# Properties

 Label 324.3.f.e.271.1 Level $324$ Weight $3$ Character 324.271 Analytic conductor $8.828$ Analytic rank $0$ Dimension $2$ CM discriminant -4 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$324 = 2^{2} \cdot 3^{4}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 324.f (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$8.82836056527$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{6})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{25}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 36) Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

## Embedding invariants

 Embedding label 271.1 Root $$0.500000 + 0.866025i$$ of defining polynomial Character $$\chi$$ $$=$$ 324.271 Dual form 324.3.f.e.55.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+(-1.00000 - 1.73205i) q^{2} +(-2.00000 + 3.46410i) q^{4} +(4.00000 - 6.92820i) q^{5} +8.00000 q^{8} +O(q^{10})$$ $$q+(-1.00000 - 1.73205i) q^{2} +(-2.00000 + 3.46410i) q^{4} +(4.00000 - 6.92820i) q^{5} +8.00000 q^{8} -16.0000 q^{10} +(5.00000 - 8.66025i) q^{13} +(-8.00000 - 13.8564i) q^{16} +16.0000 q^{17} +(16.0000 + 27.7128i) q^{20} +(-19.5000 - 33.7750i) q^{25} -20.0000 q^{26} +(-20.0000 - 34.6410i) q^{29} +(-16.0000 + 27.7128i) q^{32} +(-16.0000 - 27.7128i) q^{34} -70.0000 q^{37} +(32.0000 - 55.4256i) q^{40} +(40.0000 - 69.2820i) q^{41} +(-24.5000 + 42.4352i) q^{49} +(-39.0000 + 67.5500i) q^{50} +(20.0000 + 34.6410i) q^{52} -56.0000 q^{53} +(-40.0000 + 69.2820i) q^{58} +(11.0000 + 19.0526i) q^{61} +64.0000 q^{64} +(-40.0000 - 69.2820i) q^{65} +(-32.0000 + 55.4256i) q^{68} +110.000 q^{73} +(70.0000 + 121.244i) q^{74} -128.000 q^{80} -160.000 q^{82} +(64.0000 - 110.851i) q^{85} +160.000 q^{89} +(65.0000 + 112.583i) q^{97} +98.0000 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{2} - 4q^{4} + 8q^{5} + 16q^{8} + O(q^{10})$$ $$2q - 2q^{2} - 4q^{4} + 8q^{5} + 16q^{8} - 32q^{10} + 10q^{13} - 16q^{16} + 32q^{17} + 32q^{20} - 39q^{25} - 40q^{26} - 40q^{29} - 32q^{32} - 32q^{34} - 140q^{37} + 64q^{40} + 80q^{41} - 49q^{49} - 78q^{50} + 40q^{52} - 112q^{53} - 80q^{58} + 22q^{61} + 128q^{64} - 80q^{65} - 64q^{68} + 220q^{73} + 140q^{74} - 256q^{80} - 320q^{82} + 128q^{85} + 320q^{89} + 130q^{97} + 196q^{98} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/324\mathbb{Z}\right)^\times$$.

 $$n$$ $$163$$ $$245$$ $$\chi(n)$$ $$-1$$ $$e\left(\frac{1}{3}\right)$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.00000 1.73205i −0.500000 0.866025i
$$3$$ 0 0
$$4$$ −2.00000 + 3.46410i −0.500000 + 0.866025i
$$5$$ 4.00000 6.92820i 0.800000 1.38564i −0.119615 0.992820i $$-0.538166\pi$$
0.919615 0.392820i $$-0.128501\pi$$
$$6$$ 0 0
$$7$$ 0 0 −0.500000 0.866025i $$-0.666667\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$8$$ 8.00000 1.00000
$$9$$ 0 0
$$10$$ −16.0000 −1.60000
$$11$$ 0 0 −0.500000 0.866025i $$-0.666667\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$12$$ 0 0
$$13$$ 5.00000 8.66025i 0.384615 0.666173i −0.607100 0.794625i $$-0.707667\pi$$
0.991716 + 0.128452i $$0.0410008\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ −8.00000 13.8564i −0.500000 0.866025i
$$17$$ 16.0000 0.941176 0.470588 0.882353i $$-0.344042\pi$$
0.470588 + 0.882353i $$0.344042\pi$$
$$18$$ 0 0
$$19$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$20$$ 16.0000 + 27.7128i 0.800000 + 1.38564i
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 0 0 0.500000 0.866025i $$-0.333333\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$24$$ 0 0
$$25$$ −19.5000 33.7750i −0.780000 1.35100i
$$26$$ −20.0000 −0.769231
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −20.0000 34.6410i −0.689655 1.19452i −0.971949 0.235190i $$-0.924429\pi$$
0.282294 0.959328i $$-0.408905\pi$$
$$30$$ 0 0
$$31$$ 0 0 0.500000 0.866025i $$-0.333333\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$32$$ −16.0000 + 27.7128i −0.500000 + 0.866025i
$$33$$ 0 0
$$34$$ −16.0000 27.7128i −0.470588 0.815083i
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −70.0000 −1.89189 −0.945946 0.324324i $$-0.894863\pi$$
−0.945946 + 0.324324i $$0.894863\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 32.0000 55.4256i 0.800000 1.38564i
$$41$$ 40.0000 69.2820i 0.975610 1.68981i 0.297702 0.954659i $$-0.403780\pi$$
0.677908 0.735147i $$-0.262887\pi$$
$$42$$ 0 0
$$43$$ 0 0 −0.500000 0.866025i $$-0.666667\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 0 0 −0.500000 0.866025i $$-0.666667\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$48$$ 0 0
$$49$$ −24.5000 + 42.4352i −0.500000 + 0.866025i
$$50$$ −39.0000 + 67.5500i −0.780000 + 1.35100i
$$51$$ 0 0
$$52$$ 20.0000 + 34.6410i 0.384615 + 0.666173i
$$53$$ −56.0000 −1.05660 −0.528302 0.849057i $$-0.677171\pi$$
−0.528302 + 0.849057i $$0.677171\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 0 0
$$58$$ −40.0000 + 69.2820i −0.689655 + 1.19452i
$$59$$ 0 0 0.500000 0.866025i $$-0.333333\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$60$$ 0 0
$$61$$ 11.0000 + 19.0526i 0.180328 + 0.312337i 0.941992 0.335635i $$-0.108951\pi$$
−0.761664 + 0.647972i $$0.775617\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 64.0000 1.00000
$$65$$ −40.0000 69.2820i −0.615385 1.06588i
$$66$$ 0 0
$$67$$ 0 0 0.500000 0.866025i $$-0.333333\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$68$$ −32.0000 + 55.4256i −0.470588 + 0.815083i
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$72$$ 0 0
$$73$$ 110.000 1.50685 0.753425 0.657534i $$-0.228401\pi$$
0.753425 + 0.657534i $$0.228401\pi$$
$$74$$ 70.0000 + 121.244i 0.945946 + 1.63843i
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 0 0 −0.500000 0.866025i $$-0.666667\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$80$$ −128.000 −1.60000
$$81$$ 0 0
$$82$$ −160.000 −1.95122
$$83$$ 0 0 −0.500000 0.866025i $$-0.666667\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$84$$ 0 0
$$85$$ 64.0000 110.851i 0.752941 1.30413i
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 160.000 1.79775 0.898876 0.438202i $$-0.144385\pi$$
0.898876 + 0.438202i $$0.144385\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 65.0000 + 112.583i 0.670103 + 1.16065i 0.977875 + 0.209192i $$0.0670835\pi$$
−0.307771 + 0.951460i $$0.599583\pi$$
$$98$$ 98.0000 1.00000
$$99$$ 0 0
$$100$$ 156.000 1.56000
$$101$$ −20.0000 34.6410i −0.198020 0.342980i 0.749866 0.661589i $$-0.230118\pi$$
−0.947886 + 0.318609i $$0.896784\pi$$
$$102$$ 0 0
$$103$$ 0 0 0.500000 0.866025i $$-0.333333\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$104$$ 40.0000 69.2820i 0.384615 0.666173i
$$105$$ 0 0
$$106$$ 56.0000 + 96.9948i 0.528302 + 0.915046i
$$107$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$108$$ 0 0
$$109$$ 182.000 1.66972 0.834862 0.550459i $$-0.185547\pi$$
0.834862 + 0.550459i $$0.185547\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 112.000 193.990i 0.991150 1.71672i 0.380616 0.924733i $$-0.375712\pi$$
0.610534 0.791990i $$-0.290955\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 160.000 1.37931
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −60.5000 + 104.789i −0.500000 + 0.866025i
$$122$$ 22.0000 38.1051i 0.180328 0.312337i
$$123$$ 0 0
$$124$$ 0 0
$$125$$ −112.000 −0.896000
$$126$$ 0 0
$$127$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$128$$ −64.0000 110.851i −0.500000 0.866025i
$$129$$ 0 0
$$130$$ −80.0000 + 138.564i −0.615385 + 1.06588i
$$131$$ 0 0 0.500000 0.866025i $$-0.333333\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 128.000 0.941176
$$137$$ 88.0000 + 152.420i 0.642336 + 1.11256i 0.984910 + 0.173067i $$0.0553679\pi$$
−0.342574 + 0.939491i $$0.611299\pi$$
$$138$$ 0 0
$$139$$ 0 0 0.500000 0.866025i $$-0.333333\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ −320.000 −2.20690
$$146$$ −110.000 190.526i −0.753425 1.30497i
$$147$$ 0 0
$$148$$ 140.000 242.487i 0.945946 1.63843i
$$149$$ −140.000 + 242.487i −0.939597 + 1.62743i −0.173374 + 0.984856i $$0.555467\pi$$
−0.766223 + 0.642574i $$0.777866\pi$$
$$150$$ 0 0
$$151$$ 0 0 −0.500000 0.866025i $$-0.666667\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −85.0000 + 147.224i −0.541401 + 0.937735i 0.457423 + 0.889249i $$0.348773\pi$$
−0.998824 + 0.0484851i $$0.984561\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 128.000 + 221.703i 0.800000 + 1.38564i
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$164$$ 160.000 + 277.128i 0.975610 + 1.68981i
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 0 0 0.500000 0.866025i $$-0.333333\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$168$$ 0 0
$$169$$ 34.5000 + 59.7558i 0.204142 + 0.353584i
$$170$$ −256.000 −1.50588
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 52.0000 + 90.0666i 0.300578 + 0.520616i 0.976267 0.216570i $$-0.0694871\pi$$
−0.675689 + 0.737187i $$0.736154\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 0 0
$$178$$ −160.000 277.128i −0.898876 1.55690i
$$179$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$180$$ 0 0
$$181$$ 38.0000 0.209945 0.104972 0.994475i $$-0.466525\pi$$
0.104972 + 0.994475i $$0.466525\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ −280.000 + 484.974i −1.51351 + 2.62148i
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 0 0 −0.500000 0.866025i $$-0.666667\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$192$$ 0 0
$$193$$ 95.0000 164.545i 0.492228 0.852564i −0.507732 0.861515i $$-0.669516\pi$$
0.999960 + 0.00895123i $$0.00284930\pi$$
$$194$$ 130.000 225.167i 0.670103 1.16065i
$$195$$ 0 0
$$196$$ −98.0000 169.741i −0.500000 0.866025i
$$197$$ −56.0000 −0.284264 −0.142132 0.989848i $$-0.545396\pi$$
−0.142132 + 0.989848i $$0.545396\pi$$
$$198$$ 0 0
$$199$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$200$$ −156.000 270.200i −0.780000 1.35100i
$$201$$ 0 0
$$202$$ −40.0000 + 69.2820i −0.198020 + 0.342980i
$$203$$ 0 0
$$204$$ 0 0
$$205$$ −320.000 554.256i −1.56098 2.70369i
$$206$$ 0 0
$$207$$ 0 0
$$208$$ −160.000 −0.769231
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 0 0 0.500000 0.866025i $$-0.333333\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$212$$ 112.000 193.990i 0.528302 0.915046i
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 0 0
$$218$$ −182.000 315.233i −0.834862 1.44602i
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 80.0000 138.564i 0.361991 0.626987i
$$222$$ 0 0
$$223$$ 0 0 −0.500000 0.866025i $$-0.666667\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ −448.000 −1.98230
$$227$$ 0 0 −0.500000 0.866025i $$-0.666667\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$228$$ 0 0
$$229$$ 221.000 382.783i 0.965066 1.67154i 0.255627 0.966776i $$-0.417718\pi$$
0.709439 0.704767i $$-0.248948\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ −160.000 277.128i −0.689655 1.19452i
$$233$$ −416.000 −1.78541 −0.892704 0.450644i $$-0.851194\pi$$
−0.892704 + 0.450644i $$0.851194\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 0 0 0.500000 0.866025i $$-0.333333\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$240$$ 0 0
$$241$$ 209.000 + 361.999i 0.867220 + 1.50207i 0.864826 + 0.502072i $$0.167429\pi$$
0.00239399 + 0.999997i $$0.499238\pi$$
$$242$$ 242.000 1.00000
$$243$$ 0 0
$$244$$ −88.0000 −0.360656
$$245$$ 196.000 + 339.482i 0.800000 + 1.38564i
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 112.000 + 193.990i 0.448000 + 0.775959i
$$251$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 0 0
$$256$$ −128.000 + 221.703i −0.500000 + 0.866025i
$$257$$ −32.0000 + 55.4256i −0.124514 + 0.215664i −0.921543 0.388277i $$-0.873070\pi$$
0.797029 + 0.603941i $$0.206404\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 320.000 1.23077
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 0 0 −0.500000 0.866025i $$-0.666667\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$264$$ 0 0
$$265$$ −224.000 + 387.979i −0.845283 + 1.46407i
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 520.000 1.93309 0.966543 0.256506i $$-0.0825712\pi$$
0.966543 + 0.256506i $$0.0825712\pi$$
$$270$$ 0 0
$$271$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$272$$ −128.000 221.703i −0.470588 0.815083i
$$273$$ 0 0
$$274$$ 176.000 304.841i 0.642336 1.11256i
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −115.000 199.186i −0.415162 0.719082i 0.580283 0.814415i $$-0.302942\pi$$
−0.995445 + 0.0953324i $$0.969609\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 160.000 + 277.128i 0.569395 + 0.986221i 0.996626 + 0.0820785i $$0.0261558\pi$$
−0.427231 + 0.904143i $$0.640511\pi$$
$$282$$ 0 0
$$283$$ 0 0 0.500000 0.866025i $$-0.333333\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −33.0000 −0.114187
$$290$$ 320.000 + 554.256i 1.10345 + 1.91123i
$$291$$ 0 0
$$292$$ −220.000 + 381.051i −0.753425 + 1.30497i
$$293$$ −68.0000 + 117.779i −0.232082 + 0.401978i −0.958421 0.285359i $$-0.907887\pi$$
0.726339 + 0.687337i $$0.241220\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ −560.000 −1.89189
$$297$$ 0 0
$$298$$ 560.000 1.87919
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 176.000 0.577049
$$306$$ 0 0
$$307$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 0 0 0.500000 0.866025i $$-0.333333\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$312$$ 0 0
$$313$$ −25.0000 43.3013i −0.0798722 0.138343i 0.823322 0.567574i $$-0.192118\pi$$
−0.903195 + 0.429231i $$0.858785\pi$$
$$314$$ 340.000 1.08280
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −308.000 533.472i −0.971609 1.68288i −0.690700 0.723141i $$-0.742697\pi$$
−0.280909 0.959734i $$-0.590636\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 256.000 443.405i 0.800000 1.38564i
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 0 0
$$325$$ −390.000 −1.20000
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 320.000 554.256i 0.975610 1.68981i
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 0 0 −0.500000 0.866025i $$-0.666667\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −175.000 + 303.109i −0.519288 + 0.899433i 0.480461 + 0.877016i $$0.340469\pi$$
−0.999749 + 0.0224168i $$0.992864\pi$$
$$338$$ 69.0000 119.512i 0.204142 0.353584i
$$339$$ 0 0
$$340$$ 256.000 + 443.405i 0.752941 + 1.30413i
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 104.000 180.133i 0.300578 0.520616i
$$347$$ 0 0 0.500000 0.866025i $$-0.333333\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$348$$ 0 0
$$349$$ 299.000 + 517.883i 0.856734 + 1.48391i 0.875027 + 0.484073i $$0.160843\pi$$
−0.0182939 + 0.999833i $$0.505823\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −272.000 471.118i −0.770538 1.33461i −0.937268 0.348609i $$-0.886654\pi$$
0.166730 0.986003i $$-0.446679\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ −320.000 + 554.256i −0.898876 + 1.55690i
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$360$$ 0 0
$$361$$ 361.000 1.00000
$$362$$ −38.0000 65.8179i −0.104972 0.181817i
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 440.000 762.102i 1.20548 2.08795i
$$366$$ 0 0
$$367$$ 0 0 −0.500000 0.866025i $$-0.666667\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 1120.00 3.02703
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 275.000 476.314i 0.737265 1.27698i −0.216457 0.976292i $$-0.569450\pi$$
0.953722 0.300689i $$-0.0972166\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −400.000 −1.06101
$$378$$ 0 0
$$379$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 0 0 0.500000 0.866025i $$-0.333333\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −380.000 −0.984456
$$387$$ 0 0
$$388$$ −520.000 −1.34021
$$389$$ 340.000 + 588.897i 0.874036 + 1.51387i 0.857786 + 0.514007i $$0.171839\pi$$
0.0162499 + 0.999868i $$0.494827\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ −196.000 + 339.482i −0.500000 + 0.866025i
$$393$$ 0 0
$$394$$ 56.0000 + 96.9948i 0.142132 + 0.246180i
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 650.000 1.63728 0.818640 0.574307i $$-0.194729\pi$$
0.818640 + 0.574307i $$0.194729\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ −312.000 + 540.400i −0.780000 + 1.35100i
$$401$$ 40.0000 69.2820i 0.0997506 0.172773i −0.811831 0.583893i $$-0.801529\pi$$
0.911581 + 0.411120i $$0.134862\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 160.000 0.396040
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ −391.000 + 677.232i −0.955990 + 1.65582i −0.223905 + 0.974611i $$0.571880\pi$$
−0.732086 + 0.681213i $$0.761453\pi$$
$$410$$ −640.000 + 1108.51i −1.56098 + 2.70369i
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 160.000 + 277.128i 0.384615 + 0.666173i
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 0 0 0.500000 0.866025i $$-0.333333\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$420$$ 0 0
$$421$$ 29.0000 + 50.2295i 0.0688836 + 0.119310i 0.898410 0.439157i $$-0.144723\pi$$
−0.829527 + 0.558467i $$0.811390\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ −448.000 −1.05660
$$425$$ −312.000 540.400i −0.734118 1.27153i
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$432$$ 0 0
$$433$$ 290.000 0.669746 0.334873 0.942263i $$-0.391307\pi$$
0.334873 + 0.942263i $$0.391307\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ −364.000 + 630.466i −0.834862 + 1.44602i
$$437$$ 0 0
$$438$$ 0 0
$$439$$ 0 0 −0.500000 0.866025i $$-0.666667\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ −320.000 −0.723982
$$443$$ 0 0 −0.500000 0.866025i $$-0.666667\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$444$$ 0 0
$$445$$ 640.000 1108.51i 1.43820 2.49104i
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −560.000 −1.24722 −0.623608 0.781737i $$-0.714334\pi$$
−0.623608 + 0.781737i $$0.714334\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 448.000 + 775.959i 0.991150 + 1.71672i
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 425.000 + 736.122i 0.929978 + 1.61077i 0.783353 + 0.621577i $$0.213508\pi$$
0.146625 + 0.989192i $$0.453159\pi$$
$$458$$ −884.000 −1.93013
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −380.000 658.179i −0.824295 1.42772i −0.902457 0.430780i $$-0.858238\pi$$
0.0781619 0.996941i $$-0.475095\pi$$
$$462$$ 0 0
$$463$$ 0 0 0.500000 0.866025i $$-0.333333\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$464$$ −320.000 + 554.256i −0.689655 + 1.19452i
$$465$$ 0 0
$$466$$ 416.000 + 720.533i 0.892704 + 1.54621i
$$467$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 0 0 −0.500000 0.866025i $$-0.666667\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$480$$ 0 0
$$481$$ −350.000 + 606.218i −0.727651 + 1.26033i
$$482$$ 418.000 723.997i 0.867220 1.50207i
$$483$$ 0 0
$$484$$ −242.000 419.156i −0.500000 0.866025i
$$485$$ 1040.00 2.14433
$$486$$ 0 0
$$487$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$488$$ 88.0000 + 152.420i 0.180328 + 0.312337i
$$489$$ 0 0
$$490$$ 392.000 678.964i 0.800000 1.38564i
$$491$$ 0 0 0.500000 0.866025i $$-0.333333\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$492$$ 0 0
$$493$$ −320.000 554.256i −0.649087 1.12425i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 0 0 0.500000 0.866025i $$-0.333333\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$500$$ 224.000 387.979i 0.448000 0.775959i
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$504$$ 0 0
$$505$$ −320.000 −0.633663
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 220.000 381.051i 0.432220 0.748627i −0.564844 0.825198i $$-0.691064\pi$$
0.997064 + 0.0765706i $$0.0243970\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 512.000 1.00000
$$513$$ 0 0
$$514$$ 128.000 0.249027
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 0 0
$$520$$ −320.000 554.256i −0.615385 1.06588i
$$521$$ 880.000 1.68906 0.844530 0.535509i $$-0.179880\pi$$
0.844530 + 0.535509i $$0.179880\pi$$
$$522$$ 0 0
$$523$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 0 0
$$528$$ 0 0
$$529$$ −264.500 458.127i −0.500000 0.866025i
$$530$$ 896.000 1.69057
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −400.000 692.820i −0.750469 1.29985i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 0 0
$$538$$ −520.000 900.666i −0.966543 1.67410i
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −682.000 −1.26063 −0.630314 0.776340i $$-0.717074\pi$$
−0.630314 + 0.776340i $$0.717074\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ −256.000 + 443.405i −0.470588 + 0.815083i
$$545$$ 728.000 1260.93i 1.33578 2.31364i
$$546$$ 0 0
$$547$$ 0 0 −0.500000 0.866025i $$-0.666667\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$548$$ −704.000 −1.28467
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 0 0
$$553$$ 0 0
$$554$$ −230.000 + 398.372i −0.415162 + 0.719082i
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −1064.00 −1.91023 −0.955117 0.296230i $$-0.904271\pi$$
−0.955117 + 0.296230i $$0.904271\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 320.000 554.256i 0.569395 0.986221i
$$563$$ 0 0 0.500000 0.866025i $$-0.333333\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$564$$ 0 0
$$565$$ −896.000 1551.92i −1.58584 2.74676i
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 520.000 + 900.666i 0.913884 + 1.58289i 0.808527 + 0.588459i $$0.200265\pi$$
0.105357 + 0.994434i $$0.466401\pi$$
$$570$$ 0 0
$$571$$ 0 0 0.500000 0.866025i $$-0.333333\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −1150.00 −1.99307 −0.996534 0.0831889i $$-0.973490\pi$$
−0.996534 + 0.0831889i $$0.973490\pi$$
$$578$$ 33.0000 + 57.1577i 0.0570934 + 0.0988887i
$$579$$ 0 0
$$580$$ 640.000 1108.51i 1.10345 1.91123i
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 880.000 1.50685
$$585$$ 0 0
$$586$$ 272.000 0.464164
$$587$$ 0 0 −0.500000 0.866025i $$-0.666667\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 560.000 + 969.948i 0.945946 + 1.63843i
$$593$$ 736.000 1.24115 0.620573 0.784148i $$-0.286900\pi$$
0.620573 + 0.784148i $$0.286900\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −560.000 969.948i −0.939597 1.62743i
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 0 0 0.500000 0.866025i $$-0.333333\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$600$$ 0 0
$$601$$ 551.000 + 954.360i 0.916805 + 1.58795i 0.804236 + 0.594309i $$0.202575\pi$$
0.112569 + 0.993644i $$0.464092\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 484.000 + 838.313i 0.800000 + 1.38564i
$$606$$ 0 0
$$607$$ 0 0 0.500000 0.866025i $$-0.333333\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ −176.000 304.841i −0.288525 0.499739i
$$611$$ 0 0
$$612$$ 0 0
$$613$$ −70.0000 −0.114192 −0.0570962 0.998369i $$-0.518184\pi$$
−0.0570962 + 0.998369i $$0.518184\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −608.000 + 1053.09i −0.985413 + 1.70679i −0.345328 + 0.938482i $$0.612232\pi$$
−0.640085 + 0.768304i $$0.721101\pi$$
$$618$$ 0 0
$$619$$ 0 0 −0.500000 0.866025i $$-0.666667\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 39.5000 68.4160i 0.0632000 0.109466i
$$626$$ −50.0000 + 86.6025i −0.0798722 + 0.138343i
$$627$$ 0 0
$$628$$ −340.000 588.897i −0.541401 0.937735i
$$629$$ −1120.00 −1.78060
$$630$$ 0 0
$$631$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ −616.000 + 1066.94i −0.971609 + 1.68288i
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 245.000 + 424.352i 0.384615 + 0.666173i
$$638$$ 0 0
$$639$$ 0 0
$$640$$ −1024.00 −1.60000
$$641$$ −200.000 346.410i −0.312012 0.540421i 0.666785 0.745250i $$-0.267670\pi$$
−0.978798 + 0.204828i $$0.934336\pi$$
$$642$$ 0 0
$$643$$ 0 0 0.500000 0.866025i $$-0.333333\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 390.000 + 675.500i 0.600000 + 1.03923i
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −572.000 + 990.733i −0.875957 + 1.51720i −0.0202175 + 0.999796i $$0.506436\pi$$
−0.855740 + 0.517407i $$0.826897\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ −1280.00 −1.95122
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 0 0 −0.500000 0.866025i $$-0.666667\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$660$$ 0 0
$$661$$ −589.000 + 1020.18i −0.891074 + 1.54339i −0.0524847 + 0.998622i $$0.516714\pi$$
−0.838589 + 0.544764i $$0.816619\pi$$
$$662$$ 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
$$673$$ −385.000 666.840i −0.572065 0.990846i −0.996354 0.0853191i $$-0.972809\pi$$
0.424288 0.905527i $$-0.360524\pi$$
$$674$$ 700.000 1.03858
$$675$$ 0 0
$$676$$ −276.000 −0.408284
$$677$$ 52.0000 + 90.0666i 0.0768095 + 0.133038i 0.901872 0.432004i $$-0.142193\pi$$
−0.825062 + 0.565042i $$0.808860\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 512.000 886.810i 0.752941 1.30413i
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$684$$ 0 0
$$685$$ 1408.00 2.05547
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ −280.000 + 484.974i −0.406386 + 0.703881i
$$690$$ 0 0
$$691$$ 0 0 −0.500000 0.866025i $$-0.666667\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$692$$ −416.000 −0.601156
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 640.000 1108.51i 0.918221 1.59041i
$$698$$ 598.000 1035.77i 0.856734 1.48391i
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 520.000 0.741797 0.370899 0.928673i $$-0.379050\pi$$
0.370899 + 0.928673i $$0.379050\pi$$
$$702$$ 0 0
$$703$$ 0 0
$$704$$ 0 0
$$705$$ 0 0
$$706$$ −544.000 + 942.236i −0.770538 + 1.33461i
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −259.000 448.601i −0.365303 0.632724i 0.623522 0.781806i $$-0.285701\pi$$
−0.988825 + 0.149082i $$0.952368\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 1280.00 1.79775
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ −361.000 625.270i −0.500000 0.866025i
$$723$$ 0 0
$$724$$ −76.0000 + 131.636i −0.104972 + 0.181817i
$$725$$ −780.000 + 1351.00i −1.07586 + 1.86345i
$$726$$ 0 0
$$727$$ 0 0 −0.500000 0.866025i $$-0.666667\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ −1760.00 −2.41096
$$731$$ 0 0
$$732$$ 0 0
$$733$$ 725.000 1255.74i 0.989086 1.71315i 0.366943 0.930243i $$-0.380404\pi$$
0.622143 0.782904i $$-0.286262\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$740$$ −1120.00 1939.90i −1.51351 2.62148i
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 0 0 0.500000 0.866025i $$-0.333333\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$744$$ 0 0
$$745$$ 1120.00 + 1939.90i 1.50336 + 2.60389i
$$746$$ −1100.00 −1.47453
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 0 0 0.500000 0.866025i $$-0.333333\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 400.000 + 692.820i 0.530504 + 0.918860i
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 1190.00 1.57199 0.785997 0.618230i $$-0.212150\pi$$
0.785997 + 0.618230i $$0.212150\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 760.000 1316.36i 0.998686 1.72977i 0.454961 0.890512i $$-0.349653\pi$$
0.543725 0.839263i $$-0.317013\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ −481.000 + 833.116i −0.625488 + 1.08338i 0.362959 + 0.931805i $$0.381767\pi$$
−0.988446 + 0.151571i $$0.951567\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 380.000 + 658.179i 0.492228 + 0.852564i
$$773$$ −1496.00 −1.93532 −0.967658 0.252264i $$-0.918825\pi$$
−0.967658 + 0.252264i $$0.918825\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 520.000 + 900.666i 0.670103 + 1.16065i
$$777$$ 0 0
$$778$$ 680.000 1177.79i 0.874036 1.51387i
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 784.000 1.00000
$$785$$ 680.000 + 1177.79i 0.866242 + 1.50038i
$$786$$ 0 0
$$787$$ 0 0 0.500000 0.866025i $$-0.333333\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$788$$ 112.000 193.990i 0.142132 0.246180i
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 220.000 0.277427
$$794$$ −650.000 1125.83i −0.818640 1.41793i
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −572.000 + 990.733i −0.717691 + 1.24308i 0.244221 + 0.969720i $$0.421468\pi$$
−0.961912 + 0.273358i $$0.911866\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 1248.00 1.56000
$$801$$ 0 0
$$802$$ −160.000 −0.199501
$$803$$ 0