# Properties

 Label 729.2.e.f.649.1 Level $729$ Weight $2$ Character 729.649 Analytic conductor $5.821$ Analytic rank $0$ Dimension $6$ CM discriminant -3 Inner twists $12$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$729 = 3^{6}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 729.e (of order $$9$$, degree $$6$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.82109430735$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: $$\Q(\zeta_{18})$$ Defining polynomial: $$x^{6} - x^{3} + 1$$ x^6 - x^3 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 27) Sato-Tate group: $\mathrm{U}(1)[D_{9}]$

## Embedding invariants

 Embedding label 649.1 Root $$-0.766044 - 0.642788i$$ of defining polynomial Character $$\chi$$ $$=$$ 729.649 Dual form 729.2.e.f.82.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+(1.87939 + 0.684040i) q^{4} +(0.939693 - 0.342020i) q^{7} +O(q^{10})$$ $$q+(1.87939 + 0.684040i) q^{4} +(0.939693 - 0.342020i) q^{7} +(0.868241 + 4.92404i) q^{13} +(3.06418 + 2.57115i) q^{16} +(3.50000 - 6.06218i) q^{19} +(-0.868241 + 4.92404i) q^{25} +2.00000 q^{28} +(3.75877 + 1.36808i) q^{31} +(-5.50000 - 9.52628i) q^{37} +(6.12836 + 5.14230i) q^{43} +(-4.59627 + 3.85673i) q^{49} +(-1.73648 + 9.84808i) q^{52} +(0.939693 - 0.342020i) q^{61} +(4.00000 + 6.92820i) q^{64} +(0.868241 + 4.92404i) q^{67} +(3.50000 - 6.06218i) q^{73} +(10.7246 - 8.99903i) q^{76} +(2.95202 - 16.7417i) q^{79} +(2.50000 + 4.33013i) q^{91} +(-14.5548 - 12.2130i) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q+O(q^{10})$$ 6 * q $$6 q + 21 q^{19} + 12 q^{28} - 33 q^{37} + 24 q^{64} + 21 q^{73} + 15 q^{91}+O(q^{100})$$ 6 * q + 21 * q^19 + 12 * q^28 - 33 * q^37 + 24 * q^64 + 21 * q^73 + 15 * q^91

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/729\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$e\left(\frac{5}{9}\right)$$

## 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 −0.984808 0.173648i $$-0.944444\pi$$
0.984808 + 0.173648i $$0.0555556\pi$$
$$3$$ 0 0
$$4$$ 1.87939 + 0.684040i 0.939693 + 0.342020i
$$5$$ 0 0 −0.642788 0.766044i $$-0.722222\pi$$
0.642788 + 0.766044i $$0.277778\pi$$
$$6$$ 0 0
$$7$$ 0.939693 0.342020i 0.355170 0.129271i −0.158272 0.987396i $$-0.550592\pi$$
0.513442 + 0.858124i $$0.328370\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 0 0 0.642788 0.766044i $$-0.277778\pi$$
−0.642788 + 0.766044i $$0.722222\pi$$
$$12$$ 0 0
$$13$$ 0.868241 + 4.92404i 0.240807 + 1.36568i 0.830033 + 0.557714i $$0.188322\pi$$
−0.589226 + 0.807968i $$0.700567\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 3.06418 + 2.57115i 0.766044 + 0.642788i
$$17$$ 0 0 0.866025 0.500000i $$-0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$18$$ 0 0
$$19$$ 3.50000 6.06218i 0.802955 1.39076i −0.114708 0.993399i $$-0.536593\pi$$
0.917663 0.397360i $$-0.130073\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 0 0 0.342020 0.939693i $$-0.388889\pi$$
−0.342020 + 0.939693i $$0.611111\pi$$
$$24$$ 0 0
$$25$$ −0.868241 + 4.92404i −0.173648 + 0.984808i
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 2.00000 0.377964
$$29$$ 0 0 −0.984808 0.173648i $$-0.944444\pi$$
0.984808 + 0.173648i $$0.0555556\pi$$
$$30$$ 0 0
$$31$$ 3.75877 + 1.36808i 0.675095 + 0.245715i 0.656740 0.754117i $$-0.271935\pi$$
0.0183550 + 0.999832i $$0.494157\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −5.50000 9.52628i −0.904194 1.56611i −0.821995 0.569495i $$-0.807139\pi$$
−0.0821995 0.996616i $$-0.526194\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 0 0 0.984808 0.173648i $$-0.0555556\pi$$
−0.984808 + 0.173648i $$0.944444\pi$$
$$42$$ 0 0
$$43$$ 6.12836 + 5.14230i 0.934565 + 0.784194i 0.976631 0.214921i $$-0.0689495\pi$$
−0.0420659 + 0.999115i $$0.513394\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 0 0 −0.342020 0.939693i $$-0.611111\pi$$
0.342020 + 0.939693i $$0.388889\pi$$
$$48$$ 0 0
$$49$$ −4.59627 + 3.85673i −0.656610 + 0.550961i
$$50$$ 0 0
$$51$$ 0 0
$$52$$ −1.73648 + 9.84808i −0.240807 + 1.36568i
$$53$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 0 0 −0.642788 0.766044i $$-0.722222\pi$$
0.642788 + 0.766044i $$0.277778\pi$$
$$60$$ 0 0
$$61$$ 0.939693 0.342020i 0.120315 0.0437912i −0.281161 0.959661i $$-0.590719\pi$$
0.401476 + 0.915869i $$0.368497\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 4.00000 + 6.92820i 0.500000 + 0.866025i
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 0.868241 + 4.92404i 0.106073 + 0.601567i 0.990787 + 0.135433i $$0.0432425\pi$$
−0.884714 + 0.466134i $$0.845646\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 0 0 0.866025 0.500000i $$-0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$72$$ 0 0
$$73$$ 3.50000 6.06218i 0.409644 0.709524i −0.585206 0.810885i $$-0.698986\pi$$
0.994850 + 0.101361i $$0.0323196\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 10.7246 8.99903i 1.23020 1.03226i
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 2.95202 16.7417i 0.332128 1.88359i −0.121802 0.992554i $$-0.538867\pi$$
0.453930 0.891038i $$-0.350022\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 0 0 −0.984808 0.173648i $$-0.944444\pi$$
0.984808 + 0.173648i $$0.0555556\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 0 0 −0.866025 0.500000i $$-0.833333\pi$$
0.866025 + 0.500000i $$0.166667\pi$$
$$90$$ 0 0
$$91$$ 2.50000 + 4.33013i 0.262071 + 0.453921i
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −14.5548 12.2130i −1.47782 1.24004i −0.908474 0.417941i $$-0.862752\pi$$
−0.569346 0.822098i $$-0.692804\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ −5.00000 + 8.66025i −0.500000 + 0.866025i
$$101$$ 0 0 −0.342020 0.939693i $$-0.611111\pi$$
0.342020 + 0.939693i $$0.388889\pi$$
$$102$$ 0 0
$$103$$ −9.95858 + 8.35624i −0.981248 + 0.823365i −0.984277 0.176631i $$-0.943480\pi$$
0.00302937 + 0.999995i $$0.499036\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$108$$ 0 0
$$109$$ 2.00000 0.191565 0.0957826 0.995402i $$-0.469465\pi$$
0.0957826 + 0.995402i $$0.469465\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 3.75877 + 1.36808i 0.355170 + 0.129271i
$$113$$ 0 0 −0.642788 0.766044i $$-0.722222\pi$$
0.642788 + 0.766044i $$0.277778\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −1.91013 10.8329i −0.173648 0.984808i
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 6.12836 + 5.14230i 0.550343 + 0.461792i
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −10.0000 + 17.3205i −0.887357 + 1.53695i −0.0443678 + 0.999015i $$0.514127\pi$$
−0.842989 + 0.537931i $$0.819206\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 0 0 0.342020 0.939693i $$-0.388889\pi$$
−0.342020 + 0.939693i $$0.611111\pi$$
$$132$$ 0 0
$$133$$ 1.21554 6.89365i 0.105400 0.597756i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 0 0 −0.984808 0.173648i $$-0.944444\pi$$
0.984808 + 0.173648i $$0.0555556\pi$$
$$138$$ 0 0
$$139$$ −21.6129 7.86646i −1.83318 0.667225i −0.991962 0.126536i $$-0.959614\pi$$
−0.841223 0.540689i $$-0.818164\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 0 0
$$148$$ −3.82026 21.6658i −0.314023 1.78092i
$$149$$ 0 0 0.984808 0.173648i $$-0.0555556\pi$$
−0.984808 + 0.173648i $$0.944444\pi$$
$$150$$ 0 0
$$151$$ −14.5548 12.2130i −1.18446 0.993877i −0.999939 0.0110477i $$-0.996483\pi$$
−0.184517 0.982829i $$-0.559072\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 10.7246 8.99903i 0.855918 0.718201i −0.105167 0.994455i $$-0.533538\pi$$
0.961085 + 0.276254i $$0.0890931\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −25.0000 −1.95815 −0.979076 0.203497i $$-0.934769\pi$$
−0.979076 + 0.203497i $$0.934769\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 0 0 −0.642788 0.766044i $$-0.722222\pi$$
0.642788 + 0.766044i $$0.277778\pi$$
$$168$$ 0 0
$$169$$ −11.2763 + 4.10424i −0.867409 + 0.315711i
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 8.00000 + 13.8564i 0.609994 + 1.05654i
$$173$$ 0 0 0.642788 0.766044i $$-0.277778\pi$$
−0.642788 + 0.766044i $$0.722222\pi$$
$$174$$ 0 0
$$175$$ 0.868241 + 4.92404i 0.0656328 + 0.372222i
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 0 0 0.866025 0.500000i $$-0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$180$$ 0 0
$$181$$ 3.50000 6.06218i 0.260153 0.450598i −0.706129 0.708083i $$-0.749560\pi$$
0.966282 + 0.257485i $$0.0828937\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 0 0 −0.984808 0.173648i $$-0.944444\pi$$
0.984808 + 0.173648i $$0.0555556\pi$$
$$192$$ 0 0
$$193$$ −21.6129 7.86646i −1.55573 0.566240i −0.585979 0.810326i $$-0.699290\pi$$
−0.969754 + 0.244086i $$0.921512\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ −11.2763 + 4.10424i −0.805451 + 0.293160i
$$197$$ 0 0 −0.866025 0.500000i $$-0.833333\pi$$
0.866025 + 0.500000i $$0.166667\pi$$
$$198$$ 0 0
$$199$$ −5.50000 9.52628i −0.389885 0.675300i 0.602549 0.798082i $$-0.294152\pi$$
−0.992434 + 0.122782i $$0.960818\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 0 0
$$208$$ −10.0000 + 17.3205i −0.693375 + 1.20096i
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −9.95858 + 8.35624i −0.685577 + 0.575267i −0.917630 0.397436i $$-0.869900\pi$$
0.232053 + 0.972703i $$0.425456\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 4.00000 0.271538
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 26.3114 9.57656i 1.76194 0.641294i 0.761961 0.647623i $$-0.224237\pi$$
0.999980 + 0.00632846i $$0.00201443\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 0 0 0.642788 0.766044i $$-0.277778\pi$$
−0.642788 + 0.766044i $$0.722222\pi$$
$$228$$ 0 0
$$229$$ −3.82026 21.6658i −0.252450 1.43171i −0.802535 0.596606i $$-0.796516\pi$$
0.550085 0.835109i $$-0.314595\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 0 0 0.866025 0.500000i $$-0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 0 0 0.342020 0.939693i $$-0.388889\pi$$
−0.342020 + 0.939693i $$0.611111\pi$$
$$240$$ 0 0
$$241$$ 2.95202 16.7417i 0.190156 1.07843i −0.728993 0.684521i $$-0.760011\pi$$
0.919150 0.393909i $$-0.128877\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 2.00000 0.128037
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 32.8892 + 11.9707i 2.09269 + 0.761678i
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 0 0 −0.866025 0.500000i $$-0.833333\pi$$
0.866025 + 0.500000i $$0.166667\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 2.77837 + 15.7569i 0.173648 + 0.984808i
$$257$$ 0 0 0.984808 0.173648i $$-0.0555556\pi$$
−0.984808 + 0.173648i $$0.944444\pi$$
$$258$$ 0 0
$$259$$ −8.42649 7.07066i −0.523597 0.439350i
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 0 0 −0.342020 0.939693i $$-0.611111\pi$$
0.342020 + 0.939693i $$0.388889\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 0 0
$$268$$ −1.73648 + 9.84808i −0.106073 + 0.601567i
$$269$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$270$$ 0 0
$$271$$ 29.0000 1.76162 0.880812 0.473466i $$-0.156997\pi$$
0.880812 + 0.473466i $$0.156997\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −24.4320 + 8.89252i −1.46798 + 0.534300i −0.947550 0.319606i $$-0.896449\pi$$
−0.520427 + 0.853906i $$0.674227\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 0 0 0.642788 0.766044i $$-0.277778\pi$$
−0.642788 + 0.766044i $$0.722222\pi$$
$$282$$ 0 0
$$283$$ 5.55674 + 31.5138i 0.330314 + 1.87330i 0.469344 + 0.883016i $$0.344491\pi$$
−0.139030 + 0.990288i $$0.544398\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 8.50000 14.7224i 0.500000 0.866025i
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 10.7246 8.99903i 0.627611 0.526628i
$$293$$ 0 0 0.342020 0.939693i $$-0.388889\pi$$
−0.342020 + 0.939693i $$0.611111\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 7.51754 + 2.73616i 0.433304 + 0.157710i
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 26.3114 9.57656i 1.50906 0.549254i
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 8.00000 + 13.8564i 0.456584 + 0.790827i 0.998778 0.0494267i $$-0.0157394\pi$$
−0.542194 + 0.840254i $$0.682406\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 0 0 0.984808 0.173648i $$-0.0555556\pi$$
−0.984808 + 0.173648i $$0.944444\pi$$
$$312$$ 0 0
$$313$$ 26.8116 + 22.4976i 1.51548 + 1.27164i 0.852134 + 0.523324i $$0.175308\pi$$
0.663345 + 0.748314i $$0.269136\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 17.0000 29.4449i 0.956325 1.65640i
$$317$$ 0 0 −0.342020 0.939693i $$-0.611111\pi$$
0.342020 + 0.939693i $$0.388889\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 0 0
$$325$$ −25.0000 −1.38675
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 0.939693 0.342020i 0.0516502 0.0187991i −0.316066 0.948737i $$-0.602362\pi$$
0.367716 + 0.929938i $$0.380140\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 0.868241 + 4.92404i 0.0472961 + 0.268229i 0.999281 0.0379157i $$-0.0120718\pi$$
−0.951985 + 0.306145i $$0.900961\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ −6.50000 + 11.2583i −0.350967 + 0.607893i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 0 0 0.342020 0.939693i $$-0.388889\pi$$
−0.342020 + 0.939693i $$0.611111\pi$$
$$348$$ 0 0
$$349$$ −6.42498 + 36.4379i −0.343921 + 1.95048i −0.0350017 + 0.999387i $$0.511144\pi$$
−0.308920 + 0.951088i $$0.599967\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 0 0 −0.984808 0.173648i $$-0.944444\pi$$
0.984808 + 0.173648i $$0.0555556\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 0 0 −0.866025 0.500000i $$-0.833333\pi$$
0.866025 + 0.500000i $$0.166667\pi$$
$$360$$ 0 0
$$361$$ −15.0000 25.9808i −0.789474 1.36741i
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 1.73648 + 9.84808i 0.0910164 + 0.516180i
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 26.8116 + 22.4976i 1.39955 + 1.17436i 0.961298 + 0.275512i $$0.0888475\pi$$
0.438254 + 0.898851i $$0.355597\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −9.95858 + 8.35624i −0.515636 + 0.432670i −0.863107 0.505021i $$-0.831485\pi$$
0.347472 + 0.937691i $$0.387040\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 29.0000 1.48963 0.744815 0.667271i $$-0.232538\pi$$
0.744815 + 0.667271i $$0.232538\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 0 0 −0.642788 0.766044i $$-0.722222\pi$$
0.642788 + 0.766044i $$0.277778\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 0 0
$$388$$ −19.0000 32.9090i −0.964579 1.67070i
$$389$$ 0 0 0.642788 0.766044i $$-0.277778\pi$$
−0.642788 + 0.766044i $$0.722222\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 17.0000 29.4449i 0.853206 1.47780i −0.0250943 0.999685i $$-0.507989\pi$$
0.878300 0.478110i $$-0.158678\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ −15.3209 + 12.8558i −0.766044 + 0.642788i
$$401$$ 0 0 0.342020 0.939693i $$-0.388889\pi$$
−0.342020 + 0.939693i $$0.611111\pi$$
$$402$$ 0 0
$$403$$ −3.47296 + 19.6962i −0.173001 + 0.981135i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ 29.1305 + 10.6026i 1.44041 + 0.524266i 0.939895 0.341463i $$-0.110922\pi$$
0.500514 + 0.865729i $$0.333144\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ −24.4320 + 8.89252i −1.20368 + 0.438103i
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 0 0 0.984808 0.173648i $$-0.0555556\pi$$
−0.984808 + 0.173648i $$0.944444\pi$$
$$420$$ 0 0
$$421$$ −14.5548 12.2130i −0.709360 0.595223i 0.215060 0.976601i $$-0.431005\pi$$
−0.924419 + 0.381377i $$0.875450\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 0.766044 0.642788i 0.0370715 0.0311067i
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$432$$ 0 0
$$433$$ 2.00000 0.0961139 0.0480569 0.998845i $$-0.484697\pi$$
0.0480569 + 0.998845i $$0.484697\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 3.75877 + 1.36808i 0.180012 + 0.0655192i
$$437$$ 0 0
$$438$$ 0 0
$$439$$ 26.3114 9.57656i 1.25577 0.457064i 0.373425 0.927660i $$-0.378183\pi$$
0.882349 + 0.470596i $$0.155961\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 0 0 0.642788 0.766044i $$-0.277778\pi$$
−0.642788 + 0.766044i $$0.722222\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 6.12836 + 5.14230i 0.289538 + 0.242951i
$$449$$ 0 0 0.866025 0.500000i $$-0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −1.73648 + 9.84808i −0.0812292 + 0.460674i 0.916878 + 0.399169i $$0.130701\pi$$
−0.998107 + 0.0615051i $$0.980410\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 0 0 −0.984808 0.173648i $$-0.944444\pi$$
0.984808 + 0.173648i $$0.0555556\pi$$
$$462$$ 0 0
$$463$$ −21.6129 7.86646i −1.00444 0.365586i −0.213144 0.977021i $$-0.568370\pi$$
−0.791294 + 0.611435i $$0.790592\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 0 0 −0.866025 0.500000i $$-0.833333\pi$$
0.866025 + 0.500000i $$0.166667\pi$$
$$468$$ 0 0
$$469$$ 2.50000 + 4.33013i 0.115439 + 0.199947i
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 26.8116 + 22.4976i 1.23020 + 1.03226i
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 0 0 −0.342020 0.939693i $$-0.611111\pi$$
0.342020 + 0.939693i $$0.388889\pi$$
$$480$$ 0 0
$$481$$ 42.1324 35.3533i 1.92107 1.61197i
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 3.82026 21.6658i 0.173648 0.984808i
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −25.0000 −1.13286 −0.566429 0.824110i $$-0.691675\pi$$
−0.566429 + 0.824110i $$0.691675\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 0 0 −0.642788 0.766044i $$-0.722222\pi$$
0.642788 + 0.766044i $$0.277778\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 8.00000 + 13.8564i 0.359211 + 0.622171i
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 5.55674 + 31.5138i 0.248754 + 1.41075i 0.811610 + 0.584199i $$0.198591\pi$$
−0.562857 + 0.826555i $$0.690298\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 0 0 0.866025 0.500000i $$-0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 0 0
$$508$$ −30.6418 + 25.7115i −1.35951 + 1.14076i
$$509$$ 0 0 0.342020 0.939693i $$-0.388889\pi$$
−0.342020 + 0.939693i $$0.611111\pi$$
$$510$$ 0 0
$$511$$ 1.21554 6.89365i 0.0537722 0.304957i
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 0 0 −0.866025 0.500000i $$-0.833333\pi$$
0.866025 + 0.500000i $$0.166667\pi$$
$$522$$ 0 0
$$523$$ 21.5000 + 37.2391i 0.940129 + 1.62835i 0.765222 + 0.643767i $$0.222629\pi$$
0.174908 + 0.984585i $$0.444037\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 0 0
$$528$$ 0 0
$$529$$ −17.6190 14.7841i −0.766044 0.642788i
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 7.00000 12.1244i 0.303488 0.525657i
$$533$$ 0 0
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 29.0000 1.24681 0.623404 0.781900i $$-0.285749\pi$$
0.623404 + 0.781900i $$0.285749\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 0.939693 0.342020i 0.0401784 0.0146237i −0.321853 0.946790i $$-0.604306\pi$$
0.362031 + 0.932166i $$0.382083\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 0 0
$$553$$ −2.95202 16.7417i −0.125533 0.711931i
$$554$$ 0 0
$$555$$ 0 0
$$556$$ −35.2380 29.5682i −1.49443 1.25397i
$$557$$ 0 0 0.866025 0.500000i $$-0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$558$$ 0 0
$$559$$ −20.0000 + 34.6410i −0.845910 + 1.46516i
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 0 0 0.342020 0.939693i $$-0.388889\pi$$
−0.342020 + 0.939693i $$0.611111\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 0 0 −0.984808 0.173648i $$-0.944444\pi$$
0.984808 + 0.173648i $$0.0555556\pi$$
$$570$$ 0 0
$$571$$ 29.1305 + 10.6026i 1.21907 + 0.443706i 0.869842 0.493331i $$-0.164221\pi$$
0.349231 + 0.937037i $$0.386443\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −5.50000 9.52628i −0.228968 0.396584i 0.728535 0.685009i $$-0.240202\pi$$
−0.957503 + 0.288425i $$0.906868\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 0 0 −0.342020 0.939693i $$-0.611111\pi$$
0.342020 + 0.939693i $$0.388889\pi$$
$$588$$ 0 0
$$589$$ 21.4492 17.9981i 0.883801 0.741597i
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 7.64052 43.3315i 0.314023 1.78092i
$$593$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 0 0 −0.642788 0.766044i $$-0.722222\pi$$
0.642788 + 0.766044i $$0.277778\pi$$
$$600$$ 0 0
$$601$$ −24.4320 + 8.89252i −0.996602 + 0.362734i −0.788273 0.615325i $$-0.789025\pi$$
−0.208329 + 0.978059i $$0.566802\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ −19.0000 32.9090i −0.773099 1.33905i
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −8.50876 48.2556i −0.345360 1.95863i −0.276531 0.961005i $$-0.589185\pi$$
−0.0688294 0.997628i $$-0.521926\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ −23.5000 + 40.7032i −0.949156 + 1.64399i −0.201948 + 0.979396i $$0.564727\pi$$
−0.747208 + 0.664590i $$0.768606\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 0 0 0.342020 0.939693i $$-0.388889\pi$$
−0.342020 + 0.939693i $$0.611111\pi$$
$$618$$ 0 0
$$619$$ 2.95202 16.7417i 0.118652 0.672907i −0.866226 0.499653i $$-0.833461\pi$$
0.984877 0.173254i $$-0.0554281\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −23.4923 8.55050i −0.939693 0.342020i
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 26.3114 9.57656i 1.04994 0.382147i
$$629$$ 0 0
$$630$$ 0 0
$$631$$ 21.5000 + 37.2391i 0.855901 + 1.48246i 0.875806 + 0.482663i $$0.160330\pi$$
−0.0199047 + 0.999802i $$0.506336\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −22.9813 19.2836i −0.910554 0.764045i
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 0 0 −0.342020 0.939693i $$-0.611111\pi$$
0.342020 + 0.939693i $$0.388889\pi$$
$$642$$ 0 0
$$643$$ −30.6418 + 25.7115i −1.20839 + 1.01396i −0.209044 + 0.977906i $$0.567035\pi$$
−0.999350 + 0.0360565i $$0.988520\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −46.9846 17.1010i −1.84006 0.669727i
$$653$$ 0 0 −0.642788 0.766044i $$-0.722222\pi$$
0.642788 + 0.766044i $$0.277778\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 0 0 0.642788 0.766044i $$-0.277778\pi$$
−0.642788 + 0.766044i $$0.722222\pi$$
$$660$$ 0 0
$$661$$ −8.50876 48.2556i −0.330952 1.87692i −0.464031 0.885819i $$-0.653597\pi$$
0.133078 0.991106i $$-0.457514\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ −6.42498 + 36.4379i −0.247665 + 1.40458i 0.566557 + 0.824023i $$0.308275\pi$$
−0.814221 + 0.580554i $$0.802836\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ −24.0000 −0.923077
$$677$$ 0 0 −0.984808 0.173648i $$-0.944444\pi$$
0.984808 + 0.173648i $$0.0555556\pi$$
$$678$$ 0 0
$$679$$ −17.8542 6.49838i −0.685180 0.249385i
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 0 0 −0.866025 0.500000i $$-0.833333\pi$$
0.866025 + 0.500000i $$0.166667\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 5.55674 + 31.5138i 0.211849 + 1.20145i
$$689$$ 0 0
$$690$$ 0 0
$$691$$ 6.12836 + 5.14230i 0.233134 + 0.195622i 0.751869 0.659313i $$-0.229153\pi$$
−0.518735 + 0.854935i $$0.673597\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 0 0
$$698$$ 0 0
$$699$$ 0 0
$$700$$ −1.73648 + 9.84808i −0.0656328 + 0.372222i
$$701$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$702$$ 0 0
$$703$$ −77.0000 −2.90411
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −49.8037 + 18.1271i −1.87042 + 0.680776i −0.901837 + 0.432077i $$0.857781\pi$$
−0.968581 + 0.248700i $$0.919997\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 0 0 0.866025 0.500000i $$-0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$720$$ 0 0
$$721$$ −6.50000 + 11.2583i −0.242073 + 0.419282i
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 10.7246 8.99903i 0.398577 0.334446i
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 7.64052 43.3315i 0.283371 1.60708i −0.427675 0.903933i $$-0.640667\pi$$
0.711046 0.703145i $$-0.248222\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 0 0
$$733$$ −46.9846 17.1010i −1.73542 0.631640i −0.736425 0.676520i $$-0.763487\pi$$
−0.998992 + 0.0448796i $$0.985710\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ 8.00000 + 13.8564i 0.294285 + 0.509716i 0.974818 0.223001i $$-0.0715853\pi$$
−0.680534 + 0.732717i $$0.738252\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 0 0 0.984808 0.173648i $$-0.0555556\pi$$
−0.984808 + 0.173648i $$0.944444\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 31.4078 26.3543i 1.14609 0.961682i 0.146467 0.989216i $$-0.453210\pi$$
0.999621 + 0.0275338i $$0.00876539\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 29.0000 1.05402 0.527011 0.849858i $$-0.323312\pi$$
0.527011 + 0.849858i $$0.323312\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 0 0 −0.642788 0.766044i $$-0.722222\pi$$
0.642788 + 0.766044i $$0.277778\pi$$
$$762$$ 0 0
$$763$$ 1.87939 0.684040i 0.0680383 0.0247639i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ −8.50876 48.2556i −0.306834 1.74014i −0.614745 0.788726i $$-0.710741\pi$$
0.307912 0.951415i $$-0.400370\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −35.2380 29.5682i −1.26824 1.06418i
$$773$$ 0 0 0.866025 0.500000i $$-0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$774$$ 0 0
$$775$$ −10.0000 + 17.3205i −0.359211 + 0.622171i
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 0 0
$$784$$ −24.0000 −0.857143
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 29.1305 + 10.6026i 1.03839 + 0.377943i 0.804269 0.594265i $$-0.202557\pi$$
0.234120 + 0.972208i $$0.424779\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 2.50000 + 4.33013i 0.0887776 + 0.153767i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ −3.82026 21.6658i −0.135406 0.767923i
$$797$$ 0 0 0.984808 0.173648i $$-0.0555556\pi$$
−0.984808 + 0.173648i $$0.944444\pi$$
$$798$$ 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