LMFDB - Embedded newform 216.8.i.a.145.8

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			145.8
Root			$0.500000 + 186.740i$ of defining polynomial
Character	$\chi$	$=$	216.145
Dual form			216.8.i.a.73.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(167.971 - 290.935i) q^{5} +(410.729 + 711.403i) q^{7} +O(q^{10})$	$q+(167.971 - 290.935i) q^{5} +(410.729 + 711.403i) q^{7} +(-1316.87 - 2280.88i) q^{11} +(199.836 - 346.125i) q^{13} +19223.8 q^{17} +4493.95 q^{19} +(40302.1 - 69805.3i) q^{23} +(-17366.3 - 30079.4i) q^{25} +(109218. + 189170. i) q^{29} +(-33807.4 + 58556.2i) q^{31} +275963. q^{35} -34749.4 q^{37} +(-275387. + 476985. i) q^{41} +(-414835. - 718515. i) q^{43} +(-385771. - 668174. i) q^{47} +(74375.1 - 128821. i) q^{49} +954526. q^{53} -884783. q^{55} +(128918. - 223293. i) q^{59} +(-1.07657e6 - 1.86468e6i) q^{61} +(-67133.4 - 116278. i) q^{65} +(412737. - 714881. i) q^{67} +3.51073e6 q^{71} +5.76196e6 q^{73} +(1.08175e6 - 1.87364e6i) q^{77} +(3.81036e6 + 6.59974e6i) q^{79} +(-3.67669e6 - 6.36822e6i) q^{83} +(3.22905e6 - 5.59288e6i) q^{85} +5.14346e6 q^{89} +328313. q^{91} +(754855. - 1.30745e6i) q^{95} +(-5.78300e6 - 1.00164e7i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	167.971	−	290.935i	0.600953	−	1.04088i	−0.391724	−	0.920083i	$-0.628121\pi$
$5$	167.971	−	290.935i	0.600953	−	1.04088i	0.992677	−	0.120798i	$-0.0385454\pi$
$6$		0			0
$7$	410.729	+	711.403i	0.452597	+	0.783922i	0.998547	−	0.0538966i	$-0.0171641\pi$
$7$	410.729	+	711.403i	0.452597	+	0.783922i	−0.545949	+	0.837818i	$0.683831\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−1316.87	−	2280.88i	−0.298309	−	0.516687i	0.677440	−	0.735578i	$-0.263089\pi$
$11$	−1316.87	−	2280.88i	−0.298309	−	0.516687i	−0.975749	+	0.218891i	$0.929756\pi$
$12$		0			0
$13$	199.836	−	346.125i	0.0252273	−	0.0436950i	−0.853136	−	0.521688i	$-0.825302\pi$
$13$	199.836	−	346.125i	0.0252273	−	0.0436950i	0.878363	+	0.477993i	$0.158636\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	19223.8			0.949003			0.474502	−	0.880255i	$-0.342628\pi$
$17$	19223.8			0.949003			0.474502	+	0.880255i	$0.342628\pi$
$18$		0			0
$19$	4493.95			0.150311			0.0751554	−	0.997172i	$-0.476055\pi$
$19$	4493.95			0.150311			0.0751554	+	0.997172i	$0.476055\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	40302.1	−	69805.3i	0.690685	−	1.19630i	−0.280929	−	0.959729i	$-0.590642\pi$
$23$	40302.1	−	69805.3i	0.690685	−	1.19630i	0.971614	−	0.236573i	$-0.0760243\pi$
$24$		0			0
$25$	−17366.3	−	30079.4i	−0.222289	−	0.385016i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	109218.	+	189170.i	0.831571	+	1.44032i	0.896792	+	0.442453i	$0.145892\pi$
$29$	109218.	+	189170.i	0.831571	+	1.44032i	−0.0652203	+	0.997871i	$0.520775\pi$
$30$		0			0
$31$	−33807.4	+	58556.2i	−0.203820	+	0.353026i	−0.949756	−	0.312991i	$-0.898669\pi$
$31$	−33807.4	+	58556.2i	−0.203820	+	0.353026i	0.745936	+	0.666017i	$0.232002\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	275963.			1.08796
$36$		0			0
$37$	−34749.4			−0.112782			−0.0563912	−	0.998409i	$-0.517959\pi$
$37$	−34749.4			−0.112782			−0.0563912	+	0.998409i	$0.517959\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−275387.	+	476985.i	−0.624023	+	1.08084i	0.364706	+	0.931123i	$0.381169\pi$
$41$	−275387.	+	476985.i	−0.624023	+	1.08084i	−0.988729	+	0.149717i	$0.952164\pi$
$42$		0			0
$43$	−414835.	−	718515.i	−0.795675	−	1.37815i	−0.922409	−	0.386213i	$-0.873783\pi$
$43$	−414835.	−	718515.i	−0.795675	−	1.37815i	0.126734	−	0.991937i	$-0.459551\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−385771.	−	668174.i	−0.541984	−	0.938744i	−0.998790	−	0.0491778i	$-0.984340\pi$
$47$	−385771.	−	668174.i	−0.541984	−	0.938744i	0.456806	−	0.889566i	$-0.348993\pi$
$48$		0			0
$49$	74375.1	−	128821.i	0.0903111	−	0.156423i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	954526.			0.880688			0.440344	−	0.897829i	$-0.354857\pi$
$53$	954526.			0.880688			0.440344	+	0.897829i	$0.354857\pi$
$54$		0			0
$55$	−884783.			−0.717080
$56$		0			0
$57$		0			0
$58$		0			0
$59$	128918.	−	223293.i	0.0817209	−	0.141545i	−0.822268	−	0.569100i	$-0.807292\pi$
$59$	128918.	−	223293.i	0.0817209	−	0.141545i	0.903989	+	0.427555i	$0.140625\pi$
$60$		0			0
$61$	−1.07657e6	−	1.86468e6i	−0.607280	−	1.05184i	−0.991687	−	0.128677i	$-0.958927\pi$
$61$	−1.07657e6	−	1.86468e6i	−0.607280	−	1.05184i	0.384406	−	0.923164i	$-0.374406\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−67133.4	−	116278.i	−0.0303209	−	0.0525173i
$66$		0			0
$67$	412737.	−	714881.i	0.167653	−	0.290384i	−0.769941	−	0.638115i	$-0.779714\pi$
$67$	412737.	−	714881.i	0.167653	−	0.290384i	0.937594	+	0.347731i	$0.113048\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	3.51073e6			1.16411			0.582055	−	0.813150i	$-0.302249\pi$
$71$	3.51073e6			1.16411			0.582055	+	0.813150i	$0.302249\pi$
$72$		0			0
$73$	5.76196e6			1.73356			0.866782	−	0.498687i	$-0.166184\pi$
$73$	5.76196e6			1.73356			0.866782	+	0.498687i	$0.166184\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	1.08175e6	−	1.87364e6i	0.270028	−	0.467702i
$78$		0			0
$79$	3.81036e6	+	6.59974e6i	0.869504	+	1.50602i	0.862505	+	0.506049i	$0.168895\pi$
$79$	3.81036e6	+	6.59974e6i	0.869504	+	1.50602i	0.00699922	+	0.999976i	$0.497772\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−3.67669e6	−	6.36822e6i	−0.705804	−	1.22249i	−0.966401	−	0.257041i	$-0.917252\pi$
$83$	−3.67669e6	−	6.36822e6i	−0.705804	−	1.22249i	0.260596	−	0.965448i	$-0.416081\pi$
$84$		0			0
$85$	3.22905e6	−	5.59288e6i	0.570306	−	0.987800i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	5.14346e6			0.773375			0.386688	−	0.922211i	$-0.373619\pi$
$89$	5.14346e6			0.773375			0.386688	+	0.922211i	$0.373619\pi$
$90$		0			0
$91$	328313.			0.0456713
$92$		0			0
$93$		0			0
$94$		0			0
$95$	754855.	−	1.30745e6i	0.0903297	−	0.156456i
$96$		0			0
$97$	−5.78300e6	−	1.00164e7i	−0.643357	−	1.11433i	−0.984678	−	0.174380i	$-0.944208\pi$
$97$	−5.78300e6	−	1.00164e7i	−0.643357	−	1.11433i	0.341322	−	0.939947i	$-0.389126\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	1.84847e6	+	3.20164e6i	0.178520	+	0.309206i	0.941374	−	0.337365i	$-0.109536\pi$
$101$	1.84847e6	+	3.20164e6i	0.178520	+	0.309206i	−0.762854	+	0.646571i	$0.776202\pi$
$102$		0			0
$103$	8.17415e6	−	1.41580e7i	0.737076	−	1.27665i	−0.216730	−	0.976232i	$-0.569539\pi$
$103$	8.17415e6	−	1.41580e7i	0.737076	−	1.27665i	0.953806	−	0.300422i	$-0.0971274\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	8.80321e6			0.694701			0.347350	−	0.937735i	$-0.387081\pi$
$107$	8.80321e6			0.694701			0.347350	+	0.937735i	$0.387081\pi$
$108$		0			0
$109$	−1.72455e7			−1.27551			−0.637753	−	0.770241i	$-0.720136\pi$
$109$	−1.72455e7			−1.27551			−0.637753	+	0.770241i	$0.720136\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	1.13312e7	−	1.96262e7i	0.738755	−	1.27956i	−0.214301	−	0.976768i	$-0.568747\pi$
$113$	1.13312e7	−	1.96262e7i	0.738755	−	1.27956i	0.953056	−	0.302794i	$-0.0979195\pi$
$114$		0			0
$115$	−1.35392e7	−	2.34506e7i	−0.830139	−	1.43784i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	7.89577e6	+	1.36759e7i	0.429517	+	0.743944i
$120$		0			0
$121$	6.27532e6	−	1.08692e7i	0.322023	−	0.557760i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	1.45774e7			0.667565
$126$		0			0
$127$	−2.72263e7			−1.17944			−0.589719	−	0.807609i	$-0.700761\pi$
$127$	−2.72263e7			−1.17944			−0.589719	+	0.807609i	$0.700761\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	4.70586e6	−	8.15079e6i	0.182890	−	0.316775i	−0.759974	−	0.649954i	$-0.774788\pi$
$131$	4.70586e6	−	8.15079e6i	0.182890	−	0.316775i	0.942863	+	0.333179i	$0.108121\pi$
$132$		0			0
$133$	1.84579e6	+	3.19701e6i	0.0680303	+	0.117832i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	1.50051e7	+	2.59896e7i	0.498560	+	0.863532i	0.999999	−	0.00166167i	$-0.000528925\pi$
$137$	1.50051e7	+	2.59896e7i	0.498560	+	0.863532i	−0.501438	+	0.865193i	$0.667196\pi$
$138$		0			0
$139$	2.09596e7	−	3.63032e7i	0.661961	−	1.14655i	−0.318139	−	0.948044i	$-0.603058\pi$
$139$	2.09596e7	−	3.63032e7i	0.661961	−	1.14655i	0.980100	−	0.198505i	$-0.0636087\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−1.05263e6			−0.0301022
$144$		0			0
$145$	7.33817e7			1.99894
$146$		0			0
$147$		0			0
$148$		0			0
$149$	3.11700e7	−	5.39880e7i	0.771943	−	1.33704i	−0.164554	−	0.986368i	$-0.552619\pi$
$149$	3.11700e7	−	5.39880e7i	0.771943	−	1.33704i	0.936497	−	0.350676i	$-0.114048\pi$
$150$		0			0
$151$	−1.12868e7	−	1.95494e7i	−0.266780	−	0.462076i	0.701249	−	0.712917i	$-0.252626\pi$
$151$	−1.12868e7	−	1.95494e7i	−0.266780	−	0.462076i	−0.968028	+	0.250841i	$0.919293\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	1.13574e7	+	1.96715e7i	0.244972	+	0.424304i
$156$		0			0
$157$	−1.78770e7	+	3.09639e7i	−0.368677	+	0.638568i	−0.989359	−	0.145495i	$-0.953523\pi$
$157$	−1.78770e7	+	3.09639e7i	−0.368677	+	0.638568i	0.620682	+	0.784063i	$0.286856\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	6.62129e7			1.25041
$162$		0			0
$163$	−6.44432e7			−1.16552			−0.582761	−	0.812644i	$-0.698028\pi$
$163$	−6.44432e7			−1.16552			−0.582761	+	0.812644i	$0.698028\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−8.40546e6	+	1.45587e7i	−0.139654	+	0.241888i	−0.927366	−	0.374156i	$-0.877932\pi$
$167$	−8.40546e6	+	1.45587e7i	−0.139654	+	0.241888i	0.787712	+	0.616044i	$0.211266\pi$
$168$		0			0
$169$	3.12944e7	+	5.42035e7i	0.498727	+	0.863821i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−3.23437e7	−	5.60209e7i	−0.474929	−	0.822601i	0.524659	−	0.851312i	$-0.324193\pi$
$173$	−3.23437e7	−	5.60209e7i	−0.474929	−	0.822601i	−0.999588	+	0.0287119i	$0.990859\pi$
$174$		0			0
$175$	1.42657e7	−	2.47089e7i	0.201215	−	0.348514i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−8.56650e7			−1.11639			−0.558197	−	0.829708i	$-0.688507\pi$
$179$	−8.56650e7			−1.11639			−0.558197	+	0.829708i	$0.688507\pi$
$180$		0			0
$181$	3.87872e7			0.486199			0.243099	−	0.970001i	$-0.421836\pi$
$181$	3.87872e7			0.486199			0.243099	+	0.970001i	$0.421836\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−5.83690e6	+	1.01098e7i	−0.0677769	+	0.117393i
$186$		0			0
$187$	−2.53151e7	−	4.38471e7i	−0.283097	−	0.490338i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−3.49883e7	−	6.06015e7i	−0.363334	−	0.629313i	0.625173	−	0.780486i	$-0.285028\pi$
$191$	−3.49883e7	−	6.06015e7i	−0.363334	−	0.629313i	−0.988507	+	0.151173i	$0.951695\pi$
$192$		0			0
$193$	−5.58414e6	+	9.67201e6i	−0.0559120	+	0.0968425i	−0.892627	−	0.450797i	$-0.851140\pi$
$193$	−5.58414e6	+	9.67201e6i	−0.0559120	+	0.0968425i	0.836715	+	0.547639i	$0.184473\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−1.15789e8			−1.07903			−0.539516	−	0.841975i	$-0.681393\pi$
$197$	−1.15789e8			−1.07903			−0.539516	+	0.841975i	$0.681393\pi$
$198$		0			0
$199$	−3.77550e7			−0.339616			−0.169808	−	0.985477i	$-0.554315\pi$
$199$	−3.77550e7			−0.339616			−0.169808	+	0.985477i	$0.554315\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−8.97176e7	+	1.55395e8i	−0.752734	+	1.30377i
$204$		0			0
$205$	9.25144e7	+	1.60240e8i	0.750017	+	1.29907i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−5.91792e6	−	1.02501e7i	−0.0448391	−	0.0776637i
$210$		0			0
$211$	−110132.	+	190755.i	−0.000807098	+	0.00139794i	−0.866429	−	0.499301i	$-0.833590\pi$
$211$	−110132.	+	190755.i	−0.000807098	+	0.00139794i	0.865622	+	0.500699i	$0.166924\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−2.78722e8			−1.91265
$216$		0			0
$217$	−5.55428e7			−0.368993
$218$		0			0
$219$		0			0
$220$		0			0
$221$	3.84160e6	−	6.65384e6i	0.0239408	−	0.0414667i
$222$		0			0
$223$	1.44105e8	+	2.49598e8i	0.870187	+	1.50721i	0.861802	+	0.507244i	$0.169336\pi$
$223$	1.44105e8	+	2.49598e8i	0.870187	+	1.50721i	0.00838496	+	0.999965i	$0.497331\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	6.62525e7	+	1.14753e8i	0.375934	+	0.651137i	0.990466	−	0.137755i	$-0.0439886\pi$
$227$	6.62525e7	+	1.14753e8i	0.375934	+	0.651137i	−0.614532	+	0.788892i	$0.710655\pi$
$228$		0			0
$229$	−2.45247e7	+	4.24780e7i	−0.134952	+	0.233743i	−0.925579	−	0.378554i	$-0.876421\pi$
$229$	−2.45247e7	+	4.24780e7i	−0.134952	+	0.233743i	0.790627	+	0.612298i	$0.209755\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	3.50099e8			1.81320			0.906600	−	0.421991i	$-0.138669\pi$
$233$	3.50099e8			1.81320			0.906600	+	0.421991i	$0.138669\pi$
$234$		0			0
$235$	−2.59194e8			−1.30283
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−1.14369e8	+	1.98093e8i	−0.541896	+	0.938591i	0.456899	+	0.889518i	$0.348960\pi$
$239$	−1.14369e8	+	1.98093e8i	−0.541896	+	0.938591i	−0.998795	+	0.0490728i	$0.984373\pi$
$240$		0			0
$241$	9.36268e7	+	1.62166e8i	0.430864	+	0.746279i	0.996948	−	0.0780690i	$-0.0248755\pi$
$241$	9.36268e7	+	1.62166e8i	0.430864	+	0.746279i	−0.566084	+	0.824348i	$0.691542\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−2.49858e7	−	4.32766e7i	−0.108545	−	0.188006i
$246$		0			0
$247$	898051.	−	1.55547e6i	0.00379194	−	0.00656783i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	1.75641e8			0.701081			0.350540	−	0.936548i	$-0.385998\pi$
$251$	1.75641e8			0.701081			0.350540	+	0.936548i	$0.385998\pi$
$252$		0			0
$253$	−2.12290e8			−0.824151
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−9.66802e7	+	1.67455e8i	−0.355281	+	0.615364i	−0.987166	−	0.159698i	$-0.948948\pi$
$257$	−9.66802e7	+	1.67455e8i	−0.355281	+	0.615364i	0.631885	+	0.775062i	$0.282281\pi$
$258$		0			0
$259$	−1.42726e7	−	2.47208e7i	−0.0510450	−	0.0884125i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−2.09952e8	−	3.63648e8i	−0.711665	−	1.23264i	−0.964232	−	0.265061i	$-0.914608\pi$
$263$	−2.09952e8	−	3.63648e8i	−0.711665	−	1.23264i	0.252566	−	0.967580i	$-0.418725\pi$
$264$		0			0
$265$	1.60333e8	−	2.77705e8i	0.529252	−	0.916691i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	1.01581e7			0.0318185			0.0159092	−	0.999873i	$-0.494936\pi$
$269$	1.01581e7			0.0318185			0.0159092	+	0.999873i	$0.494936\pi$
$270$		0			0
$271$	−2.32624e8			−0.710005			−0.355003	−	0.934865i	$-0.615520\pi$
$271$	−2.32624e8			−0.710005			−0.355003	+	0.934865i	$0.615520\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−4.57382e7	+	7.92209e7i	−0.132622	+	0.229708i
$276$		0			0
$277$	2.65265e8	+	4.59452e8i	0.749895	+	1.29886i	0.947873	+	0.318649i	$0.103229\pi$
$277$	2.65265e8	+	4.59452e8i	0.749895	+	1.29886i	−0.197978	+	0.980206i	$0.563437\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	1.51008e8	+	2.61553e8i	0.406002	+	0.703215i	0.994437	−	0.105329i	$-0.0335894\pi$
$281$	1.51008e8	+	2.61553e8i	0.406002	+	0.703215i	−0.588436	+	0.808544i	$0.700256\pi$
$282$		0			0
$283$	−2.72331e8	+	4.71691e8i	−0.714240	+	1.23710i	0.249012	+	0.968501i	$0.419894\pi$
$283$	−2.72331e8	+	4.71691e8i	−0.714240	+	1.23710i	−0.963252	+	0.268600i	$0.913439\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−4.52438e8			−1.12972
$288$		0			0
$289$	−4.07846e7			−0.0993925
$290$		0			0
$291$		0			0
$292$		0			0
$293$	4.84448e7	−	8.39089e7i	0.112515	−	0.194882i	−0.804269	−	0.594266i	$-0.797443\pi$
$293$	4.84448e7	−	8.39089e7i	0.112515	−	0.194882i	0.916784	+	0.399384i	$0.130776\pi$
$294$		0			0
$295$	−4.33092e7	−	7.50138e7i	−0.0982208	−	0.170123i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−1.61076e7	−	2.78992e7i	−0.0348483	−	0.0603590i
$300$		0			0
$301$	3.40770e8	−	5.90230e8i	0.720241	−	1.24749i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−7.23335e8			−1.45979
$306$		0			0
$307$	−4.26776e8			−0.841814			−0.420907	−	0.907104i	$-0.638288\pi$
$307$	−4.26776e8			−0.841814			−0.420907	+	0.907104i	$0.638288\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−1.33300e8	+	2.30882e8i	−0.251286	+	0.435241i	−0.963880	−	0.266336i	$-0.914187\pi$
$311$	−1.33300e8	+	2.30882e8i	−0.251286	+	0.435241i	0.712594	+	0.701577i	$0.247520\pi$
$312$		0			0
$313$	3.69561e8	+	6.40099e8i	0.681211	+	1.17989i	0.974612	+	0.223902i	$0.0718795\pi$
$313$	3.69561e8	+	6.40099e8i	0.681211	+	1.17989i	−0.293401	+	0.955989i	$0.594787\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	3.38371e8	+	5.86077e8i	0.596604	+	1.03335i	0.993318	+	0.115407i	$0.0368171\pi$
$317$	3.38371e8	+	5.86077e8i	0.596604	+	1.03335i	−0.396714	+	0.917942i	$0.629850\pi$
$318$		0			0
$319$	2.87650e8	−	4.98224e8i	0.496131	−	0.859324i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	8.63907e7			0.142645
$324$		0			0
$325$	−1.38816e7			−0.0224310
$326$		0			0
$327$		0			0
$328$		0			0
$329$	3.16894e8	−	5.48877e8i	0.490601	−	0.849747i
$330$		0			0
$331$	−3.28805e8	−	5.69506e8i	−0.498356	−	0.863179i	0.501642	−	0.865075i	$-0.332730\pi$
$331$	−3.28805e8	−	5.69506e8i	−0.498356	−	0.863179i	−0.999998	+	0.00189681i	$0.999396\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−1.38656e8	−	2.40159e8i	−0.201503	−	0.349014i
$336$		0			0
$337$	−2.16527e8	+	3.75036e8i	−0.308182	+	0.533788i	−0.977965	−	0.208770i	$-0.933054\pi$
$337$	−2.16527e8	+	3.75036e8i	−0.308182	+	0.533788i	0.669782	+	0.742557i	$0.266387\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	1.78079e8			0.243205
$342$		0			0
$343$	7.98698e8			1.06869
$344$		0			0
$345$		0			0
$346$		0			0
$347$	1.84387e8	−	3.19368e8i	0.236907	−	0.410335i	−0.722918	−	0.690934i	$-0.757200\pi$
$347$	1.84387e8	−	3.19368e8i	0.236907	−	0.410335i	0.959825	+	0.280599i	$0.0905330\pi$
$348$		0			0
$349$	1.27150e8	+	2.20231e8i	0.160114	+	0.277325i	0.934909	−	0.354887i	$-0.115481\pi$
$349$	1.27150e8	+	2.20231e8i	0.160114	+	0.277325i	−0.774796	+	0.632212i	$0.782147\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−4.57240e8	−	7.91962e8i	−0.553264	−	0.958282i	−0.998036	−	0.0626379i	$-0.980049\pi$
$353$	−4.57240e8	−	7.91962e8i	−0.553264	−	0.958282i	0.444772	−	0.895644i	$-0.353285\pi$
$354$		0			0
$355$	5.89703e8	−	1.02140e9i	0.699575	−	1.21170i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	2.93290e8			0.334554			0.167277	−	0.985910i	$-0.446503\pi$
$359$	2.93290e8			0.334554			0.167277	+	0.985910i	$0.446503\pi$
$360$		0			0
$361$	−8.73676e8			−0.977407
$362$		0			0
$363$		0			0
$364$		0			0
$365$	9.67844e8	−	1.67636e9i	1.04179	−	1.80443i
$366$		0			0
$367$	5.90997e8	+	1.02364e9i	0.624099	+	1.08097i	0.988714	+	0.149814i	$0.0478673\pi$
$367$	5.90997e8	+	1.02364e9i	0.624099	+	1.08097i	−0.364615	+	0.931158i	$0.618799\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	3.92051e8	+	6.79053e8i	0.398597	+	0.690390i
$372$		0			0
$373$	6.22724e8	−	1.07859e9i	0.621319	−	1.07616i	−0.367921	−	0.929857i	$-0.619930\pi$
$373$	6.22724e8	−	1.07859e9i	0.621319	−	1.07616i	0.989240	−	0.146300i	$-0.0467363\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	8.73023e7			0.0839133
$378$		0			0
$379$	−3.63588e8			−0.343062			−0.171531	−	0.985179i	$-0.554871\pi$
$379$	−3.63588e8			−0.343062			−0.171531	+	0.985179i	$0.554871\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	4.47251e8	−	7.74662e8i	0.406777	−	0.704558i	−0.587750	−	0.809043i	$-0.699986\pi$
$383$	4.47251e8	−	7.74662e8i	0.406777	−	0.704558i	0.994526	+	0.104485i	$0.0333194\pi$
$384$		0			0
$385$	−3.63406e8	−	6.29438e8i	−0.324548	−	0.562134i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	6.27871e8	+	1.08750e9i	0.540812	+	0.936714i	0.998858	+	0.0477855i	$0.0152164\pi$
$389$	6.27871e8	+	1.08750e9i	0.540812	+	0.936714i	−0.458045	+	0.888929i	$0.651450\pi$
$390$		0			0
$391$	7.74759e8	−	1.34192e9i	0.655463	−	1.13529i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	2.56013e9			2.09012
$396$		0			0
$397$	−3.60213e7			−0.0288930			−0.0144465	−	0.999896i	$-0.504599\pi$
$397$	−3.60213e7			−0.0288930			−0.0144465	+	0.999896i	$0.504599\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−3.53976e8	+	6.13104e8i	−0.274138	+	0.474820i	−0.969917	−	0.243435i	$-0.921726\pi$
$401$	−3.53976e8	+	6.13104e8i	−0.274138	+	0.474820i	0.695780	+	0.718255i	$0.255059\pi$
$402$		0			0
$403$	1.35119e7	+	2.34032e7i	0.0102837	+	0.0178118i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	4.57603e7	+	7.92591e7i	0.0336440	+	0.0582732i
$408$		0			0
$409$	1.15668e8	−	2.00344e8i	0.0835956	−	0.144792i	−0.821196	−	0.570646i	$-0.806693\pi$
$409$	1.15668e8	−	2.00344e8i	0.0835956	−	0.144792i	0.904792	+	0.425854i	$0.140026\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	2.11802e8			0.147947
$414$		0			0
$415$	−2.47032e9			−1.69662
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−9.86493e8	+	1.70866e9i	−0.655156	+	1.13476i	0.326699	+	0.945129i	$0.394064\pi$
$419$	−9.86493e8	+	1.70866e9i	−0.655156	+	1.13476i	−0.981855	+	0.189635i	$0.939270\pi$
$420$		0			0
$421$	−1.12855e9	−	1.95471e9i	−0.737114	−	1.27672i	−0.953790	−	0.300475i	$-0.902855\pi$
$421$	−1.12855e9	−	1.95471e9i	−0.737114	−	1.27672i	0.216676	−	0.976244i	$-0.430479\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−3.33847e8	−	5.78239e8i	−0.210953	−	0.365381i
$426$		0			0
$427$	8.84360e8	−	1.53176e9i	0.549707	−	0.952121i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	2.01080e9			1.20976			0.604878	−	0.796318i	$-0.293222\pi$
$431$	2.01080e9			1.20976			0.604878	+	0.796318i	$0.293222\pi$
$432$		0			0
$433$	−2.97329e9			−1.76007			−0.880036	−	0.474908i	$-0.842481\pi$
$433$	−2.97329e9			−1.76007			−0.880036	+	0.474908i	$0.842481\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	1.81115e8	−	3.13701e8i	0.103817	−	0.179817i
$438$		0			0
$439$	1.05748e9	+	1.83161e9i	0.596549	+	1.03325i	0.993326	+	0.115339i	$0.0367953\pi$
$439$	1.05748e9	+	1.83161e9i	0.596549	+	1.03325i	−0.396777	+	0.917915i	$0.629871\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−2.78233e8	−	4.81913e8i	−0.152053	−	0.263364i	0.779929	−	0.625868i	$-0.215255\pi$
$443$	−2.78233e8	−	4.81913e8i	−0.152053	−	0.263364i	−0.931982	+	0.362504i	$0.881922\pi$
$444$		0			0
$445$	8.63954e8	−	1.49641e9i	0.464762	−	0.804992i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	2.23937e8			0.116752			0.0583759	−	0.998295i	$-0.481408\pi$
$449$	2.23937e8			0.116752			0.0583759	+	0.998295i	$0.481408\pi$
$450$		0			0
$451$	1.45059e9			0.744607
$452$		0			0
$453$		0			0
$454$		0			0
$455$	5.51472e7	−	9.55178e7i	0.0274463	−	0.0475384i
$456$		0			0
$457$	−6.02163e8	−	1.04298e9i	−0.295126	−	0.511173i	0.679888	−	0.733316i	$-0.262028\pi$
$457$	−6.02163e8	−	1.04298e9i	−0.295126	−	0.511173i	−0.975014	+	0.222143i	$0.928695\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−4.70201e8	−	8.14413e8i	−0.223527	−	0.387161i	0.732349	−	0.680929i	$-0.238424\pi$
$461$	−4.70201e8	−	8.14413e8i	−0.223527	−	0.387161i	−0.955877	+	0.293768i	$0.905091\pi$
$462$		0			0
$463$	−3.06390e8	+	5.30684e8i	−0.143464	+	0.248486i	−0.928799	−	0.370585i	$-0.879157\pi$
$463$	−3.06390e8	+	5.30684e8i	−0.143464	+	0.248486i	0.785335	+	0.619071i	$0.212491\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−7.93008e8			−0.360303			−0.180152	−	0.983639i	$-0.557659\pi$
$467$	−7.93008e8			−0.360303			−0.180152	+	0.983639i	$0.557659\pi$
$468$		0			0
$469$	6.78092e8			0.303517
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−1.09256e9	+	1.89238e9i	−0.474715	+	0.822230i
$474$		0			0
$475$	−7.80433e7	−	1.35175e8i	−0.0334124	−	0.0578720i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−6.84301e8	−	1.18524e9i	−0.284494	−	0.492758i	0.687992	−	0.725718i	$-0.258492\pi$
$479$	−6.84301e8	−	1.18524e9i	−0.284494	−	0.492758i	−0.972486	+	0.232960i	$0.925159\pi$
$480$		0			0
$481$	−6.94417e6	+	1.20276e7i	−0.00284520	+	0.00492803i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−3.88551e9			−1.54651
$486$		0			0
$487$	4.88560e8			0.191676			0.0958378	−	0.995397i	$-0.469447\pi$
$487$	4.88560e8			0.191676			0.0958378	+	0.995397i	$0.469447\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	2.49577e9	−	4.32279e9i	0.951521	−	1.64808i	0.209386	−	0.977833i	$-0.432853\pi$
$491$	2.49577e9	−	4.32279e9i	0.951521	−	1.64808i	0.742135	−	0.670250i	$-0.233813\pi$
$492$		0			0
$493$	2.09958e9	+	3.63657e9i	0.789164	+	1.36687i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	1.44196e9	+	2.49755e9i	0.526873	+	0.912570i
$498$		0			0
$499$	−1.74415e9	+	3.02096e9i	−0.628394	+	1.08841i	0.359480	+	0.933153i	$0.382954\pi$
$499$	−1.74415e9	+	3.02096e9i	−0.628394	+	1.08841i	−0.987874	+	0.155257i	$0.950379\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	3.04493e9			1.06681			0.533407	−	0.845858i	$-0.320911\pi$
$503$	3.04493e9			1.06681			0.533407	+	0.845858i	$0.320911\pi$
$504$		0			0
$505$	1.24196e9			0.429129
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−1.15421e9	+	1.99915e9i	−0.387946	+	0.671943i	−0.992173	−	0.124870i	$-0.960149\pi$
$509$	−1.15421e9	+	1.99915e9i	−0.387946	+	0.671943i	0.604227	+	0.796812i	$0.293482\pi$
$510$		0			0
$511$	2.36660e9	+	4.09907e9i	0.784607	+	1.35898i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−2.74605e9	−	4.75629e9i	−0.885896	−	1.53442i
$516$		0			0
$517$	−1.01602e9	+	1.75979e9i	−0.323358	+	0.560072i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−2.29851e9			−0.712058			−0.356029	−	0.934475i	$-0.615870\pi$
$521$	−2.29851e9			−0.712058			−0.356029	+	0.934475i	$0.615870\pi$
$522$		0			0
$523$	2.19073e9			0.669628			0.334814	−	0.942284i	$-0.391327\pi$
$523$	2.19073e9			0.669628			0.334814	+	0.942284i	$0.391327\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−6.49907e8	+	1.12567e9i	−0.193426	+	0.335023i
$528$		0			0
$529$	−1.54610e9	−	2.67793e9i	−0.454092	−	0.786510i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	1.10064e8	+	1.90637e8i	0.0314849	+	0.0545334i
$534$		0			0
$535$	1.47869e9	−	2.56116e9i	0.417483	−	0.723101i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−3.91768e8			−0.107763
$540$		0			0
$541$	2.95717e9			0.802946			0.401473	−	0.915871i	$-0.368498\pi$
$541$	2.95717e9			0.802946			0.401473	+	0.915871i	$0.368498\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−2.89675e9	+	5.01732e9i	−0.766519	+	1.32765i
$546$		0			0
$547$	3.01083e9	+	5.21491e9i	0.786557	+	1.36236i	0.928065	+	0.372419i	$0.121472\pi$
$547$	3.01083e9	+	5.21491e9i	0.786557	+	1.36236i	−0.141508	+	0.989937i	$0.545195\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	4.90818e8	+	8.50122e8i	0.124994	+	0.216496i
$552$		0			0
$553$	−3.13005e9	+	5.42141e9i	−0.787070	+	1.36325i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−6.02161e9			−1.47645			−0.738227	−	0.674553i	$-0.764336\pi$
$557$	−6.02161e9			−1.47645			−0.738227	+	0.674553i	$0.764336\pi$
$558$		0			0
$559$	−3.31595e8			−0.0802910
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−1.95236e9	+	3.38159e9i	−0.461085	+	0.798623i	−0.999015	−	0.0443663i	$-0.985873\pi$
$563$	−1.95236e9	+	3.38159e9i	−0.461085	+	0.798623i	0.537930	+	0.842989i	$0.319206\pi$
$564$		0			0
$565$	−3.80663e9	−	6.59328e9i	−0.887915	−	1.53791i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	3.00366e9	+	5.20250e9i	0.683531	+	1.18391i	0.973896	+	0.226995i	$0.0728900\pi$
$569$	3.00366e9	+	5.20250e9i	0.683531	+	1.18391i	−0.290365	+	0.956916i	$0.593777\pi$
$570$		0			0
$571$	−3.72333e9	+	6.44900e9i	−0.836961	+	1.44966i	0.0554630	+	0.998461i	$0.482337\pi$
$571$	−3.72333e9	+	6.44900e9i	−0.836961	+	1.44966i	−0.892424	+	0.451198i	$0.850997\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−2.79960e9			−0.614127
$576$		0			0
$577$	−6.30828e9			−1.36708			−0.683542	−	0.729911i	$-0.739562\pi$
$577$	−6.30828e9			−1.36708			−0.683542	+	0.729911i	$0.739562\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	3.02025e9	−	5.23123e9i	0.638890	−	1.10659i
$582$		0			0
$583$	−1.25698e9	−	2.17716e9i	−0.262717	−	0.455040i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	1.43421e8	+	2.48412e8i	0.0292670	+	0.0506919i	0.880288	−	0.474440i	$-0.157349\pi$
$587$	1.43421e8	+	2.48412e8i	0.0292670	+	0.0506919i	−0.851021	+	0.525132i	$0.824016\pi$
$588$		0			0
$589$	−1.51929e8	+	2.63148e8i	−0.0306363	+	0.0530637i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−5.50867e9			−1.08481			−0.542407	−	0.840116i	$-0.682487\pi$
$593$	−5.50867e9			−1.08481			−0.542407	+	0.840116i	$0.682487\pi$
$594$		0			0
$595$	5.30505e9			1.03248
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−1.83687e9	+	3.18155e9i	−0.349208	+	0.604846i	−0.986109	−	0.166100i	$-0.946883\pi$
$599$	−1.83687e9	+	3.18155e9i	−0.349208	+	0.604846i	0.636901	+	0.770946i	$0.280216\pi$
$600$		0			0
$601$	3.20076e9	+	5.54388e9i	0.601441	+	1.04173i	0.992603	+	0.121404i	$0.0387398\pi$
$601$	3.20076e9	+	5.54388e9i	0.601441	+	1.04173i	−0.391162	+	0.920322i	$0.627927\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−2.10815e9	−	3.65142e9i	−0.387041	−	0.670375i
$606$		0			0
$607$	−2.14999e9	+	3.72389e9i	−0.390189	+	0.675827i	−0.992474	−	0.122454i	$-0.960924\pi$
$607$	−2.14999e9	+	3.72389e9i	−0.390189	+	0.675827i	0.602285	+	0.798281i	$0.294257\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−3.08363e8			−0.0546913
$612$		0			0
$613$	−1.68039e9			−0.294645			−0.147323	−	0.989089i	$-0.547066\pi$
$613$	−1.68039e9			−0.294645			−0.147323	+	0.989089i	$0.547066\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	3.65672e9	−	6.33363e9i	0.626749	−	1.08556i	−0.361450	−	0.932391i	$-0.617718\pi$
$617$	3.65672e9	−	6.33363e9i	0.626749	−	1.08556i	0.988200	−	0.153170i	$-0.0489483\pi$
$618$		0			0
$619$	−4.85665e9	−	8.41196e9i	−0.823037	−	1.42554i	−0.903411	−	0.428777i	$-0.858945\pi$
$619$	−4.85665e9	−	8.41196e9i	−0.823037	−	1.42554i	0.0803737	−	0.996765i	$-0.474389\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	2.11257e9	+	3.65907e9i	0.350028	+	0.606266i
$624$		0			0
$625$	3.80532e9	−	6.59101e9i	0.623464	−	1.07987i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−6.68015e8			−0.107031
$630$		0			0
$631$	−9.07513e9			−1.43797			−0.718986	−	0.695025i	$-0.755393\pi$
$631$	−9.07513e9			−1.43797			−0.718986	+	0.695025i	$0.755393\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−4.57324e9	+	7.92108e9i	−0.708787	+	1.22765i
$636$		0			0
$637$	−2.97256e7	−	5.14862e7i	−0.00455662	−	0.00789229i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	4.32096e9	+	7.48412e9i	0.648003	+	1.12237i	0.983599	+	0.180369i	$0.0577290\pi$
$641$	4.32096e9	+	7.48412e9i	0.648003	+	1.12237i	−0.335596	+	0.942006i	$0.608938\pi$
$642$		0			0
$643$	−1.90085e9	+	3.29237e9i	−0.281975	+	0.488394i	−0.971871	−	0.235514i	$-0.924323\pi$
$643$	−1.90085e9	+	3.29237e9i	−0.281975	+	0.488394i	0.689896	+	0.723908i	$0.257656\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	8.46777e9			1.22915			0.614574	−	0.788859i	$-0.289328\pi$
$647$	8.46777e9			1.22915			0.614574	+	0.788859i	$0.289328\pi$
$648$		0			0
$649$	−6.79073e8			−0.0975124
$650$		0			0
$651$		0			0
$652$		0			0
$653$	5.89711e9	−	1.02141e10i	0.828788	−	1.43550i	−0.0702016	−	0.997533i	$-0.522364\pi$
$653$	5.89711e9	−	1.02141e10i	0.828788	−	1.43550i	0.898990	−	0.437970i	$-0.144302\pi$
$654$		0			0
$655$	−1.58090e9	−	2.73820e9i	−0.219816	−	0.380733i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	9.08551e8	+	1.57366e9i	0.123666	+	0.214196i	0.921211	−	0.389064i	$-0.127202\pi$
$659$	9.08551e8	+	1.57366e9i	0.123666	+	0.214196i	−0.797545	+	0.603260i	$0.793868\pi$
$660$		0			0
$661$	−4.24501e9	+	7.35258e9i	−0.571707	+	0.990226i	0.424683	+	0.905342i	$0.360385\pi$
$661$	−4.24501e9	+	7.35258e9i	−0.571707	+	0.990226i	−0.996391	+	0.0848844i	$0.972948\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	1.24016e9			0.163532
$666$		0			0
$667$	1.76068e10			2.29742
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−2.83540e9	+	4.91107e9i	−0.362315	+	0.627548i
$672$		0			0
$673$	3.97752e9	+	6.88927e9i	0.502991	+	0.871206i	0.999994	+	0.00345699i	$0.00110040\pi$
$673$	3.97752e9	+	6.88927e9i	0.502991	+	0.871206i	−0.497003	+	0.867749i	$0.665566\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−2.70292e9	−	4.68159e9i	−0.334790	−	0.579873i	0.648654	−	0.761083i	$-0.275332\pi$
$677$	−2.70292e9	−	4.68159e9i	−0.334790	−	0.579873i	−0.983444	+	0.181210i	$0.941999\pi$
$678$		0			0
$679$	4.75049e9	−	8.22809e9i	0.582363	−	1.00868i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	6.09733e9			0.732263			0.366131	−	0.930563i	$-0.380682\pi$
$683$	6.09733e9			0.732263			0.366131	+	0.930563i	$0.380682\pi$
$684$		0			0
$685$	1.00817e10			1.19845
$686$		0			0
$687$		0			0
$688$		0			0
$689$	1.90748e8	−	3.30386e8i	0.0222174	−	0.0384817i
$690$		0			0
$691$	−1.32275e9	−	2.29107e9i	−0.152512	−	0.264159i	0.779638	−	0.626230i	$-0.215403\pi$
$691$	−1.32275e9	−	2.29107e9i	−0.152512	−	0.264159i	−0.932150	+	0.362071i	$0.882070\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−7.04124e9	−	1.21958e10i	−0.795614	−	1.37804i
$696$		0			0
$697$	−5.29399e9	+	9.16946e9i	−0.592200	+	1.02572i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	3.29736e9			0.361537			0.180769	−	0.983526i	$-0.442141\pi$
$701$	3.29736e9			0.361537			0.180769	+	0.983526i	$0.442141\pi$
$702$		0			0
$703$	−1.56162e8			−0.0169524
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−1.51844e9	+	2.63001e9i	−0.161596	+	0.279892i
$708$		0			0
$709$	4.00423e9	+	6.93553e9i	0.421946	+	0.730833i	0.996130	−	0.0878941i	$-0.0280137\pi$
$709$	4.00423e9	+	6.93553e9i	0.421946	+	0.730833i	−0.574183	+	0.818727i	$0.694680\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	2.72502e9	+	4.71987e9i	0.281551	+	0.487660i
$714$		0			0
$715$	−1.76811e8	+	3.06246e8i	−0.0180900	+	0.0313328i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−1.78101e9			−0.178696			−0.0893481	−	0.996000i	$-0.528478\pi$
$719$	−1.78101e9			−0.178696			−0.0893481	+	0.996000i	$0.528478\pi$
$720$		0			0
$721$	1.34294e10			1.33440
$722$		0			0
$723$		0			0
$724$		0			0
$725$	3.79342e9	−	6.57039e9i	0.369698	−	0.640336i
$726$		0			0
$727$	−9.80634e9	−	1.69851e10i	−0.946535	−	1.63945i	−0.752648	−	0.658423i	$-0.771224\pi$
$727$	−9.80634e9	−	1.69851e10i	−0.946535	−	1.63945i	−0.193887	−	0.981024i	$-0.562109\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−7.97470e9	−	1.38126e10i	−0.755099	−	1.30787i
$732$		0			0
$733$	7.18804e9	−	1.24500e10i	0.674134	−	1.16763i	−0.302587	−	0.953122i	$-0.597850\pi$
$733$	7.18804e9	−	1.24500e10i	0.674134	−	1.16763i	0.976721	−	0.214513i	$-0.0688164\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−2.17408e9			−0.200050
$738$		0			0
$739$	−9.53732e9			−0.869303			−0.434651	−	0.900599i	$-0.643128\pi$
$739$	−9.53732e9			−0.869303			−0.434651	+	0.900599i	$0.643128\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	4.08431e9	−	7.07424e9i	0.365307	−	0.632730i	−0.623518	−	0.781809i	$-0.714297\pi$
$743$	4.08431e9	−	7.07424e9i	0.365307	−	0.632730i	0.988825	+	0.149078i	$0.0476306\pi$
$744$		0			0
$745$	−1.04713e10	−	1.81369e10i	−0.927802	−	1.60700i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	3.61573e9	+	6.26263e9i	0.314420	+	0.544591i
$750$		0			0
$751$	−3.81414e8	+	6.60629e8i	−0.0328592	+	0.0569138i	−0.881987	−	0.471273i	$-0.843795\pi$
$751$	−3.81414e8	+	6.60629e8i	−0.0328592	+	0.0569138i	0.849128	+	0.528187i	$0.177128\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−7.58347e9			−0.641289
$756$		0			0
$757$	7.97879e9			0.668500			0.334250	−	0.942484i	$-0.391517\pi$
$757$	7.97879e9			0.668500			0.334250	+	0.942484i	$0.391517\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−1.10004e9	+	1.90532e9i	−0.0904819	+	0.156719i	−0.907714	−	0.419589i	$-0.862174\pi$
$761$	−1.10004e9	+	1.90532e9i	−0.0904819	+	0.156719i	0.817232	+	0.576309i	$0.195507\pi$
$762$		0			0
$763$	−7.08322e9	−	1.22685e10i	−0.577291	−	0.999897i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−5.15250e7	−	8.92439e7i	−0.00412320	−	0.00714159i
$768$		0			0
$769$	−4.50755e9	+	7.80730e9i	−0.357436	+	0.619097i	−0.987532	−	0.157420i	$-0.949682\pi$
$769$	−4.50755e9	+	7.80730e9i	−0.357436	+	0.619097i	0.630096	+	0.776517i	$0.283016\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	9.06498e9			0.705893			0.352946	−	0.935644i	$-0.385180\pi$
$773$	9.06498e9			0.705893			0.352946	+	0.935644i	$0.385180\pi$
$774$		0			0
$775$	2.34844e9			0.181227
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−1.23758e9	+	2.14354e9i	−0.0937974	+	0.162462i
$780$		0			0
$781$	−4.62316e9	−	8.00755e9i	−0.347265	−	0.601480i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	6.00566e9	+	1.04021e10i	0.443115	+	0.767498i
$786$		0			0
$787$	−4.24052e9	+	7.34480e9i	−0.310104	+	0.537117i	−0.978385	−	0.206793i	$-0.933697\pi$
$787$	−4.24052e9	+	7.34480e9i	−0.310104	+	0.537117i	0.668280	+	0.743910i	$0.267031\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	1.86162e10			1.33744
$792$		0			0
$793$	−8.60551e8			−0.0612803
$794$		0			0
$795$		0			0
$796$		0			0
$797$	7.62874e9	−	1.32134e10i	0.533763	−	0.924505i	−0.465459	−	0.885070i	$-0.654111\pi$
$797$	7.62874e9	−	1.32134e10i	0.533763	−	0.924505i	0.999222	−	0.0394358i	$-0.0125561\pi$
$798$		0			0
$799$	−7.41597e9	−	1.28448e10i	−0.514345	−	0.890872i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−7.58772e9	−	1.31423e10i	−0.517138	−	0.895710i
$804$		0			0
$805$	1.11219e10	−	1.92637e10i	0.751437	−	1.30153i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	2.03731e10			1.35281			0.676405	−	0.736530i	$-0.263537\pi$
$809$	2.03731e10			1.35281			0.676405	+	0.736530i	$0.263537\pi$
$810$		0			0
$811$	−1.82217e8			−0.0119954			−0.00599772	−	0.999982i	$-0.501909\pi$
$811$	−1.82217e8			−0.0119954			−0.00599772	+	0.999982i	$0.501909\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−1.08246e10	+	1.87488e10i	−0.700424	+	1.21317i
$816$		0			0
$817$	−1.86425e9	−	3.22897e9i	−0.119599	−	0.207151i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−1.05331e10	−	1.82439e10i	−0.664286	−	1.15058i	−0.979478	−	0.201549i	$-0.935403\pi$
$821$	−1.05331e10	−	1.82439e10i	−0.664286	−	1.15058i	0.315193	−	0.949028i	$-0.397931\pi$
$822$		0			0
$823$	−7.33844e9	+	1.27106e10i	−0.458886	+	0.794813i	−0.998902	−	0.0468411i	$-0.985085\pi$
$823$	−7.33844e9	+	1.27106e10i	−0.458886	+	0.794813i	0.540017	+	0.841654i	$0.318418\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	1.64329e10			1.01029			0.505143	−	0.863036i	$-0.331440\pi$
$827$	1.64329e10			1.01029			0.505143	+	0.863036i	$0.331440\pi$
$828$		0			0
$829$	2.82341e10			1.72121			0.860603	−	0.509276i	$-0.170087\pi$
$829$	2.82341e10			1.72121			0.860603	+	0.509276i	$0.170087\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	1.42977e9	−	2.47644e9i	0.0857056	−	0.148446i
$834$		0			0
$835$	2.82375e9	+	4.89089e9i	0.167851	+	0.290727i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	1.24437e10	+	2.15532e10i	0.727419	+	1.25993i	0.957971	+	0.286866i	$0.0926135\pi$
$839$	1.24437e10	+	2.15532e10i	0.727419	+	1.25993i	−0.230552	+	0.973060i	$0.574053\pi$
$840$		0			0
$841$	−1.52320e10	+	2.63826e10i	−0.883022	+	1.52944i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	2.10263e10			1.19885
$846$		0			0
$847$	1.03098e10			0.582987
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−1.40047e9	+	2.42569e9i	−0.0778971	+	0.134922i
$852$		0			0
$853$	6.27700e9	+	1.08721e10i	0.346283	+	0.599779i	0.985586	−	0.169176i	$-0.0541105\pi$
$853$	6.27700e9	+	1.08721e10i	0.346283	+	0.599779i	−0.639303	+	0.768955i	$0.720777\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	1.33766e10	+	2.31690e10i	0.725963	+	1.25740i	0.958577	+	0.284835i	$0.0919388\pi$
$857$	1.33766e10	+	2.31690e10i	0.725963	+	1.25740i	−0.232614	+	0.972569i	$0.574728\pi$
$858$		0			0
$859$	1.47350e10	−	2.55219e10i	0.793187	−	1.37384i	−0.130797	−	0.991409i	$-0.541754\pi$
$859$	1.47350e10	−	2.55219e10i	0.793187	−	1.37384i	0.923984	−	0.382431i	$-0.124913\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−2.04758e10			−1.08443			−0.542216	−	0.840239i	$-0.682415\pi$
$863$	−2.04758e10			−1.08443			−0.542216	+	0.840239i	$0.682415\pi$
$864$		0			0
$865$	−2.17313e10			−1.14164
$866$		0			0
$867$		0			0
$868$		0			0
$869$	1.00355e10	−	1.73819e10i	0.518762	−	0.898523i
$870$		0			0
$871$	−1.64959e8	−	2.85717e8i	−0.00845888	−	0.0146512i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	5.98734e9	+	1.03704e10i	0.302138	+	0.523319i
$876$		0			0
$877$	1.18076e10	−	2.04513e10i	0.591101	−	1.02382i	−0.402983	−	0.915207i	$-0.632027\pi$
$877$	1.18076e10	−	2.04513e10i	0.591101	−	1.02382i	0.994084	−	0.108610i	$-0.0346399\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−3.80845e10			−1.87643			−0.938216	−	0.346050i	$-0.887523\pi$
$881$	−3.80845e10			−1.87643			−0.938216	+	0.346050i	$0.887523\pi$
$882$		0			0
$883$	−1.82305e10			−0.891120			−0.445560	−	0.895252i	$-0.646995\pi$
$883$	−1.82305e10			−0.891120			−0.445560	+	0.895252i	$0.646995\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	9.41134e9	−	1.63009e10i	0.452813	−	0.784295i	−0.545746	−	0.837950i	$-0.683754\pi$
$887$	9.41134e9	−	1.63009e10i	0.452813	−	0.784295i	0.998560	+	0.0536551i	$0.0170872\pi$
$888$		0			0
$889$	−1.11826e10	−	1.93689e10i	−0.533810	−	0.924587i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−1.73363e9	−	3.00274e9i	−0.0814661	−	0.141103i
$894$		0			0
$895$	−1.43893e10	+	2.49229e10i	−0.670901	+	1.16203i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−1.47695e10			−0.677963
$900$		0			0
$901$	1.83496e10			0.835776
$902$		0			0
$903$		0			0
$904$		0			0
$905$	6.51515e9	−	1.12846e10i	0.292183	−	0.506075i
$906$		0			0
$907$	4.18114e9	+	7.24194e9i	0.186067	+	0.322277i	0.943935	−	0.330130i	$-0.107093\pi$
$907$	4.18114e9	+	7.24194e9i	0.186067	+	0.322277i	−0.757869	+	0.652407i	$0.773759\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−7.62570e9	−	1.32081e10i	−0.334169	−	0.578797i	0.649156	−	0.760655i	$-0.275122\pi$
$911$	−7.62570e9	−	1.32081e10i	−0.334169	−	0.578797i	−0.983325	+	0.181858i	$0.941789\pi$
$912$		0			0
$913$	−9.68342e9	+	1.67722e10i	−0.421096	+	0.729360i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	7.73134e9			0.331102
$918$		0			0
$919$	−1.16450e9			−0.0494921			−0.0247461	−	0.999694i	$-0.507878\pi$
$919$	−1.16450e9			−0.0494921			−0.0247461	+	0.999694i	$0.507878\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	7.01570e8	−	1.21515e9i	0.0293674	−	0.0508658i
$924$		0			0
$925$	6.03469e8	+	1.04524e9i	0.0250703	+	0.0434230i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	5.77060e9	+	9.99497e9i	0.236138	+	0.409003i	0.959603	−	0.281358i	$-0.0907849\pi$
$929$	5.77060e9	+	9.99497e9i	0.236138	+	0.409003i	−0.723465	+	0.690361i	$0.757452\pi$
$930$		0			0
$931$	3.34238e8	−	5.78917e8i	0.0135747	−	0.0235121i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−1.70089e10			−0.680511
$936$		0			0
$937$	−4.43673e10			−1.76187			−0.880936	−	0.473235i	$-0.843086\pi$
$937$	−4.43673e10			−1.76187			−0.880936	+	0.473235i	$0.843086\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−1.12095e10	+	1.94154e10i	−0.438553	+	0.759597i	−0.997578	−	0.0695541i	$-0.977842\pi$
$941$	−1.12095e10	+	1.94154e10i	−0.438553	+	0.759597i	0.559025	+	0.829151i	$0.311176\pi$
$942$		0			0
$943$	2.21974e10	+	3.84470e10i	0.862007	+	1.49304i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	2.39906e9	+	4.15530e9i	0.0917945	+	0.158993i	0.908266	−	0.418393i	$-0.137406\pi$
$947$	2.39906e9	+	4.15530e9i	0.0917945	+	0.158993i	−0.816472	+	0.577385i	$0.804073\pi$
$948$		0			0
$949$	1.15144e9	−	1.99436e9i	0.0437332	−	0.0757481i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−4.07049e10			−1.52343			−0.761714	−	0.647913i	$-0.775642\pi$
$953$	−4.07049e10			−1.52343			−0.761714	+	0.647913i	$0.775642\pi$
$954$		0			0
$955$	−2.35081e10			−0.873387
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−1.23261e10	+	2.13494e10i	−0.451294	+	0.781664i
$960$		0			0
$961$	1.14704e10	+	1.98674e10i	0.416915	+	0.722118i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	1.87595e9	+	3.24924e9i	0.0672010	+	0.116396i
$966$		0			0
$967$	1.17064e10	−	2.02761e10i	0.416323	−	0.721092i	−0.579244	−	0.815155i	$-0.696652\pi$
$967$	1.17064e10	−	2.02761e10i	0.416323	−	0.721092i	0.995566	+	0.0940623i	$0.0299853\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−1.76566e10			−0.618928			−0.309464	−	0.950911i	$-0.600150\pi$
$971$	−1.76566e10			−0.618928			−0.309464	+	0.950911i	$0.600150\pi$
$972$		0			0
$973$	3.44349e10			1.19841
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−2.67152e10	+	4.62721e10i	−0.916490	+	1.58741i	−0.111784	+	0.993732i	$0.535657\pi$
$977$	−2.67152e10	+	4.62721e10i	−0.916490	+	1.58741i	−0.804705	+	0.593674i	$0.797677\pi$
$978$		0			0
$979$	−6.77324e9	−	1.17316e10i	−0.230705	−	0.399593i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−1.70551e10	−	2.95404e10i	−0.572688	−	0.991924i	−0.996289	−	0.0860752i	$-0.972567\pi$
$983$	−1.70551e10	−	2.95404e10i	−0.572688	−	0.991924i	0.423601	−	0.905849i	$-0.360766\pi$
$984$		0			0
$985$	−1.94492e10	+	3.36870e10i	−0.648447	+	1.12314i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−6.68749e10			−2.19824
$990$		0			0
$991$	1.88897e10			0.616548			0.308274	−	0.951298i	$-0.400249\pi$
$991$	1.88897e10			0.616548			0.308274	+	0.951298i	$0.400249\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−6.34176e9	+	1.09842e10i	−0.204093	+	0.353500i
$996$		0			0
$997$	1.70404e10	+	2.95148e10i	0.544561	+	0.943207i	0.998634	+	0.0522427i	$0.0166370\pi$
$997$	1.70404e10	+	2.95148e10i	0.544561	+	0.943207i	−0.454074	+	0.890964i	$0.650030\pi$
$998$		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.145.8		20
3.2	odd	2		72.8.i.a.49.2	yes	20
4.3	odd	2		432.8.i.e.145.8		20
9.2	odd	6		72.8.i.a.25.2	✓	20
9.7	even	3	inner	216.8.i.a.73.8		20
12.11	even	2		144.8.i.e.49.9		20
36.7	odd	6		432.8.i.e.289.8		20
36.11	even	6		144.8.i.e.97.9		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.8.i.a.25.2	✓	20	9.2	odd	6
72.8.i.a.49.2	yes	20	3.2	odd	2
144.8.i.e.49.9		20	12.11	even	2
144.8.i.e.97.9		20	36.11	even	6
216.8.i.a.73.8		20	9.7	even	3	inner
216.8.i.a.145.8		20	1.1	even	1	trivial
432.8.i.e.145.8		20	4.3	odd	2
432.8.i.e.289.8		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)

\(f(q)\)	\(=\)	\(q+(167.971 - 290.935i) q^{5} +(410.729 + 711.403i) q^{7} +O(q^{10})\)	\(q+(167.971 - 290.935i) q^{5} +(410.729 + 711.403i) q^{7} +(-1316.87 - 2280.88i) q^{11} +(199.836 - 346.125i) q^{13} +19223.8 q^{17} +4493.95 q^{19} +(40302.1 - 69805.3i) q^{23} +(-17366.3 - 30079.4i) q^{25} +(109218. + 189170. i) q^{29} +(-33807.4 + 58556.2i) q^{31} +275963. q^{35} -34749.4 q^{37} +(-275387. + 476985. i) q^{41} +(-414835. - 718515. i) q^{43} +(-385771. - 668174. i) q^{47} +(74375.1 - 128821. i) q^{49} +954526. q^{53} -884783. q^{55} +(128918. - 223293. i) q^{59} +(-1.07657e6 - 1.86468e6i) q^{61} +(-67133.4 - 116278. i) q^{65} +(412737. - 714881. i) q^{67} +3.51073e6 q^{71} +5.76196e6 q^{73} +(1.08175e6 - 1.87364e6i) q^{77} +(3.81036e6 + 6.59974e6i) q^{79} +(-3.67669e6 - 6.36822e6i) q^{83} +(3.22905e6 - 5.59288e6i) q^{85} +5.14346e6 q^{89} +328313. q^{91} +(754855. - 1.30745e6i) q^{95} +(-5.78300e6 - 1.00164e7i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 125 q^{5} + 1245 q^{7}+O(q^{10}) \) 20 * q + 125 * q^5 + 1245 * q^7	\( 20 q + 125 q^{5} + 1245 q^{7} - 6106 q^{11} + 4937 q^{13} - 48722 q^{17} - 26882 q^{19} - 19387 q^{23} - 218957 q^{25} + 46791 q^{29} - 185039 q^{31} - 83094 q^{35} + 108420 q^{37} + 638112 q^{41} + 892628 q^{43} + 230883 q^{47} - 1034741 q^{49} + 2872940 q^{53} + 1089998 q^{55} - 2172454 q^{59} - 1878325 q^{61} - 1239133 q^{65} + 531496 q^{67} - 3723056 q^{71} - 1804522 q^{73} - 6276543 q^{77} + 3607847 q^{79} - 10794491 q^{83} + 3597658 q^{85} - 32214888 q^{89} - 16117530 q^{91} + 756868 q^{95} + 17951260 q^{97}+O(q^{100}) \) 20 * q + 125 * q^5 + 1245 * q^7 - 6106 * q^11 + 4937 * q^13 - 48722 * q^17 - 26882 * q^19 - 19387 * q^23 - 218957 * q^25 + 46791 * q^29 - 185039 * q^31 - 83094 * q^35 + 108420 * q^37 + 638112 * q^41 + 892628 * q^43 + 230883 * q^47 - 1034741 * q^49 + 2872940 * q^53 + 1089998 * q^55 - 2172454 * q^59 - 1878325 * q^61 - 1239133 * q^65 + 531496 * q^67 - 3723056 * q^71 - 1804522 * q^73 - 6276543 * q^77 + 3607847 * q^79 - 10794491 * q^83 + 3597658 * q^85 - 32214888 * q^89 - 16117530 * q^91 + 756868 * q^95 + 17951260 * q^97

Introduction

L-functions

Modular forms

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Database

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