# Properties

 Label 912.3.o.c.721.2 Level $912$ Weight $3$ Character 912.721 Analytic conductor $24.850$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$912 = 2^{4} \cdot 3 \cdot 19$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 912.o (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$24.8502001097$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.219615408.1 Defining polynomial: $$x^{6} - 3 x^{5} + 22 x^{4} - 39 x^{3} + 112 x^{2} - 93 x + 39$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{5}\cdot 3$$ Twist minimal: no (minimal twist has level 228) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 721.2 Root $$0.500000 - 3.19918i$$ of defining polynomial Character $$\chi$$ $$=$$ 912.721 Dual form 912.3.o.c.721.5

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.73205i q^{3} +0.556406 q^{5} +9.52587 q^{7} -3.00000 q^{9} +O(q^{10})$$ $$q-1.73205i q^{3} +0.556406 q^{5} +9.52587 q^{7} -3.00000 q^{9} +13.5259 q^{11} +10.5348i q^{13} -0.963723i q^{15} +2.63868 q^{17} +(-0.112811 + 18.9997i) q^{19} -16.4993i q^{21} +22.0212 q^{23} -24.6904 q^{25} +5.19615i q^{27} +48.7824i q^{29} -0.248288i q^{31} -23.4275i q^{33} +5.30025 q^{35} -59.3172i q^{37} +18.2468 q^{39} +69.0705i q^{41} +27.5869 q^{43} -1.66922 q^{45} -44.8855 q^{47} +41.7421 q^{49} -4.57032i q^{51} -3.85489i q^{53} +7.52587 q^{55} +(32.9084 + 0.195394i) q^{57} -59.3539i q^{59} +96.4554 q^{61} -28.5776 q^{63} +5.86162i q^{65} -10.7831i q^{67} -38.1418i q^{69} +52.6373i q^{71} +13.7362 q^{73} +42.7650i q^{75} +128.846 q^{77} -51.3224i q^{79} +9.00000 q^{81} +63.7744 q^{83} +1.46818 q^{85} +84.4936 q^{87} +44.9275i q^{89} +100.353i q^{91} -0.430048 q^{93} +(-0.0627687 + 10.5715i) q^{95} -72.6403i q^{97} -40.5776 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 2 q^{5} + 2 q^{7} - 18 q^{9} + O(q^{10})$$ $$6 q - 2 q^{5} + 2 q^{7} - 18 q^{9} + 26 q^{11} - 50 q^{17} + 10 q^{19} - 28 q^{23} + 28 q^{25} - 2 q^{35} - 72 q^{39} + 210 q^{43} + 6 q^{45} - 22 q^{47} - 36 q^{49} - 10 q^{55} + 48 q^{57} + 214 q^{61} - 6 q^{63} + 102 q^{73} + 266 q^{77} + 54 q^{81} + 404 q^{83} + 370 q^{85} + 144 q^{87} - 120 q^{93} - 358 q^{95} - 78 q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/912\mathbb{Z}\right)^\times$$.

 $$n$$ $$97$$ $$229$$ $$305$$ $$799$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$ $$1$$

## 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
$$3$$ 1.73205i 0.577350i
$$4$$ 0 0
$$5$$ 0.556406 0.111281 0.0556406 0.998451i $$-0.482280\pi$$
0.0556406 + 0.998451i $$0.482280\pi$$
$$6$$ 0 0
$$7$$ 9.52587 1.36084 0.680419 0.732823i $$-0.261798\pi$$
0.680419 + 0.732823i $$0.261798\pi$$
$$8$$ 0 0
$$9$$ −3.00000 −0.333333
$$10$$ 0 0
$$11$$ 13.5259 1.22962 0.614812 0.788674i $$-0.289232\pi$$
0.614812 + 0.788674i $$0.289232\pi$$
$$12$$ 0 0
$$13$$ 10.5348i 0.810370i 0.914235 + 0.405185i $$0.132793\pi$$
−0.914235 + 0.405185i $$0.867207\pi$$
$$14$$ 0 0
$$15$$ 0.963723i 0.0642482i
$$16$$ 0 0
$$17$$ 2.63868 0.155216 0.0776082 0.996984i $$-0.475272\pi$$
0.0776082 + 0.996984i $$0.475272\pi$$
$$18$$ 0 0
$$19$$ −0.112811 + 18.9997i −0.00593742 + 0.999982i
$$20$$ 0 0
$$21$$ 16.4993i 0.785680i
$$22$$ 0 0
$$23$$ 22.0212 0.957443 0.478722 0.877967i $$-0.341100\pi$$
0.478722 + 0.877967i $$0.341100\pi$$
$$24$$ 0 0
$$25$$ −24.6904 −0.987617
$$26$$ 0 0
$$27$$ 5.19615i 0.192450i
$$28$$ 0 0
$$29$$ 48.7824i 1.68215i 0.540917 + 0.841076i $$0.318077\pi$$
−0.540917 + 0.841076i $$0.681923\pi$$
$$30$$ 0 0
$$31$$ 0.248288i 0.00800930i −0.999992 0.00400465i $$-0.998725\pi$$
0.999992 0.00400465i $$-0.00127472\pi$$
$$32$$ 0 0
$$33$$ 23.4275i 0.709924i
$$34$$ 0 0
$$35$$ 5.30025 0.151436
$$36$$ 0 0
$$37$$ 59.3172i 1.60317i −0.597882 0.801584i $$-0.703991\pi$$
0.597882 0.801584i $$-0.296009\pi$$
$$38$$ 0 0
$$39$$ 18.2468 0.467867
$$40$$ 0 0
$$41$$ 69.0705i 1.68465i 0.538974 + 0.842323i $$0.318812\pi$$
−0.538974 + 0.842323i $$0.681188\pi$$
$$42$$ 0 0
$$43$$ 27.5869 0.641557 0.320778 0.947154i $$-0.396056\pi$$
0.320778 + 0.947154i $$0.396056\pi$$
$$44$$ 0 0
$$45$$ −1.66922 −0.0370937
$$46$$ 0 0
$$47$$ −44.8855 −0.955011 −0.477505 0.878629i $$-0.658459\pi$$
−0.477505 + 0.878629i $$0.658459\pi$$
$$48$$ 0 0
$$49$$ 41.7421 0.851881
$$50$$ 0 0
$$51$$ 4.57032i 0.0896142i
$$52$$ 0 0
$$53$$ 3.85489i 0.0727338i −0.999339 0.0363669i $$-0.988422\pi$$
0.999339 0.0363669i $$-0.0115785\pi$$
$$54$$ 0 0
$$55$$ 7.52587 0.136834
$$56$$ 0 0
$$57$$ 32.9084 + 0.195394i 0.577340 + 0.00342797i
$$58$$ 0 0
$$59$$ 59.3539i 1.00600i −0.864287 0.503000i $$-0.832230\pi$$
0.864287 0.503000i $$-0.167770\pi$$
$$60$$ 0 0
$$61$$ 96.4554 1.58124 0.790618 0.612309i $$-0.209759\pi$$
0.790618 + 0.612309i $$0.209759\pi$$
$$62$$ 0 0
$$63$$ −28.5776 −0.453613
$$64$$ 0 0
$$65$$ 5.86162i 0.0901788i
$$66$$ 0 0
$$67$$ 10.7831i 0.160942i −0.996757 0.0804708i $$-0.974358\pi$$
0.996757 0.0804708i $$-0.0256424\pi$$
$$68$$ 0 0
$$69$$ 38.1418i 0.552780i
$$70$$ 0 0
$$71$$ 52.6373i 0.741371i 0.928759 + 0.370685i $$0.120877\pi$$
−0.928759 + 0.370685i $$0.879123\pi$$
$$72$$ 0 0
$$73$$ 13.7362 0.188167 0.0940835 0.995564i $$-0.470008\pi$$
0.0940835 + 0.995564i $$0.470008\pi$$
$$74$$ 0 0
$$75$$ 42.7650i 0.570201i
$$76$$ 0 0
$$77$$ 128.846 1.67332
$$78$$ 0 0
$$79$$ 51.3224i 0.649651i −0.945774 0.324826i $$-0.894694\pi$$
0.945774 0.324826i $$-0.105306\pi$$
$$80$$ 0 0
$$81$$ 9.00000 0.111111
$$82$$ 0 0
$$83$$ 63.7744 0.768366 0.384183 0.923257i $$-0.374483\pi$$
0.384183 + 0.923257i $$0.374483\pi$$
$$84$$ 0 0
$$85$$ 1.46818 0.0172726
$$86$$ 0 0
$$87$$ 84.4936 0.971191
$$88$$ 0 0
$$89$$ 44.9275i 0.504804i 0.967622 + 0.252402i $$0.0812205\pi$$
−0.967622 + 0.252402i $$0.918780\pi$$
$$90$$ 0 0
$$91$$ 100.353i 1.10278i
$$92$$ 0 0
$$93$$ −0.430048 −0.00462417
$$94$$ 0 0
$$95$$ −0.0627687 + 10.5715i −0.000660723 + 0.111279i
$$96$$ 0 0
$$97$$ 72.6403i 0.748870i −0.927253 0.374435i $$-0.877837\pi$$
0.927253 0.374435i $$-0.122163\pi$$
$$98$$ 0 0
$$99$$ −40.5776 −0.409875
$$100$$ 0 0
$$101$$ 116.783 1.15627 0.578133 0.815943i $$-0.303782\pi$$
0.578133 + 0.815943i $$0.303782\pi$$
$$102$$ 0 0
$$103$$ 72.3921i 0.702836i −0.936219 0.351418i $$-0.885700\pi$$
0.936219 0.351418i $$-0.114300\pi$$
$$104$$ 0 0
$$105$$ 9.18029i 0.0874314i
$$106$$ 0 0
$$107$$ 154.057i 1.43979i −0.694085 0.719893i $$-0.744191\pi$$
0.694085 0.719893i $$-0.255809\pi$$
$$108$$ 0 0
$$109$$ 159.670i 1.46487i −0.680839 0.732433i $$-0.738385\pi$$
0.680839 0.732433i $$-0.261615\pi$$
$$110$$ 0 0
$$111$$ −102.740 −0.925590
$$112$$ 0 0
$$113$$ 79.6420i 0.704796i 0.935850 + 0.352398i $$0.114634\pi$$
−0.935850 + 0.352398i $$0.885366\pi$$
$$114$$ 0 0
$$115$$ 12.2527 0.106545
$$116$$ 0 0
$$117$$ 31.6044i 0.270123i
$$118$$ 0 0
$$119$$ 25.1357 0.211224
$$120$$ 0 0
$$121$$ 61.9491 0.511976
$$122$$ 0 0
$$123$$ 119.634 0.972630
$$124$$ 0 0
$$125$$ −27.6480 −0.221184
$$126$$ 0 0
$$127$$ 30.2528i 0.238211i 0.992882 + 0.119106i $$0.0380027\pi$$
−0.992882 + 0.119106i $$0.961997\pi$$
$$128$$ 0 0
$$129$$ 47.7820i 0.370403i
$$130$$ 0 0
$$131$$ 142.842 1.09040 0.545199 0.838306i $$-0.316454\pi$$
0.545199 + 0.838306i $$0.316454\pi$$
$$132$$ 0 0
$$133$$ −1.07462 + 180.988i −0.00807987 + 1.36081i
$$134$$ 0 0
$$135$$ 2.89117i 0.0214161i
$$136$$ 0 0
$$137$$ −165.488 −1.20795 −0.603973 0.797005i $$-0.706416\pi$$
−0.603973 + 0.797005i $$0.706416\pi$$
$$138$$ 0 0
$$139$$ 169.324 1.21816 0.609079 0.793110i $$-0.291539\pi$$
0.609079 + 0.793110i $$0.291539\pi$$
$$140$$ 0 0
$$141$$ 77.7440i 0.551376i
$$142$$ 0 0
$$143$$ 142.492i 0.996450i
$$144$$ 0 0
$$145$$ 27.1428i 0.187192i
$$146$$ 0 0
$$147$$ 72.2995i 0.491833i
$$148$$ 0 0
$$149$$ −139.406 −0.935612 −0.467806 0.883831i $$-0.654955\pi$$
−0.467806 + 0.883831i $$0.654955\pi$$
$$150$$ 0 0
$$151$$ 152.742i 1.01154i −0.862669 0.505769i $$-0.831209\pi$$
0.862669 0.505769i $$-0.168791\pi$$
$$152$$ 0 0
$$153$$ −7.91603 −0.0517388
$$154$$ 0 0
$$155$$ 0.138149i 0.000891284i
$$156$$ 0 0
$$157$$ −295.932 −1.88492 −0.942459 0.334321i $$-0.891493\pi$$
−0.942459 + 0.334321i $$0.891493\pi$$
$$158$$ 0 0
$$159$$ −6.67687 −0.0419929
$$160$$ 0 0
$$161$$ 209.771 1.30293
$$162$$ 0 0
$$163$$ 112.731 0.691602 0.345801 0.938308i $$-0.387607\pi$$
0.345801 + 0.938308i $$0.387607\pi$$
$$164$$ 0 0
$$165$$ 13.0352i 0.0790011i
$$166$$ 0 0
$$167$$ 297.205i 1.77967i 0.456284 + 0.889834i $$0.349180\pi$$
−0.456284 + 0.889834i $$0.650820\pi$$
$$168$$ 0 0
$$169$$ 58.0179 0.343301
$$170$$ 0 0
$$171$$ 0.338433 56.9990i 0.00197914 0.333327i
$$172$$ 0 0
$$173$$ 262.849i 1.51936i −0.650299 0.759678i $$-0.725357\pi$$
0.650299 0.759678i $$-0.274643\pi$$
$$174$$ 0 0
$$175$$ −235.198 −1.34399
$$176$$ 0 0
$$177$$ −102.804 −0.580814
$$178$$ 0 0
$$179$$ 73.9185i 0.412953i −0.978452 0.206476i $$-0.933800\pi$$
0.978452 0.206476i $$-0.0661996\pi$$
$$180$$ 0 0
$$181$$ 34.8893i 0.192759i 0.995345 + 0.0963793i $$0.0307262\pi$$
−0.995345 + 0.0963793i $$0.969274\pi$$
$$182$$ 0 0
$$183$$ 167.066i 0.912928i
$$184$$ 0 0
$$185$$ 33.0044i 0.178402i
$$186$$ 0 0
$$187$$ 35.6904 0.190858
$$188$$ 0 0
$$189$$ 49.4979i 0.261893i
$$190$$ 0 0
$$191$$ −111.001 −0.581156 −0.290578 0.956851i $$-0.593848\pi$$
−0.290578 + 0.956851i $$0.593848\pi$$
$$192$$ 0 0
$$193$$ 83.5616i 0.432962i 0.976287 + 0.216481i $$0.0694579\pi$$
−0.976287 + 0.216481i $$0.930542\pi$$
$$194$$ 0 0
$$195$$ 10.1526 0.0520648
$$196$$ 0 0
$$197$$ −365.319 −1.85441 −0.927205 0.374554i $$-0.877796\pi$$
−0.927205 + 0.374554i $$0.877796\pi$$
$$198$$ 0 0
$$199$$ 10.6303 0.0534184 0.0267092 0.999643i $$-0.491497\pi$$
0.0267092 + 0.999643i $$0.491497\pi$$
$$200$$ 0 0
$$201$$ −18.6769 −0.0929197
$$202$$ 0 0
$$203$$ 464.695i 2.28914i
$$204$$ 0 0
$$205$$ 38.4312i 0.187469i
$$206$$ 0 0
$$207$$ −66.0636 −0.319148
$$208$$ 0 0
$$209$$ −1.52587 + 256.987i −0.00730080 + 1.22960i
$$210$$ 0 0
$$211$$ 230.194i 1.09097i 0.838122 + 0.545483i $$0.183654\pi$$
−0.838122 + 0.545483i $$0.816346\pi$$
$$212$$ 0 0
$$213$$ 91.1705 0.428031
$$214$$ 0 0
$$215$$ 15.3495 0.0713932
$$216$$ 0 0
$$217$$ 2.36516i 0.0108994i
$$218$$ 0 0
$$219$$ 23.7918i 0.108638i
$$220$$ 0 0
$$221$$ 27.7980i 0.125783i
$$222$$ 0 0
$$223$$ 263.814i 1.18302i −0.806297 0.591511i $$-0.798532\pi$$
0.806297 0.591511i $$-0.201468\pi$$
$$224$$ 0 0
$$225$$ 74.0712 0.329206
$$226$$ 0 0
$$227$$ 213.928i 0.942414i 0.882023 + 0.471207i $$0.156182\pi$$
−0.882023 + 0.471207i $$0.843818\pi$$
$$228$$ 0 0
$$229$$ −63.4698 −0.277161 −0.138580 0.990351i $$-0.544254\pi$$
−0.138580 + 0.990351i $$0.544254\pi$$
$$230$$ 0 0
$$231$$ 223.167i 0.966092i
$$232$$ 0 0
$$233$$ 288.449 1.23798 0.618989 0.785400i $$-0.287543\pi$$
0.618989 + 0.785400i $$0.287543\pi$$
$$234$$ 0 0
$$235$$ −24.9745 −0.106275
$$236$$ 0 0
$$237$$ −88.8931 −0.375076
$$238$$ 0 0
$$239$$ −279.128 −1.16790 −0.583950 0.811790i $$-0.698493\pi$$
−0.583950 + 0.811790i $$0.698493\pi$$
$$240$$ 0 0
$$241$$ 242.687i 1.00700i −0.863995 0.503500i $$-0.832045\pi$$
0.863995 0.503500i $$-0.167955\pi$$
$$242$$ 0 0
$$243$$ 15.5885i 0.0641500i
$$244$$ 0 0
$$245$$ 23.2256 0.0947982
$$246$$ 0 0
$$247$$ −200.158 1.18844i −0.810355 0.00481151i
$$248$$ 0 0
$$249$$ 110.460i 0.443616i
$$250$$ 0 0
$$251$$ 162.678 0.648118 0.324059 0.946037i $$-0.394952\pi$$
0.324059 + 0.946037i $$0.394952\pi$$
$$252$$ 0 0
$$253$$ 297.856 1.17730
$$254$$ 0 0
$$255$$ 2.54295i 0.00997237i
$$256$$ 0 0
$$257$$ 256.629i 0.998555i 0.866442 + 0.499277i $$0.166401\pi$$
−0.866442 + 0.499277i $$0.833599\pi$$
$$258$$ 0 0
$$259$$ 565.048i 2.18165i
$$260$$ 0 0
$$261$$ 146.347i 0.560718i
$$262$$ 0 0
$$263$$ −71.2027 −0.270733 −0.135366 0.990796i $$-0.543221\pi$$
−0.135366 + 0.990796i $$0.543221\pi$$
$$264$$ 0 0
$$265$$ 2.14488i 0.00809390i
$$266$$ 0 0
$$267$$ 77.8168 0.291449
$$268$$ 0 0
$$269$$ 219.411i 0.815653i 0.913059 + 0.407827i $$0.133713\pi$$
−0.913059 + 0.407827i $$0.866287\pi$$
$$270$$ 0 0
$$271$$ −323.913 −1.19525 −0.597626 0.801775i $$-0.703889\pi$$
−0.597626 + 0.801775i $$0.703889\pi$$
$$272$$ 0 0
$$273$$ 173.817 0.636691
$$274$$ 0 0
$$275$$ −333.959 −1.21440
$$276$$ 0 0
$$277$$ −37.4953 −0.135362 −0.0676810 0.997707i $$-0.521560\pi$$
−0.0676810 + 0.997707i $$0.521560\pi$$
$$278$$ 0 0
$$279$$ 0.744865i 0.00266977i
$$280$$ 0 0
$$281$$ 177.483i 0.631613i −0.948824 0.315806i $$-0.897725\pi$$
0.948824 0.315806i $$-0.102275\pi$$
$$282$$ 0 0
$$283$$ −72.4063 −0.255853 −0.127926 0.991784i $$-0.540832\pi$$
−0.127926 + 0.991784i $$0.540832\pi$$
$$284$$ 0 0
$$285$$ 18.3104 + 0.108719i 0.0642470 + 0.000381469i
$$286$$ 0 0
$$287$$ 657.956i 2.29253i
$$288$$ 0 0
$$289$$ −282.037 −0.975908
$$290$$ 0 0
$$291$$ −125.817 −0.432360
$$292$$ 0 0
$$293$$ 57.9819i 0.197891i 0.995093 + 0.0989453i $$0.0315469\pi$$
−0.995093 + 0.0989453i $$0.968453\pi$$
$$294$$ 0 0
$$295$$ 33.0249i 0.111949i
$$296$$ 0 0
$$297$$ 70.2825i 0.236641i
$$298$$ 0 0
$$299$$ 231.989i 0.775883i
$$300$$ 0 0
$$301$$ 262.790 0.873055
$$302$$ 0 0
$$303$$ 202.274i 0.667570i
$$304$$ 0 0
$$305$$ 53.6683 0.175962
$$306$$ 0 0
$$307$$ 484.604i 1.57851i −0.614062 0.789257i $$-0.710466\pi$$
0.614062 0.789257i $$-0.289534\pi$$
$$308$$ 0 0
$$309$$ −125.387 −0.405782
$$310$$ 0 0
$$311$$ −241.304 −0.775898 −0.387949 0.921681i $$-0.626816\pi$$
−0.387949 + 0.921681i $$0.626816\pi$$
$$312$$ 0 0
$$313$$ −313.749 −1.00239 −0.501196 0.865334i $$-0.667107\pi$$
−0.501196 + 0.865334i $$0.667107\pi$$
$$314$$ 0 0
$$315$$ −15.9007 −0.0504785
$$316$$ 0 0
$$317$$ 184.282i 0.581331i 0.956825 + 0.290665i $$0.0938767\pi$$
−0.956825 + 0.290665i $$0.906123\pi$$
$$318$$ 0 0
$$319$$ 659.825i 2.06842i
$$320$$ 0 0
$$321$$ −266.835 −0.831261
$$322$$ 0 0
$$323$$ −0.297672 + 50.1340i −0.000921585 + 0.155214i
$$324$$ 0 0
$$325$$ 260.109i 0.800334i
$$326$$ 0 0
$$327$$ −276.557 −0.845741
$$328$$ 0 0
$$329$$ −427.573 −1.29961
$$330$$ 0 0
$$331$$ 460.335i 1.39074i −0.718652 0.695370i $$-0.755241\pi$$
0.718652 0.695370i $$-0.244759\pi$$
$$332$$ 0 0
$$333$$ 177.952i 0.534389i
$$334$$ 0 0
$$335$$ 5.99977i 0.0179098i
$$336$$ 0 0
$$337$$ 493.633i 1.46479i −0.680882 0.732393i $$-0.738403\pi$$
0.680882 0.732393i $$-0.261597\pi$$
$$338$$ 0 0
$$339$$ 137.944 0.406914
$$340$$ 0 0
$$341$$ 3.35831i 0.00984843i
$$342$$ 0 0
$$343$$ −69.1373 −0.201567
$$344$$ 0 0
$$345$$ 21.2223i 0.0615140i
$$346$$ 0 0
$$347$$ 415.426 1.19719 0.598596 0.801051i $$-0.295725\pi$$
0.598596 + 0.801051i $$0.295725\pi$$
$$348$$ 0 0
$$349$$ 34.1332 0.0978030 0.0489015 0.998804i $$-0.484428\pi$$
0.0489015 + 0.998804i $$0.484428\pi$$
$$350$$ 0 0
$$351$$ −54.7405 −0.155956
$$352$$ 0 0
$$353$$ −154.833 −0.438620 −0.219310 0.975655i $$-0.570381\pi$$
−0.219310 + 0.975655i $$0.570381\pi$$
$$354$$ 0 0
$$355$$ 29.2877i 0.0825005i
$$356$$ 0 0
$$357$$ 43.5363i 0.121950i
$$358$$ 0 0
$$359$$ −565.544 −1.57533 −0.787665 0.616103i $$-0.788710\pi$$
−0.787665 + 0.616103i $$0.788710\pi$$
$$360$$ 0 0
$$361$$ −360.975 4.28674i −0.999929 0.0118746i
$$362$$ 0 0
$$363$$ 107.299i 0.295589i
$$364$$ 0 0
$$365$$ 7.64289 0.0209394
$$366$$ 0 0
$$367$$ −197.695 −0.538678 −0.269339 0.963045i $$-0.586805\pi$$
−0.269339 + 0.963045i $$0.586805\pi$$
$$368$$ 0 0
$$369$$ 207.211i 0.561548i
$$370$$ 0 0
$$371$$ 36.7212i 0.0989789i
$$372$$ 0 0
$$373$$ 198.960i 0.533404i 0.963779 + 0.266702i $$0.0859339\pi$$
−0.963779 + 0.266702i $$0.914066\pi$$
$$374$$ 0 0
$$375$$ 47.8878i 0.127701i
$$376$$ 0 0
$$377$$ −513.913 −1.36317
$$378$$ 0 0
$$379$$ 432.419i 1.14095i 0.821316 + 0.570474i $$0.193240\pi$$
−0.821316 + 0.570474i $$0.806760\pi$$
$$380$$ 0 0
$$381$$ 52.3994 0.137531
$$382$$ 0 0
$$383$$ 543.758i 1.41973i −0.704336 0.709867i $$-0.748755\pi$$
0.704336 0.709867i $$-0.251245\pi$$
$$384$$ 0 0
$$385$$ 71.6904 0.186209
$$386$$ 0 0
$$387$$ −82.7608 −0.213852
$$388$$ 0 0
$$389$$ 109.783 0.282218 0.141109 0.989994i $$-0.454933\pi$$
0.141109 + 0.989994i $$0.454933\pi$$
$$390$$ 0 0
$$391$$ 58.1069 0.148611
$$392$$ 0 0
$$393$$ 247.410i 0.629542i
$$394$$ 0 0
$$395$$ 28.5561i 0.0722939i
$$396$$ 0 0
$$397$$ −540.791 −1.36219 −0.681097 0.732193i $$-0.738497\pi$$
−0.681097 + 0.732193i $$0.738497\pi$$
$$398$$ 0 0
$$399$$ 313.481 + 1.86130i 0.785666 + 0.00466492i
$$400$$ 0 0
$$401$$ 224.500i 0.559849i 0.960022 + 0.279925i $$0.0903095\pi$$
−0.960022 + 0.279925i $$0.909691\pi$$
$$402$$ 0 0
$$403$$ 2.61567 0.00649049
$$404$$ 0 0
$$405$$ 5.00765 0.0123646
$$406$$ 0 0
$$407$$ 802.317i 1.97129i
$$408$$ 0 0
$$409$$ 397.737i 0.972463i 0.873830 + 0.486231i $$0.161629\pi$$
−0.873830 + 0.486231i $$0.838371\pi$$
$$410$$ 0 0
$$411$$ 286.634i 0.697408i
$$412$$ 0 0
$$413$$ 565.398i 1.36900i
$$414$$ 0 0
$$415$$ 35.4844 0.0855046
$$416$$ 0 0
$$417$$ 293.278i 0.703304i
$$418$$ 0 0
$$419$$ 769.815 1.83727 0.918634 0.395111i $$-0.129294\pi$$
0.918634 + 0.395111i $$0.129294\pi$$
$$420$$ 0 0
$$421$$ 515.309i 1.22401i 0.790853 + 0.612006i $$0.209637\pi$$
−0.790853 + 0.612006i $$0.790363\pi$$
$$422$$ 0 0
$$423$$ 134.656 0.318337
$$424$$ 0 0
$$425$$ −65.1501 −0.153294
$$426$$ 0 0
$$427$$ 918.822 2.15181
$$428$$ 0 0
$$429$$ 246.804 0.575301
$$430$$ 0 0
$$431$$ 484.686i 1.12456i −0.826946 0.562281i $$-0.809924\pi$$
0.826946 0.562281i $$-0.190076\pi$$
$$432$$ 0 0
$$433$$ 734.040i 1.69524i −0.530602 0.847621i $$-0.678034\pi$$
0.530602 0.847621i $$-0.321966\pi$$
$$434$$ 0 0
$$435$$ 47.0127 0.108075
$$436$$ 0 0
$$437$$ −2.48423 + 418.395i −0.00568475 + 0.957426i
$$438$$ 0 0
$$439$$ 108.108i 0.246260i −0.992391 0.123130i $$-0.960707\pi$$
0.992391 0.123130i $$-0.0392933\pi$$
$$440$$ 0 0
$$441$$ −125.226 −0.283960
$$442$$ 0 0
$$443$$ 519.843 1.17346 0.586730 0.809782i $$-0.300415\pi$$
0.586730 + 0.809782i $$0.300415\pi$$
$$444$$ 0 0
$$445$$ 24.9979i 0.0561751i
$$446$$ 0 0
$$447$$ 241.459i 0.540176i
$$448$$ 0 0
$$449$$ 810.462i 1.80504i 0.430651 + 0.902519i $$0.358284\pi$$
−0.430651 + 0.902519i $$0.641716\pi$$
$$450$$ 0 0
$$451$$ 934.238i 2.07148i
$$452$$ 0 0
$$453$$ −264.557 −0.584012
$$454$$ 0 0
$$455$$ 55.8371i 0.122719i
$$456$$ 0 0
$$457$$ 231.412 0.506373 0.253186 0.967418i $$-0.418521\pi$$
0.253186 + 0.967418i $$0.418521\pi$$
$$458$$ 0 0
$$459$$ 13.7110i 0.0298714i
$$460$$ 0 0
$$461$$ 766.366 1.66240 0.831200 0.555974i $$-0.187654\pi$$
0.831200 + 0.555974i $$0.187654\pi$$
$$462$$ 0 0
$$463$$ −152.091 −0.328490 −0.164245 0.986420i $$-0.552519\pi$$
−0.164245 + 0.986420i $$0.552519\pi$$
$$464$$ 0 0
$$465$$ −0.239281 −0.000514583
$$466$$ 0 0
$$467$$ −123.992 −0.265508 −0.132754 0.991149i $$-0.542382\pi$$
−0.132754 + 0.991149i $$0.542382\pi$$
$$468$$ 0 0
$$469$$ 102.718i 0.219016i
$$470$$ 0 0
$$471$$ 512.570i 1.08826i
$$472$$ 0 0
$$473$$ 373.137 0.788874
$$474$$ 0 0
$$475$$ 2.78535 469.110i 0.00586390 0.987599i
$$476$$ 0 0
$$477$$ 11.5647i 0.0242446i
$$478$$ 0 0
$$479$$ 443.502 0.925892 0.462946 0.886387i $$-0.346792\pi$$
0.462946 + 0.886387i $$0.346792\pi$$
$$480$$ 0 0
$$481$$ 624.895 1.29916
$$482$$ 0 0
$$483$$ 363.334i 0.752244i
$$484$$ 0 0
$$485$$ 40.4175i 0.0833350i
$$486$$ 0 0
$$487$$ 628.314i 1.29017i 0.764110 + 0.645086i $$0.223179\pi$$
−0.764110 + 0.645086i $$0.776821\pi$$
$$488$$ 0 0
$$489$$ 195.256i 0.399296i
$$490$$ 0 0
$$491$$ −877.289 −1.78674 −0.893370 0.449322i $$-0.851666\pi$$
−0.893370 + 0.449322i $$0.851666\pi$$
$$492$$ 0 0
$$493$$ 128.721i 0.261098i
$$494$$ 0 0
$$495$$ −22.5776 −0.0456113
$$496$$ 0 0
$$497$$ 501.416i 1.00889i
$$498$$ 0 0
$$499$$ −206.303 −0.413433 −0.206716 0.978401i $$-0.566278\pi$$
−0.206716 + 0.978401i $$0.566278\pi$$
$$500$$ 0 0
$$501$$ 514.773 1.02749
$$502$$ 0 0
$$503$$ 276.617 0.549934 0.274967 0.961454i $$-0.411333\pi$$
0.274967 + 0.961454i $$0.411333\pi$$
$$504$$ 0 0
$$505$$ 64.9786 0.128671
$$506$$ 0 0
$$507$$ 100.490i 0.198205i
$$508$$ 0 0
$$509$$ 728.875i 1.43197i −0.698114 0.715987i $$-0.745977\pi$$
0.698114 0.715987i $$-0.254023\pi$$
$$510$$ 0 0
$$511$$ 130.849 0.256065
$$512$$ 0 0
$$513$$ −98.7252 0.586183i −0.192447 0.00114266i
$$514$$ 0 0
$$515$$ 40.2793i 0.0782123i
$$516$$ 0 0
$$517$$ −607.115 −1.17430
$$518$$ 0 0
$$519$$ −455.267 −0.877201
$$520$$ 0 0
$$521$$ 14.2470i 0.0273455i 0.999907 + 0.0136727i $$0.00435230\pi$$
−0.999907 + 0.0136727i $$0.995648\pi$$
$$522$$ 0 0
$$523$$ 535.561i 1.02402i 0.858981 + 0.512008i $$0.171098\pi$$
−0.858981 + 0.512008i $$0.828902\pi$$
$$524$$ 0 0
$$525$$ 407.374i 0.775951i
$$526$$ 0 0
$$527$$ 0.655153i 0.00124317i
$$528$$ 0 0
$$529$$ −44.0669 −0.0833023
$$530$$ 0 0
$$531$$ 178.062i 0.335333i
$$532$$ 0 0
$$533$$ −727.644 −1.36519
$$534$$ 0 0
$$535$$ 85.7182i 0.160221i
$$536$$ 0 0
$$537$$ −128.031 −0.238418
$$538$$ 0 0
$$539$$ 564.599 1.04749
$$540$$ 0 0
$$541$$ 1000.18 1.84875 0.924376 0.381482i $$-0.124586\pi$$
0.924376 + 0.381482i $$0.124586\pi$$
$$542$$ 0 0
$$543$$ 60.4300 0.111289
$$544$$ 0 0
$$545$$ 88.8415i 0.163012i
$$546$$ 0 0
$$547$$ 675.400i 1.23473i −0.786675 0.617367i $$-0.788199\pi$$
0.786675 0.617367i $$-0.211801\pi$$
$$548$$ 0 0
$$549$$ −289.366 −0.527079
$$550$$ 0 0
$$551$$ −926.850 5.50320i −1.68212 0.00998765i
$$552$$ 0 0
$$553$$ 488.891i 0.884070i
$$554$$ 0 0
$$555$$ −57.1654 −0.103001
$$556$$ 0 0
$$557$$ −366.913 −0.658730 −0.329365 0.944203i $$-0.606835\pi$$
−0.329365 + 0.944203i $$0.606835\pi$$
$$558$$ 0 0
$$559$$ 290.623i 0.519898i
$$560$$ 0 0
$$561$$ 61.8176i 0.110192i
$$562$$ 0 0
$$563$$ 50.3339i 0.0894029i −0.999000 0.0447015i $$-0.985766\pi$$
0.999000 0.0447015i $$-0.0142337\pi$$
$$564$$ 0 0
$$565$$ 44.3132i 0.0784305i
$$566$$ 0 0
$$567$$ 85.7328 0.151204
$$568$$ 0 0
$$569$$ 61.6165i 0.108289i −0.998533 0.0541446i $$-0.982757\pi$$
0.998533 0.0541446i $$-0.0172432\pi$$
$$570$$ 0 0
$$571$$ −256.504 −0.449218 −0.224609 0.974449i $$-0.572111\pi$$
−0.224609 + 0.974449i $$0.572111\pi$$
$$572$$ 0 0
$$573$$ 192.259i 0.335530i
$$574$$ 0 0
$$575$$ −543.712 −0.945587
$$576$$ 0 0
$$577$$ 429.278 0.743983 0.371991 0.928236i $$-0.378675\pi$$
0.371991 + 0.928236i $$0.378675\pi$$
$$578$$ 0 0
$$579$$ 144.733 0.249970
$$580$$ 0 0
$$581$$ 607.506 1.04562
$$582$$ 0 0
$$583$$ 52.1407i 0.0894352i
$$584$$ 0 0
$$585$$ 17.5849i 0.0300596i
$$586$$ 0 0
$$587$$ −491.725 −0.837691 −0.418845 0.908058i $$-0.637565\pi$$
−0.418845 + 0.908058i $$0.637565\pi$$
$$588$$ 0 0
$$589$$ 4.71739 + 0.0280097i 0.00800916 + 4.75546e-5i
$$590$$ 0 0
$$591$$ 632.751i 1.07064i
$$592$$ 0 0
$$593$$ 954.134 1.60899 0.804497 0.593956i $$-0.202435\pi$$
0.804497 + 0.593956i $$0.202435\pi$$
$$594$$ 0 0
$$595$$ 13.9856 0.0235053
$$596$$ 0 0
$$597$$ 18.4122i 0.0308411i
$$598$$ 0 0
$$599$$ 533.310i 0.890334i −0.895448 0.445167i $$-0.853144\pi$$
0.895448 0.445167i $$-0.146856\pi$$
$$600$$ 0 0
$$601$$ 830.453i 1.38179i 0.722957 + 0.690893i $$0.242782\pi$$
−0.722957 + 0.690893i $$0.757218\pi$$
$$602$$ 0 0
$$603$$ 32.3493i 0.0536472i
$$604$$ 0 0
$$605$$ 34.4688 0.0569732
$$606$$ 0 0
$$607$$ 584.294i 0.962594i −0.876558 0.481297i $$-0.840166\pi$$
0.876558 0.481297i $$-0.159834\pi$$
$$608$$ 0 0
$$609$$ 804.875 1.32163
$$610$$ 0 0
$$611$$ 472.860i 0.773912i
$$612$$ 0 0
$$613$$ −152.216 −0.248314 −0.124157 0.992263i $$-0.539623\pi$$
−0.124157 + 0.992263i $$0.539623\pi$$
$$614$$ 0 0
$$615$$ 66.5648 0.108235
$$616$$ 0 0
$$617$$ 647.334 1.04916 0.524582 0.851360i $$-0.324222\pi$$
0.524582 + 0.851360i $$0.324222\pi$$
$$618$$ 0 0
$$619$$ 104.700 0.169145 0.0845723 0.996417i $$-0.473048\pi$$
0.0845723 + 0.996417i $$0.473048\pi$$
$$620$$ 0 0
$$621$$ 114.425i 0.184260i
$$622$$ 0 0
$$623$$ 427.974i 0.686956i
$$624$$ 0 0
$$625$$ 601.877 0.963003
$$626$$ 0 0
$$627$$ 445.114 + 2.64288i 0.709911 + 0.00421512i
$$628$$ 0 0
$$629$$ 156.519i 0.248838i
$$630$$ 0 0
$$631$$ −628.425 −0.995919 −0.497959 0.867200i $$-0.665917\pi$$
−0.497959 + 0.867200i $$0.665917\pi$$
$$632$$ 0 0
$$633$$ 398.707 0.629870
$$634$$ 0 0
$$635$$ 16.8328i 0.0265084i
$$636$$ 0 0
$$637$$ 439.745i 0.690338i
$$638$$ 0 0
$$639$$ 157.912i 0.247124i
$$640$$ 0 0
$$641$$ 462.053i 0.720832i 0.932792 + 0.360416i $$0.117365\pi$$
−0.932792 + 0.360416i $$0.882635\pi$$
$$642$$ 0 0
$$643$$ 898.660 1.39761 0.698803 0.715314i $$-0.253716\pi$$
0.698803 + 0.715314i $$0.253716\pi$$
$$644$$ 0 0
$$645$$ 26.5862i 0.0412189i
$$646$$ 0 0
$$647$$ 773.813 1.19600 0.598001 0.801495i $$-0.295962\pi$$
0.598001 + 0.801495i $$0.295962\pi$$
$$648$$ 0 0
$$649$$ 802.814i 1.23700i
$$650$$ 0 0
$$651$$ −4.09658 −0.00629275
$$652$$ 0 0
$$653$$ −701.376 −1.07408 −0.537041 0.843556i $$-0.680458\pi$$
−0.537041 + 0.843556i $$0.680458\pi$$
$$654$$ 0 0
$$655$$ 79.4782 0.121341
$$656$$ 0 0
$$657$$ −41.2086 −0.0627223
$$658$$ 0 0
$$659$$ 1255.87i 1.90572i 0.303415 + 0.952858i $$0.401873\pi$$
−0.303415 + 0.952858i $$0.598127\pi$$
$$660$$ 0 0
$$661$$ 399.716i 0.604714i −0.953195 0.302357i $$-0.902227\pi$$
0.953195 0.302357i $$-0.0977735\pi$$
$$662$$ 0 0
$$663$$ 48.1475 0.0726206
$$664$$ 0 0
$$665$$ −0.597926 + 100.703i −0.000899137 + 0.151433i
$$666$$ 0 0
$$667$$ 1074.25i 1.61057i
$$668$$ 0 0
$$669$$ −456.939 −0.683018
$$670$$ 0 0
$$671$$ 1304.64 1.94433
$$672$$ 0 0
$$673$$ 116.954i 0.173779i 0.996218 + 0.0868897i $$0.0276928\pi$$
−0.996218 + 0.0868897i $$0.972307\pi$$
$$674$$ 0 0
$$675$$ 128.295i 0.190067i
$$676$$ 0 0
$$677$$ 497.520i 0.734890i 0.930045 + 0.367445i $$0.119767\pi$$
−0.930045 + 0.367445i $$0.880233\pi$$
$$678$$ 0 0
$$679$$ 691.962i 1.01909i
$$680$$ 0 0
$$681$$ 370.534 0.544103
$$682$$ 0 0
$$683$$ 803.013i 1.17572i −0.808964 0.587858i $$-0.799972\pi$$
0.808964 0.587858i $$-0.200028\pi$$
$$684$$ 0 0
$$685$$ −92.0787 −0.134421
$$686$$ 0 0
$$687$$ 109.933i 0.160019i
$$688$$ 0 0
$$689$$ 40.6105 0.0589413
$$690$$ 0 0
$$691$$ −891.667 −1.29040 −0.645200 0.764014i $$-0.723226\pi$$
−0.645200 + 0.764014i $$0.723226\pi$$
$$692$$ 0 0
$$693$$ −386.537 −0.557773
$$694$$ 0 0
$$695$$ 94.2128 0.135558
$$696$$ 0 0
$$697$$ 182.255i 0.261485i
$$698$$ 0 0
$$699$$ 499.608i 0.714746i
$$700$$ 0 0
$$701$$ 1110.82 1.58463 0.792313 0.610114i $$-0.208876\pi$$
0.792313 + 0.610114i $$0.208876\pi$$
$$702$$ 0 0
$$703$$ 1127.01 + 6.69164i 1.60314 + 0.00951869i
$$704$$ 0 0
$$705$$ 43.2572i 0.0613577i
$$706$$ 0 0
$$707$$ 1112.46 1.57349
$$708$$ 0 0
$$709$$ −9.63354 −0.0135875 −0.00679375 0.999977i $$-0.502163\pi$$
−0.00679375 + 0.999977i $$0.502163\pi$$
$$710$$ 0 0
$$711$$ 153.967i 0.216550i
$$712$$ 0 0
$$713$$ 5.46760i 0.00766845i
$$714$$ 0 0
$$715$$ 79.2835i 0.110886i
$$716$$ 0 0
$$717$$ 483.464i 0.674287i
$$718$$ 0 0
$$719$$ −97.4316 −0.135510 −0.0677549 0.997702i $$-0.521584\pi$$
−0.0677549 + 0.997702i $$0.521584\pi$$
$$720$$ 0 0
$$721$$ 689.597i 0.956445i
$$722$$ 0 0
$$723$$ −420.346 −0.581392
$$724$$ 0 0
$$725$$ 1204.46i 1.66132i
$$726$$ 0 0
$$727$$ 466.358 0.641483 0.320742 0.947167i $$-0.396068\pi$$
0.320742 + 0.947167i $$0.396068\pi$$
$$728$$ 0 0
$$729$$ −27.0000 −0.0370370
$$730$$ 0 0
$$731$$ 72.7931 0.0995801
$$732$$ 0 0
$$733$$ 452.839 0.617789 0.308895 0.951096i $$-0.400041\pi$$
0.308895 + 0.951096i $$0.400041\pi$$
$$734$$ 0 0
$$735$$ 40.2279i 0.0547318i
$$736$$ 0 0
$$737$$ 145.851i 0.197898i
$$738$$ 0 0
$$739$$ 1346.02 1.82140 0.910701 0.413066i $$-0.135542\pi$$
0.910701 + 0.413066i $$0.135542\pi$$
$$740$$ 0 0
$$741$$ −2.05844 + 346.683i −0.00277793 + 0.467859i
$$742$$ 0 0
$$743$$ 787.553i 1.05996i 0.848009 + 0.529982i $$0.177801\pi$$
−0.848009 + 0.529982i $$0.822199\pi$$
$$744$$ 0 0
$$745$$ −77.5664 −0.104116
$$746$$ 0 0
$$747$$ −191.323 −0.256122
$$748$$ 0 0
$$749$$ 1467.53i 1.95932i
$$750$$ 0 0
$$751$$ 120.158i 0.159997i −0.996795 0.0799986i $$-0.974508\pi$$
0.996795 0.0799986i $$-0.0254916\pi$$
$$752$$ 0 0
$$753$$ 281.766i 0.374191i
$$754$$ 0 0
$$755$$ 84.9866i 0.112565i
$$756$$ 0 0
$$757$$ 263.516 0.348105 0.174053 0.984736i $$-0.444314\pi$$
0.174053 + 0.984736i $$0.444314\pi$$
$$758$$ 0 0
$$759$$ 515.901i 0.679712i
$$760$$ 0 0
$$761$$ −60.7691 −0.0798543 −0.0399271 0.999203i $$-0.512713\pi$$
−0.0399271 + 0.999203i $$0.512713\pi$$
$$762$$ 0 0
$$763$$ 1521.00i 1.99345i
$$764$$ 0 0
$$765$$ −4.40453 −0.00575755
$$766$$ 0 0
$$767$$ 625.282 0.815231
$$768$$ 0 0
$$769$$ −713.480 −0.927802 −0.463901 0.885887i $$-0.653551\pi$$
−0.463901 + 0.885887i $$0.653551\pi$$
$$770$$ 0 0
$$771$$ 444.494 0.576516
$$772$$ 0 0
$$773$$ 971.062i 1.25622i −0.778123 0.628112i $$-0.783828\pi$$
0.778123 0.628112i $$-0.216172\pi$$
$$774$$ 0 0
$$775$$ 6.13034i 0.00791012i
$$776$$ 0 0
$$777$$ −978.692 −1.25958
$$778$$ 0 0
$$779$$ −1312.32 7.79191i −1.68462 0.0100025i
$$780$$ 0 0
$$781$$ 711.965i 0.911607i
$$782$$ 0 0
$$783$$ −253.481 −0.323730
$$784$$ 0 0
$$785$$ −164.658 −0.209756
$$786$$ 0 0
$$787$$ 903.790i 1.14840i −0.818716 0.574199i $$-0.805313\pi$$
0.818716 0.574199i $$-0.194687\pi$$
$$788$$ 0 0
$$789$$ 123.327i 0.156308i
$$790$$ 0 0
$$791$$ 758.659i 0.959114i
$$792$$ 0 0
$$793$$ 1016.14i 1.28139i
$$794$$ 0 0
$$795$$ −3.71505 −0.00467301
$$796$$ 0 0
$$797$$ 836.376i 1.04941i −0.851286 0.524703i $$-0.824177\pi$$
0.851286 0.524703i $$-0.175823\pi$$
$$798$$ 0 0
$$799$$ −118.438 −0.148233
$$800$$ 0 0
$$801$$ 134.783i 0.168268i
$$802$$ 0 0