# Properties

 Label 648.1.j.a.269.1 Level $648$ Weight $1$ Character 648.269 Analytic conductor $0.323$ Analytic rank $0$ Dimension $2$ Projective image $D_{3}$ CM discriminant -24 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$648 = 2^{3} \cdot 3^{4}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 648.j (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.323394128186$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{6})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 216) Projective image: $$D_{3}$$ Projective field: Galois closure of 3.1.216.1 Artin image: $C_3\times S_3$ Artin field: Galois closure of 6.0.10077696.3

## Embedding invariants

 Embedding label 269.1 Root $$0.500000 + 0.866025i$$ of defining polynomial Character $$\chi$$ $$=$$ 648.269 Dual form 648.1.j.a.53.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+(-0.500000 + 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(0.500000 + 0.866025i) q^{5} +(0.500000 - 0.866025i) q^{7} +1.00000 q^{8} +O(q^{10})$$ $$q+(-0.500000 + 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(0.500000 + 0.866025i) q^{5} +(0.500000 - 0.866025i) q^{7} +1.00000 q^{8} -1.00000 q^{10} +(0.500000 - 0.866025i) q^{11} +(0.500000 + 0.866025i) q^{14} +(-0.500000 + 0.866025i) q^{16} +(0.500000 - 0.866025i) q^{20} +(0.500000 + 0.866025i) q^{22} -1.00000 q^{28} +(-1.00000 + 1.73205i) q^{29} +(0.500000 + 0.866025i) q^{31} +(-0.500000 - 0.866025i) q^{32} +1.00000 q^{35} +(0.500000 + 0.866025i) q^{40} -1.00000 q^{44} -1.00000 q^{53} +1.00000 q^{55} +(0.500000 - 0.866025i) q^{56} +(-1.00000 - 1.73205i) q^{58} +(-1.00000 - 1.73205i) q^{59} -1.00000 q^{62} +1.00000 q^{64} +(-0.500000 + 0.866025i) q^{70} -1.00000 q^{73} +(-0.500000 - 0.866025i) q^{77} +(-1.00000 + 1.73205i) q^{79} -1.00000 q^{80} +(0.500000 - 0.866025i) q^{83} +(0.500000 - 0.866025i) q^{88} +(0.500000 - 0.866025i) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{2} - q^{4} + q^{5} + q^{7} + 2q^{8} + O(q^{10})$$ $$2q - q^{2} - q^{4} + q^{5} + q^{7} + 2q^{8} - 2q^{10} + q^{11} + q^{14} - q^{16} + q^{20} + q^{22} - 2q^{28} - 2q^{29} + q^{31} - q^{32} + 2q^{35} + q^{40} - 2q^{44} - 2q^{53} + 2q^{55} + q^{56} - 2q^{58} - 2q^{59} - 2q^{62} + 2q^{64} - q^{70} - 2q^{73} - q^{77} - 2q^{79} - 2q^{80} + q^{83} + q^{88} + q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/648\mathbb{Z}\right)^\times$$.

 $$n$$ $$325$$ $$487$$ $$569$$ $$\chi(n)$$ $$-1$$ $$1$$ $$e\left(\frac{1}{6}\right)$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −0.500000 + 0.866025i −0.500000 + 0.866025i
$$3$$ 0 0
$$4$$ −0.500000 0.866025i −0.500000 0.866025i
$$5$$ 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 $$0$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$6$$ 0 0
$$7$$ 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i $$-0.666667\pi$$
1.00000 $$0$$
$$8$$ 1.00000 1.00000
$$9$$ 0 0
$$10$$ −1.00000 −1.00000
$$11$$ 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i $$-0.666667\pi$$
1.00000 $$0$$
$$12$$ 0 0
$$13$$ 0 0 −0.500000 0.866025i $$-0.666667\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$14$$ 0.500000 + 0.866025i 0.500000 + 0.866025i
$$15$$ 0 0
$$16$$ −0.500000 + 0.866025i −0.500000 + 0.866025i
$$17$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$18$$ 0 0
$$19$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$20$$ 0.500000 0.866025i 0.500000 0.866025i
$$21$$ 0 0
$$22$$ 0.500000 + 0.866025i 0.500000 + 0.866025i
$$23$$ 0 0 −0.500000 0.866025i $$-0.666667\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 0 0
$$28$$ −1.00000 −1.00000
$$29$$ −1.00000 + 1.73205i −1.00000 + 1.73205i −0.500000 + 0.866025i $$0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$30$$ 0 0
$$31$$ 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 $$0$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$32$$ −0.500000 0.866025i −0.500000 0.866025i
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 1.00000 1.00000
$$36$$ 0 0
$$37$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0.500000 + 0.866025i 0.500000 + 0.866025i
$$41$$ 0 0 −0.500000 0.866025i $$-0.666667\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$42$$ 0 0
$$43$$ 0 0 0.500000 0.866025i $$-0.333333\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$44$$ −1.00000 −1.00000
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 0 0 0.500000 0.866025i $$-0.333333\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ −1.00000 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$54$$ 0 0
$$55$$ 1.00000 1.00000
$$56$$ 0.500000 0.866025i 0.500000 0.866025i
$$57$$ 0 0
$$58$$ −1.00000 1.73205i −1.00000 1.73205i
$$59$$ −1.00000 1.73205i −1.00000 1.73205i −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 0.866025i $$-0.666667\pi$$
$$60$$ 0 0
$$61$$ 0 0 0.500000 0.866025i $$-0.333333\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$62$$ −1.00000 −1.00000
$$63$$ 0 0
$$64$$ 1.00000 1.00000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 0 0 −0.500000 0.866025i $$-0.666667\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ −0.500000 + 0.866025i −0.500000 + 0.866025i
$$71$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$72$$ 0 0
$$73$$ −1.00000 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −0.500000 0.866025i −0.500000 0.866025i
$$78$$ 0 0
$$79$$ −1.00000 + 1.73205i −1.00000 + 1.73205i −0.500000 + 0.866025i $$0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$80$$ −1.00000 −1.00000
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i $$-0.666667\pi$$
1.00000 $$0$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0.500000 0.866025i 0.500000 0.866025i
$$89$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i $$-0.666667\pi$$
1.00000 $$0$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i $$-0.666667\pi$$
1.00000 $$0$$
$$102$$ 0 0
$$103$$ −1.00000 1.73205i −1.00000 1.73205i −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 0.866025i $$-0.666667\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0.500000 0.866025i 0.500000 0.866025i
$$107$$ −1.00000 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$108$$ 0 0
$$109$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$110$$ −0.500000 + 0.866025i −0.500000 + 0.866025i
$$111$$ 0 0
$$112$$ 0.500000 + 0.866025i 0.500000 + 0.866025i
$$113$$ 0 0 −0.500000 0.866025i $$-0.666667\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 2.00000 2.00000
$$117$$ 0 0
$$118$$ 2.00000 2.00000
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 0 0
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0.500000 0.866025i 0.500000 0.866025i
$$125$$ 1.00000 1.00000
$$126$$ 0 0
$$127$$ −1.00000 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$128$$ −0.500000 + 0.866025i −0.500000 + 0.866025i
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 $$0$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 0 0 0.500000 0.866025i $$-0.333333\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$138$$ 0 0
$$139$$ 0 0 −0.500000 0.866025i $$-0.666667\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$140$$ −0.500000 0.866025i −0.500000 0.866025i
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ −2.00000 −2.00000
$$146$$ 0.500000 0.866025i 0.500000 0.866025i
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 $$0$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$150$$ 0 0
$$151$$ 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i $$-0.666667\pi$$
1.00000 $$0$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 1.00000 1.00000
$$155$$ −0.500000 + 0.866025i −0.500000 + 0.866025i
$$156$$ 0 0
$$157$$ 0 0 −0.500000 0.866025i $$-0.666667\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$158$$ −1.00000 1.73205i −1.00000 1.73205i
$$159$$ 0 0
$$160$$ 0.500000 0.866025i 0.500000 0.866025i
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0.500000 + 0.866025i 0.500000 + 0.866025i
$$167$$ 0 0 −0.500000 0.866025i $$-0.666667\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$168$$ 0 0
$$169$$ −0.500000 + 0.866025i −0.500000 + 0.866025i
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i $$-0.666667\pi$$
1.00000 $$0$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0.500000 + 0.866025i 0.500000 + 0.866025i
$$177$$ 0 0
$$178$$ 0 0
$$179$$ −1.00000 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$180$$ 0 0
$$181$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$182$$ 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 0.500000 0.866025i $$-0.333333\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$192$$ 0 0
$$193$$ 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 $$0$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$194$$ 0.500000 + 0.866025i 0.500000 + 0.866025i
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −1.00000 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$198$$ 0 0
$$199$$ −1.00000 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0.500000 + 0.866025i 0.500000 + 0.866025i
$$203$$ 1.00000 + 1.73205i 1.00000 + 1.73205i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 2.00000 2.00000
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 0 0 −0.500000 0.866025i $$-0.666667\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$212$$ 0.500000 + 0.866025i 0.500000 + 0.866025i
$$213$$ 0 0
$$214$$ 0.500000 0.866025i 0.500000 0.866025i
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 1.00000 1.00000
$$218$$ 0 0
$$219$$ 0 0
$$220$$ −0.500000 0.866025i −0.500000 0.866025i
$$221$$ 0 0
$$222$$ 0 0
$$223$$ −1.00000 + 1.73205i −1.00000 + 1.73205i −0.500000 + 0.866025i $$0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$224$$ −1.00000 −1.00000
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −1.00000 + 1.73205i −1.00000 + 1.73205i −0.500000 + 0.866025i $$0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$228$$ 0 0
$$229$$ 0 0 −0.500000 0.866025i $$-0.666667\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ −1.00000 + 1.73205i −1.00000 + 1.73205i
$$233$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ −1.00000 + 1.73205i −1.00000 + 1.73205i
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 0 0 −0.500000 0.866025i $$-0.666667\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$240$$ 0 0
$$241$$ −1.00000 + 1.73205i −1.00000 + 1.73205i −0.500000 + 0.866025i $$0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0.500000 + 0.866025i 0.500000 + 0.866025i
$$249$$ 0 0
$$250$$ −0.500000 + 0.866025i −0.500000 + 0.866025i
$$251$$ 2.00000 2.00000 1.00000 $$0$$
1.00000 $$0$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0.500000 0.866025i 0.500000 0.866025i
$$255$$ 0 0
$$256$$ −0.500000 0.866025i −0.500000 0.866025i
$$257$$ 0 0 −0.500000 0.866025i $$-0.666667\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 0 0
$$262$$ −1.00000 −1.00000
$$263$$ 0 0 0.500000 0.866025i $$-0.333333\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$264$$ 0 0
$$265$$ −0.500000 0.866025i −0.500000 0.866025i
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 2.00000 2.00000 1.00000 $$0$$
1.00000 $$0$$
$$270$$ 0 0
$$271$$ −1.00000 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 0 0 0.500000 0.866025i $$-0.333333\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 1.00000 1.00000
$$281$$ 0 0 0.500000 0.866025i $$-0.333333\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$282$$ 0 0
$$283$$ 0 0 −0.500000 0.866025i $$-0.666667\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 1.00000 1.00000
$$290$$ 1.00000 1.73205i 1.00000 1.73205i
$$291$$ 0 0
$$292$$ 0.500000 + 0.866025i 0.500000 + 0.866025i
$$293$$ −1.00000 1.73205i −1.00000 1.73205i −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 0.866025i $$-0.666667\pi$$
$$294$$ 0 0
$$295$$ 1.00000 1.73205i 1.00000 1.73205i
$$296$$ 0 0
$$297$$ 0 0
$$298$$ −1.00000 −1.00000
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0.500000 + 0.866025i 0.500000 + 0.866025i
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$308$$ −0.500000 + 0.866025i −0.500000 + 0.866025i
$$309$$ 0 0
$$310$$ −0.500000 0.866025i −0.500000 0.866025i
$$311$$ 0 0 −0.500000 0.866025i $$-0.666667\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$312$$ 0 0
$$313$$ 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i $$-0.666667\pi$$
1.00000 $$0$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 2.00000 2.00000
$$317$$ 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i $$-0.666667\pi$$
1.00000 $$0$$
$$318$$ 0 0
$$319$$ 1.00000 + 1.73205i 1.00000 + 1.73205i
$$320$$ 0.500000 + 0.866025i 0.500000 + 0.866025i
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 0 0 0.500000 0.866025i $$-0.333333\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$332$$ −1.00000 −1.00000
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −1.00000 1.73205i −1.00000 1.73205i −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 0.866025i $$-0.666667\pi$$
$$338$$ −0.500000 0.866025i −0.500000 0.866025i
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 1.00000 1.00000
$$342$$ 0 0
$$343$$ 1.00000 1.00000
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0.500000 + 0.866025i 0.500000 + 0.866025i
$$347$$ 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 $$0$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$348$$ 0 0
$$349$$ 0 0 0.500000 0.866025i $$-0.333333\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ −1.00000 −1.00000
$$353$$ 0 0 0.500000 0.866025i $$-0.333333\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0.500000 0.866025i 0.500000 0.866025i
$$359$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$360$$ 0 0
$$361$$ 1.00000 1.00000
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −0.500000 0.866025i −0.500000 0.866025i
$$366$$ 0 0
$$367$$ 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i $$-0.666667\pi$$
1.00000 $$0$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −0.500000 + 0.866025i −0.500000 + 0.866025i
$$372$$ 0 0
$$373$$ 0 0 −0.500000 0.866025i $$-0.666667\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 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 $$-0.666667\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$384$$ 0 0
$$385$$ 0.500000 0.866025i 0.500000 0.866025i
$$386$$ −1.00000 −1.00000
$$387$$ 0 0
$$388$$ −1.00000 −1.00000
$$389$$ 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i $$-0.666667\pi$$
1.00000 $$0$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0.500000 0.866025i 0.500000 0.866025i
$$395$$ −2.00000 −2.00000
$$396$$ 0 0
$$397$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$398$$ 0.500000 0.866025i 0.500000 0.866025i
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 0 0 −0.500000 0.866025i $$-0.666667\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ −1.00000 −1.00000
$$405$$ 0 0
$$406$$ −2.00000 −2.00000
$$407$$ 0 0
$$408$$ 0 0
$$409$$ 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 $$0$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ −1.00000 + 1.73205i −1.00000 + 1.73205i
$$413$$ −2.00000 −2.00000
$$414$$ 0 0
$$415$$ 1.00000 1.00000
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ −1.00000 1.73205i −1.00000 1.73205i −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 0.866025i $$-0.666667\pi$$
$$420$$ 0 0
$$421$$ 0 0 0.500000 0.866025i $$-0.333333\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ −1.00000 −1.00000
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0.500000 + 0.866025i 0.500000 + 0.866025i
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$432$$ 0 0
$$433$$ −1.00000 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$434$$ −0.500000 + 0.866025i −0.500000 + 0.866025i
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 0 0
$$438$$ 0 0
$$439$$ 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i $$-0.666667\pi$$
1.00000 $$0$$
$$440$$ 1.00000 1.00000
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −1.00000 + 1.73205i −1.00000 + 1.73205i −0.500000 + 0.866025i $$0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ −1.00000 1.73205i −1.00000 1.73205i
$$447$$ 0 0
$$448$$ 0.500000 0.866025i 0.500000 0.866025i
$$449$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ 0 0
$$454$$ −1.00000 1.73205i −1.00000 1.73205i
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i $$-0.666667\pi$$
1.00000 $$0$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i $$-0.666667\pi$$
1.00000 $$0$$
$$462$$ 0 0
$$463$$ 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 $$0$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$464$$ −1.00000 1.73205i −1.00000 1.73205i
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −1.00000 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 0 0
$$472$$ −1.00000 1.73205i −1.00000 1.73205i
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 0 0 0.500000 0.866025i $$-0.333333\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ −1.00000 1.73205i −1.00000 1.73205i
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 1.00000 1.00000
$$486$$ 0 0
$$487$$ 2.00000 2.00000 1.00000 $$0$$
1.00000 $$0$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 $$0$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ 0 0
$$495$$ 0 0
$$496$$ −1.00000 −1.00000
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 0 0 −0.500000 0.866025i $$-0.666667\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$500$$ −0.500000 0.866025i −0.500000 0.866025i
$$501$$ 0 0
$$502$$ −1.00000 + 1.73205i −1.00000 + 1.73205i
$$503$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$504$$ 0 0
$$505$$ 1.00000 1.00000
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0.500000 + 0.866025i 0.500000 + 0.866025i
$$509$$ 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 $$0$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$510$$ 0 0
$$511$$ −0.500000 + 0.866025i −0.500000 + 0.866025i
$$512$$ 1.00000 1.00000
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 1.00000 1.73205i 1.00000 1.73205i
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$522$$ 0 0
$$523$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$524$$ 0.500000 0.866025i 0.500000 0.866025i
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 0 0
$$528$$ 0 0
$$529$$ −0.500000 + 0.866025i −0.500000 + 0.866025i
$$530$$ 1.00000 1.00000
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ −0.500000 0.866025i −0.500000 0.866025i
$$536$$ 0 0
$$537$$ 0 0
$$538$$ −1.00000 + 1.73205i −1.00000 + 1.73205i
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$542$$ 0.500000 0.866025i 0.500000 0.866025i
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 0 0 0.500000 0.866025i $$-0.333333\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 0 0
$$553$$ 1.00000 + 1.73205i 1.00000 + 1.73205i
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −1.00000 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ −0.500000 + 0.866025i −0.500000 + 0.866025i
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 $$0$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 0 0 0.500000 0.866025i $$-0.333333\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$570$$ 0 0
$$571$$ 0 0 −0.500000 0.866025i $$-0.666667\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 2.00000 2.00000 1.00000 $$0$$
1.00000 $$0$$
$$578$$ −0.500000 + 0.866025i −0.500000 + 0.866025i
$$579$$ 0 0
$$580$$ 1.00000 + 1.73205i 1.00000 + 1.73205i
$$581$$ −0.500000 0.866025i −0.500000 0.866025i
$$582$$ 0 0
$$583$$ −0.500000 + 0.866025i −0.500000 + 0.866025i
$$584$$ −1.00000 −1.00000
$$585$$ 0 0
$$586$$ 2.00000 2.00000
$$587$$ 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i $$-0.666667\pi$$
1.00000 $$0$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 1.00000 + 1.73205i 1.00000 + 1.73205i
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0.500000 0.866025i 0.500000 0.866025i
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 0 0 −0.500000 0.866025i $$-0.666667\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$600$$ 0 0
$$601$$ 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i $$-0.666667\pi$$
1.00000 $$0$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ −1.00000 −1.00000
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −1.00000 1.73205i −1.00000 1.73205i −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 0.866025i $$-0.666667\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ −0.500000 0.866025i −0.500000 0.866025i
$$617$$ 0 0 −0.500000 0.866025i $$-0.666667\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$618$$ 0 0
$$619$$ 0 0 0.500000 0.866025i $$-0.333333\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$620$$ 1.00000 1.00000
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 0.500000 + 0.866025i 0.500000 + 0.866025i
$$626$$ 0.500000 + 0.866025i 0.500000 + 0.866025i
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 0 0
$$630$$ 0 0
$$631$$ −1.00000 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$632$$ −1.00000 + 1.73205i −1.00000 + 1.73205i
$$633$$ 0 0
$$634$$ 0.500000 + 0.866025i 0.500000 + 0.866025i
$$635$$ −0.500000 0.866025i −0.500000 0.866025i
$$636$$ 0 0
$$637$$ 0 0
$$638$$ −2.00000 −2.00000
$$639$$ 0 0
$$640$$ −1.00000 −1.00000
$$641$$ 0 0 0.500000 0.866025i $$-0.333333\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$642$$ 0 0
$$643$$ 0 0 −0.500000 0.866025i $$-0.666667\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$648$$ 0 0
$$649$$ −2.00000 −2.00000
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 $$0$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$654$$ 0 0
$$655$$ −0.500000 + 0.866025i −0.500000 + 0.866025i
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i $$-0.666667\pi$$
1.00000 $$0$$
$$660$$ 0 0
$$661$$ 0 0 −0.500000 0.866025i $$-0.666667\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0.500000 0.866025i 0.500000 0.866025i
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i $$-0.666667\pi$$
1.00000 $$0$$
$$674$$ 2.00000 2.00000
$$675$$ 0 0
$$676$$ 1.00000 1.00000
$$677$$ −1.00000 + 1.73205i −1.00000 + 1.73205i −0.500000 + 0.866025i $$0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$678$$ 0 0
$$679$$ −0.500000 0.866025i −0.500000 0.866025i
$$680$$ 0 0
$$681$$ 0 0
$$682$$ −0.500000 + 0.866025i −0.500000 + 0.866025i
$$683$$ 2.00000 2.00000 1.00000 $$0$$
1.00000 $$0$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ −0.500000 + 0.866025i −0.500000 + 0.866025i
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 0 0
$$690$$ 0 0
$$691$$ 0 0 0.500000 0.866025i $$-0.333333\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$692$$ −1.00000 −1.00000
$$693$$ 0 0
$$694$$ −1.00000 −1.00000
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 0 0
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −1.00000 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$702$$ 0 0
$$703$$ 0 0
$$704$$ 0.500000 0.866025i 0.500000 0.866025i
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −0.500000 0.866025i −0.500000 0.866025i
$$708$$ 0 0
$$709$$ 0 0 0.500000 0.866025i $$-0.333333\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0.500000 + 0.866025i 0.500000 + 0.866025i
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$720$$ 0 0
$$721$$ −2.00000 −2.00000
$$722$$ −0.500000 + 0.866025i −0.500000 + 0.866025i
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i $$-0.666667\pi$$
1.00000 $$0$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 1.00000 1.00000
$$731$$ 0 0
$$732$$ 0 0
$$733$$ 0 0 −0.500000 0.866025i $$-0.666667\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$734$$ 0.500000 + 0.866025i 0.500000 + 0.866025i
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ −0.500000 0.866025i −0.500000 0.866025i
$$743$$ 0 0 −0.500000 0.866025i $$-0.666667\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$744$$ 0 0
$$745$$ −0.500000 + 0.866025i −0.500000 + 0.866025i
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ −0.500000 + 0.866025i −0.500000 + 0.866025i
$$750$$ 0 0
$$751$$ 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 $$0$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 1.00000 1.00000
$$756$$ 0 0
$$757$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 0 0 −0.500000 0.866025i $$-0.666667\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 $$0$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$770$$ 0.500000 + 0.866025i 0.500000 + 0.866025i
$$771$$ 0 0
$$772$$ 0.500000 0.866025i 0.500000 0.866025i
$$773$$ 2.00000 2.00000 1.00000 $$0$$
1.00000 $$0$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0.500000 0.866025i 0.500000 0.866025i
$$777$$ 0 0
$$778$$ 0.500000 + 0.866025i 0.500000 + 0.866025i
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 0 0 −0.500000 0.866025i $$-0.666667\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$788$$ 0.500000 + 0.866025i 0.500000 + 0.866025i
$$789$$ 0 0
$$790$$ 1.00000 1.73205i 1.00000 1.73205i
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0.500000 + 0.866025i 0.500000 + 0.866025i
$$797$$ 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 $$0$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ −0.500000 + 0.866025i −0.500000 + 0.866025i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0.500000 0.866025i 0.500000