LMFDB - Embedded newform 216.8.i.a.145.4

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			145.4
Root			$0.500000 - 73.9766i$ of defining polynomial
Character	$\chi$	$=$	216.145
Dual form			216.8.i.a.73.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-57.8156 + 100.140i) q^{5} +(783.840 + 1357.65i) q^{7} +O(q^{10})$	$q+(-57.8156 + 100.140i) q^{5} +(783.840 + 1357.65i) q^{7} +(-2003.92 - 3470.90i) q^{11} +(-5895.01 + 10210.5i) q^{13} -21266.7 q^{17} -9451.99 q^{19} +(-12182.8 + 21101.2i) q^{23} +(32377.2 + 56079.0i) q^{25} +(5037.15 + 8724.60i) q^{29} +(57584.7 - 99739.6i) q^{31} -181273. q^{35} +533259. q^{37} +(309299. - 535722. i) q^{41} +(-211503. - 366334. i) q^{43} +(-298495. - 517008. i) q^{47} +(-817040. + 1.41515e6i) q^{49} -1.82706e6 q^{53} +463432. q^{55} +(-774292. + 1.34111e6i) q^{59} +(-888359. - 1.53868e6i) q^{61} +(-681648. - 1.18065e6i) q^{65} +(-481416. + 833837. i) q^{67} +4.96018e6 q^{71} -4.31942e6 q^{73} +(3.14151e6 - 5.44126e6i) q^{77} +(-1.85784e6 - 3.21787e6i) q^{79} +(511377. + 885731. i) q^{83} +(1.22955e6 - 2.12964e6i) q^{85} -4.01824e6 q^{89} -1.84830e7 q^{91} +(546473. - 946519. i) q^{95} +(4.08399e6 + 7.07368e6i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−57.8156	+	100.140i	−0.206848	+	0.358270i	−0.950720	−	0.310051i	$-0.899654\pi$
$5$	−57.8156	+	100.140i	−0.206848	+	0.358270i	0.743872	+	0.668322i	$0.232987\pi$
$6$		0			0
$7$	783.840	+	1357.65i	0.863743	+	1.49605i	0.868290	+	0.496057i	$0.165219\pi$
$7$	783.840	+	1357.65i	0.863743	+	1.49605i	−0.00454707	+	0.999990i	$0.501447\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−2003.92	−	3470.90i	−0.453948	−	0.786262i	0.544679	−	0.838645i	$-0.316652\pi$
$11$	−2003.92	−	3470.90i	−0.453948	−	0.786262i	−0.998627	+	0.0523830i	$0.983318\pi$
$12$		0			0
$13$	−5895.01	+	10210.5i	−0.744189	+	1.28897i	0.206384	+	0.978471i	$0.433830\pi$
$13$	−5895.01	+	10210.5i	−0.744189	+	1.28897i	−0.950573	+	0.310501i	$0.899503\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−21266.7			−1.04985			−0.524927	−	0.851147i	$-0.675908\pi$
$17$	−21266.7			−1.04985			−0.524927	+	0.851147i	$0.675908\pi$
$18$		0			0
$19$	−9451.99			−0.316145			−0.158072	−	0.987428i	$-0.550528\pi$
$19$	−9451.99			−0.316145			−0.158072	+	0.987428i	$0.550528\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−12182.8	+	21101.2i	−0.208785	+	0.361626i	−0.951332	−	0.308168i	$-0.900284\pi$
$23$	−12182.8	+	21101.2i	−0.208785	+	0.361626i	0.742547	+	0.669794i	$0.233618\pi$
$24$		0			0
$25$	32377.2	+	56079.0i	0.414428	+	0.717811i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	5037.15	+	8724.60i	0.0383523	+	0.0664282i	0.884564	−	0.466418i	$-0.154456\pi$
$29$	5037.15	+	8724.60i	0.0383523	+	0.0664282i	−0.846212	+	0.532846i	$0.821122\pi$
$30$		0			0
$31$	57584.7	−	99739.6i	0.347169	−	0.601314i	−0.638576	−	0.769559i	$-0.720476\pi$
$31$	57584.7	−	99739.6i	0.347169	−	0.601314i	0.985745	+	0.168244i	$0.0538097\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−181273.			−0.714652
$36$		0			0
$37$	533259.			1.73074			0.865370	−	0.501133i	$-0.167083\pi$
$37$	533259.			1.73074			0.865370	+	0.501133i	$0.167083\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	309299.	−	535722.i	0.700867	−	1.21394i	−0.267296	−	0.963615i	$-0.586130\pi$
$41$	309299.	−	535722.i	0.700867	−	1.21394i	0.968163	−	0.250322i	$-0.0805366\pi$
$42$		0			0
$43$	−211503.	−	366334.i	−0.405674	−	0.702647i	0.588726	−	0.808333i	$-0.299630\pi$
$43$	−211503.	−	366334.i	−0.405674	−	0.702647i	−0.994400	+	0.105685i	$0.966296\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−298495.	−	517008.i	−0.419367	−	0.726365i	0.576509	−	0.817091i	$-0.304415\pi$
$47$	−298495.	−	517008.i	−0.419367	−	0.726365i	−0.995876	+	0.0907257i	$0.971081\pi$
$48$		0			0
$49$	−817040.	+	1.41515e6i	−0.992104	+	1.71837i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−1.82706e6			−1.68572			−0.842862	−	0.538129i	$-0.819131\pi$
$53$	−1.82706e6			−1.68572			−0.842862	+	0.538129i	$0.819131\pi$
$54$		0			0
$55$	463432.			0.375592
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−774292.	+	1.34111e6i	−0.490820	+	0.850126i	−0.999944	−	0.0105674i	$-0.996636\pi$
$59$	−774292.	+	1.34111e6i	−0.490820	+	0.850126i	0.509124	+	0.860693i	$0.329970\pi$
$60$		0			0
$61$	−888359.	−	1.53868e6i	−0.501111	−	0.867950i	−0.999999	−	0.00128332i	$-0.999592\pi$
$61$	−888359.	−	1.53868e6i	−0.501111	−	0.867950i	0.498888	−	0.866666i	$-0.333742\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−681648.	−	1.18065e6i	−0.307867	−	0.533242i
$66$		0			0
$67$	−481416.	+	833837.i	−0.195550	+	0.338703i	−0.947081	−	0.320995i	$-0.895983\pi$
$67$	−481416.	+	833837.i	−0.195550	+	0.338703i	0.751530	+	0.659698i	$0.229316\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	4.96018e6			1.64472			0.822362	−	0.568965i	$-0.192656\pi$
$71$	4.96018e6			1.64472			0.822362	+	0.568965i	$0.192656\pi$
$72$		0			0
$73$	−4.31942e6			−1.29956			−0.649779	−	0.760123i	$-0.725139\pi$
$73$	−4.31942e6			−1.29956			−0.649779	+	0.760123i	$0.725139\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	3.14151e6	−	5.44126e6i	0.784190	−	1.35826i
$78$		0			0
$79$	−1.85784e6	−	3.21787e6i	−0.423949	−	0.734301i	0.572373	−	0.819993i	$-0.306023\pi$
$79$	−1.85784e6	−	3.21787e6i	−0.423949	−	0.734301i	−0.996322	+	0.0856928i	$0.972690\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	511377.	+	885731.i	0.0981675	+	0.170031i	0.910926	−	0.412569i	$-0.135369\pi$
$83$	511377.	+	885731.i	0.0981675	+	0.170031i	−0.812759	+	0.582601i	$0.802035\pi$
$84$		0			0
$85$	1.22955e6	−	2.12964e6i	0.217160	−	0.376132i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−4.01824e6			−0.604186			−0.302093	−	0.953278i	$-0.597685\pi$
$89$	−4.01824e6			−0.604186			−0.302093	+	0.953278i	$0.597685\pi$
$90$		0			0
$91$	−1.84830e7			−2.57115
$92$		0			0
$93$		0			0
$94$		0			0
$95$	546473.	−	946519.i	0.0653937	−	0.113265i
$96$		0			0
$97$	4.08399e6	+	7.07368e6i	0.454343	+	0.786945i	0.998650	−	0.0519412i	$-0.0165408\pi$
$97$	4.08399e6	+	7.07368e6i	0.454343	+	0.786945i	−0.544307	+	0.838886i	$0.683208\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−4.08931e6	−	7.08288e6i	−0.394934	−	0.684046i	0.598159	−	0.801378i	$-0.295899\pi$
$101$	−4.08931e6	−	7.08288e6i	−0.394934	−	0.684046i	−0.993093	+	0.117332i	$0.962566\pi$
$102$		0			0
$103$	−251113.	+	434941.i	−0.0226433	+	0.0392193i	−0.877125	−	0.480262i	$-0.840542\pi$
$103$	−251113.	+	434941.i	−0.0226433	+	0.0392193i	0.854482	+	0.519481i	$0.173875\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−2.12243e7			−1.67490			−0.837451	−	0.546513i	$-0.815955\pi$
$107$	−2.12243e7			−1.67490			−0.837451	+	0.546513i	$0.815955\pi$
$108$		0			0
$109$	−1.68267e7			−1.24453			−0.622267	−	0.782805i	$-0.713788\pi$
$109$	−1.68267e7			−1.24453			−0.622267	+	0.782805i	$0.713788\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	8.73550e6	−	1.51303e7i	0.569525	−	0.986447i	−0.427088	−	0.904210i	$-0.640460\pi$
$113$	8.73550e6	−	1.51303e7i	0.569525	−	0.986447i	0.996613	−	0.0822364i	$-0.0262063\pi$
$114$		0			0
$115$	−1.40871e6	−	2.43996e6i	−0.0863734	−	0.149603i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−1.66697e7	−	2.88728e7i	−0.906805	−	1.57063i
$120$		0			0
$121$	1.71217e6	−	2.96557e6i	0.0878615	−	0.152181i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−1.65213e7			−0.756589
$126$		0			0
$127$	5.27310e6			0.228430			0.114215	−	0.993456i	$-0.463565\pi$
$127$	5.27310e6			0.228430			0.114215	+	0.993456i	$0.463565\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−5.36227e6	+	9.28772e6i	−0.208401	+	0.360960i	−0.951211	−	0.308542i	$-0.900159\pi$
$131$	−5.36227e6	+	9.28772e6i	−0.208401	+	0.360960i	0.742810	+	0.669502i	$0.233492\pi$
$132$		0			0
$133$	−7.40885e6	−	1.28325e7i	−0.273068	−	0.472967i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−2.03191e7	−	3.51937e7i	−0.675122	−	1.16935i	−0.976433	−	0.215820i	$-0.930757\pi$
$137$	−2.03191e7	−	3.51937e7i	−0.675122	−	1.16935i	0.301311	−	0.953526i	$-0.402576\pi$
$138$		0			0
$139$	−2.16336e7	+	3.74705e7i	−0.683246	+	1.18342i	0.290738	+	0.956803i	$0.406099\pi$
$139$	−2.16336e7	+	3.74705e7i	−0.683246	+	1.18342i	−0.973984	+	0.226615i	$0.927234\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	4.72526e7			1.35129
$144$		0			0
$145$	−1.16490e6			−0.0317323
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−3.46416e6	+	6.00011e6i	−0.0857920	+	0.148596i	−0.905728	−	0.423859i	$-0.860675\pi$
$149$	−3.46416e6	+	6.00011e6i	−0.0857920	+	0.148596i	0.819936	+	0.572455i	$0.194009\pi$
$150$		0			0
$151$	9.57196e6	+	1.65791e7i	0.226246	+	0.391870i	0.956693	−	0.291100i	$-0.0940212\pi$
$151$	9.57196e6	+	1.65791e7i	0.226246	+	0.391870i	−0.730446	+	0.682970i	$0.760688\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	6.65859e6	+	1.15330e7i	0.143622	+	0.248761i
$156$		0			0
$157$	2.87286e7	−	4.97595e7i	0.592470	−	1.02619i	−0.401429	−	0.915890i	$-0.631486\pi$
$157$	2.87286e7	−	4.97595e7i	0.592470	−	1.02619i	0.993899	−	0.110298i	$-0.0351804\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−3.81975e7			−0.721347
$162$		0			0
$163$	−3.30819e7			−0.598321			−0.299160	−	0.954203i	$-0.596707\pi$
$163$	−3.30819e7			−0.598321			−0.299160	+	0.954203i	$0.596707\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	1.54554e7	−	2.67695e7i	0.256786	−	0.444767i	−0.708593	−	0.705617i	$-0.750670\pi$
$167$	1.54554e7	−	2.67695e7i	0.256786	−	0.444767i	0.965379	+	0.260851i	$0.0840031\pi$
$168$		0			0
$169$	−3.81281e7	−	6.60398e7i	−0.607633	−	1.05245i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−9.47336e6	−	1.64083e7i	−0.139105	−	0.240937i	0.788053	−	0.615607i	$-0.211089\pi$
$173$	−9.47336e6	−	1.64083e7i	−0.139105	−	0.240937i	−0.927158	+	0.374670i	$0.877756\pi$
$174$		0			0
$175$	−5.07571e7	+	8.79139e7i	−0.715919	+	1.24001i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	6.35461e6			0.0828139			0.0414070	−	0.999142i	$-0.486816\pi$
$179$	6.35461e6			0.0828139			0.0414070	+	0.999142i	$0.486816\pi$
$180$		0			0
$181$	−2.62225e7			−0.328699			−0.164349	−	0.986402i	$-0.552552\pi$
$181$	−2.62225e7			−0.328699			−0.164349	+	0.986402i	$0.552552\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−3.08307e7	+	5.34003e7i	−0.357999	+	0.620073i
$186$		0			0
$187$	4.26169e7	+	7.38146e7i	0.476580	+	0.825461i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	6.72202e7	+	1.16429e8i	0.698045	+	1.20905i	0.969144	+	0.246497i	$0.0792796\pi$
$191$	6.72202e7	+	1.16429e8i	0.698045	+	1.20905i	−0.271099	+	0.962551i	$0.587387\pi$
$192$		0			0
$193$	5.09264e7	−	8.82072e7i	0.509909	−	0.883188i	−0.490025	−	0.871708i	$-0.663012\pi$
$193$	5.09264e7	−	8.82072e7i	0.509909	−	0.883188i	0.999934	−	0.0114799i	$-0.00365423\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	6.06236e7			0.564950			0.282475	−	0.959275i	$-0.408845\pi$
$197$	6.06236e7			0.564950			0.282475	+	0.959275i	$0.408845\pi$
$198$		0			0
$199$	−8.00356e7			−0.719942			−0.359971	−	0.932963i	$-0.617213\pi$
$199$	−8.00356e7			−0.719942			−0.359971	+	0.932963i	$0.617213\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−7.89664e6	+	1.36774e7i	−0.0662531	+	0.114754i
$204$		0			0
$205$	3.57647e7	+	6.19462e7i	0.289945	+	0.502200i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	1.89411e7	+	3.28069e7i	0.143513	+	0.248572i
$210$		0			0
$211$	7.41096e7	−	1.28362e8i	0.543108	−	0.940690i	−0.455616	−	0.890177i	$-0.650581\pi$
$211$	7.41096e7	−	1.28362e8i	0.543108	−	0.940690i	0.998723	−	0.0505137i	$-0.0160859\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	4.89127e7			0.335650
$216$		0			0
$217$	1.80549e8			1.19946
$218$		0			0
$219$		0			0
$220$		0			0
$221$	1.25368e8	−	2.17143e8i	0.781290	−	1.35323i
$222$		0			0
$223$	4.58867e6	+	7.94780e6i	0.0277089	+	0.0479932i	0.879547	−	0.475811i	$-0.157846\pi$
$223$	4.58867e6	+	7.94780e6i	0.0277089	+	0.0479932i	−0.851838	+	0.523805i	$0.824512\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	1.41938e8	+	2.45843e8i	0.805391	+	1.39498i	0.916027	+	0.401118i	$0.131378\pi$
$227$	1.41938e8	+	2.45843e8i	0.805391	+	1.39498i	−0.110635	+	0.993861i	$0.535289\pi$
$228$		0			0
$229$	1.30295e8	−	2.25678e8i	0.716976	−	1.24184i	−0.245217	−	0.969468i	$-0.578859\pi$
$229$	1.30295e8	−	2.25678e8i	0.716976	−	1.24184i	0.962193	−	0.272370i	$-0.0878075\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	1.77251e8			0.918001			0.459000	−	0.888436i	$-0.348208\pi$
$233$	1.77251e8			0.918001			0.459000	+	0.888436i	$0.348208\pi$
$234$		0			0
$235$	6.90307e7			0.346980
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−1.99948e8	+	3.46321e8i	−0.947382	+	1.64091i	−0.196472	+	0.980509i	$0.562949\pi$
$239$	−1.99948e8	+	3.46321e8i	−0.947382	+	1.64091i	−0.750910	+	0.660405i	$0.770385\pi$
$240$		0			0
$241$	1.48258e8	+	2.56791e8i	0.682275	+	1.18173i	0.974285	+	0.225320i	$0.0723426\pi$
$241$	1.48258e8	+	2.56791e8i	0.682275	+	1.18173i	−0.292010	+	0.956415i	$0.594324\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−9.44754e7	−	1.63636e8i	−0.410428	−	0.710883i
$246$		0			0
$247$	5.57196e7	−	9.65092e7i	0.235271	−	0.407502i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−2.34718e8			−0.936889			−0.468444	−	0.883493i	$-0.655185\pi$
$251$	−2.34718e8			−0.936889			−0.468444	+	0.883493i	$0.655185\pi$
$252$		0			0
$253$	9.76535e7			0.379111
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−1.50908e8	+	2.61381e8i	−0.554559	+	0.960524i	0.443379	+	0.896334i	$0.353780\pi$
$257$	−1.50908e8	+	2.61381e8i	−0.554559	+	0.960524i	−0.997938	+	0.0641897i	$0.979554\pi$
$258$		0			0
$259$	4.17990e8	+	7.23980e8i	1.49492	+	2.58927i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	2.61705e8	+	4.53286e8i	0.887087	+	1.53648i	0.843302	+	0.537439i	$0.180608\pi$
$263$	2.61705e8	+	4.53286e8i	0.887087	+	1.53648i	0.0437851	+	0.999041i	$0.486058\pi$
$264$		0			0
$265$	1.05632e8	−	1.82961e8i	0.348688	−	0.603945i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−5.97138e8			−1.87043			−0.935215	−	0.354080i	$-0.884794\pi$
$269$	−5.97138e8			−1.87043			−0.935215	+	0.354080i	$0.884794\pi$
$270$		0			0
$271$	−5.81954e8			−1.77622			−0.888108	−	0.459635i	$-0.847980\pi$
$271$	−5.81954e8			−1.77622			−0.888108	+	0.459635i	$0.847980\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	1.29763e8	−	2.24756e8i	0.376258	−	0.651698i
$276$		0			0
$277$	1.17522e8	+	2.03555e8i	0.332232	+	0.575442i	0.982949	−	0.183878i	$-0.0588651\pi$
$277$	1.17522e8	+	2.03555e8i	0.332232	+	0.575442i	−0.650717	+	0.759320i	$0.725532\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−2.14334e8	−	3.71238e8i	−0.576261	−	0.998113i	−0.995903	−	0.0904237i	$-0.971178\pi$
$281$	−2.14334e8	−	3.71238e8i	−0.576261	−	0.998113i	0.419642	−	0.907689i	$-0.362155\pi$
$282$		0			0
$283$	9.28504e7	−	1.60822e8i	0.243518	−	0.421786i	−0.718196	−	0.695841i	$-0.755032\pi$
$283$	9.28504e7	−	1.60822e8i	0.243518	−	0.421786i	0.961714	+	0.274055i	$0.0883651\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	9.69765e8			2.42148
$288$		0			0
$289$	4.19347e7			0.102195
$290$		0			0
$291$		0			0
$292$		0			0
$293$	9.01793e7	−	1.56195e8i	0.209445	−	0.362769i	−0.742095	−	0.670295i	$-0.766168\pi$
$293$	9.01793e7	−	1.56195e8i	0.209445	−	0.362769i	0.951540	+	0.307525i	$0.0995009\pi$
$294$		0			0
$295$	−8.95324e7	−	1.55075e8i	−0.203050	−	0.351693i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−1.43635e8	−	2.48784e8i	−0.310751	−	0.538237i
$300$		0			0
$301$	3.31569e8	−	5.74294e8i	0.700795	−	1.21381i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	2.05444e8			0.414614
$306$		0			0
$307$	5.84485e8			1.15289			0.576447	−	0.817135i	$-0.304439\pi$
$307$	5.84485e8			1.15289			0.576447	+	0.817135i	$0.304439\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−2.75970e8	+	4.77994e8i	−0.520236	+	0.901076i	0.479487	+	0.877549i	$0.340823\pi$
$311$	−2.75970e8	+	4.77994e8i	−0.520236	+	0.901076i	−0.999723	+	0.0235267i	$0.992511\pi$
$312$		0			0
$313$	−1.64716e8	−	2.85296e8i	−0.303620	−	0.525885i	0.673333	−	0.739339i	$-0.264862\pi$
$313$	−1.64716e8	−	2.85296e8i	−0.303620	−	0.525885i	−0.976953	+	0.213454i	$0.931529\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	4.60398e8	+	7.97434e8i	0.811758	+	1.40601i	0.911633	+	0.411006i	$0.134822\pi$
$317$	4.60398e8	+	7.97434e8i	0.811758	+	1.40601i	−0.0998750	+	0.995000i	$0.531844\pi$
$318$		0			0
$319$	2.01881e7	−	3.49668e7i	0.0348200	−	0.0603100i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	2.01013e8			0.331906
$324$		0			0
$325$	−7.63456e8			−1.23365
$326$		0			0
$327$		0			0
$328$		0			0
$329$	4.67945e8	−	8.10504e8i	0.724451	−	1.25479i
$330$		0			0
$331$	−2.01926e8	−	3.49746e8i	−0.306052	−	0.530097i	0.671443	−	0.741056i	$-0.265675\pi$
$331$	−2.01926e8	−	3.49746e8i	−0.306052	−	0.530097i	−0.977495	+	0.210959i	$0.932341\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−5.56668e7	−	9.64177e7i	−0.0808983	−	0.140120i
$336$		0			0
$337$	−2.24570e8	+	3.88967e8i	−0.319630	+	0.553616i	−0.980411	−	0.196963i	$-0.936892\pi$
$337$	−2.24570e8	+	3.88967e8i	−0.319630	+	0.553616i	0.660781	+	0.750579i	$0.270225\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−4.61581e8			−0.630387
$342$		0			0
$343$	−1.27066e9			−1.70020
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−4.31294e8	+	7.47023e8i	−0.554141	+	0.959800i	0.443829	+	0.896112i	$0.353620\pi$
$347$	−4.31294e8	+	7.47023e8i	−0.554141	+	0.959800i	−0.997970	+	0.0636888i	$0.979713\pi$
$348$		0			0
$349$	5.51462e6	+	9.55161e6i	0.00694427	+	0.0120278i	0.869477	−	0.493974i	$-0.164456\pi$
$349$	5.51462e6	+	9.55161e6i	0.00694427	+	0.0120278i	−0.862532	+	0.506002i	$0.831123\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−1.38717e8	−	2.40264e8i	−0.167848	−	0.290722i	0.769815	−	0.638267i	$-0.220349\pi$
$353$	−1.38717e8	−	2.40264e8i	−0.167848	−	0.290722i	−0.937663	+	0.347545i	$0.887015\pi$
$354$		0			0
$355$	−2.86776e8	+	4.96710e8i	−0.340207	+	0.589256i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−6.29161e8			−0.717681			−0.358840	−	0.933399i	$-0.616828\pi$
$359$	−6.29161e8			−0.717681			−0.358840	+	0.933399i	$0.616828\pi$
$360$		0			0
$361$	−8.04532e8			−0.900053
$362$		0			0
$363$		0			0
$364$		0			0
$365$	2.49730e8	−	4.32546e8i	0.268810	−	0.465593i
$366$		0			0
$367$	1.74230e8	+	3.01775e8i	0.183989	+	0.318678i	0.943235	−	0.332125i	$-0.107766\pi$
$367$	1.74230e8	+	3.01775e8i	0.183989	+	0.318678i	−0.759247	+	0.650803i	$0.774432\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−1.43212e9	−	2.48051e9i	−1.45603	−	2.52192i
$372$		0			0
$373$	−2.45354e8	+	4.24966e8i	−0.244801	+	0.424007i	−0.962076	−	0.272783i	$-0.912056\pi$
$373$	−2.45354e8	+	4.24966e8i	−0.244801	+	0.424007i	0.717275	+	0.696790i	$0.245389\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−1.18776e8			−0.114165
$378$		0			0
$379$	−1.01844e9			−0.960945			−0.480472	−	0.877010i	$-0.659535\pi$
$379$	−1.01844e9			−0.960945			−0.480472	+	0.877010i	$0.659535\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	8.07126e8	−	1.39798e9i	0.734084	−	1.27147i	−0.221040	−	0.975265i	$-0.570945\pi$
$383$	8.07126e8	−	1.39798e9i	0.734084	−	1.27147i	0.955124	−	0.296206i	$-0.0957215\pi$
$384$		0			0
$385$	3.63257e8	+	6.29180e8i	0.324415	+	0.561904i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	9.57064e8	+	1.65768e9i	0.824361	+	1.42783i	0.902407	+	0.430885i	$0.141798\pi$
$389$	9.57064e8	+	1.65768e9i	0.824361	+	1.42783i	−0.0780463	+	0.996950i	$0.524868\pi$
$390$		0			0
$391$	2.59088e8	−	4.48754e8i	0.219194	−	0.379655i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	4.29649e8			0.350771
$396$		0			0
$397$	−7.45741e8			−0.598166			−0.299083	−	0.954227i	$-0.596681\pi$
$397$	−7.45741e8			−0.598166			−0.299083	+	0.954227i	$0.596681\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−3.74226e8	+	6.48179e8i	−0.289820	+	0.501984i	−0.973767	−	0.227549i	$-0.926929\pi$
$401$	−3.74226e8	+	6.48179e8i	−0.289820	+	0.501984i	0.683946	+	0.729532i	$0.260262\pi$
$402$		0			0
$403$	6.78925e8	+	1.17593e9i	0.516718	+	0.894983i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−1.06861e9	−	1.85089e9i	−0.785667	−	1.36082i
$408$		0			0
$409$	1.12357e9	−	1.94607e9i	0.812021	−	1.40646i	−0.0994264	−	0.995045i	$-0.531701\pi$
$409$	1.12357e9	−	1.94607e9i	0.812021	−	1.40646i	0.911447	−	0.411417i	$-0.134966\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−2.42768e9			−1.69577
$414$		0			0
$415$	−1.18262e8			−0.0812228
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−2.10913e8	+	3.65311e8i	−0.140073	+	0.242613i	−0.927524	−	0.373764i	$-0.878067\pi$
$419$	−2.10913e8	+	3.65311e8i	−0.140073	+	0.242613i	0.787451	+	0.616377i	$0.211400\pi$
$420$		0			0
$421$	−1.24966e8	−	2.16447e8i	−0.0816213	−	0.141372i	0.822325	−	0.569018i	$-0.192676\pi$
$421$	−1.24966e8	−	2.16447e8i	−0.0816213	−	0.141372i	−0.903946	+	0.427646i	$0.859343\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−6.88557e8	−	1.19262e9i	−0.435089	−	0.753597i
$426$		0			0
$427$	1.39266e9	−	2.41216e9i	0.865662	−	1.49937i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−2.15394e8			−0.129587			−0.0647937	−	0.997899i	$-0.520639\pi$
$431$	−2.15394e8			−0.129587			−0.0647937	+	0.997899i	$0.520639\pi$
$432$		0			0
$433$	−2.11963e9			−1.25473			−0.627367	−	0.778724i	$-0.715867\pi$
$433$	−2.11963e9			−1.25473			−0.627367	+	0.778724i	$0.715867\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	1.15152e8	−	1.99449e8i	0.0660063	−	0.114326i
$438$		0			0
$439$	6.33704e8	+	1.09761e9i	0.357487	+	0.619186i	0.987540	−	0.157366i	$-0.0503002\pi$
$439$	6.33704e8	+	1.09761e9i	0.357487	+	0.619186i	−0.630053	+	0.776552i	$0.716967\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	6.69670e8	+	1.15990e9i	0.365972	+	0.633882i	0.988932	−	0.148372i	$-0.0474032\pi$
$443$	6.69670e8	+	1.15990e9i	0.365972	+	0.633882i	−0.622960	+	0.782254i	$0.714070\pi$
$444$		0			0
$445$	2.32317e8	−	4.02385e8i	0.124974	−	0.216462i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	3.22718e9			1.68252			0.841262	−	0.540628i	$-0.181813\pi$
$449$	3.22718e9			1.68252			0.841262	+	0.540628i	$0.181813\pi$
$450$		0			0
$451$	−2.47925e9			−1.27263
$452$		0			0
$453$		0			0
$454$		0			0
$455$	1.06861e9	−	1.85088e9i	0.531836	−	0.921167i
$456$		0			0
$457$	1.25454e9	+	2.17293e9i	0.614863	+	1.06497i	0.990408	+	0.138171i	$0.0441222\pi$
$457$	1.25454e9	+	2.17293e9i	0.614863	+	1.06497i	−0.375545	+	0.926804i	$0.622544\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−1.67493e9	−	2.90107e9i	−0.796240	−	1.37913i	−0.922049	−	0.387074i	$-0.873486\pi$
$461$	−1.67493e9	−	2.90107e9i	−0.796240	−	1.37913i	0.125808	−	0.992055i	$-0.459848\pi$
$462$		0			0
$463$	3.42072e8	−	5.92486e8i	0.160171	−	0.277425i	−0.774759	−	0.632257i	$-0.782129\pi$
$463$	3.42072e8	−	5.92486e8i	0.160171	−	0.277425i	0.934930	+	0.354832i	$0.115462\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−3.17657e9			−1.44328			−0.721638	−	0.692271i	$-0.756610\pi$
$467$	−3.17657e9			−1.44328			−0.721638	+	0.692271i	$0.756610\pi$
$468$		0			0
$469$	−1.50941e9			−0.675621
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−8.47671e8	+	1.46821e9i	−0.368310	+	0.637931i
$474$		0			0
$475$	−3.06029e8	−	5.30058e8i	−0.131019	−	0.226932i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	3.45236e8	+	5.97966e8i	0.143530	+	0.248600i	0.928823	−	0.370523i	$-0.120821\pi$
$479$	3.45236e8	+	5.97966e8i	0.143530	+	0.248600i	−0.785294	+	0.619123i	$0.787488\pi$
$480$		0			0
$481$	−3.14357e9	+	5.44482e9i	−1.28800	+	2.23088i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−9.44474e8			−0.375919
$486$		0			0
$487$	4.75789e9			1.86665			0.933326	−	0.359030i	$-0.116892\pi$
$487$	4.75789e9			1.86665			0.933326	+	0.359030i	$0.116892\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−2.15434e8	+	3.73143e8i	−0.0821351	+	0.142262i	−0.904167	−	0.427179i	$-0.859507\pi$
$491$	−2.15434e8	+	3.73143e8i	−0.0821351	+	0.142262i	0.822032	+	0.569442i	$0.192841\pi$
$492$		0			0
$493$	−1.07124e8	−	1.85544e8i	−0.0402644	−	0.0697400i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	3.88799e9	+	6.73419e9i	1.42062	+	2.46058i
$498$		0			0
$499$	7.11074e8	−	1.23162e9i	0.256190	−	0.443735i	−0.709028	−	0.705181i	$-0.750866\pi$
$499$	7.11074e8	−	1.23162e9i	0.256190	−	0.443735i	0.965218	+	0.261446i	$0.0841992\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	9.03048e8			0.316390			0.158195	−	0.987408i	$-0.449433\pi$
$503$	9.03048e8			0.316390			0.158195	+	0.987408i	$0.449433\pi$
$504$		0			0
$505$	9.45703e8			0.326765
$506$		0			0
$507$		0			0
$508$		0			0
$509$	9.02477e8	−	1.56314e9i	0.303336	−	0.525393i	−0.673553	−	0.739139i	$-0.735233\pi$
$509$	9.02477e8	−	1.56314e9i	0.303336	−	0.525393i	0.976889	+	0.213745i	$0.0685662\pi$
$510$		0			0
$511$	−3.38574e9	−	5.86427e9i	−1.12248	−	1.94420i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−2.90366e7	−	5.02928e7i	−0.00936742	−	0.0162248i
$516$		0			0
$517$	−1.19632e9	+	2.07209e9i	−0.380742	+	0.659465i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	2.14791e9			0.665402			0.332701	−	0.943032i	$-0.392040\pi$
$521$	2.14791e9			0.665402			0.332701	+	0.943032i	$0.392040\pi$
$522$		0			0
$523$	1.14615e9			0.350337			0.175169	−	0.984538i	$-0.443953\pi$
$523$	1.14615e9			0.350337			0.175169	+	0.984538i	$0.443953\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−1.22464e9	+	2.12113e9i	−0.364477	+	0.631293i
$528$		0			0
$529$	1.40557e9	+	2.43452e9i	0.412818	+	0.715021i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	3.64665e9	+	6.31618e9i	1.04315	+	1.80680i
$534$		0			0
$535$	1.22709e9	−	2.12539e9i	0.346449	−	0.600068i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	6.54914e9			1.80146
$540$		0			0
$541$	2.83440e9			0.769611			0.384805	−	0.922998i	$-0.374269\pi$
$541$	2.83440e9			0.769611			0.384805	+	0.922998i	$0.374269\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	9.72847e8	−	1.68502e9i	0.257429	−	0.445879i
$546$		0			0
$547$	−2.79292e9	−	4.83747e9i	−0.729629	−	1.26375i	−0.957040	−	0.289956i	$-0.906359\pi$
$547$	−2.79292e9	−	4.83747e9i	−0.729629	−	1.26375i	0.227411	−	0.973799i	$-0.426974\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−4.76111e7	−	8.24648e7i	−0.0121249	−	0.0210009i
$552$		0			0
$553$	2.91250e9	−	5.04460e9i	0.732365	−	1.26849i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	7.96757e8			0.195359			0.0976793	−	0.995218i	$-0.468858\pi$
$557$	7.96757e8			0.195359			0.0976793	+	0.995218i	$0.468858\pi$
$558$		0			0
$559$	4.98725e9			1.20759
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−2.17886e9	+	3.77389e9i	−0.514576	+	0.891272i	0.485281	+	0.874358i	$0.338717\pi$
$563$	−2.17886e9	+	3.77389e9i	−0.514576	+	0.891272i	−0.999857	+	0.0169139i	$0.994616\pi$
$564$		0			0
$565$	1.01010e9	+	1.74954e9i	0.235610	+	0.408088i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−1.20460e9	−	2.08643e9i	−0.274127	−	0.474801i	0.695788	−	0.718247i	$-0.255055\pi$
$569$	−1.20460e9	−	2.08643e9i	−0.274127	−	0.474801i	−0.969914	+	0.243446i	$0.921722\pi$
$570$		0			0
$571$	−1.43211e9	+	2.48048e9i	−0.321921	+	0.557583i	−0.980884	−	0.194592i	$-0.937662\pi$
$571$	−1.43211e9	+	2.48048e9i	−0.321921	+	0.557583i	0.658964	+	0.752175i	$0.270995\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−1.57778e9			−0.346106
$576$		0			0
$577$	4.63688e9			1.00487			0.502436	−	0.864614i	$-0.332437\pi$
$577$	4.63688e9			1.00487			0.502436	+	0.864614i	$0.332437\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−8.01676e8	+	1.38854e9i	−0.169583	+	0.293726i
$582$		0			0
$583$	3.66128e9	+	6.34153e9i	0.765232	+	1.32542i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−2.56255e9	−	4.43847e9i	−0.522925	−	0.905732i	−0.999644	−	0.0266765i	$-0.991508\pi$
$587$	−2.56255e9	−	4.43847e9i	−0.522925	−	0.905732i	0.476720	−	0.879055i	$-0.341826\pi$
$588$		0			0
$589$	−5.44290e8	+	9.42738e8i	−0.109756	+	0.190102i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−3.56850e9			−0.702739			−0.351369	−	0.936237i	$-0.614284\pi$
$593$	−3.56850e9			−0.702739			−0.351369	+	0.936237i	$0.614284\pi$
$594$		0			0
$595$	3.85508e9			0.750281
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−2.56710e9	+	4.44635e9i	−0.488033	+	0.845299i	−0.999905	−	0.0137631i	$-0.995619\pi$
$599$	−2.56710e9	+	4.44635e9i	−0.488033	+	0.845299i	0.511872	+	0.859062i	$0.328952\pi$
$600$		0			0
$601$	2.21338e9	+	3.83368e9i	0.415905	+	0.720369i	0.995523	−	0.0945194i	$-0.0301314\pi$
$601$	2.21338e9	+	3.83368e9i	0.415905	+	0.720369i	−0.579618	+	0.814889i	$0.696798\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	1.97981e8	+	3.42913e8i	0.0363479	+	0.0629564i
$606$		0			0
$607$	−9.82949e8	+	1.70252e9i	−0.178390	+	0.308981i	−0.941329	−	0.337490i	$-0.890422\pi$
$607$	−9.82949e8	+	1.70252e9i	−0.178390	+	0.308981i	0.762939	+	0.646470i	$0.223756\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	7.03853e9			1.24835
$612$		0			0
$613$	−1.88232e9			−0.330051			−0.165026	−	0.986289i	$-0.552771\pi$
$613$	−1.88232e9			−0.330051			−0.165026	+	0.986289i	$0.552771\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	5.20439e7	−	9.01427e7i	0.00892014	−	0.0154501i	−0.861531	−	0.507705i	$-0.830494\pi$
$617$	5.20439e7	−	9.01427e7i	0.00892014	−	0.0154501i	0.870451	+	0.492255i	$0.163827\pi$
$618$		0			0
$619$	5.38259e9	+	9.32291e9i	0.912166	+	1.57992i	0.810999	+	0.585048i	$0.198924\pi$
$619$	5.38259e9	+	9.32291e9i	0.912166	+	1.57992i	0.101167	+	0.994869i	$0.467742\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−3.14966e9	−	5.45536e9i	−0.521861	−	0.903890i
$624$		0			0
$625$	−1.57428e9	+	2.72673e9i	−0.257930	+	0.446747i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−1.13407e10			−1.81703
$630$		0			0
$631$	−5.58223e8			−0.0884514			−0.0442257	−	0.999022i	$-0.514082\pi$
$631$	−5.58223e8			−0.0884514			−0.0442257	+	0.999022i	$0.514082\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−3.04868e8	+	5.28046e8i	−0.0472501	+	0.0818396i
$636$		0			0
$637$	−9.63292e9	−	1.66847e10i	−1.47662	−	2.55759i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	5.15081e9	+	8.92146e9i	0.772453	+	1.33793i	0.936215	+	0.351428i	$0.114304\pi$
$641$	5.15081e9	+	8.92146e9i	0.772453	+	1.33793i	−0.163762	+	0.986500i	$0.552363\pi$
$642$		0			0
$643$	3.06381e9	−	5.30667e9i	0.454489	−	0.787197i	−0.544170	−	0.838975i	$-0.683155\pi$
$643$	3.06381e9	−	5.30667e9i	0.454489	−	0.787197i	0.998659	+	0.0517777i	$0.0164887\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	2.28533e9			0.331730			0.165865	−	0.986148i	$-0.446958\pi$
$647$	2.28533e9			0.331730			0.165865	+	0.986148i	$0.446958\pi$
$648$		0			0
$649$	6.20648e9			0.891229
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−1.33765e8	+	2.31689e8i	−0.0187996	+	0.0325618i	−0.875272	−	0.483631i	$-0.839318\pi$
$653$	−1.33765e8	+	2.31689e8i	−0.0187996	+	0.0325618i	0.856473	+	0.516193i	$0.172651\pi$
$654$		0			0
$655$	−6.20046e8	−	1.07395e9i	−0.0862143	−	0.149327i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−3.69424e8	−	6.39860e8i	−0.0502835	−	0.0870936i	0.839788	−	0.542914i	$-0.182679\pi$
$659$	−3.69424e8	−	6.39860e8i	−0.0502835	−	0.0870936i	−0.890072	+	0.455821i	$0.849346\pi$
$660$		0			0
$661$	3.43258e9	−	5.94541e9i	0.462292	−	0.800713i	−0.536783	−	0.843720i	$-0.680361\pi$
$661$	3.43258e9	−	5.94541e9i	0.462292	−	0.800713i	0.999075	+	0.0430075i	$0.0136939\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	1.71339e9			0.225933
$666$		0			0
$667$	−2.45466e8			−0.0320296
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−3.56040e9	+	6.16680e9i	−0.454957	+	0.788009i
$672$		0			0
$673$	−7.65777e8	−	1.32637e9i	−0.0968389	−	0.167730i	0.813536	−	0.581515i	$-0.197540\pi$
$673$	−7.65777e8	−	1.32637e9i	−0.0968389	−	0.167730i	−0.910375	+	0.413785i	$0.864206\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	3.29185e9	+	5.70166e9i	0.407737	+	0.706221i	0.994636	−	0.103439i	$-0.0329846\pi$
$677$	3.29185e9	+	5.70166e9i	0.407737	+	0.706221i	−0.586899	+	0.809660i	$0.699651\pi$
$678$		0			0
$679$	−6.40239e9	+	1.10893e10i	−0.784871	+	1.35944i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	8.77936e9			1.05436			0.527182	−	0.849752i	$-0.323249\pi$
$683$	8.77936e9			1.05436			0.527182	+	0.849752i	$0.323249\pi$
$684$		0			0
$685$	4.69905e9			0.558590
$686$		0			0
$687$		0			0
$688$		0			0
$689$	1.07705e10	−	1.86551e10i	1.25450	−	2.17285i
$690$		0			0
$691$	−1.77165e9	−	3.06858e9i	−0.204270	−	0.353805i	0.745630	−	0.666360i	$-0.232149\pi$
$691$	−1.77165e9	−	3.06858e9i	−0.204270	−	0.353805i	−0.949900	+	0.312555i	$0.898815\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−2.50152e9	−	4.33277e9i	−0.282656	−	0.489574i
$696$		0			0
$697$	−6.57778e9	+	1.13931e10i	−0.735809	+	1.27446i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−5.36664e9			−0.588422			−0.294211	−	0.955740i	$-0.595057\pi$
$701$	−5.36664e9			−0.588422			−0.294211	+	0.955740i	$0.595057\pi$
$702$		0			0
$703$	−5.04036e9			−0.547164
$704$		0			0
$705$		0			0
$706$		0			0
$707$	6.41072e9	−	1.11037e10i	0.682243	−	1.18168i
$708$		0			0
$709$	−3.12434e9	−	5.41151e9i	−0.329228	−	0.570239i	0.653131	−	0.757245i	$-0.273455\pi$
$709$	−3.12434e9	−	5.41151e9i	−0.329228	−	0.570239i	−0.982359	+	0.187006i	$0.940122\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	1.40308e9	+	2.43021e9i	0.144967	+	0.251091i
$714$		0			0
$715$	−2.73194e9	+	4.73186e9i	−0.279512	+	0.484128i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−7.84824e9			−0.787446			−0.393723	−	0.919229i	$-0.628813\pi$
$719$	−7.84824e9			−0.787446			−0.393723	+	0.919229i	$0.628813\pi$
$720$		0			0
$721$	−7.87331e8			−0.0782319
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−3.26178e8	+	5.64956e8i	−0.0317886	+	0.0550594i
$726$		0			0
$727$	−4.60370e9	−	7.97385e9i	−0.444362	−	0.769657i	0.553646	−	0.832752i	$-0.313236\pi$
$727$	−4.60370e9	−	7.97385e9i	−0.444362	−	0.769657i	−0.998008	+	0.0630950i	$0.979903\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	4.49797e9	+	7.79072e9i	0.425898	+	0.737678i
$732$		0			0
$733$	4.21455e9	−	7.29981e9i	0.395264	−	0.684617i	−0.597871	−	0.801592i	$-0.703986\pi$
$733$	4.21455e9	−	7.29981e9i	0.395264	−	0.684617i	0.993135	+	0.116975i	$0.0373198\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	3.85888e9			0.355079
$738$		0			0
$739$	−2.02547e10			−1.84616			−0.923081	−	0.384605i	$-0.874338\pi$
$739$	−2.02547e10			−1.84616			−0.923081	+	0.384605i	$0.874338\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−7.07235e9	+	1.22497e10i	−0.632562	+	1.09563i	0.354464	+	0.935070i	$0.384663\pi$
$743$	−7.07235e9	+	1.22497e10i	−0.632562	+	1.09563i	−0.987026	+	0.160560i	$0.948670\pi$
$744$		0			0
$745$	−4.00566e8	−	6.93800e8i	−0.0354917	−	0.0614734i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−1.66364e10	−	2.88151e10i	−1.44668	−	2.50573i
$750$		0			0
$751$	−6.55445e9	+	1.13526e10i	−0.564672	+	0.978041i	0.432408	+	0.901678i	$0.357664\pi$
$751$	−6.55445e9	+	1.13526e10i	−0.564672	+	0.978041i	−0.997080	+	0.0763631i	$0.975669\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−2.21364e9			−0.187194
$756$		0			0
$757$	−8.93042e9			−0.748232			−0.374116	−	0.927382i	$-0.622054\pi$
$757$	−8.93042e9			−0.748232			−0.374116	+	0.927382i	$0.622054\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	1.59439e9	−	2.76157e9i	0.131144	−	0.227148i	−0.792974	−	0.609256i	$-0.791468\pi$
$761$	1.59439e9	−	2.76157e9i	0.131144	−	0.227148i	0.924118	+	0.382107i	$0.124802\pi$
$762$		0			0
$763$	−1.31895e10	−	2.28448e10i	−1.07496	−	1.86188i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−9.12892e9	−	1.58118e10i	−0.730526	−	1.26531i
$768$		0			0
$769$	−8.87387e9	+	1.53700e10i	−0.703673	+	1.21880i	0.263496	+	0.964661i	$0.415124\pi$
$769$	−8.87387e9	+	1.53700e10i	−0.703673	+	1.21880i	−0.967168	+	0.254136i	$0.918209\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−2.20141e10			−1.71424			−0.857121	−	0.515116i	$-0.827749\pi$
$773$	−2.20141e10			−1.71424			−0.857121	+	0.515116i	$0.827749\pi$
$774$		0			0
$775$	7.45772e9			0.575507
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−2.92349e9	+	5.06364e9i	−0.221575	+	0.383780i
$780$		0			0
$781$	−9.93981e9	−	1.72163e10i	−0.746620	−	1.29318i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	3.32193e9	+	5.75375e9i	0.245102	+	0.424529i
$786$		0			0
$787$	−1.14141e10	+	1.97698e10i	−0.834698	+	1.44574i	0.0595780	+	0.998224i	$0.481024\pi$
$787$	−1.14141e10	+	1.97698e10i	−0.834698	+	1.44574i	−0.894276	+	0.447516i	$0.852309\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	2.73889e10			1.96769
$792$		0			0
$793$	2.09475e10			1.49168
$794$		0			0
$795$		0			0
$796$		0			0
$797$	1.76593e9	−	3.05868e9i	0.123558	−	0.214008i	−0.797611	−	0.603173i	$-0.793903\pi$
$797$	1.76593e9	−	3.05868e9i	0.123558	−	0.214008i	0.921168	+	0.389165i	$0.127236\pi$
$798$		0			0
$799$	6.34801e9	+	1.09951e10i	0.440275	+	0.762578i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	8.65579e9	+	1.49923e10i	0.589933	+	1.02179i
$804$		0			0
$805$	2.20841e9	−	3.82508e9i	0.149209	−	0.258437i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−1.56526e10			−1.03936			−0.519682	−	0.854360i	$-0.673950\pi$
$809$	−1.56526e10			−1.03936			−0.519682	+	0.854360i	$0.673950\pi$
$810$		0			0
$811$	2.25921e10			1.48725			0.743624	−	0.668598i	$-0.233105\pi$
$811$	2.25921e10			1.48725			0.743624	+	0.668598i	$0.233105\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	1.91265e9	−	3.31281e9i	0.123761	−	0.214361i
$816$		0			0
$817$	1.99912e9	+	3.46258e9i	0.128251	+	0.222138i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−1.42536e10	−	2.46880e10i	−0.898927	−	1.55699i	−0.828868	−	0.559444i	$-0.811015\pi$
$821$	−1.42536e10	−	2.46880e10i	−0.898927	−	1.55699i	−0.0700584	−	0.997543i	$-0.522319\pi$
$822$		0			0
$823$	5.06244e9	−	8.76841e9i	0.316563	−	0.548304i	−0.663205	−	0.748438i	$-0.730804\pi$
$823$	5.06244e9	−	8.76841e9i	0.316563	−	0.548304i	0.979769	+	0.200134i	$0.0641377\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−1.36727e10			−0.840592			−0.420296	−	0.907387i	$-0.638074\pi$
$827$	−1.36727e10			−0.840592			−0.420296	+	0.907387i	$0.638074\pi$
$828$		0			0
$829$	−1.29537e10			−0.789683			−0.394842	−	0.918749i	$-0.629201\pi$
$829$	−1.29537e10			−0.789683			−0.394842	+	0.918749i	$0.629201\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	1.73758e10	−	3.00957e10i	1.04156	−	1.80404i
$834$		0			0
$835$	1.78712e9	+	3.09539e9i	0.106231	+	0.183998i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	4.37464e9	+	7.57710e9i	0.255726	+	0.442931i	0.965093	−	0.261909i	$-0.0843521\pi$
$839$	4.37464e9	+	7.57710e9i	0.255726	+	0.442931i	−0.709366	+	0.704840i	$0.751019\pi$
$840$		0			0
$841$	8.57419e9	−	1.48509e10i	0.497058	−	0.860930i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	8.81760e9			0.502750
$846$		0			0
$847$	5.36828e9			0.303559
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−6.49658e9	+	1.12524e10i	−0.361353	+	0.625882i
$852$		0			0
$853$	−5.70790e9	−	9.88638e9i	−0.314887	−	0.545400i	0.664526	−	0.747265i	$-0.268633\pi$
$853$	−5.70790e9	−	9.88638e9i	−0.314887	−	0.545400i	−0.979413	+	0.201864i	$0.935300\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	3.27396e9	+	5.67066e9i	0.177681	+	0.307752i	0.941086	−	0.338168i	$-0.109807\pi$
$857$	3.27396e9	+	5.67066e9i	0.177681	+	0.307752i	−0.763405	+	0.645920i	$0.776474\pi$
$858$		0			0
$859$	−1.18486e10	+	2.05224e10i	−0.637811	+	1.10472i	0.348102	+	0.937457i	$0.386826\pi$
$859$	−1.18486e10	+	2.05224e10i	−0.637811	+	1.10472i	−0.985912	+	0.167263i	$0.946507\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	3.46394e9			0.183457			0.0917283	−	0.995784i	$-0.470761\pi$
$863$	3.46394e9			0.183457			0.0917283	+	0.995784i	$0.470761\pi$
$864$		0			0
$865$	2.19083e9			0.115094
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−7.44593e9	+	1.28967e10i	−0.384902	+	0.666669i
$870$		0			0
$871$	−5.67591e9	−	9.83096e9i	−0.291053	−	0.504118i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−1.29501e10	−	2.24302e10i	−0.653498	−	1.13189i
$876$		0			0
$877$	−9.96986e9	+	1.72683e10i	−0.499103	+	0.864472i	−0.999999	−	0.00103527i	$-0.999670\pi$
$877$	−9.96986e9	+	1.72683e10i	−0.499103	+	0.864472i	0.500896	+	0.865507i	$0.333004\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−4.41512e9			−0.217534			−0.108767	−	0.994067i	$-0.534690\pi$
$881$	−4.41512e9			−0.217534			−0.108767	+	0.994067i	$0.534690\pi$
$882$		0			0
$883$	−9.26148e9			−0.452708			−0.226354	−	0.974045i	$-0.572681\pi$
$883$	−9.26148e9			−0.452708			−0.226354	+	0.974045i	$0.572681\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	2.44968e9	−	4.24297e9i	0.117863	−	0.204144i	−0.801058	−	0.598587i	$-0.795729\pi$
$887$	2.44968e9	−	4.24297e9i	0.117863	−	0.204144i	0.918921	+	0.394443i	$0.129062\pi$
$888$		0			0
$889$	4.13327e9	+	7.15903e9i	0.197305	+	0.341742i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	2.82137e9	+	4.88676e9i	0.132581	+	0.229636i
$894$		0			0
$895$	−3.67396e8	+	6.36348e8i	−0.0171299	+	0.0296698i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	1.16025e9			0.0532590
$900$		0			0
$901$	3.88555e10			1.76977
$902$		0			0
$903$		0			0
$904$		0			0
$905$	1.51607e9	−	2.62591e9i	0.0679906	−	0.117763i
$906$		0			0
$907$	2.81935e9	+	4.88325e9i	0.125465	+	0.217312i	0.921915	−	0.387393i	$-0.126624\pi$
$907$	2.81935e9	+	4.88325e9i	0.125465	+	0.217312i	−0.796449	+	0.604705i	$0.793291\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	1.53446e10	+	2.65776e10i	0.672419	+	1.16466i	0.977216	+	0.212247i	$0.0680781\pi$
$911$	1.53446e10	+	2.65776e10i	0.672419	+	1.16466i	−0.304797	+	0.952417i	$0.598589\pi$
$912$		0			0
$913$	2.04952e9	−	3.54987e9i	0.0891260	−	0.154371i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−1.68126e10			−0.720018
$918$		0			0
$919$	−1.56839e10			−0.666575			−0.333287	−	0.942825i	$-0.608158\pi$
$919$	−1.56839e10			−0.666575			−0.333287	+	0.942825i	$0.608158\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−2.92403e10	+	5.06457e10i	−1.22398	+	2.12000i
$924$		0			0
$925$	1.72654e10	+	2.99046e10i	0.717268	+	1.24234i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	2.41333e9	+	4.18000e9i	0.0987554	+	0.171049i	0.911170	−	0.412031i	$-0.135181\pi$
$929$	2.41333e9	+	4.18000e9i	0.0987554	+	0.171049i	−0.812414	+	0.583081i	$0.801847\pi$
$930$		0			0
$931$	7.72265e9	−	1.33760e10i	0.313648	−	0.543255i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−9.85569e9			−0.394318
$936$		0			0
$937$	−1.18345e9			−0.0469959			−0.0234980	−	0.999724i	$-0.507480\pi$
$937$	−1.18345e9			−0.0469959			−0.0234980	+	0.999724i	$0.507480\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−1.39947e10	+	2.42395e10i	−0.547518	+	0.948329i	0.450926	+	0.892562i	$0.351094\pi$
$941$	−1.39947e10	+	2.42395e10i	−0.547518	+	0.948329i	−0.998444	+	0.0557678i	$0.982239\pi$
$942$		0			0
$943$	7.53626e9	+	1.30532e10i	0.292661	+	0.506904i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	4.72685e8	+	8.18714e8i	0.0180862	+	0.0313262i	0.874927	−	0.484255i	$-0.160909\pi$
$947$	4.72685e8	+	8.18714e8i	0.0180862	+	0.0313262i	−0.856841	+	0.515581i	$0.827576\pi$
$948$		0			0
$949$	2.54631e10	−	4.41033e10i	0.967117	−	1.67510i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	4.97383e10			1.86151			0.930755	−	0.365643i	$-0.119151\pi$
$953$	4.97383e10			1.86151			0.930755	+	0.365643i	$0.119151\pi$
$954$		0			0
$955$	−1.55455e10			−0.577555
$956$		0			0
$957$		0			0
$958$		0			0
$959$	3.18539e10	−	5.51725e10i	1.16626	−	2.02003i
$960$		0			0
$961$	7.12432e9	+	1.23397e10i	0.258947	+	0.448510i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	5.88869e9	+	1.01995e10i	0.210947	+	0.365371i
$966$		0			0
$967$	2.27616e10	−	3.94242e10i	0.809486	−	1.40207i	−0.103735	−	0.994605i	$-0.533079\pi$
$967$	2.27616e10	−	3.94242e10i	0.809486	−	1.40207i	0.913221	−	0.407465i	$-0.133587\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	3.32399e10			1.16518			0.582589	−	0.812767i	$-0.302040\pi$
$971$	3.32399e10			1.16518			0.582589	+	0.812767i	$0.302040\pi$
$972$		0			0
$973$	−6.78292e10			−2.36060
$974$		0			0
$975$		0			0
$976$		0			0
$977$	5.04057e9	−	8.73052e9i	0.172921	−	0.299508i	−0.766519	−	0.642222i	$-0.778013\pi$
$977$	5.04057e9	−	8.73052e9i	0.172921	−	0.299508i	0.939440	+	0.342714i	$0.111346\pi$
$978$		0			0
$979$	8.05224e9	+	1.39469e10i	0.274269	+	0.475048i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	7.77753e9	+	1.34711e10i	0.261159	+	0.452340i	0.966550	−	0.256478i	$-0.0825621\pi$
$983$	7.77753e9	+	1.34711e10i	0.261159	+	0.452340i	−0.705391	+	0.708818i	$0.749229\pi$
$984$		0			0
$985$	−3.50499e9	+	6.07082e9i	−0.116858	+	0.202405i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	1.03068e10			0.338795
$990$		0			0
$991$	3.35662e10			1.09558			0.547791	−	0.836615i	$-0.315469\pi$
$991$	3.35662e10			1.09558			0.547791	+	0.836615i	$0.315469\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	4.62731e9	−	8.01474e9i	0.148918	−	0.257934i
$996$		0			0
$997$	−1.98765e10	−	3.44270e10i	−0.635193	−	1.10019i	−0.986474	−	0.163917i	$-0.947587\pi$
$997$	−1.98765e10	−	3.44270e10i	−0.635193	−	1.10019i	0.351281	−	0.936270i	$-0.385746\pi$
$998$		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.145.4		20
3.2	odd	2		72.8.i.a.49.5	yes	20
4.3	odd	2		432.8.i.e.145.4		20
9.2	odd	6		72.8.i.a.25.5	✓	20
9.7	even	3	inner	216.8.i.a.73.4		20
12.11	even	2		144.8.i.e.49.6		20
36.7	odd	6		432.8.i.e.289.4		20
36.11	even	6		144.8.i.e.97.6		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.8.i.a.25.5	✓	20	9.2	odd	6
72.8.i.a.49.5	yes	20	3.2	odd	2
144.8.i.e.49.6		20	12.11	even	2
144.8.i.e.97.6		20	36.11	even	6
216.8.i.a.73.4		20	9.7	even	3	inner
216.8.i.a.145.4		20	1.1	even	1	trivial
432.8.i.e.145.4		20	4.3	odd	2
432.8.i.e.289.4		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)

\(f(q)\)	\(=\)	\(q+(-57.8156 + 100.140i) q^{5} +(783.840 + 1357.65i) q^{7} +O(q^{10})\)	\(q+(-57.8156 + 100.140i) q^{5} +(783.840 + 1357.65i) q^{7} +(-2003.92 - 3470.90i) q^{11} +(-5895.01 + 10210.5i) q^{13} -21266.7 q^{17} -9451.99 q^{19} +(-12182.8 + 21101.2i) q^{23} +(32377.2 + 56079.0i) q^{25} +(5037.15 + 8724.60i) q^{29} +(57584.7 - 99739.6i) q^{31} -181273. q^{35} +533259. q^{37} +(309299. - 535722. i) q^{41} +(-211503. - 366334. i) q^{43} +(-298495. - 517008. i) q^{47} +(-817040. + 1.41515e6i) q^{49} -1.82706e6 q^{53} +463432. q^{55} +(-774292. + 1.34111e6i) q^{59} +(-888359. - 1.53868e6i) q^{61} +(-681648. - 1.18065e6i) q^{65} +(-481416. + 833837. i) q^{67} +4.96018e6 q^{71} -4.31942e6 q^{73} +(3.14151e6 - 5.44126e6i) q^{77} +(-1.85784e6 - 3.21787e6i) q^{79} +(511377. + 885731. i) q^{83} +(1.22955e6 - 2.12964e6i) q^{85} -4.01824e6 q^{89} -1.84830e7 q^{91} +(546473. - 946519. i) q^{95} +(4.08399e6 + 7.07368e6i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 125 q^{5} + 1245 q^{7}+O(q^{10}) \) 20 * q + 125 * q^5 + 1245 * q^7	\( 20 q + 125 q^{5} + 1245 q^{7} - 6106 q^{11} + 4937 q^{13} - 48722 q^{17} - 26882 q^{19} - 19387 q^{23} - 218957 q^{25} + 46791 q^{29} - 185039 q^{31} - 83094 q^{35} + 108420 q^{37} + 638112 q^{41} + 892628 q^{43} + 230883 q^{47} - 1034741 q^{49} + 2872940 q^{53} + 1089998 q^{55} - 2172454 q^{59} - 1878325 q^{61} - 1239133 q^{65} + 531496 q^{67} - 3723056 q^{71} - 1804522 q^{73} - 6276543 q^{77} + 3607847 q^{79} - 10794491 q^{83} + 3597658 q^{85} - 32214888 q^{89} - 16117530 q^{91} + 756868 q^{95} + 17951260 q^{97}+O(q^{100}) \) 20 * q + 125 * q^5 + 1245 * q^7 - 6106 * q^11 + 4937 * q^13 - 48722 * q^17 - 26882 * q^19 - 19387 * q^23 - 218957 * q^25 + 46791 * q^29 - 185039 * q^31 - 83094 * q^35 + 108420 * q^37 + 638112 * q^41 + 892628 * q^43 + 230883 * q^47 - 1034741 * q^49 + 2872940 * q^53 + 1089998 * q^55 - 2172454 * q^59 - 1878325 * q^61 - 1239133 * q^65 + 531496 * q^67 - 3723056 * q^71 - 1804522 * q^73 - 6276543 * q^77 + 3607847 * q^79 - 10794491 * q^83 + 3597658 * q^85 - 32214888 * q^89 - 16117530 * q^91 + 756868 * q^95 + 17951260 * q^97

Introduction

L-functions

Modular forms

Varieties

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Database

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