LMFDB - Embedded newform 216.8.i.a.73.9

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			73.9
Root			$0.500000 - 238.457i$ of defining polynomial
Character	$\chi$	$=$	216.73
Dual form			216.8.i.a.145.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(212.760 + 368.511i) q^{5} +(612.894 - 1061.56i) q^{7} +O(q^{10})$	$q+(212.760 + 368.511i) q^{5} +(612.894 - 1061.56i) q^{7} +(3650.27 - 6322.45i) q^{11} +(-2143.53 - 3712.71i) q^{13} +1986.56 q^{17} -27453.3 q^{19} +(-55504.6 - 96136.7i) q^{23} +(-51471.1 + 89150.6i) q^{25} +(-71682.9 + 124158. i) q^{29} +(-115887. - 200722. i) q^{31} +521598. q^{35} -294885. q^{37} +(161521. + 279763. i) q^{41} +(108274. - 187537. i) q^{43} +(-233918. + 405158. i) q^{47} +(-339508. - 588044. i) q^{49} +572586. q^{53} +3.10652e6 q^{55} +(-981852. - 1.70062e6i) q^{59} +(-296339. + 513274. i) q^{61} +(912117. - 1.57983e6i) q^{65} +(-821791. - 1.42338e6i) q^{67} -3.22227e6 q^{71} -2.12920e6 q^{73} +(-4.47445e6 - 7.74998e6i) q^{77} +(2.75968e6 - 4.77990e6i) q^{79} +(-4.52800e6 + 7.84273e6i) q^{83} +(422660. + 732069. i) q^{85} -6.85191e6 q^{89} -5.25504e6 q^{91} +(-5.84097e6 - 1.01169e7i) q^{95} +(5.55762e6 - 9.62609e6i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/216\mathbb{Z}\right)^\times$.

$n$	$55$	$109$	$137$
$\chi(n)$	$1$	$1$	$e\left(\frac{2}{3}\right)$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	212.760	+	368.511i	0.761193	+	1.31843i	0.942236	+	0.334950i	$0.108720\pi$
$5$	212.760	+	368.511i	0.761193	+	1.31843i	−0.181042	+	0.983475i	$0.557947\pi$
$6$		0			0
$7$	612.894	−	1061.56i	0.675371	−	1.16978i	−0.300989	−	0.953628i	$-0.597317\pi$
$7$	612.894	−	1061.56i	0.675371	−	1.16978i	0.976360	−	0.216150i	$-0.0693499\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	3650.27	−	6322.45i	0.826894	−	1.43222i	−0.0735684	−	0.997290i	$-0.523439\pi$
$11$	3650.27	−	6322.45i	0.826894	−	1.43222i	0.900463	−	0.434933i	$-0.143228\pi$
$12$		0			0
$13$	−2143.53	−	3712.71i	−0.270601	−	0.468694i	0.698415	−	0.715693i	$-0.253889\pi$
$13$	−2143.53	−	3712.71i	−0.270601	−	0.468694i	−0.969016	+	0.246999i	$0.920556\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	1986.56			0.0980687			0.0490343	−	0.998797i	$-0.484386\pi$
$17$	1986.56			0.0980687			0.0490343	+	0.998797i	$0.484386\pi$
$18$		0			0
$19$	−27453.3			−0.918242			−0.459121	−	0.888374i	$-0.651836\pi$
$19$	−27453.3			−0.918242			−0.459121	+	0.888374i	$0.651836\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−55504.6	−	96136.7i	−0.951221	−	1.64756i	−0.742789	−	0.669525i	$-0.766498\pi$
$23$	−55504.6	−	96136.7i	−0.951221	−	1.64756i	−0.208431	−	0.978037i	$-0.566836\pi$
$24$		0			0
$25$	−51471.1	+	89150.6i	−0.658831	+	1.14113i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−71682.9	+	124158.i	−0.545787	+	0.945330i	0.452770	+	0.891627i	$0.350436\pi$
$29$	−71682.9	+	124158.i	−0.545787	+	0.945330i	−0.998557	+	0.0537028i	$0.982898\pi$
$30$		0			0
$31$	−115887.	−	200722.i	−0.698664	−	1.21012i	−0.968930	−	0.247336i	$-0.920445\pi$
$31$	−115887.	−	200722.i	−0.698664	−	1.21012i	0.270266	−	0.962786i	$-0.412888\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	521598.			2.05635
$36$		0			0
$37$	−294885.			−0.957076			−0.478538	−	0.878067i	$-0.658833\pi$
$37$	−294885.			−0.957076			−0.478538	+	0.878067i	$0.658833\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	161521.	+	279763.i	0.366004	+	0.633938i	0.988937	−	0.148338i	$-0.0473923\pi$
$41$	161521.	+	279763.i	0.366004	+	0.633938i	−0.622933	+	0.782275i	$0.714059\pi$
$42$		0			0
$43$	108274.	−	187537.i	0.207676	−	0.359705i	−0.743306	−	0.668951i	$-0.766743\pi$
$43$	108274.	−	187537.i	0.207676	−	0.359705i	0.950982	+	0.309246i	$0.100077\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−233918.	+	405158.i	−0.328640	+	0.569222i	−0.982242	−	0.187617i	$-0.939924\pi$
$47$	−233918.	+	405158.i	−0.328640	+	0.569222i	0.653602	+	0.756838i	$0.273257\pi$
$48$		0			0
$49$	−339508.	−	588044.i	−0.412252	−	0.714042i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	572586.			0.528293			0.264147	−	0.964483i	$-0.414910\pi$
$53$	572586.			0.528293			0.264147	+	0.964483i	$0.414910\pi$
$54$		0			0
$55$	3.10652e6			2.51771
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−981852.	−	1.70062e6i	−0.622392	−	1.07801i	−0.989039	−	0.147654i	$-0.952828\pi$
$59$	−981852.	−	1.70062e6i	−0.622392	−	1.07801i	0.366647	−	0.930360i	$-0.380506\pi$
$60$		0			0
$61$	−296339.	+	513274.i	−0.167161	+	0.289531i	−0.937421	−	0.348199i	$-0.886793\pi$
$61$	−296339.	+	513274.i	−0.167161	+	0.289531i	0.770260	+	0.637730i	$0.220127\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	912117.	−	1.57983e6i	0.411959	−	0.713534i
$66$		0			0
$67$	−821791.	−	1.42338e6i	−0.333810	−	0.578176i	0.649445	−	0.760408i	$-0.275001\pi$
$67$	−821791.	−	1.42338e6i	−0.333810	−	0.578176i	−0.983256	+	0.182232i	$0.941668\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−3.22227e6			−1.06846			−0.534230	−	0.845339i	$-0.679398\pi$
$71$	−3.22227e6			−1.06846			−0.534230	+	0.845339i	$0.679398\pi$
$72$		0			0
$73$	−2.12920e6			−0.640598			−0.320299	−	0.947317i	$-0.603783\pi$
$73$	−2.12920e6			−0.640598			−0.320299	+	0.947317i	$0.603783\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−4.47445e6	−	7.74998e6i	−1.11692	−	1.93456i
$78$		0			0
$79$	2.75968e6	−	4.77990e6i	0.629743	−	1.09075i	−0.357860	−	0.933775i	$-0.616494\pi$
$79$	2.75968e6	−	4.77990e6i	0.629743	−	1.09075i	0.987603	−	0.156971i	$-0.0501730\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−4.52800e6	+	7.84273e6i	−0.869228	+	1.50555i	−0.00644030	+	0.999979i	$0.502050\pi$
$83$	−4.52800e6	+	7.84273e6i	−0.869228	+	1.50555i	−0.862787	+	0.505567i	$0.831283\pi$
$84$		0			0
$85$	422660.	+	732069.i	0.0746492	+	0.129296i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−6.85191e6			−1.03026			−0.515130	−	0.857112i	$-0.672256\pi$
$89$	−6.85191e6			−1.03026			−0.515130	+	0.857112i	$0.672256\pi$
$90$		0			0
$91$	−5.25504e6			−0.731023
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−5.84097e6	−	1.01169e7i	−0.698960	−	1.21063i
$96$		0			0
$97$	5.55762e6	−	9.62609e6i	0.618284	−	1.07090i	−0.371515	−	0.928427i	$-0.621161\pi$
$97$	5.55762e6	−	9.62609e6i	0.618284	−	1.07090i	0.989799	−	0.142473i	$-0.0455053\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	6.80611e6	−	1.17885e7i	0.657316	−	1.13850i	−0.323992	−	0.946060i	$-0.605025\pi$
$101$	6.80611e6	−	1.17885e7i	0.657316	−	1.13850i	0.981308	−	0.192444i	$-0.0616414\pi$
$102$		0			0
$103$	4.13063e6	+	7.15445e6i	0.372465	+	0.645128i	0.989944	−	0.141459i	$-0.0451793\pi$
$103$	4.13063e6	+	7.15445e6i	0.372465	+	0.645128i	−0.617479	+	0.786587i	$0.711846\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	9.46943e6			0.747275			0.373638	−	0.927575i	$-0.378110\pi$
$107$	9.46943e6			0.747275			0.373638	+	0.927575i	$0.378110\pi$
$108$		0			0
$109$	2.44794e7			1.81054			0.905269	−	0.424839i	$-0.139669\pi$
$109$	2.44794e7			1.81054			0.905269	+	0.424839i	$0.139669\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−2.71070e6	−	4.69507e6i	−0.176729	−	0.306103i	0.764029	−	0.645181i	$-0.223218\pi$
$113$	−2.71070e6	−	4.69507e6i	−0.176729	−	0.306103i	−0.940758	+	0.339078i	$0.889885\pi$
$114$		0			0
$115$	2.36183e7	−	4.09081e7i	1.44813	−	2.50823i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	1.21755e6	−	2.10886e6i	0.0662328	−	0.114719i
$120$		0			0
$121$	−1.69053e7	−	2.92808e7i	−0.867509	−	1.50257i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−1.05603e7			−0.483603
$126$		0			0
$127$	−1.72866e7			−0.748855			−0.374427	−	0.927256i	$-0.622161\pi$
$127$	−1.72866e7			−0.748855			−0.374427	+	0.927256i	$0.622161\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	5.45173e6	+	9.44267e6i	0.211877	+	0.366983i	0.952302	−	0.305157i	$-0.0987089\pi$
$131$	5.45173e6	+	9.44267e6i	0.211877	+	0.366983i	−0.740425	+	0.672139i	$0.765376\pi$
$132$		0			0
$133$	−1.68260e7	+	2.91435e7i	−0.620154	+	1.07414i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	1.56402e6	−	2.70897e6i	0.0519662	−	0.0900082i	−0.838872	−	0.544329i	$-0.816785\pi$
$137$	1.56402e6	−	2.70897e6i	0.0519662	−	0.0900082i	0.890838	+	0.454320i	$0.150118\pi$
$138$		0			0
$139$	−2.08304e6	−	3.60794e6i	−0.0657880	−	0.113948i	0.831255	−	0.555891i	$-0.187623\pi$
$139$	−2.08304e6	−	3.60794e6i	−0.0657880	−	0.113948i	−0.897043	+	0.441943i	$0.854289\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−3.12979e7			−0.895033
$144$		0			0
$145$	−6.10050e7			−1.66180
$146$		0			0
$147$		0			0
$148$		0			0
$149$	2.06272e6	+	3.57273e6i	0.0510843	+	0.0884806i	0.890437	−	0.455107i	$-0.150399\pi$
$149$	2.06272e6	+	3.57273e6i	0.0510843	+	0.0884806i	−0.839353	+	0.543587i	$0.817066\pi$
$150$		0			0
$151$	7.12520e6	−	1.23412e7i	0.168414	−	0.291701i	−0.769448	−	0.638709i	$-0.779469\pi$
$151$	7.12520e6	−	1.23412e7i	0.168414	−	0.291701i	0.937862	+	0.347008i	$0.112802\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	4.93122e7	−	8.54112e7i	1.06364	−	1.84227i
$156$		0			0
$157$	−2.98979e7	−	5.17847e7i	−0.616584	−	1.06796i	−0.990104	−	0.140333i	$-0.955183\pi$
$157$	−2.98979e7	−	5.17847e7i	−0.616584	−	1.06796i	0.373520	−	0.927622i	$-0.378151\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−1.36074e8			−2.56971
$162$		0			0
$163$	4.85200e7			0.877535			0.438768	−	0.898601i	$-0.355415\pi$
$163$	4.85200e7			0.877535			0.438768	+	0.898601i	$0.355415\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	1.25911e7	+	2.18084e7i	0.209197	+	0.362340i	0.951462	−	0.307767i	$-0.0995816\pi$
$167$	1.25911e7	+	2.18084e7i	0.209197	+	0.362340i	−0.742265	+	0.670107i	$0.766248\pi$
$168$		0			0
$169$	2.21848e7	−	3.84252e7i	0.353551	−	0.612368i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	1.27704e7	−	2.21190e7i	0.187518	−	0.324790i	−0.756904	−	0.653526i	$-0.773289\pi$
$173$	1.27704e7	−	2.21190e7i	0.187518	−	0.324790i	0.944422	+	0.328735i	$0.106622\pi$
$174$		0			0
$175$	6.30928e7	+	1.09280e8i	0.889911	+	1.54137i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	8.55453e7			1.11483			0.557417	−	0.830232i	$-0.311792\pi$
$179$	8.55453e7			1.11483			0.557417	+	0.830232i	$0.311792\pi$
$180$		0			0
$181$	8.68707e7			1.08893			0.544463	−	0.838785i	$-0.316733\pi$
$181$	8.68707e7			1.08893			0.544463	+	0.838785i	$0.316733\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−6.27397e7	−	1.08668e8i	−0.728520	−	1.26183i
$186$		0			0
$187$	7.25147e6	−	1.25599e7i	0.0810925	−	0.140456i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−6.70081e7	+	1.16061e8i	−0.695842	+	1.20523i	0.274054	+	0.961714i	$0.411635\pi$
$191$	−6.70081e7	+	1.16061e8i	−0.695842	+	1.20523i	−0.969896	+	0.243519i	$0.921698\pi$
$192$		0			0
$193$	3.33292e7	+	5.77278e7i	0.333714	+	0.578009i	0.983237	−	0.182333i	$-0.0583648\pi$
$193$	3.33292e7	+	5.77278e7i	0.333714	+	0.578009i	−0.649523	+	0.760342i	$0.725031\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	1.15508e8			1.07641			0.538206	−	0.842813i	$-0.319102\pi$
$197$	1.15508e8			1.07641			0.538206	+	0.842813i	$0.319102\pi$
$198$		0			0
$199$	4.58693e7			0.412607			0.206303	−	0.978488i	$-0.433857\pi$
$199$	4.58693e7			0.412607			0.206303	+	0.978488i	$0.433857\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	8.78681e7	+	1.52192e8i	0.737217	+	1.27690i
$204$		0			0
$205$	−6.87305e7	+	1.19045e8i	−0.557200	+	0.965098i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−1.00212e8	+	1.73572e8i	−0.759289	+	1.31513i
$210$		0			0
$211$	5.00297e6	+	8.66539e6i	0.0366639	+	0.0635038i	0.883775	−	0.467912i	$-0.154994\pi$
$211$	5.00297e6	+	8.66539e6i	0.0366639	+	0.0635038i	−0.847111	+	0.531416i	$0.821660\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	9.21458e7			0.632326
$216$		0			0
$217$	−2.84106e8			−1.88743
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−4.25826e6	−	7.37552e6i	−0.0265374	−	0.0459642i
$222$		0			0
$223$	7.84805e7	−	1.35932e8i	0.473909	−	0.820834i	−0.525645	−	0.850704i	$-0.676176\pi$
$223$	7.84805e7	−	1.35932e8i	0.473909	−	0.820834i	0.999554	+	0.0298702i	$0.00950941\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−8.52409e7	+	1.47642e8i	−0.483679	+	0.837757i	−0.999824	−	0.0187442i	$-0.994033\pi$
$227$	−8.52409e7	+	1.47642e8i	−0.483679	+	0.837757i	0.516145	+	0.856501i	$0.327367\pi$
$228$		0			0
$229$	1.70403e8	+	2.95146e8i	0.937675	+	1.62410i	0.769794	+	0.638293i	$0.220359\pi$
$229$	1.70403e8	+	2.95146e8i	0.937675	+	1.62410i	0.167881	+	0.985807i	$0.446308\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	2.41202e8			1.24921			0.624605	−	0.780941i	$-0.285260\pi$
$233$	2.41202e8			1.24921			0.624605	+	0.780941i	$0.285260\pi$
$234$		0			0
$235$	−1.99073e8			−1.00064
$236$		0			0
$237$		0			0
$238$		0			0
$239$	5.23990e7	+	9.07577e7i	0.248274	+	0.430022i	0.963047	−	0.269334i	$-0.0868035\pi$
$239$	5.23990e7	+	9.07577e7i	0.248274	+	0.430022i	−0.714773	+	0.699356i	$0.753470\pi$
$240$		0			0
$241$	6.33571e6	−	1.09738e7i	0.0291565	−	0.0505006i	−0.851079	−	0.525038i	$-0.824051\pi$
$241$	6.33571e6	−	1.09738e7i	0.0291565	−	0.0505006i	0.880235	+	0.474537i	$0.157385\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	1.44467e8	−	2.50225e8i	0.627608	−	1.08705i
$246$		0			0
$247$	5.88471e7	+	1.01926e8i	0.248477	+	0.430374i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	1.47818e8			0.590023			0.295012	−	0.955494i	$-0.404676\pi$
$251$	1.47818e8			0.590023			0.295012	+	0.955494i	$0.404676\pi$
$252$		0			0
$253$	−8.10426e8			−3.14624
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−1.32308e8	−	2.29165e8i	−0.486207	−	0.842135i	0.513667	−	0.857989i	$-0.328287\pi$
$257$	−1.32308e8	−	2.29165e8i	−0.486207	−	0.842135i	−0.999874	+	0.0158541i	$0.994953\pi$
$258$		0			0
$259$	−1.80733e8	+	3.13039e8i	−0.646382	+	1.11957i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−4.02150e7	+	6.96544e7i	−0.136315	+	0.236104i	−0.926099	−	0.377281i	$-0.876859\pi$
$263$	−4.02150e7	+	6.96544e7i	−0.136315	+	0.236104i	0.789784	+	0.613385i	$0.210193\pi$
$264$		0			0
$265$	1.21823e8	+	2.11004e8i	0.402133	+	0.696516i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−1.30520e8			−0.408832			−0.204416	−	0.978884i	$-0.565530\pi$
$269$	−1.30520e8			−0.408832			−0.204416	+	0.978884i	$0.565530\pi$
$270$		0			0
$271$	4.18106e8			1.27613			0.638063	−	0.769984i	$-0.279736\pi$
$271$	4.18106e8			1.27613			0.638063	+	0.769984i	$0.279736\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	3.75767e8	+	6.50847e8i	1.08957	+	1.88719i
$276$		0			0
$277$	6.80469e7	−	1.17861e8i	0.192366	−	0.333188i	−0.753668	−	0.657256i	$-0.771717\pi$
$277$	6.80469e7	−	1.17861e8i	0.192366	−	0.333188i	0.946034	+	0.324068i	$0.105050\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	6.33582e7	−	1.09740e8i	0.170346	−	0.295047i	−0.768195	−	0.640216i	$-0.778845\pi$
$281$	6.33582e7	−	1.09740e8i	0.170346	−	0.295047i	0.938541	+	0.345169i	$0.112178\pi$
$282$		0			0
$283$	1.01360e8	+	1.75561e8i	0.265837	+	0.460442i	0.967782	−	0.251788i	$-0.0810185\pi$
$283$	1.01360e8	+	1.75561e8i	0.265837	+	0.460442i	−0.701946	+	0.712230i	$0.747685\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	3.95982e8			0.988754
$288$		0			0
$289$	−4.06392e8			−0.990383
$290$		0			0
$291$		0			0
$292$		0			0
$293$	2.24787e8	+	3.89342e8i	0.522076	+	0.904262i	0.999670	+	0.0256818i	$0.00817567\pi$
$293$	2.24787e8	+	3.89342e8i	0.522076	+	0.904262i	−0.477594	+	0.878581i	$0.658491\pi$
$294$		0			0
$295$	4.17798e8	−	7.23647e8i	0.947521	−	1.64115i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−2.37952e8	+	4.12145e8i	−0.514802	+	0.891663i
$300$		0			0
$301$	−1.32721e8	−	2.29880e8i	−0.280517	−	0.485869i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−2.52196e8			−0.508967
$306$		0			0
$307$	1.29994e7			0.0256413			0.0128206	−	0.999918i	$-0.495919\pi$
$307$	1.29994e7			0.0256413			0.0128206	+	0.999918i	$0.495919\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−2.77591e8	−	4.80802e8i	−0.523293	−	0.906369i	−0.999633	−	0.0271081i	$-0.991370\pi$
$311$	−2.77591e8	−	4.80802e8i	−0.523293	−	0.906369i	0.476340	−	0.879261i	$-0.341963\pi$
$312$		0			0
$313$	−1.55894e8	+	2.70016e8i	−0.287358	+	0.497719i	−0.973178	−	0.230052i	$-0.926110\pi$
$313$	−1.55894e8	+	2.70016e8i	−0.287358	+	0.497719i	0.685820	+	0.727771i	$0.259444\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	7.65506e7	−	1.32590e8i	0.134971	−	0.233777i	−0.790615	−	0.612313i	$-0.790239\pi$
$317$	7.65506e7	−	1.32590e8i	0.134971	−	0.233777i	0.925586	+	0.378536i	$0.123572\pi$
$318$		0			0
$319$	5.23323e8	+	9.06423e8i	0.902616	+	1.56338i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−5.45376e7			−0.0900508
$324$		0			0
$325$	4.41321e8			0.713120
$326$		0			0
$327$		0			0
$328$		0			0
$329$	2.86734e8	+	4.96638e8i	0.443909	+	0.768872i
$330$		0			0
$331$	3.26644e8	−	5.65764e8i	0.495081	−	0.857506i	−0.504903	−	0.863176i	$-0.668472\pi$
$331$	3.26644e8	−	5.65764e8i	0.495081	−	0.857506i	0.999984	+	0.00567025i	$0.00180491\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	3.49688e8	−	6.05678e8i	0.508188	−	0.880207i
$336$		0			0
$337$	3.47844e8	+	6.02484e8i	0.495086	+	0.857513i	0.999984	−	0.00566548i	$-0.00180339\pi$
$337$	3.47844e8	+	6.02484e8i	0.495086	+	0.857513i	−0.504898	+	0.863179i	$0.668470\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−1.69207e9			−2.31089
$342$		0			0
$343$	1.77160e8			0.237049
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−7.41613e8	−	1.28451e9i	−0.952849	−	1.65038i	−0.739215	−	0.673470i	$-0.764803\pi$
$347$	−7.41613e8	−	1.28451e9i	−0.952849	−	1.65038i	−0.213635	−	0.976914i	$-0.568530\pi$
$348$		0			0
$349$	2.09696e8	−	3.63204e8i	0.264059	−	0.457364i	−0.703258	−	0.710935i	$-0.748272\pi$
$349$	2.09696e8	−	3.63204e8i	0.264059	−	0.457364i	0.967317	+	0.253571i	$0.0816052\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−5.41754e7	+	9.38346e7i	−0.0655527	+	0.113541i	−0.896939	−	0.442154i	$-0.854214\pi$
$353$	−5.41754e7	+	9.38346e7i	−0.0655527	+	0.113541i	0.831386	+	0.555695i	$0.187548\pi$
$354$		0			0
$355$	−6.85571e8	−	1.18744e9i	−0.813304	−	1.40868i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−1.07778e9			−1.22942			−0.614708	−	0.788755i	$-0.710726\pi$
$359$	−1.07778e9			−1.22942			−0.614708	+	0.788755i	$0.710726\pi$
$360$		0			0
$361$	−1.40187e8			−0.156832
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−4.53008e8	−	7.84632e8i	−0.487619	−	0.844581i
$366$		0			0
$367$	−1.24759e8	+	2.16089e8i	−0.131747	+	0.228192i	−0.924350	−	0.381546i	$-0.875392\pi$
$367$	−1.24759e8	+	2.16089e8i	−0.131747	+	0.228192i	0.792603	+	0.609738i	$0.208725\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	3.50935e8	−	6.07837e8i	0.356794	−	0.617986i
$372$		0			0
$373$	−2.11397e8	−	3.66150e8i	−0.210920	−	0.365324i	0.741083	−	0.671414i	$-0.234313\pi$
$373$	−2.11397e8	−	3.66150e8i	−0.210920	−	0.365324i	−0.952003	+	0.306090i	$0.900979\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	6.14619e8			0.590761
$378$		0			0
$379$	−1.40362e9			−1.32438			−0.662191	−	0.749335i	$-0.730373\pi$
$379$	−1.40362e9			−1.32438			−0.662191	+	0.749335i	$0.730373\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−2.15171e8	−	3.72687e8i	−0.195699	−	0.338960i	0.751431	−	0.659812i	$-0.229364\pi$
$383$	−2.15171e8	−	3.72687e8i	−0.195699	−	0.338960i	−0.947129	+	0.320852i	$0.896031\pi$
$384$		0			0
$385$	1.90397e9	−	3.29777e9i	1.70039	−	2.94516i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	8.42828e8	−	1.45982e9i	0.725964	−	1.25741i	−0.232612	−	0.972570i	$-0.574727\pi$
$389$	8.42828e8	−	1.45982e9i	0.725964	−	1.25741i	0.958576	−	0.284837i	$-0.0919395\pi$
$390$		0			0
$391$	−1.10263e8	−	1.90981e8i	−0.0932850	−	0.161574i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	2.34859e9			1.91742
$396$		0			0
$397$	−1.10230e7			−0.00884168			−0.00442084	−	0.999990i	$-0.501407\pi$
$397$	−1.10230e7			−0.00884168			−0.00442084	+	0.999990i	$0.501407\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−1.73022e7	−	2.99683e7i	−0.0133997	−	0.0232090i	0.859248	−	0.511560i	$-0.170932\pi$
$401$	−1.73022e7	−	2.99683e7i	−0.0133997	−	0.0232090i	−0.872648	+	0.488351i	$0.837599\pi$
$402$		0			0
$403$	−4.96815e8	+	8.60509e8i	−0.378118	+	0.654919i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−1.07641e9	+	1.86439e9i	−0.791401	+	1.37075i
$408$		0			0
$409$	1.11555e9	+	1.93220e9i	0.806230	+	1.39643i	0.915457	+	0.402415i	$0.131829\pi$
$409$	1.11555e9	+	1.93220e9i	0.806230	+	1.39643i	−0.109227	+	0.994017i	$0.534838\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−2.40709e9			−1.68138
$414$		0			0
$415$	−3.85351e9			−2.64660
$416$		0			0
$417$		0			0
$418$		0			0
$419$	7.98396e7	+	1.38286e8i	0.0530236	+	0.0918395i	0.891319	−	0.453377i	$-0.149781\pi$
$419$	7.98396e7	+	1.38286e8i	0.0530236	+	0.0918395i	−0.838295	+	0.545216i	$0.816447\pi$
$420$		0			0
$421$	1.14223e9	−	1.97840e9i	0.746048	−	1.29219i	−0.203655	−	0.979043i	$-0.565282\pi$
$421$	1.14223e9	−	1.97840e9i	0.746048	−	1.29219i	0.949703	−	0.313151i	$-0.101385\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−1.02251e8	+	1.77103e8i	−0.0646107	+	0.111909i
$426$		0			0
$427$	3.63249e8	+	6.29166e8i	0.225791	+	0.391082i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−2.25009e9			−1.35372			−0.676860	−	0.736112i	$-0.736660\pi$
$431$	−2.25009e9			−1.35372			−0.676860	+	0.736112i	$0.736660\pi$
$432$		0			0
$433$	1.29723e9			0.767908			0.383954	−	0.923352i	$-0.374562\pi$
$433$	1.29723e9			0.767908			0.383954	+	0.923352i	$0.374562\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	1.52378e9	+	2.63927e9i	0.873451	+	1.51286i
$438$		0			0
$439$	1.46335e9	−	2.53460e9i	0.825512	−	1.42983i	−0.0760161	−	0.997107i	$-0.524220\pi$
$439$	1.46335e9	−	2.53460e9i	0.825512	−	1.42983i	0.901528	−	0.432721i	$-0.142447\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	5.72835e8	−	9.92179e8i	0.313052	−	0.542222i	−0.665970	−	0.745979i	$-0.731982\pi$
$443$	5.72835e8	−	9.92179e8i	0.313052	−	0.542222i	0.979021	+	0.203757i	$0.0653153\pi$
$444$		0			0
$445$	−1.45781e9	−	2.52501e9i	−0.784227	−	1.35832i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	3.70743e9			1.93291			0.966453	−	0.256844i	$-0.0826828\pi$
$449$	3.70743e9			1.93291			0.966453	+	0.256844i	$0.0826828\pi$
$450$		0			0
$451$	2.35838e9			1.21059
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−1.11806e9	−	1.93654e9i	−0.556450	−	0.963800i
$456$		0			0
$457$	−1.35620e9	+	2.34901e9i	−0.664688	+	1.15127i	0.314682	+	0.949197i	$0.398102\pi$
$457$	−1.35620e9	+	2.34901e9i	−0.664688	+	1.15127i	−0.979370	+	0.202076i	$0.935231\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−3.08709e8	+	5.34700e8i	−0.146756	+	0.254189i	−0.930027	−	0.367492i	$-0.880217\pi$
$461$	−3.08709e8	+	5.34700e8i	−0.146756	+	0.254189i	0.783271	+	0.621681i	$0.213550\pi$
$462$		0			0
$463$	1.72622e9	+	2.98989e9i	0.808280	+	1.39998i	0.914055	+	0.405591i	$0.132934\pi$
$463$	1.72622e9	+	2.98989e9i	0.808280	+	1.39998i	−0.105775	+	0.994390i	$0.533732\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	1.57059e8			0.0713600			0.0356800	−	0.999363i	$-0.488640\pi$
$467$	1.57059e8			0.0713600			0.0356800	+	0.999363i	$0.488640\pi$
$468$		0			0
$469$	−2.01468e9			−0.901783
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−7.90460e8	−	1.36912e9i	−0.343452	−	0.594876i
$474$		0			0
$475$	1.41305e9	−	2.44748e9i	0.604966	−	1.04783i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	7.39297e6	−	1.28050e7i	0.00307358	−	0.00532360i	−0.864485	−	0.502659i	$-0.832355\pi$
$479$	7.39297e6	−	1.28050e7i	0.00307358	−	0.00532360i	0.867558	+	0.497336i	$0.165688\pi$
$480$		0			0
$481$	6.32096e8	+	1.09482e9i	0.258985	+	0.448576i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	4.72976e9			1.88254
$486$		0			0
$487$	−2.57789e9			−1.01138			−0.505689	−	0.862716i	$-0.668762\pi$
$487$	−2.57789e9			−1.01138			−0.505689	+	0.862716i	$0.668762\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−2.51718e9	−	4.35988e9i	−0.959685	−	1.66222i	−0.723263	−	0.690572i	$-0.757359\pi$
$491$	−2.51718e9	−	4.35988e9i	−0.959685	−	1.66222i	−0.236421	−	0.971651i	$-0.575975\pi$
$492$		0			0
$493$	−1.42402e8	+	2.46648e8i	−0.0535246	+	0.0927073i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−1.97491e9	+	3.42065e9i	−0.721607	+	1.24986i
$498$		0			0
$499$	−1.74673e9	−	3.02543e9i	−0.629324	−	1.09002i	−0.987688	−	0.156439i	$-0.949999\pi$
$499$	−1.74673e9	−	3.02543e9i	−0.629324	−	1.09002i	0.358364	−	0.933582i	$-0.383335\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	6.91317e8			0.242209			0.121104	−	0.992640i	$-0.461356\pi$
$503$	6.91317e8			0.242209			0.121104	+	0.992640i	$0.461356\pi$
$504$		0			0
$505$	5.79227e9			2.00138
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−1.02607e9	−	1.77721e9i	−0.344878	−	0.597346i	0.640454	−	0.767997i	$-0.278746\pi$
$509$	−1.02607e9	−	1.77721e9i	−0.344878	−	0.597346i	−0.985332	+	0.170651i	$0.945413\pi$
$510$		0			0
$511$	−1.30497e9	+	2.26028e9i	−0.432641	+	0.749357i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−1.75766e9	+	3.04436e9i	−0.567036	+	0.982135i
$516$		0			0
$517$	1.70772e9	+	2.95787e9i	0.543502	+	0.941373i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−5.56810e9			−1.72494			−0.862472	−	0.506105i	$-0.831085\pi$
$521$	−5.56810e9			−1.72494			−0.862472	+	0.506105i	$0.831085\pi$
$522$		0			0
$523$	−7.46790e8			−0.228267			−0.114133	−	0.993465i	$-0.536409\pi$
$523$	−7.46790e8			−0.228267			−0.114133	+	0.993465i	$0.536409\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−2.30216e8	−	3.98746e8i	−0.0685171	−	0.118675i
$528$		0			0
$529$	−4.45910e9	+	7.72339e9i	−1.30964	+	2.26837i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	6.92452e8	−	1.19936e9i	0.198082	−	0.343088i
$534$		0			0
$535$	2.01472e9	+	3.48959e9i	0.568821	+	0.985227i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−4.95717e9			−1.36356
$540$		0			0
$541$	−2.12164e9			−0.576078			−0.288039	−	0.957619i	$-0.593003\pi$
$541$	−2.12164e9			−0.576078			−0.288039	+	0.957619i	$0.593003\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	5.20823e9	+	9.02092e9i	1.37817	+	2.38706i
$546$		0			0
$547$	−3.71015e9	+	6.42617e9i	−0.969251	+	1.67879i	−0.271518	+	0.962433i	$0.587526\pi$
$547$	−3.71015e9	+	6.42617e9i	−0.969251	+	1.67879i	−0.697733	+	0.716358i	$0.745808\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	1.96793e9	−	3.40856e9i	0.501164	−	0.868042i
$552$		0			0
$553$	−3.38278e9	−	5.85915e9i	−0.850620	−	1.47332i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	1.25261e9			0.307130			0.153565	−	0.988139i	$-0.450924\pi$
$557$	1.25261e9			0.307130			0.153565	+	0.988139i	$0.450924\pi$
$558$		0			0
$559$	−9.28359e8			−0.224789
$560$		0			0
$561$		0			0
$562$		0			0
$563$	3.16638e9	+	5.48434e9i	0.747798	+	1.29522i	0.948876	+	0.315649i	$0.102222\pi$
$563$	3.16638e9	+	5.48434e9i	0.747798	+	1.29522i	−0.201078	+	0.979575i	$0.564445\pi$
$564$		0			0
$565$	1.15346e9	−	1.99785e9i	0.269049	−	0.466007i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	1.33997e9	−	2.32089e9i	0.304931	−	0.528155i	−0.672315	−	0.740265i	$-0.734700\pi$
$569$	1.33997e9	−	2.32089e9i	0.304931	−	0.528155i	0.977246	+	0.212110i	$0.0680334\pi$
$570$		0			0
$571$	−5.08207e8	−	8.80240e8i	−0.114239	−	0.197868i	0.803236	−	0.595661i	$-0.203110\pi$
$571$	−5.08207e8	−	8.80240e8i	−0.114239	−	0.197868i	−0.917475	+	0.397793i	$0.869776\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	1.14275e10			2.50677
$576$		0			0
$577$	6.18175e9			1.33966			0.669832	−	0.742513i	$-0.266366\pi$
$577$	6.18175e9			1.33966			0.669832	+	0.742513i	$0.266366\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	5.55038e9	+	9.61354e9i	1.17410	+	2.03361i
$582$		0			0
$583$	2.09009e9	−	3.62014e9i	0.436843	−	0.756634i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−2.67158e9	+	4.62731e9i	−0.545173	+	0.944268i	0.453423	+	0.891296i	$0.350203\pi$
$587$	−2.67158e9	+	4.62731e9i	−0.545173	+	0.944268i	−0.998596	+	0.0529722i	$0.983131\pi$
$588$		0			0
$589$	3.18148e9	+	5.51048e9i	0.641542	+	1.11118i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−2.09814e9			−0.413183			−0.206592	−	0.978427i	$-0.566237\pi$
$593$	−2.09814e9			−0.413183			−0.206592	+	0.978427i	$0.566237\pi$
$594$		0			0
$595$	1.03618e9			0.201664
$596$		0			0
$597$		0			0
$598$		0			0
$599$	1.00995e9	+	1.74929e9i	0.192002	+	0.332558i	0.945914	−	0.324418i	$-0.105168\pi$
$599$	1.00995e9	+	1.74929e9i	0.192002	+	0.332558i	−0.753911	+	0.656976i	$0.771835\pi$
$600$		0			0
$601$	2.05248e9	−	3.55500e9i	0.385673	−	0.668005i	−0.606190	−	0.795320i	$-0.707303\pi$
$601$	2.05248e9	−	3.55500e9i	0.385673	−	0.668005i	0.991862	+	0.127316i	$0.0406361\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	7.19354e9	−	1.24596e10i	1.32068	−	2.28749i
$606$		0			0
$607$	−3.41410e9	−	5.91340e9i	−0.619607	−	1.07319i	−0.989557	−	0.144139i	$-0.953959\pi$
$607$	−3.41410e9	−	5.91340e9i	−0.619607	−	1.07319i	0.369951	−	0.929051i	$-0.379375\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	2.00564e9			0.355721
$612$		0			0
$613$	1.07489e9			0.188475			0.0942374	−	0.995550i	$-0.469959\pi$
$613$	1.07489e9			0.188475			0.0942374	+	0.995550i	$0.469959\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	2.29801e9	+	3.98027e9i	0.393871	+	0.682205i	0.992956	−	0.118481i	$-0.0378023\pi$
$617$	2.29801e9	+	3.98027e9i	0.393871	+	0.682205i	−0.599085	+	0.800685i	$0.704469\pi$
$618$		0			0
$619$	−5.27032e9	+	9.12846e9i	−0.893140	+	1.54696i	−0.0570494	+	0.998371i	$0.518169\pi$
$619$	−5.27032e9	+	9.12846e9i	−0.893140	+	1.54696i	−0.836090	+	0.548592i	$0.815164\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−4.19950e9	+	7.27375e9i	−0.695808	+	1.20517i
$624$		0			0
$625$	1.77438e9	+	3.07332e9i	0.290715	+	0.503533i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−5.85807e8			−0.0938592
$630$		0			0
$631$	−7.66298e9			−1.21421			−0.607107	−	0.794620i	$-0.707670\pi$
$631$	−7.66298e9			−1.21421			−0.607107	+	0.794620i	$0.707670\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−3.67791e9	−	6.37032e9i	−0.570023	−	0.987309i
$636$		0			0
$637$	−1.45549e9	+	2.52099e9i	−0.223112	+	0.386441i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	3.37525e8	−	5.84610e8i	0.0506177	−	0.0876724i	−0.839606	−	0.543195i	$-0.817214\pi$
$641$	3.37525e8	−	5.84610e8i	0.0506177	−	0.0876724i	0.890224	+	0.455523i	$0.150548\pi$
$642$		0			0
$643$	4.03272e9	+	6.98488e9i	0.598218	+	1.03614i	0.993084	+	0.117406i	$0.0374578\pi$
$643$	4.03272e9	+	6.98488e9i	0.598218	+	1.03614i	−0.394866	+	0.918739i	$0.629209\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	2.84442e8			0.0412884			0.0206442	−	0.999787i	$-0.493428\pi$
$647$	2.84442e8			0.0412884			0.0206442	+	0.999787i	$0.493428\pi$
$648$		0			0
$649$	−1.43361e10			−2.05861
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−3.62505e9	−	6.27876e9i	−0.509468	−	0.882425i	−0.999940	−	0.0109679i	$-0.996509\pi$
$653$	−3.62505e9	−	6.27876e9i	−0.509468	−	0.882425i	0.490471	−	0.871457i	$-0.336825\pi$
$654$		0			0
$655$	−2.31982e9	+	4.01805e9i	−0.322559	+	0.558689i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	4.41480e8	−	7.64667e8i	0.0600914	−	0.104081i	−0.834415	−	0.551137i	$-0.814194\pi$
$659$	4.41480e8	−	7.64667e8i	0.0600914	−	0.104081i	0.894506	+	0.447056i	$0.147527\pi$
$660$		0			0
$661$	−5.34861e9	−	9.26406e9i	−0.720337	−	1.24766i	−0.960865	−	0.277018i	$-0.910654\pi$
$661$	−5.34861e9	−	9.26406e9i	−0.720337	−	1.24766i	0.240527	−	0.970642i	$-0.422680\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−1.43196e10			−1.88823
$666$		0			0
$667$	1.59149e10			2.07665
$668$		0			0
$669$		0			0
$670$		0			0
$671$	2.16343e9	+	3.74717e9i	0.276449	+	0.478823i
$672$		0			0
$673$	7.33483e8	−	1.27043e9i	0.0927551	−	0.160656i	−0.815914	−	0.578173i	$-0.803766\pi$
$673$	7.33483e8	−	1.27043e9i	0.0927551	−	0.160656i	0.908669	+	0.417516i	$0.137099\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	1.45404e9	−	2.51848e9i	0.180102	−	0.311945i	−0.761813	−	0.647797i	$-0.775691\pi$
$677$	1.45404e9	−	2.51848e9i	0.180102	−	0.311945i	0.941915	+	0.335852i	$0.109024\pi$
$678$		0			0
$679$	−6.81247e9	−	1.17996e10i	−0.835143	−	1.44651i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	1.44171e9			0.173143			0.0865717	−	0.996246i	$-0.472409\pi$
$683$	1.44171e9			0.173143			0.0865717	+	0.996246i	$0.472409\pi$
$684$		0			0
$685$	1.33105e9			0.158225
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−1.22736e9	−	2.12585e9i	−0.142957	−	0.247608i
$690$		0			0
$691$	−4.63280e9	+	8.02425e9i	−0.534159	+	0.925191i	0.465044	+	0.885287i	$0.346038\pi$
$691$	−4.63280e9	+	8.02425e9i	−0.534159	+	0.925191i	−0.999204	+	0.0399035i	$0.987295\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	8.86377e8	−	1.53525e9i	0.100155	−	0.173473i
$696$		0			0
$697$	3.20871e8	+	5.55765e8i	0.0358935	+	0.0621694i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	9.92592e7			0.0108832			0.00544162	−	0.999985i	$-0.498268\pi$
$701$	9.92592e7			0.0108832			0.00544162	+	0.999985i	$0.498268\pi$
$702$		0			0
$703$	8.09557e9			0.878828
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−8.34285e9	−	1.44502e10i	−0.887864	−	1.53783i
$708$		0			0
$709$	4.30255e9	−	7.45224e9i	0.453382	−	0.785281i	−0.545212	−	0.838299i	$-0.683551\pi$
$709$	4.30255e9	−	7.45224e9i	0.453382	−	0.785281i	0.998594	+	0.0530178i	$0.0168840\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−1.28645e10	+	2.22820e10i	−1.32917	+	2.30218i
$714$		0			0
$715$	−6.65894e9	−	1.15336e10i	−0.681293	−	1.18003i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	1.90115e10			1.90750			0.953750	−	0.300602i	$-0.0971876\pi$
$719$	1.90115e10			1.90750			0.953750	+	0.300602i	$0.0971876\pi$
$720$		0			0
$721$	1.01265e10			1.00621
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−7.37921e9	−	1.27812e10i	−0.719162	−	1.24562i
$726$		0			0
$727$	3.03468e9	−	5.25623e9i	0.292916	−	0.507345i	−0.681582	−	0.731742i	$-0.738708\pi$
$727$	3.03468e9	−	5.25623e9i	0.292916	−	0.507345i	0.974498	+	0.224396i	$0.0720410\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	2.15093e8	−	3.72553e8i	0.0203665	−	0.0352758i
$732$		0			0
$733$	−6.21034e8	−	1.07566e9i	−0.0582441	−	0.100882i	0.835433	−	0.549592i	$-0.185217\pi$
$733$	−6.21034e8	−	1.07566e9i	−0.0582441	−	0.100882i	−0.893677	+	0.448710i	$0.851883\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−1.19990e10			−1.10410
$738$		0			0
$739$	1.86706e10			1.70178			0.850889	−	0.525346i	$-0.176064\pi$
$739$	1.86706e10			1.70178			0.850889	+	0.525346i	$0.176064\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−6.73747e9	−	1.16696e10i	−0.602610	−	1.04375i	−0.992424	−	0.122857i	$-0.960794\pi$
$743$	−6.73747e9	−	1.16696e10i	−0.602610	−	1.04375i	0.389814	−	0.920893i	$-0.372539\pi$
$744$		0			0
$745$	−8.77727e8	+	1.52027e9i	−0.0777701	+	0.134702i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	5.80376e9	−	1.00524e10i	0.504688	−	0.874145i
$750$		0			0
$751$	4.09947e9	+	7.10049e9i	0.353173	+	0.611714i	0.986804	−	0.161922i	$-0.0517693\pi$
$751$	4.09947e9	+	7.10049e9i	0.353173	+	0.611714i	−0.633630	+	0.773636i	$0.718436\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	6.06383e9			0.512782
$756$		0			0
$757$	−1.46091e9			−0.122402			−0.0612008	−	0.998125i	$-0.519493\pi$
$757$	−1.46091e9			−0.122402			−0.0612008	+	0.998125i	$0.519493\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	3.95201e9	+	6.84509e9i	0.325067	+	0.563032i	0.981526	−	0.191330i	$-0.0612800\pi$
$761$	3.95201e9	+	6.84509e9i	0.325067	+	0.563032i	−0.656459	+	0.754361i	$0.727947\pi$
$762$		0			0
$763$	1.50033e10	−	2.59864e10i	1.22278	−	2.11793i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−4.20927e9	+	7.29066e9i	−0.336839	+	0.583423i
$768$		0			0
$769$	6.66991e9	+	1.15526e10i	0.528905	+	0.916091i	0.999432	+	0.0337048i	$0.0107306\pi$
$769$	6.66991e9	+	1.15526e10i	0.528905	+	0.916091i	−0.470527	+	0.882386i	$0.655936\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−1.88966e10			−1.47148			−0.735741	−	0.677263i	$-0.763166\pi$
$773$	−1.88966e10			−1.47148			−0.735741	+	0.677263i	$0.763166\pi$
$774$		0			0
$775$	2.38593e10			1.84120
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−4.43429e9	−	7.68041e9i	−0.336080	−	0.582108i
$780$		0			0
$781$	−1.17621e10	+	2.03726e10i	−0.883503	+	1.53027i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	1.27222e10	−	2.20354e10i	0.938680	−	1.62584i
$786$		0			0
$787$	−6.04802e9	−	1.04755e10i	−0.442284	−	0.766059i	0.555574	−	0.831467i	$-0.312498\pi$
$787$	−6.04802e9	−	1.04755e10i	−0.442284	−	0.766059i	−0.997859	+	0.0654082i	$0.979165\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−6.64549e9			−0.477430
$792$		0			0
$793$	2.54085e9			0.180935
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−4.94534e9	−	8.56558e9i	−0.346013	−	0.599311i	0.639525	−	0.768771i	$-0.279131\pi$
$797$	−4.94534e9	−	8.56558e9i	−0.346013	−	0.599311i	−0.985537	+	0.169459i	$0.945798\pi$
$798$		0			0
$799$	−4.64692e8	+	8.04870e8i	−0.0322293	+	0.0558228i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−7.77213e9	+	1.34617e10i	−0.529707	+	0.917479i
$804$		0			0
$805$	−2.89511e10	−	5.01447e10i	−1.95604	−	3.38797i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	9.57415e9			0.635742			0.317871	−	0.948134i	$-0.397032\pi$
$809$	9.57415e9			0.635742			0.317871	+	0.948134i	$0.397032\pi$
$810$		0			0
$811$	1.69050e10			1.11287			0.556434	−	0.830892i	$-0.312169\pi$
$811$	1.69050e10			1.11287			0.556434	+	0.830892i	$0.312169\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	1.03231e10	+	1.78802e10i	0.667974	+	1.15696i
$816$		0			0
$817$	−2.97249e9	+	5.14850e9i	−0.190697	+	0.330296i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	5.30321e9	−	9.18543e9i	0.334455	−	0.579293i	−0.648925	−	0.760852i	$-0.724781\pi$
$821$	5.30321e9	−	9.18543e9i	0.334455	−	0.579293i	0.983380	+	0.181559i	$0.0581144\pi$
$822$		0			0
$823$	−4.34705e9	−	7.52931e9i	−0.271829	−	0.470821i	0.697502	−	0.716583i	$-0.254295\pi$
$823$	−4.34705e9	−	7.52931e9i	−0.271829	−	0.470821i	−0.969330	+	0.245762i	$0.920962\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−3.92667e9			−0.241410			−0.120705	−	0.992688i	$-0.538516\pi$
$827$	−3.92667e9			−0.241410			−0.120705	+	0.992688i	$0.538516\pi$
$828$		0			0
$829$	−1.74002e10			−1.06075			−0.530377	−	0.847762i	$-0.677950\pi$
$829$	−1.74002e10			−1.06075			−0.530377	+	0.847762i	$0.677950\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−6.74452e8	−	1.16819e9i	−0.0404291	−	0.0700252i
$834$		0			0
$835$	−5.35776e9	+	9.27991e9i	−0.318479	+	0.551622i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−1.29958e9	+	2.25093e9i	−0.0759688	+	0.131582i	−0.901507	−	0.432764i	$-0.857538\pi$
$839$	−1.29958e9	+	2.25093e9i	−0.0759688	+	0.131582i	0.825538	+	0.564346i	$0.190872\pi$
$840$		0			0
$841$	−1.65195e9	−	2.86126e9i	−0.0957658	−	0.165871i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	1.88801e10			1.07648
$846$		0			0
$847$	−4.14446e10			−2.34356
$848$		0			0
$849$		0			0
$850$		0			0
$851$	1.63675e10	+	2.83493e10i	0.910391	+	1.57684i
$852$		0			0
$853$	−7.36915e9	+	1.27637e10i	−0.406533	+	0.704136i	−0.994499	−	0.104750i	$-0.966596\pi$
$853$	−7.36915e9	+	1.27637e10i	−0.406533	+	0.704136i	0.587966	+	0.808886i	$0.299929\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	8.01446e9	−	1.38814e10i	0.434952	−	0.753359i	−0.562340	−	0.826906i	$-0.690099\pi$
$857$	8.01446e9	−	1.38814e10i	0.434952	−	0.753359i	0.997292	+	0.0735475i	$0.0234321\pi$
$858$		0			0
$859$	1.44662e9	+	2.50561e9i	0.0778713	+	0.134877i	0.902331	−	0.431043i	$-0.141854\pi$
$859$	1.44662e9	+	2.50561e9i	0.0778713	+	0.134877i	−0.824460	+	0.565920i	$0.808521\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−2.23190e10			−1.18205			−0.591027	−	0.806651i	$-0.701277\pi$
$863$	−2.23190e10			−1.18205			−0.591027	+	0.806651i	$0.701277\pi$
$864$		0			0
$865$	1.08681e10			0.570949
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−2.01471e10	−	3.48958e10i	−1.04146	−	1.80386i
$870$		0			0
$871$	−3.52307e9	+	6.10214e9i	−0.180658	+	0.312910i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−6.47232e9	+	1.12104e10i	−0.326612	+	0.565708i
$876$		0			0
$877$	−1.86183e10	−	3.22479e10i	−0.932056	−	1.61437i	−0.779801	−	0.626027i	$-0.784680\pi$
$877$	−1.86183e10	−	3.22479e10i	−0.932056	−	1.61437i	−0.152255	−	0.988341i	$-0.548653\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−3.05610e10			−1.50575			−0.752873	−	0.658166i	$-0.771333\pi$
$881$	−3.05610e10			−1.50575			−0.752873	+	0.658166i	$0.771333\pi$
$882$		0			0
$883$	−2.06651e10			−1.01012			−0.505062	−	0.863083i	$-0.668530\pi$
$883$	−2.06651e10			−1.01012			−0.505062	+	0.863083i	$0.668530\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−8.46864e9	−	1.46681e10i	−0.407456	−	0.705735i	0.587148	−	0.809480i	$-0.300251\pi$
$887$	−8.46864e9	−	1.46681e10i	−0.407456	−	0.705735i	−0.994604	+	0.103745i	$0.966917\pi$
$888$		0			0
$889$	−1.05949e10	+	1.83509e10i	−0.505755	+	0.875993i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	6.42182e9	−	1.11229e10i	0.301771	−	0.522683i
$894$		0			0
$895$	1.82006e10	+	3.15244e10i	0.848605	+	1.46983i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	3.32284e10			1.52529
$900$		0			0
$901$	1.13748e9			0.0518090
$902$		0			0
$903$		0			0
$904$		0			0
$905$	1.84826e10	+	3.20128e10i	0.828883	+	1.43567i
$906$		0			0
$907$	1.52327e10	−	2.63839e10i	0.677879	−	1.17412i	−0.297739	−	0.954647i	$-0.596233\pi$
$907$	1.52327e10	−	2.63839e10i	0.677879	−	1.17412i	0.975618	−	0.219474i	$-0.0704340\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−2.57545e9	+	4.46081e9i	−0.112860	+	0.195479i	−0.916922	−	0.399066i	$-0.869334\pi$
$911$	−2.57545e9	+	4.46081e9i	−0.112860	+	0.195479i	0.804062	+	0.594545i	$0.202668\pi$
$912$		0			0
$913$	3.30568e10	+	5.72561e10i	1.43752	+	2.48986i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	1.33653e10			0.572384
$918$		0			0
$919$	3.89705e10			1.65627			0.828135	−	0.560529i	$-0.189402\pi$
$919$	3.89705e10			1.65627			0.828135	+	0.560529i	$0.189402\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	6.90705e9	+	1.19634e10i	0.289126	+	0.500780i
$924$		0			0
$925$	1.51781e10	−	2.62892e10i	0.630551	−	1.09215i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−8.15154e9	+	1.41189e10i	−0.333568	+	0.577757i	−0.983209	−	0.182484i	$-0.941586\pi$
$929$	−8.15154e9	+	1.41189e10i	−0.333568	+	0.577757i	0.649640	+	0.760242i	$0.274919\pi$
$930$		0			0
$931$	9.32061e9	+	1.61438e10i	0.378547	+	0.655664i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	6.17129e9			0.246908
$936$		0			0
$937$	3.38911e10			1.34585			0.672926	−	0.739710i	$-0.265037\pi$
$937$	3.38911e10			1.34585			0.672926	+	0.739710i	$0.265037\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	9.63397e9	+	1.66865e10i	0.376914	+	0.652833i	0.990611	−	0.136708i	$-0.0436521\pi$
$941$	9.63397e9	+	1.66865e10i	0.376914	+	0.652833i	−0.613698	+	0.789541i	$0.710319\pi$
$942$		0			0
$943$	1.79303e10	−	3.10562e10i	0.696301	−	1.20603i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	1.70188e10	−	2.94774e10i	0.651183	−	1.12788i	−0.331653	−	0.943401i	$-0.607606\pi$
$947$	1.70188e10	−	2.94774e10i	0.651183	−	1.12788i	0.982836	−	0.184480i	$-0.0590602\pi$
$948$		0			0
$949$	4.56400e9	+	7.90509e9i	0.173346	+	0.300244i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	1.16499e10			0.436010			0.218005	−	0.975948i	$-0.430045\pi$
$953$	1.16499e10			0.436010			0.218005	+	0.975948i	$0.430045\pi$
$954$		0			0
$955$	−5.70266e10			−2.11868
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−1.91716e9	−	3.32062e9i	−0.0701930	−	0.121578i
$960$		0			0
$961$	−1.31032e10	+	2.26955e10i	−0.476262	+	0.824911i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−1.41822e10	+	2.45643e10i	−0.508041	+	0.879953i
$966$		0			0
$967$	−8.00017e9	−	1.38567e10i	−0.284516	−	0.492796i	0.687976	−	0.725734i	$-0.258500\pi$
$967$	−8.00017e9	−	1.38567e10i	−0.284516	−	0.492796i	−0.972492	+	0.232938i	$0.925166\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	5.65391e9			0.198190			0.0990949	−	0.995078i	$-0.468405\pi$
$971$	5.65391e9			0.198190			0.0990949	+	0.995078i	$0.468405\pi$
$972$		0			0
$973$	−5.10674e9			−0.177725
$974$		0			0
$975$		0			0
$976$		0			0
$977$	1.54763e10	+	2.68057e10i	0.530929	+	0.919596i	0.999349	+	0.0360895i	$0.0114901\pi$
$977$	1.54763e10	+	2.68057e10i	0.530929	+	0.919596i	−0.468420	+	0.883506i	$0.655177\pi$
$978$		0			0
$979$	−2.50113e10	+	4.33208e10i	−0.851916	+	1.47556i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	8.26864e9	−	1.43217e10i	0.277649	−	0.480903i	−0.693151	−	0.720793i	$-0.743778\pi$
$983$	8.26864e9	−	1.43217e10i	0.277649	−	0.480903i	0.970800	+	0.239890i	$0.0771113\pi$
$984$		0			0
$985$	2.45754e10	+	4.25658e10i	0.819358	+	1.41917i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−2.40389e10			−0.790182
$990$		0			0
$991$	2.21260e10			0.722178			0.361089	−	0.932531i	$-0.382405\pi$
$991$	2.21260e10			0.722178			0.361089	+	0.932531i	$0.382405\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	9.75915e9	+	1.69033e10i	0.314073	+	0.543991i
$996$		0			0
$997$	7.78399e9	−	1.34823e10i	0.248753	−	0.430853i	−0.714427	−	0.699710i	$-0.753312\pi$
$997$	7.78399e9	−	1.34823e10i	0.248753	−	0.430853i	0.963180	+	0.268857i	$0.0866458\pi$
$998$		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.73.9		20
3.2	odd	2		72.8.i.a.25.9	✓	20
4.3	odd	2		432.8.i.e.289.9		20
9.4	even	3	inner	216.8.i.a.145.9		20
9.5	odd	6		72.8.i.a.49.9	yes	20
12.11	even	2		144.8.i.e.97.2		20
36.23	even	6		144.8.i.e.49.2		20
36.31	odd	6		432.8.i.e.145.9		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.8.i.a.25.9	✓	20	3.2	odd	2
72.8.i.a.49.9	yes	20	9.5	odd	6
144.8.i.e.49.2		20	36.23	even	6
144.8.i.e.97.2		20	12.11	even	2
216.8.i.a.73.9		20	1.1	even	1	trivial
216.8.i.a.145.9		20	9.4	even	3	inner
432.8.i.e.145.9		20	36.31	odd	6
432.8.i.e.289.9		20	4.3	odd	2

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)

\(f(q)\)	\(=\)	\(q+(212.760 + 368.511i) q^{5} +(612.894 - 1061.56i) q^{7} +O(q^{10})\)	\(q+(212.760 + 368.511i) q^{5} +(612.894 - 1061.56i) q^{7} +(3650.27 - 6322.45i) q^{11} +(-2143.53 - 3712.71i) q^{13} +1986.56 q^{17} -27453.3 q^{19} +(-55504.6 - 96136.7i) q^{23} +(-51471.1 + 89150.6i) q^{25} +(-71682.9 + 124158. i) q^{29} +(-115887. - 200722. i) q^{31} +521598. q^{35} -294885. q^{37} +(161521. + 279763. i) q^{41} +(108274. - 187537. i) q^{43} +(-233918. + 405158. i) q^{47} +(-339508. - 588044. i) q^{49} +572586. q^{53} +3.10652e6 q^{55} +(-981852. - 1.70062e6i) q^{59} +(-296339. + 513274. i) q^{61} +(912117. - 1.57983e6i) q^{65} +(-821791. - 1.42338e6i) q^{67} -3.22227e6 q^{71} -2.12920e6 q^{73} +(-4.47445e6 - 7.74998e6i) q^{77} +(2.75968e6 - 4.77990e6i) q^{79} +(-4.52800e6 + 7.84273e6i) q^{83} +(422660. + 732069. i) q^{85} -6.85191e6 q^{89} -5.25504e6 q^{91} +(-5.84097e6 - 1.01169e7i) q^{95} +(5.55762e6 - 9.62609e6i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 125 q^{5} + 1245 q^{7}+O(q^{10}) \) 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

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