LMFDB - Embedded newform 177.11.c.a.58.20

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			58.20
Character	$\chi$	$=$	177.58
Dual form			177.11.c.a.58.81

$q$-expansion

$f(q)$	$=$	$q-44.5197i q^{2} +140.296 q^{3} -958.005 q^{4} -404.139 q^{5} -6245.94i q^{6} -20369.6 q^{7} -2938.10i q^{8} +19683.0 q^{9} +O(q^{10})$	\(q-44.5197i q^{2} +140.296 q^{3} -958.005 q^{4} -404.139 q^{5} -6245.94i q^{6} -20369.6 q^{7} -2938.10i q^{8} +19683.0 q^{9} +17992.1i q^{10} +217841. i q^{11} -134404. q^{12} +504544. i q^{13} +906847. i q^{14} -56699.1 q^{15} -1.11180e6 q^{16} +782310. q^{17} -876281. i q^{18} +1.77007e6 q^{19} +387167. q^{20} -2.85777e6 q^{21} +9.69824e6 q^{22} -1.01853e7i q^{23} -412204. i q^{24} -9.60230e6 q^{25} +2.24622e7 q^{26} +2.76145e6 q^{27} +1.95141e7 q^{28} +3.48342e6 q^{29} +2.52423e6i q^{30} -3.18716e7i q^{31} +4.64884e7i q^{32} +3.05623e7i q^{33} -3.48282e7i q^{34} +8.23213e6 q^{35} -1.88564e7 q^{36} +6.84669e7i q^{37} -7.88030e7i q^{38} +7.07856e7i q^{39} +1.18740e6i q^{40} +1.22130e8 q^{41} +1.27227e8i q^{42} +3.24896e7i q^{43} -2.08693e8i q^{44} -7.95466e6 q^{45} -4.53445e8 q^{46} -4.40737e8i q^{47} -1.55981e8 q^{48} +1.32444e8 q^{49} +4.27491e8i q^{50} +1.09755e8 q^{51} -4.83355e8i q^{52} +7.76026e8 q^{53} -1.22939e8i q^{54} -8.80382e7i q^{55} +5.98478e7i q^{56} +2.48334e8 q^{57} -1.55081e8i q^{58} +(-5.68858e8 + 4.33033e8i) q^{59} +5.43180e7 q^{60} -1.05370e9i q^{61} -1.41891e9 q^{62} -4.00934e8 q^{63} +9.31167e8 q^{64} -2.03906e8i q^{65} +1.36063e9 q^{66} +7.80318e8i q^{67} -7.49456e8 q^{68} -1.42895e9i q^{69} -3.66492e8i q^{70} +9.51934e8 q^{71} -5.78306e7i q^{72} -1.89948e9i q^{73} +3.04812e9 q^{74} -1.34716e9 q^{75} -1.69573e9 q^{76} -4.43734e9i q^{77} +3.15135e9 q^{78} -1.34494e9 q^{79} +4.49321e8 q^{80} +3.87420e8 q^{81} -5.43720e9i q^{82} -8.04031e8i q^{83} +2.73776e9 q^{84} -3.16162e8 q^{85} +1.44643e9 q^{86} +4.88711e8 q^{87} +6.40040e8 q^{88} -8.55267e9i q^{89} +3.54139e8i q^{90} -1.02773e10i q^{91} +9.75752e9i q^{92} -4.47146e9i q^{93} -1.96215e10 q^{94} -7.15354e8 q^{95} +6.52214e9i q^{96} -1.17842e10i q^{97} -5.89638e9i q^{98} +4.28777e9i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$		−	44.5197i		−	1.39124i	−0.718409	−	0.695620i	$-0.755130\pi$
$2$		−	44.5197i		−	1.39124i	0.718409	−	0.695620i	$-0.244870\pi$
$3$	140.296			0.577350
$3$	140.296			0.577350
$4$	−958.005			−0.935551
$5$	−404.139			−0.129324			−0.0646622	−	0.997907i	$-0.520597\pi$
$5$	−404.139			−0.129324			−0.0646622	+	0.997907i	$0.520597\pi$
$6$		−	6245.94i		−	0.803233i
$7$	−20369.6			−1.21197			−0.605985	−	0.795476i	$-0.707221\pi$
$7$	−20369.6			−1.21197			−0.605985	+	0.795476i	$0.707221\pi$
$8$		−	2938.10i		−	0.0896637i
$9$	19683.0			0.333333
$10$			17992.1i			0.179921i
$11$			217841.i			1.35262i	0.736615	+	0.676312i	$0.236423\pi$
$11$			217841.i			1.35262i	−0.736615	+	0.676312i	$0.763577\pi$
$12$	−134404.			−0.540141
$13$			504544.i			1.35888i	0.733729	+	0.679442i	$0.237778\pi$
$13$			504544.i			1.35888i	−0.733729	+	0.679442i	$0.762222\pi$
$14$			906847.i			1.68614i
$15$	−56699.1			−0.0746655
$16$	−1.11180e6			−1.06030
$17$	782310.			0.550978			0.275489	−	0.961304i	$-0.411160\pi$
$17$	782310.			0.550978			0.275489	+	0.961304i	$0.411160\pi$
$18$		−	876281.i		−	0.463747i
$19$	1.77007e6			0.714862			0.357431	−	0.933940i	$-0.383653\pi$
$19$	1.77007e6			0.714862			0.357431	+	0.933940i	$0.383653\pi$
$20$	387167.			0.120990
$21$	−2.85777e6			−0.699731
$22$	9.69824e6			1.88183
$23$		−	1.01853e7i		−	1.58246i	−0.611518	−	0.791230i	$-0.709441\pi$
$23$		−	1.01853e7i		−	1.58246i	0.611518	−	0.791230i	$-0.290559\pi$
$24$		−	412204.i		−	0.0517674i
$25$	−9.60230e6			−0.983275
$26$	2.24622e7			1.89053
$27$	2.76145e6			0.192450
$28$	1.95141e7			1.13386
$29$	3.48342e6			0.169831			0.0849154	−	0.996388i	$-0.472938\pi$
$29$	3.48342e6			0.169831			0.0849154	+	0.996388i	$0.472938\pi$
$30$			2.52423e6i			0.103878i
$31$		−	3.18716e7i		−	1.11326i	−0.830762	−	0.556628i	$-0.812095\pi$
$31$		−	3.18716e7i		−	1.11326i	0.830762	−	0.556628i	$-0.187905\pi$
$32$			4.64884e7i			1.38546i
$33$			3.05623e7i			0.780938i
$34$		−	3.48282e7i		−	0.766543i
$35$	8.23213e6			0.156737
$36$	−1.88564e7			−0.311850
$37$			6.84669e7i			0.987351i	0.869646	+	0.493676i	$0.164347\pi$
$37$			6.84669e7i			0.987351i	−0.869646	+	0.493676i	$0.835653\pi$
$38$		−	7.88030e7i		−	0.994545i
$39$			7.07856e7i			0.784552i
$40$			1.18740e6i			0.0115957i
$41$	1.22130e8			1.05415			0.527077	−	0.849818i	$-0.323288\pi$
$41$	1.22130e8			1.05415			0.527077	+	0.849818i	$0.323288\pi$
$42$			1.27227e8i			0.973494i
$43$			3.24896e7i			0.221005i	0.993876	+	0.110502i	$0.0352460\pi$
$43$			3.24896e7i			0.221005i	−0.993876	+	0.110502i	$0.964754\pi$
$44$		−	2.08693e8i		−	1.26545i
$45$	−7.95466e6			−0.0431081
$46$	−4.53445e8			−2.20158
$47$		−	4.40737e8i		−	1.92172i	−0.277034	−	0.960860i	$-0.589351\pi$
$47$		−	4.40737e8i		−	1.92172i	0.277034	−	0.960860i	$-0.410649\pi$
$48$	−1.55981e8			−0.612162
$49$	1.32444e8			0.468870
$50$			4.27491e8i			1.36797i
$51$	1.09755e8			0.318107
$52$		−	4.83355e8i		−	1.27131i
$53$	7.76026e8			1.85565			0.927827	−	0.373011i	$-0.121675\pi$
$53$	7.76026e8			1.85565			0.927827	+	0.373011i	$0.121675\pi$
$54$		−	1.22939e8i		−	0.267744i
$55$		−	8.80382e7i		−	0.174927i
$56$			5.98478e7i			0.108670i
$57$	2.48334e8			0.412726
$58$		−	1.55081e8i		−	0.236276i
$59$	−5.68858e8	+	4.33033e8i	−0.795689	+	0.605705i
$59$	−5.68858e8	+	4.33033e8i	−0.795689	+	0.605705i
$60$	5.43180e7			0.0698534
$61$		−	1.05370e9i		−	1.24758i	−0.781593	−	0.623788i	$-0.785593\pi$
$61$		−	1.05370e9i		−	1.24758i	0.781593	−	0.623788i	$-0.214407\pi$
$62$	−1.41891e9			−1.54881
$63$	−4.00934e8			−0.403990
$64$	9.31167e8			0.867217
$65$		−	2.03906e8i		−	0.175737i
$66$	1.36063e9			1.08647
$67$			7.80318e8i			0.577960i	0.957335	+	0.288980i	$0.0933161\pi$
$67$			7.80318e8i			0.577960i	−0.957335	+	0.288980i	$0.906684\pi$
$68$	−7.49456e8			−0.515468
$69$		−	1.42895e9i		−	0.913634i
$70$		−	3.66492e8i		−	0.218059i
$71$	9.51934e8			0.527612			0.263806	−	0.964576i	$-0.415022\pi$
$71$	9.51934e8			0.527612			0.263806	+	0.964576i	$0.415022\pi$
$72$		−	5.78306e7i		−	0.0298879i
$73$		−	1.89948e9i		−	0.916266i	−0.888884	−	0.458133i	$-0.848518\pi$
$73$		−	1.89948e9i		−	0.916266i	0.888884	−	0.458133i	$-0.151482\pi$
$74$	3.04812e9			1.37364
$75$	−1.34716e9			−0.567694
$76$	−1.69573e9			−0.668790
$77$		−	4.43734e9i		−	1.63934i
$78$	3.15135e9			1.09150
$79$	−1.34494e9			−0.437085			−0.218542	−	0.975827i	$-0.570130\pi$
$79$	−1.34494e9			−0.437085			−0.218542	+	0.975827i	$0.570130\pi$
$80$	4.49321e8			0.137122
$81$	3.87420e8			0.111111
$82$		−	5.43720e9i		−	1.46658i
$83$		−	8.04031e8i		−	0.204119i	−0.994778	−	0.102059i	$-0.967457\pi$
$83$		−	8.04031e8i		−	0.204119i	0.994778	−	0.102059i	$-0.0325431\pi$
$84$	2.73776e9			0.654634
$85$	−3.16162e8			−0.0712549
$86$	1.44643e9			0.307471
$87$	4.88711e8			0.0980519
$88$	6.40040e8			0.121281
$89$		−	8.55267e9i		−	1.53162i	−0.643065	−	0.765811i	$-0.722338\pi$
$89$		−	8.55267e9i		−	1.53162i	0.643065	−	0.765811i	$-0.277662\pi$
$90$			3.54139e8i			0.0599738i
$91$		−	1.02773e10i		−	1.64693i
$92$			9.75752e9i			1.48047i
$93$		−	4.47146e9i		−	0.642739i
$94$	−1.96215e10			−2.67358
$95$	−7.15354e8			−0.0924491
$96$			6.52214e9i			0.799897i
$97$		−	1.17842e10i		−	1.37227i	−0.727474	−	0.686135i	$-0.759306\pi$
$97$		−	1.17842e10i		−	1.37227i	0.727474	−	0.686135i	$-0.240694\pi$
$98$		−	5.89638e9i		−	0.652311i
$99$			4.28777e9i			0.450875i
$100$	9.19904e9			0.919904
$101$			1.28240e10i			1.22016i	0.792339	+	0.610082i	$0.208863\pi$
$101$			1.28240e10i			1.22016i	−0.792339	+	0.610082i	$0.791137\pi$
$102$		−	4.88626e9i		−	0.442564i
$103$			1.49310e10i			1.28796i	0.765040	+	0.643982i	$0.222719\pi$
$103$			1.49310e10i			1.28796i	−0.765040	+	0.643982i	$0.777281\pi$
$104$	1.48240e9			0.121843
$105$	1.15494e9			0.0904923
$106$		−	3.45485e10i		−	2.58166i
$107$	2.44078e10			1.74025			0.870123	−	0.492835i	$-0.164039\pi$
$107$	2.44078e10			1.74025			0.870123	+	0.492835i	$0.164039\pi$
$108$	−2.64548e9			−0.180047
$109$			5.84380e9i			0.379807i	0.981803	+	0.189904i	$0.0608175\pi$
$109$			5.84380e9i			0.379807i	−0.981803	+	0.189904i	$0.939182\pi$
$110$	−3.91943e9			−0.243366
$111$			9.60563e9i			0.570048i
$112$	2.26469e10			1.28505
$113$		−	1.61853e10i		−	0.878474i	−0.898371	−	0.439237i	$-0.855249\pi$
$113$		−	1.61853e10i		−	0.878474i	0.898371	−	0.439237i	$-0.144751\pi$
$114$		−	1.10557e10i		−	0.574201i
$115$			4.11626e9i			0.204651i
$116$	−3.33714e9			−0.158885
$117$			9.93094e9i			0.452961i
$118$	1.92785e10	+	2.53254e10i	0.842682	+	1.10700i
$119$	−1.59353e10			−0.667768
$120$			1.66588e8i			0.00669478i
$121$	−2.15175e10			−0.829592
$122$	−4.69104e10			−1.73568
$123$	1.71344e10			0.608616
$124$			3.05331e10i			1.04151i
$125$	7.82733e9			0.256486
$126$			1.78495e10i			0.562047i
$127$	3.67473e10			1.11226			0.556130	−	0.831095i	$-0.312286\pi$
$127$	3.67473e10			1.11226			0.556130	+	0.831095i	$0.312286\pi$
$128$			6.14885e9i			0.178955i
$129$			4.55816e9i			0.127597i
$130$	−9.07783e9			−0.244492
$131$		−	8.52392e9i		−	0.220944i	−0.993879	−	0.110472i	$-0.964764\pi$
$131$		−	8.52392e9i		−	0.220944i	0.993879	−	0.110472i	$-0.0352363\pi$
$132$		−	2.92788e10i		−	0.730607i
$133$	−3.60555e10			−0.866391
$134$	3.47395e10			0.804082
$135$	−1.11601e9			−0.0248885
$136$		−	2.29850e9i		−	0.0494027i
$137$	2.32685e10			0.482132			0.241066	−	0.970509i	$-0.422503\pi$
$137$	2.32685e10			0.482132			0.241066	+	0.970509i	$0.422503\pi$
$138$	−6.36165e10			−1.27108
$139$	−7.85021e10			−1.51289			−0.756445	−	0.654057i	$-0.773066\pi$
$139$	−7.85021e10			−1.51289			−0.756445	+	0.654057i	$0.773066\pi$
$140$	−7.88642e9			−0.146636
$141$		−	6.18337e10i		−	1.10951i
$142$		−	4.23798e10i		−	0.734036i
$143$	−1.09911e11			−1.83806
$144$	−2.18836e10			−0.353432
$145$	−1.40779e9			−0.0219633
$146$	−8.45645e10			−1.27475
$147$	1.85814e10			0.270702
$148$		−	6.55916e10i		−	0.923718i
$149$		−	4.42223e10i		−	0.602157i	−0.953599	−	0.301079i	$-0.902653\pi$
$149$		−	4.42223e10i		−	0.602157i	0.953599	−	0.301079i	$-0.0973467\pi$
$150$			5.99754e10i			0.789799i
$151$		−	4.09572e10i		−	0.521729i	−0.965375	−	0.260865i	$-0.915992\pi$
$151$		−	4.09572e10i		−	0.521729i	0.965375	−	0.260865i	$-0.0840077\pi$
$152$		−	5.20064e9i		−	0.0640972i
$153$	1.53982e10			0.183659
$154$	−1.97549e11			−2.28072
$155$			1.28805e10i			0.143971i
$156$		−	6.78129e10i		−	0.733988i
$157$		−	7.64414e10i		−	0.801365i	−0.916217	−	0.400682i	$-0.868773\pi$
$157$		−	7.64414e10i		−	0.801365i	0.916217	−	0.400682i	$-0.131227\pi$
$158$			5.98761e10i			0.608090i
$159$	1.08873e11			1.07136
$160$		−	1.87878e10i		−	0.179174i
$161$			2.07469e11i			1.91789i
$162$		−	1.72478e10i		−	0.154582i
$163$	−1.37094e11			−1.19146			−0.595731	−	0.803184i	$-0.703137\pi$
$163$	−1.37094e11			−1.19146			−0.595731	+	0.803184i	$0.703137\pi$
$164$	−1.17001e11			−0.986215
$165$		−	1.23514e10i		−	0.100994i
$166$	−3.57952e10			−0.283978
$167$	−1.91516e11			−1.47443			−0.737213	−	0.675660i	$-0.763859\pi$
$167$	−1.91516e11			−1.47443			−0.737213	+	0.675660i	$0.763859\pi$
$168$			8.39642e9i			0.0627405i
$169$	−1.16706e11			−0.846565
$170$			1.40754e10i			0.0991327i
$171$	3.48403e10			0.238287
$172$		−	3.11252e10i		−	0.206761i
$173$		−	4.21931e10i		−	0.272277i	−0.990690	−	0.136139i	$-0.956531\pi$
$173$		−	4.21931e10i		−	0.272277i	0.990690	−	0.136139i	$-0.0434692\pi$
$174$		−	2.17573e10i		−	0.136414i
$175$	1.95595e11			1.19170
$176$		−	2.42196e11i		−	1.43418i
$177$	−7.98085e10	+	6.07529e10i	−0.459391	+	0.349704i
$178$	−3.80763e11			−2.13086
$179$			2.23288e11i			1.21507i	0.794294	+	0.607534i	$0.207841\pi$
$179$			2.23288e11i			1.21507i	−0.794294	+	0.607534i	$0.792159\pi$
$180$	7.62060e9			0.0403299
$181$	1.43418e11			0.738265			0.369132	−	0.929377i	$-0.379655\pi$
$181$	1.43418e11			0.738265			0.369132	+	0.929377i	$0.379655\pi$
$182$	−4.57544e11			−2.29127
$183$		−	1.47830e11i		−	0.720289i
$184$	−2.99253e10			−0.141889
$185$		−	2.76701e10i		−	0.127689i
$186$	−1.99068e11			−0.894205
$187$			1.70420e11i			0.745266i
$188$			4.22228e11i			1.79787i
$189$	−5.62495e10			−0.233244
$190$			3.18473e10i			0.128619i
$191$		−	2.93853e11i		−	1.15602i	−0.816031	−	0.578008i	$-0.803830\pi$
$191$		−	2.93853e11i		−	1.15602i	0.816031	−	0.578008i	$-0.196170\pi$
$192$	1.30639e11			0.500688
$193$	1.32219e11			0.493751			0.246876	−	0.969047i	$-0.420596\pi$
$193$	1.32219e11			0.493751			0.246876	+	0.969047i	$0.420596\pi$
$194$	−5.24627e11			−1.90916
$195$		−	2.86072e10i		−	0.101462i
$196$	−1.26882e11			−0.438652
$197$	4.33826e11			1.46213			0.731063	−	0.682310i	$-0.239024\pi$
$197$	4.33826e11			1.46213			0.731063	+	0.682310i	$0.239024\pi$
$198$	1.90890e11			0.627275
$199$	5.60523e11			1.79609			0.898045	−	0.439904i	$-0.144988\pi$
$199$	5.60523e11			1.79609			0.898045	+	0.439904i	$0.144988\pi$
$200$			2.82125e10i			0.0881641i
$201$			1.09476e11i			0.333685i
$202$	5.70922e11			1.69754
$203$	−7.09559e10			−0.205830
$204$	−1.05146e11			−0.297606
$205$	−4.93576e10			−0.136328
$206$	6.64725e11			1.79187
$207$		−	2.00476e11i		−	0.527487i
$208$		−	5.60952e11i		−	1.44082i
$209$			3.85594e11i			0.966939i
$210$		−	5.14174e10i		−	0.125897i
$211$		−	4.46975e11i		−	1.06874i	−0.845252	−	0.534368i	$-0.820550\pi$
$211$		−	4.46975e11i		−	1.06874i	0.845252	−	0.534368i	$-0.179450\pi$
$212$	−7.43436e11			−1.73606
$213$	1.33553e11			0.304617
$214$		−	1.08663e12i		−	2.42110i
$215$		−	1.31303e10i		−	0.0285813i
$216$		−	8.11341e9i		−	0.0172558i
$217$			6.49211e11i			1.34923i
$218$	2.60164e11			0.528403
$219$		−	2.66490e11i		−	0.529006i
$220$			8.43410e10i			0.163653i
$221$			3.94710e11i			0.748715i
$222$	4.27640e11			0.793074
$223$	−1.83058e11			−0.331944			−0.165972	−	0.986130i	$-0.553076\pi$
$223$	−1.83058e11			−0.331944			−0.165972	+	0.986130i	$0.553076\pi$
$224$		−	9.46949e11i		−	1.67914i
$225$	−1.89002e11			−0.327758
$226$	−7.20566e11			−1.22217
$227$			5.28624e11i			0.877037i	0.898722	+	0.438518i	$0.144497\pi$
$227$			5.28624e11i			0.877037i	−0.898722	+	0.438518i	$0.855503\pi$
$228$	−2.37905e11			−0.386126
$229$		−	1.02826e12i		−	1.63277i	−0.577509	−	0.816384i	$-0.695975\pi$
$229$		−	1.02826e12i		−	1.63277i	0.577509	−	0.816384i	$-0.304025\pi$
$230$	1.83255e11			0.284718
$231$		−	6.22541e11i		−	0.946473i
$232$		−	1.02347e10i		−	0.0152277i
$233$			5.29944e10i			0.0771703i	0.999255	+	0.0385852i	$0.0122851\pi$
$233$			5.29944e10i			0.0771703i	−0.999255	+	0.0385852i	$0.987715\pi$
$234$	4.42123e11			0.630178
$235$			1.78119e11i			0.248525i
$236$	5.44968e11	−	4.14848e11i	0.744408	−	0.566668i
$237$	−1.88689e11			−0.252351
$238$			7.09436e11i			0.929027i
$239$	−7.56160e11			−0.969670			−0.484835	−	0.874606i	$-0.661120\pi$
$239$	−7.56160e11			−0.969670			−0.484835	+	0.874606i	$0.661120\pi$
$240$	6.30381e10			0.0791674
$241$	1.24623e12			1.53290			0.766451	−	0.642302i	$-0.222021\pi$
$241$	1.24623e12			1.53290			0.766451	+	0.642302i	$0.222021\pi$
$242$			9.57952e11i			1.15416i
$243$	5.43536e10			0.0641500
$244$			1.00945e12i			1.16717i
$245$	−5.35258e10			−0.0606363
$246$		−	7.62819e11i		−	0.846732i
$247$			8.93078e11i			0.971414i
$248$	−9.36419e10			−0.0998187
$249$		−	1.12802e11i		−	0.117848i
$250$		−	3.48470e11i		−	0.356834i
$251$	−6.99905e11			−0.702539			−0.351270	−	0.936274i	$-0.614250\pi$
$251$	−6.99905e11			−0.702539			−0.351270	+	0.936274i	$0.614250\pi$
$252$	3.84097e11			0.377953
$253$	2.21877e12			2.14047
$254$		−	1.63598e12i		−	1.54742i
$255$	−4.43563e10			−0.0411390
$256$	1.22726e12			1.11619
$257$	−1.71467e12			−1.52938			−0.764688	−	0.644401i	$-0.777107\pi$
$257$	−1.71467e12			−1.52938			−0.764688	+	0.644401i	$0.777107\pi$
$258$	2.02928e11			0.177519
$259$		−	1.39464e12i		−	1.19664i
$260$			1.95343e11i			0.164411i
$261$	6.85643e10			0.0566103
$262$	−3.79483e11			−0.307387
$263$	1.23966e11			0.0985200			0.0492600	−	0.998786i	$-0.484314\pi$
$263$	1.23966e11			0.0985200			0.0492600	+	0.998786i	$0.484314\pi$
$264$	8.97951e10			0.0700218
$265$	−3.13622e11			−0.239981
$266$			1.60518e12i			1.20536i
$267$		−	1.19991e12i		−	0.884283i
$268$		−	7.47549e11i		−	0.540711i
$269$		−	2.34489e10i		−	0.0166479i	−0.999965	−	0.00832397i	$-0.997350\pi$
$269$		−	2.34489e10i		−	0.0166479i	0.999965	−	0.00832397i	$-0.00264963\pi$
$270$			4.96844e10i			0.0346259i
$271$	1.77300e12			1.21300			0.606501	−	0.795083i	$-0.292573\pi$
$271$	1.77300e12			1.21300			0.606501	+	0.795083i	$0.292573\pi$
$272$	−8.69772e11			−0.584199
$273$		−	1.44187e12i		−	0.950853i
$274$		−	1.03591e12i		−	0.670761i
$275$		−	2.09178e12i		−	1.33000i
$276$			1.36894e12i			0.854751i
$277$	2.94048e12			1.80310			0.901549	−	0.432677i	$-0.142431\pi$
$277$	2.94048e12			1.80310			0.901549	+	0.432677i	$0.142431\pi$
$278$			3.49489e12i			2.10480i
$279$		−	6.27329e11i		−	0.371086i
$280$		−	2.41868e10i		−	0.0140536i
$281$	7.51601e11			0.428999			0.214499	−	0.976724i	$-0.431188\pi$
$281$	7.51601e11			0.428999			0.214499	+	0.976724i	$0.431188\pi$
$282$	−2.75282e12			−1.54359
$283$			8.77176e11i			0.483231i	0.970372	+	0.241615i	$0.0776772\pi$
$283$			8.77176e11i			0.483231i	−0.970372	+	0.241615i	$0.922323\pi$
$284$	−9.11957e11			−0.493608
$285$	−1.00361e11			−0.0533755
$286$			4.89319e12i			2.55718i
$287$	−2.48774e12			−1.27760
$288$			9.15031e11i			0.461821i
$289$	−1.40399e12			−0.696423
$290$			6.26743e10i			0.0305562i
$291$		−	1.65327e12i		−	0.792281i
$292$			1.81971e12i			0.857214i
$293$	4.81109e11			0.222795			0.111398	−	0.993776i	$-0.464467\pi$
$293$	4.81109e11			0.222795			0.111398	+	0.993776i	$0.464467\pi$
$294$		−	8.27239e11i		−	0.376612i
$295$	2.29897e11	−	1.75006e11i	0.102902	−	0.0783325i
$296$	2.01162e11			0.0885296
$297$			6.01558e11i			0.260313i
$298$	−1.96876e12			−0.837746
$299$	5.13891e12			2.15038
$300$	1.29059e12			0.531107
$301$		−	6.61799e11i		−	0.267851i
$302$	−1.82340e12			−0.725851
$303$			1.79916e12i			0.704461i
$304$	−1.96796e12			−0.757965
$305$			4.25841e11i			0.161342i
$306$		−	6.85524e11i		−	0.255514i
$307$	2.86946e12			1.05222			0.526112	−	0.850415i	$-0.323649\pi$
$307$	2.86946e12			1.05222			0.526112	+	0.850415i	$0.323649\pi$
$308$			4.25099e12i			1.53369i
$309$			2.09477e12i			0.743607i
$310$	5.73438e11			0.200299
$311$	−3.23107e12			−1.11057			−0.555284	−	0.831661i	$-0.687390\pi$
$311$	−3.23107e12			−1.11057			−0.555284	+	0.831661i	$0.687390\pi$
$312$	2.07975e11			0.0703458
$313$		−	4.31323e12i		−	1.43576i	−0.696168	−	0.717879i	$-0.745113\pi$
$313$		−	4.31323e12i		−	1.43576i	0.696168	−	0.717879i	$-0.254887\pi$
$314$	−3.40315e12			−1.11489
$315$	1.62033e11			0.0522457
$316$	1.28845e12			0.408915
$317$	−3.32947e12			−1.04011			−0.520054	−	0.854133i	$-0.674088\pi$
$317$	−3.32947e12			−1.04011			−0.520054	+	0.854133i	$0.674088\pi$
$318$		−	4.84701e12i		−	1.49052i
$319$			7.58834e11i			0.229717i
$320$	−3.76321e11			−0.112152
$321$	3.42433e12			1.00473
$322$	9.23647e12			2.66825
$323$	1.38474e12			0.393873
$324$	−3.71151e11			−0.103950
$325$		−	4.84478e12i		−	1.33616i
$326$			6.10338e12i			1.65761i
$327$			8.19863e11i			0.219282i
$328$		−	3.58831e11i		−	0.0945193i
$329$			8.97762e12i			2.32907i
$330$	−5.49881e11			−0.140507
$331$	6.08638e11			0.153186			0.0765929	−	0.997062i	$-0.475596\pi$
$331$	6.08638e11			0.153186			0.0765929	+	0.997062i	$0.475596\pi$
$332$			7.70266e11i			0.190963i
$333$			1.34763e12i			0.329117i
$334$			8.52625e12i			2.05128i
$335$		−	3.15357e11i		−	0.0747443i
$336$	3.17727e12			0.741921
$337$			3.59578e12i			0.827264i	0.910444	+	0.413632i	$0.135740\pi$
$337$			3.59578e12i			0.827264i	−0.910444	+	0.413632i	$0.864260\pi$
$338$			5.19572e12i			1.17778i
$339$		−	2.27074e12i		−	0.507187i
$340$	3.02884e11			0.0666626
$341$	6.94295e12			1.50582
$342$		−	1.55108e12i		−	0.331515i
$343$	3.05607e12			0.643713
$344$	9.54577e10			0.0198161
$345$			5.77495e11i			0.118155i
$346$	−1.87843e12			−0.378803
$347$		−	9.14707e11i		−	0.181817i	−0.995859	−	0.0909085i	$-0.971023\pi$
$347$		−	9.14707e11i		−	0.181817i	0.995859	−	0.0909085i	$-0.0289771\pi$
$348$	−4.68187e11			−0.0917325
$349$			9.57475e12i			1.84927i	0.380854	+	0.924635i	$0.375630\pi$
$349$			9.57475e12i			1.84927i	−0.380854	+	0.924635i	$0.624370\pi$
$350$		−	8.70782e12i		−	1.65794i
$351$			1.39327e12i			0.261517i
$352$	−1.01271e13			−1.87401
$353$		−	6.77934e12i		−	1.23684i	−0.785847	−	0.618420i	$-0.787773\pi$
$353$		−	6.77934e12i		−	1.23684i	0.785847	−	0.618420i	$-0.212227\pi$
$354$	2.70470e12	+	3.55305e12i	0.486523	+	0.639124i
$355$	−3.84713e11			−0.0682331
$356$			8.19350e12i			1.43291i
$357$	−2.23566e12			−0.385536
$358$	9.94072e12			1.69045
$359$	1.09240e13			1.83193			0.915963	−	0.401263i	$-0.131429\pi$
$359$	1.09240e13			1.83193			0.915963	+	0.401263i	$0.131429\pi$
$360$			2.33716e10i			0.00386523i
$361$	−2.99792e12			−0.488972
$362$		−	6.38495e12i		−	1.02710i
$363$	−3.01882e12			−0.478965
$364$			9.84574e12i			1.54078i
$365$			7.67655e11i			0.118496i
$366$	−6.58134e12			−1.00210
$367$		−	5.47292e12i		−	0.822032i	−0.911628	−	0.411016i	$-0.865174\pi$
$367$		−	5.47292e12i		−	0.822032i	0.911628	−	0.411016i	$-0.134826\pi$
$368$			1.13240e13i			1.67787i
$369$	2.40389e12			0.351385
$370$	−1.23187e12			−0.177646
$371$	−1.58073e13			−2.24900
$372$			4.28368e12i			0.601315i
$373$	−4.26707e12			−0.590997			−0.295499	−	0.955343i	$-0.595486\pi$
$373$	−4.26707e12			−0.590997			−0.295499	+	0.955343i	$0.595486\pi$
$374$	7.58703e12			1.03684
$375$	1.09814e12			0.148082
$376$	−1.29493e12			−0.172309
$377$			1.75754e12i			0.230780i
$378$			2.50421e12i			0.324498i
$379$	−7.08379e12			−0.905877			−0.452939	−	0.891542i	$-0.649624\pi$
$379$	−7.08379e12			−0.905877			−0.452939	+	0.891542i	$0.649624\pi$
$380$	6.85312e11			0.0864909
$381$	5.15550e12			0.642164
$382$	−1.30823e13			−1.60830
$383$	1.13036e13			1.37158			0.685791	−	0.727799i	$-0.259456\pi$
$383$	1.13036e13			1.37158			0.685791	+	0.727799i	$0.259456\pi$
$384$			8.62659e11i			0.103320i
$385$			1.79330e12i			0.212007i
$386$		−	5.88636e12i		−	0.686927i
$387$			6.39493e11i			0.0736683i
$388$			1.12893e13i			1.28383i
$389$	1.56295e13			1.75467			0.877337	−	0.479874i	$-0.159318\pi$
$389$	1.56295e13			1.75467			0.877337	+	0.479874i	$0.159318\pi$
$390$	−1.27358e12			−0.141158
$391$		−	7.96803e12i		−	0.871901i
$392$		−	3.89134e11i		−	0.0420406i
$393$		−	1.19587e12i		−	0.127562i
$394$		−	1.93138e13i		−	2.03417i
$395$	5.43540e11			0.0565258
$396$		−	4.10771e12i		−	0.421816i
$397$		−	3.48262e12i		−	0.353145i	−0.984288	−	0.176573i	$-0.943499\pi$
$397$		−	3.48262e12i		−	0.353145i	0.984288	−	0.176573i	$-0.0565011\pi$
$398$		−	2.49543e13i		−	2.49879i
$399$	−5.05845e12			−0.500211
$400$	1.06758e13			1.04256
$401$		−	6.73309e12i		−	0.649370i	−0.945822	−	0.324685i	$-0.894742\pi$
$401$		−	6.73309e12i		−	0.649370i	0.945822	−	0.324685i	$-0.105258\pi$
$402$	4.87382e12			0.464237
$403$	1.60806e13			1.51279
$404$		−	1.22855e13i		−	1.14153i
$405$	−1.56572e11			−0.0143694
$406$			3.15893e12i			0.286359i
$407$	−1.49149e13			−1.33552
$408$		−	3.22471e11i		−	0.0285227i
$409$			5.82257e12i			0.508742i	0.967107	+	0.254371i	$0.0818685\pi$
$409$			5.82257e12i			0.508742i	−0.967107	+	0.254371i	$0.918131\pi$
$410$			2.19739e12i			0.189665i
$411$	3.26448e12			0.278359
$412$		−	1.43040e13i		−	1.20496i
$413$	1.15874e13	−	8.82070e12i	0.964351	−	0.734096i
$414$	−8.92515e12			−0.733861
$415$			3.24940e11i			0.0263975i
$416$	−2.34554e13			−1.88268
$417$	−1.10135e13			−0.873468
$418$	1.71666e13			1.34525
$419$		−	1.84198e13i		−	1.42632i	−0.701004	−	0.713158i	$-0.747264\pi$
$419$		−	1.84198e13i		−	1.42632i	0.701004	−	0.713158i	$-0.252736\pi$
$420$	−1.10643e12			−0.0846602
$421$		−	1.05826e13i		−	0.800170i	−0.916478	−	0.400085i	$-0.868981\pi$
$421$		−	1.05826e13i		−	0.800170i	0.916478	−	0.400085i	$-0.131019\pi$
$422$	−1.98992e13			−1.48687
$423$		−	8.67503e12i		−	0.640573i
$424$		−	2.28004e12i		−	0.166385i
$425$	−7.51197e12			−0.541763
$426$		−	5.94572e12i		−	0.423796i
$427$			2.14634e13i			1.51203i
$428$	−2.33828e13			−1.62809
$429$	−1.54200e13			−1.06120
$430$	−5.84557e11			−0.0397635
$431$			4.63964e12i			0.311959i	0.987760	+	0.155980i	$0.0498535\pi$
$431$			4.63964e12i			0.311959i	−0.987760	+	0.155980i	$0.950147\pi$
$432$	−3.07018e12			−0.204054
$433$	2.71254e13			1.78212			0.891058	−	0.453889i	$-0.149964\pi$
$433$	2.71254e13			1.78212			0.891058	+	0.453889i	$0.149964\pi$
$434$	2.89027e13			1.87711
$435$	−1.97507e11			−0.0126805
$436$		−	5.59839e12i		−	0.355329i
$437$		−	1.80286e13i		−	1.13124i
$438$	−1.18641e13			−0.735975
$439$	9.17569e11			0.0562751			0.0281375	−	0.999604i	$-0.491042\pi$
$439$	9.17569e11			0.0562751			0.0281375	+	0.999604i	$0.491042\pi$
$440$	−2.58665e11			−0.0156846
$441$	2.60690e12			0.156290
$442$	1.75724e13			1.04164
$443$		−	1.91563e13i		−	1.12278i	−0.827552	−	0.561389i	$-0.810267\pi$
$443$		−	1.91563e13i		−	1.12278i	0.827552	−	0.561389i	$-0.189733\pi$
$444$		−	9.20224e12i		−	0.533309i
$445$			3.45647e12i			0.198076i
$446$			8.14969e12i			0.461814i
$447$		−	6.20422e12i		−	0.347656i
$448$	−1.89675e13			−1.05104
$449$	2.20311e13			1.20727			0.603634	−	0.797262i	$-0.293719\pi$
$449$	2.20311e13			1.20727			0.603634	+	0.797262i	$0.293719\pi$
$450$			8.41431e12i			0.455991i
$451$			2.66050e13i			1.42587i
$452$			1.55056e13i			0.821858i
$453$		−	5.74613e12i		−	0.301221i
$454$	2.35342e13			1.22017
$455$			4.15347e12i			0.212988i
$456$		−	7.29630e11i		−	0.0370065i
$457$			2.51973e13i			1.26408i	0.774938	+	0.632038i	$0.217781\pi$
$457$			2.51973e13i			1.26408i	−0.774938	+	0.632038i	$0.782219\pi$
$458$	−4.57777e13			−2.27157
$459$	2.16031e12			0.106036
$460$		−	3.94339e12i		−	0.191461i
$461$	2.32158e13			1.11501			0.557504	−	0.830174i	$-0.311759\pi$
$461$	2.32158e13			1.11501			0.557504	+	0.830174i	$0.311759\pi$
$462$	−2.77154e13			−1.31677
$463$		−	1.16250e13i		−	0.546371i	−0.961961	−	0.273186i	$-0.911923\pi$
$463$		−	1.16250e13i		−	0.546371i	0.961961	−	0.273186i	$-0.0880773\pi$
$464$	−3.87287e12			−0.180071
$465$			1.80709e12i			0.0831218i
$466$	2.35929e12			0.107362
$467$		−	3.39054e13i		−	1.52645i	−0.646130	−	0.763227i	$-0.723614\pi$
$467$		−	3.39054e13i		−	1.52645i	0.646130	−	0.763227i	$-0.276386\pi$
$468$		−	9.51388e12i		−	0.423768i
$469$		−	1.58948e13i		−	0.700470i
$470$	7.92980e12			0.345759
$471$		−	1.07244e13i		−	0.462668i
$472$	1.27230e12	+	1.67136e12i	0.0543098	+	0.0713444i
$473$	−7.07758e12			−0.298937
$474$			8.40039e12i			0.351081i
$475$	−1.69967e13			−0.702906
$476$	1.52661e13			0.624732
$477$	1.52745e13			0.618551
$478$			3.36640e13i			1.34905i
$479$	−1.20587e13			−0.478213			−0.239106	−	0.970993i	$-0.576854\pi$
$479$	−1.20587e13			−0.478213			−0.239106	+	0.970993i	$0.576854\pi$
$480$		−	2.63585e12i		−	0.103446i
$481$	−3.45445e13			−1.34170
$482$		−	5.54820e13i		−	2.13264i
$483$			2.91071e13i			1.10730i
$484$	2.06138e13			0.776126
$485$			4.76243e12i			0.177468i
$486$		−	2.41981e12i		−	0.0892481i
$487$	−2.37666e13			−0.867606			−0.433803	−	0.901008i	$-0.642829\pi$
$487$	−2.37666e13			−0.867606			−0.433803	+	0.901008i	$0.642829\pi$
$488$	−3.09587e12			−0.111862
$489$	−1.92337e13			−0.687891
$490$			2.38295e12i			0.0843598i
$491$	−6.53918e12			−0.229148			−0.114574	−	0.993415i	$-0.536550\pi$
$491$	−6.53918e12			−0.229148			−0.114574	+	0.993415i	$0.536550\pi$
$492$	−1.64148e13			−0.569392
$493$	2.72512e12			0.0935730
$494$	3.97596e13			1.35147
$495$		−	1.73286e12i		−	0.0583091i
$496$			3.54348e13i			1.18038i
$497$	−1.93905e13			−0.639450
$498$	−5.02193e12			−0.163955
$499$	4.86138e13			1.57129			0.785646	−	0.618676i	$-0.212331\pi$
$499$	4.86138e13			1.57129			0.785646	+	0.618676i	$0.212331\pi$
$500$	−7.49862e12			−0.239956
$501$	−2.68690e13			−0.851261
$502$			3.11596e13i			0.977401i
$503$			1.05896e13i			0.328883i	0.986387	+	0.164441i	$0.0525821\pi$
$503$			1.05896e13i			0.328883i	−0.986387	+	0.164441i	$0.947418\pi$
$504$			1.17799e12i			0.0362232i
$505$		−	5.18269e12i		−	0.157797i
$506$		−	9.87790e13i		−	2.97791i
$507$	−1.63734e13			−0.488764
$508$	−3.52040e13			−1.04058
$509$			2.81554e13i			0.824087i	0.911164	+	0.412044i	$0.135185\pi$
$509$			2.81554e13i			0.824087i	−0.911164	+	0.412044i	$0.864815\pi$
$510$			1.97473e12i			0.0572343i
$511$			3.86917e13i			1.11049i
$512$		−	4.83408e13i		−	1.37393i
$513$	4.88795e12			0.137575
$514$			7.63365e13i			2.12773i
$515$		−	6.03421e12i		−	0.166565i
$516$		−	4.36674e12i		−	0.119374i
$517$	9.60108e13			2.59936
$518$	−6.20890e13			−1.66481
$519$		−	5.91953e12i		−	0.157199i
$520$	−5.99096e11			−0.0157572
$521$	−3.10161e13			−0.807975			−0.403987	−	0.914765i	$-0.632376\pi$
$521$	−3.10161e13			−0.807975			−0.403987	+	0.914765i	$0.632376\pi$
$522$		−	3.05246e12i		−	0.0787585i
$523$	−2.84566e13			−0.727235			−0.363617	−	0.931548i	$-0.618458\pi$
$523$	−2.84566e13			−0.727235			−0.363617	+	0.931548i	$0.618458\pi$
$524$			8.16596e12i			0.206705i
$525$	2.74412e13			0.688028
$526$		−	5.51894e12i		−	0.137065i
$527$		−	2.49335e13i		−	0.613380i
$528$		−	3.39792e13i		−	0.828025i
$529$	−6.23130e13			−1.50418
$530$			1.39624e13i			0.333872i
$531$	−1.11968e13	+	8.52339e12i	−0.265230	+	0.201902i
$532$	3.45414e13			0.810553
$533$			6.16201e13i			1.43247i
$534$	−5.34195e13			−1.23025
$535$	−9.86416e12			−0.225056
$536$	2.29265e12			0.0518220
$537$			3.13264e13i			0.701519i
$538$	−1.04394e12			−0.0231613
$539$			2.88518e13i			0.634205i
$540$	1.06914e12			0.0232845
$541$			4.88003e12i			0.105302i	0.998613	+	0.0526509i	$0.0167671\pi$
$541$			4.88003e12i			0.105302i	−0.998613	+	0.0526509i	$0.983233\pi$
$542$		−	7.89333e13i		−	1.68758i
$543$	2.01211e13			0.426237
$544$			3.63683e13i			0.763359i
$545$		−	2.36171e12i		−	0.0491183i
$546$	−6.41917e13			−1.32287
$547$	7.70756e12			0.157391			0.0786955	−	0.996899i	$-0.474925\pi$
$547$	7.70756e12			0.157391			0.0786955	+	0.996899i	$0.474925\pi$
$548$	−2.22913e13			−0.451059
$549$		−	2.07400e13i		−	0.415859i
$550$	−9.31254e13			−1.85035
$551$	6.16590e12			0.121406
$552$	−4.19840e12			−0.0819198
$553$	2.73958e13			0.529734
$554$		−	1.30909e14i		−	2.50854i
$555$		−	3.88201e12i		−	0.0737211i
$556$	7.52054e13			1.41539
$557$	−8.10191e13			−1.51116			−0.755582	−	0.655054i	$-0.772646\pi$
$557$	−8.10191e13			−1.51116			−0.755582	+	0.655054i	$0.772646\pi$
$558$	−2.79285e13			−0.516269
$559$	−1.63924e13			−0.300320
$560$	−9.15249e12			−0.166188
$561$			2.39092e13i			0.430279i
$562$		−	3.34611e13i		−	0.596840i
$563$			1.43723e13i			0.254089i	0.991897	+	0.127044i	$0.0405491\pi$
$563$			1.43723e13i			0.254089i	−0.991897	+	0.127044i	$0.959451\pi$
$564$			5.92369e13i			1.03800i
$565$			6.54111e12i			0.113608i
$566$	3.90516e13			0.672290
$567$	−7.89159e12			−0.134663
$568$		−	2.79688e12i		−	0.0473077i
$569$			3.15550e13i			0.529063i	0.964377	+	0.264532i	$0.0852173\pi$
$569$			3.15550e13i			0.529063i	−0.964377	+	0.264532i	$0.914783\pi$
$570$			4.46806e12i			0.0742582i
$571$		−	6.11208e13i		−	1.00695i	−0.864010	−	0.503475i	$-0.832054\pi$
$571$		−	6.11208e13i		−	1.00695i	0.864010	−	0.503475i	$-0.167946\pi$
$572$	1.05295e14			1.71960
$573$		−	4.12265e13i		−	0.667426i
$574$			1.10754e14i			1.77745i
$575$			9.78019e13i			1.55599i
$576$	1.83282e13			0.289072
$577$	7.26221e13			1.13551			0.567754	−	0.823199i	$-0.307813\pi$
$577$	7.26221e13			1.13551			0.567754	+	0.823199i	$0.307813\pi$
$578$			6.25050e13i			0.968893i
$579$	1.85499e13			0.285067
$580$	1.34867e12			0.0205478
$581$			1.63778e13i			0.247385i
$582$	−7.36032e13			−1.10225
$583$			1.69051e14i			2.51000i
$584$	−5.58088e12			−0.0821558
$585$		−	4.01348e12i		−	0.0585789i
$586$		−	2.14189e13i		−	0.309962i
$587$			1.17608e14i			1.68750i	0.536733	+	0.843752i	$0.319658\pi$
$587$			1.17608e14i			1.68750i	−0.536733	+	0.843752i	$0.680342\pi$
$588$	−1.78011e13			−0.253256
$589$		−	5.64149e13i		−	0.795825i
$590$	−7.79120e12	−	1.02350e13i	−0.108979	−	0.143162i
$591$	6.08642e13			0.844159
$592$		−	7.61214e13i		−	1.04688i
$593$	5.71849e13			0.779845			0.389922	−	0.920848i	$-0.372502\pi$
$593$	5.71849e13			0.779845			0.389922	+	0.920848i	$0.372502\pi$
$594$	2.67812e13			0.362158
$595$	6.44008e12			0.0863588
$596$			4.23652e13i			0.563349i
$597$	7.86392e13			1.03697
$598$		−	2.28783e14i		−	2.99170i
$599$	−1.04476e13			−0.135482			−0.0677412	−	0.997703i	$-0.521579\pi$
$599$	−1.04476e13			−0.135482			−0.0677412	+	0.997703i	$0.521579\pi$
$600$			3.95811e12i			0.0509016i
$601$		−	3.38326e13i		−	0.431483i	−0.976451	−	0.215741i	$-0.930783\pi$
$601$		−	3.38326e13i		−	0.431483i	0.976451	−	0.215741i	$-0.0692168\pi$
$602$	−2.94631e13			−0.372646
$603$			1.53590e13i			0.192653i
$604$			3.92372e13i			0.488105i
$605$	8.69604e12			0.107286
$606$	8.00982e13			0.980076
$607$	8.28195e12			0.100505			0.0502526	−	0.998737i	$-0.483997\pi$
$607$	8.28195e12			0.100505			0.0502526	+	0.998737i	$0.483997\pi$
$608$			8.22877e13i			0.990414i
$609$	−9.95483e12			−0.118836
$610$	1.89583e13			0.224466
$611$	2.22371e14			2.61139
$612$	−1.47515e13			−0.171823
$613$		−	1.35326e14i		−	1.56343i	−0.623636	−	0.781715i	$-0.714345\pi$
$613$		−	1.35326e14i		−	1.56343i	0.623636	−	0.781715i	$-0.285655\pi$
$614$		−	1.27748e14i		−	1.46390i
$615$	−6.92468e12			−0.0787089
$616$	−1.30373e13			−0.146989
$617$	−1.17189e14			−1.31058			−0.655289	−	0.755379i	$-0.727453\pi$
$617$	−1.17189e14			−1.31058			−0.655289	+	0.755379i	$0.727453\pi$
$618$	9.32584e13			1.03454
$619$	−3.39180e13			−0.373231			−0.186615	−	0.982433i	$-0.559752\pi$
$619$	−3.39180e13			−0.373231			−0.186615	+	0.982433i	$0.559752\pi$
$620$		−	1.23396e13i		−	0.134692i
$621$		−	2.81261e13i		−	0.304545i
$622$			1.43847e14i			1.54507i
$623$			1.74214e14i			1.85628i
$624$		−	7.86994e13i		−	0.831856i
$625$	9.06091e13			0.950105
$626$	−1.92024e14			−1.99749
$627$			5.40974e13i			0.558263i
$628$			7.32312e13i			0.749718i
$629$			5.35623e13i			0.544009i
$630$		−	7.21367e12i		−	0.0726864i
$631$	−1.41485e13			−0.141437			−0.0707186	−	0.997496i	$-0.522529\pi$
$631$	−1.41485e13			−0.141437			−0.0707186	+	0.997496i	$0.522529\pi$
$632$			3.95155e12i			0.0391907i
$633$		−	6.27088e13i		−	0.617035i
$634$			1.48227e14i			1.44704i
$635$	−1.48510e13			−0.143842
$636$	−1.04301e14			−1.00231
$637$			6.68239e13i			0.637140i
$638$	3.37831e13			0.319592
$639$	1.87369e13			0.175871
$640$		−	2.48499e12i		−	0.0231432i
$641$	−7.49349e13			−0.692459			−0.346229	−	0.938150i	$-0.612538\pi$
$641$	−7.49349e13			−0.692459			−0.346229	+	0.938150i	$0.612538\pi$
$642$		−	1.52450e14i		−	1.39782i
$643$	7.12517e13			0.648247			0.324123	−	0.946015i	$-0.394931\pi$
$643$	7.12517e13			0.648247			0.324123	+	0.946015i	$0.394931\pi$
$644$		−	1.98757e14i		−	1.79429i
$645$		−	1.84213e12i		−	0.0165014i
$646$		−	6.16483e13i		−	0.547972i
$647$	1.07039e14			0.944105			0.472052	−	0.881571i	$-0.343513\pi$
$647$	1.07039e14			0.944105			0.472052	+	0.881571i	$0.343513\pi$
$648$		−	1.13828e12i		−	0.00996263i
$649$	−9.43326e13	−	1.23921e14i	−0.819291	−	1.07627i
$650$	−2.15688e14			−1.85892
$651$			9.10817e13i			0.778980i
$652$	1.31337e14			1.11467
$653$	4.66881e12			0.0393224			0.0196612	−	0.999807i	$-0.493741\pi$
$653$	4.66881e12			0.0393224			0.0196612	+	0.999807i	$0.493741\pi$
$654$	3.65001e13			0.305074
$655$			3.44485e12i			0.0285735i
$656$	−1.35784e14			−1.11771
$657$		−	3.73876e13i		−	0.305422i
$658$	3.99681e14			3.24029
$659$		−	1.26192e14i		−	1.01533i	−0.861556	−	0.507663i	$-0.830509\pi$
$659$		−	1.26192e14i		−	1.01533i	0.861556	−	0.507663i	$-0.169491\pi$
$660$			1.18327e13i			0.0944854i
$661$	2.02053e14			1.60125			0.800624	−	0.599167i	$-0.204501\pi$
$661$	2.02053e14			1.60125			0.800624	+	0.599167i	$0.204501\pi$
$662$		−	2.70964e13i		−	0.213118i
$663$			5.53762e13i			0.432271i
$664$	−2.36232e12			−0.0183020
$665$	1.45714e13			0.112045
$666$	5.99962e13			0.457881
$667$		−	3.54796e13i		−	0.268751i
$668$	1.83473e14			1.37940
$669$	−2.56823e13			−0.191648
$670$	−1.40396e13			−0.103987
$671$	2.29539e14			1.68750
$672$		−	1.32853e14i		−	0.969451i
$673$			1.94663e14i			1.40997i	0.709223	+	0.704984i	$0.249046\pi$
$673$			1.94663e14i			1.40997i	−0.709223	+	0.704984i	$0.750954\pi$
$674$	1.60083e14			1.15092
$675$	−2.65162e13			−0.189231
$676$	1.11805e14			0.792005
$677$	4.09581e13			0.288003			0.144001	−	0.989577i	$-0.454003\pi$
$677$	4.09581e13			0.288003			0.144001	+	0.989577i	$0.454003\pi$
$678$	−1.01093e14			−0.705620
$679$			2.40038e14i			1.66315i
$680$			9.28915e11i			0.00638898i
$681$			7.41639e13i			0.506357i
$682$		−	3.09098e14i		−	2.09496i
$683$		−	2.42465e14i		−	1.63135i	−0.578513	−	0.815673i	$-0.696367\pi$
$683$		−	2.42465e14i		−	1.63135i	0.578513	−	0.815673i	$-0.303633\pi$
$684$	−3.33771e13			−0.222930
$685$	−9.40371e12			−0.0623514
$686$		−	1.36055e14i		−	0.895560i
$687$		−	1.44261e14i		−	0.942680i
$688$		−	3.61219e13i		−	0.234330i
$689$			3.91539e14i			2.52162i
$690$	2.57099e13			0.164382
$691$			1.97068e14i			1.25091i	0.780259	+	0.625456i	$0.215087\pi$
$691$			1.97068e14i			1.25091i	−0.780259	+	0.625456i	$0.784913\pi$
$692$			4.04212e13i			0.254729i
$693$		−	8.73401e13i		−	0.546446i
$694$	−4.07225e13			−0.252951
$695$	3.17258e13			0.195654
$696$		−	1.43588e12i		−	0.00879169i
$697$	9.55437e13			0.580816
$698$	4.26265e14			2.57278
$699$			7.43491e12i			0.0445543i
$700$	−1.87381e14			−1.11490
$701$			2.17248e14i			1.28341i	0.766952	+	0.641705i	$0.221773\pi$
$701$			2.17248e14i			1.28341i	−0.766952	+	0.641705i	$0.778227\pi$
$702$	6.20281e13			0.363834
$703$			1.21191e14i			0.705820i
$704$			2.02847e14i			1.17302i
$705$			2.49894e13i			0.143486i
$706$	−3.01814e14			−1.72074
$707$		−	2.61220e14i		−	1.47880i
$708$	7.64569e13	−	5.82015e13i	0.429784	−	0.327166i
$709$	6.87942e13			0.383991			0.191995	−	0.981396i	$-0.438504\pi$
$709$	6.87942e13			0.383991			0.191995	+	0.981396i	$0.438504\pi$
$710$			1.71273e13i			0.0949287i
$711$	−2.64724e13			−0.145695
$712$	−2.51286e13			−0.137331
$713$	−3.24620e14			−1.76168
$714$			9.95311e13i			0.536374i
$715$	4.44191e13			0.237706
$716$		−	2.13911e14i		−	1.13676i
$717$	−1.06086e14			−0.559839
$718$		−	4.86332e14i		−	2.54865i
$719$			1.36484e14i			0.710291i	0.934811	+	0.355145i	$0.115569\pi$
$719$			1.36484e14i			0.710291i	−0.934811	+	0.355145i	$0.884431\pi$
$720$	8.84399e12			0.0457073
$721$		−	3.04139e14i		−	1.56097i
$722$			1.33467e14i			0.680278i
$723$	1.74842e14			0.885022
$724$	−1.37396e14			−0.690685
$725$	−3.34489e13			−0.166990
$726$			1.34397e14i			0.666356i
$727$	1.12919e14			0.556028			0.278014	−	0.960577i	$-0.410324\pi$
$727$	1.12919e14			0.556028			0.278014	+	0.960577i	$0.410324\pi$
$728$	−3.01959e13			−0.147669
$729$	7.62560e12			0.0370370
$730$	3.41758e13			0.164856
$731$			2.54169e13i			0.121769i
$732$			1.41622e14i			0.673867i
$733$	−1.55631e14			−0.735488			−0.367744	−	0.929927i	$-0.619870\pi$
$733$	−1.55631e14			−0.735488			−0.367744	+	0.929927i	$0.619870\pi$
$734$	−2.43653e14			−1.14365
$735$	−7.50947e12			−0.0350084
$736$	4.73496e14			2.19244
$737$	−1.69986e14			−0.781763
$738$		−	1.07020e14i		−	0.488861i
$739$			3.63072e13i			0.164729i	0.996602	+	0.0823646i	$0.0262472\pi$
$739$			3.63072e13i			0.164729i	−0.996602	+	0.0823646i	$0.973753\pi$
$740$			2.65081e13i			0.119459i
$741$			1.25295e14i			0.560846i
$742$			7.03737e14i			3.12889i
$743$	6.28873e13			0.277728			0.138864	−	0.990311i	$-0.455655\pi$
$743$	6.28873e13			0.277728			0.138864	+	0.990311i	$0.455655\pi$
$744$	−1.31376e13			−0.0576304
$745$			1.78719e13i			0.0778736i
$746$			1.89969e14i			0.822220i
$747$		−	1.58257e13i		−	0.0680395i
$748$		−	1.63263e14i		−	0.697235i
$749$	−4.97177e14			−2.10912
$750$		−	4.88890e13i		−	0.206018i
$751$		−	1.72446e13i		−	0.0721863i	−0.999348	−	0.0360931i	$-0.988509\pi$
$751$		−	1.72446e13i		−	0.0721863i	0.999348	−	0.0360931i	$-0.0114913\pi$
$752$			4.90011e14i			2.03759i
$753$	−9.81940e13			−0.405611
$754$	7.82452e13			0.321071
$755$			1.65524e13i			0.0674723i
$756$	5.38873e13			0.218211
$757$	−1.41562e13			−0.0569466			−0.0284733	−	0.999595i	$-0.509065\pi$
$757$	−1.41562e13			−0.0569466			−0.0284733	+	0.999595i	$0.509065\pi$
$758$			3.15368e14i			1.26029i
$759$	3.11285e14			1.23580
$760$			2.10178e12i			0.00828933i
$761$	−3.21252e14			−1.25870			−0.629350	−	0.777122i	$-0.716679\pi$
$761$	−3.21252e14			−1.25870			−0.629350	+	0.777122i	$0.716679\pi$
$762$		−	2.29521e14i		−	0.893404i
$763$		−	1.19036e14i		−	0.460315i
$764$			2.81513e14i			1.08151i
$765$	−6.22301e12			−0.0237516
$766$		−	5.03232e14i		−	1.90820i
$767$	−2.18484e14	−	2.87014e14i	−0.823083	−	1.08125i
$768$	1.72180e14			0.644430
$769$			1.72173e14i			0.640225i	0.947380	+	0.320113i	$0.103721\pi$
$769$			1.72173e14i			0.640225i	−0.947380	+	0.320113i	$0.896279\pi$
$770$	7.98372e13			0.294952
$771$	−2.40561e14			−0.882985
$772$	−1.26667e14			−0.461930
$773$			3.04500e14i			1.10329i	0.834079	+	0.551644i	$0.185999\pi$
$773$			3.04500e14i			1.10329i	−0.834079	+	0.551644i	$0.814001\pi$
$774$	2.84700e13			0.102490
$775$			3.06040e14i			1.09464i
$776$	−3.46230e13			−0.123043
$777$		−	1.95663e14i		−	0.690880i
$778$		−	6.95820e14i		−	2.44118i
$779$	2.16179e14			0.753575

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)

\(f(q)\)	\(=\)	\(q-44.5197i q^{2} +140.296 q^{3} -958.005 q^{4} -404.139 q^{5} -6245.94i q^{6} -20369.6 q^{7} -2938.10i q^{8} +19683.0 q^{9} +O(q^{10})\)	\(q-44.5197i q^{2} +140.296 q^{3} -958.005 q^{4} -404.139 q^{5} -6245.94i q^{6} -20369.6 q^{7} -2938.10i q^{8} +19683.0 q^{9} +17992.1i q^{10} +217841. i q^{11} -134404. q^{12} +504544. i q^{13} +906847. i q^{14} -56699.1 q^{15} -1.11180e6 q^{16} +782310. q^{17} -876281. i q^{18} +1.77007e6 q^{19} +387167. q^{20} -2.85777e6 q^{21} +9.69824e6 q^{22} -1.01853e7i q^{23} -412204. i q^{24} -9.60230e6 q^{25} +2.24622e7 q^{26} +2.76145e6 q^{27} +1.95141e7 q^{28} +3.48342e6 q^{29} +2.52423e6i q^{30} -3.18716e7i q^{31} +4.64884e7i q^{32} +3.05623e7i q^{33} -3.48282e7i q^{34} +8.23213e6 q^{35} -1.88564e7 q^{36} +6.84669e7i q^{37} -7.88030e7i q^{38} +7.07856e7i q^{39} +1.18740e6i q^{40} +1.22130e8 q^{41} +1.27227e8i q^{42} +3.24896e7i q^{43} -2.08693e8i q^{44} -7.95466e6 q^{45} -4.53445e8 q^{46} -4.40737e8i q^{47} -1.55981e8 q^{48} +1.32444e8 q^{49} +4.27491e8i q^{50} +1.09755e8 q^{51} -4.83355e8i q^{52} +7.76026e8 q^{53} -1.22939e8i q^{54} -8.80382e7i q^{55} +5.98478e7i q^{56} +2.48334e8 q^{57} -1.55081e8i q^{58} +(-5.68858e8 + 4.33033e8i) q^{59} +5.43180e7 q^{60} -1.05370e9i q^{61} -1.41891e9 q^{62} -4.00934e8 q^{63} +9.31167e8 q^{64} -2.03906e8i q^{65} +1.36063e9 q^{66} +7.80318e8i q^{67} -7.49456e8 q^{68} -1.42895e9i q^{69} -3.66492e8i q^{70} +9.51934e8 q^{71} -5.78306e7i q^{72} -1.89948e9i q^{73} +3.04812e9 q^{74} -1.34716e9 q^{75} -1.69573e9 q^{76} -4.43734e9i q^{77} +3.15135e9 q^{78} -1.34494e9 q^{79} +4.49321e8 q^{80} +3.87420e8 q^{81} -5.43720e9i q^{82} -8.04031e8i q^{83} +2.73776e9 q^{84} -3.16162e8 q^{85} +1.44643e9 q^{86} +4.88711e8 q^{87} +6.40040e8 q^{88} -8.55267e9i q^{89} +3.54139e8i q^{90} -1.02773e10i q^{91} +9.75752e9i q^{92} -4.47146e9i q^{93} -1.96215e10 q^{94} -7.15354e8 q^{95} +6.52214e9i q^{96} -1.17842e10i q^{97} -5.89638e9i q^{98} +4.28777e9i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 100q - 51200q^{4} + 18392q^{7} + 1968300q^{9} + O(q^{10}) \)	\( 100q - 51200q^{4} + 18392q^{7} + 1968300q^{9} + 620136q^{12} + 662904q^{15} + 27925160q^{16} - 2053136q^{17} - 5169828q^{19} - 1076324q^{20} - 1829224q^{22} + 215378180q^{25} - 22082700q^{26} - 102921320q^{28} - 112503588q^{29} - 76491392q^{35} - 1007769600q^{36} + 473464516q^{41} + 1215405588q^{46} - 1676272320q^{48} + 1975297276q^{49} + 733970808q^{51} + 3267506728q^{53} + 591502824q^{57} + 508142200q^{59} + 1264196808q^{60} - 6538206968q^{62} + 362009736q^{63} - 10324137972q^{64} - 2764346616q^{66} + 9997685952q^{68} + 14908523204q^{71} + 4863508712q^{74} + 1890481680q^{75} + 2044437240q^{76} - 758396196q^{78} - 3599839500q^{79} - 23217941144q^{80} + 38742048900q^{81} - 13094894808q^{84} + 23360564412q^{85} + 12186923752q^{86} + 7965322272q^{87} + 32415437996q^{88} - 22098322280q^{94} + 7834510028q^{95} + O(q^{100}) \)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Properties

Related objects

Downloads

Learn more about