LMFDB - Embedded newform 216.8.i.a.145.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [216,8,Mod(73,216)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(216, base_ring=CyclotomicField(6))

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

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("216.73");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 216 = 2^{3} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	216.i (of order $3$, degree $2$, not 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:	$67.4751655046$
Analytic rank:	$0$
Dimension:	$20$
Relative dimension:	$10$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 10 x^{19} + 332929 x^{18} - 2996076 x^{17} + 44578211685 x^{16} - 356557783716 x^{15} + \cdots + 79\!\cdots\!67 $ x^20 - 10x^19 + 332929x^18 - 2996076x^17 + 44578211685x^16 - 356557783716x^15 + 3097483289700118x^14 - 21676143504368032x^13 + 119894285805340530265x^12 - 719083941188982737718x^11 + 2563232515499525310010527x^10 - 12809571491275660354252428x^9 + 27741346343998271210179496359x^8 - 110888535856083235142883207868x^7 + 121645138768923372291810301132558x^6 - 364547360223565095382569624026928x^5 + 164798224123857530371927578258953667x^4 - 328988998658352429191508377804789922x^3 + 18304547169687091044739805564504070921x^2 - 18140113409773705053928159582048276332*x + 79919855971481307718190429644423027167
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{48}\cdot 3^{45} $
Twist minimal:	no (minimal twist has level 72)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			145.6
Root			$0.500000 + 2.08484i$ of defining polynomial
Character	$\chi$	$=$	216.145
Dual form			216.8.i.a.73.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(8.05553 - 13.9526i) q^{5} +(-629.010 - 1089.48i) q^{7} +O(q^{10})$	$q+(8.05553 - 13.9526i) q^{5} +(-629.010 - 1089.48i) q^{7} +(-3591.86 - 6221.29i) q^{11} +(-4460.66 + 7726.08i) q^{13} -4941.76 q^{17} -34592.2 q^{19} +(37022.5 - 64124.8i) q^{23} +(38932.7 + 67433.4i) q^{25} +(83109.0 + 143949. i) q^{29} +(50746.9 - 87896.2i) q^{31} -20268.0 q^{35} -360076. q^{37} +(198048. - 343029. i) q^{41} +(431833. + 747956. i) q^{43} +(-212800. - 368580. i) q^{47} +(-379535. + 657374. i) q^{49} +634468. q^{53} -115737. q^{55} +(-880298. + 1.52472e6i) q^{59} +(-380043. - 658255. i) q^{61} +(71865.9 + 124475. i) q^{65} +(942841. - 1.63305e6i) q^{67} -2.99804e6 q^{71} +2.45405e6 q^{73} +(-4.51863e6 + 7.82650e6i) q^{77} +(3.91237e6 + 6.77642e6i) q^{79} +(2.74085e6 + 4.74729e6i) q^{83} +(-39808.4 + 68950.3i) q^{85} +1.40387e6 q^{89} +1.12232e7 q^{91} +(-278659. + 482651. i) q^{95} +(2.17082e6 + 3.75996e6i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 125 q^{5} + 1245 q^{7}+O(q^{10}) $ 20 * q + 125 * q^5 + 1245 * q^7	$ 20 q + 125 q^{5} + 1245 q^{7} - 6106 q^{11} + 4937 q^{13} - 48722 q^{17} - 26882 q^{19} - 19387 q^{23} - 218957 q^{25} + 46791 q^{29} - 185039 q^{31} - 83094 q^{35} + 108420 q^{37} + 638112 q^{41} + 892628 q^{43} + 230883 q^{47} - 1034741 q^{49} + 2872940 q^{53} + 1089998 q^{55} - 2172454 q^{59} - 1878325 q^{61} - 1239133 q^{65} + 531496 q^{67} - 3723056 q^{71} - 1804522 q^{73} - 6276543 q^{77} + 3607847 q^{79} - 10794491 q^{83} + 3597658 q^{85} - 32214888 q^{89} - 16117530 q^{91} + 756868 q^{95} + 17951260 q^{97}+O(q^{100}) $ 20 * q + 125 * q^5 + 1245 * q^7 - 6106 * q^11 + 4937 * q^13 - 48722 * q^17 - 26882 * q^19 - 19387 * q^23 - 218957 * q^25 + 46791 * q^29 - 185039 * q^31 - 83094 * q^35 + 108420 * q^37 + 638112 * q^41 + 892628 * q^43 + 230883 * q^47 - 1034741 * q^49 + 2872940 * q^53 + 1089998 * q^55 - 2172454 * q^59 - 1878325 * q^61 - 1239133 * q^65 + 531496 * q^67 - 3723056 * q^71 - 1804522 * q^73 - 6276543 * q^77 + 3607847 * q^79 - 10794491 * q^83 + 3597658 * q^85 - 32214888 * q^89 - 16117530 * q^91 + 756868 * q^95 + 17951260 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/216\mathbb{Z}\right)^\times$.

$n$	$55$	$109$	$137$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{3}\right)$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	8.05553	−	13.9526i	0.0288203	−	0.0499183i	−0.851255	−	0.524751i	$-0.824158\pi$
$5$	8.05553	−	13.9526i	0.0288203	−	0.0499183i	0.880076	+	0.474833i	$0.157492\pi$
$6$		0			0
$7$	−629.010	−	1089.48i	−0.693129	−	1.20054i	−0.970807	−	0.239861i	$-0.922898\pi$
$7$	−629.010	−	1089.48i	−0.693129	−	1.20054i	0.277678	−	0.960674i	$-0.410435\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−3591.86	−	6221.29i	−0.813664	−	1.40931i	−0.910283	−	0.413986i	$-0.864136\pi$
$11$	−3591.86	−	6221.29i	−0.813664	−	1.40931i	0.0966189	−	0.995321i	$-0.469197\pi$
$12$		0			0
$13$	−4460.66	+	7726.08i	−0.563115	+	0.975343i	0.434108	+	0.900861i	$0.357064\pi$
$13$	−4460.66	+	7726.08i	−0.563115	+	0.975343i	−0.997222	+	0.0744824i	$0.976270\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−4941.76			−0.243955			−0.121978	−	0.992533i	$-0.538924\pi$
$17$	−4941.76			−0.243955			−0.121978	+	0.992533i	$0.538924\pi$
$18$		0			0
$19$	−34592.2			−1.15702			−0.578510	−	0.815675i	$-0.696366\pi$
$19$	−34592.2			−1.15702			−0.578510	+	0.815675i	$0.696366\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	37022.5	−	64124.8i	0.634480	−	1.09895i	−0.352145	−	0.935946i	$-0.614547\pi$
$23$	37022.5	−	64124.8i	0.634480	−	1.09895i	0.986625	−	0.163007i	$-0.0521192\pi$
$24$		0			0
$25$	38932.7	+	67433.4i	0.498339	+	0.863148i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	83109.0	+	143949.i	0.632784	+	1.09601i	0.986980	+	0.160842i	$0.0514210\pi$
$29$	83109.0	+	143949.i	0.632784	+	1.09601i	−0.354197	+	0.935171i	$0.615246\pi$
$30$		0			0
$31$	50746.9	−	87896.2i	0.305945	−	0.529913i	−0.671526	−	0.740981i	$-0.734361\pi$
$31$	50746.9	−	87896.2i	0.305945	−	0.529913i	0.977471	+	0.211068i	$0.0676942\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−20268.0			−0.0799049
$36$		0			0
$37$	−360076.			−1.16866			−0.584329	−	0.811517i	$-0.698642\pi$
$37$	−360076.			−1.16866			−0.584329	+	0.811517i	$0.698642\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	198048.	−	343029.i	0.448773	−	0.777297i	−0.549534	−	0.835472i	$-0.685195\pi$
$41$	198048.	−	343029.i	0.448773	−	0.777297i	0.998306	+	0.0581742i	$0.0185279\pi$
$42$		0			0
$43$	431833.	+	747956.i	0.828277	+	1.43462i	0.899389	+	0.437150i	$0.144012\pi$
$43$	431833.	+	747956.i	0.828277	+	1.43462i	−0.0711112	+	0.997468i	$0.522655\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−212800.	−	368580.i	−0.298971	−	0.517832i	0.676930	−	0.736047i	$-0.263310\pi$
$47$	−212800.	−	368580.i	−0.298971	−	0.517832i	−0.975901	+	0.218215i	$0.929977\pi$
$48$		0			0
$49$	−379535.	+	657374.i	−0.460856	+	0.798227i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	634468.			0.585389			0.292694	−	0.956206i	$-0.405448\pi$
$53$	634468.			0.585389			0.292694	+	0.956206i	$0.405448\pi$
$54$		0			0
$55$	−115737.			−0.0938003
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−880298.	+	1.52472e6i	−0.558017	+	0.966514i	0.439645	+	0.898172i	$0.355104\pi$
$59$	−880298.	+	1.52472e6i	−0.558017	+	0.966514i	−0.997662	+	0.0683426i	$0.978229\pi$
$60$		0			0
$61$	−380043.	−	658255.i	−0.214377	−	0.371312i	0.738702	−	0.674032i	$-0.235439\pi$
$61$	−380043.	−	658255.i	−0.214377	−	0.371312i	−0.953080	+	0.302719i	$0.902106\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	71865.9	+	124475.i	0.0324583	+	0.0562194i
$66$		0			0
$67$	942841.	−	1.63305e6i	0.382981	−	0.663342i	−0.608506	−	0.793549i	$-0.708231\pi$
$67$	942841.	−	1.63305e6i	0.382981	−	0.663342i	0.991487	+	0.130207i	$0.0415642\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−2.99804e6			−0.994106			−0.497053	−	0.867720i	$-0.665585\pi$
$71$	−2.99804e6			−0.994106			−0.497053	+	0.867720i	$0.665585\pi$
$72$		0			0
$73$	2.45405e6			0.738334			0.369167	−	0.929363i	$-0.379643\pi$
$73$	2.45405e6			0.738334			0.369167	+	0.929363i	$0.379643\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−4.51863e6	+	7.82650e6i	−1.12795	+	1.95366i
$78$		0			0
$79$	3.91237e6	+	6.77642e6i	0.892780	+	1.54634i	0.836528	+	0.547924i	$0.184582\pi$
$79$	3.91237e6	+	6.77642e6i	0.892780	+	1.54634i	0.0562523	+	0.998417i	$0.482085\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	2.74085e6	+	4.74729e6i	0.526153	+	0.911324i	0.999536	+	0.0304670i	$0.00969944\pi$
$83$	2.74085e6	+	4.74729e6i	0.526153	+	0.911324i	−0.473383	+	0.880857i	$0.656967\pi$
$84$		0			0
$85$	−39808.4	+	68950.3i	−0.00703087	+	0.0121778i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	1.40387e6			0.211088			0.105544	−	0.994415i	$-0.466342\pi$
$89$	1.40387e6			0.211088			0.105544	+	0.994415i	$0.466342\pi$
$90$		0			0
$91$	1.12232e7			1.56125
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−278659.	+	482651.i	−0.0333457	+	0.0577565i
$96$		0			0
$97$	2.17082e6	+	3.75996e6i	0.241503	+	0.418295i	0.961143	−	0.276053i	$-0.0890264\pi$
$97$	2.17082e6	+	3.75996e6i	0.241503	+	0.418295i	−0.719640	+	0.694348i	$0.755693\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	3.48961e6	+	6.04418e6i	0.337017	+	0.583731i	0.983870	−	0.178884i	$-0.0572487\pi$
$101$	3.48961e6	+	6.04418e6i	0.337017	+	0.583731i	−0.646853	+	0.762615i	$0.723915\pi$
$102$		0			0
$103$	−3.65411e6	+	6.32910e6i	−0.329497	+	0.570705i	−0.982412	−	0.186726i	$-0.940212\pi$
$103$	−3.65411e6	+	6.32910e6i	−0.329497	+	0.570705i	0.652915	+	0.757431i	$0.273546\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−2.43091e7			−1.91834			−0.959170	−	0.282830i	$-0.908727\pi$
$107$	−2.43091e7			−1.91834			−0.959170	+	0.282830i	$0.908727\pi$
$108$		0			0
$109$	−8.25424e6			−0.610498			−0.305249	−	0.952273i	$-0.598740\pi$
$109$	−8.25424e6			−0.610498			−0.305249	+	0.952273i	$0.598740\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−7.55175e6	+	1.30800e7i	−0.492349	+	0.852773i	−0.999961	−	0.00881220i	$-0.997195\pi$
$113$	−7.55175e6	+	1.30800e7i	−0.492349	+	0.852773i	0.507612	+	0.861586i	$0.330528\pi$
$114$		0			0
$115$	−596471.	−	1.03312e6i	−0.0365719	−	0.0633443i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	3.10841e6	+	5.38393e6i	0.169092	+	0.292877i
$120$		0			0
$121$	−1.60594e7	+	2.78156e7i	−0.824099	+	1.42738i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	2.51317e6			0.115090
$126$		0			0
$127$	2.81769e7			1.22062			0.610310	−	0.792162i	$-0.291045\pi$
$127$	2.81769e7			1.22062			0.610310	+	0.792162i	$0.291045\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	6.78835e6	−	1.17578e7i	0.263824	−	0.456957i	−0.703431	−	0.710764i	$-0.748349\pi$
$131$	6.78835e6	−	1.17578e7i	0.263824	−	0.456957i	0.967255	+	0.253807i	$0.0816828\pi$
$132$		0			0
$133$	2.17589e7	+	3.76874e7i	0.801965	+	1.38904i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	5.49999e6	+	9.52626e6i	0.182743	+	0.316520i	0.942814	−	0.333321i	$-0.108169\pi$
$137$	5.49999e6	+	9.52626e6i	0.182743	+	0.316520i	−0.760071	+	0.649840i	$0.774836\pi$
$138$		0			0
$139$	−6.95480e6	+	1.20461e7i	−0.219651	+	0.380447i	−0.954701	−	0.297566i	$-0.903825\pi$
$139$	−6.95480e6	+	1.20461e7i	−0.219651	+	0.380447i	0.735050	+	0.678013i	$0.237159\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	6.40882e7			1.83275
$144$		0			0
$145$	2.67795e6			0.0729481
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−2.60354e7	+	4.50947e7i	−0.644782	+	1.11679i	0.339570	+	0.940581i	$0.389718\pi$
$149$	−2.60354e7	+	4.50947e7i	−0.644782	+	1.11679i	−0.984352	+	0.176214i	$0.943615\pi$
$150$		0			0
$151$	2.95863e7	+	5.12449e7i	0.699311	+	1.21124i	0.968706	+	0.248213i	$0.0798433\pi$
$151$	2.95863e7	+	5.12449e7i	0.699311	+	1.21124i	−0.269394	+	0.963030i	$0.586823\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−817586.	−	1.41610e6i	−0.0176349	−	0.0305445i
$156$		0			0
$157$	3.09069e7	−	5.35323e7i	0.637392	−	1.10399i	−0.348611	−	0.937267i	$-0.613347\pi$
$157$	3.09069e7	−	5.35323e7i	0.637392	−	1.10399i	0.986003	−	0.166727i	$-0.0533200\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−9.31500e7			−1.75911
$162$		0			0
$163$	−7.98394e7			−1.44398			−0.721989	−	0.691904i	$-0.756772\pi$
$163$	−7.98394e7			−1.44398			−0.721989	+	0.691904i	$0.756772\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−1.87461e7	+	3.24692e7i	−0.311460	+	0.539465i	−0.978679	−	0.205397i	$-0.934151\pi$
$167$	−1.87461e7	+	3.24692e7i	−0.311460	+	0.539465i	0.667218	+	0.744862i	$0.267485\pi$
$168$		0			0
$169$	−8.42064e6	−	1.45850e7i	−0.134197	−	0.232435i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−1.50788e7	−	2.61172e7i	−0.221414	−	0.383500i	0.733824	−	0.679340i	$-0.237734\pi$
$173$	−1.50788e7	−	2.61172e7i	−0.221414	−	0.383500i	−0.955238	+	0.295840i	$0.904401\pi$
$174$		0			0
$175$	4.89781e7	−	8.48326e7i	0.690826	−	1.19655i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	2.92111e7			0.380682			0.190341	−	0.981718i	$-0.439041\pi$
$179$	2.92111e7			0.380682			0.190341	+	0.981718i	$0.439041\pi$
$180$		0			0
$181$	−9.33823e7			−1.17055			−0.585275	−	0.810835i	$-0.699013\pi$
$181$	−9.33823e7			−1.17055			−0.585275	+	0.810835i	$0.699013\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−2.90060e6	+	5.02399e6i	−0.0336811	+	0.0583374i
$186$		0			0
$187$	1.77501e7	+	3.07441e7i	0.198498	+	0.343808i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−2.09571e7	−	3.62988e7i	−0.217628	−	0.376942i	0.736454	−	0.676487i	$-0.236499\pi$
$191$	−2.09571e7	−	3.62988e7i	−0.217628	−	0.376942i	−0.954082	+	0.299545i	$0.903165\pi$
$192$		0			0
$193$	1.39456e7	−	2.41545e7i	0.139633	−	0.241851i	−0.787725	−	0.616027i	$-0.788741\pi$
$193$	1.39456e7	−	2.41545e7i	0.139633	−	0.241851i	0.927358	+	0.374176i	$0.122074\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−3.00157e7			−0.279716			−0.139858	−	0.990172i	$-0.544665\pi$
$197$	−3.00157e7			−0.279716			−0.139858	+	0.990172i	$0.544665\pi$
$198$		0			0
$199$	1.28960e8			1.16003			0.580016	−	0.814605i	$-0.303046\pi$
$199$	1.28960e8			1.16003			0.580016	+	0.814605i	$0.303046\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	1.04553e8	−	1.81091e8i	0.877202	−	1.51936i
$204$		0			0
$205$	−3.19076e6	−	5.52656e6i	−0.0258676	−	0.0448039i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	1.24251e8	+	2.15208e8i	0.941426	+	1.63060i
$210$		0			0
$211$	8.17536e7	−	1.41601e8i	0.599126	−	1.03772i	−0.393824	−	0.919186i	$-0.628848\pi$
$211$	8.17536e7	−	1.41601e8i	0.599126	−	1.03772i	0.992950	−	0.118531i	$-0.0378186\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	1.39146e7			0.0954849
$216$		0			0
$217$	−1.27681e8			−0.848239
$218$		0			0
$219$		0			0
$220$		0			0
$221$	2.20435e7	−	3.81804e7i	0.137375	−	0.237940i
$222$		0			0
$223$	−7.27567e7	−	1.26018e8i	−0.439345	−	0.760968i	0.558294	−	0.829643i	$-0.311456\pi$
$223$	−7.27567e7	−	1.26018e8i	−0.439345	−	0.760968i	−0.997639	+	0.0686751i	$0.978123\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−1.63056e8	−	2.82421e8i	−0.925223	−	1.60253i	−0.791201	−	0.611556i	$-0.790544\pi$
$227$	−1.63056e8	−	2.82421e8i	−0.925223	−	1.60253i	−0.134022	−	0.990978i	$-0.542789\pi$
$228$		0			0
$229$	−1.05969e8	+	1.83543e8i	−0.583115	+	1.00998i	0.411993	+	0.911187i	$0.364833\pi$
$229$	−1.05969e8	+	1.83543e8i	−0.583115	+	1.00998i	−0.995108	+	0.0987973i	$0.968500\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−2.31310e7			−0.119798			−0.0598988	−	0.998204i	$-0.519078\pi$
$233$	−2.31310e7			−0.119798			−0.0598988	+	0.998204i	$0.519078\pi$
$234$		0			0
$235$	−6.85686e6			−0.0344657
$236$		0			0
$237$		0			0
$238$		0			0
$239$	3.58609e7	−	6.21128e7i	0.169914	−	0.294299i	−0.768476	−	0.639879i	$-0.778984\pi$
$239$	3.58609e7	−	6.21128e7i	0.169914	−	0.294299i	0.938389	+	0.345580i	$0.112318\pi$
$240$		0			0
$241$	−2.00073e8	−	3.46537e8i	−0.920724	−	1.59474i	−0.798298	−	0.602263i	$-0.794266\pi$
$241$	−2.00073e8	−	3.46537e8i	−0.920724	−	1.59474i	−0.122426	−	0.992478i	$-0.539067\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	6.11471e6	+	1.05910e7i	0.0265641	+	0.0460103i
$246$		0			0
$247$	1.54304e8	−	2.67262e8i	0.651535	−	1.12849i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	4.64954e8			1.85589			0.927945	−	0.372716i	$-0.121574\pi$
$251$	4.64954e8			1.85589			0.927945	+	0.372716i	$0.121574\pi$
$252$		0			0
$253$	−5.31919e8			−2.06502
$254$		0			0
$255$		0			0
$256$		0			0
$257$	1.24827e8	−	2.16207e8i	0.458715	−	0.794518i	−0.540178	−	0.841551i	$-0.681643\pi$
$257$	1.24827e8	−	2.16207e8i	0.458715	−	0.794518i	0.998893	+	0.0470326i	$0.0149765\pi$
$258$		0			0
$259$	2.26491e8	+	3.92294e8i	0.810032	+	1.40302i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−1.97113e8	−	3.41410e8i	−0.668144	−	1.15726i	−0.978423	−	0.206614i	$-0.933756\pi$
$263$	−1.97113e8	−	3.41410e8i	−0.668144	−	1.15726i	0.310279	−	0.950646i	$-0.399578\pi$
$264$		0			0
$265$	5.11098e6	−	8.85247e6i	0.0168711	−	0.0292216i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	2.97222e8			0.930997			0.465499	−	0.885049i	$-0.345875\pi$
$269$	2.97222e8			0.930997			0.465499	+	0.885049i	$0.345875\pi$
$270$		0			0
$271$	−2.39731e8			−0.731698			−0.365849	−	0.930674i	$-0.619221\pi$
$271$	−2.39731e8			−0.731698			−0.365849	+	0.930674i	$0.619221\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	2.79682e8	−	4.84423e8i	0.810961	−	1.40463i
$276$		0			0
$277$	−6.11716e6	−	1.05952e7i	−0.0172930	−	0.0299524i	0.857249	−	0.514901i	$-0.172171\pi$
$277$	−6.11716e6	−	1.05952e7i	−0.0172930	−	0.0299524i	−0.874542	+	0.484949i	$0.838838\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−2.00760e8	−	3.47727e8i	−0.539766	−	0.934903i	−0.998916	−	0.0465438i	$-0.985179\pi$
$281$	−2.00760e8	−	3.47727e8i	−0.539766	−	0.934903i	0.459150	−	0.888359i	$-0.348154\pi$
$282$		0			0
$283$	−1.51229e8	+	2.61936e8i	−0.396626	+	0.686977i	−0.993307	−	0.115501i	$-0.963152\pi$
$283$	−1.51229e8	+	2.61936e8i	−0.396626	+	0.686977i	0.596681	+	0.802479i	$0.296486\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−4.98296e8			−1.24423
$288$		0			0
$289$	−3.85918e8			−0.940486
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−2.15237e8	+	3.72802e8i	−0.499897	+	0.865848i	−1.00000		0.000118525i	$-0.999962\pi$
$293$	−2.15237e8	+	3.72802e8i	−0.499897	+	0.865848i	0.500103	+	0.865966i	$0.333296\pi$
$294$		0			0
$295$	1.41825e7	+	2.45649e7i	0.0321645	+	0.0557105i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	3.30289e8	+	5.72078e8i	0.714571	+	1.23767i
$300$		0			0
$301$	5.43254e8	−	9.40943e8i	1.14821	−	1.98875i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−1.22458e7			−0.0247137
$306$		0			0
$307$	3.90378e8			0.770018			0.385009	−	0.922913i	$-0.374198\pi$
$307$	3.90378e8			0.770018			0.385009	+	0.922913i	$0.374198\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−1.82625e8	+	3.16316e8i	−0.344270	+	0.596294i	−0.985221	−	0.171288i	$-0.945207\pi$
$311$	−1.82625e8	+	3.16316e8i	−0.344270	+	0.596294i	0.640951	+	0.767582i	$0.278540\pi$
$312$		0			0
$313$	3.99671e8	+	6.92250e8i	0.736711	+	1.27602i	0.953969	+	0.299906i	$0.0969555\pi$
$313$	3.99671e8	+	6.92250e8i	0.736711	+	1.27602i	−0.217258	+	0.976114i	$0.569711\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	1.89033e8	+	3.27414e8i	0.333295	+	0.577284i	0.983156	−	0.182769i	$-0.0585061\pi$
$317$	1.89033e8	+	3.27414e8i	0.333295	+	0.577284i	−0.649861	+	0.760053i	$0.725173\pi$
$318$		0			0
$319$	5.97032e8	−	1.03409e9i	1.02975	−	1.78357i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	1.70946e8			0.282261
$324$		0			0
$325$	−6.94662e8			−1.12249
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−2.67706e8	+	4.63681e8i	−0.414451	+	0.717850i
$330$		0			0
$331$	9.05939e7	+	1.56913e8i	0.137310	+	0.237827i	0.926477	−	0.376350i	$-0.122821\pi$
$331$	9.05939e7	+	1.56913e8i	0.137310	+	0.237827i	−0.789168	+	0.614178i	$0.789488\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−1.51902e7	−	2.63101e7i	−0.0220753	−	0.0382355i
$336$		0			0
$337$	8.10544e7	−	1.40390e8i	0.115364	−	0.199817i	−0.802561	−	0.596570i	$-0.796530\pi$
$337$	8.10544e7	−	1.40390e8i	0.115364	−	0.199817i	0.917925	+	0.396753i	$0.129863\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−7.29104e8			−0.995747
$342$		0			0
$343$	−8.11083e7			−0.108526
$344$		0			0
$345$		0			0
$346$		0			0
$347$	5.72711e7	−	9.91964e7i	0.0735838	−	0.127451i	−0.826886	−	0.562370i	$-0.809890\pi$
$347$	5.72711e7	−	9.91964e7i	0.0735838	−	0.127451i	0.900470	+	0.434919i	$0.143223\pi$
$348$		0			0
$349$	7.38174e8	+	1.27855e9i	0.929543	+	1.61002i	0.784087	+	0.620651i	$0.213132\pi$
$349$	7.38174e8	+	1.27855e9i	0.929543	+	1.61002i	0.145456	+	0.989365i	$0.453535\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	4.61441e8	+	7.99240e8i	0.558348	+	0.967087i	0.997635	+	0.0687402i	$0.0218980\pi$
$353$	4.61441e8	+	7.99240e8i	0.558348	+	0.967087i	−0.439287	+	0.898347i	$0.644769\pi$
$354$		0			0
$355$	−2.41508e7	+	4.18304e7i	−0.0286505	+	0.0496241i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−2.04373e8			−0.233127			−0.116564	−	0.993183i	$-0.537188\pi$
$359$	−2.04373e8			−0.233127			−0.116564	+	0.993183i	$0.537188\pi$
$360$		0			0
$361$	3.02751e8			0.338696
$362$		0			0
$363$		0			0
$364$		0			0
$365$	1.97686e7	−	3.42403e7i	0.0212790	−	0.0368564i
$366$		0			0
$367$	4.57810e8	+	7.92951e8i	0.483453	+	0.837365i	0.999819	−	0.0190026i	$-0.00604908\pi$
$367$	4.57810e8	+	7.92951e8i	0.483453	+	0.837365i	−0.516366	+	0.856368i	$0.672716\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−3.99087e8	−	6.91239e8i	−0.405750	−	0.702780i
$372$		0			0
$373$	−3.11520e8	+	5.39569e8i	−0.310818	+	0.538352i	−0.978540	−	0.206059i	$-0.933936\pi$
$373$	−3.11520e8	+	5.39569e8i	−0.310818	+	0.538352i	0.667722	+	0.744411i	$0.267269\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−1.48288e9			−1.42532
$378$		0			0
$379$	−2.07365e9			−1.95658			−0.978290	−	0.207242i	$-0.933551\pi$
$379$	−2.07365e9			−1.95658			−0.978290	+	0.207242i	$0.933551\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−8.84294e8	+	1.53164e9i	−0.804269	+	1.39303i	0.112515	+	0.993650i	$0.464109\pi$
$383$	−8.84294e8	+	1.53164e9i	−0.804269	+	1.39303i	−0.916784	+	0.399384i	$0.869224\pi$
$384$		0			0
$385$	7.27999e7	+	1.26093e8i	0.0650157	+	0.112611i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−2.18262e8	−	3.78041e8i	−0.187999	−	0.325623i	0.756584	−	0.653896i	$-0.226867\pi$
$389$	−2.18262e8	−	3.78041e8i	−0.187999	−	0.325623i	−0.944583	+	0.328273i	$0.893533\pi$
$390$		0			0
$391$	−1.82956e8	+	3.16889e8i	−0.154785	+	0.268095i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	1.26065e8			0.102921
$396$		0			0
$397$	1.93370e9			1.55104			0.775519	−	0.631325i	$-0.217488\pi$
$397$	1.93370e9			1.55104			0.775519	+	0.631325i	$0.217488\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	7.70049e8	−	1.33376e9i	0.596366	−	1.03294i	−0.396986	−	0.917825i	$-0.629944\pi$
$401$	7.70049e8	−	1.33376e9i	0.596366	−	1.03294i	0.993352	−	0.115112i	$-0.0367228\pi$
$402$		0			0
$403$	4.52729e8	+	7.84150e8i	0.344565	+	0.596803i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	1.29334e9	+	2.24013e9i	0.950896	+	1.64700i
$408$		0			0
$409$	2.41538e8	−	4.18356e8i	0.174564	−	0.302353i	−0.765446	−	0.643500i	$-0.777482\pi$
$409$	2.41538e8	−	4.18356e8i	0.174564	−	0.302353i	0.940010	+	0.341146i	$0.110815\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	2.21486e9			1.54711
$414$		0			0
$415$	8.83160e7			0.0606556
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−8.97234e8	+	1.55405e9i	−0.595877	+	1.03209i	0.397545	+	0.917582i	$0.369862\pi$
$419$	−8.97234e8	+	1.55405e9i	−0.595877	+	1.03209i	−0.993422	+	0.114507i	$0.963471\pi$
$420$		0			0
$421$	3.89591e8	+	6.74791e8i	0.254461	+	0.440739i	0.964749	−	0.263172i	$-0.0847686\pi$
$421$	3.89591e8	+	6.74791e8i	0.254461	+	0.440739i	−0.710288	+	0.703911i	$0.751435\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−1.92396e8	−	3.33240e8i	−0.121572	−	0.210569i
$426$		0			0
$427$	−4.78102e8	+	8.28097e8i	−0.297182	+	0.514735i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	2.69529e9			1.62157			0.810785	−	0.585344i	$-0.199040\pi$
$431$	2.69529e9			1.62157			0.810785	+	0.585344i	$0.199040\pi$
$432$		0			0
$433$	1.51823e9			0.898731			0.449366	−	0.893348i	$-0.351650\pi$
$433$	1.51823e9			0.898731			0.449366	+	0.893348i	$0.351650\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−1.28069e9	+	2.21822e9i	−0.734107	+	1.27151i
$438$		0			0
$439$	1.03683e9	+	1.79585e9i	0.584901	+	1.01308i	0.994888	+	0.100987i	$0.0322000\pi$
$439$	1.03683e9	+	1.79585e9i	0.584901	+	1.01308i	−0.409987	+	0.912092i	$0.634467\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−8.72205e8	−	1.51070e9i	−0.476657	−	0.825593i	0.522986	−	0.852342i	$-0.324818\pi$
$443$	−8.72205e8	−	1.51070e9i	−0.476657	−	0.825593i	−0.999642	+	0.0267480i	$0.991485\pi$
$444$		0			0
$445$	1.13089e7	−	1.95877e7i	0.00608362	−	0.0105371i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−2.40235e9			−1.25249			−0.626244	−	0.779627i	$-0.715409\pi$
$449$	−2.40235e9			−1.25249			−0.626244	+	0.779627i	$0.715409\pi$
$450$		0			0
$451$	−2.84544e9			−1.46060
$452$		0			0
$453$		0			0
$454$		0			0
$455$	9.04087e7	−	1.56592e8i	0.0449956	−	0.0779347i
$456$		0			0
$457$	6.99854e8	+	1.21218e9i	0.343005	+	0.594103i	0.984989	−	0.172615i	$-0.0552218\pi$
$457$	6.99854e8	+	1.21218e9i	0.343005	+	0.594103i	−0.641984	+	0.766718i	$0.721888\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−3.51935e8	−	6.09569e8i	−0.167305	−	0.289781i	0.770166	−	0.637843i	$-0.220173\pi$
$461$	−3.51935e8	−	6.09569e8i	−0.167305	−	0.289781i	−0.937471	+	0.348062i	$0.886840\pi$
$462$		0			0
$463$	−4.10315e8	+	7.10686e8i	−0.192125	+	0.332770i	−0.945954	−	0.324300i	$-0.894871\pi$
$463$	−4.10315e8	+	7.10686e8i	−0.192125	+	0.332770i	0.753829	+	0.657070i	$0.228205\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−1.85482e9			−0.842737			−0.421368	−	0.906890i	$-0.638450\pi$
$467$	−1.85482e9			−0.842737			−0.421368	+	0.906890i	$0.638450\pi$
$468$		0			0
$469$	−2.37223e9			−1.06182
$470$		0			0
$471$		0			0
$472$		0			0
$473$	3.10217e9	−	5.37311e9i	1.34788	−	2.33460i
$474$		0			0
$475$	−1.34677e9	−	2.33267e9i	−0.576588	−	0.998680i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−4.94650e8	−	8.56758e8i	−0.205647	−	0.356192i	0.744691	−	0.667409i	$-0.232597\pi$
$479$	−4.94650e8	−	8.56758e8i	−0.205647	−	0.356192i	−0.950339	+	0.311217i	$0.899263\pi$
$480$		0			0
$481$	1.60617e9	−	2.78197e9i	0.658089	−	1.13984i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	6.99483e7			0.0278408
$486$		0			0
$487$	7.40241e8			0.290417			0.145208	−	0.989401i	$-0.453615\pi$
$487$	7.40241e8			0.290417			0.145208	+	0.989401i	$0.453615\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−2.41638e9	+	4.18529e9i	−0.921255	+	1.59566i	−0.123779	+	0.992310i	$0.539501\pi$
$491$	−2.41638e9	+	4.18529e9i	−0.921255	+	1.59566i	−0.797476	+	0.603351i	$0.793832\pi$
$492$		0			0
$493$	−4.10704e8	−	7.11361e8i	−0.154371	−	0.267378i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	1.88579e9	+	3.26629e9i	0.689044	+	1.19346i
$498$		0			0
$499$	1.65538e9	−	2.86720e9i	0.596412	−	1.03302i	−0.396934	−	0.917847i	$-0.629926\pi$
$499$	1.65538e9	−	2.86720e9i	0.596412	−	1.03302i	0.993346	−	0.115168i	$-0.0367408\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−4.51666e9			−1.58245			−0.791225	−	0.611526i	$-0.790556\pi$
$503$	−4.51666e9			−1.58245			−0.791225	+	0.611526i	$0.790556\pi$
$504$		0			0
$505$	1.12443e8			0.0388518
$506$		0			0
$507$		0			0
$508$		0			0
$509$	2.35555e9	−	4.07994e9i	0.791737	−	1.37133i	−0.133154	−	0.991095i	$-0.542510\pi$
$509$	2.35555e9	−	4.07994e9i	0.791737	−	1.37133i	0.924891	−	0.380233i	$-0.124156\pi$
$510$		0			0
$511$	−1.54362e9	−	2.67363e9i	−0.511761	−	0.886396i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	5.88715e7	+	1.01969e8i	0.0189924	+	0.0328958i
$516$		0			0
$517$	−1.52869e9	+	2.64778e9i	−0.486524	+	0.842684i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	1.36592e9			0.423150			0.211575	−	0.977362i	$-0.432141\pi$
$521$	1.36592e9			0.423150			0.211575	+	0.977362i	$0.432141\pi$
$522$		0			0
$523$	3.67627e9			1.12370			0.561852	−	0.827238i	$-0.310089\pi$
$523$	3.67627e9			1.12370			0.561852	+	0.827238i	$0.310089\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−2.50779e8	+	4.34362e8i	−0.0746369	+	0.129275i
$528$		0			0
$529$	−1.03892e9	−	1.79946e9i	−0.305131	−	0.528502i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	1.76685e9	+	3.06027e9i	0.505421	+	0.875415i
$534$		0			0
$535$	−1.95823e8	+	3.39175e8i	−0.0552872	+	0.0957602i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	5.45295e9			1.49993
$540$		0			0
$541$	2.21813e9			0.602277			0.301139	−	0.953580i	$-0.402633\pi$
$541$	2.21813e9			0.602277			0.301139	+	0.953580i	$0.402633\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−6.64923e7	+	1.15168e8i	−0.0175948	+	0.0304750i
$546$		0			0
$547$	9.60119e8	+	1.66297e9i	0.250824	+	0.434440i	0.963753	−	0.266796i	$-0.0859651\pi$
$547$	9.60119e8	+	1.66297e9i	0.250824	+	0.434440i	−0.712929	+	0.701236i	$0.752632\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−2.87493e9	−	4.97952e9i	−0.732143	−	1.26811i
$552$		0			0
$553$	4.92183e9	−	8.52486e9i	1.23762	−	2.14363i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	1.27139e9			0.311736			0.155868	−	0.987778i	$-0.450183\pi$
$557$	1.27139e9			0.311736			0.155868	+	0.987778i	$0.450183\pi$
$558$		0			0
$559$	−7.70503e9			−1.86566
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−1.21064e9	+	2.09689e9i	−0.285915	+	0.495219i	−0.972831	−	0.231518i	$-0.925631\pi$
$563$	−1.21064e9	+	2.09689e9i	−0.285915	+	0.495219i	0.686916	+	0.726737i	$0.258964\pi$
$564$		0			0
$565$	1.21667e8	+	2.10733e8i	0.0283793	+	0.0491544i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−2.80973e9	−	4.86659e9i	−0.639399	−	1.10747i	−0.985565	−	0.169298i	$-0.945850\pi$
$569$	−2.80973e9	−	4.86659e9i	−0.639399	−	1.10747i	0.346166	−	0.938173i	$-0.387483\pi$
$570$		0			0
$571$	3.80308e8	−	6.58714e8i	0.0854889	−	0.148071i	−0.820111	−	0.572205i	$-0.806088\pi$
$571$	3.80308e8	−	6.58714e8i	0.0854889	−	0.148071i	0.905599	+	0.424134i	$0.139421\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	5.76554e9			1.26474
$576$		0			0
$577$	−7.92531e9			−1.71752			−0.858758	−	0.512381i	$-0.828763\pi$
$577$	−7.92531e9			−1.71752			−0.858758	+	0.512381i	$0.828763\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	3.44804e9	−	5.97219e9i	0.729384	−	1.26333i
$582$		0			0
$583$	−2.27892e9	−	3.94721e9i	−0.476310	−	0.824993i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−1.84157e9	−	3.18969e9i	−0.375798	−	0.650902i	0.614648	−	0.788802i	$-0.289298\pi$
$587$	−1.84157e9	−	3.18969e9i	−0.375798	−	0.650902i	−0.990446	+	0.137900i	$0.955965\pi$
$588$		0			0
$589$	−1.75545e9	+	3.04053e9i	−0.353985	+	0.613120i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−6.41737e9			−1.26376			−0.631882	−	0.775065i	$-0.717717\pi$
$593$	−6.41737e9			−1.26376			−0.631882	+	0.775065i	$0.717717\pi$
$594$		0			0
$595$	1.00160e8			0.0194932
$596$		0			0
$597$		0			0
$598$		0			0
$599$	2.72849e9	−	4.72589e9i	0.518715	−	0.898441i	−0.481048	−	0.876694i	$-0.659744\pi$
$599$	2.72849e9	−	4.72589e9i	0.518715	−	0.898441i	0.999764	−	0.0217468i	$-0.00692278\pi$
$600$		0			0
$601$	−3.33717e8	−	5.78016e8i	−0.0627073	−	0.108612i	0.832967	−	0.553322i	$-0.186640\pi$
$601$	−3.33717e8	−	5.78016e8i	−0.0627073	−	0.108612i	−0.895675	+	0.444710i	$0.853307\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	2.58733e8	+	4.48139e8i	0.0475016	+	0.0822752i
$606$		0			0
$607$	−3.87497e9	+	6.71164e9i	−0.703246	+	1.21806i	0.264074	+	0.964502i	$0.414934\pi$
$607$	−3.87497e9	+	6.71164e9i	−0.703246	+	1.21806i	−0.967321	+	0.253556i	$0.918400\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	3.79691e9			0.673419
$612$		0			0
$613$	5.23247e9			0.917477			0.458738	−	0.888571i	$-0.348301\pi$
$613$	5.23247e9			0.917477			0.458738	+	0.888571i	$0.348301\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	3.38936e9	−	5.87054e9i	0.580924	−	1.00619i	−0.414446	−	0.910074i	$-0.636025\pi$
$617$	3.38936e9	−	5.87054e9i	0.580924	−	1.00619i	0.995370	−	0.0961166i	$-0.0306422\pi$
$618$		0			0
$619$	−3.60531e9	−	6.24458e9i	−0.610977	−	1.05824i	−0.991076	−	0.133298i	$-0.957443\pi$
$619$	−3.60531e9	−	6.24458e9i	−0.610977	−	1.05824i	0.380099	−	0.924946i	$-0.375890\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−8.83051e8	−	1.52949e9i	−0.146311	−	0.253418i
$624$		0			0
$625$	−3.02137e9	+	5.23317e9i	−0.495022	+	0.857403i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	1.77941e9			0.285100
$630$		0			0
$631$	−3.24048e9			−0.513460			−0.256730	−	0.966483i	$-0.582645\pi$
$631$	−3.24048e9			−0.513460			−0.256730	+	0.966483i	$0.582645\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	2.26980e8	−	3.93141e8i	0.0351787	−	0.0609313i
$636$		0			0
$637$	−3.38595e9	−	5.86464e9i	−0.519030	−	0.898986i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−2.41139e9	−	4.17665e9i	−0.361630	−	0.626362i	0.626599	−	0.779342i	$-0.284446\pi$
$641$	−2.41139e9	−	4.17665e9i	−0.361630	−	0.626362i	−0.988229	+	0.152980i	$0.951113\pi$
$642$		0			0
$643$	−2.87064e9	+	4.97210e9i	−0.425835	+	0.737567i	−0.996498	−	0.0836170i	$-0.973353\pi$
$643$	−2.87064e9	+	4.97210e9i	−0.425835	+	0.737567i	0.570663	+	0.821184i	$0.306686\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	4.26278e9			0.618768			0.309384	−	0.950937i	$-0.399877\pi$
$647$	4.26278e9			0.618768			0.309384	+	0.950937i	$0.399877\pi$
$648$		0			0
$649$	1.26476e10			1.81616
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−3.20410e9	+	5.54967e9i	−0.450308	+	0.779957i	−0.998405	−	0.0564580i	$-0.982019\pi$
$653$	−3.20410e9	+	5.54967e9i	−0.450308	+	0.779957i	0.548097	+	0.836415i	$0.315353\pi$
$654$		0			0
$655$	−1.09367e8	−	1.89430e8i	−0.0152070	−	0.0263393i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	4.74480e9	+	8.21823e9i	0.645831	+	1.11861i	0.984109	+	0.177566i	$0.0568222\pi$
$659$	4.74480e9	+	8.21823e9i	0.645831	+	1.11861i	−0.338278	+	0.941046i	$0.609844\pi$
$660$		0			0
$661$	−3.96599e9	+	6.86929e9i	−0.534129	+	0.925138i	0.465076	+	0.885271i	$0.346027\pi$
$661$	−3.96599e9	+	6.86929e9i	−0.534129	+	0.925138i	−0.999205	+	0.0398677i	$0.987306\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	7.01116e8			0.0924515
$666$		0			0
$667$	1.23076e10			1.60596
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−2.73013e9	+	4.72872e9i	−0.348862	+	0.604247i
$672$		0			0
$673$	−3.39566e9	−	5.88145e9i	−0.429409	−	0.743758i	0.567412	−	0.823434i	$-0.307945\pi$
$673$	−3.39566e9	−	5.88145e9i	−0.429409	−	0.743758i	−0.996821	+	0.0796763i	$0.974611\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	6.50731e9	+	1.12710e10i	0.806012	+	1.39605i	0.915606	+	0.402077i	$0.131712\pi$
$677$	6.50731e9	+	1.12710e10i	0.806012	+	1.39605i	−0.109594	+	0.993976i	$0.534955\pi$
$678$		0			0
$679$	2.73093e9	−	4.73011e9i	0.334785	−	0.579865i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−9.51975e9			−1.14328			−0.571640	−	0.820504i	$-0.693693\pi$
$683$	−9.51975e9			−1.14328			−0.571640	+	0.820504i	$0.693693\pi$
$684$		0			0
$685$	1.77221e8			0.0210668
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−2.83014e9	+	4.90195e9i	−0.329641	+	0.570955i
$690$		0			0
$691$	−2.34333e8	−	4.05877e8i	−0.0270185	−	0.0467973i	0.852200	−	0.523216i	$-0.175268\pi$
$691$	−2.34333e8	−	4.05877e8i	−0.0270185	−	0.0467973i	−0.879218	+	0.476419i	$0.841935\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	1.12049e8	+	1.94075e8i	0.0126608	+	0.0219292i
$696$		0			0
$697$	−9.78704e8	+	1.69516e9i	−0.109480	+	0.189626i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−1.05093e10			−1.15229			−0.576145	−	0.817348i	$-0.695444\pi$
$701$	−1.05093e10			−1.15229			−0.576145	+	0.817348i	$0.695444\pi$
$702$		0			0
$703$	1.24558e10			1.35216
$704$		0			0
$705$		0			0
$706$		0			0
$707$	4.38999e9	−	7.60369e9i	0.467193	−	0.809201i
$708$		0			0
$709$	−2.92228e8	−	5.06154e8i	−0.0307936	−	0.0533361i	0.850218	−	0.526431i	$-0.176470\pi$
$709$	−2.92228e8	−	5.06154e8i	−0.0307936	−	0.0533361i	−0.881012	+	0.473095i	$0.843137\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−3.75756e9	−	6.50828e9i	−0.388233	−	0.672439i
$714$		0			0
$715$	5.16264e8	−	8.94196e8i	0.0528203	−	0.0914875i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	4.03702e9			0.405051			0.202526	−	0.979277i	$-0.435085\pi$
$719$	4.03702e9			0.405051			0.202526	+	0.979277i	$0.435085\pi$
$720$		0			0
$721$	9.19388e9			0.913535
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−6.47132e9	+	1.12087e10i	−0.630681	+	1.09237i
$726$		0			0
$727$	2.29032e9	+	3.96695e9i	0.221068	+	0.382901i	0.955133	−	0.296179i	$-0.0957124\pi$
$727$	2.29032e9	+	3.96695e9i	0.221068	+	0.382901i	−0.734065	+	0.679080i	$0.762379\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−2.13401e9	−	3.69622e9i	−0.202063	−	0.349983i
$732$		0			0
$733$	−6.80858e9	+	1.17928e10i	−0.638547	+	1.10600i	0.347205	+	0.937789i	$0.387131\pi$
$733$	−6.80858e9	+	1.17928e10i	−0.638547	+	1.10600i	−0.985752	+	0.168207i	$0.946202\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−1.35462e10			−1.24647
$738$		0			0
$739$	2.25285e9			0.205341			0.102671	−	0.994715i	$-0.467261\pi$
$739$	2.25285e9			0.205341			0.102671	+	0.994715i	$0.467261\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−3.93509e9	+	6.81578e9i	−0.351961	+	0.609614i	−0.986593	−	0.163200i	$-0.947818\pi$
$743$	−3.93509e9	+	6.81578e9i	−0.351961	+	0.609614i	0.634632	+	0.772814i	$0.281152\pi$
$744$		0			0
$745$	4.19458e8	+	7.26523e8i	0.0371656	+	0.0643728i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	1.52907e10	+	2.64842e10i	1.32966	+	2.30303i
$750$		0			0
$751$	−2.54147e9	+	4.40195e9i	−0.218950	+	0.379232i	−0.954487	−	0.298252i	$-0.903596\pi$
$751$	−2.54147e9	+	4.40195e9i	−0.218950	+	0.379232i	0.735537	+	0.677484i	$0.236930\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	9.53331e8			0.0806175
$756$		0			0
$757$	−1.51049e10			−1.26556			−0.632780	−	0.774332i	$-0.718086\pi$
$757$	−1.51049e10			−1.26556			−0.632780	+	0.774332i	$0.718086\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−2.88888e9	+	5.00369e9i	−0.237620	+	0.411570i	−0.960031	−	0.279894i	$-0.909701\pi$
$761$	−2.88888e9	+	5.00369e9i	−0.237620	+	0.411570i	0.722411	+	0.691464i	$0.243034\pi$
$762$		0			0
$763$	5.19200e9	+	8.99280e9i	0.423154	+	0.732925i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−7.85341e9	−	1.36025e10i	−0.628456	−	1.08852i
$768$		0			0
$769$	1.19009e9	−	2.06130e9i	0.0943712	−	0.163456i	−0.814975	−	0.579496i	$-0.803249\pi$
$769$	1.19009e9	−	2.06130e9i	0.0943712	−	0.163456i	0.909346	+	0.416041i	$0.136583\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−7.06529e9			−0.550176			−0.275088	−	0.961419i	$-0.588707\pi$
$773$	−7.06529e9			−0.550176			−0.275088	+	0.961419i	$0.588707\pi$
$774$		0			0
$775$	7.90286e9			0.609858
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−6.85092e9	+	1.18661e10i	−0.519239	+	0.899349i
$780$		0			0
$781$	1.07685e10	+	1.86517e10i	0.808869	+	1.40100i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−4.97942e8	−	8.62461e8i	−0.0367397	−	0.0636350i
$786$		0			0
$787$	6.76448e9	−	1.17164e10i	0.494679	−	0.856809i	−0.505303	−	0.862942i	$-0.668619\pi$
$787$	6.76448e9	−	1.17164e10i	0.494679	−	0.856809i	0.999981	+	0.00613374i	$0.00195244\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	1.90005e10			1.36505
$792$		0			0
$793$	6.78097e9			0.482876
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−2.90848e9	+	5.03763e9i	−0.203499	+	0.352470i	−0.949653	−	0.313302i	$-0.898565\pi$
$797$	−2.90848e9	+	5.03763e9i	−0.203499	+	0.352470i	0.746155	+	0.665773i	$0.231898\pi$
$798$		0			0
$799$	1.05160e9	+	1.82143e9i	0.0729355	+	0.126328i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−8.81460e9	−	1.52673e10i	−0.600756	−	1.04054i
$804$		0			0
$805$	−7.50373e8	+	1.29968e9i	−0.0506981	+	0.0878116i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	1.81965e9			0.120828			0.0604140	−	0.998173i	$-0.480758\pi$
$809$	1.81965e9			0.120828			0.0604140	+	0.998173i	$0.480758\pi$
$810$		0			0
$811$	9.75638e9			0.642267			0.321133	−	0.947034i	$-0.395936\pi$
$811$	9.75638e9			0.642267			0.321133	+	0.947034i	$0.395936\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−6.43148e8	+	1.11397e9i	−0.0416159	+	0.0720809i
$816$		0			0
$817$	−1.49381e10	−	2.58735e10i	−0.958334	−	1.65988i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	1.19115e10	+	2.06314e10i	0.751220	+	1.30115i	0.947232	+	0.320550i	$0.103868\pi$
$821$	1.19115e10	+	2.06314e10i	0.751220	+	1.30115i	−0.196012	+	0.980602i	$0.562799\pi$
$822$		0			0
$823$	−7.89296e9	+	1.36710e10i	−0.493561	+	0.854872i	−0.999972	−	0.00741953i	$-0.997638\pi$
$823$	−7.89296e9	+	1.36710e10i	−0.493561	+	0.854872i	0.506412	+	0.862292i	$0.330972\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	7.54697e9			0.463985			0.231992	−	0.972718i	$-0.425476\pi$
$827$	7.54697e9			0.463985			0.231992	+	0.972718i	$0.425476\pi$
$828$		0			0
$829$	−2.77361e10			−1.69085			−0.845424	−	0.534096i	$-0.820652\pi$
$829$	−2.77361e10			−1.69085			−0.845424	+	0.534096i	$0.820652\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	1.87557e9	−	3.24858e9i	0.112428	−	0.194732i
$834$		0			0
$835$	3.02019e8	+	5.23112e8i	0.0179528	+	0.0310951i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	2.06404e8	+	3.57502e8i	0.0120657	+	0.0208983i	0.871995	−	0.489514i	$-0.162826\pi$
$839$	2.06404e8	+	3.57502e8i	0.0120657	+	0.0208983i	−0.859930	+	0.510413i	$0.829493\pi$
$840$		0			0
$841$	−5.18928e9	+	8.98810e9i	−0.300830	+	0.521053i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−2.71331e8			−0.0154704
$846$		0			0
$847$	4.04060e10			2.28483
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−1.33309e10	+	2.30898e10i	−0.741491	+	1.28430i
$852$		0			0
$853$	1.24159e10	+	2.15050e10i	0.684947	+	1.18636i	0.973453	+	0.228885i	$0.0735081\pi$
$853$	1.24159e10	+	2.15050e10i	0.684947	+	1.18636i	−0.288506	+	0.957478i	$0.593159\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−4.21928e9	−	7.30801e9i	−0.228984	−	0.396612i	0.728523	−	0.685021i	$-0.240207\pi$
$857$	−4.21928e9	−	7.30801e9i	−0.228984	−	0.396612i	−0.957507	+	0.288409i	$0.906874\pi$
$858$		0			0
$859$	−1.37913e10	+	2.38873e10i	−0.742387	+	1.28585i	0.209018	+	0.977912i	$0.432973\pi$
$859$	−1.37913e10	+	2.38873e10i	−0.742387	+	1.28585i	−0.951406	+	0.307941i	$0.900360\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−5.47588e9			−0.290012			−0.145006	−	0.989431i	$-0.546320\pi$
$863$	−5.47588e9			−0.290012			−0.145006	+	0.989431i	$0.546320\pi$
$864$		0			0
$865$	−4.85870e8			−0.0255249
$866$		0			0
$867$		0			0
$868$		0			0
$869$	2.81054e10	−	4.86799e10i	1.45285	−	2.51640i
$870$		0			0
$871$	8.41138e9	+	1.45689e10i	0.431324	+	0.747075i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−1.58081e9	−	2.73804e9i	−0.0797721	−	0.138169i
$876$		0			0
$877$	−1.30792e10	+	2.26538e10i	−0.654761	+	1.13408i	0.327193	+	0.944958i	$0.393897\pi$
$877$	−1.30792e10	+	2.26538e10i	−0.654761	+	1.13408i	−0.981954	+	0.189122i	$0.939436\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−7.51788e9			−0.370408			−0.185204	−	0.982700i	$-0.559295\pi$
$881$	−7.51788e9			−0.370408			−0.185204	+	0.982700i	$0.559295\pi$
$882$		0			0
$883$	6.87631e9			0.336119			0.168060	−	0.985777i	$-0.446250\pi$
$883$	6.87631e9			0.336119			0.168060	+	0.985777i	$0.446250\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	8.65691e9	−	1.49942e10i	0.416515	−	0.721425i	−0.579071	−	0.815277i	$-0.696585\pi$
$887$	8.65691e9	−	1.49942e10i	0.416515	−	0.721425i	0.995586	+	0.0938521i	$0.0299181\pi$
$888$		0			0
$889$	−1.77236e10	−	3.06981e10i	−0.846048	−	1.46540i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	7.36122e9	+	1.27500e10i	0.345915	+	0.599143i
$894$		0			0
$895$	2.35311e8	−	4.07571e8i	0.0109714	−	0.0190030i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	1.68701e10			0.774389
$900$		0			0
$901$	−3.13539e9			−0.142809
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−7.52244e8	+	1.30292e9i	−0.0337356	+	0.0584318i
$906$		0			0
$907$	−1.37840e10	−	2.38747e10i	−0.613410	−	1.06246i	−0.990661	−	0.136346i	$-0.956464\pi$
$907$	−1.37840e10	−	2.38747e10i	−0.613410	−	1.06246i	0.377251	−	0.926111i	$-0.376869\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−5.21133e9	−	9.02628e9i	−0.228367	−	0.395544i	0.728957	−	0.684559i	$-0.240005\pi$
$911$	−5.21133e9	−	9.02628e9i	−0.228367	−	0.395544i	−0.957324	+	0.289016i	$0.906672\pi$
$912$		0			0
$913$	1.96895e10	−	3.41032e10i	0.856224	−	1.48302i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−1.70798e10			−0.731457
$918$		0			0
$919$	−4.43234e10			−1.88377			−0.941887	−	0.335929i	$-0.890950\pi$
$919$	−4.43234e10			−1.88377			−0.941887	+	0.335929i	$0.890950\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	1.33732e10	−	2.31631e10i	0.559796	−	0.969595i
$924$		0			0
$925$	−1.40187e10	−	2.42811e10i	−0.582388	−	1.00873i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	8.15099e9	+	1.41179e10i	0.333546	+	0.577718i	0.983204	−	0.182508i	$-0.0584215\pi$
$929$	8.15099e9	+	1.41179e10i	0.333546	+	0.577718i	−0.649659	+	0.760226i	$0.725088\pi$
$930$		0			0
$931$	1.31290e10	−	2.27400e10i	0.533220	−	0.923564i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	5.71946e8			0.0228831
$936$		0			0
$937$	1.40475e10			0.557843			0.278921	−	0.960314i	$-0.410023\pi$
$937$	1.40475e10			0.557843			0.278921	+	0.960314i	$0.410023\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−2.23540e10	+	3.87182e10i	−0.874563	+	1.51479i	−0.0173363	+	0.999850i	$0.505519\pi$
$941$	−2.23540e10	+	3.87182e10i	−0.874563	+	1.51479i	−0.857227	+	0.514938i	$0.827815\pi$
$942$		0			0
$943$	−1.46644e10	−	2.53996e10i	−0.569475	−	0.986360i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−1.80653e10	−	3.12901e10i	−0.691228	−	1.19724i	−0.971436	−	0.237302i	$-0.923737\pi$
$947$	−1.80653e10	−	3.12901e10i	−0.691228	−	1.19724i	0.280208	−	0.959939i	$-0.409597\pi$
$948$		0			0
$949$	−1.09467e10	+	1.89602e10i	−0.415767	+	0.720129i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−2.92408e10			−1.09437			−0.547185	−	0.837011i	$-0.684301\pi$
$953$	−2.92408e10			−1.09437			−0.547185	+	0.837011i	$0.684301\pi$
$954$		0			0
$955$	−6.75282e8			−0.0250884
$956$		0			0
$957$		0			0
$958$		0			0
$959$	6.91909e9	−	1.19842e10i	0.253329	−	0.438778i
$960$		0			0
$961$	8.60581e9	+	1.49057e10i	0.312795	+	0.541777i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−2.24679e8	−	3.89155e8i	−0.00804852	−	0.0139404i
$966$		0			0
$967$	1.93427e10	−	3.35026e10i	0.687900	−	1.19148i	−0.284616	−	0.958642i	$-0.591866\pi$
$967$	1.93427e10	−	3.35026e10i	0.687900	−	1.19148i	0.972516	−	0.232836i	$-0.0748006\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−6.44519e9			−0.225927			−0.112964	−	0.993599i	$-0.536034\pi$
$971$	−6.44519e9			−0.225927			−0.112964	+	0.993599i	$0.536034\pi$
$972$		0			0
$973$	1.74986e10			0.608986
$974$		0			0
$975$		0			0
$976$		0			0
$977$	9.78292e9	−	1.69445e10i	0.335612	−	0.581297i	−0.647990	−	0.761649i	$-0.724390\pi$
$977$	9.78292e9	−	1.69445e10i	0.335612	−	0.581297i	0.983602	+	0.180352i	$0.0577235\pi$
$978$		0			0
$979$	−5.04252e9	−	8.73391e9i	−0.171755	−	0.297488i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	1.84494e10	+	3.19553e10i	0.619505	+	1.07301i	0.989576	+	0.144011i	$0.0460001\pi$
$983$	1.84494e10	+	3.19553e10i	0.619505	+	1.07301i	−0.370071	+	0.929004i	$0.620667\pi$
$984$		0			0
$985$	−2.41792e8	+	4.18797e8i	−0.00806150	+	0.0139629i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	6.39501e10			2.10210
$990$		0			0
$991$	−8.90183e9			−0.290550			−0.145275	−	0.989391i	$-0.546407\pi$
$991$	−8.90183e9			−0.290550			−0.145275	+	0.989391i	$0.546407\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	1.03884e9	−	1.79933e9i	0.0334325	−	0.0579068i
$996$		0			0
$997$	1.30622e10	+	2.26244e10i	0.417430	+	0.723009i	0.995680	−	0.0928501i	$-0.0295977\pi$
$997$	1.30622e10	+	2.26244e10i	0.417430	+	0.723009i	−0.578251	+	0.815859i	$0.696264\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	216.8.i.a.145.6		20
3.2	odd	2		72.8.i.a.49.8	yes	20
4.3	odd	2		432.8.i.e.145.6		20
9.2	odd	6		72.8.i.a.25.8	✓	20
9.7	even	3	inner	216.8.i.a.73.6		20
12.11	even	2		144.8.i.e.49.3		20
36.7	odd	6		432.8.i.e.289.6		20
36.11	even	6		144.8.i.e.97.3		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.8.i.a.25.8	✓	20	9.2	odd	6
72.8.i.a.49.8	yes	20	3.2	odd	2
144.8.i.e.49.3		20	12.11	even	2
144.8.i.e.97.3		20	36.11	even	6
216.8.i.a.73.6		20	9.7	even	3	inner
216.8.i.a.145.6		20	1.1	even	1	trivial
432.8.i.e.145.6		20	4.3	odd	2
432.8.i.e.289.6		20	36.7	odd	6

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:	\( N \)	\(=\)	\( 216 = 2^{3} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	216.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(8.05553 - 13.9526i) q^{5} +(-629.010 - 1089.48i) q^{7} +O(q^{10})\)	\(q+(8.05553 - 13.9526i) q^{5} +(-629.010 - 1089.48i) q^{7} +(-3591.86 - 6221.29i) q^{11} +(-4460.66 + 7726.08i) q^{13} -4941.76 q^{17} -34592.2 q^{19} +(37022.5 - 64124.8i) q^{23} +(38932.7 + 67433.4i) q^{25} +(83109.0 + 143949. i) q^{29} +(50746.9 - 87896.2i) q^{31} -20268.0 q^{35} -360076. q^{37} +(198048. - 343029. i) q^{41} +(431833. + 747956. i) q^{43} +(-212800. - 368580. i) q^{47} +(-379535. + 657374. i) q^{49} +634468. q^{53} -115737. q^{55} +(-880298. + 1.52472e6i) q^{59} +(-380043. - 658255. i) q^{61} +(71865.9 + 124475. i) q^{65} +(942841. - 1.63305e6i) q^{67} -2.99804e6 q^{71} +2.45405e6 q^{73} +(-4.51863e6 + 7.82650e6i) q^{77} +(3.91237e6 + 6.77642e6i) q^{79} +(2.74085e6 + 4.74729e6i) q^{83} +(-39808.4 + 68950.3i) q^{85} +1.40387e6 q^{89} +1.12232e7 q^{91} +(-278659. + 482651. i) q^{95} +(2.17082e6 + 3.75996e6i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 125 q^{5} + 1245 q^{7}+O(q^{10}) \) 20 * q + 125 * q^5 + 1245 * q^7	\( 20 q + 125 q^{5} + 1245 q^{7} - 6106 q^{11} + 4937 q^{13} - 48722 q^{17} - 26882 q^{19} - 19387 q^{23} - 218957 q^{25} + 46791 q^{29} - 185039 q^{31} - 83094 q^{35} + 108420 q^{37} + 638112 q^{41} + 892628 q^{43} + 230883 q^{47} - 1034741 q^{49} + 2872940 q^{53} + 1089998 q^{55} - 2172454 q^{59} - 1878325 q^{61} - 1239133 q^{65} + 531496 q^{67} - 3723056 q^{71} - 1804522 q^{73} - 6276543 q^{77} + 3607847 q^{79} - 10794491 q^{83} + 3597658 q^{85} - 32214888 q^{89} - 16117530 q^{91} + 756868 q^{95} + 17951260 q^{97}+O(q^{100}) \) 20 * q + 125 * q^5 + 1245 * q^7 - 6106 * q^11 + 4937 * q^13 - 48722 * q^17 - 26882 * q^19 - 19387 * q^23 - 218957 * q^25 + 46791 * q^29 - 185039 * q^31 - 83094 * q^35 + 108420 * q^37 + 638112 * q^41 + 892628 * q^43 + 230883 * q^47 - 1034741 * q^49 + 2872940 * q^53 + 1089998 * q^55 - 2172454 * q^59 - 1878325 * q^61 - 1239133 * q^65 + 531496 * q^67 - 3723056 * q^71 - 1804522 * q^73 - 6276543 * q^77 + 3607847 * q^79 - 10794491 * q^83 + 3597658 * q^85 - 32214888 * q^89 - 16117530 * q^91 + 756868 * q^95 + 17951260 * q^97

Introduction

L-functions

Modular forms

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