# Properties

 Label 3024.2.k.g.1889.3 Level 3024 Weight 2 Character 3024.1889 Analytic conductor 24.147 Analytic rank 0 Dimension 4 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3024 = 2^{4} \cdot 3^{3} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3024.k (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$24.1467615712$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{3})$$ Defining polynomial: $$x^{4} + 4 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 189) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 1889.3 Root $$1.93185i$$ of defining polynomial Character $$\chi$$ $$=$$ 3024.1889 Dual form 3024.2.k.g.1889.4

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.73205 q^{5} +(-1.00000 - 2.44949i) q^{7} +O(q^{10})$$ $$q+1.73205 q^{5} +(-1.00000 - 2.44949i) q^{7} +1.41421i q^{11} +2.44949i q^{13} -5.19615 q^{17} +7.34847i q^{19} -2.82843i q^{23} -2.00000 q^{25} +7.07107i q^{29} -2.44949i q^{31} +(-1.73205 - 4.24264i) q^{35} +5.00000 q^{37} -8.66025 q^{41} -5.00000 q^{43} -8.66025 q^{47} +(-5.00000 + 4.89898i) q^{49} +11.3137i q^{53} +2.44949i q^{55} +8.66025 q^{59} -2.44949i q^{61} +4.24264i q^{65} -2.00000 q^{67} +14.1421i q^{71} +(3.46410 - 1.41421i) q^{77} +13.0000 q^{79} +1.73205 q^{83} -9.00000 q^{85} +10.3923 q^{89} +(6.00000 - 2.44949i) q^{91} +12.7279i q^{95} -17.1464i q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{7} + O(q^{10})$$ $$4q - 4q^{7} - 8q^{25} + 20q^{37} - 20q^{43} - 20q^{49} - 8q^{67} + 52q^{79} - 36q^{85} + 24q^{91} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3024\mathbb{Z}\right)^\times$$.

 $$n$$ $$757$$ $$785$$ $$1135$$ $$2593$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$ $$-1$$

## 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
$$3$$ 0 0
$$4$$ 0 0
$$5$$ 1.73205 0.774597 0.387298 0.921954i $$-0.373408\pi$$
0.387298 + 0.921954i $$0.373408\pi$$
$$6$$ 0 0
$$7$$ −1.00000 2.44949i −0.377964 0.925820i
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 1.41421i 0.426401i 0.977008 + 0.213201i $$0.0683888\pi$$
−0.977008 + 0.213201i $$0.931611\pi$$
$$12$$ 0 0
$$13$$ 2.44949i 0.679366i 0.940540 + 0.339683i $$0.110320\pi$$
−0.940540 + 0.339683i $$0.889680\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −5.19615 −1.26025 −0.630126 0.776493i $$-0.716997\pi$$
−0.630126 + 0.776493i $$0.716997\pi$$
$$18$$ 0 0
$$19$$ 7.34847i 1.68585i 0.538028 + 0.842927i $$0.319170\pi$$
−0.538028 + 0.842927i $$0.680830\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 2.82843i 0.589768i −0.955533 0.294884i $$-0.904719\pi$$
0.955533 0.294884i $$-0.0952810\pi$$
$$24$$ 0 0
$$25$$ −2.00000 −0.400000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 7.07107i 1.31306i 0.754298 + 0.656532i $$0.227977\pi$$
−0.754298 + 0.656532i $$0.772023\pi$$
$$30$$ 0 0
$$31$$ 2.44949i 0.439941i −0.975506 0.219971i $$-0.929404\pi$$
0.975506 0.219971i $$-0.0705962\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −1.73205 4.24264i −0.292770 0.717137i
$$36$$ 0 0
$$37$$ 5.00000 0.821995 0.410997 0.911636i $$-0.365181\pi$$
0.410997 + 0.911636i $$0.365181\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −8.66025 −1.35250 −0.676252 0.736670i $$-0.736397\pi$$
−0.676252 + 0.736670i $$0.736397\pi$$
$$42$$ 0 0
$$43$$ −5.00000 −0.762493 −0.381246 0.924473i $$-0.624505\pi$$
−0.381246 + 0.924473i $$0.624505\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −8.66025 −1.26323 −0.631614 0.775283i $$-0.717607\pi$$
−0.631614 + 0.775283i $$0.717607\pi$$
$$48$$ 0 0
$$49$$ −5.00000 + 4.89898i −0.714286 + 0.699854i
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 11.3137i 1.55406i 0.629465 + 0.777029i $$0.283274\pi$$
−0.629465 + 0.777029i $$0.716726\pi$$
$$54$$ 0 0
$$55$$ 2.44949i 0.330289i
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 8.66025 1.12747 0.563735 0.825956i $$-0.309364\pi$$
0.563735 + 0.825956i $$0.309364\pi$$
$$60$$ 0 0
$$61$$ 2.44949i 0.313625i −0.987628 0.156813i $$-0.949878\pi$$
0.987628 0.156813i $$-0.0501218\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 4.24264i 0.526235i
$$66$$ 0 0
$$67$$ −2.00000 −0.244339 −0.122169 0.992509i $$-0.538985\pi$$
−0.122169 + 0.992509i $$0.538985\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 14.1421i 1.67836i 0.543852 + 0.839181i $$0.316965\pi$$
−0.543852 + 0.839181i $$0.683035\pi$$
$$72$$ 0 0
$$73$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 3.46410 1.41421i 0.394771 0.161165i
$$78$$ 0 0
$$79$$ 13.0000 1.46261 0.731307 0.682048i $$-0.238911\pi$$
0.731307 + 0.682048i $$0.238911\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 1.73205 0.190117 0.0950586 0.995472i $$-0.469696\pi$$
0.0950586 + 0.995472i $$0.469696\pi$$
$$84$$ 0 0
$$85$$ −9.00000 −0.976187
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 10.3923 1.10158 0.550791 0.834643i $$-0.314326\pi$$
0.550791 + 0.834643i $$0.314326\pi$$
$$90$$ 0 0
$$91$$ 6.00000 2.44949i 0.628971 0.256776i
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 12.7279i 1.30586i
$$96$$ 0 0
$$97$$ 17.1464i 1.74096i −0.492207 0.870478i $$-0.663810\pi$$
0.492207 0.870478i $$-0.336190\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 3.46410 0.344691 0.172345 0.985037i $$-0.444865\pi$$
0.172345 + 0.985037i $$0.444865\pi$$
$$102$$ 0 0
$$103$$ 9.79796i 0.965422i −0.875780 0.482711i $$-0.839652\pi$$
0.875780 0.482711i $$-0.160348\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 14.1421i 1.36717i 0.729870 + 0.683586i $$0.239581\pi$$
−0.729870 + 0.683586i $$0.760419\pi$$
$$108$$ 0 0
$$109$$ −7.00000 −0.670478 −0.335239 0.942133i $$-0.608817\pi$$
−0.335239 + 0.942133i $$0.608817\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 7.07107i 0.665190i 0.943070 + 0.332595i $$0.107924\pi$$
−0.943070 + 0.332595i $$0.892076\pi$$
$$114$$ 0 0
$$115$$ 4.89898i 0.456832i
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 5.19615 + 12.7279i 0.476331 + 1.16677i
$$120$$ 0 0
$$121$$ 9.00000 0.818182
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ −12.1244 −1.08444
$$126$$ 0 0
$$127$$ −5.00000 −0.443678 −0.221839 0.975083i $$-0.571206\pi$$
−0.221839 + 0.975083i $$0.571206\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −17.3205 −1.51330 −0.756650 0.653820i $$-0.773165\pi$$
−0.756650 + 0.653820i $$0.773165\pi$$
$$132$$ 0 0
$$133$$ 18.0000 7.34847i 1.56080 0.637193i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 1.41421i 0.120824i −0.998174 0.0604122i $$-0.980758\pi$$
0.998174 0.0604122i $$-0.0192415\pi$$
$$138$$ 0 0
$$139$$ 12.2474i 1.03882i 0.854527 + 0.519408i $$0.173847\pi$$
−0.854527 + 0.519408i $$0.826153\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ −3.46410 −0.289683
$$144$$ 0 0
$$145$$ 12.2474i 1.01710i
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 7.07107i 0.579284i 0.957135 + 0.289642i $$0.0935363\pi$$
−0.957135 + 0.289642i $$0.906464\pi$$
$$150$$ 0 0
$$151$$ −5.00000 −0.406894 −0.203447 0.979086i $$-0.565214\pi$$
−0.203447 + 0.979086i $$0.565214\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 4.24264i 0.340777i
$$156$$ 0 0
$$157$$ 17.1464i 1.36843i 0.729279 + 0.684217i $$0.239856\pi$$
−0.729279 + 0.684217i $$0.760144\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −6.92820 + 2.82843i −0.546019 + 0.222911i
$$162$$ 0 0
$$163$$ −5.00000 −0.391630 −0.195815 0.980641i $$-0.562735\pi$$
−0.195815 + 0.980641i $$0.562735\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −12.1244 −0.938211 −0.469105 0.883142i $$-0.655424\pi$$
−0.469105 + 0.883142i $$0.655424\pi$$
$$168$$ 0 0
$$169$$ 7.00000 0.538462
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ −6.92820 −0.526742 −0.263371 0.964695i $$-0.584834\pi$$
−0.263371 + 0.964695i $$0.584834\pi$$
$$174$$ 0 0
$$175$$ 2.00000 + 4.89898i 0.151186 + 0.370328i
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 7.07107i 0.528516i −0.964452 0.264258i $$-0.914873\pi$$
0.964452 0.264258i $$-0.0851271\pi$$
$$180$$ 0 0
$$181$$ 14.6969i 1.09241i −0.837650 0.546207i $$-0.816071\pi$$
0.837650 0.546207i $$-0.183929\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 8.66025 0.636715
$$186$$ 0 0
$$187$$ 7.34847i 0.537373i
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 14.1421i 1.02329i 0.859197 + 0.511645i $$0.170964\pi$$
−0.859197 + 0.511645i $$0.829036\pi$$
$$192$$ 0 0
$$193$$ −1.00000 −0.0719816 −0.0359908 0.999352i $$-0.511459\pi$$
−0.0359908 + 0.999352i $$0.511459\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 1.41421i 0.100759i −0.998730 0.0503793i $$-0.983957\pi$$
0.998730 0.0503793i $$-0.0160430\pi$$
$$198$$ 0 0
$$199$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 17.3205 7.07107i 1.21566 0.496292i
$$204$$ 0 0
$$205$$ −15.0000 −1.04765
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ −10.3923 −0.718851
$$210$$ 0 0
$$211$$ 16.0000 1.10149 0.550743 0.834675i $$-0.314345\pi$$
0.550743 + 0.834675i $$0.314345\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ −8.66025 −0.590624
$$216$$ 0 0
$$217$$ −6.00000 + 2.44949i −0.407307 + 0.166282i
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 12.7279i 0.856173i
$$222$$ 0 0
$$223$$ 12.2474i 0.820150i −0.912052 0.410075i $$-0.865503\pi$$
0.912052 0.410075i $$-0.134497\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −13.8564 −0.919682 −0.459841 0.888001i $$-0.652094\pi$$
−0.459841 + 0.888001i $$0.652094\pi$$
$$228$$ 0 0
$$229$$ 4.89898i 0.323734i −0.986813 0.161867i $$-0.948248\pi$$
0.986813 0.161867i $$-0.0517515\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 7.07107i 0.463241i 0.972806 + 0.231621i $$0.0744028\pi$$
−0.972806 + 0.231621i $$0.925597\pi$$
$$234$$ 0 0
$$235$$ −15.0000 −0.978492
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 18.3848i 1.18921i 0.804017 + 0.594606i $$0.202692\pi$$
−0.804017 + 0.594606i $$0.797308\pi$$
$$240$$ 0 0
$$241$$ 2.44949i 0.157786i −0.996883 0.0788928i $$-0.974862\pi$$
0.996883 0.0788928i $$-0.0251385\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ −8.66025 + 8.48528i −0.553283 + 0.542105i
$$246$$ 0 0
$$247$$ −18.0000 −1.14531
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −5.19615 −0.327978 −0.163989 0.986462i $$-0.552436\pi$$
−0.163989 + 0.986462i $$0.552436\pi$$
$$252$$ 0 0
$$253$$ 4.00000 0.251478
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −3.46410 −0.216085 −0.108042 0.994146i $$-0.534458\pi$$
−0.108042 + 0.994146i $$0.534458\pi$$
$$258$$ 0 0
$$259$$ −5.00000 12.2474i −0.310685 0.761019i
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 1.41421i 0.0872041i 0.999049 + 0.0436021i $$0.0138834\pi$$
−0.999049 + 0.0436021i $$0.986117\pi$$
$$264$$ 0 0
$$265$$ 19.5959i 1.20377i
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 25.9808 1.58408 0.792038 0.610472i $$-0.209020\pi$$
0.792038 + 0.610472i $$0.209020\pi$$
$$270$$ 0 0
$$271$$ 14.6969i 0.892775i 0.894840 + 0.446388i $$0.147290\pi$$
−0.894840 + 0.446388i $$0.852710\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 2.82843i 0.170561i
$$276$$ 0 0
$$277$$ 5.00000 0.300421 0.150210 0.988654i $$-0.452005\pi$$
0.150210 + 0.988654i $$0.452005\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 14.1421i 0.843649i −0.906677 0.421825i $$-0.861390\pi$$
0.906677 0.421825i $$-0.138610\pi$$
$$282$$ 0 0
$$283$$ 12.2474i 0.728035i 0.931392 + 0.364018i $$0.118595\pi$$
−0.931392 + 0.364018i $$0.881405\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 8.66025 + 21.2132i 0.511199 + 1.25218i
$$288$$ 0 0
$$289$$ 10.0000 0.588235
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ −19.0526 −1.11306 −0.556531 0.830827i $$-0.687868\pi$$
−0.556531 + 0.830827i $$0.687868\pi$$
$$294$$ 0 0
$$295$$ 15.0000 0.873334
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 6.92820 0.400668
$$300$$ 0 0
$$301$$ 5.00000 + 12.2474i 0.288195 + 0.705931i
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 4.24264i 0.242933i
$$306$$ 0 0
$$307$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −12.1244 −0.687509 −0.343755 0.939060i $$-0.611699\pi$$
−0.343755 + 0.939060i $$0.611699\pi$$
$$312$$ 0 0
$$313$$ 24.4949i 1.38453i −0.721642 0.692267i $$-0.756612\pi$$
0.721642 0.692267i $$-0.243388\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 5.65685i 0.317721i −0.987301 0.158860i $$-0.949218\pi$$
0.987301 0.158860i $$-0.0507819\pi$$
$$318$$ 0 0
$$319$$ −10.0000 −0.559893
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 38.1838i 2.12460i
$$324$$ 0 0
$$325$$ 4.89898i 0.271746i
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 8.66025 + 21.2132i 0.477455 + 1.16952i
$$330$$ 0 0
$$331$$ 25.0000 1.37412 0.687062 0.726599i $$-0.258900\pi$$
0.687062 + 0.726599i $$0.258900\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ −3.46410 −0.189264
$$336$$ 0 0
$$337$$ −7.00000 −0.381314 −0.190657 0.981657i $$-0.561062\pi$$
−0.190657 + 0.981657i $$0.561062\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 3.46410 0.187592
$$342$$ 0 0
$$343$$ 17.0000 + 7.34847i 0.917914 + 0.396780i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 7.07107i 0.379595i −0.981823 0.189797i $$-0.939217\pi$$
0.981823 0.189797i $$-0.0607831\pi$$
$$348$$ 0 0
$$349$$ 19.5959i 1.04895i 0.851427 + 0.524473i $$0.175738\pi$$
−0.851427 + 0.524473i $$0.824262\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 8.66025 0.460939 0.230469 0.973080i $$-0.425974\pi$$
0.230469 + 0.973080i $$0.425974\pi$$
$$354$$ 0 0
$$355$$ 24.4949i 1.30005i
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 11.3137i 0.597115i −0.954392 0.298557i $$-0.903495\pi$$
0.954392 0.298557i $$-0.0965054\pi$$
$$360$$ 0 0
$$361$$ −35.0000 −1.84211
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 4.89898i 0.255725i −0.991792 0.127862i $$-0.959188\pi$$
0.991792 0.127862i $$-0.0408116\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 27.7128 11.3137i 1.43878 0.587378i
$$372$$ 0 0
$$373$$ −25.0000 −1.29445 −0.647225 0.762299i $$-0.724071\pi$$
−0.647225 + 0.762299i $$0.724071\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −17.3205 −0.892052
$$378$$ 0 0
$$379$$ −23.0000 −1.18143 −0.590715 0.806880i $$-0.701154\pi$$
−0.590715 + 0.806880i $$0.701154\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ −1.73205 −0.0885037 −0.0442518 0.999020i $$-0.514090\pi$$
−0.0442518 + 0.999020i $$0.514090\pi$$
$$384$$ 0 0
$$385$$ 6.00000 2.44949i 0.305788 0.124838i
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 32.5269i 1.64918i 0.565731 + 0.824590i $$0.308594\pi$$
−0.565731 + 0.824590i $$0.691406\pi$$
$$390$$ 0 0
$$391$$ 14.6969i 0.743256i
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 22.5167 1.13294
$$396$$ 0 0
$$397$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 19.7990i 0.988714i 0.869259 + 0.494357i $$0.164597\pi$$
−0.869259 + 0.494357i $$0.835403\pi$$
$$402$$ 0 0
$$403$$ 6.00000 0.298881
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 7.07107i 0.350500i
$$408$$ 0 0
$$409$$ 19.5959i 0.968956i −0.874804 0.484478i $$-0.839010\pi$$
0.874804 0.484478i $$-0.160990\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ −8.66025 21.2132i −0.426143 1.04383i
$$414$$ 0 0
$$415$$ 3.00000 0.147264
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ −1.73205 −0.0846162 −0.0423081 0.999105i $$-0.513471\pi$$
−0.0423081 + 0.999105i $$0.513471\pi$$
$$420$$ 0 0
$$421$$ −4.00000 −0.194948 −0.0974740 0.995238i $$-0.531076\pi$$
−0.0974740 + 0.995238i $$0.531076\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 10.3923 0.504101
$$426$$ 0 0
$$427$$ −6.00000 + 2.44949i −0.290360 + 0.118539i
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 5.65685i 0.272481i 0.990676 + 0.136241i $$0.0435020\pi$$
−0.990676 + 0.136241i $$0.956498\pi$$
$$432$$ 0 0
$$433$$ 36.7423i 1.76572i 0.469632 + 0.882862i $$0.344387\pi$$
−0.469632 + 0.882862i $$0.655613\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 20.7846 0.994263
$$438$$ 0 0
$$439$$ 19.5959i 0.935262i −0.883924 0.467631i $$-0.845108\pi$$
0.883924 0.467631i $$-0.154892\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 32.5269i 1.54540i −0.634771 0.772700i $$-0.718906\pi$$
0.634771 0.772700i $$-0.281094\pi$$
$$444$$ 0 0
$$445$$ 18.0000 0.853282
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 24.0416i 1.13459i 0.823513 + 0.567297i $$0.192011\pi$$
−0.823513 + 0.567297i $$0.807989\pi$$
$$450$$ 0 0
$$451$$ 12.2474i 0.576710i
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 10.3923 4.24264i 0.487199 0.198898i
$$456$$ 0 0
$$457$$ 32.0000 1.49690 0.748448 0.663193i $$-0.230799\pi$$
0.748448 + 0.663193i $$0.230799\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −12.1244 −0.564688 −0.282344 0.959313i $$-0.591112\pi$$
−0.282344 + 0.959313i $$0.591112\pi$$
$$462$$ 0 0
$$463$$ −11.0000 −0.511213 −0.255607 0.966781i $$-0.582275\pi$$
−0.255607 + 0.966781i $$0.582275\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 20.7846 0.961797 0.480899 0.876776i $$-0.340311\pi$$
0.480899 + 0.876776i $$0.340311\pi$$
$$468$$ 0 0
$$469$$ 2.00000 + 4.89898i 0.0923514 + 0.226214i
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 7.07107i 0.325128i
$$474$$ 0 0
$$475$$ 14.6969i 0.674342i
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 32.9090 1.50365 0.751825 0.659363i $$-0.229174\pi$$
0.751825 + 0.659363i $$0.229174\pi$$
$$480$$ 0 0
$$481$$ 12.2474i 0.558436i
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 29.6985i 1.34854i
$$486$$ 0 0
$$487$$ 22.0000 0.996915 0.498458 0.866914i $$-0.333900\pi$$
0.498458 + 0.866914i $$0.333900\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 15.5563i 0.702048i −0.936366 0.351024i $$-0.885834\pi$$
0.936366 0.351024i $$-0.114166\pi$$
$$492$$ 0 0
$$493$$ 36.7423i 1.65479i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 34.6410 14.1421i 1.55386 0.634361i
$$498$$ 0 0
$$499$$ −17.0000 −0.761025 −0.380512 0.924776i $$-0.624252\pi$$
−0.380512 + 0.924776i $$0.624252\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ −36.3731 −1.62179 −0.810897 0.585188i $$-0.801021\pi$$
−0.810897 + 0.585188i $$0.801021\pi$$
$$504$$ 0 0
$$505$$ 6.00000 0.266996
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ −8.66025 −0.383859 −0.191930 0.981409i $$-0.561474\pi$$
−0.191930 + 0.981409i $$0.561474\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 16.9706i 0.747812i
$$516$$ 0 0
$$517$$ 12.2474i 0.538642i
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 5.19615 0.227648 0.113824 0.993501i $$-0.463690\pi$$
0.113824 + 0.993501i $$0.463690\pi$$
$$522$$ 0 0
$$523$$ 36.7423i 1.60663i −0.595554 0.803315i $$-0.703067\pi$$
0.595554 0.803315i $$-0.296933\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 12.7279i 0.554437i
$$528$$ 0 0
$$529$$ 15.0000 0.652174
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 21.2132i 0.918846i
$$534$$ 0 0
$$535$$ 24.4949i 1.05901i
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ −6.92820 7.07107i −0.298419 0.304572i
$$540$$ 0 0
$$541$$ 5.00000 0.214967 0.107483 0.994207i $$-0.465721\pi$$
0.107483 + 0.994207i $$0.465721\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ −12.1244 −0.519350
$$546$$ 0 0
$$547$$ 13.0000 0.555840 0.277920 0.960604i $$-0.410355\pi$$
0.277920 + 0.960604i $$0.410355\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −51.9615 −2.21364
$$552$$ 0 0
$$553$$ −13.0000 31.8434i −0.552816 1.35412i
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 14.1421i 0.599222i −0.954062 0.299611i $$-0.903143\pi$$
0.954062 0.299611i $$-0.0968568\pi$$
$$558$$ 0 0
$$559$$ 12.2474i 0.518012i
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 34.6410 1.45994 0.729972 0.683477i $$-0.239533\pi$$
0.729972 + 0.683477i $$0.239533\pi$$
$$564$$ 0 0
$$565$$ 12.2474i 0.515254i
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 39.5980i 1.66003i −0.557738 0.830017i $$-0.688331\pi$$
0.557738 0.830017i $$-0.311669\pi$$
$$570$$ 0 0
$$571$$ 19.0000 0.795125 0.397563 0.917575i $$-0.369856\pi$$
0.397563 + 0.917575i $$0.369856\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 5.65685i 0.235907i
$$576$$ 0 0
$$577$$ 29.3939i 1.22368i 0.790980 + 0.611842i $$0.209571\pi$$
−0.790980 + 0.611842i $$0.790429\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −1.73205 4.24264i −0.0718576 0.176014i
$$582$$ 0 0
$$583$$ −16.0000 −0.662652
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −13.8564 −0.571915 −0.285958 0.958242i $$-0.592312\pi$$
−0.285958 + 0.958242i $$0.592312\pi$$
$$588$$ 0 0
$$589$$ 18.0000 0.741677
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 15.5885 0.640141 0.320071 0.947394i $$-0.396293\pi$$
0.320071 + 0.947394i $$0.396293\pi$$
$$594$$ 0 0
$$595$$ 9.00000 + 22.0454i 0.368964 + 0.903774i
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 18.3848i 0.751182i 0.926786 + 0.375591i $$0.122560\pi$$
−0.926786 + 0.375591i $$0.877440\pi$$
$$600$$ 0 0
$$601$$ 46.5403i 1.89842i −0.314645 0.949209i $$-0.601886\pi$$
0.314645 0.949209i $$-0.398114\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 15.5885 0.633761
$$606$$ 0 0
$$607$$ 12.2474i 0.497109i 0.968618 + 0.248554i $$0.0799554\pi$$
−0.968618 + 0.248554i $$0.920045\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 21.2132i 0.858194i
$$612$$ 0 0
$$613$$ −4.00000 −0.161558 −0.0807792 0.996732i $$-0.525741\pi$$
−0.0807792 + 0.996732i $$0.525741\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 48.0833i 1.93576i −0.251414 0.967880i $$-0.580896\pi$$
0.251414 0.967880i $$-0.419104\pi$$
$$618$$ 0 0
$$619$$ 26.9444i 1.08299i −0.840705 0.541493i $$-0.817859\pi$$
0.840705 0.541493i $$-0.182141\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ −10.3923 25.4558i −0.416359 1.01987i
$$624$$ 0 0
$$625$$ −11.0000 −0.440000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ −25.9808 −1.03592
$$630$$ 0 0
$$631$$ −29.0000 −1.15447 −0.577236 0.816577i $$-0.695869\pi$$
−0.577236 + 0.816577i $$0.695869\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ −8.66025 −0.343672
$$636$$ 0 0
$$637$$ −12.0000 12.2474i −0.475457 0.485262i
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 1.41421i 0.0558581i −0.999610 0.0279290i $$-0.991109\pi$$
0.999610 0.0279290i $$-0.00889125\pi$$
$$642$$ 0 0
$$643$$ 24.4949i 0.965984i −0.875625 0.482992i $$-0.839550\pi$$
0.875625 0.482992i $$-0.160450\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 20.7846 0.817127 0.408564 0.912730i $$-0.366030\pi$$
0.408564 + 0.912730i $$0.366030\pi$$
$$648$$ 0 0
$$649$$ 12.2474i 0.480754i
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 18.3848i 0.719452i −0.933058 0.359726i $$-0.882870\pi$$
0.933058 0.359726i $$-0.117130\pi$$
$$654$$ 0 0
$$655$$ −30.0000 −1.17220
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 11.3137i 0.440720i −0.975419 0.220360i $$-0.929277\pi$$
0.975419 0.220360i $$-0.0707231\pi$$
$$660$$ 0 0
$$661$$ 17.1464i 0.666919i 0.942764 + 0.333459i $$0.108216\pi$$
−0.942764 + 0.333459i $$0.891784\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 31.1769 12.7279i 1.20899 0.493568i
$$666$$ 0 0
$$667$$ 20.0000 0.774403
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 3.46410 0.133730
$$672$$ 0 0
$$673$$ −40.0000 −1.54189 −0.770943 0.636904i $$-0.780215\pi$$
−0.770943 + 0.636904i $$0.780215\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −17.3205 −0.665681 −0.332841 0.942983i $$-0.608007\pi$$
−0.332841 + 0.942983i $$0.608007\pi$$
$$678$$ 0 0
$$679$$ −42.0000 + 17.1464i −1.61181 + 0.658020i
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 24.0416i 0.919927i −0.887938 0.459964i $$-0.847862\pi$$
0.887938 0.459964i $$-0.152138\pi$$
$$684$$ 0 0
$$685$$ 2.44949i 0.0935902i
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ −27.7128 −1.05577
$$690$$ 0 0
$$691$$ 26.9444i 1.02501i −0.858683 0.512506i $$-0.828717\pi$$
0.858683 0.512506i $$-0.171283\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 21.2132i 0.804663i
$$696$$ 0 0
$$697$$ 45.0000 1.70450
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 1.41421i 0.0534141i −0.999643 0.0267071i $$-0.991498\pi$$
0.999643 0.0267071i $$-0.00850213\pi$$
$$702$$ 0 0
$$703$$ 36.7423i 1.38576i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −3.46410 8.48528i −0.130281 0.319122i
$$708$$ 0 0
$$709$$ 5.00000 0.187779 0.0938895 0.995583i $$-0.470070\pi$$
0.0938895 + 0.995583i $$0.470070\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ −6.92820 −0.259463
$$714$$ 0 0
$$715$$ −6.00000 −0.224387
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 15.5885 0.581351 0.290676 0.956822i $$-0.406120\pi$$
0.290676 + 0.956822i $$0.406120\pi$$
$$720$$ 0 0
$$721$$ −24.0000 + 9.79796i −0.893807 + 0.364895i
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 14.1421i 0.525226i
$$726$$ 0 0
$$727$$ 12.2474i 0.454233i −0.973868 0.227116i $$-0.927070\pi$$
0.973868 0.227116i $$-0.0729298\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 25.9808 0.960933
$$732$$ 0 0
$$733$$ 39.1918i 1.44758i 0.690018 + 0.723792i $$0.257602\pi$$
−0.690018 + 0.723792i $$0.742398\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 2.82843i 0.104186i
$$738$$ 0 0
$$739$$ 22.0000 0.809283 0.404642 0.914475i $$-0.367396\pi$$
0.404642 + 0.914475i $$0.367396\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 7.07107i 0.259412i −0.991552 0.129706i $$-0.958597\pi$$
0.991552 0.129706i $$-0.0414034\pi$$
$$744$$ 0 0
$$745$$ 12.2474i 0.448712i
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 34.6410 14.1421i 1.26576 0.516742i
$$750$$ 0 0
$$751$$ 34.0000 1.24068 0.620339 0.784334i $$-0.286995\pi$$
0.620339 + 0.784334i $$0.286995\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ −8.66025 −0.315179
$$756$$ 0 0
$$757$$ 47.0000 1.70824 0.854122 0.520073i $$-0.174095\pi$$
0.854122 + 0.520073i $$0.174095\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 12.1244 0.439508 0.219754 0.975555i $$-0.429475\pi$$
0.219754 + 0.975555i $$0.429475\pi$$
$$762$$ 0 0
$$763$$ 7.00000 + 17.1464i 0.253417 + 0.620742i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 21.2132i 0.765964i
$$768$$ 0 0
$$769$$ 4.89898i 0.176662i −0.996091 0.0883309i $$-0.971847\pi$$
0.996091 0.0883309i $$-0.0281533\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 15.5885 0.560678 0.280339 0.959901i $$-0.409553\pi$$
0.280339 + 0.959901i $$0.409553\pi$$
$$774$$ 0 0
$$775$$ 4.89898i 0.175977i
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 63.6396i 2.28013i
$$780$$ 0 0
$$781$$ −20.0000 −0.715656
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 29.6985i 1.05998i
$$786$$ 0 0
$$787$$ 4.89898i 0.174630i 0.996181 + 0.0873149i $$0.0278286\pi$$
−0.996181 + 0.0873149i $$0.972171\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 17.3205 7.07107i 0.615846 0.251418i
$$792$$ 0 0
$$793$$ 6.00000 0.213066
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 38.1051 1.34975 0.674876 0.737931i $$-0.264197\pi$$
0.674876 + 0.737931i $$0.264197\pi$$
$$798$$ 0 0
$$799$$ 45.0000 1.59199
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 0 0
$$804$$ 0 0
$$805$$ −12.0000 + 4.89898i −0.422944 + 0.172666i
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 14.1421i 0.497211i −0.968605 0.248606i $$-0.920028\pi$$
0.968605 0.248606i $$-0.0799723\pi$$
$$810$$ 0 0
$$811$$ 36.7423i 1.29020i 0.764099 + 0.645099i $$0.223184\pi$$
−0.764099 + 0.645099i $$0.776816\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ −8.66025 −0.303355
$$816$$