LMFDB - Embedded newform 177.5.c.a.58.29

Newspace parameters

Level:	$ N $	$=$	$ 177 = 3 \cdot 59 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	177.c (of order $2$, degree $1$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$18.2964834658$
Analytic rank:	$0$
Dimension:	$40$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			58.29
Character	$\chi$	$=$	177.58
Dual form			177.5.c.a.58.12

$q$-expansion

$f(q)$	$=$	$q+4.05068i q^{2} -5.19615 q^{3} -0.408022 q^{4} -16.4107 q^{5} -21.0480i q^{6} +47.3137 q^{7} +63.1581i q^{8} +27.0000 q^{9} +O(q^{10})$	\(q+4.05068i q^{2} -5.19615 q^{3} -0.408022 q^{4} -16.4107 q^{5} -21.0480i q^{6} +47.3137 q^{7} +63.1581i q^{8} +27.0000 q^{9} -66.4744i q^{10} +25.6339i q^{11} +2.12014 q^{12} +105.588i q^{13} +191.653i q^{14} +85.2723 q^{15} -262.362 q^{16} +441.346 q^{17} +109.368i q^{18} -560.726 q^{19} +6.69591 q^{20} -245.849 q^{21} -103.835 q^{22} +764.114i q^{23} -328.179i q^{24} -355.690 q^{25} -427.703 q^{26} -140.296 q^{27} -19.3050 q^{28} -1381.28 q^{29} +345.411i q^{30} -950.873i q^{31} -52.2142i q^{32} -133.198i q^{33} +1787.75i q^{34} -776.449 q^{35} -11.0166 q^{36} +632.407i q^{37} -2271.32i q^{38} -548.650i q^{39} -1036.47i q^{40} +1029.59 q^{41} -995.857i q^{42} -2959.97i q^{43} -10.4592i q^{44} -443.088 q^{45} -3095.18 q^{46} +4322.20i q^{47} +1363.27 q^{48} -162.413 q^{49} -1440.79i q^{50} -2293.30 q^{51} -43.0822i q^{52} -2110.03 q^{53} -568.295i q^{54} -420.669i q^{55} +2988.25i q^{56} +2913.62 q^{57} -5595.14i q^{58} +(-3445.38 + 496.740i) q^{59} -34.7930 q^{60} +4274.01i q^{61} +3851.68 q^{62} +1277.47 q^{63} -3986.29 q^{64} -1732.77i q^{65} +539.541 q^{66} -867.430i q^{67} -180.079 q^{68} -3970.45i q^{69} -3145.15i q^{70} -236.548 q^{71} +1705.27i q^{72} -6211.26i q^{73} -2561.68 q^{74} +1848.22 q^{75} +228.788 q^{76} +1212.84i q^{77} +2222.41 q^{78} +8916.22 q^{79} +4305.53 q^{80} +729.000 q^{81} +4170.56i q^{82} +10864.4i q^{83} +100.312 q^{84} -7242.78 q^{85} +11989.9 q^{86} +7177.37 q^{87} -1618.99 q^{88} +6262.72i q^{89} -1794.81i q^{90} +4995.75i q^{91} -311.775i q^{92} +4940.88i q^{93} -17507.9 q^{94} +9201.88 q^{95} +271.313i q^{96} +7894.97i q^{97} -657.884i q^{98} +692.115i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 40q - 320q^{4} + 80q^{7} + 1080q^{9} + O(q^{10}) $	$ 40q - 320q^{4} + 80q^{7} + 1080q^{9} + 360q^{12} + 144q^{15} + 3944q^{16} - 528q^{17} + 444q^{19} + 444q^{20} + 1304q^{22} + 4880q^{25} - 1452q^{26} - 1160q^{28} - 996q^{29} + 10320q^{35} - 8640q^{36} - 5196q^{41} - 10476q^{46} + 576q^{48} + 5104q^{49} + 936q^{51} - 2184q^{53} - 2520q^{57} - 11736q^{59} - 11448q^{60} + 15240q^{62} + 2160q^{63} - 81012q^{64} + 17352q^{66} + 29568q^{68} - 5964q^{71} + 14376q^{74} - 2736q^{75} + 3480q^{76} + 37692q^{78} + 19020q^{79} + 33096q^{80} + 29160q^{81} + 25128q^{84} + 20220q^{85} - 65880q^{86} + 1512q^{87} - 14932q^{88} - 17864q^{94} + 11004q^{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 $

$2$			4.05068i			1.01267i	0.862337	+	0.506335i	$0.169000\pi$
$2$			4.05068i			1.01267i	−0.862337	+	0.506335i	$0.831000\pi$
$3$	−5.19615			−0.577350
$3$	−5.19615			−0.577350
$4$	−0.408022			−0.0255014
$5$	−16.4107			−0.656426			−0.328213	−	0.944604i	$-0.606446\pi$
$5$	−16.4107			−0.656426			−0.328213	+	0.944604i	$0.606446\pi$
$6$		−	21.0480i		−	0.584666i
$7$	47.3137			0.965586			0.482793	−	0.875735i	$-0.339622\pi$
$7$	47.3137			0.965586			0.482793	+	0.875735i	$0.339622\pi$
$8$			63.1581i			0.986846i
$9$	27.0000			0.333333
$10$		−	66.4744i		−	0.664744i
$11$			25.6339i			0.211850i	0.994374	+	0.105925i	$0.0337804\pi$
$11$			25.6339i			0.211850i	−0.994374	+	0.105925i	$0.966220\pi$
$12$	2.12014			0.0147232
$13$			105.588i			0.624780i	0.949954	+	0.312390i	$0.101130\pi$
$13$			105.588i			0.624780i	−0.949954	+	0.312390i	$0.898870\pi$
$14$			191.653i			0.977820i
$15$	85.2723			0.378988
$16$	−262.362			−1.02485
$17$	441.346			1.52715			0.763575	−	0.645719i	$-0.223442\pi$
$17$	441.346			1.52715			0.763575	+	0.645719i	$0.223442\pi$
$18$			109.368i			0.337557i
$19$	−560.726			−1.55326			−0.776629	−	0.629959i	$-0.783072\pi$
$19$	−560.726			−1.55326			−0.776629	+	0.629959i	$0.783072\pi$
$20$	6.69591			0.0167398
$21$	−245.849			−0.557481
$22$	−103.835			−0.214535
$23$			764.114i			1.44445i	0.691658	+	0.722225i	$0.256880\pi$
$23$			764.114i			1.44445i	−0.691658	+	0.722225i	$0.743120\pi$
$24$		−	328.179i		−	0.569756i
$25$	−355.690			−0.569104
$26$	−427.703			−0.632696
$27$	−140.296			−0.192450
$28$	−19.3050			−0.0246238
$29$	−1381.28			−1.64243			−0.821216	−	0.570618i	$-0.806704\pi$
$29$	−1381.28			−1.64243			−0.821216	+	0.570618i	$0.806704\pi$
$30$			345.411i			0.383790i
$31$		−	950.873i		−	0.989462i	−0.869046	−	0.494731i	$-0.835267\pi$
$31$		−	950.873i		−	0.989462i	0.869046	−	0.494731i	$-0.164733\pi$
$32$		−	52.2142i		−	0.0509904i
$33$		−	133.198i		−	0.122312i
$34$			1787.75i			1.54650i
$35$	−776.449			−0.633836
$36$	−11.0166			−0.00850046
$37$			632.407i			0.461948i	0.972960	+	0.230974i	$0.0741913\pi$
$37$			632.407i			0.461948i	−0.972960	+	0.230974i	$0.925809\pi$
$38$		−	2271.32i		−	1.57294i
$39$		−	548.650i		−	0.360717i
$40$		−	1036.47i		−	0.647792i
$41$	1029.59			0.612489			0.306245	−	0.951953i	$-0.400927\pi$
$41$	1029.59			0.612489			0.306245	+	0.951953i	$0.400927\pi$
$42$		−	995.857i		−	0.564545i
$43$		−	2959.97i		−	1.60085i	−0.599434	−	0.800424i	$-0.704608\pi$
$43$		−	2959.97i		−	1.60085i	0.599434	−	0.800424i	$-0.295392\pi$
$44$		−	10.4592i		−	0.00540248i
$45$	−443.088			−0.218809
$46$	−3095.18			−1.46275
$47$			4322.20i			1.95663i	0.207119	+	0.978316i	$0.433591\pi$
$47$			4322.20i			1.95663i	−0.207119	+	0.978316i	$0.566409\pi$
$48$	1363.27			0.591698
$49$	−162.413			−0.0676439
$50$		−	1440.79i		−	0.576315i
$51$	−2293.30			−0.881700
$52$		−	43.0822i		−	0.0159327i
$53$	−2110.03			−0.751168			−0.375584	−	0.926788i	$-0.622558\pi$
$53$	−2110.03			−0.751168			−0.375584	+	0.926788i	$0.622558\pi$
$54$		−	568.295i		−	0.194889i
$55$		−	420.669i		−	0.139064i
$56$			2988.25i			0.952884i
$57$	2913.62			0.896773
$58$		−	5595.14i		−	1.66324i
$59$	−3445.38	+	496.740i	−0.989766	+	0.142700i
$59$	−3445.38	+	496.740i	−0.989766	+	0.142700i
$60$	−34.7930			−0.00966471
$61$			4274.01i			1.14862i	0.818638	+	0.574309i	$0.194729\pi$
$61$			4274.01i			1.14862i	−0.818638	+	0.574309i	$0.805271\pi$
$62$	3851.68			1.00200
$63$	1277.47			0.321862
$64$	−3986.29			−0.973215
$65$		−	1732.77i		−	0.410122i
$66$	539.541			0.123862
$67$		−	867.430i		−	0.193234i	−0.995322	−	0.0966172i	$-0.969198\pi$
$67$		−	867.430i		−	0.193234i	0.995322	−	0.0966172i	$-0.0308023\pi$
$68$	−180.079			−0.0389444
$69$		−	3970.45i		−	0.833953i
$70$		−	3145.15i		−	0.641867i
$71$	−236.548			−0.0469249			−0.0234624	−	0.999725i	$-0.507469\pi$
$71$	−236.548			−0.0469249			−0.0234624	+	0.999725i	$0.507469\pi$
$72$			1705.27i			0.328949i
$73$		−	6211.26i		−	1.16556i	−0.812631	−	0.582779i	$-0.801965\pi$
$73$		−	6211.26i		−	1.16556i	0.812631	−	0.582779i	$-0.198035\pi$
$74$	−2561.68			−0.467801
$75$	1848.22			0.328573
$76$	228.788			0.0396102
$77$			1212.84i			0.204560i
$78$	2222.41			0.365287
$79$	8916.22			1.42865			0.714326	−	0.699813i	$-0.246733\pi$
$79$	8916.22			1.42865			0.714326	+	0.699813i	$0.246733\pi$
$80$	4305.53			0.672739
$81$	729.000			0.111111
$82$			4170.56i			0.620250i
$83$			10864.4i			1.57706i	0.614994	+	0.788532i	$0.289158\pi$
$83$			10864.4i			1.57706i	−0.614994	+	0.788532i	$0.710842\pi$
$84$	100.312			0.0142165
$85$	−7242.78			−1.00246
$86$	11989.9			1.62113
$87$	7177.37			0.948258
$88$	−1618.99			−0.209064
$89$			6262.72i			0.790647i	0.918542	+	0.395324i	$0.129368\pi$
$89$			6262.72i			0.790647i	−0.918542	+	0.395324i	$0.870632\pi$
$90$		−	1794.81i		−	0.221581i
$91$			4995.75i			0.603279i
$92$		−	311.775i		−	0.0368355i
$93$			4940.88i			0.571266i
$94$	−17507.9			−1.98142
$95$	9201.88			1.01960
$96$			271.313i			0.0294393i
$97$			7894.97i			0.839087i	0.907735	+	0.419543i	$0.137810\pi$
$97$			7894.97i			0.839087i	−0.907735	+	0.419543i	$0.862190\pi$
$98$		−	657.884i		−	0.0685010i
$99$			692.115i			0.0706168i
$100$	145.129			0.0145129
$101$			6756.86i			0.662372i	0.943566	+	0.331186i	$0.107449\pi$
$101$			6756.86i			0.662372i	−0.943566	+	0.331186i	$0.892551\pi$
$102$		−	9289.44i		−	0.892872i
$103$			19989.4i			1.88420i	0.335338	+	0.942098i	$0.391149\pi$
$103$			19989.4i			1.88420i	−0.335338	+	0.942098i	$0.608851\pi$
$104$	−6668.73			−0.616562
$105$	4034.55			0.365945
$106$		−	8547.07i		−	0.760686i
$107$	14259.8			1.24550			0.622751	−	0.782420i	$-0.286015\pi$
$107$	14259.8			1.24550			0.622751	+	0.782420i	$0.286015\pi$
$108$	57.2439			0.00490774
$109$			219.395i			0.0184660i	0.999957	+	0.00923302i	$0.00293900\pi$
$109$			219.395i			0.0184660i	−0.999957	+	0.00923302i	$0.997061\pi$
$110$	1704.00			0.140826
$111$		−	3286.08i		−	0.266706i
$112$	−12413.3			−0.989582
$113$		−	12616.9i		−	0.988089i	−0.869437	−	0.494044i	$-0.835518\pi$
$113$		−	12616.9i		−	0.988089i	0.869437	−	0.494044i	$-0.164482\pi$
$114$			11802.1i			0.908136i
$115$		−	12539.6i		−	0.948175i
$116$	563.595			0.0418843
$117$			2850.87i			0.208260i
$118$	−2012.14	−	13956.1i	−0.144508	−	1.00231i
$119$	20881.7			1.47459
$120$			5385.64i			0.374003i
$121$	13983.9			0.955119
$122$	−17312.7			−1.16317
$123$	−5349.93			−0.353621
$124$			387.977i			0.0252326i
$125$	16093.8			1.03000
$126$			5174.62i			0.325940i
$127$	3740.54			0.231914			0.115957	−	0.993254i	$-0.463007\pi$
$127$	3740.54			0.231914			0.115957	+	0.993254i	$0.463007\pi$
$128$		−	16982.6i		−	1.03654i
$129$			15380.4i			0.924250i
$130$	7018.88			0.415318
$131$		−	21977.1i		−	1.28064i	−0.768108	−	0.640320i	$-0.778802\pi$
$131$		−	21977.1i		−	1.28064i	0.768108	−	0.640320i	$-0.221198\pi$
$132$			54.3476i			0.00311912i
$133$	−26530.0			−1.49980
$134$	3513.68			0.195683
$135$	2302.35			0.126329
$136$			27874.6i			1.50706i
$137$	−8143.37			−0.433873			−0.216937	−	0.976186i	$-0.569606\pi$
$137$	−8143.37			−0.433873			−0.216937	+	0.976186i	$0.569606\pi$
$138$	16083.0			0.844520
$139$	13508.7			0.699172			0.349586	−	0.936904i	$-0.386322\pi$
$139$	13508.7			0.699172			0.349586	+	0.936904i	$0.386322\pi$
$140$	316.808			0.0161637
$141$		−	22458.8i		−	1.12966i
$142$		−	958.182i		−	0.0475194i
$143$	−2706.63			−0.132360
$144$	−7083.77			−0.341617
$145$	22667.8			1.07814
$146$	25159.8			1.18033
$147$	843.923			0.0390542
$148$		−	258.036i		−	0.0117803i
$149$			5748.17i			0.258915i	0.991585	+	0.129457i	$0.0413235\pi$
$149$			5748.17i			0.258915i	−0.991585	+	0.129457i	$0.958676\pi$
$150$			7486.55i			0.332736i
$151$		−	39105.9i		−	1.71510i	−0.514404	−	0.857548i	$-0.671987\pi$
$151$		−	39105.9i		−	1.71510i	0.514404	−	0.857548i	$-0.328013\pi$
$152$		−	35414.4i		−	1.53283i
$153$	11916.3			0.509050
$154$	−4912.81			−0.207152
$155$			15604.5i			0.649509i
$156$			223.861i			0.00919878i
$157$			995.408i			0.0403833i	0.999796	+	0.0201917i	$0.00642764\pi$
$157$			995.408i			0.0403833i	−0.999796	+	0.0201917i	$0.993572\pi$
$158$			36116.8i			1.44675i
$159$	10964.0			0.433687
$160$			856.869i			0.0334715i
$161$			36153.1i			1.39474i
$162$			2952.95i			0.112519i
$163$	−51373.5			−1.93359			−0.966794	−	0.255559i	$-0.917741\pi$
$163$	−51373.5			−1.93359			−0.966794	+	0.255559i	$0.917741\pi$
$164$	−420.097			−0.0156193
$165$			2185.86i			0.0802888i
$166$	−44008.2			−1.59705
$167$	−17563.4			−0.629760			−0.314880	−	0.949132i	$-0.601964\pi$
$167$	−17563.4			−0.629760			−0.314880	+	0.949132i	$0.601964\pi$
$168$		−	15527.4i		−	0.550148i
$169$	17412.2			0.609650
$170$		−	29338.2i		−	1.01516i
$171$	−15139.6			−0.517752
$172$			1207.73i			0.0408238i
$173$		−	59732.1i		−	1.99579i	−0.0648356	−	0.997896i	$-0.520652\pi$
$173$		−	59732.1i		−	1.99579i	0.0648356	−	0.997896i	$-0.479348\pi$
$174$			29073.2i			0.960273i
$175$	−16829.0			−0.549519
$176$		−	6725.36i		−	0.217115i
$177$	17902.7	−	2581.14i	0.571442	−	0.0823881i
$178$	−25368.3			−0.800665
$179$		−	52499.1i		−	1.63850i	−0.573439	−	0.819248i	$-0.694391\pi$
$179$		−	52499.1i		−	1.63850i	0.573439	−	0.819248i	$-0.305609\pi$
$180$	180.790			0.00557993
$181$	−1001.39			−0.0305665			−0.0152832	−	0.999883i	$-0.504865\pi$
$181$	−1001.39			−0.0305665			−0.0152832	+	0.999883i	$0.504865\pi$
$182$	−20236.2			−0.610923
$183$		−	22208.4i		−	0.663155i
$184$	−48260.0			−1.42545
$185$		−	10378.2i		−	0.303235i
$186$	−20013.9			−0.578504
$187$			11313.4i			0.323527i
$188$		−	1763.55i		−	0.0498968i
$189$	−6637.93			−0.185827
$190$			37273.9i			1.03252i
$191$			65871.0i			1.80563i	0.430034	+	0.902813i	$0.358502\pi$
$191$			65871.0i			1.80563i	−0.430034	+	0.902813i	$0.641498\pi$
$192$	20713.4			0.561886
$193$	46752.7			1.25514			0.627570	−	0.778560i	$-0.284050\pi$
$193$	46752.7			1.25514			0.627570	+	0.778560i	$0.284050\pi$
$194$	−31980.0			−0.849718
$195$			9003.72i			0.236784i
$196$	66.2681			0.00172501
$197$	56429.6			1.45403			0.727017	−	0.686619i	$-0.240906\pi$
$197$	56429.6			1.45403			0.727017	+	0.686619i	$0.240906\pi$
$198$	−2803.54			−0.0715116
$199$	4451.64			0.112412			0.0562061	−	0.998419i	$-0.482100\pi$
$199$	4451.64			0.112412			0.0562061	+	0.998419i	$0.482100\pi$
$200$		−	22464.7i		−	0.561618i
$201$			4507.30i			0.111564i
$202$	−27369.9			−0.670764
$203$	−65353.7			−1.58591
$204$	935.718			0.0224846
$205$	−16896.3			−0.402054
$206$	−80970.8			−1.90807
$207$			20631.1i			0.481483i
$208$		−	27702.2i		−	0.640306i
$209$		−	14373.6i		−	0.329058i
$210$			16342.7i			0.370582i
$211$			42832.4i			0.962071i	0.876701	+	0.481036i	$0.159739\pi$
$211$			42832.4i			0.962071i	−0.876701	+	0.481036i	$0.840261\pi$
$212$	860.939			0.0191558
$213$	1229.14			0.0270921
$214$			57761.7i			1.26128i
$215$			48575.0i			1.05084i
$216$		−	8860.84i		−	0.189919i
$217$		−	44989.3i		−	0.955411i
$218$	−888.699			−0.0187000
$219$			32274.6i			0.672935i
$220$			171.642i			0.00354633i
$221$			46600.8i			0.954132i
$222$	13310.9			0.270085
$223$	−14776.1			−0.297132			−0.148566	−	0.988902i	$-0.547466\pi$
$223$	−14776.1			−0.297132			−0.148566	+	0.988902i	$0.547466\pi$
$224$		−	2470.45i		−	0.0492356i
$225$	−9603.64			−0.189701
$226$	51107.1			1.00061
$227$		−	37257.5i		−	0.723040i	−0.932364	−	0.361520i	$-0.882258\pi$
$227$		−	37257.5i		−	0.723040i	0.932364	−	0.361520i	$-0.117742\pi$
$228$	−1188.82			−0.0228690
$229$			36389.1i			0.693906i	0.937883	+	0.346953i	$0.112784\pi$
$229$			36389.1i			0.693906i	−0.937883	+	0.346953i	$0.887216\pi$
$230$	50794.0			0.960189
$231$		−	6302.08i		−	0.118103i
$232$		−	87239.4i		−	1.62083i
$233$			20634.6i			0.380088i	0.981776	+	0.190044i	$0.0608630\pi$
$233$			20634.6i			0.380088i	−0.981776	+	0.190044i	$0.939137\pi$
$234$	−11548.0			−0.210899
$235$		−	70930.1i		−	1.28438i
$236$	1405.79	−	202.681i	0.0252404	−	0.00363906i
$237$	−46330.0			−0.824833
$238$			84585.2i			1.49328i
$239$	−18302.2			−0.320411			−0.160205	−	0.987084i	$-0.551216\pi$
$239$	−18302.2			−0.320411			−0.160205	+	0.987084i	$0.551216\pi$
$240$	−22372.2			−0.388406
$241$	73211.9			1.26051			0.630257	−	0.776386i	$-0.282949\pi$
$241$	73211.9			1.26051			0.630257	+	0.776386i	$0.282949\pi$
$242$			56644.3i			0.967221i
$243$	−3788.00			−0.0641500
$244$		−	1743.89i		−	0.0292914i
$245$	2665.31			0.0444033
$246$		−	21670.9i		−	0.358101i
$247$		−	59205.8i		−	0.970444i
$248$	60055.4			0.976447
$249$		−	56453.0i		−	0.910518i
$250$			65190.8i			1.04305i
$251$	66492.8			1.05542			0.527712	−	0.849423i	$-0.323050\pi$
$251$	66492.8			1.05542			0.527712	+	0.849423i	$0.323050\pi$
$252$	−521.236			−0.00820792
$253$	−19587.2			−0.306007
$254$			15151.7i			0.234852i
$255$	37634.6			0.578771
$256$	5010.54			0.0764548
$257$	−48002.7			−0.726774			−0.363387	−	0.931638i	$-0.618380\pi$
$257$	−48002.7			−0.726774			−0.363387	+	0.931638i	$0.618380\pi$
$258$	−62301.3			−0.935961
$259$			29921.5i			0.446051i
$260$			707.007i			0.0104587i
$261$	−37294.7			−0.547477
$262$	89022.1			1.29687
$263$	23476.9			0.339413			0.169707	−	0.985495i	$-0.445718\pi$
$263$	23476.9			0.339413			0.169707	+	0.985495i	$0.445718\pi$
$264$	8412.52			0.120703
$265$	34627.0			0.493087
$266$		−	107465.i		−	1.51881i
$267$		−	32542.0i		−	0.456480i
$268$			353.930i			0.00492775i
$269$			86186.2i			1.19106i	0.803334	+	0.595529i	$0.203058\pi$
$269$			86186.2i			1.19106i	−0.803334	+	0.595529i	$0.796942\pi$
$270$			9326.09i			0.127930i
$271$	−18673.8			−0.254270			−0.127135	−	0.991885i	$-0.540578\pi$
$271$	−18673.8			−0.254270			−0.127135	+	0.991885i	$0.540578\pi$
$272$	−115792.			−1.56510
$273$		−	25958.7i		−	0.348303i
$274$		−	32986.2i		−	0.439371i
$275$		−	9117.73i		−	0.120565i
$276$			1620.03i			0.0212670i
$277$	132710.			1.72959			0.864797	−	0.502122i	$-0.167447\pi$
$277$	132710.			1.72959			0.864797	+	0.502122i	$0.167447\pi$
$278$			54719.4i			0.708030i
$279$		−	25673.6i		−	0.329821i
$280$		−	49039.1i		−	0.625499i
$281$	42447.9			0.537581			0.268790	−	0.963199i	$-0.413376\pi$
$281$	42447.9			0.537581			0.268790	+	0.963199i	$0.413376\pi$
$282$	90973.5			1.14398
$283$		−	1417.90i		−	0.0177041i	−0.999961	−	0.00885205i	$-0.997182\pi$
$283$		−	1417.90i		−	0.0177041i	0.999961	−	0.00885205i	$-0.00281773\pi$
$284$	96.5169			0.00119665
$285$	−47814.4			−0.588666
$286$		−	10963.7i		−	0.134037i
$287$	48713.9			0.591411
$288$		−	1409.78i		−	0.0169968i
$289$	111265.			1.33219
$290$			91820.0i			1.09180i
$291$		−	41023.5i		−	0.484447i
$292$			2534.33i			0.0297233i
$293$	44835.8			0.522264			0.261132	−	0.965303i	$-0.415904\pi$
$293$	44835.8			0.522264			0.261132	+	0.965303i	$0.415904\pi$
$294$			3418.46i			0.0395491i
$295$	56540.9	−	8151.83i	0.649708	−	0.0936723i
$296$	−39941.7			−0.455872
$297$		−	3596.34i		−	0.0407706i
$298$	−23284.0			−0.262195
$299$	−80681.1			−0.902463
$300$	−754.115			−0.00837905
$301$		−	140047.i		−	1.54576i
$302$	158406.			1.73683
$303$		−	35109.7i		−	0.382421i
$304$	147113.			1.59186
$305$		−	70139.3i		−	0.753984i
$306$			48269.3i			0.515500i
$307$	−15561.6			−0.165112			−0.0825559	−	0.996586i	$-0.526308\pi$
$307$	−15561.6			−0.165112			−0.0825559	+	0.996586i	$0.526308\pi$
$308$		−	494.863i		−	0.00521656i
$309$		−	103868.i		−	1.08784i
$310$	−63208.7			−0.657739
$311$	−160952.			−1.66409			−0.832043	−	0.554711i	$-0.812829\pi$
$311$	−160952.			−1.66409			−0.832043	+	0.554711i	$0.812829\pi$
$312$	34651.7			0.355972
$313$			79828.7i			0.814836i	0.913242	+	0.407418i	$0.133571\pi$
$313$			79828.7i			0.814836i	−0.913242	+	0.407418i	$0.866429\pi$
$314$	−4032.08			−0.0408950
$315$	−20964.1			−0.211279
$316$	−3638.01			−0.0364326
$317$	108281.			1.07754			0.538770	−	0.842453i	$-0.318889\pi$
$317$	108281.			1.07754			0.538770	+	0.842453i	$0.318889\pi$
$318$			44411.9i			0.439182i
$319$		−	35407.7i		−	0.347950i
$320$	65417.6			0.638844
$321$	−74095.9			−0.719091
$322$	−146445.			−1.41241
$323$	−247474.			−2.37206
$324$	−297.448			−0.00283349
$325$		−	37556.6i		−	0.355565i
$326$		−	208098.i		−	1.95809i
$327$		−	1140.01i		−	0.0106614i
$328$			65027.2i			0.604432i
$329$			204499.i			1.88930i
$330$	−8854.23			−0.0813061
$331$	124431.			1.13573			0.567864	−	0.823123i	$-0.307770\pi$
$331$	124431.			1.13573			0.567864	+	0.823123i	$0.307770\pi$
$332$		−	4432.91i		−	0.0402173i
$333$			17075.0i			0.153983i
$334$		−	71143.6i		−	0.637739i
$335$			14235.1i			0.126844i
$336$	64501.5			0.571335
$337$			88676.9i			0.780820i	0.920641	+	0.390410i	$0.127667\pi$
$337$			88676.9i			0.780820i	−0.920641	+	0.390410i	$0.872333\pi$
$338$			70531.3i			0.617374i
$339$			65559.4i			0.570473i
$340$	2955.21			0.0255641
$341$	24374.6			0.209618
$342$		−	61325.7i		−	0.524313i
$343$	−121285.			−1.03090
$344$	186946.			1.57979
$345$			65157.7i			0.547429i
$346$	241956.			2.02108
$347$			55831.8i			0.463685i	0.972753	+	0.231842i	$0.0744754\pi$
$347$			55831.8i			0.463685i	−0.972753	+	0.231842i	$0.925525\pi$
$348$	−2928.52			−0.0241819
$349$			66212.9i			0.543616i	0.962352	+	0.271808i	$0.0876215\pi$
$349$			66212.9i			0.543616i	−0.962352	+	0.271808i	$0.912378\pi$
$350$		−	68169.0i		−	0.556482i
$351$		−	14813.6i		−	0.120239i
$352$	1338.45			0.0108023
$353$			119306.i			0.957439i	0.877968	+	0.478719i	$0.158899\pi$
$353$			119306.i			0.957439i	−0.877968	+	0.478719i	$0.841101\pi$
$354$	10455.4	+	72518.1i	0.0834320	+	0.578682i
$355$	3881.91			0.0308027
$356$		−	2555.33i		−	0.0201626i
$357$	−108505.			−0.851357
$358$	212657.			1.65926
$359$	−164492.			−1.27631			−0.638155	−	0.769908i	$-0.720302\pi$
$359$	−164492.			−1.27631			−0.638155	+	0.769908i	$0.720302\pi$
$360$		−	27984.6i		−	0.215931i
$361$	184092.			1.41261
$362$		−	4056.30i		−	0.0309538i
$363$	−72662.5			−0.551438
$364$		−	2038.38i		−	0.0153844i
$365$			101931.i			0.765103i
$366$	89959.2			0.671558
$367$		−	10621.8i		−	0.0788613i	−0.999222	−	0.0394307i	$-0.987446\pi$
$367$		−	10621.8i		−	0.0788613i	0.999222	−	0.0394307i	$-0.0125544\pi$
$368$		−	200474.i		−	1.48035i
$369$	27799.0			0.204163
$370$	42038.9			0.307077
$371$	−99833.4			−0.725318
$372$		−	2015.99i		−	0.0145681i
$373$	49988.6			0.359297			0.179648	−	0.983731i	$-0.442504\pi$
$373$	49988.6			0.359297			0.179648	+	0.983731i	$0.442504\pi$
$374$	−45827.1			−0.327627
$375$	−83625.7			−0.594672
$376$	−272982.			−1.93089
$377$		−	145847.i		−	1.02616i
$378$		−	26888.1i		−	0.188182i
$379$	−23033.2			−0.160352			−0.0801762	−	0.996781i	$-0.525548\pi$
$379$	−23033.2			−0.160352			−0.0801762	+	0.996781i	$0.525548\pi$
$380$	−3754.57			−0.0260012
$381$	−19436.4			−0.133895
$382$	−266823.			−1.82850
$383$	−35518.2			−0.242133			−0.121067	−	0.992644i	$-0.538631\pi$
$383$	−35518.2			−0.242133			−0.121067	+	0.992644i	$0.538631\pi$
$384$			88244.2i			0.598444i
$385$		−	19903.4i		−	0.134278i
$386$			189380.i			1.27104i
$387$		−	79919.1i		−	0.533616i
$388$		−	3221.32i		−	0.0213979i
$389$	157436.			1.04041			0.520206	−	0.854041i	$-0.325855\pi$
$389$	157436.			1.04041			0.520206	+	0.854041i	$0.325855\pi$
$390$	−36471.2			−0.239784
$391$			337239.i			2.20589i
$392$		−	10257.7i		−	0.0667541i
$393$			114196.i			0.739378i
$394$			228578.i			1.47246i
$395$	−146321.			−0.937805
$396$		−	282.398i		−	0.00180083i
$397$		−	163490.i		−	1.03731i	−0.854983	−	0.518656i	$-0.826432\pi$
$397$		−	163490.i		−	1.03731i	0.854983	−	0.518656i	$-0.173568\pi$
$398$			18032.2i			0.113837i
$399$	137854.			0.865912
$400$	93319.6			0.583247
$401$		−	74973.1i		−	0.466248i	−0.972447	−	0.233124i	$-0.925105\pi$
$401$		−	74973.1i		−	0.466248i	0.972447	−	0.233124i	$-0.0748948\pi$
$402$	−18257.6			−0.112978
$403$	100401.			0.618196
$404$		−	2756.95i		−	0.0168914i
$405$	−11963.4			−0.0729363
$406$		−	264727.i		−	1.60600i
$407$	−16211.1			−0.0978640
$408$		−	144841.i		−	0.870102i
$409$			216548.i			1.29452i	0.762271	+	0.647258i	$0.224084\pi$
$409$			216548.i			1.29452i	−0.762271	+	0.647258i	$0.775916\pi$
$410$		−	68441.6i		−	0.407148i
$411$	42314.2			0.250497
$412$		−	8156.13i		−	0.0480496i
$413$	−163013.	+	23502.6i	−0.955704	+	0.137789i
$414$	−83569.9			−0.487584
$415$		−	178292.i		−	1.03523i
$416$	5513.18			0.0318578
$417$	−70193.2			−0.403667
$418$	58222.8			0.333228
$419$		−	221474.i		−	1.26152i	−0.775977	−	0.630761i	$-0.782743\pi$
$419$		−	221474.i		−	1.26152i	0.775977	−	0.630761i	$-0.217257\pi$
$420$	−1646.18			−0.00933211
$421$			315604.i			1.78065i	0.455329	+	0.890323i	$0.349522\pi$
$421$			315604.i			1.78065i	−0.455329	+	0.890323i	$0.650478\pi$
$422$	−173500.			−0.974261
$423$			116699.i			0.652211i
$424$		−	133266.i		−	0.741287i
$425$	−156983.			−0.869107
$426$			4978.86i			0.0274353i
$427$			202219.i			1.10909i
$428$	−5818.29			−0.0317620
$429$	14064.1			0.0764180
$430$	−196762.			−1.06415
$431$			37329.5i			0.200954i	0.994939	+	0.100477i	$0.0320369\pi$
$431$			37329.5i			0.200954i	−0.994939	+	0.100477i	$0.967963\pi$
$432$	36808.4			0.197233
$433$	60211.0			0.321144			0.160572	−	0.987024i	$-0.448666\pi$
$433$	60211.0			0.321144			0.160572	+	0.987024i	$0.448666\pi$
$434$	182237.			0.967516
$435$	−117785.			−0.622462
$436$		−	89.5180i		−	0.000470909i
$437$		−	428458.i		−	2.24360i
$438$	−130734.			−0.681461
$439$	−185108.			−0.960495			−0.480248	−	0.877133i	$-0.659453\pi$
$439$	−185108.			−0.960495			−0.480248	+	0.877133i	$0.659453\pi$
$440$	26568.7			0.137235
$441$	−4385.15			−0.0225480
$442$	−188765.			−0.966222
$443$			234937.i			1.19714i	0.801071	+	0.598569i	$0.204264\pi$
$443$			234937.i			1.19714i	−0.801071	+	0.598569i	$0.795736\pi$
$444$			1340.79i			0.00680137i
$445$		−	102775.i		−	0.519002i
$446$		−	59853.3i		−	0.300897i
$447$		−	29868.3i		−	0.149484i
$448$	−188606.			−0.939722
$449$	−69120.8			−0.342859			−0.171430	−	0.985196i	$-0.554839\pi$
$449$	−69120.8			−0.342859			−0.171430	+	0.985196i	$0.554839\pi$
$450$		−	38901.3i		−	0.192105i
$451$			26392.5i			0.129756i
$452$			5147.98i			0.0251976i
$453$			203200.i			0.990211i
$454$	150918.			0.732201
$455$		−	81983.6i		−	0.396008i
$456$			184019.i			0.884977i
$457$		−	307630.i		−	1.47298i	−0.676449	−	0.736489i	$-0.736482\pi$
$457$		−	307630.i		−	1.47298i	0.676449	−	0.736489i	$-0.263518\pi$
$458$	−147401.			−0.702698
$459$	−61919.2			−0.293900
$460$			5116.44i			0.0241798i
$461$	20386.5			0.0959272			0.0479636	−	0.998849i	$-0.484727\pi$
$461$	20386.5			0.0959272			0.0479636	+	0.998849i	$0.484727\pi$
$462$	25527.7			0.119599
$463$		−	50749.8i		−	0.236740i	−0.992970	−	0.118370i	$-0.962233\pi$
$463$		−	50749.8i		−	0.236740i	0.992970	−	0.118370i	$-0.0377669\pi$
$464$	362396.			1.68325
$465$		−	81083.1i		−	0.374994i
$466$	−83584.1			−0.384904
$467$			9160.46i			0.0420033i	0.999779	+	0.0210017i	$0.00668553\pi$
$467$			9160.46i			0.0420033i	−0.999779	+	0.0210017i	$0.993314\pi$
$468$		−	1163.22i		−	0.00531092i
$469$		−	41041.3i		−	0.186584i
$470$	287315.			1.30066
$471$		−	5172.29i		−	0.0233153i
$472$	−31373.2	−	217603.i	−0.140823	−	0.976746i
$473$	75875.5			0.339140
$474$		−	187668.i		−	0.835284i
$475$	199445.			0.883965
$476$	−8520.20			−0.0376042
$477$	−56970.9			−0.250389
$478$		−	74136.3i		−	0.324471i
$479$	281255.			1.22583			0.612913	−	0.790150i	$-0.289997\pi$
$479$	281255.			1.22583			0.612913	+	0.790150i	$0.289997\pi$
$480$		−	4452.42i		−	0.0193248i
$481$	−66774.5			−0.288616
$482$			296558.i			1.27649i
$483$		−	187857.i		−	0.805254i
$484$	−5705.74			−0.0243569
$485$		−	129562.i		−	0.550799i
$486$		−	15344.0i		−	0.0649628i
$487$	−12464.0			−0.0525534			−0.0262767	−	0.999655i	$-0.508365\pi$
$487$	−12464.0			−0.0525534			−0.0262767	+	0.999655i	$0.508365\pi$
$488$	−269939.			−1.13351
$489$	266944.			1.11636
$490$			10796.3i			0.0449659i
$491$	−267488.			−1.10954			−0.554768	−	0.832005i	$-0.687193\pi$
$491$	−267488.			−1.10954			−0.554768	+	0.832005i	$0.687193\pi$
$492$	2182.89			0.00901781
$493$	−609625.			−2.50824
$494$	239824.			0.982740
$495$		−	11358.1i		−	0.0463547i
$496$			249473.i			1.01405i
$497$	−11192.0			−0.0453100
$498$	228673.			0.922054
$499$	−211721.			−0.850283			−0.425142	−	0.905127i	$-0.639776\pi$
$499$	−211721.			−0.850283			−0.425142	+	0.905127i	$0.639776\pi$
$500$	−6566.61			−0.0262665
$501$	91261.9			0.363592
$502$			269341.i			1.06880i
$503$			46231.2i			0.182725i	0.995818	+	0.0913627i	$0.0291223\pi$
$503$			46231.2i			0.182725i	−0.995818	+	0.0913627i	$0.970878\pi$
$504$			80682.6i			0.317628i
$505$		−	110884.i		−	0.434798i
$506$		−	79341.6i		−	0.309885i
$507$	−90476.5			−0.351982
$508$	−1526.22			−0.00591412
$509$		−	118612.i		−	0.457819i	−0.973448	−	0.228909i	$-0.926484\pi$
$509$		−	118612.i		−	0.457819i	0.973448	−	0.228909i	$-0.0735159\pi$
$510$			152446.i			0.586105i
$511$		−	293878.i		−	1.12545i
$512$		−	251426.i		−	0.959113i
$513$	78667.7			0.298924
$514$		−	194444.i		−	0.735982i
$515$		−	328040.i		−	1.23684i
$516$		−	6275.56i		−	0.0235696i
$517$	−110795.			−0.414513
$518$	−121203.			−0.451702
$519$			310377.i			1.15227i
$520$	109438.			0.404727
$521$	−20361.9			−0.0750140			−0.0375070	−	0.999296i	$-0.511942\pi$
$521$	−20361.9			−0.0750140			−0.0375070	+	0.999296i	$0.511942\pi$
$522$		−	151069.i		−	0.554414i
$523$	36769.4			0.134426			0.0672130	−	0.997739i	$-0.478589\pi$
$523$	36769.4			0.134426			0.0672130	+	0.997739i	$0.478589\pi$
$524$			8967.13i			0.0326581i
$525$	87446.2			0.317265
$526$			95097.4i			0.343714i
$527$		−	419664.i		−	1.51106i
$528$			34946.0i			0.125351i
$529$	−304029.			−1.08643
$530$			140263.i			0.499334i
$531$	−93025.1	+	13412.0i	−0.329922	+	0.0475668i
$532$	10824.8			0.0382470
$533$			108713.i			0.382671i
$534$	131817.			0.462264
$535$	−234012.			−0.817581
$536$	54785.2			0.190693
$537$			272793.i			0.945986i
$538$	−349113.			−1.20615
$539$		−	4163.28i		−	0.0143304i
$540$	−939.410			−0.00322157
$541$			136111.i			0.465049i	0.972590	+	0.232525i	$0.0746986\pi$
$541$			136111.i			0.465049i	−0.972590	+	0.232525i	$0.925301\pi$
$542$		−	75641.7i		−	0.257491i
$543$	5203.37			0.0176476
$544$		−	23044.5i		−	0.0778700i
$545$		−	3600.42i		−	0.0121216i
$546$	105150.			0.352716
$547$	532541.			1.77983			0.889915	−	0.456125i	$-0.150763\pi$
$547$	532541.			1.77983			0.889915	+	0.456125i	$0.150763\pi$
$548$	3322.67			0.0110644
$549$			115398.i			0.382873i
$550$	36933.0			0.122093
$551$	774522.			2.55112
$552$	250766.			0.822983
$553$	421859.			1.37949
$554$			537566.i			1.75151i
$555$			53926.8i			0.175073i
$556$	−5511.84			−0.0178298
$557$	−166657.			−0.537172			−0.268586	−	0.963256i	$-0.586556\pi$
$557$	−166657.			−0.537172			−0.268586	+	0.963256i	$0.586556\pi$
$558$	103995.			0.334000
$559$	312537.			1.00018
$560$	203711.			0.649588
$561$		−	58786.3i		−	0.186789i
$562$			171943.i			0.544392i
$563$			111610.i			0.352116i	0.984380	+	0.176058i	$0.0563347\pi$
$563$			111610.i			0.352116i	−0.984380	+	0.176058i	$0.943665\pi$
$564$			9163.69i			0.0288079i
$565$			207052.i			0.648608i
$566$	5743.47			0.0179284
$567$	34491.7			0.107287
$568$		−	14939.9i		−	0.0463076i
$569$		−	112850.i		−	0.348560i	−0.984696	−	0.174280i	$-0.944240\pi$
$569$		−	112850.i		−	0.348560i	0.984696	−	0.174280i	$-0.0557598\pi$
$570$		−	193681.i		−	0.596124i
$571$			441479.i			1.35406i	0.735956	+	0.677029i	$0.236733\pi$
$571$			441479.i			1.35406i	−0.735956	+	0.677029i	$0.763267\pi$
$572$	1104.36			0.00337536
$573$		−	342276.i		−	1.04248i
$574$			197325.i			0.598904i
$575$		−	271788.i		−	0.822043i
$576$	−107630.			−0.324405
$577$	214151.			0.643234			0.321617	−	0.946870i	$-0.395774\pi$
$577$	214151.			0.643234			0.321617	+	0.946870i	$0.395774\pi$
$578$			450701.i			1.34906i
$579$	−242934.			−0.724655
$580$	−9248.96			−0.0274939
$581$			514034.i			1.52279i
$582$	166173.			0.490585
$583$		−	54088.4i		−	0.159135i
$584$	392291.			1.15023
$585$		−	46784.7i		−	0.136707i
$586$			181616.i			0.528881i
$587$			229548.i			0.666189i	0.942893	+	0.333094i	$0.108093\pi$
$587$			229548.i			0.666189i	−0.942893	+	0.333094i	$0.891907\pi$
$588$	−344.339			−0.000995937
$589$			533179.i			1.53689i
$590$	33020.5	+	229029.i	0.0948592	+	0.657941i
$591$	−293217.			−0.839487
$592$		−	165920.i		−	0.473428i
$593$	161537.			0.459371			0.229686	−	0.973265i	$-0.426230\pi$
$593$	161537.			0.459371			0.229686	+	0.973265i	$0.426230\pi$
$594$	14567.6			0.0412872
$595$	−342683.			−0.967962
$596$		−	2345.38i		−	0.00660268i
$597$	−23131.4			−0.0649012
$598$		−	326813.i		−	0.913898i
$599$	−259778.			−0.724017			−0.362008	−	0.932175i	$-0.617909\pi$
$599$	−259778.			−0.724017			−0.362008	+	0.932175i	$0.617909\pi$
$600$			116730.i			0.324251i
$601$		−	609538.i		−	1.68753i	−0.536713	−	0.843765i	$-0.680334\pi$
$601$		−	609538.i		−	1.68753i	0.536713	−	0.843765i	$-0.319666\pi$
$602$	567286.			1.56534
$603$		−	23420.6i		−	0.0644115i
$604$			15956.1i			0.0437373i
$605$	−229485.			−0.626966
$606$	142218.			0.387266
$607$	123432.			0.335005			0.167502	−	0.985872i	$-0.446430\pi$
$607$	123432.			0.335005			0.167502	+	0.985872i	$0.446430\pi$
$608$			29277.8i			0.0792012i
$609$	339588.			0.915625
$610$	284112.			0.763537
$611$	−456372.			−1.22246
$612$	−4862.13			−0.0129815
$613$		−	182402.i		−	0.485409i	−0.970100	−	0.242705i	$-0.921965\pi$
$613$		−	182402.i		−	0.485409i	0.970100	−	0.242705i	$-0.0780346\pi$
$614$		−	63035.2i		−	0.167204i
$615$	87795.8			0.232126
$616$	−76600.4			−0.201869
$617$	550891.			1.44709			0.723544	−	0.690278i	$-0.242512\pi$
$617$	550891.			1.44709			0.723544	+	0.690278i	$0.242512\pi$
$618$	420737.			1.10162
$619$	172751.			0.450856			0.225428	−	0.974260i	$-0.427622\pi$
$619$	172751.			0.450856			0.225428	+	0.974260i	$0.427622\pi$
$620$		−	6366.96i		−	0.0165634i
$621$		−	107202.i		−	0.277984i
$622$		−	651966.i		−	1.68517i
$623$			296312.i			0.763438i
$624$			143945.i			0.369681i
$625$	−41803.1			−0.107016
$626$	−323361.			−0.825160
$627$			74687.4i			0.189982i
$628$		−	406.149i		−	0.00102983i
$629$			279111.i			0.705464i
$630$		−	84919.0i		−	0.213956i
$631$	−6939.21			−0.0174282			−0.00871408	−	0.999962i	$-0.502774\pi$
$631$	−6939.21			−0.0174282			−0.00871408	+	0.999962i	$0.502774\pi$
$632$			563132.i			1.40986i
$633$		−	222563.i		−	0.555452i
$634$			438611.i			1.09119i
$635$	−61384.7			−0.152234
$636$	−4473.57			−0.0110596
$637$		−	17148.8i		−	0.0422626i
$638$	143425.			0.352358
$639$	−6386.80			−0.0156416
$640$			278696.i			0.680410i
$641$	−213083.			−0.518601			−0.259300	−	0.965797i	$-0.583492\pi$
$641$	−213083.			−0.518601			−0.259300	+	0.965797i	$0.583492\pi$
$642$		−	300139.i		−	0.728202i
$643$	−323868.			−0.783333			−0.391666	−	0.920107i	$-0.628101\pi$
$643$	−323868.			−0.783333			−0.391666	+	0.920107i	$0.628101\pi$
$644$		−	14751.2i		−	0.0355678i
$645$		−	252403.i		−	0.606702i
$646$		−	1.00244e6i		−	2.40211i
$647$	−268558.			−0.641548			−0.320774	−	0.947156i	$-0.603943\pi$
$647$	−268558.			−0.641548			−0.320774	+	0.947156i	$0.603943\pi$
$648$			46042.3i			0.109650i
$649$	−12733.4	−	88318.4i	−0.0302311	−	0.209682i
$650$	152130.			0.360070
$651$			233771.i			0.551607i
$652$	20961.5			0.0493091
$653$	−43918.8			−0.102997			−0.0514985	−	0.998673i	$-0.516400\pi$
$653$	−43918.8			−0.102997			−0.0514985	+	0.998673i	$0.516400\pi$
$654$	4617.82			0.0107965
$655$			360658.i			0.840646i
$656$	−270126.			−0.627710
$657$		−	167704.i		−	0.388519i
$658$	−828361.			−1.91323
$659$		−	246753.i		−	0.568187i	−0.958797	−	0.284093i	$-0.908307\pi$
$659$		−	246753.i		−	0.568187i	0.958797	−	0.284093i	$-0.0916925\pi$
$660$		−	891.880i		−	0.00204747i
$661$	91190.3			0.208711			0.104356	−	0.994540i	$-0.466722\pi$
$661$	91190.3			0.208711			0.104356	+	0.994540i	$0.466722\pi$
$662$			504032.i			1.15012i
$663$		−	242145.i		−	0.550869i
$664$	−686175.			−1.55632
$665$	435375.			0.984510
$666$	−69165.4			−0.155934
$667$		−	1.05546e6i		−	2.37241i
$668$	7166.24			0.0160597
$669$	76778.8			0.171549
$670$	−57661.8			−0.128451
$671$	−109560.			−0.243335
$672$			12836.8i			0.0284262i
$673$		−	377278.i		−	0.832973i	−0.909142	−	0.416486i	$-0.863261\pi$
$673$		−	377278.i		−	0.832973i	0.909142	−	0.416486i	$-0.136739\pi$
$674$	−359202.			−0.790713
$675$	49902.0			0.109524
$676$	−7104.57			−0.0155469
$677$	301726.			0.658317			0.329159	−	0.944275i	$-0.393235\pi$
$677$	301726.			0.658317			0.329159	+	0.944275i	$0.393235\pi$
$678$	−265560.			−0.577701
$679$			373540.i			0.810210i
$680$		−	457441.i		−	0.989275i
$681$			193596.i			0.417447i
$682$			98733.7i			0.212274i
$683$			44408.5i			0.0951974i	0.998867	+	0.0475987i	$0.0151569\pi$
$683$			44408.5i			0.0951974i	−0.998867	+	0.0475987i	$0.984843\pi$
$684$	6177.29			0.0132034
$685$	133638.			0.284806
$686$		−	491285.i		−	1.04396i
$687$		−	189083.i		−	0.400627i
$688$			776583.i			1.64063i
$689$		−	222794.i		−	0.469315i
$690$	−263933.			−0.554365
$691$			428913.i			0.898284i	0.893460	+	0.449142i	$0.148270\pi$
$691$			428913.i			0.898284i	−0.893460	+	0.449142i	$0.851730\pi$
$692$			24372.0i			0.0508954i
$693$			32746.5i			0.0681866i
$694$	−226157.			−0.469560
$695$	−221687.			−0.458955
$696$			453309.i			0.935785i
$697$	454407.			0.935362
$698$	−268208.			−0.550504
$699$		−	107220.i		−	0.219444i
$700$	6866.61			0.0140135
$701$		−	142828.i		−	0.290654i	−0.989384	−	0.145327i	$-0.953577\pi$
$701$		−	142828.i		−	0.290654i	0.989384	−	0.145327i	$-0.0464234\pi$
$702$	60005.0			0.121762
$703$		−	354607.i		−	0.717525i
$704$		−	102184.i		−	0.206176i
$705$			368564.i			0.741540i
$706$	−483269.			−0.969570
$707$			319692.i			0.639577i
$708$	−7304.69	+	1053.16i	−0.0145725	+	0.00210101i
$709$	361262.			0.718670			0.359335	−	0.933209i	$-0.383004\pi$
$709$	361262.			0.718670			0.359335	+	0.933209i	$0.383004\pi$
$710$			15724.4i			0.0311930i
$711$	240738.			0.476217
$712$	−395542.			−0.780247
$713$	726575.			1.42923
$714$		−	439518.i		−	0.862144i
$715$	44417.6			0.0868845
$716$			21420.8i			0.0417839i
$717$	95101.0			0.184989
$718$		−	666305.i		−	1.29248i
$719$			364020.i			0.704154i	0.935971	+	0.352077i	$0.114525\pi$
$719$			364020.i			0.704154i	−0.935971	+	0.352077i	$0.885475\pi$
$720$	116249.			0.224246
$721$			945774.i			1.81935i
$722$			745700.i			1.43051i
$723$	−380420.			−0.727758
$724$	408.588			0.000779487
$725$	491309.			0.934715
$726$		−	294333.i		−	0.558425i
$727$	295247.			0.558619			0.279310	−	0.960201i	$-0.409894\pi$
$727$	295247.			0.558619			0.279310	+	0.960201i	$0.409894\pi$
$728$	−315522.			−0.595343
$729$	19683.0			0.0370370
$730$	−412889.			−0.774797
$731$		−	1.30637e6i		−	2.44473i
$732$			9061.52i			0.0169114i
$733$	203200.			0.378195			0.189097	−	0.981958i	$-0.439444\pi$
$733$	203200.			0.378195			0.189097	+	0.981958i	$0.439444\pi$
$734$	43025.3			0.0798605
$735$	−13849.3			−0.0256362
$736$	39897.6			0.0736531
$737$	22235.6			0.0409368
$738$			112605.i			0.206750i
$739$		−	1.00932e6i		−	1.84816i	−0.382195	−	0.924082i	$-0.624832\pi$
$739$		−	1.00932e6i		−	1.84816i	0.382195	−	0.924082i	$-0.375168\pi$
$740$			4234.54i			0.00773291i
$741$			307642.i			0.560286i
$742$		−	404393.i		−	0.734508i
$743$	−535342.			−0.969737			−0.484869	−	0.874587i	$-0.661133\pi$
$743$	−535342.			−0.969737			−0.484869	+	0.874587i	$0.661133\pi$
$744$	−312057.			−0.563752
$745$		−	94331.2i		−	0.169958i
$746$			202488.i			0.363849i
$747$			293338.i			0.525688i
$748$		−	4616.13i		−	0.00825039i
$749$	674682.			1.20264
$750$		−	338741.i		−	0.602206i
$751$		−	808120.i		−	1.43283i	−0.697672	−	0.716417i	$-0.745781\pi$
$751$		−	808120.i		−	1.43283i	0.697672	−	0.716417i	$-0.254219\pi$
$752$		−	1.13398e6i		−	2.00526i
$753$	−345507.			−0.609350
$754$	590779.			1.03916
$755$			641754.i			1.12583i
$756$	2708.42			0.00473885
$757$	−597248.			−1.04223			−0.521114	−	0.853487i	$-0.674483\pi$
$757$	−597248.			−1.04223			−0.521114	+	0.853487i	$0.674483\pi$
$758$		−	93300.0i		−	0.162384i
$759$	101778.			0.176673
$760$			581174.i			1.00619i
$761$	337522.			0.582818			0.291409	−	0.956599i	$-0.405876\pi$
$761$	337522.			0.582818			0.291409	+	0.956599i	$0.405876\pi$
$762$		−	78730.7i		−	0.135592i
$763$			10380.4i			0.0178305i
$764$		−	26876.8i		−	0.0460459i
$765$	−195555.			−0.334154
$766$		−	143873.i		−	0.245201i
$767$	−52449.7	−	363790.i	−0.0891564	−	0.618386i
$768$	−26035.5			−0.0441412
$769$		−	135631.i		−	0.229354i	−0.993403	−	0.114677i	$-0.963417\pi$
$769$		−	135631.i		−	0.229354i	0.993403	−	0.114677i	$-0.0365833\pi$
$770$	80622.4			0.135980
$771$	249429.			0.419603
$772$	−19076.1			−0.0320078
$773$		−	320257.i		−	0.535969i	−0.963423	−	0.267985i	$-0.913642\pi$
$773$		−	320257.i		−	0.535969i	0.963423	−	0.267985i	$-0.0863576\pi$
$774$	323727.			0.540377
$775$			338216.i			0.563107i
$776$	−498631.			−0.828049
$777$		−	155477.i		−	0.257528i
$778$			637723.i			1.05359i
$779$	−577320.			−0.951353
$780$		−	3673.71i		−	0.00603832i
$781$		−	6063.65i		−	0.00994105i
$782$	−1.36605e6			−2.23384
$783$	193789.			0.316086
$784$	42611.0			0.0693249
$785$		−	16335.3i		−	0.0265087i
$786$	−462573.			−0.748746
$787$	−584994.			−0.944500			−0.472250	−	0.881465i	$-0.656558\pi$
$787$	−584994.			−0.944500			−0.472250	+	0.881465i	$0.656558\pi$
$788$	−23024.5			−0.0370799

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:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	177.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+4.05068i q^{2} -5.19615 q^{3} -0.408022 q^{4} -16.4107 q^{5} -21.0480i q^{6} +47.3137 q^{7} +63.1581i q^{8} +27.0000 q^{9} +O(q^{10})\)	\(q+4.05068i q^{2} -5.19615 q^{3} -0.408022 q^{4} -16.4107 q^{5} -21.0480i q^{6} +47.3137 q^{7} +63.1581i q^{8} +27.0000 q^{9} -66.4744i q^{10} +25.6339i q^{11} +2.12014 q^{12} +105.588i q^{13} +191.653i q^{14} +85.2723 q^{15} -262.362 q^{16} +441.346 q^{17} +109.368i q^{18} -560.726 q^{19} +6.69591 q^{20} -245.849 q^{21} -103.835 q^{22} +764.114i q^{23} -328.179i q^{24} -355.690 q^{25} -427.703 q^{26} -140.296 q^{27} -19.3050 q^{28} -1381.28 q^{29} +345.411i q^{30} -950.873i q^{31} -52.2142i q^{32} -133.198i q^{33} +1787.75i q^{34} -776.449 q^{35} -11.0166 q^{36} +632.407i q^{37} -2271.32i q^{38} -548.650i q^{39} -1036.47i q^{40} +1029.59 q^{41} -995.857i q^{42} -2959.97i q^{43} -10.4592i q^{44} -443.088 q^{45} -3095.18 q^{46} +4322.20i q^{47} +1363.27 q^{48} -162.413 q^{49} -1440.79i q^{50} -2293.30 q^{51} -43.0822i q^{52} -2110.03 q^{53} -568.295i q^{54} -420.669i q^{55} +2988.25i q^{56} +2913.62 q^{57} -5595.14i q^{58} +(-3445.38 + 496.740i) q^{59} -34.7930 q^{60} +4274.01i q^{61} +3851.68 q^{62} +1277.47 q^{63} -3986.29 q^{64} -1732.77i q^{65} +539.541 q^{66} -867.430i q^{67} -180.079 q^{68} -3970.45i q^{69} -3145.15i q^{70} -236.548 q^{71} +1705.27i q^{72} -6211.26i q^{73} -2561.68 q^{74} +1848.22 q^{75} +228.788 q^{76} +1212.84i q^{77} +2222.41 q^{78} +8916.22 q^{79} +4305.53 q^{80} +729.000 q^{81} +4170.56i q^{82} +10864.4i q^{83} +100.312 q^{84} -7242.78 q^{85} +11989.9 q^{86} +7177.37 q^{87} -1618.99 q^{88} +6262.72i q^{89} -1794.81i q^{90} +4995.75i q^{91} -311.775i q^{92} +4940.88i q^{93} -17507.9 q^{94} +9201.88 q^{95} +271.313i q^{96} +7894.97i q^{97} -657.884i q^{98} +692.115i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40q - 320q^{4} + 80q^{7} + 1080q^{9} + O(q^{10}) \)	\( 40q - 320q^{4} + 80q^{7} + 1080q^{9} + 360q^{12} + 144q^{15} + 3944q^{16} - 528q^{17} + 444q^{19} + 444q^{20} + 1304q^{22} + 4880q^{25} - 1452q^{26} - 1160q^{28} - 996q^{29} + 10320q^{35} - 8640q^{36} - 5196q^{41} - 10476q^{46} + 576q^{48} + 5104q^{49} + 936q^{51} - 2184q^{53} - 2520q^{57} - 11736q^{59} - 11448q^{60} + 15240q^{62} + 2160q^{63} - 81012q^{64} + 17352q^{66} + 29568q^{68} - 5964q^{71} + 14376q^{74} - 2736q^{75} + 3480q^{76} + 37692q^{78} + 19020q^{79} + 33096q^{80} + 29160q^{81} + 25128q^{84} + 20220q^{85} - 65880q^{86} + 1512q^{87} - 14932q^{88} - 17864q^{94} + 11004q^{95} + O(q^{100}) \)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Properties

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