# Properties

 Label 731.2.s.a.214.1 Level 731 Weight 2 Character 731.214 Analytic conductor 5.837 Analytic rank 0 Dimension 16 CM discriminant -43 Inner twists 4

# Learn more about

## Newspace parameters

 Level: $$N$$ = $$731 = 17 \cdot 43$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 731.s (of order $$16$$, degree $$8$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.83706438776$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$2$$ over $$\Q(\zeta_{16})$$ Coefficient field: 16.0.3289935900927224469054816256.1 Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{16}]$

## Embedding invariants

 Embedding label 214.1 Root $$-0.792772 + 3.22048i$$ Character $$\chi$$ = 731.214 Dual form 731.2.s.a.386.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+(-1.41421 - 1.41421i) q^{4} +(1.14805 + 2.77164i) q^{9} +O(q^{10})$$ $$q+(-1.41421 - 1.41421i) q^{4} +(1.14805 + 2.77164i) q^{9} +(-0.165842 - 0.248200i) q^{11} +(-4.77872 + 4.77872i) q^{13} +4.00000i q^{16} +(3.56441 + 2.07243i) q^{17} +(7.86853 - 5.25759i) q^{23} +(4.61940 - 1.91342i) q^{25} +(-2.59764 + 3.88764i) q^{31} +(2.29610 - 5.54328i) q^{36} +(11.3414 + 2.25594i) q^{41} +(2.50942 + 6.05828i) q^{43} +(-0.116472 + 0.585544i) q^{44} +(-7.48531 + 7.48531i) q^{47} +(6.46716 + 2.67878i) q^{49} +13.5163 q^{52} +(-3.63588 + 8.77779i) q^{53} +(-11.4652 + 4.74906i) q^{59} +(5.65685 - 5.65685i) q^{64} -11.7985i q^{67} +(-2.10997 - 7.97170i) q^{68} +(-2.35438 - 3.52358i) q^{79} +(-6.36396 + 6.36396i) q^{81} +(-6.05828 - 2.50942i) q^{83} +(-18.5631 - 3.69244i) q^{92} +(3.08269 + 15.4977i) q^{97} +(0.497526 - 0.744600i) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + O(q^{10})$$ $$16q - 24q^{13} + 56q^{23} - 64q^{59} + 96q^{79} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/731\mathbb{Z}\right)^\times$$.

 $$n$$ $$173$$ $$562$$ $$\chi(n)$$ $$e\left(\frac{3}{16}\right)$$ $$-1$$

## 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 0.382683 0.923880i $$-0.375000\pi$$
−0.382683 + 0.923880i $$0.625000\pi$$
$$3$$ 0 0 −0.831470 0.555570i $$-0.812500\pi$$
0.831470 + 0.555570i $$0.187500\pi$$
$$4$$ −1.41421 1.41421i −0.707107 0.707107i
$$5$$ 0 0 0.980785 0.195090i $$-0.0625000\pi$$
−0.980785 + 0.195090i $$0.937500\pi$$
$$6$$ 0 0
$$7$$ 0 0 −0.980785 0.195090i $$-0.937500\pi$$
0.980785 + 0.195090i $$0.0625000\pi$$
$$8$$ 0 0
$$9$$ 1.14805 + 2.77164i 0.382683 + 0.923880i
$$10$$ 0 0
$$11$$ −0.165842 0.248200i −0.0500032 0.0748351i 0.805626 0.592425i $$-0.201829\pi$$
−0.855629 + 0.517590i $$0.826829\pi$$
$$12$$ 0 0
$$13$$ −4.77872 + 4.77872i −1.32538 + 1.32538i −0.416025 + 0.909353i $$0.636577\pi$$
−0.909353 + 0.416025i $$0.863423\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 4.00000i 1.00000i
$$17$$ 3.56441 + 2.07243i 0.864496 + 0.502639i
$$18$$ 0 0
$$19$$ 0 0 0.382683 0.923880i $$-0.375000\pi$$
−0.382683 + 0.923880i $$0.625000\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 7.86853 5.25759i 1.64070 1.09628i 0.729800 0.683660i $$-0.239613\pi$$
0.910902 0.412622i $$-0.135387\pi$$
$$24$$ 0 0
$$25$$ 4.61940 1.91342i 0.923880 0.382683i
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 0 0 −0.195090 0.980785i $$-0.562500\pi$$
0.195090 + 0.980785i $$0.437500\pi$$
$$30$$ 0 0
$$31$$ −2.59764 + 3.88764i −0.466550 + 0.698241i −0.987898 0.155103i $$-0.950429\pi$$
0.521349 + 0.853344i $$0.325429\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 2.29610 5.54328i 0.382683 0.923880i
$$37$$ 0 0 −0.831470 0.555570i $$-0.812500\pi$$
0.831470 + 0.555570i $$0.187500\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 11.3414 + 2.25594i 1.77123 + 0.352319i 0.969447 0.245299i $$-0.0788863\pi$$
0.801781 + 0.597619i $$0.203886\pi$$
$$42$$ 0 0
$$43$$ 2.50942 + 6.05828i 0.382683 + 0.923880i
$$44$$ −0.116472 + 0.585544i −0.0175588 + 0.0882740i
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −7.48531 + 7.48531i −1.09185 + 1.09185i −0.0965136 + 0.995332i $$0.530769\pi$$
−0.995332 + 0.0965136i $$0.969231\pi$$
$$48$$ 0 0
$$49$$ 6.46716 + 2.67878i 0.923880 + 0.382683i
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 13.5163 1.87437
$$53$$ −3.63588 + 8.77779i −0.499426 + 1.20572i 0.450367 + 0.892844i $$0.351293\pi$$
−0.949793 + 0.312878i $$0.898707\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ −11.4652 + 4.74906i −1.49265 + 0.618274i −0.971891 0.235431i $$-0.924350\pi$$
−0.520756 + 0.853706i $$0.674350\pi$$
$$60$$ 0 0
$$61$$ 0 0 0.195090 0.980785i $$-0.437500\pi$$
−0.195090 + 0.980785i $$0.562500\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 5.65685 5.65685i 0.707107 0.707107i
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 11.7985i 1.44142i −0.693236 0.720710i $$-0.743816\pi$$
0.693236 0.720710i $$-0.256184\pi$$
$$68$$ −2.10997 7.97170i −0.255872 0.966711i
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 0 0 −0.831470 0.555570i $$-0.812500\pi$$
0.831470 + 0.555570i $$0.187500\pi$$
$$72$$ 0 0
$$73$$ 0 0 0.980785 0.195090i $$-0.0625000\pi$$
−0.980785 + 0.195090i $$0.937500\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −2.35438 3.52358i −0.264889 0.396434i 0.675053 0.737769i $$-0.264121\pi$$
−0.939942 + 0.341335i $$0.889121\pi$$
$$80$$ 0 0
$$81$$ −6.36396 + 6.36396i −0.707107 + 0.707107i
$$82$$ 0 0
$$83$$ −6.05828 2.50942i −0.664983 0.275445i 0.0245507 0.999699i $$-0.492184\pi$$
−0.689534 + 0.724254i $$0.742184\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ −18.5631 3.69244i −1.93534 0.384963i
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 3.08269 + 15.4977i 0.313000 + 1.57356i 0.742093 + 0.670297i $$0.233833\pi$$
−0.429093 + 0.903260i $$0.641167\pi$$
$$98$$ 0 0
$$99$$ 0.497526 0.744600i 0.0500032 0.0748351i
$$100$$ −9.23880 3.82683i −0.923880 0.382683i
$$101$$ 1.21270i 0.120668i 0.998178 + 0.0603342i $$0.0192166\pi$$
−0.998178 + 0.0603342i $$0.980783\pi$$
$$102$$ 0 0
$$103$$ 2.90887 0.286620 0.143310 0.989678i $$-0.454225\pi$$
0.143310 + 0.989678i $$0.454225\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 15.6772 3.11839i 1.51557 0.301466i 0.633932 0.773389i $$-0.281440\pi$$
0.881640 + 0.471923i $$0.156440\pi$$
$$108$$ 0 0
$$109$$ −20.2630 4.03056i −1.94084 0.386057i −0.998708 0.0508181i $$-0.983817\pi$$
−0.942133 0.335239i $$-0.891183\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 0 0 −0.555570 0.831470i $$-0.687500\pi$$
0.555570 + 0.831470i $$0.312500\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ −18.7311 7.75867i −1.73169 0.717290i
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 4.17542 10.0804i 0.379583 0.916396i
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 9.17157 1.82434i 0.823631 0.163830i
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 20.6805 8.56613i 1.83509 0.760121i 0.872818 0.488046i $$-0.162290\pi$$
0.962276 0.272075i $$-0.0877098\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 0 0 −0.195090 0.980785i $$-0.562500\pi$$
0.195090 + 0.980785i $$0.437500\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$138$$ 0 0
$$139$$ 0.0999825 + 0.0668062i 0.00848041 + 0.00566643i 0.559803 0.828626i $$-0.310877\pi$$
−0.551323 + 0.834292i $$0.685877\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 1.97859 + 0.393566i 0.165458 + 0.0329116i
$$144$$ −11.0866 + 4.59220i −0.923880 + 0.382683i
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$150$$ 0 0
$$151$$ 0 0 −0.923880 0.382683i $$-0.875000\pi$$
0.923880 + 0.382683i $$0.125000\pi$$
$$152$$ 0 0
$$153$$ −1.65191 + 12.2585i −0.133549 + 0.991042i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 0 0 0.195090 0.980785i $$-0.437500\pi$$
−0.195090 + 0.980785i $$0.562500\pi$$
$$164$$ −12.8488 19.2295i −1.00332 1.50157i
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −6.28278 + 9.40284i −0.486176 + 0.727613i −0.990742 0.135760i $$-0.956653\pi$$
0.504566 + 0.863373i $$0.331653\pi$$
$$168$$ 0 0
$$169$$ 32.6723i 2.51326i
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 5.01885 12.1166i 0.382683 0.923880i
$$173$$ −21.0658 14.0757i −1.60160 1.07016i −0.950300 0.311335i $$-0.899224\pi$$
−0.651300 0.758820i $$-0.725776\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0.992800 0.663368i 0.0748351 0.0500032i
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 0 0 −0.382683 0.923880i $$-0.625000\pi$$
0.382683 + 0.923880i $$0.375000\pi$$
$$180$$ 0 0
$$181$$ 5.32439 + 7.96852i 0.395759 + 0.592295i 0.974821 0.222988i $$-0.0715812\pi$$
−0.579062 + 0.815283i $$0.696581\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −0.0767508 1.22838i −0.00561258 0.0898282i
$$188$$ 21.1717 1.54410
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$192$$ 0 0
$$193$$ −15.6371 + 10.4484i −1.12558 + 0.752090i −0.971751 0.236007i $$-0.924161\pi$$
−0.153831 + 0.988097i $$0.549161\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ −5.35757 12.9343i −0.382683 0.923880i
$$197$$ −5.33864 + 26.8391i −0.380362 + 1.91221i 0.0284595 + 0.999595i $$0.490940\pi$$
−0.408822 + 0.912614i $$0.634060\pi$$
$$198$$ 0 0
$$199$$ 0 0 −0.195090 0.980785i $$-0.562500\pi$$
0.195090 + 0.980785i $$0.437500\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 23.6056 + 15.7728i 1.64070 + 1.09628i
$$208$$ −19.1149 19.1149i −1.32538 1.32538i
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 0 0 −0.980785 0.195090i $$-0.937500\pi$$
0.980785 + 0.195090i $$0.0625000\pi$$
$$212$$ 17.5556 7.27176i 1.20572 0.499426i
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −26.9369 + 7.12974i −1.81197 + 0.479598i
$$222$$ 0 0
$$223$$ 0 0 0.382683 0.923880i $$-0.375000\pi$$
−0.382683 + 0.923880i $$0.625000\pi$$
$$224$$ 0 0
$$225$$ 10.6066 + 10.6066i 0.707107 + 0.707107i
$$226$$ 0 0
$$227$$ 0 0 0.831470 0.555570i $$-0.187500\pi$$
−0.831470 + 0.555570i $$0.812500\pi$$
$$228$$ 0 0
$$229$$ 8.65964 3.58694i 0.572245 0.237032i −0.0777462 0.996973i $$-0.524772\pi$$
0.649992 + 0.759941i $$0.274772\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 0 0 −0.195090 0.980785i $$-0.562500\pi$$
0.195090 + 0.980785i $$0.437500\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 22.9305 + 9.49811i 1.49265 + 0.618274i
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −10.5254 −0.680830 −0.340415 0.940275i $$-0.610568\pi$$
−0.340415 + 0.940275i $$0.610568\pi$$
$$240$$ 0 0
$$241$$ 0 0 −0.831470 0.555570i $$-0.812500\pi$$
0.831470 + 0.555570i $$0.187500\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 12.2213 12.2213i 0.771400 0.771400i −0.206951 0.978351i $$-0.566354\pi$$
0.978351 + 0.206951i $$0.0663540\pi$$
$$252$$ 0 0
$$253$$ −2.60986 1.08104i −0.164081 0.0679645i
$$254$$ 0 0
$$255$$ 0 0
$$256$$ −16.0000 −1.00000
$$257$$ 0 0 0.382683 0.923880i $$-0.375000\pi$$
−0.382683 + 0.923880i $$0.625000\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 0 0 0.923880 0.382683i $$-0.125000\pi$$
−0.923880 + 0.382683i $$0.875000\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 0 0
$$268$$ −16.6856 + 16.6856i −1.01924 + 1.01924i
$$269$$ 10.5820 15.8370i 0.645195 0.965602i −0.354341 0.935116i $$-0.615295\pi$$
0.999535 0.0304855i $$-0.00970535\pi$$
$$270$$ 0 0
$$271$$ 15.3188i 0.930548i −0.885167 0.465274i $$-0.845956\pi$$
0.885167 0.465274i $$-0.154044\pi$$
$$272$$ −8.28973 + 14.2576i −0.502639 + 0.864496i
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −1.24100 0.829209i −0.0748351 0.0500032i
$$276$$ 0 0
$$277$$ 0 0 0.980785 0.195090i $$-0.0625000\pi$$
−0.980785 + 0.195090i $$0.937500\pi$$
$$278$$ 0 0
$$279$$ −13.7574 2.73651i −0.823631 0.163830i
$$280$$ 0 0
$$281$$ −12.6171 30.4605i −0.752676 1.81712i −0.543884 0.839161i $$-0.683047\pi$$
−0.208792 0.977960i $$-0.566953\pi$$
$$282$$ 0 0
$$283$$ −18.6923 27.9750i −1.11114 1.66294i −0.555732 0.831362i $$-0.687562\pi$$
−0.555409 0.831578i $$-0.687438\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 8.41004 + 14.7740i 0.494708 + 0.869059i
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ −2.11488 2.11488i −0.123552 0.123552i 0.642627 0.766179i $$-0.277845\pi$$
−0.766179 + 0.642627i $$0.777845\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ −12.4770 + 62.7260i −0.721563 + 3.62754i
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 34.3208 1.95879 0.979395 0.201954i $$-0.0647291\pi$$
0.979395 + 0.201954i $$0.0647291\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 19.6541 3.90944i 1.11448 0.221684i 0.396697 0.917950i $$-0.370156\pi$$
0.717784 + 0.696265i $$0.245156\pi$$
$$312$$ 0 0
$$313$$ 0 0 −0.980785 0.195090i $$-0.937500\pi$$
0.980785 + 0.195090i $$0.0625000\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ −1.65350 + 8.31270i −0.0930166 + 0.467626i
$$317$$ 18.3606 + 27.4786i 1.03123 + 1.54335i 0.825228 + 0.564799i $$0.191046\pi$$
0.206005 + 0.978551i $$0.433954\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 18.0000 1.00000
$$325$$ −12.9311 + 31.2185i −0.717290 + 1.73169i
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 0 0 0.923880 0.382683i $$-0.125000\pi$$
−0.923880 + 0.382683i $$0.875000\pi$$
$$332$$ 5.01885 + 12.1166i 0.275445 + 0.664983i
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 19.0240 28.4714i 1.03630 1.55093i 0.218163 0.975912i $$-0.429994\pi$$
0.818138 0.575022i $$-0.195006\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 1.39571 0.0755819
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 0 0 −0.980785 0.195090i $$-0.937500\pi$$
0.980785 + 0.195090i $$0.0625000\pi$$
$$348$$ 0 0
$$349$$ 0 0 −0.382683 0.923880i $$-0.625000\pi$$
0.382683 + 0.923880i $$0.375000\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −21.7787 + 21.7787i −1.15916 + 1.15916i −0.174509 + 0.984656i $$0.555834\pi$$
−0.984656 + 0.174509i $$0.944166\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −3.73079 + 9.00692i −0.196903 + 0.475367i −0.991234 0.132119i $$-0.957822\pi$$
0.794330 + 0.607486i $$0.207822\pi$$
$$360$$ 0 0
$$361$$ −13.4350 13.4350i −0.707107 0.707107i
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 6.38915 32.1204i 0.333511 1.67667i −0.342296 0.939592i $$-0.611204\pi$$
0.675806 0.737079i $$-0.263796\pi$$
$$368$$ 21.0303 + 31.4741i 1.09628 + 1.64070i
$$369$$ 6.76783 + 34.0242i 0.352319 + 1.77123i
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 38.1805 7.59458i 1.96120 0.390107i 0.976819 0.214065i $$-0.0686705\pi$$
0.984383 0.176042i $$-0.0563295\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 0 0 −0.382683 0.923880i $$-0.625000\pi$$
0.382683 + 0.923880i $$0.375000\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −13.9104 + 13.9104i −0.707107 + 0.707107i
$$388$$ 17.5575 26.2767i 0.891349 1.33400i
$$389$$ 0 0 −0.923880 0.382683i $$-0.875000\pi$$
0.923880 + 0.382683i $$0.125000\pi$$
$$390$$ 0 0
$$391$$ 38.9427 2.43319i 1.96942 0.123051i
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 0 0
$$396$$ −1.75663 + 0.349416i −0.0882740 + 0.0175588i
$$397$$ 28.7196 19.1898i 1.44139 0.963109i 0.443634 0.896208i $$-0.353689\pi$$
0.997760 0.0669005i $$-0.0213110\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 7.65367 + 18.4776i 0.382683 + 0.923880i
$$401$$ −7.14894 + 35.9402i −0.357001 + 1.79477i 0.217281 + 0.976109i $$0.430281\pi$$
−0.574283 + 0.818657i $$0.694719\pi$$
$$402$$ 0 0
$$403$$ −6.16456 30.9913i −0.307079 1.54379i
$$404$$ 1.71502 1.71502i 0.0853254 0.0853254i
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ −4.11377 4.11377i −0.202671 0.202671i
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 0 0 −0.555570 0.831470i $$-0.687500\pi$$
0.555570 + 0.831470i $$0.312500\pi$$
$$420$$ 0 0
$$421$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$422$$ 0 0
$$423$$ −29.3401 12.1531i −1.42656 0.590902i
$$424$$ 0 0
$$425$$ 20.4309 + 2.75319i 0.991042 + 0.133549i
$$426$$ 0 0
$$427$$ 0 0
$$428$$ −26.5810 17.7608i −1.28484 0.858502i
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 31.3742 20.9635i 1.51124 1.00978i 0.523789 0.851848i $$-0.324518\pi$$
0.987450 0.157930i $$-0.0504821\pi$$
$$432$$ 0 0
$$433$$ 0 0 0.923880 0.382683i $$-0.125000\pi$$
−0.923880 + 0.382683i $$0.875000\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 22.9561 + 34.3562i 1.09940 + 1.64537i
$$437$$ 0 0
$$438$$ 0 0
$$439$$ −18.0289 + 26.9822i −0.860473 + 1.28779i 0.0958262 + 0.995398i $$0.469451\pi$$
−0.956299 + 0.292391i $$0.905549\pi$$
$$440$$ 0 0
$$441$$ 21.0000i 1.00000i
$$442$$ 0 0
$$443$$ 37.5579 1.78443 0.892215 0.451612i $$-0.149151\pi$$
0.892215 + 0.451612i $$0.149151\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 0 0 −0.980785 0.195090i $$-0.937500\pi$$
0.980785 + 0.195090i $$0.0625000\pi$$
$$450$$ 0 0
$$451$$ −1.32095 3.18906i −0.0622012 0.150167i
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 0 0 −0.923880 0.382683i $$-0.875000\pi$$
0.923880 + 0.382683i $$0.125000\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 10.0377 24.2331i 0.467502 1.12865i −0.497748 0.867322i $$-0.665840\pi$$
0.965250 0.261328i $$-0.0841604\pi$$
$$462$$ 0 0
$$463$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 0 0 0.923880 0.382683i $$-0.125000\pi$$
−0.923880 + 0.382683i $$0.875000\pi$$
$$468$$ 15.5173 + 37.4622i 0.717290 + 1.73169i
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 1.08750 1.62756i 0.0500032 0.0748351i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ −28.5030 −1.30506
$$478$$ 0 0
$$479$$ −28.9678 19.3556i −1.32357 0.884382i −0.325444 0.945561i $$-0.605514\pi$$
−0.998127 + 0.0611793i $$0.980514\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 0 0
$$483$$ 0 0
$$484$$ −20.1607 + 8.35084i −0.916396 + 0.379583i
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 19.2710 + 28.8410i 0.873251 + 1.30691i 0.950762 + 0.309921i $$0.100303\pi$$
−0.0775113 + 0.996991i $$0.524697\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 0 0 −0.923880 0.382683i $$-0.875000\pi$$
0.923880 + 0.382683i $$0.125000\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ 0 0
$$495$$ 0 0
$$496$$ −15.5506 10.3906i −0.698241 0.466550i
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 0 0 0.831470 0.555570i $$-0.187500\pi$$
−0.831470 + 0.555570i $$0.812500\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 0 0 0.195090 0.980785i $$-0.437500\pi$$
−0.195090 + 0.980785i $$0.562500\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 0 0
$$508$$ −41.3609 17.1323i −1.83509 0.760121i
$$509$$ 16.0932i 0.713316i 0.934235 + 0.356658i $$0.116084\pi$$
−0.934235 + 0.356658i $$0.883916\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 3.09923 + 0.616476i 0.136304 + 0.0271126i
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 0 0 −0.555570 0.831470i $$-0.687500\pi$$
0.555570 + 0.831470i $$0.312500\pi$$
$$522$$ 0 0
$$523$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −17.3159 + 8.47372i −0.754294 + 0.369121i
$$528$$ 0 0
$$529$$ 25.4699 61.4898i 1.10739 2.67347i
$$530$$ 0 0
$$531$$ −26.3253 26.3253i −1.14242 1.14242i
$$532$$ 0 0
$$533$$ −64.9778 + 43.4168i −2.81450 + 1.88059i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ −0.407651 2.04940i −0.0175588 0.0882740i
$$540$$ 0 0
$$541$$ −5.12397 + 7.66857i −0.220297 + 0.329698i −0.925113 0.379693i $$-0.876030\pi$$
0.704816 + 0.709390i $$0.251030\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 35.8915 + 23.9819i 1.53461 + 1.02539i 0.981315 + 0.192406i $$0.0616291\pi$$
0.553293 + 0.832987i $$0.313371\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 0 0
$$556$$ −0.0469184 0.235875i −0.00198978 0.0100033i
$$557$$ 19.6071 19.6071i 0.830780 0.830780i −0.156844 0.987623i $$-0.550132\pi$$
0.987623 + 0.156844i $$0.0501319\pi$$
$$558$$ 0 0
$$559$$ −40.9427 16.9590i −1.73169 0.717290i
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −15.6567 + 37.7986i −0.659851 + 1.59302i 0.138182 + 0.990407i $$0.455874\pi$$
−0.798033 + 0.602614i $$0.794126\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −21.4943 + 8.90324i −0.901088 + 0.373243i −0.784639 0.619953i $$-0.787151\pi$$
−0.116450 + 0.993197i $$0.537151\pi$$
$$570$$ 0 0
$$571$$ 0 0 0.195090 0.980785i $$-0.437500\pi$$
−0.195090 + 0.980785i $$0.562500\pi$$
$$572$$ −2.24156 3.35473i −0.0937244 0.140268i
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 26.2879 39.3427i 1.09628 1.64070i
$$576$$ 22.1731 + 9.18440i 0.923880 + 0.382683i
$$577$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 2.78163 0.553300i 0.115203 0.0229154i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 0 0 −0.382683 0.923880i $$-0.625000\pi$$
0.382683 + 0.923880i $$0.375000\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 0 0 −0.923880 0.382683i $$-0.875000\pi$$
0.923880 + 0.382683i $$0.125000\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −32.4577 32.4577i −1.32618 1.32618i −0.908671 0.417514i $$-0.862902\pi$$
−0.417514 0.908671i $$-0.637098\pi$$
$$600$$ 0 0
$$601$$ 0 0 0.831470 0.555570i $$-0.187500\pi$$
−0.831470 + 0.555570i $$0.812500\pi$$
$$602$$ 0 0
$$603$$ 32.7013 13.5453i 1.33170 0.551608i
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 0 0 −0.195090 0.980785i $$-0.562500\pi$$
0.195090 + 0.980785i $$0.437500\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 71.5404i 2.89422i
$$612$$ 19.6723 15.0000i 0.795206 0.606339i
$$613$$ 8.16043 0.329597 0.164798 0.986327i $$-0.447303\pi$$
0.164798 + 0.986327i $$0.447303\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 9.51744 1.89314i 0.383158 0.0762148i 0.000246592 1.00000i $$-0.499922\pi$$
0.382911 + 0.923785i $$0.374922\pi$$
$$618$$ 0 0
$$619$$ −46.2899 9.20763i −1.86055 0.370086i −0.868474 0.495735i $$-0.834899\pi$$
−0.992075 + 0.125649i $$0.959899\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 17.6777 17.6777i 0.707107 0.707107i
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 0 0
$$630$$ 0 0
$$631$$ 0 0 0.382683 0.923880i $$-0.375000\pi$$
−0.382683 + 0.923880i $$0.625000\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −43.7059 + 18.1036i −1.73169 + 0.717290i
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 0 0 −0.195090 0.980785i $$-0.562500\pi$$
0.195090 + 0.980785i $$0.437500\pi$$
$$642$$ 0 0
$$643$$ 28.0519 41.9827i 1.10626 1.65563i 0.475281 0.879834i $$-0.342346\pi$$
0.630978 0.775800i $$-0.282654\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$648$$ 0 0
$$649$$ 3.08013 + 2.05808i 0.120906 + 0.0807866i
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 0 0 −0.980785 0.195090i $$-0.937500\pi$$
0.980785 + 0.195090i $$0.0625000\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ −9.02377 + 45.3656i −0.352319 + 1.77123i
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −2.60446 + 2.60446i −0.101455 + 0.101455i −0.756013 0.654557i $$-0.772855\pi$$
0.654557 + 0.756013i $$0.272855\pi$$
$$660$$ 0 0
$$661$$ −6.05828 2.50942i −0.235640 0.0976052i 0.261739 0.965139i $$-0.415704\pi$$
−0.497379 + 0.867533i $$0.665704\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 22.1828 4.41243i 0.858279 0.170722i
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 0 0 0.195090 0.980785i $$-0.437500\pi$$
−0.195090 + 0.980785i $$0.562500\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ −46.2056 + 46.2056i −1.77714 + 1.77714i
$$677$$ 0 0 0.555570 0.831470i $$-0.312500\pi$$
−0.555570 + 0.831470i $$0.687500\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 20.1855 + 13.4875i 0.772378 + 0.516086i 0.878197 0.478299i $$-0.158747\pi$$
−0.105819 + 0.994385i $$0.533747\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 0 0
$$688$$ −24.2331 + 10.0377i −0.923880 + 0.382683i
$$689$$ −24.5717 59.3214i −0.936109 2.25997i
$$690$$ 0 0
$$691$$ 0 0 −0.555570 0.831470i $$-0.687500\pi$$
0.555570 + 0.831470i $$0.312500\pi$$
$$692$$ 9.88545 + 49.6975i 0.375788 + 1.88922i
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 35.7501 + 31.5454i 1.35413 + 1.19487i
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −36.1149 36.1149i −1.36404 1.36404i −0.868698 0.495342i $$-0.835043\pi$$
−0.495342 0.868698i $$-0.664957\pi$$
$$702$$ 0 0
$$703$$ 0 0
$$704$$ −2.34217 0.465887i −0.0882740 0.0175588i
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −6.91327 34.7554i −0.259633 1.30526i −0.861944 0.507003i $$-0.830753\pi$$
0.602311 0.798262i $$-0.294247\pi$$
$$710$$ 0 0
$$711$$ 7.06315 10.5708i 0.264889 0.396434i
$$712$$ 0 0
$$713$$ 44.2473i 1.65708i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 2.66595 0.530291i 0.0994233 0.0197765i −0.145128 0.989413i $$-0.546359\pi$$
0.244551 + 0.969636i $$0.421359\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 3.73936 18.7990i 0.138972 0.698660i
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$728$$ 0 0
$$729$$ −24.9447 10.3325i −0.923880 0.382683i
$$730$$ 0 0
$$731$$ −3.61077 + 26.7948i −0.133549 + 0.991042i
$$732$$ 0 0
$$733$$ 0 0 0.382683 0.923880i $$-0.375000\pi$$
−0.382683 + 0.923880i $$0.625000\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −2.92840 + 1.95669i −0.107869 + 0.0720756i
$$738$$ 0 0
$$739$$ 0 0 0.923880 0.382683i $$-0.125000\pi$$
−0.923880 + 0.382683i $$0.875000\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 0 0 −0.195090 0.980785i $$-0.562500\pi$$
0.195090 + 0.980785i $$0.437500\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 19.6723i 0.719772i
$$748$$ −1.62865 + 1.84574i −0.0595495 + 0.0674868i
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 0 0 −0.831470 0.555570i $$-0.812500\pi$$
0.831470 + 0.555570i $$0.187500\pi$$
$$752$$ −29.9413 29.9413i −1.09185 1.09185i
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 0 0 −0.382683 0.923880i $$-0.625000\pi$$
0.382683 + 0.923880i $$0.375000\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 32.0947 77.4836i 1.15887 2.79777i
$$768$$ 0 0
$$769$$ −39.1416 39.1416i −1.41148 1.41148i −0.749670 0.661812i $$-0.769788\pi$$
−0.661812 0.749670i $$-0.730212\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 36.8904 + 7.33796i 1.32771 + 0.264099i
$$773$$ 0 0 0.923880 0.382683i $$-0.125000\pi$$
−0.923880 + 0.382683i $$0.875000\pi$$
$$774$$ 0 0
$$775$$ −4.56085 + 22.9289i −0.163830 + 0.823631i
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 0 0
$$784$$ −10.7151 + 25.8686i −0.382683 + 0.923880i
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 30.9493 6.15620i 1.10322 0.219445i 0.390301 0.920687i $$-0.372371\pi$$
0.712923 + 0.701242i $$0.247371\pi$$
$$788$$ 45.5062 30.4063i 1.62109 1.08318i
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 51.9513 + 21.5189i 1.84021 + 0.762240i 0.954664 + 0.297687i $$0.0962151\pi$$
0.885545 + 0.464553i $$0.153785\pi$$
$$798$$ 0 0
$$799$$ −42.1935 + 11.1679i −1.49270 + 0.395093i
$$800$$ 0 0
$$801$$ 0 0
\(80