LMFDB - Embedded newform 177.7.c.a.58.13

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			58.13
Character	$\chi$	$=$	177.58
Dual form			177.7.c.a.58.48

$q$-expansion

$f(q)$	$=$	$q-10.1526i q^{2} -15.5885 q^{3} -39.0745 q^{4} -104.301 q^{5} +158.263i q^{6} -654.718 q^{7} -253.057i q^{8} +243.000 q^{9} +O(q^{10})$	\(q-10.1526i q^{2} -15.5885 q^{3} -39.0745 q^{4} -104.301 q^{5} +158.263i q^{6} -654.718 q^{7} -253.057i q^{8} +243.000 q^{9} +1058.93i q^{10} -442.991i q^{11} +609.112 q^{12} +3003.02i q^{13} +6647.07i q^{14} +1625.90 q^{15} -5069.95 q^{16} -6614.72 q^{17} -2467.07i q^{18} -2022.49 q^{19} +4075.53 q^{20} +10206.1 q^{21} -4497.50 q^{22} -7860.31i q^{23} +3944.78i q^{24} -4746.22 q^{25} +30488.4 q^{26} -3788.00 q^{27} +25582.8 q^{28} +24340.1 q^{29} -16507.0i q^{30} +13844.9i q^{31} +35277.3i q^{32} +6905.55i q^{33} +67156.4i q^{34} +68288.0 q^{35} -9495.11 q^{36} -73303.9i q^{37} +20533.5i q^{38} -46812.5i q^{39} +26394.2i q^{40} -5255.31 q^{41} -103618. i q^{42} +72295.0i q^{43} +17309.7i q^{44} -25345.2 q^{45} -79802.3 q^{46} -175954. i q^{47} +79032.7 q^{48} +311007. q^{49} +48186.3i q^{50} +103113. q^{51} -117342. i q^{52} -23803.5 q^{53} +38457.9i q^{54} +46204.6i q^{55} +165681. i q^{56} +31527.5 q^{57} -247114. i q^{58} +(-114946. - 170200. i) q^{59} -63531.2 q^{60} +28156.5i q^{61} +140561. q^{62} -159097. q^{63} +33678.3 q^{64} -313219. i q^{65} +70109.0 q^{66} -320803. i q^{67} +258467. q^{68} +122530. i q^{69} -693299. i q^{70} +554734. q^{71} -61493.0i q^{72} +125135. i q^{73} -744223. q^{74} +73986.2 q^{75} +79027.9 q^{76} +290035. i q^{77} -475266. q^{78} -858752. q^{79} +528803. q^{80} +59049.0 q^{81} +53354.8i q^{82} +520126. i q^{83} -398797. q^{84} +689925. q^{85} +733979. q^{86} -379424. q^{87} -112102. q^{88} +300665. i q^{89} +257319. i q^{90} -1.96613e6i q^{91} +307138. i q^{92} -215820. i q^{93} -1.78639e6 q^{94} +210949. q^{95} -549919. i q^{96} -1.22552e6i q^{97} -3.15752e6i q^{98} -107647. i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$		−	10.1526i		−	1.26907i	−0.772894	−	0.634535i	$-0.781192\pi$
$2$		−	10.1526i		−	1.26907i	0.772894	−	0.634535i	$-0.218808\pi$
$3$	−15.5885			−0.577350
$3$	−15.5885			−0.577350
$4$	−39.0745			−0.610539
$5$	−104.301			−0.834411			−0.417206	−	0.908812i	$-0.636990\pi$
$5$	−104.301			−0.834411			−0.417206	+	0.908812i	$0.636990\pi$
$6$			158.263i			0.732698i
$7$	−654.718			−1.90880			−0.954400	−	0.298531i	$-0.903503\pi$
$7$	−654.718			−1.90880			−0.954400	+	0.298531i	$0.903503\pi$
$8$		−	253.057i		−	0.494253i
$9$	243.000			0.333333
$10$			1058.93i			1.05893i
$11$		−	442.991i		−	0.332826i	−0.986056	−	0.166413i	$-0.946782\pi$
$11$		−	442.991i		−	0.332826i	0.986056	−	0.166413i	$-0.0532185\pi$
$12$	609.112			0.352495
$13$			3003.02i			1.36687i	0.730010	+	0.683437i	$0.239515\pi$
$13$			3003.02i			1.36687i	−0.730010	+	0.683437i	$0.760485\pi$
$14$			6647.07i			2.42240i
$15$	1625.90			0.481747
$16$	−5069.95			−1.23778
$17$	−6614.72			−1.34637			−0.673186	−	0.739474i	$-0.735074\pi$
$17$	−6614.72			−1.34637			−0.673186	+	0.739474i	$0.735074\pi$
$18$		−	2467.07i		−	0.423023i
$19$	−2022.49			−0.294867			−0.147433	−	0.989072i	$-0.547101\pi$
$19$	−2022.49			−0.294867			−0.147433	+	0.989072i	$0.547101\pi$
$20$	4075.53			0.509441
$21$	10206.1			1.10205
$22$	−4497.50			−0.422379
$23$		−	7860.31i		−	0.646035i	−0.946393	−	0.323018i	$-0.895303\pi$
$23$		−	7860.31i		−	0.646035i	0.946393	−	0.323018i	$-0.104697\pi$
$24$			3944.78i			0.285357i
$25$	−4746.22			−0.303758
$26$	30488.4			1.73466
$27$	−3788.00			−0.192450
$28$	25582.8			1.16540
$29$	24340.1			0.997993			0.498997	−	0.866604i	$-0.333702\pi$
$29$	24340.1			0.997993			0.498997	+	0.866604i	$0.333702\pi$
$30$		−	16507.0i		−	0.611371i
$31$			13844.9i			0.464733i	0.972628	+	0.232367i	$0.0746469\pi$
$31$			13844.9i			0.464733i	−0.972628	+	0.232367i	$0.925353\pi$
$32$			35277.3i			1.07658i
$33$			6905.55i			0.192157i
$34$			67156.4i			1.70864i
$35$	68288.0			1.59272
$36$	−9495.11			−0.203513
$37$		−	73303.9i		−	1.44718i	−0.690231	−	0.723589i	$-0.742491\pi$
$37$		−	73303.9i		−	1.44718i	0.690231	−	0.723589i	$-0.257509\pi$
$38$			20533.5i			0.374207i
$39$		−	46812.5i		−	0.789165i
$40$			26394.2i			0.412410i
$41$	−5255.31			−0.0762512			−0.0381256	−	0.999273i	$-0.512139\pi$
$41$	−5255.31			−0.0762512			−0.0381256	+	0.999273i	$0.512139\pi$
$42$		−	103618.i		−	1.39857i
$43$			72295.0i			0.909291i	0.890673	+	0.454645i	$0.150234\pi$
$43$			72295.0i			0.909291i	−0.890673	+	0.454645i	$0.849766\pi$
$44$			17309.7i			0.203203i
$45$	−25345.2			−0.278137
$46$	−79802.3			−0.819864
$47$		−	175954.i		−	1.69475i	−0.530993	−	0.847376i	$-0.678181\pi$
$47$		−	175954.i		−	1.69475i	0.530993	−	0.847376i	$-0.321819\pi$
$48$	79032.7			0.714633
$49$	311007.			2.64352
$50$			48186.3i			0.385490i
$51$	103113.			0.777328
$52$		−	117342.i		−	0.834530i
$53$	−23803.5			−0.159887			−0.0799435	−	0.996799i	$-0.525474\pi$
$53$	−23803.5			−0.159887			−0.0799435	+	0.996799i	$0.525474\pi$
$54$			38457.9i			0.244233i
$55$			46204.6i			0.277714i
$56$			165681.i			0.943430i
$57$	31527.5			0.170242
$58$		−	247114.i		−	1.26652i
$59$	−114946.	−	170200.i	−0.559676	−	0.828711i
$59$	−114946.	−	170200.i	−0.559676	−	0.828711i
$60$	−63531.2			−0.294126
$61$			28156.5i			0.124048i	0.998075	+	0.0620240i	$0.0197555\pi$
$61$			28156.5i			0.124048i	−0.998075	+	0.0620240i	$0.980244\pi$
$62$	140561.			0.589779
$63$	−159097.			−0.636267
$64$	33678.3			0.128473
$65$		−	313219.i		−	1.14053i
$66$	70109.0			0.243861
$67$		−	320803.i		−	1.06663i	−0.845917	−	0.533315i	$-0.820946\pi$
$67$		−	320803.i		−	1.06663i	0.845917	−	0.533315i	$-0.179054\pi$
$68$	258467.			0.822013
$69$			122530.i			0.372989i
$70$		−	693299.i		−	2.02128i
$71$	554734.			1.54992			0.774961	−	0.632009i	$-0.217769\pi$
$71$	554734.			1.54992			0.774961	+	0.632009i	$0.217769\pi$
$72$		−	61493.0i		−	0.164751i
$73$			125135.i			0.321671i	0.986981	+	0.160835i	$0.0514188\pi$
$73$			125135.i			0.321671i	−0.986981	+	0.160835i	$0.948581\pi$
$74$	−744223.			−1.83657
$75$	73986.2			0.175375
$76$	79027.9			0.180028
$77$			290035.i			0.635298i
$78$	−475266.			−1.00151
$79$	−858752.			−1.74175			−0.870877	−	0.491501i	$-0.836448\pi$
$79$	−858752.			−1.74175			−0.870877	+	0.491501i	$0.836448\pi$
$80$	528803.			1.03282
$81$	59049.0			0.111111
$82$			53354.8i			0.0967681i
$83$			520126.i			0.909650i	0.890581	+	0.454825i	$0.150298\pi$
$83$			520126.i			0.909650i	−0.890581	+	0.454825i	$0.849702\pi$
$84$	−398797.			−0.672843
$85$	689925.			1.12343
$86$	733979.			1.15395
$87$	−379424.			−0.576192
$88$	−112102.			−0.164500
$89$			300665.i			0.426495i	0.976998	+	0.213247i	$0.0684040\pi$
$89$			300665.i			0.426495i	−0.976998	+	0.213247i	$0.931596\pi$
$90$			257319.i			0.352975i
$91$		−	1.96613e6i		−	2.60909i
$92$			307138.i			0.394430i
$93$		−	215820.i		−	0.268314i
$94$	−1.78639e6			−2.15076
$95$	210949.			0.246040
$96$		−	549919.i		−	0.621563i
$97$		−	1.22552e6i		−	1.34278i	−0.741103	−	0.671391i	$-0.765697\pi$
$97$		−	1.22552e6i		−	1.34278i	0.741103	−	0.671391i	$-0.234303\pi$
$98$		−	3.15752e6i		−	3.35481i
$99$		−	107647.i		−	0.110942i
$100$	185456.			0.185456
$101$			927913.i			0.900624i	0.892871	+	0.450312i	$0.148687\pi$
$101$			927913.i			0.900624i	−0.892871	+	0.450312i	$0.851313\pi$
$102$		−	1.04686e6i		−	0.986484i
$103$			412082.i			0.377114i	0.982062	+	0.188557i	$0.0603809\pi$
$103$			412082.i			0.377114i	−0.982062	+	0.188557i	$0.939619\pi$
$104$	759937.			0.675581
$105$	−1.06451e6			−0.919560
$106$			241666.i			0.202908i
$107$	−879511.			−0.717943			−0.358971	−	0.933349i	$-0.616872\pi$
$107$	−879511.			−0.717943			−0.358971	+	0.933349i	$0.616872\pi$
$108$	148014.			0.117498
$109$			1.22209e6i			0.943676i	0.881685	+	0.471838i	$0.156409\pi$
$109$			1.22209e6i			0.943676i	−0.881685	+	0.471838i	$0.843591\pi$
$110$	469095.			0.352438
$111$			1.14270e6i			0.835529i
$112$	3.31939e6			2.36268
$113$			1.16448e6i			0.807040i	0.914971	+	0.403520i	$0.132213\pi$
$113$			1.16448e6i			0.807040i	−0.914971	+	0.403520i	$0.867787\pi$
$114$		−	320085.i		−	0.216048i
$115$			819841.i			0.539059i
$116$	−951076.			−0.609314
$117$			729734.i			0.455624i
$118$	−1.72796e6	+	1.16699e6i	−1.05169	+	0.710269i
$119$	4.33078e6			2.56995
$120$		−	411446.i		−	0.238105i
$121$	1.57532e6			0.889227
$122$	285861.			0.157426
$123$	81922.1			0.0440236
$124$		−	540982.i		−	0.283738i
$125$	2.12475e6			1.08787
$126$			1.61524e6i			0.807467i
$127$	−1.36949e6			−0.668571			−0.334285	−	0.942472i	$-0.608495\pi$
$127$	−1.36949e6			−0.668571			−0.334285	+	0.942472i	$0.608495\pi$
$128$			1.91583e6i			0.913538i
$129$		−	1.12697e6i		−	0.524979i
$130$	−3.17998e6			−1.44742
$131$			3.89962e6i			1.73464i	0.497754	+	0.867318i	$0.334158\pi$
$131$			3.89962e6i			1.73464i	−0.497754	+	0.867318i	$0.665842\pi$
$132$		−	269831.i		−	0.117319i
$133$	1.32416e6			0.562842
$134$	−3.25697e6			−1.35363
$135$	395093.			0.160582
$136$			1.67390e6i			0.665448i
$137$	−538204.			−0.209308			−0.104654	−	0.994509i	$-0.533373\pi$
$137$	−538204.			−0.209308			−0.104654	+	0.994509i	$0.533373\pi$
$138$	1.24399e6			0.473349
$139$	3.79961e6			1.41480			0.707400	−	0.706813i	$-0.249868\pi$
$139$	3.79961e6			1.41480			0.707400	+	0.706813i	$0.249868\pi$
$140$	−2.66832e6			−0.972421
$141$			2.74286e6i			0.978466i
$142$		−	5.63198e6i		−	1.96696i
$143$	1.33031e6			0.454931
$144$	−1.23200e6			−0.412594
$145$	−2.53870e6			−0.832737
$146$	1.27044e6			0.408223
$147$	−4.84812e6			−1.52624
$148$			2.86432e6i			0.883560i
$149$			2.19953e6i			0.664921i	0.943117	+	0.332461i	$0.107879\pi$
$149$			2.19953e6i			0.664921i	−0.943117	+	0.332461i	$0.892121\pi$
$150$		−	751150.i		−	0.222563i
$151$		−	5.72520e6i		−	1.66288i	−0.555617	−	0.831438i	$-0.687518\pi$
$151$		−	5.72520e6i		−	1.66288i	0.555617	−	0.831438i	$-0.312482\pi$
$152$			511807.i			0.145739i
$153$	−1.60738e6			−0.448790
$154$	2.94459e6			0.806238
$155$		−	1.44404e6i		−	0.387779i
$156$			1.82917e6i			0.481816i
$157$			332770.i			0.0859893i	0.999075	+	0.0429947i	$0.0136898\pi$
$157$			332770.i			0.0859893i	−0.999075	+	0.0429947i	$0.986310\pi$
$158$			8.71854e6i			2.21041i
$159$	371060.			0.0923108
$160$		−	3.67947e6i		−	0.898309i
$161$			5.14629e6i			1.23315i
$162$		−	599499.i		−	0.141008i
$163$	3.50429e6			0.809165			0.404582	−	0.914502i	$-0.367417\pi$
$163$	3.50429e6			0.809165			0.404582	+	0.914502i	$0.367417\pi$
$164$	205349.			0.0465544
$165$		−	720258.i		−	0.160338i
$166$	5.28061e6			1.15441
$167$	4.97745e6			1.06870			0.534352	−	0.845262i	$-0.320556\pi$
$167$	4.97745e6			1.06870			0.534352	+	0.845262i	$0.320556\pi$
$168$		−	2.58272e6i		−	0.544690i
$169$	−4.19132e6			−0.868343
$170$		−	7.00450e6i		−	1.42571i
$171$	−491466.			−0.0982890
$172$		−	2.82489e6i		−	0.555158i
$173$			7.61411e6i			1.47055i	0.677767	+	0.735276i	$0.262948\pi$
$173$			7.61411e6i			1.47055i	−0.677767	+	0.735276i	$0.737052\pi$
$174$			3.85213e6i			0.731228i
$175$	3.10744e6			0.579813
$176$			2.24594e6i			0.411966i
$177$	1.79183e6	+	2.65315e6i	0.323129	+	0.478457i
$178$	3.05252e6			0.541252
$179$		−	1.59663e6i		−	0.278384i	−0.990265	−	0.139192i	$-0.955549\pi$
$179$		−	1.59663e6i		−	0.278384i	0.990265	−	0.139192i	$-0.0444505\pi$
$180$	990353.			0.169814
$181$	−1.02310e7			−1.72537			−0.862686	−	0.505741i	$-0.831219\pi$
$181$	−1.02310e7			−1.72537			−0.862686	+	0.505741i	$0.831219\pi$
$182$	−1.99613e7			−3.31112
$183$		−	438917.i		−	0.0716191i
$184$	−1.98911e6			−0.319305
$185$			7.64570e6i			1.20754i
$186$	−2.19113e6			−0.340509
$187$			2.93026e6i			0.448107i
$188$			6.87533e6i			1.03471i
$189$	2.48007e6			0.367349
$190$		−	2.14167e6i		−	0.312242i
$191$		−	1.13201e7i		−	1.62462i	−0.583227	−	0.812309i	$-0.698210\pi$
$191$		−	1.13201e7i		−	1.62462i	0.583227	−	0.812309i	$-0.301790\pi$
$192$	−524993.			−0.0741736
$193$	1.96778e6			0.273719			0.136860	−	0.990590i	$-0.456299\pi$
$193$	1.96778e6			0.273719			0.136860	+	0.990590i	$0.456299\pi$
$194$	−1.24422e7			−1.70409
$195$			4.88260e6i			0.658488i
$196$	−1.21525e7			−1.61397
$197$	−6.29328e6			−0.823149			−0.411574	−	0.911376i	$-0.635021\pi$
$197$	−6.29328e6			−0.823149			−0.411574	+	0.911376i	$0.635021\pi$
$198$	−1.09289e6			−0.140793
$199$	8.10498e6			1.02847			0.514236	−	0.857649i	$-0.328076\pi$
$199$	8.10498e6			1.02847			0.514236	+	0.857649i	$0.328076\pi$
$200$			1.20107e6i			0.150133i
$201$			5.00082e6i			0.615819i
$202$	9.42070e6			1.14295
$203$	−1.59359e7			−1.90497
$204$	−4.02910e6			−0.474589
$205$	548136.			0.0636248
$206$	4.18369e6			0.478584
$207$		−	1.91006e6i		−	0.215345i
$208$		−	1.52252e7i		−	1.69189i
$209$			895947.i			0.0981394i
$210$			1.08075e7i			1.16699i
$211$			1.19788e7i			1.27516i	0.770382	+	0.637582i	$0.220065\pi$
$211$			1.19788e7i			1.27516i	−0.770382	+	0.637582i	$0.779935\pi$
$212$	930110.			0.0976173
$213$	−8.64745e6			−0.894848
$214$			8.92929e6i			0.911120i
$215$		−	7.54047e6i		−	0.758722i
$216$			958580.i			0.0951190i
$217$		−	9.06449e6i		−	0.887083i
$218$	1.24073e7			1.19759
$219$		−	1.95067e6i		−	0.185717i
$220$		−	1.80542e6i		−	0.169555i
$221$		−	1.98641e7i		−	1.84032i
$222$	1.16013e7			1.06035
$223$	−6.83511e6			−0.616355			−0.308178	−	0.951329i	$-0.599719\pi$
$223$	−6.83511e6			−0.616355			−0.308178	+	0.951329i	$0.599719\pi$
$224$		−	2.30967e7i		−	2.05497i
$225$	−1.15333e6			−0.101253
$226$	1.18224e7			1.02419
$227$			9.63771e6i			0.823941i	0.911197	+	0.411971i	$0.135159\pi$
$227$			9.63771e6i			0.823941i	−0.911197	+	0.411971i	$0.864841\pi$
$228$	−1.23192e6			−0.103939
$229$			1.02822e7i			0.856212i	0.903728	+	0.428106i	$0.140819\pi$
$229$			1.02822e7i			0.856212i	−0.903728	+	0.428106i	$0.859181\pi$
$230$	8.32349e6			0.684104
$231$		−	4.52119e6i		−	0.366790i
$232$		−	6.15943e6i		−	0.493261i
$233$		−	195801.i		−	0.0154791i	−0.999970	−	0.00773956i	$-0.997536\pi$
$233$		−	195801.i		−	0.0154791i	0.999970	−	0.00773956i	$-0.00246360\pi$
$234$	7.40867e6			0.578219
$235$			1.83523e7i			1.41412i
$236$	4.49145e6	+	6.65048e6i	0.341705	+	0.505961i
$237$	1.33866e7			1.00560
$238$		−	4.39685e7i		−	3.26145i
$239$	3.73985e6			0.273943			0.136972	−	0.990575i	$-0.456263\pi$
$239$	3.73985e6			0.273943			0.136972	+	0.990575i	$0.456263\pi$
$240$	−8.24322e6			−0.596298
$241$	−3.69307e6			−0.263837			−0.131919	−	0.991261i	$-0.542114\pi$
$241$	−3.69307e6			−0.263837			−0.131919	+	0.991261i	$0.542114\pi$
$242$		−	1.59935e7i		−	1.12849i
$243$	−920483.			−0.0641500
$244$		−	1.10020e6i		−	0.0757362i
$245$	−3.24385e7			−2.20578
$246$		−	831720.i		−	0.0558691i
$247$		−	6.07359e6i		−	0.403046i
$248$	3.50355e6			0.229696
$249$		−	8.10797e6i		−	0.525187i
$250$		−	2.15716e7i		−	1.38058i
$251$	1.05621e7			0.667928			0.333964	−	0.942586i	$-0.391614\pi$
$251$	1.05621e7			0.667928			0.333964	+	0.942586i	$0.391614\pi$
$252$	6.21662e6			0.388466
$253$	−3.48205e6			−0.215017
$254$			1.39038e7i			0.848463i
$255$	−1.07549e7			−0.648611
$256$	2.16060e7			1.28782
$257$	1.61502e6			0.0951432			0.0475716	−	0.998868i	$-0.484852\pi$
$257$	1.61502e6			0.0951432			0.0475716	+	0.998868i	$0.484852\pi$
$258$	−1.14416e7			−0.666236
$259$			4.79934e7i			2.76237i
$260$			1.22389e7i			0.696341i
$261$	5.91463e6			0.332664
$262$	3.95911e7			2.20138
$263$	−3.39349e6			−0.186543			−0.0932715	−	0.995641i	$-0.529732\pi$
$263$	−3.39349e6			−0.186543			−0.0932715	+	0.995641i	$0.529732\pi$
$264$	1.74750e6			0.0949742
$265$	2.48274e6			0.133411
$266$		−	1.34437e7i		−	0.714286i
$267$		−	4.68691e6i		−	0.246237i
$268$			1.25352e7i			0.651219i
$269$		−	669790.i		−	0.0344098i	−0.999852	−	0.0172049i	$-0.994523\pi$
$269$		−	669790.i		−	0.0344098i	0.999852	−	0.0172049i	$-0.00547676\pi$
$270$		−	4.01121e6i		−	0.203790i
$271$	−9.11749e6			−0.458107			−0.229054	−	0.973414i	$-0.573563\pi$
$271$	−9.11749e6			−0.458107			−0.229054	+	0.973414i	$0.573563\pi$
$272$	3.35363e7			1.66651
$273$			3.06490e7i			1.50636i
$274$			5.46415e6i			0.265626i
$275$			2.10253e6i			0.101099i
$276$		−	4.78781e6i		−	0.227724i
$277$	2.33110e7			1.09679			0.548393	−	0.836221i	$-0.315240\pi$
$277$	2.33110e7			1.09679			0.548393	+	0.836221i	$0.315240\pi$
$278$		−	3.85758e7i		−	1.79548i
$279$			3.36430e6i			0.154911i
$280$		−	1.72808e7i		−	0.787208i
$281$	−1.64403e7			−0.740954			−0.370477	−	0.928842i	$-0.620806\pi$
$281$	−1.64403e7			−0.740954			−0.370477	+	0.928842i	$0.620806\pi$
$282$	2.78470e7			1.24174
$283$		−	2.24174e7i		−	0.989066i	−0.869159	−	0.494533i	$-0.835339\pi$
$283$		−	2.24174e7i		−	0.989066i	0.869159	−	0.494533i	$-0.164661\pi$
$284$	−2.16760e7			−0.946289
$285$	−3.28837e6			−0.142051
$286$		−	1.35061e7i		−	0.577339i
$287$	3.44075e6			0.145548
$288$			8.57239e6i			0.358859i
$289$	1.96170e7			0.812715
$290$			2.57743e7i			1.05680i
$291$			1.91040e7i			0.775256i
$292$		−	4.88960e6i		−	0.196393i
$293$	3.78032e7			1.50289			0.751443	−	0.659798i	$-0.229358\pi$
$293$	3.78032e7			1.50289			0.751443	+	0.659798i	$0.229358\pi$
$294$			4.92209e7i			1.93690i
$295$	1.19890e7	+	1.77521e7i	0.467000	+	0.691486i
$296$	−1.85501e7			−0.715272
$297$			1.67805e6i			0.0640524i
$298$	2.23308e7			0.843832
$299$	2.36047e7			0.883048
$300$	−2.89098e6			−0.107073
$301$		−	4.73329e7i		−	1.73565i
$302$	−5.81255e7			−2.11031
$303$		−	1.44647e7i		−	0.519975i
$304$	1.02539e7			0.364981
$305$		−	2.93677e6i		−	0.103507i
$306$			1.63190e7i			0.569547i
$307$	−1.78309e7			−0.616252			−0.308126	−	0.951345i	$-0.599702\pi$
$307$	−1.78309e7			−0.616252			−0.308126	+	0.951345i	$0.599702\pi$
$308$		−	1.13330e7i		−	0.387875i
$309$		−	6.42372e6i		−	0.217727i
$310$	−1.46607e7			−0.492118
$311$	−4.32981e7			−1.43942			−0.719710	−	0.694274i	$-0.755725\pi$
$311$	−4.32981e7			−1.43942			−0.719710	+	0.694274i	$0.755725\pi$
$312$	−1.18462e7			−0.390047
$313$		−	3.12259e7i		−	1.01831i	−0.860674	−	0.509157i	$-0.829957\pi$
$313$		−	3.12259e7i		−	1.01831i	0.860674	−	0.509157i	$-0.170043\pi$
$314$	3.37846e6			0.109127
$315$	1.65940e7			0.530908
$316$	3.35553e7			1.06341
$317$	344531.			0.0108156			0.00540780	−	0.999985i	$-0.498279\pi$
$317$	344531.			0.0108156			0.00540780	+	0.999985i	$0.498279\pi$
$318$		−	3.76721e6i		−	0.117149i
$319$		−	1.07824e7i		−	0.332158i
$320$	−3.51269e6			−0.107199
$321$	1.37102e7			0.414505
$322$	5.22480e7			1.56496
$323$	1.33782e7			0.397000
$324$	−2.30731e6			−0.0678377
$325$		−	1.42530e7i		−	0.415199i
$326$		−	3.55775e7i		−	1.02689i
$327$		−	1.90505e7i		−	0.544831i
$328$			1.32989e6i			0.0376874i
$329$			1.15201e8i			3.23494i
$330$	−7.31247e6			−0.203480
$331$	1.63978e7			0.452171			0.226086	−	0.974107i	$-0.427407\pi$
$331$	1.63978e7			0.452171			0.226086	+	0.974107i	$0.427407\pi$
$332$		−	2.03237e7i		−	0.555377i
$333$		−	1.78129e7i		−	0.482393i
$334$		−	5.05339e7i		−	1.35626i
$335$			3.34602e7i			0.890007i
$336$	−5.17442e7			−1.36409
$337$		−	7.51217e7i		−	1.96280i	−0.191976	−	0.981400i	$-0.561490\pi$
$337$		−	7.51217e7i		−	1.96280i	0.191976	−	0.981400i	$-0.438510\pi$
$338$			4.25527e7i			1.10199i
$339$		−	1.81524e7i		−	0.465945i
$340$	−2.69585e7			−0.685896
$341$	6.13315e6			0.154675
$342$			4.98964e6i			0.124736i
$343$	−1.26595e8			−3.13715
$344$	1.82948e7			0.449420
$345$		−	1.27801e7i		−	0.311226i
$346$	7.73027e7			1.86623
$347$		−	5.29666e7i		−	1.26769i	−0.773459	−	0.633846i	$-0.781475\pi$
$347$		−	5.29666e7i		−	1.26769i	0.773459	−	0.633846i	$-0.218525\pi$
$348$	1.48258e7			0.351788
$349$			7.42339e7i			1.74633i	0.487426	+	0.873164i	$0.337936\pi$
$349$			7.42339e7i			1.74633i	−0.487426	+	0.873164i	$0.662064\pi$
$350$		−	3.15485e7i		−	0.735824i
$351$		−	1.13754e7i		−	0.263055i
$352$	1.56275e7			0.358313
$353$			6.31527e6i			0.143571i	0.997420	+	0.0717857i	$0.0228698\pi$
$353$			6.31527e6i			0.143571i	−0.997420	+	0.0717857i	$0.977130\pi$
$354$	2.69363e7	−	1.81916e7i	0.607195	−	0.410074i
$355$	−5.78596e7			−1.29327
$356$		−	1.17484e7i		−	0.260392i
$357$	−6.75102e7			−1.48376
$358$	−1.62099e7			−0.353289
$359$	−3.59250e7			−0.776450			−0.388225	−	0.921565i	$-0.626912\pi$
$359$	−3.59250e7			−0.776450			−0.388225	+	0.921565i	$0.626912\pi$
$360$			6.41380e6i			0.137470i
$361$	−4.29554e7			−0.913053
$362$			1.03871e8i			2.18962i
$363$	−2.45568e7			−0.513395
$364$			7.68257e7i			1.59295i
$365$		−	1.30518e7i		−	0.268405i
$366$	−4.45613e6			−0.0908897
$367$			1.84700e7i			0.373653i	0.982393	+	0.186827i	$0.0598203\pi$
$367$			1.84700e7i			0.373653i	−0.982393	+	0.186827i	$0.940180\pi$
$368$			3.98514e7i			0.799650i
$369$	−1.27704e6			−0.0254171
$370$	7.76235e7			1.53246
$371$	1.55846e7			0.305192
$372$			8.43307e6i			0.163816i
$373$	−7.31675e7			−1.40991			−0.704955	−	0.709252i	$-0.749033\pi$
$373$	−7.31675e7			−1.40991			−0.704955	+	0.709252i	$0.749033\pi$
$374$	2.97497e7			0.568679
$375$	−3.31215e7			−0.628082
$376$	−4.45266e7			−0.837636
$377$			7.30937e7i			1.36413i
$378$		−	2.51791e7i		−	0.466191i
$379$	−2.43432e7			−0.447157			−0.223579	−	0.974686i	$-0.571774\pi$
$379$	−2.43432e7			−0.447157			−0.223579	+	0.974686i	$0.571774\pi$
$380$	−8.24272e6			−0.150217
$381$	2.13482e7			0.386000
$382$	−1.14928e8			−2.06176
$383$	3.71024e7			0.660399			0.330199	−	0.943911i	$-0.392884\pi$
$383$	3.71024e7			0.660399			0.330199	+	0.943911i	$0.392884\pi$
$384$		−	2.98648e7i		−	0.527431i
$385$		−	3.02510e7i		−	0.530100i
$386$		−	1.99781e7i		−	0.347369i
$387$			1.75677e7i			0.303097i
$388$			4.78867e7i			0.819822i
$389$	−3.94013e7			−0.669362			−0.334681	−	0.942331i	$-0.608629\pi$
$389$	−3.94013e7			−0.669362			−0.334681	+	0.942331i	$0.608629\pi$
$390$	4.95709e7			0.835667
$391$			5.19938e7i			0.869803i
$392$		−	7.87027e7i		−	1.30657i
$393$		−	6.07891e7i		−	1.00149i
$394$			6.38929e7i			1.04463i
$395$	8.95691e7			1.45334
$396$			4.20625e6i			0.0677344i
$397$			8.86748e7i			1.41719i	0.705614	+	0.708596i	$0.250671\pi$
$397$			8.86748e7i			1.41719i	−0.705614	+	0.708596i	$0.749329\pi$
$398$		−	8.22863e7i		−	1.30520i
$399$	−2.06417e7			−0.324957
$400$	2.40631e7			0.375986
$401$		−	1.10931e8i		−	1.72036i	−0.509990	−	0.860181i	$-0.670351\pi$
$401$		−	1.10931e8i		−	1.72036i	0.509990	−	0.860181i	$-0.329649\pi$
$402$	5.07711e7			0.781517
$403$	−4.15764e7			−0.635231
$404$		−	3.62578e7i		−	0.549866i
$405$	−6.15889e6			−0.0927123
$406$			1.61790e8i			2.41754i
$407$	−3.24730e7			−0.481658
$408$		−	2.60936e7i		−	0.384196i
$409$		−	4.60916e7i		−	0.673677i	−0.941562	−	0.336839i	$-0.890642\pi$
$409$		−	4.60916e7i		−	0.673677i	0.941562	−	0.336839i	$-0.109358\pi$
$410$		−	5.56498e6i		−	0.0807444i
$411$	8.38977e6			0.120844
$412$		−	1.61019e7i		−	0.230243i
$413$	7.52571e7	+	1.11433e8i	1.06831	+	1.58184i
$414$	−1.93920e7			−0.273288
$415$		−	5.42499e7i		−	0.759022i
$416$	−1.05939e8			−1.47155
$417$	−5.92301e7			−0.816835
$418$	9.09615e6			0.124546
$419$		−	6.43202e7i		−	0.874390i	−0.899367	−	0.437195i	$-0.855972\pi$
$419$		−	6.43202e7i		−	0.874390i	0.899367	−	0.437195i	$-0.144028\pi$
$420$	4.15950e7			0.561427
$421$		−	1.80783e7i		−	0.242276i	−0.992636	−	0.121138i	$-0.961346\pi$
$421$		−	1.80783e7i		−	0.242276i	0.992636	−	0.121138i	$-0.0386543\pi$
$422$	1.21616e8			1.61827
$423$		−	4.27569e7i		−	0.564918i
$424$			6.02365e6i			0.0790246i
$425$	3.13949e7			0.408971
$426$			8.77938e7i			1.13563i
$427$		−	1.84346e7i		−	0.236783i
$428$	3.43665e7			0.438333
$429$	−2.07375e7			−0.262654
$430$	−7.65551e7			−0.962872
$431$			1.40777e7i			0.175833i	0.996128	+	0.0879166i	$0.0280209\pi$
$431$			1.40777e7i			0.175833i	−0.996128	+	0.0879166i	$0.971979\pi$
$432$	1.92050e7			0.238211
$433$	−9.56225e7			−1.17787			−0.588934	−	0.808181i	$-0.700452\pi$
$433$	−9.56225e7			−1.17787			−0.588934	+	0.808181i	$0.700452\pi$
$434$	−9.20278e7			−1.12577
$435$	3.95744e7			0.480781
$436$		−	4.77525e7i		−	0.576151i
$437$			1.58974e7i			0.190494i
$438$	−1.98043e7			−0.235687
$439$	8.07733e7			0.954716			0.477358	−	0.878709i	$-0.341595\pi$
$439$	8.07733e7			0.954716			0.477358	+	0.878709i	$0.341595\pi$
$440$	1.16924e7			0.137261
$441$	7.55748e7			0.881173
$442$	−2.01672e8			−2.33549
$443$			8.66157e7i			0.996290i	0.867094	+	0.498145i	$0.165985\pi$
$443$			8.66157e7i			0.996290i	−0.867094	+	0.498145i	$0.834015\pi$
$444$		−	4.46503e7i		−	0.510123i
$445$		−	3.13598e7i		−	0.355872i
$446$			6.93939e7i			0.782198i
$447$		−	3.42872e7i		−	0.383892i
$448$	−2.20498e7			−0.245228
$449$	1.10404e8			1.21968			0.609841	−	0.792524i	$-0.291233\pi$
$449$	1.10404e8			1.21968			0.609841	+	0.792524i	$0.291233\pi$
$450$			1.17093e7i			0.128497i
$451$			2.32806e6i			0.0253784i
$452$		−	4.55013e7i		−	0.492730i
$453$			8.92471e7i			0.960062i
$454$	9.78474e7			1.04564
$455$			2.05070e8i			2.17705i
$456$		−	7.97828e6i		−	0.0841424i
$457$			1.11206e8i			1.16514i	0.812779	+	0.582572i	$0.197953\pi$
$457$			1.11206e8i			1.16514i	−0.812779	+	0.582572i	$0.802047\pi$
$458$	1.04391e8			1.08659
$459$	2.50565e7			0.259109
$460$		−	3.20349e7i		−	0.329117i
$461$	8.07065e7			0.823770			0.411885	−	0.911236i	$-0.364871\pi$
$461$	8.07065e7			0.823770			0.411885	+	0.911236i	$0.364871\pi$
$462$	−4.59017e7			−0.465482
$463$			1.13852e8i			1.14709i	0.819173	+	0.573547i	$0.194433\pi$
$463$			1.13852e8i			1.14709i	−0.819173	+	0.573547i	$0.805567\pi$
$464$	−1.23403e8			−1.23530
$465$			2.25103e7i			0.223884i
$466$	−1.98788e6			−0.0196441
$467$		−	1.34224e8i		−	1.31789i	−0.752192	−	0.658944i	$-0.771003\pi$
$467$		−	1.34224e8i		−	1.31789i	0.752192	−	0.658944i	$-0.228997\pi$
$468$		−	2.85140e7i		−	0.278177i
$469$			2.10035e8i			2.03598i
$470$	1.86323e8			1.79462
$471$		−	5.18736e6i		−	0.0496460i
$472$	−4.30704e7	+	2.90879e7i	−0.409593	+	0.276622i
$473$	3.20260e7			0.302635
$474$		−	1.35909e8i		−	1.27618i
$475$	9.59920e6			0.0895682
$476$	−1.69223e8			−1.56906
$477$	−5.78425e6			−0.0532957
$478$		−	3.79691e7i		−	0.347653i
$479$	1.89590e8			1.72508			0.862538	−	0.505992i	$-0.168874\pi$
$479$	1.89590e8			1.72508			0.862538	+	0.505992i	$0.168874\pi$
$480$			5.73573e7i			0.518639i
$481$	2.20133e8			1.97811
$482$			3.74941e7i			0.334828i
$483$		−	8.02227e7i		−	0.711961i
$484$	−6.15549e7			−0.542908
$485$			1.27824e8i			1.12043i
$486$			9.34526e6i			0.0814109i
$487$	−1.40587e7			−0.121719			−0.0608595	−	0.998146i	$-0.519384\pi$
$487$	−1.40587e7			−0.121719			−0.0608595	+	0.998146i	$0.519384\pi$
$488$	7.12522e6			0.0613111
$489$	−5.46264e7			−0.467171
$490$			3.29334e8i			2.79929i
$491$	1.36388e8			1.15221			0.576104	−	0.817377i	$-0.304572\pi$
$491$	1.36388e8			1.15221			0.576104	+	0.817377i	$0.304572\pi$
$492$	−3.20107e6			−0.0268782
$493$	−1.61003e8			−1.34367
$494$	−6.16625e7			−0.511493
$495$			1.12277e7i			0.0925712i
$496$		−	7.01928e7i		−	0.575238i
$497$	−3.63195e8			−2.95849
$498$	−8.23166e7			−0.666499
$499$	1.80444e8			1.45225			0.726123	−	0.687565i	$-0.241320\pi$
$499$	1.80444e8			1.45225			0.726123	+	0.687565i	$0.241320\pi$
$500$	−8.30235e7			−0.664188
$501$	−7.75907e7			−0.617016
$502$		−	1.07232e8i		−	0.847647i
$503$		−	4.39678e7i		−	0.345486i	−0.984967	−	0.172743i	$-0.944737\pi$
$503$		−	4.39678e7i		−	0.345486i	0.984967	−	0.172743i	$-0.0552631\pi$
$504$			4.02606e7i			0.314477i
$505$		−	9.67827e7i		−	0.751490i
$506$			3.53517e7i			0.272872i
$507$	6.53363e7			0.501338
$508$	5.35121e7			0.408189
$509$			4.22630e7i			0.320484i	0.987078	+	0.160242i	$0.0512275\pi$
$509$			4.22630e7i			0.320484i	−0.987078	+	0.160242i	$0.948773\pi$
$510$			1.09189e8i			0.823133i
$511$		−	8.19284e7i		−	0.614005i
$512$		−	9.67430e7i		−	0.720791i
$513$	7.66119e6			0.0567472
$514$		−	1.63966e7i		−	0.120743i
$515$		−	4.29807e7i		−	0.314668i
$516$			4.40357e7i			0.320521i
$517$	−7.79462e7			−0.564058
$518$	4.87256e8			3.50565
$519$		−	1.18692e8i		−	0.849024i
$520$	−7.92625e7			−0.563712
$521$	2.74322e7			0.193976			0.0969880	−	0.995286i	$-0.469079\pi$
$521$	2.74322e7			0.193976			0.0969880	+	0.995286i	$0.469079\pi$
$522$		−	6.00487e7i		−	0.422175i
$523$	7.55469e7			0.528095			0.264047	−	0.964510i	$-0.414943\pi$
$523$	7.55469e7			0.528095			0.264047	+	0.964510i	$0.414943\pi$
$524$		−	1.52376e8i		−	1.05906i
$525$	−4.84402e7			−0.334755
$526$			3.44526e7i			0.236736i
$527$		−	9.15799e7i		−	0.625703i
$528$		−	3.50108e7i		−	0.237848i
$529$	8.62514e7			0.582638
$530$		−	2.52061e7i		−	0.169309i
$531$	−2.79318e7	−	4.13586e7i	−0.186559	−	0.276237i
$532$	−5.17410e7			−0.343637
$533$		−	1.57818e7i		−	0.104226i
$534$	−4.75842e7			−0.312492
$535$	9.17342e7			0.599060
$536$	−8.11815e7			−0.527185
$537$			2.48890e7i			0.160725i
$538$	−6.80009e6			−0.0436684
$539$		−	1.37773e8i		−	0.879831i
$540$	−1.54381e7			−0.0980419
$541$		−	2.82600e8i		−	1.78476i	−0.451285	−	0.892380i	$-0.649034\pi$
$541$		−	2.82600e8i		−	1.78476i	0.451285	−	0.892380i	$-0.350966\pi$
$542$			9.25659e7i			0.581371i
$543$	1.59486e8			0.996143
$544$		−	2.33350e8i		−	1.44947i
$545$		−	1.27465e8i		−	0.787414i
$546$	3.11166e8			1.91167
$547$	7.30113e7			0.446095			0.223048	−	0.974808i	$-0.428399\pi$
$547$	7.30113e7			0.446095			0.223048	+	0.974808i	$0.428399\pi$
$548$	2.10301e7			0.127791
$549$			6.84204e6i			0.0413493i
$550$	2.13461e7			0.128301
$551$	−4.92276e7			−0.294275
$552$	3.10072e7			0.184351
$553$	5.62241e8			3.32466
$554$		−	2.36667e8i		−	1.39190i
$555$		−	1.19185e8i		−	0.697175i
$556$	−1.48468e8			−0.863791
$557$	−3.01043e8			−1.74206			−0.871031	−	0.491229i	$-0.836548\pi$
$557$	−3.01043e8			−1.74206			−0.871031	+	0.491229i	$0.836548\pi$
$558$	3.41563e7			0.196593
$559$	−2.17103e8			−1.24289
$560$	−3.46217e8			−1.97144
$561$		−	4.56783e7i		−	0.258715i
$562$			1.66911e8i			0.940323i
$563$			1.26927e7i			0.0711262i	0.999367	+	0.0355631i	$0.0113225\pi$
$563$			1.26927e7i			0.0711262i	−0.999367	+	0.0355631i	$0.988678\pi$
$564$		−	1.07176e8i		−	0.597392i
$565$		−	1.21456e8i		−	0.673403i
$566$	−2.27594e8			−1.25519
$567$	−3.86605e7			−0.212089
$568$		−	1.40380e8i		−	0.766054i
$569$			2.72018e8i			1.47659i	0.674476	+	0.738297i	$0.264370\pi$
$569$			2.72018e8i			1.47659i	−0.674476	+	0.738297i	$0.735630\pi$
$570$			3.33853e7i			0.180273i
$571$			2.66755e8i			1.43286i	0.697658	+	0.716431i	$0.254226\pi$
$571$			2.66755e8i			1.43286i	−0.697658	+	0.716431i	$0.745774\pi$
$572$	−5.19813e7			−0.277753
$573$			1.76463e8i			0.937974i
$574$		−	3.49324e7i		−	0.184711i
$575$			3.73068e7i			0.196238i
$576$	8.18383e6			0.0428242
$577$	−2.29588e8			−1.19515			−0.597575	−	0.801813i	$-0.703869\pi$
$577$	−2.29588e8			−1.19515			−0.597575	+	0.801813i	$0.703869\pi$
$578$		−	1.99162e8i		−	1.03139i
$579$	−3.06747e7			−0.158032
$580$	9.91986e7			0.508419
$581$		−	3.40536e8i		−	1.73634i
$582$	1.93954e8			0.983854
$583$			1.05447e7i			0.0532145i
$584$	3.16664e7			0.158987
$585$		−	7.61123e7i		−	0.380178i
$586$		−	3.83800e8i		−	1.90727i
$587$		−	3.03353e8i		−	1.49980i	−0.661550	−	0.749901i	$-0.730101\pi$
$587$		−	3.03353e8i		−	1.49980i	0.661550	−	0.749901i	$-0.269899\pi$
$588$	1.89438e8			0.931827
$589$		−	2.80011e7i		−	0.137034i
$590$	1.80229e8	−	1.21719e8i	0.877544	−	0.592656i
$591$	9.81025e7			0.475245
$592$			3.71647e8i			1.79129i
$593$	3.40728e7			0.163397			0.0816985	−	0.996657i	$-0.473966\pi$
$593$	3.40728e7			0.163397			0.0816985	+	0.996657i	$0.473966\pi$
$594$	1.70365e7			0.0812870
$595$	−4.51706e8			−2.14440
$596$		−	8.59454e7i		−	0.405961i
$597$	−1.26344e8			−0.593789
$598$		−	2.39648e8i		−	1.12065i
$599$	2.96815e8			1.38104			0.690520	−	0.723314i	$-0.257382\pi$
$599$	2.96815e8			1.38104			0.690520	+	0.723314i	$0.257382\pi$
$600$		−	1.87228e7i		−	0.0866795i
$601$			1.19578e8i			0.550845i	0.961323	+	0.275423i	$0.0888178\pi$
$601$			1.19578e8i			0.550845i	−0.961323	+	0.275423i	$0.911182\pi$
$602$	−4.80550e8			−2.20267
$603$		−	7.79550e7i		−	0.355543i
$604$			2.23710e8i			1.01525i
$605$	−1.64308e8			−0.741981
$606$	−1.46854e8			−0.659885
$607$	−4.34360e8			−1.94216			−0.971078	−	0.238763i	$-0.923258\pi$
$607$	−4.34360e8			−1.94216			−0.971078	+	0.238763i	$0.923258\pi$
$608$		−	7.13481e7i		−	0.317447i
$609$	2.48416e8			1.09983
$610$	−2.98157e7			−0.131358
$611$	5.28394e8			2.31651
$612$	6.28075e7			0.274004
$613$		−	1.91315e8i		−	0.830552i	−0.909696	−	0.415276i	$-0.863685\pi$
$613$		−	1.91315e8i		−	0.830552i	0.909696	−	0.415276i	$-0.136315\pi$
$614$			1.81030e8i			0.782068i
$615$	−8.54459e6			−0.0367338
$616$	7.33954e7			0.313998
$617$	1.22851e8			0.523027			0.261513	−	0.965200i	$-0.415778\pi$
$617$	1.22851e8			0.523027			0.261513	+	0.965200i	$0.415778\pi$
$618$	−6.52173e7			−0.276310
$619$	2.76299e8			1.16495			0.582474	−	0.812849i	$-0.302085\pi$
$619$	2.76299e8			1.16495			0.582474	+	0.812849i	$0.302085\pi$
$620$			5.64251e7i			0.236754i
$621$			2.97748e7i			0.124330i
$622$			4.39587e8i			1.82673i
$623$		−	1.96851e8i		−	0.814093i
$624$			2.37337e8i			0.976813i
$625$	−1.47454e8			−0.603973
$626$	−3.17022e8			−1.29231
$627$		−	1.39664e7i		−	0.0566608i
$628$		−	1.30028e7i		−	0.0524999i
$629$			4.84885e8i			1.94844i
$630$		−	1.68472e8i		−	0.673760i
$631$	−4.17250e7			−0.166077			−0.0830383	−	0.996546i	$-0.526462\pi$
$631$	−4.17250e7			−0.166077			−0.0830383	+	0.996546i	$0.526462\pi$
$632$			2.17314e8i			0.860867i
$633$		−	1.86731e8i		−	0.736217i
$634$		−	3.49787e6i		−	0.0137258i
$635$	1.42840e8			0.557863
$636$	−1.44990e7			−0.0563594
$637$			9.33961e8i			3.61335i
$638$	−1.09469e8			−0.421532
$639$	1.34800e8			0.516641
$640$		−	1.99823e8i		−	0.762266i
$641$	4.30270e8			1.63368			0.816840	−	0.576864i	$-0.195724\pi$
$641$	4.30270e8			1.63368			0.816840	+	0.576864i	$0.195724\pi$
$642$		−	1.39194e8i		−	0.526035i
$643$	5.22591e7			0.196575			0.0982877	−	0.995158i	$-0.468663\pi$
$643$	5.22591e7			0.196575			0.0982877	+	0.995158i	$0.468663\pi$
$644$		−	2.01089e8i		−	0.752888i
$645$			1.17544e8i			0.438048i
$646$		−	1.35823e8i		−	0.503821i
$647$	1.95145e8			0.720516			0.360258	−	0.932853i	$-0.382689\pi$
$647$	1.95145e8			0.720516			0.360258	+	0.932853i	$0.382689\pi$
$648$		−	1.49428e7i		−	0.0549170i
$649$	−7.53971e7	+	5.09200e7i	−0.275817	+	0.186275i
$650$	−1.44704e8			−0.526916
$651$			1.41301e8i			0.512157i
$652$	−1.36928e8			−0.494027
$653$	−2.87811e8			−1.03364			−0.516818	−	0.856095i	$-0.672884\pi$
$653$	−2.87811e8			−1.03364			−0.516818	+	0.856095i	$0.672884\pi$
$654$	−1.93411e8			−0.691429
$655$		−	4.06736e8i		−	1.44740i
$656$	2.66442e7			0.0943823
$657$			3.04079e7i			0.107224i
$658$	1.16958e9			4.10537
$659$		−	4.19404e8i		−	1.46547i	−0.680516	−	0.732734i	$-0.738244\pi$
$659$		−	4.19404e8i		−	1.46547i	0.680516	−	0.732734i	$-0.261756\pi$
$660$			2.81438e7i			0.0978927i
$661$	−1.64917e8			−0.571032			−0.285516	−	0.958374i	$-0.592165\pi$
$661$	−1.64917e8			−0.571032			−0.285516	+	0.958374i	$0.592165\pi$
$662$		−	1.66480e8i		−	0.573837i
$663$			3.09651e8i			1.06251i
$664$	1.31622e8			0.449597
$665$	−1.38112e8			−0.469642
$666$	−1.80846e8			−0.612190
$667$		−	1.91320e8i		−	0.644739i
$668$	−1.94491e8			−0.652486
$669$	1.06549e8			0.355853
$670$	3.39706e8			1.12948
$671$	1.24731e7			0.0412864
$672$			3.60042e8i			1.18644i
$673$			4.61298e8i			1.51334i	0.653796	+	0.756671i	$0.273175\pi$
$673$			4.61298e8i			1.51334i	−0.653796	+	0.756671i	$0.726825\pi$
$674$	−7.62678e8			−2.49093
$675$	1.79787e7			0.0584583
$676$	1.63774e8			0.530157
$677$	4.24564e7			0.136829			0.0684143	−	0.997657i	$-0.478206\pi$
$677$	4.24564e7			0.136829			0.0684143	+	0.997657i	$0.478206\pi$
$678$	−1.84293e8			−0.591317
$679$			8.02371e8i			2.56310i
$680$		−	1.74591e8i		−	0.555257i
$681$		−	1.50237e8i		−	0.475703i
$682$		−	6.22672e7i		−	0.196294i
$683$		−	6.61697e7i		−	0.207681i	−0.994594	−	0.103841i	$-0.966887\pi$
$683$		−	6.61697e7i		−	0.207681i	0.994594	−	0.103841i	$-0.0331132\pi$
$684$	1.92038e7			0.0600093
$685$	5.61355e7			0.174649
$686$			1.28527e9i			3.98126i
$687$		−	1.60284e8i		−	0.494334i
$688$		−	3.66532e8i		−	1.12550i
$689$		−	7.14824e7i		−	0.218545i
$690$	−1.29750e8			−0.394968
$691$		−	9.68980e7i		−	0.293684i	−0.989160	−	0.146842i	$-0.953089\pi$
$691$		−	9.68980e7i		−	0.293684i	0.989160	−	0.146842i	$-0.0469109\pi$
$692$		−	2.97518e8i		−	0.897830i
$693$			7.04784e7i			0.211766i
$694$	−5.37747e8			−1.60879
$695$	−3.96305e8			−1.18053
$696$			9.60161e7i			0.284784i
$697$	3.47624e7			0.102662
$698$	7.53664e8			2.21621
$699$			3.05223e6i			0.00893688i
$700$	−1.21422e8			−0.353999
$701$			5.03940e8i			1.46294i	0.681876	+	0.731468i	$0.261164\pi$
$701$			5.03940e8i			1.46294i	−0.681876	+	0.731468i	$0.738836\pi$
$702$	−1.15490e8			−0.333835
$703$			1.48257e8i			0.426725i
$704$		−	1.49192e7i		−	0.0427590i
$705$		−	2.86084e8i		−	0.816443i
$706$	6.41162e7			0.182202
$707$		−	6.07522e8i		−	1.71911i
$708$	−7.00148e7	−	1.03671e8i	−0.197283	−	0.292117i
$709$	1.17112e8			0.328596			0.164298	−	0.986411i	$-0.447464\pi$
$709$	1.17112e8			0.328596			0.164298	+	0.986411i	$0.447464\pi$
$710$			5.87423e8i			1.64125i
$711$	−2.08677e8			−0.580585
$712$	7.60856e7			0.210796
$713$	1.08825e8			0.300234
$714$			6.85401e8i			1.88300i
$715$	−1.38753e8			−0.379599
$716$			6.23874e7i			0.169964i
$717$	−5.82985e7			−0.158161
$718$			3.64731e8i			0.985369i
$719$		−	5.86668e8i		−	1.57836i	−0.614162	−	0.789180i	$-0.710506\pi$
$719$		−	5.86668e8i		−	1.57836i	0.614162	−	0.789180i	$-0.289494\pi$
$720$	1.28499e8			0.344273
$721$		−	2.69798e8i		−	0.719834i
$722$			4.36107e8i			1.15873i
$723$	5.75693e7			0.152327
$724$	3.99772e8			1.05341
$725$	−1.15523e8			−0.303149
$726$			2.49314e8i			0.651535i
$727$	−1.83589e8			−0.477797			−0.238898	−	0.971045i	$-0.576786\pi$
$727$	−1.83589e8			−0.477797			−0.238898	+	0.971045i	$0.576786\pi$
$728$	−4.97545e8			−1.28955
$729$	1.43489e7			0.0370370
$730$	−1.32509e8			−0.340625
$731$		−	4.78211e8i		−	1.22424i
$732$			1.71505e7i			0.0437263i
$733$	−1.27061e8			−0.322626			−0.161313	−	0.986903i	$-0.551573\pi$
$733$	−1.27061e8			−0.322626			−0.161313	+	0.986903i	$0.551573\pi$
$734$	1.87518e8			0.474192
$735$	5.05666e8			1.27351
$736$	2.77291e8			0.695508
$737$	−1.42113e8			−0.355002
$738$			1.29652e7i			0.0322560i
$739$		−	6.00659e8i		−	1.48831i	−0.668005	−	0.744157i	$-0.732851\pi$
$739$		−	6.00659e8i		−	1.48831i	0.668005	−	0.744157i	$-0.267149\pi$
$740$		−	2.98752e8i		−	0.737252i
$741$			9.46779e7i			0.232699i
$742$		−	1.58223e8i		−	0.387310i
$743$	3.85539e8			0.939944			0.469972	−	0.882681i	$-0.344264\pi$
$743$	3.85539e8			0.939944			0.469972	+	0.882681i	$0.344264\pi$
$744$	−5.46149e7			−0.132615
$745$		−	2.29414e8i		−	0.554818i
$746$			7.42838e8i			1.78928i
$747$			1.26391e8i			0.303217i
$748$		−	1.14499e8i		−	0.273587i
$749$	5.75832e8			1.37041
$750$			3.36268e8i			0.797080i
$751$			1.49633e8i			0.353270i	0.984276	+	0.176635i	$0.0565213\pi$
$751$			1.49633e8i			0.353270i	−0.984276	+	0.176635i	$0.943479\pi$
$752$			8.92080e8i			2.09773i
$753$	−1.64647e8			−0.385628
$754$	7.42088e8			1.73118
$755$			5.97147e8i			1.38752i
$756$	−9.69076e7			−0.224281
$757$	3.76913e7			0.0868868			0.0434434	−	0.999056i	$-0.486167\pi$
$757$	3.76913e7			0.0868868			0.0434434	+	0.999056i	$0.486167\pi$
$758$			2.47146e8i			0.567474i
$759$	5.42798e7			0.124140
$760$		−	5.33822e7i		−	0.121606i
$761$	−3.09488e8			−0.702248			−0.351124	−	0.936329i	$-0.614200\pi$
$761$	−3.09488e8			−0.702248			−0.351124	+	0.936329i	$0.614200\pi$
$762$		−	2.16739e8i		−	0.489861i
$763$		−	8.00123e8i		−	1.80129i
$764$			4.42329e8i			0.991894i
$765$	1.67652e8			0.374476
$766$		−	3.76685e8i		−	0.838092i
$767$	5.11114e8	−	3.45185e8i	1.13274	−	0.765007i
$768$	−3.36804e8			−0.743521
$769$			1.14414e8i			0.251594i	0.992056	+	0.125797i	$0.0401488\pi$
$769$			1.14414e8i			0.251594i	−0.992056	+	0.125797i	$0.959851\pi$
$770$	−3.07125e8			−0.672734
$771$	−2.51756e7			−0.0549310
$772$	−7.68902e7			−0.167116
$773$			3.74267e8i			0.810294i	0.914252	+	0.405147i	$0.132780\pi$
$773$			3.74267e8i			0.810294i	−0.914252	+	0.405147i	$0.867220\pi$
$774$	1.78357e8			0.384651
$775$		−	6.57108e7i		−	0.141166i
$776$	−3.10127e8			−0.663674
$777$		−	7.48144e8i		−	1.59486i
$778$			4.00024e8i			0.849468i
$779$	1.06288e7			0.0224840
$780$		−	1.90785e8i		−	0.402033i
$781$		−	2.45742e8i		−	0.515854i
$782$	5.27870e8			1.10384
$783$	−9.22000e7			−0.192064
$784$	−1.57679e9			−3.27210
$785$		−	3.47083e7i		−	0.0717505i

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)

\(f(q)\)	\(=\)	\(q-10.1526i q^{2} -15.5885 q^{3} -39.0745 q^{4} -104.301 q^{5} +158.263i q^{6} -654.718 q^{7} -253.057i q^{8} +243.000 q^{9} +O(q^{10})\)	\(q-10.1526i q^{2} -15.5885 q^{3} -39.0745 q^{4} -104.301 q^{5} +158.263i q^{6} -654.718 q^{7} -253.057i q^{8} +243.000 q^{9} +1058.93i q^{10} -442.991i q^{11} +609.112 q^{12} +3003.02i q^{13} +6647.07i q^{14} +1625.90 q^{15} -5069.95 q^{16} -6614.72 q^{17} -2467.07i q^{18} -2022.49 q^{19} +4075.53 q^{20} +10206.1 q^{21} -4497.50 q^{22} -7860.31i q^{23} +3944.78i q^{24} -4746.22 q^{25} +30488.4 q^{26} -3788.00 q^{27} +25582.8 q^{28} +24340.1 q^{29} -16507.0i q^{30} +13844.9i q^{31} +35277.3i q^{32} +6905.55i q^{33} +67156.4i q^{34} +68288.0 q^{35} -9495.11 q^{36} -73303.9i q^{37} +20533.5i q^{38} -46812.5i q^{39} +26394.2i q^{40} -5255.31 q^{41} -103618. i q^{42} +72295.0i q^{43} +17309.7i q^{44} -25345.2 q^{45} -79802.3 q^{46} -175954. i q^{47} +79032.7 q^{48} +311007. q^{49} +48186.3i q^{50} +103113. q^{51} -117342. i q^{52} -23803.5 q^{53} +38457.9i q^{54} +46204.6i q^{55} +165681. i q^{56} +31527.5 q^{57} -247114. i q^{58} +(-114946. - 170200. i) q^{59} -63531.2 q^{60} +28156.5i q^{61} +140561. q^{62} -159097. q^{63} +33678.3 q^{64} -313219. i q^{65} +70109.0 q^{66} -320803. i q^{67} +258467. q^{68} +122530. i q^{69} -693299. i q^{70} +554734. q^{71} -61493.0i q^{72} +125135. i q^{73} -744223. q^{74} +73986.2 q^{75} +79027.9 q^{76} +290035. i q^{77} -475266. q^{78} -858752. q^{79} +528803. q^{80} +59049.0 q^{81} +53354.8i q^{82} +520126. i q^{83} -398797. q^{84} +689925. q^{85} +733979. q^{86} -379424. q^{87} -112102. q^{88} +300665. i q^{89} +257319. i q^{90} -1.96613e6i q^{91} +307138. i q^{92} -215820. i q^{93} -1.78639e6 q^{94} +210949. q^{95} -549919. i q^{96} -1.22552e6i q^{97} -3.15752e6i q^{98} -107647. i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 60q - 1920q^{4} - 408q^{7} + 14580q^{9} + O(q^{10}) \)	\( 60q - 1920q^{4} - 408q^{7} + 14580q^{9} - 1944q^{12} - 4536q^{15} + 56616q^{16} + 8480q^{17} + 11376q^{19} + 40796q^{20} - 8232q^{22} + 197940q^{25} + 147252q^{26} + 71640q^{28} + 63456q^{29} - 364432q^{35} - 466560q^{36} + 99632q^{41} - 470316q^{46} + 171072q^{48} + 1737420q^{49} + 60912q^{51} + 92240q^{53} + 186624q^{57} + 917264q^{59} + 1063368q^{60} - 115768q^{62} - 99144q^{63} - 1107444q^{64} + 1172232q^{66} - 4247232q^{68} + 1498048q^{71} + 1161448q^{74} - 1477440q^{75} - 1045320q^{76} - 1060452q^{78} - 90600q^{79} + 77096q^{80} + 3542940q^{81} - 2225880q^{84} - 693408q^{85} - 1567768q^{86} + 1821528q^{87} + 62892q^{88} + 5268696q^{94} + 296128q^{95} + O(q^{100}) \)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Properties

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