LMFDB - Embedded newform 177.4.d.c.176.18

Newspace parameters

Level:	$ N $	$=$	$ 177 = 3 \cdot 59 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	177.d (of order $2$, degree $1$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$10.4433380710$
Analytic rank:	$0$
Dimension:	$52$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			176.18
Character	$\chi$	$=$	177.176
Dual form			177.4.d.c.176.17

$q$-expansion

$f(q)$	$=$	$q-2.62059 q^{2} +(5.14935 + 0.695815i) q^{3} -1.13250 q^{4} +14.1294i q^{5} +(-13.4944 - 1.82345i) q^{6} +10.3672 q^{7} +23.9326 q^{8} +(26.0317 + 7.16599i) q^{9} +O(q^{10})$	\(q-2.62059 q^{2} +(5.14935 + 0.695815i) q^{3} -1.13250 q^{4} +14.1294i q^{5} +(-13.4944 - 1.82345i) q^{6} +10.3672 q^{7} +23.9326 q^{8} +(26.0317 + 7.16599i) q^{9} -37.0275i q^{10} -30.0340 q^{11} +(-5.83163 - 0.788009i) q^{12} -8.23644i q^{13} -27.1683 q^{14} +(-9.83148 + 72.7575i) q^{15} -53.6575 q^{16} +83.4833i q^{17} +(-68.2184 - 18.7791i) q^{18} -30.5066 q^{19} -16.0016i q^{20} +(53.3845 + 7.21367i) q^{21} +78.7068 q^{22} -41.7480 q^{23} +(123.237 + 16.6526i) q^{24} -74.6412 q^{25} +21.5844i q^{26} +(129.060 + 55.0135i) q^{27} -11.7409 q^{28} -73.8768i q^{29} +(25.7643 - 190.668i) q^{30} +289.520i q^{31} -50.8461 q^{32} +(-154.656 - 20.8981i) q^{33} -218.776i q^{34} +146.483i q^{35} +(-29.4808 - 8.11548i) q^{36} +191.214i q^{37} +79.9454 q^{38} +(5.73104 - 42.4124i) q^{39} +338.154i q^{40} -36.3424i q^{41} +(-139.899 - 18.9041i) q^{42} +491.069i q^{43} +34.0134 q^{44} +(-101.252 + 367.813i) q^{45} +109.405 q^{46} +26.4171 q^{47} +(-276.301 - 37.3357i) q^{48} -235.521 q^{49} +195.604 q^{50} +(-58.0889 + 429.885i) q^{51} +9.32776i q^{52} +181.825i q^{53} +(-338.214 - 144.168i) q^{54} -424.363i q^{55} +248.114 q^{56} +(-157.089 - 21.2270i) q^{57} +193.601i q^{58} +(-187.180 - 412.726i) q^{59} +(11.1341 - 82.3977i) q^{60} -775.225i q^{61} -758.713i q^{62} +(269.876 + 74.2915i) q^{63} +562.507 q^{64} +116.376 q^{65} +(405.289 + 54.7654i) q^{66} -176.188i q^{67} -94.5447i q^{68} +(-214.975 - 29.0489i) q^{69} -383.873i q^{70} -447.258i q^{71} +(623.005 + 171.501i) q^{72} +417.573i q^{73} -501.095i q^{74} +(-384.354 - 51.9364i) q^{75} +34.5487 q^{76} -311.369 q^{77} +(-15.0187 + 111.145i) q^{78} +574.434 q^{79} -758.150i q^{80} +(626.297 + 373.086i) q^{81} +95.2385i q^{82} +998.011 q^{83} +(-60.4579 - 8.16947i) q^{84} -1179.57 q^{85} -1286.89i q^{86} +(51.4046 - 380.418i) q^{87} -718.790 q^{88} +1179.83 q^{89} +(265.339 - 963.888i) q^{90} -85.3891i q^{91} +47.2796 q^{92} +(-201.452 + 1490.84i) q^{93} -69.2284 q^{94} -431.041i q^{95} +(-261.825 - 35.3795i) q^{96} -806.397i q^{97} +617.203 q^{98} +(-781.835 - 215.223i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 52q - 8q^{3} + 268q^{4} - 16q^{7} - 4q^{9} + O(q^{10}) $	$ 52q - 8q^{3} + 268q^{4} - 16q^{7} - 4q^{9} + 28q^{12} + 114q^{15} + 484q^{16} - 184q^{19} - 758q^{21} - 60q^{22} + 36q^{25} + 742q^{27} - 4q^{28} - 888q^{36} + 1402q^{45} - 660q^{46} - 488q^{48} - 924q^{49} - 1772q^{51} - 630q^{57} - 1880q^{60} - 212q^{63} + 7648q^{64} + 1316q^{66} - 1556q^{75} - 5680q^{76} + 3224q^{78} - 1504q^{79} - 276q^{81} + 1228q^{84} - 848q^{85} + 3598q^{87} + 5760q^{88} + 888q^{94} + 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 $

$2$	−2.62059			−0.926519			−0.463260	−	0.886223i	$-0.653320\pi$
$2$	−2.62059			−0.926519			−0.463260	+	0.886223i	$0.653320\pi$
$3$	5.14935	+	0.695815i	0.990994	+	0.133910i
$3$	5.14935	+	0.695815i	0.990994	+	0.133910i
$4$	−1.13250			−0.141562
$5$			14.1294i			1.26378i	0.775060	+	0.631888i	$0.217720\pi$
$5$			14.1294i			1.26378i	−0.775060	+	0.631888i	$0.782280\pi$
$6$	−13.4944	−	1.82345i	−0.918174	−	0.124070i
$7$	10.3672			0.559778			0.279889	−	0.960032i	$-0.409702\pi$
$7$	10.3672			0.559778			0.279889	+	0.960032i	$0.409702\pi$
$8$	23.9326			1.05768
$9$	26.0317	+	7.16599i	0.964136	+	0.265407i
$10$		−	37.0275i		−	1.17091i
$11$	−30.0340			−0.823235			−0.411617	−	0.911357i	$-0.635036\pi$
$11$	−30.0340			−0.823235			−0.411617	+	0.911357i	$0.635036\pi$
$12$	−5.83163	−	0.788009i	−0.140287	−	0.0189566i
$13$		−	8.23644i		−	0.175721i	−0.996133	−	0.0878607i	$-0.971997\pi$
$13$		−	8.23644i		−	0.175721i	0.996133	−	0.0878607i	$-0.0280030\pi$
$14$	−27.1683			−0.518645
$15$	−9.83148	+	72.7575i	−0.169232	+	1.25239i
$16$	−53.6575			−0.838398
$17$			83.4833i			1.19104i	0.803341	+	0.595520i	$0.203054\pi$
$17$			83.4833i			1.19104i	−0.803341	+	0.595520i	$0.796946\pi$
$18$	−68.2184	−	18.7791i	−0.893291	−	0.245905i
$19$	−30.5066			−0.368352			−0.184176	−	0.982893i	$-0.558962\pi$
$19$	−30.5066			−0.368352			−0.184176	+	0.982893i	$0.558962\pi$
$20$		−	16.0016i		−	0.178903i
$21$	53.3845	+	7.21367i	0.554736	+	0.0749597i
$22$	78.7068			0.762743
$23$	−41.7480			−0.378481			−0.189241	−	0.981931i	$-0.560603\pi$
$23$	−41.7480			−0.378481			−0.189241	+	0.981931i	$0.560603\pi$
$24$	123.237	+	16.6526i	1.04815	+	0.141633i
$25$	−74.6412			−0.597129
$26$			21.5844i			0.162809i
$27$	129.060	+	55.0135i	0.919912	+	0.392124i
$28$	−11.7409			−0.0792434
$29$		−	73.8768i		−	0.473054i	−0.971625	−	0.236527i	$-0.923991\pi$
$29$		−	73.8768i		−	0.473054i	0.971625	−	0.236527i	$-0.0760092\pi$
$30$	25.7643	−	190.668i	0.156796	−	1.16037i
$31$			289.520i			1.67740i	0.544597	+	0.838698i	$0.316683\pi$
$31$			289.520i			1.67740i	−0.544597	+	0.838698i	$0.683317\pi$
$32$	−50.8461			−0.280888
$33$	−154.656	−	20.8981i	−0.815821	−	0.110239i
$34$		−	218.776i		−	1.10352i
$35$			146.483i			0.707434i
$36$	−29.4808	−	8.11548i	−0.136485	−	0.0375716i
$37$			191.214i			0.849607i	0.905286	+	0.424803i	$0.139657\pi$
$37$			191.214i			0.849607i	−0.905286	+	0.424803i	$0.860343\pi$
$38$	79.9454			0.341286
$39$	5.73104	−	42.4124i	0.0235308	−	0.174139i
$40$			338.154i			1.33667i
$41$		−	36.3424i		−	0.138432i	−0.997602	−	0.0692162i	$-0.977950\pi$
$41$		−	36.3424i		−	0.138432i	0.997602	−	0.0692162i	$-0.0220498\pi$
$42$	−139.899	−	18.9041i	−0.513974	−	0.0694516i
$43$			491.069i			1.74156i	0.491669	+	0.870782i	$0.336387\pi$
$43$			491.069i			1.74156i	−0.491669	+	0.870782i	$0.663613\pi$
$44$	34.0134			0.116539
$45$	−101.252	+	367.813i	−0.335415	+	1.21845i
$46$	109.405			0.350670
$47$	26.4171			0.0819857			0.0409929	−	0.999159i	$-0.486948\pi$
$47$	26.4171			0.0819857			0.0409929	+	0.999159i	$0.486948\pi$
$48$	−276.301	−	37.3357i	−0.830847	−	0.112270i
$49$	−235.521			−0.686649
$50$	195.604			0.553252
$51$	−58.0889	+	429.885i	−0.159492	+	1.18031i
$52$			9.32776i			0.0248755i
$53$			181.825i			0.471238i	0.971846	+	0.235619i	$0.0757117\pi$
$53$			181.825i			0.471238i	−0.971846	+	0.235619i	$0.924288\pi$
$54$	−338.214	−	144.168i	−0.852316	−	0.363310i
$55$		−	424.363i		−	1.04038i
$56$	248.114			0.592065
$57$	−157.089	−	21.2270i	−0.365035	−	0.0493260i
$58$			193.601i			0.438294i
$59$	−187.180	−	412.726i	−0.413029	−	0.910718i
$59$	−187.180	−	412.726i	−0.413029	−	0.910718i
$60$	11.1341	−	82.3977i	0.0239568	−	0.177292i
$61$		−	775.225i		−	1.62717i	−0.581446	−	0.813585i	$-0.697513\pi$
$61$		−	775.225i		−	1.62717i	0.581446	−	0.813585i	$-0.302487\pi$
$62$		−	758.713i		−	1.55414i
$63$	269.876	+	74.2915i	0.539702	+	0.148569i
$64$	562.507			1.09865
$65$	116.376			0.222072
$66$	405.289	+	54.7654i	0.755873	+	0.102139i
$67$		−	176.188i		−	0.321265i	−0.987014	−	0.160632i	$-0.948647\pi$
$67$		−	176.188i		−	0.321265i	0.987014	−	0.160632i	$-0.0513533\pi$
$68$		−	94.5447i		−	0.168606i
$69$	−214.975	−	29.0489i	−0.375072	−	0.0506823i
$70$		−	383.873i		−	0.655451i
$71$		−	447.258i		−	0.747602i	−0.927509	−	0.373801i	$-0.878054\pi$
$71$		−	447.258i		−	0.747602i	0.927509	−	0.373801i	$-0.121946\pi$
$72$	623.005	+	171.501i	1.01975	+	0.280716i
$73$			417.573i			0.669496i	0.942308	+	0.334748i	$0.108651\pi$
$73$			417.573i			0.669496i	−0.942308	+	0.334748i	$0.891349\pi$
$74$		−	501.095i		−	0.787177i
$75$	−384.354	−	51.9364i	−0.591751	−	0.0799614i
$76$	34.5487			0.0521448
$77$	−311.369			−0.460829
$78$	−15.0187	+	111.145i	−0.0218017	+	0.161343i
$79$	574.434			0.818086			0.409043	−	0.912515i	$-0.365863\pi$
$79$	574.434			0.818086			0.409043	+	0.912515i	$0.365863\pi$
$80$		−	758.150i		−	1.05955i
$81$	626.297	+	373.086i	0.859118	+	0.511777i
$82$			95.2385i			0.128260i
$83$	998.011			1.31983			0.659916	−	0.751340i	$-0.270592\pi$
$83$	998.011			1.31983			0.659916	+	0.751340i	$0.270592\pi$
$84$	−60.4579	−	8.16947i	−0.0785297	−	0.0106115i
$85$	−1179.57			−1.50521
$86$		−	1286.89i		−	1.61359i
$87$	51.4046	−	380.418i	0.0633466	−	0.468794i
$88$	−718.790			−0.870719
$89$	1179.83			1.40518			0.702592	−	0.711593i	$-0.252026\pi$
$89$	1179.83			1.40518			0.702592	+	0.711593i	$0.252026\pi$
$90$	265.339	−	963.888i	0.310769	−	1.12892i
$91$		−	85.3891i		−	0.0983649i
$92$	47.2796			0.0535786
$93$	−201.452	+	1490.84i	−0.224619	+	1.66229i
$94$	−69.2284			−0.0759613
$95$		−	431.041i		−	0.465515i
$96$	−261.825	−	35.3795i	−0.278358	−	0.0376136i
$97$		−	806.397i		−	0.844095i	−0.906574	−	0.422048i	$-0.861312\pi$
$97$		−	806.397i		−	0.844095i	0.906574	−	0.422048i	$-0.138688\pi$
$98$	617.203			0.636193
$99$	−781.835	−	215.223i	−0.793711	−	0.218492i
$100$	84.5310			0.0845310
$101$	−891.133			−0.877931			−0.438966	−	0.898504i	$-0.644655\pi$
$101$	−891.133			−0.877931			−0.438966	+	0.898504i	$0.644655\pi$
$102$	152.227	−	1126.55i	0.147772	−	1.09358i
$103$			950.143i			0.908936i	0.890763	+	0.454468i	$0.150171\pi$
$103$			950.143i			0.908936i	−0.890763	+	0.454468i	$0.849829\pi$
$104$		−	197.119i		−	0.185857i
$105$	−101.925	+	754.294i	−0.0947322	+	0.701062i
$106$		−	476.489i		−	0.436611i
$107$		−	1066.57i		−	0.963637i	−0.876271	−	0.481818i	$-0.839976\pi$
$107$		−	1066.57i		−	0.963637i	0.876271	−	0.481818i	$-0.160024\pi$
$108$	−146.160	−	62.3027i	−0.130225	−	0.0555100i
$109$		−	1354.53i		−	1.19028i	−0.803622	−	0.595140i	$-0.797097\pi$
$109$		−	1354.53i		−	1.19028i	0.803622	−	0.595140i	$-0.202903\pi$
$110$			1112.08i			0.963936i
$111$	−133.050	+	984.630i	−0.113771	+	0.841955i
$112$	−556.279			−0.469317
$113$	−1014.43			−0.844506			−0.422253	−	0.906478i	$-0.638761\pi$
$113$	−1014.43			−0.844506			−0.422253	+	0.906478i	$0.638761\pi$
$114$	411.667	+	55.6272i	0.338212	+	0.0457014i
$115$		−	589.876i		−	0.478315i
$116$			83.6654i			0.0669667i
$117$	59.0223	−	214.408i	0.0466377	−	0.169419i
$118$	490.522	+	1081.59i	0.382680	+	0.843797i
$119$			865.491i			0.666718i
$120$	−235.292	+	1741.27i	−0.178993	+	1.32463i
$121$	−428.960			−0.322284
$122$			2031.55i			1.50760i
$123$	25.2876	−	187.140i	0.0185374	−	0.137186i
$124$		−	327.881i		−	0.237456i
$125$			711.542i			0.509138i
$126$	−707.236	−	194.688i	−0.500044	−	0.137652i
$127$	2012.85			1.40639			0.703195	−	0.710997i	$-0.251756\pi$
$127$	2012.85			1.40639			0.703195	+	0.710997i	$0.251756\pi$
$128$	−1067.33			−0.737029
$129$	−341.693	+	2528.69i	−0.233212	+	1.72588i
$130$	−304.975			−0.205754
$131$	2285.52			1.52433			0.762164	−	0.647384i	$-0.224137\pi$
$131$	2285.52			1.52433			0.762164	+	0.647384i	$0.224137\pi$
$132$	175.147	+	23.6671i	0.115489	+	0.0156057i
$133$	−316.269			−0.206196
$134$			461.716i			0.297658i
$135$	−777.310	+	1823.55i	−0.495557	+	1.16256i
$136$			1997.97i			1.25974i
$137$			187.297i			0.116802i	0.998293	+	0.0584009i	$0.0186002\pi$
$137$			187.297i			0.116802i	−0.998293	+	0.0584009i	$0.981400\pi$
$138$	563.363	+	76.1253i	0.347512	+	0.0469581i
$139$	1350.81			0.824274			0.412137	−	0.911122i	$-0.364782\pi$
$139$	1350.81			0.824274			0.412137	+	0.911122i	$0.364782\pi$
$140$		−	165.892i		−	0.100146i
$141$	136.031	+	18.3814i	0.0812473	+	0.0109787i
$142$			1172.08i			0.692667i
$143$			247.373i			0.144660i
$144$	−1396.79	−	384.509i	−0.808330	−	0.222517i
$145$	1043.84			0.597835
$146$		−	1094.29i		−	0.620301i
$147$	−1212.78	−	163.879i	−0.680464	−	0.0919489i
$148$		−	216.550i		−	0.120272i
$149$	412.444			0.226770			0.113385	−	0.993551i	$-0.463831\pi$
$149$	412.444			0.226770			0.113385	+	0.993551i	$0.463831\pi$
$150$	1007.23	+	136.104i	0.548269	+	0.0740857i
$151$		−	1168.07i		−	0.629509i	−0.949173	−	0.314755i	$-0.898078\pi$
$151$		−	1168.07i		−	0.629509i	0.949173	−	0.314755i	$-0.101922\pi$
$152$	−730.101			−0.389599
$153$	−598.241	+	2173.21i	−0.316111	+	1.14832i
$154$	815.972			0.426967
$155$	−4090.75			−2.11985
$156$	−6.49039	+	48.0319i	−0.00333107	+	0.0246515i
$157$		−	1079.86i		−	0.548932i	−0.961597	−	0.274466i	$-0.911499\pi$
$157$		−	1079.86i		−	0.548932i	0.961597	−	0.274466i	$-0.0885010\pi$
$158$	−1505.36			−0.757973
$159$	−126.517	+	936.281i	−0.0631033	+	0.466993i
$160$		−	718.427i		−	0.354979i
$161$	−432.811			−0.211865
$162$	−1641.27	−	977.706i	−0.795989	−	0.474172i
$163$	2225.21			1.06928			0.534638	−	0.845081i	$-0.320448\pi$
$163$	2225.21			1.06928			0.534638	+	0.845081i	$0.320448\pi$
$164$			41.1577i			0.0195968i
$165$	295.278	−	2185.20i	0.139318	−	1.03101i
$166$	−2615.38			−1.22285
$167$			1986.92i			0.920675i	0.887744	+	0.460337i	$0.152272\pi$
$167$			1986.92i			0.920675i	−0.887744	+	0.460337i	$0.847728\pi$
$168$	1277.63	+	172.642i	0.586733	+	0.0792833i
$169$	2129.16			0.969122
$170$	3091.18			1.39460
$171$	−794.139	−	218.610i	−0.355142	−	0.0977634i
$172$		−	556.134i		−	0.246540i
$173$	1605.80			0.705702			0.352851	−	0.935679i	$-0.385212\pi$
$173$	1605.80			0.705702			0.352851	+	0.935679i	$0.385212\pi$
$174$	−134.710	+	996.920i	−0.0586918	+	0.434347i
$175$	−773.822			−0.334260
$176$	1611.55			0.690198
$177$	−676.674	−	2255.51i	−0.287356	−	0.957824i
$178$	−3091.84			−1.30193
$179$	−2114.17			−0.882795			−0.441397	−	0.897312i	$-0.645517\pi$
$179$	−2114.17			−0.882795			−0.441397	+	0.897312i	$0.645517\pi$
$180$	114.667	−	416.548i	0.0474821	−	0.172487i
$181$	43.0123			0.0176634			0.00883172	−	0.999961i	$-0.497189\pi$
$181$	43.0123			0.0176634			0.00883172	+	0.999961i	$0.497189\pi$
$182$			223.770i			0.0911370i
$183$	539.413	−	3991.91i	0.217894	−	1.61252i
$184$	−999.137			−0.400312
$185$	−2701.75			−1.07371
$186$	527.924	−	3906.88i	0.208114	−	1.54014i
$187$		−	2507.34i		−	0.980506i
$188$	−29.9173			−0.0116061
$189$	1338.00	+	570.337i	0.514947	+	0.219502i
$190$			1129.58i			0.431309i
$191$	3484.72			1.32013			0.660066	−	0.751207i	$-0.270528\pi$
$191$	3484.72			1.32013			0.660066	+	0.751207i	$0.270528\pi$
$192$	2896.55	+	391.400i	1.08875	+	0.147119i
$193$	2185.66			0.815166			0.407583	−	0.913168i	$-0.366372\pi$
$193$	2185.66			0.815166			0.407583	+	0.913168i	$0.366372\pi$
$194$			2113.24i			0.782070i
$195$	599.263	+	80.9764i	0.220072	+	0.0297376i
$196$	266.727			0.0972036
$197$			2739.35i			0.990713i	0.868690	+	0.495357i	$0.164963\pi$
$197$			2739.35i			0.990713i	−0.868690	+	0.495357i	$0.835037\pi$
$198$	2048.87	+	564.012i	0.735388	+	0.202437i
$199$	−5466.83			−1.94740			−0.973701	−	0.227829i	$-0.926837\pi$
$199$	−5466.83			−1.94740			−0.973701	+	0.227829i	$0.926837\pi$
$200$	−1786.35			−0.631571
$201$	122.594	−	907.252i	0.0430204	−	0.318371i
$202$	2335.30			0.813420
$203$		−	765.898i		−	0.264805i
$204$	65.7856	−	486.844i	0.0225780	−	0.167088i
$205$	513.498			0.174947
$206$		−	2489.94i		−	0.842147i
$207$	−1086.77	−	299.166i	−0.364907	−	0.100452i
$208$			441.947i			0.147324i
$209$	916.235			0.303241
$210$	267.104	−	1976.70i	0.0877712	−	0.649548i
$211$			2434.64i			0.794349i	0.917743	+	0.397174i	$0.130009\pi$
$211$			2434.64i			0.794349i	−0.917743	+	0.397174i	$0.869991\pi$
$212$		−	205.917i		−	0.0667095i
$213$	311.209	−	2303.09i	0.100111	−	0.740868i
$214$			2795.04i			0.892828i
$215$	−6938.52			−2.20095
$216$	3088.74	+	1316.61i	0.972972	+	0.414741i
$217$			3001.52i			0.938969i
$218$			3549.67i			1.10282i
$219$	−290.554	+	2150.23i	−0.0896520	+	0.663466i
$220$			480.591i			0.147279i
$221$	687.605			0.209291
$222$	348.669	−	2580.31i	0.105411	−	0.780087i
$223$	−1839.93			−0.552516			−0.276258	−	0.961084i	$-0.589094\pi$
$223$	−1839.93			−0.552516			−0.276258	+	0.961084i	$0.589094\pi$
$224$	−527.133			−0.157235
$225$	−1943.03	−	534.878i	−0.575714	−	0.158482i
$226$	2658.40			0.782451
$227$	5447.94			1.59292			0.796459	−	0.604693i	$-0.206704\pi$
$227$	5447.94			1.59292			0.796459	+	0.604693i	$0.206704\pi$
$228$	177.903	+	24.0395i	0.0516752	+	0.00698269i
$229$		−	514.199i		−	0.148381i	−0.997244	−	0.0741904i	$-0.976363\pi$
$229$		−	514.199i		−	0.148381i	0.997244	−	0.0741904i	$-0.0236373\pi$
$230$			1545.83i			0.443168i
$231$	−1603.35	−	216.655i	−0.456678	−	0.0617094i
$232$		−	1768.06i		−	0.500340i
$233$	168.140			0.0472756			0.0236378	−	0.999721i	$-0.492475\pi$
$233$	168.140			0.0472756			0.0236378	+	0.999721i	$0.492475\pi$
$234$	−154.673	+	561.877i	−0.0432107	+	0.156970i
$235$			373.259i			0.103612i
$236$	211.981	+	467.411i	0.0584694	+	0.128923i
$237$	2957.96	+	399.700i	0.810718	+	0.109550i
$238$		−	2268.10i		−	0.617727i
$239$			4028.24i			1.09023i	0.838361	+	0.545116i	$0.183514\pi$
$239$			4028.24i			1.09023i	−0.838361	+	0.545116i	$0.816486\pi$
$240$	527.532	−	3903.98i	0.141884	−	1.05000i
$241$	−3105.01			−0.829922			−0.414961	−	0.909839i	$-0.636205\pi$
$241$	−3105.01			−0.829922			−0.414961	+	0.909839i	$0.636205\pi$
$242$	1124.13			0.298602
$243$	2965.43	+	2356.94i	0.782848	+	0.622212i
$244$			877.941i			0.230346i
$245$		−	3327.77i		−	0.867770i
$246$	−66.2684	+	490.417i	−0.0171753	+	0.127105i
$247$			251.266i			0.0647274i
$248$			6928.94i			1.77415i
$249$	5139.11	+	694.431i	1.30794	+	0.176738i
$250$		−	1864.66i		−	0.471726i
$251$		−	6148.67i		−	1.54622i	−0.634274	−	0.773108i	$-0.718701\pi$
$251$		−	6148.67i		−	1.54622i	0.634274	−	0.773108i	$-0.281299\pi$
$252$	−305.635	−	84.1350i	−0.0764015	−	0.0210318i
$253$	1253.86			0.311579
$254$	−5274.86			−1.30305
$255$	−6074.04	−	820.764i	−1.49165	−	0.201562i
$256$	−1703.01			−0.415775
$257$		−	5810.82i		−	1.41038i	−0.709016	−	0.705192i	$-0.750861\pi$
$257$		−	5810.82i		−	1.41038i	0.709016	−	0.705192i	$-0.249139\pi$
$258$	895.437	−	6626.65i	0.216076	−	1.59906i
$259$			1982.36i			0.475591i
$260$	−131.796			−0.0314371
$261$	529.401	−	1923.14i	0.125552	−	0.456089i
$262$	−5989.42			−1.41232
$263$			1977.98i			0.463756i	0.972745	+	0.231878i	$0.0744870\pi$
$263$			1977.98i			0.463756i	−0.972745	+	0.231878i	$0.925513\pi$
$264$	−3701.30	−	500.145i	−0.862877	−	0.116598i
$265$	−2569.09			−0.595539
$266$	828.812			0.191044
$267$	6075.34	+	820.941i	1.39253	+	0.188168i
$268$			199.532i			0.0454790i
$269$	3386.28			0.767529			0.383765	−	0.923431i	$-0.374627\pi$
$269$	3386.28			0.767529			0.383765	+	0.923431i	$0.374627\pi$
$270$	2037.01	−	4778.77i	0.459143	−	1.07714i
$271$	−1085.64			−0.243351			−0.121676	−	0.992570i	$-0.538827\pi$
$271$	−1085.64			−0.243351			−0.121676	+	0.992570i	$0.538827\pi$
$272$		−	4479.50i		−	0.998565i
$273$	59.4150	−	439.699i	0.0131720	−	0.0974790i
$274$		−	490.828i		−	0.108219i
$275$	2241.77			0.491578
$276$	243.459	+	32.8978i	0.0530961	+	0.00717470i
$277$	482.947			0.104756			0.0523781	−	0.998627i	$-0.483320\pi$
$277$	482.947			0.104756			0.0523781	+	0.998627i	$0.483320\pi$
$278$	−3539.92			−0.763706
$279$	−2074.70	+	7536.68i	−0.445193	+	1.61724i
$280$			3505.72i			0.748238i
$281$		−	8460.57i		−	1.79614i	−0.439853	−	0.898070i	$-0.644969\pi$
$281$		−	8460.57i		−	1.79614i	0.439853	−	0.898070i	$-0.355031\pi$
$282$	−356.482	−	48.1702i	−0.0752772	−	0.0101720i
$283$			3950.67i			0.829834i	0.909859	+	0.414917i	$0.136189\pi$
$283$			3950.67i			0.829834i	−0.909859	+	0.414917i	$0.863811\pi$
$284$			506.518i			0.105832i
$285$	299.925	−	2219.58i	0.0623369	−	0.461322i
$286$		−	648.264i		−	0.134030i
$287$		−	376.770i		−	0.0774914i
$288$	−1323.61	−	364.363i	−0.270814	−	0.0745496i
$289$	−2056.46			−0.418576
$290$	−2735.47			−0.553905
$291$	561.103	−	4152.42i	0.113032	−	0.836493i
$292$		−	472.901i		−	0.0947754i
$293$		−	3996.29i		−	0.796812i	−0.917209	−	0.398406i	$-0.869563\pi$
$293$		−	3996.29i		−	0.796812i	0.917209	−	0.398406i	$-0.130437\pi$
$294$	3178.20	+	429.459i	0.630463	+	0.0851924i
$295$	5831.59	−	2644.75i	1.15094	−	0.521977i
$296$			4576.25i			0.898611i
$297$	−3876.19	−	1652.27i	−0.757304	−	0.322810i
$298$	−1080.85			−0.210107
$299$			343.855i			0.0665072i
$300$	435.280	+	58.8179i	0.0837697	+	0.0113195i
$301$			5091.02i			0.974889i
$302$			3061.03i			0.583252i
$303$	−4588.76	−	620.064i	−0.870024	−	0.117563i
$304$	1636.91			0.308826
$305$	10953.5			2.05638
$306$	1567.75	−	5695.10i	0.292882	−	1.06394i
$307$	3399.89			0.632058			0.316029	−	0.948750i	$-0.397650\pi$
$307$	3399.89			0.632058			0.316029	+	0.948750i	$0.397650\pi$
$308$	352.625			0.0652360
$309$	−661.124	+	4892.62i	−0.121715	+	0.900750i
$310$	10720.2			1.96408
$311$			9897.44i			1.80460i	0.431104	+	0.902302i	$0.358124\pi$
$311$			9897.44i			1.80460i	−0.431104	+	0.902302i	$0.641876\pi$
$312$	137.158	−	1015.04i	0.0248880	−	0.184183i
$313$		−	2022.75i		−	0.365280i	−0.983180	−	0.182640i	$-0.941536\pi$
$313$		−	2022.75i		−	0.365280i	0.983180	−	0.182640i	$-0.0584643\pi$
$314$			2829.87i			0.508596i
$315$	−1049.70	+	3813.20i	−0.187758	+	0.682063i
$316$	−650.545			−0.115810
$317$		−	4543.64i		−	0.805036i	−0.915412	−	0.402518i	$-0.868135\pi$
$317$		−	4543.64i		−	0.805036i	0.915412	−	0.402518i	$-0.131865\pi$
$318$	331.548	−	2453.61i	0.0584664	−	0.432678i
$319$			2218.81i			0.389435i
$320$			7947.90i			1.38844i
$321$	742.135	−	5492.14i	0.129040	−	0.954958i
$322$	1134.22			0.196297
$323$		−	2546.79i		−	0.438722i
$324$	−709.280	−	422.519i	−0.121619	−	0.0724484i
$325$			614.778i			0.104928i
$326$	−5831.37			−0.990705
$327$	942.502	−	6974.96i	0.159390	−	1.17956i
$328$		−	869.766i		−	0.146417i
$329$	273.872			0.0458938
$330$	−773.804	+	5726.51i	−0.129080	+	0.955254i
$331$	3902.76			0.648082			0.324041	−	0.946043i	$-0.394958\pi$
$331$	3902.76			0.648082			0.324041	+	0.946043i	$0.394958\pi$
$332$	−1130.25			−0.186838
$333$	−1370.24	+	4977.63i	−0.225492	+	0.819137i
$334$		−	5206.91i		−	0.853023i
$335$	2489.43			0.406007
$336$	−2864.48	−	387.067i	−0.465090	−	0.0628460i
$337$		−	4345.21i		−	0.702369i	−0.936306	−	0.351185i	$-0.885779\pi$
$337$		−	4345.21i		−	0.702369i	0.936306	−	0.351185i	$-0.114221\pi$
$338$	−5579.66			−0.897910
$339$	−5223.64	−	705.852i	−0.836900	−	0.113087i
$340$	1335.86			0.213081
$341$		−	8695.43i		−	1.38089i
$342$	2081.11	+	572.888i	0.329046	+	0.0905797i
$343$	−5997.66			−0.944149
$344$			11752.5i			1.84202i
$345$	410.445	−	3037.48i	0.0640510	−	0.474007i
$346$	−4208.14			−0.653847
$347$	−8675.42			−1.34213			−0.671067	−	0.741396i	$-0.734164\pi$
$347$	−8675.42			−1.34213			−0.671067	+	0.741396i	$0.734164\pi$
$348$	−58.2156	+	430.822i	−0.00896748	+	0.0663635i
$349$		−	1901.71i		−	0.291679i	−0.989308	−	0.145840i	$-0.953412\pi$
$349$		−	1901.71i		−	0.291679i	0.989308	−	0.145840i	$-0.0465883\pi$
$350$	2027.87			0.309698
$351$	453.115	−	1063.00i	0.0689046	−	0.161648i
$352$	1527.11			0.231237
$353$	3640.14			0.548853			0.274427	−	0.961608i	$-0.411512\pi$
$353$	3640.14			0.548853			0.274427	+	0.961608i	$0.411512\pi$
$354$	1773.29	+	5910.78i	0.266241	+	0.887442i
$355$	6319.50			0.944801
$356$	−1336.15			−0.198921
$357$	−602.221	+	4456.72i	−0.0892799	+	0.660713i
$358$	5540.37			0.817926
$359$			3116.66i			0.458193i	0.973404	+	0.229096i	$0.0735771\pi$
$359$			3116.66i			0.458193i	−0.973404	+	0.229096i	$0.926423\pi$
$360$	−2423.21	+	8802.71i	−0.354762	+	1.28873i
$361$	−5928.35			−0.864316
$362$	−112.718			−0.0163655
$363$	−2208.87	−	298.477i	−0.319382	−	0.0431570i
$364$			96.7030i			0.0139248i
$365$	−5900.07			−0.846093
$366$	−1413.58	+	10461.2i	−0.201883	+	1.49403i
$367$		−	12568.8i		−	1.78770i	−0.448363	−	0.893852i	$-0.647993\pi$
$367$		−	12568.8i		−	1.78770i	0.448363	−	0.893852i	$-0.352007\pi$
$368$	2240.09			0.317318
$369$	260.429	−	946.053i	0.0367409	−	0.133468i
$370$	7080.19			0.994815
$371$			1885.02i			0.263788i
$372$	228.144	−	1688.37i	0.0317976	−	0.235317i
$373$	−7837.46			−1.08796			−0.543979	−	0.839099i	$-0.683083\pi$
$373$	−7837.46			−1.08796			−0.543979	+	0.839099i	$0.683083\pi$
$374$			6570.70i			0.908457i
$375$	−495.102	+	3663.98i	−0.0681785	+	0.504553i
$376$	632.228			0.0867146
$377$	−608.482			−0.0831258
$378$	−3506.34	−	1494.62i	−0.477108	−	0.203373i
$379$	−958.789			−0.129946			−0.0649732	−	0.997887i	$-0.520696\pi$
$379$	−958.789			−0.129946			−0.0649732	+	0.997887i	$0.520696\pi$
$380$			488.154i			0.0658994i
$381$	10364.9	+	1400.57i	1.39372	+	0.188329i
$382$	−9132.02			−1.22313
$383$		−	4651.79i		−	0.620615i	−0.950636	−	0.310307i	$-0.899568\pi$
$383$		−	4651.79i		−	0.620615i	0.950636	−	0.310307i	$-0.100432\pi$
$384$	−5496.07	−	742.665i	−0.730391	−	0.0986952i
$385$		−	4399.47i		−	0.582384i
$386$	−5727.72			−0.755267
$387$	−3518.99	+	12783.3i	−0.462224	+	1.67911i
$388$			913.243i			0.119492i
$389$			4534.02i			0.590961i	0.955349	+	0.295481i	$0.0954797\pi$
$389$			4534.02i			0.590961i	−0.955349	+	0.295481i	$0.904520\pi$
$390$	−1570.42	−	212.206i	−0.203901	−	0.0275525i
$391$		−	3485.26i		−	0.450786i
$392$	−5636.61			−0.726254
$393$	11769.0	+	1590.30i	1.51060	+	0.204122i
$394$		−	7178.72i		−	0.917915i
$395$			8116.43i			1.03388i
$396$	885.427	+	243.740i	0.112360	+	0.0309303i
$397$		−	12116.2i		−	1.53172i	−0.643006	−	0.765861i	$-0.722313\pi$
$397$		−	12116.2i		−	1.53172i	0.643006	−	0.765861i	$-0.277687\pi$
$398$	14326.3			1.80431
$399$	−1628.58	−	220.065i	−0.204338	−	0.0276116i
$400$	4005.06			0.500632
$401$	6566.79			0.817780			0.408890	−	0.912584i	$-0.365916\pi$
$401$	6566.79			0.817780			0.408890	+	0.912584i	$0.365916\pi$
$402$	−321.269	+	2377.54i	−0.0398593	+	0.294977i
$403$	2384.61			0.294754
$404$	1009.21			0.124282
$405$	−5271.49	+	8849.23i	−0.646772	+	1.08573i
$406$			2007.11i			0.245347i
$407$		−	5742.93i		−	0.699426i
$408$	−1390.22	+	10288.2i	−0.168691	+	1.24839i
$409$			13854.7i			1.67498i	0.546449	+	0.837492i	$0.315979\pi$
$409$			13854.7i			1.67498i	−0.546449	+	0.837492i	$0.684021\pi$
$410$	−1345.67			−0.162092
$411$	−130.324	+	964.457i	−0.0156409	+	0.115750i
$412$		−	1076.04i		−	0.128671i
$413$	−1940.54	−	4278.83i	−0.231205	−	0.509800i
$414$	2847.98	+	783.992i	0.338094	+	0.0930703i
$415$			14101.3i			1.66797i
$416$			418.791i			0.0493580i
$417$	6955.79	+	939.913i	0.816850	+	0.110378i
$418$	−2401.08			−0.280958
$419$	−13610.4			−1.58690			−0.793452	−	0.608633i	$-0.791718\pi$
$419$	−13610.4			−1.58690			−0.793452	+	0.608633i	$0.791718\pi$
$420$	115.430	−	854.236i	0.0134105	−	0.0992440i
$421$			6670.28i			0.772185i	0.922460	+	0.386092i	$0.126175\pi$
$421$			6670.28i			0.772185i	−0.922460	+	0.386092i	$0.873825\pi$
$422$		−	6380.20i		−	0.735979i
$423$	687.681	+	189.305i	0.0790454	+	0.0217596i
$424$			4351.54i			0.498418i
$425$		−	6231.29i		−	0.711205i
$426$	−815.551	+	6035.45i	−0.0927548	+	0.686429i
$427$		−	8036.94i		−	0.910854i
$428$			1207.89i			0.136415i
$429$	−172.126	+	1273.81i	−0.0193714	+	0.143357i
$430$	18183.0			2.03922
$431$	3357.81			0.375266			0.187633	−	0.982239i	$-0.439918\pi$
$431$	3357.81			0.375266			0.187633	+	0.982239i	$0.439918\pi$
$432$	−6925.04	−	2951.88i	−0.771253	−	0.328756i
$433$	−13001.8			−1.44301			−0.721507	−	0.692408i	$-0.756550\pi$
$433$	−13001.8			−1.44301			−0.721507	+	0.692408i	$0.756550\pi$
$434$		−	7865.75i		−	0.869973i
$435$	5375.09	+	726.318i	0.592450	+	0.0800558i
$436$			1534.00i			0.168499i
$437$	1273.59			0.139414
$438$	761.422	−	5634.88i	0.0830643	−	0.614714i
$439$	−6978.94			−0.758740			−0.379370	−	0.925245i	$-0.623859\pi$
$439$	−6978.94			−0.758740			−0.379370	+	0.925245i	$0.623859\pi$
$440$		−	10156.1i		−	1.10039i
$441$	−6131.00	−	1687.74i	−0.662023	−	0.182242i
$442$	−1801.93			−0.193912
$443$	12822.4			1.37519			0.687595	−	0.726094i	$-0.258666\pi$
$443$	12822.4			1.37519			0.687595	+	0.726094i	$0.258666\pi$
$444$	150.679	−	1115.09i	0.0161056	−	0.119189i
$445$			16670.3i			1.77584i
$446$	4821.72			0.511917
$447$	2123.82	+	286.985i	0.224728	+	0.0303667i
$448$	5831.64			0.614998
$449$		−	11068.9i		−	1.16342i	−0.813397	−	0.581709i	$-0.802384\pi$
$449$		−	11068.9i		−	1.16342i	0.813397	−	0.581709i	$-0.197616\pi$
$450$	5091.90	+	1401.70i	0.533410	+	0.146837i
$451$			1091.51i			0.113962i
$452$	1148.84			0.119550
$453$	812.758	−	6014.79i	0.0842974	−	0.623840i
$454$	−14276.8			−1.47587
$455$	1206.50			0.124311
$456$	−3759.55	−	508.015i	−0.386090	−	0.0521710i
$457$		−	14195.7i		−	1.45306i	−0.687135	−	0.726530i	$-0.741132\pi$
$457$		−	14195.7i		−	1.45306i	0.687135	−	0.726530i	$-0.258868\pi$
$458$			1347.50i			0.137478i
$459$	−4592.71	+	10774.4i	−0.467035	+	1.09565i
$460$			668.034i			0.0677114i
$461$		−	12340.2i		−	1.24672i	−0.781934	−	0.623361i	$-0.785767\pi$
$461$		−	12340.2i		−	1.24672i	0.781934	−	0.623361i	$-0.214233\pi$
$462$	4201.73	+	567.765i	0.423121	+	0.0571750i
$463$			12455.4i			1.25022i	0.780536	+	0.625111i	$0.214946\pi$
$463$			12455.4i			1.25022i	−0.780536	+	0.625111i	$0.785054\pi$
$464$			3964.04i			0.396608i
$465$	−21064.7	−	2846.41i	−2.10076	−	0.283869i
$466$	−440.626			−0.0438018
$467$	14655.7			1.45221			0.726106	−	0.687583i	$-0.241328\pi$
$467$	14655.7			1.45221			0.726106	+	0.687583i	$0.241328\pi$
$468$	−66.8426	+	242.817i	−0.00660214	+	0.0239834i
$469$		−	1826.58i		−	0.179837i
$470$		−	978.159i		−	0.0959981i
$471$	751.383	−	5560.58i	0.0735072	−	0.543988i
$472$	−4479.69	−	9877.58i	−0.436853	−	0.963247i
$473$		−	14748.7i		−	1.43372i
$474$	−7751.61	−	1047.45i	−0.751146	−	0.101500i
$475$	2277.05			0.219954
$476$		−	980.167i		−	0.0943821i
$477$	−1302.96	+	4733.21i	−0.125070	+	0.454337i
$478$		−	10556.4i		−	1.01012i
$479$			19253.9i			1.83660i	0.395883	+	0.918301i	$0.370439\pi$
$479$			19253.9i			1.83660i	−0.395883	+	0.918301i	$0.629561\pi$
$480$	499.892	−	3699.43i	0.0475351	−	0.351782i
$481$	1574.93			0.149294
$482$	8136.96			0.768939
$483$	−2228.70	−	301.157i	−0.209957	−	0.0283708i
$484$	485.797			0.0456233
$485$	11393.9			1.06675
$486$	−7771.17	−	6176.57i	−0.725324	−	0.576492i
$487$	−6730.13			−0.626225			−0.313112	−	0.949716i	$-0.601372\pi$
$487$	−6730.13			−0.626225			−0.313112	+	0.949716i	$0.601372\pi$
$488$		−	18553.1i		−	1.72102i
$489$	11458.4	+	1548.34i	1.05965	+	0.143186i
$490$			8720.74i			0.804006i
$491$			13349.0i			1.22695i	0.789715	+	0.613473i	$0.210228\pi$
$491$			13349.0i			1.22695i	−0.789715	+	0.613473i	$0.789772\pi$
$492$	−28.6381	+	211.935i	−0.00262420	+	0.0194203i
$493$	6167.48			0.563427
$494$		−	658.466i		−	0.0599712i
$495$	3040.99	−	11046.9i	0.276126	−	1.00307i
$496$		−	15534.9i		−	1.40633i
$497$		−	4636.82i		−	0.418491i
$498$	−13467.5	−	1819.82i	−1.21184	−	0.163751i
$499$	13091.9			1.17449			0.587247	−	0.809408i	$-0.300212\pi$
$499$	13091.9			1.17449			0.587247	+	0.809408i	$0.300212\pi$
$500$		−	805.820i		−	0.0720748i
$501$	−1382.53	+	10231.4i	−0.123287	+	0.912383i
$502$			16113.1i			1.43260i
$503$	−22274.5			−1.97449			−0.987246	−	0.159205i	$-0.949107\pi$
$503$	−22274.5			−1.97449			−0.987246	+	0.159205i	$0.949107\pi$
$504$	6458.83	+	1777.99i	0.570832	+	0.157138i
$505$		−	12591.2i		−	1.10951i
$506$	−3285.85			−0.288684
$507$	10963.8	+	1481.50i	0.960394	+	0.129775i
$508$	−2279.55			−0.199092
$509$	2733.09			0.238000			0.119000	−	0.992894i	$-0.462031\pi$
$509$	2733.09			0.238000			0.119000	+	0.992894i	$0.462031\pi$
$510$	15917.6	+	2150.89i	1.38204	+	0.186751i
$511$			4329.08i			0.374769i
$512$	13001.6			1.12225
$513$	−3937.19	−	1678.27i	−0.338852	−	0.144440i
$514$			15227.8i			1.30675i
$515$	−13425.0			−1.14869
$516$	386.966	−	2863.73i	0.0330141	−	0.244319i
$517$	−793.410			−0.0674935
$518$		−	5194.96i		−	0.440644i
$519$	8268.82	+	1117.34i	0.699346	+	0.0945003i
$520$	2785.18			0.234881
$521$		−	9376.92i		−	0.788504i	−0.919002	−	0.394252i	$-0.871004\pi$
$521$		−	9376.92i		−	0.788504i	0.919002	−	0.394252i	$-0.128996\pi$
$522$	−1387.34	+	5039.76i	−0.116326	+	0.422575i
$523$	−5235.01			−0.437688			−0.218844	−	0.975760i	$-0.570229\pi$
$523$	−5235.01			−0.437688			−0.218844	+	0.975760i	$0.570229\pi$
$524$	−2588.35			−0.215787
$525$	−3984.68	−	538.437i	−0.331249	−	0.0447606i
$526$		−	5183.49i		−	0.429679i
$527$	−24170.1			−1.99784
$528$	8298.43	+	1121.34i	0.683982	+	0.0924242i
$529$	−10424.1			−0.856752
$530$	6732.53			0.551778
$531$	−1915.01	−	12085.3i	−0.156506	−	0.987677i
$532$	358.174			0.0291895
$533$	−299.332			−0.0243255
$534$	−15921.0	−	2151.35i	−1.29020	−	0.174341i
$535$	15070.0			1.21782
$536$		−	4216.62i		−	0.339795i
$537$	−10886.6	−	1471.07i	−0.874844	−	0.118215i
$538$	−8874.07			−0.711131
$539$	7073.62			0.565273
$540$	880.302	−	2065.16i	0.0701522	−	0.164575i
$541$			11705.8i			0.930261i	0.885242	+	0.465131i	$0.153993\pi$
$541$			11705.8i			0.930261i	−0.885242	+	0.465131i	$0.846007\pi$
$542$	2845.03			0.225469
$543$	221.486	+	29.9286i	0.0175043	+	0.00236530i
$544$		−	4244.80i		−	0.334548i
$545$	19138.8			1.50425
$546$	−155.702	+	1152.27i	−0.0122041	+	0.0903162i
$547$	24163.1			1.88874			0.944369	−	0.328889i	$-0.106674\pi$
$547$	24163.1			1.88874			0.944369	+	0.328889i	$0.106674\pi$
$548$		−	212.113i		−	0.0165347i
$549$	5555.26	−	20180.4i	0.431863	−	1.56881i
$550$	−5874.77			−0.455456
$551$			2253.73i			0.174251i
$552$	−5144.91	−	695.214i	−0.396706	−	0.0536056i
$553$	5955.29			0.457947
$554$	−1265.61			−0.0970586
$555$	−13912.3	−	1879.92i	−1.06404	−	0.143780i
$556$	−1529.79			−0.116686
$557$			1473.30i			0.112075i	0.998429	+	0.0560374i	$0.0178466\pi$
$557$			1473.30i			0.112075i	−0.998429	+	0.0560374i	$0.982153\pi$
$558$	5436.93	−	19750.6i	0.412480	−	1.49840i
$559$	4044.66			0.306030
$560$		−	7859.92i		−	0.593111i
$561$	1744.64	−	12911.2i	0.131299	−	0.971675i
$562$			22171.7i			1.66416i
$563$	2473.95			0.185194			0.0925971	−	0.995704i	$-0.470483\pi$
$563$	2473.95			0.185194			0.0925971	+	0.995704i	$0.470483\pi$
$564$	−154.055	−	20.8169i	−0.0115016	−	0.00155417i
$565$		−	14333.3i		−	1.06727i
$566$		−	10353.1i		−	0.768857i
$567$	6492.97	+	3867.87i	0.480915	+	0.286482i
$568$		−	10704.0i		−	0.790723i
$569$	−5358.46			−0.394795			−0.197398	−	0.980324i	$-0.563249\pi$
$569$	−5358.46			−0.394795			−0.197398	+	0.980324i	$0.563249\pi$
$570$	−785.981	+	5816.63i	−0.0577564	+	0.427424i
$571$			14870.4i			1.08985i	0.838483	+	0.544927i	$0.183443\pi$
$571$			14870.4i			1.08985i	−0.838483	+	0.544927i	$0.816557\pi$
$572$		−	280.150i		−	0.0204784i
$573$	17944.0	+	2424.72i	1.30824	+	0.176779i
$574$			987.360i			0.0717972i
$575$	3116.12			0.226002
$576$	14643.0	+	4030.92i	1.05924	+	0.291588i
$577$	13682.7			0.987206			0.493603	−	0.869687i	$-0.335680\pi$
$577$	13682.7			0.987206			0.493603	+	0.869687i	$0.335680\pi$
$578$	5389.15			0.387818
$579$	11254.7	+	1520.81i	0.807824	+	0.109159i
$580$	−1182.14			−0.0846309
$581$	10346.6			0.738812
$582$	−1470.42	+	10881.8i	−0.104727	+	0.775027i
$583$		−	5460.93i		−	0.387939i
$584$			9993.59i			0.708112i
$585$	3029.47	+	833.952i	0.214108	+	0.0589396i
$586$			10472.7i			0.738262i
$587$	−9648.79			−0.678446			−0.339223	−	0.940706i	$-0.610164\pi$
$587$	−9648.79			−0.678446			−0.339223	+	0.940706i	$0.610164\pi$
$588$	1373.47	+	185.592i	0.0963281	+	0.0130165i
$589$		−	8832.27i		−	0.617873i
$590$	−15282.2	+	6930.80i	−1.06637	+	0.483621i
$591$	−1906.08	+	14105.9i	−0.132666	+	0.981791i
$592$		−	10260.1i		−	0.712308i
$593$		−	18246.6i		−	1.26357i	−0.775143	−	0.631786i	$-0.782322\pi$
$593$		−	18246.6i		−	1.26357i	0.775143	−	0.631786i	$-0.217678\pi$
$594$	10157.9	+	4329.93i	0.701657	+	0.299090i
$595$	−12228.9			−0.842582
$596$	−467.092			−0.0321021
$597$	−28150.6	−	3803.90i	−1.92986	−	0.260776i
$598$		−	901.104i		−	0.0616202i
$599$			10841.3i			0.739506i	0.929130	+	0.369753i	$0.120558\pi$
$599$			10841.3i			0.739506i	−0.929130	+	0.369753i	$0.879442\pi$
$600$	−9198.56	−	1242.97i	−0.625883	−	0.0845735i
$601$		−	12538.7i		−	0.851022i	−0.904953	−	0.425511i	$-0.860094\pi$
$601$		−	12538.7i		−	0.851022i	0.904953	−	0.425511i	$-0.139906\pi$
$602$		−	13341.5i		−	0.903253i
$603$	1262.56	−	4586.46i	0.0852660	−	0.309743i
$604$			1322.83i			0.0891148i
$605$		−	6060.97i		−	0.407295i
$606$	12025.3	+	1624.93i	0.806094	+	0.108925i
$607$	17732.8			1.18575			0.592876	−	0.805294i	$-0.297993\pi$
$607$	17732.8			1.18575			0.592876	+	0.805294i	$0.297993\pi$
$608$	1551.14			0.103466
$609$	532.923	−	3943.88i	0.0354600	−	0.262420i
$610$	−28704.6			−1.90527
$611$		−	217.583i		−	0.0144066i
$612$	677.507	−	2461.16i	0.0447493	−	0.162559i
$613$			20181.4i			1.32972i	0.746967	+	0.664861i	$0.231509\pi$
$613$			20181.4i			1.32972i	−0.746967	+	0.664861i	$0.768491\pi$
$614$	−8909.72			−0.585614
$615$	2644.18	+	357.299i	0.173372	+	0.0234272i
$616$	−7451.86			−0.487409
$617$			12638.9i			0.824672i	0.911032	+	0.412336i	$0.135287\pi$
$617$			12638.9i			0.824672i	−0.911032	+	0.412336i	$0.864713\pi$
$618$	1732.54	−	12821.6i	0.112772	−	0.834562i
$619$	−20291.8			−1.31760			−0.658802	−	0.752316i	$-0.728937\pi$
$619$	−20291.8			−1.31760			−0.658802	+	0.752316i	$0.728937\pi$
$620$	4632.77			0.300091
$621$	−5388.01	−	2296.70i	−0.348169	−	0.148412i
$622$		−	25937.1i		−	1.67200i
$623$	12231.5			0.786591
$624$	−307.513	+	2275.74i	−0.0197282	+	0.145998i
$625$	−19383.8			−1.24057
$626$			5300.80i			0.338439i
$627$	4718.02	+	637.530i	0.300510	+	0.0406069i
$628$			1222.94i			0.0777080i
$629$	−15963.2			−1.01192
$630$	2750.83	−	9992.85i	0.173961	−	0.631944i
$631$	−16311.4			−1.02907			−0.514537	−	0.857468i	$-0.672036\pi$
$631$	−16311.4			−1.02907			−0.514537	+	0.857468i	$0.672036\pi$
$632$	13747.7			0.865273
$633$	−1694.06	+	12536.8i	−0.106371	+	0.787194i
$634$			11907.0i			0.745882i
$635$			28440.5i			1.77736i
$636$	143.280	−	1060.34i	0.00893304	−	0.0661086i
$637$			1939.85i			0.120659i
$638$		−	5814.61i		−	0.360819i
$639$	3205.05	−	11642.9i	0.198419	−	0.720790i
$640$		−	15080.8i		−	0.931439i
$641$		−	39.0296i		−	0.00240496i	−0.999999	−	0.00120248i	$-0.999617\pi$
$641$		−	39.0296i		−	0.00240496i	0.999999	−	0.00120248i	$-0.000382761\pi$
$642$	−1944.83	+	14392.7i	−0.119558	+	0.884787i
$643$	−16730.8			−1.02612			−0.513062	−	0.858352i	$-0.671489\pi$
$643$	−16730.8			−1.02612			−0.513062	+	0.858352i	$0.671489\pi$
$644$	490.158			0.0299921
$645$	−35728.9	−	4827.93i	−2.18112	−	0.294728i
$646$			6674.10i			0.406485i
$647$			1223.42i			0.0743392i	0.999309	+	0.0371696i	$0.0118342\pi$
$647$			1223.42i			0.0743392i	−0.999309	+	0.0371696i	$0.988166\pi$
$648$	14988.9	+	8928.89i	0.908671	+	0.541296i
$649$	5621.75	+	12395.8i	0.340020	+	0.749735i
$650$		−	1611.08i		−	0.0972182i
$651$	−2088.50	+	15455.9i	−0.125737	+	0.930512i
$652$	−2520.05			−0.151369
$653$			17337.5i			1.03900i	0.854470	+	0.519501i	$0.173882\pi$
$653$			17337.5i			1.03900i	−0.854470	+	0.519501i	$0.826118\pi$
$654$	−2469.91	+	18278.5i	−0.147678	+	1.09288i
$655$			32293.1i			1.92641i
$656$			1950.04i			0.116061i
$657$	−2992.33	+	10870.1i	−0.177689	+	0.645486i
$658$	−717.707			−0.0425215
$659$	2323.76			0.137361			0.0686805	−	0.997639i	$-0.478121\pi$
$659$	2323.76			0.137361			0.0686805	+	0.997639i	$0.478121\pi$
$660$	−334.402	+	2474.73i	−0.0197221	+	0.145953i
$661$	12976.1			0.763556			0.381778	−	0.924254i	$-0.375312\pi$
$661$	12976.1			0.763556			0.381778	+	0.924254i	$0.375312\pi$
$662$	−10227.5			−0.600460
$663$	3540.72	+	478.446i	0.207406	+	0.0280261i
$664$	23885.0			1.39596
$665$		−	4468.71i		−	0.260585i
$666$	3590.84	−	13044.3i	0.208922	−	0.758946i
$667$			3084.21i			0.179042i
$668$		−	2250.19i		−	0.130333i
$669$	−9474.47	−	1280.25i	−0.547540	−	0.0739872i
$670$	−6523.78			−0.376173
$671$			23283.1i			1.33954i
$672$	−2714.40	−	366.787i	−0.155819	−	0.0210552i
$673$			25426.3i			1.45633i	0.685400	+	0.728166i	$0.259627\pi$
$673$			25426.3i			1.45633i	−0.685400	+	0.728166i	$0.740373\pi$
$674$			11387.0i			0.650759i
$675$	−9633.20	−	4106.27i	−0.549307	−	0.234149i
$676$	−2411.27			−0.137191
$677$			28904.4i			1.64090i	0.571721	+	0.820448i	$0.306276\pi$
$677$			28904.4i			1.64090i	−0.571721	+	0.820448i	$0.693724\pi$
$678$	13689.0	+	1849.75i	0.775403	+	0.104778i
$679$		−	8360.11i		−	0.472506i
$680$	−28230.2			−1.59203
$681$	28053.3	+	3790.76i	1.57857	+	0.213307i
$682$			22787.2i			1.27942i
$683$	30973.2			1.73522			0.867611	−	0.497244i	$-0.165655\pi$
$683$	30973.2			1.73522			0.867611	+	0.497244i	$0.165655\pi$
$684$	899.360	+	247.576i	0.0502747	+	0.0138396i
$685$	−2646.40			−0.147611
$686$	15717.4			0.874772
$687$	357.787	−	2647.79i	0.0198696	−	0.147044i
$688$		−	26349.5i		−	1.46012i
$689$	1497.59			0.0828065
$690$	−1075.61	+	7960.00i	−0.0593445	+	0.439177i
$691$		−	29027.7i		−	1.59807i	−0.601286	−	0.799034i	$-0.705345\pi$
$691$		−	29027.7i		−	1.59807i	0.601286	−	0.799034i	$-0.294655\pi$
$692$	−1818.56			−0.0999008
$693$	−8105.46	−	2231.27i	−0.444302	−	0.122307i
$694$	22734.7			1.24351
$695$			19086.2i			1.04170i
$696$	1230.24	−	9104.37i	0.0670003	−	0.495834i
$697$	3033.98			0.164878
$698$			4983.59i			0.270246i
$699$	865.812	+	116.994i	0.0468498	+	0.00633066i
$700$	876.352			0.0473186
$701$	−14844.2			−0.799800			−0.399900	−	0.916559i	$-0.630955\pi$
$701$	−14844.2			−0.799800			−0.399900	+	0.916559i	$0.630955\pi$
$702$	−1187.43	+	2785.68i	−0.0638414	+	0.149770i
$703$		−	5833.30i		−	0.312955i
$704$	−16894.3			−0.904444
$705$	−259.719	+	1922.04i	−0.0138746	+	0.102678i
$706$	−9539.33			−0.508523
$707$	−9238.58			−0.491446
$708$	766.332	+	2554.37i	0.0406787	+	0.135592i
$709$	6585.05			0.348811			0.174405	−	0.984674i	$-0.444200\pi$
$709$	6585.05			0.348811			0.174405	+	0.984674i	$0.444200\pi$
$710$	−16560.8			−0.875376
$711$	14953.5	+	4116.39i	0.788747	+	0.217126i
$712$	28236.3			1.48623
$713$		−	12086.9i		−	0.634863i
$714$	1578.18	−	11679.2i	0.0827196	−	0.612163i
$715$	−3495.24			−0.182818
$716$	2394.29			0.124970
$717$	−2802.91	+	20742.8i	−0.145992	+	1.08041i
$718$		−	8167.51i		−	0.424525i
$719$	−37101.0			−1.92439			−0.962193	−	0.272368i	$-0.912193\pi$
$719$	−37101.0			−1.92439			−0.962193	+	0.272368i	$0.912193\pi$
$720$	5432.90	−	19735.9i	0.281211	−	1.02155i
$721$			9850.36i			0.508802i
$722$	15535.8			0.800806
$723$	−15988.8	−	2160.51i	−0.822448	−	0.111135i
$724$	−48.7114			−0.00250048
$725$			5514.25i			0.282475i
$726$	5788.54	+	782.186i	0.295913	+	0.0399857i
$727$	−25067.3			−1.27881			−0.639404	−	0.768871i	$-0.720819\pi$
$727$	−25067.3			−1.27881			−0.639404	+	0.768871i	$0.720819\pi$
$728$		−	2043.58i		−	0.104039i
$729$	13630.0	+	14200.1i	0.692478	+	0.721439i
$730$	15461.7			0.783921
$731$	−40996.0			−2.07427
$732$	−610.884	+	4520.83i	−0.0308455	+	0.228271i
$733$	30886.1			1.55635			0.778176	−	0.628047i	$-0.216145\pi$
$733$	30886.1			1.55635			0.778176	+	0.628047i	$0.216145\pi$
$734$			32937.8i			1.65634i
$735$	2315.51	−	17135.9i	0.116203	−	0.859955i
$736$	2122.72			0.106311
$737$			5291.61i			0.264476i
$738$	−682.479	+	2479.22i	−0.0340412	+	0.123660i
$739$		−	13226.2i		−	0.658368i	−0.944266	−	0.329184i	$-0.893226\pi$
$739$		−	13226.2i		−	0.658368i	0.944266	−	0.329184i	$-0.106774\pi$
$740$	3059.73			0.151997
$741$	−174.835	+	1293.86i	−0.00866762	+	0.0641444i
$742$		−	4939.87i		−	0.244405i
$743$		−	13180.4i		−	0.650797i	−0.945577	−	0.325399i	$-0.894502\pi$
$743$		−	13180.4i		−	0.650797i	0.945577	−	0.325399i	$-0.105498\pi$
$744$	−4821.26	+	35679.6i	−0.237575	+	1.75817i
$745$			5827.60i			0.286586i
$746$	20538.8			1.00801
$747$	25979.9	+	7151.74i	1.27250	+	0.350293i
$748$			2839.55i			0.138803i
$749$		−	11057.4i		−	0.539423i
$750$	1297.46	−	9601.80i	0.0631687	−	0.467478i
$751$			11157.1i			0.542117i	0.962563	+	0.271058i	$0.0873737\pi$
$751$			11157.1i			0.542117i	−0.962563	+	0.271058i	$0.912626\pi$
$752$	−1417.47			−0.0687366
$753$	4278.33	−	31661.7i	0.207053	−	1.53229i
$754$	1594.58			0.0770176
$755$	16504.1			0.795558
$756$	−1515.28	−	645.906i	−0.0728970	−	0.0310732i
$757$	6916.46			0.332078			0.166039	−	0.986119i	$-0.446902\pi$
$757$	6916.46			0.332078			0.166039	+	0.986119i	$0.446902\pi$
$758$	2512.60			0.120398
$759$	6456.56	+	872.454i	0.308773	+	0.0417234i
$760$		−	10315.9i		−	0.492366i
$761$			16380.6i			0.780283i	0.920755	+	0.390141i	$0.127574\pi$
$761$			16380.6i			0.780283i	−0.920755	+	0.390141i	$0.872426\pi$
$762$	−27162.1	−	3670.33i	−1.29131	−	0.174491i
$763$		−	14042.7i		−	0.666292i
$764$	−3946.44			−0.186881
$765$	−30706.3	−	8452.81i	−1.45122	−	0.399493i
$766$			12190.5i			0.575012i
$767$	−3399.39	+	1541.70i	−0.160033	+	0.0725781i
$768$	−8769.41	−	1184.98i	−0.412030	−	0.0556762i
$769$		−	39727.7i		−	1.86296i	−0.363792	−	0.931480i	$-0.618518\pi$
$769$		−	39727.7i		−	1.86296i	0.363792	−	0.931480i	$-0.381482\pi$
$770$			11529.2i			0.539590i
$771$	4043.25	−	29922.0i	0.188864	−	1.39768i
$772$	−2475.25			−0.115397
$773$	−21849.0			−1.01663			−0.508315	−	0.861171i	$-0.669732\pi$
$773$	−21849.0			−1.01663			−0.508315	+	0.861171i	$0.669732\pi$
$774$	9221.85	−	33499.9i	0.428259	−	1.55572i
$775$		−	21610.1i		−	1.00162i
$776$		−	19299.1i		−	0.892782i
$777$	−1379.36	+	10207.9i	−0.0636862	+	0.471308i
$778$		−	11881.8i		−	0.547537i
$779$			1108.68i			0.0509919i
$780$	−678.664	−	91.7056i	−0.0311539	−	0.00420973i
$781$			13432.9i			0.615452i
$782$			9133.45i			0.417662i
$783$	4064.22	−	9534.55i	0.185496	−	0.435169i

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:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	177.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.62059 q^{2} +(5.14935 + 0.695815i) q^{3} -1.13250 q^{4} +14.1294i q^{5} +(-13.4944 - 1.82345i) q^{6} +10.3672 q^{7} +23.9326 q^{8} +(26.0317 + 7.16599i) q^{9} +O(q^{10})\)	\(q-2.62059 q^{2} +(5.14935 + 0.695815i) q^{3} -1.13250 q^{4} +14.1294i q^{5} +(-13.4944 - 1.82345i) q^{6} +10.3672 q^{7} +23.9326 q^{8} +(26.0317 + 7.16599i) q^{9} -37.0275i q^{10} -30.0340 q^{11} +(-5.83163 - 0.788009i) q^{12} -8.23644i q^{13} -27.1683 q^{14} +(-9.83148 + 72.7575i) q^{15} -53.6575 q^{16} +83.4833i q^{17} +(-68.2184 - 18.7791i) q^{18} -30.5066 q^{19} -16.0016i q^{20} +(53.3845 + 7.21367i) q^{21} +78.7068 q^{22} -41.7480 q^{23} +(123.237 + 16.6526i) q^{24} -74.6412 q^{25} +21.5844i q^{26} +(129.060 + 55.0135i) q^{27} -11.7409 q^{28} -73.8768i q^{29} +(25.7643 - 190.668i) q^{30} +289.520i q^{31} -50.8461 q^{32} +(-154.656 - 20.8981i) q^{33} -218.776i q^{34} +146.483i q^{35} +(-29.4808 - 8.11548i) q^{36} +191.214i q^{37} +79.9454 q^{38} +(5.73104 - 42.4124i) q^{39} +338.154i q^{40} -36.3424i q^{41} +(-139.899 - 18.9041i) q^{42} +491.069i q^{43} +34.0134 q^{44} +(-101.252 + 367.813i) q^{45} +109.405 q^{46} +26.4171 q^{47} +(-276.301 - 37.3357i) q^{48} -235.521 q^{49} +195.604 q^{50} +(-58.0889 + 429.885i) q^{51} +9.32776i q^{52} +181.825i q^{53} +(-338.214 - 144.168i) q^{54} -424.363i q^{55} +248.114 q^{56} +(-157.089 - 21.2270i) q^{57} +193.601i q^{58} +(-187.180 - 412.726i) q^{59} +(11.1341 - 82.3977i) q^{60} -775.225i q^{61} -758.713i q^{62} +(269.876 + 74.2915i) q^{63} +562.507 q^{64} +116.376 q^{65} +(405.289 + 54.7654i) q^{66} -176.188i q^{67} -94.5447i q^{68} +(-214.975 - 29.0489i) q^{69} -383.873i q^{70} -447.258i q^{71} +(623.005 + 171.501i) q^{72} +417.573i q^{73} -501.095i q^{74} +(-384.354 - 51.9364i) q^{75} +34.5487 q^{76} -311.369 q^{77} +(-15.0187 + 111.145i) q^{78} +574.434 q^{79} -758.150i q^{80} +(626.297 + 373.086i) q^{81} +95.2385i q^{82} +998.011 q^{83} +(-60.4579 - 8.16947i) q^{84} -1179.57 q^{85} -1286.89i q^{86} +(51.4046 - 380.418i) q^{87} -718.790 q^{88} +1179.83 q^{89} +(265.339 - 963.888i) q^{90} -85.3891i q^{91} +47.2796 q^{92} +(-201.452 + 1490.84i) q^{93} -69.2284 q^{94} -431.041i q^{95} +(-261.825 - 35.3795i) q^{96} -806.397i q^{97} +617.203 q^{98} +(-781.835 - 215.223i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 52q - 8q^{3} + 268q^{4} - 16q^{7} - 4q^{9} + O(q^{10}) \)	\( 52q - 8q^{3} + 268q^{4} - 16q^{7} - 4q^{9} + 28q^{12} + 114q^{15} + 484q^{16} - 184q^{19} - 758q^{21} - 60q^{22} + 36q^{25} + 742q^{27} - 4q^{28} - 888q^{36} + 1402q^{45} - 660q^{46} - 488q^{48} - 924q^{49} - 1772q^{51} - 630q^{57} - 1880q^{60} - 212q^{63} + 7648q^{64} + 1316q^{66} - 1556q^{75} - 5680q^{76} + 3224q^{78} - 1504q^{79} - 276q^{81} + 1228q^{84} - 848q^{85} + 3598q^{87} + 5760q^{88} + 888q^{94} + O(q^{100}) \)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Properties

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