−0.816730	−	0.577020i	$-0.804215\pi$
$389$	−1.66098e10	+	9.58967e9i	−0.725381	+	0.418799i	0.0913491	+	0.995819i	$0.470882\pi$
$390$		0			0
$391$	−8.62728e9			−0.369120
$392$		0			0
$393$	−4.29392e9	−	2.61382e10i	−0.180005	−	1.09574i
$394$		0			0
$395$	1.51656e9	−	8.75586e8i	0.0622976	−	0.0359675i
$396$		0			0
$397$	−9.78644e8	+	1.69506e9i	−0.0393970	+	0.0682376i	−0.885052	−	0.465493i	$-0.845877\pi$
$397$	−9.78644e8	+	1.69506e9i	−0.0393970	+	0.0682376i	0.845655	+	0.533731i	$0.179210\pi$
$398$		0			0
$399$	−1.72177e10	−	1.99008e10i	−0.679334	−	0.785198i
$400$		0			0
$401$	4.95809e9	+	2.86256e9i	0.191751	+	0.110707i	0.592802	−	0.805348i	$-0.298022\pi$
$401$	4.95809e9	+	2.86256e9i	0.191751	+	0.110707i	−0.401051	+	0.916056i	$0.631355\pi$
$402$		0			0
$403$	9.27376e9	+	1.60626e10i	0.351589	+	0.608970i
$404$		0			0
$405$	−2.76909e10	−	3.63280e10i	−1.02924	−	1.35027i
$406$		0			0
$407$			2.06787e9i			0.0753608i
$408$		0			0
$409$	−1.00640e10	−	1.74313e10i	−0.359646	−	0.622925i	0.628256	−	0.778007i	$-0.283769\pi$
$409$	−1.00640e10	−	1.74313e10i	−0.359646	−	0.622925i	−0.987902	+	0.155082i	$0.950436\pi$
$410$		0			0
$411$	−3.45147e10	−	1.30220e10i	−1.20959	−	0.456363i
$412$		0			0
$413$	2.81708e10	+	1.52723e10i	0.968274	+	0.524933i
$414$		0			0
$415$	2.10990e10	−	3.65445e10i	0.711327	−	1.23205i
$416$		0			0
$417$	2.49702e10	+	3.04761e10i	0.825807	+	1.00789i
$418$		0			0
$419$		−	2.99362e10i		−	0.971272i	−0.874161	−	0.485636i	$-0.838588\pi$
$419$		−	2.99362e10i		−	0.971272i	0.874161	−	0.485636i	$-0.161412\pi$
$420$		0			0
$421$	−4.22444e10			−1.34475			−0.672374	−	0.740212i	$-0.734725\pi$
$421$	−4.22444e10			−1.34475			−0.672374	+	0.740212i	$0.734725\pi$
$422$		0			0
$423$	1.36357e10	−	1.54983e10i	0.425908	−	0.484087i
$424$		0			0
$425$	1.11166e10	+	6.41818e9i	0.340735	+	0.196723i
$426$		0			0
$427$	−2.60396e9	+	6.98447e7i	−0.0783291	+	0.00210098i
$428$		0			0
$429$	9.32880e10	+	3.51965e10i	2.75421	+	1.03913i
$430$		0			0
$431$	1.33948e10	−	7.73348e9i	0.388174	−	0.224112i	−0.293195	−	0.956053i	$-0.594718\pi$
$431$	1.33948e10	−	7.73348e9i	0.388174	−	0.224112i	0.681369	+	0.731940i	$0.261385\pi$
$432$		0			0
$433$	−1.60149e10			−0.455588			−0.227794	−	0.973709i	$-0.573151\pi$
$433$	−1.60149e10			−0.455588			−0.227794	+	0.973709i	$0.573151\pi$
$434$		0			0
$435$	−3.80133e10	+	6.24473e9i	−1.06164	+	0.174404i
$436$		0			0
$437$	−5.79167e10	+	3.34382e10i	−1.58810	+	0.916890i
$438$		0			0
$439$	−3.36434e10	+	5.82720e10i	−0.905819	+	1.56892i	−0.0860042	+	0.996295i	$0.527410\pi$
$439$	−3.36434e10	+	5.82720e10i	−0.905819	+	1.56892i	−0.819814	+	0.572629i	$0.805923\pi$
$440$		0			0
$441$	−2.96952e10	−	2.34257e10i	−0.785112	−	0.619353i
$442$		0			0
$443$	1.50960e10	+	8.71565e9i	0.391963	+	0.226300i	0.683011	−	0.730409i	$-0.260670\pi$
$443$	1.50960e10	+	8.71565e9i	0.391963	+	0.226300i	−0.291047	+	0.956709i	$0.594004\pi$
$444$		0			0
$445$	−1.75505e10	−	3.03984e10i	−0.447559	−	0.775195i
$446$		0			0
$447$	5.10602e9	+	3.10817e10i	0.127895	+	0.778528i
$448$		0			0
$449$			1.07566e10i			0.264662i	0.991206	+	0.132331i	$0.0422461\pi$
$449$			1.07566e10i			0.264662i	−0.991206	+	0.132331i	$0.957754\pi$
$450$		0			0
$451$	1.25147e10	+	2.16761e10i	0.302493	+	0.523933i
$452$		0			0
$453$	1.33374e10	−	3.53507e10i	0.316723	−	0.839471i
$454$		0			0
$455$	−3.29862e9	−	1.22979e11i	−0.0769638	−	2.86937i
$456$		0			0
$457$	3.94664e10	−	6.83578e10i	0.904821	−	1.56720i	0.0836646	−	0.996494i	$-0.473338\pi$
$457$	3.94664e10	−	6.83578e10i	0.904821	−	1.56720i	0.821157	−	0.570703i	$-0.193329\pi$
$458$		0			0
$459$	−4.91757e9	+	7.86586e9i	−0.110790	+	0.177213i
$460$		0			0
$461$		−	2.33933e10i		−	0.517949i	−0.965884	−	0.258975i	$-0.916615\pi$
$461$		−	2.33933e10i		−	0.517949i	0.965884	−	0.258975i	$-0.0833846\pi$
$462$		0			0
$463$	7.91719e9			0.172285			0.0861425	−	0.996283i	$-0.472546\pi$
$463$	7.91719e9			0.172285			0.0861425	+	0.996283i	$0.472546\pi$
$464$		0			0
$465$	−2.55379e10	+	2.09242e10i	−0.546226	+	0.447544i
$466$		0			0
$467$	2.23524e10	+	1.29051e10i	0.469954	+	0.271328i	0.716221	−	0.697874i	$-0.245870\pi$
$467$	2.23524e10	+	1.29051e10i	0.469954	+	0.271328i	−0.246266	+	0.969202i	$0.579204\pi$
$468$		0			0
$469$	2.89611e10	−	5.34207e10i	0.598582	−	1.10413i
$470$		0			0
$471$	−1.39759e10	+	3.70431e10i	−0.283986	+	0.752703i
$472$		0			0
$473$	2.72696e10	−	1.57441e10i	0.544796	−	0.314538i
$474$		0			0
$475$	9.95040e10			1.95464
$476$		0			0
$477$	1.47855e9	+	4.37872e9i	0.0285603	+	0.0845812i
$478$		0			0
$479$	−5.66500e10	+	3.27069e10i	−1.07611	+	0.621295i	−0.929845	−	0.367950i	$-0.880060\pi$
$479$	−5.66500e10	+	3.27069e10i	−1.07611	+	0.621295i	−0.146268	+	0.989245i	$0.546726\pi$
$480$		0			0
$481$	−1.95843e9	+	3.39209e9i	−0.0365870	+	0.0633705i
$482$		0			0
$483$	−7.26916e10	+	6.28909e10i	−1.33566	+	1.15558i
$484$		0			0
$485$	−9.67905e10	−	5.58820e10i	−1.74931	−	1.00996i
$486$		0			0
$487$	−2.25828e10	−	3.91146e10i	−0.401479	−	0.695381i	0.592426	−	0.805625i	$-0.298170\pi$
$487$	−2.25828e10	−	3.91146e10i	−0.401479	−	0.695381i	−0.993905	+	0.110244i	$0.964837\pi$
$488$		0			0
$489$	7.02974e10	−	1.15483e10i	1.22943	−	0.201968i
$490$		0			0
$491$			8.59974e10i			1.47965i	0.672798	+	0.739826i	$0.265092\pi$
$491$			8.59974e10i			1.47965i	−0.672798	+	0.739826i	$0.734908\pi$
$492$		0			0
$493$	3.91168e9	+	6.77523e9i	0.0662180	+	0.114693i
$494$		0			0
$495$	−3.49038e10	+	1.74015e11i	−0.581370	+	2.89846i
$496$		0			0
$497$	7.46005e10	−	4.57805e10i	1.22269	−	0.750335i
$498$		0			0
$499$	2.15422e10	−	3.73121e10i	0.347446	−	0.601794i	−0.638349	−	0.769747i	$-0.720382\pi$
$499$	2.15422e10	−	3.73121e10i	0.347446	−	0.601794i	0.985795	+	0.167953i	$0.0537157\pi$
$500$		0			0
$501$	4.06980e10	−	3.33455e10i	0.645985	−	0.529281i
$502$		0			0
$503$			4.51381e10i			0.705134i	0.935787	+	0.352567i	$0.114691\pi$
$503$			4.51381e10i			0.705134i	−0.935787	+	0.352567i	$0.885309\pi$
$504$		0			0
$505$	−1.01868e11			−1.56629
$506$		0			0
$507$	7.78181e10	+	9.49767e10i	1.17774	+	1.43742i
$508$		0			0
$509$	−3.76753e9	−	2.17518e9i	−0.0561287	−	0.0324059i	0.471673	−	0.881774i	$-0.343650\pi$
$509$	−3.76753e9	−	2.17518e9i	−0.0561287	−	0.0324059i	−0.527802	+	0.849368i	$0.676984\pi$
$510$		0			0
$511$	−1.07148e10	+	1.97643e10i	−0.157146	+	0.289866i
$512$		0			0
$513$	−2.52559e9	+	7.18649e10i	−0.0364664	+	1.03764i
$514$		0			0
$515$	3.32430e10	−	1.91929e10i	0.472576	−	0.272842i
$516$		0			0
$517$	−8.02073e10			−1.12267
$518$		0			0
$519$	−1.02953e10	−	6.26703e10i	−0.141896	−	0.863758i
$520$		0			0
$521$	3.06485e10	−	1.76949e10i	0.415967	−	0.240159i	−0.277383	−	0.960759i	$-0.589467\pi$
$521$	3.06485e10	−	1.76949e10i	0.415967	−	0.240159i	0.693350	+	0.720601i	$0.256134\pi$
$522$		0			0
$523$	5.86284e10	−	1.01547e11i	0.783613	−	1.35726i	−0.146211	−	0.989253i	$-0.546708\pi$
$523$	5.86284e10	−	1.01547e11i	0.783613	−	1.35726i	0.929824	−	0.368004i	$-0.119959\pi$
$524$		0			0
$525$	1.40453e11	−	2.69594e10i	1.84882	−	0.354873i
$526$		0			0
$527$	5.80658e9	+	3.35243e9i	0.0752797	+	0.0434628i
$528$		0			0
$529$	8.29841e10	+	1.43733e11i	1.05967	+	1.83541i
$530$		0			0
$531$	−2.80137e10	−	8.29624e10i	−0.352365	−	1.04353i
$532$		0			0
$533$			4.74095e10i			0.587431i
$534$		0			0
$535$	2.48057e10	+	4.29646e10i	0.302786	+	0.524440i
$536$		0			0
$537$	−1.29794e11	−	4.89699e10i	−1.56084	−	0.588887i
$538$		0			0
$539$	7.87796e9	+	1.46748e11i	0.0933381	+	1.73867i
$540$		0			0
$541$	1.21675e9	−	2.10747e9i	0.0142041	−	0.0246021i	−0.858836	−	0.512251i	$-0.828812\pi$
$541$	1.21675e9	−	2.10747e9i	0.0142041	−	0.0246021i	0.873040	+	0.487648i	$0.162145\pi$
$542$		0			0
$543$	−1.52910e10	−	1.86626e10i	−0.175888	−	0.214671i
$544$		0			0
$545$			4.35528e10i			0.493663i
$546$		0			0
$547$	5.90988e10			0.660130			0.330065	−	0.943958i	$-0.392929\pi$
$547$	5.90988e10			0.660130			0.330065	+	0.943958i	$0.392929\pi$
$548$		0			0
$549$	5.34420e9	+	4.70191e9i	0.0588292	+	0.0517589i
$550$		0			0
$551$	5.25198e10	+	3.03223e10i	0.569793	+	0.328970i
$552$		0			0
$553$	−3.96091e9	+	1.06241e8i	−0.0423540	+	0.00113604i
$554$		0			0
$555$	−6.52328e9	−	2.46116e9i	−0.0687534	−	0.0259399i
$556$		0			0
$557$	−6.37318e10	+	3.67955e10i	−0.662118	+	0.382274i	−0.793083	−	0.609113i	$-0.791526\pi$
$557$	−6.37318e10	+	3.67955e10i	−0.662118	+	0.382274i	0.130966	+	0.991387i	$0.458192\pi$
$558$		0			0
$559$	5.96433e10			0.610822
$560$		0			0
$561$	3.55669e10	−	5.84284e9i	0.359083	−	0.0589892i
$562$		0			0
$563$	−1.04603e11	+	6.03927e10i	−1.04115	+	0.601106i	−0.920158	−	0.391548i	$-0.871940\pi$
$563$	−1.04603e11	+	6.03927e10i	−1.04115	+	0.601106i	−0.120988	+	0.992654i	$0.538606\pi$
$564$		0			0
$565$	1.14815e11	−	1.98865e11i	1.12669	−	1.95148i
$566$		0			0
$567$	1.59060e10	+	1.02124e11i	0.153896	+	0.988087i
$568$		0			0
$569$	4.21469e10	+	2.43336e10i	0.402084	+	0.232143i	0.687383	−	0.726295i	$-0.258759\pi$
$569$	4.21469e10	+	2.43336e10i	0.402084	+	0.232143i	−0.285299	+	0.958439i	$0.592093\pi$
$570$		0			0
$571$	−5.39066e10	−	9.33689e10i	−0.507104	−	0.878330i	−0.999966	−	0.00822288i	$-0.997383\pi$
$571$	−5.39066e10	−	9.33689e10i	−0.507104	−	0.878330i	0.492862	−	0.870108i	$-0.335951\pi$
$572$		0			0
$573$	5.06576e9	+	3.08366e10i	0.0469923	+	0.286054i
$574$		0			0
$575$		−	3.63458e11i		−	3.32493i
$576$		0			0
$577$	8.42746e10	+	1.45968e11i	0.760315	+	1.31690i	0.942688	+	0.333675i	$0.108289\pi$
$577$	8.42746e10	+	1.45968e11i	0.760315	+	1.31690i	−0.182373	+	0.983229i	$0.558378\pi$
$578$		0			0
$579$	1.41482e10	−	3.74996e10i	0.125889	−	0.333666i
$580$		0			0
$581$	−8.13787e10	+	4.99401e10i	−0.714177	+	0.438274i
$582$		0			0
$583$	8.97855e9	−	1.55513e10i	0.0777199	−	0.134615i
$584$		0			0
$585$	−2.22061e11	+	2.52395e11i	−1.89605	+	2.15505i
$586$		0			0
$587$			9.01435e10i			0.759245i	0.925141	+	0.379623i	$0.123946\pi$
$587$			9.01435e10i			0.759245i	−0.925141	+	0.379623i	$0.876054\pi$
$588$		0			0
$589$	5.19743e10			0.431845
$590$		0			0
$591$	1.02953e11	−	8.43530e10i	0.843893	−	0.691434i
$592$		0			0
$593$	3.87120e9	+	2.23504e9i	0.0313060	+	0.0180745i	0.515571	−	0.856847i	$-0.327580\pi$
$593$	3.87120e9	+	2.23504e9i	0.0313060	+	0.0180745i	−0.484265	+	0.874921i	$0.660913\pi$
$594$		0			0
$595$	−2.32610e10	−	3.79044e10i	−0.185593	−	0.302428i
$596$		0			0
$597$	−2.10795e10	+	5.58710e10i	−0.165945	+	0.439834i
$598$		0			0
$599$	−1.23569e10	+	7.13425e9i	−0.0959846	+	0.0554167i	−0.547224	−	0.836986i	$-0.684315\pi$
$599$	−1.23569e10	+	7.13425e9i	−0.0959846	+	0.0554167i	0.451239	+	0.892403i	$0.350982\pi$
$600$		0			0
$601$	−1.45261e11			−1.11340			−0.556700	−	0.830714i	$-0.687933\pi$
$601$	−1.45261e11			−1.11340			−0.556700	+	0.830714i	$0.687933\pi$
$602$		0			0
$603$	−1.57323e11	+	5.31230e10i	−1.18994	+	0.401803i
$604$		0			0
$605$	4.00219e11	−	2.31067e11i	2.98728	−	1.72471i
$606$		0			0
$607$	6.81122e10	−	1.17974e11i	0.501730	−	0.869022i	−0.498268	−	0.867023i	$-0.666030\pi$
$607$	6.81122e10	−	1.17974e11i	0.501730	−	0.869022i	0.999998	−	0.00199861i	$-0.000636177\pi$
$608$		0			0
$609$	8.23489e10	+	2.85714e10i	0.598672	+	0.207712i
$610$		0			0
$611$	−1.31571e11	−	7.59623e10i	−0.944048	−	0.545046i
$612$		0			0
$613$	−6.52558e10	−	1.13026e11i	−0.462144	−	0.800456i	0.536924	−	0.843631i	$-0.319586\pi$
$613$	−6.52558e10	−	1.13026e11i	−0.462144	−	0.800456i	−0.999068	+	0.0431743i	$0.986253\pi$
$614$		0			0
$615$	−8.32743e10	+	1.36801e10i	−0.582118	+	0.0956288i
$616$		0			0
$617$			8.16488e10i			0.563390i	0.959504	+	0.281695i	$0.0908966\pi$
$617$			8.16488e10i			0.563390i	−0.959504	+	0.281695i	$0.909103\pi$
$618$		0			0
$619$	−6.03552e10	−	1.04538e11i	−0.411104	−	0.712053i	0.583907	−	0.811821i	$-0.301523\pi$
$619$	−6.03552e10	−	1.04538e11i	−0.411104	−	0.712053i	−0.995011	+	0.0997675i	$0.968190\pi$
$620$		0			0
$621$	2.62501e11	+	9.22521e9i	1.76508	+	0.0620311i
$622$		0			0
$623$	2.12954e9	+	7.93938e10i	0.0141362	+	0.527029i
$624$		0			0
$625$	−5.04680e10	+	8.74132e10i	−0.330747	+	0.572871i
$626$		0			0
$627$	2.16122e11	−	1.77077e11i	1.39839	−	1.14575i
$628$		0			0
$629$			1.41593e9i			0.00904562i
$630$		0			0
$631$	−2.00502e11			−1.26474			−0.632369	−	0.774667i	$-0.717917\pi$
$631$	−2.00502e11			−1.26474			−0.632369	+	0.774667i	$0.717917\pi$
$632$		0			0
$633$	−1.11178e11	−	1.35692e11i	−0.692473	−	0.845161i
$634$		0			0
$635$	−8.81451e10	−	5.08906e10i	−0.542130	−	0.312999i
$636$		0			0
$637$	−1.26058e11	+	2.48184e11i	−0.765621	+	1.50735i
$638$		0			0
$639$	−2.34508e11	−	4.70374e10i	−1.40655	−	0.282124i
$640$		0			0
$641$	1.00770e11	−	5.81796e10i	0.596897	−	0.344619i	−0.170923	−	0.985284i	$-0.554675\pi$
$641$	1.00770e11	−	5.81796e10i	0.596897	−	0.344619i	0.767820	+	0.640666i	$0.221342\pi$
$642$		0			0
$643$	1.01456e11			0.593520			0.296760	−	0.954952i	$-0.404094\pi$
$643$	1.01456e11			0.593520			0.296760	+	0.954952i	$0.404094\pi$
$644$		0			0
$645$	1.72102e10	+	1.04763e11i	0.0994367	+	0.605297i
$646$		0			0
$647$	−1.62313e11	+	9.37115e10i	−0.926267	+	0.534781i	−0.885629	−	0.464393i	$-0.846272\pi$
$647$	−1.62313e11	+	9.37115e10i	−0.926267	+	0.534781i	−0.0406381	+	0.999174i	$0.512939\pi$
$648$		0			0
$649$	−1.70114e11	+	2.94646e11i	−0.958874	+	1.66082i
$650$		0			0
$651$	7.33634e10	−	1.40818e10i	0.408466	−	0.0784033i
$652$		0			0
$653$	−8.56477e10	−	4.94487e10i	−0.471045	−	0.271958i	0.245632	−	0.969363i	$-0.421005\pi$
$653$	−8.56477e10	−	4.94487e10i	−0.471045	−	0.271958i	−0.716677	+	0.697405i	$0.754338\pi$
$654$		0			0
$655$	−1.73506e11	−	3.00520e11i	−0.942645	−	1.63271i
$656$		0			0
$657$	5.82055e10	−	1.96541e10i	0.312394	−	0.105485i
$658$		0			0
$659$		−	2.12301e10i		−	0.112567i	−0.998415	−	0.0562835i	$-0.982075\pi$
$659$		−	2.12301e10i		−	0.112567i	0.998415	−	0.0562835i	$-0.0179251\pi$
$660$		0			0
$661$	−7.28948e10	−	1.26258e11i	−0.381848	−	0.661381i	0.609478	−	0.792803i	$-0.291379\pi$
$661$	−7.28948e10	−	1.26258e11i	−0.381848	−	0.661381i	−0.991326	+	0.131422i	$0.958046\pi$
$662$		0			0
$663$	6.38769e10	+	2.41000e10i	0.330590	+	0.124728i
$664$		0			0
$665$	−3.03068e11	−	1.64303e11i	−1.54972	−	0.840154i
$666$		0			0
$667$	1.10758e11	−	1.91839e11i	0.559594	−	0.969245i
$668$		0			0
$669$	1.56847e10	+	1.91431e10i	0.0783016	+	0.0955669i
$670$		0			0
$671$		−	2.76574e10i		−	0.136433i
$672$		0			0
$673$	−2.32849e9			−0.0113505			−0.00567525	−	0.999984i	$-0.501806\pi$
$673$	−2.32849e9			−0.0113505			−0.00567525	+	0.999984i	$0.501806\pi$
$674$		0			0
$675$	−3.31380e11	−	2.07172e11i	−1.59629	−	0.997964i
$676$		0			0
$677$	−1.98662e11	−	1.14698e11i	−0.945715	−	0.546009i	−0.0539676	−	0.998543i	$-0.517187\pi$
$677$	−1.98662e11	−	1.14698e11i	−0.945715	−	0.546009i	−0.891747	+	0.452534i	$0.850520\pi$
$678$		0			0
$679$	1.32270e11	+	2.15536e11i	0.622273	+	1.01401i
$680$		0			0
$681$	−2.42878e11	−	9.16351e10i	−1.12928	−	0.426063i
$682$		0			0
$683$	1.49702e11	−	8.64304e10i	0.687930	−	0.397177i	−0.114906	−	0.993376i	$-0.536657\pi$
$683$	1.49702e11	−	8.64304e10i	0.687930	−	0.397177i	0.802836	+	0.596200i	$0.203323\pi$
$684$		0			0
$685$	−4.83268e11			−2.19496
$686$		0			0
$687$	6.64165e10	−	1.09107e10i	0.298160	−	0.0489810i
$688$		0			0
$689$	2.94565e10	−	1.70067e10i	0.130709	−	0.0754647i
$690$		0			0
$691$	−3.67751e10	+	6.36963e10i	−0.161303	+	0.279384i	−0.935336	−	0.353760i	$-0.884903\pi$
$691$	−3.67751e10	+	6.36963e10i	−0.161303	+	0.279384i	0.774033	+	0.633145i	$0.218236\pi$
$692$		0			0
$693$	2.57086e11	−	3.08505e11i	1.11467	−	1.33761i
$694$		0			0
$695$	4.46996e11	+	2.58074e11i	1.91586	+	1.10613i
$696$		0			0
$697$	8.56919e9	+	1.48423e10i	0.0363085	+	0.0628882i
$698$		0			0
$699$	3.41030e10	+	2.07594e11i	0.142851	+	0.869572i
$700$		0			0
$701$			3.38530e11i			1.40193i	0.713197	+	0.700963i	$0.247246\pi$
$701$			3.38530e11i			1.40193i	−0.713197	+	0.700963i	$0.752754\pi$
$702$		0			0
$703$	5.48795e9	+	9.50541e9i	0.0224693	+	0.0389179i
$704$		0			0
$705$	9.54621e10	−	2.53021e11i	0.386433	−	1.02424i
$706$		0			0
$707$	2.02632e11	+	1.09854e11i	0.811020	+	0.439680i
$708$		0			0
$709$	−1.45941e11	+	2.52777e11i	−0.577553	+	1.00035i	0.418207	+	0.908352i	$0.362659\pi$
$709$	−1.45941e11	+	2.52777e11i	−0.577553	+	1.00035i	−0.995759	+	0.0919984i	$0.970675\pi$
$710$		0			0
$711$	8.12911e9	+	7.15212e9i	0.0318101	+	0.0279870i
$712$		0			0
$713$		−	1.89846e11i		−	0.734588i
$714$		0			0
$715$	1.30620e12			4.99787
$716$		0			0
$717$	−1.81213e11	+	1.48475e11i	−0.685666	+	0.561792i
$718$		0			0
$719$	2.25682e11	+	1.30298e11i	0.844465	+	0.487552i	0.858779	−	0.512346i	$-0.171223\pi$
$719$	2.25682e11	+	1.30298e11i	0.844465	+	0.487552i	−0.0143148	+	0.999898i	$0.504557\pi$
$720$		0			0
$721$	−8.68232e10	+	2.32882e9i	−0.321288	+	0.00861776i
$722$		0			0
$723$	−2.93691e10	+	7.78426e10i	−0.107482	+	0.284881i
$724$		0			0
$725$	−2.85433e11	+	1.64795e11i	−1.03312	+	0.596474i
$726$		0			0
$727$	−1.50112e11			−0.537374			−0.268687	−	0.963228i	$-0.586590\pi$
$727$	−1.50112e11			−0.537374			−0.268687	+	0.963228i	$0.586590\pi$
$728$		0			0
$729$	1.58037e11	−	2.34074e11i	0.559562	−	0.828789i
$730$		0			0
$731$	1.86722e10	−	1.07804e10i	0.0653923	−	0.0377543i
$732$		0			0
$733$	2.14124e10	−	3.70874e10i	0.0741736	−	0.128472i	−0.826553	−	0.562859i	$-0.809701\pi$
$733$	2.14124e10	−	3.70874e10i	0.0741736	−	0.128472i	0.900727	+	0.434386i	$0.143035\pi$
$734$		0			0
$735$	−4.72306e11	−	1.49806e11i	−1.61836	−	0.513312i
$736$		0			0
$737$	5.58743e11	+	3.22591e11i	1.89384	+	1.09341i
$738$		0			0
$739$	1.43125e11	+	2.47900e11i	0.479885	+	0.831186i	0.999734	−	0.0230727i	$-0.00734494\pi$
$739$	1.43125e11	+	2.47900e11i	0.479885	+	0.831186i	−0.519848	+	0.854258i	$0.674012\pi$
$740$		0			0
$741$	5.22227e11	−	8.57902e10i	1.73215	−	0.284554i
$742$		0			0
$743$		−	3.36516e9i		−	0.0110421i	−0.999985	−	0.00552104i	$-0.998243\pi$
$743$		−	3.36516e9i		−	0.0110421i	0.999985	−	0.00552104i	$-0.00175741\pi$
$744$		0			0
$745$	2.06320e11	+	3.57357e11i	0.669756	+	1.16005i
$746$		0			0
$747$	2.55816e11	+	5.13113e10i	0.821570	+	0.164790i
$748$		0			0
$749$	−3.00986e9	−	1.12214e11i	−0.00956354	−	0.356549i
$750$		0			0
$751$	−2.76841e11	+	4.79503e11i	−0.870304	+	1.50741i	−0.00862178	+	0.999963i	$0.502744\pi$
$751$	−2.76841e11	+	4.79503e11i	−0.870304	+	1.50741i	−0.861682	+	0.507448i	$0.830589\pi$
$752$		0			0
$753$	4.29982e11	−	3.52300e11i	1.33743	−	1.09580i
$754$		0			0
$755$		−	4.94974e11i		−	1.52333i
$756$		0			0
$757$	4.42352e11			1.34705			0.673525	−	0.739164i	$-0.264779\pi$
$757$	4.42352e11			1.34705			0.673525	+	0.739164i	$0.264779\pi$
$758$		0			0
$759$	−6.46807e11	−	7.89426e11i	−1.94898	−	2.37873i
$760$		0			0
$761$	−2.59312e11	−	1.49714e11i	−0.773185	−	0.446399i	0.0608246	−	0.998148i	$-0.480627\pi$
$761$	−2.59312e11	−	1.49714e11i	−0.773185	−	0.446399i	−0.834010	+	0.551750i	$0.813960\pi$
$762$		0			0
$763$	4.69670e10	−	8.66338e10i	0.138578	−	0.255617i
$764$		0			0
$765$	−2.38996e10	+	1.19153e11i	−0.0697823	+	0.347904i
$766$		0			0
$767$	−5.58104e11	+	3.22222e11i	−1.61263	+	0.931051i
$768$		0			0
$769$	−2.59483e11			−0.742000			−0.371000	−	0.928633i	$-0.620985\pi$
$769$	−2.59483e11			−0.742000			−0.371000	+	0.928633i	$0.620985\pi$
$770$		0			0
$771$	8.30909e10	+	5.05796e11i	0.235145	+	1.43139i
$772$		0			0
$773$	−3.89800e11	+	2.25051e11i	−1.09175	+	0.630323i	−0.934042	−	0.357163i	$-0.883744\pi$
$773$	−3.89800e11	+	2.25051e11i	−1.09175	+	0.630323i	−0.157709	+	0.987486i	$0.550411\pi$
$774$		0			0
$775$	−1.41234e11	+	2.44625e11i	−0.391501	+	0.678100i
$776$		0			0
$777$	1.03218e10	+	1.19303e10i	0.0283185	+	0.0327316i
$778$		0			0
$779$	1.15053e11	+	6.64261e10i	0.312428	+	0.180380i
$780$		0			0
$781$	4.64660e11	+	8.04815e11i	1.24891	+	2.16318i
$782$		0			0
$783$	−1.11775e11	−	2.10331e11i	−0.297371	−	0.559574i
$784$		0			0
$785$			5.18670e11i			1.36588i
$786$		0			0
$787$	−9.64843e10	−	1.67116e11i	−0.251511	−	0.435631i	0.712431	−	0.701742i	$-0.247594\pi$
$787$	−9.64843e10	−	1.67116e11i	−0.251511	−	0.435631i	−0.963942	+	0.266112i	$0.914261\pi$
$788$		0			0
$789$	−4.54239e11	−	1.71379e11i	−1.17213	−	0.442232i
$790$		0			0
$791$	−4.42839e11	+	2.71760e11i	−1.13120	+	0.694191i
$792$		0			0
$793$	2.61936e10	−	4.53687e10i	0.0662373	−	0.114726i
$794$		0			0
$795$	3.83719e10	+	4.68327e10i	0.0960604	+	0.117241i
$796$		0			0
$797$		−	6.58238e11i		−	1.63136i	−0.578503	−	0.815680i	$-0.696363\pi$
$797$		−	6.58238e11i		−	1.63136i	0.578503	−	0.815680i	$-0.303637\pi$
$798$		0			0
$799$	−5.49202e10			−0.134755
$800$		0			0
$801$	1.43359e11	−	1.62942e11i	0.348254	−	0.395826i
$802$		0			0
$803$	−2.06720e11	−	1.19350e11i	−0.497189	−	0.287052i
$804$		0			0
$805$	−6.00149e11	+	1.10702e12i	−1.42914	+	2.63615i
$806$		0			0
$807$	7.82852e10	+	2.95361e10i	0.184580	+	0.0696401i
$808$		0			0
$809$	1.94021e11	−	1.12018e11i	0.452956	−	0.261514i	−0.256122	−	0.966644i	$-0.582445\pi$
$809$	1.94021e11	−	1.12018e11i	0.452956	−	0.261514i	0.709078	+	0.705130i	$0.249112\pi$
$810$		0			0
$811$	3.37816e11			0.780903			0.390452	−	0.920623i	$-0.372319\pi$
$811$	3.37816e11			0.780903			0.390452	+	0.920623i	$0.372319\pi$
$812$		0			0
$813$	3.70185e11	−	6.08131e10i	0.847338	−	0.139199i
$814$		0			0
$815$	8.08234e11	−	4.66634e11i	1.83192	−	1.05766i
$816$		0			0
$817$	8.35670e10	−	1.44742e11i	0.187563	−	0.324868i
$818$		0			0
$819$	7.13896e11	−	2.62587e11i	1.58672	−	0.583629i
$820$		0			0
$821$	−5.47981e11	−	3.16377e11i	−1.20613	−	0.696358i	−0.244216	−	0.969721i	$-0.578531\pi$
$821$	−5.47981e11	−	3.16377e11i	−1.20613	−	0.696358i	−0.961911	+	0.273363i	$0.911864\pi$
$822$		0			0
$823$	−1.68609e11	−	2.92040e11i	−0.367521	−	0.636564i	0.621657	−	0.783290i	$-0.286460\pi$
$823$	−1.68609e11	−	2.92040e11i	−0.367521	−	0.636564i	−0.989177	+	0.146726i	$0.953127\pi$
$824$		0			0
$825$	2.46152e11	+	1.49839e12i	0.531359	+	3.23452i
$826$		0			0
$827$		−	7.90389e11i		−	1.68974i	−0.534975	−	0.844868i	$-0.679679\pi$
$827$		−	7.90389e11i		−	1.68974i	0.534975	−	0.844868i	$-0.320321\pi$
$828$		0			0
$829$	−4.43484e11	−	7.68136e11i	−0.938987	−	1.62637i	−0.767365	−	0.641211i	$-0.778433\pi$
$829$	−4.43484e11	−	7.68136e11i	−0.938987	−	1.62637i	−0.171622	−	0.985163i	$-0.554901\pi$
$830$		0			0
$831$	−2.68109e11	+	7.10621e11i	−0.562222	+	1.49016i
$832$		0			0
$833$	5.39426e9	+	1.00482e11i	0.0112035	+	0.208694i
$834$		0			0
$835$	3.44634e11	−	5.96923e11i	0.708944	−	1.22793i
$836$		0			0
$837$	−1.73091e11	−	1.08213e11i	−0.352673	−	0.220484i
$838$		0			0
$839$			8.67757e9i			0.0175126i	0.999962	+	0.00875629i	$0.00278725\pi$
$839$			8.67757e9i			0.0175126i	−0.999962	+	0.00875629i	$0.997213\pi$
$840$		0			0
$841$	2.99372e11			0.598448
$842$		0			0
$843$	4.84158e11	−	3.96690e11i	0.958688	−	0.785490i
$844$		0			0
$845$	1.39303e12	+	8.04268e11i	2.73234	+	1.57752i
$846$		0			0
$847$	−1.04528e12	+	2.80371e10i	−2.03095	+	0.0544752i
$848$		0			0
$849$	−1.58068e11	+	4.18958e11i	−0.304238	+	0.806380i
$850$		0			0
$851$	3.47204e10	−	2.00458e10i	0.0662012	−	0.0382213i
$852$		0			0
$853$	6.07053e11			1.14665			0.573325	−	0.819328i	$-0.305653\pi$
$853$	6.07053e11			1.14665			0.573325	+	0.819328i	$0.305653\pi$
$854$		0			0
$855$	3.01379e11	+	8.92531e11i	0.563960	+	1.67016i
$856$		0			0
$857$	−2.10613e11	+	1.21597e11i	−0.390446	+	0.225424i	−0.682354	−	0.731022i	$-0.739044\pi$
$857$	−2.10613e11	+	1.21597e11i	−0.390446	+	0.225424i	0.291907	+	0.956447i	$0.405710\pi$
$858$		0			0
$859$	4.61093e11	−	7.98636e11i	0.846868	−	1.46682i	−0.0371216	−	0.999311i	$-0.511819\pi$
$859$	4.61093e11	−	7.98636e11i	0.846868	−	1.46682i	0.883989	−	0.467507i	$-0.154848\pi$
$860$		0			0
$861$	1.80399e11	+	6.25903e10i	0.328262	+	0.113892i
$862$		0			0
$863$	−5.34535e11	−	3.08614e11i	−0.963680	−	0.556381i	−0.0663762	−	0.997795i	$-0.521144\pi$
$863$	−5.34535e11	−	3.08614e11i	−0.963680	−	0.556381i	−0.897304	+	0.441414i	$0.854477\pi$
$864$		0			0
$865$	−4.16005e11	−	7.20542e11i	−0.743078	−	1.28705i
$866$		0			0
$867$	−5.33209e11	+	8.75943e10i	−0.943673	+	0.155024i
$868$		0			0
$869$		−	4.20699e10i		−	0.0737722i
$870$		0			0
$871$	6.11035e11	+	1.05834e12i	1.06168	+	1.83888i
$872$		0			0
$873$	1.35901e11	−	6.77543e11i	0.233973	−	1.16649i
$874$		0			0
$875$	7.48629e11	−	4.59416e11i	1.27713	−	0.783743i
$876$		0			0
$877$	−1.27012e11	+	2.19991e11i	−0.214707	+	0.371883i	−0.953182	−	0.302398i	$-0.902213\pi$
$877$	−1.27012e11	+	2.19991e11i	−0.214707	+	0.371883i	0.738475	+	0.674281i	$0.235546\pi$
$878$		0			0
$879$	4.67591e10	−	3.83116e10i	0.0783269	−	0.0641762i
$880$		0			0
$881$			4.71203e11i			0.782175i	0.920353	+	0.391088i	$0.127901\pi$
$881$			4.71203e11i			0.782175i	−0.920353	+	0.391088i	$0.872099\pi$
$882$		0			0
$883$	4.44667e11			0.731463			0.365732	−	0.930720i	$-0.380819\pi$
$883$	4.44667e11			0.731463			0.365732	+	0.930720i	$0.380819\pi$
$884$		0			0
$885$	−7.27021e11	−	8.87327e11i	−1.18515	−	1.44647i
$886$		0			0
$887$	−9.45508e11	−	5.45890e11i	−1.52746	−	0.881882i	−0.999467	−	0.0326349i	$-0.989610\pi$
$887$	−9.45508e11	−	5.45890e11i	−1.52746	−	0.881882i	−0.527996	−	0.849247i	$-0.677057\pi$
$888$		0			0
$889$	1.20455e11	+	1.96285e11i	0.192849	+	0.314253i
$890$		0			0
$891$	−1.08843e12	+	1.39747e11i	−1.72700	+	0.221734i
$892$		0			0
$893$	−3.68690e11	+	2.12864e11i	−0.579770	+	0.334731i
$894$		0			0
$895$	−1.81735e12			−2.83235
$896$		0			0
$897$	−3.13365e11	−	1.90754e12i	−0.484040	−	2.94648i
$898$		0			0
$899$	−1.49091e11	+	8.60779e10i	−0.228252	+	0.131781i
$900$		0			0
$901$	6.14787e9	−	1.06484e10i	0.00932879	−	0.0161579i
$902$		0			0
$903$	7.87414e10	−	2.26950e11i	0.118427	−	0.341333i
$904$		0			0
$905$	−2.73726e11	−	1.58036e11i	−0.408058	−	0.235592i
$906$		0			0
$907$	−4.26148e11	−	7.38110e11i	−0.629696	−	1.09067i	−0.987613	−	0.156912i	$-0.949846\pi$
$907$	−4.26148e11	−	7.38110e11i	−0.629696	−	1.09067i	0.357916	−	0.933754i	$-0.383487\pi$
$908$		0			0
$909$	−2.01503e11	−	5.96750e11i	−0.295139	−	0.874051i
$910$		0			0
$911$			6.71635e11i			0.975124i	0.873088	+	0.487562i	$0.162114\pi$
$911$			6.71635e11i			0.975124i	−0.873088	+	0.487562i	$0.837886\pi$
$912$		0			0
$913$	−5.06879e11	−	8.77941e11i	−0.729494	−	1.26352i
$914$		0			0
$915$	8.72478e10	+	3.29176e10i	0.124471	+	0.0469617i
$916$		0			0
$917$	2.10527e10	+	7.84891e11i	0.0297736	+	1.11002i
$918$		0			0
$919$	2.26505e11	−	3.92318e11i	0.317552	−	0.550016i	−0.662425	−	0.749129i	$-0.730472\pi$
$919$	2.26505e11	−	3.92318e11i	0.317552	−	0.550016i	0.979977	+	0.199112i	$0.0638058\pi$
$920$		0			0
$921$	2.98618e11	+	3.64463e11i	0.415029	+	0.506541i
$922$		0			0
$923$			1.76027e12i			2.42534i
$924$		0			0
$925$	−5.96514e10			−0.0814806
$926$		0			0
$927$	1.78190e11	+	1.56775e11i	0.241304	+	0.212303i
$928$		0			0
$929$	−5.08083e11	−	2.93342e11i	−0.682137	−	0.393832i	0.118522	−	0.992951i	$-0.462184\pi$
$929$	−5.08083e11	−	2.93342e11i	−0.682137	−	0.393832i	−0.800660	+	0.599119i	$0.795518\pi$
$930$		0			0
$931$	4.25670e11	+	6.53652e11i	0.566597	+	0.870057i
$932$		0			0
$933$	1.62733e11	+	6.13972e10i	0.214757	+	0.0810255i
$934$		0			0
$935$	4.08925e11	−	2.36093e11i	0.535054	−	0.308913i
$936$		0			0
$937$	−8.06204e11			−1.04589			−0.522946	−	0.852366i	$-0.675167\pi$
$937$	−8.06204e11			−1.04589			−0.522946	+	0.852366i	$0.675167\pi$
$938$		0			0
$939$	−8.71140e11	+	1.43109e11i	−1.12054	+	0.184079i
$940$		0			0
$941$	1.14040e12	−	6.58411e11i	1.45445	−	0.839727i	0.455721	−	0.890123i	$-0.349381\pi$
$941$	1.14040e12	−	6.58411e11i	1.45445	−	0.839727i	0.998729	+	0.0503950i	$0.0160480\pi$
$942$		0			0
$943$	2.42634e11	−	4.20255e11i	0.306835	−	0.531454i
$944$		0			0
$945$	6.67226e11	+	1.17818e12i	0.836654	+	1.47736i
$946$		0			0
$947$	1.44702e11	+	8.35440e10i	0.179919	+	0.103876i	0.587254	−	0.809402i	$-0.300209\pi$
$947$	1.44702e11	+	8.35440e10i	0.179919	+	0.103876i	−0.407336	+	0.913278i	$0.633542\pi$
$948$		0			0
$949$	−2.26067e11	−	3.91559e11i	−0.278723	−	0.482762i
$950$		0			0
$951$	−2.57488e11	−	1.56740e12i	−0.314801	−	1.91627i
$952$		0			0
$953$		−	7.84730e11i		−	0.951369i	−0.879616	−	0.475684i	$-0.842200\pi$
$953$		−	7.84730e11i		−	0.951369i	0.879616	−	0.475684i	$-0.157800\pi$
$954$		0			0
$955$	2.04693e11	+	3.54539e11i	0.246088	+	0.426237i
$956$		0			0
$957$	−3.26689e11	+	8.65888e11i	−0.389482	+	1.03232i
$958$		0			0
$959$	9.61300e11	+	5.21152e11i	1.13654	+	0.616154i
$960$		0			0
$961$	3.52674e11	−	6.10850e11i	0.413504	−	0.716211i
$962$		0			0
$963$	−2.02622e11	+	2.30300e11i	−0.235603	+	0.267787i
$964$		0			0
$965$		−	5.25062e11i		−	0.605482i
$966$		0			0
$967$	4.41871e11			0.505347			0.252674	−	0.967552i	$-0.418690\pi$
$967$	4.41871e11			0.505347			0.252674	+	0.967552i	$0.418690\pi$
$968$		0			0
$969$	1.47985e11	−	1.21249e11i	0.167850	−	0.137526i
$970$		0			0
$971$	2.33297e11	+	1.34694e11i	0.262442	+	0.151521i	0.625448	−	0.780266i	$-0.284916\pi$
$971$	2.33297e11	+	1.34694e11i	0.262442	+	0.151521i	−0.363006	+	0.931787i	$0.618250\pi$
$972$		0			0
$973$	−6.10846e11	−	9.95387e11i	−0.681522	−	1.11056i
$974$		0			0
$975$	−1.01531e12	+	2.69106e12i	−1.12351	+	2.97787i
$976$		0			0
$977$	1.17941e12	−	6.80935e11i	1.29446	−	0.747356i	0.315017	−	0.949086i	$-0.397990\pi$
$977$	1.17941e12	−	6.80935e11i	1.29446	−	0.747356i	0.979441	+	0.201730i	$0.0646563\pi$
$978$		0			0
$979$	−8.43263e11			−0.917978
$980$		0			0
$981$	−2.55135e11	+	8.61509e10i	−0.275483	+	0.0930216i
$982$		0			0
$983$	−2.24052e11	+	1.29357e11i	−0.239958	+	0.138540i	−0.615157	−	0.788404i	$-0.710908\pi$
$983$	−2.24052e11	+	1.29357e11i	−0.239958	+	0.138540i	0.375199	+	0.926944i	$0.377574\pi$
$984$		0			0
$985$	8.71809e11	−	1.51002e12i	0.926140	−	1.60412i
$986$		0			0
$987$	−4.62746e11	+	4.00356e11i	−0.487611	+	0.421869i
$988$		0			0
$989$	−5.28699e11	−	3.05245e11i	−0.552616	−	0.319053i
$990$		0			0
$991$	−1.28622e11	−	2.22781e11i	−0.133359	−	0.230984i	0.791610	−	0.611026i	$-0.209243\pi$
$991$	−1.28622e11	−	2.22781e11i	−0.133359	−	0.230984i	−0.924969	+	0.380042i	$0.875910\pi$
$992$		0			0
$993$	1.55091e12	−	2.54780e11i	1.59511	−	0.262041i
$994$		0			0
$995$			7.82295e11i			0.798138i
$996$		0			0
$997$	8.26866e11	+	1.43217e12i	0.836863	+	1.44949i	0.892504	+	0.451039i	$0.148946\pi$
$997$	8.26866e11	+	1.43217e12i	0.836863	+	1.44949i	−0.0556413	+	0.998451i	$0.517720\pi$
$998$		0			0
$999$	1.51406e9	−	4.30821e10i	0.00152013	−	0.0432549i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	84.9.p.b.65.14	yes	40
3.2	odd	2	inner	84.9.p.b.65.13	yes	40
7.4	even	3	inner	84.9.p.b.53.13	✓	40
21.11	odd	6	inner	84.9.p.b.53.14	yes	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.9.p.b.53.13	✓	40	7.4	even	3	inner
84.9.p.b.53.14	yes	40	21.11	odd	6	inner
84.9.p.b.65.13	yes	40	3.2	odd	2	inner
84.9.p.b.65.14	yes	40	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+(51.3357 + 62.6550i) q^{3} +(918.968 + 530.566i) q^{5} +(-1255.82 - 2046.39i) q^{7} +(-1290.30 + 6432.87i) q^{9} +O(q^{10})\)	\(q+(51.3357 + 62.6550i) q^{3} +(918.968 + 530.566i) q^{5} +(-1255.82 - 2046.39i) q^{7} +(-1290.30 + 6432.87i) q^{9} +(22077.2 - 12746.3i) q^{11} +48286.6 q^{13} +(13933.2 + 84814.9i) q^{15} +(15116.9 - 8727.72i) q^{17} +(67655.0 - 117182. i) q^{19} +(63748.2 - 183736. i) q^{21} +(-428030. - 247123. i) q^{23} +(367689. + 636856. i) q^{25} +(-469290. + 249392. i) q^{27} +448191. i q^{29} +(192056. + 332652. i) q^{31} +(1.93196e6 + 728907. i) q^{33} +(-68313.3 - 2.54686e6i) q^{35} +(-40558.3 + 70249.1i) q^{37} +(2.47883e6 + 3.02540e6i) q^{39} +981836. i q^{41} +1.23519e6 q^{43} +(-4.59881e6 + 5.22701e6i) q^{45} +(-2.72478e6 - 1.57315e6i) q^{47} +(-2.61063e6 + 5.13980e6i) q^{49} +(1.32287e6 + 499103. i) q^{51} +(610035. - 352204. i) q^{53} +2.70509e7 q^{55} +(1.08151e7 - 1.77669e6i) q^{57} +(-1.15582e7 + 6.67310e6i) q^{59} +(542461. - 939570. i) q^{61} +(1.47846e7 - 5.43808e6i) q^{63} +(4.43739e7 + 2.56193e7i) q^{65} +(1.26543e7 + 2.19179e7i) q^{67} +(-6.48969e6 - 3.95044e7i) q^{69} +3.64547e7i q^{71} +(-4.68177e6 - 8.10906e6i) q^{73} +(-2.10267e7 + 5.57310e7i) q^{75} +(-5.38088e7 - 2.91715e7i) q^{77} +(825142. - 1.42919e6i) q^{79} +(-3.97170e7 - 1.66007e7i) q^{81} -3.97669e7i q^{83} +1.85225e7 q^{85} +(-2.80814e7 + 2.30082e7i) q^{87} +(-2.86471e7 - 1.65394e7i) q^{89} +(-6.06394e7 - 9.88133e7i) q^{91} +(-1.09829e7 + 2.91102e7i) q^{93} +(1.24346e8 - 7.17910e7i) q^{95} -1.05325e8 q^{97} +(5.35089e7 + 1.58466e8i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 81 q^{3} - 34 q^{7} + 4771 q^{9}+O(q^{10}) \) 40 * q + 81 * q^3 - 34 * q^7 + 4771 * q^9	\( 40 q + 81 q^{3} - 34 q^{7} + 4771 q^{9} - 55464 q^{13} + 68482 q^{15} + 311690 q^{19} - 172343 q^{21} + 1766792 q^{25} - 3451932 q^{27} + 31596 q^{31} + 1874885 q^{33} - 1853482 q^{37} + 11217526 q^{39} - 13372600 q^{43} - 527785 q^{45} - 12653462 q^{49} - 1103461 q^{51} + 71577224 q^{55} - 17195214 q^{57} - 21761970 q^{61} + 21945045 q^{63} - 26337350 q^{67} - 5588722 q^{69} + 41115682 q^{73} - 17971730 q^{75} - 120916932 q^{79} - 24550133 q^{81} + 139250060 q^{85} - 16321046 q^{87} + 345074940 q^{91} + 25774675 q^{93} - 707216948 q^{97} - 94510994 q^{99}+O(q^{100}) \) 40 * q + 81 * q^3 - 34 * q^7 + 4771 * q^9 - 55464 * q^13 + 68482 * q^15 + 311690 * q^19 - 172343 * q^21 + 1766792 * q^25 - 3451932 * q^27 + 31596 * q^31 + 1874885 * q^33 - 1853482 * q^37 + 11217526 * q^39 - 13372600 * q^43 - 527785 * q^45 - 12653462 * q^49 - 1103461 * q^51 + 71577224 * q^55 - 17195214 * q^57 - 21761970 * q^61 + 21945045 * q^63 - 26337350 * q^67 - 5588722 * q^69 + 41115682 * q^73 - 17971730 * q^75 - 120916932 * q^79 - 24550133 * q^81 + 139250060 * q^85 - 16321046 * q^87 + 345074940 * q^91 + 25774675 * q^93 - 707216948 * q^97 - 94510994 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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