LMFDB - Embedded newform 177.11.c.a.58.4

Newspace parameters

Level:	$ N $	$=$	$ 177 = 3 \cdot 59 $
Weight:	$ k $	$=$	$ 11 $
Character orbit:	$[\chi]$	$=$	177.c (of order $2$, degree $1$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$112.458233723$
Analytic rank:	$0$
Dimension:	$100$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			58.4
Character	$\chi$	$=$	177.58
Dual form			177.11.c.a.58.97

$q$-expansion

$f(q)$	$=$	$q-59.5712i q^{2} +140.296 q^{3} -2524.73 q^{4} -4763.75 q^{5} -8357.61i q^{6} +8815.37 q^{7} +89400.5i q^{8} +19683.0 q^{9} +O(q^{10})$	\(q-59.5712i q^{2} +140.296 q^{3} -2524.73 q^{4} -4763.75 q^{5} -8357.61i q^{6} +8815.37 q^{7} +89400.5i q^{8} +19683.0 q^{9} +283782. i q^{10} -307379. i q^{11} -354210. q^{12} -68920.8i q^{13} -525142. i q^{14} -668335. q^{15} +2.74037e6 q^{16} +1.05697e6 q^{17} -1.17254e6i q^{18} +4.26559e6 q^{19} +1.20272e7 q^{20} +1.23676e6 q^{21} -1.83109e7 q^{22} -3.74201e6i q^{23} +1.25425e7i q^{24} +1.29277e7 q^{25} -4.10570e6 q^{26} +2.76145e6 q^{27} -2.22564e7 q^{28} +1.29563e7 q^{29} +3.98136e7i q^{30} -5.38973e7i q^{31} -7.17012e7i q^{32} -4.31241e7i q^{33} -6.29652e7i q^{34} -4.19942e7 q^{35} -4.96943e7 q^{36} -2.46831e7i q^{37} -2.54107e8i q^{38} -9.66932e6i q^{39} -4.25881e8i q^{40} +1.00932e8 q^{41} -7.36754e7i q^{42} -1.57771e8i q^{43} +7.76050e8i q^{44} -9.37648e7 q^{45} -2.22916e8 q^{46} +1.16768e8i q^{47} +3.84463e8 q^{48} -2.04765e8 q^{49} -7.70117e8i q^{50} +1.48289e8 q^{51} +1.74007e8i q^{52} +6.16971e8 q^{53} -1.64503e8i q^{54} +1.46428e9i q^{55} +7.88098e8i q^{56} +5.98446e8 q^{57} -7.71821e8i q^{58} +(-2.89308e8 - 6.53772e8i) q^{59} +1.68737e9 q^{60} +9.65398e7i q^{61} -3.21073e9 q^{62} +1.73513e8 q^{63} -1.46519e9 q^{64} +3.28321e8i q^{65} -2.56896e9 q^{66} +1.44219e9i q^{67} -2.66857e9 q^{68} -5.24990e8i q^{69} +2.50165e9i q^{70} +1.91425e9 q^{71} +1.75967e9i q^{72} +1.00218e9i q^{73} -1.47040e9 q^{74} +1.81370e9 q^{75} -1.07695e10 q^{76} -2.70966e9i q^{77} -5.76013e8 q^{78} +4.68097e8 q^{79} -1.30544e10 q^{80} +3.87420e8 q^{81} -6.01266e9i q^{82} +3.18460e9i q^{83} -3.12249e9 q^{84} -5.03515e9 q^{85} -9.39859e9 q^{86} +1.81771e9 q^{87} +2.74798e10 q^{88} -7.27168e9i q^{89} +5.58569e9i q^{90} -6.07562e8i q^{91} +9.44757e9i q^{92} -7.56159e9i q^{93} +6.95600e9 q^{94} -2.03202e10 q^{95} -1.00594e10i q^{96} +1.55181e10i q^{97} +1.21981e10i q^{98} -6.05014e9i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 100q - 51200q^{4} + 18392q^{7} + 1968300q^{9} + O(q^{10}) $	$ 100q - 51200q^{4} + 18392q^{7} + 1968300q^{9} + 620136q^{12} + 662904q^{15} + 27925160q^{16} - 2053136q^{17} - 5169828q^{19} - 1076324q^{20} - 1829224q^{22} + 215378180q^{25} - 22082700q^{26} - 102921320q^{28} - 112503588q^{29} - 76491392q^{35} - 1007769600q^{36} + 473464516q^{41} + 1215405588q^{46} - 1676272320q^{48} + 1975297276q^{49} + 733970808q^{51} + 3267506728q^{53} + 591502824q^{57} + 508142200q^{59} + 1264196808q^{60} - 6538206968q^{62} + 362009736q^{63} - 10324137972q^{64} - 2764346616q^{66} + 9997685952q^{68} + 14908523204q^{71} + 4863508712q^{74} + 1890481680q^{75} + 2044437240q^{76} - 758396196q^{78} - 3599839500q^{79} - 23217941144q^{80} + 38742048900q^{81} - 13094894808q^{84} + 23360564412q^{85} + 12186923752q^{86} + 7965322272q^{87} + 32415437996q^{88} - 22098322280q^{94} + 7834510028q^{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 $

$2$		−	59.5712i		−	1.86160i	−0.365527	−	0.930801i	$-0.619111\pi$
$2$		−	59.5712i		−	1.86160i	0.365527	−	0.930801i	$-0.380889\pi$
$3$	140.296			0.577350
$3$	140.296			0.577350
$4$	−2524.73			−2.46556
$5$	−4763.75			−1.52440			−0.762200	−	0.647342i	$-0.775881\pi$
$5$	−4763.75			−1.52440			−0.762200	+	0.647342i	$0.775881\pi$
$6$		−	8357.61i		−	1.07480i
$7$	8815.37			0.524506			0.262253	−	0.964999i	$-0.415535\pi$
$7$	8815.37			0.524506			0.262253	+	0.964999i	$0.415535\pi$
$8$			89400.5i			2.72829i
$9$	19683.0			0.333333
$10$			283782.i			2.83782i
$11$		−	307379.i		−	1.90858i	−0.298879	−	0.954291i	$-0.596613\pi$
$11$		−	307379.i		−	1.90858i	0.298879	−	0.954291i	$-0.403387\pi$
$12$	−354210.			−1.42349
$13$		−	68920.8i		−	0.185624i	−0.995684	−	0.0928119i	$-0.970414\pi$
$13$		−	68920.8i		−	0.185624i	0.995684	−	0.0928119i	$-0.0295855\pi$
$14$		−	525142.i		−	0.976420i
$15$	−668335.			−0.880112
$16$	2.74037e6			2.61342
$17$	1.05697e6			0.744422			0.372211	−	0.928148i	$-0.378600\pi$
$17$	1.05697e6			0.744422			0.372211	+	0.928148i	$0.378600\pi$
$18$		−	1.17254e6i		−	0.620534i
$19$	4.26559e6			1.72271			0.861354	−	0.508006i	$-0.169617\pi$
$19$	4.26559e6			1.72271			0.861354	+	0.508006i	$0.169617\pi$
$20$	1.20272e7			3.75850
$21$	1.23676e6			0.302823
$22$	−1.83109e7			−3.55302
$23$		−	3.74201e6i		−	0.581388i	−0.956816	−	0.290694i	$-0.906114\pi$
$23$		−	3.74201e6i		−	0.581388i	0.956816	−	0.290694i	$-0.0938861\pi$
$24$			1.25425e7i			1.57518i
$25$	1.29277e7			1.32379
$26$	−4.10570e6			−0.345557
$27$	2.76145e6			0.192450
$28$	−2.22564e7			−1.29320
$29$	1.29563e7			0.631670			0.315835	−	0.948814i	$-0.397715\pi$
$29$	1.29563e7			0.631670			0.315835	+	0.948814i	$0.397715\pi$
$30$			3.98136e7i			1.63842i
$31$		−	5.38973e7i		−	1.88260i	−0.337567	−	0.941302i	$-0.609604\pi$
$31$		−	5.38973e7i		−	1.88260i	0.337567	−	0.941302i	$-0.390396\pi$
$32$		−	7.17012e7i		−	2.13686i
$33$		−	4.31241e7i		−	1.10192i
$34$		−	6.29652e7i		−	1.38582i
$35$	−4.19942e7			−0.799556
$36$	−4.96943e7			−0.821853
$37$		−	2.46831e7i		−	0.355952i	−0.984035	−	0.177976i	$-0.943045\pi$
$37$		−	2.46831e7i		−	0.355952i	0.984035	−	0.177976i	$-0.0569549\pi$
$38$		−	2.54107e8i		−	3.20699i
$39$		−	9.66932e6i		−	0.107170i
$40$		−	4.25881e8i		−	4.15900i
$41$	1.00932e8			0.871186			0.435593	−	0.900144i	$-0.356539\pi$
$41$	1.00932e8			0.871186			0.435593	+	0.900144i	$0.356539\pi$
$42$		−	7.36754e7i		−	0.563736i
$43$		−	1.57771e8i		−	1.07321i	−0.843834	−	0.536604i	$-0.819707\pi$
$43$		−	1.57771e8i		−	1.07321i	0.843834	−	0.536604i	$-0.180293\pi$
$44$			7.76050e8i			4.70572i
$45$	−9.37648e7			−0.508133
$46$	−2.22916e8			−1.08231
$47$			1.16768e8i			0.509136i	0.967055	+	0.254568i	$0.0819333\pi$
$47$			1.16768e8i			0.509136i	−0.967055	+	0.254568i	$0.918067\pi$
$48$	3.84463e8			1.50886
$49$	−2.04765e8			−0.724894
$50$		−	7.70117e8i		−	2.46437i
$51$	1.48289e8			0.429792
$52$			1.74007e8i			0.457666i
$53$	6.16971e8			1.47532			0.737659	−	0.675174i	$-0.235931\pi$
$53$	6.16971e8			1.47532			0.737659	+	0.675174i	$0.235931\pi$
$54$		−	1.64503e8i		−	0.358265i
$55$			1.46428e9i			2.90944i
$56$			7.88098e8i			1.43100i
$57$	5.98446e8			0.994606
$58$		−	7.71821e8i		−	1.17592i
$59$	−2.89308e8	−	6.53772e8i	−0.404670	−	0.914463i
$59$	−2.89308e8	−	6.53772e8i	−0.404670	−	0.914463i
$60$	1.68737e9			2.16997
$61$			9.65398e7i			0.114303i	0.998366	+	0.0571515i	$0.0182018\pi$
$61$			9.65398e7i			0.114303i	−0.998366	+	0.0571515i	$0.981798\pi$
$62$	−3.21073e9			−3.50466
$63$	1.73513e8			0.174835
$64$	−1.46519e9			−1.36456
$65$			3.28321e8i			0.282965i
$66$	−2.56896e9			−2.05134
$67$			1.44219e9i			1.06819i	0.845423	+	0.534097i	$0.179348\pi$
$67$			1.44219e9i			1.06819i	−0.845423	+	0.534097i	$0.820652\pi$
$68$	−2.66857e9			−1.83542
$69$		−	5.24990e8i		−	0.335664i
$70$			2.50165e9i			1.48845i
$71$	1.91425e9			1.06098			0.530491	−	0.847691i	$-0.322008\pi$
$71$	1.91425e9			1.06098			0.530491	+	0.847691i	$0.322008\pi$
$72$			1.75967e9i			0.909429i
$73$			1.00218e9i			0.483429i	0.970347	+	0.241714i	$0.0777097\pi$
$73$			1.00218e9i			0.483429i	−0.970347	+	0.241714i	$0.922290\pi$
$74$	−1.47040e9			−0.662640
$75$	1.81370e9			0.764292
$76$	−1.07695e10			−4.24744
$77$		−	2.70966e9i		−	1.00106i
$78$	−5.76013e8			−0.199508
$79$	4.68097e8			0.152125			0.0760625	−	0.997103i	$-0.475765\pi$
$79$	4.68097e8			0.152125			0.0760625	+	0.997103i	$0.475765\pi$
$80$	−1.30544e10			−3.98390
$81$	3.87420e8			0.111111
$82$		−	6.01266e9i		−	1.62180i
$83$			3.18460e9i			0.808472i	0.914655	+	0.404236i	$0.132463\pi$
$83$			3.18460e9i			0.808472i	−0.914655	+	0.404236i	$0.867537\pi$
$84$	−3.12249e9			−0.746629
$85$	−5.03515e9			−1.13480
$86$	−9.39859e9			−1.99788
$87$	1.81771e9			0.364695
$88$	2.74798e10			5.20716
$89$		−	7.27168e9i		−	1.30222i	−0.758983	−	0.651110i	$-0.774304\pi$
$89$		−	7.27168e9i		−	1.30222i	0.758983	−	0.651110i	$-0.225696\pi$
$90$			5.58569e9i			0.945941i
$91$		−	6.07562e8i		−	0.0973607i
$92$			9.44757e9i			1.43345i
$93$		−	7.56159e9i		−	1.08692i
$94$	6.95600e9			0.947808
$95$	−2.03202e10			−2.62609
$96$		−	1.00594e10i		−	1.23372i
$97$			1.55181e10i			1.80709i	0.428495	+	0.903544i	$0.359044\pi$
$97$			1.55181e10i			1.80709i	−0.428495	+	0.903544i	$0.640956\pi$
$98$			1.21981e10i			1.34946i
$99$		−	6.05014e9i		−	0.636194i
$100$	−3.26389e10			−3.26389
$101$		−	6.63900e9i		−	0.631678i	−0.948813	−	0.315839i	$-0.897714\pi$
$101$		−	6.63900e9i		−	0.631678i	0.948813	−	0.315839i	$-0.102286\pi$
$102$		−	8.83377e9i		−	0.800102i
$103$			1.17829e9i			0.101640i	0.998708	+	0.0508202i	$0.0161836\pi$
$103$			1.17829e9i			0.101640i	−0.998708	+	0.0508202i	$0.983816\pi$
$104$	6.16155e9			0.506435
$105$	−5.89162e9			−0.461624
$106$		−	3.67537e10i		−	2.74645i
$107$	1.76272e10			1.25679			0.628397	−	0.777893i	$-0.283711\pi$
$107$	1.76272e10			1.25679			0.628397	+	0.777893i	$0.283711\pi$
$108$	−6.97192e9			−0.474497
$109$			5.60648e8i			0.0364383i	0.999834	+	0.0182191i	$0.00579966\pi$
$109$			5.60648e8i			0.0364383i	−0.999834	+	0.0182191i	$0.994200\pi$
$110$	8.72287e10			5.41622
$111$		−	3.46294e9i		−	0.205509i
$112$	2.41574e10			1.37075
$113$		−	8.03621e9i		−	0.436174i	−0.975929	−	0.218087i	$-0.930018\pi$
$113$		−	8.03621e9i		−	0.436174i	0.975929	−	0.218087i	$-0.0699816\pi$
$114$		−	3.56502e10i		−	1.85156i
$115$			1.78260e10i			0.886267i
$116$	−3.27111e10			−1.55742
$117$		−	1.35657e9i		−	0.0618746i
$118$	−3.89460e10	+	1.72344e10i	−1.70237	+	0.753334i
$119$	9.31760e9			0.390454
$120$		−	5.97495e10i		−	2.40120i
$121$	−6.85445e10			−2.64269
$122$	5.75100e9			0.212786
$123$	1.41604e10			0.502979
$124$			1.36076e11i			4.64167i
$125$	−1.50632e10			−0.493590
$126$		−	1.03364e10i		−	0.325473i
$127$	−6.34339e10			−1.92001			−0.960003	−	0.279990i	$-0.909669\pi$
$127$	−6.34339e10			−1.92001			−0.960003	+	0.279990i	$0.909669\pi$
$128$			1.38611e10i			0.403411i
$129$		−	2.21346e10i		−	0.619617i
$130$	1.95585e10			0.526767
$131$		−	1.65942e10i		−	0.430130i	−0.976600	−	0.215065i	$-0.931004\pi$
$131$		−	1.65942e10i		−	0.430130i	0.976600	−	0.215065i	$-0.0689963\pi$
$132$			1.08877e11i			2.71685i
$133$	3.76028e10			0.903570
$134$	8.59133e10			1.98855
$135$	−1.31548e10			−0.293371
$136$			9.44939e10i			2.03100i
$137$	5.76929e9			0.119542			0.0597709	−	0.998212i	$-0.480963\pi$
$137$	5.76929e9			0.119542			0.0597709	+	0.998212i	$0.480963\pi$
$138$	−3.12743e10			−0.624873
$139$	−7.84083e10			−1.51108			−0.755541	−	0.655102i	$-0.772626\pi$
$139$	−7.84083e10			−1.51108			−0.755541	+	0.655102i	$0.772626\pi$
$140$	1.06024e11			1.97135
$141$			1.63821e10i			0.293950i
$142$		−	1.14034e11i		−	1.97512i
$143$	−2.11848e10			−0.354278
$144$	5.39387e10			0.871140
$145$	−6.17204e10			−0.962917
$146$	5.97013e10			0.899952
$147$	−2.87277e10			−0.418518
$148$			6.23182e10i			0.877620i
$149$		−	1.01063e11i		−	1.37613i	−0.725648	−	0.688066i	$-0.758460\pi$
$149$		−	1.01063e11i		−	1.37613i	0.725648	−	0.688066i	$-0.241540\pi$
$150$		−	1.08044e11i		−	1.42281i
$151$		−	7.82666e10i		−	0.996993i	−0.866892	−	0.498496i	$-0.833886\pi$
$151$		−	7.82666e10i		−	0.996993i	0.866892	−	0.498496i	$-0.166114\pi$
$152$			3.81346e11i			4.70004i
$153$	2.08044e10			0.248141
$154$	−1.61418e11			−1.86358
$155$			2.56753e11i			2.86984i
$156$			2.44124e10i			0.264234i
$157$			6.29703e9i			0.0660143i	0.999455	+	0.0330071i	$0.0105084\pi$
$157$			6.29703e9i			0.0660143i	−0.999455	+	0.0330071i	$0.989492\pi$
$158$		−	2.78851e10i		−	0.283196i
$159$	8.65586e10			0.851775
$160$			3.41566e11i			3.25743i
$161$		−	3.29872e10i		−	0.304941i
$162$		−	2.30791e10i		−	0.206845i
$163$	1.18731e11			1.03187			0.515936	−	0.856627i	$-0.327444\pi$
$163$	1.18731e11			1.03187			0.515936	+	0.856627i	$0.327444\pi$
$164$	−2.54827e11			−2.14796
$165$			2.05432e11i			1.67977i
$166$	1.89711e11			1.50505
$167$	1.46766e11			1.12991			0.564953	−	0.825123i	$-0.308894\pi$
$167$	1.46766e11			1.12991			0.564953	+	0.825123i	$0.308894\pi$
$168$			1.10567e11i			0.826189i
$169$	1.33108e11			0.965544
$170$			2.99950e11i			2.11254i
$171$	8.39597e10			0.574236
$172$			3.98328e11i			2.64606i
$173$			2.48584e11i			1.60414i	0.597228	+	0.802071i	$0.296269\pi$
$173$			2.48584e11i			1.60414i	−0.597228	+	0.802071i	$0.703731\pi$
$174$		−	1.08283e11i		−	0.678916i
$175$	1.13962e11			0.694337
$176$		−	8.42333e11i		−	4.98793i
$177$	−4.05888e10	−	9.17216e10i	−0.233636	−	0.527965i
$178$	−4.33183e11			−2.42422
$179$		−	1.83522e11i		−	0.998672i	−0.866408	−	0.499336i	$-0.833577\pi$
$179$		−	1.83522e11i		−	0.998672i	0.866408	−	0.499336i	$-0.166423\pi$
$180$	2.36731e11			1.25283
$181$	2.33072e11			1.19977			0.599885	−	0.800086i	$-0.295213\pi$
$181$	2.33072e11			1.19977			0.599885	+	0.800086i	$0.295213\pi$
$182$	−3.61932e10			−0.181247
$183$			1.35442e10i			0.0659928i
$184$	3.34537e11			1.58619
$185$			1.17584e11i			0.542613i
$186$	−4.50453e11			−2.02341
$187$		−	3.24891e11i		−	1.42079i
$188$		−	2.94807e11i		−	1.25531i
$189$	2.43432e10			0.100941
$190$			1.21050e12i			4.88874i
$191$		−	6.46864e10i		−	0.254476i	−0.991872	−	0.127238i	$-0.959389\pi$
$191$		−	6.46864e10i		−	0.254476i	0.991872	−	0.127238i	$-0.0406111\pi$
$192$	−2.05560e11			−0.787831
$193$	−3.39025e11			−1.26604			−0.633018	−	0.774137i	$-0.718184\pi$
$193$	−3.39025e11			−1.26604			−0.633018	+	0.774137i	$0.718184\pi$
$194$	9.24431e11			3.36408
$195$			4.60622e10i			0.163370i
$196$	5.16976e11			1.78727
$197$	−1.17018e11			−0.394385			−0.197192	−	0.980365i	$-0.563182\pi$
$197$	−1.17018e11			−0.394385			−0.197192	+	0.980365i	$0.563182\pi$
$198$	−3.60414e11			−1.18434
$199$	−9.16201e9			−0.0293579			−0.0146790	−	0.999892i	$-0.504673\pi$
$199$	−9.16201e9			−0.0293579			−0.0146790	+	0.999892i	$0.504673\pi$
$200$			1.15574e12i			3.61169i
$201$			2.02334e11i			0.616722i
$202$	−3.95493e11			−1.17593
$203$	1.14214e11			0.331314
$204$	−3.74391e11			−1.05968
$205$	−4.80816e11			−1.32804
$206$	7.01923e10			0.189214
$207$		−	7.36540e10i		−	0.193796i
$208$		−	1.88869e11i		−	0.485113i
$209$		−	1.31115e12i		−	3.28793i
$210$			3.50971e11i			0.859359i
$211$			3.54835e11i			0.848425i	0.905563	+	0.424213i	$0.139449\pi$
$211$			3.54835e11i			0.848425i	−0.905563	+	0.424213i	$0.860551\pi$
$212$	−1.55769e12			−3.63748
$213$	2.68562e11			0.612558
$214$		−	1.05007e12i		−	2.33965i
$215$			7.51579e11i			1.63600i
$216$			2.46875e11i			0.525059i
$217$		−	4.75125e11i		−	0.987436i
$218$	3.33985e10			0.0678336
$219$			1.40602e11i			0.279108i
$220$		−	3.69691e12i		−	7.17340i
$221$		−	7.28474e10i		−	0.138182i
$222$	−2.06292e11			−0.382576
$223$	1.19156e11			0.216068			0.108034	−	0.994147i	$-0.465544\pi$
$223$	1.19156e11			0.216068			0.108034	+	0.994147i	$0.465544\pi$
$224$		−	6.32072e11i		−	1.12080i
$225$	2.54455e11			0.441264
$226$	−4.78727e11			−0.811981
$227$		−	6.14134e10i		−	0.101891i	−0.998701	−	0.0509453i	$-0.983777\pi$
$227$		−	6.14134e10i		−	0.101891i	0.998701	−	0.0509453i	$-0.0162234\pi$
$228$	−1.51092e12			−2.45226
$229$		−	8.61736e11i		−	1.36835i	−0.729318	−	0.684175i	$-0.760163\pi$
$229$		−	8.61736e11i		−	1.36835i	0.729318	−	0.684175i	$-0.239837\pi$
$230$	1.06192e12			1.64988
$231$		−	3.80155e11i		−	0.577963i
$232$			1.15830e12i			1.72338i
$233$		−	6.75198e10i		−	0.0983222i	−0.998791	−	0.0491611i	$-0.984345\pi$
$233$		−	6.75198e10i		−	0.0983222i	0.998791	−	0.0491611i	$-0.0156548\pi$
$234$	−8.08124e10			−0.115186
$235$		−	5.56252e11i		−	0.776127i
$236$	7.30426e11	+	1.65060e12i	0.997737	+	2.25466i
$237$	6.56722e10			0.0878294
$238$		−	5.55061e11i		−	0.726869i
$239$	2.58785e11			0.331857			0.165928	−	0.986138i	$-0.446938\pi$
$239$	2.58785e11			0.331857			0.165928	+	0.986138i	$0.446938\pi$
$240$	−1.83149e12			−2.30010
$241$	−1.33896e12			−1.64696			−0.823478	−	0.567348i	$-0.807969\pi$
$241$	−1.33896e12			−1.64696			−0.823478	+	0.567348i	$0.807969\pi$
$242$			4.08328e12i			4.91963i
$243$	5.43536e10			0.0641500
$244$		−	2.43737e11i		−	0.281821i
$245$	9.75447e11			1.10503
$246$		−	8.43553e11i		−	0.936347i
$247$		−	2.93988e11i		−	0.319775i
$248$	4.81845e12			5.13628
$249$			4.46787e11i			0.466771i
$250$			8.97332e11i			0.918868i
$251$	−1.54586e12			−1.55168			−0.775838	−	0.630932i	$-0.782673\pi$
$251$	−1.54586e12			−1.55168			−0.775838	+	0.630932i	$0.782673\pi$
$252$	−4.38073e11			−0.431066
$253$	−1.15022e12			−1.10963
$254$			3.77883e12i			3.57428i
$255$	−7.06412e11			−0.655175
$256$	−6.74631e11			−0.613573
$257$	1.34163e12			1.19665			0.598324	−	0.801254i	$-0.295834\pi$
$257$	1.34163e12			1.19665			0.598324	+	0.801254i	$0.295834\pi$
$258$	−1.31859e12			−1.15348
$259$		−	2.17591e11i		−	0.186699i
$260$		−	8.28923e11i		−	0.697666i
$261$	2.55018e11			0.210557
$262$	−9.88536e11			−0.800730
$263$	−1.31680e12			−1.04650			−0.523252	−	0.852178i	$-0.675281\pi$
$263$	−1.31680e12			−1.04650			−0.523252	+	0.852178i	$0.675281\pi$
$264$	3.85531e12			3.00635
$265$	−2.93909e12			−2.24897
$266$		−	2.24004e12i		−	1.68209i
$267$		−	1.02019e12i		−	0.751837i
$268$		−	3.64116e12i		−	2.63369i
$269$		−	1.94110e12i		−	1.37812i	−0.724703	−	0.689061i	$-0.758023\pi$
$269$		−	1.94110e12i		−	1.37812i	0.724703	−	0.689061i	$-0.241977\pi$
$270$			7.83650e11i			0.546139i
$271$	9.78319e11			0.669321			0.334660	−	0.942339i	$-0.391378\pi$
$271$	9.78319e11			0.669321			0.334660	+	0.942339i	$0.391378\pi$
$272$	2.89650e12			1.94549
$273$		−	8.52386e10i		−	0.0562112i
$274$		−	3.43684e11i		−	0.222539i
$275$		−	3.97369e12i		−	2.52657i
$276$			1.32546e12i			0.827600i
$277$	−1.22302e11			−0.0749957			−0.0374978	−	0.999297i	$-0.511939\pi$
$277$	−1.22302e11			−0.0749957			−0.0374978	+	0.999297i	$0.511939\pi$
$278$			4.67088e12i			2.81303i
$279$		−	1.06086e12i		−	0.627534i
$280$		−	3.75430e12i		−	2.18142i
$281$	−2.67988e12			−1.52962			−0.764811	−	0.644254i	$-0.777168\pi$
$281$	−2.67988e12			−1.52962			−0.764811	+	0.644254i	$0.777168\pi$
$282$	9.75900e11			0.547217
$283$			2.07320e12i			1.14211i	0.820910	+	0.571057i	$0.193466\pi$
$283$			2.07320e12i			1.14211i	−0.820910	+	0.571057i	$0.806534\pi$
$284$	−4.83298e12			−2.61591
$285$	−2.85085e12			−1.51618
$286$			1.26201e12i			0.659525i
$287$	8.89755e11			0.456942
$288$		−	1.41129e12i		−	0.712287i
$289$	−8.98802e11			−0.445836
$290$			3.67676e12i			1.79257i
$291$			2.17713e12i			1.04332i
$292$		−	2.53024e12i		−	1.19192i
$293$	−1.16615e11			−0.0540027			−0.0270014	−	0.999635i	$-0.508596\pi$
$293$	−1.16615e11			−0.0540027			−0.0270014	+	0.999635i	$0.508596\pi$
$294$			1.71134e12i			0.779113i
$295$	1.37819e12	+	3.11440e12i	0.616878	+	1.39401i
$296$	2.20668e12			0.971138
$297$		−	8.48811e11i		−	0.367307i
$298$	−6.02044e12			−2.56181
$299$	−2.57902e11			−0.107919
$300$	−4.57911e12			−1.88441
$301$		−	1.39081e12i		−	0.562903i
$302$	−4.66244e12			−1.85600
$303$		−	9.31426e11i		−	0.364699i
$304$	1.16893e13			4.50216
$305$		−	4.59891e11i		−	0.174243i
$306$		−	1.23934e12i		−	0.461939i
$307$	−1.00943e12			−0.370155			−0.185077	−	0.982724i	$-0.559254\pi$
$307$	−1.00943e12			−0.370155			−0.185077	+	0.982724i	$0.559254\pi$
$308$			6.84116e12i			2.46818i
$309$			1.65310e11i			0.0586822i
$310$	1.52951e13			5.34250
$311$	4.42123e12			1.51964			0.759821	−	0.650133i	$-0.225287\pi$
$311$	4.42123e12			1.51964			0.759821	+	0.650133i	$0.225287\pi$
$312$	8.64442e11			0.292390
$313$			1.33939e12i			0.445847i	0.974836	+	0.222924i	$0.0715600\pi$
$313$			1.33939e12i			0.445847i	−0.974836	+	0.222924i	$0.928440\pi$
$314$	3.75122e11			0.122892
$315$	−8.26571e11			−0.266519
$316$	−1.18182e12			−0.375073
$317$	2.96670e12			0.926782			0.463391	−	0.886154i	$-0.346632\pi$
$317$	2.96670e12			0.926782			0.463391	+	0.886154i	$0.346632\pi$
$318$		−	5.15641e12i		−	1.58567i
$319$		−	3.98249e12i		−	1.20559i
$320$	6.97979e12			2.08014
$321$	2.47303e12			0.725611
$322$	−1.96509e12			−0.567679
$323$	4.50862e12			1.28242
$324$	−9.78133e11			−0.273951
$325$		−	8.90985e11i		−	0.245727i
$326$		−	7.07294e12i		−	1.92093i
$327$			7.86568e10i			0.0210377i
$328$			9.02339e12i			2.37684i
$329$			1.02935e12i			0.267045i
$330$	1.22379e13			3.12706
$331$	4.58261e12			1.15338			0.576691	−	0.816962i	$-0.304344\pi$
$331$	4.58261e12			1.15338			0.576691	+	0.816962i	$0.304344\pi$
$332$		−	8.04027e12i		−	1.99333i
$333$		−	4.85838e11i		−	0.118651i
$334$		−	8.74301e12i		−	2.10343i
$335$		−	6.87025e12i		−	1.62835i
$336$	3.38919e12			0.791405
$337$			4.93314e12i			1.13494i	0.823393	+	0.567471i	$0.192078\pi$
$337$			4.93314e12i			1.13494i	−0.823393	+	0.567471i	$0.807922\pi$
$338$		−	7.92943e12i		−	1.79746i
$339$		−	1.12745e12i		−	0.251825i
$340$	1.27124e13			2.79791
$341$	−1.65669e13			−3.59310
$342$		−	5.00158e12i		−	1.06900i
$343$	−4.29520e12			−0.904716
$344$	1.41048e13			2.92802
$345$			2.50092e12i			0.511686i
$346$	1.48085e13			2.98627
$347$			6.30374e12i			1.25300i	0.779421	+	0.626500i	$0.215513\pi$
$347$			6.30374e12i			1.25300i	−0.779421	+	0.626500i	$0.784487\pi$
$348$	−4.58924e12			−0.899176
$349$			1.03542e12i			0.199981i	0.994988	+	0.0999905i	$0.0318812\pi$
$349$			1.03542e12i			0.199981i	−0.994988	+	0.0999905i	$0.968119\pi$
$350$		−	6.78886e12i		−	1.29258i
$351$		−	1.90321e11i		−	0.0357233i
$352$	−2.20394e13			−4.07838
$353$			3.93173e12i			0.717315i	0.933469	+	0.358657i	$0.116765\pi$
$353$			3.93173e12i			0.717315i	−0.933469	+	0.358657i	$0.883235\pi$
$354$	−5.46397e12	+	2.41793e12i	−0.982861	+	0.434937i
$355$	−9.11902e12			−1.61736
$356$			1.83590e13i			3.21070i
$357$	1.30722e12			0.225428
$358$	−1.09326e13			−1.85913
$359$	−8.10572e11			−0.135931			−0.0679656	−	0.997688i	$-0.521651\pi$
$359$	−8.10572e11			−0.135931			−0.0679656	+	0.997688i	$0.521651\pi$
$360$		−	8.38262e12i		−	1.38633i
$361$	1.20642e13			1.96772
$362$		−	1.38844e13i		−	2.23349i
$363$	−9.61652e12			−1.52576
$364$			1.53393e12i			0.240049i
$365$		−	4.77414e12i		−	0.736939i
$366$	8.06842e11			0.122852
$367$		−	5.56208e12i		−	0.835424i	−0.908579	−	0.417712i	$-0.862832\pi$
$367$		−	5.56208e12i		−	0.835424i	0.908579	−	0.417712i	$-0.137168\pi$
$368$		−	1.02545e13i		−	1.51941i
$369$	1.98665e12			0.290395
$370$	7.00463e12			1.01013
$371$	5.43883e12			0.773812
$372$			1.90910e13i			2.67987i
$373$	−1.16740e13			−1.61687			−0.808433	−	0.588589i	$-0.799684\pi$
$373$	−1.16740e13			−1.61687			−0.808433	+	0.588589i	$0.799684\pi$
$374$	−1.93542e13			−2.64495
$375$	−2.11330e12			−0.284974
$376$	−1.04391e13			−1.38907
$377$		−	8.92956e11i		−	0.117253i
$378$		−	1.45015e12i		−	0.187912i
$379$	−8.15197e12			−1.04248			−0.521239	−	0.853411i	$-0.674530\pi$
$379$	−8.15197e12			−1.04248			−0.521239	+	0.853411i	$0.674530\pi$
$380$	5.13031e13			6.47479
$381$	−8.89952e12			−1.10852
$382$	−3.85345e12			−0.473732
$383$	−3.46373e11			−0.0420291			−0.0210146	−	0.999779i	$-0.506690\pi$
$383$	−3.46373e11			−0.0420291			−0.0210146	+	0.999779i	$0.506690\pi$
$384$			1.94466e12i			0.232909i
$385$			1.29081e13i			1.52602i
$386$			2.01962e13i			2.35685i
$387$		−	3.10540e12i		−	0.357736i
$388$		−	3.91790e13i		−	4.45548i
$389$	−4.90715e12			−0.550911			−0.275455	−	0.961314i	$-0.588829\pi$
$389$	−4.90715e12			−0.550911			−0.275455	+	0.961314i	$0.588829\pi$
$390$	2.74398e12			0.304129
$391$		−	3.95520e12i		−	0.432798i
$392$		−	1.83060e13i		−	1.97772i
$393$		−	2.32810e12i		−	0.248335i
$394$			6.97089e12i			0.734187i
$395$	−2.22990e12			−0.231899
$396$			1.52750e13i			1.56857i
$397$			1.52710e13i			1.54852i	0.632870	+	0.774258i	$0.281877\pi$
$397$			1.52710e13i			1.54852i	−0.632870	+	0.774258i	$0.718123\pi$
$398$			5.45792e11i			0.0546528i
$399$	5.27552e12			0.521676
$400$	3.54266e13			3.45963
$401$			1.28285e13i			1.23724i	0.785691	+	0.618619i	$0.212308\pi$
$401$			1.28285e13i			1.23724i	−0.785691	+	0.618619i	$0.787692\pi$
$402$	1.20533e13			1.14809
$403$	−3.71465e12			−0.349456
$404$			1.67617e13i			1.55744i
$405$	−1.84557e12			−0.169378
$406$		−	6.80388e12i		−	0.616775i
$407$	−7.58707e12			−0.679363
$408$			1.32571e13i			1.17260i
$409$			3.71099e12i			0.324245i	0.986771	+	0.162122i	$0.0518339\pi$
$409$			3.71099e12i			0.324245i	−0.986771	+	0.162122i	$0.948166\pi$
$410$			2.86428e13i			2.47227i
$411$	8.09409e11			0.0690175
$412$		−	2.97487e12i		−	0.250601i
$413$	−2.55036e12	−	5.76324e12i	−0.212252	−	0.479641i
$414$	−4.38766e12			−0.360771
$415$		−	1.51706e13i		−	1.23243i
$416$	−4.94170e12			−0.396652
$417$	−1.10004e13			−0.872423
$418$	−7.81071e13			−6.12081
$419$		−	1.91177e13i		−	1.48035i	−0.672412	−	0.740177i	$-0.734742\pi$
$419$		−	1.91177e13i		−	1.48035i	0.672412	−	0.740177i	$-0.265258\pi$
$420$	1.48748e13			1.13816
$421$		−	3.47260e12i		−	0.262570i	−0.991345	−	0.131285i	$-0.958090\pi$
$421$		−	3.47260e12i		−	0.262570i	0.991345	−	0.131285i	$-0.0419102\pi$
$422$	2.11379e13			1.57943
$423$			2.29834e12i			0.169712i
$424$			5.51575e13i			4.02509i
$425$	1.36642e13			0.985461
$426$		−	1.59986e13i		−	1.14034i
$427$			8.51034e11i			0.0599525i
$428$	−4.45039e13			−3.09870
$429$	−2.97215e12			−0.204543
$430$	4.47725e13			3.04557
$431$		−	8.12194e12i		−	0.546102i	−0.962000	−	0.273051i	$-0.911967\pi$
$431$		−	8.12194e12i		−	0.546102i	0.962000	−	0.273051i	$-0.0880327\pi$
$432$	7.56739e12			0.502953
$433$	−3.57079e12			−0.234598			−0.117299	−	0.993097i	$-0.537424\pi$
$433$	−3.57079e12			−0.234598			−0.117299	+	0.993097i	$0.537424\pi$
$434$	−2.83038e13			−1.83821
$435$	−8.65913e12			−0.555940
$436$		−	1.41549e12i		−	0.0898407i
$437$		−	1.59619e13i		−	1.00156i
$438$	8.37585e12			0.519587
$439$	−6.08525e12			−0.373212			−0.186606	−	0.982435i	$-0.559749\pi$
$439$	−6.08525e12			−0.373212			−0.186606	+	0.982435i	$0.559749\pi$
$440$	−1.30907e14			−7.93779
$441$	−4.03038e12			−0.241631
$442$	−4.33961e12			−0.257241
$443$		−	8.84077e12i		−	0.518169i	−0.965855	−	0.259084i	$-0.916579\pi$
$443$		−	8.84077e12i		−	0.518169i	0.965855	−	0.259084i	$-0.0834208\pi$
$444$			8.74301e12i			0.506694i
$445$			3.46404e13i			1.98510i
$446$		−	7.09825e12i		−	0.402232i
$447$		−	1.41787e13i		−	0.794511i
$448$	−1.29162e13			−0.715721
$449$	2.04538e13			1.12084			0.560419	−	0.828209i	$-0.310640\pi$
$449$	2.04538e13			1.12084			0.560419	+	0.828209i	$0.310640\pi$
$450$		−	1.51582e13i		−	0.821458i
$451$		−	3.10245e13i		−	1.66273i
$452$			2.02893e13i			1.07541i
$453$		−	1.09805e13i		−	0.575614i
$454$	−3.65847e12			−0.189680
$455$			2.89427e12i			0.148417i
$456$			5.35014e13i			2.71357i
$457$		−	2.70573e12i		−	0.135739i	−0.997694	−	0.0678693i	$-0.978380\pi$
$457$		−	2.70573e12i		−	0.135739i	0.997694	−	0.0678693i	$-0.0216201\pi$
$458$	−5.13347e13			−2.54732
$459$	2.91878e12			0.143264
$460$		−	4.50059e13i		−	2.18514i
$461$	−6.63428e11			−0.0318632			−0.0159316	−	0.999873i	$-0.505071\pi$
$461$	−6.63428e11			−0.0318632			−0.0159316	+	0.999873i	$0.505071\pi$
$462$	−2.26463e13			−1.07594
$463$		−	2.63519e13i		−	1.23853i	−0.785182	−	0.619266i	$-0.787430\pi$
$463$		−	2.63519e13i		−	1.23853i	0.785182	−	0.619266i	$-0.212570\pi$
$464$	3.55050e13			1.65082
$465$			3.60215e13i			1.65690i
$466$	−4.02224e12			−0.183037
$467$			4.60839e12i			0.207475i	0.994605	+	0.103737i	$0.0330801\pi$
$467$			4.60839e12i			0.207475i	−0.994605	+	0.103737i	$0.966920\pi$
$468$			3.42497e12i			0.152555i
$469$			1.27135e13i			0.560273i
$470$	−3.31366e13			−1.44484
$471$			8.83449e11i			0.0381134i
$472$	5.84475e13	−	2.58643e13i	2.49492	−	1.10405i
$473$	−4.84954e13			−2.04830
$474$		−	3.91218e12i		−	0.163503i
$475$	5.51442e13			2.28051
$476$	−2.35245e13			−0.962686
$477$	1.21438e13			0.491772
$478$		−	1.54162e13i		−	0.617785i
$479$	4.03658e13			1.60080			0.800399	−	0.599468i	$-0.204621\pi$
$479$	4.03658e13			1.60080			0.800399	+	0.599468i	$0.204621\pi$
$480$			4.79204e13i			1.88068i
$481$	−1.70118e12			−0.0660731
$482$			7.97634e13i			3.06598i
$483$		−	4.62797e12i		−	0.176058i
$484$	1.73056e14			6.51570
$485$		−	7.39242e13i		−	2.75472i
$486$		−	3.23791e12i		−	0.119422i
$487$	1.09680e13			0.400391			0.200196	−	0.979756i	$-0.435842\pi$
$487$	1.09680e13			0.400391			0.200196	+	0.979756i	$0.435842\pi$
$488$	−8.63071e12			−0.311851
$489$	1.66575e13			0.595751
$490$		−	5.81086e13i		−	2.05712i
$491$	−2.14337e13			−0.751085			−0.375542	−	0.926805i	$-0.622544\pi$
$491$	−2.14337e13			−0.751085			−0.375542	+	0.926805i	$0.622544\pi$
$492$	−3.57512e13			−1.24013
$493$	1.36944e13			0.470229
$494$	−1.75132e13			−0.595294
$495$			2.88213e13i			0.969814i
$496$		−	1.47699e14i		−	4.92004i
$497$	1.68749e13			0.556491
$498$	2.66157e13			0.868942
$499$	−3.10709e13			−1.00427			−0.502136	−	0.864789i	$-0.667452\pi$
$499$	−3.10709e13			−1.00427			−0.502136	+	0.864789i	$0.667452\pi$
$500$	3.80305e13			1.21697
$501$	2.05907e13			0.652351
$502$			9.20887e13i			2.88860i
$503$		−	2.00583e13i		−	0.622952i	−0.950254	−	0.311476i	$-0.899177\pi$
$503$		−	2.00583e13i		−	0.622952i	0.950254	−	0.311476i	$-0.100823\pi$
$504$			1.55121e13i			0.477000i
$505$			3.16265e13i			0.962930i
$506$			6.85198e13i			2.06568i
$507$	1.86746e13			0.557457
$508$	1.60153e14			4.73389
$509$			2.00516e13i			0.586894i	0.955975	+	0.293447i	$0.0948024\pi$
$509$			2.00516e13i			0.586894i	−0.955975	+	0.293447i	$0.905198\pi$
$510$			4.20819e13i			1.21967i
$511$			8.83461e12i			0.253561i
$512$			5.43824e13i			1.54564i
$513$	1.17792e13			0.331535
$514$		−	7.99224e13i		−	2.22768i
$515$		−	5.61308e12i		−	0.154941i
$516$			5.58839e13i			1.52770i
$517$	3.58920e13			0.971728
$518$	−1.29621e13			−0.347559
$519$			3.48754e13i			0.926152i
$520$	−2.93521e13			−0.772009
$521$	−2.12115e13			−0.552564			−0.276282	−	0.961077i	$-0.589102\pi$
$521$	−2.12115e13			−0.552564			−0.276282	+	0.961077i	$0.589102\pi$
$522$		−	1.51918e13i		−	0.391972i
$523$	3.90091e13			0.996913			0.498457	−	0.866915i	$-0.333900\pi$
$523$	3.90091e13			0.996913			0.498457	+	0.866915i	$0.333900\pi$
$524$			4.18959e13i			1.06051i
$525$	1.59884e13			0.400876
$526$			7.84434e13i			1.94817i
$527$		−	5.69680e13i		−	1.40145i
$528$		−	1.18176e14i		−	2.87978i
$529$	2.74239e13			0.661988
$530$			1.75086e14i			4.18669i
$531$	−5.69445e12	−	1.28682e13i	−0.134890	−	0.304821i
$532$	−9.49369e13			−2.22780
$533$		−	6.95633e12i		−	0.161713i
$534$	−6.07739e13			−1.39962
$535$	−8.39715e13			−1.91586
$536$	−1.28933e14			−2.91434
$537$		−	2.57474e13i		−	0.576583i
$538$	−1.15634e14			−2.56551
$539$			6.29403e13i			1.38352i
$540$	3.32125e13			0.723323
$541$		−	2.97252e13i		−	0.641415i	−0.947178	−	0.320707i	$-0.896079\pi$
$541$		−	2.97252e13i		−	0.641415i	0.947178	−	0.320707i	$-0.103921\pi$
$542$		−	5.82797e13i		−	1.24601i
$543$	3.26992e13			0.692688
$544$		−	7.57862e13i		−	1.59073i
$545$		−	2.67079e12i		−	0.0555465i
$546$	−5.07777e12			−0.104643
$547$	1.57133e12			0.0320871			0.0160436	−	0.999871i	$-0.494893\pi$
$547$	1.57133e12			0.0320871			0.0160436	+	0.999871i	$0.494893\pi$
$548$	−1.45659e13			−0.294737
$549$			1.90019e12i			0.0381010i
$550$	−2.36718e14			−4.70346
$551$	5.52662e13			1.08818
$552$	4.69343e13			0.915788
$553$	4.12645e12			0.0797904
$554$			7.28571e12i			0.139612i
$555$			1.64966e13i			0.313278i
$556$	1.97960e14			3.72566
$557$	1.01564e14			1.89436			0.947181	−	0.320699i	$-0.103918\pi$
$557$	1.01564e14			1.89436			0.947181	+	0.320699i	$0.103918\pi$
$558$	−6.31968e13			−1.16822
$559$	−1.08737e13			−0.199213
$560$	−1.15080e14			−2.08958
$561$		−	4.55810e13i		−	0.820294i
$562$			1.59644e14i			2.84755i
$563$		−	2.62954e13i		−	0.464877i	−0.972611	−	0.232438i	$-0.925330\pi$
$563$		−	2.62954e13i		−	0.464877i	0.972611	−	0.232438i	$-0.0746704\pi$
$564$		−	4.13603e13i		−	0.724751i
$565$			3.82825e13i			0.664903i
$566$	1.23503e14			2.12616
$567$	3.41525e12			0.0582784
$568$			1.71135e14i			2.89466i
$569$			9.01712e13i			1.51184i	0.654662	+	0.755921i	$0.272811\pi$
$569$			9.01712e13i			1.51184i	−0.654662	+	0.755921i	$0.727189\pi$
$570$			1.69828e14i			2.82252i
$571$		−	9.16721e13i		−	1.51028i	−0.655565	−	0.755138i	$-0.727570\pi$
$571$		−	9.16721e13i		−	1.51028i	0.655565	−	0.755138i	$-0.272430\pi$
$572$	5.34860e13			0.873494
$573$		−	9.07525e12i		−	0.146922i
$574$		−	5.30038e13i		−	0.850644i
$575$		−	4.83755e13i		−	0.769637i
$576$	−2.88393e13			−0.454855
$577$	−9.46419e13			−1.47980			−0.739902	−	0.672714i	$-0.765128\pi$
$577$	−9.46419e13			−1.47980			−0.739902	+	0.672714i	$0.765128\pi$
$578$			5.35427e13i			0.829968i
$579$	−4.75640e13			−0.730946
$580$	1.55827e14			2.37413
$581$			2.80734e13i			0.424048i
$582$	1.29694e14			1.94225
$583$		−	1.89644e14i		−	2.81576i
$584$	−8.95956e13			−1.31893
$585$			6.46235e12i			0.0943216i
$586$			6.94689e12i			0.100532i
$587$			3.82820e13i			0.549292i	0.961545	+	0.274646i	$0.0885607\pi$
$587$			3.82820e13i			0.549292i	−0.961545	+	0.274646i	$0.911439\pi$
$588$	7.25297e13			1.03188
$589$		−	2.29904e14i		−	3.24317i
$590$	1.85529e14	−	8.21006e13i	2.59508	−	1.14838i
$591$	−1.64171e13			−0.227698
$592$		−	6.76409e13i		−	0.930252i
$593$	−1.13813e14			−1.55209			−0.776044	−	0.630678i	$-0.782777\pi$
$593$	−1.13813e14			−1.55209			−0.776044	+	0.630678i	$0.782777\pi$
$594$	−5.05647e13			−0.683779
$595$	−4.43867e13			−0.595207
$596$			2.55157e14i			3.39294i
$597$	−1.28539e12			−0.0169498
$598$			1.53636e13i			0.200903i
$599$	−2.06483e13			−0.267763			−0.133881	−	0.990997i	$-0.542744\pi$
$599$	−2.06483e13			−0.267763			−0.133881	+	0.990997i	$0.542744\pi$
$600$			1.62146e14i			2.08521i
$601$			1.03016e14i			1.31380i	0.753976	+	0.656902i	$0.228133\pi$
$601$			1.03016e14i			1.31380i	−0.753976	+	0.656902i	$0.771867\pi$
$602$	−8.28520e13			−1.04790
$603$			2.83867e13i			0.356065i
$604$			1.97602e14i			2.45814i
$605$	3.26528e14			4.02851
$606$	−5.54862e13			−0.678925
$607$	9.74616e13			1.18274			0.591371	−	0.806400i	$-0.298587\pi$
$607$	9.74616e13			1.18274			0.591371	+	0.806400i	$0.298587\pi$
$608$		−	3.05848e14i		−	3.68119i
$609$	1.60238e13			0.191284
$610$	−2.73963e13			−0.324371
$611$	8.04773e12			0.0945078
$612$	−5.25255e13			−0.611805
$613$			1.46820e14i			1.69622i	0.529820	+	0.848110i	$0.322260\pi$
$613$			1.46820e14i			1.69622i	−0.529820	+	0.848110i	$0.677740\pi$
$614$			6.01329e13i			0.689081i
$615$	−6.74566e13			−0.766741
$616$	2.42245e14			2.73118
$617$	−5.74060e13			−0.641995			−0.320997	−	0.947080i	$-0.604018\pi$
$617$	−5.74060e13			−0.641995			−0.320997	+	0.947080i	$0.604018\pi$
$618$	9.84771e12			0.109243
$619$	1.74878e14			1.92434			0.962170	−	0.272448i	$-0.0878334\pi$
$619$	1.74878e14			1.92434			0.962170	+	0.272448i	$0.0878334\pi$
$620$		−	6.48233e14i		−	7.07576i
$621$		−	1.03334e13i		−	0.111888i
$622$		−	2.63378e14i		−	2.82897i
$623$		−	6.41025e13i		−	0.683022i
$624$		−	2.64975e13i		−	0.280080i
$625$	−5.44896e13			−0.571365
$626$	7.97892e13			0.829990
$627$		−	1.83950e14i		−	1.89829i
$628$		−	1.58983e13i		−	0.162762i
$629$		−	2.60894e13i		−	0.264978i
$630$			4.92399e13i			0.496151i
$631$	−4.23221e13			−0.423078			−0.211539	−	0.977370i	$-0.567848\pi$
$631$	−4.23221e13			−0.423078			−0.211539	+	0.977370i	$0.567848\pi$
$632$			4.18481e13i			0.415040i
$633$			4.97819e13i			0.489839i
$634$		−	1.76730e14i		−	1.72530i
$635$	3.02183e14			2.92686
$636$	−2.18537e14			−2.10010
$637$			1.41125e13i			0.134558i
$638$	−2.37242e14			−2.24433
$639$	3.76783e13			0.353661
$640$		−	6.60308e13i		−	0.614959i
$641$	−1.07607e14			−0.994379			−0.497190	−	0.867642i	$-0.665635\pi$
$641$	−1.07607e14			−0.994379			−0.497190	+	0.867642i	$0.665635\pi$
$642$		−	1.47321e14i		−	1.35080i
$643$	−4.82800e13			−0.439251			−0.219625	−	0.975584i	$-0.570483\pi$
$643$	−4.82800e13			−0.439251			−0.219625	+	0.975584i	$0.570483\pi$
$644$			8.32838e13i			0.751850i
$645$			1.05444e14i			0.944543i
$646$		−	2.68584e14i		−	2.38736i
$647$	1.99080e14			1.75593			0.877963	−	0.478728i	$-0.158902\pi$
$647$	1.99080e14			1.75593			0.877963	+	0.478728i	$0.158902\pi$
$648$			3.46356e13i			0.303143i
$649$	−2.00956e14	+	8.89273e13i	−1.74533	+	0.772345i
$650$	−5.30771e13			−0.457446
$651$		−	6.66582e13i		−	0.570096i
$652$	−2.99764e14			−2.54414
$653$	−1.58076e14			−1.33138			−0.665689	−	0.746229i	$-0.731862\pi$
$653$	−1.58076e14			−1.33138			−0.665689	+	0.746229i	$0.731862\pi$
$654$	4.68568e12			0.0391637
$655$			7.90505e13i			0.655689i
$656$	2.76592e14			2.27678
$657$			1.97260e13i			0.161143i
$658$	6.13197e13			0.497131
$659$			4.18691e13i			0.336874i	0.985712	+	0.168437i	$0.0538719\pi$
$659$			4.18691e13i			0.336874i	−0.985712	+	0.168437i	$0.946128\pi$
$660$		−	5.18661e14i		−	4.14156i
$661$	1.58756e14			1.25812			0.629062	−	0.777355i	$-0.283439\pi$
$661$	1.58756e14			1.25812			0.629062	+	0.777355i	$0.283439\pi$
$662$		−	2.72992e14i		−	2.14714i
$663$		−	1.02202e13i		−	0.0797797i
$664$	−2.84705e14			−2.20574
$665$	−1.79130e14			−1.37740
$666$	−2.89420e13			−0.220880
$667$		−	4.84825e13i		−	0.367245i
$668$	−3.70544e14			−2.78585
$669$	1.67171e13			0.124747
$670$	−4.09269e14			−3.03134
$671$	2.96743e13			0.218156
$672$		−	8.86773e13i		−	0.647092i
$673$			1.29737e13i			0.0939697i	0.998896	+	0.0469849i	$0.0149613\pi$
$673$			1.29737e13i			0.0939697i	−0.998896	+	0.0469849i	$0.985039\pi$
$674$	2.93873e14			2.11281
$675$	3.56991e13			0.254764
$676$	−3.36063e14			−2.38061
$677$	−1.52016e14			−1.06892			−0.534462	−	0.845192i	$-0.679486\pi$
$677$	−1.52016e14			−1.06892			−0.534462	+	0.845192i	$0.679486\pi$
$678$	−6.71636e13			−0.468798
$679$			1.36798e14i			0.947828i
$680$		−	4.50145e14i		−	3.09605i
$681$		−	8.61606e12i		−	0.0588265i
$682$			9.86911e14i			6.68892i
$683$			2.93309e14i			1.97343i	0.162451	+	0.986717i	$0.448060\pi$
$683$			2.93309e14i			1.97343i	−0.162451	+	0.986717i	$0.551940\pi$
$684$	−2.11976e14			−1.41581
$685$	−2.74834e13			−0.182229
$686$			2.55870e14i			1.68422i
$687$		−	1.20898e14i		−	0.790017i
$688$		−	4.32350e14i		−	2.80474i
$689$		−	4.25221e13i		−	0.273854i
$690$	1.48983e14			0.952556
$691$			2.21613e13i			0.140671i	0.997523	+	0.0703356i	$0.0224070\pi$
$691$			2.21613e13i			0.140671i	−0.997523	+	0.0703356i	$0.977593\pi$
$692$		−	6.27608e14i		−	3.95511i
$693$		−	5.33342e13i		−	0.333687i
$694$	3.75522e14			2.33259
$695$	3.73517e14			2.30349
$696$			1.62505e14i			0.994991i
$697$	1.06683e14			0.648530
$698$	6.16812e13			0.372285
$699$		−	9.47277e12i		−	0.0567664i
$700$	−2.87724e14			−1.71193
$701$			1.12954e14i			0.667285i	0.942700	+	0.333643i	$0.108278\pi$
$701$			1.12954e14i			0.667285i	−0.942700	+	0.333643i	$0.891722\pi$
$702$	−1.13377e13			−0.0665026
$703$		−	1.05288e14i		−	0.613201i
$704$			4.50368e14i			2.60438i
$705$		−	7.80401e13i		−	0.448097i
$706$	2.34218e14			1.33535
$707$		−	5.85252e13i		−	0.331319i
$708$	1.02476e14	+	2.31573e14i	0.576044	+	1.30173i
$709$	2.71154e14			1.51351			0.756755	−	0.653698i	$-0.226783\pi$
$709$	2.71154e14			1.51351			0.756755	+	0.653698i	$0.226783\pi$
$710$			5.43232e14i			3.01088i
$711$	9.21356e12			0.0507083
$712$	6.50091e14			3.55283
$713$	−2.01684e14			−1.09452
$714$		−	7.78729e13i		−	0.419658i
$715$	1.00919e14			0.540061
$716$			4.63344e14i			2.46228i
$717$	3.63066e13			0.191597
$718$			4.82868e13i			0.253050i
$719$		−	3.12625e13i		−	0.162697i	−0.996686	−	0.0813485i	$-0.974077\pi$
$719$		−	3.12625e13i		−	0.162697i	0.996686	−	0.0813485i	$-0.0259227\pi$
$720$	−2.56950e14			−1.32797
$721$			1.03871e13i			0.0533110i
$722$		−	7.18681e14i		−	3.66311i
$723$	−1.87851e14			−0.950870
$724$	−5.88445e14			−2.95810
$725$	1.67494e14			0.836200
$726$			5.72868e14i			2.84035i
$727$	1.46683e13			0.0722284			0.0361142	−	0.999348i	$-0.488502\pi$
$727$	1.46683e13			0.0722284			0.0361142	+	0.999348i	$0.488502\pi$
$728$	5.43163e13			0.265628
$729$	7.62560e12			0.0370370
$730$	−2.84402e14			−1.37189
$731$		−	1.66759e14i		−	0.798919i
$732$		−	3.41954e13i		−	0.162709i
$733$	−1.72116e14			−0.813393			−0.406696	−	0.913563i	$-0.633319\pi$
$733$	−1.72116e14			−0.813393			−0.406696	+	0.913563i	$0.633319\pi$
$734$	−3.31340e14			−1.55523
$735$	1.36851e14			0.637988
$736$	−2.68307e14			−1.24235
$737$	4.43300e14			2.03873
$738$		−	1.18347e14i		−	0.540600i
$739$		−	1.77557e14i		−	0.805595i	−0.915289	−	0.402797i	$-0.868038\pi$
$739$		−	1.77557e14i		−	0.805595i	0.915289	−	0.402797i	$-0.131962\pi$
$740$		−	2.96868e14i		−	1.33784i
$741$		−	4.12454e13i		−	0.184622i
$742$		−	3.23998e14i		−	1.44053i
$743$	2.33449e14			1.03097			0.515487	−	0.856897i	$-0.327611\pi$
$743$	2.33449e14			1.03097			0.515487	+	0.856897i	$0.327611\pi$
$744$	6.76009e14			2.96543
$745$			4.81438e14i			2.09778i
$746$			6.95432e14i			3.00996i
$747$			6.26825e13i			0.269491i
$748$			8.20264e14i			3.50304i
$749$	1.55390e14			0.659196
$750$			1.25892e14i			0.530508i
$751$		−	8.63557e13i		−	0.361486i	−0.983530	−	0.180743i	$-0.942150\pi$
$751$		−	8.63557e13i		−	0.361486i	0.983530	−	0.180743i	$-0.0578502\pi$
$752$			3.19987e14i			1.33059i
$753$	−2.16878e14			−0.895861
$754$	−5.31945e13			−0.218278
$755$			3.72843e14i			1.51981i
$756$	−6.14600e13			−0.248876
$757$	−1.30911e14			−0.526621			−0.263311	−	0.964711i	$-0.584814\pi$
$757$	−1.30911e14			−0.526621			−0.263311	+	0.964711i	$0.584814\pi$
$758$			4.85623e14i			1.94068i
$759$	−1.61371e14			−0.640643
$760$		−	1.81664e15i		−	7.16474i
$761$	3.64240e14			1.42713			0.713566	−	0.700588i	$-0.247079\pi$
$761$	3.64240e14			1.42713			0.713566	+	0.700588i	$0.247079\pi$
$762$			5.30156e14i			2.06361i
$763$			4.94232e12i			0.0191121i
$764$			1.63316e14i			0.627425i
$765$	−9.91069e13			−0.378266
$766$			2.06339e13i			0.0782415i
$767$	−4.50585e13	+	1.99394e13i	−0.169746	+	0.0751163i
$768$	−9.46481e13			−0.354247
$769$		−	1.91581e14i		−	0.712393i	−0.934411	−	0.356197i	$-0.884073\pi$
$769$		−	1.91581e14i		−	0.712393i	0.934411	−	0.356197i	$-0.115927\pi$
$770$	7.68953e14			2.84084
$771$	1.88225e14			0.690885
$772$	8.55948e14			3.12148
$773$			2.83544e14i			1.02736i	0.857982	+	0.513680i	$0.171718\pi$
$773$			2.83544e14i			1.02736i	−0.857982	+	0.513680i	$0.828282\pi$
$774$	−1.84992e14			−0.665961
$775$		−	6.96767e14i		−	2.49218i
$776$	−1.38732e15			−4.93025
$777$		−	3.05271e13i		−	0.107791i
$778$			2.92325e14i			1.02558i
$779$	4.30536e14			1.50080

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:	\( k \)	\(=\)	\( 11 \)
Character orbit:	\([\chi]\)	\(=\)	177.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-59.5712i q^{2} +140.296 q^{3} -2524.73 q^{4} -4763.75 q^{5} -8357.61i q^{6} +8815.37 q^{7} +89400.5i q^{8} +19683.0 q^{9} +O(q^{10})\)	\(q-59.5712i q^{2} +140.296 q^{3} -2524.73 q^{4} -4763.75 q^{5} -8357.61i q^{6} +8815.37 q^{7} +89400.5i q^{8} +19683.0 q^{9} +283782. i q^{10} -307379. i q^{11} -354210. q^{12} -68920.8i q^{13} -525142. i q^{14} -668335. q^{15} +2.74037e6 q^{16} +1.05697e6 q^{17} -1.17254e6i q^{18} +4.26559e6 q^{19} +1.20272e7 q^{20} +1.23676e6 q^{21} -1.83109e7 q^{22} -3.74201e6i q^{23} +1.25425e7i q^{24} +1.29277e7 q^{25} -4.10570e6 q^{26} +2.76145e6 q^{27} -2.22564e7 q^{28} +1.29563e7 q^{29} +3.98136e7i q^{30} -5.38973e7i q^{31} -7.17012e7i q^{32} -4.31241e7i q^{33} -6.29652e7i q^{34} -4.19942e7 q^{35} -4.96943e7 q^{36} -2.46831e7i q^{37} -2.54107e8i q^{38} -9.66932e6i q^{39} -4.25881e8i q^{40} +1.00932e8 q^{41} -7.36754e7i q^{42} -1.57771e8i q^{43} +7.76050e8i q^{44} -9.37648e7 q^{45} -2.22916e8 q^{46} +1.16768e8i q^{47} +3.84463e8 q^{48} -2.04765e8 q^{49} -7.70117e8i q^{50} +1.48289e8 q^{51} +1.74007e8i q^{52} +6.16971e8 q^{53} -1.64503e8i q^{54} +1.46428e9i q^{55} +7.88098e8i q^{56} +5.98446e8 q^{57} -7.71821e8i q^{58} +(-2.89308e8 - 6.53772e8i) q^{59} +1.68737e9 q^{60} +9.65398e7i q^{61} -3.21073e9 q^{62} +1.73513e8 q^{63} -1.46519e9 q^{64} +3.28321e8i q^{65} -2.56896e9 q^{66} +1.44219e9i q^{67} -2.66857e9 q^{68} -5.24990e8i q^{69} +2.50165e9i q^{70} +1.91425e9 q^{71} +1.75967e9i q^{72} +1.00218e9i q^{73} -1.47040e9 q^{74} +1.81370e9 q^{75} -1.07695e10 q^{76} -2.70966e9i q^{77} -5.76013e8 q^{78} +4.68097e8 q^{79} -1.30544e10 q^{80} +3.87420e8 q^{81} -6.01266e9i q^{82} +3.18460e9i q^{83} -3.12249e9 q^{84} -5.03515e9 q^{85} -9.39859e9 q^{86} +1.81771e9 q^{87} +2.74798e10 q^{88} -7.27168e9i q^{89} +5.58569e9i q^{90} -6.07562e8i q^{91} +9.44757e9i q^{92} -7.56159e9i q^{93} +6.95600e9 q^{94} -2.03202e10 q^{95} -1.00594e10i q^{96} +1.55181e10i q^{97} +1.21981e10i q^{98} -6.05014e9i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 100q - 51200q^{4} + 18392q^{7} + 1968300q^{9} + O(q^{10}) \)	\( 100q - 51200q^{4} + 18392q^{7} + 1968300q^{9} + 620136q^{12} + 662904q^{15} + 27925160q^{16} - 2053136q^{17} - 5169828q^{19} - 1076324q^{20} - 1829224q^{22} + 215378180q^{25} - 22082700q^{26} - 102921320q^{28} - 112503588q^{29} - 76491392q^{35} - 1007769600q^{36} + 473464516q^{41} + 1215405588q^{46} - 1676272320q^{48} + 1975297276q^{49} + 733970808q^{51} + 3267506728q^{53} + 591502824q^{57} + 508142200q^{59} + 1264196808q^{60} - 6538206968q^{62} + 362009736q^{63} - 10324137972q^{64} - 2764346616q^{66} + 9997685952q^{68} + 14908523204q^{71} + 4863508712q^{74} + 1890481680q^{75} + 2044437240q^{76} - 758396196q^{78} - 3599839500q^{79} - 23217941144q^{80} + 38742048900q^{81} - 13094894808q^{84} + 23360564412q^{85} + 12186923752q^{86} + 7965322272q^{87} + 32415437996q^{88} - 22098322280q^{94} + 7834510028q^{95} + O(q^{100}) \)

Introduction

L-functions

Modular forms

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Fields

Representations

Groups

Properties

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