LMFDB - Embedded newform 177.7.c.a.58.7

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			58.7
Character	$\chi$	$=$	177.58
Dual form			177.7.c.a.58.54

$q$-expansion

$f(q)$	$=$	$q-13.2670i q^{2} +15.5885 q^{3} -112.012 q^{4} +130.160 q^{5} -206.811i q^{6} +304.309 q^{7} +636.976i q^{8} +243.000 q^{9} +O(q^{10})$	\(q-13.2670i q^{2} +15.5885 q^{3} -112.012 q^{4} +130.160 q^{5} -206.811i q^{6} +304.309 q^{7} +636.976i q^{8} +243.000 q^{9} -1726.83i q^{10} -302.342i q^{11} -1746.10 q^{12} +3866.28i q^{13} -4037.26i q^{14} +2029.00 q^{15} +1281.96 q^{16} +4992.75 q^{17} -3223.87i q^{18} +11773.0 q^{19} -14579.5 q^{20} +4743.71 q^{21} -4011.16 q^{22} +17075.4i q^{23} +9929.48i q^{24} +1316.68 q^{25} +51293.8 q^{26} +3788.00 q^{27} -34086.3 q^{28} +34405.8 q^{29} -26918.6i q^{30} -35505.8i q^{31} +23758.8i q^{32} -4713.05i q^{33} -66238.6i q^{34} +39608.9 q^{35} -27219.0 q^{36} -32831.2i q^{37} -156193. i q^{38} +60269.3i q^{39} +82909.0i q^{40} -63683.4 q^{41} -62934.6i q^{42} -54194.2i q^{43} +33866.0i q^{44} +31628.9 q^{45} +226539. q^{46} +14109.8i q^{47} +19983.7 q^{48} -25045.0 q^{49} -17468.4i q^{50} +77829.2 q^{51} -433071. i q^{52} +86435.8 q^{53} -50255.2i q^{54} -39352.9i q^{55} +193838. i q^{56} +183524. q^{57} -456461. i q^{58} +(-55030.6 + 197869. i) q^{59} -227272. q^{60} +21873.5i q^{61} -471054. q^{62} +73947.1 q^{63} +397252. q^{64} +503236. i q^{65} -62527.8 q^{66} +351365. i q^{67} -559249. q^{68} +266180. i q^{69} -525490. i q^{70} -350770. q^{71} +154785. i q^{72} -216142. i q^{73} -435570. q^{74} +20525.0 q^{75} -1.31873e6 q^{76} -92005.5i q^{77} +799591. q^{78} -797387. q^{79} +166860. q^{80} +59049.0 q^{81} +844885. i q^{82} -710474. i q^{83} -531353. q^{84} +649857. q^{85} -718992. q^{86} +536334. q^{87} +192585. q^{88} -1.09404e6i q^{89} -419620. i q^{90} +1.17654e6i q^{91} -1.91266e6i q^{92} -553480. i q^{93} +187194. q^{94} +1.53238e6 q^{95} +370363. i q^{96} -1.57323e6i q^{97} +332271. i q^{98} -73469.2i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$		−	13.2670i		−	1.65837i	−0.558974	−	0.829185i	$-0.688805\pi$
$2$		−	13.2670i		−	1.65837i	0.558974	−	0.829185i	$-0.311195\pi$
$3$	15.5885			0.577350
$3$	15.5885			0.577350
$4$	−112.012			−1.75019
$5$	130.160			1.04128			0.520641	−	0.853776i	$-0.325693\pi$
$5$	130.160			1.04128			0.520641	+	0.853776i	$0.325693\pi$
$6$		−	206.811i		−	0.957460i
$7$	304.309			0.887198			0.443599	−	0.896225i	$-0.353701\pi$
$7$	304.309			0.887198			0.443599	+	0.896225i	$0.353701\pi$
$8$			636.976i			1.24409i
$9$	243.000			0.333333
$10$		−	1726.83i		−	1.72683i
$11$		−	302.342i		−	0.227154i	−0.993529	−	0.113577i	$-0.963769\pi$
$11$		−	302.342i		−	0.227154i	0.993529	−	0.113577i	$-0.0362309\pi$
$12$	−1746.10			−1.01047
$13$			3866.28i			1.75980i	0.475159	+	0.879900i	$0.342391\pi$
$13$			3866.28i			1.75980i	−0.475159	+	0.879900i	$0.657609\pi$
$14$		−	4037.26i		−	1.47130i
$15$	2029.00			0.601184
$16$	1281.96			0.312978
$17$	4992.75			1.01623			0.508116	−	0.861289i	$-0.330342\pi$
$17$	4992.75			1.01623			0.508116	+	0.861289i	$0.330342\pi$
$18$		−	3223.87i		−	0.552790i
$19$	11773.0			1.71644			0.858219	−	0.513284i	$-0.171571\pi$
$19$	11773.0			1.71644			0.858219	+	0.513284i	$0.171571\pi$
$20$	−14579.5			−1.82244
$21$	4743.71			0.512224
$22$	−4011.16			−0.376706
$23$			17075.4i			1.40342i	0.712461	+	0.701711i	$0.247580\pi$
$23$			17075.4i			1.40342i	−0.712461	+	0.701711i	$0.752420\pi$
$24$			9929.48i			0.718278i
$25$	1316.68			0.0842676
$26$	51293.8			2.91840
$27$	3788.00			0.192450
$28$	−34086.3			−1.55277
$29$	34405.8			1.41071			0.705355	−	0.708854i	$-0.250787\pi$
$29$	34405.8			1.41071			0.705355	+	0.708854i	$0.250787\pi$
$30$		−	26918.6i		−	0.996986i
$31$		−	35505.8i		−	1.19183i	−0.803048	−	0.595915i	$-0.796790\pi$
$31$		−	35505.8i		−	1.19183i	0.803048	−	0.595915i	$-0.203210\pi$
$32$			23758.8i			0.725062i
$33$		−	4713.05i		−	0.131148i
$34$		−	66238.6i		−	1.68529i
$35$	39608.9			0.923823
$36$	−27219.0			−0.583397
$37$		−	32831.2i		−	0.648158i	−0.946030	−	0.324079i	$-0.894946\pi$
$37$		−	32831.2i		−	0.648158i	0.946030	−	0.324079i	$-0.105054\pi$
$38$		−	156193.i		−	2.84649i
$39$			60269.3i			1.01602i
$40$			82909.0i			1.29545i
$41$	−63683.4			−0.924006			−0.462003	−	0.886878i	$-0.652869\pi$
$41$	−63683.4			−0.924006			−0.462003	+	0.886878i	$0.652869\pi$
$42$		−	62934.6i		−	0.849457i
$43$		−	54194.2i		−	0.681628i	−0.940131	−	0.340814i	$-0.889297\pi$
$43$		−	54194.2i		−	0.681628i	0.940131	−	0.340814i	$-0.110703\pi$
$44$			33866.0i			0.397563i
$45$	31628.9			0.347094
$46$	226539.			2.32739
$47$			14109.8i			0.135902i	0.997689	+	0.0679512i	$0.0216462\pi$
$47$			14109.8i			0.135902i	−0.997689	+	0.0679512i	$0.978354\pi$
$48$	19983.7			0.180698
$49$	−25045.0			−0.212879
$50$		−	17468.4i		−	0.139747i
$51$	77829.2			0.586722
$52$		−	433071.i		−	3.07999i
$53$	86435.8			0.580585			0.290293	−	0.956938i	$-0.406247\pi$
$53$	86435.8			0.580585			0.290293	+	0.956938i	$0.406247\pi$
$54$		−	50255.2i		−	0.319153i
$55$		−	39352.9i		−	0.236532i
$56$			193838.i			1.10376i
$57$	183524.			0.990986
$58$		−	456461.i		−	2.33948i
$59$	−55030.6	+	197869.i	−0.267947	+	0.963434i
$59$	−55030.6	+	197869.i	−0.267947	+	0.963434i
$60$	−227272.			−1.05219
$61$			21873.5i			0.0963669i	0.998839	+	0.0481834i	$0.0153432\pi$
$61$			21873.5i			0.0963669i	−0.998839	+	0.0481834i	$0.984657\pi$
$62$	−471054.			−1.97649
$63$	73947.1			0.295733
$64$	397252.			1.51540
$65$			503236.i			1.83245i
$66$	−62527.8			−0.217491
$67$			351365.i			1.16824i	0.811666	+	0.584122i	$0.198561\pi$
$67$			351365.i			1.16824i	−0.811666	+	0.584122i	$0.801439\pi$
$68$	−559249.			−1.77860
$69$			266180.i			0.810266i
$70$		−	525490.i		−	1.53204i
$71$	−350770.			−0.980047			−0.490024	−	0.871709i	$-0.663012\pi$
$71$	−350770.			−0.980047			−0.490024	+	0.871709i	$0.663012\pi$
$72$			154785.i			0.414698i
$73$		−	216142.i		−	0.555611i	−0.960637	−	0.277805i	$-0.910393\pi$
$73$		−	216142.i		−	0.555611i	0.960637	−	0.277805i	$-0.0896070\pi$
$74$	−435570.			−1.07489
$75$	20525.0			0.0486519
$76$	−1.31873e6			−3.00409
$77$		−	92005.5i		−	0.201531i
$78$	799591.			1.68494
$79$	−797387.			−1.61729			−0.808645	−	0.588297i	$-0.799799\pi$
$79$	−797387.			−1.61729			−0.808645	+	0.588297i	$0.799799\pi$
$80$	166860.			0.325898
$81$	59049.0			0.111111
$82$			844885.i			1.53234i
$83$		−	710474.i		−	1.24255i	−0.783593	−	0.621275i	$-0.786615\pi$
$83$		−	710474.i		−	1.24255i	0.783593	−	0.621275i	$-0.213385\pi$
$84$	−531353.			−0.896490
$85$	649857.			1.05818
$86$	−718992.			−1.13039
$87$	536334.			0.814474
$88$	192585.			0.282601
$89$		−	1.09404e6i		−	1.55190i	−0.630793	−	0.775951i	$-0.717270\pi$
$89$		−	1.09404e6i		−	1.55190i	0.630793	−	0.775951i	$-0.282730\pi$
$90$		−	419620.i		−	0.575610i
$91$			1.17654e6i			1.56129i
$92$		−	1.91266e6i		−	2.45626i
$93$		−	553480.i		−	0.688103i
$94$	187194.			0.225376
$95$	1.53238e6			1.78730
$96$			370363.i			0.418614i
$97$		−	1.57323e6i		−	1.72377i	−0.507107	−	0.861883i	$-0.669285\pi$
$97$		−	1.57323e6i		−	1.72377i	0.507107	−	0.861883i	$-0.330715\pi$
$98$			332271.i			0.353032i
$99$		−	73469.2i		−	0.0757181i
$100$	−147484.			−0.147484
$101$			613316.i			0.595279i	0.954678	+	0.297639i	$0.0961993\pi$
$101$			613316.i			0.595279i	−0.954678	+	0.297639i	$0.903801\pi$
$102$		−	1.03256e6i		−	0.973001i
$103$		−	1.07926e6i		−	0.987679i	−0.869553	−	0.493839i	$-0.835593\pi$
$103$		−	1.07926e6i		−	0.987679i	0.869553	−	0.493839i	$-0.164407\pi$
$104$	−2.46273e6			−2.18936
$105$	617442.			0.533370
$106$		−	1.14674e6i		−	0.962825i
$107$	−1.82635e6			−1.49085			−0.745424	−	0.666591i	$-0.767753\pi$
$107$	−1.82635e6			−1.49085			−0.745424	+	0.666591i	$0.767753\pi$
$108$	−424302.			−0.336824
$109$			1.34684e6i			1.04001i	0.854164	+	0.520004i	$0.174069\pi$
$109$			1.34684e6i			1.04001i	−0.854164	+	0.520004i	$0.825931\pi$
$110$	−522094.			−0.392257
$111$		−	511787.i		−	0.374214i
$112$	390111.			0.277673
$113$			34952.4i			0.0242238i	0.999927	+	0.0121119i	$0.00385543\pi$
$113$			34952.4i			0.0242238i	−0.999927	+	0.0121119i	$0.996145\pi$
$114$		−	2.43480e6i		−	1.64342i
$115$			2.22254e6i			1.46136i
$116$	−3.85387e6			−2.46901
$117$			939506.i			0.586600i
$118$	2.62512e6	+	730089.i	1.59773	+	0.444355i
$119$	1.51934e6			0.901599
$120$			1.29242e6i			0.747930i
$121$	1.68015e6			0.948401
$122$	290194.			0.159812
$123$	−992726.			−0.533475
$124$			3.97708e6i			2.08593i
$125$	−1.86237e6			−0.953535
$126$		−	981053.i		−	0.490434i
$127$	−3.48711e6			−1.70237			−0.851185	−	0.524865i	$-0.824116\pi$
$127$	−3.48711e6			−1.70237			−0.851185	+	0.524865i	$0.824116\pi$
$128$		−	3.74977e6i		−	1.78803i
$129$		−	844804.i		−	0.393538i
$130$	6.67641e6			3.03888
$131$			396214.i			0.176245i	0.996110	+	0.0881223i	$0.0280866\pi$
$131$			396214.i			0.176245i	−0.996110	+	0.0881223i	$0.971913\pi$
$132$			527919.i			0.229533i
$133$	3.58264e6			1.52282
$134$	4.66154e6			1.93738
$135$	493046.			0.200395
$136$			3.18026e6i			1.26429i
$137$	2.29114e6			0.891024			0.445512	−	0.895276i	$-0.353022\pi$
$137$	2.29114e6			0.891024			0.445512	+	0.895276i	$0.353022\pi$
$138$	3.53140e6			1.34372
$139$	4.70914e6			1.75347			0.876733	−	0.480978i	$-0.159718\pi$
$139$	4.70914e6			1.75347			0.876733	+	0.480978i	$0.159718\pi$
$140$	−4.43668e6			−1.61687
$141$			219950.i			0.0784632i
$142$			4.65365e6i			1.62528i
$143$	1.16894e6			0.399746
$144$	311515.			0.104326
$145$	4.47827e6			1.46895
$146$	−2.86755e6			−0.921408
$147$	−390413.			−0.122906
$148$			3.67749e6i			1.13440i
$149$			2.45902e6i			0.743366i	0.928360	+	0.371683i	$0.121219\pi$
$149$			2.45902e6i			0.743366i	−0.928360	+	0.371683i	$0.878781\pi$
$150$		−	272305.i		−	0.0806829i
$151$			629431.i			0.182817i	0.995813	+	0.0914086i	$0.0291369\pi$
$151$			629431.i			0.182817i	−0.995813	+	0.0914086i	$0.970863\pi$
$152$			7.49915e6i			2.13541i
$153$	1.21324e6			0.338744
$154$	−1.22063e6			−0.334213
$155$		−	4.62144e6i		−	1.24103i
$156$		−	6.75090e6i		−	1.77823i
$157$		−	4.88779e6i		−	1.26303i	−0.775364	−	0.631514i	$-0.782434\pi$
$157$		−	4.88779e6i		−	1.26303i	0.775364	−	0.631514i	$-0.217566\pi$
$158$			1.05789e7i			2.68207i
$159$	1.34740e6			0.335201
$160$			3.09245e6i			0.754993i
$161$			5.19621e6i			1.24511i
$162$		−	783401.i		−	0.184263i
$163$	7.35509e6			1.69834			0.849171	−	0.528117i	$-0.177102\pi$
$163$	7.35509e6			1.69834			0.849171	+	0.528117i	$0.177102\pi$
$164$	7.13332e6			1.61719
$165$		−	613451.i		−	0.136562i
$166$	−9.42583e6			−2.06061
$167$	−3.16416e6			−0.679375			−0.339687	−	0.940538i	$-0.610321\pi$
$167$	−3.16416e6			−0.679375			−0.339687	+	0.940538i	$0.610321\pi$
$168$			3.02163e6i			0.637255i
$169$	−1.01213e7			−2.09690
$170$		−	8.62162e6i		−	1.75486i
$171$	2.86085e6			0.572146
$172$			6.07041e6i			1.19298i
$173$			4.86437e6i			0.939481i	0.882805	+	0.469740i	$0.155652\pi$
$173$			4.86437e6i			0.939481i	−0.882805	+	0.469740i	$0.844348\pi$
$174$		−	7.11552e6i		−	1.35070i
$175$	400678.			0.0747621
$176$		−	387590.i		−	0.0710942i
$177$	−857842.	+	3.08447e6i	−0.154699	+	0.556239i
$178$	−1.45146e7			−2.57363
$179$		−	3.40368e6i		−	0.593458i	−0.954962	−	0.296729i	$-0.904104\pi$
$179$		−	3.40368e6i		−	0.593458i	0.954962	−	0.296729i	$-0.0958957\pi$
$180$	−3.54283e6			−0.607481
$181$	6.00041e6			1.01192			0.505959	−	0.862557i	$-0.331139\pi$
$181$	6.00041e6			1.01192			0.505959	+	0.862557i	$0.331139\pi$
$182$	1.56092e7			2.58920
$183$			340973.i			0.0556375i
$184$	−1.08767e7			−1.74599
$185$		−	4.27331e6i		−	0.674915i
$186$	−7.34300e6			−1.14113
$187$		−	1.50952e6i		−	0.230841i
$188$		−	1.58047e6i		−	0.237855i
$189$	1.15272e6			0.170741
$190$		−	2.03301e7i		−	2.96400i
$191$		−	5.95294e6i		−	0.854342i	−0.904171	−	0.427171i	$-0.859510\pi$
$191$		−	5.95294e6i		−	0.854342i	0.904171	−	0.427171i	$-0.140490\pi$
$192$	6.19255e6			0.874915
$193$	8.14665e6			1.13320			0.566601	−	0.823992i	$-0.308258\pi$
$193$	8.14665e6			1.13320			0.566601	+	0.823992i	$0.308258\pi$
$194$	−2.08720e7			−2.85864
$195$			7.84467e6i			1.05796i
$196$	2.80535e6			0.372579
$197$	−6.88372e6			−0.900377			−0.450189	−	0.892934i	$-0.648643\pi$
$197$	−6.88372e6			−0.900377			−0.450189	+	0.892934i	$0.648643\pi$
$198$	−974712.			−0.125569
$199$	−3.86736e6			−0.490744			−0.245372	−	0.969429i	$-0.578910\pi$
$199$	−3.86736e6			−0.490744			−0.245372	+	0.969429i	$0.578910\pi$
$200$			838695.i			0.104837i
$201$			5.47723e6i			0.674486i
$202$	8.13684e6			0.987192
$203$	1.04700e7			1.25158
$204$	−8.71782e6			−1.02687
$205$	−8.28905e6			−0.962151
$206$	−1.43185e7			−1.63794
$207$			4.14933e6i			0.467808i
$208$			4.95640e6i			0.550778i
$209$		−	3.55949e6i		−	0.389896i
$210$		−	8.19158e6i		−	0.884524i
$211$			1.02663e7i			1.09287i	0.837502	+	0.546435i	$0.184015\pi$
$211$			1.02663e7i			1.09287i	−0.837502	+	0.546435i	$0.815985\pi$
$212$	−9.68187e6			−1.01614
$213$	−5.46796e6			−0.565831
$214$			2.42301e7i			2.47238i
$215$		−	7.05393e6i		−	0.709767i
$216$			2.41286e6i			0.239426i
$217$		−	1.08047e7i		−	1.05739i
$218$	1.78685e7			1.72472
$219$		−	3.36932e6i		−	0.320782i
$220$			4.40801e6i			0.413975i
$221$			1.93034e7i			1.78836i
$222$	−6.78986e6			−0.620586
$223$	1.02204e7			0.921621			0.460811	−	0.887498i	$-0.347559\pi$
$223$	1.02204e7			0.921621			0.460811	+	0.887498i	$0.347559\pi$
$224$			7.23002e6i			0.643273i
$225$	319954.			0.0280892
$226$	463713.			0.0401720
$227$			4.14140e6i			0.354054i	0.984206	+	0.177027i	$0.0566480\pi$
$227$			4.14140e6i			0.354054i	−0.984206	+	0.177027i	$0.943352\pi$
$228$	−2.05569e7			−1.73441
$229$			3.11497e6i			0.259387i	0.991554	+	0.129693i	$0.0413993\pi$
$229$			3.11497e6i			0.259387i	−0.991554	+	0.129693i	$0.958601\pi$
$230$	2.94864e7			2.42347
$231$		−	1.43422e6i		−	0.116354i
$232$			2.19157e7i			1.75506i
$233$		−	3.13161e6i		−	0.247571i	−0.992309	−	0.123786i	$-0.960496\pi$
$233$		−	3.13161e6i		−	0.247571i	0.992309	−	0.123786i	$-0.0395035\pi$
$234$	1.24644e7			0.972800
$235$			1.83653e6i			0.141513i
$236$	6.16410e6	−	2.21638e7i	0.468958	−	1.68619i
$237$	−1.24300e7			−0.933743
$238$		−	2.01570e7i		−	1.49518i
$239$	3.41855e6			0.250408			0.125204	−	0.992131i	$-0.460042\pi$
$239$	3.41855e6			0.250408			0.125204	+	0.992131i	$0.460042\pi$
$240$	2.60109e6			0.188157
$241$	−7.48627e6			−0.534829			−0.267414	−	0.963582i	$-0.586169\pi$
$241$	−7.48627e6			−0.534829			−0.267414	+	0.963582i	$0.586169\pi$
$242$		−	2.22905e7i		−	1.57280i
$243$	920483.			0.0641500
$244$		−	2.45009e6i		−	0.168660i
$245$	−3.25986e6			−0.221667
$246$			1.31705e7i			0.884699i
$247$			4.55179e7i			3.02059i
$248$	2.26163e7			1.48275
$249$		−	1.10752e7i		−	0.717387i
$250$			2.47080e7i			1.58131i
$251$	1.02216e7			0.646395			0.323197	−	0.946332i	$-0.395242\pi$
$251$	1.02216e7			0.646395			0.323197	+	0.946332i	$0.395242\pi$
$252$	−8.28298e6			−0.517589
$253$	5.16263e6			0.318793
$254$			4.62633e7i			2.82316i
$255$	1.01303e7			0.610942
$256$	−2.43239e7			−1.44982
$257$	−1.51696e7			−0.893663			−0.446832	−	0.894618i	$-0.647448\pi$
$257$	−1.51696e7			−0.893663			−0.446832	+	0.894618i	$0.647448\pi$
$258$	−1.12080e7			−0.652632
$259$		−	9.99082e6i		−	0.575045i
$260$		−	5.63686e7i		−	3.20713i
$261$	8.36062e6			0.470237
$262$	5.25656e6			0.292279
$263$	−1.18718e6			−0.0652601			−0.0326301	−	0.999467i	$-0.510388\pi$
$263$	−1.18718e6			−0.0652601			−0.0326301	+	0.999467i	$0.510388\pi$
$264$	3.00210e6			0.163160
$265$	1.12505e7			0.604553
$266$		−	4.75308e7i		−	2.52540i
$267$		−	1.70544e7i		−	0.895991i
$268$		−	3.93571e7i		−	2.04465i
$269$			2.00074e7i			1.02786i	0.857833	+	0.513929i	$0.171811\pi$
$269$			2.00074e7i			1.02786i	−0.857833	+	0.513929i	$0.828189\pi$
$270$		−	6.54122e6i		−	0.332329i
$271$	−2.02889e7			−1.01941			−0.509706	−	0.860348i	$-0.670246\pi$
$271$	−2.02889e7			−1.01941			−0.509706	+	0.860348i	$0.670246\pi$
$272$	6.40048e6			0.318058
$273$			1.83405e7i			0.901412i
$274$		−	3.03964e7i		−	1.47765i
$275$		−	398088.i		−	0.0191417i
$276$		−	2.98154e7i		−	1.41812i
$277$	5.83030e6			0.274316			0.137158	−	0.990549i	$-0.456203\pi$
$277$	5.83030e6			0.274316			0.137158	+	0.990549i	$0.456203\pi$
$278$		−	6.24760e7i		−	2.90789i
$279$		−	8.62790e6i		−	0.397276i
$280$			2.52299e7i			1.14932i
$281$	1.21058e7			0.545601			0.272800	−	0.962071i	$-0.412050\pi$
$281$	1.21058e7			0.545601			0.272800	+	0.962071i	$0.412050\pi$
$282$	2.91807e6			0.130121
$283$		−	1.30713e7i		−	0.576711i	−0.957523	−	0.288356i	$-0.906891\pi$
$283$		−	1.30713e7i		−	0.576711i	0.957523	−	0.288356i	$-0.0931085\pi$
$284$	3.92905e7			1.71527
$285$	2.38875e7			1.03190
$286$		−	1.55083e7i		−	0.662927i
$287$	−1.93794e7			−0.819777
$288$			5.77339e6i			0.241687i
$289$	789935.			0.0327264
$290$		−	5.94130e7i		−	2.43606i
$291$		−	2.45243e7i		−	0.995217i
$292$			2.42105e7i			0.972425i
$293$	2.58060e7			1.02593			0.512966	−	0.858409i	$-0.328547\pi$
$293$	2.58060e7			1.02593			0.512966	+	0.858409i	$0.328547\pi$
$294$			5.17959e6i			0.203823i
$295$	−7.16280e6	+	2.57547e7i	−0.279008	+	1.00321i
$296$	2.09127e7			0.806370
$297$		−	1.14527e6i		−	0.0437158i
$298$	3.26237e7			1.23278
$299$	−6.60184e7			−2.46974
$300$	−2.29905e6			−0.0851502
$301$		−	1.64918e7i		−	0.604739i
$302$	8.35063e6			0.303179
$303$			9.56066e6i			0.343684i
$304$	1.50925e7			0.537207
$305$			2.84705e6i			0.100345i
$306$		−	1.60960e7i		−	0.561763i
$307$	−2.55575e7			−0.883291			−0.441645	−	0.897190i	$-0.645605\pi$
$307$	−2.55575e7			−0.883291			−0.441645	+	0.897190i	$0.645605\pi$
$308$			1.03057e7i			0.352717i
$309$		−	1.68241e7i		−	0.570237i
$310$	−6.13125e7			−2.05809
$311$	−3.58308e7			−1.19117			−0.595587	−	0.803291i	$-0.703080\pi$
$311$	−3.58308e7			−1.19117			−0.595587	+	0.803291i	$0.703080\pi$
$312$	−3.83901e7			−1.26403
$313$			2.94627e7i			0.960815i	0.877045	+	0.480408i	$0.159511\pi$
$313$			2.94627e7i			0.960815i	−0.877045	+	0.480408i	$0.840489\pi$
$314$	−6.48461e7			−2.09457
$315$	9.62497e6			0.307941
$316$	8.93171e7			2.83057
$317$	−2.81452e7			−0.883541			−0.441770	−	0.897128i	$-0.645649\pi$
$317$	−2.81452e7			−0.883541			−0.441770	+	0.897128i	$0.645649\pi$
$318$		−	1.78759e7i		−	0.555887i
$319$		−	1.04023e7i		−	0.320449i
$320$	5.17065e7			1.57796
$321$	−2.84700e7			−0.860741
$322$	6.89379e7			2.06486
$323$	5.87798e7			1.74430
$324$	−6.61421e6			−0.194466
$325$			5.09066e6i			0.148294i
$326$		−	9.75797e7i		−	2.81648i
$327$			2.09952e7i			0.600449i
$328$		−	4.05648e7i		−	1.14955i
$329$			4.29374e6i			0.120572i
$330$	−8.13864e6			−0.226470
$331$	8.53611e6			0.235384			0.117692	−	0.993050i	$-0.462451\pi$
$331$	8.53611e6			0.235384			0.117692	+	0.993050i	$0.462451\pi$
$332$			7.95818e7i			2.17470i
$333$		−	7.97797e6i		−	0.216053i
$334$			4.19788e7i			1.12665i
$335$			4.57337e7i			1.21647i
$336$	6.08123e6			0.160315
$337$			5.17480e7i			1.35209i	0.736862	+	0.676043i	$0.236307\pi$
$337$			5.17480e7i			1.35209i	−0.736862	+	0.676043i	$0.763693\pi$
$338$			1.34279e8i			3.47743i
$339$			544855.i			0.0139856i
$340$	−7.27919e7			−1.85202
$341$	−1.07349e7			−0.270729
$342$		−	3.79548e7i		−	0.948830i
$343$	−4.34231e7			−1.07606
$344$	3.45204e7			0.848009
$345$			3.46460e7i			0.843716i
$346$	6.45353e7			1.55801
$347$			4.80773e7i			1.15067i	0.817917	+	0.575336i	$0.195129\pi$
$347$			4.80773e7i			1.15067i	−0.817917	+	0.575336i	$0.804871\pi$
$348$	−6.00759e7			−1.42549
$349$			4.31976e7i			1.01621i	0.861295	+	0.508104i	$0.169654\pi$
$349$			4.31976e7i			1.01621i	−0.861295	+	0.508104i	$0.830346\pi$
$350$		−	5.31578e6i		−	0.123983i
$351$			1.46455e7i			0.338674i
$352$	7.18329e6			0.164701
$353$		−	5.00314e7i		−	1.13741i	−0.822540	−	0.568707i	$-0.807444\pi$
$353$		−	5.00314e7i		−	1.13741i	0.822540	−	0.568707i	$-0.192556\pi$
$354$	4.09216e7	+	1.13810e7i	0.922450	+	0.256548i
$355$	−4.56563e7			−1.02051
$356$			1.22546e8i			2.71613i
$357$	2.36841e7			0.520538
$358$	−4.51565e7			−0.984172
$359$	8.77345e6			0.189621			0.0948106	−	0.995495i	$-0.469775\pi$
$359$	8.77345e6			0.189621			0.0948106	+	0.995495i	$0.469775\pi$
$360$			2.01469e7i			0.431818i
$361$	9.15587e7			1.94616
$362$		−	7.96072e7i		−	1.67813i
$363$	2.61909e7			0.547560
$364$		−	1.31787e8i		−	2.73256i
$365$		−	2.81331e7i		−	0.578547i
$366$	4.52368e6			0.0922675
$367$		−	6.37952e7i		−	1.29060i	−0.763931	−	0.645298i	$-0.776733\pi$
$367$		−	6.37952e7i		−	1.29060i	0.763931	−	0.645298i	$-0.223267\pi$
$368$			2.18900e7i			0.439240i
$369$	−1.54751e7			−0.308002
$370$	−5.66938e7			−1.11926
$371$	2.63032e7			0.515094
$372$			6.19966e7i			1.20431i
$373$	−7.33491e7			−1.41341			−0.706705	−	0.707508i	$-0.749819\pi$
$373$	−7.33491e7			−1.41341			−0.706705	+	0.707508i	$0.749819\pi$
$374$	−2.00267e7			−0.382820
$375$	−2.90315e7			−0.550524
$376$	−8.98760e6			−0.169075
$377$			1.33023e8i			2.48257i
$378$		−	1.52931e7i		−	0.283152i
$379$	4.24353e7			0.779489			0.389745	−	0.920923i	$-0.372563\pi$
$379$	4.24353e7			0.779489			0.389745	+	0.920923i	$0.372563\pi$
$380$	−1.71646e8			−3.12811
$381$	−5.43586e7			−0.982864
$382$	−7.89775e7			−1.41681
$383$	−7.62200e7			−1.35667			−0.678333	−	0.734755i	$-0.737297\pi$
$383$	−7.62200e7			−1.35667			−0.678333	+	0.734755i	$0.737297\pi$
$384$		−	5.84531e7i		−	1.03232i
$385$		−	1.19755e7i		−	0.209850i
$386$		−	1.08081e8i		−	1.87927i
$387$		−	1.31692e7i		−	0.227209i
$388$			1.76222e8i			3.01692i
$389$	−1.78271e7			−0.302854			−0.151427	−	0.988468i	$-0.548387\pi$
$389$	−1.78271e7			−0.302854			−0.151427	+	0.988468i	$0.548387\pi$
$390$	1.04075e8			1.75450
$391$			8.52533e7i			1.42620i
$392$		−	1.59531e7i		−	0.264842i
$393$			6.17637e6i			0.101755i
$394$			9.13260e7i			1.49316i
$395$	−1.03788e8			−1.68405
$396$			8.22944e6i			0.132521i
$397$		−	4.58059e7i		−	0.732065i	−0.930602	−	0.366033i	$-0.880716\pi$
$397$		−	4.58059e7i		−	0.732065i	0.930602	−	0.366033i	$-0.119284\pi$
$398$			5.13081e7i			0.813836i
$399$	5.58479e7			0.879201
$400$	1.68793e6			0.0263739
$401$		−	1.68612e7i		−	0.261491i	−0.991416	−	0.130745i	$-0.958263\pi$
$401$		−	1.68612e7i		−	0.261491i	0.991416	−	0.130745i	$-0.0417370\pi$
$402$	7.26662e7			1.11855
$403$	1.37275e8			2.09738
$404$		−	6.86989e7i		−	1.04185i
$405$	7.68583e6			0.115698
$406$		−	1.38905e8i		−	2.07558i
$407$	−9.92625e6			−0.147232
$408$			4.95754e7i			0.729937i
$409$		−	7.74091e7i		−	1.13142i	−0.824606	−	0.565708i	$-0.808603\pi$
$409$		−	7.74091e7i		−	1.13142i	0.824606	−	0.565708i	$-0.191397\pi$
$410$			1.09970e8i			1.59560i
$411$	3.57153e7			0.514433
$412$			1.20891e8i			1.72863i
$413$	−1.67463e7	+	6.02133e7i	−0.237722	+	0.854757i
$414$	5.50490e7			0.775798
$415$		−	9.24755e7i		−	1.29384i
$416$	−9.18582e7			−1.27596
$417$	7.34082e7			1.01236
$418$	−4.72236e7			−0.646592
$419$		−	6.40310e7i		−	0.870459i	−0.900319	−	0.435230i	$-0.856667\pi$
$419$		−	6.40310e7i		−	0.870459i	0.900319	−	0.435230i	$-0.143333\pi$
$420$	−6.91611e7			−0.933499
$421$		−	1.06216e8i		−	1.42346i	−0.702454	−	0.711730i	$-0.747912\pi$
$421$		−	1.06216e8i		−	1.42346i	0.702454	−	0.711730i	$-0.252088\pi$
$422$	1.36203e8			1.81238
$423$			3.42868e6i			0.0453008i
$424$			5.50576e7i			0.722303i
$425$	6.57385e6			0.0856354
$426$			7.25432e7i			0.938357i
$427$			6.65629e6i			0.0854965i
$428$	2.04574e8			2.60927
$429$	1.82220e7			0.230793
$430$	−9.35842e7			−1.17706
$431$			4.74955e7i			0.593227i	0.954998	+	0.296613i	$0.0958573\pi$
$431$			4.74955e7i			0.593227i	−0.954998	+	0.296613i	$0.904143\pi$
$432$	4.85605e6			0.0602326
$433$	−3.56448e7			−0.439068			−0.219534	−	0.975605i	$-0.570454\pi$
$433$	−3.56448e7			−0.439068			−0.219534	+	0.975605i	$0.570454\pi$
$434$	−1.43346e8			−1.75354
$435$	6.98093e7			0.848097
$436$		−	1.50863e8i		−	1.82021i
$437$			2.01030e8i			2.40889i
$438$	−4.47006e7			−0.531975
$439$	1.02332e8			1.20954			0.604770	−	0.796401i	$-0.293265\pi$
$439$	1.02332e8			1.20954			0.604770	+	0.796401i	$0.293265\pi$
$440$	2.50669e7			0.294268
$441$	−6.08594e6			−0.0709597
$442$	2.56097e8			2.96577
$443$		−	3.68919e6i		−	0.0424345i	−0.999775	−	0.0212173i	$-0.993246\pi$
$443$		−	3.68919e6i		−	0.0424345i	0.999775	−	0.0212173i	$-0.00675417\pi$
$444$			5.73264e7i			0.654947i
$445$		−	1.42401e8i		−	1.61597i
$446$		−	1.35593e8i		−	1.52839i
$447$			3.83323e7i			0.429183i
$448$	1.20888e8			1.34446
$449$	−7.44766e7			−0.822775			−0.411387	−	0.911461i	$-0.634956\pi$
$449$	−7.44766e7			−0.822775			−0.411387	+	0.911461i	$0.634956\pi$
$450$		−	4.24481e6i		−	0.0465823i
$451$			1.92542e7i			0.209892i
$452$		−	3.91510e6i		−	0.0423963i
$453$			9.81186e6i			0.105550i
$454$	5.49438e7			0.587152
$455$			1.53139e8i			1.62574i
$456$			1.16900e8i			1.23288i
$457$		−	5.98613e6i		−	0.0627188i	−0.999508	−	0.0313594i	$-0.990016\pi$
$457$		−	5.98613e6i		−	0.0627188i	0.999508	−	0.0313594i	$-0.00998364\pi$
$458$	4.13262e7			0.430159
$459$	1.89125e7			0.195574
$460$		−	2.48952e8i		−	2.55766i
$461$	−5.41854e7			−0.553069			−0.276534	−	0.961004i	$-0.589186\pi$
$461$	−5.41854e7			−0.553069			−0.276534	+	0.961004i	$0.589186\pi$
$462$	−1.90278e7			−0.192958
$463$			6.23169e6i			0.0627860i	0.999507	+	0.0313930i	$0.00999435\pi$
$463$			6.23169e6i			0.0627860i	−0.999507	+	0.0313930i	$0.990006\pi$
$464$	4.41068e7			0.441521
$465$		−	7.20411e7i		−	0.716509i
$466$	−4.15470e7			−0.410565
$467$			1.39127e8i			1.36603i	0.730405	+	0.683014i	$0.239331\pi$
$467$			1.39127e8i			1.36603i	−0.730405	+	0.683014i	$0.760669\pi$
$468$		−	1.05236e8i		−	1.02666i
$469$			1.06923e8i			1.03646i
$470$	2.43652e7			0.234680
$471$		−	7.61930e7i		−	0.729210i
$472$	−1.26038e8	−	3.50532e7i	−1.19860	−	0.333351i
$473$	−1.63852e7			−0.154835
$474$			1.64909e8i			1.54849i
$475$	1.55013e7			0.144640
$476$	−1.70184e8			−1.57797
$477$	2.10039e7			0.193528
$478$		−	4.53537e7i		−	0.415269i
$479$	−2.07373e7			−0.188688			−0.0943442	−	0.995540i	$-0.530075\pi$
$479$	−2.07373e7			−0.188688			−0.0943442	+	0.995540i	$0.530075\pi$
$480$			4.82066e7i			0.435896i
$481$	1.26934e8			1.14063
$482$			9.93201e7i			0.886944i
$483$			8.10009e7i			0.718867i
$484$	−1.88197e8			−1.65988
$485$		−	2.04773e8i		−	1.79493i
$486$		−	1.22120e7i		−	0.106384i
$487$	−1.57737e8			−1.36567			−0.682835	−	0.730573i	$-0.739253\pi$
$487$	−1.57737e8			−1.36567			−0.682835	+	0.730573i	$0.739253\pi$
$488$	−1.39329e7			−0.119890
$489$	1.14655e8			0.980539
$490$			4.32485e7i			0.367606i
$491$	−1.05877e8			−0.894451			−0.447226	−	0.894421i	$-0.647588\pi$
$491$	−1.05877e8			−0.894451			−0.447226	+	0.894421i	$0.647588\pi$
$492$	1.11197e8			0.933683
$493$	1.71780e8			1.43361
$494$	6.03884e8			5.00925
$495$		−	9.56276e6i		−	0.0788438i
$496$		−	4.55169e7i		−	0.373016i
$497$	−1.06742e8			−0.869496
$498$	−1.46934e8			−1.18969
$499$	1.18761e8			0.955808			0.477904	−	0.878412i	$-0.341397\pi$
$499$	1.18761e8			0.955808			0.477904	+	0.878412i	$0.341397\pi$
$500$	2.08609e8			1.66887
$501$	−4.93244e7			−0.392237
$502$		−	1.35610e8i		−	1.07196i
$503$		−	1.90458e8i		−	1.49657i	−0.663380	−	0.748283i	$-0.730878\pi$
$503$		−	1.90458e8i		−	1.49657i	0.663380	−	0.748283i	$-0.269122\pi$
$504$			4.71025e7i			0.367919i
$505$			7.98294e7i			0.619853i
$506$		−	6.84924e7i		−	0.528677i
$507$	−1.57776e8			−1.21064
$508$	3.90599e8			2.97947
$509$			2.50692e8i			1.90102i	0.310690	+	0.950511i	$0.399440\pi$
$509$			2.50692e8i			1.90102i	−0.310690	+	0.950511i	$0.600560\pi$
$510$		−	1.34398e8i		−	1.01317i
$511$		−	6.57740e7i		−	0.492937i
$512$			8.27186e7i			0.616302i
$513$	4.45962e7			0.330329
$514$			2.01254e8i			1.48202i
$515$		−	1.40477e8i		−	1.02845i
$516$			9.46283e7i			0.688767i
$517$	4.26598e6			0.0308708
$518$	−1.32548e8			−0.953637
$519$			7.58280e7i			0.542409i
$520$	−3.20549e8			−2.27974
$521$	3.29248e7			0.232815			0.116407	−	0.993202i	$-0.462862\pi$
$521$	3.29248e7			0.232815			0.116407	+	0.993202i	$0.462862\pi$
$522$		−	1.10920e8i		−	0.779827i
$523$	−8.55987e7			−0.598360			−0.299180	−	0.954197i	$-0.596713\pi$
$523$	−8.55987e7			−0.598360			−0.299180	+	0.954197i	$0.596713\pi$
$524$		−	4.43808e7i		−	0.308462i
$525$	6.24595e6			0.0431639
$526$			1.57502e7i			0.108225i
$527$		−	1.77271e8i		−	1.21117i
$528$		−	6.04192e6i		−	0.0410463i
$529$	−1.43535e8			−0.969595
$530$		−	1.49260e8i		−	1.00257i
$531$	−1.33724e7	+	4.80822e7i	−0.0893155	+	0.321145i
$532$	−4.01300e8			−2.66523
$533$		−	2.46218e8i		−	1.62607i
$534$	−2.26261e8			−1.48589
$535$	−2.37718e8			−1.55239
$536$	−2.23811e8			−1.45341
$537$		−	5.30581e7i		−	0.342633i
$538$	2.65437e8			1.70457
$539$			7.57216e6i			0.0483564i
$540$	−5.52272e7			−0.350729
$541$		−	2.89276e8i		−	1.82693i	−0.406920	−	0.913464i	$-0.633397\pi$
$541$		−	2.89276e8i		−	1.82693i	0.406920	−	0.913464i	$-0.366603\pi$
$542$			2.69172e8i			1.69056i
$543$	9.35372e7			0.584231
$544$			1.18622e8i			0.736830i
$545$			1.75305e8i			1.08294i
$546$	2.43323e8			1.49487
$547$	3.21322e8			1.96327			0.981633	−	0.190779i	$-0.0611014\pi$
$547$	3.21322e8			1.96327			0.981633	+	0.190779i	$0.0611014\pi$
$548$	−2.56635e8			−1.55946
$549$			5.31525e6i			0.0321223i
$550$	−5.28142e6			−0.0317441
$551$	4.05061e8			2.42140
$552$	−1.69550e8			−1.00805
$553$	−2.42652e8			−1.43486
$554$		−	7.73503e7i		−	0.454918i
$555$		−	6.66143e7i		−	0.389663i
$556$	−5.27481e8			−3.06890
$557$	1.42874e8			0.826778			0.413389	−	0.910555i	$-0.364345\pi$
$557$	1.42874e8			0.826778			0.413389	+	0.910555i	$0.364345\pi$
$558$	−1.14466e8			−0.658831
$559$	2.09530e8			1.19953
$560$	5.07769e7			0.289136
$561$		−	2.35311e7i		−	0.133276i
$562$		−	1.60607e8i		−	0.904808i
$563$			9.47478e7i			0.530938i	0.964119	+	0.265469i	$0.0855268\pi$
$563$			9.47478e7i			0.530938i	−0.964119	+	0.265469i	$0.914473\pi$
$564$		−	2.46371e7i		−	0.137326i
$565$			4.54942e6i			0.0252238i
$566$	−1.73416e8			−0.956401
$567$	1.79691e7			0.0985776
$568$		−	2.23432e8i		−	1.21927i
$569$			1.29523e8i			0.703087i	0.936172	+	0.351543i	$0.114343\pi$
$569$			1.29523e8i			0.703087i	−0.936172	+	0.351543i	$0.885657\pi$
$570$		−	3.16914e8i		−	1.71126i
$571$		−	4.87058e7i		−	0.261621i	−0.991407	−	0.130811i	$-0.958242\pi$
$571$		−	4.87058e7i		−	0.261621i	0.991407	−	0.130811i	$-0.0417580\pi$
$572$	−1.30936e8			−0.699632
$573$		−	9.27972e7i		−	0.493254i
$574$			2.57106e8i			1.35949i
$575$			2.24829e7i			0.118263i
$576$	9.65324e7			0.505133
$577$	4.43975e7			0.231117			0.115558	−	0.993301i	$-0.463134\pi$
$577$	4.43975e7			0.231117			0.115558	+	0.993301i	$0.463134\pi$
$578$		−	1.04800e7i		−	0.0542724i
$579$	1.26994e8			0.654254
$580$	−5.01621e8			−2.57094
$581$		−	2.16204e8i		−	1.10239i
$582$	−3.25363e8			−1.65044
$583$		−	2.61332e7i		−	0.131882i
$584$	1.37677e8			0.691232
$585$			1.22286e8i			0.610816i
$586$		−	3.42368e8i		−	1.70137i
$587$		−	2.32647e8i		−	1.15022i	−0.818075	−	0.575112i	$-0.804958\pi$
$587$		−	2.32647e8i		−	1.15022i	0.818075	−	0.575112i	$-0.195042\pi$
$588$	4.37310e7			0.215109
$589$		−	4.18011e8i		−	2.04570i
$590$	3.41686e8	+	9.50285e7i	1.66369	+	0.462698i
$591$	−1.07307e8			−0.519833
$592$		−	4.20881e7i		−	0.202859i
$593$	−7.02595e7			−0.336931			−0.168466	−	0.985708i	$-0.553881\pi$
$593$	−7.02595e7			−0.336931			−0.168466	+	0.985708i	$0.553881\pi$
$594$	−1.51943e7			−0.0724970
$595$	1.97757e8			0.938818
$596$		−	2.75440e8i		−	1.30103i
$597$	−6.02862e7			−0.283331
$598$			8.75864e8i			4.09575i
$599$	1.26550e8			0.588817			0.294408	−	0.955680i	$-0.404877\pi$
$599$	1.26550e8			0.588817			0.294408	+	0.955680i	$0.404877\pi$
$600$			1.30740e7i			0.0605276i
$601$		−	1.51823e8i		−	0.699381i	−0.936865	−	0.349690i	$-0.886287\pi$
$601$		−	1.51823e8i		−	0.699381i	0.936865	−	0.349690i	$-0.113713\pi$
$602$	−2.18796e8			−1.00288
$603$			8.53816e7i			0.389415i
$604$		−	7.05040e7i		−	0.319965i
$605$	2.18689e8			0.987553
$606$	1.26841e8			0.569956
$607$	2.49091e8			1.11376			0.556881	−	0.830593i	$-0.311998\pi$
$607$	2.49091e8			1.11376			0.556881	+	0.830593i	$0.311998\pi$
$608$			2.79714e8i			1.24452i
$609$	1.63211e8			0.722600
$610$	3.77717e7			0.166409
$611$	−5.45524e7			−0.239161
$612$	−1.35897e8			−0.592866
$613$			1.65012e8i			0.716363i	0.933652	+	0.358181i	$0.116603\pi$
$613$			1.65012e8i			0.716363i	−0.933652	+	0.358181i	$0.883397\pi$
$614$			3.39071e8i			1.46482i
$615$	−1.29213e8			−0.555498
$616$	5.86053e7			0.250723
$617$	−3.28492e8			−1.39852			−0.699262	−	0.714866i	$-0.746488\pi$
$617$	−3.28492e8			−1.39852			−0.699262	+	0.714866i	$0.746488\pi$
$618$	−2.23204e8			−0.945663
$619$	−1.14198e6			−0.00481490			−0.00240745	−	0.999997i	$-0.500766\pi$
$619$	−1.14198e6			−0.00481490			−0.00240745	+	0.999997i	$0.500766\pi$
$620$			5.17658e8i			2.17204i
$621$			6.46817e7i			0.270089i
$622$			4.75365e8i			1.97541i
$623$		−	3.32927e8i		−	1.37685i
$624$			7.72627e7i			0.317992i
$625$	−2.62980e8			−1.07717
$626$	3.90881e8			1.59339
$627$		−	5.54869e7i		−	0.225107i
$628$			5.47492e8i			2.21054i
$629$		−	1.63918e8i		−	0.658679i
$630$		−	1.27694e8i		−	0.510680i
$631$	−4.22819e8			−1.68293			−0.841465	−	0.540312i	$-0.818306\pi$
$631$	−4.22819e8			−1.68293			−0.841465	+	0.540312i	$0.818306\pi$
$632$		−	5.07917e8i		−	2.01206i
$633$			1.60036e8i			0.630969i
$634$			3.73401e8i			1.46524i
$635$	−4.53883e8			−1.77265
$636$	−1.50925e8			−0.586666
$637$		−	9.68310e7i		−	0.374625i
$638$	−1.38007e8			−0.531423
$639$	−8.52371e7			−0.326682
$640$		−	4.88071e8i		−	1.86184i
$641$	1.89312e8			0.718792			0.359396	−	0.933185i	$-0.382983\pi$
$641$	1.89312e8			0.718792			0.359396	+	0.933185i	$0.382983\pi$
$642$			3.77711e8i			1.42743i
$643$	2.27620e7			0.0856205			0.0428102	−	0.999083i	$-0.486369\pi$
$643$	2.27620e7			0.0856205			0.0428102	+	0.999083i	$0.486369\pi$
$644$		−	5.82039e8i		−	2.17919i
$645$		−	1.09960e8i		−	0.409784i
$646$		−	7.79829e8i		−	2.89269i
$647$	−1.90979e8			−0.705134			−0.352567	−	0.935787i	$-0.614691\pi$
$647$	−1.90979e8			−0.705134			−0.352567	+	0.935787i	$0.614691\pi$
$648$			3.76128e7i			0.138233i
$649$	5.98242e7	+	1.66381e7i	0.218848	+	0.0608652i
$650$	6.75376e7			0.245926
$651$		−	1.68429e8i		−	0.610484i
$652$	−8.23860e8			−2.97242
$653$	2.90721e8			1.04409			0.522043	−	0.852919i	$-0.325170\pi$
$653$	2.90721e8			1.04409			0.522043	+	0.852919i	$0.325170\pi$
$654$	2.78542e8			0.995766
$655$			5.15713e7i			0.183520i
$656$	−8.16394e7			−0.289193
$657$		−	5.25225e7i		−	0.185204i
$658$	5.69648e7			0.199954
$659$			4.08241e7i			0.142646i	0.997453	+	0.0713231i	$0.0227222\pi$
$659$			4.08241e7i			0.142646i	−0.997453	+	0.0713231i	$0.977278\pi$
$660$			6.87141e7i			0.239009i
$661$	−4.20379e8			−1.45558			−0.727791	−	0.685799i	$-0.759453\pi$
$661$	−4.20379e8			−1.45558			−0.727791	+	0.685799i	$0.759453\pi$
$662$		−	1.13248e8i		−	0.390353i
$663$			3.00909e8i			1.03251i
$664$	4.52555e8			1.54585
$665$	4.66318e8			1.58569
$666$	−1.05843e8			−0.358295
$667$			5.87495e8i			1.97982i
$668$	3.54425e8			1.18904
$669$	1.59320e8			0.532098
$670$	6.06747e8			2.01736
$671$	6.61327e6			0.0218901
$672$			1.12705e8i			0.371394i
$673$		−	3.44287e8i		−	1.12947i	−0.825271	−	0.564737i	$-0.808978\pi$
$673$		−	3.44287e8i		−	1.12947i	0.825271	−	0.564737i	$-0.191022\pi$
$674$	6.86539e8			2.24226
$675$	4.98758e6			0.0162173
$676$	1.13371e9			3.66997
$677$	−5.11470e8			−1.64837			−0.824184	−	0.566322i	$-0.808366\pi$
$677$	−5.11470e8			−1.64837			−0.824184	+	0.566322i	$0.808366\pi$
$678$	7.22856e6			0.0231933
$679$		−	4.78750e8i		−	1.52932i
$680$			4.13943e8i			1.31648i
$681$			6.45580e7i			0.204413i
$682$			1.42419e8i			0.448969i
$683$			1.47287e8i			0.462276i	0.972921	+	0.231138i	$0.0742448\pi$
$683$			1.47287e8i			0.462276i	−0.972921	+	0.231138i	$0.925755\pi$
$684$	−3.20450e8			−1.00136
$685$	2.98215e8			0.927807
$686$			5.76092e8i			1.78451i
$687$			4.85576e7i			0.149757i
$688$		−	6.94746e7i		−	0.213334i
$689$			3.34185e8i			1.02171i
$690$	4.59647e8			1.39919
$691$			2.22376e8i			0.673989i	0.941507	+	0.336995i	$0.109410\pi$
$691$			2.22376e8i			0.673989i	−0.941507	+	0.336995i	$0.890590\pi$
$692$		−	5.44868e8i		−	1.64427i
$693$		−	2.23573e7i		−	0.0671769i
$694$	6.37840e8			1.90824
$695$	6.12943e8			1.82585
$696$			3.41632e8i			1.01328i
$697$	−3.17955e8			−0.939004
$698$	5.73100e8			1.68525
$699$		−	4.88170e7i		−	0.142935i
$700$	−4.48808e7			−0.130848
$701$			1.91012e8i			0.554507i	0.960797	+	0.277254i	$0.0894242\pi$
$701$			1.91012e8i			0.554507i	−0.960797	+	0.277254i	$0.910576\pi$
$702$	1.94301e8			0.561646
$703$		−	3.86523e8i		−	1.11252i
$704$		−	1.20106e8i		−	0.344229i
$705$			2.86287e7i			0.0817023i
$706$	−6.63764e8			−1.88625
$707$			1.86638e8i			0.528130i
$708$	9.60888e7	−	3.45499e8i	0.270753	−	0.973524i
$709$	−9.36384e7			−0.262733			−0.131367	−	0.991334i	$-0.541937\pi$
$709$	−9.36384e7			−0.262733			−0.131367	+	0.991334i	$0.541937\pi$
$710$			6.05720e8i			1.69238i
$711$	−1.93765e8			−0.539097
$712$	6.96880e8			1.93071
$713$	6.06277e8			1.67264
$714$		−	3.14216e8i		−	0.863245i
$715$	1.52149e8			0.416248
$716$			3.81254e8i			1.03866i
$717$	5.32898e7			0.144573
$718$		−	1.16397e8i		−	0.314462i
$719$		−	5.79765e8i		−	1.55979i	−0.625913	−	0.779893i	$-0.715273\pi$
$719$		−	5.79765e8i		−	1.55979i	0.625913	−	0.779893i	$-0.284727\pi$
$720$	4.05469e7			0.108633
$721$		−	3.28430e8i		−	0.876267i
$722$		−	1.21471e9i		−	3.22745i
$723$	−1.16699e8			−0.308783
$724$	−6.72120e8			−1.77105
$725$	4.53015e7			0.118877
$726$		−	3.47474e8i		−	0.908056i
$727$	−6.49571e7			−0.169053			−0.0845266	−	0.996421i	$-0.526938\pi$
$727$	−6.49571e7			−0.169053			−0.0845266	+	0.996421i	$0.526938\pi$
$728$	−7.49431e8			−1.94239
$729$	1.43489e7			0.0370370
$730$	−3.73241e8			−0.959445
$731$		−	2.70578e8i		−	0.692692i
$732$		−	3.81932e7i		−	0.0973762i
$733$	2.57152e7			0.0652946			0.0326473	−	0.999467i	$-0.489606\pi$
$733$	2.57152e7			0.0652946			0.0326473	+	0.999467i	$0.489606\pi$
$734$	−8.46369e8			−2.14028
$735$	−5.08162e7			−0.127980
$736$	−4.05692e8			−1.01757
$737$	1.06232e8			0.265372
$738$			2.05307e8i			0.510781i
$739$			6.36740e6i			0.0157772i	0.999969	+	0.00788858i	$0.00251104\pi$
$739$			6.36740e6i			0.0157772i	−0.999969	+	0.00788858i	$0.997489\pi$
$740$			4.78663e8i			1.18123i
$741$			7.09554e8i			1.74394i
$742$		−	3.48963e8i		−	0.854217i
$743$	−6.88384e8			−1.67828			−0.839140	−	0.543916i	$-0.816941\pi$
$743$	−6.88384e8			−1.67828			−0.839140	+	0.543916i	$0.816941\pi$
$744$	3.52554e8			0.856065
$745$			3.20066e8i			0.774054i
$746$			9.73119e8i			2.34396i
$747$		−	1.72645e8i		−	0.414183i
$748$			1.69084e8i			0.404016i
$749$	−5.55776e8			−1.32268
$750$			3.85160e8i			0.912972i
$751$		−	6.76662e8i		−	1.59754i	−0.601637	−	0.798770i	$-0.705484\pi$
$751$		−	6.76662e8i		−	1.59754i	0.601637	−	0.798770i	$-0.294516\pi$
$752$			1.80881e7i			0.0425344i
$753$	1.59339e8			0.373196
$754$	1.76480e9			4.11702
$755$			8.19269e7i			0.190364i
$756$	−1.29119e8			−0.298830
$757$	2.15394e8			0.496530			0.248265	−	0.968692i	$-0.420140\pi$
$757$	2.15394e8			0.496530			0.248265	+	0.968692i	$0.420140\pi$
$758$		−	5.62988e8i		−	1.29268i
$759$	8.04774e7			0.184055
$760$			9.76091e8i			2.22356i
$761$	3.83354e8			0.869852			0.434926	−	0.900466i	$-0.356774\pi$
$761$	3.83354e8			0.869852			0.434926	+	0.900466i	$0.356774\pi$
$762$			7.21174e8i			1.62995i
$763$			4.09856e8i			0.922693i
$764$			6.66802e8i			1.49526i
$765$	1.57915e8			0.352728
$766$			1.01121e9i			2.24985i
$767$	−7.65017e8	−	2.12764e8i	−1.69545	−	0.471532i
$768$	−3.79172e8			−0.837051
$769$			4.83594e8i			1.06341i	0.846929	+	0.531706i	$0.178449\pi$
$769$			4.83594e8i			1.06341i	−0.846929	+	0.531706i	$0.821551\pi$
$770$	−1.58878e8			−0.348010
$771$	−2.36470e8			−0.515957
$772$	−9.12524e8			−1.98332
$773$			1.06726e7i			0.0231064i	0.999933	+	0.0115532i	$0.00367758\pi$
$773$			1.06726e7i			0.0231064i	−0.999933	+	0.0115532i	$0.996322\pi$
$774$	−1.74715e8			−0.376797
$775$		−	4.67498e7i		−	0.100433i
$776$	1.00211e9			2.14453
$777$		−	1.55741e8i		−	0.332002i
$778$			2.36512e8i			0.502244i
$779$	−7.49748e8			−1.58600
$780$		−	8.78699e8i		−	1.85164i
$781$			1.06053e8i			0.222622i
$782$	1.13105e9			2.36517
$783$	1.30329e8			0.271491
$784$	−3.21066e7			−0.0666264
$785$		−	6.36195e8i		−	1.31517i
$786$	8.19416e7

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)

\(f(q)\)	\(=\)	\(q-13.2670i q^{2} +15.5885 q^{3} -112.012 q^{4} +130.160 q^{5} -206.811i q^{6} +304.309 q^{7} +636.976i q^{8} +243.000 q^{9} +O(q^{10})\)	\(q-13.2670i q^{2} +15.5885 q^{3} -112.012 q^{4} +130.160 q^{5} -206.811i q^{6} +304.309 q^{7} +636.976i q^{8} +243.000 q^{9} -1726.83i q^{10} -302.342i q^{11} -1746.10 q^{12} +3866.28i q^{13} -4037.26i q^{14} +2029.00 q^{15} +1281.96 q^{16} +4992.75 q^{17} -3223.87i q^{18} +11773.0 q^{19} -14579.5 q^{20} +4743.71 q^{21} -4011.16 q^{22} +17075.4i q^{23} +9929.48i q^{24} +1316.68 q^{25} +51293.8 q^{26} +3788.00 q^{27} -34086.3 q^{28} +34405.8 q^{29} -26918.6i q^{30} -35505.8i q^{31} +23758.8i q^{32} -4713.05i q^{33} -66238.6i q^{34} +39608.9 q^{35} -27219.0 q^{36} -32831.2i q^{37} -156193. i q^{38} +60269.3i q^{39} +82909.0i q^{40} -63683.4 q^{41} -62934.6i q^{42} -54194.2i q^{43} +33866.0i q^{44} +31628.9 q^{45} +226539. q^{46} +14109.8i q^{47} +19983.7 q^{48} -25045.0 q^{49} -17468.4i q^{50} +77829.2 q^{51} -433071. i q^{52} +86435.8 q^{53} -50255.2i q^{54} -39352.9i q^{55} +193838. i q^{56} +183524. q^{57} -456461. i q^{58} +(-55030.6 + 197869. i) q^{59} -227272. q^{60} +21873.5i q^{61} -471054. q^{62} +73947.1 q^{63} +397252. q^{64} +503236. i q^{65} -62527.8 q^{66} +351365. i q^{67} -559249. q^{68} +266180. i q^{69} -525490. i q^{70} -350770. q^{71} +154785. i q^{72} -216142. i q^{73} -435570. q^{74} +20525.0 q^{75} -1.31873e6 q^{76} -92005.5i q^{77} +799591. q^{78} -797387. q^{79} +166860. q^{80} +59049.0 q^{81} +844885. i q^{82} -710474. i q^{83} -531353. q^{84} +649857. q^{85} -718992. q^{86} +536334. q^{87} +192585. q^{88} -1.09404e6i q^{89} -419620. i q^{90} +1.17654e6i q^{91} -1.91266e6i q^{92} -553480. i q^{93} +187194. q^{94} +1.53238e6 q^{95} +370363. i q^{96} -1.57323e6i q^{97} +332271. i q^{98} -73469.2i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 60q - 1920q^{4} - 408q^{7} + 14580q^{9} + O(q^{10}) \)	\( 60q - 1920q^{4} - 408q^{7} + 14580q^{9} - 1944q^{12} - 4536q^{15} + 56616q^{16} + 8480q^{17} + 11376q^{19} + 40796q^{20} - 8232q^{22} + 197940q^{25} + 147252q^{26} + 71640q^{28} + 63456q^{29} - 364432q^{35} - 466560q^{36} + 99632q^{41} - 470316q^{46} + 171072q^{48} + 1737420q^{49} + 60912q^{51} + 92240q^{53} + 186624q^{57} + 917264q^{59} + 1063368q^{60} - 115768q^{62} - 99144q^{63} - 1107444q^{64} + 1172232q^{66} - 4247232q^{68} + 1498048q^{71} + 1161448q^{74} - 1477440q^{75} - 1045320q^{76} - 1060452q^{78} - 90600q^{79} + 77096q^{80} + 3542940q^{81} - 2225880q^{84} - 693408q^{85} - 1567768q^{86} + 1821528q^{87} + 62892q^{88} + 5268696q^{94} + 296128q^{95} + O(q^{100}) \)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Properties

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