# Properties

 Label 177.4.d.b.176.4 Level $177$ Weight $4$ Character 177.176 Analytic conductor $10.443$ Analytic rank $0$ Dimension $4$ CM discriminant -59 Inner twists $4$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$10.4433380710$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{-59})$$ Defining polynomial: $$x^{4} - x^{3} - 14 x^{2} - 15 x + 225$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2\cdot 3$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## Embedding invariants

 Embedding label 176.4 Root $$-3.07603 - 2.35330i$$ of defining polynomial Character $$\chi$$ $$=$$ 177.176 Dual form 177.4.d.b.176.3

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+(1.57603 + 4.95138i) q^{3} -8.00000 q^{4} +14.3460i q^{5} +5.45620 q^{7} +(-22.0322 + 15.6071i) q^{9} +O(q^{10})$$ $$q+(1.57603 + 4.95138i) q^{3} -8.00000 q^{4} +14.3460i q^{5} +5.45620 q^{7} +(-22.0322 + 15.6071i) q^{9} +(-12.6083 - 39.6110i) q^{12} +(-71.0322 + 22.6097i) q^{15} +64.0000 q^{16} -61.4492i q^{17} -159.281 q^{19} -114.768i q^{20} +(8.59916 + 27.0157i) q^{21} -80.8066 q^{25} +(-112.000 - 84.4926i) q^{27} -43.6496 q^{28} -3.27799i q^{29} +78.2745i q^{35} +(176.258 - 124.857i) q^{36} +450.823i q^{41} +(-223.898 - 316.074i) q^{45} +(100.866 + 316.888i) q^{48} -313.230 q^{49} +(304.258 - 96.8460i) q^{51} -66.3018i q^{53} +(-251.032 - 788.660i) q^{57} +453.188i q^{59} +(568.258 - 180.878i) q^{60} +(-120.212 + 85.1553i) q^{63} -512.000 q^{64} +491.593i q^{68} +1182.90i q^{71} +(-127.354 - 400.104i) q^{75} +1274.25 q^{76} +1395.35 q^{79} +918.141i q^{80} +(241.839 - 687.717i) q^{81} +(-68.7933 - 216.126i) q^{84} +881.547 q^{85} +(16.2306 - 5.16622i) q^{87} -2285.04i q^{95} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 7q^{3} - 32q^{4} - 58q^{7} + 5q^{9} + O(q^{10})$$ $$4q - 7q^{3} - 32q^{4} - 58q^{7} + 5q^{9} + 56q^{12} - 191q^{15} + 256q^{16} - 238q^{19} + 367q^{21} - 882q^{25} - 448q^{27} + 464q^{28} - 40q^{36} + 49q^{45} - 448q^{48} + 1062q^{49} + 472q^{51} - 911q^{57} + 1528q^{60} - 1931q^{63} - 2048q^{64} + 3402q^{75} + 1904q^{76} + 1670q^{79} + 1433q^{81} - 2936q^{84} - 944q^{85} + 637q^{87} + O(q^{100})$$

## 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 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$3$$ 1.57603 + 4.95138i 0.303308 + 0.952893i
$$4$$ −8.00000 −1.00000
$$5$$ 14.3460i 1.28314i 0.767064 + 0.641571i $$0.221717\pi$$
−0.767064 + 0.641571i $$0.778283\pi$$
$$6$$ 0 0
$$7$$ 5.45620 0.294607 0.147304 0.989091i $$-0.452941\pi$$
0.147304 + 0.989091i $$0.452941\pi$$
$$8$$ 0 0
$$9$$ −22.0322 + 15.6071i −0.816009 + 0.578040i
$$10$$ 0 0
$$11$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$12$$ −12.6083 39.6110i −0.303308 0.952893i
$$13$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$14$$ 0 0
$$15$$ −71.0322 + 22.6097i −1.22270 + 0.389187i
$$16$$ 64.0000 1.00000
$$17$$ 61.4492i 0.876683i −0.898808 0.438342i $$-0.855566\pi$$
0.898808 0.438342i $$-0.144434\pi$$
$$18$$ 0 0
$$19$$ −159.281 −1.92324 −0.961620 0.274384i $$-0.911526\pi$$
−0.961620 + 0.274384i $$0.911526\pi$$
$$20$$ 114.768i 1.28314i
$$21$$ 8.59916 + 27.0157i 0.0893567 + 0.280729i
$$22$$ 0 0
$$23$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$24$$ 0 0
$$25$$ −80.8066 −0.646453
$$26$$ 0 0
$$27$$ −112.000 84.4926i −0.798311 0.602245i
$$28$$ −43.6496 −0.294607
$$29$$ 3.27799i 0.0209899i −0.999945 0.0104950i $$-0.996659\pi$$
0.999945 0.0104950i $$-0.00334071\pi$$
$$30$$ 0 0
$$31$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 78.2745i 0.378023i
$$36$$ 176.258 124.857i 0.816009 0.578040i
$$37$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 450.823i 1.71724i 0.512616 + 0.858618i $$0.328677\pi$$
−0.512616 + 0.858618i $$0.671323\pi$$
$$42$$ 0 0
$$43$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$44$$ 0 0
$$45$$ −223.898 316.074i −0.741707 1.04705i
$$46$$ 0 0
$$47$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$48$$ 100.866 + 316.888i 0.303308 + 0.952893i
$$49$$ −313.230 −0.913207
$$50$$ 0 0
$$51$$ 304.258 96.8460i 0.835385 0.265905i
$$52$$ 0 0
$$53$$ 66.3018i 0.171835i −0.996302 0.0859175i $$-0.972618\pi$$
0.996302 0.0859175i $$-0.0273822\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ −251.032 788.660i −0.583334 1.83264i
$$58$$ 0 0
$$59$$ 453.188i 1.00000i
$$60$$ 568.258 180.878i 1.22270 0.389187i
$$61$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$62$$ 0 0
$$63$$ −120.212 + 85.1553i −0.240402 + 0.170295i
$$64$$ −512.000 −1.00000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$68$$ 491.593i 0.876683i
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 1182.90i 1.97724i 0.150437 + 0.988620i $$0.451932\pi$$
−0.150437 + 0.988620i $$0.548068\pi$$
$$72$$ 0 0
$$73$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$74$$ 0 0
$$75$$ −127.354 400.104i −0.196074 0.616000i
$$76$$ 1274.25 1.92324
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 1395.35 1.98721 0.993605 0.112913i $$-0.0360180\pi$$
0.993605 + 0.112913i $$0.0360180\pi$$
$$80$$ 918.141i 1.28314i
$$81$$ 241.839 687.717i 0.331740 0.943371i
$$82$$ 0 0
$$83$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$84$$ −68.7933 216.126i −0.0893567 0.280729i
$$85$$ 881.547 1.12491
$$86$$ 0 0
$$87$$ 16.2306 5.16622i 0.0200011 0.00636640i
$$88$$ 0 0
$$89$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 2285.04i 2.46779i
$$96$$ 0 0
$$97$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 646.453 0.646453
$$101$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$102$$ 0 0
$$103$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$104$$ 0 0
$$105$$ −387.566 + 123.363i −0.360215 + 0.114657i
$$106$$ 0 0
$$107$$ 1276.32i 1.15314i 0.817047 + 0.576570i $$0.195609\pi$$
−0.817047 + 0.576570i $$0.804391\pi$$
$$108$$ 896.000 + 675.941i 0.798311 + 0.602245i
$$109$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 349.197 0.294607
$$113$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 26.2239i 0.0209899i
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 335.279i 0.258277i
$$120$$ 0 0
$$121$$ −1331.00 −1.00000
$$122$$ 0 0
$$123$$ −2232.19 + 710.512i −1.63634 + 0.520851i
$$124$$ 0 0
$$125$$ 633.997i 0.453651i
$$126$$ 0 0
$$127$$ −2785.29 −1.94610 −0.973049 0.230600i $$-0.925931\pi$$
−0.973049 + 0.230600i $$0.925931\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$132$$ 0 0
$$133$$ −869.069 −0.566601
$$134$$ 0 0
$$135$$ 1212.13 1606.75i 0.772765 1.02435i
$$136$$ 0 0
$$137$$ 1994.05i 1.24353i 0.783204 + 0.621765i $$0.213584\pi$$
−0.783204 + 0.621765i $$0.786416\pi$$
$$138$$ 0 0
$$139$$ −1960.00 −1.19601 −0.598004 0.801493i $$-0.704039\pi$$
−0.598004 + 0.801493i $$0.704039\pi$$
$$140$$ 626.196i 0.378023i
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 0 0
$$144$$ −1410.06 + 998.852i −0.816009 + 0.578040i
$$145$$ 47.0259 0.0269330
$$146$$ 0 0
$$147$$ −493.661 1550.92i −0.276983 0.870188i
$$148$$ 0 0
$$149$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$150$$ 0 0
$$151$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$152$$ 0 0
$$153$$ 959.041 + 1353.86i 0.506758 + 0.715381i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$158$$ 0 0
$$159$$ 328.285 104.494i 0.163740 0.0521189i
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −4048.00 −1.94518 −0.972588 0.232533i $$-0.925299\pi$$
−0.972588 + 0.232533i $$0.925299\pi$$
$$164$$ 3606.58i 1.71724i
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 4045.40i 1.87451i −0.348648 0.937254i $$-0.613359\pi$$
0.348648 0.937254i $$-0.386641\pi$$
$$168$$ 0 0
$$169$$ 2197.00 1.00000
$$170$$ 0 0
$$171$$ 3509.32 2485.91i 1.56938 1.11171i
$$172$$ 0 0
$$173$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$174$$ 0 0
$$175$$ −440.897 −0.190450
$$176$$ 0 0
$$177$$ −2243.90 + 714.239i −0.952893 + 0.303308i
$$178$$ 0 0
$$179$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$180$$ 1791.19 + 2528.59i 0.741707 + 1.04705i
$$181$$ 3135.88 1.28778 0.643890 0.765118i $$-0.277320\pi$$
0.643890 + 0.765118i $$0.277320\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ −611.095 461.009i −0.235188 0.177426i
$$190$$ 0 0
$$191$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$192$$ −806.929 2535.10i −0.303308 0.952893i
$$193$$ 3362.00 1.25390 0.626948 0.779061i $$-0.284304\pi$$
0.626948 + 0.779061i $$0.284304\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 2505.84 0.913207
$$197$$ 4086.37i 1.47788i 0.673773 + 0.738939i $$0.264673\pi$$
−0.673773 + 0.738939i $$0.735327\pi$$
$$198$$ 0 0
$$199$$ −4002.65 −1.42583 −0.712916 0.701249i $$-0.752626\pi$$
−0.712916 + 0.701249i $$0.752626\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 17.8854i 0.00618378i
$$204$$ −2434.06 + 774.768i −0.835385 + 0.265905i
$$205$$ −6467.49 −2.20346
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$212$$ 530.415i 0.171835i
$$213$$ −5856.96 + 1864.28i −1.88410 + 0.599712i
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 5852.00 1.75730 0.878652 0.477462i $$-0.158443\pi$$
0.878652 + 0.477462i $$0.158443\pi$$
$$224$$ 0 0
$$225$$ 1780.35 1261.15i 0.527511 0.373675i
$$226$$ 0 0
$$227$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$228$$ 2008.26 + 6309.28i 0.583334 + 1.83264i
$$229$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 3625.50i 1.00000i
$$237$$ 2199.12 + 6908.92i 0.602736 + 1.89360i
$$238$$ 0 0
$$239$$ 5457.37i 1.47702i 0.674242 + 0.738511i $$0.264471\pi$$
−0.674242 + 0.738511i $$0.735529\pi$$
$$240$$ −4546.06 + 1447.02i −1.22270 + 0.389187i
$$241$$ 6992.52 1.86900 0.934498 0.355969i $$-0.115849\pi$$
0.934498 + 0.355969i $$0.115849\pi$$
$$242$$ 0 0
$$243$$ 3786.29 + 113.569i 0.999550 + 0.0299814i
$$244$$ 0 0
$$245$$ 4493.58i 1.17177i
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 2329.28i 0.585749i −0.956151 0.292875i $$-0.905388\pi$$
0.956151 0.292875i $$-0.0946118\pi$$
$$252$$ 961.699 681.243i 0.240402 0.170295i
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 1389.35 + 4364.87i 0.341194 + 1.07192i
$$256$$ 4096.00 1.00000
$$257$$ 8191.93i 1.98832i −0.107907 0.994161i $$-0.534415\pi$$
0.107907 0.994161i $$-0.465585\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 51.1598 + 72.2214i 0.0121330 + 0.0171279i
$$262$$ 0 0
$$263$$ 3158.18i 0.740462i −0.928940 0.370231i $$-0.879279\pi$$
0.928940 0.370231i $$-0.120721\pi$$
$$264$$ 0 0
$$265$$ 951.163 0.220489
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$270$$ 0 0
$$271$$ 2152.19 0.482421 0.241210 0.970473i $$-0.422456\pi$$
0.241210 + 0.970473i $$0.422456\pi$$
$$272$$ 3932.75i 0.876683i
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 5553.35 1.20458 0.602290 0.798277i $$-0.294255\pi$$
0.602290 + 0.798277i $$0.294255\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 8289.78i 1.75988i 0.475083 + 0.879941i $$0.342418\pi$$
−0.475083 + 0.879941i $$0.657582\pi$$
$$282$$ 0 0
$$283$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$284$$ 9463.17i 1.97724i
$$285$$ 11314.1 3601.30i 2.35154 0.748500i
$$286$$ 0 0
$$287$$ 2459.78i 0.505910i
$$288$$ 0 0
$$289$$ 1137.00 0.231427
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 10021.4i 1.99815i 0.0430278 + 0.999074i $$0.486300\pi$$
−0.0430278 + 0.999074i $$0.513700\pi$$
$$294$$ 0 0
$$295$$ −6501.41 −1.28314
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 1018.83 + 3200.83i 0.196074 + 0.616000i
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 0 0
$$304$$ −10194.0 −1.92324
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 10591.3 1.96898 0.984488 0.175450i $$-0.0561380\pi$$
0.984488 + 0.175450i $$0.0561380\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 10124.2i 1.84595i −0.384863 0.922974i $$-0.625751\pi$$
0.384863 0.922974i $$-0.374249\pi$$
$$312$$ 0 0
$$313$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$314$$ 0 0
$$315$$ −1221.63 1724.56i −0.218512 0.308470i
$$316$$ −11162.8 −1.98721
$$317$$ 215.072i 0.0381062i 0.999818 + 0.0190531i $$0.00606515\pi$$
−0.999818 + 0.0190531i $$0.993935\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 7345.13i 1.28314i
$$321$$ −6319.52 + 2011.52i −1.09882 + 0.349757i
$$322$$ 0 0
$$323$$ 9787.69i 1.68607i
$$324$$ −1934.71 + 5501.74i −0.331740 + 0.943371i
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −6706.97 −1.11374 −0.556870 0.830599i $$-0.687998\pi$$
−0.556870 + 0.830599i $$0.687998\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 550.346 + 1729.01i 0.0893567 + 0.280729i
$$337$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ −7052.38 −1.12491
$$341$$ 0 0
$$342$$ 0 0
$$343$$ −3580.52 −0.563644
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$348$$ −129.844 + 41.3298i −0.0200011 + 0.00636640i
$$349$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$354$$ 0 0
$$355$$ −16969.8 −2.53708
$$356$$ 0 0
$$357$$ 1660.09 528.411i 0.246110 0.0783375i
$$358$$ 0 0
$$359$$ 5516.23i 0.810963i −0.914103 0.405481i $$-0.867104\pi$$
0.914103 0.405481i $$-0.132896\pi$$
$$360$$ 0 0
$$361$$ 18511.4 2.69885
$$362$$ 0 0
$$363$$ −2097.70 6590.28i −0.303308 0.952893i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$368$$ 0 0
$$369$$ −7036.02 9932.63i −0.992631 1.40128i
$$370$$ 0 0
$$371$$ 361.756i 0.0506238i
$$372$$ 0 0
$$373$$ 4898.00 0.679916 0.339958 0.940441i $$-0.389587\pi$$
0.339958 + 0.940441i $$0.389587\pi$$
$$374$$ 0 0
$$375$$ −3139.16 + 999.201i −0.432281 + 0.137596i
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ −505.683 −0.0685361 −0.0342681 0.999413i $$-0.510910\pi$$
−0.0342681 + 0.999413i $$0.510910\pi$$
$$380$$ 18280.3i 2.46779i
$$381$$ −4389.71 13791.0i −0.590267 1.85442i
$$382$$ 0 0
$$383$$ 14025.8i 1.87124i −0.353014 0.935618i $$-0.614843\pi$$
0.353014 0.935618i $$-0.385157\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 11829.0i 1.54178i 0.636968 + 0.770890i $$0.280188\pi$$
−0.636968 + 0.770890i $$0.719812\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 20017.7i 2.54987i
$$396$$ 0 0
$$397$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$398$$ 0 0
$$399$$ −1369.68 4303.09i −0.171854 0.539909i
$$400$$ −5171.62 −0.646453
$$401$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 0 0
$$405$$ 9865.96 + 3469.41i 1.21048 + 0.425670i
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$410$$ 0 0
$$411$$ −9873.31 + 3142.70i −1.18495 + 0.377172i
$$412$$ 0 0
$$413$$ 2472.68i 0.294607i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ −3089.03 9704.70i −0.362758 1.13967i
$$418$$ 0 0
$$419$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$420$$ 3100.53 986.905i 0.360215 0.114657i
$$421$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 4965.50i 0.566734i
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 10210.5i 1.15314i
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$432$$ −7168.00 5407.53i −0.798311 0.602245i
$$433$$ 16431.5 1.82366 0.911832 0.410563i $$-0.134668\pi$$
0.911832 + 0.410563i $$0.134668\pi$$
$$434$$ 0 0
$$435$$ 74.1144 + 232.843i 0.00816900 + 0.0256643i
$$436$$ 0 0
$$437$$ 0 0
$$438$$ 0 0
$$439$$ −3220.00 −0.350073 −0.175037 0.984562i $$-0.556004\pi$$
−0.175037 + 0.984562i $$0.556004\pi$$
$$440$$ 0 0
$$441$$ 6901.15 4888.60i 0.745185 0.527870i
$$442$$ 0 0
$$443$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 0 0
$$448$$ −2793.58 −0.294607
$$449$$ 19027.9i 1.99996i −0.00634411 0.999980i $$-0.502019\pi$$
0.00634411 0.999980i $$-0.497981\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$458$$ 0 0
$$459$$ −5192.00 + 6882.31i −0.527978 + 0.699866i
$$460$$ 0 0
$$461$$ 13488.1i 1.36270i 0.731959 + 0.681348i $$0.238606\pi$$
−0.731959 + 0.681348i $$0.761394\pi$$
$$462$$ 0 0
$$463$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$464$$ 209.791i 0.0209899i
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 12871.0 1.24328
$$476$$ 2682.23i 0.258277i
$$477$$ 1034.78 + 1460.78i 0.0993274 + 0.140219i
$$478$$ 0 0
$$479$$ 1459.42i 0.139212i 0.997575 + 0.0696059i $$0.0221742\pi$$
−0.997575 + 0.0696059i $$0.977826\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 10648.0 1.00000
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −17264.3 −1.60640 −0.803201 0.595708i $$-0.796872\pi$$
−0.803201 + 0.595708i $$0.796872\pi$$
$$488$$ 0 0
$$489$$ −6379.78 20043.2i −0.589987 1.85354i
$$490$$ 0 0
$$491$$ 21733.7i 1.99762i 0.0488092 + 0.998808i $$0.484457\pi$$
−0.0488092 + 0.998808i $$0.515543\pi$$
$$492$$ 17857.5 5684.09i 1.63634 0.520851i
$$493$$ −201.430 −0.0184015
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 6454.12i 0.582509i
$$498$$ 0 0
$$499$$ −12706.2 −1.13989 −0.569946 0.821682i $$-0.693036\pi$$
−0.569946 + 0.821682i $$0.693036\pi$$
$$500$$ 5071.98i 0.453651i
$$501$$ 20030.3 6375.69i 1.78620 0.568553i
$$502$$ 0 0
$$503$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 3462.55 + 10878.2i 0.303308 + 0.952893i
$$508$$ 22282.3 1.94610
$$509$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ 17839.5 + 13458.1i 1.53534 + 1.15826i
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 11552.4i 0.971443i 0.874114 + 0.485721i $$0.161443\pi$$
−0.874114 + 0.485721i $$0.838557\pi$$
$$522$$ 0 0
$$523$$ 16932.6 1.41570 0.707849 0.706364i $$-0.249666\pi$$
0.707849 + 0.706364i $$0.249666\pi$$
$$524$$ 0 0
$$525$$ −694.869 2183.05i −0.0577649 0.181478i
$$526$$ 0 0
$$527$$ 0 0
$$528$$ 0 0
$$529$$ −12167.0 −1.00000
$$530$$ 0 0
$$531$$ −7072.93 9984.74i −0.578040 0.816009i
$$532$$ 6952.55 0.566601
$$533$$ 0 0
$$534$$ 0 0
$$535$$ −18310.0 −1.47964
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 0 0
$$540$$ −9697.02 + 12854.0i −0.772765 + 1.02435i
$$541$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$542$$ 0 0
$$543$$ 4942.25 + 15526.9i 0.390594 + 1.22712i
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −12616.0 −0.986145 −0.493072 0.869988i $$-0.664126\pi$$
−0.493072 + 0.869988i $$0.664126\pi$$
$$548$$ 15952.4i 1.24353i
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 522.121i 0.0403686i
$$552$$ 0 0
$$553$$ 7613.33 0.585446
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 15680.0 1.19601
$$557$$ 25691.9i 1.95440i 0.212328 + 0.977198i $$0.431895\pi$$
−0.212328 + 0.977198i $$0.568105\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 5009.57i 0.378023i
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 1319.52 3752.32i 0.0977331 0.277924i
$$568$$ 0 0
$$569$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$570$$ 0 0
$$571$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 11280.5 7990.82i 0.816009 0.578040i
$$577$$ 17746.1 1.28038 0.640189 0.768218i $$-0.278856\pi$$
0.640189 + 0.768218i $$0.278856\pi$$
$$578$$ 0 0
$$579$$ 5298.62 + 16646.5i 0.380316 + 1.19483i
$$580$$ −376.207 −0.0269330
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$588$$ 3949.29 + 12407.3i 0.276983 + 0.870188i
$$589$$ 0 0
$$590$$ 0 0
$$591$$ −20233.1 + 6440.26i −1.40826 + 0.448252i
$$592$$ 0 0
$$593$$ 20931.9i 1.44952i −0.688999 0.724762i $$-0.741950\pi$$
0.688999 0.724762i $$-0.258050\pi$$
$$594$$ 0 0
$$595$$ 4809.90 0.331406
$$596$$ 0 0
$$597$$ −6308.32 19818.6i −0.432466 1.35866i
$$598$$ 0 0
$$599$$ 1763.61i 0.120299i −0.998189 0.0601496i $$-0.980842\pi$$
0.998189 0.0601496i $$-0.0191578\pi$$
$$600$$ 0 0
$$601$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 19094.5i 1.28314i
$$606$$ 0 0
$$607$$ 1130.24 0.0755764 0.0377882 0.999286i $$-0.487969\pi$$
0.0377882 + 0.999286i $$0.487969\pi$$
$$608$$ 0 0
$$609$$ 88.5572 28.1879i 0.00589248 0.00187559i
$$610$$ 0 0
$$611$$ 0 0
$$612$$ −7672.33 10830.9i −0.506758 0.715381i
$$613$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$614$$ 0 0
$$615$$ −10193.0 32022.9i −0.668326 2.09966i
$$616$$ 0 0
$$617$$ 14451.4i 0.942937i 0.881883 + 0.471469i $$0.156276\pi$$
−0.881883 + 0.471469i $$0.843724\pi$$
$$618$$ 0 0
$$619$$ −19054.1 −1.23724 −0.618619 0.785691i $$-0.712308\pi$$
−0.618619 + 0.785691i $$0.712308\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −19196.1 −1.22855
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 0 0
$$630$$ 0 0
$$631$$ −29860.0 −1.88385 −0.941924 0.335827i $$-0.890984\pi$$
−0.941924 + 0.335827i $$0.890984\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 39957.6i 2.49712i
$$636$$ −2626.28 + 835.951i −0.163740 + 0.0521189i
$$637$$ 0 0
$$638$$ 0 0
$$639$$ −18461.5 26061.9i −1.14292 1.61344i
$$640$$ 0 0
$$641$$ 17205.8i 1.06020i −0.847936 0.530099i $$-0.822155\pi$$
0.847936 0.530099i $$-0.177845\pi$$
$$642$$ 0 0
$$643$$ 32463.0 1.99100 0.995502 0.0947388i $$-0.0302016\pi$$
0.995502 + 0.0947388i $$0.0302016\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 17340.3i 1.05366i 0.849971 + 0.526830i $$0.176620\pi$$
−0.849971 + 0.526830i $$0.823380\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 32384.0 1.94518
$$653$$ 31696.4i 1.89951i −0.313000 0.949753i $$-0.601334\pi$$
0.313000 0.949753i $$-0.398666\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 28852.7i 1.71724i
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$660$$ 0 0
$$661$$ −26389.4 −1.55284 −0.776422 0.630214i $$-0.782967\pi$$
−0.776422 + 0.630214i $$0.782967\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 12467.6i 0.727029i
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 32363.2i 1.87451i
$$669$$ 9222.95 + 28975.4i 0.533004 + 1.67452i
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$674$$ 0 0
$$675$$ 9050.34 + 6827.56i 0.516071 + 0.389323i
$$676$$ −17576.0 −1.00000
$$677$$ 33397.6i 1.89597i 0.318308 + 0.947987i $$0.396885\pi$$
−0.318308 + 0.947987i $$0.603115\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$684$$ −28074.5 + 19887.3i −1.56938 + 1.11171i
$$685$$ −28606.6 −1.59562
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 0 0
$$690$$ 0 0
$$691$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 28118.1i 1.53465i
$$696$$ 0 0
$$697$$ 27702.7 1.50547
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 3527.18 0.190450
$$701$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$702$$ 0 0
$$703$$ 0 0
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 17951.2 5713.91i 0.952893 0.303308i
$$709$$ −22569.6 −1.19551 −0.597756 0.801678i $$-0.703941\pi$$
−0.597756 + 0.801678i $$0.703941\pi$$
$$710$$ 0 0
$$711$$ −30742.8 + 21777.4i −1.62158 + 1.14869i
$$712$$ 0 0
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ −27021.5 + 8601.00i −1.40744 + 0.447992i
$$718$$ 0 0
$$719$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$720$$ −14329.5 20228.7i −0.741707 1.04705i
$$721$$ 0 0
$$722$$ 0 0
$$723$$ 11020.4 + 34622.6i 0.566881 + 1.78095i
$$724$$ −25087.0 −1.28778
$$725$$ 264.883i 0.0135690i
$$726$$ 0 0
$$727$$ 39116.0 1.99551 0.997753 0.0670071i $$-0.0213450\pi$$
0.997753 + 0.0670071i $$0.0213450\pi$$
$$728$$ 0 0
$$729$$ 5405.00 + 18926.3i 0.274602 + 0.961558i
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 0 0
$$733$$ −9898.00 −0.498760 −0.249380 0.968406i $$-0.580227\pi$$
−0.249380 + 0.968406i $$0.580227\pi$$
$$734$$ 0 0
$$735$$ 22249.4 7082.04i 1.11657 0.355408i
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 35594.4i 1.75751i −0.477269 0.878757i $$-0.658373\pi$$
0.477269 0.878757i $$-0.341627\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 6963.84i 0.339724i
$$750$$ 0 0
$$751$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$752$$ 0 0
$$753$$ 11533.2 3671.03i 0.558156 0.177662i
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 4888.76 + 3688.07i 0.235188 + 0.177426i
$$757$$ −21779.9 −1.04571 −0.522856 0.852421i $$-0.675133\pi$$
−0.522856 + 0.852421i $$0.675133\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 3613.97i 0.172150i −0.996289 0.0860752i $$-0.972567\pi$$
0.996289 0.0860752i $$-0.0274325\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ −19422.5 + 13758.4i −0.917935 + 0.650242i
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 6455.43 + 20280.8i 0.303308 + 0.952893i
$$769$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$770$$ 0 0
$$771$$ 40561.3 12910.8i 1.89466 0.603074i
$$772$$ −26896.0 −1.25390
$$773$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 71807.5i 3.30266i
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ −276.966 + 367.135i −0.0126411 + 0.0167565i
$$784$$ −20046.7 −0.913207
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −42784.0 −1.93785 −0.968923 0.247362i $$-0.920436\pi$$
−0.968923 + 0.247362i $$0.920436\pi$$
$$788$$ 32691.0i 1.47788i
$$789$$ 15637.3 4977.39i 0.705581 0.224588i
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 0 0
$$795$$ 1499.07 + 4709.57i 0.0668759 + 0.210102i
$$796$$ 32021.2 1.42583
$$797$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 0 0