LMFDB - Embedded newform 81.9.d.g.26.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [81,9,Mod(26,81)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(81, base_ring=CyclotomicField(6))

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

N = Newforms(chi, 9, names="a")

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

chi := DirichletCharacter("81.26");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 81 = 3^{4} $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	81.d (of order $6$, 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:	$32.9976674150$
Analytic rank:	$0$
Dimension:	$32$
Relative dimension:	$16$ over $\Q(\zeta_{6})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			26.6
Character	$\chi$	$=$	81.26
Dual form			81.9.d.g.53.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-8.67350 - 5.00765i) q^{2} +(-77.8470 - 134.835i) q^{4} +(-514.754 + 297.193i) q^{5} +(-2204.76 + 3818.75i) q^{7} +4123.23i q^{8} +5952.96 q^{10} +(3013.23 + 1739.69i) q^{11} +(9131.48 + 15816.2i) q^{13} +(38245.9 - 22081.3i) q^{14} +(718.880 - 1245.14i) q^{16} -86637.8i q^{17} +76109.5 q^{19} +(80144.1 + 46271.2i) q^{20} +(-17423.5 - 30178.4i) q^{22} +(-453885. + 262051. i) q^{23} +(-18664.8 + 32328.3i) q^{25} -182909. i q^{26} +686534. q^{28} +(-671532. - 387709. i) q^{29} +(462712. + 801441. i) q^{31} +(901661. - 520574. i) q^{32} +(-433851. + 751453. i) q^{34} -2.62095e6i q^{35} +237767. q^{37} +(-660136. - 381130. i) q^{38} +(-1.22540e6 - 2.12245e6i) q^{40} +(-446772. + 257944. i) q^{41} +(-1.97918e6 + 3.42804e6i) q^{43} -541718. i q^{44} +5.24903e6 q^{46} +(-4.61257e6 - 2.66307e6i) q^{47} +(-6.83949e6 - 1.18463e7i) q^{49} +(323777. - 186933. i) q^{50} +(1.42172e6 - 2.46248e6i) q^{52} -1.30260e7i q^{53} -2.06810e6 q^{55} +(-1.57456e7 - 9.09072e6i) q^{56} +(3.88302e6 + 6.72559e6i) q^{58} +(1.08626e7 - 6.27153e6i) q^{59} +(-5.94218e6 + 1.02922e7i) q^{61} -9.26840e6i q^{62} -1.07955e7 q^{64} +(-9.40093e6 - 5.42763e6i) q^{65} +(-6.76578e6 - 1.17187e7i) q^{67} +(-1.16818e7 + 6.74449e6i) q^{68} +(-1.31248e7 + 2.27328e7i) q^{70} +133345. i q^{71} +5.24749e7 q^{73} +(-2.06227e6 - 1.19065e6i) q^{74} +(-5.92489e6 - 1.02622e7i) q^{76} +(-1.32869e7 + 7.67118e6i) q^{77} +(2.39249e7 - 4.14392e7i) q^{79} +854585. i q^{80} +5.16676e6 q^{82} +(5.06857e7 + 2.92634e7i) q^{83} +(2.57482e7 + 4.45971e7i) q^{85} +(3.43328e7 - 1.98220e7i) q^{86} +(-7.17315e6 + 1.24243e7i) q^{88} -4.62606e7i q^{89} -8.05307e7 q^{91} +(7.06671e7 + 4.07997e7i) q^{92} +(2.66714e7 + 4.61962e7i) q^{94} +(-3.91777e7 + 2.26192e7i) q^{95} +(4.41196e7 - 7.64173e7i) q^{97} +1.36999e8i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q + 2048 q^{4} - 3692 q^{7} + 21504 q^{10} - 63860 q^{13} - 95116 q^{16} + 370216 q^{19} + 691980 q^{22} + 541712 q^{25} - 1994264 q^{28} - 571136 q^{31} - 1027656 q^{34} + 8708536 q^{37} + 2973768 q^{40}+ \cdots + 133878688 q^{97}+O(q^{100}) $ 32 * q + 2048 * q^4 - 3692 * q^7 + 21504 * q^10 - 63860 * q^13 - 95116 * q^16 + 370216 * q^19 + 691980 * q^22 + 541712 * q^25 - 1994264 * q^28 - 571136 * q^31 - 1027656 * q^34 + 8708536 * q^37 + 2973768 * q^40 - 5453084 * q^43 - 8468088 * q^46 - 14602560 * q^49 + 25384960 * q^52 - 28905144 * q^55 - 1213068 * q^58 - 31037996 * q^61 + 48506000 * q^64 + 62356828 * q^67 - 11075124 * q^70 - 42547328 * q^73 - 69593876 * q^76 + 151045036 * q^79 - 80402280 * q^82 - 160995564 * q^85 - 366976068 * q^88 + 747361592 * q^91 + 544741584 * q^94 + 133878688 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/81\mathbb{Z}\right)^\times$.

$n$	$2$
$\chi(n)$	$e\left(\frac{1}{6}\right)$

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 $

$2$	−8.67350	−	5.00765i	−0.542094	−	0.312978i	0.203833	−	0.979006i	$-0.434660\pi$
$2$	−8.67350	−	5.00765i	−0.542094	−	0.312978i	−0.745927	+	0.666028i	$0.767993\pi$
$3$		0			0
$3$		0			0
$4$	−77.8470	−	134.835i	−0.304090	−	0.526699i
$5$	−514.754	+	297.193i	−0.823606	+	0.475509i	−0.851658	−	0.524097i	$-0.824403\pi$
$5$	−514.754	+	297.193i	−0.823606	+	0.475509i	0.0280522	+	0.999606i	$0.491070\pi$
$6$		0			0
$7$	−2204.76	+	3818.75i	−0.918265	+	1.59048i	−0.116217	+	0.993224i	$0.537077\pi$
$7$	−2204.76	+	3818.75i	−0.918265	+	1.59048i	−0.802049	+	0.597258i	$0.796257\pi$
$8$			4123.23i			1.00665i
$9$		0			0
$10$	5952.96			0.595296
$11$	3013.23	+	1739.69i	0.205808	+	0.118823i	0.599362	−	0.800478i	$-0.295421\pi$
$11$	3013.23	+	1739.69i	0.205808	+	0.118823i	−0.393554	+	0.919302i	$0.628755\pi$
$12$		0			0
$13$	9131.48	+	15816.2i	0.319718	+	0.553768i	0.980429	−	0.196872i	$-0.0630784\pi$
$13$	9131.48	+	15816.2i	0.319718	+	0.553768i	−0.660711	+	0.750641i	$0.729745\pi$
$14$	38245.9	−	22081.3i	0.995572	−	0.574794i
$15$		0			0
$16$	718.880	−	1245.14i	0.0109692	−	0.0189993i
$17$		−	86637.8i		−	1.03732i	−0.854981	−	0.518659i	$-0.826432\pi$
$17$		−	86637.8i		−	1.03732i	0.854981	−	0.518659i	$-0.173568\pi$
$18$		0			0
$19$	76109.5			0.584016			0.292008	−	0.956416i	$-0.405677\pi$
$19$	76109.5			0.584016			0.292008	+	0.956416i	$0.405677\pi$
$20$	80144.1	+	46271.2i	0.500900	+	0.289195i
$21$		0			0
$22$	−17423.5	−	30178.4i	−0.0743780	−	0.128826i
$23$	−453885.	+	262051.i	−1.62194	+	0.936427i	−0.635538	+	0.772070i	$0.719222\pi$
$23$	−453885.	+	262051.i	−1.62194	+	0.936427i	−0.986401	+	0.164357i	$0.947445\pi$
$24$		0			0
$25$	−18664.8	+	32328.3i	−0.0477818	+	0.0827605i
$26$		−	182909.i		−	0.400259i
$27$		0			0
$28$	686534.			1.11694
$29$	−671532.	−	387709.i	−0.949455	−	0.548168i	−0.0565435	−	0.998400i	$-0.518008\pi$
$29$	−671532.	−	387709.i	−0.949455	−	0.548168i	−0.892912	+	0.450232i	$0.851341\pi$
$30$		0			0
$31$	462712.	+	801441.i	0.501031	+	0.867811i	0.999999	+	0.00119055i	$0.000378963\pi$
$31$	462712.	+	801441.i	0.501031	+	0.867811i	−0.498969	+	0.866620i	$0.666288\pi$
$32$	901661.	−	520574.i	0.859891	−	0.496458i
$33$		0			0
$34$	−433851.	+	751453.i	−0.324657	+	0.562323i
$35$		−	2.62095e6i		−	1.74658i
$36$		0			0
$37$	237767.			0.126866			0.0634329	−	0.997986i	$-0.479795\pi$
$37$	237767.			0.126866			0.0634329	+	0.997986i	$0.479795\pi$
$38$	−660136.	−	381130.i	−0.316591	−	0.182784i
$39$		0			0
$40$	−1.22540e6	−	2.12245e6i	−0.478671	−	0.829083i
$41$	−446772.	+	257944.i	−0.158107	+	0.0912829i	−0.576966	−	0.816768i	$-0.695763\pi$
$41$	−446772.	+	257944.i	−0.158107	+	0.0912829i	0.418859	+	0.908051i	$0.362430\pi$
$42$		0			0
$43$	−1.97918e6	+	3.42804e6i	−0.578910	+	1.00270i	0.416695	+	0.909046i	$0.363188\pi$
$43$	−1.97918e6	+	3.42804e6i	−0.578910	+	1.00270i	−0.995605	+	0.0936549i	$0.970145\pi$
$44$		−	541718.i		−	0.144532i
$45$		0			0
$46$	5.24903e6			1.17232
$47$	−4.61257e6	−	2.66307e6i	−0.945260	−	0.545746i	−0.0536550	−	0.998560i	$-0.517087\pi$
$47$	−4.61257e6	−	2.66307e6i	−0.945260	−	0.545746i	−0.891605	+	0.452813i	$0.850420\pi$
$48$		0			0
$49$	−6.83949e6	−	1.18463e7i	−1.18642	−	2.05494i
$50$	323777.	−	186933.i	0.0518044	−	0.0299093i
$51$		0			0
$52$	1.42172e6	−	2.46248e6i	0.194446	−	0.336791i
$53$		−	1.30260e7i		−	1.65085i	−0.564514	−	0.825424i	$-0.690936\pi$
$53$		−	1.30260e7i		−	1.65085i	0.564514	−	0.825424i	$-0.309064\pi$
$54$		0			0
$55$	−2.06810e6			−0.226006
$56$	−1.57456e7	−	9.09072e6i	−1.60106	−	0.924371i
$57$		0			0
$58$	3.88302e6	+	6.72559e6i	0.343129	+	0.594317i
$59$	1.08626e7	−	6.27153e6i	0.896450	−	0.517566i	0.0204032	−	0.999792i	$-0.493505\pi$
$59$	1.08626e7	−	6.27153e6i	0.896450	−	0.517566i	0.876047	+	0.482226i	$0.160172\pi$
$60$		0			0
$61$	−5.94218e6	+	1.02922e7i	−0.429167	+	0.743340i	−0.996799	−	0.0799427i	$-0.974526\pi$
$61$	−5.94218e6	+	1.02922e7i	−0.429167	+	0.743340i	0.567632	+	0.823282i	$0.307860\pi$
$62$		−	9.26840e6i		−	0.627246i
$63$		0			0
$64$	−1.07955e7			−0.643460
$65$	−9.40093e6	−	5.42763e6i	−0.526644	−	0.304058i
$66$		0			0
$67$	−6.76578e6	−	1.17187e7i	−0.335752	−	0.581540i	0.647877	−	0.761745i	$-0.275657\pi$
$67$	−6.76578e6	−	1.17187e7i	−0.335752	−	0.581540i	−0.983629	+	0.180205i	$0.942324\pi$
$68$	−1.16818e7	+	6.74449e6i	−0.546354	+	0.315438i
$69$		0			0
$70$	−1.31248e7	+	2.27328e7i	−0.546639	+	0.946807i
$71$			133345.i			0.00524738i	0.999997	+	0.00262369i	$0.000835147\pi$
$71$			133345.i			0.00524738i	−0.999997	+	0.00262369i	$0.999165\pi$
$72$		0			0
$73$	5.24749e7			1.84782			0.923910	−	0.382609i	$-0.124974\pi$
$73$	5.24749e7			1.84782			0.923910	+	0.382609i	$0.124974\pi$
$74$	−2.06227e6	−	1.19065e6i	−0.0687731	−	0.0397062i
$75$		0			0
$76$	−5.92489e6	−	1.02622e7i	−0.177593	−	0.307600i
$77$	−1.32869e7	+	7.67118e6i	−0.377972	+	0.218222i
$78$		0			0
$79$	2.39249e7	−	4.14392e7i	0.614246	−	1.06391i	−0.376270	−	0.926510i	$-0.622794\pi$
$79$	2.39249e7	−	4.14392e7i	0.614246	−	1.06391i	0.990516	−	0.137395i	$-0.0438731\pi$
$80$			854585.i			0.0208639i
$81$		0			0
$82$	5.16676e6			0.114278
$83$	5.06857e7	+	2.92634e7i	1.06800	+	0.616612i	0.927635	−	0.373487i	$-0.121838\pi$
$83$	5.06857e7	+	2.92634e7i	1.06800	+	0.616612i	0.140369	+	0.990099i	$0.455171\pi$
$84$		0			0
$85$	2.57482e7	+	4.45971e7i	0.493254	+	0.854341i
$86$	3.43328e7	−	1.98220e7i	0.627647	−	0.362372i
$87$		0			0
$88$	−7.17315e6	+	1.24243e7i	−0.119613	+	0.207176i
$89$		−	4.62606e7i		−	0.737312i	−0.929566	−	0.368656i	$-0.879818\pi$
$89$		−	4.62606e7i		−	0.737312i	0.929566	−	0.368656i	$-0.120182\pi$
$90$		0			0
$91$	−8.05307e7			−1.17435
$92$	7.06671e7	+	4.07997e7i	0.986430	+	0.569515i
$93$		0			0
$94$	2.66714e7	+	4.61962e7i	0.341613	+	0.591691i
$95$	−3.91777e7	+	2.26192e7i	−0.480999	+	0.277705i
$96$		0			0
$97$	4.41196e7	−	7.64173e7i	0.498361	−	0.863187i	−0.501637	−	0.865078i	$-0.667269\pi$
$97$	4.41196e7	−	7.64173e7i	0.498361	−	0.863187i	0.999998	+	0.00189124i	$0.000602002\pi$
$98$			1.36999e8i			1.48530i
$99$		0			0
$100$	5.81198e6			0.0581198
$101$	−4.97112e7	−	2.87007e7i	−0.477714	−	0.275809i	0.241749	−	0.970339i	$-0.422279\pi$
$101$	−4.97112e7	−	2.87007e7i	−0.477714	−	0.275809i	−0.719464	+	0.694530i	$0.755612\pi$
$102$		0			0
$103$	3.94562e7	+	6.83401e7i	0.350563	+	0.607193i	0.986348	−	0.164673i	$-0.0526569\pi$
$103$	3.94562e7	+	6.83401e7i	0.350563	+	0.607193i	−0.635785	+	0.771866i	$0.719324\pi$
$104$	−6.52138e7	+	3.76512e7i	−0.557450	+	0.321844i
$105$		0			0
$106$	−6.52295e7	+	1.12981e8i	−0.516679	+	0.894914i
$107$			2.50821e8i			1.91350i	0.290907	+	0.956751i	$0.406043\pi$
$107$			2.50821e8i			1.91350i	−0.290907	+	0.956751i	$0.593957\pi$
$108$		0			0
$109$	−7.76440e7			−0.550050			−0.275025	−	0.961437i	$-0.588686\pi$
$109$	−7.76440e7			−0.550050			−0.275025	+	0.961437i	$0.588686\pi$
$110$	1.79376e7	+	1.03563e7i	0.122516	+	0.0707349i
$111$		0			0
$112$	3.16991e6	+	5.49044e6i	0.0201453	+	0.0348927i
$113$	5.33353e7	−	3.07931e7i	0.327115	−	0.188860i	−0.327444	−	0.944870i	$-0.606187\pi$
$113$	5.33353e7	−	3.07931e7i	0.327115	−	0.188860i	0.654560	+	0.756010i	$0.272854\pi$
$114$		0			0
$115$	1.55759e8	−	2.69783e8i	0.890559	−	1.54249i
$116$			1.20728e8i			0.666769i
$117$		0			0
$118$	−1.25622e8			−0.647946
$119$	3.30848e8	+	1.91015e8i	1.64983	+	0.952533i
$120$		0			0
$121$	−1.01126e8	−	1.75156e8i	−0.471762	−	0.817116i
$122$	1.03079e8	−	5.95127e7i	0.465298	−	0.268640i
$123$		0			0
$124$	7.20415e7	−	1.24780e8i	0.304717	−	0.527785i
$125$		−	2.54370e8i		−	1.04190i
$126$		0			0
$127$	−3.58325e8			−1.37741			−0.688703	−	0.725044i	$-0.741819\pi$
$127$	−3.58325e8			−1.37741			−0.688703	+	0.725044i	$0.741819\pi$
$128$	−1.37191e8	−	7.92071e7i	−0.511075	−	0.295069i
$129$		0			0
$130$	5.43593e7	+	9.41530e7i	0.190327	+	0.329656i
$131$	1.62075e8	−	9.35739e7i	0.550339	−	0.317738i	−0.198920	−	0.980016i	$-0.563743\pi$
$131$	1.62075e8	−	9.35739e7i	0.550339	−	0.317738i	0.749259	+	0.662277i	$0.230410\pi$
$132$		0			0
$133$	−1.67803e8	+	2.90643e8i	−0.536281	+	0.928867i
$134$			1.35523e8i			0.420332i
$135$		0			0
$136$	3.57228e8			1.04421
$137$	1.30060e8	+	7.50900e7i	0.369199	+	0.213157i	0.673108	−	0.739544i	$-0.264959\pi$
$137$	1.30060e8	+	7.50900e7i	0.369199	+	0.213157i	−0.303909	+	0.952701i	$0.598292\pi$
$138$		0			0
$139$	−6.49820e7	−	1.12552e8i	−0.174074	−	0.301505i	0.765766	−	0.643119i	$-0.222360\pi$
$139$	−6.49820e7	−	1.12552e8i	−0.174074	−	0.301505i	−0.939840	+	0.341614i	$0.889027\pi$
$140$	−3.53396e8	+	2.04033e8i	−0.919919	+	0.531115i
$141$		0			0
$142$	667743.	−	1.15656e6i	0.00164231	−	0.00284457i
$143$			6.35437e7i			0.151960i
$144$		0			0
$145$	4.60898e8			1.04264
$146$	−4.55140e8	−	2.62775e8i	−1.00169	−	0.578327i
$147$		0			0
$148$	−1.85094e7	−	3.20593e7i	−0.0385786	−	0.0668200i
$149$	1.30697e8	−	7.54578e7i	0.265167	−	0.153094i	−0.361522	−	0.932363i	$-0.617743\pi$
$149$	1.30697e8	−	7.54578e7i	0.265167	−	0.153094i	0.626689	+	0.779269i	$0.284410\pi$
$150$		0			0
$151$	−8.56221e7	+	1.48302e8i	−0.164694	+	0.285258i	−0.936547	−	0.350543i	$-0.885997\pi$
$151$	−8.56221e7	+	1.48302e8i	−0.164694	+	0.285258i	0.771853	+	0.635802i	$0.219330\pi$
$152$			3.13817e8i			0.587899i
$153$		0			0
$154$	1.53658e8			0.273195
$155$	−4.76366e8	−	2.75030e8i	−0.825304	−	0.476490i
$156$		0			0
$157$	3.60934e8	+	6.25156e8i	0.594059	+	1.02894i	0.993679	+	0.112259i	$0.0358087\pi$
$157$	3.60934e8	+	6.25156e8i	0.594059	+	1.02894i	−0.399620	+	0.916681i	$0.630858\pi$
$158$	−4.15026e8	+	2.39615e8i	−0.665958	+	0.384491i
$159$		0			0
$160$	−3.09422e8	+	5.35935e8i	−0.472141	+	0.817772i
$161$		−	2.31103e9i		−	3.43955i
$162$		0			0
$163$	−1.90963e8			−0.270520			−0.135260	−	0.990810i	$-0.543187\pi$
$163$	−1.90963e8			−0.270520			−0.135260	+	0.990810i	$0.543187\pi$
$164$	6.95596e7	+	4.01603e7i	0.0961572	+	0.0555164i
$165$		0			0
$166$	−2.93081e8	−	5.07632e8i	−0.385972	−	0.668523i
$167$	4.84429e8	−	2.79685e8i	0.622822	−	0.359586i	−0.155145	−	0.987892i	$-0.549584\pi$
$167$	4.84429e8	−	2.79685e8i	0.622822	−	0.359586i	0.777967	+	0.628305i	$0.216251\pi$
$168$		0			0
$169$	2.41098e8	−	4.17593e8i	0.295560	−	0.511926i
$170$		−	5.15751e8i		−	0.617510i
$171$		0			0
$172$	6.16292e8			0.704162
$173$	−5.25797e8	−	3.03569e8i	−0.586994	−	0.338901i	0.176914	−	0.984226i	$-0.443389\pi$
$173$	−5.25797e8	−	3.03569e8i	−0.586994	−	0.338901i	−0.763908	+	0.645325i	$0.776722\pi$
$174$		0			0
$175$	−8.23024e7	−	1.42552e8i	−0.0877527	−	0.151992i
$176$	4.33230e6	−	2.50125e6i	0.00451510	−	0.00260680i
$177$		0			0
$178$	−2.31657e8	+	4.01241e8i	−0.230762	+	0.399692i
$179$		−	4.18834e8i		−	0.407971i	−0.978974	−	0.203986i	$-0.934610\pi$
$179$		−	4.18834e8i		−	0.407971i	0.978974	−	0.203986i	$-0.0653896\pi$
$180$		0			0
$181$	1.10216e9			1.02690			0.513451	−	0.858119i	$-0.328367\pi$
$181$	1.10216e9			1.02690			0.513451	+	0.858119i	$0.328367\pi$
$182$	6.98483e8	+	4.03269e8i	0.636605	+	0.367544i
$183$		0			0
$184$	−1.08050e9	−	1.87147e9i	−0.942653	−	1.63272i
$185$	−1.22391e8	+	7.06627e7i	−0.104487	+	0.0603258i
$186$		0			0
$187$	1.50723e8	−	2.61060e8i	0.123257	−	0.213488i
$188$			8.29247e8i			0.663823i
$189$		0			0
$190$	4.53077e8			0.347662
$191$	−1.54781e9	−	8.93626e8i	−1.16301	−	0.671463i	−0.210985	−	0.977489i	$-0.567667\pi$
$191$	−1.54781e9	−	8.93626e8i	−1.16301	−	0.671463i	−0.952023	+	0.306026i	$0.901001\pi$
$192$		0			0
$193$	3.58643e8	+	6.21188e8i	0.258484	+	0.447707i	0.965836	−	0.259154i	$-0.0834438\pi$
$193$	3.58643e8	+	6.21188e8i	0.258484	+	0.447707i	−0.707352	+	0.706861i	$0.750110\pi$
$194$	−7.65342e8	+	4.41870e8i	−0.540317	+	0.311952i
$195$		0			0
$196$	−1.06487e9	+	1.84440e9i	−0.721558	+	1.24977i
$197$			1.81939e9i			1.20798i	0.796992	+	0.603990i	$0.206423\pi$
$197$			1.81939e9i			1.20798i	−0.796992	+	0.603990i	$0.793577\pi$
$198$		0			0
$199$	−1.90300e9			−1.21346			−0.606730	−	0.794908i	$-0.707519\pi$
$199$	−1.90300e9			−1.21346			−0.606730	+	0.794908i	$0.707519\pi$
$200$	−1.33297e8	−	7.69592e7i	−0.0833108	−	0.0480995i
$201$		0			0
$202$	2.87446e8	+	4.97872e8i	0.172644	+	0.299028i
$203$	2.96113e9	−	1.70961e9i	1.74370	−	1.00673i
$204$		0			0
$205$	1.53318e8	−	2.65555e8i	0.0868118	−	0.150362i
$206$		−	7.90331e8i		−	0.438874i
$207$		0			0
$208$	2.62577e7			0.0140283
$209$	2.29335e8	+	1.32407e8i	0.120195	+	0.0693946i
$210$		0			0
$211$	−7.99513e7	−	1.38480e8i	−0.0403363	−	0.0698645i	0.845152	−	0.534525i	$-0.179510\pi$
$211$	−7.99513e7	−	1.38480e8i	−0.0403363	−	0.0698645i	−0.885489	+	0.464661i	$0.846176\pi$
$212$	−1.75636e9	+	1.01403e9i	−0.869499	+	0.502006i
$213$		0			0
$214$	1.25602e9	−	2.17550e9i	0.598884	−	1.03730i
$215$		−	2.35279e9i		−	1.10111i
$216$		0			0
$217$	−4.08067e9			−1.84032
$218$	6.73445e8	+	3.88814e8i	0.298179	+	0.172153i
$219$		0			0
$220$	1.60995e8	+	2.78851e8i	0.0687261	+	0.119037i
$221$	1.37028e9	−	7.91131e8i	0.574434	−	0.331649i
$222$		0			0
$223$	3.06593e8	−	5.31035e8i	0.123977	−	0.214735i	−0.797355	−	0.603510i	$-0.793768\pi$
$223$	3.06593e8	−	5.31035e8i	0.123977	−	0.214735i	0.921333	+	0.388775i	$0.127102\pi$
$224$			4.59096e9i			1.82352i
$225$		0			0
$226$	−6.16805e8			−0.236436
$227$	2.56740e9	+	1.48229e9i	0.966919	+	0.558251i	0.898296	−	0.439392i	$-0.144806\pi$
$227$	2.56740e9	+	1.48229e9i	0.966919	+	0.558251i	0.0686235	+	0.997643i	$0.478139\pi$
$228$		0			0
$229$	1.89482e9	+	3.28193e9i	0.689013	+	1.19340i	0.972158	+	0.234328i	$0.0752889\pi$
$229$	1.89482e9	+	3.28193e9i	0.689013	+	1.19340i	−0.283145	+	0.959077i	$0.591378\pi$
$230$	−2.70196e9	+	1.55998e9i	−0.965533	+	0.557451i
$231$		0			0
$232$	1.59862e9	−	2.76888e9i	0.551813	−	0.955768i
$233$			3.76374e9i			1.27701i	0.769616	+	0.638507i	$0.220448\pi$
$233$			3.76374e9i			1.27701i	−0.769616	+	0.638507i	$0.779552\pi$
$234$		0			0
$235$	3.16578e9			1.03803
$236$	−1.69124e9	−	9.76439e8i	−0.545202	−	0.314773i
$237$		0			0
$238$	−1.91307e9	−	3.31354e9i	−0.596243	−	1.03272i
$239$	−4.58786e8	+	2.64880e8i	−0.140611	+	0.0811816i	−0.568655	−	0.822576i	$-0.692536\pi$
$239$	−4.58786e8	+	2.64880e8i	−0.140611	+	0.0811816i	0.428044	+	0.903758i	$0.359203\pi$
$240$		0			0
$241$	4.19453e8	−	7.26513e8i	0.124341	−	0.215365i	−0.797134	−	0.603802i	$-0.793652\pi$
$241$	4.19453e8	−	7.26513e8i	0.124341	−	0.215365i	0.921475	+	0.388437i	$0.126985\pi$
$242$			2.02562e9i			0.590604i
$243$		0			0
$244$	1.85032e9			0.522021
$245$	7.04131e9	+	4.06530e9i	1.95429	+	1.12831i
$246$		0			0
$247$	6.94992e8	+	1.20376e9i	0.186721	+	0.323409i
$248$	−3.30453e9	+	1.90787e9i	−0.873581	+	0.504362i
$249$		0			0
$250$	−1.27380e9	+	2.20628e9i	−0.326092	+	0.564808i
$251$		−	2.40283e9i		−	0.605380i	−0.953089	−	0.302690i	$-0.902115\pi$
$251$		−	2.40283e9i		−	0.605380i	0.953089	−	0.302690i	$-0.0978847\pi$
$252$		0			0
$253$	−1.82355e9			−0.445076
$254$	3.10793e9	+	1.79436e9i	0.746683	+	0.431097i
$255$		0			0
$256$	2.17510e9	+	3.76739e9i	0.506431	+	0.877164i
$257$	4.15154e9	−	2.39689e9i	0.951648	−	0.549434i	0.0580555	−	0.998313i	$-0.481510\pi$
$257$	4.15154e9	−	2.39689e9i	0.951648	−	0.549434i	0.893592	+	0.448879i	$0.148177\pi$
$258$		0			0
$259$	−5.24218e8	+	9.07972e8i	−0.116496	+	0.201778i
$260$			1.69010e9i			0.369844i
$261$		0			0
$262$	−1.87434e9			−0.397780
$263$	−1.94790e9	−	1.12462e9i	−0.407139	−	0.235062i	0.282421	−	0.959291i	$-0.408863\pi$
$263$	−1.94790e9	−	1.12462e9i	−0.407139	−	0.235062i	−0.689560	+	0.724229i	$0.742196\pi$
$264$		0			0
$265$	3.87123e9	+	6.70517e9i	0.784993	+	1.35965i
$266$	2.91088e9	−	1.68059e9i	0.581429	−	0.335688i
$267$		0			0
$268$	−1.05339e9	+	1.82453e9i	−0.204197	+	0.353680i
$269$		−	2.01512e9i		−	0.384851i	−0.981312	−	0.192425i	$-0.938365\pi$
$269$		−	2.01512e9i		−	0.384851i	0.981312	−	0.192425i	$-0.0616353\pi$
$270$		0			0
$271$	−5.85840e9			−1.08618			−0.543090	−	0.839675i	$-0.682746\pi$
$271$	−5.85840e9			−1.08618			−0.543090	+	0.839675i	$0.682746\pi$
$272$	−1.07876e8	−	6.22821e7i	−0.0197083	−	0.0113786i
$273$		0			0
$274$	−7.52049e8	−	1.30259e9i	−0.133427	−	0.231102i
$275$	−1.12482e8	+	6.49417e7i	−0.0196677	+	0.0113552i
$276$		0			0
$277$	3.40794e9	−	5.90273e9i	0.578859	−	1.00261i	−0.416751	−	0.909021i	$-0.636831\pi$
$277$	3.40794e9	−	5.90273e9i	0.578859	−	1.00261i	0.995611	−	0.0935933i	$-0.0298353\pi$
$278$			1.30163e9i			0.217925i
$279$		0			0
$280$	1.08068e10			1.75819
$281$	−1.79419e9	−	1.03588e9i	−0.287768	−	0.166143i	0.349167	−	0.937061i	$-0.386465\pi$
$281$	−1.79419e9	−	1.03588e9i	−0.287768	−	0.166143i	−0.636935	+	0.770918i	$0.719798\pi$
$282$		0			0
$283$	−4.80878e8	−	8.32906e8i	−0.0749704	−	0.129852i	0.826103	−	0.563519i	$-0.190553\pi$
$283$	−4.80878e8	−	8.32906e8i	−0.0749704	−	0.129852i	−0.901073	+	0.433667i	$0.857220\pi$
$284$	1.79795e7	−	1.03805e7i	0.00276379	−	0.00159567i
$285$		0			0
$286$	3.18204e8	−	5.51146e8i	0.0475600	−	0.0823764i
$287$		−	2.27481e9i		−	0.335288i
$288$		0			0
$289$	−5.30347e8			−0.0760272
$290$	−3.99760e9	−	2.30801e9i	−0.565207	−	0.326322i
$291$		0			0
$292$	−4.08501e9	−	7.07544e9i	−0.561903	−	0.973245i
$293$	−1.05291e10	+	6.07899e9i	−1.42864	+	0.824823i	−0.997013	−	0.0772309i	$-0.975392\pi$
$293$	−1.05291e10	+	6.07899e9i	−1.42864	+	0.824823i	−0.431623	+	0.902054i	$0.642059\pi$
$294$		0			0
$295$	−3.72771e9	+	6.45659e9i	−0.492215	+	0.852541i
$296$			9.80369e8i			0.127709i
$297$		0			0
$298$	−1.51146e9			−0.191660
$299$	−8.28928e9	−	4.78582e9i	−1.03713	−	0.598786i
$300$		0			0
$301$	−8.72720e9	−	1.51160e10i	−1.06319	−	1.84149i
$302$	1.48529e9	−	8.57530e8i	0.178559	−	0.103091i
$303$		0			0
$304$	5.47136e7	−	9.47667e7i	0.00640620	−	0.0110959i
$305$		−	7.06391e9i		−	0.816292i
$306$		0			0
$307$	4.91035e9			0.552788			0.276394	−	0.961044i	$-0.410860\pi$
$307$	4.91035e9			0.552788			0.276394	+	0.961044i	$0.410860\pi$
$308$	2.06868e9	+	1.19436e9i	0.229875	+	0.132718i
$309$		0			0
$310$	2.75451e9	+	4.77094e9i	0.298261	+	0.516604i
$311$	2.32924e9	−	1.34479e9i	0.248984	−	0.143751i	−0.370315	−	0.928906i	$-0.620750\pi$
$311$	2.32924e9	−	1.34479e9i	0.248984	−	0.143751i	0.619299	+	0.785155i	$0.287417\pi$
$312$		0			0
$313$	2.56653e9	−	4.44536e9i	0.267405	−	0.463158i	−0.700786	−	0.713371i	$-0.747167\pi$
$313$	2.56653e9	−	4.44536e9i	0.267405	−	0.463158i	0.968191	+	0.250213i	$0.0805007\pi$
$314$		−	7.22972e9i		−	0.743709i
$315$		0			0
$316$	−7.44993e9			−0.747144
$317$	−1.33196e10	−	7.69006e9i	−1.31903	−	0.761540i	−0.335455	−	0.942056i	$-0.608890\pi$
$317$	−1.33196e10	−	7.69006e9i	−1.31903	−	0.761540i	−0.983572	+	0.180516i	$0.942223\pi$
$318$		0			0
$319$	−1.34899e9	−	2.33651e9i	−0.130270	−	0.225634i
$320$	5.55701e9	−	3.20834e9i	0.529958	−	0.305971i
$321$		0			0
$322$	−1.15728e10	+	2.00447e10i	−1.07650	+	1.86456i
$323$		−	6.59396e9i		−	0.605810i
$324$		0			0
$325$	−6.81747e8			−0.0611068
$326$	1.65632e9	+	9.56278e8i	0.146647	+	0.0846669i
$327$		0			0
$328$	−1.06356e9	−	1.84214e9i	−0.0918899	−	0.159158i
$329$	2.03392e10	−	1.17428e10i	1.73600	−	1.00228i
$330$		0			0
$331$	3.70488e9	−	6.41704e9i	0.308647	−	0.534593i	−0.669419	−	0.742885i	$-0.733457\pi$
$331$	3.70488e9	−	6.41704e9i	0.308647	−	0.534593i	0.978067	+	0.208292i	$0.0667904\pi$
$332$		−	9.11226e9i		−	0.750022i
$333$		0			0
$334$	−5.60225e9			−0.450170
$335$	6.96542e9	+	4.02149e9i	0.553055	+	0.319306i
$336$		0			0
$337$	−2.35250e9	−	4.07466e9i	−0.182394	−	0.315916i	0.760301	−	0.649571i	$-0.225051\pi$
$337$	−2.35250e9	−	4.07466e9i	−0.182394	−	0.315916i	−0.942695	+	0.333655i	$0.891718\pi$
$338$	−4.18232e9	+	2.41466e9i	−0.320443	+	0.185008i
$339$		0			0
$340$	4.00883e9	−	6.94350e9i	0.299987	−	0.519593i
$341$			3.21990e9i			0.238136i
$342$		0			0
$343$	3.48977e10			2.52127
$344$	−1.41346e10	−	8.16061e9i	−1.00937	−	0.582759i
$345$		0			0
$346$	3.04033e9	+	5.26601e9i	0.212137	+	0.367432i
$347$	−4.21934e9	+	2.43604e9i	−0.291023	+	0.168022i	−0.638403	−	0.769702i	$-0.720405\pi$
$347$	−4.21934e9	+	2.43604e9i	−0.291023	+	0.168022i	0.347380	+	0.937724i	$0.387071\pi$
$348$		0			0
$349$	−1.18998e10	+	2.06110e10i	−0.802117	+	1.38931i	0.116104	+	0.993237i	$0.462959\pi$
$349$	−1.18998e10	+	2.06110e10i	−0.802117	+	1.38931i	−0.918221	+	0.396070i	$0.870374\pi$
$350$			1.64857e9i			0.109859i
$351$		0			0
$352$	3.62255e9			0.235963
$353$	−6.84869e9	−	3.95409e9i	−0.441071	−	0.254652i	0.262981	−	0.964801i	$-0.415294\pi$
$353$	−6.84869e9	−	3.95409e9i	−0.441071	−	0.254652i	−0.704052	+	0.710149i	$0.748628\pi$
$354$		0			0
$355$	−3.96291e7	−	6.86397e7i	−0.00249518	−	0.00432177i
$356$	−6.23754e9	+	3.60125e9i	−0.388341	+	0.224209i
$357$		0			0
$358$	−2.09737e9	+	3.63275e9i	−0.127686	+	0.221158i
$359$			7.90247e9i			0.475757i	0.971295	+	0.237879i	$0.0764520\pi$
$359$			7.90247e9i			0.475757i	−0.971295	+	0.237879i	$0.923548\pi$
$360$		0			0
$361$	−1.11909e10			−0.658926
$362$	−9.55954e9	−	5.51920e9i	−0.556676	−	0.321397i
$363$		0			0
$364$	6.26907e9	+	1.08583e10i	0.357106	+	0.618526i
$365$	−2.70116e10	+	1.55952e10i	−1.52188	+	0.878656i
$366$		0			0
$367$	6.94556e9	−	1.20301e10i	0.382863	−	0.663137i	−0.608608	−	0.793471i	$-0.708272\pi$
$367$	6.94556e9	−	1.20301e10i	0.382863	−	0.663137i	0.991470	+	0.130334i	$0.0416049\pi$
$368$			7.53531e8i			0.0410875i
$369$		0			0
$370$	1.41542e9			0.0755226
$371$	4.97429e10	+	2.87191e10i	2.62564	+	1.51592i
$372$		0			0
$373$	−1.08855e10	−	1.88542e10i	−0.562356	−	0.974029i	−0.997290	−	0.0735671i	$-0.976562\pi$
$373$	−1.08855e10	−	1.88542e10i	−0.562356	−	0.974029i	0.434934	−	0.900462i	$-0.356772\pi$
$374$	−2.61459e9	+	1.50953e9i	−0.133634	+	0.0771536i
$375$		0			0
$376$	1.09805e10	−	1.90187e10i	0.549375	−	0.951546i
$377$		−	1.41614e10i		−	0.701038i
$378$		0			0
$379$	−1.86928e10			−0.905975			−0.452988	−	0.891517i	$-0.649642\pi$
$379$	−1.86928e10			−0.905975			−0.452988	+	0.891517i	$0.649642\pi$
$380$	6.09973e9	+	3.52168e9i	0.292534	+	0.168894i
$381$		0			0
$382$	8.94992e9	+	1.55017e10i	0.420306	+	0.727992i
$383$	−1.29018e10	+	7.44885e9i	−0.599590	+	0.346174i	−0.768880	−	0.639393i	$-0.779186\pi$
$383$	−1.29018e10	+	7.44885e9i	−0.599590	+	0.346174i	0.169290	+	0.985566i	$0.445852\pi$
$384$		0			0
$385$	4.55965e9	−	7.89754e9i	0.207533	−	0.359459i
$386$		−	7.18383e9i		−	0.323599i
$387$		0			0
$388$	−1.37383e10			−0.606186
$389$	−1.45384e10	−	8.39374e9i	−0.634918	−	0.366570i	0.147736	−	0.989027i	$-0.452801\pi$
$389$	−1.45384e10	−	8.39374e9i	−0.634918	−	0.366570i	−0.782654	+	0.622457i	$0.786135\pi$
$390$		0			0
$391$	2.27035e10	+	3.93236e10i	0.971372	+	1.68246i
$392$	4.88453e10	−	2.82008e10i	2.06861	−	1.19431i
$393$		0			0
$394$	9.11084e9	−	1.57804e10i	0.378071	−	0.654838i
$395$			2.84413e10i			1.16832i
$396$		0			0
$397$	3.63950e10			1.46514			0.732570	−	0.680692i	$-0.238321\pi$
$397$	3.63950e10			1.46514			0.732570	+	0.680692i	$0.238321\pi$
$398$	1.65056e10	+	9.52953e9i	0.657809	+	0.379786i
$399$		0			0
$400$	2.68354e7	+	4.64803e7i	0.00104826	+	0.00181564i
$401$	−2.67932e9	+	1.54691e9i	−0.103621	+	0.0598255i	−0.550915	−	0.834561i	$-0.685721\pi$
$401$	−2.67932e9	+	1.54691e9i	−0.103621	+	0.0598255i	0.447294	+	0.894387i	$0.352388\pi$
$402$		0			0
$403$	−8.45049e9	+	1.46367e10i	−0.320377	+	0.554910i
$404$			8.93706e9i			0.335482i
$405$		0			0
$406$	−3.42444e10			−1.26033
$407$	7.16446e8	+	4.13640e8i	0.0261099	+	0.0150746i
$408$		0			0
$409$	2.89420e9	+	5.01291e9i	0.103427	+	0.179142i	0.913095	−	0.407748i	$-0.133686\pi$
$409$	2.89420e9	+	5.01291e9i	0.103427	+	0.179142i	−0.809667	+	0.586889i	$0.800352\pi$
$410$	−2.65961e9	+	1.53553e9i	−0.0941202	+	0.0543403i
$411$		0			0
$412$	6.14309e9	−	1.06401e10i	0.213205	−	0.369282i
$413$			5.53087e10i			1.90105i
$414$		0			0
$415$	−3.47875e10			−1.17282
$416$	1.64670e10	+	9.50722e9i	0.549846	+	0.317454i
$417$		0			0
$418$	−1.32609e9	−	2.29686e9i	−0.0434379	−	0.0752367i
$419$	−2.09245e10	+	1.20808e10i	−0.678889	+	0.391957i	−0.799436	−	0.600751i	$-0.794868\pi$
$419$	−2.09245e10	+	1.20808e10i	−0.678889	+	0.391957i	0.120547	+	0.992708i	$0.461535\pi$
$420$		0			0
$421$	−5.83472e9	+	1.01060e10i	−0.185734	+	0.321701i	−0.943824	−	0.330450i	$-0.892800\pi$
$421$	−5.83472e9	+	1.01060e10i	−0.185734	+	0.321701i	0.758090	+	0.652150i	$0.226133\pi$
$422$			1.60147e9i			0.0504975i
$423$		0			0
$424$	5.37092e10			1.66182
$425$	2.80085e9	+	1.61707e9i	0.0858489	+	0.0495649i
$426$		0			0
$427$	−2.62021e10	−	4.53834e10i	−0.788179	−	1.36517i
$428$	3.38194e10	−	1.95257e10i	1.00784	−	0.581876i
$429$		0			0
$430$	−1.17820e10	+	2.04069e10i	−0.344622	+	0.596904i
$431$			6.44251e10i			1.86701i	0.358568	+	0.933504i	$0.383265\pi$
$431$			6.44251e10i			1.86701i	−0.358568	+	0.933504i	$0.616735\pi$
$432$		0			0
$433$	−5.35776e10			−1.52416			−0.762082	−	0.647480i	$-0.775823\pi$
$433$	−5.35776e10			−1.52416			−0.762082	+	0.647480i	$0.775823\pi$
$434$	3.53937e10	+	2.04346e10i	0.997624	+	0.575978i
$435$		0			0
$436$	6.04435e9	+	1.04691e10i	0.167265	+	0.289711i
$437$	−3.45450e10	+	1.99445e10i	−0.947238	+	0.546888i
$438$		0			0
$439$	2.13976e9	−	3.70617e9i	0.0576111	−	0.0997854i	−0.835781	−	0.549062i	$-0.814985\pi$
$439$	2.13976e9	−	3.70617e9i	0.0576111	−	0.0997854i	0.893393	+	0.449277i	$0.148318\pi$
$440$		−	8.52725e9i		−	0.227509i
$441$		0			0
$442$	−1.58468e10			−0.415196
$443$	−1.04572e10	−	6.03744e9i	−0.271518	−	0.156761i	0.358059	−	0.933699i	$-0.383439\pi$
$443$	−1.04572e10	−	6.03744e9i	−0.271518	−	0.156761i	−0.629577	+	0.776938i	$0.716772\pi$
$444$		0			0
$445$	1.37483e10	+	2.38128e10i	0.350599	+	0.607255i
$446$	−5.31847e9	+	3.07062e9i	−0.134415	+	0.0776044i
$447$		0			0
$448$	2.38014e10	−	4.12252e10i	0.590867	−	1.02341i
$449$		−	3.17911e10i		−	0.782205i	−0.920347	−	0.391103i	$-0.872094\pi$
$449$		−	3.17911e10i		−	0.782205i	0.920347	−	0.391103i	$-0.127906\pi$
$450$		0			0
$451$	−1.79497e9			−0.0433861
$452$	−8.30398e9	−	4.79431e9i	−0.198945	−	0.114861i
$453$		0			0
$454$	−1.48456e10	−	2.57133e10i	−0.349440	−	0.605249i
$455$	4.14535e10	−	2.39332e10i	0.967198	−	0.558412i
$456$		0			0
$457$	4.15200e9	−	7.19148e9i	0.0951903	−	0.164874i	−0.814498	−	0.580167i	$-0.802987\pi$
$457$	4.15200e9	−	7.19148e9i	0.0951903	−	0.164874i	0.909688	+	0.415292i	$0.136321\pi$
$458$		−	3.79544e10i		−	0.862583i
$459$		0			0
$460$	−4.85016e10			−1.08324
$461$	−2.76292e10	−	1.59518e10i	−0.611738	−	0.353187i	0.161908	−	0.986806i	$-0.448235\pi$
$461$	−2.76292e10	−	1.59518e10i	−0.611738	−	0.353187i	−0.773645	+	0.633619i	$0.781569\pi$
$462$		0			0
$463$	−1.37770e10	−	2.38625e10i	−0.299800	−	0.519269i	0.676290	−	0.736636i	$-0.263587\pi$
$463$	−1.37770e10	−	2.38625e10i	−0.299800	−	0.519269i	−0.976090	+	0.217366i	$0.930253\pi$
$464$	−9.65501e8	+	5.57432e8i	−0.0208296	+	0.0120260i
$465$		0			0
$466$	1.88475e10	−	3.26448e10i	0.399677	−	0.692261i
$467$		−	2.60699e10i		−	0.548114i	−0.961713	−	0.274057i	$-0.911634\pi$
$467$		−	2.60699e10i		−	0.548114i	0.961713	−	0.274057i	$-0.0883658\pi$
$468$		0			0
$469$	5.96675e10			1.23324
$470$	−2.74584e10	−	1.58531e10i	−0.562709	−	0.324880i
$471$		0			0
$472$	2.58590e10	+	4.47891e10i	0.521007	+	0.902411i
$473$	−1.19274e10	+	6.88631e9i	−0.238288	+	0.137576i
$474$		0			0
$475$	−1.42057e9	+	2.46049e9i	−0.0279053	+	0.0483334i
$476$		−	5.94798e10i		−	1.15862i
$477$		0			0
$478$	5.30570e9			0.101632
$479$	1.92371e9	+	1.11066e9i	0.0365425	+	0.0210978i	0.518160	−	0.855284i	$-0.326617\pi$
$479$	1.92371e9	+	1.11066e9i	0.0365425	+	0.0210978i	−0.481617	+	0.876382i	$0.659951\pi$
$480$		0			0
$481$	2.17116e9	+	3.76056e9i	0.0405613	+	0.0702542i
$482$	−7.27624e9	+	4.20094e9i	−0.134809	+	0.0778321i
$483$		0			0
$484$	−1.57448e10	+	2.72707e10i	−0.286916	+	0.496953i
$485$			5.24482e10i			0.947902i
$486$		0			0
$487$	8.62519e10			1.53339			0.766696	−	0.642011i	$-0.221900\pi$
$487$	8.62519e10			1.53339			0.766696	+	0.642011i	$0.221900\pi$
$488$	−4.24370e10	−	2.45010e10i	−0.748282	−	0.432021i
$489$		0			0
$490$	−4.07152e10	−	7.05208e10i	−0.706272	−	1.22330i
$491$	3.81434e9	−	2.20221e9i	0.0656286	−	0.0378907i	−0.466827	−	0.884349i	$-0.654603\pi$
$491$	3.81434e9	−	2.20221e9i	0.0656286	−	0.0378907i	0.532455	+	0.846458i	$0.321269\pi$
$492$		0			0
$493$	−3.35902e10	+	5.81800e10i	−0.568624	+	0.984886i
$494$		−	1.39211e10i		−	0.233758i
$495$		0			0
$496$	1.33054e9			0.0219837
$497$	−5.09210e8	−	2.93992e8i	−0.00834586	−	0.00481848i
$498$		0			0
$499$	1.38520e10	+	2.39924e10i	0.223414	+	0.386965i	0.955843	−	0.293879i	$-0.0949464\pi$
$499$	1.38520e10	+	2.39924e10i	0.223414	+	0.386965i	−0.732428	+	0.680844i	$0.761613\pi$
$500$	−3.42980e10	+	1.98020e10i	−0.548768	+	0.316831i
$501$		0			0
$502$	−1.20325e10	+	2.08409e10i	−0.189471	+	0.328173i
$503$		−	2.37306e10i		−	0.370712i	−0.982671	−	0.185356i	$-0.940656\pi$
$503$		−	2.37306e10i		−	0.370712i	0.982671	−	0.185356i	$-0.0593438\pi$
$504$		0			0
$505$	3.41187e10			0.524598
$506$	1.58165e10	+	9.13167e9i	0.241273	+	0.139299i
$507$		0			0
$508$	2.78945e10	+	4.83147e10i	0.418855	+	0.725478i
$509$	−3.31689e10	+	1.91501e10i	−0.494152	+	0.285299i	−0.726295	−	0.687383i	$-0.758759\pi$
$509$	−3.31689e10	+	1.91501e10i	−0.494152	+	0.285299i	0.232143	+	0.972682i	$0.425426\pi$
$510$		0			0
$511$	−1.15694e11	+	2.00388e11i	−1.69679	+	2.93893i
$512$		−	3.01455e9i		−	0.0438674i
$513$		0			0
$514$	−4.80111e10			−0.687843
$515$	−4.06205e10	−	2.34522e10i	−0.577452	−	0.333392i
$516$		0			0
$517$	−9.26582e9	−	1.60489e10i	−0.129695	−	0.224638i
$518$	9.09360e9	−	5.25019e9i	0.126304	−	0.0729216i
$519$		0			0
$520$	2.23794e10	−	3.87622e10i	0.306080	−	0.530146i
$521$		−	1.29094e11i		−	1.75208i	−0.482238	−	0.876040i	$-0.660176\pi$
$521$		−	1.29094e11i		−	1.75208i	0.482238	−	0.876040i	$-0.339824\pi$
$522$		0			0
$523$	1.37220e11			1.83405			0.917024	−	0.398831i	$-0.130584\pi$
$523$	1.37220e11			1.83405			0.917024	+	0.398831i	$0.130584\pi$
$524$	−2.52341e10	−	1.45689e10i	−0.334705	−	0.193242i
$525$		0			0
$526$	1.12634e10	+	1.95087e10i	0.147138	+	0.254851i
$527$	6.94351e10	−	4.00884e10i	0.900195	−	0.519728i
$528$		0			0
$529$	9.81855e10	−	1.70062e11i	1.25379	−	2.17163i
$530$		−	7.75431e10i		−	0.982742i
$531$		0			0
$532$	5.22518e10			0.652311
$533$	−8.15937e9	−	4.71081e9i	−0.101099	−	0.0583696i
$534$		0			0
$535$	−7.45424e10	−	1.29111e11i	−0.909888	−	1.57597i
$536$	4.83188e10	−	2.78969e10i	0.585406	−	0.337984i
$537$		0			0
$538$	−1.00910e10	+	1.74782e10i	−0.120450	+	0.208625i
$539$		−	4.75944e10i		−	0.563898i
$540$		0			0
$541$	−7.71547e10			−0.900686			−0.450343	−	0.892856i	$-0.648698\pi$
$541$	−7.71547e10			−0.900686			−0.450343	+	0.892856i	$0.648698\pi$
$542$	5.08128e10	+	2.93368e10i	0.588811	+	0.339950i
$543$		0			0
$544$	−4.51014e10	−	7.81179e10i	−0.514985	−	0.891980i
$545$	3.99676e10	−	2.30753e10i	0.453025	−	0.261554i
$546$		0			0
$547$	2.66534e10	−	4.61651e10i	0.297717	−	0.515661i	−0.677896	−	0.735157i	$-0.737108\pi$
$547$	2.66534e10	−	4.61651e10i	0.297717	−	0.515661i	0.975613	+	0.219497i	$0.0704415\pi$
$548$		−	2.33821e10i		−	0.259276i
$549$		0			0
$550$	1.30082e9			0.0142157
$551$	−5.11099e10	−	2.95083e10i	−0.554497	−	0.320139i
$552$		0			0
$553$	1.05497e11	+	1.82727e11i	1.12808	+	1.95390i
$554$	−5.91175e10	+	3.41315e10i	−0.627592	+	0.362340i
$555$		0			0
$556$	−1.01173e10	+	1.75237e10i	−0.105868	+	0.183369i
$557$			1.56229e11i			1.62309i	0.584291	+	0.811544i	$0.301373\pi$
$557$			1.56229e11i			1.62309i	−0.584291	+	0.811544i	$0.698627\pi$
$558$		0			0
$559$	−7.22912e10			−0.740352
$560$	−3.26344e9	−	1.88415e9i	−0.0331836	−	0.0191586i
$561$		0			0
$562$	1.03746e10	+	1.79693e10i	0.103998	+	0.180130i
$563$	−4.20078e10	+	2.42532e10i	−0.418116	+	0.241399i	−0.694271	−	0.719714i	$-0.744273\pi$
$563$	−4.20078e10	+	2.42532e10i	−0.418116	+	0.241399i	0.276155	+	0.961113i	$0.410940\pi$
$564$		0			0
$565$	−1.83030e10	+	3.17018e10i	−0.179609	+	0.311093i
$566$			9.63228e9i			0.0938563i
$567$		0			0
$568$	−5.49811e8			−0.00528227
$569$	−4.20475e10	−	2.42762e10i	−0.401136	−	0.231596i	0.285838	−	0.958278i	$-0.407728\pi$
$569$	−4.20475e10	−	2.42762e10i	−0.401136	−	0.231596i	−0.686974	+	0.726682i	$0.741061\pi$
$570$		0			0
$571$	−8.27408e10	−	1.43311e11i	−0.778351	−	1.34814i	−0.932892	−	0.360156i	$-0.882723\pi$
$571$	−8.27408e10	−	1.43311e11i	−0.778351	−	1.34814i	0.154541	−	0.987986i	$-0.450610\pi$
$572$	8.56791e9	−	4.94668e9i	0.0800370	−	0.0462094i
$573$		0			0
$574$	−1.13914e10	+	1.97306e10i	−0.104938	+	0.181757i
$575$		−	1.95644e10i		−	0.178977i
$576$		0			0
$577$	−1.25738e11			−1.13439			−0.567194	−	0.823584i	$-0.691971\pi$
$577$	−1.25738e11			−1.13439			−0.567194	+	0.823584i	$0.691971\pi$
$578$	4.59997e9	+	2.65579e9i	0.0412139	+	0.0237948i
$579$		0			0
$580$	−3.58795e10	−	6.21451e10i	−0.317055	−	0.549155i
$581$	−2.23499e11	+	1.29037e11i	−1.96142	+	1.13243i
$582$		0			0
$583$	2.26612e10	−	3.92503e10i	0.196159	−	0.339757i
$584$			2.16366e11i			1.86011i
$585$		0			0
$586$	1.21766e11			1.03261
$587$	5.90985e10	+	3.41205e10i	0.497765	+	0.287385i	0.727790	−	0.685800i	$-0.240548\pi$
$587$	5.90985e10	+	3.41205e10i	0.497765	+	0.287385i	−0.230025	+	0.973185i	$0.573881\pi$
$588$		0			0
$589$	3.52168e10	+	6.09973e10i	0.292610	+	0.506815i
$590$	6.46646e10	−	3.73341e10i	0.533653	−	0.308105i
$591$		0			0
$592$	1.70926e8	−	2.96052e8i	0.00139162	−	0.00241036i
$593$			2.22279e10i			0.179755i	0.995953	+	0.0898774i	$0.0286475\pi$
$593$			2.22279e10i			0.179755i	−0.995953	+	0.0898774i	$0.971352\pi$
$594$		0			0
$595$	−2.27074e11			−1.81175
$596$	−2.03487e10	−	1.17483e10i	−0.161269	−	0.0931088i
$597$		0			0
$598$	4.79314e10	+	8.30195e10i	0.374813	+	0.649196i
$599$	1.84741e11	−	1.06660e11i	1.43501	−	0.828503i	0.437513	−	0.899212i	$-0.355859\pi$
$599$	1.84741e11	−	1.06660e11i	1.43501	−	0.828503i	0.997497	+	0.0707089i	$0.0225261\pi$
$600$		0			0
$601$	−1.74980e10	+	3.03074e10i	−0.134119	+	0.232301i	−0.925261	−	0.379332i	$-0.876154\pi$
$601$	−1.74980e10	+	3.03074e10i	−0.134119	+	0.232301i	0.791142	+	0.611633i	$0.209487\pi$
$602$			1.74811e11i			1.33101i
$603$		0			0
$604$	2.66617e10			0.200327
$605$	1.04110e11	+	6.01082e10i	0.777093	+	0.448655i
$606$		0			0
$607$	5.53846e10	+	9.59289e10i	0.407975	+	0.706634i	0.994663	−	0.103179i	$-0.0329015\pi$
$607$	5.53846e10	+	9.59289e10i	0.407975	+	0.706634i	−0.586687	+	0.809814i	$0.699568\pi$
$608$	6.86250e10	−	3.96207e10i	0.502190	−	0.289939i
$609$		0			0
$610$	−3.53736e10	+	6.12688e10i	−0.255481	+	0.442507i
$611$		−	9.72710e10i		−	0.697940i
$612$		0			0
$613$	−4.84434e10			−0.343078			−0.171539	−	0.985177i	$-0.554874\pi$
$613$	−4.84434e10			−0.343078			−0.171539	+	0.985177i	$0.554874\pi$
$614$	−4.25899e10	−	2.45893e10i	−0.299663	−	0.173011i
$615$		0			0
$616$	−3.16301e10	−	5.47849e10i	−0.219673	−	0.380485i
$617$	6.64633e10	−	3.83726e10i	0.458607	−	0.264777i	−0.252851	−	0.967505i	$-0.581368\pi$
$617$	6.64633e10	−	3.83726e10i	0.458607	−	0.264777i	0.711459	+	0.702728i	$0.248035\pi$
$618$		0			0
$619$	1.05958e11	−	1.83524e11i	0.721721	−	1.25006i	−0.238589	−	0.971121i	$-0.576685\pi$
$619$	1.05958e11	−	1.83524e11i	0.721721	−	1.25006i	0.960310	−	0.278936i	$-0.0899818\pi$
$620$			8.56410e10i			0.579582i
$621$		0			0
$622$	−2.69368e10			−0.179964
$623$	1.76658e11	+	1.01993e11i	1.17268	+	0.677048i
$624$		0			0
$625$	6.83063e10	+	1.18310e11i	0.447652	+	0.775356i
$626$	−4.45216e10	+	2.57045e10i	−0.289917	+	0.167383i
$627$		0			0
$628$	5.61953e10	−	9.73330e10i	0.361294	−	0.625780i
$629$		−	2.05996e10i		−	0.131600i
$630$		0			0
$631$	−1.44761e10			−0.0913130			−0.0456565	−	0.998957i	$-0.514538\pi$
$631$	−1.44761e10			−0.0913130			−0.0456565	+	0.998957i	$0.514538\pi$
$632$	1.70864e11	+	9.86481e10i	1.07098	+	0.618330i
$633$		0			0
$634$	7.70182e10	+	1.33400e11i	0.476691	+	0.825652i
$635$	1.84449e11	−	1.06492e11i	1.13444	−	0.654969i
$636$		0			0
$637$	1.24909e11	−	2.16349e11i	0.758642	−	1.31401i
$638$			2.70210e10i			0.163087i
$639$		0			0
$640$	9.41593e10			0.561233
$641$	2.06878e10	+	1.19441e10i	0.122541	+	0.0707494i	0.560018	−	0.828481i	$-0.310794\pi$
$641$	2.06878e10	+	1.19441e10i	0.122541	+	0.0707494i	−0.437476	+	0.899230i	$0.644128\pi$
$642$		0			0
$643$	−7.57675e10	−	1.31233e11i	−0.443240	−	0.767714i	0.554688	−	0.832059i	$-0.312838\pi$
$643$	−7.57675e10	−	1.31233e11i	−0.443240	−	0.767714i	−0.997928	+	0.0643445i	$0.979504\pi$
$644$	−3.11607e11	+	1.79907e11i	−1.81161	+	1.04593i
$645$		0			0
$646$	−3.30202e10	+	5.71927e10i	−0.189605	+	0.328406i
$647$			1.51614e11i			0.865208i	0.901584	+	0.432604i	$0.142405\pi$
$647$			1.51614e11i			0.865208i	−0.901584	+	0.432604i	$0.857595\pi$
$648$		0			0
$649$	4.36420e10			0.245995
$650$	5.91313e9	+	3.41395e9i	0.0331256	+	0.0191251i
$651$		0			0
$652$	1.48659e10	+	2.57485e10i	0.0822624	+	0.142483i
$653$	−1.58425e11	+	9.14665e10i	−0.871305	+	0.503048i	−0.867782	−	0.496946i	$-0.834455\pi$
$653$	−1.58425e11	+	9.14665e10i	−0.871305	+	0.503048i	−0.00352329	+	0.999994i	$0.501122\pi$
$654$		0			0
$655$	−5.56191e10	+	9.63351e10i	−0.302175	+	0.523383i
$656$			7.41722e8i			0.00400521i
$657$		0			0
$658$	−2.35216e11			−1.25477
$659$	2.57869e11	+	1.48881e11i	1.36728	+	0.789398i	0.990580	−	0.136938i	$-0.0437260\pi$
$659$	2.57869e11	+	1.48881e11i	1.36728	+	0.789398i	0.376698	+	0.926336i	$0.377059\pi$
$660$		0			0
$661$	−5.59595e10	−	9.69247e10i	−0.293135	−	0.507725i	0.681414	−	0.731898i	$-0.261365\pi$
$661$	−5.59595e10	−	9.69247e10i	−0.293135	−	0.507725i	−0.974549	+	0.224173i	$0.928032\pi$
$662$	−6.42686e10	+	3.71055e10i	−0.334631	+	0.193199i
$663$		0			0
$664$	−1.20660e11	+	2.08989e11i	−0.620712	+	1.07511i
$665$		−	1.99480e11i		−	1.02003i
$666$		0			0
$667$	4.06397e11			2.05328
$668$	−7.54226e10	−	4.35452e10i	−0.378787	−	0.218693i
$669$		0			0
$670$	−4.02764e10	−	6.97607e10i	−0.199872	−	0.346188i
$671$	−3.58103e10	+	2.06751e10i	−0.176652	+	0.101990i
$672$		0			0
$673$	2.33548e10	−	4.04517e10i	0.113845	−	0.197186i	−0.803472	−	0.595342i	$-0.797016\pi$
$673$	2.33548e10	−	4.04517e10i	0.113845	−	0.197186i	0.917318	+	0.398156i	$0.130350\pi$
$674$			4.71220e10i			0.228341i
$675$		0			0
$676$	−7.50749e10			−0.359507
$677$	−1.48801e11	−	8.59105e10i	−0.708357	−	0.408970i	0.102095	−	0.994775i	$-0.467445\pi$
$677$	−1.48801e11	−	8.59105e10i	−0.708357	−	0.408970i	−0.810452	+	0.585804i	$0.800779\pi$
$678$		0			0
$679$	1.94546e11	+	3.36963e11i	0.915256	+	1.58527i
$680$	−1.83884e11	+	1.06166e11i	−0.860022	+	0.496534i
$681$		0			0
$682$	1.61241e10	−	2.79278e10i	0.0745313	−	0.129092i
$683$			6.74141e10i			0.309790i	0.987931	+	0.154895i	$0.0495040\pi$
$683$			6.74141e10i			0.309790i	−0.987931	+	0.154895i	$0.950496\pi$
$684$		0			0
$685$	−8.92650e10			−0.405433
$686$	−3.02685e11	−	1.74755e11i	−1.36677	−	0.789103i
$687$		0			0
$688$	2.84558e9	+	4.92869e9i	0.0127004	+	0.0219977i
$689$	2.06021e11	−	1.18946e11i	0.914187	−	0.527806i
$690$		0			0
$691$	−3.43078e10	+	5.94228e10i	−0.150481	+	0.260640i	−0.931404	−	0.363987i	$-0.881415\pi$
$691$	−3.43078e10	+	5.94228e10i	−0.150481	+	0.260640i	0.780924	+	0.624626i	$0.214749\pi$
$692$			9.45277e10i			0.412225i
$693$		0			0
$694$	4.87953e10			0.210349
$695$	6.68995e10	+	3.86244e10i	0.286737	+	0.165548i
$696$		0			0
$697$	2.23477e10	+	3.87073e10i	0.0946894	+	0.164007i
$698$	2.06426e11	−	1.19180e11i	0.869644	−	0.502089i
$699$		0			0
$700$	−1.28140e10	+	2.21945e10i	−0.0533694	+	0.0924385i
$701$		−	3.47803e11i		−	1.44033i	−0.693804	−	0.720164i	$-0.744066\pi$
$701$		−	3.47803e11i		−	1.44033i	0.693804	−	0.720164i	$-0.255934\pi$
$702$		0			0
$703$	1.80963e10			0.0740916
$704$	−3.25292e10	−	1.87808e10i	−0.132429	−	0.0764580i
$705$		0			0
$706$	3.96014e10	+	6.85916e10i	0.159401	+	0.276091i
$707$	2.19202e11	−	1.26556e11i	0.877337	−	0.506531i
$708$		0			0
$709$	1.32405e11	−	2.29332e11i	0.523986	−	0.907570i	−0.475624	−	0.879648i	$-0.657778\pi$
$709$	1.32405e11	−	2.29332e11i	0.523986	−	0.907570i	0.999610	−	0.0279213i	$-0.00888879\pi$
$710$			7.93795e8i			0.00312374i
$711$		0			0
$712$	1.90743e11			0.742215
$713$	−4.20036e11	−	2.42508e11i	−1.62528	−	0.938357i
$714$		0			0
$715$	−1.88848e10	−	3.27094e10i	−0.0722583	−	0.125155i
$716$	−5.64734e10	+	3.26049e10i	−0.214878	+	0.124060i
$717$		0			0
$718$	3.95728e10	−	6.85421e10i	0.148901	−	0.257905i
$719$		−	2.48269e11i		−	0.928981i	−0.885578	−	0.464491i	$-0.846237\pi$
$719$		−	2.48269e11i		−	0.928981i	0.885578	−	0.464491i	$-0.153763\pi$
$720$		0			0
$721$	−3.47965e11			−1.28764
$722$	9.70643e10	+	5.60401e10i	0.357199	+	0.206229i
$723$		0			0
$724$	−8.57995e10	−	1.48609e11i	−0.312270	−	0.540868i
$725$	2.50679e10	−	1.44730e10i	0.0907333	−	0.0523849i
$726$		0			0
$727$	−1.47750e11	+	2.55911e11i	−0.528921	+	0.916117i	0.470511	+	0.882394i	$0.344070\pi$
$727$	−1.47750e11	+	2.55911e11i	−0.528921	+	0.916117i	−0.999431	+	0.0337230i	$0.989264\pi$
$728$		−	3.32047e11i		−	1.18215i
$729$		0			0
$730$	3.12380e11			1.10000
$731$	2.96997e11	+	1.71472e11i	1.04012	+	0.600513i
$732$		0			0
$733$	−2.13954e11	−	3.70578e11i	−0.741146	−	1.28370i	−0.951974	−	0.306179i	$-0.900949\pi$
$733$	−2.13954e11	−	3.70578e11i	−0.741146	−	1.28370i	0.210828	−	0.977523i	$-0.432384\pi$
$734$	−1.20485e11	+	6.95618e10i	−0.415095	+	0.239655i
$735$		0			0
$736$	−2.72834e11	+	4.72562e11i	−0.929794	+	1.61045i
$737$		−	4.70814e10i		−	0.159580i
$738$		0			0
$739$	−1.19830e10			−0.0401781			−0.0200890	−	0.999798i	$-0.506395\pi$
$739$	−1.19830e10			−0.0401781			−0.0200890	+	0.999798i	$0.506395\pi$
$740$	1.90556e10	+	1.10018e10i	0.0635471	+	0.0366889i
$741$		0			0
$742$	−2.87630e11	−	4.98190e11i	−0.948896	−	1.64354i
$743$	2.40378e11	−	1.38782e11i	0.788749	−	0.455384i	−0.0507731	−	0.998710i	$-0.516169\pi$
$743$	2.40378e11	−	1.38782e11i	0.788749	−	0.455384i	0.839522	+	0.543326i	$0.182835\pi$
$744$		0			0
$745$	−4.48511e10	+	7.76844e10i	−0.145595	+	0.252179i
$746$			2.18042e11i			0.704020i
$747$		0			0
$748$	−4.69332e10			−0.149925
$749$	−9.57823e11	−	5.52999e11i	−3.04339	−	1.75710i
$750$		0			0
$751$	−7.24572e9	−	1.25499e10i	−0.0227783	−	0.0394532i	0.854412	−	0.519597i	$-0.173918\pi$
$751$	−7.24572e9	−	1.25499e10i	−0.0227783	−	0.0394532i	−0.877190	+	0.480144i	$0.840585\pi$
$752$	−6.63176e9	+	3.82885e9i	−0.0207376	+	0.0119728i
$753$		0			0
$754$	−7.09154e10	+	1.22829e11i	−0.219409	+	0.380028i
$755$		−	1.01785e11i		−	0.313254i
$756$		0			0
$757$	1.94269e11			0.591590			0.295795	−	0.955251i	$-0.404415\pi$
$757$	1.94269e11			0.591590			0.295795	+	0.955251i	$0.404415\pi$
$758$	1.62132e11	+	9.36067e10i	0.491123	+	0.283550i
$759$		0			0
$760$	−9.32644e10	−	1.61539e11i	−0.279551	−	0.484197i
$761$	−3.88024e11	+	2.24026e11i	−1.15696	+	0.667974i	−0.950575	−	0.310496i	$-0.899505\pi$
$761$	−3.88024e11	+	2.24026e11i	−1.15696	+	0.667974i	−0.206390	+	0.978470i	$0.566172\pi$
$762$		0			0
$763$	1.71186e11	−	2.96503e11i	0.505092	−	0.874845i
$764$			2.78264e11i			0.816740i
$765$		0			0
$766$	1.49205e11			0.433379
$767$	1.98383e11	+	1.14537e11i	0.573223	+	0.330950i
$768$		0			0
$769$	−1.13694e11	−	1.96924e11i	−0.325113	−	0.563111i	0.656423	−	0.754393i	$-0.272069\pi$
$769$	−1.13694e11	−	1.96924e11i	−0.325113	−	0.563111i	−0.981535	+	0.191282i	$0.938736\pi$
$770$	−7.90961e10	+	4.56662e10i	−0.225005	+	0.129907i
$771$		0			0
$772$	5.58386e10	−	9.67152e10i	0.157205	−	0.272286i
$773$		−	3.48003e11i		−	0.974686i	−0.873211	−	0.487343i	$-0.837966\pi$
$773$		−	3.48003e11i		−	0.974686i	0.873211	−	0.487343i	$-0.162034\pi$
$774$		0			0
$775$	−3.45457e10			−0.0957605
$776$	3.15087e11	+	1.81915e11i	0.868926	+	0.501675i
$777$		0			0
$778$	8.40657e10	+	1.45606e11i	0.229457	+	0.397431i
$779$	−3.40036e10	+	1.96320e10i	−0.0923368	+	0.0533107i
$780$		0			0
$781$	−2.31978e8	+	4.01798e8i	−0.000623509	+	0.00107995i
$782$		−	4.54764e11i		−	1.21607i
$783$		0			0
$784$	−1.96671e10			−0.0520566
$785$	−3.71585e11	−	2.14534e11i	−0.978541	−	0.564961i
$786$		0			0
$787$	−1.27120e11	−	2.20178e11i	−0.331370	−	0.573950i	0.651411	−	0.758725i	$-0.274178\pi$
$787$	−1.27120e11	−	2.20178e11i	−0.331370	−	0.573950i	−0.982781	+	0.184775i	$0.940844\pi$
$788$	2.45317e11	−	1.41634e11i	0.636242	−	0.367334i
$789$		0			0
$790$	1.42424e11	−	2.46686e11i	0.365658	−	0.633338i
$791$			2.71565e11i			0.693695i
$792$		0			0
$793$	−2.17044e11			−0.548851
$794$	−3.15672e11	−	1.82253e11i	−0.794243	−	0.458557i
$795$		0			0
$796$	1.48143e11	+	2.56590e11i	0.369001	+	0.639128i
$797$	4.79540e11	−	2.76863e11i	1.18848	−	0.686169i	0.230519	−	0.973068i	$-0.425958\pi$
$797$	4.79540e11	−	2.76863e11i	1.18848	−	0.686169i	0.957961	+	0.286898i	$0.0926242\pi$
$798$		0			0
$799$	−2.30722e11	+	3.99623e11i	−0.566112	+	0.980535i
$800$			3.88656e10i			0.0948866i
$801$		0			0
$802$	3.09854e10			0.0748962
$803$	1.58119e11	+	9.12899e10i	0.380296	+	0.219564i
$804$		0			0
$805$	6.86823e11	+	1.18961e12i	1.63554	+	2.83284i
$806$	1.46591e11	−	8.46342e10i	0.347349	−	0.200542i
$807$		0			0
$808$	1.18340e11	−	2.04971e11i	0.277642	−	0.480891i
$809$			1.00027e11i			0.233519i	0.993160	+	0.116759i	$0.0372506\pi$
$809$			1.00027e11i			0.233519i	−0.993160	+	0.116759i	$0.962749\pi$
$810$		0			0
$811$	9.06002e10			0.209433			0.104717	−	0.994502i	$-0.466606\pi$
$811$	9.06002e10			0.209433			0.104717	+	0.994502i	$0.466606\pi$
$812$	−4.61029e11	−	2.66175e11i	−1.06048	−	0.612271i
$813$		0			0
$814$	−4.14273e9	−	7.17542e9i	−0.00943602	−	0.0163437i
$815$	9.82992e10	−	5.67531e10i	0.222802	−	0.128635i
$816$		0			0
$817$	−1.50634e11	+	2.60906e11i	−0.338092	+	0.585593i
$818$		−	5.79726e10i		−	0.129482i
$819$		0			0
$820$	−4.77415e10			−0.105594
$821$	5.36582e11	+	3.09796e11i	1.18104	+	0.681871i	0.956254	−	0.292538i	$-0.0944997\pi$
$821$	5.36582e11	+	3.09796e11i	1.18104	+	0.681871i	0.224782	+	0.974409i	$0.427833\pi$
$822$		0			0
$823$	−1.81861e10	−	3.14993e10i	−0.0396407	−	0.0686597i	0.845524	−	0.533937i	$-0.179288\pi$
$823$	−1.81861e10	−	3.14993e10i	−0.0396407	−	0.0686597i	−0.885165	+	0.465277i	$0.845955\pi$
$824$	−2.81782e11	+	1.62687e11i	−0.611231	+	0.352894i
$825$		0			0
$826$	2.76967e11	−	4.79720e11i	0.594987	−	1.03055i
$827$			6.85214e10i			0.146489i	0.997314	+	0.0732444i	$0.0233353\pi$
$827$			6.85214e10i			0.146489i	−0.997314	+	0.0732444i	$0.976665\pi$
$828$		0			0
$829$	−6.13133e11			−1.29819			−0.649093	−	0.760709i	$-0.724851\pi$
$829$	−6.13133e11			−1.29819			−0.649093	+	0.760709i	$0.724851\pi$
$830$	3.01730e11	+	1.74204e11i	0.635778	+	0.367067i
$831$		0			0
$832$	−9.85786e10	−	1.70743e11i	−0.205726	−	0.356328i
$833$	−1.02634e12	+	5.92558e11i	−2.13163	+	1.23070i
$834$		0			0
$835$	−1.66241e11	+	2.87938e11i	−0.341973	+	0.592315i
$836$		−	4.12299e10i		−	0.0844087i
$837$		0			0
$838$	2.41985e11			0.490695
$839$	3.19993e11	+	1.84748e11i	0.645792	+	0.372848i	0.786842	−	0.617154i	$-0.211715\pi$
$839$	3.19993e11	+	1.84748e11i	0.645792	+	0.372848i	−0.141050	+	0.990002i	$0.545048\pi$
$840$		0			0
$841$	5.05133e10	+	8.74916e10i	0.100977	+	0.174897i
$842$	1.01215e11	−	5.84364e10i	0.201370	−	0.116261i
$843$		0			0
$844$	−1.24479e10	+	2.15605e10i	−0.0245317	+	0.0424901i
$845$			2.86610e11i			0.562167i
$846$		0			0
$847$	8.91836e11			1.73281
$848$	−1.62191e10	−	9.36411e9i	−0.0313649	−	0.0181085i
$849$		0			0
$850$	−1.61955e10	−	2.80514e10i	−0.0310254	−	0.0537376i
$851$	−1.07919e11	+	6.23069e10i	−0.205768	+	0.118800i
$852$		0			0
$853$	−1.02690e11	+	1.77864e11i	−0.193969	+	0.335963i	−0.946562	−	0.322522i	$-0.895469\pi$
$853$	−1.02690e11	+	1.77864e11i	−0.193969	+	0.335963i	0.752593	+	0.658486i	$0.228803\pi$
$854$			5.24844e11i			0.986730i
$855$		0			0
$856$	−1.03419e12			−1.92623
$857$	4.89607e11	+	2.82675e11i	0.907662	+	0.524039i	0.879678	−	0.475570i	$-0.157758\pi$
$857$	4.89607e11	+	2.82675e11i	0.907662	+	0.524039i	0.0279838	+	0.999608i	$0.491091\pi$
$858$		0			0
$859$	−3.17033e10	−	5.49116e10i	−0.0582279	−	0.100854i	0.835442	−	0.549579i	$-0.185212\pi$
$859$	−3.17033e10	−	5.49116e10i	−0.0582279	−	0.100854i	−0.893670	+	0.448725i	$0.851878\pi$
$860$	−3.17239e11	+	1.83158e11i	−0.579952	+	0.334836i
$861$		0			0
$862$	3.22618e11	−	5.58791e11i	0.584332	−	1.01209i
$863$			7.20037e11i			1.29811i	0.760742	+	0.649055i	$0.224835\pi$
$863$			7.20037e11i			1.29811i	−0.760742	+	0.649055i	$0.775165\pi$
$864$		0			0
$865$	3.60875e11			0.644602
$866$	4.64705e11	+	2.68298e11i	0.826240	+	0.477030i
$867$		0			0
$868$	3.17668e11	+	5.50217e11i	0.559621	+	0.969293i
$869$	1.44183e11	−	8.32439e10i	0.252833	−	0.145973i
$870$		0			0
$871$	1.23563e11	−	2.14018e11i	0.214692	−	0.371858i
$872$		−	3.20145e11i		−	0.553707i
$873$		0			0
$874$	3.99501e11			0.684655
$875$	9.71377e11	+	5.60825e11i	1.65713	+	0.956742i
$876$		0			0
$877$	−2.43477e11	−	4.21715e11i	−0.411586	−	0.712887i	0.583478	−	0.812129i	$-0.301692\pi$
$877$	−2.43477e11	−	4.21715e11i	−0.411586	−	0.712887i	−0.995063	+	0.0992421i	$0.968358\pi$
$878$	−3.71184e10	+	2.14303e10i	−0.0624612	+	0.0360620i
$879$		0			0
$880$	−1.48671e9	+	2.57506e9i	−0.00247911	+	0.00429395i
$881$			1.10625e12i			1.83633i	0.396196	+	0.918166i	$0.370330\pi$
$881$			1.10625e12i			1.83633i	−0.396196	+	0.918166i	$0.629670\pi$
$882$		0			0
$883$	7.45262e10			0.122593			0.0612966	−	0.998120i	$-0.480476\pi$
$883$	7.45262e10			0.122593			0.0612966	+	0.998120i	$0.480476\pi$
$884$	−2.13344e11	−	1.23174e11i	−0.349359	−	0.201702i
$885$		0			0
$886$	6.04667e10	+	1.04731e11i	0.0981254	+	0.169958i
$887$	1.00427e12	−	5.79818e11i	1.62240	−	0.936692i	0.636123	−	0.771588i	$-0.280537\pi$
$887$	1.00427e12	−	5.79818e11i	1.62240	−	0.936692i	0.986276	−	0.165105i	$-0.0527962\pi$
$888$		0			0
$889$	7.90018e11	−	1.36835e12i	1.26482	−	2.19074i
$890$		−	2.75387e11i		−	0.438919i
$891$		0			0
$892$	−9.54693e10			−0.150801
$893$	−3.51060e11	−	2.02685e11i	−0.552047	−	0.318724i
$894$		0			0
$895$	1.24475e11	+	2.15596e11i	0.193994	+	0.336008i
$896$	6.04944e11	−	3.49265e11i	0.938606	−	0.541904i
$897$		0			0
$898$	−1.59199e11	+	2.75740e11i	−0.244813	+	0.424028i
$899$		−	7.17591e11i		−	1.09860i
$900$		0			0
$901$	−1.12854e12			−1.71245
$902$	1.55686e10	+	8.98856e9i	0.0235193	+	0.0135789i
$903$		0			0
$904$	1.26967e11	+	2.19914e11i	0.190116	+	0.329290i
$905$	−5.67339e11	+	3.27553e11i	−0.845762	+	0.488301i
$906$		0			0
$907$	−6.30868e11	+	1.09270e12i	−0.932201	+	1.61462i	−0.152650	+	0.988280i	$0.548781\pi$
$907$	−6.30868e11	+	1.09270e12i	−0.932201	+	1.61462i	−0.779551	+	0.626339i	$0.784553\pi$
$908$		−	4.61567e11i		−	0.679034i
$909$		0			0
$910$	−4.79396e11			−0.699083
$911$	−6.43775e11	−	3.71684e11i	−0.934675	−	0.539635i	−0.0463878	−	0.998924i	$-0.514771\pi$
$911$	−6.43775e11	−	3.71684e11i	−0.934675	−	0.539635i	−0.888287	+	0.459289i	$0.848104\pi$
$912$		0			0
$913$	1.01818e11	+	1.76355e11i	0.146536	+	0.253807i
$914$	−7.20248e10	+	4.15835e10i	−0.103204	+	0.0595849i
$915$		0			0
$916$	2.95013e11	−	5.10977e11i	0.419043	−	0.725804i
$917$			8.25230e11i			1.16707i
$918$		0			0
$919$	−8.99130e10			−0.126055			−0.0630276	−	0.998012i	$-0.520076\pi$
$919$	−8.99130e10			−0.126055			−0.0630276	+	0.998012i	$0.520076\pi$
$920$	1.11238e12	+	6.42232e11i	1.55275	+	0.896481i
$921$		0			0
$922$	1.59761e11	+	2.76715e11i	0.221079	+	0.382921i
$923$	−2.10900e9	+	1.21763e9i	−0.00290583	+	0.00167768i
$924$		0			0
$925$	−4.43786e9	+	7.68660e9i	−0.00606187	+	0.0104995i
$926$			2.75962e11i			0.375323i
$927$		0			0
$928$	−8.07325e11			−1.08857
$929$	−7.43680e11	−	4.29364e11i	−0.998442	−	0.576451i	−0.0906554	−	0.995882i	$-0.528896\pi$
$929$	−7.43680e11	−	4.29364e11i	−0.998442	−	0.576451i	−0.907787	+	0.419431i	$0.862230\pi$
$930$		0			0
$931$	−5.20550e11	−	9.01620e11i	−0.692890	−	1.20012i
$932$	5.07483e11	−	2.92995e11i	0.672602	−	0.388327i
$933$		0			0
$934$	−1.30549e11	+	2.26117e11i	−0.171548	+	0.297129i
$935$			1.79175e11i			0.234440i
$936$		0			0
$937$	1.25416e11			0.162702			0.0813512	−	0.996685i	$-0.474076\pi$
$937$	1.25416e11			0.162702			0.0813512	+	0.996685i	$0.474076\pi$
$938$	−5.17526e11	−	2.98794e11i	−0.668530	−	0.385976i
$939$		0			0
$940$	−2.46447e11	−	4.26858e11i	−0.315654	−	0.546729i
$941$	−8.77244e11	+	5.06477e11i	−1.11882	+	0.645954i	−0.941101	−	0.338127i	$-0.890207\pi$
$941$	−8.77244e11	+	5.06477e11i	−1.11882	+	0.645954i	−0.177724	+	0.984080i	$0.556873\pi$
$942$		0			0
$943$	1.35189e11	−	2.34154e11i	0.170960	−	0.296111i
$944$		−	1.80339e10i		−	0.0227092i
$945$		0			0
$946$	1.37937e11			0.172233
$947$	−2.58946e11	−	1.49503e11i	−0.321966	−	0.185887i	0.330303	−	0.943875i	$-0.392849\pi$
$947$	−2.58946e11	−	1.49503e11i	−0.321966	−	0.185887i	−0.652269	+	0.757988i	$0.726183\pi$
$948$		0			0
$949$	4.79173e11	+	8.29952e11i	0.590782	+	1.02326i
$950$	2.46425e10	−	1.42274e10i	0.0302546	−	0.0174675i
$951$		0			0
$952$	−7.87600e11	+	1.36416e12i	−0.958866	+	1.66080i
$953$			9.57954e11i			1.16138i	0.814126	+	0.580688i	$0.197217\pi$
$953$			9.57954e11i			1.16138i	−0.814126	+	0.580688i	$0.802783\pi$
$954$		0			0
$955$	1.06232e12			1.27715
$956$	7.14301e10	+	4.12402e10i	0.0855165	+	0.0493730i
$957$		0			0
$958$	−1.11235e10	−	1.92665e10i	−0.0132063	−	0.0228740i
$959$	−5.73500e11	+	3.31110e11i	−0.678045	+	0.391470i
$960$		0			0
$961$	−1.75995e9	+	3.04832e9i	−0.00206351	+	0.00357410i
$962$		−	4.34896e10i		−	0.0507792i
$963$		0			0
$964$	−1.30612e11			−0.151243
$965$	−3.69226e11	−	2.13173e11i	−0.425778	−	0.245823i
$966$		0			0
$967$	−5.92952e11	−	1.02702e12i	−0.678131	−	1.17456i	−0.975543	−	0.219808i	$-0.929457\pi$
$967$	−5.92952e11	−	1.02702e12i	−0.678131	−	1.17456i	0.297412	−	0.954749i	$-0.403876\pi$
$968$	7.22210e11	−	4.16968e11i	0.822549	−	0.474899i
$969$		0			0
$970$	2.62642e11	−	4.54909e11i	0.296672	−	0.513851i
$971$			5.47759e10i			0.0616187i	0.999525	+	0.0308093i	$0.00980847\pi$
$971$			5.47759e10i			0.0616187i	−0.999525	+	0.0308093i	$0.990192\pi$
$972$		0			0
$973$	5.73077e11			0.639384
$974$	−7.48106e11	−	4.31919e11i	−0.831242	−	0.479918i
$975$		0			0
$976$	8.54343e9	+	1.47977e10i	0.00941527	+	0.0163077i
$977$	−8.01165e11	+	4.62553e11i	−0.879313	+	0.507672i	−0.870432	−	0.492289i	$-0.836160\pi$
$977$	−8.01165e11	+	4.62553e11i	−0.879313	+	0.507672i	−0.00888112	+	0.999961i	$0.502827\pi$
$978$		0			0
$979$	8.04791e10	−	1.39394e11i	0.0876097	−	0.151744i
$980$		−	1.26589e12i		−	1.37243i
$981$		0			0
$982$	−4.41116e10			−0.0474358
$983$	−5.00770e11	−	2.89120e11i	−0.536321	−	0.309645i	0.207266	−	0.978285i	$-0.433543\pi$
$983$	−5.00770e11	−	2.89120e11i	−0.536321	−	0.309645i	−0.743586	+	0.668640i	$0.766877\pi$
$984$		0			0
$985$	−5.40709e11	−	9.36536e11i	−0.574406	−	0.994900i
$986$	5.82690e11	−	3.36416e11i	0.616495	−	0.355934i
$987$		0			0
$988$	1.08206e11	−	1.87418e11i	0.113560	−	0.196691i
$989$		−	2.07458e12i		−	2.16843i
$990$		0			0
$991$	−1.41284e11			−0.146487			−0.0732435	−	0.997314i	$-0.523335\pi$
$991$	−1.41284e11			−0.146487			−0.0732435	+	0.997314i	$0.523335\pi$
$992$	8.34419e11	+	4.81752e11i	0.861664	+	0.497482i
$993$		0			0
$994$	2.94442e9	+	5.09988e9i	0.00301616	+	0.00522414i
$995$	9.79575e11	−	5.65558e11i	0.999414	−	0.577012i
$996$		0			0
$997$	6.20994e11	−	1.07559e12i	0.628502	−	1.08860i	−0.359351	−	0.933203i	$-0.617002\pi$
$997$	6.20994e11	−	1.07559e12i	0.628502	−	1.08860i	0.987852	−	0.155395i	$-0.0496649\pi$
$998$		−	2.77464e11i		−	0.279695i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	81.9.d.g.26.6		32
3.2	odd	2	inner	81.9.d.g.26.11		32
9.2	odd	6		81.9.b.b.80.6	✓	16
9.4	even	3	inner	81.9.d.g.53.11		32
9.5	odd	6	inner	81.9.d.g.53.6		32
9.7	even	3		81.9.b.b.80.11	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
81.9.b.b.80.6	✓	16	9.2	odd	6
81.9.b.b.80.11	yes	16	9.7	even	3
81.9.d.g.26.6		32	1.1	even	1	trivial
81.9.d.g.26.11		32	3.2	odd	2	inner
81.9.d.g.53.6		32	9.5	odd	6	inner
81.9.d.g.53.11		32	9.4	even	3	inner

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:	\( N \)	\(=\)	\( 81 = 3^{4} \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	81.d (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-8.67350 - 5.00765i) q^{2} +(-77.8470 - 134.835i) q^{4} +(-514.754 + 297.193i) q^{5} +(-2204.76 + 3818.75i) q^{7} +4123.23i q^{8} +5952.96 q^{10} +(3013.23 + 1739.69i) q^{11} +(9131.48 + 15816.2i) q^{13} +(38245.9 - 22081.3i) q^{14} +(718.880 - 1245.14i) q^{16} -86637.8i q^{17} +76109.5 q^{19} +(80144.1 + 46271.2i) q^{20} +(-17423.5 - 30178.4i) q^{22} +(-453885. + 262051. i) q^{23} +(-18664.8 + 32328.3i) q^{25} -182909. i q^{26} +686534. q^{28} +(-671532. - 387709. i) q^{29} +(462712. + 801441. i) q^{31} +(901661. - 520574. i) q^{32} +(-433851. + 751453. i) q^{34} -2.62095e6i q^{35} +237767. q^{37} +(-660136. - 381130. i) q^{38} +(-1.22540e6 - 2.12245e6i) q^{40} +(-446772. + 257944. i) q^{41} +(-1.97918e6 + 3.42804e6i) q^{43} -541718. i q^{44} +5.24903e6 q^{46} +(-4.61257e6 - 2.66307e6i) q^{47} +(-6.83949e6 - 1.18463e7i) q^{49} +(323777. - 186933. i) q^{50} +(1.42172e6 - 2.46248e6i) q^{52} -1.30260e7i q^{53} -2.06810e6 q^{55} +(-1.57456e7 - 9.09072e6i) q^{56} +(3.88302e6 + 6.72559e6i) q^{58} +(1.08626e7 - 6.27153e6i) q^{59} +(-5.94218e6 + 1.02922e7i) q^{61} -9.26840e6i q^{62} -1.07955e7 q^{64} +(-9.40093e6 - 5.42763e6i) q^{65} +(-6.76578e6 - 1.17187e7i) q^{67} +(-1.16818e7 + 6.74449e6i) q^{68} +(-1.31248e7 + 2.27328e7i) q^{70} +133345. i q^{71} +5.24749e7 q^{73} +(-2.06227e6 - 1.19065e6i) q^{74} +(-5.92489e6 - 1.02622e7i) q^{76} +(-1.32869e7 + 7.67118e6i) q^{77} +(2.39249e7 - 4.14392e7i) q^{79} +854585. i q^{80} +5.16676e6 q^{82} +(5.06857e7 + 2.92634e7i) q^{83} +(2.57482e7 + 4.45971e7i) q^{85} +(3.43328e7 - 1.98220e7i) q^{86} +(-7.17315e6 + 1.24243e7i) q^{88} -4.62606e7i q^{89} -8.05307e7 q^{91} +(7.06671e7 + 4.07997e7i) q^{92} +(2.66714e7 + 4.61962e7i) q^{94} +(-3.91777e7 + 2.26192e7i) q^{95} +(4.41196e7 - 7.64173e7i) q^{97} +1.36999e8i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 2048 q^{4} - 3692 q^{7} + 21504 q^{10} - 63860 q^{13} - 95116 q^{16} + 370216 q^{19} + 691980 q^{22} + 541712 q^{25} - 1994264 q^{28} - 571136 q^{31} - 1027656 q^{34} + 8708536 q^{37} + 2973768 q^{40}+ \cdots + 133878688 q^{97}+O(q^{100}) \) 32 * q + 2048 * q^4 - 3692 * q^7 + 21504 * q^10 - 63860 * q^13 - 95116 * q^16 + 370216 * q^19 + 691980 * q^22 + 541712 * q^25 - 1994264 * q^28 - 571136 * q^31 - 1027656 * q^34 + 8708536 * q^37 + 2973768 * q^40 - 5453084 * q^43 - 8468088 * q^46 - 14602560 * q^49 + 25384960 * q^52 - 28905144 * q^55 - 1213068 * q^58 - 31037996 * q^61 + 48506000 * q^64 + 62356828 * q^67 - 11075124 * q^70 - 42547328 * q^73 - 69593876 * q^76 + 151045036 * q^79 - 80402280 * q^82 - 160995564 * q^85 - 366976068 * q^88 + 747361592 * q^91 + 544741584 * q^94 + 133878688 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more