LMFDB - Embedded newform 177.7.c.a.58.1

Newspace parameters

Level:	$ N $	$=$	$ 177 = 3 \cdot 59 $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	177.c (of order $2$, degree $1$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$40.7195728007$
Analytic rank:	$0$
Dimension:	$60$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			58.1
Character	$\chi$	$=$	177.58
Dual form			177.7.c.a.58.60

$q$-expansion

$f(q)$	$=$	$q-15.3575i q^{2} +15.5885 q^{3} -171.853 q^{4} -142.367 q^{5} -239.400i q^{6} +584.638 q^{7} +1656.35i q^{8} +243.000 q^{9} +O(q^{10})$	\(q-15.3575i q^{2} +15.5885 q^{3} -171.853 q^{4} -142.367 q^{5} -239.400i q^{6} +584.638 q^{7} +1656.35i q^{8} +243.000 q^{9} +2186.40i q^{10} +1773.45i q^{11} -2678.92 q^{12} -2047.24i q^{13} -8978.57i q^{14} -2219.28 q^{15} +14438.8 q^{16} +3646.36 q^{17} -3731.87i q^{18} +562.351 q^{19} +24466.2 q^{20} +9113.60 q^{21} +27235.7 q^{22} -12118.1i q^{23} +25819.9i q^{24} +4643.38 q^{25} -31440.5 q^{26} +3788.00 q^{27} -100472. q^{28} +29947.3 q^{29} +34082.6i q^{30} -4051.38i q^{31} -115738. i q^{32} +27645.3i q^{33} -55998.9i q^{34} -83233.1 q^{35} -41760.2 q^{36} +43754.1i q^{37} -8636.31i q^{38} -31913.4i q^{39} -235810. i q^{40} +99607.1 q^{41} -139962. i q^{42} -20260.9i q^{43} -304772. i q^{44} -34595.2 q^{45} -186104. q^{46} +114033. i q^{47} +225079. q^{48} +224152. q^{49} -71310.6i q^{50} +56841.1 q^{51} +351825. i q^{52} -220188. q^{53} -58174.1i q^{54} -252480. i q^{55} +968364. i q^{56} +8766.19 q^{57} -459916. i q^{58} +(-168079. + 118025. i) q^{59} +381390. q^{60} -443485. i q^{61} -62219.1 q^{62} +142067. q^{63} -853356. q^{64} +291460. i q^{65} +424562. q^{66} -249807. i q^{67} -626637. q^{68} -188903. i q^{69} +1.27825e6i q^{70} +605052. q^{71} +402493. i q^{72} -199631. i q^{73} +671954. q^{74} +72383.1 q^{75} -96641.7 q^{76} +1.03682e6i q^{77} -490110. q^{78} +541051. q^{79} -2.05561e6 q^{80} +59049.0 q^{81} -1.52972e6i q^{82} -742610. i q^{83} -1.56620e6 q^{84} -519121. q^{85} -311157. q^{86} +466832. q^{87} -2.93745e6 q^{88} +827595. i q^{89} +531296. i q^{90} -1.19690e6i q^{91} +2.08253e6i q^{92} -63154.8i q^{93} +1.75126e6 q^{94} -80060.3 q^{95} -1.80417e6i q^{96} -497661. i q^{97} -3.44242e6i q^{98} +430947. i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 60q - 1920q^{4} - 408q^{7} + 14580q^{9} + O(q^{10}) $	$ 60q - 1920q^{4} - 408q^{7} + 14580q^{9} - 1944q^{12} - 4536q^{15} + 56616q^{16} + 8480q^{17} + 11376q^{19} + 40796q^{20} - 8232q^{22} + 197940q^{25} + 147252q^{26} + 71640q^{28} + 63456q^{29} - 364432q^{35} - 466560q^{36} + 99632q^{41} - 470316q^{46} + 171072q^{48} + 1737420q^{49} + 60912q^{51} + 92240q^{53} + 186624q^{57} + 917264q^{59} + 1063368q^{60} - 115768q^{62} - 99144q^{63} - 1107444q^{64} + 1172232q^{66} - 4247232q^{68} + 1498048q^{71} + 1161448q^{74} - 1477440q^{75} - 1045320q^{76} - 1060452q^{78} - 90600q^{79} + 77096q^{80} + 3542940q^{81} - 2225880q^{84} - 693408q^{85} - 1567768q^{86} + 1821528q^{87} + 62892q^{88} + 5268696q^{94} + 296128q^{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 $

$2$		−	15.3575i		−	1.91969i	−0.280535	−	0.959844i	$-0.590512\pi$
$2$		−	15.3575i		−	1.91969i	0.280535	−	0.959844i	$-0.409488\pi$
$3$	15.5885			0.577350
$3$	15.5885			0.577350
$4$	−171.853			−2.68520
$5$	−142.367			−1.13894			−0.569468	−	0.822013i	$-0.692851\pi$
$5$	−142.367			−1.13894			−0.569468	+	0.822013i	$0.692851\pi$
$6$		−	239.400i		−	1.10833i
$7$	584.638			1.70448			0.852242	−	0.523149i	$-0.175243\pi$
$7$	584.638			1.70448			0.852242	+	0.523149i	$0.175243\pi$
$8$			1656.35i			3.23506i
$9$	243.000			0.333333
$10$			2186.40i			2.18640i
$11$			1773.45i			1.33242i	0.745766	+	0.666208i	$0.232084\pi$
$11$			1773.45i			1.33242i	−0.745766	+	0.666208i	$0.767916\pi$
$12$	−2678.92			−1.55030
$13$		−	2047.24i		−	0.931836i	−0.884828	−	0.465918i	$-0.845724\pi$
$13$		−	2047.24i		−	0.931836i	0.884828	−	0.465918i	$-0.154276\pi$
$14$		−	8978.57i		−	3.27207i
$15$	−2219.28			−0.657565
$16$	14438.8			3.52510
$17$	3646.36			0.742185			0.371093	−	0.928596i	$-0.378983\pi$
$17$	3646.36			0.742185			0.371093	+	0.928596i	$0.378983\pi$
$18$		−	3731.87i		−	0.639896i
$19$	562.351			0.0819874			0.0409937	−	0.999159i	$-0.486948\pi$
$19$	562.351			0.0819874			0.0409937	+	0.999159i	$0.486948\pi$
$20$	24466.2			3.05827
$21$	9113.60			0.984084
$22$	27235.7			2.55782
$23$		−	12118.1i		−	0.995982i	−0.867182	−	0.497991i	$-0.834071\pi$
$23$		−	12118.1i		−	0.995982i	0.867182	−	0.497991i	$-0.165929\pi$
$24$			25819.9i			1.86776i
$25$	4643.38			0.297176
$26$	−31440.5			−1.78883
$27$	3788.00			0.192450
$28$	−100472.			−4.57688
$29$	29947.3			1.22790			0.613951	−	0.789344i	$-0.289579\pi$
$29$	29947.3			1.22790			0.613951	+	0.789344i	$0.289579\pi$
$30$			34082.6i			1.26232i
$31$		−	4051.38i		−	0.135993i	−0.997686	−	0.0679967i	$-0.978339\pi$
$31$		−	4051.38i		−	0.135993i	0.997686	−	0.0679967i	$-0.0216608\pi$
$32$		−	115738.i		−	3.53203i
$33$			27645.3i			0.769271i
$34$		−	55998.9i		−	1.42476i
$35$	−83233.1			−1.94130
$36$	−41760.2			−0.895067
$37$			43754.1i			0.863802i	0.901921	+	0.431901i	$0.142157\pi$
$37$			43754.1i			0.863802i	−0.901921	+	0.431901i	$0.857843\pi$
$38$		−	8636.31i		−	0.157390i
$39$		−	31913.4i		−	0.537996i
$40$		−	235810.i		−	3.68452i
$41$	99607.1			1.44524			0.722618	−	0.691248i	$-0.242939\pi$
$41$	99607.1			1.44524			0.722618	+	0.691248i	$0.242939\pi$
$42$		−	139962.i		−	1.88913i
$43$		−	20260.9i		−	0.254832i	−0.991849	−	0.127416i	$-0.959332\pi$
$43$		−	20260.9i		−	0.254832i	0.991849	−	0.127416i	$-0.0406683\pi$
$44$		−	304772.i		−	3.57780i
$45$	−34595.2			−0.379645
$46$	−186104.			−1.91197
$47$			114033.i			1.09834i	0.835710	+	0.549171i	$0.185056\pi$
$47$			114033.i			1.09834i	−0.835710	+	0.549171i	$0.814944\pi$
$48$	225079.			2.03522
$49$	224152.			1.90526
$50$		−	71310.6i		−	0.570485i
$51$	56841.1			0.428501
$52$			351825.i			2.50217i
$53$	−220188.			−1.47899			−0.739495	−	0.673162i	$-0.764936\pi$
$53$	−220188.			−1.47899			−0.739495	+	0.673162i	$0.764936\pi$
$54$		−	58174.1i		−	0.369444i
$55$		−	252480.i		−	1.51754i
$56$			968364.i			5.51410i
$57$	8766.19			0.0473354
$58$		−	459916.i		−	2.35719i
$59$	−168079.	+	118025.i	−0.818385	+	0.574671i
$59$	−168079.	+	118025.i	−0.818385	+	0.574671i
$60$	381390.			1.76569
$61$		−	443485.i		−	1.95384i	−0.213597	−	0.976922i	$-0.568518\pi$
$61$		−	443485.i		−	1.95384i	0.213597	−	0.976922i	$-0.431482\pi$
$62$	−62219.1			−0.261065
$63$	142067.			0.568161
$64$	−853356.			−3.25529
$65$			291460.i			1.06130i
$66$	424562.			1.47676
$67$		−	249807.i		−	0.830577i	−0.909690	−	0.415288i	$-0.863681\pi$
$67$		−	249807.i		−	0.830577i	0.909690	−	0.415288i	$-0.136319\pi$
$68$	−626637.			−1.99292
$69$		−	188903.i		−	0.575031i
$70$			1.27825e6i			3.72668i
$71$	605052.			1.69051			0.845255	−	0.534363i	$-0.179448\pi$
$71$	605052.			1.69051			0.845255	+	0.534363i	$0.179448\pi$
$72$			402493.i			1.07835i
$73$		−	199631.i		−	0.513169i	−0.966522	−	0.256584i	$-0.917403\pi$
$73$		−	199631.i		−	0.513169i	0.966522	−	0.256584i	$-0.0825971\pi$
$74$	671954.			1.65823
$75$	72383.1			0.171575
$76$	−96641.7			−0.220152
$77$			1.03682e6i			2.27108i
$78$	−490110.			−1.03278
$79$	541051.			1.09738			0.548690	−	0.836026i	$-0.315127\pi$
$79$	541051.			1.09738			0.548690	+	0.836026i	$0.315127\pi$
$80$	−2.05561e6			−4.01486
$81$	59049.0			0.111111
$82$		−	1.52972e6i		−	2.77440i
$83$		−	742610.i		−	1.29875i	−0.760467	−	0.649377i	$-0.775030\pi$
$83$		−	742610.i		−	1.29875i	0.760467	−	0.649377i	$-0.224970\pi$
$84$	−1.56620e6			−2.64246
$85$	−519121.			−0.845302
$86$	−311157.			−0.489197
$87$	466832.			0.708930
$88$	−2.93745e6			−4.31044
$89$			827595.i			1.17395i	0.809607	+	0.586973i	$0.199680\pi$
$89$			827595.i			1.17395i	−0.809607	+	0.586973i	$0.800320\pi$
$90$			531296.i			0.728801i
$91$		−	1.19690e6i		−	1.58830i
$92$			2.08253e6i			2.67441i
$93$		−	63154.8i		−	0.0785159i
$94$	1.75126e6			2.10847
$95$	−80060.3			−0.0933784
$96$		−	1.80417e6i		−	2.03922i
$97$		−	497661.i		−	0.545279i	−0.962116	−	0.272640i	$-0.912103\pi$
$97$		−	497661.i		−	0.545279i	0.962116	−	0.272640i	$-0.0878966\pi$
$98$		−	3.44242e6i		−	3.65751i
$99$			430947.i			0.444139i
$100$	−797977.			−0.797977
$101$		−	1.36463e6i		−	1.32450i	−0.749285	−	0.662248i	$-0.769603\pi$
$101$		−	1.36463e6i		−	1.32450i	0.749285	−	0.662248i	$-0.230397\pi$
$102$		−	872937.i		−	0.822588i
$103$			149108.i			0.136455i	0.997670	+	0.0682274i	$0.0217343\pi$
$103$			149108.i			0.136455i	−0.997670	+	0.0682274i	$0.978266\pi$
$104$	3.39095e6			3.01454
$105$	−1.29748e6			−1.12081
$106$			3.38153e6i			2.83920i
$107$	−1.17390e6			−0.958254			−0.479127	−	0.877746i	$-0.659047\pi$
$107$	−1.17390e6			−0.958254			−0.479127	+	0.877746i	$0.659047\pi$
$108$	−650978.			−0.516767
$109$		−	535435.i		−	0.413454i	−0.978399	−	0.206727i	$-0.933719\pi$
$109$		−	535435.i		−	0.413454i	0.978399	−	0.206727i	$-0.0662812\pi$
$110$	−3.87747e6			−2.91320
$111$			682060.i			0.498716i
$112$	8.44147e6			6.00847
$113$			1.93878e6i			1.34367i	0.740701	+	0.671835i	$0.234494\pi$
$113$			1.93878e6i			1.34367i	−0.740701	+	0.671835i	$0.765506\pi$
$114$		−	134627.i		−	0.0908692i
$115$			1.72522e6i			1.13436i
$116$	−5.14653e6			−3.29716
$117$		−	497480.i		−	0.310612i
$118$	1.81257e6	+	2.58127e6i	1.10319	+	1.57104i
$119$	2.13180e6			1.26504
$120$		−	3.67591e6i		−	2.12726i
$121$	−1.37355e6			−0.775333
$122$	−6.81083e6			−3.75077
$123$	1.55272e6			0.834407
$124$			696241.i			0.365170i
$125$	1.56342e6			0.800472
$126$		−	2.18179e6i		−	1.09069i
$127$	1.74558e6			0.852173			0.426087	−	0.904682i	$-0.359892\pi$
$127$	1.74558e6			0.852173			0.426087	+	0.904682i	$0.359892\pi$
$128$			5.69821e6i			2.71712i
$129$		−	315836.i		−	0.147127i
$130$	4.47610e6			2.03737
$131$		−	1.26031e6i		−	0.560612i	−0.959911	−	0.280306i	$-0.909564\pi$
$131$		−	1.26031e6i		−	0.560612i	0.959911	−	0.280306i	$-0.0904360\pi$
$132$		−	4.75092e6i		−	2.06565i
$133$	328772.			0.139746
$134$	−3.83641e6			−1.59445
$135$	−539286.			−0.219188
$136$			6.03964e6i			2.40101i
$137$	3.60282e6			1.40114			0.700569	−	0.713585i	$-0.252930\pi$
$137$	3.60282e6			1.40114			0.700569	+	0.713585i	$0.252930\pi$
$138$	−2.90107e6			−1.10388
$139$	4.34147e6			1.61656			0.808281	−	0.588797i	$-0.200398\pi$
$139$	4.34147e6			1.61656			0.808281	+	0.588797i	$0.200398\pi$
$140$	1.43038e7			5.21277
$141$			1.77760e6i			0.634128i
$142$		−	9.29209e6i		−	3.24525i
$143$	3.63068e6			1.24159
$144$	3.50863e6			1.17503
$145$	−4.26351e6			−1.39850
$146$	−3.06584e6			−0.985123
$147$	3.49419e6			1.10000
$148$		−	7.51927e6i		−	2.31948i
$149$			1.18051e6i			0.356871i	0.983952	+	0.178435i	$0.0571036\pi$
$149$			1.18051e6i			0.356871i	−0.983952	+	0.178435i	$0.942896\pi$
$150$		−	1.11162e6i		−	0.329370i
$151$		−	1.80511e6i		−	0.524292i	−0.965028	−	0.262146i	$-0.915570\pi$
$151$		−	1.80511e6i		−	0.524292i	0.965028	−	0.262146i	$-0.0844301\pi$
$152$			931450.i			0.265234i
$153$	886065.			0.247395
$154$	1.59230e7			4.35977
$155$			576783.i			0.154888i
$156$			5.48440e6i			1.44463i
$157$		−	6.24085e6i		−	1.61267i	−0.591462	−	0.806333i	$-0.701449\pi$
$157$		−	6.24085e6i		−	1.61267i	0.591462	−	0.806333i	$-0.298551\pi$
$158$		−	8.30919e6i		−	2.10663i
$159$	−3.43238e6			−0.853895
$160$			1.64772e7i			4.02276i
$161$		−	7.08471e6i		−	1.69764i
$162$		−	906845.i		−	0.213299i
$163$	3.89198e6			0.898686			0.449343	−	0.893359i	$-0.351658\pi$
$163$	3.89198e6			0.898686			0.449343	+	0.893359i	$0.351658\pi$
$164$	−1.71178e7			−3.88075
$165$		−	3.93578e6i		−	0.876151i
$166$	−1.14046e7			−2.49320
$167$	3.76388e6			0.808140			0.404070	−	0.914728i	$-0.367595\pi$
$167$	3.76388e6			0.808140			0.404070	+	0.914728i	$0.367595\pi$
$168$			1.50953e7i			3.18357i
$169$	635601.			0.131681
$170$			7.97240e6i			1.62272i
$171$	136651.			0.0273291
$172$			3.48189e6i			0.684274i
$173$		−	6.13706e6i		−	1.18528i	−0.805467	−	0.592641i	$-0.798085\pi$
$173$		−	6.13706e6i		−	1.18528i	0.805467	−	0.592641i	$-0.201915\pi$
$174$		−	7.16938e6i		−	1.36092i
$175$	2.71469e6			0.506532
$176$			2.56064e7i			4.69690i
$177$	−2.62009e6	+	1.83983e6i	−0.472495	+	0.331786i
$178$	1.27098e7			2.25361
$179$			4.36255e6i			0.760643i	0.924854	+	0.380322i	$0.124187\pi$
$179$			4.36255e6i			0.760643i	−0.924854	+	0.380322i	$0.875813\pi$
$180$	5.94528e6			1.01942
$181$	−6.52017e6			−1.09957			−0.549785	−	0.835306i	$-0.685290\pi$
$181$	−6.52017e6			−1.09957			−0.549785	+	0.835306i	$0.685290\pi$
$182$	−1.83813e7			−3.04904
$183$		−	6.91325e6i		−	1.12805i
$184$	2.00718e7			3.22206
$185$		−	6.22915e6i		−	0.983815i
$186$	−969900.			−0.150726
$187$			6.46662e6i			0.988900i
$188$		−	1.95969e7i		−	2.94927i
$189$	2.21460e6			0.328028
$190$			1.22953e6i			0.179257i
$191$			7.81540e6i			1.12163i	0.827940	+	0.560817i	$0.189513\pi$
$191$			7.81540e6i			1.12163i	−0.827940	+	0.560817i	$0.810487\pi$
$192$	−1.33025e7			−1.87945
$193$	−3.91093e6			−0.544012			−0.272006	−	0.962296i	$-0.587687\pi$
$193$	−3.91093e6			−0.544012			−0.272006	+	0.962296i	$0.587687\pi$
$194$	−7.64284e6			−1.04677
$195$			4.54341e6i			0.612743i
$196$	−3.85212e7			−5.11601
$197$	4.95293e6			0.647833			0.323917	−	0.946086i	$-0.395000\pi$
$197$	4.95293e6			0.647833			0.323917	+	0.946086i	$0.395000\pi$
$198$	6.61827e6			0.852608
$199$	−1.11977e6			−0.142091			−0.0710457	−	0.997473i	$-0.522634\pi$
$199$	−1.11977e6			−0.142091			−0.0710457	+	0.997473i	$0.522634\pi$
$200$			7.69105e6i			0.961382i
$201$		−	3.89410e6i		−	0.479534i
$202$	−2.09573e7			−2.54262
$203$	1.75083e7			2.09294
$204$	−9.76830e6			−1.15061
$205$	−1.41808e7			−1.64603
$206$	2.28992e6			0.261951
$207$		−	2.94470e6i		−	0.331994i
$208$		−	2.95598e7i		−	3.28481i
$209$			997300.i			0.109241i
$210$			1.99260e7i			2.15160i
$211$		−	1.81512e6i		−	0.193223i	−0.995322	−	0.0966115i	$-0.969200\pi$
$211$		−	1.81512e6i		−	0.193223i	0.995322	−	0.0966115i	$-0.0308004\pi$
$212$	3.78398e7			3.97138
$213$	9.43183e6			0.976017
$214$			1.80282e7i			1.83955i
$215$			2.88449e6i			0.290237i
$216$			6.27424e6i			0.622587i
$217$		−	2.36859e6i		−	0.231799i
$218$	−8.22295e6			−0.793703
$219$		−	3.11194e6i		−	0.296278i
$220$			4.33894e7i			4.07489i
$221$		−	7.46498e6i		−	0.691595i
$222$	1.04747e7			0.957379
$223$	−7.76823e6			−0.700499			−0.350249	−	0.936657i	$-0.613903\pi$
$223$	−7.76823e6			−0.700499			−0.350249	+	0.936657i	$0.613903\pi$
$224$		−	6.76645e7i		−	6.02028i
$225$	1.12834e6			0.0990587
$226$	2.97748e7			2.57943
$227$			1.84475e7i			1.57710i	0.614971	+	0.788550i	$0.289168\pi$
$227$			1.84475e7i			1.57710i	−0.614971	+	0.788550i	$0.710832\pi$
$228$	−1.50649e6			−0.127105
$229$		−	1.23698e7i		−	1.03005i	−0.857176	−	0.515024i	$-0.827783\pi$
$229$		−	1.23698e7i		−	1.03005i	0.857176	−	0.515024i	$-0.172217\pi$
$230$	2.64951e7			2.17762
$231$			1.61625e7i			1.31121i
$232$			4.96032e7i			3.97233i
$233$			1.92702e7i			1.52342i	0.647919	+	0.761709i	$0.275639\pi$
$233$			1.92702e7i			1.52342i	−0.647919	+	0.761709i	$0.724361\pi$
$234$	−7.64005e6			−0.596278
$235$		−	1.62346e7i		−	1.25094i
$236$	2.88849e7	−	2.02830e7i	2.19753	−	1.54311i
$237$	8.43415e6			0.633572
$238$		−	3.27391e7i		−	2.42849i
$239$	−9.18556e6			−0.672840			−0.336420	−	0.941712i	$-0.609216\pi$
$239$	−9.18556e6			−0.672840			−0.336420	+	0.941712i	$0.609216\pi$
$240$	−3.20438e7			−2.31798
$241$	−3.82811e6			−0.273485			−0.136743	−	0.990607i	$-0.543663\pi$
$241$	−3.82811e6			−0.273485			−0.136743	+	0.990607i	$0.543663\pi$
$242$			2.10943e7i			1.48840i
$243$	920483.			0.0641500
$244$			7.62142e7i			5.24646i
$245$	−3.19119e7			−2.16997
$246$		−	2.38459e7i		−	1.60180i
$247$		−	1.15127e6i		−	0.0763988i
$248$	6.71050e6			0.439947
$249$		−	1.15761e7i		−	0.749835i
$250$		−	2.40102e7i		−	1.53666i
$251$	−7.67636e6			−0.485438			−0.242719	−	0.970097i	$-0.578039\pi$
$251$	−7.67636e6			−0.485438			−0.242719	+	0.970097i	$0.578039\pi$
$252$	−2.44146e7			−1.52563
$253$	2.14908e7			1.32706
$254$		−	2.68077e7i		−	1.63591i
$255$	−8.09230e6			−0.488035
$256$	3.28955e7			1.96072
$257$	−2.93455e7			−1.72879			−0.864393	−	0.502816i	$-0.832297\pi$
$257$	−2.93455e7			−1.72879			−0.864393	+	0.502816i	$0.832297\pi$
$258$	−4.85046e6			−0.282438
$259$			2.55803e7i			1.47234i
$260$		−	5.00882e7i		−	2.84981i
$261$	7.27720e6			0.409301
$262$	−1.93552e7			−1.07620
$263$	−1.49416e6			−0.0821351			−0.0410675	−	0.999156i	$-0.513076\pi$
$263$	−1.49416e6			−0.0821351			−0.0410675	+	0.999156i	$0.513076\pi$
$264$	−4.57902e7			−2.48864
$265$	3.13475e7			1.68448
$266$		−	5.04911e6i		−	0.268269i
$267$			1.29009e7i			0.677778i
$268$			4.29300e7i			2.23026i
$269$			1.42313e7i			0.731116i	0.930789	+	0.365558i	$0.119122\pi$
$269$			1.42313e7i			0.731116i	−0.930789	+	0.365558i	$0.880878\pi$
$270$			8.28208e6i			0.420773i
$271$	8.71158e6			0.437713			0.218856	−	0.975757i	$-0.429767\pi$
$271$	8.71158e6			0.437713			0.218856	+	0.975757i	$0.429767\pi$
$272$	5.26490e7			2.61628
$273$		−	1.86578e7i		−	0.917005i
$274$		−	5.53303e7i		−	2.68975i
$275$			8.23478e6i			0.395962i
$276$			3.24635e7i			1.54407i
$277$	4.17642e6			0.196501			0.0982505	−	0.995162i	$-0.468675\pi$
$277$	4.17642e6			0.196501			0.0982505	+	0.995162i	$0.468675\pi$
$278$		−	6.66741e7i		−	3.10329i
$279$		−	984486.i		−	0.0453312i
$280$		−	1.37863e8i		−	6.28021i
$281$	−1.72008e7			−0.775228			−0.387614	−	0.921822i	$-0.626701\pi$
$281$	−1.72008e7			−0.775228			−0.387614	+	0.921822i	$0.626701\pi$
$282$	2.72995e7			1.21733
$283$			2.70788e7i			1.19473i	0.801968	+	0.597366i	$0.203786\pi$
$283$			2.70788e7i			1.19473i	−0.801968	+	0.597366i	$0.796214\pi$
$284$	−1.03980e8			−4.53936
$285$	−1.24802e6			−0.0539120
$286$		−	5.57581e7i		−	2.38347i
$287$	5.82341e7			2.46338
$288$		−	2.81242e7i		−	1.17734i
$289$	−1.08416e7			−0.449161
$290$			6.54769e7i			2.68469i
$291$		−	7.75777e6i		−	0.314817i
$292$			3.43072e7i			1.37796i
$293$	4.01420e7			1.59586			0.797932	−	0.602747i	$-0.205927\pi$
$293$	4.01420e7			1.59586			0.797932	+	0.602747i	$0.205927\pi$
$294$		−	5.36620e7i		−	2.11166i
$295$	2.39289e7	−	1.68029e7i	0.932088	−	0.654513i
$296$	−7.24721e7			−2.79445
$297$			6.71781e6i			0.256424i
$298$	1.81297e7			0.685081
$299$	−2.48087e7			−0.928092
$300$	−1.24392e7			−0.460712
$301$		−	1.18453e7i		−	0.434356i
$302$	−2.77220e7			−1.00648
$303$		−	2.12725e7i		−	0.764698i
$304$	8.11968e6			0.289014
$305$			6.31377e7i			2.22530i
$306$		−	1.36077e7i		−	0.474921i
$307$	2.64835e7			0.915295			0.457647	−	0.889134i	$-0.348692\pi$
$307$	2.64835e7			0.915295			0.457647	+	0.889134i	$0.348692\pi$
$308$		−	1.78181e8i		−	6.09831i
$309$			2.32436e6i			0.0787822i
$310$	8.85795e6			0.297336
$311$	−8.46832e6			−0.281524			−0.140762	−	0.990043i	$-0.544955\pi$
$311$	−8.46832e6			−0.281524			−0.140762	+	0.990043i	$0.544955\pi$
$312$	5.28597e7			1.74045
$313$			1.51004e7i			0.492443i	0.969214	+	0.246222i	$0.0791891\pi$
$313$			1.51004e7i			0.492443i	−0.969214	+	0.246222i	$0.920811\pi$
$314$	−9.58438e7			−3.09581
$315$	−2.02257e7			−0.647099
$316$	−9.29811e7			−2.94668
$317$	−8.85633e6			−0.278020			−0.139010	−	0.990291i	$-0.544392\pi$
$317$	−8.85633e6			−0.278020			−0.139010	+	0.990291i	$0.544392\pi$
$318$			5.27128e7i			1.63921i
$319$			5.31099e7i			1.63608i
$320$	1.21490e8			3.70757
$321$	−1.82993e7			−0.553248
$322$	−1.08803e8			−3.25893
$323$	2.05053e6			0.0608498
$324$	−1.01477e7			−0.298356
$325$		−	9.50612e6i		−	0.276919i
$326$		−	5.97711e7i		−	1.72520i
$327$		−	8.34661e6i		−	0.238708i
$328$			1.64984e8i			4.67542i
$329$			6.66681e7i			1.87210i
$330$	−6.04437e7			−1.68194
$331$	5.61297e7			1.54778			0.773889	−	0.633322i	$-0.218309\pi$
$331$	5.61297e7			1.54778			0.773889	+	0.633322i	$0.218309\pi$
$332$			1.27620e8i			3.48741i
$333$			1.06323e7i			0.287934i
$334$		−	5.78038e7i		−	1.55138i
$335$			3.55643e7i			0.945974i
$336$	1.31589e8			3.46899
$337$			3.64613e7i			0.952670i	0.879264	+	0.476335i	$0.158035\pi$
$337$			3.64613e7i			0.952670i	−0.879264	+	0.476335i	$0.841965\pi$
$338$		−	9.76124e6i		−	0.252787i
$339$			3.02226e7i			0.775768i
$340$	8.92124e7			2.26980
$341$	7.18491e6			0.181200
$342$		−	2.09862e6i		−	0.0524634i
$343$	6.22658e7			1.54300
$344$	3.35591e7			0.824395
$345$			2.68935e7i			0.654923i
$346$	−9.42498e7			−2.27537
$347$		−	2.81353e7i		−	0.673385i	−0.941615	−	0.336693i	$-0.890692\pi$
$347$		−	2.81353e7i		−	0.673385i	0.941615	−	0.336693i	$-0.109308\pi$
$348$	−8.02265e7			−1.90362
$349$			2.03122e7i			0.477837i	0.971040	+	0.238919i	$0.0767929\pi$
$349$			2.03122e7i			0.477837i	−0.971040	+	0.238919i	$0.923207\pi$
$350$		−	4.16909e7i		−	0.972382i
$351$		−	7.75495e6i		−	0.179332i
$352$	2.05254e8			4.70613
$353$		−	5.37321e6i		−	0.122155i	−0.998133	−	0.0610773i	$-0.980546\pi$
$353$		−	5.37321e6i		−	0.122155i	0.998133	−	0.0610773i	$-0.0194536\pi$
$354$	2.82552e7	+	4.02381e7i	0.636926	+	0.907042i
$355$	−8.61395e7			−1.92538
$356$		−	1.42225e8i		−	3.15228i
$357$	3.32314e7			0.730373
$358$	6.69978e7			1.46020
$359$	5.22643e7			1.12959			0.564796	−	0.825231i	$-0.308955\pi$
$359$	5.22643e7			1.12959			0.564796	+	0.825231i	$0.308955\pi$
$360$		−	5.73017e7i		−	1.22817i
$361$	−4.67296e7			−0.993278
$362$			1.00133e8i			2.11083i
$363$	−2.14115e7			−0.447639
$364$			2.05690e8i			4.26490i
$365$			2.84209e7i			0.584466i
$366$	−1.06170e8			−2.16551
$367$			1.30899e6i			0.0264812i	0.999912	+	0.0132406i	$0.00421474\pi$
$367$			1.30899e6i			0.0264812i	−0.999912	+	0.0132406i	$0.995785\pi$
$368$		−	1.74971e8i		−	3.51094i
$369$	2.42045e7			0.481745
$370$	−9.56641e7			−1.88862
$371$	−1.28730e8			−2.52091
$372$			1.08533e7i			0.210831i
$373$	−2.14407e7			−0.413154			−0.206577	−	0.978430i	$-0.566232\pi$
$373$	−2.14407e7			−0.413154			−0.206577	+	0.978430i	$0.566232\pi$
$374$	9.93111e7			1.89838
$375$	2.43713e7			0.462153
$376$	−1.88879e8			−3.55320
$377$		−	6.13095e7i		−	1.14420i
$378$		−	3.40108e7i		−	0.629711i
$379$	4.05156e7			0.744227			0.372113	−	0.928187i	$-0.378633\pi$
$379$	4.05156e7			0.744227			0.372113	+	0.928187i	$0.378633\pi$
$380$	1.37586e7			0.250740
$381$	2.72109e7			0.492002
$382$	1.20025e8			2.15319
$383$	7.37761e7			1.31317			0.656583	−	0.754254i	$-0.272001\pi$
$383$	7.37761e7			1.31317			0.656583	+	0.754254i	$0.272001\pi$
$384$			8.88263e7i			1.56873i
$385$		−	1.47609e8i		−	2.58662i
$386$			6.00621e7i			1.04433i
$387$		−	4.92340e6i		−	0.0849439i
$388$			8.55245e7i			1.46418i
$389$	2.38734e7			0.405570			0.202785	−	0.979223i	$-0.435001\pi$
$389$	2.38734e7			0.405570			0.202785	+	0.979223i	$0.435001\pi$
$390$	6.97755e7			1.17628
$391$		−	4.41870e7i		−	0.739204i
$392$			3.71274e8i			6.16363i
$393$		−	1.96463e7i		−	0.323670i
$394$		−	7.60646e7i		−	1.24364i
$395$	−7.70278e7			−1.24985
$396$		−	7.40595e7i		−	1.19260i
$397$			1.10820e8i			1.77111i	0.464533	+	0.885556i	$0.346222\pi$
$397$			1.10820e8i			1.77111i	−0.464533	+	0.885556i	$0.653778\pi$
$398$			1.71968e7i			0.272771i
$399$	5.12505e6			0.0806824
$400$	6.70448e7			1.04757
$401$			9.67278e7i			1.50009i	0.661385	+	0.750047i	$0.269969\pi$
$401$			9.67278e7i			1.50009i	−0.661385	+	0.750047i	$0.730031\pi$
$402$	−5.98037e7			−0.920555
$403$	−8.29417e6			−0.126724
$404$			2.34515e8i			3.55653i
$405$	−8.40663e6			−0.126548
$406$		−	2.68884e8i		−	4.01779i
$407$	−7.75956e7			−1.15094
$408$			9.41487e7i			1.38623i
$409$			3.32272e7i			0.485651i	0.970070	+	0.242825i	$0.0780742\pi$
$409$			3.32272e7i			0.485651i	−0.970070	+	0.242825i	$0.921926\pi$
$410$			2.17781e8i			3.15987i
$411$	5.61624e7			0.808947
$412$		−	2.56246e7i		−	0.366409i
$413$	−9.82653e7	+	6.90020e7i	−1.39492	+	0.979516i
$414$	−4.52233e7			−0.637325
$415$			1.05723e8i			1.47920i
$416$	−2.36943e8			−3.29127
$417$	6.76768e7			0.933322
$418$	1.53160e7			0.209709
$419$		−	5.48195e7i		−	0.745234i	−0.927985	−	0.372617i	$-0.878461\pi$
$419$		−	5.48195e7i		−	0.745234i	0.927985	−	0.372617i	$-0.121539\pi$
$420$	2.22975e8			3.00960
$421$		−	8.66580e7i		−	1.16135i	−0.814136	−	0.580674i	$-0.802789\pi$
$421$		−	8.66580e7i		−	1.16135i	0.814136	−	0.580674i	$-0.197211\pi$
$422$	−2.78758e7			−0.370928
$423$			2.77100e7i			0.366114i
$424$		−	3.64707e8i		−	4.78462i
$425$	1.69314e7			0.220560
$426$		−	1.44849e8i		−	1.87365i
$427$		−	2.59278e8i		−	3.33029i
$428$	2.01738e8			2.57310
$429$	5.65967e7			0.716834
$430$	4.42985e7			0.557165
$431$			7.11138e7i			0.888223i	0.895972	+	0.444111i	$0.146481\pi$
$431$			7.11138e7i			0.888223i	−0.895972	+	0.444111i	$0.853519\pi$
$432$	5.46941e7			0.678405
$433$	−4.47048e7			−0.550669			−0.275335	−	0.961348i	$-0.588789\pi$
$433$	−4.47048e7			−0.550669			−0.275335	+	0.961348i	$0.588789\pi$
$434$	−3.63756e7			−0.444981
$435$	−6.64616e7			−0.807426
$436$			9.20160e7i			1.11021i
$437$		−	6.81464e6i		−	0.0816580i
$438$	−4.77917e7			−0.568761
$439$	−9.88814e7			−1.16875			−0.584374	−	0.811484i	$-0.698660\pi$
$439$	−9.88814e7			−1.16875			−0.584374	+	0.811484i	$0.698660\pi$
$440$	4.18195e8			4.90932
$441$	5.44690e7			0.635088
$442$	−1.14643e8			−1.32765
$443$		−	1.34618e8i		−	1.54844i	−0.632919	−	0.774218i	$-0.718143\pi$
$443$		−	1.34618e8i		−	1.54844i	0.632919	−	0.774218i	$-0.281857\pi$
$444$		−	1.17214e8i		−	1.33915i
$445$		−	1.17822e8i		−	1.33705i
$446$			1.19301e8i			1.34474i
$447$			1.84023e7i			0.206040i
$448$	−4.98904e8			−5.54860
$449$	−9.71018e6			−0.107272			−0.0536362	−	0.998561i	$-0.517081\pi$
$449$	−9.71018e6			−0.107272			−0.0536362	+	0.998561i	$0.517081\pi$
$450$		−	1.73285e7i		−	0.190162i
$451$			1.76648e8i			1.92566i
$452$		−	3.33184e8i		−	3.60802i
$453$		−	2.81389e7i		−	0.302700i
$454$	2.83307e8			3.02754
$455$			1.70399e8i			1.80897i
$456$			1.45199e7i			0.153133i
$457$			1.07666e8i			1.12805i	0.825756	+	0.564027i	$0.190749\pi$
$457$			1.07666e8i			1.12805i	−0.825756	+	0.564027i	$0.809251\pi$
$458$	−1.89970e8			−1.97737
$459$	1.38124e7			0.142834
$460$		−	2.96484e8i		−	3.04598i
$461$	−1.61768e8			−1.65117			−0.825583	−	0.564281i	$-0.809153\pi$
$461$	−1.61768e8			−1.65117			−0.825583	+	0.564281i	$0.809153\pi$
$462$	2.48215e8			2.51711
$463$		−	6.83555e7i		−	0.688701i	−0.938841	−	0.344350i	$-0.888099\pi$
$463$		−	6.83555e7i		−	0.688701i	0.938841	−	0.344350i	$-0.111901\pi$
$464$	4.32403e8			4.32848
$465$			8.99116e6i			0.0894246i
$466$	2.95942e8			2.92449
$467$		−	3.74836e7i		−	0.368037i	−0.982923	−	0.184018i	$-0.941089\pi$
$467$		−	3.74836e7i		−	0.368037i	0.982923	−	0.184018i	$-0.0589106\pi$
$468$			8.54934e7i			0.834055i
$469$		−	1.46046e8i		−	1.41570i
$470$	−2.49322e8			−2.40142
$471$		−	9.72852e7i		−	0.931073i
$472$	−1.95491e8	−	2.78398e8i	−1.85909	−	2.64752i
$473$	3.59316e7			0.339542
$474$		−	1.29527e8i		−	1.21626i
$475$	2.61121e6			0.0243647
$476$	−3.66355e8			−3.39689
$477$	−5.35056e7			−0.492997
$478$			1.41067e8i			1.29164i
$479$	−9.43319e7			−0.858326			−0.429163	−	0.903227i	$-0.641191\pi$
$479$	−9.43319e7			−0.858326			−0.429163	+	0.903227i	$0.641191\pi$
$480$			2.56854e8i			2.32254i
$481$	8.95754e7			0.804921
$482$			5.87902e7i			0.525006i
$483$		−	1.10440e8i		−	0.980130i
$484$	2.36048e8			2.08192
$485$			7.08506e7i			0.621038i
$486$		−	1.41363e7i		−	0.123148i
$487$	7.84183e7			0.678938			0.339469	−	0.940617i	$-0.389753\pi$
$487$	7.84183e7			0.678938			0.339469	+	0.940617i	$0.389753\pi$
$488$	7.34567e8			6.32080
$489$	6.06700e7			0.518856
$490$			4.90087e8i			4.16567i
$491$	1.69057e8			1.42820			0.714100	−	0.700044i	$-0.246836\pi$
$491$	1.69057e8			1.42820			0.714100	+	0.700044i	$0.246836\pi$
$492$	−2.66839e8			−2.24055
$493$	1.09199e8			0.911331
$494$	−1.76806e7			−0.146662
$495$		−	6.13527e7i		−	0.505846i
$496$		−	5.84971e7i		−	0.479390i
$497$	3.53736e8			2.88145
$498$	−1.77781e8			−1.43945
$499$	−1.51272e8			−1.21747			−0.608734	−	0.793375i	$-0.708322\pi$
$499$	−1.51272e8			−1.21747			−0.608734	+	0.793375i	$0.708322\pi$
$500$	−2.68678e8			−2.14943
$501$	5.86731e7			0.466580
$502$			1.17890e8i			0.931890i
$503$		−	1.39770e8i		−	1.09827i	−0.835732	−	0.549137i	$-0.814957\pi$
$503$		−	1.39770e8i		−	1.09827i	0.835732	−	0.549137i	$-0.185043\pi$
$504$			2.35312e8i			1.83803i
$505$			1.94278e8i			1.50852i
$506$		−	3.30045e8i		−	2.54755i
$507$	9.90804e6			0.0760263
$508$	−2.99982e8			−2.28826
$509$			1.39563e8i			1.05832i	0.848521	+	0.529161i	$0.177493\pi$
$509$			1.39563e8i			1.05832i	−0.848521	+	0.529161i	$0.822507\pi$
$510$			1.24277e8i			0.936875i
$511$		−	1.16712e8i		−	0.874687i
$512$		−	1.40507e8i		−	1.04686i
$513$	2.13018e6			0.0157785
$514$			4.50673e8i			3.31873i
$515$		−	2.12280e7i		−	0.155413i
$516$			5.42773e7i			0.395066i
$517$	−2.02232e8			−1.46345
$518$	3.92850e8			2.82642
$519$		−	9.56672e7i		−	0.684323i
$520$	−4.82760e8			−3.43337
$521$	−2.47764e8			−1.75196			−0.875980	−	0.482347i	$-0.839785\pi$
$521$	−2.47764e8			−1.75196			−0.875980	+	0.482347i	$0.839785\pi$
$522$		−	1.11760e8i		−	0.785730i
$523$	−1.38810e8			−0.970324			−0.485162	−	0.874424i	$-0.661239\pi$
$523$	−1.38810e8			−0.970324			−0.485162	+	0.874424i	$0.661239\pi$
$524$			2.16587e8i			1.50536i
$525$	4.23179e7			0.292446
$526$			2.29465e7i			0.157674i
$527$		−	1.47728e7i		−	0.100932i
$528$			3.99165e8i			2.71176i
$529$	1.18712e6			0.00801915
$530$		−	4.81418e8i		−	3.23367i
$531$	−4.08432e7	+	2.86801e7i	−0.272795	+	0.191557i
$532$	−5.65004e7			−0.375246
$533$		−	2.03920e8i		−	1.34672i
$534$	1.98126e8			1.30112
$535$	1.67125e8			1.09139
$536$	4.13767e8			2.68696
$537$			6.80054e7i			0.439158i
$538$	2.18556e8			1.40351
$539$			3.97522e8i			2.53860i
$540$	9.26777e7			0.588565
$541$		−	2.45184e8i		−	1.54846i	−0.632902	−	0.774232i	$-0.718136\pi$
$541$		−	2.45184e8i		−	1.54846i	0.632902	−	0.774232i	$-0.281864\pi$
$542$		−	1.33788e8i		−	0.840271i
$543$	−1.01639e8			−0.634837
$544$		−	4.22020e8i		−	2.62142i
$545$			7.62283e7i			0.470898i
$546$	−2.86537e8			−1.76036
$547$	1.41249e8			0.863025			0.431513	−	0.902107i	$-0.357980\pi$
$547$	1.41249e8			0.863025			0.431513	+	0.902107i	$0.357980\pi$
$548$	−6.19155e8			−3.76233
$549$		−	1.07767e8i		−	0.651281i
$550$	1.26466e8			0.760124
$551$	1.68409e7			0.100672
$552$	3.12889e8			1.86026
$553$	3.16319e8			1.87046
$554$		−	6.41393e7i		−	0.377220i
$555$		−	9.71028e7i		−	0.568006i
$556$	−7.46093e8			−4.34079
$557$	−4.71404e7			−0.272790			−0.136395	−	0.990655i	$-0.543552\pi$
$557$	−4.71404e7			−0.272790			−0.136395	+	0.990655i	$0.543552\pi$
$558$	−1.51192e7			−0.0870217
$559$	−4.14790e7			−0.237461
$560$	−1.20179e9			−6.84327
$561$			1.00805e8i			0.570942i
$562$			2.64161e8i			1.48820i
$563$		−	7.33285e7i		−	0.410911i	−0.978666	−	0.205455i	$-0.934132\pi$
$563$		−	7.33285e7i		−	0.410911i	0.978666	−	0.205455i	$-0.0658676\pi$
$564$		−	3.05486e8i		−	1.70276i
$565$		−	2.76018e8i		−	1.53036i
$566$	4.15863e8			2.29351
$567$	3.45223e7			0.189387
$568$			1.00218e9i			5.46890i
$569$		−	1.74218e8i		−	0.945708i	−0.881141	−	0.472854i	$-0.843224\pi$
$569$		−	1.74218e8i		−	0.945708i	0.881141	−	0.472854i	$-0.156776\pi$
$570$			1.91664e7i			0.103494i
$571$			1.41847e8i			0.761925i	0.924590	+	0.380963i	$0.124407\pi$
$571$			1.41847e8i			0.761925i	−0.924590	+	0.380963i	$0.875593\pi$
$572$	−6.23942e8			−3.33393
$573$			1.21830e8i			0.647575i
$574$		−	8.94329e8i		−	4.72892i
$575$		−	5.62690e7i		−	0.295982i
$576$	−2.07366e8			−1.08510
$577$	1.44952e8			0.754563			0.377282	−	0.926099i	$-0.376859\pi$
$577$	1.44952e8			0.754563			0.377282	+	0.926099i	$0.376859\pi$
$578$			1.66501e8i			0.862248i
$579$	−6.09654e7			−0.314085
$580$	7.32696e8			3.75526
$581$		−	4.34158e8i		−	2.21370i
$582$	−1.19140e8			−0.604350
$583$		−	3.90491e8i		−	1.97063i
$584$	3.30659e8			1.66013
$585$			7.08248e7i			0.353767i
$586$		−	6.16480e8i		−	3.06356i
$587$			3.09187e8i			1.52865i	0.644834	+	0.764323i	$0.276927\pi$
$587$			3.09187e8i			1.52865i	−0.644834	+	0.764323i	$0.723073\pi$
$588$	−6.00486e8			−2.95373
$589$		−	2.27830e6i		−	0.0111497i
$590$	−2.58051e8	−	3.67488e8i	−1.25646	−	1.78932i
$591$	7.72085e7			0.374027
$592$			6.31757e8i			3.04499i
$593$	−2.12830e8			−1.02063			−0.510316	−	0.859987i	$-0.670472\pi$
$593$	−2.12830e8			−1.02063			−0.510316	+	0.859987i	$0.670472\pi$
$594$	1.03169e8			0.492253
$595$	−3.03498e8			−1.44080
$596$		−	2.02874e8i		−	0.958270i
$597$	−1.74554e7			−0.0820365
$598$			3.81000e8i			1.78165i
$599$	−1.53295e8			−0.713260			−0.356630	−	0.934246i	$-0.616074\pi$
$599$	−1.53295e8			−0.713260			−0.356630	+	0.934246i	$0.616074\pi$
$600$			1.19892e8i			0.555054i
$601$			1.13259e8i			0.521735i	0.965375	+	0.260867i	$0.0840086\pi$
$601$			1.13259e8i			0.521735i	−0.965375	+	0.260867i	$0.915991\pi$
$602$	−1.81914e8			−0.833828
$603$		−	6.07031e7i		−	0.276859i
$604$			3.10213e8i			1.40783i
$605$	1.95548e8			0.883055
$606$	−3.26692e8			−1.46798
$607$	−3.65896e7			−0.163603			−0.0818016	−	0.996649i	$-0.526067\pi$
$607$	−3.65896e7			−0.163603			−0.0818016	+	0.996649i	$0.526067\pi$
$608$		−	6.50852e7i		−	0.289582i
$609$	2.72928e8			1.20836
$610$	9.69637e8			4.27189
$611$	2.33454e8			1.02347
$612$	−1.52273e8			−0.664305
$613$			4.36665e8i			1.89569i	0.318735	+	0.947844i	$0.396742\pi$
$613$			4.36665e8i			1.89569i	−0.318735	+	0.947844i	$0.603258\pi$
$614$		−	4.06721e8i		−	1.75708i
$615$	−2.21056e8			−0.950337
$616$	−1.71734e9			−7.34708
$617$	2.76981e8			1.17922			0.589610	−	0.807688i	$-0.299282\pi$
$617$	2.76981e8			1.17922			0.589610	+	0.807688i	$0.299282\pi$
$618$	3.56964e7			0.151237
$619$	1.28857e8			0.543296			0.271648	−	0.962397i	$-0.412431\pi$
$619$	1.28857e8			0.543296			0.271648	+	0.962397i	$0.412431\pi$
$620$		−	9.91218e7i		−	0.415905i
$621$		−	4.59034e7i		−	0.191677i
$622$			1.30052e8i			0.540439i
$623$			4.83843e8i			2.00097i
$624$		−	4.60791e8i		−	1.89649i
$625$	−2.95132e8			−1.20886
$626$	2.31905e8			0.945337
$627$			1.55464e7i			0.0630705i
$628$			1.07251e9i			4.33033i
$629$			1.59543e8i			0.641101i
$630$			3.10615e8i			1.24223i
$631$	6.88588e7			0.274076			0.137038	−	0.990566i	$-0.456242\pi$
$631$	6.88588e7			0.274076			0.137038	+	0.990566i	$0.456242\pi$
$632$			8.96169e8i			3.55009i
$633$		−	2.82950e7i		−	0.111557i
$634$			1.36011e8i			0.533712i
$635$	−2.48513e8			−0.970571
$636$	5.89865e8			2.29288
$637$		−	4.58894e8i		−	1.77539i
$638$	8.15636e8			3.14076
$639$	1.47028e8			0.563503
$640$		−	8.11238e8i		−	3.09463i
$641$	−2.85061e8			−1.08234			−0.541170	−	0.840913i	$-0.682018\pi$
$641$	−2.85061e8			−1.08234			−0.541170	+	0.840913i	$0.682018\pi$
$642$			2.81032e8i			1.06206i
$643$	−1.78023e8			−0.669642			−0.334821	−	0.942282i	$-0.608676\pi$
$643$	−1.78023e8			−0.669642			−0.334821	+	0.942282i	$0.608676\pi$
$644$			1.21753e9i			4.55849i
$645$			4.49647e7i			0.167568i
$646$		−	3.14911e7i		−	0.116813i
$647$	−1.25538e8			−0.463514			−0.231757	−	0.972774i	$-0.574447\pi$
$647$	−1.25538e8			−0.463514			−0.231757	+	0.972774i	$0.574447\pi$
$648$			9.78058e7i			0.359451i
$649$	−2.09311e8	−	2.98079e8i	−0.765701	−	1.09043i
$650$	−1.45990e8			−0.531599
$651$		−	3.69227e7i		−	0.133829i
$652$	−6.68848e8			−2.41315
$653$	1.87020e8			0.671659			0.335830	−	0.941923i	$-0.390983\pi$
$653$	1.87020e8			0.671659			0.335830	+	0.941923i	$0.390983\pi$
$654$	−1.28183e8			−0.458245
$655$			1.79426e8i			0.638502i
$656$	1.43821e9			5.09460
$657$		−	4.85104e7i		−	0.171056i
$658$	1.02385e9			3.59386
$659$		−	4.16762e8i		−	1.45623i	−0.685453	−	0.728117i	$-0.740396\pi$
$659$		−	4.16762e8i		−	1.45623i	0.685453	−	0.728117i	$-0.259604\pi$
$660$			6.76374e8i			2.35264i
$661$	9.85540e7			0.341248			0.170624	−	0.985336i	$-0.445422\pi$
$661$	9.85540e7			0.341248			0.170624	+	0.985336i	$0.445422\pi$
$662$		−	8.62011e8i		−	2.97125i
$663$		−	1.16368e8i		−	0.399293i
$664$	1.23002e9			4.20154
$665$	−4.68063e7			−0.159162
$666$	1.63285e8			0.552743
$667$		−	3.62905e8i		−	1.22297i
$668$	−6.46833e8			−2.17002
$669$	−1.21095e8			−0.404433
$670$	5.46178e8			1.81597
$671$	7.86497e8			2.60333
$672$		−	1.05479e9i		−	3.47581i
$673$		−	1.98818e8i		−	0.652244i	−0.945328	−	0.326122i	$-0.894258\pi$
$673$		−	1.98818e8i		−	0.652244i	0.945328	−	0.326122i	$-0.105742\pi$
$674$	5.59955e8			1.82883
$675$	1.75891e7			0.0571916
$676$	−1.09230e8			−0.353591
$677$	1.57413e8			0.507313			0.253656	−	0.967294i	$-0.418367\pi$
$677$	1.57413e8			0.507313			0.253656	+	0.967294i	$0.418367\pi$
$678$	4.64143e8			1.48923
$679$		−	2.90952e8i		−	0.929419i
$680$		−	8.59846e8i		−	2.73460i
$681$			2.87567e8i			0.910539i
$682$		−	1.10342e8i		−	0.347847i
$683$		−	2.84746e8i		−	0.893707i	−0.894607	−	0.446854i	$-0.852545\pi$
$683$		−	2.84746e8i		−	0.893707i	0.894607	−	0.446854i	$-0.147455\pi$
$684$	−2.34839e7			−0.0733842
$685$	−5.12923e8			−1.59581
$686$		−	9.56247e8i		−	2.96209i
$687$		−	1.92827e8i		−	0.594698i
$688$		−	2.92543e8i		−	0.898307i
$689$			4.50778e8i			1.37818i
$690$	4.13017e8			1.25725
$691$			9.90109e7i			0.300088i	0.988679	+	0.150044i	$0.0479416\pi$
$691$			9.90109e7i			0.300088i	−0.988679	+	0.150044i	$0.952058\pi$
$692$			1.05467e9i			3.18272i
$693$			2.51948e8i			0.757027i
$694$	−4.32088e8			−1.29269
$695$	−6.18082e8			−1.84116
$696$			7.73237e8i			2.29343i
$697$	3.63203e8			1.07263
$698$	3.11944e8			0.917298
$699$			3.00393e8i			0.879546i
$700$	−4.66528e8			−1.36014
$701$		−	4.25678e8i		−	1.23574i	−0.786280	−	0.617870i	$-0.787996\pi$
$701$		−	4.25678e8i		−	1.23574i	0.786280	−	0.617870i	$-0.212004\pi$
$702$	−1.19097e8			−0.344261
$703$			2.46052e7i			0.0708208i
$704$		−	1.51338e9i		−	4.33741i
$705$		−	2.53072e8i		−	0.722231i
$706$	−8.25190e7			−0.234499
$707$		−	7.97813e8i		−	2.25758i
$708$	4.50270e8	−	3.16180e8i	1.26874	−	0.890912i
$709$	−3.64346e8			−1.02229			−0.511146	−	0.859494i	$-0.670779\pi$
$709$	−3.64346e8			−1.02229			−0.511146	+	0.859494i	$0.670779\pi$
$710$			1.32289e9i			3.69614i
$711$	1.31475e8			0.365793
$712$	−1.37079e9			−3.79778
$713$	−4.90951e7			−0.135447
$714$		−	5.10352e8i		−	1.40209i
$715$	−5.16889e8			−1.41410
$716$		−	7.49716e8i		−	2.04248i
$717$	−1.43189e8			−0.388464
$718$		−	8.02648e8i		−	2.16846i
$719$			1.67040e8i			0.449401i	0.974428	+	0.224700i	$0.0721403\pi$
$719$			1.67040e8i			0.449401i	−0.974428	+	0.224700i	$0.927860\pi$
$720$	−4.99513e8			−1.33829
$721$			8.71741e7i			0.232585i
$722$			7.17650e8i			1.90678i
$723$	−5.96744e7			−0.157897
$724$	1.12051e9			2.95257
$725$	1.39057e8			0.364903
$726$			3.28827e8i			0.859327i
$727$	3.16596e8			0.823953			0.411977	−	0.911194i	$-0.364839\pi$
$727$	3.16596e8			0.823953			0.411977	+	0.911194i	$0.364839\pi$
$728$	1.98248e9			5.13824
$729$	1.43489e7			0.0370370
$730$	4.36474e8			1.12199
$731$		−	7.38785e7i		−	0.189132i
$732$			1.18806e9i			3.02905i
$733$	−2.05904e8			−0.522822			−0.261411	−	0.965228i	$-0.584188\pi$
$733$	−2.05904e8			−0.522822			−0.261411	+	0.965228i	$0.584188\pi$
$734$	2.01028e7			0.0508357
$735$	−4.97457e8			−1.25283
$736$	−1.40252e9			−3.51784
$737$	4.43019e8			1.10667
$738$		−	3.71721e8i		−	0.924800i
$739$			4.82162e7i			0.119470i	0.998214	+	0.0597351i	$0.0190256\pi$
$739$			4.82162e7i			0.119470i	−0.998214	+	0.0597351i	$0.980974\pi$
$740$			1.07050e9i			2.64174i
$741$		−	1.79465e7i		−	0.0441089i
$742$			1.97697e9i			4.83936i
$743$	−5.06474e8			−1.23478			−0.617392	−	0.786656i	$-0.711811\pi$
$743$	−5.06474e8			−1.23478			−0.617392	+	0.786656i	$0.711811\pi$
$744$	1.04606e8			0.254003
$745$		−	1.68066e8i		−	0.406453i
$746$			3.29275e8i			0.793126i
$747$		−	1.80454e8i		−	0.432918i
$748$		−	1.11131e9i		−	2.65539i
$749$	−6.86308e8			−1.63333
$750$		−	3.74283e8i		−	0.887188i
$751$			4.53729e8i			1.07122i	0.844467	+	0.535608i	$0.179918\pi$
$751$			4.53729e8i			1.07122i	−0.844467	+	0.535608i	$0.820082\pi$
$752$			1.64650e9i			3.87176i
$753$	−1.19663e8			−0.280268
$754$	−9.41560e8			−2.19651
$755$			2.56988e8i			0.597135i
$756$	−3.80586e8			−0.880821
$757$	1.29246e8			0.297940			0.148970	−	0.988842i	$-0.452404\pi$
$757$	1.29246e8			0.297940			0.148970	+	0.988842i	$0.452404\pi$
$758$		−	6.22219e8i		−	1.42868i
$759$	3.35009e8			0.766180
$760$		−	1.32608e8i		−	0.302084i
$761$	−1.51435e8			−0.343615			−0.171808	−	0.985131i	$-0.554961\pi$
$761$	−1.51435e8			−0.343615			−0.171808	+	0.985131i	$0.554961\pi$
$762$		−	4.17891e8i		−	0.944491i
$763$		−	3.13036e8i		−	0.704726i
$764$		−	1.34310e9i		−	3.01181i
$765$	−1.26146e8			−0.281767
$766$		−	1.13302e9i		−	2.52087i
$767$	2.41627e8	+	3.44099e8i	0.535499	+	0.762600i
$768$	5.12790e8			1.13202
$769$		−	6.47028e8i		−	1.42280i	−0.702787	−	0.711401i	$-0.748061\pi$
$769$		−	6.47028e8i		−	1.42280i	0.702787	−	0.711401i	$-0.251939\pi$
$770$	−2.26691e9			−4.96550
$771$	−4.57450e8			−0.998116
$772$	6.72104e8			1.46078
$773$		−	5.11737e8i		−	1.10792i	−0.832544	−	0.553959i	$-0.813116\pi$
$773$		−	5.11737e8i		−	1.10792i	0.832544	−	0.553959i	$-0.186884\pi$
$774$	−7.56111e7			−0.163066
$775$		−	1.88121e7i		−	0.0404140i
$776$	8.24301e8			1.76401
$777$			3.98758e8i			0.850053i
$778$		−	3.66636e8i		−	0.778567i
$779$	5.60142e7			0.118491
$780$		−	7.80798e8i		−	1.64534i
$781$			1.07303e9i			2.25246i
$782$	−6.78602e8			−1.41904
$783$	1.13440e8			0.236310
$784$	3.23649e9			6.71624
$785$			8.88491e8i

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:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	177.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-15.3575i q^{2} +15.5885 q^{3} -171.853 q^{4} -142.367 q^{5} -239.400i q^{6} +584.638 q^{7} +1656.35i q^{8} +243.000 q^{9} +O(q^{10})\)	\(q-15.3575i q^{2} +15.5885 q^{3} -171.853 q^{4} -142.367 q^{5} -239.400i q^{6} +584.638 q^{7} +1656.35i q^{8} +243.000 q^{9} +2186.40i q^{10} +1773.45i q^{11} -2678.92 q^{12} -2047.24i q^{13} -8978.57i q^{14} -2219.28 q^{15} +14438.8 q^{16} +3646.36 q^{17} -3731.87i q^{18} +562.351 q^{19} +24466.2 q^{20} +9113.60 q^{21} +27235.7 q^{22} -12118.1i q^{23} +25819.9i q^{24} +4643.38 q^{25} -31440.5 q^{26} +3788.00 q^{27} -100472. q^{28} +29947.3 q^{29} +34082.6i q^{30} -4051.38i q^{31} -115738. i q^{32} +27645.3i q^{33} -55998.9i q^{34} -83233.1 q^{35} -41760.2 q^{36} +43754.1i q^{37} -8636.31i q^{38} -31913.4i q^{39} -235810. i q^{40} +99607.1 q^{41} -139962. i q^{42} -20260.9i q^{43} -304772. i q^{44} -34595.2 q^{45} -186104. q^{46} +114033. i q^{47} +225079. q^{48} +224152. q^{49} -71310.6i q^{50} +56841.1 q^{51} +351825. i q^{52} -220188. q^{53} -58174.1i q^{54} -252480. i q^{55} +968364. i q^{56} +8766.19 q^{57} -459916. i q^{58} +(-168079. + 118025. i) q^{59} +381390. q^{60} -443485. i q^{61} -62219.1 q^{62} +142067. q^{63} -853356. q^{64} +291460. i q^{65} +424562. q^{66} -249807. i q^{67} -626637. q^{68} -188903. i q^{69} +1.27825e6i q^{70} +605052. q^{71} +402493. i q^{72} -199631. i q^{73} +671954. q^{74} +72383.1 q^{75} -96641.7 q^{76} +1.03682e6i q^{77} -490110. q^{78} +541051. q^{79} -2.05561e6 q^{80} +59049.0 q^{81} -1.52972e6i q^{82} -742610. i q^{83} -1.56620e6 q^{84} -519121. q^{85} -311157. q^{86} +466832. q^{87} -2.93745e6 q^{88} +827595. i q^{89} +531296. i q^{90} -1.19690e6i q^{91} +2.08253e6i q^{92} -63154.8i q^{93} +1.75126e6 q^{94} -80060.3 q^{95} -1.80417e6i q^{96} -497661. i q^{97} -3.44242e6i q^{98} +430947. i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 60q - 1920q^{4} - 408q^{7} + 14580q^{9} + O(q^{10}) \)	\( 60q - 1920q^{4} - 408q^{7} + 14580q^{9} - 1944q^{12} - 4536q^{15} + 56616q^{16} + 8480q^{17} + 11376q^{19} + 40796q^{20} - 8232q^{22} + 197940q^{25} + 147252q^{26} + 71640q^{28} + 63456q^{29} - 364432q^{35} - 466560q^{36} + 99632q^{41} - 470316q^{46} + 171072q^{48} + 1737420q^{49} + 60912q^{51} + 92240q^{53} + 186624q^{57} + 917264q^{59} + 1063368q^{60} - 115768q^{62} - 99144q^{63} - 1107444q^{64} + 1172232q^{66} - 4247232q^{68} + 1498048q^{71} + 1161448q^{74} - 1477440q^{75} - 1045320q^{76} - 1060452q^{78} - 90600q^{79} + 77096q^{80} + 3542940q^{81} - 2225880q^{84} - 693408q^{85} - 1567768q^{86} + 1821528q^{87} + 62892q^{88} + 5268696q^{94} + 296128q^{95} + O(q^{100}) \)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Properties

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