LMFDB - Embedded newform 177.7.c.a.58.5

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			58.5
Character	$\chi$	$=$	177.58
Dual form			177.7.c.a.58.56

$q$-expansion

$f(q)$	$=$	$q-13.8068i q^{2} +15.5885 q^{3} -126.629 q^{4} +76.4303 q^{5} -215.227i q^{6} -5.71721 q^{7} +864.707i q^{8} +243.000 q^{9} +O(q^{10})$	\(q-13.8068i q^{2} +15.5885 q^{3} -126.629 q^{4} +76.4303 q^{5} -215.227i q^{6} -5.71721 q^{7} +864.707i q^{8} +243.000 q^{9} -1055.26i q^{10} -701.846i q^{11} -1973.95 q^{12} -170.934i q^{13} +78.9366i q^{14} +1191.43 q^{15} +3834.62 q^{16} -6750.64 q^{17} -3355.06i q^{18} +1487.36 q^{19} -9678.28 q^{20} -89.1224 q^{21} -9690.27 q^{22} -23106.5i q^{23} +13479.4i q^{24} -9783.41 q^{25} -2360.05 q^{26} +3788.00 q^{27} +723.963 q^{28} -30946.9 q^{29} -16449.9i q^{30} +16180.0i q^{31} +2397.24i q^{32} -10940.7i q^{33} +93205.0i q^{34} -436.968 q^{35} -30770.8 q^{36} +66868.3i q^{37} -20535.8i q^{38} -2664.59i q^{39} +66089.8i q^{40} -50288.8 q^{41} +1230.50i q^{42} -62873.3i q^{43} +88873.9i q^{44} +18572.6 q^{45} -319028. q^{46} -73060.6i q^{47} +59775.8 q^{48} -117616. q^{49} +135078. i q^{50} -105232. q^{51} +21645.1i q^{52} -3796.98 q^{53} -52300.2i q^{54} -53642.3i q^{55} -4943.71i q^{56} +23185.7 q^{57} +427279. i q^{58} +(-24789.9 - 203877. i) q^{59} -150870. q^{60} +275330. i q^{61} +223395. q^{62} -1389.28 q^{63} +278514. q^{64} -13064.5i q^{65} -151056. q^{66} +22908.3i q^{67} +854826. q^{68} -360195. i q^{69} +6033.15i q^{70} +405836. q^{71} +210124. i q^{72} -156729. i q^{73} +923240. q^{74} -152508. q^{75} -188343. q^{76} +4012.60i q^{77} -36789.6 q^{78} +412146. q^{79} +293081. q^{80} +59049.0 q^{81} +694330. i q^{82} -95524.7i q^{83} +11285.5 q^{84} -515954. q^{85} -868082. q^{86} -482415. q^{87} +606891. q^{88} -187842. i q^{89} -256428. i q^{90} +977.263i q^{91} +2.92595e6i q^{92} +252222. i q^{93} -1.00874e6 q^{94} +113680. q^{95} +37369.2i q^{96} -706432. i q^{97} +1.62391e6i q^{98} -170548. i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 60q - 1920q^{4} - 408q^{7} + 14580q^{9} + O(q^{10}) $	$ 60q - 1920q^{4} - 408q^{7} + 14580q^{9} - 1944q^{12} - 4536q^{15} + 56616q^{16} + 8480q^{17} + 11376q^{19} + 40796q^{20} - 8232q^{22} + 197940q^{25} + 147252q^{26} + 71640q^{28} + 63456q^{29} - 364432q^{35} - 466560q^{36} + 99632q^{41} - 470316q^{46} + 171072q^{48} + 1737420q^{49} + 60912q^{51} + 92240q^{53} + 186624q^{57} + 917264q^{59} + 1063368q^{60} - 115768q^{62} - 99144q^{63} - 1107444q^{64} + 1172232q^{66} - 4247232q^{68} + 1498048q^{71} + 1161448q^{74} - 1477440q^{75} - 1045320q^{76} - 1060452q^{78} - 90600q^{79} + 77096q^{80} + 3542940q^{81} - 2225880q^{84} - 693408q^{85} - 1567768q^{86} + 1821528q^{87} + 62892q^{88} + 5268696q^{94} + 296128q^{95} + O(q^{100}) $

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/177\mathbb{Z}\right)^\times$.

$n$	$61$	$119$
$\chi(n)$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		−	13.8068i		−	1.72586i	−0.505328	−	0.862928i	$-0.668628\pi$
$2$		−	13.8068i		−	1.72586i	0.505328	−	0.862928i	$-0.331372\pi$
$3$	15.5885			0.577350
$3$	15.5885			0.577350
$4$	−126.629			−1.97858
$5$	76.4303			0.611443			0.305721	−	0.952121i	$-0.401102\pi$
$5$	76.4303			0.611443			0.305721	+	0.952121i	$0.401102\pi$
$6$		−	215.227i		−	0.996423i
$7$	−5.71721			−0.0166682			−0.00833412	−	0.999965i	$-0.502653\pi$
$7$	−5.71721			−0.0166682			−0.00833412	+	0.999965i	$0.502653\pi$
$8$			864.707i			1.68888i
$9$	243.000			0.333333
$10$		−	1055.26i		−	1.05526i
$11$		−	701.846i		−	0.527307i	−0.964617	−	0.263653i	$-0.915072\pi$
$11$		−	701.846i		−	0.527307i	0.964617	−	0.263653i	$-0.0849275\pi$
$12$	−1973.95			−1.14233
$13$		−	170.934i		−	0.0778032i	−0.999243	−	0.0389016i	$-0.987614\pi$
$13$		−	170.934i		−	0.0778032i	0.999243	−	0.0389016i	$-0.0123859\pi$
$14$			78.9366i			0.0287670i
$15$	1191.43			0.353017
$16$	3834.62			0.936187
$17$	−6750.64			−1.37404			−0.687018	−	0.726640i	$-0.741081\pi$
$17$	−6750.64			−1.37404			−0.687018	+	0.726640i	$0.741081\pi$
$18$		−	3355.06i		−	0.575285i
$19$	1487.36			0.216849			0.108424	−	0.994105i	$-0.465419\pi$
$19$	1487.36			0.216849			0.108424	+	0.994105i	$0.465419\pi$
$20$	−9678.28			−1.20979
$21$	−89.1224			−0.00962341
$22$	−9690.27			−0.910055
$23$		−	23106.5i		−	1.89911i	−0.313593	−	0.949557i	$-0.601533\pi$
$23$		−	23106.5i		−	1.89911i	0.313593	−	0.949557i	$-0.398467\pi$
$24$			13479.4i			0.975075i
$25$	−9783.41			−0.626138
$26$	−2360.05			−0.134277
$27$	3788.00			0.192450
$28$	723.963			0.0329794
$29$	−30946.9			−1.26889			−0.634444	−	0.772969i	$-0.718771\pi$
$29$	−30946.9			−1.26889			−0.634444	+	0.772969i	$0.718771\pi$
$30$		−	16449.9i		−	0.609255i
$31$			16180.0i			0.543118i	0.962422	+	0.271559i	$0.0875392\pi$
$31$			16180.0i			0.543118i	−0.962422	+	0.271559i	$0.912461\pi$
$32$			2397.24i			0.0731578i
$33$		−	10940.7i		−	0.304441i
$34$			93205.0i			2.37139i
$35$	−436.968			−0.0101917
$36$	−30770.8			−0.659525
$37$			66868.3i			1.32013i	0.751211	+	0.660063i	$0.229470\pi$
$37$			66868.3i			1.32013i	−0.751211	+	0.660063i	$0.770530\pi$
$38$		−	20535.8i		−	0.374249i
$39$		−	2664.59i		−	0.0449197i
$40$			66089.8i			1.03265i
$41$	−50288.8			−0.729659			−0.364829	−	0.931074i	$-0.618873\pi$
$41$	−50288.8			−0.729659			−0.364829	+	0.931074i	$0.618873\pi$
$42$			1230.50i			0.0166086i
$43$		−	62873.3i		−	0.790790i	−0.918511	−	0.395395i	$-0.870608\pi$
$43$		−	62873.3i		−	0.790790i	0.918511	−	0.395395i	$-0.129392\pi$
$44$			88873.9i			1.04332i
$45$	18572.6			0.203814
$46$	−319028.			−3.27760
$47$		−	73060.6i		−	0.703703i	−0.936056	−	0.351852i	$-0.885552\pi$
$47$		−	73060.6i		−	0.703703i	0.936056	−	0.351852i	$-0.114448\pi$
$48$	59775.8			0.540508
$49$	−117616.			−0.999722
$50$			135078.i			1.08062i
$51$	−105232.			−0.793300
$52$			21645.1i			0.153940i
$53$	−3796.98			−0.0255042			−0.0127521	−	0.999919i	$-0.504059\pi$
$53$	−3796.98			−0.0255042			−0.0127521	+	0.999919i	$0.504059\pi$
$54$		−	52300.2i		−	0.332141i
$55$		−	53642.3i		−	0.322418i
$56$		−	4943.71i		−	0.0281507i
$57$	23185.7			0.125198
$58$			427279.i			2.18992i
$59$	−24789.9	−	203877.i	−0.120703	−	0.992689i
$59$	−24789.9	−	203877.i	−0.120703	−	0.992689i
$60$	−150870.			−0.698470
$61$			275330.i			1.21301i	0.795080	+	0.606504i	$0.207429\pi$
$61$			275330.i			1.21301i	−0.795080	+	0.606504i	$0.792571\pi$
$62$	223395.			0.937343
$63$	−1389.28			−0.00555608
$64$	278514.			1.06245
$65$		−	13064.5i		−	0.0475722i
$66$	−151056.			−0.525421
$67$			22908.3i			0.0761671i	0.999275	+	0.0380836i	$0.0121253\pi$
$67$			22908.3i			0.0761671i	−0.999275	+	0.0380836i	$0.987875\pi$
$68$	854826.			2.71864
$69$		−	360195.i		−	1.09645i
$70$			6033.15i			0.0175893i
$71$	405836.			1.13390			0.566951	−	0.823751i	$-0.308123\pi$
$71$	405836.			1.13390			0.566951	+	0.823751i	$0.308123\pi$
$72$			210124.i			0.562960i
$73$		−	156729.i		−	0.402885i	−0.979500	−	0.201442i	$-0.935437\pi$
$73$		−	156729.i		−	0.402885i	0.979500	−	0.201442i	$-0.0645629\pi$
$74$	923240.			2.27834
$75$	−152508.			−0.361501
$76$	−188343.			−0.429052
$77$			4012.60i			0.00878928i
$78$	−36789.6			−0.0775249
$79$	412146.			0.835930			0.417965	−	0.908463i	$-0.362743\pi$
$79$	412146.			0.835930			0.417965	+	0.908463i	$0.362743\pi$
$80$	293081.			0.572424
$81$	59049.0			0.111111
$82$			694330.i			1.25929i
$83$		−	95524.7i		−	0.167063i	−0.996505	−	0.0835317i	$-0.973380\pi$
$83$		−	95524.7i		−	0.167063i	0.996505	−	0.0835317i	$-0.0266200\pi$
$84$	11285.5			0.0190407
$85$	−515954.			−0.840145
$86$	−868082.			−1.36479
$87$	−482415.			−0.732593
$88$	606891.			0.890558
$89$		−	187842.i		−	0.266454i	−0.991086	−	0.133227i	$-0.957466\pi$
$89$		−	187842.i		−	0.266454i	0.991086	−	0.133227i	$-0.0425339\pi$
$90$		−	256428.i		−	0.351754i
$91$			977.263i			0.00129684i
$92$			2.92595e6i			3.75754i
$93$			252222.i			0.313569i
$94$	−1.00874e6			−1.21449
$95$	113680.			0.132590
$96$			37369.2i			0.0422377i
$97$		−	706432.i		−	0.774025i	−0.922074	−	0.387012i	$-0.873507\pi$
$97$		−	706432.i		−	0.774025i	0.922074	−	0.387012i	$-0.126493\pi$
$98$			1.62391e6i			1.72538i
$99$		−	170548.i		−	0.175769i
$100$	1.23886e6			1.23886
$101$			1.02971e6i			0.999431i	0.866190	+	0.499715i	$0.166562\pi$
$101$			1.02971e6i			0.999431i	−0.866190	+	0.499715i	$0.833438\pi$
$102$			1.45292e6i			1.36912i
$103$			720650.i			0.659497i	0.944069	+	0.329749i	$0.106964\pi$
$103$			720650.i			0.659497i	−0.944069	+	0.329749i	$0.893036\pi$
$104$	147807.			0.131400
$105$	−6811.66			−0.00588416
$106$			52424.3i			0.0440165i
$107$	−159709.			−0.130370			−0.0651850	−	0.997873i	$-0.520764\pi$
$107$	−159709.			−0.130370			−0.0651850	+	0.997873i	$0.520764\pi$
$108$	−479669.			−0.380777
$109$		−	1.78282e6i		−	1.37666i	−0.725395	−	0.688332i	$-0.758343\pi$
$109$		−	1.78282e6i		−	1.37666i	0.725395	−	0.688332i	$-0.241657\pi$
$110$	−740630.			−0.556447
$111$			1.04237e6i			0.762175i
$112$	−21923.3			−0.0156046
$113$			446781.i			0.309642i	0.987943	+	0.154821i	$0.0494800\pi$
$113$			446781.i			0.309642i	−0.987943	+	0.154821i	$0.950520\pi$
$114$		−	320122.i		−	0.216073i
$115$		−	1.76604e6i		−	1.16120i
$116$	3.91877e6			2.51059
$117$		−	41536.9i		−	0.0259344i
$118$	−2.81490e6	+	342271.i	−1.71324	+	0.208317i
$119$	38594.8			0.0229028
$120$			1.03024e6i			0.596203i
$121$	1.27897e6			0.721947
$122$	3.80143e6			2.09348
$123$	−783925.			−0.421269
$124$		−	2.04886e6i		−	1.07460i
$125$	−1.94197e6			−0.994290
$126$			19181.6i			0.00958899i
$127$	−3.17988e6			−1.55239			−0.776194	−	0.630494i	$-0.782852\pi$
$127$	−3.17988e6			−1.55239			−0.776194	+	0.630494i	$0.782852\pi$
$128$		−	3.69197e6i		−	1.76047i
$129$		−	980098.i		−	0.456563i
$130$	−180380.			−0.0821027
$131$			585371.i			0.260386i	0.991489	+	0.130193i	$0.0415597\pi$
$131$			585371.i			0.260386i	−0.991489	+	0.130193i	$0.958440\pi$
$132$			1.38541e6i			0.602359i
$133$	−8503.57			−0.00361449
$134$	316291.			0.131453
$135$	289518.			0.117672
$136$		−	5.83733e6i		−	2.32058i
$137$	−1.59790e6			−0.621425			−0.310713	−	0.950504i	$-0.600568\pi$
$137$	−1.59790e6			−0.621425			−0.310713	+	0.950504i	$0.600568\pi$
$138$	−4.97316e6			−1.89232
$139$	−3.19302e6			−1.18893			−0.594466	−	0.804121i	$-0.702637\pi$
$139$	−3.19302e6			−1.18893			−0.594466	+	0.804121i	$0.702637\pi$
$140$	55332.7			0.0201650
$141$		−	1.13890e6i		−	0.406283i
$142$		−	5.60332e6i		−	1.95695i
$143$	−119969.			−0.0410262
$144$	931813.			0.312062
$145$	−2.36528e6			−0.775852
$146$	−2.16393e6			−0.695321
$147$	−1.83346e6			−0.577190
$148$		−	8.46746e6i		−	2.61197i
$149$		−	2.29518e6i		−	0.693837i	−0.937895	−	0.346919i	$-0.887228\pi$
$149$		−	2.29518e6i		−	0.693837i	0.937895	−	0.346919i	$-0.112772\pi$
$150$			2.10566e6i			0.623898i
$151$		−	458534.i		−	0.133181i	−0.997780	−	0.0665903i	$-0.978788\pi$
$151$		−	458534.i		−	0.133181i	0.997780	−	0.0665903i	$-0.0212121\pi$
$152$			1.28613e6i			0.366231i
$153$	−1.64041e6			−0.458012
$154$	55401.3			0.0151690
$155$			1.23665e6i			0.332086i
$156$			337414.i			0.0888770i
$157$		−	2.13679e6i		−	0.552157i	−0.961135	−	0.276078i	$-0.910965\pi$
$157$		−	2.13679e6i		−	0.552157i	0.961135	−	0.276078i	$-0.0890350\pi$
$158$		−	5.69043e6i		−	1.44269i
$159$	−59189.1			−0.0147248
$160$			183221.i			0.0447318i
$161$			132105.i			0.0316549i
$162$		−	815280.i		−	0.191762i
$163$	64149.9			0.0148127			0.00740633	−	0.999973i	$-0.497642\pi$
$163$	64149.9			0.0148127			0.00740633	+	0.999973i	$0.497642\pi$
$164$	6.36802e6			1.44369
$165$		−	836200.i		−	0.186148i
$166$	−1.31889e6			−0.288327
$167$	5.06205e6			1.08687			0.543434	−	0.839452i	$-0.317124\pi$
$167$	5.06205e6			1.08687			0.543434	+	0.839452i	$0.317124\pi$
$168$		−	77064.8i		−	0.0162528i
$169$	4.79759e6			0.993947
$170$			7.12369e6i			1.44997i
$171$	361430.			0.0722829
$172$			7.96158e6i			1.56464i
$173$		−	2.06522e6i		−	0.398867i	−0.979911	−	0.199434i	$-0.936090\pi$
$173$		−	2.06522e6i		−	0.398867i	0.979911	−	0.199434i	$-0.0639102\pi$
$174$			6.66062e6i			1.26435i
$175$	55933.8			0.0104366
$176$		−	2.69131e6i		−	0.493658i
$177$	−386437.	−	3.17813e6i	−0.0696881	−	0.573129i
$178$	−2.59350e6			−0.459860
$179$		−	4.01842e6i		−	0.700643i	−0.936630	−	0.350321i	$-0.886072\pi$
$179$		−	4.01842e6i		−	0.700643i	0.936630	−	0.350321i	$-0.113928\pi$
$180$	−2.35182e6			−0.403262
$181$	−5.34348e6			−0.901133			−0.450566	−	0.892743i	$-0.648778\pi$
$181$	−5.34348e6			−0.901133			−0.450566	+	0.892743i	$0.648778\pi$
$182$	13492.9			0.00223816
$183$			4.29197e6i			0.700331i
$184$	1.99804e7			3.20738
$185$			5.11077e6i			0.807181i
$186$	3.48239e6			0.541175
$187$			4.73791e6i			0.724539i
$188$			9.25158e6i			1.39233i
$189$	−21656.8			−0.00320780
$190$		−	1.56956e6i		−	0.228832i
$191$			901349.i			0.129358i	0.997906	+	0.0646789i	$0.0206023\pi$
$191$			901349.i			0.129358i	−0.997906	+	0.0646789i	$0.979398\pi$
$192$	4.34160e6			0.613404
$193$	−3.15087e6			−0.438287			−0.219144	−	0.975693i	$-0.570326\pi$
$193$	−3.15087e6			−0.438287			−0.219144	+	0.975693i	$0.570326\pi$
$194$	−9.75359e6			−1.33585
$195$		−	203656.i		−	0.0274658i
$196$	1.48936e7			1.97803
$197$	−5.44558e6			−0.712271			−0.356136	−	0.934434i	$-0.615906\pi$
$197$	−5.44558e6			−0.712271			−0.356136	+	0.934434i	$0.615906\pi$
$198$	−2.35474e6			−0.303352
$199$	−1.06370e7			−1.34977			−0.674884	−	0.737924i	$-0.735806\pi$
$199$	−1.06370e7			−1.34977			−0.674884	+	0.737924i	$0.735806\pi$
$200$		−	8.45978e6i		−	1.05747i
$201$			357104.i			0.0439751i
$202$	1.42171e7			1.72487
$203$	176930.			0.0211501
$204$	1.33254e7			1.56961
$205$	−3.84359e6			−0.446145
$206$	9.94990e6			1.13820
$207$		−	5.61489e6i		−	0.633038i
$208$		−	655465.i		−	0.0728383i
$209$		−	1.04390e6i		−	0.114346i
$210$			94047.4i			0.0101552i
$211$			2.58627e6i			0.275313i	0.990480	+	0.137656i	$0.0439570\pi$
$211$			2.58627e6i			0.275313i	−0.990480	+	0.137656i	$0.956043\pi$
$212$	480808.			0.0504619
$213$	6.32636e6			0.654659
$214$			2.20507e6i			0.225000i
$215$		−	4.80543e6i		−	0.483522i
$216$			3.27550e6i			0.325025i
$217$		−	92504.6i		−	0.00905282i
$218$	−2.46151e7			−2.37592
$219$		−	2.44316e6i		−	0.232606i
$220$			6.79266e6i			0.637928i
$221$			1.15391e6i			0.106904i
$222$	1.43919e7			1.31540
$223$	6.79409e6			0.612656			0.306328	−	0.951926i	$-0.400900\pi$
$223$	6.79409e6			0.612656			0.306328	+	0.951926i	$0.400900\pi$
$224$		−	13705.5i		−	0.00121941i
$225$	−2.37737e6			−0.208713
$226$	6.16863e6			0.534397
$227$		−	4.99197e6i		−	0.426771i	−0.976968	−	0.213385i	$-0.931551\pi$
$227$		−	4.99197e6i		−	0.426771i	0.976968	−	0.213385i	$-0.0684490\pi$
$228$	−2.93598e6			−0.247713
$229$		−	6.25405e6i		−	0.520781i	−0.965503	−	0.260391i	$-0.916149\pi$
$229$		−	6.25405e6i		−	0.520781i	0.965503	−	0.260391i	$-0.0838513\pi$
$230$	−2.43834e7			−2.00406
$231$			62550.2i			0.00507449i
$232$		−	2.67600e7i		−	2.14300i
$233$		−	1.77129e7i		−	1.40030i	−0.713994	−	0.700152i	$-0.753116\pi$
$233$		−	1.77129e7i		−	1.40030i	0.713994	−	0.700152i	$-0.246884\pi$
$234$	−573493.			−0.0447590
$235$		−	5.58404e6i		−	0.430274i
$236$	3.13912e6	+	2.58168e7i	0.238821	+	1.96411i
$237$	6.42472e6			0.482624
$238$		−	532873.i		−	0.0395269i
$239$	1.64830e7			1.20738			0.603689	−	0.797220i	$-0.293697\pi$
$239$	1.64830e7			1.20738			0.603689	+	0.797220i	$0.293697\pi$
$240$	4.56868e6			0.330489
$241$	2.38086e7			1.70092			0.850458	−	0.526043i	$-0.176325\pi$
$241$	2.38086e7			1.70092			0.850458	+	0.526043i	$0.176325\pi$
$242$		−	1.76586e7i		−	1.24598i
$243$	920483.			0.0641500
$244$		−	3.48647e7i		−	2.40003i
$245$	−8.98945e6			−0.611273
$246$			1.08235e7i			0.727049i
$247$		−	254241.i		−	0.0168715i
$248$	−1.39910e7			−0.917261
$249$		−	1.48908e6i		−	0.0964541i
$250$			2.68125e7i			1.71600i
$251$	1.81000e7			1.14461			0.572306	−	0.820040i	$-0.306049\pi$
$251$	1.81000e7			1.14461			0.572306	+	0.820040i	$0.306049\pi$
$252$	175923.			0.0109931
$253$	−1.62172e7			−1.00142
$254$			4.39042e7i			2.67920i
$255$	−8.04292e6			−0.485058
$256$	−3.31496e7			−1.97587
$257$	7.51102e6			0.442486			0.221243	−	0.975219i	$-0.428989\pi$
$257$	7.51102e6			0.442486			0.221243	+	0.975219i	$0.428989\pi$
$258$	−1.35321e7			−0.787961
$259$		−	382300.i		−	0.0220042i
$260$			1.65434e6i			0.0941252i
$261$	−7.52010e6			−0.422963
$262$	8.08212e6			0.449388
$263$	−3.00298e7			−1.65077			−0.825383	−	0.564573i	$-0.809041\pi$
$263$	−3.00298e7			−1.65077			−0.825383	+	0.564573i	$0.809041\pi$
$264$	9.46049e6			0.514164
$265$	−290205.			−0.0155943
$266$			117407.i			0.00623808i
$267$		−	2.92816e6i		−	0.153837i
$268$		−	2.90085e6i		−	0.150702i
$269$			9.50277e6i			0.488195i	0.969751	+	0.244097i	$0.0784917\pi$
$269$			9.50277e6i			0.488195i	−0.969751	+	0.244097i	$0.921508\pi$
$270$		−	3.99732e6i		−	0.203085i
$271$	−3.58711e6			−0.180234			−0.0901170	−	0.995931i	$-0.528724\pi$
$271$	−3.58711e6			−0.180234			−0.0901170	+	0.995931i	$0.528724\pi$
$272$	−2.58861e7			−1.28635
$273$			15234.0i			0.000748732i
$274$			2.20620e7i			1.07249i
$275$			6.86644e6i			0.330167i
$276$			4.56111e7i			2.16942i
$277$	1.06356e7			0.500406			0.250203	−	0.968193i	$-0.419503\pi$
$277$	1.06356e7			0.500406			0.250203	+	0.968193i	$0.419503\pi$
$278$			4.40855e7i			2.05192i
$279$			3.93175e6i			0.181039i
$280$		−	377849.i		−	0.0172125i
$281$	194792.			0.00877915			0.00438957	−	0.999990i	$-0.498603\pi$
$281$	194792.			0.00877915			0.00438957	+	0.999990i	$0.498603\pi$
$282$	−1.57246e7			−0.701186
$283$		−	4.00849e7i		−	1.76857i	−0.466952	−	0.884283i	$-0.654648\pi$
$283$		−	4.00849e7i		−	1.76857i	0.466952	−	0.884283i	$-0.345352\pi$
$284$	−5.13906e7			−2.24351
$285$	1.77209e6			0.0765512
$286$			1.65639e6i			0.0708052i
$287$	287512.			0.0121621
$288$			582528.i			0.0243859i
$289$	2.14336e7			0.887977
$290$			3.26571e7i			1.33901i
$291$		−	1.10122e7i		−	0.446883i
$292$			1.98464e7i			0.797138i
$293$	4.09477e7			1.62790			0.813949	−	0.580937i	$-0.197314\pi$
$293$	4.09477e7			1.62790			0.813949	+	0.580937i	$0.197314\pi$
$294$			2.53142e7i			0.996146i
$295$	−1.89470e6	−	1.55824e7i	−0.0738032	−	0.606972i
$296$	−5.78215e7			−2.22953
$297$		−	2.65859e6i		−	0.101480i
$298$	−3.16892e7			−1.19746
$299$	−3.94968e6			−0.147757
$300$	1.93119e7			0.715257
$301$			359460.i			0.0131811i
$302$	−6.33091e6			−0.229851
$303$			1.60517e7i			0.577022i
$304$	5.70348e6			0.203011
$305$			2.10435e7i			0.741685i
$306$			2.26488e7i			0.790463i
$307$	−1.52761e6			−0.0527955			−0.0263978	−	0.999652i	$-0.508404\pi$
$307$	−1.52761e6			−0.0527955			−0.0263978	+	0.999652i	$0.508404\pi$
$308$		−	508110.i		−	0.0173903i
$309$			1.12338e7i			0.380761i
$310$	1.70742e7			0.573132
$311$	7.04493e6			0.234205			0.117102	−	0.993120i	$-0.462639\pi$
$311$	7.04493e6			0.234205			0.117102	+	0.993120i	$0.462639\pi$
$312$	2.30409e6			0.0758640
$313$			1.84231e7i			0.600801i	0.953813	+	0.300400i	$0.0971203\pi$
$313$			1.84231e7i			0.600801i	−0.953813	+	0.300400i	$0.902880\pi$
$314$	−2.95023e7			−0.952943
$315$	−106183.			−0.00339722
$316$	−5.21896e7			−1.65395
$317$	3.56499e7			1.11913			0.559565	−	0.828786i	$-0.310968\pi$
$317$	3.56499e7			1.11913			0.559565	+	0.828786i	$0.310968\pi$
$318$			817215.i			0.0254129i
$319$			2.17199e7i			0.669093i
$320$	2.12869e7			0.649625
$321$	−2.48961e6			−0.0752691
$322$	1.82395e6			0.0546318
$323$	−1.00407e7			−0.297958
$324$	−7.47731e6			−0.219842
$325$			1.67231e6i			0.0487155i
$326$		−	885707.i		−	0.0255645i
$327$		−	2.77914e7i		−	0.794818i
$328$		−	4.34851e7i		−	1.23231i
$329$			417702.i			0.0117295i
$330$	−1.15453e7			−0.321265
$331$	4.97150e7			1.37089			0.685447	−	0.728123i	$-0.259607\pi$
$331$	4.97150e7			1.37089			0.685447	+	0.728123i	$0.259607\pi$
$332$			1.20962e7i			0.330548i
$333$			1.62490e7i			0.440042i
$334$		−	6.98909e7i		−	1.87578i
$335$			1.75088e6i			0.0465718i
$336$	−341751.			−0.00900931
$337$			3.81569e6i			0.0996973i	0.998757	+	0.0498486i	$0.0158739\pi$
$337$			3.81569e6i			0.0996973i	−0.998757	+	0.0498486i	$0.984126\pi$
$338$		−	6.62396e7i		−	1.71541i
$339$			6.96463e6i			0.178772i
$340$	6.53346e7			1.66229
$341$	1.13559e7			0.286390
$342$		−	4.99020e6i		−	0.124750i
$343$	1.34506e6			0.0333319
$344$	5.43670e7			1.33555
$345$		−	2.75298e7i		−	0.670419i
$346$	−2.85142e7			−0.688387
$347$		−	6.39758e7i		−	1.53118i	−0.643326	−	0.765592i	$-0.722446\pi$
$347$		−	6.39758e7i		−	1.53118i	0.643326	−	0.765592i	$-0.277554\pi$
$348$	6.10876e7			1.44949
$349$		−	5.89910e7i		−	1.38774i	−0.720098	−	0.693872i	$-0.755903\pi$
$349$		−	5.89910e7i		−	1.38774i	0.720098	−	0.693872i	$-0.244097\pi$
$350$		−	772268.i		−	0.0180121i
$351$		−	647496.i		−	0.0149732i
$352$	1.68249e6			0.0385766
$353$		−	3.35328e7i		−	0.762335i	−0.924506	−	0.381167i	$-0.875522\pi$
$353$		−	3.35328e7i		−	0.762335i	0.924506	−	0.381167i	$-0.124478\pi$
$354$	−4.38800e7	+	5.33547e6i	−0.989138	+	0.120272i
$355$	3.10182e7			0.693316
$356$			2.37862e7i			0.527199i
$357$	601634.			0.0132229
$358$	−5.54817e7			−1.20921
$359$	4.94040e7			1.06777			0.533886	−	0.845556i	$-0.320731\pi$
$359$	4.94040e7			1.06777			0.533886	+	0.845556i	$0.320731\pi$
$360$			1.60598e7i			0.344218i
$361$	−4.48336e7			−0.952977
$362$			7.37766e7i			1.55522i
$363$	1.99372e7			0.416816
$364$		−	123750.i		−	0.00256590i
$365$		−	1.19789e7i		−	0.246341i
$366$	5.92585e7			1.20867
$367$		−	4.35649e7i		−	0.881331i	−0.897671	−	0.440665i	$-0.854743\pi$
$367$		−	4.35649e7i		−	0.881331i	0.897671	−	0.440665i	$-0.145257\pi$
$368$		−	8.86048e7i		−	1.77793i
$369$	−1.22202e7			−0.243220
$370$	7.05635e7			1.39308
$371$	21708.1			0.000425110
$372$		−	3.19386e7i		−	0.620421i
$373$	−3.99825e7			−0.770448			−0.385224	−	0.922823i	$-0.625876\pi$
$373$	−3.99825e7			−0.770448			−0.385224	+	0.922823i	$0.625876\pi$
$374$	6.54156e7			1.25045
$375$	−3.02724e7			−0.574054
$376$	6.31760e7			1.18847
$377$			5.28987e6i			0.0987235i
$378$			299011.i			0.00553621i
$379$	−8.61848e6			−0.158312			−0.0791558	−	0.996862i	$-0.525222\pi$
$379$	−8.61848e6			−0.158312			−0.0791558	+	0.996862i	$0.525222\pi$
$380$	−1.43951e7			−0.262340
$381$	−4.95695e7			−0.896271
$382$	1.24448e7			0.223253
$383$	8.27357e7			1.47264			0.736320	−	0.676633i	$-0.236562\pi$
$383$	8.27357e7			1.47264			0.736320	+	0.676633i	$0.236562\pi$
$384$		−	5.75522e7i		−	1.01641i
$385$			306684.i			0.00537414i
$386$			4.35036e7i			0.756420i
$387$		−	1.52782e7i		−	0.263597i
$388$			8.94546e7i			1.53147i
$389$	6.58216e7			1.11820			0.559100	−	0.829100i	$-0.311147\pi$
$389$	6.58216e7			1.11820			0.559100	+	0.829100i	$0.311147\pi$
$390$	−2.81184e6			−0.0474020
$391$			1.55984e8i			2.60945i
$392$		−	1.01704e8i		−	1.68841i
$393$			9.12503e6i			0.150334i
$394$			7.51862e7i			1.22928i
$395$	3.15005e7			0.511123
$396$			2.15964e7i			0.347772i
$397$			1.83419e7i			0.293139i	0.989200	+	0.146570i	$0.0468232\pi$
$397$			1.83419e7i			0.293139i	−0.989200	+	0.146570i	$0.953177\pi$
$398$			1.46863e8i			2.32950i
$399$	−132558.			−0.00208682
$400$	−3.75156e7			−0.586182
$401$		−	9.06964e7i		−	1.40656i	−0.710915	−	0.703278i	$-0.751719\pi$
$401$		−	9.06964e7i		−	1.40656i	0.710915	−	0.703278i	$-0.248281\pi$
$402$	4.93048e6			0.0758947
$403$	2.76571e6			0.0422563
$404$		−	1.30392e8i		−	1.97745i
$405$	4.51313e6			0.0679381
$406$		−	2.44284e6i		−	0.0365021i
$407$	4.69312e7			0.696111
$408$		−	9.09949e7i		−	1.33979i
$409$		−	2.75611e7i		−	0.402835i	−0.979506	−	0.201417i	$-0.935445\pi$
$409$		−	2.75611e7i		−	0.402835i	0.979506	−	0.201417i	$-0.0645547\pi$
$410$			5.30678e7i			0.769981i
$411$	−2.49088e7			−0.358780
$412$		−	9.12551e7i		−	1.30487i
$413$	141729.	+	1.16561e6i	0.00201191	+	0.0165464i
$414$	−7.75238e7			−1.09253
$415$		−	7.30099e6i		−	0.102150i
$416$	409768.			0.00569191
$417$	−4.97742e7			−0.686430
$418$	−1.44130e7			−0.197344
$419$		−	1.36523e8i		−	1.85594i	−0.372656	−	0.927969i	$-0.621553\pi$
$419$		−	1.36523e8i		−	1.85594i	0.372656	−	0.927969i	$-0.378447\pi$
$420$	862552.			0.0116423
$421$			3.01688e7i			0.404308i	0.979354	+	0.202154i	$0.0647942\pi$
$421$			3.01688e7i			0.404308i	−0.979354	+	0.202154i	$0.935206\pi$
$422$	3.57082e7			0.475150
$423$		−	1.77537e7i		−	0.234568i
$424$		−	3.28328e6i		−	0.0430735i
$425$	6.60443e7			0.860337
$426$		−	8.73470e7i		−	1.12985i
$427$		−	1.57412e6i		−	0.0202187i
$428$	2.02237e7			0.257947
$429$	−1.87013e6			−0.0236865
$430$	−6.63478e7			−0.834490
$431$			1.17705e8i			1.47015i	0.677983	+	0.735077i	$0.262854\pi$
$431$			1.17705e8i			1.47015i	−0.677983	+	0.735077i	$0.737146\pi$
$432$	1.45255e7			0.180169
$433$	−7.96409e7			−0.981007			−0.490504	−	0.871439i	$-0.663187\pi$
$433$	−7.96409e7			−0.981007			−0.490504	+	0.871439i	$0.663187\pi$
$434$	−1.27720e6			−0.0156239
$435$	−3.68711e7			−0.447938
$436$			2.25757e8i			2.72384i
$437$		−	3.43678e7i		−	0.411821i
$438$	−3.37324e7			−0.401444
$439$	−1.60921e8			−1.90204			−0.951022	−	0.309125i	$-0.899964\pi$
$439$	−1.60921e8			−1.90204			−0.951022	+	0.309125i	$0.899964\pi$
$440$	4.63848e7			0.544525
$441$	−2.85808e7			−0.333241
$442$	1.59319e7			0.184502
$443$			8.84927e7i			1.01788i	0.860802	+	0.508940i	$0.169962\pi$
$443$			8.84927e7i			1.01788i	−0.860802	+	0.508940i	$0.830038\pi$
$444$		−	1.31995e8i		−	1.50802i
$445$		−	1.43568e7i		−	0.162921i
$446$		−	9.38049e7i		−	1.05736i
$447$		−	3.57783e7i		−	0.400587i
$448$	−1.59232e6			−0.0177091
$449$	4.69931e7			0.519153			0.259577	−	0.965723i	$-0.416417\pi$
$449$	4.69931e7			0.519153			0.259577	+	0.965723i	$0.416417\pi$
$450$			3.28239e7i			0.360208i
$451$			3.52950e7i			0.384754i
$452$		−	5.65754e7i		−	0.612649i
$453$		−	7.14785e6i		−	0.0768919i
$454$	−6.89234e7			−0.736544
$455$			74692.5i			0.000792945i
$456$			2.00488e7i			0.211444i
$457$			8.23265e7i			0.862563i	0.902217	+	0.431282i	$0.141938\pi$
$457$			8.23265e7i			0.862563i	−0.902217	+	0.431282i	$0.858062\pi$
$458$	−8.63487e7			−0.898793
$459$	−2.55714e7			−0.264433
$460$			2.23632e8i			2.29752i
$461$	3.09968e7			0.316384			0.158192	−	0.987408i	$-0.449434\pi$
$461$	3.09968e7			0.316384			0.158192	+	0.987408i	$0.449434\pi$
$462$	863620.			0.00875784
$463$		−	1.96462e8i		−	1.97941i	−0.143109	−	0.989707i	$-0.545710\pi$
$463$		−	1.96462e8i		−	1.97941i	0.143109	−	0.989707i	$-0.454290\pi$
$464$	−1.18670e8			−1.18792
$465$			1.92774e7i			0.191730i
$466$	−2.44559e8			−2.41672
$467$			6.29578e7i			0.618157i	0.951037	+	0.309078i	$0.100021\pi$
$467$			6.29578e7i			0.618157i	−0.951037	+	0.309078i	$0.899979\pi$
$468$			5.25977e6i			0.0513132i
$469$		−	130971.i		−	0.00126957i
$470$	−7.70980e7			−0.742591
$471$		−	3.33092e7i		−	0.318788i
$472$	1.76294e8	−	2.14360e7i	1.67653	−	0.203854i
$473$	−4.41274e7			−0.416989
$474$		−	8.87051e7i		−	0.832940i
$475$	−1.45515e7			−0.135777
$476$	−4.88722e6			−0.0453149
$477$	−922667.			−0.00850139
$478$		−	2.27578e8i		−	2.08376i
$479$	1.93655e8			1.76207			0.881034	−	0.473052i	$-0.156848\pi$
$479$	1.93655e8			1.76207			0.881034	+	0.473052i	$0.156848\pi$
$480$			2.85614e6i			0.0258259i
$481$	1.14300e7			0.102710
$482$		−	3.28722e8i		−	2.93554i
$483$			2.05931e6i			0.0182760i
$484$	−1.61955e8			−1.42843
$485$		−	5.39928e7i		−	0.473272i
$486$		−	1.27090e7i		−	0.110714i
$487$	−1.45856e8			−1.26281			−0.631403	−	0.775455i	$-0.717521\pi$
$487$	−1.45856e8			−1.26281			−0.631403	+	0.775455i	$0.717521\pi$
$488$	−2.38080e8			−2.04863
$489$	999998.			0.00855209
$490$			1.24116e8i			1.05497i
$491$	−1.59916e8			−1.35098			−0.675489	−	0.737370i	$-0.736068\pi$
$491$	−1.59916e8			−1.35098			−0.675489	+	0.737370i	$0.736068\pi$
$492$	9.92675e7			0.833512
$493$	2.08912e8			1.74350
$494$	−3.51026e6			−0.0291178
$495$		−	1.30351e7i		−	0.107473i
$496$			6.20443e7i			0.508460i
$497$	−2.32025e6			−0.0189002
$498$	−2.05595e7			−0.166466
$499$	6.15997e7			0.495766			0.247883	−	0.968790i	$-0.420265\pi$
$499$	6.15997e7			0.495766			0.247883	+	0.968790i	$0.420265\pi$
$500$	2.45910e8			1.96728
$501$	7.89095e7			0.627504
$502$		−	2.49904e8i		−	1.97543i
$503$		−	1.08230e8i		−	0.850444i	−0.905089	−	0.425222i	$-0.860196\pi$
$503$		−	1.08230e8i		−	0.850444i	0.905089	−	0.425222i	$-0.139804\pi$
$504$		−	1.20132e6i		−	0.00938355i
$505$			7.87014e7i			0.611095i
$506$			2.23909e8i			1.72830i
$507$	7.47870e7			0.573855
$508$	4.02665e8			3.07152
$509$			1.22346e8i			0.927761i	0.885898	+	0.463881i	$0.153543\pi$
$509$			1.22346e8i			0.927761i	−0.885898	+	0.463881i	$0.846457\pi$
$510$			1.11047e8i			0.837139i
$511$			896052.i			0.00671538i
$512$			2.21405e8i			1.64960i
$513$	5.63413e6			0.0417325
$514$		−	1.03703e8i		−	0.763667i
$515$			5.50795e7i			0.403245i
$516$			1.24109e8i			0.903344i
$517$	−5.12772e7			−0.371068
$518$	−5.27835e6			−0.0379760
$519$		−	3.21936e7i		−	0.230286i
$520$	1.12970e7			0.0803437
$521$	−1.00276e8			−0.709063			−0.354532	−	0.935044i	$-0.615360\pi$
$521$	−1.00276e8			−0.709063			−0.354532	+	0.935044i	$0.615360\pi$
$522$			1.03829e8i			0.729972i
$523$	−9.60867e7			−0.671673			−0.335837	−	0.941920i	$-0.609019\pi$
$523$	−9.60867e7			−0.671673			−0.335837	+	0.941920i	$0.609019\pi$
$524$		−	7.41249e7i		−	0.515193i
$525$	871921.			0.00602558
$526$			4.14617e8i			2.84898i
$527$		−	1.09226e8i		−	0.746264i
$528$		−	4.19534e7i		−	0.285013i
$529$	−3.85876e8			−2.60664
$530$			4.00681e6i			0.0269136i
$531$	−6.02396e6	−	4.95422e7i	−0.0402345	−	0.330896i
$532$	1.07680e6			0.00715153
$533$			8.59605e6i			0.0567698i
$534$	−4.04286e7			−0.265500
$535$	−1.22066e7			−0.0797137
$536$	−1.98089e7			−0.128637
$537$		−	6.26410e7i		−	0.404516i
$538$	1.31203e8			0.842554
$539$			8.25485e7i			0.527160i
$540$	−3.66613e7			−0.232823
$541$			2.95188e8i			1.86426i	0.362118	+	0.932132i	$0.382054\pi$
$541$			2.95188e8i			1.86426i	−0.362118	+	0.932132i	$0.617946\pi$
$542$			4.95267e7i			0.311058i
$543$	−8.32967e7			−0.520269
$544$		−	1.61829e7i		−	0.100522i
$545$		−	1.36262e8i		−	0.841752i
$546$	210334.			0.00129220
$547$	−6.18129e7			−0.377674			−0.188837	−	0.982008i	$-0.560472\pi$
$547$	−6.18129e7			−0.377674			−0.188837	+	0.982008i	$0.560472\pi$
$548$	2.02341e8			1.22954
$549$			6.69051e7i			0.404336i
$550$	9.48039e7			0.569820
$551$	−4.60293e7			−0.275157
$552$	3.11463e8			1.85178
$553$	−2.35632e6			−0.0139335
$554$		−	1.46844e8i		−	0.863628i
$555$			7.96689e7i			0.466026i
$556$	4.04328e8			2.35239
$557$	−6.46982e6			−0.0374392			−0.0187196	−	0.999825i	$-0.505959\pi$
$557$	−6.46982e6			−0.0374392			−0.0187196	+	0.999825i	$0.505959\pi$
$558$	5.42850e7			0.312448
$559$	−1.07472e7			−0.0615260
$560$	−1.67561e6			−0.00954131
$561$			7.38567e7i			0.418313i
$562$		−	2.68946e6i		−	0.0151515i
$563$			1.69126e8i			0.947733i	0.880597	+	0.473866i	$0.157142\pi$
$563$			1.69126e8i			0.947733i	−0.880597	+	0.473866i	$0.842858\pi$
$564$			1.44218e8i			0.803862i
$565$			3.41476e7i			0.189328i
$566$	−5.53445e8			−3.05229
$567$	−337595.			−0.00185203
$568$			3.50929e8i			1.91503i
$569$			2.47807e8i			1.34517i	0.740021	+	0.672584i	$0.234816\pi$
$569$			2.47807e8i			1.34517i	−0.740021	+	0.672584i	$0.765184\pi$
$570$		−	2.44670e7i		−	0.132116i
$571$		−	4.00196e7i		−	0.214963i	−0.994207	−	0.107482i	$-0.965721\pi$
$571$		−	4.00196e7i		−	0.214963i	0.994207	−	0.107482i	$-0.0342787\pi$
$572$	1.51915e7			0.0811734
$573$			1.40506e7i			0.0746848i
$574$		−	3.96963e6i		−	0.0209901i
$575$			2.26061e8i			1.18911i
$576$	6.76789e7			0.354149
$577$	2.75991e8			1.43670			0.718351	−	0.695681i	$-0.244897\pi$
$577$	2.75991e8			1.43670			0.718351	+	0.695681i	$0.244897\pi$
$578$		−	2.95930e8i		−	1.53252i
$579$	−4.91172e7			−0.253045
$580$	2.99513e8			1.53508
$581$			546135.i			0.00278465i
$582$	−1.52043e8			−0.771256
$583$			2.66490e6i			0.0134485i
$584$	1.35525e8			0.680424
$585$		−	3.17468e6i		−	0.0158574i
$586$		−	5.65359e8i		−	2.80952i
$587$			183611.i			0.000907790i	1.00000		0.000453895i	$0.000144479\pi$
$587$			183611.i			0.000907790i	−1.00000		0.000453895i	$0.999856\pi$
$588$	2.32169e8			1.14201
$589$			2.40656e7i			0.117774i
$590$	−2.15144e8	+	2.61599e7i	−1.04755	+	0.127374i
$591$	−8.48882e7			−0.411230
$592$			2.56415e8i			1.23588i
$593$	−3.23810e8			−1.55284			−0.776420	−	0.630216i	$-0.782966\pi$
$593$	−3.23810e8			−1.55284			−0.776420	+	0.630216i	$0.782966\pi$
$594$	−3.67067e7			−0.175140
$595$	2.94981e6			0.0140037
$596$			2.90636e8i			1.37281i
$597$	−1.65814e8			−0.779289
$598$			5.45326e7i			0.255008i
$599$	8.19183e7			0.381154			0.190577	−	0.981672i	$-0.438964\pi$
$599$	8.19183e7			0.381154			0.190577	+	0.981672i	$0.438964\pi$
$600$		−	1.31875e8i		−	0.610532i
$601$			1.65135e8i			0.760706i	0.924841	+	0.380353i	$0.124198\pi$
$601$			1.65135e8i			0.760706i	−0.924841	+	0.380353i	$0.875802\pi$
$602$	4.96300e6			0.0227486
$603$			5.56670e6i			0.0253890i
$604$			5.80637e7i			0.263508i
$605$	9.77524e7			0.441429
$606$	2.21623e8			0.995856
$607$	1.07036e8			0.478589			0.239294	−	0.970947i	$-0.423084\pi$
$607$	1.07036e8			0.478589			0.239294	+	0.970947i	$0.423084\pi$
$608$			3.56556e6i			0.0158642i
$609$	2.75806e6			0.0122110
$610$	2.90545e8			1.28004
$611$	−1.24885e7			−0.0547504
$612$	2.07723e8			0.906212
$613$		−	1.49782e8i		−	0.650246i	−0.945672	−	0.325123i	$-0.894594\pi$
$613$		−	1.49782e8i		−	0.650246i	0.945672	−	0.325123i	$-0.105406\pi$
$614$			2.10915e7i			0.0911175i
$615$	−5.99157e7			−0.257582
$616$	−3.46972e6			−0.0148440
$617$	3.30963e8			1.40904			0.704522	−	0.709682i	$-0.251162\pi$
$617$	3.30963e8			1.40904			0.704522	+	0.709682i	$0.251162\pi$
$618$	1.55104e8			0.657138
$619$	−2.05287e8			−0.865545			−0.432772	−	0.901503i	$-0.642465\pi$
$619$	−2.05287e8			−0.865545			−0.432772	+	0.901503i	$0.642465\pi$
$620$		−	1.56595e8i		−	0.657056i
$621$		−	8.75274e7i		−	0.365485i
$622$		−	9.72682e7i		−	0.404203i
$623$			1.07393e6i			0.00444131i
$624$		−	1.02177e7i		−	0.0420532i
$625$	4.44014e6			0.0181868
$626$	2.54365e8			1.03690
$627$		−	1.62728e7i		−	0.0660176i
$628$			2.70579e8i			1.09248i
$629$		−	4.51404e8i		−	1.81390i
$630$			1.46605e6i			0.00586312i
$631$	−9.56680e7			−0.380784			−0.190392	−	0.981708i	$-0.560976\pi$
$631$	−9.56680e7			−0.380784			−0.190392	+	0.981708i	$0.560976\pi$
$632$			3.56385e8i			1.41179i
$633$			4.03159e7i			0.158952i
$634$		−	4.92213e8i		−	1.93146i
$635$	−2.43040e8			−0.949196
$636$	7.49505e6			0.0291342
$637$			2.01046e7i			0.0777816i
$638$	2.99884e8			1.15476
$639$	9.86182e7			0.377968
$640$		−	2.82179e8i		−	1.07643i
$641$	−2.42343e8			−0.920146			−0.460073	−	0.887881i	$-0.652177\pi$
$641$	−2.42343e8			−0.920146			−0.460073	+	0.887881i	$0.652177\pi$
$642$			3.43737e7i			0.129904i
$643$	2.67124e8			1.00480			0.502400	−	0.864635i	$-0.332450\pi$
$643$	2.67124e8			1.00480			0.502400	+	0.864635i	$0.332450\pi$
$644$		−	1.67283e7i		−	0.0626316i
$645$		−	7.49092e7i		−	0.279162i
$646$			1.38630e8i			0.514232i
$647$	9.87605e7			0.364645			0.182322	−	0.983239i	$-0.441639\pi$
$647$	9.87605e7			0.364645			0.182322	+	0.983239i	$0.441639\pi$
$648$			5.10601e7i			0.187653i
$649$	−1.43090e8	+	1.73987e7i	−0.523452	+	0.0636477i
$650$	2.30894e7			0.0840760
$651$		−	1.44200e6i		−	0.00522665i
$652$	−8.12323e6			−0.0293080
$653$	−7.27488e7			−0.261268			−0.130634	−	0.991431i	$-0.541701\pi$
$653$	−7.27488e7			−0.261268			−0.130634	+	0.991431i	$0.541701\pi$
$654$	−3.83712e8			−1.37174
$655$			4.47401e7i			0.159211i
$656$	−1.92839e8			−0.683097
$657$		−	3.80852e7i		−	0.134295i
$658$	5.76715e6			0.0202434
$659$			4.04397e8i			1.41303i	0.707698	+	0.706515i	$0.249734\pi$
$659$			4.04397e8i			1.41303i	−0.707698	+	0.706515i	$0.750266\pi$
$660$			1.05887e8i			0.368308i
$661$	−1.92868e8			−0.667815			−0.333908	−	0.942606i	$-0.608367\pi$
$661$	−1.92868e8			−0.667815			−0.333908	+	0.942606i	$0.608367\pi$
$662$		−	6.86408e8i		−	2.36596i
$663$			1.79877e7i			0.0617213i
$664$	8.26009e7			0.282150
$665$	−649931.			−0.00221005
$666$	2.24347e8			0.759448
$667$			7.15076e8i			2.40976i
$668$	−6.41002e8			−2.15045
$669$	1.05909e8			0.353717
$670$	2.41742e7			0.0803762
$671$	1.93239e8			0.639628
$672$		−	213647.i		−	0.000704028i
$673$		−	1.29241e8i		−	0.423989i	−0.977271	−	0.211994i	$-0.932004\pi$
$673$		−	1.29241e8i		−	0.423989i	0.977271	−	0.211994i	$-0.0679959\pi$
$674$	5.26826e7			0.172063
$675$	−3.70595e7			−0.120500
$676$	−6.07513e8			−1.96660
$677$	−3.69410e8			−1.19053			−0.595267	−	0.803528i	$-0.702954\pi$
$677$	−3.69410e8			−1.19053			−0.595267	+	0.803528i	$0.702954\pi$
$678$	9.61595e7			0.308534
$679$			4.03882e6i			0.0129016i
$680$		−	4.46149e8i		−	1.41890i
$681$		−	7.78171e7i		−	0.246396i
$682$		−	1.56789e8i		−	0.494268i
$683$		−	4.64586e8i		−	1.45816i	−0.684430	−	0.729079i	$-0.739949\pi$
$683$		−	4.64586e8i		−	1.45816i	0.684430	−	0.729079i	$-0.260051\pi$
$684$	−4.57674e7			−0.143017
$685$	−1.22128e8			−0.379966
$686$		−	1.85710e7i		−	0.0575259i
$687$		−	9.74911e7i		−	0.300673i
$688$		−	2.41095e8i		−	0.740327i
$689$			649032.i			0.00198431i
$690$	−3.80100e8			−1.15705
$691$			5.05263e8i			1.53138i	0.643209	+	0.765691i	$0.277603\pi$
$691$			5.05263e8i			1.53138i	−0.643209	+	0.765691i	$0.722397\pi$
$692$			2.61517e8i			0.789189i
$693$			975061.i			0.00292976i
$694$	−8.83304e8			−2.64260
$695$	−2.44043e8			−0.726964
$696$		−	4.17147e8i		−	1.23726i
$697$	3.39482e8			1.00258
$698$	−8.14479e8			−2.39505
$699$		−	2.76117e8i		−	0.808466i
$700$	−7.08283e6			−0.0206496
$701$		−	3.46156e8i		−	1.00489i	−0.864610	−	0.502444i	$-0.832434\pi$
$701$		−	3.46156e8i		−	1.00489i	0.864610	−	0.502444i	$-0.167566\pi$
$702$	−8.93987e6			−0.0258416
$703$			9.94576e7i			0.286267i
$704$		−	1.95474e8i		−	0.560235i
$705$		−	8.70466e7i		−	0.248419i
$706$	−4.62982e8			−1.31568
$707$		−	5.88709e6i		−	0.0166588i
$708$	4.89341e7	+	4.02443e8i	0.137883	+	1.13398i
$709$	−4.52162e8			−1.26869			−0.634345	−	0.773050i	$-0.718730\pi$
$709$	−4.52162e8			−1.26869			−0.634345	+	0.773050i	$0.718730\pi$
$710$		−	4.28263e8i		−	1.19656i
$711$	1.00151e8			0.278643
$712$	1.62428e8			0.450008
$713$	3.73864e8			1.03144
$714$		−	8.30666e6i		−	0.0228208i
$715$	−9.16927e6			−0.0250851
$716$			5.08848e8i			1.38627i
$717$	2.56945e8			0.697080
$718$		−	6.82113e8i		−	1.84282i
$719$			1.71324e8i			0.460925i	0.973081	+	0.230463i	$0.0740239\pi$
$719$			1.71324e8i			0.460925i	−0.973081	+	0.230463i	$0.925976\pi$
$720$	7.12187e7			0.190808
$721$		−	4.12011e6i		−	0.0109927i
$722$			6.19011e8i			1.64470i
$723$	3.71140e8			0.982024
$724$	6.76639e8			1.78296
$725$	3.02766e8			0.794499
$726$		−	2.75270e8i		−	0.719365i
$727$	3.72134e8			0.968493			0.484247	−	0.874932i	$-0.339094\pi$
$727$	3.72134e8			0.968493			0.484247	+	0.874932i	$0.339094\pi$
$728$	−845046.			−0.00219021
$729$	1.43489e7			0.0370370
$730$	−1.65390e8			−0.425149
$731$			4.24435e8i			1.08657i
$732$		−	5.43487e8i		−	1.38566i
$733$	1.44204e8			0.366155			0.183078	−	0.983098i	$-0.441394\pi$
$733$	1.44204e8			0.366155			0.183078	+	0.983098i	$0.441394\pi$
$734$	−6.01494e8			−1.52105
$735$	−1.40132e8			−0.352918
$736$	5.53918e7			0.138935
$737$	1.60781e7			0.0401635
$738$			1.68722e8i			0.419762i
$739$		−	5.87245e7i		−	0.145508i	−0.997350	−	0.0727538i	$-0.976821\pi$
$739$		−	5.87245e7i		−	0.145508i	0.997350	−	0.0727538i	$-0.0231787\pi$
$740$		−	6.47170e8i		−	1.59707i
$741$		−	3.96322e6i		−	0.00974078i
$742$		−	299721.i		−	0.000733677i
$743$	−6.34696e8			−1.54739			−0.773694	−	0.633560i	$-0.781593\pi$
$743$	−6.34696e8			−1.54739			−0.773694	+	0.633560i	$0.781593\pi$
$744$	−2.18098e8			−0.529581
$745$		−	1.75421e8i		−	0.424242i
$746$			5.52032e8i			1.32968i
$747$		−	2.32125e7i		−	0.0556878i
$748$		−	5.99956e8i		−	1.43356i
$749$	913088.			0.00217304
$750$			4.17966e8i			0.990733i
$751$			7.55144e8i			1.78283i	0.453187	+	0.891415i	$0.350287\pi$
$751$			7.55144e8i			1.78283i	−0.453187	+	0.891415i	$0.649713\pi$
$752$		−	2.80160e8i		−	0.658797i
$753$	2.82152e8			0.660842
$754$	7.30363e7			0.170383
$755$		−	3.50459e7i		−	0.0814323i
$756$	2.74237e6			0.00634688
$757$	6.07614e8			1.40068			0.700342	−	0.713807i	$-0.253031\pi$
$757$	6.07614e8			1.40068			0.700342	+	0.713807i	$0.253031\pi$
$758$			1.18994e8i			0.273223i
$759$	−2.52801e8			−0.578168
$760$			9.82997e7i			0.223929i
$761$	5.84643e8			1.32659			0.663295	−	0.748358i	$-0.269157\pi$
$761$	5.84643e8			1.32659			0.663295	+	0.748358i	$0.269157\pi$
$762$			6.84398e8i			1.54683i
$763$			1.01928e7i			0.0229466i
$764$		−	1.14137e8i		−	0.255944i
$765$	−1.25377e8			−0.280048
$766$		−	1.14232e9i		−	2.54156i
$767$	−3.48495e7	+	4.23743e6i	−0.0772343	+	0.00939111i
$768$	−5.16751e8			−1.14077
$769$			2.44620e8i			0.537914i	0.963152	+	0.268957i	$0.0866789\pi$
$769$			2.44620e8i			0.537914i	−0.963152	+	0.268957i	$0.913321\pi$
$770$	4.23434e6			0.00927499
$771$	1.17085e8			0.255469
$772$	3.98991e8			0.867185
$773$			2.90161e8i			0.628203i	0.949389	+	0.314102i	$0.101703\pi$
$773$			2.90161e8i			0.628203i	−0.949389	+	0.314102i	$0.898297\pi$
$774$	−2.10944e8			−0.454929
$775$		−	1.58296e8i		−	0.340067i
$776$	6.10856e8			1.30724
$777$		−	5.95947e6i		−	0.0127041i
$778$		−	9.08789e8i		−	1.92985i
$779$	−7.47978e7			−0.158226
$780$			2.57887e7i			0.0543432i
$781$		−	2.84834e8i		−	0.597915i
$782$	2.15365e9			4.50354
$783$	−1.17227e8			−0.244198
$784$	−4.51014e8			−0.935926
$785$		−	1.63315e8i		−	0.337612i

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)

\(f(q)\)	\(=\)	\(q-13.8068i q^{2} +15.5885 q^{3} -126.629 q^{4} +76.4303 q^{5} -215.227i q^{6} -5.71721 q^{7} +864.707i q^{8} +243.000 q^{9} +O(q^{10})\)	\(q-13.8068i q^{2} +15.5885 q^{3} -126.629 q^{4} +76.4303 q^{5} -215.227i q^{6} -5.71721 q^{7} +864.707i q^{8} +243.000 q^{9} -1055.26i q^{10} -701.846i q^{11} -1973.95 q^{12} -170.934i q^{13} +78.9366i q^{14} +1191.43 q^{15} +3834.62 q^{16} -6750.64 q^{17} -3355.06i q^{18} +1487.36 q^{19} -9678.28 q^{20} -89.1224 q^{21} -9690.27 q^{22} -23106.5i q^{23} +13479.4i q^{24} -9783.41 q^{25} -2360.05 q^{26} +3788.00 q^{27} +723.963 q^{28} -30946.9 q^{29} -16449.9i q^{30} +16180.0i q^{31} +2397.24i q^{32} -10940.7i q^{33} +93205.0i q^{34} -436.968 q^{35} -30770.8 q^{36} +66868.3i q^{37} -20535.8i q^{38} -2664.59i q^{39} +66089.8i q^{40} -50288.8 q^{41} +1230.50i q^{42} -62873.3i q^{43} +88873.9i q^{44} +18572.6 q^{45} -319028. q^{46} -73060.6i q^{47} +59775.8 q^{48} -117616. q^{49} +135078. i q^{50} -105232. q^{51} +21645.1i q^{52} -3796.98 q^{53} -52300.2i q^{54} -53642.3i q^{55} -4943.71i q^{56} +23185.7 q^{57} +427279. i q^{58} +(-24789.9 - 203877. i) q^{59} -150870. q^{60} +275330. i q^{61} +223395. q^{62} -1389.28 q^{63} +278514. q^{64} -13064.5i q^{65} -151056. q^{66} +22908.3i q^{67} +854826. q^{68} -360195. i q^{69} +6033.15i q^{70} +405836. q^{71} +210124. i q^{72} -156729. i q^{73} +923240. q^{74} -152508. q^{75} -188343. q^{76} +4012.60i q^{77} -36789.6 q^{78} +412146. q^{79} +293081. q^{80} +59049.0 q^{81} +694330. i q^{82} -95524.7i q^{83} +11285.5 q^{84} -515954. q^{85} -868082. q^{86} -482415. q^{87} +606891. q^{88} -187842. i q^{89} -256428. i q^{90} +977.263i q^{91} +2.92595e6i q^{92} +252222. i q^{93} -1.00874e6 q^{94} +113680. q^{95} +37369.2i q^{96} -706432. i q^{97} +1.62391e6i q^{98} -170548. i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 60q - 1920q^{4} - 408q^{7} + 14580q^{9} + O(q^{10}) \)	\( 60q - 1920q^{4} - 408q^{7} + 14580q^{9} - 1944q^{12} - 4536q^{15} + 56616q^{16} + 8480q^{17} + 11376q^{19} + 40796q^{20} - 8232q^{22} + 197940q^{25} + 147252q^{26} + 71640q^{28} + 63456q^{29} - 364432q^{35} - 466560q^{36} + 99632q^{41} - 470316q^{46} + 171072q^{48} + 1737420q^{49} + 60912q^{51} + 92240q^{53} + 186624q^{57} + 917264q^{59} + 1063368q^{60} - 115768q^{62} - 99144q^{63} - 1107444q^{64} + 1172232q^{66} - 4247232q^{68} + 1498048q^{71} + 1161448q^{74} - 1477440q^{75} - 1045320q^{76} - 1060452q^{78} - 90600q^{79} + 77096q^{80} + 3542940q^{81} - 2225880q^{84} - 693408q^{85} - 1567768q^{86} + 1821528q^{87} + 62892q^{88} + 5268696q^{94} + 296128q^{95} + O(q^{100}) \)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Properties

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