LMFDB - Embedded newform 216.8.i.a.145.1

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			145.1
Root			$0.500000 - 315.199i$ of defining polynomial
Character	$\chi$	$=$	216.145
Dual form			216.8.i.a.73.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-266.720 + 461.973i) q^{5} +(-66.5594 - 115.284i) q^{7} +O(q^{10})$	$q+(-266.720 + 461.973i) q^{5} +(-66.5594 - 115.284i) q^{7} +(-731.541 - 1267.07i) q^{11} +(1045.78 - 1811.34i) q^{13} -21961.2 q^{17} -34405.2 q^{19} +(-32866.5 + 56926.4i) q^{23} +(-103217. - 178777. i) q^{25} +(74878.5 + 129693. i) q^{29} +(17144.3 - 29694.7i) q^{31} +71010.9 q^{35} -219233. q^{37} +(371671. - 643752. i) q^{41} +(246801. + 427472. i) q^{43} +(661565. + 1.14586e6i) q^{47} +(402911. - 697863. i) q^{49} +421155. q^{53} +780467. q^{55} +(1.23603e6 - 2.14086e6i) q^{59} +(-693435. - 1.20107e6i) q^{61} +(557861. + 966244. i) q^{65} +(-447311. + 774766. i) q^{67} -2.89163e6 q^{71} +115265. q^{73} +(-97381.9 + 168670. i) q^{77} +(-2.15801e6 - 3.73779e6i) q^{79} +(-2.20388e6 - 3.81724e6i) q^{83} +(5.85749e6 - 1.01455e7i) q^{85} -3.58129e6 q^{89} -278426. q^{91} +(9.17655e6 - 1.58943e7i) q^{95} +(4.45354e6 + 7.71376e6i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−266.720	+	461.973i	−0.954247	+	1.65281i	−0.218167	+	0.975912i	$0.570008\pi$
$5$	−266.720	+	461.973i	−0.954247	+	1.65281i	−0.736081	+	0.676894i	$0.763326\pi$
$6$		0			0
$7$	−66.5594	−	115.284i	−0.0733443	−	0.127036i	0.827021	−	0.562171i	$-0.190034\pi$
$7$	−66.5594	−	115.284i	−0.0733443	−	0.127036i	−0.900365	+	0.435135i	$0.856701\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−731.541	−	1267.07i	−0.165716	−	0.287028i	0.771193	−	0.636601i	$-0.219660\pi$
$11$	−731.541	−	1267.07i	−0.165716	−	0.287028i	−0.936909	+	0.349573i	$0.886327\pi$
$12$		0			0
$13$	1045.78	−	1811.34i	0.132020	−	0.228665i	−0.792435	−	0.609956i	$-0.791187\pi$
$13$	1045.78	−	1811.34i	0.132020	−	0.228665i	0.924455	+	0.381291i	$0.124520\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−21961.2			−1.08414			−0.542069	−	0.840334i	$-0.682359\pi$
$17$	−21961.2			−1.08414			−0.542069	+	0.840334i	$0.682359\pi$
$18$		0			0
$19$	−34405.2			−1.15076			−0.575382	−	0.817885i	$-0.695146\pi$
$19$	−34405.2			−1.15076			−0.575382	+	0.817885i	$0.695146\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−32866.5	+	56926.4i	−0.563256	+	0.975588i	0.433954	+	0.900935i	$0.357118\pi$
$23$	−32866.5	+	56926.4i	−0.563256	+	0.975588i	−0.997210	+	0.0746525i	$0.976215\pi$
$24$		0			0
$25$	−103217.	−	178777.i	−1.32118	−	2.28834i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	74878.5	+	129693.i	0.570118	+	0.987473i	0.996553	+	0.0829547i	$0.0264357\pi$
$29$	74878.5	+	129693.i	0.570118	+	0.987473i	−0.426436	+	0.904518i	$0.640231\pi$
$30$		0			0
$31$	17144.3	−	29694.7i	0.103360	−	0.179025i	−0.809707	−	0.586834i	$-0.800374\pi$
$31$	17144.3	−	29694.7i	0.103360	−	0.179025i	0.913067	+	0.407810i	$0.133707\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	71010.9			0.279954
$36$		0			0
$37$	−219233.			−0.711542			−0.355771	−	0.934573i	$-0.615782\pi$
$37$	−219233.			−0.711542			−0.355771	+	0.934573i	$0.615782\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	371671.	−	643752.i	0.842199	−	1.45873i	−0.0458325	−	0.998949i	$-0.514594\pi$
$41$	371671.	−	643752.i	0.842199	−	1.45873i	0.888032	−	0.459782i	$-0.152073\pi$
$42$		0			0
$43$	246801.	+	427472.i	0.473377	+	0.819913i	0.999536	−	0.0304733i	$-0.00970144\pi$
$43$	246801.	+	427472.i	0.473377	+	0.819913i	−0.526158	+	0.850387i	$0.676368\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	661565.	+	1.14586e6i	0.929458	+	1.60987i	0.784230	+	0.620471i	$0.213058\pi$
$47$	661565.	+	1.14586e6i	0.929458	+	1.60987i	0.145228	+	0.989398i	$0.453608\pi$
$48$		0			0
$49$	402911.	−	697863.i	0.489241	−	0.847391i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	421155.			0.388577			0.194288	−	0.980944i	$-0.437760\pi$
$53$	421155.			0.388577			0.194288	+	0.980944i	$0.437760\pi$
$54$		0			0
$55$	780467.			0.632536
$56$		0			0
$57$		0			0
$58$		0			0
$59$	1.23603e6	−	2.14086e6i	0.783513	−	1.35708i	−0.146371	−	0.989230i	$-0.546759\pi$
$59$	1.23603e6	−	2.14086e6i	0.783513	−	1.35708i	0.929884	−	0.367854i	$-0.119907\pi$
$60$		0			0
$61$	−693435.	−	1.20107e6i	−0.391157	−	0.677504i	0.601445	−	0.798914i	$-0.294592\pi$
$61$	−693435.	−	1.20107e6i	−0.391157	−	0.677504i	−0.992602	+	0.121410i	$0.961259\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	557861.	+	966244.i	0.251959	+	0.436405i
$66$		0			0
$67$	−447311.	+	774766.i	−0.181697	+	0.314709i	−0.942459	−	0.334323i	$-0.891492\pi$
$67$	−447311.	+	774766.i	−0.181697	+	0.314709i	0.760761	+	0.649032i	$0.224826\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−2.89163e6			−0.958823			−0.479411	−	0.877590i	$-0.659150\pi$
$71$	−2.89163e6			−0.958823			−0.479411	+	0.877590i	$0.659150\pi$
$72$		0			0
$73$	115265.			0.0346790			0.0173395	−	0.999850i	$-0.494480\pi$
$73$	115265.			0.0346790			0.0173395	+	0.999850i	$0.494480\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−97381.9	+	168670.i	−0.0243086	+	0.0421038i
$78$		0			0
$79$	−2.15801e6	−	3.73779e6i	−0.492447	−	0.852943i	0.507515	−	0.861643i	$-0.330564\pi$
$79$	−2.15801e6	−	3.73779e6i	−0.492447	−	0.852943i	−0.999962	+	0.00869992i	$0.997231\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−2.20388e6	−	3.81724e6i	−0.423073	−	0.732784i	0.573165	−	0.819440i	$-0.305715\pi$
$83$	−2.20388e6	−	3.81724e6i	−0.423073	−	0.732784i	−0.996238	+	0.0866560i	$0.972382\pi$
$84$		0			0
$85$	5.85749e6	−	1.01455e7i	1.03454	−	1.79187i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−3.58129e6			−0.538485			−0.269243	−	0.963072i	$-0.586773\pi$
$89$	−3.58129e6			−0.538485			−0.269243	+	0.963072i	$0.586773\pi$
$90$		0			0
$91$	−278426.			−0.0387315
$92$		0			0
$93$		0			0
$94$		0			0
$95$	9.17655e6	−	1.58943e7i	1.09811	−	1.90199i
$96$		0			0
$97$	4.45354e6	+	7.71376e6i	0.495456	+	0.858154i	0.999986	−	0.00523960i	$-0.00166782\pi$
$97$	4.45354e6	+	7.71376e6i	0.495456	+	0.858154i	−0.504531	+	0.863394i	$0.668334\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	8.89730e6	+	1.54106e7i	0.859277	+	1.48831i	0.872619	+	0.488401i	$0.162420\pi$
$101$	8.89730e6	+	1.54106e7i	0.859277	+	1.48831i	−0.0133420	+	0.999911i	$0.504247\pi$
$102$		0			0
$103$	859264.	−	1.48829e6i	0.0774812	−	0.134201i	−0.824681	−	0.565598i	$-0.808646\pi$
$103$	859264.	−	1.48829e6i	0.0774812	−	0.134201i	0.902163	+	0.431396i	$0.141979\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	2.23386e7			1.76284			0.881419	−	0.472336i	$-0.156589\pi$
$107$	2.23386e7			1.76284			0.881419	+	0.472336i	$0.156589\pi$
$108$		0			0
$109$	9.63597e6			0.712693			0.356347	−	0.934354i	$-0.384022\pi$
$109$	9.63597e6			0.712693			0.356347	+	0.934354i	$0.384022\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	7.84603e6	−	1.35897e7i	0.511535	−	0.886005i	−0.488375	−	0.872634i	$-0.662410\pi$
$113$	7.84603e6	−	1.35897e7i	0.511535	−	0.886005i	0.999911	−	0.0133713i	$-0.00425635\pi$
$114$		0			0
$115$	−1.75323e7	−	3.03668e7i	−1.07497	−	1.86190i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	1.46172e6	+	2.53178e6i	0.0795153	+	0.137724i
$120$		0			0
$121$	8.67328e6	−	1.50226e7i	0.445076	−	0.770895i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	6.84451e7			3.13442
$126$		0			0
$127$	1.36446e7			0.591081			0.295541	−	0.955330i	$-0.404500\pi$
$127$	1.36446e7			0.591081			0.295541	+	0.955330i	$0.404500\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−2.13096e7	+	3.69094e7i	−0.828183	+	1.43446i	0.0712787	+	0.997456i	$0.477292\pi$
$131$	−2.13096e7	+	3.69094e7i	−0.828183	+	1.43446i	−0.899462	+	0.436999i	$0.856041\pi$
$132$		0			0
$133$	2.28999e6	+	3.96637e6i	0.0844019	+	0.146188i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−2.41545e7	−	4.18368e7i	−0.802557	−	1.39007i	−0.917928	−	0.396747i	$-0.870139\pi$
$137$	−2.41545e7	−	4.18368e7i	−0.802557	−	1.39007i	0.115371	−	0.993322i	$-0.463194\pi$
$138$		0			0
$139$	1.60723e7	−	2.78380e7i	0.507605	−	0.879198i	−0.492356	−	0.870394i	$-0.663864\pi$
$139$	1.60723e7	−	2.78380e7i	0.507605	−	0.879198i	0.999961	−	0.00880394i	$-0.00280242\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−3.06012e6			−0.0875110
$144$		0			0
$145$	−7.98865e7			−2.17613
$146$		0			0
$147$		0			0
$148$		0			0
$149$	8.12294e6	−	1.40693e7i	0.201169	−	0.348435i	−0.747736	−	0.663996i	$-0.768859\pi$
$149$	8.12294e6	−	1.40693e7i	0.201169	−	0.348435i	0.948905	+	0.315561i	$0.102193\pi$
$150$		0			0
$151$	−2.46903e7	−	4.27649e7i	−0.583590	−	1.01081i	−0.995050	−	0.0993794i	$-0.968314\pi$
$151$	−2.46903e7	−	4.27649e7i	−0.583590	−	1.01081i	0.411460	−	0.911428i	$-0.365019\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	9.14544e6	+	1.58404e7i	0.197262	+	0.341668i
$156$		0			0
$157$	3.22880e7	−	5.59244e7i	0.665874	−	1.15333i	−0.313173	−	0.949696i	$-0.601392\pi$
$157$	3.22880e7	−	5.59244e7i	0.665874	−	1.15333i	0.979047	−	0.203632i	$-0.0652747\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	8.75029e6			0.165246
$162$		0			0
$163$	−1.67560e7			−0.303050			−0.151525	−	0.988453i	$-0.548418\pi$
$163$	−1.67560e7			−0.303050			−0.151525	+	0.988453i	$0.548418\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−4.24653e7	+	7.35521e7i	−0.705549	+	1.22205i	0.260944	+	0.965354i	$0.415966\pi$
$167$	−4.24653e7	+	7.35521e7i	−0.705549	+	1.22205i	−0.966493	+	0.256692i	$0.917367\pi$
$168$		0			0
$169$	2.91869e7	+	5.05533e7i	0.465142	+	0.805649i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−6.06023e6	−	1.04966e7i	−0.0889872	−	0.154130i	0.818096	−	0.575082i	$-0.195030\pi$
$173$	−6.06023e6	−	1.04966e7i	−0.0889872	−	0.154130i	−0.907083	+	0.420951i	$0.861696\pi$
$174$		0			0
$175$	−1.37401e7	+	2.37986e7i	−0.193801	+	0.335674i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−5.17162e7			−0.673970			−0.336985	−	0.941510i	$-0.609407\pi$
$179$	−5.17162e7			−0.673970			−0.336985	+	0.941510i	$0.609407\pi$
$180$		0			0
$181$	9.59020e7			1.20213			0.601067	−	0.799199i	$-0.294743\pi$
$181$	9.59020e7			1.20213			0.601067	+	0.799199i	$0.294743\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	5.84739e7	−	1.01280e8i	0.678987	−	1.17604i
$186$		0			0
$187$	1.60655e7	+	2.78263e7i	0.179659	+	0.311178i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−4.79064e7	−	8.29763e7i	−0.497481	−	0.861662i	0.502515	−	0.864569i	$-0.332408\pi$
$191$	−4.79064e7	−	8.29763e7i	−0.497481	−	0.861662i	−0.999996	+	0.00290627i	$0.999075\pi$
$192$		0			0
$193$	−6.93095e7	+	1.20048e8i	−0.693972	+	1.20200i	0.276553	+	0.960999i	$0.410808\pi$
$193$	−6.93095e7	+	1.20048e8i	−0.693972	+	1.20200i	−0.970526	+	0.240997i	$0.922526\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	6.13963e7			0.572151			0.286075	−	0.958207i	$-0.407649\pi$
$197$	6.13963e7			0.572151			0.286075	+	0.958207i	$0.407649\pi$
$198$		0			0
$199$	3.17586e7			0.285677			0.142839	−	0.989746i	$-0.454377\pi$
$199$	3.17586e7			0.285677			0.142839	+	0.989746i	$0.454377\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	9.96774e6	−	1.72646e7i	0.0836297	−	0.144851i
$204$		0			0
$205$	1.98264e8	+	3.43404e8i	1.60733	+	2.78398i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	2.51688e7	+	4.35936e7i	0.190700	+	0.330302i
$210$		0			0
$211$	3.33699e7	−	5.77983e7i	0.244549	−	0.423571i	−0.717456	−	0.696604i	$-0.754693\pi$
$211$	3.33699e7	−	5.77983e7i	0.244549	−	0.423571i	0.962005	+	0.273033i	$0.0880267\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−2.63307e8			−1.80688
$216$		0			0
$217$	−4.56445e6			−0.0303235
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−2.29665e7	+	3.97792e7i	−0.143127	+	0.247904i
$222$		0			0
$223$	−1.93530e7	−	3.35204e7i	−0.116864	−	0.202415i	0.801659	−	0.597781i	$-0.203951\pi$
$223$	−1.93530e7	−	3.35204e7i	−0.116864	−	0.202415i	−0.918523	+	0.395366i	$0.870618\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	5.07190e7	+	8.78479e7i	0.287793	+	0.498472i	0.973283	−	0.229610i	$-0.0737452\pi$
$227$	5.07190e7	+	8.78479e7i	0.287793	+	0.498472i	−0.685490	+	0.728082i	$0.740412\pi$
$228$		0			0
$229$	4.64786e7	−	8.05033e7i	0.255758	−	0.442986i	−0.709343	−	0.704863i	$-0.751008\pi$
$229$	4.64786e7	−	8.05033e7i	0.255758	−	0.442986i	0.965101	+	0.261878i	$0.0843417\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−2.57010e8			−1.33108			−0.665540	−	0.746362i	$-0.731799\pi$
$233$	−2.57010e8			−1.33108			−0.665540	+	0.746362i	$0.731799\pi$
$234$		0			0
$235$	−7.05811e8			−3.54773
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−2.59061e7	+	4.48707e7i	−0.122747	+	0.212604i	−0.920850	−	0.389917i	$-0.872504\pi$
$239$	−2.59061e7	+	4.48707e7i	−0.122747	+	0.212604i	0.798103	+	0.602521i	$0.205837\pi$
$240$		0			0
$241$	−7.44669e6	−	1.28980e7i	−0.0342692	−	0.0593559i	0.848382	−	0.529384i	$-0.177577\pi$
$241$	−7.44669e6	−	1.28980e7i	−0.0342692	−	0.0593559i	−0.882651	+	0.470028i	$0.844244\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	2.14929e8	+	3.72268e8i	0.933714	+	1.61724i
$246$		0			0
$247$	−3.59802e7	+	6.23195e7i	−0.151923	+	0.263139i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−2.63741e8			−1.05274			−0.526368	−	0.850257i	$-0.676447\pi$
$251$	−2.63741e8			−1.05274			−0.526368	+	0.850257i	$0.676447\pi$
$252$		0			0
$253$	9.61727e7			0.373362
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−9.92140e7	+	1.71844e8i	−0.364592	+	0.631492i	−0.988711	−	0.149838i	$-0.952125\pi$
$257$	−9.92140e7	+	1.71844e8i	−0.364592	+	0.631492i	0.624119	+	0.781329i	$0.285458\pi$
$258$		0			0
$259$	1.45920e7	+	2.52741e7i	0.0521875	+	0.0903914i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−1.51106e8	−	2.61724e8i	−0.512197	−	0.887151i	−0.999900	−	0.0141417i	$-0.995498\pi$
$263$	−1.51106e8	−	2.61724e8i	−0.512197	−	0.887151i	0.487703	−	0.873010i	$-0.337835\pi$
$264$		0			0
$265$	−1.12331e8	+	1.94562e8i	−0.370798	+	0.642241i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−1.05057e8			−0.329072			−0.164536	−	0.986371i	$-0.552613\pi$
$269$	−1.05057e8			−0.329072			−0.164536	+	0.986371i	$0.552613\pi$
$270$		0			0
$271$	−2.58838e8			−0.790016			−0.395008	−	0.918678i	$-0.629258\pi$
$271$	−2.58838e8			−0.790016			−0.395008	+	0.918678i	$0.629258\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−1.51015e8	+	2.61565e8i	−0.437880	+	0.758431i
$276$		0			0
$277$	−6.16771e7	−	1.06828e8i	−0.174359	−	0.301999i	0.765580	−	0.643340i	$-0.222452\pi$
$277$	−6.16771e7	−	1.06828e8i	−0.174359	−	0.301999i	−0.939939	+	0.341342i	$0.889119\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	2.09130e7	+	3.62224e7i	0.0562269	+	0.0973878i	0.892769	−	0.450515i	$-0.148760\pi$
$281$	2.09130e7	+	3.62224e7i	0.0562269	+	0.0973878i	−0.836542	+	0.547903i	$0.815426\pi$
$282$		0			0
$283$	1.84191e8	−	3.19029e8i	0.483078	−	0.836715i	−0.516734	−	0.856146i	$-0.672852\pi$
$283$	1.84191e8	−	3.19029e8i	0.483078	−	0.836715i	0.999811	+	0.0194313i	$0.00618558\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−9.89527e7			−0.247082
$288$		0			0
$289$	7.19546e7			0.175354
$290$		0			0
$291$		0			0
$292$		0			0
$293$	2.70598e8	−	4.68689e8i	0.628474	−	1.08855i	−0.359384	−	0.933190i	$-0.617013\pi$
$293$	2.70598e8	−	4.68689e8i	0.628474	−	1.08855i	0.987858	−	0.155360i	$-0.0496536\pi$
$294$		0			0
$295$	6.59347e8	+	1.14202e9i	1.49533	+	2.58999i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	6.87422e7	+	1.19065e8i	0.148722	+	0.257593i
$300$		0			0
$301$	3.28538e7	−	5.69045e7i	0.0694390	−	0.120272i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	7.39813e8			1.49304
$306$		0			0
$307$	−1.85659e7			−0.0366210			−0.0183105	−	0.999832i	$-0.505829\pi$
$307$	−1.85659e7			−0.0366210			−0.0183105	+	0.999832i	$0.505829\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−1.35014e8	+	2.33850e8i	−0.254517	+	0.440836i	−0.964764	−	0.263116i	$-0.915250\pi$
$311$	−1.35014e8	+	2.33850e8i	−0.254517	+	0.440836i	0.710247	+	0.703952i	$0.248583\pi$
$312$		0			0
$313$	3.06051e8	+	5.30096e8i	0.564142	+	0.977123i	0.997129	+	0.0757227i	$0.0241264\pi$
$313$	3.06051e8	+	5.30096e8i	0.564142	+	0.977123i	−0.432987	+	0.901400i	$0.642540\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−4.38932e8	−	7.60252e8i	−0.773908	−	1.34045i	−0.935406	−	0.353575i	$-0.884966\pi$
$317$	−4.38932e8	−	7.60252e8i	−0.773908	−	1.34045i	0.161498	−	0.986873i	$-0.448368\pi$
$318$		0			0
$319$	1.09553e8	−	1.89752e8i	0.188955	−	0.327280i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	7.55578e8			1.24759
$324$		0			0
$325$	−4.31768e8			−0.697685
$326$		0			0
$327$		0			0
$328$		0			0
$329$	8.80667e7	−	1.52536e8i	0.136341	−	0.236149i
$330$		0			0
$331$	−6.31951e7	−	1.09457e8i	−0.0957824	−	0.165900i	0.814152	−	0.580651i	$-0.197202\pi$
$331$	−6.31951e7	−	1.09457e8i	−0.0957824	−	0.165900i	−0.909935	+	0.414751i	$0.863869\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−2.38614e8	−	4.13292e8i	−0.346768	−	0.600620i
$336$		0			0
$337$	4.11001e8	−	7.11875e8i	0.584977	−	1.01321i	−0.409902	−	0.912130i	$-0.634437\pi$
$337$	4.11001e8	−	7.11875e8i	0.584977	−	1.01321i	0.994878	−	0.101079i	$-0.0322296\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−5.01669e7			−0.0685137
$342$		0			0
$343$	−2.16899e8			−0.290221
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−4.01462e8	+	6.95352e8i	−0.515811	+	0.893411i	0.484020	+	0.875057i	$0.339176\pi$
$347$	−4.01462e8	+	6.95352e8i	−0.515811	+	0.893411i	−0.999832	+	0.0183545i	$0.994157\pi$
$348$		0			0
$349$	6.02303e7	+	1.04322e8i	0.0758449	+	0.131367i	0.901453	−	0.432876i	$-0.142501\pi$
$349$	6.02303e7	+	1.04322e8i	0.0758449	+	0.131367i	−0.825609	+	0.564243i	$0.809168\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−8.43578e7	−	1.46112e8i	−0.102074	−	0.176797i	0.810465	−	0.585787i	$-0.199214\pi$
$353$	−8.43578e7	−	1.46112e8i	−0.102074	−	0.176797i	−0.912539	+	0.408990i	$0.865881\pi$
$354$		0			0
$355$	7.71256e8	−	1.33585e9i	0.914954	−	1.58475i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−4.90600e8			−0.559625			−0.279812	−	0.960055i	$-0.590272\pi$
$359$	−4.90600e8			−0.559625			−0.279812	+	0.960055i	$0.590272\pi$
$360$		0			0
$361$	2.89843e8			0.324256
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−3.07434e7	+	5.32492e7i	−0.0330923	+	0.0573176i
$366$		0			0
$367$	−4.60920e8	−	7.98338e8i	−0.486737	−	0.843054i	0.513146	−	0.858301i	$-0.328480\pi$
$367$	−4.60920e8	−	7.98338e8i	−0.486737	−	0.843054i	−0.999884	+	0.0152472i	$0.995146\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−2.80318e7	−	4.85525e7i	−0.0284999	−	0.0493632i
$372$		0			0
$373$	−1.05604e8	+	1.82911e8i	−0.105365	+	0.182498i	−0.913887	−	0.405968i	$-0.866935\pi$
$373$	−1.05604e8	+	1.82911e8i	−0.105365	+	0.182498i	0.808522	+	0.588466i	$0.200268\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	3.13226e8			0.301067
$378$		0			0
$379$	−1.54128e9			−1.45427			−0.727136	−	0.686494i	$-0.759149\pi$
$379$	−1.54128e9			−1.45427			−0.727136	+	0.686494i	$0.759149\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	4.86985e8	−	8.43482e8i	0.442914	−	0.767150i	−0.554990	−	0.831857i	$-0.687278\pi$
$383$	4.86985e8	−	8.43482e8i	0.442914	−	0.767150i	0.997904	+	0.0647071i	$0.0206113\pi$
$384$		0			0
$385$	−5.19474e7	−	8.99756e7i	−0.0463929	−	0.0803549i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	9.29227e7	+	1.60947e8i	0.0800384	+	0.138631i	0.903266	−	0.429081i	$-0.141162\pi$
$389$	9.29227e7	+	1.60947e8i	0.0800384	+	0.138631i	−0.823228	+	0.567711i	$0.807829\pi$
$390$		0			0
$391$	7.21786e8	−	1.25017e9i	0.610647	−	1.05767i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	2.30234e9			1.87966
$396$		0			0
$397$	−2.02132e9			−1.62132			−0.810659	−	0.585519i	$-0.800891\pi$
$397$	−2.02132e9			−1.62132			−0.810659	+	0.585519i	$0.800891\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−6.97914e7	+	1.20882e8i	−0.0540501	+	0.0936175i	−0.891785	−	0.452460i	$-0.850546\pi$
$401$	−6.97914e7	+	1.20882e8i	−0.0540501	+	0.0936175i	0.837734	+	0.546078i	$0.183880\pi$
$402$		0			0
$403$	−3.58582e7	−	6.21083e7i	−0.0272911	−	0.0472696i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	1.60378e8	+	2.77783e8i	0.117914	+	0.204233i
$408$		0			0
$409$	1.10712e9	−	1.91758e9i	0.800133	−	1.38587i	−0.119395	−	0.992847i	$-0.538095\pi$
$409$	1.10712e9	−	1.91758e9i	0.800133	−	1.38587i	0.919528	−	0.393025i	$-0.128571\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−3.29077e8			−0.229865
$414$		0			0
$415$	2.35128e9			1.61486
$416$		0			0
$417$		0			0
$418$		0			0
$419$	9.12567e8	−	1.58061e9i	0.606060	−	1.04973i	−0.385823	−	0.922573i	$-0.626083\pi$
$419$	9.12567e8	−	1.58061e9i	0.606060	−	1.04973i	0.991883	−	0.127154i	$-0.0405842\pi$
$420$		0			0
$421$	1.04524e8	+	1.81040e8i	0.0682697	+	0.118247i	0.898140	−	0.439710i	$-0.144919\pi$
$421$	1.04524e8	+	1.81040e8i	0.0682697	+	0.118247i	−0.829870	+	0.557957i	$0.811586\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	2.26676e9	+	3.92615e9i	1.43234	+	2.48088i
$426$		0			0
$427$	−9.23093e7	+	1.59884e8i	−0.0573783	+	0.0993821i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	3.22269e9			1.93887			0.969435	−	0.245350i	$-0.0789029\pi$
$431$	3.22269e9			1.93887			0.969435	+	0.245350i	$0.0789029\pi$
$432$		0			0
$433$	−3.34353e8			−0.197923			−0.0989617	−	0.995091i	$-0.531552\pi$
$433$	−3.34353e8			−0.197923			−0.0989617	+	0.995091i	$0.531552\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	1.13078e9	−	1.95856e9i	0.648174	−	1.12267i
$438$		0			0
$439$	−7.81986e8	−	1.35444e9i	−0.441137	−	0.764071i	0.556637	−	0.830756i	$-0.312091\pi$
$439$	−7.81986e8	−	1.35444e9i	−0.441137	−	0.764071i	−0.997774	+	0.0666843i	$0.978758\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−4.66035e7	−	8.07196e7i	−0.0254686	−	0.0441129i	0.853010	−	0.521894i	$-0.174774\pi$
$443$	−4.66035e7	−	8.07196e7i	−0.0254686	−	0.0441129i	−0.878479	+	0.477781i	$0.841441\pi$
$444$		0			0
$445$	9.55201e8	−	1.65446e9i	0.513848	−	0.890011i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−1.77844e9			−0.927208			−0.463604	−	0.886042i	$-0.653444\pi$
$449$	−1.77844e9			−0.927208			−0.463604	+	0.886042i	$0.653444\pi$
$450$		0			0
$451$	−1.08757e9			−0.558263
$452$		0			0
$453$		0			0
$454$		0			0
$455$	7.42618e7	−	1.28625e8i	0.0369595	−	0.0640156i
$456$		0			0
$457$	2.13606e8	+	3.69977e8i	0.104691	+	0.181329i	0.913612	−	0.406588i	$-0.133281\pi$
$457$	2.13606e8	+	3.69977e8i	0.104691	+	0.181329i	−0.808921	+	0.587917i	$0.799948\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−1.03434e9	−	1.79154e9i	−0.491713	−	0.851672i	0.508241	−	0.861215i	$-0.330296\pi$
$461$	−1.03434e9	−	1.79154e9i	−0.491713	−	0.851672i	−0.999954	+	0.00954253i	$0.996962\pi$
$462$		0			0
$463$	−1.67864e9	+	2.90750e9i	−0.786004	+	1.36140i	0.142393	+	0.989810i	$0.454520\pi$
$463$	−1.67864e9	+	2.90750e9i	−0.786004	+	1.36140i	−0.928397	+	0.371589i	$0.878813\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−4.30858e8			−0.195760			−0.0978802	−	0.995198i	$-0.531206\pi$
$467$	−4.30858e8			−0.195760			−0.0978802	+	0.995198i	$0.531206\pi$
$468$		0			0
$469$	1.19091e8			0.0533058
$470$		0			0
$471$		0			0
$472$		0			0
$473$	3.61090e8	−	6.25427e8i	0.156892	−	0.271745i
$474$		0			0
$475$	3.55119e9	+	6.15085e9i	1.52036	+	2.63334i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	8.48173e6	+	1.46908e7i	0.00352623	+	0.00610760i	0.867783	−	0.496943i	$-0.165544\pi$
$479$	8.48173e6	+	1.46908e7i	0.00352623	+	0.00610760i	−0.864257	+	0.503051i	$0.832211\pi$
$480$		0			0
$481$	−2.29270e8	+	3.97107e8i	−0.0939374	+	0.162704i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−4.75140e9			−1.89115
$486$		0			0
$487$	4.04688e9			1.58770			0.793852	−	0.608111i	$-0.208073\pi$
$487$	4.04688e9			1.58770			0.793852	+	0.608111i	$0.208073\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−6.56052e8	+	1.13631e9i	−0.250123	+	0.433225i	−0.963559	−	0.267495i	$-0.913804\pi$
$491$	−6.56052e8	+	1.13631e9i	−0.250123	+	0.433225i	0.713437	+	0.700720i	$0.247138\pi$
$492$		0			0
$493$	−1.64442e9	−	2.84822e9i	−0.618086	−	1.07056i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	1.92465e8	+	3.33359e8i	0.0703241	+	0.121805i
$498$		0			0
$499$	2.06483e9	−	3.57639e9i	0.743930	−	1.28853i	−0.206763	−	0.978391i	$-0.566293\pi$
$499$	2.06483e9	−	3.57639e9i	0.743930	−	1.28853i	0.950693	−	0.310134i	$-0.100374\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−3.12317e9			−1.09423			−0.547115	−	0.837058i	$-0.684274\pi$
$503$	−3.12317e9			−1.09423			−0.547115	+	0.837058i	$0.684274\pi$
$504$		0			0
$505$	−9.49236e9			−3.27985
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−6.44861e8	+	1.11693e9i	−0.216747	+	0.375418i	−0.953812	−	0.300405i	$-0.902878\pi$
$509$	−6.44861e8	+	1.11693e9i	−0.216747	+	0.375418i	0.737064	+	0.675823i	$0.236211\pi$
$510$		0			0
$511$	−7.67195e6	−	1.32882e7i	−0.00254350	−	0.00440548i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	4.58366e8	+	7.93913e8i	0.147872	+	0.256123i
$516$		0			0
$517$	9.67924e8	−	1.67649e9i	0.308052	−	0.533562i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−4.74687e9			−1.47053			−0.735267	−	0.677777i	$-0.762943\pi$
$521$	−4.74687e9			−1.47053			−0.735267	+	0.677777i	$0.762943\pi$
$522$		0			0
$523$	1.53189e9			0.468243			0.234122	−	0.972207i	$-0.424779\pi$
$523$	1.53189e9			0.468243			0.234122	+	0.972207i	$0.424779\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−3.76508e8	+	6.52131e8i	−0.112057	+	0.194088i
$528$		0			0
$529$	−4.57997e8	−	7.93274e8i	−0.134514	−	0.232985i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−7.77371e8	−	1.34645e9i	−0.222374	−	0.385162i
$534$		0			0
$535$	−5.95815e9	+	1.03198e10i	−1.68218	+	2.91363i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−1.17898e9			−0.324300
$540$		0			0
$541$	4.15229e9			1.12745			0.563725	−	0.825962i	$-0.309368\pi$
$541$	4.15229e9			1.12745			0.563725	+	0.825962i	$0.309368\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−2.57011e9	+	4.45156e9i	−0.680086	+	1.17794i
$546$		0			0
$547$	5.17003e8	+	8.95476e8i	0.135063	+	0.233937i	0.925622	−	0.378450i	$-0.123543\pi$
$547$	5.17003e8	+	8.95476e8i	0.135063	+	0.233937i	−0.790558	+	0.612387i	$0.790210\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−2.57621e9	−	4.46212e9i	−0.656070	−	1.13635i
$552$		0			0
$553$	−2.87272e8	+	4.97570e8i	−0.0722363	+	0.125117i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	1.16070e9			0.284594			0.142297	−	0.989824i	$-0.454551\pi$
$557$	1.16070e9			0.284594			0.142297	+	0.989824i	$0.454551\pi$
$558$		0			0
$559$	1.03240e9			0.249980
$560$		0			0
$561$		0			0
$562$		0			0
$563$	6.24836e8	−	1.08225e9i	0.147566	−	0.255592i	−0.782761	−	0.622322i	$-0.786189\pi$
$563$	6.24836e8	−	1.08225e9i	0.147566	−	0.255592i	0.930327	+	0.366730i	$0.119523\pi$
$564$		0			0
$565$	4.18539e9	+	7.24931e9i	0.976262	+	1.69094i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	5.77827e8	+	1.00083e9i	0.131494	+	0.227754i	0.924253	−	0.381782i	$-0.124689\pi$
$569$	5.77827e8	+	1.00083e9i	0.131494	+	0.227754i	−0.792759	+	0.609535i	$0.791356\pi$
$570$		0			0
$571$	−2.40144e9	+	4.15942e9i	−0.539816	+	0.934988i	0.459098	+	0.888386i	$0.348173\pi$
$571$	−2.40144e9	+	4.15942e9i	−0.539816	+	0.934988i	−0.998914	+	0.0466025i	$0.985161\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	1.35695e10			2.97664
$576$		0			0
$577$	3.18167e9			0.689510			0.344755	−	0.938693i	$-0.387962\pi$
$577$	3.18167e9			0.689510			0.344755	+	0.938693i	$0.387962\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−2.93378e8	+	5.08146e8i	−0.0620599	+	0.107491i
$582$		0			0
$583$	−3.08092e8	−	5.33632e8i	−0.0643933	−	0.111533i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−4.16528e9	−	7.21448e9i	−0.849984	−	1.47222i	−0.881221	−	0.472704i	$-0.843278\pi$
$587$	−4.16528e9	−	7.21448e9i	−0.849984	−	1.47222i	0.0312374	−	0.999512i	$-0.490055\pi$
$588$		0			0
$589$	−5.89851e8	+	1.02165e9i	−0.118943	+	0.206015i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−5.80401e9			−1.14297			−0.571487	−	0.820611i	$-0.693633\pi$
$593$	−5.80401e9			−1.14297			−0.571487	+	0.820611i	$0.693633\pi$
$594$		0			0
$595$	−1.55948e9			−0.303509
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−4.89357e9	+	8.47592e9i	−0.930320	+	1.61136i	−0.147546	+	0.989055i	$0.547137\pi$
$599$	−4.89357e9	+	8.47592e9i	−0.930320	+	1.61136i	−0.782774	+	0.622306i	$0.786196\pi$
$600$		0			0
$601$	−3.04825e9	−	5.27972e9i	−0.572783	−	0.992089i	−0.996279	−	0.0861907i	$-0.972531\pi$
$601$	−3.04825e9	−	5.27972e9i	−0.572783	−	0.992089i	0.423496	−	0.905898i	$-0.360803\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	4.62668e9	+	8.01364e9i	0.849426	+	1.47125i
$606$		0			0
$607$	4.08870e9	−	7.08184e9i	0.742036	−	1.28524i	−0.209531	−	0.977802i	$-0.567194\pi$
$607$	4.08870e9	−	7.08184e9i	0.742036	−	1.28524i	0.951567	−	0.307442i	$-0.0994731\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	2.76740e9			0.490827
$612$		0			0
$613$	3.02808e9			0.530953			0.265477	−	0.964117i	$-0.414471\pi$
$613$	3.02808e9			0.530953			0.265477	+	0.964117i	$0.414471\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−4.38870e8	+	7.60146e8i	−0.0752208	+	0.130286i	−0.901182	−	0.433440i	$-0.857300\pi$
$617$	−4.38870e8	+	7.60146e8i	−0.0752208	+	0.130286i	0.825961	+	0.563727i	$0.190633\pi$
$618$		0			0
$619$	2.65741e9	+	4.60277e9i	0.450340	+	0.780013i	0.998407	−	0.0564222i	$-0.0179693\pi$
$619$	2.65741e9	+	4.60277e9i	0.450340	+	0.780013i	−0.548067	+	0.836435i	$0.684636\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	2.38368e8	+	4.12866e8i	0.0394948	+	0.0684070i
$624$		0			0
$625$	−1.01919e10	+	1.76529e10i	−1.66984	+	2.89224i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	4.81462e9			0.771409
$630$		0			0
$631$	−3.01146e9			−0.477171			−0.238586	−	0.971121i	$-0.576684\pi$
$631$	−3.01146e9			−0.477171			−0.238586	+	0.971121i	$0.576684\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−3.63929e9	+	6.30343e9i	−0.564038	+	0.976942i
$636$		0			0
$637$	−8.42713e8	−	1.45962e9i	−0.129179	−	0.223744i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−2.65767e9	−	4.60322e9i	−0.398564	−	0.690334i	0.594985	−	0.803737i	$-0.297158\pi$
$641$	−2.65767e9	−	4.60322e9i	−0.398564	−	0.690334i	−0.993549	+	0.113403i	$0.963825\pi$
$642$		0			0
$643$	2.99268e9	−	5.18347e9i	0.443937	−	0.768922i	−0.554040	−	0.832490i	$-0.686915\pi$
$643$	2.99268e9	−	5.18347e9i	0.443937	−	0.768922i	0.997978	+	0.0635680i	$0.0202480\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−3.24427e9			−0.470925			−0.235463	−	0.971883i	$-0.575661\pi$
$647$	−3.24427e9			−0.470925			−0.235463	+	0.971883i	$0.575661\pi$
$648$		0			0
$649$	−3.61682e9			−0.519362
$650$		0			0
$651$		0			0
$652$		0			0
$653$	3.93554e9	−	6.81656e9i	0.553106	−	0.958008i	−0.444942	−	0.895559i	$-0.646776\pi$
$653$	3.93554e9	−	6.81656e9i	0.553106	−	0.958008i	0.998048	−	0.0624488i	$-0.0198910\pi$
$654$		0			0
$655$	−1.13674e10	−	1.96890e10i	−1.58058	−	2.73765i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−2.20083e9	−	3.81195e9i	−0.299563	−	0.518858i	0.676473	−	0.736467i	$-0.263507\pi$
$659$	−2.20083e9	−	3.81195e9i	−0.299563	−	0.518858i	−0.976036	+	0.217609i	$0.930174\pi$
$660$		0			0
$661$	2.90322e9	−	5.02853e9i	0.390999	−	0.677230i	−0.601583	−	0.798810i	$-0.705463\pi$
$661$	2.90322e9	−	5.02853e9i	0.390999	−	0.677230i	0.992582	+	0.121581i	$0.0387964\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−2.44314e9			−0.322161
$666$		0			0
$667$	−9.84397e9			−1.28449
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−1.01455e9	+	1.75726e9i	−0.129642	+	0.224547i
$672$		0			0
$673$	−1.57613e9	−	2.72994e9i	−0.199315	−	0.345223i	0.748992	−	0.662579i	$-0.230538\pi$
$673$	−1.57613e9	−	2.72994e9i	−0.199315	−	0.345223i	−0.948306	+	0.317356i	$0.897205\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	3.95539e8	+	6.85094e8i	0.0489924	+	0.0848574i	0.889482	−	0.456971i	$-0.151066\pi$
$677$	3.95539e8	+	6.85094e8i	0.0489924	+	0.0848574i	−0.840489	+	0.541828i	$0.817732\pi$
$678$		0			0
$679$	5.92850e8	−	1.02685e9i	0.0726776	−	0.125881i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−1.05529e10			−1.26735			−0.633677	−	0.773598i	$-0.718455\pi$
$683$	−1.05529e10			−1.26735			−0.633677	+	0.773598i	$0.718455\pi$
$684$		0			0
$685$	2.57700e10			3.06335
$686$		0			0
$687$		0			0
$688$		0			0
$689$	4.40435e8	−	7.62856e8i	0.0512997	−	0.0888537i
$690$		0			0
$691$	2.35912e9	+	4.08611e9i	0.272005	+	0.471126i	0.969375	−	0.245585i	$-0.0789800\pi$
$691$	2.35912e9	+	4.08611e9i	0.272005	+	0.471126i	−0.697370	+	0.716711i	$0.745647\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	8.57361e9	+	1.48499e10i	0.968762	+	1.67794i
$696$		0			0
$697$	−8.16232e9	+	1.41376e10i	−0.913060	+	1.58147i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−2.06877e9			−0.226829			−0.113414	−	0.993548i	$-0.536179\pi$
$701$	−2.06877e9			−0.226829			−0.113414	+	0.993548i	$0.536179\pi$
$702$		0			0
$703$	7.54275e9			0.818816
$704$		0			0
$705$		0			0
$706$		0			0
$707$	1.18440e9	−	2.05144e9i	0.126046	−	0.218318i
$708$		0			0
$709$	−8.45547e9	−	1.46453e10i	−0.890996	−	1.54325i	−0.838683	−	0.544620i	$-0.816674\pi$
$709$	−8.45547e9	−	1.46453e10i	−0.890996	−	1.54325i	−0.0523138	−	0.998631i	$-0.516660\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	1.12694e9	+	1.95192e9i	0.116436	+	0.201674i
$714$		0			0
$715$	8.16197e8	−	1.41369e9i	0.0835072	−	0.144639i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−5.93452e9			−0.595435			−0.297718	−	0.954654i	$-0.596225\pi$
$719$	−5.93452e9			−0.595435			−0.297718	+	0.954654i	$0.596225\pi$
$720$		0			0
$721$	−2.28768e8			−0.0227312
$722$		0			0
$723$		0			0
$724$		0			0
$725$	1.54575e10	−	2.67731e10i	1.50645	−	2.60925i
$726$		0			0
$727$	3.86981e9	+	6.70270e9i	0.373524	+	0.646963i	0.990105	−	0.140329i	$-0.0448160\pi$
$727$	3.86981e9	+	6.70270e9i	0.373524	+	0.646963i	−0.616581	+	0.787292i	$0.711483\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−5.42004e9	−	9.38778e9i	−0.513206	−	0.888899i
$732$		0			0
$733$	1.84980e9	−	3.20395e9i	0.173485	−	0.300484i	−0.766151	−	0.642660i	$-0.777831\pi$
$733$	1.84980e9	−	3.20395e9i	0.173485	−	0.300484i	0.939636	+	0.342176i	$0.111164\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	1.30891e9			0.120440
$738$		0			0
$739$	1.77671e10			1.61942			0.809711	−	0.586829i	$-0.199624\pi$
$739$	1.77671e10			1.61942			0.809711	+	0.586829i	$0.199624\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	3.85189e9	−	6.67166e9i	0.344518	−	0.596723i	−0.640748	−	0.767752i	$-0.721376\pi$
$743$	3.85189e9	−	6.67166e9i	0.344518	−	0.596723i	0.985266	+	0.171028i	$0.0547089\pi$
$744$		0			0
$745$	4.33311e9	+	7.50516e9i	0.383930	+	0.664987i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−1.48684e9	−	2.57529e9i	−0.129294	−	0.223944i
$750$		0			0
$751$	−3.42522e9	+	5.93266e9i	−0.295086	+	0.511104i	−0.975005	−	0.222183i	$-0.928682\pi$
$751$	−3.42522e9	+	5.93266e9i	−0.295086	+	0.511104i	0.679919	+	0.733288i	$0.262015\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	2.63417e10			2.22756
$756$		0			0
$757$	7.84916e9			0.657639			0.328820	−	0.944393i	$-0.393349\pi$
$757$	7.84916e9			0.657639			0.328820	+	0.944393i	$0.393349\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	7.77512e9	−	1.34669e10i	0.639530	−	1.10770i	−0.346006	−	0.938232i	$-0.612462\pi$
$761$	7.77512e9	−	1.34669e10i	0.639530	−	1.10770i	0.985536	−	0.169466i	$-0.0542043\pi$
$762$		0			0
$763$	−6.41364e8	−	1.11088e9i	−0.0522720	−	0.0905377i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−2.58522e9	−	4.47774e9i	−0.206878	−	0.358323i
$768$		0			0
$769$	1.09657e10	−	1.89932e10i	0.869553	−	1.50611i	0.00709847	−	0.999975i	$-0.497740\pi$
$769$	1.09657e10	−	1.89932e10i	0.869553	−	1.50611i	0.862454	−	0.506135i	$-0.168926\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	1.66645e10			1.29767			0.648836	−	0.760929i	$-0.275256\pi$
$773$	1.66645e10			1.29767			0.648836	+	0.760929i	$0.275256\pi$
$774$		0			0
$775$	−7.07831e9			−0.546228
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−1.27874e10	+	2.21484e10i	−0.969172	+	1.67865i
$780$		0			0
$781$	2.11535e9	+	3.66389e9i	0.158892	+	0.275209i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	1.72237e10	+	2.98324e10i	1.27082	+	2.20112i
$786$		0			0
$787$	4.71586e9	−	8.16810e9i	0.344865	−	0.597323i	−0.640464	−	0.767988i	$-0.721258\pi$
$787$	4.71586e9	−	8.16810e9i	0.344865	−	0.597323i	0.985329	+	0.170664i	$0.0545914\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−2.08891e9			−0.150073
$792$		0			0
$793$	−2.90072e9			−0.206562
$794$		0			0
$795$		0			0
$796$		0			0
$797$	8.85452e9	−	1.53365e10i	0.619528	−	1.07305i	−0.370044	−	0.929014i	$-0.620658\pi$
$797$	8.85452e9	−	1.53365e10i	0.619528	−	1.07305i	0.989572	−	0.144039i	$-0.0460091\pi$
$798$		0			0
$799$	−1.45287e10	−	2.51645e10i	−1.00766	−	1.74532i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−8.43209e7	−	1.46048e8i	−0.00574686	−	0.00995386i
$804$		0			0
$805$	−2.33388e9	+	4.04240e9i	−0.157686	+	0.273120i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	3.34937e9			0.222404			0.111202	−	0.993798i	$-0.464530\pi$
$809$	3.34937e9			0.222404			0.111202	+	0.993798i	$0.464530\pi$
$810$		0			0
$811$	7.29930e8			0.0480517			0.0240258	−	0.999711i	$-0.492352\pi$
$811$	7.29930e8			0.0480517			0.0240258	+	0.999711i	$0.492352\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	4.46916e9	−	7.74082e9i	0.289184	−	0.500882i
$816$		0			0
$817$	−8.49123e9	−	1.47072e10i	−0.544745	−	0.943526i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	5.21782e9	+	9.03754e9i	0.329070	+	0.569966i	0.982328	−	0.187170i	$-0.0599314\pi$
$821$	5.21782e9	+	9.03754e9i	0.329070	+	0.569966i	−0.653258	+	0.757136i	$0.726598\pi$
$822$		0			0
$823$	−7.40738e9	+	1.28300e10i	−0.463196	+	0.802279i	−0.999118	−	0.0419882i	$-0.986631\pi$
$823$	−7.40738e9	+	1.28300e10i	−0.463196	+	0.802279i	0.535922	+	0.844268i	$0.319964\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−1.25726e10			−0.772957			−0.386478	−	0.922298i	$-0.626309\pi$
$827$	−1.25726e10			−0.772957			−0.386478	+	0.922298i	$0.626309\pi$
$828$		0			0
$829$	1.18391e10			0.721737			0.360869	−	0.932617i	$-0.382480\pi$
$829$	1.18391e10			0.721737			0.360869	+	0.932617i	$0.382480\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−8.84840e9	+	1.53259e10i	−0.530405	+	0.918688i
$834$		0			0
$835$	−2.26527e10	−	3.92357e10i	−1.34654	−	2.33227i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−9.27141e9	−	1.60586e10i	−0.541975	−	0.938728i	−0.998791	−	0.0491664i	$-0.984344\pi$
$839$	−9.27141e9	−	1.60586e10i	−0.541975	−	0.938728i	0.456816	−	0.889561i	$-0.348990\pi$
$840$		0			0
$841$	−2.58865e9	+	4.48368e9i	−0.150068	+	0.259925i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−3.11390e10			−1.77544
$846$		0			0
$847$	−2.30915e9			−0.130575
$848$		0			0
$849$		0			0
$850$		0			0
$851$	7.20542e9	−	1.24802e10i	0.400780	−	0.694171i
$852$		0			0
$853$	4.31684e9	+	7.47699e9i	0.238147	+	0.412482i	0.960183	−	0.279373i	$-0.0901267\pi$
$853$	4.31684e9	+	7.47699e9i	0.238147	+	0.412482i	−0.722036	+	0.691856i	$0.756793\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	9.36326e9	+	1.62176e10i	0.508153	+	0.880146i	0.999955	+	0.00943966i	$0.00300478\pi$
$857$	9.36326e9	+	1.62176e10i	0.508153	+	0.880146i	−0.491803	+	0.870707i	$0.663662\pi$
$858$		0			0
$859$	−7.89141e9	+	1.36683e10i	−0.424794	+	0.735765i	−0.996401	−	0.0847627i	$-0.972987\pi$
$859$	−7.89141e9	+	1.36683e10i	−0.424794	+	0.735765i	0.571607	+	0.820527i	$0.306320\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	2.21027e10			1.17060			0.585300	−	0.810817i	$-0.300977\pi$
$863$	2.21027e10			1.17060			0.585300	+	0.810817i	$0.300977\pi$
$864$		0			0
$865$	6.46554e9			0.339663
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−3.15735e9	+	5.46869e9i	−0.163213	+	0.282692i
$870$		0			0
$871$	9.35578e8	+	1.62047e9i	0.0479752	+	0.0830954i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−4.55567e9	−	7.89064e9i	−0.229892	−	0.398184i
$876$		0			0
$877$	5.63562e9	−	9.76117e9i	0.282126	−	0.488656i	−0.689782	−	0.724017i	$-0.742294\pi$
$877$	5.63562e9	−	9.76117e9i	0.282126	−	0.488656i	0.971908	+	0.235361i	$0.0756271\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	2.22841e10			1.09794			0.548970	−	0.835842i	$-0.315020\pi$
$881$	2.22841e10			1.09794			0.548970	+	0.835842i	$0.315020\pi$
$882$		0			0
$883$	−3.02051e10			−1.47645			−0.738224	−	0.674555i	$-0.764335\pi$
$883$	−3.02051e10			−1.47645			−0.738224	+	0.674555i	$0.764335\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−1.32998e10	+	2.30360e10i	−0.639903	+	1.10834i	0.345551	+	0.938400i	$0.387692\pi$
$887$	−1.32998e10	+	2.30360e10i	−0.639903	+	1.10834i	−0.985454	+	0.169944i	$0.945641\pi$
$888$		0			0
$889$	−9.08175e8	−	1.57300e9i	−0.0433524	−	0.0750886i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−2.27612e10	−	3.94236e10i	−1.06959	−	1.85258i
$894$		0			0
$895$	1.37938e10	−	2.38915e10i	0.643135	−	1.11394i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	5.13495e9			0.235710
$900$		0			0
$901$	−9.24906e9			−0.421270
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−2.55790e10	+	4.43041e10i	−1.14713	+	1.98689i
$906$		0			0
$907$	1.34288e9	+	2.32594e9i	0.0597602	+	0.103508i	0.894358	−	0.447353i	$-0.147633\pi$
$907$	1.34288e9	+	2.32594e9i	0.0597602	+	0.103508i	−0.834597	+	0.550860i	$0.814300\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−3.82911e9	−	6.63221e9i	−0.167797	−	0.290632i	0.769848	−	0.638227i	$-0.220332\pi$
$911$	−3.82911e9	−	6.63221e9i	−0.167797	−	0.290632i	−0.937645	+	0.347595i	$0.886999\pi$
$912$		0			0
$913$	−3.22446e9	+	5.58493e9i	−0.140220	+	0.242868i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	5.67342e9			0.242970
$918$		0			0
$919$	2.78330e9			0.118292			0.0591462	−	0.998249i	$-0.481162\pi$
$919$	2.78330e9			0.118292			0.0591462	+	0.998249i	$0.481162\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−3.02401e9	+	5.23773e9i	−0.126583	+	0.219249i
$924$		0			0
$925$	2.26286e10	+	3.91938e10i	0.940072	+	1.62825i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	1.97854e10	+	3.42692e10i	0.809634	+	1.40233i	0.913117	+	0.407697i	$0.133668\pi$
$929$	1.97854e10	+	3.42692e10i	0.809634	+	1.40233i	−0.103483	+	0.994631i	$0.532999\pi$
$930$		0			0
$931$	−1.38622e10	+	2.40101e10i	−0.563001	+	0.975146i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−1.71400e10			−0.685756
$936$		0			0
$937$	−4.48366e9			−0.178051			−0.0890255	−	0.996029i	$-0.528375\pi$
$937$	−4.48366e9			−0.178051			−0.0890255	+	0.996029i	$0.528375\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	1.37920e10	−	2.38885e10i	0.539591	−	0.934599i	−0.459335	−	0.888263i	$-0.651912\pi$
$941$	1.37920e10	−	2.38885e10i	0.539591	−	0.934599i	0.998926	−	0.0463355i	$-0.0147543\pi$
$942$		0			0
$943$	2.44310e10	+	4.23157e10i	0.948747	+	1.64328i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	1.71646e9	+	2.97299e9i	0.0656762	+	0.113755i	0.896994	−	0.442043i	$-0.145746\pi$
$947$	1.71646e9	+	2.97299e9i	0.0656762	+	0.113755i	−0.831318	+	0.555798i	$0.812413\pi$
$948$		0			0
$949$	1.20541e8	−	2.08784e8i	0.00457831	−	0.00792986i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	3.24967e10			1.21623			0.608113	−	0.793851i	$-0.291927\pi$
$953$	3.24967e10			1.21623			0.608113	+	0.793851i	$0.291927\pi$
$954$		0			0
$955$	5.11104e10			1.89888
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−3.21542e9	+	5.56926e9i	−0.117726	+	0.203907i
$960$		0			0
$961$	1.31685e10	+	2.28084e10i	0.478633	+	0.829017i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−3.69725e10	−	6.40383e10i	−1.32444	−	2.29400i
$966$		0			0
$967$	−1.29956e10	+	2.25091e10i	−0.462173	+	0.800508i	−0.999069	−	0.0431408i	$-0.986264\pi$
$967$	−1.29956e10	+	2.25091e10i	−0.462173	+	0.800508i	0.536896	+	0.843649i	$0.319597\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	1.28654e9			0.0450980			0.0225490	−	0.999746i	$-0.492822\pi$
$971$	1.28654e9			0.0450980			0.0225490	+	0.999746i	$0.492822\pi$
$972$		0			0
$973$	−4.27905e9			−0.148920
$974$		0			0
$975$		0			0
$976$		0			0
$977$	5.73369e9	−	9.93105e9i	0.196700	−	0.340694i	−0.750757	−	0.660579i	$-0.770311\pi$
$977$	5.73369e9	−	9.93105e9i	0.196700	−	0.340694i	0.947456	+	0.319885i	$0.103644\pi$
$978$		0			0
$979$	2.61986e9	+	4.53773e9i	0.0892356	+	0.154561i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−2.37066e10	−	4.10609e10i	−0.796033	−	1.37877i	−0.922181	−	0.386758i	$-0.873595\pi$
$983$	−2.37066e10	−	4.10609e10i	−0.796033	−	1.37877i	0.126148	−	0.992011i	$-0.459739\pi$
$984$		0			0
$985$	−1.63756e10	+	2.83634e10i	−0.545973	+	0.945653i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−3.24459e10			−1.06653
$990$		0			0
$991$	4.06382e10			1.32641			0.663203	−	0.748440i	$-0.269197\pi$
$991$	4.06382e10			1.32641			0.663203	+	0.748440i	$0.269197\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−8.47066e9	+	1.46716e10i	−0.272607	+	0.472169i
$996$		0			0
$997$	−9.07756e8	−	1.57228e9i	−0.0290092	−	0.0502455i	0.851156	−	0.524912i	$-0.175902\pi$
$997$	−9.07756e8	−	1.57228e9i	−0.0290092	−	0.0502455i	−0.880166	+	0.474667i	$0.842569\pi$
$998$		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.145.1		20
3.2	odd	2		72.8.i.a.49.1	yes	20
4.3	odd	2		432.8.i.e.145.1		20
9.2	odd	6		72.8.i.a.25.1	✓	20
9.7	even	3	inner	216.8.i.a.73.1		20
12.11	even	2		144.8.i.e.49.10		20
36.7	odd	6		432.8.i.e.289.1		20
36.11	even	6		144.8.i.e.97.10		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.8.i.a.25.1	✓	20	9.2	odd	6
72.8.i.a.49.1	yes	20	3.2	odd	2
144.8.i.e.49.10		20	12.11	even	2
144.8.i.e.97.10		20	36.11	even	6
216.8.i.a.73.1		20	9.7	even	3	inner
216.8.i.a.145.1		20	1.1	even	1	trivial
432.8.i.e.145.1		20	4.3	odd	2
432.8.i.e.289.1		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)

\(f(q)\)	\(=\)	\(q+(-266.720 + 461.973i) q^{5} +(-66.5594 - 115.284i) q^{7} +O(q^{10})\)	\(q+(-266.720 + 461.973i) q^{5} +(-66.5594 - 115.284i) q^{7} +(-731.541 - 1267.07i) q^{11} +(1045.78 - 1811.34i) q^{13} -21961.2 q^{17} -34405.2 q^{19} +(-32866.5 + 56926.4i) q^{23} +(-103217. - 178777. i) q^{25} +(74878.5 + 129693. i) q^{29} +(17144.3 - 29694.7i) q^{31} +71010.9 q^{35} -219233. q^{37} +(371671. - 643752. i) q^{41} +(246801. + 427472. i) q^{43} +(661565. + 1.14586e6i) q^{47} +(402911. - 697863. i) q^{49} +421155. q^{53} +780467. q^{55} +(1.23603e6 - 2.14086e6i) q^{59} +(-693435. - 1.20107e6i) q^{61} +(557861. + 966244. i) q^{65} +(-447311. + 774766. i) q^{67} -2.89163e6 q^{71} +115265. q^{73} +(-97381.9 + 168670. i) q^{77} +(-2.15801e6 - 3.73779e6i) q^{79} +(-2.20388e6 - 3.81724e6i) q^{83} +(5.85749e6 - 1.01455e7i) q^{85} -3.58129e6 q^{89} -278426. q^{91} +(9.17655e6 - 1.58943e7i) q^{95} +(4.45354e6 + 7.71376e6i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 125 q^{5} + 1245 q^{7}+O(q^{10}) \) 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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