LMFDB - Embedded newform 177.9.c.a.58.10

Newspace parameters

Level:	$ N $	$=$	$ 177 = 3 \cdot 59 $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	177.c (of order $2$, degree $1$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$72.1060139808$
Analytic rank:	$0$
Dimension:	$80$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			58.10
Character	$\chi$	$=$	177.58
Dual form			177.9.c.a.58.71

$q$-expansion

$f(q)$	$=$	$q-25.6289i q^{2} +46.7654 q^{3} -400.842 q^{4} -239.734 q^{5} -1198.55i q^{6} -1755.35 q^{7} +3712.14i q^{8} +2187.00 q^{9} +O(q^{10})$	\(q-25.6289i q^{2} +46.7654 q^{3} -400.842 q^{4} -239.734 q^{5} -1198.55i q^{6} -1755.35 q^{7} +3712.14i q^{8} +2187.00 q^{9} +6144.13i q^{10} +3302.81i q^{11} -18745.5 q^{12} +13138.8i q^{13} +44987.8i q^{14} -11211.3 q^{15} -7477.27 q^{16} +119793. q^{17} -56050.5i q^{18} +148304. q^{19} +96095.5 q^{20} -82089.8 q^{21} +84647.5 q^{22} +144769. i q^{23} +173600. i q^{24} -333153. q^{25} +336735. q^{26} +102276. q^{27} +703620. q^{28} +397853. q^{29} +287332. i q^{30} +1.82954e6i q^{31} +1.14194e6i q^{32} +154457. i q^{33} -3.07016e6i q^{34} +420818. q^{35} -876641. q^{36} -278533. i q^{37} -3.80086e6i q^{38} +614443. i q^{39} -889927. i q^{40} +1.92610e6 q^{41} +2.10387e6i q^{42} -3.71138e6i q^{43} -1.32390e6i q^{44} -524298. q^{45} +3.71028e6 q^{46} -4.04454e6i q^{47} -349677. q^{48} -2.68353e6 q^{49} +8.53834e6i q^{50} +5.60215e6 q^{51} -5.26660e6i q^{52} -1.28463e6 q^{53} -2.62122e6i q^{54} -791796. i q^{55} -6.51613e6i q^{56} +6.93547e6 q^{57} -1.01966e7i q^{58} +(3.67157e6 - 1.15477e7i) q^{59} +4.49394e6 q^{60} -2.00520e7i q^{61} +4.68892e7 q^{62} -3.83896e6 q^{63} +2.73526e7 q^{64} -3.14983e6i q^{65} +3.95857e6 q^{66} -6.13210e6i q^{67} -4.80179e7 q^{68} +6.77018e6i q^{69} -1.07851e7i q^{70} +3.64592e7 q^{71} +8.11846e6i q^{72} -1.88627e7i q^{73} -7.13849e6 q^{74} -1.55800e7 q^{75} -5.94463e7 q^{76} -5.79760e6i q^{77} +1.57475e7 q^{78} +7.26205e7 q^{79} +1.79256e6 q^{80} +4.78297e6 q^{81} -4.93639e7i q^{82} +5.12745e7i q^{83} +3.29050e7 q^{84} -2.87184e7 q^{85} -9.51187e7 q^{86} +1.86058e7 q^{87} -1.22605e7 q^{88} +4.51465e7i q^{89} +1.34372e7i q^{90} -2.30633e7i q^{91} -5.80296e7i q^{92} +8.55591e7i q^{93} -1.03657e8 q^{94} -3.55534e7 q^{95} +5.34034e7i q^{96} +6.88077e7i q^{97} +6.87761e7i q^{98} +7.22324e6i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 80q - 10240q^{4} + 160q^{7} + 174960q^{9} + O(q^{10}) $	$ 80q - 10240q^{4} + 160q^{7} + 174960q^{9} - 22680q^{12} - 59616q^{15} + 1199848q^{16} - 10608q^{17} - 27516q^{19} - 146436q^{20} - 974696q^{22} + 5718040q^{25} - 797484q^{26} - 3133000q^{28} + 1725924q^{29} + 4318800q^{35} - 22394880q^{36} - 732180q^{41} + 22752084q^{46} + 8703936q^{48} + 55899176q^{49} - 10373832q^{51} - 39265944q^{53} - 11408040q^{57} - 33575112q^{59} - 18034488q^{60} + 13038600q^{62} + 349920q^{63} - 241654260q^{64} - 35711928q^{66} + 36772608q^{68} - 235272660q^{71} - 63050712q^{74} + 74363184q^{75} + 9454680q^{76} - 10865988q^{78} + 17252580q^{79} + 318203976q^{80} + 382637520q^{81} - 20743128q^{84} - 27245820q^{85} + 105666984q^{86} + 29437992q^{87} + 82079788q^{88} + 121215992q^{94} - 690837276q^{95} + O(q^{100}) $

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/177\mathbb{Z}\right)^\times$.

$n$	$61$	$119$
$\chi(n)$	$-1$	$1$

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 $

$2$		−	25.6289i		−	1.60181i	−0.598793	−	0.800904i	$-0.704353\pi$
$2$		−	25.6289i		−	1.60181i	0.598793	−	0.800904i	$-0.295647\pi$
$3$	46.7654			0.577350
$3$	46.7654			0.577350
$4$	−400.842			−1.56579
$5$	−239.734			−0.383574			−0.191787	−	0.981437i	$-0.561428\pi$
$5$	−239.734			−0.383574			−0.191787	+	0.981437i	$0.561428\pi$
$6$		−	1198.55i		−	0.924804i
$7$	−1755.35			−0.731093			−0.365546	−	0.930793i	$-0.619118\pi$
$7$	−1755.35			−0.731093			−0.365546	+	0.930793i	$0.619118\pi$
$8$			3712.14i			0.906285i
$9$	2187.00			0.333333
$10$			6144.13i			0.614413i
$11$			3302.81i			0.225586i	0.993618	+	0.112793i	$0.0359797\pi$
$11$			3302.81i			0.225586i	−0.993618	+	0.112793i	$0.964020\pi$
$12$	−18745.5			−0.904009
$13$			13138.8i			0.460027i	0.973187	+	0.230014i	$0.0738771\pi$
$13$			13138.8i			0.460027i	−0.973187	+	0.230014i	$0.926123\pi$
$14$			44987.8i			1.17107i
$15$	−11211.3			−0.221457
$16$	−7477.27			−0.114094
$17$	119793.			1.43428			0.717141	−	0.696928i	$-0.245450\pi$
$17$	119793.			1.43428			0.717141	+	0.696928i	$0.245450\pi$
$18$		−	56050.5i		−	0.533936i
$19$	148304.			1.13799			0.568994	−	0.822342i	$-0.307333\pi$
$19$	148304.			1.13799			0.568994	+	0.822342i	$0.307333\pi$
$20$	96095.5			0.600597
$21$	−82089.8			−0.422097
$22$	84647.5			0.361346
$23$			144769.i			0.517327i	0.965968	+	0.258663i	$0.0832820\pi$
$23$			144769.i			0.517327i	−0.965968	+	0.258663i	$0.916718\pi$
$24$			173600.i			0.523244i
$25$	−333153.			−0.852871
$26$	336735.			0.736876
$27$	102276.			0.192450
$28$	703620.			1.14474
$29$	397853.			0.562511			0.281255	−	0.959633i	$-0.409249\pi$
$29$	397853.			0.562511			0.281255	+	0.959633i	$0.409249\pi$
$30$			287332.i			0.354731i
$31$			1.82954e6i			1.98105i	0.137337	+	0.990524i	$0.456146\pi$
$31$			1.82954e6i			1.98105i	−0.137337	+	0.990524i	$0.543854\pi$
$32$			1.14194e6i			1.08904i
$33$			154457.i			0.130242i
$34$		−	3.07016e6i		−	2.29745i
$35$	420818.			0.280429
$36$	−876641.			−0.521930
$37$		−	278533.i		−	0.148617i	−0.997235	−	0.0743086i	$-0.976325\pi$
$37$		−	278533.i		−	0.148617i	0.997235	−	0.0743086i	$-0.0236750\pi$
$38$		−	3.80086e6i		−	1.82284i
$39$			614443.i			0.265597i
$40$		−	889927.i		−	0.347628i
$41$	1.92610e6			0.681622			0.340811	−	0.940132i	$-0.389298\pi$
$41$	1.92610e6			0.681622			0.340811	+	0.940132i	$0.389298\pi$
$42$			2.10387e6i			0.676118i
$43$		−	3.71138e6i		−	1.08558i	−0.839869	−	0.542790i	$-0.817368\pi$
$43$		−	3.71138e6i		−	1.08558i	0.839869	−	0.542790i	$-0.182632\pi$
$44$		−	1.32390e6i		−	0.353221i
$45$	−524298.			−0.127858
$46$	3.71028e6			0.828658
$47$		−	4.04454e6i		−	0.828853i	−0.910083	−	0.414427i	$-0.863982\pi$
$47$		−	4.04454e6i		−	0.828853i	0.910083	−	0.414427i	$-0.136018\pi$
$48$	−349677.			−0.0658723
$49$	−2.68353e6			−0.465503
$50$			8.53834e6i			1.36614i
$51$	5.60215e6			0.828083
$52$		−	5.26660e6i		−	0.720306i
$53$	−1.28463e6			−0.162808			−0.0814038	−	0.996681i	$-0.525940\pi$
$53$	−1.28463e6			−0.162808			−0.0814038	+	0.996681i	$0.525940\pi$
$54$		−	2.62122e6i		−	0.308268i
$55$		−	791796.i		−	0.0865292i
$56$		−	6.51613e6i		−	0.662579i
$57$	6.93547e6			0.657017
$58$		−	1.01966e7i		−	0.901035i
$59$	3.67157e6	−	1.15477e7i	0.303000	−	0.952990i
$59$	3.67157e6	−	1.15477e7i	0.303000	−	0.952990i
$60$	4.49394e6			0.346755
$61$		−	2.00520e7i		−	1.44823i	−0.689679	−	0.724115i	$-0.742248\pi$
$61$		−	2.00520e7i		−	1.44823i	0.689679	−	0.724115i	$-0.257752\pi$
$62$	4.68892e7			3.17326
$63$	−3.83896e6			−0.243698
$64$	2.73526e7			1.63034
$65$		−	3.14983e6i		−	0.176455i
$66$	3.95857e6			0.208623
$67$		−	6.13210e6i		−	0.304306i	−0.988357	−	0.152153i	$-0.951379\pi$
$67$		−	6.13210e6i		−	0.304306i	0.988357	−	0.152153i	$-0.0486206\pi$
$68$	−4.80179e7			−2.24578
$69$			6.77018e6i			0.298679i
$70$		−	1.07851e7i		−	0.449193i
$71$	3.64592e7			1.43474			0.717372	−	0.696691i	$-0.245345\pi$
$71$	3.64592e7			1.43474			0.717372	+	0.696691i	$0.245345\pi$
$72$			8.11846e6i			0.302095i
$73$		−	1.88627e7i		−	0.664221i	−0.943240	−	0.332111i	$-0.892239\pi$
$73$		−	1.88627e7i		−	0.664221i	0.943240	−	0.332111i	$-0.107761\pi$
$74$	−7.13849e6			−0.238056
$75$	−1.55800e7			−0.492405
$76$	−5.94463e7			−1.78185
$77$		−	5.79760e6i		−	0.164925i
$78$	1.57475e7			0.425435
$79$	7.26205e7			1.86445			0.932225	−	0.361879i	$-0.117865\pi$
$79$	7.26205e7			1.86445			0.932225	+	0.361879i	$0.117865\pi$
$80$	1.79256e6			0.0437636
$81$	4.78297e6			0.111111
$82$		−	4.93639e7i		−	1.09183i
$83$			5.12745e7i			1.08041i	0.841533	+	0.540205i	$0.181653\pi$
$83$			5.12745e7i			1.08041i	−0.841533	+	0.540205i	$0.818347\pi$
$84$	3.29050e7			0.660914
$85$	−2.87184e7			−0.550154
$86$	−9.51187e7			−1.73889
$87$	1.86058e7			0.324766
$88$	−1.22605e7			−0.204446
$89$			4.51465e7i			0.719555i	0.933038	+	0.359778i	$0.117147\pi$
$89$			4.51465e7i			0.719555i	−0.933038	+	0.359778i	$0.882853\pi$
$90$			1.34372e7i			0.204804i
$91$		−	2.30633e7i		−	0.336323i
$92$		−	5.80296e7i		−	0.810024i
$93$			8.55591e7i			1.14376i
$94$	−1.03657e8			−1.32766
$95$	−3.55534e7			−0.436503
$96$			5.34034e7i			0.628759i
$97$			6.88077e7i			0.777230i	0.921400	+	0.388615i	$0.127046\pi$
$97$			6.88077e7i			0.777230i	−0.921400	+	0.388615i	$0.872954\pi$
$98$			6.87761e7i			0.745647i
$99$			7.22324e6i			0.0751954i
$100$	1.33542e8			1.33542
$101$		−	1.23394e8i		−	1.18579i	−0.805280	−	0.592895i	$-0.797985\pi$
$101$		−	1.23394e8i		−	1.18579i	0.805280	−	0.592895i	$-0.202015\pi$
$102$		−	1.43577e8i		−	1.32643i
$103$		−	4.13868e7i		−	0.367716i	−0.982953	−	0.183858i	$-0.941141\pi$
$103$		−	4.13868e7i		−	0.367716i	0.982953	−	0.183858i	$-0.0588587\pi$
$104$	−4.87733e7			−0.416916
$105$	1.96797e7			0.161906
$106$			3.29237e7i			0.260787i
$107$	−1.55501e8			−1.18631			−0.593156	−	0.805088i	$-0.702118\pi$
$107$	−1.55501e8			−1.18631			−0.593156	+	0.805088i	$0.702118\pi$
$108$	−4.09965e7			−0.301336
$109$		−	1.54559e8i		−	1.09493i	−0.836827	−	0.547467i	$-0.815592\pi$
$109$		−	1.54559e8i		−	1.09493i	0.836827	−	0.547467i	$-0.184408\pi$
$110$	−2.02929e7			−0.138603
$111$		−	1.30257e7i		−	0.0858042i
$112$	1.31253e7			0.0834134
$113$			6.02902e7i			0.369771i	0.982760	+	0.184885i	$0.0591914\pi$
$113$			6.02902e7i			0.369771i	−0.982760	+	0.184885i	$0.940809\pi$
$114$		−	1.77749e8i		−	1.05242i
$115$		−	3.47061e7i		−	0.198433i
$116$	−1.59476e8			−0.880773
$117$			2.87347e7i			0.153342i
$118$	−2.95956e8	−	9.40983e7i	−1.52651	−	0.485349i
$119$	−2.10279e8			−1.04859
$120$		−	4.16178e7i		−	0.200703i
$121$	2.03450e8			0.949111
$122$	−5.13911e8			−2.31979
$123$	9.00749e7			0.393535
$124$		−	7.33356e8i		−	3.10190i
$125$	1.73514e8			0.710714
$126$			9.83884e7i			0.390357i
$127$	−5.45221e7			−0.209584			−0.104792	−	0.994494i	$-0.533418\pi$
$127$	−5.45221e7			−0.209584			−0.104792	+	0.994494i	$0.533418\pi$
$128$		−	4.08680e8i		−	1.52245i
$129$		−	1.73564e8i		−	0.626760i
$130$	−8.07267e7			−0.282647
$131$		−	1.30574e8i		−	0.443375i	−0.975118	−	0.221687i	$-0.928844\pi$
$131$		−	1.30574e8i		−	0.443375i	0.975118	−	0.221687i	$-0.0711565\pi$
$132$		−	6.19129e7i		−	0.203932i
$133$	−2.60325e8			−0.831974
$134$	−1.57159e8			−0.487440
$135$	−2.45190e7			−0.0738189
$136$			4.44688e8i			1.29987i
$137$	2.49809e8			0.709131			0.354566	−	0.935031i	$-0.384629\pi$
$137$	2.49809e8			0.709131			0.354566	+	0.935031i	$0.384629\pi$
$138$	1.73513e8			0.478426
$139$	−1.91233e8			−0.512277			−0.256138	−	0.966640i	$-0.582450\pi$
$139$	−1.91233e8			−0.512277			−0.256138	+	0.966640i	$0.582450\pi$
$140$	−1.68682e8			−0.439092
$141$		−	1.89144e8i		−	0.478539i
$142$		−	9.34411e8i		−	2.29818i
$143$	−4.33951e7			−0.103776
$144$	−1.63528e7			−0.0380314
$145$	−9.53790e7			−0.215765
$146$	−4.83431e8			−1.06395
$147$	−1.25496e8			−0.268758
$148$			1.11648e8i			0.232703i
$149$			8.24265e8i			1.67233i	0.548478	+	0.836165i	$0.315207\pi$
$149$			8.24265e8i			1.67233i	−0.548478	+	0.836165i	$0.684793\pi$
$150$			3.99299e8i			0.788738i
$151$			5.40833e8i			1.04029i	0.854077	+	0.520147i	$0.174123\pi$
$151$			5.40833e8i			1.04029i	−0.854077	+	0.520147i	$0.825877\pi$
$152$			5.50524e8i			1.03134i
$153$	2.61987e8			0.478094
$154$	−1.48586e8			−0.264177
$155$		−	4.38603e8i		−	0.759880i
$156$		−	2.46295e8i		−	0.415869i
$157$			1.06979e9i			1.76076i	0.474272	+	0.880379i	$0.342711\pi$
$157$			1.06979e9i			1.76076i	−0.474272	+	0.880379i	$0.657289\pi$
$158$		−	1.86119e9i		−	2.98649i
$159$	−6.00762e7			−0.0939970
$160$		−	2.73763e8i		−	0.417729i
$161$		−	2.54121e8i		−	0.378214i
$162$		−	1.22582e8i		−	0.177979i
$163$	1.06110e9			1.50316			0.751579	−	0.659643i	$-0.229293\pi$
$163$	1.06110e9			1.50316			0.751579	+	0.659643i	$0.229293\pi$
$164$	−7.72062e8			−1.06728
$165$		−	3.70286e7i		−	0.0499576i
$166$	1.31411e9			1.73061
$167$	6.71744e6			0.00863650			0.00431825	−	0.999991i	$-0.498625\pi$
$167$	6.71744e6			0.00863650			0.00431825	+	0.999991i	$0.498625\pi$
$168$		−	3.04729e8i		−	0.382540i
$169$	6.43101e8			0.788375
$170$			7.36022e8i			0.881241i
$171$	3.24340e8			0.379329
$172$			1.48768e9i			1.69979i
$173$			1.61132e9i			1.79887i	0.437060	+	0.899433i	$0.356020\pi$
$173$			1.61132e9i			1.79887i	−0.437060	+	0.899433i	$0.643980\pi$
$174$		−	4.76846e8i		−	0.520213i
$175$	5.84801e8			0.623528
$176$		−	2.46960e7i		−	0.0257381i
$177$	1.71702e8	−	5.40034e8i	0.174937	−	0.550209i
$178$	1.15706e9			1.15259
$179$		−	4.00114e7i		−	0.0389737i	−0.999810	−	0.0194869i	$-0.993797\pi$
$179$		−	4.00114e7i		−	0.0389737i	0.999810	−	0.0194869i	$-0.00620325\pi$
$180$	2.10161e8			0.200199
$181$	7.36437e8			0.686153			0.343077	−	0.939307i	$-0.388531\pi$
$181$	7.36437e8			0.686153			0.343077	+	0.939307i	$0.388531\pi$
$182$	−5.91088e8			−0.538725
$183$		−	9.37738e8i		−	0.836136i
$184$	−5.37404e8			−0.468845
$185$			6.67737e7i			0.0570058i
$186$	2.19279e9			1.83208
$187$			3.95652e8i			0.323555i
$188$			1.62122e9i			1.29781i
$189$	−1.79530e8			−0.140699
$190$			9.11196e8i			0.699194i
$191$		−	1.70160e9i		−	1.27857i	−0.768971	−	0.639283i	$-0.779231\pi$
$191$		−	1.70160e9i		−	1.27857i	0.768971	−	0.639283i	$-0.220769\pi$
$192$	1.27915e9			0.941278
$193$	2.42920e9			1.75079			0.875394	−	0.483410i	$-0.160602\pi$
$193$	2.42920e9			1.75079			0.875394	+	0.483410i	$0.160602\pi$
$194$	1.76347e9			1.24497
$195$		−	1.47303e8i		−	0.101876i
$196$	1.07567e9			0.728880
$197$	−2.28650e9			−1.51812			−0.759059	−	0.651022i	$-0.774341\pi$
$197$	−2.28650e9			−1.51812			−0.759059	+	0.651022i	$0.774341\pi$
$198$	1.85124e8			0.120449
$199$	2.26762e9			1.44597			0.722983	−	0.690865i	$-0.242770\pi$
$199$	2.26762e9			1.44597			0.722983	+	0.690865i	$0.242770\pi$
$200$		−	1.23671e9i		−	0.772944i
$201$		−	2.86770e8i		−	0.175691i
$202$	−3.16245e9			−1.89941
$203$	−6.98373e8			−0.411248
$204$	−2.24558e9			−1.29660
$205$	−4.61752e8			−0.261453
$206$	−1.06070e9			−0.589011
$207$			3.16610e8i			0.172442i
$208$		−	9.82427e7i		−	0.0524864i
$209$			4.89819e8i			0.256714i
$210$		−	5.04370e8i		−	0.259342i
$211$			1.01729e9i			0.513234i	0.966513	+	0.256617i	$0.0826080\pi$
$211$			1.01729e9i			0.513234i	−0.966513	+	0.256617i	$0.917392\pi$
$212$	5.14934e8			0.254922
$213$	1.70503e9			0.828349
$214$			3.98533e9i			1.90024i
$215$			8.89744e8i			0.416401i
$216$			3.79663e8i			0.174415i
$217$		−	3.21149e9i		−	1.44833i
$218$	−3.96118e9			−1.75387
$219$		−	8.82122e8i		−	0.383488i
$220$			3.17385e8i			0.135486i
$221$			1.57394e9i			0.659809i
$222$	−3.33834e8			−0.137442
$223$	3.43376e9			1.38852			0.694258	−	0.719727i	$-0.255733\pi$
$223$	3.43376e9			1.38852			0.694258	+	0.719727i	$0.255733\pi$
$224$		−	2.00451e9i		−	0.796191i
$225$	−7.28605e8			−0.284290
$226$	1.54517e9			0.592302
$227$		−	8.60197e8i		−	0.323962i	−0.986794	−	0.161981i	$-0.948212\pi$
$227$		−	8.60197e8i		−	0.323962i	0.986794	−	0.161981i	$-0.0517884\pi$
$228$	−2.78003e9			−1.02875
$229$		−	1.87481e9i		−	0.681734i	−0.940112	−	0.340867i	$-0.889279\pi$
$229$		−	1.87481e9i		−	0.681734i	0.940112	−	0.340867i	$-0.110721\pi$
$230$	−8.89480e8			−0.317852
$231$		−	2.71127e8i		−	0.0952192i
$232$			1.47689e9i			0.509795i
$233$		−	2.13434e9i		−	0.724167i	−0.932146	−	0.362084i	$-0.882065\pi$
$233$		−	2.13434e9i		−	0.724167i	0.932146	−	0.362084i	$-0.117935\pi$
$234$	7.36438e8			0.245625
$235$			9.69614e8i			0.317927i
$236$	−1.47172e9	+	4.62881e9i	−0.474435	+	1.49218i
$237$	3.39612e9			1.07644
$238$			5.38922e9i			1.67965i
$239$	−1.40442e9			−0.430434			−0.215217	−	0.976566i	$-0.569046\pi$
$239$	−1.40442e9			−0.430434			−0.215217	+	0.976566i	$0.569046\pi$
$240$	8.38296e7			0.0252669
$241$	−2.58892e9			−0.767452			−0.383726	−	0.923447i	$-0.625359\pi$
$241$	−2.58892e9			−0.767452			−0.383726	+	0.923447i	$0.625359\pi$
$242$		−	5.21421e9i		−	1.52029i
$243$	2.23677e8			0.0641500
$244$			8.03767e9i			2.26762i
$245$	6.43334e8			0.178555
$246$		−	2.30852e9i		−	0.630367i
$247$			1.94854e9i			0.523505i
$248$	−6.79152e9			−1.79540
$249$			2.39787e9i			0.623775i
$250$		−	4.44698e9i		−	1.13843i
$251$	3.85692e9			0.971731			0.485866	−	0.874033i	$-0.338504\pi$
$251$	3.85692e9			0.971731			0.485866	+	0.874033i	$0.338504\pi$
$252$	1.53882e9			0.381579
$253$	−4.78145e8			−0.116702
$254$			1.39734e9i			0.335713i
$255$	−1.34303e9			−0.317632
$256$	−3.47177e9			−0.808335
$257$	7.36020e9			1.68716			0.843582	−	0.537000i	$-0.180443\pi$
$257$	7.36020e9			1.68716			0.843582	+	0.537000i	$0.180443\pi$
$258$	−4.44826e9			−1.00395
$259$			4.88923e8i			0.108653i
$260$			1.26258e9i			0.276291i
$261$	8.70105e8			0.187504
$262$	−3.34647e9			−0.710201
$263$	5.56299e9			1.16275			0.581373	−	0.813637i	$-0.302516\pi$
$263$	5.56299e9			1.16275			0.581373	+	0.813637i	$0.302516\pi$
$264$	−5.73367e8			−0.118037
$265$	3.07970e8			0.0624489
$266$			6.67186e9i			1.33266i
$267$			2.11129e9i			0.415436i
$268$			2.45800e9i			0.476479i
$269$			1.88061e9i			0.359162i	0.983743	+	0.179581i	$0.0574741\pi$
$269$			1.88061e9i			0.359162i	−0.983743	+	0.179581i	$0.942526\pi$
$270$			6.28396e8i			0.118244i
$271$	−5.55098e9			−1.02918			−0.514592	−	0.857435i	$-0.672056\pi$
$271$	−5.55098e9			−1.02918			−0.514592	+	0.857435i	$0.672056\pi$
$272$	−8.95722e8			−0.163643
$273$		−	1.07856e9i		−	0.194176i
$274$		−	6.40235e9i		−	1.13589i
$275$		−	1.10034e9i		−	0.192396i
$276$		−	2.71377e9i		−	0.467668i
$277$	−7.43367e9			−1.26265			−0.631327	−	0.775517i	$-0.717489\pi$
$277$	−7.43367e9			−1.26265			−0.631327	+	0.775517i	$0.717489\pi$
$278$			4.90111e9i			0.820569i
$279$			4.00120e9i			0.660350i
$280$			1.56214e9i			0.254148i
$281$	2.25008e9			0.360888			0.180444	−	0.983585i	$-0.442247\pi$
$281$	2.25008e9			0.360888			0.180444	+	0.983585i	$0.442247\pi$
$282$	−4.84757e9			−0.766527
$283$		−	1.89174e9i		−	0.294928i	−0.989067	−	0.147464i	$-0.952889\pi$
$283$		−	1.89174e9i		−	0.294928i	0.989067	−	0.147464i	$-0.0471111\pi$
$284$	−1.46144e10			−2.24650
$285$	−1.66267e9			−0.252015
$286$			1.11217e9i			0.166229i
$287$	−3.38099e9			−0.498329
$288$			2.49743e9i			0.363014i
$289$	7.37453e9			1.05717
$290$			2.44446e9i			0.345614i
$291$			3.21782e9i			0.448734i
$292$			7.56097e9i			1.04003i
$293$	2.55240e9			0.346320			0.173160	−	0.984894i	$-0.444602\pi$
$293$	2.55240e9			0.346320			0.173160	+	0.984894i	$0.444602\pi$
$294$			3.21634e9i			0.430499i
$295$	−8.80199e8	+	2.76838e9i	−0.116223	+	0.365543i
$296$	1.03395e9			0.134690
$297$			3.37798e8i			0.0434141i
$298$	2.11250e10			2.67875
$299$	−1.90210e9			−0.237984
$300$	6.24512e9			0.771002
$301$			6.51479e9i			0.793660i
$302$	1.38610e10			1.66635
$303$		−	5.77056e9i		−	0.684616i
$304$	−1.10891e9			−0.129838
$305$			4.80714e9i			0.555504i
$306$		−	6.71444e9i		−	0.765815i
$307$	8.22450e9			0.925883			0.462942	−	0.886389i	$-0.346794\pi$
$307$	8.22450e9			0.925883			0.462942	+	0.886389i	$0.346794\pi$
$308$			2.32392e9i			0.258237i
$309$		−	1.93547e9i		−	0.212301i
$310$	−1.12409e10			−1.21718
$311$	−4.06228e9			−0.434239			−0.217119	−	0.976145i	$-0.569666\pi$
$311$	−4.06228e9			−0.434239			−0.217119	+	0.976145i	$0.569666\pi$
$312$	−2.28090e9			−0.240707
$313$		−	1.18792e9i		−	0.123768i	−0.998083	−	0.0618840i	$-0.980289\pi$
$313$		−	1.18792e9i		−	0.123768i	0.998083	−	0.0618840i	$-0.0197109\pi$
$314$	2.74175e10			2.82039
$315$	9.20329e8			0.0934762
$316$	−2.91093e10			−2.91934
$317$	−4.81904e9			−0.477225			−0.238613	−	0.971115i	$-0.576693\pi$
$317$	−4.81904e9			−0.477225			−0.238613	+	0.971115i	$0.576693\pi$
$318$			1.53969e9i			0.150565i
$319$			1.31403e9i			0.126895i
$320$	−6.55735e9			−0.625358
$321$	−7.27207e9			−0.684917
$322$	−6.51285e9			−0.605826
$323$	1.77657e10			1.63220
$324$	−1.91721e9			−0.173977
$325$		−	4.37724e9i		−	0.392344i
$326$		−	2.71948e10i		−	2.40777i
$327$		−	7.22801e9i		−	0.632161i
$328$			7.14997e9i			0.617744i
$329$			7.09960e9i			0.605969i
$330$	−9.49004e8			−0.0800225
$331$	−7.42414e9			−0.618492			−0.309246	−	0.950982i	$-0.600077\pi$
$331$	−7.42414e9			−0.618492			−0.309246	+	0.950982i	$0.600077\pi$
$332$		−	2.05530e10i		−	1.69170i
$333$		−	6.09151e8i		−	0.0495391i
$334$		−	1.72161e8i		−	0.0138340i
$335$			1.47007e9i			0.116724i
$336$	6.13807e8			0.0481587
$337$		−	1.50440e10i		−	1.16639i	−0.812332	−	0.583196i	$-0.801802\pi$
$337$		−	1.50440e10i		−	1.16639i	0.812332	−	0.583196i	$-0.198198\pi$
$338$		−	1.64820e10i		−	1.26282i
$339$			2.81949e9i			0.213487i
$340$	1.15115e10			0.861425
$341$	−6.04262e9			−0.446898
$342$		−	8.31249e9i		−	0.607612i
$343$	1.48298e10			1.07142
$344$	1.37772e10			0.983845
$345$		−	1.62304e9i		−	0.114565i
$346$	4.12965e10			2.88144
$347$		−	1.82555e10i		−	1.25915i	−0.776941	−	0.629573i	$-0.783230\pi$
$347$		−	1.82555e10i		−	1.25915i	0.776941	−	0.629573i	$-0.216770\pi$
$348$	−7.45797e9			−0.508515
$349$		−	3.56777e9i		−	0.240489i	−0.992744	−	0.120245i	$-0.961632\pi$
$349$		−	3.56777e9i		−	0.240489i	0.992744	−	0.120245i	$-0.0383679\pi$
$350$		−	1.49878e10i		−	0.998772i
$351$			1.34379e9i			0.0885323i
$352$	−3.77162e9			−0.245673
$353$			4.28882e9i			0.276210i	0.990418	+	0.138105i	$0.0441011\pi$
$353$			4.28882e9i			0.276210i	−0.990418	+	0.138105i	$0.955899\pi$
$354$	−1.38405e10	−	4.40054e9i	−0.881330	−	0.280216i
$355$	−8.74052e9			−0.550331
$356$		−	1.80966e10i		−	1.12667i
$357$	−9.83376e9			−0.605406
$358$	−1.02545e9			−0.0624284
$359$	−6.19358e9			−0.372876			−0.186438	−	0.982467i	$-0.559694\pi$
$359$	−6.19358e9			−0.372876			−0.186438	+	0.982467i	$0.559694\pi$
$360$		−	1.94627e9i		−	0.115876i
$361$	5.01040e9			0.295015
$362$		−	1.88741e10i		−	1.09909i
$363$	9.51443e9			0.547969
$364$			9.24475e9i			0.526611i
$365$			4.52203e9i			0.254778i
$366$	−2.40332e10			−1.33933
$367$			2.89548e10i			1.59609i	0.602599	+	0.798044i	$0.294132\pi$
$367$			2.89548e10i			1.59609i	−0.602599	+	0.798044i	$0.705868\pi$
$368$		−	1.08248e9i		−	0.0590239i
$369$	4.21238e9			0.227207
$370$	1.71134e9			0.0913123
$371$	2.25498e9			0.119028
$372$		−	3.42957e10i		−	1.79089i
$373$	2.97339e10			1.53609			0.768045	−	0.640395i	$-0.221230\pi$
$373$	2.97339e10			1.53609			0.768045	+	0.640395i	$0.221230\pi$
$374$	1.01401e10			0.518272
$375$	8.11445e9			0.410331
$376$	1.50139e10			0.751177
$377$			5.22733e9i			0.258770i
$378$			4.60117e9i			0.225373i
$379$	−2.01521e10			−0.976707			−0.488353	−	0.872646i	$-0.662402\pi$
$379$	−2.01521e10			−0.976707			−0.488353	+	0.872646i	$0.662402\pi$
$380$	1.42513e10			0.683471
$381$	−2.54975e9			−0.121003
$382$	−4.36101e10			−2.04802
$383$	1.58914e10			0.738529			0.369265	−	0.929324i	$-0.379610\pi$
$383$	1.58914e10			0.738529			0.369265	+	0.929324i	$0.379610\pi$
$384$		−	1.91121e10i		−	0.878989i
$385$			1.38988e9i			0.0632608i
$386$		−	6.22577e10i		−	2.80443i
$387$		−	8.11679e9i		−	0.361860i
$388$		−	2.75810e10i		−	1.21698i
$389$	2.81843e10			1.23086			0.615431	−	0.788191i	$-0.288982\pi$
$389$	2.81843e10			1.23086			0.615431	+	0.788191i	$0.288982\pi$
$390$	−3.77522e9			−0.163186
$391$			1.73423e10i			0.741992i
$392$		−	9.96166e9i		−	0.421879i
$393$		−	6.10634e9i		−	0.255983i
$394$			5.86004e10i			2.43173i
$395$	−1.74096e10			−0.715155
$396$		−	2.89538e9i		−	0.117740i
$397$			3.92674e10i			1.58078i	0.612607	+	0.790388i	$0.290121\pi$
$397$			3.92674e10i			1.58078i	−0.612607	+	0.790388i	$0.709879\pi$
$398$		−	5.81167e10i		−	2.31616i
$399$	−1.21742e10			−0.480341
$400$	2.49107e9			0.0973075
$401$		−	1.00645e10i		−	0.389239i	−0.980879	−	0.194619i	$-0.937653\pi$
$401$		−	1.00645e10i		−	0.389239i	0.980879	−	0.194619i	$-0.0623471\pi$
$402$	−7.34961e9			−0.281423
$403$	−2.40380e10			−0.911337
$404$			4.94614e10i			1.85670i
$405$	−1.14664e9			−0.0426194
$406$			1.78986e10i			0.658740i
$407$	9.19940e8			0.0335260
$408$			2.07960e10i			0.750480i
$409$		−	1.66826e10i		−	0.596171i	−0.954539	−	0.298085i	$-0.903652\pi$
$409$		−	1.66826e10i		−	0.596171i	0.954539	−	0.298085i	$-0.0963480\pi$
$410$			1.18342e10i			0.418797i
$411$	1.16824e10			0.409417
$412$			1.65896e10i			0.575766i
$413$	−6.44490e9	+	2.02704e10i	−0.221521	+	0.696725i
$414$	8.11438e9			0.276219
$415$		−	1.22922e10i		−	0.414418i
$416$	−1.50038e10			−0.500989
$417$	−8.94310e9			−0.295763
$418$	1.25535e10			0.411207
$419$			1.14293e9i			0.0370820i	0.999828	+	0.0185410i	$0.00590212\pi$
$419$			1.14293e9i			0.0370820i	−0.999828	+	0.0185410i	$0.994098\pi$
$420$	−7.88846e9			−0.253510
$421$			2.00529e10i			0.638334i	0.947698	+	0.319167i	$0.103403\pi$
$421$			2.00529e10i			0.638334i	−0.947698	+	0.319167i	$0.896597\pi$
$422$	2.60721e10			0.822103
$423$		−	8.84541e9i		−	0.276284i
$424$		−	4.76873e9i		−	0.147550i
$425$	−3.99093e10			−1.22326
$426$		−	4.36981e10i		−	1.32686i
$427$			3.51983e10i			1.05879i
$428$	6.23314e10			1.85751
$429$	−2.02939e9			−0.0599150
$430$	2.28032e10			0.666994
$431$			3.68452e10i			1.06776i	0.845562	+	0.533878i	$0.179266\pi$
$431$			3.68452e10i			1.06776i	−0.845562	+	0.533878i	$0.820734\pi$
$432$	−7.64744e8			−0.0219574
$433$	−1.93863e10			−0.551498			−0.275749	−	0.961230i	$-0.588926\pi$
$433$	−1.93863e10			−0.551498			−0.275749	+	0.961230i	$0.588926\pi$
$434$	−8.23071e10			−2.31995
$435$	−4.46043e9			−0.124572
$436$			6.19537e10i			1.71444i
$437$			2.14698e10i			0.588711i
$438$	−2.26078e10			−0.614275
$439$	−3.11626e10			−0.839025			−0.419513	−	0.907750i	$-0.637799\pi$
$439$	−3.11626e10			−0.839025			−0.419513	+	0.907750i	$0.637799\pi$
$440$	2.93926e9			0.0784201
$441$	−5.86889e9			−0.155168
$442$	4.03383e10			1.05689
$443$			6.59173e9i			0.171153i	0.996332	+	0.0855766i	$0.0272732\pi$
$443$			6.59173e9i			0.171153i	−0.996332	+	0.0855766i	$0.972727\pi$
$444$			5.22124e9i			0.134351i
$445$		−	1.08232e10i		−	0.276003i
$446$		−	8.80036e10i		−	2.22413i
$447$			3.85471e10i			0.965520i
$448$	−4.80135e10			−1.19193
$449$	−4.41656e10			−1.08667			−0.543336	−	0.839515i	$-0.682839\pi$
$449$	−4.41656e10			−1.08667			−0.543336	+	0.839515i	$0.682839\pi$
$450$			1.86734e10i			0.455378i
$451$			6.36155e9i			0.153765i
$452$		−	2.41668e10i		−	0.578983i
$453$			2.52923e10i			0.600614i
$454$	−2.20459e10			−0.518925
$455$			5.52906e9i			0.129005i
$456$			2.57455e10i			0.595445i
$457$		−	4.56812e10i		−	1.04730i	−0.851933	−	0.523651i	$-0.824569\pi$
$457$		−	4.56812e10i		−	1.04730i	0.851933	−	0.523651i	$-0.175431\pi$
$458$	−4.80493e10			−1.09201
$459$	1.22519e10			0.276028
$460$			1.39117e10i			0.310705i
$461$	−1.89312e10			−0.419156			−0.209578	−	0.977792i	$-0.567209\pi$
$461$	−1.89312e10			−0.419156			−0.209578	+	0.977792i	$0.567209\pi$
$462$	−6.94869e9			−0.152523
$463$		−	1.11150e10i		−	0.241872i	−0.992660	−	0.120936i	$-0.961410\pi$
$463$		−	1.11150e10i		−	0.241872i	0.992660	−	0.120936i	$-0.0385897\pi$
$464$	−2.97486e9			−0.0641792
$465$		−	2.05114e10i		−	0.438717i
$466$	−5.47007e10			−1.15998
$467$			2.77925e10i			0.584332i	0.956368	+	0.292166i	$0.0943760\pi$
$467$			2.77925e10i			0.584332i	−0.956368	+	0.292166i	$0.905624\pi$
$468$		−	1.15181e10i		−	0.240102i
$469$			1.07640e10i			0.222476i
$470$	2.48502e10			0.509258
$471$			5.00291e10i			1.01657i
$472$	4.28668e10	+	1.36294e10i	0.863681	+	0.274605i
$473$	1.22580e10			0.244892
$474$		−	8.70390e10i		−	1.72425i
$475$	−4.94077e10			−0.970556
$476$	8.42885e10			1.64188
$477$	−2.80949e9			−0.0542692
$478$			3.59939e10i			0.689473i
$479$	−8.70053e10			−1.65274			−0.826368	−	0.563130i	$-0.809597\pi$
$479$	−8.70053e10			−1.65274			−0.826368	+	0.563130i	$0.809597\pi$
$480$		−	1.28026e10i		−	0.241176i
$481$	3.65960e9			0.0683680
$482$			6.63514e10i			1.22931i
$483$		−	1.18841e10i		−	0.218362i
$484$	−8.15514e10			−1.48611
$485$		−	1.64955e10i		−	0.298126i
$486$		−	5.73261e9i		−	0.102756i
$487$	1.01454e11			1.80365			0.901825	−	0.432101i	$-0.142228\pi$
$487$	1.01454e11			1.80365			0.901825	+	0.432101i	$0.142228\pi$
$488$	7.44358e10			1.31251
$489$	4.96226e10			0.867848
$490$		−	1.64880e10i		−	0.286011i
$491$	−3.52846e10			−0.607099			−0.303549	−	0.952816i	$-0.598172\pi$
$491$	−3.52846e10			−0.607099			−0.303549	+	0.952816i	$0.598172\pi$
$492$	−3.61058e10			−0.616192
$493$	4.76599e10			0.806800
$494$	4.99389e10			0.838555
$495$		−	1.73166e9i		−	0.0288431i
$496$		−	1.36800e10i		−	0.226026i
$497$	−6.39989e10			−1.04893
$498$	6.14548e10			0.999169
$499$	8.78804e10			1.41739			0.708696	−	0.705514i	$-0.249284\pi$
$499$	8.78804e10			1.41739			0.708696	+	0.705514i	$0.249284\pi$
$500$	−6.95517e10			−1.11283
$501$	3.14144e8			0.00498629
$502$		−	9.88488e10i		−	1.55653i
$503$			1.18462e11i			1.85057i	0.379269	+	0.925287i	$0.376176\pi$
$503$			1.18462e11i			1.85057i	−0.379269	+	0.925287i	$0.623824\pi$
$504$		−	1.42508e10i		−	0.220860i
$505$			2.95817e10i			0.454839i
$506$			1.22543e10i			0.186934i
$507$	3.00749e10			0.455168
$508$	2.18548e10			0.328164
$509$		−	8.56499e10i		−	1.27601i	−0.770030	−	0.638007i	$-0.779759\pi$
$509$		−	8.56499e10i		−	1.27601i	0.770030	−	0.638007i	$-0.220241\pi$
$510$			3.44203e10i			0.508785i
$511$			3.31107e10i			0.485607i
$512$		−	1.56443e10i		−	0.227655i
$513$	1.51679e10			0.219006
$514$		−	1.88634e11i		−	2.70251i
$515$			9.92183e9i			0.141047i
$516$			6.95718e10i			0.981373i
$517$	1.33583e10			0.186978
$518$	1.25306e10			0.174041
$519$			7.53542e10i			1.03858i
$520$	1.16926e10			0.159918
$521$	−1.11648e11			−1.51530			−0.757652	−	0.652658i	$-0.773654\pi$
$521$	−1.11648e11			−1.51530			−0.757652	+	0.652658i	$0.773654\pi$
$522$		−	2.22999e10i		−	0.300345i
$523$	−7.64824e9			−0.102224			−0.0511122	−	0.998693i	$-0.516277\pi$
$523$	−7.64824e9			−0.102224			−0.0511122	+	0.998693i	$0.516277\pi$
$524$			5.23395e10i			0.694231i
$525$	2.73484e10			0.359994
$526$		−	1.42573e11i		−	1.86250i
$527$			2.19166e11i			2.84138i
$528$		−	1.15492e9i		−	0.0148599i
$529$	5.73529e10			0.732373
$530$		−	7.89293e9i		−	0.100031i
$531$	8.02972e9	−	2.52549e10i	0.101000	−	0.317663i
$532$	1.04349e11			1.30270
$533$			2.53068e10i			0.313565i
$534$	5.41102e10			0.665448
$535$	3.72789e10			0.455039
$536$	2.27633e10			0.275788
$537$		−	1.87115e9i		−	0.0225015i
$538$	4.81980e10			0.575308
$539$		−	8.86320e9i		−	0.105011i
$540$	9.82825e9			0.115585
$541$			2.00200e10i			0.233709i	0.993149	+	0.116855i	$0.0372811\pi$
$541$			2.00200e10i			0.233709i	−0.993149	+	0.116855i	$0.962719\pi$
$542$			1.42266e11i			1.64855i
$543$	3.44397e10			0.396151
$544$			1.36796e11i			1.56199i
$545$			3.70530e10i			0.419989i
$546$	−2.76425e10			−0.311033
$547$	−4.77890e9			−0.0533800			−0.0266900	−	0.999644i	$-0.508497\pi$
$547$	−4.77890e9			−0.0533800			−0.0266900	+	0.999644i	$0.508497\pi$
$548$	−1.00134e11			−1.11035
$549$		−	4.38537e10i		−	0.482744i
$550$	−2.82005e10			−0.308181
$551$	5.90031e10			0.640130
$552$	−2.51319e10			−0.270688
$553$	−1.27475e11			−1.36309
$554$			1.90517e11i			2.02253i
$555$			3.12270e9i			0.0329123i
$556$	7.66544e10			0.802117
$557$	2.70838e10			0.281378			0.140689	−	0.990054i	$-0.455068\pi$
$557$	2.70838e10			0.281378			0.140689	+	0.990054i	$0.455068\pi$
$558$	1.02547e11			1.05775
$559$	4.87633e10			0.499396
$560$	−3.14657e9			−0.0319952
$561$			1.85028e10i			0.186804i
$562$		−	5.76671e10i		−	0.578073i
$563$			9.90305e10i			0.985678i	0.870120	+	0.492839i	$0.164041\pi$
$563$			9.90305e10i			0.985678i	−0.870120	+	0.492839i	$0.835959\pi$
$564$			7.58170e10i			0.749291i
$565$		−	1.44536e10i		−	0.141835i
$566$	−4.84834e10			−0.472419
$567$	−8.39580e9			−0.0812325
$568$			1.35342e11i			1.30029i
$569$		−	2.36985e10i		−	0.226085i	−0.993590	−	0.113042i	$-0.963940\pi$
$569$		−	2.36985e10i		−	0.226085i	0.993590	−	0.113042i	$-0.0360596\pi$
$570$			4.26124e10i			0.403680i
$571$		−	1.46131e11i		−	1.37467i	−0.726341	−	0.687334i	$-0.758781\pi$
$571$		−	1.46131e11i		−	1.37467i	0.726341	−	0.687334i	$-0.241219\pi$
$572$	1.73946e10			0.162491
$573$		−	7.95758e10i		−	0.738181i
$574$			8.66512e10i			0.798228i
$575$		−	4.82302e10i		−	0.441213i
$576$	5.98201e10			0.543447
$577$	−5.60882e10			−0.506021			−0.253011	−	0.967464i	$-0.581421\pi$
$577$	−5.60882e10			−0.506021			−0.253011	+	0.967464i	$0.581421\pi$
$578$		−	1.89001e11i		−	1.69338i
$579$	1.13602e11			1.01082
$580$	3.82319e10			0.337842
$581$		−	9.00049e10i		−	0.789881i
$582$	8.24692e10			0.718786
$583$		−	4.24289e9i		−	0.0367272i
$584$	7.00211e10			0.601974
$585$		−	6.88867e9i		−	0.0588183i
$586$		−	6.54153e10i		−	0.554739i
$587$		−	2.14359e10i		−	0.180546i	−0.995917	−	0.0902732i	$-0.971226\pi$
$587$		−	2.14359e10i		−	0.180546i	0.995917	−	0.0902732i	$-0.0287740\pi$
$588$	5.03042e10			0.420819
$589$			2.71327e11i			2.25441i
$590$	7.09507e10	+	2.25586e10i	0.585529	+	0.186167i
$591$	−1.06929e11			−0.876486
$592$			2.08266e9i			0.0169563i
$593$	1.87169e10			0.151361			0.0756805	−	0.997132i	$-0.475887\pi$
$593$	1.87169e10			0.151361			0.0756805	+	0.997132i	$0.475887\pi$
$594$	8.65739e9			0.0695411
$595$	5.04109e10			0.402214
$596$		−	3.30400e11i		−	2.61852i
$597$	1.06046e11			0.834829
$598$			4.87488e10i			0.381205i
$599$	−8.71690e10			−0.677103			−0.338552	−	0.940948i	$-0.609937\pi$
$599$	−8.71690e10			−0.677103			−0.338552	+	0.940948i	$0.609937\pi$
$600$		−	5.78352e10i		−	0.446259i
$601$			4.72599e10i			0.362238i	0.983461	+	0.181119i	$0.0579720\pi$
$601$			4.72599e10i			0.362238i	−0.983461	+	0.181119i	$0.942028\pi$
$602$	1.66967e11			1.27129
$603$		−	1.34109e10i		−	0.101435i
$604$		−	2.16789e11i		−	1.62888i
$605$	−4.87740e10			−0.364055
$606$	−1.47893e11			−1.09662
$607$	−4.28839e10			−0.315893			−0.157946	−	0.987448i	$-0.550487\pi$
$607$	−4.28839e10			−0.315893			−0.157946	+	0.987448i	$0.550487\pi$
$608$			1.69354e11i			1.23932i
$609$	−3.26597e10			−0.237434
$610$	1.23202e11			0.889811
$611$	5.31406e10			0.381295
$612$	−1.05015e11			−0.748595
$613$		−	1.02665e11i		−	0.727077i	−0.931579	−	0.363538i	$-0.881569\pi$
$613$		−	1.02665e11i		−	0.727077i	0.931579	−	0.363538i	$-0.118431\pi$
$614$		−	2.10785e11i		−	1.48309i
$615$	−2.15940e10			−0.150950
$616$	2.15215e10			0.149469
$617$	−1.21352e11			−0.837345			−0.418673	−	0.908137i	$-0.637505\pi$
$617$	−1.21352e11			−0.837345			−0.418673	+	0.908137i	$0.637505\pi$
$618$	−4.96040e10			−0.340066
$619$	−8.51181e10			−0.579775			−0.289887	−	0.957061i	$-0.593618\pi$
$619$	−8.51181e10			−0.579775			−0.289887	+	0.957061i	$0.593618\pi$
$620$			1.75811e11i			1.18981i
$621$			1.48064e10i			0.0995595i
$622$			1.04112e11i			0.695567i
$623$		−	7.92481e10i		−	0.526062i
$624$		−	4.59436e9i		−	0.0303030i
$625$	8.85405e10			0.580259
$626$	−3.04450e10			−0.198252
$627$			2.29066e10i			0.148214i
$628$		−	4.28816e11i		−	2.75697i
$629$		−	3.33662e10i		−	0.213159i
$630$		−	2.35871e10i		−	0.149731i
$631$	1.10368e11			0.696186			0.348093	−	0.937460i	$-0.386829\pi$
$631$	1.10368e11			0.696186			0.348093	+	0.937460i	$0.386829\pi$
$632$			2.69578e11i			1.68972i
$633$			4.75740e10i			0.296316i
$634$			1.23507e11i			0.764423i
$635$	1.30708e10			0.0803910
$636$	2.40811e10			0.147180
$637$		−	3.52585e10i		−	0.214144i
$638$	3.36773e10			0.203261
$639$	7.97363e10			0.478248
$640$			9.79746e10i			0.583974i
$641$	2.92806e11			1.73440			0.867198	−	0.497964i	$-0.165919\pi$
$641$	2.92806e11			1.73440			0.867198	+	0.497964i	$0.165919\pi$
$642$			1.86375e11i			1.09711i
$643$	−8.22453e10			−0.481135			−0.240568	−	0.970632i	$-0.577334\pi$
$643$	−8.22453e10			−0.481135			−0.240568	+	0.970632i	$0.577334\pi$
$644$			1.01862e11i			0.592203i
$645$			4.16092e10i			0.240409i
$646$		−	4.55316e11i		−	2.61446i
$647$	−3.20501e11			−1.82899			−0.914497	−	0.404592i	$-0.867413\pi$
$647$	−3.20501e11			−1.82899			−0.914497	+	0.404592i	$0.867413\pi$
$648$			1.77551e10i			0.100698i
$649$	3.81399e10	+	1.21265e10i	0.214982	+	0.0683528i
$650$	−1.12184e11			−0.628460
$651$		−	1.50187e11i		−	0.836194i
$652$	−4.25332e11			−2.35363
$653$	−1.66879e11			−0.917804			−0.458902	−	0.888487i	$-0.651757\pi$
$653$	−1.66879e11			−0.917804			−0.458902	+	0.888487i	$0.651757\pi$
$654$	−1.85246e11			−1.01260
$655$			3.13030e10i			0.170067i
$656$	−1.44020e10			−0.0777691
$657$		−	4.12528e10i		−	0.221407i
$658$	1.81955e11			0.970646
$659$		−	1.80669e11i		−	0.957947i	−0.877829	−	0.478974i	$-0.841009\pi$
$659$		−	1.80669e11i		−	0.957947i	0.877829	−	0.478974i	$-0.158991\pi$
$660$			1.48426e10i			0.0782231i
$661$	−1.56896e11			−0.821874			−0.410937	−	0.911664i	$-0.634798\pi$
$661$	−1.56896e11			−0.821874			−0.410937	+	0.911664i	$0.634798\pi$
$662$			1.90273e11i			0.990705i
$663$			7.36058e10i			0.380941i
$664$	−1.90338e11			−0.979160
$665$	6.24089e10			0.319124
$666$	−1.56119e10			−0.0793521
$667$			5.75969e10i			0.291002i
$668$	−2.69263e9			−0.0135229
$669$	1.60581e11			0.801660
$670$	3.76764e10			0.186969
$671$	6.62278e10			0.326701
$672$		−	9.37419e10i		−	0.459681i
$673$		−	2.73935e9i		−	0.0133532i	−0.999978	−	0.00667662i	$-0.997875\pi$
$673$		−	2.73935e9i		−	0.0133532i	0.999978	−	0.00667662i	$-0.00212525\pi$
$674$	−3.85562e11			−1.86834
$675$	−3.40735e10			−0.164135
$676$	−2.57782e11			−1.23443
$677$	−2.83127e11			−1.34781			−0.673903	−	0.738820i	$-0.735383\pi$
$677$	−2.83127e11			−1.34781			−0.673903	+	0.738820i	$0.735383\pi$
$678$	7.22606e10			0.341966
$679$		−	1.20782e11i		−	0.568228i
$680$		−	1.06607e11i		−	0.498597i
$681$		−	4.02274e10i		−	0.187040i
$682$			1.54866e11i			0.715844i
$683$			3.34373e11i			1.53655i	0.640117	+	0.768277i	$0.278886\pi$
$683$			3.34373e11i			1.53655i	−0.640117	+	0.768277i	$0.721114\pi$
$684$	−1.30009e11			−0.593949
$685$	−5.98878e10			−0.272005
$686$		−	3.80072e11i		−	1.71621i
$687$		−	8.76761e10i		−	0.393599i
$688$			2.77510e10i			0.123858i
$689$		−	1.68786e10i		−	0.0748960i
$690$	−4.15969e10			−0.183512
$691$		−	3.07698e11i		−	1.34962i	−0.737990	−	0.674812i	$-0.764225\pi$
$691$		−	3.07698e11i		−	1.34962i	0.737990	−	0.674812i	$-0.235775\pi$
$692$		−	6.45886e11i		−	2.81664i
$693$		−	1.26794e10i		−	0.0549749i
$694$	−4.67869e11			−2.01691
$695$	4.58451e10			0.196496
$696$			6.90673e10i			0.294330i
$697$	2.30733e11			0.977639
$698$	−9.14381e10			−0.385217
$699$		−	9.98130e10i		−	0.418098i
$700$	−2.34413e11			−0.976313
$701$			7.07046e10i			0.292803i	0.989225	+	0.146401i	$0.0467691\pi$
$701$			7.07046e10i			0.292803i	−0.989225	+	0.146401i	$0.953231\pi$
$702$	3.44398e10			0.141812
$703$		−	4.13074e10i		−	0.169124i
$704$			9.03404e10i			0.367783i
$705$			4.53444e10i			0.183555i
$706$	1.09918e11			0.442435
$707$			2.16600e11i			0.866923i
$708$	−6.88254e10	+	2.16468e11i	−0.273915	+	0.861512i
$709$	3.94533e11			1.56134			0.780671	−	0.624942i	$-0.214877\pi$
$709$	3.94533e11			1.56134			0.780671	+	0.624942i	$0.214877\pi$
$710$			2.24010e11i			0.881524i
$711$	1.58821e11			0.621483
$712$	−1.67590e11			−0.652122
$713$	−2.64861e11			−1.02485
$714$			2.52029e11i			0.969744i
$715$	1.04033e10			0.0398058
$716$			1.60383e10i			0.0610246i
$717$	−6.56784e10			−0.248511
$718$			1.58735e11i			0.597275i
$719$			2.33339e11i			0.873114i	0.899676	+	0.436557i	$0.143802\pi$
$719$			2.33339e11i			0.873114i	−0.899676	+	0.436557i	$0.856198\pi$
$720$	3.92032e9			0.0145879
$721$			7.26485e10i			0.268835i
$722$		−	1.28411e11i		−	0.472557i
$723$	−1.21072e11			−0.443089
$724$	−2.95195e11			−1.07437
$725$	−1.32546e11			−0.479749
$726$		−	2.43845e11i		−	0.877742i
$727$	3.33581e11			1.19416			0.597081	−	0.802181i	$-0.296327\pi$
$727$	3.33581e11			1.19416			0.597081	+	0.802181i	$0.296327\pi$
$728$	8.56144e10			0.304804
$729$	1.04604e10			0.0370370
$730$	1.15895e11			0.408106
$731$		−	4.44596e11i		−	1.55703i
$732$			3.75885e11i			1.30921i
$733$	1.35099e10			0.0467989			0.0233995	−	0.999726i	$-0.492551\pi$
$733$	1.35099e10			0.0467989			0.0233995	+	0.999726i	$0.492551\pi$
$734$	7.42081e11			2.55663
$735$	3.00858e10			0.103089
$736$	−1.65318e11			−0.563390
$737$	2.02532e10			0.0686472
$738$		−	1.07959e11i		−	0.363943i
$739$			2.13081e11i			0.714441i	0.934020	+	0.357221i	$0.116276\pi$
$739$			2.13081e11i			0.714441i	−0.934020	+	0.357221i	$0.883724\pi$
$740$		−	2.67657e10i		−	0.0892590i
$741$			9.11241e10i			0.302246i
$742$		−	5.77928e10i		−	0.190659i
$743$	1.07130e11			0.351524			0.175762	−	0.984433i	$-0.443761\pi$
$743$	1.07130e11			0.351524			0.175762	+	0.984433i	$0.443761\pi$
$744$	−3.17608e11			−1.03657
$745$		−	1.97604e11i		−	0.641463i
$746$		−	7.62049e11i		−	2.46052i
$747$			1.12137e11i			0.360137i
$748$		−	1.58594e11i		−	0.506618i
$749$	2.72960e11			0.867304
$750$		−	2.07965e11i		−	0.657271i
$751$		−	1.88581e11i		−	0.592841i	−0.955058	−	0.296421i	$-0.904207\pi$
$751$		−	1.88581e11i		−	0.592841i	0.955058	−	0.296421i	$-0.0957930\pi$
$752$			3.02421e10i			0.0945673i
$753$	1.80370e11			0.561029
$754$	1.33971e11			0.414501
$755$		−	1.29656e11i		−	0.399030i
$756$	7.19633e10			0.220305
$757$	3.24459e11			0.988044			0.494022	−	0.869449i	$-0.335526\pi$
$757$	3.24459e11			0.988044			0.494022	+	0.869449i	$0.335526\pi$
$758$			5.16478e11i			1.56450i
$759$	−2.23606e10			−0.0673778
$760$		−	1.31979e11i		−	0.395596i
$761$	3.60109e11			1.07373			0.536866	−	0.843668i	$-0.319608\pi$
$761$	3.60109e11			1.07373			0.536866	+	0.843668i	$0.319608\pi$
$762$			6.53473e10i			0.193824i
$763$			2.71306e11i			0.800499i
$764$			6.82071e11i			2.00196i
$765$	−6.28071e10			−0.183385
$766$		−	4.07280e11i		−	1.18298i
$767$	1.51724e11	+	4.82401e10i	0.438402	+	0.139389i
$768$	−1.62359e11			−0.466693
$769$		−	1.48428e10i		−	0.0424434i	−0.999775	−	0.0212217i	$-0.993244\pi$
$769$		−	1.48428e10i		−	0.0424434i	0.999775	−	0.0212217i	$-0.00675559\pi$
$770$	3.56212e10			0.101332
$771$	3.44203e11			0.974085
$772$	−9.73724e11			−2.74136
$773$		−	1.95676e11i		−	0.548049i	−0.961723	−	0.274024i	$-0.911645\pi$
$773$		−	1.95676e11i		−	0.548049i	0.961723	−	0.274024i	$-0.0883549\pi$
$774$	−2.08025e11			−0.579630
$775$		−	6.09516e11i		−	1.68958i
$776$	−2.55424e11			−0.704392
$777$			2.28647e10i			0.0627308i
$778$		−	7.22334e11i		−	1.97160i
$779$	2.85648e11			0.775678
$780$			5.90452e10i			0.159517i
$781$			1.20418e11i			0.323658i
$782$	4.44464e11			1.18853

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:	\( N \)	\(=\)	\( 177 = 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	177.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-25.6289i q^{2} +46.7654 q^{3} -400.842 q^{4} -239.734 q^{5} -1198.55i q^{6} -1755.35 q^{7} +3712.14i q^{8} +2187.00 q^{9} +O(q^{10})\)	\(q-25.6289i q^{2} +46.7654 q^{3} -400.842 q^{4} -239.734 q^{5} -1198.55i q^{6} -1755.35 q^{7} +3712.14i q^{8} +2187.00 q^{9} +6144.13i q^{10} +3302.81i q^{11} -18745.5 q^{12} +13138.8i q^{13} +44987.8i q^{14} -11211.3 q^{15} -7477.27 q^{16} +119793. q^{17} -56050.5i q^{18} +148304. q^{19} +96095.5 q^{20} -82089.8 q^{21} +84647.5 q^{22} +144769. i q^{23} +173600. i q^{24} -333153. q^{25} +336735. q^{26} +102276. q^{27} +703620. q^{28} +397853. q^{29} +287332. i q^{30} +1.82954e6i q^{31} +1.14194e6i q^{32} +154457. i q^{33} -3.07016e6i q^{34} +420818. q^{35} -876641. q^{36} -278533. i q^{37} -3.80086e6i q^{38} +614443. i q^{39} -889927. i q^{40} +1.92610e6 q^{41} +2.10387e6i q^{42} -3.71138e6i q^{43} -1.32390e6i q^{44} -524298. q^{45} +3.71028e6 q^{46} -4.04454e6i q^{47} -349677. q^{48} -2.68353e6 q^{49} +8.53834e6i q^{50} +5.60215e6 q^{51} -5.26660e6i q^{52} -1.28463e6 q^{53} -2.62122e6i q^{54} -791796. i q^{55} -6.51613e6i q^{56} +6.93547e6 q^{57} -1.01966e7i q^{58} +(3.67157e6 - 1.15477e7i) q^{59} +4.49394e6 q^{60} -2.00520e7i q^{61} +4.68892e7 q^{62} -3.83896e6 q^{63} +2.73526e7 q^{64} -3.14983e6i q^{65} +3.95857e6 q^{66} -6.13210e6i q^{67} -4.80179e7 q^{68} +6.77018e6i q^{69} -1.07851e7i q^{70} +3.64592e7 q^{71} +8.11846e6i q^{72} -1.88627e7i q^{73} -7.13849e6 q^{74} -1.55800e7 q^{75} -5.94463e7 q^{76} -5.79760e6i q^{77} +1.57475e7 q^{78} +7.26205e7 q^{79} +1.79256e6 q^{80} +4.78297e6 q^{81} -4.93639e7i q^{82} +5.12745e7i q^{83} +3.29050e7 q^{84} -2.87184e7 q^{85} -9.51187e7 q^{86} +1.86058e7 q^{87} -1.22605e7 q^{88} +4.51465e7i q^{89} +1.34372e7i q^{90} -2.30633e7i q^{91} -5.80296e7i q^{92} +8.55591e7i q^{93} -1.03657e8 q^{94} -3.55534e7 q^{95} +5.34034e7i q^{96} +6.88077e7i q^{97} +6.87761e7i q^{98} +7.22324e6i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80q - 10240q^{4} + 160q^{7} + 174960q^{9} + O(q^{10}) \)	\( 80q - 10240q^{4} + 160q^{7} + 174960q^{9} - 22680q^{12} - 59616q^{15} + 1199848q^{16} - 10608q^{17} - 27516q^{19} - 146436q^{20} - 974696q^{22} + 5718040q^{25} - 797484q^{26} - 3133000q^{28} + 1725924q^{29} + 4318800q^{35} - 22394880q^{36} - 732180q^{41} + 22752084q^{46} + 8703936q^{48} + 55899176q^{49} - 10373832q^{51} - 39265944q^{53} - 11408040q^{57} - 33575112q^{59} - 18034488q^{60} + 13038600q^{62} + 349920q^{63} - 241654260q^{64} - 35711928q^{66} + 36772608q^{68} - 235272660q^{71} - 63050712q^{74} + 74363184q^{75} + 9454680q^{76} - 10865988q^{78} + 17252580q^{79} + 318203976q^{80} + 382637520q^{81} - 20743128q^{84} - 27245820q^{85} + 105666984q^{86} + 29437992q^{87} + 82079788q^{88} + 121215992q^{94} - 690837276q^{95} + O(q^{100}) \)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Properties

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