# Properties

 Label 177.2.d.b.176.1 Level $177$ Weight $2$ Character 177.176 Analytic conductor $1.413$ Analytic rank $0$ Dimension $6$ CM discriminant -59 Inner twists $4$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$1.41335211578$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.149721291.1 Defining polynomial: $$x^{6} - 7 x^{3} + 27$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$3$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## Embedding invariants

 Embedding label 176.1 Root $$-1.24353 - 1.20567i$$ of defining polynomial Character $$\chi$$ $$=$$ 177.176 Dual form 177.2.d.b.176.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+(-1.24353 - 1.20567i) q^{3} -2.00000 q^{4} +3.58579i q^{5} +5.15952 q^{7} +(0.0927106 + 2.99857i) q^{9} +O(q^{10})$$ $$q+(-1.24353 - 1.20567i) q^{3} -2.00000 q^{4} +3.58579i q^{5} +5.15952 q^{7} +(0.0927106 + 2.99857i) q^{9} +(2.48705 + 2.41134i) q^{12} +(4.32329 - 4.45902i) q^{15} +4.00000 q^{16} +7.68115i q^{17} -2.30163 q^{19} -7.17158i q^{20} +(-6.41600 - 6.22069i) q^{21} -7.85789 q^{25} +(3.50000 - 3.84057i) q^{27} -10.3190 q^{28} -3.64824i q^{29} +18.5010i q^{35} +(-0.185421 - 5.99713i) q^{36} +0.0624503i q^{41} +(-10.7522 + 0.332441i) q^{45} +(-4.97410 - 4.82269i) q^{48} +19.6207 q^{49} +(9.26094 - 9.55170i) q^{51} -14.4056i q^{53} +(2.86213 + 2.77501i) q^{57} -7.68115i q^{59} +(-8.64657 + 8.91804i) q^{60} +(0.478343 + 15.4712i) q^{63} -8.00000 q^{64} -15.3623i q^{68} +7.68115i q^{71} +(9.77149 + 9.47404i) q^{75} +4.60326 q^{76} +13.1769 q^{79} +14.3432i q^{80} +(-8.98281 + 0.555998i) q^{81} +(12.8320 + 12.4414i) q^{84} -27.5430 q^{85} +(-4.39858 + 4.53668i) q^{87} -8.25316i q^{95} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q - 12q^{4} + O(q^{10})$$ $$6q - 12q^{4} + 3q^{15} + 24q^{16} - 15q^{21} - 30q^{25} + 21q^{27} - 33q^{45} + 42q^{49} + 39q^{57} - 6q^{60} + 12q^{63} - 48q^{64} - 24q^{75} + 30q^{84} - 51q^{87} + O(q^{100})$$

## 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 $$0.5\pi$$
$$3$$ −1.24353 1.20567i −0.717950 0.696095i
$$4$$ −2.00000 −1.00000
$$5$$ 3.58579i 1.60361i 0.597583 + 0.801807i $$0.296128\pi$$
−0.597583 + 0.801807i $$0.703872\pi$$
$$6$$ 0 0
$$7$$ 5.15952 1.95012 0.975058 0.221950i $$-0.0712420\pi$$
0.975058 + 0.221950i $$0.0712420\pi$$
$$8$$ 0 0
$$9$$ 0.0927106 + 2.99857i 0.0309035 + 0.999522i
$$10$$ 0 0
$$11$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$12$$ 2.48705 + 2.41134i 0.717950 + 0.696095i
$$13$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$14$$ 0 0
$$15$$ 4.32329 4.45902i 1.11627 1.15131i
$$16$$ 4.00000 1.00000
$$17$$ 7.68115i 1.86295i 0.363803 + 0.931476i $$0.381478\pi$$
−0.363803 + 0.931476i $$0.618522\pi$$
$$18$$ 0 0
$$19$$ −2.30163 −0.528030 −0.264015 0.964519i $$-0.585047\pi$$
−0.264015 + 0.964519i $$0.585047\pi$$
$$20$$ 7.17158i 1.60361i
$$21$$ −6.41600 6.22069i −1.40009 1.35747i
$$22$$ 0 0
$$23$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$24$$ 0 0
$$25$$ −7.85789 −1.57158
$$26$$ 0 0
$$27$$ 3.50000 3.84057i 0.673575 0.739119i
$$28$$ −10.3190 −1.95012
$$29$$ 3.64824i 0.677461i −0.940883 0.338731i $$-0.890002\pi$$
0.940883 0.338731i $$-0.109998\pi$$
$$30$$ 0 0
$$31$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 18.5010i 3.12723i
$$36$$ −0.185421 5.99713i −0.0309035 0.999522i
$$37$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 0.0624503i 0.00975310i 0.999988 + 0.00487655i $$0.00155226\pi$$
−0.999988 + 0.00487655i $$0.998448\pi$$
$$42$$ 0 0
$$43$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$44$$ 0 0
$$45$$ −10.7522 + 0.332441i −1.60285 + 0.0495574i
$$46$$ 0 0
$$47$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$48$$ −4.97410 4.82269i −0.717950 0.696095i
$$49$$ 19.6207 2.80295
$$50$$ 0 0
$$51$$ 9.26094 9.55170i 1.29679 1.33751i
$$52$$ 0 0
$$53$$ 14.4056i 1.97876i −0.145342 0.989382i $$-0.546428\pi$$
0.145342 0.989382i $$-0.453572\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 2.86213 + 2.77501i 0.379099 + 0.367559i
$$58$$ 0 0
$$59$$ 7.68115i 1.00000i
$$60$$ −8.64657 + 8.91804i −1.11627 + 1.15131i
$$61$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$62$$ 0 0
$$63$$ 0.478343 + 15.4712i 0.0602655 + 1.94918i
$$64$$ −8.00000 −1.00000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$68$$ 15.3623i 1.86295i
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 7.68115i 0.911584i 0.890086 + 0.455792i $$0.150644\pi$$
−0.890086 + 0.455792i $$0.849356\pi$$
$$72$$ 0 0
$$73$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$74$$ 0 0
$$75$$ 9.77149 + 9.47404i 1.12831 + 1.09397i
$$76$$ 4.60326 0.528030
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 13.1769 1.48252 0.741261 0.671217i $$-0.234228\pi$$
0.741261 + 0.671217i $$0.234228\pi$$
$$80$$ 14.3432i 1.60361i
$$81$$ −8.98281 + 0.555998i −0.998090 + 0.0617776i
$$82$$ 0 0
$$83$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$84$$ 12.8320 + 12.4414i 1.40009 + 1.35747i
$$85$$ −27.5430 −2.98746
$$86$$ 0 0
$$87$$ −4.39858 + 4.53668i −0.471577 + 0.486383i
$$88$$ 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
$$95$$ 8.25316i 0.846756i
$$96$$ 0 0
$$97$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 15.7158 1.57158
$$101$$ 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
$$105$$ 22.3061 23.0064i 2.17685 2.24520i
$$106$$ 0 0
$$107$$ 10.8823i 1.05203i −0.850476 0.526014i $$-0.823686\pi$$
0.850476 0.526014i $$-0.176314\pi$$
$$108$$ −7.00000 + 7.68115i −0.673575 + 0.739119i
$$109$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 20.6381 1.95012
$$113$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 7.29648i 0.677461i
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 39.6310i 3.63297i
$$120$$ 0 0
$$121$$ −11.0000 −1.00000
$$122$$ 0 0
$$123$$ 0.0752946 0.0776586i 0.00678908 0.00700224i
$$124$$ 0 0
$$125$$ 10.2478i 0.916592i
$$126$$ 0 0
$$127$$ −18.3365 −1.62710 −0.813549 0.581496i $$-0.802467\pi$$
−0.813549 + 0.581496i $$0.802467\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$132$$ 0 0
$$133$$ −11.8753 −1.02972
$$134$$ 0 0
$$135$$ 13.7715 + 12.5503i 1.18526 + 1.08015i
$$136$$ 0 0
$$137$$ 10.6949i 0.913729i −0.889536 0.456864i $$-0.848972\pi$$
0.889536 0.456864i $$-0.151028\pi$$
$$138$$ 0 0
$$139$$ 5.00000 0.424094 0.212047 0.977259i $$-0.431987\pi$$
0.212047 + 0.977259i $$0.431987\pi$$
$$140$$ 37.0019i 3.12723i
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0.370843 + 11.9943i 0.0309035 + 0.999522i
$$145$$ 13.0818 1.08639
$$146$$ 0 0
$$147$$ −24.3988 23.6561i −2.01238 1.95112i
$$148$$ 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
$$153$$ −23.0324 + 0.712124i −1.86206 + 0.0575718i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$158$$ 0 0
$$159$$ −17.3684 + 17.9137i −1.37741 + 1.42065i
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 11.0000 0.861586 0.430793 0.902451i $$-0.358234\pi$$
0.430793 + 0.902451i $$0.358234\pi$$
$$164$$ 0.124901i 0.00975310i
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 25.1630i 1.94717i −0.228325 0.973585i $$-0.573325\pi$$
0.228325 0.973585i $$-0.426675\pi$$
$$168$$ 0 0
$$169$$ 13.0000 1.00000
$$170$$ 0 0
$$171$$ −0.213386 6.90159i −0.0163180 0.527778i
$$172$$ 0 0
$$173$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$174$$ 0 0
$$175$$ −40.5430 −3.06476
$$176$$ 0 0
$$177$$ −9.26094 + 9.55170i −0.696095 + 0.717950i
$$178$$ 0 0
$$179$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$180$$ 21.5045 0.664882i 1.60285 0.0495574i
$$181$$ 12.0644 0.896741 0.448370 0.893848i $$-0.352005\pi$$
0.448370 + 0.893848i $$0.352005\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ 18.0583 19.8155i 1.31355 1.44137i
$$190$$ 0 0
$$191$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$192$$ 9.94820 + 9.64538i 0.717950 + 0.696095i
$$193$$ −3.41416 −0.245756 −0.122878 0.992422i $$-0.539212\pi$$
−0.122878 + 0.992422i $$0.539212\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ −39.2413 −2.80295
$$197$$ 7.68115i 0.547259i 0.961835 + 0.273629i $$0.0882242\pi$$
−0.961835 + 0.273629i $$0.911776\pi$$
$$198$$ 0 0
$$199$$ 28.0992 1.99190 0.995951 0.0898948i $$-0.0286531\pi$$
0.995951 + 0.0898948i $$0.0286531\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 18.8232i 1.32113i
$$204$$ −18.5219 + 19.1034i −1.29679 + 1.33751i
$$205$$ −0.223934 −0.0156402
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$212$$ 28.8112i 1.97876i
$$213$$ 9.26094 9.55170i 0.634549 0.654472i
$$214$$ 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
$$223$$ −19.0000 −1.27233 −0.636167 0.771551i $$-0.719481\pi$$
−0.636167 + 0.771551i $$0.719481\pi$$
$$224$$ 0 0
$$225$$ −0.728510 23.5624i −0.0485674 1.57083i
$$226$$ 0 0
$$227$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$228$$ −5.72427 5.55002i −0.379099 0.367559i
$$229$$ 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
$$236$$ 15.3623i 1.00000i
$$237$$ −16.3859 15.8871i −1.06438 1.03198i
$$238$$ 0 0
$$239$$ 18.1163i 1.17185i −0.810367 0.585923i $$-0.800732\pi$$
0.810367 0.585923i $$-0.199268\pi$$
$$240$$ 17.2931 17.8361i 1.11627 1.15131i
$$241$$ 21.1944 1.36525 0.682624 0.730770i $$-0.260839\pi$$
0.682624 + 0.730770i $$0.260839\pi$$
$$242$$ 0 0
$$243$$ 11.8407 + 10.1389i 0.759581 + 0.650412i
$$244$$ 0 0
$$245$$ 70.3556i 4.49486i
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 17.8665i 1.12772i 0.825869 + 0.563862i $$0.190685\pi$$
−0.825869 + 0.563862i $$0.809315\pi$$
$$252$$ −0.956685 30.9424i −0.0602655 1.94918i
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 34.2504 + 33.2078i 2.14484 + 2.07955i
$$256$$ 16.0000 1.00000
$$257$$ 25.2879i 1.57741i 0.614769 + 0.788707i $$0.289249\pi$$
−0.614769 + 0.788707i $$0.710751\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 10.9395 0.338231i 0.677138 0.0209360i
$$262$$ 0 0
$$263$$ 32.3970i 1.99769i −0.0480951 0.998843i $$-0.515315\pi$$
0.0480951 0.998843i $$-0.484685\pi$$
$$264$$ 0 0
$$265$$ 51.6555 3.17317
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$270$$ 0 0
$$271$$ 13.7332 0.834233 0.417116 0.908853i $$-0.363041\pi$$
0.417116 + 0.908853i $$0.363041\pi$$
$$272$$ 30.7246i 1.86295i
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −33.2588 −1.99833 −0.999163 0.0409112i $$-0.986974\pi$$
−0.999163 + 0.0409112i $$0.986974\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 28.8737i 1.72246i −0.508216 0.861230i $$-0.669695\pi$$
0.508216 0.861230i $$-0.330305\pi$$
$$282$$ 0 0
$$283$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$284$$ 15.3623i 0.911584i
$$285$$ −9.95060 + 10.2630i −0.589423 + 0.607929i
$$286$$ 0 0
$$287$$ 0.322214i 0.0190197i
$$288$$ 0 0
$$289$$ −42.0000 −2.47059
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 32.3346i 1.88901i 0.328504 + 0.944503i $$0.393456\pi$$
−0.328504 + 0.944503i $$0.606544\pi$$
$$294$$ 0 0
$$295$$ 27.5430 1.60361
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 0 0
$$300$$ −19.5430 18.9481i −1.12831 1.09397i
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 0 0
$$304$$ −9.20652 −0.528030
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 4.04699 0.230974 0.115487 0.993309i $$-0.463157\pi$$
0.115487 + 0.993309i $$0.463157\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 10.6325i 0.602912i 0.953480 + 0.301456i $$0.0974727\pi$$
−0.953480 + 0.301456i $$0.902527\pi$$
$$312$$ 0 0
$$313$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$314$$ 0 0
$$315$$ −55.4764 + 1.71524i −3.12574 + 0.0966426i
$$316$$ −26.3539 −1.48252
$$317$$ 30.7246i 1.72566i 0.505490 + 0.862832i $$0.331312\pi$$
−0.505490 + 0.862832i $$0.668688\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 28.6863i 1.60361i
$$321$$ −13.1204 + 13.5324i −0.732312 + 0.755304i
$$322$$ 0 0
$$323$$ 17.6792i 0.983694i
$$324$$ 17.9656 1.11200i 0.998090 0.0617776i
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −18.8927 −1.03844 −0.519219 0.854641i $$-0.673777\pi$$
−0.519219 + 0.854641i $$0.673777\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 0 0
$$336$$ −25.6640 24.8828i −1.40009 1.35747i
$$337$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 55.0860 2.98746
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 65.1167 3.51597
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$348$$ 8.79716 9.07336i 0.471577 0.486383i
$$349$$ 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
$$355$$ −27.5430 −1.46183
$$356$$ 0 0
$$357$$ 47.7820 49.2822i 2.52889 2.60829i
$$358$$ 0 0
$$359$$ 36.0453i 1.90240i 0.308581 + 0.951198i $$0.400146\pi$$
−0.308581 + 0.951198i $$0.599854\pi$$
$$360$$ 0 0
$$361$$ −13.7025 −0.721184
$$362$$ 0 0
$$363$$ 13.6788 + 13.2624i 0.717950 + 0.696095i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$368$$ 0 0
$$369$$ −0.187261 + 0.00578981i −0.00974844 + 0.000301405i
$$370$$ 0 0
$$371$$ 74.3261i 3.85882i
$$372$$ 0 0
$$373$$ −31.0000 −1.60512 −0.802560 0.596572i $$-0.796529\pi$$
−0.802560 + 0.596572i $$0.796529\pi$$
$$374$$ 0 0
$$375$$ −12.3555 + 12.7434i −0.638035 + 0.658067i
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ −24.6851 −1.26799 −0.633994 0.773338i $$-0.718586\pi$$
−0.633994 + 0.773338i $$0.718586\pi$$
$$380$$ 16.5063i 0.846756i
$$381$$ 22.8019 + 22.1078i 1.16817 + 1.13261i
$$382$$ 0 0
$$383$$ 15.3623i 0.784976i −0.919757 0.392488i $$-0.871614\pi$$
0.919757 0.392488i $$-0.128386\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 38.4057i 1.94725i −0.228159 0.973624i $$-0.573271\pi$$
0.228159 0.973624i $$-0.426729\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 47.2497i 2.37739i
$$396$$ 0 0
$$397$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$398$$ 0 0
$$399$$ 14.7672 + 14.3177i 0.739287 + 0.716783i
$$400$$ −31.4316 −1.57158
$$401$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 0 0
$$405$$ −1.99369 32.2105i −0.0990674 1.60055i
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$410$$ 0 0
$$411$$ −12.8946 + 13.2994i −0.636042 + 0.656011i
$$412$$ 0 0
$$413$$ 39.6310i 1.95012i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ −6.21763 6.02836i −0.304478 0.295210i
$$418$$ 0 0
$$419$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$420$$ −44.6122 + 46.0128i −2.17685 + 2.24520i
$$421$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 60.3576i 2.92777i
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 21.7645i 1.05203i
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$432$$ 14.0000 15.3623i 0.673575 0.739119i
$$433$$ −40.7199 −1.95687 −0.978437 0.206545i $$-0.933778\pi$$
−0.978437 + 0.206545i $$0.933778\pi$$
$$434$$ 0 0
$$435$$ −16.2676 15.7724i −0.779971 0.756228i
$$436$$ 0 0
$$437$$ 0 0
$$438$$ 0 0
$$439$$ 35.0000 1.67046 0.835229 0.549902i $$-0.185335\pi$$
0.835229 + 0.549902i $$0.185335\pi$$
$$440$$ 0 0
$$441$$ 1.81905 + 58.8339i 0.0866212 + 2.80161i
$$442$$ 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
$$448$$ −41.2762 −1.95012
$$449$$ 32.5219i 1.53480i 0.641166 + 0.767402i $$0.278451\pi$$
−0.641166 + 0.767402i $$0.721549\pi$$
$$450$$ 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
$$459$$ 29.5000 + 26.8840i 1.37694 + 1.25484i
$$460$$ 0 0
$$461$$ 30.7246i 1.43099i 0.698620 + 0.715493i $$0.253798\pi$$
−0.698620 + 0.715493i $$0.746202\pi$$
$$462$$ 0 0
$$463$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$464$$ 14.5930i 0.677461i
$$465$$ 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
$$475$$ 18.0860 0.829841
$$476$$ 79.2621i 3.63297i
$$477$$ 43.1962 1.33555i 1.97782 0.0611508i
$$478$$ 0 0
$$479$$ 38.4057i 1.75480i −0.479757 0.877401i $$-0.659275\pi$$
0.479757 0.877401i $$-0.340725\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 22.0000 1.00000
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −16.6677 −0.755284 −0.377642 0.925952i $$-0.623265\pi$$
−0.377642 + 0.925952i $$0.623265\pi$$
$$488$$ 0 0
$$489$$ −13.6788 13.2624i −0.618576 0.599746i
$$490$$ 0 0
$$491$$ 6.98423i 0.315194i −0.987504 0.157597i $$-0.949625\pi$$
0.987504 0.157597i $$-0.0503747\pi$$
$$492$$ −0.150589 + 0.155317i −0.00678908 + 0.00700224i
$$493$$ 28.0227 1.26208
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 39.6310i 1.77770i
$$498$$ 0 0
$$499$$ 35.0041 1.56700 0.783500 0.621392i $$-0.213432\pi$$
0.783500 + 0.621392i $$0.213432\pi$$
$$500$$ 20.4956i 0.916592i
$$501$$ −30.3383 + 31.2908i −1.35542 + 1.39797i
$$502$$ 0 0
$$503$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ −16.1658 15.6737i −0.717950 0.696095i
$$508$$ 36.6729 1.62710
$$509$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ −8.05570 + 8.83958i −0.355668 + 0.390277i
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 7.68115i 0.336517i 0.985743 + 0.168259i $$0.0538144\pi$$
−0.985743 + 0.168259i $$0.946186\pi$$
$$522$$ 0 0
$$523$$ −3.97042 −0.173614 −0.0868072 0.996225i $$-0.527666\pi$$
−0.0868072 + 0.996225i $$0.527666\pi$$
$$524$$ 0 0
$$525$$ 50.4162 + 48.8815i 2.20034 + 2.13336i
$$526$$ 0 0
$$527$$ 0 0
$$528$$ 0 0
$$529$$ −23.0000 −1.00000
$$530$$ 0 0
$$531$$ 23.0324 0.712124i 0.999522 0.0309035i
$$532$$ 23.7506 1.02972
$$533$$ 0 0
$$534$$ 0 0
$$535$$ 39.0215 1.68705
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 0 0
$$540$$ −27.5430 25.1005i −1.18526 1.08015i
$$541$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$542$$ 0 0
$$543$$ −15.0024 14.5457i −0.643815 0.624217i
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 8.00000 0.342055 0.171028 0.985266i $$-0.445291\pi$$
0.171028 + 0.985266i $$0.445291\pi$$
$$548$$ 21.3898i 0.913729i
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 8.39690i 0.357720i
$$552$$ 0 0
$$553$$ 67.9867 2.89109
$$554$$ 0 0
$$555$$ 0 0
$$556$$ −10.0000 −0.424094
$$557$$ 43.0919i 1.82586i 0.408112 + 0.912932i $$0.366187\pi$$
−0.408112 + 0.912932i $$0.633813\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 74.0039i 3.12723i
$$561$$ 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
$$567$$ −46.3470 + 2.86869i −1.94639 + 0.120473i
$$568$$ 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
$$576$$ −0.741685 23.9885i −0.0309035 0.999522i
$$577$$ 44.1341 1.83733 0.918663 0.395043i $$-0.129270\pi$$
0.918663 + 0.395043i $$0.129270\pi$$
$$578$$ 0 0
$$579$$ 4.24559 + 4.11635i 0.176441 + 0.171070i
$$580$$ −26.1637 −1.08639
$$581$$ 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 $$0.5\pi$$
$$588$$ 48.7976 + 47.3122i 2.01238 + 1.95112i
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 9.26094 9.55170i 0.380944 0.392904i
$$592$$ 0 0
$$593$$ 3.83559i 0.157509i −0.996894 0.0787544i $$-0.974906\pi$$
0.996894 0.0787544i $$-0.0250943\pi$$
$$594$$ 0 0
$$595$$ −142.109 −5.82589
$$596$$ 0 0
$$597$$ −34.9421 33.8785i −1.43009 1.38655i
$$598$$ 0 0
$$599$$ 32.2097i 1.31605i −0.752996 0.658026i $$-0.771392\pi$$
0.752996 0.658026i $$-0.228608\pi$$
$$600$$ 0 0
$$601$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 39.4437i 1.60361i
$$606$$ 0 0
$$607$$ −32.1462 −1.30478 −0.652388 0.757885i $$-0.726233\pi$$
−0.652388 + 0.757885i $$0.726233\pi$$
$$608$$ 0 0
$$609$$ −22.6946 + 23.4071i −0.919631 + 0.948504i
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 46.0649 1.42425i 1.86206 0.0575718i
$$613$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$614$$ 0 0
$$615$$ 0.278467 + 0.269991i 0.0112289 + 0.0108871i
$$616$$ 0 0
$$617$$ 46.8651i 1.88672i −0.331775 0.943358i $$-0.607648\pi$$
0.331775 0.943358i $$-0.392352\pi$$
$$618$$ 0 0
$$619$$ −26.9101 −1.08161 −0.540805 0.841148i $$-0.681880\pi$$
−0.540805 + 0.841148i $$0.681880\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −2.54298 −0.101719
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 0 0
$$630$$ 0 0
$$631$$ 20.0000 0.796187 0.398094 0.917345i $$-0.369672\pi$$
0.398094 + 0.917345i $$0.369672\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 65.7507i 2.60924i
$$636$$ 34.7369 35.8275i 1.37741 1.42065i
$$637$$ 0 0
$$638$$ 0 0
$$639$$ −23.0324 + 0.712124i −0.911149 + 0.0281712i
$$640$$ 0 0
$$641$$ 38.4057i 1.51694i −0.651711 0.758468i $$-0.725948\pi$$
0.651711 0.758468i $$-0.274052\pi$$
$$642$$ 0 0
$$643$$ −48.1811 −1.90008 −0.950038 0.312134i $$-0.898956\pi$$
−0.950038 + 0.312134i $$0.898956\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 24.9756i 0.981893i −0.871190 0.490947i $$-0.836651\pi$$
0.871190 0.490947i $$-0.163349\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −22.0000 −0.861586
$$653$$ 14.1558i 0.553960i 0.960876 + 0.276980i $$0.0893335\pi$$
−0.960876 + 0.276980i $$0.910666\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0.249801i 0.00975310i
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$660$$ 0 0
$$661$$ 51.0390 1.98519 0.992593 0.121489i $$-0.0387669\pi$$
0.992593 + 0.121489i $$0.0387669\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 42.5824i 1.65127i
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 50.3260i 1.94717i
$$669$$ 23.6270 + 22.9078i 0.913472 + 0.885665i
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$674$$ 0 0
$$675$$ −27.5026 + 30.1788i −1.05858 + 1.16158i
$$676$$ −26.0000 −1.00000
$$677$$ 30.7246i 1.18084i 0.807096 + 0.590421i $$0.201038\pi$$
−0.807096 + 0.590421i $$0.798962\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$684$$ 0.426771 + 13.8032i 0.0163180 + 0.527778i
$$685$$ 38.3497 1.46527
$$686$$ 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
$$695$$ 17.9290i 0.680084i
$$696$$ 0 0
$$697$$ −0.479690 −0.0181696
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 81.0860 3.06476
$$701$$ 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
$$708$$ 18.5219 19.1034i 0.696095 0.717950i
$$709$$ −49.2936 −1.85126 −0.925630 0.378430i $$-0.876464\pi$$
−0.925630 + 0.378430i $$0.876464\pi$$
$$710$$ 0 0
$$711$$ 1.22164 + 39.5119i 0.0458152 + 1.48181i
$$712$$ 0 0
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ −21.8423 + 22.5281i −0.815716 + 0.841327i
$$718$$ 0 0
$$719$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$720$$ −43.0089 + 1.32976i −1.60285 + 0.0495574i
$$721$$ 0 0
$$722$$ 0 0
$$723$$ −26.3557 25.5534i −0.980180 0.950343i
$$724$$ −24.1288 −0.896741
$$725$$ 28.6675i 1.06468i
$$726$$ 0 0
$$727$$ −28.0000 −1.03846 −0.519231 0.854634i $$-0.673782\pi$$
−0.519231 + 0.854634i $$0.673782\pi$$
$$728$$ 0 0
$$729$$ −2.50000 26.8840i −0.0925926 0.995704i
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 0 0
$$733$$ −49.0000 −1.80986 −0.904928 0.425564i $$-0.860076\pi$$
−0.904928 + 0.425564i $$0.860076\pi$$
$$734$$ 0 0
$$735$$ 84.8258 87.4890i 3.12885 3.22708i
$$736$$ 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
$$743$$ 53.7680i 1.97256i 0.165089 + 0.986279i $$0.447209\pi$$
−0.165089 + 0.986279i $$0.552791\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 56.1473i 2.05158i
$$750$$ 0 0
$$751$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$752$$ 0 0
$$753$$ 21.5411 22.2174i 0.785003 0.809649i
$$754$$ 0 0
$$755$$ 0 0
$$756$$ −36.1167 + 39.6310i −1.31355 + 1.44137i
$$757$$ 42.4653 1.54343 0.771713 0.635970i $$-0.219400\pi$$
0.771713 + 0.635970i $$0.219400\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 54.0367i 1.95883i 0.201863 + 0.979414i $$0.435300\pi$$
−0.201863 + 0.979414i $$0.564700\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ −2.55353 82.5895i −0.0923230 2.98603i
$$766$$ 0 0
$$767$$ 0 0
$$768$$ −19.8964 19.2908i −0.717950 0.696095i
$$769$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$770$$ 0 0
$$771$$ 30.4889 31.4461i 1.09803 1.13250i
$$772$$ 6.82831 0.245756
$$773$$ 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
$$779$$ 0.143737i 0.00514993i
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ −14.0113 12.7688i −0.500724 0.456321i
$$784$$ 78.4827 2.80295
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 32.0000 1.14068 0.570338 0.821410i $$-0.306812\pi$$
0.570338 + 0.821410i $$0.306812\pi$$
$$788$$ 15.3623i 0.547259i
$$789$$ −39.0602 + 40.2865i −1.39058 + 1.43424i
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 0 0
$$795$$ −64.2349 62.2796i −2.27818 2.20883i
$$796$$ −56.1985 −1.99190
$$797$$ 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
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0