LMFDB - Embedded newform 216.8.i.a.145.10

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			145.10
Root			$0.500000 + 271.860i$ of defining polynomial
Character	$\chi$	$=$	216.145
Dual form			216.8.i.a.73.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(241.687 - 418.615i) q^{5} +(-351.089 - 608.105i) q^{7} +O(q^{10})$	$q+(241.687 - 418.615i) q^{5} +(-351.089 - 608.105i) q^{7} +(-2817.09 - 4879.34i) q^{11} +(7414.47 - 12842.2i) q^{13} -24277.6 q^{17} +19398.7 q^{19} +(-16631.4 + 28806.4i) q^{23} +(-77763.1 - 134690. i) q^{25} +(-38775.7 - 67161.5i) q^{29} +(35685.1 - 61808.4i) q^{31} -339416. q^{35} +300968. q^{37} +(-232593. + 402864. i) q^{41} +(465496. + 806263. i) q^{43} +(243582. + 421897. i) q^{47} +(165244. - 286211. i) q^{49} +1.59113e6 q^{53} -2.72342e6 q^{55} +(-651954. + 1.12922e6i) q^{59} +(180759. + 313083. i) q^{61} +(-3.58397e6 - 6.20762e6i) q^{65} +(-267454. + 463244. i) q^{67} +2.28563e6 q^{71} +145959. q^{73} +(-1.97810e6 + 3.42617e6i) q^{77} +(-1.64791e6 - 2.85426e6i) q^{79} +(-3.69100e6 - 6.39301e6i) q^{83} +(-5.86758e6 + 1.01630e7i) q^{85} -1.18136e7 q^{89} -1.04126e7 q^{91} +(4.68841e6 - 8.12057e6i) q^{95} +(-1.88710e6 - 3.26855e6i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	241.687	−	418.615i	0.864687	−	1.49768i	−0.00267002	−	0.999996i	$-0.500850\pi$
$5$	241.687	−	418.615i	0.864687	−	1.49768i	0.867357	−	0.497686i	$-0.165817\pi$
$6$		0			0
$7$	−351.089	−	608.105i	−0.386878	−	0.670093i	0.605149	−	0.796112i	$-0.293113\pi$
$7$	−351.089	−	608.105i	−0.386878	−	0.670093i	−0.992028	+	0.126019i	$0.959780\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−2817.09	−	4879.34i	−0.638155	−	1.10532i	−0.985837	−	0.167704i	$-0.946365\pi$
$11$	−2817.09	−	4879.34i	−0.638155	−	1.10532i	0.347683	−	0.937612i	$-0.386969\pi$
$12$		0			0
$13$	7414.47	−	12842.2i	0.936006	−	1.62121i	0.163175	−	0.986597i	$-0.447826\pi$
$13$	7414.47	−	12842.2i	0.936006	−	1.62121i	0.772831	−	0.634612i	$-0.218840\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−24277.6			−1.19849			−0.599245	−	0.800566i	$-0.704532\pi$
$17$	−24277.6			−1.19849			−0.599245	+	0.800566i	$0.704532\pi$
$18$		0			0
$19$	19398.7			0.648835			0.324417	−	0.945914i	$-0.394832\pi$
$19$	19398.7			0.648835			0.324417	+	0.945914i	$0.394832\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−16631.4	+	28806.4i	−0.285024	+	0.493676i	−0.972615	−	0.232422i	$-0.925335\pi$
$23$	−16631.4	+	28806.4i	−0.285024	+	0.493676i	0.687591	+	0.726098i	$0.258668\pi$
$24$		0			0
$25$	−77763.1	−	134690.i	−0.995368	−	1.72403i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−38775.7	−	67161.5i	−0.295234	−	0.511361i	0.679805	−	0.733393i	$-0.262064\pi$
$29$	−38775.7	−	67161.5i	−0.295234	−	0.511361i	−0.975039	+	0.222032i	$0.928731\pi$
$30$		0			0
$31$	35685.1	−	61808.4i	0.215140	−	0.372633i	−0.738176	−	0.674608i	$-0.764313\pi$
$31$	35685.1	−	61808.4i	0.215140	−	0.372633i	0.953316	+	0.301975i	$0.0976459\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−339416.			−1.33812
$36$		0			0
$37$	300968.			0.976821			0.488410	−	0.872614i	$-0.337577\pi$
$37$	300968.			0.976821			0.488410	+	0.872614i	$0.337577\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−232593.	+	402864.i	−0.527053	+	0.912882i	0.472450	+	0.881357i	$0.343370\pi$
$41$	−232593.	+	402864.i	−0.527053	+	0.912882i	−0.999503	+	0.0315247i	$0.989964\pi$
$42$		0			0
$43$	465496.	+	806263.i	0.892845	+	1.54645i	0.836449	+	0.548045i	$0.184628\pi$
$43$	465496.	+	806263.i	0.892845	+	1.54645i	0.0563966	+	0.998408i	$0.482039\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	243582.	+	421897.i	0.342218	+	0.592739i	0.984844	−	0.173441i	$-0.0554885\pi$
$47$	243582.	+	421897.i	0.342218	+	0.592739i	−0.642626	+	0.766180i	$0.722155\pi$
$48$		0			0
$49$	165244.	−	286211.i	0.200650	−	0.347536i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	1.59113e6			1.46805			0.734023	−	0.679124i	$-0.237640\pi$
$53$	1.59113e6			1.46805			0.734023	+	0.679124i	$0.237640\pi$
$54$		0			0
$55$	−2.72342e6			−2.20722
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−651954.	+	1.12922e6i	−0.413271	+	0.715807i	−0.995245	−	0.0974007i	$-0.968947\pi$
$59$	−651954.	+	1.12922e6i	−0.413271	+	0.715807i	0.581974	+	0.813207i	$0.302280\pi$
$60$		0			0
$61$	180759.	+	313083.i	0.101964	+	0.176606i	0.912494	−	0.409091i	$-0.134154\pi$
$61$	180759.	+	313083.i	0.101964	+	0.176606i	−0.810530	+	0.585697i	$0.800821\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−3.58397e6	−	6.20762e6i	−1.61870	−	2.80368i
$66$		0			0
$67$	−267454.	+	463244.i	−0.108640	+	0.188169i	−0.915219	−	0.402956i	$-0.867983\pi$
$67$	−267454.	+	463244.i	−0.108640	+	0.188169i	0.806580	+	0.591125i	$0.201316\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	2.28563e6			0.757884			0.378942	−	0.925421i	$-0.376288\pi$
$71$	2.28563e6			0.757884			0.378942	+	0.925421i	$0.376288\pi$
$72$		0			0
$73$	145959.			0.0439136			0.0219568	−	0.999759i	$-0.493010\pi$
$73$	145959.			0.0439136			0.0219568	+	0.999759i	$0.493010\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−1.97810e6	+	3.42617e6i	−0.493777	+	0.855246i
$78$		0			0
$79$	−1.64791e6	−	2.85426e6i	−0.376044	−	0.651327i	0.614439	−	0.788964i	$-0.289382\pi$
$79$	−1.64791e6	−	2.85426e6i	−0.376044	−	0.651327i	−0.990483	+	0.137638i	$0.956049\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−3.69100e6	−	6.39301e6i	−0.708551	−	1.22725i	−0.965395	−	0.260794i	$-0.916016\pi$
$83$	−3.69100e6	−	6.39301e6i	−0.708551	−	1.22725i	0.256843	−	0.966453i	$-0.417318\pi$
$84$		0			0
$85$	−5.86758e6	+	1.01630e7i	−1.03632	+	1.79496i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−1.18136e7			−1.77631			−0.888155	−	0.459544i	$-0.848013\pi$
$89$	−1.18136e7			−1.77631			−0.888155	+	0.459544i	$0.848013\pi$
$90$		0			0
$91$	−1.04126e7			−1.44848
$92$		0			0
$93$		0			0
$94$		0			0
$95$	4.68841e6	−	8.12057e6i	0.561039	−	0.971748i
$96$		0			0
$97$	−1.88710e6	−	3.26855e6i	−0.209939	−	0.363625i	0.741756	−	0.670670i	$-0.233993\pi$
$97$	−1.88710e6	−	3.26855e6i	−0.209939	−	0.363625i	−0.951695	+	0.307045i	$0.900660\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−795542.	−	1.37792e6i	−0.0768313	−	0.133076i	0.825050	−	0.565060i	$-0.191147\pi$
$101$	−795542.	−	1.37792e6i	−0.0768313	−	0.133076i	−0.901881	+	0.431984i	$0.857814\pi$
$102$		0			0
$103$	−5.35308e6	+	9.27181e6i	−0.482696	+	0.836054i	−0.999803	−	0.0198669i	$-0.993676\pi$
$103$	−5.35308e6	+	9.27181e6i	−0.482696	+	0.836054i	0.517107	+	0.855921i	$0.327009\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	3.87045e6			0.305435			0.152717	−	0.988270i	$-0.451198\pi$
$107$	3.87045e6			0.305435			0.152717	+	0.988270i	$0.451198\pi$
$108$		0			0
$109$	586326.			0.0433657			0.0216829	−	0.999765i	$-0.493098\pi$
$109$	586326.			0.0433657			0.0216829	+	0.999765i	$0.493098\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	2.16760e6	−	3.75439e6i	0.141320	−	0.244774i	−0.786674	−	0.617369i	$-0.788199\pi$
$113$	2.16760e6	−	3.75439e6i	0.141320	−	0.244774i	0.927994	+	0.372595i	$0.121532\pi$
$114$		0			0
$115$	8.03920e6	+	1.39243e7i	0.492913	+	0.853750i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	8.52360e6	+	1.47633e7i	0.463670	+	0.803099i
$120$		0			0
$121$	−6.12838e6	+	1.06147e7i	−0.314483	+	0.544700i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−3.74138e7			−1.71335
$126$		0			0
$127$	−2.08109e7			−0.901526			−0.450763	−	0.892644i	$-0.648848\pi$
$127$	−2.08109e7			−0.901526			−0.450763	+	0.892644i	$0.648848\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−8.51991e6	+	1.47569e7i	−0.331120	+	0.573517i	−0.982732	−	0.185036i	$-0.940760\pi$
$131$	−8.51991e6	+	1.47569e7i	−0.331120	+	0.573517i	0.651612	+	0.758553i	$0.274093\pi$
$132$		0			0
$133$	−6.81066e6	−	1.17964e7i	−0.251020	−	0.434780i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−1.95826e7	−	3.39181e7i	−0.650653	−	1.12696i	−0.982965	−	0.183794i	$-0.941162\pi$
$137$	−1.95826e7	−	3.39181e7i	−0.650653	−	1.12696i	0.332312	−	0.943170i	$-0.392171\pi$
$138$		0			0
$139$	1.19880e7	−	2.07638e7i	0.378612	−	0.655776i	−0.612248	−	0.790666i	$-0.709735\pi$
$139$	1.19880e7	−	2.07638e7i	0.378612	−	0.655776i	0.990861	+	0.134890i	$0.0430680\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−8.35489e7			−2.38927
$144$		0			0
$145$	−3.74864e7			−1.02114
$146$		0			0
$147$		0			0
$148$		0			0
$149$	2.32172e6	−	4.02133e6i	0.0574986	−	0.0995905i	−0.835843	−	0.548968i	$-0.815021\pi$
$149$	2.32172e6	−	4.02133e6i	0.0574986	−	0.0995905i	0.893342	+	0.449378i	$0.148354\pi$
$150$		0			0
$151$	3.80890e7	+	6.59721e7i	0.900286	+	1.55934i	0.827123	+	0.562021i	$0.189976\pi$
$151$	3.80890e7	+	6.59721e7i	0.900286	+	1.55934i	0.0731635	+	0.997320i	$0.476691\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−1.72493e7	−	2.98766e7i	−0.372058	−	0.644423i
$156$		0			0
$157$	3.21026e7	−	5.56033e7i	0.662050	−	1.14670i	−0.318026	−	0.948082i	$-0.603020\pi$
$157$	3.21026e7	−	5.56033e7i	0.662050	−	1.14670i	0.980076	−	0.198622i	$-0.0636467\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	2.33564e7			0.441078
$162$		0			0
$163$	−1.80810e7			−0.327014			−0.163507	−	0.986542i	$-0.552281\pi$
$163$	−1.80810e7			−0.327014			−0.163507	+	0.986542i	$0.552281\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	1.55965e7	−	2.70140e7i	0.259132	−	0.448829i	−0.706878	−	0.707336i	$-0.749897\pi$
$167$	1.55965e7	−	2.70140e7i	0.259132	−	0.448829i	0.966010	+	0.258506i	$0.0832303\pi$
$168$		0			0
$169$	−7.85745e7	−	1.36095e8i	−1.25221	−	2.16890i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	3.08477e7	+	5.34298e7i	0.452962	+	0.784553i	0.998568	−	0.0534883i	$-0.0170340\pi$
$173$	3.08477e7	+	5.34298e7i	0.452962	+	0.784553i	−0.545607	+	0.838041i	$0.683701\pi$
$174$		0			0
$175$	−5.46036e7	+	9.45763e7i	−0.770173	+	1.33398i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	6.17316e7			0.804493			0.402246	−	0.915531i	$-0.368230\pi$
$179$	6.17316e7			0.804493			0.402246	+	0.915531i	$0.368230\pi$
$180$		0			0
$181$	2.61264e7			0.327495			0.163747	−	0.986502i	$-0.447642\pi$
$181$	2.61264e7			0.327495			0.163747	+	0.986502i	$0.447642\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	7.27403e7	−	1.25990e8i	0.844644	−	1.46297i
$186$		0			0
$187$	6.83920e7	+	1.18459e8i	0.764821	+	1.32471i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	6.55659e7	+	1.13564e8i	0.680866	+	1.17929i	0.974717	+	0.223443i	$0.0717296\pi$
$191$	6.55659e7	+	1.13564e8i	0.680866	+	1.17929i	−0.293851	+	0.955851i	$0.594937\pi$
$192$		0			0
$193$	−9.62032e6	+	1.66629e7i	−0.0963250	+	0.166840i	−0.910161	−	0.414255i	$-0.864042\pi$
$193$	−9.62032e6	+	1.66629e7i	−0.0963250	+	0.166840i	0.813836	+	0.581095i	$0.197375\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	1.13660e6			0.0105919			0.00529596	−	0.999986i	$-0.498314\pi$
$197$	1.13660e6			0.0105919			0.00529596	+	0.999986i	$0.498314\pi$
$198$		0			0
$199$	−8.49483e7			−0.764133			−0.382066	−	0.924135i	$-0.624787\pi$
$199$	−8.49483e7			−0.764133			−0.382066	+	0.924135i	$0.624787\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−2.72275e7	+	4.71594e7i	−0.228440	+	0.395669i
$204$		0			0
$205$	1.12430e8	+	1.94734e8i	0.911472	+	1.57871i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−5.46477e7	−	9.46526e7i	−0.414057	−	0.717168i
$210$		0			0
$211$	9.09046e7	−	1.57451e8i	0.666189	−	1.15387i	−0.312773	−	0.949828i	$-0.601258\pi$
$211$	9.09046e7	−	1.57451e8i	0.666189	−	1.15387i	0.978962	−	0.204045i	$-0.0654088\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	4.50018e8			3.08813
$216$		0			0
$217$	−5.01147e7			−0.332932
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−1.80005e8	+	3.11778e8i	−1.12179	+	1.94300i
$222$		0			0
$223$	3.75995e7	+	6.51243e7i	0.227047	+	0.393257i	0.956932	−	0.290314i	$-0.0937596\pi$
$223$	3.75995e7	+	6.51243e7i	0.227047	+	0.393257i	−0.729885	+	0.683570i	$0.760426\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	7.76132e7	+	1.34430e8i	0.440398	+	0.762791i	0.997719	−	0.0675059i	$-0.0215041\pi$
$227$	7.76132e7	+	1.34430e8i	0.440398	+	0.762791i	−0.557321	+	0.830297i	$0.688171\pi$
$228$		0			0
$229$	1.51501e8	−	2.62407e8i	0.833663	−	1.44395i	−0.0614514	−	0.998110i	$-0.519573\pi$
$229$	1.51501e8	−	2.62407e8i	0.833663	−	1.44395i	0.895114	−	0.445837i	$-0.147094\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−2.61547e8			−1.35458			−0.677290	−	0.735716i	$-0.736846\pi$
$233$	−2.61547e8			−1.35458			−0.677290	+	0.735716i	$0.736846\pi$
$234$		0			0
$235$	2.35483e8			1.18365
$236$		0			0
$237$		0			0
$238$		0			0
$239$	6.19423e7	−	1.07287e8i	0.293491	−	0.508342i	−0.681142	−	0.732152i	$-0.738516\pi$
$239$	6.19423e7	−	1.07287e8i	0.293491	−	0.508342i	0.974633	+	0.223810i	$0.0718496\pi$
$240$		0			0
$241$	−9.59273e7	−	1.66151e8i	−0.441451	−	0.764616i	0.556346	−	0.830951i	$-0.312203\pi$
$241$	−9.59273e7	−	1.66151e8i	−0.441451	−	0.764616i	−0.997797	+	0.0663348i	$0.978869\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−7.98748e7	−	1.38347e8i	−0.346999	−	0.601020i
$246$		0			0
$247$	1.43831e8	−	2.49122e8i	0.607313	−	1.05190i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	2.72095e8			1.08608			0.543040	−	0.839707i	$-0.317273\pi$
$251$	2.72095e8			1.08608			0.543040	+	0.839707i	$0.317273\pi$
$252$		0			0
$253$	1.87408e8			0.727557
$254$		0			0
$255$		0			0
$256$		0			0
$257$	2.80495e7	−	4.85831e7i	0.103076	−	0.178533i	−0.809874	−	0.586603i	$-0.800465\pi$
$257$	2.80495e7	−	4.85831e7i	0.103076	−	0.178533i	0.912951	+	0.408070i	$0.133798\pi$
$258$		0			0
$259$	−1.05667e8	−	1.83020e8i	−0.377911	−	0.654561i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−3.11763e7	−	5.39989e7i	−0.105677	−	0.183037i	0.808338	−	0.588719i	$-0.200368\pi$
$263$	−3.11763e7	−	5.39989e7i	−0.105677	−	0.183037i	−0.914014	+	0.405682i	$0.867034\pi$
$264$		0			0
$265$	3.84556e8	−	6.66070e8i	1.26940	−	2.19867i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−2.24846e7			−0.0704291			−0.0352145	−	0.999380i	$-0.511211\pi$
$269$	−2.24846e7			−0.0704291			−0.0352145	+	0.999380i	$0.511211\pi$
$270$		0			0
$271$	−1.85370e8			−0.565778			−0.282889	−	0.959153i	$-0.591293\pi$
$271$	−1.85370e8			−0.565778			−0.282889	+	0.959153i	$0.591293\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−4.38131e8	+	7.58865e8i	−1.27040	+	2.20039i
$276$		0			0
$277$	1.27482e8	+	2.20805e8i	0.360387	+	0.624208i	0.988024	−	0.154297i	$-0.0493113\pi$
$277$	1.27482e8	+	2.20805e8i	0.360387	+	0.624208i	−0.627638	+	0.778506i	$0.715978\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	1.16699e8	+	2.02129e8i	0.313758	+	0.543445i	0.979173	−	0.203029i	$-0.0650785\pi$
$281$	1.16699e8	+	2.02129e8i	0.313758	+	0.543445i	−0.665414	+	0.746474i	$0.731745\pi$
$282$		0			0
$283$	−3.30060e8	+	5.71680e8i	−0.865645	+	1.49934i	0.000760174		1.00000i	$0.499758\pi$
$283$	−3.30060e8	+	5.71680e8i	−0.865645	+	1.49934i	−0.866405	+	0.499342i	$0.833575\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	3.26644e8			0.815621
$288$		0			0
$289$	1.79062e8			0.436376
$290$		0			0
$291$		0			0
$292$		0			0
$293$	2.56096e8	−	4.43571e8i	0.594794	−	1.03021i	−0.398782	−	0.917046i	$-0.630567\pi$
$293$	2.56096e8	−	4.43571e8i	0.594794	−	1.03021i	0.993576	−	0.113167i	$-0.0360996\pi$
$294$		0			0
$295$	3.15138e8	+	5.45836e8i	0.714701	+	1.23790i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	2.46626e8	+	4.27169e8i	0.533568	+	0.924166i
$300$		0			0
$301$	3.26861e8	−	5.66140e8i	0.690845	−	1.19658i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	1.74748e8			0.352666
$306$		0			0
$307$	8.48332e7			0.167333			0.0836665	−	0.996494i	$-0.473337\pi$
$307$	8.48332e7			0.167333			0.0836665	+	0.996494i	$0.473337\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	3.57936e8	−	6.19964e8i	0.674752	−	1.16871i	−0.301789	−	0.953375i	$-0.597584\pi$
$311$	3.57936e8	−	6.19964e8i	0.674752	−	1.16871i	0.976541	−	0.215330i	$-0.0690828\pi$
$312$		0			0
$313$	4.57941e8	+	7.93176e8i	0.844120	+	1.46206i	0.886384	+	0.462952i	$0.153210\pi$
$313$	4.57941e8	+	7.93176e8i	0.844120	+	1.46206i	−0.0422639	+	0.999106i	$0.513457\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−2.41063e7	−	4.17533e7i	−0.0425033	−	0.0736179i	0.843991	−	0.536357i	$-0.180200\pi$
$317$	−2.41063e7	−	4.17533e7i	−0.0425033	−	0.0736179i	−0.886494	+	0.462739i	$0.846867\pi$
$318$		0			0
$319$	−2.18469e8	+	3.78400e8i	−0.376810	+	0.652655i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−4.70952e8			−0.777621
$324$		0			0
$325$	−2.30629e9			−3.72668
$326$		0			0
$327$		0			0
$328$		0			0
$329$	1.71038e8	−	2.96247e8i	0.264794	−	0.458636i
$330$		0			0
$331$	1.61331e8	+	2.79433e8i	0.244522	+	0.423525i	0.961997	−	0.273059i	$-0.0880355\pi$
$331$	1.61331e8	+	2.79433e8i	0.244522	+	0.423525i	−0.717475	+	0.696584i	$0.754702\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	1.29281e8	+	2.23921e8i	0.187878	+	0.325415i
$336$		0			0
$337$	−1.70382e8	+	2.95110e8i	−0.242504	+	0.420029i	−0.961427	−	0.275060i	$-0.911302\pi$
$337$	−1.70382e8	+	2.95110e8i	−0.242504	+	0.420029i	0.718923	+	0.695090i	$0.244635\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−4.02112e8			−0.549170
$342$		0			0
$343$	−8.10336e8			−1.08427
$344$		0			0
$345$		0			0
$346$		0			0
$347$	3.95367e8	−	6.84797e8i	0.507981	−	0.879850i	−0.491976	−	0.870609i	$-0.663725\pi$
$347$	3.95367e8	−	6.84797e8i	0.507981	−	0.879850i	0.999957	−	0.00924074i	$-0.00294146\pi$
$348$		0			0
$349$	−3.11684e8	−	5.39853e8i	−0.392487	−	0.679808i	0.600290	−	0.799783i	$-0.295052\pi$
$349$	−3.11684e8	−	5.39853e8i	−0.392487	−	0.679808i	−0.992777	+	0.119975i	$0.961719\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	1.10681e8	+	1.91706e8i	0.133925	+	0.231965i	0.925186	−	0.379513i	$-0.123908\pi$
$353$	1.10681e8	+	1.91706e8i	0.133925	+	0.231965i	−0.791261	+	0.611478i	$0.790575\pi$
$354$		0			0
$355$	5.52409e8	−	9.56801e8i	0.655332	−	1.13507i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−1.31320e9			−1.49796			−0.748980	−	0.662593i	$-0.769456\pi$
$359$	−1.31320e9			−1.49796			−0.748980	+	0.662593i	$0.769456\pi$
$360$		0			0
$361$	−5.17564e8			−0.579013
$362$		0			0
$363$		0			0
$364$		0			0
$365$	3.52763e7	−	6.11004e7i	0.0379716	−	0.0657687i
$366$		0			0
$367$	1.73902e8	+	3.01208e8i	0.183643	+	0.318079i	0.943118	−	0.332457i	$-0.107878\pi$
$367$	1.73902e8	+	3.01208e8i	0.183643	+	0.318079i	−0.759475	+	0.650536i	$0.774544\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−5.58629e8	−	9.67573e8i	−0.567956	−	0.983728i
$372$		0			0
$373$	3.90213e8	−	6.75869e8i	0.389333	−	0.674345i	−0.603027	−	0.797721i	$-0.706039\pi$
$373$	3.90213e8	−	6.75869e8i	0.389333	−	0.674345i	0.992360	+	0.123376i	$0.0393722\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−1.15001e9			−1.10536
$378$		0			0
$379$	2.89498e8			0.273155			0.136577	−	0.990629i	$-0.456390\pi$
$379$	2.89498e8			0.273155			0.136577	+	0.990629i	$0.456390\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	1.60002e8	−	2.77131e8i	0.145522	−	0.252052i	−0.784045	−	0.620703i	$-0.786847\pi$
$383$	1.60002e8	−	2.77131e8i	0.145522	−	0.252052i	0.929568	+	0.368652i	$0.120180\pi$
$384$		0			0
$385$	9.56163e8	+	1.65612e9i	0.853925	+	1.47904i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−8.11991e8	−	1.40641e9i	−0.699403	−	1.21140i	−0.968674	−	0.248337i	$-0.920116\pi$
$389$	−8.11991e8	−	1.40641e9i	−0.699403	−	1.21140i	0.269270	−	0.963065i	$-0.413218\pi$
$390$		0			0
$391$	4.03770e8	−	6.99350e8i	0.341598	−	0.591665i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−1.59311e9			−1.30064
$396$		0			0
$397$	−5.58913e8			−0.448309			−0.224154	−	0.974554i	$-0.571962\pi$
$397$	−5.58913e8			−0.448309			−0.224154	+	0.974554i	$0.571962\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−6.38618e8	+	1.10612e9i	−0.494579	+	0.856636i	−0.999980	−	0.00624801i	$-0.998011\pi$
$401$	−6.38618e8	+	1.10612e9i	−0.494579	+	0.856636i	0.505401	+	0.862884i	$0.331345\pi$
$402$		0			0
$403$	−5.29172e8	−	9.16554e8i	−0.402744	−	0.697574i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−8.47854e8	−	1.46853e9i	−0.623363	−	1.07970i
$408$		0			0
$409$	−9.63193e8	+	1.66830e9i	−0.696116	+	1.20571i	0.273686	+	0.961819i	$0.411757\pi$
$409$	−9.63193e8	+	1.66830e9i	−0.696116	+	1.20571i	−0.969803	+	0.243890i	$0.921576\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	9.15577e8			0.639543
$414$		0			0
$415$	−3.56828e9			−2.45070
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−8.51570e8	+	1.47496e9i	−0.565551	+	0.979562i	0.431448	+	0.902138i	$0.358003\pi$
$419$	−8.51570e8	+	1.47496e9i	−0.565551	+	0.979562i	−0.996998	+	0.0774244i	$0.975330\pi$
$420$		0			0
$421$	−6.66198e8	−	1.15389e9i	−0.435127	−	0.753663i	0.562179	−	0.827016i	$-0.309963\pi$
$421$	−6.66198e8	−	1.15389e9i	−0.435127	−	0.753663i	−0.997306	+	0.0733532i	$0.976630\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	1.88790e9	+	3.26994e9i	1.19294	+	2.06623i
$426$		0			0
$427$	1.26925e8	−	2.19840e8i	0.0788950	−	0.136650i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	2.64975e9			1.59417			0.797085	−	0.603867i	$-0.206374\pi$
$431$	2.64975e9			1.59417			0.797085	+	0.603867i	$0.206374\pi$
$432$		0			0
$433$	1.98266e9			1.17366			0.586829	−	0.809711i	$-0.300376\pi$
$433$	1.98266e9			1.17366			0.586829	+	0.809711i	$0.300376\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−3.22627e8	+	5.58806e8i	−0.184933	+	0.320314i
$438$		0			0
$439$	1.69568e9	+	2.93700e9i	0.956572	+	1.65683i	0.730729	+	0.682667i	$0.239180\pi$
$439$	1.69568e9	+	2.93700e9i	0.956572	+	1.65683i	0.225842	+	0.974164i	$0.427487\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−1.70292e9	−	2.94954e9i	−0.930638	−	1.61191i	−0.782234	−	0.622985i	$-0.785920\pi$
$443$	−1.70292e9	−	2.94954e9i	−0.930638	−	1.61191i	−0.148403	−	0.988927i	$-0.547413\pi$
$444$		0			0
$445$	−2.85521e9	+	4.94537e9i	−1.53595	+	2.66035i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−1.12986e8			−0.0589066			−0.0294533	−	0.999566i	$-0.509377\pi$
$449$	−1.12986e8			−0.0589066			−0.0294533	+	0.999566i	$0.509377\pi$
$450$		0			0
$451$	2.62094e9			1.34536
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−2.51659e9	+	4.35886e9i	−1.25248	+	2.16937i
$456$		0			0
$457$	1.57960e8	+	2.73595e8i	0.0774177	+	0.134091i	0.902135	−	0.431454i	$-0.141999\pi$
$457$	1.57960e8	+	2.73595e8i	0.0774177	+	0.134091i	−0.824717	+	0.565545i	$0.808666\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	2.41606e8	+	4.18475e8i	0.114856	+	0.198937i	0.917722	−	0.397222i	$-0.130026\pi$
$461$	2.41606e8	+	4.18475e8i	0.114856	+	0.198937i	−0.802866	+	0.596160i	$0.796693\pi$
$462$		0			0
$463$	2.03136e9	−	3.51841e9i	0.951158	−	1.64745i	0.208233	−	0.978079i	$-0.433229\pi$
$463$	2.03136e9	−	3.51841e9i	0.951158	−	1.64745i	0.742925	−	0.669375i	$-0.233438\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	1.91056e9			0.868065			0.434032	−	0.900897i	$-0.357090\pi$
$467$	1.91056e9			0.868065			0.434032	+	0.900897i	$0.357090\pi$
$468$		0			0
$469$	3.75602e8			0.168121
$470$		0			0
$471$		0			0
$472$		0			0
$473$	2.62269e9	−	4.54262e9i	1.13955	−	1.97375i
$474$		0			0
$475$	−1.50850e9	−	2.61280e9i	−0.645830	−	1.11861i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−1.83119e9	−	3.17172e9i	−0.761307	−	1.31862i	−0.942177	−	0.335116i	$-0.891225\pi$
$479$	−1.83119e9	−	3.17172e9i	−0.761307	−	1.31862i	0.180870	−	0.983507i	$-0.442109\pi$
$480$		0			0
$481$	2.23152e9	−	3.86511e9i	0.914310	−	1.58363i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−1.82435e9			−0.726127
$486$		0			0
$487$	−4.60080e9			−1.80502			−0.902510	−	0.430668i	$-0.858278\pi$
$487$	−4.60080e9			−1.80502			−0.902510	+	0.430668i	$0.858278\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	9.65018e8	−	1.67146e9i	0.367917	−	0.637252i	−0.621322	−	0.783555i	$-0.713404\pi$
$491$	9.65018e8	−	1.67146e9i	0.367917	−	0.637252i	0.989240	+	0.146303i	$0.0467376\pi$
$492$		0			0
$493$	9.41381e8	+	1.63052e9i	0.353835	+	0.612861i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−8.02462e8	−	1.38990e9i	−0.293209	−	0.507853i
$498$		0			0
$499$	9.88938e7	−	1.71289e8i	0.0356301	−	0.0617132i	−0.847660	−	0.530539i	$-0.821990\pi$
$499$	9.88938e7	−	1.71289e8i	0.0356301	−	0.0617132i	0.883291	+	0.468826i	$0.155323\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−1.25495e8			−0.0439683			−0.0219842	−	0.999758i	$-0.506998\pi$
$503$	−1.25495e8			−0.0439683			−0.0219842	+	0.999758i	$0.506998\pi$
$504$		0			0
$505$	−7.69090e8			−0.265740
$506$		0			0
$507$		0			0
$508$		0			0
$509$	7.99791e8	−	1.38528e9i	0.268822	−	0.465613i	−0.699736	−	0.714401i	$-0.746699\pi$
$509$	7.99791e8	−	1.38528e9i	0.268822	−	0.465613i	0.968558	+	0.248788i	$0.0800324\pi$
$510$		0			0
$511$	−5.12445e7	−	8.87581e7i	−0.0169892	−	0.0294262i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	2.58755e9	+	4.48176e9i	0.834762	+	1.44585i
$516$		0			0
$517$	1.37238e9	−	2.37704e9i	0.436776	−	0.756518i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	4.95631e9			1.53542			0.767708	−	0.640800i	$-0.221397\pi$
$521$	4.95631e9			1.53542			0.767708	+	0.640800i	$0.221397\pi$
$522$		0			0
$523$	−5.04830e9			−1.54308			−0.771542	−	0.636179i	$-0.780514\pi$
$523$	−5.04830e9			−1.54308			−0.771542	+	0.636179i	$0.780514\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−8.66348e8	+	1.50056e9i	−0.257843	+	0.446597i
$528$		0			0
$529$	1.14921e9	+	1.99048e9i	0.337523	+	0.584607i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	3.44912e9	+	5.97404e9i	0.986649	+	1.70893i
$534$		0			0
$535$	9.35440e8	−	1.62023e9i	0.264106	−	0.457444i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−1.86203e9			−0.512183
$540$		0			0
$541$	3.40361e9			0.924165			0.462082	−	0.886837i	$-0.347102\pi$
$541$	3.40361e9			0.924165			0.462082	+	0.886837i	$0.347102\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	1.41708e8	−	2.45445e8i	0.0374978	−	0.0649481i
$546$		0			0
$547$	1.21549e9	+	2.10529e9i	0.317538	+	0.549993i	0.979974	−	0.199126i	$-0.0638104\pi$
$547$	1.21549e9	+	2.10529e9i	0.317538	+	0.549993i	−0.662435	+	0.749119i	$0.730477\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−7.52197e8	−	1.30284e9i	−0.191558	−	0.331789i
$552$		0			0
$553$	−1.15713e9	+	2.00420e9i	−0.290966	+	0.503968i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−2.29520e9			−0.562767			−0.281383	−	0.959595i	$-0.590793\pi$
$557$	−2.29520e9			−0.562767			−0.281383	+	0.959595i	$0.590793\pi$
$558$		0			0
$559$	1.38056e10			3.34283
$560$		0			0
$561$		0			0
$562$		0			0
$563$	1.88607e9	−	3.26677e9i	0.445429	−	0.771506i	−0.552653	−	0.833412i	$-0.686384\pi$
$563$	1.88607e9	−	3.26677e9i	0.445429	−	0.771506i	0.998082	+	0.0619054i	$0.0197177\pi$
$564$		0			0
$565$	−1.04776e9	−	1.81478e9i	−0.244395	−	0.423305i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	2.43318e9	+	4.21439e9i	0.553709	+	0.959051i	0.998003	+	0.0631705i	$0.0201212\pi$
$569$	2.43318e9	+	4.21439e9i	0.553709	+	0.959051i	−0.444294	+	0.895881i	$0.646545\pi$
$570$		0			0
$571$	−2.56161e9	+	4.43683e9i	−0.575819	+	0.997348i	0.420133	+	0.907463i	$0.361983\pi$
$571$	−2.56161e9	+	4.43683e9i	−0.575819	+	0.997348i	−0.995952	+	0.0898855i	$0.971350\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	5.17324e9			1.13481
$576$		0			0
$577$	−1.78410e9			−0.386638			−0.193319	−	0.981136i	$-0.561925\pi$
$577$	−1.78410e9			−0.386638			−0.193319	+	0.981136i	$0.561925\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−2.59174e9	+	4.48903e9i	−0.548246	+	0.949591i
$582$		0			0
$583$	−4.48235e9	−	7.76366e9i	−0.936841	−	1.62266i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−8.28199e8	−	1.43448e9i	−0.169006	−	0.292726i	0.769065	−	0.639171i	$-0.220722\pi$
$587$	−8.28199e8	−	1.43448e9i	−0.169006	−	0.292726i	−0.938071	+	0.346444i	$0.887389\pi$
$588$		0			0
$589$	6.92243e8	−	1.19900e9i	0.139590	−	0.241777i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	2.65457e9			0.522761			0.261380	−	0.965236i	$-0.415822\pi$
$593$	2.65457e9			0.522761			0.261380	+	0.965236i	$0.415822\pi$
$594$		0			0
$595$	8.24019e9			1.60372
$596$		0			0
$597$		0			0
$598$		0			0
$599$	6.54199e8	−	1.13311e9i	0.124370	−	0.215415i	−0.797116	−	0.603826i	$-0.793642\pi$
$599$	6.54199e8	−	1.13311e9i	0.124370	−	0.215415i	0.921487	+	0.388410i	$0.126976\pi$
$600$		0			0
$601$	3.27381e9	+	5.67040e9i	0.615166	+	1.06550i	0.990355	+	0.138551i	$0.0442444\pi$
$601$	3.27381e9	+	5.67040e9i	0.615166	+	1.06550i	−0.375189	+	0.926948i	$0.622422\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	2.96230e9	+	5.13086e9i	0.543858	+	0.941990i
$606$		0			0
$607$	4.39100e9	−	7.60544e9i	0.796899	−	1.38027i	−0.124727	−	0.992191i	$-0.539805\pi$
$607$	4.39100e9	−	7.60544e9i	0.796899	−	1.38027i	0.921626	−	0.388079i	$-0.126861\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	7.22413e9			1.28127
$612$		0			0
$613$	7.20663e9			1.26363			0.631816	−	0.775118i	$-0.282310\pi$
$613$	7.20663e9			1.26363			0.631816	+	0.775118i	$0.282310\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	3.19935e9	−	5.54144e9i	0.548357	−	0.949783i	−0.450030	−	0.893013i	$-0.648587\pi$
$617$	3.19935e9	−	5.54144e9i	0.548357	−	0.949783i	0.998387	−	0.0567692i	$-0.0180799\pi$
$618$		0			0
$619$	−3.89757e9	−	6.75078e9i	−0.660505	−	1.14403i	−0.980483	−	0.196604i	$-0.937009\pi$
$619$	−3.89757e9	−	6.75078e9i	−0.660505	−	1.14403i	0.319978	−	0.947425i	$-0.396325\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	4.14764e9	+	7.18393e9i	0.687216	+	1.19029i
$624$		0			0
$625$	−2.96721e9	+	5.13936e9i	−0.486148	+	0.842033i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−7.30678e9			−1.17071
$630$		0			0
$631$	−5.27269e9			−0.835467			−0.417733	−	0.908570i	$-0.637175\pi$
$631$	−5.27269e9			−0.835467			−0.417733	+	0.908570i	$0.637175\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−5.02974e9	+	8.71177e9i	−0.779538	+	1.35020i
$636$		0			0
$637$	−2.45039e9	−	4.24421e9i	−0.375619	−	0.650592i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−1.61321e9	−	2.79417e9i	−0.241929	−	0.419034i	0.719334	−	0.694664i	$-0.244447\pi$
$641$	−1.61321e9	−	2.79417e9i	−0.241929	−	0.419034i	−0.961264	+	0.275630i	$0.911114\pi$
$642$		0			0
$643$	−3.02440e8	+	5.23842e8i	−0.0448643	+	0.0777072i	−0.887586	−	0.460643i	$-0.847619\pi$
$643$	−3.02440e8	+	5.23842e8i	−0.0448643	+	0.0777072i	0.842721	+	0.538350i	$0.180952\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−1.16182e10			−1.68646			−0.843228	−	0.537556i	$-0.819348\pi$
$647$	−1.16182e10			−1.68646			−0.843228	+	0.537556i	$0.819348\pi$
$648$		0			0
$649$	7.34645e9			1.05492
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−1.44688e9	+	2.50607e9i	−0.203346	+	0.352206i	−0.949605	−	0.313451i	$-0.898515\pi$
$653$	−1.44688e9	+	2.50607e9i	−0.203346	+	0.352206i	0.746258	+	0.665656i	$0.231848\pi$
$654$		0			0
$655$	4.11831e9	+	7.13313e9i	0.572631	+	0.991826i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−2.40766e9	−	4.17020e9i	−0.327715	−	0.567620i	0.654343	−	0.756198i	$-0.272945\pi$
$659$	−2.40766e9	−	4.17020e9i	−0.327715	−	0.567620i	−0.982058	+	0.188578i	$0.939612\pi$
$660$		0			0
$661$	−1.56280e8	+	2.70685e8i	−0.0210474	+	0.0364551i	−0.876357	−	0.481662i	$-0.840033\pi$
$661$	−1.56280e8	+	2.70685e8i	−0.0210474	+	0.0364551i	0.855310	+	0.518117i	$0.173367\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−6.58421e9			−0.868216
$666$		0			0
$667$	2.57958e9			0.336595
$668$		0			0
$669$		0			0
$670$		0			0
$671$	1.01843e9	−	1.76397e9i	0.130137	−	0.225404i
$672$		0			0
$673$	−6.70912e9	−	1.16205e10i	−0.848424	−	1.46951i	−0.882614	−	0.470098i	$-0.844219\pi$
$673$	−6.70912e9	−	1.16205e10i	−0.848424	−	1.46951i	0.0341908	−	0.999415i	$-0.489115\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	2.60444e8	+	4.51102e8i	0.0322592	+	0.0558746i	0.881704	−	0.471802i	$-0.156396\pi$
$677$	2.60444e8	+	4.51102e8i	0.0322592	+	0.0558746i	−0.849445	+	0.527677i	$0.823063\pi$
$678$		0			0
$679$	−1.32508e9	+	2.29511e9i	−0.162442	+	0.281358i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	3.27118e9			0.392855			0.196428	−	0.980518i	$-0.437066\pi$
$683$	3.27118e9			0.392855			0.196428	+	0.980518i	$0.437066\pi$
$684$		0			0
$685$	−1.89315e10			−2.25045
$686$		0			0
$687$		0			0
$688$		0			0
$689$	1.17974e10	−	2.04337e10i	1.37410	−	2.38001i
$690$		0			0
$691$	−2.66214e9	−	4.61097e9i	−0.306943	−	0.531641i	0.670749	−	0.741685i	$-0.265973\pi$
$691$	−2.66214e9	−	4.61097e9i	−0.306943	−	0.531641i	−0.977692	+	0.210043i	$0.932640\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−5.79470e9	−	1.00367e10i	−0.654763	−	1.13408i
$696$		0			0
$697$	5.64681e9	−	9.78055e9i	0.631667	−	1.09408i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−4.17788e9			−0.458081			−0.229041	−	0.973417i	$-0.573559\pi$
$701$	−4.17788e9			−0.458081			−0.229041	+	0.973417i	$0.573559\pi$
$702$		0			0
$703$	5.83838e9			0.633795
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−5.58613e8	+	9.67545e8i	−0.0594487	+	0.102968i
$708$		0			0
$709$	−3.08696e9	−	5.34678e9i	−0.325289	−	0.563417i	0.656282	−	0.754516i	$-0.272128\pi$
$709$	−3.08696e9	−	5.34678e9i	−0.325289	−	0.563417i	−0.981571	+	0.191099i	$0.938795\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	1.18699e9	+	2.05592e9i	0.122640	+	0.212419i
$714$		0			0
$715$	−2.01927e10	+	3.49748e10i	−2.06597	+	3.57836i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	7.57603e8			0.0760134			0.0380067	−	0.999277i	$-0.487899\pi$
$719$	7.57603e8			0.0760134			0.0380067	+	0.999277i	$0.487899\pi$
$720$		0			0
$721$	7.51764e9			0.746979
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−6.03065e9	+	1.04454e10i	−0.587734	+	1.01799i
$726$		0			0
$727$	6.17240e9	+	1.06909e10i	0.595776	+	1.03192i	0.993437	+	0.114382i	$0.0364889\pi$
$727$	6.17240e9	+	1.06909e10i	0.595776	+	1.03192i	−0.397660	+	0.917533i	$0.630178\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−1.13011e10	−	1.95741e10i	−1.07007	−	1.85341i
$732$		0			0
$733$	−5.34368e9	+	9.25552e9i	−0.501160	+	0.868035i	0.498839	+	0.866695i	$0.333760\pi$
$733$	−5.34368e9	+	9.25552e9i	−0.501160	+	0.868035i	−0.999999	+	0.00133994i	$0.999573\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	3.01377e9			0.277315
$738$		0			0
$739$	9.62358e9			0.877165			0.438582	−	0.898691i	$-0.355481\pi$
$739$	9.62358e9			0.877165			0.438582	+	0.898691i	$0.355481\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−6.69015e9	+	1.15877e10i	−0.598377	+	1.03642i	0.394684	+	0.918817i	$0.370854\pi$
$743$	−6.69015e9	+	1.15877e10i	−0.598377	+	1.03642i	−0.993061	+	0.117602i	$0.962479\pi$
$744$		0			0
$745$	−1.12226e9	−	1.94381e9i	−0.0994366	−	0.172229i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−1.35887e9	−	2.35364e9i	−0.118166	−	0.204670i
$750$		0			0
$751$	−7.38544e9	+	1.27920e10i	−0.636263	+	1.10204i	0.349983	+	0.936756i	$0.386187\pi$
$751$	−7.38544e9	+	1.27920e10i	−0.636263	+	1.10204i	−0.986246	+	0.165284i	$0.947146\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	3.68226e10			3.11386
$756$		0			0
$757$	−1.90520e10			−1.59627			−0.798133	−	0.602482i	$-0.794179\pi$
$757$	−1.90520e10			−1.59627			−0.798133	+	0.602482i	$0.794179\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−2.99516e9	+	5.18776e9i	−0.246362	+	0.426711i	−0.962514	−	0.271233i	$-0.912568\pi$
$761$	−2.99516e9	+	5.18776e9i	−0.246362	+	0.426711i	0.716152	+	0.697944i	$0.245902\pi$
$762$		0			0
$763$	−2.05853e8	−	3.56548e8i	−0.0167773	−	0.0290591i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	9.66779e9	+	1.67451e10i	0.773648	+	1.34000i
$768$		0			0
$769$	−6.01785e9	+	1.04232e10i	−0.477199	+	0.826533i	−0.999659	−	0.0261313i	$-0.991681\pi$
$769$	−6.01785e9	+	1.04232e10i	−0.477199	+	0.826533i	0.522460	+	0.852664i	$0.325015\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−6.70596e9			−0.522195			−0.261098	−	0.965312i	$-0.584084\pi$
$773$	−6.70596e9			−0.522195			−0.261098	+	0.965312i	$0.584084\pi$
$774$		0			0
$775$	−1.10999e10			−0.856574
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−4.51200e9	+	7.81502e9i	−0.341970	+	0.592310i
$780$		0			0
$781$	−6.43883e9	−	1.11524e10i	−0.483647	−	0.837701i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−1.55176e10	−	2.68772e10i	−1.14493	−	1.98308i
$786$		0			0
$787$	−1.12942e9	+	1.95621e9i	−0.0825930	+	0.143055i	−0.904363	−	0.426764i	$-0.859653\pi$
$787$	−1.12942e9	+	1.95621e9i	−0.0825930	+	0.143055i	0.821770	+	0.569819i	$0.192987\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−3.04408e9			−0.218695
$792$		0			0
$793$	5.36092e9			0.381754
$794$		0			0
$795$		0			0
$796$		0			0
$797$	1.12692e10	−	1.95189e10i	0.788480	−	1.36569i	−0.138418	−	0.990374i	$-0.544202\pi$
$797$	1.12692e10	−	1.95189e10i	0.788480	−	1.36569i	0.926898	−	0.375313i	$-0.122465\pi$
$798$		0			0
$799$	−5.91358e9	−	1.02426e10i	−0.410145	−	0.710391i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−4.11178e8	−	7.12181e8i	−0.0280237	−	0.0485385i
$804$		0			0
$805$	5.64495e9	−	9.77734e9i	0.381395	−	0.660595i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−7.67964e9			−0.509942			−0.254971	−	0.966949i	$-0.582066\pi$
$809$	−7.67964e9			−0.509942			−0.254971	+	0.966949i	$0.582066\pi$
$810$		0			0
$811$	−1.11394e10			−0.733315			−0.366658	−	0.930356i	$-0.619498\pi$
$811$	−1.11394e10			−0.733315			−0.366658	+	0.930356i	$0.619498\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−4.36995e9	+	7.56898e9i	−0.282765	+	0.489763i
$816$		0			0
$817$	9.03000e9	+	1.56404e10i	0.579309	+	1.00339i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−7.09049e9	−	1.22811e10i	−0.447172	−	0.774525i	0.551028	−	0.834487i	$-0.314236\pi$
$821$	−7.09049e9	−	1.22811e10i	−0.447172	−	0.774525i	−0.998201	+	0.0599612i	$0.980902\pi$
$822$		0			0
$823$	−6.32825e9	+	1.09609e10i	−0.395717	+	0.685402i	−0.993192	−	0.116485i	$-0.962837\pi$
$823$	−6.32825e9	+	1.09609e10i	−0.395717	+	0.685402i	0.597476	+	0.801887i	$0.296170\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−6.39608e9			−0.393228			−0.196614	−	0.980481i	$-0.562995\pi$
$827$	−6.39608e9			−0.393228			−0.196614	+	0.980481i	$0.562995\pi$
$828$		0			0
$829$	2.54339e10			1.55050			0.775250	−	0.631655i	$-0.217624\pi$
$829$	2.54339e10			1.55050			0.775250	+	0.631655i	$0.217624\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−4.01172e9	+	6.94851e9i	−0.240477	+	0.416518i
$834$		0			0
$835$	−7.53897e9	−	1.30579e10i	−0.448136	−	0.776194i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−8.86165e9	−	1.53488e10i	−0.518022	−	0.897240i	−0.999781	−	0.0209361i	$-0.993335\pi$
$839$	−8.86165e9	−	1.53488e10i	−0.518022	−	0.897240i	0.481759	−	0.876304i	$-0.339998\pi$
$840$		0			0
$841$	5.61782e9	−	9.73035e9i	0.325673	−	0.564083i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−7.59619e10			−4.33109
$846$		0			0
$847$	8.60643e9			0.486666
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−5.00552e9	+	8.66982e9i	−0.278417	+	0.482233i
$852$		0			0
$853$	−8.87525e9	−	1.53724e10i	−0.489620	−	0.848046i	0.510309	−	0.859991i	$-0.329531\pi$
$853$	−8.87525e9	−	1.53724e10i	−0.489620	−	0.848046i	−0.999929	+	0.0119447i	$0.996198\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	1.09509e10	+	1.89675e10i	0.594314	+	1.02938i	0.993643	+	0.112574i	$0.0359096\pi$
$857$	1.09509e10	+	1.89675e10i	0.594314	+	1.02938i	−0.399330	+	0.916807i	$0.630757\pi$
$858$		0			0
$859$	1.17639e10	−	2.03756e10i	0.633249	−	1.09682i	−0.353635	−	0.935384i	$-0.615054\pi$
$859$	1.17639e10	−	2.03756e10i	0.633249	−	1.09682i	0.986883	−	0.161435i	$-0.0516123\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	2.67719e9			0.141789			0.0708944	−	0.997484i	$-0.477415\pi$
$863$	2.67719e9			0.141789			0.0708944	+	0.997484i	$0.477415\pi$
$864$		0			0
$865$	2.98220e10			1.56668
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−9.28460e9	+	1.60814e10i	−0.479948	+	0.831294i
$870$		0			0
$871$	3.96607e9	+	6.86943e9i	0.203374	+	0.352255i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	1.31356e10	+	2.27515e10i	0.662860	+	1.14811i
$876$		0			0
$877$	8.41146e8	−	1.45691e9i	0.0421088	−	0.0729345i	−0.844203	−	0.536024i	$-0.819926\pi$
$877$	8.41146e8	−	1.45691e9i	0.0421088	−	0.0729345i	0.886312	+	0.463089i	$0.153259\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	1.23651e10			0.609229			0.304615	−	0.952476i	$-0.401472\pi$
$881$	1.23651e10			0.609229			0.304615	+	0.952476i	$0.401472\pi$
$882$		0			0
$883$	−1.37918e10			−0.674151			−0.337075	−	0.941478i	$-0.609438\pi$
$883$	−1.37918e10			−0.674151			−0.337075	+	0.941478i	$0.609438\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	1.76027e10	−	3.04888e10i	0.846929	−	1.46692i	−0.0370059	−	0.999315i	$-0.511782\pi$
$887$	1.76027e10	−	3.04888e10i	0.846929	−	1.46692i	0.883935	−	0.467609i	$-0.154885\pi$
$888$		0			0
$889$	7.30650e9	+	1.26552e10i	0.348781	+	0.604107i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	4.72517e9	+	8.18423e9i	0.222043	+	0.384590i
$894$		0			0
$895$	1.49198e10	−	2.58418e10i	0.695635	−	1.20487i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−5.53487e9			−0.254067
$900$		0			0
$901$	−3.86288e10			−1.75944
$902$		0			0
$903$		0			0
$904$		0			0
$905$	6.31442e9	−	1.09369e10i	0.283181	−	0.490483i
$906$		0			0
$907$	8.89827e9	+	1.54123e10i	0.395986	+	0.685868i	0.993227	−	0.116194i	$-0.0370695\pi$
$907$	8.89827e9	+	1.54123e10i	0.395986	+	0.685868i	−0.597240	+	0.802062i	$0.703736\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	5.55822e9	+	9.62712e9i	0.243569	+	0.421874i	0.961728	−	0.274005i	$-0.0883485\pi$
$911$	5.55822e9	+	9.62712e9i	0.243569	+	0.421874i	−0.718159	+	0.695879i	$0.755015\pi$
$912$		0			0
$913$	−2.07958e10	+	3.60193e10i	−0.904330	+	1.56635i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	1.19650e10			0.512413
$918$		0			0
$919$	−2.22162e10			−0.944202			−0.472101	−	0.881544i	$-0.656504\pi$
$919$	−2.22162e10			−0.944202			−0.472101	+	0.881544i	$0.656504\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	1.69468e10	−	2.93527e10i	0.709383	−	1.22869i
$924$		0			0
$925$	−2.34042e10	−	4.05373e10i	−0.972296	−	1.68407i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	1.56911e10	+	2.71779e10i	0.642096	+	1.11214i	0.984964	+	0.172759i	$0.0552682\pi$
$929$	1.56911e10	+	2.71779e10i	0.642096	+	1.11214i	−0.342868	+	0.939383i	$0.611399\pi$
$930$		0			0
$931$	3.20551e9	−	5.55211e9i	0.130189	−	0.225494i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	6.61180e10			2.64533
$936$		0			0
$937$	4.63144e10			1.83919			0.919596	−	0.392865i	$-0.128516\pi$
$937$	4.63144e10			1.83919			0.919596	+	0.392865i	$0.128516\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	1.26733e10	−	2.19508e10i	0.495821	−	0.858788i	−0.504167	−	0.863606i	$-0.668200\pi$
$941$	1.26733e10	−	2.19508e10i	0.495821	−	0.858788i	0.999988	+	0.00481836i	$0.00153374\pi$
$942$		0			0
$943$	−7.73670e9	−	1.34004e10i	−0.300445	−	0.520386i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	4.82354e9	+	8.35461e9i	0.184561	+	0.319670i	0.943429	−	0.331576i	$-0.107580\pi$
$947$	4.82354e9	+	8.35461e9i	0.184561	+	0.319670i	−0.758867	+	0.651245i	$0.774247\pi$
$948$		0			0
$949$	1.08221e9	−	1.87443e9i	0.0411034	−	0.0711932i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	4.16627e10			1.55927			0.779637	−	0.626232i	$-0.215404\pi$
$953$	4.16627e10			1.55927			0.779637	+	0.626232i	$0.215404\pi$
$954$		0			0
$955$	6.33858e10			2.35494
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−1.37505e10	+	2.38166e10i	−0.503447	+	0.871996i
$960$		0			0
$961$	1.12095e10	+	1.94153e10i	0.407430	+	0.705689i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	4.65022e9	+	8.05442e9i	0.166582	+	0.288529i
$966$		0			0
$967$	−1.86979e10	+	3.23858e10i	−0.664968	+	1.15176i	0.314325	+	0.949315i	$0.398222\pi$
$967$	−1.86979e10	+	3.23858e10i	−0.664968	+	1.15176i	−0.979294	+	0.202444i	$0.935112\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−5.30808e10			−1.86067			−0.930337	−	0.366705i	$-0.880486\pi$
$971$	−5.30808e10			−1.86067			−0.930337	+	0.366705i	$0.880486\pi$
$972$		0			0
$973$	−1.68354e10			−0.585908
$974$		0			0
$975$		0			0
$976$		0			0
$977$	1.94353e10	−	3.36629e10i	0.666745	−	1.15484i	−0.312064	−	0.950061i	$-0.601020\pi$
$977$	1.94353e10	−	3.36629e10i	0.666745	−	1.15484i	0.978809	−	0.204775i	$-0.0656464\pi$
$978$		0			0
$979$	3.32801e10	+	5.76428e10i	1.13356	+	1.96338i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	1.04190e10	+	1.80463e10i	0.349857	+	0.605970i	0.986224	−	0.165417i	$-0.0528969\pi$
$983$	1.04190e10	+	1.80463e10i	0.349857	+	0.605970i	−0.636367	+	0.771386i	$0.719564\pi$
$984$		0			0
$985$	2.74701e8	−	4.75796e8i	0.00915870	−	0.0158633i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−3.09674e10			−1.01793
$990$		0			0
$991$	1.62339e9			0.0529865			0.0264932	−	0.999649i	$-0.491566\pi$
$991$	1.62339e9			0.0529865			0.0264932	+	0.999649i	$0.491566\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−2.05309e10	+	3.55606e10i	−0.660736	+	1.14443i
$996$		0			0
$997$	−1.03970e10	−	1.80081e10i	−0.332257	−	0.575487i	0.650697	−	0.759338i	$-0.274477\pi$
$997$	−1.03970e10	−	1.80081e10i	−0.332257	−	0.575487i	−0.982954	+	0.183851i	$0.941144\pi$
$998$		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.145.10		20
3.2	odd	2		72.8.i.a.49.7	yes	20
4.3	odd	2		432.8.i.e.145.10		20
9.2	odd	6		72.8.i.a.25.7	✓	20
9.7	even	3	inner	216.8.i.a.73.10		20
12.11	even	2		144.8.i.e.49.4		20
36.7	odd	6		432.8.i.e.289.10		20
36.11	even	6		144.8.i.e.97.4		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.8.i.a.25.7	✓	20	9.2	odd	6
72.8.i.a.49.7	yes	20	3.2	odd	2
144.8.i.e.49.4		20	12.11	even	2
144.8.i.e.97.4		20	36.11	even	6
216.8.i.a.73.10		20	9.7	even	3	inner
216.8.i.a.145.10		20	1.1	even	1	trivial
432.8.i.e.145.10		20	4.3	odd	2
432.8.i.e.289.10		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)

\(f(q)\)	\(=\)	\(q+(241.687 - 418.615i) q^{5} +(-351.089 - 608.105i) q^{7} +O(q^{10})\)	\(q+(241.687 - 418.615i) q^{5} +(-351.089 - 608.105i) q^{7} +(-2817.09 - 4879.34i) q^{11} +(7414.47 - 12842.2i) q^{13} -24277.6 q^{17} +19398.7 q^{19} +(-16631.4 + 28806.4i) q^{23} +(-77763.1 - 134690. i) q^{25} +(-38775.7 - 67161.5i) q^{29} +(35685.1 - 61808.4i) q^{31} -339416. q^{35} +300968. q^{37} +(-232593. + 402864. i) q^{41} +(465496. + 806263. i) q^{43} +(243582. + 421897. i) q^{47} +(165244. - 286211. i) q^{49} +1.59113e6 q^{53} -2.72342e6 q^{55} +(-651954. + 1.12922e6i) q^{59} +(180759. + 313083. i) q^{61} +(-3.58397e6 - 6.20762e6i) q^{65} +(-267454. + 463244. i) q^{67} +2.28563e6 q^{71} +145959. q^{73} +(-1.97810e6 + 3.42617e6i) q^{77} +(-1.64791e6 - 2.85426e6i) q^{79} +(-3.69100e6 - 6.39301e6i) q^{83} +(-5.86758e6 + 1.01630e7i) q^{85} -1.18136e7 q^{89} -1.04126e7 q^{91} +(4.68841e6 - 8.12057e6i) q^{95} +(-1.88710e6 - 3.26855e6i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 125 q^{5} + 1245 q^{7}+O(q^{10}) \) 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

Varieties

Fields

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Database

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