LMFDB - Embedded newform 63.8.p.a.17.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [63,8,Mod(17,63)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(63, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 1]))

N = Newforms(chi, 8, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("63.17");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 63 = 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	63.p (of order $6$, degree $2$, 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:	$19.6802566055$
Analytic rank:	$0$
Dimension:	$36$
Relative dimension:	$18$ over $\Q(\zeta_{6})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			17.13
Character	$\chi$	$=$	63.17
Dual form			63.8.p.a.26.13

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(6.95719 - 4.01673i) q^{2} +(-31.7317 + 54.9609i) q^{4} +(-79.8320 - 138.273i) q^{5} +(726.865 - 543.333i) q^{7} +1538.12i q^{8} +(-1110.81 - 641.328i) q^{10} +(263.758 + 152.281i) q^{11} -7181.21i q^{13} +(2874.51 - 6699.69i) q^{14} +(2116.54 + 3665.96i) q^{16} +(11932.8 - 20668.2i) q^{17} +(41545.4 - 23986.3i) q^{19} +10132.8 q^{20} +2446.69 q^{22} +(-36696.9 + 21186.9i) q^{23} +(26316.2 - 45581.0i) q^{25} +(-28845.0 - 49961.0i) q^{26} +(6797.39 + 57190.0i) q^{28} -225444. i q^{29} +(89168.9 + 51481.7i) q^{31} +(-141052. - 81436.2i) q^{32} -191724. i q^{34} +(-133155. - 57130.6i) q^{35} +(217757. + 377165. i) q^{37} +(192693. - 333754. i) q^{38} +(212680. - 122791. i) q^{40} -262195. q^{41} -26102.5 q^{43} +(-16739.0 + 9664.26i) q^{44} +(-170205. + 294803. i) q^{46} +(-147468. - 255422. i) q^{47} +(233123. - 789859. i) q^{49} -422821. i q^{50} +(394686. + 227872. i) q^{52} +(749984. + 433003. i) q^{53} -48627.6i q^{55} +(835708. + 1.11800e6i) q^{56} +(-905550. - 1.56846e6i) q^{58} +(-624984. + 1.08250e6i) q^{59} +(-570860. + 329586. i) q^{61} +827153. q^{62} -1.85026e6 q^{64} +(-992968. + 573290. i) q^{65} +(-490387. + 849376. i) q^{67} +(757297. + 1.31168e6i) q^{68} +(-1.15587e6 + 137382. i) q^{70} +1.65029e6i q^{71} +(1.73722e6 + 1.00298e6i) q^{73} +(3.02995e6 + 1.74934e6i) q^{74} +3.04450e6i q^{76} +(274456. - 32620.8i) q^{77} +(558983. + 968188. i) q^{79} +(337936. - 585322. i) q^{80} +(-1.82414e6 + 1.05317e6i) q^{82} -7.99427e6 q^{83} -3.81048e6 q^{85} +(-181600. + 104847. i) q^{86} +(-234226. + 405690. i) q^{88} +(4.33239e6 + 7.50393e6i) q^{89} +(-3.90179e6 - 5.21977e6i) q^{91} -2.68919e6i q^{92} +(-2.05192e6 - 1.18468e6i) q^{94} +(-6.63331e6 - 3.82974e6i) q^{95} +8.18016e6i q^{97} +(-1.55078e6 - 6.43159e6i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 36 q + 1024 q^{4} - 2074 q^{7} + 1248 q^{10} - 59708 q^{16} - 105330 q^{19} + 544 q^{22} - 56250 q^{25} + 556220 q^{28} + 86862 q^{31} + 591034 q^{37} - 3036324 q^{40} - 837332 q^{43} + 3896752 q^{46} + 6626454 q^{49}+ \cdots + 96093708 q^{94}+O(q^{100}) $ 36 * q + 1024 * q^4 - 2074 * q^7 + 1248 * q^10 - 59708 * q^16 - 105330 * q^19 + 544 * q^22 - 56250 * q^25 + 556220 * q^28 + 86862 * q^31 + 591034 * q^37 - 3036324 * q^40 - 837332 * q^43 + 3896752 * q^46 + 6626454 * q^49 - 11085900 * q^52 - 134084 * q^58 + 4324644 * q^61 - 11555464 * q^64 + 8750706 * q^67 + 8823972 * q^70 - 744402 * q^73 + 1136238 * q^79 - 29281932 * q^82 + 18719472 * q^85 - 4365916 * q^88 + 21779742 * q^91 + 96093708 * q^94

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/63\mathbb{Z}\right)^\times$.

$n$	$10$	$29$
$\chi(n)$	$e\left(\frac{1}{6}\right)$	$-1$

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 $

$2$	6.95719	−	4.01673i	0.614934	−	0.355032i	−0.159960	−	0.987124i	$-0.551136\pi$
$2$	6.95719	−	4.01673i	0.614934	−	0.355032i	0.774894	+	0.632091i	$0.217803\pi$
$3$		0			0
$3$		0			0
$4$	−31.7317	+	54.9609i	−0.247904	+	0.429382i
$5$	−79.8320	−	138.273i	−0.285616	−	0.494701i	0.687143	−	0.726523i	$-0.258865\pi$
$5$	−79.8320	−	138.273i	−0.285616	−	0.494701i	−0.972758	+	0.231822i	$0.925531\pi$
$6$		0			0
$7$	726.865	−	543.333i	0.800960	−	0.598718i
$7$	726.865	−	543.333i	0.800960	−	0.598718i
$8$			1538.12i			1.06212i
$9$		0			0
$10$	−1110.81	−	641.328i	−0.351270	−	0.202806i
$11$	263.758	+	152.281i	0.0597491	+	0.0344962i	0.529577	−	0.848262i	$-0.322351\pi$
$11$	263.758	+	152.281i	0.0597491	+	0.0344962i	−0.469828	+	0.882758i	$0.655684\pi$
$12$		0			0
$13$		−	7181.21i		−	0.906559i	−0.891369	−	0.453279i	$-0.850254\pi$
$13$		−	7181.21i		−	0.906559i	0.891369	−	0.453279i	$-0.149746\pi$
$14$	2874.51	−	6699.69i	0.279973	−	0.652539i
$15$		0			0
$16$	2116.54	+	3665.96i	0.129184	+	0.223752i
$17$	11932.8	−	20668.2i	0.589076	−	1.02031i	−0.405277	−	0.914194i	$-0.632825\pi$
$17$	11932.8	−	20668.2i	0.589076	−	1.02031i	0.994354	−	0.106116i	$-0.0338416\pi$
$18$		0			0
$19$	41545.4	−	23986.3i	1.38959	−	0.802279i	0.396319	−	0.918113i	$-0.370288\pi$
$19$	41545.4	−	23986.3i	1.38959	−	0.802279i	0.993269	+	0.115834i	$0.0369542\pi$
$20$	10132.8			0.283221
$21$		0			0
$22$	2446.69			0.0489891
$23$	−36696.9	+	21186.9i	−0.628900	+	0.363095i	−0.780326	−	0.625373i	$-0.784947\pi$
$23$	−36696.9	+	21186.9i	−0.628900	+	0.363095i	0.151426	+	0.988469i	$0.451613\pi$
$24$		0			0
$25$	26316.2	−	45581.0i	0.336847	−	0.583437i
$26$	−28845.0	−	49961.0i	−0.321858	−	0.557474i
$27$		0			0
$28$	6797.39	+	57190.0i	0.0585179	+	0.492342i
$29$		−	225444.i		−	1.71651i	−0.513223	−	0.858255i	$-0.671549\pi$
$29$		−	225444.i		−	1.71651i	0.513223	−	0.858255i	$-0.328451\pi$
$30$		0			0
$31$	89168.9	+	51481.7i	0.537585	+	0.310375i	0.744100	−	0.668068i	$-0.232879\pi$
$31$	89168.9	+	51481.7i	0.537585	+	0.310375i	−0.206514	+	0.978444i	$0.566212\pi$
$32$	−141052.	−	81436.2i	−0.760945	−	0.439332i
$33$		0			0
$34$		−	191724.i		−	0.836565i
$35$	−133155.	−	57130.6i	−0.524953	−	0.225232i
$36$		0			0
$37$	217757.	+	377165.i	0.706749	+	1.22413i	0.966057	+	0.258330i	$0.0831723\pi$
$37$	217757.	+	377165.i	0.706749	+	1.22413i	−0.259308	+	0.965795i	$0.583494\pi$
$38$	192693.	−	333754.i	0.569670	−	0.986697i
$39$		0			0
$40$	212680.	−	122791.i	0.525432	−	0.303358i
$41$	−262195.			−0.594129			−0.297065	−	0.954857i	$-0.596008\pi$
$41$	−262195.			−0.594129			−0.297065	+	0.954857i	$0.596008\pi$
$42$		0			0
$43$	−26102.5			−0.0500659			−0.0250329	−	0.999687i	$-0.507969\pi$
$43$	−26102.5			−0.0500659			−0.0250329	+	0.999687i	$0.507969\pi$
$44$	−16739.0	+	9664.26i	−0.0296241	+	0.0171035i
$45$		0			0
$46$	−170205.	+	294803.i	−0.257821	+	0.446560i
$47$	−147468.	−	255422.i	−0.207183	−	0.358852i	0.743643	−	0.668577i	$-0.233096\pi$
$47$	−147468.	−	255422.i	−0.207183	−	0.358852i	−0.950826	+	0.309725i	$0.899763\pi$
$48$		0			0
$49$	233123.	−	789859.i	0.283073	−	0.959098i
$50$		−	422821.i		−	0.478367i
$51$		0			0
$52$	394686.	+	227872.i	0.389260	+	0.224739i
$53$	749984.	+	433003.i	0.691968	+	0.399508i	0.804349	−	0.594157i	$-0.202514\pi$
$53$	749984.	+	433003.i	0.691968	+	0.399508i	−0.112381	+	0.993665i	$0.535848\pi$
$54$		0			0
$55$		−	48627.6i		−	0.0394106i
$56$	835708.	+	1.11800e6i	0.635911	+	0.850716i
$57$		0			0
$58$	−905550.	−	1.56846e6i	−0.609417	−	1.05554i
$59$	−624984.	+	1.08250e6i	−0.396175	+	0.686195i	−0.993250	−	0.115990i	$-0.962996\pi$
$59$	−624984.	+	1.08250e6i	−0.396175	+	0.686195i	0.597075	+	0.802185i	$0.296329\pi$
$60$		0			0
$61$	−570860.	+	329586.i	−0.322014	+	0.185915i	−0.652290	−	0.757969i	$-0.726192\pi$
$61$	−570860.	+	329586.i	−0.322014	+	0.185915i	0.330276	+	0.943884i	$0.392858\pi$
$62$	827153.			0.440773
$63$		0			0
$64$	−1.85026e6			−0.882275
$65$	−992968.	+	573290.i	−0.448475	+	0.258927i
$66$		0			0
$67$	−490387.	+	849376.i	−0.199195	+	0.345015i	−0.948267	−	0.317472i	$-0.897166\pi$
$67$	−490387.	+	849376.i	−0.199195	+	0.345015i	0.749073	+	0.662487i	$0.230499\pi$
$68$	757297.	+	1.31168e6i	0.292069	+	0.505878i
$69$		0			0
$70$	−1.15587e6	+	137382.i	−0.402776	+	0.0478725i
$71$			1.65029e6i			0.547213i	0.961842	+	0.273607i	$0.0882166\pi$
$71$			1.65029e6i			0.547213i	−0.961842	+	0.273607i	$0.911783\pi$
$72$		0			0
$73$	1.73722e6	+	1.00298e6i	0.522665	+	0.301761i	0.738025	−	0.674774i	$-0.235759\pi$
$73$	1.73722e6	+	1.00298e6i	0.522665	+	0.301761i	−0.215359	+	0.976535i	$0.569092\pi$
$74$	3.02995e6	+	1.74934e6i	0.869208	+	0.501838i
$75$		0			0
$76$			3.04450e6i			0.795552i
$77$	274456.	−	32620.8i	0.0685101	−	0.00814285i
$78$		0			0
$79$	558983.	+	968188.i	0.127557	+	0.220935i	0.922730	−	0.385448i	$-0.125953\pi$
$79$	558983.	+	968188.i	0.127557	+	0.220935i	−0.795173	+	0.606383i	$0.792620\pi$
$80$	337936.	−	585322.i	0.0737937	−	0.127814i
$81$		0			0
$82$	−1.82414e6	+	1.05317e6i	−0.365350	+	0.210935i
$83$	−7.99427e6			−1.53464			−0.767318	−	0.641267i	$-0.778409\pi$
$83$	−7.99427e6			−1.53464			−0.767318	+	0.641267i	$0.778409\pi$
$84$		0			0
$85$	−3.81048e6			−0.672998
$86$	−181600.	+	104847.i	−0.0307872	+	0.0177750i
$87$		0			0
$88$	−234226.	+	405690.i	−0.0366391	+	0.0634608i
$89$	4.33239e6	+	7.50393e6i	0.651423	+	1.12830i	0.982778	+	0.184791i	$0.0591608\pi$
$89$	4.33239e6	+	7.50393e6i	0.651423	+	1.12830i	−0.331355	+	0.943506i	$0.607506\pi$
$90$		0			0
$91$	−3.90179e6	−	5.21977e6i	−0.542773	−	0.726117i
$92$		−	2.68919e6i		−	0.360051i
$93$		0			0
$94$	−2.05192e6	−	1.18468e6i	−0.254808	−	0.147114i
$95$	−6.63331e6	−	3.82974e6i	−0.793776	−	0.458287i
$96$		0			0
$97$			8.18016e6i			0.910040i	0.890481	+	0.455020i	$0.150368\pi$
$97$			8.18016e6i			0.910040i	−0.890481	+	0.455020i	$0.849632\pi$
$98$	−1.55078e6	−	6.43159e6i	−0.166440	−	0.690283i
$99$		0			0
$100$	1.67012e6	+	2.89272e6i	0.167012	+	0.289272i
$101$	3.68757e6	−	6.38706e6i	0.356136	−	0.616845i	−0.631176	−	0.775640i	$-0.717427\pi$
$101$	3.68757e6	−	6.38706e6i	0.356136	−	0.616845i	0.987312	+	0.158794i	$0.0507607\pi$
$102$		0			0
$103$	1.82216e7	−	1.05203e7i	1.64307	−	0.948630i	0.663343	−	0.748315i	$-0.269137\pi$
$103$	1.82216e7	−	1.05203e7i	1.64307	−	0.948630i	0.979732	−	0.200315i	$-0.0641964\pi$
$104$	1.10455e7			0.962875
$105$		0			0
$106$	6.95703e6			0.567353
$107$	−8.09275e6	+	4.67235e6i	−0.638636	+	0.368716i	−0.784089	−	0.620649i	$-0.786869\pi$
$107$	−8.09275e6	+	4.67235e6i	−0.638636	+	0.368716i	0.145453	+	0.989365i	$0.453536\pi$
$108$		0			0
$109$	1.06731e7	−	1.84863e7i	0.789401	−	1.36728i	−0.136934	−	0.990580i	$-0.543725\pi$
$109$	1.06731e7	−	1.84863e7i	0.789401	−	1.36728i	0.926335	−	0.376702i	$-0.122942\pi$
$110$	−195324.	−	338311.i	−0.0139920	−	0.0242349i
$111$		0			0
$112$	3.53028e6	+	1.51467e6i	0.237435	+	0.101872i
$113$		−	2.69564e7i		−	1.75747i	−0.477311	−	0.878735i	$-0.658388\pi$
$113$		−	2.69564e7i		−	1.75747i	0.477311	−	0.878735i	$-0.341612\pi$
$114$		0			0
$115$	5.85917e6	+	3.38279e6i	0.359247	+	0.207411i
$116$	1.23906e7	+	7.15373e6i	0.737039	+	0.425530i
$117$		0			0
$118$			1.00416e7i			0.562620i
$119$	−2.55618e6	−	2.15065e7i	−0.139052	−	1.16992i
$120$		0			0
$121$	−9.69721e6	−	1.67961e7i	−0.497620	−	0.861903i
$122$	−2.64772e6	+	4.58599e6i	−0.132012	+	0.228651i
$123$		0			0
$124$	−5.65896e6	+	3.26720e6i	−0.266539	+	0.153886i
$125$	−2.08773e7			−0.956067
$126$		0			0
$127$	−1.47613e7			−0.639456			−0.319728	−	0.947509i	$-0.603592\pi$
$127$	−1.47613e7			−0.639456			−0.319728	+	0.947509i	$0.603592\pi$
$128$	5.18197e6	−	2.99181e6i	0.218404	−	0.126095i
$129$		0			0
$130$	−4.60551e6	+	7.97698e6i	−0.183855	+	0.318447i
$131$	4.04219e6	+	7.00128e6i	0.157097	+	0.272100i	0.933821	−	0.357742i	$-0.116453\pi$
$131$	4.04219e6	+	7.00128e6i	0.157097	+	0.272100i	−0.776724	+	0.629841i	$0.783120\pi$
$132$		0			0
$133$	1.71654e7	−	4.00078e7i	0.632664	−	1.47456i
$134$			7.87902e6i			0.282882i
$135$		0			0
$136$	3.17901e7	+	1.83540e7i	1.08369	+	0.625670i
$137$	−2.20103e7	−	1.27077e7i	−0.731315	−	0.422225i	0.0875880	−	0.996157i	$-0.472084\pi$
$137$	−2.20103e7	−	1.27077e7i	−0.731315	−	0.422225i	−0.818903	+	0.573932i	$0.805417\pi$
$138$		0			0
$139$			2.69509e7i			0.851181i	0.904916	+	0.425590i	$0.139934\pi$
$139$			2.69509e7i			0.851181i	−0.904916	+	0.425590i	$0.860066\pi$
$140$	7.36519e6	−	5.50549e6i	0.226849	−	0.169570i
$141$		0			0
$142$	6.62878e6	+	1.14814e7i	0.194278	+	0.336500i
$143$	1.09356e6	−	1.89410e6i	0.0312728	−	0.0541661i
$144$		0			0
$145$	−3.11729e7	+	1.79977e7i	−0.849159	+	0.490262i
$146$	1.61149e7			0.428540
$147$		0			0
$148$	−2.76391e7			−0.700823
$149$	−4.89787e7	+	2.82779e7i	−1.21299	+	0.700318i	−0.963409	−	0.268037i	$-0.913625\pi$
$149$	−4.89787e7	+	2.82779e7i	−1.21299	+	0.700318i	−0.249577	+	0.968355i	$0.580292\pi$
$150$		0			0
$151$	−1.22411e7	+	2.12022e7i	−0.289335	+	0.501143i	−0.973651	−	0.228042i	$-0.926768\pi$
$151$	−1.22411e7	+	2.12022e7i	−0.289335	+	0.501143i	0.684316	+	0.729186i	$0.260101\pi$
$152$	3.68936e7	+	6.39017e7i	0.852117	+	1.47591i
$153$		0			0
$154$	1.77841e6	−	1.32936e6i	0.0392383	−	0.0293306i
$155$		−	1.64395e7i		−	0.354592i
$156$		0			0
$157$	−1.26855e7	−	7.32399e6i	−0.261613	−	0.151042i	0.363457	−	0.931611i	$-0.381596\pi$
$157$	−1.26855e7	−	7.32399e6i	−0.261613	−	0.151042i	−0.625070	+	0.780569i	$0.714930\pi$
$158$	7.77791e6	+	4.49058e6i	0.156878	+	0.0905737i
$159$		0			0
$160$			2.60049e7i			0.501920i
$161$	−1.51621e7	+	3.53386e7i	−0.286331	+	0.667359i
$162$		0			0
$163$	−2.68526e7	−	4.65100e7i	−0.485657	−	0.841182i	0.514207	−	0.857666i	$-0.328086\pi$
$163$	−2.68526e7	−	4.65100e7i	−0.485657	−	0.841182i	−0.999864	+	0.0164836i	$0.994753\pi$
$164$	8.31989e6	−	1.44105e7i	0.147287	−	0.255108i
$165$		0			0
$166$	−5.56176e7	+	3.21108e7i	−0.943700	+	0.544846i
$167$	5.46056e7			0.907255			0.453628	−	0.891191i	$-0.350130\pi$
$167$	5.46056e7			0.907255			0.453628	+	0.891191i	$0.350130\pi$
$168$		0			0
$169$	1.11787e7			0.178151
$170$	−2.65102e7	+	1.53057e7i	−0.413849	+	0.238936i
$171$		0			0
$172$	828276.	−	1.43462e6i	0.0124115	−	0.0214974i
$173$	3.36215e7	+	5.82342e7i	0.493692	+	0.855100i	0.999974	−	0.00726861i	$-0.00231369\pi$
$173$	3.36215e7	+	5.82342e7i	0.493692	+	0.855100i	−0.506282	+	0.862368i	$0.668980\pi$
$174$		0			0
$175$	−5.63731e6	−	4.74297e7i	−0.0795131	−	0.668986i
$176$			1.28924e6i			0.0178253i
$177$		0			0
$178$	6.02826e7	+	3.48041e7i	0.801164	+	0.462552i
$179$	8.34579e7	+	4.81845e7i	1.08763	+	0.627945i	0.932945	−	0.360019i	$-0.117230\pi$
$179$	8.34579e7	+	4.81845e7i	1.08763	+	0.627945i	0.154687	+	0.987964i	$0.450563\pi$
$180$		0			0
$181$		−	4.99829e7i		−	0.626537i	−0.949665	−	0.313269i	$-0.898576\pi$
$181$		−	4.99829e7i		−	0.626537i	0.949665	−	0.313269i	$-0.101424\pi$
$182$	−4.81119e7	−	2.06425e7i	−0.591565	−	0.253812i
$183$		0			0
$184$	−3.25879e7	−	5.64440e7i	−0.385651	−	0.667967i
$185$	3.47679e7	−	6.02197e7i	0.403717	−	0.699259i
$186$		0			0
$187$	6.29475e6	−	3.63428e6i	0.0703936	−	0.0406418i
$188$	1.87176e7			0.205446
$189$		0			0
$190$	−6.15323e7			−0.650827
$191$	−1.44720e8	+	8.35540e7i	−1.50283	+	0.867662i	−0.502840	+	0.864380i	$0.667711\pi$
$191$	−1.44720e8	+	8.35540e7i	−1.50283	+	0.867662i	−0.999995	+	0.00328204i	$0.998955\pi$
$192$		0			0
$193$	−9.90822e6	+	1.71615e7i	−0.0992076	+	0.171833i	−0.911357	−	0.411617i	$-0.864964\pi$
$193$	−9.90822e6	+	1.71615e7i	−0.0992076	+	0.171833i	0.812149	+	0.583450i	$0.198297\pi$
$194$	3.28575e7	+	5.69109e7i	0.323094	+	0.559615i
$195$		0			0
$196$	3.60140e7	+	3.78762e7i	0.341645	+	0.359311i
$197$		−	6.22113e6i		−	0.0579746i	−0.999580	−	0.0289873i	$-0.990772\pi$
$197$		−	6.22113e6i		−	0.0579746i	0.999580	−	0.0289873i	$-0.00922824\pi$
$198$		0			0
$199$	5.38158e6	+	3.10706e6i	0.0484088	+	0.0279488i	0.524009	−	0.851713i	$-0.324436\pi$
$199$	5.38158e6	+	3.10706e6i	0.0484088	+	0.0279488i	−0.475600	+	0.879662i	$0.657769\pi$
$200$	7.01088e7	+	4.04773e7i	0.619680	+	0.357773i
$201$		0			0
$202$		−	5.92480e7i		−	0.505759i
$203$	−1.22491e8	−	1.63868e8i	−1.02771	−	1.37486i
$204$		0			0
$205$	2.09315e7	+	3.62545e7i	0.169693	+	0.293916i
$206$	8.45143e7	−	1.46383e8i	0.673589	−	1.16669i
$207$		0			0
$208$	2.63260e7	−	1.51993e7i	0.202845	−	0.117112i
$209$	1.46106e7			0.110702
$210$		0			0
$211$	2.38902e8			1.75078			0.875388	−	0.483421i	$-0.160606\pi$
$211$	2.38902e8			1.75078			0.875388	+	0.483421i	$0.160606\pi$
$212$	−4.75965e7	+	2.74798e7i	−0.343083	+	0.198079i
$213$		0			0
$214$	−3.75352e7	+	6.50129e7i	−0.261813	+	0.453473i
$215$	2.08381e6	+	3.60927e6i	0.0142996	+	0.0247676i
$216$		0			0
$217$	9.27854e7	−	1.10281e7i	0.616411	−	0.0732643i
$218$		−	1.71484e8i		−	1.12105i
$219$		0			0
$220$	2.67261e6	+	1.54303e6i	0.0169222	+	0.00977004i
$221$	−1.48423e8	−	8.56920e7i	−0.924971	−	0.534032i
$222$		0			0
$223$			1.65228e8i			0.997741i	0.866677	+	0.498870i	$0.166252\pi$
$223$			1.65228e8i			0.997741i	−0.866677	+	0.498870i	$0.833748\pi$
$224$	−1.46772e8	+	1.74448e7i	−0.872522	+	0.103705i
$225$		0			0
$226$	−1.08277e8	−	1.87541e8i	−0.623959	−	1.08073i
$227$	1.72318e8	−	2.98463e8i	0.977777	−	1.69356i	0.307327	−	0.951604i	$-0.400566\pi$
$227$	1.72318e8	−	2.98463e8i	0.977777	−	1.69356i	0.670450	−	0.741955i	$-0.266101\pi$
$228$		0			0
$229$	−2.64418e8	+	1.52662e8i	−1.45501	+	0.840053i	−0.998759	−	0.0497957i	$-0.984143\pi$
$229$	−2.64418e8	+	1.52662e8i	−1.45501	+	0.840053i	−0.456255	+	0.889849i	$0.650810\pi$
$230$	5.43511e7			0.294551
$231$		0			0
$232$	3.46759e8			1.82314
$233$	−7.08819e7	+	4.09237e7i	−0.367104	+	0.211948i	−0.672193	−	0.740376i	$-0.734647\pi$
$233$	−7.08819e7	+	4.09237e7i	−0.367104	+	0.211948i	0.305088	+	0.952324i	$0.401314\pi$
$234$		0			0
$235$	−2.35453e7	+	4.07817e7i	−0.118350	+	0.204988i
$236$	−3.96636e7	−	6.86994e7i	−0.196427	−	0.340221i
$237$		0			0
$238$	−1.04170e8	−	1.39357e8i	−0.500867	−	0.670055i
$239$			1.50479e8i			0.712990i	0.934297	+	0.356495i	$0.116028\pi$
$239$			1.50479e8i			0.712990i	−0.934297	+	0.356495i	$0.883972\pi$
$240$		0			0
$241$	2.74860e8	+	1.58690e8i	1.26489	+	0.730283i	0.974016	−	0.226480i	$-0.0727217\pi$
$241$	2.74860e8	+	1.58690e8i	1.26489	+	0.730283i	0.290871	+	0.956762i	$0.406055\pi$
$242$	−1.34931e8	−	7.79022e7i	−0.612007	−	0.353343i
$243$		0			0
$244$		−	4.18333e7i		−	0.184356i
$245$	−1.27827e8	+	3.08214e7i	−0.555317	+	0.133897i
$246$		0			0
$247$	−1.72250e8	−	2.98347e8i	−0.727313	−	1.25974i
$248$	−7.91848e7	+	1.37152e8i	−0.329656	+	0.570981i
$249$		0			0
$250$	−1.45247e8	+	8.38584e7i	−0.587918	+	0.339435i
$251$	1.79103e8			0.714898			0.357449	−	0.933933i	$-0.383647\pi$
$251$	1.79103e8			0.714898			0.357449	+	0.933933i	$0.383647\pi$
$252$		0			0
$253$	−1.29055e7			−0.0501016
$254$	−1.02697e8	+	5.92921e7i	−0.393223	+	0.227028i
$255$		0			0
$256$	1.42452e8	−	2.46733e8i	0.530673	−	0.919153i
$257$	2.01176e8	+	3.48446e8i	0.739281	+	1.28047i	0.952820	+	0.303537i	$0.0981677\pi$
$257$	2.01176e8	+	3.48446e8i	0.739281	+	1.28047i	−0.213539	+	0.976935i	$0.568499\pi$
$258$		0			0
$259$	3.63206e8	+	1.55834e8i	1.29898	+	0.557331i
$260$		−	7.27659e7i		−	0.256756i
$261$		0			0
$262$	5.62445e7	+	3.24728e7i	0.193208	+	0.111549i
$263$	3.59842e8	+	2.07755e8i	1.21974	+	0.704216i	0.964862	−	0.262757i	$-0.0846319\pi$
$263$	3.59842e8	+	2.07755e8i	1.21974	+	0.704216i	0.254876	+	0.966974i	$0.417965\pi$
$264$		0			0
$265$		−	1.38270e8i		−	0.456423i
$266$	−4.12776e7	−	3.47290e8i	−0.134471	−	1.13138i
$267$		0			0
$268$	−3.11216e7	−	5.39043e7i	−0.0987622	−	0.171061i
$269$	1.59733e8	−	2.76666e8i	0.500337	−	0.866610i	−0.499663	−	0.866220i	$-0.666543\pi$
$269$	1.59733e8	−	2.76666e8i	0.500337	−	0.866610i	1.00000		0.000389513i	$-0.000123986\pi$
$270$		0			0
$271$	−4.03209e8	+	2.32793e8i	−1.23066	+	0.710521i	−0.967167	−	0.254141i	$-0.918207\pi$
$271$	−4.03209e8	+	2.32793e8i	−1.23066	+	0.710521i	−0.263491	+	0.964662i	$0.584874\pi$
$272$	1.01025e8			0.304396
$273$		0			0
$274$	−2.04173e8			−0.599614
$275$	1.38822e7	−	8.01491e6i	0.0402527	−	0.0232399i
$276$		0			0
$277$	−1.34447e8	+	2.32869e8i	−0.380078	+	0.658314i	−0.991073	−	0.133320i	$-0.957436\pi$
$277$	−1.34447e8	+	2.32869e8i	−0.380078	+	0.658314i	0.610995	+	0.791634i	$0.290769\pi$
$278$	1.08255e8	+	1.87503e8i	0.302197	+	0.523420i
$279$		0			0
$280$	8.78734e7	−	2.04808e8i	0.239224	−	0.557564i
$281$			4.37912e8i			1.17737i	0.808361	+	0.588687i	$0.200355\pi$
$281$			4.37912e8i			1.17737i	−0.808361	+	0.588687i	$0.799645\pi$
$282$		0			0
$283$	−3.77294e8	−	2.17831e8i	−0.989526	−	0.571303i	−0.0843931	−	0.996433i	$-0.526895\pi$
$283$	−3.77294e8	−	2.17831e8i	−0.989526	−	0.571303i	−0.905132	+	0.425130i	$0.860228\pi$
$284$	−9.07015e7	−	5.23665e7i	−0.234963	−	0.135656i
$285$		0			0
$286$		−	1.75702e7i		−	0.0444115i
$287$	−1.90580e8	+	1.42459e8i	−0.475873	+	0.355716i
$288$		0			0
$289$	−7.96147e7	−	1.37897e8i	−0.194022	−	0.336056i
$290$	−1.44584e8	+	2.50426e8i	−0.348118	+	0.602958i
$291$		0			0
$292$	−1.10250e8	+	6.36526e7i	−0.259142	+	0.149615i
$293$	6.77845e8			1.57432			0.787162	−	0.616747i	$-0.211550\pi$
$293$	6.77845e8			1.57432			0.787162	+	0.616747i	$0.211550\pi$
$294$		0			0
$295$	1.99575e8			0.452615
$296$	−5.80124e8	+	3.34935e8i	−1.30017	+	0.750653i
$297$		0			0
$298$	−2.27170e8	+	3.93469e8i	−0.497271	+	0.861299i
$299$	1.52148e8	+	2.63528e8i	0.329167	+	0.570135i
$300$		0			0
$301$	−1.89730e7	+	1.41823e7i	−0.0401008	+	0.0299754i
$302$			1.96677e8i			0.410894i
$303$		0			0
$304$	1.75865e8	+	1.01536e8i	0.359023	+	0.207282i
$305$	9.11458e7	+	5.26231e7i	0.183945	+	0.106201i
$306$		0			0
$307$			4.20196e8i			0.828833i	0.910087	+	0.414417i	$0.136014\pi$
$307$			4.20196e8i			0.828833i	−0.910087	+	0.414417i	$0.863986\pi$
$308$	−6.91608e6	+	1.61194e7i	−0.0134875	+	0.0314357i
$309$		0			0
$310$	−6.60333e7	−	1.14373e8i	−0.125892	−	0.218051i
$311$	−7.09821e6	+	1.22945e7i	−0.0133810	+	0.0231765i	−0.872638	−	0.488367i	$-0.837593\pi$
$311$	−7.09821e6	+	1.22945e7i	−0.0133810	+	0.0231765i	0.859257	+	0.511543i	$0.170926\pi$
$312$		0			0
$313$	−9.76677e7	+	5.63885e7i	−0.180030	+	0.103941i	−0.587307	−	0.809364i	$-0.699812\pi$
$313$	−9.76677e7	+	5.63885e7i	−0.180030	+	0.103941i	0.407277	+	0.913305i	$0.366479\pi$
$314$	−1.17674e8			−0.214500
$315$		0			0
$316$	−7.09500e7			−0.126487
$317$	−4.66188e8	+	2.69154e8i	−0.821965	+	0.474562i	−0.851094	−	0.525014i	$-0.824060\pi$
$317$	−4.66188e8	+	2.69154e8i	−0.821965	+	0.474562i	0.0291284	+	0.999576i	$0.490727\pi$
$318$		0			0
$319$	3.43309e7	−	5.94628e7i	0.0592131	−	0.102560i
$320$	1.47710e8	+	2.55842e8i	0.251992	+	0.436462i
$321$		0			0
$322$	3.64603e7	+	3.06760e8i	0.0608590	+	0.512039i
$323$		−	1.14489e9i		−	1.89041i
$324$		0			0
$325$	−3.27327e8	−	1.88982e8i	−0.528920	−	0.305372i
$326$	−3.73637e8	−	2.15719e8i	−0.597294	−	0.344848i
$327$		0			0
$328$		−	4.03286e8i		−	0.631037i
$329$	−2.45968e8	−	1.05533e8i	−0.380797	−	0.163382i
$330$		0			0
$331$	5.74472e8	+	9.95014e8i	0.870704	+	1.50810i	0.861270	+	0.508148i	$0.169670\pi$
$331$	5.74472e8	+	9.95014e8i	0.870704	+	1.50810i	0.00943450	+	0.999955i	$0.496997\pi$
$332$	2.53672e8	−	4.39372e8i	0.380442	−	0.658945i
$333$		0			0
$334$	3.79901e8	−	2.19336e8i	0.557903	−	0.322105i
$335$	1.56594e8			0.227572
$336$		0			0
$337$	−8.39038e8			−1.19420			−0.597100	−	0.802167i	$-0.703680\pi$
$337$	−8.39038e8			−1.19420			−0.597100	+	0.802167i	$0.703680\pi$
$338$	7.77725e7	−	4.49020e7i	0.109551	−	0.0632495i
$339$		0			0
$340$	1.20913e8	−	2.09428e8i	0.166839	−	0.288973i
$341$	1.56794e7	+	2.71574e7i	0.0214135	+	0.0370893i
$342$		0			0
$343$	−2.59707e8	−	7.00784e8i	−0.347500	−	0.937680i
$344$		−	4.01486e7i		−	0.0531760i
$345$		0			0
$346$	4.67823e8	+	2.70097e8i	0.607176	+	0.350553i
$347$	2.28377e8	+	1.31854e8i	0.293426	+	0.169410i	0.639486	−	0.768803i	$-0.279147\pi$
$347$	2.28377e8	+	1.31854e8i	0.293426	+	0.169410i	−0.346060	+	0.938213i	$0.612481\pi$
$348$		0			0
$349$			2.59046e8i			0.326203i	0.986609	+	0.163102i	$0.0521498\pi$
$349$			2.59046e8i			0.326203i	−0.986609	+	0.163102i	$0.947850\pi$
$350$	−2.29732e8	−	3.07334e8i	−0.286407	−	0.383153i
$351$		0			0
$352$	−2.48023e7	−	4.29589e7i	−0.0303105	−	0.0524994i
$353$	1.02389e7	−	1.77344e7i	0.0123892	−	0.0214588i	−0.859764	−	0.510691i	$-0.829390\pi$
$353$	1.02389e7	−	1.77344e7i	0.0123892	−	0.0214588i	0.872154	+	0.489232i	$0.162723\pi$
$354$		0			0
$355$	2.28191e8	−	1.31746e8i	0.270707	−	0.156293i
$356$	−5.49897e8			−0.645961
$357$		0			0
$358$	7.74177e8			0.891763
$359$	1.09229e9	−	6.30633e8i	1.24597	−	0.719360i	0.275666	−	0.961254i	$-0.411102\pi$
$359$	1.09229e9	−	6.30633e8i	1.24597	−	0.719360i	0.970303	+	0.241893i	$0.0777684\pi$
$360$		0			0
$361$	7.03747e8	−	1.21893e9i	0.787302	−	1.36365i
$362$	−2.00768e8	−	3.47741e8i	−0.222441	−	0.385279i
$363$		0			0
$364$	4.10694e8	−	4.88135e7i	0.446337	−	0.0530499i
$365$		−	3.20280e8i		−	0.344751i
$366$		0			0
$367$	1.33421e9	+	7.70308e8i	1.40894	+	0.813454i	0.995286	−	0.0969786i	$-0.0309179\pi$
$367$	1.33421e9	+	7.70308e8i	1.40894	+	0.813454i	0.413657	+	0.910433i	$0.364251\pi$
$368$	−1.55341e8	−	8.96861e7i	−0.162487	−	0.0938119i
$369$		0			0
$370$		−	5.58613e8i		−	0.573331i
$371$	7.80402e8	−	9.27556e7i	0.793431	−	0.0943042i
$372$		0			0
$373$	−7.48555e8	−	1.29654e9i	−0.746866	−	1.29361i	−0.949318	−	0.314318i	$-0.898224\pi$
$373$	−7.48555e8	−	1.29654e9i	−0.746866	−	1.29361i	0.202452	−	0.979292i	$-0.435109\pi$
$374$	2.91959e7	−	5.05687e7i	0.0288583	−	0.0499840i
$375$		0			0
$376$	3.92868e8	−	2.26822e8i	0.381144	−	0.220054i
$377$	−1.61896e9			−1.55612
$378$		0			0
$379$	−1.37753e8			−0.129976			−0.0649880	−	0.997886i	$-0.520701\pi$
$379$	−1.37753e8			−0.129976			−0.0649880	+	0.997886i	$0.520701\pi$
$380$	4.20972e8	−	2.43049e8i	0.393560	−	0.227222i
$381$		0			0
$382$	−6.71228e8	+	1.16260e9i	−0.616096	+	1.06711i
$383$	−7.13509e8	−	1.23583e9i	−0.648939	−	1.12400i	−0.983377	−	0.181578i	$-0.941880\pi$
$383$	−7.13509e8	−	1.23583e9i	−0.648939	−	1.12400i	0.334437	−	0.942418i	$-0.391454\pi$
$384$		0			0
$385$	−2.64209e7	−	3.53457e7i	−0.0235958	−	0.0315663i
$386$			1.59195e8i			0.140888i
$387$		0			0
$388$	−4.49589e8	−	2.59570e8i	−0.390755	−	0.225603i
$389$	2.67684e8	+	1.54547e8i	0.230568	+	0.133118i	0.610834	−	0.791759i	$-0.290834\pi$
$389$	2.67684e8	+	1.54547e8i	0.230568	+	0.133118i	−0.380266	+	0.924877i	$0.624168\pi$
$390$		0			0
$391$			1.01128e9i			0.855564i
$392$	1.21489e9	+	3.58569e8i	1.01868	+	0.300657i
$393$		0			0
$394$	−2.49886e7	−	4.32816e7i	−0.0205829	−	0.0356506i
$395$	8.92495e7	−	1.54585e8i	0.0728645	−	0.126205i
$396$		0			0
$397$	1.17021e9	−	6.75624e8i	0.938640	−	0.541924i	0.0491060	−	0.998794i	$-0.484363\pi$
$397$	1.17021e9	−	6.75624e8i	0.938640	−	0.541924i	0.889534	+	0.456870i	$0.151029\pi$
$398$	4.99209e7			0.0396910
$399$		0			0
$400$	2.22797e8			0.174060
$401$	−1.38715e9	+	8.00873e8i	−1.07428	+	0.620238i	−0.929349	−	0.369203i	$-0.879631\pi$
$401$	−1.38715e9	+	8.00873e8i	−1.07428	+	0.620238i	−0.144935	+	0.989441i	$0.546297\pi$
$402$		0			0
$403$	3.69701e8	−	6.40341e8i	0.281373	−	0.487353i
$404$	2.34026e8	+	4.05345e8i	0.176575	+	0.305837i
$405$		0			0
$406$	−1.51041e9	−	6.48043e8i	−1.12009	−	0.480577i
$407$			1.32641e8i			0.0975206i
$408$		0			0
$409$	−1.05325e9	−	6.08091e8i	−0.761199	−	0.439478i	0.0685273	−	0.997649i	$-0.478170\pi$
$409$	−1.05325e9	−	6.08091e8i	−0.761199	−	0.439478i	−0.829726	+	0.558171i	$0.811503\pi$
$410$	2.91249e8	+	1.68153e8i	0.208700	+	0.120493i
$411$		0			0
$412$			1.33530e9i			0.940676i
$413$	1.33881e8	+	1.12641e9i	0.0935174	+	0.786812i
$414$		0			0
$415$	6.38198e8	+	1.10539e9i	0.438316	+	0.759186i
$416$	−5.84810e8	+	1.01292e9i	−0.398280	+	0.689841i
$417$		0			0
$418$	1.01649e8	−	5.86869e7i	0.0680746	−	0.0393029i
$419$	−7.97541e8			−0.529668			−0.264834	−	0.964294i	$-0.585317\pi$
$419$	−7.97541e8			−0.529668			−0.264834	+	0.964294i	$0.585317\pi$
$420$		0			0
$421$	−1.28327e9			−0.838170			−0.419085	−	0.907947i	$-0.637649\pi$
$421$	−1.28327e9			−0.838170			−0.419085	+	0.907947i	$0.637649\pi$
$422$	1.66208e9	−	9.59605e8i	1.07661	−	0.621583i
$423$		0			0
$424$	−6.66009e8	+	1.15356e9i	−0.424326	+	0.734954i
$425$	−6.28053e8	−	1.08782e9i	−0.396858	−	0.687378i
$426$		0			0
$427$	−2.35863e8	+	5.49732e8i	−0.146610	+	0.341706i
$428$		−	5.93047e8i		−	0.365625i
$429$		0			0
$430$	2.89949e7	+	1.67402e7i	0.0175866	+	0.0101536i
$431$	−2.10960e8	−	1.21798e8i	−0.126920	−	0.0732773i	0.435196	−	0.900336i	$-0.356679\pi$
$431$	−2.10960e8	−	1.21798e8i	−0.126920	−	0.0732773i	−0.562116	+	0.827059i	$0.690012\pi$
$432$		0			0
$433$			1.56537e9i			0.926634i	0.886193	+	0.463317i	$0.153341\pi$
$433$			1.56537e9i			0.926634i	−0.886193	+	0.463317i	$0.846659\pi$
$434$	6.01229e8	−	4.49419e8i	0.353041	−	0.263899i
$435$		0			0
$436$	6.77350e8	+	1.17321e9i	0.391391	+	0.677909i
$437$	−1.01639e9	+	1.76044e9i	−0.582607	+	1.00911i
$438$		0			0
$439$	2.35511e8	−	1.35972e8i	0.132857	−	0.0767051i	−0.432098	−	0.901826i	$-0.642227\pi$
$439$	2.35511e8	−	1.35972e8i	0.132857	−	0.0767051i	0.564955	+	0.825121i	$0.308893\pi$
$440$	7.47948e7			0.0418588
$441$		0			0
$442$	−1.37681e9			−0.758395
$443$	1.39258e9	−	8.04009e8i	0.761041	−	0.439387i	−0.0686281	−	0.997642i	$-0.521862\pi$
$443$	1.39258e9	−	8.04009e8i	0.761041	−	0.439387i	0.829670	+	0.558255i	$0.188529\pi$
$444$		0			0
$445$	6.91727e8	−	1.19811e9i	0.372113	−	0.644519i
$446$	6.63679e8	+	1.14953e9i	0.354230	+	0.613545i
$447$		0			0
$448$	−1.34489e9	+	1.00531e9i	−0.706667	+	0.528234i
$449$			2.83900e9i			1.48014i	0.672528	+	0.740072i	$0.265208\pi$
$449$			2.83900e9i			1.48014i	−0.672528	+	0.740072i	$0.734792\pi$
$450$		0			0
$451$	−6.91561e7	−	3.99273e7i	−0.0354987	−	0.0204952i
$452$	1.48155e9	+	8.55373e8i	0.754626	+	0.435683i
$453$		0			0
$454$		−	2.76862e9i		−	1.38857i
$455$	−4.10267e8	+	9.56217e8i	−0.204186	+	0.475901i
$456$		0			0
$457$	3.38203e8	+	5.85785e8i	0.165757	+	0.287099i	0.936924	−	0.349534i	$-0.113660\pi$
$457$	3.38203e8	+	5.85785e8i	0.165757	+	0.287099i	−0.771167	+	0.636633i	$0.780327\pi$
$458$	−1.22641e9	+	2.12420e9i	−0.596492	+	1.03316i
$459$		0			0
$460$	−3.71843e8	+	2.14683e8i	−0.178118	+	0.102836i
$461$	−2.87578e9			−1.36711			−0.683553	−	0.729901i	$-0.739566\pi$
$461$	−2.87578e9			−1.36711			−0.683553	+	0.729901i	$0.739566\pi$
$462$		0			0
$463$	−1.89874e9			−0.889062			−0.444531	−	0.895763i	$-0.646630\pi$
$463$	−1.89874e9			−0.889062			−0.444531	+	0.895763i	$0.646630\pi$
$464$	8.26470e8	−	4.77163e8i	0.384073	−	0.221745i
$465$		0			0
$466$	−3.28759e8	+	5.69427e8i	−0.150497	+	0.260668i
$467$	2.67750e8	+	4.63757e8i	0.121652	+	0.210708i	0.920419	−	0.390932i	$-0.127847\pi$
$467$	2.67750e8	+	4.63757e8i	0.121652	+	0.210708i	−0.798767	+	0.601640i	$0.794514\pi$
$468$		0			0
$469$	1.05048e8	+	8.83825e8i	0.0470200	+	0.395604i
$470$			3.78301e8i			0.168072i
$471$		0			0
$472$	−1.66502e9	−	9.61297e8i	−0.728822	−	0.420786i
$473$	−6.88474e6	−	3.97491e6i	−0.00299139	−	0.00172708i
$474$		0			0
$475$		−	2.52491e9i		−	1.08098i
$476$	1.26313e9	+	5.41948e8i	0.536813	+	0.230321i
$477$		0			0
$478$	6.04434e8	+	1.04691e9i	0.253134	+	0.438442i
$479$	1.26132e8	−	2.18467e8i	0.0524386	−	0.0908263i	−0.838615	−	0.544725i	$-0.816634\pi$
$479$	1.26132e8	−	2.18467e8i	0.0524386	−	0.0908263i	0.891053	+	0.453899i	$0.149967\pi$
$480$		0			0
$481$	2.70850e9	−	1.56376e9i	1.10974	−	0.640709i
$482$	2.54967e9			1.03710
$483$		0			0
$484$	1.23084e9			0.493448
$485$	1.13110e9	−	6.53038e8i	0.450198	−	0.259922i
$486$		0			0
$487$	9.22439e8	−	1.59771e9i	0.361898	−	0.626826i	−0.626375	−	0.779522i	$-0.715462\pi$
$487$	9.22439e8	−	1.59771e9i	0.361898	−	0.626826i	0.988273	+	0.152696i	$0.0487955\pi$
$488$	−5.06942e8	−	8.78048e8i	−0.197464	−	0.342018i
$489$		0			0
$490$	−7.65514e8	+	7.27877e8i	−0.293946	+	0.279494i
$491$		−	1.15345e9i		−	0.439758i	−0.975527	−	0.219879i	$-0.929434\pi$
$491$		−	1.15345e9i		−	0.439758i	0.975527	−	0.219879i	$-0.0705663\pi$
$492$		0			0
$493$	−4.65954e9	−	2.69019e9i	−1.75137	−	1.01116i
$494$	−2.39676e9	−	1.38377e9i	−0.894499	−	0.516439i
$495$		0			0
$496$			4.35853e8i			0.160381i
$497$	8.96657e8	+	1.19954e9i	0.327627	+	0.438296i
$498$		0			0
$499$	2.30062e9	+	3.98480e9i	0.828884	+	1.43567i	0.898914	+	0.438125i	$0.144357\pi$
$499$	2.30062e9	+	3.98480e9i	0.828884	+	1.43567i	−0.0700301	+	0.997545i	$0.522310\pi$
$500$	6.62471e8	−	1.14743e9i	0.237013	−	0.410518i
$501$		0			0
$502$	1.24605e9	−	7.19407e8i	0.439615	−	0.253812i
$503$	2.63052e9			0.921623			0.460812	−	0.887498i	$-0.347558\pi$
$503$	2.63052e9			0.921623			0.460812	+	0.887498i	$0.347558\pi$
$504$		0			0
$505$	−1.17755e9			−0.406872
$506$	−8.97857e7	+	5.18378e7i	−0.0308092	+	0.0177877i
$507$		0			0
$508$	4.68400e8	−	8.11293e8i	0.158524	−	0.274571i
$509$	−2.82600e9	−	4.89477e9i	−0.949860	−	1.64521i	−0.745714	−	0.666266i	$-0.767891\pi$
$509$	−2.82600e9	−	4.89477e9i	−0.949860	−	1.64521i	−0.204146	−	0.978940i	$-0.565442\pi$
$510$		0			0
$511$	1.80767e9	−	2.14853e8i	0.599304	−	0.0712310i
$512$		−	1.52286e9i		−	0.501435i
$513$		0			0
$514$	2.79923e9	+	1.61614e9i	0.909218	+	0.524937i
$515$	−2.90934e9	−	1.67971e9i	−0.938576	−	0.541887i
$516$		0			0
$517$		−	8.98261e7i		−	0.0285881i
$518$	3.15284e9	−	3.74734e8i	0.996660	−	0.118459i
$519$		0			0
$520$	−8.81787e8	−	1.52730e9i	−0.275012	−	0.476335i
$521$	−1.79442e9	+	3.10803e9i	−0.555895	+	0.962839i	0.441938	+	0.897045i	$0.354291\pi$
$521$	−1.79442e9	+	3.10803e9i	−0.555895	+	0.962839i	−0.997833	+	0.0657930i	$0.979042\pi$
$522$		0			0
$523$	2.27933e9	−	1.31597e9i	0.696707	−	0.402244i	−0.109413	−	0.993996i	$-0.534897\pi$
$523$	2.27933e9	−	1.31597e9i	0.696707	−	0.402244i	0.806120	+	0.591752i	$0.201564\pi$
$524$	−5.13062e8			−0.155780
$525$		0			0
$526$	3.33798e9			1.00008
$527$	2.12807e9	−	1.22864e9i	0.633358	−	0.365669i
$528$		0			0
$529$	−8.04640e8	+	1.39368e9i	−0.236323	+	0.409324i
$530$	−5.55394e8	−	9.61971e8i	−0.162045	−	0.280670i
$531$		0			0
$532$	1.65418e9	+	2.21294e9i	0.476311	+	0.637205i
$533$			1.88288e9i			0.538613i
$534$		0			0
$535$	1.29212e9	+	7.46007e8i	0.364809	+	0.210622i
$536$	−1.30644e9	−	7.54272e8i	−0.366448	−	0.211569i
$537$		0			0
$538$		−	2.56643e9i		−	0.710544i
$539$	1.81768e8	−	1.72832e8i	0.0499986	−	0.0475404i
$540$		0			0
$541$	−8.25492e8	−	1.42979e9i	−0.224142	−	0.388225i	0.731920	−	0.681391i	$-0.238625\pi$
$541$	−8.25492e8	−	1.42979e9i	−0.224142	−	0.388225i	−0.956062	+	0.293166i	$0.905291\pi$
$542$	−1.87013e9	+	3.23916e9i	−0.504516	+	0.873847i
$543$		0			0
$544$	−3.36628e9	+	1.94353e9i	−0.896509	+	0.517600i
$545$	−3.40822e9			−0.901861
$546$		0			0
$547$	1.64567e9			0.429919			0.214959	−	0.976623i	$-0.431038\pi$
$547$	1.64567e9			0.429919			0.214959	+	0.976623i	$0.431038\pi$
$548$	1.39685e9	−	8.06472e8i	0.362592	−	0.209342i
$549$		0			0
$550$	6.43875e7	−	1.11522e8i	0.0165018	−	0.0285820i
$551$	−5.40757e9	−	9.36619e9i	−1.37712	−	2.38524i
$552$		0			0
$553$	9.32353e8	+	4.00028e8i	0.234446	+	0.100589i
$554$			2.16015e9i			0.539760i
$555$		0			0
$556$	−1.48125e9	−	8.55198e8i	−0.365482	−	0.211011i
$557$	−6.27401e8	−	3.62230e8i	−0.153834	−	0.0888160i	0.421107	−	0.907011i	$-0.361642\pi$
$557$	−6.27401e8	−	3.62230e8i	−0.153834	−	0.0888160i	−0.574941	+	0.818195i	$0.694975\pi$
$558$		0			0
$559$			1.87447e8i			0.0453877i
$560$	−7.23907e7	−	6.09061e8i	−0.0174191	−	0.146556i
$561$		0			0
$562$	1.75898e9	+	3.04664e9i	0.418006	+	0.724008i
$563$	3.19980e9	−	5.54222e9i	0.755690	−	1.30889i	−0.189341	−	0.981911i	$-0.560635\pi$
$563$	3.19980e9	−	5.54222e9i	0.755690	−	1.30889i	0.945031	−	0.326982i	$-0.106032\pi$
$564$		0			0
$565$	−3.72735e9	+	2.15199e9i	−0.869421	+	0.501961i
$566$	−3.49987e9			−0.811324
$567$		0			0
$568$	−2.53834e9			−0.581206
$569$	5.32872e9	−	3.07654e9i	1.21263	−	0.700115i	0.249302	−	0.968426i	$-0.419799\pi$
$569$	5.32872e9	−	3.07654e9i	1.21263	−	0.700115i	0.963332	+	0.268311i	$0.0864655\pi$
$570$		0			0
$571$	7.05090e8	−	1.22125e9i	0.158496	−	0.274523i	−0.775831	−	0.630941i	$-0.782669\pi$
$571$	7.05090e8	−	1.22125e9i	0.158496	−	0.274523i	0.934326	+	0.356418i	$0.116002\pi$
$572$	6.94011e7	+	1.20206e8i	0.0155053	+	0.0268560i
$573$		0			0
$574$	−7.53683e8	+	1.75662e9i	−0.166340	+	0.387692i
$575$			2.23024e9i			0.489231i
$576$		0			0
$577$	−1.05307e9	−	6.07993e8i	−0.228215	−	0.131760i	0.381533	−	0.924355i	$-0.375396\pi$
$577$	−1.05307e9	−	6.07993e8i	−0.228215	−	0.131760i	−0.609748	+	0.792595i	$0.708729\pi$
$578$	−1.10779e9	−	6.39582e8i	−0.238621	−	0.137768i
$579$		0			0
$580$		−	2.28439e9i		−	0.486152i
$581$	−5.81075e9	+	4.34355e9i	−1.22918	+	0.918815i
$582$		0			0
$583$	1.31876e8	+	2.28416e8i	0.0275630	+	0.0477405i
$584$	−1.54270e9	+	2.67204e9i	−0.320507	+	0.555134i
$585$		0			0
$586$	4.71590e9	−	2.72272e9i	0.968105	−	0.558936i
$587$	7.81812e9			1.59540			0.797698	−	0.603057i	$-0.206051\pi$
$587$	7.81812e9			1.59540			0.797698	+	0.603057i	$0.206051\pi$
$588$		0			0
$589$	4.93942e9			0.996029
$590$	1.38848e9	−	8.01639e8i	0.278329	−	0.160693i
$591$		0			0
$592$	−9.21782e8	+	1.59657e9i	−0.182601	+	0.316274i
$593$	2.46375e9	+	4.26734e9i	0.485183	+	0.840361i	0.999855	−	0.0170260i	$-0.00541981\pi$
$593$	2.46375e9	+	4.26734e9i	0.485183	+	0.840361i	−0.514672	+	0.857387i	$0.672086\pi$
$594$		0			0
$595$	−2.76971e9	+	2.07036e9i	−0.539044	+	0.402936i
$596$		−	3.58922e9i		−	0.694446i
$597$		0			0
$598$	2.11704e9	+	1.22228e9i	0.404833	+	0.233730i
$599$	6.19146e9	+	3.57464e9i	1.17706	+	0.679577i	0.955333	−	0.295531i	$-0.0954966\pi$
$599$	6.19146e9	+	3.57464e9i	1.17706	+	0.679577i	0.221729	+	0.975108i	$0.428830\pi$
$600$		0			0
$601$		−	4.83270e9i		−	0.908091i	−0.890978	−	0.454046i	$-0.849980\pi$
$601$		−	4.83270e9i		−	0.908091i	0.890978	−	0.454046i	$-0.150020\pi$
$602$	−7.50319e7	+	1.74878e8i	−0.0140171	+	0.0326700i
$603$		0			0
$604$	−7.76862e8	−	1.34556e9i	−0.143455	−	0.248471i
$605$	−1.54829e9	+	2.68173e9i	−0.284256	+	0.492346i
$606$		0			0
$607$	−3.22185e9	+	1.86014e9i	−0.584716	+	0.337586i	−0.763005	−	0.646392i	$-0.776277\pi$
$607$	−3.22185e9	+	1.86014e9i	−0.584716	+	0.337586i	0.178289	+	0.983978i	$0.442944\pi$
$608$	−7.81340e9			−1.40987
$609$		0			0
$610$	8.45491e8			0.150819
$611$	−1.83424e9	+	1.05900e9i	−0.325320	+	0.187824i
$612$		0			0
$613$	−2.90251e7	+	5.02729e7i	−0.00508934	+	0.00881500i	−0.868559	−	0.495586i	$-0.834953\pi$
$613$	−2.90251e7	+	5.02729e7i	−0.00508934	+	0.00881500i	0.863470	+	0.504401i	$0.168287\pi$
$614$	1.68781e9	+	2.92338e9i	0.294263	+	0.509678i
$615$		0			0
$616$	5.01745e7	+	4.22145e8i	0.00864869	+	0.0727660i
$617$			6.85458e9i			1.17485i	0.809278	+	0.587425i	$0.199858\pi$
$617$			6.85458e9i			1.17485i	−0.809278	+	0.587425i	$0.800142\pi$
$618$		0			0
$619$	−8.14512e9	−	4.70259e9i	−1.38032	−	0.796929i	−0.388124	−	0.921607i	$-0.626877\pi$
$619$	−8.14512e9	−	4.70259e9i	−1.38032	−	0.796929i	−0.992197	+	0.124679i	$0.960210\pi$
$620$	9.03532e8	+	5.21655e8i	0.152255	+	0.0879047i
$621$		0			0
$622$			1.14047e8i			0.0190027i
$623$	7.22619e9	+	3.10041e9i	1.19730	+	0.513702i
$624$		0			0
$625$	−3.89280e8	−	6.74253e8i	−0.0637797	−	0.110470i
$626$	−4.52995e8	+	7.84611e8i	−0.0738046	+	0.127833i
$627$		0			0
$628$	8.05066e8	−	4.64805e8i	0.129710	−	0.0748880i
$629$	1.03938e10			1.66532
$630$		0			0
$631$	2.44762e9			0.387830			0.193915	−	0.981018i	$-0.437881\pi$
$631$	2.44762e9			0.387830			0.193915	+	0.981018i	$0.437881\pi$
$632$	−1.48918e9	+	8.59781e8i	−0.234660	+	0.135481i
$633$		0			0
$634$	−2.16224e9	+	3.74511e9i	−0.336970	+	0.583649i
$635$	1.17842e9	+	2.04109e9i	0.182639	+	0.316339i
$636$		0			0
$637$	−5.67214e9	−	1.67410e9i	−0.869479	−	0.256622i
$638$		−	5.51592e8i		−	0.0840902i
$639$		0			0
$640$	−8.27374e8	−	4.77684e8i	−0.124759	−	0.0720296i
$641$	−2.36346e9	−	1.36454e9i	−0.354441	−	0.204637i	0.312198	−	0.950017i	$-0.398935\pi$
$641$	−2.36346e9	−	1.36454e9i	−0.354441	−	0.204637i	−0.666640	+	0.745380i	$0.732268\pi$
$642$		0			0
$643$			1.18675e10i			1.76044i	0.474564	+	0.880221i	$0.342606\pi$
$643$			1.18675e10i			1.76044i	−0.474564	+	0.880221i	$0.657394\pi$
$644$	−1.46112e9	−	1.95468e9i	−0.215569	−	0.288386i
$645$		0			0
$646$	−4.59874e9	−	7.96525e9i	−0.671158	−	1.16248i
$647$	5.67492e9	−	9.82924e9i	0.823748	−	1.42677i	−0.0791240	−	0.996865i	$-0.525212\pi$
$647$	5.67492e9	−	9.82924e9i	0.823748	−	1.42677i	0.902872	−	0.429909i	$-0.141454\pi$
$648$		0			0
$649$	−3.29689e8	+	1.90346e8i	−0.0473422	+	0.0273330i
$650$	−3.03636e9			−0.433668
$651$		0			0
$652$	3.40831e9			0.481585
$653$	−6.36961e9	+	3.67750e9i	−0.895193	+	0.516840i	−0.875638	−	0.482969i	$-0.839558\pi$
$653$	−6.36961e9	+	3.67750e9i	−0.895193	+	0.516840i	−0.0195554	+	0.999809i	$0.506225\pi$
$654$		0			0
$655$	6.45392e8	−	1.11785e9i	0.0897386	−	0.155432i
$656$	−5.54947e8	−	9.61196e8i	−0.0767517	−	0.132938i
$657$		0			0
$658$	−2.13514e9	+	2.53775e8i	−0.292171	+	0.0347263i
$659$			1.12262e10i			1.52804i	0.645192	+	0.764020i	$0.276777\pi$
$659$			1.12262e10i			1.52804i	−0.645192	+	0.764020i	$0.723223\pi$
$660$		0			0
$661$	−1.82029e8	−	1.05094e8i	−0.0245151	−	0.0141538i	0.487692	−	0.873016i	$-0.337839\pi$
$661$	−1.82029e8	−	1.05094e8i	−0.0245151	−	0.0141538i	−0.512207	+	0.858862i	$0.671172\pi$
$662$	7.99341e9	+	4.61500e9i	1.07085	+	0.618257i
$663$		0			0
$664$		−	1.22961e10i		−	1.62997i
$665$	−6.90235e9	+	8.20387e8i	−0.910167	+	0.108179i
$666$		0			0
$667$	4.77648e9	+	8.27310e9i	0.623257	+	1.07951i
$668$	−1.73273e9	+	3.00117e9i	−0.224912	+	0.389559i
$669$		0			0
$670$	1.08946e9	−	6.28998e8i	0.139942	−	0.0807956i
$671$	−2.00759e8			−0.0256534
$672$		0			0
$673$	−1.43053e10			−1.80903			−0.904513	−	0.426447i	$-0.859765\pi$
$673$	−1.43053e10			−1.80903			−0.904513	+	0.426447i	$0.859765\pi$
$674$	−5.83734e9	+	3.37019e9i	−0.734354	+	0.423979i
$675$		0			0
$676$	−3.54720e8	+	6.14393e8i	−0.0441644	+	0.0764950i
$677$	2.58103e9	+	4.47047e9i	0.319692	+	0.553723i	0.980424	−	0.196899i	$-0.0630871\pi$
$677$	2.58103e9	+	4.47047e9i	0.319692	+	0.553723i	−0.660731	+	0.750622i	$0.729754\pi$
$678$		0			0
$679$	4.44455e9	+	5.94587e9i	0.544858	+	0.728906i
$680$		−	5.86096e9i		−	0.714805i
$681$		0			0
$682$	2.18168e8	+	1.25960e8i	0.0263358	+	0.0152050i
$683$	1.30876e9	+	7.55611e8i	0.157176	+	0.0907457i	0.576525	−	0.817079i	$-0.304408\pi$
$683$	1.30876e9	+	7.55611e8i	0.157176	+	0.0907457i	−0.419349	+	0.907825i	$0.637742\pi$
$684$		0			0
$685$			4.05792e9i			0.482376i
$686$	−4.62169e9	−	3.83231e9i	−0.546597	−	0.453238i
$687$		0			0
$688$	−5.52470e7	−	9.56906e7i	−0.00646769	−	0.0112024i
$689$	3.10949e9	−	5.38579e9i	0.362178	−	0.627310i
$690$		0			0
$691$	4.35968e8	−	2.51706e8i	0.0502668	−	0.0290216i	−0.474656	−	0.880171i	$-0.657428\pi$
$691$	4.35968e8	−	2.51706e8i	0.0502668	−	0.0290216i	0.524923	+	0.851150i	$0.324094\pi$
$692$	−4.26747e9			−0.489553
$693$		0			0
$694$	2.11848e9			0.240584
$695$	3.72659e9	−	2.15155e9i	0.421080	−	0.243111i
$696$		0			0
$697$	−3.12872e9	+	5.41911e9i	−0.349987	+	0.606196i
$698$	1.04052e9	+	1.80223e9i	0.115813	+	0.200594i
$699$		0			0
$700$	2.78566e9	+	1.19519e9i	0.306962	+	0.131703i
$701$		−	7.97507e9i		−	0.874422i	−0.899359	−	0.437211i	$-0.855966\pi$
$701$		−	7.97507e9i		−	0.874422i	0.899359	−	0.437211i	$-0.144034\pi$
$702$		0			0
$703$	1.80936e10	+	1.04463e10i	1.96418	+	1.13402i
$704$	−4.88023e8	−	2.81760e8i	−0.0527152	−	0.0304351i
$705$		0			0
$706$		−	1.64509e8i		−	0.0175943i
$707$	−7.89931e8	−	6.64611e9i	−0.0840662	−	0.707293i
$708$		0			0
$709$	−2.59137e9	−	4.48839e9i	−0.273066	−	0.472965i	0.696579	−	0.717480i	$-0.254705\pi$
$709$	−2.59137e9	−	4.48839e9i	−0.273066	−	0.472965i	−0.969645	+	0.244515i	$0.921371\pi$
$710$	1.05838e9	−	1.83316e9i	0.110978	−	0.192219i
$711$		0			0
$712$	−1.15419e10	+	6.66372e9i	−1.19839	+	0.691890i
$713$	−4.36296e9			−0.450783
$714$		0			0
$715$	−3.49205e8			−0.0357280
$716$	−5.29652e9	+	3.05795e9i	−0.539256	+	0.311340i
$717$		0			0
$718$	5.06617e9	−	8.77487e9i	0.510793	−	0.884719i
$719$	−2.43940e9	−	4.22517e9i	−0.244755	−	0.423929i	0.717307	−	0.696757i	$-0.245374\pi$
$719$	−2.43940e9	−	4.22517e9i	−0.244755	−	0.423929i	−0.962063	+	0.272828i	$0.912041\pi$
$720$		0			0
$721$	7.52867e9	−	1.75472e10i	0.748075	−	1.74355i
$722$		−	1.13071e10i		−	1.11807i
$723$		0			0
$724$	2.74711e9	+	1.58604e9i	0.269024	+	0.155321i
$725$	−1.02760e10	−	5.93284e9i	−1.00148	−	0.578202i
$726$		0			0
$727$			1.25987e10i			1.21606i	0.793915	+	0.608029i	$0.208040\pi$
$727$			1.25987e10i			1.21606i	−0.793915	+	0.608029i	$0.791960\pi$
$728$	8.02861e9	−	6.00139e9i	0.771224	−	0.576491i
$729$		0			0
$730$	−1.28648e9	−	2.22825e9i	−0.122398	−	0.211999i
$731$	−3.11476e8	+	5.39492e8i	−0.0294926	+	0.0510827i
$732$		0			0
$733$	9.42671e9	−	5.44251e9i	0.884089	−	0.510429i	0.0120847	−	0.999927i	$-0.496153\pi$
$733$	9.42671e9	−	5.44251e9i	0.884089	−	0.510429i	0.872005	+	0.489498i	$0.162820\pi$
$734$	1.23765e10			1.15521
$735$		0			0
$736$	6.90153e9			0.638077
$737$	−2.58687e8	+	1.49353e8i	−0.0238034	+	0.0137429i
$738$		0			0
$739$	−8.17785e8	+	1.41645e9i	−0.0745390	+	0.129105i	−0.900886	−	0.434056i	$-0.857082\pi$
$739$	−8.17785e8	+	1.41645e9i	−0.0745390	+	0.129105i	0.826347	+	0.563162i	$0.190415\pi$
$740$	2.20649e9	+	3.82175e9i	0.200166	+	0.346698i
$741$		0			0
$742$	5.05683e9	−	3.77998e9i	0.454427	−	0.339685i
$743$		−	1.01867e10i		−	0.911110i	−0.890208	−	0.455555i	$-0.849441\pi$
$743$		−	1.01867e10i		−	0.911110i	0.890208	−	0.455555i	$-0.150559\pi$
$744$		0			0
$745$	7.82014e9	+	4.51496e9i	0.692896	+	0.400043i
$746$	−1.04157e10	−	6.01349e9i	−0.918547	−	0.530324i
$747$		0			0
$748$			4.61287e8i			0.0403010i
$749$	−3.34370e9	+	7.79323e9i	−0.290764	+	0.677690i
$750$		0			0
$751$	−5.96289e9	−	1.03280e10i	−0.513709	−	0.889769i	−0.999874	−	0.0159023i	$-0.994938\pi$
$751$	−5.96289e9	−	1.03280e10i	−0.513709	−	0.889769i	0.486165	−	0.873867i	$-0.338395\pi$
$752$	6.24244e8	−	1.08122e9i	0.0535293	−	0.0927155i
$753$		0			0
$754$	−1.12634e10	+	6.50295e9i	−0.956910	+	0.552472i
$755$	3.90893e9			0.330555
$756$		0			0
$757$	−2.63914e8			−0.0221119			−0.0110560	−	0.999939i	$-0.503519\pi$
$757$	−2.63914e8			−0.0221119			−0.0110560	+	0.999939i	$0.503519\pi$
$758$	−9.58372e8	+	5.53316e8i	−0.0799267	+	0.0461457i
$759$		0			0
$760$	5.89059e9	−	1.02028e10i	0.486756	−	0.843086i
$761$	7.19322e9	+	1.24590e10i	0.591667	+	1.02480i	0.994008	+	0.109307i	$0.0348633\pi$
$761$	7.19322e9	+	1.24590e10i	0.591667	+	1.02480i	−0.402341	+	0.915490i	$0.631803\pi$
$762$		0			0
$763$	−2.28633e9	−	1.92361e10i	−0.186339	−	1.56777i
$764$		−	1.06052e10i		−	0.860387i
$765$		0			0
$766$	−9.92804e9	−	5.73195e9i	−0.798110	−	0.460789i
$767$	7.77369e9	+	4.48814e9i	0.622076	+	0.359156i
$768$		0			0
$769$			8.88927e9i			0.704894i	0.935832	+	0.352447i	$0.114650\pi$
$769$			8.88927e9i			0.704894i	−0.935832	+	0.352447i	$0.885350\pi$
$770$	−3.25790e8	−	1.39781e8i	−0.0257170	−	0.0110339i
$771$		0			0
$772$	−6.28809e8	−	1.08913e9i	−0.0491879	−	0.0851959i
$773$	2.11439e9	−	3.66223e9i	0.164648	−	0.285179i	−0.771882	−	0.635766i	$-0.780684\pi$
$773$	2.11439e9	−	3.66223e9i	0.164648	−	0.285179i	0.936530	+	0.350587i	$0.114018\pi$
$774$		0			0
$775$	4.69317e9	−	2.70960e9i	0.362168	−	0.209098i
$776$	−1.25820e10			−0.966573
$777$		0			0
$778$	2.48310e9			0.189045
$779$	−1.08930e10	+	6.28908e9i	−0.825594	+	0.476657i
$780$		0			0
$781$	−2.51308e8	+	4.35278e8i	−0.0188768	+	0.0326955i
$782$	4.06204e9	+	7.03566e9i	0.303753	+	0.526116i
$783$		0			0
$784$	3.38900e9	−	8.17152e8i	0.251169	−	0.0605615i
$785$			2.33875e9i			0.172560i
$786$		0			0
$787$	4.97780e9	+	2.87393e9i	0.364020	+	0.210167i	0.670843	−	0.741599i	$-0.265932\pi$
$787$	4.97780e9	+	2.87393e9i	0.364020	+	0.210167i	−0.306823	+	0.951767i	$0.599266\pi$
$788$	3.41919e8	+	1.97407e8i	0.0248933	+	0.0143721i
$789$		0			0
$790$		−	1.43397e9i		−	0.103477i
$791$	−1.46463e10	−	1.95937e10i	−1.05223	−	1.40766i
$792$		0			0
$793$	2.36683e9	+	4.09947e9i	0.168543	+	0.291925i
$794$	5.42760e9	−	9.40088e9i	0.384801	−	0.666495i
$795$		0			0
$796$	−3.41534e8	+	1.97184e8i	−0.0240015	+	0.0138572i
$797$	2.43832e10			1.70603			0.853015	−	0.521886i	$-0.174771\pi$
$797$	2.43832e10			1.70603			0.853015	+	0.521886i	$0.174771\pi$
$798$		0			0
$799$	−7.03882e9			−0.488187
$800$	−7.42389e9	+	4.28618e9i	−0.512645	+	0.295975i
$801$		0			0
$802$	−6.43379e9	+	1.11436e10i	−0.440409	+	0.762811i
$803$	3.05470e8	+	5.29090e8i	0.0208192	+	0.0360599i
$804$		0			0
$805$	6.09680e9	−	7.24643e8i	0.411924	−	0.0489597i
$806$		−	5.93996e9i		−	0.399587i
$807$		0			0
$808$	9.82404e9	+	5.67191e9i	0.655164	+	0.378259i
$809$	1.28248e10	+	7.40442e9i	0.851593	+	0.491667i	0.861188	−	0.508286i	$-0.169721\pi$
$809$	1.28248e10	+	7.40442e9i	0.851593	+	0.491667i	−0.00959498	+	0.999954i	$0.503054\pi$
$810$		0			0
$811$		−	4.46501e9i		−	0.293934i	−0.989141	−	0.146967i	$-0.953049\pi$
$811$		−	4.46501e9i		−	0.293934i	0.989141	−	0.146967i	$-0.0469510\pi$
$812$	1.28932e10	−	1.53243e9i	0.845111	−	0.100447i
$813$		0			0
$814$	5.32782e8	+	9.22806e8i	0.0346230	+	0.0599687i
$815$	−4.28739e9	+	7.42598e9i	−0.277422	+	0.480510i
$816$		0			0
$817$	−1.08444e9	+	6.26101e8i	−0.0695709	+	0.0401668i
$818$	−9.77017e9			−0.624116
$819$		0			0
$820$	−2.65677e9			−0.168270
$821$	−6.94286e9	+	4.00846e9i	−0.437862	+	0.252800i	−0.702690	−	0.711496i	$-0.748018\pi$
$821$	−6.94286e9	+	4.00846e9i	−0.437862	+	0.252800i	0.264828	+	0.964296i	$0.414685\pi$
$822$		0			0
$823$	1.83110e9	−	3.17156e9i	0.114502	−	0.198323i	−0.803079	−	0.595873i	$-0.796806\pi$
$823$	1.83110e9	−	3.17156e9i	0.114502	−	0.198323i	0.917581	+	0.397550i	$0.130139\pi$
$824$	1.61814e10	+	2.80270e10i	1.00756	+	1.74514i
$825$		0			0
$826$	5.45592e9	+	7.29887e9i	0.336851	+	0.450636i
$827$			1.72516e10i			1.06062i	0.847804	+	0.530309i	$0.177924\pi$
$827$			1.72516e10i			1.06062i	−0.847804	+	0.530309i	$0.822076\pi$
$828$		0			0
$829$	3.98486e9	+	2.30066e9i	0.242925	+	0.140253i	0.616520	−	0.787339i	$-0.288542\pi$
$829$	3.98486e9	+	2.30066e9i	0.242925	+	0.140253i	−0.373595	+	0.927592i	$0.621875\pi$
$830$	8.88013e9	+	5.12695e9i	0.539071	+	0.311233i
$831$		0			0
$832$			1.32871e10i			0.799834i
$833$	−1.35432e10	−	1.42435e10i	−0.811826	−	0.853804i
$834$		0			0
$835$	−4.35928e9	−	7.55049e9i	−0.259126	−	0.448820i
$836$	−4.63619e8	+	8.03012e8i	−0.0274435	+	0.0475335i
$837$		0			0
$838$	−5.54864e9	+	3.20351e9i	−0.325711	+	0.188049i
$839$	3.40305e9			0.198931			0.0994654	−	0.995041i	$-0.468287\pi$
$839$	3.40305e9			0.198931			0.0994654	+	0.995041i	$0.468287\pi$
$840$		0			0
$841$	−3.35753e10			−1.94641
$842$	−8.92798e9	+	5.15457e9i	−0.515420	+	0.297578i
$843$		0			0
$844$	−7.58076e9	+	1.31303e10i	−0.434024	+	0.751752i
$845$	−8.92421e8	−	1.54572e9i	−0.0508828	−	0.0881316i
$846$		0			0
$847$	−1.61744e10	−	6.93966e9i	−0.914611	−	0.392415i
$848$			3.66588e9i			0.206439i
$849$		0			0
$850$	−8.73896e9	−	5.04544e9i	−0.488083	−	0.281795i
$851$	−1.59820e10	−	9.22719e9i	−0.888949	−	0.513235i
$852$		0			0
$853$			2.15871e10i			1.19089i	0.803395	+	0.595446i	$0.203025\pi$
$853$			2.15871e10i			1.19089i	−0.803395	+	0.595446i	$0.796975\pi$
$854$	5.67180e8	+	4.77199e9i	0.0311615	+	0.262178i
$855$		0			0
$856$	−7.18662e9	−	1.24476e10i	−0.391621	−	0.678308i
$857$	3.33532e9	−	5.77695e9i	0.181011	−	0.313520i	−0.761214	−	0.648501i	$-0.775396\pi$
$857$	3.33532e9	−	5.77695e9i	0.181011	−	0.313520i	0.942225	+	0.334980i	$0.108730\pi$
$858$		0			0
$859$	−1.62268e10	+	9.36854e9i	−0.873487	+	0.504308i	−0.868505	−	0.495680i	$-0.834919\pi$
$859$	−1.62268e10	+	9.36854e9i	−0.873487	+	0.504308i	−0.00498155	+	0.999988i	$0.501586\pi$
$860$	−2.64492e8			−0.0141797
$861$		0			0
$862$	−1.95692e9			−0.104063
$863$	−2.17945e10	+	1.25831e10i	−1.15427	+	0.666420i	−0.949925	−	0.312478i	$-0.898841\pi$
$863$	−2.17945e10	+	1.25831e10i	−1.15427	+	0.666420i	−0.204349	+	0.978898i	$0.565508\pi$
$864$		0			0
$865$	5.36815e9	−	9.29790e9i	0.282012	−	0.488460i
$866$	6.28766e9	+	1.08905e10i	0.328985	+	0.569819i
$867$		0			0
$868$	−2.33812e9	+	5.44951e9i	−0.121352	+	0.282839i
$869$			3.40490e8i			0.0176009i
$870$		0			0
$871$	6.09955e9	+	3.52157e9i	0.312776	+	0.180582i
$872$	2.84341e10	+	1.64164e10i	1.45222	+	0.838439i
$873$		0			0
$874$			1.63303e10i			0.827378i
$875$	−1.51749e10	+	1.13433e10i	−0.765771	+	0.572415i
$876$		0			0
$877$	−4.67159e9	−	8.09143e9i	−0.233865	−	0.405067i	0.725077	−	0.688668i	$-0.241804\pi$
$877$	−4.67159e9	−	8.09143e9i	−0.233865	−	0.405067i	−0.958942	+	0.283601i	$0.908471\pi$
$878$	1.09233e9	−	1.89197e9i	0.0544656	−	0.0943372i
$879$		0			0
$880$	1.78267e8	−	1.02922e8i	0.00881821	−	0.00509120i
$881$	−3.97329e9			−0.195765			−0.0978825	−	0.995198i	$-0.531207\pi$
$881$	−3.97329e9			−0.195765			−0.0978825	+	0.995198i	$0.531207\pi$
$882$		0			0
$883$	1.04774e10			0.512143			0.256072	−	0.966658i	$-0.417572\pi$
$883$	1.04774e10			0.512143			0.256072	+	0.966658i	$0.417572\pi$
$884$	9.41942e9	−	5.43831e9i	0.458608	−	0.264777i
$885$		0			0
$886$	6.45898e9	−	1.11873e10i	0.311994	−	0.540389i
$887$	1.07525e10	+	1.86238e10i	0.517339	+	0.896057i	0.999797	+	0.0201380i	$0.00641055\pi$
$887$	1.07525e10	+	1.86238e10i	0.517339	+	0.896057i	−0.482459	+	0.875919i	$0.660256\pi$
$888$		0			0
$889$	−1.07294e10	+	8.02028e9i	−0.512178	+	0.382854i
$890$		−	1.11139e10i		−	0.528449i
$891$		0			0
$892$	−9.08111e9	−	5.24298e9i	−0.428412	−	0.247344i
$893$	−1.22532e10	−	7.07441e9i	−0.575799	−	0.332437i
$894$		0			0
$895$		−	1.53866e10i		−	0.717403i
$896$	2.14104e9	−	4.99017e9i	0.0994368	−	0.231759i
$897$		0			0
$898$	1.14035e10	+	1.97515e10i	0.525499	+	0.910191i
$899$	1.16063e10	−	2.01026e10i	0.532762	−	0.922771i
$900$		0			0
$901$	1.78988e10	−	1.03339e10i	0.815244	−	0.470682i
$902$	−6.41509e8			−0.0291058
$903$		0			0
$904$	4.14621e10			1.86664
$905$	−6.91130e9	+	3.99024e9i	−0.309948	+	0.178949i
$906$		0			0
$907$	−4.43980e8	+	7.68996e8i	−0.0197578	+	0.0342215i	−0.875735	−	0.482792i	$-0.839623\pi$
$907$	−4.43980e8	+	7.68996e8i	−0.0197578	+	0.0342215i	0.855977	+	0.517013i	$0.172956\pi$
$908$	1.09359e10	+	1.89415e10i	0.484789	+	0.839679i
$909$		0			0
$910$	9.86567e8	+	8.30051e9i	0.0433992	+	0.365140i
$911$			3.55236e9i			0.155669i	0.996966	+	0.0778346i	$0.0248006\pi$
$911$			3.55236e9i			0.155669i	−0.996966	+	0.0778346i	$0.975199\pi$
$912$		0			0
$913$	−2.10855e9	−	1.21737e9i	−0.0916932	−	0.0529391i
$914$	4.70589e9	+	2.71695e9i	0.203859	+	0.117698i
$915$		0			0
$916$		−	1.93769e10i		−	0.833010i
$917$	6.74215e9	+	2.89273e9i	0.288739	+	0.123884i
$918$		0			0
$919$	−1.01152e10	−	1.75200e10i	−0.429902	−	0.744612i	0.566962	−	0.823744i	$-0.308119\pi$
$919$	−1.01152e10	−	1.75200e10i	−0.429902	−	0.744612i	−0.996864	+	0.0791317i	$0.974785\pi$
$920$	−5.20312e9	+	9.01207e9i	−0.220296	+	0.381564i
$921$		0			0
$922$	−2.00073e10	+	1.15512e10i	−0.840680	+	0.485367i
$923$	1.18511e10			0.496081
$924$		0			0
$925$	2.29221e10			0.952266
$926$	−1.32099e10	+	7.62674e9i	−0.546715	+	0.315646i
$927$		0			0
$928$	−1.83593e10	+	3.17993e10i	−0.754117	+	1.30617i
$929$	−2.21346e10	−	3.83383e10i	−0.905768	−	1.56884i	−0.819883	−	0.572531i	$-0.805962\pi$
$929$	−2.21346e10	−	3.83383e10i	−0.905768	−	1.56884i	−0.0858848	−	0.996305i	$-0.527372\pi$
$930$		0			0
$931$	−9.26059e9	−	3.84068e10i	−0.376110	−	1.55985i
$932$		−	5.19431e9i		−	0.210171i
$933$		0			0
$934$	3.72558e9	+	2.15096e9i	0.149616	+	0.0863811i
$935$	−1.00505e9	−	5.80263e8i	−0.0402110	−	0.0232158i
$936$		0			0
$937$		−	1.52361e10i		−	0.605042i	−0.953143	−	0.302521i	$-0.902172\pi$
$937$		−	1.52361e10i		−	0.605042i	0.953143	−	0.302521i	$-0.0978283\pi$
$938$	4.28093e9	+	5.72698e9i	0.169367	+	0.226577i
$939$		0			0
$940$	−1.49426e9	−	2.58814e9i	−0.0586786	−	0.101634i
$941$	1.55682e10	−	2.69650e10i	0.609082	−	1.05496i	−0.382310	−	0.924034i	$-0.624871\pi$
$941$	1.55682e10	−	2.69650e10i	0.609082	−	1.05496i	0.991392	−	0.130927i	$-0.0417952\pi$
$942$		0			0
$943$	9.62173e9	−	5.55511e9i	0.373648	−	0.215726i
$944$	−5.29122e9			−0.204717
$945$		0			0
$946$	−6.38646e7			−0.00245268
$947$	4.14053e10	−	2.39053e10i	1.58428	−	0.914682i	0.590051	−	0.807366i	$-0.299108\pi$
$947$	4.14053e10	−	2.39053e10i	1.58428	−	0.914682i	0.994225	−	0.107316i	$-0.0342257\pi$
$948$		0			0
$949$	7.20263e9	−	1.24753e10i	0.273564	−	0.473827i
$950$	−1.01419e10	−	1.75663e10i	−0.383784	−	0.664733i
$951$		0			0
$952$	3.30795e10	−	3.93170e9i	1.24259	−	0.147690i
$953$			2.23812e10i			0.837640i	0.908069	+	0.418820i	$0.137556\pi$
$953$			2.23812e10i			0.837640i	−0.908069	+	0.418820i	$0.862444\pi$
$954$		0			0
$955$	2.31065e10	+	1.33406e10i	0.858466	+	0.495636i
$956$	−8.27046e9	−	4.77495e9i	−0.306145	−	0.176753i
$957$		0			0
$958$		−	2.02655e9i		−	0.0744696i
$959$	−2.29030e10	+	2.72217e9i	−0.838548	+	0.0996666i
$960$		0			0
$961$	−8.45558e9	−	1.46455e10i	−0.307335	−	0.532319i
$962$	1.25624e10	−	2.17587e10i	0.454945	−	0.787988i
$963$		0			0
$964$	−1.74435e10	+	1.00710e10i	−0.627141	+	0.362080i
$965$	3.16397e9			0.113341
$966$		0			0
$967$	−2.29570e10			−0.816437			−0.408218	−	0.912884i	$-0.633850\pi$
$967$	−2.29570e10			−0.816437			−0.408218	+	0.912884i	$0.633850\pi$
$968$	2.58343e10	−	1.49154e10i	0.915445	−	0.528533i
$969$		0			0
$970$	5.24616e9	−	9.08662e9i	0.184561	−	0.319670i
$971$	−1.71932e10	−	2.97795e10i	−0.602684	−	1.04388i	−0.992413	−	0.122950i	$-0.960765\pi$
$971$	−1.71932e10	−	2.97795e10i	−0.602684	−	1.04388i	0.389729	−	0.920930i	$-0.372569\pi$
$972$		0			0
$973$	1.46433e10	+	1.95897e10i	0.509618	+	0.681762i
$974$		−	1.48208e10i		−	0.513942i
$975$		0			0
$976$	−2.41650e9	−	1.39517e9i	−0.0831979	−	0.0480343i
$977$	−1.61184e10	−	9.30598e9i	−0.552957	−	0.319250i	0.197357	−	0.980332i	$-0.436764\pi$
$977$	−1.61184e10	−	9.30598e9i	−0.552957	−	0.319250i	−0.750314	+	0.661082i	$0.770098\pi$
$978$		0			0
$979$			2.63896e9i			0.0898864i
$980$	2.36219e9	−	8.00350e9i	0.0801721	−	0.271637i
$981$		0			0
$982$	−4.63311e9	−	8.02478e9i	−0.156129	−	0.270423i
$983$	−2.47227e10	+	4.28211e10i	−0.830156	+	1.43787i	0.0677590	+	0.997702i	$0.478415\pi$
$983$	−2.47227e10	+	4.28211e10i	−0.830156	+	1.43787i	−0.897915	+	0.440170i	$0.854918\pi$
$984$		0			0
$985$	−8.60215e8	+	4.96646e8i	−0.0286801	+	0.0165585i
$986$	−4.32230e10			−1.43597
$987$		0			0
$988$	2.18632e10			0.721214
$989$	9.57879e8	−	5.53031e8i	0.0314864	−	0.0181787i
$990$		0			0
$991$	4.05406e9	−	7.02184e9i	0.132322	−	0.229189i	−0.792249	−	0.610198i	$-0.791090\pi$
$991$	4.05406e9	−	7.02184e9i	0.132322	−	0.229189i	0.924571	+	0.381009i	$0.124423\pi$
$992$	−8.38494e9	−	1.45231e10i	−0.272715	−	0.472357i
$993$		0			0
$994$	1.10564e10	+	4.74379e9i	0.357078	+	0.153205i
$995$		−	9.92171e8i		−	0.0319305i
$996$		0			0
$997$	3.72231e10	+	2.14907e10i	1.18954	+	0.686781i	0.958202	−	0.286093i	$-0.0923565\pi$
$997$	3.72231e10	+	2.14907e10i	1.18954	+	0.686781i	0.231338	+	0.972874i	$0.425690\pi$
$998$	3.20118e10	+	1.84820e10i	1.01942	+	0.588562i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	63.8.p.a.17.13	yes	36
3.2	odd	2	inner	63.8.p.a.17.6	✓	36
7.3	odd	6		441.8.c.a.440.16		36
7.4	even	3		441.8.c.a.440.22		36
7.5	odd	6	inner	63.8.p.a.26.6	yes	36
21.5	even	6	inner	63.8.p.a.26.13	yes	36
21.11	odd	6		441.8.c.a.440.15		36
21.17	even	6		441.8.c.a.440.21		36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.8.p.a.17.6	✓	36	3.2	odd	2	inner
63.8.p.a.17.13	yes	36	1.1	even	1	trivial
63.8.p.a.26.6	yes	36	7.5	odd	6	inner
63.8.p.a.26.13	yes	36	21.5	even	6	inner
441.8.c.a.440.15		36	21.11	odd	6
441.8.c.a.440.16		36	7.3	odd	6
441.8.c.a.440.21		36	21.17	even	6
441.8.c.a.440.22		36	7.4	even	3

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:	\( N \)	\(=\)	\( 63 = 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	63.p (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(6.95719 - 4.01673i) q^{2} +(-31.7317 + 54.9609i) q^{4} +(-79.8320 - 138.273i) q^{5} +(726.865 - 543.333i) q^{7} +1538.12i q^{8} +(-1110.81 - 641.328i) q^{10} +(263.758 + 152.281i) q^{11} -7181.21i q^{13} +(2874.51 - 6699.69i) q^{14} +(2116.54 + 3665.96i) q^{16} +(11932.8 - 20668.2i) q^{17} +(41545.4 - 23986.3i) q^{19} +10132.8 q^{20} +2446.69 q^{22} +(-36696.9 + 21186.9i) q^{23} +(26316.2 - 45581.0i) q^{25} +(-28845.0 - 49961.0i) q^{26} +(6797.39 + 57190.0i) q^{28} -225444. i q^{29} +(89168.9 + 51481.7i) q^{31} +(-141052. - 81436.2i) q^{32} -191724. i q^{34} +(-133155. - 57130.6i) q^{35} +(217757. + 377165. i) q^{37} +(192693. - 333754. i) q^{38} +(212680. - 122791. i) q^{40} -262195. q^{41} -26102.5 q^{43} +(-16739.0 + 9664.26i) q^{44} +(-170205. + 294803. i) q^{46} +(-147468. - 255422. i) q^{47} +(233123. - 789859. i) q^{49} -422821. i q^{50} +(394686. + 227872. i) q^{52} +(749984. + 433003. i) q^{53} -48627.6i q^{55} +(835708. + 1.11800e6i) q^{56} +(-905550. - 1.56846e6i) q^{58} +(-624984. + 1.08250e6i) q^{59} +(-570860. + 329586. i) q^{61} +827153. q^{62} -1.85026e6 q^{64} +(-992968. + 573290. i) q^{65} +(-490387. + 849376. i) q^{67} +(757297. + 1.31168e6i) q^{68} +(-1.15587e6 + 137382. i) q^{70} +1.65029e6i q^{71} +(1.73722e6 + 1.00298e6i) q^{73} +(3.02995e6 + 1.74934e6i) q^{74} +3.04450e6i q^{76} +(274456. - 32620.8i) q^{77} +(558983. + 968188. i) q^{79} +(337936. - 585322. i) q^{80} +(-1.82414e6 + 1.05317e6i) q^{82} -7.99427e6 q^{83} -3.81048e6 q^{85} +(-181600. + 104847. i) q^{86} +(-234226. + 405690. i) q^{88} +(4.33239e6 + 7.50393e6i) q^{89} +(-3.90179e6 - 5.21977e6i) q^{91} -2.68919e6i q^{92} +(-2.05192e6 - 1.18468e6i) q^{94} +(-6.63331e6 - 3.82974e6i) q^{95} +8.18016e6i q^{97} +(-1.55078e6 - 6.43159e6i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q + 1024 q^{4} - 2074 q^{7} + 1248 q^{10} - 59708 q^{16} - 105330 q^{19} + 544 q^{22} - 56250 q^{25} + 556220 q^{28} + 86862 q^{31} + 591034 q^{37} - 3036324 q^{40} - 837332 q^{43} + 3896752 q^{46} + 6626454 q^{49}+ \cdots + 96093708 q^{94}+O(q^{100}) \) 36 * q + 1024 * q^4 - 2074 * q^7 + 1248 * q^10 - 59708 * q^16 - 105330 * q^19 + 544 * q^22 - 56250 * q^25 + 556220 * q^28 + 86862 * q^31 + 591034 * q^37 - 3036324 * q^40 - 837332 * q^43 + 3896752 * q^46 + 6626454 * q^49 - 11085900 * q^52 - 134084 * q^58 + 4324644 * q^61 - 11555464 * q^64 + 8750706 * q^67 + 8823972 * q^70 - 744402 * q^73 + 1136238 * q^79 - 29281932 * q^82 + 18719472 * q^85 - 4365916 * q^88 + 21779742 * q^91 + 96093708 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more