LMFDB - Embedded newform 177.11.c.a.58.9

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			58.9
Character	$\chi$	$=$	177.58
Dual form			177.11.c.a.58.92

$q$-expansion

$f(q)$	$=$	$q-55.6543i q^{2} +140.296 q^{3} -2073.41 q^{4} -1651.80 q^{5} -7808.09i q^{6} -6427.48 q^{7} +58404.0i q^{8} +19683.0 q^{9} +O(q^{10})$	\(q-55.6543i q^{2} +140.296 q^{3} -2073.41 q^{4} -1651.80 q^{5} -7808.09i q^{6} -6427.48 q^{7} +58404.0i q^{8} +19683.0 q^{9} +91929.7i q^{10} -42364.0i q^{11} -290891. q^{12} -179652. i q^{13} +357717. i q^{14} -231741. q^{15} +1.12727e6 q^{16} -2.44515e6 q^{17} -1.09544e6i q^{18} -2.94996e6 q^{19} +3.42485e6 q^{20} -901750. q^{21} -2.35774e6 q^{22} +4.47755e6i q^{23} +8.19385e6i q^{24} -7.03719e6 q^{25} -9.99843e6 q^{26} +2.76145e6 q^{27} +1.33268e7 q^{28} +3.67308e7 q^{29} +1.28974e7i q^{30} +595402. i q^{31} -2.93168e6i q^{32} -5.94350e6i q^{33} +1.36083e8i q^{34} +1.06169e7 q^{35} -4.08108e7 q^{36} -1.07486e7i q^{37} +1.64178e8i q^{38} -2.52045e7i q^{39} -9.64716e7i q^{40} +1.47249e8 q^{41} +5.01863e7i q^{42} +2.36868e8i q^{43} +8.78377e7i q^{44} -3.25123e7 q^{45} +2.49195e8 q^{46} -3.78784e8i q^{47} +1.58151e8 q^{48} -2.41163e8 q^{49} +3.91650e8i q^{50} -3.43046e8 q^{51} +3.72492e8i q^{52} -2.09357e8 q^{53} -1.53687e8i q^{54} +6.99767e7i q^{55} -3.75390e8i q^{56} -4.13868e8 q^{57} -2.04423e9i q^{58} +(2.07411e8 + 6.84177e8i) q^{59} +4.80493e8 q^{60} -3.23023e8i q^{61} +3.31367e7 q^{62} -1.26512e8 q^{63} +9.91162e8 q^{64} +2.96749e8i q^{65} -3.30782e8 q^{66} -1.67595e9i q^{67} +5.06980e9 q^{68} +6.28183e8i q^{69} -5.90876e8i q^{70} +1.42921e9 q^{71} +1.14957e9i q^{72} +2.75324e9i q^{73} -5.98206e8 q^{74} -9.87291e8 q^{75} +6.11647e9 q^{76} +2.72293e8i q^{77} -1.40274e9 q^{78} +4.89865e9 q^{79} -1.86202e9 q^{80} +3.87420e8 q^{81} -8.19506e9i q^{82} -4.90223e8i q^{83} +1.86969e9 q^{84} +4.03890e9 q^{85} +1.31827e10 q^{86} +5.15318e9 q^{87} +2.47422e9 q^{88} -6.61963e9i q^{89} +1.80945e9i q^{90} +1.15471e9i q^{91} -9.28378e9i q^{92} +8.35325e7i q^{93} -2.10810e10 q^{94} +4.87274e9 q^{95} -4.11304e8i q^{96} -1.12096e9i q^{97} +1.34218e10i q^{98} -8.33850e8i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$		−	55.6543i		−	1.73920i	−0.493758	−	0.869599i	$-0.664377\pi$
$2$		−	55.6543i		−	1.73920i	0.493758	−	0.869599i	$-0.335623\pi$
$3$	140.296			0.577350
$3$	140.296			0.577350
$4$	−2073.41			−2.02481
$5$	−1651.80			−0.528575			−0.264288	−	0.964444i	$-0.585137\pi$
$5$	−1651.80			−0.528575			−0.264288	+	0.964444i	$0.585137\pi$
$6$		−	7808.09i		−	1.00413i
$7$	−6427.48			−0.382429			−0.191214	−	0.981548i	$-0.561243\pi$
$7$	−6427.48			−0.382429			−0.191214	+	0.981548i	$0.561243\pi$
$8$			58404.0i			1.78235i
$9$	19683.0			0.333333
$10$			91929.7i			0.919297i
$11$		−	42364.0i		−	0.263047i	−0.991313	−	0.131523i	$-0.958013\pi$
$11$		−	42364.0i		−	0.263047i	0.991313	−	0.131523i	$-0.0419869\pi$
$12$	−290891.			−1.16902
$13$		−	179652.i		−	0.483856i	−0.970294	−	0.241928i	$-0.922220\pi$
$13$		−	179652.i		−	0.483856i	0.970294	−	0.241928i	$-0.0777798\pi$
$14$			357717.i			0.665119i
$15$	−231741.			−0.305173
$16$	1.12727e6			1.07505
$17$	−2.44515e6			−1.72211			−0.861056	−	0.508509i	$-0.830197\pi$
$17$	−2.44515e6			−1.72211			−0.861056	+	0.508509i	$0.830197\pi$
$18$		−	1.09544e6i		−	0.579733i
$19$	−2.94996e6			−1.19137			−0.595687	−	0.803216i	$-0.703120\pi$
$19$	−2.94996e6			−1.19137			−0.595687	+	0.803216i	$0.703120\pi$
$20$	3.42485e6			1.07026
$21$	−901750.			−0.220795
$22$	−2.35774e6			−0.457491
$23$			4.47755e6i			0.695667i	0.937556	+	0.347834i	$0.113083\pi$
$23$			4.47755e6i			0.695667i	−0.937556	+	0.347834i	$0.886917\pi$
$24$			8.19385e6i			1.02904i
$25$	−7.03719e6			−0.720608
$26$	−9.99843e6			−0.841521
$27$	2.76145e6			0.192450
$28$	1.33268e7			0.774345
$29$	3.67308e7			1.79077			0.895385	−	0.445292i	$-0.146900\pi$
$29$	3.67308e7			1.79077			0.895385	+	0.445292i	$0.146900\pi$
$30$			1.28974e7i			0.530756i
$31$			595402.i			0.0207970i	0.999946	+	0.0103985i	$0.00331001\pi$
$31$			595402.i			0.0207970i	−0.999946	+	0.0103985i	$0.996690\pi$
$32$		−	2.93168e6i		−	0.0873710i
$33$		−	5.94350e6i		−	0.151870i
$34$			1.36083e8i			2.99510i
$35$	1.06169e7			0.202142
$36$	−4.08108e7			−0.674937
$37$		−	1.07486e7i		−	0.155004i	−0.996992	−	0.0775021i	$-0.975306\pi$
$37$		−	1.07486e7i		−	0.155004i	0.996992	−	0.0775021i	$-0.0246944\pi$
$38$			1.64178e8i			2.07204i
$39$		−	2.52045e7i		−	0.279354i
$40$		−	9.64716e7i		−	0.942105i
$41$	1.47249e8			1.27097			0.635483	−	0.772115i	$-0.280801\pi$
$41$	1.47249e8			1.27097			0.635483	+	0.772115i	$0.280801\pi$
$42$			5.01863e7i			0.384007i
$43$			2.36868e8i			1.61126i	0.592422	+	0.805628i	$0.298172\pi$
$43$			2.36868e8i			1.61126i	−0.592422	+	0.805628i	$0.701828\pi$
$44$			8.78377e7i			0.532620i
$45$	−3.25123e7			−0.176192
$46$	2.49195e8			1.20990
$47$		−	3.78784e8i		−	1.65159i	−0.563969	−	0.825796i	$-0.690726\pi$
$47$		−	3.78784e8i		−	1.65159i	0.563969	−	0.825796i	$-0.309274\pi$
$48$	1.58151e8			0.620679
$49$	−2.41163e8			−0.853748
$50$			3.91650e8i			1.25328i
$51$	−3.43046e8			−0.994262
$52$			3.72492e8i			0.979717i
$53$	−2.09357e8			−0.500620			−0.250310	−	0.968166i	$-0.580532\pi$
$53$	−2.09357e8			−0.500620			−0.250310	+	0.968166i	$0.580532\pi$
$54$		−	1.53687e8i		−	0.334709i
$55$			6.99767e7i			0.139040i
$56$		−	3.75390e8i		−	0.681621i
$57$	−4.13868e8			−0.687840
$58$		−	2.04423e9i		−	3.11451i
$59$	2.07411e8	+	6.84177e8i	0.290116	+	0.956992i
$59$	2.07411e8	+	6.84177e8i	0.290116	+	0.956992i
$60$	4.80493e8			0.617917
$61$		−	3.23023e8i		−	0.382458i	−0.981545	−	0.191229i	$-0.938753\pi$
$61$		−	3.23023e8i		−	0.382458i	0.981545	−	0.191229i	$-0.0612473\pi$
$62$	3.31367e7			0.0361702
$63$	−1.26512e8			−0.127476
$64$	9.91162e8			0.923091
$65$			2.96749e8i			0.255754i
$66$	−3.30782e8			−0.264132
$67$		−	1.67595e9i		−	1.24133i	−0.784075	−	0.620666i	$-0.786862\pi$
$67$		−	1.67595e9i		−	1.24133i	0.784075	−	0.620666i	$-0.213138\pi$
$68$	5.06980e9			3.48695
$69$			6.28183e8i			0.401644i
$70$		−	5.90876e8i		−	0.351565i
$71$	1.42921e9			0.792143			0.396071	−	0.918220i	$-0.370373\pi$
$71$	1.42921e9			0.792143			0.396071	+	0.918220i	$0.370373\pi$
$72$			1.14957e9i			0.594116i
$73$			2.75324e9i			1.32810i	0.747689	+	0.664049i	$0.231164\pi$
$73$			2.75324e9i			1.32810i	−0.747689	+	0.664049i	$0.768836\pi$
$74$	−5.98206e8			−0.269583
$75$	−9.87291e8			−0.416043
$76$	6.11647e9			2.41231
$77$			2.72293e8i			0.100597i
$78$	−1.40274e9			−0.485853
$79$	4.89865e9			1.59199			0.795996	−	0.605301i	$-0.206947\pi$
$79$	4.89865e9			1.59199			0.795996	+	0.605301i	$0.206947\pi$
$80$	−1.86202e9			−0.568243
$81$	3.87420e8			0.111111
$82$		−	8.19506e9i		−	2.21046i
$83$		−	4.90223e8i		−	0.124452i	−0.998062	−	0.0622261i	$-0.980180\pi$
$83$		−	4.90223e8i		−	0.124452i	0.998062	−	0.0622261i	$-0.0198200\pi$
$84$	1.86969e9			0.447068
$85$	4.03890e9			0.910266
$86$	1.31827e10			2.80229
$87$	5.15318e9			1.03390
$88$	2.47422e9			0.468841
$89$		−	6.61963e9i		−	1.18545i	−0.805405	−	0.592725i	$-0.798052\pi$
$89$		−	6.61963e9i		−	1.18545i	0.805405	−	0.592725i	$-0.201948\pi$
$90$			1.80945e9i			0.306432i
$91$			1.15471e9i			0.185040i
$92$		−	9.28378e9i		−	1.40859i
$93$			8.35325e7i			0.0120072i
$94$	−2.10810e10			−2.87245
$95$	4.87274e9			0.629731
$96$		−	4.11304e8i		−	0.0504437i
$97$		−	1.12096e9i		−	0.130537i	−0.997868	−	0.0652683i	$-0.979210\pi$
$97$		−	1.12096e9i		−	0.130537i	0.997868	−	0.0652683i	$-0.0207903\pi$
$98$			1.34218e10i			1.48484i
$99$		−	8.33850e8i		−	0.0876823i
$100$	1.45910e10			1.45910
$101$			4.89302e9i			0.465554i	0.972530	+	0.232777i	$0.0747813\pi$
$101$			4.89302e9i			0.465554i	−0.972530	+	0.232777i	$0.925219\pi$
$102$			1.90920e10i			1.72922i
$103$			6.05488e8i			0.0522300i	0.999659	+	0.0261150i	$0.00831360\pi$
$103$			6.05488e8i			0.0522300i	−0.999659	+	0.0261150i	$0.991686\pi$
$104$	1.04924e10			0.862400
$105$	1.48951e9			0.116707
$106$			1.16516e10i			0.870677i
$107$	−1.34731e10			−0.960612			−0.480306	−	0.877101i	$-0.659474\pi$
$107$	−1.34731e10			−0.960612			−0.480306	+	0.877101i	$0.659474\pi$
$108$	−5.72560e9			−0.389675
$109$		−	2.45190e10i		−	1.59357i	−0.604265	−	0.796783i	$-0.706533\pi$
$109$		−	2.45190e10i		−	1.59357i	0.604265	−	0.796783i	$-0.293467\pi$
$110$	3.89451e9			0.241818
$111$		−	1.50799e9i		−	0.0894917i
$112$	−7.24549e9			−0.411129
$113$			3.11336e10i			1.68981i	0.534917	+	0.844904i	$0.320343\pi$
$113$			3.11336e10i			1.68981i	−0.534917	+	0.844904i	$0.679657\pi$
$114$			2.30336e10i			1.19629i
$115$		−	7.39601e9i		−	0.367712i
$116$	−7.61578e10			−3.62597
$117$		−	3.53610e9i		−	0.161285i
$118$	3.80774e10	−	1.15433e10i	1.66440	−	0.504569i
$119$	1.57162e10			0.658585
$120$		−	1.35346e10i		−	0.543925i
$121$	2.41427e10			0.930806
$122$	−1.79776e10			−0.665171
$123$	2.06585e10			0.733792
$124$		−	1.23451e9i		−	0.0421101i
$125$	2.77548e10			0.909471
$126$			7.04094e9i			0.221706i
$127$	−2.06395e10			−0.624713			−0.312356	−	0.949965i	$-0.601118\pi$
$127$	−2.06395e10			−0.624713			−0.312356	+	0.949965i	$0.601118\pi$
$128$		−	5.81645e10i		−	1.69281i
$129$			3.32317e10i			0.930259i
$130$	1.65154e10			0.444807
$131$			5.93554e10i			1.53852i	0.638934	+	0.769261i	$0.279376\pi$
$131$			5.93554e10i			1.53852i	−0.638934	+	0.769261i	$0.720624\pi$
$132$			1.23233e10i			0.307508i
$133$	1.89608e10			0.455616
$134$	−9.32741e10			−2.15892
$135$	−4.56135e9			−0.101724
$136$		−	1.42807e11i		−	3.06941i
$137$	−6.49128e10			−1.34502			−0.672508	−	0.740090i	$-0.734783\pi$
$137$	−6.49128e10			−1.34502			−0.672508	+	0.740090i	$0.734783\pi$
$138$	3.49611e10			0.698538
$139$	8.69158e10			1.67504			0.837519	−	0.546408i	$-0.184005\pi$
$139$	8.69158e10			1.67504			0.837519	+	0.546408i	$0.184005\pi$
$140$	−2.20131e10			−0.409300
$141$		−	5.31420e10i		−	0.953547i
$142$		−	7.95416e10i		−	1.37769i
$143$	−7.61078e9			−0.127277
$144$	2.21880e10			0.358349
$145$	−6.06718e10			−0.946557
$146$	1.53230e11			2.30983
$147$	−3.38342e10			−0.492912
$148$			2.22862e10i			0.313854i
$149$		−	7.01159e10i		−	0.954741i	−0.878702	−	0.477370i	$-0.841590\pi$
$149$		−	7.01159e10i		−	0.954741i	0.878702	−	0.477370i	$-0.158410\pi$
$150$			5.49470e10i			0.723582i
$151$			3.25291e10i			0.414369i	0.978302	+	0.207184i	$0.0664300\pi$
$151$			3.25291e10i			0.414369i	−0.978302	+	0.207184i	$0.933570\pi$
$152$		−	1.72290e11i		−	2.12344i
$153$	−4.81280e10			−0.574038
$154$	1.51543e10			0.174958
$155$		−	9.83483e8i		−	0.0109928i
$156$			5.22592e10i			0.565640i
$157$		−	1.35643e11i		−	1.42200i	−0.703194	−	0.710998i	$-0.748244\pi$
$157$		−	1.35643e11i		−	1.42200i	0.703194	−	0.710998i	$-0.251756\pi$
$158$		−	2.72631e11i		−	2.76879i
$159$	−2.93720e10			−0.289033
$160$			4.84255e9i			0.0461821i
$161$		−	2.87794e10i		−	0.266043i
$162$		−	2.15616e10i		−	0.193244i
$163$	−2.53362e10			−0.220193			−0.110097	−	0.993921i	$-0.535116\pi$
$163$	−2.53362e10			−0.220193			−0.110097	+	0.993921i	$0.535116\pi$
$164$	−3.05307e11			−2.57346
$165$			9.81746e9i			0.0802748i
$166$	−2.72830e10			−0.216447
$167$	−8.34235e10			−0.642253			−0.321126	−	0.947036i	$-0.604061\pi$
$167$	−8.34235e10			−0.642253			−0.321126	+	0.947036i	$0.604061\pi$
$168$		−	5.26658e10i		−	0.393534i
$169$	1.05584e11			0.765883
$170$		−	2.24782e11i		−	1.58313i
$171$	−5.80641e10			−0.397125
$172$		−	4.91124e11i		−	3.26249i
$173$			2.37750e11i			1.53423i	0.641509	+	0.767115i	$0.278309\pi$
$173$			2.37750e11i			1.53423i	−0.641509	+	0.767115i	$0.721691\pi$
$174$		−	2.86797e11i		−	1.79816i
$175$	4.52314e10			0.275581
$176$		−	4.77556e10i		−	0.282788i
$177$	2.90989e10	+	9.59873e10i	0.167498	+	0.552519i
$178$	−3.68411e11			−2.06173
$179$		−	2.07663e11i		−	1.13004i	−0.825077	−	0.565020i	$-0.808868\pi$
$179$		−	2.07663e11i		−	1.13004i	0.825077	−	0.565020i	$-0.191132\pi$
$180$	6.74112e10			0.356755
$181$	8.92034e10			0.459186			0.229593	−	0.973287i	$-0.426260\pi$
$181$	8.92034e10			0.459186			0.229593	+	0.973287i	$0.426260\pi$
$182$	6.42647e10			0.321822
$183$		−	4.53188e10i		−	0.220812i
$184$	−2.61507e11			−1.23992
$185$			1.77545e10i			0.0819313i
$186$	4.64895e9			0.0208829
$187$			1.03586e11i			0.452996i
$188$			7.85374e11i			3.34416i
$189$	−1.77491e10			−0.0735984
$190$		−	2.71189e11i		−	1.09523i
$191$			8.84099e10i			0.347804i	0.984763	+	0.173902i	$0.0556375\pi$
$191$			8.84099e10i			0.347804i	−0.984763	+	0.173902i	$0.944362\pi$
$192$	1.39056e11			0.532947
$193$	−2.58894e11			−0.966796			−0.483398	−	0.875401i	$-0.660598\pi$
$193$	−2.58894e11			−0.966796			−0.483398	+	0.875401i	$0.660598\pi$
$194$	−6.23864e10			−0.227029
$195$			4.16328e10i			0.147660i
$196$	5.00028e11			1.72868
$197$	2.60482e11			0.877904			0.438952	−	0.898510i	$-0.355350\pi$
$197$	2.60482e11			0.877904			0.438952	+	0.898510i	$0.355350\pi$
$198$	−4.64074e10			−0.152497
$199$	1.43116e11			0.458589			0.229294	−	0.973357i	$-0.426358\pi$
$199$	1.43116e11			0.458589			0.229294	+	0.973357i	$0.426358\pi$
$200$		−	4.11000e11i		−	1.28438i
$201$		−	2.35130e11i		−	0.716683i
$202$	2.72318e11			0.809692
$203$	−2.36086e11			−0.684842
$204$	7.11273e11			2.01319
$205$	−2.43226e11			−0.671801
$206$	3.36981e10			0.0908383
$207$			8.81317e10i			0.231889i
$208$		−	2.02516e11i		−	0.520168i
$209$			1.24972e11i			0.313387i
$210$		−	8.28976e10i		−	0.202976i
$211$			5.72585e11i			1.36908i	0.728977	+	0.684538i	$0.239996\pi$
$211$			5.72585e11i			1.36908i	−0.728977	+	0.684538i	$0.760004\pi$
$212$	4.34082e11			1.01366
$213$	2.00512e11			0.457344
$214$			7.49835e11i			1.67069i
$215$		−	3.91258e11i		−	0.851670i
$216$			1.61280e11i			0.343013i
$217$		−	3.82693e9i		−	0.00795338i
$218$	−1.36459e12			−2.77153
$219$			3.86269e11i			0.766778i
$220$		−	1.45090e11i		−	0.281530i
$221$			4.39278e11i			0.833255i
$222$	−8.39260e10			−0.155644
$223$	1.04640e12			1.89746			0.948731	−	0.316085i	$-0.102368\pi$
$223$	1.04640e12			1.89746			0.948731	+	0.316085i	$0.102368\pi$
$224$			1.88433e10i			0.0334132i
$225$	−1.38513e11			−0.240203
$226$	1.73272e12			2.93891
$227$		−	7.11857e11i		−	1.18104i	−0.807024	−	0.590519i	$-0.798923\pi$
$227$		−	7.11857e11i		−	1.18104i	0.807024	−	0.590519i	$-0.201077\pi$
$228$	8.58117e11			1.39275
$229$			1.23805e12i			1.96589i	0.183900	+	0.982945i	$0.441128\pi$
$229$			1.23805e12i			1.96589i	−0.183900	+	0.982945i	$0.558872\pi$
$230$	−4.11620e11			−0.639525
$231$			3.82017e10i			0.0580795i
$232$			2.14522e12i			3.19178i
$233$		−	4.46825e11i		−	0.650666i	−0.945599	−	0.325333i	$-0.894524\pi$
$233$		−	4.46825e11i		−	0.650666i	0.945599	−	0.325333i	$-0.105476\pi$
$234$	−1.96799e11			−0.280507
$235$			6.25675e11i			0.872991i
$236$	−4.30047e11	−	1.41858e12i	−0.587429	−	1.93773i
$237$	6.87262e11			0.919138
$238$		−	8.74673e11i		−	1.14541i
$239$	−8.29032e11			−1.06312			−0.531559	−	0.847021i	$-0.678394\pi$
$239$	−8.29032e11			−1.06312			−0.531559	+	0.847021i	$0.678394\pi$
$240$	−2.61234e11			−0.328075
$241$	−4.35002e11			−0.535065			−0.267532	−	0.963549i	$-0.586208\pi$
$241$	−4.35002e11			−0.535065			−0.267532	+	0.963549i	$0.586208\pi$
$242$		−	1.34365e12i		−	1.61886i
$243$	5.43536e10			0.0641500
$244$			6.69757e11i			0.774405i
$245$	3.98352e11			0.451270
$246$		−	1.14973e12i		−	1.27621i
$247$			5.29967e11i			0.576454i
$248$	−3.47738e10			−0.0370676
$249$		−	6.87763e10i		−	0.0718526i
$250$		−	1.54468e12i		−	1.58175i
$251$	6.03262e11			0.605532			0.302766	−	0.953065i	$-0.402090\pi$
$251$	6.03262e11			0.605532			0.302766	+	0.953065i	$0.402090\pi$
$252$	2.62311e11			0.258115
$253$	1.89687e11			0.182993
$254$			1.14868e12i			1.08650i
$255$	5.66642e11			0.525542
$256$	−2.22216e12			−2.02104
$257$	−1.71456e12			−1.52928			−0.764640	−	0.644458i	$-0.777083\pi$
$257$	−1.71456e12			−1.52928			−0.764640	+	0.644458i	$0.777083\pi$
$258$	1.84949e12			1.61790
$259$			6.90864e10i			0.0592780i
$260$		−	6.15282e11i		−	0.517854i
$261$	7.22972e11			0.596924
$262$	3.30339e12			2.67580
$263$	1.07745e12			0.856288			0.428144	−	0.903710i	$-0.359168\pi$
$263$	1.07745e12			0.856288			0.428144	+	0.903710i	$0.359168\pi$
$264$	3.47124e11			0.270686
$265$	3.45815e11			0.264615
$266$		−	1.05525e12i		−	0.792406i
$267$		−	9.28708e11i		−	0.684420i
$268$			3.47493e12i			2.51346i
$269$			1.28198e12i			0.910165i	0.890449	+	0.455083i	$0.150390\pi$
$269$			1.28198e12i			0.910165i	−0.890449	+	0.455083i	$0.849610\pi$
$270$			2.53859e11i			0.176919i
$271$	4.50951e11			0.308520			0.154260	−	0.988030i	$-0.450701\pi$
$271$	4.50951e11			0.308520			0.154260	+	0.988030i	$0.450701\pi$
$272$	−2.75634e12			−1.85135
$273$			1.62001e11i			0.106833i
$274$			3.61268e12i			2.33925i
$275$			2.98123e11i			0.189554i
$276$		−	1.30248e12i		−	0.813252i
$277$	1.91181e12			1.17232			0.586160	−	0.810195i	$-0.300639\pi$
$277$	1.91181e12			1.17232			0.586160	+	0.810195i	$0.300639\pi$
$278$		−	4.83724e12i		−	2.91322i
$279$			1.17193e10i			0.00693235i
$280$			6.20069e11i			0.360288i
$281$	−4.01203e11			−0.228998			−0.114499	−	0.993423i	$-0.536526\pi$
$281$	−4.01203e11			−0.228998			−0.114499	+	0.993423i	$0.536526\pi$
$282$	−2.95758e12			−1.65841
$283$		−	3.77846e11i		−	0.208153i	−0.994569	−	0.104077i	$-0.966811\pi$
$283$		−	3.77846e11i		−	0.208153i	0.994569	−	0.104077i	$-0.0331887\pi$
$284$	−2.96333e12			−1.60394
$285$	6.83626e11			0.363575
$286$			4.23573e11i			0.221360i
$287$	−9.46441e11			−0.486053
$288$		−	5.77043e10i		−	0.0291237i
$289$	3.96278e12			1.96567
$290$			3.37665e12i			1.64625i
$291$		−	1.57267e11i		−	0.0753653i
$292$		−	5.70859e12i		−	2.68915i
$293$	1.67521e12			0.775767			0.387883	−	0.921708i	$-0.373206\pi$
$293$	1.67521e12			0.775767			0.387883	+	0.921708i	$0.373206\pi$
$294$			1.88302e12i			0.857271i
$295$	−3.42601e11	−	1.13012e12i	−0.153348	−	0.505842i
$296$	6.27761e11			0.276271
$297$		−	1.16986e11i		−	0.0506234i
$298$	−3.90226e12			−1.66048
$299$	8.04403e11			0.336603
$300$	2.04705e12			0.842409
$301$		−	1.52246e12i		−	0.616190i
$302$	1.81038e12			0.720669
$303$			6.86472e11i			0.268788i
$304$	−3.32540e12			−1.28078
$305$			5.33568e11i			0.202158i
$306$			2.67853e12i			0.998365i
$307$	−1.64568e12			−0.603466			−0.301733	−	0.953392i	$-0.597565\pi$
$307$	−1.64568e12			−0.603466			−0.301733	+	0.953392i	$0.597565\pi$
$308$		−	5.64575e11i		−	0.203689i
$309$			8.49477e10i			0.0301550i
$310$	−5.47351e10			−0.0191187
$311$	4.42075e11			0.151948			0.0759739	−	0.997110i	$-0.475793\pi$
$311$	4.42075e11			0.151948			0.0759739	+	0.997110i	$0.475793\pi$
$312$	1.47204e12			0.497907
$313$			4.05890e12i			1.35110i	0.737316	+	0.675548i	$0.236093\pi$
$313$			4.05890e12i			1.35110i	−0.737316	+	0.675548i	$0.763907\pi$
$314$	−7.54911e12			−2.47313
$315$	2.08972e11			0.0673807
$316$	−1.01569e13			−3.22348
$317$	−4.01843e12			−1.25534			−0.627669	−	0.778481i	$-0.715991\pi$
$317$	−4.01843e12			−1.25534			−0.627669	+	0.778481i	$0.715991\pi$
$318$			1.63468e12i			0.502686i
$319$		−	1.55606e12i		−	0.471057i
$320$	−1.63720e12			−0.487923
$321$	−1.89022e12			−0.554610
$322$	−1.60170e12			−0.462702
$323$	7.21311e12			2.05168
$324$	−8.03280e11			−0.224979
$325$			1.26425e12i			0.348671i
$326$			1.41007e12i			0.382960i
$327$		−	3.43992e12i		−	0.920046i
$328$			8.59994e12i			2.26530i
$329$			2.43463e12i			0.631616i
$330$	5.46384e11			0.139614
$331$	4.62890e11			0.116503			0.0582516	−	0.998302i	$-0.481447\pi$
$331$	4.62890e11			0.116503			0.0582516	+	0.998302i	$0.481447\pi$
$332$			1.01643e12i			0.251992i
$333$		−	2.11565e11i		−	0.0516680i
$334$			4.64288e12i			1.11700i
$335$			2.76833e12i			0.656137i
$336$	−1.01651e12			−0.237365
$337$			2.91225e11i			0.0670007i	0.999439	+	0.0335003i	$0.0106655\pi$
$337$			2.91225e11i			0.0670007i	−0.999439	+	0.0335003i	$0.989335\pi$
$338$		−	5.87618e12i		−	1.33202i
$339$			4.36793e12i			0.975612i
$340$	−8.37428e12			−1.84312
$341$	2.52236e10			0.00547060
$342$			3.23152e12i			0.690679i
$343$	3.36567e12			0.708926
$344$	−1.38340e13			−2.87182
$345$		−	1.03763e12i		−	0.212299i
$346$	1.32318e13			2.66833
$347$			9.42217e12i			1.87285i	0.350864	+	0.936427i	$0.385888\pi$
$347$			9.42217e12i			1.87285i	−0.350864	+	0.936427i	$0.614112\pi$
$348$	−1.06846e13			−2.09346
$349$			2.22714e12i			0.430150i	0.976598	+	0.215075i	$0.0689996\pi$
$349$			2.22714e12i			0.430150i	−0.976598	+	0.215075i	$0.931000\pi$
$350$		−	2.51732e12i		−	0.479290i
$351$		−	4.96101e11i		−	0.0931181i
$352$	−1.24198e11			−0.0229827
$353$		−	1.90174e11i		−	0.0346959i	−0.999850	−	0.0173479i	$-0.994478\pi$
$353$		−	1.90174e11i		−	0.0346959i	0.999850	−	0.0173479i	$-0.00552230\pi$
$354$	5.34211e12	−	1.61948e12i	0.960941	−	0.291313i
$355$	−2.36076e12			−0.418707
$356$			1.37252e13i			2.40031i
$357$	2.20492e12			0.380234
$358$	−1.15573e13			−1.96536
$359$	−5.83565e12			−0.978627			−0.489314	−	0.872108i	$-0.662753\pi$
$359$	−5.83565e12			−0.978627			−0.489314	+	0.872108i	$0.662753\pi$
$360$		−	1.89885e12i		−	0.314035i
$361$	2.57121e12			0.419373
$362$		−	4.96456e12i		−	0.798616i
$363$	3.38713e12			0.537401
$364$		−	2.39418e12i		−	0.374672i
$365$		−	4.54780e12i		−	0.702000i
$366$	−2.52219e12			−0.384036
$367$		−	4.45054e12i		−	0.668471i	−0.942490	−	0.334236i	$-0.891522\pi$
$367$		−	4.45054e12i		−	0.668471i	0.942490	−	0.334236i	$-0.108478\pi$
$368$			5.04740e12i			0.747875i
$369$	2.89831e12			0.423655
$370$	9.88115e11			0.142495
$371$	1.34564e12			0.191451
$372$		−	1.73197e11i		−	0.0243123i
$373$	−6.13757e12			−0.850066			−0.425033	−	0.905178i	$-0.639737\pi$
$373$	−6.13757e12			−0.850066			−0.425033	+	0.905178i	$0.639737\pi$
$374$	5.76503e12			0.787851
$375$	3.89390e12			0.525083
$376$	2.21225e13			2.94371
$377$		−	6.59877e12i		−	0.866475i
$378$			9.87817e11i			0.128002i
$379$	2.88638e12			0.369111			0.184555	−	0.982822i	$-0.440915\pi$
$379$	2.88638e12			0.369111			0.184555	+	0.982822i	$0.440915\pi$
$380$	−1.01032e13			−1.27509
$381$	−2.89564e12			−0.360678
$382$	4.92039e12			0.604899
$383$	9.06675e12			1.10017			0.550083	−	0.835110i	$-0.314596\pi$
$383$	9.06675e12			1.10017			0.550083	+	0.835110i	$0.314596\pi$
$384$		−	8.16025e12i		−	0.977344i
$385$		−	4.49774e11i		−	0.0531729i
$386$			1.44086e13i			1.68145i
$387$			4.66228e12i			0.537085i
$388$			2.32421e12i			0.264312i
$389$	1.21502e13			1.36406			0.682030	−	0.731324i	$-0.261097\pi$
$389$	1.21502e13			1.36406			0.682030	+	0.731324i	$0.261097\pi$
$390$	2.31704e12			0.256810
$391$		−	1.09483e13i		−	1.19802i
$392$		−	1.40849e13i		−	1.52168i
$393$			8.32733e12i			0.888266i
$394$		−	1.44970e13i		−	1.52685i
$395$	−8.09158e12			−0.841488
$396$			1.72891e12i			0.177540i
$397$		−	3.53287e12i		−	0.358241i	−0.983827	−	0.179121i	$-0.942675\pi$
$397$		−	3.53287e12i		−	0.358241i	0.983827	−	0.179121i	$-0.0573252\pi$
$398$		−	7.96504e12i		−	0.797577i
$399$	2.66013e12			0.263050
$400$	−7.93280e12			−0.774688
$401$			8.52789e12i			0.822469i	0.911530	+	0.411234i	$0.134902\pi$
$401$			8.52789e12i			0.822469i	−0.911530	+	0.411234i	$0.865098\pi$
$402$	−1.30860e13			−1.24645
$403$	1.06965e11			0.0100628
$404$		−	1.01452e13i		−	0.942660i
$405$	−6.39940e11			−0.0587306
$406$			1.31392e13i			1.19108i
$407$	−4.55353e11			−0.0407734
$408$		−	2.00352e13i		−	1.77212i
$409$		−	6.06947e12i		−	0.530316i	−0.964205	−	0.265158i	$-0.914576\pi$
$409$		−	6.06947e12i		−	0.530316i	0.964205	−	0.265158i	$-0.0854241\pi$
$410$			1.35366e13i			1.16839i
$411$	−9.10701e12			−0.776545
$412$		−	1.25542e12i		−	0.105756i
$413$	−1.33313e12	−	4.39753e12i	−0.110949	−	0.365981i
$414$	4.90491e12			0.403301
$415$			8.09748e11i			0.0657824i
$416$	−5.26684e11			−0.0422750
$417$	1.21939e13			0.967084
$418$	6.95524e12			0.545043
$419$			8.83996e12i			0.684511i	0.939607	+	0.342255i	$0.111191\pi$
$419$			8.83996e12i			0.684511i	−0.939607	+	0.342255i	$0.888809\pi$
$420$	−3.08836e12			−0.236309
$421$			1.43920e12i			0.108820i	0.998519	+	0.0544102i	$0.0173279\pi$
$421$			1.43920e12i			0.108820i	−0.998519	+	0.0544102i	$0.982672\pi$
$422$	3.18668e13			2.38109
$423$		−	7.45561e12i		−	0.550531i
$424$		−	1.22273e13i		−	0.892279i
$425$	1.72070e13			1.24097
$426$		−	1.11594e13i		−	0.795412i
$427$			2.07622e12i			0.146263i
$428$	2.79352e13			1.94506
$429$	−1.06776e12			−0.0734833
$430$	−2.17752e13			−1.48122
$431$		−	4.30204e12i		−	0.289260i	−0.989486	−	0.144630i	$-0.953801\pi$
$431$		−	4.30204e12i		−	0.289260i	0.989486	−	0.144630i	$-0.0461991\pi$
$432$	3.11289e12			0.206893
$433$	2.66770e13			1.75266			0.876330	−	0.481712i	$-0.159985\pi$
$433$	2.66770e13			1.75266			0.876330	+	0.481712i	$0.159985\pi$
$434$	−2.12985e11			−0.0138325
$435$	−8.51201e12			−0.546495
$436$			5.08378e13i			3.22667i
$437$		−	1.32086e13i		−	0.828800i
$438$	2.14976e13			1.33358
$439$	1.51588e13			0.929701			0.464850	−	0.885389i	$-0.346108\pi$
$439$	1.51588e13			0.929701			0.464850	+	0.885389i	$0.346108\pi$
$440$	−4.08692e12			−0.247818
$441$	−4.74681e12			−0.284583
$442$	2.44477e13			1.44919
$443$			1.75776e13i			1.03024i	0.857117	+	0.515122i	$0.172253\pi$
$443$			1.75776e13i			1.03024i	−0.857117	+	0.515122i	$0.827747\pi$
$444$			3.12667e12i			0.181204i
$445$			1.09343e13i			0.626600i
$446$		−	5.82367e13i		−	3.30006i
$447$		−	9.83699e12i		−	0.551220i
$448$	−6.37067e12			−0.353016
$449$	3.46668e11			0.0189968			0.00949842	−	0.999955i	$-0.496977\pi$
$449$	3.46668e11			0.0189968			0.00949842	+	0.999955i	$0.496977\pi$
$450$			7.70885e12i			0.417760i
$451$		−	6.23806e12i		−	0.334323i
$452$		−	6.45527e13i		−	3.42154i
$453$			4.56370e12i			0.239236i
$454$	−3.96179e13			−2.05406
$455$		−	1.90735e12i		−	0.0978077i
$456$		−	2.41716e13i		−	1.22597i
$457$			2.42767e13i			1.21789i	0.793212	+	0.608945i	$0.208407\pi$
$457$			2.42767e13i			1.21789i	−0.793212	+	0.608945i	$0.791593\pi$
$458$	6.89026e13			3.41907
$459$	−6.75217e12			−0.331421
$460$			1.53349e13i			0.744548i
$461$	−3.10274e12			−0.149019			−0.0745093	−	0.997220i	$-0.523739\pi$
$461$	−3.10274e12			−0.149019			−0.0745093	+	0.997220i	$0.523739\pi$
$462$	2.12609e12			0.101012
$463$			2.46401e13i			1.15808i	0.815300	+	0.579038i	$0.196572\pi$
$463$			2.46401e13i			1.15808i	−0.815300	+	0.579038i	$0.803428\pi$
$464$	4.14054e13			1.92516
$465$		−	1.37979e11i		−	0.00634670i
$466$	−2.48678e13			−1.13164
$467$			3.96954e13i			1.78713i	0.448937	+	0.893564i	$0.351803\pi$
$467$			3.96954e13i			1.78713i	−0.448937	+	0.893564i	$0.648197\pi$
$468$			7.33176e12i			0.326572i
$469$			1.07722e13i			0.474721i
$470$	3.48215e13			1.51830
$471$		−	1.90301e13i		−	0.820989i
$472$	−3.99586e13	+	1.21136e13i	−1.70569	+	0.517087i
$473$	1.00347e13			0.423836
$474$		−	3.82491e13i		−	1.59856i
$475$	2.07594e13			0.858514
$476$	−3.25860e13			−1.33351
$477$	−4.12077e12			−0.166873
$478$			4.61392e13i			1.84897i
$479$	2.53641e13			1.00587			0.502934	−	0.864325i	$-0.332254\pi$
$479$	2.53641e13			1.00587			0.502934	+	0.864325i	$0.332254\pi$
$480$			6.79391e11i			0.0266633i
$481$	−1.93101e12			−0.0749997
$482$			2.42098e13i			0.930584i
$483$		−	4.03763e12i		−	0.153600i
$484$	−5.00577e13			−1.88471
$485$			1.85160e12i			0.0689984i
$486$		−	3.02501e12i		−	0.111570i
$487$	−1.22072e13			−0.445628			−0.222814	−	0.974861i	$-0.571524\pi$
$487$	−1.22072e13			−0.445628			−0.222814	+	0.974861i	$0.571524\pi$
$488$	1.88658e13			0.681674
$489$	−3.55457e12			−0.127129
$490$		−	2.21700e13i		−	0.784848i
$491$	−4.37652e13			−1.53363			−0.766817	−	0.641866i	$-0.778160\pi$
$491$	−4.37652e13			−1.53363			−0.766817	+	0.641866i	$0.778160\pi$
$492$	−4.28334e13			−1.48579
$493$	−8.98124e13			−3.08391
$494$	2.94950e13			1.00257
$495$			1.37735e12i			0.0463467i
$496$			6.71177e11i			0.0223578i
$497$	−9.18620e12			−0.302938
$498$	−3.82770e12			−0.124966
$499$	−3.80257e13			−1.22906			−0.614531	−	0.788893i	$-0.710655\pi$
$499$	−3.80257e13			−1.22906			−0.614531	+	0.788893i	$0.710655\pi$
$500$	−5.75471e13			−1.84151
$501$	−1.17040e13			−0.370805
$502$		−	3.35741e13i		−	1.05314i
$503$			7.64477e12i			0.237424i	0.992929	+	0.118712i	$0.0378765\pi$
$503$			7.64477e12i			0.237424i	−0.992929	+	0.118712i	$0.962124\pi$
$504$		−	7.38881e12i		−	0.227207i
$505$		−	8.08228e12i		−	0.246081i
$506$		−	1.05569e13i		−	0.318261i
$507$	1.48130e13			0.442183
$508$	4.27940e13			1.26493
$509$		−	6.27268e13i		−	1.83596i	−0.396621	−	0.917982i	$-0.629817\pi$
$509$		−	6.27268e13i		−	1.83596i	0.396621	−	0.917982i	$-0.370183\pi$
$510$		−	3.15361e13i		−	0.914022i
$511$		−	1.76964e13i		−	0.507903i
$512$			6.41123e13i			1.82218i
$513$	−8.14617e12			−0.229280
$514$			9.54227e13i			2.65972i
$515$		−	1.00014e12i		−	0.0276075i
$516$		−	6.89028e13i		−	1.88360i
$517$	−1.60468e13			−0.434446
$518$	3.84496e12			0.103096
$519$			3.33555e13i			0.885788i
$520$	−1.73313e13			−0.455843
$521$	6.93620e12			0.180690			0.0903448	−	0.995911i	$-0.471203\pi$
$521$	6.93620e12			0.180690			0.0903448	+	0.995911i	$0.471203\pi$
$522$		−	4.02365e13i		−	1.03817i
$523$	3.19441e13			0.816360			0.408180	−	0.912901i	$-0.366164\pi$
$523$	3.19441e13			0.816360			0.408180	+	0.912901i	$0.366164\pi$
$524$		−	1.23068e14i		−	3.11522i
$525$	6.34579e12			0.159107
$526$		−	5.99650e13i		−	1.48926i
$527$		−	1.45585e12i		−	0.0358149i
$528$		−	6.69992e12i		−	0.163268i
$529$	2.13780e13			0.516047
$530$		−	1.92461e13i		−	0.460218i
$531$	4.08247e12	+	1.34666e13i	0.0967052	+	0.318997i
$532$	−3.93135e13			−0.922535
$533$		−	2.64537e13i		−	0.614964i
$534$	−5.16866e13			−1.19034
$535$	2.22548e13			0.507756
$536$	9.78824e13			2.21249
$537$		−	2.91343e13i		−	0.652429i
$538$	7.13478e13			1.58296
$539$			1.02166e13i			0.224576i
$540$	9.45754e12			0.205972
$541$			2.47793e13i			0.534691i	0.963601	+	0.267345i	$0.0861464\pi$
$541$			2.47793e13i			0.534691i	−0.963601	+	0.267345i	$0.913854\pi$
$542$		−	2.50974e13i		−	0.536577i
$543$	1.25149e13			0.265111
$544$			7.16842e12i			0.150463i
$545$			4.05004e13i			0.842319i
$546$	9.01609e12			0.185804
$547$	−3.88238e13			−0.792797			−0.396398	−	0.918079i	$-0.629740\pi$
$547$	−3.88238e13			−0.792797			−0.396398	+	0.918079i	$0.629740\pi$
$548$	1.34591e14			2.72340
$549$		−	6.35806e12i		−	0.127486i
$550$	1.65919e13			0.329672
$551$	−1.08354e14			−2.13348
$552$	−3.66884e13			−0.715869
$553$	−3.14860e13			−0.608824
$554$		−	1.06401e14i		−	2.03890i
$555$			2.49089e12i			0.0473031i
$556$	−1.80212e14			−3.39163
$557$	7.83101e13			1.46064			0.730318	−	0.683107i	$-0.239372\pi$
$557$	7.83101e13			1.46064			0.730318	+	0.683107i	$0.239372\pi$
$558$	6.52229e11			0.0120567
$559$	4.25539e13			0.779616
$560$	1.19681e13			0.217312
$561$			1.45328e13i			0.261538i
$562$			2.23287e13i			0.398274i
$563$			6.47119e13i			1.14404i	0.820239	+	0.572021i	$0.193840\pi$
$563$			6.47119e13i			1.14404i	−0.820239	+	0.572021i	$0.806160\pi$
$564$			1.10185e14i			1.93075i
$565$		−	5.14264e13i		−	0.893191i
$566$	−2.10288e13			−0.362020
$567$	−2.49014e12			−0.0424921
$568$			8.34714e13i			1.41187i
$569$		−	3.52782e13i		−	0.591487i	−0.955267	−	0.295743i	$-0.904433\pi$
$569$		−	3.52782e13i		−	0.591487i	0.955267	−	0.295743i	$-0.0955673\pi$
$570$		−	3.80468e13i		−	0.632330i
$571$			9.85801e13i			1.62409i	0.583598	+	0.812043i	$0.301644\pi$
$571$			9.85801e13i			1.62409i	−0.583598	+	0.812043i	$0.698356\pi$
$572$	1.57802e13			0.257711
$573$			1.24036e13i			0.200804i
$574$			5.26735e13i			0.845343i
$575$		−	3.15094e13i		−	0.501304i
$576$	1.95090e13			0.307697
$577$	−2.85789e13			−0.446855			−0.223427	−	0.974721i	$-0.571725\pi$
$577$	−2.85789e13			−0.446855			−0.223427	+	0.974721i	$0.571725\pi$
$578$		−	2.20546e14i		−	3.41869i
$579$	−3.63218e13			−0.558180
$580$	1.25797e14			1.91660
$581$			3.15089e12i			0.0475941i
$582$	−8.75257e12			−0.131075
$583$			8.86919e12i			0.131686i
$584$	−1.60800e14			−2.36713
$585$			5.84091e12i			0.0852514i
$586$		−	9.32327e13i		−	1.34921i
$587$			1.17700e14i			1.68883i	0.535692	+	0.844413i	$0.320051\pi$
$587$			1.17700e14i			1.68883i	−0.535692	+	0.844413i	$0.679949\pi$
$588$	7.01520e13			0.998053
$589$		−	1.75641e12i		−	0.0247771i
$590$	−6.28961e13	+	1.90672e13i	−0.879759	+	0.266702i
$591$	3.65447e13			0.506858
$592$		−	1.21166e13i		−	0.166637i
$593$	−5.84651e13			−0.797302			−0.398651	−	0.917103i	$-0.630522\pi$
$593$	−5.84651e13			−0.797302			−0.398651	+	0.917103i	$0.630522\pi$
$594$	−6.51077e12			−0.0880441
$595$	−2.59599e13			−0.348112
$596$			1.45379e14i			1.93317i
$597$	2.00786e13			0.264766
$598$		−	4.47685e13i		−	0.585419i
$599$	−1.34740e14			−1.74728			−0.873639	−	0.486575i	$-0.838246\pi$
$599$	−1.34740e14			−1.74728			−0.873639	+	0.486575i	$0.838246\pi$
$600$		−	5.76617e13i		−	0.741534i
$601$			1.67071e13i			0.213073i	0.994309	+	0.106536i	$0.0339761\pi$
$601$			1.67071e13i			0.213073i	−0.994309	+	0.106536i	$0.966024\pi$
$602$	−8.47318e13			−1.07168
$603$		−	3.29878e13i		−	0.413777i
$604$		−	6.74459e13i		−	0.839018i
$605$	−3.98789e13			−0.492001
$606$	3.82052e13			0.467476
$607$	2.65267e13			0.321914			0.160957	−	0.986961i	$-0.448542\pi$
$607$	2.65267e13			0.321914			0.160957	+	0.986961i	$0.448542\pi$
$608$			8.64836e12i			0.104092i
$609$	−3.31220e13			−0.395394
$610$	2.96954e13			0.351593
$611$	−6.80495e13			−0.799133
$612$	9.97888e13			1.16232
$613$			2.40505e13i			0.277857i	0.990302	+	0.138928i	$0.0443658\pi$
$613$			2.40505e13i			0.277857i	−0.990302	+	0.138928i	$0.955634\pi$
$614$			9.15892e13i			1.04955i
$615$	−3.41236e13			−0.387864
$616$	−1.59030e13			−0.179298
$617$	−5.25545e13			−0.587738			−0.293869	−	0.955846i	$-0.594943\pi$
$617$	−5.25545e13			−0.587738			−0.293869	+	0.955846i	$0.594943\pi$
$618$	4.72771e12			0.0524455
$619$	3.62492e13			0.398882			0.199441	−	0.979910i	$-0.436087\pi$
$619$	3.62492e13			0.398882			0.199441	+	0.979910i	$0.436087\pi$
$620$			2.03916e12i			0.0222583i
$621$			1.23645e13i			0.133881i
$622$		−	2.46034e13i		−	0.264267i
$623$			4.25475e13i			0.453350i
$624$		−	2.84123e13i		−	0.300319i
$625$	2.28772e13			0.239885
$626$	2.25895e14			2.34982
$627$			1.75331e13i			0.180934i
$628$			2.81242e14i			2.87927i
$629$			2.62820e13i			0.266935i
$630$		−	1.16302e13i		−	0.117188i
$631$	−2.24031e13			−0.223956			−0.111978	−	0.993711i	$-0.535719\pi$
$631$	−2.24031e13			−0.223956			−0.111978	+	0.993711i	$0.535719\pi$
$632$			2.86101e14i			2.83749i
$633$			8.03314e13i			0.790436i
$634$			2.23643e14i			2.18328i
$635$	3.40923e13			0.330208
$636$	6.09000e13			0.585237
$637$			4.33255e13i			0.413091i
$638$	−8.66015e13			−0.819261
$639$	2.81311e13			0.264048
$640$			9.60760e13i			0.894777i
$641$	−2.86399e13			−0.264656			−0.132328	−	0.991206i	$-0.542245\pi$
$641$	−2.86399e13			−0.264656			−0.132328	+	0.991206i	$0.542245\pi$
$642$			1.05199e14i			0.964576i
$643$	−1.20323e13			−0.109470			−0.0547350	−	0.998501i	$-0.517431\pi$
$643$	−1.20323e13			−0.109470			−0.0547350	+	0.998501i	$0.517431\pi$
$644$			5.96713e13i			0.538687i
$645$		−	5.48920e13i		−	0.491712i
$646$		−	4.01441e14i		−	3.56828i
$647$	3.55855e13			0.313871			0.156936	−	0.987609i	$-0.449838\pi$
$647$	3.55855e13			0.313871			0.156936	+	0.987609i	$0.449838\pi$
$648$			2.26269e13i			0.198039i
$649$	2.89844e13	−	8.78674e12i	0.251734	−	0.0763140i
$650$	7.03609e13			0.606407
$651$		−	5.36903e11i		−	0.00459189i
$652$	5.25323e13			0.445849
$653$	−6.82597e13			−0.574908			−0.287454	−	0.957794i	$-0.592809\pi$
$653$	−6.82597e13			−0.574908			−0.287454	+	0.957794i	$0.592809\pi$
$654$	−1.91446e14			−1.60014
$655$		−	9.80431e13i		−	0.813225i
$656$	1.65989e14			1.36635
$657$			5.41921e13i			0.442700i
$658$	1.35498e14			1.09851
$659$			1.49491e14i			1.20278i	0.798954	+	0.601392i	$0.205387\pi$
$659$			1.49491e14i			1.20278i	−0.798954	+	0.601392i	$0.794613\pi$
$660$		−	2.03556e13i		−	0.162541i
$661$	1.75015e14			1.38697			0.693487	−	0.720469i	$-0.256073\pi$
$661$	1.75015e14			1.38697			0.693487	+	0.720469i	$0.256073\pi$
$662$		−	2.57618e13i		−	0.202622i
$663$			6.16289e13i			0.481080i
$664$	2.86310e13			0.221817
$665$	−3.13194e13			−0.240827
$666$	−1.17745e13			−0.0898610
$667$			1.64464e14i			1.24578i
$668$	1.72971e14			1.30044
$669$	1.46806e14			1.09550
$670$	1.54070e14			1.14115
$671$	−1.36845e13			−0.100604
$672$			2.64365e12i			0.0192911i
$673$		−	2.32901e13i		−	0.168693i	−0.996437	−	0.0843463i	$-0.973120\pi$
$673$		−	2.32901e13i		−	0.168693i	0.996437	−	0.0843463i	$-0.0268802\pi$
$674$	1.62079e13			0.116527
$675$	−1.94328e13			−0.138681
$676$	−2.18918e14			−1.55077
$677$	−9.46918e13			−0.665838			−0.332919	−	0.942955i	$-0.608034\pi$
$677$	−9.46918e13			−0.665838			−0.332919	+	0.942955i	$0.608034\pi$
$678$	2.43094e14			1.69678
$679$			7.20496e12i			0.0499209i
$680$			2.35888e14i			1.62241i
$681$		−	9.98708e13i		−	0.681872i
$682$		−	1.40380e12i		−	0.00951445i
$683$			2.43084e13i			0.163551i	0.996651	+	0.0817754i	$0.0260590\pi$
$683$			2.43084e13i			0.163551i	−0.996651	+	0.0817754i	$0.973941\pi$
$684$	1.20390e14			0.804103
$685$	1.07223e14			0.710942
$686$		−	1.87314e14i		−	1.23296i
$687$			1.73693e14i			1.13501i
$688$			2.67014e14i			1.73218i
$689$			3.76115e13i			0.242228i
$690$	−5.77487e13			−0.369230
$691$		−	2.82671e14i		−	1.79428i	−0.441745	−	0.897141i	$-0.645640\pi$
$691$		−	2.82671e14i		−	1.79428i	0.441745	−	0.897141i	$-0.354360\pi$
$692$		−	4.92953e14i		−	3.10653i
$693$			5.35955e12i			0.0335322i
$694$	5.24385e14			3.25726
$695$	−1.43567e14			−0.885384
$696$			3.00966e14i			1.84277i
$697$	−3.60047e14			−2.18875
$698$	1.23950e14			0.748116
$699$		−	6.26879e13i		−	0.375662i
$700$	−9.37830e13			−0.558000
$701$		−	1.30823e14i		−	0.772850i	−0.922321	−	0.386425i	$-0.873710\pi$
$701$		−	1.30823e14i		−	0.772850i	0.922321	−	0.386425i	$-0.126290\pi$
$702$	−2.76102e13			−0.161951
$703$			3.17080e13i			0.184668i
$704$		−	4.19895e13i		−	0.242816i
$705$			8.77798e13i			0.504021i
$706$	−1.05840e13			−0.0603430
$707$		−	3.14498e13i		−	0.178041i
$708$	−6.03339e13	−	1.99021e14i	−0.339152	−	1.11875i
$709$	−2.90493e14			−1.62145			−0.810726	−	0.585425i	$-0.800928\pi$
$709$	−2.90493e14			−1.62145			−0.810726	+	0.585425i	$0.800928\pi$
$710$			1.31387e14i			0.728214i
$711$	9.64202e13			0.530664
$712$	3.86613e14			2.11289
$713$	−2.66594e12			−0.0144678
$714$		−	1.22713e14i		−	0.661303i
$715$	1.25715e13			0.0672754
$716$			4.30569e14i			2.28812i
$717$	−1.16310e14			−0.613792
$718$			3.24779e14i			1.70203i
$719$			2.13909e14i			1.11323i	0.830770	+	0.556616i	$0.187900\pi$
$719$			2.13909e14i			1.11323i	−0.830770	+	0.556616i	$0.812100\pi$
$720$	−3.66501e13			−0.189414
$721$		−	3.89176e12i		−	0.0199742i
$722$		−	1.43099e14i		−	0.729374i
$723$	−6.10291e13			−0.308920
$724$	−1.84955e14			−0.929765
$725$	−2.58481e14			−1.29044
$726$		−	1.88508e14i		−	0.934647i
$727$	−2.68389e13			−0.132158			−0.0660789	−	0.997814i	$-0.521049\pi$
$727$	−2.68389e13			−0.132158			−0.0660789	+	0.997814i	$0.521049\pi$
$728$	−6.74397e13			−0.329806
$729$	7.62560e12			0.0370370
$730$	−2.53105e14			−1.22092
$731$		−	5.79179e14i		−	2.77476i
$732$			9.39644e13i			0.447103i
$733$	3.37680e14			1.59582			0.797912	−	0.602774i	$-0.205938\pi$
$733$	3.37680e14			1.59582			0.797912	+	0.602774i	$0.205938\pi$
$734$	−2.47692e14			−1.16260
$735$	5.58872e13			0.260541
$736$	1.31268e13			0.0607811
$737$	−7.10000e13			−0.326528
$738$		−	1.61303e14i		−	0.736820i
$739$		−	1.09944e14i		−	0.498828i	−0.968397	−	0.249414i	$-0.919762\pi$
$739$		−	1.09944e14i		−	0.498828i	0.968397	−	0.249414i	$-0.0802380\pi$
$740$		−	3.68123e13i		−	0.165895i
$741$			7.43524e13i			0.332816i
$742$		−	7.48905e13i		−	0.332972i
$743$	−3.17357e14			−1.40154			−0.700769	−	0.713389i	$-0.747159\pi$
$743$	−3.17357e14			−1.40154			−0.700769	+	0.713389i	$0.747159\pi$
$744$	−4.87863e12			−0.0214010
$745$			1.15817e14i			0.504652i
$746$			3.41583e14i			1.47843i
$747$		−	9.64905e12i		−	0.0414841i
$748$		−	2.14777e14i		−	0.917232i
$749$	8.65979e13			0.367365
$750$		−	2.16712e14i		−	0.913224i
$751$			7.95007e13i			0.332791i	0.986059	+	0.166395i	$0.0532128\pi$
$751$			7.95007e13i			0.332791i	−0.986059	+	0.166395i	$0.946787\pi$
$752$		−	4.26992e14i		−	1.77554i
$753$	8.46353e13			0.349604
$754$	−3.67250e14			−1.50697
$755$		−	5.37314e13i		−	0.219025i
$756$	3.68012e13			0.149023
$757$	−2.85972e13			−0.115039			−0.0575193	−	0.998344i	$-0.518319\pi$
$757$	−2.85972e13			−0.115039			−0.0575193	+	0.998344i	$0.518319\pi$
$758$		−	1.60639e14i		−	0.641957i
$759$	2.66123e13			0.105651
$760$			2.84587e14i			1.12240i
$761$	−6.02955e12			−0.0236245			−0.0118122	−	0.999930i	$-0.503760\pi$
$761$	−6.02955e12			−0.0236245			−0.0118122	+	0.999930i	$0.503760\pi$
$762$			1.61155e14i			0.627291i
$763$			1.57595e14i			0.609425i
$764$		−	1.83310e14i		−	0.704236i
$765$	7.94976e13			0.303422
$766$		−	5.04604e14i		−	1.91341i
$767$	1.22914e14	−	3.72618e13i	0.463046	−	0.140374i
$768$	−3.11760e14			−1.16685
$769$			2.20971e14i			0.821681i	0.911707	+	0.410841i	$0.134765\pi$
$769$			2.20971e14i			0.821681i	−0.911707	+	0.410841i	$0.865235\pi$
$770$	−2.50318e13			−0.0924782
$771$	−2.40546e14			−0.882930
$772$	5.36792e14			1.95758
$773$		−	4.20413e14i		−	1.52328i	−0.648003	−	0.761638i	$-0.724395\pi$
$773$		−	4.20413e14i		−	1.52328i	0.648003	−	0.761638i	$-0.275605\pi$
$774$	2.59476e14			0.934098
$775$		−	4.18995e12i		−	0.0149865i
$776$	6.54687e13			0.232662
$777$			9.69255e12i			0.0342242i
$778$		−	6.76209e14i		−	2.37237i
$779$	−4.34379e14			−1.51420

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)

\(f(q)\)	\(=\)	\(q-55.6543i q^{2} +140.296 q^{3} -2073.41 q^{4} -1651.80 q^{5} -7808.09i q^{6} -6427.48 q^{7} +58404.0i q^{8} +19683.0 q^{9} +O(q^{10})\)	\(q-55.6543i q^{2} +140.296 q^{3} -2073.41 q^{4} -1651.80 q^{5} -7808.09i q^{6} -6427.48 q^{7} +58404.0i q^{8} +19683.0 q^{9} +91929.7i q^{10} -42364.0i q^{11} -290891. q^{12} -179652. i q^{13} +357717. i q^{14} -231741. q^{15} +1.12727e6 q^{16} -2.44515e6 q^{17} -1.09544e6i q^{18} -2.94996e6 q^{19} +3.42485e6 q^{20} -901750. q^{21} -2.35774e6 q^{22} +4.47755e6i q^{23} +8.19385e6i q^{24} -7.03719e6 q^{25} -9.99843e6 q^{26} +2.76145e6 q^{27} +1.33268e7 q^{28} +3.67308e7 q^{29} +1.28974e7i q^{30} +595402. i q^{31} -2.93168e6i q^{32} -5.94350e6i q^{33} +1.36083e8i q^{34} +1.06169e7 q^{35} -4.08108e7 q^{36} -1.07486e7i q^{37} +1.64178e8i q^{38} -2.52045e7i q^{39} -9.64716e7i q^{40} +1.47249e8 q^{41} +5.01863e7i q^{42} +2.36868e8i q^{43} +8.78377e7i q^{44} -3.25123e7 q^{45} +2.49195e8 q^{46} -3.78784e8i q^{47} +1.58151e8 q^{48} -2.41163e8 q^{49} +3.91650e8i q^{50} -3.43046e8 q^{51} +3.72492e8i q^{52} -2.09357e8 q^{53} -1.53687e8i q^{54} +6.99767e7i q^{55} -3.75390e8i q^{56} -4.13868e8 q^{57} -2.04423e9i q^{58} +(2.07411e8 + 6.84177e8i) q^{59} +4.80493e8 q^{60} -3.23023e8i q^{61} +3.31367e7 q^{62} -1.26512e8 q^{63} +9.91162e8 q^{64} +2.96749e8i q^{65} -3.30782e8 q^{66} -1.67595e9i q^{67} +5.06980e9 q^{68} +6.28183e8i q^{69} -5.90876e8i q^{70} +1.42921e9 q^{71} +1.14957e9i q^{72} +2.75324e9i q^{73} -5.98206e8 q^{74} -9.87291e8 q^{75} +6.11647e9 q^{76} +2.72293e8i q^{77} -1.40274e9 q^{78} +4.89865e9 q^{79} -1.86202e9 q^{80} +3.87420e8 q^{81} -8.19506e9i q^{82} -4.90223e8i q^{83} +1.86969e9 q^{84} +4.03890e9 q^{85} +1.31827e10 q^{86} +5.15318e9 q^{87} +2.47422e9 q^{88} -6.61963e9i q^{89} +1.80945e9i q^{90} +1.15471e9i q^{91} -9.28378e9i q^{92} +8.35325e7i q^{93} -2.10810e10 q^{94} +4.87274e9 q^{95} -4.11304e8i q^{96} -1.12096e9i q^{97} +1.34218e10i q^{98} -8.33850e8i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 100q - 51200q^{4} + 18392q^{7} + 1968300q^{9} + O(q^{10}) \)	\( 100q - 51200q^{4} + 18392q^{7} + 1968300q^{9} + 620136q^{12} + 662904q^{15} + 27925160q^{16} - 2053136q^{17} - 5169828q^{19} - 1076324q^{20} - 1829224q^{22} + 215378180q^{25} - 22082700q^{26} - 102921320q^{28} - 112503588q^{29} - 76491392q^{35} - 1007769600q^{36} + 473464516q^{41} + 1215405588q^{46} - 1676272320q^{48} + 1975297276q^{49} + 733970808q^{51} + 3267506728q^{53} + 591502824q^{57} + 508142200q^{59} + 1264196808q^{60} - 6538206968q^{62} + 362009736q^{63} - 10324137972q^{64} - 2764346616q^{66} + 9997685952q^{68} + 14908523204q^{71} + 4863508712q^{74} + 1890481680q^{75} + 2044437240q^{76} - 758396196q^{78} - 3599839500q^{79} - 23217941144q^{80} + 38742048900q^{81} - 13094894808q^{84} + 23360564412q^{85} + 12186923752q^{86} + 7965322272q^{87} + 32415437996q^{88} - 22098322280q^{94} + 7834510028q^{95} + O(q^{100}) \)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Properties

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