LMFDB - Embedded newform 84.9.p.b.53.13

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			53.13
Character	$\chi$	$=$	84.53
Dual form			84.9.p.b.65.13

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(28.5930 - 75.7855i) q^{3} +(-918.968 + 530.566i) q^{5} +(-1255.82 + 2046.39i) q^{7} +(-4925.88 - 4333.87i) q^{9} +O(q^{10})$	\(q+(28.5930 - 75.7855i) q^{3} +(-918.968 + 530.566i) q^{5} +(-1255.82 + 2046.39i) q^{7} +(-4925.88 - 4333.87i) q^{9} +(-22077.2 - 12746.3i) q^{11} +48286.6 q^{13} +(13933.2 + 84814.9i) q^{15} +(-15116.9 - 8727.72i) q^{17} +(67655.0 + 117182. i) q^{19} +(119179. + 153685. i) q^{21} +(428030. - 247123. i) q^{23} +(367689. - 636856. i) q^{25} +(-469290. + 249392. i) q^{27} +448191. i q^{29} +(192056. - 332652. i) q^{31} +(-1.59723e6 + 1.30867e6i) q^{33} +(68313.3 - 2.54686e6i) q^{35} +(-40558.3 - 70249.1i) q^{37} +(1.38066e6 - 3.65943e6i) q^{39} +981836. i q^{41} +1.23519e6 q^{43} +(6.82613e6 + 1.36918e6i) q^{45} +(2.72478e6 - 1.57315e6i) q^{47} +(-2.61063e6 - 5.13980e6i) q^{49} +(-1.09367e6 + 896087. i) q^{51} +(-610035. - 352204. i) q^{53} +2.70509e7 q^{55} +(1.08151e7 - 1.77669e6i) q^{57} +(1.15582e7 + 6.67310e6i) q^{59} +(542461. + 939570. i) q^{61} +(1.50548e7 - 4.63771e6i) q^{63} +(-4.43739e7 + 2.56193e7i) q^{65} +(1.26543e7 - 2.19179e7i) q^{67} +(-6.48969e6 - 3.95044e7i) q^{69} +3.64547e7i q^{71} +(-4.68177e6 + 8.10906e6i) q^{73} +(-3.77511e7 - 4.60751e7i) q^{75} +(5.38088e7 - 2.91715e7i) q^{77} +(825142. + 1.42919e6i) q^{79} +(5.48189e6 + 4.26962e7i) q^{81} -3.97669e7i q^{83} +1.85225e7 q^{85} +(3.39664e7 + 1.28151e7i) q^{87} +(2.86471e7 - 1.65394e7i) q^{89} +(-6.06394e7 + 9.88133e7i) q^{91} +(-1.97187e7 - 2.40666e7i) q^{93} +(-1.24346e8 - 7.17910e7i) q^{95} -1.05325e8 q^{97} +(5.35089e7 + 1.58466e8i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/84\mathbb{Z}\right)^\times$.

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

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
$3$	28.5930	−	75.7855i	0.353000	−	0.935623i
$3$	28.5930	−	75.7855i	0.353000	−	0.935623i
$4$		0			0
$5$	−918.968	+	530.566i	−1.47035	+	0.848906i	−0.999446	−	0.0332787i	$-0.989405\pi$
$5$	−918.968	+	530.566i	−1.47035	+	0.848906i	−0.470903	+	0.882185i	$0.656072\pi$
$6$		0			0
$7$	−1255.82	+	2046.39i	−0.523041	+	0.852308i
$7$	−1255.82	+	2046.39i	−0.523041	+	0.852308i
$8$		0			0
$9$	−4925.88	−	4333.87i	−0.750782	−	0.660550i
$10$		0			0
$11$	−22077.2	−	12746.3i	−1.50790	−	0.870586i	−0.999958	−	0.00919529i	$-0.997073\pi$
$11$	−22077.2	−	12746.3i	−1.50790	−	0.870586i	−0.507942	−	0.861391i	$-0.669594\pi$
$12$		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	$-0.366657\pi$
$17$	−15116.9	−	8727.72i	−0.180995	−	0.104497i	−0.587760	+	0.809035i	$0.699990\pi$
$18$		0			0
$19$	67655.0	+	117182.i	0.519141	+	0.899179i	0.999753	+	0.0222452i	$0.00708146\pi$
$19$	67655.0	+	117182.i	0.519141	+	0.899179i	−0.480611	+	0.876934i	$0.659585\pi$
$20$		0			0
$21$	119179.	+	153685.i	0.612805	+	0.790234i
$22$		0			0
$23$	428030.	−	247123.i	1.52955	−	0.883084i	0.530165	−	0.847894i	$-0.322130\pi$
$23$	428030.	−	247123.i	1.52955	−	0.883084i	0.999381	−	0.0351895i	$-0.0112035\pi$
$24$		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	−0.743111	−	0.669168i	$-0.766651\pi$
$31$	192056.	−	332652.i	0.207961	−	0.360199i	0.951072	+	0.308969i	$0.0999839\pi$
$32$		0			0
$33$	−1.59723e6	+	1.30867e6i	−1.34683	+	1.10351i
$34$		0			0
$35$	68313.3	−	2.54686e6i	0.0455232	−	1.69720i
$36$		0			0
$37$	−40558.3	−	70249.1i	−0.0216408	−	0.0374830i	0.855002	−	0.518624i	$-0.173556\pi$
$37$	−40558.3	−	70249.1i	−0.0216408	−	0.0374830i	−0.876643	+	0.481141i	$0.840222\pi$
$38$		0			0
$39$	1.38066e6	−	3.65943e6i	0.596799	−	1.58181i
$40$		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
$45$	6.82613e6	+	1.36918e6i	1.66466	+	0.333895i
$46$		0			0
$47$	2.72478e6	−	1.57315e6i	0.558394	−	0.322389i	−0.194107	−	0.980980i	$-0.562181\pi$
$47$	2.72478e6	−	1.57315e6i	0.558394	−	0.322389i	0.752501	+	0.658592i	$0.228847\pi$
$48$		0			0
$49$	−2.61063e6	−	5.13980e6i	−0.452857	−	0.891583i
$50$		0			0
$51$	−1.09367e6	+	896087.i	−0.161661	+	0.132455i
$52$		0			0
$53$	−610035.	−	352204.i	−0.0773127	−	0.0446365i	0.460845	−	0.887480i	$-0.347546\pi$
$53$	−610035.	−	352204.i	−0.0773127	−	0.0446365i	−0.538158	+	0.842844i	$0.680880\pi$
$54$		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	$-0.147692\pi$
$59$	1.15582e7	+	6.67310e6i	0.953851	+	0.550706i	0.0595758	+	0.998224i	$0.481025\pi$
$60$		0			0
$61$	542461.	+	939570.i	0.0391786	+	0.0678594i	0.884950	−	0.465686i	$-0.154193\pi$
$61$	542461.	+	939570.i	0.0391786	+	0.0678594i	−0.845771	+	0.533546i	$0.820859\pi$
$62$		0			0
$63$	1.50548e7	−	4.63771e6i	0.955681	−	0.294403i
$64$		0			0
$65$	−4.43739e7	+	2.56193e7i	−2.48584	+	1.43520i
$66$		0			0
$67$	1.26543e7	−	2.19179e7i	0.627971	−	1.08768i	−0.359987	−	0.932957i	$-0.617219\pi$
$67$	1.26543e7	−	2.19179e7i	0.627971	−	1.08768i	0.987958	−	0.154721i	$-0.0494479\pi$
$68$		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	−0.936606	−	0.350384i	$-0.886051\pi$
$73$	−4.68177e6	+	8.10906e6i	−0.164861	+	0.285548i	0.771745	+	0.635932i	$0.219384\pi$
$74$		0			0
$75$	−3.77511e7	−	4.60751e7i	−1.19312	−	1.45620i
$76$		0			0
$77$	5.38088e7	−	2.91715e7i	1.53070	−	0.829842i
$78$		0			0
$79$	825142.	+	1.42919e6i	0.0211846	+	0.0366928i	0.876423	−	0.481541i	$-0.159923\pi$
$79$	825142.	+	1.42919e6i	0.0211846	+	0.0366928i	−0.855239	+	0.518234i	$0.826590\pi$
$80$		0			0
$81$	5.48189e6	+	4.26962e7i	0.127347	+	0.991858i
$82$		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
$87$	3.39664e7	+	1.28151e7i	0.592887	+	0.223689i
$88$		0			0
$89$	2.86471e7	−	1.65394e7i	0.456584	−	0.263609i	−0.254023	−	0.967198i	$-0.581754\pi$
$89$	2.86471e7	−	1.65394e7i	0.456584	−	0.263609i	0.710607	+	0.703589i	$0.248420\pi$
$90$		0			0
$91$	−6.06394e7	+	9.88133e7i	−0.884278	+	1.44095i
$92$		0			0
$93$	−1.97187e7	−	2.40666e7i	−0.263601	−	0.321724i
$94$		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	$-0.180728\pi$
$101$	8.31379e7	+	4.79997e7i	0.798939	+	0.461268i	−0.0441610	+	0.999024i	$0.514061\pi$
$102$		0			0
$103$	1.80871e7	+	3.13279e7i	0.160702	+	0.278344i	0.935121	−	0.354330i	$-0.115291\pi$
$103$	1.80871e7	+	3.13279e7i	0.160702	+	0.278344i	−0.774419	+	0.632673i	$0.781958\pi$
$104$		0			0
$105$	−1.91062e8	−	7.79996e7i	−1.57187	−	0.641705i
$106$		0			0
$107$	−4.04894e7	+	2.33766e7i	−0.308892	+	0.178339i	−0.646430	−	0.762973i	$-0.723739\pi$
$107$	−4.04894e7	+	2.33766e7i	−0.308892	+	0.178339i	0.337539	+	0.941312i	$0.390406\pi$
$108$		0			0
$109$	2.05219e7	−	3.55449e7i	0.145382	−	0.251809i	−0.784133	−	0.620592i	$-0.786892\pi$
$109$	2.05219e7	−	3.55449e7i	0.145382	−	0.251809i	0.929515	+	0.368783i	$0.120226\pi$
$110$		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
$117$	−2.37854e8	−	2.09268e8i	−1.26931	−	1.11676i
$118$		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
$123$	7.44089e7	+	2.80736e7i	0.325091	+	0.122653i
$124$		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
$129$	3.53179e7	−	9.36097e7i	0.127537	−	0.338035i
$130$		0			0
$131$	2.83207e8	−	1.63510e8i	0.961654	−	0.555211i	0.0649724	−	0.997887i	$-0.479304\pi$
$131$	2.83207e8	−	1.63510e8i	0.961654	−	0.555211i	0.896682	+	0.442676i	$0.145971\pi$
$132$		0			0
$133$	−3.24763e8	−	8.71094e6i	−1.03791	−	0.0278393i
$134$		0			0
$135$	2.98944e8	−	4.78173e8i	0.900024	−	1.43963i
$136$		0			0
$137$	3.94411e8	+	2.27713e8i	1.11961	+	0.646407i	0.941301	−	0.337568i	$-0.109604\pi$
$137$	3.94411e8	+	2.27713e8i	1.11961	+	0.646407i	0.178308	+	0.983975i	$0.442938\pi$
$138$		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
$147$	−4.64168e8	+	5.08854e7i	−0.994045	+	0.108974i
$148$		0			0
$149$	−3.36769e8	+	1.94434e8i	−0.683262	+	0.394482i	−0.801083	−	0.598553i	$-0.795742\pi$
$149$	−3.36769e8	+	1.94434e8i	−0.683262	+	0.394482i	0.117821	+	0.993035i	$0.462409\pi$
$150$		0			0
$151$	−2.33229e8	+	4.03964e8i	−0.448616	+	0.777026i	−0.998296	−	0.0583495i	$-0.981416\pi$
$151$	−2.33229e8	+	4.03964e8i	−0.448616	+	0.777026i	0.549680	+	0.835375i	$0.314750\pi$
$152$		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	−0.591750	−	0.806122i	$-0.701563\pi$
$157$	2.44394e8	−	4.23303e8i	0.402247	−	0.696712i	0.993997	+	0.109410i	$0.0348961\pi$
$158$		0			0
$159$	−4.41346e7	+	3.61612e7i	−0.0690544	+	0.0565789i
$160$		0			0
$161$	−3.18184e7	+	1.18626e9i	−0.0473560	+	1.76553i
$162$		0			0
$163$	4.39751e8	+	7.61671e8i	0.622955	+	1.07899i	0.988933	+	0.148366i	$0.0474012\pi$
$163$	4.39751e8	+	7.61671e8i	0.622955	+	1.07899i	−0.365978	+	0.930624i	$0.619265\pi$
$164$		0			0
$165$	7.73468e8	−	2.05007e9i	1.04353	−	2.76588i
$166$		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
$171$	1.74590e8	−	8.70432e8i	0.204191	−	1.01801i
$172$		0			0
$173$	6.79031e8	−	3.92039e8i	0.758063	−	0.437668i	−0.0705369	−	0.997509i	$-0.522471\pi$
$173$	6.79031e8	−	3.92039e8i	0.758063	−	0.437668i	0.828600	+	0.559841i	$0.189138\pi$
$174$		0			0
$175$	8.41505e8	+	1.55221e9i	0.897231	+	1.65500i
$176$		0			0
$177$	8.36207e8	−	6.85136e8i	0.851963	−	0.698046i
$178$		0			0
$179$	1.48320e9	+	8.56326e8i	1.44473	+	0.834117i	0.998160	−	0.0606300i	$-0.0193110\pi$
$179$	1.48320e9	+	8.56326e8i	1.44473	+	0.834117i	0.446573	+	0.894747i	$0.352644\pi$
$180$		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
$189$	7.89909e7	−	1.27354e9i	0.0619056	−	0.998082i
$190$		0			0
$191$	−3.34114e8	+	1.92901e8i	−0.251051	+	0.144944i	−0.620245	−	0.784408i	$-0.712967\pi$
$191$	−3.34114e8	+	1.92901e8i	−0.251051	+	0.144944i	0.369195	+	0.929352i	$0.379634\pi$
$192$		0			0
$193$	−2.47406e8	+	4.28520e8i	−0.178312	+	0.308846i	−0.941303	−	0.337564i	$-0.890397\pi$
$193$	−2.47406e8	+	4.28520e8i	−0.178312	+	0.308846i	0.762990	+	0.646410i	$0.223730\pi$
$194$		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	−0.724238	−	0.689550i	$-0.757808\pi$
$199$	3.68613e8	−	6.38456e8i	0.235049	−	0.407117i	0.959287	+	0.282433i	$0.0911416\pi$
$200$		0			0
$201$	−1.29924e9	−	1.58571e9i	−0.795983	−	0.971495i
$202$		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
$207$	−3.17942e9	−	6.37726e8i	−1.73168	−	0.347338i
$208$		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
$213$	2.76273e9	+	1.04235e9i	1.34221	+	0.506401i
$214$		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
$219$	4.80684e8	+	5.86673e8i	0.208969	+	0.255047i
$220$		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	$-0.127110\pi$
$227$	2.77545e9	+	1.60240e9i	1.04527	+	0.603488i	0.123950	+	0.992288i	$0.460444\pi$
$228$		0			0
$229$	4.15474e8	+	7.19622e8i	0.151078	+	0.261675i	0.931624	−	0.363423i	$-0.118392\pi$
$229$	4.15474e8	+	7.19622e8i	0.151078	+	0.261675i	−0.780546	+	0.625098i	$0.785059\pi$
$230$		0			0
$231$	−6.72221e8	−	4.91203e9i	−0.236083	−	1.72509i
$232$		0			0
$233$	−2.24927e9	+	1.29862e9i	−0.763165	+	0.440614i	−0.830431	−	0.557122i	$-0.811906\pi$
$233$	−2.24927e9	+	1.29862e9i	−0.763165	+	0.440614i	0.0672659	+	0.997735i	$0.478572\pi$
$234$		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	−0.779810	−	0.626017i	$-0.784684\pi$
$241$	5.13572e8	−	8.89532e8i	0.152241	−	0.263690i	0.932051	+	0.362327i	$0.118018\pi$
$242$		0			0
$243$	3.39250e9	+	8.05366e8i	0.972959	+	0.230977i
$244$		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
$249$	−3.01376e9	−	1.13706e9i	−0.783990	−	0.295791i
$250$		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
$255$	5.29615e8	−	1.40374e9i	0.125256	−	0.331991i
$256$		0			0
$257$	−5.48029e9	+	3.16405e9i	−1.25624	+	0.725288i	−0.972341	−	0.233568i	$-0.924960\pi$
$257$	−5.48029e9	+	3.16405e9i	−1.25624	+	0.725288i	−0.283895	+	0.958855i	$0.591627\pi$
$258$		0			0
$259$	1.94691e8	+	5.22210e6i	0.0432660	+	0.00116050i
$260$		0			0
$261$	1.94240e9	−	2.20773e9i	0.418578	−	0.475757i
$262$		0			0
$263$	5.19074e9	+	2.99687e9i	1.08494	+	0.626391i	0.932225	−	0.361879i	$-0.117865\pi$
$263$	5.19074e9	+	2.99687e9i	1.08494	+	0.626391i	0.152716	+	0.988270i	$0.451198\pi$
$264$		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	$-0.364783\pi$
$269$	−8.94591e8	−	5.16492e8i	−0.170850	−	0.0986404i	−0.582987	+	0.812482i	$0.698116\pi$
$270$		0			0
$271$	2.31572e9	+	4.01095e9i	0.429348	+	0.743652i	0.996815	−	0.0797434i	$-0.0254101\pi$
$271$	2.31572e9	+	4.01095e9i	0.429348	+	0.743652i	−0.567468	+	0.823396i	$0.692077\pi$
$272$		0			0
$273$	5.75475e9	+	7.42095e9i	1.03604	+	1.33601i
$274$		0			0
$275$	−1.62351e10	+	9.37332e9i	−2.83872	+	1.63894i
$276$		0			0
$277$	4.68837e9	−	8.12050e9i	0.796348	−	1.37931i	−0.125632	−	0.992077i	$-0.540096\pi$
$277$	4.68837e9	−	8.12050e9i	0.796348	−	1.37931i	0.921980	−	0.387238i	$-0.126571\pi$
$278$		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	−0.566022	−	0.824390i	$-0.691518\pi$
$283$	2.76411e9	−	4.78757e9i	0.430932	−	0.746397i	0.996954	+	0.0779939i	$0.0248515\pi$
$284$		0			0
$285$	−8.99612e9	+	7.37087e9i	−1.36357	+	1.11722i
$286$		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
$291$	−3.01156e9	+	7.98212e9i	−0.419972	+	1.11313i
$292$		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
$297$	1.35394e10	+	4.75823e8i	1.74010	+	0.0611533i
$298$		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
$303$	6.01484e9	−	4.92819e9i	0.713598	−	0.584678i
$304$		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	$-0.369945\pi$
$311$	−1.85960e9	−	1.07364e9i	−0.198782	−	0.114767i	−0.596087	+	0.802920i	$0.703279\pi$
$312$		0			0
$313$	−5.44948e9	−	9.43878e9i	−0.567777	−	0.983419i	−0.996785	−	0.0801184i	$-0.974470\pi$
$313$	−5.44948e9	−	9.43878e9i	−0.567777	−	0.983419i	0.429008	−	0.903301i	$-0.358863\pi$
$314$		0			0
$315$	−1.13743e10	+	1.22495e10i	−1.15527	+	1.24416i
$316$		0			0
$317$	1.69827e10	−	9.80498e9i	1.68179	−	0.970979i	0.721310	−	0.692612i	$-0.243540\pi$
$317$	1.69827e10	−	9.80498e9i	1.68179	−	0.970979i	0.960475	−	0.278367i	$-0.0897931\pi$
$318$		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
$327$	−2.10701e9	−	2.57159e9i	−0.184279	−	0.224911i
$328$		0			0
$329$	−2.02552e8	+	7.55157e9i	−0.0172883	+	0.644546i
$330$		0			0
$331$	9.70186e9	+	1.68041e10i	0.808245	+	1.39992i	0.914078	+	0.405538i	$0.132916\pi$
$331$	9.70186e9	+	1.68041e10i	0.808245	+	1.39992i	−0.105833	+	0.994384i	$0.533751\pi$
$332$		0			0
$333$	−1.04665e8	+	5.21813e8i	−0.00851185	+	0.0424364i
$334$		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
$339$	−1.64000e10	−	6.18753e9i	−1.24178	−	0.468509i
$340$		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
$345$	2.69235e10	+	3.28601e10i	1.90045	+	2.31949i
$346$		0			0
$347$	−1.26518e10	−	7.30451e9i	−0.872638	−	0.503818i	−0.00441390	−	0.999990i	$-0.501405\pi$
$347$	−1.26518e10	−	7.30451e9i	−0.872638	−	0.503818i	−0.868224	+	0.496173i	$0.834738\pi$
$348$		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	$-0.495159\pi$
$353$	−1.30913e10	−	7.55829e9i	−0.843112	−	0.486771i	−0.858321	+	0.513113i	$0.828492\pi$
$354$		0			0
$355$	−1.93416e10	−	3.35007e10i	−1.21781	−	2.10931i
$356$		0			0
$357$	−4.60288e8	−	3.36340e9i	−0.0283372	−	0.207065i
$358$		0			0
$359$	5.65058e9	−	3.26236e9i	0.340185	−	0.196406i	−0.320169	−	0.947360i	$-0.603740\pi$
$359$	5.65058e9	−	3.26236e9i	0.340185	−	0.196406i	0.660354	+	0.750955i	$0.270406\pi$
$360$		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	−0.290303	−	0.956935i	$-0.593756\pi$
$367$	1.24009e10	−	2.14790e10i	0.683578	−	1.18399i	0.973881	−	0.227058i	$-0.0729105\pi$
$368$		0			0
$369$	4.25515e9	−	4.83641e9i	0.229514	−	0.260866i
$370$		0			0
$371$	1.48684e9	−	8.06064e8i	0.0784818	−	0.0425475i
$372$		0			0
$373$	−6.93425e9	−	1.20105e10i	−0.358232	−	0.620476i	0.629434	−	0.777054i	$-0.283287\pi$
$373$	−6.93425e9	−	1.20105e10i	−0.358232	−	0.620476i	−0.987666	+	0.156578i	$0.949954\pi$
$374$		0			0
$375$	2.77245e10	+	1.04601e10i	1.40197	+	0.528948i
$376$		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
$381$	−2.74257e9	+	7.26915e9i	−0.130154	+	0.344972i
$382$		0			0
$383$	4.79035e9	−	2.76571e9i	0.222624	−	0.128532i	−0.384541	−	0.923108i	$-0.625640\pi$
$383$	4.79035e9	−	2.76571e9i	0.222624	−	0.128532i	0.607165	+	0.794576i	$0.292307\pi$
$384$		0			0
$385$	−3.39711e10	+	5.53568e10i	−1.54621	+	2.51958i
$386$		0			0
$387$	−6.08442e9	−	5.35317e9i	−0.271253	−	0.238653i
$388$		0			0
$389$	1.66098e10	+	9.58967e9i	0.725381	+	0.418799i	0.816730	−	0.577020i	$-0.195785\pi$
$389$	1.66098e10	+	9.58967e9i	0.725381	+	0.418799i	−0.0913491	+	0.995819i	$0.529118\pi$
$390$		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	0.845655	−	0.533731i	$-0.179210\pi$
$397$	−9.78644e8	−	1.69506e9i	−0.0393970	−	0.0682376i	−0.885052	+	0.465493i	$0.845877\pi$
$398$		0			0
$399$	−9.94610e9	+	2.43632e10i	−0.392429	+	0.961265i
$400$		0			0
$401$	−4.95809e9	+	2.86256e9i	−0.191751	+	0.110707i	−0.592802	−	0.805348i	$-0.701978\pi$
$401$	−4.95809e9	+	2.86256e9i	−0.191751	+	0.110707i	0.401051	+	0.916056i	$0.368645\pi$
$402$		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	−0.987902	−	0.155082i	$-0.950436\pi$
$409$	−1.00640e10	+	1.74313e10i	−0.359646	+	0.622925i	0.628256	+	0.778007i	$0.283769\pi$
$410$		0			0
$411$	2.85347e10	−	2.33796e10i	1.00002	−	0.819351i
$412$		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
$417$	1.39080e10	−	3.68629e10i	0.459959	−	1.21912i
$418$		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
$423$	−2.02398e10	−	4.05968e9i	−0.632186	−	0.126803i
$424$		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
$429$	−7.71250e10	+	6.31915e10i	−2.27702	+	1.86565i
$430$		0			0
$431$	−1.33948e10	−	7.73348e9i	−0.388174	−	0.224112i	0.293195	−	0.956053i	$-0.405282\pi$
$431$	−1.33948e10	−	7.73348e9i	−0.388174	−	0.224112i	−0.681369	+	0.731940i	$0.738615\pi$
$432$		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	−0.819814	−	0.572629i	$-0.805923\pi$
$439$	−3.36434e10	−	5.82720e10i	−0.905819	−	1.56892i	−0.0860042	−	0.996295i	$-0.527410\pi$
$440$		0			0
$441$	−9.41558e9	+	3.66322e10i	−0.248939	+	0.968519i
$442$		0			0
$443$	−1.50960e10	+	8.71565e9i	−0.391963	+	0.226300i	−0.683011	−	0.730409i	$-0.739330\pi$
$443$	−1.50960e10	+	8.71565e9i	−0.391963	+	0.226300i	0.291047	+	0.956709i	$0.405996\pi$
$444$		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
$453$	2.39459e10	+	2.92259e10i	0.568642	+	0.694026i
$454$		0			0
$455$	3.29862e9	−	1.22979e11i	0.0769638	−	2.86937i
$456$		0			0
$457$	3.94664e10	+	6.83578e10i	0.904821	+	1.56720i	0.821157	+	0.570703i	$0.193329\pi$
$457$	3.94664e10	+	6.83578e10i	0.904821	+	1.56720i	0.0836646	+	0.996494i	$0.473338\pi$
$458$		0			0
$459$	9.27081e9	+	3.25810e8i	0.208866	+	0.00734029i
$460$		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
$465$	3.08898e10	+	1.16544e10i	0.660698	+	0.249274i
$466$		0			0
$467$	−2.23524e10	+	1.29051e10i	−0.469954	+	0.271328i	−0.716221	−	0.697874i	$-0.754130\pi$
$467$	−2.23524e10	+	1.29051e10i	−0.469954	+	0.271328i	0.246266	+	0.969202i	$0.420796\pi$
$468$		0			0
$469$	2.89611e10	+	5.34207e10i	0.598582	+	1.10413i
$470$		0			0
$471$	−2.50923e10	−	3.06251e10i	−0.509867	−	0.622291i
$472$		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	$-0.119940\pi$
$479$	5.66500e10	+	3.27069e10i	1.07611	+	0.621295i	0.146268	+	0.989245i	$0.453274\pi$
$480$		0			0
$481$	−1.95843e9	−	3.39209e9i	−0.0365870	−	0.0633705i
$482$		0			0
$483$	8.89914e10	+	3.63301e10i	1.63516	+	0.667540i
$484$		0			0
$485$	9.67905e10	−	5.58820e10i	1.74931	−	1.00996i
$486$		0			0
$487$	−2.25828e10	+	3.91146e10i	−0.401479	+	0.695381i	−0.993905	−	0.110244i	$-0.964837\pi$
$487$	−2.25828e10	+	3.91146e10i	−0.401479	+	0.695381i	0.592426	+	0.805625i	$0.298170\pi$
$488$		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
$495$	−1.33250e11	−	1.17235e11i	−2.21945	−	1.95271i
$496$		0			0
$497$	−7.46005e10	−	4.57805e10i	−1.22269	−	0.750335i
$498$		0			0
$499$	2.15422e10	+	3.73121e10i	0.347446	+	0.601794i	0.985795	−	0.167953i	$-0.0537157\pi$
$499$	2.15422e10	+	3.73121e10i	0.347446	+	0.601794i	−0.638349	+	0.769747i	$0.720382\pi$
$500$		0			0
$501$	−4.92271e10	−	1.85728e10i	−0.781363	−	0.294799i
$502$		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
$507$	4.33432e10	−	1.14881e11i	0.655978	−	1.73866i
$508$		0			0
$509$	3.76753e9	−	2.17518e9i	0.0561287	−	0.0324059i	−0.471673	−	0.881774i	$-0.656350\pi$
$509$	3.76753e9	−	2.17518e9i	0.0561287	−	0.0324059i	0.527802	+	0.849368i	$0.323016\pi$
$510$		0			0
$511$	−1.07148e10	−	1.97643e10i	−0.157146	−	0.289866i
$512$		0			0
$513$	−6.09740e10	−	3.81197e10i	−0.880391	−	0.550402i
$514$		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	$-0.410533\pi$
$521$	−3.06485e10	−	1.76949e10i	−0.415967	−	0.240159i	−0.693350	+	0.720601i	$0.743866\pi$
$522$		0			0
$523$	5.86284e10	+	1.01547e11i	0.783613	+	1.35726i	0.929824	+	0.368004i	$0.119959\pi$
$523$	5.86284e10	+	1.01547e11i	0.783613	+	1.35726i	−0.146211	+	0.989253i	$0.546708\pi$
$524$		0			0
$525$	1.41696e11	−	1.93914e10i	1.86518	−	0.255254i
$526$		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
$537$	1.07306e11	−	8.79201e10i	1.29041	−	1.05728i
$538$		0			0
$539$	−7.87796e9	+	1.46748e11i	−0.0933381	+	1.73867i
$540$		0			0
$541$	1.21675e9	+	2.10747e9i	0.0142041	+	0.0246021i	0.873040	−	0.487648i	$-0.162145\pi$
$541$	1.21675e9	+	2.10747e9i	0.0142041	+	0.0246021i	−0.858836	+	0.512251i	$0.828812\pi$
$542$		0			0
$543$	−8.51678e9	+	2.25737e10i	−0.0979662	+	0.259659i
$544$		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
$549$	1.39987e9	−	6.97916e9i	0.0154099	−	0.0768270i
$550$		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
$555$	5.39307e9	−	4.41875e9i	0.0568413	−	0.0465723i
$556$		0			0
$557$	6.37318e10	+	3.67955e10i	0.662118	+	0.382274i	0.793083	−	0.609113i	$-0.208474\pi$
$557$	6.37318e10	+	3.67955e10i	0.662118	+	0.382274i	−0.130966	+	0.991387i	$0.541808\pi$
$558$		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	$-0.128060\pi$
$563$	1.04603e11	+	6.03927e10i	1.04115	+	0.601106i	0.120988	+	0.992654i	$0.461394\pi$
$564$		0			0
$565$	1.14815e11	+	1.98865e11i	1.12669	+	1.95148i
$566$		0			0
$567$	−9.42575e10	−	4.24008e10i	−0.911976	−	0.410243i
$568$		0			0
$569$	−4.21469e10	+	2.43336e10i	−0.402084	+	0.232143i	−0.687383	−	0.726295i	$-0.741241\pi$
$569$	−4.21469e10	+	2.43336e10i	−0.402084	+	0.232143i	0.285299	+	0.958439i	$0.407907\pi$
$570$		0			0
$571$	−5.39066e10	+	9.33689e10i	−0.507104	+	0.878330i	0.492862	+	0.870108i	$0.335951\pi$
$571$	−5.39066e10	+	9.33689e10i	−0.507104	+	0.878330i	−0.999966	+	0.00822288i	$0.997383\pi$
$572$		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	−0.182373	−	0.983229i	$-0.558378\pi$
$577$	8.42746e10	−	1.45968e11i	0.760315	−	1.31690i	0.942688	−	0.333675i	$-0.108289\pi$
$578$		0			0
$579$	2.54015e10	+	3.10025e10i	0.226019	+	0.275856i
$580$		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
$585$	3.29611e11	+	6.61131e10i	2.81435	+	0.564500i
$586$		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
$591$	−1.24528e11	−	4.69831e10i	−1.02075	−	0.385116i
$592$		0			0
$593$	−3.87120e9	+	2.23504e9i	−0.0313060	+	0.0180745i	−0.515571	−	0.856847i	$-0.672420\pi$
$593$	−3.87120e9	+	2.23504e9i	−0.0313060	+	0.0180745i	0.484265	+	0.874921i	$0.339087\pi$
$594$		0			0
$595$	−2.32610e10	+	3.79044e10i	−0.185593	+	0.302428i
$596$		0			0
$597$	−3.78460e10	−	4.61909e10i	−0.297936	−	0.363629i
$598$		0			0
$599$	1.23569e10	+	7.13425e9i	0.0959846	+	0.0554167i	0.547224	−	0.836986i	$-0.315685\pi$
$599$	1.23569e10	+	7.13425e9i	0.0959846	+	0.0554167i	−0.451239	+	0.892403i	$0.649018\pi$
$600$		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	0.999998	+	0.00199861i	$0.000636177\pi$
$607$	6.81122e10	+	1.17974e11i	0.501730	+	0.869022i	−0.498268	+	0.867023i	$0.666030\pi$
$608$		0			0
$609$	−6.88804e10	+	5.34149e10i	−0.500756	+	0.388323i
$610$		0			0
$611$	1.31571e11	−	7.59623e10i	0.944048	−	0.545046i
$612$		0			0
$613$	−6.52558e10	+	1.13026e11i	−0.462144	+	0.800456i	−0.999068	−	0.0431743i	$-0.986253\pi$
$613$	−6.52558e10	+	1.13026e11i	−0.462144	+	0.800456i	0.536924	+	0.843631i	$0.319586\pi$
$614$		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	−0.995011	−	0.0997675i	$-0.968190\pi$
$619$	−6.03552e10	+	1.04538e11i	−0.411104	+	0.712053i	0.583907	+	0.811821i	$0.301523\pi$
$620$		0			0
$621$	−1.39240e11	+	2.22720e11i	−0.936260	+	1.49759i
$622$		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
$627$	−2.61414e11	−	9.86285e10i	−1.69145	−	0.638164i
$628$		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
$633$	−6.19239e10	+	1.64129e11i	−0.385694	+	1.02228i
$634$		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
$639$	1.57990e11	−	1.79571e11i	0.947600	−	1.07704i
$640$		0			0
$641$	−1.00770e11	−	5.81796e10i	−0.596897	−	0.344619i	0.170923	−	0.985284i	$-0.445325\pi$
$641$	−1.00770e11	−	5.81796e10i	−0.596897	−	0.344619i	−0.767820	+	0.640666i	$0.778658\pi$
$642$		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	$-0.153728\pi$
$647$	1.62313e11	+	9.37115e10i	0.926267	+	0.534781i	0.0406381	+	0.999174i	$0.487061\pi$
$648$		0			0
$649$	−1.70114e11	−	2.94646e11i	−0.958874	−	1.66082i
$650$		0			0
$651$	7.40128e10	−	1.01288e10i	0.412081	−	0.0563942i
$652$		0			0
$653$	8.56477e10	−	4.94487e10i	0.471045	−	0.271958i	−0.245632	−	0.969363i	$-0.578995\pi$
$653$	8.56477e10	−	4.94487e10i	0.471045	−	0.271958i	0.716677	+	0.697405i	$0.245662\pi$
$654$		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	−0.991326	−	0.131422i	$-0.958046\pi$
$661$	−7.28948e10	+	1.26258e11i	−0.381848	+	0.661381i	0.609478	+	0.792803i	$0.291379\pi$
$662$		0			0
$663$	−5.28097e10	+	4.32690e10i	−0.273312	+	0.223935i
$664$		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
$669$	8.73606e9	−	2.31549e10i	0.0436125	−	0.115595i
$670$		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
$675$	−1.37260e10	+	3.90569e11i	−0.0661194	+	1.88141i
$676$		0			0
$677$	1.98662e11	−	1.14698e11i	0.945715	−	0.546009i	0.0539676	−	0.998543i	$-0.482813\pi$
$677$	1.98662e11	−	1.14698e11i	0.945715	−	0.546009i	0.891747	+	0.452534i	$0.149480\pi$
$678$		0			0
$679$	1.32270e11	−	2.15536e11i	0.622273	−	1.01401i
$680$		0			0
$681$	2.00797e11	−	1.64521e11i	0.933619	−	0.764950i
$682$		0			0
$683$	−1.49702e11	−	8.64304e10i	−0.687930	−	0.397177i	0.114906	−	0.993376i	$-0.463343\pi$
$683$	−1.49702e11	−	8.64304e10i	−0.687930	−	0.397177i	−0.802836	+	0.596200i	$0.796677\pi$
$684$		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	0.774033	−	0.633145i	$-0.218236\pi$
$691$	−3.67751e10	−	6.36963e10i	−0.161303	−	0.279384i	−0.935336	+	0.353760i	$0.884903\pi$
$692$		0			0
$693$	−3.91481e11	−	8.95050e10i	−1.69737	−	0.388074i
$694$		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
$705$	1.71392e11	+	2.09183e11i	0.693799	+	0.846780i
$706$		0			0
$707$	−2.02632e11	+	1.09854e11i	−0.811020	+	0.439680i
$708$		0			0
$709$	−1.45941e11	−	2.52777e11i	−0.577553	−	1.00035i	−0.995759	−	0.0919984i	$-0.970675\pi$
$709$	−1.45941e11	−	2.52777e11i	−0.577553	−	1.00035i	0.418207	−	0.908352i	$-0.362659\pi$
$710$		0			0
$711$	2.12936e9	−	1.06161e10i	0.00833242	−	0.0415418i
$712$		0			0
$713$		−	1.89846e11i		−	0.734588i
$714$		0			0
$715$	1.30620e12			4.99787
$716$		0			0
$717$	2.19189e11	+	8.26976e10i	0.829359	+	0.312908i
$718$		0			0
$719$	−2.25682e11	+	1.30298e11i	−0.844465	+	0.487552i	−0.858779	−	0.512346i	$-0.828777\pi$
$719$	−2.25682e11	+	1.30298e11i	−0.844465	+	0.487552i	0.0143148	+	0.999898i	$0.495443\pi$
$720$		0			0
$721$	−8.68232e10	−	2.32882e9i	−0.321288	−	0.00861776i
$722$		0			0
$723$	−5.27291e10	−	6.43557e10i	−0.192973	−	0.235523i
$724$		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	0.900727	−	0.434386i	$-0.143035\pi$
$733$	2.14124e10	+	3.70874e10i	0.0741736	+	0.128472i	−0.826553	+	0.562859i	$0.809701\pi$
$734$		0			0
$735$	3.99557e11	−	2.93034e11i	1.36908	−	1.00408i
$736$		0			0
$737$	−5.58743e11	+	3.22591e11i	−1.89384	+	1.09341i
$738$		0			0
$739$	1.43125e11	−	2.47900e11i	0.479885	−	0.831186i	−0.519848	−	0.854258i	$-0.674012\pi$
$739$	1.43125e11	−	2.47900e11i	0.479885	−	0.831186i	0.999734	+	0.0230727i	$0.00734494\pi$
$740$		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
$747$	−1.72345e11	+	1.95887e11i	−0.553497	+	0.629106i
$748$		0			0
$749$	3.00986e9	−	1.12214e11i	0.00956354	−	0.356549i
$750$		0			0
$751$	−2.76841e11	−	4.79503e11i	−0.870304	−	1.50741i	−0.861682	−	0.507448i	$-0.830589\pi$
$751$	−2.76841e11	−	4.79503e11i	−0.870304	−	1.50741i	−0.00862178	−	0.999963i	$-0.502744\pi$
$752$		0			0
$753$	−5.20092e11	−	1.96225e11i	−1.61771	−	0.610343i
$754$		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
$759$	−3.60260e11	+	9.54865e11i	−1.08555	+	2.87723i
$760$		0			0
$761$	2.59312e11	−	1.49714e11i	0.773185	−	0.446399i	−0.0608246	−	0.998148i	$-0.519373\pi$
$761$	2.59312e11	−	1.49714e11i	0.773185	−	0.446399i	0.834010	+	0.551750i	$0.186040\pi$
$762$		0			0
$763$	4.69670e10	+	8.66338e10i	0.138578	+	0.255617i
$764$		0			0
$765$	−9.12398e10	−	8.02743e10i	−0.266403	−	0.234385i
$766$		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	$-0.116256\pi$
$773$	3.89800e11	+	2.25051e11i	1.09175	+	0.630323i	0.157709	+	0.987486i	$0.449589\pi$
$774$		0			0
$775$	−1.41234e11	−	2.44625e11i	−0.391501	−	0.678100i
$776$		0			0
$777$	5.96256e9	−	1.46054e10i	0.0163587	−	0.0400711i
$778$		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	−0.963942	−	0.266112i	$-0.914261\pi$
$787$	−9.64843e10	+	1.67116e11i	−0.251511	+	0.435631i	0.712431	+	0.701742i	$0.247594\pi$
$788$		0			0
$789$	3.75538e11	−	3.07693e11i	0.969050	−	0.793980i
$790$		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
$795$	2.13724e10	−	5.66474e10i	0.0535038	−	0.141811i
$796$		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
$801$	−2.12792e11	−	4.26816e10i	−0.516922	−	0.103684i
$802$		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
$807$	−6.47216e10	+	5.30289e10i	−0.152600	+	0.125031i
$808$		0			0
$809$	−1.94021e11	−	1.12018e11i	−0.452956	−	0.261514i	0.256122	−	0.966644i	$-0.417555\pi$
$809$	−1.94021e11	−	1.12018e11i	−0.452956	−	0.261514i	−0.709078	+	0.705130i	$0.750888\pi$
$810$		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
$819$	7.26946e11	−	2.23940e11i	1.61572	−	0.497731i
$820$		0			0
$821$	5.47981e11	−	3.16377e11i	1.20613	−	0.696358i	0.244216	−	0.969721i	$-0.421469\pi$
$821$	5.47981e11	−	3.16377e11i	1.20613	−	0.696358i	0.961911	+	0.273363i	$0.0881361\pi$
$822$		0			0
$823$	−1.68609e11	+	2.92040e11i	−0.367521	+	0.636564i	−0.989177	−	0.146726i	$-0.953127\pi$
$823$	−1.68609e11	+	2.92040e11i	−0.367521	+	0.636564i	0.621657	+	0.783290i	$0.286460\pi$
$824$		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	−0.171622	+	0.985163i	$0.554901\pi$
$829$	−4.43484e11	+	7.68136e11i	−0.938987	+	1.62637i	−0.767365	+	0.641211i	$0.778433\pi$
$830$		0			0
$831$	−4.81361e11	−	5.87500e11i	−1.00941	−	1.23198i
$832$		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
$837$	−7.16955e9	+	2.04007e11i	−0.0146080	+	0.415666i
$838$		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
$843$	−5.85622e11	−	2.20949e11i	−1.15960	−	0.437503i
$844$		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
$849$	−2.83794e11	−	3.46370e11i	−0.546227	−	0.666668i
$850$		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	$-0.260956\pi$
$857$	2.10613e11	+	1.21597e11i	0.390446	+	0.225424i	−0.291907	+	0.956447i	$0.594290\pi$
$858$		0			0
$859$	4.61093e11	+	7.98636e11i	0.846868	+	1.46682i	0.883989	+	0.467507i	$0.154848\pi$
$859$	4.61093e11	+	7.98636e11i	0.846868	+	1.46682i	−0.0371216	+	0.999311i	$0.511819\pi$
$860$		0			0
$861$	−1.50894e11	+	1.17014e11i	−0.274574	+	0.212925i
$862$		0			0
$863$	5.34535e11	−	3.08614e11i	0.963680	−	0.556381i	0.0663762	−	0.997795i	$-0.478856\pi$
$863$	5.34535e11	−	3.08614e11i	0.963680	−	0.556381i	0.897304	+	0.441414i	$0.145523\pi$
$864$		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
$873$	5.18819e11	+	4.56465e11i	0.893221	+	0.785870i
$874$		0			0
$875$	−7.48629e11	−	4.59416e11i	−1.27713	−	0.783743i
$876$		0			0
$877$	−1.27012e11	−	2.19991e11i	−0.214707	−	0.371883i	0.738475	−	0.674281i	$-0.235546\pi$
$877$	−1.27012e11	−	2.19991e11i	−0.214707	−	0.371883i	−0.953182	+	0.302398i	$0.902213\pi$
$878$		0			0
$879$	−5.65583e10	−	2.13388e10i	−0.0947417	−	0.0357450i
$880$		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
$885$	−4.04937e11	+	1.07328e12i	−0.660107	+	1.74961i
$886$		0			0
$887$	9.45508e11	−	5.45890e11i	1.52746	−	0.881882i	0.527996	−	0.849247i	$-0.322943\pi$
$887$	9.45508e11	−	5.45890e11i	1.52746	−	0.881882i	0.999467	−	0.0326349i	$-0.0103899\pi$
$888$		0			0
$889$	1.20455e11	−	1.96285e11i	0.192849	−	0.314253i
$890$		0			0
$891$	4.23193e11	−	1.01249e12i	0.671471	−	1.60649i
$892$		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
$903$	1.47209e11	+	1.89831e11i	0.221403	+	0.285507i
$904$		0			0
$905$	2.73726e11	−	1.58036e11i	0.408058	−	0.235592i
$906$		0			0
$907$	−4.26148e11	+	7.38110e11i	−0.629696	+	1.09067i	0.357916	+	0.933754i	$0.383487\pi$
$907$	−4.26148e11	+	7.38110e11i	−0.629696	+	1.09067i	−0.987613	+	0.156912i	$0.949846\pi$
$908$		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
$915$	−7.21313e10	+	5.91000e10i	−0.102906	+	0.0843146i
$916$		0			0
$917$	−2.10527e10	+	7.84891e11i	−0.0297736	+	1.11002i
$918$		0			0
$919$	2.26505e11	+	3.92318e11i	0.317552	+	0.550016i	0.979977	−	0.199112i	$-0.0638058\pi$
$919$	2.26505e11	+	3.92318e11i	0.317552	+	0.550016i	−0.662425	+	0.749129i	$0.730472\pi$
$920$		0			0
$921$	1.66325e11	−	4.40842e11i	0.231163	−	0.612696i
$922$		0			0
$923$			1.76027e12i			2.42534i
$924$		0			0
$925$	−5.96514e10			−0.0814806
$926$		0			0
$927$	4.66757e10	−	2.32705e11i	0.0632079	−	0.315127i
$928$		0			0
$929$	5.08083e11	−	2.93342e11i	0.682137	−	0.393832i	−0.118522	−	0.992951i	$-0.537816\pi$
$929$	5.08083e11	−	2.93342e11i	0.682137	−	0.393832i	0.800660	+	0.599119i	$0.204482\pi$
$930$		0			0
$931$	4.25670e11	−	6.53652e11i	0.566597	−	0.870057i
$932$		0			0
$933$	−1.34538e11	+	1.10232e11i	−0.177549	+	0.145473i
$934$		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	$-0.650619\pi$
$941$	−1.14040e12	−	6.58411e11i	−1.45445	−	0.839727i	−0.998729	+	0.0503950i	$0.983952\pi$
$942$		0			0
$943$	2.42634e11	+	4.20255e11i	0.306835	+	0.531454i
$944$		0			0
$945$	6.03109e11	+	1.21225e12i	0.756255	+	1.52008i
$946$		0			0
$947$	−1.44702e11	+	8.35440e10i	−0.179919	+	0.103876i	−0.587254	−	0.809402i	$-0.699791\pi$
$947$	−1.44702e11	+	8.35440e10i	−0.179919	+	0.103876i	0.407336	+	0.913278i	$0.366458\pi$
$948$		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
$957$	−5.86536e11	−	7.15865e11i	−0.699273	−	0.853461i
$958$		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
$963$	3.00757e11	+	6.03256e10i	0.349712	+	0.0701449i
$964$		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
$969$	−1.78997e11	−	6.75337e10i	−0.203026	−	0.0765994i
$970$		0			0
$971$	−2.33297e11	+	1.34694e11i	−0.262442	+	0.151521i	−0.625448	−	0.780266i	$-0.715084\pi$
$971$	−2.33297e11	+	1.34694e11i	−0.262442	+	0.151521i	0.363006	+	0.931787i	$0.381750\pi$
$972$		0			0
$973$	−6.10846e11	+	9.95387e11i	−0.681522	+	1.11056i
$974$		0			0
$975$	−1.82287e12	−	2.22481e12i	−2.01715	−	2.46193i
$976$		0			0
$977$	−1.17941e12	−	6.80935e11i	−1.29446	−	0.747356i	−0.315017	−	0.949086i	$-0.602010\pi$
$977$	−1.17941e12	−	6.80935e11i	−1.29446	−	0.747356i	−0.979441	+	0.201730i	$0.935344\pi$
$978$		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	$-0.289092\pi$
$983$	2.24052e11	+	1.29357e11i	0.239958	+	0.138540i	−0.375199	+	0.926944i	$0.622426\pi$
$984$		0			0
$985$	8.71809e11	+	1.51002e12i	0.926140	+	1.60412i
$986$		0			0
$987$	5.66508e11	+	2.31273e11i	0.596949	+	0.243700i
$988$		0			0
$989$	5.28699e11	−	3.05245e11i	0.552616	−	0.319053i
$990$		0			0
$991$	−1.28622e11	+	2.22781e11i	−0.133359	+	0.230984i	−0.924969	−	0.380042i	$-0.875910\pi$
$991$	−1.28622e11	+	2.22781e11i	−0.133359	+	0.230984i	0.791610	+	0.611026i	$0.209243\pi$
$992$		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	−0.0556413	−	0.998451i	$-0.517720\pi$
$997$	8.26866e11	−	1.43217e12i	0.836863	−	1.44949i	0.892504	−	0.451039i	$-0.148946\pi$
$998$		0			0
$999$	3.65532e10	+	2.28523e10i	0.0366998	+	0.0229439i

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.9.p.b.53.13	✓	40	1.1	even	1	trivial
84.9.p.b.53.14	yes	40	3.2	odd	2	inner
84.9.p.b.65.13	yes	40	21.2	odd	6	inner
84.9.p.b.65.14	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)

\(f(q)\)	\(=\)	\(q+(28.5930 - 75.7855i) q^{3} +(-918.968 + 530.566i) q^{5} +(-1255.82 + 2046.39i) q^{7} +(-4925.88 - 4333.87i) q^{9} +O(q^{10})\)	\(q+(28.5930 - 75.7855i) q^{3} +(-918.968 + 530.566i) q^{5} +(-1255.82 + 2046.39i) q^{7} +(-4925.88 - 4333.87i) q^{9} +(-22077.2 - 12746.3i) q^{11} +48286.6 q^{13} +(13933.2 + 84814.9i) q^{15} +(-15116.9 - 8727.72i) q^{17} +(67655.0 + 117182. i) q^{19} +(119179. + 153685. i) q^{21} +(428030. - 247123. i) q^{23} +(367689. - 636856. i) q^{25} +(-469290. + 249392. i) q^{27} +448191. i q^{29} +(192056. - 332652. i) q^{31} +(-1.59723e6 + 1.30867e6i) q^{33} +(68313.3 - 2.54686e6i) q^{35} +(-40558.3 - 70249.1i) q^{37} +(1.38066e6 - 3.65943e6i) q^{39} +981836. i q^{41} +1.23519e6 q^{43} +(6.82613e6 + 1.36918e6i) q^{45} +(2.72478e6 - 1.57315e6i) q^{47} +(-2.61063e6 - 5.13980e6i) q^{49} +(-1.09367e6 + 896087. i) q^{51} +(-610035. - 352204. i) q^{53} +2.70509e7 q^{55} +(1.08151e7 - 1.77669e6i) q^{57} +(1.15582e7 + 6.67310e6i) q^{59} +(542461. + 939570. i) q^{61} +(1.50548e7 - 4.63771e6i) q^{63} +(-4.43739e7 + 2.56193e7i) q^{65} +(1.26543e7 - 2.19179e7i) q^{67} +(-6.48969e6 - 3.95044e7i) q^{69} +3.64547e7i q^{71} +(-4.68177e6 + 8.10906e6i) q^{73} +(-3.77511e7 - 4.60751e7i) q^{75} +(5.38088e7 - 2.91715e7i) q^{77} +(825142. + 1.42919e6i) q^{79} +(5.48189e6 + 4.26962e7i) q^{81} -3.97669e7i q^{83} +1.85225e7 q^{85} +(3.39664e7 + 1.28151e7i) q^{87} +(2.86471e7 - 1.65394e7i) q^{89} +(-6.06394e7 + 9.88133e7i) q^{91} +(-1.97187e7 - 2.40666e7i) q^{93} +(-1.24346e8 - 7.17910e7i) q^{95} -1.05325e8 q^{97} +(5.35089e7 + 1.58466e8i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 81 q^{3} - 34 q^{7} + 4771 q^{9}+O(q^{10}) \) 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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